diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzanjv b/data_all_eng_slimpj/shuffled/split2/finalzzanjv new file mode 100644 index 0000000000000000000000000000000000000000..1ce74ae551ceaba82b5a57584cee440200bea2d9 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzanjv @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\label{s-1}\nIn 2012 a new particle was discovered by the ATLAS and CMS experiments at the LHC~\\cite{Aad:2012,Chatrchyan:2012}.\nThis new particle, with a mass of about 125 GeV, is consistent with the Standard Model \nHiggs boson. With this discovery the Higgs mechanism~\\cite{Englert:1964,Higgs:1964,Guralnik:1964,Higgs:1966}\nhas been validated. \nThe Standard Model of Particle Physics introduces the Higgs mechanism in order to explain particle masses by means of the\nso-called spontaneous symmetry breaking mechanism. Spontaneous symmetry breaking can be obtained by introducing a scalar field, the Higgs field, with a specific potential. Actually, the Higgs mechanism had emerged as the only mechanism capable of reconciling gauge field theories\n with the observed mass spectrum. Indeed, the striking conceptual and empirical success of the Standard Model established \n increasing trust in the viability of the Higgs mechanism. \\\\\nUsually the spontaneous symmetry breaking in the Standard Model is implemented\nwithin the perturbation theory which leads to predict that the Higgs boson mass squared is proportional to $\\lambda \\, v^2$,\nwhere $\\lambda$ is the renormalised scalar self-coupling and $v \\simeq 246$ GeV is the known weak scale. \n On the other hand, it is known that, within the non-perturbative description of spontaneous symmetry breaking,\n self-interacting scalar fields are subject to the triviality problem~\\cite{Fernandez:1992}, namely the renormalised self-coupling\n$\\lambda \\rightarrow 0$ when the ultraviolet cutoff is sent to infinity. Strictly speaking, there are no rigorous proof \nof triviality~\\footnote{It is worthwhile to mention interesting progresses in this direction offered by the recent proof of the Gaussian structure\nof the scaling limits of the critical Ising and $\\phi^4$ fields in the marginal case of four dimensions~\\cite{Aizenman:2021}.}.\nOn the other hand, there exist several numerical studies which leave little doubt on the triviality conjecture. As a consequence, \nwithin the perturbative approach, the scalar sector of the Standard Model must be considered just an effective description \n valid only up to some (unknown) cut-off scale. \nNotwithstanding, extensive numerical simulations showed that even without self-interactions the scalar bosons could trigger spontaneous\n symmetry breaking. Moreover, precise non-perturbative numerical simulations~\\cite{Cea:2004,Cea:2012} indicated that the excitation\nof the Bose-Einstein scalar condensate should be a rather heavy scalar particle with mass of about 750(30) GeV~\\cite{Cea:2012}. \\\\\nTo reconcile the overwhelming evidence of a rather light Higgs boson with mass of 125 GeV (indicated with h) with\nthe indications from non-perturbative studies of a heavy Higgs boson with mass around 750 GeV (indicated with H),\nin a recent article~\\cite{Cea:2020} we advanced the proposal that the Higgs condensate of the Standard Model should be considered\nlike a relativistic quantum fluid analogous to superfluid helium. As discussed at length in Ref.~\\cite{Cea:2020}, we found that\n there are two different kind of Higgs condensate excitations that are similar to phonon and rotons in He II. Moreover,\n in the dilute gas approximation these two Higgs condensate excitations behave like the perturbative Standard Model Higgs \n boson and a heavy Standard Model Higgs boson. \nIndeed, in Ref.~\\cite{Cea:2019} we presented some phenomenological consequences of the Standard Model heavy Higgs boson\nproposal. In particular, we discussed the couplings of the H Higgs boson to the massive vector bosons and to fermions,\n the expected production mechanisms, and the main decay modes. We also attempted a quantitative comparison in the so-called \n golden channel $H \\rightarrow \\ell^+ \\ell^- \\ell'^+ \\ell'^-$ where $\\ell , \\ell' = e$ or $\\mu$.\n More precisely, by means of an unofficial combination of the preliminary Run 2 data collected by the ATLAS and CMS \n experiments, we found some evidence of a broad scalar resonance that looked consistent \nwith our Standard Model heavy Higgs boson~\\cite{Cea:2019,Cea:2020}. Unfortunately, this preliminary\nevidence has not been corroborated by the full data sets collected by the ATLAS Collaboration during the LHC Run 2.\n In Ref.~\\cite{Cea:2021} we critically compared our proposal to the full Run 2 data sets released by the ATLAS Collaboration. \n A search for a new high-mass resonance has been performed by the ATLAS experiment using data collected at $\\sqrt{s}$ = 13 TeV\n corresponding to an integrated luminosity of 139 fb$^{-1}$ both in the golden channel $H \\rightarrow \\ell^+ \\ell^- \\ell'^+ \\ell'^-$, \n $\\ell , \\ell' = e$ or $ \\mu$, and in the decays into WW or ZZ with production mechanisms and branching ratios that mimic the ones \n of a heavy Standard Model Higgs boson. We do not found a clear statistical evidence for our heavy Higgs boson. \n At beast we found a hint of a signal in the gluon-gluon fusion Higgs production mechanism.\n As a matter of fact, we reached the conclusion that there \n was not enough sensitivity to detect the signal in vector-boson fusion mechanism mainly due to tight event-selection cuts.\n In any case, we concluded that our theoretical proposal was still not ruled out by the ATLAS observations. \nHowever, we must admit that it is problematic the absence of experimental evidence in the decays of the heavy Higgs boson into\ntwo W vector bosons. It is known that the main decay mode of a heavy Higgs boson is the decay into WW. \nTherefore, the most stringent constraints should arise from the experimental searches for a heavy Higgs boson decaying into two W gauge bosons.\nFortunately, the CMS Collaboration recently reported a search for high-mass resonances decaying into a pair of W bosons into\n the fully leptonic final state using the full Run 2 data set corresponding to an integrated luminosity of \n 138 fb$^{-1}$~\\cite{CMS:2022}. The aim of the present note is to contrast our theoretical expectations to the LHC Run 2 data \n from the CMS Collaboration in the above specified decay channel. We will show that the CMS data display a broad excess\n that seems to compare favourably with our proposal. \\\\\nThis paper is organised as follows. In Sect.~\\ref{s-2}, for the reader's convenience, we briefly discuss our theoretical\nproposal for two Standard Model Higgs bosons together with the main production mechanisms and decay modes.\nSect.~\\ref{s-3} is devoted to the comparison with the latest available CMS data.\nFinally, in Sect.~\\ref{s-4} we summarise the main result of the present paper and draw our conclusions.\n\\section{Heavy and light Higgs bosons}\n\\label{s-2}\nIn the present Section we would like to illustrate very briefly the proposal advanced in Ref.~\\cite{Cea:2020} to look at the Higgs condensate \nas a quantum liquid analogous to the Bose-Einstein condensate in superfluid He II. We found that the low-lying excitations of the Higgs condensate \n resembled two Higgs bosons that were the relativistic version of the phonons and rotons in superfluid He II. \n Actually, in the dilute gas approximation that is \nthe relevant regime for the LHC physics, these low-lying excitations of the Higgs condensate resembled two Standard Model Higgs bosons\n with masses around 100 GeV and 750 GeV, respectively.\nThe lighter Higgs boson h was identified with the new LHC particle with mass $M_h \\simeq 125$ GeV that seemed to behave\n consistently with the Standard Model perturbative Higgs boson. On the other hand, the heavy Standard Model Higgs boson H, in accordance with\n the phenomenological analyses presented in Ref.~\\cite{Cea:2019}, was assumed to have mass $M_H \\simeq 730$ GeV.\nNote that this mass value is in accordance with previous extensive numerical studies~\\cite{Cea:2004,Cea:2012}.\nWe emphasise that we are not saying that there are two different elementary quantum Higgs fields.\nOn the contrary, we have a unique quantum Higgs field. However, since the scalar condensate\nbehaves like the He II quantum liquid, when the Higgs field acts on the condensate\nit can give rise to two elementary excitations, namely the phonon-like and roton-like excitations\ncorresponding to long-range collective and localised disturbances of the condensate, respectively.\nThese elementary condensate excitations behave as weakly interacting scalar\nfields with vastly different mass. Moreover, one remarkable aspect of our approach is that the Higgs boson\nmasses are not free parameters, but these can be estimated from first principles. \\\\\nOnce established that the perturbations of the scalar condensate due to the quantum Higgs field behave as two independent\nweakly interacting massive scalar fields, we need to investigate the experimental signatures and the interactions of these\nHiggs condensate elementary excitations. Obviously, the most striking consequence of\nour approach is the prevision of an additional heavy Higgs boson. As we already said, the light Higgs boson is the\nnatural candidate for the new LHC scalar resonance at 125 GeV. On the other hand, our previous phenomenological analysis of the\npreliminary LHC Run 2 data in the golden channel~\\cite{Cea:2019} corroborated the presence of a broad scalar resonance with\ncentral mass at 730 GeV. These two Higgs bosons will interact with the gauge vector bosons.\nWe already pointed out~\\cite{Cea:2020,Cea:2019} that the couplings of the Higgs condensate elementary excitations to the\ngauge vector bosons are fixed by the gauge symmetries. As a consequence, both the Higgs bosons h and H will be\ncoupled to gauge bosons as in the usual perturbative approximation of the Standard Model. Moreover, these scalar\nbosons have an effective finite self-coupling $\\lambda_{eff}$ that, in general, is smaller than the perturbative\n renormalised scalar self-coupling $\\lambda$.\nAs concern the coupling to fermion fields, if we admit the presence of the Yukawa terms in the Lagrangian, then we are led to an effective\nYukawa lagrangian:\n\\begin{equation}\n\\label{2.1}\n{\\cal L}_{Y}^{eff}(x) = \n\\sqrt{Z^h_{wf}} \\; \\frac{\\lambda_f}{\\sqrt{2}} \\; \\hat{\\bar{\\psi}}_f(x) \\, \\hat{\\psi}_f(x) \\; \\hat{h}(x) + \n\\sqrt{Z^H_{wf}} \\; \\frac{\\lambda_f}{\\sqrt{2}} \\; \\hat{\\bar{\\psi}}_f(x) \\, \\hat{\\psi}_f(x) \\; \\hat{H}(x) \\; , \n\\end{equation}\nwhere $\\hat{\\psi}_f(x)$ indicates a generic fermion quantum field and the Yukawa coupling satisfies the usual relation:\n\\begin{equation}\n\\label{2.2}\n\\lambda_f \\; = \\; \\frac{\\sqrt{2} \\, m_f }{v} \\; .\n\\end{equation}\nIn Eq.~(\\ref{2.1}) $Z^h_{wf}$ and $Z^H_{wf}$ are wavefunction renormalisation constant that, roughly, take care\nof the eventual mismatch in the overlap between the fermion and quasiparticle wavefunctions. \nA direct calculation of the wavefunction renormalisation constants is not easy. Nevertheless, in Ref.~\\cite{Cea:2020} \nwe fixed these constants from a comparison with the experimental observations. As a result we argued \nthat~\\footnote{The wavefunction renormalisation constant $Z^H_{wf}$ coincides with the phenomenological parameter $\\kappa$\nintroduced in Ref.~\\cite{Cea:2019}.}:\n\\begin{equation}\n\\label{2.3}\nZ^h_{wf} \\; \\simeq \\; 1 \\; \\; \\; , \\; \\; \\; Z^H_{wf} \\; \\simeq \\; \\frac{m_h}{m_H} \\; .\n\\end{equation}\nNote that Eq.~(\\ref{2.3}) has the remarkable consequence that our light Higgs boson h is practically indistinguishable from the perturbative Higgs boson. As a consequence, in the following we shall concentrate on the hypothetical heavy Higgs boson. \\\\\nIn our previous papers~\\cite{Cea:2020,Cea:2019} we argued that for large Higgs masses the main\nproduction processes are by vector-boson fusion and gluon-gluon fusion processes.\nTo evaluate the Higgs event production at LHC we need the inclusive Higgs production cross section. As in perturbation\ntheory, for large Higgs masses the main production processes are by vector-boson fusion (VBF) and gluon-gluon fusion (GGF). \nIn fact, since the couplings of the H boson to vector bosons are the same as those of a Standard Model Higgs boson, the H boson\n production cross section by vector-boson fusion is the same as in the perturbative Standard Model calculations.\nAs concern the gluon fusion production mechanism, it is known that the gluon coupling to the Higgs boson in the Standard Model is \nmediated mainly by triangular loops of top and bottom quarks. Indeed, in perturbation theory the Yukawa couplings \nof the Higgs particle to heavy quarks grows with quark mass, thus balancing the decrease of the triangle amplitude so that\nthe effective gluon coupling approaches a non-zero value for large loop-quark masses. This means that for\nheavy Higgs the gluon fusion inclusive cross section is almost completely determined by the top quark.\nTherefore, the total inclusive cross section for the production of the H Higgs boson\ncan be written as:\n %\n\\begin{equation}\n\\label{2.4}\n\\sigma(p \\; p \\; \\rightarrow \\; H) \\; \\simeq \\; \\sigma_{VBF}(p \\; p \\; \\rightarrow \\; H)\n\\; + \\; \\sigma_{GGF}(p \\; p \\; \\rightarrow \\; H) \\; ,\n\\end{equation}\nwhere $\\sigma_{VBF}$ and $\\sigma_{GGF}$ are the vector-boson fusion and gluon-gluon fusion inclusive cross\nsections, respectively. \nIn the Standard Model the calculations of the cross sections computed at next-to-next-to-leading \nand next-to-leading order for a high mass Higgs boson with Standard Model-like couplings\n at $\\sqrt{s} = $ 13 TeV are provided by the LHC Higgs Cross Section Working Group~\\cite{Hcross13Tev}.\nAs concern the Standard Model gluon-gluon fusion cross section we found~\\cite{Cea:2019} that this cross section can be\n usefully interpolated by:\n\\begin{equation}\n\\label{2.5}\n \\sigma_{GGF}^{SM}(p \\; p \\rightarrow H) \\; \\simeq \\; \n \\left\\{ \\begin{array}{ll}\n \\; \\left ( \\frac{ a_1}{ M_H} \n \\; + \\; a_2 \\; M_{H}^3 \\right ) \\; \\exp (- a_3 M_{H}) \\; \\; & M_{H} \\; \\leq \\; 300 \\; GeV \n \\\\\n \\; \\; \\; \\; a_4 \\; & 300 \\; GeV \\leq M_{H} \\leq 400 \\; GeV\n \\\\\n \\; \\; \\; \\;a_4 \\; \\exp \\big [ - a_5 ( M_{H} - 400 \\; GeV) \\big ] \\; \\; & 400 \\; GeV \\; \\leq \\; M_{H}\n\\end{array}\n \\right.\n\\end{equation}\nwhere $M_{H} $ is expressed in GeV and\n\\begin{eqnarray}\n\\label{2.6}\na_1 \\simeq 1.24 \\, 10^4 \\; pb \\, GeV \\; \\; , \\; \\; a_2 \\simeq 1.49 \\, 10^{-6} \\; pb \\, GeV^{-3} \\; , \\; \n\\nonumber \\\\\na_3 \\simeq 7.06 \\, 10^{-3} \\; GeV^{-1} \\; , \\; \\; a_4 \\simeq 9.80 \\;\\, pb \\; , \\hspace{2.2 cm}\n\\\\ \\nonumber\na_5 \\simeq 7.63 \\, 10^{-3} \\; GeV^{-1} \\; . \\hspace{5.05 cm}\n\\end{eqnarray}\nLikewise the Standard Model vector-boson fusion cross section can be parametrised as:\n\\begin{equation}\n\\label{2.7}\n \\sigma_{VBF}^{SM}(p \\; p \\rightarrow H) \\; \\simeq \\; \\bigg ( b_1 \\; + \\; \\frac{ b_2}{ M_{H}} \n \\; + \\; \\frac{b_3}{ M_{H}^2} \\bigg ) \\; \\exp (- b_4 \\; M_{H} ) \\; ,\n\\end{equation}\nwith:\n\\begin{eqnarray}\n\\label{2.8}\nb_1 \\simeq - 2.69 \\, 10^{-6} \\; pb \\; \\; , \\; \\; b_2 \\simeq 8.08 \\, 10^{2} \\; pb \\, GeV \\; , \\hspace{1.15 cm}\n \\nonumber \\\\\nb_3 \\simeq - 1.98 \\, 10^{4} \\; pb \\, GeV^{2} \\; \\; , \\; b_4 \\simeq 2.26 \\, 10^{-3} \\; GeV^{-1} \\; . \\; \\; \\,\n\\end{eqnarray}\n\\begin{figure}\n\\vspace{-0.5cm}\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth,clip]{Fig1.eps}\n\\end{center}\n\\caption{\\label{Fig1} \nInclusive H Higgs boson production cross sections for the VBF processes (blue continuous line) and\nGGF processes (red continuous line) as a function of $M_H$ at $\\sqrt{s} = $ 13 TeV.}\n\\end{figure}\nOur previous discussion lead us to assume that to a good approximation we can write:\n\\begin{equation}\n\\label{2.9}\n \\sigma_{VBF}(p p \\rightarrow H) \\simeq \\sigma_{VBF}^{SM}(p p \\rightarrow H) \\; \\; , \\; \\; \n \\sigma_{GGF}(p p \\rightarrow H) \\simeq Z^H_{wf} \\; \\sigma_{GGF}^{SM}(p p \\rightarrow H) \\; .\n\\end{equation}\nIn Fig.~\\ref{Fig1} we compare the VBF and GGF production cross sections given by Eq.~(\\ref{2.9}) after taking into\naccount the value of $Z^H_{wf}$ in Eq.~(\\ref{2.3}). From Fig.~\\ref{Fig1} we see that the main production mechanism\nof the heavy H Higgs boson is by the VBF processes since:\n\\begin{equation}\n\\label{2.10}\n \\sigma_{VBF}(p p \\rightarrow H) \\; \\simeq \\; 2 \\; \\sigma_{GGF}(p p \\rightarrow H) \\; \\; , \\; \\; M_H \\; \\simeq \\; 730 \\; GeV \\; .\n\\end{equation}\n %\nIn order to determine the phenomenological signatures of the massive H Higgs boson we need to examine the decay modes.\nGiven the rather large mass of the heavy Higgs boson, the main decay modes are the decays into two massive\nvector bosons (see, e.g., Refs.~\\cite{Gunion:1990,Djouadi:2008}):\n\\begin{equation}\n\\label{2.11}\n\\Gamma( H \\; \\rightarrow \\; W^+ \\, W^-) \\; \\simeq \\; \\frac{G_F \\, M^3_{H}}{8 \\pi \\sqrt{2}} \\;\n \\sqrt{1 - \\frac{4 m^2_W}{M^2_{H}}} \\; \\bigg ( 1 - 4 \\, \\frac{m^2_W}{M^2_{H}} + 12 \\, \\frac{ m^4_W}{M^4_{H}}\n \\bigg ) \\; \n\\end{equation}\nand\n\\begin{equation}\n\\label{2.12}\n\\Gamma( H \\; \\rightarrow \\; Z^0 \\, Z^0) \\; \\simeq \\; \\frac{G_F \\, M^3_{H}}{16 \\pi \\sqrt{2}} \\;\n \\sqrt{1 - \\frac{4 m^2_Z}{M^2_{H}}} \\; \\bigg ( 1 - 4 \\, \\frac{m^2_Z}{M^2_{H}} + 12 \\, \\frac{ m^4_Z}{M^4_{H}}\n \\bigg ) \\; . \n\\end{equation}\nThe couplings of the H Higgs boson to the fermions are given by the Yukawa couplings. For heavy Higgs the only relevant fermion coupling is \nthe top Yukawa coupling. The width for the decays of the H boson into a $t \\bar{t}$ pairs is easily found~\\cite{Gunion:1990,Djouadi:2008}:\n\\begin{equation}\n\\label{2.13}\n\\Gamma( H \\rightarrow \\; t \\, \\bar{t}) \\; \\simeq \\; Z^H_{wf} \\; \\frac{3 \\, G_F \\, M_{H} m^2_t}{4 \\pi \\, \\sqrt{2}} \\;\n\\bigg ( 1 - 4 \\, \\frac{m^2_t}{M^2_{H}} \\bigg )^{\\frac{3}{2}} \\; ,\n\\end{equation}\nwhere we have taken into account Eq.~(\\ref{2.1}). So that, to a good approximation, the heavy Higgs boson total width is given by:\n\\begin{equation}\n\\label{2.14}\n\\Gamma_{H} \\; \\simeq \\; \\Gamma( H \\rightarrow W^+ \\, W^-) \\; + \\; \\Gamma( H \\rightarrow Z^0 \\, Z^0) \\; + \\;\n \\Gamma( H \\rightarrow t \\, \\bar{t}) \\; .\n\\end{equation}\n\\section{Comparison with the CMS Run 2 dataset}\n\\label{s-3}\nThe discussion in the previous Section showed that our heavy Higgs boson is a rather broad resonance \nand that almost all the decay modes are given by the decays into W$^+$W$^-$ and Z$^0$Z$^0$ with:\n\\begin{equation}\n\\label{3.1}\nBr(H \\rightarrow W^+ W^-) \\; \\simeq \\; 2 \\; Br(H \\rightarrow Z^0 Z^0) \\; .\n\\end{equation}\nIn our previous papers we found some evidence of a broad scalar resonance that looks consistent \nwith our Standard Model heavy Higgs boson in the golden channel. Actually, the decay channels $H \\rightarrow ZZ \\rightarrow 4 \\ell$ \n have very low branching ratios, but the presence of leptons allows to efficiently reduce the huge background due mainly to diboson production.\nIn fact, the four-lepton channel, albeit rare, has the clearest and cleanest signature of all the\npossible Higgs boson decay modes due to the small background contamination. \nNevertheless, from one hand we did not find convincingly evidence of a heavy Standard Model Higgs boson in the\ncomparison with the full LHC Run 2 data sets released by the ATLAS Collaboration. On the other hand, according to\nEq.~(\\ref{3.1}), the main decay mode of a heavy Higgs boson is the decays into two\n W vector bosons. As a consequence, the most stringent constraints should arise from the experimental\n searches for a heavy Higgs boson decaying into two W gauge bosons. The lack, at least up to now, of experimental evidences \nin this decay channel looks problematic. In fact, if this situation should persist our theoretical\nproposal should not be in agreement with observations and, therefore, should be rejected. \nThe aim of the present Section is to contrast our proposal with the recent document Ref.~\\cite{CMS:2022} where\n the CMS Collaboration presented a search for a high mass resonance decaying into a pair \nof W bosons, using the full data set recorded by CMS during the LHC Run 2\ncorresponding to an integrated luminosity of 138 fb$^{-1}$.\n The search strategy for $H \\rightarrow W^+W^-$ was based on the final state in which\n both W bosons decay leptonically, resulting in a signature with two isolated, \n oppositely charged, high $p_T$ leptons (electrons or muons) and large missing \n transverse momentum, due to the undetected neutrinos. So that,\n the bulk of the signal comes from direct W decays to electrons or muons of opposite \n charge. However, even if not explicitly mentioned in Ref.~\\cite{CMS:2022}, the small contributions \n proceeding through an intermediate $\\tau$ lepton are implicitly included.\n Therefore, in Ref.~\\cite{CMS:2022} it is always included the $W$ boson decays into all three lepton types, \n so that the $\\ell$ symbol in $W \\rightarrow \\ell + \\nu$ comprises all three leptons\n $e, \\mu, \\tau$.\nTo increase the signal sensitivity event categorisation optimised for the gluon-gluon fusion and\nvector-boson fusion production mechanisms were used. To this end, it was introduced\na parameter $f_{VBF}$ corresponding to the fraction of the VBF production cross section \n with respect to the total cross section. In this way, $f_{VBF} = 0$ corresponds to GGF production signal,\n while $f_{VBF} = 1$ considers only the VBF production signal. For a heavy scalar resonance\n with Standard Model-like couplings $f_{VBF}$ was set to the expectation using\n the cross sections provided by the LHC Higgs Cross SectionWorking Group~\\cite{Hcross13Tev}.\n\\begin{figure}\n\\vspace{-0.5cm}\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth,clip]{Fig2.eps}\n\\end{center}\n\\caption{\\label{Fig2} \nProduct of the cross section $\\sigma(p p \\rightarrow H)$ with the branching ratio $Br(H \\rightarrow WW \\rightarrow 2 \\ell 2 \\nu)$\nfor a heavy Higgs boson with Standard Model $f_{VBF}$ versus the Higgs mass $M_H$ obtained by combining the Run 2 data sets.\nThe data have been adapted from Fig.~4, bottom right panel, in Ref.~\\cite{CMS:2022}. Full black circles correspond to the observed\nsignal, the red dashed line is the expected Standard Model background together with the 68 \\% CL limits (red dotted lines).\nThe full green line corresponds to the product of the cross section and branching ratio for our heavy Higgs boson with central mass\n$M_H \\simeq 730$ GeV. }\n\\end{figure}\nThe results are presented as upper limits on the product of the cross section with the relevant branching ratio on the production of a\n high mass scalar resonance. The 68 \\% and 95 \\% confidence level upper limits \n on $\\sigma(p p \\rightarrow H \\rightarrow WW \\rightarrow 2 \\ell 2 \\nu)$ \nare displayed in Fig.~4 of Ref.~\\cite{CMS:2022} for four different scenarios, $f_{VBF} = 0$, $f_{VBF} = 1$, floating $f_{VBF}$ and\nStandard Model $f_{VBF} $. Interestingly enough, the CMS Collaboration reported a small excess of data over\nthe Standard Model background expectations for heavy Higgs boson masses ranging in the interval 500 GeV - 1000 GeV.\nMoreover, the signal hypothesis with the highest local significance was found in the VBF production mechanism ( $f_{VBF} = 1$)\naround the Higgs mass $M_H \\simeq 650$ GeV. \\\\\nIn order to compare with our theoretical expectations, for definiteness,\n in Fig.~\\ref{Fig2} we report $\\sigma(p p \\rightarrow H \\rightarrow WW \\rightarrow 2 \\ell 2 \\nu)$ as a function of the Higgs mass $M_H$ \nin the case of a heavy scalar resonance with Standard Model-like couplings (Standard Model $f_{VBF}$). The data have been extracted\nfrom Fig.~4, bottom right panel, of Ref.~\\cite{CMS:2022}. \nLooking at Fig.~\\ref{Fig2} we see that the observed signals display a sizeable broad excess with respect to expected Standard Model signal in \nthe mass range 600 GeV - 800 GeV. Clearly, these excesses cannot be accounted for by a heavy scalar resonance with a narrow width.\nIn addition, for a heavy Standard Model Higgs boson the main production mechanism would be by gluon-fusion processes for \n$M_H \\lesssim 1000$ GeV, so that the resulting production cross section would led to a signal greater by at least a factor of two with respect\nto the observed signal (see red line in Fig.~4, bottom right panel, of Ref.~\\cite{CMS:2022}).\n On the other hand, in our theoretical proposal the heavy Standard Model Higgs boson has a rather large width. So that, the expected main\n signal extends on the mass range 600 GeV - 800 GeV with a broad peak around $M_H \\simeq 700$ GeV. Moreover, as we said in\n Sect.~\\ref{s-2}, the main production mechanism is by vector-boson fusion since the gluon-gluon fusion processes are strongly suppressed\n (see Fig.~\\ref{Fig1}). To be quantitative, using Eqs.~(\\ref{2.4}) and (\\ref{2.10}) we may easily evaluate the inclusive production cross\n section for our heavy Higgs boson. The result, displayed in Fig.~\\ref{Fig2}, seems to compare reasonable well to the observed signal\n in the relevant Higgs mass range 600 GeV $ \\lesssim M_H \\lesssim$ 800 GeV.\nIt should be emphasised that the rejection of the background-only hypothesis in a statistical sense will depend \n in general on the plausibility of the new signal hypothesis and the degree to which it can describe the data. \nIn this respect, the presence of a rather broad excess around $M_H \\simeq 700$ GeV is perfectly consistent\nwith the fact that our Standard Model heavy Higgs boson has a central mass at $M_H \\simeq 730$ GeV and\na huge width. Moreover, we have estimate that the cumulative effects of the excesses in the mass range \n600 GeV - 800 GeV reach a statistical significance of about eight standard deviations. However, when searching for a new resonance \nsomewhere in a possible mass range, the significance of observing a local excess of events must take into account \nthe probability of observing such an excess anywhere in the range. This is the so called \"look elsewhere effect\"~\\cite{Gross:2010}.\nEven taking into account the look elsewhere effect, the cumulative statistical significance is at level of five standard deviations.\nTo obtain more precise statements it should be necessary to implement our heavy Higgs boson in the Monte Carlo numerical\nsimulations. However, we are aware that the implementation of a heavy resonance with a large width is still problematic.\n\\section{Conclusions}\n\\label{s-4}\nIn our previous papers we pictured the Higgs condensate of the Standard Model as a quantum liquid analogous to\nthe superfluid He II. We found that the low-lying Higgs condensate excitations behave as two Standard Model Higgs bosons.\nThe light Higgs boson, identified with the LHC narrow resonance at 125 GeV, turned out to practically indistinguishable from \nthe perturbative Standard Model Higgs boson. As concern the heavy Higgs boson, we found some evidence in \nour previous phenomenological analysis in the golden channel of the preliminary LHC Run 2 data from ATLAS and CMS Collaborations. \nHowever, that evidence was not corroborated by the full Run 2 data sets recently released by the ATLAS Collaboration.\nMoreover, considering that the main decay mode of a heavy Higgs boson is the decay into two W vector bosons, the absence \n of experimental evidences of a heavy Higgs boson in this decay channel constituted a serious problem for our proposal.\nIn the present note we compared our theoretical proposal to the CMS full Run 2 data set dealing with the search for\n a high mass Higgs boson decaying into a pair of W bosons in the dileptonic channel. The main results of the present paper, \n showed in Fig.~\\ref{Fig2}, indicated that our prevision of an additional heavy Standard Model Higgs boson compared \n in a satisfying way to the experimental findings.\nThe agreement between our estimate of the inclusive production cross section and the observed signal in the Higgs mass range\n600 GeV $ \\lesssim M_H \\lesssim$ 800 GeV seems to us particularly significative. Indeed, once one fixes the Higgs masses,\n$M_h \\simeq 125$ GeV and $M_H \\simeq 730$ GeV, in our theory there are no more free parameters. Obviously, to further validate\n our theoretical proposal we must wait for further LHC Run 2 data, in particular the release by the CMS Collaboration of the full Run 2 dataset\n in the so-called golden channel.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nDiamond and silicon are wide-gapped semiconductors \/ insulators which exhibit indirect energy gaps of about 5.5\\,eV (diamond) and 1.1\\,eV (silicon). They are well-known for their outstanding physical properties and technical applications, e.\\,g.\\ the excellent heat conductivity of diamond, its withstanding of high electric fields, or the numerous applications of silicon in semiconductor technologies. It is well-known, too, that the physical properties of these and other semiconductors can be influenced by charge-carrier doping either by donor or acceptor atoms which changes their resistivity many orders of magnitude leading to intriguing properties. Small doping concentrations are widely used in the application of semiconductors. At higher doping levels the systems undergo a semiconductor-to-metal transition above a certain critical doping level, i.\\,e.\\ charge-carrier concentration, and further doping might even lead to superconductivity. From the theoretical and experimental point of view superconductivity in doped semiconductors is an outstanding issue. The prediction of superconductivity in Ge and GeSi and the suggestion that other semiconductor-based compounds may also exhibit superconductivity at very low temperatures were given by Cohen already in 1964.\\cite{cohen64a} Indeed, some examples have been reported so far, e.\\,g.\\ self-doped Ge$_x$Te,\\cite{hein64a} Sn$_x$Te,\\cite{hein69a} doped SrTiO$_3$,\\cite{schooley65a} or more recently doped silicon clathrates.\\cite{kawaji95a,grosche01a,connetable03a} The superconductivity of the doped silicon clathrates is the first example of compounds exhibiting superconductivity in a covalent tetrahedral $sp^3$ network with bond lengths similar to those in diamond.\n\nHowever, before 2004 (C:B)\\cite{ekimov04a} and 2006 (Si:B)\\cite{bustarret06a} superconductivity was never reported for diamond and cubic silicon in the diamond structure although there are several studies available concerning hole-doped induced metallicity in carbon and silicon using boron, nitrogen, or phosphorus, e.\\,g.\\ Refs.~\\onlinecite{dai91a}, \\onlinecite{bustarret03a} and the references therein. Therefore, it was an important progress to find superconductivity in these compounds upon boron doping, which attracted a lot of interest and stimulated many theoretical and experimental studies in the last four years.\\cite{baskaran04a,blase04a,boeri04a,bustarret04a,kwlee04a,takano04a,xiang04a,umezawa05a,yokoya05a,sacepe06a,wu06a,bourgeois07a,shirakawa07a,takano07a} Boron has one partially filled electron less than carbon or silicon and hence acts as an acceptor leading to hole doping. Both compounds are type-II superconductors with \\ensuremath{T_{\\rm c}}\\ values of 11.4\\,K (C:B) and 0.35\\,K (Si:B). The upper critical fields are $\\ensuremath{H_{\\rm c2}}\\approx 8.7$\\,T and 0.4\\,T, respectively.\\cite{umezawa05a,bustarret06a} \n\nIn order to explain the superconductivity in diamond theoretical studies point towards two different scenarios: (i) it is the result of a simple electron-phonon interaction\\cite{blase04a,boeri04a,kwlee04a,xiang04a} and (ii) it is caused by a resonating valence-bond mechanism.\\cite{baskaran03a,baskaran06a,baskaran04a} The former model is based on a conventional electron-phonon mechanism where the charge carriers are introduced into intrinsic diamond bands leading to a three-dimensional analog of the two-dimensional superconductor MgB$_2$. The superconductivity is attributed to holes located at the top of the zone-centered $\\sigma$-bonding valence bands which couple strongly to optical bond-stretching phonon modes.\\cite{boeri04a,kwlee04a} The latter model attributes the superconductivity to holes in the impurity bands rather than in the intrinsic diamond bands.\\cite{baskaran04a} With the premise that the doping level in superconducting diamond is close to the Mott limit the randomly distributed boron atoms, i.\\,e.\\ their random Coulomb potential, may lift the degeneracy of the boron acceptor states leading to a narrow half-filled band from which superconductivity develops. However, spectroscopical studies seem to support the former explanation and rule out the latter suggestion,\\cite{kacmarcik05a,sacepe06b} although a complete understanding of the superconducting phase is not yet obtained.\\cite{nakamura04b,baskaran06a,wu06a} Recently, a theoretical study suggested the possibility to achieve superconducting transition temperatures on the order of 100\\,K in C:B due to the exceptionally high Debye temperature of diamond and under the precondition that the doped boron atoms are ordered.\\cite{shirakawa07a} \n\\begin{figure}\n\\centering\n\\includegraphics[width=8.5cm,clip]{fig1.pdf}\n\\caption[]{(color online) (a) Unit cell of diamond-related cubic 3C-SiC. The three bilayers consisting of C and Si layers are emphasized. The stacking sequence is ABC\\,--\\,\\dots The green arrow denotes the $\\left<111\\right>$ direction whereas the gray rods refer to the tetrahedral bond alignment of diamond. The cube defines the conventional unit cell of 3C-SiC, which consists of four formula units SiC. (b) Four unit cells of hexagonal 6H-SiC. The cuboid defines one unit cell. The six bilayers of the stacking sequence ABCACB\\,--\\,\\dots are emphasized. The green arrow denotes the $\\left<001\\right>$ direction. One unit cell of 6H-SiC consists of six formula units SiC. For the drawings the software \\textit{Vesta} was used.\\cite{Vesta}} \\label{structureSiC}\n\\end{figure}\n\nIn Ref.~\\onlinecite{ren07a}, we reported the discovery of superconductivity in a closely related system originating from a well-known and widely used semiconductor, namely boron-doped SiC, the stoichiometric ''mixture'' of the two afore discussed ''parent'' materials. SiC is used increasingly for high-temperature, high-power, and high-frequency applications due to its high thermal conductivity, the existence of large band gaps, strong covalent bondings, chemical inertness, or its high tolerance to radiation and heat. Another hallmark of this system is the huge number (about\\cite{casady96a} 200) of crystal modifications with cubic (''C''), hexagonal (''H''), or rombohedral (''R'') symmetry of the unit cell.\\cite{SiCinfo,casady96a} They are usually referred to as $m$C-SiC, $m$H-SiC, and $m$R-SiC, respectively.\n\\begin{table}[t]\n\\centering\n\\begin{ruledtabular}\n\\caption{Basic parameters of 3C-SiC and 6H-SiC at room temperature.\\cite{landolt} The parameter $V_0$ denotes the volume of the conventional unit cell, $\\ensuremath{V_{\\rm mol}}=V_{0}\\cdot \\ensuremath{N_{\\rm A}}\/t$ the molar volume where $t$ is the number of formula units SiC in the unit cell (''f.\\,u.\\ \/ unit cell''). In our analysis we will use the average of $\\ensuremath{V_{\\rm mol}}^{\\rm 3C-SiC}$ and $\\ensuremath{V_{\\rm mol}}^{\\rm 6H-SiC}$ because the sample used contains both polytypes; see text.}\n\\begin{tabular}{lcc}\n\\toprule\n & 3C-SiC & 6H-SiC \\\\ \n & $\\beta$-SiC & $\\alpha$-SiC \\\\ \n\\addlinespace[0.25em]\\hline \\addlinespace[0.75em]\n\\multirow{2}{*}{symmetry} & cubic & hexagonal \\\\ \n & zincblende & moissanite-6H\\\\ \n\\addlinespace[0.25em]\nspace group & F$\\bar{4}$3m (T$_{\\rm d}^2$) & P6$_3$mc (C$^{4}_{6v}$)\\\\ \n\\addlinespace[0.25em]\n\\multirow{2}{*}{bilayer stacking} & ABC\\,--\\,\\dots & ABCACB\\,--\\dots\\\\ \n & along $\\left<111\\right>$ & along $\\left<001\\right>$\\\\ \n\\addlinespace[0.25em]\n f.\\,u.\\ \/ unit cell & $t=4$ & $t=6$\\\\ \n \\addlinespace[0.25em]\n\\multirow{2}{*}{lattice constants (\\AA)} & \\multirow{2}{*}{$a_{\\rm cub}= 4.3596$} & $a_{\\rm hex}= 3.0806$\\\\ \n & & $c_{\\rm hex}= 15.1173$\\\\ \n\\addlinespace[0.25em]\n$V_0$ (\\AA$^3$) & 82.859 & 124.244 \\\\ \n\\addlinespace[0.25em]\n\\ensuremath{V_{\\rm mol}}\\,$\\left(\\frac{\\rm cm^3}{\\rm mol}\\right)$ & 12.475 & 12.470\\\\\n\\addlinespace[0.25em]\nenergy gap (eV) & 2.2 & 3.02 \\\\ \n\\addlinespace[0.25em]\n\\ensuremath{{\\it \\Theta}_{\\rm D}}\\ (K) & 1270 & 1200\\\\ \n\\addlinespace[0.25em]\n\\bottomrule\n\\end{tabular}\n\\label{SiCbasic}\n\\end{ruledtabular}\n\\end{table}\nThe variable $m$ gives the number of Si\\,--\\,C bilayers consisting of a C and a Si layer stacking in the unit cell. However, most of the available studies refer to the following polytypes:\\cite{SiCinfo} cubic 3C- (zincblende structure, space group F$\\bar{4}$3m (T$_{\\rm d}^2$); ''ordered'' diamond) and hexagonal 2H-, 4H-, and 6H-SiC (wurtzite, moissanite-4H, and -6H structure, all space group P6$_3$mc (C$^{4}_{6v}$)). The 3C- (2H-) polytype is the only ''pure'' cubic (hexagonal) modification, all other mH-SiC polytypes consist of hexagonal and cubic bonds.\\cite{pensl93a} The cubic 3C-modification is also labeled as $\\beta$-SiC, whereas the hexagonal polytypes are generally denoted as $\\alpha$-SiC. Fig.~\\ref{structureSiC} gives a sketch of (a) the diamond-related modification 3C-SiC and (b) the hexagonal 6H-SiC. The C\\,--\\,Si bilayers are emphasized. In 3C-SiC both elements form face-centered cubic sublattices which are shifted by (1\/4, 1\/4, 1\/4) with respect to each other. Along the $\\left<111\\right>$ direction the bilayer stacking in 3C-SiC is ABC\\,--\\dots For the polytypes 2H-, 4H-, and 6H-SiC it is along the $\\left<001\\right>$ direction ABAB\\,--\\dots, ABAC\\,--\\dots, and ABCACB\\,--\\dots\\ Some basic parameters of undoped 3C- and 6H-SiC at room temperature are summarized in Table\\,\\ref{SiCbasic}.\n\nDepending on the crystal modification, pure SiC exhibits an indirect energy gap between $\\sim 2$\\,eV (3C-SiC) and $\\sim 3.3$\\,eV (2H-SiC).\\cite{SiCinfo} Slightly doped SiC with donors and acceptors was intensely studied for nitrogen, phosphorus, boron, aluminum, etc.\\ by ion-implantation or thermo-diffusion doping. Compared with other dopants, boron was found to have a much faster diffusion rate in SiC. Diffusion processes mediated by the silicon interstitials and by carbon vacancies have been proposed to explain such fast diffusion rates.\\cite{bracht00a,rurali02a,gao03a,gao04a} Under silicon-rich conditions the carbon-site substitution is dominating.\\cite{bockstedte04a} Among other dopants, the insulator-to-metal transition was observed recently in nitrogen-doped 4H-SiC at carrier concentrations above $10^{19}$\\,cm$^{-3}$.\\cite{dasilva06a}\n\nIn this paper we report a specific-heat study on SiC:B and give a detailed analysis. Moreover, the density-of-states, band dispersions, and two- and three-dimensional plots of the Fermi surfaces for 3C-SiC are presented. We estimate the basic superconducting parameters and compare our findings with the reported results for C:B and Si:B. Finally, we comment on possible origins of the differences between the three superconducting systems.\n\n\\section{Experiment}\nThe preparation and characterization of our samples is described in detail in Ref.~\\onlinecite{ren07a}. We studied several samples from different growth processes reproducing the general findings presented in this paper. The particular sample used in this study is identical to that used to map out the $H$\\,--\\,$T$\\ phase diagram in our previous study, namely ''sample 1'', referred to as ''SiC-1'' in this paper. The hole-doping charge-carrier concentration of SiC-1 was estimated to be $n=1.91\\cdot 10^{21}$\\,cm$^{-3}$ by a Hall-effect measurement.\\cite{nVmolcomment} We note, that all of our so-far prepared samples are polycrystalline materials and not single phase. We found phase fractions of 3C-SiC, 6H-SiC, and Si. In spite of this result the residual resistivity for specimen SiC-1 is already as low as 60\\,\\ensuremath{\\muup\\Omega{\\rm cm}}. The residual-resistivity ratio ${\\rm RRR}=\\rho_{\\rm 300\\,K}\/\\rho_{\\rm 1.5\\,K}$ amounts to 10. SiC-1 undergoes a sharp superconducting transition around 1.45\\,K. The thermodynamic critical field is estimated to be $\\sim 115$\\,Oe. In contrast to the type-II superconductivity in C:B and Si:B the nature of the superconductivity in SiC:B is type-I: We find a clear hysteresis in the temperature (field) dependence of the AC susceptibility between cooling (field-down sweep) and subsequent warming (field-up sweep) runs indicating the hallmark of type-I superconductivity as discussed in Ref.\\,\\onlinecite{ren07a}. \n\nSpecific-heat data was taken by a relaxation-time method using a commercial system (Quantum Design, PPMS). First, we applied a degaussing procedure before the measurement in order to reduce the remanent field of the magnet. Second, the addendum heat capacity was measured at 0\\,Oe. Next, specific-heat data was taken in $H=0$\\,Oe and subsequently in 200\\,Oe, for which the addenda data was not measured, because the difference is expected to be negligibly small. However, this procedure lead to a small but visible artifact in the in-field normal-state data for $0.45\\,{\\rm K}\\leq T \\leq 0.6$\\,K. Thus, the corresponding data points were removed and not used for the analysis.\n\n\\section{Results and Analysis}\n\\subsection{Specific heat}\n\\begin{figure}\n\\centering\n\\includegraphics[width=8.5cm,clip]{fig2.pdf}\n\\caption[]{(color online) Specific heat of SiC-1: The (red) {$\\medbullet$} symbols denote the data in zero magnetic field. The (blue) $\\blacktriangle$ refer to data measured in a magnetic field $H=200\\,{\\rm Oe}>\\ensuremath{H_{\\rm c}}$, representing the normal-state specific heat. (a) Specific heat \\ensuremath{c_{p}}\\ as measured: The line is a fit to the in-field data using the standard Debye formula (Eq.~\\ref{GlDebye}). The inset shows the specific heat up to 360\\,K. (b) Electronic specific heat $\\ensuremath{c_{\\rm el}}\/T$: The lines are an entropy-conserving construction in order to estimate the intrinsic jump height; see text.} \\label{cp_SiC-1}\n\\end{figure}\nFig.~\\ref{cp_SiC-1}\\,(a) summarizes the temperature dependence of the specific heat \\ensuremath{c_{p}}\\ for specimen SiC-1 in the superconducting state ($H=0$\\,Oe) and in the normal-conducting state (achieved by applying a magnetic field $H=200\\,{\\rm Oe}> \\ensuremath{H_{\\rm c}}$). The raw data of this figure is the same as that used for Fig.~4 in Ref.~\\onlinecite{ren07a} except the removed data points which were affected by an experimental artifact. SiC:B is a bulk superconductor as indicated by the clear jump of \\ensuremath{c_{p}}\\ at \\ensuremath{T_{\\rm c}}. The inset of Fig.~\\ref{cp_SiC-1}\\,(a) shows the specific heat up to $\\sim 360$\\,K for comparison. The room temperature value is still clearly below the classical high-$T$ Dulong-Petit limit (49.88\\,J\/molK).\n\nThe solid curve in Fig.~\\ref{cp_SiC-1}\\,(a) is a fit to the in-field data for $0.6\\,{\\rm K} < T < 2$\\,K applying the conventional Debye formula\n\\begin{equation}\\label{GlDebye}\n\\ensuremath{c_{p}}=\\ensuremath{c_{\\rm ph}}+\\ensuremath{c_{\\rm el}} = \\ensuremath{\\gamma_{\\rm n}} T+\\beta T^3\n\\end{equation}\nwith the Sommerfeld coefficient of the normal-state specific heat \\ensuremath{\\gamma_{\\rm n}}\\ and the coefficient of the phononic contribution $\\beta$ as adjustable parameters. The fit yields a very good description of the data below 2\\,K. The obtained values are $\\ensuremath{\\gamma_{\\rm n}}=0.29$\\,mJ\/molK$^2$ and $\\beta = 0.02$\\,mJ\/molK$^4$. From the latter value we determined the Debye temperature \\ensuremath{{\\it \\Theta}_{\\rm D}}\\ using $\\beta = (12\/5)\\,\\pi^4N \\ensuremath{N_{\\rm A}}\\ensuremath{k_{\\rm B}}\/\\ensuremath{{\\it \\Theta}_{\\rm D}}^3$ with the number of atoms per formula unit $N=2$, the Avogadro number \\ensuremath{N_{\\rm A}}, and Boltzmann's constant \\ensuremath{k_{\\rm B}}, yielding $\\ensuremath{{\\it \\Theta}_{\\rm D}}=590$\\,K. This is surprisingly low, only half of the value reported for undoped SiC (cf.\\,Table\\,\\ref{SiCbasic}): $\\ensuremath{{\\it \\Theta}_{\\rm D}}^{\\rm SiC}\\approx 1200\\,{\\rm K}-1300$\\,K. We note that in the case of C:B a significant reduction of the Debye temperature to about 75\\,\\% of the pure diamond value ($\\ensuremath{{\\it \\Theta}_{\\rm D}}^{\\rm C}\\approx 1860$\\,K, $\\ensuremath{{\\it \\Theta}_{\\rm D}}^{\\rm Si}\\approx 625$\\,K)\\cite{ashcroft76} is reported, too.\\cite{sidorov05a} In SiC:B this effect turns out to be even more pronounced.\n\nTo further analyze the data we plot the electronic specific heat $\\ensuremath{c_{\\rm el}}=\\ensuremath{c_{p}}-\\ensuremath{c_{\\rm ph}}$ in Fig.~\\ref{cp_SiC-1}\\,(b). The specific-heat jump starts slightly below $\\ensuremath{T_{\\rm c}}\\approx 1.5$\\,K coinciding with the results obtained by our AC susceptibility and resistivity measurements. However, the superconducting transition in \\ensuremath{c_{p}}\\ is rather broad. The lines in Fig.~\\ref{cp_SiC-1}\\,(b) are an entropy-conserving construction in order to estimate the intrinsic jump hight \\ensuremath{\\Delta c_{\\rm el}}. The ''jump'' temperature $\\ensuremath{T_{\\rm c}}^*= 1.31$\\,K indicated by the perpendicular (green) line in Fig.~\\ref{cp_SiC-1}\\,(b) is lower than the onset temperature \\ensuremath{T_{\\rm c}}\\ reflecting the broadness of the transition. The jump height is estimated to $\\ensuremath{\\Delta c_{\\rm el}}\/\\ensuremath{\\gamma_{\\rm n}}\\ensuremath{T_{\\rm c}}^* \\approx 1$. The obtained value is only two thirds of the weak-coupling BCS expectation, namely 1.43. For C:B and other semicondcutor-based superconductors, e.\\,g.\\ Ge$_{0.95}$Te, an even smaller jump height as low as 0.5 is reported.\\cite{sidorov05a,finegold64a} However, the overall shape of the C:B specific-heat data given in Fig.~3 of Ref.~\\onlinecite{sidorov05a} is qualitatively different compared to our data. The authors report a very broad transition consisting of two well-separated transitions.\n\nNext we focus on the question of the superconducting gap symmetry. We try the following two models to describe our experimental data: \n\nModel (i) \\textit{Assuming an isotropic gap structure:}\n\nThe simplest approach to obtain information about the superconducting gap is given by the conventional BCS text-book formula\\cite{tinkham96} $\\ensuremath{c_{\\rm el}}(T)\/T\\propto \\exp(-\\Delta(0)\/T)\/T$. However, paying respect to the facts that on one hand the exponential behavior is only expected well below \\ensuremath{T_{\\rm c}}\\ and on the other hand data below approximately 0.45\\,K is lacking leads to the idea to replace the exponential term by tabulated numerical specific-heat data calculated in the standard weak-coupling BCS framework,\\cite{muehlschlegel59a} which is in principal valid up to \\ensuremath{T_{\\rm c}}. Therefore, we fitted the tabulated data with a polynomial $\\ensuremath{c_{\\rm el}}^{\\rm BCS}$ (15$^{\\rm th}$ order) leading to\n\\begin{equation}\\label{GlBCS_neu}\n\\ensuremath{c_{\\rm el}}(T)\/T \\propto \\ensuremath{c_{\\rm el}}^{\\rm BCS}(T)\/T.\n\\end{equation}\nNext, considering that the samples are not single phase, it is reasonable to assume an additional $T$-linear term $\\ensuremath{\\gamma_{\\rm res}} T$ reflecting a residual density of states originating from non-superconducting metallic inclusions. This modifies Eq.\\,\\ref{GlBCS_neu} as follows:\n\\begin{equation}\\label{GlBCS_res}\n\\ensuremath{c_{\\rm el}}(T)\/T = \\ensuremath{\\gamma_{\\rm res}} + \\ensuremath{\\gamma_{\\rm s}}\\cdot \\ensuremath{c_{\\rm el}}^{\\rm BCS}(T)\/T.\n\\end{equation}\nSince the entropy related to a residual term $\\ensuremath{\\gamma_{\\rm res}}\\ensuremath{T_{\\rm c}}^*$ does not contribute to the specific-heat jump, the prefactor of the BCS term is given by $\\ensuremath{\\gamma_{\\rm s}}=\\ensuremath{\\gamma_{\\rm n}}-\\ensuremath{\\gamma_{\\rm res}}$. Therefore \\ensuremath{\\gamma_{\\rm res}}\\ is the only adjustable parameter in this approach.\n\nModel (ii) \\textit{Assuming a power-law behavior of the electronic specific heat:}\n\nAt temperatures well below \\ensuremath{T_{\\rm c}}, a superconducting gap structure with nodes leads to the power-law behavior\n\\begin{equation}\\label{Glpower}\n\\ensuremath{c_{\\rm el}}(T)\/T = \\ensuremath{\\gamma_{\\rm res}} + a\\cdot T^b\n\\end{equation}\nwith $b=1$ or 2 for line or point nodes.\\cite{volovik85a,tinkham96} Here, we pay respect to a residual contribution, too.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=7.5cm,clip]{fig3.pdf}\n\\caption[]{(color online) Electronic specific heat of SiC-1: The (red) {$\\medbullet$} \/ blue $\\blacktriangle$ symbols denote again zero-field \/ in-field data. The lines are fits to the data assuming an isotropic gap (dotted line) and a nodal gap (dahsed line). The dashed-dotted horizontal line denotes the value of the Sommerfeld parameter in the normal-conducting state \\ensuremath{\\gamma_{\\rm n}}. The two other $\\gamma$ values are estimates related to model (i) and denote the superconducting \\ensuremath{\\gamma_{\\rm s}}\\ and a possible residual Sommerfeld parameter \\ensuremath{\\gamma_{\\rm res}}; see text.} \\label{cel_SiC-1_sc_fits}\n\\end{figure}\nThe results obtained by applying both models are summarized in Fig.~\\ref{cel_SiC-1_sc_fits}. Applying Eq.\\,(\\ref{GlBCS_res}) to the data yields the green dashed curve, Eq.\\,(\\ref{Glpower}) the black dotted line. The estimated $\\gamma$ factors for model (i) are given, too: \\ensuremath{\\gamma_{\\rm n}}\\ (normal-conducting state), \\ensuremath{\\gamma_{\\rm s}}\\ (superconducting state), \\ensuremath{\\gamma_{\\rm res}}\\ (residual contribution).\n\n\\textbf{Model (i)}\nApplying Eq.\\,(\\ref{GlBCS_res}) to the data for $T<0.7$\\,K gives a reasonable description as seen in Fig.~\\ref{cel_SiC-1_sc_fits} (green dashed line). The residual $\\gamma$ coefficient amounts to $\\ensuremath{\\gamma_{\\rm res}}=0.14$\\,mJ\/molK$^2$. Above 0.7\\,K the fit undershoots the data corresponding to a distribution of \\ensuremath{T_{\\rm c}}\\ values reflected in the broadness of the transition. An entropy-conserving construction using the fit result instead of the linear approximation shown in Fig.~\\ref{cp_SiC-1}\\,(b) yields a slightly higher ''jump'' temperature $\\ensuremath{T_{\\rm c}}^*=1.35$\\,K due to the downwards curvature around \\ensuremath{T_{\\rm c}}. Paying respect to the residual contribution $\\ensuremath{\\gamma_{\\rm res}}\\ensuremath{T_{\\rm c}}^*$ and evaluating the jump height with the Sommerfeld parameter of the superconducting part of the sample, $\\ensuremath{\\gamma_{\\rm s}}=0.16$\\,mJ\/molK$^2$, gives almost the value predicted by the BCS theory: $\\ensuremath{\\Delta c_{\\rm el}}\/\\ensuremath{\\gamma_{\\rm s}}\\ensuremath{T_{\\rm c}}^*=1.48$.\n\n\\textbf{Model (ii)} \nThe assumption of a power-law behavior in order to describe the data yields an even better description: Assuming a linear temperature dependence of $\\ensuremath{c_{\\rm el}}\/T$ reproduces the experimental data in the whole temperature range below $\\approx 1.1$\\,K, i.\\,e.\\ below the transition down to 0.45\\,K and extrapolates further down to 0\\,K without any indication of a residual contribution $\\ensuremath{\\gamma_{\\rm res}}\\ensuremath{T_{\\rm c}}^*$ in contrast to the results obtained by applying model (i) to the data. The resulting fitting curve is shown in Fig.~\\ref{cel_SiC-1_sc_fits} (dotted black line). It was obtained by adjusting only the prefactor $a$ and fixing $b=1$ and $\\ensuremath{\\gamma_{\\rm res}}=0$ in Eq.\\,(\\ref{Glpower}). The attempt to include a residual term to the fit gave $\\ensuremath{\\gamma_{\\rm res}}\\approx 0$. A $T$-linear behavior of $\\ensuremath{c_{\\rm el}}\/T$ is expected in the case of a gap containing line nodes, but only well below \\ensuremath{T_{\\rm c}}\\ where the superconducting gap is nearly independent of temperature. It is expected that at higher temperatures $T\\rightarrow \\ensuremath{T_{\\rm c}}$ the specific heat is affected by the reduction of the gap magnitude and therefore deviates from the linear extrapolation.\\cite{hasselbach93a} A complex balance of different effects is needed to cause an apparent linear temperature dependence up to \\ensuremath{T_{\\rm c}}. We note, that a $T$-linear behavior of $\\ensuremath{c_{\\rm el}}\/T$ up to \\ensuremath{T_{\\rm c}}\\ has been reported for e.\\,g.\\ the heavy-fermion compounds URu$_2$Si$_2$ and UPt$_3$.\\cite{hasselbach93a,fisher89a}\n\nThe obtained jump height using a linear entropy-conserving construction, $\\ensuremath{\\Delta c_{\\rm el}}\/\\ensuremath{\\gamma_{\\rm n}}\\ensuremath{T_{\\rm c}}^* \\approx 1$ (Fig.~\\ref{cp_SiC-1}\\,(b)), is similar to the value predicted theoretically for a superconductor with a nodal gap structure.\\cite{hasselbach93a,nishizaki99a} We note that in such a model $\\ensuremath{c_{\\rm el}}\/T$ should exhibit a rounded maximum at \\ensuremath{T_{\\rm c}}\\ rather than the triangle-like peak used in the simplified entropy-conserving construction shown in Fig.~\\ref{cp_SiC-1}\\,(b). Therefore the jump height would be even smaller than 1 and $\\ensuremath{T_{\\rm c}}^*$ slightly higher.\n\nHowever, in spite of the satisfying description of the data following model (ii) it is still necessary to obtain data down to several 10\\,mK to clarify the true nature of the superconducting gap. It would not be surprising if the specific heat of the multi-phase sample used consists of an additional $T$-linear term due to a residual \\ensuremath{\\gamma_{\\rm res}}\\ as suggested by the result of model (i). Therefore the apparent power-law behavior of the experimental electronic specific heat for $0.45\\,{\\rm K} < T < 1.1$\\,K extrapolating to 0 for $T\\rightarrow 0$\\,K is rather striking.\n\n\\subsection{Superconducting Parameters}\nTogether with the resistivity, Hall-effect, and AC susceptibility data published in Ref.~\\onlinecite{ren07a} we are able to estimate the basic superconducting parameters. They are summarized in Table~\\ref{SiCprop} along with the derived normal-state parameters. For comparison the so-far known corresponding parameters for C:B and Si:B are listed, too.\n\\begin{table}[t]\n\\centering\n\\begin{ruledtabular}\n\\caption{Normal-state and superconducting properties of SiC:B compared to those reported for C:B (Refs.~\\onlinecite{ekimov04a} and \\onlinecite{sidorov05a}) and Si:B (Ref.~\\onlinecite{bustarret06a}). Note that the highest \\ensuremath{T_{\\rm c}}\\ (\\ensuremath{H_{\\rm c2}}) for C:B reported so far is 11.4\\,K ($8.7\\cdot 10^4$\\,Oe).\\cite{umezawa05a} The asterisked ''*$\\dots$*'' values are preliminary because they depend on the value of $\\rho_0$ which we believe is still not the intrinsic one; see text. The coherence length $\\xi$ in the case of the type-II superconductor C:B was estimated using Eq.\\,(\\ref{cohlength1}) (i.\\,e.\\ from $n$, \\ensuremath{\\gamma_{\\rm n}}, and \\ensuremath{T_{\\rm c}}) for better comparison with the type-I superconductor SiC:B. The numbers given in parantheses are the published values from Refs.~\\onlinecite{bustarret06a} (Si:B) and \\onlinecite{sidorov05a} (C:B) calculated with Eq.\\,(\\ref{cohlength2}) (i.\\,e.\\ from \\ensuremath{H_{\\rm c2}}).}\n\\label{SiCprop}\n\\begin{tabular}{lccc}\n\\toprule\n & SiC:B & C:B & Si:B \\\\ \\hline \\addlinespace[0.75em]\n$n$ (cm$^{-3}$) & $1.91\\cdot 10^{21}$ & $1.80\\cdot 10^{21}$ & $2.80\\cdot 10^{21}$ \\\\ \\addlinespace[0.1em]\n$\\ensuremath{\\gamma_{\\rm n}}$ (mJ\/molK$^2$) & 0.294 & 0.113 & -- \\\\ \\addlinespace[0.1em]\n$\\beta$ (mJ\/molK$^4$) & 0.0193 & 0.0007 & -- \\\\ \\addlinespace[0.1em]\n\\ensuremath{{\\it \\Theta}_{\\rm D}}\\ (K) & 590 & 1440 & -- \\\\ \\addlinespace[0.1em]\n$\\ensuremath{\\Delta c_{\\rm el}}\/\\ensuremath{\\gamma_{\\rm n}}\\ensuremath{T_{\\rm c}}^*$ & 1 & 0.50 & -- \\\\ \\addlinespace[0.1em]\n$\\rho_0$ (\\ensuremath{\\muup\\Omega{\\rm cm}}) & *60* & 2500 & 130 \\\\ \\addlinespace[0.1em]\nRRR & *10.0* & 0.9 & 1.2 \\\\ \\addlinespace[0.1em]\n$\\ensuremath{T_{\\rm c}}(0)$ (K) (onset) & 1.45 & 4.50 & 0.35 \\\\ \\addlinespace[0.1em]\n$\\ensuremath{H_{\\rm c}}(0)$ (Oe) & 115 & -- & -- \\\\ \\addlinespace[0.1em]\n$\\ensuremath{H_{\\rm sc}}(0)$ (Oe) & 80 & -- & -- \\\\ \\addlinespace[0.1em]\n$\\ensuremath{H_{\\rm c2}}(0)$ (Oe) & type-I & $4.2\\cdot10^4$ & 4000 \\\\ \\addlinespace[0.1em]\n\\ensuremath{k_{\\rm F}}\\ (nm$^{-1}$) & 3.8 & 3.8 & -- \\\\ \\addlinespace[0.1em]\n$m^*$ (\\ensuremath{m_{\\rm el}}) & 1.2 & 1.7 & -- \\\\ \\addlinespace[0.1em]\n$\\tau(0)$ (fs) & *37* & -- & -- \\\\ \\addlinespace[0.1em]\n\\ensuremath{v_{\\rm F}}\\ (m\/s) & $3.8\\cdot 10^5$ & -- & -- \\\\ \\addlinespace[0.1em]\n$\\ell$ (nm) & *14* & 0.34 & -- \\\\ \\addlinespace[0.1em]\n$\\xi(0)$ (nm) & 360 & 80 (9) & (20) \\\\ \\addlinespace[0.1em]\n$\\lambda(0)$ (nm) & 130 & 160 & -- \\\\ \\addlinespace[0.1em]\n\\ensuremath{\\kappa_{\\rm GL}}(0) & 0.35 & 2 (18) & -- \\\\ \\addlinespace[0.1em]\n\\bottomrule\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\nFrom the charge-carrier concentration\\cite{nVmolcomment} ($n=1.91\\cdot 10^{21}$\\,cm$^{-3}$) assuming a single spherical Fermi surface we obtain the Fermi-wave number $\\ensuremath{k_{\\rm F}} = (3\\pi^ 2n)^{1\/3}=3.8$\\,nm$^{-1}$. The effective mass is evaluated as $m^*=(3\\hbar^2\\ensuremath{\\gamma_{\\rm n}})\/(\\ensuremath{V_{\\rm mol}} \\ensuremath{k_{\\rm B}}^2 \\ensuremath{k_{\\rm F}})=1.2\\ensuremath{m_{\\rm el}}$ with the bare-electron mass \\ensuremath{m_{\\rm el}}\\ and the molar volume\\cite{nVmolcomment} \\ensuremath{V_{\\rm mol}}. The Fermi velocity $\\ensuremath{v_{\\rm F}}=\\hbar\\ensuremath{k_{\\rm F}}\/m^*$ amounts to about 0.1\\,\\% of the speed of light. The mean-free-path is estimated to be $\\ell = \\hbar\\ensuremath{k_{\\rm F}}\/(\\rho_0 n e^2)=14$\\,nm with the elementary charge $e$. The superconducting penetration depth amounts to $\\lambda(0)= \\sqrt{m^*\/(\\mu_0 n e^2)}=130$\\,nm. The coherence length is estimated using the BCS expression\\cite{tinkham96}\n\\begin{equation}\n\\xi(0)=0.18\\hbar\\ensuremath{v_{\\rm F}}\/(\\ensuremath{k_{\\rm B}}\\ensuremath{T_{\\rm c}}),\n\\label{cohlength1}\n\\end{equation}\nwhich yields $\\xi(0)=360$\\,nm. Hence, the Ginzburg-Landau parameter\\cite{kGLcomment} is $\\ensuremath{\\kappa_{\\rm GL}}=0.96\\lambda(0)\/\\xi(0)=0.35< 1\/\\sqrt{2}$, clearly placing SiC:B in the type-I regime. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=7.5cm,clip]{fig4.pdf}\n\\caption[]{(color online) Ratio of $\\ensuremath{H_{\\rm c}}\/\\ensuremath{H_{\\rm sc}}$ according to Eq.\\,(\\ref{GLsc}): The solid line is a linear fit to the data; see text.}\n\\label{HTanalysis}\n\\end{figure}\nAn independent approach to estimate the GL parameter \\ensuremath{\\kappa_{\\rm GL}}\\ is to start with the two phase lines given in the $H$\\,--\\,$T$\\ phase diagram (Fig.\\,5 in Ref.\\,\\onlinecite{ren07a}), i.\\,e.\\ the supercooling field \\ensuremath{H_{\\rm sc}}\\ and the critical field strength \\ensuremath{H_{\\rm c}}. Following the GL theory the analysis of the observed supercooling behavior in AC susceptibility provides an upper limit of the GL parameter. Upon decreasing an external applied magnetic field at constant temperature the superconducting {\\it nucleation field} is given by \\ensuremath{H_{\\rm c2}}. For a type-I superconductor this is smaller than the thermodynamic {\\it critical field} \\ensuremath{H_{\\rm c}}, which leads to the effect of {\\it supercooling} if moreover the GL parameter satisfies $\\ensuremath{\\kappa_{\\rm GL}}<0.417$. However, Saint-James and de~Gennes showed that the nucleation of superconducting parts of a sample sets in near to the surface at the {\\it surface nucleation field} $\\ensuremath{H_{\\rm c3}} = 1.695\\ensuremath{H_{\\rm c2}} = 1.695 \\sqrt{2}\\ensuremath{\\kappa_{\\rm GL}}\\ensuremath{H_{\\rm c}}$, larger than \\ensuremath{H_{\\rm c2}}\\ if the sample is placed in vacuum.\\cite{saintjames63a,Hc3comment} In real experiments, however, the onset of superconductivity will be observed at a field \\ensuremath{H_{\\rm sc}}, which is larger than the ideal supercooling \\ensuremath{H_{\\rm c3}}: the experimentally derived supercooling phase line in an $H$\\,--\\,$T$\\ phase diagram will satisfy the inequality $\\ensuremath{H_{\\rm sc}} \\geq \\ensuremath{H_{\\rm c3}}$. Hence, the ''real'' GL parameter has to be smaller than the ''supercooling'' $\\kappa_{\\rm sc}$:\\cite{feder69a,tinkham96}\n\\begin{equation}\\label{GLsc}\n\\ensuremath{\\kappa_{\\rm GL}}\\leq \\kappa_{\\rm sc}= \\frac{1}{1.695\\sqrt{2}}\\cdot\\frac{\\ensuremath{H_{\\rm sc}}}{\\ensuremath{H_{\\rm c}}}=0.417\\frac{\\ensuremath{H_{\\rm sc}}}{\\ensuremath{H_{\\rm c}}}.\n\\end{equation}\nWe would like to emphasize that the qualitative observation of supercooling in the field-dependence of our AC susceptibility data already confirms that the GL parameter has to be {\\it smaller} than 0.417.\n\nWe calculated the ratio of $\\ensuremath{H_{\\rm sc}}$ and $\\ensuremath{H_{\\rm c}}$ deduced from field-sweep measurements at several temperatures. The result is shown in Fig.~\\ref{HTanalysis}. Following the procedure used by Feder and McLachlan,\\cite{feder69a} i.\\,e.\\ extrapolating the data to $T=\\ensuremath{T_{\\rm c}}$, yields in our case $\\ensuremath{\\kappa_{\\rm GL}} < 0.3$, which is even smaller than the afore reported estimate derived from $n$, \\ensuremath{T_{\\rm c}}, and \\ensuremath{\\gamma_{\\rm n}}, supporting the conclusion that SiC:B is a type-I superconductor.\n\nHowever, our results also imply that SiC:B is a dirty-limit superconductor because the coherence length is much larger than the mean-free path: $\\xi(0)\\gg \\ell$. The Ginzburg-Landau parameter for a dirty-limit superconductor\\cite{tinkham96} is given by $\\tilde{\\kappa}_{\\rm GL}=0.715\\lambda(0)\/\\ell$ which is $\\gg 1\/\\sqrt{2}$ in the sample used due to the small $\\ell$ value. The large $\\tilde{\\kappa}_{\\rm GL}$ \/ small $\\ell$ is mainly caused by the residual resistivity $\\rho_0$. Among our samples the residual resistivity varies from 60\\,\\ensuremath{\\muup\\Omega{\\rm cm}}\\ to the order of m$\\Omega$cm, all of them exhibiting a type-I behavior in the AC susceptibility. With this experimental finding and keeping in mind that the so-far prepared crystals are polycrystalline multi-phase materials, it is reasonable to assume that the intrinsic value of $\\rho_0$ (and hence $\\ell$) could be much lower (larger) than even the 60\\,\\ensuremath{\\muup\\Omega{\\rm cm}}\\ (14\\,nm) found for the sample SiC-1. The quantities given in Table~\\ref{SiCprop} which are related to the value of $\\rho_0$ are not very reliable and therefore asterisked ''*$\\dots$*''. A decrease of the residual resistivity to a few \\ensuremath{\\muup\\Omega{\\rm cm}}\\ would be sufficient to shift $\\tilde{\\kappa}_{\\rm GL}$ below the critical value of $0.417<1\/\\sqrt{2}$ in accordance with our experimental finding of a supercooled type-I superconductor.\n\n\\subsection{Band structure of 3C-SiC}\n\\begin{figure}\n\\centering\n\\includegraphics[width=8.5cm,clip]{fig5.pdf}\n\\caption[]{(color online) Calculated density of states (DOS) vs. energy for zincblende 3C-SiC. The inset gives an enlarged view of the energy range near to the Fermi level \\ensuremath{E_{\\rm F}}. The origin of energy is taken at the valence-band maximum without doping. The dotted lines in both panels mark the experimental value of the Sommerfeld parameter $\\ensuremath{\\gamma_{\\rm n}}=0.29$\\,J\/molK$^2$\\,$=0.5$\\,\/(eV unit cell) and the respective energy shift due to the boron doping $\\Delta E=0.56$\\,eV assuming rigid bands; see text for details.}\n\\label{3C-SiC_DOS}\n\\end{figure}\nCalculated density of states (DOS) and band structure data provide another possibility to determine an upper limit of the Fermi-wave number and hence the GL parameter \\ensuremath{\\kappa_{\\rm GL}}\\ using the experimental value of the Sommerfeld coefficient and \n\\begin{equation}\n\\ensuremath{\\gamma_{\\rm n}} = \\frac{\\pi^2\\ensuremath{k_{\\rm B}}^2}{3}\\cdot \\sum_{i=1}^{3}{\\rm DOS}_{i}(E_{\\rm F})=0.29\\,{\\rm mJ\/molK}^2.\n\\label{DOS}\n\\end{equation}\n\nFor simplicity, we will focus only on the 3C-modification of SiC. We approximate, that all three valence bands are free-electron like. Moreover, we assume rigid bands, i.\\,e.\\ the band structure is independent of charge-carrier doping. \n\nThe electronic band structure of 3C-SiC is calculated within the local density approximation (LDA) to the density functional theory. The all-electron full-potential linear-augmented-plane-wave method is used to solve one-electron Kohn-Sham equations. All the relativistic effects including spin-orbit coupling are included to every self-consistent-field iteration. The results for the total DOS and the band dispersions are shown in Figs.\\,\\ref{3C-SiC_DOS} and \\ref{3C-SiC_BS}, respectively. Two- and three-dimensional plots of the three Fermi surfaces corresponding to the upper three valence bands of 3C-SiC are given in Fig.\\,\\ref{3C-SiC_FS}. Therein panel (a), (b), and (c) display three-dimensional representations of the heavy-hole (hh), the light-hole (lh), and the split-hole (sh) bands. Panel (d) gives their cross sections.\\cite{persson97a}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=8.5cm,clip]{fig6.pdf}\n\\caption[]{(color online) Band-structure calculation of 3C-SiC. Spin-orbit coupling is included. The upper three valence bands are the heavy-hole (hh, green), the light-hole (lh, red), and the split-hole (sh, blue) band. Panel (a) summarizes the band-structure calculation based on the DOS calculation given in Fig.\\,\\ref{3C-SiC_DOS}. Panel (b) gives an enlarged view along the [110] direction from the Brillouin zone center $\\Gamma$ towards the K point. The dotted lines mark the energy shift $\\Delta E=0.56$\\,eV estimated from the data shown in Fig.\\,\\ref{3C-SiC_DOS} and the corresponding Fermi-wave number $\\ensuremath{k_{\\rm F}}=0.5\\cdot\\ensuremath{k_{\\rm F}}(\\Gamma-{\\rm K})= 0.5\\cdot \\sqrt{18}\/4\\cdot 2\\pi\/a$; see text for details. Panel (c) gives an enlarged view of the zone center at $\\Gamma$. The bands are split due to the spin-orbit coupling.}\n\\label{3C-SiC_BS}\n\\end{figure}\nThe DOS corresponding to the experimental \\ensuremath{\\gamma_{\\rm n}}\\ value is indicated by a dotted line in Fig.\\,\\ref{3C-SiC_DOS}. The inset gives an expanded view of the relevant energy range. Using Eq.\\,(\\ref{DOS}), the molar volume \\ensuremath{V_{\\rm mol}}\\ and the volume of the unit cell $V_0$ (cf.\\ Table\\,\\ref{SiCbasic}) to convert the corresponding units, $\\ensuremath{\\gamma_{\\rm n}} =0.29$\\,mJ\/molK corresponds to ${\\rm DOS}\\approx 6\\cdot 10^{21}$\\,states\\,\/\\,(eV$\\cdot$cm$^3$) = 0.5\\,states\\,\/\\,(eV$\\cdot$ unit cell). The energy shift of the Fermi energy due to the charge-carrier doping was estimated to $\\Delta E= 0.56$\\,eV also indicated by a dotted line in Fig.\\,\\ref{3C-SiC_DOS}.\n\nThe band structure of 3C-SiC is shown in Fig.\\,\\ref{3C-SiC_BS}\\,(a). Panel (b) provides an enlarged view of the relevant bands near the $\\Gamma$ point. In panel (c) the effect of spin-orbit coupling at the $\\Gamma$ point is shown, which is about 10\\,meV.\n\nNext we plot the estimated energy shift $\\Delta E$ into the band structure plot as marked by a dotted line in Figs.\\,\\ref{3C-SiC_BS}\\,(a) and (b). The corresponding Fermi-wave number for the heavy-hole band was estimated to $\\ensuremath{k_{\\rm F}}\\approx 50$\\,\\% of the distance from $\\Gamma$ to K in the fcc Brillouin zone which equals $\\sqrt{18}\/4\\cdot 2\\pi\/a$. The parameter $a$ denotes the lattice constant of 3C-SiC $a=4.3596$\\,\\AA\\ (cf.\\ Table\\,\\ref{SiCbasic}). This yields an upper limit of $\\ensuremath{k_{\\rm F}}< 7.6$\\,nm$^{-1}$, which is more than double the value evaluated before (3.8\\,nm$^{-1}$) assuming a single spherical Fermi surface. \n\nPossible origins of the discrepancy between the two estimates of \\ensuremath{k_{\\rm F}}\\ could be the neglect of the 6H-SiC phase fraction. Moreover, the assumption of a single spherical Fermi surface is too simple as can be seen in Fig\\,\\ref{3C-SiC_FS}. It is not known how the heavy-boron doping modifies the real DOS and band structure, either, leading back to the question if the superconductivity in this material evolves from intrinsic or impurity bands as discussed in literature for C:B. \n\\begin{figure}\n\\centering\n\\includegraphics[width=8.5cm,clip]{fig7.pdf}\n\\caption[]{(color online) Plots of the Fermi \nsurfaces of 3C-SiC around the $\\Gamma$ point of the fcc Brillouin zone (given in black) for a Fermi energy below the top of the valence band by $-0.56$\\,eV. Only three instead of six bands are shown due to the smallness of the spin-orbit splitting. Panels (a), (b), and (c) display three-dimensional representations of the heavy-hole (hh), the light-hole (lh), and the split-hole (sh) band. Panel (d) contains their cross sections. The colors of the respective band plots are the same as used in Fig.\\,\\ref{3C-SiC_BS}.}\n\\label{3C-SiC_FS}\n\\end{figure}\n\n \n\\section{Comparison and Discussion}\nAfter the analysis of the experimental data we now focus on the question why the ''mixed'' compound SiC is a type-I superconductor whereas the ''pure'' parent compounds silicon and diamond exhibit type-II superconductivity upon boron doping. Let us therefore briefly compare the obtained parameters of SiC:B with those reported for C:B:\\cite{sidorov05a} For Si:B no specific-heat study is available so far, hence a comparison is not possible for all parameters. \n\nThe charge-carrier concentrations for all three specimen are comparable and on the order of $2\\cdot 10^{21}$\\,cm$^{-3}$. However, the temperature dependence and the absolute values of the resistivity are different: SiC:B turns out to be a much better conductor exhibiting a metallic $\\rho(T)$ for $\\ensuremath{T_{\\rm c}}\\leq T \\leq 300$\\,K with an RRR value of about 10, whereas the resistivity of C:B and Si:B decreases slightly above \\ensuremath{T_{\\rm c}}. For Si:B the slope of the resistivity becomes positive above $\\sim 50$\\,K, for C:B a slightly positive slope is observed only above 200\\,K.\\cite{sidorov05a,bustarret06a} In both cases the resistivity is almost temperature independent resulting in RRR values of about 1.\n\nThe Sommerfeld parameter $\\ensuremath{\\gamma_{\\rm n}}$ is somewhat smaller for C:B compared to SiC:B, the coefficient of the phononic contribution is much smaller for C:B resulting in a higher Debye temperature in the latter case. This is not surprising since the Debye temperature of pure diamond is much higher than that of pure SiC. For both compounds the jump height of the specific heat (Fig.~\\ref{cp_SiC-1}\\,(b)) is much smaller than the BCS expectation for a weak-coupling superconductor. The superconducting penetration depths are similar for C:B and SiC:B ($\\lambda(0)\\approx 150$\\,nm) but the coherence lengths make the essential difference. The published values from Refs.\\,\\onlinecite{bustarret06a} (Si:B) and \\onlinecite{sidorov05a} (C:B) have been estimated from the upper critical field strength \\ensuremath{H_{\\rm c2}}\\ using the GL expression\n\\begin{equation}\n\\xi = \\sqrt{\\Phi_0\/2\\pi\\ensuremath{H_{\\rm c2}}(0)}\n\\label{cohlength2}\n\\end{equation}\nwith the flux quanta $\\Phi_0$. Applying this formula the coherence lengths of C:B and Si:B are both on the order of 10\\,nm. These values of $\\xi$ are given in parantheses in Table~\\ref{SiCprop}. In the case of C:B we calculated $\\xi(0)$ using Eq.\\,(\\ref{cohlength1}) for a better comparability with SiC:B, too. The latter yields $\\xi(0)=80$\\,nm, whereas for SiC:B $\\xi(0)$ amounts to 360\\,nm resulting in different GL parameters $\\ensuremath{\\kappa_{\\rm GL}}= 0.35$ for SiC:B and 2 for C:B (which is 18 using the published value of $\\xi=9$\\,nm), cf.\\ Table~\\ref{SiCprop}. Hence, SiC:B is a type-I and C:B a type-II superconductor.\n\nAt the current state of research we can only speculate about the physical reasons for this different nature of superconductivity in C:B \/ Si:B and SiC:B. In the case of C:B one apparent reason leading to a smaller coherence length and hence a larger GL parameter is the higher critical temperature of this superconductor: $\\xi(0)\\propto \\ensuremath{T_{\\rm c}}^{-1}$. However, this argument does not hold for Si:B, \\ensuremath{T_{\\rm c}}\\ of which is much smaller than that of SiC:B. One can argue that SiC:B is a much cleaner system than C:B and Si:C. Hence, the coherence length $\\xi(0)$ of Si:B (thin film and diffuse doping) might be limited by a very short mean-free path $\\ell$ and thus the GL parameter is larger, leading to the speculation that ''clean'' Si:B could be a type-I superconductor, too.\n\nFinally, we would like to mention a couple of apparent differences between the systems: \n\n\\textbf{Si\\,--\\,C bilayers:} SiC is in a certain sense a ''layered'' system consisting of Si\\,--\\,C bilayers. Many polytypes are known distinguished by the stacking sequence of these bilayers in the crystal unit cell. In this sense one may refer to SiC as an ''ordered system''. Our results\\cite{ren07a} suggest that boron is introduced only into the carbon sites in SiC:B and hence only half of the crystal sites are directly affected by the disorder due to the hole-doping process, whereas in C:B and Si:B in principle all sites can be randomly involved. \n\n\\textbf{structure:} For silicon and diamond the cubic crystal structure seems to be important or a precondition for the appearance of superconductivity upon doping. In SiC the situation is different. The multi-phase crystal used in this study contains cubic 3C-SiC {\\it and} hexagonal 6H-SiC. At the moment we cannot rule out the possibility that {\\it both} phase fractions contribute to the superconductivity which would be a clear difference compared to the two parent compounds. Moreover, Cohen suggested in 1964 that {\\it hexagonal} SiC could exhibit superconductivity.\\cite{cohen64b} However, the same author predicts that most of the many-valley semiconductor based superconductors should be type-II rather than type-I.\\cite{cohen64a} \n\n\\textbf{band structure:} In contrast to cubic diamond and silicon the ''mixed'' compound SiC breaks inversion symmetry. In crystals with inversion-symmetry, states with different spin orientations are degenerated. This is not true in general for crystals with broken inversion symmetry. The degeneracy might be lifted by the spin-orbit interaction. In 3C-SiC (zincblende structure) the degeneracy is preserved only along the $\\left[100\\right]$ direction. Along e.\\,g.\\ the $\\left[110\\right]$ direction (i.\\,e.\\ from the $\\Gamma$ point to the K point in the Brillouin zone) the states with different spins split up.\\cite{cardona88a,theodorou99a} Using the above estimate of the Fermi-wave number for the heavy-hole band $\\ensuremath{k_{\\rm F}}<7.6$\\,nm$^{-1}$ we can give a rough estimate of the spin-orbit splitting for this band along $\\Gamma$\\,--\\,K in 3C-SiC:\\cite{theodorou99a} $\\Delta_{\\rm SO}=0.02$\\,meV, which is a rather small value. We note, that the light-hole, split-hole, and (in the case of electron doping) the lowest-conduction band exhibit larger values.\\cite{theodorou99a} Using as an example $\\ensuremath{k_{\\rm F}}=3.8$\\,nm$^{-1}$ gives the same spin splitting because $\\Delta_{\\rm SO}$ of the heavy-hole band is almost constant in the interval $0.25\\cdot 2\\pi\/a$ to $0.85\\cdot 2\\pi\/a$ along the [110] direction of the Brillouin zone.\\cite{theodorou99a}\n\n\\textbf{inversion symmetry:} Moreover, a broken inversion symmetry is known to give rise to a highly interesting nature of the superconducting ground state, including a spin singlet -- triplet mixture.\\cite{frigeri04a,fujimoto07a} In SiC:B we do not expect any unconventional scenario based on the broken inversion symmetry because of the comparably light elements silicon and carbon without strong electron-electron interaction. \n\n\\textbf{charge-carrier concentration:}\nIn diamond, cubic silicon, and 3C-SiC the indirect band gaps are between the zone center ($\\Gamma$ point) and the X point of the Brillouin zone. For all other SiC polytypes the valence-band maximum is located at the $\\Gamma$ point, too, but the conduction-band minimum differs. For 6H-SiC it occurs at the M point.\\cite{pensl93a,park94a,harris95} If one takes into account spin-orbit interaction, the splitting of the band structure of diamond and silicon is not affected, but for 3C-SiC the splitting will cause a shift of the valence-band maximum.\\cite{burns77} However, the boron doping in SiC removes electrons from the valence bands and therefore the difference in the semiconducting gaps might be of minor relevance.\n\nNevertheless, it underlines again the importance of answering the question whether the holes induced by boron doping in SiC reside in the intrinsic bands or form an impurity band, i.\\,e.\\ what is the nature of the metallic ground state, from which superconductivity develops? \n\n\\section{Summary}\nIn summary we present a specific-heat study of heavily boron-doped silicon carbide SiC:B using the same crystal used in our recent publication reporting the discovery of superconductivity. In contrast to the type-II superconductivity in the two parent compounds, boron-doped diamond C:B and boron-doped silicon Si:B, the bulk superconductivity in SiC:B is type I. This is reflected in rather different values of the superconducting coherence length, i.\\,e.\\ 360\\,nm for SiC:B and only 80\\,nm for C:B, whereas the penetration depths are of the same order of magnitude. We presented two different approaches to describe the data: assuming (i) an isotropic gap structure and (ii) a power-law behavior. The electronic specific heat in the superconducting state is well reproduced by the former assumption with a residual density of states or by the latter assumption of a quadratic temperature dependence. The specific-heat jump height $\\Delta \\ensuremath{c_{\\rm el}}\/T$ at \\ensuremath{T_{\\rm c}}\\ is about 1 or even smaller in the latter model and hence far away from the expectation in a BCS framework. However, due to the lack of data points below 0.45\\,K it is difficult to give a final conclusion about the superconducting gap structure. To further clarify the gap structure a specific-heat study in a dilution refrigerator system is desired.\n\nThe origin of the different nature between the type-II superconductors C:B and Si:B on the one hand and the type-I superconductor SiC:B on the other hand remains at this state of research unclear. To clarify this intriguing issue further experimental and theoretical work is needed. From the experimental point of view single crystalline samples are highly desirable. Moreover samples with only one phase fraction, either 3C-SiC or 6H-SiC, are eligible to answer the question which phase fraction is liable for the occurrence of superconductivity in SiC:B. This work is currently under way.\n\n\\section{Acknowledgments}\nWe acknowledge fruitful discussions with S.~Yonezawa.\n\nThis work was supported by the 21st century COE programs, ''High-Tech Research Center Project for Private Universities: matching fund subsidy'', as well as ''Diversity and Universality of Physics'' from the Ministry of Education, Culture, Sports, Science, and Technology (MEXT), and by a Grand-in-Aid for Scientific Research on Priority Area from MEXT. MK is supported by MEXT.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Languages}\nWe show a detailed overview of languages in the cross-lingual benchmark including interesting typological differences in Table \\ref{tab:languages}. Wikipedia information is taken from Wikipedia\\footnote{\\url{https:\/\/meta.wikimedia.org\/wiki\/List_of_Wikipedias}} and linguistic information from WALS Online\\footnote{\\url{https:\/\/wals.info\/languoid}}. \\textsc{xtreme}\\xspace includes members of the Afro-Asiatic, Austro-Asiatic, Austronesian, Dravidian, Indo-European, Japonic, Kartvelian, Kra-Dai, Niger-Congo, Sino-Tibetan, Turkic, and Uralic language families as well as of two isolates, Basque and Korean.\n\n\\begin{table*}[]\n\\centering\n\\caption{Statistics about languages in the cross-lingual benchmark. Languages belong to 12 language families and two isolates, with Indo-European (IE) having the most members. Diacritics \/ special characters: Language adds diacritics (additional symbols to letters). Compounding: Language makes extensive use of word compounds. Bound words \/ clitics: Function words attach to other words. Inflection: Words are inflected to represent grammatical meaning (e.g.~case marking). Derivation: A single token can represent entire phrases or sentences.}\n\\label{tab:languages}\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{l c c l l ccccc c}\n\\toprule\nLanguage & \\begin{tabular}[c]{@{}l@{}}ISO\\\\ 639-1\\\\ code\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}\\# Wikipedia\\\\ articles (in\\\\ millions)\\end{tabular} & Script & \\begin{tabular}[c]{@{}l@{}}Language\\\\ family\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}Diacritics \/\\\\ special\\\\ characters\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}Extensive\\\\ compound-\\\\ ing\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}Bound\\\\ words \/\\\\ clitics\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}Inflec-\\\\ tion\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}Deriva-\\\\ tion\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}\\# datasets\\\\ with\\\\ language\\end{tabular} \\\\ \\midrule\nAfrikaans & af & 0.09 & Latin & IE: Germanic & & X & & & & 3 \\\\\nArabic & ar & 1.02 & Arabic & Afro-Asiatic & X & & X & X & & 7 \\\\\nBasque & eu & 0.34 & Latin & Basque & X & & X & X & X & 3 \\\\\nBengali & bn & 0.08 & Brahmic & IE: Indo-Aryan & X & X & X & X & X & 3 \\\\\nBulgarian & bg & 0.26 & Cyrillic & IE: Slavic & X & & X & X & & 4 \\\\\nBurmese & my & 0.05 & Brahmic & Sino-Tibetan & X & X & & & & 1 \\\\\nDutch & nl & 1.99 & Latin & IE: Germanic & & X & & & & 3 \\\\\nEnglish & en & 5.98 & Latin & IE: Germanic & & & & & & 9 \\\\\nEstonian & et & 0.20 & Latin & Uralic & X & X & & X & X & 3 \\\\\nFinnish & fi & 0.47 & Latin & Uralic & & & & X & X & 3 \\\\\nFrench & fr & 2.16 & Latin & IE: Romance & X & & X & & & 6 \\\\\nGeorgian & ka & 0.13 & Georgian & Kartvelian & & & & X & X & 2 \\\\\nGerman & de & 2.37 & Latin & IE: Germanic & & X & & X & & 8 \\\\\nGreek & el & 0.17 & Greek & IE: Greek & X & X & & X & & 5 \\\\\nHebrew & he & 0.25 & Hebrew & Afro-Asiatic & & & & X & & 3 \\\\\nHindi & hi & 0.13 & Devanagari & IE: Indo-Aryan & X & X & X & X & X & 6 \\\\\nHungarian & hu & 0.46 & Latin & Uralic & X & X & & X & X & 4 \\\\\nIndonesian & id & 0.51 & Latin & Austronesian & & & X & X & X & 4 \\\\\nItalian & it & 1.57 & Latin & IE: Romance & X & & X & & & 3 \\\\\nJapanese & ja & 1.18 & Ideograms & Japonic & & & X & X & & 4 \\\\\nJavanese & jv & 0.06 & Brahmic & Austronesian & X & & X & & & 1 \\\\\nKazakh & kk & 0.23 & Arabic & Turkic & X & & & X & X & 1 \\\\\nKorean & ko & 0.47 & Hangul & Koreanic & & X & & X & X & 5 \\\\\nMalay & ms & 0.33 & Latin & Austronesian & & & X & X & & 2 \\\\\nMalayalam & ml & 0.07 & Brahmic & Dravidian & X & X & X & X & & 2 \\\\\nMandarin & zh & 1.09 & Chinese ideograms & Sino-Tibetan & & X & & & & 8 \\\\\nMarathi & mr & 0.06 & Devanagari & IE: Indo-Aryan & & & X & X & & 3 \\\\\nPersian & fa & 0.70 & Perso-Arabic & IE: Iranian & & X & & & & 2 \\\\\nPortuguese & pt & 1.02 & Latin & IE: Romance & X & & X & & & 3 \\\\\nRussian & ru & 1.58 & Cyrillic & IE: Slavic & & & & X & & 7 \\\\\nSpanish & es & 1.56 & Latin & IE: Romance & X & & X & & & 7 \\\\\nSwahili & sw & 0.05 & Latin & Niger-Congo & & & X & X & X & 3 \\\\\nTagalog & tl & 0.08 & Brahmic & Austronesian & X & & X & X & & 1 \\\\\nTamil & ta & 0.12 & Brahmic & Dravidian & X & X & X & X & X & 3 \\\\\nTelugu & te & 0.07 & Brahmic & Dravidian & X & X & X & X & X & 4 \\\\\nThai & th & 0.13 & Brahmic & Kra-Dai & X & & & & & 4 \\\\\nTurkish & tr & 0.34 & Latin & Turkic & X & X & & X & X & 5 \\\\\nUrdu & ur & 0.15 & Perso-Arabic & IE: Indo-Aryan & X & X & X & X & X & 4 \\\\\nVietnamese & vi & 1.24 & Latin & Austro-Asiatic & X & & & & & 6 \\\\\nYoruba & yo & 0.03 & Arabic & Niger-Congo & X & & & & & 1 \\\\\n\\bottomrule\n\\end{tabular}%\n}\n\\end{table*}\n\n\\section{Hyper-parameters} \\label{app:hyper-parameters}\nTable~\\ref{tab:params} summarizes the hyper-parameters of baseline and state-of-the-art models. We refer to XLM-100 as XLM, and XLM-R-large as XLM-R in our paper to simplify the notation.\n\n\\noindent \\textbf{mBERT} $\\:$ We use the cased version, which covers 104 languages, has 12 layers, 768 hidden units per layer, 12 attention heads, a 110k shared WordPiece vocabulary, and 110M parameters.\\footnote{\\url{https:\/\/github.com\/google-research\/bert\/blob\/master\/multilingual.md}} The model was trained using Wikipedia data in all 104 languages, oversampling low-resource languages with an exponential smoothing factor of 0.7. We generally fine-tune mBERT for two epochs, with a training batch size of 32 and a learning rate of 2e-5. For training BERT models on the QA tasks, we use the original BERT codebase. For all other tasks, we use the Transformers library \\cite{wolf2019huggingface}.\n\n\\noindent \\textbf{XLM and XLM-R} $\\:$ We use the XLM and XLM-R Large versions that cover 100 languages, use a 200k shared BPE vocabulary, and that have been trained with masked language modelling.\\footnote{\\url{https:\/\/github.com\/facebookresearch\/XLM}} We fine-tune both for two epochs with a learning rate of 3e-5 and an effective batch size of 16. In contrast to XLM, XLM-R does not use language embeddings. We use the Transformers library for training XLM and XLM-R models on all tasks. \n\n\\begin{table}[]\n\\caption{Hyper-parameters of baseline and state-of-the-art models. We do not use XLM-15 and XLM-R-Base in our experiments.}\n\\label{tab:params}\n\\resizebox{\\columnwidth}{!}{%\n\\begin{tabular}{l cccc}\n\\toprule\nModel & Parameters & Langs & Vocab size & Layers \\\\ \\midrule\nBERT-large & 364,353,862 & 1 & 28,996 & 24 \\\\\nmBERT & 178,566,653 & 104 & 119,547 & 12 \\\\ \nMMTE & 191,733,123 & 103 & 64,000 & 6 \\\\ \nXLM-15 & 346,351,384 & 15 & 95,000 & 12 \\\\ \nXLM-100 & 827,696,960 & 100 & 200,000 & 12 \\\\ \nXLM-R-Base & 470,295,954 & 100 & 250,002 & 12 \\\\ \nXLM-R-Large & 816,143,506 & 100 & 250,002 & 24 \\\\ \n\\bottomrule\n\\end{tabular}\n}\n\\end{table}\n\n\\section{Translations for QA datasets} \\label{app:qa_translations} \n\nWe use an in-house translation tool to obtain translations for our datasets. For the question answering tasks (XQuAD and MLQA), the answer span is often not recoverable if the context is translated directly. We experimented with enclosing the answer span in the English context in quotes \\cite{Lee2018semi-supervised,Lewis2019mlqa} but found that quotes were often dropped in translations (at different rates depending on the language). We found that enclosing the answer span in HTML tags (e.g. \\texttt{} and \\texttt{<\/b>}) worked more reliably. If this fails, as a back-off we fuzzy match the translated answer with the context similar to \\cite{Hsu2019zero-shot}. If the minimal edit distance between the closest match and the translated answer is larger than $\\min(10, \\texttt{answer\\_len}\/2)$, we drop the example. On the whole, using this combination, we recover more than 97\\% of all answer spans in training and test data.\n\n\\section{Performance on translated test sets}\n\nWe show results comparing the performance of mBERT and translate-train (mBERT) baselines on the XQuAD test sets with automatically translated test sets in Table \\ref{tab:xquad-gold-auto-translation-comparison}. Performance on the automatically translated test sets underestimates the performance of mBERT by 2.9 F1 \/ 0.2 EM points but overestimates the performance of the translate-train baseline by 4.0 F1 \/ 6.7 EM points. The biggest part of this margin is explained by the difference in scores on the Thai test set. Overall, this indicates that automatically translated test sets are useful as a proxy for cross-lingual performance but may not be reliable for evaluating models that have been trained on translations as these have learnt to exploit the biases of \\emph{translationese}.\n\n\\begin{table*}[]\n\\centering\n\\caption{Comparison of F1 and EM scores of mBERT and translate-train (mBERT) baselines on XQuAD test sets (gold), which were translated by professional translators and automatically translated test sets (auto).}\n\\label{tab:xquad-gold-auto-translation-comparison}\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{l c ccccccccccc}\n\\toprule\n & Test set & es & de & el & ru & tr & ar & vi & th & zh & hi & avg \\\\ \\midrule\n\\multirow{2}{*}{mBERT} & gold & 75.6 \/ 56.9 & 70.6 \/ 54.0 & 62.6 \/ 44.9 & 71.3 \/ 53.3 & 55.4 \/ 40.1 & 61.5 \/ 45.1 & 69.5 \/ 49.6 & 42.7 \/ 33.5 & 58.0 \/ 48.3 & 59.2 \/ 46.0 & 62.6 \/ 47.2 \\\\\n & auto & 76.1 \/ 58.7 & 64.3 \/ 49.9 & 57.9 \/ 42.5 & 68.3 \/ 51.8 & 55.6 \/ 42.9 & 62.1 \/ 48.6 & 68.6 \/ 54.3 & 41.1 \/ 32.6 & 48.5 \/ 47.7 & 54.1 \/ 40.9 & 59.7 \/ 47.0 \\\\\n\\multirow{2}{*}{translate-train} & gold & 80.2 \/ 63.1 & 75.6 \/ 60.7 & 70.0 \/ 53.0 & 75.0 \/ 59.7 & 68.9 \/ 54.8 & 68.0 \/ 51.1 & 75.6 \/ 56.2 & 36.9 \/ 33.5 & 66.2 \/ 56.6 & 69.6 \/ 55.4 & 68.7 \/ 54.6 \\\\\n & auto & 80.7 \/ 66.0 & 71.1 \/ 58.9 & 69.3 \/ 54.5 & 75.7 \/ 61.5 & 71.2 \/ 59.1 & 74.3 \/ 60.7 & 76.8 \/ 64.0 & 79.5 \/ 74.8 & 59.3 \/ 58.0 & 69.1 \/ 55.2 & 72.7 \/ 61.3 \\\\\n\\bottomrule\n\\end{tabular}%\n}\n\\end{table*}\n\n\\begin{table*}[]\n\\caption{Comparison of accuracy scores of mBERT baseline on XNLI test sets (gold), which were translated by professional translators and automatically translated test sets (auto) in 14 languages. BLEU and chrF scores are computed to measure the translation quality between gold and automatically translated test sets.}\n\\label{tab:xnli-translated-gold-comparison}\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{l cccccccccccccc c}\n\\toprule\nLanguages & zh & es & de & ar & ur & ru & bg & el & fr & hi & sw & th & tr & vi & avg \\\\ \\midrule\nauto Acc. & 69.1 & 74.7 & 72.8 & 66.5 & 64.5 & 71.6 & 70.2 & 67.7 & 74.3 & 65.1 & 50.2 & 54.5 & 60.0 & 72.7 & 66.7 \\\\ \ngold Acc. & 67.8 & 73.5 & 70.0 & 64.3 & 57.2 & 67.8 & 68.0 & 65.3 & 73.4 & 58.9 & 49.7 & 54.1 & 60.9 & 69.3 & 64.3 \\\\ \\midrule\nBLEU & 40.92 & 43.46 & 30.94 & 32.35 & 20.13 & 22.62 & 45.04 & 60.29 & 47.91 & 29.55 & 31.25 & 10.65 & 15.39 & 56.93 & 34.82 \\\\ \nchrF & 35.96 & 67.92 & 60.28 & 59.64 & 48.21 & 50.38 & 67.52 & 75.34 & 69.58 & 53.85 & 59.84 & 54.89 & 51.46 & 69.37 & 58.87 \\\\ \\bottomrule\n\\end{tabular}}\n\\end{table*}\n\n\\section{mBERT performance across tasks and languages}\n\nWe show the performance of mBERT across all tasks and languages of \\textsc{xtreme}\\xspace in Table \\ref{fig:scores_vs_tasks_mbert}.\n\n\\begin{figure}[!t]\n \\centering\n \\includegraphics[width=1.0\\linewidth]{scores_vs_tasks_mbert.png}\n \\vspace{-1.5mm}\n \\caption{An overview of mBERT's performance on the \\textsc{xtreme}\\xspace tasks for the languages of each task. We highlight an estimate of human performance, performance on the English test set, the average of all languages excluding English, and the family of each language. Performance on pseudo test sets for XNLI and XQuAD is shown with slightly transparent markers.}\n \\label{fig:scores_vs_tasks_mbert}\n\\end{figure}\n\n\\section{Correlation with pretraining data size}\n\nWe show the Pearson correlation coefficient $\\rho$ of mBERT, XLM, and XLM-R with the number of Wikipedia articles in Table \\ref{tab:dataset-size-correlations}. XLM and mBERT were pretrained on Wikipedia, while XLM-R was pretrained on data from the web.\n\n\\begin{table}[]\n\\centering\n\\caption{Pearson correlation coefficients ($\\rho$) of zero-shot transfer performance and Wikipedia size across datasets and models.}\n\\label{tab:dataset-size-correlations}\n\\resizebox{\\columnwidth}{!}{%\n\\begin{tabular}{l c c c c c c c c c}\n\\toprule\n & XNLI & PAWS-X & POS & NER & XQuAD & MLQA & TyDiQA-GoldP & BUCC & Tatoeba \\\\ \\midrule\nmBERT & 0.79 & 0.81 & 0.36 & 0.35 & 0.80 & 0.87 & 0.82 & 0.95 & 0.68 \\\\\nXLM & 0.80 & 0.76 & 0.32 & 0.29 & 0.74 & 0.73 & 0.52 & 0.61 & 0.68 \\\\\nXLM-R & 0.75 & 0.79 & 0.22 & 0.27 & 0.50 & 0.76 & 0.14 & 0.36 & 0.49 \\\\ \\bottomrule\n\\end{tabular}%\n}\n\\end{table}\n\n\\section{Generalization to unseen tag combinations}\n\n\\begin{table}[]\n\\centering\n\\caption{Accuracy of mBERT on the target language dev data on POS tag trigrams and 4-grams that appeared and did not appear in the English training data. We show the average performance across all non-English languages and the difference of said average compared to the English performance on the bottom.}\n\\label{tab:pos-tag-generalization}\n\\begin{tabular}{lllll}\n\\toprule\n & \\begin{tabular}[c]{@{}l@{}}trigram,\\\\ seen\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}trigram,\\\\ unseen\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}4-gram,\\\\ seen\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}4-gram,\\\\ unseen\\end{tabular} \\\\ \\midrule\nen & 90.3 & 63.0 & 88.1 & 67.5 \\\\ \\midrule\naf & 68.1 & 8.2 & 64.1 & 24.2 \\\\\nar & 22.0 & 0.7 & 14.9 & 4.6 \\\\\nbg & 63.1 & 14.6 & 56.1 & 23.9 \\\\\nde & 77.8 & 47.2 & 73.0 & 48.7 \\\\\nel & 59.6 & 9.1 & 52.5 & 14.2 \\\\\nes & 68.6 & 10.6 & 62.4 & 24.9 \\\\\net & 60.7 & 14.4 & 53.1 & 31.9 \\\\\neu & 32.8 & 7.1 & 28.7 & 8.1 \\\\\nhe & 52.7 & 35.7 & 44.0 & 27.4 \\\\\nhi & 38.7 & 13.0 & 32.6 & 12.5 \\\\\nhu & 55.5 & 28.8 & 46.9 & 23.7 \\\\\nid & 60.8 & 16.6 & 54.7 & 21.6 \\\\\nit & 75.5 & 12.8 & 71.8 & 23.5 \\\\\nja & 16.3 & 0.0 & 12.3 & 1.0 \\\\\nko & 22.0 & 2.9 & 14.7 & 3.8 \\\\\nmr & 31.7 & 0.0 & 25.5 & 3.3 \\\\\nnl & 75.5 & 24.1 & 71.0 & 37.8 \\\\\npt & 76.2 & 14.9 & 71.2 & 30.6 \\\\\nru & 69.1 & 4.8 & 63.8 & 20.6 \\\\\nta & 30.3 & 0.0 & 24.5 & 4.2 \\\\\nte & 57.8 & 0.0 & 48.7 & 24.7 \\\\\ntr & 41.2 & 6.2 & 33.9 & 10.1 \\\\\nur & 30.6 & 18.3 & 22.3 & 10.9 \\\\\nzh & 29.0 & 0.0 & 21.7 & 3.9 \\\\ \\midrule\navg & 50.6 & 12.1 & 44.3 & 18.3 \\\\\ndiff & 39.7 & 50.9 & 43.7 & 49.2 \\\\ \\bottomrule\n\\end{tabular}%\n\\end{table}\n\nWe show the performance of mBERT on POS tag trigrams and 4-grams that were seen and not seen in the English training data in Table \\ref{tab:pos-tag-generalization}.\n\n\\section{Generalization to unseen entities}\n\n\\begin{table*}[]\n\\centering\n\\caption{Comparison of accuracies for entities in the target language NER dev data that were seen in the English NER training data (a); were not seen in the English NER training data (b); only consist of up to two tokens (c); only consist of Latin characters (d); and occur at least twice in the dev data (e). We only show languages where the sets (a--e) contain at least 100 entities each. We show the difference between (a) and (b) and the minimum difference between (a) and (c-e).}\n\\label{tab:mbert-entities-ner}\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{llllllllllllllllllllll}\n\\toprule\n& af & de & el & en & es & et & eu & fi & fr & he & hu & id & it & ka & ms & nl & pt & ru & sw & tr & vi \\\\ \\midrule\n(a) Seen & 94.7 & 88.3 & 91.4 & 91.9 & 76.3 & 88.3 & 83.6 & 85.3 & 90.5 & 78.2 & 90.7 & 89.4 & 88.4 & 92.3 & 88.6 & 93.5 & 88.6 & 83.9 & 96.3 & 85.2 & 91.4 \\\\\n(b) Not seen & 82.1 & 80.2 & 74.8 & 84.6 & 80.4 & 78.9 & 69.4 & 79.8 & 80.1 & 56.5 & 78.3 & 58.0 & 81.5 & 70.2 & 75.0 & 82.9 & 82.3 & 68.5 & 66.6 & 73.7 & 73.4 \\\\ \\midrule\n(a) $-$ (b) & 12.6 & 8.1 & 16.5 & 7.2 & -4.1 & 9.4 & 14.1 & 5.5 & 10.4 & 21.7 & 12.3 & 31.5 & 6.9 & 22.1 & 13.6 & 10.6 & 6.4 & 15.4 & 29.7 & 11.6 & 18.0 \\\\ \\midrule\n(c) Short & 86.5 & 82.9 & 80.3 & 88.2 & 86.6 & 81.7 & 72.5 & 83.9 & 88.6 & 66.3 & 83.7 & 85.8 & 87.2 & 72.5 & 89.1 & 87.6 & 87.8 & 78.0 & 65.7 & 83.1 & 84.6 \\\\\n(d) Latin & 83.6 & 81.2 & 87.5 & 86.2 & 80.0 & 79.5 & 70.3 & 80.3 & 81.1 & 77.2 & 79.9 & 61.8 & 82.6 & 89.6 & 76.3 & 84.2 & 83.0 & 83.8 & 70.0 & 75.0 & 74.9 \\\\\n(e) Freq & 87.3 & 80.6 & 81.9 & 91.6 & 83.4 & 79.4 & 68.8 & 85.7 & 77.3 & 66.8 & 86.0 & 56.5 & 88.8 & 74.3 & 81.3 & 87.1 & 84.4 & 76.5 & 49.1 & 81.9 & 78.6 \\\\ \\bottomrule\n$\\min($(a) $-$ (c--e)$)$ & 7.4 & 5.4 & 3.9 & 0.3 & 3.7 & 6.6 & 11.0 & 0.4 & 1.9 & 1.0 & 4.7 & 3.6 & 0.4 & 2.7 & 0.5 & 5.9 & 0.8 & 0.1 & 26.4 & 2.2 & 6.8 \\\\ \\bottomrule\n\\end{tabular}%\n}\n\\end{table*}\n\n\\begin{table*}[!h]\n\\caption{XNLI accuracy scores for each language.}\n\\label{tab:xnli_results}\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{l|ccccccccccccccc|c}\n\\toprule\nModel & en & ar & bg & de & el & es & fr & hi & ru & sw & th & tr & ur & vi & zh & \\textbf{avg} \\\\\n\\midrule\nmBERT & 80.8 & 64.3 & 68.0 & 70.0 & 65.3 & 73.5 & 73.4 & 58.9 & 67.8 & 49.7 & 54.1 & 60.9 & 57.2 & 69.3 & 67.8 & 65.4 \\\\\nXLM & 82.8 & 66.0 & 71.9 & 72.7 & 70.4 & 75.5 & 74.3 & 62.5 & 69.9 & 58.1 & 65.5 & 66.4 & 59.8 & 70.7 & 70.2 & 69.1 \\\\\nXLMR & \\textbf{88.7} & \\textbf{77.2} & \\textbf{83.0} & \\textbf{82.5} & \\textbf{80.8} & \\textbf{83.7} & \\textbf{82.2} & \\textbf{75.6} & \\textbf{79.1} & \\textbf{71.2} & \\textbf{77.4} & \\textbf{78.0} & \\textbf{71.7} & \\textbf{79.3} & \\textbf{78.2} & \\textbf{79.2} \\\\\nMMTE & 79.6 & 64.9 & 70.4 & 68.2 & 67.3 & 71.6 & 69.5 & 63.5 & 66.2 & 61.9 & 66.2 & 63.6 & 60.0 & 69.7 & 69.2 & 67.5 \\\\\n\\midrule\n\\textit{\\begin{tabular}[c]{@{}l@{}}Translate-train\\\\ (multi-task)\\end{tabular}} & 81.9 & \\textbf{73.8} & \\textbf{77.6} & \\textbf{77.6} & \\textbf{75.9} & \\textbf{79.1} & \\textbf{77.8} & 70.7 & \\textbf{75.4} & \\textbf{70.5} & 70.0 & 74.3 & \\textbf{67.4} & \\textbf{77.0} & \\textbf{77.6} & \\textbf{75.1} \\\\\n\\textit{Translate-train} & 80.8 & 73.6 & 76.6 & 77.4 & 75.7 & 78.1 & 77.4 & \\textbf{71.9} & 75.2 & 69.4 & \\textbf{70.9} & \\textbf{75.3} & 67.2 & 75.0 & 74.1 & 74.6 \\\\\n\\textit{Translate-test} & \\textbf{85.9} & 73.1 & 76.6 & 76.9 & 75.3 & 78.0 & 77.5 & 69.1 & 74.8 & 68.0 & 67.1 & 73.5 & 66.4 & 76.6 & 76.3 & 76.8 \n\\\\\n\\bottomrule\n\\end{tabular}}\n\\end{table*}\n\n\\begin{table*}[]\n\\centering\n\\caption{Tatoeba results (Accuracy) for each language}\n\\label{tab:tatoeba_results}\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{l|cccccccccccccccccc}\n\\toprule\nLang. & af & ar & bg & bn & de & el & es & et & eu & fa & fi & fr & he & hi & hu & id & it & ja \\\\\nBERT & 42.7 & 25.8 & 49.3 & 17 & 77.2 & 29.8 & 68.7 & 29.3 & 25.5 & 46.1 & 39 & 66.3 & 41.9 & 34.8 & 38.7 & 54.6 & 58.4 & 42 \\\\\nXLM & 43.2 & 18.2 & 40 & 13.5 & 66.2 & 25.6 & 58.4 & 24.8 & 17.1 & 32.2 & 32.2 & 54.5 & 32.1 & 26.5 & 30.1 & 45.9 & 56.5 & 40 \\\\\nXLMR & \\textbf{58.2} & \\textbf{47.5} & \\textbf{71.6} & \\textbf{43} & \\textbf{88.8} & \\textbf{61.8} & \\textbf{75.7} & \\textbf{52.2} & \\textbf{35.8} & \\textbf{70.5} & \\textbf{71.6} & \\textbf{73.7} & \\textbf{66.4} & \\textbf{72.2} & \\textbf{65.4} & \\textbf{77} & \\textbf{68.3} & \\textbf{60.6} \\\\\n\\midrule\n & jv & ka & kk & ko & ml & mr & nl & pt & ru & sw & ta & te & th & tl & tr & ur & vi & zh \\\\\n \\midrule\nBERT & 17.6 & 20.5 & 27.1 & 38.5 & 19.8 & 20.9 & 68 & 69.9 & 61.2 & 11.5 & 14.3 & 16.2 & 13.7 & 16 & 34.8 & \\textbf{31.6} & 62 & \\textbf{71.6} \\\\\nXLM & \\textbf{22.4} & 22.9 & 17.9 & 25.5 & 20.1 & 13.9 & 59.6 & 63.9 & 44.8 & 12.6 & 20.2 & 12.4 & \\textbf{31.8} & 14.8 & 26.2 & 18.1 & 47.1 & 42.2 \\\\\nXLMR & 14.1 & \\textbf{52.1} & \\textbf{48.5} & \\textbf{61.4} & \\textbf{65.4} & \\textbf{56.8} & \\textbf{80.8} & \\textbf{82.2} & \\textbf{74.1} & \\textbf{20.3} & \\textbf{26.4} & \\textbf{35.9} & 29.4 & \\textbf{36.7} & \\textbf{65.7} & 24.3 & \\textbf{74.7} & 68.3 \\\\\n\\bottomrule\n\\end{tabular}}\n\\end{table*}\n\n\nWe show the performance of mBERT on entities in the target language NER dev data that were seen and not seen in the English NER training data in Table \\ref{tab:mbert-entities-ner}. For simplicity, we count an entity as occurring in the English training data if a subset of at least two tokens matches with an entity in the English training data. As most matching entities in the target language data only consist of up to two tokens, are somewhat frequent, and consist only of Latin characters, we provide the performance on all entities fitting each criterion respectively for comparison. For all target languages in the table except Spanish, entities that appeared in the English training data are more likely to be tagged correctly than ones that did not. The differences are largest for two languages that are typologically distant to English, Indonesian (id) and Swahili (sw). For most languages, entities that appear in the English training data are similarly likely to be correctly classified as entities that are either frequent, appear in Latin characters, or are short. However, for Swahili and Basque (eu), mBERT does much better on entities that appeared in the English training data compared to the comparison entities. Another interesting case is Georgian (ka), which uses a unique script. The NER model is very good at recognizing entities that are written in Latin script but performs less well on entities in Georgian script.\n\n\\section{Sentence representations across all layers} $\\:$ For sentence retrieval tasks, we analyze whether the multilingual sentence representations obtained from all layers are well-aligned in the embedding spaces. Without fine-tuning on any parallel sentences at all, we explore three ways of extracting the sentence representations from all the models: (1) the embeddings of the first token in the last layer, also known as [CLS] token; (2) the average word embeddings in each layer; (3) the concatenation of the average word embeddings in the bottom, middle, and top 4 layers, i.e., Layer 1 to 4 (bottom), Layer 5 to 8 (middle), Layer 9 to 12 (top). Figure~\\ref{fig:sent_embed} shows the F1 scores of the average word embeddings in each layer of mBERT in the BUCC task. We observe that the average word embeddings in the middle layers, e.g., Layer 6 to 8, perform better than that in the bottom or the top layers. In Table~\\ref{tab:sent_emb}, we show the performance of these three types of sentence embeddings in the BUCC task. The embeddings of the CLS token perform relatively bad in cross-lingual retrieval tasks. We conjecture that the CLS embeddings highly abstract the semantic meaning of a sentence, while they lose the token-level information which is important for matching two translated sentences in two languages. With respect to the concatenation of average word embeddings from four continuous layers, We also observe that embeddings from the middle layers perform better than that from the bottom and top layers. Average word embeddings in the middle individual layer perform comparative to the concatenated embeddings from the middle four layers.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=1.0\\linewidth]{sentence_embed_mbert_fill.png}\n \\caption{Comparison of mBERT's sentence representations by averaging word embeddings in each layer in the BUCC task.}\n \\label{fig:sent_embed}\n\\end{figure}\n\n\\begin{table}[]\n\\caption{Three types of sentence embeddings from mBERT in BUCC tasks: (1) CLS token embeddings in the last layer; (2) Average word embeddings in the middle layers, i.e., Layer 6, 7, 8; (3) the concatenation of average word embeddings in the continuous four layers, i.e., Layer 1-4 (bottom layers), Layer 5-8 (middle layers), Layer 9-12 (top layers).}\n\\label{tab:sent_emb}\n\\centering\n\\begin{tabular}{lrrrr}\n\\toprule\nType & de & fr & zh & ru \\\\ \\midrule\nCLS & 3.88 & 4.73 & 0.89 & 2.15 \\\\\nLayer 6 & 51.29 & 56.32 & 41.38 & 38.81 \\\\\nLayer 7 & 62.51 & 62.62 & 49.99 & 51.84 \\\\\nLayer 8 & 64.32 & 62.46 & 50.49 & 53.58 \\\\\nLayer 1-4 & 6.98 & 12.3 & 12.05 & 4.33 \\\\\nLayer 5-8 & 63.12 & 63.42 & 52.84 & 51.67 \\\\\nLayer 9-12 & 53.97 & 52.68 & 44.18 & 43.13 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\n\\subsection{Language Families and Scripts}\nWe also report the performance of XLM-R in all the tasks across different language families and writing scripts in Figure~\\ref{fig:xlmr_correlation_lang_family_script}.\n\\begin{figure*}[!]\n\\centering\n\\begin{subfigure}{.5\\textwidth}\n \\centering\n \\includegraphics[width=\\linewidth]{scores_vs_lang_families_single_xlmr_fill.png}\n \\label{fig:xlmr_lang_families}\n\\end{subfigure}%\n\\begin{subfigure}{.5\\textwidth}\n \\centering\n \\includegraphics[width=\\linewidth]{scores_vs_lang_scripts_single_xlmr_fill.png}\n \\label{fig:xlmr_lang_scripts}\n\\end{subfigure}\n\\vspace{-0.8cm}\n\\caption{Performance of XLM-R across tasks grouped by language families (left) and scripts (right). The number of languages per group is in brackets and the groups are from low-resource to high-resource on the x-axis. We additionally plot the 3rd order polynomial fit for the minimum and maximum values for each group.}\n\\label{fig:xlmr_correlation_lang_family_script}\n\\end{figure*}\n\\begin{table}[!ht]\n\\caption{PAWS-X accuracy scores for each language.}\n\\label{tab:paws-x-results}\n\\resizebox{\\columnwidth}{!}{\n\\begin{tabular}{l|ccccccc|c}\n\\toprule\nModel & en & de & es & fr & ja & ko & zh & \\textbf{avg} \\\\\n\\midrule\nmBERT & 94.0 & 85.7 & 87.4 & 87.0 & 73.0 & 69.6 & 77.0 & 81.9 \\\\\nXLM & 94.0 & 85.9 & 88.3 & 87.4 & 69.3 & 64.8 & 76.5 & 80.9 \\\\\nXLMR & \\textbf{94.7} & \\textbf{89.7} & \\textbf{90.1} & \\textbf{90.4} & \\textbf{78.7} & \\textbf{79.0} & \\textbf{82.3} & \\textbf{86.4} \\\\\nMMTE & 93.1 & 85.1 & 87.2 & 86.9 & 72.0 & 69.2 & 75.9 & 81.3 \\\\\n\\midrule\n\\textit{Translate-train} & 94.0 & 87.5 & 89.4 & 89.6 & 78.6 & 81.6 & 83.5 & 86.3 \\\\\n\\textit{\\begin{tabular}[c]{@{}l@{}}Translate-train\\\\ (multi-task)\\end{tabular}} & \\textbf{94.5} & \\textbf{90.5} & \\textbf{91.6} & \\textbf{91.7} & \\textbf{84.4} & \\textbf{83.9} & \\textbf{85.8} & \\textbf{88.9} \\\\\n\\textit{Translate-test} & 93.5 & 88.2 & 89.3 & 87.4 & 78.4 & 76.6 & 77.6 & 84.4 \\\\\n\\bottomrule\n\\end{tabular}}\n\\end{table}\n\n\\begin{table}[h]\n\\centering\n\\caption{BUCC results (F1 scores) for each languages.}\n\\label{tab:bucc_results}\n\\begin{tabular}{l|cccc|c}\n\\toprule\nModel & de & fr & ru & zh & avg \\\\ \n\\midrule\nBERT & 62.5 & 62.6 & 51.8 & 50.0 & 56.7 \\\\\nXLM & 56.3 & 63.9 & 60.6 & 46.6 & 56.8 \\\\\nXLMR & 67.5 & \\textbf{66.5} & \\textbf{73.5} & \\textbf{56.7} & \\textbf{66.0} \\\\\nMMTE & \\textbf{67.9} & 63.9 & 54.3 & 53.3 & 59.8 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\n\\section{Results for each task and language}\n\nWe show the detailed results for all tasks and languages in Tables \\ref{tab:xnli_results} (XNLI), \\ref{tab:paws-x-results} (PAWS-X), \\ref{tab:pos_results} (POS), \\ref{tab:ner_results} (NER), \\ref{tab:xquad-results} (XQuAD), \\ref{tab:mlqa_results} (MLQA), \\ref{tab:tydiqa_results} (TyDiQA-GoldP), \\ref{tab:bucc_results} (BUCC), and \\ref{tab:tatoeba_results} (Tatoeba).\n\n\\begin{table*}[]\n\\caption{XQuAD results (F1 \/ EM) for each language.}\n\\label{tab:xquad-results}\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{l|ccccccccccc|c}\n\\toprule\nModel & en & ar & de & el & es & hi & ru & th & tr & vi & zh & \\textbf{avg} \\\\\n\\midrule\nmBERT & 83.5 \/ 72.2 & 61.5 \/ 45.1 & 70.6 \/ 54.0 & 62.6 \/ 44.9 & 75.5 \/ 56.9 & 59.2 \/ 46.0 & 71.3 \/ 53.3 & 42.7 \/ 33.5 & 55.4 \/ 40.1 & 69.5 \/ 49.6 & 58.0 \/ 48.3 & 64.5 \/ 49.4 \\\\\nXLM & 74.2 \/ 62.1 & 61.4 \/ 44.7 & 66.0 \/ 49.7 & 57.5 \/ 39.1 & 68.2 \/ 49.8 & 56.6 \/ 40.3 & 65.3 \/ 48.2 & 35.4 \/ 24.5 & 57.9 \/ 41.2 & 65.8 \/ 47.6 & 49.7 \/ 39.7 & 59.8 \/ 44.3 \\\\\nXLMR & \\textbf{86.5 \/ 75.7} & \\textbf{68.6 \/ 49.0} & \\textbf{80.4 \/ 63.4} & \\textbf{79.8 \/ 61.7} & \\textbf{82.0 \/ 63.9} & \\textbf{76.7 \/ 59.7} & \\textbf{80.1 \/ 64.3} & \\textbf{74.2 \/ 62.8} & \\textbf{75.9 \/ 59.3} & \\textbf{79.1 \/ 59.0} & \\textbf{59.3 \/ 50.0} & \\textbf{76.6 \/ 60.8} \\\\\nMMTE & 80.1 \/ 68.1 & 63.2 \/ 46.2 & 68.8 \/ 50.3 & 61.3 \/ 35.9 & 72.4 \/ 52.5 & 61.3 \/ 47.2 & 68.4 \/ 45.2 & 48.4 \/ 35.9 & 58.1 \/ 40.9 & 70.9 \/ 50.1 & 55.8 \/ 36.4 & 64.4 \/ 46.2 \\\\\n\\midrule\n\\textit{Translate-train} & 83.5 \/ 72.2 & 68.0 \/ 51.1 & 75.6 \/ 60.7 & 70.0 \/ 53.0 & 80.2 \/ 63.1 & 69.6 \/ 55.4 & 75.0 \/ 59.7 & 36.9 \/ 33.5 & 68.9 \/ 54.8 & 75.6 \/ 56.2 & 66.2 \/ 56.6 & 70.0 \/ 56.0 \\\\\n\\textit{\\begin{tabular}[c]{@{}l@{}}Translate-train\\\\ (multi-task)\\end{tabular}} & 86.0 \/ 74.5 & 71.0 \/ 54.1 & 78.8 \/ 63.9 & 74.2 \/ 56.1 & 82.4 \/ 66.2 & 71.3 \/ 56.2 & 78.1 \/ 63.0 & 38.1 \/ 34.5 & 70.6 \/ 55.7 & 78.5 \/ 58.8 & 67.7 \/ 58.7 & 72.4 \/ 58.3 \\\\\n\\textit{Translate-test} & \\textbf{87.9 \/ 77.1} & \\textbf{73.7 \/ 58.8} & \\textbf{79.8 \/ 66.7} & \\textbf{79.4 \/ 65.5} & \\textbf{82.0 \/ 68.4} & \\textbf{74.9 \/ 60.1} & \\textbf{79.9 \/ 66.7} & \\textbf{64.6 \/ 50.0} & \\textbf{67.4 \/ 49.6} & \\textbf{76.3 \/ 61.5} & \\textbf{73.7 \/ 59.1} & \\textbf{76.3 \/ 62.1} \\\\\n\\bottomrule\n\\end{tabular}}\n\n\\end{table*}\n\n\\begin{table*}[]\n\\centering\n\\caption{TyDiQA-GoldP results (F1 \/ EM) for each language.}\n\\label{tab:tydiqa_results}\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{l c c c c c c c c c c}\n\\toprule\nModel & en & ar & bn & fi & id & ko & ru & sw & te & avg \\\\ \\midrule\nmBERT & \\textbf{75.3 \/ 63.6} & 62.2 \/ 42.8 & 49.3 \/ 32.7 & 59.7 \/ 45.3 & 64.8 \/ 45.8 & \\textbf{58.8 \/ 50.0} & 60.0 \/ 38.8 & 57.5 \/ 37.9 & 49.6 \/ 38.4 & 59.7 \/ 43.9 \\\\\nXLM & 66.9 \/ 53.9 & 59.4 \/ 41.2 & 27.2 \/ 15.0 & 58.2 \/ 41.4 & 62.5 \/ 45.8 & 14.2 \/ 5.1 & 49.2 \/ 30.7 & 39.4 \/ 21.6 & 15.5 \/ 6.9 & 43.6 \/ 29.1 \\\\\nXLM-R & 71.5 \/ 56.8 & \\textbf{67.6 \/ 40.4} & \\textbf{64.0 \/ 47.8} & \\textbf{70.5 \/ 53.2} & \\textbf{77.4 \/ 61.9} & 31.9 \/ 10.9 & \\textbf{67.0 \/ 42.1} & \\textbf{66.1 \/ 48.1} & \\textbf{70.1 \/ 43.6} & \\textbf{65.1 \/ 45.0} \\\\\nMMTE & 62.9 \/ 49.8 & 63.1 \/ 39.2 & 55.8 \/ 41.9 & 53.9 \/ 42.1 & 60.9 \/ 47.6 & 49.9 \/ 42.6 & 58.9 \/ 37.9 & 63.1 \/ 47.2 & 54.2 \/ 45.8 & 58.1 \/ 43.8 \\\\ \\midrule\n\\textit{Translate-train} & 75.3 \/ 63.6 & 61.5 \/ 44.1 & 31.9 \/ 31.9 & 62.6 \/ 49.0 & 68.6 \/ 52.0 & 53.2 \/ 41.3 & 53.1 \/ 33.9 & 61.9 \/ 45.5 & 27.4 \/ 17.5 & 55.1 \/ 42.1 \\\\\n\\textit{\\begin{tabular}[c]{@{}l@{}}Translate-train\\\\ (multi-task)\\end{tabular}} & 73.2 \/ 62.5 & \\textbf{71.8 \/ 54.2} & 49.7 \/ 36.3 & 68.1 \/ 53.6 & 72.3 \/ 55.2 & 58.6 \/ 47.8 & 64.3 \/ 45.3 & 66.8 \/ 48.9 & 53.3 \/ 40.2 & 64.2 \/ 49.3 \\\\\n\\textit{Translate-test} & \\textbf{75.9 \/ 65.9} & 68.8 \/ 49.6 & \\textbf{66.7 \/ 48.1} & \\textbf{72.0 \/ 56.6} & \\textbf{76.8 \/ 60.9} & \\textbf{69.2 \/ 55.7} & \\textbf{71.4 \/ 54.3} & \\textbf{73.3 \/ 53.8} & \\textbf{75.1 \/ 59.2} & \\textbf{72.1 \/ 56.0} \\\\ \\midrule\n\\textit{Monolingual} & 75.3 \/ 63.6 & 80.5 \/ 67.0 & 71.1 \/ 60.2 & 75.6 \/ 64.1 & \\textbf{81.3 \/ 70.4} & 59.0 \/ 49.6 & 72.1 \/ 56.2 & 75.0 \/ 66.7 & 80.2 \/ 66.4 & 74.5 \/ 62.7 \\\\\n\\textit{\\begin{tabular}[c]{@{}l@{}}Monolingual\\\\ few-shot\\end{tabular}} & 63.1 \/ 50.9 & 61.3 \/ 44.8 & 58.7 \/ 49.6 & 51.4 \/ 38.1 & 70.4 \/ 58.1 & 45.4 \/ 38.4 & 56.9 \/ 42.6 & 55.4 \/ 46.3 & 65.2 \/ 49.6 & 58.7 \/ 46.5 \\\\\n\\textit{\\begin{tabular}[c]{@{}l@{}}Joint\\\\ monolingual\\end{tabular}} & \\textbf{77.6 \/ 69.3} & \\textbf{82.7 \/ 69.4} & \\textbf{79.6 \/ 69.9} & \\textbf{79.2 \/ 67.8} & 68.9 \/ 72.7 & \\textbf{68.9 \/ 59.4} & \\textbf{75.8 \/ 59.2} & \\textbf{81.9 \/ 74.3} & \\textbf{83.4 \/ 70.3} & \\textbf{77.6 \/ 68.0} \\\\\n\\bottomrule\n\\end{tabular}%\n}\n\\end{table*}\n\n\n\\begin{table*}[]\n\\centering\n\\caption{MLQA results (F1 \/ EM) for each language.}\n\\label{tab:mlqa_results}\n\\resizebox{0.8\\textwidth}{!}{\n\\begin{tabular}{l|ccccccc|c}\n\\toprule\nModel & en & ar & de & es & hi & vi & zh & avg \\\\\n\\toprule\nmBERT & 80.2 \/ 67.0 & 52.3 \/ 34.6 & 59.0 \/ 43.8 & 67.4 \/ 49.2 & 50.2 \/ 35.3 & 61.2 \/ 40.7 & 59.6 \/ 38.6 & 61.4 \/ 44.2 \\\\\nXLM & 68.6 \/ 55.2 & 42.5 \/ 25.2 & 50.8 \/ 37.2 & 54.7 \/ 37.9 & 34.4 \/ 21.1 & 48.3 \/ 30.2 & 40.5 \/ 21.9 & 48.5 \/ 32.6 \\\\\nXLM-R & \\textbf{83.5 \/ 70.6} & \\textbf{66.6 \/ 47.1} & \\textbf{70.1 \/ 54.9} & \\textbf{74.1 \/ 56.6} & \\textbf{70.6 \/ 53.1} & \\textbf{74 \/ 52.9} & \\textbf{62.1 \/ 37.0} & \\textbf{71.6 \/ 53.2} \\\\\nMMTE & 78.5 \/ -- & 56.1 \/ -- & 58.4 \/ -- & 64.9 \/ -- & 46.2 \/ -- & 59.4 \/ -- & 58.3 \/ -- & 60.3 \/ 41.4 \\\\\n\\midrule\n\\textit{Translate-train} & 80.2 \/ 67.0 & 55.0 \/ 35.6 & 64.4 \/ 49.4 & 70.0 \/ 52.0 & 60.1 \/ 43.4 & 65.7 \/ 45.5 & 63.9 \/ 42.7 & 65.6 \/ 47.9 \\\\\n\\textit{\\begin{tabular}[c]{@{}l@{}}Translate-train\\\\ (multi-task)\\end{tabular}} & 80.7 \/ 67.7 & 58.9 \/ 39.0 & 66.0 \/ 51.6 & 71.3 \/ 53.7 & 62.4 \/ 45.0 & 67.9 \/ 47.6 & 66.0 \/ 43.9 & 67.6 \/ 49.8 \\\\\n\\textit{Translate-test} & \\textbf{83.8 \/ 71.0} & \\textbf{65.3 \/ 46.4} & \\textbf{71.2 \/ 54.0} & \\textbf{73.9 \/ 55.9} & \\textbf{71.0 \/ 55.1} & \\textbf{70.6 \/ 54.0} & \\textbf{67.2 \/ 50.6} & \\textbf{71.9 \/ 55.3} \\\\\n\\bottomrule\n\\end{tabular}}\n\\end{table*}\n\n\\begin{table*}[]\n\\centering\n\\caption{POS results (Accuracy) for each language}\n\\label{tab:pos_results}\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{l|ccccccccccccccccc}\n\\toprule\nLang. & af & ar & bg & de & el & en & es & et & eu & fa & fi & fr & he & hi & hu & id & it \\\\\n\\midrule\nmBERT & 86.6 & 56.2 & 85.0 & 85.2 & 81.1 & 95.5 & 86.9 & 79.1 & 60.7 & 66.7 & 78.9 & 84.2 & 56.2 & 67.2 & 78.3 & 71.0 & 88.4 \\\\\nXLM & 88.5 & 63.1 & 85.0 & 85.8 & 84.3 & 95.4 & 85.8 & 78.3 & 62.8 & 64.7 & 78.4 & 82.8 & 65.9 & 66.2 & 77.3 & 70.2 & 87.4 \\\\\nXLMR & \\textbf{89.8} & \\textbf{67.5} & \\textbf{88.1} & \\textbf{88.5} & \\textbf{86.3} & 96.1 & \\textbf{88.3} & \\textbf{86.5} & \\textbf{72.5} & \\textbf{70.6} & \\textbf{85.8} & \\textbf{87.2} & \\textbf{68.3} & \\textbf{76.4} & \\textbf{82.6} & 72.4 & \\textbf{89.4} \\\\\nMMTE & 86.2 & 65.9 & 87.2 & 85.8 & 77.7 & \\textbf{96.6} & 85.8 & 81.6 & 61.9 & 67.3 & 81.1 & 84.3 & 57.3 & 76.4 & 78.1 & \\textbf{73.5} & 89.2 \\\\\n\\midrule\n & ja & kk & ko & mr & nl & pt & ru & ta & te & th & tl & tr & ur & vi & yo & zh & avg \\\\\n \\midrule\nmBERT & \\textbf{49.2} & 70.5 & 49.6 & 69.4 & 88.6 & 86.2 & 85.5 & 59.0 & 75.9 & 41.7 & 81.4 & 68.5 & 57.0 & 53.2 & \\textbf{55.7} & 61.6 & 71.5 \\\\\nXLM & 49.0 & 70.2 & 50.1 & 68.7 & 88.1 & 84.9 & 86.5 & 59.8 & 76.8 & 55.2 & 76.3 & 66.4 & 61.2 & 52.4 & 20.5 & 65.4 & 71.3 \\\\\nXLMR & 15.9 & \\textbf{78.1} & 53.9 & \\textbf{80.8} & \\textbf{89.5} & \\textbf{87.6} & \\textbf{89.5} & \\textbf{65.2} & \\textbf{86.6} & \\textbf{47.2} & \\textbf{92.2} & \\textbf{76.3} & \\textbf{70.3} & \\textbf{56.8} & 24.6 & 25.7 & \\textbf{73.8} \\\\\nMMTE & 48.6 & 70.5 & \\textbf{59.3} & 74.4 & 83.2 & 86.1 & 88.1 & 63.7 & 81.9 & 43.1 & 80.3 & 71.8 & 61.1 & 56.2 & 51.9 & \\textbf{68.1} & 73.5 \\\\\n\\bottomrule\n\\end{tabular}}\n\\end{table*}\n\n\\begin{table*}[]\n\\centering\n\\caption{NER results (F1 Score) for each language}\n\\label{tab:ner_results}\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{l|cccccccccccccccccccc}\n\\toprule\nLang. & en & af & ar & bg & bn & de & el & es & et & eu & fa & fi & fr & he & hi & hu & id & it & ja & jv \\\\\n\\midrule\nmBERT & \\textbf{85.2} & 77.4 & 41.1 & 77.0 & 70.0 & 78.0 & 72.5 & 77.4 & 75.4 & \\textbf{66.3} & 46.2 & 77.2 & 79.6 & 56.6 & 65.0 & 76.4 & \\textbf{53.5} & \\textbf{81.5} & 29.0 & \\textbf{66.4} \\\\\nXLM & 82.6 & 74.9 & 44.8 & 76.7 & 70.0 & 78.1 & 73.5 & 74.8 & 74.8 & 62.3 & 49.2 & 79.6 & 78.5 & 57.7 & 66.1 & 76.5 & 53.1 & 80.7 & 23.6 & 63.0 \\\\\nXLMR & 84.7 & \\textbf{78.9} & \\textbf{53.0} & \\textbf{81.4} & \\textbf{78.8} & \\textbf{78.8} & \\textbf{79.5} & \\textbf{79.6} & \\textbf{79.1} & 60.9 & \\textbf{61.9} & \\textbf{79.2} & \\textbf{80.5} & \\textbf{56.8} & \\textbf{73.0} & \\textbf{79.8} & 53.0 & 81.3 & 23.2 & 62.5 \\\\\nMMTE & 77.9 & 74.9 & 41.8 & 75.1 & 64.9 & 71.9 & 68.3 & 71.8 & 74.9 & 62.6 & 45.6 & 75.2 & 73.9 & 54.2 & 66.2 & 73.8 & 47.9 & 74.1 & \\textbf{31.2} & 63.9 \\\\\n\\midrule\n & ka & kk & ko & ml & mr & ms & my & nl & pt & ru & sw & ta & te & th & tl & tr & ur & vi & yo & zh \\\\\n \\midrule\nmBERT & 64.6 & 45.8 & 59.6 & 52.3 & 58.2 & \\textbf{72.7} & 45.2 & 81.8 & 80.8 & 64.0 & 67.5 & 50.7 & 48.5 & 3.6 & 71.7 & 71.8 & 36.9 & 71.8 & \\textbf{44.9} & \\textbf{42.7} \\\\\nXLM & 67.7 & \\textbf{57.2} & 26.3 & 59.4 & 62.4 & 69.6 & 47.6 & 81.2 & 77.9 & 63.5 & 68.4 & 53.6 & 49.6 & 0.3 & \\textbf{78.6} & 71.0 & 43.0 & 70.1 & 26.5 & 32.4 \\\\\nXLMR & \\textbf{71.6} & 56.2 & \\textbf{60.0} & \\textbf{67.8} & \\textbf{68.1} & 57.1 & \\textbf{54.3} & \\textbf{84.0} & \\textbf{81.9} & \\textbf{69.1} & \\textbf{70.5} & \\textbf{59.5} & \\textbf{55.8} & 1.3 & 73.2 & \\textbf{76.1} & \\textbf{56.4} & \\textbf{79.4} & 33.6 & 33.1 \\\\\nMMTE & 60.9 & 43.9 & 58.2 & 44.8 & 58.5 & 68.3 & 42.9 & 74.8 & 72.9 & 58.2 & 66.3 & 48.1 & 46.9 & \\textbf{3.9} & 64.1 & 61.9 & 37.2 & 68.1 & 32.1 & 28.9 \\\\\n\\bottomrule\n\\end{tabular}}\n\\end{table*}\n\n\n\n\n\n\n\n\n\\section{Introduction}\n\nIn natural language processing (NLP), there is a pressing urgency to build systems that serve \\emph{all} of the world's approximately 6,900 languages to overcome language barriers and enable universal information access for the world's citizens \\cite{Ruder2019survey,aharoni2019massively, Arivazhagan2019massively_multilingual}.\nAt the same time, building NLP systems for most of these languages is challenging due to a stark lack of data.\nLuckily, many languages have similarities in syntax or vocabulary, and multilingual learning approaches that train on multiple languages while leveraging the shared structure of the input space have begun to show promise as ways to alleviate data sparsity. Early work in this direction focused on single tasks, such as grammar induction \\cite{Snyder2009unsupervised}, part-of-speech (POS) tagging \\cite{Tackstrom2013token_and_type}, parsing \\cite{McDonald2011delexicalized}, and text classification \\cite{Klementiev2012inducing}. Over the last few years, there has been a move towards \\emph{general-purpose multilingual representations} that are applicable to many tasks, both\non the word level \\cite{mikolov2013exploiting,faruqui2014improving,artetxe2017learning} or the full-sentence level \\cite{Devlin2019bert,Lample2019xlm}. Despite the fact that such representations are intended to be general-purpose, evaluation of them has often been performed on a very limited and often disparate set of tasks---typically focusing on translation \\cite{Glavas2019,Lample2019xlm} and classification \\cite{Schwenk2018mldoc,Conneau2018xnli}---and typologically similar languages \\cite{conneau2018word}.\n\n\n\n\nTo address this problem and incentivize research on truly general-purpose cross-lingual representation and transfer learning, we introduce the Cross-lingual TRansfer Evaluation of Multilingual Encoders (\\textsc{xtreme}\\xspace) benchmark. \\textsc{xtreme}\\xspace covers 40 typologically diverse languages spanning 12 language families and includes 9 tasks that require reasoning about different levels of syntax or semantics.\\footnote{By typologically diverse, we mean languages that span a wide set of linguistic phenomena such as compounding, inflection, derivation, etc.~which occur in many of the world's languages.} In addition, we introduce \\emph{pseudo} test sets as diagnostics that cover all 40 languages by automatically translating the English test set of the natural language inference and question-answering dataset to the remaining languages. \n\n\\textsc{xtreme}\\xspace focuses on the \\emph{zero-shot cross-lingual transfer} scenario, where annotated training data is provided in English but none is provided in the language to which systems must transfer.%\n\\footnote{This is done both for efficiency purposes (as it only requires testing, not training, on each language) and practical considerations (as annotated training data is not available for many languages).}\nWe evaluate a range of state-of-the-art machine translation (MT) and multilingual representation-based approaches to performing this transfer.\nWe find that while state-of-the-art models come close to human performance in English on many of the tasks we consider, performance drops significantly when evaluated on other languages.\nOverall, performance differences are highest for syntactic and sentence retrieval tasks.\nFurther, while models do reasonably well in most languages in the Indo-European family, we observe lower performance particularly for Sino-Tibetan, Japonic, Koreanic, and Niger-Congo languages. \n\nIn sum, our contributions are the following: (i) We release a suite of 9 cross-lingual benchmark tasks covering 40 typologically diverse languages. (ii) We provide an online platform and leaderboard for the evaluation of multilingual models. (iii) We provide a set of strong baselines, which we evaluate across all tasks, and release code to facilitate adoption. (iv) We provide an extensive analysis of limitations of state-of-the-art cross-lingual models.\n\n\\section{Related Work}\n\n\\noindent \\textbf{Cross-lingual representations} $\\:$ Early work focused on learning cross-lingual representations using either parallel corpora \\cite{gouws2015bilbowa,luong2015bilingual} or a bilingual dictionary to learn a linear transformation \\cite{mikolov2013exploiting,faruqui2014improving}. Later approaches reduced the amount of supervision required using self-training \\cite{artetxe2017learning} and unsupervised strategies such as adversarial training \\cite{conneau2018word}, heuristic initialisation \\cite{artetxe2018robust}, and optimal transport \\cite{zhang2017earth}. Building on advances in monolingual transfer learning \\cite{mccann2017learned,howard2018universal,peters2018deep,Devlin2019bert}, multilingual extensions of pretrained encoders have recently been shown to be effective for learning deep cross-lingual representations \\cite{eriguchi2018zero,Pires2019,Wu2019,Lample2019xlm,Siddhant2019evaluating}. \n\n\\noindent \\textbf{Cross-lingual evaluation} $\\:$ One pillar of the evaluation of cross-lingual representations has been translation, either on the word level (\\emph{bilingual lexicon induction}) or on the sentence level (\\emph{machine translation}). In most cases, evaluation has been restricted to typologically related languages and similar domains; approaches have been shown to fail in less favorable conditions \\cite{Glavas2019,Vulic2019,Guzman2019}. Past work has also reported issues with common datasets for bilingual lexicon induction \\cite{czarnowska2019dont,kementchedjhieva2019lost} and a weak correlation with certain downstream tasks \\cite{Glavas2019}. Translation, however, only covers one facet of a model's cross-lingual generalization ability. For instance, it does not capture differences in classification performance that are due to cultural differences \\cite{mohammad2016translation,smith2016does}.\n\nOn the other hand, cross-lingual approaches have been evaluated on a wide range of tasks, including dependency parsing \\cite{Schuster2019}, named entity recognition \\cite{Rahimi2019}, sentiment analysis \\citep{barnes2018bilingual}, natural language inference \\cite{Conneau2018xnli}, document classification \\cite{Schwenk2018mldoc}, and question answering \\cite{artetxe2019cross,Lewis2019mlqa}. Evaluation on a single task is problematic as past work has noted potential issues with standard datasets: MLDoc \\cite{Schwenk2018mldoc} can be solved by matching keywords \\cite{artetxe2019cross}, while MultiNLI, the dataset from which XNLI \\cite{Conneau2018xnli} was derived, contains superficial cues that can be exploited \\cite{Gururangan2018}. Evaluation on multiple tasks is thus necessary to fairly compare cross-lingual models. Benchmarks covering multiple tasks like GLUE \\cite{Wang2019glue} and SuperGLUE \\cite{wang2019superglue} have arguably spurred research in monolingual transfer learning. In the cross-lingual setting, such a benchmark not only needs to cover a diverse set of tasks but also languages. \\textsc{xtreme}\\xspace aims to fill this gap.\n\n\\section{\\textsc{xtreme}\\xspace}\n\n\\subsection{Design principles}\n\\label{sec:design}\n\n\\begin{table*}[]\n\\centering\n\\caption{Characteristics of the datasets in \\textsc{xtreme}\\xspace for the zero-shot transfer setting. For tasks that have training and dev sets in other languages, we only report the English numbers. We report the number of test examples per target language and the nature of the test sets (whether they are translations of English data or independently annotated). The number in brackets is the size of the intersection with our selected languages. For NER and POS, sizes are in sentences. Struct. pred.: structured prediction. Sent. retrieval: sentence retrieval.}\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{l l r r r r r l l l}\n\\toprule\nTask & Corpus & $|$Train$|$ & $|$Dev$|$ & $|$Test$|$ & Test sets & $|$Lang.$|$ & Task & Metric & Domain \\\\ \\midrule\n\\multirow{2}{*}{Classification} & XNLI & 392,702 & 2,490 & 5,010 & translations & 15 & NLI & Acc. & Misc. \\\\\n& PAWS-X & 49,401 & 2,000 & 2,000 & translations & 7 & Paraphrase & Acc. & Wiki \/ Quora \\\\ \\midrule\n\\multirow{2}{*}{Struct. pred.} & POS & 21,253 & 3,974 & 47-20,436 & ind. annot. & 33 (90) & POS & F1 & Misc. \\\\\n& NER & 20,000 & 10,000 & 1,000-10,000 & ind. annot. & 40 (176) & NER & F1 & Wikipedia \\\\\n\\midrule\n\\multirow{3}{*}{QA} & XQuAD & \\multirow{2}{*}{87,599} & \\multirow{2}{*}{34,726} & 1,190 & translations & 11 & Span extraction & F1 \/ EM & Wikipedia \\\\\n& MLQA & & & 4,517--11,590 & translations & 7 & Span extraction & F1 \/ EM & Wikipedia \\\\ \n& TyDiQA-GoldP & 3,696 & 634 & 323--2,719 & ind. annot. & 9 & Span extraction & F1 \/ EM & Wikipedia \\\\ \\midrule\n\\multirow{2}{*}{Retrieval} & BUCC & - & - & 1,896--14,330 & - & 5 & Sent. retrieval & F1 & Wiki \/ news \\\\\n& Tatoeba & - & - & 1,000 & - & 33 (122) & Sent. retrieval & Acc. & misc. \\\\\n\\bottomrule\n\\end{tabular}%\n}\n\\label{tab:tasks}\n\\end{table*}\n\nGiven \\textsc{xtreme}\\xspace's goal of providing an accessible benchmark for the evaluation of cross-lingual transfer learning on a diverse and representative set of tasks and languages, we select the tasks and languages that make up the benchmark based on the following principles:\n\n\\noindent \\textbf{Task difficulty} $\\:$ Tasks should be sufficiently challenging so that cross-language performance falls short of human performance.\n\n\\noindent \\textbf{Task diversity} $\\:$ Tasks should require multilingual models to transfer their meaning representations at different levels, e.g.~words, phrases and sentences. For example, while classification tasks require sentence-level transfer of meaning, sequence labeling tasks like part-of-speech (POS) tagging or named entity recognition (NER) test the model's transfer capabilities at the word level.\n\n\\noindent \\textbf{Training efficiency} $\\:$ Tasks should be trainable on a single GPU for less than a day. This is to make the benchmark accessible, in particular to practitioners working with low-resource languages under resource constraints.\n\n\\noindent \\textbf{Multilinguality} $\\:$ We prefer tasks that cover as many languages and language families as possible.\n\n\\noindent \\textbf{Sufficient monolingual data} $\\:$ Languages should have sufficient monolingual data for learning useful pre-trained representations.\n\n\\noindent \\textbf{Accessibility} $\\:$ Each task should be available under a permissive license that allows the use and redistribution of the data for research purposes.\n\n\\subsection{Tasks}\n\n\\textsc{xtreme}\\xspace consists of nine tasks that fall into four different categories requiring reasoning on different levels of meaning. \nWe give an overview of all tasks in Table \\ref{tab:tasks}, and describe the task details as follows.\n\n\n\n\n\n\\noindent \\textbf{XNLI} $\\:$ The Cross-lingual Natural Language Inference corpus \\cite{Conneau2018xnli} asks whether a premise sentence entails, contradicts, or is neutral toward a hypothesis sentence. Crowd-sourced English data is translated to ten other languages by professional translators and used for evaluation, while the MultiNLI \\cite{Williams2018multinli} training data is used for training.\n\n\\noindent \\textbf{PAWS-X} $\\:$ The Cross-lingual Paraphrase Adversaries from Word Scrambling \\cite{Yang2019paws-x} dataset requires to determine whether two sentences are paraphrases. A subset of the PAWS dev and test sets \\cite{Zhang2019paws} was translated to six other languages by professional translators and is used for evaluation, while the PAWS training set is used for training.\n\n\n\\noindent \\textbf{POS} $\\:$ We use POS tagging data from the Universal Dependencies v2.5 \\cite{nivre2018universal} treebanks, which cover 90 languages. Each word is assigned one of 17 universal POS tags. We use the English training data for training and evaluate on the test sets of the target languages.\n\n\n\\noindent \\textbf{NER} $\\:$ For NER, we use the \\texttt{Wikiann} \\cite{Pan2017} dataset. Named entities in Wikipedia were automatically annotated with \\texttt{LOC}, \\texttt{PER}, and \\texttt{ORG} tags in IOB2 format using a combination of knowledge base properties, cross-lingual and anchor links, self-training, and data selection. \nWe use the balanced train, dev, and test splits from \\citet{Rahimi2019}.\n\n\n\n\\noindent \\textbf{XQuAD} $\\:$ The Cross-lingual Question Answering Dataset \\cite{artetxe2019cross} requires identifying the answer to a question as a span in the corresponding paragraph. A subset of the English SQuAD v1.1 \\cite{Rajpurkar2016squad} dev set was translated into ten other languages by professional translators and is used for evaluation.\n\n\\noindent \\textbf{MLQA} $\\:$ The Multilingual Question Answering \\cite{Lewis2019mlqa} dataset is another cross-lingual question answering dataset similar to XQuAD. The evaluation data for English and six other languages was obtained by automatically mining target language sentences that are parallel to sentences in English from Wikipedia, crowd-sourcing annotations in English, and translating the question and aligning the answer spans in the target languages. For both XQuAD and MLQA, we use the SQuAD v1.1 training data for training and evaluate on the test data of the corresponding task.\n\n\\noindent \\textbf{TyDiQA-GoldP} $\\:$ We use the gold passage version of the Typologically Diverse Question Answering \\cite{Clark2020tydiqa} dataset, a benchmark for information-seeking question answering, which covers nine languages. The gold passage version is a simplified version of the primary task, which uses only the gold passage as context and excludes unanswerable questions. It is thus similar to XQuAD and MLQA, while being more challenging as questions have been written without seeing the answers, leading to $3\\times$ and $2\\times$ less lexical overlap compared to XQuAD and MLQA respectively. We use the English training data for training and evaluate on the test sets of the target languages.\n\n\n\\noindent \\textbf{BUCC} $\\:$ The goal of the second and third shared task of the workshop on Building and Using Parallel Corpora \\cite{zweigenbaum2017overview,zweigenbaum2018overview} is to extract parallel sentences from a comparable corpus between English and four other languages. The dataset provides train and test splits for each language. For simplicity, we evaluate representations on the test sets directly without fine-tuning and calculate similarity using cosine similarity.\\footnote{Results can be improved using more sophisticated similarity metrics \\cite{Artetxe2019massively}.}\n\n\n\\noindent \\textbf{Tatoeba} $\\:$ We use the Tatoeba dataset \\cite{Artetxe2019massively}, which consists of up to 1,000 English-aligned sentence pairs covering 122 languages. We find the nearest neighbour using cosine similarity and calculate error rate.\n\n\\subsection{Languages}\n\nAs noted in Section \\ref{sec:design}, we choose our target languages based on availability of monolingual data, and typological diversity.\nWe use the number of articles in Wikipedia as a proxy for the amount of monolingual data available online. In order to strike a balance between language diversity and availability of monolingual data, we select all languages out of the top 100 Wikipedias\\footnote{\\url{https:\/\/meta.wikimedia.org\/wiki\/List_of_Wikipedias}} with the most articles as of December 2019.\\footnote{This also has the benefit that they are covered by state-of-the-art methods such as mBERT and XLM.} We first select all languages that appear in at least three of our benchmark datasets. This leaves us with 19 languages, most of which are Indo-European or major world languages. We now select 21 additional languages that appear in at least one dataset and come from less represented language families. Wherever possible, we choose at least two languages per family.\\footnote{For the Austro-Asiatic, Kartvelian, and Kra-Dai families as well as for isolates, we only obtain one language.}\n\nIn total, \\textsc{xtreme}\\xspace covers the following 40 languages (shown with their ISO 639-1 codes for brevity) belonging to 12 language families and two isolates: af, ar, bg, bn, de, el, en, es, et, eu, fa, fi, fr, he, hi, hu, id, it, ja, jv, ka, kk, ko, ml, mr, ms, my, nl, pt, ru, sw, ta, te, th, tl, tr, ur, vi, yo, and zh.\nWe provide a detailed overview of these languages in terms of their number of Wikipedia articles, linguistic features, and coverage in \\textsc{xtreme}\\xspace in the appendix.\n\nWhile \\textsc{xtreme}\\xspace covers these languages in the sense that there is gold standard data in at least one task in each language, this does not mean that it covers all aspects of each language that are necessary for transfer. Languages may reveal different characteristics based on the task, domain, and register in which they are used. \\textsc{xtreme}\\xspace thus only serves as a glimpse into a model's true cross-lingual generalization capability.\n\n\\subsection{Pseudo test data for analyses}\n\n\\textsc{xtreme}\\xspace covers 40 languages overall. Evaluation across the majority of languages is only possible for a subset of tasks, i.e. POS, NER, and Tatoeba.\nAs additional diagnositics and to enable a broader comparison across languages for a more diverse set of tasks, we automatically translate the English portions of a representative classification and QA task to the remaining languages using an in-house translation system.\\footnote{Details of our translation system are provided in the appendix.} We choose XNLI and XQuAD as both have test sets that are translations of the English data by professional translators.\n\nWe first verify that performance on the translated test sets is a good proxy for performance on the gold standard test sets. We report the detailed results in the appendix. For XQuAD, the automatically translated test sets underestimate mBERT's true performance by 3.0 F1 \/ 0.2 EM points, similar to the 2.6 F1 points reported by \\citet{Agic2018baselines} when translating the test data to other languages.\\footnote{Note that even human translated test sets may underestimate a model's true cross-lingual generalization ability as such \\emph{translationese} has been shown to be less lexically diverse than naturally composed language \\cite{koppel2011translationese}.} For XNLI, the automatically translated test sets overestimate the true prediction accuracy by 2.4 points. In order to measure the translation quality between the human-translated test data and our pseudo test data, we compute the BLEU score, and the chrF score~\\cite{popovic-2015-chrf}, which is suitable for measuring the translation quality of some languages such as Chinese and Russian. For the 14 languages in XNLI, we obtain average scores of 34.2 BLEU and 58.9 chrF scores on our pseudo test data compared to the reference translations, which correlate with a Pearson's $\\rho$ of 0.57 and 0.28 respectively with mBERT performance.\n\nTranslating the English data to the remaining languages yields 40-way parallel pseudo test data that we employ for analyses in Section \\ref{sec:analyses}.\n\n\\section{Experiments}\n\n\\subsection{Training and evaluation setup}\n\n\\textsc{xtreme}\\xspace focuses on the evaluation of multilingual representations. We do not place any restriction on the amount or nature of the monolingual data used for pretraining multilingual representations. However, we request authors to be explicit about the data they use for training, in particular any cross-lingual signal. In addition, we suggest authors should not use any additional labelled data in the target task beyond the one that is provided.\n\nFor evaluation, we focus on \\emph{zero-shot cross-lingual transfer} with English as the source language as this is the most common setting for the evaluation of multilingual representations and as many tasks only have training data available in English. Although English is not generally the best source language for cross-lingual transfer for all target languages \\cite{Lin2019transfer_languages}, this is still the most practically useful setting. A single source language also facilitates evaluation as models only need to be trained once and can be evaluated on all other languages.\\footnote{Future work may also consider multi-source transfer, which is interesting particularly for low-resource languages, and transfer to unknown languages or unknown language-task combinations.}\n\nConcretely, pretrained multilingual representations are fine-tuned on English labelled data of an \\textsc{xtreme}\\xspace task. The model is then evaluated on the test data of the task in the target languages.\n\n\\subsection{Baselines}\n\nWe evaluate a number of strong baselines and state-of-the-art models. The approaches we consider learn multilingual representations via self-supervision or leverage translations---either for representation learning or for training models in the source or target language. We focus on models that learn deep contextual representations as these have achieved state-of-the-art results on many tasks. For comparability among the representation learning approaches, we focus on models that learn a multilingual embedding space between all languages in \\textsc{xtreme}\\xspace. We encourage future work to focus on these languages to capture as much language diversity as possible. We report hyper-parameters in the appendix. All hyper-parameter tuning is done on English validation data. We encourage authors evaluating on \\textsc{xtreme}\\xspace to do the same.\n\n\\noindent \\textbf{mBERT} $\\:$ Multilingual BERT \\cite{Devlin2019bert} is a transformer model \\cite{Vaswani2017attention} that has been pretrained on the Wikipedias of 104 languages using masked language modelling (MLM).\n\n\\noindent \\textbf{XLM} $\\:$ XLM \\cite{Lample2019xlm} uses a similar pretraining objective as mBERT with a larger model, a larger shared vocabulary, and trained on the same Wikipedia data covering 100 languages.\n\n\\noindent \\textbf{XLM-R} $\\:$ XLM-R Large \\cite{Conneau2019xlm-r} is similar to XLM but was trained on more than a magnitude more data from the web covering 100 languages.\n\n\\noindent \\textbf{MMTE} $\\:$ The massively multilingual translation encoder is part of an NMT model that has been trained on in-house parallel data of 103 languages extracted from the web \\cite{Arivazhagan2019massively_multilingual}. For transfer, we fine-tune the encoder of the model \\cite{Siddhant2019evaluating}.\n\n\n\\noindent \\textbf{Translate-train} $\\:$ For many language pairs, an MT model may be available, which can be used to obtain data in the target language. To evaluate the impact of using such data, we translate the English training data into the target language using our in-house MT system. We then fine-tune mBERT on the translated data. We provide details on how we align answer spans in the source and target language for the QA tasks in the appendix. We do not provide translation-based baselines for structured prediction tasks due to an abundance of in-language data and a requirement for annotation projection.\n\n\\noindent \\textbf{Translate-train multi-task} $\\:$ We also experiment with a multi-task version of the translate-train setting where we fine-tune mBERT on the combined translated training data of all languages jointly.\n\n\\noindent \\textbf{Translate-test} $\\:$ Alternatively, we train the English BERT-Large \\cite{Devlin2019bert} model on the English training data and evaluate it on test data that we translated from the target language to English using our in-house MT system. \n\n\\noindent \\textbf{In-language model} $\\:$ For the POS, NER, and TyDiQA-GoldP tasks where target-language training data is available, we fine-tune mBERT on monolingual data in the target language to estimate how useful target language labelled data is compared to labelled data in a source language.\n\n\\noindent \\textbf{In-language few-shot} $\\:$ In many cases, it may be possible to procure a small number of labelled examples in the target language \\cite{Eisenschlos2019multifit}. To evaluate the viability of such an approach, we additionally compare against an mBERT model fine-tuned on 1,000 target language examples for the tasks where monolingual training data is available in the target languages.\n\n\\begin{table*}[]\n\\caption{Overall results of baselines across all \\textsc{xtreme}\\xspace tasks. Translation-based baselines are not meaningful for sentence retrieval. We provide in-language baselines where target language training data is available. Note that for the QA tasks, translate-test performance is not directly comparable to the other scores as a small number of test questions were discarded and alignment is measured on the English data.\n}\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{l c cc cc ccc cc}\n\\toprule\n\\multirow{2}{*}{Model} & \\multirow{2}{*}{Avg} & \\multicolumn{2}{c}{Pair sentence} & \\multicolumn{2}{c}{Structured prediction} & \\multicolumn{3}{c}{Question answering} & \\multicolumn{2}{c}{Sentence retrieval} \\\\\n & & XNLI & PAWS-X & POS & NER & XQuAD & MLQA & TyDiQA-GoldP & BUCC & Tatoeba \\\\\n\\midrule\nMetrics & & Acc. & Acc. & F1 & F1 & F1 \/ EM & F1 \/ EM & F1 \/ EM & F1 & Acc. \\\\ \\midrule\n\\multicolumn{11}{l}{\\emph{Cross-lingual zero-shot transfer (models are trained on English data)}} \\\\ \\midrule\nmBERT & 59.8 & 65.4 & 81.9 & 71.5 & 62.2 & 64.5 \/ 49.4 & 61.4 \/ 44.2 & 59.7 \/ 43.9 & 56.7 & 38.7\\\\\nXLM & 55.7 & 69.1 & 80.9 & 71.3 & 61.2 & 59.8 \/ 44.3 & 48.5 \/ 32.6 & 43.6 \/ 29.1 & 56.8 & 32.6\\\\\nXLM-R Large & 68.2 & 79.2 & 86.4 & 73.8 & 65.4 & 76.6 \/ 60.8 & 71.6 \/ 53.2 & 65.1 \/ 45.0 & 66.0 & 57.3\\\\\nMMTE & 59.5 & 67.4 & 81.3 & 73.5 & 58.3 & 64.4 \/ 46.2 & 60.3 \/ 41.4 & 58.1 \/ 43.8 & 59.8 & 37.9 \\\\ \\midrule\n\\multicolumn{11}{l}{\\emph{Translate-train (models are trained on English training data translated to the target language)}} \\\\ \\midrule\nmBERT & - & 74.6 & 86.3 & - & - & 70.0 \/ 56.0 & 65.6 \/ 48.0 & 55.1 \/ 42.1 & - & - \\\\\nmBERT, multi-task & - & 75.1 & 88.9 & - & - & 72.4 \/ 58.3 & 67.6 \/ 49.8 & 64.2 \/ 49.3 & - & - \\\\ \\midrule\n\\multicolumn{11}{l}{\\emph{Translate-test (models are trained on English data and evaluated on target language data translated to English)}} \\\\ \\midrule\nBERT-large & - & 76.8 & 84.4 & - & - & 76.3 \/ 62.1 & 72.9 \/ 55.3 & 72.1 \/ 56.0 & - & - \\\\\n\\midrule\n\\multicolumn{11}{l}{\\emph{In-language models (models are trained on the target language training data)}} \\\\ \\midrule\nmBERT, 1000 examples & - & - & - & 87.6 & 77.9 & - & - & 58.7 \/ 46.5 & - & - \\\\\nmBERT & - & - & - & 89.8 & 88.3 & - & - & 74.5 \/ 62.7 & - & - \\\\\nmBERT, multi-task & - & - & - & 91.5 & 89.1 & - & - & 77.6 \/ 68.0 & - & - \\\\\n\\midrule\nHuman & - & 92.8 & 97.5 & 97.0 & - & 91.2 \/ 82.3 & 91.2 \/ 82.3 & 90.1 \/ - & - & - \\\\\n\\bottomrule\n\\end{tabular}%\n}\n\\label{tab:main-results}\n\\end{table*}\n\n\n\\noindent \\textbf{In-language multi-task} $\\:$ For the tasks where monolingual training data is available, we additionally compare against an mBERT model that is jointly trained on the combined training data of all languages.\n\n\\noindent \\textbf{Human performance} $\\:$ For XNLI, PAWS-X, and XQuAD, we obtain human performance estimates from the English datasets they are derived from, MNLI, PAWS-X, and SQuAD respectively \\cite{Nangia2019human_vs_muppet,Zhang2019paws,Rajpurkar2016squad}.\\footnote{Performance may differ across languages due to many factors but English performance still serves as a reasonable proxy.} For TyDiQA-GoldP, we use the performance estimate of \\citet{Clark2020tydiqa}. For MLQA, as answers are annotated using the same format as SQuAD, we employ the same human performance estimate. For POS tagging, we adopt 97\\% as a canonical estimate of human performance based on \\citet{manning2011part}. We are not able to obtain human performance estimates for NER as annotations have been automatically generated and for sentence retrieval as identifying a translation among a large number of documents is too time-consuming.\n\n\\subsection{Results}\n\n\\noindent \\textbf{Overall results} $\\:$ We show the main results in Table \\ref{tab:main-results}. XLM-R is the best-performing zero-shot transfer model and generally improves upon mBERT significantly. The improvement is smaller, however, for the structured prediction tasks. MMTE achieves performance competitive with mBERT on most tasks, with stronger results on XNLI, POS, and BUCC.\n\nIf a strong MT system is available, translating the training sets provides improvements over using the same model with zero-shot transfer. Translating the test data provides similar benefits compared to translating the training data and is particularly effective for the more complex QA tasks, while being more expensive during inference time. While using an MT system as a black box leads to strong baselines, the MT system could be further improved in the context of data augmentation.\n\n\nFor the tasks where in-language training data is available, multilingual models trained on in-language data outperform zero-shot transfer models. However, zero-shot transfer models nevertheless outperform multilingual models trained on only 1,000 in-language examples on the complex QA tasks as long as more samples in English are available. For the structured prediction tasks, 1,000 in-language examples enable the model to achieve performance that is similar to being trained on the full labelled dataset, similar to findings for classification \\cite{Eisenschlos2019multifit}. Finally, multi-task learning on the Translate-train and In-language setting generally improves upon single language training. \n\n\\noindent \\textbf{Cross-lingual transfer gap} $\\:$ For a number of representative models, we show the cross-lingual transfer gap, i.e. the difference between the performance on the English test set and all other languages in Table \\ref{tab:transfer-gap}.\\footnote{This comparison should be taken with a grain of salt, as scores across languages are not directly comparable for the tasks where test sets differ, i.e. POS, NER, MLQA, and TyDiQA-GoldP and differences in scores may not be linearly related.} While powerful models such as XLM-R reduce the gap significantly compared to mBERT for challenging tasks such as XQuAD and MLQA, they do not have the same impact on the syntactic structured prediction tasks. On the classification tasks, the transfer learning gap is lowest, indicating that there may be less headroom for progress on these tasks. The use of MT reduces the gap across all tasks. Overall, a large gap remains for all approaches, which indicates much potential for work on cross-lingual transfer.\n\n\\begin{table}[]\n\\centering\n\\caption{The cross-lingual transfer gap (lower is better) of different models on \\textsc{xtreme}\\xspace tasks. The transfer gap is the difference between performance on the English test set and the average performance on the other languages. A transfer gap of 0 indicates perfect cross-lingual transfer. For the QA datasets, we only show EM scores. The average gaps are computed over the sentence classification and QA tasks.}\n\\resizebox{\\columnwidth}{!}{%\n\\begin{tabular}{l@{~}@{~}c@{~}c@{~}c@{~}c@{~}c@{~}@{~}|c@{~}@{~}|c@{~}c@{~}}\n\\toprule\nModel & XNLI & PAWS-X & XQuAD & MLQA & TyDiQA-GoldP & Avg & POS & NER \\\\\n\\midrule\nmBERT & 16.5 & 14.1 & 25.0 & 27.5 & 22.2 & 21.1 & 25.5 & 23.6 \\\\\nXLM-R & 10.2 & 12.4 & 16.3 & 19.1 & 13.3 & 14.3 & 24.3 & 19.8 \\\\\nTranslate-train & 7.3 & 9.0 & 17.6 & 22.2 & 24.2 & 16.1 & - & - \\\\\nTranslate-test & 6.7 & 12.0 & 16.3 & 18.3 & 11.2 & 12.9 & - & -\\\\\n\\bottomrule\n\\end{tabular}%\n}\n\\label{tab:transfer-gap}\n\\end{table}\n\n\\section{Analyses} \\label{sec:analyses}\n\nWe conduct a series of analyses investigating the limitations of state-of-the-art cross-lingual models.\n\n\\noindent \\textbf{Best zero-shot model analysis} $\\:$ We show the performance of the best zero-shot transfer model, XLM-R Large broken down by task and language in Figure \\ref{fig:scores_vs_tasks}. The figure illustrates why it is important to evaluate general-purpose multilingual representations across a diverse range of tasks and languages: On XNLI, probably the most common standard cross-lingual evaluation task, and PAWS-X, scores cluster in a relatively small range---even considering pseudo test sets for XNLI. However, scores for the remaining tasks have significantly wider spread, particularly as we include pseudo test sets. For TyDiQA-GoldP, English performance is lowest in comparison; the high performance on members of the Austronesian and Uralic language families (Indonesian and Finnish) may be due to less complex Wikipedia context passages for these languages. \nAcross tasks, we generally observe higher performance on Indo-European languages and lower performance for other language families, particularly for Sino-Tibetan, Japonic, Koreanic, and Niger-Congo languages. Some of these difficulties may be due to tokenisation and an under-representation of ideograms in the joint sentencepiece vocabulary, which has been shown to be important in a cross-lingual model's performance \\cite{artetxe2019cross,Conneau2019xlm-r}. We observe similar trends for mBERT, for which we show the same graph in the appendix.\n\n\\begin{figure}[!t]\n \\centering\n \\includegraphics[width=\\linewidth]{scores_vs_tasks_xlmr.png}\n \\vspace{-0.5cm}\n \\caption{An overview of XLM-R's performance on the \\textsc{xtreme}\\xspace tasks across all languages in each task. We highlight an estimate of human performance, performance on the English test set, the average of all languages excluding English, and the family of each language. Performance on pseudo test sets for XNLI and XQuAD is shown with slightly transparent markers.}\n \\label{fig:scores_vs_tasks}\n\\end{figure}\n\n\\begin{figure*}[!h]\n\\centering\n\\begin{subfigure}{.5\\textwidth}\n \\centering\n \\includegraphics[width=\\linewidth]{scores_vs_doc_counts_corr_mbert.png}\n \\label{fig:sub1}\n\\end{subfigure}%\n\\begin{subfigure}{.5\\textwidth}\n \\centering\n \\includegraphics[width=\\linewidth]{scores_vs_doc_counts_no_corr_mbert.png}\n \\label{fig:sub2}\n\\end{subfigure}\n\\vspace{-0.8cm}\n\\caption{Performance of mBERT across tasks and languages in comparison to the number of Wikipedia articles for each language. We show tasks with a Pearson correlation coefficient $\\rho > 0.7$ on the left and others on the right. Numbers across tasks are not directly comparable. We remove the $x$ axis labels of overlapping languages for clarity. We additionally plot the linear fit for each task (curved due to the logarithmic scale of the $x$ axis).}\n\\label{fig:mbert_correlation_dataset_size}\n\\end{figure*}\n\n\\noindent \\textbf{Correlation with pretraining data size} $\\:$ We calculate the Pearson correlation coefficient $\\rho$ of the model performance and the number of Wikipedia articles (see appendix) in each language and show results in Figure \\ref{fig:mbert_correlation_dataset_size}.\\footnote{We observe similar correlations when using the number of tokens in Wikipedia instead.} For mBERT, which was pretrained on Wikipedia, we observe a high correlation for most tasks ($\\rho \\approx 0.8$) except for the structured prediction tasks where $\\rho \\approx 0.35$.\nWe observe similar trends for XLM and XLM-R, with lower numbers for XLM-R due to the different pretraining domain (see the appendix). This indicates that current models are not able to fully leverage the information extracted from the pretraining data to transfer to syntactic tasks.\n\n\\noindent \\textbf{Analysis of language characteristics} $\\:$ We analyze results based on different language families and writing scripts in Figure~\\ref{fig:mbert_correlation_lang_family_script}. For mBERT, we observe the best transfer performance on branches of the Indo-European language family such as Germanic, Romance and Slavic languages. In contrast, cross-lingual transfer performance on low-resource language families such as Niger-Congo and Kra-Dai is still low. Looking at scripts, we find that the performance on syntactic tasks differs among popular scripts such as Latin and ideogram scripts. For example in the NER task, mBERT performs better on data in Latin script than that in Chinese or Japanese ideograms. This indicates that the current models still have difficulty transferring word-level syntactic information across languages written in different scripts.\n\n\\begin{figure*}[!]\n\\centering\n\\begin{subfigure}{.5\\textwidth}\n \\centering\n \\includegraphics[width=\\linewidth]{scores_vs_lang_families_single_mbert_fill.png}\n \\label{fig:lang_families}\n\\end{subfigure}%\n\\begin{subfigure}{.5\\textwidth}\n \\centering\n \\includegraphics[width=\\linewidth]{scores_vs_lang_scripts_single_mbert_fill.png}\n \\label{fig:lang_scripts}\n\\end{subfigure}\n\\vspace{-0.8cm}\n\\caption{Performance of mBERT across tasks grouped by language families (left) and scripts (right). The number of languages per group is in brackets and the groups are from low-resource to high-resource on the x-axis. We additionally plot the 3rd order polynomial fit for the minimum and maximum values for each group.}\n\\label{fig:mbert_correlation_lang_family_script}\n\\end{figure*}\n\n\\noindent \\textbf{Errors across languages} $\\:$ For XNLI and XQuAD where the other test sets are translations from English, we analyze whether approaches make the same type of errors in the source and target languages. To this end, we explore whether examples that are correctly and incorrectly predicted in English are correctly predicted in other languages. On the XNLI dev set, mBERT correctly predicts on average 71.8\\% of examples that were correctly predicted in English. For examples that were misclassified, the model's performance is about random. On average, predictions on XNLI are consistent between English and another language for 68.3\\% of examples. On the XQuAD test set, mBERT correctly predicts around 60\\% of examples that were correclty predicted in English and 20\\% of examples that were incorrectly predicted. While some of these are plausible spans, more work needs to focus on achieving consistent predictions across languages.\n\n\\begin{table}[]\n\\centering\n\\caption{Accuracy of mBERT on POS tag trigrams and 4-grams in the target language dev data that appeared and did not appear in the English training data. We show the performance on English, the average across all other languages, and their difference.}\n\\label{tab:tag-ngrams-mbert}\n\\resizebox{\\columnwidth}{!}{%\n\\begin{tabular}{l c c c c}\n\\toprule\n & \\begin{tabular}[c]{@{}l@{}}trigram,\\\\ seen\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}trigram,\\\\ unseen\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}4-gram,\\\\ seen\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}4-gram,\\\\ unseen\\end{tabular} \\\\ \\midrule\nen & 90.3 & 63.0 & 88.1 & 67.5 \\\\\navg w\/o en & 50.6 & 12.1 & 44.3 & 18.3 \\\\ \\midrule\ndifference & 39.7 & 50.9 & 43.7 & 49.2 \\\\ \\bottomrule\n\\end{tabular}%\n}\n\\end{table}\n\n\\noindent \\textbf{Generalization to unseen tag combinations and entities} $\\:$ We analyze possible reasons for the less successful transfer on structured prediction tasks. The Universal Dependencies dataset used for POS tagging uses a common set of 17 POS tags for all languages, so a model is not required to generalize to unseen tags at test time. However, a model may be required to generalize to unseen tag \\emph{combinations} at test time, for instance due to differences in word order between languages. We gauge how challenging such generalization is by computing a model's accuracy for POS tag n-grams in the target language dev data that were not seen in the English training data. We calculate values for tag trigrams and 4-grams and show accuracy scores for mBERT in Table \\ref{tab:tag-ngrams-mbert}. We observe the largest differences in performance for unseen trigrams and 4-grams, which highlights that existing cross-lingual models struggle to transfer to the syntactic characteristics of other languages. For NER, we estimate how well models generalize to unseen entities at test time. We compute mBERT's accuracy on entities in the target language dev data that were not seen in the English training data. We observe the largest difference between performance on seen and unseen entities for Indonesian and Swahili. Isolating for confounding factors such as entity length, frequency, and Latin script, we find the largest differences in performance for Swahili and Basque. Together, this indicates that the model may struggle to generalize to entities that are more characteristic of the target language. We show the detailed results for both analyses in the appendix.\n\n\n\n\n\n\\section{Conclusions}\n\nAs we have highlighted in our analysis, a model's cross-lingual transfer performance varies significantly both between tasks and languages. \\textsc{xtreme}\\xspace is a first step towards obtaining a more accurate estimate of a model's cross-lingual generalization ability. While \\textsc{xtreme}\\xspace is still inherently limited by the data coverage of its constituent tasks for many low-resource languages, \\textsc{xtreme}\\xspace nevertheless provides significantly broader coverage and more fine-grained analysis tools to encourage research on cross-lingual generalization ability of models. We have released the code for \\textsc{xtreme}\\xspace and scripts for fine-tuning models on tasks in \\textsc{xtreme}\\xspace, which should be to catalyze future research.\n\n\n\\section*{Acknowledgements}\n\nWe'd like to thank Jon Clark for sharing with us the TyDiQA Gold Passage data and for valuable feedback. We would also like to thank Sam Bowman, Sebastian Goodman, and Tal Linzen for their feedback. JH and GN are sponsored by the Air Force Research Laboratory under agreement number FA8750-19-2-0200. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction \\& Notation}\n\\label{sec:introduction}\nRecurrent Neural Networks (RNNs) are a type of neural network architecture which is mainly used to detect patterns in a sequence of data. Such data can be handwriting, genomes, text or numerical time series which are often produced in industry settings (e.g. stock markets or sensors) \\cite{chung2015gated, bworld}. However, they are also applicable to images if these get respectively decomposed into a series of patches and treated as a sequence \\cite{bworld}. On a higher level, RNNs find applications in \\textit{Language Modelling \\& Generating Text}, \\textit{Speech Recognition}, \\textit{Generating Image Descriptions} or \\textit{Video Tagging}. What differentiates Recurrent Neural Networks from Feedforward Neural Networks also known as Multi-Layer Perceptrons (MLPs) is how information gets passed through the network. While Feedforward Networks pass information through the network without cycles, the RNN has cycles and transmits information back into itself. This enables them to extend the functionality of Feedforward Networks to also take into account previous inputs $\\mathbf{X}_{0:t-1}$ and not only the current input $\\mathbf{X}_t$. This difference is visualised on a high level in Figure \\ref{fig:rnn}. Note, that here the option of having multiple hidden layers is aggregated to one Hidden Layer block $\\mathbf{H}$. This block can obviously be extended to multiple hidden layers.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\textwidth]{presentation\/assets\/rnn-cropped.pdf}\n \\caption{Visualisation of differences between Feedfoward NNs und Recurrent NNs}\n \\label{fig:rnn}\n\\end{figure}\n\nWe can describe this process of passing information from the previous iteration to the hidden layer with the mathematical notation proposed in \\cite{zhang2019dive}. For that, we denote the hidden state and the input at time step $t$ respecively as $\\mathbf{H}_{t} \\in \\mathbb{R}^{n \\times h}$ and $\\mathbf{X}_t \\in \\mathbb{R}^{n \\times d}$ where $n$ is number of samples, $d$ is the number of inputs of each sample and $h$ is the number of hidden units. Further, we use a weight matrix $\\mathbf{W}_{x h} \\in \\mathbb{R}^{d \\times h}$, hidden-state-to-hidden-state matrix $\\mathbf{W}_{h h} \\in \\mathbb{R}^{h \\times h}$ and a bias parameter $\\mathbf{b}_{h} \\in \\mathbb{R}^{1 \\times h}$. Lastly, all these informations get passed to a activation function $\\phi$ which is usually a logistic sigmoid or tanh function to prepair the gradients for usage in backpropagation. Putting all these notations together yields Equation \\ref{eq:not} as the hidden variable and Equation \\ref{eq:outrnn} as the output variable.\n\\begin{equation}\n\\label{eq:not}\n \\mathbf{H}_{t}=\\phi_h\\left(\\mathbf{X}_{t} \\mathbf{W}_{x h}+\\mathbf{H}_{t-1} \\mathbf{W}_{h h}+\\mathbf{b}_{h}\\right)\n\\end{equation}\n\\begin{equation}\n\\label{eq:outrnn}\n \\mathbf{O}_{t}=\\phi_o\\left(\\mathbf{H}_{t} \\mathbf{W}_{h o}+\\mathbf{b}_{o}\\right)\n\\end{equation}\n\nSince $\\mathbf{H}_{t}$ recursively includes $\\mathbf{H}_{t-1}$ and this process occurs for every time step the RNN includes traces of all hidden states that preceded $\\mathbf{H}_{t-1}$ as well as $\\mathbf{H}_{t-1}$ itself.\n\nIf we compare that notation for RNNs with similar notation for Feedforward Neural Networks we can clearly see the difference we described earlier. In Equation \\ref{eq:Hidden} we can see the computation for the hidden variable while Equation \\ref{eq:output} shows the output variable.\n\\begin{equation}\n\\label{eq:Hidden}\n \\mathbf{H} =\\phi_h\\left(\\mathbf{X} \\mathbf{W}_{x h}+\\mathbf{b}_{h}\\right)\n\\end{equation}\n\\begin{equation}\n\\label{eq:output}\n \\mathbf{O}=\\phi_o \\left(\\mathbf{H W}_{h o}+\\mathbf{b}_{o}\\right)\n\\end{equation}\n\nIf you are familiar with training techniques for Feedforward Neural Networks such as backpropagation one question which might arise is how to properly backpropagate the error through a RNN. Here, a technique called Backpropagation Through Time (BPTT) is used which gets described in detail in the next section.\n\n\\section{Backpropagation Through Time (BPTT) \\& Truncated BPTT}\nBackpropagation Through Time (BPTT) is the adaption of the backpropagation algorithm for RNNs \\cite{zhang2019dive}. In theory, this unfolds the RNN to construct a traditional Feedfoward Neural Network where we can apply backpropagation. For that, we use the same notations for the RNN as proposed before.\n\nWhen we forward pass our input $\\mathbf{X}_t$ through the network we compute the hidden state $\\mathbf{H}_t$ and the output state $\\mathbf{O}_t$ one step at a time. We can then define a loss function $\\mathcal{L}\\left(\\mathbf{O}, \\mathbf{Y}\\right)$ to describe the difference between all outputs $\\mathbf{O}_t$ and target values $\\mathbf{Y}_t$ as shown in Equation \\ref{eq:loss}. This basically sums up every loss term $\\ell_t$ of each update step so far. This loss term $\\ell_t$ can have different definitions based on the specific problem (e.g. Mean Squared Error, Hinge Loss, Cross Entropy Loss, etc.).\n\\begin{equation}\n\\label{eq:loss}\n \\mathcal{L}\\left(\\mathbf{O}, \\mathbf{Y}\\right)=\\sum_{t=1}^{T} \\ell_t\\left(\\mathbf{O}_{t}, \\mathbf{Y}_{t}\\right)\n\\end{equation}\nSince we have three weight matrices $\\mathbf{W}_{x h}$, $\\mathbf{W}_{h h}$ and $\\mathbf{W}_{h o}$ we need to compute the partial derivative w.r.t. to each of these weight matrices. With the chain rule which is also used in normal backpropagation we get to the result for $\\mathbf{W}_{h o}$ shown in Equation \\ref{eq:derive_ho}.\n\\begin{equation}\n\\label{eq:derive_ho}\n \\frac{\\partial \\mathcal{L}}{\\partial \\mathbf{W}_{ho}} = \\sum_{t=1}^{T} \\frac{\\partial \\ell_t}{\\partial \\mathbf{O}_t} \\cdot \\frac{\\partial \\mathbf{O}_t}{\\partial \\phi_o} \\cdot \\frac{\\partial \\phi_o}{\\mathbf{W}_{ho}} = \\sum_{t=1}^{T} \\frac{\\partial \\ell_t}{\\partial \\mathbf{O}_t} \\cdot \\frac{\\partial \\mathbf{O}_t}{\\partial \\phi_o} \\cdot \\mathbf{H_t}\n\\end{equation}\n\nFor the partial derivative with respect to $\\mathbf{W}_{hh}$ we get the result shown in Equation \\ref{eq:derive_hh}.\n\\begin{equation}\n\\label{eq:derive_hh}\n \\frac{\\partial \\mathcal{L}}{\\partial \\mathbf{W}_{hh}} = \\sum_{t=1}^{T} \\frac{\\partial \\ell_t}{\\partial \\mathbf{O}_t} \\cdot \\frac{\\partial \\mathbf{O}_t}{\\partial \\phi_o} \\cdot \\frac{\\partial \\phi_o}{\\partial \\mathbf{H}_t} \\cdot \\frac{\\partial \\mathbf{H}_t}{\\partial \\phi_h} \\cdot \\frac{\\partial \\phi_h}{\\partial \\mathbf{W}_{hh}} = \\sum_{t=1}^{T} \\frac{\\partial \\ell_t}{\\partial \\mathbf{O}_t} \\cdot \\frac{\\partial \\mathbf{O}_t}{\\partial \\phi_o} \\cdot \\mathbf{W}_{ho} \\cdot \\frac{\\partial \\mathbf{H}_t}{\\partial \\phi_h} \\cdot \\frac{\\partial \\phi_h}{\\partial \\mathbf{W}_{hh}}\n\\end{equation}\n\nFor the partial derivative with respect to $\\mathbf{W}_{xh}$ we get the result shown in Equation \\ref{eq:derive_xh}.\n\\begin{equation}\n\\label{eq:derive_xh}\n \\frac{\\partial \\mathcal{L}}{\\partial \\mathbf{W}_{xh}} = \\sum_{t=1}^{T} \\frac{\\partial \\ell_t}{\\partial \\mathbf{O}_t} \\cdot \\frac{\\partial \\mathbf{O}_t}{\\partial \\phi_o} \\cdot \\frac{\\partial \\phi_o}{\\partial \\mathbf{H}_t} \\cdot \\frac{\\partial \\mathbf{H}_t}{\\partial \\phi_h} \\cdot \\frac{\\partial \\phi_h}{\\partial \\mathbf{W}_{xh}} = \\sum_{t=1}^{T} \\frac{\\partial \\ell_t}{\\partial \\mathbf{O}_t} \\cdot \\frac{\\partial \\mathbf{O}_t}{\\partial \\phi_o} \\cdot \\mathbf{W}_{ho} \\cdot \\frac{\\partial \\mathbf{H}_t}{\\partial \\phi_h} \\cdot \\frac{\\partial \\phi_h}{\\partial \\mathbf{W}_{xh}}\n\\end{equation}\n\nSince each $\\mathbf{H}_t$ depends on the previous time step we can substitute the last part from above equations to get Equation \\ref{trick_hh} and Equation \\ref{trick_xh}.\n\\begin{equation}\n\\label{trick_hh}\n \\frac{\\partial \\mathcal{L}}{\\partial \\mathbf{W}_{hh}} = \\sum_{t=1}^{T} \\frac{\\partial \\ell_t}{\\partial \\mathbf{O}_t} \\cdot \\frac{\\partial \\mathbf{O}_t}{\\partial \\phi_o} \\cdot \\mathbf{W}_{ho} \\sum_{k=1}^t \\frac{\\partial \\mathbf{H}_t}{\\partial \\mathbf{H}_k} \\cdot \\frac{\\partial \\mathbf{H}_k}{\\partial \\mathbf{W}_{hh}}\n\\end{equation}\n\\begin{equation}\n\\label{trick_xh}\n \\frac{\\partial \\mathcal{L}}{\\partial \\mathbf{W}_{xh}} = \\sum_{t=1}^{T} \\frac{\\partial \\ell_t}{\\partial \\mathbf{O}_t} \\cdot \\frac{\\partial \\mathbf{O}_t}{\\partial \\phi_o} \\cdot \\mathbf{W}_{ho} \\sum_{k=1}^t \\frac{\\partial \\mathbf{H}_t}{\\partial \\mathbf{H}_k} \\cdot \\frac{\\partial \\mathbf{H}_k}{\\partial \\mathbf{W}_{xh}}\n\\end{equation}\n\nThe adapted part can then further be written as shown in Equation \\ref{finish_hh} and Equation \\ref{finish_xh}.\n\\begin{equation}\n\\label{finish_hh}\n \\frac{\\partial \\mathcal{L}}{\\partial \\mathbf{W}_{hh}} = \\sum_{t=1}^{T} \\frac{\\partial \\ell_t}{\\partial \\mathbf{O}_t} \\cdot \\frac{\\partial \\mathbf{O}_t}{\\partial \\phi_o} \\cdot \\mathbf{W}_{ho} \\sum_{k=1}^t \\left( \\mathbf{W}_{hh}^\\top \\right)^{t-k} \\cdot \\mathbf{H}_k\n\\end{equation}\n\\begin{equation}\n\\label{finish_xh}\n \\frac{\\partial \\mathcal{L}}{\\partial \\mathbf{W}_{xh}} = \\sum_{t=1}^{T} \\frac{\\partial \\ell_t}{\\partial \\mathbf{O}_t} \\cdot \\frac{\\partial \\mathbf{O}_t}{\\partial \\phi_o} \\cdot \\mathbf{W}_{ho} \\sum_{k=1}^t \\left( \\mathbf{W}_{hh}^\\top \\right)^{t-k} \\cdot \\mathbf{X}_k\n\\end{equation}\n\nFrom here, we can see that we need to store powers of $\\mathbf{W}_{hh}^k$ as we proceed through each loss term $\\ell_t$ of the overall loss function $\\mathcal{L}$ which can become very large. For these large values this method becomes numerically unstable since eigenvalues smaller than $1$ vanish and eigenvalues larger than $1$ diverge \\cite{longterm}. One method of solving this problem is truncate the sum at\na computationally convenient size \\cite{zhang2019dive}. When you do this, you're using Truncated BPTT \\cite{trunc}. This basically establishes an upper bound for the number of time steps the gradient can flow back to \\cite{Sutskever:2013:TRN:2604780}. One can think of this upper bound as a moving window of past time steps which the RNN considers. Anything before the cut-off time step doesn't get taken into account. Since BPTT basically unfolds the RNN to create a new layer for each time step we can also think of this procedure as limiting the number of hidden layers. \n\n\\section{Problems of RNNs: Vanishing \\& Exploding Gradients}\nAs in most neural networks, vanishing or exploding gradients is a key problem of RNNs \\cite{bworld}. In Equation \\ref{trick_hh} and Equation \\ref{trick_xh} we can see $\\frac{\\partial \\mathbf{H}_t}{\\partial \\mathbf{H}_k}$ which basically introduces matrix multiplication over the (potentially very long) sequence, if there are small values ($< 1$) in the matrix multiplication this causes the gradient to decrease with each layer (or time step) and finally vanish \\cite{chen2016gentle}. This basically stops the contribution of states that happened far earlier than the current time step towards the current time step \\cite{chen2016gentle}. Similarly, this can happen in the opposite direction if we have large values ($>1$) during matrix multiplication causing an exploding gradient which in result values each weight too much and changes it heavily \\cite{chen2016gentle}.\n\nThis problem motivated the introduction of the long short term memory units (LSTMs) to particularly handle the vanishing gradient problem. This approach was able to outperform traditional RNNs on a variety of tasks \\cite{chen2016gentle}. In the next section we want to go deeper on the proposed structure of LSTMs.\n\n\\section{Long Short-Term Memory Units (LSTMs)}\n\\label{sec:lstms}\nLong Short-Term Memory Units (LSTMs) \\cite{lstms} were designed to properly handle the vanishing gradient problem. Since they use a more constant error, they allow RNNs to learn over a lot more time steps (way over $1000$) \\cite{bworld}. To achieve that, LSTMs store more information outside of the traditional neural network flow in structures called gated cells \\cite{chen2016gentle, bworld}. To make things work in an LSTM we use an output gate $\\mathbf{O}_{t}$ to read entries of the cell, an input gate $\\mathbf{I}_{t}$ to read data into the cell and a forget gate $\\mathbf{F}_{t}$ to reset the content of the cell. The computations for these gates are shown in Equation \\ref{eq:output_gate}, Equation \\ref{eq:input_gate} and Equation \\ref{eq:forget_gate}. For a more visual approach please see Figure \\ref{fig:gates} in Appendix \\ref{app:vis}. \n\\begin{equation}\n\\label{eq:output_gate}\n\\mathbf{O}_{t}=\\sigma\\left(\\mathbf{X}_{t} \\mathbf{W}_{x o}+\\mathbf{H}_{t-1} \\mathbf{W}_{h o}+\\mathbf{b}_{o}\\right)\n\\end{equation}\n\\begin{equation}\n\\label{eq:input_gate}\n \\mathbf{I}_{t}=\\sigma\\left(\\mathbf{X}_{t} \\mathbf{W}_{x i}+\\mathbf{H}_{t-1} \\mathbf{W}_{h i}+\\mathbf{b}_{i}\\right)\n\\end{equation}\n\\begin{equation}\n\\label{eq:forget_gate}\n\\mathbf{F}_{t}=\\sigma\\left(\\mathbf{X}_{t} \\mathbf{W}_{x f}+\\mathbf{H}_{t-1} \\mathbf{W}_{h f}+\\mathbf{b}_{f}\\right)\n\\end{equation}\n\nThe shown equations use $\\mathbf{W}_{x i}, \\mathbf{W}_{x f}, \\mathbf{W}_{x o} \\in \\mathbb{R}^{d \\times h}$ and $\\mathbf{W}_{h i}, \\mathbf{W}_{h f}, \\mathbf{W}_{h o} \\in \\mathbb{R}^{h \\times h}$ as weight matrices while $\\mathbf{b}_{i}, \\mathbf{b}_{f}, \\mathbf{b}_{o} \\in \\mathbb{R}^{1 \\times h}$ are their respective biases. Further, they use the sigmoid activation function $\\sigma$ to transform the output $\\in (0,1)$ which each results in a vector with entries $\\in (0,1)$. \n\nNext, we need a candidate memory cell $\\tilde{\\mathbf{C}}_{t} \\in \\mathbb{R}^{n \\times h}$ which has a similar computation as the previously mentioned gates but instead uses a tanh activation function to have an output $\\in (-1,1)$. Further, it again has its own weights $\\mathbf{W}_{x c} \\in \\mathbb{R}^{d \\times h}$, $\\mathbf{W}_{h c} \\in \\mathbb{R}^{h \\times h}$ and biases $\\mathbf{b}_{c} \\in \\mathbb{R}^{1 \\times h}$. The respective computation is shown in Equation \\ref{eq:candidate}. See Figure \\ref{fig:candidate} in Appendix \\ref{app:vis} for a visualisation of this enhancement.\n\\begin{equation}\n\\label{eq:candidate}\n\\tilde{\\mathbf{C}}_{t}=\\tanh \\left(\\mathbf{X}_{t} \\mathbf{W}_{x c}+\\mathbf{H}_{t-1} \\mathbf{W}_{h c}+\\mathbf{b}_{c}\\right)\n\\end{equation}\n\nTo plug some things together we introduce old memory content $\\mathbf{C}_{t-1} \\in \\mathbb{R}^{n \\times h}$ which together with the introduced gates controls how much of the old memory content we want to preserve to get to the new memory content $\\mathbf{C}_t$. This is shown in Equation \\ref{eq:memory} where $\\odot$ denotes element-wise multiplication. The structure so far can be seen in Figure \\ref{fig:memory} in Appendix \\ref{app:vis}.\n\\begin{equation}\n\\label{eq:memory}\n\\mathbf{C}_{t}=\\mathbf{F}_{t} \\odot \\mathbf{C}_{t-1}+\\mathbf{I}_{t} \\odot \\tilde{\\mathbf{C}}_{t}\n\\end{equation}\n\nThe last step is to introduce the computation for the hidden states $\\mathbf{H}_{t} \\in \\mathbb{R}^{n \\times h}$ into the framework. This can be seen in Equation \\ref{eq:lstm_hidden}.\n\\begin{equation}\n\\label{eq:lstm_hidden}\n\\mathbf{H}_{t}=\\mathbf{O}_{t} \\odot \\tanh \\left(\\mathbf{C}_{t}\\right)\n\\end{equation}\nWith the tanh function we ensure that each element of $\\mathbf{H}_{t}$ is $\\in (-1,1)$. The full LSTM framework can be seen in Figure \\ref{fig:hidden} in Appendix \\ref{app:vis}.\n\n\\section{Deep Recurrent Neural Networks (DRNNs)}\nDeep Recurrent Neural Networks (DRNNs) are in theory a really easy concept. To construct a deep RNN with $L$ hidden layers we simply stack ordinary RNNs of any type on top of each other. Each hidden state $\\mathbf{H}_t^{(\\ell)} \\in \\mathbb{R}^{n \\times h}$ is passed to the next time step of the current layer $\\mathbf{H}_{t+1}^{(\\ell)}$ as well as the current time step of the next layer $\\mathbf{H}_{t}^{(\\ell+1)}$. For the first layer we compute the hidden state as proposed in the previous models shown shown in Equation \\ref{eq:layer1} while for the subsequent layer we use Equation \\ref{eq:conslayer} where the hidden state from the previous layer is treated as input. \n\\begin{equation}\n\\label{eq:layer1}\n\\mathbf{H}_{t}^{(1)}=\\phi_{1}\\left(\\mathbf{X}_{t}, \\mathbf{H}_{t-1}^{(1)}\\right)\n\\end{equation}\n\\begin{equation}\n\\label{eq:conslayer}\n\\mathbf{H}_{t}^{(\\ell)}=\\phi_{\\ell}\\left(\\mathbf{H}_{t}^{(\\ell-1)}, \\mathbf{H}_{t-1}^{(\\ell)}\\right)\n\\end{equation}\n\nThe output $\\mathbf{O}_t \\in \\mathbb{R}^{n \\times o}$ where $o$ is the number of outputs is then computed as shown in Equation \\ref{outputdnn} where we only use the hidden state of layer $L$.\n\\begin{equation}\n\\label{outputdnn}\n \\mathbf{O}_t = \\phi_o \\left(\\mathbf{H}_t^{(L)} \\mathbf{W}_{h o}+\\mathbf{b}_{o}\\right)\n\\end{equation}\n\n\\section{Bidirectional Recurrent Neural Networks (BRNNs)}\nLets take an example of language modeling for now. Based on our current models we are able to reliably predict the next sequence element (i.e. the next word) based on what we have seen so far. However, there scenarios where we might want to fill in a gap in a sentence and the part of the sentence after the gap conveys significant information. This information is necessary to take into account to perform well on this kind of task \\cite{zhang2019dive}. On a more generalised level we want to incorporate a look-ahead property for sequences.\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=0.8\\textwidth]{presentation\/assets\/brnn2.PNG}\n \\caption{Architecture of a bidirectional recurrent neural network}\n \\label{fig:brnn}\n\\end{figure}\n\nTo achieve this look-ahead property Bidirectional Recurrent Neural Networks (BRNNs) \\cite{Schuster1997BidirectionalRN} got introduced which basically add another hidden layer which run the sequence backwards starting from the last element \\cite{zhang2019dive}. An architectural overview can is visualised in Figure \\ref{fig:brnn}. Here, we introduce a forward hidden state $\\overrightarrow{\\mathbf{H}}_{t} \\in \\mathbb{R}^{n \\times h}$ and a backward hidden state $\\overleftarrow{\\mathbf{H}}_{t} \\in \\mathbb{R}^{n \\times h}$. Their respective calculations are shown in Equation \\ref{eq:forward} and Equation \\ref{eq:backward}.\n\\begin{equation}\n\\label{eq:forward}\n\\overrightarrow{\\mathbf{H}}_{t}=\\phi\\left(\\mathbf{X}_{t} \\mathbf{W}_{x h}^{(f)}+\\overrightarrow{\\mathbf{H}}_{t-1} \\mathbf{W}_{h h}^{(f)}+\\mathbf{b}_{h}^{(f)}\\right)\n\\end{equation}\n\\begin{equation}\n\\label{eq:backward}\n\\overleftarrow{\\mathbf{H}}_{t}=\\phi\\left(\\mathbf{X}_{t} \\mathbf{W}_{x h}^{(b)}+\\overleftarrow{\\mathbf{H}}_{t+1} \\mathbf{W}_{h h}^{(b)}+\\mathbf{b}_{h}^{(b)}\\right)\n\\end{equation}\n\nFor that, we have similar weight matrices as in definitions before but now they are seperated into two sets. One set of weight matrices is for the forward hidden states $\\mathbf{W}_{x h}^{(f)} \\in \\mathbb{R}^{d \\times h}$ and $\\mathbf{W}_{h h}^{(f)} \\in \\mathbb{R}^{h \\times h}$ while the other one is for the backward hidden states $\\mathbf{W}_{x h}^{(b)} \\in \\mathbb{R}^{d \\times h}$ and $\\mathbf{W}_{h h}^{(b)} \\in \\mathbb{R}^{h \\times h}$. They also have their respective biases $\\mathbf{b}_{h}^{(f)} \\in \\mathbb{R}^{1 \\times h}$ and $\\mathbf{b}_{h}^{(b)} \\in \\mathbb{R}^{1 \\times h}$. With that, we can compute the output $\\mathbf{O}_{t} \\in \\mathbb{R}^{n \\times o}$ with $o$ being the number of outputs and $\\frown$ denoting the concatenation of the two matrices on axis $0$ (stacking them on top of each other).\n\\begin{equation}\n\\mathbf{O}_{t}=\\phi\\left(\\left[\\overrightarrow{\\mathbf{H}}_{t}^\\frown \\overleftarrow{\\mathbf{H}}_{t}\\right] \\mathbf{W}_{h o}+\\mathbf{b}_{o}\\right)\n\\end{equation}\n\nAgain, we have weight matrices $\\mathbf{W}_{h o} \\in \\mathbb{R}^{2 h \\times o}$ and bias parameters $\\mathbf{b}_{o} \\in \\mathbb{R}^{1 \\times o}$. Keep in mind that the two directions can have different number of hidden units.\n\n\\section{Encoder-Decoder Architecture \\& Sequence to Sequence (seq2seq)}\n\\label{seq2}\nThe Encoder-Decoder architecture is a type of neural network architecture where the network is twofold. It consists of encoder network and a decoder network whose respective roles are to \\textit{encode} the input into a state and \\textit{decode} the state to an output. This state usually has shape of a vector or a tensor \\cite{zhang2019dive}. A visualisation of this structure is shown in Figure \\ref{fig:encoder-dec}.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.85\\textwidth]{presentation\/assets\/encoder-decoder-cropped.pdf}\n \\caption{Encoder-Decoder Architecture Overview alternated from: \\cite{zhang2019dive}}\n \\label{fig:encoder-dec}\n\\end{figure}\n\nBased on this Encoder-Decoder architecture a model called Sequence to Sequence (seq2seq) \\cite{seq2seq} got proposed for generating a sequence output based on a sequence input. This model uses RNNs for the encoder as well as the decoder where the hidden state of the encoder gets passed to the hidden state of the decoder. Common applications of the model are Google Translate \\cite{seq2seq, wu2016googles}, voice-enabled devices \\cite{seq2seqspeech} or labeling video data \\cite{venugopalan2015sequence}. It mainly focuses on mapping a fixed length input sequence of size $n$ to an fixed length output sequence of size $m$ where $n \\neq m$ can be true but isn't a necessity.\n\nA de-rellod visualisation of the proposed architecture is shown in Figure \\ref{fig:seq}. Here, we have a encoder which consists of a RNN accepting a single element of the sequence $\\mathbf{X}_t$ where $t$ is the order of the sequence element. These RNNs can be LSTMs or Gated Recurrent Units (GRUs) to further improve performance \\cite{seq2seq}. Further, the hidden states $\\mathbf{H}_t$ are computed according to the definition of the hidden states in the used RNN type (e.g. LSTM or GRU). The Encoder Vector (context) is a representation of the last hidden state of the encoder network which aims to aggregate all information from all previous input elements. This functions as initial hidden state of the decoder network of the model and enables the decoder to make accurate predictions. The decoder network again is built of a RNN which predicts an output $\\mathbf{Y}_t$ at a time step $t$. The produced output is again a sequence where each $\\mathbf{Y}_t$ is a sequence element with order $t$. At each time step the RNN accepts a hidden state from the previous unit and itself produces an output as well as a new hidden state.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.8\\textwidth]{presentation\/assets\/seq2seq_test.pdf}\n \\caption{Visualisation of the Sequence to Sequence (seq2seq) Model}\n \\label{fig:seq}\n\\end{figure}\n\nThe Encoder Vector (context) was shown to be a bottleneck for these type of models since it needed to contain all all the necessary information of a source sentence in a fixed-length vector which was particularly problematic for long sequences. There have been approaches to solve this problem by introducing Attention in for example \\cite{attention1} or \\cite{attention2}. In the next section, we take a closer look at the proposed solutions.\n\n\\section{Attention Mechanism \\& Transformer}\n\nThe Attention Mechanism for RNNs is partly motivated by human visual focus and the peripheral perception \\cite{attention_blog}. It allows humans to focus on a certain region to achieve high resolution while adjacent objects are perceived with a rather low resolution. Based on these focus points and adjacent perception, we can make inference about what we expect to perceive when shifting our focus point. Similarly, we can transfer this method on our sequence of words where we are able to perform inference based on observed words. For example, if we perceive the word \\textit{eating} in the sequence ``She is eating a green apple'' we assume to observe a food object in the near future \\cite{attention_blog}.\n\n\nGenerally, Attention takes two sentences and transforms them into a matrix where each sequence element (i.e. a word) corresponds to a row or column. Based on this matrix layout we can fill in the entries to identify relevant context or correlations between them. An example of this process can be seen in Figure \\ref{fig:matrix} where white denotes high correlation while black denotes low correlation. This method isn't limited to two sentences of a different languages as seen the example but can also be applied to the same sentence which is then called self-attention.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.35\\textwidth]{presentation\/assets\/attention_matrix.PNG}\n \\caption{Example of an Alignment matrix of ``L'accord sur la zone \\'{e}conomique europ\\'{e}en a \\'{e}t\\'{e} sign\\'{e} en ao\\^{u}t 1992'' (French) and its English translation ``The agreement on the European Economic Area was signed in August 1992'': \\cite{attention1}}\n \\label{fig:matrix}\n\\end{figure}\n\n\\subsection{Definition}\n\nTo help the seq2seq model to better deal with long sequences the attention mechanism got introduced. Instead of constructing the Encoder Vector out of the last hidden state of the encoder network, attention introduces shortcuts between context vector and the entire source input. A visualisation of this process can be seen in Figure \\ref{fig:attention_arch}. Here, we have source sequence $\\mathbf{X}$ of length $n$ and try to output a target sequence $\\mathbf{Y}$ of size $m$. In that regard the formulation is rather similar to the one we described before in Section \\ref{seq2}. We have an overall hidden state $\\mathbf{H}_{t^\\prime}$ which is the concatenated version of the forward and backward pass as shown in Equation \\ref{eq:concatenated}. Also, the hidden state of the decoder network is denoted as $\\mathbf{S}_t$ while the encoder vector (context vector) is denoted as $\\mathbf{C}_t$. Both of these are shown in Equation \\ref{eq:decoder_state} and Equation \\ref{eq:context_vector} respectively.\n\\begin{equation}\n\\label{eq:concatenated}\n \\mathbf{H}_{t^\\prime} = \\left[\\overrightarrow{\\mathbf{H}}_{t^\\prime}^\\frown \\overleftarrow{\\mathbf{H}}_{t^\\prime}\\right]\n\\end{equation}\n\\begin{equation}\n\\label{eq:decoder_state}\n \\mathbf{S}_t = \\phi_d \\left(\\mathbf{S}_{t-1}, \\mathbf{Y_{t-1}, \\mathbf{C}_t} \\right)\n\\end{equation}\n\nThe context vector $\\mathbf{C}_t$ is a sum of hidden states of the input sequence each weighted with an alignment score $\\alpha_{t, t^\\prime}$ where $\\sum_{t^\\prime=1}^T \\alpha_{t, t^\\prime} = 1$. This is shown in Equation \\ref{eq:context_vector} as well as Equation \\ref{eq:alignment}.\n\\begin{equation}\n\\label{eq:context_vector}\n \\mathbf{C}_t = \\sum_{t^\\prime=1}^T \\alpha_{t, t^\\prime} \\cdot \\mathbf{H}_{t^\\prime}\n\\end{equation}\n\\begin{equation}\n\\label{eq:alignment}\n \\alpha_{t, t^\\prime} = \\operatorname{align}(\\mathbf{Y}_t, \\mathbf{X}_{t^\\prime}) = \\frac{\\operatorname{exp}\\left(\\operatorname{score}(\\mathbf{S}_{t-1}, \\mathbf{H}_{t^\\prime})\\right)}{\\sum_{t^\\prime = 1}^T\\operatorname{exp}(\\operatorname{score}(\\mathbf{S}_{t-1}, \\mathbf{H}_{t^\\prime}))}\n\\end{equation}\n\nThe alignment $\\alpha_{t, t^\\prime}$ connects an alignment score for the input at position $t^\\prime$ and the output at position $t$. This score is based on how well this pair matches \\cite{attention_blog}. The set of all alignment scores defines how much each source hidden state should be considered for each output \\cite{attention_blog}. Please see Appendix \\ref{app:seq2seq_attention} for a more easy and visual explanation of the attention mechanism in the seq2seq model.\n\n\\begin{figure}[!ht]\n \\centering\n \\includegraphics[width=0.5\\textwidth]{presentation\/assets\/Attention-cropped.pdf}\n \\caption{Encoder-Decoder architecture with additive attention mechanism alternated from: \\cite{attention1}}\n \\label{fig:attention_arch}\n\\end{figure}\n\n\\subsection{Different types of score functions}\nGenerally, there are different implementations for this score function which have been used in various works. Table \\ref{tab:score} gives an overview over their respective name, equation and the usage in publications. Here, we have two trainable weight matrices in the alignment model denoted as $\\mathbf{v}_a$ and $\\mathbf{W}_a$.\n\\begin{table}[!ht]\n \\centering\n \\begin{tabular}{c|c|c}\n \\toprule\n \\textbf{Name} & \\textbf{Equation for:} $\\operatorname{score}(\\mathbf{S}_t, \\mathbf{H}_{t^\\prime})$ & \\textbf{Used In}\\\\\n \\midrule\n Content-base & $\\operatorname{cosine}[\\mathbf{S}_t, \\mathbf{H}_{t^\\prime}]$ & \\cite{graves2014neural} \\\\\n \\midrule\n Additive & $\\mathbf{v}_a^\\top \\tanh{\\mathbf{W}_a[\\mathbf{S}_t; \\mathbf{H}_{t^\\prime}]}$ & \\cite{bahdanau2014neural} \\\\\n \\midrule\n Location-Base & $\\operatorname{softmax}(\\mathbf{W}_a\\mathbf{S}_t)$ & \\cite{luong-etal-2015-effective} \\\\\n \\midrule\n General & $\\mathbf{S}_t^\\top \\mathbf{W}_a \\mathbf{H}_{t^\\prime}$ & \\cite{luong-etal-2015-effective} \\\\\n \\midrule\n Dot-Product & $\\mathbf{S}_t^\\top\\mathbf{H}_{t^\\prime}$ & \\cite{luong-etal-2015-effective} \\\\\n \\midrule\n Scaled Dot-Product & $\\frac{\\mathbf{S}_t^\\top\\mathbf{H}_{t^\\prime}}{\\sqrt{n_{source}}}$ & \\cite{attention} \\\\\n \\bottomrule\n \\end{tabular}\n \\caption{Different score functions with their respective equations and usage alternated from: \\cite{attention_blog}}\n \\label{tab:score}\n\\end{table}\n\nThe Scaled-Dot-Product used in \\cite{attention} scales the dot-product by the number of characters of the current word which is motivated by the problem that when the input is large, the softmax function may have an extremely small gradient which is a problem for efficient learning.\n\n\\subsection{Transformer}\nBy encorporating this Attention Mechanism the Transformer \\cite{attention} got introduced which achieves parallelization by capturing recurrence sequence with attention but at the same time encoding each item's position in the sequence based on the encoder-decoder architecture \\cite{zhang2019dive}. In fact, for that it doesn't use any recurrent network units and entirely relies on the self-attention mechanism to improve performance. The encoding part of the architecture is made out of several encoders (e.g. six encoders in \\cite{attention}) while the decoder part consists out of decoders with the same amount as the encoders. A general overview over the architecture is illustrated in Figure \\ref{fig:transformer}.\n\nHere, each encoder component consists out of two sub-layers which are Self-Attention and a Feed Forward Neural Network. Similarly, those two sub-layers are found in each decoder component but with a Encoder-Decoder Attention sub-layer in between them which works similarly to the Attention used in the seq2seq model. The deployed Attention layers are not your ordinary attention layers but a method called Multi-Headed Attention which improves performance of the attention layer. This allows the model to jointly attend to information from different representation\nsubspaces at different positions which in easier terms runs different chunks in parallel and concatenates the results \\cite{attention}. Unfortunately, explaining the design choices and mathematical formulations contained in multi-headed attention would be to much details at this point. Please refer to the original paper \\cite{attention} for more information. The architecture shown in Figure \\ref{fig:transformer} also deploys skip connections and layer normalisation for each sub-layer of the encoder as well as the decoder. One thing to note is that the input as well as the output get embedded and a positional encoding is applied which represents the proximity of sequence elements (see Appendix \\ref{app:pos}).\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.6\\textwidth]{presentation\/assets\/transformer.PNG}\n \\caption{Model Architecture of the Transformer: \\cite{attention}}\n \\label{fig:transformer}\n\\end{figure}\n\nThe final linear and softmax layer turn the vector of floats which is the output of the decoder stack into a word. This is done by transforming the vector through the linear layer into a much larger vector called a logits vector \\cite{transformer_blog}. This logits vector has the size of the learned vocabulary from the training dataset where each cell corresponds to the score of a unique word \\cite{transformer_blog}. By applying a softmax function we turn those scores into probabilities which sum up to $1$ and therefore we can choose the cell (i.e. the word) with the highest probability as output for this particular time step.\n\n\n\n\\section{Pointer Networks (Ptr-Nets)}\nPointer Networks (Ptr-Nets) \\cite{pointer} adapt the seq2seq model with attention to improve it by not fixing the discrete categories (i.e. elements) of the output dictionary \\textit{a priori}. Instead of yielding an output sequence generated from an input sequence, a pointer network creates a succession of pointers to the elements of the input series \\cite{pointer_blog}. In \\cite{pointer} they show that by using Pointer Networks they can solve combinatorial optimization problems such as computing planar convex hulls, Delaunay triangulations and the symmetric planar Travelling Salesman Problem (TSP).\n\nGenerally, we apply additive attention (from Table \\ref{tab:score}) between states and then normalize it by applying the softmax function to model the output conditional probability as seen in Equation \\ref{eq:pointer}.\n\\begin{equation}\n\\label{eq:pointer}\n \\mathbf{Y}_t = \\operatorname{softmax}\\left(\\operatorname{score}\\left(\\mathbf{S}_t, \\mathbf{H}_{t^\\prime}\\right)\\right) = \\operatorname{softmax}\\left( \\mathbf{v}_a^\\top \\tanh{\\mathbf{W}_a[\\mathbf{S}_t; \\mathbf{H}_{t^\\prime}]}\\right)\n\\end{equation} \n\nThe attention mechanism is simplified, as Ptr-Net does not blend the encoder states into the output with attention weights. In this way, the output only responds to the positions but not the input content \\cite{attention_blog}.\n\n\\section{Conlusion \\& Outlook}\nIn this work we gave an introduction into fundamentals for Recurrent Neural Networks (RNNs). This includes the general framework for RNNs, Backpropagation through time, problems of traditional RNNs, LSTMS, Deep and Bidirectional RNNs as well as more recent advances such as the Encoder-Decoder Architecture, seq2seq model, Attention, Transformer and Pointer Networks. Most topics are only covered conceptionally and don't go too deep into implementation specifications. To get a broader understanding of the covered topics, we recommend looking into some of the cited original papers. Additionally, most recent publications use some of the presented concepts so we recommend taking a look at such papers.\n\nOne recent publication which uses many of the presented concepts is ``Grandmaster level in StarCraft II using multi-agent reinforcement learning'' by Vinyals et al. \\cite{Vinyals2019}. Here, they present their approach to train agents to play the real-time strategy game Starcraft II with great success. If the presented concepts were a little too theoretical for you we recommend reading that paper to see LSTMs, the Transformer or Pointer Networks in a setting which can be deployed in a more practical environment. \n\n\\printbibliography[]\n\n\\begin{appendices}\n\\section{Visual Representation of LSTMs}\n\\label{app:vis}\nIn this section we consecutively construct the full architecture of Long Short-Term Memory Units (LSTMs) explained in Section \\ref{sec:lstms}. For a description what is changing between each step please read Section \\ref{sec:lstms} or refer to the source of the illustrations \\cite{zhang2019dive}.\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[height=6cm]{presentation\/assets\/gates.PNG}\n \\caption{Calculation of input, forget, and output gates in an LSTM: \\cite{zhang2019dive}}\n \\label{fig:gates}\n\\end{figure}\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[height=6cm]{presentation\/assets\/candidate.PNG}\n \\caption{Computation of candidate memory cells in LSTM: \\cite{zhang2019dive}}\n \\label{fig:candidate}\n\\end{figure}\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[height=6cm]{presentation\/assets\/memory.PNG}\n \\caption{Computation of memory cells in an LSTM: \\cite{zhang2019dive}}\n \\label{fig:memory}\n\\end{figure}\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[height=6cm]{presentation\/assets\/hidden.PNG}\n \\caption{Computation of the hidden state in an LSTM: \\cite{zhang2019dive}}\n \\label{fig:hidden}\n\\end{figure}\n\n\\section{Visual Representation of seq2seq with Attention}\n\\label{app:seq2seq_attention}\nThe seq2seq model with attention passes a lot more data from the encoder to the decoder than the regular seq2seq model. Instead of passing the last hidden state of the encoding stage, the encoder passes all the hidden states to the decoder. The first step of the decoder part in the seq2seq model with attention is illustrated in Figure \\ref{fig:seq2seqframe1} where we pass ``\\textit{I am a student}'' to the encoder and expect a translation to french producing ``\\textit{je suis un \\'{e}tudiant}''. Here, all the hidden states of the encoder $\\mathbf{H}_1$, $\\mathbf{H}_2$, $\\mathbf{H}_3$ are passed to the attention decoder as well as the embedding from the $$ token and an initial decoder hidden state $\\mathbf{H}_{init}$.\n\n \\begin{figure}[H]\n \\centering\n \\includegraphics[width=\\textwidth, height=6.3cm]{presentation\/assets\/videos\/seq2seq_attention\/frame1.pdf}\n \\caption{Seq2Seq Model with Attention Mechanism Step 1 alternated from: \\cite{seq2seqBlog}}\n \\label{fig:seq2seqframe1}\n \\end{figure}\n\nNext, we produce an output and a new hidden state vector $\\mathbf{H}_4$. However, the output is discarded. This can be seen in Figure \\ref{fig:seq2seqframe2}.\n\n \\begin{figure}[H]\n \\centering\n \\includegraphics[width=\\textwidth, height=6.3cm]{presentation\/assets\/videos\/seq2seq_attention\/frame2.pdf}\n \\caption{Seq2Seq Model with Attention Mechanism Step 2 alternated from: \\cite{seq2seqBlog}}\n \\label{fig:seq2seqframe2}\n \\end{figure}\n\nFor the attention step we use this produced hidden state vector $\\mathbf{H}_4$ and the hidden states from the encoder $\\mathbf{H}_1$, $\\mathbf{H}_2$, $\\mathbf{H}_3$ to produce a context vector $\\mathbf{C}_4$ (blue). This process can be seen in Figure \\ref{fig:seq2seqframe4}. Each encoder hidden state is most associated with a certain word in the input sentence \\cite{seq2seqBlog}. When we give these hidden states scores and apply a softmax to it we generate probability values. These probabilities are represented with by the three-element pink vector where light values stand for high probabilities while dark values denote low probabilities. Next, we apply each hidden state vector $\\mathbf{H}_1$, $\\mathbf{H}_2$, $\\mathbf{H}_3$ by its softmaxed score which increases hidden states with high scores, and decreases hidden states with low scores. This is visualised by graying out the hidden states $\\mathbf{H}_2$ and $\\mathbf{H}_3$ while keeping $\\mathbf{H}_1$ in solid color.\n\n \\begin{figure}[H]\n \\centering\n \\includegraphics[width=\\textwidth, height=6.3cm]{presentation\/assets\/videos\/seq2seq_attention\/frame4.pdf}\n \\caption{Seq2Seq Model with Attention Mechanism Step 4 alternated from: \\cite{seq2seqBlog}}\n \\label{fig:seq2seqframe4}\n \\end{figure}\n\nAfter that, we concatenate this produced context vector $\\mathbf{C}_4$ with the produced hidden state $\\mathbf{H}_4$. One can see this process in Figure \\ref{fig:seq2seqframe5}. This process just stacks the two vectors on top of each other.\n\n \\begin{figure}[H]\n \\centering\n \\includegraphics[width=\\textwidth, height=6.3cm]{presentation\/assets\/videos\/seq2seq_attention\/frame5.pdf}\n \\caption{Seq2Seq Model with Attention Mechanism Step 5 alternated from: \\cite{seq2seqBlog}}\n \\label{fig:seq2seqframe5}\n \\end{figure}\n\nThis concatenated version of hidden state $\\mathbf{H}_4$ and context vector $\\mathbf{C}_4$ is then passed into a jointly trained Feedforward Neural Network. This network is visualised by the red box with round edges in Figure \\ref{fig:seq2seqframe6}. The output of this network then represents the output of the current time step $t$ which in this case represents the word ``\\textit{I}''. This basically concludes all the steps needed at each iteration step.\n\n \\begin{figure}[H]\n \\centering\n \\includegraphics[width=\\textwidth, height=6.3cm]{presentation\/assets\/videos\/seq2seq_attention\/frame6.pdf}\n \\caption{Seq2Seq Model with Attention Mechanism Step 6 alternated from: \\cite{seq2seqBlog}}\n \\label{fig:seq2seqframe6}\n \\end{figure}\n\nIf we take a look at the next iteration step in Figure \\ref{fig:seq2seqframe7} we can see that the output from the previous hidden state $\\mathbf{H}_4$ is passed instead of the $$ token. All the other steps are equal from the previous iteration. However, we can see that the hidden state $\\mathbf{H}_2$ has the best score during the attention stage. Again, this is represented by the lightest shade of pink in the score vector. By multiplying the scores with the hidden states we achieve two reduced hidden states $\\mathbf{H}_1$ and $\\mathbf{H}_3$ while keeping $\\mathbf{H}_2$ as the most active hidden state. This results in the word ``\\textit{am}'' being produced as the output of the Feedforward Neural Network for this time step.\n\n \\begin{figure}[H]\n \\centering\n \\includegraphics[width=\\textwidth, height=6.3cm]{presentation\/assets\/videos\/seq2seq_attention\/frame7.pdf}\n \\caption{Seq2Seq Model with Attention Mechanism Step 7 alternated from: \\cite{seq2seqBlog}}\n \\label{fig:seq2seqframe7}\n \\end{figure}\n\nObviously, there are still two more attention decoder time steps which are omitted here for illustration purposes. The functionality of each of those steps however would still be equivalent to the already seen time steps.\n\n\\section{Visual Representation of Positional Encodings used in the Transformer}\n\\label{app:pos}\nOn example of a positional encoding used inside the transformer is applying trigonometric functions as seen in Figure \\ref{fig:pos_encoding}. Here, we have multiple trigonometric functions with different frequency. We also show the encoding for three words i.e. $\\mathbf{X}_1$, $\\mathbf{X}_2$, $\\mathbf{X}_3$. \n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\textwidth]{presentation\/assets\/positional_encodings.pdf}\n \\caption{Positional Encoding Example based on trigonometric functions}\n \\label{fig:pos_encoding}\n\\end{figure}\n\nIn principal the encoding for $\\mathbf{X}_1$ is therefore high for the first curve (blue), mid for the second curve (red) and low for the last curve (green). Similarly, this applies for the other words as well. What we can see here is that close words have closer encodings while distant words have more different encodings. Generally, this is a method for binary encoding the position of a given sequence. \n\nThe choice of such a positional encoding algorithm definitely is not the main contribution of \\cite{attention} but it is a relevant concept to at least understand in theory since this boosts performance.\n\n\\end{appendices}\n\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nNamed entity recognition (NER) is commonly dealt with as a sequence labeling job and has been one of the most successfully tackled NLP tasks. Inspired by the similarity between NER and object detection in computer vision and the success of two-stage object detection methods, it's natural to ask the question, can we borrow some ideas from there and simply copy the success. This work aims at pushing further our understanding in NER by applying a two-stage object-detection-like approach and evaluating its effects in precision, recall and different types of errors in variant circumstances. \n\nAs a prototype for many other following object detection models, Faster R-CNN~\\cite{ren2015faster} is a good example to illustrate how the two stages work. In the first stage, a region proposal network is responsible for generating candidate regions of interest from the input image and narrowing down the search space of locations for a more complex and comprehensive investigation process. The second stage work is then done by the regression and classification models, which look at finer feature maps, further adjust the size and position of the bounding box and tell whether there is an object and what type it belongs to. Admittedly a two-stage pipeline is often blamed for error propagation, however in the case of object detection it apparently brings much more than that.\n\nWe can easily see the analogy here in NER. To do NER in a similar way, first we find candidate ranges of tokens that are more likely to be entities. Second, we scrutinize into each of them, make a judgment if this is a true entity or not, and then do entity classification and also regression if necessary. Even though the search space in 1-D NER problem is significantly smaller than in a 2-D image, the benefits of a two-stage approach are still obvious. Firstly, with the better division of labor, the two components can focus on their specialized tasks which are quite different. More specifically, the region proposal part takes care of the context information and entity signals regardless of entity type, while the second part covers more on entity integrity and type characteristics. Secondly, since the region prediction has been given by the first stage, the second part of the model can take the global information of all the tokens in the region, instead of looking separately as in sequence labeling. Although a single token vector can also encode context information as in BERT~\\cite{devlin2018bert} and other similar LMs~\\cite{elmo2018, Yang2019XLNet, Yinhan2019Roberta, Vaswani2017transformer, radford2019GPT2}, having a global view of all relevant tokens is definitely an advantage and may provide more hints for the final decision. Thirdly, the model gets better interpretability, since it separates the concepts of entityness (how likely this mention is a qualified named entity) and entity classification, you get an interpretable probability for each entity prediction. Finally, another benefit of this method is that each entity prediction is independent, which makes it possible to be used in a nested entity recognition problem that can not be easily handled by sequence labeling approach, which can only predict one label for each token.\n\nThis paper is structured as follows. In Section \\ref{section:model} we explain the model architecture, including two stages, region proposal and entity classification. Section \\ref{section:related} reviews the past related works in NER, especially region based NER that are most similar to our work. Section \\ref{section:experiment} explains the training process and evaluation results in detail on flat NER tasks. Section ~\\ref{section:nestedexp} shows the training and evaluation on nested NER tasks. In Section \\ref{section:ablation} we evaluate the importance of different parts of the model by ablation. And finally in Section ~\\ref{section:error} we show the error analysis of the new method and compared it with the traditional sequence labeling methodology.\n\n\\section{Model Design}\n\\label{section:model}\nWe propose a two-stage method for NER. In the first stage, an entity region proposal network predicts highly suspected entity regions. In the second stage, another model takes the output candidates, extracts more features and then makes a prediction whether this candidate is a qualified entity or not and what type should be attached to this entity. Precision and recall would be evaluated end-to-end, in the same manner as traditional NER models.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=1\\linewidth]{Figures\/proposal_diagram.pdf}\n\t\\caption{\\small{Region proposal network is made by adding a linear layer on top of the BERT output. The start prediction is independent from the end prediction, both of which use cross entropy loss over two classes. For the end position, we assign it to the first token of the word right immediate after the entity. We only predict for the first token of every word if the tokenizer outputs WordPieces or other subword tokens, and all other tokens will be masked (the mask is omitted from the diagram).}}\n\t\\label{fig:proposalNetwork}\n\\end{figure}\n\n\\subsection{Entity Region Proposal}\nFor the first stage, we used a simple model similar to sequence labeling ones. But instead of predicting IOB-format entity labels~\\cite{ramshaw-marcus-1995-text}, we predict $\\langle$Start$\\rangle$ and $\\langle$End$\\rangle$ labels. We appended another linear layer on top of the hidden state outputs from the BERT model~\\cite{devlin2018bert}, to predict the probability for a token to be $\\langle$Start$\\rangle$ and $\\langle$End$\\rangle$, we only consider the starting token in each word and mask out all trailing tokens within that word. And we used cross entropy over two classes for the prediction. The model structure is demonstrated in Figure \\ref{fig:proposalNetwork}. The goal of the first-stage model is high recall and acceptable precision, so we could tilt the weights a little bit towards the positive labels to favor better recall numbers. After extracting the start and end tokens, we then select pairs to form a complete region, with the simple rule that the length of the entity cannot exceed some limit. Admittedly, there is an apparent disadvantage here that is we discard longer candidates without even a try. But in fact, those longer ones only compose quite a small portion and are usually very difficult to get right anyway. In practice we have used 6 and 12 as the length limit for different datasets and can easily cover 98$\\%$-99$\\%$ of ground-truth entities. \n\n\\subsection{Entity Discrimination and Classification}\n\n\\begin{figure*}[!h]\n\t\\centering\n\t\\includegraphics[ width=0.92\\linewidth ]{Figures\/model_diagram.pdf}\n\t\\caption{\\small{The second-stage model is responsible for re-examining the entity region proposals generated by the first stage. The model trains with 4 tasks simultaneously. The entityness part and type classification part takes global view of all tokens within the range, while the two boundary losses only zoom into the two tokens across the boundary to make a double check of the exact range.}\n\t}\n\t\\label{fig:mainModel}\n\\end{figure*}\n\nThe second stage is also using BERT~\\cite{devlin2018bert} as the encoding model. The second stage has two main tasks, discrimination and classification, which defines the two major components in our loss function, the entityness loss and the type classification loss. Contrary to a typical object detection model, we don't do regression for the bounding box. The model will only tell if this range is correct or not and if not, discard it directly. One reason is that we want to keep it as simple as possible and another reason is that the problem is much easier than the 2-D object detection, and our proposals are usually accurate enough. Compared to the sequence labeling method, our model focuses more on aggregate features across all tokens spanning the entity, which can be seen from the max pooling layer in both the entityness and classification components. To make the model more sensitive to boundary errors, especially when coming across long entities, we added another two losses, the start loss and the end loss, which predict if the boundaries are correct, so that the prediction won't be dominated by the bulk part but also pay attention to the boundary. These start\/end logits are also concatenated with the entityness feature to predict the entityness score. The total loss function can be written as below:\n\n\\scriptsize\n\\begin{align}\n\\textbf{L} &= \\alpha \\left(\\textbf{L}_{\\text{start}} + \\textbf{L}_{\\text{end}}\\right) + \\beta \\textbf{L}_{\\text{entityness}} + \\textbf{L}_{\\text{type}}\n\\\\\n\\textbf{L}_{\\text{s}} &= -\\frac{1}{N}\\sum_{i = 1}^{N}\\left[\\mathlarger{\\mathbbm{1}}_{i}^{\\text{s}} \\log (p_{i}^{\\text{s}}) + (1-\\mathlarger{\\mathbbm{1}}_{i}^{\\text{s}}) \\log (1-p_{i}^{\\text{s}})\\right], \\nonumber\n\\\\\n& \\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\text{s} = \\text{start, end, entityness}\n\\\\\n\\textbf{L}_{\\text{type}} &= -\\frac{1}{N}\\sum_{i = 1}^{N}\\mathlarger{\\mathbbm{1}}_{i}^{\\text{entity}}\\left[\\sum_{c \\in \\textrm{classes}}\\mathlarger{\\mathbbm{1}}_{i}^{\\text{c}}\\log (p_{i}^{\\text{c}})\\right]\n\\end{align}\n\\normalsize\n\nIn Equation 1, $\\alpha$ and $\\beta$ are hyperparameters that control the weights of the boundary loss and the total entityness loss, and we used 0.5 and 1.0 as default values. In Equation 2-3, $i$ iterates through from sample 1 to N where each sample can be seen as a tuple $(sentence, index_{start}, index_{end})$. $\\mathlarger{\\mathbbm{1}}_{i}^{\\text{entity}}$ is an indicator that the sample $i$ is an entity. $\\mathlarger{\\mathbbm{1}}_{i}^{\\text{c}}$ indicates if the entity belongs to type c. Same idea for $\\mathlarger{\\mathbbm{1}}_{i}^{\\text{s}}$. The start loss, end loss and entityness loss are all cross entropy loss over two classes. The type classification loss is only applied when the region proposal matches a true entity, and would be ignored otherwise.\n\nThe model structure is illustrated in Figure \\ref{fig:mainModel}. For the start and end loss, we used a model that calculate multi-head dot products between the two tokens across the boundary. The intuition is that it needs to catch the relation signal between the two sides. After dot products we put an extra fully connected layer to transform the feature vector to the logits of a two-class classification. All heads have independent weights. \n\nFor the entityness and type classification loss, we used the same architecture with separate weights, that is a fully connected layer followed by a max pooling layer, and then another fully connected layer to get the final logits. And we use ReLU activation function after each linear layer. The same as region proposal, we only consider the starting token in each word and mask out all trailing tokens within that word.\n\nDuring inference, we look at the entityness probability first. Only if it's above a specified threshold will we look at the classification results and output the most likely type. \n\n\\section{Related Work}\n\\label{section:related}\nNamed entity recognition (NER) is a classic problem in NLP. The goal of NER is to extract named entities from free text and these entities can be classified into several categories, for example person (PER), location (LOC) and geo-political entity (GPE). Traditionally NER is tackled by sequence labeling method. Many different models are developed along this direction, such as CRFs in \\cite{lafferty2001conditional, sutton2007dynamic}, LSTM in ~\\cite{hammerton2003named}, LSTM-CRF in ~\\cite{lample2016neural} etc. More recently, people start using large-scale language models, such as BERT~\\cite{devlin2018bert} and ELMo~\\cite{elmo2018}. \n\nNested named entity recognition takes the overlapping between entities into consideration~\\cite{kim2003genia}. This cannot be easily done by traditional sequence labeling in that one can only assign one tag for each token. There have been many different approaches proposed for this problem. One important branch is the region-based method and our work can also be classified into this category. ~\\cite{finkel2009nested} leveraged parsing trees to extract subsequences as candidates of entities. ~\\cite{xu2017fofe,sohrab-miwa-2018-deep} considers all subsequences of a sentence as candidates. A more recent paper by ~\\cite{hongyu2019nugget} developed a model that locates anchor word first and then searches for boundaries of the entity, with the assumption that nested entities have different anchor words. Another work that is very close to ours is done by~\\cite{zheng2019boundary}. In their paper they also proposed a two stage method. Our work is different than theirs from several perspectives. We have an entityness prediction in the second stage like the objectness in object detection, and thus we don't completely depend on the first stage to determine the region. And our model is built on BERT language model and finetunes all lower layers, while theirs is using LSTM plus pretrained word embedding. Another branch of researches is trying to design more expressive tagging schemas, some representative works are done by ~\\cite{lu2015joint, kati2018nested, luwei2018overlap}. The current state of the art is ~\\cite{li2019unified}, where they viewed the NER problem as a question answering problem and naturally solved the nested issue. Their model showed impressive power in both flat and nested NER tests. A major difference between our model and theirs is that they predict a paring score for any start and end index pair, which makes the feature matrix and the computational complexity a big issue. We instead predict entityness only for very few candidates, and we don't need to duplicate training examples for multiple queries, thus our training process takes much less time.\n\nThe main contribution of our work is that we bring up the idea to use a high-recall and relatively low-precision first stage model to select regions and use a more complicated model to predict the entityness and classification at the same time, with a global view of all the tokens spanning the candidate entity. Besides that, our model is much simpler and more lightweight than other similar models designed for nested NER tasks, and both training and inference run as fast as the plain BERT sequence labeling model. Our core model architecture is nothing but a linear layer plus a max pooling layer, but gives pretty good performance, especially on the ACE2005 dataset.\n\n\\section{Flat NER Experiments}\n\\label{section:experiment}\n\nFor the flat NER experiment, we used the CoNLL2003~\\cite{conll2003ner} and OntoNotes 5.0~\\cite{ontonotes5}. CoNLL2003 is an\nEnglish dataset with four types of named entities, namely Location, Organization, Person and Miscellaneous. And OntoNotes 5.0 is an English dataset containing text from many sources and including 18 types of named entity,\n\n\n\\begin{table*}[!h]\n\t\\small\n\t\\begin{center}\n\t\t\\begin{tabular}{lrrrr}\n\t\t\t& Dev Precision & Dev Recall & Test Precision & Test Recall \\\\\n\t\t\t\\hline\n\t\t\t\\textbf{CoNLL2003 Region Proposal} & & & & \\\\\n\t\t\t\\hline\n\t\t\tBERT base & 71.3 & 98.0 & 70.7 & 96.2 \\\\\n\t\t\tBERT large & 71.4 & 98.0 & 70.6 & 96.5 \\\\\n\t\t\t\\hline\n\t\t\t\\hline\n\t\t\t\\textbf{OntoNotes 5.0 Region Proposal} & & & & \\\\\n\t\t\t\\hline\n\t\t\tBERT base (weight 0.5:0.5) & 69.3 & 90.8 & 69.3 & 89.6 \\\\\n\t\t\tBERT base (weight 0.3:0.7) & 67.8 & 92.7 & 67.4 & 92.2 \\\\\n\t\t\tBERT base (weight 0.2:0.8) & 66.3 & 93.9 & 65.8 & 93.7 \\\\\n\t\t\tBERT base (weight 0.1:0.9) & 63.8 & 95.1 & 63.1 & 95.5 \n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{\\small \\textbf{Region Proposal Model Results.} Precision and recall numbers are region metrics regardless of entity type. A prediction is correct as long as the region predicted matches the start and end of the ground-truth entity. Region precision and recall are reported for both dev and test sets.}\n\t\\label{tab:rpn}\n\\end{table*}\n\n\n\\begin{table*}[!h]\n\t\\small\n\t\\begin{center}\n\t\t\\begin{tabular}{lrrrrrr}\n\t\t\t& Dev Precision & Dev Recall & Dev F1 & Test Precision & Test Recall & Test F1 \\\\\n\t\t\t\\hline\n\t\t\t\\textbf{CoNLL2003} & & & & & & \\\\\n\t\t\t\\hline\n\t\t\tBERT base & 95.1 & 95.1 & 95.1 & 91.9 & \\textbf{91.7} & \\textbf{91.8} \\\\\n\t\t\tBERT large & 96.1 & 95.4 & 95.8 & \\textbf{92.0} & 91.6 & \\textbf{91.8} \\\\\n\t\t\t\\hline\n\t\t\t\\hline\n\t\t\t\\textbf{OntoNotes 5.0} & & & & & & \\\\\n\t\t\t\\hline\n\t\t\tBERT base & 87.7 & 87.3 & 87.5 & \\textbf{86.9} & 86.5 & 86.7 \\\\\n\t\t\tBERT large & 88.0 & 88.3 & 88.2 & 86.8 & \\textbf{87.3} & \\textbf{87.0} \n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{\\small \\textbf{Flat NER Results.} Standard NER precision and recall are reported here for both Dev and Test sets. We only showed the best model for each combination of dataset and the size of BERT encoder.}\n\t\\label{tab:e2e}\n\\end{table*}\n\n\\subsection{First-Stage Training}\nFor region prediction, we used the default training parameters provided by HuggingFace Transformers~\\cite{Wolf2019HuggingFacesTS} for token classification task, i.e. AdamW optimizer with learning rate=$5\\times10^{-5}$, $\\beta_{1}$=$0.9$ and $\\beta_{2}$=$0.999$, hidden state dropout probability 0.1 etc. We finetuned both BERT-base-cased and BERT-large-cased models for 3 epochs with batch size of 64. The regions with length equal or less than 6 were selected as candidates for the second stage. We chose the threshold 6 because most named entites are shorter than 6 (99.9$\\%$ in CoNLL2003 and 99.2$\\%$ in OntoNotes 5.0). Since we ignore entity type in the first stage model, the precision and recall are based only on region proposals regardless of type. We keep the model to be as simple as possible because the only goal is to get high recall in the first stage. For the OntoNotes 5.0 model, a default training gave a model with pretty low recall, only 89.6$\\%$, so we changed the weights in the cross entropy loss to raise the recall with a little trade-off of precision. Therefore precision and recall were reported at different weights for OntoNotes. The training results can be found in Table~\\ref{tab:rpn}.\n\n\nFrom the result, we can see that for CoNLL2003~\\cite{conll2003ner}, base and large models gave pretty close p\/r numbers, so we used BERT base in the following experiments. For OntoNotes dataset, we tried several different weights, it turned out that weights 0.4:0.6 and 0.3:0.7 both gave pretty good end-to-end results.\n\n\\begin{table*}[t]\n\t\\small\n\t\\begin{center}\n\t\t\\begin{tabular}{lrrrrrr}\n\t\t\t&&\\textbf{Genia}&&&\\textbf{ACE2005}&\\\\\n\t\t\t\\hline\n\t\t\t\\textbf{Model} & \\textbf{Precision($\\%$)} & \\textbf{Recall($\\%$)} & \\textbf{F1($\\%$)} & \\textbf{Precision($\\%$)} & \\textbf{Recall($\\%$)} & \\textbf{ F1($\\%$)}\\\\\n\t\t\t\\hline\n\t\t\tARN~\\cite{hongyu2019nugget} & 75.8 & 73.9 & 74.8 & 76.2 & 73.6 & 74.9 \\\\\n\t\t\tBoundary-aware Neural~\\cite{zheng2019boundary} & 75.8 & 73.6 & 74.7 & - & - & - \\\\\n\t\t\tMerge-BERT~\\cite{joseph2019merge} & - & - & - & 82.7 & 82.1 & 82.4 \\\\\n\t\t\tSeq2seq-BERT~\\cite{strakova2019nested} & 80.1 & 76.6 & 78.3 & 83.5 & 85.2 & 84.3 \\\\\n\t\t\tPath-BERT~\\cite{shibuya2019second} & 77.81 & 76.94 & 77.36 & 83.83 & 84.87 & 84.34 \\\\\n\t\t\tBERT-MRC~\\cite{li2019unified} & 85.18 & 81.12 & \\textbf{83.75} & 87.16 & 86.59 & \\textbf{86.88} \\\\\n\t\t\t\\hline\n\t\t\t\\textbf{Our Model} &&&&&&\\\\\n\t\t\t\\hline\n\t\t\tBERT Base Model & 76.6 & 75.1 & 75.9 & 82.8 & 84.9 & 83.8 \\\\\n\t\t\tBERT Large Model & 77.4 & 76.3 & 76.8 & 85.2 & 85.9 & 85.6 \\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{\\small \\textbf{Nested NER Results on Genia and ACE2005.} }\n\t\\label{tab:nested}\n\\end{table*}\n\n\\subsection{Second-Stage Training}\n\nThere are quite a few differences between our second-stage training and the traditional sequence labeling NER training, in the sense that our training is on region proposal level while sequence labeling is on sentence level. Each training example is now a combination of a sentence, a proposal start index and a proposal end index, and one sentence could emit multiple training examples. And the labels are no longer token labels, but labels designed for our specific loss, specifying if the start index is correct, if the end index is correct, if the region is corresponding to a true entity, and at last what the entity type is if all the previous answers are positive. The training samples are those region proposals output from the first model and we added more randomly selected negative regions to make it more robust, which turned out to be very important and will be explained with more details in the following paragraphs. Finally we evaluated the model performance in an end-to-end manner. Precision, recall and F1 score on CoNLL2003 and OntoNotes 5.0 can be found in Table \\ref{tab:e2e}.\n\nFor the CoNLL2003 dataset~\\cite{conll2003ner}, the best F1 score we got with base and large models are both 91.8, which is comparable but a little lower than the reported sequence labeling BERT results (BERT-Tagger). We didn't spend too much time on finetuning hyperparameters, it's possible that there is still room for the model. Another possible reason could be in our model we only predict for one entity at a time and didn't consider the interaction between entities when there are multiple in one sentence. For the OntoNotes 5.0 dataset we got F1 score 87.0. \n\nWe tried a few tricks to improve the p\/r number. The most effective one is to add more randomly selected regions as negative samples during the second stage. For each sentence, we generate one random region that has length in range [1, 6] and that doesn't fall into the existing candidates or true entities. We then labeled them as wrong samples and fed into the training process together with other candidates. With more random negative samples, the model is more robust when there is a big gap between the train and test set distributions, especially when we have a very strong stage-one model with high precision, which could have a strong bias on the distribution of negative samples. By adding more negatives, we have almost 0.7$\\%$ gain in the F1 score, which will be shown with more details in the following ablation study section \\ref{section:ablation}.\n\nWe also tried adding an extra loss to take into account the classification error for those regions that were not predicted completely correct but have a large overlap with one of the true entities. They were not exactly matching the ground true region, therefore we assigned a weight less than 1 (0.2 in our experiment). Intuitively, this solution is equivalent to adding more classification training data. However we didn't see significant improvement after this change. We also explored changing hyperparameters, for example the numbers of channels in the second-stage model. In the default settings we set the number of heads in dot product to be 32, and the dimension of feature vector in both entityness and classification to be 64. We tried half or double these numbers but the model performance degraded in both cases. More details can be found in the next ablation study subsection \\ref{section:ablation}.\n\n\\section{Nested NER Experiments}\n\\label{section:nestedexp}\nFor nested NER experiments, we chose two mainstream datasets, the ACE2005 dataset~\\cite{ace2005ner} and Genia dataset~\\cite{kim2003genia}. The ACE2005 dataset contains 7 entity categories. For comparison, we follow the data preprocessing procedure in ~\\cite{lu2015joint, luwei2018overlap, kati2018nested} by keeping files from bw, bn, nw and wl, and splitting these files randomly into train, dev and test sets by 8:1:1, respectively. For the Genia dataset, again we use the same preprocessing as ~\\cite{finkel2009nested, lu2015joint} and we also consider 5 entity types - DNA, RNA, protein, cell line\nand cell type.\n\nOur training process for nested NER is basically the same as the previous section, since our model doesn't differentiate between flat and nested and will simply treat all entity regions in the same way even if they are overlapped. Considering the different distribution of entity lengths, we increased the length limit for region candidates from 6 words to 12 for ACE2005 and 8 for Genia. For the region proposal model, we again trained on top of BERT-base-cased for 3 epochs, and in the cross entropy loss we used weights 0.1:0.9. For the second stage, we trained both BERT-base and BERT-large for 10 epochs, the results are reported in Table~\\ref{tab:nested}. With the BERT large model we got an average F1 score of 85.6 on ACE2005 and 76.8 on Genia. This is not as good as the current state-of-the-art result, but is quite competitive, and especially on ACE2005 dataset it's better than all other models except the BERT-MRC model.\n\n\\begin{table}[h]\n\t\\small\n\t\\begin{center}\n\t\t\\begin{tabular}{|c|c|c|}\n\t\t\t\\hline\n\t\t\t\\textbf{Entity Type} & \\textbf{Test Precision} & \\textbf{Test Recall} \\\\\n\t\t\t\\hline\n\t\t\tDNA & 74.8 & 72.6 \\\\\n\t\t\t\\hline\n\t\t\tRNA & 90.6 & 82.1 \\\\\n\t\t\t\\hline\n\t\t\tcell line & 78.5 & 67.1 \\\\\n\t\t\t\\hline\n\t\t\tcell type & 73.6 & 72.2 \\\\\n\t\t\t\\hline\n\t\t\tprotein & 78.5 & 79.7 \\\\\n\t\t\t\\hline\n\t\t\tOverall & 77.4 & 76.3 \\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{\\small \\textbf{Precision and Recall by Category. (Genia)} }\n\t\\label{tab:geniatype}\n\\end{table}\n\n\\begin{table}[h]\n\t\\small\n\t\\begin{center}\n\t\t\\begin{tabular}{|c|c|c|}\n\t\t\t\\hline\n\t\t\t\\textbf{Entity Type} & \\textbf{Test Precision} & \\textbf{Test Recall} \\\\\n\t\t\t\\hline\n\t\t\tPER & 88.6 & 89.9 \\\\\n\t\t\t\\hline\n\t\t\tLOC & 77.0 & 76.2 \\\\\n\t\t\t\\hline\n\t\t\tORG & 74.8 & 77.8 \\\\\n\t\t\t\\hline\n\t\t\tGPE & 87.7 & 86.7 \\\\\n\t\t\t\\hline\n\t\t\tFAC & 82.0 & 77.8 \\\\\n\t\t\t\\hline\n\t\t\tVEH & 72.3 & 71.4 \\\\\n\t\t\t\\hline\n\t\t\tWEA & 75.0 & 73.8 \\\\\n\t\t\t\\hline\n\t\t\tOverall & 85.2 & 85.9 \\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{\\small \\textbf{Precision and Recall by Category. (ACE2005)} }\n\t\\label{tab:acetype}\n\\end{table}\n\nA detailed P\/R analysis by category for Genia and ACE2005 datasets are given in Table~\\ref{tab:geniatype} and Table~\\ref{tab:acetype}.\n\n\\section{Ablation Study}\n\\label{section:ablation}\nIn this part, we did an ablation study to assess the contribution from each component of the new model. In the first experiment, we removed the start\/end logits from the concatenated vector for entityness prediction and also removed the start\/end loss from the total loss, to see if they are helpful to resolve boundary errors. Then, we evaluated the effects of the max pooling layer over the whole entity and used only the first token's feature vector instead. In the following experiments we also tried removing random negative samples and reducing the model size by using less channels or dimensions for the start\/end unit, entityness prediction and entity classification etc. The evaluation results on the CoNLL2003 test set is shown in Table \\ref{tab:ablation}.\n\nFrom the results, we can see that the start\/end prediction only has limited effect for the final F1, but with a more closer look at the errors we found that with start\/end prediction, the boundary errors dropped from 149 (out of 5711 test samples) to 127 while the type classification errors changed from 222 to 230. The start\/end loss did change the pattern of errors and corrected some boundary mistakes. Removing the max pooling layer brought an F1 drop greater than 1.0$\\%$ and removing random negative samples brought 0.7$\\%$ drop. After reducing the number of channels to half of the default, we got a drop of 0.5$\\%$ and when cutting further, keeping only 25$\\%$ channels, the model performance degraded pretty fast to 65$\\%$, indicating a strong underfit.\n\n\\section{Error Analysis}\n\\label{section:error}\nTo have a deeper understanding what type of errors our model is making, we did a further error profiling and also compared it with the result of a standard sequence labeling model based on BERT. Inspired by the methodology from Object Detection research, We divided the NER errors into the following types: (1) the region is correct but the type classification is wrong; (2) one of the left\/right boundaries of the region is wrong, but classification is correct; (3) one of the left\/right boundaries of the region is wrong, and the classification is also wrong; (4) over-triggering: the predicted entity has no overlap with any ground-truth entity; (5) under-triggering: for a true entity, either no overlapping region is proposed or the entityness model predicts no entity; (6) another type of error is that both boundaries are wrong but the region has overlap with at least one of true entities, in the evaluation we found that this type of error occurs only once, so we ignored it in the pie plot. We displayed the error composition for the new two-stage model as well as a traditional sequence labeling BERT model side by side, as shown in Figure \\ref{fig:error}.\n\n\\begin{table}[t]\n\t\\small\n\t\\begin{center}\n\t\t\\begin{tabular}{lrrr}\n\t\t\t& \\textbf{Test P} & \\textbf{Test R} & \\textbf{F1}\\\\\n\t\t\t\\hline\n\t\t\t\\textbf{Original setting} & \\textbf{91.9} & 91.7 & \\textbf{91.8} \\\\\n\t\t\tRemove start\/end prediction & 90.4 & \\textbf{91.9} & 91.7 \\\\\n\t\t\tRemove max pooling & 90.0 & 90.7 & 90.4 \\\\\n\t\t\tRemove random negatives & 90.6 & 91.6 & 91.1\\\\\n\t\t\tRemove 50$\\%$ channels & 91.2 & 91.3 & 91.3 \\\\\n\t\t\tRemove 75$\\%$ channels & 65.0 & 65.4 & 65.2\\\\\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{\\small \\textbf{Ablation Study using CoNLL2003 Dataset.} All experiments are using the BERT base model and training with the same epochs. P, R and F1 are reported on test set.}\n\t\\label{tab:ablation}\n\\end{table}\n\n\\begin{figure}[h]\n\t\\centering\n\t\\includegraphics[width=1.05\\linewidth]{Figures\/error_analysis.png}\n\t\\caption{\\small{\\textbf{Error Analysis. } We have divided the entity recognition error to 5 classes, more details can be found in the text. We profiled the errors for traditional sequence labeling BERT model (the left panel) and also the two-stage model proposed in this paper (the right panel), to provide a deeper insight of the composition of model errors.}}\n\t\\label{fig:error}\n\\end{figure}\n\nWe ran analysis on two models both trained from BERT-base and having similar precision and recall. As we can see, for both models the dominant part is the type error (region is correct). We could consider using larger and more complicated type prediction models since there is a lot of room in that direction reading from the plot. The new model made significantly less over-triggering errors, which means the precision of the entityness prediction is good. And two models have similar amounts of single-boundary errors and under-triggering errors.\n\n\n\\section{Conclusion}\n\nIn this paper, we proposed a new two-stage model for named entity recognition. And many of the ideas are coming from the inspiration of object detection in computer vision. More specifically, with a coarse first-stage model to provide region proposals, we rely on a second-stage model to predict entityness and entity types at the same time. Through sufficient experiments in both flat and nested NER tasks, we found that it has better performance on nested NER, and we got F1 85.6 on ACE2005 dataset and F1 76.8 on Genia datase, better than many more complicated models. On flat NER tasks, it's still a few points behind the current SOTA results. In the future we are going to improve the model further to see where is the real limit of two-stage region-based named entity recognition.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzauas b/data_all_eng_slimpj/shuffled/split2/finalzzauas new file mode 100644 index 0000000000000000000000000000000000000000..a4cb62d1c69f89a711a76c762ecf4b55f2ce0a20 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzauas @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\label{sec:intro+results}\n\nThe main theme of the paper is the dynamics of Hamiltonian\ndiffeomorphisms of ${\\mathbb C}{\\mathbb P}^n$, equipped with the standard symplectic\nstructure, with exactly $n+1$ periodic points, called here Hamiltonian\npseudo-rotations. By Arnold's conjecture established in this case by\nFortune and Weinstein, \\cite{Fo, FW}, and by Floer, \\cite{Fl}, this is\nthe minimal possible number of fixed points and, in particular, of\nperiodic points.\n\nWe prove several results on the dynamics of such maps. One is the\nexistence of invariant sets in arbitrarily small punctured\nneighborhoods of the fixed points, partially extending the results of\nLe Calvez and Yoccoz and of Franks and Misiurewicz from \\cite{Fr99, FM,\n LCY} to higher dimensions. Another result is a strong variant of the\nLagrangian Poincar\\'e recurrence conjecture for pseudo-rotations. We\nalso prove the $C^0$-rigidity of pseudo-rotations with exponentially\nLiouville mean index vector. This is a $2n$-dimensional analog of\nBramham's theorem from \\cite{Br}.\n\nPerhaps the most important general point of this paper is that rather\nunexpectedly one can obtain a lot information about the dynamics of\npseudo-rotations in dimensions greater than two, going far beyond\nperiodic orbits, by purely Floer theoretical methods. Before turning\nto the precise statements of sample results we need to discuss the\nnotion of a Hamiltonian pseudo-rotation -- the key player in this work\n-- in a greater generality and more detail.\n\n\\subsection{Hamiltonian pseudo-rotations}\n\\label{sec:PRs}\nIn the framework of two-dimensional dynamics, pseudo-rotations are\narea preserving diffeomorphisms of $D^2$ or $S^2$ with exactly one or\ntwo periodic points, respectively, which are then automatically the\nfixed points. There are several ways of extending this notion to\nhigher dimensions in the context of Hamiltonian dynamics and\nsymplectic topology.\n\nFor instance, one can define a Hamiltonian pseudo-rotation of a closed\nsymplectic manifold $(M^{2n},\\omega)$ as a Hamiltonian diffeomorphism\n$\\varphi$ with finite and minimal possible number of periodic points,\nand such that the periodic points are the fixed points. This is the\ndefinition we prefer to use here even though the notion of the minimal\npossible number is ambiguous. However, when $M$ is sufficiently nice\nand Arnold's conjecture is known to hold for $M$, this can be the sum\nof Betti numbers of $M$ when $\\varphi$ is non-degenerate, and a\nLusternik--Schnirelmann type lower bound (e.g., the category or the\ncup-length plus one) in general. For ${\\mathbb C}{\\mathbb P}^n$ both lower bounds are\n$n+1$; hence Definition \\ref{def:PR}.\n\nOne can also define a pseudo-rotation as a Hamiltonian diffeomorphism\nwith finitely many periodic points. These are the so-called perfect\nHamiltonian diffeomorphisms studied in the context of the Conley\nconjecture; see, e.g., \\cite{CGG, GG:survey} and references\ntherein. As follows from the results in \\cite{Fr92, Fr96, FH, LeC}, in\ndimension two this definition is equivalent to the one adopted\nhere. We will further discuss the difference or a lack thereof between\npseudo-rotations and perfect Hamiltonian diffeomorphisms below, but at\nthe moment we only mention that in all known examples perfect\nHamiltonian diffeomorphisms are non-degenerate pseudo-rotations.\n\nIt is believed that rather few manifolds admit\npseudo-rotations. Namely, the Conley conjecture asserts that for a\nbroad class of closed symplectic manifolds every Hamiltonian\ndiffeomorphism has infinitely many simple (a.k.a.\\ prime, i.e., not\niterated) periodic orbits. At the time of writing, the conjecture has\nbeen established for all symplectic manifolds $(M^{2n},\\omega)$ such\nthat there is no class $A\\in\\pi_2(M)$ with $\\omega(A)>0$ and\n$\\left>0$; see \\cite{GG:Rev}. In particular, the\nconjecture holds for all symplectic CY manifolds, \\cite{GG:gaps,\n He:irr}, and all negative monotone symplectic manifolds, \\cite{CGG,\n GG:nm}. (See also \\cite{FH, Gi:CC, Hi, LeC, SZ} for some milestone\nintermediate results and \\cite{GG:survey} for a general survey and\nfurther references.) One can also show that $N\\leq 2n$ when $M$ admits\na pseudo-rotation, where $N$ is the minimal Chern number of $M$,\nalthough it is not yet known if the Conley conjecture holds whenever\n$N>2n$.\n\nOn the other hand, all symplectic manifolds $M$ which carry\nHamiltonian circle (or torus) actions with isolated fixed points also\nadmit pseudo-rotations. Indeed, then a generic element of the circle\nor torus induces a pseudo-rotation of $M$. In particular, ${\\mathbb C}{\\mathbb P}^n$, the\ncomplex Grassmannians and flag manifolds, or more generally a majority\nof coadjoint orbits of compact Lie groups, and symplectic toric\nmanifolds admit pseudo-rotations. For ${\\mathbb C}{\\mathbb P}^n$ identified with the\nquotient of the unit sphere $S^{2n+1}\\subset{\\mathbb C}^{n+1}$, such a\npseudo-rotation is a true rotation, by which we mean an element of\n$\\operatorname{SU}(n)$, and can be generated by a quadratic Hamiltonian\n$Q=\\sum a_i|z_i|^2$ where $a_i-a_j\\not\\in{\\mathbb Q}$ for $i \\neq j$. (The\ncombinatorics of rotations is not entirely straightforward and we will\nexamine it more closely in \\cite{GG:PR2}; see also Examples\n\\ref{ex:cpn} and \\ref{ex:cpn-2}.)\n\nHowever, at least for $S^2$, not every pseudo-rotation is a true\nrotation. Indeed, diffeomorphisms arising as generic elements of\n$S^1$-actions have simple, essentially trivial dynamics. This is in\ngeneral not the case for pseudo-rotations, and pseudo-rotations occupy\na special place among low-dimensional dynamical systems. For instance,\nin \\cite{AK} Anosov and Katok constructed area preserving\ndiffeomorphisms $\\varphi$ of $S^2$ with exactly three ergodic\nmeasures, the area form and the two fixed points, by developing what\nis now known as the conjugation method; see also \\cite{FK} and\nreferences therein. Such a diffeomorphism $\\varphi$ is automatically a\npseudo-rotation. Indeed, $\\varphi$ is area preserving and hence\nHamiltonian, and $\\varphi$ has exactly two periodic orbits, which are\nits fixed points. Furthermore, $\\varphi$ is ergodic, necessarily has\ndense orbits, and thus is not conjugate to a true rotation. As a\nconsequence, the products $(S^2)^n$ also admit pseudo-rotations which\nare not conjugate to true rotations.\n\n\nThe situation is more complicated for projective spaces. It is\nbelieved that ${\\mathbb C}{\\mathbb P}^n$ and other toric manifolds admit dynamically\ninteresting pseudo-rotations, e.g., ergodic or even with finite number\nof ergodic measures. However, in the Hamiltonian setting, the\nconjugation method encounters a conceptual difficulty along the lines\nof the symplectic packing or flexibility questions. This difficulty\ndisappears for ${\\mathbb C}{\\mathbb P}^2$ due to a theorem of McDuff asserting that any\ntwo symplectic embeddings of any fixed collection of closed balls\ninto ${\\mathbb C}{\\mathbb P}^2$ are Hamiltonian isotopic; see \\cite{McD98,\n McD:ellipsoids}. As a consequence, in this case the problem becomes\nmore technical than conceptual, although still quite non-trivial due\nto the effect of the fixed points. A parallel question in the\nsymplectomorphism case was studied in \\cite{HCP} where a different\nvariant of the conjugation method introduced in \\cite{FaHe} was used\nto construct minimal symplectomorphisms of symplectic manifolds\nadmitting symplectic $S^1$-actions without fixed points.\n\nWe conclude this section by elaborating on the conjecture that every\nperfect Hamiltonian diffeomorphism, i.e., a Hamiltonian diffeomorphism\nwith finitely many periodic points, is automatically a\npseudo-rotation, i.e., it has the minimal possible number of periodic\npoints, and all such points are fixed points. (Moreover,\nhypothetically every perfect Hamiltonian diffeomorphism is strongly\nnon-degenerate and all its periodic orbits, which are then fixed\npoints, are elliptic.) For instance, for ${\\mathbb C}{\\mathbb P}^n$, according to this\nconjecture every Hamiltonian diffeomorphism with more than $n+1$ fixed\npoints must have infinitely many periodic orbits. As has been pointed\nout above, in dimension two this fact is essentially the combination\nof the celebrated theorem of Franks (see \\cite{Fr92, Fr96, LeC})\nasserting that every area preserving diffeomorphism of $S^2$ with more\nthan two fixed points has infinitely many periodic orbits and the\nConley conjecture for surfaces proved in \\cite{FH}. (See also\n\\cite{CKRTZ} for a Floer theoretical proof of Franks' theorem and\n\\cite{BH} for an approach utilizing finite energy foliations.) The\nearliest mentioning of this hypothetical generalization of Franks'\ntheorem known to us is on p.\\ 263 in~\\cite{HZ}.\n\nOne can also make similar conjectures for symplectomorphisms and other\ntypes of ``Hamiltonian dynamical systems''. Furthermore, looking at\nthe question from a broader perspective one may expect that the\npresence of a periodic orbit which is homologically or geometrically\nunnecessary (e.g., degenerate, non-contractible, hyperbolic) forces\nthe system to have infinitely many periodic orbits. We refer the\nreader to \\cite{Ba1, Ba2, GG:hyperbolic, GG:nc, Gu:nc, Gu:hq, Or1,\n Or2} (and also to Theorem \\ref{thm:isolated}) for some recent\nresults in this direction in dimensions greater than~two.\n\n\n\n\\subsection{Sample results}\n\\label{sec:results} In this section, we state without detailed\ndiscussion and in some instances in a simplified form the key results\nof the paper. This is a ``sampler plate'' and a much more thorough\ntreatment is given in Sections \\ref{sec:sets} and\n\\ref{sec:PR+PO}. Here and throughout the paper, ${\\mathbb C}{\\mathbb P}^n$ is equipped\nwith the standard (sometimes up to a factor) Fubini--Study symplectic\nstructure.\n\n\\begin{Definition}\n\\label{def:PR}\nA pseudo-rotation of ${\\mathbb C}{\\mathbb P}^n$ is a Hamiltonian diffeomorphism of ${\\mathbb C}{\\mathbb P}^n$\nwith exactly $n+1$ periodic points.\n\\end{Definition}\n\nOur first theorem concerns invariant sets of pseudo-rotations. As has\nbeen mentioned above, the pseudo-rotations $\\varphi$ of $S^2$\nconstructed by the conjugation method in \\cite{AK} have exactly three\nergodic measures: the two fixed points and the area form. As a\nconsequence, such pseudo-rotations are very close to being uniquely\nergodic: $\\varphi$ is uniquely ergodic on the complement to the two\nfixed points. Recall also that volume preserving uniquely ergodic maps\nof compact manifolds are necessarily minimal, i.e., all orbits are\ndense; see, e.g., \\cite{Wa}. These facts suggest that every orbit of\n$\\varphi$ other than a fixed point should be dense. (Furthermore,\nminimal symplectomorphisms exist in abundance and can also be\nconstructed by the conjugation method; \\cite{HCP}.) However, this\nturns out to be false and $\\varphi$ must have many proper, closed\ninvariant subsets as proved in \\cite[Prop.\\ 5.5]{FM}, which is in turn\ninspired by \\cite{LCY} and based on the approach developed in\n\\cite{Fr99}. Our first theorem is a partial generalization of these\nresults to higher dimensions:\n\n\\begin{Theorem}\n\\label{thm:inv_sets0}\nNo fixed point of a pseudo-rotation of ${\\mathbb C}{\\mathbb P}^n$ is isolated as an\ninvariant set.\n\\end{Theorem}\n\nThis theorem is proved in Section \\ref{sec:sets}. The proof hinges on\na result of independent interest asserting that a Hamiltonian\ndiffeomorphism of ${\\mathbb C}{\\mathbb P}^n$ with a fixed point which is isolated as an\ninvariant set and has non-vanishing local Floer homology must have\ninfinitely many periodic orbits. This is Theorem \\ref{thm:isolated}\nextending \\cite[Thm.\\ 1.1]{GG:hyperbolic} to degenerate periodic\norbits and proved in Section \\ref{sec:energy-isolated}.\n \nOur next result concerns the Lagrangian Poincar\\'e recurrence\nconjecture put forth by the first author and independently by Viterbo\naround 2010. According to this conjecture the images of a Lagrangian\nsubmanifold $L$ under the iterations of a Hamiltonian diffeomorphism\nwith compact support cannot be all disjoint. In dimension two, this is\nan easy consequence of the standard Poincar\\'e recurrence combined\nwith the observation that when $L$ does not bound, it cannot be\ndisplaced by a Hamiltonian diffeomorphism. However, in higher\ndimensions the assertion is not obvious even for a particular\nHamiltonian diffeomorphism unless, of course, it is periodic. We\ndiscuss Lagrangian Poincar\\'e recurrence in more detail in Section\n\\ref{sec:PR}, where we establish the conjecture for pseudo-rotations\nof ${\\mathbb C}{\\mathbb P}^n$ and a sufficiently broad class of Lagrangians $L$. In\nparticular, we prove the following.\n\n\\begin{Theorem}\n\\label{thm:LPR0}\nLet $\\varphi$ be a pseudo-rotation of ${\\mathbb C}{\\mathbb P}^n$ and let $L\\subset {\\mathbb C}{\\mathbb P}^n$\nbe a closed Lagrangian submanifold admitting a metric without\ncontractible closed geodesics. Then $\\varphi^k(L)\\cap L\\neq \\emptyset$\nfor infinitely many $k\\in {\\mathbb N}$.\n\\end{Theorem}\n\nThis theorem is a consequence of Corollary \\ref{sec:gamma}\n($\\gamma$-norm convergence) asserting that\n$\\gamma(\\varphi^{k_i})\\to 0$ for some sequence $k_i\\to\\infty$, where\n$\\gamma$ is the $\\gamma$-norm. In fact, in Theorem \\ref{thm:LPR} we\nestablish a stronger version of the Lagrangian Poincar\\'e recurrence\nconjecture relating the return rate, i.e., the frequency of\nintersections $\\varphi^k(L)\\cap L$, and the homological capacity\n$\\operatorname{c_{\\scriptscriptstyle{hom}}}(L)$ of $L$. Moreover, the result holds for all compact subsets\n$L$ with $\\operatorname{c_{\\scriptscriptstyle{hom}}}(L)>0$. The proof is based on a quantitative version of\nthe $\\gamma$-norm convergence; see Section \\ref{sec:gamma} and Theorem\n\\ref{thm:gamma}.\n\nFinally, our third result is $C^0$-rigidity of pseudo-rotations of\n${\\mathbb C}{\\mathbb P}^n$. This is a higher-dimensional counterpart of the main theorem\nof \\cite{Br}. Let $\\varphi$ be a Hamiltonian pseudo-rotation of\n${\\mathbb C}{\\mathbb P}^n$ with fixed points $x_0,\\ldots,x_n$. These are also its\nperiodic points. The mean index ``vector'' of $\\varphi$ is by\ndefinition\n$$\n\\vec{\\Delta}=\\big(\\hmu(x_0),\\ldots,\\hmu(x_n)\\big)\\in\n{\\mathbb T}^{n+1}={\\mathbb R}^{n+1}\/2(n+1){\\mathbb Z}^{n+1},\n$$\nwhere we treat the mean indices $\\hmu(x_i)$ of uncapped one-periodic\norbits as elements of the circle ${\\mathbb R}\/ 2(n+1){\\mathbb Z}$; see Section\n\\ref{sec:background} for more details. We say that $\\vec{\\Delta}$ is\nexponentially Liouville if the iterates $k\\vec{\\Delta}$ approximate\n$0\\in {\\mathbb T}^{n+1}$ exponentially accurately: for any $c>0$ there exists\n$k\\in {\\mathbb N}$ such that $\\|k\\vec{\\Delta}\\|$. In other words, $\\lambda$ is\nthe positive generator of $\\left< [\\omega],\\pi_2(M)\\right>\\subset {\\mathbb R}$,\nthe rationality constant. For the standard Fubini--Study normalization\n$\\lambda=\\pi$. Recall also that the minimal Chern number $N$, i.e.,\nthe positive generator of $\\left< c_1(TM),\\pi_2(M)\\right>$, is $n+1$\nfor ${\\mathbb C}{\\mathbb P}^n$. On several occasions we will work with more general\nsymplectic manifolds. Then $M$ is always assumed to be \\emph{strictly\n monotone}, i.e.,\n$[\\omega]\\!\\mid_{\\pi_2(M)}=\\tau c_1(TM)\\!\\mid_{\\pi_2(M)}\\neq 0$ for\nsome $\\tau\\in{\\mathbb R}$, and hence $M$ is rational and $N<\\infty$. In\nparticular, $\\tau=\\lambda\/N$.\n\nAll Hamiltonians $H$ considered in this paper are assumed to be\n$k$-periodic in time, i.e., $H\\colon S^1_k\\times M\\to{\\mathbb R}$, where\n$S^1_k={\\mathbb R}\/k{\\mathbb Z}$ and $k\\in{\\mathbb N}$. When the period $k$ is not specified, it\nis equal to one and $S^1=S^1_1={\\mathbb R}\/{\\mathbb Z}$. We set $H_t = H(t,\\cdot)$ for\n$t\\in S^1_k$. The Hamiltonian vector field $X_H$ of $H$ is defined by\n$i_{X_H}\\omega=-dH$. The (time-dependent) flow of $X_H$ is denoted by\n$\\varphi_H^t$ and its time-one map by $\\varphi_H$. Such time-one maps\nare referred to as \\emph{Hamiltonian diffeomorphisms}. A one-periodic\nHamiltonian $H$ can always be treated as $k$-periodic, which we will\nthen denote by $H^{{\\natural} k}$ and, abusing terminology, call\n$H^{{\\natural} k}$ the $k$th iterate of $H$.\n\nLet $H$ and $K$ be one-periodic Hamiltonians such that $H_1=K_0$\ntogether with $t$-derivatives of all orders. We denote by $H {\\natural} K$\nthe two-periodic Hamiltonian equal to $H_t$ for $t\\in [0,\\,1]$ and\n$K_{t-1}$ for $t\\in [1,\\,2]$. Thus $H^{{\\natural} k}=H {\\natural} \\ldots {\\natural} H$\n($k$ times). More generally, when $H$ is $l$-periodic and $K$ is\n$k$-periodic, $H{\\natural} K$ is $(l+k)$-periodic. (Strictly speaking, here\nwe need to assume that $H_l=K_0$ again together with all\n$t$-derivatives.)\n\nLet $x\\colon S^1_k\\to M$ be a contractible loop. A \\emph{capping} of\n$x$ is an equivalence class of maps $A\\colon D^2\\to M$ such that\n$A\\mid_{S^1_k}=x$. Two cappings $A$ and $A'$ of $x$ are equivalent if\nthe integrals of $\\omega$ and $c_1(TM)$ over the sphere obtained by\nattaching $A$ to $A'$ are equal to zero. A capped closed curve\n$\\bar{x}$ is, by definition, a closed curve $x$ equipped with an\nequivalence class of cappings. In what follows, the presence of\ncapping is always indicated by a bar.\n\n\nThe action of a Hamiltonian $H$ on a capped closed curve\n$\\bar{x}=(x,A)$ is\n$$\n{\\mathcal A}_H(\\bar{x})=-\\int_A\\omega+\\int_{S^1} H_t(x(t))\\,dt.\n$$\nThe space of capped closed curves is a covering space of the space of\ncontractible loops, and the critical points of ${\\mathcal A}_H$ on this space\nare exactly the capped one-periodic orbits of $X_H$. The \\emph{action\n spectrum} ${\\mathcal S}(H)$ of $H$ is the set of critical values of\n${\\mathcal A}_H$. This is a zero measure set; see, e.g., \\cite{HZ}.\n\n\nThe $k$-periodic \\emph{points} of $\\varphi_H$ are in one-to-one\ncorrespondence with the $k$-periodic \\emph{orbits} of $H$, i.e., of\nthe time-dependent flow $\\varphi_H^t$, which we denote by\n${\\mathcal P}_k(H)$. (Thus, for instance, ${\\mathcal P}_1(H)$ can be identified with the\nfixed point set of $\\varphi_H$.) The collections of all periodic\norbits and all capped periodic orbits of $H$ will be denoted by\n${\\mathcal P}(H)$ and, respectively, $\\bar{\\mathcal{P}}(H)$. Recall also that a $k$-periodic\norbit of $H$ is called \\emph{simple} or \\emph{prime} if it is not\niterated. The definition of the action of $H$ extends to $k$-periodic\norbits and Hamiltonians in an obvious way. Clearly, the action\nfunctional is homogeneous with respect to iteration:\n$$\n{\\mathcal A}_{H^{{\\natural} k}}\\big(\\bar{x}^k\\big)=k{\\mathcal A}_H(\\bar{x}),\n$$\nwhere $\\bar{x}^k$ is the $k$th iteration of the capped orbit $\\bar{x}$. (The\ncapping of $\\bar{x}^k$ is obtained from the capping of $\\bar{x}$ by taking its\n$k$-fold cover branched at the origin.)\n\nA $k$-periodic orbit $x$ of $H$ is said to be \\emph{non-degenerate} if\nthe linearized return map $d\\varphi_H^k \\colon T_{x(0)}M\\to T_{x(0)}M$\nhas no eigenvalues equal to one. Following \\cite{SZ}, we call $x$\n\\emph{weakly non-degenerate} if at least one of the eigenvalues is\ndifferent from one and \\emph{totally degenerate} if all eigenvalues\nare equal to one. A Hamiltonian $H$ is (weakly) non-degenerate if all\nits one-periodic orbits are (weakly) non-degenerate and $H$ is\n\\emph{strongly non-degenerate} if all periodic orbits of $H$ (of all\nperiods) are non-degenerate.\n\nLet $\\bar{x}=(x,A)$ be a non-degenerate capped periodic orbit. The\n\\emph{Conley--Zehnder index} $\\mu(\\bar{x})\\in{\\mathbb Z}$ is defined, up to a\nsign, as in \\cite{Sa,SZ}. In this paper, we normalize $\\mu$ so that\n$\\mu(\\bar{x})=n$ when $x$ is a non-degenerate maximum (with trivial\ncapping) of an autonomous Hamiltonian with small Hessian. The\n\\emph{mean index} $\\hmu(\\bar{x})\\in{\\mathbb R}$ measures, roughly speaking, the\ntotal angle swept by certain unit eigenvalues of the linearized flow\n$d\\varphi^t_H|_x$ with respect to the trivialization associated with\nthe capping; see \\cite{Lo,SZ}. The mean index is defined even when $x$\nis degenerate and depends continuously on $H$ and $\\bar{x}$ in the obvious\nsense. Furthermore,\n\\begin{equation}\n\\label{eq:mean-cz}\n\\big|\\hmu(\\bar{x})-\\mu(\\bar{x})\\big|\\leq n\n\\end{equation} \nand the inequality is strict when $x$ is weakly non-degenerate. The\nmean index is homogeneous with respect to iteration:\n$\\hmu\\big(\\bar{x}^k\\big)=k\\hmu(\\bar{x})$. For an uncapped orbit $x$, the mean\nindex $\\hmu(x)$ and the action ${\\mathcal A}_H(x)$ are well defined as elements\nof $S^1_{2N}={\\mathbb R}\/2N{\\mathbb Z}$ and, respectively, $S^1_\\lambda={\\mathbb R}\/\\lambda{\\mathbb Z}$;\nsee \\eqref{eq:recap} and \\eqref{eq:delta}. Likewise, when $x$ is\nnon-degenerate, the Conley--Zehnder index $\\mu(x)$ is well defined as\nan element of ${\\mathbb Z}\/2N{\\mathbb Z}$.\n\n\\subsection{Global and local Floer homology}\n\\label{sec:FH}\nIn this subsection, we very briefly discuss, mainly to set notation,\nthe constructions of filtered and local Floer homology. We refer the\nreader to, e.g., \\cite{GG:gaps, MS, Sa, SZ} for detailed accounts and\nadditional references.\n\n\\subsubsection{Filtered Floer homology}\n\\label{sec:FFH}\nFix a ground field ${\\mathbb F}$. Let $H$ be a non-degenerate Hamiltonian on\n$M$. Denote by $\\operatorname{CF}^{(-\\infty,\\, b)}_m(H)$, with\n$b\\in (-\\infty,\\,\\infty]\\setminus{\\mathcal S}(H)$, the vector space of formal\nfinite linear combinations\n$$ \n\\sigma=\\sum_{\\bar{x}\\in \\bar{\\mathcal{P}}(H)} \\sigma_{\\bar{x}}\\bar{x}, \n$$\nwhere $\\sigma_{\\bar{x}}\\in{\\mathbb F}$ and $\\mu(\\bar{x})=m$ and\n${\\mathcal A}(\\bar{x}),\n$$\nwhere $A\\in\\pi_2(M)$. Thus $I_\\omega=\\tau I_{c_1}\/2$ since $M$ is\nstrictly monotone.\n\n\nLet $\\Gamma$ be the quotient of $\\pi_2(M)$ by the equivalence relation\nwhere the two spheres $A$ and $A'$ are considered to be equivalent if\n$I_\\omega(A)=I_\\omega(A')$ or, equivalently, since $M$ is strictly\nmonotone, $I_{c_1}(A)=I_{c_1}(A')$. Clearly, $\\Gamma\\simeq {\\mathbb Z}$. The\nhomomorphisms $I_\\omega$ and $I_{c_1}$ descend to $\\Gamma$ and become\nisomorphisms onto the image.\n\nThe group $\\Gamma$ acts on $\\operatorname{CF}_*(H)$ and $\\operatorname{HF}_*(H)$ by recapping: an\nelement $A\\in \\Gamma$ acts on a capped one-periodic orbit $\\bar{x}$ of\n$H$ by attaching the sphere $A$ to the original capping. Denoting the\nresulting capped orbit by $\\bar{x}\\# A$, we have\n\\begin{equation}\n\\label{eq:recap}\n\\mu(\\bar{x}\\# A)=\\mu(\\bar{x})+ I_{c_1}(A)\n\\end{equation}\nwhen $x$ is non-degenerate. In a similar vein,\n\\begin{equation}\n\\label{eq:delta}\n{\\mathcal A}_H(\\bar{x}\\# A)={\\mathcal A}_H(\\bar{x})+I_\\omega(A)\n\\text{ and } \\hmu(\\bar{x}\\# A)=\\hmu(\\bar{x})+ I_{c_1}(A)\n\\end{equation}\nregardless of whether $x$ is non-degenerate or not.\n\nIn general, the Novikov ring $\\Lambda$ is a certain completion of the\ngroup ring of $\\Gamma$ over ${\\mathbb F}$ but for our purposes, since $M$ is\nmonotone, we can take as $\\Lambda$ the group ring itself, i.e., the\ncollection of finite linear combinations $\\sum \\alpha_A e^A$, where\n$\\alpha_A\\in{\\mathbb F}$ and $A\\in \\Gamma$. The Novikov ring $\\Lambda$ is\ngraded by setting $|e^A|=I_{c_1}(A)$. The action of $\\Gamma$ turns\n$\\operatorname{CF}_*(H)$ and $\\operatorname{HF}_*(H)$ into $\\Lambda$-modules. The maps $I_{c_1}$\nand $I_\\omega$ naturally extend to $\\Lambda$. We set $\\mathrm{q}=e^{A_0}$,\nwhere $A_0$ is the generator of $\\Gamma$ with $I_{c_1}(A_0)=-2N$. Thus\n$|\\mathrm{q}|=-2N$ and $I_\\omega(\\mathrm{q})=-\\lambda$.\n\nThe construction of the filtered Floer homology extends to all, not\nnecessarily non-degenerate, Hamiltonians by continuity. Namely, let\n$H$ be an arbitrary (one-periodic in time) Hamiltonian on $M$ and let\nthe end-points $a$ and $b$ of the action interval $I$ be outside\n${\\mathcal S}(H)$. By definition, we set\n$$\n\\operatorname{HF}^{I}_*(H)=\\operatorname{HF}^{I}_*(\\tilde{H}),\n$$\nwhere $\\tilde{H}$ is a non-degenerate, small perturbation of $H$. The\nright-hand side is independent of $\\tilde{H}$ once $\\tilde{H}$ is sufficiently\nclose to $H$. The total Floer homology is independent of the\nHamiltonian and, up to a shift of the grading and the effect of\nrecapping, is isomorphic to the homology of $M$. More precisely, we\nhave\n\\begin{equation}\n \\label{eq:H-M}\n\\operatorname{HF}_*(H)\\cong \\operatorname{H}_ {*+n}(M;{\\mathbb F})\\otimes \\Lambda\n\\end{equation}\nas graded $\\Lambda$-modules.\n\n\\begin{Example}[Projective Spaces]\n Let $H$ be a Hamiltonian on ${\\mathbb C}{\\mathbb P}^n$. Then, by \\eqref{eq:H-M},\n $\\operatorname{HF}_m(H)={\\mathbb F}$ when $m$ has the same parity as $n$ and $\\operatorname{HF}_m(H)=0$\n otherwise. To see this directly, one can, for instance, take a\n non-degenerate quadratic Hamiltonian as $H$ and show by a\n calculation that all fixed points of $\\varphi_H$ are elliptic and\n hence their indices have the same parity as $n$. (In particular, the\n Floer differential vanishes.) Recapping by the generator $A_0$ of\n $\\Gamma\\cong{\\mathbb Z}$ decreases the index by $2N=2(n+1)$.\n\\end{Example}\n\n\\subsubsection{Local Floer homology}\n\\label{sec:LFH}\nThe notion of \\emph{local Floer homology} goes back to the original\nwork of Floer and it has been revisited a number of times since\nthen. Here we only briefly recall the definition following mainly\n\\cite{Gi:CC,GG:gaps,GG:gap} where the reader can find a much more\nthorough discussion and further references.\n\nLet $x$ be an isolated one-periodic orbit of a Hamiltonian\n$H\\colon S^1\\times M\\to {\\mathbb R}$. The local Floer homology $\\operatorname{HF}(x)$ is the\nhomology of the Floer complex $\\operatorname{CF}_*(\\tilde{H},x)$ generated by the orbits\n$x_i$ which $x$ splits into under a $C^2$-small non-degenerate\nperturbation $\\tilde{H}$ of $H$. This homology is well defined, i.e.,\nindependent of the perturbation. The homology $\\operatorname{HF}(x)$ is only\nrelatively graded and to fix an absolute grading one needs to pick a\ntrivialization of $TM$ along $x$. This can be done by using, for\ninstance, a capping of $x$ and in this case we write\n$\\operatorname{HF}_*(\\bar{x})$. Note that then the orbits $x_i$ inherit a capping from\n$\\bar{x}$.\n\nFor example, if $x$ is non-degenerate and $\\mu(\\bar{x})=m$, we have\n$\\operatorname{HF}_l(\\bar{x})={\\mathbb F}$ when $l=m$ and $\\operatorname{HF}_l(\\bar{x})=0$ otherwise. This\nconstruction is local: it requires $H$ to be defined only on a\nneighborhood of $x$.\n\nBy definition, the \\emph{support} of $\\operatorname{HF}_*(\\bar{x})$, denoted by\n$\\operatorname{supp}\\operatorname{HF}_*(\\bar{x})$, is the collection of integers $m$ such that\n$\\operatorname{HF}_m(\\bar{x})\\neq 0$. By \\eqref{eq:mean-cz} and continuity of the mean\nindex,\n\\begin{equation}\n\\label{eq:supp}\n\\operatorname{supp} \\operatorname{HF}_*(\\bar{x})\\subset [\\hmu(\\bar{x})-n,\\, \\hmu(\\bar{x})+n].\n\\end{equation}\nMoreover, when $x$ is weakly non-degenerate, the closed interval can\nbe replaced by the open interval.\n\nThe local Floer homology groups are building blocks for the filtered\nFloer homology. For instance, assume that $c$ is the only point of\n${\\mathcal S}(H)$ in an interval $I$ and that all one-periodic orbits of $H$\nwith action $c$ are isolated. Then, as is easy to see,\n\\begin{equation}\n\\label{eq:split}\n\\operatorname{HF}_*^I(H)=\\bigoplus_{{\\mathcal A}_H(\\bar{x})=c}\\operatorname{HF}_*(\\bar{x}).\n\\end{equation}\nTo prove this and also with Lemma \\ref{lemma:spec} in mind, first\nobserve that, when the orbits are isolated, a finite energy solution\nof the Floer equation $u$ is automatically asymptotic to unique\none-periodic orbits as $s\\to\\pm\\infty$ and there is an \\emph{a priori}\nlower bound $\\epsilon$ on the energy of $u$; see, e.g., \\cite[Sect.\\\n1.5]{Sa}. (Hence, we have, in this case, the notion of a solution\nconnecting two orbits.) Then one can shrink $I$ to an interval with\nlength below $\\epsilon$ and notice that for a sufficiently small\nnon-degenerate perturbation $\\tilde{H}$ the Floer complex splits; see\n\\cite[Sect.\\ 2.5]{GG:gaps}.\n\nWe have the following simple result sharpening \\eqref{eq:split}, which\nwe state and prove in a slightly more general form than needed\nhere. Denote by $d(\\cdot,\\,\\cdot)$ the distance between two subsets of\n${\\mathbb R}$, i.e.,\n$$ \nd(A,\\,B)=\\inf\\{|a-b|\\mid a\\in A,\\, b\\in B\\}.\n$$\n\n\\begin{Lemma}\n\\label{lemma:spec}\nAssume that all capped one-periodic orbits $\\bar{x}$ of $H$ with action in\n$I$ are isolated. Then there exists a spectral sequence with\n\\begin{equation}\n\\label{eq:E1}\nE^1=\\bigoplus_{\\bar{x}}\\operatorname{HF}_*(\\bar{x})\n\\end{equation}\nconverging to $\\operatorname{HF}_*^I(H)$. Furthermore, $\\operatorname{HF}_*(\\bar{x})$ is a direct\nsummand in $\\operatorname{HF}_*^I(H)$ when one of the following two conditions is\nmet:\n\\begin{itemize}\n\\item[(i)] $\\bar{x}$ is not connected to any other capped periodic orbit\n of $H$ with action in $I$ by a solution of the Floer equation;\n\\item[(ii)] for any $\\bar{y}$ with ${\\mathcal A}_H(\\bar{y})\\in I$, we have\n$$\nd\\big(\\operatorname{supp}\\operatorname{HF}_*(\\bar{x}), \\operatorname{supp}\\operatorname{HF}_*(\\bar{y})\\big)>1.\n$$\n\\end{itemize}\nMoreover, assume that (i) holds or the following condition is\nsatisfied:\n\\begin{itemize}\n\\item[(ii')] for any $\\bar{y}$ with ${\\mathcal A}_H(\\bar{y})\\in I$, we have\n$$\n\\big|\\hmu(\\bar{x})-\\hmu(\\bar{y})\\big|> 2n+1.\n$$\n\\end{itemize}\nThen, when $\\tilde{H}$ is sufficiently $C^2$-close to $H$, the complex\n$\\operatorname{CF}_*(\\tilde{H},\\bar{x})$ is a direct summand in $\\operatorname{CF}_*^I(\\tilde{H})$.\n\\end{Lemma}\n\nNote that while requirement (ii') is more restrictive than (ii), the\nthird assertion of the lemma, that $\\operatorname{CF}_*(\\tilde{H},\\bar{x})$ enters\n$\\operatorname{CF}_*^I(\\tilde{H})$ as a direct summand, is stronger than the second\nconcerning a similar fact on the level of homology. The proof of the\nlemma is routine and below we merely comment on the argument.\n\n\\begin{proof}[Outline of the proof]\n The required spectral sequence is associated with the action\n filtration on the Floer complex of a small non-degenerate\n perturbation of $H$ and \\eqref{eq:E1} is essentially a rephrasing of\n \\eqref{eq:split}. (It is a variant of the Morse--Bott spectral\n sequence.) To prove the second assertion, it is enough to show that\n the parts of the differentials $d_j$, $j\\geq 2$, connecting\n $\\operatorname{HF}_*(\\bar{x})$ with the rest of the spectral sequence vanish when (i)\n or (ii) holds. When (i) is satisfied, this is a consequence of the\n third assertion proved below. When (ii) holds, the vanishing of the\n differentials follows from the fact that $|\\mu(\\bar{x}_i)-\\mu(\\bar{y}_j)|>1$\n for all capped orbits $\\bar{x}_i$ and $\\bar{y}_j$ which $\\bar{x}$ and,\n respectively, $\\bar{y}$ split into. This argument also shows that\n $\\operatorname{CF}_*(\\tilde{H},\\bar{x})$ is a direct summand when (ii') is satisfied.\n\n Finally, let us prove that $\\operatorname{CF}_*(\\tilde{H},\\bar{x})$ is a direct summand in\n $\\operatorname{CF}_*^I(\\tilde{H})$ when (i) holds. Arguing by contradiction assume that\n there is a sequence of Floer trajectories $u$ connecting $\\bar{x}_i$ and\n $\\bar{y}_j$ for some sequence of perturbations $\\tilde{H}\\to H$. (Say, $u$ is\n asymptotic to $\\bar{x}_i$ at $-\\infty$ and $\\bar{y}_j$ at $\\infty$.) Fix a\n closed neighborhood $U$ of $x$ which contains no other one-periodic\n orbits of $H$. Without loss of generality we may assume that\n $u(0,0)\\in\\partial U$ and that the half-cylinder\n $(-\\infty,\\, 0]\\times S^1$ is mapped into $U$. Applying the\n target-local compactness theorem from \\cite{Fi} to the restrictions\n of $u$ to a finite cylinder $[-L,\\,L]\\times S^1$ and then the\n diagonal process as $L\\to\\infty$, we obtain a solution $v$ of the\n Floer equation for $H$ mapping the half-cylinder\n $(-\\infty,\\, 0]\\times S^1$ into $U$. Hence, $v$ is asymptotic to\n $\\bar{x}$ at $-\\infty$. It is easy to see that the capped orbit which\n $v$ is asymptotic to at $\\infty$ has action in $I$. This contradicts\n (i).\n\\end{proof}\n\n\n\n\\subsection{Cap product}\n\\label{sec:cap-product}\nThe algebraic structure on the Floer homology, which is crucial for\nwhat follows, is that of a module over the (small) \\emph{quantum\n homology} of $M$. The quantum homology $\\operatorname{HQ}_*(M)$ of $M$ is an\nalgebra over the Novikov ring $\\Lambda$ defined above and\n$\\operatorname{HQ}_*(M)=\\operatorname{H}_*(M)[-n]\\otimes \\Lambda$ as $\\Lambda$-modules. We refer\nthe reader to, e.g., \\cite[Chap.\\ 11]{MS} for the definition of the\nproduct $*$ in $\\operatorname{HQ}_*(M)$. The fundamental class $[M]$ is the unit\nwith respect to this product. The maps $I_{c_1}$ and $I_\\omega$\nnaturally extend to $\\operatorname{HQ}_*(M)$. For instance,\n$$\nI_\\omega(\\alpha) =\\max\\,\\{I_\\omega(A)\\mid \\alpha_A\\neq 0\\}\n=\\max\\,\\{-\\lambda_0 k \\mid \\alpha_k\\neq 0\\},\n$$\nwhere $\\alpha=\\sum \\alpha_A e^A=\\sum \\alpha_k \\mathrm{q}^k\\in \\operatorname{HQ}_*(M)$.\n\n\\begin{Example} \n\\label{ex:cpn}\nWe will make an extensive use of the product structure\non $\\operatorname{HQ}_*({\\mathbb C}{\\mathbb P}^n)$. In this case $N=n+1$ and $\\operatorname{HQ}_*({\\mathbb C}{\\mathbb P}^n)$ is generated by\nthe hyperplane class $[{\\mathbb C}{\\mathbb P}^{n-1}]$. To be more precise, we have\n$[{\\mathbb C}{\\mathbb P}^{n-1}]^{\\ell}=[{\\mathbb C}{\\mathbb P}^{n-\\ell}]$ when $\\ell\\leq n$ and\n\\begin{equation}\n\\label{eq:qp-cpn}\n[{\\mathbb C}{\\mathbb P}^{n-1}]^{n+1}=\\mathrm{q} [{\\mathbb C}{\\mathbb P}^n],\n\\end{equation}\nwhere $|\\mathrm{q}|=-2(n+1)$; see, e.g., \\cite[Sect.\\ 11.3]{MS} and\nreferences therein. In other words,\n$[{\\mathbb C}{\\mathbb P}^{n-1}] * \\alpha_\\ell=\\alpha_{\\ell-1}$, where $\\alpha_\\ell$ is a generator\nof $\\operatorname{HQ}_{2\\ell-n}({\\mathbb C}{\\mathbb P}^n)$. For instance, $[{\\mathit pt}]*[{\\mathbb C}{\\mathbb P}^{n-1}]=\\mathrm{q}\n[{\\mathbb C}{\\mathbb P}^n]$, which reflects the fact that there is exactly one\nline passing through any two distinct points of ${\\mathbb C}{\\mathbb P}^n$ and when \na point ${\\mathit pt}$ is fixed every line through ${\\mathit pt}$ intersects\n${\\mathbb C}{\\mathbb P}^{n-1}\\not\\ni{\\mathit pt}$ exactly once. \n\\end{Example}\n\nIdentifying the global Floer homology with $\\operatorname{HQ}_*(M)$, we can view\n$\\operatorname{HF}_*(H)$ as an $\\operatorname{HQ}_*(M)$-module. It is important for our purposes\nthat this module structure extends in a certain way to the filtered\nFloer homology, and we refer to the resulting ``$\\operatorname{HQ}_*(M)$-action'',\n\\eqref{eq:cap-action}, as the \\emph{cap product}. (Strictly speaking,\nthe cap product is not a true algebra action due to a filtration\nshift; rather it should be thought of as an ``action'' of $\\operatorname{HQ}_*(M)$\non the entire collection of the filtered Floer homology groups.) The\ndefinition of the cap product, recalled below, goes back to\n\\cite{Vi:product} and \\cite{LO}. Here we closely follow \\cite[Sect.\\\n2.3]{GG:hyperbolic}; see also \\cite[Rmk.\\ 12.3.3]{MS}.\n\nOn the level of cycles, the action of a pseudo-cycle $\\zeta$ with\n$[\\zeta]\\in \\operatorname{H}_\\ell(M)$ on $\\bar{x}$ is given by counting the solutions\n$u$ of the Floer equation with $u(0,0)\\in\\zeta$. More precisely, pick\na generic almost complex structure and a generic $\\zeta$ and set\n$$\n\\Phi_{\\zeta}(\\bar{x}):=\\sum_{\\bar{y}} m(\\bar{x},\\bar{y};\\zeta)\\bar{y}.\n$$\nHere $m(\\bar{x},\\bar{y};\\zeta)$ is the number of the elements, taken with\nappropriate signs, in the moduli space ${\\mathcal M}(\\bar{x},\\bar{y};\\zeta)$ of\nsolutions $u$ asymptotic to $\\bar{x}$ at $-\\infty$ and\n$\\bar{y}$ at $\\infty$ and such that $u(0,0)\\in \\zeta$. This moduli space\nhas dimension $|\\bar{x}|-|\\bar{y}|-\\operatorname{codim}(\\zeta)$ and, by definition,\n$m(\\bar{x},\\bar{y};\\zeta)=0$ when $\\dim {\\mathcal M}(\\bar{x},\\bar{y};\\zeta)>0$.\n\nThe map $\\Phi_\\zeta$ commutes with the Floer differential and gives\nrise to a well-defined map\n$$\n\\Phi_{[\\zeta]} \\colon \\operatorname{HF}^{(a,\\,b)}_*(H)\\to\n\\operatorname{HF}^{(a,\\,b)}_{*-\\operatorname{codim}(\\zeta)}(H).\n$$\nThe analytical details of this construction and complete proofs can be\nfound in, e.g., \\cite{LO}, in much greater generality than is needed\nhere. Clearly,\n$$\n\\Phi_{[M]}={\\mathit id}.\n$$\n\nThe action of the class $\\alpha=\\mathrm{q}^\\ell[\\zeta]\\in \\operatorname{HQ}_*(M)$ is\ninduced by the map\n$$\n\\Phi_{\\mathrm{q}^\\ell\\zeta}(\\bar{x}):=\\sum_{\\bar{y}} m(\\mathrm{q}^\\ell\\bar{x},\\bar{y};\\zeta)\\bar{y}.\n$$\nHere, as in Section \\ref{sec:FFH}, $\\mathrm{q}=e^{A_0}$ where $A_0$ is the\ngenerator of $\\Gamma$ with $I_{c_1}(A_0)=-2N$. It is routine to check\nthat $\\Phi_{\\mathrm{q}^\\ell[\\zeta]}=\\mathrm{q}^\\ell\\Phi_{[\\zeta]}$. (This is a\nconsequence of the fact that\n$ {\\mathcal M}(\\mathrm{q}^\\ell\\bar{x},\\bar{y};\\zeta)={\\mathcal M}(\\bar{x},\\mathrm{q}^{-\\ell}\\bar{y};\\zeta)$.) The\nresulting map shifts the action interval by $I_\\omega(\\alpha)$, i.e.,\n\\begin{equation}\n\\label{eq:cap-action}\n\\Phi_\\alpha \\colon \\operatorname{HF}^{(a,\\,b)}_*(H)\\to \n\\operatorname{HF}^{(a,\\,b)+I_\\omega(\\alpha)}_{*+|\\alpha|-2n}(H),\n\\end{equation}\nwhere $(a,\\,b)+c$ stands for $(a+c,\\,b+c)$.\n\nBy linearity over $\\Lambda$, we extend $\\Phi_\\alpha$ to all\n$\\alpha\\in \\operatorname{HQ}_*(M)$ so that \\eqref{eq:cap-action} still holds. The\nmaps $\\Phi_\\alpha$ are linear in $\\alpha$ once the shift of the action\nfiltration is taken into account; see \\cite[Sect.\\\n2.3]{GG:hyperbolic}. These maps fit together to form an action of the\nquantum homology on the collection of the filtered Floer homology\ngroups. The action is multiplicative. In other words, we have\n\\begin{equation}\n\\label{eq:qh-fh-action}\n\\Phi_\\alpha\\Phi_\\beta=\\Phi_{\\alpha*\\beta},\n\\end{equation}\nwhich can be thought of as a form of associativity of the quantum\nproduct. Strictly speaking, in \\eqref{eq:qh-fh-action}, as in the\ncase of additivity, the maps on the two sides of the identity have\ndifferent target spaces, which can be accounted for by considering the\nshifts of the action filtration. We refer the reader to \\cite[Sect.\\\n2.3]{GG:hyperbolic} for the precise statement. The identity\n\\eqref{eq:qh-fh-action} was essentially established in \\cite{LO} and\n\\cite{PSS}; see also \\cite[Rmk.\\ 12.3.3]{MS}.\n\nThe cap product action also extends to the local Floer\nhomology. Namely, let $\\bar{x}$ be an isolated one-periodic orbit of $H$ and\nlet $\\tilde{H}$ be a $C^2$-small non-degenerate perturbation of\n$H$. Applying the above construction word-for-word to the complex\n$\\operatorname{CF}_*(\\tilde{H},x)$ from Section \\ref{sec:LFH}, we obtain an action (i.e.,\na module structure) of $\\operatorname{HQ}_*(M)$ on $\\operatorname{HF}_*(\\bar{x})$. In other words, for\nevery $\\alpha\\in \\operatorname{HQ}_*(M)$ we have a map\n$\\Phi_\\alpha\\colon \\operatorname{HF}_*(\\bar{x})\\to\\operatorname{HF}_{*+|\\alpha|-2n}(\\bar{x})$ and\n\\eqref{eq:qh-fh-action} is satisfied. However, the resulting action is\ntrivial unless of course $\\alpha$ is a multiple of $[M]$:\n\n\\begin{Lemma}\n\\label{lemma:trivial}\nAssume that $|\\alpha|<2n$. Then $\\Phi_\\alpha=0$ on $\\operatorname{HF}_*(\\bar{x})$.\n\\end{Lemma}\n\nThe proof of the lemma is standard and we just outline the argument.\n\n\\begin{proof}\n We argue as in the proof of the fact that $\\operatorname{CF}_*(\\tilde{H},\\bar{x})$ is a\n subcomplex in $\\operatorname{CF}_*(\\tilde{H})$; cf.\\ \\cite{Gi:CC, GG:gaps}. Let $B$ be\n a small ball centered at the fixed point $x(0)$ such that its\n closure $\\bar{B}$ contains no other fixed points, and let $\\tilde{H}$ be a\n $C^2$-small, non-degenerate perturbation of $H$. Denote by $\\bar{x}_i$\n the capped one-periodic orbits which $\\bar{x}$ splits into. Clearly,\n $x_i(0)\\in B$. Every class $\\alpha$ with $|\\alpha|<2n$ can be\n represented by a cycle avoiding $B$. Thus it is enough to show that\n for every solution $u$ of the Floer equation for $\\tilde{H}$ asymptotic to\n some orbits $\\bar{x}_i$ and $\\bar{x}_j$, we have $u(s,0)\\in B$ for all\n $s\\in {\\mathbb R}$ when $\\tilde{H}$ is close to $H$. Arguing by contradiction\n assume that there is a sequence of $C^2$-small, non-degenerate\n perturbations $\\tilde{H}_l\\to H$ and solutions $u_l$ of the Floer equation\n for $\\tilde{H}_l$ such that $u_l(s_l,0)\\in \\partial B=\\bar{B}\\setminus\n B$. Without loss of generality we can assume that $s_l=0$ by\n applying a shift in $s$. Clearly, $E(u_l)\\to 0$ and, as a\n consequence, the loops $t\\mapsto u_l(0,t)$ converge to an integral\n curve of $\\varphi_H^t$; see, e.g., \\cite[Sect.\\ 1.5]{Sa}. Passing to\n a subsequence, we conclude that $p=\\lim u_l(0,0)$ is a fixed point\n of $\\varphi_H$ which contradicts our choice of $B$.\n\\end{proof}\n\n\\begin{Remark}\n Lemma \\ref{lemma:trivial} has a counterpart in the context of the\n algebra structure and the pair-of-pants product in Floer\n homology. This is the fact established in \\cite{Ci} that, unless $x$\n is a so-called SDM, the local Floer homology algebra\n $\\bigoplus_{k>0}\\operatorname{HF}_*(\\bar{x}^k)$ is non-uniformly nilpotent.\n\\end{Remark}\n\n\\subsection{Spectral invariants and action carriers}\n\\label{sec:spec}\nIn this section, we briefly discuss spectral invariants and action\ncarriers to the extent necessary for our purposes. The theory of\nHamiltonian \\emph{spectral invariants} was developed in its present\nFloer theoretical form in \\cite{Oh:constr, Sc}, although the first\nversions of the theory go back to \\cite{HZ, Vi:gen}. Action carriers\nwere introduced in \\cite{GG:gaps} and then studied in \\cite{CGG,\n GG:nm}.\n\nLet $H$ be a Hamiltonian on a closed monotone (or even rational)\nsymplectic manifold $M^{2n}$. The \\emph{spectral invariant} or\n\\emph{action selector} $\\operatorname{c}_\\alpha$ associated with a class\n$\\alpha\\in \\operatorname{HF}_*(H)=\\operatorname{HQ}_*(M)$ is defined as\n$$\n\\operatorname{c}_\\alpha(H)= \\inf\\{ a\\in {\\mathbb R}\\setminus {\\mathcal S}(H)\\mid \\alpha \\in \\operatorname{im}(i^a)\\}\n=\\inf\\{ a\\in {\\mathbb R}\\setminus {\\mathcal S}(H)\\mid j^a\\left( \\alpha \\right)=0\\},\n$$\nwhere $i^a\\colon \\operatorname{HF}_*^{(-\\infty,\\,a)}(H)\\to \\operatorname{HF}_*(H)$ and\n$j^a\\colon \\operatorname{HF}_*(H)\\to\\operatorname{HF}_*^{(a,\\, \\infty)}(H)$ are the natural\n``inclusion'' and ``quotient'' maps. It is easy to see that\n$\\operatorname{c}_\\alpha(H)>-\\infty$ when $\\alpha\\neq 0$ and one can show that\n$\\operatorname{c}_\\alpha(H)\\in S(H)$. In other words, there exists a capped\none-periodic orbit $\\bar{x}$ of $H$ such that $\\operatorname{c}_\\alpha(H)={\\mathcal A}_H(\\bar{x})$. As an\nimmediate consequence of the definition,\n$$\n\\operatorname{c}_\\alpha(H+a(t))=\\operatorname{c}_\\alpha(H)+\\int_{S^1}a(t)\\,dt,\n$$\nwhere $a\\colon S^1\\to{\\mathbb R}$.\n\nAction selectors have several important properties. The function\n$\\operatorname{c}_\\alpha$ is homotopy invariant: $\\operatorname{c}_\\alpha(H)=\\operatorname{c}_\\alpha(K)$ when\n$\\varphi_H=\\varphi_K$ in ${\\widetilde{\\mathit{Ham}}}(M)$ and $H$ and $K$ have the same mean\nvalue. Furthermore, it is sub-additive:\n$$\n\\operatorname{c}_{\\alpha * \\beta}(H {\\natural} K)\\leq\\operatorname{c}_{\\alpha}(H)+\\operatorname{c}_{\\beta}(K).\n$$\nIn particular,\n$$\n\\operatorname{c}_{[M]}(H {\\natural} K)\\leq\\operatorname{c}_{[M]}(H)+\\operatorname{c}_{[M]}(K).\n$$\nFinally, $\\operatorname{c}_\\alpha$ is monotone and Lipschitz in the $C^0$-topology\nas a function of $H$.\n\nWhen $H$ is non-degenerate, the action selector can also be evaluated\nas\n$$\n\\operatorname{c}_\\alpha(H)=\\inf_{[\\sigma]=\\alpha}\\operatorname{c}_\\sigma(H),\n$$\nwhere we set \n\\begin{equation}\n\\label{eq:cycle-action}\n\\operatorname{c}_\\sigma(H)=\\max\\big\\{{\\mathcal A}_H(\\bar{x})\\,\\big|\\, \\sigma_{\\bar{x}} \n\\neq 0\\big\\}\\text{ for }\n\\sigma=\\sum\\sigma_{\\bar{x}} \\bar{x}\\in\\operatorname{CF}_*(H).\n\\end{equation}\nThe infimum here is attained, since $M$ is rational and thus ${\\mathcal S}(H)$\nis closed. Hence there exists a cycle\n$\\sigma=\\sum\\sigma_{\\bar{x}} \\bar{x}\\in\\operatorname{CF}_{|\\alpha|}(H)$, representing the\nclass $\\alpha$, such that $\\operatorname{c}_\\alpha(H)={\\mathcal A}_H(\\bar{x})$ for an orbit $\\bar{x}$\nentering $\\sigma$. In other words, $\\bar{x}$ maximizes the action on\n$\\sigma$ and the cycle $\\sigma$ minimizes the action over all cycles\nin the homology class $\\alpha$. Such an orbit $\\bar{x}$ is called a\n\\emph{carrier} of the action selector. This is a stronger requirement\nthan just that $\\operatorname{c}_\\alpha(H)={\\mathcal A}_H(\\bar{x})$ and $\\mu(\\bar{x})=|\\alpha|$. When\n$H$ is possibly degenerate, a capped one-periodic orbit $\\bar{x}$ of $H$\nis a carrier of the action selector if there exists a sequence of\n$C^2$-small, non-degenerate perturbations $\\tilde{H}_i\\to H$ such that one\nof the capped orbits $\\bar{x}$ splits into is a carrier for $\\tilde{H}_i$. An\norbit (without capping) is said to be a carrier if it turns into one\nfor a suitable choice of capping.\n\nIt is easy to see that a carrier necessarily exists, but, in\ngeneral, is not unique. However, it becomes unique when all one-periodic\norbits of $H$ have distinct action values.\n\nAs consequence of the definition of the carrier and continuity of the\naction and the mean index, we have\n$$\n\\operatorname{c}_\\alpha(H)\n={\\mathcal A}_H(\\bar{x})\\text{ and } \\big|\\hmu(\\bar{x})-|\\alpha|\\big| \\leq n,\n$$\nand the inequality is strict when $x$ is weakly non-degenerate.\nFurthermore, a carrier $\\bar{x}$ for $\\operatorname{c}_\\alpha$ is in some sense\nhomologically essential as the following result asserts.\n\n\\begin{Lemma}\n\\label{lemma:ac}\nLet $\\bar{x}$ be an action carrier for $\\operatorname{c}_\\alpha$. Then\n$\\operatorname{HF}_{|\\alpha|}(\\bar{x})\\neq 0$ when $x$ is isolated.\n\\end{Lemma}\n\nThis lemma is proved in \\cite[Lemma 3.2]{GG:nm} for $\\operatorname{c}_{[M]}$, but\nthe proof goes through word-for-word for any homology class. For the\nsake of brevity, we omit the argument.\n\n\n\\section{Background results on pseudo-rotations}\n\\label{sec:background}\nIn this section, we assemble several known symplectic topological\nresults on pseudo-rotations of projective spaces, crucial for what\nfollows.\n\nLet $\\varphi=\\varphi_H$ be a pseudo-rotation of ${\\mathbb C}{\\mathbb P}^n$, which we do\nnot assume to be non-degenerate. Denote by $\\alpha_l$ the generator in\n$\\operatorname{HF}_{2l-n}(H)={\\mathbb F}$, $l\\in{\\mathbb Z}$, and let $\\bar{x}_{l}\\in\\bar{\\mathcal{P}}_1(H)$ be an action\ncarrier for $\\alpha_l$. We will write $\\operatorname{c}_l:=\\operatorname{c}_{\\alpha_l}$. Then, in\nparticular,\n$$\n\\operatorname{c}_{l}(H)={\\mathcal A}_H(\\bar{x}_l)\\textrm{ and } \\operatorname{HF}_{2l-n}(\\bar{x}_l)\\neq 0.\n$$\nThroughout the paper it is convenient to occasionally rescale the\nstandard Fubini--Study symplectic structure $\\omega$ on\n${\\mathbb C}{\\mathbb P}^n$. Hence, we state the results in a slightly more general form\nassuming that $\\omega$ is proportional to the Fubini--Study form and\n$\\lambda=\\left<\\omega,{\\mathbb C}{\\mathbb P}^1\\right>$ is the rationality constant. (For\nthe Fubini--Study form, $\\lambda=\\pi$.)\n\n\\begin{Theorem}[\\cite{GG:gaps}]\n\\label{thm:bijection}\nFor every $l\\in {\\mathbb Z}$ the action carrier $\\bar{x}_l$ is unique and the\nresulting map\n\\begin{equation}\n\\label{eq:map-x_i}\n{\\mathbb Z}\\to\\bar{\\mathcal{P}}_1(H), \\quad l\\mapsto\\bar{x}_l\n\\end{equation}\nis a bijection. Furthermore, the map\n\\begin{equation}\n\\label{eq:map-s}\n{\\mathbb Z}\\to{\\mathcal S}(H), \\quad l\\mapsto \\operatorname{c}_{l}(H)={\\mathcal A}_H(\\bar{x}_l)\n\\end{equation}\nis strictly monotone, i.e., $l>l'$ if and only if\n${\\mathcal A}_H(\\bar{x}_l)>{\\mathcal A}_H(\\bar{x}_{l'})$. \n\\end{Theorem}\n\nOne important consequence of the theorem is that distinct capped\none-periodic orbits of $\\varphi_H$ necessarily have different\nactions. Another is that, by Lemma \\ref{lemma:ac},\n$\\operatorname{HF}_{2l-n}(\\bar{x}_l)\\neq 0$. In particular, $\\operatorname{HF}(x)\\neq 0$ for all\n$x\\in{\\mathcal P}(H)$.\n\nWhen $H$ is non-degenerate we have $\\mu(\\bar{x}_l)=2l-n$, and the proof of\nthe theorem is rather straightforward. The general case of the theorem\nis established in \\cite[Sect.\\ 6]{GG:gaps}. First, one shows that the\nmap \\eqref{eq:map-s} is strictly monotone. This is a consequence of\nthe facts that the orbits are isolated and\n$[{\\mathbb C}{\\mathbb P}^{n-1}] * \\alpha_l=\\alpha_{l-1}$; see \\cite[Prop.\\\n6.2]{GG:gaps}. Next, we have the relation\n\\begin{equation}\n\\label{eq:lambda-bound}\n\\operatorname{c}_{l+(n+1)}(H)=\\operatorname{c}_l(H)+\\lambda,\n\\end{equation}\nwhich follows from \\eqref{eq:qp-cpn}. Now the chain of the\ninequalities\n$$\n\\cdots<\\operatorname{c}_0(H)<\\dots<\\operatorname{c}_n(H)<\\operatorname{c}_{n+1}(H)=\\operatorname{c}_0(H)+\\lambda<\\cdots,\n$$\nis used to infer the uniqueness of $\\bar{x}_l$ and the assertion that the\nmap \\eqref{eq:map-x_i} is a bijection. (The argument is similar to the\nproof of the degenerate case of Arnold's conjecture for ${\\mathbb C}{\\mathbb P}^n$; see\n\\cite{Fl, Fo, FW} and also \\cite{Oh:constr, Sc:invent} and a detailed\naccount in \\cite[Sect.\\ 6.2.3]{GG:gaps}.) Note also that we\nautomatically have the identity $\\bar{x}_{l+ (n+1)}=\\bar{x}_l\\# {\\mathbb C}{\\mathbb P}^1$.\n\n\\begin{Example}[Rotations of ${\\mathbb C}{\\mathbb P}^n$, I]\n\\label{ex:cpn}\nIn the context of this paper, a rotation of ${\\mathbb C}{\\mathbb P}^n$ is a Hamiltonian\ndiffeomorphism $\\varphi_Q$ generated by a quadratic Hamiltonian\n$Q=\\sum a_i|z_i|^2$, where we identified ${\\mathbb C}{\\mathbb P}^n$ with the quotient of\nthe unit sphere $S^{2n+1}\\subset {\\mathbb C}^{n+1}$ by the diagonal (Hopf)\n$S^1$-action. A calculation shows that $\\varphi_Q$ is strongly\nnon-degenerate if and only if it is a pseudo-rotation and if and only\nif $a_j-a_i\\not\\in{\\mathbb Q}$ for all pairs $j\\neq i$. Then\n${\\mathcal P}(Q)={\\mathcal P}_1(Q)=\\{x_0,\\ldots,x_n\\}$ is the set of the coordinate axes\nand, without loss of generality, we may assume that\n$a_0<\\ldots0$ and a non-negative integer $d\\leq n$, both\ndepending only on $\\vec{\\Delta}(\\varphi)$, such that for every $\\epsilon$ such\nthat $0<\\epsilon\\leq \\lambda$, we have\n\\begin{equation}\n\\label{eq:gamma}\n\\liminf_{k\\to\\infty} \\frac{|\\{\\ell\\leq k \\mid \n\\gamma(\\varphi^\\ell)<\\epsilon\\}|}{k}\\geq C\\epsilon^d.\n\\end{equation}\nIn particular, the limit inferior is positive.\n\\end{Theorem}\n\n\n\\begin{Corollary}[$\\gamma$-norm convergence]\n\\label{cor:gamma}\nLet $\\varphi$ be a pseudo-rotation of ${\\mathbb C}{\\mathbb P}^n$. Then\n$\\gamma(\\varphi^{k_i})\\to 0$ for some sequence $k_i\\to\\infty$.\n\\end{Corollary}\n\n\\begin{Remark}\n\\label{rmk:gamma}\nTo the best of our knowledge, both the theorem and the corollary are\nnew even when $n=1$. It might be possible to relax the conditions of\nTheorem \\ref{thm:gamma} by combining its proof with the proof of\n\\cite[Thm.\\ 1.1]{CGG} and extend the theorem, or at least Corollary\n\\ref{cor:gamma}, to perfect Hamiltonian diffeomorphisms of ${\\mathbb C}{\\mathbb P}^n$.\nFurthermore, recall that $\\gamma(\\varphi)$ is \\emph{a priori} bounded\nfor all $\\varphi\\in\\operatorname{Ham}({\\mathbb C}{\\mathbb P}^n)$ and, in fact,\n$\\gamma(\\varphi)\\leq \\lambda$; see \\cite{EP} and also \\cite{McD}. A\nsimple way to see this in the context of the paper is to use\n\\eqref{eq:H^inv} and \\eqref{eq:gamma(H)} together with the fact that\n$\\operatorname{c}_{[{\\mathit pt}]}(H)\\geq \\operatorname{c}_{[M]}(H)-\\lambda$ by\n\\eqref{eq:lambda-bound}. This shows that the condition that\n$\\epsilon\\leq \\lambda$ is not really restrictive: when $\\epsilon>\\lambda$ the\ndensity on the left-hand side of \\eqref{eq:gamma} is equal to one. In\nfact, the theorem is most interesting for small values of $\\epsilon$.\n\n The constant $d$ in Theorem \\ref{thm:gamma} and also Theorem\n \\ref{thm:LPR} below is the difference $d=n+1-r$, where $r$ is the\n number of linearly independent resonance relations the mean indices\n $\\hmu(x_i)$ satisfy; see \\cite{GK} or Section \\ref{sec:background}.\n\\end{Remark}\n\n\\begin{Remark}\n Few manifolds $M$ are expected to admit Hamiltonian diffeomorphisms\n $\\varphi$ such that $\\gamma\\big(\\varphi^{k_i}\\big)\\to 0$ for some\n sequence $k_i\\to\\infty$. For instance, hypothetically, this is never\n the case when $M$ is symplectically aspherical. (As far as we know,\n this question\/conjecture is due to L. Polterovich;\n we learned of it from Seyfaddini.) There is also a similar\n question for the $C^0$- or $C^1$-norm and in this instance some partial\n results are available. For example, it is not hard to see that one\n can never have $\\varphi^{k_i}\\to {\\mathit id}$ in the $C^1$-sense when $M$ is\n symplectically aspherical; cf.\\ \\cite{Po}.\n\\end{Remark}\n\n\\begin{proof}\n Throughout the proof, it will be convenient to rescale the\n symplectic structure on ${\\mathbb C}{\\mathbb P}^n$ so that $[\\omega]=2c_1(T{\\mathbb C}{\\mathbb P}^n)$ and\n hence $\\lambda=2(n+1)$, and to normalize the Hamiltonian to ensure\n that all fixed points have zero augmented action. Then\n \\eqref{eq:equal-spectra} holds, ${\\check{\\mathcal S}}(H)={\\check{\\mathcal S}}_{\\mathit ind}(\\varphi)$, or, in\n other words,\n$$\n{\\mathcal A}_H(\\bar{x})=\\hmu(\\bar{x})\n$$\nfor every capped one-periodic orbit $\\bar{x}$ of $\\varphi$. Since the\naugmented action is homogeneous this is also true for all iterates\n$H^{{\\natural} k}$. Furthermore, since\n$\\gamma(\\varphi^k)\\leq \\gamma\\big(H^{{\\natural} k}\\big)$, it suffices to\nprove the theorem for $\\gamma\\big(H^{{\\natural} k}\\big)$ in place of\n$\\gamma(\\varphi^k)$.\n\nObserve also that we only need to prove the theorem for small\n$\\epsilon>0$, e.g., for every $\\epsilon< 4$. Then the result for the entire range of\n$\\epsilon$ from $0$ to $\\lambda=2(n+1)$ will follow by adjusting the value\nof $C$.\n\nConsider the mean index vector $\\vec{\\Delta}$ defined by \\eqref{eq:vDelta}\nand let $\\Gamma\\subset {\\mathbb T}^{n+1}$ be the subgroup topologically\ngenerated by this vector. Thus $\\Gamma$ is the closure of the positive\nsemi-orbit ${\\mathcal O}=\\{k\\vec{\\Delta}\\mid k\\in{\\mathbb N}\\}$. The connected component of\nthe identity in $\\Gamma$ is a torus, and $\\Gamma$ is isomorphic to the\ndirect product of this torus and a cyclic group of order\n$k_0$. Replacing $\\varphi$ by $\\varphi^{k_0}$ we can assume that\n$\\Gamma$ is connected, and hence isomorphic to a torus. Set\n$d=\\dim\\Gamma$. Since the components of $\\vec{\\Delta}$ satisfy at least one\nresonance relation by Theorem \\ref{thm:index-RR}, we have $d\\leq n$.\n\nThe volume of the intersection of the $\\epsilon\/2$-neighborhood $B(\\epsilon)$\nof $0$ in ${\\mathbb T}^{n+1}$ with $\\Gamma$ is bounded from below by $C\\epsilon^d$,\nwhere $C$ is determined by the geometry of $\\Gamma$:\n\\begin{equation}\n\\label{eq:vol}\n\\operatorname{vol}\\big(B(\\epsilon)\\cap\\Gamma\\big)\\geq C\\epsilon^d.\n\\end{equation}\nThe semi-orbit ${\\mathcal O}$ is uniformly distributed in $\\Gamma$ and hence to\nprove \\eqref{eq:gamma} it is enough to show that\n\\begin{equation}\n\\label{eq:eps}\n\\gamma\\big(H^{{\\natural} k}\\big)<\\epsilon\\textrm{ whenever }\nk\\vec{\\Delta}\\in B(\\epsilon).\n\\end{equation}\n\nTo this end, it is convenient to equip ${\\mathbb T}^{n+1}$ with the metric\ngenerated by the norm (the distance to the origin)\n$$\n\\|\\vec{\\theta}\\|=\\max_i\\|\\theta_i\\|,\n$$\nwhere $\\vec{\\theta}=(\\theta_0,\\ldots,\\theta_n)\\in{\\mathbb T}^{n+1}$ and $\\|\\cdot\\|$\non the right-hand side stands for the distance to zero in\n${\\mathbb R}\/2(n+1){\\mathbb Z}$. Thus $B(\\epsilon)$ is actually a cube with faces\nperpendicular to the ``coordinate axes'' and the diameter of\n${\\mathbb T}^{n+1}$ is $2(n+1)$. (The choice of a norm on ${\\mathbb T}^{n+1}$ effects\nonly the value of the constant $C$ in \\eqref{eq:vol} and\n\\eqref{eq:gamma}.)\n\nAssume that $k\\vec{\\Delta}\\in B(\\epsilon)$. Then the spectrum\n${\\mathcal S}\\big(H^{{\\natural} k}\\big)={\\mathcal S}_{\\mathit ind}(\\varphi^k)$ is contained in the\n$\\epsilon\/2$-neighborhood of $2(n+1){\\mathbb Z}$. We will call the part of this\nspectrum lying in the $\\epsilon\/2$-neighborhood of one point of $2(n+1){\\mathbb Z}$\na cluster. To finish the proof we simply need to show that\n$\\operatorname{c}_{[M]}\\big(H^{{\\natural} k}\\big)$, where $M={\\mathbb C}{\\mathbb P}^n$, and\n$\\operatorname{c}_{[{\\mathit pt}]}\\big(H^{{\\natural} k}\\big)$ are in the same cluster. Indeed, then\n$$\n\\gamma\\big(H^{{\\natural} k}\\big)=\\operatorname{c}_{[M]}\\big(H^{{\\natural}\n k}\\big)-\\operatorname{c}_{[{\\mathit pt}]}\\big(H^{{\\natural} k}\\big)<\\epsilon.\n$$\nWe will prove that these action values are in fact in the cluster\ncentered at $0$. (Up to this point we could have worked directly with\nthe ``action vector'' instead of $\\vec{\\Delta}$, an element of ${\\mathbb T}^{n+1}$\nwhose components are the actions of uncapped one-periodic orbits\nviewed as points in ${\\mathbb R}\/\\lambda{\\mathbb Z}$. However, in the next step, the role\nof $\\vec{\\Delta}$ becomes essential because of \\eqref{eq:supp}.)\n\n\nFocusing on $\\operatorname{c}_{[{\\mathit pt}]}\\big(H^{{\\natural} k}\\big)$, denote by $2(n+1)q$ the\ncenter of the cluster containing this point. Our goal is to show that\n$q=0$. Let $\\bar{x}$ be the action carrier for $[{\\mathit pt}]$, i.e., $\\bar{x}$ is the\ncapped $k$-periodic orbit uniquely determined by the condition\n$$\n\\operatorname{c}_{[{\\mathit pt}]}\\big(H^{{\\natural} k}\\big)={\\mathcal A}_{H^{{\\natural} k}}(\\bar{x}).\n$$\nThis orbit has LS-index $-n$. (Recall that in the non-degenerate case\nthis is simply the Conley--Zehnder index.) Since\n$$\n\\mu(\\bar{x})=-n\\in\n\\operatorname{supp} \\operatorname{HF}(\\bar{x})\\subset [\\hmu(\\bar{x})-n,\\,\\hmu(\\bar{x})+n]\n$$\nby \\eqref{eq:supp}, we have $-2n\\leq \\hmu(\\bar{x})\\leq 0$. On the other\nhand, $\\big|\\hmu(\\bar{x})-2(n+1)q\\big|<\\epsilon\/2$, and thus $q=0$ due to the\nassumption that $\\epsilon<4$. A similar argument shows that\n$\\operatorname{c}_{[M]}\\big(H^{{\\natural} k}\\big)$ is also in the cluster centered at\n$0$. This proves \\eqref{eq:eps} and completes the proof of the\ntheorem.\n\\end{proof}\n\n\\begin{Remark}\n\\label{rmk:Delta-gamma}\nIt is clear from the proof that we have also established the inequality\n$$\n\\gamma\\big(\\varphi^k\\big)\\leq {\\mathit const}\\, \\|\\vec{\\Delta}\\big(\\varphi^k\\big)\\|,\n$$\nwhere ${\\mathit const}>0$ depends only on $\\vec{\\Delta}(\\varphi)$. In particular, if\n$\\|\\vec{\\Delta}\\big(\\varphi^{k_i}\\big)\\|$ converges to zero, the sequence\n$\\gamma\\big(\\varphi^{k_i}\\big)$ converges to zero at least as fast.\n\\end{Remark}\n\n\\subsubsection{Lagrangian Poincar\\'e recurrence}\nConsider a compactly supported Hamiltonian diffeomorphism $\\varphi$ of\na symplectic manifold $M^{2n}$. The following conjecture was put forth\nby the first author and independently by Claude Viterbo around 2010.\n\n\\begin{Conjecture}[Lagrangian Poincar\\'e Recurrence]\n\\label{conj:LPR}\nFor any closed Lagrangian submanifold $L\\subset M$ there exists a\nsequence of iterations $k_i\\to\\infty$ such that\n$$\\varphi^{k_i}(L)\\cap L\\neq \\emptyset.$$ \nMoreover, the density of the sequence $k_i$ is related to a symplectic\ncapacity of $L$.\n\\end{Conjecture}\n\nThe requirement that $\\varphi$ is Hamiltonian is essential: the\nconjecture fails for symplectomorphisms of ${\\mathbb T}^2$, e.g., for an\nirrational shift. Furthermore, the conjecture is most interesting when\n$L$ is ``small''. When it is not, e.g., if $L$ is not displaceable,\nthe assertion is often obvious. In dimension two (i.e., for $n=1$),\nthe conjecture readily follows from the standard Poincar\\'e recurrence\ntheorem when $L$ bounds and the observation that otherwise $L$ is not\ndisplaceable by a Hamiltonian diffeomorphism. On the other hand, to\nthe best of our knowledge, beyond $n=1$ the question has been\ncompletely open prior to now. Surprisingly, the conjecture is not\nstraightforward to prove even for a given Hamiltonian diffeomorphism\n$\\varphi$ unless, of course, it is periodic, i.e., $\\varphi^k={\\mathit id}$ for\nsome $k$. The difficulty is present already when $\\varphi$ is as\nsimple as an irrational rotation of ${\\mathbb C}{\\mathbb P}^n$, i.e., the Hamiltonian\ndiffeomorphisms generated by a quadratic Hamiltonian.\n\nIt is also worth pointing out that it is sufficient to prove the\nexistence of one iteration $k=k(\\varphi)>1$, the first return time,\nsuch that $\\varphi^k(L)\\cap L\\neq 0$ for every $\\varphi$ or at least\nfor the iterates of a fixed map. Then the existence of infinitely many\nsuch iterates will follow by replacing $\\varphi$ by $\\varphi^k$ and\nrepeating the process.\n\nTo state our main result on Lagrangian Poincar\\'e recurrence, we need\nto recall several definitions. Let $U$ be an open subset of a closed,\nrational, weakly monotone symplectic manifold $M$. The\n\\emph{homological capacity} of $U$ is defined as\n\\begin{equation}\n\\label{eq:c-hom}\n\\operatorname{c_{\\scriptscriptstyle{hom}}}(U)=\\sup_F\\gamma(\\varphi_F),\n\\end{equation}\nwhere $F$ ranges through all Hamiltonians $F$ supported in $S^1\\times U$; see,\ne.g., \\cite{Sc, Us:ineq, Vi:gen} and references therein. This function\nof $U$ has all the expected properties of a symplectic capacity and it\nis a standard fact that\n\\begin{equation}\n\\label{eq:chom-gamma}\n\\operatorname{c_{\\scriptscriptstyle{hom}}}(U)\\leq \\gamma(\\varphi) \\textrm{ when } \\varphi(U)\\cap U\n=\\emptyset;\n\\end{equation}\nsee, e.g., \\cite[Prop.\\ 3.1]{Us:ineq}. (In applications, sometimes it\nis convenient to replace $\\gamma(\\varphi_F)$ in \\eqref{eq:c-hom} by\n$\\operatorname{c}_{[M]}(F)$; the resulting capacity has the same properties as the\none defined above; see \\cite{Gi:We}.) We extend $\\operatorname{c_{\\scriptscriptstyle{hom}}}$ to closed\nsubsets $L$ of $M$ (for instance, to Lagrangian submanifolds) by\nsetting\n$$\n\\operatorname{c_{\\scriptscriptstyle{hom}}}(L)=\\inf_U\\operatorname{c_{\\scriptscriptstyle{hom}}}(U),\n$$\nwhere the infimum is taken over all open sets $U\\supset L$. Note that\n$\\operatorname{c_{\\scriptscriptstyle{hom}}}(L)\\leq \\lambda$; see Remark \\ref{rmk:gamma} and the references\ntherein.\n\n\n\n\\begin{Example}\n\\label{lem:chom}\nAssume that $L\\subset M$ is a closed Lagrangian admitting a Riemannian\nmetric without contractible closed geodesics. Then\n\\begin{equation}\n\\label{eq:chom}\n\\operatorname{c_{\\scriptscriptstyle{hom}}}(L)>0.\n\\end{equation}\nThe proof of this well-known fact, which we omit here, is quite\nstandard and implicitly contained in, e.g., the proof of \\cite[Thm.\\\n8.2]{Us:BD}. Moreover, the same is true for certain classes of\ncoisotropic manifolds (contact type, stable or with totally geodesic\ncharacteristic foliation); see \\cite{Gi:coiso, Us:BD}. Conjecturally,\n\\eqref{eq:chom} holds for all closed Lagrangians, but this, to the\nbest of our understanding, is unknown.\n\\end{Example}\n\nNow we are in a position to state and prove the key result of this\nsection.\n\n\\begin{Theorem}\n\\label{thm:LPR}\nLet $\\varphi$ be a pseudo-rotation of ${\\mathbb C}{\\mathbb P}^n$ and let $L\\subset {\\mathbb C}{\\mathbb P}^n$\nbe a closed subset (e.g., a Lagrangian submanifold). Then there exists\na constant $C>0$ and a non-negative integer $d\\leq n$, both depending\nonly on $\\vec{\\Delta}(\\varphi)$ but not $L$, such that\n$$\n\\liminf_{k\\to\\infty} \\frac{|\\{\\ell\\leq k \\mid \\varphi^\\ell(L)\\cap L\\neq\n \\emptyset\\}|}{k}\\geq C\\cdot\\operatorname{c_{\\scriptscriptstyle{hom}}}(L)^d.\n$$\nIn particular, the limit inferior is positive when $\\operatorname{c_{\\scriptscriptstyle{hom}}}(L)>0$.\n\\end{Theorem}\n\nAs a consequence, we obtain Theorem \\ref{thm:LPR0} from the\nintroduction:\n\n\\begin{Corollary}\n\\label{cor:LPR}\nIn the setting of Theorem \\ref{thm:LPR}, assume that $\\operatorname{c_{\\scriptscriptstyle{hom}}}(L)>0$\n(e.g., $L$ is as in Example \\ref{lem:chom}). Then\n$\\varphi^{k_i}(L)\\cap L\\neq \\emptyset$ for some sequence\n$k_i\\to\\infty$.\n\\end{Corollary}\n\n\\begin{Remark}[Multiplicity of Intersections] \n\\label{rmk:mult-inter}\nIn the general framework of the Lagrangian Poincar\\'e recurrence\nconjecture, we see no reason to expect a lower bound on the number of\nintersections of $\\varphi^k(L)$ and $L$ -- after all this is a\ndynamics rather than a symplectic topological question. However, in\nthe setting considered here where the recurrence is a consequence of\nthe $\\gamma$-convergence, the situation is different. Namely, assume\nfor the sake of simplicity that $\\varphi^k(L)$ and $L$ are transverse\nfor all $k$, which is a generic condition on $L$. Then, in Corollary\n\\ref{cor:LPR}, the number of intersections is bounded from below by\n$\\dim\\operatorname{H}(L)$, provided that one can replace the upper bound on the\nHofer norm in Chekanov's theorem, \\cite{Ch:Lagr}, by an upper bound on\nthe $\\gamma$-norm. Some results in this direction have been recently\nannounced in \\cite{KS}.\n\\end{Remark}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:LPR}]\n Without loss of generality we can assume that $\\operatorname{c_{\\scriptscriptstyle{hom}}}(L)>0$ --\n otherwise the assertion is void. Set $\\epsilon=\\operatorname{c_{\\scriptscriptstyle{hom}}}(L)$; then\n $\\epsilon\\leq \\lambda$. Consider the set\n$$\nK=\\{k\\mid \\gamma(\\varphi^k)<\\epsilon\\} \\subset {\\mathbb N}.\n$$\nLet $U$ be a neighborhood of $L$. Clearly,\n$\\operatorname{c_{\\scriptscriptstyle{hom}}}(U)\\geq \\epsilon$ and, by \\eqref{eq:chom-gamma}, \n$$\n\\varphi^k(U)\\cap U\\neq \\emptyset\n$$\nfor every $k\\in K$. Since this holds for all $U\\supset L$, we\nhave\n $$\n\\varphi^k(L)\\cap L\\neq \\emptyset\n$$\nfor all $k\\in K$ and the result now follows from Theorem\n\\ref{thm:gamma}.\n\\end{proof}\n\n\\begin{Remark}[Return Frequency for Small Balls] \n Let $U$ be a small ball of radius $\\delta>0$ in ${\\mathbb C}{\\mathbb P}^n$. Then\n $\\operatorname{c_{\\scriptscriptstyle{hom}}}(U)\\geq{\\mathit const}\\cdot \\delta^2$. Arguing as in the proof of\n Theorem \\ref{thm:LPR}, it is easy to see that the return frequency\n for $\\varphi$ and $U$ is bounded from below by $C\\delta^{2d}$:\n\\begin{equation}\n\\label{eq:freq}\n\\lim_{k\\to\\infty} \\frac{|\\{\\ell\\leq k \\mid \n\\varphi^\\ell(U)\\cap U\\neq \\emptyset\\}|}{k}\\geq\nC\\delta^{2d}.\n\\end{equation}\nWhen $d=n$, \\eqref{eq:freq} is exactly the lower bound guaranteed by\nthe standard Poincar\\'e recurrence theorem. In general, $d=n+1-r$,\nwhere $r$ is the number of linearly independent resonance relations\nwhich the mean indices $\\hmu(x_i)$ satisfy. Thus \\eqref{eq:freq}\nprovides a stronger lower bound when $r\\geq 2$.\n\\end{Remark}\n\n\\begin{Remark}\n\\label{rmk:packing}\nThe Lagrangian Poincar\\'e recurrence can also be thought of as a\nconsequence of a hypothetical obstruction to Lagrangian packing in the\nsame way as the standard Poincar\\'e recurrence can be viewed as coming\nfrom the volume obstruction to ball packing. For instance, one might\nconjecture that for a compact symplectic manifold $M$ (possibly with\nboundary) and a closed Lagrangian $L\\subset M$, one can embed into $M$\nonly a finite number of disjoint Lagrangian submanifolds Hamiltonian\nisotopic to $L$. We do not have any counterexamples to this more\ngeneral conjecture; nor are we aware of any results in this direction.\n\\end{Remark}\n\n\\begin{Remark}\n It is interesting to compare the Lagrangian Poincar\\'e recurrence\n conjecture with Arnold's Legendrian chord conjecture. While at first\n glance the two questions appear to be similar, there are some\n fundamental differences. For instance, in many cases the first\n return time in the chord conjecture is independent of the Legendrian\n submanifold and completely determined by the Reeb flow (see, e.g.,\n \\cite{Mo}), but the first return time in the Lagrangian Poincar\\'e\n recurrence must, clearly, depend on $L$. (Ultimately, the reason is\n that small, localized, Legendrian submanifolds have localized chords\n unrelated to global dynamics. In other words, the chord conjecture\n is a symplectic topological fact while Lagrangian Poincar\\'e\n recurrence relates to dynamics.) On a more technical level,\n Legendrian chords can be treated in the framework of a Floer- or\n SFT-type homology theory (see, e.g., \\cite{Ch, EGH}), but no such \n theory for the Lagrangian Poincar\\'e recurrence is known.\n\\end{Remark}\n\n\\subsection{$C^0$-rigidity}\n\\label{sec:PO}\nIt turns out that under suitable additional conditions on the mean\nindex vector $\\vec{\\Delta}$, $\\gamma$-convergence in Corollary\n\\ref{cor:gamma} can be replaced by $C^0$-convergence. For\npseudo-rotations in dimension two this is the $C^0$-rigidity\nestablished in \\cite{Br}, and our goal is to extend this result to\nhigher dimensions. Let us begin with several preliminary observations.\n\nConsider a compact abelian group $\\Gamma$ equipped with some\nbi-invariant metric. As above we denote the norm of\n$\\vec{\\theta}\\in\\Gamma$, i.e., the distance to the origin, by $\\|\\vec{\\theta}\\|$.\n\n\\begin{Definition}\n\\label{def:exp-L}\nA ``vector'' $\\vec{\\theta}\\in\\Gamma$ is \\emph{exponentially Liouville} if\nfor every constant $c>0$ there exists $k\\in{\\mathbb N}$ such that\n$\\|k\\vec{\\theta}\\|8\\| H\\|_{C^2}$. Furthermore, the convergence is exponential:\n $\\|\\varphi^{k_i}\\|_{C^0}\\leq e^{-a k_i}$ for some $a>0$ which can be\n made arbitrarily large by choosing a large $c$. Finally, note that\n in this theorem we impose no non-degeneracy requirements on~$H$.\n\\end{Remark}\n\n\n\\begin{proof} The argument closely follows the proof from \\cite{Br},\n although we make several shortcuts and use Floer theory instead of\n pseudo-holomorphic curves.\n \n Let us first assume that $\\varphi$ is strongly non-degenerate, i.e.,\n all its iterates are non-degenerate. It is easy to see that since\n $\\varphi$ is a pseudo-rotation the Floer differential vanishes and\n hence we can identify the Floer homology of $\\varphi$ with the Floer\n complex. Let $\\bar{x}$ and $\\bar{y}$ be the capped $k$-periodic orbits\n representing the fundamental class $[{\\mathbb C}{\\mathbb P}^n]$ and the class of the\n point $[{\\mathit pt}]$ in the Floer complex\/homology of $\\varphi^k$. Denote\n by ${\\mathcal M}(\\bar{x},\\bar{y})$ the moduli space of Floer trajectories\n $u\\colon S^1_k\\times {\\mathbb R} \\to {\\mathbb C}{\\mathbb P}^n$, where $S^1_k={\\mathbb R}\/k{\\mathbb Z}$, from $\\bar{x}$\n to $\\bar{y}$. The image $U$ of the evaluation map\n$$\n{\\mathcal M}(\\bar{x},\\bar{y})\\to{\\mathbb C}{\\mathbb P}^n, \\quad u\\mapsto u(0,0)\n$$ \ncontains an open and dense subset in ${\\mathbb C}{\\mathbb P}^n$. (This is true for any\nclosed rational symplectic manifold $M$ and any non-degenerate\nHamiltonian when $\\bar{x}$ and $\\bar{y}$ are replaced by action carriers for\n$[M]$ and $[{\\mathit pt}]$.) This is an immediate consequence of the standard\nfact that for a generic $p\\in{\\mathbb C}{\\mathbb P}^n$ the number of $u\\in{\\mathcal M}(\\bar{x},\\bar{y})$\nwith $u(0,0)=p$, taken with appropriate signs and viewed as an element\nof ${\\mathbb F}$, represents the action of $[p]=[{\\mathit pt}]\\in\\operatorname{HQ}_*({\\mathbb C}{\\mathbb P}^n)$ on\n$\\operatorname{HF}_*(\\varphi^k)$ and $[{\\mathit pt}]*[{\\mathbb C}{\\mathbb P}^n]=[{\\mathit pt}]$; see\nSection~\\ref{sec:cap-product}.\n\nNext, recall that\n\\begin{equation}\n\\label{eq:velocity-energy}\n\\big\\|\\partial_s u \\big\\|_{L^\\infty}\\leq O\\big(E(u)^{1\/4}\\big),\n\\end{equation}\nwhere $s$ is the coordinate on ${\\mathbb R}$, and the energy $E(u)$ of $u$ is\nsufficiently small; see \\cite[Sect.\\ 1.5]{Sa} or \\cite{Br} for a\nsimple self-contained proof. The upper bound on the right in\n\\eqref{eq:velocity-energy} is uniform in $k$, and in fact independent\nof $k$, and completely determined by the $C^2$-norm of $H$ and the\nalmost complex structure $J$. (It is essential here that we view the\niterated flow not as $\\varphi_H^{kt}$ but as the flow $\\varphi_H^t$\nwith $t\\in [0,k]$.)\n\nDenote by $d$ the distance in ${\\mathbb C}{\\mathbb P}^n$. We claim that\n\\begin{equation}\n\\label{eq:distance}\nd\\big(p,\\varphi^k(p)\\big) \\leq e^{Ck} O\\big(E(u)^{1\/4}\\big)\n\\end{equation}\nfor every $p\\in {\\mathbb C}{\\mathbb P}^n$, where we can take any $C>2\\| H\\|_{C^2}$,\nprovided that $E(u)$ is small enough. To prove this, assume first that\n$p\\in U$. Set $z(t)=u(0,t)$ for some $u$ with $p=u(0,0)$ and\n$\\zeta(t)=\\big(\\varphi^t\\big)^{-1}\\big(z(t)\\big)$. Then\n$$\n\\dot{z}(t)=X_H\\big(z(t)\\big)+ D\\varphi^t\\big(\\dot{\\zeta}(t)\\big),\n$$\nand hence\n$$\n\\dot{\\zeta}(t)=\\big(D\\varphi^t\\big)^{-1}\\big(\n\\dot{z}(t)-X_H(z(t))\\big).\n$$\nIt is easy to see that \n\\begin{equation}\n\\label{eq:exp}\n\\big\\|\\big(D\\varphi^t\\big)^{-1}\\big\\|\\leq e^{C_0t},\n\\end{equation}\nwhere we can take $C_0=\\|H\\|_{C^2}$. Moreover, from the Floer equation\nand \\eqref{eq:velocity-energy}, we infer that\n$$\n\\big\\|X_H\\big(z(t)\\big)-\\dot{z}(t)\\big\\|\\leq O\\big(E(u)^{1\/4}\\big),\n$$\nfor all $t\\in [0,k]$. Therefore,\n$$\n\\big\\| \\dot{\\zeta}(t)\\big\\|\\leq e^{C_0t} O\\big(E(u)^{1\/4}\\big)\n$$\n and thus\n$$\nd\\big(\\zeta(0),\\zeta(k)\\big)\\leq e^{C_0 k} O\\big(E(u)^{1\/4}\\big).\n$$\nClearly, $\\zeta(0)=p$ and, since $z$ is a loop,\n$\\zeta(k)=\\varphi^{-k}(p)$. Applying $\\varphi^k$ to $\\zeta(0)$ and\n$\\zeta(k)$ and using \\eqref{eq:exp} again, we obtain\n\\eqref{eq:distance} with $C>2C_0$. Finally, the upper bound is uniform\nin $p$ and $U$ is dense in ${\\mathbb C}{\\mathbb P}^n$. Therefore, \\eqref{eq:distance}\nholds on ${\\mathbb C}{\\mathbb P}^n$.\n\nNext, let us find an upper bound on $E(u)$. This is where the\nrequirement that $\\vec{\\Delta}$ is exponentially Liouville enters the proof\nand the argument is quite similar to the end of the proof of Theorem\n\\ref{thm:gamma}. As in that proof, it is convenient to rescale the\nsymplectic structure on ${\\mathbb C}{\\mathbb P}^n$ so that $[\\omega]=2c_1(T{\\mathbb C}{\\mathbb P}^n)$ and\nnormalize $H$ to ensure that ${\\mathcal S}(H)={\\mathcal S}_{\\mathit ind}(\\varphi)$. Then, in\nparticular,\n$$\n{\\mathcal A}_{H^{{\\natural} k}}(\\bar{x})=\\hmu(\\bar{x}) \\textrm{ and } \n{\\mathcal A}_{H^{{\\natural} k}}(\\bar{y})=\\hmu(\\bar{y}).\n$$\n\nAssume that $k=k_i$ for the sequence $k_i$ from Definition\n\\ref{def:exp-L} with $\\vec{\\theta}=\\vec{\\Delta}$ and a sufficiently large $c$ to\nbe specified later. Then, by \\eqref{eq:exp-L}, the spectrum\n${\\mathcal S}\\big(H^{{\\natural} k}\\big)={\\mathcal S}_{\\mathit ind}(\\varphi^k)$ is located in a neighborhood\nof $2(n+1){\\mathbb Z}$ of size $e^{-ck}$. Arguing exactly as in the proof of\nTheorem \\ref{thm:gamma}, it is easy to show that $\\hmu(\\bar{x})$ and\n$\\hmu(\\bar{y})$ both lie in the cluster centered at 0. Thus\n\n$$\n|\\hmu(x)|\\leq e^{-ck}\\textrm{ and } |\\hmu(y)|\\leq e^{-ck}\n$$\nand\n$$\nE(u)={\\mathcal A}_{H^{{\\natural} k}}(\\bar{x})-{\\mathcal A}_{H^{{\\natural}\n k}}(\\bar{y})=\\hmu(\\bar{x})-\\hmu(\\bar{y})\\leq 2e^{-ck}.\n$$\n\nSet $c>4C$. (For instance, $C=2.1\\|H\\|_{C^2}$ and\n$c=8.5\\| H\\|_{C^2}$.) Then, by \\eqref{eq:distance} for $k=k_i$,\nwe have\n$$\nd\\big(p,\\varphi^{k_i}(p)\\big) \\leq O\\big(e^{(C-c\/4)k_i}\\big)\\to 0\n$$\nas $k_i\\to\\infty$. Thus\n$$\n\\|\\varphi^{k_i}\\|_{C^0}\\to 0,\n$$\nwhich proves the theorem when $\\varphi$ is strongly non-degenerate.\n\nDealing with the degenerate case, consider a $C^2$-small\nnon-degenerate perturbation $F$ of $H^{{\\natural} k}$. As above, let $\\bar{x}$\nand $\\bar{y}$ be the action carriers for $[{\\mathbb C}{\\mathbb P}^n]$ and, respectively,\n$[{\\mathit pt}]$.\n\n\\begin{Lemma}\n\\label{lemma:deg-energy}\nAssume that $F$ is sufficiently $C^2$-close to $H^{{\\natural} k}$. Then, for\na generic point $p\\in{\\mathbb C}{\\mathbb P}^n$, there exists a solution $u$ of the Floer\nequation for $F$ with $u(0,0)=p$ such that\n\\begin{equation}\n\\label{eq:deg-energy}\nE(u)\\leq 2 \\big( {\\mathcal A}_{H^{{\\natural} k}}(\\bar{x})- {\\mathcal A}_{H^{{\\natural} k}}(\\bar{y})\\big).\n\\end{equation}\n\\end{Lemma}\n\n\\begin{proof} We start with several general remarks. Fix\n $\\epsilon>0$. Then, when $F$ is sufficiently $C^2$-close to\n $H^{{\\natural} k}$, every capped $k$-periodic orbit $\\bar{z}$ of $H^{{\\natural} k}$\n splits under the perturbation $F$ into several capped orbits $\\bar{z}_i$\n located near $\\bar{z}$ with actions and mean indices $\\epsilon$-close to the\n action and the mean index of $\\bar{z}$. All $k$-periodic orbits of $F$\n arise in this way. (We view both $F$ and $H^{{\\natural} k}$ as\n $k$-periodic Hamiltonians.) Moreover, for two $k$-periodic orbits\n $\\bar{z}$ and $\\bar{z}'$ we have\n$$\n{\\mathcal A}_{H^{{\\natural} k}}(\\bar{z})>{\\mathcal A}_{H^{{\\natural} k}}(\\bar{z}')\n\\textrm{ iff }\n{\\mathcal A}_{F}(\\bar{z}_i)>{\\mathcal A}_{F}(\\bar{z}'_j)\n\\textrm{ for all (or just one pair of) $i$ and $j$}.\n$$\n\nLet $\\bar{z}$ be the action carrier for some class\n$\\alpha\\in\\operatorname{HF}\\big(H^{{\\natural} k}\\big)$. Then, in the notation from Section\n\\ref{sec:spec} and, in particular, \\eqref{eq:cycle-action}, for every\ncycle $\\sigma\\in \\operatorname{CF}_*(F)$ representing~$\\alpha$,\n\\begin{equation}\n\\label{eq:A}\n\\operatorname{c}_\\sigma(F)\\geq \\operatorname{c}_\\alpha\\big( H^{{\\natural} k}\\big)-\\epsilon={\\mathcal A}_{H^{{\\natural}\n k}}(\\bar{z})-\\epsilon,\n\\end{equation}\nand there exists a representative $\\sigma_{\\min}$ such that\n\\begin{equation}\n\\label{eq:Amin}\n\\operatorname{c}_{\\sigma_{\\min}}(F)\\leq \\operatorname{c}_\\alpha\\big( H^{{\\natural} k}\\big)+\\epsilon={\\mathcal A}_{H^{{\\natural}\n k}}(\\bar{z})+\\epsilon.\n\\end{equation}\nIn particular, at least one of the orbits $\\bar{z}_i$ enters\n$\\sigma_{\\min}$ with non-zero coefficient.\n\nLet us pick such a cycle $\\sigma_{\\min}$ representing $[{\\mathbb C}{\\mathbb P}^n]$ for\n$F$ and satisfying \\eqref{eq:Amin}. Thus\n$$\n\\operatorname{c}_{\\sigma_{\\min}}(F)\\leq {\\mathcal A}_{H^{{\\natural}\n k}}(\\bar{x})+\\epsilon.\n$$\n\nNext, we pick a generic $p\\in{\\mathbb C}{\\mathbb P}^n$ and view it as a cycle\nrepresenting $[{\\mathit pt}]\\in \\operatorname{HQ}_*({\\mathbb C}{\\mathbb P}^n)$. Then acting by $p$ on\n$\\sigma_{\\min}$, we obtain a cycle $P\\in \\operatorname{CF}_{-n}(F)$ also\nrepresenting $[{\\mathit pt}]$. By \\eqref{eq:A},\n$$\n\\operatorname{c}_P(F)\\geq {\\mathcal A}_{H^{{\\natural}\n k}}(\\bar{y})-\\epsilon.\n$$\nA point $\\bar{y}_j$ with action $\\operatorname{c}_P(F)$ enters the cycle $P$, and hence\nthere exists a solution $u$ of the Floer equation for $F$ connecting a\npoint from $\\sigma_{\\min}$ to $\\bar{y}_j$. This solution has energy\n$$\nE(u)\\leq \\operatorname{c}_{\\sigma_{\\min}}(F)-\\operatorname{c}_P(F)\\leq {\\mathcal A}_{H^{{\\natural}\n k}}(\\bar{x})-{\\mathcal A}_{H^{{\\natural} k}}(\\bar{y})+2\\epsilon\n$$\nand passes through $p$, i.e., $u(0,0)=p$. Making $\\epsilon>0$ small enough\nwe obtain \\eqref{eq:deg-energy}, which completes the proof of the\nlemma.\n\\end{proof}\n\nNow the proof of the theorem is finished essentially in the same way\nas in the non-degenerate case. First, note that $\\| F\\|_{C^2}$ can be\nmade arbitrarily close to $\\| H^{{\\natural} k}\\|_{C^2}=\\| H\\|_{C^2}$. (It is\nagain essential here that $H$ is periodic in time and $H^{{\\natural} k}_t$\nis simply the Hamiltonian $H_t$ but with $t\\in S^1_k$.) Then\n\\eqref{eq:velocity-energy} still holds, where the upper bound on the\nright is uniform in $k$ and completely determined by $\\| H\\|_{C^2}$.\n\nNext, the bound \\eqref{eq:distance} turns into\n$$\nd\\big(p,\\varphi_F(p)\\big) \\leq e^{Ck} O\\big(E(u)^{1\/4}\\big)\n$$\nfor a generic $p$, where we can again take any $C>2\\| H\\|_{C^2}$. By\nmaking $F$ sufficiently close to $H^{{\\natural} k}$ we can make sure that\n$d\\big(\\varphi^k(p),\\varphi_F(p)\\big)$ is arbitrarily small for all\n$p$, and hence \\eqref{eq:distance} holds in its original form for all\n$p$ uniformly in $k$, provided that $E(u)$ is small enough (depending\non $\\| H\\|_{C^2}$).\n\nFinally, when $k=k_i$, we have\n$$\nE(u)\\leq 2\\big({\\mathcal A}_{H^{{\\natural} k}}(\\bar{x})-{\\mathcal A}_{H^{{\\natural}\n k}}(\\bar{y})\\big)=2\\big(\\hmu(\\bar{x})-\\hmu(\\bar{y})\\big)\\leq 4e^{-ck}\n$$\nby Lemma \\ref{lemma:deg-energy} and again\n$ \\|\\varphi^{k_i}\\|_{C^0}\\to 0 $ exponentially fast when $c>4C$.\n\\end{proof}\n\n\\begin{Remark}\n As has been pointed out to us by Seyfaddini, one can also derive\n Theorem \\ref{thm:exp-L} from Theorem \\ref{thm:gamma} and Remark\n \\ref{rmk:Delta-gamma} by employing a ``H\\\"older type'' inequality\n relating the $C^0$-norm, the $\\gamma$-norm and the $C^0$-norm of the\n derivative. However, the direct proof given here is of independent\n interest and may have other applications.\n\\end{Remark}\n\n\nJust as in \\cite{Br} we have the following corollary of Theorem\n\\ref{thm:exp-L}:\n\n\\begin{Corollary}\n\\label{cor:mixing}\nLet $\\varphi$ be a pseudo-rotation of ${\\mathbb C}{\\mathbb P}^n$ with exponentially\nLiouville mean index vector. Then $\\varphi$ is not topologically\nmixing and, in particular, not mixing with respect to the Lebesgue\nmeasure.\n\\end{Corollary}\n\nWe conclude this section with several remarks. First, note that in\ndimension two the $\\gamma$-norm is continuous with respect to the\n$C^0$-norm; \\cite{Se}. Furthermore, similar results in higher\ndimensions for symplectically aspherical manifolds and in some other\nsettings have been recently obtained in \\cite{BHS}. Thus, for $S^2$\nand more generally when such continuity is established, Theorem\n\\ref{thm:exp-L} implies Corollary \\ref{cor:gamma} in the exponentially\nLiouville case. Finally, we conjecture that a variant of Theorem\n\\ref{thm:exp-L} holds without the assumption that $\\Delta$ is\nexponentially Liouville and in this case one may also have an analog\nof the frequency bound similar to that in Theorem~\\ref{thm:gamma}.\n\nIt is also worth mentioning that strictly speaking the results in\n\\cite{Br} are established for exponentially Liouville pseudo-rotations\nof $D^2$ while, for $n=1$, our Theorem \\ref{thm:exp-L} concerns\nexponentially Liouville pseudo-rotations of $S^2$. The difference in\nthe domains is rather technical than conceptual, although it does\neffect what symplectic topological tools are better suited for the\ntask (holomorphic curves vs.\\ Floer homology). In any event, the\nresults for $S^2$ can be derived from those for $D^2$ and vice\nversa. In one direction, from $D^2$ to $S^2$, this can be done by\nsimply applying an oriented real blow-up to one of the fixed\npoints. In the opposite direction, the argument is considerably more\nsubtle and requires more work. The difficulty lies in the fact that a\npseudo-rotation of $S^2$ obtained from one of $D^2$ by, e.g.,\ncollapsing $\\partial D^2$ to a point or doubling $D^2$, is not\nsmooth. However, it is Lipschitz and the proof seems to go through\nwith only minor modifications including extending the resonance\nrelations to this setting. Finally, the requirement in \\cite{Br} that\nthe pseudo-rotation is irrational is more of terminological than of\nmathematical nature: a pseudo-rotation of $D^2$ is automatically\nirrational, for otherwise it would have periodic orbits on $\\partial D^2$.\nOne can interpret this requirement as an implicit non-degeneracy\ncondition. Then similarly, it is also automatically satisfied for\npseudo-rotations of $S^2$; see Corollary \\ref{cor:S^2-nondeg}.\n\n\n\n\\section{Crossing energy and the proof of Theorem \\ref{thm:isolated}}\n\\label{sec:energy-isolated}\n\nThe proof of Theorem \\ref{thm:isolated} hinges on a technical result\ngeneralizing \\cite[Thm.\\ 3.1]{GG:hyperbolic} and asserting, roughly\nspeaking, that the energy of a Floer trajectory asymptotic to $x^k$\nand crossing a fixed isolating neighborhood of $x$ is bounded from\nbelow by a constant independent of $k$. With future applications in\nmind we start by proving this result in a form more general than\nneeded for the proof of the theorem.\n\n\\subsection{Crossing energy}\n\\label{sec:cross-energy}\nLet $K\\subset M$ be a compact invariant set of a Hamiltonian\ndiffeomorphism $\\varphi=\\varphi_H$ of a symplectic manifold\n$M$. Recall that $K$ is said to be isolated (as an invariant set) if\nthere exists a neighborhood $U\\supset K$ such that for no initial\ncondition $p\\in U\\setminus K$ the orbit through $p$ is contained in\n$U$, i.e., there exists $k\\in{\\mathbb Z}$, possibly depending on $p$, such that\n$\\varphi^k(p)\\not\\in U$. The neighborhood $U$ is called an isolating\nneighborhood of $K$. Then any neighborhood of $K$ contained in $U$ is\nalso isolating, and hence such neighborhoods can be made arbitrarily\nsmall. (Note that a fixed point may be isolated for all iterations as\na fixed point while not isolated as an invariant set.)\n\nConsider solutions $u\\colon \\Sigma\\to M$ of the Floer equation for\n$H^{{\\natural} k}$, where $\\Sigma\\subset {\\mathbb R}\\times S^1_k$ is a closed domain,\ni.e., a closed subset with non-empty interior. Note that the period\n$k$ is not fixed, and the domain $\\Sigma$ of $u$ need not be the\nentire cylinder ${\\mathbb R}\\times S^1_k$. By definition, the energy of $u$ is\n$$\nE(u)=\\int_\\Sigma \\|\\partial_s u\\|^2 \\, ds dt.\n$$\nHere $\\|\\cdot\\|$ stands for the norm with respect to\n$\\left<\\cdot\\,,\\cdot\\right>=\\omega(\\cdot,J\\cdot)$, and hence\n$\\|\\cdot\\|$ also depends on the background almost complex structure\n$J$. We say that $u$ is asymptotic to $K$ as $s\\to\\infty$ (or as\n$s\\to-\\infty$) if for any neighborhood $V$ of $K$ the domain $\\Sigma$\ncontains a cylinder $[s_V,\\,\\infty)\\times S^1_k$ (or\n$(-\\infty,\\,s_V]\\times S^1_k$) which is mapped into $V$ by $u$.\n\nFinally, let $U$ be a fixed (sufficiently small) isolating\nneighborhood of $K$. Set $\\partial U:=\\bar{U}\\setminus U$.\n\n\\begin{Theorem}[Crossing Energy Theorem]\n\\label{thm:energy}\nThere exists a constant $c_\\infty>0$, independent of $k$ and $\\Sigma$,\nsuch that for any solution $u$ of the Floer equation with\n$u(\\partial \\Sigma)\\subset \\partial U$ and $\\partial \\Sigma\\neq\\emptyset$, which is\nasymptotic to $K$ as $s\\to\\infty$ or $s\\to-\\infty$, we have\n\\begin{equation}\n\\label{eq:energy}\nE(u)>c_\\infty .\n\\end{equation}\nMoreover, the constant $c_\\infty$ can be chosen to make\n\\eqref{eq:energy} hold for all $k$-periodic almost complex structures\n(with varying $k$) $C^\\infty$-close to $J$ uniformly on ${\\mathbb R}\\times U$.\n\\end{Theorem}\n\nThe key point of this result is that the lower bound $c_\\infty>0$ can\nbe taken independent of $k$. The requirements of the theorem are met\nwhen $K$ is just one fixed point $x$ of $\\varphi$ and $x$ is\nhyperbolic -- this is the setting of \\cite[Thm.\\ 3.1]{GG:hyperbolic}\n-- or more generally when $K=\\{x\\}$ is isolated as an invariant set as\nin Theorem \\ref{thm:isolated}. The requirements are also satisfied\nwhen $K$ is a hyperbolic invariant subset (e.g., a Smale's horseshoe).\nThe condition that $K$ is isolated is essential and cannot be removed;\nsee \\cite[Rmk.\\ 3.4]{GG:hyperbolic}. Also note that periodic orbits\nof $\\varphi_H^t$ in $M$ corresponding to the fixed points in $K$ need\nnot be contractible. The proof of Theorem \\ref{thm:energy} follows\nclosely the line of reasoning establishing \\cite[Thm.\\\n3.1]{GG:hyperbolic}. However, Theorem \\ref{thm:energy} is considerably\nmore general than that result and we feel that giving a detailed proof\nis justified.\n\n\\begin{proof}\n We will focus on solutions $u$ asymptotic to $K$ as\n $s\\to\\infty$. The case of $s\\to-\\infty$ can be handled in a similar\n fashion. Before going into details of the proof, let us spell out\n the idea. Note first that since $U$ is an isolating neighborhood,\n there exists $T_0>1$, an escape time, such that every integral curve\n $\\varphi_H^t(p)$, $t\\in [-T_0,\\, T_0]$, touching $\\partial U$ at some\n moment $\\tau\\in [0,\\,1]$ cannot be entirely contained in\n $\\bar{U}$. Arguing by contradiction, assume that there exists a\n sequence of solutions $u_i$ of the Floer equation with $E(u_i)\\to 0$\n asymptotic to $K$ at $+\\infty$ and defined on\n $\\Sigma_i\\subset {\\mathbb R}\\times S^1_{k_i}$. Then for every $T>T_0$ one can\n also find a sequence of solutions $v_i$ defined on a subset of a\n rectangle $[-a,\\,a]\\times [-T,\\,T]\\subset {\\mathbb C}$ containing\n $[-a,\\,a]\\times [0,\\,T]$, mapping this half-rectangle into $\\bar{U}$\n and such that $v_i(0, \\tau_i)\\in \\partial U$ for some $\\tau_i\\in [0,\\,1]$.\n As $i\\to\\infty$, these solutions converge to an integral curve\n $(s,t)\\mapsto \\varphi_H^t(p)$ in the sense of the target-local\n compactness theorem from \\cite{Fi}. This curve is parametrized by\n $[-T',\\,T']$ with $T>T'>T_0$, touches $\\partial U$ at some moment\n $\\tau\\in [0,1]$ and is entirely contained in $\\bar{U}$, which is\n impossible by the definition of $T_0$.\n\n Throughout the proof, it will be convenient to work in $M\\times S^1$\n rather than in $M$. Thus let $\\tilde{\\varphi}^t$ be the flow on\n $M\\times S^1$ induced by the isotopy $\\varphi_H^t$, i.e.,\n $\\tilde{\\varphi}^t(p,\\theta)=(\\varphi_H^t(p),\\theta+t\\, \\mathrm{mod}\\,\n 1)$. This is indeed a true flow since $H$ is one-periodic in\n time. In a similar vein, the map $u$ gives rise to the map\n$$\n\\tilde{u}\\colon \\Sigma_i\\to M\\times S^1, \\quad (s,t)\\mapsto\n\\big(u(s,t),t\\,\\mathrm{mod}\\, 1\\big).\n$$ \nSet $\\tilde{K}=K\\times S^1$, and let $\\tilde{B}$ and $\\tilde{U}$ be closed isolating\nneighborhoods of $\\tilde{K}$ such that $\\tilde{B}\\subset \\mathrm{int}(\\tilde{U})$. (For\ninstance, we can initially take $\\tilde{U}=\\bar{U}\\times S^1$.)\n\nArguing by contradiction, assume that there exists a sequence of\niterations $k_i\\to\\infty$, a sequence of $k_i$-periodic almost complex\nstructures $J_i$ on $M$, compatible with $\\omega$ and\n$C^\\infty$-converging to $J$ uniformly on ${\\mathbb R}\\times U$, and a sequence\n$u_i\\colon\\Sigma_i\\to U$ of solutions of the Floer equation for $J_i$\nand $H^{{\\natural} k_i}$, satisfying the hypotheses of Theorem\n\\ref{thm:energy} and such that $E(u_i)\\to 0$.\n\nTo proceed, let us first make several simplifying assumptions. Namely,\nwithout loss of generality we can assume that the boundaries $\\partial \\tilde{U}$\nand $\\partial \\Sigma_i$ are smooth. Indeed, to this end we can shrink $\\tilde{U}$\nslightly and simultaneously make sure that $\\partial \\tilde{U}$ is transverse to\nthe maps $\\tilde{u}_i$. At this stage we do not need $\\tilde{U}$ to be a direct\nproduct.\n\nFurthermore, we can assume that $[0,\\infty)\\times S^1_{k_i}$ is the\nlargest half-cylinder in $\\Sigma_i$ mapped by $\\tilde{u}_i$ into $\\tilde{B}$,\ni.e., $\\tilde{u}_i\\big([0,\\infty)\\times S^1_{k_i}\\big)\\subset \\tilde{B}$ and\n$\\tilde{u}_i\\big(\\{0\\}\\times S^1_{k_i}\\big)$ touches $\\partial \\tilde{B}$ at at least\none point $\\tilde{u}_i(0,\\tau_i)$ with $0\\leq \\tau_i\\leq 1$. Here the first\nassertion readily follows since $H$ is independent of $s$, and hence\nthe Floer equation is translation invariant. As a consequence, we can\nchange $u_i$ by applying a translation in $s$ without affecting the\nenergy. To ensure that $0\\leq \\tau_i\\leq 1$, we apply an integer\ntranslation in $t$ to $u_i$ and $J_i$. Since the almost complex\nstructures $J_i$ $C^\\infty$-converge to $J$ uniformly in $t\\in{\\mathbb R}$, the\nsame is true for the translated almost complex structures. (This\nchanges the almost complex structures $J_i$ and the solutions $u_i$ by\ntranslation, but again does not affect the energy of $u_i$.)\n\nFinally, by passing if necessary to a subsequence, we may assume that\nthe sequence $\\tau_i$ converges.\n\nAs has been pointed out above, since $\\tilde{B}$ is an isolating\nneighborhood, there exists a constant $T_0>1$ (an escape time),\ndepending only on $\\tilde{B}$ and $H$, such that no integral curve of\n$\\tilde{\\varphi}^t$ passing through a point of $\\partial \\tilde{B}$ (or near $\\partial \\tilde{B}$) at\na moment $\\tau\\in [0,\\,1]$ can stay in $\\tilde{B}$ for all $t$ with\n$|t|T_0$ and $a>0$ and set\n$$\n\\Pi=[-a,\\,a]\\times [- T,\\, T]\\subset {\\mathbb C}.\n$$\nFrom now on we will focus on the restrictions $v_i:=u_i|_\\Pi$ and\n$\\tilde{v}_i:=\\tilde{u}_i|_\\Pi$. Let $S_i$ be the graph of $v_i$. Denoting by\n$\\tilde{U}_T$ the part of the lift of $\\tilde{U}$ to the covering\n$M\\times {\\mathbb R}\\to M\\times S^1$ lying over $[-T,\\,T]\\subset {\\mathbb R}$, we have\n$$\nS_i=\\Gamma_i\\cap P,\\textrm{ where } P=\\tilde{U}_T\\times [-a,\\,a]\\subset\nM\\times {\\mathbb C}.\n$$\nClearly, $\\partial S_i\\subset \\partial P$ and\n$$\n\\operatorname{Area}(S_i)\\leq\\operatorname{Area}(\\Pi)+E(u_i)<{\\mathit const},\n$$ \nwhere the constant on the right is independent of $i$. Let us now\nshrink $\\Pi$ and $\\tilde{U}$ slightly. To be more precise, set\n$$\n\\Pi'=[-a',\\,a']\\times [-T',\\,T']\\subset \\Pi,\n$$\nwhere $00\\}\\cap\\Pi'$.\nIndeed, $\\tilde{u}_i$ maps $\\Pi_+$, the $\\{s\\geq 0\\}$-part of $\\Pi$, into\n$\\tilde{B}\\subset \\textrm{int}(\\tilde{U}')$. Hence the projection of $S'_j$ to\n$\\Pi$ contains $\\Pi_+$ or, in other words, $\\Pi_+$ is in the domain of\n$v$. Furthermore, $D$ also contains the closure of $\\Pi'_+$ and, in\nparticular, the point $(0,\\tau)$. In fact,\n$(0,\\tau)\\in\\textrm{int}(D)$ since the distance from the points\n$\\tilde{v}_i(0,\\tau)\\in \\tilde{B}$ to $\\partial \\tilde{U}'$ stays bounded away from zero. Let\n$\\tilde{v}$ be the natural lift of $v$ to a map to $\\tilde{U}'$. Then\n$$\n\\tilde{v}(0,\\tau)=p:=\\lim \\tilde{v}_i(0,\\tau_i)\\in \\partial \\tilde{B}.\n$$\n\nSince $E(u_i)\\to 0$, we have $E(v)=0$. Thus $\\partial_s v(s,t)=0$\nidentically on $D$, and hence $v(s,t)$ is an integral curve\n$\\gamma(t)$ of the flow $\\tilde{\\varphi}^t$ on $M\\times S^1$. This integral\ncurve passes through the point $p\\in \\partial \\tilde{B}$ at the moment $\\tau$, and\n$\\gamma(t)\\in \\tilde{B}$ for all $t\\in [-T',\\,T']$, which is impossible due\nto our choice of $T_0$ and the fact that $T'>T_0$. This contradiction\ncompletes the proof of the theorem.\n\\end{proof}\n\n\\subsection{Proof of Theorem \\ref{thm:isolated}}\n\\label{sec:pf-isolated}\nWith the crossing energy lower bound established, we are now in a\nposition to prove Theorem \\ref{thm:isolated}. In fact, we prove a\nslightly more precise result. To state it, recall that the normalized\naugmented action of a $k$-periodic orbit $y^k$ is simply\n$\\tilde{\\mathcal A}_{H^{{\\natural} k}}(y^k)\/k$, where the augmented action is defined by\n\\eqref{eq:aug-action}. Thus all iterations of an orbit have the same\nnormalized augmented action.\n\n\n\\begin{Theorem}\n\\label{thm:isolated2}\nLet $M^{2n}$ be a strictly monotone symplectic manifold.\nAssume that $N\\geq n+1$ and \n\\begin{equation}\n\\label{eq:hom-relation}\n\\alpha *\\beta = \\mathrm{q} [M]\n\\end{equation}\nin $\\operatorname{HQ}_*(M)$ for some homology classes $\\alpha\\in \\operatorname{H}_*(M)$ and\n$\\beta\\in \\operatorname{H}_*(M)$ with $|\\alpha|c_\\infty$,\n and pick $\\epsilon=\\lambda$ and\n $\\left< c_1(TM),A_0\\right> = 2N$. The action of $\\bar{y}$ is either in\n $(-\\epsilon,\\,\\epsilon)$ or in $(-\\epsilon,\\,\\epsilon)-\\lambda$, and hence either\n $E(u)<\\epsilon$ or $E(v)<\\epsilon$, which is impossibly by Theorem\n \\ref{thm:energy} due to the assumption that $\\epsilon0$.\n\nFix a one-periodic in time almost complex structure $J^0$. Let $U$ be\nan isolating neighborhood of $x$ such that, in addition, no periodic\norbit of $\\varphi_H$ with normalized augmented action in $\\tilde{I}$ other\nthan $x$ enters $U$. By Theorem \\ref{thm:energy} applied to\n$K=\\{x(0)\\}$, there exists a constant $c_\\infty>0$ such that, for all\n$k\\in{\\mathbb N}$, every non-trivial $k$-periodic solution of the Floer\nequation for the pair $(H,J)$ asymptotic to $x^k$ as $s\\to\\pm\\infty$\nhas energy greater than~$c_\\infty$, where $J$ is $k$-periodic and\nsufficiently $C^\\infty$-close to $J^0$. (In what follows, such an\nalmost complex structure $J$ is chosen to be generic and is suppressed\nin the notaion.)\n\nLet $c>0$ be outside the union of the action spectra\n${\\mathcal S}\\big(H^{{\\natural} k}\\big)$, $k\\in {\\mathbb N}$, and set $I=(-c,\\,c)$. Next we\npick a large constant $C>0$ and a small constant $\\epsilon>0$ depending on\n$c$, to be specified later. As is easy to show using the Kronecker\ntheorem, there exists an arbitrarily large $k\\in{\\mathbb N}$ such that for all\n$i$\n\\begin{equation}\n\\label{eq:T1}\n\\| k a_i\\|_{\\lambda}<\\epsilon\n\\end{equation}\nand \n\\begin{equation}\n\\label{eq:T2}\n\\textrm{either }\\tilde{a}_i=\\tilde{a}_0\\textrm{ or }k|\\tilde{a}_i-\\tilde{a}_0|>C.\n\\end{equation}\nHere $\\|a\\|_{b}\\in [0,\\, b\/2]$ stands for the distance from\n$a\\in S^1_{b}={\\mathbb R}\/b{\\mathbb Z}$ to $0$. We will also require that\n\\begin{equation}\n\\label{eq:eta-C}\nk>C\/\\eta.\n\\end{equation}\n\n\n\\begin{Lemma}\n\\label{lemma:homology}\nAssume that $C>0$ is sufficiently large and $\\epsilon>0$ is sufficiently\nsmall and that $k\\in{\\mathbb N}$ satisfies the requirements \\eqref{eq:T1},\n\\eqref{eq:T2} and \\eqref{eq:eta-C}, and let $F$ be a $k$-periodic,\nnon-degenerate $C^2$-small perturbation of $H^{{\\natural} k}$. Then, in the\nnotation from Section \\ref{sec:LFH},\n$\\operatorname{CF}_*\\big(F,\\bar{x}^k\\#(\\ell A_0)\\big)$ is a direct summand in\n$\\operatorname{CF}_*^I(F)$ whenever\n${\\mathcal A}_{H^{{\\natural} k}}\\big(\\bar{x}^k\\#(\\ell A_0)\\big)\\in I$. In particular,\n$$\n\\bigoplus_{|\\ell|0$ and $\\epsilon>0$ are \\eqref{eq:C},\n\\eqref{eq:eps1} and \\eqref{eq:eps2} below.\n\n\\begin{proof}\n Set $\\bar{x}_k=\\bar{x}^k\\# (\\ell A_0)$, where $|\\ell| 2n+1.\n\\end{equation}\nOnce this is established, Lemma \\ref{lemma:homology} will follow from\nLemma \\ref{lemma:spec}.\n\nAssume first that $y_k$ is one of the orbits $x_i^k$, e.g.,\n$y_k=x_1^k$. Then, by \\eqref{eq:T2}, there are two cases to consider.\n\nIf $k|\\tilde{a}_1-\\tilde{a}_0|>C$, we have\n$$\n\\big|{\\mathcal A}_{H^{{\\natural} k}}(\\bar{x}_k)-{\\mathcal A}_{H^{{\\natural} k}}(\\bar{y}_k)\\big|\n+\\frac{\\lambda}{2N}\\big|\\hmu(\\bar{x}_k)-\\hmu(\\bar{y}_k)\\big|> C.\n$$\nHere the first term is bounded from above by $2c=|I|$ since both\norbits have action in $I$. Hence, when $C$ is sufficiently large, we\ninfer that\n\\begin{equation}\n\\label{eq:2(n+1)}\n\\big|\\hmu(\\bar{x}_k)-\\hmu(\\bar{y}_k)\\big|> 2(n+1)\n\\end{equation}\nand \\eqref{eq:diff-index} follows. Note that here it suffices to take\n\\begin{equation}\n\\label{eq:C}\nC> \\frac{4N}{\\lambda}(c+n+1).\n\\end{equation}\nIn fact, we can make the right-hand side in \\eqref{eq:diff-index}\narbitrarily large by taking a sufficiently large $C$.\n\nThe second case is when $\\tilde{a}_1=\\tilde{a}_0$. Then\n$$\n\\big|{\\mathcal A}_{H^{{\\natural} k}}(\\bar{x}_k)-{\\mathcal A}_{H^{{\\natural} k}}(\\bar{y}_k)\\big|\n=\\frac{\\lambda}{2N}\\big|\\hmu(\\bar{x}_k)-\\hmu(\\bar{y}_k)\\big|.\n$$\nAssume that \n\\begin{equation}\n\\label{eq:eps1}\n\\epsilon\\epsilon.\n$$\nThis, by the first inequality of \\eqref{eq:T1}, implies that\n\n$$\n\\big|{\\mathcal A}_{H^{{\\natural} k}}(\\bar{x}_k)-{\\mathcal A}_{H^{{\\natural} k}}(\\bar{y}_k)\\big|>\\lambda-\\epsilon,\n$$\nand hence\n$$\n\\big|\\hmu(\\bar{x}_k)-\\hmu(\\bar{y}_k)\\big|>{2N}\\frac{\\lambda-\\epsilon}{\\lambda}.\n$$\nRecall that $N\\geq n+1$. Therefore, \\eqref{eq:diff-index} holds when\n$\\epsilon>0$ is sufficiently small, e.g., such that\n\\begin{equation}\n\\label{eq:eps2}\n\\epsilon<\\lambda\/2(n+1).\n\\end{equation}\n\n\nFinally, if $y_k$ is not one of the orbits $x_i^k$, i.e., its\nnormalized augmented action is not in $\\tilde{I}$, we have\n$$\n\\big|\\tilde{\\mathcal A}_{H^{{\\natural} k}}(y_k)-\\tilde{\\mathcal A}_{H^{{\\natural} k}}(x^k)\\big|>k\\eta>C\n$$\nby \\eqref{eq:eta-C}. Then, exactly as in the first case, we have\n\\eqref{eq:2(n+1)} and hence \\eqref{eq:diff-index}.\n\\end{proof}\n\nNote also that the last argument shows that for any $k$-periodic orbit\n$\\bar{y}_k$ with action in $I$, but normalized augmented action outside\n$\\tilde{I}$, we have\n\\begin{equation}\n\\label{eq:mu-O}\n\\big|\\hmu(\\bar{x}^k)-\\hmu(\\bar{y}_k)\\big|\\geq O(k)\n\\end{equation}\nwhere the lower bound on the right is independent of the orbit.\n\nThroughout the rest of the proof, we assume that $c>\\lambda$, and\nhence both $\\bar{x}^k$ and $\\bar{x}^k\\# A_0$ have action in $I$; that $C>0$ is\nsufficiently large and $\\epsilon>0$ is sufficiently small and, in\nparticular, \\eqref{eq:eps1} holds; and $k$ satisfying \\eqref{eq:T1}\nand \\eqref{eq:T2} is also sufficiently large. As a consequence, the\nrequirements of Lemma \\ref{lemma:homology} are met.\n\nLet $F$ be $C^2$-close to $H^{{\\natural} k}$. Then the orbits which $x^k$\nsplits into under the perturbation $F$ lie in the isolating\nneighborhood $U$ of $x$.\n\nPick a non-zero class \n$$\n\\gamma\\in\\operatorname{HF}_*(\\bar{x}^k)\\subset \\operatorname{HF}_*^I\\big(H^{{\\natural} k}\\big).\n$$\nThen the class \n$$\n\\mathrm{q} \\gamma=\\Phi_\\alpha\\big(\\Phi_\\beta(\\gamma)\\big)\\in \\operatorname{HF}_*(\\bar{x}^k\\#\nA_0)\\subset \\operatorname{HF}_*\\big(H^{{\\natural} k}\\big)\n$$\nis also non-zero. (Here we use \\eqref{eq:qh-fh-action} and\n\\eqref{eq:hom-relation}.) Moreover, by Lemma \\ref{lemma:trivial}, the\nintermediate class $\\Phi_\\beta(\\gamma)$ is not in $\\operatorname{HF}_*(\\bar{x}^k)$ or\n$\\operatorname{HF}_*(\\bar{x}_k\\# A_0)$.\n\nNow, it is a formal algebraic consequence of Lemmas\n\\ref{lemma:homology} and \\ref{lemma:trivial} that there exists a\ncapped $k$-periodic orbit $\\bar{z}$ of $F$, one of the orbits that $\\bar{x}^k$\nsplits into, connected by a solution $u_F$ of the Floer equation to a\ncapped $k$-periodic orbit $\\bar{z}_*$, which, in turn, is connected to a\nclosed orbit $\\bar{z}'\\# A_0$ by a solution $v_F$, where again $\\bar{z}'$ is\none of the orbits which $\\bar{x}^k$ splits into. Furthermore, $z_*$ is not\namong the orbits arising from $x$, and $z_*$ does not enter the\nneighborhood $U$ of $x$.\n\nPassing to the limit as $F\\to H^{{\\natural} k}$ for a suitably chosen\nsequence of perturbations and using the target-local compactness\ntheorem from \\cite{Fi} exactly as in the proof of Lemma\n\\ref{lemma:spec}, it is easy to show that $\\bar{x}^k$ is connected by a\nsolution of the Floer equation for $H^{{\\natural} k}$ to some $k$-periodic\norbit $\\bar{y}_+$ lying entirely outside $U$ and such that, by Theorem\n\\ref{thm:energy},\n$$\n{\\mathcal A}_{H^{{\\natural} k}}(\\bar{x}^k)-c_\\infty>{\\mathcal A}_{H^{{\\natural} k}}(\\bar{y}_+)\n$$\nBy passing if necessary to a subsequence of perturbations\n$F\\to H^{{\\natural} k}$, we can also assume that the capped orbits $\\bar{z}_*$\nconverge to a capped $k$-periodic orbit $\\bar{y}_*$ of $H$.\n\nWe claim that, when $k$ is sufficiently large, the orbit $y_*$ is\nnecessarily one of the orbits $x_i^k$. (In other words, $z_*$ is among\nthe orbits which $y_*=x_i^k$ splits into under the perturbation $F$.)\nIndeed, this follows from \\eqref{eq:mu-O} and the fact that $\\bar{y}_*$\nhas action in $I$; for $\\mu(\\bar{z}_*)=\\mu(\\bar{z})-|\\beta|$ and hence\n$$\n\\big|\\hmu(\\bar{y}_*)-\\hmu(\\bar{x}^k)\\big|\\leq 2n+|\\beta|.\n$$\n\nThe orbit $\\bar{y}_*$ might be different from $\\bar{y}_+$, but\n$$\n{\\mathcal A}_{H^{{\\natural} k}}(\\bar{y}_+)\\geq {\\mathcal A}_{H^{{\\natural} k}}(\\bar{y}_*)\\geq {\\mathcal A}_{H^{{\\natural}\n k}}(\\bar{x}^k\\# A_0)= {\\mathcal A}_{H^{{\\natural} k}}(\\bar{x}^k)-\\lambda.\n$$\n\nIf $c_\\infty>\\lambda$, we have arrived at a contradiction and the\nproof is finished. Thus we can assume that $c_\\infty\\leq \\lambda$.\nThen, by \\eqref{eq:T1}, we have\n$$\n\\big|{\\mathcal A}_{H^{{\\natural} k}}(\\bar{y}_*)-{\\mathcal A}_{H^{{\\natural} k}}(\\bar{x}^k\\# A_0)\\big|<\\epsilon.\n$$\n\nHowever, applying the same argument to the Floer trajectories $v_F$,\nwe find a $k$-periodic orbit $\\bar{y}_-$ of $H^{{\\natural} k}$ which does not\nenter $U$, is connected to $\\bar{x}^k\\# A_0$ by a solution of the Floer\nequation, and such that, again by Theorem \\ref{thm:energy},\n$$\n{\\mathcal A}_{H^{{\\natural} k}}(\\bar{y}_-)>{\\mathcal A}_{H^{{\\natural} k}}(\\bar{x}^k\\# A_0)+c_\\infty \n$$\nand\n$$\n{\\mathcal A}_{H^{{\\natural} k}}(\\bar{x}^k\\# A_0)+\\epsilon>\n{\\mathcal A}_{H^{{\\natural} k}}(\\bar{y}_*)\\geq{\\mathcal A}_{H^{{\\natural} k}}(\\bar{y}_-)\n$$\nwhich is impossible by \\eqref{eq:eps1}. \n\nThis contradiction concludes the proof of the theorem.\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{Sec1}\n\t\\par\t{\\em Graeffe Iteration} was \\oldtext{certainly}\n\t\t\\newtext{one of}{A6\\S 2(2)} the most prestigious \n\t\tXIX century algorithm for finding roots of polynomials.\n\t\tAt that time, computations were performed by hand, by people payed \n\t\tspecifically to perform those computations. They\n\t\twere called {\\em calculateurs}~\\cite{OSTROWSKII} or\n {\\em computers}~\\cite{BABBAGE}.\n\t\\par Let $f$ be a univariate polynomial, of degree $d$. Its\n\t\tGraeffe iterate is defined as:\n\t\\begin{equation}\n\t\tGf(x) = (-1)^d f(\\sqrt{x}) f(-\\sqrt{x}) \n\t\\end{equation}\t\n\t\\par \tThis defines a many-to-one mapping in the space\n\t\tof all degree-$d$ polynomials (real or complex,\n\t\tas wish). The effect of this mapping is to\n\t\tsquare each root of $f$.\n\t\\par \tAfter a few Graeffe iterations, the roots of $G^k f$ have\n\t\t(hopefully) incommensurate moduli. This is not true for\n\t\t\\newtext{complex-}{A5(1)}conjugate roots, \n\t\twhich can be worked out in a different way.\n\t\\par \tAssume, for simplicity, that $f$ is a complex polynomial,\n\t\twith no two roots of the same modulus. Then, $G^k f$ can\n\t\tbe written\n\t\\begin{equation}\n\t\tG^k f(x) = \\sum_{i=0}^d a^{(k)}_i x^i\n\t\\end{equation}\n\t\twhere $a^{(k)}_i$ is given by the \\oldtext{$d-i$-symmetric}\n\t\t\\newtext{$(d-i)$-th symmetric}{A7(1)} function\n\t\tof the roots of $G^k f$. Therefore, $a^{(k)}_i$ is dominated\n\t\tby:\n\t\\begin{equation}\n\t\t(-1)^{d-i} {\\zeta_1}^{2^k} {\\zeta_2}^{2^k} \n\t\t\\dots {\\zeta_{d-i}}^{2^k}\n\t\\end{equation}\n\t\twhere $\\zeta_1$, \\dots, $\\zeta_d$ are the roots of $f$\n\t\tordered with decreasing modulus. (A more rigorous \n\t\tstatement of this will appear in Section~\n\t\t\\ref{Sec4})\n\t\t\\par\n\t\tTherefore, $-a^{(k)}_{i-1} \/ a^{(k)}_{i}$ is a good approximation\n\t\tfor ${\\zeta_{d-i+1}}^{2^k}$.\n\t\t\\par\n\t\t\\oldtext{Hence, we can easily obtain $|\\zeta_i|$ for all $i$.\n\t\tThere are many classical algorithms to recover the\n\t\tactual value of $\\zeta_i$.}\n\t\t\\newtext{Hence it is computationally easy to approximate\n\t\t$|\\zeta_i|$ for all $i$. Although we also obtain\n\t\t$\\arg \\zeta_i \\mod 2^{1-k} \\pi$, we will discard this\n\t\tinformation in this paper to avoid additional complications.\n\t\tThere are many classical algorithms\n\t\tto recover the actual value of $\\zeta_i$. \n\t\t}{A5(2-3)}\n\t\tSee Pan~\\cite{PAN} for\n\t\ta discussion. \n\t\t\\footnote{Added in the revised version: We deal\n\t\twith this issue in ~\\cite{TANGRA}.}\n\t\\medskip\n\t\\par \t\n\t\\oldtext{In this note, we claim that Graeffe iteration\n\t\t(after a suitable change of coordinates) is\n\t\ta globally convergent algorithm with probability $1$\n\t\t(Theorem~1 below).}\n\t\\newtext{In this note, we apply a suitable non-uniform\n\t change of coordinates (renormalization)\n\t\tto the Graeffe iteration \n\t\toperator, to make it `convergent' with probability $1$.\n\\par\n\t\tThe algorithm obtained by this change of coordinates will\n\t\tbe called {\\em Renormalized Graeffe Iteration}. \n\t\tWe use the following systems of coordinates for\n\t\teach iterate $G^{k} f$ of the Graeffe method applied to\n\t\t$f$:\n\t\n\t\tThe coefficients of $G^k(f)$ and all related intermediate\n\t\tcomputations will be represented in scaled polar coordinates,\n\t\twhere a complex number $w$ is represented by `magnitude'\n\t\t$2^{-k} \\log_2 |w|$ and `argument' $\\arg w \\in [-\\pi, \\pi]$.\n\t\tCalculations will always be performed `in coordinates'. \n\t\tThe `magnitude' variables of $G^k f$ `in coordinates' will\n\t\tconverge with probability~1 (Theorem~\\ref{th1} below).\n\\par\n\t\tThe precise construction of the Renormalized Graeffe Operator\n\t\tis postponed to Section~\\ref{Sec2}.\n\t\\medskip\n}{A2\\S -1, A5(4-5) and B\\S 3}\n\t\\par We also claim that the Renormalized Graeffe algorithm \n\t\tcompares well with\n\t\tavailable numerical software or theoretical algorithms.\n\\newtext{\\par\n However, in this paper, we are considering a modified problem:\n\tour algorithm is designed to find the absolute values of the\n\troots, not the actual roots. Therefore we will compare its\n\tcomplexity to the complexity of finding the absolute value\n\tof the roots by other existing algorithms. \n\t\\par\n\tIn \\cite{TANGRA}, we explain how to\n\tmodify this algorithm to obtain the actual roots, without\n\tendangering the complexity estimates.\n\t\\par\n\t} {A2(1)}\n\t\tThe algorithm presented here has arithmetic complexity\n\t\t $ {\\mathcal{O}} (d^{2})$ for each iteration, \n\t\tand memory size $ {\\mathcal{O}} (d)$, where $d$ is the degree\n\t\tof the polynomial.\n\t\\par\n\t\t\\oldtext{Complexity}\\newtext{The number of iterations}{A2(2)}\n will be bounded also in terms of a probability of\n\t\tfailure (Theorem~\\ref{th2} below). \n\t\t\\oldtext{This will be possible due to}\n\t\t\\newtext{The authors personally believe that this is only\n\t\tpossible due to}{A5(6)}\n\t\tthe clean mathematical structure of Graeffe iteration.\n\t\tOur bound improves previous probabilistic bounds \n\t\t\\newtext{(in the sense of probability of success)}{A7(2)}\n\t\ton the\n\t\tcomplexity of solving polynomials. \n\t\t(See Renegar~ \\cite{RENEGAR} and Shub-Smale~\\cite{BEZIV}).\n\t\\par\n\t\tAlso, our algorithm compares well with practical software,\n\t\tlike for instance the algorithm\n\t\tin Matlab (running time $ {\\mathcal{O}} (d^{3})$ and memory $ {\\mathcal{O}} (d^{2})$).\n\t\t\\newtext{Instead, Renormalized Graeffe Iteration\n seems to run in time $ {\\mathcal{O}} (d^2)$. It also}\n\t\t{A2(2)}\n\t\t\\oldtext{Renormalized Graeffe iteration} seems much more stable,\n\t\tfor \\newtext{most (in a probabilistic sense)}{A3\\S 1}\n\t\t\\oldtext{generic}\n\t\tcomplex polynomials of degree, say, 1000. (See\n\t\tSection~\\ref{Sec6} for a discussion of preliminary\n experimental results).\n\t\\par\n\\newtext{In order to compare with deterministic algorithms one should\n bear in mind that the algorithm presented here is probabilistic\n\t and may be quite slow on a set of non-zero measure. Also, \n\t the complexity of deterministic algorithms such as\n\t \\cite{NEFF-REIF,PAN,SCHONHAGE0,SCHONHAGE}\n\t can be given in terms of number of arithmetic operations or\n\t in number of bit operations.\n\t We believe \n\t that estimates of the order of $ {\\mathcal{O}} (d^{1+\\alpha})$, $\\alpha >0$\n\t in the number of operations are made at the expense of a\n\t large increase in the working precision, of the order of\n\t $ {\\mathcal{O}} (d^{2+\\alpha})$ bits. However, this is not a lower bound\n\t and it may as well happen that those algorithms can work\n\t with substantially smaller precision on a large set of\n\t inputs. \n\\par\n\tIn comparison, our Renormalized Graeffe Iteration was designed\n\tfor floating-point arithmetic (our results assume an arbitrary,\n\tbut fixed floating-point precision).}{A2(3--5)}\n\\oldtext{\n\t\tIn order to compare it with theoretical algorithms\n\t\tof arithmetic complexity \n\t\t$ {\\mathcal{O}} (d^{1+\\alpha})$, $\\alpha >0$, \n\t\t(see \\cite{NEFF-REIF,PAN,SCHONHAGE0,SCHONHAGE}) \n\t\tone has to keep in mind that the bit complexity of\n\t\tthose fast algorithms is (crudely) $ {\\mathcal{O}} (d^{3+\\alpha})$, \n\t\tsince they \n\t\trequire at least $ {\\mathcal{O}} (d^{2+\\alpha})$ bits of precision \n\t\t(We are omitting here several factors). In comparison, our\n\t\tRenormalized Graeffe Iteration is, by construction, suitable \n\t\tto usual floating-point arithmetic. \n}\n\t\tHowever, the details of the rounding-off\n\t\tanalysis of renormalized Graeffe iteration will not be \n\t\tdiscussed in this paper. \n\t\\par \tInstead, some preliminary \n\t\tnumerical results are presented in Section~\\ref{Sec6}.\n\t\tThey support the \\newtext{empirical}{A5(7)}\n\t\tfact that \\oldtext{generic}\n\t\t\\newtext{typical random}{A3\\S 1} degree 1000 polynomials\n\t\tcan be solved within a precision of \\oldtext{64}\n\t\t\\newtext{64}{A5(7)} bits of mantissa \\newtext{(IEEE 854 \n\t\tlong double)}{A5(7)}. We\n\t\texpect a factor of a \\oldtext{polylog of $d$}\n\t\t\\newtext{polynomial in $\\log d$}{A5(8)} \n\t\tbits of mantissa to be\n\t\t\\oldtext{necessary} \\newtext{sufficient}{A5(8)} \n\t\tin general, with probability 1. (This is a conjecture).\n\\newtext{\n \\par The result in Theorem~\\ref{theo1} holds for ``reasonable''\n probability distributions on the space of polynomials. By\n reasonable, we mean all probabilities with\n\t\tbounded Radon-Nikodym derivative with respect to\n Lebesgue probability in the projectivization of\n\t\tcoefficient space.\n}{Changed.}\n\n \n\n\\begin{theorem} \\label{th1}\n There is \\oldtext{(we construct)}\n\t\ta renormalization of the Graeffe\n\t\titeration, such that if $f$ is a \\oldtext{generic}\n\t\tdegree-$d$ polynomial\n\t\t(in a measure theoretical sense) then \n\t\t\\newtext{with probability 1}{A3\\S 1}\n\t\tthis renormalized Graeffe\n\t\titeration produces $d+1$ sequences, each one converging to\n\t\tsome $h_i$, s.t. $\\log |\\zeta_{i}| = h_i - h_{i+1}$, \n and \\oldtext{$\\zeta_{i}$ is a root}\n\t \\newtext{$\\zeta_1$, $\\cdots$, $\\zeta_d$ are roots}{A5(10)}\t\n\t\tof $f$. \n\t\tMoreover, each iteration can be performed in \n\t\t$ {\\mathcal{O}} (d^2)$ arithmetical operations and all iterations can be \n\t\tperformed with memory $ {\\mathcal{O}} (d)$.\n\\label{theo1}\n\\end{theorem}\n\\par\n\\newtext{ This theorem is constructive, in the sense that an explicit \n construction of the renormalized Graeffe\n iteration will be given.}{A5(9)}\n\t\\par\tIn Section~\\ref{Sec2}, we discuss the precise meaning of \n\t\trenormalization in our context. \n\t\tIts main consequence, will be to produce an\n\t\talgorithm operating on a bounded set of numbers. This solves\n\t\tthe main stability problem of classical Graeffe iteration that\n\t\tprevented it from finding all roots at once. See for example\n\t\tHenrici's comments on \\oldtext{Graeffe~\\cite{HENRICI}}\n\t\t\\newtext{FFT-based Graeffe iteration~\\cite{HENRICI}\n\t\t(Vol~III, last paragraph of p. 69)}{A5(11)}. \n\\oldtext{\n\\par The definition of renormalization will outlaw, for instance,\n\t\tthe following fast but unstable algorithm: Perform\n\t\t$k$ steps of Graeffe using FFT-based polynomial multiplication,\n where $k$ is given in Theorem~\\ref{th2}\n\t\t}\n\\newtext{\n The definition of renormalization will outlaw FFT-based\n Graeffe iteration. Indeed, although FFT is known to be\n stable with respect to vector norms, it is not component-wise\n stable with respect to the relative error. This means that\n some of the coefficients of the $k$-th Graeffe iterate may have\n a large relative error, and hence some of the roots will be\n extremely inaccurate. This may be disastrous if one wants to \n retrieve all the roots at the same time.}{B\\S 2(1)}\n \n\\medskip\n\\par\n\n\n\n\n\\newtext{\n\\begin{theorem}\\label{th2}\nLet $f$ be a random complex polynomial of degree $d$. Let $b \\ge 1 + \\log_2 d$.\nThen, with probability $1-\\delta$, $k$ steps of the Renormalized\nGraeffe Iteration will approximate the $\\log |\\zeta_i|$'s with relative\nprecision $2^{-b}$, where\n\\[\nk \\ge c_1 + c_2 \\log_2 b - c_3 \\log_2 \\delta\n\\]\nand $c_2$, $c_3$ are universal constants. The constant $c_1$\ndepends on the choice of the probability distribution, and on\n$d$.\n\\end{theorem}\n\n\n\nWhenever speaking of random polynomials, we like to \nconsider the normally invariant probability density introduced by\nKostlan~\\cite{KOSTLAN} (See also Section~\\ref{Sec5}). \nHowever, the above mentioned result is true for any reasonable \nprobability distribution.\n\nThe experimental results in figure~\\ref{results4} support the conjecture that,\nunder Kostlan's probability distribution, we can fix\n\\[\nc_1 = c_4 \\log_2 d\n\\]\nwhere $c_4 \\approx 2$ is a universal constant.\n\n}{}\n\\par\n\\newtext{}{LARGE SECTION DELETED}\n\\medskip\n\\par\n We will briefly discuss the real case, and how to deal with\n \\newtext{complex-}{A5(1)}conjugate \n roots or roots with same modulus in Section ~\\ref{Sec4}.\n\n\n\n\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\\medskip\n\t\\par\n\n\t{\\bf Historical remarks}. The Graeffe iteration was \n\t\tdeveloped independently by Dandelin (1826), by\n\t\tGraeffe (1837) and by Lobachevsky (1834). We call\n\t\tit Graeffe iteration to conform with most of\n\t\tthe literature. See Householder~\\cite{HOUSEHOLDER}\n\t\tfor early references and priority questions.\n\t\tSee Dedieu~\\cite{DEDIEU} for an application of Graeffe\n\t\t\\newtext{'s}{A7(4)}\n\t\talgorithm. \n\t\\par \tImportant theoretical results were obtained by\n\t\tOstrowskii~\\cite{OSTROWSKII} in 1940. Also, by that\n\t\ttime, numerical analysis books mentioned Graeffe\n\t\titeration as the preferred algorithm for \n zero-finding (see e.g. Uspensky ~\\cite{USPENSKII}\n\t\t\\newtext{Page 318}{A6\\S 2(1)}.\n\t\tFor another early \\oldtext{computational}\n\t\t\\newtext{computer implemented}{A7(5)} algorithm, see \n\t\tBareiss~\\cite{BAREISS1,BAREISS2} and also Blish and\n\t\tCurry~\\cite{BC}).\n\t\\par \tWith the advent of digital computing, the practical\n\t\tuse of Graeffe iteration seems to have been forgotten.\n\n\t\tMost popular zero-finding algorithms seem to be based\n\t\tnow on QR iteration (Matlab) or in a several steps,\n\t\troot-finding\n\t\tplus deflation scheme. (e.g. Jenkins and Traub \n\t\t\\cite{JENKINS-TRAUB}). \\oldtext{However} \n\t\t\\newtext{See, however,}{} Cardinal~\\cite{CARDINAL},\n\t\tEdelman and Murakami~\\cite{EDELMAN-MURAKAMI},\n\t\tEmiris, Galligo and Lombardi ~\\cite{EGH},\n\t\tand Toh and Trefethen~\\cite{TREFETHEN-TOH}. \n\n\t\\par \tIn a more theoretical perspective, Graeffe iteration is\n\t\tconsidered as a sort of pre-conditioning for polynomial\n\t\tsplitting. Splitting a polynomial means factorizing it\n\t\tinto one factor with \\oldtext{very}\\newtext{}{A6\\S 2(1)}\n\t\tlarge roots, and another with\n\t\t\\oldtext{very} small roots. \n\t\tSplitting is used to obtain extremely\n\t\tfast theoretical algorithms (see Sch\\\"onhage~\\cite{SCHONHAGE0,\n\t\tSCHONHAGE}, \\newtext{Kirrinnis~\\cite{KIRRINIS},\n\t\tNeff and Reif~\\cite{NEFF-REIF},}{A7(6)}\n\t\tBini and Pan~\\cite{BP2},\n\t\tMourrain and Pan~\\cite{MP},\n\t\tPan~\\cite{PAN2,PAN3,PAN}, Pan et \\oldtext{alli}\n\t\t\\newtext{al.}{A7(7)}~\\cite{PANETALLI}, \n\t\tMalajovich and Zubelli\n\t\t~\\cite{SPLITTING}). The main practical difficulty for those \n algorithms seems to be the large precision required by \n\t\tGraeffe iteration. Also, those algorithms are quite close\n\t\tto known lower bounds on \\oldtext{arithmetic}\n\t\t\\newtext{topological}{A5(15)} complexity \n (See Vassiliev~\\cite{VASSILIEV}). For a \n\t\trelated lower bound see Novak and Wo\\'zniakowski~\\cite{NW}. \n\t\\par \tAn important paper by Grau in 1963~\\cite{GRAU}\n\t\tlaid some of the bases for a version of Graeffe iteration\n\t\tadapted to digital computers. He identified the problem\n\t\tof the increasing {\\em numerical range}. During Graeffe\n\t\titeration, some of the coefficients can become so large\n\t\tthat the floating point system cannot accommodate them\n\t\tanymore.\n\t\\par\tWhile most of\n\t\tthe literature suggests to find one root at a time and\n\t\tthen use deflation, disregarding some stability problems\n\t\t(See e.g. Henrici~\\cite{HENRICI}), Grau proposed a \n\t\tglobally convergent algorithm. Grau's algorithm would\n\t\tinvolve only bounded quantities. \n\t\\par\tAs far as we know, that\n\t\tpaper was completely forgotten. The algorithm suggested\n\t\tby Grau has complexity $ {\\mathcal{O}} (d^2)$ and memory usage of\n\t\t$ {\\mathcal{O}} (d^2)$. It\n\t\tmay be considered as the precursor of the one we \n\t\tshall introduce below.\n\n\n\n\\section{Iterative Algorithms and Renormalization}\n\\label{Sec2}\n\n\n\n\t\\par\tIn this paper, we will produce a version of Graeffe\n\t\titeration that has {\\em bounded} numerical range, for\n\t\tmost input polynomials. The crucial concept in the\n\t\tconstruction of this algorithm is the idea of\n\t\t{\\em renormalization.}\n \t\\medskip\n\t\\par \tRenormalization is a tool used in understanding the qualitative\n\t\tbehavior of iterative phenomena that range over\n\t\tdifferent scales. A \\oldtext{very}\\newtext{}{A6\\S 2(1)}\n\t\trich theory of renormalization\n\t\texists for one-dimensional dynamical systems.\n\t\tSee Feigenbaum~\\cite{FEIGENBAUM}, McMullen~\\cite{MCMULLEN},\n\t\tand De Melo-Strien~\\cite{MELO_STRIEN}. As for the \n\t\tmulti-dimensional case see Palis-Takens~\\cite{G_D}.\n\t\\par \t\n\t\\oldtext{ Although we do not want to propose a general theory of\n\t\trenormalization of iterative algorithms, we may \n\t\tillustrate what renormalization means by a well-known\n\t\texample and then give some tentative definitions.\n\t\t}\n \\newtext{We shall illustrate what we mean by renormalization in\n\t\tthe setting of iterative algorithms by a well-known\n\t\texample and then proceed with the definitions}{A5(16)}\n\t\\par\n\\begin{example}\n\t\tLet $f: \\mathbb C \\rightarrow \\mathbb C$ \n\t\tbe a smooth function. \\oldtext{$\\mathbb C \\rightarrow \\mathbb C$}\n\t\tNewton iteration associates to a point $x$ the point\n\t\t$N(x) = x - f(x) \/ f'(x)$. The sequence of Newton iterates \n\t\tconverges to a zero of $f$, provided that $x$ is picked\n\t\tclose enough to a non-degenerate zero. \n\t\\par\tWe assume now that $f(x) = 0 + f'(0) x + \\text{h.o.t.}$\n\t\t\\newtext{}{B\\S 4(3)},\n\t\tand that $f'(0) \\ne 0$. We will consider now the \n renormalization of the {\\em iterative algorithm} (or\n mapping) $x \\mapsto N(x)$. Although we are dealing here\n\t with a simple example (the answer is always $0$),\n\t its renormalization will have some of the main features\n of the renormalized Graeffe iteration, yet to be defined.\n\t\\par \t\\oldtext{Therefore, we should} \n\t\t\\newtext{The basic idea, therefore, is to}{for clarity} \n\t\tlook at Newton iteration with a variable\n {\\em microscope} of variable lens. \\oldtext{In our example we are\n\t\tperforming Newton iteration $N$ of a univariate\n\t\tpolynomial $f$ with a root located\n\t\tat the origin.}\\newtext{}{A5(17)} We look at the mapping:\n\t\\begin{equation}\n\t\tN_{\\epsilon} = h_{\\epsilon ^{-2}} \n\t\t\t\t\\circ N\n\t\t\t\t\\circ h_{\\epsilon }\n\t\\end{equation} \n\t\twhere $h_{\\epsilon}$ means the homothety of ratio $\\epsilon$.\n \\par \tWhen $\\epsilon$ tends to zero, $N_{\\epsilon}$ tends \n\t\tto the map: $y \\mapsto \\gamma y^2$, defined from the disk \n\t\t$D(|\\gamma^{-1}|)$ of\n\t\tradius $|\\gamma^{-1}|$ onto itself.\n\t\tHere, $\\gamma$ is a parameter\n\t\tchosen equal to \\oldtext{$f^{(2)}(0) \/ f'(0)$}\n\t\t\\newtext{$f^{(2)}(0) \/ 2 f'(0)$}{B\\S 4(4)}.\t\n\t\t(Proof of this fact: Taylor's Theorem.\n\t\tThe choice of the radius $|\\gamma|^{-1}$ makes\n\t\tthe limiting map surjective).\n\\end{example}\n\t\\par \tThis process (which we call renormalization) gives us\n\t\tqualitative information on the dynamics of Newton \n\t\titeration near a root. We can summarize this information\n\t\tin an {\\em eventually commutative} diagram (commutative in the \n\t\tlimit):\n\\oldtext{\n \\begin{equation}\n\t\\begin{CD}\n D(|\\gamma^{-1}|\\epsilon) @>\\left({h_{\\epsilon}}\\right)^{-1}\n\t\t>> D(|\\gamma^{-1}|) \\\\\n\t\t@V{N}VV @VV{y \\mapsto \\gamma y^2}V \\\\\n D(|\\gamma^{-1}|\\epsilon^{-2}) \n\t\t@>{\\left(h_{\\epsilon} \\right)}^2>> D(|\\gamma^{-1}| )\n\t\\end{CD}\n\t\\end{equation}}\n\\newtext{\n \\begin{equation}\n\t\\begin{CD}\n D(|\\gamma^{-1}|\\epsilon) & \\ni & x\n\t @>\\left({h_{\\epsilon}}\\right)^{-1} >> \n\t y & \\in & D(|\\gamma^{-1}|) \\\\\n& &\t@V{N}VV @VV{y \\mapsto \\gamma y^2}V \\\\\n D(|\\gamma^{-1}|\\epsilon^{-2}) & \\ni & N(x)\n\t@>{\\left(h_{\\epsilon} \\right)}^{-2}>> \n\t\t\\gamma y^2 & \\in & D(|\\gamma^{-1}| )\n\t\\end{CD}\n\t\\end{equation}}{A3\\S 3(1) and A6\\S 1(18)}\n\n\t\\par \tIn the example above, the homothety $\\left(h_{\\epsilon}\n\t\t\\right)^{-1}$ is\n\t\tgenerating the renormalization group. In general, \n\t\twe want to consider renormalizations that lead to\n\t\ta commutative diagram, on the limit:\n\t\\begin{equation}\n\t\\begin{CD}\n y @>R>> R(y)\\\\\n\t\t@VGVV @VV{G^R}V \\\\\n G(y) @>>R^2> R^2(G(y)) \n\t\\end{CD}\n\t\\end{equation}\n\t\\par \tAbove, $G$ is the original algorithm (possibly after \n\t\\oldtext{eventually}\\newtext{}{A7(8)}\n\t\tsome uniform change of coordinates). We denote by\n\t\t$R$ the renormalization map, and by $G^R$ the renormalized\n\t\tversion of our iteration. Usually, $G^R$ depends on\n\t\ta renormalization parameter. However, we want $G^R$ to \n\t\tconverge to a limiting map, with a simple dynamics. Moreover,\n computation of $G^R$ should be `stable', in a \\oldtext{very}\n\t\t\\newtext{}{A6\\S 2(2)}\n\t\tprecise sense. \n\t\\medskip\n\t\\par \t\n\t\tThis suggests the following heuristics, in the case\n of Graeffe iteration:\n\t\twe would like to consider\n\t\ta polynomial $f$ represented by its roots. More precisely,\n\t\ta polynomial $f$ can be represented by the vector\n\t\t$(\\log |\\zeta_i|, \\arg(\\zeta_i))$ $\\in$ $\\mathbb R^d \\times\n\t\t\\mathbb T^d$, where $\\zeta_i$ is the $i$-th root of $f$,\n\t\tand $\\mathbb T$ denotes the \n\t\tadditive group $\\mathbb R \/ 2 \\pi \\mathbb Z$. In that case,\n\t\tGraeffe iteration is just multiplication by 2. \n\t\\par \tRenormalization would be a division by 2 of the\n\t\tlog of the radii of the roots. We would have:\n\\oldtext{\n\\begin{equation}\n\t\\begin{CD}\n \\mathbb R^d \\times \\mathbb T^d \n\t\t@>R^k>>\n\t\t\\mathbb R^d \\times \\mathbb T^d \\\\\n\t @V{\\times 2}VV \n\t\t@VV(r,\\theta)\\mapsto(r,2\\theta)V\\\\\n \\mathbb R^d \\times \\mathbb T^d \n\t\t@>>R^{k+1}> \n\t\t\\mathbb R^d \\times \\mathbb T^d \n\t\\end{CD}\n\t\\end{equation}\n}\n\\newtext{\n\\begin{equation}\n\\label{CDlim}\n\t\\begin{CD}\n \\mathbb R^d \\times \\mathbb T^d \n\t\t& \\ni & (\\log |\\zeta_i|, \\arg \\zeta_i)\n\t\t@>R^k>>\n\t\t(r,\\theta) & \\in & \n\t\t\\mathbb R^d \\times \\mathbb T^d \\\\\n\t & & @V{\\times 2}VV \n\t\t@VV(r,\\theta)\\mapsto(r,2\\theta)V\\\\\n \\mathbb R^d \\times \\mathbb T^d \n\t\t& \\ni & (2 \\log |\\zeta_i|, 2 \\arg \\zeta_i \\mod 2\\pi)\n\t\t@>>R^{k+1}> \n\t\t(r, 2 \\theta \\mod 2 \\pi) & \\in &\n\t\t\\mathbb R^d \\times \\mathbb T^d \n\t\\end{CD}\n\t\\end{equation}\n}{A3\\S 3(1)}\n\n\\par \tTherefore, the limit of the renormalized map should be\n the map: $(r,\\theta) \\mapsto (r,2 \\theta)$. \n Of course, we do not know in advance the roots of the\n\t\tpolynomial. Instead, we will produce a chain of\n\t\tcommutative diagrams `converging' to diagram~(\\ref{CDlim}).\n\t\tIn order to do that,\n\t \tlet us assume that a polynomial\n\t\\begin{equation}\n\t\ta_0 + a_1 x + a_2 x^2 + \\dots + a_d x^d\n\t\\end{equation} \n\t\tis represented by the vector\n\\newtext{\n \\begin{equation}\n\t\t(\\hat a_i, \\alpha_i) =\n\t\t\\left(\n\t \\log \\left| a_i \\right|,\t\n\t\t\\arg( a_i)\n\t\t\\right)\n \\end{equation} \n\t}{A6\\S 1(19)}\n\t\\newtext{\n\tIn this paper, for clarity of exposition, we will ignore\n\tthe case where some of the $a_i$ is zero. We can do that\n\tbecause most of our results hold `with probability 1'. In\n\tpractice, polynomials with a zero coefficient do arise.\n\tSee Section ~\\ref{Sec6} for implementation comments.\n\t}{B\\S 4(5,6)}\n\t\\par \n\t If the operator $G$ represents Graeffe iteration in this\n\t\tnew system of coordinates, we will have:\n\\oldtext{\n\t\\begin{equation}\n \\begin{CD}\n @VVV \n\t\t@VVV \\\\\n G^k (f) \\in \\mathbb R^{d+1} \\times \\mathbb T^{d+1} \n\t\t@>R^k>>\n (\\hat a^k, \\alpha^k) \\in \\mathbb R^{d+1} \\times \\mathbb T^{d+1} \n\t\t\\\\\n\t\t@V{G}VV \n @VV(G^{R,k})V \n\t\t\\\\\n\t\tG^{k+1}(f) \\in \\mathbb R^{d+1} \\times \\mathbb T^{d+1} \n @>>R^{k+1}> \n (\\hat a^{k+1}, \\alpha^{k+1}) \\in \n \\mathbb R^{d+1} \\times \\mathbb T^{d+1} \n\t\t\\\\\n @VVV \n\t\t@VVV\n \\end{CD}\n\t\\end{equation}\n\t}\n\\newtext{\n\t\\begin{equation}\n\\label{cd}\n \\begin{CD}\n & & @VVV \n\t\t@VVV \\\\\n\t\t\\mathbb R^{d+1} \\times \\mathbb T^{d+1} \n & \\ni & G^k (f) \n\t\t@>R^k>>\n (\\hat a^k, \\alpha^k) & \\in & \\mathbb R^{d+1} \\times \\mathbb T^{d+1} \n\t\t\\\\\n\t\t& & @V{G}VV \n @VV(G_k)V \n\t\t\\\\\n\t\t\\mathbb R^{d+1} \\times \\mathbb T^{d+1} \n\t\t& \\ni & G^{k+1}(f) \n @>>R^{k+1}> \n (\\hat a^{k+1}, \\alpha^{k+1}) & \\in &\n \\mathbb R^{d+1} \\times \\mathbb T^{d+1} \n\t\t\\\\\n & & @VVV \n\t\t@VVV\n \\end{CD}\n\t\\end{equation}\n\t}{A3\\S 3(1,6)}\n\\newtext{\n\t\\par\n\tAbove, $\\hat a^k = (\\hat a_0^k, \\cdots, \\hat a_d^k)$ and\n\t$\\alpha^k = (\\alpha_0^k, \\cdots, \\alpha_d^k)$. Also,\n\t$k$ is a superscript, not an exponent. However, \n\t$R^{k}$ denotes the $k$-th iterate of $R$.}{A6\\S 1(20)}\n\t\\medskip\n\t\\par\tIn this diagram, $R$ maps $(r,\\theta)$ into $(r\/2, \\theta)$.\n Although this defines \\oldtext{$G^{R,k}$}\n\t\t\\newtext{$G_k$}{A3\\S 3(6)} as a mapping, the \n\t\talgorithmic construction of \\oldtext{$G^{R,k}$} \n\t\t\\newtext{$G_k$}{A3\\S 3(6)} is postponed to Section\n\t\t~\\ref{Sec3}. On the limit, \\oldtext{$G^{R,k}$} \n\t\t\\newtext{$G_k$}{A3\\S 3(6)} `converges' to\n\t\tthe mapping: \\oldtext{$a, \\theta \\mapsto a, 2 \\theta$}\n\t\t\\newtext{$(a, \\theta) \\mapsto (a, 2 \\theta)$}{B\\S 4(7)}. \n\t\tThis will\n\t\t\\oldtext{assure}\\newtext{ensure}{A6\\S 1(21)}\n\t\tthat $\\lim \\hat a^k$ exists. We will \n\t\t\\oldtext{claim}\\newtext{show}{A6\\S 1(21)} that\n\t\tthis limit satisfies \\oldtext{(generically)}\n\t\t\\newtext{with probability 1}{A3\\S 1}\n\\[\n\t\t\\lim_{k \\rightarrow \\infty} \\hat a^k_j \n\t - \\lim_{k \\rightarrow \\infty} \\hat a^k_{j+1} \n\t\t= \\log |\\zeta_{d-j}|\n\\]\n \t\twhere $\\zeta_{d-j}$ is the \\oldtext{$d-j$-th}\n\t\t\\newtext{$(d-j)$-th}{A7(10)} root of the\n\t\toriginal polynomial, the roots being ordered by decreasing\n\t\tmodulus.. \n\t\\par \tAlso, by using the renormalized Graeffe iteration\n\t\t\\oldtext{$G^{R,k}$} \n\t\t\\newtext{$G_k$}{A3\\S 3(6)} \n\t\tinstead of the classical Graeffe iteration\n $G$, we will \n\t\tbe able to bound all the intermediate calculations to\n\t\ta compact, depending on the input $f$. This will \n\t\timply several stability results.\n\t\\medskip\n\t\\par\n\t\tIn order to define formally what we mean by a renormalized\n\talgorithm, we need to state the conditions that we expect\n\t\\oldtext{$G^{R,k}$} \n\t\\newtext{$G_k$}{A3\\S 3(6)} \n\tto satisfy. These conditions will define a class\n\tof algorithms, that we call renormalized iterative algorithms.\n\tSome examples: usual Newton, renormalized Newton, all reasonable\n\talgorithms based on a contraction principle and, of course,\n\tRenormalized Graeffe Iteration (To be constructed). \n\t\\medskip\n\t\\par\n\n\nBefore defining what we mean by a renormalized algorithm, we will\nbriefly discuss the notion of algorithm. There are many possible\ndefinitions (See Blum, Cucker, Shub and Smale~\\cite{BCSS}).\n\n\\begin{definition} \\label{itealg}\nAn iterative algorithm $M$ is a Blum-Shub-Smale (BSS) machine\nover ${\\mathbb R}$, modified as follows: \n\\begin{enumerate}\n\\item If the input is in ${\\mathbb R}^{l}\\times {\\mathbb T}^{m}$ \n for a pair \\oldtext {$(l,k)$}\n \\newtext{$(l,m)$}{A7(11) and B\\S 4(8)} \n, then the output\nis in ${\\mathbb R}^{l}\\times {\\mathbb T}^{m}$. \nThe integer $l+m$ is called the input size.\nIf the input is denoted by $x$, the output is denoted \n$M(x)$, and $M^k $ means the composition $M\\circ M \\circ \\cdots \n$\\newtext{$\\circ$}{A7(12)}$M$\n\\newtext{\\par Also, we say that the iterative algorithm `computes'\nthe function $x \\mapsto \\lim_{k \\rightarrow \\infty} M^k(x)$}{Avoid\nconfusion between iterate and limit.}\n\\item Computation nodes are allowed to perform \n\\newtext{the following}{A4\\S 1(1)} \nelementary functions\\oldtext{, i.e.}\\newtext{:}{}\npolynomial evaluation, \\newtext{absolute value,}{}\n(real) logarithms, (real) exponential, sine, \ncosine, rational\nfunctions, and compositions of those.\n\n\\item $M_{\\epsilon}(x)$ will denote the result of approximating \n$M(x)$ by allowing each operation with non-integer parameters \nto be performed with relative precision $\\epsilon$. (This is also\nknown in numerical analysis as the $(1+\\epsilon)$ property.)\n\\newtext{The parameter $\\epsilon$ is allowed to vary in $(0, \\frac{1}{2})$.}\n{A4\\S 1(2)}\nThe machine is supposed to ``know\" $\\epsilon$. \n\\newtext{This means that the value of $\\epsilon$ can be used in intermediate\ncalculations.} {A4\\S 1(2)}\nAlso, the approximation\nis performed by some prescribed algorithm. (e.g., IEEE arithmetic\nwith $- \\log_2 \\epsilon$ bits of mantissa). \n\n\\item \\newtext{For all $\\epsilon$,}{A4\\S 1(2)}\nthe arithmetic complexity of $M$ applied to the input $x$ is the\nnumber of \\oldtext{arithmetic operations,}\nelementary function evaluations \n\\newtext{(from the list of item~1)}{A4\\S 1(1)} and\nbranchings performed with input $x$. We require\nthe arithmetic complexity of the approximate algorithm \n\\oldtext{\n$M_{\\epsilon}(x)$\nto be the same as the arithmetic complexity of the original algorithm $M(x)$.\n}\n\\newtext{\n$M_{\\epsilon}$ applied to the input $x$ \nto be the same as the arithmetic complexity of the original algorithm $M$\napplied to input $x$.\n}{A3\\S 3(3)}\n\\end{enumerate}\n\\end{definition}\n\n\\begin{remark}\nItem~1 will allow us to distinguish between real and angular variables,\nthe latter one being defined modulo $2\\pi$. \nThis will simplify notation when we speak of the distance between\npoints in ${\\mathbb R}^{l}\\times {\\mathbb T}^{m}$. \\newtext{Let ${ \\mathrm{d}}$ be\nthat distance.}{A4\\S 1(7)}\n\\end{remark}\n\n\\newtext{\n\\begin{remark}\n\tIt has been argued that the outcome of a branching node would\n\tnot be well-defined in the presence of numerical error. This is\n\tnot true in the definition above. The branching nodes of \n\tthe machine $M$\n\tcan be assumed, without loss of generality, \n\tto branch on queries of the form $y>0$ or $y \\ge 0$ or $y = 0$. \n\tWhen the machine $M$ gets replaced by $M_{\\epsilon}$, the\n\tbranching nodes stay formally the same. The value of $y$, however, is\n\tcontaminated with a certain numerical error. It is still\n\ta perfectly defined real number, and it may be compared to zero.\n\tThus, the branching nodes branch `correctly', for a \n\tslightly perturbed input.\n\t\\par\n\tThis would lead to disastrous results if an approximate machine\n\tenters a `loop' due to numerical errors. Item 4 in the\n\tdefinition above is there to preclude that sort of loop, by\n\tensuring that the arithmetic complexity of each iteration\n\tdoes not depend on the working precision. \\oldtext{ This is the case\n\tfor all iterative algorithms that the authors know about.}\n\\end{remark}}{A4\\S 1(3) and B\\S 2(3,4)}\n\\newtext{\n\\begin{definition} A function $\\varphi$ can be computed in {\\em finite time}\nif and only if there is a BSS machine over $\\mathbb R$ \nmodified as above items 2, 3 and 4,\nthat computes $\\varphi$.\n\\end{definition}\n}{Define the notion finite time precisley.}\nOne consequence of the previous definition is the following: suppose\na certain function $\\varphi$ can be computed in finite time. Then, its branching\nset is given by a finite set of equations. Outside the branching set,\n$\\varphi$ can be written (locally) as a composition of elementary functions.\nMoreover, only a finite number of such compositions may appear,\none corresponding to each set of possible branchings. \n\n\\medskip\n\\par\nA few definitions \\oldtext{that will be used in the sequel }\nare in order now. In the sequel we will need to use \nalgorithms depending effectively on a parameter $k \\in \\mathbb N$. \n\\oldtext{ Let $M$ be an iterative algorithm, with input $(f,k)$, \nwhere the parameter $k$ will keep track of iterations.\nWe write $M_{k}(f)=M(f,k)$.} \\newtext{\n Those will\n be given by a machine $M$ with two inputs, say $k$ and $f$.\n However, we will denote the output as $M_k(f)$ and we\n will write $M_k$ for each of the input-output mappings\n obtained by restricting that machine to some fixed value\n of $k$.\n Also, in that situation we will speak explicitly of the\n algorithm $(M_k)_{k \\in \\mathbb N}$ or $M_k$ for short.\n\\par\n}{A4\\S 1(5)}\n\n\\par\nThe sequence $\\left( M_{k} \\right)_{k=1}^{\\infty}$ can be considered\n\\newtext{as a sequence of mappings}{Syntax.} \nfrom $\\RT{l}{m}$ into itself. \nThe orbit of $f$ by the sequence $\\left( M_{k} \\right)_{k=1}^{\\infty}$ \nis the set \n\\[ {\\mathrm{orb}} f \\stackrel{\\mbox{\\scriptsize def}}{=} \\left( f, M_{1}(f),M_{2}\\circ M_{1}(f),\\dots \\right) \n\\subset \\RT{l}{m} \\mbox{ .} \\] \n\nThis set should be understood as the orbit of $f$ by the non-autonomous\ndynamical system $M_k$ (Not the semi-group !)\nThe closure of ${\\mathrm{orb}} f$ will be denoted by\n$\\overline{{\\mathrm{orb}} f}$.\n\nWe shall say that a subset of $\\RT{l}{m}$ \\oldtext{is generic}\n\\newtext{has full-measure}{A3\\S 1} if its complement is \ncontained in a set of null measure. \\newtext{We say that some property\nis true almost everywhere (a.e.) if that property is true in a full-measure\nset.}{}\nWe can now define a {\\em renormalized algorithm} and make precise our\nconcept of renormalization:\n\n\n\\begin{definition} \\label{defren2}\nThe algorithm \\newtext{$(M_k)_{k \\in \\mathbb N}$ (or\n$M_k$ for short)}{A4\\S 1(5)}\\oldtext{$M_k$}\nis said to be a {\\em renormalized iterative algorithm \nto compute} \n\\oldtext{$\\varphi(f): \\RT{l}{m} \\rightarrow {\\mathbb R}^{s}$}\n\\newtext{$\\varphi : \\RT{l}{m} \\rightarrow {\\mathbb R}^{s}$, \n$f \\mapsto \\varphi(f)$}{A3\\S 3(4)}, where $0\\le s \\le l$,\n if, and only if, it satisfies the Axioms 1 through 4 below. \n\\end{definition}\n\\begin{axiom}[Consistency] \n\tFor almost every $f\\in {\\mathbb R}^{l} \\times {\\mathbb T}^m$ we have \n\\oldtext{\n\\[ \n\t\\lim_{k\\rightarrow \\infty} \n\t\\pi \\circ M_{k}\\circ M_{k-1}\\circ \\cdots \\circ M_{1}(f)\n\t= \\varphi(f) \n\\]\n}\n\\newtext{\n\\[ \n\t\\lim_{k\\rightarrow \\infty} \n\t\\left( \\pi \\circ M_{k}\\circ M_{k-1}\\circ \\cdots \\circ M_{1} \\right)(f)\n\t= \\varphi(f) \n\\]}{A3\\S 3(5)}\n\twhere $\\pi$ is the projection of $\\RT{l}{m}$ onto the first $s$ \n\tcoordinates of ${\\mathbb R}^{l}$.\n\\end{axiom}\n\n\\begin{axiom}[Arithmetic Complexity] \n\tThe arithmetic complexity of \\oldtext{$M_{k}(f)$}\n\t\\newtext{$M_k$ with input $f$}{A3\\S 3(4)} is bounded in terms of the\n\tsize \\newtext{$l+m$}{Clarity.} of $f$, independently of \n\t$k$ and the coefficients of $f$.\n\\end{axiom}\n\n\\begin{axiom}[Propagation] \n\tFor \\oldtext{generic $f\\in \\RT{l}{m}$ } \n\t\\newtext{almost every $f \\in \\RT{l}{m}$,}{A3\\S 1}\n\tthere exists a compact neighborhood\n\t$V \\subset \\RT{l}{m}$ \n\tof ${\\mathrm{orb}} f$ (under $\\{ M_{k} \\}$) \n\tand \\oldtext{$C \\ge 1$}\\newtext{C}{A4\\S 1(6)}\n\tsuch that $M_{k}|_{V}$ is local diffeomorphism with derivative bounded in norm by $C_{k}$\n\tand eventually $C_{k}0$}\n\twe have that \\newtext{for sufficiently small $\\epsilon>0$}{} \n\\[\n { \\mathrm{d}} \\left(\n M_{k,\\epsilon} \\circ \\cdots \\circ M_{1,\\epsilon} (f)\n , M_{k} \\circ \\cdots \\circ M_{1} (f)\n \\right) \n < k A^k B \\epsilon \\mbox{ . } \\]\n\t\\newtext{Here, $A$ and $B$, \n\t depend on $f$, but not on $k$.}{} \n \\oldtext{Moreover, one can take}\n\t\\newtext{In particular, that will be true for values of\n\t$\\epsilon$ of the form}{} \n\\[ \\epsilon \\le \\frac{\\rho}{kA^kB} \\mbox{ ,} \\]\n\twhere $\\rho$ is defined below.\n\\end{lemma}\nAs an immediate consequence of Lemma~\\ref{lemE} we have that\n\\[\n \\| \\left( \\pi \\circ \n M_{k,\\epsilon} \\circ \\cdots \\circ M_{1,\\epsilon} )(f)\n - ( \\pi \\circ M_{k} \\circ \\cdots \\circ M_{1} )(f)\n \\right) \\|_2\n < k A^k B \\epsilon \\mbox{ .}\n\\]\n\n\n\n\\begin{proof}[Proof of Lemma~\\ref{lemE}]\nWe now describe the construction of the constants in the statement of\nLemma~\\ref{lemE}.\nCombining Axioms 3 and 4, there exists a compact neighborhood $W$ of ${\\mathrm{orb}} f$\nsuch that:\n\\begin{enumerate}\n\\item \nEvery mapping $M_{k}$ is Lipschitz on $W$. Moreover, there\nexists a constant $A$ such that for every $k$, the norm of the Lipschitz\nconstant \nof $M_{k}$ is uniformly bounded by \n\\oldtext{$A = \\sup C_k$}\n\\newtext{$A = \\max \\left( 1, \\sup_k C_k \\right)$,}{A4\\S 1(6)} \n$C_k$ as in Axiom~3. \n\\item \n\\oldtext{For $g\\in W$, there exists}\n\\newtext{There exists}{A3\\S 2(1)}\n$\\epsilon_{0}$ such that\n$\\forall \\epsilon < \\epsilon_{0}$ we have\n\\[ {{ \\mathrm{d}}}(M_{k,\\epsilon}(g),M_{k}(g)) < \\epsilon B\n\\mbox{ ,} \\hspace{1cm} \\forall g \\in W \\mbox{ .} \\] \n\nWe then define $\\rho$ as the minimum of $\\epsilon_{0}$ and\nthe distance of ${\\mathrm{orb}} f$ to the boundary\nof $W$.\n\\end{enumerate}\n\nWe may now conclude the proof of Lemma~\\ref{lemE}.\nSet \n\\[ g=(M_{k-1}\\circ \\dots \\circ M_{1})(f) \\mbox{ ,} \\]\nand\n\\[ h=(M_{k-1,\\epsilon} \\circ \\dots \\circ M_{1,\\epsilon})(f) \\mbox{ .} \\]\nIn that case, we want to bound\n\\[ { \\mathrm{d}}(M_{k,\\epsilon}(h),M_{k}(g)) \\le { \\mathrm{d}} (M_{k,\\epsilon}(h),M_{k}(h))\n+ { \\mathrm{d}}(M_{k}(h),M_{k}(g)) \\]\nIf $h\\in W$, one can bound \n\\[ { \\mathrm{d}}(M_{k}(h),M_{k,\\epsilon}(h)) \\le B\\epsilon \\]\nand\n\\[ { \\mathrm{d}}(M_{k}(h),M_{k}(g)) \\le A \\ { \\mathrm{d}}(g,h) \\mbox{ .} \\]\nBy induction, \n\\[ { \\mathrm{d}}(g,h) \\le (k-1) A^{k-1} B \\epsilon \\mbox{ ,} \\]\nand hence\n\\[ { \\mathrm{d}}(M_{k,\\epsilon} (h), M_{k}(g) )\\le k A^{k} B \\epsilon \\mbox{ .} \\]\nThe condition \n\\[ \\epsilon < \\frac{\\rho}{k A^{k} B} \\]\nguarantees that $h \\in W$. \n\\end{proof}\n\n\n\\medskip\n\\par\n\tIf we use a Turing machine model (or any other classical\n\tdiscrete complexity model), we can perform all the operations of\n\t$M$ in finite precision, and obtain:\n\n\\begin{lemma} \\label{lemT}\nLet $M$ be a renormalized iterative algorithm.\n\\oldtext{ Fix a generic} \\newtext{For almost every $f$,\nassume that the truncation error is bounded by}{A3\\S 1}\n\\oldtext{\n\\[ \\| \\pi \\left(\n M_{k} \\circ \\cdots \\circ M_{1} \\right) (f)\n - \\varphi(f)\n \\|_2 \\]\n\t}\n\\newtext{\n\\[ \\| \\left( \\pi \\circ\n M_{k} \\circ \\cdots \\circ M_{1} \\right) (f)\n - \\varphi(f)\n \\|_2 < E^{2^{k}} \\mbox{ ,} \\] \nwhere $ E = E(f) \\in (0,1) $.}{A3\\S 3(5)}\n\\oldtext{ is bounded by $E^{2^{k}}$} \nThen, the \\oldtext{Turing} complexity of approximating $\\varphi(f)$ with\nprecision $\\delta$ is $ {\\mathcal{O}} ((\\log_{2} \\frac{1}{\\delta})^{1+\\alpha})$\nwhere $\\alpha > 0 $ is arbitrarily \\newtext{\nsmall.}{}\n\\end{lemma}\n\\newtext{\nIt is assumed above that the cost of arithmetic with $l$ bits\nof mantissa is $O(l^{1+\\alpha})$, for all $\\alpha > 0$. \nSee~\\cite{BP} pp~78-79 for a sharper bound on the complexity\nof long integer multiplication.}{A6\\S 1(22)}\n\n\n\\begin{proof}[Proof of Lemma~\\ref{lemT}]\n\\par\n\tLet \n\\[\n\tk = \\left\\lceil \\log_2 \\frac{-\\log_2 \\frac{\\delta}{2}}{-\\log_2 E} \n\t\\right\\rceil\n\t\\mbox{ ,}\n\\]\n\tso that \n\\[\n\tE^{2^k} \\le \\frac{\\delta}{2} \\mbox{ .}\n\\]\n\n\n\tChoose $\\epsilon$ so that Lemma~\\ref{lemE} holds, i.e.,\n\t$\\epsilon < \\rho\/k A^k B$, and such that\n\t$k A^k B \\epsilon < \\delta\/2$. This can be done\n\tfor\n\\oldtext{\n\\[\n\t\\epsilon \\in\n\t {\\mathrm{o}} \n\t\\left(\n\t \\frac{\\delta}{\\log_2 \\log_2 \\delta^{-1} (\\log_2 \\delta^{-1})^\n\t\t{\\log_2 A}}\n\t\\right)\n \\subset\t\n\t {\\mathrm{o}} (\\delta^{1 + \\alpha})\n\\]\n}\n\\newtext{\n\\[\n\\epsilon < c_1 \\delta (\\log \\delta^{-1})^{c_2}\n\\]\nfor some constants $c_1$ and $c_2$ dependent of $f$.\n}{}\n\tThe total cost of computing $M_k$ is \n $ {\\mathcal{O}} \\left( a (\\log_2 \\epsilon^{-1})^{1 + \\alpha} \\right)$,\n\twhere $a$ is the arithmetic complexity of the algorithm and\n\t$\\alpha$ is arbitrarily small. \\oldtext{Hence,}\n\t\\newtext{Since $f$ is fixed, $a$ is a constant and \n\t$k < \\log_2 \\log_2 \\delta^{-1}$}{}\n\t the total cost is\n\\[\n\t {\\mathcal{O}} \\left( (\\log_2 \\delta^{-1})^{1+\\alpha}\\right)\n\\]\n\\newtext{for all $\\alpha$ arbitrarily small.}{}\n\\end{proof}\n\n\n\n\\medskip\n\\par\tThe complexity bound of Lemma~\\ref{lemT} is non-uniform in $f$.\n\tIt is also non-effective, in the sense that we give no procedure\n\tto estimate $k$ and \\oldtext{$m$}\\newtext{$\\epsilon$}{}\n\twithout the knowledge of $\\rho$, \n\t$B$ and $A$. Indeed, those quantities may depend on $f$.\n\tIt \\newtext{would be of some interest}{} to bound those \n\tquantities in a probabilistic\n\tsetting, similar to Theorem~\\ref{th2}.\n\n\\begin{definition}\n\tLet $M$ be an iterative algorithm to compute \\oldtext{$\\Phi(F)$}\n\t\\newtext{$\\Phi: F \\mapsto \\Phi(F)$,}{A3\\S 3(4)} \n\\newtext{i.e., $\\lim_{n \\rightarrow \\infty} M^n(F) = \\Phi(F)$).\n }{A4\\S1 (8)}\t\n\tWe say\n\tthat $M_k$ is a renormalization of $M$ if \n\\begin{enumerate}\n\t\\item $M_k$ is a renormalized iterative algorithm to compute $\\varphi(f)$.\n\t\\item There are functions $\\psi$ and $\\eta$, defined almost everywhere\n and computable in finite time,\n\tsuch that the diagram\n\\[\n\\begin{CD}\n\tF @>\\psi>> \\psi(F) \\\\\n@V{\\Phi}VV\t@VV{\\varphi}V\\\\\n\t\\Phi(F) @<\\eta<< \\varphi(\\psi(F))\n \\end{CD}\n\\]\n\tcommutes.\n\t\\item There is a function $R$, \n\t\tcomputable in finite time, so that the diagrams:\n\\[\n \\begin{CD}\n F @>R^k \\circ \\psi>> R^k(\\psi(F)) \\\\\n @V{M}VV @VV{M_k}V \\\\\n M(F) @>R^{k+1} \\circ \\psi>> R^{k+1}(\\psi(M(F))) \\\\\n \\end{CD}\n\\]\n\tcommute.\n\\end{enumerate}\n\\end{definition} \t\n\n\tTheorem~\\ref{th1} can be stated now in a more concise way:\n\tLet \\oldtext{$z(f)$}\\newtext{$ {\\zeta} : f \\mapsto {\\zeta} (f)$}{A3\\S 3(4)}\n\tbe the function that associates, to any univariate\n\tdegree $d$ polynomial $f$, its roots $\\zeta_1 , \\dots , \\zeta_d$ ordered by\n\tdecreasing modulus. We have the following: \n\n\\begin{theoquo}\n There is \\oldtext{(we construct)}\n\t\ta renormalization of the Graeffe\n\t\titeration to compute $|\\zeta(f)|=\\left( |\\zeta_1(f)|, \n \\dots, |\\zeta_d(f)|\\right)$.\n\t\tEach iteration has arithmetic complexity \n\t\t $ {\\mathcal{O}} (d^2)$ and needs\n\t\tmemory $ {\\mathcal{O}} (d)$.\n\\end{theoquo}\n\n\n\n\\section{Recurrence Relations and the Renormalization of Graeffe}\n\\label{Sec3}\n\n\n\t\tIt is time to construct the renormalized Graeffe\n\t\t\\oldtext{operator} \n\t\t\\newtext{iteration}{A7(3)}.\n\t\tLet $f(x) = f_0 + f_1 x + \\dots + f_d x^d$. Let $h = Gf$ be\n\t\tits Graeffe iterate. The coefficients of $h$ can be written\n\t\tas~:\n\t\\begin{equation}\n\\label{g1}\n\t\th_i = (-1)^d \\sum\n\t\t\t_{\\substack{\n\t\t\t\t0\\le i-j \\le d \\\\\n\t\t\t\t0\\le i+j \\le d \\\\ }}\n\t\t\t (-1)^{i-j} f_{i-j} f_{i+j}\n\t\\end{equation} \n\t\\par\tFor convenience, we rewrite (\\ref{g1}) as: \n\t\\begin{equation}\n\\label{g2}\n\t\th_i = (-1)^{d+i} {f_{i}}^2 + \n\t\t\t2 \\sum\n\t\t\t_{ 1\\le j \\le \\min(i,d-i)}\n\t\t\t (-1)^{d+i-j} f_{i-j} f_{i+j}\n\t\\end{equation} \n\n\n\t\\medskip\n\t\\par\tThe next step is to write those equations in terms\n\t\tof the log of the coefficients. More precisely, we will\n\t\thave to deal with the following two quantities:\n\t\\begin{equation}\n\\label{g7}\n\t\tf^{\\log}_i \\stackrel{\\mbox{\\scriptsize def}}{=} \\log \\left| f_i \\right|\n\t\\end{equation} \n\t\\begin{equation}\n\\label{g8}\n\t\tf^{\\arg}_i \\stackrel{\\mbox{\\scriptsize def}}{=} \\arg f_i \n\t\\end{equation} \n\t\\medskip\n\t\\par\tIt is possible now to construct the renormalized Graeffe\n\t\t\\oldtext{operator} \n\t\t\\newtext{iteration}{A7(3)}\n\t\t\\oldtext{$G^{R,k}$} \n\t\t\\newtext{$G_k$}{A3\\S 3(6)}. \n\t\tRecall from diagram (\\ref{cd}) that \n\t\tthis \\oldtext{operator} \n\t\t\\newtext{iteration}{A7(3)} will map \n\t\t$2^{-k} f^{\\log}$ and $f^{\\arg}$ into $2^{-k-1} h^{\\log}$\n\t\tand $h^{\\arg}$.\n\t\\par\tFor that purpose, we introduce the notation:\n\t\\par\n\\begin{equation} \\label{defren}\n\\left\\{ \\begin{array}{lll}\nf^{k} & \\stackrel{\\mbox{\\scriptsize def}}{=} & (2^{-k} f^{\\log},f^{\\arg}) \\\\\nh^{k+1} & \\stackrel{\\mbox{\\scriptsize def}}{=} & (2^{-k-1} h^{\\log},h^{\\arg}) \\\\\n\\end{array}\n\\right.\n\\end{equation}\n\nWe also introduce operators:\n\\begin{eqnarray}\n (x,\\alpha) \\rentimes{k} (y, \\beta) &\\stackrel{\\mbox{\\scriptsize def}}{=}& \n (x + y, \\alpha + \\beta) \\\\\n (x,\\alpha) \\renpow{k}{\\lambda} &\\stackrel{\\mbox{\\scriptsize def}}{=}& \n (\\lambda x , \\lambda \\alpha) \\\\\n \\renscal{k}{z} (x,\\alpha) &\\stackrel{\\mbox{\\scriptsize def}}{=}& \n (x + 2^{-k} \\log |z|, \\alpha + \\arg z) \n\\end{eqnarray}\nand\n\\begin{small}\n\\begin{eqnarray}\n (x,\\alpha) \\renplus{k} (y, \\beta) &\\stackrel{\\mbox{\\scriptsize def}}{=}& \n \\left(\n 2^{-k} \\log \n \t\\left|\n\te^{i \\alpha + 2^k x}\n\t+\n\te^{i \\beta + 2^k y}\n\t\\right|\n ,\n \\arg\n \\left(\n\te^{i \\alpha + 2^k x}\n\t+\n\te^{i \\beta + 2^k y}\n \\right)\n \\right) \\label{deflast} \n\\end{eqnarray} \\end{small}\\newtext{}{A6\\S 1(24) Added parenthesis}\n\\par\nWe remark that the purpose of the sub-index $k$ in\nthe above formulae is to keep track of the degree of the \nrenormalization. For operations which do not change,\nwe omitted the sub-index. \\newtext{The operator\n$\\renscal{k}{z}$ stands for the multiplication of\na renormalized value by a (non-renormalized) constant $z$}{A6\\S 1(23)} \n\\par\n Also, binary renormalized operations are defined for operands\n with the same renormalization index. Therefore, one should\n first convert $f_i^k$ to $f_i^{k+1}$ before attempting to\n \\oldtext{to}\\newtext{}{B\\S 4(9)}\n `multiply' it with a factor of renormalization index\n of order $k+1$. This conversion will be implicit in the\n formulae below.\n\\par \n\tEquation~(\\ref{g2}) becomes:\n\\begin{small}\n\\begin{equation}\n\\label{g666}\n{h_i}^{k+1} = \n\\left( \\renscal{k+1}{ s_{i0}}\n\\left({f_i}^k \\right)\n\\renpow{k+1}{2} \\right)\n\\renplus{k+1}\n\\left( \\renscal{k+1}{2}\n\\overset{\\min(i, d-i)}{\\underset{j=1}{\\ \\ \\ \\ \\bigrenplus{k+1}} } \n\\left(\n\\renscal{k+1}{s_{ij}}\n{f_{i-j}}^k\n\\rentimes{k+1}\n{f_{i+j}}^k\n\\right) \\right) \\mbox{ ,}\n\\end{equation} \\end{small}\nwhere $s_{ij}= (-1)^{d+i-j}$.\n\t\\newtext{Recall that above, $k$ is a superscript, not an\n\texponent.}{A6\\S 2(3)}\n\tThe `renormalized operations' above are \\oldtext{very}\n\t\\newtext{}{A6\\S 2(1)} easy to\n\timplement in terms of the classical ones. The most \n delicate being the renormalized sum, so we give here our\n\tpreferred algorithm \n \n\n\\begin{example} \\label{ex2} How to compute the `Renormalized sum':\n\\[\n\t(c,\\gamma) := (a,\\alpha) \\renplus{k} (b,\\beta)\n\\]\\newtext{}{A7(13)}\n\\begin{tt} \n\\begin{tabbing}\n\\ \\ \\ \\ \\= \\\\\n\\> If $a > b$, \\= do: \\\\\n\\>\t\t\\>$ s = \\exp{i\\alpha} + \\exp{ (i \\beta + 2^{k} (b-a)) } $ \\\\\n\\>\t\t\\>$ c = a + 2^{-k} \\ln |s| $ \\\\\n\\>\t\t\\>$ \\gamma = \\arg(s) $ \\\\\n\\> else \\\\\n\\>\t\t\\> $ s = \\exp{i\\beta} + \\exp{ (i \\alpha + 2^{k} (a-b)) }$ \\\\\n\\>\t\t\\>$ c = b + 2^{-k} \\ln |s| $ \\\\\n\\>\t\t\\>$ \\gamma = \\arg(s) $ \\\\\n\\> endif\\\\\n\\end{tabbing}\n\\end{tt}\n\\end{example}\n\\oldtext{$ c = a + 2^{-k} \\ln |s| $}\nWe remark that in the above formula, the complex arithmetic \noperations can be performed in terms of real elementary \nones. Moreover, in numerical implementations if $k$ is large \nenough, as compared to $\\epsilon$, it may be faster to approximate\n$(c,\\gamma)$ with $(a,\\alpha)$ \\newtext{or $(b,\\beta)$, whichever is\nlarger}{A7(14)}. \n\n\n\t\\medskip\n\t\\par\tIn order to finish\n\t\tthe proof of Theorem~\\ref{th1}, we\n\t\tstill need a few remarks about the renormalized Graeffe \n \t\\oldtext{operator} \n\t \\newtext{iteration, which}{A7(3)}\n\t\twe just constructed. \n\t\tClearly, in order to compute formula~(\\ref{g666}),\n\t\twe only need memory space of the order $ {\\mathcal{O}} (d)$\n\t\t\\newtext{and time of the order $ {\\mathcal{O}} (d^2)$}{A6\\S 1(25)}.\n\t\\par\tWhen $k \\rightarrow \\infty$, the quantities \n $|f^k_{i}|$ are all convergent.\n\n\t\\medskip\n\t\\par\n \tIf $G_k$ is the renormalized Graeffe iteration, we \n \tdefine $G_{\\infty}$ as the limit of $G_k$ when $k$ goes to infinity.\n \tThis means that renormalized sums $\\renplus{k}$ are replaced by\n \ttheir limit $\\renplus{\\infty}$, where:\n\\[\n (a, \\alpha) \\renplus{\\infty} (b, \\beta) =\n \\left\\{\n \\begin{array}{ll}\n (a, \\alpha) & \\text{if } a>b \\\\\n (b, \\beta) & \\text{if } a 0$.} \\newtext{ For almost every $f$ and \nany $\\delta>0$,}{A3\\S 1} there exist a \n compact neighborhood $W \\subseteq \\mathbb R^l \\times \\mathbb T^m$\n of $\\text{orb} (f)$ and an integer $k_0$\n such that, $G_k$ is a local diffeomorphism $W \\rightarrow G_k(W)$,\n and such that for $k_0 \\le k \\le \\infty$, the derivative of $G_k$ is\n bounded by $2+\\delta$\n}\n\n\\begin{proof}[Proof of Lemma~\\ref{technical}]\n The proof is divided in several steps.\n\\begin{itemize}\n\\item [Step 1:]For any $j \\in \\mathbb N$, there is a \\oldtext{generic}\n \\newtext{full-measure set}{A3\\S 1} $U_j$ such that\n $G_j \\circ \\cdots \\circ G_1$ is well-defined, and a local\n diffeomorphism in $U_j$. Hence, there is a \\oldtext{generic}\n \\newtext{full-measure set}{A3\\S 1} $U_{j,k}$ such that\n\\oldtext{ \n $G_k$ \n is well-defined and a local diffeomorphism near\n $G_j \\circ \\cdots \\circ G_1 (f)$.}\n\\newtext{$G_k \\circ \\left( G_j \\circ \\cdots \\circ G_1 \\right)$ \n is defined on $U_{j,k}$ and for all $f \\in U_{j,k}$ there is a\n neighborhood $V_f$ of $G_j \\circ \\cdots \\circ G_1 (f)$ such that\n $G_k$ is a diffeomorphism $V_f \\rightarrow G_k(V_f)$.}\n {A3\\S 2(2)}\n\\item [Step 2:]Let $U_{\\infty}$ be the set of all $f$ such that \n \\oldtext{\n $G_k(f)$ is\n a local diffeomorphism in $(\\pi ^{-1} \\circ \\varphi) (f)$, for\n $k$ large enough.}\n \\newtext{$G_k$ is a diffeomorphism near \n $(\\pi ^{-1} \\circ \\varphi) (f)$ for all values of $k$ that\n are large enough.}{A4\\S 1(9)}\n Then $U_\\infty$ contains the set of complex \n polynomials without roots of the same modulus. Hence $U_\\infty$ \n \\oldtext{is generic.} \\newtext{has full measure.}{A3\\S 1}\n\\item [Step 3:]Let $U = U_\\infty \\cap \\left( \\bigcap_j U_j \\right) \n \\cap \\left( \\bigcap_{jk} U_{jk} \\right)$. Then $U$ \n \\oldtext{is generic.} \\newtext{has full measure}{A3\\S 1}.\n Moreover, let $f \\in U$. Then $G_k$ is a local\n diffeomorphism with derivative \\newtext{of norm}\n {This is more precise.} $< 2+\\delta \/ 2$ in an open\n neighborhood $V_k(g)$ of every $g \\in \\overline{\\text{orb} f}$,\n for all $k \\ge k_0 (g)$. \n \\newtext{Indeed, if we write $G_k = G_{\\infty} + \n \\left( G_k - G_{\\infty} \\right)$, we can make the\n $C^1$ norm of the second term arbitrarily small, namely\n less than $\\delta \/2$. We know that the norm of the derivative\n of $G_{\\infty}$ is precisely 2, hence the bound $2+\\delta \/ 2$\n }{A4\\S 2(1)}\n In the particular case $\\delta = \n \\infty$ we can set $k_0 = 1$.\n\\item [Step 4:]Since $G_k \\rightarrow G_{\\infty}$ \\oldtext{in a locally\n\t$C^1$ sense}\\newtext{pointwise in the $C^1$ topology and }{A4\\S 2(2)}\n\tfor $g$ almost everywhere, we can assume \n\tthat\n $\\bigcap_{k \\ge k_0} V_k(g)$ contains an open ball $V(g)$ of\n center $g$, where $G_k$ is a local diffeomorphism with \n derivative bounded by $2+\\delta \/ 2$. \n\\item [Step 5:]Since $\\overline {\\text{orb}(f)}$ is compact, the union\n $\\bigcup_{g \\in \\overline{\\text{orb}(f)}} V(g)$ has a finite \n sub-cover $\\bigcup_{g \\in \\Gamma} V(g)$, and we set $W =\n \\bigcup_{g \\in \\Gamma} \\overline{V(g)}$. Then we set $k_0\n = \\max_{g \\in \\Gamma} k_0(g)$, and we obtain that for any\n $k \\ge k_0$, $G_k$ is a local diffeomorphism in $W$, with\n derivative of norm bounded by $2 + \\delta$. \n\\end{itemize}\n\\end{proof}\n\n\n\n\\medskip\n\\par\n\tThe proof that our algorithm satisfies Axiom 4 is divided in two parts\n\t\\newtext{, dealing (respectively) with small and large values of $k$}\n\t{For clarity.}.\n\\oldtext{\n\\begin{lemma*}\n\tFor a.e. $g$ given, $|G_{k,\\epsilon}(g) - G_{k}(g) | < B \\epsilon$,\n\twhere $B$ depends on $g$ and $k$, and $\\epsilon$ is small\n\tenough.\n\\end{lemma*}\n\n\n\\begin{lemma*} \n\tFor a.e. $g$ given, there is $k_0$ such that for any \n\t$k > k_0$\n\\[\n{ \\mathrm{d}} \\left(G_{k,\\epsilon}(g) , G_{k}(g) \\right) < B \\epsilon\n\\]\n\twhere $B$ depends on $\\varphi(g)$ only.\n\\end{lemma*}\n}\n\\newtext{\n\\begin{lemma}\n\t\\label{lp1}\n\tFor almost every $f$ and for all $k$,\n\tthere exist an open neighborhood\n\t$U$ containing $f$ and $B>0$ such that for all\n\t$g \\in U$ and for all $\\epsilon$ small enough\n\\[\n{ \\mathrm{d}} \\left(G_{k,\\epsilon}(g) - G_{k}(g) \\right) < B \\epsilon \\mbox{ .} \n\\]\n\\end{lemma}\n\n\n\\begin{lemma} \n\t\\label{lp2}\n\tFor almost every $f$, there exist $k_0$, $B>0$, \n\tand an open neighborhood\n\t$U$ containing $f$, such that for all\n\t$g \\in U$ and $k \\ge k_0$, for all $\\epsilon$ small enough,\n\\oldtext{\n\\[\n|G_{k,\\epsilon}(g) - G_{k}(g) | < B \\epsilon\n\\]\n\tFor a.e. $g$ given, there is $k_0$ such that for any \n\t$k > k_0$\n}\n\\[\n{ \\mathrm{d}} \\left(G_{k,\\epsilon}(g) , G_{k}(g) \\right) < B \\epsilon\n\\]\n\\end{lemma}\n}{Restated for clarity and also A3\\S 3(7).}\n\n\\medskip\n\\par\t\\oldtext{Since $G_k$ is an almost everywhere local diffeomorphism, \n and hence the preimage\n\tof a zero measure set has zero measure,} Lemmas~\\ref{lp1} and\n\t\\ref{lp2} together imply that for almost all \\oldtext{$g$}\n\t\\newtext{$f$, there is a neighborhood $U$ containing $f$ \n\tsuch that for all $g \\in U$}{},\n\\[\n{ \\mathrm{d}} \\left( G_{k,\\epsilon}(g) , G_{k}(g) \\right) < B \\epsilon\n\\]\n\tindependently of $k$.\n\\par\n\n\t\\oldtext{\n\tThis result can be extended to an open neighborhood\n\tof any $g$ generic.}\n\tSince \\oldtext{(generically)} $\\text{orb}(f)$ is defined\n \\newtext{for almost every $f$}{A3\\S 1} \n\tand admits a compact neighborhood $V$, we can select a finite\n subcover of those neighborhoods, and hence find a finite $B$\n valid for all $g \\in V$. Therefore, our algorithm satisfies\n Axiom 4. \n\\medskip\n\\par\t\n\\begin{proof}[Proof of Lemma~\\ref{lp1}:]\n\tOur iteration $G_k$ can be written in terms of the following\n\treal operations: $+$, $-$, $\\cos$, $\\sin$, $\\arctan$, \n\tmultiplication by $2^k$, by $2^{-k}$, absolute value,\n\t$\\exp$, $\\log$.\n\\par\tThe set of inputs such that a `log of zero' or an `absolute\n\tvalue of zero' occurs has zero measure. \\oldtext{Therefore, if $g$ is\n\tgeneric,}\n\t\\newtext{Therefore, for almost every $f$,}{A3\\S 1}\t\n\t$G_k(f)$ is computed by a composition of analytic\n\tfunctions. Also, \\oldtext{if $g$ is generic}\n\t\\newtext{for almost every $f$,}{A3\\S 1} \n\tnone of the output values\n\tis zero.\n\\par\tTherefore, for every intermediate quantity $x_l$, the derivative\n\tof any output $y_m$ with respect to $x_l$ is finite (say $\\le D_{lm}$).\n\\oldtext{\n\tTherefore, a relative perturbation of $\\epsilon$ in $x$ leads\n\tto a perturbation of $xD\\epsilon$ in $y$. Thus, we set\n\t$B = \\sum D_{lm}x_l$ and we are done. \n\t}\n\\newtext{\n\tTherefore, a relative perturbation of $\\epsilon$ in $x_l$ leads\n\tto a perturbation of size $|x_l| D_{lm}\\epsilon$ in $y_m$. Thus, we set\n\t$B = \\frac{1}{2} \\sum_{l,m} D_{lm}|x_l|$. By continuity,\n\ta perturbation of $\\epsilon$ in each intermediate value $x_l$\n\tleads to a perturbation smaller than $B \\epsilon$, for input\n\t$g$ in a certain neighborhood of $f$. \n\t}{A6\\S 1(27)}\n\\end{proof}\n\\medskip\n\\par\t\n\\begin{proof}[Proof of Lemma~\\ref{lp2}:]\n\\oldtext{\n\tWe claim that there is a full-measure set $W$ such that \n\tif $\\varphi(g) \\in W$, then there are $k_0$ and $B$ such \n\tthat \n\\[\n{ \\mathrm{d}} \\left( G_{k,\\epsilon}(g) , G_{k}(g) \\right) < B \\epsilon\n\\]\n\tfor $k > k_0$. Since $\\varphi ^{-1}$ maps zero\n\tmeasure sets into zero measure sets, Lemma~\\ref{lp2} will\n\tbe true in the full-measure set \n\t$\\varphi^{-1}(W) \\cap W'$. where $W'$ is the set of\n\t$g$ such that $G_k \\circ \\dots \\circ G_1 \\rightarrow \\varphi(g)$.\n\tAccording to Axiom 1, the set $W'$ has full measure.\n}\n\\newtext{\n\tLet $W$ be the set of all $f$ such that $G_{\\infty}(f)$ is\n\twell-defined. Recall that\n\\[\n\t\\begin{array}{lccll}\n\t\\renplus{\\infty} : & (a,\\alpha), (b,\\beta) &\n\t\\mapsto & (a,\\alpha) & \\text{if } a>b \\\\\n\t&&& (b,\\beta) & \\text{if } b>a \n\t\\end{array}\t\n\\]\n\tand that the operator $\\renplus{\\infty}$ is not defined \n\tfor $a=b$. Therefore, $W$ is open and has full measure.}{Clarity.}\n\\oldtext{\n\\par\tIn order to prove our claim, we consider the `limit' iteration\n\t$G_{\\infty}$, where $\\renplus{k}$ was replaced by \n\t$\\renplus{\\infty}$:\n\\[\n\t\\begin{array}{lccll}\n\t\\renplus{\\infty} : & (a,\\alpha), (b,\\beta) &\n\t\\mapsto & (a,\\alpha) & \\text{if } a>b \\\\\n\t&&& (b,\\beta) & \\text{if } b>a \n\t\\end{array}\t\n\\]\n\\par\tThe operator $\\renplus{\\infty}$ is not defined for $a=b$.\n\tIn the same way, we replace all the other renormalized operators by\n\ttheir limit values.\n\tWherever defined, $G_{\\infty}$ fixes scalar quantities, and\n\tdoubles angles.\n\\par\tLet $W$ be the domain of definition of $G_{\\infty}$. Clearly,\n\t$W$ is open and has full measure.}\n\\newtext{\n\tLet $f \\in W$ and $U$ be a small connected\n\tneighborhood containing $f$.\n\tLet $g \\in U$, then\n\tby taking $U$ small enough and $k$ large enough, we\n\tcan guarantee that $G_k(g)$ and $G_{k,\\epsilon}(g)$\n\tare well-defined. Since $U$ is connected, all the\n\tbranching outcomes in the computation of $G_k(g)$ and\n\t$G_{k,\\epsilon}$ are\n\tthe same, hence $G_k$ restricted to $U$ is a composition\n\tof locally analytic functions. We can assume without loss\n\tof generality that all derivatives are bounded, hence\n\tthere is a constant $B$ such that \n\\[\n\t{ \\mathrm{d}} \\left(G_{k,\\epsilon}(g) , G_{k}(g) \\right) < B \\epsilon\n\\]\n\tfor $\\epsilon$ small enough, but still independent\n\tof the choice of $g$.\n\t}{Fixes A3\\S 3(7). The following remarks are no more relevant:\n A3\\S 3(8), A6\\S 1(28), A6\\S 1(30), A3\\S 2(3)}\n\t\n\\oldtext{\t\n\tFor $y \\in W$, there is $k$\n\tsuch that every operation $\\renplus{k}$ can be replaced by\n\t$\\renplus{\\infty}$, with relative error $\\epsilon$. Indeed,\n\tfor $k$ large enough,\n\\[\n\ts = \\left| 1 + e^{i(\\beta - \\alpha) + 2^k (b-a)} \\right| \\ , \\ a > b\n\\]\n\tis bounded away from zero, so that \n\t$2^{-k} \\log s$ can be\n\tmade smaller than $|a| \\epsilon$. The error in the angle may be\n\testimated the same way. \n\\par\tThis last property holds in an open neighborhood $U$ of $y$,\n\t$U \\subset W$. Once we take $k > k_0$, with $k_0$ large enough,\n $(G_k \\circ \\dots \\circ G_1) (f) \\in U$. Since all of our\n\toperators $\\renplus{k}$ and $\\rentimes{k}$ are \n\tlocally Lipschitz in\n\t$W$, there is $B$ s.t. \n for $\\epsilon$ small enough, for all $z \\in U$,\n$\\| G_{k \\epsilon}(z) - G_k(z) \\|$\nis uniformly bounded by $B \\epsilon$}\n\\end{proof}\t\n\\medskip\n\\par\n\\oldtext{\n\\begin{proof}[End of the proof of Theorem~\\ref{th1}:]\n\tWe define:\n\\begin{eqnarray*}\n\\psi(f_0, \\cdots, f_d) \n\t&=&\n\t\\left( \\log|f_0|, \\cdots, \\log|f_d| ;\n\t\\arg|f_0|, \\cdots, \\arg|f_d| \\right) \\\\\n\\eta(r_0, \\cdots, r_d ; a_0, \\cdots, a_d) \n\t&=&\n\t\\left( \\exp {(r_0-r_1)}, \\cdots, \\exp {(r_{d-1} - r_d)} \n\t\\right) \\\\\nR(r_0, \\cdots, r_d ; a_0, \\cdots, a_d) \n\t&=&\n\t\\left( \\frac{r_0}{2}, \\cdots, \\frac{r_d}{2}\n ; a_0, \\cdots, a_d \\right) \\\\\n\\end{eqnarray*}\n\\medskip\n\\par\n\tWe define $\\Phi$ as the function that associates\n\tto a polynomial $f$ the values $|\\zeta_1|, \\dots, |zeta_d|$\n where $\\zeta_i$ are the roots of $f$, ordered by decreasing\n\tmodulus. \n\\par\n\tThen we set\n\\[\n\\varphi = \\eta^{-1} \\circ \\Phi \\circ \\psi^{-1}\n\\]\n \n\\medskip\n\\par\n\tWith the definitions above, $G^{R,k}$ \n\tis indeed a renormalization\n\tof the classical Graeffe algorithm to compute $\\Phi$. \n\\end{proof}\n}\n\\newtext{This concludes the proof of Proposition~\\ref{prop1} and\nhence of Theorem~\\ref{th1}}{A6\\S 1(26)}\n\n\\section{Probability of Success}\n\\label{Sec5}\n\n\n\\oldtext{MOST OF THIS SECTION WAS REWRITTEN FROM SCRATCH}\n\nThrough this section, $\\|.\\|_d$ will denote Weyl's unitary\ninvariant norm (\\cite{WEYL} ~III--7, pp~137--140) \nin the space $\\mathcal P_d$ of complex \npolynomials of degree at most $d$ (See~\\cite{BCSS}). \nIf $f(x) = \\sum_{i=0}^d f_i x^i$,\nthen\n\\[\n\\| f \\|_d = \\sqrt{ \\sum_{i=0}^d \\frac{|f_i|}{\\binomial{d}{i}}}\n\\]\n\\par\nThis norm is invariant under the following action of the group\n$U(2)$ of unitary $2\\times 2$ matrices: if \n$\\varphi = \\left[ \\begin{array}{cc} \\alpha & \\beta \\\\ \\gamma & \\delta\n\\end{array} \\right]$ is unitary, define\n\\[\nf^{\\varphi} (y) = (\\gamma y + \\delta)^d f\\left( \\frac{\\alpha y + \\beta}\n{\\gamma y + \\delta} \\right)\n\\]\nThus, $\\| f^{\\varphi} \\|_d = \\| f \\|$. (See Ch.~12, Th.~1 of \\cite{BCSS}.)\n\n\nThis is in some sense the most `natural' norm in the space of\nall polynomials. \\newtext{More information about that norm, its associated\nprobability distribution and its applications can be found in}{A5(14)}\n\\cite{BD,DEDIEU2,DEDIEU3,KOSTLAN,TCS,SPLITTING,BEZI,BEZII,BEZIII, BEZIV,\nBEZV}.\n\n\n\n\nLet $\\mathbb P \\mathcal P_d$ be the projectivization of normed complex vector\nspace $(\\mathcal P_d, \\|.\\|_d)$. We can define the `sine' distance in\n$\\mathbb P \\mathcal P_d$ by:\n\\[\nd_{\\mathbb P} (f,g) = \\min_{\\lambda \\in \\mathbb C} \\frac{\\|f - \\lambda g\\|}\n{\\|f\\|} \n=\n\\sin \\varrho(f,g)\n\\]\nwhere $\\varrho$ is the usual Riemann metric in $\\mathbb P \\mathcal P_d$.\nAlso, there is a natural volume form in $\\mathbb P \\mathcal P_d$.\nNormal invariant distributed polynomials in $\\mathcal P_d$ correspond\nto uniformly distributed `polynomials' in $\\mathbb P \\mathcal P_d$.\n\nLet $f$ be a random polynomial (in any of the two equivalent senses above).\nThen we may consider its roots $\\zeta_1, \\cdots, \\zeta_d$ as random variables.\nThe joint distribution of $\\zeta_1, \\cdots, \\zeta_d$ was studied by Kostlan in\n~\\cite{KOSTLAN}. However, the result below will rely on elementary\nestimates:\n\n\\begin{lemma} \\label{lem6}\n Let $f$ be random in the sense above.\n The probability that \n\\[\n\\min_{|\\zeta_i|>|\\zeta_j|} \\frac{|\\zeta_i|}{|\\zeta_j|} > 1+\\epsilon\n\\]\n is larger than $1-M \\epsilon$, where $M$ is a positive constant depending\n on $d$.\n\\end{lemma}\n\n\n This result is also true if we chose $f$ random with respect to any \n other probability distribution \\oldtext{\n \n that is absolutely continuous\n with respect to the volume form of $\\mathbb P \\mathcal P_d$.}\n \\newtext{with bounded Radon-Nikodym derivative with respect to\n the volume form in $\\mathbb P \\mathcal P_d$.}{Precise definition.}\n The constant $M$ will have to be multiplied by the maximum of the\n Radon-Nikodym derivative of the new distribution with respect to\n the volume form.\n\n\nLemma~\\ref{lem6} will be a consequence of the `Condition Number \nTheorem' below. Let\n\\[\n\\rho(f) = \\min_{|\\zeta_i| < |\\zeta_j|} 1 - \\frac{|\\zeta_i|}{|\\zeta_j|}\n\\]\n\nWe will interpret $\\rho(f)^{-1}$ as a condition number. Let\n$\\Sigma_G$ be the locus of ill-posed problems, i.e., the set of\npolynomials such that $|\\zeta_i| = |\\zeta_j|$ for some $i \\ne j$. Then,\n\\begin{theorem}[Condition Number Theorem for Graeffe Iteration]\\label{cnt}\n\\[\n\\rho(f) \\ge \\frac{ d_{\\mathbb P} (f, \\Sigma_G) }{\\sqrt{d}}\n\\]\n\\end{theorem}\n\n\nTherefore, the probability that $\\rho(f)>\\epsilon$ is no\nless than $1 - \\mathbf Vol\\ V_{\\epsilon \\sqrt{d}} \\Sigma_G$ where \n$V_{\\epsilon \\sqrt{d}}$ denotes an $\\epsilon \\sqrt{d}$-neighborhood. \nThis volume is of $O(\\epsilon \\sqrt{d} ) \\le M \\epsilon$. \n\n\n\n\n\\begin{lemma}\\label {norms1}\n\tLet $f(x) = (x-\\zeta_1) g(x) \\in \\mathcal P_d$, where\n\t$g \\in \\mathcal P_{d-1}$. Then\n\\[\n\\| g \\|_{d-1} \\le \\sqrt{ \\frac{d}{1+|\\zeta_1|^2}} \\| f \\|_d\n\\]\n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma~\\ref{norms1}]\n\nWe start with the easy case and assume that $\\zeta_1=0$. Set\n$g(x) = \\sum_{i=0}^{d-1} g_i x^i$. Then, $f(x) = \\sum_{i=1}^{d} g_{i-1} x^i$.\nNow,\n\\[\n\\|g\\|_{d-1}^2 \n=\n\\sum _{i=0}^{d-1} \\frac{|g_i|^2}{\\binomial{d-1}{i}}\n=\n\\sum _{i=1}^{d} \\frac{|g_{i-1}|^2}{\\binomial{d-1}{i-1}}\n=\n\\sum _{i=1}^{d} \\frac{d}{i} \\frac{|g_{i-1}|^2}{\\binomial{d}{i}}\n\\le\nd \\|f\\|_d ^2\n\\]\nand hence $\\|g\\|_{d-1} \\le \\sqrt{d} \\|f\\|_d$.\n\nFor the general case, we will use $U(2)$-invariance of\n$\\|.\\|_d$ and $\\|.\\|_{d-1}$. Let $\\varphi$ be a convenient\nunitary matrix:\n\\[\n\\varphi = \\frac{1}{\\sqrt{1+|\\zeta_1|^2}}\n\\left[\n\\begin{array}{cc} 1 & \\zeta_1 \\\\ - \\bar{\\zeta_1} & 1 \\end{array} \\right]\n\\]\n\nSet $f^{\\varphi} = f \\circ \\varphi$ and $g^{\\varphi} = g \\circ \\varphi$.\nThe choice of $\\varphi$ has the particularity that $f^{\\varphi}(0) =\nf(\\zeta_1) = 0$. We can compute $f^{\\varphi}$ in terms of $g^{\\varphi}$:\n\\[\nf^{\\varphi}(y) = \\sqrt{1 + |\\zeta_1|^2 } \\ y \\ g^{\\varphi} (y)\n\\]\n\nUsing the easy case,\n\\[ \n\\| \\sqrt{1 + |\\zeta_1|^2 } g^{\\varphi} \\|_{d-1}\n\\le\n\\sqrt{d} \\|f^{\\varphi}\\|_d\n\\]\n\nBy $U(2)$-invariance,\n\n\\begin{eqnarray*}\n\\| g \\|_{d-1} &=& \\| g^{\\varphi} \\| _{d-1} \\\\\n&=& \\frac{1}{ \\sqrt{1 + |\\zeta_1|^2 }} \\| \\sqrt{1 + |\\zeta_1|^2 } g^{\\varphi} \\| _{d-1} \\\\\n&\\le& \\frac{\\sqrt{d}}{ \\sqrt{1 + |\\zeta_1|^2 }} \\| f^{\\varphi} \\|_d \\\\\n&=&\n\\frac{\\sqrt{d}}{ \\sqrt{1 + |\\zeta_1|^2 }} \\| f \\|_d\n\\end{eqnarray*}\n\\end{proof}\n\n\\begin{lemma} \\label{norms2}\n Let $g \\in \\mathcal P_{d-1}$. Then $\\|g\\|_d \\le \\|g\\|_{d-1}$.\n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma~\\ref{norms2}]\n\\[\n\\|g\\|_d^2 = \n\\sum_{i=0}^{d-1} \\frac{|g_i|^2}{\\binomial{d}{i}}\n=\n\\sum_{i=0}^{d-1} \\frac{d-i}{d} \\frac{|g_i|^2}{\\binomial{d-1}{i}}\n\\le\n\\|g\\|_{d-1}^2 \n\\]\n\\end{proof}\n\nPutting Lemma~\\ref{norms1} and Lemma~\\ref{norms2} together,\n\n\\begin{lemma}\\label {norms3}\n\tLet $f(x) = (x-\\zeta_1) g(x) \\in \\mathcal P_d$, where\n\t$g \\in \\mathcal P_{d-1}$. Then\n\\[\n\\| g \\|_{d} \\le \\frac{\\sqrt{d}}{\\sqrt{1+|\\zeta_1|^2}} \\| f \\|_d\n\\]\n\\end{lemma}\n\n\n\\begin{proof}[Proof of Theorem~\\ref{cnt}]\n\nLet $f(x) = (x-\\zeta_1) (x-\\zeta_2) \\cdots (x-\\zeta_d)$ and order the $\\zeta_i$'s\nsuch that\n\\[\n\\rho(f) = \\min_{|\\zeta_i| < |\\zeta_j|} 1- \\frac{|\\zeta_i|}{|\\zeta_j|} \n= 1 - \\frac{|\\zeta_2|}{|\\zeta_1|} \n\\]\n\n\nDefine $h(z) = (x - \\zeta_1 (1-\\rho)) (x-\\zeta_2) \\cdots (x-\\zeta_d) \\in \\Sigma_G$.\nThen\n\\begin{eqnarray*}\n\\|f - h\\|_d\n&=&\n\\| \\zeta_1 \\rho (x-\\zeta_2) \\cdots (x-\\zeta_d) \\|_d \\\\\n&=& |\\zeta_1| \\rho \\| (x-\\zeta_2) \\cdots (x-\\zeta_d) \\|_d\\\\\n&\\le&\n|\\zeta_1| \\rho \\frac{\\sqrt{d}}{\\sqrt{1+|\\zeta_1|^2}} \\|f\\|_d\n\\end{eqnarray*}\n\n\nHence,\n\n\\[\n\\frac{\\|f-h\\|_d}{\\|f\\|_d} \\le \\frac{|\\zeta_1|}{\\sqrt{1+|\\zeta_1|^2}} \n\\rho \\sqrt{d} \\le \\rho \\sqrt{d} \n\\]\n\nHence,\n\\[\nd_{\\mathbb P} (f, \\Sigma_G) \\le \\rho \\sqrt{d}\n\\]\n\\end{proof}\n\n\n\n\n\n\\begin{proof}[Proof of Theorem~\\ref{th2}]\n\n\n\n\nWe set $\\delta = M \\epsilon$, where $M$ is the constant such that\nthe volume of an $\\epsilon \\sqrt{d}$ neighborhood of $\\Sigma_G$ is less\nthan $M \\epsilon$. With probability\nlarger than $1-\\delta$,\n\\[ \n\\max_{|\\zeta_{i}| < |\\zeta_{j}|} \\frac{|\\zeta_{i}|}{|\\zeta_{j}|}\n> 1 - \\epsilon \n\\mbox{ .} \\]\n\\par\nWe now use the fact that, for any $N > (\\log 2) \/ \\epsilon$, \nwe have $(1-\\epsilon)^N < 1\/2$. We set $k_1 = 1 + \\lceil \\log_2 \\epsilon^{-1}\n\\rceil$ and using $N=2^{k_1}$ in the previous formula, we obtain:\n\\[\n\\max_{|\\zeta_{i}| < |\\zeta_{j}|}\n\\frac{|\\zeta_{i}|^{2^{k_1}}}{|\\zeta_{j}|^{2^{k_1}}}\n<\n(1-\\epsilon)^{2^{k_1}}\n< \\frac{1}{2}\n\\mbox{ .} \\]\n\\par\nAn extra $1+\\log_2 b$ iterations ensures that \n\\[\n\\max_{|\\zeta_{i}| < |\\zeta_{j}|}\n\\frac{|\\zeta_{i}|^{2^{k}}}{|\\zeta_{j}|^{2^{k}}}\n<\n2^{-1-b}\n\\mbox{ ,} \\]\nprovided that \n\\[ k = k_1 + 1 + \\log_2 b \\mbox{ .} \\]\nWe claim that we have in this case that the sum\n\\[ \\sum_{i_1 < \\dots < i_r} \\zeta_{i_1}^{2^{k}}\\dots \\zeta_{i_r}^{2^{k}}= \n\\zeta_{1}^{2^{k}} \\dots \\zeta_{r}^{2^{k}} \n(1+\\varepsilon) \\mbox{ ,} \\] \nwhere \n\\[| \\varepsilon |< 2^{-b} \\mbox{ .} \\]\nIndeed, assume we have reordered the roots so that \n$|\\zeta_1| > \\dots > |\\zeta_{d}| $.\nIt then follows that\n\\[ | \\zeta_{i_1}^{2^k} \\dots \\zeta_{i_r}^{2^k} | < |\\zeta_{1}^{2^k} \\dots \\zeta_{r}^{2^k}| \n2^{-(1+b)(i_1+\\dots + i_{r}- r (r+1)\/2)} \\mbox{ .} \\]\n\n\\oldtext{\nNow, let \n\\[ {\\cal S}_{k} = \\{(i_1,\\dots,i_n) \n\\ \\ | \\ \\ i_1 + \\dots + i_n - n(n+1)\/2 =k \\} \\mbox{ .} \\]\nNote that ${\\cal S}_{0}=\\{(1,\\dots,n)\\}$, and\n${\\cal S}_{1}=\\{(1,\\dots,n-1,n+1)\\}$. Also,\n${\\cal S}_{2}=\\{(1,\\dots,n-1,n+2) ; (1,\\dots,n-2,n,n+1)\\}$\n\n\nBut, $\\#{\\cal S}_{i+1} \\le d \\# {\\cal S}_i$, and hence\n\\[ \\sum_{i\\ge 1} 2^{-b i}\\#{\\cal S}_i \\le 2^{-b} (1+2 2^{-b} + \n2 d 2^{-b} + 2 d^2 2^{-2b}\\dots ) \n\\le 2^{-b+1}\\mbox{ ,} \\]\nunder the assumption that \n$d \\le 2^b$.\nThis concludes the proof of Theorem~\\ref{th2}.\n}\n\\newtext{\nNow, for $r$ fixed, let \n\\[ {\\cal S}_{k} = \\{(i_1,\\dots,i_r) \n\\ \\ | \\ i_1 < \\cdots < i_r \\text{ and }\ni_1 + \\dots i_r - r(r+1)\/2 =k \\} \\mbox{ .} \\]\nNote that ${\\cal S}_{0}=\\{(1,\\dots,r)\\}$, and\n${\\cal S}_{1}=\\{(1,\\dots,r-1,r+1)\\}$. Also,\n${\\cal S}_{2}=\\{(1,\\dots,r-1,r+2) ; (1,\\dots,r-2,r,r+1)\\}$\n\\par\nIn general, every multi-index $i_1 < \\cdots < i_r$ may be\nobtained by starting from $1<2< \\cdots -1+\\epsilon$ for all $i$;\n\n\\item[(b)] \\emph{$\\epsilon$-log canonical} (\\emph{$\\epsilon$-lc}, for\nshort) if $a_i\\geq -1+\\epsilon$ for all $i$;\n\n\\item[(c)] \\emph{terminal} if $a_i> 0$ for all $f$-exceptional divisors $F_i$. \n\\end{itemize}\nUsually we write $X$ instead of $(X,0)$ in the case $\\Delta=0$.\n\\end{definition}\nNote that $0$-klt (resp. $0$-lc) is just klt (resp. lc) in the usual sense. \n\n\n\\begin{definition}\nA variety $X$ is \\emph{of $\\epsilon$-Fano type} if there exists an effective $\\mathbb{Q}$-divisor $\\Delta$ such that $(X, \\Delta)$ is an $\\epsilon$-klt log Fano pair. \n\\end{definition}\n\nWe are mainly interested in the boundedness of varieties of $\\epsilon$-Fano type. \n\n\\begin{definition}\nA collection of varieties $\\{X_\\lambda\\}_{\\lambda\\in \\Lambda}$ is\nsaid to be \\emph{bounded} if there exists $h:\\mathcal{X}\\rightarrow\nS$ a projective morphism between schemes of finite type such that for\neach $X_\\lambda$, $X_\\lambda\\simeq \\mathcal{X}_s$ for some $s\\in S$.\n\\end{definition}\n\nOur motivation is the following BAB\nConjecture due to A. Borisov, L. Borisov, and V. Alexeev.\n\n\\begin{conj}[BAB Conjecture]\nFix $0<\\epsilon<1$, an integer $n>0$. \n\nThen the set of all\n$n$-dimensional varieties of $\\epsilon$-Fano type is bounded.\n\\end{conj}\nBAB Conjecture is one of the most important conjecture in birational geometry and it is related to the termination of flips. As the approach to this conjecture, \nwe\nare interested in the following much weak conjecture for\nanti-canonical volumes which is a consequence of BAB Conjecture.\n\n\\begin{conj}[Weak BAB Conjecture]\\label{conj2}\nFix $0<\\epsilon<1$, an integer $n>0$. \n\nThen there exists a number $M(n,\\epsilon)$ depending only\non $n$ and $\\epsilon$ with the following property:\n\nIf $X$ is an \n$n$-dimensional variety of $\\epsilon$-Fano type, then \n$${\\rm Vol}(-K_X) \\leq M(n,\\epsilon).$$\n\\end{conj}\n\nBAB Conjecture was proved in dimension two by Alexeev \\cite{AK2} with a simplified argument by Alexeev--Mori \\cite{AM}. In dimension three or higher, BAB Conjecture is still open. There are only some partial boundedness results. For example, we have boundedness of smooth Fano manifolds by Koll\\'ar--Miyaoka--Mori \\cite{KMM92}, \nthat of terminal $\\mathbb{Q}$-Fano $\\mathbb{Q}$-factorial threefolds of Picard number one\nby Kawamata \\cite{K}, that of canonical $\\mathbb{Q}$-Fano threefolds by Koll\\'ar--Miyaoka--Mori--Takagi \\cite{KMMT}, and that of toric varieties by Borisov--Borisov \\cite{BB}.\n\nWeak BAB Conjecture in dimension two was treated by Alexeev \\cite{AK2}, Alexeev--Mori \\cite{AM}, and Lai \\cite{Lai}. Recently, the author \\cite{J1} gave an optimal value for the number $M(2,\\epsilon)$ (see also Corollary \\ref{2bab}). For Weak BAB Conjecture in dimension three assuming that Picard number of $X$ is one, an effective value of $M(3, \\epsilon)$ was announced by Lai \\cite{Lai}. For general case in dimension three and higher, Weak BAB Conjecture is still open.\n\nAs the main theorem of this paper, we prove Weak BAB Conjecture in dimension three.\n\n\\begin{thm}\\label{BAB3main}\nWeak BAB Conjecture holds for $n=3$.\n\\end{thm}\n\nAs a consequence, we get a different proof of a result on the boundedness\nof log Fano varieties of fixed index in dimension three which was conjectured by Batyrev, and proved by A. Borisov \\cite{Bor} in dimension three and Hacon--M$^c$Kernan--Xu \\cite[Corollary 1.8]{HMX} in arbitrary dimension.\n\\begin{cor}\\label{baty}\nFix a positive integer $r$.\n\nLet $\\mathcal{D}$ be the set of all normal projective varieties $X$, where\n$\\dim X=3$, $K_X$ is $\\mathbb{Q}$-Cartier, and there exists an effective $\\mathbb{Q}$-divisor $\\Delta$ such that $(X,\\Delta)$ is klt and $-r(K_X + \\Delta)$ is \nCartier and ample. \n\nThen $\\mathcal{D}$ forms a bounded family.\n\\end{cor}\n \n\n{\\it Acknowledgments.} The author would like to express his \ngratitude to his supervisor Professor Yujiro Kawamata, for \nsuggestions, discussions, encouragement, and support. \nThe author is indebted to Professors Florin Ambro, Caucher Birkar, Yoshinori Gongyo, Dr. Yusuke Nakamura, and Mr. Pu Cao for effective conversations. \nA part of this paper was done during the author's visit to University of Cambridge in 2013 and he would like to thank Professors Caucher Birkar and Yifei Chen for the hospitality. \n\n\n\\section{Description of the proof}\nFirstly, we give an approach to Weak BAB Conjecture via Mori fiber spaces. \n\n\\begin{definition}\nA projective morphism $X\\rightarrow T$ between normal varieties is called a \\emph{Mori fiber space} if the following conditions hold:\n\\begin{itemize}\n\\item[(i)] $X$ is $\\mathbb{Q}$-factorial with terminal singularities; \n\n\n\\item[(ii)] $f$ is a {\\it contraction}, i.e., $f_*\\OO_X=\\OO_T$; \n\n\n\\item[(iii)] $-K_X$ is ample over $T$;\n\n\\item[(iv)] $\\rho(X\/T)=1$;\n\n\\item[(v)] $\\dim X > \\dim T$. \n\\end{itemize}\nAt this time, we say that $X$ is with a \\emph{Mori fiber structure}. \n\\end{definition}\n\n\n\nWe raise the following conjecture for Mori fiber spaces. \n\\begin{conj}[Weak BAB Conjecture for Mori fiber spaces]\nFix $0<\\epsilon<1$, an integer $n>0$. \n\n\nThen there exists a number $M(n,\\epsilon)$ depending only\non $n$ and $\\epsilon$ with the following property:\n\nIf $X$ is an\n$n$-dimensional variety of $\\epsilon$-Fano type with a Mori fiber structure, then \n$${\\rm Vol}(-K_X) \\leq M(n,\\epsilon).$$ \n\\end{conj}\n\nWe prove the following theorem by using Minimal Model Program.\n\\begin{thm}\\label{main thm mfs}\nWeak BAB Conjecture holds for fixed $\\epsilon$ and $n$ if and only if Weak BAB Conjecture for Mori fiber spaces holds for fixed $\\epsilon$ an $n$.\n\\end{thm}\n\nBy Theorem \\ref{main thm mfs}, to consider the boundedness of anti-canonical volumes of log Fano pairs, we only need to consider the ones with better singularities ($\\mathbb{Q}$-factorial terminal singularities) and with an additional structures (Mori fiber structures). This is the advantage of this theorem. In dimension two, this theorem appears as a crucial step to get the optimal value of $M(2,\\epsilon)$ (c.f. \\cite{J1}).\n\nRestricting our interest to dimension three, we prove the following theorem.\n\\begin{thm}\\label{mfs3}\nWeak BAB Conjecture for Mori fiber spaces holds for $n=3$.\n\\end{thm}\n\nTheorem \\ref{BAB3main} follows from Theorems \\ref{main thm mfs} and \\ref{mfs3} directly.\n\n\nTo prove Theorem \\ref{mfs3}, we need to consider $3$-fold $X$ of $\\epsilon$-Fano type with a Mori fiber structure $X\\rightarrow T$. There are $3$ cases:\n\\begin{itemize}\n\\item[(1)] $\\dim T=0$, $X$ is a $\\mathbb{Q}$-factorial terminal $\\mathbb{Q}$-Fano $3$-folds with $\\rho=1$;\n\\item[(2)] $\\dim T=1$, $X\\rightarrow T\\simeq\\mathbb{P}^1$ is a {\\it del Pezzo fibration}, i.e. a general fiber is a smooth del Pezzo surface;\n\\item[(3)] $\\dim T=2$, $X\\rightarrow T$ is a {\\it conic bundle}, i.e. a general fiber is a smooth rational curve.\n\\end{itemize}\n\nThe second statement is implied by the following fact: if $(X, \\Delta)$ is a klt log Fano pair, then $X$ is rationally connected (see \\cite[Theorem 1]{ZQ}), in particular, for any surjective morphism $X\\rightarrow T$ to a normal curve, $T\\simeq \\mathbb{P}^1$.\n\nIn Case (1), $X$ is bounded by Kawamata \\cite{K}, and the optimal bound of ${\\rm Vol}(-K_X)=(-K_X)^3$ is $64$ due to the classification on smooth Fano $3$-folds of Iskovskikh and Mori--Mukai and by Namikawa's result \\cite{Nami} (Gorenstein case) and Prokhorov \\cite{ProkK} (non-Gorenstein case).\n\nWe will mainly treat Cases (2) and (3). \n\nOne basic idea is to construct singular pairs which is not klt along fibers of $X\\rightarrow T$. Then by Connectedness Lemma, we may find a non-klt center intersecting with the fibers. Finally by restricting on a general fiber, we get the bound after some arguments on lower dimensional varieties. But several difficulties arise here.\n \nIn Case (3), the difficulty arises in the construction of singular pair\n because we need to avoid components which are vertical over $T$. To do this, we need a good understanding of the singularities and boundedness of the surface $T$, which was done by several papers as \\cite{AK2}, \\cite{MP}, and \\cite{Birkar}.\n\nIn Case (2), the difficulty arises in the last step. After restricting on a general fiber, we need to bound the (generalized) log canonical thresholds on surfaces. So we are done by proving the following (generalized) Ambro's conjecture in dimension two.\n\n\\begin{definition}\nLet $(X, B)$ be an lc pair and $D\\geq 0$ be a $\\mathbb{Q}$-Cartier $\\mathbb{Q}$-divisor. The\n{\\it log canonical threshold} of $D$ with respect to $(X, B)$ is\n$$\\lct(X, B; D) = \\sup\\{t\\in \\mathbb{Q} \\mid (X, B+ tD) \\text{ is lc}\\}.$$\nFor the use of this paper, we need to consider the case when $D$ is not effective. \nLet $G$ be a $\\mathbb{Q}$-Cartier $\\mathbb{Q}$-divisor satisfying $G+B\\geq 0$, The\n{\\it generalized log canonical threshold} of $G$ with respect to $(X, B)$ is\n$$\\glct(X, B; G) = \\sup\\{t\\in [0,1] \\cap \\mathbb{Q} \\mid (X, B+ tG) \\text{ is lc}\\}.$$\nNote that the assumption $t\\in [0,1]$ guarantees that $B+ tG\\geq 0$.\n\\end{definition}\n\\begin{conj}[Ambro's conjecture]\\label{ac}\nFix $0<\\epsilon<1$ and integer $n>0$. \n\nThen there exists a number $\\mu(n,\\epsilon)>0$ depending only on $n$ and $\\epsilon$ with the following property:\n \nIf $(Y, B)$ is an $\\epsilon$-klt log Fano pair of dimension $n$, then\n$$\n\\inf\\{\\lct(Y,B;D)\\mid D\\sim_\\mathbb{Q}-(K_Y+B), D\\geq 0 \\}\\geq \\mu(n,\\epsilon).\n$$\n\\end{conj}\nNote that we do not assume any special conditions on the coefficients of $B$. The left-hand side of the inequality is called {\\it $\\alpha$-invariant} of $(Y, B)$ which generalizes the concept of $\\alpha$-invariant of Tian for Fano manifolds in differential geometry (see \\cite{CGM, Dem, Tian}). Recently Ambro \\cite{Am} announced a proof of this conjecture assuming that $(Y, B)$ is a toric pair where an explicit sharp number $\\mu(n, \\epsilon)$ was given.\nFor the use of this paper, we need a stronger version of this conjecture where $D$ may not be effective.\n\\begin{conj}[generalized Ambro's conjecture]\\label{gac}\nFix $0<\\epsilon<1$ and integer $n>0$. \n\nThen there exists a number $\\mu(n,\\epsilon)>0$ depending only on $n$ and $\\epsilon$ with the following property:\n\n If $(Y, B)$ is an $\\epsilon$-klt weak log Fano pair of dimension $n$ and $Y$ has at worst terminal singularities, then\n$$\n\\inf\\{\\glct(Y,B;G)\\mid G\\sim_\\mathbb{Q}-(K_Y+B), G+B\\geq 0 \\}\\geq \\mu(n,\\epsilon).\n$$\n\\end{conj}\nNote that Conjecture \\ref{ac} follows from Conjecture \\ref{gac} easily after taking terminalization of $(Y,B)$.\n \n\nWe prove the conjecture in dimension two by following some ideas in the proof of BAB Conjecture in dimension two (\\cite{AK2, AM}). But it seems that this conjecture does not follow from BAB Conjecture trivially.\n \\begin{thm}\\label{gac2}\nConjecture \\ref{gac} holds for $n=2$.\n\\end{thm} \n\n\nFor the proof of Corollary \\ref{baty}, we basically follow the idea in \\cite{Bor} to bound the Hilbert polynomials by \\cite{Matsusaka}.\n\n\n\n\n\nThis paper is organized as follows. In Section \\ref{section reduction}, we prove the reduction step to Mori fiber spaces (Theorem \\ref{main thm mfs}). In Section \\ref{section ambro}, we prove generalized Ambro's conjecture in dimension two (Theorem \\ref{gac2}). In Section \\ref{section mfs}, we prove Weak BAB Conjecture for Mori fiber spaces in dimension three (Theorem \\ref{mfs3}). In Section \\ref{section bat}, we prove the boundedness of log Fano threefolds of fixed index (Corollary \\ref{baty}).\n\n\\section{Preliminaries}\n\n\\subsection{Volumes}\\\n\n\\begin{definition}\nLet $X$ be an $n$-dimensional projective variety and $D$ be a Cartier divisor on $X$. The {\\it volume} of $D$ is the real number\n$$\n{\\rm Vol}(D)=\\limsup_{m\\rightarrow \\infty}\\frac{h^0(X,\\OO_X(mD))}{m^n\/n!}.\n$$\nNote that the limsup is actually a limit. Moreover by the homogenous property of the volume, we can extend the definition to $\\mathbb{Q}$-Cartier $\\mathbb{Q}$-divisors. Note that if $D$ is a nef $\\mathbb{Q}$-divisor, then $\\Vol(D)=D^n$. If $D$ is a non-$\\mathbb{Q}$-Cartier $\\mathbb{Q}$-divisors, we may take a $\\mathbb{Q}$-factorialization of $X$, i.e., a birational morphism $\\phi:Y\\to X$ which is isomorphic in codimension one and $Y$ is $\\mathbb{Q}$-factorial, then $\\Vol(D):=\\Vol(\\phi^{-1}_*D)$. Note that $\\mathbb{Q}$-factorialization always exists for klt pairs (cf. \\cite[Theorem 1.4.3]{BCHM}).\n\\end{definition}\n\nFor more background on volumes, see \\cite[11.4.A]{Positivity2}.\n\n\\subsection{Hirzebruch surfaces}\\\n\nWe\nrecall some basic properties of the Hirzebruch surfaces\n$\\mathbb{F}_n=\\mathbb{P}_{\\mathbb{P}^1}(\\mathcal{O}_{\\mathbb{P}^1}\\oplus\\mathcal{O}_{\n\\mathbb{P}^1}(n))$, $n\\geq 0$. Denote by $h$ (resp. $f$) the\nclass in $\\textrm{Pic }\\mathbb{F}_n$ of the tautological bundle\n$\\mathcal{O}_{\\mathbb{F}_n}(1)$ (resp. of a fiber). Then\n$\\textrm{Pic }\\mathbb{F}_n=\\mathbb{Z}h\\oplus \\mathbb{Z}f$ with\n$f^2=0$, $f\\cdot h=1$, $h^2=n$. If $n>0$, there is a unique\nirreducible curve $\\sigma_n\\subset \\mathbb{F}_n$ such that $\\sigma_n \\sim h-nf$, $\\sigma_n^2=-n$. For\n$n=0$, we can also choose one curve whose class in $\\textrm{Pic\n}\\mathbb{F}_0$ is $h$ and denote it by $\\sigma_0$. \nNote that \n$$\n-K_{\\mathbb{F}_n}\\sim 2h-(n-2)f\\sim 2\\sigma_n+(n+2)f.\n$$\n\\begin{lem}\\label{multif}\nFor an effective $\\mathbb{Q}$-divisor $D\\sim_\\mathbb{Q} -K_{\\mathbb{F}_n}$ and a fiber $f$, $\\mult_f D\\leq n+2$.\n\\end{lem}\n\\begin{proof}\nSince $D-(\\mult_f D)f$ is effective, $(D-(\\mult_f D)f)\\cdot h\\geq 0$. On the other hand, $(D-(\\mult_f D)f)\\cdot h=n+2-\\mult_f D$.\n\\end{proof}\n\\begin{lem}\\label{multQ}\nLet $T=\\mathbb{P}^2$ or $\\mathbb{F}_n$, then for an effective $\\mathbb{Q}$-divisor $D\\sim_\\mathbb{Q} -K_{T}$ and a point $Q$,\n$\\mult_Q D\\leq n+4$ holds. Moreover, if we write $D=\\sum_j b_jD_j$ by its components and assume that $b_j\\leq 1$ for all $j$, then $\\sum_j b_j\\leq 4.$\n\\end{lem}\n\\begin{proof}\nIf $T=\\mathbb{P}^2$, taking a general line $L$ through $Q$, we have\n$$\n3=(D\\cdot L)\\geq \\mult_{Q}(D).\n$$\n\nIf $T=\\mathbb{F}_n$, take $f$ be the fiber passing through $Q$, by Lemma \\ref{multif} and intersection theory, we have\n$$\n2=D\\cdot f\\geq \\mult_{Q}D- \\mult_{f}D\\geq \\mult_{Q}D-n-2.\n$$\n\n\n\nFor the latter statement, if $T=\\mathbb{F}_n$, then the conclusion follows by \\cite[Lemma 1.4]{AM}. If $T=\\mathbb{P}^2$, then $\\sum b_j\\leq 3$ by degree computation. \n\\end{proof}\n\n\\subsection{Non-klt centers and connectedness lemma}\n\\begin{definition}\nLet $X$ be a normal projective variety and $\\Delta$ be a $\\mathbb{Q}$-divisor on $X$ such that $K_X+\\Delta$ is $\\mathbb{Q}$-Cartier. Let $f: Y\\rightarrow X$ be a log\nresolution of $(X, \\Delta)$, write\n$$\nK_Y =f^*(K_X+\\Delta)+\\sum a_iF_i,\n$$\nwhere $F_i$ is a prime divisor. $F_i$ is called a {\\it non-klt place} of $(X, \\Delta)$ if $a_i\\leq -1$.\nA subvariety $V\\subset X$ is called a {\\it non-klt center} of $(X, \\Delta)$ if it is the image of a non-klt place. The {\\it non-klt locus} $\\text{Nklt}(X, \\Delta)$ is the union of all non-klt centers of $(X, \\Delta)$. A non-klt center is {\\it maximal} if it is an irreducible component of $\\text{Nklt}(X, \\Delta)$.\n\\end{definition}\n\nThe following lemma suggests a standard way to construct non-klt centers.\n\\begin{lem}[{cf. \\cite[Lemma 2.29]{KM}}]\\label{dimk}\nLet $(X, \\Delta)$ be a pair and $Z\\subset X$ be a close subvariety of codimesion $k$ such that $Z$ is not contained in the singular locus of $X$. If $\\mult_Z \\Delta\\geq k$, then $Z$ is a non-klt center of $(X, \\Delta)$. \n\\end{lem}\nRecall that the {\\it multiplicity} $\\mult_ZF$ of a divisor $F$ along a subvariety $Z$ is defined by the multiplicity $\\mult_xF$ of $F$ at a general point $x\\in Z$.\n\nUnfortunately, the converse of Lemma \\ref{dimk} is not true unless $k=1$. Usually we do not have good estimations for the multiplicity along a non-klt center but the following lemma.\n\n\\begin{lem}[{cf. \\cite[Theorem 9.5.13]{Positivity2}}]\nLet $(X, \\Delta)$ be a pair and $Z\\subset X$ be a non-klt center of $(X, \\Delta)$ such that $Z$ is not contained in the singular locus of $X$. Then $\\mult_Z \\Delta\\geq 1$. \n\\end{lem}\n\nIf we assume some simple normal crossing condition on the boundary, we can get more information on the multiplicity along a non-klt center. For simplicity, we just consider surfaces.\n\\begin{lem}[{cf. \\cite[4.1 Lemma]{McK}}]\\label{multi}\nFix $0 1-e$.\n\\end{lem}\n\\begin{proof}\nBy taking a sequence of point blow-ups, we can get the divisor $E$. Consider the blow-up at $P$, we have $f:S_1\\rightarrow S$ with $K_{S_1}+B_1+D_1+mE_1=f^*(K_S+B+D)$ where $B_1$ and $D_1$ are the strict transforms of $B$ and $D$ respectively, and $E_1$ is the exceptional divisor with $m=\\mult_P(B+D)-1\\leq 1-e+2e-1=e$. \nNow $D_1+mE_1$ is again simple normal crossing supported and $\\mult_QB_1\\leq \\mult_PB$ for $Q\\in E_1$. Hence by induction on the number of blow-ups, we conclude that the coefficient of $E$ is at most $e$ and hence $a_{E}(S, B+D)\\geq -e$. \n\\end{proof}\n\n\n\n\n\nWe have the following connectedness lemma of Koll\\'{a}r and Shokurov for non-klt locus (cf. \nShokurov \\cite{Shokurov}, Koll\\'{a}r \\cite[17.4]{Kol92}).\n\\begin{thm}[Connectedness Lemma]\nLet $f:X\\rightarrow Z$ be a proper morphism of normal varieties with connected fibers and $D$ is a $\\mathbb{Q}$-divisor such that $-(K_X+D)$ is $\\mathbb{Q}$-Cartier, $f$-nef and $f$-big. Write $D=D^+-D^-$ where $D^+$ and $D^-$ are effective with no common components. If $D^-$ is $f$-exceptional (i.e. all of its components have image of codimension at least $2$), then ${\\rm Nklt} (X,D)\\cap f^{-1}(z)$ is connected for any $z\\in Z$. \n\\end{thm}\n\\begin{remark}\nThere are two main cases of interest of Connectedness Lemma:\n\\begin{itemize}\n\\item[(i)] $Z$ is a point and $(X,D)$ is a weak log Fano pair. Then $\\Nklt(X,D)$ is connected.\n\\item[(ii)] $f:X\\rightarrow Z$ is birational, $(Z,B)$ is a log pair and $K_X+D=f^*(K_Z+B)$.\n\\end{itemize}\n\\end{remark}\n\n\n\n\n\\section{Reduction to Mori fiber spaces}\\label{section reduction}\nIn this section, we prove the reduction step to Mori fiber spaces (Theorem \\ref{main thm mfs}).\n\nThe ``only if'' direction is trivial, we only need to prove the ``if'' direction.\n\nFix $0<\\epsilon<1$, an integer $n>0$. Let $(X,\\Delta)$ be an $\\epsilon$-klt log Fano pair of dimension $n$.\nBy \\cite[Corollary 1.4.3]{BCHM}, taking $\\mathbb{Q}$-factorialization of $(X, \\Delta)$, we have $\\phi: X_0 \\rightarrow X$ where\n$K_{X_0}+\\phi^{-1}_*\\Delta=\\phi^*(K_X+\\Delta)$, $X_0$ is $\\mathbb{Q}$-factorial, and $\\phi$ is isomorphic in codimension one. Note that $\\Vol(-K_{X_0})=\\Vol(-K_{X})$.\n\n\n\nAgain by \\cite[Corollary 1.4.3]{BCHM}, taking terminalization of $X_1$, \nwe have $\\pi: X_1 \\rightarrow X_0$ where\n$K_{X_1}+\\Delta_{X_1}=\\pi^*(K_{X_0}+\\phi^{-1}_*\\Delta)$, \n$\\Delta_{{X_1}}$ is an effective $\\mathbb{Q}$-divisor, $X_1$ is $\\mathbb{Q}$-factorial terminal, and $(X_1,\\Delta_{X_1})$ is $\\epsilon$-klt.\nHere $-(K_{X_1}+\\Delta_{X_1})$ is nef and big. \n\n\nBy Kodaira's lemma (cf. \\cite[Proposition 2.61]{KM}) there exist a $\\mathbb{Q}$-divisor $\\Delta_1$ such that $\\Delta_1\\geq \\Delta_{X_1}$, $-(K_{X_1}+\\Delta_1)$ is ample, and $(X, \\Delta_1)$ is $\\epsilon$-klt. In particular, $X_1$ is $\\mathbb{Q}$-factorial terminal and of $\\epsilon$-Fano type. \n\nRunning $K$-MMP on $X_1$, we get a sequence of normal projective varieties: \n$$\nX_1\\dashrightarrow X_2 \\dashrightarrow X_3\\dashrightarrow \\cdots\\dashrightarrow X_r\\rightarrow T. \n$$ \nSince $-K_{X_1}$ is big, this sequence ends up with a Mori fiber space $X_r\\rightarrow T$ (cf. \\cite[Corollary 1.3.3]{BCHM}). In particular, $X_r$ is $\\mathbb{Q}$-factorial terminal. \n\nBeing of $\\epsilon$-Fano type is preserved by MMP according to the following lemma.\n\\begin{lem}[{cf. \\cite[Lemma 3.1]{GOST}}]\\label{mmp lem}\nLet $Y$ be a projective normal variety and $f: Y\\rightarrow Z$ be a projective birational contraction. \n\\begin{itemize}\n\\item[(1)] If $Y$ is of $\\epsilon$-Fano type, so is $Z$;\n\n\\item[(2)] Assume that $f$ is small, then $Y$ is of $\\epsilon$-Fano type if and only if so is $Z$.\n\\end{itemize}\nIn particular, minimal model program preserves $\\epsilon$-Fano type.\n\\end{lem}\n\n\n\\begin{proof}\nThe proof is almost the same as \\cite[Lemma 3.1]{GOST} where $0$-Fano type is considered.\nFirst we assume that $Y$ is of $\\epsilon$-Fano type, that is, there exists an effective $\\mathbb{Q}$-divisor $\\Delta$ on $Y$ such that $(Y, \\Delta)$ is $\\epsilon$-klt log Fano pair. Let $H$ be a general effective ample divisor on $Z$ and take a sufficiently small rational number $\\delta>0$ such that $-(K_Y+\\Delta+\\delta f^*H)$ is ample and $(Y, \\Delta+\\delta f^*H)$ is $\\epsilon$-klt. Then take a general effective ample $\\mathbb{Q}$-divisor $A$ on $Y$ such that $(Y, \\Delta+\\delta f^*H+A)$ is $\\epsilon$-klt and \n$$\nK_Y+\\Delta+\\delta f^*H+A\\sim_{\\mathbb{Q}}0.\n$$\nThen \n$$\nK_Z+f_*\\Delta+\\delta H+f_*A=f_*(K_Y+\\Delta+\\delta f^*H+A)\\sim_{\\mathbb{Q}}0,\n$$\nand \n$$\nf^*(K_Z+f_*\\Delta+\\delta H+f_*A)=K_Y+\\Delta+\\delta f^*H+A.\n$$\nTherefore, $(Z,f_*\\Delta+\\delta H+f_*A)$ is $\\epsilon$-klt.\nHence $(Z,f_*\\Delta+f_*A)$ is $\\epsilon$-klt and $-(K_Z+f_*\\Delta+f_*A)\\sim_{\\mathbb{Q}}\\delta H$ is ample, that is, $Z$ is of $\\epsilon$-Fano type.\n\nNext we assume that $f$ is small and $Z$ is of $\\epsilon$-Fano type. Let $\\Gamma$ be an effective $\\mathbb{Q}$-divisor on $Z$ such that $(Z, \\Gamma)$ is $\\epsilon$-klt log Fano pair. Let $\\Gamma_Y$ be the strict transform of $\\Gamma$ on $Y$. Since $f$ is small, \n$$\nK_Y+\\Gamma_Y=f^*(Z+\\Gamma).\n$$\nHence $(Y, \\Gamma_Y)$ is $\\epsilon$-klt and $-(K_Y+\\Gamma_Y)$ is nef and big. By Kodaira's lemma, there exist a $\\mathbb{Q}$-divisor $\\Gamma^\\prime$ such that $\\Gamma^\\prime\\geq\\Gamma_Y$, $-(K_{Y}+\\Gamma^\\prime)$ is ample and $(Y, \\Gamma^\\prime)$ is $\\epsilon$-klt, that is, $Y$ is of $\\epsilon$-Fano type. \n\nWe proved the lemma.\n\\end{proof}\n\nBy Lemma \\ref{mmp lem}, for all $i$, $X_i$ is of $\\epsilon$-Fano type. To compare the volumes between these varieties, we have the following lemma.\n\n\\begin{lem}\\label{vol lem}\nLet $X_i\\dashrightarrow X_{i+1}$ be one step of $K$-MMP. Then\n$$\n{\\rm Vol}(-K_{X_i})\\leq {\\rm Vol}(-K_{X_{i+1}}).\n$$\n\\end{lem}\n\n\\begin{proof}\nTake a common resolution $p:W \\rightarrow X_i$, $q: W\\rightarrow X_{i+1}$. Then\n$$\np^*(K_{X_i})=q^*(K_{X_{i+1}})+E,\n$$\nwhere $E$ is an effective $q$-exceptional $\\mathbb{Q}$-divisor. \nHence\n\\begin{align*}\n {\\rm Vol}(-K_{X_i}) \n={}&{\\rm Vol}(-p^*(K_{X_i}))\\\\\n={}&{\\rm Vol}(-q^*(K_{X_{i+1}})-E)\\\\\n\\leq {}&{\\rm Vol}(-q^*(K_{X_{i+1}}))\\\\\n= {}&{\\rm Vol}(-K_{X_{i+1}}).\n\\end{align*}\nWe proved the lemma.\n\\end{proof}\n\nTherefore we can compare the volumes on $X$ and $X_r$.\nRecall that we take $X_1$ as the terminalization of $X_0$, we have $K_{X_1}+F=\\pi^*K_{X_0}$ with $F$ an effective $\\mathbb{Q}$-divisor. Hence\n$$\n{\\rm Vol}(-K_{X_0})\\leq {\\rm Vol}(-K_{X_{1}}). \n$$\nBy Lemma \\ref{vol lem},\n\\begin{align*}\n{\\rm Vol}(-K_{X})\n={}&{\\rm Vol}(-K_{X_0})\\\\\n\\leq{}&{\\rm Vol}(-K_{X_1})\\\\\n\\leq {}&{\\rm Vol}(-K_{X_{r}}).\n\\end{align*}\n\nNow $X_r$ is an $n$-dimensional variety of $\\epsilon$-Fano type with a Mori fiber structure by construction. Assuming Weak BAB Conjecture for Mori fiber spaces, there exists $M(n, \\epsilon)$ such that \n$$\n{\\rm Vol}(-K_{X_{r}})\\leq M(n,\\epsilon). \n$$\nHence \n$$\n{\\rm Vol}(-K_{X})\\leq M(n,\\epsilon). \n$$\nWe complete the proof of Theorem \\ref{main thm mfs}. \n\nAs a direct corollary, we recover the main result in \\cite{J1} on Weak BAB Conjecture in dimension two. \n\\begin{cor}\\label{2bab}\nFix $0<\\epsilon<1$. \n\nThen there exists a number \n$$M(2,\\epsilon):=\\max\\Big\\{9, \\rounddown{2\/\\epsilon}+4+\\frac{4}{\\rounddown{2\/\\epsilon}}\\Big\\}$$\n with the following property:\n\nIf $X$ is a surface of $\\epsilon$-klt del Pezzo type, then \n$${\\rm Vol}(-K_X) \\leq M(2,\\epsilon).$$\n\\end{cor}\n\\begin{proof}\nBy Theorem \\ref{main thm mfs}, we only need to consider the cases when $X=\\mathbb{P}^2$ or $\\mathbb{F}_n$ with $n\\leq 2\/\\epsilon$ (see \\cite[Lemma 1.4]{AM} or \\cite[Lemma 3.1]{J1}). And the result follows by volume computation directly. \n\\end{proof}\n\n\\section{Generalized Ambro's conjecture in dimension two}\\label{section ambro}\nIn this section, we prove generalized Ambro's conjecture in dimension two (Theorem \\ref{gac2}).\n\nFix an $\\epsilon$-klt weak log del Pezzo pair $(S,B)$ with $S$ smooth and a $\\mathbb{Q}$-divisor $G\\sim_\\mathbb{Q} -(K_S+B)$ such that $G+B\\geq 0$. Set $a:=\\glct(S,B;G)$. Since we work on $\\mathbb{Q}$-divisors, $a$ is a positive rational number. The problem is to bound $a$ from below. We may assume that $a<1$. Set $D=G+B\\geq 0$. Then $(S, B+aG)=(S,(1-a)B+aD)$ is not klt.\nNote that $D\\sim_\\mathbb{Q} -K_S$.\n\nBy Base Point Free Theorem (cf. \\cite[Theorem 3.3]{KM}),\n$-(K_S+B)$ is semi-ample. Hence there exists an effective\n$\\mathbb{Q}$-divisor $M$ such that\n$K_S+B+M\\sim_\\mathbb{Q} 0$ and $(S, B+M)$ is $\\epsilon$-klt. \nFor any birational morphism $f:S\\rightarrow T$ between smooth surfaces, we have\n\\begin{align*}\nK_S+B+M={}&f^*(K_T+f_*B+f_*M), \\\\ \nK_S+(1-a)(B+M)+aD={}&f^*(K_T+(1-a)(f_*B+f_*M)+af_*D).\n\\end{align*}\nHence $(T, f_*B+f_*M)$ is $\\epsilon$-klt and $(T, (1-a)(f_*B+f_*M)+af_*D)$ is not klt with \n$$\nK_T+f_*B+f_*M\\sim_\\mathbb{Q} K_T+(1-a)(f_*B+f_*M)+af_*D\\sim_\\mathbb{Q} 0.\n$$\nRecall that either $S\\simeq \\mathbb{P}^2$ or there exists a birational\nmorphism $g:S\\rightarrow \\mathbb{F}_n$ with $n\\leq 2\/\\epsilon$ by \\cite[Lemma 1.4]{AM} or \\cite[Lemma 3.1]{J1}. \n\nHence by replacing $S$ by $T=\\mathbb{P}^2$ or $\\mathbb{F}_n$, we may assume that there exists a triple $(T, B_T, D_T)$ satisfying the following conditions:\n\\begin{itemize}\n\\item[(i)] $T=\\mathbb{P}^2$ or $\\mathbb{F}_n$ with $n\\leq 2\/\\epsilon$;\n\\item[(ii)] $B_T, D_T$ are effective $\\mathbb{Q}$-divisors on $T$;\n\\item[(iii)] $(T, B_T)$ is $\\epsilon$-klt and $(T, (1-a)B_T+aD_T)$ is not klt; \n\\item[(iv)] $K_T+B_T\\sim_\\mathbb{Q} K_T+(1-a)B_T+aD_T\\sim_\\mathbb{Q} 0$, equivalently, $B_T\\sim_\\mathbb{Q} D_T\\sim_\\mathbb{Q} -K_T$.\n\\end{itemize}\nSince $(T, (1-a)B_T+aD_T)$ is not klt, we may take a sequence of point blow-ups \n$$\nT_{r+1}\\rightarrow T_{r}\\rightarrow \\cdots \\rightarrow T_{2}\\rightarrow T_{1}=T\n$$\nwhere $T_{i+1}\\rightarrow T_i$ is the blow-up at a non-klt center $P_i\\in \\Nklt(T_i, (1-a)B_i+aD_i+E_i)$ where $B_i$ and $D_i$\nare the strict transforms of $B_T$ and $D_T$ respectively and \n$$\nK_{T_i}+(1-a)B_i+aD_i+E_i=\\pi_i^*(K_T+(1-a)B_T+aD_T),\n$$\nwhere $\\pi_i:T_i\\rightarrow T$ is the composition map and $E_i$ is a $\\pi_i$-exceptional $\\mathbb{Q}$-divisor. \nWe stop this process at $T_{r+1}$ if \n$$\\dim\\Nklt(T_{r+1}, (1-a)B_{r+1}+aD_{r+1}+E_{r+1})>0.$$\nSince $P_i$ is a non-klt center of $(T_i, (1-a)B_i+aD_i+E_i)$, $\\mult_{P_i}((1-a)B_i+aD_i+E_i)\\geq 1$. Note that the coefficients of $E_i$ are $(\\mult_{P_j}((1-a)B_j+aD_j+E_j)-1)$ for $j k$.\nWrite $B_T=\\sum_j b_jB^j$ and $B_i=\\sum_j b_jB_i^j$ by components. We have $b_j<1-\\epsilon$ since $(T, B_T)$ is $\\epsilon$-klt. Recall that $\\sum_j b_j\\leq 4$ by Lemma \\ref{multQ}.\n\n\n\\begin{claim}\nIf $\\mult_{B^j}(aD_T)>\\epsilon\/2$ for some $j$, then $a\\geq \\epsilon^2\/({4+4\\epsilon})$.\n\\end{claim}\n\\begin{proof}\nRecall that $T=\\mathbb{P}^2$ or $\\mathbb{F}_n$ with $n\\leq 2\/\\epsilon$.\n\nIf $T=\\mathbb{P}^2$, then $\\mult_{B^j}D_T\\leq 3$ be degree counting.\nIf $T= \\mathbb{F}_n$ and $B^j$ is a fiber, then $\\mult_{B^j}D_T\\leq n+2\\leq 2\/{\\epsilon}+2$ by Lemma \\ref{multif}. \nIf $T= \\mathbb{F}_n$ and $B^j$ is not a fiber, then $\\mult_{B^j}D_T\\leq D_T\\cdot f=2$ where $f$ is a fiber.\nHence \n$$\na\\geq \\frac{\\epsilon}{2\\mult_{B^j}D_T}\\geq \\frac{\\epsilon^2}{4+4\\epsilon}.\n$$\nWe proved the claim.\n\\end{proof}\n\nSince we need a lower bound of $a$, from now on, we may assume that $\\mult_{B^j}(aD_T)\\leq \\epsilon\/{2}$ for all $j$. In particular, $\\mult_{B_i^j}(aD_i)\\leq \\epsilon\/2$ and $$\\mult_{B_i^j}((1-a)B_i+aD_i) < 1-\\epsilon\/2$$ for all $i$ and $j$.\n\n\\begin{claim}\n$(B_{k+1}^j)^2\\geq -{4}\/{\\epsilon}$ for all $j$.\n\\end{claim}\n\\begin{proof}\nIf $(B_{k+1}^j)^2<0$, then\n\\begin{align*}\n-2\\leq{}&\n2p_a(B_{k+1}^j)-2=(K_{T_{k+1}}+B_{k+1}^j)\\cdot B_{k+1}^j\\\\\n={}&\\frac{\\epsilon}{2} (B_{k+1}^j)^2+(K_{T_{k+1}}+(1-\\frac{\\epsilon}{2})B_{k+1}^j)\\cdot B_{k+1}^j\\\\\n\\leq{}&\\frac{\\epsilon}{2} (B_{k+1}^j)^2+(K_{T_{k+1}}+ (1-a)B_{k+1}+aD_{k+1}+E_{k+1})\\cdot B_{k+1}^j\\\\\n={}&\\frac{\\epsilon}{2} (B_{k+1}^j)^2<0,\n\\end{align*}\nHence we proved the claim.\n\\end{proof}\n\n\n\nNow we can bound the number $k$. On $T_{k+1}$, we have\n\\begin{align*}\n(B_{k+1})^2={}&(\\sum_j b_jB_{k+1}^j)^2\\geq \\sum_j b_j^2(B_{k+1}^j)^2\\geq (\\sum_j b_j^2)(-4\/\\epsilon)\\\\\n\\geq{}& (\\sum_j b_j)(1-\\epsilon)(-4\/\\epsilon)\\geq 16-\\frac{16}{\\epsilon}\n\\end{align*}\nand $(B_1)^2=(K_T)^2\\leq 9$.\nOn the other hand, at each blow-up, $(B_i)^2$ decreases by at least $\\epsilon^2\/4$ by the assumption $\\mult_{P_i}B_i\\geq \\epsilon\/2$ for $i\\leq k$. Hence \n$$\n{k}\\leq \\frac{9-(16-16\/\\epsilon)}{\\epsilon^2\/4}\\leq \\frac{64}{\\epsilon^3}.\n$$\n\nNow we consider $\\pi_{k+1}^*(aD_T)$ on $T_{k+1}$. \n\\begin{claim}\\label{QQQ}There exists a point $Q$ on $T_{k+1}$ such that $\\mult_Q\\pi_{k+1}^*(aD_T)\\geq \\epsilon\/4$.\n\\end{claim}\n\\begin{proof}\nConsider the pair $(T_{k+1}, (1-a)B_{k+1}+aD_{k+1}+E_{k+1})$. Note that $E_{k+1}$ is simple normal crossing supported.\n\n\nAssume that there exists a curve $E$ with coefficient at least $1-3\\epsilon\/4$ in $E_{k+1}$, that is, $$\\mult_E(K_{T_{k+1}}-\\pi_{k+1}^*(K_T+(1-a)B_T+aD_T))\\leq -1+3\\epsilon\/4.$$\n On the other hand, since $(T, (1-a)B_T)$ is $\\epsilon$-klt, \n$$\\mult_E(K_{T_{k+1}}-\\pi_{k+1}^*(K_T+(1-a)B_T))> -1+\\epsilon.$$\nHence $\\mult_E\\pi_{k+1}^*(aD_T)\\geq {\\epsilon}\/{4}$.\n\nIf all coefficients of $E_{k+1}$ are smaller than $1-3\\epsilon\/4$, then $k2$.\n\nThe following lemma allows us to construct non-klt centers. \n\\begin{claim}\nFor a general fiber $F$ of $g$, $-K_Y-sF$ is $\\mathbb{Q}$-effective. In particular, there exists an effective $\\mathbb{Q}$-divisor $B_F\\sim_\\mathbb{Q} -\\frac{1}{s}K_Y$ such that $F$ is a non-klt center of $(Y, B_F)$.\n\\end{claim}\n\\begin{proof}\nFor a positive integer $p$ and a sufficiently divisible positive integer $m$, we have exact sequence\n$$\n0\\rightarrow \\OO_Y(-mK_Y-pF)\\rightarrow \\OO_Y(-mK_Y-(p-1)F)\\rightarrow \\OO_F(-mK_Y-(p-1)F)\\rightarrow 0.\n$$\nNote that $\\OO_F(-mK_Y-(p-1)F)= \\OO_F(-mK_F)$. Hence\n$$\nh^0(Y, \\OO_Y(-mK_Y-pF))\\geq h^0(Y, \\OO_Y(-mK_Y-(p-1)F))- h^0(F, \\OO_F(-mK_F)).\n$$\nInductively, we have\n$$\nh^0(Y, \\OO_Y(-mK_Y-pF))\\geq h^0(Y, \\OO_Y(-mK_Y))- ph^0(F, \\OO_F(-mK_F)).\n$$\nWe may take $p=sm$ since $m$ is sufficiently divisible. By the definition of volume, we have\n\\begin{align*}\n{}&\\limsup_{m\\rightarrow \\infty}\\frac{h^0(Y, \\OO_Y(-mK_Y))- smh^0(F, \\OO_F(-mK_F))}{m^n}\\\\\n={}&\\frac{1}{n!}\\Vol(-K_Y)-\\frac{s}{(n-1)!}\\Vol(-K_F)>0.\n\\end{align*}\nHence $h^0(Y, \\OO_Y(-mK_Y-smF))>0$ for $m$ sufficiently divisible, that is, $-K_Y-sF$ is $\\mathbb{Q}$-effective. In particular, there exists an effective $\\mathbb{Q}$-divisor $B_F\\sim_\\mathbb{Q} -\\frac{1}{s}K_Y$ such that $B_F-F\\geq 0$, and hence $F$ is a non-klt center of $(Y, B_F)$.\n\nWe proved the claim.\n\\end{proof}\n\n\n\nNow for two general fibers $F_1$ and $F_2$, consider $B'=B_{F_1}+B_{F_2}$. By construction, $F_1\\cup F_2\\subset \\Nklt(Y, (1-\\frac{2}{s})B+B')$. Note that \n$$\n-(K_Y+(1-\\frac{2}{s})B+B')\\sim_\\mathbb{Q} -(1-\\frac{2}{s})(K_Y+B)\n$$\nis ample, by Connectedness Lemma, $\\Nklt(Y, (1-\\frac{2}{s})B+B')$ is connected. \nHence there is a non-klt center $W\\subset \\Nklt(Y, (1-\\frac{2}{s})B+B')$ connecting $F_1$ and $F_2$. In particular, $W$ dominates $\\mathbb{P}^1$. Restricting on a general fiber $F$, by adjunction formula, we have\n$(F, B|_F)$ is $\\epsilon$-klt log Fano with $F$ terminal and $(F, (1-\\frac{2}{s})B|_F+B'|_F)$ is not klt (see \\cite[Lemma 5.17, Lemma 5.50]{KM}) with $B'|_F\\sim_\\mathbb{Q} -\\frac{2}{s}K_F$. Hence \n$$\\frac{2}{s}\\geq \\glct(F, B|_F; \\frac{s}{2}B'|_F-B|_F).$$\nTo bound $s$ from above, generalized Ambro's conjecture arises naturally. By generalized Ambro's conjecture in dimension $n-1$, \n$$\ns\\leq \\frac{2}{\\mu(n-1,\\epsilon)},\n$$\nand hence \n$$\n\\Vol(-K_Y)\\leq \\frac{2nM(n-1,\\epsilon)}{\\mu(n-1,\\epsilon)}.\n$$\n\nWe completed the proof.\n\\end{proof}\n\nIn particular, by Corollary \\ref{2bab} and Theorem \\ref{gac2}, Weak BAB Conjecture and generalized Ambro's conjecture hold in dimension $2$, and hence the following corollary holds.\n\n\\begin{cor}\nLet $(X, \\Delta)$ be an $\\epsilon$-klt log Fano pair of dimension $3$ with a contraction $f:X\\rightarrow \\mathbb{P}^1$ and $X$ having terminal singularities. Then $$\\Vol(-K_X)\\leq \\frac{6M(2,\\epsilon)}{\\mu(2,\\epsilon)}.$$\n\\end{cor}\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Conic bundles}\\\n\nIn this subsection, we treat the case that $X$ has a conic bundle structure $f:X\\rightarrow S$. Firstly we collect some facts about singularities of the surface $S$.\n\\begin{thm}\\label{thmS}\nLet $(X,\\Delta)$ be an $\\epsilon$-klt log Fano pair of dimension $3$ and $f:X\\rightarrow S$ be a Mori fiber space to a surface $S$, then \n\\begin{itemize}\n\\item[(i)] $S$ has only Du Val singularities of type A;\n\\item[(ii)] There exists an effective $\\mathbb{Q}$-divisor $\\Delta_S$ on $S$, such that $(S, \\Delta_S)$ is klt log del Pezzo;\n\\item[(iii)] $S$ is a Mori dream space;\n\\item[(iv)] There exists an effective $\\mathbb{Q}$-divisor $\\Delta'_S$ on $S$, such that $(S, \\Delta'_S)$ is $\\delta(\\epsilon)$-klt and $K_S+\\Delta'_S\\sim_\\mathbb{Q} 0$, where $\\delta(\\epsilon)$ is a positive real number depending only on $\\epsilon$;\n\\item[(v)] The family of such $S$ is bounded, in particular, the Picard number of minimal resolution of $S$ is bounded by $128\/\\delta(\\epsilon)^5$.\n\\item[(vi)] $S$ is $N(\\epsilon)$-factorial, i.e. for a Weil divisor $D$ on $S$, $N(\\epsilon)D$ is Cartier, where $N(\\epsilon)$ is a positive integer depending only on $\\epsilon$. \n\\end{itemize}\n\\end{thm}\n\\begin{proof}\n(i) is by \\cite[(1.2.7) Theorem]{MP}. (ii) is by \\cite[Corollary 3.3]{FG}. (iii) is by (ii) and \\cite[Corollary 1.3.2]{BCHM}.\n(iv) is by \\cite[Corollary 1.7]{Birkar} since we may find a boundary $\\Delta'\\geq \\Delta$ such that $(X,\\Delta')$ is $\\epsilon$-klt and $K_X+\\Delta'\\sim_\\mathbb{Q} 0$. (v) is by (iv), \\cite[Theorem 6.8]{AK2}, and \\cite[Theorem 1.8]{AM}. (vi) is a direct consequence of (i) and (v).\n\\end{proof}\nFor the definition and properties of {\\it Mori dream spaces} we refer to the famous paper by Hu--Keel \\cite{HK}.\nWe will use the following property of Mori dream spaces: every nef divisor on $S$ is semi-ample and there are finitely many irreducible curves with negative self intersection. In particular, a curve through a general point is nef. By a curve we always mean an irreducible reduced one.\n\nIf there is a curve $C$ on $S$ satisfying $(C)^2=0$, then $C$ is semi-ample. In particular, a multiple of $C$ induces a contraction $S\\rightarrow \\mathbb{P}^1$ and we are done by Subsection \\ref{curve section}. Hence we may assume that there does not exist such curve $C$ on $S$ satisfying $(C)^2=0$.\n\n\nFix a positive rational number $t$ satisfying $$\\frac{\\Vol(-K_X)}{24}-\\frac{1}{A} {768N(\\epsilon)}\/{\\epsilon}$.\n\\begin{lem}\nFor a general fiber $F$ of $f$, $$h^0(X, \\OO_X(-mK_X)\\otimes \\mathcal{I}_F^{2tm})>0$$ for $m$ sufficiently divisible, where $\\mathcal{I}_F$ is the ideal sheaf of $F$. In particular, there exists an effective $\\mathbb{Q}$-divisor $\\Delta_F\\sim_\\mathbb{Q} -\\frac{1}{t}K_X$ such that $\\mult_F\\Delta_F \\geq 2$.\n\\end{lem}\n\\begin{proof}\nFor a positive integer $p$ and a sufficiently divisible positive integer $m$, we have exact sequence\n$$\n0\\rightarrow \\OO_X(-mK_X)\\otimes \\mathcal{I}_F^{p}\n\\rightarrow \\OO_X(-mK_X)\\otimes \\mathcal{I}_F^{p-1}\n\\rightarrow \\OO_X(-mK_X)\\otimes \\mathcal{I}_F^{p-1}\/\\mathcal{I}_F^{p}\n\\rightarrow 0.\n$$\nNote that $\\mathcal{I}_F^{p-1}\/\\mathcal{I}_F^{p}=S^{p-1}(\\mathcal{I}_F\/\\mathcal{I}_F^{2})$ (see \\cite[II. Theorem 8.24]{H}) and $\\mathcal{I}_F\/\\mathcal{I}_F^{2}=\\OO_{F}^{\\oplus 2}$. Hence\n$$\nh^0(X, \\OO_X(-mK_X)\\otimes \\mathcal{I}_F^{p})\n\\geq h^0(X, \\OO_X(-mK_X)\\otimes \\mathcal{I}_F^{p-1})- ph^0(F, \\OO_F(-mK_F)).\n$$\nInductively, we have\n$$\nh^0(X, \\OO_X(-mK_X)\\otimes \\mathcal{I}_F^{p})\n\\geq h^0(X, \\OO_X(-mK_X))- \\frac{p(p+1)}{2}h^0(F, \\OO_F(-mK_F)).\n$$\nWe may take $p=2tm$ since $m$ is sufficiently divisible. By the definition of volume, we have\n\\begin{align*}\n{}&\\limsup_{m\\rightarrow \\infty}\\frac{h^0(X, \\OO_X(-mK_X))- (2t^2m^2+tm)h^0(F, \\OO_F(-mK_F))}{m^3}\\\\\n={}&\\frac{1}{6}\\Vol(-K_X)-2t^2\\Vol(-K_F)>0.\n\\end{align*}\nNote that $F\\simeq \\mathbb{P}^1$ and $\\Vol(-K_F)=2$.\nHence $h^0(X, \\OO_X(-mK_X)\\otimes \\mathcal{I}_F^{2tm})>0$ for $m$ sufficiently divisible. In particular, there exists an effective $\\mathbb{Q}$-divisor $\\Delta_F\\sim_\\mathbb{Q} -\\frac{1}{t}K_X$ such that $\\mult_F\\Delta_F\\geq 2$.\n\\end{proof}\n\nA prime divisor $V$ on $X$ is {\\it vertical} if $f(V)$ is a curve or $V$ does not dominate $S$. Note that for a curve $C$ on $S$ passing through a general point, there is only one vertical prime divisor contained in $f^{-1}(C)$ and we denote it by $V_C$. It is easy to see that $f^*C=V_C$ as Weil divisors is well defined (since removing finitely many points of $S$, $f$ is flat). For a general point $P\\in S$, denote $F_P$ be the fiber at $P$. \n\nWe can modify the $\\mathbb{Q}$-divisor $\\Delta_F$ to control vertical divisors by the following lemma. \n\\begin{lem}\\label{lemma vertical}\nFor a general point $P\\in S$, there exist an effective $\\mathbb{Q}$-divisor $B_P\\sim_\\mathbb{Q} -\\frac{a_P}{t}K_X$ for some $a_P\\leq {384N(\\epsilon)}\/{\\epsilon}$ such that \n\\begin{itemize}\n\\item[(i)] $\\mult_{F_P}B_P\\geq 2$, and\n\\item[(ii)] For any curve $C$ passing through $P$, $\\mult_{V_C}B_P\\leq {\\epsilon}\/{2}$. \n\\end{itemize}\n\\end{lem}\n\\begin{proof}\nWrite $\\Delta_{F_P}=\\Delta_0+\\sum_i b_i V_{C_i}$ where $\\Delta_0$ does not contain vertical divisors passing through $F_P$ and $C_i$ is a curve passing through $P$. If $b_i\\leq {\\epsilon}\/{2}$ for all $i$, then we can take $B_P=\\Delta_{F_P}$ with $a_P=1$. \n\nNow we assume that $b_1> {\\epsilon}\/{2}$. \nNote that by assumption, $(C_1)^2>0$ and $N(\\epsilon)C_1$ is Cartier. So $N(\\epsilon)C_1$ is a nef and big Cartier divisor. Hence by Koll\\'{a}r's Effective Base Point Free Theorem (see \\cite[1.1 Theorem]{EBPF}), $|96N(\\epsilon)C_1|$ is base point free. It follows that $|96N(\\epsilon)C_1|$ defines a generically finite map $\\Phi:S\\rightarrow \\mathbb{P}(|96N(\\epsilon)C_1|)$. Since $P\\in C_1$, $P$ is not on the contracted curves of $\\Phi$. Hence by taking the pull back of general hyperplanes passing through $\\Phi(P)$, we can write $96N(\\epsilon)C_1\\sim_\\mathbb{Q} \\sum_jh_jH_j$ with $96N(\\epsilon)C_1\\sim H_j$ a general curve passing through $P$, $01$, since $C_i$ is semi-ample, we may take $C_i\\sim_\\mathbb{Q} D_i$ such that $D_i$ is an effective $\\mathbb{Q}$-divisor on $S$ passing through a general point but not containing $P$.\nNow define \n$$B_P:=\\frac{192N(\\epsilon)}{b_1}\\Big(\\Delta_0+\\sum_{i>1}b_if^* D_i\\Big)+2\\sum_jh_jf^*H_j\\sim_\\mathbb{Q} -\\frac{192N(\\epsilon)}{b_1t}K_X,$$\nand we can take $a_P= {192N(\\epsilon)}\/{b_1}\\leq {384N(\\epsilon)}\/{\\epsilon}$. Note that \n$$\\mult_{F_P}B_P\\geq \\mult_P \\big(2\\sum_jh_jH_j\\big)\\geq 2.$$\nAnd by construction, for every curve $C$ passing through $P$, if $C=H_j$ for some $j$, then $\\mult_{V_C}B_P=2h_j\\leq {\\epsilon}\/{2}$; otherwise $\\mult_{V_C}B_P=0$.\n\nHence we proved the lemma.\n\\end{proof}\n\nTake two general points $P_1$ and $P_2$ on $S$. For simplicity, for $i=1,2$, we denote $F_{P_i}=F_i$, $a_{P_i}=a_i$, and $B_{P_i}=B_i$. Note that by construction, \n$$F_i\\subset \\Nklt(X, B_i).$$\n\n\n\n{\\bf Case 1.} There exists a non-klt center $E$ of dimension $2$ of $(X, B_i)$ containing $F_i$ for some $i=1$ or $2$.\n\nIn this case, $$\\mult_E(B_i)\\geq 1.$$ \n By construction of $B_i$, $E$ is not vertical. Restricting on a general fiber $F$ of $f$, we have\n $$\n \\frac{2a_i}{t}= -\\frac{a_i}{t}K_X\\cdot F=B_i\\cdot F\\geq E\\cdot F\\geq 1.\n $$\nHence \n$$\nt\\leq 2a_i\\leq \\frac{768N(\\epsilon)}{\\epsilon}.\n$$\n\n{\\bf Case 2.} $F_i$ is a maximal non-klt center of $(X, B_i)$ for $i=1$ and $2$.\n\n\nSince $P_1$ is a general point, we may assume \n$F_1\\not \\subset {\\rm Supp}(\\Delta+B_2)$. Hence $F_1$ is a maximal non-klt center of $(X, (1-\\frac{a_1+a_2}{t})\\Delta+B_1+B_2)$ and $F_2$ is a non-klt center. Note that \n$$\n-(K_X+ (1-\\frac{a_1+a_2}{t})\\Delta+B_1+B_2)\\sim_\\mathbb{Q} -(1-\\frac{a_1+a_2}{t})(K_X+\\Delta)\n$$\nis ample by the assumption $t> {768N(\\epsilon)}\/{\\epsilon}$. By Connectedness Lemma, $\\Nklt(X, (1-\\frac{a_1+a_2}{t})\\Delta+B_1+B_2)$ is connected. Hence there is a non-klt center $W$ intersecting with $F_1$. \nHence we have $$\\mult_W\\Big(\\Big(1-\\frac{a_1+a_2}{t}\\Big)\\Delta+B_1+B_2\\Big)\\geq 1.$$ \n\nIf $\\dim W=2$, since $(X, (1-\\frac{a_1+a_2}{t})\\Delta)$ is $\\epsilon$-klt, $$\\mult_W\\Big(\\Big(1-\\frac{a_1+a_2}{t}\\Big)\\Delta\\Big)<1-\\epsilon.$$ \nHence $$\\mult_W(B_1+B_2)\\geq \\epsilon.$$\nSince $F_1\\not\\subset W$ by the maximality of $F_1$, $W$ is not vertical. Restricting on a general fiber $F$ of $f$, we have\n $$\n \\frac{2(a_1+a_2)}{t}= -\\frac{a_1+a_2}{t}K_X\\cdot F=(B_1+B_2)\\cdot F\\geq \\epsilon W\\cdot F\\geq \\epsilon.\n $$\nHence \n$$\nt\\leq \\frac{2(a_1+a_2)}{\\epsilon}\\leq \\frac{1536N(\\epsilon)}{\\epsilon^2}.\n$$\n\nIf $\\dim W=1$, then since $P_1$ is general, we may assume that for each point $Q\\in F_1\\cap {\\rm Supp} (\\Delta+B_2)$, $Q$ is not contained in the singular locus of ${\\rm Supp} (\\Delta+B_2)$. This is because the singular locus of ${\\rm Supp} (\\Delta+B_2)$ has dimension at most $1$ and hence does not dominate $S$. Now if $W\\subset {\\rm Supp}\\Delta$, then $W$ is contained in exactly one component of $\\Delta$ since ${\\rm Supp}\\Delta$ is smooth at points in $F_1\\cap W$. Since $(X, \\Delta)$ is $\\epsilon$-klt, the coefficients of $\\Delta$ is smaller than $1-\\epsilon$. Hence \n$$\\mult_W\\Delta< 1-\\epsilon.$$\nOf course this inequality also holds if $W\\not \\subset {\\rm Supp}\\Delta$. So we have\n$$\\mult_W(B_1+B_2)\\geq \\epsilon.$$ \nNote that to compute the intersection number $(B_1+B_2)\\cdot F$ for some fiber $F$ by $\\mult_W(B_1+B_2)$, it is necessary to avoid $V_{f(W)}$ in $B_1+B_2$.\nDenote $V_{f(W)}$ by $V$.\nBy construction of $B_1$, $\\mult_{V}B_1\\leq {\\epsilon}\/{2}$. On the other hand, $\\mult_{V}B_2=0$ since $F_1\\not\\subset {\\rm Supp}B_2$ but $F_1\\subset V$.\nWe can write $B_1+B_2=B+\\lambda V$ where the support of $B$ does not contain $V$. \nThen $\\lambda\\leq {\\epsilon}\/{2}$. It is easy to see that $\\mult_WV=1$ and hence \n$$\n\\mult_W B = \\mult_W (B_1+B_2)-\\lambda\\geq \\frac{\\epsilon}{2}.\n$$\nRestricting on a fiber $F$ of $f$ at a general point of $f(W)$, we have\n $$\n \\frac{2(a_1+a_2)}{t}= -\\frac{a_1+a_2}{t}K_X\\cdot F=(B_1+B_2)\\cdot F= B\\cdot F\\geq \\frac{\\epsilon}{2}.\n $$\nHence \n$$\nt\\leq \\frac{4(a_1+a_2)}{\\epsilon}\\leq \\frac{3072N(\\epsilon)}{\\epsilon^2}.\n$$\n\nIn summary, we have \n$$\nt\\leq \\frac{4(a_1+a_2)}{\\epsilon}\\leq \\frac{3072N(\\epsilon)}{\\epsilon^2},\n$$\nand hence \n$$\n\\Vol(-K_X)\\leq \\frac{24\\cdot 3072^2N(\\epsilon)^2}{\\epsilon^4}.\n$$\n\nWe have completed the proof of Theorem \\ref{mfs3}.\n\n\n\n\n\n\\section{Boundedness of log Fano threefolds of fixed index}\\label{section bat}\n\nIn this section, we prove the boundedness of log Fano threefolds of fixed index (Corollary \\ref{baty}).\n Corollary \\ref{baty} follows directly by Theorem \\ref{BAB3main} and the following more general theorem which might be well known to experts. \n\n\n\\begin{thm}\nFix positive integers $r$ and $n$. Assume Weak BAB Conjecture holds in dimension $n$. \n\nLet $\\mathcal{D}$ be the set of all normal projective varieties $X$, where\n$\\dim X=n$, $K_X$ is $\\mathbb{Q}$-Cartier, and there exists an effective $\\mathbb{Q}$-divisor $\\Delta$ such that $(X,\\Delta)$ is klt and $-r(K_X + \\Delta)$ is \nCartier and ample. \n\nThen $\\mathcal{D}$ forms a bounded family.\n\\end{thm}\n\n\\begin{proof}\nConsider a klt log Fano pair $(X, \\Delta)$ of dimension $n$ such that $K_X$ is $\\mathbb{Q}$-Cartier and $-r(K_X + \\Delta)$ is \nCartier. \n\nNote that $(X, \\Delta)$ is $\\epsilon$-klt with $\\epsilon=1\/2r$ by the assumption. It follows that $(-(K_X+\\Delta))^n\\leq M(n,\\epsilon)$ by Weak BAB Conjecture in dimension $n$. \n\nSince $-r(K_X + \\Delta)$ is \nCartier and ample, by Koll\\'ar's Effective Base Point Free Theorem \\cite[1.1 Theorem, 1.2 Lemma]{EBPF}, $G:=-Nr(K_X+\\Delta)$ is very ample for $N=2n\\cdot(n+3)!$. \n\nNote that $(K_X+G)\\cdot C\\geq 0$ for all curves $C$ satisfying $K_X\\cdot C\\geq -2n$. Hence by Cone Theorem (see \\cite[Theorem 3.7]{KM}), $K_X+G$ is nef.\n\nNow we can bound $G^n$ and $|-K_X\\cdot G^{n-1}|$ from above. Clearly $G^n\\leq N^nr^nM(n,\\epsilon)$ by definition and $-K_X\\cdot G^{n-1}>0$ since $-K_X$ is big. On the other hand, \n\\begin{align*}\n{}&-K_X\\cdot G^{n-1} \\\\\n={}& -(K_X+G)\\cdot G^{n-1}+G\\cdot G^{n-1}\\\\\n\\leq {}& N^nr^nM(n,\\epsilon).\n\\end{align*}\nHence $G^n$ and $|-K_X\\cdot G^{n-1}|$ are bounded from above. By \\cite{Matsusaka}, the coefficients of the Hilbert polynomial \n$P(t)=\\chi(X, \\OO_X(tG))$ is bounded and hence there are only finitely many Hilbert polynomials for the polarized variety $(X, G)$. And hence $X$ is in a bounded family.\n\nWe complete the proof.\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{sec1}\n\n Gamma ray bursts (GRBs) are among the most amazing transients known. In a few second a GRB emits the energy that a star like our sun emits in its whole life time. Their origin has puzzled astronomers since their serendipitous discovery in the late sixties. After two decades in which it was believed that the GRBs are Galactic\nit was realized, in the early nineties, that they have a cosmological origin \\cite{Meegan+92,P92,MaoPac92}. The distance scale set immediately the energy scale to be $\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$>$}}} 10^{51}$ erg (including beaming corrections that were realized towards the end of the nineties \\cite{Rhoads97,Rhoads99,SPH99}). Together with the short time scale this has led inevitably to the conclusion that the events involve the formation of a newborn compact object, most likely a black hole. This conclusion has left practically just two progenitor candidates: A collapsing massive star or the merger of two neutron stars (or a neutron star and a black hole).\n\n\n\nThe observations of a few (long) GRB afterglows in 1997 revealed that those bursts arose in star forming regions. Paczynski (1998) who noticed that, quickly suggested that long GRBs (LGRBs) are related to core collapse events. At roughly the same time MacFadyen \\& Woosley (1999) suggested the Collapsar model. {\nAccording to this model a GRB is produced by a relativistic jet that emerges from the center of a massive collapsing star and penetrates the stellar envelope. By now the name Collapsar is used with different variations\\footnote{In some cases the term Collapsar is used generically for any model that involve a collapsing star, regardless whether there is a jet or not. In other cases it is used more restrictively to imply a situation in which the collapsing star produces an accreting black hole as a central engine that drives a relativistic jet.}. We stress that here we use this original definition for a Collapsar:} {\\it A jet that penetrates the envelope of a Collapsing star.} Using numerical simulations MacFadyen \\& Woosley (1999)\ndemonstrated that a relativistic jet can indeed penetrate a stellar envelope. Again roughly at the same time Galama et al. (1998) discovered that GRB 980425 was associated with the powerful type Ic supernova: SN 1998bw. However GRB 940425 was a strange GRB. It was very weak, with an energy a few orders of magnitude less than the energy of a typical GRB. Additionally it was single peaked and smooth and it had a very soft spectrum. It was not clear that this was a regular GRB and hence the association of GRB 980425 with SN 1998bw was not sufficient to demonstrate a GRB-SNe association. Shortly after that Bloom et al. (1999) and others discovered red bumps in the afterglows of more distant aregular GRBs. These red bumps were interpreted as the signatures of 1998bw-like SNe, supporting the GRB-SNe association. However the evidence for a GRB-SNe association was inconclusive until SN 2003dh was discovered in association with the regular GRB 030329 \\cite{Hjorth+03,Stanek+03}. Since then a few other GRB-SNe associations were discovered. Even though most of these GRB-SNe associations are with weak, smooth, single peaked GRBs\\footnote{Another association of a regular regular GRB and an SN, GRB 101219B and SN 2010ma, was discovered recently.} this is generally considered as a ``proof'' of the Collapsar model for LGRBs.\n\n\n\n\n\nAn inspection of BATSE's GRBs' temporal distribution revealed \\cite{Kouveliotou+93} two groups: short ($T_{90} < 2$ sec) and long ($T_{90} > 2$ sec.). Already in 1995\nit was pointed out \\cite{CP95,P96} that the two groups have a different spatial distribution. The observed short GRBs (SGRBs) are significantly nearer (and weaker). This suggested the possibility of different physical origin for the two populations. As it takes time (and energy) to cross the relatively large stellar envelope it was argued that SGRBs cannot be produced by Collapsars \\cite{Matzner03}. In most cases Collapsars produce LGRBs, but by now we know that in some cases Collapsars produce SGRBs (see \\S \\ref{sec:Collapsar} below). However a variant on this original argument that we discuss here (in \\S \\ref{sec:llGRB} and \\ref{sec:Collapsar} below) shows that most SGRBs cannot be produced by Collapsars.\nLack of detection of SGRB afterglows\nleft the situation inconclusive until 2005, when {\\it Swift} localized the first short bursts and the first SGRBs' afterglows were detected. It turned out that SGRBs are not associated with star forming regions (some arise in elliptical galaxies) and as such they are not associated with deaths of massive stars. The progenitors could be neutron star mergers (as suggested already in 1989 \\cite{Eichler+89}). However as yet there is no conclusive demonstration of this origin \\cite{Nakar07}.\n\nWe describe here new results, derived by Bromberg et al. (2011a, 2011b, 2012a,2012b) concerning the nature of GRB progenitors. We briefly discuss, in \\S \\ref{sec:jet}, some recent analytic results concerning relativistic jet penetration through the stellar envelope \\cite{Bromberg1}. We then consider their implications on this picture. In \\S \\ref{sec:llGRB} we demonstrate that {\\it}GRBs, those that appear in most GRB-SNe associations, cannot be produced by Collapsars \\cite{Bromberg2}. While this weakens the case for the association of regular LGRBs with SNe, we show in \\S \\ref{sec:Collapsar} that when combined with the GRBs' temporal distribution these considerations demonstrate that the LGRBs originate from Collapsars \\cite{Bromberg3}, providing a direct observational indication for jets that puncture the stellar envelope. Further inspection of the temporal distribution enables us to estimate (in \\S \\ref{sec:SGRBs}), for the first time, the fraction of Collapsars among SGRBs as a function of the observed duration \\cite{Bromberg4}. We show that this fraction depends strongly on the detector (in particular on its spectral window). In particular the standard limit of 2 sec is invalid for {\\it Swift}'s observations, for which a limit of 0.8 sec is much more appropriate. \n\n\\section{Jet Propagation}\n\\label{sec:jet}\n\nA schematic picture of a relativistic jet propagating within a stellar envelope is depicted in Fig. \\ref{fig:Jet}. There are a few critical components. A a double shock system appears at the head of the jet \\cite{Matzner03}. While the jet is highly relativistic these shocks slow down the head and it typically propagates with a sub or mildly relativistic velocity. The hot material that streams sideways out of the jet's head produces a cocoon that engulfs the jet.\nWhile expanding sideways into the rest of the stellar envelope (this expansion will eventually blow out a fraction of the stellar envelope) it also squeezes the jet and produces a (radiative) collimation shock within the jet \\cite{BrombergLevinson}.\n\n\\begin{figure}[!h]\n\\centering{\\includegraphics[width=90mm]{Jet_cocoon3.pdf}}\n\\vskip -1cm\n\\caption{ A schematic description of a jet propagating in a stellar atmosphere superimposed on a numerical jet simulation of \\cite{Morsony+07}.\n\\label{fig:Jet}}\n \\end{figure}\n\nAs long as the jet is within the stellar atmosphere all its energy is dissipated at the jet's head. The total dissipated energy equals therefore the jet's luminosity times the time it takes to cross the envelope. Since the inner engine is much smaller than the envelope it is decoupled from the jet that crosses the envelope on a much larger scale and one can expect that the luminosity before and after the jet breaks out are comparable. Using the observed GRB luminosity to estimate the jet power before breakout we can estimate the duration of the dissipation phase as \\cite{Bromberg1}:\n\\begin{equation}\\label{eq:tB_GRB}\nt_B \\simeq15~{\\rm sec} \\cdot \\left( \\frac{ L_{iso}} {10^{51} {\\rm~\nerg\/sec}}\\right)^{-1\/3} \\left (\\frac{\\theta}{10^\\circ}\\right)^{2\/3}\n\\left ( \\frac{R_{*}} {5 R_\\odot}\\right)^{2\/3} \\left\n(\\frac{M_{*}}{15M_\\odot}\\right)^{1\/3},\n\\end{equation}\nwhere $ L_{iso}$ is the isotropic equivalent jet luminosity,\n$\\theta$ is the jet half opening angle and we have used typical\nvalues for a LGRB. $ R_{*}$ and $M_{*}$ are the radius and the\nmass of the progenitor star, where we have normalized their value\naccording to the typical radius and mass inferred from observations\nof the few SNe associated with LGRBs.\nFor the jet to break out, the central engine must continue operating for a duration longer than $t_B$. If the inner engine stops before the jet's head crosses the envelope the jet won't produce a regular GRB.\n\n\n\\section{ Low luminosity GRBs}\n\\label{sec:llGRB}\n\nThe duration of the prompt emission, approximated by $T_{90}$, is given simply by:\n\\begin{equation}\nT_{90} = t_{e} - t_B ,\n\\label{eq:diff}\n\\end{equation}\nwhere $t_{e}$ is the total time that the engine powering the jet is active.\nWithin the Collapsar model, without fine tuning, only a small fractions of the bursts should have $T_{90}\/ t_B \\ll 1$ (see \\S \\ref{sec:Collapsar}). Namely, it is\nunlikely that the engine operates just long enough\nto let the jet break out of the star and then stops right after\nbreakout. This argument was used by Matzner (2003) to argue that Collapsars cannot produce SGRBs, for which $T_{90}\/ t_B \\ll 1$. This is indeed confirmed in Fig \\ref{fig:T90_tB} in which the distribution of $T_{90}\/ t_B$ is shown for both LGRBs and SGRBs. One\ncan clearly see two distinct populations LGRBs for the majority of which $T_{90}\/ t_B > 1$ and SGRBs, all of which satisfy $T_{90}\/ t_B <1$. \n\n\\begin{figure}[!h]\n\\includegraphics[width=120mm]{T90_vs_TB.pdf}\n \\caption{The distribution of $T_{90}\/t_B$ for LGRBs,\n {\\it ll}GRBs and SGRBs (from \\cite{Bromberg2}).\n\n }\n\\label{fig:T90_tB}\n\\end{figure}\n\n\n\nFig \\ref{fig:T90_tB} depicts also a third group of GRBs, low luminosity ({\\it ll}GRBs).\nLike SGRBs, the observed duration distribution of {\\it ll}GRBs is inconsistent\nwith the predictions of the collapsar model. In particular a large\nfraction of {\\it ll}GRBs have $T_{90}\/t_B\\ll 1$. The probability that\nthe observed {\\it ll}GRBs $T_{90}\/t_B$ distribution is consistent with\nthe LGRBs distribution is smaller than $5\\%$\nimplying that {\\it ll}GRBs cannot be generated by Collapsars and they must have a different origin. \n\n{\\it ll}GRBs are a group of six GRBs whose luminosities are around $10^{47}-10^{49}$ ergs\/sec, at least two orders of magnitude below the average luminosity of a typical GRB.\nRemarkably {\\it ll}GRBs are not characterized just by their low luminosity. They are also single peaked, smooth and soft.\n{\\it ll}GRBs include GRB9890425 (the first GRB detected to accompany a Supernova - 1998bw) as well as a few other GRB-SN pairs: GRB 031203\/SN2003lw; GRB060218\/SN2006aj; GRB100316D\/SN2010bh. GRB051109B shows all the common features of {\\it ll}GRBs but it lacks a reported SN. It is associated with a star forming region in a spiral galaxy at $z=0.08$ \\cite{Perley06}.\n All {\\it ll}GRBs are at very low redshifts. With such low luminosities they couldn't have been detected from further out. While only six {\\it ll}GRBs have been observed so far, given these distances, the {\\it ll}GRBs inferred rate per unit volume is much larger than the rate of regular LGRBs \\cite{Soderberg+06}. In fact this rate is so large that {\\it ll}GRBs cannot be significantly beamed as even with a modest beaming corrections the rate would have exceed the rates of their associated SNe - broad line type Ibc.\n\n\n\nAn interesting and likely possibility is that\n {\\it ll}GRBs' jets are weak and fail to break out from their progenitors. A``failed jet'' dissipates all\nits energy into the surrounding cocoon and drives its expansion. As\nthe cocoon reaches the edge of the star its forward shock may become\nmildly or even ultra relativistic emitting the $\\gamma$-rays observed in {\\it ll}GRBs\nwhen it breaks out. This idea that {\\it ll}GRBs arise from\nshock breakouts was suggested shortly following the observations of GRB980425\/SN1998bw\n\\cite{Kulkarni+98,MacFdyen+01,Tan+01}. It drew much more attention\nfollowing the observation of additional {\\it ll}GRBs with similar\nproperties and especially with the observation of a thermal\ncomponent in the spectrum of {{\\it ll}GRB} 060218 \\cite{Campana+06,Wang+07,Waxman+07}. Yet, it was hard to explain how\nshock breakout releases enough energy in the form of $\\gamma$-rays.\nKatz, Budnik \\& Waxman (2010) realized that the deviation of the breakout\nradiation from thermal equilibrium provides a natural explanation to\nthe observed $\\gamma$-rays. More recently, Nakar \\& Sari (2012) calculated\nthe emission from mildly and ultra-relativistic shock breakouts,\nincluding the post breakout dynamics and gas-radiation coupling.\nThey find that the total energy, spectral peak and duration of all\n{\\it ll}GRBs can be well explained by relativistic shock\nbreakouts. Moreover, they find that such breakouts must satisfy a\nspecific relation between the observed total energy, spectral peak\nand duration, and that all observed {\\it ll}GRBs satisfy this relation. These\nresults lend a strong support to the idea that {\\it ll}GRBs are\nrelativistic shock breakouts. From a historical point of view this\nunderstanding closes the loop with Colgate's\n(1968) original idea, that preceded the detection of\nGRBs, that a SN shock breakout will produce a GRB.\n\nAs we discuss in the following section, the observed GRB duration distribution indicates the existence\nof many \"failed jets\" in which the engine time is shorter than the breakout time. This is consistent with the\nobservations that the rate of {\\it ll}GRBs is much higher than the rate of regular LGRBs.\n\n\\section{Long GRBs and Collapsars}\n\\label{sec:Collapsar}\n\nAs most of the GRBs associated with SNe are {\\it ll}GRBs one might think at first that this new understanding rules out the Collapsar model for LGRBs. However, on the contrary, these arguments provide a new and unexpected direct observational confirmation of the Collapsar origin of LGRBs.\nConsider again Eq. \\ref{eq:diff}. Under very general conditions this equation results in a flat duration distribution for durations significantly shorter than the typical breakout time.\n\nIt follows from Eq. \\ref{eq:diff} that the distribution, $p_\\gamma(T_{90})$ of the observed GRB durations is a convolution of $p_e(t_e)$, the distribution of engine operating times, and $p_B(t_B)$ the distribution of jet breakout times. Under quite general conditions (more specifically unless $p_e$ varies very rapidly around $t_B$, an unlikely situation) the following limits hold:\n\\begin{equation}\\label{tg}\np_\\gamma (T_{90}) \\approx \\left\\{\n\\begin{array}{cc}\n p_e(t_B) & T_{90} \\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} t_B \\\\\n p_e(T_{90}) & T_{90} \\gg t_B\n\\end{array}\n\\right. .\n\\end{equation}\n\nParticularly interesting for our purpose here is the appearance at short durations $T_{90} \\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} t_B$ of a flat region, a plateau, in which the rate of events is independent of the duration. Remarkably such plateaus exist in all the observed GRB duration distributions (see Fig. \\ref{fig:T-dist}). They weren't noticed so far because the \"canonical\" distribution plot \\cite{Kouveliotou+93} depicts $d N\/d\\log(T)$ instead of $dN\/dT$. \n\nThe higher end of the plateau enables us to estimate $t_B$ and from this to infer some the basic properties of the collapsing stars. We find, for example, that a typical progenitor size is $ \\sim 5 R_\\odot$. \nAnother interesting feature seen in Fig. \\ref{fig:T-dist} is the rapid decline at durations longerr than $t_B$. At this regime\naccording to Eq. \\ref{tg} the distribution is dominated by $p_e(t_e)$, thus $p_e(T_{90})\\approx p_{\\gamma}(T_{90})$. An extrapolation of this distribution to shorter engine operating times suggests that there are numerous cases in which $t_e < t_B$ and the jet fails to break out. This is in a very nice agreement with the very large inferred event rate of {\\it ll}GRB if these are interpreted as``failed jets''. \n\n\n\n\\begin{figure}[!h]\n\\includegraphics[width=120mm]{Durations_distribution.pdf}\n \\caption{The duration distributions, $dN\/dT_{90}$, of BATSE (red), {\\it Swift} (blue) and Fermi GBM (green) GRBs.\n Also plotted is the distribution of the soft (hardness ratio $< 2.6$) BATSE bursts (magenta).\n For clarity the {\\it Swift} values are divided by a factor of 5 and the Fermi GBM by 15.\n Note that the quantity $dN\/dT$ is depicted and\n not ${dN}\/{d\\log T}$ as traditionally shown in such plots \\cite{Kouveliotou+93}.\n The black lines show the best fitted flat interval in each data set:\n $5-25$ sec (BATSE), $0.7-21$ sec ({\\it Swift}), and $2.5-31$ sec (Fermi). The upper limits of this range\n indicate a typical breakout time of a few\n dozens seconds, in agreement with the prediction of the Collapsar model.\n Soft BATSE bursts show a considerably longer plateau\n ($0.4-25$ sec), indicating that most of the soft short bursts are in fact\n Collapsars (from \\cite{Bromberg3}.)}\n\\label{fig:T-dist}\n\\end{figure}\n\nThe appearance of these plateau (and their dependence on the observed hardness) is the first direct observational confirmation of the Collapsar model. More specifically these plateau demonstrates the fact that the duration of an LGRB is the difference between two independent time scales. This confirms a basic prediction of the Collapsar model: the overall duration is the difference between the time that the engine operates and the time it takes the jet to penetrate the stellar envelope.\n\nAt very short durations the plateau doesn't extend all the way to zero. This does not rule out the model. At this regime non-Collapsar SGRBs, that have a different origin, appear and dominate the distribution. As different detectors have different relative sensitivities to long (and soft) vs. short (and hard) GRBs the duration at which short non-Collapsars begin to dominate varies from one detector to another\\footnote{In principle we should have worked with a redshift corrected sample. However such samples of SGRBs are too small and we are forced to use the observed durations. This imply that we have to worry also about time dilation which, in turn depend on the sensitivity of the detector and the corresponding depth of the samples. However, as SGRBs are detected only from relatively small redshifts this time dilation correction is not significant.}. To demonstrate this dependence on the detector's spectral window we artificially change BATSE's effectiveness for detection of hard SGRBs by considering only softer BATSE bursts (hardness ratio $< 2.6$). As expected, for this softer BATSE sample the non-Collapsar peak shrinks and the plateau extends down to shorter durations.\n\n\n\\section{Short non - Collapsar GRBs}\n\\label{sec:SGRBs}\n\nShortly after Kouveliotou et al. (1993) demonstrated that there are two populations of GRBs: long and short ones it became clear that these two populations have different spatial distributions and a different origin \\cite{CP95,P96}. By now we know that \nLGRBs arise from Collapsars. SGRBs have other projenitors, most likely neutron star mergers. As we are still uncertain concerning the origin of SGRBs we denote them here as non-Collapsars. So far it was implicitly assumed that the division line between long and SGRB is at 2 sec regardless of the observing satellite. The existence of a plateau in the observed LGRBs (of Collapsar origin) duration distribution enables us to determine, for the first time, the fractions of Collapsars vs. non-Collapsars as a function of the observed time for every specific detector. While we cannot determine if a specific bursts is a Collapsar or not we can give now\na probabilistic estimate for a given duration and hardness.\n\nThe basic idea is very simple. For a given detector we determine the rate of detection of bursts within the plateau. { This provides an estimate for the detection rate\nof short duration Collapsars by this detector}. Now we can compare this rate to the rate of SGRBs at any given duration and obtain the Collapsar and non-Collapsar fractions as a function of duration. This estimate is performed for different detectors or even for different detection windows (hardness) for a specific detector.\n\n{\nWe have fitted the different duration distributions with a plateau (representing Collapsars) and a lognormal distribution (for non-Collapsars). The fit is remarkably good and it enables us to estimate the fraction, $f_{NC}$ of non-Collapsars from the total number of observed GRBs as a\nfunction of the observed duration, $T_{90}$ (see Fig. \\ref{fig:Short_fraction}).\nFor Batse $T_{90} < 2$ sec is a reasonable threshold to identify non-Collapsars.\nThis limit results in a probability $> 70\\%$ for a correct classification for BATSE bursts.\nHowever, this condition is misleading for {\\it Swift} bursts.\n{At $T_{90}=2$ sec a {\\it Swift} burst has a $84{\\pm14}\\%$ probability to be a Collapsar!\nClearly, for {\\it Swift} a $2$ sec division line results in a large number of misidentified Collapsars as non-Collapsars.\nWe propose to draw the division line between collapsars and non-Collapsars at the duration where\nthe probability that a GRB is a non-Collapsar is 50\\%.\nWith this condition, a BATSE GRB can be classified as a non-Collapsar if its $T_{90}<3.1{\\pm0.5}$ sec.\n{\\it Swift} bursts can be identified as non-collapsars only if their duration $T_{90} < 0.8{\\pm 0.3}$ sec,\nwhile the corresponding limit for GBM is $T_{90} < 1.7^{+0.5}_{-0.5}$ sec.}\nThe results shown here can be expanded and improved when we consider the hardness of the bursts.\n}\n\\begin{figure}[!h]\n\\includegraphics[width=120mm]{NC_f.pdf}\n \\caption{ From top to bottom: BATSE, {\\it Swift} and\nFermi GBM fractions, $f_{NC}$ of non-Collapsars from the total number of observed GRBs as a\nfunction of the observed duration, $T_{90}$. The shaded regions represent 67\\% confidence limits of $f_{NC}$.\nAlso plotted {in red} are the $T_{90}$ values for which $f_{NC} = 0.5$ (from \\cite{Bromberg4}).}\n\\label{fig:Short_fraction}\n\\end{figure}\n\nUsing these results one can go back and examine various studies of short bursts that attempted to compare \nvarious features of short (standing for non-Collapsars) with long (Collapsars) and check the samples used.\nA preliminary inspection of such studies\nreveals that while in some cases the short sample has only a small number of potential Collapsars in other cases the short\nsample used was heavily contaminated by high probability potential Collapsars (with observed duration shorter than 2 sec) and it is possible and even likely that these have dominated the results. { Note in particular that our results are not inconsistent with those of Berger (2011), who finds a larger dispersion in properties of {\\it Swift} SGRBs (with $T_{20}< 2 $ sec host galaxies and the positions of the bursts within the hosts, as compared with the more homogenous properties of LGRBs host and their position within the hosts. A mixed sample that contains both Collapsars and non-Collapsars is expected to show such larger dispersion (see \\cite{Bromberg4} for further details). }\n\n\n\\section{Conclusions}\n\nTo conclude we summarize our basic findings. We define\\footnote{Note that the term Collapsars has also been used in both wider and narrower contents.} a Collapsar as a collapsing massive star that produces in it center a relativistic jet. The jet penetrates the stellar envelope and produces the GRB ones it has left the star. The duration of the envelope penetration phase depends on the jet's luminosity, its opening angle and the size and density profile of the stellar envelope.\n\nOne can expect that the duration of a burst is typically comparable to or longer than the jet breakout time. Indeed, a comparison of the estimated jet breakout time of a typical LGRB shows that it is shorter than the observed duration of the burst. On the other hand the jet breakout time is much\nlonger than the duration of a short burst. This provides the first indication that SGRBs are not produced by Collapsars. We have shown that\na third group of low luminosity GRBs also don't satisfy this condition. This implies that {\\it ll}GRBs don't arise from Collapsars.\n\nWithin the Collapsar model the observed duration of a GRB is the difference between the time its central engine operates, producing the jet, and the jet's breakout time. This directly implies that at short durations the rate of bursts produced by Collapsars\nshould be independent of their duration. We have shown that such a behavior is observed in the duration distributions of all GRB satellites: BATSE, {\\it Swift}\nand GBM. This provides the a direct observational confirmation of the basic prediction of the Collapsar model and as such demonstrates the Collapsar origin of LGRBs.\n\nThis last feature also enables us to determine the fraction of Collapsars within the observed SGRBs. This fraction depends on the characteristics of the detector. For BATSE the standard division between Collapsar and non-Collapsars is indeed at $\\sim 2 $ sec. However for the softer {\\it Swift}\nmany bursts shorter than 2 sec are of Collapsar origin. This might have led to some confusion in the past in interpreting observations of these short bursts as indications for properties of non-Collapsar GRBs.\n\n\n\\vskip 1cm\nThis research was supported by an Advanced ERC\n grant and by the Israeli center for Excellence for\nHigh Energy AstroPhysics (TP), by an ERC starting grant and an ISF grant (EN) and by\nPackard, Guggenheim and Radcliffe fellowships (RS).\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nUser behavior prediction is one of the most important tasks in the current e-commerce system, where its main purpose is to predict the probability of a user taking a certain behavior toward the candidate item. It is obvious that accurate prediction improves the feeling of the experience of users and optimizes the search system's ability to precisely match materials. In general, the algorithm for modeling and predicting user behavior has a wide range of applications and has attracted lots of attention both from industry and academia.\nHowever, when users search for products in the local consumer service platform, the local context information they are in largely affects and restricts potential interest. For example, a user may want to browse coffee and breakfast on the way to work in the morning, buy a work meal near the company at noon, and purchase fruit on the way back home. It can be seen that the change in the context would lead to a huge variation in the potential interest of users. Therefore, creating a model to accurately understand the potential interests of users under different contexts has become a key urgent in predicting user click behavior in the local consumer service platform. It can be concluded that the main challenges of modeling context-aware interest representation come from the following two aspects:\n\\begin{itemize}\n\\item How to uniformly represent the heterogeneous behavior of users and the contextual information contained in it?\n\\item How to dynamically generate the context-aware interest representation of users?\n\\end{itemize}\nNext, we try to employ existing methods to solve the above challenges:\n\\paragraph{The User Representation Methods}\nMany works have adopted user behavior sequences for representing user behaviors and modeling user's potential interest for future click prediction, such as DIN\\cite{zhou2018deep} or DIEN\\cite{zhou2019deep}. However, this type of approach only makes use of information in the time dimension for interest generation, and it is difficult to distinguish the correlation between historical behavior and the current candidate items under different contexts. Moreover, some works\\cite{10.1145\/3292500.3330836} input artificially designed contextual features into the model to build context-sensitive predictions, but such approaches only model the cross-relationship between user interests and the context. In general, it is difficult for the existing user representation methods to naturally integrate the historical heterogeneous behaviors of users and to perform accurate user interest modeling in a frequently changing context.\n\n\n\n\n\n\n\\paragraph{The Graph Methods}\n From the perspective of heterogeneous behavior recording and representation, the commonly used method in the industry is to first establish a heterogeneous interest network of users and then decompose the vertices of the heterogeneous graph with tag matching\\cite{10.1145\/3292500.3330673}. However, such methods cannot describe contextual information and incorporate it into the process of graph aggregation. Moreover, the graph attention model is commonly used in the industry to capture vertex preferences in the graph. And most of the existing methods choose GAT or HAN as benchmarks(\\cite{velivckovic2017graph},\\cite{han2019}), then change the aggregation method in their respective application scenarios. But the original attention mechanism in GAT is performed on a complete graph that has been constructed and is utilized to compare its correlation with all neighbors on the graph. Leaving such attention methods poor performance to integrate real-time contextual information for graph aggregation. \n\n\n\n\n\n\nBased on the above analysis, the existing methods cannot naturally solve the above two challenges. Therefore, we propose the \\textbf{C}ontext-aware \\textbf{H}eterogeneous \\textbf{G}raph \\textbf{A}ttention ne\\textbf{T}work(\\textbf{CHGAT}) and introduce the whole pipeline of our model in this paper, which is described in Fig.(\\ref{model_framework}). Specifically, the user's own behavior and similar crowds' behavior are firstly utilized to construct heterogeneous graphs for the current user. Then, the uniform knowledge representation is proposed to perform a unified semantic mapping and transformation both on graph vertices and contextual features in the ranking system. Moreover, we introduce a context-aware multi-level attention mechanism to dynamically aggregate graphs by fusing real-time context information. And the detailed aggregation process of our constructed heterogeneous graph is then formulated. Through the multi-level attention mechanism proposed in our research, the heterogeneous behavior graph is able to aggregate the potential user interest according to the current scene to the great extent, which increases the generalization ability of user interest modeling in different scenarios. In general, the main contributions of this paper are as follows:\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{itemize}\n\n\\item We introduce meta-path based heterogeneous graphs to describe the heterogeneous behavior of users in different contexts, which contains search scenarios and locations. Considering the fact that the limited scale of the user's own behavior may affect the performance of the model, the behavior of similar crowds is innovatively introduced to connect with the pre-constructed behavior graph. In order to capture the characteristics of similar people, the portrait feature of users is chosen as the context feature of similar crowds based behavior graph.\n\n\\item We raise a Unified Knowledge Representation(UKR) method based on the knowledge graph to express uniformly the vertices and context features of heterogeneous graphs at the semantic level. UKR can not only provide basic representations for vertices on meta-paths in the process of aggregating user interest but also provide generalized semantic information for external real-time context features. The vertex representation designed in this way reduces the overall parameter amount of the model and adds semantic information to the cold start scenario as well.\n\\item We propose a heterogeneous graph attention model that is sensitive to the real-time context. Different from the previous approaches, we create a novel multi-level attention mechanism that can be calculated during the graph generation process. Specifically, multiple type-specific vertex-level attention network select vertices that are more related to the current context. The path-level attention network is able to select meta-path based on the relevance of context information at the semantic level. Applying the multi-level attention mechanism, the model proposed in our research is able to aggregate user interest expressions related to specific context scenarios to increase the prediction performance of the ranking system.\n\\item We collect the real data of the Koubei platform to verify the actual performance of the proposed model. Experimental results show that the proposed model achieves the best performance. In order to verify the auctual application value of CHGAT, we conducted several A\/B compare groups, and accumulated millions of browsing data for confident statistical results. The online experiment results prove that CHGAT can well capture contextual information for dynamic aggregation of heterogeneous graphs and return better-matched prediction results.\n\n\\end{itemize}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{figure*}[htbp]\n\\centering\n\\subfigure[Graph Construction]{\n\\centering\n\\includegraphics[scale=0.35]{figure\/graph_construction.png}\n}\\hspace{0.5mm}\n\\subfigure[CHGAT]{\n\\centering\n\\includegraphics[scale=0.35]{figure\/chgat.png}\n \n}\\hspace{0.5mm}\n\\subfigure[Prediction Model]{\n\\centering\n\\includegraphics[scale=0.35]{figure\/model.png}\n \n}\\hspace{0.5mm}\n\\centering\n\\setlength\\abovecaptionskip{-0.1cm}\n\\setlength\\belowcaptionskip{-0.1cm}\n\\caption{The whole pipeline of our model:(a)Graph Construction,(b)Context-aware Heterogeneous Graph Attention Network,(c)The Complete Prediction Model}\n\\label{model_framework}\n\\vspace{-1em}\n\\end{figure*}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Preliminary}\n\n\nIn this section, we begin by introducing the formulation of the problem that we want to solve. Then, a detailed description of definitions and examples of some key concepts in heterogeneous graphs is given.\n\n\n\\vspace{-0.3cm}\n\\subsection{Problem Formulaion}\n\n\\begin{dfn}[User behavior prediction for the ranking sysytem]\nGiven a set $<\\mathcal{U},\\mathcal{Q},\\mathcal{I},\\mathcal{C},\\mathcal{K}>$, where $\\mathcal{U}=\\{u_1,\\cdots,u_P\\}$ stands for the set of $P$ users, $\\mathcal{Q}=\\{q_1,\\cdots,q_R\\}$ denotes the set of $R$ queries searched by users, $\\mathcal{I}=\\{i_1,\\cdots,i_M\\}$ represents the set of $M$ candidate items that remains to be predicted, $\\mathcal{C}=\\{c_1,\\cdots,c_N\\}$ stands for the set of $N$ different contexts, and $\\mathcal{K}=\\{k_1,\\cdots,k_S\\}$ denotes the set of $S$ uniform knowledge representations. Generally, the overall goal of predicting user behavior for the ranking system in the location-based search(LBS) system is to return more matching items to the user, when they search in specific contexts.\n\\end{dfn}\n\nIn the LBS system of the local consumer service platform, when a user $u\\in \\mathcal{U}$ searches for a query $q\\in \\mathcal{Q}$, the ranking algorithm needs to match the user $u$ with the most interesting store $i\\in \\mathcal{I}$ utilizing the information of search context $c\\in \\mathcal{C}$, which composed of the real-time location and time of $u$. For example, the user would like to go to a nearer store for a quick lunch at noon on weekdays, and may wish to visit the famous restaurant in the city center to taste unique cuisines on weekend evenings. It can be seen from the above example that the user's interest dynamically changes according to the contextual information. And the ranking algorithm needs to predict a higher score for the candidate item in which the user is more interested in a specific context.\n\n\n\n\n\\vspace{-0.3cm}\n\\subsection{Heterogeneous Graph}\\label{2}\n\n\nIn order to generate dynamic interest that changes with the contexts, we propose to utilize heterogeneous interest network as the basic data structure to describe the user behavior, which is also known as a heterogeneous graph $\\mathcal{G}=(\\mathcal{V},\\mathcal{E})$. The heterogeneous graph consists of a vertex set $\\mathcal{V}$ and an edge set $\\mathcal{E}$. And the whole graph $\\mathcal{G}$ can also be associated with a vertex type mapping function $\\psi:\\mathcal{V}\\xrightarrow{}\\mathcal{A}$ and a edge type mapping function $\\varphi:\\mathcal{E}\\xrightarrow{}\\mathcal{B}$. In the definition of heterogeneous graph, $\\mathcal{A}$ and $\\mathcal{B}$ represent different types of nodes and edges, respectively, and it should satisfy $\\left|\\mathcal{A}\\right|+\\left|\\mathcal{B}\\right|>2$.\n\n\n\n\\begin{dfn}[Meta-path]\n\\vspace{-0.1cm}\nMeta-path $\\phi$ is defined as the assembly method of different types of vertices and edges in the heterogeneous graph $\\mathcal{G}$, which also be considered to include the sequence relationship of the order of vertices.\n\\end{dfn}\n\n\\vspace{-0.1cm}\n Considering that the meta-path represents the semantic meaning of the path formed by relations between $\\mathcal{V}$ and $\\mathcal{E}$, in this paper, the natural behavior sequences of users are particularly employed to generate the meta-path $\\phi$. Therefore, we propose several meta-paths originating from the heterogeneous behavior link, which consists of four types of vertices including $User(U)$, $Item(I)$, $Query(Q)$,$Context(C)$, and their rich interactions. For example, $User-Context-Query-Item(UCQI)$ indicates that the user enters a query in a certain context and then interacts with several items returned by the search engine. And $User-Context-Item(UCI)$ indicates that a user directly interacts with multiple items under a certain context, where the form of interactions include clicking, purchasing, and adding to the shopping cart, etc. Given the premise of the meta-path $\\phi$, further analysis of its semantics information requires the definition of neighbor vertices along the meta-path.\n\n\n\n\n\n\\begin{dfn}[Meta-path based Neighbor Vertices]\nMeta-path based neighbor vertices $\\mathcal{N}_{v}^\\phi$ is defined as the set of neighbor vertices of a vertex $v$ on the meta-path $\\phi$. \n\\end{dfn}\n\n\n\n\n\nAs shown in the Fig.(\\ref{model_framework}), for example, in the meta-path $\\phi_{UCQI}$, we can get the following neighbor nodes, $\\mathcal{N}_{u}^{\\phi_{UCQI}}=\\{c_1,c_2\\}$ represents that the user $u$ has historical behaviors under context $c_1$ and context $c_2$. $\\mathcal{N}_{c_1}^{\\phi_{UCQI}}=\\{q_1,q_2\\}$ denotes that user has searched query $q_1$ and query $q_2$ under the context $c_1$, and $\\mathcal{N}_{q_1}^{\\phi_{UCQI}}=\\{i_1,i_2\\}$ and represents that the user interacts with item $i_1$ and item $i_2$ after searching for $q_1$. \n\n\n\n\n\n\n\n\n\n\n\\section{The Proposed Model}\n\n\n\nIn this section, we propose a novel supervised graph neural network, which named context-aware heterogeneous graph attention network(CHGAT). For each step of the whole pipeline of our model, we introduce its specific technical details from the background to the target. In the end, we design and assemble a prediction network and define the overall loss function.\n\n\n\n\\subsection{Overview}\nThe basic idea of the proposed CHGAT is to design a graph neural network to capture the context-aware potential interest in heterogeneous behaviors and to provide broadened semantic representations for users. As shown in Fig.(\\ref{model_framework}), firstly, to prepare the data required by CHGAT, we employ a variety of user behaviors to construct heterogeneous graphs and define contextual features corresponding to different meta-paths. After building the heterogeneous graph, we introduce a unified knowledge representation method that assembles multiple knowledge units to provide unified transformations for different types of vertices in the heterogeneous graph, which can greatly reduce the overall amount of model parameters and represent user interests with clearer semantic information. Moreover, a variety of attention networks that are sensitive to external contextual scenarios are designed, and the calculated attention coefficients are utilized to dynamically aggregate vertices from the same layer to the upper layers in heterogeneous graphs. After the above procedures, the user embedding vector aggregated in the heterogeneous graph is sent to the subsequent deep network for predicting the probability of clicking the candidate $i$ when the user searches for query $q$.\n\n \n\n\n\n\n\n \n\n\n\n\n\n\\vspace{-1em}\n\n\n\n\\subsection{Graph Construction}\n\\label{graph_construction_section}\n\n\\paragraph{Heterogeneous graph constructed from the self-behavior}\nFor the graph representation method, the information that the model can aggregate is closely related to the way that the graph is constructed. Here, we first apply the historical self behaviors of users to construct the heterogeneous graph according to the two meta-paths of $\\phi_{UCQI}$ and $\\phi_{UCI}$, and only retain the historical behavior related edges during the construction. And considering the requirements of the LBS system, we set the location where the user's historical behavior occurred as the context vertex $C$, and merge the similar behaviors that occurred in the same context into the same meta-path. Meanwhile, the central vertex in the graph is defined as the root vertex in the meta-path, which is conducive to the subsequent graph aggregation process to maximize the retention of hidden interest information in the original behavior sequence. It is worth noting that our graph is constructed from users, so we set the user as the root vertex $v_r$ of the constructed graph. \n\n\n\\paragraph{Heterogeneous graph transferred from similar crowds} However, when the user has a small number of historical behaviors, the scale of the constructed heterogeneous graph is limited, which in turn will affect the prediction effect of the whole model. In response to this problem, we additionally introduce a heterogeneous graph constructed based on heterogeneous behaviors of similar crowds. Considering the spatial characteristics of the LBS scenario, we treat users within a certain distance from the current user's search location as similar crowds and utilize their most recent interaction behavior as a historical behavior database of similar crowds to build the graph. At the same time, we set the feature of the user portrait as the context vertex $C$ to facilitate the selection of the user expression that is most similar to the current user from the behavior of other people during subsequent graph aggregation.\n\n\n\n\nAfter establishing the user's own heterogeneous graph and the heterogeneous graph of similar crowds, we connect them to the shared root vertex, which is the user to be represented in the current ranking system. At this time, the complete heterogeneous graph contains four types of meta-paths, which names $\\phi_{UCQI_{self}}$, $\\phi_{UCI_{self}}$, $\\phi_{UCQI_{sim}}$, and $\\phi_{UCI_{sim}}$. And before the subsequent aggregation, all the vertices in the graph used their original ID for recording.\n\n\n\n\\begin{figure}[] \n\\centering\n\\includegraphics[width=0.25\\textwidth]{figure\/UKR.png} %\n\\setlength{\\abovecaptionskip}{0cm}\\caption{An example of the uniform knowledge representation} \n\\label{UKR} \n\\vspace{-2em}\n\\end{figure}\n\n\\subsection{Uniform knowledge representation}\n\\label{Uniform knowledge representation}\nOne of the key challenges of applying heterogeneous graphs in ranking algorithms is to comprehend the heterogeneous vertices in the graph. But if directly employing the original ID of the vertex for embedding or use its pre-trained embedding as the expression of the vertices in the graph, there may exist following two problems: \n\n\\begin{itemize}\n\\item With tens of millions of vertices in the heterogeneous graph, a separate embedding expression for each node in the graph will cause the model to have a tremendous amount of parameters, which in turn affect the availability and time-consuming performance of the model.\n\\item In the heterogeneous graph, the vertex to be aggregated and its neighbor vertices in the same meta-path are often heterogeneous, which indicates that the embedding of original vertices is not very semantically related. It also denotes that during the process of graph aggregation, there would be conflicts of the heterogeneous information between neighbor vertices and the vertex to be aggregated, which eventually increases the difficulty of the converging of model parameters.\n\\end{itemize}\n\nIn order to avoid the impact of the above problems on the performance of the model, we introduce the uniform knowledge representation(UKR) to comprehend vertices in the heterogeneous behavior graph. The conversion range of the unified knowledge expression includes all types of vertices in the meta-path except the root vertex. For example, in our research, the main vertices are query, item, and search scene lies both in constructed meta-paths and the input of the ranking system. \n\n\n\nSpecifically, we first utilize the knowledge graph\\cite{wang2014knowledge} to extract key knowledge about the store in our platform, which includes the primary business category of the store, the main tag in the title of the store, and the name of commodities in the store. Next, the search comprehension engine is used to predict and retain the key knowledge of search intent, search category, text entity, and so on. And at the aspect of comprehending search scenarios, locations, nearby high-frequency shops, weather, and time, are selected as the knowledge representation. \n\n\nIt is worth noting that the knowledge representation extracted from the original vertex is first retained in the form of several texts, and the knowledge expressed in the same space can be extracted from different heterogeneous materials. As shown in the Fig.(\\ref{UKR}), a fried chicken shop $i_1$ contains the main knowledge of hamburger $k_1$, fried chicken $k_2$. And the query $q_1$ search for fried chicken legs also contains fried chicken $k_2$. At this time, the knowledge unit $k_2$ of representing fried chicken for these two heterogeneous vertices is the same.\n\n Then we apply the conversion relationship between all heterogeneous materials and knowledge representations to construct a key-value pair knowledge dictionary $\\mathcal{K}$, the key in the dictionary is the original id of vertices in the graph, and value is multiple converted knowledge units. Taking Fig.(\\ref{UKR}) as an example, $\\mathcal{K}=\\{i_{1}:\\{k_{1},k_{2}\\},q_{1}:\\{k_{2},k_{3}\\}\\}$ represents the knowledge dictionary.\n\n After mapping the original ID of the vertex in the heterogeneous graph to the knowledge unit $k$, a function $m:\\mathcal{K}\\rightarrow\\mathbb{R}^d$ is designed to respectively map the $k$ after the one-hot transformation to the $d$-dimensional embedding vector $e$. Note that each original vertex in the heterogeneous graph is composed of multiple knowledge units, so it is necessary to introduce an knowledge aggregation function $g_k$ to fuse multiple knowledge embedding vectors as the embedding vector of the current vertex. In the above example, the embedding of the fried chicken shop $i_1$ and the embedding of the query $q_1$ of fried chicken legd are represented as following\n\\setlength\\abovedisplayskip{1.5pt}\n\\setlength\\belowdisplayskip{1.5pt}\n\\begin{equation}\n\\begin{split}\ne_{i_1}&={g_k}(e_{k_1},e_{k_2})={g_k}(m({k_1}),m({k_2}))\\\\\ne_{q_1}&={g_k}(e_{k_2},e_{k_3})={g_k}(m({k_2}),m({k_3}))\n \\end{split}\n\\end{equation}\nwhere the knowledge aggregation function $g_k$, which can be assembled by the neural network, weighted sum, and other methods, determines the current vertex knowledge assembly method and knowledge focus. In order to incorporate more information, an element-wise average function is selected as the function $g_k$ in this paper.\n\n\n\n\n\n By applying the above approach, it is convenient for us to uniformly express large-scale original materials in the form of assembly knowledge units, which greatly reduces the scale of model parameters and solve the problem of the semantic gap between adjacent vertices to a certain extent. Meanwhile, for a new vertex in a graph, we can also utilize the trained knowledge embeddings to quickly comprehend it. It is worth noting that the embedding parameters here are part of the model and are also trained with the CHGAT main model. Referring to the Table.(\\ref{auc_table}), this design further improves the overall prediction accuracy of the model. \n\n\n\n\n\n\n\n\\subsection{Context-aware Heterogeneous Graph Attention Network}\n\\label{CHGAT section}\n\n\nIn the process of the whole pipeline, another key challenge of utilizing the meta-path guided heterogeneous graphs is the selection and aggregation of vertices in the heterogeneous graph. Different from other graph aggregation methods that utilize the relationship between neighbor vertices in the complete graph\\cite{velivckovic2017graph}, we propose a new multi-level attention based graph aggregation mechanism, which fuses the outside contextual features to aggregate vertices in the constructed graph. Especially, considering the fact that vertices on the meta-path are divided into the root vertex and its neighbor vertices, we introduce two design schemes of attention mechanism for the topological characteristics of the heterogeneous graph. It is also worth noting that the external real-time outside features utilized below include the real-time search scenario of the user, the current search query in the ranking system, and the candidate item to be predicted. For the convenience of expression, we call such features as outside vertices $v_o$.\n\n\n\n\nWe first introduce a novel vertex-level attention mechanism that can learn the importance of different vertices in the current contextual scene and gradually aggregate the meaningful information in the vertices along the meta-path. Given a vertices pair $(v_i,v_j),{v_j}\\in \\mathcal{N}_{v_i}^\\phi$, the vertex-level attention aims to get the attention weight $\\alpha_{v_i,v_j}^\\phi$ that determines the importance of vertice $v_i$ to vertice $v_j$. Different from methods such as HAN\\cite{han2019} which directly utilize the vertex pair $(v_i,v_j)$ to calculate the weight, we introduce the real-time outside vertex $v_o$ of the same type with vertex $v_j$ to calculate the attention coefficient:\n\\setlength\\abovedisplayskip{1.5pt}\n\\setlength\\belowdisplayskip{1.5pt}\n\\begin{equation}\n\\alpha_{v_i,v_j}^\\phi=Attention_{\\psi(v_j)}(v_j,v_o)\n\\end{equation}\nwhere $v_i$ stands for the vertex to be aggregated, and $Attention_{\\psi(v_j)}$ refers to a specially designed vertex-level attention network for the type of vertex $v_j$, and $\\phi\\in \\Phi$ is one of the meta-paths in the heterogeneous graph. \n\n\n\nIn order to better construct the correlation between external vertex and vertices in the graph, the uniform knowledge representation is applied to map and transform these vertices before inputted to the attention network. And the detailed formulation of the attention weight can be calculated as follows:\n\\begin{equation}\\label{1}\n\\alpha_{v_i,v_j}^\\phi=\\frac{exp(\\sigma(MLP_{\\psi(v_j)}(e_{v_j},e_{v_o})))}{\\sum_{v\\in\\mathcal{N}_{v_i}^\\phi}exp(\\sigma(MLP_{\\psi(v)}(e_v,e_{v_o})))}\n\\end{equation}\nwhere $\\mathcal{N}_{v_i}^{\\phi}$ stands for the meta-path based neighbor upstream vertices of $v_i$, and $MLP_{\\psi(v)}$ denotes a multilayer perceptron for specific type of the current vertex, $\\sigma$ represents the activation function. Then, introducing the vertex-level aggregation function $g_r$, the aggregated representation of the vertex $v_i$ can be obtained as:\n\\setlength\\abovedisplayskip{1.5pt}\n\\setlength\\belowdisplayskip{1.5pt}\n\\begin{equation}\n\\begin{split}\ne_{v_i}^\\phi&=g_{r}(\\alpha_{v_i,v}^\\phi,e_v), {v\\in \\mathcal{N}_{v_i}^\\phi}\\\\\n\\end{split}\n\\end{equation}\nwhere $e_{v_i}^\\phi$ is the aggregated embedding vector for vertex $v_i$ along the meta-path $\\phi$, and the aggregation function $g_{r}$ can be any other approaches that can incorporate the attention weight coefficients, such as attention-based LSTM\\cite{wang2016attention}, etc.\n\n\n\n\n\n\n\nFurthermore, after utilizing the vertex-level attention to obtain the correlation weight of each vertex and aggregate the representation along the meta-path $\\phi$, the number of aggregated representations $e^\\phi, \\phi\\in \\Phi$ that connected with the root vertex $v_r$ is consistent with the number of pre-defined meta-paths in the graph, which is $\\left|\\Phi\\right|_{\\mathcal{G}}$. And in this paper, as described in Subection\\ref{graph_construction_section}, each meta-path $\\phi$ has a unique context feature $c_\\phi$. Therefore the path representation $e_{v_r}^\\phi$ equals to the embedding vector aggregated to the vertices of the context $e_{c}^\\phi$ along the meta-path $\\phi$. In order to obtain the unique representation of the root vertex, we then propose the path-level attention to fuse multiple meta-path representations in the graph, which can be defined as:\n\\setlength\\abovedisplayskip{1.5pt}\n\\setlength\\belowdisplayskip{1.5pt}\n\\begin{equation}\n\\alpha_{v_r}^\\phi=Attention_{\\psi(c_\\phi)}(c_\\phi,c_o)\n\\end{equation}\n\nwhere $\\alpha_{v_r}^\\phi$ is the attention weight of meta-path $\\phi$ for the root vertex $v_r$, and $c_o$ is the outside context feature, and $\\psi(c_\\phi)$ is the type of the context feature $c_\\phi$. \nThe goal of the path-level attention is to use the information of external context characteristics to filter the most relevant path of the heterogeneous graph and choose the representation of the meta-path that is most similar to the current context. In order to achieve this target, and considering that users' interests do not explicitly include contextual features, we use the self-representation $e_{c_{\\phi}}$ of the context feature $c_\\phi$ to calculate the weight coefficient, and utilize the aggregated representation $e_{v_r}^{\\phi}$ from the meta-path $\\phi$ as the path representation together with the weight coefficient $\\alpha_{v_r}^\\phi$ to get the final representation of the root vertex $v_r$. Hence, the detailed formula of the path-level attention network is as follows:\n\\setlength\\abovedisplayskip{1.5pt}\n\\setlength\\belowdisplayskip{1.5pt}\n\\begin{equation}\\label{3}\n\\alpha_{v_r}^\\phi=\\frac{exp(\\sigma(MLP_{\\psi(c_\\phi)}(e_{c_\\phi},e_{c_o})))}{\\sum_{c_{\\phi'}\\in\\mathcal{N}_{v_r}}exp(\\sigma(MLP_{\\psi(c_\\phi')}(e_{c_{\\phi'}},e_{c_o})))}\n\\end{equation}\n\n\nwhere $c_{\\phi'}$ are context vertices that belong to the neighbor of root vertex $v_r$. Therefore, in the design concept of the path-level attention, the representation of the root vertex can be calculated as the aggregation from all meta-paths, where we introduce $g_p$ as the aggregate function:\n\\setlength\\abovedisplayskip{1.5pt}\n\\setlength\\belowdisplayskip{1.5pt}\n\\begin{equation}\\label{4}\ne_{v_r}=g_p(\\alpha_{v_r}^\\phi,e_{v_r}^\\phi),{\\phi\\in\\Phi}\n\\end{equation}\nwhere $\\Phi$ denotes all meta-paths in the heterogeneous graph, and $e_{v_r}^\\phi$ refers to the representation of different meta-paths of the root vertex $v_r$, and $e_{v_r}$ stands for the final representation of the root vertex. Next, we apply the above two attention mechanisms to the aggregation process of heterogeneous behavior graphs constructed in Subsection\\ref{graph_construction_section}.\n\n\n\n\n\\paragraph{Aggregation process in $\\phi_{UCQI_{self}}$ and $\\phi_{UCI_{self}}$ based graph} \nAt this time, the behavior in the heterogeneous graph based on $\\phi_{UCQI_{self}}$ and $\\phi_{UCI_{self}}$ comes from the user itself and records the user's active interaction behavior in different contexts. Considering that the historical behavior structure in this type of meta-path is similar to the current prediction scene, so the real-time input query and the item to be predicted in the ranking algorithm are utilized as the outside vertex $v_o$ of the vertex-level attention. For example, as shown in Fig.(\\ref{model_framework}), when the information of vertex $v_{i_1}$ is aggregated to vertex $v_{q_1}$ along the meta-path $\\phi_{UCQI_{self}}$, the outside item $v_{i_o}$ to be predicted and the current vertex $v_{i_1}$ are used as the input of the attention network $Attention_{item}$, which can be detailed as:\n\\setlength\\abovedisplayskip{1.5pt}\n\\setlength\\belowdisplayskip{1.5pt}\n\\begin{equation}\n\\label{vertex-example}\n\\alpha_{q_1,i_1}^{UCQI_{self}}=\\frac{exp(\\sigma(MLP_{item}(e_{i_1},e_{i_o})))}{\\sum_{i\\in\\mathcal{N}_{q_1}^{UCQI_{self}}}exp(\\sigma(MLP_{item}(e_i,e_{i_o})))}\n\\end{equation}\nAfter multi-layer vertex-level aggregation along the meta-path, the representation $e_u^{\\phi}$ of each meta-path $\\phi$ can be obtained. Considering the characteristics of the LBS scenario, we directly apply the search location as the context feature of each meta-path, and at this time $c_\\phi$ represents the recorded user search location for each meta-path $\\phi$ in the historical behavior graph. Then the importance of meta-path $\\phi$ to the current user $u$ can be obtained by the Eq.(\\ref{3}). And using the weights generated by path-level attention to fuse the representation of multiple meta-paths, the user representation $e_u^{chgat}$ of the $\\phi_{UCQI_{self}}$ and $\\phi_{UCI_{self}}$ based graph can be obtained.\n\n\n\n\n\\paragraph{Aggregation process in $\\phi_{UCQI_{sim}}$ and $\\phi_{UCI_{sim}}$ based graph}\n\nFor heterogeneous graphs constructed from other similar crowds behaviors, which contains the $\\phi_{UCQI_{sim}}$ and $\\phi_{UCI_{sim}}$, the aggregation at the vertex-level is consistent with the Eq.(\\ref{vertex-example}), and the basic representations of each path can be obtained, which is also the representation of similar crowds. In order to select the user representation most similar to the current user among similar crowds, we apply the basic portrait feature of users as the context feature $c$ in Eq.(\\ref{3}), for example, the path-level attention weight of the meta-path $\\phi$ to the present user $u$ can be obtained as:\n\\setlength\\abovedisplayskip{1.5pt}\n\\setlength\\belowdisplayskip{1.5pt}\n\\begin{equation}\n\\alpha_{u}^{\\phi}=\\frac{exp(\\sigma(MLP_{user}(e_{c_{\\phi}},e_{c_u})))}{\\sum_{c_{\\phi'}\\in\\mathcal{N}_{u}}exp(\\sigma(MLP_{user}(e_{c_{\\phi'}},e_{c_u})))}\n\\end{equation}\nwhere $e_{c_u}$ is the basic portrait feature of the present user $u$. Same as the Eq.(\\ref{4}), applying the weighted sum as the function to aggregate embeddings of meta-paths:\n\\setlength\\abovedisplayskip{-2pt}\n\\setlength\\belowdisplayskip{-2pt}\n\\begin{equation}\\label{weighted_sum}\ne_{u}=\\sum_{\\phi\\in \\Phi}(\\alpha_{u}^\\phi,e_{u}^\\phi)\n\\end{equation}\nwhere $e_{u}^\\phi$ is the aggregated embedding vector for meta-path $\\phi$, $e_{u}$ is the final representation originated from similar crowds, which is also named $e_u^{sim-chgat}$ in our paper.\n\n\n\nIn conclusion, the focus of the vertex-level attention network is to capture the correlation between the vertices to be aggregated and outside vertex of the same type and to select the vertex that is most similar to the current outside feature as the vertex-level aggregation information passed down. From the perspective of the user interest, the vertex-level attention is able to choose the item that most similar to the current outside item from the historical behavior as the basic representation of interest. Moreover, the purpose of the path-level attention network is to predict the correlation between path-level contextual information and current contextual attributes in the ranking system, and then choose the meta-path that is most similar to the current outside context feature as the primary representation of the root vertex in the graph.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\vspace{-1em}\n\n\n\\subsection{The Loss Function}\n\n\n\n\nAfter obtaining $e_{u}^{chgat}$ and $e_{u}^{sim-chgat}$, we introduce two networks $f_{chgat}(\\cdot)$ and $f_{sim-chgat}(\\cdot)$ to separately obtain their logits, which are then accumulated with the logits of the original network as the final output of the network. Therefore, the probability ${\\widehat y}_{u,q_o,i_o,c_o}$ denotes that user $u$ searches for the query $q_o$ and clicks the candidate $i_o$ in a certain outside search context $c_o$, which is predicted by our proposed CHGAT model, as Fig.(\\ref{model_framework}) shows, is established as:\n\\begin{equation}\n\\label{concat_embedding}\n\\begin{split}\n{\\widehat y}_{u,q_o,i_o,c_o}&=sigmoid(f_{chgat}(e_u^{chgat})\\vert\\vert f_{attri}(u,q_o,i_o,c_o) \\\\&+ \\frac\\beta{\\left|\\Phi_{chgat}\\right|}\\cdot f_{sim-chgat}(e_u^{sim-chgat})\\vert\\vert f_{attri}(u,q_o,i_o,c_o) \\\\&+ f_{oriDNN}(f_{attri}(u,q_o,i_o,c_o)))\n\\end{split}\n\\end{equation}\nwhere $\\{u,q_o,i_o,c_o\\}$ denotes the basic element of the search prediction system, $f(\\cdot)$ can be any form of deep neural network, $f_{attri}(\\cdot)$ represents multiple types of attributes, such as the features described for $u$, $q_o$, $i_o$, and $c_o$. It is worth noting that $\\frac\\beta{\\left|\\Phi_{chgat}\\right|}$ measures the number of meta-paths in the self-behavior graph, especially when the user's original heterogeneous behavior graph is small, it will improve the influence of similar crowds to our model. And $\\beta$ is a hyperparameter for the SIM-CHGAT part.\n\n\nThe loss function $\\mathcal L$ defines the discrepancy between the predicted probability ${\\widehat y}_{u,q_o,i_o,c_o}$ of the model and the true probability ${y}_{u,q_o,i_o,c_o}$. Here we utilize the cross entropy function as the loss function:\n\\begin{equation}\n\\begin{split}\n\\mathcal L =&-\\sum_{(u,q_o,i_o,c_o)}\\lbrack y_{(u,q_o,i_o,c_o)}\\log({\\widehat y}_{(u,q_o,i_o,c_o)})\\\\&+(1-y_{(u,q_o,i_o,c_o)})\\log(1-{\\widehat y}_{(u,q_o,i_o,c_o)})\\rbrack \\\\&+ \\lambda R(\\theta)\n\\end{split}\n\\end{equation}\nwhere $R(\\theta)$ denotes the regularization function for parameters of the whole model, and $\\lambda$ is the hyperparameter for the regularization part. And in the model training process, the Adam optimizer is utilized to minimize the loss function\\cite{kingma2014adam}.\n\n\n\n\n\\section{Experiments}\n\n\n\nIn this section, we first employ experiments on datasets of the Koubei platform to compare the proposed CHGAT with up-to-date state-of-the-art methods. Next, the sensitivity of the model performance to multiple hyperparameters is also verified, including the number of uniform knowledge units in the UKR part and the $\\beta$ in Eq.(\\ref{concat_embedding}). Moreover, we also examine and obtain the A\/B results of the model on the actual online local e-commerce system.\n\n\n\\begin{table}[]\n\\scalebox{0.7}{\n\\begin{tabular}{c|c|c|c|c}\n\\hline\n\\hline\nDatasets & \\# Sample & \\# Positive sample & \\# $E(\\left|\\Phi\\right|_{chgat})$ & \\# $E(\\left|\\mathcal{V}\\right|_{chgat})$ \\\\ \\hline\nFull-week Train & $3.26\\times10^8$ & $6.27\\times10^7$ & $5.35$ & $32.41$ \\\\ \nFull-day Test & $7.24\\times10^6$ & $1.38\\times10^6$ & $5.17$ & $30.94$ \\\\ \nFull-week Test & $8.16\\times10^6$ & $1.51\\times10^6$ & $5.41$ & $33.18$ \\\\ \nFull-week Hard Test & $2.29\\times10^6$ & $1.78\\times10^5$ & $0.85$ & $4.71$ \\\\ \n\\hline\n\\hline\n\\end{tabular}}\n\\caption{Basic Description of Datasets}\n\\label{Basic_Description_of_Datasets}\n\\vspace{-3em}\n\\end{table}\n\n\n\\subsection{Datasets}\nWe collect real online data from the leading local consumer service platform in China, the Koubei app\\footnote{https:\/\/www.koubei.com}. Specifically, the offline dataset covers a consecutive week's true behavior, which is taken measures such as negative sampling and noise filtering before further employed. Then the dataset can be described from the following multiple perspectives:\n\n\n\\begin{itemize}\n\\item From the perspective of feature generation, we have constructed the attribute features, statistical features, sequence features, and category features of users, queries, scenes, and shops. These basic features constitute the attribute feature parts in the Eq.(\\ref{concat_embedding}).\n\n\\item From the perspective of the user's own heterogeneous graph, we utilize the user's historical behavior in the past 30 days to construct a heterogeneous behavior graph, which mainly contains the meta-path $\\phi_{UCQI_{self}}$ and the meta-path $\\phi_{UCI_{self}}$.\n\n\\item From the perspective of the heterogeneous graph of similar crowds, we make use of search behaviors of people within three kilometers from the user's current search position to construct a heterogeneous graph. The meta-path in the graph mainly includes $\\phi_{UCQI_{sim}}$ and $\\phi_{UCI_{sim}}$, where the number of each type of meta-path is limited up to $20$.\n\\end{itemize}\n\nAs shown in the Table.(\\ref{Basic_Description_of_Datasets}), we randomly sample a major part of a whole week's dataset as the full-week training dataset and another small part as the full-week test dataset. And the purpose of doing like this is to reduce the impact of different dates on the sample distribution as much as possible. Moreover, the full-day test dataset utilizes random sampling throughout the day on a certain day after a week to simulate actual prediction scenarios. As for the statistic attributes of the heterogeneous graph in Table.(\\ref{Basic_Description_of_Datasets}), $E(\\left|\\Phi\\right|_{chgat})$ refers to the expectation value of the number of meta-paths per sample and $E(\\left|\\mathcal{V}\\right|_{chgat})$ denotes the the expectation value of the number of edges in the graph. It is worth noting that we select the sample of few-behaving users to construct a full-week hard test dataset. The $E(\\left|\\Phi\\right|_{chgat})$ in this dataset is much smaller than other datasets, which aims to verify the effect of the proposed SIM-CHGAT.\n\n\n\n\\begin{table*}[ht]\n\\scalebox{1}{\n\\begin{tabular}{c|c|c|c|c|c|c}\n\\hline\n\\hline\n\\multirow{2}{*}{Method} & \\multicolumn{2}{c|}{Full-day} & \\multicolumn{2}{c|}{Full-week} & \\multicolumn{2}{c}{Full-week hard} \\\\ \\cline{2-7} \n & AUC & NDCG & AUC & NDCG & AUC & NDCG \\\\ \\hline\nLR & $0.7620$ & $0.5120$ & $0.7709$ & $0.5097$ & $0.7204$ & $0.4830$ \\\\ \\hline\nDNN & $0.7763$ & $0.5266$ & $0.7811$ & $0.5241$ & $0.7362$ & $0.4975$ \\\\ \\hline\nWide\\&Deep(WD) & $0.7795$ & $0.5289$ & $0.7859$ & $0.5308$ & $0.7366$ & $0.5017$ \\\\ \\hline \\hline\nDIN-WD & $0.7816$ & $0.5407$ & $0.7938$ & $0.5382$ & $0.7371$ & $0.5094$ \\\\ \\hline\nQuery-DIN-WD & $0.7860$ & $0.5396$ & $0.7951$ & $0.5401$ & $0.7354$ & $0.5088$ \\\\ \\hline \\hline\nHAN-WD & $0.7741$ & $0.5325$ & $0.7826$ & $0.5327$ & $0.7309$ & $0.5051$ \\\\ \\hline\nMEIRec-WD & $0.7807$ & $0.5364$ & $0.7919$ & $0.5412$ & $0.7317$ & $0.5092$ \\\\ \\hline \\hline\nnoUKR-CHGAT & $0.7871$ & $0.5390$ & $0.7942$ & $0.5327$ & $0.7345$ & $0.5063$ \\\\ \\hline\nCHGAT & \\bm{$0.8018$} & $0.5453$ & $0.8059$ & $0.5420$ & $0.7360$ & $0.5097$ \\\\ \\hline\nSIM-CHGAT & $0.8006$ & \\bm{$0.5471$} & \\bm{$0.8120$} & \\bm{$0.5494$} & \\bm{$0.7591$} & \\bm{$0.5215$} \\\\ \\hline\n\nImprovement & $2.01\\%$ & $1.18\\%$ & $2.12\\%$ & $1.51\\%$ & $2.97\\%$ & $2.31\\%$ \\\\ \n\\hline\n\\hline\n\\end{tabular}}\n\\setlength\\belowcaptionskip{0.1cm}\n\\caption{The ranking metrics of different methods. The best results are indicated in bold, and the last row denotes the improvement of the method proposed in our research compared to the best baseline method}\n\\label{auc_table}\n\\vspace{-0.5em}\n\\end{table*}\n\n\n\n\\subsection{Baseline Methods and Experimental Settings}\n\nIn order to verify the performance of the proposed model, we utilized the latest prediction model in the industry and methods related to our model as baseline methods to create offline compare groups, which can be described as:\n\n\\begin{itemize}\n\\item \\textbf{Logistic Regression}\\cite{hosmer2013applied} is a basic linear model, which employs statistical features and one-hot features predict the probability of classification tasks.\n\\item \\textbf{Deep Neural Network}\\cite{45530} is a neural network with multiple layers, which is able to transform categorical features into embedding vectors.\n\\item \\textbf{Wide\\&Deep}\\cite{DBLP:journalscorrChengKHSCAACCIA16} combines LR and DNN to balance memory performance and generalization performance, which is choosed as the original model $f_{oriDNN}(\\cdot)$ for following models.\n\\item \\textbf{DIN-WD}\\cite{zhou2018deep} utilizes the item sequence of the users' past interactions to model interest representation. In our experiments, we combine it with WD to predict the click-through rate.\n\\item \\textbf{Query-DIN-WD} adds a query sequence more than the original DIN-WD model.\n\\item \\textbf{MEIRec-WD}\\cite{10.1145\/3292500.3330673} builds a heterogeneous graph based on multiple artificial meta-paths, which is selected as the comparison of graph aggregation methods. \n\\item \\textbf{HAN-WD}\\cite{han2019} employs the correlation between neighbor nodes of the heterogeneous graph to aggregate. In our experiments, we treat it as a comparison from the perspective of the graph attention network.\n\\item \\textbf{noUKR-CHGAT} is the proposed model in our research. However, it is short of the uniform knowledge representation part, which is detailed in Section\\ref{Uniform knowledge representation}.\n\\item \\textbf{CHGAT} is the proposed model in Section\\ref{CHGAT section}, which lacks the embedding aggregated from the similar crowds graph.\n\\item \\textbf{SIM-CHGAT} is the complete version of the model proposed in our research.\n\n\\end{itemize}\n\n\n\n\n\n\n\n\n\n\n\\begin{figure*}[htbp]\n\\vspace{-0.8cm}\n\\centering\n\\subfigure[Full-day]{\n\\centering\n\\includegraphics[scale=0.2]{figure\/UKR_number_auc1.png}\n}\\hspace{0.5mm}\n\\subfigure[Full-week]{\n\\centering\n\\includegraphics[scale=0.2]{figure\/UKR_number_auc2.png}\n \n}\\hspace{0.5mm}\n\\subfigure[Full-week hard]{\n\\centering\n\\includegraphics[scale=0.2]{figure\/UKR_number_auc3.png}\n}\\hspace{0.5mm}\n\\centering\n\\setlength{\\abovecaptionskip}{-0.1cm}\n\\caption{Parameter sensitivity of the number of uniform knowledge id}\n\\label{UKR_number}\n\\vspace{-1em}\n\\end{figure*}\n\n\n\n\n\n\n\\begin{figure*}[htbp]\n\\centering\n\\subfigure[Full-day]{\n\\centering\n\\includegraphics[scale=0.18]{figure\/beta_auc1.png}\n}\\hspace{0.5mm}\n\\subfigure[Full-week]{\n\\centering\n\\includegraphics[scale=0.18]{figure\/beta_auc2.png}\n \n}\\hspace{0.5mm}\n\\subfigure[Full-week hard]{\n\\centering\n\\includegraphics[scale=0.18]{figure\/beta_auc3.png}\n}\\hspace{0.5mm}\n\\centering\n\\setlength{\\abovecaptionskip}{-0.1cm}\n\\caption{Parameter sensitivity of the $\\beta$ for SIM-CHGAT part}\n\\label{beta_auc}\n\\vspace{-1em}\n\\end{figure*}\n\nFor the above-selected methods, the shared data, features, and hyperparameters of the model are all kept consistent to establish corresponding control groups. And the aggregation function $g$ in our model is set to be the weighted sum. And we utilize $Relu$ as the activation function\\cite{agarap2018deep} and set the $\\lambda$ equals $0.01$. In order to compare the performance of the model on user behavior prediction tasks, we leverage AUC\\cite{lobo2008auc} and average NDCG\\cite{valizadegan2009learning} as the evaluation indicators.\n\n\n\\subsection{Overall Performance}\n\n\nIn order to evaluate the prediction performance of the proposed model in different scenarios, we verified the performance of the baseline and our algorithm in three test datasets, and the obtained results are shown in the Table.(\\ref{auc_table}). According to the detailed data, we now give analyses of experiment results::\n\\begin{itemize}\n\\vspace{-0.3em}\n\\item Under evaluate datasets employed in our paper, performances of CHGAT and SIM-CHGAT are both significantly better than other baseline methods. Specifically, on the Full-day dataset that simulates the actual online prediction scenario, CHGAT improves the AUC by $2.01\\%$ compared to QUERY-DIN-WD, indicating that the performance of the method proposed in this article is better than other methods in the actual prediction scenario. With a longer time distribution, the full-week dataset performance of CHGAT and SIM-CHGAT is significantly better than other graph methods or behavior sequence methods, indicating that the proposed CHGAT model is able to describe dynamic user interest which varifies with different contexts, and returns more matching items for users.\n\n\\item The noUKR-CHGAT employs graphs that have not been transformed with the unified knowledge representation. As can be seen in Table.(\\ref{auc_table}), in the prediction results of all test datasets, noUKR-CHGAT is significantly weaker than CHGAT with UKR whether in the aspect of AUC or NDCG. In addition, the number of knowledge IDs in UKR represents the amount of information contained in the transformed vertices, where the noUKR-CHGAT has no knowledge IDs. As we can see in the Fig.(\\ref{UKR_number}), as the number of knowledge id increases, the AUC of CHGAT in multiple test datasets can be significantly improved, indicating that our proposed UKR method is able to effectively establish the semantic relationship between heterogeneous behavior and heterogeneous materials. It is also worth noting that the employ of UKR can reduce the model size from $1.72G$ to $0.59G$, which improves the training convergence effect of the model. And we set the number of knowledge id equals to $5$.\n\n\n\\item We verify the performance of SIM-CHGAT model under multiple test conditions. It can be seen from the right column in Table.(\\ref{auc_table}) that in the hard test dataset, the performance of SIM-CHGAT is better than other methods, which proves that the introduction of similar crowd behavior has a better impact on cold-start user behavior prediction. In addition, as shown in the Fig.(\\ref{beta_auc}), we also find that when the $\\beta$ in Eq.(\\ref{concat_embedding}) is equal to $1$, the prediction effect of SIM-CHGAT is the best, but if the $\\beta$ is larger, the prediction performance will decrease to some extent, which can be explained that when the $\\beta$ is too large, the behavior of other users will affect the user's own representation of interest. Hence, before the online experiment, we set $\\beta$ to $1$.\n\n\n\n\n\n\n\\end{itemize}\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\\subsection{Online A\/B Experiment}\n\\label{Online_AB_Experiment}\n\n\nWe apply CHGAT and SIM-CHGAT respectively in the actual online search scenarios of the Koubei app\\footnote{https:\/\/www.koubei.com}. Consistent with the construction method of offline datasets, we employ the real historical behavior of users to construct heterogeneous graphs and update them in real-time, which is detailed in section\\ref{SUPPLEMENT}. Under the framework of the A\/B tests, we set one of the buckets as the experimental group and the other bucket as the baseline group and conduct two sets of A\/B tests. A\/B testS would hit all users who use the search function of the Koubei app. In order to increase the confidence level of the A\/B experiment as much as possible, we randomly divide users into buckets with the granularity of days. After 14 days of cumulative testing, the effective data volume in each test bucket is close to two million.\n\nCompared with the online baseline method without CHGAT, the CHGAT proposed in this article has increased the click rate of unique visitors per day(uvCTR) by $3.85\\%$, the purchase rate of unique visitors per day(uvCVR) has increased by $2.95\\%$, the average revenue per user(ARPU) has been promoted by $7.99\\%$, and the average click position of the list of the search result has risen by $4.24\\%$. And compared with CHGAT, SIM-CHGAT improves $1.74\\%$ on uvCTR, among which the uvCTR of new customers increased by $3.01\\%$. These results all show that the algorithm proposed in this article can increase customer flow and overall revenue for merchants on the Koubei platform, and can also help users find more interesting items. In conclusion, our proposed CHGAT has both high application potential and economic value.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\n\nIn this paper, in order to solve the challenge that fusing multiple heterogeneous contextual information for dynamic user representation, we propose a context-aware heterogeneous graph attention model to predict user behavior based on dynamic modeling of user interests. Specifically, a variety of different sources of behavioral data is employed to construct heterogeneous graphs of users, and similar crowds behavior graphs are build to solve the problem of limited self-graph scale. Later the proposed unified knowledge representation in CHGAT is able to map multiple vertices in a heterogeneous graph to a similar semantic space. Moreover, the newly designed vertex-level and path-level attention mechanisms are capable of selecting vertices most relevant to outside features in the graph for aggregation. In order to verify the performance of our proposed CHGAT in different scenarios, we employ extensive experiments in large scale offline evaluation datasets and also conduct several two-week online A\/B tests. Experimental results demonstrate that the proposed CHGAT achieves obvious advantages compared to other user representation approaches or heterogeneous graph methods and significantly improves the revenue of merchants and the enthusiasm of users on Koubei app. In the future, we will explore more flexible unified knowledge conversion methods.\n\n\n\n\n\\bibliographystyle{ACM-Reference-Format}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzaybx b/data_all_eng_slimpj/shuffled/split2/finalzzaybx new file mode 100644 index 0000000000000000000000000000000000000000..f4fde657a426bf199cdbe39fb44ce73ca198d69c --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzaybx @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nGiven the abundance of hydrogen in the universe, the Lyman $\\alpha$ (Ly$\\alpha$) line is an important component of radiation fields in a wide range of astrophysical settings. \\lya radiation transport is an active area of research in the study of planets, stars, galaxies, and cosmology \\citep{2019SAAS...46....1D}. An example application motivating our work is the role of \\lya in planetary atmospheres. The outer layers of the atmosphere are central to a planet's evolution, since they can shelter the lower atmosphere from high energy radiation as well as regulate the escape of gas into space. There are two sources of Ly$\\alpha$: the star, and recombinations in the planet's atmosphere. \\lya may ionize atoms and dissociate molecules, as well as exert pressure forces that drive an outflow \\citep{2018A&A...620A.147B}. \\lya can also excite H atoms to the 2p state, creating a population of Balmer-line absorbers that can be observed via transmission spectroscopy \\citep{2017ApJ...851..150H, 2021ApJ...907L..47Y}. Due to the low gas densities in the upper atmosphere, collisional de-excitation and broadening are of secondary importance and \\lya may undergo ``resonant scattering''.\n\nHubble Space Telescope (HST) observations with the STIS have found large \\lya transit depths around a handful of exoplanets \\citep{2003Natur.422..143V, 2012A&A...543L...4L, 2012A&A...547A..18E, 2015Natur.522..459E, 2017A&A...597A..26B, 2017A&A...599L...3B, 2017A&A...602A.106B, 2018A&A...620A.147B, 2019AJ....158...50W, 2019EPSC...13.1928L, 2020ApJ...888L..21G,2021arXiv210309864B}. These observations have revealed a population of atoms extending out to distances of order a few planetary radii or more for several planets around bright, nearby stars, motivating a study of the physics of \\lya interactions with the H atom population. The transition from the atomic to the molecular layer in these hot upper atmospheres may take place at pressures of order ${\\sim}\\ 10\\ \\mu$bar (see discussion in \\citealt{2017ApJ...851..150H} for details). This suggests the presence of a thick layer of atomic H which can have a line center optical depth of $\\tau_0\\ {\\sim}\\ 10^8$. A careful treatment of resonant scattering is necessary in order to construct accurate models of H atom excitation, heating, and radiative forces. \n\nDue to the technical challenge of including resonant scattering, the fully three-dimensional geometry, and the presence of an outflow, numerical simulations may be required to fully understand the dynamics of these irradiated exoplanet atmospheres. The large optical depths at \\lya line center impose a steep computational cost for solving radiative transfer with Monte Carlo methods directly coupled with fluid dynamics \\citep{2017MNRAS.464.2963S}. The number of scatterings a photon undergoes is proportional to the line center optical depth, $\\tau_0$, of the domain \\citep{1972ApJ...174..439A}. Near the base of the atomic layer, the line center optical depth is ${\\sim}10^8$, so most of the time is spent following photons in these cells. A method that can accurately characterize transfer through these zones without following every photon scattering has the potential to greatly accelerate the calculation (see e.g. \\citealt{1968ApJ...153..783A,2002ApJ...567..922A}).\n\nApproximate analytic solutions for resonant scattering exist in certain limits. \\citet{1973MNRAS.162...43H} showed that when most of the radiation is in the damping wings, the transfer equation reduces to the Poisson equation. However, their solution uses an ansatz to handle the boundary condition. To our knowledge, the errors introduced by this treatment have never been quantified. They attempt a separation of variables as $J(\\tau,\\sigma) = \\theta(\\tau) j(\\sigma)$ in spatial variable $\\tau$ and frequency variable $\\sigma$ (their Equations 16 and 23). The solutions for the eigenfunctions $\\theta(\\tau)$ and $j(\\sigma)$ then depend explicitly on the separation constant $\\lambda$. In order to satisfy the boundary conditions, the separation constant is shown to satisfy an eigenvalue equation of the form\n\\begin{eqnarray}\n\\lambda \\tan(\\lambda B) & = & \\frac{3}{2} \\phi \\Delta,\n\\label{eq:evalue}\n\\end{eqnarray}\nwhere $2B$ is the slab optical depth at line center, $\\phi$ is the line profile, and $\\Delta$ is the Doppler width. The key point is that the line profile depends on one of the coordinates: frequency. This causes the eigenvalues of the separation constant to be frequency-dependent. Thus, the separation ``constant'' is not constant, and the function does not satisfy the Poisson equation since the frequency derivatives will act on the separation ``constant\", giving extra terms. In the limit of large optical depth $B$, they approximate the eigenvalues as $\\lambda_n B \\simeq \\pi (n-1\/2)$, which gives zero mean intensity at the surface. Their Equation 34 subsequently allows $\\lambda$ to have a small deviation from the above expression, which is explicitly frequency dependent. This allows a nonzero intensity at the surface, but at the cost of rendering the separation of variables assumption invalid. Our treatment, using the correct boundary condition, quantifies the errors in this ansatz.\n\nSeveral other works have followed \\citet{1973MNRAS.162...43H}. \\citet{1990ApJ...350..216N} extends the solution to media of intermediate optical depth, including the effects of scattering in the Doppler core of the line. \\citet{2006ApJ...649...14D} generalize the same problem to spherical geometry, as is used here. \\cite{2020MNRAS.497.3925L} generalize both the slab and sphere solutions to arbitrary power-law density and emissivity profiles. Each of these works, and several others \\citep{2020ApJS..250....9S, 2021MNRAS.504...89T}, use either the same surface boundary condition and ansatz as \\citet{1973MNRAS.162...43H}, or use a solution that does not handle the frequency-dependence of the boundary condition. Our novel steady-state solution involves a frequency-dependent correction to the solution that fixes an observed excess at the spectral peaks as compared with Monte Carlo, which is present in many of the works cited above.\n\nThe motivation for including time-dependence in the transfer equation is to characterize the distribution of photon escape times, which is needed to calculate the radiation moments in the Monte Carlo simulation. Additionally, steady-state solutions to this problem are not always sufficient to describe all the physics of \\lya transport. Time-variable, optically-thick environments necessitate a time-dependent solution to include the dynamic effects of \\lya transfer. These include the optical afterglow of gamma-ray bursts \\citep{2010ApJ...716..604R} and \\lya sources redshifted by cosmological expansion \\citep{2011MNRAS.418..853X}, among others.\n\n\\section{STEADY-STATE SOLUTION}\n\\label{sec:steadystate}\n\nConsider a sphere of radius $R$ with uniform density $n_{\\rm sc}$, luminosity $L$, and line-center optical depth $\\tau_0$, containing a point source of photons. We aim to find the intensity within the sphere as a function of radius and photon frequency. The point source is assumed to be a delta function in space and photon frequency. Photons of frequency $\\nu$ near the line center frequency $\\nu_0$ are considered. The photon frequency of the source is $\\nu_s$. The Doppler width is $\\Delta = \\nu_0 v_{\\rm th}\/c$, where $v_{\\rm th}=\\sqrt{2k_{\\rm B}T\/m_{\\rm H}}$ is the thermal speed of hydrogen atoms of mass $m_{\\rm H}$ and temperature $T$, and $c$ is the speed of light. The photon frequency in Doppler units is $x = (\\nu-\\nu_0)\/\\Delta$, and $x_s = (\\nu_s - \\nu_0)\/\\Delta$ is the corresponding source frequency. For upper-state de-excitation rate $\\Gamma$, the ratio of natural to Doppler broadening is $a=\\Gamma\/(4\\pi \\Delta)$. For the \\lya transition and T=$10^4$ K, $a = 4.72\\times 10^{-4}$. $\\mathcal{H}(x,a)$ is the Voigt function, and the Voigt line profile is $\\phi = \\mathcal{H}(x,a)\/(\\sqrt{\\pi} \\Delta)$, which is normalized as $\\int d\\nu\\, \\phi(\\nu) = 1$. The line center optical depth is $\\tau_0 = kR\/(\\sqrt{\\pi}\\Delta)$, where $k = n_{\\rm sc} \\pi e^2 f\/(m_e c)$. Here, $e$ and $m_e$ are the charge and mass of the electron, and $f$ is the oscillator strength of the transition, which is 0.4162 for \\lya \\citep{1986rpa..book.....R}.\n\nAppendix \\ref{app:rteqn_derivation} contains a derivation of the transfer equation for convenience. Starting with the full transfer equation, Equation (\\ref{eq:finaleqn}), ignoring photon destruction and including a photon emission term given by Equation (\\ref{eq:jem}), the steady-state transfer equation is\n\\begin{eqnarray}\n\\nabla^2 J + \\left( \\frac{k}{\\Delta} \\right)^2 \\frac{\\partial^2 J}{\\partial \\sigma^2} & = & \n- \\frac{ \\sqrt{6} kL}{4\\pi \\Delta^2} \\delta^3(\\vec{x} - \\vec{x}_s) \\delta (\\sigma - \\sigma_{\\rm s}).\n\\label{eq:rt_no_destr}\n\\end{eqnarray}\nwhere $J$ is the mean intensity, the spatial variable is $\\vec{x}$, and $\\vec{x}_s$ is the position of the source. We will consider only the case where $\\vec{x}_s=0$. Following \\citet{1973MNRAS.162...43H}, we have used a change of variables in photon frequency from $x$ to $\\sigma$,\n\\begin{eqnarray} \\label{eq:int_change_of_variables}\n\\sigma(x) = \\sqrt{\\frac{2}{3}}\\int_0^x \\frac{dx}{\\phi(x) \\Delta} \\approx \\sqrt{\\frac{2}{3}}\\frac{\\pi}{a}\\frac{x^3}{3}, \n\\end{eqnarray}\nwhere the approximation is applicable in the damping wing. From Equation (\\ref{eq:app:line_profile_wing}), the line profile is then approximately \n\\begin{eqnarray} \\label{eq:line_profile_approx}\n\\phi \\approx \\frac{a}{\\pi x^2 \\Delta} \\approx \\frac{1}{3 \\Delta}\\left(\\frac{2a}{\\pi}\\right)^{1\/3}|\\sigma|^{-2\/3}.\n\\end{eqnarray}\nIn Equation (\\ref{eq:rt_no_destr}), $\\sigma_s \\equiv \\sigma(x_s)$ is the photon frequency of the source. $\\sigma_s$ is interchangeable with $x_s$ and $\\nu_s$ in Doppler widths or Hz, respectively. Balancing the two terms on the left-hand side of Equation (\\ref{eq:rt_no_destr}) gives $\\sigma \\ {\\sim}\\ \\tau_0$, or $x_{\\rm peak}\\ {\\sim}\\ (a\\tau_0)^{1\/3}$. The boundary condition of no incoming intensity at the surface \\citep{1986rpa..book.....R} is\n\\begin{eqnarray}\nJ & = & \\sqrt{3} H\n\\label{eq:bc}\n\\end{eqnarray}\nat $r=R$. \n\nA solution for the mean intensity $J_d$ which is divergent at the origin\nand $\\sigma=\\sigma_s$ and is zero at infinity is presented in \\citet{1990ApJ...350..216N}. Here it is extended to spherical geometry and generalized to allow emission frequencies away from line center: \n\\begin{eqnarray}\nJ_{\\rm d} & = & \n\\left(\\frac{\\sqrt{6}k^2L}{16\\pi^3 \\Delta^3}\\right)\\left(\\frac{1}{(kr\/\\Delta)^2 + (\\sigma - \\sigma_{\\rm s})^2}\\right)\n\\label{eq:Jd}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nH_{\\rm d} & = & - \\frac{1}{3k\\phi} \\frac{\\partial J_d}{\\partial r}\n= \\left( \\frac{1}{3k\\phi} \\right) \n\\left( \\frac{ \\sqrt{6}k^3L }{ 8\\pi^3 \\Delta^4} \\right)\n\\left( \\frac{k r\/\\Delta}{ \\left[ (kr\/\\Delta)^2 + (\\sigma-\\sigma_{\\rm s})^2 \\right]^2 } \\right).\n\\label{eq:Hd}\n\\end{eqnarray}\nThis solution is useful as a simple analytic formula. However, it is not a good approximation to the true solution, as it is too large at $r=R$ by a factor of $J_{\\rm d}(R,\\sigma)\/ H_{\\rm d}(R,\\sigma) \\sim a\\tau_0\/x^2 \\sim (a\\tau_0)^{1\/3} \\gg 1$ and does not adhere to the correct boundary condition. This solution is included in Figure \\ref{fig:sol_mc_residual_0} for illustration.\n\nA better approximation to the true solution has been derived by \\citet{2006ApJ...649...14D}, who generalized the closed-form solution in slab geometry found in \\citet{1973MNRAS.162...43H}. It satisfies a $J=0$ boundary condition at $r=R$. Again, we generalize their solution to allow emission at frequency $\\sigma_{\\rm s}$ away from line center. The result can be written as a sum over spatial modes,\n\\begin{eqnarray} \\label{eq:J0_sum}\nJ_0 = \\frac{\\sqrt{6}L}{16\\pi \\Delta} \\frac{1}{R^2}\\sum_{n=1}^{\\infty}n\\frac{\\sin{\\kappa_n r}}{\\kappa_n r}\\exp{\\left(\\frac{-\\kappa_n \\Delta}{k}|\\sigma - \\sigma_s|\\right)},\n\\end{eqnarray}\nand\n\\begin{eqnarray} \\label{eq:H0_sum}\nH_0 = - \\frac{1}{3k\\phi} \\frac{\\partial J_0}{\\partial r} = -\\frac{1}{3k\\phi}\\frac{\\sqrt{6}L}{16\\pi\\Delta} \\frac{1}{R^2}\\sum_{n=1}^{\\infty}n\\left(\\frac{\\cos{\\kappa_n r}}{r} - \\frac{\\sin{\\kappa_n r}}{\\kappa_n r^2}\\right)\\exp{\\left(\\frac{-\\kappa_n \\Delta}{k}|\\sigma - \\sigma_s|\\right)},\n\\end{eqnarray}\nwhere $\\kappa_n=n\\pi\/R$. These can be summed to give the closed form expressions\n\\begin{eqnarray}\nJ_0 & = & \\frac{\\sqrt{6}L}{32\\pi^2 \\Delta}\n\\frac{1}{Rr}\n\\left( \n\\frac{ \\sin(\\pi r\/R) }{ \\cosh \\left[ \\frac{\\pi \\Delta}{k R} (\\sigma - \\sigma_s) \\right] - \\cos(\\pi r\/R)}\n\\right)\n\\label{eq:J0}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\nH_0 & = &\\frac{1}{3k\\phi}\n\\frac{\\sqrt{6}L}{32\\pi^2 \\Delta}\n\\frac{1}{Rr^2}\n\\left( \n\\frac{ \\sin(\\pi r\/R) }{ \\cosh \\left[ \\frac{\\pi \\Delta}{k R} (\\sigma - \\sigma_s) \\right] - \\cos(\\pi r\/R)}\n\\right. \\nonumber \\\\ & & \\left. - \\left( \\frac{\\pi r}{R} \\right)\n\\frac{ \\cos(\\pi r\/R) }{ \\cosh \\left[ \\frac{\\pi \\Delta}{k R} (\\sigma - \\sigma_s) \\right] - \\cos(\\pi r\/R)}\n+ \\left( \\frac{\\pi r}{R} \\right)\n\\frac{ \\sin^2(\\pi r\/R) }{ \\left[ \\cosh \\left[ \\frac{\\pi \\Delta}{k R} (\\sigma - \\sigma_s) \\right] - \\cos(\\pi r\/R) \\right]^2 }\n\\right).\n\\label{eq:H0}\n\\end{eqnarray}\nThese solutions agree with Equations (\\ref{eq:Jd}) and (\\ref{eq:Hd}) when the arguments of the trigonometric and hyperbolic functions are small. Again $J_0 \\gg H_0$, except near $r=R$, where it goes to zero. The flux at $r=R$ can be written\n\\begin{eqnarray}\n\\nonumber\nH_0(R, \\sigma) & = & - \\frac{1}{3k\\phi}\n\\frac{\\sqrt{6}L}{16\\pi \\Delta}\n\\frac{1}{R^3}\n\\sum_{n=1}^{\\infty} \nn (-1)^n \\exp{\\left(\\frac{-\\kappa_n \\Delta}{k}|\\sigma - \\sigma_s|\\right)}\\\\\n& = & \\frac{1}{3k\\phi}\n\\frac{\\sqrt{6}L}{32\\pi \\Delta}\n\\frac{1}{R^3}\n\\left( \n\\frac{ 1 }{ \\cosh \\left[ \\frac{\\pi \\Delta}{k R} (\\sigma - \\sigma_s) \\right] +1 }\n\\right).\n\\label{eq:H0surf}\n\\end{eqnarray}\nEquation (\\ref{eq:H0}) will be shown to be a better approximation to the solution than Equation (\\ref{eq:Hd}). It is still valid near the delta function at $r=0$, but is also a better approximation at $r=R$. $J_0$ decreases exponentially, rather than as a power-law in frequency as it does for $J_d$, giving a much smaller flux in the line wings as compared to the divergent solution. \n\nIn order to enforce the boundary conditions, a different solution method is attempted here, namely a continuous Fourier expansion in the frequency variable $\\sigma$. The solution of this problem is split into two pieces: $J_0$ which includes the delta function source and satisfies $J=0$ at $r=R$, and $J_{\\rm bc}$ which allows the boundary condition $J=\\sqrt{3}H$ to be satisfied at $r=R$. The total solution is\n\\begin{eqnarray}\nJ(r,\\sigma) & = & J_0(r,\\sigma) + J_{\\rm bc}(r,\\sigma)\n\\end{eqnarray}\nand\n\\begin{eqnarray} \\label{eq:totalflux}\nH(r,\\sigma) & = & H_0(r,\\sigma) + H_{\\rm bc}(r,\\sigma).\n\\end{eqnarray}\nThe additional term $J_{\\rm bc}$ must then be a solution of the homogeneous equation\n\\begin{eqnarray} \\label{eq:diffeq}\n\\frac{\\partial^2J_{\\rm bc}}{\\partial r^2} + \\frac{2}{r} \\frac{\\partial J_{\\rm bc}}{\\partial r}\n+ \\left( \\frac{k}{\\Delta} \\right)^2 \\frac{\\partial^2 J_{\\rm bc}}{\\partial \\sigma^2} &= & 0\n\\end{eqnarray}\nwith no delta function source term, and it must allow the boundary conditions to be satisfied at the surface. Since $J_0(R,\n\\sigma)=0$, the surface boundary condition becomes\n\\begin{eqnarray}\nJ_{\\rm bc}(R,\\sigma) - \\sqrt{3} H_{\\rm bc}(R,\\sigma) & = \n\\sqrt{3} H_0(R,\\sigma).\n\\label{eq:bc2}\n\\end{eqnarray}\nInserting a frequency dependence $J_{\\rm bc} \\propto e^{is\\sigma}$, for ``wavenumber\" $s$, gives the equation for modified spherical Bessel functions of the first kind, $i_0(z)=\\sinh(z)\/z$ for the radial dependence. The solution can then be represented as\n\\begin{eqnarray}\nJ_{\\rm bc}(r,\\sigma) & = & \n\\int_{-\\infty}^\\infty \\frac{ds}{2\\pi} e^{is\\sigma} A(s) \n\\frac{i_0(krs\/\\Delta)}{i_0(kRs\/\\Delta)},\n\\label{eq:Jbc}\n\\end{eqnarray}\nwhere $A(s)$ is the Fourier amplitude. Inserting Equation (\\ref{eq:Jbc}) into Equation (\\ref{eq:bc2}) leads to the following equation for the Fourier amplitudes,\n\\begin{eqnarray}\n\\int_{-\\infty}^\\infty \\frac{ds}{2\\pi} e^{is\\sigma} A(s)\n\\left[ 1 + \\left( \\frac{s}{\\sqrt{3} \\Delta \\phi} \\right) \\left( \\frac{i_0^\\prime(kRs\/\\Delta)}{i_0(kRs\/\\Delta)} \\right) \\right]\n& = & \\sqrt{3} H_0(R,\\sigma).\n\\label{eq:bc3}\n\\end{eqnarray}\nDiscretization of Equation (\\ref{eq:bc3}) for frequency variables $\\sigma_i$ and wavenumbers $s_j$\nleads to a set of coupled linear equations for the $A(s_j)$. We use equally-spaced points $\\delta \\sigma = 2\\sigma_{\\rm max}\/(N-1)$ and $\\delta s = 2\\pi\/(N\\delta \\sigma)$, where $N$ is the number of points for each grid. The maximum frequency is set as $\\sigma_{\\rm max} = {\\rm constant} \\times \\tau_0$, for a large enough constant that the end of the frequency grid is at such small intensities that it does not affect the solution except close to the boundaries. The number of points was increased until the solution was well-resolved near line center, and only became inaccurate close to the boundaries. We found that values of $N=4097$ and $\\sigma_{\\rm max} = 60 \\tau_0$ were sufficient. Given the Fourier amplitudes $A(s)$, $J_{\\rm bc}$ is computed using Equation (\\ref{eq:Jbc}), and the flux is given by\n\\begin{eqnarray}\nH_{\\rm bc}(r,\\sigma) & = & -\\frac{1}{3k\\phi}\n\\frac{\\partial J_{\\rm bc}(r,\\sigma)}{\\partial r}\n= -\\frac{1}{3k\\phi}\n\\int_{-\\infty}^\\infty \\frac{ds}{2\\pi} e^{is\\sigma} A(s) \n\\left( \\frac{ks}{\\Delta} \\right) \n\\left( \\frac{i_0^\\prime(krs\/\\Delta)}{i_0(kRs\/\\Delta)} \\right).\n\\label{eq:Hbc}\n\\end{eqnarray}\nThe Bessel functions are finite at the center and rise steeply toward the surface when $kRs\/\\Delta \\gg 1$. \n\n\\subsection{Scaling with Line Center Optical Depth $\\tau_0$}\n\nWe now estimate the scaling of $H_{\\rm bc}$ with $\\tau_0$. In the limit $J_{\\rm bc} \\gg H_{\\rm bc}$, we find that $J_{\\rm bc} \\approx \\sqrt{3} H_0$ from Equation (\\ref{eq:bc2}). We estimate $H_{\\rm bc}$ from $J_{\\rm bc}$ using Equation (\\ref{eq:Hbc}) as\n\\begin{eqnarray}\nH_{\\rm bc}(R, \\sigma) \\approx \\frac{1}{\\sqrt{3}k\\phi}\\frac{ks}{\\Delta}H_0\\ {\\sim}\\ H_0 s \\frac{x^2}{a}\\ {\\sim}\\ H_0 \\frac{1}{\\tau_0}\\frac{(a\\tau_0)^{2\/3}}{a}\\ {\\sim}\\ H_0 (a\\tau_0)^{-1\/3},\n\\end{eqnarray}\nwhere we have used $s\\ {\\sim}\\ 1\/\\sigma\\ {\\sim}\\ 1\/\\tau_0$ so that\n\\begin{eqnarray} \\label{eq:hbc_scaling}\n\\frac{H_{\\rm bc}(R, \\sigma)}{H_0(R, \\sigma)} \\propto (a\\tau_0)^{-1\/3}.\n\\end{eqnarray}\nAt large $\\tau_0$, it is expected that the correction term will be small, but it will become increasingly important as $\\tau_0$ decreases. Our solution of the transfer equation is only valid when the peaks of the spectral energy distribution lie well outside of the Doppler core, i.e., for large $\\tau_0$. The value of $x$ at which the Doppler and Lorentzian components of the line profile are equal is $x_{\\rm cw}=3.3$. Setting $x_{\\rm cw} = x_{\\rm peak}$ and solving for $\\tau_0$ gives the value at which the peak of the spectrum falls at the Doppler core boundary, which is $\\tau_{\\rm cp} \\approx 10^5$ (``core-peak'' optical depth). Hence $H_{\\rm bc}\/H_0 {\\sim} (\\tau_{\\rm cp}\/\\tau_0)^{1\/3}$ is large at $\\tau_0 \\leq \\tau_{\\rm cp}$ and decreases relatively slowly as $\\tau_0$ increases. Additionally, the optical depth at $x_{\\rm peak}$ is proportional to $(a\\tau_0)^{1\/3}$, so photons here become optically thin when $a\\tau_0 {\\sim} 1$.\n\n\\begin{figure}\n \\centering\n \\includegraphics{final_residual.pdf}\n \\caption{Spectrum $P(x)$ vs. frequency $x$ for $\\tau_0 = 10^7$ with $x_s = 0$. At $T=10^4$ K, this corresponds to $a\\tau_0=4.7 \\times 10^3$. The legend to the right describes each line style. For reference, $\\rm H_{0}$: fiducial solution, $\\rm H_{d}$: divergent solution, $\\rm H_{bc}$: our boundary condition correction to the fiducial solution. The top panel is linear scale, the middle panel is log scale with $|\\rm H_{bc}|$ shown instead of $\\rm H_{bc}$, and the bottom panel is the residual of each solution with Monte Carlo.} \n \\label{fig:sol_mc_residual_0}\n\\end{figure}\n\n\\subsection{Comparison to Monte Carlo}\nThe Monte Carlo method is used to solve the transfer equation numerically in order to compare the analytic approximation to an ``exact'' solution. This method is valid at all $\\tau_0$, being restricted only by the computational demand, which grows proportionally to the number of photons used and $\\tau_0$. For each simulation, a total of ${\\sim}10^6$ photon packets are initialized at a monochromatic source frequency $x_s$ and are allowed to propagate through the sphere until escaping, at which point their positions, outgoing angles, and escape frequencies are tabulated to obtain the spectrum at the surface of the spherical simulation domain. A constant temperature of $T=10^4\\ \\rm K$ is set for the gas. Frequency redistribution is calculated at each scattering. In the comparisons shown in this section, the raw photon data is binned in frequency to obtain spectra. Further details of the Monte Carlo implementation are discussed in \\citet{2017ApJ...851..150H}.\n\nWe now compare each of the previously-discussed solutions for surface flux to the Monte Carlo results. The spectrum $P(x)$ is defined as the specific luminosity at the surface divided by the source luminosity, or\n\\begin{eqnarray} \\label{eq:prob_spectrum}\nP(x) = \\frac{16\\pi^2R^2H(R, x)\\Delta}{L}.\n\\end{eqnarray}\nThis is normalized so that $\\int P(x)dx = 1$. Since $H(R, x)$ is per $d\\nu$, a factor of $\\Delta$ gives the expression the correct units. \n\nIn Figure \\ref{fig:sol_mc_residual_0}, the Monte Carlo spectrum is shown along with that of the solutions $H_{\\rm d}$, $H_0$, and $H_0 + H_{\\rm bc}$ for an optical depth of $\\tau_0 = 10^7$ and photons emitted at line center $\\rm x_s = 0$. Note that the errorbars shown on the Monte Carlo data points are proportional to $\\sqrt{N}$, with $N$ being the photon count in each frequency bin, since the photons are all equally weighted. The $H_{\\rm bc}$ term is negative at the peak of the spectrum and positive in the line wing such that, when added to $H_0$, it corrects for the apparent excess of flux in the peaks of the spectrum. The solution with the correct frequency-dependent boundary condition enforced, $H_0 + H_{\\rm bc}$, has lower residuals to Monte Carlo results than the other solutions, especially in the line wing. The boundary term corrects the deficit of $H_0$ in the line wings, further improving agreement with the numerical result. The residuals to the $H_0$ solution are a close match to the $H_{\\rm bc}$ term, since the Monte Carlo represents the ``true'' solution, $H$, and $H_{\\rm bc} = H - H_0$. It is evident that the divergent solution $H_{\\rm d}$ fails in the line wings. Also note that the ``V'' shape of the solution in the line core is due to the low number of points plotted, as the analytic solutions are not valid in this frequency regime since they utilize the damping wing approximation of the Voigt line profile.\n\n \\begin{figure}\n \\centering\n \\includegraphics[width=\\textwidth]{tau_threepanel.pdf}\n \\caption{The same as Figure \\ref{fig:sol_mc_residual_0}, but for $\\tau_0 = 10^5$ (top panel), $10^6$ (middle panel), and $10^7$ (bottom panel). The $x$ and $y$ axes are scaled by $(a\\tau_0)^{1\/3}$.}\n \\label{fig:sol_mc_tau}\n\\end{figure}\n\nThe size of $H_{\\rm bc}$ is dependent on $\\tau_0$. $H_{\\rm bc}$ is significant even at $\\tau_0 {\\sim} 10^7$ where the $H_0$ solution is expected to perform well, i.e., photons are pushed further out into the wing where the simplifying assumptions made in the derivation of the differential equation are a better approximation. \n\nIn Figure \\ref{fig:sol_mc_tau} we show the solutions alongside Monte Carlo, now for three different optical depths $\\tau_0=10^5, 10^6$, and $10^7$. From Equation (\\ref{eq:hbc_scaling}), the size of the term $H_{\\rm bc}$ should become smaller with larger optical depths, following a $(a\\tau_0)^{-1\/3}$ scaling. Indeed, agreement between Equation (\\ref{eq:H0surf}) and the Monte Carlo points in Figure \\ref{fig:sol_mc_tau} improves as $\\tau_0$ increases, with $H_{\\rm bc}$ providing a fractionally smaller correction to $H_0$. One factor of $(a\\tau_0)^{1\/3}$ has been scaled out of the x-axis such that the peaks of the distributions are horizontally aligned. This scaling has also been applied to the y-axis to preserve normalization of the escape probability. At lower $\\tau_0$, the scattering of photons within the Doppler core of the line becomes important, but our analytic solution does not include this effect. The effects of line core scattering can be seen in the Monte Carlo data for $\\tau_0=10^5$ and, to a lesser extent, $\n\\tau_0=10^6$.\n \n \\begin{figure}\n \\centering\n \\includegraphics[width=\\textwidth]{xinit_threepanel.pdf}\n \\caption{The same as Figure \\ref{fig:sol_mc_residual_0}, but for $\\tau_0=10^7$ and $\\rm x_s = 0$ (top panel), $6$ (middle panel), and $12$ (bottom panel). The optical depth at each of these source frequencies is $\\tau_s = 10^7, 77,$ and $19$, respectively.} \n \\label{fig:sol_mc_xinit}\n\\end{figure}\n\n \\begin{figure}\n \\centering\n \\includegraphics[width=\\textwidth]{xinit_threepanel_tau1e6.pdf}\n \\caption{The same as Figure \\ref{fig:sol_mc_xinit}, but at a lower optical depth $\\tau_0 = 10^6$. The shift $x_s$ is a much larger fraction of the distance to the spectral peak $(a\\tau_0)^{1\/3}$, and thus the asymmetry in the spectrum is much larger. The optical depth at the source frequency is $\\tau_s = 10^6$, $7.7$, and $1.9$ for $x_s=0, 6,$ and $12$, respectively. } \n \\label{fig:sol_mc_xinit_lowtau}\n\\end{figure}\n\nNext, we show $P(x)$ for $x_s \\neq 0$. Photons initialized further out in the line wing have larger mean free paths. The larger spatial diffusion implies greater escape probability for these photons. In the limit that $|\\rm x_s|$ becomes large, the distribution becomes a delta function at $x_s$ as all photons escape the sphere without scattering. \n\nFigure \\ref{fig:sol_mc_xinit} shows calculations performed for $x_s = 0, 6$, and $12$ and $\\tau_0=10^7$. The asymmetry of the spectrum is slight for $\\rm x_s = 6$ where $\\tau(x_s)=77$, but is larger for $\\rm x_s=12$ outside the line core where $\\tau(x_s)=19$. It is seen here that the difference between the Monte Carlo data and $H_0$ becomes larger as $\\rm x_s$ increases. Thus, for large $|x_s|$, inclusion of $H_{\\rm bc}$ is more important.\n\nFigure \\ref{fig:sol_mc_xinit_lowtau} shows emission away from line center at the same values of $x_s$ as in Figure \\ref{fig:sol_mc_xinit}, but for $\\tau_0=10^6$ rather than $10^7$. The difference between the left and right side of the escaping spectrum is now substantial since $(a\\tau_0)^{-1\/3}$ increased by a factor of ${\\sim}2$. It is clear from the figure that as $x_s$ extends further into the wing, the spectrum becomes more strongly peaked in frequency. Additionally, since the sphere is increasingly optically thin in the wing, we expect there to be a stronger disagreement with the Monte Carlo as the analytic solution assumed large optical depths.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{xinit.pdf}\n \\caption{$P_{\\rm bc}(x) = 16\\pi^2R^2H_{\\rm bc}(R, x)\\Delta\/L$ (Equation (\\ref{eq:prob_spectrum}) with $H \\rightarrow H_{\\rm bc}$) vs. frequency $x$. Each panel shows $\\sigma_s$ further from line center, labeled by $\\sigma_s=0, \\tau_0, 2\\tau_0$. The solution is shown at a range of line center optical depths between $\\tau_0=10^5$ and $10^9$ with solid lines. The location of the core-wing boundary $x_{cw}$ for each $\\tau_0$ is shown with dashed vertical lines which match the color of the corresponding solution. The axes are scaled to show that the size of the correction factor agrees with the predicted $(a\\tau_0)^{-1\/3}$ scaling.}\n \\label{fig:xinit}\n\\end{figure}\n\nFigure \\ref{fig:xinit} shows how the correction $P_{\\rm bc}(x)$, Equation (\\ref{eq:prob_spectrum}) with $H \\rightarrow H_{\\rm bc}$, scales with both $\\tau_0$ and $x_s$. In this figure, $\\sigma_s$ is shifted by integer multiples of $\\tau_0$ (Equation \\ref{eq:change_of_variables}) in each panel such that the source falls near the peak of the spectrum for each $\\tau_0$. For clarity, only the $x > 0$ side of the spectrum is shown. From Equation (\\ref{eq:hbc_scaling}), it is expected that the fractional size of $H_{\\rm bc}$ relative to $H_0$ should become smaller with larger optical depths following $(a\\tau_0)^{-1\/3}$. This factor has been scaled out of the figure such that solutions for different $\\tau_0$ and the same $\\sigma_s$ should show close agreement in scale on the figure's vertical axis if the relation holds. Indeed, the scaled solutions converge as $\\tau_0$ becomes larger, indicating agreement with the $(a\\tau_0)^{-1\/3}$ scaling. The remaining discrepancy present in the vertical axis for fixed $\\sigma_s$ results from $x_{\\rm peak}$ becoming close to $x_{\\rm cw}$; at lower $\\tau_0$, this causes the line profile approximation in the wing, Equation (\\ref{eq:app:line_profile_wing}), to break down. From this, we conclude that the errors introduced by the incorrect separation of variables in \\citet{1973MNRAS.162...43H}, \\citet{1990ApJ...350..216N}, \\citet{2006ApJ...649...14D} and others are indeed proportional to $(a\\tau_0)^{-1\/3}$.\n\n\\section{Time-Dependent Diffusion}\n\\label{sec:time_dependent}\n\nIn order to understand how long it takes for the photons to escape the uniform sphere of gas we must reintroduce the time dependence of the diffusion equation, which was ignored in the steady-state calculations in Section \\ref{sec:steadystate}. To obtain the radiative intensity $I=dE\/(dAdtd\\Omega d\\nu)$ on timescales comparable to the light-crossing time $t_{\\rm lc} = R\/c$, the time-dependent response to a delta function impulse is found. This allows the distribution of photon escape times (the ``wait time distribution'') to be characterized. For simplicity, a $J=0$ boundary condition will be used in the following derivations, which is a rough approximation for $a\\tau_0 \\gg 1$. \n\n\\subsection{Derivation of the time-dependent solution}\n\\label{subsec:time_dependent:background}\n\nThe emissivity for an impulsive source with energy E, source position $\\vec{x}_s$, and frequency $\\nu_s$ is derived in Appendix \\ref{app:rteqn_derivation}. Considering a photon source at $\\vec{x}_s=0$, we have (Equation \\ref{eq:jem})\n\\begin{eqnarray}\nj_{\\rm em} & = & \\frac{E}{4\\pi} \\delta^3(\\vec{x}) \\delta(\\nu-\\nu_{\\rm s})\\delta (t) ,\n\\label{eq:jem2}\n\\end{eqnarray} \nThe resulting equation for $J(r,\\sigma,t)$ is\n\\begin{eqnarray}\n \\frac{-3k\\phi}{c} \\frac{\\partial J}{\\partial t} + \\nabla^2 J + \\left( \\frac{k}{\\Delta} \\right)^2 \\frac{\\partial^2 J}{\\partial \\sigma^2}\n& = & - \\frac{\\sqrt{6} kE}{4\\pi \\Delta^2} \\delta^3(\\vec{x}) \\delta (\\sigma - \\sigma_s ) \\delta (t).\n\\label{eq:diffusion_eqn}\n\\end{eqnarray}\nWe employ an expansion in terms of spherical Bessel functions in $r$ and Fourier transform in time. The zeroth spherical Bessel function is $j_0(x) = \\sin{x}\/x$. The expansion for $J(r, \\sigma, t)$ is then\n\\begin{eqnarray}\n\\label{eq:jrsigmat_expansion}\nJ(r, \\sigma, t) = \\sum_{n=1}^{\\infty} \\int_{-\\infty}^\\infty \\frac{d\\omega}{2\\pi} e^{-i\\omega t} j_0\\left(\\kappa_n r\\right) J(n, \\sigma, \\omega),\n\\end{eqnarray}\nwith\n\\begin{eqnarray} \\label{eq:jnsigmaomega}\nJ(n, \\sigma, \\omega) = \\frac{2\\kappa_n^2}{R} \\int_0^R dr\\ r^2 j_0(\\kappa_n r) \\int_{-\\infty}^\\infty dt\\ e^{i\\omega t} J(r, \\sigma, t).\n\\end{eqnarray}\nHere, $\\kappa_n = n\\pi\/R$ and $\\omega$ describes the time-dependence of $J$. Though it is written as a function of the photon frequency variable $\\sigma$, $J(r, \\sigma, t)$ is the specific mean intensity $dE\/(dA dt d\\nu)$ and is a distribution in $\\nu$. The Fourier coefficient $J(n, \\sigma, \\omega)$ has units $dE\/(dA d\\nu)$. Using Equation (\\ref{eq:jnsigmaomega}), we obtain\n\\begin{eqnarray} \\label{eq:diffusion_plugged_in}\n \\left( \\frac{3k\\phi}{c}i\\omega - \\kappa_n^2 \\right) J(n,\\sigma,\\omega) &+& \\left( \\frac{k}{\\Delta} \\right)^2 \\frac{\\partial^2J(n,\\sigma,\\omega)}{\\partial\\sigma^2} = -\\frac{2\\kappa_n^2}{R} \\frac{\\sqrt{6}}{4\\pi} \\frac{kE}{\\Delta^2} \\frac{1}{4\\pi} \\delta(\\sigma - \\sigma_s).\n\\end{eqnarray}\nAt $\\sigma=\\sigma_s$, continuity must be enforced,\n\\begin{eqnarray} \\label{eq:matching_condition_1}\nJ(n, \\sigma^-, \\omega) = J(n, \\sigma^+, \\omega),\n\\end{eqnarray}\nand the discontinuity in $dJ\/d\\sigma$ due to the source is\n\\begin{eqnarray} \\label{eq:matching_condition_2}\n\\frac{\\partial J(n, \\sigma^+, \\omega)}{\\partial \\sigma} - \\frac{\\partial J(n, \\sigma^-, \\omega)}{\\partial \\sigma} & = & \n- \\frac{\\sqrt{6}}{8} n^2 \\frac{E}{kR^3}.\n\\end{eqnarray}\nAt large values of $\\sigma$ the line profile $\\phi$ is small and Equation (\\ref{eq:diffusion_plugged_in}) becomes\n\\begin{eqnarray} \\label{eq:diffusion_at_large_sigma}\n\\frac{\\partial^2J}{\\partial\\sigma^2} \\approx \\frac{\\Delta^2\\kappa_n^2}{k^2} J,\n\\end{eqnarray}\nwhich has solutions \n\\begin{eqnarray}\nJ(n, \\sigma, \\omega)\\ {\\sim}\\ e^{\\pm \\kappa_n \\sigma \\Delta \/ k}.\n\\end{eqnarray}\nThis approximate solution implies a boundary condition at large $|\\sigma|$\n\\begin{eqnarray} \\label{eq:single_j_derivative}\n\\frac{\\partial J}{\\partial \\sigma} = \\mp \\frac{\\kappa_n\\Delta}{k} J,\n\\end{eqnarray}\nwhere a negative sign is taken for large $+\\sigma$ and a positive sign is taken for large $-\\sigma$ to choose the finite solution as $|\\sigma|\\to \\infty$. Numerical integrations are performed inward toward $\\sigma_s$ over several domains: from large $|\\sigma|$ to $\\sigma_s$, from large $|\\sigma|$ to 0, and from 0 to $\\sigma_s$, depending on whether $\\sigma_s$ is positive or negative. If $\\sigma_s=0$, just two integrations are performed inward from large $|\\sigma|$ to 0. Initial values for integration are obtained either by setting $J=1$ and $dJ\/d\\sigma$ from Equation (\\ref{eq:single_j_derivative}) at large $|\\sigma|$ or by matching $J$ and $dJ\/d\\sigma$ at 0. This gives $J$ and $J'$ on either side of $\\sigma_s$, where a prime indicates the derivative $\\partial\/\\partial \\sigma$. By enforcing the matching conditions, Equations (\\ref{eq:matching_condition_1}) and (\\ref{eq:matching_condition_2}), the eigenfunctions $J(n, \\sigma, \\omega)$ are obtained over the domain of photon frequencies $\\sigma$. Since the solutions are linear in the starting conditions, only two integrations with different starting values are necessary.\n\nWe now wish to reconstruct the specific mean intensity $J(r, \\sigma, t)$. While one might expect this could be expressed as a sum over eigenmodes, the analysis presented in Appendix \\ref{app:wkb} suggests this treatment is incomplete in the case where $x_s \\neq 0$ and the solution is asymmetric about the line center. This ansatz does, however, roughly agree with Monte Carlo results for $x_s=0$ based on numerical calculations of this result.\n\nLet us define the damping rate to be $\\gamma \\equiv i\\omega$, which is real and positive for damped solutions. At the eigenvalues $\\gamma=\\gamma_{nm}$, the response $J(n, \\sigma, \\omega)$ is resonant. We find that near these $\\gamma_{nm}$ poles an approximate expression for the resonant response of the eigenfunctions is\n\\begin{eqnarray} \\label{eq:jnsigmaomega_approx}\nJ(n,\\sigma,-i\\gamma) & \\simeq \\frac{ J_{nm}(\\sigma) }{\\gamma_{nm} - \\gamma} + C(\\gamma, \\sigma),\n\\end{eqnarray}\nwhere $C(\\gamma, \\sigma)$ varies slowly in $\\gamma$. If the $\\omega$-integral in Equation (\\ref{eq:jrsigmat_expansion}) could be closed at infinity and evaluated using the residue theorem, the result would be\n\\begin{eqnarray}\nJ(r,\\sigma,t) & \\simeq & j_0(\\kappa_n r) J_{nm}(\\sigma) e^{-\\gamma_{nm}t}.\n\\end{eqnarray}\nSumming over all spatial modes $n$ and over all eigenmodes $m$ for a given $n$, we obtain\n\\begin{eqnarray} \\label{eq:Jrsigmat}\nJ(r,\\sigma,t) & = & \\sum_{n=1}^\\infty j_0(\\kappa_n r) \\sum_{m=1}^{\\infty} J_{nm}(\\sigma) e^{-\\gamma_{nm}t}.\n\\end{eqnarray}\nThis ansatz captures the contributions from $n \\times m$ simple poles. Taking a derivative with respect to $r$ and evaluating at the surface $r=R$, we use\n\\begin{eqnarray}\n\\frac{dj_0(\\kappa_n r)}{dr} \\bigg\\rvert_R & =& \\frac{d}{dr} \\left[ \\frac{\\sin(\\kappa_n r)}{\\kappa_n r} \\right]\\bigg\\rvert_R\n= \\left( \\frac{\\cos(\\kappa_n R)}{R} - \\frac{\\sin(\\kappa_n R)}{\\kappa_n R^2} \\right) = \\frac{(-1)^n}{R}\n\\end{eqnarray}\nto obtain the flux, which is\n\\begin{eqnarray}\nF(R,\\sigma,t) & =& - \\frac{4\\pi}{3k\\phi} \\frac{dJ(R,\\sigma,t)}{dr} \n= - \\frac{4\\pi}{3k\\phi R} \\sum_{nm} (-1)^n J_{nm}(\\sigma) e^{-\\gamma_{nm}t}.\n\\end{eqnarray}\nMultiplying by $4\\pi R^2$ gives the energy per time per frequency emerging from the sphere to be\n\\begin{eqnarray}\n\\frac{dE}{dtd\\nu} & = & - \\frac{16\\pi^2 R}{3k\\phi} \\sum_{nm} (-1)^n J_{nm}(\\sigma) e^{-\\gamma_{nm}t}.\n\\label{eq:dEdtdnu}\n\\end{eqnarray}\nIntegrating over time yields a factor $1\/\\gamma_{nm}$, and by integrating over $d\\nu$ we find\n\\begin{eqnarray} \\label{eq:sum_rule}\nE & = & \\sqrt{ \\frac{3}{2} } \\frac{16\\pi^2R\\Delta^2}{3k} \\sum_{nm} (-1)^{n+1} \\gamma_{nm}^{-1} \\int d\\sigma J_{nm}(\\sigma).\n\\end{eqnarray}\nThis non-trivial ``sum rule'' provides a check on the values of $\\gamma_{nm}$ and $J_{nm}(\\sigma)$. This expression can also be written as\n\\begin{eqnarray}\n1 & =& \\sum_{nm} P_{nm},\n\\label{eq:sumrule}\n\\end{eqnarray}\nwhere the contribution of each mode is\n\\begin{eqnarray} \\label{eq:pnmsoln}\nP_{nm} & \\equiv & \\sqrt{ \\frac{3}{2} } \\frac{16\\pi^2R\\Delta^2}{3kE} (-1)^{n+1} \\gamma_{nm}^{-1} \\int d\\sigma J_{nm}(\\sigma).\n\\end{eqnarray}\nThese coefficients $P_{nm}$ are negative for odd values of $n$ and positive for even $n$. The size of each contribution scales roughly as $0.5\/(m-7\/8)^{2\/3}$, with a weak dependence on $n$. This indicates the need for a large number of $n$ and $m$ to converge, in that it takes roughly ten times as many $m$ modes for a given $n$ to reduce the size of $P_{nm}$ by a factor of ${\\sim}$5. The physical intuition for the convergence of these terms is that the $n$ spatial terms must provide sufficient spatial resolution to resolve the steep falloff in intensity at the surface of the sphere. Additionally, the function falls off steeply in frequency in the line wing, which requires more $m$ terms in the Fourier sum to resolve (also see the discussion of Figure \\ref{fig:steadystate} in Section \\ref{subsec:steadystatemc}). \n\n\\subsection{Numerical calculation}\n\nWe seek now to calculate the eigenmodes $J_{nm}(\\sigma)$ and eigenfrequencies $\\gamma_{nm}$ for a given spatial $n$. These will be labelled by an index $m=1, 2, ...$. To measure the size of the response to detect where resonances occur, we sum the absolute value of $J(n,\\sigma,-i\\gamma)$ over the array $\\sigma$. We call this response $f$, and use the index $j$ to represent the value of the response at discrete points $\\gamma_j$ over a range of $\\gamma$. In places where $f_j > f_{j-1}$ and $f_j>f_{j+1}$, we have bracketed a resonance that occurs in the interval $(\\gamma_{j-1},\\gamma_{j+1})$. To refine the value of the eigenfrequency before continuing the sweep in $\\gamma$, we evaluate $f_{j-1}$, $f_j$, and $f_{j+1}$ at the points $(\\gamma_{j-1},\\gamma_{j},\\gamma_{j+1})$. Assuming the form in Equation (\\ref{eq:jnsigmaomega_approx}), a guess at the correct eigenvalue $\\gamma_{nm}$ can be calculated by linear interpolation from\n\\begin{eqnarray}\n\\gamma_{\\rm guess} &=& \\frac{b\\gamma_{j-1} - \\gamma_{j+1}}{b - 1},\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nb &=& \\left(\\frac{f_{j} - f_{j-1}}{f_{j} - f_{j+1}}\\right)\\left(\\frac{\\gamma_{j+1}-\\gamma_{j}}{\\gamma_{j-1}-\\gamma_{j}}\\right).\n\\end{eqnarray}\nThe error of the current guess is $|\\gamma_{\\rm guess} - \\gamma_{j}|$. This error is reduced iteratively by replacing initial points $(\\gamma_{j-1},\\gamma_{j},\\gamma_{j+1})$ with closer estimates while the size of the response grows as the resonance is approached. After iterating an eigenvalue $\\gamma_{nm}$ to convergence, we now find the corresponding eigenfunction $J_{nm}(\\sigma)$. We evaluate Equation \\ref{eq:jnsigmaomega_approx} at two points $\\gamma_1$ and $\\gamma_2$ near the resonance, subtracting them and solving for $J_{nm}(\\sigma)$ to find\n\\begin{eqnarray}\nJ_{nm}(\\sigma) & \\simeq & \\frac{ J(n,\\sigma,-i\\gamma_1) - J(n,\\sigma,-i\\gamma_2) }{ 1\/(\\gamma_{nm}-\\gamma_1) - 1\/(\\gamma_{nm}-\\gamma_2)},\n\\end{eqnarray}\n\\noindent where $C(\\gamma, \\sigma)$ has cancelled in the difference.\n\n\\begin{figure}\n \\centering\n \\includegraphics{Jsoln_n1_m100.pdf}\n \\caption{Eigenfunctions $J_{nm}(x)$ for the lowest-order spatial eigenmode $n=1$, and $m=1, ..., 100$ with $x_s=0$ and $\\tau_0=10^7$. The scale of the $J_{nm}(\\sigma)$ are set by the factor $E\/(kR^3)$, which here is ${\\sim}10^{-37}$ in ergs\/(cm$^2$ Hz).}\n \\label{fig:jsoln}\n\\end{figure}\n\nThe form of a single eigenmode $J_{nm}(\\sigma)$ is oscillatory out to some turning point, $\\sigma_{\\rm tp}$, at which point the function becomes evanescent. The location of the turning point can be found by ignoring the delta-function discontinuity at the source frequency $\\sigma_s$ in Equation (\\ref{eq:diffusion_plugged_in}) and examining the resulting homogeneous differential equation. We obtain\n\\begin{eqnarray} \\label{eq:wkb_differential_eqn}\n\\frac{d^2J}{d\\sigma^2} & = & \\left[ \\left( \\frac{\\kappa_n \\Delta }{k} \\right)^2 - \\frac{3\\phi \\gamma\\Delta^2}{ck}\\right] J,\n\\end{eqnarray}\nwhere the line profile is approximated as in Equation (\\ref{eq:app:line_profile_wing}). When the coefficient on the right hand side is positive, exponential growth or decaying evanescent solutions are found. This occurs in the line wings. When the coefficient on the right hand side is negative, oscillatory solutions are found (propagation), which occurs near the line core. The boundary between propagation and evanescence occurs at the turning point, given by\n\\begin{eqnarray} \\label{eq:sigma_tp}\n\\sigma_{\\rm tp} & = & \\sqrt{\\frac{2a}{\\pi}}\\left( \\frac{k \\gamma}{ \\kappa_n^2 c \\Delta} \\right)^{3\/2}\n\\end{eqnarray}\nThus, to ensure accuracy in each term of Equation (\\ref{eq:Jrsigmat}), the bounds of $\\sigma$ must be set sufficiently far outside of $\\sigma_{\\rm tp}$ such that the function is small at the edges. The scale of an $e$-folding in $J_{nm}(\\sigma)$ is $k\/(\\kappa_n \\Delta) = \\tau_0 \/ (\\sqrt{\\pi} n)$, so a grid of $\\sigma$ is chosen that spans a large enough number of $e$-foldings that no oscillatory behavior is present at the boundaries of the domain.\n\nThe eigenfunction's oscillatory forms have varying amplitudes which sum in Equation (\\ref{eq:Jrsigmat}) to create the final form of the mean intensity. The largest contribution at late times always comes from the ($n=1, m=1$) lowest-order eigenfunction. Figure \\ref{fig:jsoln} shows a set of eigenfunctions $J_{nm}(\\sigma)$ to illustrate their relative scales for different $m$ at a fixed spatial eigenmode $n$. The overall scale of the $J_{nm}(\\sigma)$ are set by the factor $E\/(kR^3)$ with $E$ arbitrarily set to 1. For H atoms with $T=10^4$ K and $\\tau_0=10^7$, an eigenfunction has typical size ${\\sim} E a \/ \\left(R^2 \\Delta \\right) = 10^{-37}$ in units of specific mean intensity times time. Additional terms add smaller-magnitude, faster-oscillating components that lead to higher accuracy upon summation with the lower-order terms in Equation (\\ref{eq:Jrsigmat}). The oscillations of various modes must cancel in the Fourier sum, so many modes $m$ and $n$ are required for convergence to the solution. \n\\begin{figure}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{gamma_nm.pdf}\n \\caption{Dimensionless decay time vs. mode number. Overlapping solid lines are plotted for each $n$. The thickness of the lines decrease with $n$ in order to clearly show where they are in agreement. These numerically-obtained resonant frequencies are compared with the analytic expression in Equation (\\ref{eq:gamma_nm}), shown as a dashed line. The parameters for the calculation are $\\tau_0=10^7$ and $x_s=0$. The value of $n$ is given by the color bar on the right-hand side.}\n \\label{fig:gamma_nm}\n\\end{figure}\nThe values of the $\\gamma_{nm}$ can be described approximately with Equation (\\ref{eq:gamma_nm}). Their values depend on $m$, $n$, and other physical parameters according to \n\\begin{eqnarray} \\label{eq:gamma_nm}\n\\gamma_{nm} = 2^{-1\/3} \\pi^{13\/6} n^{4\/3}\\left(m-\\frac{7}{8}\\right)^{2\/3}\\frac{c}{R}(a\\tau_0)^{-1\/3}\n\\end{eqnarray}\nas shown by setting the denominator of Equation \\ref{eq:a3\/a2} to zero in WKB approximation of Appendix \\ref{app:wkb}. The power law in $m$ is weak, requiring up to $m=1000$ to reduce the scale of $\\gamma_{nm}^{\\ \\ -1}$ by two orders of magnitude. When sweeping through to find resonances, Equation (\\ref{eq:gamma_nm}) is used to set the scale of the sweep points $\\gamma_j$ to ensure no $\\gamma_{nm}$ are missed. The close agreement with the analytic expression shown in Figure \\ref{fig:gamma_nm} indicate that the numerical solutions are accurate.\n\n\\subsection{Comparison with Steady State and Monte Carlo} \\label{subsec:steadystatemc}\n\nWe now calculate the wait time distribution for escape from the sphere. This is obtained by integrating Equation (\\ref{eq:dEdtdnu}) over all frequencies. We find\n\\begin{eqnarray}\nP(t) & = & \\sqrt{\\frac{3}{2}} \\frac{16\\pi^2 R \\Delta^2 }{3kE} \\sum_{nm} (-1)^{n+1} e^{-\\gamma_{nm}t} \\int d\\sigma J_{nm}(\\sigma) \n\\nonumber \\\\ & = & \\sum_{nm} P_{nm} \\gamma_{nm} e^{-\\gamma_{nm}t},\n\\label{eq:waittime}\n\\end{eqnarray}\nwhich is normalized to unity. For a sufficiently large number of spatial modes $n$ and frequency modes $m$, the result of this sum can agree with Monte Carlo escape time distributions when $x_s=0$. The late-time distribution is simply an exponential falloff. The rate constant of the falloff is the lowest-order eigenfrequency, $\\gamma_{11}$, and its scale is determined by the coefficient $P_{11}$ as in Equation (\\ref{eq:pnmsoln}). Thus, an approximate ``fitting function'' that captures both the peak of the escape time distribution and the exponential falloff is\n\\begin{eqnarray} \\label{eq:fitting_function}\nP(t) = \\exp{\\left[-\\left(\\frac{t_{\\rm diff}}{t}\\right)^2\\right]} \\times \\gamma_{11} P_{11} e^{-\\gamma_{11}t}.\n\\end{eqnarray}\nThe first term represents the early-time distribution, which then transitions to an exponential falloff past a point $ct_{\\rm diff}\/R = (a\\tau_0)^{1\/3}$, where $t_{\\rm diff}$ is the characteristic diffusion timescale.\n\nIn Figure \\ref{fig:tau_scaling}, the late-time decay timescale of the wait time distribution is shown as a function of $\\tau_0$. It is shown that the time constant of exponential decay in fitted Monte Carlo escape time distributions converges with $\\gamma_{11}^{-1}$ at sufficiently high $\\tau_0$, following a $t\\propto(a\\tau_0)^{1\/3}$ scaling. The coefficient of this scaling ($0.51$) is within a factor of 2 of the approximate ``light-trapping time'' defined in \\citet{2020MNRAS.497.3925L}, which predicts $ct\/R=0.901(a\\tau_0)^{1\/3}$. At lower $\\tau_0$, the effects of line core scattering are most important, leading to a larger discrepancy in the characteristic escape timescale. Here, the Monte Carlo accurately includes the photons which scatter in the core many times before escaping, while the semi-analytic solution does not capture this behavior as it uses only the Lorentzian piece of the line profile, and also does not use enough spatial modes to accurately model the frequency regime near line center. However, as $\\tau_0$ grows, the effect of core scattering becomes smaller and the approximations hold, agreeing better with the expected $(a\\tau_0)^{1\/3}$ scaling \\citep{1975ApJ...201..350A} for the rate constant at late times. The excess in the Monte Carlo data points due to core scattering decreases exponentially at higher $\\tau_0$, and though these points are not shown at $\\tau_0=10^8$ and $10^9$ due to computational expense, it is expected that the fractional error between the Monte Carlo and the $(a\\tau_0)^{1\/3}$ scaling would be less than 2\\% at $\\tau_0=10^9$.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.75\\textwidth]{tau_scaling.pdf}\n \\caption{Late-time exponential decay timescale as a function of $\\tau_0$. The Monte Carlo points on this figure are obtained by fitting only the exponential piece of the escape time distribution to obtain the rate constant.}\n \\label{fig:tau_scaling}\n\\end{figure}\n\nWe now evaluate the time-integrated spectrum (fluence) of the response to an impulse and compare it with the solution for the $H_0$ steady-state spectrum (Equation \\ref{eq:H0surf}). Integrating Equation (\\ref{eq:dEdtdnu}) over all times and dividing by the energy $E$, we find the fluence\n\\begin{eqnarray} \\label{eq:spectrum}\nP(x) & = & \\frac{16\\pi^2 R \\Delta}{3k\\phi E} \\sum_{nm} (-1)^{n+1} \\gamma_{nm}^{-1} J_{nm}(\\sigma).\n\\end{eqnarray}\nIntegrating over $\\nu$ then gives unity as required by the sum rule in Equation (\\ref{eq:sumrule}). \n\nIn Figure \\ref{fig:steadystate}, the fluence for $x_s=0$ and $\\tau_0=10^7$ is shown for a sum up to $n=20$ and $m=500$, labelled ``Time-integrated'', and is compared with two analytic solutions: the steady-state $H_0$ solution (Equation \\ref{eq:H0surf}), labelled ``Steady State'', and the result for summing a finite number of spatial modes in the steady-state eigenfunction expansion as in the first line of Equation (\\ref{eq:H0surf}), labelled ``Partial Sum''. Additional spatial modes $n$ increase the solutions' accuracy in the core of the line. If more spatial modes are included, the agreement with the steady-state spectrum extends further toward the line core. If additional frequency modes are included, faster-oscillating terms are incorporated into the Fourier sum over eigenmodes which create more perfect cancellations with the lower-order terms, reducing the ``ringing'' seen in the time-integrated spectrum. Extending the calculation deep into the line core by adding additional spatial modes could have an impact on the accuracy of the escape time distribution, but this would primarily affect the distribution at early times since the late time distribution is determined by the lowest order modes. This was the motivation for choosing a comparatively low number of spatial eigenmodes with respect to the number of frequency eigenmodes calculated.\n\n\\begin{figure}\n \\centering\n \\includegraphics{steadystate.pdf}\n \\caption{Fluence $P(x)$ vs. $x$ (see Equation \\ref{eq:spectrum}). Fluence is the radiation flux integrated over time. Steady-state and time-integrated spectra for $n=1, ..., 20$ and $m=1, ..., 500$ are shown with $x_s=0$ and $\\tau_0=10^7$. Note that the x-axis begins near the edge of the line core, as we are only concerned with the solutions' accuracy near the line wing.}\n \\label{fig:steadystate}\n\\end{figure}\n\n\nIn Figure \\ref{fig:escape_time}, the escape time distributions calculated from Equation (\\ref{eq:waittime}) are shown alongside Monte Carlo and the fitting function Equation (\\ref{eq:fitting_function}) for $\\tau_0=10^6, 10^7$ with $x_s=0$. The disagreement between the tail of the distribution and the Monte Carlo data is due to line core scattering which is not modeled by the eigenfunction solution, but improves for larger optical depth as seen in the figure. A large number of scatterings in the Doppler core affects the tail of the escape-time distribution, since photons with frequencies near line center will take longer to escape. Thus, the rate constant for the exponential falloff is overestimated slightly in the eigenfunction solution as compared with the Monte Carlo. The error in this rate constant is a function of $\\tau_0$ since the effect from the Doppler core is greatest when it extends into the peak of the spectrum.\n\n\\begin{figure}\n \\centering\n \\includegraphics{waittime.pdf}\n \\caption{Wait-time distribution of escaping photons $P(t)$ vs. t for $\\tau_0=10^6$ and $10^7$, including the fitting function from Equation (\\ref{eq:fitting_function}). A sum over 20 spatial eigenmodes and 500 frequency eigenmodes is labeled ``Eigenfunctions''. All calculations were performed with a monochromatic source of photons at line center ($x_s = 0$).}\n \\label{fig:escape_time}\n\\end{figure}\n\n\\section{Discussion}\n\n\\subsection{Steady-State Source}\nA primary goal of this work is to present a solution for resonant scattering of photons near the line-center frequency $\\nu_0$ in a uniform sphere. We have generalized a spherically symmetric solution derived by \\citet{2006ApJ...649...14D} (called $H_0$ here) to allow a monochromatic source of photons with frequencies away from line center. We introduce a new term to this solution, $J_{\\rm bc}$, which allows the boundary condition $J=\\sqrt{3}H$ to be satisfied at the surface of the sphere. This is solved using a continuous Fourier expansion in frequency. The integrals are discretized and the Fourier coefficients solved for numerically. The resulting flux correction, $H_{\\rm bc}$, scales as $H_0(a\\tau_0)^{-1\/3}$. Thus, for large $a\\tau_0$, only a small correction to $H_0$ is needed, while larger errors are present in calculations performed at lower $a\\tau_0$. Since the Laplacian form for frequency redistribution in the differential equation is only correct for photons in the wing where the line profile is $\\phi \\approx a\/(\\pi x^2 \\Delta)$, our solutions do not accurately model the Doppler core of the \\lya line. Because the peak of the spectral energy distribution of escaping photons is $x_{\\rm peak} {\\sim} (a\\tau_0)^{1\/3}$, calculations performed at small $a\\tau_0$ are inaccurate due to the close proximity of the spectral peak and the Doppler core of the line.\n\nBy comparison with Monte Carlo simulations, we have shown that the enforcement of the correct frequency-dependent boundary conditions improves the accuracy of these analytic solutions for $a\\tau_0 \\gg 1$. Specifically, this solution shows improvement over previous solutions that utilized a $J=0$ surface boundary condition presented in \\citet{1973MNRAS.162...43H}, \\citet{1990ApJ...350..216N}, and \\citet{2006ApJ...649...14D}. Several papers have previously compared these analytic models to Monte Carlo and seen discrepancies on the order of this correction. For example, in the top-left panel of Figure 1 from \\citet{2006ApJ...649...14D}, the \\lya spectrum emergent from a sphere of uniform optical depth is shown for $\\tau_0=10^5, 10^6,$ and $10^7$ at a temperature of $T=10$ K, corresponding to $a=1.5 \\times 10^{-2}$. The dotted line showing their theoretically-derived spectrum ($H_0$) displays an excess at the peak of at least 5-10 percent as compared with the Monte Carlo for $\\tau_0=10^5$ and $10^6$. Another example is in \\citet{2015MNRAS.449.4336S}, where the peak excess in the \\lya spectrum is particularly noticeable for line center optical depths of $\\tau_0=10^6$ and $10^7$ in the top panel of their Figure 5, which used slab geometry and a gas temperature of $T=10^4$ K. Again, the error in their solution is of order 5-10 percent. Both of these solutions are too large at the spectral peaks and too small further out in the wing, and the error scales approximately as $(a\\tau_0)^{-1\/3}$. We show in our Figure \\ref{fig:sol_mc_tau} that the error present in $H_0$ is corrected by our treatment of the boundary condition at $\\tau_0 = 10^7$ for T$=10^4$ K, corresponding to $a=4.72 \\times 10^{-4}$. We note that our correction term $H_{\\rm bc}$ is positive in the line wing and negative at the peak of the spectrum, which matches with the discrepancies noted in the aforementioned solutions.\n\n\\subsection{Impulsive Source}\nThe time-dependent transfer equation is solved in order to characterize the distribution of photon escape times. A semi-analytic approach is used, utilizing an expansion in space, time, and photon frequency. This boundary value problem in frequency $\\sigma$ is solved to find the flux at the surface of the sphere as a function of $t$ and $\\nu$. This solution is expressed as a sum over spatial and frequency modes $n$ and $m$, respectively. Calculating additional spatial eigenmodes increases the accuracy nearer to line center, but convergence is slow due to each eigenmode's weak dependence on $n$. Additional frequency eigenmodes introduce fast-oscillating terms that improve the accuracy of the Fourier sum, as their contributions cancel with components of lower-order terms to better represent the true solution. Integrating the solution over time produces a fluence that is shown to broadly agree with the steady-state calculations in Section \\ref{sec:steadystate}, provided a sufficient number of terms in the sum and emission at the line-center frequency $\\nu_0$. Integrating the solution over frequency leads to a distribution of photon escape time, which can be compared directly with Monte Carlo simulations. The sum over eigenmodes produces an escape-time distribution that broadly captures the behavior shown by Monte Carlo data---a rise at early times, transitioning to exponential decay in the tail of the distribution. It is expected that the accuracy of the rate constant for the tail of the distribution is limited by the effect of the Doppler core, which can trap photons at high optical depths until they diffuse outward in frequency, weighting the distribution toward later times. This physics is not modeled by our solution for two reasons: 1) our calculations ignored the Gaussian component of the Voigt line profile, leaving the Lorentzian piece which is accurate only in the line wing, and 2) knowing the core is not modeled accurately, we do not include a large enough number of spatial eigenmodes in the sum to resolve it. However, an approximate fitting function dependent on parameters $a$ and $\\tau_0$ is found that adequately represents the escape time distribution of the Monte Carlo results within these constraints.\n\nOur characterization of the escape time distribution leads to a possible application of this work. Models of the interaction of stellar \\lya with the upper atmosphere of exoplanets and the associated transmission spectrum can be constructed with a treatment of resonant scattering in spherical geometry \\citep{2017ApJ...851..150H, 2021ApJ...907L..47Y}. The Monte Carlo method can be used for this problem, but is limited by its high computational demand for large $\\tau_0$ where there are many photon scatterings before escape. We seek to develop a method to accelerate the radiative transfer calculation. \n\nThere are several methods that are commonly used to accelerate Monte Carlo radiation transfer calculations, including core skipping methods \\citep{1968ApJ...153..783A,2002ApJ...567..922A} and hybrid diffusion methods \\citep{2018MNRAS.479.2065S}. Another approach with wide application is modified random walk methods, such as those discussed in \\citet{1984JCoPh..54..508F, 2009A&A...497..155M, 2010A&A...520A..70R}. In this approach, an outgoing photon is randomly sampled on the surface of the outgoing sphere by drawing its properties from distributions in outgoing frequencies, directions, and escape times, based on solutions to the diffusion equation. A method similar to this has been applied by \\citet{2006ApJ...645..792T} to Lyman $\\alpha$ transfer using the \\citet{1990ApJ...350..216N} solution, but this solution of course does not utilize the frequency-dependent boundary condition at the surface of the sphere. Furthermore, to perform a full radiation hydrodynamic simulation with Monte Carlo acceleration, it will be necessary to calculate radiation forces within each cell due to \\lya transfer. Similar calculations have been done in \\citet{1976ApJ...208..286W} in plane-parallel geometry. However, these solutions are limited to optical depths below $2.5 \\times 10^3$. For this work, it would be necessary to model line center optical depths of up to 1 million or more.\n\n\\section{Summary}\n\nWe have examined previous solutions to \\lya transfer including resonant scattering in the limit of large optical depth, noting that the separation of variables and treatment of the boundary condition in \\citet{1973MNRAS.162...43H}, \\citet{1990ApJ...350..216N}, \\citet{2006ApJ...649...14D} and others produces a discrepancy in the outgoing spectrum as compared with Monte Carlo. Here, we have derived the solution in spherical geometry with an appropriate treatment of the surface boundary condition. The key result is that the errors in the previously-cited works have been quantified via a correction term, $H_{\\rm bc}$, which explains an excess in flux at the spectral peak and a deficit in the line wing of the calculated spectrum of \\lya radiation as compared with Monte Carlo. The size of $H_{\\rm bc}\/H_0$ is of order unity when the spectral peaks are near the Doppler core, and diminishes at larger $\\tau_0$ following a $(a\\tau_0)^{-1\/3}$ scaling. \n\nThe time-dependent transfer equation for the impulsive source is solved numerically with an eigenfunction expansion. We demonstrate that it agrees with the steady-state spectrum for $x_s=0$ when integrated over time, though its rate of numerical convergence is slow and requires a sum over many modes to become accurate. The time-dependent solution is utilized to create wait-time distributions for photons escaping the sphere of optically-thick hydrogen gas. We compare the calculations from the time-dependent solution with Monte Carlo for a sample of $\\tau_0$, noting general agreement in the resulting escape time distributions. The solution derived in our work here may be used as the basis for a novel implementation of the modified random walk method, which would accelerate Monte Carlo \\lya transfer at large optical depths with potential applications in radiation hydrodynamic simulations of the atmospheres of exoplanets. \n\n\\acknowledgments\n\nThis research was funded by NASA ATP grant 80NSSC18K0696, ``Exoplanetary MHD Outflows Driven by EUV Heating, Lyman alpha Radiation Forces and Stellar Tides\". Resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center. We thank the referee for a detailed report that helped improve the presentation of our work.\n\\restartappendixnumbering\n\n\\software{\\texttt{numpy} \\citep{2020NumPy-Array}, \\texttt{scipy} \\citep{2020SciPy-NMeth}, Coblis - Color Blindness Simulator (\\href{https:\/\/www.color-blindness.com\/coblis-color-blindness-simulator\/}{color-blindness.com})}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{Introd}\n\n\nIn the last two decades, soft-glassy materials (SGM) have gained growing attention due to their applications in several industrial sectors. \nIn particular, emulsions and foams are employed to design novel soft mesoscale materials for chemical, food processing, manufacturing, and biomedical purposes \\cite{fernandez2016fluids,piazza2011soft,mezzenga2005understanding}.\nBesides the technological relevance, their major significant theoretical interest stems from their intriguing non-equilibrium effects, \nincluding long-time relaxation, yield-stress behavior, and highly non-Newtonian dynamics. \n\nIn this context, computational fluid dynamics (CFD) provides a valuable tool to improve the knowledge of the underlying physics of SGM.\nTo that purpose, a reliable SGM model alongside its software implementation is of apparent interest for the rational designing and shaping up of novel soft porous materials.\n\nIn this paper, we present and make available the CUDA Fortran code \\textbf{LBcuda}, specifically designed to simulate on GPUs bi-continuous systems with colloidal particles under a variety of different conditions. \\textbf{LBcuda} is a direct port of the LBsoft code \\cite{lbsoft}, an open-source software for simulations of soft glassy emulsions originally developed for CPU-architectures, which successfully combines the lattice-Boltzmann method (LBM) \\cite{succi2018lattice,kruger2017lattice,benzi1992lattice} with a Lagrangian solver to tackle the multi-scale coupling of fluids and particles \\cite{bernaschi2019mesoscopic}.\n\nNowadays, the straightforward parallelization of LBM makes the lattice-Boltzmann algorithm an excellent candidate for high-performance CFD, especially on GPU-based architectures, given the relative simplicity and locality of its underlying algorithm.\nAs a consequence, several LBM implementations have been developed for GPU architectures, both academic packages \nsuch as the GPU-enabled versions of WaLBerla \\cite{bauer2021walberla,holzer2021highly}, Palabos \\cite{latt2021palabos}, Ludwig \\cite{desplat2001ludwig}, MUPHY \\cite{bernaschi2009muphy}, and commercially \nlicensed software such as XFlow 2021 \\cite{holman2012solution}, \nto name a few. \n\nAs aforementioned, the model to describe colloidal particles is derived from the previous CPU-based LBSoft code, to which the reader is referred for further details \\cite{lbsoft}. Briefly, SGM modeling requires specific implementations of LBM and Lagrangian solvers to include the hydrodynamic interactions between solid particles and fluids, following several strategies reported in the literature \\cite{ladd2015lattice,ladd2001lattice,aidun1998direct,ladd1994numericala}. \nThis extension has opened the possibility to simulate complex colloidal systems, also referred to as Pickering emulsions \\cite{pickering1907cxcvi} which are of primary interest for the rational design of SGM \\cite{xie2017direct,liu2016multiphase, frijters2012effects,jansen2011bijels}. This intrinsically multi-scale approach can catch, for example, the dynamical transition from a bi-continuous interfacially jammed emulsion gel, also referred to as bijel (see Fig. \\ref{fig:bijel}), capturing the associated mechanical and spatial properties \\cite{sun2021pickering}.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.8\\linewidth]{figure01.pdf}\n\\caption{A typical bijel configuration with colloidal particles (blue spheres) entrapping the interfaces between the two fluids (red and transparent green). }\n\\label{fig:bijel}\n\\end{center}\n\\end{figure}\n\n\nThe paper is structured as follows. In Section \\ref{sec:method} we report a very brief description of the underlying method, referring our previous paper for a deeper explanation of the details.\nIn Section \\ref{sec:impl} we describe the details of the data structures on the GPU, while in Section \\ref{sec:parallel} we explain the parallelization strategy and its impact on performance.\nIn Section \\ref{perfomance} we report a set of tests used to validate the implementation and we investigate the performance against the reference CPU version (LBsoft). Finally, conclusion and outlook on future development directions are discussed.\n\n\\section{Method}\n\\label{sec:method}\n\nIn this section, we briefly review the approach to the simulation of SGM implemented in LBcuda alongside the more significant algorithmic adaptations required by the GPU-based hardware. A more detailed illustration of the underlying algorithms can be found in Ref. \\cite{lbsoft}. \n\nThe code combines two different levels of description: the first exploits a continuum approach for the dynamics of immiscible fluids, whereas the second manages individual rigid bodies representing colloidal particles or other suspended species. \nThe two levels exchange information at each step of the time integration scheme to describe the concurrent interaction among particles and surrounding fluids.\n\nIn the first level, the LBM exploits a fully discretized analog of the Boltzmann kinetic equation to model flows and hydrodynamic interactions in fluids.\n\nIn the LBM approach, the fundamental quantity is $f_{i}(\\vec{r};t)$, namely the probability of finding a \"fluid particle\" at the spatial point mesh $\\vec{r}$ and at time $t$ with velocity $\\vec{c}_i$ selected from a \nfinite set of possible speeds. The LBcuda code implements the 3-dimension 19-speed cubic lattice scheme (D3Q19) with the discrete velocities $\\vec{c}_i$ with $i \\in [0,...,18]$ connecting mesh points with spacing $\\Delta x$ (length lattice unit) to first and second mesh neighbours, located at distance $\\Delta x$ and $\\sqrt 2 \\Delta x$, respectively (in other words, D3Q19 neglects 8 out of 27 possible velocities: those having distance $\\sqrt 3\\Delta x$).\n\n\n\nDenoted $\\rho(\\vec{r};t)$ and $\\vec{u}(\\vec{r};t)$ respectively the fluid density and the fluid velocity, the lattice-Boltzmann equation is implemented in single-relaxation time (Bhatnagar-Gross-Krook equation) as follows:\n\\begin{equation}\\label{eq:bgk}\nf_{i}(\\vec{r}+\\vec{c}_{i};t+1)=(1-\\omega)f_{i}(\\vec{r};t)+\\omega f_{i}^{eq}(\\rho(\\vec{r};t),\\vec{u}(\\vec{r};t))\n\\end{equation}\nwhere $f^{eq}$ is the lattice local equilibrium, basically the local Maxwell-Boltzmann distribution (see Appendix A), and $\\omega$ is a frequency tuning the relaxation towards the local equilibrium on a timescale $\\tau=1\/\\omega$. The relaxation frequency $\\omega$ controls the kinematic viscosity of the fluids according to the relation:\n\\begin{equation}\n\\nu=c_{s}^{2}\\frac{\\Delta x^{2}}{\\Delta t} \\left(\\frac{1}{\\omega}-\\frac{1}{2}\\right),\n\\label{eq:viscosity}\n\\end{equation}\nwhere $\\Delta x$ and $\\Delta t$ are the physical length and time of the correspondent counterparts in lattice units. Note that the positivity of the kinematic viscosity requires the condition $0<\\omega<2$.\n\n\nIn order to model a two component systems we adopted, a color gradient (CG) algorithm, which enforces a diffuse interface between the two fluids \\cite{leclaire2017generalized}. \nIn short, in the update phase of the populations, the CG collision contains three sub-steps: a plain BGK collision, a perturbation operator, and a final recoloring step. It is worth stressing that the last two sub-steps act only near the interface between the two fluids. Further details are reported in Appendix.\n\nThe second level of description involves a Lagrangian solver for the particle evolution, where each particle (colloid) is represented by a closed surface ${\\mathcal S}$, taken, for simplicity, as a rigid sphere in the following. \n\nThe LBcuda code adopts the formulation given by Jansen and Harting \\cite{jansen2011bijels}, where only the exterior regions are filled with fluid, whereas the interior parts of the particles are solid nodes.\nThe solid--fluid interaction is managed via a simple generalization of the bounce-back rule including the correction due to the relative motion of the solid particle with respect to the surrounding fluid medium.\n\nHence, the particle position, speed $\\vec{v}_p$ and angular momentum\\index{angular!momentum} $\\vec{\\omega}_p$ are updated according to Newton's equations of motion:\n\n\\begin{equation}\n\\label{PARDYN}\n\\left\\{ \\begin{array}{lll}\n\\frac{d \\vec{r}_p}{d t} = \\vec{v}_p,\\\\\nm_p \\frac{d \\vec{v}_p}{d t} = \\vec{F}_p,\\\\\nI_p \\frac{d \\vec{\\omega}_p}{d t} = \\vec{T}_p,\n\\end{array} \n\\right.\n\\end{equation}\nwhere $m_p$ and $I_p$ are the particle mass and moment of inertia, respectively.\n\nFollowing Ladd's seminal works \\cite{ladd1994numericala,ladd1994numericalb}, we advance in time eq. \\ref{PARDYN} with a leap-frog scheme, which is second order accurate in time.\nThis set of equations considers the full many-body hydrodynamic interactions since the forces and torques are computed with the actual flow, as dictated by the presence of all $N$ particles simultaneously.\n\n\n\\section{Implementation}\n\\label{sec:impl}\n\nThe code is implemented in CUDA Fortran, using modules to minimize code cluttering. The LBcuda code requires no external libraries besides the CUDA runtime and compiles using a simple Makefile. The code is written for the nvfortran compiler, with the GPU kernels confined in {\\em cuf} extension files, whereas the I\/O part and the main are coded in standard FORTRAN files.\n\nThe code is composed of 6 files:\n\\begin{itemize}\n\\item \\textbf{dimension.cuf}, which sets constants for the LB algorithm and the physical values of the simulation\n\\item \\textbf{kernels\\_fluid.cuf}, containing all GPU variables\n\\item \\textbf{kernels\\_fluid\\_CG.cuf}, containing the color-gradient GPU code \n\\item \\textbf{kernels\\_fluid\\_PART.cuf}, containing the particles GPU code\n\\item \\textbf{write\\_output.f90}, which outputs the VTK and VTI files for external visualization by graphical programs (e.g, ParaView) \n\\item \\textbf{main.f90}, finally contains the driving code of all the subroutines.\n\\end{itemize}\n\nMost of the input for the simulation is defined by setting Fortran \\textbf{parameters} in dimension.cuf. In contrast, other runtime parameters, such as print frequency of VTK output files and average statistical quantities, can be set up without recompilation in a plain-text input file.\n\nThe data for each fluid component are organized in a five dimension matrix having x,y,z, then the population index, and finally two possible values for \\textit{switching} between old and new values during the collide-stream phases of the LB algorithm, also referred to as one-step two-grid algorithm \\cite{wittmann2013comparison}.\n\nAll data residing on the GPU are defined in kernels\\_fluid.cuf, whereas writing output files requires just a few memory passages from GPU to CPU at the printing frequency for fluid densities, flow field, and particle positions.\n\n\n\nIn order to solve the lattice-Boltzmann equation with particle dynamics, the algorithm proceeds executing the following sequence of subroutine calls:\n\\begin{itemize}\n \\item Each thread of the GPU device computes fluid and particle quantities at a single spatial point, say located at the (i, j, k) node;\n \\item In each node, the code proceeds according to three different cases:\n \\begin{enumerate}\n \\item If the node contains fluid far away from any particles, the thread will only advance the LB algorithm;\n \\item If at that point a fluid touches a particle, the thread computes its part for the LB algorithm, then it computes its contribute for the force\/toque integral. Note that when there are touching particles, a point can contribute to more than one particle.\n \\item If the point is inside a particle, no computation is performed and the following steps are skipped;\n \\end{enumerate}\n \\item Apply the collision step of Eq. \\ref{eq:bgk};\n \\item Apply the halfway bounce-back rule at particle surface and the relative force terms on particles;\n \\item Evolve position and angular velocity of particles (if present);\n \\item Apply the stream step of Eq. \\ref{eq:bgk}.\n\\end{itemize}\nIt is worth stressing that the net force and torque exerted from the fluid on the particle center of mass is obtained by summing over all the particle surface nodes.\n\n\n\\section{Parallelization strategy: CUDA and MPI}\n\\label{sec:parallel}\n\nFor the LB part of the algorithm, the CUDA porting decomposes the global domain according to a 3D block distribution among the CUDA threads (see Fig. \\ref{Figdec}). The selection of the block distribution is fixed at compile-time, and it can be tuned to obtain the best performance given the global grid dimension and the compute capability of the GPU device. Usually, a 3D decomposition that emphasizes the x-axis dimension achieves the best performance since it exploits the high memory bandwidth due to the data continuity in the column-major order of the FORTRAN language. Each thread will be responsible for only one grid point of the fluid box in each CUDA block. In the LB part, the thread iterates over all the fluid populations in most kernels. On the other hand, the thread computes the contribution of the fluid grid point to the force and torque of the overlapping particles, defined as particles whose surface overlaps the fluid node owned by the thread.\n\n\nThe LBcuda code is designed to exploit multiple GPU devices.\nTo that purpose, the code resorts to MPI having one GPU card associated to each MPI task. The LB domain is divided into sub-domains of equal size, whereas the variables related to the particles are replicated in all the MPI processes. Thus, the LB solver proceeds locally on each GPU device, with the extra computational cost due to the communication of border information among the local sub-domains of the neighbor MPI tasks. \n\nEach MPI task computes the part (section) of the particles falling in its sub-domain in the particle solver. \nIn this framework, particles evolution is crucial for achieving a high computational throughput by avoiding excessive communication (or memory conflicts) among threads while integrating the particle quantities.\nThus, we have adopted a single particle list, which is stored on GPU devices. In particular, whenever the multi GPU is used, each MPI sub-domain has its list of owned particles on its GPU card. When a LB time step is completed, each thread in the sub-domain checks if it needs to compute the contribution of its fluid node to the computation of the surface integral for the force and torque that the fluid exerts on each particle surface node and vice versa.\nHence, a global MPI reduction is used to compute the corresponding total force and torque acting on the center of mass of each particle, so that particle positions, orientations, and velocities can be advanced in time on all the MPI processes. \nThe selection of the sub-domain particle list on each MPI process is made by a CUDA kernel, leveraging the parallel computing power of each GPU card that makes the required computing time almost negligible.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0\\linewidth]{figure02.pdf}\n\\end{center}\n\\caption{Sketch showing the domain decomposition strategy used for the particle data on the GPU device. On the left, each sub-domain (thread block) has a list of owned particles. On the right, the global GPU data vector stores all the particles in contiguous way over the sub-domains.\n}\n\\label{Figdec}\n\\end{figure}\n\nIt is worth highlighting that the overlap between particle and fluid node provides an unbalance in the work performed by the CUDA threads, notably wasting computational power at the surface particle node to treat the boundary conditions and the momentum exchange between the surrounding fluid and the particle. Nonetheless, we found that the overhead for the fluid solver is quite marginal, retaining an acceptable code scalability.\n\n\n\n\n\n\n\n\n\\section{Performance results}\n\\label{perfomance}\n\nIn order to analyze the computational performance of LBcuda, we consider two cubic boxes of side 128 and 256 lattice nodes\nwith periodic boundary along the three Cartesian axes. \nWe limited the size to a 256 cubic box, which is the largest grid that fits in the memory of a single GPU device.\nWe perform three different test cases. First, we examine the case of a single fluid in a cubic box with an initial density equal to one and zero velocity flow ($case \\; 1$).\nAs a second test ($case \\; 2$), a bi-component system is considered where all the fluid nodes are randomly filled with fluid mass density of the two fluids, red and blue component, to achieve the value of the order parameter, $\\phi=\\frac{\\rho_r-\\rho_b}{\\rho_r+\\rho_b}$, equal to 1 or -1 with zero velocity flow field with the subscripts r and b standing for \"red\" and \"blue\" fluids, respectively. The third test ($case \\; 3$) checks the entire LBcuda algorithm with\nthe two-component fluid combined with the particle solver \ndescribing the colloids in a rapid demixing emulsion.\nIn particular, we defined three sub-setups with different numbers of particles to assess the performance of the particle solver.\nHence, three values of the volume fraction occupied by particles are considered:\n0.1\\%, 1.0\\% and 10\\%, labeled $case \\; 3a$, $case \\; 3b$, and $case \\; 3c$, respectively.\n\nTo evaluate the performance of LBcuda, we compare the theoretical peak performance to the actual one achieved by our code. In particular, the roofline model \\cite{williams2009roofline} is used to rank the achievable computational performance in terms of Operational Intensity (OI), defined as the ratio between flops performed and data that need to be loaded\/stored from\/to memory. \nAt low OI (say, $ < 10 $), the performance is limited by the memory bandwidth, whereas for higher OI values, the limitation comes from the availability of floating-point units. \nIt is well known that LB is a bandwidth-limited numerical scheme, like most CFD models \\cite{towards}. The OI index for LB schemes is around 0.7 for double-precision (DP) simulations using a D3Q19 lattice. As a matter of fact, for a single fluid, since the number of floating-point operations per lattice site and time step is $F \\simeq 200 \\div 250 $ and the load\/store burden in bytes is $B= 19 \\times 2 \\times 8=304 $ (using double precision), the operational intensity is $F\/B \\sim 0.7$, whereas in single precision is $F\/B \\sim 1.4$ confirming that the code is bandwidth limited (see also Figure \\ref{fig:roof}).\n\nIn the following, we assess the efficiency of the LBcuda code by means of the Giga Lattice Updates Per Second (GLUPS) metrics. In particular, \nthe definition of GLUPS reads:\n\\begin{equation}\n\\label{eq:glups}\n\\text{GLUPS}=\\frac{L_x L_y L_z}{10^9 t_{\\text{s}}},\n\\label{eq:mlups}\n\\end{equation}\nwhere $L_x$, $L_y$, and $L_z$ are the domain sizes in the\n$x-$, $y-$, and $z-$ axis, and $t_{\\text{s}}$ is the run (wall-clock) time (in seconds) per single time step iteration.\n\n\n\\subsection{Single fluid}\n\nThe $case \\; 1$ with one fluid is tested using a plain BGK collision and a fused implementation (in which the collision and streaming step are performed simultaneously) of the LB time-advancing, the latter being the most popular approach for major LB codes. The results are reported for two different GPUs: a V100 and a GeForce RTX 2060S in Table \\ref{tab:1}.\n\n\\FloatBarrier\n\\begin{table}[h]\n \\centering\n \\begin{tabular}{|c|c|c|c|c|}\n \\hline \n \\textbf{Size} & \\textbf{GPU} & \\textbf{Time} & \\textbf{GLUPS} & \\textbf{Approach} \\\\\n \\hline \n $128^3$ & V100 & 1.09 ms\/iter & 1.923 & fused \\\\\n $128^3$ & 2060S & 1.64 ms\/iter & 1.278 & fused \\\\\n \\hline\n $256^3$ & V100 & 8.34 ms\/iter & 2.011 & fused \\\\\n $256^3$ & 2060S & 15.6 ms\/iter & 1.075 & fused \\\\\n \\hline \n $128^3$ & V100 & 2.09 ms\/iter & 1.003 & plain \\\\\n $128^3$ & 2060S & 3.14 ms\/iter & 0.667 & plain \\\\\n \\hline\n $256^3$ & V100 & 15.34 ms\/iter & 1.093 & plain \\\\\n $256^3$ & 2060S & 30.1 ms\/iter & 0.557 & plain \\\\\n \\hline \n \\end{tabular}\n \\caption{Timings alongside with GLUPS of $128^3$ and $256^3$ cubic boxes for the $case \\; 1$ using both the optimized fused and plain approach single fluid on a Tesla V100 and a GeForce RTX 2060S in single precision.}\n \\label{tab:1}\n\\end{table}\n\\FloatBarrier\n\nThe key differences between the two GPUs are: the V100 has 5132 Cuda cores offering a peak performance of 14TFlop\/s and a memory bandwidth of 900GB\/s, whereas the GeForce RTX 2060S has 2176 cores for a peak of 6.4TFlop\/s and a memory bandwidth of 448GB\/s. On the other hand, we observe that the obtained 2.011 GLUPS for a cubic box of side 256 is comparable with the state-of-the-art represented, for instance, by the highly optimized code by G. Falcucci et al. in Ref. \\cite{falcucci2021extreme} which reaches 3.406 GLUPS on a single V100 with the fused implementation on the same cubic box size.\n\nAlthough the fused approach reduces of a factor two the number of memory accesses, it is worth highlighting that the particle solver requires the mandatory use of the plain approach, showing a decrease of the performance of about a factor two for $case \\; 1$.\nIndeed, the particle boundaries require using the plain Lattice-Boltzmann algorithm, where the collision is first computed for all the fluid lattice nodes. Then, the boundary conditions are applied (internal walls or particles), and finally the streaming of the populations is carried out (see Section \\ref{sec:impl}).\n\n\n\\begin{figure} \\begin{center}\n\\includegraphics[width=1.0\\linewidth]{figure03.pdf}\n\\end{center}\n\\caption{Roofline model for single fluid using a GeForce RTX 2060S, as measured by NVIDIA NSight in the plain LB approach and in single precision. Bandwidth and Float point computation limits were obtained performing {\\em memory-stream} and High Performance Computing Linpack benchmark. Note that the LBcuda code lies on the left part of the plot showing that it is bandwidth limited.}\n\\label{fig:roof}\n\\end{figure}\n\n\nFrom an in-depth analysis using the NVidia dedicated tool (NSight), we observe that on the GeForce RTX 2060S the main kernel (the LB time-stepping before the streaming substep)128x1x1 achieves better performance achieves almost $\\sim 520 GFlop\/s$ with an arithmetic intensity of 2.0 in single precision (see Figure \\ref{fig:roof}). This result shows that we are far from an intensity $\\sim 15.0$ which should give the peak performance, and in the $case \\; 1$ we attain about 60\\% of memory bandwidth utilization due to its non-optimal use following the plain approach.\n \n\n\\subsection{Two fluid test}\n\nThe $case \\; 2$ is related to the simulation of a two-component system by the color-gradient model (see Section \\ref{sec:method}). We remark that the time integration is implemented using the plain approach with a standard collide-stream 2-pass algorithm. Table \\ref{tab:cg} highlights the measured performance for the $128^3$ and $256^3$ cubic box. In particular, we observe an increase of about a factor 3 with respect to the previous $case \\; 1$, which is mainly due to the larger number of operations, more than doubled, in the color gradient collision operator containing three steps (see Section \\ref{sec:method}) instead of the single step of the plain BGK single fluid case. \n\n\\FloatBarrier\n\\begin{table}[h]\n \\centering\n \\begin{tabular}{|c|c|c|c|c|}\n \\hline \n \\textbf{Size} & \\textbf{GPU} & \\textbf{CUDA Block} & \\textbf{Time} & \\textbf{GLUPS} \\\\\n \\hline \n $128^3$ & V100 & 8x4x4 & 4.5 ms\/iter & 0.466 \\\\\n $128^3$ & 2060S & 8x4x4 & 11 ms\/iter & 0.190 \\\\\n \\hline\n $256^3$ & V100 & 8x4x4 & 34.1 ms\/iter & 0.492 \\\\\n $256^3$ & 2060S & 8x4x4 & 82.3 ms\/iter & 0.204 \\\\\n \\hline \n \\hline \n $128^3$ & V100 & 128x1x1 & 3.96 ms\/iter & 0.554 \\\\\n $128^3$ & 2060S & 128x1x1 & 8.96 ms\/iter & 0.245 \\\\\n \\hline\n $256^3$ & V100 & 128x1x1 & 27.9 ms\/iter & 0.601 \\\\\n $256^3$ & 2060S & 128x1x1 & 64.5 ms\/iter & 0.261 \\\\\n \\hline \n \\end{tabular}\n \\caption{Timings alongside with GLUPS for $128^3$ and $256^3$, 2 fluids with CG using a Tesla V100 and a GeForce RTX 2060S with the plain approach in single precision with two different decompositions of threads in CUDA block. Note that the CUDA block configuration 128x1x1 achieves better performance exploiting the contiguous data over $x$ in the population arrays.}\n \\label{tab:cg}\n\\end{table}\n\\FloatBarrier\n\n\n\n\\subsection{Particles}\n\nThe entire LBcuda algorithm is evaluated with the two-component colour gradient method (see Section \\ref{sec:method}) combined\nwith the particle solver to model the colloids in a rapid demixing emulsion. To assess\nthe performance of the particle solver, we prepared three simulation setups with different numbers of particles with radius equal to 5.5 lattice units in a cubic box of side 256 lattice points.\nThe three cases, in the following labelled $case \\; 3a$, $3b$, and $3c$, differ in the ratio between the volume occupied by the particles compared to the box volume equal to $\\varphi=$ 0.1\\%, 1.0\\% and 10\\%, respectively, in order to study the impact of the particle evolution on the simulation time.\nThe performance impact of having particles goes from negligible in $case \\; 3a$ to be comparable with the LB computation in $ case \\; 3c$ in which the time for each iteration almost doubles, as shown in table \\ref{tab:partic}.\n\n\\FloatBarrier\n\\begin{table}[h]\n \\centering\n \\begin{tabular}{|c|c|c|c|c|}\n \\hline \n \\textit{\\textbf{case}} & \\textbf{GPU} & $\\varphi$ & \\textbf{Time} & \\textbf{GLUPS} \\\\\n \\hline \n $3a$ & V100 & 0.1 \\% & 34.6 ms\/iter & 0.484 \\\\\n $3a$ & 2060S & 0.1 \\% & 84 ms\/iter & 0.199 \\\\\n \\hline \n $3b$ & V100 & 1.0 \\% & 37.6 ms\/iter & 0.446 \\\\\n $3b$ & 2060S & 1.0 \\% & 88 ms\/iter & 0.190 \\\\\n \\hline \n $3c$ & V100 & 10 \\% & 56 ms\/iter & 0.299 \\\\\n $3c$ & 2060S & 10 \\% & 113 ms\/iter & 0.148 \\\\\n \\hline\n \\end{tabular}\n \\caption{Timings alongside with GLUPS for the three cases with particle volume fraction, $\\varphi$, equal to 0.1\\%, 1.0\\%, and 10\\%, respectively, in a cubic box of $256^3$ lattice nodes in single precision.}\n \\label{tab:partic}\n\\end{table}\n\\FloatBarrier\n\n\\begin{figure}[h!]\n\\includegraphics[width=1.0\\linewidth]{figure04.pdf}\n\\caption{Renderings for $256^3$ simulation without (plot d) and with particles (plot a,b,c). From top-left to bottom-right) a) Initial condition for particle simulation. b) Density field after 50k iterations with 10\\% volume fraction. c) Density field after 100k iterations (10\\% vol. fraction). d) Density field after 100k iterations without particles starting from a mixed bi-component fluid system (similar to plot a).}\n\\label{fig:stopInterface}\n\\end{figure}\n\nIt is worth highlighting that LBcuda code implements a double precision accumulator because of floating point accuracy problems related to the momentum transfer from the fluid to each particle. In particular, the particle force and torque computation suffer from floating accuracy problems due to strong cancellation between addends of alternating sign over the nodes of the two-fluids interface.\n\nThe benchmark results can be also compared to the corresponding box size without particles reported in Table \\ref{tab:cg}.\nFor the test in $case \\; 3$ the particle radius is equal to 5.5 lattice units, and\nthe particle positions are randomly distributed in the box. \nThe initial particle velocity is zero in all runs.\nThe particle wettability is tuned to set an angle equal to $90^{\\circ}$ \nwith respect to the axis $\\vec{x}^{\\star}$ in the local reference frame \nof each particle. The impenetrability among particles is avoided by an hertzian repulsive contact force computed by means of neighbor's lists (see Section \\ref{sec:parallel}).\nThe lubrication force is also considered by adding an extra force term whenever two particles are located at a mutual distance lower than 2\/3 lattice unit, as reported in previous simulations \\cite{jansen2011bijels}. \nThe particle mass was estimated as the weight corresponding to a particle \nmade of silica \\cite{herzig2007bicontinuous}. \n\n\n\n\nAll runs were simulated on both the CPU and GPU architectures using the previous LBSoft code and the corresponding GPU ported version, LBcuda. In all the $cases \\; 3$ we observe only a small deviation in the position always lower than $10^{-4}$ in lattice units, mainly due to the aforementioned floating point accuracy problem. Indeed, the order of the addends over the particle surface nodes is completely random on the GPU device. \nIn the $case \\; 3c$ we observe the arrest of the phase demixing process with particles located at the fluid-fluid interface entrapping the demixing process into a metastable state, the bi-continuous jammed gel state \\cite{stratford2005colloidal}. In Figure \\ref{fig:stopInterface} we show two of these numerical experiments, in which a random mixture of two fluids evolves in a very different way in the presence of a high number of particles ($case \\; 3c$) and without particles ($case \\; 2$). Indeed, the particles stop the complete spinodal decomposition of the two fluids corresponding to the condition of minimal energy and minimal interface between them. Whenever a high volume fraction of particles is in the simulation box, the separation surface becomes way more \"corrugated\" showing the formation of the bi-continuous jammed gel state.\nFigure \\ref{fig:stopInterface} shows the initial condition, the fluids after 50k (with particles), and final configurations of both simulations after 100k iterations ($case \\; 3c$ and $case \\; 2$).\n\n\n\n\n\\subsection{Multi GPUs Performance}\n\nThe LBcuda code resorts to the Message Passing Interface (MPI) library to exchange data among GPU devices running in parallel.\nThe performance obtained using multiple GPUs show a good scaling behavior as reported in Tables \\ref{tab:multiGPU} and \\ref{tab:multiGPU2}. The benchmark in Table \\ref{tab:multiGPU} was carried out on Marconi100 at CINECA, a cluster of V100 cards, each with 16 GB of global memory using a different number of GPU devices, for three cubic boxes: $256^3$, $512^3$, and $1024^3$. The cluster is made of nodes, each endowed with four GPU cards so that the MPI communication does not incur network latency unless the job is using more than 4 GPU cards. The benchmark in Table \\ref{tab:multiGPU2} ran on a cluster made of NVIDIA DGX A100. Each NVIDIA DGX A100 is equipped with 8 A100 NVIDIA GPU with 80 GB of RAM interconnected intra-node through the NVswitch and 8 NVIDIA Infiniband HDR (one NIC for each GPU) for multi-node scaling.\n\n\\begin{table}[h!]\n \\centering\n \\begin{tabular}{|c|c|c|c|c|c|c|}\n \\hline \n \\textbf{Grid} & \\textbf{1} & \\textbf{2} & \\textbf{4} & \\textbf{8} & \\textbf{16} & \\textbf{32} \\\\\n \\hline \n \n $256^3 V100\\ 16GB$ & 0.60 & 1.17 & 2.24 & 2.93 & 2.71 & 3.15 \\\\\n \\hline \n $512^3 V100\\ 16GB$ & x & x & 2.67 & 4.57 & 6.23 & 9.26 \\\\\n \\hline\n $1024^3 V100\\ 16GB$ & x & x & x & x & x & 12.48 \\\\\n \\hline\n \\end{tabular}\n \\caption{Performance for $256^3$, $512^3$, $1024^3$, measured in GLUPS running on multiple NVIDIA Volta V100 GPUs (each card equipped with 16 GB of RAM). Note that the symbol (x) denotes a grid size too big to fit in the local GPU memory. For the $256^3$, performance degrade dramatically when using more than 8 V100 GPUs and more than 64 A100 because each local domain becomes too small (256x256x8) to offset the cost of scheduling the GPU task.}\n \\label{tab:multiGPU}\n\\end{table}\n\n\n\\begin{table}[h!]\n \\centering\n \\begin{tabular}{|c|c|c|c|c|c|c|c|}\n \\hline\n \\textbf{Grid} & \\textbf{8} & \\textbf{16} & \\textbf{32} & \\textbf{64} & \\textbf{128} & \\textbf{256} & \\textbf{512} \\\\\n \\hline\n $256^3 A100\\ 80GB$ & 6.17 & 9.42 & 12.38 & 15.38 & & & \\\\\n \\hline \n $512^3 A100\\ 80GB$ & 7.63 & 14.13 & 23.13 & 36.05 & & &\\\\\n \\hline\n $1024^3 A100\\ 80GB$ & 11.65 & 20.57 & 28.59 & 49.11 & 76.91 & & \\\\\n \\hline\n $2048^3 A100\\ 80GB$ & x & x & x & 55.22 & 102.3 & 160.9 & 204.5 \\\\\n \\hline\n \n \\end{tabular}\n \\caption{Performance for $256^3$, $512^3$, $1024^3$, $2048^3$ measured in GLUPS running on multiple NVIDIA Volta A100 GPUs (each card equipped with 80 GByte of RAM). Note that the symbol (x) denotes a grid size too big to fit in the local GPU memory.}\n \\label{tab:multiGPU2}\n\\end{table}\n\nAll benchmarks reported in Tables \\ref{tab:multiGPU} and \\ref{tab:multiGPU2} were carried out on a bi-component fluid system without particles ($case \\; 2$). \nThe data for the cubic box sizes $256^3$, $512^3$ are also plotted in Figure \\ref{fig:GPUMlups} with an evident decrease in the scalability whenever the code starts to run on more than one node (more than 4 GPUs).\nHence, the three tests, $case \\; 3a$, $3b$, and $3c$, with different values in the particle volume fraction, $\\varphi$, were performed to probe the efficiency of the particles solver as a function of the particles number in the system. All benchmarks were carried out on eight GPU cards: both V100 cards with a 16 GB RAM and A100 cards with a 40 GB RAM. In Table \\ref{tab:partmultiGPU}, we observe a clear communication penalty as the particles number increases that is mainly due to the replicated data strategy used for the particle solver parallelization. Indeed, the replicated data parallel approach replicates the physical quantities of all particles across the MPI processes, performing local updating and global MPI sum reductions in order to advance the system in time, which decreases the performance as the quantity of particles data increases.\nNonetheless, the analysis of the performance as a function of the number of particles shows that the code is able to reach about 3.80 GLUPS in a system with 10\\% in the particle volume fraction.\n\n\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.5\\linewidth]{figure05.pdf}\n\\caption{Top panel: Measured GLUPS for different numbers of V100 cards with a 16 GB RAM, running two cubic boxes of side 256 and 512, respectively. Bottom panel: Measured GLUPS for different numbers of A100 cards with a 80 GB RAM, running two cubic boxes of side 1024 and 2048, respectively.}\n\\label{fig:GPUMlups}\n\\end{center}\n\\end{figure}\n\n\n\\begin{table}[h!]\n \\centering\n \\begin{tabular}{|c|c|c|c|c|}\n \\hline \n \\textbf{GPU} & \\textbf{No particles} & \\bm{$\\varphi=0.1\\%$} & \\bm{$\\varphi=1.0\\%$}& \\bm{$\\varphi=10\\%$} \\\\\n \\hline \n 8 V100@16 & 30 ms (4.47) & 40 ms (3.35) & 51 ms (2.63) & 95 ms (1.41) \\\\\n \\hline\n 8 A100@40 & 18 ms (7.55) & 19 ms (7.06) & 22 ms (6.10) & 35 ms (3.83) \\\\\n \\hline\n \\end{tabular}\n \\caption{Time per single iteration alongside with GLUPS in parenthesis for $512^3$ without and with particles (at different particle volume fraction $\\varphi$) on 2 different machines: using eight V100 with 16 GB of RAM on 2 nodes (connected by InfiniBand) and using eight A100 with 40 GB in a single node (without the latency time due to the InfiniBand communication).}\n \\label{tab:partmultiGPU}\n\\end{table}\n\n\n\n\n\\subsection{Comparing LBcuda with LBsoft}\n\nFor the sake of completeness, we probe the gain provided by the CUDA port reported in the present article. Thus, the $case \\; 2$ bi-component system was initialized with the same values in density and flow field in both LBcuda and LBsoft code. The cubic box size is equal to $512^3$ lattice points. Although it is difficult to compare two completely different computing architectures, we measure the wall-clock time obtained on a GPU cluster made of V100 cards with 16 GB RAM and a CPU cluster containing two Intel Cascade Lake 8260 CPUs at 2.40 GHz with 48 cores and 384 GB of RAM per node.\nWe note that the code produces the same results in single precision unless a slight difference of $10^{-4}$ order of magnitude in the particle positions due to floating point accuracy problems (mainly due to different order of summation). The wall-clock time for iteration results in 12 ms on 32 V100 cards of LBcuda versus 88 ms on 528 cores of LBSoft, confirming the clear advantage in running the CUDA ersion on a GPU HPC cluster.\n\n\n\\section{Conclusion}\n\n\nWe have presented LBcuda, a CUDA port of LBsoft, an open-source software aimed at simulating specifically colloidal systems. \nLBcuda is written in CUDA Fortran and\npermits to simulate large system sizes running on multiple GPU devices by exploiting an efficient parallel domain decomposition implementation.\n\nIn particular, the code shows good scaling behavior of the fluid solver achieving the performance peak of 200 GLUPS on 512 NVIDIA A100 cards with a grid of eight billion lattice points.\nOn the other hand, the particle solver combined with the LB approach shows a very satisfactory performance in terms of scalability in both system size and number of processing cores, especially using the Nvidia Ampere A100 cards.\n\nIn this work, the main structure of LBcuda has been outlined along with the key steps of its implementation. \nFurthermore, several cases have been introduced to test the code over typical problems that the LBcuda code can deal with. \nIn particular, the simulations with particles demonstrate the capabilities of the present code to reproduce the complex dynamics of bi-jel systems in a rapid de-mixing emulsion.\n\nThe LBcuda code is open source and completely accessible at the public repository GitHub, which is in line with the spirit of open-source software, mainly to promote the contribution of independent developers.\n\n\n\\section{Acknowledgments}\nThe research leading to these results has received\nfunding from the European Research Council under the European\nUnion's Horizon 2020 Framework Programme (No. FP\/2014-\n2020)\/ERC Grant Agreement No. 739964 ``COPMAT'' and from MIUR under the project \"3D-Phys\" (No. PRIN 2017PHRM8X). The CINECA is acknowledged for the support granted by the ISCRA project ``porting LBSOft in CUda on multi-node GPUs (LBSOCU)''.\n\n\n\n\\section*{Appendix}\n\nFor the details of the color gradient (GC) model of the Lattice Boltzmann method employed in the bi-component systems \\cite{leclaire2017generalized}, we recall some notions.\n\nIn the color gradient LB for two-component flows, two\nsets of distribution functions are defined to track the evolution of the two fluid components, which occurs via a streaming-collision algorithm:\n\n\\begin{equation} \\label{CGLBE}\nf_{i}^{k} \\left(\\vec{x}+\\vec{c}_{i}\\Delta t,\\,t+\\Delta t\\right) =f_{i}^{k}\\left(\\vec{x},\\,t\\right)+\\Omega_{i}^{k}( f_{i}^{k}\\left(\\vec{x},\\,t\\right)),\n\\end{equation}\n\nwhere $f_{i}^{k}$ is the discrete distribution function, representing\nthe probability of finding a particle of the $k^{th}$ component at position $\\vec{x}$ and time\n$t$ with discrete velocity $\\vec{c}_{i}$ . \n\nIn the last Eq. $i$ is the index running over the lattice discrete directions $i = 0,...,b$, where $b=18$ for a three dimensional 19 speed lattice (D3Q19) implemented in LBcuda.\nThe lattice time step $\\Delta t$ has been taken as $1$ (in lattice units) for convenience.\nThe density $\\rho^{k}$ of the $k^{th}$ component is given by the zeroth moment of the distribution functions:\n\\begin{equation}\n\\rho^{k}\\left(\\vec{x},\\,t\\right) = \\sum_i f_{i}^{k}\\left(\\vec{x},\\,t\\right),\n\\end{equation}\nwhile the total fluid density is assessed as $\\rho=\\sum_k \\rho^k$, and the total momentum of the mixture is given as the sum of the linear momentum of the two components:\n\\begin{equation}\n\\rho \\vec{u} = \\sum_k \\sum_i f_{i}^{k}\\left(\\vec{x},\\,t\\right) \\vec{c}_{i}.\n\\end{equation}\n\nThe collision operator in the CG model is made of three parts: \n\n\\begin{equation}\n\\Omega_{i}^{k} = \\left(\\Omega_{i}^{k}\\right)^{(3)}\\left[\\left(\\Omega_{i}^{k}\\right)^{(1)}+\\left(\\Omega_{i}^{k}\\right)^{(2)}\\right].\n\\end{equation}\n\nIn the above, $\\left(\\Omega_{i}^{k}\\right)^{(1)}$ stands for the standard collisional relaxation which reads:\n\\begin{equation}\n\\left(\\Omega_{i}^{k}\\right)^{(1)}=\\omega(f_i^{k,eq} - f_i^k),\n\\label{1coll}\n\\end{equation}\nwhere $\\omega=2\/(6\\bar{\\nu} -1)$ is the effective relaxation parameter being $\\bar{\\nu}$ the mean viscosity of the bi-component system computed as\n$\\frac{1}{\\bar{\\nu}}=\\frac{\\rho_1}{(\\rho_1+\\rho_2)}\\frac{1}{\\nu_1} + \\frac{\\rho_2}{(\\rho_1+\\rho_2)}\\frac{1}{\\nu_2}$ ($\\nu_1$ and $\\nu_2$ are the kinematic viscosities of the two pure components in the bulk). \nThe equilibrium distribution function of the $k^{th}$ component $f_i^{k,eq}$ is given by a low-Mach, second-order, expansion\nof a local Maxwellian, namely:\n\\begin{equation}\nf_i^{k,eq}=w_i \\rho^k (1 + \\frac{ \\vec{c_i} \\cdot \\vec{u}}{c_s^2} +\\frac{(\\vec{c_i} \\cdot \\vec{u})^2}{2c_s^4} - \\frac{\\vec{u} \\cdot \\vec{u}}{2 c_s^2}),\n\\label{eq:LEQ}\n\\end{equation}\nwhere $c_s=1\/\\sqrt{3}$ is the sound speed of the model.\nThe symbol $\\left(\\Omega_{i}^{k}\\right)^{(2)}$ denotes the perturbation step, which \ncontributes to the build up of an interfacial tension. Finally, $\\left(\\Omega_{i}^{k}\\right)^{(3)}$ is the recoloring step, which promotes the segregation between species, so as to minimize their mutual diffusion.\n\nIn order to reproduce the correct form of the stress tensor, the perturbation operator can be constructed by exploiting the concept of the continuum surface force.\nFirstly, the perturbation operator must satisfy the following conservation constraints:\n\n\\begin{eqnarray} \\label{consconstr}\n\\sum_i \\left(\\Omega_{i}^{k}\\right)^{(2)}=0 \\\\\n\\sum_k \\sum_i \\left(\\Omega_{i}^{k}\\right)^{(2)} \\vec{c}_i=0\n\\end{eqnarray}\n\nBy performing a Chapman-Enskog expansion, it can be shown that the hydrodynamic limit of Eq.\\ref{CGLBE} is represented by a\nset of equations for the conservation of mass and linear momentum:\n\n\\begin{eqnarray} \\label{NSE}\n\\frac{\\partial \\rho}{\\partial t} + \\nabla \\cdot {\\rho \\vec{u}}=0 \\\\\n\\frac{\\partial \\rho \\vec{u}}{\\partial t} + \\nabla \\cdot {\\rho \\vec{u}\\vec{u}}=-\\nabla p + \\nabla \\cdot [\\rho \\nu (\\nabla \\vec{u} + \\nabla \\vec{u}^T)] + \\nabla \\cdot \\bm{\\Sigma}\n\\end{eqnarray}\n\nwhere $p=\\sum_k p_k$ is the pressure and $\\nu=c_s^2(\\tau-1\/2)$ is the kinematic viscosity of the mixture, being $\\tau$ the single relaxation time.\n\n\nThe stress tensor in the momentum equation is given by:\n\n\\begin{equation}\n\\bm{\\Sigma}=-\\tau \\sum_i \\sum_k \\left( \\Omega_{i}^{k} \\right)^{(2)} \\vec{c}_i \\vec{c}_i\n\\end{equation}\n\n\n\nSince the perturbation operator is responsible for generating interfacial tension, the following \nrelation must hold:\n\n\\begin{equation} \n\\nabla \\cdot \\bm{\\Sigma} = \\vec{F}\n\\label{SeqF}\n\\end{equation}\n\nDenoting $\\Theta =(\\rho^1-\\rho^2)\/(\\rho^1+\\rho^2)$ \nthe phase field, by choosing the second operator as:\n\\begin{equation} \n\\left(\\Omega_{i}^{k}\\right)^{(2)}= \\frac{A_k}{2} |\\nabla \\Theta|\\left[w_i \\frac{(\\vec{c}_i \\cdot \\nabla \\Theta)^2}{|\\nabla \\Theta|^2} -B_i \\right], \n\\label{2coll}\n\\end{equation}\nsubstituting it into Eqs \\ref{consconstr} and \\ref{SeqF} and by imposing that the set $B_i$ must satisfy the following isotropy constraints\n\n\\begin{eqnarray}\n\\sum_i B_i= \\frac{1}{3} \\; ; \\sum_i B_i \\vec{c}_i=0 \\; ; \\sum_i B_i \\vec{c}_i \\vec{c}_i= \\frac{1}{3} \\mathbf{I},\n\\end{eqnarray}\n\nwe obtain an equation for the surface tension of the model:\n\\begin{equation}\\label{sigmaA}\n\\sigma=\\frac{2}{9}(A_1+A_2)\\frac{1}{\\omega}=\\frac{4}{9}A\\frac{1}{\\omega}.\n\\end{equation}\nThe above relation shows a direct link between the surface tension and the parameter $A$ with $A_1=A_2$.\nIn actual practice, after choosing the viscosity of the two components and the surface tension of the model, at each time step, one locally computes the $A$ coefficient by using the formula reported in eq. (\\ref{sigmaA}).\n\n\nAs pointed out above, the perturbation operator generates an interfacial tension in compliance with the capillary-stress tensor of the Navier-Stokes equations for a multicomponent fluid system.\n\nNonetheless, the perturbation operator alone does not guarantee the immiscibility of different fluid components.\nFor this reason, a further step is needed (i.e. the recoloring step) to minimize the mutual diffusion between components.\n\nFollowing the work of Latva-Kokko and\nRothman, the recoloring operator for the two sets of distributions takes the following form:\n\n\\begin{eqnarray}\n\\left(\\Omega_{i}^{1}\\right)^{(3)} =\\frac{\\rho^1}{\\rho} f_i^* + \\beta \\frac{\\rho^1\\rho^2}{\\rho^2} \\cos{\\phi_i} f_i^{eq,0} \\\\\n\\left(\\Omega_{i}^{2}\\right)^{(3)} =\\frac{\\rho^2}{\\rho} f_i^* - \\beta \\frac{\\rho^1\\rho^2}{\\rho^2} \\cos{\\phi_i} f_i^{eq,0},\n\\label{3coll}\n\\end{eqnarray}\n\nwhere $f_i^*=\\sum_k f_i^{k,*}$ denotes the set of post-perturbation distributions, $\\rho=\\rho^1 + \\rho^2$, $\\cos{\\phi_i}$ is the angle between the phase field gradient and the $i^{th}$ lattice vector and $f_i^{eq,0}=f_i(\\rho,\\vec{u}=0)^{eq}=\\sum_k f_i^k(\\rho,\\vec{u}=0)^{eq}$ is the total zero-velocity equilibrium distribution function. In Eq. \\ref{3coll}, the coefficient $\\beta$ is a free parameter which tunes the interface width, thus playing the role of an inverse diffusion length scale. The coefficients used in Eqs \\ref{eq:LEQ} and \\ref{2coll} are reported in Table.\n\n\\begin{table}\n \\centering\n \\begin{tabular}{ c c c c }\n \\hline \n \\hline \n D3Q19 & $\\{ i:\\left| c \\right|^2 = 0 \\}$ & $\\{ i:\\left| c \\right|^2 = 1 \\}$ & $\\{ i:\\left| c \\right|^2 = 2 \\}$ \\\\\n \\hline \n \\hline \n $w_i$ & 1\/3 & 1\/18 & 1\/36 \\\\\n \n $B_i$ & -2\/9 & 1\/54 & 1\/27 \\\\\n \n \\hline\n \\end{tabular}\n \\caption*{D3Q19 lattice velocity and weights \\cite{leclaire2017generalized}.}\n \\label{tab:coef}\n\\end{table}\n\nFor the details of the particle time evolution, a full explanation is in Ref. \\citep{lbsoft}. For the sake of completeness, the main key steps are outlined in the following.\n\n\nThe velocity at a boundary node of $p$-th particle is given by:\n\\begin{equation}\n\\label{UPI}\n\\vec{u}_{b} = \\vec{v}_p + \\vec{r}_b \\times \\vec{\\omega}_p,\n\\end{equation}\nwhere, $r_{b} = \\vec{r}_s + \\frac {1}{2} \\vec{c}_i$ is the \nlocation of the moving wall along the $i$-th link connecting\nthe solid node $\\vec{r}_s$ to the \nfluid node $\\vec{r}_i = \\vec{r}_s +\\vec{c}_i$.\nAll coordinates are relative to the center of the $p$-th particle, located at position $\\vec{r}_p$ and moving with translation and angular velocities $\\vec{v}_p$ and $\\vec{\\omega}_p$, respectively.\n\nThe timestep is made unit for simplicity. \nThis velocity sets the bias between colliding pairs:\n\\begin{eqnarray}\n\\label{LADDFLBE}\nf_i \\left( \\vec{r}+\\vec{c}_i, t+1 \\right) =\nf_{\\bar i} \\left( \\vec{r}+\\vec{c}_i, t' \\right) + 2 \\rho w_i u_{bi}, \\\\\nf_{\\bar i} \\left( \\vec{r}, t+1 \\right) = f_i \\left( \\vec{r}, t' \\right) \n- 2 \\rho w_i u_{bi}\n\\end{eqnarray}\nwhere $t'$ denotes post-collisional states and we have set\n$$\nu_{bi}= \\frac{\\vec{u}_b \\cdot \\vec{c}_i}{c_s^2}\n$$\nNote that these rules reduce to the usual bounce-back conditions for a solid at rest, $\\vec{v}_p=\\vec{\\omega}_p=0$.\n\nThese collision rules produce a net momentum transfer between the fluid and the solid site:\n\\begin{equation}\n\\label{FORCEPP}\n\\vec{F}_i(\\vec{r}_b,t+\\frac{1}{2}) = 2 \\vec{c}_i \n[f_i (\\vec{r},t') - f_{\\bar i}(\\vec{r}_i,t') - 2 \\rho w_i u_{bi}]\n\\end{equation}\n\nThe net force acting upon particle $p$ is obtained by summing over all boundary sites $\\vec{r}_b$ and associated interacting links, namely:\n\\begin{equation}\n\\label{FORCEPARTEQ}\n\\vec{F}_p = \\sum_{{\\vec{r}_b,i} \\in \\Sigma_p} \\vec{F}_{i}(\\vec{r}_b).\n\\end{equation}\nSimilarly, the total torque\\index{torque} $\\vec{T}_p$ is computed as\n\\begin{equation}\n\\label{TORQUEPARTEQ}\n\\vec{T}_p = \\sum_{{\\vec{r}_b,i} \\in \\Sigma_p} \\vec{F}_{i}(\\vec{r}_b)\n\\times \\vec{r}_b,\n\\end{equation}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\nThe main goal of this article is to study the uniqueness property of the following mean field equations with singularities:\n\\begin{equation}\\label{m-equ}\n\\Delta_g v+\\rho\\bigg(\\frac{he^v}{\\int_M h e^v{\\rm d}\\mu}-\\frac{1}{vol_g(M)}\\bigg)=\\sum_{j=1}^N 4\\pi \\alpha_j (\\delta_{q_j}-\\frac{1}{vol_g(M)}) \\quad {\\rm in} \\ \\; M,\n\\end{equation}\nwhere $(M,g)$ be a Riemann surface with the metric $g$, $\\Delta_g$ is the Laplace-Beltrami operator ($-\\Delta_g\\ge 0$), $h$ is a positive smooth function on $M$, $q_1,\\cdots,q_N$ are distinct points on $M$, $\\rho>0,\\alpha_j>-1$ are constants, $\\delta_{q_j}$ is the Dirac measure at $q_j\\in M$. Equation (\\ref{m-equ}) is one of the most extensively studied elliptic PDE in the past few decades, partly due to its immense and profound connections with many branches of mathematics and Physics. In conformal geometry, (\\ref{m-equ}) represents a metric on M with conic singularity (see \\cite{fang-lai,troy,wei-zhang-pacific}). Also it is derived from the mean\nfield limit of point vortices in the Euler flow \\cite{caglioti-1,caglioti-2} and serves as a model equation\nin the Chern-Simons-Higgs theory \\cite{spruck-yang,taran-1,y-yang} and in the electroweak theory \\cite{ambjorn}, etc. The literature for the study of various form of (\\ref{m-equ}) is just too numerous to be listed in any reasonable way.\n\n\nRecently it was found by Lin-Yan \\cite{lin-yan-uniq} that the uniqueness property is particularly important for equations with concentration phenomenon. In their work \\cite{lin-yan-uniq} they proved the first uniqueness property for bubbling solutions of Chern-Simon-Higgs equation and computed the exact number of solutions in certain special cases. In an important work \\cite{bart-4} Bartolucci, et. al, extended Lin-Yan's result for mean field equation (\\ref{m-equ}) if the blowup points are not singular sources. Our goal in this article is to further extend the uniqueness property to the case that some singular sources coincide with blowup points.\n\n\\smallskip\n\nTo write the main equation in an equivalent form, we invoke the standard Green's function$G(x,p)$:\n\\begin{equation}\\label{gf}\n\\left\\{\\begin{array}{ll}\n-\\Delta_g G(x,p)=\\delta_p-1\\quad {\\rm in}\\ \\; M\n\\\\\n\\int_{M}G(x,p){\\rm d}\\mu=0,\n\\end{array}\n\\right.\n\\end{equation}\nwhere the volume of $M$ is assumed to be $1$ for convenience. Then it is well known that in a neighborhood of $p$, $G(x,p)$ can be written as\n$$G(x,p)=-\\frac 1{2\\pi }\\log dist(x,p)+R(x,p)$$\nwhere $dist(x,p)$ is the geodesic distance from $p$ to $x$ for $x$ close to $p$.\n\nUsing $G(x,p)$ we write (\\ref{m-equ}) as\n\\begin{equation}\\label{r-equ}\n\t\\Delta_g w+\\rho\\bigg(\\frac{He^w}{\\int_M H e^w{\\rm d}\\mu}-1\\bigg)=0 \\quad {\\rm in}\\ \\; M,\n\\end{equation}\nwhere\n\\begin{equation}\\label{r-sol}\n\tw(x)=v(x)+4\\pi \\sum_{j=1}^N \\alpha_j G(x,q_j),\n\\end{equation}\nand\n\\begin{equation}\\label{H1}\n\tH(x)=h(x)\\prod_{j=1}^N e^{-4\\pi\\alpha_j G(x,q_j)}.\n\\end{equation}\nNote that in a local coordinate near $q_j$,\n\\begin{equation}\\label{H2}\nH(x)=h_j(x)|x-q_j|^{2\\alpha_j},\\quad |x-q_j|\\ll 1,\\quad 1\\leq j\\leq N,\n\\end{equation}\nfor some $h_j(x)>0$.\n\nWe say that $\\{v_k\\}$ is a sequence of bubbling solutions of (\\ref{m-equ}) if the corresponding $w_k$ defined by (\\ref{r-sol}) tends to infinity as $k$ goes to infinity. The places that $w_k$ tends to infinity are called blowup points of $v_k$ or $w_k$. In this article we use $p_1,...,p_m$ to denote blowup points. Let $q_1,...,q_N$ be the location of singular sources. If none of $p_1,...p_m$ is a singular source, Bartolucci, et. al have obtained the uniqueness of the blow up solution in \\cite{bart-4}. Thus in this article we consider two cases: either all blowup points are singular sources or part of blowup points coincide with singular sources. In more precise terms let\n\\begin{equation}\\label{pq}\n\\left\\{\\begin{array}{ll}\np_j=q_j\\quad &{\\rm if} \\ 1\\leq j \\leq \\tau,\n\\\\\np_j \\notin \\{q_1,\\cdots,q_N\\}\\quad &{\\rm if} \\ \\tau+1\\leq j\\leq m,\n\\end{array}\n\\right.\n\\end{equation}\nfor some $1\\le \\tau\\le m$. Thus if $\\tau=m$ all blowup points are singular sources, if $1\\le \\tau\\tau$. Since the largest $\\alpha_j$ matters the most we require the first $t$ of them to have this strength:\n\\begin{equation}\n\\alpha_1=\\cdots =\\alpha_t>\\alpha_l, \\quad l\\geq t+1, \\quad \\mbox{where } 1\\le t\\le \\tau.\n\\end{equation}\n\nIt is well known that equation (\\ref{r-equ}) is the Euler-Lagrange equation of the variational form:\n$$I_{\\rho}(w)=\\frac 12 \\int_M |\\nabla w|^2+\\rho\\int_Mw-\\rho \\log \\int_M He^w, $$\nfor $w\\in H^1(M)$. Since adding a constant to any solution of (\\ref{r-equ}) certainly gives to another solution, the space of solutions for (\\ref{r-equ}) is the set of all $H^1(M)$ function with average equal to $0$. The discussion on the variational structure of (\\ref{r-equ}) can be found in \\cite{machiodi-1}.\n\n\\smallskip\n\nTo state the main results we use the following notations:\n\\begin{align}\n&G_j^*(x)=8\\pi (1+\\alpha_j)R(x,p_j)+8\\pi \\sum_{l\\neq j}^{1,\\cdots,m}(1+\\alpha_l)G(x,p_l), \\label{G_j*}\n\\\\\n&L(\\mathbf{p})=\\sum_{j=1}^t \\big[\\Delta \\log h(p_j)+\\rho_*-N^*-2K(p_j)\\big] (h_j(p_j))^{\\frac{1}{1+\\alpha_1}}e^{\\frac{G_j^*(p_j)}{1+\\alpha_1}}, \\label{L}\n\\\\\n&D(\\mathbf{p})=\n\\begin{pmatrix}\n\\nabla(\\log h_1+G_1^*)(p_1) \\\\\n\\cdots \\\\\n\\nabla(\\log h_t+G_t^*)(p_t)\n\\end{pmatrix}\n, \\label{D}\n\\end{align}\nwhere $h_j$ is defined in (\\ref{H2}), and\n\\begin{equation*}\n \\rho_*=8\\pi\\sum_{j=1}^m(1+\\alpha_j),\\quad N^*=4\\pi\\sum_{j=1}^N\\alpha_j\n\\end{equation*}\n\nOur first result is when all blowup points are singular sources:\n\\begin{thm}\\label{main-theorem}\n\t\n\tLet $v_k^{(1)}$ and $v_k^{(2)}$ be two sequences of bubbling solutions of (\\ref{m-equ}) with $\\rho_k^{(1)}=\\rho_k=\\rho_k^{(1)}$ and $\\alpha_j\\in\\mathbb{R}^+\\setminus\\mathbb{N}\\,(1\\leq j\\leq m)$. If $L(\\mathbf{p})\\neq0$ and $D(\\mathbf{p})=0$, then $v_k^{(1)}=v_k^{(2)}$ for $k$ large enough.\n\t\n\\end{thm}\n\nNote that we use $\\mathbb N$ to denote the set of positive integers. The assumption that $\\alpha_j\\in \\mathbb{R}^+\\setminus\\mathbb{N}$ implies that all blowup points are singular sources. It is also very essential to require $\\alpha_j$ to be non-integer, since quantized singular sources ( if the strength is $4\\pi N$) exhibit non-simple blowup phenomenon \\cite{kuo-lin}\\cite{wei-zhang-19}\nthat has to be studied in a separate work in the future.\n\nThe assumption of $D({\\bf p})$ is also very interesting. It is well known that if $p$ is not a singular source, the vanishing rate of $D({\\bf p})$ is very fast for a regular blowup point ( \\cite{gluck},\\cite{chen-lin-sharp}).\n\\medskip\n\nOur second main result is about the uniqueness of bubbling solutions when some blowup points are non-quantized singular sources and some are regular points. So in this case we require $1\\le \\tau0$ is a $C^1$ funation in $\\Omega$, $q_1,\\cdots,q_N$ are distinct points in $\\Omega$, $\\rho>0$, $\\alpha_j>0$ are constants.\n\nLet $\\{v_k\\}$ be a sequence of solutions to (\\ref{equ-flat}) with $\\rho=\\rho_k$. We say\n\\begin{equation}\\label{blowup-flat}\nv_k \\ {\\rm blows} \\ {\\rm up} \\ {\\rm at} \\ p_j\\in\\Omega,\\quad 1\\leq j\\leq m,\n\\end{equation}\nif $\\rho \\frac{he^v}{\\int_{\\Omega} h e^v{\\rm d}x}\\rightharpoonup8\\pi\\sum_{j=1}^N(1+\\alpha_j)\\delta_{p_j}$ in $\\Omega$ in the sense of measure, where $\\alpha_j=0$ if $p_j\\notin\\{q_1\\cdots,q_N\\}$. Similar to notations for the first part, we assume there exist $1\\leq t\\leq\\tau\\leq m$ such that $\\alpha_1=\\cdots\\alpha_t>\\alpha_i$, $i\\ge t+1$ and $\\alpha_{\\tau+1}=\\cdots\\alpha_m$.\n\nLet $G_{\\Omega}$ be the Green's function defined by\n\\begin{equation*}\n\\left\\{\\begin{array}{lll}\n-\\Delta G_{\\Omega}(x,p)=\\delta_{p} &{\\rm in} \\;\\ \\Omega,\n\\\\\nG_{\\Omega}(x,p)=0 &{\\rm on} \\;\\ \\partial\\Omega,\n\\end{array}\n\\right.\n\\end{equation*}\nand $R_{\\Omega}(x,p)=G_{\\Omega}(x,p)+\\frac{1}{2\\pi}\\log |x-p|$ be the regular part of $G_{\\Omega}(x,p)$. In order to state the uniqueness results of (\\ref{equ-flat}) we denote $N^*=4\\pi\\sum_{j=1}^m\\alpha_j$ and\n\\begin{align*}\n&G_{j,\\Omega}^*(x)=8\\pi (1+\\alpha_j)R_{\\Omega}(x,p_j)+8\\pi \\sum_{l\\neq j}^{1,\\cdots,m}(1+\\alpha_l)G_{\\Omega}(x,p_l),\n\\\\\n&L_{\\Omega}(\\mathbf{p})=\\sum_{j=1}^t \\big[\\Delta \\log h(p_j)-N^*\\big] (h_j(p_j))^{\\frac{1}{1+\\alpha_1}}e^{\\frac{G_j^*(p_j)}{1+\\alpha_1}},\n\\\\\n&D_{\\Omega}(\\mathbf{p})=\n\\begin{pmatrix}\n\\nabla(\\log h_1+G_1^*)(p_1) \\\\\n\\cdots \\\\\n\\nabla(\\log h_t+G_t^*)(p_t)\n\\end{pmatrix}\n.\n\\end{align*}\n\nThen we have the following result similar to Theorem \\ref{main-theorem}.\n\n\\begin{thm}\\label{main-theorem-3}\n\t\n\tLet $v_k^{(1)}$ and $v_k^{(2)}$ be two sequences of solutions of (\\ref{equ-flat}) (\\ref{blowup-flat}) with $\\rho_k^{(1)}=\\rho_k=\\rho_k^{(1)}$ and $\\alpha_j\\in\\mathbb{R}^+\\setminus\\mathbb{N}\\,(1\\leq j\\leq m)$. If $L_{\\Omega}(\\mathbf{p})\\neq0$ and $D_{\\Omega}(\\mathbf{p})=0$, then $v_k^{(1)}=v_k^{(2)}$ for $k$ large enough.\n\t\n\\end{thm}\n\nIf the set of blowup points is a mixture of non-quantized singular sources and regular points, we also have a uniqueness result. Let\n\\begin{equation*}\nf_{\\Omega}^*(x_{\\tau+1},\\cdots,x_m)=\\sum_{j=\\tau+1}^m\\big[\\log h(x_j)+4\\pi R(x_j,x_j)\\big]+4\\pi \\sum_{l\\neq j}^{\\tau +1,\\cdots,m}G(x_l,x_j),\n\\end{equation*}\nand $D^2f_{\\Omega}^*$ be the Hessian tensor field on $M$. In this case, $(p_{\\tau+1},\\cdots,p_m)$ is a critical point of $f_{\\Omega}^*$. Then, we obtain the following result.\n\n\\begin{thm}\\label{main-theorem-4}\n\t\n\t\n\tLet $v_k^{(1)}$ and $v_k^{(2)}$ be two sequences of solutions of (\\ref{equ-flat}) (\\ref{blowup-flat}) with $\\rho_k^{(1)}=\\rho_k=\\rho_k^{(1)}$ and $0\\leq\\alpha_j<1(1\\leq j\\leq m )$. If $L_{\\Omega}(\\mathbf{p})\\neq0$, $D_{\\Omega}(\\mathbf{p})=0$ and $\\det \\big(D^2f_{\\Omega}^*(p_{\\tau+1},\\cdots,p_m)\\big)\\neq 0$, then $v_k^{(1)}=v_k^{(2)}$ for $k$ large enough.\n\t\n\\end{thm}\n\nWhen we were in the final stage of writing this article, we found that Bartolucci, et, al \\cite{bart-4-2} posted an article on arxiv.org about the same topic. Their theorem is a special case of our results and both works were carried out independently.\n\n\\smallskip\n\nThe organization of this paper is as follows. Section \\ref{preliminary} is dedicated to notations and preliminary sharp estimates for bubbling solutions of equation (\\ref{m-equ}). In section \\ref{difference} we consider the differences between two bubbling sequences and establish many estimates near each blowup point and away from all blowup points. In section \\ref{anal-pohozaev} we derive some Pohozaev-type identities and evaluate each term carefully. These Pohozaev identities play a key role in the proof of the main theorems. Finally the proof of Theorem \\ref{main-theorem} is placed in section \\ref{pf-uni-1} and that of Theorem \\ref{main-theorem-2} can be found in section \\ref{pf-uni-2}. At the end of section \\ref{pf-uni-2}, we list the brief sketch of the proof of Theorems \\ref{main-theorem-3} and \\ref{main-theorem-4} based on well known facts \\cite{ma-wei}.\n\n\n\n\\section{Preliminary Estimates}\\label{preliminary}\n\nSince the proof of the main theorems requires very delicate analysis, in this section we list some established estimates in \\cite{chen-lin-sharp,chen-lin,zhang1,zhang2}.\n\n\n\n\nLet $w_k$ be a sequence of solutions of (\\ref{r-equ}) with $\\rho =\\rho_k$. Suppose that $w_k$ blows up at $m$ points $\\{p_1 \\cdots,p_m\\}$ as we have stated in section one. To describe the bubbling profile of $w_k$ near $p_j$, we set\n\\begin{equation}\\label{n-sol}\n\tu_k=w_k-\\log\\bigg(\\int_M He^{w_k}{\\rm d}\\mu \\bigg)\n\\end{equation}\n and write the equation for $u_k$ as\n\\begin{equation}\\label{n-equ}\n\\Delta_g u_k+\\rho_k(He^{u_k}-1)=0\\quad {\\rm in} \\ \\; M.\n\\end{equation}\nIt is easy to observe from the definition of $u_k$ that $$\\int_{M}He^{u_k}{\\rm d}\\mu=1.$$\n\n\nFrom previous works of Liouville equations ( for example \\cite{chen-lin-sharp} ),\n\\begin{equation}\\label{local-cov}\n u_k-\\bar{u}_k \\ \\to \\sum_{j=1}^m 8\\pi(1+\\alpha_j)G(x,p_j) \\quad {\\rm in} \\ \\; {\\rm C}_{loc}^2(M\\backslash \\{p_1,\\cdots,p_m\\})\n\\end{equation}\nwhere $\\bar{u}_k$ is the average of $u_k$ on $M$:\n$$\\bar{u}_k=\\int_{M}u_k{\\rm d}\\mu.$$\n\nFor the convenience later we fix $r_0>0$ small and $M_j\\subset M, 1\\leq j\\leq m$ such that\n\\begin{equation}\\label{Mj}\nM=\\bigcup_{j=1}^m \\overline{M}_j;\\quad M_j\\cap M_l=\\varnothing,\\ {\\rm if}\\ j\\neq l;\\quad B(p_j,3r_0)\\subset M_j, \\ j=1,\\cdots,m.\n\\end{equation}\nAccording to this definition $M_1=M$, if $m=1$.\n\n\nThen we use $\\lambda_{k,j}$ to denote\n\\begin{equation}\\label{lambda_kj}\n\\lambda_{k,j}=\\left\\{\n\\begin{array}{lcl}\nu_k(p_j) && {\\rm if}\\ \\,\\alpha_j\\neq 0, \\\\\nu_k(p_{k,j}) \\mathrel{\\mathop:}=\\max_{B(p_j,r_0)}u_k && {\\rm if}\\ \\,\\alpha_j= 0.\n\\end{array}\n\\right.\n\\end{equation}\nand let $U_{k,j}$ be a global solution of\n\\begin{equation}\\label{U_kj-equ}\n\\Delta U_{k,j}+\\rho_kh_j(p_{k,j})|x-p_{k,j}|^{2\\alpha_j}e^{U_{k,j}}=0 \\quad {\\rm in} \\ \\; \\mathbb{R}^2\n\\end{equation}\nwith the expression ($U_{k,j}$ is called a standard bubble):\n\\begin{equation}\\label{U_kj}\nU_{k,j}(x)=\\lambda_{k,j}-2\\log\\Big(1+\\frac{\\rho_k h_j(p_{k,j})}{8(1+\\alpha_j)^2}e^{\\lambda_{k,j}}|x-p_{k,j}|^{2(1+\\alpha_j)}\\Big).\n\\end{equation}\n\nIt is well-known \\cite{li-cmp,Bartolucci-Chen-Lin-T} that $u_k$ can be approximated by the standard bubbles $U_{k,j}$ near $p_j$ with $O(1)$ error:\n\\begin{equation}\\label{standard-bubble}\n \\big|u_k(x)-U_{k,j}(x)\\big| \\leq C, \\quad x\\in B(p_j,r_0).\n\\end{equation}\n\n As a consequence,\n\\begin{equation}\\label{lambda-ij}\n|\\lambda_{k,i}-\\lambda_{k,j}|\\leq C, \\quad 1\\leq i,j \\leq m.\n\\end{equation}\nfor some $C$ independent of $k$.\nFurthermore, it is established in \\cite{bart-taran-mass} that $\\rho_*=\\lim_{k\\to +\\infty}\\rho_k$.\n\n\\medskip\n\n\nLater, sharper estimates were obtained in \\cite{zhang2,chen-lin} for $1\\leq j \\leq \\tau$ and in \\cite{chen-lin-sharp,zhang1,gluck} for $\\tau+1\\leq j\\leq m$. In order to apply those estimates, we might consider the equation in terms of the flat metric and introduce the following notations.\n\n\\medskip\n\nIn $B(p_j,r_0)$, the flat metric is ${\\rm d}s^2=e^{\\phi_j}\\big(({\\rm d}x_1)^2+({\\rm d}x_2)^2\\big)$ with $\\phi_j$ satisfying\n\\begin{equation}\\label{phi-equ}\n\\left\\{\\begin{array}{lcl}\n\\Delta \\phi_j+2Ke^{\\phi_j}=0, && {\\rm in} \\ \\; B(p_j,r_0),\n\\\\\n\\phi_j(0)=|\\nabla \\phi_j(0)|=0, && \\quad\n\\end{array}\n\\right.\n\\end{equation}\nwhere $0$ is the coordinate of $p_j$, $\\Delta =\\sum _{i=1}^{2}\\frac{\\partial^2}{\\partial x_i^2}$. In this local coordinate, equation (\\ref{n-equ}) is equivalent to\n\\begin{equation}\\label{flat-equ}\n\\Delta u_k+\\rho_ke^{\\phi_j}(He^{u_k}-1)=0\\quad {\\rm in} \\ \\; B(p_j,r_0).\n\\end{equation}\nIf we denote $\\tilde{h}_j=h_je^{\\phi_j}$, then (\\ref{flat-equ}) can be written as follows:\n\\begin{equation}\\label{flat-equ-2}\n\\Delta u_k+\\rho_k\\tilde{h}_j|x-p_j|^{2\\alpha_j}e^{u_k}-\\rho_ke^{\\phi_j}=0\\quad {\\rm in} \\ \\; B(p_j,r_0).\n\\end{equation}\n\nTo state the more refined asymptotic analysis we introduce the following notations:\n\\begin{equation}\\label{loc-mass}\n\\rho_{k,j}=\\int_{B(p_{k,j},r_0)}\\rho_kHe^{u_k}{\\rm d}\\mu,\\quad 1\\leq j\\leq m,\n\\end{equation}\n\\begin{equation}\\label{sigma1}\n\\sigma_k(x)=u_k(x)-\\bar{u}_k-\\sum_{j=1}^m\\rho_{k,j} G(x,p_{k,j}), \\quad x\\in M\\backslash \\bigcup_{j=1}^mB(p_{k,j},\\frac{r_0}{2}),\n\\end{equation}\n\\begin{equation}\\label{G_kj}\nG_{k,j}(x)=\\rho_{k,j}R(x,p_{k,j})+\\sum_{l\\neq j}^{1,\\cdots,m}\\rho_{k,l}G(x,p_{k,l}),\\quad x\\in B(p_{k,j},r_0),\n\\end{equation}\nwhere $R(x,p_{k,j})$ is the regular part of $G(x,p_{k,j})$. Finally\nfor $x\\in B(p_{k,j},r_0)$, set\n\\begin{equation*}\n\\tilde{u}_{k,j}(x)=u_k(x)-\\big(G_{k,j}(x)-G_{k,j}(p_{k,j})\\big),\n\\end{equation*}\n\\begin{equation}\\label{eta1}\n\\eta_{k,j}(x)=\\tilde{u}_{k,j}(x)-U_{k,j}(x).\n\\end{equation}\n\n\\subsection{Sharper estimates}\n\\quad\n\\medskip\n\nIf $\\alpha_j\\in \\mathbb{R}^+\\setminus\\mathbb{N}$, in order to obtain the refined estimates of the bubbling solutions, the second author considered the harmonic function $\\psi_{k,j}$ in \\cite{zhang2}, which satisfies\n\\begin{equation}\\label{psi-equ}\n\\left\\{\\begin{array}{lcl}\n\\Delta \\psi_{k,j}=0 && {\\rm in} \\ \\; B(p_{k,j},r_0),\n\\\\\n\\psi_{k,j} =\\tilde{u}_{k,j}-\\frac{1}{2\\pi r_0}\\int_{\\partial B(p_{k,j},r_0)}\\tilde{u}_{k,j} {\\rm d}s && {\\rm on} \\ \\; \\partial B(p_{k,j},r_0).\n\\end{array}\n\\right.\n\\end{equation}\n\nWith the help of $\\psi_{k,j}$, Zhang and Chen-Lin proved the following sharp estimate in \\cite{zhang2}.\n\\begin{thmA}\\label{Theorem zhang}\\cite{zhang2,chen-lin}\n\t\n\tFor $x\\in B(p_{k,j},r_0)$, it holds that\n\t\\begin{equation}\\label{zhang}\n\t\\begin{split}\n\t\\eta_{k,j}(x)=&\\psi_{k,j}(x)-\\frac{2(1+\\alpha_j)}{\\alpha_j}\\frac{\\langle a,x-p_{k,j}\\rangle}{1+\\frac{\\rho_k h_j(p_{(k,j)})}{8{(1+\\alpha_j)^2}}e^{\\lambda_{k,j}}|x-p_{k,j}|^{2(1+\\alpha_j)}}\\\\\n\t& +d_j\\log \\big(2+e^{\\frac{\\lambda_{k,j}}{2(1+\\alpha_j)}}|x-p_{k,j}|\\big)e^{-\\frac{\\lambda_{k,j}}{1+\\alpha_j}}+O(e^{-\\frac{\\lambda_{k,j}}{1+\\alpha_j}}),\n\t\\end{split}\n\t\\end{equation}\n\twhere $a=\\nabla(\\log h_j+G_{k,j})(p_{k,j}) \\in \\mathbb{R}^2$ and\n\t\\begin{equation*}\n\t\td_j=\\frac{\\pi}{(1+\\alpha_j)\\sin \\frac{\\pi}{1+\\alpha_j}}\\Big(\\frac{8(1+\\alpha_j)^2}{\\rho_kh_j(p_{k,j})}\\Big)^{\\frac{1}{1+\\alpha_j}}\\big[\\Delta\\log h(p_j)+\\rho_*-N^*-2K(p_j)\\big].\n\t\\end{equation*}\n\\end{thmA}\n\nIn \\cite{chen-lin},the following estimates for $\\psi_{k,j}$ and $\\sigma_k$ are established:\n\\begin{thmA}\\label{Theorem chen-lin1}\\cite{chen-lin}\n\t\\begin{equation}\\label{psi-est}\n\t|\\psi_{k,j}(x)|=O(e^{-\\frac{\\lambda_{k,j}}{1+\\alpha_1}}),\\quad x\\in B(p_{k,j},r_0).\n\t\\end{equation}\n\t\\begin{equation}\\label{sigma2}\n\t|\\sigma_k(x)|+|\\nabla\\sigma_k(x)|=O(e^{-\\frac{\\lambda_{k,1}}{1+\\alpha_1}}),\\quad x\\in M\\backslash \\big(\\bigcup_{j=1}^mB(p_{k,j},\\frac{r_0}{2})\\big).\n\t\\end{equation}\n\\end{thmA}\n\nThen, by Theorem \\ref{Theorem zhang} and Theorem \\ref{Theorem chen-lin1}, we have\n\\begin{equation}\\label{eta2}\n|\\eta_{k,j}(x)|=O(e^{-\\frac{\\lambda_{k,j}}{2(1+\\alpha_j)}}+e^{-\\frac{\\lambda_{k,j}}{1+\\alpha_1}}),\\quad x\\in B(p_{k,j},r_0),\\quad 1\\leq j\\leq \\tau.\n\\end{equation}\nFor the case $\\tau+1\\leq j \\leq m $, the estimate for $\\eta_{k,j}$, established in \\cite{chen-lin-sharp}\\cite{zhang1}\\cite{gluck}, is\n\\begin{equation}\\label{eta3}\n|\\eta_{k,j}(x)|=O(\\lambda_{k,j}e^{-\\lambda_{k,j}}),\\quad x\\in B(p_{k,j},r_0),\\quad \\tau+1\\leq j\\leq m.\n\\end{equation}\n\nMoreover, according to the proof of Theorem 3.5 in \\cite{chen-lin}, the following estimate holds:\n\\begin{equation}\\label{uk-ave-1}\n\\bar{u}_k+\\lambda_{k,j}+2\\log\\dfrac{\\rho_kh_j(p_{k,j})}{8(1+\\alpha_j)^2}+G_{k,j}(p_{k,j})+\\frac{d_j}{2(1+\\alpha_j)}\\lambda_{k,j} e^{-\\frac{\\lambda_{k,j}}{1+\\alpha_1}}=O(e^{-\\frac{\\lambda_{k,j}}{1+\\alpha_1}}).\n\\end{equation}\nAs a consequence, we have\n\\begin{equation}\\label{uk-ave-2}\n\\lambda_{k,j}-\\lambda_{k,1}=2\\log\\dfrac{(1+\\alpha_j)^2h_1(p_{k,1})}{(1+\\alpha_1)^2 h_j(p_{k,j})}+G_{k,1}(p_{k,1})-G_{k,j}(p_{k,j})+O(e^{-\\frac{\\lambda_{k,1}}{2(1+\\alpha_1)}}).\n\\end{equation}\n\nFor the difference between $\\rho_k$ and $\\rho_*$, $\\rho_k$ and $8\\pi(1+\\alpha_j)$, the following estimates also have been proved in \\cite{chen-lin,chen-lin-sharp}.\n\\begin{thmA}\\label{Theorem chen-lin2}\\cite{chen-lin,chen-lin-sharp}\t\n\t\\begin{align}\n\t&\\rho_{k,j}-8\\pi(1+\\alpha_j)=2\\pi d_je^{-\\frac{\\lambda_{k,j}}{1+\\alpha_j}}+O\\big(e^{-\\frac{1+\\gamma}{1+\\alpha_1}\\lambda_{k,1}}\\big), && 1\\leq j\\leq \\tau, \\label{rho-kj-1}\n\t\\\\\n\t&\\rho_{k,j}-8\\pi=O\\big(\\lambda_{k,j}e^{-\\lambda_{k,j}}\\big), &&\\tau+1\\leq j\\leq m, \\label{rho-kj-2}\n\t\\\\\n\t&\\rho_k-\\rho_*=L^*e^{-\\frac{\\lambda_{k,1}}{1+\\alpha_1}}+O\\big(e^{-\\frac{1+\\gamma}{1+\\alpha_1}\\lambda_{k,1}}\\big),&& \\label{rho-k}\n\t\\end{align}\n\twith fixed $\\gamma\\in (0,\\min({\\alpha_1,\\frac{1}{2}}))$ small and\n\t$$L^*=\\dfrac{2\\pi^2}{(1+\\alpha_1)\\sin\\frac{\\pi}{1+\\alpha_1}}e^{-\\frac{G_1^*(p_{1})}{1+\\alpha_1}}\\Big(\\frac{8(1+\\alpha_1)^2}{\\rho_* h_1(p_1)^2}\\Big)^{\\frac{1}{1+\\alpha_1}}L(\\mathbf{p}).$$\n\\end{thmA}\n\n\\smallskip\n\nIf $\\tau0 $ is not an integer, $ \\varphi $ is a $C^2$-function that satisfies\n\t\\begin{equation*}\n\t\t\\left\\{\\begin{array}{ll}\n\t\t\\Delta \\varphi+|x|^{2\\alpha}e^{U_\\alpha}\\varphi=0\\quad & {\\rm in} \\ \\; \\mathbb{R}^2,\n\t\t\\\\\n\t\t|\\varphi| \\leq (1+|x|)^{\\kappa} \\quad & {\\rm in} \\ \\;\\mathbb{R}^2,\n\t\t\\end{array}\n\t\t\\right.\n\t\\end{equation*}\n\twhere $ U_{\\alpha}(x)=\\log\\frac{8(1+\\alpha)^2}{(1+|x|^{2(1+\\alpha)})^2} $ and $\\kappa\\in(0,1)$. Then there exists some constant $b_0$ such that\n\t\\begin{equation*}\n\t\\varphi(x)= b_0\\frac{1-|x|^{2(1+\\alpha)}}{1+|x|^{2(1+\\alpha)}}.\n\t\\end{equation*}\n\\end{lemA}\n\n\nFor $\\alpha=0$, Chen-Lin proved the following lemma in \\cite{chen-lin-sharp}.\n\\begin{lemA}\\label{linear-lem-2}\n\tLet $ \\varphi $ be a $ C^2 $-function of\n\t\\begin{equation*}\n\t\\left\\{\\begin{array}{ll}\n\t\\Delta \\varphi+e^U\\varphi=0\\quad & {\\rm in} \\ \\;\\mathbb{R}^2,\n\t\\\\\n\t|\\varphi| \\leq c\\big(1+|x|\\big)^{\\kappa} \\quad & {\\rm in} \\ \\;\\mathbb{R}^2,\n\t\\end{array}\n\t\\right.\n\t\\end{equation*}\n\twhere $ U(x)=\\log\\frac{8}{(1+|x|^2)^2} $ and $ \\kappa \\in[0,1) $. Then there exist constants $b_0$, $b_1$, $b_2$ such that\n\t\\begin{equation*}\n\t\t \\varphi= b_0\\varphi_0+b_1\\varphi_1+b_2\\varphi_2,\n\t\\end{equation*}\n\twhere\n\t\\begin{equation*}\n\t\\varphi_0(x)= \\frac{1-|x|^2}{1+|x|^2},\\quad \\varphi_1(x)= \\frac{x_1}{1+|x|^2},\\quad \\varphi_2(x)= \\frac{x_2}{1+|x|^2}.\n\t\\end{equation*}\n\t\n\\end{lemA}\n\n\\section{The difference between $ u_k^{(1)} $ and $ u_k^{(2)} $}\\label{difference}\n\nThe way we prove the main theorems is by contradiction. So we assume that $ u_k^{(1)} $ and $ u_k^{(2)} $\nare two different sequences of solutions to (\\ref{r-equ}) with $\\rho_k^{(1)}=\\rho_k=\\rho_k^{(2)}$, and common blowup points located at $p_1,\\cdots,p_m$. For $i=1,2$, we use\nthe following notations\n$$\\lambda_{k,j}^{(i)}, u_{k,j}^{(i)}, v_{k,j}^{(i)}, \\rho_{k,j}^{(i)}, \\bar{u}_k^{(i)}, U_{k,j}^{(i)}, G_{k,j}^{{(i)}}, \\psi_{k,j}^{(i)}, \\eta_{k,j}^{(i)}, \\epsilon_{k,j}^{(i)}, \\sigma_k^{(i)}, p_{j}^{(i)}, $$\nwith obvious interpretations in the context.\nFinally the following three functions are defined by the difference of $u_1^k$ and $u_2^k$:\n\\begin{align}\n&\t\\varsigma_k(x)=\\dfrac{u_k^{(1)}(x)-u_k^{(2)}(x)}{\\parallel u_k^{(1)}-u_k^{(2)}\\parallel_{L^{\\infty}(M)}}, \\label{varsigma}\n\\\\\n&f_k(x)=\\rho_k H(x)\\frac{e^{u_k^{(1)}(x)}-e^{u_k^{(2)}(x)}}{\\parallel u_k^{(1)}-u_k^{(2)}\\parallel_{L^{\\infty}(M)}}, \\label{f}\n\\\\\n&c_k(x)=\\dfrac{e^{u_k^{(1)}(x)}-e^{u_k^{(2)}(x)}}{u_k^{(1)}(x)-u_k^{(2)}(x)}. \\label{c}\n\\end{align}\nClearly $\\varsigma_k$ satisfies\n\\begin{equation}\\label{sigma-equ}\n\\Delta_g\\varsigma_k(x)+f_k(x)=\\Delta_g\\varsigma_k(x)+\\rho_kH(x)c_k(x)\\varsigma_k(x)=0,\\quad x\\in M.\n\\end{equation}\n\nAs the first step of our proof, we give an initial estimate of $ \\parallel u_k^{(1)}-u_k^{(2)}\\parallel_{L^{\\infty}(M)}$ using $L({\\bf p})\\neq 0$:\n\\begin{lem}\\label{est1}\nUnder the assumption of $L(\\mathbf{p})\\neq 0$, we have\n\\begin{equation}\\label{u-est1}\n\\parallel u_k^{(1)}-u_k^{(2)}\\parallel_{L^{\\infty}(M)}=O(e^{-\\frac{\\gamma}{1+\\alpha_1}\\lambda_{k,1}^{(1)}}).\n\\end{equation}\n\n\\end{lem}\n\n\\begin{proof}[\\textbf{Proof}]\n\t\t\n\\textbf{Step 1.}\nFor $x\\in B(p_{k,j},r_0), 1\\leq j\\leq m$, by (\\ref{eta1}) (\\ref{eta2}) (\\ref{loc-mass}) (\\ref{G_kj}) and Theorem \\ref{Theorem chen-lin2}, we have\n\\begin{align*}\n&u_k^{(1)}(x)-u_k^{(2)}(x)\\\\\n=&\\,U_{k,j}^{(1)}(x)-U_{k,j}^{(2)}(x)+\\eta_{k,j}^{(1)}(x)-\\eta_{k,j}^{(2)}(x)+G_{k,j}^{(1)}(x)-G_{k,j}^{(2)}(x)\\\\\n&\\,+G_{k,j}^{(1)}(p_{k,j}^{(1)})-G_{k,j}^{(2)}(p_{k,j}^{(2)})\\\\\n=&\\,\\lambda_{k,j}^{(1)}-\\lambda_{k,j}^{(2)}-2\\log\\Big(1+\\frac{\\rho_k h_j(p_{k,j}^{(1)})}{8(1+\\alpha_j)^2}e^{\\lambda_{k,j}^{(1)}}\\big|x-p_{k,j}^{(1)}\\big|^{2(1+\\alpha_j)}\\Big)\\\\\n&\\,+2\\log\\Big(1+\\frac{\\rho_k h_j(p_{k,j}^{(2)})}{8(1+\\alpha_j)^2}e^{\\lambda_{k,j}^{(2)}}\\big|x-p_{k,j}^{(2)}\\big|^{2(1+\\alpha_j)}\\Big)+O\\Big(\\sum_{i=1}^{2}e^{-\\frac{\\lambda_{k,1}^{(i)}}{2(1+\\alpha_1)}}\\Big).\n\\end{align*}\nTheorem \\ref{Theorem chen-lin2} and $L(\\mathbf{p})\\neq 0$ give rise to\n\\begin{equation*}\ne^{-\\frac{1}{1+\\alpha_1}(\\lambda_{k,1}^{(1)}-\\lambda_{k,1}^{(2)})}=1+O\\Big(\\sum_{i=1}^{2}e^{-\\frac{\\gamma}{1+\\alpha_1}\\lambda_{k,1}^{(i)}}\\Big),\n\\end{equation*}\nwhich immediately implies\n\\begin{equation}\\label{lam-est-1}\n\\lambda_{k,1}^{(1)}-\\lambda_{k,1}^{(2)}=O\\Big(\\sum_{i=1}^{2}e^{-\\frac{\\gamma}{1+\\alpha_1}\\lambda_{k,1}^{(i)}}\\Big).\n\\end{equation}\nThen by (\\ref{lam-est-1}) and (\\ref{uk-ave-2}), what holds for one point is also true at other blowup points:\n\\begin{equation}\\label{lam-est-2}\n\\lambda_{k,j}^{(1)}-\\lambda_{k,j}^{(2)}=O\\Big(\\sum_{i=1}^{2}e^{-\\frac{\\gamma}{1+\\alpha_1}\\lambda_{k,1}^{(i)}}\\Big),\\quad 1\\leq j\\leq m.\n\\end{equation}\n\nOn the other hand, using (\\ref{p_kj-location}) in direct computation, we have,\n\\begin{align*}\n&\\log\\Big(1+\\frac{\\rho_k h_j(p_{k,j}^{(1)})}{8(1+\\alpha_j)^2}e^{\\lambda_{k,j}^{(1)}}\\big|x-p_{k,j}^{(1)}\\big|^{2(1+\\alpha_j)}\\Big)-\\log\\Big(1+\\frac{\\rho_k h_j(p_{k,j}^{(2)})}{8(1+\\alpha_j)^2}e^{\\lambda_{k,j}^{(2)}}\\big|x-p_{k,j}^{(2)}\\big|^{2(1+\\alpha_j)}\\Big)\\\\\n&=O(\\lambda_{k,j}^{(1)}-\\lambda_{k,j}^{(2)})\n\\end{align*}\nThus $u_k^{(1)}$ and $u_k^{(2)}$ are close in the interior of the ball $B(p_{k,j}^{(1)},r_0)$:\n\\begin{equation}\\label{est-1}\n\\parallel u_k^{(1)}-u_k^{(2)}\\parallel_{L^{\\infty}(B(p_{k,j}^{(1)},r_0))}=O\\Big(\\sum_{i=1}^{2}e^{-\\frac{\\gamma}{1+\\alpha_1}\\lambda_{k,1}^{(i)}}\\Big)=O(e^{-\\frac{\\gamma}{1+\\alpha_1}\\lambda_{k,1}^{(1)}}).\n\\end{equation}\n\n\n\n\\textbf{Step 2.}\nFor $x\\in M\\backslash \\bigcup_{j=1}^mB(p_{k,j}^{(1)},r_0)$, we first use the Green's representation formula to write $u_k^{(1)}-u_k^{(2)}-\\big(\\bar{u}_k^{(1)}-\\bar{u}_k^{(2)}\\big)$ in three parts:\n\n\\begin{equation}\n\\begin{split}\n & u_k^{(1)}-u_k^{(2)}-\\big(\\bar{u}_k^{(1)}-\\bar{u}_k^{(2)}\\big) \\notag\\\\\n=& \\int_{M} G(y,x)\\rho_kH(y)(e^{u_k^{(1)}(y)}-e^{u_k^{(2)}(y)}){\\rm d}\\mu(y) \\\\\n=& \\sum_{j=1}^{m}\\int_{B(p_{k,j}^{(1)},\\frac{r_0}{2})} \\big(G(y,x)-G(p_{k,j}^{(1)},x)\\big)\\rho_kH(y)(e^{u_k^{(1)}(y)}-e^{u_k^{(2)}(y)}){\\rm d}\\mu(y) \\\\\n&+ \\sum_{j=1}^{m}G(p_{k,j}^{(1)},x)\\int_{B(p_{k,j}^{(1)},\\frac{r_0}{2})}\\rho_kH(y)(e^{u_k^{(1)}(y)}-e^{u_k^{(2)}(y)}){\\rm d}\\mu(y) \\\\\n&+ \\int_{M\\backslash \\bigcup_{j=1}^mB(p_{k,j}^{(1)},\\frac{r_0}{2})} G(y,x)\\rho_kH(y)(e^{u_k^{(1)}(y)}-e^{u_k^{(2)}(y)}){\\rm d}\\mu(y) \\\\\n=&\\mathrel{\\mathop:}I_1+I_2+I_3.\n\\end{split}\n\\end{equation}\nBefore we evaluate each one of them we recall a few facts: First\n\\begin{equation*}\n\tp_{k,j}^{(1)}-p_{k,j}^{(2)}=\\left\\{\\begin{array}{ll}0, \\quad \\mbox{for}\\ 1\\le j\\le \\tau, \\\\\n\tO(\\sum_{i=1}^2\\lambda_{k,j}^{(i)}e^{-\\lambda_{k,j}^{(i)}}) \\quad \\mbox{if}\\ j>\\tau \\ \\mbox{ (see (\\ref{p_kj-location}))}.\n\t\\end{array}\n\t\\right.\n\\end{equation*}\nNext for $x\\in M\\backslash \\bigcup_{j=1}^mB(p_{k,j}^{(1)},r_0)$, $ y\\in B(p_{k,j}^{(1)},\\frac{r_0}{2}) $,\n\\begin{equation*}\n\tG(y,x)-G(p_{k,j}^{(1)},x)=\\langle\\partial_yG(y,x)\\big|_{y-p_{k,j}^{(1)}},y-p_{k,j}^{(1)}\\rangle+O(|y-p_{k,j}^{(1)}|^2)\n\\end{equation*}\nThen using symmetry, scaling, and the closeness between $u_k^{(i)}$ with standard bubbles, we have\n\n\\begin{align*}\n&I_1\n=\\sum_{j=1}^{m}\\sum_{i=1}^2\\int_{B(p_{k,j}^{(i)},\\frac{r_0}{2})}\\frac{\\langle\\partial_yG(y,x)\\big|_{y=p_{k,j}^{(i)}},y-p_{k,j}^{(i)}\\rangle\\rho_k\\tilde{h}_j(y)|y-p_{k,j}^{(i)}|^{2\\alpha_j}}{\\big(1+\\frac{\\rho_kh_j(p_{k,j}^{(i)})}{8(1+\\alpha_j)^2}e^{\\lambda_{k,j}^{(i)}}|y-p_{k,j}^{(1)}|^{2(1+\\alpha_j)}\\big)^2}\\\\\n&\\times \\Big(1+O(|y-p_{k,j}^{(i)}|)+O(e^{-\\frac{\\lambda_{k,j}^{(i)}}{2(1+\\alpha_j)}})+O(e^{-\\frac{\\lambda_{k,j}^{(i)}}{1+\\alpha_1}})\\Big){\\rm d}y+\n O\\Big(\\sum_{i=1}^{2}e^{-\\frac{\\lambda_{k,1}^{(i)}}{1+\\alpha_1}}\\Big),\\\\\n &=O\\Big(\\sum_{i=1}^{2}e^{-\\frac{\\lambda_{k,1}^{(i)}}{1+\\alpha_1}}\\Big).\n \\end{align*}\n\nThe closeness between $\\rho_{k,j}^{(1)}$ and $\\rho_{k,j}^{(2)}$ leads to the smallness of $I_2$ (see\n (\\ref{loc-mass}) (\\ref{rho-kj-1}) and (\\ref{rho-kj-2})):\n\\begin{equation}\\label{I2}\nI_2=\\sum_{j=1}^{m}G(p_{k,j}^{(1)},x)(\\rho_{k,j}^{(1)}-\\rho_{k,j}^{(2)})=O\\Big(\\sum_{i=1}^{2}e^{-\\frac{\\lambda_{k,1}^{(i)}}{1+\\alpha_1}}\\Big).\n\\end{equation}\nFor $I_3$, the magnitude of $u_k^{(i)}$ outside the bubbling area determines the smallness of $I_3$:\n$$\nI_3=\\rho_k\\int_{M\\backslash \\bigcup_{j=1}^mB(p_{k,j}^{(1)},\\frac{r_0}{2})} G(y,x)H(y)(e^{u_k^{(1)}(y)}-e^{u_k^{(2)}(y)}){\\rm d}\\mu(y)=O\\Big(\\sum_{i=1}^{2}e^{-\\lambda_{k,1}^{(i)}}\\Big).\n$$\nTherefore\n\\begin{equation}\\label{step2-2}\n u_k^{(1)}-u_k^{(2)}-\\big(\\bar{u}_k^{(1)}-\\bar{u}_k^{(2)}\\big)=O\\Big(\\sum_{i=1}^{2}e^{-\\frac{\\lambda_{k,1}^{(i)}}{1+\\alpha_1}}\\Big)\\quad \\mbox{in}\\quad\n M\\backslash \\bigcup_{j=1}^mB(p_{k,j}^{(1)},r_0).\n\\end{equation}\nTo eliminate the averages in (\\ref{step2-2}) we take advantage of (\\ref{uk-ave-1}) and (\\ref{lam-est-1}):\n\\begin{equation}\\label{step2-3}\n\\bar{u}_k^{(1)}-\\bar{u}_k^{(2)}=-(\\lambda_{k,j}^{(1)}-\\lambda_{k,j}^{(2)})+O\\Big(\\sum_{i=1}^{2}\\lambda_{k,j}^{(i)} e^{-\\frac{\\lambda_{k,1}^{(i)}}{1+\\alpha_1}}\\Big)=O\\Big(\\sum_{i=1}^{2}e^{-\\frac{\\gamma}{1+\\alpha_1}\\lambda_{k,1}^{(i)}}\\Big),\n\\end{equation}\nUsing (\\ref{step2-3}) in (\\ref{step2-2}) we arrive at\n\\begin{equation}\\label{step2-4}\nu_k^{(1)}(x)-u_k^{(2)}(x)=O\\Big(\\sum_{i=1}^{2}e^{-\\frac{\\gamma}{1+\\alpha_1}\\lambda_{k,1}^{(i)}}\\Big)=O(e^{-\\frac{\\gamma}{1+\\alpha_1}\\lambda_{k,1}^{(1)}}).\n\\end{equation}\nfor all $x\\in M\\backslash \\bigcup_{j=1}^mB(p_{k,j}^{(1)},r_0)$.\n Lemma \\ref{est1} is established.\n\n\\end{proof}\n\n\nAs an immediate application, Lemma \\ref{est1} gives ( see (\\ref{c}) )\n\\begin{equation}\\label{c-est}\n\tc_k(x)=e^{u_k^{(1)}(x)}\\big(1+O(\\parallel u_k^{(1)}-u_k^{(2)}\\parallel_{L^{\\infty}(M)})\\big)=e^{u_k^{(1)}(x)}\\big(1+O(e^{-\\frac{\\gamma}{1+\\alpha_1}\\lambda_{k,1}^{(1)}})\\big).\n\\end{equation}\n\nTo simply the notations, we set\n\\begin{equation}\n\t\\epsilon_{k,j}=\\bigg(\\frac{\\rho_kh_j(p_{k,j}^{(1)})}{8(1+\\alpha_j)^2}\\bigg)^{-\\frac{1}{2(1+\\alpha_j)}}e^{-\\frac{\\lambda_{k,j}^{(1)}}{2(1+\\alpha_j)}}.\n\\end{equation}\nand\n\\begin{equation}\\label{varsigma-kj}\n\\varsigma_{k,j}(z)=\\varsigma_k(\\epsilon_{k,j}z+p_{k,j}^{(1)}),\\quad |z|<\\frac{r_0}{\\epsilon_{k,j}},\\quad 1\\leq j\\leq m,\n\\end{equation}\nwhich satisfies\n\\begin{equation}\\label{sigma-t-equ}\n\\Delta\\varsigma_{k,j}+\\frac{8(1+\\alpha_j)^2}{\\rho_kh_j(p_{k,j}^{(1)})}\\rho_k\\tilde{h}_j(\\epsilon_{k,j}z+p_{k,j}^{(1)})e^{-\\lambda_{k,j}^{(1)}}|z|^{2\\alpha_j}c_k(\\epsilon_{k,j}z+p_{k,j}^{(1)})\\varsigma_{k,j}=0.\n\\end{equation}\nfor $ |z|-1$.\n\\end{rem}\n\n\n\\begin{lem}\\label{lem-PI1-left}\n\nFor all $1\\leq j\\leq m$,\n\t\\begin{equation}\\label{PI-1-l}\n\t{\\rm (LHS)}\\ {\\rm of}\\ (\\ref{PI-1})=-4(1+\\alpha_j)A_{k,j}+O(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}}\\sum_{l=1}^{m}|A_{k,l}|)+o(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}}).\n\t\\end{equation}\n\\end{lem}\n\n\\begin{proof}[\\textbf{Proof}]\n\t\n\tFrom (\\ref{Dv-kj-est}) and (\\ref{Dsigma_k-1}), we find that\n\t\\begin{align*}\n\t&{\\rm (LHS)}\\ {\\rm of}\\ (\\ref{PI-1})\\\\\n\t=&\\,4(1+\\alpha_j)\\int_{\\partial B(p_{k,j}^{(1)},r)}\\langle\\nu,D\\varsigma_k\\rangle{\\rm d}\\sigma+O(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}}\\parallel D\\varsigma_k\\parallel_{L^{\\infty}(\\partial B(p_{k,j}^{(1)},r))})\\\\\n\t=&\\,4(1+\\alpha_j)\\int_{\\partial B(p_{k,j}^{(1)},r)}\\langle\\nu,D\\varsigma_k\\rangle{\\rm d}\\sigma+O(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}}\\sum_{l=1}^{m}|A_{k,l}|)+o(e^{-\\frac{3}{2(1+\\alpha_1)}\\lambda_{k,j}^{(1)}}).\n\t\\end{align*}\n\tFor $x\\in\\partial B(p_{k,j}^{(1)},r)$ we use the Green's representation formula to estimate $\\varsigma_k(x)$:\n\t\\begin{align*}\n\t&\\varsigma_k(x)-\\bar{\\varsigma}_k=\\int_{M}G(y,x)f_k(y){\\rm d}\\mu(y)\\\\ =&\\sum_{l=1}^mA_{k,l}G(p_{k,l}^{(1)},x)+\\sum_{l=1}^m\\sum_{h=1}^2B_{k,l,h}\\partial_{y_h}G(y,x)\\big|_{y=p_{k,l}^{(1)}}+\\frac{1}{2}\\sum_{l=1}^m\\sum_{h,i=1}^2C_{k,l,h,i}\\partial_{y_hy_i}^2G(y,x)\\big|_{y=p_{k,l}^{(1)}}\\\\\n\t&+O(1)\\sum_{l=1}^m\\int_{M_l}|y-p_{k,j}^{(1)}|^3f_k{\\rm d}\\mu(y) \\quad {\\rm in}\\,\\ C^1\\big(B(p_{k,j}^{(1)},2r_0)\\setminus B(p_{k,j}^{(1)},\\frac{r}{2})\\big),\n\t\\end{align*}\n\twhere\n\t\\begin{align*}\n\t&\tB_{k,l,h}=\\int_{M_l}(y-p_{k,l}^{(1)})_hf_k(y){\\rm d}\\mu(y),\n\t\\\\\n\t& C_{k,l,h,i}=\\int_{M_l}(y-p_{k,l}^{(1)})_h(y-p_{k,l}^{(1)})_if_k(y){\\rm d}\\mu(y).\n\t\\end{align*}\n\t\n\tIt is easy to see that the last term is rather minor:\n\t\\begin{align*}\n\t&\\sum_{l=1}^m\\int_{M_l}|y-p_{k,j}^{(1)}|^3f_k(y){\\rm d}\\mu(y) \\\\ =&\\sum_{l=1}^m\\int_{B(p_{k,l}^{(1)},r)}\\frac{e^{\\lambda_{k,l}^{(1)}}|y-p_{k,l}^{(1)}|^{2\\alpha_l+3}}{\\big(1+e^{\\lambda_{k,l}^{(1)}}|y-p_{k,l}^{(1)}|^{2(1+\\alpha_l)}\\big)^2}{\\rm d}y +O(e^{-\\lambda_{k,1}^{(1)}}) \\\\\n\t=&\\sum_{l=1}^mO(e^{-\\frac{3}{2(1+\\alpha_l)}\\lambda_{k,l}^{(1)}})\\int_{|z|<\\frac{r}{\\epsilon_{k,l}}}\\frac{|z|^{2\\alpha_l+3}}{(1+|z|^{2(1+\\alpha_l)})^2}{\\rm d}z+O(e^{-\\lambda_{k,1}^{(1)}}) \\\\\n\t=&o(e^{-\\frac{\\lambda_{k,1}^{(1)}}{1+\\alpha_1}}).\n\t\\end{align*}\nSetting\n\t\\begin{align*}\n\t\\bar{G}_k(x)=&\\bar{\\varsigma}_k(x)+\\sum_{j=1}^m A_{k,j}G(p_{k,j}^{(1)},x)+\\sum_{l=1}^m\\sum_{h=1}^2B_{k,l,h}\\partial_{y_h}G(y,x)\\big|_{y=p_{k,l}^{(1)}}\\\\\n\t&+\\frac{1}{2}\\sum_{l=1}^m\\sum_{h,i=1}^2C_{k,l,h,i}\\partial_{y_hy_i}^2G(y,x)\\big|_{y=p_{k,l}^{(1)}},\n\t\\end{align*}\nwe now have\n\t\\begin{equation*}\n\t\t\\nabla\\varsigma_k(x)-\\nabla\\bar{G}_k(x)=o(e^{-\\frac{\\lambda_{k,1}^{(1)}}{1+\\alpha_1}}).\n\t\\end{equation*}\n\tThus\n\t\\begin{align}\\label{pi-l-1}\n\t\\begin{split}\n\t&{\\rm (LHS)}\\ {\\rm of}\\ (\\ref{PI-1})\\\\\n\t=&\\,4(1+\\alpha_j)\\int_{\\partial B(p_{k,j}^{(1)},r)}\\langle\\nu,\\nabla\\bar{G}_k\\rangle{\\rm d}\\sigma+O(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}}\\sum_{l=1}^{m}|A_{k,l}|)+o(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}}).\n\t\\end{split}\n\t\\end{align}\n\tNow we take the global cancellation property into consideration: for any fixed $\\theta\\in(0,r)$,\n\t\\begin{equation}\\label{Gbar-equ}\n\t\\Delta \\bar{G}_k=\\sum_{l=1}^{m}A_{k,l}=\\int_{M}f_k\\,{\\rm d}\\mu=0,\\quad {\\rm in}\\ \\;B(p_{k,j}^{(1)},2r_0)\\setminus B(p_{k,j}^{(1)},\\theta).\n\t\\end{equation}\n\tUsing (\\ref{G-phi-equ}) (\\ref{Gbar-equ}) and (\\ref{divergence}), we have\n\t\\begin{align*}\n\t0=&\\int_{B_r\\setminus B_{\\theta}} \\Big\\{\\Delta\\bar{G}_k\\big\\{\\nabla(\\tilde{G}_k-\\phi_{k,j})\\cdotp (x-p_{k,j}^{(1)})\\big\\}+\\Delta(\\tilde{G}_k-\\phi_{k,j})\\big\\{\\nabla \\bar{G}_k\\cdotp (x-p_{k,j}^{(1)})\\big\\}\\Big\\}{\\rm d}x\\\\\n\t=&-4\\pi(1+\\alpha_j)\\int_{\\partial(B_r\\setminus B_{\\theta})}\\frac{\\partial\\bar{G}_k}{\\partial\\nu} \\,{\\rm d}\\sigma,\n\t\\end{align*}\n\twhere $B(p_{k,j}^{(1)},r)$ and $B(p_{k,j}^{(1)},\\theta)$ are replaced by $B_r$, $B_{\\theta}$ respectively for simplicity. Therefore\n\t\\begin{equation}\\label{pi-l-2}\n\t\\int_{\\partial B_r}\\frac{\\partial\\bar{G}_k}{\\partial\\nu} {\\rm d}\\sigma=\\int_{\\partial B_{\\theta}}\\frac{\\partial\\bar{G}_k}{\\partial\\nu} {\\rm d}\\sigma.\n\t\\end{equation}\n\tFurther direct computation yields\n\t\\begin{align}\\label{pi-l-3}\n\t\\begin{split}\n\t&\\int_{\\partial B_{\\theta}}\\langle\\nu,\\sum_{l=1}^{m}A_{k,l}\\nabla_xG(p_{k,l}^{(1)},x)\\rangle{\\rm d}\\sigma\\\\\n\t=& -A_{k,j}\\int_{\\partial B_{\\theta}}\\langle\\nu,\\nabla_xG(p_{k,l}^{(1)},x)\\rangle{\\rm d}\\sigma+o_{\\theta}(1)\\\\\n\t=& -A_{k,j}\\int_{\\partial B_{\\theta}}\\langle\\nu,\\nabla_x\\frac{1}{2\\pi}\\log |x-p_{k,l}^{(1)}|\\rangle{\\rm d}\\sigma+o_{\\theta}(1) \\\\\n\t=&-A_{k,j}+o_{\\theta}(1),\n\t\\end{split}\n\t\\end{align}\n\twhere $\\lim_{\\theta\\to 0}o_{\\theta}(1)=0$, and we have used the fact that all the terms related to $l\\neq j$ are minor. Let us observe that\n\t\\begin{equation}\\label{pi-l-4}\n\t\\begin{split}\n\t&\\int_{\\partial B(0,\\theta)}\\langle\\nu,\\nabla_x\\partial_{y_h}\\log|z|\\rangle{\\rm d}\\sigma=-\\int_{\\partial B(0,\\theta)}\\sum_{i=1}^2\\frac{z_i}{|z|}\\frac{\\delta_{ih}|z|^2-2z_iz_h}{z^4}{\\rm d}\\sigma=0, \\\\\n\t&\\int_{\\partial B(0,\\theta)}\\langle\\nu,\\nabla_x\\frac{\\partial^2}{\\partial y_h^2}\\log|z|\\rangle{\\rm d}\\sigma=-\\int_{\\partial B(0,\\theta)}\\big(\\frac{2}{|z|^3}-\\frac{4z_h^2}{|z|^5}\\big){\\rm d}\\sigma=0, \\\\\n\t&\\int_{\\partial B(0,\\theta)}\\langle\\nu,\\nabla_x\\frac{\\partial^2}{\\partial y_hy_i}\\log|z|\\rangle{\\rm d}\\sigma=-\\int_{\\partial B(0,\\theta)}\\big(\\frac{4z_hz_i}{|z|^5}-\\frac{8z_hz_i}{|z|^5}\\big){\\rm d}\\sigma=0,\n\t\\end{split}\n\t\\end{equation}\n\t\n\tObviously, from (\\ref{pi-l-2})$\\sim$(\\ref{pi-l-4}) we can see that\n\t\\begin{equation*}\n\t\t\\int_{\\partial B(p_{k,j}^{(1)},r)}\\langle \\nu,\\nabla\\bar{G}_k\\rangle{\\rm d}\\sigma=-A_{k,j}+o_{\\theta}(1).\n\t\\end{equation*}\n\twhich together with (\\ref{pi-l-1}), concludes the proof of Lemma \\ref{lem-PI1-left}.\n\t\n\\end{proof}\n\n\\begin{lem}\\label{lem-PI1-right}\n\t\\begin{equation}\\label{PI-1-r-1}\n\t\\begin{split}\n\t&{\\rm (RHS)}\\ {\\rm of}\\ (\\ref{PI-1})\\\\\n\t=&-2(1+\\alpha_j)A_{k,j}-\\frac{4\\pi^2\\big[\\Delta \\log h(p_j)+\\rho_*-N^*-2K(p_j)\\big]}{\\big(\\rho_kh_j(p_{k,j}^{(1)})\\big)^{\\frac{1}{1+\\alpha_1}}(1+\\alpha_1)\\sin \\frac{\\pi}{1+\\alpha_1}}b_0e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}}\\\\\n\t&+o(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}}),\\qquad 1\\leq j\\leq t.\n\t\\end{split}\n\t\\end{equation}\n\t\\begin{align}\n\t&{\\rm (RHS)}\\ {\\rm of}\\ (\\ref{PI-1})=-2(1+\\alpha_j)A_{k,j}+o(e^{-\\frac{\\lambda_{k,j}^{(1)}}{2(1+\\alpha_1)}}),\\quad t+1\\leq j\\leq \\tau. \\label{PI-1-r-2} \\\\\n\t&{\\rm (RHS)}\\ {\\rm of}\\ (\\ref{PI-1})=-2(1+\\alpha_j)A_{k,j}+o(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}}),\\quad \\tau+1\\leq j\\leq m. \\label{PI-1-r-3}\n\t\\end{align}\n\\end{lem}\n\n\\begin{proof}[\\textbf{Proof}]\n\t\n\tWe use $K_1,K_2,K_3$ to denote the three terms on the right hand of (\\ref{PI-1}). The first two terms are quite easy to estimate:\n\t\\begin{equation}\\label{K1}\n\tK_1=\\int_{\\partial B(p_{k,j}^{(1)},r)}rf_k\\,{\\rm d}\\sigma=\\int_{\\partial B(p_{k,j}^{(1)},r)}\\rho_k\\tilde{h}_jr^{2\\alpha_j+1}e^{u_k^{(1)}}(\\varsigma_k+o(1)){\\rm d}\\sigma=O(e^{-\\lambda_{k,j}^{(1)}}),\n\t\\end{equation}\n\t\\begin{equation}\\label{K2}\n\tK_2=-2(1+\\alpha_j)A_{k,j}.\n\t\\end{equation}\n\tMore work is needed for\n\t\\begin{equation*}\n\tK_3=-\\int_{B(p_{k,j}^{(1)},r)}f_ke^{\\phi_j}\\langle \\nabla(\\log \\tilde{h}_j+\\phi_{k,j}),x-p_{k,j}^{(1)}\\rangle{\\rm d}x.\n\t\\end{equation*}\n\tFirst we use $\\nabla\\phi_j(p_{k,j}^{(1)})=0$ to write $\\nabla (\\log \\tilde h_j+\\phi_{k,j})(x)$ as\n\t\\begin{equation}\\label{expansion}\n\t\\begin{split}\n\t&\\nabla(\\log \\tilde{h}_j+\\phi_{k,j})(x)\\\\\n\t=&\\nabla(\\log h_j+\\phi_{k,j})(p_{k,j}^{(1)})+\\langle D^2(\\log \\tilde{h}_j+\\phi_{k,j})(p_{k,j}^{(1)}),x-p_{k,j}^{(1)}\\rangle+O(|x-p_{k,j}^{(1)}|^2).\n\t\\end{split}\n\t\\end{equation}\n\tThen we evaluate $K_3$ in three cases:\n\n\t$\\mathbf{Case\\ 1:}$ $1\\leq j\\leq t$ $(\\alpha_j=\\alpha_1)$. The assumption $D(\\mathbf{p})=0$ and (\\ref{phi-kj}) (\\ref{rho-k}) imply\n\t\\begin{equation*}\n\t\t\\nabla(\\log h_j+\\phi_{k,j})(p_{k,j}^{(1)})=O(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}}).\n\t\\end{equation*}\n\tThus, after the scaling $x=\\epsilon_{k,j}z+p_{k,j}^{(1)}$, the first order term can be estimated as follows:\n\t\\begin{equation}\\label{K3-first-1}\n\t\\int_{B(p_{k,j}^{(1)},r)}f_ke^{\\phi_j}\\langle \\nabla(\\log h_j+\\phi_{k,j})(p_{k,j}^{(1)}),x-p_{k,j}^{(1)}\\rangle{\\rm d}x=O(e^{-(\\frac{1}{2(1+\\alpha_j)}+\\frac{1}{1+\\alpha_1})\\lambda_{k,j}^{(1)}}).\n\t\\end{equation}\n\tFor the second order term that contains $D^2(\\log\\tilde{h}_j+\\phi_{k,j})(p_{k,j}^{(1)})$, we have\n\t\\begin{align*}\n\t&\\int_{B(p_{k,j}^{(1)},r)}f_ke^{\\phi_j} \\langle D^2(\\log \\tilde{h}_j+\\phi_{k,j})(p_{k,j}^{(1)})(x-p_{k,j}^{(1)}),x-p_{k,j}^{(1)}\\rangle{\\rm d}x \\\\\n\t=&\\int_{B(p_{k,j}^{(1)},r)}\\frac{\\rho_kh_j(p_{k,j}^{(1)})e^{\\lambda_{k,j}^{(1)}}|x-p_{k,j}^{(1)}|^{2\\alpha_j}}{\\Big(1+\\frac{\\rho_kh_j(p_{k,j}^{(1)})}{8(1+\\alpha_j)^2}e^{\\lambda_{k,j}^{(1)}}|x-p_{k,j}^{(1)}|^{2(1+\\alpha_j)}\\Big)^2}\\big(1+O(|x-p_{k,j}^{(1)}|+\\epsilon_{k,j}+\\epsilon_{k,1}^2)\\big) \\\\\n\t&\\times\\varsigma_k(x)(1+o(1))\\langle D^2(\\log \\tilde{h}_j+\\phi_{k,j})(p_{k,j}^{(1)})(x-p_{k,j}^{(1)}),x-p_{k,j}^{(1)}\\rangle{\\rm d}x \\\\\n\t=&\\epsilon_{k,j}^2\\int_{|z|<\\frac{r}{\\epsilon_{k,j}}}\\frac{8(1+\\alpha_j)^2|z|^{2\\alpha_j}}{(1+|z|^{2(1+\\alpha_j)})^2}\\langle D^2(\\log \\tilde{h}_j+\\phi_{k,j})(p_{k,j}^{(1)})z,z\\rangle\\varsigma_{k,j}(z){\\rm d}z+o(\\epsilon_{k,j}^2).\n\t\\end{align*}\n\tThe expression above can be greatly simplified by this beautiful identity:\n\t\\begin{equation*}\n\t\\int_{0}^{\\infty}\\frac{8(1+\\alpha_j)^2s^{2\\alpha_j+3}}{(1+s^{2(1+\\alpha_j)})^2}\\frac{1-s^{2(1+\\alpha_j)}}{1+s^{2(1+\\alpha_j)}}{\\rm d}s=-\\frac{4\\pi}{(1+\\alpha_j)\\sin\\frac{\\pi}{1+\\alpha_j}}.\n\t\\end{equation*}\n\tConsequently, Lemma \\ref{lem-limit-1} and the two identities above lead to\n\t\\begin{equation}\\label{K3-second-1}\n\t\\begin{split}\n\t&\\int_{B(p_{k,j}^{(1)},r)}f_ke^{\\phi_j} \\langle D^2(\\log \\tilde{h}_j+\\phi_{k,j})(p_{k,j}^{(1)})(x-p_{k,j}^{(1)}),x-p_{k,j}^{(1)}\\rangle{\\rm d}x \\\\\n\t=&\\epsilon_{k,j}^2\\pi\\Delta(\\log \\tilde{h}_j+\\phi_{k,j})(p_{k,j}^{(1)})b_0\\int_{0}^{\\infty}\\frac{8(1+\\alpha_j)^2s^{2\\alpha_j+3}}{(1+s^{2(1+\\alpha_j)})^2}\\frac{1-s^{2(1+\\alpha_j)}}{1+s^{2(1+\\alpha_j)}}{\\rm d}s+o(\\epsilon_{k,j}^2) \\\\\n\t=&-\\frac{4\\pi^2\\big[\\Delta \\log h(p_j)+\\rho_*-N^*-2K(p_j)\\big]}{\\big(\\rho_kh_j(p_{k,j}^{(1)})\\big)^{\\frac{1}{1+\\alpha_1}}(1+\\alpha_1)\\sin \\frac{\\pi}{1+\\alpha_1}}b_0e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}}+o(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}}).\n\t\\end{split}\n\t\\end{equation}\n\tAlso elementary estimate gives\n\t\\begin{equation}\\label{K3-third-1}\n\t\\begin{split}\n\t&\\int_{B(p_{k,j}^{(1)},r)}f_ke^{\\phi_j}\\langle O(|x-p_{k,j}^{(1)}|^2),x-p_{k,j}^{(1)}\\rangle{\\rm d}x \\\\\n\t=&O(1)e^{-\\frac{3}{2(1+\\alpha_j)}\\lambda_{k,j}^{(1)}}\\int_{|z|<\\frac{r}{\\epsilon_{k,j}}}\\frac{|z|^{2\\alpha_j+3}}{(1+|z|^{2(1+\\alpha_j)})^2}{\\rm d}z \\\\\n\t=& O(1)\\big(e^{-\\frac{3}{2(1+\\alpha_j)}\\lambda_{k,j}^{(1)}}+e^{-\\lambda_{k,j}^{(1)}}\\big) =\to(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_j}}).\t\n\t\\end{split}\n\t\\end{equation}\n\tTherefore, we complete the proof of (\\ref{PI-1-r-1}) by using (\\ref{K1}) (\\ref{K2}) and (\\ref{K3-first-1})$\\sim$(\\ref{K3-third-1}). Note that the leading term in the second order term is ignored at this stage, since the requirement of error in the current step is still crude.\n\t\n\\smallskip\n\t$\\mathbf{Case\\ 2:}$ $t+1\\leq j\\leq \\tau$ $(0<\\alpha_j<\\alpha_1)$. For the first term it is easy to see that\n \\begin{equation}\\label{K3-first-2}\n \\int_{B(p_{k,j}^{(1)},r)}f_ke^{\\phi_j} \\langle \\nabla(\\log h_j+\\phi_{k,j})(p_{k,j}^{(1)}),x-p_{k,j}^{(1)}\\rangle{\\rm d}x=o(e^{-\\frac{\\lambda_{k,j}^{(1)}}{2(1+\\alpha_j)}}).\n \\end{equation}\n For the second order term we have\n \\begin{equation}\\label{K3-second-2}\n \\begin{split}\n &\\int_{B(p_{k,j}^{(1)},r)}f_ke^{\\phi_j} \\langle D^2(\\log \\tilde{h}_j+\\phi_{k,j})(p_{k,j}^{(1)})(x-p_{k,j}^{(1)}),x-p_{k,j}^{(1)}\\rangle{\\rm d}x\\\\\n =&O(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_j}})=o(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}}),\n \\end{split}\n \\end{equation}\n where we used the scaling $x=\\epsilon_{k,j}z+p_{k,j}^{(1)}$ and $\\alpha_j<\\alpha_1$. Similar to (\\ref{K3-third-1}), we know\n \\begin{equation}\\label{K3-third-2}\n \\begin{split}\n \\int_{B(p_{k,j}^{(1)},r)}f_ke^{\\phi_j}\\langle O(|x-p_{k,j}^{(1)}|^2),x-p_{k,j}^{(1)}\\rangle{\\rm d}x=o(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_j}}).\t\n \\end{split}\n \\end{equation}\n Therefore (\\ref{PI-1-r-2}) follows from (\\ref{K1}), (\\ref{K2}) and (\\ref{K3-first-2})$\\sim$(\\ref{K3-third-2}).\n\n\\smallskip\n\t$\\mathbf{Case\\ 3:}$ $\\tau+1\\leq j\\leq m$ $(\\alpha_j=0)$. In view of (\\ref{first-deriv-est}), we get\n\t\\begin{equation*}\n\t\t\\nabla(\\log h_j+\\phi_{k,j})(p_{k,j}^{(1)})=\\nabla(\\log h+G_j^*)(p_j)+O(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}})=O(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}}).\n\t\\end{equation*}\n\tThe first order term is rather small:\n\t\\begin{equation}\\label{K3-first-3}\n\t\\int_{B(p_{k,j}^{(1)},r)}f_ke^{\\phi_j} \\langle D(\\log h_j+\\phi_{k,j})(p_{k,j}^{(1)}),x-p_{k,j}^{(1)}\\rangle{\\rm d}x=O(e^{-(\\frac{1}{2}+\\frac{1}{1+\\alpha_1})\\lambda_{k,j}^{(1)}}).\n\t\\end{equation}\n\tFor the second order term we have\n\t\\begin{align*}\n\t&\\int_{B(p_{k,j}^{(1)},r)}f_ke^{\\phi_j} \\langle D^2(\\log \\tilde{h}_j+\\phi_{k,j})(p_{k,j}^{(1)})(x-p_{k,j}^{(1)}),x-p_{k,j}^{(1)}\\rangle{\\rm d}x \\\\\n\t=&\\epsilon_{k,j}^2\\int_{|z|<\\frac{r}{\\epsilon_{k,j}}}\\frac{8}{(1+|z|^{2})^2}\\langle D^2(\\log \\tilde{h}_j+\\phi_{k,j})(p_{k,j}^{(1)})z,z\\rangle \\varsigma_{k,j}(z){\\rm d}z+O(e^{-\\lambda_{k,j}^{(1)}}) \\\\\n\t=&\\epsilon_{k,j}^2\\pi\\Delta(\\log \\tilde{h}_j+\\phi_{k,j})(p_{k,j}^{(1)})b_0\\int_{0}^{\\frac{r}{\\epsilon_{k,j}}}\\frac{8s^3}{(1+s^{2})^2}\\frac{1-s^{2}}{1+s^{2}}{\\rm d}s+O(e^{-\\lambda_{k,j}^{(1)}}),\n\t\\end{align*}\n\twhere we have used Lemma \\ref{lem-limit-1} and symmetry. It is easy to see\n\t\\begin{equation*}\n\t\t\\int_{0}^R\\frac{8s^3}{(1+s^{2})^2}\\frac{1-s^{2}}{1+s^{2}}{\\rm d}s=-4\\Big(\\log (1+R^2)-\\frac{1}{1+R^2}+\\frac{1}{(1+R^2)^2}\\Big),\n\t\\end{equation*}\n\tand\n\t\\begin{equation}\\label{K3-second-3}\n\t\\begin{split}\n\t\\int_{B(p_{k,j}^{(1)},r)}f_ke^{\\phi_j}\\langle D^2(\\log \\tilde{h}_j+\\phi_{k,j})(p_{k,j}^{(1)})(x-p_{k,j}^{(1)}),x-p_{k,j}^{(1)}\\rangle{\\rm d}x=O(\\lambda_{k,j}^{(1)}e^{-\\lambda_{k,j}^{(1)}}).\n\t\\end{split}\n\t\\end{equation}\n\t\n\tFinally, by scaling we immediately observe that\n\t\\begin{equation}\\label{K3-third-3}\n\t\\begin{split}\n\t&\\int_{B(p_{k,j}^{(1)},r)}f_ke^{\\phi_j}\\langle O(|x-p_{k,j}^{(1)}|^2),x-p_{k,j}^{(1)}\\rangle{\\rm d}x \\\\\n\t=&O(e^{-\\frac{3}{2}\\lambda_{k,j}^{(1)}})\\int_{|z|<\\frac{r}{\\epsilon_{k,j}}}\\frac{|z|^{3}}{(1+|z|^{2})^2}{\\rm d}z=O(e^{-\\lambda_{k,j}^{(1)}}).\n\t\\end{split}\n\t\\end{equation}\n\tTherefore, $K_3$ is small in this case as well.\n\t\\begin{equation}\\label{K3-est-2}\n\tK_3=o(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}}),\\quad \\tau+1\\leq j\\leq m,\n\t\\end{equation}\n\twhere $\\alpha_1>0$ is used. Lemma \\ref{lem-PI1-right} is established.\n\n\\end{proof}\n\nSince $|A_{k,j}|=O(1)$, (\\ref{PI-1}) along with Lemma \\ref{lem-PI1-left} and Lemma \\ref{lem-PI1-right} implies the initial estimate for $A_{k,j}$.\n\n\\begin{cor}\\label{cor-A-kj}\n\t\\begin{equation}\\label{A-kj-est}\n\t|A_{k,j}|=o(e^{-\\frac{\\lambda_{k,j}^{(1)}}{2(1+\\alpha_1)}}),\\quad 1 \\leq j \\leq m.\n\t\\end{equation}\n\t\\begin{flushright}\n\t\t\\qed\n\t\\end{flushright}\n\\end{cor}\n\n\nBased on (\\ref{A-kj-est}), we can improve the estimates in (\\ref{PI-1-l}) and (\\ref{GRF-est-1}):\n\\begin{equation}\\label{PI-1-l-re}\n\t{\\rm (LHS)}\\ {\\rm of}\\ (\\ref{PI-1})=-4(1+\\alpha_j)A_{k,j}+o(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}})\\quad 1\\leq j\\leq m.\t\n\\end{equation}\n\\begin{equation}\\label{GRF-est-2}\n\t\\varsigma_k(x)-\\bar{\\varsigma}_k=o(e^{-\\frac{\\lambda_{k,1}^{(1)}}{2(1+\\alpha_1)}}) \\quad {\\rm in}\\ \\; C^1\\Big(M\\setminus\\bigcup_{j=1}^m B(p_{k,j}^{(1)},\\theta)\\Big).\n\\end{equation}\n\nThe identity (\\ref{GRF-est-2}), which is the refined $C^1$-estimate of $\\varsigma_k$ away from the blowup points, will help to improve the estimate of RHS of Pohozaev-type identity (\\ref{PI-1}) and the estimate of $D\\varsigma_k$. The later one will play a part in section \\ref{pf-uni-2}. In order to achieve this goal, we analyse the projections of $\\varsigma_{k,j}$ in more detail.\n\nFor $1\\leq j\\leq \\tau$, we recall the equation of $\\varsigma_k$ in $B(p_{k,j}^{(1)},r_0)$:\n\\begin{equation*}\n\\begin{split}\n\\left\\{\n\\begin{array}{lcl}\n\\Delta \\varsigma_k+\\rho_k\\tilde{h}_j|x-p_{k,j}^{(1)}|^{2\\alpha_j}e^{U_{k,j}^{(1)}+G_{k,j}^{(1)}-G_{k,j}^{(1)}(p_{k,j}^{(1)})+\\eta_{k,j}^{(1)}}\\varsigma_k\\frac{1-e^{u_k^{(2)}-u_k^{(1)}}}{u_k^{(1)}-u_k^{(2)}}=0, \\\\\n|\\varsigma_k|\\leq 1,\n\\end{array}\n\\right.\n\\end{split}\n\\end{equation*}\nand set the following quantities for convenience:\n\\begin{align*}\n &a_{k,j}=\\nabla(\\log h_j+G_{k,j}^{(1)})(p_{k,j}^{(1)}), \\quad d_k=\\parallel u_k^{(1)}-u_k^{(2)}\\parallel_{L^{\\infty}(M)}, \\\\\n\t&n_0=\\max\\big\\{n\\in\\mathbb{N}:n\\leq\\frac{1}{2\\gamma} \\big\\},\\quad U_{j}(r)=\\log\\frac{8(1+\\alpha_j)^2}{(1+r^{2(1+\\alpha_j)})^2}.\n\\end{align*}\nThen the equation for $\\varsigma_k$ becomes\n\\begin{equation*}\n\\Delta \\varsigma_k+\\frac{\\rho_kh_j(p_{k,j}^{(1)})e^{\\lambda_{k,j}^{(1)}}|x-p_{k,j}^{(1)}|^{2\\alpha_j}e^{g_k(x)}}{\\big(1+\\frac{\\rho_kh_j(p_{k,j}^{(1)})}{8(1+\\alpha_j)^2}e^{\\lambda_{k,j}^{(1)}}|x-p_{k,j}^{(1)}|^{2(1+\\alpha_j)}\\big)^2}\\Big\\{\\sum_{n=0}^{n_0}\\frac{(-d_k)^{n}}{(n+1)!}\\varsigma_k^{n+1}+O(d_k^{n_0+1})\\Big\\}=0,\n\\end{equation*}\nwhere\n\\begin{align*}\n\tg_k(x)=&\\langle a_{k,j},x-p_{k,j}^{(1)}\\rangle\\Big\\{1-\\frac{2(1+\\alpha_j)}{\\alpha_j}\\Big(1+\\frac{\\rho_kh_j(p_{k,j}^{(1)})}{8(1+\\alpha_j)^2}e^{\\lambda_{k,j}^{(1)}}|x-p_{k,j}^{(1)}|^{2(1+\\alpha_j)}\\Big)^{-1}\\Big\\} \\\\\n\t&+d_j\\log\\Big(2+e^{\\frac{\\lambda_{k,j}^{(1)}}{2(1+\\alpha_j)}}|x-p_{k,j}^{(1)}|\\Big)e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_j}}+O(|x-p_{k,j}^{(1)}|^2)+O(\\epsilon_{k,1}^2)\n\\end{align*}\nAfter scaling $x=\\epsilon_{k,j}z+p_{k,j}^{(1)}$, we have\n\\begin{align}\\label{equ-varsigma_kj}\n\\Delta \\varsigma_{k,j}(z)+\\frac{8(1+\\alpha_j)^2|z|^{2\\alpha_j}}{(1+|z|^{2(1+\\alpha_j)})^2}\\varsigma_{k,j}(z)=E_{k,j}(z),\n\\end{align}\nwhere\n\\begin{align*}\n\t&E_{k,j}(z)=\\frac{8(1+\\alpha_j)^2|z|^{2\\alpha_j}}{(1+|z|^{2(1+\\alpha_j)})^2}\n\t\\bigg\\{-\\epsilon_{k,j}\\langle a_{k,j},z\\rangle\\big(1-\\frac{2(1+\\alpha_j)}{\\alpha_j}\\frac{1}{1+|z|^{2(1+\\alpha_j)}}\\big)\\varsigma_{k,j}(z) \\\\\n\t&+\n\t\\Big[1+\\epsilon_{k,j}\\langle a_{k,j},z\\rangle\\big(1-\\frac{2(1+\\alpha_j)}{\\alpha_j}\\frac{1}{1+|z|^{2(1+\\alpha_j)}}\\big)\\Big]\\sum_{n=1}^{n_0}\\frac{(-1)^{n+1}d_k^{n}}{(n+1)!}\\varsigma_{k,j}^{n+1}+o(\\epsilon_{k,1})\\bigg\\}.\n\\end{align*}\n\nFor each integer $l\\ge 0$ we define the projections of frequency $l$ as\n\\begin{align*}\n\t&\\xi_l(r)=\\frac{1}{2\\pi}\\int_{0}^{2\\pi}\\varsigma_{k,j}(r\\cos\\theta,r\\sin\\theta)\\cos(l\\theta){\\rm d}\\theta, \\\\\n\t&\\tilde{\\xi}_l(r)=\\frac{1}{2\\pi}\\int_{0}^{2\\pi}\\varsigma_{k,j}(r\\cos\\theta,r\\sin\\theta)\\sin(l\\theta){\\rm d}\\theta.\n\\end{align*}\nObviously the study of $\\xi_l$ is representative enough. (\\ref{equ-varsigma_kj}) shows that $\\xi_l$ satisfies\n\\begin{align*}\n\t\\xi_l^{''}+\\frac{1}{r}\\xi_l^{'}+\\Big(r^{2\\alpha_j}e^{U_j}-\\frac{l^2}{r^2}\\Big)\\xi_l=\\tilde{E}_{l}(r),\\quad l\\ge 1,\n\\end{align*}\nwhere\n\\begin{align*}\n\t&\\tilde{E}_{1}(r)=r^{2\\alpha_j}e^{U_j}\\Big\\{-\\frac{a_{k,j}^1}{4}\\epsilon_{k,j}r\\big(1-\\frac{2(1+\\alpha_j)}{\\alpha_j}\\frac{1}{1+r^{2(1+\\alpha_j)}}\\big)\\xi_0+O(d_k\\xi_1)+o(\\epsilon_{k,1})\\Big\\}, \\\\\n\t&\\tilde{E}_{2}(r)=r^{2\\alpha_j}e^{U_j}\\Big\\{-\\frac{a_{k,j}^1}{4}\\epsilon_{k,j}r\\big(1-\\frac{2(1+\\alpha_j)}{\\alpha_j}\\frac{1}{1+r^{2(1+\\alpha_j)}}\\big)\\xi_1+O(d_k\\xi_2)+o(\\epsilon_{k,1})\\Big\\}, \\\\\n\t&\\tilde{E}_{l}(r)=r^{2\\alpha_j}e^{U_j}\\Big\\{O(d_k\\xi_l)+o(\\epsilon_{k,1})\\Big\\},\\qquad l\\ge 3,\n\\end{align*}\nand $a_{k,j}^1$ is the first component of $a_{k,j}$. Moreover, from (\\ref{GRF-est-2}) we obtain that $\\xi_l(0)=o(1)$ for all $l\\ge 1$ and\n\\begin{equation}\\label{ode-boundary}\n\\begin{split}\n&|\\xi_l(r)|\\leq 1,\\ \\ \\quad r\\in(0,\\frac{r_0}{\\epsilon_{k,j}}),\\ \\ \\quad l\\ge 0, \\\\\n&\\xi_l(r)=o(\\epsilon_{k,1}),\\quad r\\sim e^{\\frac{\\lambda_{k,j}^{(1)}}{2(1+\\alpha_j)}},\\quad l\\ge 1.\n\\end{split}\n\\end{equation}\nFrom the equation of $\\xi_l$ and the maximum principle, we only need to consider the finite $l$. Without loss of generality, we consider $1\\leq l\\leq l_0$ in the following analysis. Let us denote $\\delta_{l,j}=\\frac{l}{1+\\alpha_j}$ and consider the homogeneous ordinary differential equation\n\\begin{equation}\\label{ode}\n\t\\xi_l^{''}+\\frac{1}{r}\\xi_l^{'}+\\Big(r^{2\\alpha_j}e^{U_j}-\\frac{l^2}{r^2}\\Big)\\xi_l=0.\n\\end{equation}\nBy direct computation, we can verify that the following two functions are two fundamental solutions of (\\ref{ode})\n\\begin{align*}\n&\\xi_{l,1}(r)=\\frac{(\\delta_{l,j}+1)r^l+(\\delta_{l,j}-1)r^{2(1+\\alpha_j)+l}}{1+r^{2(1+\\alpha_j)}},\\\\\n&\\xi_{l,2}(r)=\\frac{(\\delta_{l,j}+1)r^{2(1+\\alpha_j)-l}+(\\delta_{l,j}-1)r^{-l}}{1+r^{2(1+\\alpha_j)}}.\n\\end{align*}\nUsing $|\\xi_l|\\leq 1$ we have $C_{l,2}=0$, that is\n\\begin{equation*}\n\t\\xi_l(r)=C_{l,1}\\xi_{l,1}(r)+\\xi_{l,p}(r)\n\\end{equation*}\nwhere $C_{l,1}$ is a constant, and\n\\begin{equation}\\label{solution-p}\n\\xi_{l,p}(r)=\\Big(\\int\\frac{w_1}{w}{\\rm d}r\\Big)\\xi_{l,1}(r)+\\Big(\\int\\frac{w_2}{w}{\\rm d}r\\Big)\\xi_{l,2}(r)\n\\end{equation}\nfor\n\\begin{equation*}\nw=\n\\begin{vmatrix}\n\\xi_{l,1} & \\xi_{l,2} \\\\\n\\xi_{l,1}^{'} & \\xi_{l,2}^{'}\n\\end{vmatrix},\\quad\nw_1=\\begin{vmatrix}\n0 & \\xi_{l,2} \\\\\n\\tilde{E}_l & \\xi_{l,2}^{'}\n\\end{vmatrix},\\quad\nw_2=\\begin{vmatrix}\n\\xi_{l,1} & 0 \\\\\n\\xi_{l,1}^{'} & \\tilde{E}_l\n\\end{vmatrix}.\\quad\n\\end{equation*}\n\nIt is easy to see that $w^{'}=(\\xi_{l,1}\\xi_{l,2}^{'}-\\xi_{l,1}^{'}\\xi_{l,2})^{'}=-\\frac{1}{r}w$, which means $w(r)\\sim \\frac{1}{r}$. Next, let us estimate $\\xi_{l}$ in $(0,\\frac{r_0}{\\epsilon_{k,j}})$ for $l\\ge 1$.\n\n\\smallskip\nFor $1\\leq j\\leq t$, the assumption $D(\\mathbf{p})=0$ implies $a_{k,j}=O(\\epsilon_{k,1}^2)$. Furthermore, for $t+1\\leq j\\leq \\tau$, it is easy to see that $\\epsilon_{k,j}=o(\\epsilon_{k,1})$. Therefore, for all $1\\leq j\\leq \\tau$, we estimate $\\tilde{E}_l$ as follows\n\\begin{equation}\\label{E-1}\n\\begin{split}\n&\\tilde{E}_{l}(r)=r^{2\\alpha_j}e^{U_j}\\big\\{o(\\epsilon_{k,1})r+O(d_k\\xi_l)+o(\\epsilon_{k,1})\\big\\},\\quad l=1,2; \\\\\n&\\tilde{E}_{l}(r)=r^{2\\alpha_j}e^{U_j}\\big\\{O(d_k\\xi_l)+o(\\epsilon_{k,1})\\big\\},\\, \\qquad\\quad\\qquad l\\ge 3.\n\\end{split}\n\\end{equation}\nRoughly,\n\\begin{equation}\\label{E-2}\n\\begin{split}\n&\\tilde{E}_{l}(r)=r^{2\\alpha_j}e^{U_j}\\big\\{o(\\epsilon_{k,1})r+O(d_k)+o(\\epsilon_{k,1})\\big\\},\\quad l=1,2; \\\\\n&\\tilde{E}_{l}(r)=r^{2\\alpha_j}e^{U_j}\\big\\{O(d_k)+o(\\epsilon_{k,1})\\big\\},\\, \\qquad\\quad\\qquad l\\ge 3.\n\\end{split}\n\\end{equation}\nBy using the above estimates (\\ref{E-2}) for $\\tilde{E}_l$ and (\\ref{solution-p}), we have\n\\begin{align*}\n\\xi_{l,p}(r)=&\\big(O(d_k)+o(\\epsilon_{k,1})\\big)\\bigg\\{\\Big(\\int_{r}^{\\infty}s^{2\\alpha_j+1}e^{U_j(s)}(s+1)\\xi_{l,2}(s){\\rm d}s\\Big)\\xi_{l,1}(r)\\\\\n&+\\Big(\\int_{r}^{\\infty}s^{2\\alpha_j+1}e^{U_j(s)}(s+1)\\xi_{l,1}(s){\\rm d}s\\Big)\\xi_{l,2}(r)\\bigg\\},\\quad l=1,2.\\\\\n\\xi_{l,p}(r)=&\\big(O(d_k)+o(\\epsilon_{k,1})\\big)\\bigg\\{\\Big(\\int_{r}^{\\infty}s^{2\\alpha_j+1}e^{U_j(s)}\\xi_{l,2}(s){\\rm d}s\\Big)\\xi_{l,1}(r)\\\\\n&+\\Big(\\int_{r}^{\\infty}s^{2\\alpha_j+1}e^{U_j(s)}\\xi_{l,1}(s){\\rm d}s\\Big)\\xi_{l,2}(r)\\bigg\\},\\quad l\\ge 3.\n\\end{align*}\nDirect computation shows, for $0{\\rm d}x+o(\\epsilon_{k,1}^2). \\\\\n \\end{align*}\n where we used the fact $\\alpha_j<\\alpha_1$ and the definition of $n_0$.\n\tThen from symmetry and the estimates of high frequency of $\\varsigma_{k,j}$, which are (\\ref{xi-cos}) and (\\ref{xi-sin}), we have the following estimate\n\t\\begin{align*}\n\t\\int_{B(p_{k,j}^{(1)},r)}f_ke^{\\phi_j}\\langle \\nabla(\\log h_j+\\phi_{k,j})(p_{k,j}^{(1)}),x-p_{k,j}^{(1)}\\rangle{\\rm d}x=O(\\epsilon_{k,j})o(\\epsilon_{k,1})+o(\\epsilon_{k,1}^2)\n\t=o(\\epsilon_{k,1}^2).\n\t\\end{align*}\nTherefore, (\\ref{K3-first-2-re}) holds. Finally, combining (\\ref{K3-first-2-re}) with the proof of Lemma \\ref{lem-PI1-right}, we obtain the esstimate (\\ref{PI-1-r-4}).\n\t\n\\end{proof}\n\nBased on the Pohozaev-type identity (\\ref{PI-1}) and its refined estimates, which are (\\ref{PI-1-l-re}) (\\ref{PI-1-r-2}) (\\ref{PI-1-r-3}) and (\\ref{PI-1-r-4}), we can improve the estimate for $A_{k,j}$ and prove $b_0=0$.\n\\begin{cor}\\label{cor-A-kj-re}\n\t\\begin{equation}\\label{A-kj-est-re}\n\t|A_{k,j}|=O(e^{-\\frac{\\lambda_{k,j}^{(1)}}{1+\\alpha_1}}),\\quad 1 \\leq j \\leq m.\n\t\\end{equation}\n\t\\begin{flushright}\n\t\t\\qed\n\t\\end{flushright}\n\\end{cor}\n\\begin{prop}\\label{prop-b0}\n\t$b_0=0$. In particular, $b_{j,0}=0$, for $1\\leq j\\leq m$.\n\\end{prop}\n\\begin{proof}[\\textbf{Proof}]\n\tNow the global cancellation property of $f_k$ plays a crucial role:\n\t\\begin{equation*}\n\t\t\\sum_{j=1}^{m}A_{k,j}=\\int_{M}f_k{\\rm d}\\mu=0.\n\t\\end{equation*}\n\tFrom (\\ref{PI-1}) (\\ref{PI-1-l-re}) (\\ref{PI-1-r-2}) (\\ref{PI-1-r-3}) and (\\ref{PI-1-r-4}), we can see\n\t\\begin{align*}\n\t\tb_0e^{-\\frac{\\lambda_{k,1}^{(1)}}{1+\\alpha_1}}\\sum_{j=1}^t\\big[\\Delta h(p_j)+\\rho_*-N^*-2K(p_j)\\big]\\big(\\rho_kh_j(p_{k,j}^{(1)})\\big)^{\\frac{1}{1+\\alpha_1}}=o(e^{-\\frac{\\lambda_{k,1}^{(1)}}{1+\\alpha_1}}).\n\t\\end{align*}\n\tOn the other hand, from (\\ref{uk-ave-2}), it holds\n\t\\begin{equation*}\n\t\th_j^2(p_j)=h_1^2(p_1)e^{G_1^*(p_1)}e^{-G_j^*(p_j)}+o(1),\\quad 1\\leq j\\leq t.\n\t\\end{equation*}\n\tAs a consequence, we obtain\n\t\\begin{equation}\\label{b0-est}\n\t\\begin{split}\n\te^{-\\frac{G_1^*(p_1)}{1+\\alpha_1}}\\big(\\rho_*h_1^2(p_1)\\big)^{-\\frac{1}{1+\\alpha_1}}L(\\mathbf{p})b_0e^{-\\frac{\\lambda_{k,1}^{(1)}}{1+\\alpha_1}}=o(e^{-\\frac{\\lambda_{k,1}^{(1)}}{1+\\alpha_1}}),\n\t\\end{split}\n\t\\end{equation}\n\twhich together with the assumption $L(\\mathbf{p})\\neq 0$ implies $b_0=0$. In particular, $b_{j,0}=0$ for $1\\leq j\\leq m$.\t\n\t\n\\end{proof}\n\n\n\n\\section{Proof of Theorem \\ref{main-theorem}}\\label{pf-uni-1}\n\n\n\\begin{proof}[\\textbf{Proof of Theorem \\ref{main-theorem} }]\nLet $p_k^*$ be a maximum point of $\\varsigma_k$, which says\n\\begin{equation}\\label{max=1}\n|\\varsigma_k(p_k^*)|=1\n\\end{equation}\nIn view of Lemma \\ref{lem-limit-2} and Proposition \\ref{prop-b0}, we obtain the fact\n\\begin{equation*}\n\t\\varsigma_k\\to 0 \\quad{\\rm in}\\ \\; C_{loc}(M\\backslash\\{p_1,\\cdots,p_m\\}).\n\\end{equation*}\n Therefore,\n\\begin{equation}\\label{pk*}\n\\lim\\limits_{k\\rightarrow\\infty}p_k^*=p_j,\n\\end{equation}\nfor some $p_j\\in\\{p_1,\\cdots,p_m\\}$. Moreover, denoting $s_k=|p_k^*-p_{k,j}^{(1)}|$, by Lemma \\ref{lem-limit-1} and Proposition \\ref{prop-b0}, it holds\n\\begin{equation*}\n\\varsigma_{k,j}\\rightarrow 0 \\quad{\\rm in}\\ \\; C_{loc}(\\mathbb{R}^2).\n\\end{equation*}\nThus,\n\\begin{equation}\\label{sk}\n\\lim\\limits_{k\\rightarrow\\infty}\\epsilon_{k,j}^{-1}s_k=+\\infty\n\\end{equation}\nSetting $\\tilde{\\varsigma_k}(x)=\\varsigma_k(s_kx+p_{k,j}^{(1)})$, $|x|0$ small enough, then $\\tilde{\\varsigma_k}$ satisfies\n\\begin{align*}\n0=&\\Delta\\tilde{\\varsigma_k}(x)+\\rho_k\\tilde{h}_j(s_kx+p_{k,j}^{(1)})s_k^{2(1+\\alpha_j)}|x|^{2\\alpha_j}c_k(s_kx+p_{k,j}^{(1)})\\tilde{\\varsigma_k}(x) \\\\\n=& \\Delta\\tilde{\\varsigma_k}(x)+\\frac{8(1+\\alpha_j)^2(\\epsilon_{k,j}^{-1}s_k)^{2(1+\\alpha_j)}|x|^{2\\alpha_j}}{\\big(1+(\\epsilon_{k,j}^{-1}s_k)^{2(1+\\alpha_j)}|x|^{2(1+\\alpha_j)}\\big)^2}\\big(1+O(s_k|x|)+o(1)\\big).\n\\end{align*}\n\nOn the other hand, by (\\ref{max=1}), we also have\n\\begin{equation}\\label{scale-max=1}\n\\Big|\\tilde{\\varsigma_k}\\big(\\frac{p_k^*-p_{k,j}^{(1)}}{s_k}\\big)\\Big|=|\\varsigma_k(p_k^*)|=1.\n\\end{equation}\nIn view of (\\ref{sk}) and $|\\tilde{\\varsigma_k}|\\leq 1$, we see that $\\tilde{\\varsigma_k}\\rightarrow\\tilde{\\varsigma_0}$ in $C_{loc}(\\mathbb{R}^2\\backslash\\{0\\})$, where $\\tilde{\\varsigma_0}$ satisfies $\\Delta\\tilde{\\varsigma_0}=0$ in $\\mathbb{R}^2\\backslash\\{0\\}$. Since $|\\tilde{\\varsigma_0}|\\leq 1$, we have $\\Delta\\tilde{\\varsigma_0}=0$ in $\\mathbb{R}^2$. Hence $\\tilde{\\varsigma_0}$ is a constant.\n\nRecalling that $\\frac{|p_k^*-p_{k,j}^{(1)}|}{s_k}=1$ and (\\ref{scale-max=1}), we find that $\\tilde{\\varsigma_0}\\equiv 1$ or $\\tilde{\\varsigma_0}\\equiv -1$. Therefore, we obtain that for $ k $ large enough\n\\begin{equation}\\label{contra1}\n|\\varsigma_k(x)|\\geq\\frac{1}{2},\\quad |x-p_{k,j}^{(1)}|\\in\\big(\\frac{s_k}{2},2s_k\\big).\n\\end{equation}\n\nBy using Lemma \\ref{lem-limit-2}, we have\n\\begin{equation}\\label{contra2}\n\\varsigma_k(x)=o(1)+o(1)\\log R+O(R^{-2(1+\\alpha_j)}),\\quad |x-p_{k,j}^{(1)}|\\in(R\\epsilon_{k,j},d).\n\\end{equation}\nfor fixed $d>0$ small enough and arbitrary $R>0$ large enough.\n\n\\smallskip\nHowever, by (\\ref{sk}), $\\epsilon_{k,j}\\ll s_k$. Thus, $|\\varsigma_k(s_k)|<\\frac{1}{4}$ for $ k $ large enough, which contradicts with (\\ref{contra1}). Theorem \\ref{main-theorem} is established.\n\t\n\\end{proof}\n\n\n\\section{Proof of Theorem \\ref{main-theorem-2}}\\label{pf-uni-2}\n\nIn this section, we will analyse the behavior of $u_k^{(1)}$ and $u_k^{(2)}$ whose common blowup points include singular source(s) and regular point(s). So in this section $\\tau\\nabla_iv_{k,j}^{(1)}+<\\nu,\\nabla v_{k,j}^{(2)}>\\nabla_i\\varsigma_k \\Big) {\\rm d}\\sigma \\\\\n& \\quad -\\frac{1}{2}\\int_{\\partial B(p_{k,j}^{(1)},r)}<\\nabla(v_{k,j}^{(1)}+v_{k,j}^{(2)}),\\nabla\\varsigma_k>\\frac{(x-p_{k,j}^{(1)})_i}{|x-p_{k,j}^{(1)}|} {\\rm d}\\sigma, \\\\\n=\\ & -\\int_{\\partial B(p_{k,j}^{(1)},r)}\\rho_k\\tilde{h}_j(x)\\frac{e^{u_k^{(1)}}-e^{u_k^{(2)}}}{\\parallel u_k^{(1)}-u_k^{(2)} \\parallel_{L^{\\infty}(M)} }\\frac{(x-p_{k,j}^{(1)})_i}{|x-p_{k,j}^{(1)}|} {\\rm d}\\sigma \\\\\n& \\quad + \\int_{B(p_{k,j}^{(1)},r)} \\rho_k\\tilde{h}_j(x)\\frac{e^{u_k^{(1)}}-e^{u_k^{(2)}}}{\\parallel u_k^{(1)}-u_k^{(2)} \\parallel_{L^{\\infty}(M)}} \\nabla_i\\big(\\log \\tilde{h}_j+\\phi_{k,j}\\big) {\\rm d}x.\n\\end{split}\n\\end{align}\n\n\\end{lemA}\n\nBy Lemma 4.6 in \\cite{bart-4} and Appendix D in \\cite{lin-yan-uniq}, we have:\n\\begin{align}\\label{PI-2-r}\n{\\rm (RHS)} \\ {\\rm of}\\ (\\ref{PI-2})=e^{-\\frac{\\lambda_{k,j}^{(1)}}{2}}\\Big(\\sum_{h=1}^2 D_{h,i}^2(\\log \\tilde{h}_j+\\phi_{k,j} )(p_{k,j}^{(1)})b_{j,h}\\Big)B_j+o(e^{-\\frac{\\lambda_{k,j}^{(1)}}{2}}).\n\\end{align}\nfor $i=1,2$ and $\\tau+1\\le j\\le m$. The detail of this proof can be found in \\cite{bart-4}.\n\nThe LHS of (\\ref{PI-2}) boils down to sharp estimates of $\\nabla v_{k,j}^{(i)}$ and $\\nabla\\varsigma_k$ on $\\partial B(p_{k,j}^{(1)},r)$. The estimate for $\\nabla v_{k,j}^{(i)}$ is established in Lemma \\ref{lem-Dv-kj}, and the following lemma provides the estimates for $\\nabla\\varsigma_k$ (see (\\ref{Dsigma_k-1}) for comparison).\n\n\\begin{lem}\\label{lem-C1-est-re}\n\t\n\tFor any $\\theta\\in(0,r)$ small enough, it holds\n\t\\begin{align}\\label{GRF-est-re}\n\t\\begin{split}\n\t\\varsigma_k-\\bar{\\varsigma}_k=\\sum_{j=\\tau+1}^m e^{-\\frac{\\lambda_{k,j}^{(1)}}{2}}&\\Big(\\sum_{h=1}^2\\partial_{y_h}G(y,x)\\big|_{y=p_{k,j}^{(1)}}b_{j,h}\\Big)B_j+o(e^{-\\frac{\\lambda_{k,1}^{(1)}}{2}}) \\\\\n\t &{\\rm in}\\ \\; C^1\\Big(M\\setminus\\bigcup_{j=1}^m B(p_{k,j}^{(1)},\\theta)\\Big).\n\t\\end{split}\n\t\\end{align}\n\\end{lem}\n\n\\begin{proof}[\\textbf{Proof}]\n\tUsing the same notations in (\\ref{J1+J2+J3}) and (\\ref{J3}), now we only need to show\n\t\\begin{equation*}\n\t\tJ_1=o(e^{-\\frac{\\lambda_{k,1}^{(1)}}{2}}),\\quad J_2=o(e^{-\\frac{\\lambda_{k,1}^{(1)}}{2}}).\n\t\\end{equation*}\n\t\n\tIndeed, from (\\ref{A-kj-est-re}) and the assumption $0< \\alpha_1<1$, we have\n\t\\begin{equation}\\label{J1-re}\n\t\tJ_1=\\sum_{j=1}^m A_{k,j}G(p_{k,j}^{(1)},x)=O(e^{-\\frac{\\lambda_{k,1}^{(1)}}{1+\\alpha_1}})=o(e^{-\\frac{\\lambda_{k,1}^{(1)}}{2}})\n\t\\end{equation}\n\t\n\tRecall that\n\t\\begin{align*}\n\t\tJ_2=&\\sum_{j=1}^\\tau\\int_{M_j}\\big(G(y,x)-G(p_{k,j}^{(1)},x)\\big)f_k(y){\\rm d}\\mu(y) \\\\\n\t\t=&\\sum_{j=1}^\\tau\\int_{B(p_{k,j}^{(1)},r_0)}f_k(y)e^{\\phi_j(y)}\\langle\\partial _yG(y,x)\\big|_{y=p_{k,j}^{(1)}},y-p_{k,j}^{(1)}\\rangle {\\rm d}y +O(e^{-\\frac{\\lambda_{k,1}^{(1)}}{1+\\alpha_1}})\n\t\\end{align*}\n\tBased on (\\ref{xi-cos}) and (\\ref{xi-sin}), by the method similar to the proof of (\\ref{K3-first-2-re}) in Lemma \\ref{lem-PI1-right-re}, we have\n\t\\begin{equation*}\n\t\t\\int_{B(p_{k,j}^{(1)},r_0)}f_k(y)e^{\\phi_j(y)}\\langle\\partial _yG(y,x)\\big|_{y=p_{k,j}^{(1)}},y-p_{k,j}^{(1)}\\rangle {\\rm d}y=O(\\epsilon_{k,j})o(\\epsilon_{k,1})+o(\\epsilon_{k,1}^2)=O(\\epsilon_{k,1}^2).\n\t\\end{equation*}\n\tTherefore,\n\t\\begin{equation}\\label{J2-re}\n\t\tJ_2=O(e^{-\\frac{\\lambda_{k,1}^{(1)}}{1+\\alpha_1}})=o(e^{-\\frac{\\lambda_{k,1}^{(1)}}{2}}).\n\t\\end{equation}\n\t\n\tConsequently (\\ref{GRF-est-re}) holds in $C^1\\big(M\\setminus\\bigcup_{j=1}^m B(p_{k,j}^{(1)},\\theta)\\big)$ and the gradient estimate is\n\\begin{equation}\\label{Dsigma_k-2}\n\\begin{split}\n\\nabla\\varsigma_k(x)=\\sum_{j=\\tau+1}^m e^{-\\frac{\\lambda_{k,j}^{(1)}}{2}}\\nabla_x\\Big(\\sum_{h=1}^2\\partial_{y_h}G(y,x)\\big|_{y=p_{k,j}^{(1)}}b_{j,h}\\Big)B_j+o(e^{-\\frac{\\lambda_{k,1}^{(1)}}{2}}).\n\\end{split}\n\\end{equation}\n\t\n\\end{proof}\n\n\n\nBy the improved estimates of $\\nabla v_{k,j}^{(i)}$ and $\\nabla\\varsigma_k$ in (\\ref{Dv-kj-est}) and (\\ref{Dsigma_k-2}), we can estimate the left hand of (\\ref{PI-2}) just like Lemma 4.7 in \\cite{bart-4} or Appendix D in \\cite{lin-yan-uniq} and the result is:\n\\begin{equation}\\label{PT-l}\n\\begin{split}\n{\\rm (LHS)} \\ {\\rm of}\\ (\\ref{PI-2})=&-8\\pi\\bigg\\{\\sum_{l\\neq j}^{\\tau+1,\\cdots,m}e^{-\\frac{\\lambda_{k,l}^{(1)}}{2}}\\partial_{x_i}\\Big(\\sum_{h=1}^2\\partial_{y_h}G(y,x)\\big|_{y=p_{k,l}^{(1)}}b_{l,h}\\Big)B_l\\\\\n&\\ \\;+e^{-\\frac{\\lambda_{k,j}^{(1)}}{2}}\\partial_{x_i}\\Big(\\sum_{h=1}^2\\partial_{y_h}R(y,x)\\big|_{x=y=p_{k,j}^{(1)}}b_{j,h}\\Big)B_j\\bigg\\}+o(e^{-\\frac{\\lambda_{k,j}^{(1)}}{2}}).\n\\end{split}\n\\end{equation}\n\nFinally we prove $b_{j,1}=b_{j,2}=0$ for all $j$.\n\\begin{prop}\\label{prop-b1b2}\n\n$b_{j,1}=b_{j,2}=0$, for all $j=\\tau+1,\\cdots,m$. In particular,\n\\begin{equation*}\n\\varsigma_{k,j}\\rightarrow 0\\quad {\\rm in}\\ \\; C_{loc}(\\mathbb{R}^2),\\quad {\\rm for\\ \\, all} \\ \\, j=1,\\cdots,m.\n\\end{equation*}\n\\end{prop}\n\n\\begin{proof}[\\textbf{Proof}]\n\t\nObviously, (\\ref{PI-2}) together with (\\ref{PI-2-r}) and (\\ref{PT-l}) implies, for all $i=1,2$, and $j=\\tau +1,\\cdots,m$,\n\\begin{align}\\label{PI-final}\n\\begin{split}\n&e^{-\\frac{\\lambda_{k,j}^{(1)}}{2}}\\Big(\\sum_{h=1}^2 D_{h,i}^2(\\log \\tilde{h}_j+\\phi_{k,j})(p_{k,j}^{(1)})b_{j,h}\\Big)B_j\\\\\n=&-8\\pi\\sum_{l\\neq j}^{\\tau+1,\\cdots,m}e^{-\\frac{\\lambda_{k,l}^{(1)}}{2}}\\partial_{x_i}\\Big(\\sum_{h=1}^2\\partial_{y_h}G(y,x)\\big|_{y=p_{k,l}^{(1)}}b_{l,h}\\Big)B_l\\\\\n&-8\\pi e^{-\\frac{\\lambda_{k,j}^{(1)}}{2}}\\partial_{x_i}\\Big(\\sum_{h=1}^2\\partial_{y_h}R(y,x)\\big|_{x=y=p_{k,j}^{(1)}}b_{j,h}\\Big)B_j+o(e^{-\\frac{\\lambda_{k,j}^{(1)}}{2}}).\n\\end{split}\n\\end{align}\n\nSet $\\vec{b}=(\\tilde{b}_{\\tau+1,1}B_{\\tau+1},\\tilde{b}_{\\tau+1,2}B_{\\tau+1},\\cdots,\\tilde{b}_{m,1}B_m,\\tilde{b}_{m,2}B_m)$, where\n\\begin{equation*}\n\\tilde{b}_{l,h}=\\lim\\limits_{k\\to +\\infty}\\big(e^{\\frac{\\lambda_{k,j}^{(1)}-\\lambda_{k,1}^{(1)}}{2}}b_{l,h}\\big).\n\\end{equation*}\nThen by (\\ref{p_kj-location}) and letting $k\\to+\\infty$, we obtain that\n\\begin{equation}\\label{b-vector}\nD^2f^*(p_{\\tau+1},\\cdots,p_m)\\cdot \\vec{b}=0\n\\end{equation}\nBy using the non-degeneracy assumption $\\det \\big(D^2f^*(p_{\\tau+1},\\cdots,p_m)\\big)\\neq 0$, we conclude that\n\\begin{equation}\\label{b=0}\nb_{j,1}=b_{j,2}=0,\\quad j=\\tau+1,\\cdots,m.\n\\end{equation}\nProposition \\ref{prop-b1b2} is established.\n\n\\end{proof}\n\n\n\\begin{proof}[\\textbf{Proof of Theorem \\ref{main-theorem-2} }]\nFrom\nLemma \\ref{lem-limit-2} and Proposition \\ref{prop-b0} $\\varsigma_k$ tends to $0$ in\n$C_{loc}(M\\backslash\\{p_1,\\cdots,p_m\\})$.\nBy Lemma \\ref{lem-limit-1} and Proposition \\ref{prop-b1b2}, we have\n\\begin{equation*}\n\t\\varsigma_{k,j}\\rightarrow 0\\quad {\\rm in}\\ \\; C_{loc}(\\mathbb{R}^2),\\quad 1\\leq j\\leq m.\n\\end{equation*}\nTheorem \\ref{main-theorem-2} follows just like the last step of the proof of Theorem \\ref{main-theorem}.\n\n\\end{proof}\n\nFinally, we finish to prove Theorem \\ref{main-theorem-3} and Theorem \\ref{main-theorem-4} about Dirichlet problems.\n\n\\begin{proof}[\\textbf{Proof of Theorems \\ref{main-theorem-3}, \\ref{main-theorem-4} }]\n\tFor the blowup solutions to (\\ref{equ-flat}), the corresponding estimates as in section \\ref{preliminary} have been also obtained in \\cite{chen-lin}\\cite{zhang2} for $\\alpha_j\\in\\mathbb{R}^+\\setminus\\mathbb{N}$ and in \\cite{chen-lin-sharp}\\cite{zhang1}\\cite{gluck} for $\\alpha_j=0$. Those preliminary estimates have almost the same form except for $\\phi_j=0$ and $K\\equiv 0$, where $\\phi_j$ are the conformal factor at $p_j$ and $K$ is the Gaussian curvature of $M$.\n\t\n\tThen, under the assumption of regularity about $\\partial\\Omega$ and $q_j\\in \\Omega$ $(1\\leq j\\leq N)$, \\cite{ma-wei} has showed that the blowup points of (\\ref{equ-flat}) are far away from $\\partial\\Omega$ via the moving plane method and the Pohozaev identities. Consequently, the terms coming from the boundary of domain are included in the error term. In other words, those boundary terms do not affect our argument.\n\t\n\tOn the other hand, the vital part of estimates obtained in section \\ref{difference}, \\ref{anal-pohozaev} and \\ref{pf-uni-2} only come from local analysis, Therefore, such results still work for the Dirichlet problem (\\ref{equ-flat}).\n\t\n\tThus, Theorem \\ref{main-theorem-3} and Theorem \\ref{main-theorem-4} can be proved as Theorem \\ref{main-theorem} and Theorem \\ref{main-theorem-4}, respectively. \n\t\n\\end{proof}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\t\n\tThe COVID-19 pandemic disrupted many lives and businesses \\citep{UNcovid}. Governments around the world imposed unprecedented measures to contain the spread of the virus, which often took the form of full lockdown.\n\tThe policies implemented to contain the pandemic as well as the volume and the unbalanced shares of home production were likely to have consequences for the gender distribution of work. On one hand, women are over\u2010represented in sectors that have been defined as essential and in occupations that cannot be performed from home \\citep{OECD2021gender}. On the other hand, women tend to be over\u2010represented in service industries, such as retail, tourism, and hospitality, which have been either subject to lockdown or to strict restrictions for some time \\citep{hupkau2020work, farre2020covid}. Women are also more likely than men to be employed in the informal sector, compensated in cash with no official oversight, and no eligibility to benefits, such as the furlough scheme \\citep{EP2021gender}. Finally, even when working from home, women on average perform most of home production tasks, e.g., childcare, and more in general they bear a large share of the earning penalty associated with childbearing \\citep{sevilla2020baby, kleven2019children, yildirim2021differential}. The overall perception is that women have been hurt by the pandemic disproportionately more compared to men, but there seems to be evidence of large heterogeneity across countries \\citep{bluedorn2021gender}.\n\t\n\tWe investigate the labour market dynamics in Italy in the period 2013-2020, i.e., before and during the pandemic, using longitudinal quarterly labour force data, with a focus on gender, age and geographical differences. Italy was the first country in Europe to be hit by the COVID-19 pandemic and the first to implement a national lockdown in the beginning of March, 2020 \\citep{saglietto2020covid}. To mitigate its effect on the labour market, the Italian Government pro-actively implemented two aggressive policies: a ban on layoffs and the extension of a pre-existing furlough scheme \\citep{barbieri2021italian}. The Italian case is also particularly interesting as the labour market is largely heterogeneous, with vast, well-known and persistent regional disparities: industrial activities are mostly concentrated in the North and in the Center, while food industry and tourism are mainly concentrated in Southern regions \\citep{OECDItaly}. In addition, Italy ranks among the weakest of OECD countries regarding job quantity, defined as employment, unemployment and underemployment \\citep{OECD2018}, reflecting persistently large gender employment gaps and a remarkably low female labour force participation rate, particularly in the South \\citep{agovino2019local}. Due to these very different conditions at the outburst of the pandemic, we find that the COVID-19 shock had sizable asymmetric effects across categories of individuals but, less expected, are their size and persistence over time.\n\t\n\tIn particular, we first document how the shares of individuals across seven labour market states (permanent, temporary and self-employment, furlough scheme, unemployment, education and inactivity) before the pandemic differed across ages, gender and geographical areas. The shares of individuals in education was comparable in the Northern-Center and the Southern part of Italy among the 15-24 age cohort, but while in the North and Center the transitions from education were predominantly towards the labour force, in the South the transitions towards inactivity were already much higher. This phenomenon was particularly large among females in the South, leading to an inactivity share of above 50\\% among the 40-49 age cohort. \n\tThe COVID-19 pandemic shock lead to an increased flow of discouraged workers moving from unemployment to inactivity, across age cohorts, gender and geographical areas. We also show evidence of a large outflow of individuals who left permanent (and temporary) employment to become inactive and, more importantly, persisted outside the labour market until the fourth quarter of 2020. Females aged 30-39 with at least one young child, living in the North and Center of Italy, previously hired on a permanent contract show the largest and most persistent outflows. We find similar patterns in the South, but surprisingly the size of the impact is shown to be much smaller.\n\t\n\tThis paper fits in the growing literature which analyzes the asymmetric effects of the pandemic on different categories of individuals \\citep{caselli2021mobility}. A number of papers show that the impact of the shock has been disproportional high among vulnerable workers \\citep{chetty2020economic}. In particular, while there is quite a large consensus regarding the fact that younger low-income workers were more likely to lose their jobs, findings are more controversial regarding other demographic dimensions, such as gender. Some studies have provided evidence that the pandemic is largely affecting women's labour market outcomes. Specifically, \\cite{alon2021mancession} finds higher employment losses for women compared to men in the US; these are confirmed by \\cite{albanesi2021gendered}, who provide evidence of a substantial and persistent drop for women not only in employment, but also in labor force participation. \\cite{fabrizio2021covid} and \\cite{zamarro2020gender} find that less educated women with young children were the most adversely affected, while \\cite{shibata2020distributional} shows that women and Hispanics are the two categories who lost the most. Finally, \\cite{adams2020inequality} show that women and workers without a college degree are significantly\n\tmore likely to have lost their jobs. However, other contributions point to no gender difference in labour market outcomes as a consequence of the pandemic. \\citep{casarico2020heterogeneous} show that gender is a non-significant predictor of job loss in the aggregate, while \\cite{hupkau2020work} find no difference in outcomes between men and women at the extensive margin, and if anything smaller losses for women at the intensive margin. Overall there is evidence of large heterogeneity across countries \\citep{bluedorn2021gender, adams2020inequality, dang2021gender}.\n\t\n\tThe paper is organized as follows. Section \\ref{sec:italianPolicies} describes the policies implemented in Italy in the outburst of the pandemic, while Section \\ref{sec:ItalianLabourMarket} described the dynamics of the Italian labour market before and during the pandemic. Specifically, Section \\ref{sec:Data} describes the data, Section \\ref{sec:Pre} illustrates the shares and transition probabilities across states before the pandemic, Section \\ref{sec:Post} shows the dynamics during the pandemic and Section \\ref{sec:logit} presents the results of the parametric estimation of the probability to remain active in the labour market by gender and age. Finally, Section \\ref{sec:concludingRemarks} concludes the paper. \n\t\n\t\t\n\\section{COVID-19 policies in Italy}\\label{sec:italianPolicies}\n\t\nThe first cases of COVID-19 in Italy were registered on January 31, 2020, but the virus began to spread exponentially in the second half of February. At the beginning, the virus circulated predominantly in Northern regions but by the beginning of March, it had reached all regions. On March 10, the whole country went into a full lockdown. On March 11 the government prohibited nearly all commercial activity except for supermarkets and pharmacies and on March 21 it restricted the movement of people and closed all non-essential businesses and industries. Sectors identified as essential, which could continue operating, include mainly agriculture, some manufacturing, energy and water supply, transports and logistics, ICT, banking and insurance, professional and scientific activities, public administration, education, health care and some service activities. Non-essential sectors which were completely shut include most manufacturing, wholesale and retail trade, hotels, restaurants and bars, entertainment and sport activities \\cite{casarico2020heterogeneous}.\n\nSubsequently, on March 17 the Italian government implemented two new labor market policies to protect workers: (i) a COVID-19 furlough scheme and (ii) a ban on layoffs. The former was implemented for an initial duration of 9 weeks, and it applied retroactively starting from February 23. It represents an extension of the regular furlough scheme to all firms, independently on size. This measure aimed at preserving employment and allowed firms to cut labor costs during the lockdown period, by reducing hours of work thanks to a wage subsidy granted by the government. Firms using the COVID-19 furlough scheme could renew temporary contracts, waiving to the norms of the standard regulation. Upon completion of the furlough period, firms were allowed to dismiss employees for redundancy. The ban on layoffs prevented firm to fire workers for 60 days, starting from March 17; this ban could be applied retroactively to pending, but already validated layoffs from February 23. Two later decrees extended the validity of these measures, which were still in place until the end of 2021.\n\n\\section{The Italian labour market pre and during the COVID-19 pandemic}\\label{sec:ItalianLabourMarket}\n\n\nThe Italian labour market pre-COVID-19 presented specific characteristics, which we deem as crucial to understand the asymmetric impact of the shock on different categories of individuals. The key features are the differentials in labour market participation by gender, males versus females, and by geographical area, North and Center versus South of Italy.\nThe literature has highlighted significant gender differences, and relevant geographical differentials are also reported as a structural feature of the Italian labour market \\citep{bertola2003structure}. Women, on average, show a lower attachment to the labour force together with a lower commitment to labour market activity compared to men \\citep{schiattarella2018old}. While the North-South divide characterizes many elements of the economic and cultural life in Italy, it is particularly striking in women's work, with women from the Southern regions (and the Islands) being much less likely to work and much more likely to end up in unemployment or outside the labour force. A specific characteristic of women in\nSouthern Italy is that they are comparatively more likely not to work and not to return to the labour after marriage (or childbearing). On average in Italy 30\\% of Italian mothers in employment stop working to care for children or other relatives, and of these only about 12\\% go back to work at some point, but this number is much lower in the Italian South, due to the predominant role of the male breadwinner model \\citep{SOAS2021gender, baussola2014disadvantaged}.\n\n\n\n\nFinally, to gender and geographical characteristics, we add age as another dimension of analysis. We therefore split individuals according to 6 cohorts: the first 15-19 age cohort includes individuals who either decided to drop out high school or keeps on studying, the 20- 24 age cohorts identifies individuals who either decided to stop studying or attend university, individuals in the 25-29 age cohort are in transition between tertiary education and the labour market, the 30-39 age cohort is made of individuals who are likely to have a family with small children, while the 40-49 age cohort comprises individuals with older children. Finally, the 50- 64 age cohort includes mature adults moving towards retirement. The next section (Section \\ref{sec:Data}) describes in detail the data and the methodology used in the analysis.\n\n\n\n\\subsection{Data and methodology\\label{sec:Data}}\n\t\n\tWe use Italian quarterly longitudinal labour force data as provided by the Italian Institute of Statistics (ISTAT) for the period 2013 (quarter I) to 2020 (quarter IV).\\footnote{Data for the period 2013 (quarter I) to 2020 (quarter IV) are available upon request at: https:\/\/www.istat.it\/it\/archivio\/185540.} The Italian Labour Force Survey (LFS) follows a simple rotating sample design where households participate for two consecutive quarters, exit for the following two quarters, and come back in the sample for other two consecutive quarters. As a result, 50\\% of the households, interviewed in a quarter, are re-interviewed after three months, 50\\% after twelve months, 25\\% after nine and fifteen months. This rotation scheme allows to obtain 3 months longitudinal data, which include almost 50\\% of the original sample.\n\nThe longitudinal feature of these data is essential for achieving a complete picture of significant economic phenomena of labour market mobility. Per each individual who has been interviewed we observe a large number of individual and labour market characteristics at the time of the interview and three months before. Taking into account the structure of this database, we compute the labour market flows by calculating the quarter-on-quarter transitions made by background individuals between different labour market states. \n\nOn average approximately 70.000 individuals are interviewed each quarter, of which 45.000 are part of the working age population. The average quarterly inflow of younger individuals in the working age population is 0.3\\%, while the average quarterly outflow of older individuals from the working age population is 0.4\\%, backing our hypothesis of a (almost) constant working age population within quarters.\n\n\nThe dynamics of the labour market can be efficiently described by Markov Chains with discrete states in discrete time. Our dataset allows to consider quarters as unit of time and to define seven labour market states: permanent (PE), temporary (TE), self-employment (SE), unem- ployment (U), the furlough scheme (FS), education (EDU) and inactivity (NLFET). The NLFET state collects the working age individuals who are not in the labour force, in education or in training, therefore representing an accurate measure of inactivity \\citep{ose2017youth}. The dynamics are therefore represented through a Transition Probability Matrix (TPM), which shows both permanence in each labour market state and the probability of transition from one state to another in a given period of time, and fully characterizes the dynamics of the shares of the whole population in each state. In particular, the shares of individuals in different states provide a picture of the long-term trends, as they take longer to react to shocks, while the transition probabilities inform about the sudden impact of the (pandemic) shock. Taking into account the structure of the available database, we compute the labour market flows by calculating the quarter-on-quarter transitions made by individuals between different labour market states. In the analysis we take the first quarter of 2020, which marks the time of the initial spread of the virus, as the period when the dynamics of the Italian labour market are expected to change. The inferential analysis on the shares and \n transition probabilities is computed via bootstrap using 1000 draws from the original sample.\n\n\nImportant data limitations are to be mentioned. First, the point-in-time measurement of the worker's labour market state fails to capture transitions within the period (quarter). For instance, if an employed worker becomes unemployed and finds a new job within a quarter, we do not observe those transitions in our data. Second, the available data stop at quarter IV of 2020, while it would be desirable to have data also for 2021 to explore the further persistence of pandemic shock. Second, we do not have information about the household composition of individuals, as we only observe the household size. For this reason, in the last part of our analysis, we use data from the European Labour Force, which contains detailed information about the number and age of children. Finally, another important limit of our analysis is\nthe short longitudinal span, as we have observations about the same individuals only in two consecutive quarters, thus forcing our analysis to be based on a Markovian process of order one, which is a further limit in the study of persistence.\n\n\t\n\\subsection{Pre-COVID-19 pandemic} \\label{sec:Pre}\n \n \nIn this section we study the dynamics of the labour market before the pandemic (2013 quarter II - 2019 quarter IV), with a special attention to the long-run trends caused by the labour market reforms implemented during the period of observation. For this reason, we consider annual transition probabilities, calculated per each quarter as the product of the last four quarterly transition probabilities.\n \n \\begin{figure}[h!]\n \t\\caption{Shares of individuals aged 15-19 in NLFET and EDU in the North and Center and South of Italy.}\n \t\\label{fig:shares1519text}\n \t\\centering\n \t\\begin{subfigure}[t]{0.24\\textwidth}\n \t\t\\centering\n \t\t\\includegraphics[width=\\linewidth]{actualAverageMass_North1519EDU.eps}\n \t\t\\caption{North and Center - EDU.}\n \t\t\\label{fig:actualAverageMass_North1519EDU}\n \t\t\\vspace{0.1cm}\n \t\\end{subfigure}\n \t\\begin{subfigure}[t]{0.24\\textwidth}\n \t\t\\centering\n \t\t\\includegraphics[width=\\linewidth]{actualAverageMass_South1519EDU.eps}\n \t\t\\caption{South - EDU.}\n \t\t\\label{fig:actualAverageMass_South1519EDU}\n \t\\end{subfigure}\n \t\\begin{subfigure}[t]{0.24\\textwidth}\n \t\t\\centering\n \t\t\\includegraphics[width=\\linewidth]{actualAverageMass_North1519NLFET.eps}\n \t\t\\caption{North and Center - NLFET.}\n \t\t\\label{fig:actualAverageMass_North1519NLFET}\n \t\\end{subfigure}\n \t\\begin{subfigure}[t]{0.24\\textwidth}\n \t\t\\centering\n \t\t\\includegraphics[width=\\linewidth]{actualAverageMass_South1519NLFET.eps}\n \t\t\\caption{South - NLFET.}\n \t\t\\label{fig:actualAverageMass_South1519NLFET}\n \t\\end{subfigure}\n \t\\vspace{0.2cm}\n \t\\caption*{\\scriptsize{\\textit{Note}: Confidence intervals at 90\\% are computed using 1000 bootstraps. \\textit{Source}: LFS 3-month longitudinal data as provided by the Italian Institute of Statistics (ISTAT).}}\n \\end{figure}\n \n Specifically, a number of pension reforms increased retirement age and lead to a raise in the share of individuals aged 50-64 in permanent employment \\citep{de2017dynamics}. Moreover, a number of reforms\\footnote{The \\textit{Decreto Poletti} in 2014 reduced the hiring costs and increased the number of extensions within the same duration, the \\textit{Decreto Dignita'} in 2018 reduced the maximum length, increased the hiring costs and reduced the number of possible extensions} changed the rules for the utilization of temporary contracts, creating from time to time incentives (or disincentives) for firms to hire workers on such contract, leading to increasing or decreasing shares of workers on temporary employment. Finally, in 2015, strong fiscal incentives for the hiring of permanent employees and the introduction of a new permanent contract, with firing costs increasing with tenure, lead to a significant increase in the transitions from unemployment and temporary employment to permanent employment \\citep{boeri2019tale}. \n Net of these trends due to labour market reforms, through the analysis of the shares of individuals across states and the transition probabilities, we find large differences in the labour market choices of individuals in the North and the South of Italy. We also observe very different behaviors between males and females, across different age cohorts. We report all figures by gender, age and geographical location and detailed comments in Appendix \\ref{app:dynamicsprecovid}, while we summarize here the main findings. The different patterns between these four categories of individuals (males and females in the North and in the South) start as early as when they are in secondary education (15-19 age group). On average females stay longer in education in both geographical areas compared to males, however already in this age cohort more individuals drop out school and enter the NLFET state in the South (Figure \\ref{fig:shares1519text}). This pattern is similar among males and females, with approximately 4\\% of individuals aged 15-19 being inactive in the North and already approximately 9\\% being inactive in the South.\n \n \n \n\\begin{figure}[!h]\n\t\\caption{Annual transition probabilities of individuals aged 20-24 from education to temporary employment and the NLFET state in the North and Center and South of Italy.}\n\t\\label{fig:transprob2024femalestext}\n\t\\caption*{\\scriptsize{\\textbf{Females}}.}\n\t\\centering\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{femaleNorthAnnualtransProbFromEDUtoTE_age_class_20-24.eps}\n\t\t\\caption{TE - North and Center.}\n\t\t\\label{fig:femaleNorthAnnualtransProbFromEDUtoTE_age_class_20_I}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{femaleSouthAnnualtransProbFromEDUtoTE_age_class_20-24.eps}\n\t\t\\caption{TE - South.}\n\t\t\\label{fig:femaleSouthAnnualtransProbFromEDUtoTE_age_class_20_I}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{femaleNorthAnnualtransProbFromEDUtoNLFET_age_class_20-24.eps}\n\t\t\\caption{NLFET - North and Center.}\n\t\t\\label{fig:femaleNorthAnnualtransProbFromEDUtoNLFET_age_class_20_I}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{femaleSouthAnnualtransProbFromEDUtoNLFET_age_class_20-24.eps}\n\t\t\\caption{NLFET - South.}\n\t\t\\label{fig:femaleSouthAnnualtransProbFromEDUtoNLFET_age_class_20_I}\n\t\\end{subfigure}\n\t\\caption*{\\scriptsize{\\textbf{Males}}.}\n\t\\centering\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{maleNorthAnnualtransProbFromEDUtoTE_age_class_20-24.eps}\n\t\t\\caption{TE - North and Center.}\n\t\t\\label{fig:maleNorthAnnualtransProbFromEDUtoTE_age_class_20_I}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{maleSouthAnnualtransProbFromEDUtoTE_age_class_20-24.eps}\n\t\t\\caption{TE - South.}\n\t\t\\label{fig:maleSouthAnnualtransProbFromEDUtoTE_age_class_20_I}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{maleNorthAnnualtransProbFromEDUtoNLFET_age_class_20-24.eps}\n\t\t\\caption{NLFET - North and Center.}\n\t\t\\label{fig:maleNorthAnnualtransProbFromEDUtoNLFET_age_class_20_I}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{maleSouthAnnualtransProbFromEDUtoNLFET_age_class_20-24.eps}\n\t\t\\caption{NLFET - South.}\n\t\t\\label{fig:maleSouthAnnualtransProbFromEDUtoNLFET_age_class_20_I}\n\t\\end{subfigure}\n\t\\vspace{0.2cm}\n\t\\caption*{\\scriptsize{\\textit{Note}: Confidence intervals at 90\\% are computed using 1000 bootstraps. \\textit{Source}: LFS 3-month longitudinal data as provided by the Italian Institute of Statistics (ISTAT).}}\n\\end{figure}\n\nThe percentage of individuals (both males and females) who go to university is similar in the two geographical areas, but while in the North and Center, those who leave education are much more likely to enter the labour market, mainly with a temporary contract, in the South they are more likely to join the NLFET state (Figure \\ref{fig:transprob2024femalestext}). Among females approximately 14\\% transit from education to a temporary contract in the North and Center, compared to less than 6\\% in the South; at the same time approximately 8\\% transit to the NLFET state in the North and Center, compared to 12\\% in the South. Among males, 14\\% move from education to temporary employment and 6\\% to the NLFET state in the North and Center, compared to 8\\% and 12\\%, respectively in the South.\nThis worrying bleeding of individuals in the South from education to NLFET state, across both females and males, seems like an irreversible process: persistence in the NLFET state remains very high across all cohorts. Over time more and more individuals in the South, mostly females, keep joining the NLFET state from all other states, particularly from unemployment and temporary employment. This leads to a dramatic situation in which 35\\% of females in the 25-29 age category in the South is in the NLFET state. This percentage keeps growing as they get older, reaching 45\\% among the 30-39 age category (Figure \\ref{fig:shares3039text}) and more than 50\\% among the 40-49 age category. Hence, the percentage of 30-39 years old in permanent employment is dramatically lower in the South, with less than 40\\% of males and 22\\% of females hired on a permanent contract compared to more than 60\\% of males and 50\\% of females in the North and Center, respectively. \n\n\\begin{figure}[!ht]\n\t\\caption{Shares of individuals aged 30-39 in PE and NLFET in the North and Center and South of Italy.}\n\t\\label{fig:shares3039text}\n\t\\centering\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{actualAverageMass_North3039PE.eps}\n\t\t\\caption{PE - North and Center.}\n\t\t\\label{fig:actualAverageMass_North3039PE}\n\t\t\\vspace{0.2cm}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{actualAverageMass_South3039PE.eps}\n\t\\caption{PE - South.}\n\t\\label{fig:actualAverageMass_South3039PE}\n\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{actualAverageMass_North3039NLFET.eps}\n\t\t\\caption{NLFET - North and Center.}\n\t\t\\label{fig:actualAverageMass_North3039NLFET}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{actualAverageMass_South3039NLFET.eps}\n\t\\caption{NLFET - South.}\n\t\\label{fig:actualAverageMass_South3039NLFET}\n\\end{subfigure}\n\t\\vspace{0.2cm}\n\t\\caption*{\\scriptsize{\\textit{Note}: Confidence intervals at 90\\% are computed using 1000 bootstraps. \\textit{Source}: LFS 3-month longitudinal data as provided by the Italian Institute of Statistics (ISTAT).}}\n\\end{figure}\n\n\n\\subsection{Dynamics during the COVID-19 pandemic} \\label{sec:Post}\n\nWe analyse the impact of the pandemic on the labour market of the four categories of individuals (females and males in the North and Center and South of Italy), by age groups. Specifically, in Section \\ref{sec:dynamicsshares} we analyse the dynamics of the shares, while in Section \\ref{sec:dynamicstrans} we focus on the transition probabilities. As we are interested in the sudden impact of the pandemic and its short-run impact, we compute quarterly transition probabilities.\n\n\\subsubsection{The dynamics of shares}\\label{sec:dynamicsshares}\n\nTo investigate the way the pandemic has affected the distribution of individuals across the seven labour market states, we compare the shares of individuals by age category in quarter IV of 2020 with the same quarter one year before (Table \\ref{sharechangesqIVtext}).\\footnote{In Appendix \\ref{app:dynamicssharesduring} we report the same statistics comparing the shares of individuals by age group in quarter III of 2020 with the same quarter one year before.}\n\nIn quarter IV of 2020, the shares of males living in the North and Center of Italy in the NLFET state increased across all age groups, except for the 30-39 cohort, compared to the same quarter one year before, with larger changes among the 20-24 and 25-29 age groups. When we compare those shares with the ones of males living in the South, we notice similar patterns: across (almost) all age categories the share of males in the NLFET state is higher. We also observe a higher share of 25-29 years old in education in the South, while no significant change in the education state in the North and Center. Among females in the North and Center of Italy we also have a larger presence in the NLFET state across all age categories (except for the 15-19 and 50-64 age groups). The increase is particularly important for the 20-24 and 30-39 age categories. However, quite surprisingly, the share of females living in the South in the NFLET state is larger only among the 25-29 age group. Finally, there is an increase in the share of 15-19 and 20-24 years old females in education in the South compared to the same quarter one year before, while no significant change in education in the North and Center. \n\nOverall, we find a substantial increase in the share of males across all age categories in the NLFET state in the North and Center as well as in the South of Italy. With regards to females, while in the North and Center the NLFET share increased across all age categories, in the South it increased only for the 25-29 age category, while it did not change for all other age groups. We also observe a higher share of younger individuals in education in the South (both males and females), but no significant change in the North and Center.\n\n\\begin{table}[htbp] \\centering \n\\caption{Changes in the shares in different labour market states between quarter IV of 2019 and quarter IV of 2020 by category of individuals.} \n\\label{sharechangesqIVtext} \n\\scriptsize{\n\t\\begin{tabular}{@{\\extracolsep{5pt}} cccccccc} \n\t\\\\[-1.8ex]\\hline \\\\[-1.8ex] \n\t\\multicolumn{8}{c}{ Males - North and Center}\\\\\\\\[-1.8ex] \n\t\t\\hline \\\\[-1.8ex] \n\t\t& SE & TE & PE & U & NLFET & EDU & FS \\\\ \n\t\t\\hline \\\\[-1.8ex] \n\t\t15-19 & $0.002$ & $$-$0.012^{***}$ & $0.0004$ & $0.00001$ & $\\textbf{0.007}^{**}$ & $0.002$ & $0.001^{*}$ \\\\ \n\t\t& $(0.113)$ & $(0.002)$ & $(0.424)$ & $(0.482)$ & $(0.045)$ & $(0.383)$ & $(0.053)$ \\\\ \n\t\t20-24 & $0.002$ & $$-$0.011$ & $$-$0.011^{*}$ & $0.005$ & $\\textbf{0.018}^{***}$ & $$-$0.020^{**}$ & $0.016^{***}$ \\\\ \n\t\t& $(0.331)$ & $(0.131)$ & $(0.088)$ & $(0.212)$ & $(0.000)$ & $(0.041)$ & $(0.000)$ \\\\ \n\t\t25-29 & $$-$0.007$ & $$-$0.015^{**}$ & $$-$0.024^{***}$ & $0.003$ & $\\textbf{0.020}^{***}$ & $$-$0.003$ & $0.026^{***}$ \\\\ \n\t\t& $(0.197)$ & $(0.041)$ & $(0.002)$ & $(0.300)$ & $(0.000)$ & $(0.330)$ & $(0.000)$ \\\\ \n\t\t30-39 & $$-$0.004$ & $$-$0.010^{***}$ & $$-$0.016^{***}$ & $$-$0.002$ & $0.003$ & $0.002$ & $0.027^{***}$ \\\\ \n\t\t& $(0.222)$ & $(0.002)$ & $(0.006)$ & $(0.232)$ & $(0.127)$ & $(0.144)$ & $(0.000)$ \\\\\n\t\t40-49 & $$-$0.006$ & $$-$0.008^{***}$ & $$-$0.020^{***}$ & $$-$0.004^{**}$ & $\\textbf{0.013}^{***}$ & $$-$0.0004$ & $0.026^{***}$ \\\\ \n\t\t& $(0.121)$ & $(0.000)$ & $(0.000)$ & $(0.047)$ & $(0.000)$ & $(0.146)$ & $(0.000)$ \\\\\n\t\t50-64 & $$-$0.007^{*}$ & $$-$0.005^{***}$ & $$-$0.009^{**}$ & $$-$0.005^{***}$ & $\\textbf{0.008}^{**}$ & $0.0001$ & $0.018^{***}$ \\\\ \n\t\t& $(0.062)$ & $(0.001)$ & $(0.030)$ & $(0.000)$ & $(0.022)$ & $(0.182)$ & $(0.000)$ \\\\ \n\t\t\\hline \\\\[-1.8ex] \n \\multicolumn{8}{c}{Males - South}\\\\\\\\[-1.8ex] \n \\hline \\\\[-1.8ex] \n & SE & TE & PE & U & NLFET & EDU & FS \\\\ \n \\hline \\\\[-1.8ex] \n15-19 & $0.003$ & $0.010^{***}$ & $$-$0.003$ & $$-$0.019^{***}$ & $$-$0.0002$ & $0.009$ & $0.0003$ \\\\\n & $(0.112)$ & $(0.008)$ & $(0.214)$ & $(0.000)$ & $(0.494)$ & $(0.229)$ & $(0.167)$ \\\\ \n20-24 & $$-$0.009^{**}$ & $$-$0.020^{**}$ & $0.008$ & $$-$0.016^{**}$ & $\\textbf{0.016}^{*}$ & $0.010$ & $0.011^{***}$ \\\\\n& $(0.038)$ & $(0.014)$ & $(0.161)$ & $(0.042)$ & $(0.079)$ & $(0.229)$ & $(0.000)$ \\\\ \n25-29 & $0.002$ & $$-$0.024^{***}$ & $$-$0.018$ & $$-$0.022^{**}$ & $\\textbf{0.017}^{*}$ & $\\textbf{0.034}^{***}$ & $0.012^{***}$ \\\\ \n & $(0.409)$ & $(0.002)$ & $(0.046)$ & $(0.011)$ & $(0.065)$ & $(0.000)$ & $(0.000)$ \\\\ \n30-39 & $$-$0.004$ & $$-$0.014^{***}$ & $$-$0.019^{**}$ & $$-$0.021^{***}$ & $\\textbf{0.027}^{***}$ & $0.004$ & $0.027^{***}$ \\\\\n& $(0.285)$ & $(0.002)$ & $(0.017)$ & $(0.000)$ & $(0.000)$ & $(0.110)$ & $(0.000)$ \\\\ \n40-49 & $0.005$ & $$-$0.012^{***}$ & $$-$0.017^{**}$ & $$-$0.016^{***}$ & $\\textbf{0.016}^{***}$ & $0.0005$ & $0.025^{***}$ \\\\\n & $(0.236)$ & $(0.001)$ & $(0.018)$ & $(0.000)$ & $(0.003)$ & $(0.227)$ & $(0.000)$ \\\\ \n50-64 & $0.003$ & $0.0005$ & $$-$0.007$ & $$-$0.008^{***}$ & $$-$0.005$ & $0.0001$ & $0.017^{***}$ \\\\ \n& $(0.321)$ & $(0.459)$ & $(0.169)$ & $(0.009)$ & $(0.210)$ & $(0.222)$ & $(0.000)$ \\\\ \n\\hline \\\\[-1.8ex] \n \\multicolumn{8}{c}{Females - North and Center}\\\\\\\\[-1.8ex] \n \\hline \\\\[-1.8ex] \n & SE & TE & PE & U & NLFET & EDU & FS \\\\ \n \\hline \\\\[-1.8ex] \n15-19 & $0.001$ & $$-$0.003$ & $0.0004$ & $$-$0.002$ & $0.002$ & $0.001$ & $0.001^{*}$ \\\\\n &$(0.245)$ & $(0.204)$ & $(0.409)$ & $(0.341)$ & $(0.331)$ & $(0.450)$ & $(0.054)$ \\\\ \n20-24 & $$-$0.002$ & $$-$0.041^{***}$ & $$-$0.011^{*}$ & $0.013^{***}$ & $\\textbf{0.034}^{***}$ & $$-$0.001$ & $0.008^{***}$ \\\\\n & $(0.305)$ & $(0.000)$ & $(0.050)$ & $(0.009)$ & $(0.000)$ & $(0.467)$ & $(0.000)$ \\\\ \n25-29 & $$-$0.0001$ & $$-$0.021^{***}$ & $$-$0.027^{***}$ & $0.009^{*}$ & $\\textbf{0.017}^{**}$ & $$-$0.005$ & $0.027^{***}$ \\\\ \n & $(0.485)$ & $(0.005)$ & $(0.002)$ & $(0.077)$ & $(0.022)$ & $(0.244)$ & $(0.000)$ \\\\ \n30-39 & $$-$0.009^{**}$ & $$-$0.007^{*}$ & $$-$0.029^{***}$ & $$-$0.015^{***}$ & $\\textbf{0.034}^{***}$ & $$-$0.001^{}$ & $0.027^{***}$ \\\\ \n & $(0.012)$ & $(0.052)$ & $(0.000)$ & $(0.000)$ & $(0.000)$ & $(0.225)$ & $(0.000)$ \\\\ \n40-49 & $$-$0.009^{***}$ & $$-$0.010^{***}$ & $$-$0.020^{***}$ & $$-$0.007^{***}$ & $\\textbf{0.020}^{***}$ & $0.0004$ & $0.024^{***}$ \\\\ \n & $(0.008)$ & $(0.000)$ & $(0.000)$ & $(0.001)$ & $(0.000)$ & $(0.247)$ & $(0.000)$ \\\\\n50-64 & $$-$0.006^{**}$ & $$-$0.006^{***}$ & $$-$0.007^{*}$ & $$-$0.001$ & $0.002$ & $0.0002$ & $0.018^{***}$ \\\\ \n& $(0.010)$ & $(0.000)$ & $(0.083)$ & $(0.293)$ & $(0.358)$ & $(0.194)$ & $(0.000)$ \\\\ \n \\hline \\\\[-1.8ex] \n \\multicolumn{8}{c}{Females - South}\\\\\\\\[-1.8ex] \n \\hline \\\\[-1.8ex] \n & SE & TE & PE & U & NLFET & EDU & FS \\\\ \n \\hline \\\\[-1.8ex] \n15-19 & $0.002^{*}$ & $$-$0.002$ & $$-$0.005^{***}$ & $$-$0.015^{***}$ & $0.004$ & $\\textbf{0.017}^{**}$ & $0.0003$ \\\\\n & $(0.058)$ & $(0.229)$ & $(0.001)$ & $(0.001)$ & $(0.315)$ & $(0.049)$ & $(0.213)$ \\\\ \n20-24 & $$-$0.014^{***}$ & $$-$0.024^{***}$ & $$-$0.018^{***}$ & $$-$0.019^{**}$ & $0.013$ & $\\textbf{0.050}^{***}$ & $0.013^{***}$ \\\\\n & $(0.000)$ & $(0.000)$ & $(0.001)$ & $(0.017)$ & $(0.143)$ & $(0.000)$ & $(0.000)$ \\\\ \n25-29 & $$-$0.006$ & $$-$0.020^{***}$ & $$-$0.016^{**}$ & $$-$0.037^{***}$ & $\\textbf{0.058}^{***}$ & $0.007$ & $0.014^{***}$ \\\\\n & $(0.166)$ & $(0.003)$ & $(0.041)$ & $(0.000)$ & $(0.000)$ & $(0.254)$ & $(0.000)$ \\\\ \n30-39 & $0.001$ & $0.007$ & $$-$0.013^{**}$ & $$-$0.022^{***}$ & $0.010$ & $$-$0.001$ & $0.017^{***}$ \\\\ \n& $(0.418)$ & $(0.109)$ & $(0.042)$ & $(0.000)$ & $(0.154)$ & $(0.424)$ & $(0.000)$ \\\\\n40-49 & $0.003$ & $$-$0.001$ & $$-$0.023^{***}$ & $$-$0.005$ & $0.011$ & $0.002^{*}$ & $0.013^{***}$ \\\\\n & $(0.304)$ & $(0.461)$ & $(0.000)$ & $(0.148)$ & $(0.116)$ & $(0.086)$ & $(0.000)$ \\\\ \n50-64 & $0.001$ & $$-$0.005^{**}$ & $$-$0.013^{***}$ & $0.001$ & $0.009$ & $0.0004$ & $0.007^{***}$ \\\\\n & $(0.367)$ & $(0.024)$ & $(0.007)$ & $(0.284)$ & $(0.114)$ & $(0.143)$ & $(0.000)$ \\\\ \n \\hline \\\\[-1.8ex] \n\\end{tabular} \n}\n\\end{table} \n\n\\subsubsection{The dynamics of transition probabilities}\\label{sec:dynamicstrans}\n\nTo assess the way the pandemic has affected the transition probabilities across labour market states, we compare the actual data with the counterfactual scenario of no pandemic shock, i.e., the quarterly transition probabilities for the four categories of individuals across demographic groups against the forecasted quarterly transition probabilities for the quarters during the pandemic, i.e., quarter I of 2020 to quarter IV of 2020.\\footnote{The forecasted transition probabilities are computed using a combination of four forecasting models (ETS, TSLM, THETAF, and ARIMA) \\citep{HyndmanAthanasopoulosforecasting2021} in the period 2013 quarter I- 2019 quarter IV.} \n\nTable \\ref{fig:transProbUtoNLFET} reports the transition probabilities from unemployment to the NLFET state for all categories of individuals, thus capturing whether an increased number of discouraged workers who are pessimistic about the probability to find a job gave up on their search. Across all groups and across all age cohorts these probabilities have significantly increased in quarter II of 2020, compared to the forecasted probabilities. While for some categories the change was temporary and went back to the pre-pandemic rates in quarter III of 2020, for others, such as 20-24 and 40-49 females in the South the increased percentage persisted until quarter IV of 2020. Similar patterns are observed among 30-39 males in the South, while among both males and females in the North and Center, we do not observe persistence. Hence, while in the outburst of the pandemic all unemployed workers across different types got discouraged to some degree and left the labour market, the higher transition rates from unemployment to inactivity persisted mostly for females in the South. \n\n\\begin{figure}[htbp]\n\t\\caption{Transition probabilities from unemployment to the NLFET state by age groups.}\n\t\\label{fig:transProbUtoNLFET}\n\t\\vspace{0.1cm}\n\t\\caption*{\\scriptsize{\\textbf{15-19}}.}\n\t\\vspace{0.1cm}\n\t\\centering\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_South_transProbtoNLFETFromU_age_class_15-19.eps}\n\t\t\\caption{Female South.}\n\t\t\\label{fig:female_South_transProbtoNLFETFromU_age_class_15}\n\t\t\\vspace{0.2cm}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_North_transProbtoNLFETFromU_age_class_15-19.eps}\n\t\t\\caption{Female North and Center.}\n\t\t\\label{fig:female_North_transProbtoNLFETFromU_age_class_15}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{male_South_transProbtoNLFETFromU_age_class_15-19.eps}\n\t\t\\caption{Male South.}\n\t\t\\label{fig:male_South_transProbtoNLFETFromU_age_class_15}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_North_transProbtoNLFETFromU_age_class_15-19.eps}\n\t\t\\caption{Male North and Center.}\n\t\t\\label{fig:female_North_transProbtoNLFETFromU_age_class_15}\n\t\\end{subfigure}\n\t\\vspace{0.1cm}\n\t\\caption*{\\scriptsize{\\textbf{20-24}}.}\n\t\\vspace{0.1cm}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_South_transProbtoNLFETFromU_age_class_20-24.eps}\n\t\t\\caption{Female South.}\n\t\t\\label{fig:female_South_transProbtoNLFETFromU_age_class_20}\n\t\t\\vspace{0.2cm}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_North_transProbtoNLFETFromU_age_class_20-24.eps}\n\t\t\\caption{Female North and Center.}\n\t\t\\label{fig:female_North_transProbtoNLFETFromU_age_class_20}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{male_South_transProbtoNLFETFromU_age_class_20-24.eps}\n\t\t\\caption{Male South.}\n\t\t\\label{fig:male_South_transProbtoNLFETFromU_age_class_20}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{male_North_transProbtoNLFETFromU_age_class_20-24.eps}\n\t\t\\caption{Male North and Center.}\n\t\t\\label{fig:male_North_transProbtoNLFETFromU_age_class_20}\n\t\\end{subfigure}\n\t\\vspace{0.1cm}\n\t\\caption*{\\scriptsize{\\textbf{25-29}}.}\n\t\\vspace{0.1cm}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_South_transProbtoNLFETFromU_age_class_25-29.eps}\n\t\t\\caption{Female South.}\n\t\t\\label{fig:female_South_transProbtoNLFETFromU_age_class_25}\n\t\t\\vspace{0.2cm}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_North_transProbtoNLFETFromU_age_class_25-29.eps}\n\t\t\\caption{Female North and Center.}\n\t\t\\label{fig:female_North_transProbtoNLFETFromU_age_class_25}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{male_South_transProbtoNLFETFromU_age_class_25-29.eps}\n\t\t\\caption{Male South.}\n\t\t\\label{fig:male_South_transProbtoNLFETFromU_age_class_25}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{male_North_transProbtoNLFETFromU_age_class_25-29.eps}\n\t\t\\caption{Male North and Center.}\n\t\t\\label{fig:male_North_transProbtoNLFETFromU_age_class_25}\n\t\\end{subfigure}\n\t\\vspace{0.1cm}\n\t\\caption*{\\scriptsize{\\textbf{30-39}}.}\n\t\\vspace{0.1cm}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_South_transProbtoNLFETFromU_age_class_30-39.eps}\n\t\t\\caption{Female South.}\n\t\t\\label{fig:female_South_transProbtoNLFETFromU_age_class_30}\n\t\t\\vspace{0.2cm}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_North_transProbtoNLFETFromU_age_class_30-39.eps}\n\t\t\\caption{Female North and Center.}\n\t\t\\label{fig:female_North_transProbtoNLFETFromU_age_class_30}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{male_South_transProbtoNLFETFromU_age_class_30-39.eps}\n\t\t\\caption{Male South.}\n\t\t\\label{fig:male_South_transProbtoNLFETFromU_age_class_30}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{male_North_transProbtoNLFETFromU_age_class_30-39.eps}\n\t\t\\caption{Male North and Center.}\n\t\t\\label{fig:male_North_transProbtoNLFETFromU_age_class_30}\n\t\\end{subfigure}\n\t\\vspace{0.1cm}\n\t\\caption*{\\scriptsize{\\textbf{40-49}}.}\n\t\\vspace{0.1cm}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_South_transProbtoNLFETFromU_age_class_40-49.eps}\n\t\t\\caption{Female South.}\n\t\t\\label{fig:female_South_transProbtoNLFETFromU_age_class_40}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_North_transProbtoNLFETFromU_age_class_40-49.eps}\n\t\t\\caption{Female North and Center.}\n\t\t\\label{fig:female_North_transProbtoNLFETFromU_age_class_40}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{male_South_transProbtoNLFETFromU_age_class_40-49.eps}\n\t\t\\caption{Male South.}\n\t\t\\label{fig:male_South_transProbtoNLFETFromU_age_class_40}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{male_North_transProbtoNLFETFromU_age_class_40-49.eps}\n\t\t\\caption{Male North and Center.}\n\t\t\\label{fig:male_North_transProbtoNLFETFromU_age_class_40}\n\t\\end{subfigure}\n\t\\vspace{0.2cm}\n\t\\caption*{ \\scriptsize{\\textit{Note}: The forecasted transition probabilities are computed using a combination of four forecasting models (ETS, TSLM, THETAF, and ARIMA) \\citep{HyndmanAthanasopoulosforecasting2021} in the period 2013 (quarter I)- 2019 (quarter IV). Confidence intervals at 90\\% are computed using 1000 bootstraps and reported in parenthesis. EDU refers to education, TE to temporary employment, PE to permanent employment, U to unemployment. \\textit{Source}: LFS 3-month longitudinal data as provided by the Italian Institute of Statistics (ISTAT).}}\n\\end{figure}\n\nWe then focus on the transition rates from permanent and temporary employment to the NLFET state (Tables \\ref{fig:transProbTEtoNLFET}-\\ref{fig:transProbPEtoNLFET}) to assess the effect of the pandemic on the outflows from permanent and temporary employment. We report statistics for the 30-39 and 40-49 categories for which the results are more striking.\\footnote{Statistics for all other age categories are reported in Appendix \\ref{app:dynamicssharesduring}.} The pandemic significantly increased the transition probabilities from temporary and permanent employment to the NLFET state mostly for females, and among those mainly for the ones living in the North and Center, across both age categories. Data show that females in the North and Center are the ones who are more affected by the pandemic as compared to women in the South, as a larger percentage was active on the labour market at the time of the shock. In quarter III of 2020 females aged 30-39 (40-49) had a probability to transit from temporary employment to the NLFET state of about 25\\% (19\\%) compared to the forecasted 10\\% (12\\%). Similar patterns are found for the probability to transit from permanent employment to the NLFET state: it jumped to 2.5\\% (1.3\\%) compared to a forecasted probability of 1.7\\% (0.5\\%) for women aged 30-39 (40-49). Although these numbers seem low, they correspond to approximately 40.000 women in the age 30-39 cohort and 25.000 women in the age 40-49 cohort moving from employment to inactivity.\n\n\n\\subsubsection{Female labour market participation and household composition}\n\nDuring the pandemic due to the prolonged schools closure, women had to juggle between their jobs and the children care \\citep{qian2020covid}. Particularly women with small children might have been pushed out of the labour market due to caring responsibilities. However, in the South, where the female inactivity rate in these age cohorts was already much higher, the impact of the pandemic has been less strong. Unfortunately, our data do not provide information about the number and age of children, but only about the household size. Therefore, we use European Labour Force Survey data for Italy for 2019 to compute the shares of females in the age cohorts 30-39 and 40-49 with at least one child below the age of 11 by employment status, distinguishing between the North and Center and South of Italy (Table \\ref{sharechangesELFS2019}).\\footnote{In the Appendix we report similar statistics for 2020.} We select the age of children to be below 11, which we consider the cutoff age, below which children need the presence of their parents. \nWe find that 47.5\\% of females in the 30-39 age group with at least one young child is employed on a permanent contract in the North and Center, compared to 19.1\\% in the South. While the share on temporary employment and self-employment is comparable, we find the share in unemployment to be higher in the South (10.2\\% against 6.2\\%) while the share of inactive females is much higher in the South, i.e., 55.7\\% compared to 28.2\\%. When we focus on the 40-49 age group, we observe very similar patterns.\n\n\n\n\\begin{figure}[htbp]\n\t\\caption{Transition probabilities from temporary employment to the NLFET state by age groups.}\n\t\\label{fig:transProbTEtoNLFET}\n\t\\vspace{0.1cm}\n\\caption*{\\scriptsize{\\textbf{30-39}}.}\n\\vspace{0.1cm}\n\t\\centering\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_South_transProbtoNLFETFromTE_age_class_30-39.eps}\n\t\t\\caption{Female South.}\n\t\t\\label{fig:female_South_transProbtoNLFETFromTE_age_class_30}\n\t\t\\vspace{0.2cm}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_North_transProbtoNLFETFromTE_age_class_30-39.eps}\n\t\t\\caption{Female North and Center.}\n\t\t\\label{fig:female_North_transProbtoNLFETFromTE_age_class_30}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{male_South_transProbtoNLFETFromTE_age_class_30-39.eps}\n\t\t\\caption{Male South.}\n\t\t\\label{fig:male_South_transProbtoNLFETFromTE_age_class_30}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{male_North_transProbtoNLFETFromTE_age_class_30-39.eps}\n\t\t\\caption{Male North and Center.}\n\t\t\\label{fig:male_North_transProbtoNLFETFromTE_age_class_30}\n\t\\end{subfigure}\n\t\\vspace{0.1cm}\n\\caption*{\\scriptsize{\\textbf{40-49}}.}\n\\vspace{0.1cm}\n\t\\centering\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_South_transProbtoNLFETFromTE_age_class_40-49.eps}\n\t\t\\caption{Female South.}\n\t\t\\label{fig:female_South_transProbtoNLFETFromTE_age_class_40}\n\t\t\\vspace{0.2cm}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_North_transProbtoNLFETFromTE_age_class_40-49.eps}\n\t\t\\caption{Female North and Center.}\n\t\t\\label{fig:female_North_transProbtoNLFETFromTE_age_class_40}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{male_South_transProbtoNLFETFromTE_age_class_40-49.eps}\n\t\t\\caption{Male South.}\n\t\t\\label{fig:male_South_transProbtoNLFETFromTE_age_class_40}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{male_North_transProbtoNLFETFromTE_age_class_40-49.eps}\n\t\t\\caption{Male North and Center.}\n\t\t\\label{fig:male_North_transProbtoNLFETFromTE_age_class_40}\n\t\\end{subfigure}\n\t\\vspace{0.2cm}\n\t\\caption*{ \\scriptsize{\\textit{Note}: The forecasted transition probabilities are computed using a combination of four forecasting models (ETS, TSLM, THETAF, and ARIMA) \\citep{HyndmanAthanasopoulosforecasting2021} in the period 2013 (quarter I)- 2019 (quarter IV). Confidence intervals at 90\\% are computed using 1000 bootstraps and reported in parenthesis. EDU refers to education, TE to temporary employment, PE to permanent employment, U to unemployment. \\textit{Source}: LFS 3-month longitudinal data as provided by the Italian Institute of Statistics (ISTAT).}}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\t\\caption{Transition probabilities from permanent employment to the NLFET state by age groups.}\n\t\\label{fig:transProbPEtoNLFET}\n\t\\vspace{0.1cm}\n\\caption*{\\scriptsize{\\textbf{30-39}}.}\n\\vspace{0.1cm}\n\t\\centering\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_South_transProbtoNLFETFromPE_age_class_30-39.eps}\n\t\t\\caption{Female South.}\n\t\t\\label{fig:female_South_transProbtoNLFETFromPE_age_class_30}\n\t\t\\vspace{0.2cm}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_North_transProbtoNLFETFromPE_age_class_30-39.eps}\n\t\t\\caption{Female North and Center.}\n\t\t\\label{fig:female_North_transProbtoNLFETFromPE_age_class_30}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{male_South_transProbtoNLFETFromPE_age_class_30-39.eps}\n\t\t\\caption{Male South.}\n\t\t\\label{fig:male_South_transProbtoNLFETFromPE_age_class_30}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{male_North_transProbtoNLFETFromPE_age_class_30-39.eps}\n\t\t\\caption{Male North and Center.}\n\t\t\\label{fig:male_North_transProbtoNLFETFromPE_age_class_30}\n\t\\end{subfigure}\n\t\\vspace{0.1cm}\n\\caption*{\\scriptsize{\\textbf{40-49}}.}\n\\vspace{0.1cm}\n\t\\centering\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_South_transProbtoNLFETFromPE_age_class_40-49.eps}\n\t\t\\caption{Female South.}\n\t\t\\label{fig:transProbFromEDUtoSE_APP}\n\t\t\\vspace{0.2cm}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{female_North_transProbtoNLFETFromPE_age_class_40-49.eps}\n\t\t\\caption{Female North and Center.}\n\t\t\\label{fig:transProbFromEDUtoTE_APP}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{male_South_transProbtoNLFETFromPE_age_class_40-49.eps}\n\t\t\\caption{Male South.}\n\t\t\\label{fig:transProbFromEDUtoPE_APP}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{male_North_transProbtoNLFETFromPE_age_class_40-49.eps}\n\t\t\\caption{Male North and Center.}\n\t\t\\label{fig:transProbFromEDUtoU_APP}\n\t\\end{subfigure}\n\t\\vspace{0.2cm}\n\t\\caption*{ \\scriptsize{\\textit{Note}: The forecasted transition probabilities are computed using a combination of four forecasting models (ETS, TSLM, THETAF, and ARIMA) \\citep{HyndmanAthanasopoulosforecasting2021} in the period 2013 (quarter I)- 2019 (quarter IV). Confidence intervals at 90\\% are computed using 1000 bootstraps and reported in parenthesis. EDU refers to education, TE to temporary employment, PE to permanent employment, U to unemployment. \\textit{Source}: LFS 3-month longitudinal data as provided by the Italian Institute of Statistics (ISTAT).}}\n\\end{figure}\n\n\n\n\\begin{table}[!htbp] \\centering \n\t\\caption{Percentage of females with at least one child below the age of 11 by geographical area and employment status (2019).} h\n\t\\label{sharechangesELFS2019} \n\t\\scriptsize\n\t\\begin{tabular}{@{\\extracolsep{5pt}} ccccc} \n\t\t\\\\[-1.8ex]\\hline \\\\[-1.8ex] \n\t\t&\\multicolumn{2}{c}{Age 30-39}&\\multicolumn{2}{c}{Age 40-49}\\\\\t\n\t\t\\hline \\\\[-1.8ex]\n\t\t&North and Center&South&North and Center&South\\\\\n\t\t\\hline \\\\[-1.8ex]\n\t\tPermanent &47.5&19.1&55.0&28.1\\\\\n\t\tTemporary&8.7&7.1&6.4&5.9\\\\\n\t\tSelf-employed&8.6&6.9&12.1&9.2\\\\\n\t\tUnemployed& 6.2&10.2&5.4&8.4\\\\\n\t\tInactive&\t28.2&55.7&20.2&48.0\\\\\n\t\t\\hline \\\\[-1.8ex]\n\t\tTotal (in 000s)& 1270&736&1249&559\\\\\n\t\t\\hline \n\t\t\\multicolumn{5}{l}{\\tiny{\\textit{Source}: ELFS data.}}\n\t\\end{tabular}\t\n\\end{table}\n\nWe then split the sample of females with at least one child below the age of 11 by household size: we compute the share of females living in a household with less or more than two people, as a proxy for the presence of children, by geographical area and age cohort (Table \\ref{shareHH2ELFS2019}). \n\n\\begin{table}[!htbp] \\centering \n\t\\caption{Percentage of females with at least one child below the age of 11 by geographical area and household size.} \n\t\\label{shareHH2ELFS2019} \n\t\\scriptsize\n\t\\begin{tabular}{@{\\extracolsep{5pt}} ccccc} \n\t\t\\\\[-1.8ex]\\hline \\\\[-1.8ex] \n\t\t&\\multicolumn{2}{c}{Age 30-39}&\\multicolumn{2}{c}{Age 40-49}\\\\\t\n\t\t\\hline \\\\[-1.8ex]\n\t\t&North and Center&South&North and Center&South\\\\\n\t\t\\hline \\\\[-1.8ex]\n\t\t$>$ 2 components&81.2& 71.5&55.2&44.0\\\\\n\t\t$\\leq$ 2 components& 6.6&9.3&6.0&4.8\\\\ \n\t\t\n\t\t\\hline \n\t\t\\multicolumn{5}{l}{\\tiny{\\textit{Source}: ELFS data.}}\n\t\\end{tabular}\t\n\\end{table}\t\n\nAmong females in the 30-39 age cohort, 81.2\\% live in a household with more than two components in the North and Center, and 71.5\\% in the South. This implies that among females in the 30-39 age category the household size is a good proxy to assess whether they have children below the age of 11. We use this information to compute the transition probabilities from permanent and temporary employment to the NLFET state for the two groups of females in the North and Center and in the South using the quarterly longitudinal labour force data (Figure \\ref{fig:transProbHHsize}).\n\n\\begin{figure}[!ht]\n\t\\caption{Transition probabilities of females aged 30-39 from permanent and temporary employment, to the NLFET in the North and Center and South of Italy by household size.}\n\t\\label{fig:transProbHHsize}\n\t\\caption*{\\scriptsize{\\textbf{From permanent employment}}.}\n\t\\centering\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{FemaleNorthHHmore2_transProbtoNLFETfromPE_age_class_30-39.eps}\n\t\t\\caption{North and Center - Household $>$ 2.}\n\t\t\\label{fig:FemaleNorthHHmore2_transProbtoNLFETfromPE_age_class_30}\n\t\t\\vspace{0.1cm}\n\t\t\\vspace{0.1cm}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{FemaleNorthHHless2_transProbtoNLFETfromPE_age_class_30-39.eps}\n\t\t\\caption{North and Center - Household $\\leq$ 2.}\n\t\t\\label{fig:FemaleNorthHHless2_transProbtoNLFETfromPE_age_class_30}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{FemaleSouthHHmore2_transProbtoNLFETfromPE_age_class_30-39.eps}\n\t\t\\caption{South - Household $>$ 2.}\n\t\t\\label{fig:FemaleSouthHHmore2_transProbtoNLFETfromPE_age_class_30}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.24\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{FemaleSouthHHless2_transProbtoNLFETfromPE_age_class_30-39.eps}\n\t\t\\caption{South - Household $\\leq$ 2.}\n\t\t\\label{fig:FemaleSouthHHless2_transProbtoNLFETfromPE_age_class_30}\n\t\\end{subfigure}\n\t\\caption*{\\scriptsize{\\textbf{From temporary employment}}.}\n\t\\centering\n\\begin{subfigure}[t]{0.24\\textwidth}\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{FemaleNorthHHmore2_transProbtoNLFETfromTE_age_class_30-39.eps}\n\t\\caption{North and Center - Household $>$ 2.}\n\t\\label{fig:FemaleNorthHHmore2_transProbtoNLFETfromTE_age_class_30}\n\t\\vspace{0.1cm}\n\t\\vspace{0.1cm}\n\\end{subfigure}\n\\begin{subfigure}[t]{0.24\\textwidth}\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{FemaleNorthHHless2_transProbtoNLFETfromTE_age_class_30-39.eps}\n\t\\caption{North and Center - Household $\\leq$ 2.}\n\t\\label{fig:FemaleNorthHHless2_transProbtoNLFETfromTE_age_class_30}\n\\end{subfigure}\n\\begin{subfigure}[t]{0.24\\textwidth}\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{FemaleSouthHHmore2_transProbtoNLFETfromTE_age_class_30-39.eps}\n\t\\caption{South - Household $>$ 2.}\n\t\\label{fig:FemaleSouthHHmore2_transProbtoNLFETfromTE_age_class_30}\n\\end{subfigure}\n\\begin{subfigure}[t]{0.24\\textwidth}\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{FemaleSouthHHless2_transProbtoNLFETfromTE_age_class_30-39.eps}\n\t\\caption{South - Household $\\leq$ 2.}\n\t\\label{fig:FemaleSouthHHless2_transProbtoNLFETfromTE_age_class_30}\n\\end{subfigure}\n\t\\vspace{0.2cm}\n\t\\caption*{\\scriptsize{\\textit{Note}: Confidence intervals at 90\\% are computed using 1000 bootstraps. \\textit{Source}: LFS 3-month longitudinal data as provided by the Italian Institute of Statistics (ISTAT).}}\n\\end{figure}\n\nWe find that the transition probabilities to the NLFET state have significantly and persistently increased for females in the 30-39 age group living in the North and Center in a household with more than two people from both permanent and temporary employment. We find no statistically significant effect for females living in households with less than two components neither in the North and Center or the South, neither from permanent nor from temporary employment. Interestingly, we also find no significant effect for females living in the South in a household with more than two people from both permanent and temporary employment. These results confirm our previous evidence of a stronger pandemic impact on females with young children living in the North and Center of Italy. Those women are likely to have been forced out of the labour market to take care of their children during the prolonged school closure. The worrying part of the story is the persistence of this effect until the end of 2020, as the literature has provided large evidence of weaker attachment of females to the labour market compared to males and of high female state dependency in the inactivity state \\citep{duhautois2018state}. As explained by \\cite{honore2000panel}, this state dependency might be of two types: either because lagged decisions play a role (true state dependency) or because of different propensities across individuals to experience the event (spurious state dependency). Distinguishing between these two types is extremely important for policy reasons, however long longitudinal data on individual histories are required, which unfortunately we do not have availability of. This is therefore part of our future agenda.\n\n\\subsection{Estimation of the probability to remain active in the labour market}\\label{sec:logit}\n\nIn this section we provide additional support to our findings by using a parametric estimation technique.\nWe group observations according to whether the individuals are in or out of the labour force, as our analysis so far has shown that this is the most important feature to be attributed to the pandemic. The estimation is carried out using a \\textit{logit} model, in which the dependent variable is the probability to be active on the labour market in the next quarter conditional on being active the current quarter.\\footnote{Alternatively, we could have estimated a multinomial logit, taking into account the transitions to all seven labour market states considered in our analysis so far. However, the validity of such estimation would have depended on the fulfillment of the Independence of Irrelevant Alternatives (IIA) condition \\citep{train2009discrete}. In our case the IIA assumes that the odds-ratio is not altered with the addition or deletion of a particular alternative, which is very likely to be violated in our setting.} We split the sample into four sub-samples, according to gender and age. Specifically, we focus on the 30-39 and 40-49 age cohorts as these are the ones who have been more severely affected by the COVID-19 pandemic. Permanence in the initial state of activity is the baseline category for interpretation of results.\nThe estimation of a logit model using design-based longitudinal weights may create severe numerical issues \\citep{train2009discrete}. Hence, we run 1000 bootstraps using the longitudinal sample weights to estimate the model's coefficients and their 95\\% confidence intervals. \n\n\\begin{table}[!htbp] \\centering \n\t\\caption{Odds-ratios of remaining active on the labour market next period conditional on being active.} \n\t\\label{tab:logit} \n\t\\scriptsize{\n\t\t\\begin{tabular}{@{\\extracolsep{5pt}}lcccc} \n\t\t\t\\\\[-1.8ex]\\hline \n\t\t\n\t\t\n\t\t\t\\\\[-1.8ex] & \\multicolumn{4}{c}{Active in the labour market next quarter} \\\\\n\t\t\t\\hline \t\\\\[-1.8ex]\n\t\t\t\\\\[-1.8ex] & Female 30-39 & Male 30-39 & Female 40-49 & Male 40-49\\\\ \n\t\t\t\\hline \\\\[-1.8ex] \n\n\t\t\t2020 & \\textbf{0.509} & \\textbf{0.627} & 0.745 & 0.750 \\\\ \n\t\t\t& (0.374-0.701) & (0.460-0.834) & (0.526-1.064) & (0.543-1.022) \\\\ \n\t\t\t& & & & \\\\ \n\n\t\t\t\\\\[-1.8ex] \n\t\t\t\\textbf{2020 $\\times$(North or Center)$\\times$}& \\textbf{0.614 }& 0.761 & 0.953 & 1.272 \\\\ \n\t\t\t\\textbf{$\\times$(Household members >2)}& (0.401-0.892) & (0.473-1.113) & (0.621-1.406) & (0.850-1.854) \\\\ \n\t\t\t& & & & \\\\ \n\t\t\t(Intercept) & 18.228 & 23.247 & 25.899 & 39.220 \\\\ \n\t\t\t& (15.075-21.960) & (18.958-29.223) & (21.354-31.183) & (30.986-48.260) \\\\ \n\t\t\t& & & & \\\\ \n\t\t\t\\hline \\\\[-1.8ex] \n\t\t\tObservations & 62,950 & 78,307 & 95,990 & 120,232 \\\\ \n\t\t\tLog Likelihood & $-$14,065.210 & $-$12,152.510 & $-$16,144.900 & $-$14,945.660 \\\\ \n\t\t\tAkaike Inf. Crit. & 28,172.430 & 24,347.030 & 32,331.800 & 29,933.320 \\\\ \n\t\t\t\n\t\t\t\\hline \\\\[-1.8ex] \n\t\t\t\\multicolumn{5}{l}{\\textit{Note:} Confidence interval at 95\\% are calculated by 1000 bootstrap using sample weights.} \\\\ \n\t\t\\end{tabular} \n\t}\n\\end{table}\n\nTable \\ref{tab:logit} displays the estimated odds-ratios of the probability to remain active on the labour market, conditional on being active the quarter before given a set of explanatory variables. In particular, the odds-ratio represents the ratio between the probability that the event will occur with respect to the probability the event will not occur, conditioned to a given explanatory variable; hence, an odds-ratio grater than one implies an increased occurrence of the event, while an odds-ratio lower than one implies a decreased occurrence of the event with respect to a given explanatory variable. The reference category includes Italian individuals with a tertiary level of education living in the South in a household with less than two people.\n\nThe full regression with all the variables included is reported in Table \\ref{apptab:logit} in Appendix \\ref{app:logit}. Immigrants, both from EU and non-EU countries, are more likely to exit the labour market than Italian citizens across all four categories. Not surprisingly, primary and secondary educated individuals show a lower likelihood to persist in the active state, compared to tertiary educated individuals. The geographical location is important: living in the North and Center increases the likelihood to remain active, compared to living in the South, and the effect is much stronger for males than females. Finally, among females in both age groups, living in a household with more than two people significantly decreases the probability to remain active on the labour market. We also find evidence of seasonality, which means in quarter III among both the 30-39 and 40-49 age cohorts, for males, an increased persistence in the active state, while for females, a significant decrease in the probability to remain active. \n\nRegarding the effect of the pandemic, the probability to remain active either did not change or increased before 2020; however, with the pandemic (year 2020) we estimate a significant decrease in the probability of remaining active among individuals in the 30-39 age cohort (both males and females). Specifically, the odds-ratio is 0.5, meaning that the probability of remaining active in 2020 is half the same probability in previous years.\nImportantly, for females in the 30-39 age cohort resident in the North and Center of Italy in a household with more than two people, the pandemic has significantly reduced the probability to be active on the labour market. The odds-ratio for this category of individuals is 0.6, implying that the probability of remaining active during the pandemic is almost half than it was before the shock. This result confirms our previous finding that women with small children living in the North and Center were more likely to go out from the labour market during the pandemic. We argue that part of explanation can be found in the heavier caring responsibilities caused by the long-lasting school closure in the lockdown and post-lockdown periods in Italy. Among women in the South we do not observe any significant change, probably due to the already high inactivity rate among this category of individuals before the pandemic.\n\n \n\n\\section{Concluding remarks \\label{sec:concludingRemarks}}\n\t\nIn this paper, we assess the short-term impact of the COVID-19 pandemic on the Italian labour market by studying how the shares and transition probabilities of individuals between labour market states have changed after the country entered a full lockdown on March 10, 2020. \nWe find remarkable asymmetric effects across gender, age and geographical location, which we attribute to the very different state of the labour market different categories of individuals faced at the outburst of the pandemic. Specifically, we show evidence of an increased number of discouraged workers who exited the unemployment state and joined the NLFET state across gender, age groups and geographical location, but mostly among individuals in the South. Most worrying, we find evidence of substantial outflows of females in the North and Center of Italy in their 30s with small children leaving employment, either permanent or temporary, and becoming inactive. To appreciate \nthe magnitude of the shock, in quarter III of 2020 females aged 30-39 (40-49) had a quarterly transition probability from temporary employment to NLFET of about 25\\% (19\\%) compared to the forecasted 10\\% (12\\%) in absence of the pandemic shock. The quarterly transition probability from permanent employment to NLFET jumped to 2.5\\% (1.3\\%) compared to a forecasted probability of 1.7\\% (0.5\\%) for women aged 30-39 (40-49). Although these numbers seem low, they correspond to approximately 40.000 women in the age 30-39 cohort and 25.000 women in the age 40-49 cohort moving from employment to inactivity in a quarter.\nSurprisingly, we do not find the same strong outflows of women in the South from employment to inactivity. We argue that the already high share of females in their 30s and 40s in NLFET could have played a role, i.e., the women who are active on the labour market in Southern regions are strongly self-selected.\nThe outflow of women in the North and Center in their 30s with young children from employment to inactivity which started in the beginning of the pandemic persisted until the end of 2020, making likely the presence of long-lasting scarring effects on the Italian female labour force participation.\n\n\\newpage\n\t\n\\bibliographystyle{chicago}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nIn just a few years since its inception, the BitTorrent protocol and similar systems have become the predominant P2P file sharing model~\\cite{bittorrent}. But the recent activities of those seeking to take down P2P infrastructures have forced the file sharing community to adapt to a hostile environment~\\cite{isps}. Operators of global BitTorrent trackers now take two notable measures in order to indemnify themselves from legal action: (1) the trackers are located in countries that are not party to international copyright treaties, and (2) access to trackers is controlled by private, invite-only communities with strict membership requirements~\\cite{bittorrent-communities}. The former allows operators to ignore legal threats to shutdown their services that a law-abiding ISP would normally have to comply with. But this approach can be both prohibitively expensive and difficult to arrange. Additionally, limiting access to only privileged users only temporarily protects a site that has been made private; it takes only a single seditious user to undermine the network and provide damaging evidence to the right parties.\n\nThe aforementioned measures may protect tracker operators but they provide little protection to the average file sharing user. This is because the fundamental principle of the BitTorrent protocol is that users download and upload data directly with other untrusted users, rather than download from a single, central source~\\cite{bittorrent}. Although some P2P clients employ communication encryption and protocol obfuscation enhancements, such measures do not protect a user from malicious clients that harvest file sharing activity information for future litigation. Furthermore, it has been shown that while it may not be possible to easily view encrypted packet contents, a third-party observer can still deduce that file sharing is occurring by identifying network pairs based on a tracker's public peer list~\\cite{sandvine,isps}.\n\nAnother limitation of current BitTorrent-like models is that the networks rely on altruistic users to keep files available for others. This is problematic in an environment where users want to limit their exposure to any traffic logging clients, and thus it is in their interest to disconnect immediately once they have the successfully downloaded their desired files. Content in these networks is unavailable once all of the peers that have the complete file depart. Newly arriving clients may be able to download and share some fraction of the data (if any is available), but they must wait and hope that a client returns to the network with the rest of file. Enhancements to private trackers, such as upload\/download ratios, provide incentives for clients to continue to seed files~\\cite{bittorrent-communities}, but these economic models are difficult to initiate and do little to maintain less popular older files.\n\nIn response to the lack of user anonymity and long-term data persistence in existing P2P systems, some users may seek an alternative. But because traditional data hosting solutions are not a viable option for sharing certain content that may have legal consequences, these users must use more questionable means for sharing data. Motivated by this, we developed the \\textit{Graffiti Network} distributed file sharing protocol that uses multiple third-party storage sites as a data replication and transfer medium between clients. The Graffiti approach is to use publically available web sites to store multiple copies of shared content. We use the term \\textit{graffiti} for our work since we are storing data in a way that non-network participants may regard as unsightly or unwanted vandalism. Our approach presents several new security challenges over other existing P2P systems where clients transmit data directly with each other: (1) a newly arriving peer can still download files even if all other peers have long disconnected, (2) a peer does not need to know about the existence of other peers, and (3) a tracker does not need multiple peers to enforce tit-for-tat policies~\\cite{bittorrent}.\n\nThe layout of this paper is as follows. First, we provide an overview of the Graffiti Network file sharing model. We then discuss our experimental prototype of the Graffiti Network model that is integrated with a BitTorrent system. The results from our one-year study on the efficacy of our prototype in a real-world deployment show that the use of public storage sites in a file sharing system is possible. We then conclude with a discussion about how both administrators and software developers can guard against such a threat.\n\n\\section{Related Work}\n\\label{sec:related}\nWe motivate our work by first discussing the related background research and literature.\n\n\\subsection{BitTorrent}\n\\label{sec:related-bittorrent}\nThe BitTorrent protocol defines the operations of a P2P network that facilitates the efficient sharing of files in a distributed manner~\\cite{bittorrent}. Our model inherits many of the features of BitTorrent, but employs third-party storage sites as an intermediary for data transfers, rather than allowing clients to directly download files from each other. This indirection makes it difficult to discover the identities of users that are participating in a Graffiti Network.\n\nThe overall efficiency and throughput of BitTorrent systems has been shown to scale gracefully to accommodate many users arriving at the same time to download new and popular files~\\cite{bittorrent-modeling}. But while the model works well in the short term, it does not ensure the long term availability of esoteric content or files that become less popular over time. This problem is especially prevalent for content that is released in ``episodes'': new content is shared profusely when it is released, but the number of peers decreases as the file becomes older and newer episodes are released. In a five month study of BitTorrent network activity, it was shown that the average time that a client stays in the network to continue sharing a file after it has received the entire file set was only seven hours~\\cite{bittorrent-lifetime}. These results, however, are based on the sharing activity of copyright-free files, and therefore the clients do not have a vested interest in disconnecting immediately. In contrast, a study \\cite{bittorrent-measure2} explicitly focused on illegal file sharing activity showed that the departure rate of peers is much faster than previously assumed in \\cite{bittorrent-modeling}. The results in \\cite{bittorrent-measure1} show that the average availability of a torrent is less than nine days and that most swarms completely die out in only 13 days. Thus, without the incentives for sharing found in private communities~\\cite{bittorrent-communities}, most BitTorrent content becomes unavailable after just a short amount of time. To overcome the capricious nature of users, Graffiti Networks use storage sites that have the potential to always be available, and thus the shared files are still accessible after the initial interest in the content has subsided. With enough replication, enforced by a strict asynchronous tit-for-tat model, we believe that a Graffiti Network could provide clients with access to files months or years after it was first introduced to the Internet.\n\n\\subsection{Peer-to-Peer Storage Systems}\n\\label{sec:related-p2pstorage}\nMuch of the previous work on developing P2P storage systems that provide block storage across multiple nodes is based on distributed hash tables~\\cite{cfs,oceanstore,past}. These approaches have the same deficiencies as the BitTorrent model: peers download file blocks directly from other peers, thereby losing anonymity, and the systems do not provide mechanisms to provide long term availability for less popular files after peers disconnect from the network. Other systems are focused on providing anonymous and secure P2P data storage~\\cite{publius}. The POTSHARDS system provides secure long-term data storage when the content originator no longer exists using secret splitting and data reconstruction techniques to handle partial losses~\\cite{potshards}; their approach assumes multiple, semi-reliable storage backends that are willing to host a client's data. The Freenet anonymous storage system uses key-based routing to locate files stored on remote peers~\\cite{freenet}. As discussed in \\cite{cfs}, Freenet's anonymity limits both its reliability and performance: files are not associated with any predictable server, and thus unpopular content may disappear since no one is responsible for maintaining replicas.\n\n\\subsection{Steganographic Storage Systems}\n\\label{sec:related-stegstorage}\nAlthough the Graffiti Network model is not a pure steganographic-based storage system, it does share similar properties of this class of systems~\\cite{stegvault,mnemosyne}. The Mnemosyne storage service applies the steganography techniques from a local storage system \\cite{stegfs} to a distributed hash table~\\cite{mnemosyne}. The StegVault proposal uses secret sharing to build a secure P2P storage system on top of reliable multicast~\\cite{stegvault}. One key benefit of these systems is that users have plausible deniability of the existence of hidden data because it is concealed inside covering data~\\cite{steghide}.\n\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[width=.75\\textwidth]{images\/graffiti-diagram.eps}\n \\caption{For a given a fileset, the client communicates with the tracker in the following manner: \\textbf{(1)} the client sends the tracker the list of pieces it already has; \\textbf{(2)} the tracker responds a list of instructions on where the client should download a sub-piece and the location of where to upload a replica; \\textbf{(3)} after downloading the new sub-piece, the client then navigates the target storage site and uploads a new encrypted and encoded sub-piece payload; \\textbf{(4)} the storage site returns an HTML page and the client verifies that the upload was successful. This process repeats until the client has all the pieces of the fileset and has produced enough replicas for the tracker.}\n \\label{fig:overview}\n\\end{figure*}\n\n\\subsection{Alternative Storage Sites}\n\\label{sec:related-storagesites}\nSince the Graffiti Network model relies on gaining access to and the circumvention of third-party storage sites to host content, we consider the alternative approach of using dedicated storage services that are explicitly designed for the storage and transfer of large files. The Amazon Simple Storage Service provides a well-defined API for writing arbitrary data files, but it currently charges for both the storage space an account uses as well as the network bandwidth used to transfer data~\\cite{amazon-s3}. The Gmail Filesystem enables Google email accounts to be used as a network storage medium, but adopting approach would require users to share account information~\\cite{gmail-fs}. The Usenet news service is another potential storage system, but servers often impose a message retention time and many ISPs have discontinued providing this service to customers for free.\n\nFree web-based file-hosting sites also do not provide the robustness that we seek in our file sharing model~\\cite{rapidshare}. One limitation of these sites is that large files are broken into separate downloads and users must wait for some time period before they are allowed to retrieve the next piece. Furthermore, the user must manually enter each segment URL into their browser and repeatedly pass human-validation tests~\\cite{captcha}. These free hosting sites are also under scrutiny because many of their users post illegal content, and thus the site operators streamline the removal process for files and the disclosure of offending users' information for copyright holders in order to quickly diffuse any legal action that may disrupt the hosting site's revenue stream. Despite this, it is possible to include file-hosting sites as just one of the many options available in a Graffiti Network deployment (see Section~\\ref{sec:sites}).\n\nLastly, another proposed solution is to create a highly-volatile storage site by sending data packets to unsuspecting network entities to leverage network latency as a type of durability~\\cite{juggling}. The idea is to continuously send data to targets that relay the same data back to the source, therefore two copies of the data are always theoretically available. This approach is not practical for the Graffiti Network model because it does not allow the data to be shared amongst multiple peers. Furthermore, it requires that the original data source remain online in order to keep cycling the packets back out over the wire.\n\n\\section{Graffiti Network Model}\n\\label{sec:model}\nWe now describe how a file-sharing system based on the Graffiti Network model would operate. We discuss various measures and techniques that ensure the system is stable, usable, and scalable. Such qualities are necessary to facilitate wide-spread adoption by file-sharing participants, thereby making the threat a real possibility.\n\nTo describe the Graffiti model, we adopt the terminology of the BitTorrent protocol~\\cite{bittorrent}. We define a \\textit{fileset} as a set of one or more files that peers wish to share. The fileset's data is divided into multiple fixed-length \\textit{pieces} of $n$ bytes (the last piece can contain less than $n$ bytes) and are numbered sequentially. Each piece is divided further into fixed-length \\textit{sub-pieces}. A Graffiti Network that is deployed to distribute these pieces is comprised of three distinct components: (1) a \\textit{tracker} coordinates the replication and sharing procedures of a fileset, (2) a \\textit{client} downloads and replicates the fileset data managed by the tracker, and (3) third-party storage \\textit{sites} store and provide access to fileset data for peers. Any client that wishes to download and reconstruct the original fileset is required by the tracker to produce multiple sub-piece \\textit{replicas} on as many storage sites as possible.\n\nA high-level overview of the Graffiti Network protocol is shown in Figure~\\ref{fig:overview}. To connect to the Graffiti Network, the client first announces itself to the tracker and provides it with a list of all the pieces that the peer has already downloaded. The tracker responds with a series of sub-piece \\textit{request pairs} for a new piece that the client is missing. Each request pair consists of (1) a download location where the peer can retrieve a sub-piece and (2) instructions to produce a new replica on a different storage site for the data it just downloaded. Graffiti trackers follow a strict tit-for-tat protocol: for each sub-piece that a peer downloads, that peer is required to generate a replica for a previously downloaded sub-piece on a different storage site and send the location of this new replica back to the tracker before it can receive the next piece.\n\n\\subsection{Central Tracker}\n\\label{sec:tracker}\nThe tracker provides a directory service for peers to retrieve a fileset. For each piece of data in a fileset, the tracker maintains a table of the sub-piece replica locations on sites that were generated by clients. Each replica is annotated with three pieces of meta-data: (1) a unique encryption key for that replica, (2) a checksum for each sub-piece, and (3) the first and last byte sequences of the encrypted data block on the storage site. The tracker uses a different encryption key per entry to ensure that each replica is stored as a unique character sequence to prevent the use of tools to discover other replicas. The checksum and sequence markers also allow peers to determine whether a replica has the proper byte sequence and to locate data boundaries at the storage site location.\n\nFor each connected peer, the tracker maintains an \\textit{active piece set} (APS) of download\/upload replica pairs that are unfulfilled requests for a client. Each pair consists of a sub-piece identifier that the tracker provided for client to download and a storage site location where the tracker instructed the client to make a new replica. Once the client provides the tracker with information about a new replica for a download\/upload pair, the entry is removed from that client's APS and the client is allowed to receive new information. The size of the APS is determined by the tracker's administrator and prevents a client for downloading too many sub-pieces without producing any new replicas. As in the BitTorrent protocol, the Graffiti tracker strives for uniform availability of all data pieces~\\cite{bittorrent}. Since the tracker decrees what pieces the clients must replicate for each request in the APS, it can decide to replicate the ``rarest'' pieces first.\n\nMalicious clients in Graffiti Networks are quite different than malicious clients in BitTorrent networks~\\cite{bitthief}. A rogue Graffiti client may have other ulterior goals: (1) to discover all of the storage site locations used by a tracker in order to contact site administrators and have the replica data removed or (2) to falsely identify valid storage sites and replica locations as invalid in an attempt to disrupt operations. In the first case of trying to discover all of a fileset's replicas, the tracker can use throttling measures to prevent a client from learning too much in a short amount of time. But for the latter problem, the tracker should not actively check whether a client actually uploaded the data at the location it claims it did, due to security and economic reasons. Instead it can employ proxies or other third-party entities to determine whether a client is behaving properly. For example, the tracker can retrieve a page through the Coral Cache or Tor services to determine if the data was stored at the location claimed by a client~\\cite{freedman04,dingledine04}.\n\n\n\n\\subsection{Client}\n\\label{sec:client}\nA Graffiti client allows a user to automatically download a fileset stored on one or more storage sites. A user must first obtain a metadata file for a specific fileset uniquely identified by an ``info hash'' in order to begin downloading~\\cite{bittorrent}.\n\nAfter the client first announces itself to the tracker at the address listed in the metadata file, the tracker places the peer in an ``initialization'' mode. This is always done regardless of whether the client is connecting for the first time or if it is returning with some pieces already downloaded. The tracker sends every new client the same \\textit{initial piece set} (IPS) that will use for the first phase of downloading and replication. This initial set is the same for all clients arriving within a certain time period to prevent a client from initiating multiple new connections without ever creating new replicas. The size of the initial set is the same size as the APS and its information is changed to a different random set of sub-pieces at regular intervals (e.g., hours or days, rather than minutes). Thus, it is possible for a rogue client to retrieve a complete fileset without ever producing a new replica for the network, but it would take several days or weeks to cycle through all of the tracker's IPS combinations if there were a significantly large number of pieces. The client is required to also produce two new replicas for each sub-piece in the IPS, even if the client has already downloaded the pieces previously. This policy is akin to a new tenant paying ``last month's rent'' before moving into an apartment: it ensures that client cannot disconnect from the network without creating new replicas for each piece that it downloads.\n\nOnce the client successfully downloads and generates sufficient replicas for its IPS, it leaves the initialization phase and is then allowed to receive arbitrary pieces. The protocol works the same before: the tracker maintains an APS for each client and only gives new download locations once that particular client has produced a new replica on a storage site.\n\n\\subsection{Storage Sites}\n\\label{sec:sites}\nA potential Graffiti storage site is any accessible network entity that allows for data to be stored and retrieved using a known network protocol. In practice, peers will likely use publically available web sites that provide services that Graffiti clients repurpose to store arbitrary blocks of data. This approach has the distinction that all data movement appears as normal HTTP traffic, and thus is immune to current ISP throttling and tracking techniques~\\cite{isps}.\n\nThe ideal storage site for a Graffiti Network is one that allows for anyone to post data without CAPTCHA protections~\\cite{captcha} and is either unmoderated or has long abandoned by its owner. A popular and high-traffic wiki site, for example, would not be a good storage site candidate as it likely that non-malicious visitors would quickly notice the changes made by Graffiti clients to store replica data. With the rise of many open-source web-publishing platforms, there are many potential targets that allow for anonymous or semi-anonymous data posting. Notable examples include paste-bins, wiki sites, message boards, and blogs. An HTML-based storage site also allows the data to be disseminated to peers through disparate channels once it is online, such as through Coral Cache~\\cite{freedman04} or Tor~\\cite{dingledine04}. The data embedded in the site's pages could also be picked up by search engine caching and archiving services for longer-term storage.\n\nOther potential storage sites include any photo and file hosting sites that allow for automated data uploading. In the case of the former, the data could also be hidden inside of image files using well-known techniques~\\cite{steganography,ramkumar01}. As the Internet evolves, new targets will emerge that can be incorporated into existing networks. The system could also allow clients to use storage sites that are password protected for writing data, but where an account is not required to read back the data. This obviates the need for a client to send the tracker account information, which could then be used improperly by other clients to tamper with or destroy the data.\n\nUsing involuntary web sites as storage dumps seems counterintuitive if the main goal of the network is data persistence and availability, since replicas are promptly removed when site administrators and moderators discover them. The Graffiti model overcomes this challenge and takes advantage of ``free network storage'' through a massive replication and obfuscation process. It is not trivial, however, to automatically store arbitrary data on random web sites nor is it trivial to discover which sites are available with the properties stated above. The prevalence of popular web publishing software means that one only needs to target a small number of platforms in order to circumvent a large portion of the Internet. Furthermore, many sites, such as wikis and message boards, often display the network location of the user responsible for adding new content or making changes to their pages, which makes it difficult to deny responsibility for participating in illegal activities. We argue that by fracturing a fileset's replicas across hundreds of storage sites, it is difficult to be fully implicated when only a fraction of the evidence is available. A distributed effort to probe websites and uncover open storage paths could allow peers to draw on a nearly limitless pool of available storage.\n\n\\section{Experimental Deployment}\n\\label{sec:simulation}\nTo determine whether the Graffiti Network model is a viable and thus is a potential threat, we implemented a prototype Graffiti tracker and client as an extension to the BitTorrent protocol. We then stored a sample data set on a large number of open sites and measured the availability of our data for almost an entire year.\n\nWe built our system on top of the open-source libtorrent~\\cite{libtorrent} BitTorrent library in order to allow clients to participate in torrent swarms concurrently with Graffiti Network activities. When enough peers are available, the client operates strictly in BitTorrent mode. But if the number of distributed copies in the swarm drops below a threshold, the client begins to contact the tracker using the Graffiti protocol in conjunction with its BitTorrent operations. As new pieces are retrieved from storage sites, they are passed to libtorrent's storage manager for seeding to other peers.\n\n\\subsection{Storage Site Discovery}\nIn our experimental prototype, we target the open source MediaWiki~\\cite{mediawiki} platform as the potential storage site for the network. Due to the popularity of sites like Wikipedia that use MediaWiki, we believe that it is the most widely deployed wiki platform with a large number of less-experienced users that install the software without changing the permissive default settings. Another key characteristic is that the MediaWiki platform maintains a complete revision log for each article, which allows Graffiti peers to retrieve data even if the changes are reversed or the content is altered.\n\nWe decided to test our system on open MediaWiki sites that we do not have control over as this allows us to best measure whether our assumptions about how long the data will remain on the sites are correct. We developed a distributed web crawler to discover MediaWiki installations through search engines using keywords that are uniquely indicative of a newly installed site. The crawler purposely ignored well-known MediaWiki sites (e.g., those sites that are part of the Wikipedia Foundation) and the commercialized versions of MediaWiki (e.g., Wikia). For each site that the crawler found, we probed it to determine what kind of protection scheme it utilizes and the last time that it was updated (see Table~\\ref{tab:sites}). Of the 23,156 unique MediaWiki installations that we found, 8,483 sites allowed for anonymous editing and 5,983 allowed users to register accounts without CAPTCHA or email protections in order to make edits~\\cite{captcha}. The default MediaWiki installation provides a primitive arithmetic ``puzzle'' protection countermeasure that we found in use on 1,157 sites; this puzzle is easily broken with just a few lines of code, and thus did not prevent our system from storing data on these sites. Lastly, in order to minimize the impact of our experiments, we only targeted those sites that had not been updated within the last three months, thereby reducing our list to 5,646 sites; lowering the threshold to two months would have yielded a total of 11,987 potential storage sites.\n\n\n\\begin{table}[!t]\n \\centering\n \\begin{tabular}{lrr}\n & Sites Found & Sites Used \\\\\n Anonymous Edits & 8,483 & 3,161 \\\\\n Registration Protected & 5,983 & 2,347 \\\\\n Puzzle Protected & 1,157 & 138 \\\\\n CAPTCHA Protected & 1,586 & - \\\\\n Not Publicly Modifiable & 5,946 & - \\\\\n \\hline\n \\textbf{Total:} & \\textbf{23,156} & \\textbf{5,646} \\\\\n \\end{tabular}\n \\caption{The categories of protection used by the MediaWiki sites discovered during the collection process and the sites used in the experimental deployment.}\n \\label{tab:sites}\n\\end{table}\n\n\\begin{figure*}[!ht]\n \\centering\n \\begin{minipage}{3.2in}\n \\centering\n \\includegraphics[width=3.25in]{graphs\/visit_availability.eps}\n \\caption{Percentage of total replicas removed over time categorized by the type of failure.}\n \\label{fig:graph-visits}\n \\end{minipage}\n \\begin{minipage}{0.3in}\n \\hspace*{0.3in}\n \\end{minipage}\n \\begin{minipage}{3.2in}\n \\centering\n \\includegraphics[width=3.2in]{graphs\/total_availability.eps}\n \\caption{The availability of replicas categorized by its corresponding storage site's protection schemes.}\n \\label{fig:graph-availability}\n \\end{minipage}\n \\vspace{-.15in}\n\\end{figure*}\n\nThe Graffiti client stores data on MediaWiki sites as base64-encoded, Blowfish-encrypted blocks of text that are written in a new article titled with a random word from the dictionary. A more resilient approach would be to modify a popular page on a given site, and then immediately reverse the changes and mark the revision as vandalism. This has two significant implications compared to writing data to a newly created article. Foremost is that removing this data completely from the page's history requires administrators to delete the entire page from the database and restore the latest revision by hand, thereby losing all the previous legitimate revisions. Second, such an attack is more likely to be overlooked by a site's operators since they may only care whether the changes were reversed. We deemed this technique too malevolent for the purpose of our experiments, and thus chose to not implement it.\n\nTo retrieve a sub-piece stored on one of these storage sites, the client downloads the web page and extracts the text surrounded by the byte sequence markers provided by the tracker. The client then reverses the base64 encoding, decrypts the data, and verifies that it matches the checksum provided by the tracker.\n\n\\subsection{System Configuration}\nFor our experimental deployment, we used a Linux ISO split into 512KB pieces and 64KB sub-pieces as our sample data file that the clients want to share. Even though we were able to store up to 512KB payloads on a single MediaWiki page, we choose to use a smaller sub-piece size. Again, another more malicious approach would be to store a payload with the size that can be uploaded and retrieved but causes either a browser or the server to choke if the operator tries to access the page through the MediaWiki administrative interface. For example, we found that it was possible to store 512KB pieces that would exhaust the default 20MB memory limit of PHP if someone tried to remove the data. Thus, the only way to remove the content is to execute the proper SQL commands directly in the database, which is likely too difficult for most users.\n\nWe initiated file sharing activity on April 10th, 2009 using a tracker and five clients deployed in our departmental lab. Each client connects to the tracker and produces a full copy of a sub-piece on one of the 5,600+ MediaWiki sites. We assume that all clients are truthful about whether a replica is available and do not falsify replica URLs. We instrumented the tracker to target each storage site only once (although variations in sub-domains and URL rewriting led to some sites being used more than once).\n\nAlong with the data payload, at the top of each wiki page we stored a small paragraph with an explanation of the seemingly random text. This description also included a unique tracking link back to our web page with further information about the project. Tracking users' click-throughs from these links allows us to measure to some extent whether humans were actually discovering our payload pages before they were deleted.\n\nOnce the clients pushed out all of the data to the sites, we then used a separate tool to check daily whether the data we stored is still in place and has not been modified. We check every replica regardless if it has not been available for some time to ensure that the errors are not transient.\n\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=3.2in]{graphs\/domain_availability.eps}\n \\caption{The cumulative availability of replicas categorized by their domain type: .com (42.5\\%), .edu (3.2\\%), .org (24.1\\%), US-based other (14.0\\%), and Non-US-based other (16.1\\%). }\n \\label{fig:graph-domain}\n\\end{figure}\n\n\n\\subsection{Results}\n\\label{sec:experiments-results}\nWe now report on the availability of the 5,646 replicas that we stored in our experiments from April 10th to February 28th, 2010. For each missing replica, we categorize the replica as (1) \\textit{removed} if the site is available but the original page is missing, (2) \\textit{changed} if both the site and the original page are available, but the data does not match our stored checksum, or (3) \\textit{not found} if the site is no longer available (e.g., the domain name has expired or MediaWiki was uninstalled). Our investigation found that the missing replicas were only either removed or not found; no replica had its contents altered.\n\nOn the last day of our data collection, roughly 40\\% of the replicas were still available and hosting the original data that the prototype clients uploaded. The graph in Figure~\\ref{fig:graph-visits} shows a timeline of the percentage of replicas that are not available on each day that we checked. The first notable data point is that an initial 20\\% of the replicas were removed within the same week that they were created. The rate in which sites are removed then tapers off as time progresses. We attribute this drop-off in activity to two possible reasons. Foremost is that by default any changes to a MediaWiki site will appear on the first page of revision logs for seven days after the revision is created, and thus our actions are more likely to be discovered soon after the data is posted. The second possible reason is because a story about our project appeared on the front page of a popular technology news website on the third day of our experiment~\\cite{slashdot}. We believe that the ``notoriety'' of the project during this period may have caused administrators to examine their websites to see if they were targeted by our system. Once this initial attention diminished, the slopes of the lines in Figure~\\ref{fig:graph-visits} decrease and it takes another 35 days before another 10\\% of the replicas are removed. After about 100 days, the growth rate of replicas being removed (i.e., the lower portion of the curve in Figure~\\ref{fig:graph-visits}) tapers off and the number of sites that become unavailable begins to rise. This is expected since many of the sites were not actively used by their proprietor, and thus are taken down arbitrarily.\n\nThe graph in Figure~\\ref{fig:graph-availability} shows how the replicas were removed over time in relation to their storage site's protection scheme. The salient aspect of the result is that initially sites that employed some type of protection were faster to remove replicas. This is expected, since many of the sites that employed some protection were still being used by users despite having not been updated recently, whereas many of the completely open sites still displayed the default MediaWiki homepage message and thus were never even used once they were installed. Such sites are likely long forgotten by their owners who may never discover the replicas once they pass the default seven day revision log window. But after approximately 120 days, the percentage of missing replicas stored on sites allowing for anonymous edits surpasses sites using the basic registration protection.\n\nLastly, the graph in Figure~\\ref{fig:graph-domain} charts the availability of replicas with respect to the domain name of the storage site. We attribute greater durability of data stored on .edu and .org sites compared to other domains; such organizations are likely to use open-source software for collaboration and internal sites are often not behind corporate firewalls.\n\n\\section{Discussion}\n\\label{sec:discussion}\nThe results presented in the previous section clearly demonstrate the efficacy of the Graffiti Network model as a means for facilitating longer-term file sharing. We therefore argue that the threat of such a system does indeed exist and sites need to take measures to protect themselves from being used in such a manner that we have describe.\n\n\\subsection{Countermeasures}\nMuch of the feedback that we received on the project was from administrators that expressed their desire to provide an open wiki site that allowed anonymous contributions, despite the inevitable exposure to vandalism and spam. We counter that such sites that do not want to require users to register an account should still use CAPTCHA protections, such as before a user is allowed to edit a page. In practice, we found that the \\mbox{reCAPTCHA}~\\cite{vonahn08} project is the most effective protection as it does not require administrators to install special server-side graphics libraries and strikes a proper balance between availability and complexity. More complex CAPTCHA schemes would not deter future Graffiti clients that are able to solve CAPTCHAs (either manually or programmatically) and may only inhibit legitimate visually impaired users. If sites wish to still remain open, the CAPTCHA could be selectively enabled only when an unverified user tries to post data larger than some low default threshold or creates too many new pages in a short time span.\n\nWe also believe that other simple protection measures could be included in popular web applications to prevent abandoned or forgotten sites from being used for unintended purposes. For example, MediaWiki's default behavior could be to lock down the editing features of a site after a certain number of days if it was installed but then never actually used. This approach is similar to the one used by some blogging platforms to disable comments on older posts. Administrators could easily re-enable this functionality by simply logging into the site again. Another technique is to use a page counter that is invoked on the client-side (e.g., through JavaScript) and then compare the results with server-side logs to determine whether there are an unusually large number of users accessing pages through a non-browser client. Web application frameworks, such as Ruby on Rails and Django, could also provide similar features to protect custom-made sites.\n\n\\subsection{Variations \\& Adaptations}\nOther than for P2P activities, the Graffiti model is also of potential use for large-scale distributed systems used by criminal organizations, often referred to as \\textit{botnets}. The goal of most botnet operators is to gain access to a large supply of computational resources for purposes of network communication (e.g., sending emails or DOS attacks). If these goals shift towards more data-centric activities, then systems based on some of the principals of the Graffiti Network model may become prevalent in order to store large amounts of data for the botnet. Alternatively, instead of storing replicated data, the commandeered storage sites could also be used as a control channel for other entities in the botnet.\n\n\n\n\\section{Acknowledgments}\nThe authors would like to thank Arvid Norberg at BitTorrent, Inc. for his assistance with the libtorrent library~\\cite{libtorrent}.\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nWe have presented an overview of Graffiti Networks, a new file sharing model that allows peers to subversively use third-party storage sites as an intermediary for transferring files between users. Our client-tracker paradigm is similar to the BitTorrent protocol, but is designed to provide long term file availability to users while preserving their anonymity. We do not intend the Graffiti model to supplant BitTorrent networks, as it will never achieve the same maximum network throughput nor will it ever be as efficient. We believe, however, that our approach can have a symbiotic relationship with existing deployments: peers would use a Graffiti Network-like system to improve the long term availability of shared files, while leveraging the faster initial transfer rates of direct P2P communication for data dissemination. We have implemented a prototype and shown that data can be stored on publically accessible sites for extended periods of time, beyond what is often possible in other existing peer-to-peer systems. After almost an entire year, roughly 40\\% of the data that we stored on sites that are not under our control was still available. These results indicate that malicious users may adopt the Graffiti Network model, and thus site operators should take measures to prevent their sites from being used in this manner.\n\n\n\n\n{\\footnotesize\n \\linespread{0.85}\n\\bibliographystyle{acm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzbgtp b/data_all_eng_slimpj/shuffled/split2/finalzzbgtp new file mode 100644 index 0000000000000000000000000000000000000000..4992204362fa3f276cfd2f0e6d85acbf91e5cbe7 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzbgtp @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n A fundamental problem in symplectic 4-manifold topology is the classification of symplectic fillings of certain 3-manifolds equipped with a natural contact structure. Researchers have long studied the symplectic fillings of the link of a normal complex surface singularity. Note that Seifert 3-manifolds can be viewed as a link of weighted homogeneous surface singularities, and the link of such a normal surface singularity carries a canonical contact structure, also known as the Milnor fillable contact structure.\n For example, P.~Lisca~\\cite{Lis}, M.~Bhupal and K.~Ono~\\cite{BOn}, and the second author of this study et al.~\\cite{PPSU} completely classified all minimal symplectic fillings of lens spaces and certain small Seifert 3-manifolds coming from the link of quotient surface singularities.\n\n Topologists working on 4-manifold topology are also interested in finding a surgical interpretation for the symplectic fillings of a given 3-manifold. More specifically, topologists investigate whether a surgical description of these fillings exists. \n Indeed, a \\emph{rational blowdown} surgery, introduced by R.~Fintushel and R.~Stern~\\cite{FS} and generalized by the second author~\\cite{Par} and A.~Stipsicz, Z.~Szab{\\'o} and J.~Wahl~\\cite{SSW}, is a powerful tool used in these investigations. \nFor example, for the link of quotient surface singularities equipped with a canonical contact structure, it has been proven~\\cite{BOz}, \\cite{CP1} that every minimal symplectic filling is obtained by a sequence of rational blowdowns from the minimal resolution of the singularity. However, L.~Starkston~\\cite{Sta2} showed that the symplectic fillings of some Seifert 3-manifolds cannot be obtained by a sequence of rational blowdowns from the minimal resolution of the corresponding singularity. \nHence, knowing which Seifert 3-manifolds have a rational blowdown surgery interpretation for their minimal symplectic fillings is an intriguing question.\n\n In this study, we investigate a relation between rational blowdown surgery and the minimal symplectic fillings of a given Seifert 3-manifold with a canonical contact structure, so that we determine a necessary and sufficient condition for a minimal symplectic filling of a given Seifert 3-manifold satisfying certain conditions to be obtained by a sequence of rational blowdowns from the minimal resolution of the corresponding weighted homogeneous surface singularity. \n In general, a Seifert 3-manifold can be considered as an $S^1$-fibration over a Riemann surface and it may have any number of singular fibers. In this paper, we only consider a Seifert $3$-manifold $Y$ as an $S^1$-fibration over the $2$-sphere such that it can be described by $Y(-b; (\\alpha_1, \\beta_1), (\\alpha_2, \\beta_2),\\ldots (\\alpha_n, \\beta_n))$, whose Dehn surgery diagram is given in Figure~\\ref{Seifert} and given as a boundary of a plumbing 4-manifold of disk bundles over a $2$-sphere according to the graph $\\Gamma$ in Figure~\\ref{Seifert}.\nThe integers $b_{ij}\\geq 2$ are uniquely determined by the following continued fraction:\n $$\\frac{\\alpha_i}{\\beta_i}=[ b_{i1}, b_{i2}, \\dots, b_{ir_i} ]=b_{i1}-\\displaystyle {\\frac{1}{b_{i2}-\\displaystyle\\frac{1}{\\cdots-\\displaystyle\\frac{1}{b_{ir_i}}}}}$$\n\n\\begin{figure}[h]\n\\begin{tikzpicture}[scale=0.5]\n\\begin{scope}\n\\begin{knot}[\n\tclip width=5,\n\tclip radius = 2pt,\n\tend tolerance = 1pt,\n]\n\\strand (0,0) ellipse (3 and 1.5);\n\\strand (-1.5,-1.75) ellipse (0.25 and 0.8);\n\\strand (-.2,-1.75) ellipse (0.25 and 0.8);\n\\draw (.8,-2) node[below] {$\\cdots$};\n\\strand (1.75,-1.75) ellipse (0.25 and 0.8);\n\\draw (-3,1.5) node {$-b$};\n\\draw (-1.5,-2.55) node[below] {$-\\frac{\\alpha_1}{\\beta_1}$};\n\\draw (-.2,-2.55) node[below] {$-\\frac{\\alpha_2}{\\beta_2}$};\n\\draw (1.75,-2.55) node[below] {$-\\frac{\\alpha_n}{\\beta_n}$};\n\n\\flipcrossings{1,4,5}\n\\end{knot}\n\\end{scope}\n\\begin{scope}[shift={(8,0)}]\n\\node[bullet] at (0,1.5){};\n\\node[bullet] at (0,0){};\n\\node[bullet] at (-2,0){};\n\\node[bullet] at (3,0){};\n\\node[bullet] at (0,-1){};\n\\node[bullet] at (-2,-1){};\n\\node[bullet] at (3,-1){};\n\\node[bullet] at (0,-3){};\n\\node[bullet] at (-2,-3){};\n\\node[bullet] at (3,-3){};\n\\draw (0,1.5)--(0,-1.5);\n\\draw (0,1.5)--(3,0)--(3,-1.5);\n\\draw (0,1.5)--(-2,0)--(-2,-1.5);\n\\draw[dotted,thick](1.25,0)--(1.75,0);\n\\draw[dotted,thick](1.25,-1)--(1.75,-1);\n\\draw[dotted,thick](1.25,-3)--(1.75,-3);\n\n\\draw[dotted](0,-1.5)--(0,-2.5);\n\\draw[dotted](3,-1.5)--(3,-2.5);\n\\draw[dotted](-2,-1.5)--(-2,-2.5);\n\\draw(0,-2.5)--(0,-3);\n\\draw(3,-2.5)--(3,-3);\n\\draw(-2,-2.5)--(-2,-3);\n\\draw (0,1.5) node[above] {$-b$};\n\\draw (0,0) node[left] {$-b_{21}$};\n\\draw (0,-1) node[left] {$-b_{22}$};\n\\draw (0,-3) node[left] {$-b_{2r_2}$};\n\\draw (-2,0) node[left] {$-b_{11}$};\n\\draw (-2,-1) node[left] {$-b_{12}$};\n\\draw (-2,-3) node[left] {$-b_{1r_1}$};\n\\draw (3,0) node[right] {$-b_{n1}$};\n\\draw (3,-1) node[right] {$-b_{n2}$};\n\\draw (3,-3) node[right] {$-b_{nr_n}$};\n\\end{scope}\n\n\\end{tikzpicture}\n\\caption{\\mbox{Surgery diagram of $Y$ and its associated plumbing graph $\\Gamma$}}\n\\label{Seifert}\n\\end{figure}\n \n\n\n\n\n\n\n\n\\begin{figure}[h]\n\\begin{tikzpicture}[scale=0.5]\n\n\n\\begin{scope}[xscale=2.3,yscale=1.85, shift={(6.6,1)}]\n\n\\draw(0,0)--(6.2,0);\n\\draw (0.5,0.2)--(0,-0.7);\n\\draw (1.5,0.2)--(1,-0.7);\n\\draw (3.5,0.2)--(3,-0.7);\n\\draw (4.9,0.2)--(4.4,-0.7);\n\\draw (5.9,0.2)--(5.4,-0.7);\n\\node at (5.,-0.6) {$\\cdots$};\n\\draw[dotted,thick](2.15,-1.2)--(2.45,-1.2);\n\t\\draw [decorate,decoration={brace,amplitude=5pt,mirror},xshift=2pt,yshift=-3pt]\n\t(4.4,-0.7) -- (5.4,-0.7) node [black,midway,xshift=10pt,yshift=-10pt] \n\t{\\footnotesize $b-(n+1)$};\n\n\\draw (0,-0.3)--(0.5,-1.2);\n\\draw (1,-0.3)--(1.5,-1.2);\n\\draw (3,-0.3)--(3.5,-1.2);\n\n\\node at (0.25, -1.5) {$\\vdots$};\n\\node at (1.25, -1.5) {$\\vdots$};\n\\node at (3.25, -1.5) {$\\vdots$};\n\n\\draw (0,-1.8)--(0.5,-2.7);\n\\draw (1,-1.8)--(1.5,-2.7);\n\\draw (3,-1.8)--(3.5,-2.7);\n\n\\node[left] at (0,0) {$+1$};\n\\node[right] at (0.25, -0.25) {$-a_{11}$};\n\\node[right] at (1.25, -0.25) {$-a_{21}$};\n\\node[right] at (3.25, -0.25) {$-a_{n1}$};\n\\node[right] at (4.65, -0.25) {$-1$};\n\\node[right] at (5.65, -0.25) {$-1$};\n\n\\node[right] at (0.25, -0.75) {$-a_{12}$};\n\\node[right] at (1.25, -0.75) {$-a_{22}$};\n\\node[right] at (3.25, -0.75) {$-a_{n2}$};\n\n\\node[right] at (0.25, -2.25) {$-a_{1m_1}$};\n\\node[right] at (1.25, -2.25) {$-a_{2m_2}$};\n\\node[right] at (3.25, -2.25) {$-a_{nm_n}$};\n\n\\end{scope}\n\n\\end{tikzpicture}\n\\caption{Concave cap $K$ }\n\\label{X}\n\\end{figure}\n\n\n\n\n\nWe introduce the main results by starting with a minimal symplectic filling $W$ of a Seifert $3$-manifold $Y$ with a canonical contact structure.\nWhile $b\\geq(n+1)$, we obtain a closed rational symplectic $4$-manifold $M=W\\cup K$ by gluing a concave cap $K$ to $W$ along $Y$ (refer to Figure~\\ref{X}). Then, the image of $K$ under blowing-downs from $M$ to $\\mathbb{CP}^2$ is called a \\emph{symplectic line arrangement} $S\\subset \\mathbb{CP}^2$, which is a union of the complex line $\\mathbb{CP}^1$ with a finite number of symplectic lines, that is, symplectic $2$-spheres, each of which is homologous to $\\mathbb{CP}^1 \\subset \\mathbb{CP}^2$. \nWe call an intersection point $p$ of $S$ a \\emph{multi-intersection point} if at least three symplectic lines pass through $p$. We denote the number of multi-intersection points in a symplectic line arrangement $S$ by $N_S$. Note that we blow up all the intersection points on the symplectic lines in $S$ to obtain an embedding $K$ in $M$, because each symplectic line becomes an arm in $K$. Therefore, all intersection points of symplectic lines in $S$ correspond to an exceptional $2$-sphere whose homology class appears at the first component of the corresponding arms in $K$, implying that the homological embedding of $K$ in $M$ determines the intersection data of $S$. \n\nNow, we provide the necessary condition for $W$ to be obtained by a sequence of rational blowdowns. Assume that a minimal symplectic filling $W$ of $Y$ is obtained from another minimal symplectic filling $W'$ by rationally blowing down a negative definite star-shaped plumbing graph $G$ that is symplectically embedded in $W'$.\nIf $G$ is `nicely' embedded in $W'$, we can track the homological data of $K$ after surgery. Furthermore, we can describe a symplectic line arrangement $S$ corresponding to $W$ in terms of a symplectic line arrangement $S'$ corresponding to $W'$. In particular, we claim that the difference between the numbers $N_{S}$ and $N_{S'}$ of multi-intersection points is at most one, which is a key ingredient for obtaining the following main theorem.\n\n\n\n\\begin{thm}\n\\label{thm1}\nSuppose a Seifert $3$-manifold $Y(-b; (\\alpha_1, \\beta_1), (\\alpha_2, \\beta_2),\\ldots, (\\alpha_n, \\beta_n))$ satisfies $b\\geq n+1$. If a minimal symplectic filling $W$ of $Y$ with a canonical contact structure is obtained from the minimal resolution of the corresponding weighted homogeneous surface singularity by a sequence of rational blowdowns, then the number $N_S$ of multi-intersection points in a symplectic line arrangement $S$ corresponding to $W$ is at most one.\n\\end{thm}\n\nFurthermore, if we restrict this to the case that $b\\geq n+2$, then the condition $N_S\\leq 1$ in Theorem ~\\ref{thm1} is also a sufficient condition for a minimal symplectic filling to be obtained via rational blowdown surgeries.\n\n\\begin{thm}\n\\label{thm2}\nFor a Seifert $3$-manifold $Y(-b; (\\alpha_1, \\beta_1), (\\alpha_2, \\beta_2),\\ldots, (\\alpha_n, \\beta_n))$ with $b\\geq n+2$, any minimal symplectic filling $W$ of $Y$ with $N_S\\leq 1$ is obtained by a sequence of rational blowdowns from the minimal resolution of the corresponding weighted homogeneous surface singularity.\n\\end{thm}\n\n A strategy for proving Theorem~\\ref{thm2} is similar to that for proving Theorem 1.1 in ~\\cite{CP2}. We divide all possible minimal symplectic fillings into certain types and then show that a sequence of rational blowdowns from the minimal resolution for each type exists by using lemmas proved in Section 4 ~\\cite{CP2}.\n\n\\begin{remark}\nA family of minimal symplectic fillings of Seifert $3$-manifolds that cannot be obtained by a sequence of rational blowdowns was first provided by L.~Starkston in ~\\cite{Sta2}.\nStarkston's examples have $N_S=2$ with $b=n+2$. Hence, we can recover Starkston's result using Theorem~\\ref{thm1} above.\n\\end{remark}\n\n\n\\subsection*{Acknowledgements}\nThe authors thank all members of the 4-manifold topology group at SNU for their helpful comments during the work. Jongil Park was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (No.2020R1A5A1016126 and No.2021R1A2C1095776). \nHe also holds a joint appointment at the Research Institute of Mathematics, SNU. \n\n\n\n\n\\section{Preliminaries}\n\\label{pre}\n\n\\subsection{Weighted homogeneous surface singularities and Seifert $3$-manifolds}\nWe briefly recall the relation between a Seifert $3$-manifold $Y$ and link $L$ of a weighted homogeneous surface singularity $(X,0)$.\nWe say that a normal surface singularity $(X,0)$ is a weighted homogeneous surface singularity if $(X,0)$ is given by zero loci of weighted homogeneous polynomials of the same type. Note that a polynomial $f(z_0,\\dots, z_m)$ is called \\emph{weighted homogeneous} if there exist nonzero integers $(q_0,\\dots,q_m)$ and a positive integer $d$ that satisfy\n$$f(t^{q_0}z_0,\\dots t^{q_m}z_m)=t^df(z_0,\\dots, z_m).$$\nThen, there is a natural $\\mathbb{C}^*$-action on $(X,0)$ given by \n$$t\\cdot(z_0,\\dots, z_m)=(t^{q_0}z_0,\\dots t^{q_m}z_m),$$\nwhich induces a fixed point-free $S^1\\subset \\mathbb{C}^*$ action on link $L:=X\\cap \\partial B$ of the singularity, where $B$ is a small ball centered at the origin. Hence, link $L$ is a Seifert fibered $3$-manifold over a genus $g$ Riemann surface. In this paper, we only consider a Seifert fibered $3$-manifold over the $2$-sphere, which is denoted by $Y(-b; (\\alpha_1, \\beta_1), (\\alpha_2, \\beta_2), \\dots, (\\alpha_n, \\beta_n))$ for some integers $b, \\alpha_i$ and $\\beta_i$ with $0<\\beta_i<\\alpha_i$ and $(\\alpha_i, \\beta_i)=1$. \nNote that $n$ is the number of singular fibers, and there is an associated star-shaped plumbing graph $\\Gamma$: the central vertex has genus $0$ and weight (equivalently, degree) $-b$, and each vertex in $n$ arms has genus $0$ and weight $-b_{ij}$ uniquely determined by the continued fraction \n$$\\frac{\\alpha_i}{\\beta_i}=[ b_{i1}, b_{i2}, \\dots, b_{ir_i} ]=b_{i1}-\\displaystyle {\\frac{1}{b_{i2}-\\displaystyle\\frac{1}{\\cdots-\\displaystyle\\frac{1}{b_{ir_i}}}}}$$ \nwith $b_{ij}\\geq 2$. \nFrom P.~Orlik and P.~Wagreich~\\cite{OW}, it is well known that the plumbing graph $\\Gamma$ is a dual graph of the minimal resolution of $(X, 0)$. \nMoreover, if the intersection matrix of $\\Gamma$ is negative definite, there is a weighted homogeneous surface singularity whose dual graph of the minimal resolution is $\\Gamma$ ~\\cite{Pin}. Furthermore, if a Seifert $3$-manifold $Y$ can be viewed as the link $L$ of a weighted homogeneous surface singularity, \nthere exists a canonical contact structure $\\xi_{\\text{can}}$, called the \\emph{Milnor fillable} contact structure, on $Y$ given by complex tangencies $TL\\cap JTL$ that is known to be unique up to contactomorphism~\\cite{CNPo}.\n\n\n\\subsection{Minimal symplectic fillings of Seifert $3$-manifolds}\nIn this subsection, we briefly review well-known facts regarding the minimal symplectic fillings of a Seifert $3$-manifold $Y$ with a canonical contact structure $\\xi_{\\text{can}}$.\n\n\n \nAs mentioned in the Introduction, there is a star-shaped plumbing graph $\\Gamma$ associated to $Y$ (refer to Figure~\\ref{Seifert}).\nWhile $b\\geq(n+1)$, we can always choose a concave cap $K$ of $(Y,\\xi_{\\text{can}})$ as shown in Figure~\\ref{X}. For a minimal symplectic filling $W$ of $(Y,\\xi_{\\text{can}})$, we obtain a closed symplectic $4$-manifold $M=W\\cup K$ by gluing $K$ along $Y$ to $W$. Then, the existence of $(+1)$ $2$-sphere in $K$ implies that $M$ is a rational symplectic $4$-manifold and, after a finite number of blowing-downs, $M$ becomes $\\mathbb{CP}^2$ so that the $(+1)$ $2$-sphere in $K$ remains a complex line $\\mathbb{CP}^1 \\subset \\mathbb{CP}^2$ (see Mcduff~\\cite{McD} for details). The image of $K$ under the blowing-downs is called a \\emph{symplectic line arrangement} $S$ consisting of complex line $\\mathbb{CP}^1$ together with finite number of symplectic lines, in fact symplectic $2$-spheres, each of which is homologous to $\\mathbb{CP}^1 \\subset \\mathbb{CP}^2$~\\cite{Sta1}. \nTherefore, a minimal symplectic filling $W$ is completely determined by the homological embedding of $K$ in $M\\cong \\mathbb{CP}^2\\sharp N\\overline{\\mathbb{CP}^2}$ and the isotopy type of $S$ in $\\mathbb{CP}^2$. \nNote that the second homology group of $M$ is generated by $\\{l,e_1,\\dots, e_N\\}$, where $l$ is a homology class of $\\mathbb{CP}^1\\subset \\mathbb{CP}^2$ and $\\{e_i\\}$ are homology classes of exceptional $2$-spheres. Therefore, the homology class of each irreducible component of $K$ can be expressed in terms of this basis, which we call the \\emph{homological data of $K$} for $W$. In Theorem~\\ref{thm2}, we claim that, if the number $N_S$ of multi-intersection points of a symplectic line arrangement $S$ corresponding to $W$ is at most one, the minimal symplectic filling $W$ of $(X,0)$ is obtained from the minimal resolution of $X$ by a sequence of rational blowdowns. Because the isotopy type of a symplectic line arrangement $S$ with a fixed intersection data is known to be unique if $N_S\\leq 1$ (Proposition 4.2 in \\cite{Sta2}), the minimal symplectic filling $W$ in Theorem~\\ref{thm2} is determined uniquely by the homological data of $K$ for $W$.\n\nMoreover, the combinatorial data of a symplectic line arrangement $S$ can be described by a configuration of strands, as in Figure~\\ref{line}. Each strand represents a symplectic $2$-sphere, and an intersection between strands represents a transversely geometric intersection between the $2$-spheres.\nHence, starting from a configuration of strands representing $S$, we can draw a configuration $C$ of strands containing $K$ using the homological data of $K$ for $W$. If there are no strands with degree less than or equal to $-2$ in $C$ except for the irreducible components of $K$, we call $C$ the \\emph{curve configuration} for $W$, which is unique up to equivalence (Proposition 3.1 in~\\cite{CP2}).\n\n\\smallskip\n\n{\\it{Convention}}: We often use a terminology \\emph{configuration of strands} when we deal with an intermediate configuration between a symplectic line arrangement and a curve configuration, or a configuration containing $K$ but there are strands with degree less than or equal to $-2$ other than irreducible components of $K$.\n\n\n\\begin{figure}[h]\n\\begin{tikzpicture}[scale=1.2]\n\\begin{scope}\n\\draw (0.1,0.2)--(2.6,0.2);\n\\draw (0.5,0.5)--(1.25,-1.5);\n\\draw (2,0.5)--(0,-1.5);\n\\draw (2.5,0.3)--(-0,-.75);\n\\end{scope}\n\\begin{scope}[shift={(4.5,0)}]\n\\draw (-1,0.2)--(2,0.2);\n\\draw (1,0.5)--(1,-1.5);\n\\draw (1.5,0.5)--(0.5,-1.5);\n\\draw (0.5,0.5)--(1.5,-1.5);\n\\draw (-.5,0.5)--(2,-1.5);\n\\draw (2,0.35)--(-.5,-.5);\n\n\\draw (1.5,-1938*1.5\/4047-171\/8094)--(-1,1938\/4047-171\/8094);\n\n\\end{scope}\n\n\\end{tikzpicture}\n\\caption{Examples of symplectic line arrangements}\n\\label{line}\n\\end{figure}\n\n\n\n\\subsection{Pseudo-holomorphic curves in rational symplectic $4$-manifolds}\nAssume that a minimal symplectic filling $W$ of $Y$ is obtained from another minimal symplectic filling $W'$ by rationally blowing down a negative definite star-shaped plumbing graph $G$ that is symplectically embedded in $W'$.\nTo observe the effect of rationally blowing down $G\\subset W'$ on a symplectic line arrangement, we first need to know how $G$ is symplectically embedded in $W'$. For this, we introduce several lemmas to analyze a symplectic embedding $G$ in $M=W'\\cup K$. We assume that all irreducible components of $K$ and $G$ are $J$-holomorphic for a suitable tamed $J$. The following are some basic lemmas regarding $J$-holomorphic curves in $M$ obtained in ~\\cite{BOn}.\n\n\\begin{lem}[\\cite{BOn}]\n\\label{lem1} \nLet $L, C_1,\\dots,C_k$ be a collection of symplectic $2$-spheres in a closed symplectic $4$-manifold $M$ with $L\\cdot L=1$, $C_i\\cdot C_i \\leq 0$. Suppose that $J$ is a tame almost complex structure for which $L, C_1,\\dots,C_k$ are $J$-holomorphic. Then there exists at least one $J$-holomorphic $(-1)$ curve in $M \\setminus L$.\n\\end{lem}\n\n\n\\begin{lem}[\\cite{BOn}]\n\\label{lem2}\nLet $M$ be a closed symplectic $4$-manifold and let $L$ be a symplectically embedded $2$-sphere of self-intersection number $1$. Then, no symplectically embedded $2$-sphere of nonnegative self-intersection number is contained in $M\\setminus L$. Pseudo-holomorphic $(-1)$ curves in $M\\setminus L$ are mutually disjoint.\n\\end{lem}\n\n\n\\begin{lem}[\\cite{BOn}]\n\\label{lem3}\nLet $M$ be a closed symplectic $4$-manifold and let $L$ be a symplectically embedded $2$-sphere of self-intersection number $1$. Then, any irreducible singular or higher-genus pseudo-holomorphic curve $C$ in $M$ satisfies $C\\cdot L \\geq 3$. In particular, neither an irreducible singular nor a higher-genus pseudo-holomorphic curve is contained in $M\\setminus L$.\n\\end{lem}\n\nFrom Lemma~\\ref{lem1}, we obtain a sequence of rational symplectic $4$-manifolds $M_j$ $(0\\leq j \\leq N)$ with $M_0\\cong \\mathbb{CP}^2$ and $M_N=M \\cong \\mathbb{CP}^2 \\sharp N\\overline{\\mathbb{CP}^2}$ such that $M_{j}$ is obtained by blowing down the $J_{j+1}$-holomorphic $(-1)$ curve $e_{j+1}$ from $M_{j+1}$ for a tamed $J_{j+1}$. \nNote that for a $J$-holomorphic $(-1)$ curve $e$ and an irreducible component $C$ of $G$ and $K$ in $M$, either $C$ is disjoint from $e$ or $C$ intersects transversally once with $e$ due to Lemma~\\ref{lem3}. Hence, the image of $C$ under the blowing-downs in $M_j$ is a non-singular $J_j$-holomorphic curve. In particular, the self-intersection number of $C$ increases to $-1$. Therefore, $C$ eventually becomes the $J_j$-holomorphic curve $e_j$ under the blowing-downs unless $C$ is $C^0$ or $C^i_1$ for some $i$, which becomes an irreducible component of a symplectic line arrangement in $M_0\\cong\\mathbb{CP}^2$. Here $C^i_j$ denotes the $j^{\\text{th}}$ irreducible component of the $i^{\\text{th}}$ arm of $K$, and $C^0$ denotes the central $2$-sphere.\n\n\n\\begin{lem}\nIf there is a triple intersection between the images of the irreducible components of $K$ and $G$ during the blowing-downs, then they are the images of $C^{i_1}_1$, $C^{i_2}_1$ and $C^{i_3}_1$ for some $i_1$, $i_2$ and $i_3$ under the blowing-downs. \n\\label{triple}\n\\end{lem}\n\n\n\\begin{figure}[h]\n\\begin{tikzpicture}[scale=0.9]\n\\draw (0,1.3)--(0,-.25);\n\\draw (0.2,.2)--(-1,-1);\n\\draw (-0.2,.2)--(1,-1);\n\\filldraw (0,0) circle (1.2pt);\n\\end{tikzpicture}\n\\caption{Pseudo-holomorphic curves with a triple intersection}\n\\end{figure}\n\n\\begin{proof}\nIf one of the three pseudo-holomorphic curves does not come from $C^i_1$ of $K$, then the curve eventually becomes a $J_j$-holomorphic $(-1)$ curve $e_j$; thus, we have two pseudo-holomorphic curves with tangential intersection by blowing down $e_j$. If the other two pseudo-holomorphic curves come from the first components of $K$, then we have two symplectic lines in $S$ that do not intersect transversally, contradicting the definition of a symplectic line arrangement. Otherwise, we eventually have a singular curve intersecting the complex line $\\mathbb{CP}^1$ at most once, which contradicts Lemma~\\ref{lem3}.\n\\end{proof}\n\n\n\n\n\\section{Proof of Theorem~\\ref{thm1}}\n\\label{sec-3}\nTo prove Theorem~\\ref{thm1}, we first analyze the effect on symplectic line arrangements under a single rational blowdown surgery. In particular, we investigate the difference between two symplectic line arrangements $S$ and $S'$ corresponding to two minimal symplectic fillings $W$ and $W'$, respectively, where $W$ is obtained from $W'$ by rationally blowing down a negative definite star-shaped plumbing graph $G$ symplectically embedded in $W'$. \n\nFirst, we note how $J$-holomorphic curves intersect $K$ and $G$ in $M=W'\\cup K$ using lemmas in Section ~\\ref{pre}. \nLet $D^i_j$ be the $j^{\\text{th}}$ irreducible component of the $i^{\\text{th}}$ arm of $G$ and $D^0$ be the central $2$-sphere of $G$.\n\n\n\\begin{prop}\n\\label{prop1}\nFor the last component $D^i_{a_i}$ of each $i^{\\text{th}}$ arm in $G$, there is a $J$-holomorphic $(-1)$ curve $e_i$ and a linear chain $L_i$ (possibly empty) of the $J$-holomorphic curves in $M$ such that $D^i_{a_i}$ intersects with one end of $L_i$ and $e_i$ connects with the other end of $L_i$ and an irreducible component of $K$. Furthermore, we eliminate the $i^{\\text{th}}$ arm of $G$ by blowing down $(-1)$ curves consecutively starting from $e_i$.\n\\end{prop}\n\n\n\\begin{figure}[h]\n\\begin{tikzpicture}[scale=0.9]\n\\begin{scope}\n\\node[above] at (2.3, 0.5) {$G$};\n\n\\draw(0.8,0)--(4.8,0);\n\\draw (1.5,0.2)--(1,-0.7);\n\\draw (3.,0.2)--(2.5,-0.7);\n\n\\draw[dotted,very thick](2.15-.3,-1.2)--(2.5-.3,-1.2);\n\n\\draw (1,-0.3)--(1.5,-1.2);\n\\draw (2.5,-0.3)--(3.,-1.2);\n\n\\node at (1.25, -1.5) {$\\vdots$};\n\\node at (2.75, -1.5) {$\\vdots$};\n\n\\draw (1,-1.8)--(1.5,-2.7);\n\\draw (1.5,-2.3)--(1,-3.2);\n\\node at (1.25, -3.15) {$\\vdots$};\n\\draw (1.5,-3.2)--(1,-4.1);\n\\draw [decorate,decoration={brace,mirror,amplitude=3pt},xshift=0pt,yshift=0pt]\n\t(0.9,-2.6) -- (0.9,-4.1) node [black,midway,xshift=-10pt] \n\t{$L_i$};\n\n\\node at (0.75, -2.1) {$D^i_{a_i}$};\n\\node[red] at (4, -3.7) {$e_i$};\n\n\n\n\\draw (2.5,-1.8)--(3,-2.7);\n\\draw[red,dashdotted,very thick](4.2,0.2) to[out=-40,in=180] (6.9,-.5);\n\\draw[red,dashdotted,very thick](3.4,0.2) to[out=-60,in=200] (7.7,-.8);\n\n\\draw[red,dashdotted,very thick](2.7,-2.5)--(9,-2.1);\n\\draw[red,dashdotted,very thick](1.0,-3.85) to[out=-10,in=200](10.6,-2.5);\n\n\n\\node at(5.7,-3){$\\vdots$};\n\\end{scope}\n\n\\begin{scope}[shift={(8.5,0)}]\n\\node[above] at (1.8,0.5){$K$};\n\\draw(-2.2,0)--(3.8,0);\n\\draw (0.5,0.2)--(0,-0.7);\n\\draw (.5+1.5,0.2)--(0+1.5,-0.7);\n\\draw (3.5,0.2)--(3,-0.7);\n\n\\draw (-.5,0.2)--(-1-.2,-0.7-.2*9\/5);\n\n\\draw (-1.5,0.2)--(-2,-0.7);\n\\node at (-1.3,-0.3) {$\\cdots$};\n\\draw[dotted,very thick](2.15+.25-1.5,-1.2)--(2.5+.25-1.5,-1.2);\n\\draw[dotted,very thick](2.15+.25,-1.2)--(2.5+.25,-1.2);\n\n\n\n\n\\draw (0,-0.3)--(0.5,-1.2);\n\\draw (0+1.5,-0.3)--(0.5+1.5,-1.2);\n\n\\draw (3,-0.3)--(3.5,-1.2);\n\n\\node at (0.25, -1.5) {$\\vdots$};\n\\node at (0.25+1.5, -1.5) {$\\vdots$};\n\n\\node at (3.25, -1.5) {$\\vdots$};\n\n\\draw (0,-1.8)--(0.5,-2.7);\n\\draw (0+1.5,-1.8)--(0.5+1.5,-2.7);\n\n\\draw (3,-1.8)--(3.5,-2.7);\n\n\\node[right] at (3.8,0) {$+1$};\n\\node[above] at (-.5, 0.2) {$-1$};\n\\node[above] at (-1.5, 0.2) {$-1$};\n\n\n\n\\end{scope}\n\n\\end{tikzpicture}\n\n\\caption{$J$-holomorphic curves in $M$ intersecting $K$ and $G$}\n\\end{figure}\n\n\n\\begin{proof}\nNote that every $J$-holomorphic $(-1)$ curve in $M=W'\\cup K$ intersects some irreducible components of $K$ because $W'$ is a minimal symplectic filling. \nLet $D$ be an irreducible component of $G$ that first becomes a pseudo-holomorphic $(-1)$ curve during the blowing-downs from $M=W'\\cup K$ to $\\mathbb{CP}^2$. Then, there should exist a linear chain of $J$-holomorphic curves $D=D_0,\\dots, D_k$ in $M$ such that the last component $D_k$ is a $(-1)$ curve, and the degree of $D_i$ $(1\\leq i\\leq k-1)$ is less than that of $D_k$ because we cannot increase the degree of $D$ without such a linear chain. \nHence, we find a linear chain $L$ of $J$-holomorphic curves consisting of $D_1, \\dots, D_{k-1}$ with a $J$-holomorphic $(-1)$ curve $e=D_k$ such that $D$ intersects with one end of $L$ and $e$ intersects with the other end of $L$.\nNote that $e$ intersects only one irreducible component of $K$ due to Lemma~\\ref{triple}. Furthermore, $D$ must be the last component $D^i_{a_i}$ of some $i^{\\text{th}}$ arm of $G$. Otherwise, we would have a triple intersection consisting of the images of adjacent components of $D$ and an irreducible component of $K$ intersecting $e$, which is a contradiction.\n\nSuppose there is another linear chain $L'$ and a $(-1)$ curve $e'$ intersecting $D$ as $L$ and $e$. Subsequently, an adjacent component of $D$ with irreducible components of $K$ intersecting $e$ and $e'$ would result in a triple intersection that contradicts Lemma~\\ref{triple}. \nTherefore, starting from blowing down $e$, $D$ becomes a $(-1)$ curve under the blowing-downs along $(-1)$ curves coming from a linear chain of $J$-holomorphic curves consisting of $L$, $e$ and some irreducible components of $K$ connected to $D$ via $L$ and $e$. \nLet $G'$ be the image of $G$ under blowing-downs of the $(-1)$ curves above with the $(-1)$ curve coming from $D$. \nThen, $G'$ is still a star-shaped plumbing graph that has the same number of arms with $G$, and the number of irreducible components of $i^{\\text{th}}$ arm in $G'$ is less than that of $G$ by one. Then, using the same argument as before, we see that the last component of the $i^{\\text{th}}$ arm in $G'$ is the first irreducible component becoming a $(-1)$ curve among the irreducible components of the $i^{\\text{th}}$ arm in $G'$. \nWe repeat the same process until all irreducible components of the $i^{\\text{th}}$ arm in $G$ disappear under blowing-downs. Furthermore, by performing the same process for each arm in $G$, we conclude that $G$ eventually reduces to a single pseudo-holomorphic rational curve, which is the image of $D^0$, under the blowing-downs. \n\\end{proof}\n\n\nUnlike each arm of $G$, there may be several linear chains of $J$-holomorphic curves in $M$ intersecting $D^0$.\nThe next proposition shows how $G$ is obtained under the blowing-ups from $\\mathbb{CP}^2$ to $M=W'\\cup K$.\n \n\\begin{prop}\n\\label{prop2}\nLet $T'$ be a subset of a symplectic line arrangement $S'$ consisting of the image of arms in $K$ connected to $G$ via $J$-holomorphic curves in $M$ under the blowing-downs from $M$ to $\\mathbb{CP}^2$. Then, $T'$ has a unique intersection point, and $G$ is obtained by a sequence of blowing-ups from this point.\n\\end{prop}\n\n\n\\begin{proof}\nWe arrange a sequence of blowing-downs from $M=W'\\cup K$ to $\\mathbb{CP}^2$ into two steps: first blow down all $(-1)$ curves that only intersect $K$ and the image of $K$, and then blow down all $(-1)$ curves intersecting $G$ and the image of $G$ to obtain the image $T'\\subset S'$ of arms in $K$ connected to $G$ via $J$-holomorphic curves in $M$.\n \nFirst, note that for each arm of $K$, there is at most one arm of $G$ connected to the arm of $K$ via $J$-holomorphic curves; otherwise, we have cycles of $J$-holomorphic curves, which contradicts Lemma~\\ref{lem3}.\nNow, by the first step of the blowing-downs, the linear chain $L_i$ with $(-1)$ curve $e_i$ in Proposition~\\ref{prop1} reduces to a single $(-1)$ curve $e'_i$, and there may be several $(-1)$ curves intersecting the central curve $D^0$ of $G$.\nThen, when we blow down $e'_i$, one of the two curves intersecting $e'_i$ becomes a $(-1)$ curve. \nBecause all the irreducible components of each arm in $G$ disappear from the last to the first component, $G$ reduces to a single pseudo-holomorphic curve, which is the image of $D^0$ by blowing down all $(-1)$ curves consecutively. We further blow down $(-1)$ curves so that $D^0$ eventually becomes a $(-1)$ curve $e$. \n\nBecause of the aforementioned blowing-down process, $e$ intersects the image of arms in $K$ connected to $G$ via $J$-holomorphic curves in $M$. Moreover, $e$ corresponds to the last step in the sequence of blowing-downs from $M$ to $\\mathbb{CP}^2$, which indicates that the image $T' \\subset S'$ of the arms in $K$ connected to $G$ via $J$-holomorphic curves has a unique intersection point.\n\n\\begin{figure}[h]\n\\begin{tikzpicture}[scale=0.9]\n\\begin{scope}\n\\node[above] at (2.3, 0.5) {$G$};\n\n\\draw(0.8,0)--(4.8,0);\n\\draw (1.5,0.2)--(1,-0.7);\n\\draw (3.,0.2)--(2.5,-0.7);\n\n\\draw[dotted,very thick](2.15-.3,-1.2)--(2.5-.3,-1.2);\n\n\\draw (1,-0.3)--(1.5,-1.2);\n\\draw (2.5,-0.3)--(3.,-1.2);\n\n\\node at (1.25, -1.5) {$\\vdots$};\n\\node at (2.75, -1.5) {$\\vdots$};\n\n\\draw (1,-1.8)--(1.5,-2.7);\n\\draw (1.5,-2.3)--(1,-3.2);\n\\node at (1.25, -3.15) {$\\vdots$};\n\\draw (1.5,-3.2)--(1,-4.1);\n\\draw [decorate,decoration={brace,mirror,amplitude=3pt},xshift=0pt,yshift=0pt]\n\t(0.9,-2.6) -- (0.9,-4.1) node [black,midway,xshift=-10pt] \n\t{$L_i$};\n\n\\node at(5.7,-3){$\\vdots$};\n\\draw (2.5,-1.8)--(3,-2.7);\n\\draw[red,dashdotted,very thick](4.2,0.2) to[out=-40,in=180] (6.9,-.5);\n\\draw[red,dashdotted,very thick](3.4,0.2) to[out=-60,in=200] (7.7,-.8);\n\n\\draw[red,dashdotted,very thick](2.7,-2.5)--(9,-2.1);\n\\draw[red,dashdotted,very thick](1.0,-3.85) to[out=-10,in=200](10.6,-2.5);\n\n\n\n\n\\begin{knot}[\n\tclip width=20,\n\tclip radius = 20pt,\n\tend tolerance = 10pt,\n]\n\n\n\\draw[very thin,->](6.5,-5)--(6.5,-.8);\n\\draw[very thin,->](6.5,-5)--(7.3,-1.2);\n\\draw[very thin,->](6.5,-5)--(8.9,-2.8);\n\\draw[very thin,->](6.5,-5)--(10.4,-2.8);\n\\node at(8.7,-3.5){$\\cdots$};\n\n\\node[below] at(6.5,-5){blow down to};\n\\end{knot}\n\n\\end{scope}\n\n\n\n\\begin{scope}[shift={(4.85,-7.2)},scale=1.5]\n\\draw (-1,0.2)--(2.75,0.2);\n\\draw [decorate,decoration={brace,amplitude=3pt},xshift=0pt,yshift=5pt]\n\t(0.5,0.5) -- (1.5,0.5) node [black,midway,yshift=10pt] \n\t{$T'$};\n\\node[above] at (-1.1,0.45){$S'$};\n\\draw (1,0.5)--(1,-1.5);\n\\draw (1.75,0.5)--(0.25,-1.5);\n\\draw (0.25,0.5)--(1.75,-1.5);\n\\draw (-1,0.4)--(2,-1.5);\n\\draw (2.75,0.4)--(-.25,-1.5);\n\\node at (1.25,0.4) {$\\cdots$};\n\\node at (2.15,0.4) {$\\cdots$};\n\\node at (-0.25,0.4) {$\\cdots$};\n\\filldraw (1,-.5) circle (1.35pt);\n\\node[left] at (0.95,-0.5){$p'$};\n\\end{scope}\n\\begin{scope}[shift={(8.5,0)}]\n\\node[above] at (1.8,0.5){$K$};\n\\draw(-2.2,0)--(3.8,0);\n\\draw (0.5,0.2)--(0,-0.7);\n\\draw (.5+1.5,0.2)--(0+1.5,-0.7);\n\\draw (-.5,0.2)--(-1-.2,-0.7-.2*9\/5);\n\n\\draw (3.5,0.2)--(3,-0.7);\n\\draw (-.5,0.2)--(-1,-0.7);\n\\draw (-1.5,0.2)--(-2,-0.7);\n\\node at (-1.3,-0.3) {$\\cdots$};\n\\draw[dotted,very thick](2.15+.25-1.5,-1.2)--(2.5+.25-1.5,-1.2);\n\\draw[dotted,very thick](2.15+.25,-1.2)--(2.5+.25,-1.2);\n\n\n\n\n\\draw (0,-0.3)--(0.5,-1.2);\n\\draw (0+1.5,-0.3)--(0.5+1.5,-1.2);\n\n\\draw (3,-0.3)--(3.5,-1.2);\n\n\\node at (0.25, -1.5) {$\\vdots$};\n\\node at (0.25+1.5, -1.5) {$\\vdots$};\n\n\\node at (3.25, -1.5) {$\\vdots$};\n\n\\draw (0,-1.8)--(0.5,-2.7);\n\\draw (0+1.5,-1.8)--(0.5+1.5,-2.7);\n\n\\draw (3,-1.8)--(3.5,-2.7);\n\n\\node[right] at (3.8,0) {$+1$};\n\\node[above] at (-.5, 0.2) {$-1$};\n\\node[above] at (-1.5, 0.2) {$-1$};\n\n\n\n\\end{scope}\n\n\\end{tikzpicture}\n\n\\caption{The arms of $K$ connected to $G$ via $J-$holomorphic curves blow down to $T'\\subset S'$}\n\n\\end{figure}\n\\end{proof}\n\n\nNext, we investigate how $S'$ changes by rationally blowing down $G\\subset W'$. \nOnce we fix a sequence of blowing-downs along $J$-holomorphic $(-1)$ curves $E$ from $M=W'\\cup K$ to $\\mathbb{CP}^2$, there is a one-to-one correspondence between the set of intersection points in $S'$ and a subset of $E$ whose homology classes appear in more than one arm in $K$. \nNote that if we take another sequence of blowing-downs with the $J'$-holomorphic $(-1)$ curves $F$ from $M'$ to $\\mathbb{CP}^2$, each homology class of $f_i\\in F$ must be equal to that of some $e_j\\in E$. Therefore, the intersection data of $S'$ are determined by a homological expression of $\\{C^i_1\\}\\subset K$ in terms of a complex line $\\mathbb{CP}^1$ and some $(-1)$ 2-spheres disjoint from the complex line. \n\nNow, we arrange a sequence of blowing-downs from $M=W'\\cup K$ to $\\mathbb{CP}^2$ into two steps, as in the proof of Proposition~\\ref{prop2}. Let $E_G$ be a subset of $E$ whose homology classes appear in the homology classes of irreducible components in $G\\subset M$. If $e\\in E\\setminus E_G$ represents an intersection point of $S'$, then $e$ also represents an intersection point of a symplectic line arrangement $S$ corresponding to $W$ because $e$ is a $(-1)$ curve in $M \\setminus G$. Furthermore, since $G$ is obtained by a sequence of blowing-ups from a unique intersection point of $T'\\subset S'$, there is at most one $(-1)$ curve in $E_G$ that corresponds to an intersection point of $S'$. Then, we obtain the following relation between $N_S$ and $N_{S'}$ under rationally blowing down along $G$ in $W'$.\n\n\n\\begin{prop}\n\\label{mainthm1}\nIf a minimal symplectic filling $W$ is obtained from $W'$\nby rationally blowing down along $G$, then $N_S=N_{S'}$ or $N_S=N_{S'}-1$, where $S$ and $S'$ are symplectic line arrangements corresponding to $W$ and $W'$, respectively.\n\\end{prop}\n\n\\begin{proof}\n\nLet $K_{T'}\\subset K$ be a subset of arms in $K$ whose image under the blowing-downs is $T'$ in Proposition~\\ref{prop2}. \nThe observations above show that the intersection data of $S$ are equal to that of $S'$ except for the intersection data in $T\\subset S$, where $T$ is the image of $K_{T'}$ under a sequence of blowing-downs from $W\\cup K$ to $\\mathbb{CP}^2$. Hence, we only need to show that $T$ has at most one multi-intersection point.\n\n As we saw in Proposition~\\ref{prop2}, $G$ is obtained from the exceptional curve $e$ by blowing up at the unique intersection point $p'$ of $T'$. Therefore, the number of arms in $G$ is less than or equal to the number of points in $e$ which we blow up to get the central curve $D^0$ of $G$. Hence, the absolute value of the degree of $D^0$ is strictly larger than the number of arms in $G$, so that $G$ must be linear or $\\Gamma_{p,q,r}$ in Figure~\\ref{pqr} because of Stipsicz and Bhupal's classification result~\\cite{BS}.\n \n\\begin{figure}[h]\n\\begin{tikzpicture}[scale=0.6]\n\\node[bullet] at (0,0){};\n\\node[bullet] at (1,0){};\n\\node[bullet] at (3,0){};\n\\node[bullet] at (4,0){};\n\\node[bullet] at (5,0){};\n\\node[bullet] at (7,0){};\n\\node[bullet] at (8,0){};\n\n\\node[bullet] at (4,-1){};\n\\node[bullet] at (4,-3){};\n\\node[bullet] at (4,-4){};\n\n\n\\node[below left] at (0,0){$-(p+3)$};\n\\node[above] at (1,0){$-2$};\n\\node[above] at (3,0){$-2$};\n\\node[above] at (4,0){$-4$};\n\\node[above] at (5,0){$-2$};\n\\node[above] at (7,0){$-2$};\n\\node[below right] at (8,0){$-(q+3)$};\n\n\\node[left] at (4,-1){$-2$};\n\\node[left] at (4,-3){$-2$};\n\\node[below] at (4,-4){$-(r+3)$};\n\n\n\\node at (2,0){$\\cdots$};\n\\node at (6,0){$\\cdots$};\n\\node at (4,-1.9){$\\vdots$};\n\n\n\\draw (0,0)--(1.5,0);\n\\draw (2.5,0)--(5.5,0);\n\\draw (6.5,0)--(8,0);\n\\draw (4,0)--(4,-1.5);\n\\draw (4,-2.5)--(4,-4);\n\n\t\\draw [thick,decorate,decoration={brace,mirror,amplitude=5pt},xshift=0pt,yshift=-7pt]\n\t(1,0) -- (3,0) node [black,midway,yshift=-11pt] \n\t{$q$};\n\n\t\\draw [thick,decorate,decoration={brace,mirror,amplitude=5pt},xshift=0pt,yshift=-7pt]\n\t(5,0) -- (7,0) node [black,midway,yshift=-11pt] \n\t{$r$};\n\n\t\\draw [thick,decorate,decoration={brace,amplitude=5pt},xshift=0pt,xshift=7pt]\n\t(4,-1) -- (4,-3) node [black,midway,xshift=11pt] \n\t{$p$};\n\n\\end{tikzpicture}\n\\caption{Plumbing graph $\\Gamma_{p,q,r}$}\n\\label{pqr}\n\\end{figure}\n\n\nRecalling the blowing-down process from $G$ to a point in the proof of Proposition~\\ref{prop2}, we can observe that the effect of each blowing-down is either \nincreasing the degree of an irreducible component or decreasing the length of an arm. Conversely, under the blowing-ups from $p'$ to $G$, we obtain a star-shaped plumbing of the symplectic $2$-spheres $K_G$ consisting of the complex line in $S'$ and the image of $T'\\subset S'$. In particular, the effect of each blowing-up is either to decrease the degree of an irreducible component or to increase the length of an arm. Furthermore, the complement of $G$ in the resulting rational symplectic $4$-manifold $\\widetilde{M}$ is $K_G$, indicating that $K_G$ is a concave cap of $(\\partial G, \\xi_{\\text{can}})$. As $G$ is either linear or $\\Gamma_{p,q,r}$, $K_G$ is represented by Figure~\\ref{KG}. The degrees of unlabeled strands in $(b)$ are all $(-2)$.\n\n\\begin{figure}[h]\n\\begin{tikzpicture}[scale=0.8]\n\\begin{scope}[shift={(3.2,0)}]\n\n\\node at (2.5, -3.3) {$(a)$};\n\\draw(0,0)--(5.7,0);\n\\draw[thick] (0.5,0.2)--(0,-0.7);\n\\draw[thick] (1.25,0.2)--(0.75,-0.7);\n\n\\draw (2.25,0.2)--(1.75,-0.7);\n\\node[above] at (2.25, .2) {$-1$};\n\\draw (3.25+.25,0.2)--(2.75+.25,-0.7);\n\\node[above] at (3.25+.25, .2) {$-1$};\n\\draw (2.25,0.2)--(1.75,-0.7);\n\n\\draw (4.25+.25,0.2)--(3.75+.25,-0.7);\n\\node[above] at (4.25+.25, .2) {$-1$};\n\n\\draw (5.25+.25,0.2)--(4.75+.25,-0.7);\n\\node[above] at (5.25+.25, .2) {$-1$};\n\n\\filldraw (5-.25,-.4) circle(1pt);\n\\filldraw (4.85-.25,-.4) circle(1pt);\n\\filldraw (5.15-.25,-.4) circle(1pt);\n\n\n\\node at (2.45, -0.8) {$-2$};\n\\node at (2.45, -2.4) {$-2$};\n\n\\node at (3.7, -0.8) {$-2$};\n\\node at (3.7, -2.4) {$-2$};\n\n\n\n\\filldraw (2.5,-1.6) circle(1pt);\n\\filldraw (2.65,-1.6) circle(1pt);\n\\filldraw (2.8,-1.6) circle(1pt);\n\n\n\\draw[thick] (0,-0.3)--(0.5,-1.2);\n\\draw[thick] (0.75,-0.3)--(1.25,-1.2);\n\n\\draw (1.75,-0.3)--(2.25,-1.2);\n\\draw (2.75+.25,-0.3)--(3.25+.25,-1.2);\n\n\n\\node at (0.25, -1.5) {$\\vdots$};\n\\node at (1, -1.5) {$\\vdots$};\n\\node at (2, -1.5) {$\\vdots$};\n\\node at (3+.25, -1.5) {$\\vdots$};\n\n\\draw[thick] (0,-1.8)--(0.5,-2.7);\n\\draw[thick] (.75,-1.8)--(1.25,-2.7);\n\\draw (1.75,-1.8)--(2.25,-2.7);\n\\draw (2.75+.25,-1.8)--(3.25+.25,-2.7);\n\n\\node[left] at (0,0) {$+1$};\n\n\n\n\n\\end{scope}\n\\begin{scope}[shift={(11,0)}]\n\\node at (2.1, -3.3) {$(b)$};\n\\draw(0,0)--(4+.25,0);\n\\draw (0.5,0.2)--(0,-0.7);\n\\node[above] at (0.5, 0.2) {\\footnotesize$-(r+2)$};\n\\draw (2+.25,0.2)--(1.5+.25,-0.7);\n\\node[above] at (2+.25, 0.2) {\\footnotesize$-(p+2)$};\n\\draw (3.5+.5,0.2)--(3+.5,-0.7);\n\\node[above] at (3.5+.5, 0.2) {\\footnotesize$-(q+2)$};\n\n\n\t\\draw [decorate,decoration={brace,amplitude=5pt},yshift=10pt,xshift=5pt]\n\t(0.4,-1) -- (0.4,-2.7) node [black,midway,xshift=18.5pt, yshift=0pt] \n\t{\\footnotesize$p+1$};\n\n\n\t\\draw [decorate,decoration={brace,amplitude=5pt},yshift=10pt,xshift=5pt]\n\t(2.15,-1) -- (2.15,-2.7) node [black,midway,xshift=18.5pt, yshift=0pt] \n\t{\\footnotesize$q+1$};\n\t\\draw [decorate,decoration={brace,amplitude=5pt},yshift=10pt,xshift=5pt]\n\t(3.65+.25,-1) -- (3.65+.25,-2.7) node [black,midway,xshift=18.5pt, yshift=0pt] \n\t{\\footnotesize$r+1$};\n\n\\draw (0,-0.3)--(0.5,-1.2);\n\\draw (1.5+.25,-0.3)--(2+.25,-1.2);\n\\draw (3+.5,-0.3)--(3.5+.5,-1.2);\n\n\\node at (0.25, -1.5) {$\\vdots$};\n\\node at (2, -1.5) {$\\vdots$};\n\\node at (3.25+.25+.25, -1.5) {$\\vdots$};\n\n\\draw (0,-1.8)--(0.5,-2.7);\n\\draw (1.5+.25,-1.8)--(2+.25,-2.7);\n\\draw (3+.5,-1.8)--(3.5+.5,-2.7);\n\n\n\n\\node[left] at (0,0) {$+1$};\n\n\n\\end{scope}\n\\end{tikzpicture}\n\\caption{Concave cap $K_G$}\n\\label{KG}\n\\end{figure}\n\nMore specifically, $K_G$ is of the form (a) or (b) in Figure~\\ref{KG} depending on whether $G$ is linear or $\\Gamma_{p,q,r}$. Note that two unlabeled arms in (a) correspond to the two arms of a linear plumbing graph $G$ whereas the arms with only $(-1)$ or $(-2)$ strands in (a) contribute to the degree of $D^0$.\nLet $K'$ be an image of $S'$ in $\\widetilde{M}$ containing $K_G$ under the blowing-ups from $e$ to $G$. Then, we have a sequence of blowing-ups from $K'$ to $K$ in terms of $E\\setminus E_G$, so that the homological data of $K$ in $M$ consist of the homological data of $K_G$ in $\\widetilde{M}$ with the homological data from the blowing-ups from $K'$ to $K$. \nSimilarly, the homological data of $K$ in $W\\cup K$ consist of the homological data of $K_G$ in $(\\widetilde{M}\\setminus G) \\cup B_G$ with the homological data from the blowing-ups from $K'$ to $K$ in terms of $E\\setminus E_G$, where $B_G$ is a rational homology ball filling of $(\\partial G, \\xi_{\\text{can}})$. \nAs the arms in $K_G$ become $K_{T'}\\subset K$, the intersection data of $T$ are determined by homological data of $K_G$ in $(\\widetilde{M}\\setminus G) \\cup B_G$. Specifically, the intersection data of $T$ are equal to those of a symplectic line arrangement corresponding to $B_G$ with respect to concave cap $K_G$.\nFinally, since there are only two possible symplectic line arrangements in Figure~\\ref{possibleline} for any minimal symplectic filling of $(\\partial G, \\xi_{\\text{st}})$ with respect to $K_G$ due to the arms starting with $(-1)$ strands (refer to Proposition 3.2 in ~\\cite{CP2} for details), the number of multi-intersection points in $T$ is at most one, as required. \n\n\\begin{figure}[h]\n\\begin{tikzpicture}[scale=1.2]\n\\begin{scope}\n\\draw (0.1,0.2)--(2.6,0.2);\n\\draw (1,0.5)--(1,-1.5);\n\\draw (1.5,0.5)--(0.5,-1.5);\n\\draw (0.5,0.5)--(1.5,-1.5);\n\\draw (2,0.5)--(0,-1.5);\n\\draw (2.5,0.5)--(-.5,-1.5);\n\\node at (2.15,0.4) {$\\cdots$};\n\\end{scope}\n\\begin{scope}[shift={(4.5,0)}]\n\\draw (-1,0.2)--(2,0.2);\n\\draw (1,0.5)--(1,-1.5);\n\\draw (1.5,0.5)--(0.5,-1.5);\n\\draw (0.5,0.5)--(1.5,-1.5);\n\\draw (-1,0.5)--(2,-1.5);\n\\node at (1.25,0.4) {$\\cdots$};\n\n\\end{scope}\n\n\\end{tikzpicture}\n\\caption{Two possible symplectic line arrangements}\n\\label{possibleline}\n\\end{figure}\n\n\\end{proof}\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm1}]\nIt follows from Proposition~\\ref{mainthm1} and that the minimal resolution graph is obtained from the left-hand symplectic line arrangement in Figure~\\ref{possibleline}, which has a unique multi-intersection point.\n\\end{proof}\n\n\n\n\n\\section{Proof of Theorem~\\ref{thm2}}\n\\label{4}\n\nIn this section, we show that the converse of Theorem~\\ref{thm1} also holds for a Seifert $3$-manifold $Y(-b; (\\alpha_1, \\beta_1), (\\alpha_2, \\beta_2),\\ldots, (\\alpha_n, \\beta_n))$ with $b\\geq n+2$. As mentioned in Section 2, a minimal symplectic filling $W$ with $N_S\\leq 1$ is determined by the homological data of $K$ for $W$. Here $S$ is a symplectic line arrangement corresponding to $W$. Therefore, we need to analyze all possible curve configurations coming from $S$ with $N_S\\leq 1$ to show Theorem~\\ref{thm2}. The strategy for the proof is similar to the proof of Theorem 1.1 in ~\\cite{CP2}. We divide all possible curve configurations into certain types and then show that there are sequences of rational blowdowns from the minimal resolution for each type using lemmas in Section 4, ~\\cite{CP2}.\nFirst, when $b\\geq n+2$, we determine all possible symplectic line arrangements $S$ with $N_S\\leq 1$.\n\n\\begin{lem}\nAssume that $b\\geq n+2$. If the number $N_S$ of multi-intersection points of a symplectic line arrangement $S$ is at most 1, then $S$ is one of the two symplectic line arrangements in Figure~\\ref{possibleline}.\n\\end{lem}\n\n\\begin{proof}\n\nSince $b\\geq n+2$, there is at least one arm in $K$ that consists of a single $(-1)$ $2$-sphere. \nLet $s\\in S$ be an image of the $(-1)$ $2$-sphere under blowing-downs. Then, there are at most two intersection points on $s$ due to the degree. Because $N_S\\leq 1$, there are only two possibilities: all symplectic lines in $S$ have a common intersection point or all symplectic lines have a common intersection except one symplectic line, which are left-hand and right-hand line arrangements in Figure~\\ref{possibleline}, respectively.\n\\end{proof}\n\nWhen we attempt to obtain a curve configuration $C$ from a symplectic line arrangement $S$, we first blow up all intersection points between symplectic lines in $S$. Once we blow up an exceptional strand, we should blow up all intersection points of the strand except one to allow only strands with degree \n$\\leq -2$, if each strand represents an irreducible component of $K$. Without loss of generality, we assume that the first $n$ arms become \\emph{essential arms} in $K$ consisting of strands with degrees \n$\\leq -2$. Based on this, we can divide all the possible curve configurations obtained from $S$ with $N_S\\leq 1$ into the following three types:\n\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tikzpicture}[scale=1.3]\n\\begin{scope}[shift={(-3.5,0)}]\n\n\\draw (0,0.2)--(2,0.2);\n\\draw (0.4,0.5)--(0.4,-1.5);\n\\draw (0.8,0.5)--(0.8,-1.5);\n\\draw (1.2,0.5)--(1.2,-1.5);\n\\draw (1.6,0.5)--(1.6,-1.5);\n\\draw[red, thick, dashdotted] (0,-0.6)--(2,-0.6);\n\\node[left,red] at (0,-0.6) {$-1$};\n\\node[left] at (0,0.2) {$+1$};\n\\node at (1,-2) {$(a)$};\n\n\\filldraw (1.3,0.4) circle (.4pt);\n\\filldraw (1.4,0.4) circle (.4pt);\n\\filldraw (1.5,0.4) circle (.4pt);\n\\node[above] at (0.4,0.5) {$0$};\n\\node[above] at (0.8,0.5) {$0$};\n\\node[above] at (1.2,0.5) {$0$};\n\\node[above] at (1.6,0.5) {$0$};\n\\end{scope}\n\n\\begin{scope}\n\\node at (1,-2) {$(b)$};\n\n\\draw (0,0.2)--(2,0.2);\n\\draw (0.4,0.5)--(0,-0.5);\n\\draw (0,-0.05)--(0.4,-1.6);\n\\node[right] at (0.4,-1.6) {$c$};\n\\draw (0.8,0.5)--(0.8,-0.6);\n\\draw (1.2,0.5)--(1.2,-0.8);\n\\draw (1.6,0.5)--(1.6,-1);\n\n\\draw[red, thick, dashdotted](0,-0.7)--(0.9,-0.25);\n\\draw[red, thick, dashdotted](0.1,-1.1)--(1.3,-0.5);\n\\draw[red, thick, dashdotted](0.2,-1.5)--(1.7,-0.75);\n\n\\node[left, red] at (0,-0.7) {$e_1$};\n\\node[left, red] at (0.1,-1.1) {$e_{n-1}$};\n\n\n\\filldraw (1.3-.4,0.4) circle (.4pt);\n\\filldraw (1.4-.4,0.4) circle (.4pt);\n\\filldraw (1.5-.4,0.4) circle (.4pt);\n\n\\filldraw (1.3,0.4) circle (.4pt);\n\\filldraw (1.4,0.4) circle (.4pt);\n\\filldraw (1.5,0.4) circle (.4pt);\n\\node[above] at (0.8,0.5) {$-1$};\n\\node[above] at (1.2,0.5) {$-1$};\n\\node[above] at (1.6,0.5) {$-1$};\n\\node[left] at (0,0.2) {$+1$};\n\n\\filldraw (0.7-.25,-.95+.35) circle (.4pt);\n\\filldraw (0.75-.25,-1.+.35) circle (.4pt);\n\\filldraw (0.8-.25,-1.05+.35) circle (.4pt);\n\n\n\\filldraw (0.7,-.95) circle (.4pt);\n\\filldraw (0.75,-1.) circle (.4pt);\n\\filldraw (0.8,-1.05) circle (.4pt);\n\n\n\\end{scope}\n\n\\begin{scope}[shift={(3.5,0)}]\n\n\\filldraw (1.3-.4,0.4) circle (.4pt);\n\\filldraw (1.4-.4,0.4) circle (.4pt);\n\\filldraw (1.5-.4,0.4) circle (.4pt);\n\n\\filldraw (1.3,0.4) circle (.4pt);\n\\filldraw (1.4,0.4) circle (.4pt);\n\\filldraw (1.5,0.4) circle (.4pt);\n\\node at (1,-2) {$(c)$};\n\n\\draw (0,0.2)--(2,0.2);\n\\draw (0.4,0.5)--(0.4,-1.6);\n\\node[right] at (0.4,-1.6) {$c$};\n\n\\draw (0.8,0.5)--(0.8,-0.6);\n\\draw (1.2,0.5)--(1.2,-0.8);\n\\draw (1.6,0.5)--(1.6,-1);\n\\draw[red, thick, dashdotted] (0.6,0)--(1.8,0);\n\\draw[red, thick, dashdotted](0.3,-0.6)--(0.9,-0.3);\n\\draw[red, thick, dashdotted](0.3,-1)--(1.3,-0.5);\n\\draw[red, thick, dashdotted](0.3,-1.4)--(1.7,-0.7);\n\\node[left, red] at (0.3,-0.6) {$e_1$};\n\\node[left, red] at (0.3,-1) {$e_{n-1}$};\n\n\\filldraw (0.7-.15,-.95+.35) circle (.4pt);\n\\filldraw (0.75-.15,-1.+.35) circle (.4pt);\n\\filldraw (0.8-.15,-1.05+.35) circle (.4pt);\n\n\n\\filldraw (0.8,-.95+.05) circle (.4pt);\n\\filldraw (0.85,-1.+.05) circle (.4pt);\n\\filldraw (0.9,-1.05+.05) circle (.4pt);\n\n\\node[above] at (0.8,0.5) {$-1$};\n\\node[above] at (1.2,0.5) {$-1$};\n\\node[above] at (1.6,0.5) {$-1$};\n\\node[left] at (0,0.2) {$+1$};\n\n\\end{scope}\n\\end{tikzpicture}\n\\end{center}\n\\caption{Three configurations}\n\\label{startingposition}\n\\end{figure} \n\n\n\\begin{itemize}\n\\item{Type A:}\nCurve configurations obtained from $(a)$ in Figure~\\ref{startingposition} without blowing up the exceptional strand.\n\\item{Type B:}\nCurve configurations obtained from $(b)$ or $(c)$ in Figure~\\ref{startingposition} by blowing up at most one $e_i$ $(1\\leq i \\leq n-1)$.\n\\item{Type C:}\nCurve configurations obtained from $(b)$ or $(c)$ in Figure~\\ref{startingposition} by blowing up at least two $e_i$s $(1\\leq i \\leq n-1)$.\n\\end{itemize}\n\nNote that $(a)$ and $(c)$ are obtained from left-hand and right-hand symplectic line arrangements in Figure~\\ref{possibleline}, respectively, whereas $(b)$ is obtained from $(a)$ by blowing up the unique exceptional strand in $(a)$.\n \n We now recall several lemmas given in ~\\cite{CP2} that are useful for finding a surgical description for a minimal symplectic filling of each type. We first recall the notion of \\emph{standard blowing-ups}: for star-shaped $K'$ and $K$ of the same number of arms with central $(+1)$ vertex, we say $K'\\leq K$ if $n'_i\\leq n_i$ and $a'_{ij}\\leq a_{ij}$ for any $i$ and $j$ except for $a'_{in'_i} < a_{in'_i}$ when $n'_i,very thick] (2.25,-.5)--(2.75,-.5);\n\n\\end{scope}\n\\begin{scope}\n\\end{scope}\n\n\\begin{scope}[shift={(4.5,0)}]\n\\draw(0,0)--(4.5,0);\n\n\\node[above] at (0.5,0.2) {$-(n+1)$};\n\\draw (0.5,0.2)--(0,-0.7)--(-.5-.1,-1.6-.1*9\/5);\n\n\n\\draw[red, thick, dashdotted](-.7,-1.5-.15)--(4.35,-1.05-.225);\n\\draw (1.25+.25,0.2)--(0.75+.25,-0.7);\n\\draw[red, thick, dashdotted](0.05,-0.7-.15)--(1.5,-.1);\n\n\n\\draw (2.25+.25,0.2)--(1.75+.25+.05,-0.7+.05*9\/5);\n\n\\filldraw (2.5,-0.25) circle (.5pt);\n\\filldraw (2.6,-0.25) circle (.5pt);\n\\filldraw (2.7,-0.25) circle (.5pt);\n\n\\draw (3.25,0.2)--(2.75,-0.7);\n\n\n\\draw (4.25,0.2)--(3.75-.1,-0.7-.1*9\/5);\n\n\n\\draw (0,-0.3-.1-.1)--(0.5,-1.2-.1-.1);\n\\draw (0.75+.25,-0.3-.1-.1)--(1.25+.25-.1,-1.2-.1-.1+.1*9\/5);\n\n\\draw[red, thick, dashdotted](1.04,-.75)--(2.3,-.4);\n\\draw[red, thick, dashdotted](1.09,-1.)--(3,-.55);\n\\draw[red, thick, dashdotted](1.17,-1.2)--(3.8,-.75);\n\n\\draw (3.75,-0.3-.1-.1)--(4.25,-1.2-.1-.1);\n\n\n\t\\draw [decorate,decoration={brace,amplitude=5pt},xshift=0pt,yshift=2pt]\n\t(2.5,0.2) -- (3.25,0.2) node [black,midway,xshift=0pt,yshift=10pt] \n\t{$n-1$};\n\n\n\n\\node[left] at (0,0) {$+1$};\n\\node at (2,-2*13\/10) {$C_n$};\n\n\\draw[red, thick, dashdotted](0.05,-0.7-.15)--(1.5,-.1);\n\\draw[red, thick, dashdotted](-.7,-1.5-.15)--(4.35,-1.05-.225);\n\n\\end{scope}\n\\end{tikzpicture}\n\\caption{Curve configuration $C_n$ obtained from $S_{n+2,n+1}$}\n\\label{blowingup1}\n\\end{figure}\n\n\n\n\\begin{thm}\n\\label{thm3}\nFor each $n\\geq 3$, the minimal symplectic filling $W_n$ of $(Y_n,\\xi_{\\text{can}})$ cannot be obtained by a sequence of rational blowdowns from the minimal resolution of the corresponding weighted homogeneous surface singularity.\n\\end{thm}\n\n\nTo prove Theorem~\\ref{thm3}, we first observe the effect on symplectic line arrangements under a single rational blowdown surgery.\n\n\n\\begin{lem}\n\\label{counterexamplelem1}\nAssume that a minimal symplectic filling $W$ is obtained from $W'$ by rationally blowing down \n$G\\subset W'$. If a symplectic line arrangement corresponding to $W'$ is $S_{n,m}$, then a symplectic line arrangement corresponding to $W$ is either $S_{n,m}$ or $S_{n,m-1}$. \n\\end{lem}\n\n\n\\begin{proof}\nIn the proof of Proposition~\\ref{prop2}, we showed that $G$ is obtained by blowing-ups from a single exceptional curve $e$. If $e$ corresponds to a non-multi-intersection point, the corresponding symplectic line arrangement does not change during surgery. If $e$ corresponds to a unique multi-intersection point of $S_{n,m}$, then the symplectic line arrangement corresponding to $W$ is $S_{n,m}$ or $S_{n,m-1}$ depending on whether a symplectic line arrangement corresponding to the rational homology ball filling of $(\\partial G, \\xi_{\\text{can}})$ with respect to $K_G$ is $S_{m,m}$ or $S_{m,m-1}$.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm3}]\nWe assume that there is a sequence of rational blowdowns from the minimal resolution to $W_n$. Then there exists a minimal symplectic filling $W'_n$ of $Y_n$ such that $W_n$ is obtained from $W'_n$ by rationally blowing down $G_n\\subset W'_n$. Furthermore, $W'_n$ itself is also obtained by a sequence of rational blowdowns so that the corresponding symplectic line arrangement to $W'_n$ is $S_{n+2,n+2}$ or $S_{n+2,n+1}$, by Lemma~\\ref{counterexamplelem1}.\n\nFirst, we consider the curve configurations obtained from $S_{n+2,n+2}$. Note that each curve configuration obtained from $S_{n+2,n+2}$ is of type A or type B, as described in Section~\\ref{4}. Because of the degrees appeared in $K_n$, we can only have curve configurations of type A for the minimal symplectic fillings of $Y_n$. Furthermore, there is only one curve configuration $\\widetilde{S}_{n+2,n+2}$ of type A, standard blowing-ups of $S_{n+2,n+2}$, which corresponds to the minimal resolution of the corresponding singularity. \nIn the curve configuration $\\widetilde{S}_{n+2,n+2}$, each homology class of the $(-1)$ pseudo-holomorphic curves appears in only one arm of $K_n$ except for a pseudo-holomorphic curve $e$ corresponding to a unique multi-intersection point of $S_{n+2,n+2}$. Therefore, if a minimal symplectic filling $W_n$ is obtained from the minimal resolution by a single rational blowdown $G_n$, there is at least one $(-1)$ pseudo-holomorphic curve in the curve configuration $C_n$ of $W_n$ whose homology class appears in only one arm of $K_n$ unless $W_n$ is a rational homology ball filling. \nHowever, the homology class of every $(-1)$ pseudo-holomorphic curve in $C_n$ appears in at least two arms of $K_n$, and $W_n$ is not a rational homology ball filling unless $n=1$.\n\nNext, we show that a curve configuration $C'_n$ of $W'_n$ cannot be obtained from $S_{n+2,n+1}$. Since we should blow all intersection points among the symplectic lines of a symplectic line arrangement to obtain a curve configuration, all curve configurations obtained from $S_{n+2,n+1}$ for minimal symplectic fillings of $Y_n$ are actually obtained from $S'_{n+2,n+1}$ by blowing-ups (Figure~\\ref{blowingup1}). We can divide all curve configurations obtained from $S_{n+2,n+1}$ into two types: those with and without blowing up at an exceptional curve $e$.\n\nWe first assume that $C'_n$ is obtained from $S'_{n+2,n+1}$ without blowing up at $e$. \nThus, the homological data of $K_n$ regarding $e$ in $C'_n$ is different from that of $K_n$ regarding $e$ in $C_n$.\nSince only the homology classes of $E_{G_n}$ can change the homological data of $K_n$ for $W'_n$ under rationally blowing down $G_n\\subset W'_n$, \na symplectic embedding of $G_n$ in $W'_n$ should be obtained from $e$ (refer to the proof of Proposition~\\ref{prop2}; we blow up all intersection points of $e$ to obtain a symplectic embedding of $G_n$ from $e$), and the homology classes of $e_i$'s in $S'_{n+2,n+1}$ do not belong to $E_{G_n}$. \nHere, $E_{G_n}$ denotes the set of $(-1)$ pseudo-holomorphic curves whose homology classes appear in the irreducible components of $G_n$.\nFurthermore, since we blow up two $e_i$s to obtain $C_n$ from $S'_{n+2,n+1}$, our observation implies that we should also blow up the two $e_i$s to obtain $C'_n$ resulting from the configuration $S''_{n+2,n+1}$ in Figure~\\ref{middle1}.\nThen, the second arm of $S''_{n+2,n+1}$ becomes an arm in $K_n$ consisting of a single $(-2)$ strand in $C'_n$ because we do not blow up at intersection points of $e$ to obtain $C'_n$. This implies that there is no way of obtaining an embedding $G_n$ in $W'_n$ from $e$ by blowing-ups because $e$ intersects the single $(-2)$ arm of $K$ in $W'_n$ so that we cannot blow it up to make $e$ disjoint from $K_n$. \n\n\n\\begin{figure}[h]\n\\begin{tikzpicture}\n\\begin{scope}[scale=1.4]\n\n\\filldraw (1.3,0.4) circle (.4pt);\n\\filldraw (1.4,0.4) circle (.4pt);\n\\filldraw (1.5,0.4) circle (.4pt);\n\n\n\\draw (0,0.2)--(2.4,0.2);\n\\draw (0.4,0.5)--(0.4,-1.6);\n\n\n\\draw (0.8,0.5)--(0.8,-0.6);\n\\draw (1.2,0.5)--(1.2,-0.8);\n\\draw (1.6,0.5)--(1.6,-1);\n\\draw (2.,0.5)--(2,-1.2);\n\n\\draw (2.1,-.9)--(1.6,-1.6);\n\\node[right] at (2.1,-.9) {$-2$};\n\\draw[red, thick, dashdotted] (0.3,-1.5)--(1.8,-1.5);\n\n\n\\draw[red, thick, dashdotted] (0.6,0)--(2.2,0);\n\n\\draw[red, thick, dashdotted](0.5,-0.5)--(0.9,-0.3);\n\\draw (0.7,-0.5)--(0.3,-0.3);\n\\node[left] at (0.3,-0.3) {$-2$};\n\n\n\\draw[red, thick, dashdotted](0.3,-1)--(1.3,-0.5);\n\\draw[red, thick, dashdotted](0.3,-1.4)--(1.7,-0.7);\n\n\n\n\n\n\n\n\\filldraw (0.8,-.95+.05) circle (.4pt);\n\\filldraw (0.85,-1.+.05) circle (.4pt);\n\\filldraw (0.9,-1.05+.05) circle (.4pt);\n\\node[below] at (0.4,-1.6) {$-(n+1)$};\n\\node[above] at (0.8,0.5) {$-2$};\n\\node[above] at (1.2,0.5) {$-1$};\n\\node[above] at (1.6,0.5) {$-1$};\n\\node[above] at (2,0.5) {$-1$};\n\\node[left] at (0,0.2) {$+1$};\n\n\n\n\\end{scope}\n\\end{tikzpicture}\n\\caption{Configuration $S''_{n+2,n+1}$}\n\\label{middle1}\n\\end{figure}\n\nNext, we assume that $C'_n$ is obtained from $S'_{n+2,n+1}$ by blowing up the intersection points on $e$. Then the proper transform of $e$ is an irreducible component of $K_n$ in $C'_n$. Hence, we should blow up at least all intersection points on $e$ except one, as we obtain a curve configuration $C_n$ from $S'_{n+2,n+1}$. Since the length of each arm in $K_n$ is at most two, we should also blow up the intersection points on $e$ exactly as before, so that the first two arms in the resulting configuration $S'''_{n+2,n+1}$ (refer to Figure~\\ref{middle}) become the first and second arms of $K_n$. \nNote that we need the condition $n\\geq 3$ to guarantee that the first two arms of $S'''_{n+2,n+1}$ become the first two arms of $K_n$ in $C'_n$.\nThen, because of the degrees in $K_n$, we should reblow up again an exceptional strand in $S'''_{n+2,n+1}$ coming from one of $e_{i}$'s in $S'_{n+2,n+1}$ for $C^{n+2}_2$ of $K_n$ and an exceptional strand from $e_1$ for $C^{1}_2$ of $K_n$, so that the resulting curve configuration is equivalent to $C_n$, which contradicts the assumption. \n\n\\begin{figure}[h]\n\\begin{tikzpicture}\n\\begin{scope}[shift={(4.5,0)}]\n\\draw(0,0)--(4.5,0);\n\n\\node[above] at (0.5,0.2) {$-n$};\n\\node[above] at (1.5,0.2) {$-1$};\n\\node[above] at (2.5,0.2) {$-2$};\n\\node[above] at (3.25,0.2) {$-2$};\n\\node[above] at (4.25,0.2) {$-2$};\n\n\\draw (0.5,0.2)--(0,-0.7)--(-.5-.1-.4*5\/9,-1.6-.1*9\/5-.4);\n\n\\draw (1.25+.25,0.2)--(0.75+.25,-0.7);\n\\draw[red, thick, dashdotted](0.05,-0.2)--(1.5,-.2);\n\n\\draw (2.25+.25,0.2)--(1.75+.25+.05,-0.7+.05*9\/5);\n\n\\filldraw (2.5,-0.25) circle (.5pt);\n\\filldraw (2.6,-0.25) circle (.5pt);\n\\filldraw (2.7,-0.25) circle (.5pt);\n\n\\draw (3.25,0.2)--(2.75,-0.7);\n\n\\draw (4.25,0.2)--(3.75-.1-.9*5\/9-.4*5\/9,-0.7-.1*9\/5-.9-.4);\n\n\\draw[red, thick, dashdotted](-.9,-0.7-.1*9\/5-.9-.2)--(3.4,-0.7-.1*9\/5-.9-.2);\n\n\\draw (0.75+.25,-0.3-.1-.1)--(1.25+.25-.1,-1.2-.1-.1+.1*9\/5);\n\\node[below] at (1.25+.25-.1,-1.2-.1-.1+.1*9\/5) {$-(n+1)$};\n\n\\draw[red, thick, dashdotted](1.04,-.75)--(2.3,-.4);\n\\draw[red, thick, dashdotted](1.09,-1.)--(3,-.55);\n\\draw[red, thick, dashdotted](1.17,-1.2)--(3.8,-.75);\n\n\n\n\\node[left] at (0,0) {$+1$};\n\n\n\\end{scope}\n\\end{tikzpicture}\n\n\\caption{Configuration $S'''_{n+2,n+1}$}\n\\label{middle}\n\\end{figure}\n\n In conclusion, there is no minimal symplectic filling $W'_n$ of $Y_n$ such that $W_n$ is obtained from $W'_n$ by a single rational blowdown surgery. Hence, $W_n$ cannot be obtained by a sequence of rational blowdowns from the minimal resolution.\n \\end{proof}\n \n\n\n\n\n\n\\bigskip\n\n\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{sec:intro}\nIt is an open problem to know whether every infinite dimensional Banach space contains a monotone basic sequence (see \\cite[Problem 1.1, p. 60]{Si3}, \\cite[p. 220]{FHHMZ} and \\cite[p. 6]{HSVZ}). A sequence $(x_n)$ in a Banach space $X$ is called a basic sequence if it is a Schauder basis for its closed linear span, denoted by $[(x_n)]$. Recall also that the partial sums associated to a basic sequence $(x_n)$ is the sequence of projections $(P_n)$ on $[(x_n)]$ defined by $P_n\\colon \\sum_{i=1}^\\infty a_i x_i\\mapsto \\sum_{i=1}^n a_i x_i$. One can prove using standard arguments in Banach space theory that $\\sup_n\\| P_n\\|< \\infty$. This number is referred in the literature as the basic constant of $(x_n)$. In \\cite{Pel} Pe\\l czy\\'nski proved that for every $\\varepsilon\\in (0,1)$, every normalized weakly null sequence in $X$ has a basic subsequence with basic constant $1+\\varepsilon$. This result is also known as Bessaga-Pe\\l czy\\'nski's selection principle, after \\cite{BP} (see also, e.g., \\cite[p. 14]{AK}). A basic sequence $(x_n)$ is called monotone if $\\| P_n\\|=1$ for all $n\\in\\mathbb{N}$. A bimonotone basic sequence is a monotone basic sequence so that the norm of all tail projections is exactly $1$; that is, $\\| I - P_n\\|=1$ for all $n\\in \\mathbb{N}$. Let us recall yet that $(x_n)$ is said to be asymptotically monotone if $\\| P_n\\|\\to 1$. This slightly weaker concept of monotonicity was introduced by V. D. Milman \\cite{M} (see also \\cite[p. 93]{Day}). Naturally, one may also consider asymptotic bimonotonicity (i.e. $\\lim_n \\| I - P_n\\|=\\lim_n\\|P_n\\|=1$). \n\nMonotone basic sequences are of course asymptotically monotone, the former though being overall more hard to detect than the latter. In fact, a few important works have done regarding the existence of basic sequences with asymptotic monotonicity properties. For instance, it is well known that every Banach space has an asymptotically monotone basic sequence (cf. \\cite{Day}, see also \\cite[Theorem 1.2, p. 49]{Si3}, \\cite[Theorem 1.20]{HSVZ}). Very recently, H\\'ajek, Kania and Russo \\cite{HKR} obtained an useful refinement of the classical Mazur technique of constructing basic sequences, leading to a block version of this fact (see \\cite[Lemma 2.4]{HKR}): every basic sequence has an asymptotically monotone block basis. This statement seems also to have been, although implicitly, witnessed by Elton and Odell in \\cite[Remarks. (1)]{EO}. It is worthy to stress that asymptotic monotonicity has been proven to be very useful in many geometric and analytical problems in Banach space theory (cf. \\cite{EO, HKR, MV, Si, Si2}). For instance, in \\cite[Theorem 1]{EO} Elton and Odell used the fact that every Banach space contains an asymptotically monotone basic sequence in concert with Ramsey combinatorial methods to show that the unit sphere of every infinite dimensional normed space contains, for some $\\varepsilon>0$, a sequence $(x_n)$ such that $\\| x_n - x_m\\| > 1 + \\varepsilon$ for all $n\\neq m$. \n\nWith regarding asymptotic pre-monotonicity, it was open as to whether one could also obtain in every Banach space an asymptotically pre-monotone basic sequence (i.e. $\\| I - P_n\\|\\to 1$). It turns out that this is not true in general, as pointed out by Odell and Schlumprecht in \\cite[p. 1349]{OS}. Nevertheless, as a byproduct of James's non-distortion theorems it follows that every Banach space containing an isomorphic copy of $\\mathrm{c}_0$, also contains an asymptotically pre-monotone basic sequence (cf. \\cite[Theorem 8]{DLT}). \n\nA natural question to be addressed is whether a selection principle for asymptotically monotone sequences can be obtained. In this note we remark that every seminormalized weakly null sequence admits an asymptotically monotone basic subsequence. This result is probably well-known to experts but we were unable to locate a reference. It is worth noting that there is a lot of spaces containing seminormalized weakly null sequences. For example, non-Schur Banach spaces (because of Rosenthal's $\\ell_1$-theorem) and infinite dimensional reflexive Banach spaces (due to the theorem of Josefson-Nissenzweig, see e.g. \\cite{D}, p. 219). \n\nOur next goal concerns the existence of asymptotically monotone basic sequences which are symmetrically well-separated in unit spheres of Banach spaces containing isomorphic copies of $\\ell_1$. This topic has been extensively dealt in several contemporary and recent works (see \\cite{CP, K, HKR} and references therein). In Section \\ref{sec:3}, we will prove our first result. In Section \\ref{sec:4} we shall show the existence of symmetrically $2$-separated sequences of unit vectors of a Banach space with certain prescribed conditions related to monotonicity and nearly isometric renormings. In particular, we strengthen Propositions $5.1$ in \\cite{HKR}. We end this note in Section \\ref{sec:6} with some concluding remarks. \n\n\n\n\n\n\n\\section*{Acknowledgments}\nA significant part of this note was done when the author was attending the AMS Joint Mathematics Meeting 2019 in Baltimore, MD. The work was completed when the author was visiting UCF -- University Central of Florida, January 15--17, 2019. He would like to thank Professors Chris Lennard and Torrey Gallagher for their invitation to give a talk at JMM-2019, Professor Eduardo Teixeira for his invitation to visit the UCF and the entire Mathematics Department for its hospitality and very pleasant working environment. He is also particularly grateful to Professor Tommaso Russo for carefully reading the manuscript, for many valuable remarks related to well-separated unit sets and for the comments leading to a negative answer to the Question \\ref{qtn:Q} in the last section of this paper. \n\n\n\n\n\\section{Preliminaries}\\label{sec:2}\nThe notation used throughout this paper is quite standard and mostly follows \\cite{FHHMZ}. For a Banach space $X$ we denote by $B_X$ the closed unit ball and $S_X$ its closed unit sphere. For an infinite set $N\\subset \\mathbb{N}$, $[N]^k$ stands for the $k$-element subsets of $N$. By $[N]^\\omega$ and $[N]^{<\\omega}$ we mean respectively the family of all infinite and finite subsets of $N$. Given sets $E, F\\in[\\mathbb{N}]^{<\\omega}$, we shall use the notation $E\\leq F$ (resp. $E0$ so that $A\\leq \\| x_n\\| \\leq B$ for all $n\\in \\mathbb{N}$. A block basis of $(x_n)$ is a sequence of the form $z_n= \\sum_{i\\in E_n} \\alpha^n_i x_i$, where $(\\alpha^n_i)_{i\\in E_n}\\subset \\mathbb{R}$ and $(E_n)$ is an increasing sequence of block of integers, that is, $E_n\\in [\\mathbb{N}]^{<\\omega}$ and $E_n< E_{n+1}$ for all $n\\in \\mathbb{N}$. Every block basis of a basic sequence is a basic sequence. \n\n \n\\medskip \n\\section{Asymptotic monotone basic sequences} \\label{sec:3}\nOur first result is the following strengthening of Bessaga-Pe\\l czy\\'nski's selection principle.\n\n\\begin{lemma}\\label{thm:M1} Let $X$ be an infinite dimensional Banach space. Then every seminormalized weakly null sequence in $X$ has an asymptotically monotone basic subsequence. \n\\end{lemma}\n\n\\begin{proof} The proof involves combining Bessaga-Pe\\l czy\\'nski's selection approach together with a diagonal argument. Let $(y_i)$ be a seminormalized weakly null sequence in $X$ and fix $\\varepsilon_n\\searrow 0$ a decreasing null-sequence in $(0,1)$. We proceed by induction on $j=1, 2, 3, \\dots$. \n\nDefine $x_{n^0_i}= y_i$ for all $i\\in \\mathbb{N}$. Let $(x_{n^1_i})$ be a $(1 + \\varepsilon_1)$-basic subsequence of $(x_{n^0_i})$ given by Pe\\l czy\\'nski's selection principle (cf. \\cite{Pel}). Accordingly, we have $n^1_1=1$ and \n\\[\n\\Bigg\\| \\sum_{i=1}^m a_i x_{n^1_i}\\Bigg\\|\\leq (1 + \\varepsilon_1) \\Bigg\\| \\sum_{i=1}^n a_i x_{n^1_i}\\Bigg\\|\n\\]\nfor all integers $m\\leq n$ in $\\mathbb{N}$ and for all scalars $(a_i)_{i=1}^\\infty$ in $\\mathrm{c}_{00}$ (cf. \\cite[p. 372]{Pel}). \n\nNow assume for some $j\\geq 1$ that basic sequences $\\big\\{ (x_{n^k_i})_{i\\in\\mathbb{N}}\\big\\}_{k=1}^j$ have been already obtained so as to satisfy the next three properties, for all $k=1,\\dots, j$: \n\\begin{itemize}\n\\item $(x_{n^{k}_i})_{i\\in \\mathbb{N}}$ is a subsequence of $(x_{n^{k-1}_i})_{i\\in \\mathbb{N}}$;\n\\item $n^{k}_1=n^{1}_1$ if $k=1$,\\, and $n^k_1= n^1_1,\\, n^{k}_2= n^{2}_2,\\,\\dots, \\,n^{k}_{k-1}= n^{k-1}_{k-1}$ for $k\\geq 2$;\n\\item for all scalars $(a_i)_{i=1}^\\infty$ in $\\mathrm{c}_{00}$, one has\n\\[\n\\Bigg\\| \\sum_{i=1}^{k} a_i x_{n^{k}_i}\\Bigg\\| \\leq (1 + \\varepsilon_{k}) \\Bigg\\| \\sum_{i=1}^n a_i x_{n^{k}_i}\\Bigg\\|,\\quad \\forall \\, n\\geq k.\n\\]\n\\end{itemize}\nLet $E=\\spn\\{ x_{n^{1}_1}, x_{n^{2}_2}, \\dots, x_{n^{j-1}_{j-1}}, x_{n^{j}_{j}}\\}$ and take a decreasing null sequence $(\\delta_i)_i$ in $(0,1)$ satisfying\n\\[\n\\prod_{i=1}^\\infty (1 + \\delta_i)< 1+ \\varepsilon_{j+1}.\n\\]\nClearly the sequence $(x_{n^j_i})_{i=1}^\\infty$ is seminormalized and weakly null. From the proofs in \\cite[Proposition and lemma]{Pel} we obtain an integer $\\kappa(j,1)> j$ so that \n\\[\n\\| e\\| \\leq (1 + \\delta_{1})\\big \\| e + t x_{n^j_{\\kappa(j,1)}}\\big\\|\n\\]\nfor all $e\\in E$ and $t\\in \\mathbb{R}$. Let $\\mathcal{K}=\\prod_{i=2}^N(1 + \\delta_i)$. By making an iterated use of the arguments developed in \\cite[Proposition and lemma]{Pel} we obtain an increasing subsequence $(n^j_{\\kappa(j,i)})_{i=1}^\\infty$ of $\\{ n^j_i\\}_{i=1}^\\infty$ such that\n\\[\n\\Bigg\\| \\sum_{i=1}^j a_i x_{n^i_i} + a_{j+1} x_{n^j_{\\kappa(j,1)}}\\Bigg\\| \\leq \\mathcal{K} \\Bigg\\| \\sum_{i=1}^j a_i x_{n^i_i} + a_{j+1} x_{n^j_{\\kappa(j,1)}}+ \\sum_{i=j+2}^N a_i x_{n^j_{\\kappa(j,i)}}\\Bigg \\|\n\\]\nfor all $N\\geq j+2$ and for all scalars $(a_i)_{i=1}^\\infty \\in \\mathrm{c}_{00}$. We then define $(x_{n^{j+1}_i})$ as follows:\n\\[\nx_{n^{j+1}_i}=\\left\\{\n\\begin{split}\n&x_{n^i_i} \\hskip 1.25cm \\text{ for } i=1,\\dots, j\\\\\n&x_{n^j_{\\kappa(j, i-j)}} \\hskip .4cm \\text{ for } i\\geq j+1.\n\\end{split}\\right.\n\\]\nThe induction process succeeds. The above construction provides a countable family of basic subsequences $(x_{n^k_i})_{k, i\\in \\mathbb{N}}$ of $(y_n)$ such that\n\\[\n\\Bigg\\| \\sum_{i=1}^k a_i x_{n^i_i}\\Bigg\\| \\leq ( 1 + \\varepsilon_{k}) \\Bigg\\| \\sum_{i=1}^k a_i x_{n^i_i} + \\sum_{i=k+1}^\\infty a_i x_{n^{k+1}_i}\\Bigg\\|\n\\]\nfor all $k\\in \\mathbb{N}$ and for all $(a_i)_{i=1}^\\infty \\in \\mathrm{c}_{00}$. \n\nLet us take now the diagonal sequence $(x_i)=(x_{n^i_i})$. Since $\\{ n^i_i\\}_{i=k+1}^\\infty \\subset \\{ n^{k+1}_i\\}_{i=1}^\\infty$ for all $k\\in \\mathbb{N}$, we can make null those scalar terms $a_i$'s that are outside the diagonal and conclude directly from the previous inequality that $(x_i)$ is asymptotically monotone.\n\\end{proof}\n\n\n\n\\medskip \n\\section{Symmetric $2$-separated sequences}\\label{sec:4}\nIn connection with the ongoing study on the structure of well-separated subsets of the unit sphere of a Banach space the next question seems to be reasonable:\n\n\\begin{qtn}\\label{qtn:1} Let $\\varepsilon>0$. Under which conditions a Banach space $X$ admits a $(1+\\varepsilon)$-equivalent renorming $\\nn{\\cdot}$ so that $(X, \\nn{\\cdot})$ has a normalized asymptotically monotone basic sequence being symmetrically $2$-separated?\n\\end{qtn}\n\nA subset $A$ of a normed space is said to be symmetrically $\\delta$-separated (see \\cite{HKR}) when $\\| x\\pm y\\| \\geq \\delta$ for any distinct elements $x, y\\in A$ ($\\delta>0$). Pioneer works along the study of well-separated sets in the unit sphere of Banach spaces include J. Elton and E. Odell \\cite{EO} and C. Kottman \\cite{K}. We refer the reader to \\cite{HKR} for a recent and more detailed account about this topic. Motivation for Question \\ref{qtn:1} comes from the problem of finding symmetrically $(1+\\varepsilon)$-separated sequences of unit vectors under renorming techniques (see \\cite[Section 5]{HKR}). In \\cite[Theorem 7]{K} Kottman proved that every infinite-dimensional Banach space admits a renorming so that the new unit sphere contains a $2$-separated sequence. In \\cite[p.13]{HKR} the authors observed that Kottman's argument yields in fact a symmetrically $2$-separated sequence of norm-one vectors. However, the norm obtained in \\cite{HKR} is only $2$-equivalent to the original one. Then, in \\cite[Proposition 5.2]{HKR} they proved for every $\\varepsilon>0$, that every Banach space $X$ admits an equivalent norm $\\nn{\\cdot}$ which is $(1+\\varepsilon)$-equivalent to the original norm of $X$ and yet $S_{(X, \\nn{\\cdot})}$ contains an infinite symmetrically $(1+\\delta)$-separated subset, for some $\\delta>0$. They finally observed in \\cite[Remark 5.3]{HKR} that every separable Banach space $X$ admits a strictly convex renorming $\\nn{\\cdot}$ so that the unit sphere of $S_{(X,\\nn{\\cdot})}$ contains no $2$-separated sequences. \n\n\\smallskip \nInspired by Proposition 5.1 in \\cite{HKR} we remark the following.\n\n\\begin{prop}\\label{prop:1sec4} Let $X$ be a Banach space and $Y$ a subspace of $X$. Assume that $\\{ x_i; f_i\\}_{i=1}^\\infty$ is a biorthogonal system on $Y$ such that, for some $\\varepsilon\\in (0,1)$, one has \n\\[\n\\| x_i\\| \\leq 1 + \\varepsilon\\quad\\text{and}\\quad \\sup_{i\\neq j}\\big( | f_i(y)| + | f_j(y)|\\big)\\leq (1 + \\varepsilon) \\| y\\|\n\\]\nfor all $i\\in \\mathbb{N}$ and for all $y\\in Y$. Then there is a $(1+\\varepsilon)$-equivalent norm $\\nn{\\cdot}$ on $X$ such that $S_{(X, \\nn{\\cdot})}$ contains a symmetrically $2$-separated sequence. \n\\end{prop}\n\n\\begin{proof} Indeed, to see this we first define a new norm on $Y$ by \n\\[\n|y| =\\max\\Bigg( \\frac{1}{1 + \\varepsilon}\\| y\\|, \\sup_{i\\neq j}\\big( | f_i(y)| + | f_j(y)|\\big)\\Bigg)\\quad\\text{for }\\, y\\in Y. \n\\]\nClearly we have\n\\[\n\\frac{1}{1 + \\varepsilon}\\|y\\| \\leq | y| \\leq (1 + \\varepsilon)\\| y\\|\n\\]\nfor all $y\\in Y$, and hence $| \\cdot|$ is $(1 + \\varepsilon)$-equivalent to $\\| \\cdot\\|$ on $Y$. Moreover, notice that $| x_i| = 1$ and $| x_i \\pm x_j| \\geq 2$ for all $i, j\\in \\mathbb{N}$ with $i\\neq j$. Now, the proof of Lemma 2 in \\cite{JO} yields an equivalent norm $\\nn{\\cdot}$ on $X$ satisfying $\\nn{y}= | y|$ for all $y\\in Y$, and\n\\[\n\\frac{1}{(1 + \\varepsilon)^2}\\| x\\|\\leq \\nn{ x}\\leq (1 + \\varepsilon)\\| x\\|\\quad\\text{for all } x\\in X.\n\\]\nIt follows that $S_{(X, \\nn{\\cdot})}$ contains a symmetrically $2$-separated sequence. \n\\end{proof}\n\n\\smallskip \nProposition \\ref{prop:1sec4} provides a guide for finding affirmative answers for Question \\ref{qtn:1}. It is natural therefore to wonder which Banach spaces $X$ have the property: for every $\\varepsilon>0$, there is a biorthogonal system $\\{x_n; f_n\\}_{n=1}^\\infty$ in $(1+\\varepsilon)B(Y)\\times Y^*$, for some subspace $Y$ of $X$, so that the functionals $(f_n)_n$ satisfy \n\\[\n\\sup_{i\\neq j}\\big( |f_i(x)| + |f_j(x)|\\big)\\leq (1+\\varepsilon)\\| x\\|\\quad\\text{for all }\\, x\\in [(x_n)]. \n\\]\n\n\\smallskip \nFor the sake of simplicity,\n\n\\begin{defi} We shall call such a system as $(1+\\varepsilon)$-biorthogonal system on $X$. \n\\end{defi}\n\nThe next two results provide partial answers for the previous question when $X$ contains either isomorphic copies of $\\ell_1$. \n\n\n\\begin{prop}\\label{prop:2sec4} Let $X$ be a Banach space containing a subspace isomorphic to $\\ell_1$. Then for every $\\varepsilon\\in (0,1)$, $X$ admits a $(1+\\varepsilon)$-biorthogonal system which generates a $(1+\\varepsilon)$-equivalent norm $\\nn{\\cdot}$ under which the sphere $S_{(X, \\nn{\\cdot})}$ contains a bimonotone symmetrically $2$-separated basic sequence.\n\\end{prop}\n\n\\begin{proof} By James's non-distortion theorem \\cite[Lemma 2.1]{J} there is an isomorphic embedding $T\\colon \\ell_1\\to X$ such that $ \\| x\\|_{\\ell_1}\\leq \\| Tx \\| \\leq (1 +\\varepsilon)\\|x \\|_{\\ell_1}$ for all $x\\in \\ell_1$, where $\\| \\cdot\\|_{\\ell_1}$ stands for the usual $\\ell_1$-norm on $\\ell_1$. Let $Y=T(\\ell_1)$ and define a new norm on $Y$ as follows: \n\\[\n| y|_Y=\\inf\\{ \\| x\\|_{\\ell_1} + \\| y - Tx\\| \\colon x\\in \\ell_1\\}\\quad\\text{ for } y\\in Y. \n\\]\nIt is not hard to verify that $T\\colon \\ell_1\\to (Y, | \\cdot|_Y)$ is an isometry. In addition to this, one has\n\\begin{equation}\\label{eqn:0prop1}\n\\frac{1}{1+ \\varepsilon}\\| y\\| \\leq | y|_Y\\leq \\| y\\| \\quad\\text{ for all } y\\in Y. \n\\end{equation}\nLet now $(e_i)$ denote the canonical basis of $\\ell_1$ and define $x_i:= T(e_i)$ for $i\\in \\mathbb{N}$. The fact that $T$ is an isometry from $\\ell_1$ onto $(Y, | \\cdot|_Y)$ clearly implies \n\\begin{equation}\\label{eqn:1prop1}\n\\Bigg| \\sum_{i=1}^\\infty a_i x_i\\Bigg|_Y= \\sum_{i=1}^\\infty |a_i|\\quad\\text{ for all } (a_i)_{i=1}^\\infty \\in \\mathrm{c}_{00}. \n\\end{equation}\nHence $(x_i)$ is a normalized bimonotone basis for $(Y, | \\cdot|_Y)$. Furthermore, denoting $(f_i)$ the coefficient functionals of $(x_i)$ and using (\\ref{eqn:0prop1}) in concert with (\\ref{eqn:1prop1}) we conclude \n\\[\n \\sup_{i\\neq j}\\big( | f_i(y)| + | f_j(y)|\\big)\\leq | y|_Y\\leq \\| y\\|\\quad\\text{ for } y\\in Y. \n\\]\nNotice that (\\ref{eqn:0prop1}) also implies $\\| x_i\\|\\leq (1 + \\varepsilon)$ for all $i\\in \\mathbb{N}$. Thus $\\{ x_n; f_n\\}_{n=1}^\\infty$ defines a $(1+ \\varepsilon$)-biorthogonal system on $X$. By Proposition \\ref{prop:1sec4} the result follows. \n\\end{proof}\n\n\n \n\n\n\n\\section{Concluding remarks}\\label{sec:6}\nAs we have pointed out in the introduction, Lemma \\ref{thm:M1} is probably well-known to experts. We remark that part of our arguments concern a suitable use of \\cite[Lemma]{Pel}, which can be seen as a {\\it weakly null} version of the well-known Mazur's lemma. In turn, such a version (i.e. \\cite[Lemma]{Pel}) can also be directly deduced from \\cite[Lemma 1.5.1 and Proposition 1.5.4]{AK}. We also remark that Propositions \\ref{prop:2sec4} strengthens \\cite[Proposition 5.1]{HKR} for the cases when $X$ contains $\\ell_1$. The arguments presented in our approach were mostly inspired by the results in \\cite{HKR}. \n\nIt was remarked in \\cite[Remark 5.3]{HKR} that every separable Banach space $X$ admits a strictly convex renorming $\\nn{\\cdot}$ so that $S_{(X, \\nn{\\cdot})}$ contains no $2$-separated sequences. In fact, one can even show that, for every $\\varepsilon \\in (0,1)$, every such $X$ admits a $(1+ \\varepsilon)$-equivalent strictly convex norm for which the unit sphere contains no $2$-separated sequences. To see this take a dense sequence $\\{ f_n\\}_{n=1}^\\infty \\subset B_X$ such that $\\| x\\| =\\sup\\{ | f_n(x)| \\colon x\\in X\\}$ for all $x\\in X$. Next, define for $\\delta\\in (0,\\sqrt{1 + \\varepsilon} -1)$ a new norm on $X$ by $\\nn{x}^2= \\| x\\|^2 + \\delta \\sum_{n=1}^\\infty 2^{-n} f^2_n(x)$, $x\\in X$. Direct calculation shows that $\\nn{\\cdot}$ is strictly convex and $(1+\\varepsilon)$-equivalent to $\\| \\cdot\\|$. Using then the same reasoning as in \\cite[Remark 5.3]{HKR} one shows that $S_{(X, \\nn{\\cdot})}$ contains no $2$-separated sequences. In the opposite direction, one may naturally ask the question. \n\n\\begin{qtn}\\label{qtn:Q} Does every separable Banach space $X$ has a $(1 +\\varepsilon)$-equivalent norm $\\nn{\\cdot}$ such $S_{(X, \\nn{\\cdot})}$ contains a symmetrically $2$-separated sequence?\n\\end{qtn}\n\nLet us recall the {\\it symmetric Kottman constant} introduced in \\cite{CP}:\n\\[\nK^s(X):=\\sup\\Big\\{ \\sigma> 0\\,\\colon \\, \\exists (x_n)_{n=1}^\\infty \\subset B_X \\, \\forall n\\neq k\\,\\, \\| x_n \\pm x_k\\| \\geq \\sigma\\Big\\}.\n\\]\nThe answer for Question \\ref{qtn:Q} is in general {\\it no}. The main reason is that $K^s(X)$ is a continuous function on $X$, with respect to the Banach-Mazur distance (see the proof of Theorem 7 in \\cite{K}). Hence, if we pick $X$ so that $K^s(X)<2$ (e.g., $\\ell_p$ for $1< p< \\infty$) every $(1+\\varepsilon)$-renorming, with $\\varepsilon$ small enough, will have symmetric Kottman constant less than $2$. Therefore, Question \\ref{qtn:Q} has positive answer for $X$ exactly when $K^s(X)=2$. \n\nThe problem of describing the class of Banach spaces $X$ for which $K^s(X)=2$ has been considered by several authors. In \\cite{HKR} the authors showed that if $X$ contains a spreading model isomorphic to $\\ell_1$ then $K^s(X)=2$. The corresponding problem for $\\mathrm{c}_0$ spreading models was left as an open problem \\cite[Problem 4.11]{HKR}. It is not hard to show that $K^s(T^*)=2$, where $T^*$ is the {\\it original} Tsirelson space (cf. \\cite{FJ} for basic definition and properties of Tsirelson's spaces). On the other hand, it is a well-known fact that every spreading model of $T^*$ is isomorphic to $\\mathrm{c}_0$. These facts led us to believe that $K^s(X)=2$ for every Banach space $X$ that admit $\\mathrm{c}_0$ as a spreading model. We observe, however, that the assumption on a Banach space $X$ to admit a spreading model isomorphic to $\\mathrm{c}_0$ is not sufficient to imply that $K^s(X)=2$. We thank Professor Tommaso Russo for informing us privately about this and many other of his recent achievements on the matter.\n\n\n\\nocite{*}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nWarped product manifolds were introduced in general relativity as\na method to find general solutions to Einstein's field equations\n\\cite{BEE,ON}. Two important examples include generalized\nRobertson-Walker space-times and standard static space-times. The\nformer are obviously a generalization of Robertson-Walker\nspace-times and the latter a generalization of the Einstein static\nuniverse. In this paper, we focus on Killing vector fields for standard\nstatic space-times.\n\nWe recall that a warped product can be defined as follows\n\\cite{BEE,ON}. Let $(B,g_B)$ and $(F,g_F)$ be pseudo-Riemannian\nmanifolds and also let $b \\colon B \\to (0,\\infty)$ be a smooth\nfunction. Then the (singly) warped product, $B \\times_b F$ is the\nproduct manifold $B \\times F$ furnished with the metric tensor\n$g=g_B \\oplus b^{2}g_F$ defined by $$ g=\\pi^{\\ast}(g_B) \\oplus (b\n\\circ \\pi)^2 \\sigma^{\\ast}(g_F)$$ where $\\pi \\colon B \\times F \\to\nB$ and $\\sigma \\colon B \\times F \\to F$ are the usual projection\nmaps and ${}^\\ast$~denotes the pull-back operator on tensors. A\nstandard static space-time can be considered as a Lorentzian\nwarped product where the warping function is defined on a\nRiemannian manifold called the base and acting on the negative\ndefinite metric on an open interval of real numbers, called the\nfiber. More precisely, a standard static space-time $(t_1,t_2) _f\n\\times F$ is a Lorentzian warped product furnished with the metric\n$g=-f^2{\\rm d}t^2 \\oplus g_F,$ where $(F,g_F)$ is a Riemannian\nmanifold, $f \\colon F \\to (0,\\infty)$ is smooth and $-\\infty \\leq\nt_1 < t_2 \\leq \\infty.$ In \\cite{ON}, it was shown that any static\nspace-time is locally isometric to a standard static space-time.\n\n\\noindent Standard static space-times have been previously studied\nby many authors. Kobayashi and Obata \\cite{KO} stated the geodesic\nequation for this class of space-times and the causal structure\nand geodesic completeness was considered in \\cite{AD}, where\nsufficient conditions on the warping function for nonspacelike\ngeodesic completeness of the standard static space-time was\nobtained (see also \\cite{RASM}). In \\cite{AD1}, conditions are\nfound which guarantee that standard static space-times either\nsatisfy or else fail to satisfy certain curvature conditions from\ngeneral relativity. In \\cite{ADt}, D. Allison considered the\nglobal hyperbolicity of standard static space-times and obtained\nsufficient conditions. The problems of geodesic completeness and\nglobal hyperbolicity of standard static space-times have been also\nstudied by M. S\\'{a}nchez in \\cite{MS05} with $I=\\mathbb{R}$ where\n$(F,g_F)$ behaves at most quadratically (at infinity). The\nexistence of geodesics in standard static space-times have been\nstudied by several authors. S\\'{a}nchez \\cite{mS2} gives a good\noverview of geodesic connectedness in semi-Riemannian manifolds,\nincluding a discussion for standard static space-times. The\ngeodesic structure of standard static space-times has been studied\nin \\cite{GES} and conditions are found which imply nonreturning\nand pseudoconvex geodesic systems. As a consequence, it is shown\nthat if the complete Riemannian factor manifold $F$ satisfies the\nnonreturning property and has a pseudoconvex geodesic system and\nif the warping function $f \\colon F \\to (0,\\infty)$ is bounded\nfrom above then the standard static space-time $(a,b)_f\\times F $\nis geodesically connected. In \\cite{SSS}, some conditions for the\nRiemannian factor and the warping function of a standard static\nspace-time are obtained in order to guarantee that no nontrivial\nwarping function on the Riemannian factor can make the standard\nstatic space-time Einstein. In a recent note\n\\cite{Sanchez-Senovilla2007}, the authors discussed conditions for\nstatic Killing vector fields to be standard and then they obtained\nan interesting uniqueness result when the so called natural space\n(in the case of a standard static space-time, this is the\nRiemannian part) is compact.\n\n\\noindent Two of the most famous examples of standard static\nspace-times are the Minkowski space-time and the Einstein static\nuniverse \\cite{BEE,dewitt 2003,HE} which is $\\mathbb R \\times\n\\mathbb S^3$ equipped with the metric\n$$g=-{\\rm d}t^2+({\\rm d}r^2+\\sin^2 r {\\rm d}\\theta^2+\n\\sin^2 r \\sin^2 \\theta {\\rm d} \\phi^2)$$ where $\\mathbb S^3$ is\nthe usual 3-dimensional Euclidean sphere and the warping function\n$f \\equiv 1$ (\\textit{see Remark \\ref{rem:sharipov2007}}).\nAnother well-known example is the universal covering space of\nanti-de Sitter space-time, a standard static space-time of the\nform $\\mathbb R _f\\times \\mathbb H^3$ where $\\mathbb H^3$ is the\n3-dimensional hyperbolic space with constant negative sectional\ncurvature and the warping function $f \\colon \\mathbb H^3 \\to\n(0,\\infty)$ defined as $f(r,\\theta,\\phi)=\\cosh r$ \\cite{BEE,HE}.\nFinally, we can also mention the Exterior Schwarzschild space-time\n\\cite{BEE,HE}, a standard static space-time of the form $\\mathbb R\n_f\\times (2m,\\infty) \\times \\mathbb S^2,$ where $\\mathbb S^2$ is\nthe 2-dimensional Euclidean sphere, the warping function $f \\colon\n(2m,\\infty) \\times \\mathbb S^2 \\to (0,\\infty)$ is given by\n$f(r,\\theta,\\phi)=\\sqrt{1-2m\/r},$ $r>2m$ and the line element on\n$(2m,\\infty) \\times \\mathbb S^2$ is\n$${\\rm d}s^2=\\left(1-\\frac{2m}{r} \\right)^{-1} {\\rm d}r^2+r^2({\\rm\nd} \\theta^2+\\sin^2 \\theta {\\rm d} \\phi^2).$$\n\nIn this article, we deal with questions of existence and\ncharacterization of Killing vector fields in standard static space\ntimes\\footnote{We would like to inform the reader that some of the\nresults provided in this article were previously announced in the\narXiv preprint service as \\cite{dobarro-unal06}.}.\n\n\\noindent The problem of the existence of Killing vector fields in\nsemi-Riemannian manifolds has been analyzed by many authors\n(physicists and mathematicians) with different points of view and by\nusing several techniques. One of the recent articles of S\\'{a}nchez\n(i.e, \\cite{mS1997}) is devoted to provide a review about these\nquestions in the framework of Lorentzian geometry. Another\ninteresting results for Riemannian warped products can be found in\n\\cite{B-O} and for $4-$dimensional warped space-times in\n\\cite{Carot-da Costa 93}.\n\n\\noindent In \\cite{mSK} S\\'{a}nchez studied the structure of\nKilling vector fields on a generalized Robertson-Walker\nspace-time. He obtained necessary and sufficient conditions for a\nvector field to be Killing on generalized Robertson-Walker\nspace-times and gave a characterization of them as well as an\nexplicit list for the globally hyperbolic case.\n\nAfter this brief explanations of some of the major works in the\ngeometry of warped products, especially in the geometry of standard\nstatic space-times, we will provide an outline of the paper below.\n\nIn {\\it Section 2}, we study Killing vector fields of standard\nstatic space-times. Firstly we show necessary and sufficient\nconditions for a vector field of the form $h \\partial_t + V$ to be a\nconformal Killing (see {\\it Proposition \\ref{con-1}}).\n\n\\noindent Then adapting the techniques of S\\'{a}nchez in \\cite{mSK} to\nstandard static space-times $M=I_f \\times F$, we give the\nstructure of a generic Killing vector field on $M$ in the central\nresult of this section (see {\\it Theorem \\ref{thm:Killing ssst}}).\nEssentially, we reduce the problem to the study of a parametric\nsystem of partial differential equations (involving the Hessian) on\nthe Riemannian part $(F, g_F)$.\n\n\\noindent By studying on the latter system (see \\eqref{eq:pb\nKilling 1} and \\eqref{eq:3eqnst 4}) and applying the well known\nresults about the solutions $(\\nu, u)$ of a weighted elliptic\nproblem, with $w \\in C^\\infty_{>0}$,\n\\begin{equation*}\\label{eq1:}\n -\\Delta_{g_F} u = \\nu w\n u \\, \\textrm{ on } (F,g_F).\n\\end{equation*}\non a compact Riemannian manifold without boundary $(F,g_F)$,\nwe characterize the Killing vector fields on a standard\nstatic space-time with \\textit{compact} Riemannian part in\n{\\it Theorem \\ref{thm:killing compact fiber}}. More explicitly,\nif $(F,g_F)$ is a compact Riemannian manifold, then the set of\nKilling vector fields on a standard static space time $I _f\\times F$ is\n$$\\{a\n\\partial_t + \\tilde{K}| \\, a \\in \\mathbb{R}, \\tilde{K} \\textrm{ is a\nKilling vector field on } (F,g_F) \\textrm{ and } \\tilde{K}(f)=0\n\\}.$$\n\nIn Remark, \\ref{rem:sharipov2007} we show that a relation between\nthe result mentioned above and an article (see \\cite{Sharipov2007})\nof R. A. Sharipov about Killing vector fields of a closed homogeneous\nand isotropic universe.\n\nIn {\\it Section 3}, as an application of our results in {\\it Section 2}\nwe consider the question of the existence of non-rotating Killing\nvector fields on a standard static space-time where the Riemannian\npart is compact and simply connected.\n\n\nIn \\textit{Appendix}, we provide an equivalent expression of a\ngeneral Killing vector field on a standard static space-time and\nan application of our results to the well known case of the\nMinkowski space-time.\n\n\\section{Killing Vector Fields}\n\nWe begin by stating the formal definition of a standard static\nspace-time and then recall some elementary definitions and facts\n(see Section 3 of \\cite{mSK}). But first, we want to emphasize\nsome important notational issues. Throughout the paper $I$ will\ndenote an open connected interval of the form $I=(t_1,t_2)$ in\n$\\mathbb R,$ where $-\\infty \\leq t_1 < t_2 \\leq \\infty.$ Moreover,\nany underlying manifold is assumed to be connected. Finally, on an\narbitrary differentiable manifold $N$, $C^{\\infty}_{>0}(N)$\ndenotes the set of all strictly positive $C^{\\infty}$ functions\ndefined on $N$ and $\\mathfrak X(N)$ will denote the\n$C^{\\infty}(N)-$module of smooth vector fields on $N$.\n\n\\begin{dfn} \\label{dfna} Let $(F,g_F)$ be an $s$-dimensional\nRiemannian manifold and $f \\colon F \\to (0,\\infty)$ be a smooth\nfunction. Then $n(=s+1)$-dimensional product manifold $I \\times\nF$ furnished with the metric tensor $g=-f^2{\\rm d}t^2 \\oplus g_F$\nis called a standard static space-time and is denoted by\n$I _f \\! \\times F$, where ${\\rm d}t^2$ is the Euclidean metric\ntensor on $I.$\n\\end{dfn}\n\nWe will now define Killing and conformal-Killing vector fields on\nan arbitrary pseudo-Riemannian manifold. Let $(\\overline M,\n\\overline g)$ be an ${\\overline n}$-dimensional pseudo-Riemannian\nmanifold and $X \\in \\mathfrak X(\\overline M)$ be a vector field on\n$\\overline M.$ Then\n\n\\begin{itemize}\n\\item $X$ is said to be Killing if ${\\rm L}_X \\overline g = 0$,\n\n\\item $X$ is said to be conformal-Killing if there exists a smooth\nfunction $\\sigma \\colon \\overline M \\to \\mathbb R$ such that ${\\rm\nL}_X \\overline g = 2 \\sigma \\overline g$,\n\\end{itemize}\nwhere ${\\rm L}_X$ denotes the Lie derivative with respect to $X$.\nMoreover, for any $Y$ and $Z$ in $\\mathfrak X(\\overline M),$ we\nhave the following identity (see \\cite[p.162 and p.61]{ON})\n\n\\begin{equation}\\label{eq:Lie deriv 1}\n {\\rm L}_X \\overline g(Y,Z) = \\overline g(\\nabla_Y X,Z) + \\overline\ng(Y,\\nabla_Z X).\n\\end{equation}\n\n\n\\begin{rem} \\label{rem:Lie deriv}\nOn $(I,g_I=\\pm {\\rm d}t^2)$ any vector field is conformal Killing.\nIndeed, if $X$ is a vector field on $(I,g_I)$, then $X$ can be\nexpressed as $X = h \\partial_t$ for some smooth function\n$h \\in \\mathcal{C}^\\infty(I)$. Hence, ${\\rm L}_X g_I=2 \\sigma g_I$\nwith $\\sigma=h^\\prime$.\n\\end{rem}\n\nWe will now state a simple result which will be useful in our\nstudy (see \\cite{mSK}, \\cite{DWP} and page 126 of \\cite{PHD}).\n\n\\begin{prp} \\label{kill-m} Let $M=I_f \\times F$ be a standard\nstatic space-time with the metric $g=f^2 g_I \\oplus g_F$, where\n$g_I=-{\\rm d}t^2$. Suppose that $X,Y,Z \\in \\mathfrak X(I)$ and\n$V,W,U \\in \\mathfrak X(F).$ Then\n\\begin{eqnarray*} {\\rm L}_{X+V}g(Y+W,Z+U) & = & f^2 {\\rm L}^I_X\ng_I(Y,Z) + 2f V(f)g_I(Y,Z) \\\\\n& + & {\\rm L}^F_V g_F(W,U)\n\\end{eqnarray*}\n\\end{prp}\n\nNote that if $h \\colon I \\to \\mathbb R$ is smooth and $Y,Z \\in\n\\mathfrak X(I),$ then\n\n\\begin{equation}\\label{eq:Lie deriv 2}\n{\\rm L}_{h \\partial_t} g_I(Y,Z) = Y(h)g_I(Z,\\partial_t) + Z(h) g_I\n(Y, \\partial_t).\n\\end{equation}\n\nBy combining the previous statements we can prove the following:\n\n\\begin{prp} \\label{con-1} Let $M=I_f \\times F$ be a standard\nstatic space-time with the metric $g=-f^2{\\rm d}t^2 \\oplus g_F.$\nSuppose that $h \\colon I \\to \\mathbb R$ is smooth and $V$ is a\nvector field on $F.$ Then $h\n\\partial_t + V$ is a conformal-Killing vector\nfield on $M$ with $\\sigma \\in C^{\\infty}(M)$ if and only if the\nfollowings are satisfied:\n\\begin{enumerate}\n\\item $V$ is conformal-Killing on $F$ with $\\sigma \\in\nC^{\\infty}(F),$\n\n\\item $h$ is affine, i.e, there exist real numbers $\\mu$ and $\\nu$\nsuch that $h(t)=\\mu t + \\nu$ for any $t \\in I,$\n\n\\item $V(f) = (\\sigma-\\mu) f.$\n\\end{enumerate}\n\\end{prp}\n\n\\begin{proof}(1) follows from {\\it Proposition \\ref{kill-m}}\nby taking $Y=Z=0$ and a separation of variables argument. On the\nother hand, from {\\it Proposition \\ref{kill-m}} with $W=U=0$,\n\\eqref{eq:Lie deriv 2} and Remark \\ref{rem:Lie deriv}, we have\n$(\\sigma - h^\\prime)f=V(f)$. Hence, again by separation of\nvariables, $h^\\prime$ is constant and then (2) is obtained. Thus,\n(3) is clear.\n\nBy using similar computations described as above, the converse\nturns out to be a consequence of the decomposition of any vector\nfield on $M,$ i.e., as a sum of its horizontal and vertical parts.\n\\end{proof}\n\n\\begin{cor}\\label{cor:simple Killing}\nLet $M=I _f\\times F$ be a standard static space-time with the\nmetric $g=-f^2{\\rm d}t^2 \\oplus g_F.$ Suppose that $h \\colon I \\to\n\\mathbb R$ is smooth and $V$ is a vector field on $F$. Then $h\n\\partial_t + V$ is a Killing vector field on $M$ if and only if\nthe followings are satisfied:\n\\begin{enumerate}\n\\item $V$ is Killing on $F$,\n\n\\item $h$ is affine, i.e, there exist real numbers $\\mu$ and $\\nu$\nsuch that $h(t)=\\mu t + \\nu$ for any $t \\in I$,\n\n\\item $V(f) = - \\mu f$.\n\\end{enumerate}\n\\end{cor}\n\n\\begin{proof}\nIt is sufficient to apply Proposition \\ref{con-1} with $\\sigma\n\\equiv 0$.\n\\end{proof}\n\n\\begin{notation}\\label{notation: It0} Let $h$ be a\ngiven a continuous function defined on a real interval $I$. If\nthere exists a point $t_0 \\in I$ such that $h(t_0)\\neq 0$, then\n$I_{t_0}$ denotes the connected component of $\\{t \\in I: h(t)\\neq\n0\\}$ such that $t_0 \\in I_{t_0}$.\n\\end{notation}\n\nIn what follows, we will make use part of the arguments given in\n\\cite{mSK} (see also \\cite{Carot-da Costa 93}) about the structure\nof Killing and conformal-Killing vector fields in warped products.\nIn \\cite{mSK} by applying such arguments, M. S\\'{a}nchez obtains\nfull characterizations of the Killing and conformal-Killing vector\nfields in generalized Robertson-Walker space-times. In order to be\nmore explanatory, we begin by adapting his procedure to our set of\nframe.\n\n\\bigskip\n\nLet $(B, g_B)$ and $(F, g_F)$ be two semi-Riemannian manifolds\nwith dimensions $r$ and $s > 0$ respectively, and also let $f \\in\nC_{>0}^{\\infty}(F)$ be. Consider the warped metric $g= f^2g_B +\ng_F$ on $B \\times F$. Given a vector field $Z$ on $B \\times F$, we\nwill write $Z = Z_B + Z_F$ with $Z_B = ({\\pi_B}_*(Z), 0)$ and $Z_F\n= (0,{\\pi_F}_*(Z))$, the projections onto the natural foliations\n($B_q = B \\times \\{q\\}$, $q \\in F$ and $F_p=\\{p\\} \\times F, p\\in\nB$). Any covariant or contravariant tensor field $\\omega$ on one\nof the factors ($B$ or $F$) induces naturally a tensor field on $B\n\\times F$ (i.e. the lift), which either will be denoted by the\nsame symbol $\\omega$, or else (when necessary) will be\ndistinguished by putting a bar on it, i.e, $\\overline{\\omega}$.\n\n\\bigskip\n\n\\begin{prp}\\label{prp:sanchez99}\n(see Proposition 3.6 in \\cite{mSK}) If $K$ is a Killing vector\nfield on $M=B_f \\times F$, then $K_B$ is a conformal Killing vector\nfield on $B_q$ for any $q \\in F$ and $K_F$ is a Killing vector field\non $F_p$ for any $p \\in B$.\n\\end{prp}\n\n\\bigskip\n\nSuppose that $\\{C_{\\overline a} \\in \\mathfrak X(B) | \\, \\overline\na = 1,\\cdots, \\overline r \\}$ is a basis for the set of all\nconformal-Killing vector fields on $B$ and $\\{K_{\\overline b} \\in\n\\mathfrak X(F) | \\, \\overline b = 1,\\cdots, \\overline s \\}$ is a\nbasis for the set of all Killing vector fields on $F.$\n\nAccording to \\cite{mSK} (see also \\cite[Sections 7 and 8]{B-O}),\nKilling vector fields on a warped product of the form $M=B_f \\times F$\nwith the metric $g=f^2 g_B \\oplus g_F$ can be given as\n\n\\begin{equation} \\label{form}\nK=\\psi^{\\overline a} C_{\\overline a} + \\phi^{\\overline b}\nK_{\\overline b},\n\\end{equation}\nwhere $\\phi^{\\overline b} \\in \\mathcal C^\\infty(B)$ and\n$\\psi^{\\overline a} \\in \\mathcal C^\\infty(F).$\n\nDefine $K_B=\\psi^{\\overline a} C_{\\overline a}$ and\n$K_F=\\phi^{\\overline b} K_{\\overline b}$. Moreover,\n$\\hat{K}_{\\overline b}=g_F(K_{\\overline b},\\cdot)$ and\n$\\hat{C}_{\\overline a}=g_B(C_{\\overline a},\\cdot).$\n\nThen Proposition 3.8 of \\cite{mSK} implies that a vector field $K$\nof the form (\\ref{form}) is Killing on $B_f \\times F$ if and\nonly if the following equations are satisfied:\n\\begin{equation} \\label{2eqn}\n\\left\\{\n\\begin{array}{rcl}\n\\psi^{\\overline a} \\sigma_{\\overline a} + K_F(\\theta)& = & 0 \\\\\n {\\rm d} \\phi^{\\overline b} \\otimes \\hat{K}_{\\overline b} +\n\\hat{C}_{\\overline a} \\otimes f^2 {\\rm d} \\psi^{\\overline a} & = &\n0,\n\\end{array}\n\\right.\n\\end{equation}\nwhere $C_{\\overline a}$ is a conformal-Killing vector field on $B$\nwith $\\sigma_{\\overline a} \\in \\mathcal C^\\infty(B)$, i.e. ${\\rm\nL}_{C_{\\overline a}}^B g_B= 2\\sigma_{\\overline a}g_B$ and\n$\\theta=\\ln f$.\n\n\\medskip\n\nLet $(F,g_F)$ be a Riemannian manifold of dimension $s$ admitting\nat least one \\textit{nonzero} Killing vector field on $(F,g_F)$\nand $f \\in C^\\infty_{>0}(F)$. Thus, there exists a basis\n$\\{K_{\\overline b} \\in \\mathfrak X(F) | \\, \\overline b = 1,\\cdots,\n\\overline s \\}$ for the set of Killing vector fields on $F$.\n\nLet $I$ be an open interval of the form $I=(t_1,t_2)$ in $\\mathbb R,$\nwhere $-\\infty \\leq t_1 < t_2 \\leq \\infty$ furnished with the metric\n$-{\\rm d}t^2$. Recalling Remark \\ref{rem:Lie deriv}, we\nobserve that the dimension of the set of conformal Killing vector\nfields on $(I,-{\\rm d}t^2)$ is infinite so that one cannot apply directly\nthe above procedure due to M. S\\'{a}nchez before observing that the\nform of conformal Killing vector fields on $(I,-{\\rm d}t^2)$ is\nexplicit. Indeed, it is easy to prove that all the computations are\nvalid by considering the form of any conformal Killing vector field\non $(I,-{\\rm d}t^2)$, namely $h \\partial_t$ where $h \\in C^\\infty(I)$,\ninstead of the finite basis of conformal Killing vector fields in the\nprocedure of M. S\\'{a}nchez .\n\n\\medskip\n\nIf we apply the latter adapted technique to the standard static\nspace-time $M=I _f\\times F$ with the metric $g=f^2 g_I \\oplus g_F$\nwhere $g_I=-{\\rm d}t^2$, then a vector field $K \\in \\mathfrak\nX(M)$ is a Killing vector field if and only if $K$\ncan be written in the form\n\\begin{equation}\\label{eq:Killing structure}\n K= \\psi h \\partial_t +\n\\phi^{\\overline b} K_{\\overline b},\n\\end{equation}\nwhere\n$h$ and $\\phi^{\\overline b} \\in C^\\infty(I)$ for any $\\overline b\n\\in\\{1, \\cdots, \\overline m\\}$ and $\\psi \\in C^\\infty(F)$\nsatisfies the following version of System \\eqref{2eqn}\n\\begin{equation} \\label{2eqnst}\n\\left\\{\n\\begin{array}{rcl}\n h^\\prime \\psi+ \\phi^{\\overline\nb} K_{\\overline b}(\\ln f)& = & 0 \\\\\n{\\rm d} \\phi^{\\overline b} \\otimes g_F(K_{\\overline b},\\cdot) +\ng_I(h \\partial_t, \\cdot) \\otimes f^2 {\\rm d} \\psi & = & 0.\n\\end{array}\n\\right.\n\\end{equation}\n\nThus, in order to study Killing vector fields on standard static\nspace-times we will concentrate our attention on the existence of\nsolutions for System \\eqref{2eqnst}.\n\nSince $\\displaystyle{{\\rm d} \\phi^{\\overline b} = (\\phi^{\\overline\nb})^\\prime {\\rm d}t}$ with $\\phi^{\\overline b} \\in \\mathcal\nC^\\infty(I)$ and $g_I(h \\partial_t, \\cdot)=-h {\\rm d}t$,\n\\eqref{2eqnst} is equivalent to\n\\begin{equation} \\label{3eqnst}\n\\left\\{\n\\begin{array}{rcl}\n h^\\prime \\psi+ \\phi^{\\overline\nb} K_{\\overline b}(\\ln f)& = & 0 \\\\\n(\\phi^{\\overline b})^\\prime {\\rm d}t \\otimes g_F(K_{\\overline\nb},\\cdot) & = & h {\\rm d}t \\otimes f^2 {\\rm d} \\psi ,\n\\end{array}\n\\right.\n\\end{equation}\nand by raising indices in the second equation, it is also\nequivalent to\n\\begin{equation} \\label{3eqnst vector field}\n\\left\\{\n\\begin{array}{lrcl}\n(a)\\qquad & h^\\prime \\psi+ \\phi^{\\overline\nb} K_{\\overline b}(\\ln f)& = & 0 \\\\\n(b)\\qquad &(\\phi^{\\overline b})^\\prime \\partial_t \\otimes\nK_{\\overline b} & = & h \\partial_t \\otimes f^2 \\grad_F \\psi .\n\\end{array}\n\\right.\n\\end{equation}\n\n\nFirst of all, we will apply a separation of variables procedure to\nthe second equation in \\eqref{3eqnst vector field}. Recall that\n$\\{K_{\\overline b}\\}_{1 \\le {\\overline b} \\le {\\overline m}}$ is a\nbasis of the Killing vector fields in $(F,g_F)$. Thus by simple\ncomputations, it is possible to show that $(\\ref{3eqnst vector\nfield}-b)$ implies the following equation for $(\\phi^{\\overline\nb})^\\prime$\n\\begin{equation}\\label{eq:phi form 1}\n\\begin{array}{rcl}\n (\\phi^{\\overline b})^\\prime(t) &=&\n [h(t)-h(t_0)]\\gamma^{\\overline b}+(\\phi^{\\overline b})^\\prime (t_0)\n ,\\\\\n &=& \\gamma^{\\overline b} h(t) + \\delta^{\\overline\n b},\\\\\n\\end{array}\n\\end{equation}\nwhere $\\gamma^{\\overline b}$ and $\\delta^{\\overline b}$\n$(=-h(t_0)\\gamma^{\\overline b}+(\\phi^{\\overline b})^\\prime (t_0),\n\\textrm{ for some fixed } t_0 \\textrm{ in } I )$ are real\nconstants for any choice of ${\\overline b}$ where $1 \\le\n{\\overline b}\\le {\\overline m}$.\n\nThe solutions of the first order ordinary differential equation in\n\\eqref{eq:phi form 1} are given by\n\\begin{equation}\\label{eq:phi form 2}\n\\phi^{\\overline b}(t)=\\gamma^{\\overline b} \\int_{t_0}^t h(s)ds +\n\\delta^{\\overline b}t + \\eta^{\\overline b},\n\\end{equation}\nwhere $\\eta^{\\overline b}$ are constant for all $\\overline b$.\n\nBy introducing \\eqref{eq:phi form 1} in $(\\ref{3eqnst vector\nfield}-b)$, the last equation takes the following equivalent form:\n\\begin{equation}\\label{eq:3eqnst vector field-b-equiv}\n h\\partial_t \\otimes[\\gamma^{\\overline b} K_{\\overline b} - f^2 \\grad_F\n \\psi] = \\partial_t \\otimes [- \\delta^{\\overline b} K_{\\overline b}].\n\\end{equation}\nThus, by recalling again that $\\{K_{\\overline b}\\}_{1 \\le\n{\\overline b} \\le {\\overline m}}$ is a basis of the Killing vector\nfields in $(F,g_F)$, there results three different cases, namely.\n\\begin{description}\n \\item[$h \\equiv 0$]By \\eqref{eq:phi form 2},\n \\eqref{3eqnst vector field} takes the form\n\\begin{equation} \\label{3eqnst vector field h=0}\n\\left\\{\n\\begin{array}{lrcl}\n(a)\\qquad & (\\delta^{\\overline b}t + \\eta^{\\overline b}) K_{\\overline b}(\\ln f)& = & 0 \\\\\n(b)\\qquad &\\delta^{\\overline b} K_{\\overline b} & = & 0 .\n\\end{array}\n\\right.\n\\end{equation}\nSince $\\{K_{\\overline b}\\}_{1 \\le {\\overline b} \\le {\\overline m}}$\nis a basis, $(\\ref{3eqnst vector field h=0}-b)$ implies that for any\n$\\overline b,$ we have $\\delta^{\\overline b}=0$.\n\nThus $K = \\eta^{\\overline b} K_{\\overline b}$, so that $K$ is a\nlinear combination of the elements in the basis $\\{K_{\\overline\nb}\\}_{1 \\le {\\overline b} \\le {\\overline m}}$ and consequently, it\nis a Killing vector field on $(F,g_F)$.\n\n \\item[$h \\equiv h_0 \\neq 0 \\,\\, constant$] By \\eqref{eq:phi form 2},\n \\eqref{3eqnst vector field} takes the form\n\n\\begin{equation} \\label{3eqnst vector field h=h0}\n\\left\\{\n\\begin{array}{lrcl}\n(a)\\qquad & \\phi^{\\overline b} K_{\\overline b}(\\ln f)& = & 0 \\\\\n(b)\\qquad &\\displaystyle \\frac{1}{h_0}\\left(\\gamma^{\\overline b}\nh_0 + \\delta^{\\overline b}\\right) K_{\\overline b} & = & f^2\n\\grad_F \\psi,\n\\end{array}\n\\right.\n\\end{equation}\nwhere\n\\begin{equation}\\label{eq:phi form 2 h0}\n \\phi^{\\overline b}(t)=\\underbrace{(\\gamma^{\\overline b}\n h_0 + \\delta^{\\overline b})}_{h_0\\tau^{\\overline b}}t +\n \\underbrace{(\\eta^{\\overline b} - \\gamma^{\\overline b}\n h_0t_0)}_{\\omega^{\\overline b}}.\n\\end{equation}\n\n\\noindent Note that in particular, Equation $(\\ref{3eqnst vector\nfield h=h0}-b)$ implies that $f^2 \\grad_F \\psi$ is Killing on\n$(F,g_F)$ and gives the coefficients of $f^2 \\grad_F \\psi$ with\nrespect to the basis $\\{K_{\\overline b}\\}_{1 \\le {\\overline b} \\le\n{\\overline m}}$. On the other hand, differentiating $(\\ref{3eqnst\nvector field h=h0}-a)$ with respect to $t$ and then considering\n$(\\ref{3eqnst vector field h=h0}-b)$, we obtain\n\\begin{equation*}\\label{}\n\\begin{array}{rcl}\n 0&=&(\\gamma^{\\overline b} h_0 + \\delta^{\\overline b})\n K_{\\overline b}(\\ln f) \\\\\n &=& h_0(f^2 \\grad_F \\psi )(\\ln f) \\\\\n &=& h_0 f g_F(\\grad_F \\psi,\\grad_F\nf).\\\\\n\\end{array}\n\\end{equation*}\n\n\\noindent Furthermore, \\eqref{eq:phi form 2 h0} and $(\\ref{3eqnst\nvector field h=h0}-a)$ imply that\n\\begin{equation*}\\label{}\n \\omega^{\\overline b}K_{\\overline b}(\\ln f)=0.\n\\end{equation*}\nHence, we proved that \\eqref{3eqnst vector field}\nsuffices to the following:\n\\begin{equation}\n\\label{3eqnst vector field h=h0 2} \\left\\{\n\\begin{array}{l}\nf \\in C^\\infty_{>0}(F),\\psi \\in C^\\infty(F);\\\\\n f^{2} \\grad_F \\psi \\textrm{ is a Killing vector field on } (F,g_F)\n \\textrm{ with }\\\\\n\\textrm{ coefficients } \\{\\tau_{\\overline b}\\}_{1 \\le {\\overline\nb} \\le {\\overline m}} \\textrm{ relative to the basis }\n\\{K_{\\overline b}\\}_{1 \\le\n{\\overline b} \\le {\\overline m}};\\\\\n(f^{2} \\grad_F \\psi)(\\ln f) = 0; \\\\\n\\forall {\\overline b}: \\phi^{\\overline b}(t)=h_0 \\tau^{\\overline\nb}t+\\omega^{\\overline b} \\textrm{ with } \\omega^{\\overline b}\\in\n\\mathbb{R} : \\omega^{\\overline b}\nK_{\\overline b}(\\ln f)=0.\\\\\n\\end{array}\n\\right.\n\\end{equation}\nIt is easy to prove that \\eqref{3eqnst vector field}\nis also necessary for \\eqref{3eqnst vector field h=h0 2}.\n\n\\noindent Hence we proved that $K= \\psi h_0 \\partial_t +\n\\phi^{\\overline b} K_{\\overline b}$ is Killing if and only if\n\\eqref{3eqnst vector field h=h0 2} is satisfied.\n\n \\item[$h\\,\\, nonconstant$] The procedure for this case is a\n generalization of the previous, namely.\n\n\n \\noindent Since $h$ is nonconstant, we can take $t_0$ such that\n $h(t_0)\\neq 0$ and we will work on the subinterval $I_{t_0}$\n (see \\textit{Notation \\ref{notation: It0}}).\n\n\\noindent First of all, note that by applying the separation of\nvariables method in \\eqref{eq:3eqnst vector field-b-equiv}, the\n\\underline{non}-constancy of $h$ implies that\n\\begin{equation}\\label{eq:h no const 1}\n \\left\\{\n\\begin{array}{l}\n\\gamma^{\\overline b} K_{\\overline b} - f^2 \\grad_F\n \\psi=0\\\\\n\\forall {\\overline b}: \\delta^{\\overline b}=0 .\\\\\n\\end{array}\n \\right.\n\\end{equation}\nThus, by \\eqref{eq:phi form 2},\n\\begin{equation}\\label{eq:phi form 2 h no const}\n \\phi^{\\overline b}(t)=\\gamma^{\\overline b} \\int_{t_0}^t h(s)ds\n + \\eta^{\\overline b}.\n\\end{equation}\n\n\\noindent On the other hand, by differentiating $(\\ref{3eqnst\nvector field}-a)$ with respect to $t$ and then by considering\n\\eqref{eq:h no const 1}, we obtain\n\\begin{equation*}\\label{}\n h^{\\prime \\prime}\\psi + h (f^2 \\grad_F \\psi)(\\ln f)=0.\n\\end{equation*}\n\n\\noindent Besides, by considering \\eqref{eq:h no const 1},\n\\eqref{eq:phi form 2 h no const} and again $(\\ref{3eqnst vector\nfield}-a)$ there results\n\\begin{equation*}\\label{}\n \\left[h^\\prime - \\frac{h^{\\prime \\prime}}{h}\n \\int_{t_0}^t h(s)ds \\right]\\psi +\n \\eta^{\\overline b}K_{\\overline b}(\\ln f)=0.\n\\end{equation*}\n\n\\noindent Thus, we proved that \\eqref{3eqnst vector field} is\nsufficient to\n\\begin{equation}\n\\label{3eqnst vector field h no const} \\left\\{\n\\begin{array}{l}\nf \\in C^\\infty_{>0}(F),\\psi \\in C^\\infty(F);\\\\\n f^{2} \\grad_F \\psi \\textrm{ is a Killing vector field on }\n (F,g_F)\\textrm{ with }\\\\\n\\textrm{coefficients } \\{\\tau_{\\overline b}\\}_{1 \\le {\\overline b}\n\\le {\\overline m}} \\, \\textrm{ relative to the basis }\n\\{K_{\\overline b}\\}_{1 \\le {\\overline b} \\le {\\overline m}};\\\\\nh^{\\prime \\prime}\\psi + h (f^2 \\grad_F \\psi)(\\ln f)=0;\\\\\n\\forall {\\overline b}: \\phi^{\\overline b}(t)=\\tau^{\\overline\nb}\\displaystyle \\int_{t_0}^t h(s)ds +\\omega^{\\overline b} \\textrm{\nwith } \\omega^{\\overline b}\\in \\mathbb{R}:\\\\\n\\displaystyle \\left[h^\\prime - \\frac{h^{\\prime \\prime}}{h}\n\\int_{t_0}^t h(s)ds \\right]\\psi + \\omega^{\\overline b}\nK_{\\overline b}(\\ln f)=0 \\textrm{ on } I_{t_0}.\n\\end{array}\n\\right.\n\\end{equation}\nIt is easy to prove that \\eqref{3eqnst vector field} is also a\nnecessary condition for \\eqref{3eqnst vector field h no const}, on\nan interval where $h$ does not take the zero value.\n\n\n\n\n\\noindent It is not difficult to show that if $\\displaystyle\n-\\frac{h^{\\prime \\prime}}{h}$ is nonconstant, then $\\psi \\equiv 0$.\nThus,\n\\begin{equation}\\label{eq:h non constant}\nK= \\phi^{\\overline b} K_{\\overline b}\\textrm{ with\n}\\phi^{\\overline b}(t)=\\omega^{\\overline b}\\textrm{ and\n}\\omega^{\\overline b} \\in \\mathbb{R}: \\omega^{\\overline b}\nK_{\\overline b}(\\ln f)=0.\n\\end{equation}\n\n\\noindent On the other hand, if $\\displaystyle -\\frac{h^{\\prime\n\\prime}}{h}=\\nu$ is constant, the second statement of \\eqref{3eqnst\nvector field h no const}, namely\n\\begin{equation}\\label{eq:5.22 2}\n h^{\\prime \\prime}\\psi + h (f^2 \\grad_F \\psi)(\\ln f)=0,\n\\end{equation}\nresults equivalent to\n\\begin{equation} \\label{eq:3eqnst eq-a 1}\n\\left\\{\n\\begin{array}{rcl}\n -h^{\\prime \\prime} & = & \\nu h \\\\\n(f^2 \\grad_F \\psi)(\\ln f) & = & \\nu \\psi.\n\\end{array}\n\\right.\n\\end{equation}\nThus,\n\\begin{equation} \\label{eq:tuning 3}\nh(t)= \\left\\{\n\\begin{array}{lcl}\na e^{\\sqrt{-\\nu}\\,t} + b e^{-\\sqrt{-\\nu}\\,t}\\, \\,\n& \\textrm{ if } & \\nu \\neq 0 \\\\\na t + b \\, \\, &\\textrm{ if }& \\nu = 0,\\\\\n\\end{array}\n\\right.\n\\end{equation}\nwhere $a$ and $b$ are real constants.\n\n\n\\noindent\n\nHence, by \\eqref{3eqnst vector field h no const}, \\eqref{eq:h non\nconstant} and \\eqref{eq:tuning 3}, the problem \\eqref{3eqnst} is\nequivalent to: (see Notation \\ref{notation: It0} for the\ndefinition of the interval $I_{t_0}$ relative to the function $h$)\n\\begin{equation}\\label{eq:3eqnst 2 h no 0}\n\\left\\{\n\\begin{array}{l}\n(a)\\left\\{\n\\begin{array}{l}\nf \\in C^\\infty_{>0}(F), \\psi \\equiv 0; \\\\\n\\phi^{\\overline b}(t)=\\omega^{\\overline b} \\textrm{ on } I_{t_0}\n\\textrm{ where }\\omega^{\\overline b} \\in\n\\mathbb{R}: \\omega^{\\overline b} K_{\\overline b}(\\ln f)=0\\\\\n\\end{array}\n\\right.\n\\\\\n\\textrm{or }\\\\\n(b)\\left\\{\n\\begin{array}{l}\nf \\in C^\\infty_{>0}(F),\\psi \\in C^\\infty(F);\\\\\n f^{2} \\grad_F \\psi \\textrm{ is a Killing vector field on }\n(F,g_F)\\\\\n\\textrm{with } \\textrm{coefficients } \\{\\tau_{\\overline b}\\}_{1\n\\le {\\overline b} \\le {\\overline m}} \\, \\textrm{ relative to the } \\\\\n\\textrm{basis } \\{K_{\\overline b}\\}_{1 \\le\n{\\overline b} \\le {\\overline m}};\\\\\n(f^{2} \\grad_F \\psi)(\\ln f) = \\nu \\psi \\textrm{ where }\n\\nu \\, \\textrm{is constant} ;\\\\\nh \\textrm{ is given in } \\eqref{eq:tuning 3};\\\\\n\\forall {\\overline b}: \\phi^{\\overline b}(t)=\\tau^{\\overline\nb}\\displaystyle \\int_{t_0}^t h(s)ds +\\omega^{\\overline b} \\textrm{\nwith }\n\\omega^{\\overline b}\\in \\mathbb{R}:\\\\\n\\displaystyle h^\\prime (t_0) \\psi + \\omega^{\\overline\nb}K_{\\overline b}(\\ln f)=0 \\textrm{ on }\nI_{t_0}.\\\\\n\\end{array}\n\\right.\n\\\\\n\\end{array}\n\\right.\n\\end{equation}\n\\end{description}\n\n\\noindent It is easy to prove that if a set of functions $h$, $\\psi$\nand $\\{\\phi_{\\overline b}\\}_{1 \\le \\overline b \\le \\overline m}$,\nsatisfy \\eqref{eq:3eqnst 2 h no 0} with a real interval $I$ instead\nof $I_{t_0}$, then \\eqref{3eqnst} is verified, that is, the vector\nfield given by \\eqref{eq:Killing structure} on a standard static\nspace-time of the form $M=I _f\\times F$ is Killing.\n\nHence, in the precedent discussion we proved the following result.\n\n\n\\begin{thm}\\label{thm:Killing ssst}Let $(F,g_F)$ be a Riemannian\nmanifold, $f \\in C^\\infty_{>0}(F)$ and $\\{K_{\\overline b}\\}_{1 \\le\n\\overline b \\le \\overline m}$ a basis of Killing vector fields on\n$(F,g_F)$. Let also $I$ be an open interval of the form\n$I=(t_1,t_2)$ in $\\mathbb R,$ where $-\\infty \\leq t_1 < t_2 \\leq\n\\infty.$\nConsider the standard static space-time $I _f\\times F$ with the\nmetric $g=-f^2{\\rm d}t^2 \\oplus g_F$.\n\nThen, any Killing vector field on $I _f\\times F$ admits the\nstructure\n\\begin{equation}\\label{eq:eq:Killing 2-s no 0}\nK = \\psi h \\partial_t + \\phi^{\\overline b} K_{\\overline b}\n\\end{equation}\nwhere $h$ and $\\phi^{\\overline b} \\in C^\\infty(I)$ for any\n$\\overline b \\in\\{1, \\cdots, \\overline m\\}$ and $\\psi \\in\nC^\\infty(F).$\n\nFurthermore, assume that $K$ is a vector field on $I _f\\times F$\nwith the structure as in \\eqref{eq:eq:Killing 2-s no 0}. Then,\n\\begin{itemize}\n\\item [{\\bf (i)}] if $h \\equiv 0,$ then the vector field\n$K=\\phi^{\\overline b} K_{\\overline b}$ is Killing on the standard\nstatic space-time $I _f\\times F$ if and only if the functions\n$\\phi^{\\overline b}$ are constant and $\\phi^{\\overline b}\nK_{\\overline b}(\\ln f)=0$.\n\n\\item [{\\bf (ii)}] if $h \\equiv h_0 \\neq 0$ is constant,\nthen the vector field $K= \\psi h_0 \\partial_t + \\phi^{\\overline b}\nK_{\\overline b}$ is Killing on the standard static space-time\n$I_f \\times F$ if and only if \\eqref{3eqnst vector field h=h0 2}\nis satisfied.\n\n\\item [{\\bf (iii)}] if\n$K$ is a Killing vector field on the standard static space-time\n$I _f\\times F$ with the nonconstant function $h$, then the set of\nfunctions $h$, $\\psi$ and $\\{\\phi_{\\overline b}\\}_{1 \\le \\overline\nb \\le \\overline m}$ satisfy \\eqref{eq:3eqnst 2 h no 0} for any\n$t_0 \\in I$ with $h(t_0)\\neq 0.$\n\nConversely, if a set of functions $h$, $\\psi$ and\n$\\{\\phi_{\\overline b}\\}_{1 \\le \\overline b \\le \\overline m}$,\nsatisfy \\eqref{eq:3eqnst 2 h no 0} with an arbitrary $t_0$ in $I$\nand the entire interval $I$ (instead of $I_{t_0}$), then the\nvector field $\\tilde{K}$ on the standard static space-time\n$I _f\\times F$ associated to the set of functions as in\n\\eqref{eq:eq:Killing 2-s no 0} is Killing on $I _f\\times F$.\n\\end{itemize}\n\\end{thm}\n\n\\bigskip\n\nBy completeness we consider in the following lemma the case where\nthe Riemannian manifold $(F,g_F)$ admits no non identically zero\nKilling vector fields.\n\n\\begin{lem}\\label{lem:0 Killing}\nLet $(F,g_F)$ be a Riemannian manifold of dimension $s$ and $f \\in\nC^\\infty_{>0}(F)$. Let also $I$ be an open interval of the form\n$I=(t_1,t_2)$ in $\\mathbb R,$ where $-\\infty \\leq t_1 < t_2 \\leq\n\\infty$. Suppose that the only Killing vector field on $(F,g_F)$ is\nthe zero vector field. Then all the Killing vector fields on the\nstandard static space-time $I_f\\times F$ are given by\n$h_0 \\partial_t$ where $h_0$ is a constant.\n\\end{lem}\n\n\\begin{proof}\nIndeed, by Proposition \\ref{prp:sanchez99} if $K$ is a Killing\nvector field on $I _f\\times F$, then $K=\\psi h \\partial_t$ where\n$\\psi \\in C^\\infty(F)$ and $h \\in C^\\infty (I)$. Then, by\nsimilar arguments to those applied to the system \\eqref{3eqnst\nvector field}, a vector field of the latter form is Killing if and\nonly if the following equations are verified\n\\begin{equation*}\n\\left\\{\n\\begin{array}{l}\n(a) \\qquad h^\\prime \\psi = 0 \\\\\n(b) \\qquad h \\partial_t \\otimes f^2 \\grad_F \\psi = 0.\n\\end{array}\n\\right.\n\\end{equation*}\nAs an immediate consequence, either $h$ and $\\psi$ are constants\nor $\\psi \\equiv 0$.\n\\end{proof}\n\n\\begin{rem}\\label{rem:Killing non trivial}\nIf the Riemannian manifold $(F,g_F)$ admits a nonidentical zero\nKilling vector field, then the family of Killing vector fields\nobtained in \\textit{Corollary \\ref{cor:simple Killing}}\ncorresponds to the case of $\\psi \\equiv 1$ in \\textit{Theorem\n\\ref{thm:Killing ssst} (iii)}. Thus, \\eqref{eq:3eqnst 2 h no 0}\nimplies that $\\nu =0$ and $\\tau^{\\overline b}=0$ for any\n$\\overline b$ and also $h(t)=at+b$ is affine and $\\phi^{\\overline\nb}=\\omega^{\\overline b}$ is constant such that $\\phi^{\\overline b}\nK_{\\overline b}(\\ln f)=\\phi^{\\overline b} K_{\\overline b}(\\ln\nf)=-a$. The latter conditions agree with those in\n\\textit{Corollary \\ref{cor:simple Killing}}.\n\nIn other words, if $\\nu$ is nonzero, then the family of Killing\nvector fields in \\textit{Theorem \\ref{thm:Killing ssst} (iii)} are\ndifferent form those in \\textit{Corollary \\ref{cor:simple Killing}},\nthey correspond to the so called non-trivial Killing vector fields\nin \\cite{mSK}.\n\\end{rem}\n\n\\begin{rem}\\label{rem:subspace}\nLet $f\\in C^\\infty_{>0}(F)$ be smooth. For any $\\nu \\in\n\\mathbb{R},$ we consider the problem given by\n\\begin{equation}\n\\label{eq:pb Killing 1} \\left\\{\n\\begin{array}{l}\nf^{2} \\grad_F \\psi \\textrm{ is a Killing vector field on }\n(F,g_F);\\\\\n(f^{2} \\grad_F \\psi)(\\ln f) = \\nu \\psi ;\\\\\n\\psi \\in C^\\infty(F);\\\\\n\\end{array}\n\\right.\n\\end{equation}\nand define $\\mathcal{K}_f^\\nu= \\{ \\psi \\in C^\\infty(F): \\psi\n\\textrm{ verifies } \\eqref{eq:pb Killing 1} \\}$. It is easy to\nshow that $\\mathcal{K}_f^\\nu$ is an $\\mathbb{R}-$subspace of\n$C^\\infty(F)$. In particular, if $\\psi \\in \\mathcal{K}_f^\\nu$ then\n\\begin{equation}\\label{eq:pb Killing 2}\n(f^{2} \\grad_F \\lambda\\,\\psi)(\\ln f) = \\lambda\\nu \\,\\psi, \\,\n\\forall \\lambda \\in \\mathbb{R}\n\\end{equation}\nConsequently, if $\\{\\tau_{\\overline b}\\}_{1 \\le {\\overline b} \\le\n{\\overline m}}$ is the set of coefficients of the Killing vector\nfield $f^{2} \\grad_F \\psi$ with respect to the basis $\n\\{K_{\\overline b}\\}_{1 \\le {\\overline b} \\le {\\overline m}}$ and\n$\\lambda \\in \\mathbb{R},$ then\n\\begin{equation}\\label{eq:pb Killing 3}\n-\\lambda\\nu \\, \\psi+\\omega^{\\overline b}K_{\\overline b}(\\ln f)= 0,\n\\end{equation}\nwhere $\\omega^{\\overline b}=\\lambda \\tau^{\\overline b},$ for any\n$\\overline b.$\n\nNotice that, this is particularly useful in order to simplify the\nhypothesis of \\eqref{eq:3eqnst 2 h no 0} when $\\nu \\neq 0$.\n\\end{rem}\n\nIn order to analyze the existence of nontrivial solutions for the\nproblem \\eqref{eq:pb Killing 1} (notice that this is relevant in\n\\eqref{eq:3eqnst 2 h no 0} and \\eqref{3eqnst vector field h=h0\n2}), we introduce the following notation.\n\n\\begin{notation} Let $(F,g_F)$ be a Riemannian manifold. Suppose\nthat $Z \\in \\mathfrak {X}(F)$ is a vector field on $(F,g_F)$ and\n$\\varphi \\in C^\\infty(F)$ is a smooth function on $F.$ Then define\na (0,2)-tensor on $F$ as follows:\n\\begin{equation}\\label{eq:special Killing 2}\n B_Z^\\varphi(\\cdot,\\cdot):=\\textrm{d} \\varphi (\\cdot)\n \\otimes g_F(Z,\\cdot) + g_F(\\cdot,Z) \\otimes \\textrm{d}\n \\varphi (\\cdot).\n\\end{equation}\n\\end{notation}\n\n\\begin{prp}\\label{prp:special Killing}\nLet $(F,g_F)$ be a Riemannian manifold, $f \\in C^\\infty_{>0}(F)$\nand $\\psi \\in C^\\infty(F)$. Then the vector field $f^{2} \\grad_F\n\\psi$ is Killing on $(F,g_F)$ if and only if\n\\begin{equation}\\label{eq:special Killing 1}\n {\\rm H}_F^\\psi + \\frac{1}{f} B_{\\grad_F \\psi}^{f}=0,\n\\end{equation}\n\\end{prp}\n\n\\begin{proof} We begin by recalling two results. For all\nsmooth functions $\\varphi \\in C^\\infty(F)$ and vector fields $X,Y\n\\in \\mathfrak{X} (F),$ we have\n\\begin{equation*}\\label{}\n {\\rm H}_F^\\varphi(X,Y) =\n g_F(\\nabla_X^F \\grad_F \\varphi,Y)=\n \\frac{1}{2} {\\rm L}^F_{\\grad_F \\varphi} g_F(X,Y).\n\\end{equation*}\n\nMoreover, for any vector field $Z \\in \\mathfrak X(F),$ the\nfollowing general formula can be stated.\n\\begin{equation*}\\label{}\n {\\rm L}_{\\varphi Z}g_F(\\cdot,\\cdot)=\\varphi {\\rm L}_Z g_F(\\cdot,\\cdot) +\n \\textrm{d} \\varphi (\\cdot) \\otimes g_F(Z,\\cdot) +\n g_F(\\cdot,Z) \\otimes \\textrm{d} \\varphi (\\cdot).\n\\end{equation*}\n\nBy applying the latter formulas to the vector field $f^{2}\n\\nabla_F \\psi,$ we obtain the following:\n\\begin{equation*}\\label{}\n {\\rm L}^F_{f^2 \\grad_F \\psi} g_F = f^2 {\\rm L}_{\\grad_F \\psi}g_F +\n B_{\\grad_F \\psi}^{f^{2}} = 2 f^2 \\left[{\\rm H}^\\psi_F + \\frac{1}{f}\n B_{\\grad_F \\psi}^{f}\\right].\n\\end{equation*}\nThen one can conclude that $f^{2} \\grad_F \\psi$ is a Killing\nvector field on $(F,g_F)$ if and only if \\eqref{eq:special Killing\n1} is satisfied.\n\\end{proof}\n\nIn order to study Killing vector fields on standard static\nspace-times, notice the central role of the problem given below,\ni.e, (\\ref{eq:3eqnst 4}) which appears throughout \\textit{Theorem\n\\ref{thm:Killing ssst}}, \\textit{Proposition \\ref{prp:special\nKilling}} and the identity stated as $fg_F(\\grad_F \\psi,\\grad_F\nf)=(f \\grad_F \\psi)(f),$\n\\begin{equation} \\label{eq:3eqnst 4}\n\\left\\{\n\\begin{array}{l}\nf \\in C^\\infty_{>0}(F), \\psi \\in C^\\infty(F);\\\\\n\\displaystyle {\\rm H}_F^\\psi + \\frac{1}{f}\nB_{\\grad_F \\psi}^{f} = 0 ;\\\\\nfg_F(\\grad_F \\psi,\\grad_F f) = \\nu \\psi \\textrm{ where } \\nu\n\\textrm{ is a constant}.\n\\end{array}\n\\right.\n\\end{equation}\n\n\\begin{rem}\\label{rem:Lie algebra 1} By\n\\textit{Proposition \\ref{prp:special Killing}}, if the dimension\nof the Lie algebra of Killing vector fields of $(F,g_F)$ is zero,\nthen the system \\eqref{eq:3eqnst 4} has only the trivial solution\ngiven by a constant $\\psi $ (this constant is not $0$ only if $\\nu\n=0$). This happens, for instance when $(F,g_F)$ is a compact\nRiemannian manifold of negative-definite Ricci curvature without\nboundary, indeed it is sufficient to apply the vanishing theorem\ndue to Bochner (see for instance \\cite {Bochner46}, \\cite[Theorem\n1.84]{B} or \\cite[Proposition 6.6 of Chapter III]{RG}).\n\\end{rem}\n\n\\begin{lem}\\label{lem: laplace spectrum 1}\nLet $(F,g_F)$ be a Riemannian manifold and $f \\in\nC^\\infty_{>0}(F)$. If $(\\nu,\\psi)$ satisfies \\eqref{eq:3eqnst 4},\nthen $\\nu$ is an eigenvalue and $\\psi$ is an associated\n$\\nu-$eigenfunction of the elliptic problem:\n\\begin{equation}\\label{eq:weight Laplace-Beltrami 1}\n -\\Delta_{g_F} \\psi = \\nu \\frac{2}{f^2}\n \\psi \\, \\textrm{ on } (F,g_F).\n\\end{equation}\n\\end{lem}\n\n\\begin{proof} First of all, note that\n\\begin{equation}\\label{eq:bilinear 1}\n B_{\\grad_F \\psi}^f=\\textrm{d} f \\otimes \\textrm{d} \\psi +\n \\textrm{d} \\psi \\otimes \\textrm{d} f,\n\\end{equation}\nthen by taking the $g_F-$trace of the both sides, we have:\n\\begin{equation}\\label{eq:bilinear 2}\n \\trace B_{\\grad_F \\psi}^f= 2 g_F (\\grad_F \\psi,\\grad_F f).\n\\end{equation}\nThus, by taking the $g_F-$trace of the first equation in\n\\eqref{eq:3eqnst 4} and then by applying the second equation of\n\\eqref{eq:3eqnst 4}, we obtain \\eqref{eq:weight Laplace-Beltrami\n1}. Hence, $\\nu $ belongs to the spectrum of the weighted\neigenvalue problem \\eqref{eq:weight Laplace-Beltrami 1}.\n\\end{proof}\n\n\\begin{rem}\\label{rem:equivalent system}\nLet $(F,g_F)$ be a Riemannian manifold.\n\\begin{description}\n \\item[i]\nNotice that similar arguments to those applied in Lemma \\ref{lem:\nlaplace spectrum 1} allow us to prove that the system\n\\eqref{eq:3eqnst 4} is \\textit{equivalent} to\n\\begin{equation} \\label{eq:3eqnst 5}\n\\left\\{\n\\begin{array}{l}\nf \\in C^\\infty_{>0}(F), \\psi \\in C^\\infty(F);\\\\\n\\displaystyle {\\rm H}_F^\\psi + \\frac{1}{f}\nB_{\\grad_F \\psi}^{f} = 0 ;\\\\\n-\\Delta_{g_F} \\psi = \\displaystyle \\nu \\frac{2}{f^2}\n \\psi \\textrm{ where } \\nu\n\\textrm{ is a constant}.\n\\end{array}\n\\right.\n\\end{equation}\n \\item[ii] Assuming \\eqref{eq:3eqnst 5} (or equivalently \\eqref{eq:3eqnst\n 4}), if $p \\in F$ is a critical point of $f$ or $\\psi$, then $\\nu = 0$\n or $\\psi(p)=0$.\n\\end{description}\n\\end{rem}\n\n\n\\begin{prp}\\label{prp:laplace spectrum 1} Let $(F,g_F)$ be a compact\nRiemannian manifold and $f \\in C^\\infty_{>0}(F)$. Then $(\\nu,\\psi)$\nsatisfies \\eqref{eq:3eqnst 4} if and only if $\\nu =0$ and $\\psi$ is\nconstant.\n\\end{prp}\n\n\\begin{proof} The necessity part is clear, so we will concentrate\nour attention on the sufficiency part.\n\nFirst of all, notice that by the second equation of\n\\eqref{eq:3eqnst 4}, if $p \\in F$ is a critical point of $\\psi$,\nthen $\\nu \\psi(p)=0$. Then, since $(F,g_F)$ is compact, there\nexists a point $p_0 \\in F$ such that $\\psi(p_0)=\\inf_F \\psi$ and\nconsequently, $\\nu \\psi(p_0)=0$.\n\nOn the other hand, by applying \\textit{Lemma \\ref{lem: laplace\nspectrum 1}}, one can conclude that $\\nu$ is an eigenvalue and\n$\\psi$ is an associated $\\nu-$eigenfunction of the elliptic\nproblem \\eqref{eq:weight Laplace-Beltrami 1}. Besides, since\n$(F,g_F)$ is compact, it is well known that the eigenvalues of\n\\eqref{eq:weight Laplace-Beltrami 1} form a sequence in\n$\\mathbb{R}_{\\ge 0}$ and the only eigenfunctions without changing\nsign are the constants corresponding to the eigenvalue $0$.\n\nThus, if $\\psi(p_0) \\ge 0$, then $\\nu =0$ and $\\psi $ results a\nnonnegative constant. Alternatively, if $\\psi(p_0) < 0$, then $\\nu\n\\psi(p_0)=0$, so $\\nu =0.$ As a consequence of that, $\\psi$ is a\nnegative constant.\n\\end{proof}\n\n\\textit{Theorem \\ref{thm:Killing ssst}} and \\textit{Proposition\n\\ref{prp:laplace spectrum 1}} provide the characterization of the\nKilling vector fields on a standard static space-time when its Riemannian\npart is compact (compare the result with \\textit{Corollary\n\\ref{cor:simple Killing}}).\n\n\n\\begin{thm}\\label{thm:killing compact fiber} Let $(F,g_F)$ be a\nRiemannian manifold, $f \\in C^\\infty_{>0}(F)$\nand $I$ be an open interval of the form $I=(t_1,t_2)$ in $\\mathbb\nR,$ where $-\\infty \\leq t_1 < t_2 \\leq \\infty$.\nConsider the standard static space-time $I _f\\times F$ with the\nmetric $g=-f^2{\\rm d}t^2 \\oplus g_F$.\nIf $(F,g_F)$ is compact then, the set of all Killing vector fields\non the standard static space-time $(M,g)$ is given by\n$$\\{a\n\\partial_t + \\tilde{K}| \\, a \\in \\mathbb{R}, \\tilde{K} \\textrm{ is a\nKilling vector field on } (F,g_F) \\textrm{ and } \\tilde{K}(f)=0\n\\}.$$\n\\end{thm}\n\n\\begin{proof} If $(F,g_F)$ has only the zero\nKilling vector field, the result is an easy consequence of Lemma\n\\ref{lem:0 Killing}.\n\nLet us consider now the case there exists a basis $\\{K_{\\overline b}\\}_{1\n\\le \\overline b \\le \\overline m}$ for the space of Killing vector fields\non $(F,g_F)$. \\textit{Theorem \\ref{thm:Killing ssst}} and\n\\textit{Proposition \\ref{prp:laplace spectrum 1}} imply that a\nvector field $K$ on the standard static space-time $(M,g)$ is\nKilling if and only if it admits the structure\n\\begin{equation}\\label{eq:eq:Killing 2-s no 0 compact fiber}\nK = \\psi h \\partial_t + \\phi^{\\overline b} K_{\\overline b},\n\\end{equation}\nwhere\n\\begin{enumerate}\n\\item $h(t)=a t+b$ with constants $a$ and $b$;\n\\item $\\psi$ is constant;\n\\item $\\phi^{\\overline b}$ are constants satisfying\n$a \\psi + \\phi^{\\overline b} K_{\\overline b}(\\ln f)=0$.\n\\end{enumerate}\nSince $(F,g_F)$ is compact, then $\\inf_F \\ln f$ is reached on a\npoint $p_0 \\in F$. Set $\\tilde{K}=\\phi^{\\overline b} K_{\\overline\nb}$. So $\\tilde{K}(\\ln f)|_{p_0}=0$ and by (3) $a=0$ or $\\psi =0$.\nHence we proved that all the Killing vector fields on $I _f\\times F$\nare given by a Killing vector field on $(F,g_F)$ plus\neventually a real multiple of $\\partial_t$. Note that\n$\\displaystyle \\tilde{K}(\\ln f)=\\frac{1}{f} \\tilde{K}(f)$, so by\n(3) $\\tilde{K}(f)=0$.\n\\end{proof}\n\n\\begin{rem} \\label{rem:sharipov2007}\n\\textit{(Killing vector fields in the Einstein static universe)}\nIn \\cite{Sharipov2007}, the author studied Killing vector fields\nof a closed homogeneous and isotropic universe (for related\nquestions in quantum field theory and cosmology see \\cite{dewitt\n1975,dewitt 2003,Fulling1987, Landau Lifshits 1988}). Theorem 6.1\nof \\cite{Sharipov2007} corresponds to our Theorem \\ref{thm:killing\ncompact fiber} for the spherical universe $\\mathbb{R} \\times\n\\mathbb{S}^3$ with the pseudo-metric $\\displaystyle -( R^2 {\\rm\nd}t^2 -R^2 h_0)$, where the sphere $\\mathbb{S}^3$ endowed with the\nusual metric $h_0$ induced by the canonical Euclidean metric of\n$\\mathbb{R}^4$ and $R$ is a real constant (i.e., a stable\nuniverse).\n\\end{rem}\n\n\nAs we have already mentioned in {\\it Remark \\ref{rem:Lie algebra\n1}}, any Killing vector field of a compact Riemannian manifold of\nnegative-definite Ricci tensor is equal to zero. Thus, one can easily\nstate the following result.\n\n\n\\begin{cor} \\label{cor:killing compact fiber of neggative definite\nRicci tensor} Let $M=I _f\\times F$ be a standard static space-time\nwith the metric $g=-f^2{\\rm d}t^2 \\oplus g_F.$ Suppose that\n$(F,g_F)$ is a compact Riemannian manifold of negative-definite\nRicci tensor. Then, any Killing vector field on the standard static\nspace-time $(M,g)$ is given by $a \\partial_t$ where $a \\in \\mathbb\nR.$\n\\end{cor}\n\n\n\\section{Non-rotating Killing vector fields }\n\nIn this section we apply Theorem \\ref{thm:killing compact\nfiber} of {\\it Section 2} to the analysis of non-rotating\nKilling vector fields on standard static space-times also\ncalled \\textit{static regular predictable space-times}\nin \\cite[p.325]{HE} (also see a recent article, i.e,\n\\cite{Sanchez-Senovilla2007} for a related question).\n\nWe first recall the definition of the $\\curl$ operator on\nsemi-Riemannian manifolds as: if $V$ is a vector field on a\nsemi-Riemannian manifold $(M,g)$, then $\\curl V$ is the\n2-covariant tensor field defined by\n\\begin{equation}\\label{eq:curl}\n\\curl V (X,Y):=g(\\nabla_X V,Y)-g(\\nabla_Y V,X),\n\\end{equation}\nwhere $X,Y \\in \\mathfrak X(M)$ (see for instance\n\\cite{ON,Gutierrez-Olea2003}). Thus, it is easy to prove\n\\begin{eqnarray}\\label{eq:curl 0}\n\\curl (\\phi V) (X,Y)& = & X(\\phi) g(V,Y)- Y(\\phi)g( V,X) \\\\\n& + & \\phi \\curl V (X,Y) \\nonumber ,\n\\end{eqnarray}\nwhere $X, Y$ are as above and $\\phi \\in C^\\infty (M)$.\n\nWe will follow the next definitions (see\n\\cite{Gutierrez-Olea2003,Gutierrez-Olea2007}): A vector field $V$\non the semi-Riemannian manifold $(M,g)$ is said to be\n\\begin{description}\n \\item[non-rotating]\\footnote{In \\cite {Gutierrez-Olea2003,Gutierrez-Olea2007}\n this condition is called irrotational.}\n if $\\curl V (X,Y)=0$ for all $X,Y \\in \\mathfrak\n X(M)$.\n \\item[orthogonally irrotational]\\footnote{In \\cite{ON,Sanchez-Senovilla2007}\n this condition is called irrotational.}\n if $\\curl V (X,Y)=0$ for any $X,Y \\in \\mathfrak\n X(M)$ orthogonal to $V$. This condition is equivalent to\n ``$V$ has integrable orthogonal distribution\".\n\\end{description}\n\nNotice that being orthogonally irrotational is necessary for\nbeing non-rotating. Furthermore \\eqref{eq:curl 0} implies that\nif $V$ is orthogonally irrotational, then so is $\\phi V$ for any\n$\\phi \\in C^\\infty_{>0}(M)$. Indeed, since $\\phi$ does not vanish,\n$X$ is orthogonal to $\\phi V$ if and only if it is orthogonal to\n$V$.\n\n\\noindent However, if $V$ is non-rotating then, $\\phi V$ is not\nnecessarily non-rotating for some $\\phi \\in C^\\infty_{>0} (M)$\n(see \\eqref{eq:curl 0}).\n\n\\begin{prp}\\label{prp:time irrotational} Let $\\kappa$ be a Killing\nvector field on the standard static space-time $(\\mathbb{R} \\times\nF,g:=-f^2 dt^2 + g_F)$ such that $(\\curl \\kappa) (X,Y)=0$ for all\n$X,Y$ orthogonal vector fields to $\\partial_t$ defined on\n$(\\mathbb{R} \\times F,g)$. If $(F,g_F)$ is compact and simply\nconnected, then $\\kappa = a\n\\partial_t$ where $a$ is a real constant. In particular, $\\kappa$\nbecomes time-like if $a \\neq 0$.\n\\end{prp}\n\n\\begin{proof}\nFirst of all we observe that it is easy to prove the following\nidentity.\n\\begin{equation}\\label{eq:grad}\n -f^2 \\grad t = \\partial_t.\n\\end{equation}\nThus by using \\eqref{eq:curl 0},\n\\begin{eqnarray} \\label{eq:curl partial t}\n (\\curl \\partial_t)(X,Y) & = & \\displaystyle\n \\left(\\curl f^2 (-\\grad t)\\right) (X,Y) \\\\\n & = & \\displaystyle \\frac{X(f^2)}{f^2} g(\\partial_t,Y)-\n \\frac{Y(f^2)}{f^2}g(\\partial_t,X) \\nonumber \\\\\n & - &f^2 (\\underbrace{\\curl \\grad t}_{=0}) (X,Y) \\nonumber,\\\\\n & = & \\displaystyle \\frac{X(f^2)}{f^2} g(\\partial_t,Y)-\n \\frac{Y(f^2)}{f^2}g(\\partial_t,X) \\nonumber.\n\\end{eqnarray}\nHence,\n\\begin{equation}\\label{eq:curl partial t orth}\n\\begin{split}\n&(\\curl \\partial_t) (X,Y)=0 \\textrm{ for all } X,Y\n\\textrm{ orthogonal } \\\\\n&\\qquad\\textrm{vector fields to } \\partial_t \\textrm{ on }\n(\\mathbb{R} \\times F,g),\n\\end{split}\n\\end{equation}\ni.e., $\\partial_t$ is orthogonally irrotational. Up to this point,\nwe have not considered the compactness and the simply connectedness\nof the Riemannian part.\n\nNotice that letting $\\kappa$ be a Killing vector field on\n$(\\mathbb{R} \\times F,g)$ with $(F,g_F)$ compact, Theorem\n\\ref{thm:killing compact fiber} implies that $\\kappa = a\n\\partial_t + K$ where $a$ is a constant and $K$ is a Killing\nvector field on $(F,g_F)$ with $K(f)=0$. Hence and recalling that\n$(\\curl \\kappa) (X,Y)=0$ for all $X,Y$ orthogonal vector fields to\n$\\partial_t$ on $(\\mathbb{R} \\times F,g)$, the $\\curl$ on\n$(\\mathbb{R} \\times F,g)$ is linear and \\eqref{eq:curl partial t\north}, we obtain that\n\\begin{equation} \\label{eq:curl 1}\n \\curl K (X,Y) = 0,\n\\end{equation}\nfor all $X,Y$ orthogonal vector fields to $\\partial_t$ on\n$(\\mathbb{R} \\times F,g)$. It is clear that the lifts of the\nelements in $\\mathfrak X(F)$ verify the latter condition and as\nconsequence \\eqref{eq:curl 1}.\n\nOn the other hand, by the well known relations between the\nLevi-Civita connections on $(F,g_F)$ and on the warped product\n$(\\mathbb{R} \\times F,$ $-f^2 dt^2 + g_F)$ (see for instance\n\\cite{ON} p. 206 Proposition 35) and by the definition of the\n$\\curl$, there results $\\curl_F K = 0$ ($\\curl_F$ denotes the\n$\\curl$ on the Riemannian manifold $(F,g_F)$), i.e. $K$ is\nnon-rotating on $(F,g_F)$. Furthermore, when $(F,g_F)$ is simply\nconnected, then $K$ is a gradient (see\n\\cite{Bishop-Goldberg1980}), i.e. there exists $\\psi \\in C^\\infty\n(F)$ such that $\\grad \\psi = K$.\n\n\nSince $K$ is a Killing vector field on the Riemannian manifold\n$(F,g_F)$ and is the gradient of a function $\\psi \\in C^\\infty\n(F)$, then Proposition \\ref{prp:special Killing} (with $f\n\\equiv 1$) implies that the Hessian $H_F^\\psi \\equiv 0$ on $F$.\nBesides that if $F$ is a compact Riemannian manifold without\nboundary, then by taking the $g_F-$trace of $H_F^\\psi$ and applying\nthe maximum principle one can deduce that $\\psi$ is a constant and\nconsequently, $K \\equiv 0$.\n\nThus we have established that $\\kappa = a \\partial_t$, where $a$ is\nconstant.\n\\end{proof}\n\nNotice that \\eqref{eq:curl partial t} implies that $\\partial_t$ is\nnon-rotating on the standard static space-time of the form\n$(\\mathbb{R} \\times F,-f^2 dt^2 + g_F)$ if $f$ is a positive\nconstant. Hence, we can state and prove the following lemma.\n\n\\begin{lem}\\label{lem:partial t irrot}$\\partial_t$ is non-rotating\non the standard static space-time of the form\n$(\\mathbb{R} \\times F,g:=-f^2 dt^2 + g_F)$ if and only if $f$ is a\npositive constant.\n\\end{lem}\n\\noindent\\emph{Proof 1.} First of all we recall\n$g(\\partial_t,\\partial_t)=-f^2$. Then, by the hypothesis and the\nproperties that characterized the Levi-Civita connection, for any\nvector field $Z$ on $(\\mathbb{R} \\times F,g:=-f^2 dt^2 + g_F)$\nthe following holds\n\\begin{equation}\\label{eq:partial t - 1}\n\\begin{split}\n 0&= \\curl \\partial_t (\\partial_t,Z) = g(\\nabla _{\\partial_t}\n \\partial_t,Z) - \\underbrace{g(\\nabla _{Z}\n \\partial_t,\\partial_t)}_{\\frac{1}{2} Z\n g(\\partial_t,\\partial_t)}\\\\\n &=g(\\nabla _{\\partial_t}\n \\partial_t,Z) + \\displaystyle \\frac{1}{2} Z f^2.\n\\end{split}\n\\end{equation}\nSo, by the definition of the gradient operator denoted by $\\grad$,\n\\begin{equation}\\label{eq:partial t - 2}\n-\\nabla _{\\partial_t} \\partial_t = \\displaystyle \\frac{1}{2} \\grad\nf^2.\n\\end{equation}\nNow, by applying \\cite[p. 206 - Proposition 35]{ON} or\n\\cite[Theorem 3.4]{dobarro-unal04} as above,\n\\begin{equation}\\label{eq:partial t - 3}\n-\\nabla _{\\partial_t} \\partial_t = - \\underbrace{\\nabla^{(-dt^2)}\n_{\\partial_t}\n\\partial_t}_{0} + f \\underbrace{(-dt^2)(\\partial_t,\\partial_t)}_{-1} \\grad\nf =-\\displaystyle \\frac{1}{2} \\grad f^2.\n\\end{equation}\nThe latter and the expression \\eqref{eq:partial t - 2} imply\n$\\grad f^2 = 0$, more explicitly $f$ is a positive constant.\n\\hfill $\\square$\n\n\\medskip\n\n\\noindent\\emph{Proof 2.} An alternative and simpler proof follows\nfrom \\eqref{eq:curl partial t}. Indeed, an immediate consequence\nof the latter is\n\\begin{equation}\\label{eq:partial t - 4}\n (\\curl \\partial_t)(\\partial_t,Z) = Z (f^2),\n\\end{equation}\nfor any vector field $Z$ on $(\\mathbb{R} \\times F,g:=-f^2 dt^2 +\ng_F)$. Thus, if $\\partial_t $ is non-rotating, then $f$ is a\npositive constant.\n\\hfill $\\square$\n\n\n\n\\begin{cor}\n\\label{cor:irrotational}Let $(F,g_F)$ be a compact and simply\nconnected Riemannian manifold. If $f \\in C^\\infty_{>0} (M)$ is\nnonconstant, then there is no nontrivial (non-identically zero)\nnon-rotating Killing vector field on the standard static\nspace-time $(\\mathbb{R} \\times F,g:=-f^2 dt^2 + g_F)$.\n\\end{cor}\n\n\\begin{proof} In order to apply Proposition \\ref{prp:time irrotational},\nit is sufficient to note that if $\\kappa$ is a non-rotating\nKilling vector field on $(\\mathbb{R} \\times F,g:=-f^2 dt^2 + g_F)$,\nthen $(\\curl \\kappa) (X,Y)=0$ for all $X,Y$ orthogonal vector\nfields to $\\partial_t$ on $(\\mathbb{R} \\times F,g)$. Thus, $\\curl\n\\kappa = a \\curl \\partial_t = 0$. So, by Lemma \\ref{lem:partial t\nirrot}, $\\kappa=0$ or $f$ is a positive constant.\n\\end{proof}\n\n\\begin{rem} Notice that in Proposition \\ref{prp:time irrotational}\nthe involved vector fields necessarily turn out to be causal\n(i.e., non-space-like). The conclusion would be wrong if we\neliminated the simply connectedness. For example, consider the\nvector field $a \\partial_t + \\partial_\\theta$ where $a<1$ on\n$(\\mathbb{R} \\times \\mathbb{S}^1,-{\\rm d}t^2 + {\\rm d}\\theta^2)$.\nThis is a non-rotating Killing vector field and yet not time-like\ndue to $a<1$, so its pseudo-norm is $-a^2+1>0$.\n\\end{rem}\n\n\\section{Conclusions}\n\nIt is very well known that a space-time possesses a symmetry if it\nadmits a Killing vector fields. Thus existence and\ncharacterization problems of Killing vector fields are extremely\nimportant in the geometry of space-times. Therefore, we considered\nKilling vector fields of standard static space-times and roughly\nproved that if a vector field $K$ on the space-time is Killing,\nthen $K$ has to be of the form:\n$$ K= \\psi h \\partial_t + \\phi^{\\overline b} K_{\\overline b}\n,$$ where $\\{K_{\\overline b}\\}_{1 \\le \\overline b \\le \\overline\nm}$ is a basis of Killing vector fields on $(F,g_F)$ and $h,\n\\phi^{\\overline b} \\in \\mathcal C^\\infty(I)$ and $\\psi \\in\n\\mathcal C^\\infty(F)$ satisfy the system\n\\begin{equation*}\n\\left\\{\n\\begin{array}{rcl}\n h^\\prime \\psi+ \\phi^{\\overline\nb} K_{\\overline b}(\\ln f)& = & 0, \\\\\n{\\rm d} \\phi^{\\overline b} \\otimes g_F(K_{\\overline b},\\cdot) +\ng_I(h \\partial_t, \\cdot) \\otimes f^2 {\\rm d} \\psi & = & 0.\n\\end{array}\n\\right.\n\\end{equation*}\nIn {\\it Theorem \\ref{thm:Killing ssst}} we have characterized the\nsolutions of the latter system. We remarked the centrality of the\nproblem \\eqref{eq:pb Killing 1} or equivalently \\eqref{eq:3eqnst\n4} and \\eqref{eq:3eqnst 5}.\n\nAs a consequence of {\\it Theorem \\ref{thm:Killing ssst}} and the\nwell known results about the eigenvalues and eigenfunctions of a\npositively weighted elliptic problem on a compact Riemannian\nmanifold without boundary, we also provided a characterization of\nthe Killing vector fields on a standard static space-time with\ncompact Riemannian part in {\\it Theorem \\ref{thm:killing compact\nfiber}}. Note that by combining this theorem with the vanishing\nresults of Bochner (see \\textit{Remark \\ref{rem:Lie algebra 1}}),\nwe obtain that in a standard static space-time with compact\nRiemannian part of negative Ricci curvature without boundary, the\nunique Killing vector fields are time-like of the form $c\n\\partial_t$ where $c \\in \\mathbb{R}$ is constant.\n\nFurthermore, in Corollary \\ref{cor:irrotational} we show that there\nis no non trivial non-rotating Killing vector field on a standard\nstatic space-time where the so called natural space is compact and\nsimply connected.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sect:intro}\n\nThe purpose of ongoing and upcoming heavy ion collision experiments at the Relativistic Heavy Ion Collider (RHIC) and the Large \nHadron Collider (LHC) is to study the behavior of nuclear matter at high energy density, $\\epsilon \\gg 1\\;{\\rm GeV\/fm}^3$.\nAt such high energy densities one expects to create a deconfined quark gluon plasma (QGP). With such experiments one hopes \nto not only cross the threshold necessary to create a QGP, but to also study its properties such as transport coefficients, color \nopacity, etc. One complicating factor is that the QGP generated in such collisions lasts for only a few fm\/c and during this time \nthe bulk properties of the system, e.g. energy density and pressure, can change rapidly. Therefore, dynamical models that can \ndescribe the evolution of the system on the fm\/c timescale are necessary in order to make reliable phenomenological \npredictions.\n\nTo first approximation, it seems that the dynamics of the soft background is well-described by relativistic viscous \nhydrodynamics \\cite{Israel:1979wp,Muronga:2003ta,Baier:2006um,Romatschke:2007mq,Dusling:2007gi,Luzum:2008cw,%\nSong:2008hj,Denicol:2010tr,Schenke:2011tv,Shen:2011eg,Bozek:2011wa,Niemi:2011ix,Bozek:2012qs}. However, \nviscous corrections to the ideal energy momentum tensor cause it to become anisotropic in the local rest frame of the system. \nFor small deviations from isotropy, 2nd-order viscous hydrodynamics describes the evolution\nquite well; however, for large deviations from isotropy this is no longer the case. Large deviations from isotropy occur\nat very early times after the initial nuclear impact and near the transverse or longitudinal edges of the plasma where the\nmatter is nearly free streaming. The presence of momentum-space anisotropies seems unavoidable in dynamical \nmodels. In fact, even in the limit of infinite strong coupling, momentum-space anisotropies persist during the entire \nlifetime of the plasma \\cite{Heller:2011ju,Heller:2012je,Wu:2011yd}. Large momentum-space anisotropies pose a \nproblem for 2nd-order viscous hydrodynamics since it relies on a linearization around an isotropic\nbackground. If the linear corrections grow too large this can generate unphysical results such as negative particle \npressures, negative one-particle distribution functions, etc.~\\cite{Martinez:2009mf}.\n\nIn order to ameliorate these problems it is possible to reorganize the derivation of the necessary dynamical equations by \nlinearizing around an anisotropic instead of isotropic background. Doing so results in a dynamical framework called anisotropic \nhydrodynamics \\cite{Florkowski:2010cf,Martinez:2010sc,Ryblewski:2010bs,Martinez:2010sd,Ryblewski:2011aq,\nMartinez:2012tu,Ryblewski:2012rr}. In the limit of small deviations from isotropy, anisotropic hydrodynamics reduces\nto 2nd-order viscous hydrodynamics, but can also faithfully describe large deviations from isotropy such as those\ncreated during the initial longitudinal free streaming phase of the plasma lifetime. This framework has now been\nused to model the full (3+1)-dimensional dynamics of the QGP \\cite{Ryblewski:2012rr}. Comparison of the anisotropic\nhydrodynamics predictions for observables such as the bulk flow as a function of transverse momentum and \nrapidity with experimental data indicate that it is possible that large momentum-space anisotropies can persist for \nup to 2 -- 3 fm\/c after the initial nuclear impact. Given this, it is imperative to revisit the study of basic properties\nof the QGP in a time-evolving anisotropic background.\n\nIn this paper we study the oscillations of a uniform longitudinal chromoelectric field in a dynamically-evolving \nmomentum-space anisotropic background in the weak field limit. For simplicity, in this work we restrict ourselves \nto a (0+1)-dimensional boost-invariant background. The necessary anisotropic hydrodynamics equations are\nobtained from the first two moments of the Boltzmann-Vlasov equation using a spheroidal form for the one-particle\ndistribution function in the local rest frame \\cite{Romatschke:2003ms}. The dynamical equations in this case\nwere first obtained in Refs.~\\cite{Florkowski:2010cf,Martinez:2010sc}. In both Ref.~\\cite{Florkowski:2010cf}\nand Ref.~\\cite{Martinez:2010sc} a timescale for the \napproach to isotropic thermal equilibrium, $\\tau_{\\rm eq}$, was introduced. In Ref.~\\cite{Florkowski:2010cf}\n$\\tau_{\\rm eq}$ was assumed to be a constant, while in Ref.~\\cite{Martinez:2010sc} this time scale\nwas proportional to the average local inverse transverse momentum of the plasma constituents and was determined\nself-consistently in terms of the local plasma environment.\n\nIn the case that $\\tau_{\\rm eq}$ is constant, the late time behavior of the system is that of ideal hydrodynamics. \nIn the case that $\\tau_{\\rm eq}$ is proportional to the local inverse transverse momentum scale, the \nproportionality constant can be fixed by requiring that the late time dynamics of the system is that of 2nd-order \nviscous hydrodynamics. We will consider both cases in order to assess the impact of this choice. In either case the\nanisotropic hydrodynamics equations provide the proper-time dependence of the local transverse momentum scale,\n$\\Lambda(\\tau)$, and momentum-space anisotropy $\\xi(\\tau) = \\frac{1}{2} \\langle p_\\perp^2 \\rangle\/\\langle \np_\\parallel^2 \\rangle - 1$ where $\\langle p_\\perp^2 \\rangle$ and $\\langle p_\\parallel^2 \\rangle$ are the average transverse \nand longitudinal (beamline-direction) momenta squared in the local rest frame of the plasma constituents,\nrespectively.\n\nGiven this time evolving background, we linearize the Boltzmann-Vlasov equation in order to study the evolution\nof a uniform longitudinal chromoelectric field fluctuation, $\\mbox{\\boldmath $\\cal E$}_z$. \nWe consider the weak-field limit in which case\nwe can use the abelian dominance approximation for the color fields \\cite{Gyulassy:1986jq,Elze:1986hq,%\nElze:1986qd,Bialas:1987en} (see also Ref.~\\cite{Casher:1978wy}). In a static constant-temperature plasma, \nuniform longitudinal field fluctuations \noscillate in time with a frequency given by the plasma frequency $\\omega_{\\rm pl} = m_D\/\\sqrt{3}$ where \n$m_D^2 = (N_c\/3+ N_f\/6) g^2 T^2$ is the leading-order gluonic Debye mass. In a time evolving system, the plasma \nfrequency is time dependent and one must self-consistently solve the linearized Boltzmann-Vlasov equation together with \nthe Maxwell equations. In general, the result can be cast in the form of an integro-differential equation for the \nevolution of $\\mbox{\\boldmath $\\cal E$}_z$. \nFor the case of ideal hydrodynamical evolution, Bialas and Czyz \\cite{Bialas:1987cn} derived \nsuch an equation and solved it numerically. \n\nHere we extend the treatment of Bialas and Czyz to (i) include a dynamically evolving anisotropic background and \n(ii) include the effect of collisional damping. We will present numerical solutions to the resulting \nintegro-differential equations for small and large magnitude momentum-space anisotropies in order to assess the \nimpact of momentum-space anisotropy on plasma oscillations. The equations we obtain are applicable to an arbitrary\ntime-dependent anisotropic background. Although we consider the evolution of a stable longitudinal chromoelectric \nfield, the techniques used herein could have application to the study of the evolution of non-Abelian plasma\ninstabilities in a dynamically evolving anisotropic background \\cite{Romatschke:2006wg,Rebhan:2008uj,Rebhan:2009ku}.\nAs a cross check of our results we present a comparison with the results of Ref.~\\cite{Rebhan:2009ku} which\npresented an analysis of all stable and unstable collective modes of the QGP in the limit of a longitudinally free streaming\nbackground. We show that in this limit we obtain the same evolution and asymptotic behavior as in \nRef.~\\cite{Rebhan:2009ku}, giving us confidence in our theoretical and numerical methods.\n\nThe structure of the paper is as follows. In Section \\ref{sect:conventions} we specify the conventions we will\nuse throughout the paper. In Section \\ref{sect:semiclassical} we review the semi-classical transport equations\nfor the quark gluon plasma in the abelian dominance approximation. In Section \\ref{sect:linearization} we \nlinearize the Boltzmann-Vlasov equation to zeroth and first order in fluctuations. In Section \\ref{sect:Max}\nwe couple the fluctuations via currents to the Maxwell equations and solve the coupled Boltzmann-Vlasov-Maxwell\nsystem of equations to obtain an integro-differential equation which governs the time evolution of uniform longitudinal\nchromoelectric fluctuations. In Section \\ref{sect:results} we present the results of numerical solution to the\nevolution equations for different types of anisotropic backgrounds and compare with numerical and analytic results \navailable in the limit of longitudinal free streaming. In Section \\ref{sect:concl} we present our conclusions and an \noutlook for the future.\n\n\\section{Conventions}\n\\label{sect:conventions}\n\nBelow we use the following definitions for momentum rapidity ($y$) and spacetime rapidity ($\\eta$),\n\\begin{eqnarray}\ny = \\frac{1}{2} \\ln \\frac{E+p_\\parallel}{E-p_\\parallel}, \\quad\n\\eta = \\frac{1}{2} \\ln \\frac{t+z}{t-z}, \\label{yandeta} \n\\end{eqnarray}\nwhich come from the standard parameterization of the four-momentum and spacetime coordinates of a particle,\n\\begin{eqnarray}\np^\\mu &=& \\left(E, {\\vec p}_\\perp, p_\\parallel \\right) =\n\\left(m_\\perp \\cosh y, {\\vec p}_\\perp, m_\\perp \\sinh y \\right), \\nonumber \\\\\nx^\\mu &=& \\left( t, {\\vec x}_\\perp, z \\right) =\n\\left(\\tau \\cosh \\eta, {\\vec x}_\\perp, \\tau \\sinh \\eta \\right). \\label{pandx}\n\\end{eqnarray} \nIn Eq.~(\\ref{pandx}) the quantity $m_\\perp$ is the transverse mass\n\\begin{equation}\nm_\\perp = \\sqrt{m^2 + p_x^2 + p_y^2},\n\\label{energy}\n\\end{equation}\nand $\\tau$ is the proper time\n\\begin{equation}\n\\tau = \\sqrt{t^2 - z^2}.\n\\label{tau}\n\\end{equation} \nThroughout the paper we use natural units where $c=1$ and $\\hbar=1$.\n\n\n\\section{Semi-classical kinetic equations for quark-gluon plasma}\n\\label{sect:semiclassical}\n\nIn the abelian dominance approximation, the transport equations for quarks, antiquarks, and gluons have the form \n\\cite{Gyulassy:1986jq,Elze:1986hq,Elze:1986qd,Bialas:1987en} \n\\begin{equation}\n\\left( p^{\\mu }\\partial _{\\mu } \\pm g{\\mbox{\\boldmath $\\epsilon$}}_{i}\\cdot \n{\\bf F}^{\\mu \\nu }p_{\\nu }\\partial _{\\mu }^{p}\\right) \nQ^\\pm_{i}(x,p)= C^\\pm_{i}, \\label{kineq}\n\\end{equation}\n\\begin{equation}\n\\left( p^{\\mu }\\partial _{\\mu }+g{\\mbox{\\boldmath $\\eta$}}_{ij}\\cdot \n{\\bf F}^{\\mu \\nu }p_{\\nu }\\partial _{\\mu }^{p}\\right) \nG_{ij}(x,p)=\nC_{ij}, \\label{kineg}\n\\end{equation}\nwhere $Q^+_{i}(x,p)$, $Q^-_{i}(x,p)$, and $G_{ij}(x,p)$ are the phase-space densities of quarks, antiquarks, and charged \ngluons, respectively. Here $g$ is the strong coupling constant, and $i,j=(1,2,3)$ are color indices. The terms on the \nleft-hand-side describe the free motion of the particles and the interaction of the particles with the mean field \n$\\mathbf{F}_{\\mu \\nu }$. The latter describes neutral gluons \\cite{Huang:1982ik}. \n\nIn this work, the only non-zero components of the tensor ${\\bf F}^{\\mu \\nu }=(F^{\\mu \\nu }_{(3)},F^{\\mu \\nu }_{(8)})\n$ are those corresponding to the longitudinal chromoelectric field ${\\mbox{\\boldmath $\\cal E$}_z} = \n(F^{30}_{(3)},F^{30}_{(8)})$. The quarks couple to the chromoelectric field ${\\mbox{\\boldmath $\\cal E$}_z}$ through the \ncharges \n\\begin{equation}\n\\mbox{\\boldmath $\\epsilon$}_{1} = \\frac{1}{2}\\left(\\! 1,\\sqrt{\\frac{1}{3}}\\right) \\!, \n\\mbox{\\boldmath $\\epsilon$}_{2} = \\frac{1}{2}\\left(\\! -1,\\sqrt{\\frac{1}{3}}\\right) \\!, \n\\mbox{\\boldmath $\\epsilon$}_{3} = \\left(\\! 0,-\\sqrt{\\frac{1}{3}}\\right) .\n\\label{qcharge}\n\\end{equation}\nThe gluons couple to ${\\mbox{\\boldmath $\\cal E$}_z}$ through the charges ${\\mbox{\\boldmath $\\eta $}}_{ij}$ defined by \nthe relation \n\\begin{equation}\n{\\mbox{\\boldmath $\\eta$}}_{ij}={\\mbox{\\boldmath $\\epsilon$}}_{i}-{%\n\\mbox{\\boldmath $\\epsilon$}}_{j}. \\label{gcharge}\n\\end{equation}\nBelow, we make use of the following relations\n\\begin{eqnarray}\n\\sum_{i=1}^3 {\\mbox{\\boldmath $\\epsilon$}}^a_{i} {\\mbox{\\boldmath $\\epsilon$}}^b_{i} \n&=& \\frac{1}{2} \\, \\delta^{ab} \\, , \\nonumber \\\\\n\\sum_{i,j=1}^3 {\\mbox{\\boldmath $\\eta$}}^a_{ij} {\\mbox{\\boldmath $\\eta$}}^b_{ij} \n&=& 3 \\, \\delta^{ab} \\, , \\label{norm}\n\\end{eqnarray}\nwhere $a,b \\in \\{ 3,8 \\}$.\n\nThe terms on the right-hand-side of Eqs.~(\\ref{kineq}) and (\\ref{kineg}) are collisions terms, which we treat in the relaxation \ntime approximation\n\\begin{eqnarray}\nC^\\pm_{i} &=& - p^\\mu U_\\mu \\frac{Q^\\pm_{i}(x,p)-Q^\\pm_{\\rm eq}(x,p)}{\\tau_{\\rm eq}}, \\\\\nC_{ij} &=& - p^\\mu U_\\mu \\frac{G_{ij}(x,p)-G_{\\rm eq}(x,p)}{\\tau_{\\rm eq}}.\n\\end{eqnarray}\nHere $U^\\mu$ is the four-velocity of the local rest frame \n\\begin{equation}\nU^\\mu = \\gamma (1, v_x, v_y, v_z), \\gamma=(1-v^2)^{-1}.\n\\end{equation}\nIn this paper we consider boost-invariant longitudinal expansion and hence we set $v_x=v_y=0$ and $v_z=z\/t$.\n\n\\section{Linearization of kinetic equations}\n\\label{sect:linearization}\n\nIn the following, we seek the solutions of kinetic equations of the form\n\\begin{eqnarray}\nQ^\\pm_{i}(x,p) = Q^\\pm_{0}(x,p) + \\delta Q^\\pm_{i}(x,p), \\\\\nG_{ij}(x,p) = G_{0}(x,p) + \\delta G_{ij}(x,p),\n\\end{eqnarray}\nwhere the corrections to the background distributions are proportional to the coupling. We emphasize that the background \ndistributions $Q^\\pm_{0}(x,p)$ and $G_{0}(x,p)$ are different from the equilibrium distributions. \n\n\\subsection{Zeroth order}\n\nAt zeroth order in fluctuations one obtains \n\\begin{equation}\n p^{\\mu }\\partial_{\\mu } Q^\\pm_{0}(x,p)= \n- p^\\mu U_\\mu \\frac{Q^\\pm_{0}(x,p) - Q^\\pm_{\\rm eq}(x,p)}{\\tau_{\\rm eq}}, \n\\label{kineq0}\n\\end{equation}\n\\begin{equation}\np^{\\mu }\\partial_{\\mu } G_{0}(x,p) = \n- p^\\mu U_\\mu \\frac{G_{0}(x,p) - G_{\\rm eq}(x,p)}{\\tau_{\\rm eq}}. \n\\label{kineg0}\n\\end{equation} \nEquations (\\ref{kineq0}) and (\\ref{kineg0}) determine the evolution of $Q^\\pm_{0}(x,p)$ and $G_{0}(x,p)$. \n\nInstead of solving (\\ref{kineq0}) and (\\ref{kineg0}) directly, we take moments of these equations. \nIn order to describe (0+1)-dimensional anisotropic dynamics we take the zeroth and first moments of \nEqs.~(\\ref{kineq0}) and (\\ref{kineg0}) assuming that the distributions $Q^\\pm_{0}(x,p)$ and \n$G_{0}(x,p)$ are given by the covariant version of the \nRomatschke-Strickland distribution \\cite{Romatschke:2003ms,Florkowski:2011jg}, namely\n\\begin{eqnarray}\nQ^\\pm_{0}(x,p) = G_{0}(x,p) = f_0(x,p) ,\n\\label{f0}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nf_0(x,p) = \\exp\\left(-\\frac{1}{\\Lambda} \\sqrt{(p\\cdot U)^2 + \\xi (p\\cdot V)^2} \\right).\n\\label{RSform}\n\\end{eqnarray}\nAccordingly, we take\n\\begin{eqnarray}\nQ^\\pm_{\\rm eq}(x,p) = G_{\\rm eq}(x,p) = f_{\\rm eq}(x,p),\n\\label{feq}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nf_{\\rm eq}(x,p) = \\exp\\left(-\\frac{p\\cdot U}{T} \\right).\n\\label{eqform}\n\\end{eqnarray}\nNote that one can also use anisotropic Fermi-Dirac and Bose-Einstein distributions for the (anti-)quarks and\ngluons, respectively; however, the only change to the final result will be the precise value of the isotropic\nplasma frequency of the system. For the sake of simplicity we present the case of a Boltzmann distribution\nand generalize the results to the full quantum statistical distributions in the end.\n\nThe four-vector $V^\\mu$ appearing in (\\ref{RSform}) defines the direction of the beam ($z$-axis)\n\\begin{equation}\nV^\\mu = \\gamma_z (v_z, 0, 0, 1), \\quad \\gamma_z = (1-v_z^2)^{-1\/2}.\n\\label{V}\n\\end{equation}\nWe note that the four-vectors $U^\\mu$ and $V^\\mu$ satisfy the normalization conditions\n\\begin{eqnarray}\nU^2 = 1, \\quad V^2 = -1, \\quad U \\cdot V = 0.\n\\label{UVnorm}\n\\end{eqnarray}\nIn the local rest frame of the fluid element, $U^\\mu$ and $V^\\mu$ have simple forms\n\\begin{eqnarray}\n U^\\mu = (1,0,0,0), \\quad V^\\mu = (0,0,0,1). \n \\label{UVLRF}\n\\end{eqnarray}\nFor the (0+1)-dimensional boost-invariant expansion considered in this paper, we may use\n\\begin{eqnarray}\n U^\\mu &=& (\\cosh\\eta,0,0,\\sinh\\eta), \\nonumber \\\\\n V^\\mu &=& (\\sinh\\eta,0,0,\\cosh\\eta).\n \\label{UVbinv}\n\\end{eqnarray}\n\nWith the assumptions (\\ref{f0}) and (\\ref{feq}), the kinetic equations (\\ref{kineq0}) and (\\ref{kineg0}) are reduced to a\nsingle equation for the background distribution\n\\begin{equation}\n p^{\\mu }\\partial_{\\mu } f_{0}(x,p)= \n- p^\\mu U_\\mu \\frac{f_{0}(x,p) - f_{\\rm eq}(x,p)}{\\tau_{\\rm eq}}.\n\\label{kinef0}\n\\end{equation}\n\n\\subsubsection{Zeroth moment of the kinetic equation}\n\nIntegrating Eq.~(\\ref{kinef0}) over three-momentum and including the internal degrees of freedom we obtain\n\\begin{equation}\n\\partial_{\\mu } N_{0}^\\mu = \n \\frac{U_\\mu \\left( N_{\\rm eq}^\\mu-N_{0}^\\mu \\right)}{\\tau_{\\rm eq}},\n\\label{EQN}\n\\end{equation}\nwhere $N_0$ and $N_{\\rm eq}$ are particle currents~\\footnote{There is no term proportional to $V^\\mu$ in $N_0^\\mu$, due \nto the quadratic dependence of $f_0$ on $V^\\mu$.} \n\\begin{equation}\nN_0^\\mu = n_0 U^\\mu, \\quad N^\\mu_{\\rm eq} = n_{\\rm eq} U^\\mu.\n\\label{N0Neq}\n\\end{equation}\nA simple calculation performed in the local rest frame gives\n\\begin{eqnarray}\nn_0 = \\frac{g_0}{\\pi^2} \\frac{\\Lambda^3}{\\sqrt{1+\\xi}}, \\quad\nn_{\\rm eq} = \\frac{g_0}{\\pi^2} T^3.\n\\label{n0neq}\n\\end{eqnarray}\nHere $g_0$ is the degeneracy factor accounting for internal degrees of freedom (we show below that the equations of motion for \nthe background are insensitive to the specific choice of $g_0$). For longitudinal boost-invariant expansion one finds\n\\begin{equation}\nU^\\mu \\partial_\\mu = \\frac{d}{d\\tau}, \\quad \\partial_\\mu U^\\mu = \\frac{1}{\\tau}.\n\\end{equation}\nThus, using (\\ref{N0Neq}) and (\\ref{n0neq}) in (\\ref{EQN}), we obtain \n\\begin{eqnarray}\n\\frac{3}{\\Lambda} \\frac{d\\Lambda}{d\\tau}-\\frac{1}{2(1+\\xi)}\\frac{d\\xi}{d\\tau} + \\frac{1}{\\tau} \n= \\frac{\\left(T\/\\Lambda\\right)^3 \\sqrt{1+\\xi} -1}{\\tau_{\\rm eq}}.\n\\label{EQ1}\n\\end{eqnarray}\n\n\n\\subsubsection{First moment of the kinetic equation}\n\nIn the next step we multiply Eq.~(\\ref{kinef0}) by $p^\\nu$ and integrate over three-momentum. In this way, we obtain\n\\begin{eqnarray}\n\\partial_\\mu T_0^{\\mu \\nu} = \n\\frac{U_\\mu \\left(T_{\\rm eq}^{\\mu \\nu}-T_0^{\\mu \\nu}\\right)}{\\tau_{\\rm eq}},\n\\label{enmomcon0}\n\\end{eqnarray}\nwhere \\cite{Florkowski:2010cf,Martinez:2012tu}\n\\begin{equation}\nT_0^{\\mu \\nu} = (\\varepsilon_0+P_\\perp) U^\\mu U^\\nu - P_\\perp g^{\\mu \\nu}\n-(P_\\perp-P_\\parallel) V^\\mu V^\\nu\n\\label{Tmunu0}\n\\end{equation}\nand\n\\begin{equation}\nT_{\\rm eq}^{\\mu \\nu} = (\\varepsilon_{\\rm eq}+P_{\\rm eq}) U^\\mu U^\\nu - P_{\\rm eq} g^{\\mu \\nu}.\n\\label{Tmunu0-2}\n\\end{equation}\n\nIn order to conserve energy and momentum, the right-hand-side of (\\ref{enmomcon0}) should vanish. Hence, we obtain \nthe Landau matching condition \n\\begin{equation}\n\\varepsilon_0 = \\varepsilon_{\\rm eq},\n\\label{LM1}\n\\end{equation}\nwhere\n\\begin{eqnarray}\n\\varepsilon_0 = \\frac{3 g_0 \\Lambda^4}{\\pi^2} {\\cal R}(\\xi), \\quad\n\\varepsilon_{\\rm eq} = \\frac{3 g_0 T^4}{\\pi^2},\n\\label{eps0epseq}\n\\end{eqnarray}\nand the function ${\\cal R}(\\xi)$ has the form \\cite{Martinez:2010sc}\n\\begin{equation}\n{\\cal R}(\\xi) = \\frac{1}{2(1+\\xi)} \n\\left[1+ \\frac{ (1+\\xi) \\arctan \\sqrt{\\xi}} {\\sqrt{\\xi} } \\right].\n\\end{equation}\nEqs.~(\\ref{LM1}) and (\\ref{eps0epseq}) are used to obtain the ratio $T\/\\Lambda$ needed in (\\ref{EQ1})\n\\begin{equation}\nT = \\Lambda {\\cal R}^{1\/4}(\\xi).\n\\label{EQ2}\n\\end{equation} \n\nFor purely longitudinal boost-invariant motion, the energy-momentum conservation law $\\partial_\\mu T_0^{\\mu \\nu} =0$ takes a simple form\n\\begin{equation}\n\\frac{d\\varepsilon_0}{d\\tau} = -\\frac{\\varepsilon_0+P_\\parallel}{\\tau}.\n\\label{EQ30}\n\\end{equation}\nEq.~(\\ref{EQ30}) may be reduced to the equation\n\\begin{eqnarray}\n\\hspace{-4mm} {\\cal R}^\\prime (\\xi) \\frac{d\\xi}{d\\tau} + 4 {\\cal R}(\\xi) \\frac{d\\Lambda}{\\Lambda d\\tau}\n= -\\frac{1}{\\tau} \\left( {\\cal R}(\\xi) + \\frac{1}{3} {\\cal R}_L(\\xi) \\right) ,\n\\label{EQ3}\n\\end{eqnarray}\nwhere\n\\begin{equation}\n{\\cal R}_L(\\xi) = \\frac{3}{\\xi} \\left[ {\\cal R}(\\xi)\n- \\frac{1}{1+\\xi} \\right].\n\\end{equation}\n\n\\subsubsection{Evolution of the time-evolving background}\n\\label{subsect:background}\n\nEqs.~(\\ref{EQ1}), (\\ref{EQ2}), and (\\ref{EQ3}) provide three equations for three unknown functions: $\\Lambda(\\tau)$, \n$\\xi(\\tau)$, and $T(\\tau)$. The solutions of these equations allow us to determine the background for the plasma oscillations. \nBelow we will consider two cases: (i) a fixed relaxation time of $\\tau_{\\rm eq} = $~1~fm\/c and (ii) a relaxation time that is \nproportional to the local inverse transverse momentum scale of the plasma. In the second case the relaxation \ntime is fixed by requiring that, in the limit of small momentum-anisotropy, the linearized anisotropic hydrodynamics \nequations reproduce 2nd-order viscous hydrodynamics \\cite{Martinez:2010sc}. In this case, the relaxation time is \ngiven by\n\\begin{equation}\n\\tau_{\\rm eq}(\\tau) = \\frac{5 \\bar\\eta}{2 {\\cal R}^{1\/4}(\\xi) \\Lambda} \\, ,\n\\label{taueqdyn}\n\\end{equation}\nwhere $\\bar\\eta = \\eta_s\/{\\cal S}$ is the ratio of the shear viscosity to entropy density and\nit is implicitly understood that $\\xi$ and $\\Lambda$ depend on proper time. In what\nfollows we will assume that $\\bar\\eta$ is time independent.\n\n\\begin{figure}[t]\n\\begin{center}\n\\subfigure{\\includegraphics[angle=0,width=0.42\\textwidth]{plot_ksi1c.pdf}}\\\\\n\\subfigure{\\includegraphics[angle=0,width=0.42\\textwidth]{plot_ksi10.pdf}}\\\\\n\\subfigure{\\includegraphics[angle=0,width=0.42\\textwidth]{plot_ksi100.pdf}}\n\\end{center}\n\\caption{(Color online) Time dependence of the anisotropy parameter $\\xi$ for (a) constant $\\tau_{\\rm eq}$, (b) time varying\n$\\tau_{\\rm eq}$ with $4\\pi\\bar\\eta = 1$, and (c) time varying\n$\\tau_{\\rm eq}$ with $4\\pi\\bar\\eta = 10$. In each plot we show three different initial\nvalues of $\\xi$: $\\xi_0=99$ (dashed line), $\\xi_0=0$ (solid line), and $\\xi_0=-0.99$ (dotted line).\n}\n\\label{fig:plot_background}\n\\end{figure}\n\nIn Fig.~\\ref{fig:plot_background} we show the time dependence of the anisotropy parameter $\\xi$. The initial time of the \nhydrodynamic evolution is taken to be $\\tau_0 = 0.1$~fm\/c, and the final time is taken to be $\\tau_f = 10$~fm\/c. We consider \nthree different choices of the initial conditions corresponding to three different values of the initial anisotropy $\\xi_0=\\xi(\\tau_0)$:\n$\\xi_0=99$ (dashed lines), $\\xi_0=0$ (solid lines), and \\mbox{$\\xi_0=-0.99$} (dotted lines). The initial transverse-momentum scale \n$\\Lambda_0$ is taken to be \\mbox{$\\Lambda_0 = $~(1 GeV)$\\cdot (1+\\xi_0)^{1\/6}$} in each case. The factor of $(1+\\xi_0)^{1\/6}$ \nguarantees that the initial number density of the background is the same for all values of $\\xi_0$ considered.\nWe show the proper-time evolution of the anisotropy parameter, $\\xi$, in three different\ncases: (a) fixed $\\tau_{\\rm eq} =$~1~fm\/c, (b) $\\tau_{\\rm eq}$ given by Eq.~(\\ref{taueqdyn}) with $4\\pi\\bar\\eta =1$,\nand (c) Eq.~(\\ref{taueqdyn}) with $4\\pi\\bar\\eta =10$. In the case of $\\tau_{\\rm eq}$ = 1 fm\/c shown in \nFig.~\\ref{fig:plot_background}(a) the anisotropy vanishes at late times. When $\\tau_{\\rm eq}$ is given by \nEq.~(\\ref{taueqdyn}) (see Fig.~\\ref{fig:plot_background}(b,c)) a finite anisotropy remains at late times which is \nconsistent with 2nd-order viscous hydrodynamics. Note that the initial growth of the anisotropy is connected with the effects \nof free streaming which dominate the very early dynamics \\cite{Florkowski:2010cf,Martinez:2010sc}. \n\n\\subsection{First order}\n\nAt first order in fluctuations we obtain\n\\begin{eqnarray}\n&& p^{\\mu }\\partial _{\\mu } \\delta Q^\\pm_{i}(x,p)\n\\pm g {\\mbox{\\boldmath $\\epsilon$}}_{i}\\cdot {\\bf F}^{\\mu \\nu }\np_{\\nu } \\partial_{\\mu }^{p} Q_0^\\pm (x,p) \\nonumber \\\\\n&& \\hspace{1.0cm} = - p^\\mu U_\\mu \\frac{\\delta Q^\\pm_{i}(x,p)}{\\tau_{\\rm eq}}, \n\\label{kineq1}\n\\end{eqnarray}\n\\begin{eqnarray}\n&& p^{\\mu }\\partial _{\\mu } \\delta G_{ij}(x,p) \n+g{\\mbox{\\boldmath $\\eta$}}_{ij}\\cdot {\\bf F}^{\\mu \\nu } \np_{\\nu }\\partial _{\\mu }^{p} G_0(x,p) \\nonumber \\\\\n&& \\hspace{1.0cm} = - p^\\mu U_\\mu \\frac{\\delta G_{ij}(x,p)}{\\tau_{\\rm eq}}. \n\\label{kineg1}\n\\end{eqnarray}\n\nIn the following we use the boost-invariant variables introduced in Refs.~\\cite{Bialas:1984wv,Bialas:1984ap} \n\\begin{equation}\nw = tp_{\\Vert }-zE,\n\\label{binvv1}\n\\end{equation}\nand \n\\begin{equation}\nv = Et-p_{\\Vert }\\ z=\\sqrt{w^{2}+m_{\\perp }^{2} \\tau^2}. \n\\label{binvv2}\n\\end{equation}\nFrom these two equations one can easily find the energy and the longitudinal momentum of a particle \n\\begin{equation}\nE=p^{0}=\\frac{vt+wz}{\\tau^2},\\quad p_{\\Vert }=\\frac{wt+vz}{\\tau^2}. \\label{binvv3}\n\\end{equation}\nIn addition, we have \n\\begin{equation}\nw=\\tau m_{\\perp }\\sinh \\left( y-\\eta \\right) ,\\qquad v=\\tau m_{\\perp }\\cosh\n\\left( y-\\eta \\right) . \\label{binvv4}\n\\end{equation}\n\n\nSince the distribution functions are Lorentz scalars, they may depend only on $\\tau$, $p_\\perp$, and $w$. Therefore, we find the general boost-invariant form of the terms appearing in Eqs.~(\\ref{kineq1}) and (\\ref{kineg1})\n\\begin{eqnarray}\np^\\mu U_\\mu &=& \\frac{v}{\\tau}, \\nonumber \\\\\np^{\\mu }\\partial _{\\mu } \\delta f &=& \\frac{v}{\\tau} \\frac{\\partial}{\\partial \\tau} \\delta f, \\label{binvder} \\\\\n{\\bf F}^{\\mu \\nu } p_{\\nu } \\partial_{\\mu }^{p} f_0 &=& \n{\\mbox{\\boldmath $\\cal E$}}_z v \\frac{\\partial f_0}{\\partial w}. \\nonumber\n\\end{eqnarray}\nThe Romatschke-Strickland distribution function takes the form\n\\begin{eqnarray}\nf_0(\\tau,w) &=& \\exp\\left[-\\frac{1}{\\Lambda(\\tau) \\tau} \\sqrt{p_\\perp^2 \\tau^2 + (1+\\xi(\\tau)) w^2} \\, \\right] .\n\\nonumber \\\\\n&&\n\\label{RSformWV}\n\\end{eqnarray}\nUsing Eqs.~(\\ref{binvder}) we find the following equations\n\\begin{eqnarray}\n\\frac{\\partial}{\\partial \\tau} \\delta Q^\\pm_{i} &=& \\mp g \\tau \n{\\mbox{\\boldmath $\\epsilon$}}_{i} \\cdot {\\mbox{\\boldmath $\\cal E$}}_z\n\\frac{\\partial f_0}{\\partial w} - \\frac{\\delta Q^\\pm_{i}}{\\tau_{\\rm eq}} , \\\\\n\\frac{\\partial}{\\partial \\tau} \\delta G_{ij} &=& - g \\tau \n{\\mbox{\\boldmath $\\eta$}}_{ij} \\cdot {\\mbox{\\boldmath $\\cal E$}}_z\n\\frac{\\partial f_0}{\\partial w} - \\frac{\\delta G_{ij}}{\\tau_{\\rm eq}}.\n\\label{EQS1}\n\\end{eqnarray}\nThe formal solutions of (\\ref{EQS1}) for constant or time-dependent $\\tau_{\\rm eq}$ may be expressed as integrals\n\\begin{eqnarray}\n\\hspace{-3mm} \\delta Q^\\pm_{i} = \\mp g {\\mbox{\\boldmath $\\epsilon$}}_{i} \\cdot\n\\int\\limits_{\\tau_0}^\\tau d\\tau^\\prime \\, \\tau^\\prime \nD(\\tau,\\tau^\\prime)\n{\\mbox{\\boldmath $\\cal E$}}_z(\\tau^\\prime ) \n\\frac{\\partial f_0}{\\partial w} (\\tau^\\prime, w), \\label{SOL1q} \\\\\n\\hspace{-3mm} \\delta G_{ij} = - g {\\mbox{\\boldmath $\\eta$}}_{ij} \\cdot\n\\int\\limits_{\\tau_0}^\\tau d\\tau^\\prime \\, \\tau^\\prime \nD(\\tau,\\tau^\\prime)\n{\\mbox{\\boldmath $\\cal E$}}_z(\\tau^\\prime ) \n\\frac{\\partial f_0}{\\partial w} (\\tau^\\prime, w), \\label{SOL1g}\n\\end{eqnarray}\nwhere we have introduced the damping function\n\\begin{equation}\nD(\\tau,\\tau^\\prime) = \\exp\\left[-\\int\\limits_{\\tau^\\prime}^\\tau \\frac{d\\tau^{\\prime \\prime}}{\\tau_{\\rm eq}(\\tau^{\\prime \\prime})} \\right].\n\\end{equation}\n\n\\section{Maxwell equations}\n\\label{sect:Max}\n\nIn order to close our system of equations we have to couple the fluctuations via currents to the Maxwell equations\n\\begin{equation}\n\\partial_\\mu \\mathbf{F}^{\\mu \\nu } = \\mathbf{j}^\\nu,\n\\end{equation}\nwhere the color current is given by the expression\n\\begin{equation}\n\\mathbf{j}^\\nu = g \\int dP p^\\nu \\left[ \\sum_{i=1}^3 {\\mbox{\\boldmath $\\epsilon$}}_{i}\n\\left(\\delta Q^+_{i} - \\delta Q^-_{i} \\right) + \\sum_{i,j=1}^3 {\\mbox{\\boldmath $\\eta$}}_{ij} \n\\delta G_{ij} \\right].\n\\end{equation}\nUsing the formal solutions for the distribution functions at first order, we obtain\n\\begin{eqnarray}\n&& \\mathbf{j}^\\nu = -g^2 \\int\\limits_{\\tau_0}^\\tau d\\tau^\\prime \\, \\tau^\\prime \nD(\\tau,\\tau^\\prime)\n\\int dP \\, p^\\nu \n\\frac{\\partial f_0}{\\partial w} (\\tau^\\prime, w) \n \\\\\n&& \\times \\left[ \n\\sum_{i=1}^3 2 \\,{\\mbox{\\boldmath $\\epsilon$}}_{i} ({\\mbox{\\boldmath $\\epsilon$}}_{i} \\cdot {\\mbox{\\boldmath $\\cal E$}}_z(\\tau^\\prime )) \n+ \\sum_{i,j=1}^3 {\\mbox{\\boldmath $\\eta$}}_{ij} ({\\mbox{\\boldmath $\\eta$}}_{ij} \\cdot {\\mbox{\\boldmath $\\cal E$}}_z(\\tau^\\prime )) \n\\right] . \\nonumber\n\\end{eqnarray}\nHere we used the property $\\delta Q^-_{i}(\\tau,p_\\perp,w) = \\delta Q^+_{i}(\\tau,p_\\perp,-w)$ to eliminate the antiquark distribution function.\n\nThe invariant measure in momentum space is\n\\begin{equation}\ndP = \\nu_g d^2p_\\perp {dp_\\parallel \\over p^0} = \\nu_g d^2p_\\perp {dw \\over v},\n\\label{dP}\n\\end{equation}\nwhere $\\nu_g = \\nu_{\\rm sf}\/(2\\pi)^3$ and $\\nu_{\\rm sf}$ denotes the number of internal degrees of freedom connected with spin or flavor ($\\nu_{\\rm sf}=4$ for quarks and $\\nu_{\\rm sf}=2$ for gluons). Using Eqs.~(\\ref{norm}) and (\\ref{dP}) we find\n\\begin{eqnarray}\n&& \\mathbf{j}^\\nu = -g^2 \\int\\limits_{\\tau_0}^\\tau d\\tau^\\prime \\, \\tau^\\prime \nD(\\tau,\\tau^\\prime)\n\\int d^2p_\\perp \\int\\limits_{-\\infty}^\\infty {dw \\,p^\\nu \\over (2 \\pi)^3 \\,v} \n \\nonumber\n \\\\\n&& \\times \\frac{\\partial f_0}{\\partial w} (\\tau^\\prime, w) \\left[ \n2 \\cdot 4 \\cdot \\frac{1}{2} \\, {\\mbox{\\boldmath $\\cal E$}}_z(\\tau^\\prime ) \n+ 2 \\cdot 3 \\cdot {\\mbox{\\boldmath $\\cal E$}}_z(\\tau^\\prime ) \n\\right]. \\nonumber\n\\end{eqnarray}\n\nFor the Lorentz index $\\nu=0$, we use the symmetry of the distribution function under the change $w \\to -w$ and write\n\\begin{eqnarray}\n\\mathbf{j}^0 &=& 10 \\,z\\, g^2 \\int\\limits_{\\tau_0}^\\tau d\\tau^\\prime \\, \nD(\\tau,\\tau^\\prime) {\\mbox{\\boldmath $\\cal E$}}_z(\\tau^\\prime ) \n \\frac{ (1+\\xi^\\prime)}{\\Lambda^\\prime \\tau^2} \\label{j03} \\\\\n&& \\times \n\\int \\frac{d^2p_\\perp}{(2 \\pi)^3} \\int\\limits_{-\\infty}^\\infty {dw \\, w^2 \\over v}\n\\frac{f_0^\\prime }{ \\sqrt{p_\\perp^2 \\tau^{\\prime 2} + (1+\\xi^\\prime) w^2}}. \\nonumber\n\\end{eqnarray}\nHere, the primes denote the dependence on $\\tau^\\prime$, for example, $\\Lambda^\\prime = \\Lambda(\\tau^\\prime)$ and $f_0^\\prime=f_0(\\tau^\\prime,p_\\perp,w)$.\n\nThe zeroth component of the Maxwell equation gives\n\\begin{equation}\n\\partial_3 {\\bf F}^{3 0} = -\\frac{z}{\\tau} \n\\frac{d{\\mbox{\\boldmath $\\cal E$}}_z(\\tau) }{d\\tau},\n\\label{Max3}\n\\end{equation}\nhence, Eqs.~(\\ref{j03}) and (\\ref{Max3}) yield~\\footnote{Because of boost-invariance we obtain the same result from the Maxwell equations with $\\nu=3$.} \n\\begin{eqnarray}\n&& \\frac{d{\\mbox{\\boldmath $\\cal E$}}_z(\\tau) }{d\\tau} = - \\frac{10 g^2}{(2 \\pi)^2 \\tau} \\int\\limits_{\\tau_0}^\\tau d\\tau^\\prime \\, \nD(\\tau,\\tau^\\prime) {\\mbox{\\boldmath $\\cal E$}}_z(\\tau^\\prime ) \n \\frac{ (1+\\xi^\\prime)}{\\Lambda^\\prime} \\nonumber \\\\\n&& \\times \n\\int\\limits_0^\\infty dp^2_\\perp \\int\\limits_0^\\infty {dw \\, w^2 \\over v}\n\\frac{f_0^\\prime }{ \\sqrt{p_\\perp^2 \\tau^{\\prime 2} + (1+\\xi^\\prime) w^2}}. \\label{FE1}\n\\end{eqnarray}\nTo proceed, we introduce new variables $\\gamma$ and $\\phi$ defined by the relations\n\\begin{eqnarray}\n\\gamma \\cos\\phi &=& \\frac{p_\\perp}{\\Lambda^\\prime} , \\nonumber \\\\\n\\gamma \\sin\\phi &=& (1+\\xi^\\prime) \\frac{w}{\\Lambda^\\prime \\tau^\\prime}.\n\\end{eqnarray}\nThe integration over $\\gamma$ (from 0 to $\\infty$) and $\\phi$ (from 0 to $\\pi\/2$) can be performed analytically \n\\cite{Bialas:1987cn}, and one obtains\n\\begin{eqnarray}\n\\frac{d{\\mbox{\\boldmath $\\cal E$}}_z(\\tau) }{d\\tau} = && \n- \\frac{\\kappa g^2}{\\tau} \\int\\limits_{\\tau_0}^\\tau d\\tau^\\prime \\, \nD(\\tau,\\tau^\\prime) {\\mbox{\\boldmath $\\cal E$}}_z(\\tau^\\prime ) \n \\nonumber \\\\\n&& \\times \\tau^\\prime \\Lambda^{\\prime \\,2} J\\left(\\frac{\\tau \\sqrt{1+\\xi^\\prime} }{\\tau^\\prime}\\right), \\nonumber \\\\\n\\label{FE2}\n\\end{eqnarray}\nwhere, for a Boltzmann distribution, $\\kappa=10\/(3 \\pi^2)$ and the function $J(\\tau\\sqrt{1+\\xi^\\prime}\/\\tau^\\prime)$ is defined by the formula \\cite{Bialas:1987cn}\n\\begin{eqnarray}\nJ\\left(\\frac{\\tau \\sqrt{1+\\xi^\\prime}}{\\tau^\\prime}\\right) &=&\n\\frac{3}{4} \\frac{b+1}{b^{3\/2}} \\left[ \\frac{\\pi}{2} + \\arcsin \\left(\\frac{b-1}{b+1} \\right) \\right]\n-\\frac{3}{2b}, \\nonumber \\\\\nb &=& \\frac{\\tau^2}{\\tau^{\\prime \\,2}} (1+\\xi^\\prime)-1.\n\\end{eqnarray}\n\nEq.~(\\ref{FE2}) determines the oscillations of a uniform chromoelectric field in an arbitrary \ntime-evolving anisotropic background. It forms the \nbasis of our numerical calculations presented below. Interestingly, the form of (\\ref{FE2}) is very similar to that obtained in \n\\cite{Bialas:1987cn}. Eq.~(\\ref{FE2}) is reduced to Eq.~(4.8) in \\cite{Bialas:1987cn}, if we set $\\xi^\\prime=0$ in the \nargument of $J$, $\\Lambda^\\prime$ is replaced by the temperature $T^\\prime$, and the damping function $D$ is taken to \nbe equal to unity. \nNote that since the calculations in \\cite{Bialas:1987cn} were performed using the quantum statistical distribution functions, \nthe factor $\\kappa$ in Ref.~\\cite{Bialas:1987cn} equals $(N_c + N_f\/2)\/9 = 4\/9$ for $N_c=3$ and $N_f=2$. The calculations \npresented in this paper can be also performed using quantum statistical distribution functions and, in that case, one obtains \n$\\kappa=4\/9$. The results presented in Sec.~\\ref{sect:results} use this choice of $\\kappa$.\n\n\\subsection{Special Case: Longitudinal free streaming limit}\n\nFor the case of longitudinal free streaming it is possible to write the integro-differential equation (\\ref{FE2}) as an ordinary \ndifferential equation. In the limit $\\tau_{\\rm eq} \\rightarrow \\infty$ one has $\\xi(\\tau) = (\\tau\/\\tau_{\\rm iso})^2 - 1$ and \n$\\Lambda(\\tau) = \\Lambda_0$ with $\\tau_{\\rm iso}=\\tau_0\/\\sqrt{1+\\xi(\\tau_0)}$ being the point in time when $\\xi=0$. \nInserting these relations into (\\ref{FE2}) gives\n\\begin{equation}\n\\frac{\\tau}{J\\!\\left(\\frac{\\tau}{\\tau_{\\rm iso}}\\right)}\\frac{d{\\mbox{\\boldmath $\\cal E$}}_z(\\tau) }{d\\tau} = \n- \\omega_{\\rm pl}^2 \\int\\limits_{\\tau_0}^\\tau d\\tau^\\prime \\, \n{\\mbox{\\boldmath $\\cal E$}}_z(\\tau^\\prime ) \\tau^\\prime \\, ,\n\\label{FE2-fs}\n\\end{equation}\nwhere $\\omega_{\\rm pl}^2 = \\kappa g^2 \\Lambda_0^2$.\nTaking a derivative of both sides of this equation with respect to $\\tau$ gives an ordinary differential equation\n\\begin{equation}\n\\frac{1}{\\tau}\\frac{d}{d\\tau}\\left(\\frac{\\tau}{J\\!\\left(\\frac{\\tau}{\\tau_{\\rm iso}}\\right)}\\frac{d{\\mbox{\\boldmath $\\cal E$}}_z(\\tau) }{d\\tau}\\right) = - \\omega_{\\rm pl}^2 {\\mbox{\\boldmath $\\cal E$}}_z(\\tau ) \\, .\n\\label{FE2-fsde}\n\\end{equation}\nwhich is supplemented by the initial conditions ${\\mbox{\\boldmath $\\cal E$}}_z(\\tau_0) = \n{\\mbox{\\boldmath $\\cal E$}}_{z,0}$ and ${\\mbox{\\boldmath $\\cal E$}}_z^\\prime(\\tau_0) = 0$, where the latter\ncondition follows from (\\ref{FE2-fs}) upon setting $\\tau=\\tau_0$.\n\nThe differential equation above is nonlinear and must be solved numerically; however, at late times we can find\nthe asymptotic form of the solution by using\n\\begin{equation}\n\\lim_{\\tau \\rightarrow \\infty} J\\!\\left(\\frac{\\tau}{\\tau_{\\rm iso}}\\right) = \\frac{3 \\pi \\tau_{\\rm iso}}{4 \\tau} \n+ {\\cal O}\\left(\\left(\\frac{\\tau_{\\rm iso}}{\\tau}\\right)^2\\right) ,\n\\end{equation}\nto obtain\n\\begin{equation}\n\\frac{1}{\\tau}\\frac{d}{d\\tau}\\left(\\tau^2 \\frac{d{\\mbox{\\boldmath $\\cal E$}}_z(\\tau) }{d\\tau}\\right) = - \\mu {\\mbox{\\boldmath $\\cal E$}}_z(\\tau ) \\, ,\n\\end{equation}\nwhere $\\mu \\equiv 3 \\pi \\tau_{\\rm iso} \\omega_{\\rm pl}^2\/4$. This differential equation has a solution of the form\n\\begin{equation}\n\\lim_{\\tau \\rightarrow \\infty} \\mbox{\\boldmath $\\cal E$}_z = \\frac{1}{\\sqrt{\\mu\\tau}} \\left[ {\\bf A} J_1\\left(2 \\sqrt{\\mu \\tau}\\right) + {\\bf B} Y_1\\left(2 \\sqrt{\\mu \\tau} \\right) \\right] ,\n\\label{FE2-asymp}\n\\end{equation}\nwhere $J_1$ and $Y_1$ are Bessel functions of the first and second kind, respectively, and $\\bf A$ and $\\bf B$\ncollect undetermined constants that will be matched below. Note that this agrees with\nthe result first obtained in App.~A subsection 2 of Ref.~\\cite{Rebhan:2009ku}.\\footnote{Their result is expressed \nin terms of the longitudinal vector potential. One must compute the longitudinal electric field using \n$E^\\eta = \\Pi^\\eta =\\tau^{-1} \\partial_\\tau A_\\eta$ in order to compare with our result.}\n\n\\section{Results}\n\\label{sect:results}\n\n\\begin{figure}[th!]\n\\begin{center}\n\\subfigure{\\includegraphics[angle=0,width=0.42\\textwidth]{plot_Ez_1c_dampingOFF.pdf}}\\\\\n\\subfigure{\\includegraphics[angle=0,width=0.42\\textwidth]{plot_Ez_10_dampingOFF.pdf}}\\\\\n\\subfigure{\\includegraphics[angle=0,width=0.42\\textwidth]{plot_Ez_100_dampingOFF.pdf}}\n\\end{center}\n\\caption{(Color online) Time dependence normalized of the normalized longitudinal chromoelectric field \n$\\mbox{\\boldmath $\\cal E$}_z$ for (a) constant $\\tau_{\\rm eq}$, (b) time varying $\\tau_{\\rm eq}$ with $4\\pi\\bar\\eta = 1$, \nand (c) time varying\n$\\tau_{\\rm eq}$ with $4\\pi\\bar\\eta = 10$. We turn off the damping manually by setting the damping function\n$D(\\tau,\\tau^\\prime) = 1$. In each plot we show three different initial\nvalues of $\\xi$: $\\xi_0=99$ (dashed line), $\\xi_0=0$ (solid line), and $\\xi_0=-0.99$ (dotted line).\n}\n\\label{fig:plot_ez_nodamp}\n\\end{figure}\n\n\\begin{figure}[th!]\n\\begin{center}\n\\subfigure{\\includegraphics[angle=0,width=0.42\\textwidth]{plot_Ez_1c_dampingON.pdf}}\\\\\n\\subfigure{\\includegraphics[angle=0,width=0.42\\textwidth]{plot_Ez_10_dampingON.pdf}}\\\\\n\\subfigure{\\includegraphics[angle=0,width=0.42\\textwidth]{plot_Ez_100_dampingON.pdf}}\n\\end{center}\n\\caption{(Color online) Time dependence of the normalized longitudinal chromoelectric field $\\mbox{\\boldmath $\\cal E$}_z$ \nfor (a) constant $\\tau_{\\rm eq}$, (b) time varying $\\tau_{\\rm eq}$ with $4\\pi\\bar\\eta = 1$, and (c) time varying\n$\\tau_{\\rm eq}$ with $4\\pi\\bar\\eta = 10$. In each plot we show three different initial\nvalues of $\\xi$: $\\xi_0=99$ (dashed line), $\\xi_0=0$ (solid line), and $\\xi_0=-0.99$ (dotted line).\n}\n\\label{fig:plot_ez_damp}\n\\end{figure}\n\nFor the numerical results we choose a particular direction of the chromolectric field by aligning it initially in the `3'\ndirection, i.e. $\\mbox{\\boldmath $\\cal E$}_z(\\tau_0) = ({\\cal E}_z^{(3)}(\\tau_0),0)$. The evolution keeps \n$\\mbox{\\boldmath $\\cal E$}_z$ in the `3' direction at all times.\nIn Fig.~\\ref{fig:plot_ez_nodamp} we plot the time dependence of the normalized longitudinal chromoelectric field \n${\\cal E}_z^{(3)}$ for (a) constant $\\tau_{\\rm eq}$, (b) time varying $\\tau_{\\rm eq}$ with $4\\pi\\bar\\eta = 1$, \nand (c) time varying $\\tau_{\\rm eq}$ with $4\\pi\\bar\\eta = 10$. We turn off the damping manually by setting the damping \nfunction $D(\\tau,\\tau^\\prime) = 1$. In each plot we show three different initial values of $\\xi$: $\\xi_0=99$ (dashed line), \n$\\xi_0=0$ (solid line), and $\\xi_0=-0.99$ (dotted line). For each $\\xi_0$ we have adjusted $\\Lambda_0$ as described\nin Sec.~\\ref{subsect:background} in order to guarantee that the initial number density is held constant. This figure \ndemonstrates that, although we have varied our assumed value of $\\xi_0$ over a large range corresponding to initially \nextremely prolate ($\\xi_0 = -0.99$) to extremely oblate ($\\xi_0 = 99$), if the results are normalized such that the initial \nnumber densities are held constant, the resulting oscillations are not dramatically different. This is due to the fact that what \nsets the time scale for the plasma oscillations is $\\omega_{\\rm pl}$ and, generally, one has $\\omega_{\\rm pl} \n\\sim n(\\tau_0)\/\\Lambda_0$. However, despite being qualitatively similar, there are important quantitative differences which \nremain.\n\nIn Fig.~\\ref{fig:plot_ez_damp} we plot the time dependence of the normalized longitudinal chromoelectric field \n$\\mbox{\\boldmath $\\cal E$}_z$ for (a) constant $\\tau_{\\rm eq}$, (b) time varying $\\tau_{\\rm eq}$ with $4\\pi\\bar\\eta = 1$, \nand (c) time varying $\\tau_{\\rm eq}$ with $4\\pi\\bar\\eta = 10$. We now include the damping function in the integrand\nof the integro-differential equation. In each plot we show three different initial values of $\\xi$: $\\xi_0=99$ (dashed line), \n$\\xi_0=0$ (solid line), and $\\xi_0=-0.99$ (dotted line). Once again, for each $\\xi_0$ we have adjusted $\\Lambda_0$ as \ndescribed in Sec.~\\ref{subsect:background} in order to guarantee that the initial number density is held constant. From \nthis figure we see that the damping function $D(\\tau,\\tau^\\prime)$ has an extremely important impact on the time \nevolution of the oscillations of uniform longitudinal chromoelectric fields.\\footnote{We made a preliminary study\nof the impact of collisional damping on unstable modes and found that the damping serves only to slightly\nreduce the growth rate of unstable modes. Results from this study will be reported elsewhere.}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[angle=0,width=0.45\\textwidth]{free.pdf}\n\\end{center}\n\\caption{(Color online) Comparison of solution to the ordinary differential equation (\\ref{FE2-fsde}) (thick black line), the \nintegro-differential equation (\\ref{FE2}) (thin red line), and the asymptotic solution (\\ref{FE2-asymp}) (thick gray dashed line) \nin the case of longitudinal free streaming ($\\tau_{\\rm eq}=\\infty$). The inset shows the three results scaled by $\\tau^{3\/4}$.\n}\n\\label{fig:plot_fs_comp}\n\\end{figure}\n\nIn Fig.~\\ref{fig:plot_fs_comp} we compare solutions to the ordinary differential equation (\\ref{FE2-fsde}) (thick black line), the \nintegro-differential equation (\\ref{FE2}) (thin red line), and the asymptotic solution (\\ref{FE2-asymp}) (thick gray \ndashed line) in the case of longitudinal free streaming ($\\tau_{\\rm eq}=\\infty$). The inset shows the three results \nscaled by $\\tau^{3\/4}$ in order to make a more precise comparison of the late time behavior. For the asymptotic solution \nthe unknown constants appearing in (\\ref{FE2-asymp}) were fixed by matching numerically to the full solution at \n$\\tau = 10^4$ fm\/c. As can be seen from this figure, our numerical solution to the integro-differential equation \n(\\ref{FE2}) gives the same result as direct solution of the differential equation (\\ref{FE2-fsde}) and at late times both \nagree well with the asymptotic solution. This gives us confidence that the numerical method used for solution of the \nintegro-differential equation (\\ref{FE2}), in the general case, is reliable.\n\n\\section{Conclusions}\n\\label{sect:concl}\n\nIn this paper we studied the oscillations of a uniform longitudinal chromoelectric field in a dynamically-evolving \nmomentum-space anisotropic background. The required anisotropic hydrodynamics equations were\nobtained from the first two moments on the Boltzmann-Vlasov equation using a spheroidal form for the local\nrest frame one-particle distribution function. The resulting anisotropic hydrodynamics equations provided the proper-time \ndependence of the local transverse momentum scale, $\\Lambda(\\tau)$, and momentum-space anisotropy, $\\xi(\\tau)$.\nWe then expanded the Boltzmann-Vlasov equation to first order in fluctuations and coupled these fluctuations\nto the Maxwell equations. From this procedure we obtained an integro-differential equation which governs\nthe time evolution of a uniform longitudinal chromoelectric field. The integro-differential equation allows for\nan arbitrary time dependence of the scale $\\Lambda(\\tau)$ and momentum-space anisotropy $\\xi(\\tau)$ provided\nthat the system is boost invariant and transversely homogeneous. The integro-differential equation obtained\nalso includes the effect of collisional damping of the oscillation.\n\nHaving obtained the integro-differential equation necessary, we proceeded to solve it numerically for a variety of \ndifferent initial momentum-space anisotropies using two different assumptions for the relaxation time \n$\\tau_{\\rm eq}$. We considered (i) the case of constant $\\tau_{\\rm eq}$, in which case the late-time dynamics is \nthat of ideal hydrodynamics, and (ii) a time-dependent $\\tau_{\\rm eq}$ that is inversely proportional to the local\ninverse average transverse momentum, in which case the late-time dynamics is that of 2nd-order viscous\nhydrodynamics. We showed that for fixed initial number density the effect of time-varying momentum-space\nanisotropy is important but not overwhelming large. However, we found the effect of collisional damping to be\nquite important for the dynamics of stable chromoelectric oscillations.\n\nWe should stress that the results presented here are exploratory in the sense that we have only studied the \ndynamics of a stable uniform chromoelectric field; however, the general method used here could have a wide-ranging\napplication in the study of the dynamics of all stable and unstable modes of a longitudinally expanding QGP. Previous \nstudies in this direction have been restricted to the limiting case of a longitudinally free-streaming anisotropic background \n\\cite{Romatschke:2006wg,Rebhan:2008uj,Rebhan:2009ku}. We were able to check our results\nagainst the free-streaming results obtained by Rebhan and Steineder \\cite{Rebhan:2009ku} and found that\nwe are able to reproduce their results for the dynamics of a uniform longitudinal chromoelectric field. In addition, \nwe were able to express the free-streaming evolution of a uniform chromoelectric field as an ordinary differential \nequation, albeit a highly-nonlinear one. It would be very interesting to see if the results obtained here could be used to \nobtain modified hard-loop equations of motion that would allow one to simulate the full non-Abelian \ndynamics of stable and unstable modes in a realistically time-evolving anisotropic background. Work along these lines \nis in progress.\n\n\\begin{acknowledgments}\n\nW.F. and R.R. were supported by the Polish Ministry of Science and Higher Education under Grant No.~N N202 263438. \nM.S. was supported by NSF grant No.~PHY-1068765 and the Helmholtz International Center for FAIR LOEWE program.\n\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:paper_intro}\n\nClustering is a fundamental tool in machine learning and data analysis. It aims to partition a given set of objects into \\emph{clusters} in such a way that similar objects end up in the same cluster. \nThe classical way of approaching the clustering problem is via the $k$-means\\, formulation. \nIn this formulation, one works with a set $X$ consisting of $n$ points in the $d$-dimensional Euclidean space $\\mathbb{R}^d$ and the objective is to output a set $C$ consisting of $k$ centers so as to minimize the sum of squared distances of points in $X$ to $C$, i.e.,\n\\[\n\\phi(X, C) = \\sum_{x \\in X} \\min_{c \\in C} \\Vert x - c \\Vert^2.\n\\]\nAssigning each point to its closest cluster center naturally induces a partition of the points where nearby points tend to end up in the same partition. \n\n\\paragraph{$k$-means\\, with Outliers} One major drawback of $k$-means\\, in practice is its sensitivity to outliers \\cite{gupta2017local}. \nThis motivates optimizing a more robust version of the $k$-means\\, objective.\nProbably the simplest such formulation is the \\emph{$k$-means\\, with outliers} formulation \\cite{charikar2001algorithms}. In this formulation, one additionally receives a number $z \\in \\{0,1,\\ldots,n\\}$. The aim is to find a set $C$ consisting of $k$ cluster centers and additionally a set $X_{out} \\subseteq X$ consisting of $z$ \\emph{outliers} so as to optimize the cost of \\emph{inliers} $X_{in} := X \\setminus X_{out}$. \nThat is, we optimize \n$\n\\min_{X_{out}, C} \\phi(X \\setminus X_{out}, C). \n$ \n\n\\paragraph{Known Results for $k$-means\\, with Outliers} It is known that the problem can be approximated up to a constant factor in polynomial time by rounding a certain linear program \\cite{chen2008constant, Krishnaswamy2018}. However, the complexity of these known algorithms is a large polynomial. Moreover, as we observe in \\cref{sec:paper_nk2z}, every constant factor approximation algorithm that outputs \\emph{exactly} $z$ outliers needs to perform $\\Omega(z^2)$ queries in the query model.\n\nThis motivates to weaken our requirements and look for fast algorithms that are allowed to output slightly more than $z$ outliers. \nThere is a line of work that makes progress toward such algorithms \\cite{charikar2001algorithms, meyerson2004k, gupta2017local, Guha_distributed, bhaskara2020, nkmeans}. However, to the best of our knowledge, known algorithms either need to output $A z$ outliers, for some large constant $A \\gg 1$ to obtain a reasonable approximation guarantee, or they suffer from at least a $\\Omega(z^2)$ running time and, hence, to be truly applicable even for large $z$ they need to be sped up by coreset constructions \\cite{feldman2011unified, huang2018epsilon}.\n\n\\paragraph{Our Contribution} In this work we aim to further close the gap between theory and practice. \nWe show how to adapt a number of classical sampling-based $k$-means algorithms to the setting with outliers. The main idea of the adaptations is a reduction to the variant of the problem with penalties described below. This is a known approach \\cite{charikar2001algorithms}, but previous reductions of sampling based algorithms \\cite{bhaskara2020} necessarily need to output $Az$ outliers for some large constant $A \\gg 1$, while we obtain algorithms that need to output only $(1+\\varepsilon)z$ outliers. \nMore concretely, our contribution is threefold. \n\nFirst, building on ideas of \\cite{lattanzi2019better}, we design a simple sampling-based algorithm with running time $\\tilde{O}(nk\/\\varepsilon)$. It outputs an $O(1\/\\varepsilon)$-approximate solution while declaring at most $(1+\\varepsilon)z$ points as outliers. \nThis shows that sampling based algorithms that are known to be fast and have good practical performance can also achieve strong theoretical guarantees. \n\n\nSecond, we devise distributed algorithms for $k$-means with outliers where each machine needs to send only $\\tilde{O}(k\/\\varepsilon)$ bits. Our construction achieves an $O(1)$-approximation while outputting $(1+\\varepsilon)z$ outliers. Moreover, each machine only needs to perform polynomial-time computation. This improves on \\cite{guo2018} who achieve an $(1+\\varepsilon)$-approximation while outputting the same number of outliers as our algorithm, but their computation time is exponential. \n\nThird, we show that one can achieve an $O(1)$-approximation guarantee while discarding $O(z)$ outliers in time $\\tilde{O}(k^2 \\cdot n\/z )$.\nThis is done by speeding-up sampling with the Metropolis-Hastings algorithm as done in \\cite{bachem2016approximate, bachem2016fast} together with additional ideas. \nThis result is complemented by a matching lower bound of $\\Omega(k^2 \\cdot (n\/z) )$ for $k$-means\/$k$-median\/$k$-center algorithms that work for an arbitrary metric space accessed by distance queries.\nThis improves on \\cite{meyerson2004k,huang2018epsilon} who give algorithms in this setting with a running time of $\\tilde{O}\\left( k^2 \\cdot \\left( n \/ z \\right)^2\\right)$. \nThis is a significant improvement for $z \\ll n$. \n\n\\paragraph{Roadmap.} We overview the related work in \\cref{sec:paper_previous_work} and our approach in \\cref{sec:paper_warmup}. \nThen, we present our contributions in \\Cref{sec:paper_localsearch,sec:paper_distributed,sec:paper_nk2z} and our experiments in \\cref{sec:paper_experiments}. Technical details are mostly deferred to appendices. \n\n\n\n\n\\paragraph{Notation}\nWe define $\\phi(x, C) = \\min_{c \\in C} \\Vert x - c \\Vert^2$\nand set $\\phi(X, C) = \\sum_{x \\in X} \\phi(x, C)$. \nSimilarly, we define $\\tau_\\Theta(x, C) = \\min(\\Theta, \\phi(x, C))$ and $\\tau_\\Theta(X, C) = \\sum_{x \\in X} \\tau_\\Theta(x, C)$. \nWe call an algorithm an $(\\alpha, \\beta)$-approximation if it outputs a set of $k$ centers $C$ and a set of $z$ outliers $X_{out}$ such that $\\phi(X \\setminus X_{out}, C) \\le \\alpha \\textrm{OPT} = \\alpha \\phi(X \\setminus X_{out}^*, C^*)$, where $\\textrm{OPT}$ is the cost of a fixed optimal clustering $C^*$ with a set of outliers $X_{out}^*, |X_{out}^*| = z$. \nMoreover, we define $X_{in} := X \\setminus X_{out}$ and $X^*_{in} := X \\setminus X^*_{out}$. \n\n\n\n\n\\section{Previous Work}\n\\label{sec:paper_previous_work}\n\n\n\n\n\n\n\\paragraph{$k$-means\\, Problem}\nIt is well-known that finding the optimal solution is NP-hard \\cite{aloise2009np, mahajan2009planar}. Currently the best known approximation ratio is roughly 6.36 \\cite{ahmadian2019better} and an approximation ratio of $(1+\\varepsilon)$ can be achieved for fixed dimension $d$ \\cite{friggstad2019local} or fixed $k$ \\cite{kumar2004simple}.\nFrom the practical perspective, Lloyd's heuristic \\cite{lloyd1982least, lloyd_survey} is the algorithm of choice. \nAs Lloyd's algorithm converges only to a local optimum, it requires a careful seeding to achieve good performance. \nThe most popular seeding choice is the \\texttt{$k$-means$++$\\,} seeding \\cite{arthur2007k, ostrovsky2013effectiveness}. \nIn \\texttt{$k$-means$++$\\,} seeding (\\cref{alg:kmp_original}) one chooses the first center uniformly at random from the set of inputs points $X$. \nIn each of the following $k-1$ steps, one samples a point in $x \\in X$ as a new center with probability proportional to its current cost $\\phi(x, C)$. The seeding works well in practice and a theoretical analysis shows that even without running Lloyd's algorithm subsequently it provides an expected $O(\\log k)$-approximation to the $k$-means\\, objective \\cite{arthur2007k}. \n\n\\begin{algorithm}[]\n\t\\caption{\\texttt{$k$-means$++$\\,} seeding}\n\t\\label{alg:kmp_original}\n\tInput: $X$, $k$\n\t\\begin{algorithmic}[1]\n\t\t\\STATE Uniformly sample $c \\in X$ and set $C = \\{ c \\}$.\n\t\t\\FOR{$i \\leftarrow 2, 3, \\dots, k$}\n\t\n\t\t\\STATE Sample $c \\in X$ with probability $\\phi(c, C) \/ \\phi(X, C)$ and add it to $C$.\n\t\n\t\t\\ENDFOR\n\t\t\\STATE \\textbf{return} $C$\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\n\n\n\n\\paragraph{$k$-means\\, with Outliers}\n\n\nThere is a growing body of research related to the $k$-means\\, with outliers problem. \nOn the practical side, Lloyd's algorithm is readily adapted to the noisy setting \\cite{kmeans-}, but the output quality still remains dependent on the initial seeding. \nOn the theoretical side, constant approximation algorithms based on the method of successive local search \\cite{chen2008constant, Krishnaswamy2018} are known to provide a constant approximation guarantee. \nHowever, their running time is a large polynomial in $n$.\nWe are interested in fast algorithms that are allowed to output slightly more than $z$ outliers. Several algorithms have been proposed \\cite{charikar2001algorithms, meyerson2004k, gupta2017local, Guha_distributed, bhaskara2020, nkmeans}, but they either need at least $\\Omega(z^2)$ time or they need to output at least $Cz$ outliers for some $C \\gg 1$. \nThe algorithms can in general be sped up by coreset constructions \\cite{feldman2011unified, gupta2017local, huang2018epsilon, nkmeans}. \nHowever, there is still need for fast and simple algorithms with strong guarantees. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\paragraph{$k$-means\\, with (Uniform) Penalties}\n\n$k$-means\\, with penalties is a different way of handling outliers introduced in the seminal paper of Charikar et al.~\\cite{charikar2001algorithms}. \nIn the version of the problem with uniform penalties, we are given some positive number $\\Theta$ and the goal is to output a set of centers $C = \\{c_1, c_2, \\dots, c_k\\}$ so as to minimize the expression $\\tau_\\Theta(X, C) = \\sum_{x\\in X} \\min\\left( \\Theta, \\min_{c\\in C} \\Vert x - c \\Vert^2 \\right)$. \nThat is, the cost of any point is bounded by a threshold $\\Theta$. \n\nIt turns out that it is usually much simpler to work with $k$-means\\, with penalties than $k$-means\\, with outliers. \nThis is quite helpful because results for $k$-means\\, with penalties can be turned into results for $k$-means\\, with outliers \\cite{charikar2001algorithms, guo2018, bhaskara2020, bhaskara2020online}. \nWe also take this approach in this paper. We formalize the reduction in the next section and also present our improvement of it for sampling based algorithms. \n\n\n\n\\section{Warmup: Reducing $k$-means\\, with Outliers to $k$-means\\, with Penalties}\n\\label{sec:paper_warmup}\n\nOur approach is based on reducing the problem of $k$-means with outliers to the problem of $k$-means with penalties. \nIn this section, we first review how one can reduce the problem of $k$-means with outliers to the problem of $k$-means with penalties. \nThen we review an instance of this reduction by \\cite{bhaskara2020} and provide a refined result which is the starting point of our work. \nWe note that this can be seen as an instantiation of a general principle in optimization where one replaces a constraint with a penalty function known as Lagrangian relaxation \\cite{boyd2004convex}. \n\n\\subsection{Review of the Previously Known Reduction} \nWe start by giving the following lemma that formalizes how an $\\alpha$-approximate solution to the $k$-means\\, with penalties objective can be used to obtain an algorithm providing an $(O(\\alpha),O(\\alpha))$-approximate solution to the $k$-means\\, with outliers objective. \n\n\\begin{lemma}[\\cite{charikar2001algorithms}]\n\t\\label{lem:penalties_to_outliers}\n\tLet $C$ be an $\\alpha$-approximate solution to the $k$-means\\, with penalties objective with penalty $\\textrm{OPT}\/(2z) \\le \\Theta \\le \\textrm{OPT}\/z$. Let $X_{out}$ denote the set of points $x$ for which $\\tau_\\Theta(x, C) = \\Theta$. \n\tThen, it holds that $\\phi(X \\setminus X_{out},C) = O(\\alpha \\textrm{OPT})$ and $|X_{out}| = O(\\alpha z)$. \n \n\\end{lemma}\n\\begin{proof} \n Note that the optimal solution for $k$-means\\, with penalties has a cost upper bounded by $\\phi(X \\setminus X_{out}^*, C^*) + |X_{out}^*|\\Theta = \\textrm{OPT} + z\\Theta$.\n\tAs $C$ is an $\\alpha$-approximate solution to the $k$-means\\, with penalties objective, we have \n\t$\\phi(X \\setminus X_{out}, C) \\leq \\tau_\\Theta(X, C) \\leq \\alpha \\left(\\textrm{OPT} + z \\Theta \\right) = O(\\alpha \\textrm{OPT}).$\n\tMoreover, paying $\\Theta$ for every $x\\in X_{out}$ implies $|X_{out}| \\leq \\frac{\\tau_\\Theta(X, C)}{\\Theta} \\le \\frac{\\alpha(\\textrm{OPT} + z \\Theta) }{\\Theta} = O(\\alpha z).$\n\\end{proof}\n\nWe remark that the penalty $\\Theta$ depends on $\\textrm{OPT}$, so the algorithm for $k$-means with outliers needs to try this reduction for all $O(\\log (n\\Delta^2)) = O(\\log n)$ powers of $2$ between $1$ and $n\\Delta^2$. \n\n\nThis reduction is very helpful as it allows us to easily adapt sampling-based algorithms like \\texttt{$k$-means$++$\\,} to the setting with penalties since their analysis generalises to this setting. \nFor the case of \\texttt{$k$-means$++$\\,}, this was shown in \\cite{bhaskara2020, li_penalties2020} in the following theorem. \n\n\\begin{algorithm}[h]\n\t\\caption{\\texttt{$k$-means$++$\\,} (over)seeding with penalties}\n\t\\label{alg:kmp}\n\tInput: A set of points $X$, $k$, $\\ell$, threshold $\\Theta$\n\t\\begin{algorithmic}[1]\n\t\t\\STATE Uniformly sample $c \\in X$ and set $C = \\{ c \\}$.\n\t\t\\FOR{$i \\leftarrow 2, 3, \\dots, \\ell$}\n\t\n\t\t\\STATE Sample $c \\in X$ with probability $\\tau_\\Theta(c, C) \/ \\tau_\\Theta(X, C)$ and add it to $C$.\n\t\t\\ENDFOR\n\t\t\\STATE \\textbf{return} $C$\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{restatable}{theorem}{kmeanspp}[\\cite{arthur2007k, bhaskara2020, li_penalties2020}]\n\t\\label{thm:kmpp}\n\tSuppose we run \\cref{alg:kmp} for $\\ell = k$ steps. \n\tThen, the output set $C$ is an $O(\\log k)$-approximation to the $k$-means with penalty $\\Theta$ objective, in expectation. \n\t\n\n\\end{restatable}\n\\begin{proof}[Proof sketch]\nThe analysis of \\cite{arthur2007k} proves this guarantee for \\cref{alg:kmp_original} for any $\\phi(a,b) := d^2(a,b)$ such that $d$ is a metric. As the distance function $d'(a,b) = \\min(d(a,b), \\sqrt{\\Theta})$ still defines a metric, one can thus directly use their analysis to prove \\cref{thm:kmpp}. \n\\end{proof}\n\nMore details are given in \\cref{sec:appendix_lemmas}. \nBy \\cref{thm:kmpp}, plugging in \\cref{alg:kmp} into \\cref{lem:penalties_to_outliers} gives\nan $(O(\\log k),O(\\log k))$-approximate algorithm for $\\texttt{$k$-means$++$\\,}$ with outliers. Moreover, running \\cref{alg:kmp} for $O(k)$ steps results in an $O(1)$-approximation to the $k$-means objective in metric spaces \\cite{aggarwal2009adaptive,bhaskara2020}. \nThis gives the following tri-criteria approximation via \\cref{lem:penalties_to_outliers}: we get an $(O(1), O(1))$-approximation algorithm that needs to use $O(k)$ centers.\n\n\\subsection{Our Improved Reduction}\nThe starting point of our work is the following improvement of the tri-criteria result from the previous subsection that enables us to get a constant factor approximation algorithm that outputs only $(1+\\varepsilon)z$ outliers, which is not possible by using \\cref{lem:penalties_to_outliers} as a black box. The catch is that we need to use $O(k\/\\varepsilon)$ centers. \n\n\n\\begin{theorem}\n\\label{thm:aggarwal_informal}\n Running \\cref{alg:kmp} for $\\ell = O(k \/ \\varepsilon )$ iterations and $\\textrm{OPT} \/ (2\\varepsilon z) \\le \\Theta \\le \\textrm{OPT} \/ (\\varepsilon z)$ results in a set $C$ with $\\tau_\\Theta(X^*_{in}, C) = 20 \\textrm{OPT}$, with positive constant probability. \n\\end{theorem}\n\\begin{proof}[Proof sketch (full proof in \\cref{sec:appendix_tricriteria})]\nFor an optimal set of centers $C^* = \\{c_1^*, \\dots, c_k^*\\}$, we define $X_i^* \\subseteq X_{in}^*$ as the subset of points $x \\in X_{in}^*$ with $c_i^* = \\argmin_{c^* \\in C^*} \\phi(x, c^*)$, where ties are broken arbitrarily. \nFix one iteration of \\cref{alg:kmp} and let $C$ be its current set of centers. \nWe refer to a cluster $X_i^*$ as \\emph{unsettled} if $\\tau_\\Theta(X_i^*, C) \\ge 10 \\tau_\\Theta(X_i^*, C^*)$. \n\n\n\nSuppose that $\\tau_\\Theta(X^*_{in}, C) \\ge 20\\textrm{OPT}$, since otherwise we are already done. We sample a new point $c$ from $X^*_{in}$ with probability\n\\[\n\\frac{\\tau_\\Theta(X^*_{in}, C)}{\\tau_\\Theta(X^*_{in}, C) + \\tau_\\Theta(X^*_{out}, C)} \\ge \\frac{20\\textrm{OPT}}{20\\textrm{OPT} + \\tau_\\Theta(X^*_{out}, C)} \\ge \\frac{20\\textrm{OPT}}{20\\textrm{OPT} + \\textrm{OPT}\/\\varepsilon} \\ge \\frac{\\varepsilon}{2},\n\\]\nwhere we used $\\tau_\\Theta(X^*_{in}, C) \\ge 20\\textrm{OPT}$, $\\tau_\\Theta(X^*_{out}, C) \\le |X^*_{out}|\\Theta \\le \\textrm{OPT}\/\\varepsilon$, and that $\\varepsilon$ is small enough. \nMoreover, the cost of all settled clusters is bounded by $10 \\textrm{OPT}$, hence given that $c$ is sampled from $X^*_{in}$, the probability that it is sampled from an unsettled cluster is at least\n\\[\n\\frac{\\tau_\\Theta(X^*_{in}, C) - 10\\textrm{OPT}}{\\tau_\\Theta(X^*_{in}, C)} \n\\ge \\frac{20 \\textrm{OPT} - 10 \\textrm{OPT}}{20\\textrm{OPT}}\n= \\frac12,\n\\]\nwhere we again used $\\tau_\\Theta(X^*_{in}, C) \\ge 20\\textrm{OPT}$. \nGiven that $c$ is sampled from an unsettled cluster according to the $\\tau_\\Theta$-distribution, \\cref{cor:10apx} in \\cref{sec:appendix_lemmas} tells us that the cluster becomes settled with probability at least $\\frac15$. \nHence, we make a new cluster settled with probability at least $(\\varepsilon\/2) \\cdot \\frac12 \\cdot \\frac15 = \\varepsilon\/20$. \n\nBy a standard concentration argument, this implies that after $O(k\/\\varepsilon)$ steps of the algorithm, either all clusters are settled with positive constant probability, and hence we have $\\tau_\\Theta(X_{in}^*, C) \\le 10\\textrm{OPT}$ and we are done, or during the course of the algorithm, the condition $\\tau_\\Theta(X_{in}^*, C) \\ge 20 \\textrm{OPT}$ stopped being true, in which case we are again done. \n\\end{proof}\n\n\nNote that setting $\\varepsilon = 1$ results in the same tri-criteria result we discussed in the previous subsection: This holds as $\\tau_\n\\Theta(X^*_{in}, C) = O(\\textrm{OPT})$ implies that $\\tau_\\Theta(X, C) \\le \\tau_\\Theta(X^*_{in}, C) + z\\Theta = O(\\textrm{OPT}\/\\varepsilon)$. \n\nHowever, we can get more out of \\cref{thm:aggarwal_informal}: note that as $\\tau_\\Theta(X^*_{in}, C) = O(\\textrm{OPT})$, for the set $X_{out}$ defined as those $x \\in X$ with $\\tau_\\Theta(x, C) = \\Theta$, we have that $|X_{out} \\cap X_{in}^*| = \\frac{O(\\textrm{OPT})}{\\Theta} = O(\\varepsilon z)$. \nHence, setting $X_{out}$ as our output set of outliers gives on one hand\n\\[\n|X_{out}| \\le |X_{out}^*| + |X_{out} \\cap X_{in}^*| = z + O(\\varepsilon z). \n\\]\nOn the other hand, we can bound\n\\[\n\\phi(X \\setminus X_{out}, C) \n= \\tau_\\Theta(X \\setminus X_{out}, C) \n\\le \\tau_\\Theta(X , C) = O(\\textrm{OPT} + z \\Theta) = O(\\textrm{OPT} \/ \\varepsilon).\n\\]\n\nHence, we obtain an $O(1\/\\varepsilon)$ approximation guarantee while outputting just $(1+\\varepsilon)z$ outliers.\\footnote{In this particular case, we can even get an $O(1)$-approximation guarantee by labelling the furthest $(1+O(\\varepsilon))z$ points as outliers, since the set $X_{out}\\cup X_{out}^*$ has size at most $(1+O(\\varepsilon))z$ and $\\phi(X \\setminus (X_{out}\\cup X_{out}^*), C) \\le \\tau_\\Theta(X_{in}^*, C) = O(\\textrm{OPT})$. }\nBy itself, \\cref{thm:aggarwal_informal} is still not satisfactory, as it requires us to oversample the number of centers by a factor of $O(1\/\\varepsilon)$. \nHowever, we next show three directions of improvement that lead to more interesting results. \n\n\n\n\\iffalse\n\\section{Warm-up: reducing outliers to penalties}\n\n\n\n\\paragraph{Outliers to penalties \\cite{bhaskara2020}}\n\nOur approach is based on the one used e.g in \\cite{charikar2001algorithms, bhaskara2020} to turn an algorithm for $k$-means\\, with penalties to $k$-means\\, with outliers.\nAlthough it was not noted in the original paper, the technique readily implies several rather interesting results. \nHere, we explain the technique and its applications. Later, we show how to refine it to prove \\cref{thm:localsearch}. \n\n\\begin{lemma}\n\\label{lem:penalties_to_outliers}\\textcolor{red}{reader friendly statement and proof}\nIf there is an $\\alpha$-approximation algorithm $\\mathcal{A}(n, k, \\Theta)$ for $k$-means\\, with penalty $\\Theta$ running in time $T(n,k)$, there is also an $\\alpha$-approximation algorithm $\\mathcal{B}(n, k , z)$ for $k$-means\\, with $z$ outliers that outputs $O(z)$ outliers and runs in time $\\tilde{O}(T(n,k) + n)$. \n\\end{lemma}\n\\begin{proof}\nWe construct $\\mathcal B$ as follows. \nLet $\\textrm{OPT}$ be the optimal value of the $k$-means\\, objective for $z$ outliers. \nWe use the doubling search to get a sequence of guesses $\\textrm{OPT}^\\textrm{guess}_i = 2^i$ for $i \\in \\{1, 2, \\dots, \\log(n\\Delta^2) \\}$. \nFor each $\\textrm{OPT}^\\textrm{guess}_i$ we define $\\Theta_i = \\frac{\\beta \\textrm{OPT}^\\textrm{guess}_i}{z}$ and run $\\mathcal A(n, k, \\Theta_i)$ that outputs a set of $k$ centers $C_i$. \nWe define $X_i^{\\textrm{in}} \\subseteq X$ as the set of points $x$ for which $\\phi(x, C) < \\Theta$ and $X_i^{\\textrm{out}} = X \\setminus X_i^{\\textrm{in}}$. \nThe output of $\\mathcal B$ is $X_i$ that minimizes $\\phi(X_i^{\\textrm{in}}, C_i)$, while $|X_i^\\textrm{out}| \\le ? \\alpha z$. \n\\end{proof}\n\nThe usefulness of the reduction relies on the fact that many sampling based algorithms like \\texttt{$k$-means$++$\\,} can be readily adapted to the setting with penalties while their analysis stays essentially the same. \nFor the case of \\texttt{$k$-means$++$\\,} this was proven in \\cite{li_penalties2020, bhaskara2020}. \n\n\\begin{algorithm}[h]\n\t\\caption{\\texttt{$k$-means$++$\\,} (over)seeding with penalties}\n\t\\label{alg:kmp}\n\tInput: A set of points $X$, $k$, $\\ell$, threshold $\\Theta$\n\t\\begin{algorithmic}[1]\n\t\t\\STATE Uniformly sample $c \\in X$ and set $C = \\{ c \\}$.\n\t\t\\FOR{$i \\leftarrow 2, 3, \\dots, \\ell$}\n\t\n\t\t\\STATE Sample $c \\in X$ with probability $\\tau_\\Theta(c, C) \/ \\tau_\\Theta(X, C)}$ and add it to $C$.\n\t\t\\ENDFOR\n\t\t\\STATE \\textbf{return} $C$\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{restatable}{theorem}{kmeanspp}[? in \\cite{li_penalties2020}, ? in \\cite{bhaskara2020}]\n\\label{thm:kmpp}\nSuppose we run \\cref{alg:kmp} for $\\ell = k$ steps. \nThen the output satisfies\n\\[\n\\textrm{\\textbf{E}}[\\tau_\\Theta(X, C)] = O(\\lg k) \\tau_\\Theta(X, C^*)\n\\]\n\\end{restatable}\n\n\n\nOur first contribution is a new analysis of \\cref{alg:kmp} for $\\ell = (1+\\varepsilon)k$: we prove that setting $\\ell = (1+\\varepsilon)k$ results in expected $O(\\lg 1\/\\varepsilon)$ approximation even if one generalizes the costs from $\\phi$ to $\\tau_\\Theta$ for any $\\Theta$. \nThis improves on a result of Wei \\cite{wei2016constant}, also proven in \\cite{li_penalties2020} for general $\\tau$, that sampling $(1+\\varepsilon)k$ centers results in expected $O(1\/\\varepsilon)$ approximation.\n\nThis result is probably most interesting in the setting $\\Theta = \\infty$, i.e., for the cost function $\\phi$, since our result is tight in the sense that if we set $\\varepsilon < 1\/k$ we recover \\cref{thm:kmpp}, whose analysis is known to be tight \\cite{arthur2007k, bhattacharya2016tight, brunsch2013bad}. \n\n\n\nSince our proof in \\cref{sec:kmpp} works for any $\\tau_\\Theta$ and not just $\\phi$, we may apply \\cref{lem:penalties_to_outliers} to get a simple algorithm that samples $(1+\\varepsilon)k$ centers and provides a $(O(\\lg1\/\\varepsilon), O(\\lg1\/\\varepsilon))$-approximation guarantee. \nThis is an example of a simple use of the ``outliers to penalties'' reduction. \n\nIn the next section we describe, how running \\cref{alg:kmp} with $\\ell = O(k\/\\varepsilon)$ can help us in getting simple $(O(1\/\\varepsilon), 1+\\varepsilon)$-approximation sampling based algorithms. \n\n\n\n\n\nIn this section, we show that running the \\texttt{$k$-means$++$\\,} algorithm with correctly chosen penalty $\\Theta$ and number of steps $\\ell$, we can obtain $(O(1\/\\varepsilon), 1+\\varepsilon)$-approximation guarantee. \nThe basic result that we later generalize in several directions is the following. \n\n\n\n\\begin{theorem}\n\\label{thm:aggarwal_informal}\n\\cref{alg:kmp} run with $\\ell = O(\\frac{k}{\\varepsilon} \\log\\frac{1}{\\delta})$ and $\\Theta = \\frac{\\textrm{OPT}}{\\varepsilon z}$ outputs a set $C$ with $\\tau_\n\\Theta(X^*_{in}, C) = O(\\textrm{OPT})$ with probability at least $1-\\delta$. \n\\end{theorem}\n\nA simple proof is deferred to \\cref{sec:distributed}. \n\nLet us first compare \\cref{thm:aggarwal_informal} with \\cref{thm:bicriteria_kmpp}. \\cref{thm:bicriteria_kmpp} would in this case only guarantee that $\\tau_\\Theta(X,C) = O(\\tau_\\Theta(X, C^*)) = O(\\phi(X^*_{in}, C^*) + z \\cdot \\Theta) = O(\\textrm{OPT} \/ \\varepsilon)$. \n\\cref{thm:aggarwal_informal} guarantees more in this case. First, $\\tau_\\Theta(X^*_{in}, C) = O(\\textrm{OPT})$ also implies that $\\tau_\\Theta(X, C) = \\tau_\\Theta(X^*_{in}, C) + \\tau_\\Theta(X^*_{out}, C) = O(\\textrm{OPT} + z \\cdot \\Theta) = O(\\textrm{OPT}\/\\varepsilon)$. \nSecond, similarly as in \\cref{lem:penalties_to_outliers}, we can define a set $X' \\subseteq X^*_{in}$ of points $x$ in $X^*_{in}$ for which $\\tau_\\Theta(X') = \\Theta$. \nWe then have $|X'| \\le \\frac{\\tau_\\Theta(X^*_{in}, C)}{\\Theta} = O(\\textrm{OPT} \/ \\Theta) = O( \\varepsilon z)$. This implies $\\phi(X^*_{in} \\setminus X', C) = \\tau_\\Theta(X^*_{in} \\setminus X', C) = O(\\textrm{OPT})$, i.e., up to a small set $X'$ we can also bound $\\phi(X^*_{in}, C)$ by $O(\\textrm{OPT})$. \n\nBy itself, \\cref{thm:aggarwal_informal} is not very impressive, as it requires us to oversample the number of centers by a factor of $O(1\/\\varepsilon)$. However, we next show three directions of improvements that lead to interesting results. \n\n\\begin{itemize}\n \\item First, we use \\cref{thm:aggarwal_informal} to settle the time-complexity of the problem of $(O(1), O(1))$-approximating $k$-means with outliers. By \n \\item\n \\item\n\\end{itemize}\n\n\n\n\n\n\n\n\n\n\\fi\n\n\n\n\n\n\n\n\n\n\n\\section{Fast Sequential Algorithm}\n\\label{sec:paper_localsearch}\n\nIn this section, we present a simple sequential sampling-based algorithm for $k$-means with outliers that achieves an $O(1\/\\varepsilon)$ approximation and outputs $(1+\\varepsilon)z$ outliers: \n\\begin{theorem}\n\\label{thm:lattanzi_informal}\nFor every $ 0 < \\varepsilon < 1$, there exists an $(O(1\/\\varepsilon),1+\\varepsilon)$-approximation algorithm for $k$-means with outliers with running time $\\tilde{O}(nk\/\\varepsilon)$.\n\\end{theorem}\nThe algorithm is based on ideas of a recent paper of Lattanzi and Sohler \\cite{lattanzi2019better} who proposed augmenting \\texttt{$k$-means$++$\\,} with $\\tilde{O}(k)$ \\emph{local search} steps \\cite{kanungo2004local} as follows: \ntheir algorithm first invokes \\texttt{$k$-means$++$\\,} to obtain an initial set of $k$ centers. \nAfterwards, in each local search step, the algorithm samples a $(k+1)$-th point from the same distribution as \\texttt{$k$-means$++$\\,}. \nAfter sampling that point, the algorithm iterates over all $k+1$ current centers and takes out the one whose deletion raises the cost the least (see \\cref{alg:localsearch}). \nRunning \\texttt{$k$-means$++$\\,} followed by $O( k)$ steps of \\texttt{Local-search++ } is known to yield an $O(1)$-approximation for the cost function $\\phi = \\tau_\\infty$ \\cite{choo2020kmeans}, with a positive constant probability. \n\n\n\n\\begin{algorithm}[h]\n\t\\caption{One step of \\texttt{Local-search++ } }\n\t\\label{alg:localsearch}\n\t{\\bfseries Input:} $X$, $C$, threshold $\\Theta$\n\t\\begin{algorithmic}[1]\n\t\t\\STATE Sample $c \\in X$ with probability $\\tau_\\Theta(c, C) \/ \\tau_\\Theta(X, C)$\n\t\t\\STATE $c' \\leftarrow \\argmin_{d \\in C \\cup \\{c\\}} \\tau_\\Theta(X, C \\setminus \\{ d \\} \\cup \\{ c \\})$\n\t\n\t\n\t\n\t\t\\STATE \\textbf{return} $C \\setminus \\{ c' \\} \\cup \\{ c \\}$\n\t\\end{algorithmic}\n\\end{algorithm}\n\nWe get \\cref{thm:lattanzi_informal} by proving that $\\tilde{O}(k\/\\varepsilon)$ iterations of \\cref{alg:localsearch} with $\\textrm{OPT}\/(2\\varepsilon z) \\le \\Theta \\le \\textrm{OPT}\/(\\varepsilon z)$ result in an $(O(1\/\\varepsilon), 1+\\varepsilon)$-approximation guarantee. \nThe analysis deals with new technical challenges and is deferred to \\cref{sec:appendix_localsearch}. \nWe note that with minor changes to the original analysis of Lattanzi and Sohler, one can show that their result generalizes for arbitrary $\\tau_\\Theta$, similarly as \\cref{thm:kmpp}, and, hence, by \\cref{lem:penalties_to_outliers} one obtains an $(O(1),O(1))$-approximation algorithm.\nOur refined analysis, crucially, does \\emph{not} use \\cref{lem:penalties_to_outliers} as a black box. Instead, we use the idea of the lemma as a building stone for the rather intricate analysis of the algorithm. \n\nThe intuition behind our proof of \\cref{thm:lattanzi_informal} comes from \\cref{thm:aggarwal_informal}: the difference between running $k$-means++ for $O(k\/\\varepsilon)$ steps and the local search algorithm we described above is that the local search algorithm additionally removes one point after each sampling step. \nOne can still hope that the increase in cost due to the removals is dominated by the decrease in cost due to the newly sampled center making an unsettled cluster settled, as we have seen in the analysis of \\cref{thm:aggarwal_informal}. This is indeed the case. \nWe note that a local search based algorithm was also considered by \\cite{gupta2017local}, though with substantially weaker guarantees than our algorithm. \n\n\n\n\n\\iffalse\n\\textcolor{red}{fill in nice explanation of the algorithm}\nAs mentioned above, our algorithm is based on the idea to first optimize the $k$-means\\, with penalties objective for a suitable $\\Theta$. The resulting solution can then be converted to a solution for the $k$-means\\, with outliers objective by declaring all points whose contribution to the overall cost is $\\Theta$ as outliers. Depending on the choice of $\\Theta$, one obtains a trade-off in terms of the number of points that get declared as outliers and the resulting approximation guarantee. The value for $\\Theta$ that achieves the desired trade-off for our purposes depends both on the cost of an optimal solution for $k$-means\\, with outliers and $\\varepsilon$. Unlike $\\varepsilon$, the cost of an optimal solution is unknown in advance. However, $\\Theta$ only needs to be chosen \"correctly\" up to a constant factor. Hence, it suffices to guess the cost of an optimal solution up to a constant factor, which is done in a brute-force way by the outer-loop of \\cref{alg:noisylocalsearch}. \n\n\\fi\n\n\n\\iffalse\n\\paragraph{Our approach}\n\nAs our argument is too long to be included, we give here the overview of it. \nWe rely on two propositions that both tell us that one step of \\texttt{Local-search++ } improves the cost of our solution by a significant amount, with constant probability. \n\nThe first proposition is a simple adaptation of the original analysis of \\texttt{Local-search++ } \\cite{lattanzi2019better} and we prove it to get better running time guarantees and for completeness. \n\n\\begin{restatable}{proposition}{propone}\n\nLet $C$ denote the current set of candidate centers. Let $\\varepsilon \\in (0,1]$ be arbitrary and $\\beta := \\frac{300}{\\varepsilon}$. Suppose that $\\cost{X}{C} \\ge100\\beta \\cdot \\textrm{OPT}$ and $\\Theta \\leq 2 \\frac{\\beta \\textrm{OPT}}{z}$. Then, with probability at least $1\/100$, one local search step of \\cref{alg:noisylocalsearch} results in a new set of candidate centers $C'$ with $\\tau(X, C') \\le (1 - 1\/(100k)) \\tau(X, C)$. \n\\end{restatable}\n\nTogether with the fact that the adaptation of \\texttt{$k$-means$++$\\,} to penalties is still $O(\\log k)$ approximation with constant probability \\cite{bhaskara2020, li_penalties2020}, \\cref{lem:phase1} implies that after $O(k \\lg\\lg k)$ steps of \\texttt{Local-search++ } with outliers we get a solution of cost $O(\\beta\\textrm{OPT}) = O(\\textrm{OPT} \/ \\varepsilon)$, with constant probability. \nBut it can well be the case that this solution pays the maximum cost $\\Theta = \\frac{\\beta\\textrm{OPT}}{z}$ for $O(\\frac{\\beta\\textrm{OPT}}{\\beta\\textrm{OPT} \/ z}) = O(z)$ points. \n\nThe second proposition makes sure that while it is the case that our solution pays $\\Theta$ for more than $(1+\\varepsilon)z$ points, we still make significant progress in terms of cost, with constant probability. \nMore precisely, the cost decreases by a factor of $(1 - \\Theta(\\varepsilon \/ k))$ with probability $\\Omega(\\varepsilon)$. \n\n\n\n\n\\textcolor{red}{here we can try to move a proof sketch}\n\nIt is now a simple corollary that there can be only $O(k\/\\varepsilon^2)$ such steps, with constant probability. \nPutting the two propositions together, we conclude that $O(k\\log\\log k + k\/\\varepsilon^2)$ \\texttt{Local-search++ } with outliers steps is enough, with constant probability. \n\n\\paragraph{Running time}\nWhile one step of \\texttt{Local-search++ } is naturally implemented in time $O(nk)$, it can be sped up to $O(n\\log k)$ if one allows memory $O(nk)$ \\cite{choo2020kmeans}. \nHence, the overall time complexity can be made $O\\left( \\frac{\\log n}{\\varepsilon} \\cdot (k\\lg\\lg k + k \/\\varepsilon^2) \\cdot n\\log k \\right) = \\tilde{O}(nk\/\\varepsilon^3)$. \n\n\n\\begin{remark}\nThe term $\\log(1\/\\varepsilon)$ can be replaced by $\\lg\\lg(1\/\\varepsilon)$ by a more careful analysis. We omit proof. \n\nWe believe but do not prove that the refined analysis of \\texttt{Local-search++ } of \\cite{choo2020kmeans} can be also adapted to the setting with penalties. \nThis would imply that $O(k\/ \\varepsilon^2)$ \\texttt{Local-search++ } with outliers steps are enough, even with probability $1 - \\textrm{{e}}^{-k^{1 - o(1)}}$. \n\\end{remark}\n\n\\fi\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Distributed Algorithms for $k$-means with Outliers}\n\\label{sec:paper_distributed}\n\nTo model the distributed setting, we consider the coordinator model where the input data $X = X_1 \\sqcup X_2 \\ldots \\sqcup X_m$ is split across $m$ machines. \nEach machine can first perform some local computation. Then, each machine sends a message to a global coordinator who computes the final set of cluster centers $C$. \nThe main complexity measure is the total number of bits each machine needs to send to the coordinator. \nAn informal version of our main distributed result states the following. \n\n\\begin{theorem}\n\\label{thm:distributed_main}\nThere exists an $(O(1),1+\\varepsilon)$-approximate distributed algorithm in the coordinator model such that each machine sends at most $\\tilde{O}(k\/\\varepsilon)$ many bits. Moreover, each machine only needs to perform polynomial-time computations.\n\\end{theorem}\n\nThis improves on a recent result of \\cite{guo2018}. They give an algorithm with a $(1+\\varepsilon, 1+\\varepsilon)$-approximation guarantee, but the required local running time is exponential. Other results similar to ours include \\cite{Guha_distributed, chen2018practical}. \nBelow, we sketch the high-level idea of the constructions and leave details to \\cref{sec:appendix_distributed}. \n\n\\paragraph{Simpler Construction}\nIn this paragraph we explain the high-level intuition behind a weaker result that provides an $(O(1\/\\varepsilon), 1+\\varepsilon)$-approximation. \nThis generalizes a classical construction of \\cite{guha2003streaming} that works in the $k$-means\\, setting. \nFor simplicity, we assume here that the machines know the value $\\textrm{OPT}$ and we define $\\Theta := \\textrm{OPT}\/(\\varepsilon z)$. This assumption can be lifted at the cost of a logarithmic overhead in the time - and message complexity.\n\nEach machine starts by running \\cref{alg:kmp} for $\\tilde{O}(k\/\\varepsilon)$ steps with input $X_j$ to obtain a set of centers $Y_j$. The sets of centers satisfy the property $\\sum_j \\tau_\\Theta(X_j \\cap X_{in}^*, Y_j) = O(\\textrm{OPT})$. \nThis follows from an argument very similar to \\cref{thm:aggarwal_informal} and will be important later on.\nNow, let $X_{out,j}$ denote all points in $X_j$ that have a squared distance of at least $\\Theta$ to the closest point in $Y_j$. \nAll points in $X_{out,j}$ are declared as outliers and the remaining points in $X_j \\setminus X_{out,j}$ are moved to the closest point in $Y_j$ which creates a new, weighted instance $X'_j$. \nEach machine sends its weighted instance together with the number of declared outliers to the coordinator. The coordinator combines the weighted instances and finds an $(O(1\/\\varepsilon),1+\\varepsilon)$-approximate clustering using the algorithm guaranteed by \\cref{thm:localsearch} on the instance $X'$, but with the number of outliers equal to $z' = z - |X_{out}| + O(\\varepsilon z)$, where $X_{out} := \\bigcup_j X_{out,j}$. \nLet $C$ denote the corresponding set of cluster centers and $X_{out}'$ denote the set consisting of the $z'$ outliers. \nIn \\cref{thm:coreset_formal} in \\cref{sec:appendix_distributed} we prove that $(C, X_{out} \\cup X_{out}')$ is an $(O(1\/\\varepsilon), 1+\\varepsilon)$-approximation. \n\nThe (simple) analysis follows from two observations. \nFirst, the number of inliers incorrectly labelled as outliers in the first step, i.e., $|\\bigcup_j (X_{out, j} \\cap X^*_{in})|$, is bounded by $O(\\varepsilon z)$. This follows from $\\sum_j \\tau_\\Theta(X_j \\cap X_{in}^*, Y_j) = O(\\textrm{OPT})$ and $\\tau_\\Theta(x, Y_j) = \\Theta$ for every $x \\in X_{out,j}$. \nThis ensures that in the end we output only $O(\\varepsilon z)$ more outliers than if the coordinator ran \\cref{alg:noisylocalsearch} on the full dataset with original parameter $z$. \n\nSecond, the total movement cost of changing the instance $X$ to instance $X'$ is bounded by $\\sum_j \\phi(X_j \\setminus X_{out, j}, Y_j) \\le \\sum_j \\tau_\\Theta(X_j, Y_j) \n\\le z\\Theta + \\sum_j \\tau_\\Theta(X_j \\cap X_{in}^*, Y_j) = O(\\textrm{OPT} \/ \\varepsilon)$. \nHence, the total cost changes additively by $O(\\textrm{OPT} \/ \\varepsilon)$ compared to the case where the coordinator runs \\cref{alg:noisylocalsearch} on the full dataset. \n\n\n\\paragraph{Refined Version}\nTo improve the approximation factor from $O(1\/\\varepsilon)$ down to $O(1)$ in \\cref{thm:distributed_main}, we need to perform two changes. First, the coordinator runs the polynomial-time $(O(1), 1)$-approximation algorithm of \\cite{Krishnaswamy2018} on the weighted instance instead of \\cref{alg:noisylocalsearch}. \nSecond, each machine records not only the number of points in $X_j \\setminus X_{out,j}$ closest to each point $y \\in Y_j$, but it additionally sends for each integer $k \\in O(\\log \\Delta)$, how many of those points have a distance between $2^k$ and $2^{k+1}$ to $y$. The coordinator then constructs an instance based on this refined information. \n\n\\paragraph{$k$-means$\\|$}\nAdapting the popular $k$-means$\\|$ algorithm to the setting with outliers \\cite{bahmani2012scalable} can also be accomplished with a construction similar to the one explained above. \nThe only difference is that instead of getting the weighted instance as a union of weighted instances from each machine, it is constructed in $O(\\log n)$ sampling rounds by using the \\texttt{$k$-means$||$\\,} idea: in each round we sample from the same distribution as in \\cref{alg:kmp}, but we sample $\\tilde{O}(k\/\\varepsilon)$ points instead of just one point. \n\n\n\n\\iffalse\nAfterwards, we run the polynomial time algorithm of ... \n\n\n\nBoth of these algorithms rely on the aforementioned idea that \"a clustering of a clustering is a clustering\". More specifically, they start by finding a subset $Y$ of $O(k)$ points by oversampling such that $\\phi(X,Y) = O(\\textrm{OPT})$. Let $X'$ denote the multiset one obtains by moving each point in $X$ to the closest point in $Y$. The points in $X'$ can efficiently be represented as a weighted instance. Moreover, any constant-factor approximation for $X'$ is a constant-factor approximation for $X$. Hence, it suffices to find a good clustering with respect to $X'$. \n\n\n\\paragraph{Straightforward adaptation does not work}\n\tA straightforward variant in our framework would be the following. Let $\\Theta \\in [\\frac{\\textrm{OPT}}{2\\varepsilon z}, \\frac{\\textrm{OPT}}{\\varepsilon z}]$ be arbitrary. We first start by finding a subset $Y$ such that $\\tau_\\Theta(X,Y) = O(\\textrm{OPT})$. Then, we again denote by $X'$ the multiset one obtains by moving each point in $X$ to the closest point in $Y$. Afterwards, we find an $O(1)$-approximate clustering $C$ on the instance $X'$ with penalty $\\Theta$. Now, we output $C$ as our set of centers for $X$ and declare all points in $X$ with a squared distance of at least $\\Theta$ as outliers. This straightforward adaption does not work however. The reason is the following: In the classical $\\kmeans$-setting, the squared distance between a point $x \\in X$ and its corresponding point $x' \\in X'$ is proportional to the clustering cost of $x$ with respect to $Y$. This does not hold true in the setting with penalties, where the clustering cost of each point is bounded by $\\Theta$, even if the squared distance to the closest point in $Y$ is much larger. The construction explained in the next paragraph gets rid of this problem by declaring all points in $X$ for which we pay $\\Theta$ with respect to $Y$ as outliers. The next paragraph exlains this approach in more detail.\n\t\n\n\\paragraph{Adaptation to the setting with outliers}\n\nNext, we explain how to adapt the approach mentioned in the previous paragraph to the settings with outliers. A direct distributed realization of this approach falls short to provide the guarantees of \\cref{thm:distributed_main}. It only obtains an $(O(1\/\\varepsilon),1+\\varepsilon)$-approximation guarantee. Nevertheless, the algorithm is a good starting point to understand the overall idea.\n\nWe start by finding a set $Y$ consisting of $O(k\/\\varepsilon)$ points such that $\\tau_\\Theta(X_{in}^*,C) = O(\\textrm{OPT})$, where $\\Theta \\in [\\frac{\\textrm{OPT}}{2\\varepsilon z},\\frac{\\textrm{OPT}}{\\varepsilon z}]$ . Now, let $X_{out}$ denote all points in $X$ that have a squared distance of at least $\\Theta$ to the closest point in $Y$. All points in $X_{out}$ will later be declared as outliers. Next, all the remaining points in $X \\setminus X_{out}$ are moved to the closest point in $Y$ and we denote the resulting multiset by $X'$. Afterwards, we run the polynomial time algorithm of ... that finds an $(O(1),1)$-approximate clustering to the $k$-means\\, objective with $z'$ outliers on the instance $X'$, where $z' = z - |X_{out}| + O(\\varepsilon z)$. Let $C$ denote the corresponding set of cluster centers and $X_{out}'$ denote the set consisting of the $z'$ outliers. The final output of the algorithm is $(C,X_{out} \\cup X_{out}')$. \n\nNote that the algorithm outputs \n$$|X_{out} \\cup X_{out}'| = |X_{out}| + z' = (1 + O(\\varepsilon))z$$\n\noutliers. Thus, it remains to show that $\\tau_\\Theta(C,X \\setminus (X_{out} \\cup X_{out}')) = O(1\/\\varepsilon)\\textrm{OPT}$. We only give a brief sketch of the ingredients the proof relies on. Let $\\textrm{OPT}'$ denote the optimal objective value for the $k$-means\\, with $z'$ outliers on the instance $X'$. The first step of the proof is to show that $\\textrm{OPT}' = O(1\/\\varepsilon)\\textrm{OPT}$. The second step is to show that the points in $X'$ can be moved back to their original location without causing a big increase in the objective value. The first thing to note is that $$|X_{in}^* \\cap X_{out}| \\leq \\frac{\\tau_\\Theta(X_{in}^*,C)}{\\Theta} = \\frac{O(\\textrm{OPT})}{\\Theta} = O(\\varepsilon)z.$$ \nOn an intuitive level, this says that in the first step at most $O(\\varepsilon)z$ inliers are wrongly declared as outliers. In particular, this implies $|X_{out}^* \\setminus X_{out}| \\leq z'$. Hence, considering the set of centers $C^*$ and declaring all points in $X'$ that correspond to points in $X_{out}^* \\setminus X_{out}$ as outliers gives an upper bound on $\\textrm{OPT}'$. In this way, one can establish that $\\textrm{OPT}' \\leq O(1\/\\varepsilon)\\textrm{OPT}$. he high-level reason for this is that each point $x'$ in $X'$ has a squared distance to the corresponding point $x$ in $X$ equal to $\\tau_\\Theta(x,C)$ and $\\tau_\\Theta(X,C) \\leq O(\\textrm{OPT}) + \\Theta z = O(1\/\\varepsilon)\\textrm{OPT}$. This can also be used to show the second step of the argument, namely that the points in $X'$ can be moved back to their original location without increasing the cost by more than $O(1\/\\varepsilon)\\textrm{OPT}$.\n\n\\paragraph{Improved construction}\n\n\tHowever, this construction does not match the bound we claimed in the introduction...\n\t\n\tIn some sense, by transitioning from $X$ to the weighted instance, we lose too much distance information. Hence, we present a more sophisticated core-set construction. Instead of \n\t\n\\fi\n\n\t\n\n\n\n\n\\section{Tight Bounds for $(O(1), O(1))$-approximation Algorithms}\n\\label{sec:paper_nk2z}\n\nIn this section, we discuss tight bounds on the complexity of finding an $(O(1), O(1))$-approximation for the $k$-means with outliers objective. \nIn \\cref{sec:appendix_metropolis} we prove the following result. \n\n\\begin{restatable}{theorem}{thmmetropolis}\n\\label{thm:fast_algorithm}\nThere is an $(O(1), O(1))$-approximation algorithm for $k$-means with outliers that runs in time $\\tilde{O}(nk \\cdot \\min(1, k\/z))$ and succeeds with positive constant probability. \n\\end{restatable}\n\nThis result is based on \\cref{thm:aggarwal_informal}, but to speed up the algorithm from $\\tilde{O}(nk)$ to $\\tilde{O}(nk^2\/z)$, we use the Metropolis-Hastings algorithm (with uniform proposal distribution) which was used in \\cite{bachem2016approximate, bachem2016fast} for $k$-means. \nHere, the idea is that in one sampling step of \\cref{alg:kmp}, instead of computing $\\tau_\\Theta(x, C)$ for all the points, we first subsample $\\tilde{O}(n\/z)$ points uniformly at random and only from those points we then sample roughly proportional to their current $\\tau_\\Theta$ cost by defining a certain Markov chain. \nAfter speedup, we keep the guarantees of \\cref{thm:aggarwal_informal}, but now the running time is $\\tilde{O}(nk^2\/z)$. \n\nNote that instead of using Metropolis-Hastings algorithm, we could also just take a uniform subsample and run \\cref{alg:kmp} on it. The result of \\cite{huang2018epsilon} would give that uniform subsampling the number of points to $\\tilde{O}(k(n\/z)^2)$ would lose only a constant factor in approximation guarantees. This leads to a $\\tilde{O}(k^2(n\/z)^2)$ running time and an algorithm with similar running time also based on uniform subsampling for $k$-median was given by \\cite{meyerson2004k}. \n\n\n\n\n\n\\paragraph{Lower Bound}\nNext, we provide a matching lower bound. \nTo that end, we restrict ourselves to the class of algorithms that work in an arbitrary metric space $\\mathcal{M}$ and access the distances of the space only by asking an \\textit{oracle} that upon getting queried on two points $x,y \\in \\mathcal{M}$ returns their distance. \n\nVirtually all algorithms with guarantees we know of are of this type, possibly up to a constant loss in their approximation guarantee. In \\cref{sec:appendix_lemmas} we verify that this is the case also for algorithms that we consider here. \n\n\nFor the classical problems of $k$-means\\,, $k$-median, and $k$-center, there is an $\\Omega(nk)$ lower bound (i.e., showing that so many queries to the oracle are necessary) for metric space algorithms \\cite{mettu2004optimal, mettu2002approximation}. This essentially matches the complexity of \\texttt{$k$-means$++$\\,} \\cite{arthur2007k} or the \\texttt{Local-search++ } algorithm of Lattanzi and Sohler \\cite{lattanzi2019better}. \nThe lower bound holds also in the setting with outliers, if the output of the algorithm is an assignment of each of the $n$ points to an optimal cluster, together with the list of the outliers. \nOn the other hand, \\cref{thm:fast_algorithm} gives an $(O(1), O(1))$ algorithm with complexity $\\tilde{O}(nk^2\/z)$. \nThe catch is that the output of this algorithm is just a set of centers and it does not compute for all the points their respective assignment to a closest center. \nAlso, it does not label all the outliers. \nFor this type of algorithms that only output the set of centers, we prove a matching lower bound of $\\Omega(nk^2\/z)$ in the metric space query model. The following theorem is proved in \\cref{sec:appendix_lower_bounds}. \n\n\\begin{theorem}\n\\label{thm:lower_bound_informal}\nAny randomized algorithm for the $k$-means\/$k$-median\/$k$-center problem with outliers in the setting $k \\ge C$, $z \\ge Ck \\log k$, and $n \\ge Cz$ for an absolute constant $C$ that with probability at least $0.5$ gives an $(O(1), O(1))$-approximation in the general metric space query model, needs $\\Omega(nk^2\/z)$ queries. \n\\end{theorem}\n\nLet us provide a brief intuition of the proof. The construction that yields \\cref{thm:lower_bound_informal} is the following: we have $k\/2$ large clusters and $k\/2$ small clusters. All clusters are well separated and small clusters together contain $\\Theta(z)$ points. \nAs the algorithm can output only $\\Theta(z)$ outliers, it needs to ``find'' $\\Omega(k)$ small clusters. \nHowever, a point chosen from the input set uniformly at random is from a small cluster with probability $O(z\/n)$ and we expect to need $\\Omega(k)$ queries until we find out whether a point is from a small or a big cluster. This leads to the lower bound of $\\Omega(k \\cdot \\frac{n}{z} \\cdot k) = \\Omega(nk^2\/z)$ rounds. \n\n\\paragraph{Why it makes sense to consider bicriteria approximation}\nLet us also observe a different lower bound against $(O(1), 1)$-approximation algorithms that motivates why we are concerned with algorithms that can output $(1+\\varepsilon)z$ outliers. \nThe following construction is, e.g., in \\cite{indyk1999sublinear}. \nConsider an input metric space with $z = n-1$ and $k=1$, where any two points have a distance of $1$, up to a single pair $x_1, x_2$ whose distance is $0$ (or some small $\\varepsilon$). Any approximation algorithm (even a randomized one) that outputs exactly $z$ outliers needs to ``find'' the pair $x_1, x_2$ and $\\Omega(z^2)$ queries are needed for this. \nThe example shows that there is a fundamental limit to the speed of $(O(1), 1)$-approximation algorithms: for $z$ linear in $n$ they even need $\\Omega(n^2)$ time. \nThis should be contrasted with the $(O(1), 1+\\varepsilon)$-approximation algorithms that only need $\\mathrm{poly}(k\/\\varepsilon)$ time for $z = \\Theta(n)$ that follows from \\cite{huang2018epsilon}. \n\n\n\n\n\\section{Experiments}\n\\label{sec:paper_experiments}\n\nWe tested the following algorithms on the datasets \\emph{kdd} (KDD Cup 1999) subsampled to $10\\,000$ points with $38$ dimensions and \\emph{spam} (Spambase) with $4601$ points in $58$ dimensions \\cite{Dua:2019}. \nWe set the number of outliers $z$ to be $10$ percent of the dataset. \n\n\\iffalse\n\\texttt{Lloyd}: Variant of Lloyd's algorithm\\cite{lloyd1982least, kmeans-} that handles outliers with random initialization ($10$ iterations);\n\\texttt{$k$-means++}: $k$-means++ seeding\\cite{arthur2007k};\n\\texttt{$k$-means++ with penalties}: $k$-means++ with penalties\\cite{bhaskara2020, li_penalties2020};\n\\texttt{Metropolized $k$-means++ with penalties}: $k$-means++ with penalties, sped up by Metropolis subsampling with $100$ steps (\\cref{sec:appendix_metropolis});\n\\texttt{Distributed $k$-means++ with penalties}: simplified variant of \\cref{alg:streaming} in \\cref{sec:appendix_distributed} -- input partitioned in $10$ subinputs, $k$-means++ with penalties run on each of them with $\\ell = 2k$, weighted instances sent to coordinator who runs $k$-means++ with penalties again to obtain the final $k$ centers);\n\\texttt{Sped up local search}: $k$-means++ with penalties followed by additional $k$ local search steps (for the objective with penalties). \n\\fi\n\n\n\\begin{itemize}\n \\item \\texttt{Lloyd}: Variant of Lloyd's algorithm \\cite{lloyd1982least, kmeans-} that handles outliers with random initialization ($10$ iterations);\n \\item \\texttt{$k$-means++}: $k$-means++ seeding\\cite{arthur2007k};\n \\item \\texttt{$k$-means++ with penalties}: $k$-means++ with penalties\\cite{bhaskara2020,li_penalties2020};\n \\item \\texttt{Metropolized $k$-means++ with penalties}: $k$-means++ with penalties, sped up by Metropolis-Hastings algorithm with $100$ steps (see \\cref{sec:appendix_metropolis});\n \\item \\texttt{Distributed $k$-means++ with penalties}: simplified variant of \\cref{alg:streaming} -- input is partitioned in $10$ subinputs, $k$-means++ with penalties is run on each of them with $\\ell = 2k$, and weighted instances are sent to coordinator who runs $k$-means++ with penalties again to obtain the final $k$ centers);\n \\item \\texttt{Sped up local search}: Variant of \\cref{alg:noisylocalsearch} -- $k$-means++ with penalties followed by $k$ additional local search steps (for the objective with penalties). \n\\end{itemize}\n\n\nTo guess the value of $\\Theta$ in all except the first two algorithms, we tried $10$ values from $1$ to $10^{10}$, exponentially separated. \nThe best solution was then picked and we followed by running $10$ Lloyd iterations on it with the number of outliers for these iterations set to $z$ (the same for the second $k$-means++ algorithm). The results for this setup for $k\\in \\{5, 10, \\dots, 50\\}$ are in \\cref{fig:kdd,fig:spam}. \n\n\n\\begin{figure}[h!]\n \\centering\n \n \\centering\n \\includegraphics[width=.95\\textwidth]{kdd.png}\n \\caption{Experiments on kdd}\n \\label{fig:kdd}\n \n \n \\includegraphics[width=.95\\textwidth]{spam.png}\n \\caption{Experiments on spam}\n \\label{fig:spam}\n \n \n \\label{fig:experiments}\n\\end{figure}\n\n\\texttt{$k$-means with penalties} outperforms the first two baseline algorithms on average by around $40\\%$ in both datasets. Surprisingly, $k$-means++ seeding leads to consistently worse solutions than random initialization. We believe this indicates that the datasets indeed contain outliers that $k$-means++ picks preferably due to their large distance from other points. \nDistributed and metropolized variants are on par with \\texttt{$k$-means++ with penalties}, except for the metropolized variant on the \\emph{spam} dataset. \n\\texttt{Sped up local search} consistently outperforms \\texttt{$k$-means++ with penalties} by around $12\\%$ in both datasets. It is also significantly slower, but we implemented only its simple $\\tilde{O}(nk^2)$ implementation instead of the best possible $\\tilde{O}(nk)$. \n\n\n\n\n\n\\section{Conclusion}\n\nWe have shown that several simple sampling-based algorithms for $k$-means can be adapted to handle outliers and retain strong theoretical guarantees, while still being similarly simple to implement. \nAs a theoretical application, we settled the complexity of finding an $(O(1), O(1))$-approximation for $k$-means with outliers to $\\tilde{\\Theta}(nk^2\/z)$ in the query model. \n\n\n\\section*{Acknowledgment}\n\nWe thank Davin Choo, Mohsen Ghaffari, Saeed Ilchi, Andreas Krause, and Julian Portmann for engaging discussions. In particular, we thank Mohsen for his numerous helpful remarks and Davin and Saeed for sharing their code with us. \n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzbuhz b/data_all_eng_slimpj/shuffled/split2/finalzzbuhz new file mode 100644 index 0000000000000000000000000000000000000000..170c829e2940ea19f847d641b7849555c9e7c809 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzbuhz @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThe need to incorporate programming content into introductory physics is widely appreciated by the academic community \\cite{Fuller2006}. By some estimates, \\emph{at least 70\\%} of new STEM jobs in the US will require computer programming skills \\cite{labor2014} and in the sciences computer programming skills have become an essential part of many disciplines. In response to these shifts, groups like code.org and the ``hour of code\" have brought coding tutorials to wider and younger audiences \\cite{code}. These groups also influenced federal education legislation in the US. In particular, the Every Student Succeeds Act (ESSA), which was signed into law in December 2015, designates computer science as a ``core subject\". This is a significant change that places computer science on the same level as english and mathematics \\cite{coresubject}. The 2017-2018 school year was the first school year that this legislation was fully implemented. Yet, for physics instruction, and perhaps even more generally, the task of re-imagining STEM courses with computer science as a crucial element is still far from complete. Although there are a number of universities that use coding activities in physics with vpython \\cite{Chabay_Sherwood2008}, and there exists significant research into using these activities in calculus-based introductory physics \\cite{Caballero_etal2012}, vpython exercises (or coding using some other software framework) are much less often used in algebra-based physics and at the high school level. \n\nWe were able to find two studies that reflect the difficulty of using coding activities in algebra-based physics at the high school level\\footnote{The open source ebook by \\citet{Titus_Esquembre2016} is also notable but there seems to be no published studies examining its appropriateness for various grade levels.}. \\citet{Aiken_etal2013} describes a masters degree project by a high-school physics teacher who worked for two years to develop a vpython curriculum for a 9th grade high school physics class and found that only one third of the class successfully completed the exercises. \\citet{Aho2014} describes coding activities developed for a high school classroom that use the R programming language. Although they are not very specific in stating precisely what fraction of the students struggled with the exercises, they do indicate that a significant number of students needed extra time outside of class to complete the activities, and these students frequently needed extra practice to learn the R syntax. To mitigate this in future work, \\cite{Aho2014} proposes to set aside a week-long R programming tutorial for the students at the beginning of the year, which is a luxury most high school teachers do not have. The indications from \\citet{Aiken_etal2013} and \\citet{Aho2014} underscore the need to develop a curriculum that adds programming into \\emph{algebra-based} physics with a higher success rate.\n\n\\begin{figure*}\n\\includegraphics[width=2.4in]{Traditional_approach.pdf}\\includegraphics[width=2.4in]{Hybrid_approach.pdf}\\includegraphics[width=2.4in]{PhetPhyslet_approach_v2.pdf}\n \\caption{Illustrations of different approaches to computationally-enriched physics content. The left panel illustrates the typical structure of a code in a traditional intermediate-to-advanced level physics-major computational physics course, emphasizing that the student has control over essentially the entire code. The right panel shows the typical structure of a web interactive in which students interact with a visualization but do not see or have any control of the underlying code. The center panel shows a hybrid approach with a high degree of interactivity but the students do see and potentially modify the parts of the code that advance system variables, even though code related to visualization remains fixed and invisible to the student.}\n \\label{fig:approach}\n\\end{figure*}\n\nAs will be discussed, we use a javascript-based language called p5.js, which was designed to be a text-based (as opposed to block-based) language with a gentle learning curve for absolute beginner programmers. In principle, the exercises we describe here could be reproduced in vpython (or some other language) and used in a similar way. While there are clearly advantages and disadvantages to both vpython and javascript, the comparison of the two is not the subject of this paper. Instead we wish to emphasize the need for coding activities that would be appropriate for an algebra-based physics classroom. As discussed later, an important way to judge the appropriateness of these activities is the perceived difficulty of students who complete coding activities.\n\nThe left hand panel in Figure~\\ref{fig:approach} illustrates what we describe as the ``traditional\" computational physics approach that appears in an intermediate-to-advanced physics-major computational physics course, or in a physics-major course that has been re-tooled to include significant computational content. In this setting, the student is given complete control of the computer program, including the advance of variables (which may involve specifying forces and advancing positions and velocities, or it may involve the evolution of abstract quantities like wave functions). If visualization is needed, the student is typically given full control of a plotting program. Although there may be some template that the student is given and other advice may be provided, overall, the student has a high level of control. The drawback for this approach is that significant class time is often required for students to familiarize themselves with coding practices. Given the time constraints of a typical algebra-based college physics course, or high school physics course, this approach is in-feasible for most instructors.\n\nThe right hand panel in Figure~\\ref{fig:approach} describes interactive physics simulations in which the students do not see the code. This approach is extensively used by the PhET collaboration \\cite{PhET} and by the ``Physlet\" physics community \\cite{physlet}, and many studies have shown its utility for teaching scientific concepts \\cite[e.g.][]{Perkins_etal2006,Podolefsky_etal2010}. Largely for this reason, PhET and similar activities have been put into widespread use.\n\nThe central panel in Figure~\\ref{fig:approach} outlines the ``hybrid\" approach that we adopt in this paper in which the student does see and potentially modifies the code that evolves system variables (which is similar to the traditional approach), and there is some kind of interactive visualization that is produced in which the simulation responds to user input (which is similar to the PhET\/Physlet approach). However there are still aspects of the code, particularly related to visualization, that the student does not see in order to substantially reduce the cognitive load \\cite{Jong_2010,cogload} by shortening the length and minimizing the complexity of the program. The intention is to remove ``extraneous cognitive load\" associated with the graphical user interface among other things, and focus on the aspects of the code that directly determine or update physical quantities. Our assumption is that the ``intrinsic cognitive load\" of setting and updating the physical quantities using the target concepts and relationships is within students' abilities. As will be illustrated in this paper, the portion of the code with which the students interact can be concise both textually and conceptually, and still produce interesting game-like interactives that emphasize kinematic and diagrammatic concepts like force, acceleration, velocity, and their vector representations.\n\nTo provide some comparison to other works in the literature, there may also be some overlap with our approach and that of \\cite{kordakai2010}, who describes a graphically enriched coding environment for teaching computer science and outlines how their activities align with the educational theories of various authors. The Netlogo project \\cite{netlogo} is another comparable effort which borrows from earlier efforts to incorporate programming into schools, but we are not familiar enough with how Netlogo activities are used in introductory physics to say more than this.\n\nAlthough there are exceptions \\cite{Taub_etal2015,Weintrop_etal2016,Titus_Esquembre2016,netlogo}, interactive activities where students key-in commands and ``play\" their code like a video game, are typically not a part of programming exercises at the introductory level. In the Matter \\& Interactions curriculum that integrates vpython into calculus-based physics \\cite{Chabay_Sherwood2015}, many of these programs, such as the three-body gravitational simulation or the 3D pendulum \\cite{Chabay_Sherwood2008}, are designed for the student to perform coding tasks and then passively watch the execution of the program (except perhaps for changing the perspective). And while there are a large number of exercises currently available on the AAPT's Partnership for the Integration of Computation into Undergraduate Physics (\\href{http:\/\/compadre.org\/PICUP}{compadre.org\/PICUP}), only a few of them involve a high level of interactivity as the program is running.\nOur hypothesis is that this interactive, game-like approach with a concisely-written code will create a fun and approachable experience for students who might otherwise find a programming task to be intimidating, making it an ideal choice for engaging students in introductory courses\n\nThis paper is only the beginning of a research effort to validate this hypothesis. We will describe a set of computer programming activities designed for absolute beginner programmers in first-semester introductory physics (mechanics) classes, that were used during four semesters at Ohio State's Marion campus. Survey results will be presented that examine student perceptions from completing the first exercise, and probe the percentage of weak or absolute beginner programmers in the classroom.\n\nAlthough there is good work in the literature describing how numerical exercises can be connected with laboratory activities \\cite[e.g.][]{Serbanescu_etal2011}, we consider this out of scope for the present work. The javascript-based language p5.js does have capabilities to interact with Arduino circuit boards, making this an interesting possibility for future work.\n\n\\section{Overview of Programming Activities for Mechanics}\n\nIn a semester course of introductory physics at Ohio State University (OSU) at the regional campus in Marion, we include six required programming activities and a seventh activity that is optional or extra credit. In most other ways, the course is identical to the same course on the Columbus campus. The official description of this course is calculus-based physics I, but on all OSU campuses students only need to be concurrently enrolled in calculus in order to take the course, and as a result the calculus content in the course is rather limited. Moreover, the students at OSU's regional campuses are less prepared than their peers on OSU's Columbus campus. During the data gathering, incoming OSU Marion students had an average ACT score near 22 (in 2014 \\cite{osumarion2014} and 2015 \\cite{osumarion2015}) or 22.5 (in 2016 \\cite{osumarion2016}) whereas students admitted directly to the Columbus campus over the same time span had an average ACT score close to 29 \\cite{osucolumbus}. The limited calculus in the course and the comparably poor ACT performance of the students make interesting venue for integrating programming exercises into introductory physics with the end goal of creating a curriculum that might succeed in the high school physics classroom.\n\nEach activity is designed to take about an hour to complete. Students are not explicitly assigned to groups or pairs, but the classroom setup involves six tables of four, so students will tend to collaborate on the activities and this is not discouraged. To date, about 125 students from OSU's regional campus in Marion have completed the exercises mentioned below. The activities are graded on the completion of required steps.\n\nAll of the exercises illustrate the velocity, acceleration and force vectors. The first exercise gives the student much of the code that they will need, only asking them to make small, guided modifications, which we will describe in the next section. All of the exercises build off of each other in a way that would make it hard for a student to start in the middle of the sequence. Additionally, all of the exercises contain ``challenges\" that encourage the student to develop some functionality that often adds an interesting element to the game. The list of exercises is as follows:\n\\begin{enumerate}\n \\item Planetoids (similar to the classic game ``Asteroids\")\n \\item Lunar descent (similar to the classic game ``Lunar lander\")\n \\item Bellicose birds (similar to the popular game ``angry birds\")\n \\item Planetoids with momentum\n \\item Planetoids with torque\n \\item Planetoids with a spring (harmonic motion)\n \\item Extra credit: Bellicose birds with energy\n\\end{enumerate}\n\nThis sequence is designed to accompany a typical physics course on classical mechanics where momentum is not introduced until mid-way through the course, followed by concepts of torque and, later, harmonic motion. The ``Bellicose birds with energy\" exercise is made available to students in the middle of the course when energy is introduced, but this exercise is more difficult than the others because it is the only exercise that deals explicitly with the integration scheme. For simplicity, all the exercises adopt Euler-Cromer integration \\cite{Cromer1981} except for ``Bellicose birds with energy\", which describes the trapezoidal method in terms that an algebra-based physics student should be able to understand.\n\nIn this paper we provide a rather extensive description of the first exercise (``Planetoids\") including student survey responses. This section provides a context to this exercise since essentially all the exercises listed above are derived from this ``Planetoids\" exercise. These activities will be described in detail in later work. We will only add here that some of these additional activities use a graphing system to plot various relevant physical quantities (such as velocity) over time in the bottom right corner of the screen.\n\n\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=5in]{planetoids_fig_improved.png}\n \\caption{The code (left) and corresponding interactive (right) that the student sees at the beginning of the first exercise. This code is written with the processing javascript library p5.js. As a result, the code has a C\/C++ like syntax except that draw() replaces main() and draw() is run 60 times per second until the user stops the program. The interactive (right) shows a ship traveling towards the right with constant velocity indicated by a red velocity vector. On the left panel the student sees about 50 lines of code, but about half of these lines are spaces or comments.}\n \\label{fig:planetoids}\n\\end{figure*}\n\n\\section{The Planetoids Game}\n\nThe choice of an ``Asteroids\"-like game for the first activity is intentional. A natural environment for illustrating Newton's laws is free space, away from any sources of gravity. We are not unique in using this situation as a starting point. \\cite{white1984} found learning gains from students interacting with a video game with a similar premise, and no doubt other authors adopt a similar approach. The advantage of this environment is that objects in motion will continue with the same velocity, moving in a straight line, unless a force is acts upon them. The classic game ``Asteroids\" illustrates this well with a ship that drifts through free space, except when its rockets fire to avoid asteroids that are also drifting through free space. The net force is either zero, or constant in the direction the ship is pointing.\n\n\\subsection{Learning Goals}\n\\label{sec:learning}\n\nThe learning goals for this exercise are as follows:\n\n\\begin{enumerate}\n \\item Understand how to convert a simple 1D code into a 2D code\n \\item Understand how force, velocity and acceleration vectors relate to the motion of a ship traveling in free space\n\\end{enumerate}\n\nThe list above is intentionally short because we do not expect that during this 1-2 hour activity that the student will be able to absorb the subtleties of computational thinking \\cite{Weintrop_etal2016}\nor become proficient with the javascript-based coding framework to the point where they can comfortably make numerous modifications to the code. In the following subsections we discuss how the activity is structured to reinforce the two learning goals mentioned above, and we point out various difficulties that students often have. We discuss additional learning goals and extensions to the activity in later sections.\n\n\\subsection{Structure of the Program and Design Choices}\n\nFigure ~\\ref{fig:planetoids} shows what the student sees at the beginning of this exercise. Initially, the ship can only move in the $x$ direction and the first task is to allow the ship to rotate when the user presses the left and right arrow buttons by changing the value of $\\theta$. It is worth commenting on Fig.~\\ref{fig:planetoids} in detail because even at this stage there are a number of choices that have been made that could affect student learning. One important choice made to simplify the cognitive load for the student is to ``hide\" a significant amount of code in the \\texttt{display()} function. In this example, there are about three times more lines of code defining the \\texttt{display()} function than the $\\approx 50$ lines of code that the student sees and modifies\\footnote{We attempted to re-create this exercise in vpython using glowscript.org but found that (at least currently) there is no way of setting up a second page of code where subroutines can be defined without being in plain view by the student, which is a barrier to implementing this ``hybrid\" approach. This may or may not be a limitation with other browser-based python development environments like trinket.io or jupyter.org}.\n\nAnother important choice is to hide the variable types. There are no \\texttt{float}, \\texttt{int}, \\texttt{double} or \\texttt{var} declarations used to initialize the variables. Instead, variables are implicitly declared to be floating point decimals and the number of characters that the student sees is minimized. This syntax is essentially the same as used in Matlab, which is a popular language for absolute beginner programmers. We use the processing javascript library p5.js for these exercises and as a result the code shown in Fig.~\\ref{fig:planetoids} is javascript which does not produce an error for missing variable types. A possible drawback of postponing the discussion of variable types is that the difference between global and local variables is not explained at this stage. Students may not realize that \\texttt{accelx}, which is only used and defined inside of an \\texttt{if} statement, is a local variable while \\texttt{deltaVx} is a global variable, but this is unlikely to cause a problem at this stage. Our philosophy is to explain subtleties like these in the step-by-step tutorial only if absolutely necessary for completing a particular exercise.\n\nThe structure of the program in Fig.~\\ref{fig:planetoids} is an important choice that may affect student learning. The sections of the code are as follows:\n\\begin{enumerate}\n\\item Variable initializations \n\\item the \\texttt{draw()} function -- velocity and position advance\n\\item the \\texttt{draw()} function -- keyboard inputs\n\\item the \\texttt{draw()} function -- \\texttt{display()} function followed by other user-defined graphics\n\\end{enumerate}\nIt is understood that the \\texttt{draw()} function is run many times per second so that after the \\texttt{display()} function is executed the program will go back to the beginning of \\texttt{draw()} and advance the velocity and position again and go through the whole sequence again until the user presses stop\\footnote{An optional ``Hello world\" activity demonstrates that adding code to write a simple message to the browser console while inside of the \\texttt{draw()} function will result in that message being written many times over because the \\texttt{draw()} function is being run many times per second.}. Because \\texttt{draw()} is being run again and again, one could easily change the sequence so that, for example, the \\texttt{display()} function would be first and the velocity and position advance would be last. The drawback of this approach is that when the student parses the code for the first time they would see the physics content of the code \\emph{last}. In a physics course, our primary interest lies in directing the students' attention to how the physics content, such as $d=vt$ for example, is implemented in the code, with discussions of the programming concepts such as syntax, variable types, and the structure of the algorithm being supplementary to that.\n\nFollowing the physics section there is a line of code \\texttt{deltaVx = 0} ($\\Delta v_x = 0$) which is accompanied by a comment ``velocity is unchanged if there are no forces\". This is just a restatement of Newton's first law in a form that a computer can understand. Following this, the program checks if the user is pressing certain buttons on the keyboard.\n\nThe drawback to this physics-first, keyboard commands later approach is that the student may not fully appreciate that the program holds on to the global variable \\texttt{deltaVx}, which is determined from the keyboard command section, only using it again at the beginning of the \\emph{next} iteration of \\texttt{draw()}.\n\nIn the written step-by-step directions, the user is asked to put non-zero values in the section of the keyboard input section that changes the angle of the ship. Then the student is asked to enable motion in the $y$ direction by imitating the code for advancing the velocity and position in the $x$ direction. Finally, the student is asked to determine the correct change in velocity due to a constant force (thrust) in the $y$ direction. This involves realizing that while $\\cos \\theta$ gives the component of the force oriented in the $x$ direction, one must use $\\sin \\theta$ to obtain the component of the force in the $y$ direction. Students are given a hint that it is either a cosine, sine, or tangent function that gives the correct behavior.\n\nAt each step in the tutorial, the student can click links to see and interact with how the program should work at a particular stage, but without seeing the source code for the completed step. This is an important capability that gives the student instant guidance on whether they have completed a particular programming task correctly, leaving the instructor more time to spend on subtle issues.\n\nCommon mistakes that students make include forgetting to set \\texttt{deltaVy = 0}, in which case the ship accelerates uncontrollably in the $y$ direction. Students rarely self-diagnose this issue because the ship appears to behave correctly if the thrusters are repeatedly fired and it is only when the student stops firing the thrusters that the uncontrollable acceleration becomes obvious. When students interact with the correct version of the program (as described in the previous paragraph) they should notice this difference in behavior but the problem is subtle enough that this problem is easy to miss.\n\nAnother frequent mistake is that students tend to do a quick copy paste of the acceleration code for the $x$ direction to the $y$ direction without changing the trigonometric function from cosine to sine. This causes $\\Delta v_y = \\Delta v_x$ and as a result the ship only travels on a diagonal line regardless of the angle $\\theta$. Students have an easier time self-diagnosing this issue because the problem is easy to see and they are told that the trigonometric function in the line of code that determines $\\Delta v_y$ should be either a cosine, sine, or tangent.\n\n\\subsection{Challenges}\n\nStudents must also implement 1-2 ``challenges\". The challenges in this exercise include creating ``planetoids\" (a word play on the astronomical term planetesimals) that drift across the screen using the \\texttt{drawEllipse()} function and adding reverse thrusters when the down arrow is pressed (which can be done by copying the code from the up arrow and adding minus signs to change the direction of the force). Students can also allow the ship to shoot a projectile using the \\texttt{drawPoint()} function and the code includes an \\texttt{if} statement that detects if spacebar is pressed for this purpose. This latter task is more difficult than the others because the projectile must be launched in the same direction as the ship whereas the planetoids can be given a random velocity using the \\texttt{random()} function. One should also include the velocity of the ship when determining the velocity of the projectile as a fun illustration of Galilean invariance. Most students will just implement the reverse thrusters challenge.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=3.3in]{notorque_lab_mod_posy.png}\n \\caption{A screenshot from an activity where the student explores how changing the force of the rocket's thrust and the mass of the ship affects one's ability to avoid randomly drifting ``planetoids\". This follows code modification tasks that enable the ship to move in two dimensions (instead of one dimension as in Fig.~\\ref{fig:planetoids}).}\n \\label{fig:withplanetoids}\n\\end{figure}\n\n\nA more recent modification to this activity that was added after the study is to give the student a code that includes a number of drifting planetoids that will cause the game to end if the ship runs into one of them (Fig.~\\ref{fig:withplanetoids}). The student is asked to explore the effect of changing the force of the ship's thrust and the mass of the ship on surviving in the game. This task helps foster a discussion of how it is only the ratio of the force to the mass that matters to the acceleration of a rocket in free space.\n\n\n\\begin{figure*}\n\\begin{center}\n \\includegraphics[width=3.4in]{experience.pdf}\\includegraphics[width=3.4in]{level_of_difficulty.pdf}\n \\includegraphics[width=3.4in]{vectors.pdf}\\includegraphics[width=3.4in]{fun2.pdf}\n \\end{center}\n \\vspace{-0.6cm}\n \\caption{Survey results from Ohio State Marion students who completed the first programming exercise (rocket in free space). Results are cumulative from four semesters of students (Spring 2015 - Fall 2016).} \n \\label{fig:survey}\n\\end{figure*}\n\n\\section{Student data}\n\\label{sec:survey}\n\n\nAfter the student completes the Planetoids exercise, there is a detailed online survey that probes their experience in completing the activity. While the questions in this survey are qualitative and involve student self-reporting, the results can offer insight on whether the level of difficulty of the first exercise is appropriate and whether students find the exercises to be enjoyable to complete.\nFigure~\\ref{fig:survey} summarizes the results of the survey from four semesters of students (Spring 2015 -- Fall 2016). The upper left plot in Fig.~\\ref{fig:survey} shows that there are a significant number of absolute beginner programmers and weak programmers in the class. There were also a significant number of students who reported ``some experience\" which may have meant that they were currently enrolled in a required C++ course, but had not had significant experience with coding prior to this.\n\nThe upper right plot in Fig.~\\ref{fig:survey} shows that the difficulty level seems to be appropriate for the population of students, with a significant number of students selecting ``Easy!\". The lower right plot in Fig.~\\ref{fig:survey} indicates that many of the students found the programming activities to be enjoyable or fun. Students also have many positive things to say about the programming exercises in written evaluations at the end of the course after all of the exercises have been completed.\n\nThe bottom left plot in Fig.~\\ref{fig:survey} summarizes student responses to the question ``Did the programming lab help you understand vectors better?\" Although students can only provide a subjective estimation for how much they have learned, studies have shown that information of this kind can be valuable and even predictive other measures of student success \\cite{Sawtelle_etal2012}.\n\n\\section{Success Relative to Learning Goals}\n\nIn an earlier section (\\ref{sec:learning}) we outlined two learning goals for the exercise. The student survey data in \\ref{sec:survey} can provide some qualitative or indirect insight on whether these goals were met. In particular, the ``Level of Difficulty\" question, which is asked after the completion of the code, relates to the learning goal of ``Understand how to convert a simple 1D code into a 2D code\" since this is the main activity of the exercise. Unfortunately we do not have precise data to pinpoint the perceived difficulty for the subset of students who reported the least prior programming experience. But with only 1 student reporting ``Extremely Difficult!\" and 8 students reporting ``Difficult!\" compared to the 39 students who reported either ``No\" or ``a little bit\" of prior programming experience, the data supports the idea that students were able to complete the 1D to 2D conversion of the code without severe difficulty. Whether they fully understand the changes that were made is another important question that we can probe in future work.\n\nThe other learning goal was ``Understand how force, velocity and acceleration vectors relate to the motion of a ship traveling in free space\". Although we do not have a direct probe of this learning goal, the question ``Did the programming lab help you understand vectors better?\" relates to this learning objective in an indirect way. Many of the students found the exercise to be at least ``somewhat\" helpful in understanding vectors. As mentioned in the last section, student self-reporting can be useful and even predictive of student learning \\cite{Sawtelle_etal2012}. In retrospect, one wonders if even more students would have reported understanding vectors better if there had been a part of the exercise where the student gives the ship an initial velocity and interacts with the program from that starting point, or if we had included the activity described earlier where students change the force (thrust) and mass of the ship (Fig.~\\ref{fig:withplanetoids}) to see the effect on the motion in avoiding asteroids (an activity which was only added later).\nIt is also key to note that learning gains can only be achieved if students do actually engage with the activity. The question most closely related to this was ``Was the programming lab fun?\" An overwhelming majority of the students found the exercise to be ``enjoyable\" or ``fun\" which suggests that they did significantly play around with the simulation (which demonstrates the relationship between force, velocity and acceleration vectors in an interactive way, making it very relevant to the goal of better understanding these vectors). It is therefore reasonable that there may be sufficiently high student \"buy-in\" to warrant further study, and further optimization of the user interface to maximize learning outcomes as described above.\nNevertheless, the questions discussed here are still oblique, self-reported measures of student learning on these learning goals and we do not wish to overstate the results we obtained. \n\nIn future work we can directly probe the second learning goal using, for example, the rocket questions from either the Animated Force Concept Inventory by \\citet{Dancy2006} or the conventional Force Concept Inventory \\cite{FCI}, and other questions that ask students to identify the correct force, velocity and acceleration vectors in different situations. Importantly, we can compare results for these questions from students who complete a coding activity, and a ``control group\" of students who only play around with the interactive for that coding activity for some period of time but without actually seeing or modifying the code. This will probe whether coding activities of the kind we discuss here, which involves multiple steps where students modify the code and check the behavior of the program, cause students to look more critically at the interactives they produce than they would if they did not have to perform coding tasks.\n\n\n\n\n\n\n\n\\section{Summary and Conclusion}\n\nIn this paper we illustrate a ``hybrid\" approach to incorporating computer programming activities into introductory physics courses by describing a coding activity that resembles the classic asteroids game. The approach is so named because activities like the one described here produce interactives that bear some resemblance to web interactives that groups like PhET and Physlet have produced, but unlike PhET and Physlet, the student works with and modifies the code that evolves the system. In a ``traditional\" computational physics course the student would have a great deal of control over producing visualizations. To reduce the cognitive load for weak or absolute beginner programmers in our study, the parts of the code that are unrelated to physics are hidden away in a \\texttt{display()} function so that the student sees and works with only about 50 lines of code. In this sense our approach is a kind of ``hybrid\" between canned interactives and mature computational physics exercises that are typically used in physics-major courses.\n\nThe first exercise in our suite of activities is an interactive simulation that resembles the classic game ``asteroids\". The learning goals of this activity are to (1) understand how to convert a simple 1D code into a 2D code and (2) to understand how force, velocity and acceleration vectors relate to the motion of a ship traveling in free space. The activity includes scaffolding and hints to make the task of modifying the 1D code into 2D more manageable.\n\nIn an introductory class at OSU Marion where a substantial fraction of the students are weak or absolute beginner programmers, student survey data ($N \\approx 80-85$) confirms that most students, including those with weak or absolute beginner programming experience, are able to complete the activity without severe difficulties. We interpret this as evidence that the first learning goal is being met. \n\nWe are still only just beginning to investigate the effectiveness of the second learning goal. We discuss survey results that provide some insight into student experiences with the exercises, which in an indirect way addresses the second learning goal. However, this is no substitute for directly probing student learning with carefully chosen questions. In future work we will use the Animated Force Concept Inventory \\cite{Dancy2006}, and other assessments to probe whether students understand the relationship between velocity and acceleration vectors. Of particular importance is whether the task of making modifications to the code and checking for the effect of these modifications on the interactive program will cause students to think more critically about the physics concepts than they would by playing around with a ``canned\" interactive. This may be the real value of integrating coding at this level. \n\n\n\nWe welcome inquiries from educators who may wish to use this suite of coding activities in their courses. Individual exercises and solution sets (including the planetoids game described here) are available at \\url{http:\/\/compadre.org\/PICUP}\n\n\\acknowledgements\n\nThe authors thank Chris Britt and Michael Hardesty for their collaboration on a p5.js learning management system. Chris Orban thanks Kathy Harper, Gregory Ngirmang, and Kelly Roos for discussions. This project was made possible through a Connect and Collaborate Grant, a program supporting innovative and scholarly engagement programs that leverage academic excellence of The Ohio State University in mutually beneficial ways with external partners. Support also comes from the American Institute of Physics Meggers Award.\n\n\\bibliographystyle{apsrev}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{\\label{} Introduction}\n\nThe complex interplay between spin, charge, and lattice degrees of freedom in the quasi-two dimensional copper-oxide \nhigh temperature superconductors have been the subject of intense interest since the discovery of\nsuperconductivity in the La$_{2-x}$Ba$_x$CuO$_4$\\ system some 21 years ago\\cite{Bednorz:1986}. Both La$_{2-x}$Ba$_x$CuO$_4$\\ and La$_{2-x}$Sr$_x$CuO$_4$\\ display a fascinating series of structural, magnetic and superconducting phase transitions as a function of temperature\\cite{Kastner:1998}. While La$_{2-x}$Ba$_x$CuO$_4$\\ was the first layered cuprate high T$_{c}$ superconductor to be discovered, difficulties associated with the \ngrowth of high quality single crystals have significantly limited its study. As a result the La$_{2-x}$Ba$_x$CuO$_4$\\ family is much less studied than the La$_{2-x}$Sr$_x$CuO$_4$\\ family and other high temperature superconductors which have an extended history of being grown and characterized in single crystal form\\cite{Kastner:1998}, such as the YBa$_{2}$Cu$_3$O$_{7-\\delta }$\\ and Bi$_2$Sr$_2$CaCu$_2$O$_8$\\ families\\cite{Birgeneau:2006,Eschrig:2006,Fong:1999,Castellan:2006}. \n\nRecently, significant progress has been made in growing the La$_{2-x}$Ba$_x$CuO$_4$\\ family of materials in single crystal form, and this has enabled several important new studies of this and related systems\\cite{Fujita:2004,Reznik:2006,Tranquada:2004,Kimura:2005}. It is therefore timely to perform high resolution structural studies of these new single crystals, and to compare to previous studies on La$_{2-x}$Ba$_x$CuO$_4$\\ in polycrystalline form\\cite{Suzuki:1989,Suzuki:1989a}.\n\nOne of the many interesting properties of the La$_{2-x}$Ba$_x$CuO$_4$\\ family is the sequence of structural phase transitions which this material displays on cooling \nbelow room temperature for underdoped Ba concentrations (x$\\lesssim$ 0.18). Previous studies on polycrystalline La$_{2-x}$Ba$_x$CuO$_4$\\ shows three different structures, which proceed from High Temperature Tetragonal (HTT, $I4\/mmm$), to Middle Temperature Orthorhombic (MTO, $Cmca$) and finally to Low Temperature Tetragonal (LTT, $P4_{2}\/ncm$)\\cite{Axe:1989,Axe:1989a,Suzuki:1989,Suzuki:1989a,Adachi:2001}. The HTT$\\to$MTO and the MTO$\\to$LTT phase transition temperatures are referred to as $T_{d1}$ and $T_{d2}$, respectively. The HTT$\\to$MTO transition is \ncontinuous, while the MTO$\\to$LTT transition is known to be strongly discontinuous. These structures are closely \nrelated to the magnetic and electronic properties of the La$_{2-x}$Ba$_x$CuO$_4$\\ and La$_{2-x}$Sr$_x$CuO$_4$\\ families. The phase diagram of the La$_{2-x}$Ba$_x$CuO$_4$\\ system contains a dome\nof LTT phase, which is centred around x=0.125. This Ba-concentration corresponds to a steep depression of the superconducting $T_C$ as a function of concentration, known as the 1\/8 anomaly\\cite{Axe:1989a,Moodenbaugh:1988}. The La$_{2-x}$Sr$_x$CuO$_4$\\ system shows a much smaller $\\sim$ 10$\\%$ dip in $T_C$ at x=0.125 and the absence of the LTT phase at low temperatures\\cite{Nagano:1993,Radaelli:1994}. The 1\/8 anomaly within the LTT phase also corresponds to strong\nincommensurate magnetic long range order at temperatures just below the completion of the MTO-LTT phase transition\\cite{Fujita:2004,Tranquada:2004}. Clearly, the structural,\nmagnetic, and superconducting properties of the La$_{2-x}$Ba$_x$CuO$_4$\\ and La$_{2-x}$Sr$_x$CuO$_4$\\ systems are strongly coupled.\n\nThe critical phenomena associated with the HTT-MTO transition has been previously studied in pure La$_{2}$CuO$_{4}$ as well as in La$_{2-x}$Sr$_x$CuO$_4$\\ in single crystal and polycrystal form\\cite{Birgeneau:1987,Vaknin:1987,Boni:1988,Braden:1994,Ting:1993,Thurston:1989}, as single crystals of these materials have existed for some time. These\nstudies show the HTT$\\to$MTO phase transition to be characterized with an order parameter critical exponent $\\beta$ varying from 0.28 to 0.37\\cite{Birgeneau:1987,Vaknin:1987,Boni:1988,Braden:1994,Ting:1993,Thurston:1989}. Studies on polycrystalline samples of La$_{2-x}$Ba$_x$CuO$_4$\\ by Susuki et al\nproduced estimates for $\\beta$ $\\sim$ 0.33\\cite{Suzuki:1989,Suzuki:1989a}, and which are consistent with expectations for 3D universality\\cite{Collins:1989}.\n\nIn this paper, we report the successful growth of large La$_{2-x}$Ba$_x$CuO$_4$\\ single crystals with x=0.095 and 0.08, and a high resolution x-ray diffraction study on the x=0.125, 0.095 and 0.08 compounds in this family. This study focusses on a comparison between the structural and superconducting phase diagrams in polycrystalline and single crystal materials, critical phenomena associated with the HTT$\\to$MTO phase transition, and the nature of the LTT phase in x=0.125 and 0.095 samples at low temperatures.\n\n\n\\section{\\label{} Experiment details}\n\\subsection{\\label{} Crystal Growth}\n\n\\begin{figure}\n\\includegraphics[width=0.9\\columnwidth]{map125_arxiv.eps}\n\\caption {(a), High resolution longitudinal scans of the (3, 3, 0)$_{HTT}$ Bragg peak in single crystal La$_{2-x}$Ba$_x$CuO$_4$, x=0.125 are shown as a function of temperature. (b) Representative longitudinal scans at T=290 K, 120 K, and 20 K from which the color contour map in (a) was made.} \n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=0.9\\columnwidth]{map095_arxiv.eps}\n\\caption {(a), High resolution longitudinal scans of the (3, 3, 0)$_{HTT}$ Bragg peak in single crystal La$_{2-x}$Ba$_x$CuO$_4$, x=0.095 are shown as a function of temperature. (b) Representative longitudinal scans at T=290 K, 120 K, and 20 K from which the color contour map in (a) was made.} \n\\end{figure}\n\n\n\nWe studied three high quality La$_{2-x}$Ba$_x$CuO$_4$\\ single crystals with x=0.125, 0.095 and 0.08. All crystals were grown by using traveling solvent, floating zone image furnace techniques. The x=0.125 sample was grown separately, and the details of this growth have been previously discussed\\cite{Fujita:2004,Tranquada:2004}.\n\nThe x=0.095 and 0.08 La$_{2-x}$Ba$_x$CuO$_4$\\ single crystal growths followed similar processess, and employed polycrystalline La$_{2}$O$_{3}$, BaCO$_{3}$ and CuO as starting \nmaterials to make the initial, polycrystalline feed rod and solvent. For the production of the feed rods, the starting materials were mixed to\ngive an initial ratio of La:Ba:Cu=1.875:0.125:1. These materials were mixed, ground, and annealed at 980$^{\\circ}$C for 12 hours \nin air. This process was repeated twice in order to ensure homogeneous feed rods. To compensate for Cu evaporation during the crystal growth, the pre-annealed feed rods were mixed with extra CuO. A further 1\\% and 2\\% mol CuO was added to the starting polycrystalline materials and thoroughly mixed to prepare the two final feed rods, respectively. The final feed rods were heated to a temperature of 1190$^{\\circ}$C, at a rate of 100$^{\\circ}$C\/hour. They were held at this temperature for 12 hours. We also employed a solvent, formed from the original polycrystalline feed rod, with CuO added so as to reach a final ratio of constituent atoms (La$_{1.875}$Ba$_{0.125}$):Cu=3:7. After mixing and sintering, small disks weighing $\\sim$ 0.44 g were cut out and used as solvents in the subsequent single crystal growths.\n\nThe single crystal growths were carried out using a four-mirror image furnace (Crystal System Inc.). A small pure La$_{2}$CuO$_{4}$ single crystal was employed as the seed rod for both growths. The growths were carried out in an O$_2$ atmosphere at pressures of 165 kPa and 182 kPa for the two crystal growths. The growth rate was 1mm\/h with a counter-rotation speed of 25 rpm for feed and seed rods for both growths.\n\n\n\n\nUpon completion of the growths, the as-grown single crystals were kept above 100$^{\\circ}$C in a furnace to prevent hydrolysis of the \nmaterial, which is known to be problematic for single crystal La$_{2-x}$Ba$_x$CuO$_4$. The two crystals, which are identified in this study as being at x=0.095 and 0.08, were of almost identical dimensions of 80 mm long by 5 mm in diameter as-grown. Within the first week following completion of the growths, the initial $\\sim$ 30 mm of the crystals turned to dust as a result of hydrolysis of the second phase. The undamaged part of both crystals was stable. They had approximate dimensions of 50 mm long by 5 mm in diameter for x=0.095 and 55 mm long by 5 mm in diameter for x=0.08.\nThese volumes are sufficiently large for advanced characterization by neutron scattering techniques, and indeed a program of neutron measurements has been carried out on these samples\\cite{Dunsiger:2007}. \n\nWe note that while the two crystal growths were initiated with similar starting materials, and the growths followed similar procedures, the Ba\/La ratio, as identified by T$_{d1}$ and T$_{d2}$, were different at the $\\sim$ 15$\\%$ level. This originates from Cu evaporation during the growth. All the phase transitions observed (structural, magnetic, and superconducting) are nevertheless very sharp in temperature, indicating excellent homogeneity of concentration within the individual single crystals.\n\n\\subsection{\\label{} X-ray diffraction}\n\nSingle crystal samples with approximate dimensions 8 mm$\\times$8 mm$\\times$1 mm for x=0.125, and 5 mm$\\times$5 mm$\\times$1 mm for x=0.095 and 0.08, were cut from large single crystals of La$_{2-x}$Ba$_x$CuO$_4$. These were sequentially attached to the cold finger of a closed cycle refrigerator and mounted within a four circle x-ray \ndiffractometer. Cu K$_{\\alpha 1}$ radiation from an 18kW rotating anode x-ray generator was selected using a perfect Germanium (111) single crystal monochrometer. A Bruker Hi-Star multi-wire area detector was placed on the detector arm, 76 cm from the sample allowing an angular resolution of approximately 0.01 degrees to be achieved. All measurements focused on (3, 3, 0)$_{HTT}$ Bragg peak of the samples, using notation appropriate to the high temperature tetragonal phase. As we were interested in critical phenomena, the sample was mounted in a Be can and in the presence of a helium exchange gas and the sample temperature was stabilized to $\\sim$ 0.005 K for all measurements. \n\n\\section{\\label{} Experimental Results}\n\n\\subsection{\\label{} Identification and Nature of Phases}\n\n\n\n\\begin{figure}\n\\includegraphics[width=0.9\\columnwidth]{LattvsT_all.eps}\n\\caption {(a) The orthorhombic strain vs. temperature is plotted for La$_{2-x}$Ba$_x$CuO$_4$\\ x=0.125, 0.095 and 0.08 single crystal samples. The open and filled symbols represent warming and cooling cycles, respectively. The orthorhombic strain is obtained by fitting longitudinal scans, shown in Figs. 1 and 2. (b) The same orthorhombic strain vs. temperature as in (a) but now plotted vs T\/T$_{d1}$ and the strain has been scaled (for the x=0.125 sample, by a factor of 2.4) to emphasize universal behavior for T\/T$_{d1}$ greater than 0.8. } \n\\end{figure}\n\nTwo dimensional maps of the scattering around the (3, 3, 0)$_{HTT}$ Bragg peaks of all three x=0.125, 0.095 and 0.08 La$_{2-x}$Ba$_x$CuO$_4$\\ samples were acquired as a function of temperature. Each data set consisted of a sample angle rock through the Bragg peak which was integrated in the vertical \ndirection and plotted as a function of scattering angle, 2$\\theta$. A longitudinal cut through this two dimensional data set was performed, giving rise to the longitudinal scans shown in Fig. 1b for the x=0.125 sample, and Fig. 2b for the x=0.095 sample. Similar data sets taken over a more restricted temperature regime for the x=0.08 sample are of similar quality, but are not shown.\n\nThese data sets can be put together to display the full temperature dependence of the longitudinal scans, and this is what is shown in Figs. 1a and 2a for the x=0.125 and x=0.095 samples, respectively. These data sets clearly show the bifurcation of a single Bragg peak into two, and then back into one, as the temperature is decreased from room temperature to 20 K, signifying the sequence of phase transitions HTT$\\to$MTO$\\to$LTT. The fact that two Bragg features can be seen in a single longitudinal scan within the MTO phase is indicative of twinning within the orthorhombic phase, although the two twin domains which are observed do not possess equal volume fraction within the crystal; one Bragg feature is considerably stronger in intensity than the other. A minority and majority twin domain is clearly present, but the relevant volume fraction can change from one thermal cycle to the next. For example, the x=0.095 data set shown in Fig. 2(a) shows data from two independent thermal cycles, one ending with a lowest temperature of $\\sim$ 200 K, while the next beginning a new thermal cycle at 200 K. In the first of these, the high angle Bragg peak is the majority domain, while in the second cycle, the lower angle Bragg peak is the majority domain. \n\n\n\\begin{figure}\n\\includegraphics[width=0.9\\columnwidth]{Latt_Tzoom.eps}\n\\caption {The orthorhombic strain is plotted vs reduced temperature, (T-T$_{d1}$\/T$_{d1}$) for the x=0.125, 0.095, and 0.08 La$_{2-x}$Ba$_x$CuO$_4$\\ samples at small values of reduced temperature, near T$_{d1}$. The open and filled symbols show data from warming and cooling cycles, respectively. Fits of the data to the form of the order parameter squared vs reduced temperature, Eq. 1, \nused to extract values of $\\beta$ are shown as the solid lines.} \n\\end{figure}\n\nThe fact that we observe both twin domains in the MTO phase means that the peak positions, the lattice parameters, and consequently the\northorhombic strain, 2(a-b)\/(a+b), can be determined as a function of temperature. This is shown for all three samples in Fig. 3a. The single (3, 3, 0)$_{HTT}$ Bragg peak breaks into (6, 0, 0)$_{MTO}$ and (0, 6, 0)$_{MTO}$ near T$_{d1}$=232 K, 272 K and 305 K in the x=0.125, 0.095, and 0.08 samples, respectively, before reforming into a single (3, 3, 0)$_{LTT}$ Bragg peak near T$_{d2}$=60 K, 45 K, and 35 K, respectively.\n\nExamination of Figs 1-3 shows two qualitative features of the evolving structures. Note that for ease of comparison, the 2$\\theta$ range of the\nscattering in Figs. 1 and 2 is the same. First, the orthorhomic strain decreases quite substantially with increasing Ba concentration. The lowest\ntemperature strain, for example, in the x=0.125 sample is roughly half that of the x=0.095 sample. Secondly and more importantly, the longitudinal profile of the (3, 3, 0)$_{LTT}$ peak at the lowest temperatures measured, well within the LTT phase, is considerably broader than the\ncorresponding profile of (3, 3, 0)$_{HTT}$. This is true for both the x=0.125 sample and the x=0.095 sample as can be seen by comparing the top and bottom panels of Fig. 1b (for x=0.125) and Fig. 2b (for x=0.095). This shows that the LTT phase is either an admixture of a tetragonal and an orthorhombic phase, as was suggested by electron microscopy on \nan earlier generation of La$_{2-x}$Ba$_x$CuO$_4$\\ crystals\\cite{Zhu:1994}, or that it is itself othorhombic with a very small orthorhombic strain. In either case it is not as ``tetragonal\" as the HTT phase, and is consistent with the ``less orthorhombic\" low temperature structures proposed previously for La$_{2-x}$Sr$_{x-y}$Ba$_y$CuO$_4$ single crystals\\cite{Fujita:2002b}. \n\n\n\\subsection{\\label{} Critical Phenomena at the HTT$\\to$MTO Phase Transition}\n\nLongitudinal scans of the form shown in Fig. 1b and 2b were fit for the purpose of extracting the peak positions in 2$\\theta$ and \ntherefore the d spacings associated with the MTO phase. This is straightforward for data far removed from the HTT$\\to$MTO phase transition, as the two peaks are well defined and separated, as can be seen in the middle panels of Fig. 1b and 2b. Closer to the phase transition, one peak may appear as a shoulder to the other, and it is more difficult to ascribe unique values to the two lattice parameters. We fit these data in two different ways in order to attain robust values for the lattice parameters close to the transition. One of these was to simply fit the longitudinal scans to sums of Lorentzians or Lorentzians raised to an adjustable exponent, while a second technique was to look for zeros in the derivatives of the intensity as a function of 2$\\theta$. These gave consistent results for the lattice parameters, giving us confidence that the orthorhombic strain could be estimated accurately close to the transition. However, this technique also gives non-zero values for the orthorhombic strain, albeit relatively small ones, within the HTT phase. \n\nPrevious work on the HTT$\\to$MTO phase transition in polycrystalline La$_{2-x}$Sr$_x$CuO$_4$\\ and La$_{2-x}$Ba$_x$CuO$_4$\\ samples show the \northorhombic strain to scale as the square of the order parameter\\cite{Birgeneau:1987,Boni:1988,Suzuki:1989,Suzuki:1989a}. Consequently we examined \nthe critical behaviour of the orthorhombic strain in our La$_{2-x}$Ba$_x$CuO$_4$\\ single crystals by fitting the measured strain as a function of temperature to: \n\\begin{equation}\n\\label{ }\n\\Delta=\\Delta_{0}\\times (\\frac{T_{d1}-T} {T_{d1}})^{2\\beta}+Background\n\\end{equation}\nwhere the square of the order parameter, $\\Delta$, is the orthorhombic strain, 2(a-b)\/(a+b), and the background accounts for finite strain within the HTT phase introduced by the fitting process described above. The results of this fitting is shown in Fig. 4, which shows the orthorhombic strain as \na function of reduced temperature, (T-T$_{d1}$)\/T$_{d1}$, in the region of small reduced temperature close to T$_{d1}$. Clearly this description of the data is very good. It results in accurate estimates for both $\\beta$ and T$_{d1}$. These are T$_{d1}$=232.3 $\\pm$ 0.7 K, 271.7 $\\pm$ 1 K, and 305.4 K $\\pm$ 1 K for the x=0.125, 0.095, and 0.08 samples, respectively. The extracted values for $\\beta$ are 0.35 $\\pm$ 0.03, 0.34 $\\pm$ 0.04 and 0.28 $\\pm$ 0.06, respectively. \n\nUsing these values of T$_{d1}$ for each of the three samples, we can scale the plot of orthorhombic strain vs temperature, Fig. 3a, so as to give scaled orthorhombic strain vs T\/T$_{d1}$, which is shown in Fig. 3b. We see that above T\/T$_{d1}$ $\\sim$ 0.8 the orthorhombic strains for all three samples collapse to a single curve. We therefore expect universal behaviour in this regime, which is borne out by the similarity in the extracted values for the critical exponent $\\beta$ at all three Ba concentrations.\n\n\\begin{figure}\n\\includegraphics[width=0.9\\columnwidth]{BetaChi.eps}\n\\caption {The dependence of critical exponent $\\beta$ and goodness-of-fit parameter $\\chi^2$ are shown as a function of the assumed value of T$_{d1}$ for x=0.125 (upper panel), x=0.095 (middle panel) and x=0.08 (lower panel) La$_{2-x}$Ba$_x$CuO$_4$\\ single crystal samples. The uncertainty in $\\beta$ is largely determined by the uncertainty in critical temperature T$_{d1}$.} \n\\end{figure}\n\n\nThe uncertainties associated with the critical exponent $\\beta$ are largely determined by the uncertainties in the critical temperature, T$_{d1}$, derived from the fits to the critical behaviour. We performed fits to Eq. 1 using T$_{d1}$ set to a range of values around the approximate phase transition temperature, and then allowed the fit to adjust the other parameters in Eq. 1. This gives a monotonically increasing estimate for $\\beta$ as a function of increasing T$_{d1}$. Best estimates for $\\beta$ and T$_{d1}$ are given by the minimum in the goodness-of-fit parameter $\\chi^2$ which we define as:\n\\begin{equation}\n\\label{ }\n\\chi^{2}=\\frac{\\sum(\\Delta_{measured}-\\Delta_{calculated})^{2}} {N}\n\\end{equation}\nwhere N is the number of data points.\n\n$\\beta$ and $\\chi^2$ are shown as a function of T$_{d1}$ for the x=0.125 (top panel), x=0.095 (middle panel), and x=0.08 (bottom panel) samples in Fig. 5. The uncertainty in $\\beta$ is determined by the corresponding uncertainty in T$_{d1}$, and it is roughly 10$\\%$ for the x=0.125 and 0.095 samples where we have an extended data set throughout the MTO phase, and roughly 20$\\%$ for the x=0.08 sample where the data set is restricted to temperatures close to T$_{d1}$.\n\n\n\\begin{figure}\n\\includegraphics[width=0.9\\columnwidth]{Latt_Tlog.eps}\n\\caption {The orthorhombic strain is plotted as a function of reduced temperature, (T$_{d1}$-T)\/T$_{d1}$, on a log-log scale for the x=0.125, 0.095 and 0.08 single crystal La$_{2-x}$Ba$_x$CuO$_4$\\ samples. The open and filled symbols show results from warming and cooling cycles, respectively. For comparison power law behavior showing $\\beta$=0.35, indicative of the theoretically expected 3D XY universality class, is indicated as the straight line on this log-log plot.} \n\\end{figure}\n\nInvestigation of the critical properties at the HTT$\\to$MTO phase transition in polycrystaline La$_{2-x}$Sr$_x$CuO$_4$\\cite{Birgeneau:1987,Boni:1988}\\ and La$_{2-x}$Ba$_x$CuO$_4$\\cite{Suzuki:1989,Suzuki:1989a}\\ anticipated 3D XY universality on the basis of a Landau expansion appropriate to this ferroelastic system. These early results on polycrystalline systems were consistent with the $\\beta$=0.35 expected from 3D XY universality\\cite{Le-Guillou:1977,Le-Guillou:1980}. However, these earlier estimates for $\\beta$ spanned the range from 0.28 to 0.37, ignoring uncertainties associated with the estimates, which covers all standard 3D universality classes: Heisenberg ($\\sim$0.37), XY ($\\sim$0.35), Ising ($\\sim$0.32) and which begins to approach values consistent with tricritical phenomena (0.25)\\cite{Collins:1989}.\n\nFigure 6 shows the orthorhombic strain, 2(a-b)\/(a+b), plotted as a function of the reduced temperature, (T$_{d1}$-T)\/T$_{d1}$, on a log-log plot \nin order to identify the expected power law regime. For comparison a straight line appropriate to $\\beta$=0.35 and 3D XY universality is also plotted. For each sample, two data sets are plotted, one for a warming run and one for a cooling run. We observe very similar power law behaviour in all three samples, and behaviour which is very much consistent with 3D XY universality as anticipated theoretically. We also see, at least for the x=0.125 and 0.095 samples for which we have data over the entire MTO phase regime in temperature, that a single power law is a remarkably good descriptor of the data over a very large temperature regime. There appears to be a slight increase in slope for reduced temperatures greater than $\\sim$ 0.2, but overall, power law-like growth of the orthorhombic strain is observed over almost two decades in reduced temperature. This is in contrast to most critical phenomena, wherein asymptotic critical behaviour is expected to cross over to a mean field-like regime, as one moves away from the critical temperature.\n\nTaken together our orthorhombic strain measurements show critical behaviour at the HTT$\\to$MTO phase transition in single crystal La$_{2-x}$Ba$_x$CuO$_4$\\ over a broad range of concentration which is characterized by $\\beta$=0.34 $\\pm$ 0.04. This result clearly demonstrates 3D universality, and is consistent with 3D XY universality which is expected based on Landau theory. It is also largely consistent with previous experimental work on single crystal and polycrystal La$_{2-x}$Sr$_x$CuO$_4$\\ and polycrystalline La$_{2-x}$Ba$_x$CuO$_4$, much of which centred on measurements of superlattice Bragg peak intensities within the MTO structure, as opposed to measurements of the orthorhombic strains\\cite{Braden:1994,Thurston:1989}. Superlattice Bragg peak intensities near continuous phase transitions can be difficult to interpret, as they can be influenced by extinction and by fluctuations above the phase transition. This latter effect manifests itself in upwards curvature and difficulty identifying a precise phase transition temperature, which in turn can lead to uncertainty in critical exponents. \n\n\\begin{table}\n\\caption{\\label{tab:table1} Summary of structural and superconducting phase transition temperatures in single crystal La$_{2-x}$Ba$_x$CuO$_4$}\n\\begin{ruledtabular}\n\\begin{tabular}{ccccc}\nx&$T_{d1}$(K)&$T_{d2}$(K)&$T_{c}$(K) & $\\beta$\\\\\n\\hline\n0.125 & 232.3 & 60 & 4\\footnotemark[1] & 0.35 $\\pm$ 0.03\\\\\n0.095 & 271.7& 45\\footnotemark[2] & 27\\footnotemark[2] & 0.34 $\\pm$ 0.04 \\\\\n0.08& 305.4 & 35\\footnotemark[2] & 29\\footnotemark[2] & 0.28 $\\pm$ 0.06 \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\footnotetext[1]{From Ref.~\\onlinecite{Fujita:2004}.}\n\\footnotetext[2]{From Ref.~\\onlinecite{Dunsiger:2007}.}\n\\end{table}\n\n\n\\subsection{\\label{} Phase Diagram and Comparison to Polycrystalline Materials}\n\nIt is of interest to compare the La$_{2-x}$Ba$_x$CuO$_4$\\ phase diagram known to characterize pre-existing polycrystalline samples with that determined for the high quality single crystals\nin the present studies. A rather detailed comparison can be carried out, as two structural and one superconducting transition temperature characterize La$_{2-x}$Ba$_x$CuO$_4$\\ samples in this underdoped concentration range. The phase transitions measured for the single crystals in this study are summarized in Table 1. The critical exponent $\\beta$\nrelevant to the HTT$\\to$MTO structural transition is also shown in the same table for reference.\n\nThe superconducting transition temperatures were determined from SQUID magnetometry as reported by Dunsiger et al.\\cite{Dunsiger:2007} for the x=0.095 and x=0.08 samples, and \nby Fujita et al.\\cite{Fujita:2004} for the x=0.125 sample. The strongly first order MTO$\\to$LTT transition is measured both by the abrupt change in the orthorhombic strain seen in Fig. 1 and 2, for the x=0.125 and x=0.095 samples, respectively, as well as by the appearance of the (0, 1, 0) superlattice Bragg peak intensity as again reported by Dunsiger et al.\\cite{Dunsiger:2007} for the x=0.095 and x=0.08 samples, and by Fujita et al. for the x=0.125 sample\\cite{Fujita:2004}. \n\n\\begin{figure}\n\\includegraphics[width=0.9\\columnwidth]{PhaseLBCO_1.eps}\n\\caption {Phase boundaries identifying structural and superconducting phases of La$_{2-x}$Ba$_x$CuO$_4$\\ single crystals are plotted on the phase diagram derived from \npreviously studied polycrystalline samples. The structural transitions at T$_{d1}$ and T$_{d2}$ are indicated by filled squares, while superconducting T$_C$'s are indicated\nby open circles. The first order transition at T$_{d2}$ is indicated by a bar $\\sim$ 10 K wide, showing the onset to completion of the phase transition. Solid lines showing phase boundaries from polycrystalline La$_{2-x}$Ba$_x$CuO$_4$\\ are taken from Adachi et al.\\cite{Adachi:2001} }\n\\end{figure}\n\n\nFigure 7 shows the La$_{2-x}$Ba$_x$CuO$_4$\\ phase diagram with HTT, MTO, and LTT phases indicated. The HTT$\\to$MTO and MTO$\\to$LTT transitions are shown as filled squares for the three Ba concentrations measured. The discontinuous transition at T$_{d2}$ is indicated as a bar, in order to show the onset to completion of the transition, which is $\\sim$ 10 K wide. T$_{d2}$ in Table 1 is the midpoint of the transition. The superconducting transitions are given by the open circles, and they indicate the onset of the superconductivity, which is also what is listed in Table 1. Previous results for these same phase boundaries as determined for polycrystalline La$_{2-x}$Ba$_x$CuO$_4$\\ samples are shown as the solid lines in Fig. 7. These results were extracted from Adachi et al.\\cite{Adachi:2001} and are reproduced here.\n\nAs can be seen on inspection of Fig. 7, the agreement between the structural and superconducting phase boundaries in polycrystalline La$_{2-x}$Ba$_x$CuO$_4$\\ and the new floating zone image furnace grown single crystals is remarkably good. The absolute values for T$_{d2}$ are systematically high, at the 10$\\%$ level for the polycrystalline materials as compared to the single crystals, but overall the full level of agreement is excellent. In particular we see that good agreement between the two for T$_{d1}$ means that this transition can be used as an accurate marker for the Ba concentration in single crystal La$_{2-x}$Ba$_x$CuO$_4$, as T$_{d1}$ has such strong Ba dependence. The image furnace single crystals were grown without crucibles, and are expected to be of higher purity than the corresponding polycrystalline materials grown from a flux melt in a crucible. The similarity between the overall phase diagrams in polycrystalline and image furnace grown single crystal La$_{2-x}$Ba$_x$CuO$_4$, implies an insensitivity of these phase boundaries to this level of imperfection.\n\n\\section{\\label{} Conclusions}\n\nWe have successfully grown large single crystals of La$_{2-x}$Ba$_x$CuO$_4$\\ with x=0.095 and 0.08 using floating zone image furnace techniques. These single crystals are sufficiently large so as to enable neutron scattering studies, which will be reported separately\\cite{Dunsiger:2007}. High resolution single crystal x-ray diffraction measurements were carried out on these samples, as well as on a high quality x=0.125 single crystal. These measurements focus on the (3, 3, 0)$_{HTT}$ Bragg peak and show the HTT$\\to$MTO$\\to$LTT sequence of structural phase transitions known to be relevant to underdoped La$_{2-x}$Ba$_x$CuO$_4$. The measurements also clearly show anomolous longitudinal broadening of the (3, 3, 0)$_{LTT}$ Bragg peaks in the x=0.095 and x=0.125 samples at low temperatures, indicating that the LTT phase is not a simple tetragonal phase, but rather an admixture of tetragonal and orthorhombic phases, or an orthorhombic phase with very small orthorhombic strain. Critical\northorhombic strain measurements near the continuous HTT$\\to$MTO phase boundary show clear 3D universality, with universal behavior observed in the orthorhombic strain vs T\/T$_{d1}$ for the three x=0.125, 0.095 and 0.08 samples. The best estimate for a common critical exponent $\\beta$ for these samples is $\\beta$=0.34 $\\pm$ 0.04, which is consistent with 3D XY universality expected theoertically for such ferroelastic transitions. A detailed comparison of the La$_{2-x}$Ba$_x$CuO$_4$\\ phase diagram incorporating structural and superconding phase boundaries at this underdoped concentration regime indicates excellent agreement with pre-existing data based on polycrystalline samples.\n\nIt is a pleasure to acknowledge the contributions of Ms. Ann Kallin to the single crystal growth. This work was supported by NSERC of Canada. Gu was supported by the US Department of Energy under contract number DE-AC02-98CH10886.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nA carbon-based materials have now become the most promising candidates for nanoelectronic applications. Applications of carbon nanotubes (CNTs) and graphenes have already emerged; for example, field effect transistors (FETs) \\cite{Javey2003,Misewich2003,Ouyang2007}, electrical interconnects \\cite{Avouris2007}, and sensors \\cite{Siwy2010,Prasongkit:2011}. However, linear carbon wires; namely, cumulene and polyyne, received poor attention owing to the difficulty of getting access to the pure wire \\cite{Haley:2010jc}. The cumulene, in particular, was proposed as the ideal molecular wire \\cite{Lang:1998}. However, it is still challenging to create such a metal-molecule junction in the experiment, becuse cumulene wire is generally unstable.\n\n\n\nThe appearance of the cumulene bridging CNTs or graphene was recently observed in the experiment \\cite{Jin:2009p592,Chuvilin2009,Troiani:2003p990,Marques:2004p465} by using electron irradiation inside a high resolution transmission electron microscope (TEM), thus demonstrating a mechanical way of producing the carbon wire. An axial strain on the graphene or CNT was induced by the high-energy electron beam of TEM resulting in the fracture, and led to the establishing carbon wire-CNT junction \\cite{Jin:2009p592,Troiani:2003p990}. The observed carbon wires were more stable than those produced with previous approaches; however, the long wire ($>$10 carbon atoms) appeared to be unstable \\cite{Troiani:2003p990}. Recently, the carbon wire bridged between graphene sheets was shown to perform as a bistable switch \\cite{Standley2008,Zhang:2012}, operating for many thousands of cycles without degradation, which can be interesting from an application point of view.\n\n\nThe theoretical investigations have been carried out on the structure of carbon wire-CNT junctions \\cite{Marques:2004p465,Enyashin:2005ve}. Before breaking of the junction, the carbon wire bridging between CNTs or graphene sheets can transform into either cumulene or polyyne \\cite{Ajayan:1998p1241, Jin:2009p592,Marques:2004p465}. By using the non-equilibrium Green's function (NEGF) approach, Khoo \\emph{et. al}\\cite{Khoo:2008p688} observed negative differential resistance in the carbon wire-capped CNT junction, bonded through sp$^{3}$ bond. Recently, the carbon wire connected to graphene sheets \\cite{Zhang:2010fm,Shen2010} (representing infinite radii of CNT) have been theoretically studied, revealing different electronic functions such as molecular switches \\cite{Erdogan:2011bt}, molecular rectifiers \\cite{Zeng:2011jo}, and molecular spintronics devices\\cite{Zanolli2010}. Carbon wires connected to gold \\cite{Prasongkit:2010p780,Crljen:2007p452}, lithium \\cite{Zhang13} and fullerene \\cite{Wang:2011jf} electrodes have also been investigated. \n\n\nIn this paper, we present the transport properties of the short carbon wire suspended between CNT electrodes by employing NEGF technique based on density functional theory. Varying the gap width between the electrodes changes the wire structure and contact geometries in the junction. The latter is specific for CNT electrodes, in which the bridging site can appropriately adjust to the change of junction lengths. We have focused on the variation in conductance of the carbon wire via repeated compression\/elongation of the junction, effectively leading to a prominent difference in conductance. We consider here the zigzag (4,0)CNT and armchair (4,4)CNT, due to a very small diameter of the CNTs observed in the experiment before fracture. \\cite{Troiani:2003p990}. The influence of CNT chiralities (zigzag and armchair) on conductance has been investigated. In addition, oscillating behavior of conductance, typical for cumulenes of different lengths \\cite{Lang:1998,Lang:2000tp,Emberly2009} is reproduced. Upon junction stretching, the current-voltage characteristics of the carbon wire-zigzag junctions show the high- and low-conduction states (referred as ON and OFF states) at elevated bias, corresponding to the cumulene and polyyne structures, respectively. It should be emphasized that we produced the ON and OFF states by changing the wire configuration without breaking wires, while Standley et al.\\cite{Standley2008} have demonstrated experimentally that the switch works by breaking the cumulene wire.\n\n\nExperimental studies of the graphene and CNT edge have revealed several complex rearrangements \\cite{Caglar,Huang2009,Liu:2009iu}. The migration of carbon wire along the graphene edge was observed in the experiment \\cite{Jin:2009p592}: the carbon wire could jump along the graphene or CNT edges with a change of bonding site at the junction due to a strain accumulating along the chain. We show the conductance of the C$_5$ wire connected to all possible bridge sites on the (4,0)CNT. \nInteresting features in the electron transport properties of the carbon wires are revealed, including the possibility of current modulation via carbon wire shrinking and stretching without breaking wires. We employ a detailed analysis of transport channels in carbon wire-CNT junctions to explain our results.\n\n\n\\section{Computational Methods}\n\n\\begin{figure}[tp]\n\\begin{minipage}[ht]{1.0\\linewidth}\n\\begin{center}\n\\includegraphics[width = 8 cm]{cnt_unit.pdf}\n\\end{center}\n\\end{minipage}\n\\caption{Two-probe systems for measuring the conductance of carbon wires, connected to semi-infinite zigzag (4,0)CNT (left panel) and armchair (4,4)CNT (right panel).}\\label{grap_unit}\n\\end{figure}\n\nIn this section, we briefly describe details of the computational method and the geometrical setup procedure performed in the present work.\n\nThe carbon wire-CNT junction is referred as a two-probe system, divided into three regions: the left and right electrodes, and the central region (see Fig. \\ref{grap_unit}). To construct the carbon wire-CNT junction, an infinite cumulene and CNT were first optimized separately, and then the short wire was placed between electrodes. The central region includes the CNT on either side of the junction in order to ensure that the perturbation effect from the carbon wire edge is sufficiently screened. The carbon wire-CNT junction was optimized again, allowing all atoms in the central region to relax. Then, the gap width between electrodes was varied with a step of 0.2 \\AA.\n\n\nAll optimizations were carried out by using density functional theory (DFT) as implemented in the SIESTA code\\cite{Soler2002}. Our calculation was performed within the non spin-polarized generalized gradient approximation (GGA) \\cite{Perdew1996} method with a single-$\\zeta$ with polarization (SZP) basis, which has already been presented its validity for the short carbon wire-CNT junction \\cite{Khoo:2008p688}. The atomic core electrons were modeled with Troullier-Martins norm-conserving pseudo potential \\cite{Troullier1991}, and valence states are 2s2p for C. The real-space integrations were performed using 170 Ry cutoff, assuring the energies and forces were converged. All atoms in the central region were allowed to move until the forces were less than 0.01 eV\/\\AA.\n\n\\begin{figure}[tp]\n\\begin{minipage}[ht]{1.0\\linewidth}\n\\begin{center}\n\\includegraphics[width = 8.5 cm]{sum.pdf}\n\\end{center}\n\\end{minipage}\n\\caption{Various structures of the short carbon wires (C$_{4}$ and C$_{5}$) bridged between zigzag (4,0)CNTs and armchair (4,4)CNTs, in which the gap width is varied. Different geometrical structures and bridging sites of the carbon wires are represented by I, II, III, etc. The linear carbon wire can transform into either cumulene ``C\" or polyyne ``P\". }\\label{sum}\n\\end{figure}\n\nThe transport calculation was performed with the SMEAGOL code \\cite{Rocha:2005,Rocha:2006}, based on the combination of the NEGF and DFT. The basis set and the real-space integrations used in the electron transport calculation were the same as that of geometrical optimization part. We have performed test calculations by using a double-$\\zeta$ with polarization (DZP) basis set, resulting in only minor change of the transmission function.\n\n\nThe current through the junction was calculated using the Landauer-Buttiker formula \\cite{Datta1995,Brandbyge2002}:\n\n\\begin{equation}\nI=\\frac{2e}{h}\\int^{\\infty}_{-\\infty} dE [f(E,\\mu_L)-f(E,\\mu_R)]T(E,V),\n\\end{equation}\nwhere $\\mu_L$ and $\\mu_R$ are the electrochemical potentials of the left and right electrodes, respectively. $T(E,V)$ is the transmission coefficient at energy $E$ and bias voltage $V$, which is evaluated as\n\\begin{equation}\nT(E,V)=\\mbox{Tr}[G(E)\\Gamma_{L}G^{\\dag}(E)\\Gamma_{R}],\n\\end{equation}\nwhere $G(E)$ and $G^{\\dag}(E)$ are retarded and advanced Green's function of the central region.\n\n\n\\section{Atomic structures}\\label{sec3}\n\nWe categorize our systems into two groups: the carbon wire connected to the zigzag and armchair CNTs. We focus on the interval of 1D carbon wire existence. Pulling the wire beyond this interval results in the wire breakage, while compressing the wire more leads into folding structures. Fig.~\\ref{sum} demonstrates the various wire structures and contact geometries of the C$_{4,5}$ wire connected to the zigzag (4,0)CNT and to the armchair (4,4)CNT, varying the gap width. The zigzag and armchair junctions are labeled as (4,0)@C$_{n}$ and (4,4)@C$_{n}$ respectively, where $n$ is the number of carbon atoms in the wire. Different geometrical structures of the wires are labeled by I, II, III, etc. The linear carbon wire can take either cumulene ``C\" or polyyne ``P\" structures.\n\n\nLet us first discuss geometrical structure of the carbon wire-zigzag(4,0)CNT junction. As illustrated in Fig.~\\ref{sum}, the C$_{4}$ wire is slightly bent due to compression of the junction, labelled as (4,0)@C$_{4}$-I(C), and then becomes the straight-wire when the junction is gradually pulled apart, labelled as (4,0)@C$_{4}$-II(C). These C$_{4}$ wires, forming two bonds to each side of CNTs, are the cumulene structures with double bonds between neighbouring atoms. The central bond length of C$_{4}$ wire can vary in the range of $\\sim$ 1.23 \\AA \\ - 1.36 \\AA \\, depending on the variation of gap width. Note that the bond length alternation of the C$_4$ wire cannot be observed, since there is only one central bond in the wires. However, we have tested for C$_{6,8}$ wires, showing the bond length alternation of $\\sim$ 0.03-0.05 \\AA \\ upto the gap width. Before wire rupture, the C$_{4}$ wire forms one bond to each side of electrodes, labelled as (4,0)@C$_{4}$-III(P). At this step, the average bond lengths of the C$_{4}$ wires are 1.27 \\AA \\ and 1.40 \\AA, which is a polyyne structure due to the triple and single bond alternation.\n\n\nFor the C$_{5}$ wire, as seen in Fig.~\\ref{sum}, the bent wire is observed in the compressed junction, labelled as (4,0)@C$_{5}$-I(C). Then it becomes straight while pulling the junction, labelled as (4,0)@C$_{5}$-II(C). These C$_5$ wires, forming two bonds to each side of CNTs, show the cumulene structure. During compressing and pulling of the (4,0)@C$_{5}$-I(C) and (4,0)@C$_{5}$-II(C), the bond lengths of the cumulene wires vary in the range of $\\sim$ 1.26 \\AA \\ - 1.34 \\AA. There is no bond length alternation for the odd cumulene wire. \n\n\nIf we continue to stretch the junction, the C$_{5}$ wire forms one bond to one side of CNTs and two bonds to the other side, labelled as (4,0)@C$_{5}$-III(P). Note that this configuration is not even metastable for the C$_{4}$ wire due to broken symmetry. The average bond lengths of the C$_{5}$ wire are $\\sim$ 1.28 \\AA \\ and 1.42 \\AA, revialing the polyyne structure. In particular, the bonding geometry of the (4,0)@C$_{5}$-III(P) is similar to a transition state in the possible migration pathway for the migration of a 3\\% strained carbon chain along the graphene sheet with zigzag edge reported in Ref. \\citenum{Jin:2009p592}. We would like to emphasize that a curvature of the CNT plays an important role in the structure of the wire at this stage of elongation: the tube edge becomes capped instead of being an open-ended one. This results in a long bond along a chord across the CNT towards the next hexagon, thus defining a single bond to the wire and yielding shorter triple bonds in polyyne. The far more symmetric cumulene structure is less affected by the CNT curvature, but the bonding geometry is changed as compared to graphene, namely the wire is bonded to the site between two adjacent hexagons. Before breaking of the contact, the C$_{5}$ wire forms one bond to each side of CNTs, and becomes the cumulene structure again, (4,0)@C$_{5}$-IV(C). Typically, bond lengths of single, double, and triple bonds are 1.54 \\AA, 1.34 \\AA \\ and 1.2 \\AA, respectively \\cite{Hino2003}. Before the wire fractures, the central bond lengths of (4,0)@C$_{5}$-IV(C) are stretched so that those bond lengths can be $\\sim$ 1.40-1.43 \\AA, which are intermediate between single bond and double bond.\n\n\nIn the case of armchair junction, the wire structures and contact geometries at each step of varying the gap width are similar to that of the zigzag junction (Fig.~\\ref{sum}); thus, both zigzag and armchair junctions are labeled in the same way. Furthermore, we find that the geometrical structures of the carbon wire-graphene junctions are also similar to those of the carbon wire-CNT junction discussed above.\n\n\nIn conclusion, the gap width variation leads to a change in the wire structures and bonding geometries of the carbon wire connected to the zigzag (4,0)CNTs or armchair (4,4)CNTs. Moreover, the effect of odd-even numbered wires has played an important role for a difference in the wire structure and bonding at the junction for both cases. The observed bond lengths of the wires agree well with previous works \\cite{Senapati:2005p2333,Molder:2004dw,Khoo:2008p688}. In particular, the geometrical structure of the carbon wire bridging CNTs are similar to the experimentally observed structure of the cumulenes connected to the graphene, \\cite{Jin:2009p592} showing pathways for the migration of the carbon wire along the graphene edge. Differences in the geometry of the junctions as compared to the reported for the graphene are attributed to the finite curvature of the CNT electrodes.\n\\section{Electron transport properties}\n\\subsection{Zigzag vs. armchair junction}\n\n\n\\begin{figure*}[tp]\n\\begin{center}\n\\includegraphics[scale = 0.75]{binding-tran-cnt.pdf}\n\\end{center}\n\\vspace{-20pt}\n\\caption{Binding energies vs gap width of the C$_4$ and C$_5$ wires connected to the (4,0)CNT (left panel) and (4,4)CNT (right panel) electrodes, and corresponding zero-bias transmission on the energy-length plane $T(E, L)$ (bottom). The location of a transition state of structures is marked by red dots (shown in the upper panel). The length-dependent transmission is correlated with the binding energies and wire structures.}\\label{binding}\n\\end{figure*}\n\nFig.~\\ref{binding} presents the binding energies and the zero-bias transmission $T(E, V=0)$ projected on the energy-length plane of the C$_{4}$ and C$_{5}$ wires connected to the zigzag (4,0)CNTs and armchair (4,4)CNTs. The length-dependent transmission is correlated with the binding energies and wire structures. We show that varyation of the gap width results in a substantial change in the wire structure and contact geometries which in turn affects transport properties of the junction.\n\nLet us first discuss the zero-bias transmission $T(E,V=0)$ of the zigzag junctions in some details. We note that the effect of (4,0)CNT curvature plays an important role in the transport properties; therefore, the metallic behavior of the (4,0)CNT electrode has been observed. For the C$_5$ wire-zigzag junction, shown in Fig.~\\ref{binding}a, the cumulene wires (4,0)@C$_{5}$-I(C) and (4,0)@C$_{5}$-II(C), show a broad resonance peak at the Fermi level, resulting in the high conductance. With stretching the junction, the wire changes structure to the polyyne, (4,0)@C$_{5}$-III(P), in which the intensity of the transmission peak drops to almost zero around the Fermi level. Before breakage, the wire changes structure back to the cumulene one, in which the transmission peak around the Fermi level appears again. The average conductance values of each structure: (4,0)@C$_{5}$-I(C), (4,0)@C$_{5}$-II(C), (4,0)@C$_{5}$-III(P) and (4,0)@C$_{5}$-IV(C) are 0.9G$_{0}$, 1G$_{0}$, 0.06G$_{0}$ and 0.37G$_{0}$, respectively. From the $T(E,V=0)$ discussed above, the electronic transport properties differ for the zigzag and armchair junctions. Obviously, charge transport for the polyyne wire is strongly suppressed in the zigzag case, indicating that a switching behavior will be expected at low bias voltage. We emphasize that, due to the high curvature of the (4,0)CNT, the polyyne structure is defined with 1.28 \\AA \\ - 1.42 \\AA \\ bond alternation.\n\n\nFor the C$_4$ wire-zigzag junction, its transport properties are similar to those of the C$_5$ wire. We obtain the broadened resonance peak at the Fermi level for the cumulene; (4,0)@C${_4}$-I(C) and (4,0)CNT@C$_{4}$-II(C), but there is no resonance peak for the polyyne; (4,0)@C${_4}$-III(P). The average conductance of each structure; (4,0)@C$_{4}$-I(C), (4,0)@C$_{4}$-II(C) and (4,0)@C$_{4}$-III(P), is 0.9G$_{0}$, 0.4G$_{0}$ and 0.01G$_{0}$, respectively. Note that the conductance of (4,0)@C$_{4}$-I(C) (bent C$_4$ wire) is very high, because of a short separation between the leads of the C$_4$ compressed wire.\n\n\nNext, we discuss the zero-bias transmission function $T(E,V=0)$ of the armchair junction, as shown in Fig.~\\ref{binding}b. We find that the structural change in the wire through varying the gap width affects the value of $T(E,V=0)$. Both C$_4$ and C$_5$ wires show the transmission peaks around the Fermi energy. Unlike zigzag junction, the resonance peaks, lying below the Fermi level ($E< E_F$) of the armchair junction, do not disappear from changing the wire structure from cumulene to polyyne, but the transmission drops by a factor of 2. In the armchair case, there is not enough difference between the conductance of cumulene and polyyne wires to show the switching characteristics useful for electronic applications.\n\n\\begin{figure}[bp]\n\\begin{minipage}[tp]{1.0\\linewidth}\n\\begin{center}\n\\includegraphics[scale = 0.2]{tdc.pdf}\n\\end{center}\n\\end{minipage}\n\\caption{Zero-bias transmission on the energy-length plane $T(E, L)$ and corresponding projected density of $p_{m=1}$ and $p_{m=-1}$ states for the carbon wire stretched between armchair (4,4)CNT electrodes at the two lengths corresponding to the II(C) and III(P) configurations as indicated in the Fig. \\ref{binding}.}\\label{tdc}\n\\end{figure}\n\nThe difference of the detailed transport behavior for both junctions is due to their entirely different electronic structures and bonding geometries. The bonding geometries are different for armchair and zigzag tubes (see Fig. \\ref{sum}). It is seen that, for example, a tetragon is formed at the junction for the cumulene-zigzag junction; (4,0)@C$_{4,5}$-II(C), whereas the cumulene-armchair junction; (4,4)@C$_{4,5}$-II(C), has a pentagon. Also, the bonding geometries for the polyyne structure of both tubes are different.\n\n\n\\begin{figure}[tp]\n\\begin{center}\n\\includegraphics[width = 8 cm]{oscillate.pdf}\n\\end{center}\n\\vspace{-20pt}\n\\caption{Zero-bias conductance of (4,0)@C$_{n}$-II, n= 4-9.}\\label{oscillate}\n\\end{figure}\n\n\nTo elucidate the transport properties of the armchair junction, we show the $T(V=0,L)$ and corresponding projected density of the carbon wire stretched between armchair (4,4)CNTs (Fig.~\\ref{tdc}). For the two different widths of the gap between the electrodes, the density of $p_{m=1}$ and $p_{m=-1}$ states along the molecule calculated with DFT method. The density of states correlates well with the $T(E)$ map since the carbon wire is directly coupled to the CNT. Molecular states with $p_{m=1}$ and $p_{m=-1}$ symmetry correspond to the two conducting channels in the shorter junction. In the longer junction, one of the C-C bonds breaks, and the density distribution changes for $p_{m=-1}$ states, effectively opening a wide gap in this channel and charging the molecule with $\\approx$0.1e. We note that slight mismatch of the energies, the width of states and transmission resonances are due to the further adjusted charge of the system in NEGF calculation with open boundary conditions as compared to the neutral state of the system in DFT calculation of projected DOS.\n\n\n\n\n\nThe large difference in conductance between the cumulene and polyyne can be understood from the electronic structure of the junction: electrons tunnel through extended $p_x$ and $p_y$ orbitals delocalized both in the wire and in the CNT. In the case of armchair junction, the cumulene configuration shows the states contributing to both channels lying close to the Fermi level of the system. In the polyyne configuration, a gap over 2 eV opens in one of the channels with the $p$ orbitals lying along the CNT surface tangent plane, which reduces the number of conduction channels available close the Fermi level from two to one. In the case of the zigzag junction, the states remaining close to the Fermi level in polyyne configuration become additionally localized. Localization occurs when the defined single bond is created from last atom in the chain towards CNT. As we have discussed earlier, because of the CNT curvature, the wire connected to the tube becomes effectively capped. This leads to localization of the previously delocalized $p$-bonding along the CNT-carbon wire-CNT structure. \n\n\n\\begin{figure}[tp]\n\\begin{center}\n\\includegraphics[width = 8 cm]{iv.pdf}\n\\end{center}\n\\vspace{-20pt}\n\\caption{The $I-V$ characteristics of (a) (4,0)@C$_{4}$ (bottom panel) (b) (4,0)@C$_{5}$ (top panel) varying the gap width.}\\label{iv}\n\\end{figure}\n \nFig.\\ref{oscillate} exhibits the calculated zero-bias conductance of (4,0)@C$_{n}$-II(C), n= 4-9. Note that we selected (4,0)@C$_{n}$-II(C) in studying the oscillating conductance of the cumulene wire because that structure is the straight cumulene wire providing the highest conductance. Similar to the cumulene wire connected to metal leads\\cite{Prasongkit:2010p780, Zhang13}, we observe the oscillatory characteristics in conductance of the cumulene wire-CNT junction, resulting from the difference in the electronic properties between even- and odd-cumulene wires \\cite{Lang:1998,Prasongkit:2010p780}. We find that the conductance of the odd-wire is higher than that of even-wire by $\\sim$ 60 \\%. The conductance values do not show any pronounced dependence on the wire length when its length increased from four to nine atoms, resulting from the ballistic transport character of electrons through the short wire\\cite{Prasongkit:2010p780}.\n\n\n\n \\begin{figure}[tp]\n \\begin{center}\n \\includegraphics[width = 6 cm]{pdos_b.pdf}\n \\end{center}\n \\caption{The illustration of the C$_5$ wire bridging between (4,0)CNTs. By twisting the CNT, we can change bridging sites and bonding at the junction (top panel). The bridge sites of the left and right (4,0)CNT leads and the corresponding conductance of each structure (bottom panel).}\\label{pdosb}\n \\end{figure}\n \n \nTo compare our findings for small radii CNTs to the large ones, we note that the latter have low curvatures and are well represented by flat graphene electrodes. Graphene electrodes are also used in the recent TEM observations and electronic structure studies by Ref. \\citenum{Jin:2009p592, Erdogan:2011bt} and others. We represent infinite radii of CNTs by studying the transport properties of the carbon wire connected to zigzag- and armchair-edge graphenes. For the zigzag graphene junction, we find that no zero-bias conductance is observed due to $\\sim$ 2 eV band gap of the zigzag edge-graphene; thus, no current is expected at $V_b > 1$V. For the armchair graphene junction, the structural change of the carbon wires connected to the armchair edge graphene electrodes make the transmission resonance peak around the Fermi level shift in energy but with little change in its magnitude.\n\n\nIn the following, we will concentrate on a potential ON\/OFF switch applications of the carbon wire-CNT junction. The electron transport properties of the carbon wire-(4,0)CNT junction will be discussed in the next section.\n\n\n\\subsection{(4,0)CNT-Carbon wire-(4,0)CNT}\n\nTo confirm the possibility of electrical switching behavior, as illustrated in Fig.\\ref{iv}, we present the calculated $I-V$ characteristics (IVCs) of (a) (4,0)@C$_{5}$ (top panel) (b) (4,0)@C$_{4}$ (bottom panel), varying the gap width. We find that the ON\/OFF current ratio of the (4,0)@C$_{5}$ is higher than that of the (4,0)@C$_{4}$. At $V_b$=0.2V, the ON\/OFF ratio of (4,0)@C$_{4}$ and (4,0)@C$_{5}$ takes the value of $\\sim$ 7 and 13, respectively, indicating electrical switching characteristics at low bias, coupled to the mechanical stretching of the CNT. We note that a very small diameter of the (4,0)CNTs plays an important role in their transport properties; thus, the metallic behavior of the (4,0)CNT electrode has been observed.\n\n\nFrom the IVCs (Fig.\\ref{iv}), the charge transport of (4,0)@C$_{4}$-III(P), (4,0)@C$_{5}$-III(P) and (4,0)@C$_{5}$-IV(C) is suppressed, whereas the (4,0)@C$_{4}$-II(C), (4,0)@C$_{5}$-I(C) and (4,0)@C$_{5}$-II(C) show a high-current state. It is important to notice at this point that the cumulene wires do not always give rise to the high conductance: a variation in conductance of the cumulene wire depends on the bonding geometries.\n\n\n \\begin{figure}[tp]\n \\begin{center}\n \\includegraphics[width = 9 cm]{pdos_a.pdf}\n \\end{center}\n \\caption{The zero-bias transmission function and the PDOS of {$\\mathrm{p_x}$} and {$\\mathrm{p_y}$} orbitals of the cumulene C$_5$ wire, corresponding to the wire configuration in Fig. \\ref{pdosb}.}\\label{pdosa}\n \\end{figure}\n\n\nThere is a probability that, in the experiment, the wire can jump to other sites when the CNT is effectively twisted. Consequently, we have investigated the geometrical structures and its corresponding conductance of the C$_5$ wire connected to all possible bridge sites on the (4,0)CNT, as presented in Fig. \\ref{pdosb}. The wire structures have changed into either cumulenes or polyynes depending on the bridge site and gap width. Independently of the bridge site, we find that the polyyne wire is always a low conductance state. For the cumulene wire, interestingly, the results show a possibility to modulate the conductance of the zigzag junction by changing bridge sites, and bonding at the interface. The range of variation in conductance of the cumulene wires is $\\simeq$ 0.1G$_0$-1G$_0$.\n\n\nTo explain the variation in conductance of the cumulene wire, we analyzed the transport channels via PDOS of the wire. We observe that the alignment of PDOS is determined by bridging sites and bonding at the junction. As we have already known that the cumulene wire has two $\\pi$ orbitals around the Fermi level resulting from $\\mathrm{p_x}$ and {$\\mathrm{p_y}$} orbitals \\cite{Shen2010,Prasongkit:2010p780}, the PDOS of the wire is decomposed into $\\mathrm{p_x}$ and $\\mathrm{p_y}$ components, as demonstrated in Fig. \\ref{pdosa}. Our results can be classified into three cases according to bonding geometries at the interface. First, for the cumulene wire forming two bonds to each side of CNTs with two planes parallel to each other ((4,0)@C$_ 5$-I(C),II(C),VIII(C)), there are two states; $\\mathrm{p_x}$ and {$\\mathrm{p_y}$}, with the same energy position at Fermi level. Consequently, electrons can propagate through $\\mathrm{p_x}$ and {$\\mathrm{p_y}$} eigenchannels of the carbon wire, showing a high conductance state. Apparently, the PDOS peak positions, consisting of the two transport channels, show the transmission function close to 2$G_0$. Second, for the cumulene forming one bond to the CNT leads; ((4,0)@C$_ 5$-IV(C),V(C)), there is the only {$\\mathrm{p_y}$} state lying at the Fermi level, whereas the $\\mathrm{p_x}$ state exists above the Fermi energy. This results in a decrease of conductance owing to only one transport channel at the Fermi level. Third, for the cumulene wire forming two bonds to the CNT lead with two planes perpendicular to each other; ((4,0)@C$_ 5$-VI(C)), the PDOS projected to $\\mathrm{p_x}$ and {$\\mathrm{p_y}$} around the Fermi level is very low, causing a drop in conductance. We can therefore conclude that the variation in conductance of the cumulene wire is determined by the transport channels at the Fermi level, depending on the bridge sites and its bonding to the CNT leads.\n\n\n\\section{Summary}\nWe have performed first principles calculations to investigate the transport properties of the carbon wire between zigzag (4,0)CNTs and armchair (4,4)CNT electrodes. The gap width between the electrodes is varied and corresponding conductance variation upon the compression\/elongation of the junction is calculated. Varying the gap width make the carbon wire change the structures (cumulene or polyyne) and contact geometries. We have observed the migration pathways of the carbon wire along the edge, which agrees well with the experimental results.\n\nWe find that the transport properties of the junction are significantly affected by the choice of chirality (zigzag or armchair). For the zigzag junction, the distinguishable ON- and OFF-state is observed, corresponding to the cumulene and polyyne structure, respectively. The zigzag CNT junction can be reversibly switched between ON- and OFF-state through varying the gap width between electrodes. In contrast, for the armchair junction, there is not enough difference in conductance to perform switching. The difference of the detailed transport behavior of both junctions is due to their entirely different electronic structures and bonding geometries, in which the high and low conductance states correspond to the cumulene and polyyne structures, respectively. In the studied configuration, the cumulene usually yields higher conductance than the polyyne wire. However, the cumulene wire can show the variation in conductance in the range from 0.1G$_0$ to 1G$_0$ upon the bridging geometries and sites. Their qualitative difference in transport mechanisms, resulting in the difference in the conductance state, was discussed by means of the transport channel at the Fermi level.\n\n\nOscillatory behavior in conductance of the cumulene wires with different lengths is demonstrated. We find that odd-cumulene wires yield higher conductance than the even-cumulene wires and ballistic transport behavior. The calculated ON\/OFF ratios of the odd-wire are larger than that of the even-wire, resulting from the difference in the electronic properties between the odd- and even-wires.\n\nGraphene leads, representing infinite radii of CNTs, have also been considered. However, graphene lead with zigzag edges has a band gap, and no zero-bias conductance is observed. Due to this semiconducting character of the zigzag electrode, switching behavior will be observed at a high bias voltage. In other aspects, the results for the graphene leads (infinite radii of CNT) are similar with that of CNT leads. \n\n\nIn contrast to metal electrodes, the CNTs can act as true nanoscale electrodes and there is a possibility that the carbon wire can jump to any site of the edge. We therefore investigate how the conductance is affected by the bonding site at the junction for the C$_5$ wire connected to (4,0)CNT electrodes. The range of variation in conductance of the cumulene wires is $\\sim$ 0.1G$_0$--1G$_0$, revealing a potential to tune the conductance. The observed variation is explained by the PDOS analysis of p-orbital components of the cumulene wire. \n\nFinally, the carbon wire-zigzag CNT junctions demonstrate the prominent difference in conductance between ON and OFF states via compression\/elongation junctions without breaking the wire and may be a promising candidate for mechano-switching device in molecular and nanoelectronics.\n\n\\begin{acknowledgments}\nJ.P. has partially been supported by the Nanotech- nology Center (NANOTEC), NSTDA, Ministry of Sci- ence and Technology, Thailand, through its program of Center of Excellence Network. A.G. and R.A. gratefully acknowledge financial support from Carl Tryggers Stiftelse f\\\"or Vetenskaplig Forskning and U3MEC, Uppsala. The calculations were performed at the high performance computing centers UPPMAX within the Swedish National Infrastructure for Computing.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nLet $\\al$ be an algebraic number of degree $d \\geq 2$ over $\\Q$ with conjugates $\\al_1=\\al,\\al_2,\\dots,\\al_d$. An additive linear relation\n\\begin{equation}\\label{rysys}\nk_1\\al_1+k_2\\al_2+\\dots+k_d\\al_d=0\n\\end{equation}\nwith some $k_1,k_2,\\dots,k_d \\in \\Q$ is called {\\it nontrivial} if \n$k_i \\ne k_j$ for some $1 \\leq i1$ is called a {\\it Pisot number} if its other conjugates over $\\Q$ (if any) all lie in the open unit disc $|z| < 1$. \nThis answers two questions raised earlier in \\cite{smy0}. For instance, this implies that no two conjugates of a Pisot number can have the same imaginary part. See also a subsequent paper \\cite{DJ} for some further analysis of some simple linear relations of small length.\n\nIn the present paper, we investigate additive linear relations in conjugates of a Salem number. Recall that\nan algebraic integer $\\al>1$ is called a {\\it Salem number} if its other conjugates over $\\Q$ all lie in the closed unit disc $|z| \\le 1$ with at least one conjugate lying on the circle $|z|=1$. \n\nThroughout, if $\\al>1$ is a Salem number \nof degree $d=2s \\geq 4$ we label its conjugates\nas in the theorem below.\n\n\n\\begin{theorem}\\label{pirmoji}\nLet $\\al_1=\\al>1$ be a Salem number of degree $d=2s \\geq 4$ with conjugates $\\al_1,\\dots,\\al_d$ satisfying\n$\\al_2=1\/\\al_1$ and \n$\\al_{2j}=1\/\\al_{2j-1}=\\overline{\\al_{2j-1}}$ for $j=2,\\dots,s$. \nIf for some rational numbers $k_i$, $i=1,\\dots,d$, and for some totally real algebraic number $\\gamma$ we have \n\\begin{equation}\\label{duu}\nk_1\\al_1+k_2\\al_2+\\dots+k_d\\al_d=\\gamma,\n\\end{equation}\nthen \n$k_{2j-1}=k_{2j}$ for each $j=1,\\dots,s$.\n\\end{theorem}\n\nIn particular, the theorem obviously holds for $\\gamma=0$. So, every linear relation \\eqref{rysys} in the conjugates $\\al_i$, $i=1,\\dots,d$, of a Salem number $\\al$ is induced by the linear relation \n\\begin{equation}\\label{rysys1}\nk_1\\be_1+k_3\\be_2+\\dots+k_{2s-1}\\be_s=0\n\\end{equation}\nin conjugates of the respective totally real algebraic integer $\\be_1=\\be:=\\al+1\/\\al>2$ whose other conjugates\nare\n$$\\be_j=\\al_{2j-1}+\\al_{2j}=\\al_{2j-1}+1\/\\al_{2j-1}=\\al_{2j-1}+\\overline{\\al_{2j-1}}=2\\Re \\al_{2j-1} \\in (-2,2)$$ for $j=2,\\dots,s$. If $f$ is the minimal polynomial of a Salem number $\\al$ of degree $d=2s$ and $g$ is the minimal polynomial of \n$\\be=\\al+1\/\\al$ of degree $s$ then they are related by the identity\n$\nf(x)=x^{s} g(x+1\/x).\n$\nThen, as in \\cite{chr}, we call $g$ the {\\it trace polynomial} of $f$. \nNote that $f$ is irreducible if and only if $g$ is irreducible. Also,\n${\\rm Trace}(\\al)=\\sum_{j=1}^d \\al_j= \\sum_{i=1}^s \\be_i= {\\rm Trace}(\\be)$. \n\nBy \\cite{kurb3} (or \\cite{ds1}), the only relation with conjugates $\\be_1=\\be,\\dots,\\be_p$ \nof an irreducible polynomial of prime degree $p$ can be of the form \n$$r\\be_1+r\\be_2+\\dots+r\\be_p=0,$$ where $r \\in \\Q$. Hence, \nthe only possible linear relation with conjugates of a Salem number $\\al$\nwith degree $2p$\nis $r {\\rm Trace}(\\al)=0$, where $r \\in \\Q$. This relation is trivial.\n\nSo, in particular Theorem~\\ref{pirmoji} implies that \n\n\\begin{corollary}\\label{keturi}\nIf $p$ is a prime number then there are no nontrivial linear relations in conjugates of a Salem number of degree $d=2p$. \n\\end{corollary}\n\nBy \\cite{mcsm}, it is known that there are Salem numbers of every integral trace. The degree of a Salem number with negative trace \n$-t$ is quite large if $t \\in \\N$ is large. Earlier, in \\cite{mcsm0} \nSmyth has shown that there are Salem numbers with trace $-1$ of every even degree\n$d \\geq 8$. \n\nHere, by a similar argument, we show that\n\n\\begin{theorem}\\label{keturi-1}\nFor any even $d \\geq 6$ there is a Salem number of degree $d$\nwith trace $0$. \n\\end{theorem}\n\nIn Corollary~\\ref{hbhbhb} below, we list of all $4$ possible Salem numbers of degree $6$ and trace $0$. Note that there are no Salem numbers \nof degree $4$ and trace $0$. Indeed, otherwise the minimal polynomial of\nsuch a Salem number would be $x^4+ax^2+1$, with $a \\in \\Z$, which is impossible. \n\nOur next theorem describes the minimal length of nontrivial linear relations between conjugates of a Salem number and the minimal degree\nof a Salem number for which a nontrivial linear relation may occur. \n\n\\begin{theorem}\\label{antroji}\nSuppose $\\al>1$ is a Salem number with conjugates $\\al_1=\\al,\\al_2,\\dots, \\al_d$ over $\\Q$ labelled as in Theorem~\\ref{pirmoji}. \n\\begin{itemize}\n\\item[$(i)$]\nIf for some\nintegers $k_1,k_2,\\dots,k_d$, not all zero, the nontrivial linear relation \\eqref{rysys} holds then its length \nmust be at least $6$.\nFurthermore, there exist Salem numbers $\\al$ of degree $12$ whose six conjugates satisfy the following nontrivial linear relation of length $6$:\n$$\n\\al_1+\\al_2+\\al_3+\\al_4+\\al_5+\\al_6=0.\n$$\n\n\\item[$(ii)$] The smallest degree of a Salem number with a nontrivial linear relation between its conjugates is $8$. Furthermore, there exist Salem numbers $\\al$ of degree $8$ whose conjugates satisfy the following nontrivial linear relation:\n$$\n\\al_1+\\al_2+\\al_3+\\al_4-\\al_5-\\al_6-\\al_7-\\al_8=0.\n$$\n\\end{itemize}\n\\end{theorem}\n\n\n\n\\section{Auxiliary results}\n\nWe begin with two simple lemmas. \n\n\\begin{lemma}\\label{trecioji}\nThe cubic polynomial $x^3-ax+b \\in \\R[x]$ has three distinct roots in the interval $(-2,2)$ iff $00$. Set $x_0:=\\sqrt{a\/3}$. \nThen, the polynomial $h$\nhas three distinct roots in $(-2,+\\infty)$ iff $-2<-x_0$ (i.e., $00\n\\end{equation}\nand\n\\begin{equation}\\label{mm3}\nh(x_0)=-\\frac{2a\\sqrt{a}}{3\\sqrt{3}}+b<0.\n\\end{equation}\n\nClearly, all three roots belong to $(-2,2)$ if, in addition, we have $h(2)=8-2a+b>0$. Combined with \\eqref{mm1}, \\eqref{mm2} and \\eqref{mm3}\nthis proves \\eqref{pima}. Evidently, \\eqref{pima} is only possible for some $b$ when its left hand side does not exceed its right hand side, that is, when $02$, $\\be_2=(1-\\sqrt{1-4\\ga_1})\/2 \\in (-2,-1)$ and $\\be_3,\\dots,\\be_{2k} \\in (-1,2)$. So, $g$ has $2k-1$\nroots in $(-2,2)$ and one root greater than $2$. Clearly, by \\eqref{bbbb}, we have\n\\begin{equation}\\label{bee}\n\\be_{1}+\\be_2=\\dots=\\be_{2k-1}+\\be_{2k}=1.\n\\end{equation}\n\nNow, as the roots $\\al_1=\\al>1,\\al_2=1\/\\al,\\dots,\\al_{4k-1},\\al_{4k}=1\/\\al_{4k-1}$ of $$f(x)=x^{2k}g(x+1\/x)=(-1)^k x^{2k}h\\big((x+1\/x)(1-x-1\/x)\\big)$$ satisfy $\\be_j=\\al_{2j-1}+\\al_{2j}=\\al_{2j-1}+1\/\\al_{2j-1}$ for each $j=1,\\dots,2k$, we see that \\eqref{bee} implies \\eqref{aaa}. Furthermore, $\\al$ is a Salem number of degree $4k$ provided that $f$ is irreducible\nover $\\Q$. \n\\end{proof}\n\n\nWe made some calculations related to Lemma~\\ref{trecioji-1}. It turns out that there exactly $15$ quadratic polynomials $h$ satisfying the conditions of the lemma and thus producing $15$ Salem numbers \nof degree $8$ satisfying \\eqref{aaa} with $k=2$. For instance, $x^2+4x+1$ is such a quadratic polynomial $h$. Also, there are exactly \n$30$ cubic, $20$ quartic and $4$ quintic polynomials $h$ producing\n$30$ Salem numbers of degree $12$ (satisfying \\eqref{aaa} with $k=3$), $20$ Salem numbers of degree $16$ (satisfying \\eqref{aaa} with $k=4$) and $4$ Salem numbers of degree $20$ (satisfying \\eqref{aaa} with $k=5$), respectively. In the case $k=5$, the example of $h$ is $$x^5+9x^4+22x^3+16x^2-x-1.$$ This gives a Salem number $\\al$ of degree $20$ with minimal polynomial \n$$x^{20}-5x^{19}+11x^{18}-19x^{17}+26x^{16}-29x^{15}+\n27x^{14}-19x^{13}+8x^{12}+x^{11}$$ $$-5x^{10}+\nx^9+8x^8-19x^7+27x^6-29x^5+\n26x^4-19x^3+11x^2-5x+1$$\nwhose conjugates satisfy \\eqref{aaa} with $k=5$. \n\nThe first part of the next lemma was inspired by Lemma 1 of Beukers and Smyth in \\cite{beusmy}. Essentially, it is a version of their algorithm \\cite{beusmy} to locate cyclotomic points on curves, specialized to the case of sequences of polynomials that produce Salem numbers from Pisot numbers. Also, the second part of Lemma \\ref{Lemma_cyclotomic_factors} is loosely related to the work on irreducibility of polynomials of the type $x^nf(x)+g(x) \\in \\Z[x]$ and on the sequences and covering systems of integers by Schinzel \\cite{schi}, Filaseta et al. \\cite{fifoko, fima}, although these irreducibility results are not of direct relevance here. Throughout, $f^*(x) = x^{\\deg{f}}f(1\/x)$ stands for the \\emph{reciprocal polynomial} of $f(x)$.\n\n\\begin{lemma}\\label{Lemma_cyclotomic_factors}\nFor $n \\in \\N$, consider the sequence of polynomials\n\\[\ng_n(x) := x^nf(x) + \\eps f^{*}(x),\n\\] where $\\eps \\in \\{-1, 1\\}$ and $f(x) \\in \\Z[x]$ satisfies $f^*(x) \\ne \\pm f(x)$. \nSuppose that a root of unity $\\zeta \\in \\C$ is also a root of some polynomial $g_n(x)$. Then, $\\zeta$ must appear among the zeros of at least one of the following polynomials:\n\\[\nf(x^2)f^*(x)^2 + \\eps f(x)^2f^*(x^2), \\qquad f(x)^2f^*(-x^2) \\pm f(-x^2)f^*(x)^2,\n\\]\n\\[\nf(x)f^*(-x) \\pm f(-x)f^*(x).\n\\]\nIn particular, if none of these polynomials is identically zero, then the set of all such possible roots of unity $\\zeta$ is finite.\n\nIn addition to this, if $f(\\zeta) \\ne 0$ then the root of unity $\\zeta$ is a zero of $g_n(z)$ if and only if $n$ belongs to the arithmetic progression $\\ell k+r$, $k=0, 1, 2, \\dots$, where $r$ is a fixed integer in the range $0 \\leq r < \\ell$ and $\\ell={\\rm ord}(\\zeta)$ denotes the multiplicative order of $\\zeta$. \n\\end{lemma}\n\n\\begin{proof}\nAs $\\zeta$ is the root of unity, by Lemma 1 of \\cite{beusmy} (or Lemma 2.1 of \\cite{mcsm0}, at least one of the three numbers $\\zeta^2$, $-\\zeta^2$, $-\\zeta$ must be an algebraic conjugate of $\\zeta$ over $\\Q$.\nMultiplying $g_n(x)=x^n f(x)+\\eps f^*(x)$ by $x^n f(x)-\\eps f^*(x)$ we see that the polynomial $h(x)=x^{2n}f(x)^2 - f^*(x)^2$ has a zero at $x=\\zeta$.\n\nIf $\\zeta^2$ is conjugate of $\\zeta$, then one also has $g_n(\\zeta^2)=0$. Combining this with $h(\\zeta)=0$ yields\n\\[\n\\left\\{\n\\begin{split}\n\\zeta^{2n}f(\\zeta)^2 \t&- f^*(\\zeta)^2\t\t&= 0,\\\\\n\\zeta^{2n}f(\\zeta^2)\t&+ \\eps f^*(\\zeta^2)\t&= 0.\n\\end{split}\n\\right.\n\\]\nHence,\n\\[\n\\begin{vmatrix}\nf(\\zeta)^2 \t &- f^*(\\zeta)^2\\\\\nf(\\zeta^2) &\\eps f^*(\\zeta^2)\\\\\n\\end{vmatrix} = \\eps f(\\zeta)^2f^*(\\zeta^2) + f(\\zeta^2)f^*(\\zeta)^2 = 0.\n\\]\nThus, $\\zeta$ is the root of $f(x^2)f^*(x)^2 + \\eps f(x)^2f^*(x^2)$.\n\nSuppose next that $-\\zeta^2$ is a conjugate to $\\zeta$. Then, using $g_n(-\\zeta^2)=0$ and $h(\\zeta)=0$, one concludes that $\\zeta$ is the root of the polynomial\n $f(x)^2f^*(-x^2) +\\eps (-1)^n f(-x^2)f^*(x)^2$. \n\nIn the case when $-\\zeta$ is conjugate to $\\zeta$, from $g_n(\\zeta)=g_n(-\\zeta)=0$ one obtains $\\zeta^n f(\\zeta)+\\eps f^*(\\zeta)=0$ and $(-\\zeta)^n f(-\\zeta)+\\eps f^*(-\\zeta)=0$, which yields that $\\zeta$ is a root of $f(x)f^*(-x) +(-1)^{n+1} f(-x)f^*(x)$.\n \nFinally, if a root of unity $\\zeta$ of order $\\ell$ satisfies $g_n(\\zeta)=0$, then $g_{n+\\ell}(\\zeta)=0$. Furthermore, if $\\zeta$ is a common root of $x^{n_1}f(x) +\\eps f^*(x)$ and $x^{n_2}f(x)+ \\eps f^*(x)$, then $(\\zeta^{n_2}-\\zeta^{n_1})f(\\zeta)=(\\zeta^{n_2-n_1}-1)\\zeta^{n_1}f(\\zeta)=0$. By $f(\\zeta) \\ne 0$, it follows that $\\zeta^{n_2 - n_1}=1$. Thus, $\\ell \\mid (n_2 - n_1)$ and so all such $n$ form an arithmetic progression with difference $\\ell$, as claimed. \n\\end{proof}\n\n\\section{Proofs of the theorems}\n\n\n\n\\begin{proof}[Proof of Theorem~\\ref{pirmoji}]\nAssume that $k_{2i} \\ne k_{2i-1}$ for some $i$ in the range $1 \\le i \\le s$. Let $G$ be the Galois group of the normal extension of $\\Q(\\al,\\ga)$ over $\\Q$, and let $\\sigma$ be an automorphism of $G$ which maps $\\al_{2i-1}$ to $\\al_1=\\al$. Then, $\\sigma(\\al_{2i})=\\sigma(1\/\\al_{2i-1})=1\/\\al$, so that \\eqref{duu} maps into\n\\begin{equation}\\label{kjh}\n\\sigma(\\gamma)=k_{2i-1}\\al+k_{2i}\/\\al+ t_3\\al_3+\\dots+t_d\\al_d,\n\\end{equation}\nwhere $t_3,\\dots,t_d \\in \\Q$ is a permutation of the list obtained from the initial list \n$k_1,\\dots,k_d$ by excluding the elements $k_{2i-1}$ and $k_{2i}$. \n\nConsider the following equality which is complex conjugate to \\eqref{kjh}:\n\\begin{equation}\\label{kjh1}\n\\overline{\\sigma(\\gamma)}=k_{2i-1}\\al+k_{2i}\/\\al+ t_3\\overline{\\al_3}+\\dots+t_d\\overline{\\al_d}.\n\\end{equation}\nSince $\\overline{\\sigma(\\ga)}=\\sigma(\\ga)$ and $\\al_{2j}=\\overline{\\al_{2j-1}}$ for $j=2,\\dots,s$, by adding \\eqref{kjh} and \\eqref{kjh1}, we obtain\n$$\n2\\sigma(\\gamma) = 2k_{2i-1}\\al+2k_{2i}\/\\al+ w_2(\\al_3+\\al_4)+\n\\dots+w_{s}(\\al_{d-1}+\\al_d),\n$$\nwhere $w_{j}=t_{2j-1}+t_{2j}$ for $j=2,3,\\dots,s$. \nAdding $2(k_{2i}-k_{2i-1})\\al$ to both sides we deduce that\n$$2\\sigma(\\gamma)+ 2(k_{2i}-k_{2i-1})\\al=w_1(\\al_1+\\al_2)+w_2(\\al_3+\\al_4)+\\dots+w_{s}(\\al_{d-1}+\\al_d),$$\nwhere $w_1=2k_{2i}$.\n\nAs we already observed above, the number $\\be_1=\\be=\\al+1\/\\al=\\al_1+\\al_2$ is totally real with conjugates $\\be_2=\\al_3+\\al_4$, \\dots, $\\be_s=\\al_{d-1}+\\al_d$. Hence,\nthe number \n$$2(k_{2i}-k_{2i-1})\\al=w_1\\be_1+w_2\\be_2+\\dots+w_{s}\\be_s-2\\sigma(\\gamma)$$\nis a linear form (with rational coefficients $w_1,\\dots,w_s,-2$) in totally real algebraic numbers $\\be_1,\\dots,\\be_s,\\sigma(\\gamma)$.\nThus, it must be totally real itself. However, the number $2(k_{2i}-k_{2i-1})\\al \\ne 0$ is\nnot totally real, since it has a non-real conjugate $2(k_{2i}-k_{2i-1})\\al_3$. \nThis is a contradiction which completes the proof of the theorem. \n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Theorem \\ref{keturi-1}] Assume that there exists a smallest even degree $d$ (where $d \\geq 8$ by Corollary~\\ref{keturi-1}), such that there are no Salem numbers of that degree $d$ with trace $0$. We will track down and ultimately eliminate all such possible $d$ by considering 3 sequences of polynomials, given explicitly by Salem's original construction \\cite{salem1, salem2}.\n\nWe start with a Salem sequence \n\\[\ng_n(x)=x^n(x^3-x-1) + (-x^3-x^2+1), \\quad n \\geq 2.\n\\] Then $g_n(x)$ either posseses cyclotomic factors or it is a minimal polynomial of a Salem number of trace $0$; see \\cite{boyd, salem1, salem2}. Since we have assumed that no Salem number of degree $d$ and trace $0$ exists, the polynomial $g_n(x)$ of degree $d=\\deg{g_n}=n+3$ must be reducible, that is, it must be divisible by a cyclotomic polynomial $\\Phi_{\\ell}(x)$, where $\\ell \\in\\N$.\n\nTo find cyclotomic factors of $g_n(x)$, we apply\nLemma~\\ref{Lemma_cyclotomic_factors} with $f(x)=x^3-x-1$ and $\\eps=1$. The following candidates appear as factors of auxiliary polynomials described in\nLemma~\\ref{Lemma_cyclotomic_factors} (with $\\eps=1$):\n\\[\n\\Phi_1(x) = x-1, \\qquad \\Phi_2(x)=x+1, \\qquad \\Phi_8(x)=x^4+1\\]\n\\[\n\\Phi_{12}(x)=x^4-x^2+1, \\qquad \\Phi_{18}(x)=x^6-x^3+1,\n\\]\n\\[\n\\Phi_{30}(x) = x^8 + x^7 - x^5 - x^4 - x^3 + x + 1.\n\\] Since none of the five auxiliary polynomials is zero identically, this list is complete. \n\nTo see which of these candidates actually show up, one can apply the periodicity property stated in the second part of Lemma~\\ref{Lemma_cyclotomic_factors}. After computation of ${\\rm gcd}(g_n(x), \\Phi_{\\ell}(x))$, $0 \\leq n \\leq \\ell-1$ for $\\ell=1, 2, 8, 12, 18, 30$ it turns out that $g_n(x)$ has cyclotomic factors precisely for the degrees $d=n+3$ in one of the arithmetic progressions:\n\\[\nd \\in \\{2k+1, 8k+2, 12k+1, 18k+17, 30k+24\\},\n\\]\nwhere $k=0, 1, 2, \\dots $. As $d$ must be even, we restrict all such possible $d$ to two arithmetic progressions: $d \\in \\{8k+2, 30k+24\\}$.\n\nNext, we take the second sequence\n\\[\nh_n(x) = \\frac{x^n(x^2-x-1) - (-x^2-x+1)}{x-1}, \\quad n \\geq 2.\n\\]\nAlthough now $f(x)=x^2-x-1$ contributes the coefficient $-1$ of $x^{n+1}$ to $g_n(x)$, one regains trace $0$ after division by $x-1$. \nLet us apply \nLemma~\\ref{Lemma_cyclotomic_factors} to the polynomial $g_n(x)=(x-1)h_n(x)$ with this new choice of $f(x)$ and $\\eps=-1$. The candidate cyclotomic factors are:\n\\[\n\\Phi_1(x) = x - 1, \\qquad \\Phi_2(x) = x + 1,\\qquad \\Phi_3(x)= x^2 + x + 1,\n\\]\n\\[\n\\Phi_6(x)= x^2 - x + 1,\\qquad \\Phi_{12}(x)=x^4 - x^2 + 1.\n\\] As above, the computation of gcd's with first $12$ polynomials of the sequence yields the list of possible bad degrees $d=n+1$:\n\\[\nd \\in \\{ 2k+1, 3k+ 2, 6k+3, 12k +4\\}.\n\\]\nThis list also accounts for the single occurrence of the multiple factors, namely, $(x-1)^2$ in $g_4(x)$. Bad degrees must be even, so we are left with $d \\in \\{6k+2, 12k+4\\}$. \n\nLet us combine this with the arithmetic progressions obtained from the first sequence:\n\\[\nd \\in \\{8k+2, 30k+24\\} \\cap \\{6k+2, 12k+4\\}.\n\\]\nNotice that all integers $30k+24$ are divisible by $6$, while none of $6k+2$ or $12k+4$ are. Therefore, $d \\notin \\{30k+24\\}$, and hence $d \\in \\{8k+2\\}$. Next, notice that $12k+4$ is divisible by $4$ while $8k+2$ is not. Consequently, $d \\notin \\{12k+4\\}$. It follows that\n\\[\nd \\in \\{8k+2\\} \\cap \\{6k+2\\} = \\{24k + 2\\}.\n\\]\n\nTo eliminate this possibility, let us consider the third sequence, constructed with $f(x)=x^3-x^2-1$ and $\\eps=-1$:\n\\[\nh_n(x) = \\frac{x^n(x^3-x^2-1) - (-x^3-x+1)}{x-1}, \\quad n \\geq 2.\n\\]\nThis time, by Lemma~\\ref{Lemma_cyclotomic_factors} the candidates for cyclotomic divisors are\n\\[\n\\Phi_1(x)= x - 1, \\quad\n \\Phi_2(x)= x + 1, \\quad\n \\Phi_3(x)= x^2 + x + 1, \\quad\n \\Phi_4(x)= x^2 + 1, \\quad\n \\]\n \\[\n \\Phi_6(x)= x^2 - x + 1, \\quad\n \\Phi_{10}(x)= x^4 - x^3 + x^2 - x + 1, \\quad\n \\Phi_{18}(x)= x^6 - x^3 + 1.\n\\]\nNow, bad degrees $d=n+2$ for this sequence $h_n(x)$ are\n\\[\nd \\in \\{2k+1, 3k+1, 4k+3, 6k+4, 10k+5, 18k+ 6\\}.\n\\] This last list accounts for the factor $(x-1)^2$ of $g_5(x)$ for a single value $n=5$. Since $d$ is even, $d \\notin \\{2k+1, 4k+3, 10k+5\\}$. Since $d \\in \\{24k+2\\}$ would have remainder $2 \\pmod{3}$, we have $d \\notin \\{3k+1, 6k+4\\}$. Finally, $d \\notin \\{18k+6\\}$, since $24k+2$ is not divisible by $6$. This exhausts the list of possibilities, so no such bad degrees can exist. Hence, for each even $d \\geq 6$, we can find a Salem number of degree $d$ and trace $0$ in one of the three Salem sequences that were considered above.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Theorem~\\ref{antroji}]\nSuppose that the relation \\eqref{rysys} holds with some $k_j \\in \\Z$, not all zero, and conjugates $\\al_j$ of a Salem number $\\al$ labelled as in Theorem~\\ref{pirmoji}. Then, by Theorem~\\ref{pirmoji}, we \nhave $k_{2j}=k_{2j-1}$ for $j=1,\\dots,s$. Setting $\\beta_j=\\al_{2j-1}+1\/\\al_{2j-1}$ for $j=1,\\dots,s$ we find that\n\\eqref{rysys1} holds, namely,\n$k_{1}\\beta_1+k_3\\beta_2+\\dots+k_{2s-1}\\beta_s=0$.\n\nIn order to prove the first part of the theorem we need to show that $|k_1|+|k_3|+\\dots+|k_{2s-1}| \\geq 3$. \nFor a contradiction, assume that $$|k_1|+|k_3|+\\dots+|k_{2s-1}| \\leq 2.$$\nThe case when $|k_{2j-1}|=2$ for some $j$ (and so other $k_{2i-1}$ are all zeros) is clearly impossible, since \n$\\pm 2 \\be_j \\ne 0$. Therefore, we must have $|k_{2i-1}|=|k_{2l-1}|=1$, where $i2$ one obtains $\\be_1=-\\sigma(\\be_l)$. Here, the left hand side is a real number greater than $2$, whereas the right hand side belongs to the interval $(-2,2)$, which is a contradiction. \n\nIn order to prove the existence of a Salem number of degree $12$ with \nrequired linear relation among its conjugates we can\ntake, for instance, the following two pairs of real numbers $(a,b)$: \n$$(a_1,b_1)=(5-\\sqrt{2}, -3+2\\sqrt{2}) \\quad \\text{and} \\quad (a_2,b_2)=(5+\\sqrt{2},-3-2\\sqrt{2}).$$ \n\nHere, the first pair $(a_1,b_1)$ satisfies $0From the analysis of the radial velocity curve of a binary star one can obtain the orbital parameters of the system. In the case of the studied system the procedure \nis complicated by the pulsational variability of the Cepheid superimposed on the orbital motion. Using the RaveSpan software we have fitted a model of Keplerian orbit \n(i.e. proximity effects were ignored, which is justified by the large distance of the components even at closest approach) with an additional Fourier series \nrepresenting the pulsational radial velocity curve of the Cepheid.\n\nWe simultaneously fitted the reference time $T_0$, the eccentricity $e$, the argument of periastron $\\omega$, the velocity semi-amplitudes $K_1$ and $K_2$, \nthe systemic velocity $\\gamma$, and Nth-order Fourier series. In the beginning systemic velocities of both components were fitted, but without any improvement \nin the fit and with the values equal within the errors. Eventually only one velocity was kept.\n\nThe period $P$ was initially held fixed at the estimated value of 1550.4 days. The fitting was later repeated with a fixed value of $P=1550.354$ d and $T_0$ calculated \nfrom the photometric epoch of a primary minimum $T_{I}$ as a function of eccentricity and argument of periastron:\n$$T_0 = f(e, \\omega; P=1550.354~d, T_{I}=3959.23~d)$$\nto ensure the consistency of the model.\n\nThe error of the eccentricity turned out to be 10 times higher and the error for the argument of periastron 6 times lower (see solution 3 in Table~\\ref{tab:spec}) \nthan the ones obtained from the photometry. For this reason we have tried to solve the system with $e$ and $\\omega$ fixed (solution 1), or only $e$ fixed (solution 2). \nEventually we decided to adopt solution 2 as the final one because of the low error for $e$ from the photometry and the low error of $\\omega$ from the orbital solution. \n\nIn this way we have obtained the coefficients describing the pulsational radial velocity curve and the parameters describing the orbital motion separately. \nThe orbital radial velocity curve along with the best fitting model is shown in Fig.~\\ref{fig:rvorb}.\nTo obtain the pulsational radial velocity curve of the Cepheid we then subtracted the orbital motion from the measured velocities. The resulting radial velocity curve is shown \nin Fig.~\\ref{fig:rvpuls} together with the radius variation curve calculated with the $p$-factor 1.37 obtained from the fit. The orbital solutions are presented \nin Table~\\ref{tab:spec}.\n\nThe photometric data were analyzed using a version of the JKTEBOP code (Popper \\& Etzel 1981, Southworth et al. 2004, 2007) modified to allow the inclusion of \npulsation variability. We have previously used this package in the analysis of the OGLE-LMC-CEP-0227 system (P13), and we refer the reader \nto this work for more details.\n\nWe varied the following parameters in deriving the final model: the fractional radius of the pulsating component at phase 0.0 (pulsational), $r_1$;\nthe fractional radius of the second component, $r_2$; the orbital inclination $i$; the orbital period, $P_{orb}$; the epoch of the primary minimum, $T_{I}$; \nthe component surface brightness ratios in all three photometric bands at phase 0.0 (pulsational), $j_{21}$; and the third light in the $R_C$-band $l_{3} (R_C)$. \nThe radius change \nof the Cepheid was calculated from the pulsational radial velocity curve using the $p$-factor value of 1.37 and the change of the surface brightness ratios from \nthe instantaneous radii and out-of-eclipse pulsational light curves (for details, see P13). The third light in the $R_C$-band was introduced because we were unable \nto transform it directly to the OGLE photometric system.\n\nThe search for the best model (lowest $\\chi^2$ value) was made using the Markov chain Monte Carlo (MCMC) approach (Press et al. 2007) as described in P13. \nThe best fit photometric parameters are presented in Table~\\ref{tab:photpar}. We present two photometric solutions in this Table. In the first one, the argument\nof periastron $\\omega$ is taken from the orbital solution, in the second one it is fitted. We consider Solution 1 as the final one, being consistent with\nthe above discussion of the $\\omega$ errors. In this way we take the best advantage from the photometric and orbital radial velocity data.\nUsing these parameters we generated a model for each light curve. \nIn Fig.~\\ref{fig:ieclmodel} we show a close-up of selected eclipses for each passband. The magnitude range is the same for all plots to facilitate the comparison.\n\nMost of the parameters fitted in our approach are independent and do not exhibit any significant correlation. The only significant correlation is between \nthe orbital plane inclination $i$ and the sum of the radii $r_{1}+r_{2}$ as shown in Fig.~\\ref{fig:corr_i}.\n\n\n\\subsection{Eclipses}\nIn order to better understand the configuration of the system using the derived parameters we calculated the distances between the stars at important phases. \nAt the phase of the primary eclipse the distance between the components is about $650 R_{\\odot}$, while at the phase of the secondary eclipse it is \nabout $725 R_{\\odot}$. Both eclipses occur when the stars are relatively close to each other. The minimum and maximum separations during the orbital cycle \nare 425 and 1760 $R_{\\odot}$, respectively.\nAt the primary eclipse the projected distance between the centers of the stars is $22.9 R_{\\odot}$, and at the secondary eclipse the projected distance is $25.6 R_{\\odot}$, \nwhile the sum of the radii changes between 53.4 and 56.4 $R_{\\odot}$ depending on the instantaneous radius of the Cepheid. The configuration at both phases \nis illustrated in Fig.~\\ref{fig:config}.\n\n\\subsection{Radius and projection factor}\nTo test the results of our analysis, we have calculated the expected radius of the Cepheid from period-radius (PR) relations for classical Cepheids in the literature. \nThe relation \nof Gieren et al. (1998) for fundamental mode pulsators yields an expected mean radius value of $27.0 \\pm 1.2 R_{\\odot}$ for the pulsation period of the Cepheid in our system\nwhich agrees with our determination ($28.6 \\pm 0.2 R_{\\odot}$) within the combined 1 $\\sigma$ errors. The PR relations of Sachkov (2002) for fundamental mode\nand first overtone Cepheids predict radii of $27.4 \\pm 0.9 R_{\\odot}$ and $35.6 \\pm 5.4 R_{\\odot}$, respectively, for a pulsation period of 2.988 days. The first value \nmatches our derived radius value for the OGLE LMC562.05.9009 Cepheid much better, and is in agreement with the radius prediction from the Gieren et al. (1998) PR relation. \nWe conclude that the radius value of the Cepheid clearly supports fundamental mode pulsation, in agreement with the conclusion reached\nfrom the Fourier decomposition parameters of the I-band light curve.\nThe radius value together with the other physical parameters of the Cepheid given in Table 3, particularly its mass, \nalso leave no doubt that the pulsating star in the system is a classical (and not a Type-II) Cepheid.\n\nOur models constrain the projection factor of a Cepheid in an eclipsing binary system in the way which has been discussed in detail in P13. Briefly,\nthe shape of a Cepheid light curve in a given photometric band is determined by the change of its surface temperature and its radius. The radius change is\nparticularly important if the Cepheid resides in an eclipsing binary system. The beginning and end of an eclipse may be shifted in time according to the\ninstantaneous radius of the Cepheid, and the visible area of the eclipsed stellar disk depends on the phase of the pulsating component. In our approach\nthe Cepheid variability is a part of the model, so we can trace the influence of the related parameters on the light curve. As a base we use the raw (unscaled) \nabsolute radius change obtained from the pulsational radial velocity curve. Then we scale its amplitude with the projection factor (the $p$-factor scales\nlinearly with the amplitude of the radius variation curve). A conversion from the absolute radii to the relative radii (used in the light curve analysis)\nis done by using the orbital solution. A comparison of the resulting model light curves with the data then directly constrains the $p$-factor value. From\nour best model we obtain a radius variation amplitude of 3.04 R$_\\odot$ for the Cepheid, which corresponds to $p=1.37$ (see Figs. 6 and 9).\n \nOur current determination of the projection factor of the Cepheid in the OGLE LMC562.05.9009 system is the second reliable measurement of this important quantity\nfor a Cepheid in a binary, after the first determination made by P13 for OGLE-LMC-CEP-0227. The value of $p = 1.37 \\pm 0.07$ is smaller than the predicted p-factor value\nfrom the most recent calibration of the p-factor relation of Storm et al. (2011) which yields $p = 1.46 \\pm 0.04$ for the pulsation period of the Cepheid. However, there is possible \nagreement within the combined uncertainties of the two values. This is contrary to the finding for CEP-0227 which has a pulsation period of 3.80 days \nand $p = 1.21 \\pm 0.04$ from\nour analysis in P13, whereas its expected p-factor value from the Storm et al. calibration is $p = 1.44 \\pm 0.04$, with both values clearly discrepant within their respective\nuncertainties. The large difference of the p-factor values for the two binary Cepheids for which we could determine\nthis number so far with our method is also noteworthy (the difference is 0.16, whereas the p-factor relation of Storm et al. predicts a difference of only 0.02 for a change of the period \nfrom 2.988 to 3.80 days. Other p-factor relations, such as the theoretical relations of Neilson et al. (2012), predict an even smaller change of $p$ between the two\nperiod values). Our finding hints at the possibility that the p-factor - period relation may have an intrinsic dispersion, particularly in the short pulsation period range,\nwhere the discrepancy of the p-factor values predicted by different calibrations of the relation in the literature is largest (see discussion in Storm et al. 2011\nand Gieren et al. 2013). \n\n\\subsection{Extinction and temperature}\n\nThe extinction in the direction to the target was calculated in a similar way as described in Pilecki et al.(2015). We utilized the observed \n(not extinction-corrected) period-magnitude relations for fundamental mode Cepheids in the LMC (Soszy{\\'n}ski et al. 2008) in the optical V and I bands.\nBy comparing the observed mean magnitudes of the Cepheid with the expected magnitudes for its period,\n we determined the differential color excess (with respect to the LMC mean value) as $\\Delta E(B\\!-\\!V)=-0.016$ mag, and a total color excess \n of $E(B\\!-\\!V)=0.106$ mag using the mean extinction for the LMC given by Imara \\& Blitz (2007) - see Table~\\ref{tab:magnitud}. This color excess corresponds \n to a total extinction in the K-band of $A_K=0.036$ mag.\n\nThe mean (over the pulsation cycle of the Cepheid) observed IR magnitudes of the OGLE LMC562.05.9009 system are $J=14.232\\pm 0.018$ and $K=13.873\\pm 0.018$ mag. \nThey were transformed \nonto the 2MASS system using the equations of Carpenter (2001). We calculated an expected exctinction-free K-band magnitude of the Cepheid using relations 4 and 13 from \nRipepi et al.(2012). The observed and de-reddened magnitudes of both components in the $V$, $I_C$ and $K$ bands are given in Table~\\ref{tab:magnitud}. \nThe effective temperatures of the two stars were then calculated from their intrinsic colors, using the calibrations by Worthey \\& Lee (2011). \nThe extinction-corrected magnitudes and colors of the primary and secondary components and their temperatures are given in Table 3.\n\nIt is very interesting to note that within the uncertainties both components have the same effective temperatures, luminosities and surface gravities. \nHowever, according to the very precise OGLE-IV photometry the secondary does not show any pulsations with amplitude larger than 0.01 mag.\nThis is a striking result \nbecause, assuming the same chemical composition for both components in the system, we would expect both stars to be located \nwithin the instability strip (see discussion next section). The fact that the secondary is non-pulsating and thus outside the instability strip could imply\nthat the two components of OGLE LMC562.05.9009 have significantly different abundances, which would make this system \nunique among known binary stars. \n\nA different, and probably more likely explanation is that the secondary is just a little cooler that the Cepheid ($\\sim 70$ K), as suggested by \nour photometric Solution 2 in Table 2. In that case the Cepheid would reside almost exactly on the red boundary of the instability strip, with the secondary \nlocated just beyond the red edge. If this scenario is the correct one, the present work would provide the best known observational constraint on the exact position \nof the instability strip red edge. \n\n\\subsection{Evolutionary status and age of the Cepheid and its companion}\n\nWe computed the evolutionary tracks of the two component stars of the OGLE LMC562.05.9009 system by means of the Pisa release of the FRANEC code\n(Degl'Innocenti et al. 2008; Tognelli et al. 2011) adopting the same input physics and prescriptions described in detail in Dell'Omodarme et al. (2011).\nAn important exception is the neglecting of microscopic diffusion of helium and metals, because of their negligible impact on the evolution of\nintermediate-mass stars, as we did in our previous paper on OGLE-LMC-CEP-0227 (Prada Moroni et al. 2012). During the central hydrogen burning phase,\nwe took into account an overshooting of $l_{ov}$=$\\beta_{ov} H_p$ - where $H_p$ is the pressure height-scale and $ \\beta_{ov}=0.25 $ - beyond the Schwarzschild \nclassical border of the convective core. We computed the evolutionary tracks and isochrones adopting a value of the mixing-length parameter - which\nparametrizes the efficiency of the super-adiabatic convection - $\\alpha$ = 1.74. This value results from a solar calibration with our own\nStandard Solar Model computed with the same version of the FRANEC code used to compute the evolutionary tracks in this work. For a quantitative evaluation\nof some of the main sources of uncertainty affecting the theoretical evolutionary models of He-burning stars of intermediate mass we refer to\nValle et al. (2009).\n\nThe initial metal and helium abundances adopted for the calculations are Z=0.005 and Y=0.258, respectively.\n\nIn Figure 10, the locations of the two stars on the luminosity-effective temperature diagram from the parameters derived in this study\n(see Table 3) are shown. Also plotted are the boundaries of the classical fundamental mode Cepheid instability strip, for metallicities of Z=0.004\nand Z=0.008, taken from Bono et al. (2005). It is seen that for both metallicities, not only the Cepheid, but also the stable companion star\nare located inside the instability strip. A likely explanation is that the current uncertainty on the effective temperature of the companion star \nis somewhat underestimated and that a future, more accurate determination of the temperature will move the non-pulsating star in OGLE LMC562.05.9009\nslightly beyond the Cepheid instability strip; but there is also the possibility of significant different metallicities of the two stars.\n\nAlso shown in Fig. 10 are the evolutionary tracks computed for the masses of the two stars, using the prescripts detailed above. A isochrone for an age of\n205 Myr fits the position of both stars on the diagram reasonably well within the observational uncertainties on their luminosities and temperatures.\nThe age of the Cepheid expected from the theoretical period-age relation for fundamental mode classical Cepheids of Bono et al. (2005, their Table 4) \nfor a metallicity of Z=0.004 (very slightly smaller than our assumed metallicity of Z=0.005 for the calculation of the isochrone) is $130 \\pm 35 Myr$. Our\ncurrent age determination for the classical Cepheid in the binary system is about $2\\sigma$ larger than its age as predicted from the Bono et al.\nperiod-age relation, but given the uncertainties involved the two values are marginally consistent. We will check on this more deeply once we have\nnew data which will allow us a more accurate determination of the temperatures and luminosities of the two stars, and of their metallicities, leading to a more\naccurate age determination from the isochrone method. The current\nresults do however support the conclusion that both stars in the OGLE LMC562.05.9009 system are coeval, with an age larger than, but within the errors consistent with\nthe value predicted for the Cepheid from a theoretical period-age relation.\n\n\n\\section{Conclusions}\n\\label{sect:concl}\n\nWe have confirmed from high-resolution spectra that the eclipsing binary system OGLE LMC 562.05.9009 contains a classical Cepheid pulsating\nwith a period of 2.988 days in orbit with a stable secondary component. We performed the analysis\nof our extensive spectroscopic and photometric datasets in the same way as described in our previous analysis of the OGLE-LMC-CEP-0227\nsystem by P13, and have derived very accurate masses (to 0.8\\%) and radii (0.7\\%) for both the Cepheid and its non-pulsating companion star, which has a nearly identical\nmass and radius as the Cepheid. The orbit is highly eccentric with $e=0.61$ and a very long period of 1550 days, or 4.2 years. Our solution defines the orbital\nradial velocity curves of both components, disentangled from the pulsational velocity variations of the Cepheid, extremely well, as well as the pulsational radial\nvelocity curve of the Cepheid. Our analysis yields the second precise determination of the p-factor of a Cepheid in a binary so far in the literature,\nand was used to determine the radius variation of the Cepheid over its pulsation cycle. Our model reproduces the observed light curves extremely well, particularly the\nprimary eclipse when the companion star transits in front of the Cepheid. We calculated evolutionary tracks for the two component stars in the system\nand find that a isochrone for an age of 205 Myr fits the observed positions of both stars in the luminosity-effective temperature plane, arguing for the same age\nof the Cepheid and its red giant companion.\n\nThe p-factor value for the Cepheid is marginally consistent with the prediction of the p-factor relation of Storm et al. (2011), as opposed to the p-factor we derived \nfor OGLE-LMC-CEP-0227 in P13, which is in significant disagreement with the prediction of the Storm et al. relation. Currently the situation regarding the correct p-factor\nvalues to use in Baade-Wesselink-type Cepheid distance determinations is still very confusing. The measurements from the two binary Cepheids in this paper\nand in P13 seem to support the idea that the p-factor for classical Cepheids is not only period-dependent, but might also possess an intrinsic dispersion,\nat least for short pulsation periods in the range of a few days. Clearly more work is needed to clarify this question, and one of the very few observational approaches\nwhich promise to solve the issue is the analysis of more Cepheids in eclipsing binaries whose characteristics allow the determination of their p-factors.\nThe most\nimportant parameter in this context is the radius variation amplitude of the Cepheid; the larger the amplitude, the stronger the effect of the radius variation on \nthe binary light curve,\nand the smaller the uncertainty on the p-factor derived from our model. This was the reason why we could not measure the p-factor for the first overtone Cepheid\nin the eclipsing system OGLE-LMC-CEP-2532 whose radius variation is too small to cause a significant effect on the binary light curve, given\nthe quality of the photometric data (Pilecki et al. 2015).\nSince fundamental mode Cepheids tend to have larger radius variations, precise measurements of Cepheid projection factors with our binary method will mostly be restricted\nto eclipsing systems containing fundamental mode Cepheids.\n\nIn order to analyze the OGLE LMC562.05.9009 system, and in particular its Cepheid more fully, we plan to observe more eclipses (both primary and secondary) in the\nfuture, including coverage in near-infrared bands. A high quality out-of-eclipse the pulsational K-band\nlight curve of the Cepheid in tandem with the V-band light and pulsational radial velocity curves as determined in this paper will allow us to calculate\nthe distance to the Cepheid with the BW-type Infrared Surface Brightness Technique (Fouqu{\\'e} \\& Gieren 1997; Storm et al. 2011) and compare it to the distance\nof its companion star determined from the binary analysis and a surface brightness-color relation, as described in Pietrzynski et al. (2009, 2013). Such a comparison\nwill put further constraints on the p-factor relation valid for classical Cepheid variables.\n\nOur work has now revealed and analyzed the fifth eclipsing binary system containing a classical Cepheid in orbit with a stable giant star. Previous binary Cepheids\nanalyzed by our group are OGLE-LMC-CEP-0227 (Pietrzynski et al. 2010, P13), OGLE-LMC-CEP-1812 (Pietrzynski et al. 2011), OGLE-LMC-CEP-1718 (Gieren et al. 2014), and\nOGLE-LMC-CEP-2532 (Pilecki et al. 2015). The most exotic system is OGLE-LMC-CEP-1718 which contains two classical Cepheids in a 413-day orbit. Its analysis in\nGieren et al. (2014) has been very challenging due to the multiple superimposed variations in the light- and radial velocity curves. We hope to improve\non the analysis of that exciting system in the near future with additional data and possible improvements in our analysis code. For all systems but one,\nOGLE-LMC-CEP-1812, the mass ratio is very close to, or consistent with unity. The exception in the case of OGLE-LMC-CEP-1812 is probably explained by the result \nreported by Neilson et al. (2015a) that the Cepheid in that system is actually the product of a stellar merger of two main sequence stars. From an observational \npoint of view, there is a bias which favors the finding of systems composed of a Cepheid in orbit with a giant star of similar mass and radius which leads not only to\na higher probability to observe both eclipses, but also to observe the lines of both components in the composite spectra. For this reason, we cannot argue\nthat our results to-date on Cepheids in double-lined eclipsing binary systems in the LMC contradict results regarding the binary distribution of Cepheids\nas obtained by Evans et al. (2015), or Neilson et al. (2015b).\n\nThe binary Cepheids in the LMC, with their dynamical masses determined to better than 2\\%, will be a cornerstone for\nimproving our detailed understanding of Cepheid pulsation and post-main sequence stellar evolution, and in general of our understanding of Cepheid physics. With future\nprecise distance determinations to these systems we hope to determine from the stable binary companions, these binary Cepheids will also become excellent\nabsolute calibrators of the extragalactic distance scale.\n\n\n\n\\acknowledgments\nWe gratefully acknowledge financial support for this work from the BASAL Centro de Astrof{\\'i}sica y Tecnolog{\\'i}as Afines (CATA) PFB-06\/2007, from\nthe Polish National Science Center grant MAESTRO DEC-2012\/06\/A\/ST9\/00269, and from the Polish NCN grant DEC-2011\/03\/B\/ST9\/02573.\nWG, MG, DG, DM and MC also gratefully acknowledge support for this work from the Chilean Ministry of Economy, Development and Tourism's Millennium Science Initiative through grant IC120009 \nawarded to the Millennium Institute of Astrophysics (MAS). AG acknowledges support from FONDECYT grant 3130361, and MC from FONDECYT grant 1141141.\nThe OGLE Project has received funding \nfrom the National Science Center, Poland, grant MAESTRO 2014\/14\/A\/ST9\/00121 to AU.\n\n\nThis paper utilizes public domain data obtained by the MACHO Project, jointly funded by the US Department of Energy through the University of California, Lawrence Livermore National Laboratory\n under contract No. W-7405-Eng-48, by the National Science Foundation through the Center for Particle Astrophysics of the University of California under cooperative agreement AST-8809616, \n and by the Mount Stromlo and Siding Spring Observatory, part of the Australian National University.\n\nWe would like to thank the support staffs at the ESO Paranal and La Silla and Las Campanas Observatories for their help in obtaining the\nobservations.\nWe thank the ESO OPC and the CNTAC for generous allocation of observing time for this project.\n\nThis research has made use of NASA's Astrophysics Data System Service.\n\n{\\it Facilities:} \\facility{ESO:3.6m (HARPS)}, \\facility{ESO:NTT (SOFI)}, \\facility{VLT:Kueyen (UVES)}, \\facility{Magellan:Clay (MIKE)}, \\facility{Warsaw telescope}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \\label{sec:Intro}\n \\emph{Swift} J164449.3+573451 (hereafter, Swift J1644+57) was first discovered by the \\emph{Swift} Burst Alert Telescope (BAT) at 12:57:45 UT on 28 March 2011 \\citep{Burrows2011,Levan2011}. Some evidences suggest that Swift J1644+57 is a tidal disruption of a star by a supermassive black hole (SMBH). This phenomenon triggered the BAT three times after the initial trigger during the first few days \\citep{Burrows2011, Levan2011}. The late-time X-ray light curve was extended to a longer period by following the expected power-law decay for the tidal disruption of a star, i.e., $t^{-5\/3}$ \\citep[e.g.,][]{Rees1988}. Finally, the source of X-ray, IR, and radio emissions were well matched up with the center of the host galaxy where a SMBH resides \\citep{Levan2011,Zauderer2011}. \n\n There have been many studies performed to understand the nature of this event, such as the characteristics of the star that was disrupted. Such a question is closely connected to the SMBH mass ($M_{\\rm BH}$). The disruption of a solar-type star is possible for all $M_{\\rm BH} < 10^8\\, M_{\\odot}$ \\citep{Rees1988,Cannizzo1990,Bloom2011}, but compact stars like a white dwarf can be disrupted only if $M_{\\rm BH} < 10^{5}\\, M_{\\odot}$ \\citep{kp2011}. If so, then this kind of event provides the interesting possibility of discovering intermediate-mass black holes.\n\n\n Unfortunately, there has been controversy concerning the mass of the SMBH. \\citet{Burrows2011} provided a rough estimate of the SMBH mass of $\\sim2\\times 10^{7}\\,M_{\\odot}$ using a black hole mass -- luminosity relation and the lower limit of $\\sim10^6\\,M_{\\odot}$ based on the X-ray variability. Similarly, \\citet{Levan2011} estimated $M_{\\rm BH}$ to be $2\\times 10^{6}$ -- $10^{7}\\,M_{\\odot}$, derived from $K$-band luminosity, but at that time, $K$-band luminosity contained a significant amount of the transient light. \\citet{mg2011} utilized a relation between the black hole mass, the radio luminosity, and the X-ray luminosity, and found $M_{\\rm BH}$ $\\sim10^{5.5}\\,M_{\\odot}$. Using a quasi-periodic oscillation resonance hypothesis, \\citet{al2012} provided an $M_{\\rm BH}$ estimate of $\\sim10^{5}\\,M_{\\odot}$. \\citet{kp2011} concluded that a white dwarf was tidally disrupted and the mass of SMBH is less than $10^{5}\\,M_{\\odot}$ in light of the short timescales of the X-ray light curve. In summary, the $M_{\\rm BH}$ estimates have centered around the two discrepant values of $10^{7}\\,M_{\\odot}$ and $10^{5}\\,M_{\\odot}$ or less. A better understanding of the host galaxy properties is needed to clear up the situation. \n\n\\begin{deluxetable*}{cccccc}\n\\tabletypesize{\\scriptsize}\n\\tablewidth{0pt}\n\\tablecaption{\\emph{HST} WFC3 Data Log} \n\\tablehead{\n\\colhead{Observation date (UT)} & \\colhead{MJD\\tablenotemark{a}} & \\colhead{Days since trigger} & \\colhead{band} & \\colhead{Exptime (s)} & \\colhead{Magnitude (AB)}\n}\n\\startdata \\\\\n2011-04-04 & 55655.147614 & 6.6 & F606W & 1260 & 22.69$\\pm$0.01\\\\\n2011-08-04 & 55777.276876 & 129 & F606W & 4160 & 22.76$\\pm$0.01\\\\\n2011-12-02 & 55897.684390 & 249 & F606W & 1113 & 22.77$\\pm$0.01\\\\\n2013-04-12 & 56394.429204 & 746 & F606W & 2600 & 22.74$\\pm$0.01\\\\\n\\\\\n\\tableline \\\\\n2011-04-04 & 55655.132654 & 6.6 & F160W & 997 & 20.68$\\pm$0.01\\\\\n2011-08-04 & 55777.257148 & 129 & F160W & 1412 & 21.09$\\pm$0.01\\\\\n2011-12-02 & 55897.702220 & 249 & F160W & 1209 & 21.22$\\pm$0.02\\\\\n2013-04-12 & 56394.295795 & 746 & F160W & 2812 & 21.55$\\pm$0.02\\\\\n\\enddata\n\\tablenotetext{a}{Exposure start time in Modified Julian Date (MJD)}\n\\label{HSTtab}\n\\end{deluxetable*}\n\n\\begin{deluxetable*}{cccccc}\n\\tabletypesize{\\scriptsize}\n\\tablewidth{0pt}\n\\tablecaption{\\emph{Spitzer} IRAC Data Log} \n\\tablehead{\n\\colhead{Observation date (UT)} & \\colhead{MJD\\tablenotemark{a}} & \\colhead{Days since trigger} & \\colhead{band} & \\colhead{Exptime (s)} & \\colhead{Magnitude (AB)}\n}\n\\startdata \\\\\n2011-04-28 & 55679.975316 & 31.4 & 3.6$\\mu$m & 1250 & 19.50$\\pm$0.02 \\\\ \n2011-10-31 & 55865.023052 & 216 & 3.6$\\mu$m & 1253 & 21.49$\\pm$0.06 \\\\ \n2012-02-24 & 55981.541644 & 333 & 3.6$\\mu$m & 1252 & 21.70$\\pm$0.07 \\\\ \n\\\\\n\\tableline \\\\\n2011-04-28 & 55679.975316 & 31.4 & 4.5$\\mu$m & 1250 & 19.30$\\pm$0.01 \\\\ \n2011-10-31 & 55865.023052 & 216 & 4.5$\\mu$m & 1253 & 21.25$\\pm$0.05 \\\\ \n2012-02-24 & 55981.541644 & 333 & 4.5$\\mu$m & 1252 & 21.57$\\pm$0.08 \\\\ \n\\enddata\n\\tablenotetext{a}{MJD in UTC at data collection event (DCE) start}\n\\label{spitzertab}\n\\end{deluxetable*}\n\n In order to more accurately estimate the SMBH mass and better constrain the properties of the host galaxy, we analyze the morphology and the surface brightness profile of the host galxaxy based on high-resolution \\emph{Hubble Space Telescope} (\\emph{HST}) images and estimate the multi-band fluxes of the host galaxy using our long-term monitoring data lasting more than 2.4 years. We fit the multi-band spectral energy distribution (SED) of the host galaxy luminosity with stellar population synthesis models, and then obtain the properties of the galaxy. Finally, we provide our best estimate of $M_{\\rm BH}$ based on the host galaxy properties.\n\n This is the second of a series of two papers. In the first paper (M. Im et al. 2015 in preparation, hereafter, Im15), we present the dataset of the long-term monitoring campaign and an analysis of the late-time light curve. \n\nThroughout the paper, we selected \\emph{H$_0=70$}km s$^{-1}$Mpc$^{-1}$, $\\Omega_{\\Lambda}=0.7$, and $\\Omega_{m}=0.3$ as cosmological parameters and adopt the AB magnitude system.\n\\\\\n\n\n\n\\section{Observations and Data}\nWe observed \\emph{Swift} J1644+57 using Wide Field Camera (WFCAM) on United Kingdom Infrared Telescope (UKIRT) for nearly 2.4 years following the burst as a part of our gamma-ray burst (GRB) and transient observation program \\citep{Lee2010}. We observed intensively in the $K$ band among $Z, Y, J ,H $, and $K$ bands of WFCAM. The number of epochs of $K$ band data used for thie analysis is $101$ and the last data were observed at $884.7$ days after the initial BAT trigger. The numbers of epochs for the $Y,J,H$-band data are $3,15,28$ and the last data were observed at $\\Delta t = 712.1$, $710.1$, and $884.7$ days, respectively, where $\\Delta t$ is the number of days since the initial BAT trigger. We have only one epoch of data for the UKIRT $Z$ band which was observed at $\\Delta t = 723.0$ days. \n\n We also observed \\emph{Swift} J1644+57 using Camera for QUasars in EArly uNiverse \\citep[CQUEAN;][]{KimE2011,Park2012,Lim2013} on the 2.1m Otto-Struve telescope of the McDonald Observatory in $g, r, i, z$, and $Y$ bands. The numbers of epochs for the $g, r, i, z,$ and $Y$ band data are $2,2,14,14$, and $2$ and the last data were observed at $\\Delta t = 25.7, 217.5, 526.6, 526.6$, and $25.8$ days respectively. The UKIRT and CQUEAN observation logs and photometry results are described in Im15. \n\n In addition, we also used data from \\citet{Burrows2011} and \\citet{Levan2011} for the earlier optical and near-infrared (NIR) data. \n\n Morphology analysis requires high-resolution images because this object is so compact that it is virtually a point source in the UKIRT and CQUEAN images. For the high-resolution images, we obtained \\emph{HST} WFC3 multi-drizzled, stacked images\\footnote{Based on observations made with the NASA\/ESA Hubble Space Telescope, obtained from the data archive at the Space Telescope Science Institute. STScI is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract NAS 5-26555.} available in the MAST database. We used F606W-, F160W-band data and the number of epochs in each two band is four. These data were observed at $\\Delta t = 6.6, 129, 249,$ and $746$ days. The \\emph{HST} WFC3 data are summarized in Table~\\ref{HSTtab}.\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.225,angle=00]{Swift_Mor.eps}\n\t\\caption{Images of the host galaxy in F606W, the best-fit two-dimensional models from GALFIT, and the residuals for the single S\\'{e}rsic component model and the S\\'{e}rsic bulge $+$ exponential disk model. Right panels show one-dimensional profiles of the host galaxy and each model component.\n\t\t\\label{sbfig}}\n\\end{figure*}\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.225,angle=00]{Swift_Mor2.eps}\n\t\\caption{Same as Figure~\\ref{sbfig}, but for the single exponential disk model and the double exponential profile model. \n\t\t\\label{sb2fig}}\n\\end{figure*}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.22,angle=00]{Swift_HST.eps}\n\t\\caption{Flux fractions of the GALFIT models as a function of time. We used the model which consists of a single S\\'{e}rsic bulge with $n=3.43$ and a point source component. The upper panel shows the results for the F606W-band images, while the lower panel shows the results for the F160W-band images. Magnitudes of the model components are also shown.\n\t\t\\label{hstfig}}\n\\end{figure}\n\n \n To supplement the NIR observation data, we used the \\emph{Spitzer} IRAC 3.6, 4.5$\\mu$m post basic calibrated data (PBCD) from the NASA\/IPAC Infrared Science Archive. These were observed at $\\Delta t = 31.4, 216.5$, and $333.0$ days. A log of the \\emph{Spitzer} IRAC 3.6, 4.5$\\mu$m data are shown in Table~\\ref{spitzertab}.\n\nThe flux measurements were performed by SExtractor software\\footnote{We used aperture magnitudes with aperture correction.} \\citep{Bertin1996} except for the \\emph{HST} images for which GALFIT \\citep{Peng2010} models were used for the flux measurements. \n\n For the X-ray data, we used \\emph{Swift}\/XRT data taken from the \\emph{Swift} archive and \\emph{XMM-Newton} data\\footnote{Based on observations obtained with XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA} from the \\emph{XMM-Newton} Science Archive.\\\\\n\n\n\n\\section{Morphology of the host galaxy} \\label{sec:Mor}\n We analyzed the surface brightness profile of the host galaxy, in order to determine the bulge fraction and its nature. We used the \\emph{HST} images and GALFIT software to fit two-dimensional models to the light distribution of the host galaxy. To construct the point spread function (PSF), we selected $\\sim 5$ isolated stars with signal-to-noise ratios $\\gtrsim 300$ in the vicinity of \\emph{Swift} J1644+57, and co-added them.\n\n We used error images that are created by GALFIT for the fitting. For GALFIT to create the error image properly, we modified the unit of ADU and the image header values such that $\\mathrm{GAIN} \\times \\mathrm{ADU} \\times \\mathrm{NCOMBINE} = \\mathrm{[electrons]}$ as recommended in GALFIT website\\footnote{http:\/\/users.obs.carnegiescience.edu\/peng\/work\/galfit\/TOP10.html}.\n\n A crucial factor affecting the fitting results is background subtraction. For the background determination, we set 6 annuli with the radii logarithmically increasing between 2.5 and 9 times the radius of an ellipse for which pixel values are $1.5\\sigma$ of the background noise. We centered the annuli on the center of the host galaxy, set the minimum width of the annuli to be $\\sim1.3$ arcsec ($\\sim33$ pixel) for F606W images and $\\sim2$ arcsec ($\\sim16$ pixel) for F160W images, and augmented the widths in step with the logarithmically growing radii. We then derived the mean pixel values of each annulus. Finally, we adopted their mean value as the background value.\n\n\n\n Our surface brightness fit was carried out using a deep, stacked image of the data taken with F606W at $\\Delta t = $ 129, 249, and 749 days. It has been known that the transient component is negligible in the optical bands bluer than $i$ even at the early time \\citep{Burrows2011, Levan2011}. The use of the stacked, late-time image in the F606W band makes the transient component more negligible. On the other hand, NIR-bands, including F160W (similar to $H$ band of WFCAM), are known to contain a significant transient component which may affect the host galaxy analysis. Furthermore, the spatial resolution of the F606W images is better by a factor of three than that of F160W, which greatly helps the surface brightness fitting. The other \\emph{HST} data were also analyzed to understand the importance of the transient component, and the results for the transient component are presented later in this section.\n\n For the galaxy models, we used the S\\'{e}rsic \\citep{Sersic1968}, de Vaucouleurs \\citep{deVa1948}, and exponential disk profiles or a combination of thereof. The S\\'{e}rsic profile is described as\n\\begin{displaymath}\n \\Sigma(r)=\\Sigma_{e}\\mathrm{exp}\\Bigg[-\\kappa\\Bigg(\\bigg(\\frac{r}{r_e}\\bigg)^{1\/n}-1\\Bigg)\\Bigg],\n\\end{displaymath}\nwhere $\\Sigma_{e}$ is the surface brightness at the effective radius $r_e$, and $n$ is the S\\'{e}rsic index. $\\kappa$ is a variable parameter denpendent on $n$, where $n=4$ and $1$ correspond to the de Vaucouleurs and exponential profiles, respectively. Although $n=4$ is commonly quoted for the ellipticals and classical bulges, the S\\'{e}rsic index of ellipticals and classical bulges can assume a value in the range $2\\lesssim n \\lesssim6$, whereas pseudobulges have $n\\lesssim 2$ \\citep{Fisher2008, Fisher2010}.\n\n All of the model parameters such as ellipticity and center positions of the different components, were set free in the fitting procedure. \n\n Figures~\\ref{sbfig} and \\ref{sb2fig} show images of the host galaxy, the two-dimensional models, and the residuals (i.e., the model subtracted images), for four different models: (1) a single S\\'{e}rsic; (2) a S\\'{e}rsic bulge + exponential disk; (3) an exponential disk; and (4) a double exponential profile models. The figures also show one-dimensional profiles (along the major axis) of the host galaxy and those of each model component, which are converted through the IRAF\\footnote{IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation.} ELLIPSE task. In addition to the profiles, the differences between the data and the model profiles are shown. The results of each fit are summarized in Table~\\ref{galfittab}. Both the single S\\'{e}rsic model with $n=3.43\\pm0.05$ and the S\\'{e}rsic bulge with $n=3.39\\pm0.11$ $+$ exponential disk model fit the data well ($\\chi_{\\nu}$ $\\sim 1.2$ -- $1.3$). When the disk component is added, the bulge to total host galaxy flux ratio (B\/T) is $0.83\\pm0.03$. On the other hand, the single exponential disk model provides a poor fit to the data as shown in Figure 2 and with $\\chi_{\\nu}=6.54$. The double exponential profile model fits the data nearly as well as the single S\\'{e}rsic model and the S\\'{e}rsic bulge+disk model in terms of $\\chi_{\\nu}^{2}$. However, the analysis of the one-dimensional surface brightness profile shows that the model does not follow the outer part of the profile well, demonstrating a relatively steeper decline than that of the single S\\'{e}rsic model and the S\\'{e}rsic bulge $+$ exponential disk model. This model gives B\/T $= 0.36$, suggesting a significant bulge component. Therefore, we conclude that the host galaxy of \\emph{Swift} J1644+57 is bulge-dominant. We also conclude that the bulge is likely to have a S\\'{e}rsic index higher than $3$ regardless of the existence of the disk. This value corresponds to the range of the classical bulges \\citep{Fisher2008, Fisher2010}. We cannot completely exclude the case where the bulge is pseudobulge with $n \\sim 1$, but even in this case, the object has a significant bulge.\n\n\\begin{deluxetable*}{ccccccccccc}\n\\tabletypesize{\\scriptsize}\n\\tablewidth{0pt}\n\\tablecaption{Surface Brightness Fitting Result} \n\\tablehead{\n\\colhead{} & \\multicolumn{3}{c}{Bulge} & \\colhead{} & \\multicolumn{2}{c}{Disk}& \\colhead{} & \\colhead{} & \\colhead{} & \\colhead{} \\\\ \n\\cline{2-4}\\cline{6-7}\\\\\n\\colhead{Galaxy model} & \\colhead{$m_{\\mathrm{b}}$ [AB]} & \\colhead{$n$} & \\colhead{$r_{\\mathrm{eff}}[\\mathrm{kpc}]$} & & \\colhead{$m_{\\mathrm{d}}$ [AB]} & \\colhead{$r_{\\mathrm{s}}[\\mathrm{kpc}]$} & \\colhead{} & \\colhead {B\/T} & \\colhead {$m_{\\mathrm{t}}$ [AB]} & \\colhead{$\\chi_{\\nu}^{2}$}\\\\ \n\\colhead{(1)} & \\colhead{(2)} & \\colhead{(3)} & \\colhead{(4)} & \\colhead{} & \\colhead{(5)} & \\colhead{(6)} & \\colhead{} & \\colhead{(7)} & \\colhead{(8)} & \\colhead{(9)}\n}\n\\startdata\nB & $22.77\\pm0.01$ & $3.43\\pm0.05$ & $1.01\\pm0.01$ & & \\nodata & \\nodata & & \\nodata & $22.77\\pm0.01$ & 1.322\\\\\nB$+$D & $23.01\\pm0.03$ & $3.39\\pm0.11$ & $0.79\\pm0.03$ & & $24.75\\pm0.14$ & $1.21\\pm0.07$ & & $0.83\\pm0.03$ & $22.81\\pm0.03$ & 1.223\\\\\n\\tableline \\\\\nB$(n=4)$ & $22.72\\pm0.00$ &4 (fixed) & $1.09\\pm0.01$ & & \\nodata & \\nodata & & \\nodata & $22.72\\pm0.00$ & 1.275\\\\\nD & \\nodata & \\nodata & \\nodata & & $23.07\\pm0.00$ & $0.47\\pm0.00$ & & \\nodata & $23.07\\pm0.00$ & 6.547\\\\\nB$(n=4)$$+$D & $23.01\\pm0.03$ & 4 (fixed) & $0.85\\pm0.02$ & & $24.64\\pm0.10$ & $0.95\\pm0.03$ & & $0.82\\pm0.03$ & $22.79\\pm0.03$ & 1.228\\\\\nS$(n=1)$$+$D & $23.99\\pm0.01$ & 1 (fixed) & $0.30\\pm0.00$ & & $23.35\\pm0.00$ & $0.92\\pm0.01$ & & $0.36\\pm0.00$ & $22.87\\pm0.00$ & 1.351\n\\enddata\n\\tablecomments{Column~1: galaxy model for the two-dimensional fitting. B: S\\'{e}rsic bulge, D: exponential disk , B$(n=4)$: de Vaucouleurs bulge. S$(n=1)$: S\\'{e}rsic profile with fixed $n=1$ (exponential profile). Column~2: AB magnitude of the bulge component. Column~3: S\\'{e}rsic index for the bulge model. Column~4: effective radius. Column~5: AB magnitude of the disk component. Column~6: scale length of the disk component. Column~7: bulge to total light ratio. Column~8: total magnitude. Column~9: reduced $\\chi^2$ for the fitting model defined as\n\\begin{displaymath}\n \\chi_{\\nu}^{2}=\\frac{1}{N_\\mathrm{DOF}} \\sum_{x=1}^{nx}\\sum_{y=1}^{ny}\\frac{(f_\\mathrm{data}(x,y)-f_\\mathrm{model}(x,y))^2}{\\sigma(x,y)^2},\n\\end{displaymath} \nwhere $f_\\mathrm{data}(x,y)$ and $f_\\mathrm{model}(x,y)$ mean the input data and the model images, respectively. $N_\\mathrm{DOF}$ is the degree of freedom. $\\sigma(x,y)$ is the error image. Here, sum is only over all $nx$ and $ny$ pixels satisfying $1.5\\sigma$ of the background noise.}\n\\label{galfittab}\n\\end{deluxetable*}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.36,angle=00]{Swift_Ukirt.eps}\n\t\\caption{NIR and the X-ray light curves of the \\emph{Swift} J1644+57.\n The light curves of $Y, J ,H ,K$ bands with the early data from \\citet{Burrows2011}, \\citet{Levan2011}, and \\emph{Swift}\/XRT $0.3$ -- $10$keV are shown. All of the light curves have similar shapes except that the X-ray light curve is $\\sim$15 days ahead of the NIR light curves. After $\\sim$500 days the X-ray emission was rapidly declined as shown in with star mark for the last X-ray data from the \\emph{Swift}\/XRT.\n \\label{ukirtfig}}\n\\end{figure*}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.22,angle=00]{Swift_Xray.eps}\n\t\\caption{\\emph{Swift}\/XRT and \\emph{XMM-Newton} data as well as recent \\emph{Chandra} observation of \\emph{Swift} J1644+57. The X-ray flux abruptly declined after $\\sim$500 days since the BAT trigger. \n\t\t\\label{xfig}}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.175,angle=00]{Swift_Ukirtlinear.eps}\n\t\\caption{$H$- and $K$-band light curves in linear scale. Some of the data points at very early times are cut in order to highlight the late-time light curves. The magnitudes of the latest $H, K$ bands converge to single values, suggesting that the transient component has disappeared at $\\Delta t > 500$ days.\n\t\t\\label{linfig}}\n\\end{figure}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.30,angle=00]{Swift_lc.eps}\n\t\\caption{Light curves of CQUEAN $i$ and $z$ bands, UKIRT $Z$-band data, and \\emph{Spitzer} IRAC 3.6$\\mu$m and 4.5$\\mu$m bands. Note that the $z$-band flux decreases with time, while the $i$-band flux is almost constant with time. Considerable magnitude changes in the \\emph{Spitzer} IRAC 3.6$\\mu$m and 4.5$\\mu$m bands can be seen in the right plot.\n\t\t\\label{lcfig}}\n\\end{figure*}\n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.175,angle=00]{Swift_change.eps}\n\t\\caption{Temporal change of the SED of \\emph{Swift} J1644+57. The fluxes in the redder bands show the substantial changes with time, whereas the bluer-band fluxes do not vary with time.\n\t\t\\label{changefig}}\n\\end{figure}\n\nAdditionally we fit the observed surface brightness profile with a single de Vaucouleurs bulge model and a de Vaucouleurs bulge $+$ exponential disk model. The results of these fits are nearly identical to that of the single S\\'{e}rsic and the S\\'{e}rsic bulge+disk models.\n\n To estimate the transient component flux, we fit all the F606W and F160W images with a model containing both the point source (transient) and the host galaxy components. Here, we adopt a single S\\'{e}rsic profile with a fixed S\\'{e}rsic index ($n=3.43$) for the host galaxy component, and a PSF profile for the transient component. The compactness of the host galaxy and the bright transient component in the F160W images create a serious degeneracy, particularly between the effective radius and the S\\'{e}rsic index, when fitting multi-component models. To alleviate the degeneracy, we fixed the S\\'{e}rsic index to be $n=3.43$, similar to that of the F606W band. The flux fractions of the models as a function of time are shown in Figure~\\ref{hstfig}. In the case of F606W, the flux fraction from the transient component is very small or nonexistent, while the transient component is very bright in the earliest F160W band, even brighter than the entire host galaxy. The result reflects a very red color to the transient well, and justifies the exclusion of the point source component in the late-time F606W images during the host galaxy analysis. The point source contribution declines rapidly as time goes on in F160W, but it contributes to the total flux of the object until around $\\Delta t = 750$ days. On the other hand, the fluxes of the host galaxy component are almost constant in both bands over the entire period. The magnitude of host galaxy in the F160W band is $\\sim21.75$ mag, and as we shall see in the next section, this is the same as for the last data point of the UKIRT $H$-band light curve, suggesting that the flux of the last data point in the NIR light curve represents the host galaxy flux. \n\\\\\n\n\n\n\\section{Light Curves} \\label{sec:LC}\n In this section, we show long-term observation results and estimate multi-band fluxes of the host galaxy of \\emph{Swift} J1644+57. Figure~\\ref{ukirtfig} shows the $Y, J, H$, and $K$ light curves. The gray data points in the background show \\emph{Swift}\/XRT $0.3$ -- $10$keV data. The $J,H,K$ light curves resemble each other. The NIR light curves rapidly decline until $\\Delta t \\simeq 10$ days, turn up again with a second peak at $\\Delta t \\simeq 30$ days, and decline again steadily. The behaviors of these NIR light curves are very similar to that of the X-ray light curve except that the X-ray light curve appears shifted ahead of the NIR light curves at a time of $\\sim$15 days. The similar shapes of these light curves indicate that the origins of the X-ray emission and NIR emission are related to each other. On the other hand, the time gap between these two emissons denotes that the X-ray source and NIR source are separated from each other as much as the time gap. \\citet{Bloom2011} suggested that X-ray source is in the close vicinity of the black hole due to the fact that the X-ray emission shows very rapid, high variability, while the IR and the radio sources are located a large distance from the black hole on account of the relatively smooth and small variability. They argued that the jet generated by black hole collides with the surrounding medium where the electrons are accelerated by the jet. These high-speed electrons emit the IR to radio photons through synchrotron radiation.\n\n\n The jet seems to be nearly turned off at $\\Delta t \\simeq 500$ days in light of the fact that there is an abrupt decrease in flux of a factor of $\\sim10$ or more, which can be seen in all the \\emph{Swift}\/XRT, \\emph{Chandra}, and \\emph{XMM-Newton} data \\citep{Levan2012, Zauderer2013}. This late-stage turn-off of X-ray flux is also shown in Figure~\\ref{xfig}. If the jet was turned off, then the transient components of the NIR fluxes must be quenched following the X-ray flux, and it is expected that the fluxes of the pure host galaxy of \\emph{Swift} J1644+57 were revealed at that time. As we can see in Figure~\\ref{linfig}, which shows the $H$-, $K$- bands light curves in linear scale, the fluxes of the $H, K$ bands converge to single values at late-time. Furthermore, the latest $H$-band magnitude is nearly the same as that of the host galaxy of the F160W-band images, shown in the results of the model fitting in \\S\\ref{sec:Mor}. This evidence indicates that it is reasonable to regard the NIR ($Y, J, H$, and $K$ band) fluxes of the last data, taken at $\\Delta t = \\sim 700$ or $884$ days, as those of the pure host galaxy. \n\n\n The left panel of Figure~\\ref{lcfig} shows the light curves of the CQUEAN $i$- and $z$-band and the UKIRT $Z$-band data. In the case of the $z$ band, the fluxes from the object slightly decrease with time. We also regard the flux of the last data, that is the UKIRT $Z$-band data, as the $Z$-band flux of the host galaxy since it is observed far beyond expected quenching time of jet. On the other hand, $i$-band fluxes are virtually constant, demonstrating that the transient components are basically non-existent in the $i$ or bluer bands \\citep[Figure~\\ref{hstfig};][]{Burrows2011, Levan2011}. We take the flux of the last data of the $i$ band as that of the host galaxy. The number of CQUEAN $g$- and $r$-band data points are scarce compared to the NIR data and the last data were observed at early time ($\\Delta t =25.7, 217.5$ days, respectively). However, there is little or no change between the very early-time magnitudes from \\citet{Levan2011} and our $g$- and $r$-band magnitudes in the same way as the fluxes of the $i$ and F606W bands. Therefore, we consider the $g$- and $r$-band fluxes of the last epoch data as those from the host galaxy. Furthermore, we also consider the magnitude of the single S\\'{e}rsic model of the stacked \\emph{HST} WFC3 F606W image as that of the host galaxy since the point source contribution to whole flux is negligible.\n\n\\begin{deluxetable}{ll}\n\\tabletypesize{\\scriptsize}\n\\tablewidth{0pt}\n\\tablecaption{Magnitudes of Host Galaxy} \n\\tablehead{\n\\colhead{Band} & \\colhead{Magnitude (AB)}\n}\n\\startdata\n$B$ & 24.24$\\pm$0.10 \\citep{Levan2011}\\\\\n$g$ & 23.67$\\pm$0.19\\\\\nF606W & 22.72$\\pm$0.01\\\\\n$r$ & 22.73$\\pm$0.06\\\\\n$i$ & 22.25$\\pm$0.04\\\\\n$Z$ & 22.16$\\pm$0.06\\\\\n$Y$ & 22.18$\\pm$0.07\\\\\n$J$ & 21.96$\\pm$0.08\\\\\n$H$ & 21.74$\\pm$0.12\\\\\n$K$ & 21.55$\\pm$0.10\\\\\n3.6$\\mu$m & 21.70$\\pm$0.07 (Including transient)\\\\\n4.5$\\mu$m & 21.57$\\pm$0.08 (Including transient)\n\\enddata\n\\tablecomments{Magnitudes are corrected by Galactic extinction based on \\citet{Schlafly2011}.}\n\\label{hosttab}\n\\end{deluxetable}\n\n\\begin{deluxetable*}{rl}\n\\tabletypesize{\\scriptsize}\n\\tablewidth{0pt}\n\\tablecaption{Input Parameters for SED Fitting} \n\\tablehead{\n\\colhead{Parameter} & \\colhead{Value}\n}\n\\startdata\n$\\tau$ ($e$-folding time scale of stellar population) & $6.5\\leq$ $\\log$[$\\tau$\/yr] $\\leq11.0$ with a step size of 0.1 \\\\\n$t$ (age of stellar population) & $8.0\\leq$ $\\log$[$t$\/yr] $\\leq10.3$ with a step size of 0.1 \\\\\nIMF (initial mass function) & \\citet{Salpeter1955} \\\\\nZ (metallicity) & 0.004, 0.008, 0.020, 0.050 \\\\\nExtinction law & \\citet{Calzetti2000} \\\\\n$A_{V}$ ($V$-band attenuation for stellar population in magnitude) & $0.0\\leq$ $A_{V}$ $\\leq3.0$ with a step size of 0.1 \n\\enddata\n\\label{SEDtab}\n\\end{deluxetable*}\n\n\n\\begin{deluxetable}{rr}\n\\tabletypesize{\\scriptsize}\n\\tablewidth{0pt}\n\\tablecaption{Best-fit Parameters from SED Fitting} \n\\tablehead{\n\\colhead{Parameter} & \\colhead{Value}\n}\n\\startdata\nStellar mass [$\\log(M_{\\star}\/M_{\\odot})$] & $9.14^{+0.13}_{-0.10}$\\\\\n\\\\\nSFR [$M_{\\odot}$\/yr] & $0.03^{+0.28}_{-0.03}$\\\\\n\\\\\nSpecific SFR [$\\log(\\rm{sSFR}\/\\rm{yr}^{-1})$] & $-10.62^{+0.90}_{-\\infty}$\\\\\n\\\\\n$t$ [Gyr] & $0.63^{+0.95}_{-0.43}$\\\\\n\\\\\n$\\tau$ [Gyr] & $0.10^{+0.24}_{-0.10}$\\\\\n\\\\\n$A_{V}$ & $0.00^{+0.97}_{-0.00}$\\\\ \n\\\\\nZ & $0.050^{+0.000}_{-0.046}$\\\\\n\\\\\n$\\chi^2$ & 1.64\n\\enddata\n\\label{SED2tab}\n\\end{deluxetable}\n\n The right panel of Figure~\\ref{lcfig} shows the light curves of the \\emph{Spitzer} IRAC 3.6$\\mu$m and 4.5$\\mu$m bands. The magnitude changes in these bands are the more significant than those for the other optical\/NIR bands. The fluxes from the transient component seem to be non-negligible even in the last epoch data ($\\Delta t = 333$ days), because the last IRAC epochs were still in the rapidly decreasing phase. Therefore, we consider the fluxes of the last epoch IRAC data to be the upper limit fluxes of the host galaxy.\n\n The multi-band magnitudes of the host galaxy of \\emph{Swift} J1644+57 are shown in Table~\\ref{hosttab}. We took the Galactic extinction of photometric data into account based on \\citet{Schlafly2011}. We added the $B$-band photometric data from \\citet{Levan2011} to expand the data points to the short wavelength band.\n\n The temporal change of the observed SED of \\emph{Swift} J1644+57 are summarized in Figure~\\ref{changefig}. The fluxes in redder bands show substantial changes with time. Meanwhile, bluer band fluxes are constant. This red feature of the transient has been suggested to be due to dust extinction. From previous studies, it is known that the hydrogen column density of the source of X-ray is large ($N_{\\mathrm{H}}\\sim10^{22}$cm$^{-2}$), meaning the line of sight to the SMBH has a very large extinction value ($A_{V}=4.5$ -- $10$) \\citep{Bloom2011,Burrows2011,Levan2011,Shao2011,Saxton2012}.\\\\\n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.296,angle=00]{SED.eps}\n\t\\caption{SED fitting result of the host galaxy of \\emph{Swift} J1644+57. Plotted are the multi-band fluxes of the host galaxy in Table~\\ref{hosttab}. Two \\emph{Spitzer} data are the upper limit fluxes. The $\\chi^2$ value for the fit is 1.64.\n\t\t\\label{SEDfig}}\n\\end{figure}\n\n\n\\section{SED Fitting} \\label{sec:SED}\n\nWe performed SED model fitting of the multi-band fluxes of host galaxy of \\emph{Swift} J1644+57 in order to determine the properties of the host galaxy such as the stellar mass ($M_{\\star}$) which is an important parameter for the $M_{\\rm BH}$ estimation. We utilized the code Fitting and Assessment of Synthetic Templates \\citep[FAST;\\footnote{http:\/\/astro.berkeley.edu\/$\\sim$mariska\/FAST.html}][]{Kriek2009}, which is a public SED fitting tool for the investigation of the galaxy properties using the photometric data ranging from UV to IR. The code is based on the IDL and fits of UV to IR stellar population templates to photometric data or galaxy spectra. FAST runs with the method of $\\chi^2$ fitting and using stellar population grids to derive the best-fit model and its parameters. \n\n We used the 2003 version of the Bruzual \\& Charlot (BC03) model \\citep{BC03} for the stellar population model. There are three initial mass functions (IMFs) available in the FAST \\citep{Salpeter1955,Kroupa2001,Chabrier2003}. We chose the Salpeter IMF. To define the star formation history (SFH), we assumed an exponentially decreasing star formation rate (SFR). The stellar population was modeled with the $e$-folding time scales, $6.5\\leq$ $\\log$[$\\tau$\/yr] $\\leq11.0$ with a step size of 0.1 and ages of $8.0\\leq$ $\\log$[$t$\/yr] $\\leq10.3$ with a step size of 0.1. We used several metallicity values such as Z = 0.004, 0.008, 0.02, and 0.05. The model SEDs were attenuated by dust, for which we used attenuation curves based on \\citet{Calzetti2000}. We adopted $0.0\\leq$ $A_{V}$ $\\leq3.0$ with a step size of 0.1. All the input parameters for the SED fitting are summarized in Table~\\ref{SEDtab}. \n\n Figure~\\ref{SEDfig} shows the SED fitting result. The two \\emph{Spitzer} data were treated as the upper limit fluxes of the host galaxy. The estimated stellar mass of the host galaxy is $\\log(M_{\\star}\/M_{\\odot}) = 9.14^{+0.13}_{-0.10}$. The $e$-folding time scale is $\\tau = 0.10^{+0.24}_{-0.10}$ Gyr and the age of the stellar population is $t = 0.63^{+0.95}_{-0.43}$ Gyr. The SFR of galaxy is $0.03^{+0.28}_{-0.03} \\, M_{\\odot}$\/yr, and the specific SFR is $\\log(\\rm{sSFR}\/\\rm{yr}^{-1}) = -10.62^{+0.90}_{-\\infty}$. \\citet{Levan2011} derived SFR of $0.3$ -- $0.7 \\, M_{\\odot}$\/yr from the H$\\alpha$ and [O II] emission line luminosities. The value of $0.3 \\, M_{\\odot}$\/yr from H$\\alpha$ is consistent with our 1$\\sigma$ upper limit. The SFR from [O II] line ($0.7 \\, M_{\\odot}$\/yr) is about twice larger but the [O II] line based SFRs are known to be dependent on physical condition such as the reddening and metallicity \\citep[e.g.,][]{Kewley2004}, and less reliable than H$\\alpha$ based SFRs. Another possible cause of the discrepancy is the different timescales that are probed by different SFR indicators (emission line indicators probing recent star formation). \\citet{Levan2011} estimated $E(B-V)_{gas}=-0.01 \\pm 0.15$mag, i.e., no extinction using the intensity ratio of H$\\alpha$ and H$\\beta$ lines. This is consistent with our SED fitting result $A_{V}=0.00^{+0.97}_{-0.00}$. The $\\chi^2$ value for the fit is 1.64. \n\n\n The host galaxy of \\emph{Swift} J1644+57 is a low mass, low SFR galaxy with a low extinction. Also it seems to have experienced a rapid decline of SFR not very long ago. This fits in well with a recent suggestion by \\citet{Arcavi2014} that host galaxies of tidal disruption events are E+A galaxies with $<1$ Gyr stellar population and low or no SFRs.\n\n The output parameters are given in Table~\\ref{SED2tab}. The errors correspond to 1$\\sigma$ confidence intervals derived from 100 times Monte Carlo simulations in which the input photometric data are changed according to their errors.\n \n We also tried the Chabrier IMF instead of the Salpeter IMF for the fit. The change of the IMF influenced to the stellar mass, decreasing the stellar mass by $\\sim0.25$ dex.\n\n We also fitted the SED model with the \\citet{Maraston2005} stellar population instead of the BC03 model. We set the input parameter ranges identical to the case of the BC03 stellar populations. The results were nearly identical to the BC03 result. \n\n Our analysis of the host galaxy shows that the host galaxy is a bulge-dominated and nearly extinction free ($A_{V} \\sim 0$ mag). On the other hand, the spectral properties of the nuclear transient suggests a high extinction ($A_{V} \\sim 6$ mag). These two facts may appear contradictory, but we note that a significant amount of dust can be found easily in nuclear region of bulge-dominated galaxies when their nuclei are acitve. For example, hosts of luminous AGNs are mostly early-type, bulge-dominated galaxies \\citep[e.g.,][]{Hong2015}, and such AGNs are known to contain a significant amount of dust in nuclear region as a form of hot or warm dusty torus \\citep[e.g.,][]{Kim2015}. \n\\\\\n\n\n\\section{Discussion on Black Hole Mass} \\label{sec:BH}\n Our results on the properties of the host galaxy of \\emph{Swift} J1644+57 can be summarized as follows. It is a bulge-dominant galaxy (B\/T=$0.83\\pm0.03$). The mass of the host galaxy is somewhat low at $10^{9.14} \\,M_{\\odot}$, even though the galaxy is bulge-dominated. Now we estimate the mass of the SMBH that played the main role in the transient phenomenon.\n\n It is now generally accepted that the SMBHs ($10^6$ -- $10^{10} \\,M_{\\odot}$) reside in the bulges of all massive galaxies. Tight scaling relations have been derived between SMBH mass and several physical properties of the bulges (velocity dispersion, mass, luminosity, etc.) in many previous studies \\citep{Magorrian1998,Ferrarese2000,Gebhardt2000,McLure2002,Marconi2003,Haring2004,Aller2007,Hopkins2007,Gultekin2009,Kormendy2009,Sani2011,Kormendy2013}. Some argue that ellipticals and classical bulges follow identical relations, while the pseudobulges follow a somewhat different relation with large scatter \\citep{Hu2008,Kormendy2011,Sani2011,Kormendy2013}. We conclude from the best-fit galaxy models that the host galaxy of \\emph{Swift} J1644+57 has a classical bulge, and have also found a minor possibility of the pseudobulge with B\/T=$0.36$. For now, we consider only the best model, that is, the case of the host galaxy having a classical bulge and being bulge-dominant. \n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.175,angle=00]{Swift_BHrange.eps}\n\t\\caption{Results on the black hole mass from this work and previous studies. The red circles and dashed line represent our results, while the black squares and lines denote the results from previous studies. Error bars correspond to deviation of 1$\\sigma$ and the arrow indicates the upper bound value.\n\t\t\\label{BHfig}}\n\\end{figure}\n\n In order to esimate the central SMBH mass in the host galaxy of \\emph{Swift} J1644+57, we used the scaling relation between $M_{\\rm BH}$ and the stellar mass of the bulge ($M_{\\star,\\mathrm{bul}}$). We expect that a large part of the stellar mass derived in \\S\\ref{sec:SED} belongs to the bulge component. \\citet{Sani2011} present the $M_{\\mathrm{BH}}$ -- $M_{\\star,\\mathrm{bul}}$ relation, where $M_{\\star,\\mathrm{bul}}$ is directly obtained from the bulge luminosity ($L_{\\mathrm{bul}}$) of \\emph{Spitzer} 3.6$\\mu$m and the calibrated $M_{\\star,\\mathrm{bul}}$ -- $L_{\\mathrm{bul}}$ relation. They excluded pseudobulges when constructing the relation. The relation is \n\\begin{equation}\n\t\\mathrm{log}(M_\\mathrm{BH}\/M_{\\odot})=\\alpha+\\beta\\times[\\mathrm{log}(M_\\mathrm{\\star,bul}\/M_{\\odot})-11],\n\\end{equation}\nwhere $\\alpha=8.16\\pm0.06$, $\\beta=0.79\\pm0.08$, and the intrinsic scatter is $0.38\\pm0.05$ . The estimated mass of the SMBH is $10^{6.7\\pm0.4} \\,M_{\\odot}$ based on the stellar mass of the host galaxy and the above relation. If we consider the B\/T=0.83 and assume that the mass-to-light ratio is constant in the bulge and disk, the stellar mass is decreased by $\\sim0.1$ dex. It leads to a decrease in $M_{\\mathrm{BH}}$ by $\\sim0.1$ dex.\n\n The tight scaling relations between $M_{\\mathrm{BH}}$ and host galaxy properties suggest a close link between the SMBH growth and the galaxy evolution. There may be a cosmic evolution of the scaling relations, for which there have been various studies \\citep{Treu2004,McLure2006,Shields2006,Woo2006,Salviander2007,Treu2007,Woo2008,Jahnke2009,Bennert2010,Decarli2010,Merloni2010,Bennert2011}. The evolution of the scaling relations is still controversial in terms of the selection effects in the high redshift regime. Nevertheless, we can consider a case where the growth of $M_{\\mathrm{BH}}$ happened ahead of the assembly of the stellar mass as suggested by some of these studies. \\citet{Bennert2011} suggest the redshift evolution out to $z\\sim2$, in the form of $M_{\\mathrm{BH}}\/M_{\\star,\\mathrm{bul}}\\propto(1+z)^{1.96\\pm0.55}$, based on 11 X-ray-selected broadline AGNs. Considering this evolution effect and the redshift of \\emph{Swift} J1644+57 $z=0.35$, the mass of the SMBH could be larger by $\\sim0.3$ dex.\n\n\n We also estimated $M_{\\mathrm{BH}}$ through the $M_{\\mathrm{BH}}$ -- $K$ band luminosity of bulge ($L_{K,\\mathrm{bul}}$) relation in \\citet{Kormendy2013}, assuming that most of the NIR fluxes come from the bulge. Their relation is much improved compared with previous studies in view of serveral things. They excluded galaxies with black hole monsters which have over-massive SMBHs despite having relatively small bulges and ellipticals. They also omitted galaxies with black hole masses that are measured based on the kinematics of ionized gas without taking line widths into account, since this method may yield underestimated masses. Galaxies in the process of merging generally have low-mass black holes for their luminosities. Thus they excluded these galaxies from the relation. They also did not include pseudobulges. The relation of \\citet{Kormendy2013} is\n\\begin{equation}\n\t\\mathrm{log}(M_\\mathrm{BH}\/10^9\\,M_{\\odot})=-\\alpha-\\beta\\times(M_{K,\\mathrm{bul}}+24.21),\n\\end{equation}\nwhere the $\\alpha$ is $0.265\\pm0.050$, the $\\beta$ is $0.488\\pm0.033$, and intrinsic scatters is 0.30. $M_{K,\\mathrm{bul}}$ is the $K$-band absolute magnitude of the bulge based on the photometric system of 2MASS. Using the best-fit SED derived earlier, we applied K-correction and evolution correction. We find $M_{K} = -19.84$ Vega mag including the evolutionary correction of 1.38 mag, which is derived by the difference in the $K$-band magnitude between 0.63 Gyr old population as in Table~\\ref{SED2tab} and 4.5 Gyr old population, which is the age of this host galaxy in the local universe. Then, the mass of the SMBH is estimated to be $10^{6.6\\pm0.3}\\,M_{\\odot}$. If we take B\/T=0.83 into account and assume that this value is also applicable to the NIR bands, then $M_{\\mathrm{BH}}$ decreases by $\\sim0.1$ dex. If we use an $M_{K}$ value that has not been corrected for evolution, then $M_{\\mathrm{BH}}$ becomes $\\sim0.7$ dex larger.\n\n The tidal disruption of normal stars by a black hole is not possible for $M_\\mathrm{BH}>10^8\\,M_{\\odot}$, since the tidal radius where the disruption can occur is located inside the Schwarzschild radius \\citep{Rees1988,Cannizzo1990,Bloom2011}. Our results on the mass of the SMBH satisfy the condition for a tidal disruption event. \n\nAlthough we favor the model in which the host galaxy of \\emph{Swift} J1644+57 is a classical bulge, our analysis shows that it could be a galaxy with a pseudobulge with B\/T=0.36 (Table~\\ref{galfittab}). If so, it is rather difficult to obtain an $M_{\\rm BH}$ value, since the scaling relation is not well established for pseudobulges, especially in the low mass range of $M_{\\star}\\sim10^9\\,M_{\\odot}$ for the host galaxy. Several works have shown that the $M_{\\rm{BH}}$ -- host galaxy scaling relations are weak or zero with a large scatter for pseudobulges. Over the $M_\\mathrm{\\star,bul}$ range of $10^{9.3}$ to $10^{10.5}\\,M_{\\odot}$ where such a relation has been studied, $M_{\\rm BH}$ can have any value between $10^{6}$ to $10^{8}\\,M_{\\odot}$ \\citep[Figure 21 of][]{Kormendy2013}. To reach down to $M_{\\star,\\mathrm{bul}} \\sim 5 \\times 10^{8}\\,M_{\\odot}$ as implied from the pseudobulge fit of our data, currently one can barely do so by relying on results from low mass AGNs \\citep{Barth2005,Greene2008,Jiang2011,Xiao2011}. In such a case, an $M_{\\rm BH}$ value between $10^{5}$ to $10^{6.3}\\,M_{\\odot}$ is possible \\citep[Figure 32 of][]{Kormendy2013}. Overall, if the host galaxy harbors a pseudobulge, then we can only loosely constrain $M_{\\rm BH}$ to have a value between $10^{5}$ to $10^{7}\\,M_{\\odot}$ considering our current poor knowledge of the $M_{\\rm BH}$ value in pseudobulges. \n\nIt is also known that a small fraction of pseudobulges have a S\\'{e}rsic index of $n > 3$. The best example is Pox 52, for which $n\\sim3.6$ -- $4.3$, $M_{\\rm BH} \\sim2\\times10^{5} \\,M_{\\odot}$, and $M_{\\star} \\sim10^9 \\,M_{\\odot}$ \\citep{Barth2004,Thornton2008}. Therefore, even if we accept the S\\'{e}rsic index of $n=3.43$ as the best-fit result, we need to keep this kind of caveat in mind.\n\n Figure~\\ref{BHfig} shows our overall results on $M_{\\mathrm{BH}}$ and the results from the previous studies we mentioned in \\S\\ref{sec:Intro}. It shows that our favorite results are compatible with the previous rough estimates from \\citet{Burrows2011} and \\citet{Levan2011}, who also used scaling relations. However, our results are improved compared to the previous results, by revealing that the host galaxy has a significant bulge component through a two-dimensional bulge + disk decomposition of the surface brightness profile, and removing the transient component in NIR light using a long-term light curve. The $M_{\\rm BH}$ limit could be much looser (the dashed line) if the host galaxy harbors a pseudobulge. A critical test of the pseudobulge model would be to obtain a deep, high-resolution image to see how the surface brightness profile behaves at the outer region of the host galaxy. \n\\\\\n\n\n\n\n\n\n\n \n\n\n\\section{Summary}\n We investigated the host galaxy properties of tidal disruption event, \\emph{Swift} J1644+57 through morphology analysis, light curve analysis, and SED fitting. We also estimated $M_{\\mathrm{BH}}$ which played the main role of this phenomenon, through scaling relations.\n\n We decomposed the surface brightness profile of the host galaxy based on high-resolution \\emph{HST} WFC3 images. We found that the host galaxy of \\emph{Swift} J1644+57 is a bulge-dominated galaxy which is well described by a single S\\'{e}rsic model with the S\\'{e}rsic index, $n=3.43\\pm0.05$. If we add a disk component, the bulge to total host galaxy flux ratio (B\/T) is $0.83\\pm0.03$, still indicating a bulge-dominant galaxy. We conclude that the host galaxy of \\emph{Swift} J1644+57 has a classical bulge from the best-fit galaxy models, although we cannot completely exclude the possibility of this galaxy containing a pseudobulge with B\/T=$0.36$.\n\n The NIR light curves enabled us to isolate the fluxes from the host galaxy after $\\sim 500$ days following the dissipation of the X-ray flux. On the other hand, we found that there are no significant changes in the light curves of the short wavelength bands, supporting the red feature of the transient possibly being caused by severe dust extinction.\n\n We fit SEDs to the multi-band fluxes of the host galaxy which are derived in the light curve analysis. The estimated stellar mass of the host galaxy is $\\log(M_{\\star}\/M_{\\odot}) = 9.14^{+0.13}_{-0.10}$. The $e$-folding time scale $\\tau$ is $0.10^{+0.24}_{-0.10}$ Gyr and the age of stellar population is $0.63^{+0.95}_{-0.43}$ Gyr. The SFR of galaxy is $0.03^{+0.28}_{-0.03} \\,M_{\\odot}$\/yr. In terms of the surface brightness profile and the stellar mass, this galaxy resembles M32, a small companion galaxy of M31. \n\n We estimated the central $M_{\\mathrm{BH}}$ through scaling relations. The mass of the SMBH is estimated to be $10^{6.7\\pm0.4} \\,M_{\\odot}$ from $M_{\\mathrm{BH}}$ -- $M_{\\star,\\mathrm{bul}}$ and $M_{\\mathrm{BH}}$ -- $L_{\\mathrm{bul}}$ relations for the $K$ band. However, the limit on $M_{\\mathrm{BH}}$ can be much looser if the host galaxy has a pseudobulge. Future high-resolution, deep imaging should be able to unambiguosly distinguish the two possibilities.\n\\\\\n\n\n\\acknowledgments \nThis work was supported by the National Research Foundation of Korea (NRF) grant, No. 2008-0060544, funded by the Korea government (MSIP). We thank the observers who obtained the CQUEAN and UKIRT data that were used in our analysis. This paper includes the data taken at the McDonald Observatory of the University of Texas at Austin. At the time of the UKIRT observation, UKIRT was operated by the Joint Astronomy Centre on behalf of the Science and Technology Facilities Council of the U.K. This work is based in part on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. We acknowledge the use of public data from the Swift data archive. CP, TS, and NG acknowledge support from the NASA research grant, NNX10AF39G. MI gratefully acknowledges the hospitality and the support from the Korea Institute of Advanced Study where part of this was carried out.\n\\\\\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{\\@startsection {section}{1}{\\z@}%\n\n\n\\newcommand{\\pic}[1]{\\;\\parbox[c]{45pt}{\\begin{picture}(45,30)(0,0)\n\\SetWidth{1.0}\\SetScale{1.0} #1 \\end{picture}}\\;}\n\\newcommand{\\pics}[1]{\\;\\parbox[c]{30pt}{\\begin{picture}(30,30)(0,0)\n\\SetWidth{1.0}\\SetScale{1.0} #1 \\end{picture}}\\;}\n\\newcommand{\\pib}[1]{\\;\\parbox[c]{36pt}{\\begin{picture}(22.5,30)(0,0)\n\\SetWidth{1.0}\\SetScale{1.0} #1 \\end{picture}}\\;}\n\\newcommand{\\picb}[1]{\\;\\parbox[c]{48pt}{\\begin{picture}(45,30)(-9,0)\n\\SetWidth{1.0}\\SetScale{1.0} #1 \\end{picture}}\\;}\n\\newcommand{\\picc}[1]{\\;\\parbox[c]{45pt}{\\begin{picture}(45,30)(0,0)\n\\SetWidth{1.0}\\SetScale{1.0} #1 \\end{picture}}\\;}\n\\newcommand{\\piccb}[1]{\\;\\parbox[c]{75pt}{\\begin{picture}(75,30)(0,0)\n\\SetWidth{1.0}\\SetScale{1.0} #1 \\end{picture}}\\;}\n\\newcommand{\\pivv}[1]{\\;\\parbox[c]{36pt}{\\begin{picture}(30,15)(0,0)\n\\SetWidth{1.0}\\SetScale{1.0} #1 \\end{picture}}\\;}\n\\newcommand{\\picl}[1]{\\;\\parbox[c]{60pt}{\\begin{picture}(60,30)(0,0)\n\\SetWidth{1.0}\\SetScale{1.0} #1 \\end{picture}}\\;}\n\\newcommand{\\picss}[1]{\\;\\parbox[c]{26pt}{\\begin{picture}(20,26)(0,0)\n\\SetWidth{1.0}\\SetScale{1.0} #1 \\end{picture}}\\;}\n\n\\def1{1}\n\\special{! \/Ldensity {0.25} def}\n\\def\\Agl(#1,#2)(#3,#4,#5){\\PhotonArc(#1,#2)(#3,#4,#5){1}\n{6.283 #3 mul 360 div #4 #5 sub #4 #5 sub mul sqrt mul Ldensity mul}}\n\\def\\Lgl(#1,#2)(#3,#4){\\Photon(#1,#2)(#3,#4){1}\n{#1 #3 sub #1 #3 sub mul #2 #4 sub #2 #4 sub mul add sqrt Ldensity mul}}\n\\def\\Agh(#1,#2)(#3,#4,#5){\\DashArrowArc(#1,#2)(#3,#4,#5){1}}\n\\def\\Aagh(#1,#2)(#3,#4,#5){\\DashArrowArcn(#1,#2)(#3,#5,#4){1}}\n\\def\\Lgh(#1,#2)(#3,#4){\\DashArrowLine(#1,#2)(#3,#4){1}}\n\\def\\Lagh(#1,#2)(#3,#4){\\DashArrowLine(#3,#4)(#1,#2){1}}\n\\def\\Ahh(#1,#2)(#3,#4,#5){\\DashCArc(#1,#2)(#3,#4,#5){1}}\n\\def\\Lhh(#1,#2)(#3,#4){\\DashLine(#1,#2)(#3,#4){1}}\n\\def\\Aqu(#1,#2)(#3,#4,#5){\\ArrowArc(#1,#2)(#3,#4,#5)}\n\\def\\Aaqu(#1,#2)(#3,#4,#5){\\ArrowArcn(#1,#2)(#3,#5,#4)}\n\\def\\Lqu(#1,#2)(#3,#4){\\ArrowLine(#1,#2)(#3,#4)}\n\\def\\Laqu(#1,#2)(#3,#4){\\ArrowLine(#3,#4)(#1,#2)}\n\\def\\Aqq(#1,#2)(#3,#4,#5){\\CArc(#1,#2)(#3,#4,#5)}\n\\def\\Lqq(#1,#2)(#3,#4){\\ArrowLine(#1,#2)(#3,#4)}\n\\def\\Asc(#1,#2)(#3,#4,#5){\\ArrowArc(#1,#2)(#3,#4,#5)}\n\\def\\Lsc(#1,#2)(#3,#4){\\ArrowLine(#1,#2)(#3,#4)}\n\\def\\DAsc(#1,#2)(#3,#4,#5){\\DashCArc(#1,#2)(#3,#4,#5){3}}\n\\def\\DLsc(#1,#2)(#3,#4){\\DashLine(#1,#2)(#3,#4){3}}\n\\def\\TAsc(#1,#2)(#3,#4,#5){\\SetWidth{2.0}\\CArc(#1,#2)(#3,#4,#5)\\SetWidth{1.0}}\n\\def\\TLsc(#1,#2)(#3,#4){\\SetWidth{2.0}\\ArrowLine(#1,#2)(#3,#4)\\SetWidth{1.0}}\n\n\n\n\\begin{document}\n\n\\preprint{ECT*-06-04, HIP-2006-18\/TH, TUW-06-02}\n\\pacs{11.10.Wx, 12.38.Mh}\n\n\\title{The pressure of deconfined QCD\nfor all temperatures and\\\\ quark chemical potentials}\n\n\\author{A. Ipp}\n\\affiliation{ECT*, Villa Tambosi, Strada delle Tabarelle 286,\\\\\nI-38050 Villazzano Trento, Italy}\n\\author{K. Kajantie}\n\\affiliation{Department of Physics, P.O. Box 64, FI-00014 University of Helsinki, Finland}\n\\author{A. Rebhan}\n\\affiliation{Institut f\\\"ur Theoretische Physik, Technische\nUniversit\\\"at Wien, \\\\Wiedner Hauptstr.~8-10,\nA-1040 Vienna, Austria }\n\\author{A. Vuorinen}\n\\affiliation{Department of Physics, University of Washington, Seattle, WA 98195, U.S.A.}\n\n\n\\begin{abstract}\n We present a new method for the evaluation of the perturbative\n expansion of the QCD pressure which is\n valid at all values of the temperature and quark chemical potentials\n in the deconfined phase and which we work out up to and including\n order $g^4$ accuracy. Our calculation is manifestly four-dimensional\n and purely diagrammatic --- and thus independent of any effective\n theory descriptions of high temperature or high density QCD.\n In various limits, we recover the\n known results of dimensional reduction and the HDL and HTL\n resummation schemes, as well as the equation of state of\n zero-temperature quark matter, thereby verifying their respective\n validity. To demonstrate the overlap of the various regimes, we\n furthermore show how the predictions of dimensional reduction and\n HDL resummed perturbation theory agree in the regime $T\\sim\n \\sqrt{g}\\mu$. At parametrically smaller temperatures $T\\sim g\\mu$,\n we find that the dimensional reduction result agrees well with those\n of the nonstatic resummations down to the remarkably low value\n $T\\approx 0.2 m_\\rmi{D}$, where $m_\\rmi{D}$ is the Debye mass at $T=0$. Beyond\n this, we see that only the latter methods connect smoothly to the\n $T=0$ result of Freedman and McLerran, to which the leading\n small-$T$ corrections are given by the so-called non-Fermi-liquid terms,\n first obtained through HDL resummations. Finally, we outline the\n extension of our method to the next order, where it would include\n terms for the low-temperature entropy and specific heats\n that are unknown at present.\n\\end{abstract}\n\\maketitle\n\n\n\n\\tableofcontents\n\n\\section{Introduction}\n\nThe most fundamental thermodynamic quantity in the theory of strong\ninteractions, the QCD pressure $p_\\rmi{QCD}(T,\\mu)$, can at large\nvalues of the temperature $T$ or the quark chemical potentials $\\mu$\nbe computed in a weak coupling expansion in the gauge coupling\nconstant $g$, defined in the ${\\overline{\\mbox{\\tiny\\rm{MS}}}}$ renormalization scheme. In\nthe region where $T$ is larger than all other relevant mass scales in\nthe problem, the expansion has been extended to include terms of order\n$g^6\\log g$ \\cite{es}-\\cite{avpres}, while at $T=0$ and $\\mu$ much\ngreater than the critical chemical potential $\\mu_c$, the pressure is\nknown up to and including terms of order $g^4$ \\cite{fmcl}. In between\nthese regimes, at $01$ these become\nmore important than the two-loop terms, as $T m_\\rmi{D}^3 \\sim g^{3+x}\\mu \\gg g^{2+2x}\\mu\n\\sim g^2\\mu^2 T^2$. As we shall see, the $T$-dependent\ncontributions from ring diagrams indeed become more important\nthan the 2-loop term $g^2\\mu^2 T^2$ here, though they are not enhanced\nby a relative factor $g^{-(x-1)}$ as suggested by dimensional\nreduction, but instead only by a logarithm.\n\nFor temperatures $T \\lesssim g\\mu$, where dimensional reduction\nis no longer applicable, it becomes important to keep the nonstatic\nparts of the gluon self energy in the ring diagrams.\nAt low momenta and frequencies of the order $g\\mu$, the leading\nterms in the gluon self-energy are given by the\nso-called hard thermal loops (HTL) approximation with the overall\n$m_\\rmi{D}^2$ factor replaced by its zero-temperature value ---\na special case occasionally referred to as hard dense loops (HDL).\nIn the longitudinal gluon propagator, one can observe\nthe usual Debye screening effect at the frequency $\\omega\\ll g\\mu$,\nbut in the transverse propagator the situation\nis more complicated. At strictly zero frequency the\nmagnetostatic HDL propagator is massless, but for\nsmall but nonvanishing frequencies\n$\\omega\\ll q\\lesssim m_\\rmi{D}$ its inverse has the form\n\\begin{eqnarray}\nq^2-\\omega^2+\\Pi^\\rmi{HDL}_\\rmi{T}(\\omega,q)\n&=&q^2-{i\\pi m_\\rmi{D}^2\\04}{\\omega\\0q}+O(\\omega^2).\n\\end{eqnarray}\nThe transverse part of the propagator thus has a pole\nat imaginary $q$ and $|q|=m_\\rmi{mag}(\\omega)$,\nintroducing a new parametrically small dynamical\nscreening mass \\cite{Weldon:1982aq,Kraemmer:2003gd}\n\\begin{eqnarray}\n\\label{mmdyn}\nm_\\rmi{mag}(\\omega)&=&\\left( {\\pi m_\\rmi{D}^2 \\omega \\over 4}\n\\right)^{1\/3},\\qquad \\omega\\ll m_\\rmi{D},\n\\end{eqnarray}\nwhich represents an in-medium version of Lenz's law.\nAs soon as the temperature is small but nonvanishing, the ring\ndiagrams obtain contributions involving the Bose-Einstein\ndistribution function which leads to sensitivity\nto this additional scale.\nIn these contributions, we effectively have\n$m_\\rmi{mag}(\\omega\\sim T) \\sim g^{(2+x)\/3}\\mu$\nfor $T\\sim g^x\\mu$ and $x>1$.\nNote that\nthis is parametrically smaller than $m_\\rmi{D}\\sim g\\mu$, but always larger than\nthe magnetic mass scale of MQCD, $m_\\rmi{mag}(\\omega\\!=\\!0)=\ng^2 T \\sim g^{2+x}\\mu$.\n\nThe resummation of the nonstatic transverse gluon self-energy\ngives rise to terms nonanalytic in the temperature which to lowest\norder in a low-temperature expansion turn out to be of the order\n$g^2 \\mu^2 T^2 \\ln\\,T$. This gives rise to\nso-called anomalous or non-Fermi-liquid behavior in the\nentropy and specific heat at low $T$, because instead of\nthe usual linear behavior in $T$ the entropy then has a $T\\ln\\,T$\nterm which is the hallmark of a breakdown of the Fermi-liquid\npicture (first discussed in the context\nof nonrelativistic QED by Norton, Holstein and Pincus \\cite{Holstein:1973}).\nIndeed, inspection of the dispersion laws of\nfermionic quasiparticles reveals that there is a logarithmic\nsingularity in the group velocity at the therefore no longer\nsharply defined Fermi surface\\footnote{A systematic\ncalculation of the group velocity beyond the leading-log approximation has only\nrecently been carried out in Ref.~\\cite{Gerhold:2005uu}.}.\n\nFor a long time, only the multiplicative coefficient of the $T\\ln\\,T$ term\nin the specific heat was known. It was only rather recently \\cite{Ipp:2003cj}\nthat also the scale under the logarithm was determined\ntogether with the next order terms in the low-temperature ($T\\ll g\\mu$)\nseries which in addition involves fractional\npowers of $T$ due to the cubic root in Eq.~(\\ref{mmdyn}).\nFor the pressure, these ``anomalous'' $T$-dependent contributions\nare contained in an expression,\nwhich was first derived in Ref.~\\cite{Gerhold:2004tb}\nand which we shall label by HDL$^+$,\n\\begin{eqnarray}\\label{PHDL}\n{1\\0N_g}\\delta p^\\rmi{HDL$^+$}\n&=&-{g^2T_F\\048\\pi^2}\\mu^2T^2\n-{1\\02\\pi^3}\\int_0^\\infty dq_0\\, n_b(q_0)\n\\int_0^\\infty dq\\,q^2\\,\\biggl[ 2\\,{\\rm Im}\\, \\ln \\left( q^{2}-q_{0}^{2}+\\Pi^\\rmi{HDL} _\\rmi{T} \\over q^{2}-q_{0}^{2} \\right )\\nonumber\\\\\n&+&{\\rm Im}\\, \\ln \\left( \\frac{q^{2}-q_{0}^{2}+\\Pi^\\rmi{HDL} _\\rmi{L}}{q^{2}-q_{0}^{2}}\\right)\n\\biggr] + O(g^2T^4) + O(g^3 \\mu T^3)+ O(g^4\\mu^2T^2),\n\\end{eqnarray}\nwith $\\delta p$ denoting the temperature-dependent part of the interaction\npressure\n\\begin{eqnarray}\\label{deltaDeltap}\n\\delta p & \\equiv & \\Delta p - \\Delta p|_{T=0}, \\nonumber \\\\\n\\Delta p & \\equiv & p - p_\\rmi{SB}.\n\\end{eqnarray}\nThe expression (\\ref{PHDL}) can be viewed as a minimal\\footnote{As\nopposed to the HTL\/HDL resummation considered in \\cite{ABS,BIR}\nwhich aims at improving the convergence of the perturbative\nseries at high temperature\nby retaining higher-order effects from HTL\/HDL physics\nbeyond what is needed from a perturbative point of\nview. The + in HDL$^+$ and HTL$^+$\nis meant as a reminder that the\ncorresponding quantities are not expressed in terms\nof HTL\/HDL quantities only, but combined with\nunresummed infrared-safe contributions.}\nresummation of HDL diagrams, where the HDL self-energies are only kept in\nthe infrared sensitive part of the ring diagrams involving the distribution function $n_b$, while\ninfrared safe two-loop contributions are treated in an unresummed form.\n\n\\subsubsection{$T$ parametrically smaller than $m_\\rmi{D}$}\n\nWith $g\\ll 1$ and $x>1$ in $T\\sim g^x\\mu$, the temperature is parametrically\nsmaller than the Debye mass $m_\\rmi{D}\\sim g\\mu$ and\nEq.~(\\ref{PHDL}) contains the leading contributions\nto the temperature-dependent parts of the interaction pressure,\nwhich ignoring logarithms are of order $g^2\\mu^2 T^2\\sim g^{2+2x}\\mu^4$,\nwhile the higher-order terms in Eq.~(\\ref{PHDL}) are at least of\norder $g^{4+2x}\\mu^4$.\nThe Freedman-McLerran result for the $T=0$ pressure, Eq.~(\\ref{pFMcL}),\nis accurate to order $g^4\\mu^2$ and its error is of order $g^6\\mu^4$\n(again ignoring logarithms of $g$).\nEq.~(\\ref{PHDL}) thus represents the leading correction to the\nFreedman-McLerran result\nas long as $x<2$ (i.e., $T\\gtrsim g^2\\mu$), whereas\nin quantities such as the entropy density $s=\\6p\/\\6T$ and the various\nspecific heats, where the $T=0$ part of the pressure drops out,\nit is in fact the leading term in the interaction part for all $x\\ge1$.\n\n\\def\\bar g } %{g_{\\rm eff}{\\bar g }\n\\def\\bar\\tau } %{\\tau_{\\rm eff}{\\bar\\tau }\nIn Eq.~(\\ref{PHDL}), $g$ appears only in the\ncombination $\\bar g } %{g_{\\rm eff}^2\\equiv g^2 T_F$, and it is therefore convenient\nto define a reduced temperature variable\n\\begin{equation}\\label{bartau}\n\\bar\\tau } %{\\tau_{\\rm eff}=\\pi T\/(\\bar g } %{g_{\\rm eff}^x \\mu).\n\\end{equation}\nFor $x>1$, the perturbative content of Eq.~(\\ref{PHDL}) is that\ngiven by the low-temperature expansion worked out in\nRefs.~\\cite{Ipp:2003cj,Gerhold:2004tb}. With the above variables, this reads\n\\begin{eqnarray}\\label{PHDLx}\n{1\\0N_g}{\\delta p^\\rmi{HDL$^+$}\\0m_\\rmi{D}^4}\n&=& {\\bar\\tau } %{\\tau_{\\rm eff}^2\\bar g } %{g_{\\rm eff}^{2(x-1)}\\over72}\\left(\\ln\\left({1\\over \\bar\\tau } %{\\tau_{\\rm eff} \\bar g } %{g_{\\rm eff}^{x-1}}\\right)\n+\\ln{4\\over \\pi}\n +\\gamma_E-{6\\over\\pi^2}\\zeta^\\prime(2)-{3\\02}\\right)\\nonumber\\\\\n &-&{2^{2\/3}\\Gamma\\left({8\\over3}\\right)\\zeta\\left({8\\over3}\\right)\\over3\\sqrt{3}\\pi^{7\/3}}\n \\bar\\tau } %{\\tau_{\\rm eff}^{8\/3}\\bar g } %{g_{\\rm eff}^{8(x-1)\/3}\n +8{2^{1\/3}\\Gamma\\left({10\\over3}\\right)\\zeta\\left({10\\over3}\\right)\n \\over9\\sqrt{3}\\pi^{11\/3}}\\bar\\tau } %{\\tau_{\\rm eff}^{10\/3}\\bar g } %{g_{\\rm eff}^{10(x-1)\/3}\\nonumber\\\\\n &+&{2048-256\\pi^2-36\\pi^4+3\\pi^6\\over2160\\pi^2}\\bar\\tau } %{\\tau_{\\rm eff}^4 \\bar g } %{g_{\\rm eff}^{4(x-1)}\n \\left[\\ln\\left({1\\over \\bar\\tau } %{\\tau_{\\rm eff}\\bar g } %{g_{\\rm eff}^{x-1}}\\right)+\\ln\\pi+\\bar c\\, \\right]\n\\nonumber\\\\\n&+&{O}(\\bar g } %{g_{\\rm eff}^{14(x-1)\/3})\n+{O}(g^{2x}),\n\\end{eqnarray}\nwhere $\\bar c\\approx 4.099348\n\\ldots$ is given by a numerical integral\ndefined in Ref.~\\cite{Gerhold:2004tb}.\nThe latter of the error terms in Eq.~(\\ref{PHDLx}) corresponds to the\nleading-order terms to be expected from three- and higher-loop contributions\\footnote{There,\n$g^2$ no longer appears exclusively in combination with $T_F$.}\nproportional to $g^4\\mu^2 T^2$ which are presumably\nenhanced by logarithms of $T$ and $g$. Depending on the value of $x>1$, a finite number of terms in the\nlow-$T$ expansion remain more important than this\n(see Fig.~\\ref{fig:orders} in Sec.~\\ref{sec:summary}).\n\nWhen $x=1$, i.e.\\ $T\\sim g\\mu$, the expansion of Eq.~(\\ref{PHDLx}) clearly breaks\ndown (unless $\\bar\\tau } %{\\tau_{\\rm eff}\\ll 1$) and the HDL-resummed expression of Eq.~(\\ref{PHDL})\ntherefore needs to be evaluated numerically as in Ref.~\\cite{Gerhold:2004tb}.\nThis expression has then the form of $g^4\\mu^4$ times\na function of $T\/(g\\mu)$, and is therefore of the same order as\nthe $g^4$ term of the $T=0$ pressure of Freedman and McLerran, to\nwhich it is to be added. As displayed in Ref.~\\cite{Gerhold:2004tb}\nfor the case of the entropy,\nand as we shall see for the pressure in the plots of Section \\ref{sec:numres}\nof the present paper,\nthe $T$-dependent terms of Eq.~(\\ref{PHDLx}) smoothly\ninterpolate between a dominant $g^2 T^2\\mu^2\\ln\\,T$ behavior at low temperature\nand the terms of order $g^2T^2 \\mu^2$, $g^3 \\mu^3 T$, and $g^4\\mu^4\\ln\\,T$\nof the dimensional reduction pressure which\nshould be the dominant terms at sufficiently high temperatures and which\nremain comparable to $g^4\\mu^4$ as long as the parametric equality $T\\sim g\\mu$ holds.\n\n\n\\subsubsection{$T$ parametrically larger than $m_\\rmi{D}$}\n\n\nWhen $x<1$ in $T\\sim g^x\\mu$, \\textit{i.e.}~$T\\gg g\\mu$,\ndimensional reduction provides the most accurate\nresult available for the QCD pressure. Up to an error of the order of\nthree-loop contributions proportional to $g^4\\mu^2 T^2\\sim g^{4+2x}\\mu^4$,\none can however reproduce its prediction\nby extending the above HDL-resummed calculation\nto include the leading thermal corrections to the gluon self-energy. In practice,\nthis means replacing the HDL approximation by the HTL one and also keeping the order $g^2 T^4$\nterms originating from infrared-safe two-loop contributions to the pressure that were\nomitted in Eq.~(\\ref{PHDL}) because they were of too high order when $x\\ge1$.\nThis possibility was mentioned in Ref.~\\cite{Gerhold:2004tb}, but\nnot considered further because that work concentrated\non the region of $T\\lesssim g\\mu$. For the purposes of the present paper, we however\nwrite down the straightforward extension of Eq.~(\\ref{PHDL}) to the HTL approximation\nin the form\n\\begin{eqnarray}\\label{PHTL}\n{1\\0N_g}\\delta p^\\rmi{HDL$^+$}&=&-{g^2T_F\\048\\pi^2}\\mu^2T^2+{g^2(2C_A-T_F)\\0288}T^4\\nonumber \\\\\n&&-{1\\02\\pi^3}\\int_0^\\infty dq_0\\, n_b(q_0)\n\\int_0^\\infty dq\\,q^2\\,\\biggl[ 2\\,{\\rm Im}\\, \\ln \\left( q^{2}-q_{0}^{2}+\\Pi^\\rmi{HTL} _\\rmi{T} \\over q^{2}-q_{0}^{2} \\right )\\nonumber\\\\&&\\qquad\n+{\\rm Im}\\, \\ln \\left( \\frac{q^{2}-q_{0}^{2}+\\Pi^\\rmi{HTL} _\\rmi{L}}{q^{2}-q_{0}^{2}}\\right)\n\\biggr] + O(g^4\\mu^2T^2).\n\\end{eqnarray}\n\nCombining the above expression with the Freedman-McLerran result of Eq.~(\\ref{pFMcL}) to obtain\n\\begin{eqnarray}\\label{PHTLtot}\n\\Delta p^\\rmi{HDL$^+$} &\\equiv & p^\\rmi{HDL$^+$} - p_\\rmi{SB} \\;\\;\\equiv\\;\\;\n\\Delta p^\\rmi{FMcL}+\\delta p^\\rmi{HDL$^+$},\n\\end{eqnarray}\nwe have an expression for the interaction pressure\nwhose error is of order $g^{{\\rm min}(4+2x,6)}$\nfor all $T\\sim g^x \\mu$. This we shall compare (and thus test) in the following\nwith our new approach which resums the complete one-loop gluon self-energy\n(i.e., not only the leading HTL\/HDL contribution) in ring diagrams.\nNote that the accuracy of (\\ref{PHTLtot}) is at least of order $g^4$\nfor all parametrically small temperatures, excluding only the case of $x=0$,\nwhere $T\\sim \\mu$.\n\n\n\\section{The new approach}\\label{sec:newappr}\n\nIn this Section, we introduce our novel and strictly four-dimensional\ncalculational scheme designed to reproduce the perturbative expansion\nof the QCD pressure up to and including order $g^4$ at all values of\n$\\mu$ and $T$.\nOur guiding principle is that when faced with the necessity to\nsum up graphs with multiple self energy insertions to circumvent\ninfrared problems, we consider the entire self energy\nand not only those parts which are identified as relevant\nin some effective field theory description,\nsuch as the Debye mass in dimensional reduction or\nthe HTL\/HDL self energy in the corresponding resummation schemes.\nBecause we (at present) limit ourselves to order $g^4$ accuracy,\nit will be sufficient to resum only one-loop self-energies in the\ninfrared sensitive graphs, while IR safe diagrams will be treated\nperturbatively, using bare propagators.\nThis will introduce gauge dependence to our results, but only at orders\nbeyond $g^4$ which we will explicitly discard by either considering values\nof $g$ low enough for the higher order terms to be negligible or by performing\nnumerical series expansions up to ${\\mathcal O}(g^4)$.\n\n\nWe begin our treatment with a general diagrammatic analysis where we identify\nthe relevant classes of Feynman graphs that need to be considered. After that,\nwe describe their evaluation and show how adding them together leads to the\nfinal result displayed in Section \\ref{subsec:result}. Many details of the\ncalculations as well as the results of several individual pieces of the result\nare left to be covered in the Appendices.\n\n\\subsection{Identification of the relevant diagrams}\n\n\n\n\\def\\Elmeri(#1,#2,#3){{\\pic{#1(15,15)(15,0,180)%\n #2(15,15)(15,180,360)%\n #3(0,15)(30,15)}}}\n\n\\def\\Petteri(#1,#2,#3,#4,#5,#6){\\pic{#3(15,15)(15,-30,90)%\n #1(15,15)(15,90,210)%\n #2(15,15)(15,210,330) #5(2,7.5)(15,15) #6(15,15)(15,30) #4(15,15)(28,7.5)}}\n\n\\def\\Jalmari(#1,#2,#3,#4,#5,#6){\\picc{#1(15,15)(15,90,270)%\n #2(30,15)(15,-90,90) #4(30,30)(15,30) #3(15,0)(30,0) #5(15,0)(15,30)%\n #6(30,30)(30,0) }}\n\n\\def\\Oskari(#1,#2,#3,#4,#5,#6,#7,#8){\\picc{#1(15,15)(15,90,270)%\n #2(30,15)(15,-90,90) #4(30,30)(15,30) #3(15,0)(30,0) #6(15,0)(15,15)%\n #5(15,15)(15,30) #8(30,30)(30,15) #7(30,15)(30,0) }}\n\n\\def\\Sakari(#1,#2,#3){\\picb{#1(15,15)(15,30,150)%\n#1(15,15)(15,210,330) #2(0,15)(7.5,-90,90) #2(0,15)(7.5,90,270) %\n#3(30,15)(7.5,-90,90) #3(30,15)(7.5,90,270) }}\n\n\\def\\Maisteri(#1,#2){\\picb{#1(15,15)(15,0,150)%\n#1(15,15)(15,210,360) #2(0,15)(7.5,-90,90) #2(0,15)(7.5,90,270) #1(37.5,15)(7.5,0,360) }}\n\n\\def\\Tohtori(#1,#2){\\picb{#1(15,15)(15,0,360)#1(45,15)(15,0,360) }}\n\n\\def\\Pietari(#1){\\pic{#1(15,15)(15,0,360) #1(15,1)(20,40,140)#1(15,29)(20,220,320)}}\n\n\\def\\Ari(#1,#2){\\pic{#1(15,15)(15,0,360) #2(0,15)(27,22) #2(0,15)(27,8)}}\n\n\\def\\Kari(#1){\\pic{#1(15,15)(15,-90,270)}}\n\n\\begin{figure}[t]\n\\centering\n\\begin{eqnarray} \\nonumber\n\\begin{array}{lllll}\na)~ \\Kari(\\Agl) & b)~\n\\Kari(\\Agh)\n\\;\\;\\;\\; c)~\n\\Kari(\\Asc)\n\\nonumber \\\\\n\\nonumber \\\\\nd)~ \\Elmeri(\\Agl,\\Agl,\\Lgl) & e)~\n\\Elmeri(\\Agh,\\Agh,\\Lgl)\n\\;\\;\\;\\; f)~\n\\Elmeri(\\Asc,\\Asc,\\Lgl)\n\\;\\;\\;\\; g)~\n\\!\\!\\!\\!\\Tohtori(\\Agl,\\Agl)\n\\nonumber \\\\\n\\nonumber \\\\\nh)~\\Petteri(\\Asc,\\Agl,\\Asc,\\Lsc,\\Lsc,\\Lgl)\n&i)~\\Petteri(\\Asc,\\Asc,\\Asc,\\Lgl,\\Lgl,\\Lgl)\n\\;\\;\\;\\;\nj)~\n\\Petteri(\\Agh,\\Agh,\\Agh,\\Lgl,\\Lgl,\\Lgl)\n\\;\\;\\;\\; k)~\n\\Petteri(\\Agh,\\Agl,\\Agh,\\Lgh,\\Lgh,\\Lgl)\n\\;\\;\\;\\; l)~\n\\Jalmari(\\Asc,\\Asc,\\Lsc,\\Lsc,\\Lgl,\\Lgl)\n\\nonumber \\\\\n\\nonumber \\\\\n\\!m)~\n\\!\\Ari(\\Agl,\\Lgl)\n&\\!n)~\n\\Petteri(\\Agl,\\Agl,\\Agl,\\Lgl,\\Lgl,\\Lgl)\n\\;\\;\\;\\; o)~\n\\Pietari(\\Agl)\n\\;\\;\\;\\; p)~\n\\Jalmari(\\Agh,\\Agh,\\Lgh,\\Lgh,\\Lgl,\\Lgl)\n\\nonumber \\\\\n\\end{array}\n\\end{eqnarray}\n\\caption[a]{The one-, two- and three-loop two-gluon-irreducible (2GI) graphs of QCD. The wavy line stands for a gluon, the\ndotted line a ghost and the solid line a quark.}\n\\end{figure}\n\nTo determine the QCD pressure up to and including order $g^4$ on the entire\ndeconfined phase diagram of the theory, our first task is to identify all\nFeynman diagrams that contribute to the partition function at this order.\nThese trivially include the two-{\\em gluon}-irreducible (2GI) diagrams\nup to three-loop order, displayed in Fig.~1 which a straightforward power\ncounting as well as the explicit calculation of\nRef.~\\cite{avpres} confirms as infrared finite for all temperatures and\nchemical potentials.\n\nIn addition to these cases, there are, however, several other classes of IR\nsensitive diagrams that need to be resummed to infinite loop order, as a\npower counting reveals that the dressing of (at least some of) their gluon lines with an arbitrary\nnumber of one loop gluon polarization tensors does not increase their order beyond $g^4$. These\ndiagrams are shown in Fig.~2, where the first set corresponds to the well-known class\nof ring diagrams that leads to the known $g^3$ and $g^4\\ln\\,g$ contributions to the\npressure at high $T$ \\cite{jk,tt} and to the $g^4\\ln\\,g$ term at $T=0$ \\cite{fmcl}.\nAmong others, this class contains the set of all three-loop two particle reducible (2PR)\ngraphs of the theory which are missing from Fig.~1.\n\nAs we shall see (in contradiction to the opposite assertion in Ref.~\\cite{jk}),\nthe resummation of the ring diagrams is, however, not enough to obtain the entire\norder $g^4$ term correctly at nonzero $T$.\nAlthough without resummation starting at orders $g^6$, $g^8$ and $g^6$, respectively,\nthe classes of Fig.~2 b-d, corresponding to\nself-energy insertions in the gluonic two-loop 2GI diagrams 1d and 1g,\nhave the potential to give rise to contributions of order\n${\\mathcal O}(g^4T^2\\mu^2)$ and ${\\mathcal O}(g^4T^4)$ to the pressure.\nWhen $T$ is not parametrically larger than $m_\\rmi{D}$, it turns out that\nonly the class b gives a non-zero contribution at this order, being proportional to\n$g^2T^2m_\\rmi{D}^2$. When $T\\sim g^x\\mu$ with $x>0$, none of the three classes\ncontributes to the pressure to order $g^4\\mu^4$, but in the calculation of the\nlow-temperature entropy and specific heat they have to be taken into account already at\norder $g^4 \\mu^2 T$.\n\nFor any other classes of diagrams apart from those shown in Figs.~1 and 2, it is very straightforward to\nsee that the contributions will be beyond order $g^4$. In particular, if we were to add an additional\ngluon line with some number of self energy insertions into the graphs of Fig.~2 b-d (\\textit{i.e.~}dressing\nthe three-loop 2GI diagrams with self energies), we would notice that\nthe two extra insertions of the coupling constant due to the new vertices (vertex) ensure that these graphs only contribute\nto the pressure at order $g^6$. Similarly, one can see that the inclusion of the two-loop self energy into the ring\ndiagrams only has an effect on the pressure starting at ${\\mathcal O}(g^5)$.\n\n\n\\subsection{The 2GI diagrams}\n\nIn Feynman gauge,\nthe sum of the 2GI diagrams in Fig.~1 at arbitrary $T$ and $\\mu$ can be directly\nextracted from from Ref.~\\cite{avpres} with the result\n\\begin{eqnarray}\np_\\rmi{2GI}\n&=&\\pi^2 d_A T^4 \\Bigg(\\fr{1}{45}\\bigg\\{1+\\fr{d_F}{d_A}\\left(\\fr{7}{4}+30\\bar{\\mu}^2+60\\bar{\\mu}^4\\right)\\bigg\\}\\nonumber\n\\end{eqnarray}\n\\begin{eqnarray}\n&-&\\fr{g^2}{9(4\\pi)^2}\\bigg\\{C_A + \\fr{T_F}{2}(1+12\\bar{\\mu}^2)(5+12\\bar{\\mu}^2)\\bigg\\}\\nonumber \\\\\n&-&\\fr{g^4}{54(4\\pi)^4}\\Bigg\\{\\fr{23C_A^2-C_AT_F(29+360\\bar{\\mu}^2+720\\bar{\\mu}^4) +\n4T_F^2(5+72\\bar{\\mu}^2+144\\bar{\\mu}^4)}{\\epsilon}\\nonumber \\\\\n&+&C_A^2\\(182\\,\\ln\\fr{\\bar{\\Lambda}}{4\\pi T}+247+272\\frac{\\zeta'(-1)}{\\zeta(-1)}-90\\frac{\\zeta'(-3)}{\\zeta(-3)}\\)\\nonumber \\\\\n&+&C_AT_F\\bigg(-16\\(5+36\\bar{\\mu}^2+72\\bar{\\mu}^4\\)\\ln\\fr{\\bar{\\Lambda}}{4\\pi T}-\n\\fr{217}{5}-56\\frac{\\zeta'(-1)}{\\zeta(-1)}+\\fr{72}{5}\\frac{\\zeta'(-3)}{\\zeta(-3)} \\nonumber \\\\\n&+&24\\(9+4\\frac{\\zeta'(-1)}{\\zeta(-1)}\\)\\bar{\\mu}^2+432\\bar{\\mu}^4+144(1+4\\bar{\\mu}^2)\\aleph(1,z)+3456\\aleph(3,z)\\bigg)\\nonumber \\\\\n&+&4T_F^2\\bigg((1+12\\bar{\\mu}^2)\\(4(5+12\\bar{\\mu}^2)\\ln\\fr{\\bar{\\Lambda}}{4\\pi T}+15+8\\frac{\\zeta'(-1)}{\\zeta(-1)}+36\\bar{\\mu}^2\\)\n\\label{2pi}\\nonumber \\\\\n&+&144(1+4\\bar{\\mu}^2)\\aleph(1,z)\\bigg)\n-9C_FT_F\\bigg(\\fr{35}{2}-16\\frac{\\zeta'(-1)}{\\zeta(-1)}+4\\(59+16\\frac{\\zeta'(-1)}{\\zeta(-1)}\\)\\bar{\\mu}^2\\nonumber \\\\\n&+&664\\bar{\\mu}^4+96\\(i\\bar{\\mu}(1+4\\bar{\\mu}^2)\\aleph(0,z)+2(1+8\\bar{\\mu}^2)\\aleph(1,z)-12i\\bar{\\mu}\\aleph(2,z)\\)\n\\bigg)\\Bigg\\}\\Bigg),\n\\end{eqnarray}\nwhere $\\bar{\\mu}\\equiv\n\\mu\/(2\\pi T)$ and where we have\nrenormalized the gauge coupling using the usual zero-temperature renormalization constant\n$Z_g$. The sum, however, still contains uncanceled UV $1\/\\epsilon$ divergences and depends on the choice of gauge, so that it\nhas no separate physical significance.\n\n\\begin{figure}[t]\n\\centering\n\\begin{eqnarray} \\nonumber\n\\begin{array}{lll}\n\\nonumber \\\\\\nn\na)~\\sum_{n=2}^{\\infty}\\!\\!\\picb{\\Agl(15,15)(15,0,360)\\GCirc(0,15){2.8}{0.0} \\GCirc(7,27){2.8}{0.0} \\GCirc(7,3){2.8}{0.0}\\GCirc(23,27){2.8}{0.0} \\GCirc(23,3){2.8}{0.0}\n\\GCirc(30,15){2.8}{0.0}}\n\\;\\;\\;\\;b)~\\sum_{n_1,n_2=1}^{\\infty}\\!\\!\\picb{\\Agl(15,15)(15,0,360)\\GCirc(4,25){2.8}{0.0} \\GCirc(4,5){2.8}{0.0}\\GCirc(15,30){2.8}{0.0} \\GCirc(15,0){2.8}{0.0}\n\\GCirc(26,25){2.8}{0.0} \\GCirc(26,5){2.8}{0.0}\\Lgl(0,15)(30,15)}\n\\;\\;\\;\\;c)~\\sum_{n_1,n_2,n_3=1}^{\\infty}\\!\\!\\picb{\\Agl(15,15)(15,0,360)\\GCirc(4,25){2.8}{0.0} \\GCirc(4,5){2.8}{0.0}\\GCirc(15,30){2.8}{0.0} \\GCirc(15,0){2.8}{0.0}\n\\GCirc(26,25){2.8}{0.0} \\GCirc(26,5){2.8}{0.0}\\GCirc(9,15){2.8}{0.0}\\GCirc(21,15){2.8}{0.0}\\Lgl(0,15)(30,15)}\\nonumber \\\\\\nn\nd)~\\sum_{n_1,n_2=1}^{\\infty} \\;\\, \\piccb{\\Agl(15,15)(15,0,360)\\GCirc(0,15){2.8}{0.0} \\GCirc(7,27){2.8}{0.0} \\GCirc(7,3){2.8}{0.0}\\GCirc(23,27){2.8}{0.0}\n\\GCirc(23,3){2.8}{0.0}\\GCirc(60,15){2.8}{0.0} \\GCirc(53,27){2.8}{0.0} \\GCirc(53,3){2.8}{0.0}\\GCirc(37,27){2.8}{0.0} \\GCirc(37,3){2.8}{0.0}\\Agl(45,15)(15,0,360)}\n\n\\end{array}\n\\end{eqnarray}\n\\caption[a]{Classes of IR sensitive vacuum graphs contributing to the QCD pressure at order $g^4$.\nThe black dots represent the one-loop gluon polarization tensor given in Fig.~3a\nand the indices $n_i$ stand for the numbers of loop insertions on the respective lines.}\n\\end{figure}\n\n\\subsection{The ring sum}\n\nTo order $g^4$,\nthe ring sum of Fig.~2a can be separated into three pieces\n$p_\\rmi{VV}$, $p_\\rmi{VM}$ and $p_\\rmi{ring}$\naccording to Fig.~3 by decomposing the one-loop gluon polarization tensor\n(see Appendix \\ref{app:Pi}) into its vacuum ($T=\\mu =0$)\nand matter parts. Note that only the matter part has to be resummed, as the vacuum\nparts contribute to order $g^4$ only through the two three-loop diagrams\nin Figs.~3b and c.%\n\\footnote{Take any graph $G$ belonging to the ring sum and having four or more loops\nand at least one vacuum tensor insertion,\nand consider it in the Feynman gauge. Applying Eq.~(\\ref{polarvac}) to it and contracting\nthe Lorentz indices of the vacuum tensor with one of its neighboring gluon\npropagators, we see that $G$ is proportional to $g^2$ times a similar graph with the\nvacuum insertion removed. But this graph is nothing but one of those diagrams\nthat appeared in the original sum which implies that $G$ has to be proportional to\nat least the fifth power of the coupling. \\label{footnoteG}}\nThe evaluation of $p_\\rmi{VV}$ and $p_\\rmi{VM}$ is relatively\nstraightforward, and\nfully analytic expressions for them\nare given in Appendix \\ref{sec:anlI}.\n\n\nTo evaluate the remaining matter ring sum $p_\\rmi{ring}$ we define the standard\nlongitudinal and transverse parts of the vacuum-subtracted polarization tensor at $d=4-2\\epsilon$ by\n\\begin{eqnarray}\n\\Pi_\\rmi{L}(P)\\delta^{ab}&=&\\fr{P^2}{p^2}\\(\\Pi_{00}^{ab}(P)-\\Pi_{00}^{ab}(P)\\mid_{\\rmi{vac}}\\), \\label{pil1}\\\\\n\\Pi_\\rmi{T}(P)\\delta^{ab}&=&\\fr{1}{d-2}\\(\\Pi_{\\mu\\mu}^{ab}(P)-\\Pi_{\\mu\\mu}^{ab}(P)\\mid_{\\rmi{vac}} -\n\\fr{P^2}{p^2}\\(\\Pi_{00}^{ab}(P)-\\Pi_{00}^{ab}(P)\\mid_{\\rmi{vac}}\\)\\),\\quad\\label{pit1}\n\\end{eqnarray}\nwhere we have used the fact \\cite{Heinz:1986kz} that the one-loop\ngluon polarization tensor is transverse\nwith respect to the four-momentum $P$ in the Feynman gauge.\nIn terms of $\\Pi_\\rmi{T}$ and $\\Pi_\\rmi{L}$,\nthe sum of the ring diagrams is then readily performed with the result\n\\begin{eqnarray}\np_\\rmi{ring}&=&-\\fr{d_A}{2}\\hbox{$\\sum$}\\!\\!\\!\\!\\!\\!\\!\\int_P\\bigg\\{\\ln\\Big[1+\\Pi_\\rmi{L}(P)\/P^2\\Big]-\\Pi_\\rmi{L}(P)\/P^2\\nonumber \\\\\n&+&(d-2)\\(\\,\\ln\\Big[1+\\Pi_\\rmi{T}(P)\/P^2\\Big]-\\,\\Pi_\\rmi{T}(P)\/P^2\\)\\bigg\\}, \\label{logsi}\n\\end{eqnarray}\nwhich is now explicitly IR safe.\n\\begin{figure}[t]\n\\centering\n\\begin{eqnarray} \\nonumber\n\\begin{array}{lll}\na)\\pic{\\Lgl(0,15)(10,15)%\n \\Asc(20,15)(10,0,180) \\Asc(20,15)(10,180,360) \\Lgl(30,15)(40,15)}\\!+\\pic{\\Lgl(0,15)(10,15)%\n \\Agh(20,15)(10,0,180) \\Agh(20,15)(10,180,360) \\Lgl(30,15)(40,15)}\\!+\\pic{\\Lgl(0,15)(10,15)%\n \\Agl(20,15)(10,0,180) \\Agl(20,15)(10,180,360) \\Lgl(30,15)(40,15)}\\!+\\pic{\\Lgl(0,10)(20,10)%\n \\Agl(20,20)(10,0,360) \\Lgl(20,10)(40,10)}\\equiv\\, \\pic{\\Lgl(0,15)(10,15)%\n\\Lgl(30,15)(40,15)\\GCirc(20,15){10}{0.8} \\Text(20,15)[c]{V}} \\!\\!+\n\\pic{\\Lgl(0,15)(10,15)\\GCirc(20,15){10}{0.8}\\Text(20,15)[c]{M}%\n \\Lgl(30,15)(40,15)}\\nonumber \\\\\\nn\n b)~ p_\\rmi{VV}\\;\\equiv\\;\\picb{\\Agl(15,15)(15,30,150)%\n\\Agl(15,15)(15,210,330) \\GCirc(0,15){7.5}{0.8}\\Text(0,15)[c]{V} %\n\\GCirc(30,15){7.5}{0.8}\\Text(30,15)[c]{V}}\n\\;\\;\\;\\;\\;\\;\nc)~ p_\\rmi{VM}\\;\\equiv\\;\\picb{\\Agl(15,15)(15,30,150)%\n\\Agl(15,15)(15,210,330) \\GCirc(0,15){7.5}{0.8}\\Text(0,15)[c]{V} %\n\\GCirc(30,15){7.5}{0.8}\\Text(30.2,15)[c]{M}}\n\\;\\;\\;\\;\\;\\;\nd)~ p_\\rmi{ring}\\;\\equiv\\;\\sum_{n=2}^{\\infty}\\picb{\\Agl(15,15)(15,110,160)%\n\\Agl(15,15)(15,200,250) %\n\\GCirc(0,15){5}{0.8} \\GCirc(15,30){5}{0.8} \\GCirc(15,0){5}{0.8} \\Agl(15,15)(15,40,70)\\Agl(15,15)(15,290,320)\n\\DAsc(15,15)(15,-40,40)\\Text(15,0)[c]{{${\\mbox{\\scriptsize{M}}}$}}\n\\Text(15,30)[c]{{${\\mbox{\\scriptsize{M}}}$}}\\Text(0,15)[c]{{${\\mbox{\\scriptsize{M}}}$}} }\n\\end{array}\n\\end{eqnarray}\n\\caption[a]{a) The one-loop gluon polarization tensor $\\Pi_{\\mu\\nu}(P)$ divided\ninto its vacuum ($T=\\mu=0$) and matter (vacuum-subtracted) parts. \\\\\nb) The IR-safe Vac-Vac diagram contributing to the pressure at $\\mathcal{O}(g^4)$. \\\\\nd) The IR-safe Vac-Mat diagram contributing to the pressure at $\\mathcal{O}(g^4)$. \\\\\nd) The remaining 'matter' ring sum.}\n\\end{figure}\n\nAs the functions $\\Pi_\\rmi{L}(P)$ and $\\Pi_\\rmi{T}(P)$ behave at large $P^2$ like\n(see Sec.~B.1.2 of Ref.~\\cite{avthesis})\n\\begin{eqnarray}\n\\Pi_\\rmi{L\/T}(P)&\\xrightarrow[P^2\\rightarrow\\infty]{}&-2(1+\\epsilon)C_Ag^2\\hbox{$\\sum$}\\!\\!\\!\\!\\!\\!\\!\\int_Q \\fr{1}{Q^2}\n+\\mathcal{O}(1\/P^2)\n\\;\\;\\,\\equiv\\;\\;\\,\\Pi_\\rmi{UV}+\\mathcal{O}(1\/P^2),\n\\end{eqnarray}\nit is, however, immediately obvious that the sum-integral of Eq.~(\\ref{logsi}) is still\nlogarithmically divergent in the ultraviolet at $T\\neq 0$. To regulate the divergence, we add\nand subtract a term of the form $(1+d-2)(\\Pi_\\rmi{UV})^2\/(2(P^2+m^2)^2)$ from the integrand, with\n$m$ being an arbitrary mass parameter shielding it from IR divergences. By further adding and subtracting\nthe corresponding massless term from the counterterm, we obtain three separate contributions to $p_\\rmi{ring}$:\nan UV and IR finite (at least to order $g^4$ --- see below), $m$-dependent ring sum\n$p_\\rmi{ring}^\\rmi{finite}$, an UV finite, but IR divergent and $m$-dependent $p_\\rmi{ring}^\\rmi{IR}$ and an\nUV and IR divergent and massless $p_\\rmi{ring}^\\rmi{UV}$\n\\begin{eqnarray}\np_\\rmi{ring}&=&p_\\rmi{ring}^\\rmi{finite}+p_\\rmi{ring}^\\rmi{IR}+p_\\rmi{ring}^\\rmi{UV},\\nonumber\n\\end{eqnarray}\n\\begin{eqnarray}\np_\\rmi{ring}^\\rmi{finite}&=&-\\fr{d_A}{2}\\hbox{$\\sum$}\\!\\!\\!\\!\\!\\!\\!\\int_P\\bigg\\{\\ln\\Big[1+\\Pi_\\rmi{L}(P)\/P^2\\Big]-\n\\Pi_\\rmi{L}(P)\/P^2+C_A^2g^4T^4\/(72(P^2+m^2)^2)\\label{prf}\\nonumber \\\\\n&+&2\\(\\,\\ln\\Big[1+\\Pi_\\rmi{T}(P)\/P^2\\Big]-\\,\\Pi_\\rmi{T}(P)\/P^2+C_A^2g^4T^4\/(72(P^2+m^2)^2)\\)\\bigg\\},\n\\label{eq:pringfinite}\\\\\np_\\rmi{ring}^\\rmi{IR}&\\equiv&\\fr{d_AC_A^2g^4T^4}{48}\\hbox{$\\sum$}\\!\\!\\!\\!\\!\\!\\!\\int_P\\bigg\\{\\fr{1}{(P^2+m^2)^2}-\n\\fr{1}{P^4}\\bigg\\}\\label{pring37}\\\\\np_\\rmi{ring}^\\rmi{UV}&=&\\fr{1}{4}(d-1)d_A\\(\\Pi_\\rmi{UV}\\)^2\\hbox{$\\sum$}\\!\\!\\!\\!\\!\\!\\!\\int_P{1\\over P^4}.\n\\label{pUV}\n\\end{eqnarray}\nThe two first terms can be\nevaluated numerically at $\\epsilon=0$ while the third one needs to be regulated with finite $\\epsilon$. It is noteworthy that\none can set $\\epsilon = 0$ even in the formally divergent $p_\\rmi{ring}^\\rmi{IR}$ due to the fact\nthat its IR divergence originates solely from the zeroth Matsubara mode of its second\nterm which vanishes identically in dimensional regularization. The explicit values of $p_\\rmi{ring}^\\rmi{IR}$ and $p_\\rmi{ring}^\\rmi{UV}$\nare given in Appendix \\ref{app:ringsum}, while the numerical evaluation of\n$p_\\rmi{ring}^\\rmi{finite}$ is the subject of Appendix \\ref{app:num}.\n\n\n\n\\subsection{The double and triple sums}\n\nIf the sums in Figs.~2b--d were to start from $n=0$, these multiple resummations would clearly correspond to the dressing of the propagators in three two-loop diagrams\nwith the one-loop gluon polarization tensor. In the present case, we instead define a four-dimensionally transverse\\footnote{Thus decomposable into three-dimensionally\ntransverse and longitudinal parts.} tensor $\\Delta G_{\\mu\\nu}(P)$ by the equations\n\\begin{eqnarray}\n\\Delta G_\\rmi{L}(P)&=&\n\\fr{1}{P^2+\\Pi_\\rmi{L}(P)}-\\fr{1}{P^2}=\n-\\fr{\\Pi_\\rmi{L}(P)}{P^2(P^2+\\Pi_\\rmi{L}(P))},\\\\\n\\Delta G_\\rmi{T}(P)&=&\n\\fr{1}{P^2+\\Pi_\\rmi{T}(P)}-\\fr{1}{P^2}=\n-\\fr{\\Pi_\\rmi{T}(P)}{P^2(P^2+\\Pi_\\rmi{T}(P))},\n\\end{eqnarray}\ncorresponding to the difference of a dressed (with the vacuum-subtracted self energy) and a bare gluon propagator in the Feynman gauge. It is a straightforward\nexercise in combinatorics to show that the symmetry factors of all graphs in Figs.~2b--d equal $1\/4$ independently of $n$ --- a result particularly obvious in\n2PI formalism. To order $g^4$, these three classes of diagrams, denoted here by $p_\\rmi{b}$, $p_\\rmi{c}$ and $p_\\rmi{d}$, can then be written in the forms\n\\begin{eqnarray}\np_\\rmi{b}&=&\\fr{d_AC_A}{4}g^2\\hbox{$\\sum$}\\!\\!\\!\\!\\!\\!\\!\\int_{PQ}\\fr{\\Delta G_{\\mu\\mu '}(P)\\Delta G_{\\rho\\rho '}(Q)}{(P+Q)^2}\\nonumber \\\\\n&\\times&\\(g^{\\mu\\nu}(2P+Q)^{\\rho}-g^{\\nu\\rho}(2Q+P)^{\\mu}+g^{\\rho\\mu}(Q-P)^{\\nu}\\)\\nonumber \\\\\n&\\times&\\(g^{\\mu '\\nu}(2P+Q)^{\\rho '}-g^{\\nu\\rho '}(2Q+P)^{\\mu '}+g^{\\rho '\\mu '}(Q-P)^{\\nu}\\) + {\\mathcal O}(g^6),\\label{pb}\\\\\np_\\rmi{c}&=&\\fr{d_AC_A}{12}g^2\\hbox{$\\sum$}\\!\\!\\!\\!\\!\\!\\!\\int_{PQ}\\Delta G_{\\mu\\mu '}(P)\\Delta G_{\\rho\\rho '}(Q)\\Delta G_{\\nu\\nu '}(P+Q)\\nonumber \\\\\n&\\times&\\(g^{\\mu\\nu}(2P+Q)^{\\rho}-g^{\\nu\\rho}(2Q+P)^{\\mu}+g^{\\rho\\mu}(Q-P)^{\\nu}\\)\\nonumber \\\\\n&\\times&\\(g^{\\mu '\\nu '}(2P+Q)^{\\rho '}-g^{\\nu '\\rho '}(2Q+P)^{\\mu '}+g^{\\rho '\\mu '}(Q-P)^{\\nu '}\\)+ {\\mathcal O}(g^6),\\\\\np_\\rmi{d}&=&-\\fr{d_A C_A}{2}g^2\\hbox{$\\sum$}\\!\\!\\!\\!\\!\\!\\!\\int_{PQ}\\(\\Delta G_{\\mu\\mu}(P)\\Delta G_{\\nu\\nu}(Q)-\\Delta G_{\\mu\\nu}(P)\\Delta G_{\\mu\\nu}(Q)\\)+ {\\mathcal O}(g^6).\\label{pd}\n\\end{eqnarray}\nAll contributions involving the vacuum piece of the polarization tensor have been discarded as being of order $g^6$, following a reasoning similar to that in\nFootnote~\\ref{footnoteG}.\n\nIt is worthwhile to first perform a power counting analysis to determine at which order the above sum-integrals start to contribute to the pressure.\nIn the regime of dimensional reduction, where $T\\sim g^x \\mu$ with $x<1$, one merely needs to\nconsider the contributions of the zeroth Matsubara modes, as for the others the temperature acts as an infrared cutoff, leading to their values being proportional\nto at least $g^5T\\mu^3\\sim g^{5+x}\\mu^4$. In the region $T\\sim g\\mu$, the Debye mass is, however, of the same order as the temperature,\nimplying that all Matsubara modes give contributions to the pressure parametrically similar in magnitude. Scaling the three momenta in the integrals of\nEqs.~(\\ref{pb})--(\\ref{pd}) by $g\\mu$, one quickly sees that the results for the sum-integrals in this regime can up to ${\\mathcal O}(g^4)$ be written in the form\n$g^4T^2\\mu^2f(T\/(g\\mu))$, where the contributions of the non-static modes to the function $f$ vanish as the parameter $T\/(g\\mu)$\napproaches infinity, while in the opposite limit $T\/(g\\mu)\\rightarrow 0$ the function approaches a constant. As long as we are interested in the value of the pressure\nonly to order $g^4\\mu^4$, these graphs can clearly be altogether\nignored. They will become relevant in the determination of the ${\\mathcal O}(g^4\\mu^2 T)$ contributions to the specific heats, but this is outside the scope of the\npresent work.\n\nFor now, we can concentrate our attention to the regime of dimensional reduction and therefore to the zero Matsubara mode parts of the above sum-integrals.\nHere, we encounter an important simplification which results from the fact that only the longitudinal part of the static gluon polarization tensor has a\nnon-zero zero momentum limit at one-loop order. As the finite momentum corrections to the functions $\\Pi_\\rmi{L}(P)$ and $\\Pi_\\rmi{T}(P)$ clearly correspond\nto higher perturbative orders, we can simply replace\n\\begin{eqnarray}\n\\Delta G_{\\mu\\nu}(P)&\\rightarrow & -\\fr{m_\\rmi{D}^2}{p^2(p^2+m_\\rmi{D}^2)}\\delta_{\\mu 0}\\delta_{\\nu 0} \\label{glimit}\n\\end{eqnarray}\nin the integrals, leading to a dramatic reduction: both $p_\\rmi{c}$ and $p_\\rmi{d}$ then vanish identically. This can, however, be easily understood from the point of\nview of the three-dimensional effective theory EQCD as a demonstration of the\nfact that the $A_0^3$ and $A_0^4$ operators in its Lagrangian\nare not accompanied by couplings of order $g$ and $g^2$, respectively,\nbut only $g^3$ (at nonzero $\\mu$) and $g^4$.\n\nIn contrast to the above, for $p_\\rmi{b}$ one does obtain a non-zero value which has a direct parallel in EQCD in\nthe form of an ${\\mathcal O}(g)$ coupling between one massless $A_i$ and two massive $A_0$ fields and a corresponding two-loop diagram with one $A_i$ and two $A_0$\nlines.\nApplying the limit of Eq.~(\\ref{glimit}) to the sum-integral of Eq.~(\\ref{pb}), it is easy to see that we can reduce the expression of $p_\\rmi{b}$ (to order $g^4$) to the\nsimple form\n\\begin{eqnarray}\np_\\rmi{b}\\;\\,=\\;\\,\\fr{d_AC_A}{4}T^2m_\\rmi{D}^4g^2\\!\\int\\!\\fr{d^3p}{(2\\pi)^3}\\!\\int\\!\\fr{d^3q}{(2\\pi)^3}\n\\fr{(\\mathbf{p}-\\mathbf{q})^2}{\\mathbf{p}^2(\\mathbf{p}^2+m_\\rmi{D}^2)\\mathbf{q}^2(\\mathbf{q}^2+m_\\rmi{D}^2)(\\mathbf{p}+\\mathbf{q})^2}\n\\end{eqnarray}\nwhich can be solved straightforwardly by introducing three Feynman parameters and using standard formulae for one-loop integrals in three dimensions.\nAfter some work, we get\n\\begin{eqnarray}\np_\\rmi{b}&=&\\fr{d_AC_A}{4}T^2m_\\rmi{D}^4g^2\\bigg\\{\\!\\int\\!\\fr{d^3p}{(2\\pi)^3}\\fr{1}{\\mathbf{p}^2(\\mathbf{p}^2+m_\\rmi{D}^2)}\\int_0^1\\!\\!\\! dx\\!\\!\\int\\!\\fr{d^3q}{(2\\pi)^3}\n\\fr{1}{\\mathbf{q}^2+2x\\mathbf{q}\\cdot \\mathbf{p}+x\\mathbf{p}^2+ (1-x)m_\\rmi{D}^2}\\nonumber \\\\\n&-&\\(\\int_0^1\\!\\!\\! dx\\!\\!\\int\\!\\fr{d^3p}{(2\\pi)^3}\\fr{1}{(\\mathbf{p}^2+xm_\\rmi{D}^2)^2}\\)^2\\bigg\\}\\nonumber \\\\\n&=&\\fr{d_AC_A}{4}\\fr{T^2m_\\rmi{D}^4g^2}{(4\\pi m_\\rmi{D})^2}\\bigg\\{\\fr{1}{\\pi}\\int_0^1\\!\\!\\! dx\\fr{1}{\\sqrt{x(1-x)}}\\!\\!\\int_0^1\\!\\!\\! dy\\fr{1}{\\sqrt{y}}\\fr{1}{1-y+y\/x}\n\\!\\!\\int_0^1\\!\\!\\! dz\\fr{1}{\\sqrt{z}}-1\\bigg\\}\\nonumber \\\\\n&=&-\\fr{d_AC_A}{4}T^2m_\\rmi{D}^2\\fr{g^2}{(4\\pi)^2}(1-4\\,\\ln\\,2)\\label{pbres}\n\\end{eqnarray}\nwhich we identify as the entire contribution of the classes b-d of Fig.~2 to the QCD pressure up to order $g^4$.\n\n\n\n\n\\subsection{The result}\\label{subsec:result}\n\nWe are now ready to write down our final result for the pressure,\nvalid on the entire deconfined phase of QCD\nand accurate up to and including order $g^4$.\nAssembling all the various pieces, this function reads\n\\begin{eqnarray}\np&=&(p_\\rmi{2GI}+p_\\rmi{VV}+p_\\rmi{VM}+p_\\rmi{ring}^\\rmi{UV}+ p_\\rmi{b}) + (p_\\rmi{ring}^\\rmi{IR} + p_\\rmi{ring}^\\rmi{finite})\n+{\\mathcal O}(g^5T\\mu^3) +\n{\\mathcal O}(g^6\\mu^4)\\label{res1}\\\\\n&\\equiv &p_\\rmi{anl}+p_\\rmi{ring}^\\rmi{safe}+{\\mathcal O}(g^5T\\mu^3) +\n{\\mathcal O}(g^6\\mu^4),\n\\end{eqnarray}\nwhere $p_\\rmi{anl}$ stands for the sum of the first five terms in Eq.~(\\ref{res1}) and\n\\begin{eqnarray}\np_\\rmi{ring}^\\rmi{safe}&\\equiv& p_\\rmi{ring}^\\rmi{finite}+p_\\rmi{ring}^\\rmi{IR}\n\\end{eqnarray}\nis to be evaluated numerically. One should note that in this notation all $m$-dependence\nin contained in the two pieces of $p_\\rmi{ring}^\\rmi{safe}$, naturally canceling\nin their sum. In addition, it is worthwhile to point out that the inclusion of the term $p_b$ in Eq.~(\\ref{res1})\nis inconsistent in the region of $T\\sim g^x\\mu$, $x\\geq 1$ where we have neglected several\ncontributions of the same magnitude. As this term, however, is of order $g^{4+2x}\\mu^4$, \\textit{i.e.~}at least\nof order $g^6\\mu^4$ in the region in question, the inconsistency is in any case beyond the order to which our result is\nindicated to be valid and can therefore be ignored.\n\nCollecting the expressions for all of its parts from above and from\nAppendix \\ref{app:anl}, the function $p_\\rmi{anl}$ reads\n\\begin{eqnarray}\np_\\rmi{anl}&=&\\pi^2d_AT^4\\Bigg(\\fr{1}{45}\\bigg\\{1+\\fr{d_F}{d_A}\\(\\fr{7}{4}+30\\bar{\\mu}^2+60\\bar{\\mu}^4\\)\\bigg\\}\\nonumber \\\\\n&-&\\fr{g^2}{9(4\\pi)^2}\\bigg\\{C_A + \\fr{T_F}{2}(1+12\\bar{\\mu}^2)(5+12\\bar{\\mu}^2)\\bigg\\}\\nonumber \\\\\n&+&\\fr{g^4}{27(4\\pi)^4}\\bigg\\{-C_A^2\\bigg(22\\,\\ln\\fr{\\bar{\\Lambda}}{4\\pi T}+63-18\\gamma\n+110\\frac{\\zeta'(-1)}{\\zeta(-1)}-70\\frac{\\zeta'(-3)}{\\zeta(-3)}\n\\bigg)\\nonumber \\\\\n&-&C_AT_F\\bigg(\\(47+792\\bar{\\mu}^2+1584\\bar{\\mu}^4\\)\\ln\\fr{\\bar{\\Lambda}}{4\\pi T}+\\fr{2391}{20}+4\\frac{\\zeta'(-1)}{\\zeta(-1)}+\\fr{116}{5}\\frac{\\zeta'(-3)}{\\zeta(-3)} \\nonumber \\\\\n&+&6\\(257+88\\frac{\\zeta'(-1)}{\\zeta(-1)}\\)\\bar{\\mu}^2+2220\\bar{\\mu}^4+792(1+4\\bar{\\mu}^2)\\aleph(1,z)+3168\\aleph(3,z)\\bigg)\\nonumber \\\\\n&+&T_F^2\\bigg((1+12\\bar{\\mu}^2)\\(4(5+12\\bar{\\mu}^2)\\ln\\fr{\\bar{\\Lambda}}{4\\pi T}+16\\frac{\\zeta'(-1)}{\\zeta(-1)}\\)+\\fr{99}{5} + \\fr{16}{5}\\frac{\\zeta'(-3)}{\\zeta(-3)}\\nonumber \\\\\n&+&312\\bar{\\mu}^2+624\\bar{\\mu}^4+288(1+4\\bar{\\mu}^2)\\aleph(1,z)+1152\\aleph(3,z)\\bigg)\\nonumber \\\\\n&+&\\fr{9}{4}C_FT_F\\bigg(35-32(1-4\\bar{\\mu}^2)\\frac{\\zeta'(-1)}{\\zeta(-1)}+472\\bar{\\mu}^2+1328\\bar{\\mu}^4\\nonumber \\\\\n&+&192\\(i\\bar{\\mu}(1+4\\bar{\\mu}^2)\\aleph(0,z)+2(1+8\\bar{\\mu}^2)\\aleph(1,z)-12i\\bar{\\mu}\\aleph(2,z)\\)\\bigg)\\bigg\\}\\Bigg). \\label{panl}\n\\end{eqnarray}\nNot only have all the UV divergences canceled between the\ndifferent parts of this result,\nonce the renormalization of the gauge coupling $g$ has been taken care of,\nbut this expression actually contains all the (explicit)\nrenormalization scale dependence\nof the pressure up to the present order in perfect agreement with Ref.~\\cite{avpres},\nleaving $p_\\rmi{ring}^\\rmi{safe}$ entirely independent of the parameter $\\bar{\\Lambda}$.\nEq.~(\\ref{panl}) is also valid for all values of $T$ and $\\mu$; the limit for $\\mu\\to0$ is given in\nEq.~\\nr{mutozero} and the limit $T\\rightarrow0$ in Eq.~\\nr{ttozero}. All terms non-analytic in $g^2$\nare contained in the piece $p_\\rmi{ring}^\\rmi{safe}$ awaiting numerical evaluation.\n\n\nIn the following we shall denote our final result for the pressure\n--- which is accurate to order $g^4$ for all values of $T$ and $\\mu$\n(while also containing some incomplete contributions of higher order, to be discarded later) --- by\n\\begin{equation}\np_\\rmi{IV}=p_\\rmi{anl}+p_\\rmi{ring}^\\rmi{safe}.\n\\end{equation}\n\n\n\\subsection{Numerical infrared issues}\\label{sec:numiss}\n\nBefore moving on to examining our result by numerically evaluating the function $p_\\rmi{ring}^\\rmi{finite}$ in Eq.~(\\ref{eq:pringfinite}),\nthere is one more practical issue related to the magnetic\nmass problem \\cite{Linde:1980ts,Kalashnikov:1980tk} that needs to be dealt with. To wit, in the limit $P\\rightarrow 0$, the argument of\n$\\ln(1+\\Pi_\\rmi{T}(P)\/P^2)$ becomes negative, resulting in an unwanted imaginary\ncontribution to the integral which actually renders $p_\\rmi{ring}^\\rmi{finite}$\ninfrared singular beyond order $g^4$. This problem depends on the choice of gauge, but is present in all\ncovariant gauges (as well as the Coulomb gauge).\n\nThe origin of the problem can be traced back to the fact that when dressed with\nthe full one-loop self-energy,\nthe transverse part of the gluon propagator develops a space-like pole.\nFor $p_0=0$ this pole is determined by the equation \\cite{Kalashnikov:1980tk}\n\\begin{eqnarray}\np^2+\\Pi_\\rmi{T}(p_0=0,p)&=&p^2-g^2N_cT{8+(\\xi+1)^2\\064}p\n\\end{eqnarray}\nwhere $\\xi$ is the gauge parameter of covariant gauges.\\footnote{Replacing the ordinary one-loop gluon self-energy by one that includes\nresummation of the Debye mass does not cure the problem, but only produces a different\ngauge-dependent spacelike pole \\cite{Kalashnikov:1982sc,Kraemmer:2003gd}.}\nIt is evidently unphysical and appears only at the non-perturbative magnetic mass scale $g^2T$,\nwhich contributes to the pressure starting at order $g^6T^4$. This suggests that\nwe can in fact eliminate the entire problem by adding by hand\na magnetic mass term\nto the transverse self-energy in Eq.~(\\ref{eq:pringfinite})\n\\begin{eqnarray}\n\\Pi_\\rmi{T}(P) & \\rightarrow & \\Pi_\\rmi{T}(P) + m_\\rmi{mag}^2\n\\end{eqnarray}\nwith (for $\\xi=1$)\n\\begin{equation}\\label{mmagf}\nm_\\rmi{mag}=c_{f}\\frac{3}{32}g^{2}C_A T\n\\end{equation}\nand $c_f\\ge1$,\nwhich only has an effect on the pressure beyond $O(g^4)$.\nIndeed, comparing with the effective magnetic mass for nonzero frequencies,\nEq.~(\\ref{mmdyn}), we find that the magnetic screening behaviour\nis modified only for frequencies $p_0 \\lesssim g^4 T$ when $\\mu\\sim T$\nand even $p_0 \\lesssim g^4 T(T^2\/\\mu^2)$ when $T\\ll \\mu$.\nNote, however, that the introduction of this magnetic mass for the transverse self-energy\nalters the UV behavior of $p_\\rmi{ring}^\\rmi{finite}$, implying that both\n$p_\\rmi{ring}^\\rmi{finite}$ and $p_\\rmi{ring}^\\rmi{IR}$ have to be modified to account\nfor this reorganization. In\n$p_\\rmi{ring}^\\rmi{finite}$, this change is crucial because it renders the result finite,\nbut for the already finite $p_\\rmi{ring}^\\rmi{IR}$ the effects are beyond the order of interest\n(see App.~\\ref{app:num}).\n\nThe numerical evaluation of $p_\\rmi{ring}^\\rmi{finite}$ is performed along the lines of\nRefs.~\\cite{moore,ippreb}, with the sum over Matsubara frequencies being converted to an integration\nin the usual way (see \\textit{e.g.~}Ref.~\\cite{kap}). Contributions containing the bosonic\ndistribution function $n_b$ are best evaluated in Minkowski space, as UV problems are cut off\nby $n_b$, while the other contributions are evaluated in Euclidean space in order to\nnumerically exploit the Euclidean invariance of UV contributions.\nBy varying the parameter $c_f$ in Eq.~(\\ref{mmagf}), we can verify that the effects of this\ninfrared regulator are indeed beyond the order $g^4$ we are aiming at.\nThe remaining part,\nhowever, gives rise to yet another type of unphysical pole,\nwhich (at least in the long-wavelength limit) has been well-known since the earliest\nperturbative calculations in finite-temperature QCD \\cite{Kalashnikov:1979cy}:\nin covariant gauges, the one-loop gluon self-energy, evaluated at the location of the\npoles corresponding to time-like propagating plasmon modes, gives rise\nto a (gauge-dependent) damping constant $\\propto g^2 T$ with negative sign\n(for all gauge parameters $\\xi$, though not in Coulomb or axial gauges\n\\cite{Kajantie:1982xx,Heinz:1986kz}). A consistent systematic calculation of the\nplasmon damping constant to order $g^2T$ requires the use of a HTL-resummed gluon\nself-energy which finally leads to a positive and gauge-independent result\n\\cite{Braaten:1990it,Kraemmer:2003gd}. The corresponding pole is then on the unphysical\nsheet where it would cause no problem for the evaluation of $p_\\rmi{ring}^\\rmi{finite}$.\nWith the bare one-loop gluon self-energy appearing in our integrand we, however, have poles\non the physical sheet, connected to the light-cone by a branch cut, and we need to avoid\nthem by deforming the contour of the numerical integration in\nMinkowski space as sketched in Fig.~\\ref{fig:analyticstructure}.\nThe details of this procedure and the entire numerical calculation are described further in Appendix \\ref{app:num}.\n\n\n\n\n\n\n\\section{Numerical results}\\label{sec:numres}\n\nHaving the result of Eq.~(\\ref{res1}) for the QCD pressure now finally at hand, we move on to examine\nit numerically by evaluating the function\n$p_\\rmi{ring}^\\rmi{safe}$ using methods reviewed in Appendix \\ref{app:num} and adding to it the analytic\npart of Eq.~(\\ref{panl}).\nThe sum total we call $p_\\rmi{IV}$ as a reminder that its\naccuracy is of order $g^4$ for all $T$ and $\\mu$, while it also includes\nincomplete and gauge dependent\nhigher-order contributions. For the most part of the following analysis, we shall\nexplicitly eliminate the latter effects by either considering sufficiently small values of $g$\nor performing numerical expansions of our results in powers of $g$.\n\nWe begin by inspecting\nthe region where the temperature is parametrically larger than the Debye scale and the\nresults of dimensional reduction should be applicable,\nthen continue towards making contact with HDL results\non non-Fermi liquid behavior at $T \\lesssim m_\\rmi{D}$, and\nfinally the Freedman-McLerran result for $T\\to0$.\nIn all plots of the present section we use the values\n$N_\\rmi{c}=3$, $N_\\rmi{f}=2$. Because of the latter, we conveniently\nhave $T_F=1$ and therefore $\\tau=\\bar\\tau$ for the reduced\ntemperature variables introduced in Eqs.~(\\ref{tau}) and (\\ref{bartau}), respectively.\n\n\n\\begin{figure\n\\begin{center}\n\\includegraphics[width=10cm]{perturbative_g4_02a.eps}\n\\caption{Comparison of the $g^4\\log g$ and $g^4$ terms of the numerical computation\nand the analytic DR result, for various values of $\\mu\/T$.\nThe perturbative terms are subtracted up to order $g^3$.\n \\label{fig:X}}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{$T$ parametrically larger than $m_\\rmi{D}$}\n\nThe first non-trivial check on our result --- and that of dimensional reduction ---\nis to verify that their predictions for the pressure agree to\norder $g^4$ for all temperatures and chemical potentials that are of\nequal parametric order in $g$.\nThis is particularly important in order\nto clarify that the (entirely correct) statement in the literature about\ndimensional reduction being valid as long as $\\pi T$ is the\nlargest dynamical energy scale does not imply a condition\n$\\pi T > \\mu$, but rather $\\pi T \\gg m_\\rmi{D}$ (or even $\\pi T \\gtrsim m_{\\rmi D}$, as\nwe shall find to be sufficient below).\nTo this end, we start from the most\nwidely studied region of $\\mu=0$ by comparing our numerical result\nto that of the analytic one of dimensional reduction, and then increase the\n${\\mathcal O}(g^0)$ value of $\\mu\/T$ up to $\\mu \\gg \\pi T$\nwhile still having $\\pi T \\gg m_\\rmi{D}\\sim g\\mu$.\n\nThe results of this comparison are shown in Fig.~\\ref{fig:X}, where we plot the order $g^4\\ln\\,g$ and $g^4$\ncontributions of the ring sum of Eq.~(\\ref{prf}) to the pressure together with the same quantity\nextracted from the result of dimensional reduction (obtained by subtracting the analytic part of our result\nfrom the DR one). The agreement is perfect up to the numerical accuracy of our result, and only at larger values of $g$\ncan one see that the agreement is getting slightly worse with increasing $\\mu\/T$. This was, however, to be expected,\nsince there $\\mu\/T$ is coming closer to the value $g^{-1}$, making $m_\\rmi{D}\/T$ of order one which is parametrically\nthe limit of applicability of dimensional reduction. Our conclusion is that the result of dimensional reduction\nis valid at in principle arbitrarily large ${\\mathcal O}(g^0)$ values of $\\mu\/T$, though the expansion in $g$ only\nmakes sense at smaller and smaller values of $g$ as this parameter is increased. This statement will be made more\nconcrete in the following sections.\n\n\\begin{figure\n\\begin{center}\n\\includegraphics[width=10cm]{sqrtg_tau02_g9o2_02a.eps}\n\\caption{Comparison of the HTL+ pressure and our numerical result $p_\\rmi{IV}$ in the region of $T=\\tau \\sqrt{g} \\mu$, $\\tau=0.2$, with the known\nperturbative terms from dimensional reduction\n subtracted and the entire quantities divided by $g^{9\/2}$.\nThis plot shows that both the HTL+ result and our numerical one are accurate\nat least up to order $g^{9\/2}$. The renormalization scale has\nbeen varied between $\\mu$ and $4\\mu$. While $p_\\rmi{IV}-p_\\rmi{DR}$\nis scale independent, $p_\\rmi{HTL+}-p_\\rmi{DR}$ has a scale dependence\nat order $g^4\\mu^2 T^2\\sim g^5\\mu^4$.\n \\label{fig:ninehalf}}\n\\end{center}\n\\end{figure}\n\n\\begin{figure\n\\begin{center}\n\\includegraphics[width=10cm]{sqrtg_tau02_g5_02a.eps}\n\\caption{Same as Fig.~\\ref{fig:ninehalf}, but normalized to $g^5$.\nWhile the HTL+ result is no longer accurate to this order and diverges logarithmically,\nour numerical result still correctly reproduces the\ndimensional reduction result for the pressure at order $g^5\\mu^4$.\n \\label{fig:ninehalfb}}\n\\end{center}\n\\end{figure}\n\nThe logical next step is to test the validity of dimensional reduction at temperatures larger\nthan but now parametrically closer to the Debye scale.\nFor concreteness, we specialize to the case of $T\\sim \\sqrt{g}\\mu$, for which the prediction\nof dimensional reduction is given in Eq.~(\\ref{pDRsqrtg}).\nIn this region, the error in our result is of order $g^{11\/2}\\mu^4$ and that of the minimal\nHTL resummation $g^5 \\mu^4$,\nso that the first one should be able to reproduce the first seven and the latter the\nfirst six terms of the series (\\ref{pDRsqrtg}). And indeed, a numerical\nevaluation of both Eqs.~(\\ref{res1}) and (\\ref{PHTL}) and the subtraction of\nthe first terms of Eq.~(\\ref{pDRsqrtg}) shows the expected results: as\ndisplayed in Fig.~\\ref{fig:ninehalf}, we find perfect agreement in comparing the\ndimensional reduction result with the HTL one (\\ref{PHTLtot}) and\nwith that of our new approach up to order $g^{9\/2}$.\nIn Fig.~\\ref{fig:ninehalfb}, we see that our numerical evaluation\nof $p_\\rmi{IV}$ is accurate enough to even\nverify the $g^5\\mu^4$ term in the dimensional reduction result, while the HTL result starts deviating from the\nDR one at this order.\n\n\n\\subsection{$T$ comparable to $m_\\rmi{D}$}\n\n\n\n\n\\begin{figure\n\\begin{center}\n\\includegraphics[width=10cm]{deltap_g01_03b.eps}\n\\caption{\nThermal contribution to the interaction pressure $\\delta p$\nas a function of $T\/m_\\rmi{D}^{T=0}$ for fixed chemical potential $\\mu$\nand coupling $g=0.1$.\nFor this value of the coupling, the results of the numerical evaluation\nof $p_\\rmi{anl}+p_\\rmi{ring}^\\rmi{safe}$ and $\\rm{HTL}^{+}$\ncoincide within plot resolution.\nThe result is compared to the dimensional reduction pressure\nat orders $g^2$, $g^3$, $g^4$, and $g^5$ (where the latter\nis included only for completeness, as neither\n$p_\\rmi{IV}$ nor $p_{\\rmi{HTL}^{+}}$\ncontain contributions of order $g^5$). The effect of varying the\nrenormalization scale ${\\Lambda_{\\overline {\\mbox{\\tiny MS}} }}=\\mu \\,...\\, 4\\mu$ is not visible\nfor this value of the coupling. \\label{fig:p01}}\n\\end{center}\n\\end{figure}\n\n\\begin{figure\n\\begin{center}\n\\includegraphics[width=10cm]{deltap_g05_06a.eps}\n\\caption{\nSame as Fig.~\\ref{fig:p01}, but for $g=0.5$.\nThe results of the numerical evaluation\nof $p_\\rmi{anl}+p_\\rmi{ring}^\\rmi{safe}$ and $\\rmi{HTL}^{+}$\ncan now be distinguished due to their different content of higher-order terms.\nWhen two lines of the same type run close to each other, they differ by\nchanging the renormalization scale ${\\Lambda_{\\overline {\\mbox{\\tiny MS}} }}=\\mu \\,...\\, 4\\mu$.\n\\label{fig:p05}}\n\\end{center}\n\\end{figure}\n\n\nIn Figs.~\\ref{fig:p01}--\\ref{fig:p05parts}, we plot the the temperature-dependent contributions to the interaction pressure $\\delta p$\n(see Eq.~(\\ref{deltaDeltap})) for $T\\sim m_\\rmi{D}$ as extracted from our numerical\ncalculation of $p_\\rmi{IV}$ but with no expansions in powers of $g$. We compare this with $p_\\rmi{HTL+}$ as well\nas with the dimensional reduction result expanded to orders $g^2$, $g^3$, $g^4$ and $g^5$ which refer\nto the counting in powers of $g$ when $T\\sim\\mu$.\nFor $T\\sim g\\mu$, however,\nthe terms $g^2 \\mu^2 T^2$, $g^3 \\mu^3 T$, and $g^4\\mu^4\\ln\\,T$ all\nbecome of the same order of magnitude and together constitute the leading temperature-dependent\ncontribution to the interaction pressure $p-p_\\rmi{SB}$ which is contained in the result marked by\nthe dashed line ``$g^4$''. For completeness, we also\ninclude the complete dimensional reduction result to\n(explicit) order $g^5$, but it should be remembered that\nthe term $g^5 T\\mu^3$ is already of the same magnitude as the unknown $g^6\\mu^4$ piece\nwhen $T\\sim g\\mu$, and is therefore both incomplete and beyond our scope which also explains why the $g^4$ curve seems to produce better agreement\nwith our results than the $g^5$ one.\n\nThe different results are normalized to the leading term of the $T$-dependent part of the interaction pressure\nin the dimensional reduction result\n(\\ref{dimredpr5}),\n\\begin{equation}\\label{dpDR2}\n\\delta p^{(2)}_\\rmi{DR} = -g^2\nd_A \\biggl\\{ {T_F\\016\\pi^2} \\mu^2 T^2 + {5T_F+2C_A\\0244} T^4 \\biggr\\}.\n\\end{equation}\nTo understand the structure of these figures, note that the $g^3$ curve goes like $-1+(4\/3\\pi) m_\\rmi{D}\/T$\nfor small $T$ and like $-1+1.07g$ for large $T$. At $T\\ll m_\\rmi{D}$ it, of course, deviates from the\nexact result which is instead dominated by the\nleading $\\fr{2}{9} \\ln \\, T^{-1}$ behavior of the low-temperature series of Eq.~(\\ref{PHDLx}) when\nnormalized by the absolute value of Eq.~\\nr{dpDR2}.\n\n\n\\begin{figure\n\\begin{center}\n\\includegraphics[width=10cm]{deltapparts_g05_04b.eps}\n\\caption{\nSame as Fig.~\\ref{fig:p05}, but with $p_\\rmi{IV}$ separated into $p_\\rmi{anl}$ and $p_\\rmi{ring}^\\rmi{safe}$.\nAs the $g^4$ contribution in $\\delta p_\\rmi{anl}$\nonly amounts to a small correction\n(of effective order $g^6$),\nthe shape of the full pressure curve as a function of $T$\n(beyond the rather trivial $g^2$ contribution)\nis mainly determined by $p_\\rmi{ring}^\\rmi{safe}$.\nThe renormalization scale dependence\n${\\Lambda_{\\overline {\\mbox{\\tiny MS}} }}=\\mu\\, ...\\, 4\\mu$ is entirely due to $p_\\rmi{anl}$.\n\\label{fig:p05parts}}\n\\end{center}\n\\end{figure}\n\nFor small values of $g\\sim 0.1$, Fig.~\\ref{fig:p01} shows that the numerical evaluation\nof Eq.~(\\ref{res1}) perfectly agrees with the result of the\nHTL resummation (the two curves lie virtually on top of each other).\nAt this value of $g$, also the complete dimensional reduction result to\n(explicit) order $g^5$ is virtually indistinguishable from the\norder $g^4$ result. The dimensional reduction result\nreproduces the numerical results remarkably well down to temperatures of about\n$0.2\\,m_\\rmi{D}^{T=0}$, but at even lower $T$ severely overestimates the\nlogarithmic growth of\n$\\delta p\/T^2$ as $T\\to 0$.\nThis is to be expected, since, in the limit $T\\to0$,\nthe plasmon term of order\n$g^3\\mu^3 T$ in the pressure is clearly unphysical, as it would\nlead to a nonvanishing entropy at $T=0$; the $g^4 \\mu^4 \\ln\\,T$ term\nof the dimensional reduction result,\nwhile evidently crucial for good agreement down to $T\\approx 0.2 m_\\rmi{D}$,\nwould even lead to a diverging entropy as $T\\to0$.\nThe point at which the dimensional reduction result ceases to be a\ngood approximation for both $p_\\rmi{HTL+}$ and $p_\\rmi{IV}$ seems\nto agree rather well with the value of $T\/m_\\rmi{D}$ where $\\delta p$\nswitches sign.\n\nIn Fig.~\\ref{fig:p05} we consider a larger coupling $g=0.5$, for which we begin to see effects from\nvarying the renormalization scale ${\\Lambda_{\\overline {\\mbox{\\tiny MS}} }}$ in our result by a factor of 2 around\nthe central value $2\\mu$,\nexcept in the HTL+ result, where ${\\Lambda_{\\overline {\\mbox{\\tiny MS}} }}$ appears only\nin the $T=0$ (Freedman-McLerran) part of the result.\\footnote{\nIn Fig.~\\ref{fig:p05}, the value $g=0.5$ is kept fixed for all ${\\Lambda_{\\overline {\\mbox{\\tiny MS}} }}$,\nwhich means that the $x$-axis does not correspond to a\nrenormalization-group invariant variable. The (explicit)\ndependence of the results on ${\\Lambda_{\\overline {\\mbox{\\tiny MS}} }}$ is here shown\nonly to assess the theoretical error in the numerical comparison between the\ndifferent approaches. Taking into account the implicit\n${\\Lambda_{\\overline {\\mbox{\\tiny MS}} }}$ dependence of $g$, the scale dependence of\nall the results we are comparing is of the order of their error,\nwhich is $O(g^6\\mu^4)$ at $T\\sim g\\mu$.}\nFor small $T\/m_\\rmi{D}^{T=0}$, we find good agreement between the HTL+ result\nand $p_\\rmi{IV}$,\nwith the dimensional reduction result to order $g^4$ lying in between the two\nin the range $T\/m_\\rmi{D}^{T=0}\\approx 0.1\\ldots 10$, but deviating again\nabruptly for $T\/m_\\rmi{D}^{T=0} < 0.2$,\nwhich is where $\\delta p$ changes sign.\nAt this value of the coupling, the complete order $g^5$ result\nof dimensional reduction is still reasonably close to the order $g^4$\nresult. While it is certainly unreliable when $\\delta p>0$, the\norder $g^5$ result suggests that taking into account the next\nhigher orders in $g$ may move the onset of non-Fermi-liquid behavior\nto slightly larger $T\/m_\\rmi{D}$.\n\nFig.~\\ref{fig:p05parts} shows how the final result $\\delta p_\\rmi{IV}$\nis composed of the infrared-safe piece $p_\\rmi{anl}$ and the ring sum\n$p_\\rmi{ring}^\\rmi{safe}$. At parametrically small $T\\sim g\\mu$,\nthe $T$-dependent terms in the interaction part of $p_\\rmi{anl}$\nwhich are of effective order $g^4 \\mu^4$ come just from the terms $g^2 T^2 \\mu^2$,\nso that the shape of $\\delta p$ in the above figures is mainly\ndetermined by $\\delta p_\\rmi{ring}^\\rmi{safe}$ which is seen to\ncoincide with $\\delta p_\\rmi{HTL+}-\\delta p^{(2)}_\\rmi{DR}$ up\nto terms beyond $g^4$ accuracy.\n\nFinally, in Fig.~\\ref{fig:p1} we consider $g=1$ which is roughly the value\nof the QCD coupling at 100 GeV. Here the result for $p_\\rmi{IV}$\nstill follows $p_\\rmi{HTL+}$ for $T0$. As long as $x<1$, $T$ is parametrically\nlarger than the Debye mass $\\sim g\\mu$, and so dimensional reduction\nshould still be applicable.\nHowever, each coefficient of the original series at $x=0$ now\nhas to be expanded in powers of $T\/\\mu\\sim g^x$. The 2-loop\npressure contribution for example yields three different terms for $x>0$: one\nis proportional to $\\mu^4$ and thus is always of order $g^2$,\nanother --- proportional to $\\mu^2 T^2$ --- gives the line $y=2+2x$\nand the third term proportional to $T^4$ produces the line $y=2+4x$.\nStarting with the plasmon term which is of order $g^3$ at $x=0$,\nwe obtain an infinite series of higher-order terms for $x>0$.\nThese arise from the expansion of the third power of the Debye mass parameter in powers of $T\/\\mu$, and, for\nsubsequent terms in the dimensional reduction result, also from the expansion of the special\nfunctions $\\aleph(n,z)$. Because both the Debye mass\nand the $\\aleph$ functions can be expanded\nin even powers of $T\/\\mu$, the lines emanating from\ntheir starting points at $x=0$ come with slopes differing\nby two units. The terms proportional to $g^3$ and $g^5$\nat $x=0$ involve a single overall power of $T$, so the lines\nemanating from these have slopes 1, 3, 5, \\ldots, whereas\nthe term proportional to $g^4$ (or $g^4 \\ln\\,g$) has\nalso $T$-independent parts and thus gives rise\nto lines with slopes 0, 2, 4, \\ldots.\nIn Eq.~(\\ref{pDRsqrtg}) we have seen how this gives rise to a new series\nin $g$ at $x=1\/2$, and Fig.~\\ref{fig:orders} illustrates\nhow the individual terms of order 2, 3, $7\\over2$, 4, $9\\over2$, \\ldots\nare produced from the various coefficients of the expansion at $x=0$.\n\nMoving on to the border of applicability of the dimensional reduction\nresults, $x=1$, we see that all lines converge to points\ncorresponding to an expansion in even powers of $g$ (and\nalso involving $\\ln\\,g$). As noted before, for $x\\ge 1$ the relevant effective\ntheory is the one\ngiven by non-static hard dense loops. Their resummation\nis necessary to obtain the classic Freedman-McLerran (FMcL) result\nto order $g^4$ (again accompanied by a logarithmic term) at $T=0$\nas well as the leading thermal corrections to the interaction\npressure. In a low-$T$ expansion\nthese $T$-dependent terms start with a contribution of order\n$g^2 T^2 \\mu^2 \\ln(T\/g\\mu)$ and then involve fractional powers\n$T^{8\/3}$, $T^{10\/3}$, $T^4\\,\\ln\\, T$, $T^{14\/3}$, \\ldots$\\;$such that the corresponding lines in\nFig.~\\ref{fig:orders} (labeled by the exponent of $T$)\nmeet at $x=1$ and effective order $g^4$.\nAt this point, the leading $T$-dependent contributions are of the\nsame order as the three-loop $T=0$ (FMcL) pressure contribution\nand remain more important than the undetermined four-loop $T=0$ term even for\nparametrically lower temperatures as long as $x<2$ (i.e.\\ $T\\gg g^2 \\mu$).\nFor the entropy and specific heat, for which the zero-temperature contribution to\nthe pressure drops out, these $T$-dependent terms represent\nthe leading interaction contributions down to arbitrarily low temperatures.\nThe $T\\,\\ln\\, T$ behavior of the entropy (as well as of the specific heat) is\ncharacterized by ``anomalous'' non-Fermi-liquid behavior, caused by the only\nweakly (dynamically) screened quasi-static magnetic interactions\nwith an effective frequency-dependent screening mass, displayed in Eq.~(\\ref{mmdyn}).\n\nAs suggested by Fig.~\\ref{fig:orders} and\nshown in detail in the previous section, the HDL-resummed\nthermal pressure contributions responsible for the\nnon-Fermi-liquid behavior at $T\\ll g\\mu$ match smoothly\nto the perturbative effects at $T\\gg g\\mu$ described by\nEQCD. As the temperature is increased, electrostatic\nscreening replaces dynamical magnetic screening\nas the dominant collective phenomenon also in the $T$-dependent\ncontributions.\nFor $T$ parametrically larger than $g\\mu$ (\\textit{i.e.~}$x<1$)\nthe resummation of HDL self energies needs to be trivially extended\nto HTL self energies to avoid accuracy loss.\nWhen added to the zero temperature ${\\mathcal O}(g^4)$ result,\nthis gives an expression that gives the pressure\nfor all temperatures and chemical potentials up to\nan error of order $g^{{\\rm min}(4+2x,6)}$ (or $g^{4+x}$ throughout\nin the case of the entropy, for which the unknown four-loop T=0 pressure\ndrops out).\n\nFrom the ``flow'' of the various perturbative\ncontributions as a function of $x$ in Fig.~\\ref{fig:orders}, one notices that a single expression aiming to be\nvalid both for $x>1$ and $x<1$ needs to keep track of\ncontributions which are perhaps higher-order and irrelevant\nin some region but essential in another.\nThe novel approach we have presented here does so\nby resumming the complete one-loop gluon self-energy in all\nIR sensitive graphs while treating the infrared-safe\n2GI diagrams perturbatively. To the extent that we have worked\nit out, this procedure covers both $x>1$ and $x<1$ with an error of order\n$g^{{\\rm min}(5+x,6)}$ which improves over the HTL\/HDL result in the\nregion $x<1$ by including the contributions of all relevant three-loop\ngraphs. A drawback compared to the HTL\/HDL resummation schemes is however\nthat the resummation of the complete gluon self-energy\nleads to gauge-dependent higher-order contributions whose unphysical\nnature is highlighted by the appearance of spacelike poles in the\nlogarithmic resummation integrand with momenta $\\sim g^2T$ and also\nof an unphysical damping constant (with an incorrect sign) $\\propto g^2 T$.\nFor our expression for the pressure, the effect of these problems is, however,\nonly of the order of the nonperturbative MQCD contributions, \\textit{i.e.}~$g^6$,\nso it has not hindered us from confirming and thus validating the\nresults obtained through dimensional reduction or the HTL\/HDL approach.\n\n\n\n\\section{Conclusions and outlook}\\label{sec:concl}\n\nIn this paper, we have constructed a novel resummation scheme designed to reproduce the weak coupling expansion of the QCD pressure up to order $g^4$\non the entire $\\mu$-$T$ plane. We have used it to provide an independent check of practically all existing perturbative results. In particular,\nwe have performed\nthe first explicit test on the validity of dimensional reduction for values of $\\mu\/T$ far beyond the capability of present-day lattice\ntechniques, thus verifying that dimensionally reduced effective theories provide a solid description of the perturbative physics up to in principle arbitrarily large values\nof $\\mu\/T$ as long as $\\pi T>m_\\rmi{D}$. At temperatures parametrically smaller than the chemical potential, we have on the other hand reproduced numerically all the results of the\nHTL\/HDL resummation schemes, verifying their validity and highlighting the smooth transition taking place in the perturbative expansion of the pressure\nas one moves from the region of dimensional reduction towards the zero-temperature limit.\n\nBased on our numerical results from Section \\ref{sec:numres},\nthe dimensional reduction result for the QCD pressure appears to be provide a remarkably good\napproximation for this quantity down to the point where\nthe $T$-dependent contribution to the interaction pressure, $\\delta p$,\nceases to be negative (cf.\\ Figs.\\ \\ref{fig:p01}ff)\nwhich happens at $T\\approx 0.2 m_\\rmi{D}$. Since the\ndimensional reduction result to order $g^6\\ln\\,g$ combined\nwith optimized choices of the renormalization scale\nhas turned out to agree rather well with\nlattice results, both at zero chemical potential\n\\cite{klry,Blaizot:2003iq} and\nfor $\\mu\\sim T$ \\cite{avpres,Ipp:2003yz}, our present findings in fact suggest a remarkably wide\npractical range of applicability for the dimensional reduction method and results.\n\nProgressing down on the temperature axis to $T\\lesssim 0.2 m_\\rmi{D}$, one eventually has to switch to the nonstatic resummation schemes provided either by\nour new approach or by the calculationally much simpler HTL resummation of Eq.~(\\ref{PHTLtot}).\nAt such low temperatures, the pressure can --- up to but not including order $g^6$ --- be approximated by the Freedman-McLerran result plus positive contributions from the\nStefan-Boltzmann terms as well as the interaction pressure $\\delta p$. The latter of these is the source of the non-Fermi-liquid behavior of the entropy and specific heat.\n\nWhile we believe to have thoroughly clarified the nature of perturbative\nexpansions of the pressure in different regimes of the $\\mu$-$T$ plane,\nour new approach is, as of today, yet to produce results for the pressure beyond what has already been achieved through either dimensional reduction at\n$x<1$, the HTL\/HDL resummation schemes at $x\\geq 1$ or the Freedman-McLerran result at $T=0$. Its present relative error of order\n$g^{{\\rm min}(5+x,6)}$ can in principle be reduced through the inclusion of the two-loop gluon polarization tensor into the resummation of the\nring diagrams and in addition by taking the contributions of non-static modes into account in the multiple sums of Fig.~2.b-d. For the pressure,\nthis would bring the accuracy of our new approach up to the one currently achieved by dimensional reduction calculations (excluding the already known\n${\\mathcal O}(g^6\\ln\\,g \\,T^2(T^2+\\mu^2))$ term), so that the error, up to logarithms, would be uniformly (for all values of $x$) of order $g^6 \\ln\\,g$,\ncorresponding to the line marked ``4-loop $T=0$ pressure'' in Fig.~\\ref{fig:orders}.\nThis would then unify all existing perturbative results for the pressure of QCD, while for the entropy it would moreover lead to genuinely\nnew results. Apart from increasing \\textit{e.g.}~the accuracy of the entropy result at $x=1\/2$ to order $g^{13\/2}$\n(green open dots in Fig.~\\ref{fig:orders})\\footnote{The highest purely perturbatively calculable order at $x=1\/2$\nis $g^{15\/2}$ which would require a calculation of the contributions of order $g^6\\mu^2 T^2$ and $g^7\\mu^3T$ for the pressure.},\nit would push the error in the $T$-dependent part of the pressure up to the line denoted in Fig.~\\ref{fig:orders} by ``4-loop T contribution''\nand thus, for $x>1$, include the so far unknown order $g^4\\mu^2 T$ corrections to the non-Fermi-liquid terms in the entropy and the specific heat.\nConsidering the difficulties caused by the gauge-dependent parts of the gluon self-energy, it seems\nthat such an extension should probably aim at keeping only gauge-independent contributions such as HTL self energies\nin the ring diagrams and treating corrections to those self-energies in a perturbative manner.\nWork towards this goal is currently in progress.\n\n\n\\acknowledgments\n\nWe are grateful to Mikko Laine, York Schr\\\"oder,\nand Larry Yaffe for their helpful\ncomments and suggestions and to Dirk Rischke for discussions\non color superconductivity. This\nwork has been partially supported by the Austrian Science Foundation\nFWF, project no.\\ P16387-N08 and the Academy of Finland, project no. 109720.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Acknowledgments}\n\\noindent \nWe would especially like to thank Jenny List for many detailed\ndiscussions regarding the \\textsc{Ilc} analysis. The work has been supported\nby the Helmholtz Alliance ``Physics at the Terascale'', the \\textsc{Dfg\nSfb\/Tr9} ``Computational Particle Physics'', and the \\textsc{Dfb Sfb\/Tr33} (`The\nDark Universe'). H.K.D. would like to thank the Aspen Center for\nPhysics where part of this work was completed.\n\n\\onecolumngrid\n\\pagebreak\n\\twocolumngrid\n\n\\pagebreak\n\n\\section{Differential Cross Section for $ \\Ppositron \\Pelectron \\rightarrow \\boldsymbol{\n \\chi \\chi} \\Pphoton$}\n\n\n\\subsection{Abbreviations}\n\\label{app:xsectterms}\nWe use the following abbreviations for the final cross section list in\nTable~\\ref{tbl:2t3crosssections}:\n\\begin{align}\n\\intertext{Polarisation prefactors:}\nC_S \\equiv 1+P^+ P^-&, \\quad C_V \\equiv 1-P^+ P^-, \\\\\nC_L \\equiv (1- P^-) ( 1+P^+)&,\\quad C_R \\equiv (1+ P^-) ( 1-P^+). \\nonumber \\\\\n\\intertext{Terms with combined couplings:}\nG_{X \\pm Y} \\equiv g_{X}^2 \\pm g_{Y}^2&,\\quad G_{XY} \\equiv g_{X} g_{Y}. \\\\\n\\intertext{Relativistic velocities:}\n\\beta \\equiv \\sqrt{\\displaystyle 1-\\frac{4 M_\\chi^2}{s}}&,\\quad \\hat{\\beta} \\equiv\n\\sqrt{\\displaystyle 1-\\frac{4 M_\\chi^2}{s(1-x)}}\\;.\n\\end{align}\nKinematical functions:\n\\begin{align}\nF_{x \\theta} &\\equiv \\frac{\\alpha}{\\pi} \\frac{(x-1)^2+1}{x \\sin^2 \\theta}\\;, \\\\\nV_{x \\theta} &\\equiv \\frac{x^2\\cos(2 \\theta) + (3x -8)x + 8 }{4 \\left((x-1)^2 +\n 1\\right)}\\;.\n\\end{align}\n\nWe show terms that arise in the analytical evaluation of the differential photon cross\nsection in $\\Ppositron \\Pelectron \\rightarrow \\chi \\chi \\Pphoton$ but not in\nthe Weizs\\\"acker--Williams approximation in (\\ref{eq:analyticfirst})-(\\ref{eq:analyticlast}). They all vanish in the soft--photon\nlimit $x \\rightarrow 0$.\n\\begin{widetext}\n\\begin{align}\nA_{SF} &= \\frac{ \\left(1 - V_{x \\theta} \\right)}{4 M^2_\\Omega}\n\\frac{\\hat{s}}{1-x} \\left[(g_s+g_p)^4 C_R + (g_s-g_p)^4 C_L \\right] \\label{eq:analyticfirst}\\\\\nA_{SFr} &= \\frac{\\alpha}{8 \\pi}\n\\frac{\\hat{s}}{M_\\Omega^2} \\frac{x}{1-x} \\left[(g_s+g_p)^4 C_R + (g_s-g_p)^4 C_L \\right] \\\\\nA_{FtS} &= \\frac{(1-V_{x \\theta})}{4} \\left[ C_S (\\hat{s} - 4 M_\\chi^2) + \\frac{1}{1 - x} C_S (2 M_\\chi^2 + \\hat{s}) \\right] \\\\\nA_{VF} & = 20 G^2_{lr} C_S (1-V_{x \\theta}) \\frac{x }{1-x} (\\hat{s}^2 + 4\nM_\\chi^2 \\hat{s}-8 M_\\chi^4 )+ \\frac{(g_l^4 C_L + g_t^4 C_R)}{M_\\Omega^2 }\n\\Big[ -\\frac{1}{32}\\frac{x^4 \\sin^2(2 \\theta) \\hat{s} ( 3 \\hat{s}^2 + 26 M_\\chi^2 \\hat{s} - 32 M_\\chi^4)}{(x-1)^2\n ((x-1)^2+1)} \\nonumber \\\\\n& \\ + 6 \\frac{x}{((x-1)^2+1)} \\hat{s} ( \\hat{s}^2 + 7 M_\\chi^2 \\hat{s} - 24\nM_\\chi^4 ) - \\frac{1}{4} (1-V_{x \\theta}) (21 \\hat{s}^3 + 282 M_\\chi^2\n\\hat{s}^2 - 1144 M_\\chi^4 \\hat{s} + 160 M_\\chi^6) \\nonumber \\\\\n& \\ +\\frac{3}{2}\\frac{(1-V_{x \\theta})}{(1-x)} \\hat{s} (\\hat{s}^2-28 M_\\chi^2\n\\hat{s}+ 16 M_\\chi^4) +\\frac{1}{4} \\frac{(1-V_{x \\theta})}{(1-x)^2} \\hat{s} (7\n\\hat{s}^2-126 M_\\chi^2 \\hat{s}+ 32 M_\\chi^4) \\left. +\\frac{(1-V_{x\n \\theta})}{(1-x)^3} \\hat{s} (\\hat{s}^2 +2 M_\\chi^2 \\hat{s}+6 M_\\chi^4)\n\\right] \\\\ \nA_{VFr} & = \\frac{(g_l^4 C_L + g_r^4 C_R)}{M_\\Omega^2 } \\Big[ - \\frac{1}{32} \\frac{x^4 \\sin^2(2 \\theta)}{(x-1)^2\n ((x-1)^2+1)} \\hat{s}( \\hat{s}^2 + 32 M_\\chi^2 \\hat{s} -24 M_\\chi^4 ) + 2 \\frac{x}{((x-1)^2+1)} \\hat{s} ( \\hat{s}^2 + 12 M_\\chi^2 \\hat{s} + 56 M_\\chi^4 ) \\nonumber \\\\\n&\\ - \\frac{1}{4}(1-V_{x \\theta}) (7\\hat{s}^3 + 144 M_\\chi^2 \\hat{s}^2 -168\nM_\\chi^4 \\hat{s}+1280 M_\\chi^6) +\\frac{1}{2}\\frac{(1-V_{x \\theta})}{(1-x)} \\hat{s} (\\hat{s}^2 -48 M_\\chi^2 \\hat{s}+ 56 M_\\chi^4)\\nonumber \\\\\n&\\ +\\frac{1}{4}\\frac{(1-V_{x \\theta})}{(1-x)^2} \\hat{s} (9 \\hat{s}^2 -272\nM_\\chi^2 \\hat{s}+104 M_\\chi^4) \\left. +2 \\frac{(1-V_{x \\theta})}{(1-x)^3}\n \\hat{s} (\\hat{s}^2+2 M_\\chi^2 \\hat{s}+ 6 M_\\chi^4) \\right] \\,.\\label{eq:analyticlast}\n\\end{align}\n\\end{widetext}\n\n\\section{Astrophysical constraints}\n\\label{sec:astro}\n\nAny model which aims to describe dark matter, for example through a\n\\textsc{Wimp}, has to agree with present data. It has to give the correct relic\nabundance, and must be consistent with the bounds from direct and\nindirect detection\nsearches \\cite{Komatsu:2010fb,Aprile:2012nq,Adriani:2008zr}.\n\\allowdisplaybreaks\n\\subsection{The Relic Abundance}\nWe first consider the best measurement of the relic abundance from\n\\textsc{Wmap}-7 \\cite{Komatsu:2010fb},\n\\begin{equation}\n\\Omega^{\\text{DM}} h^2 = 0.1099 \\pm 0.0056\\,.\n\\end{equation}\nWe employ the solution of the model dependent Boltzmann equation obtained\nin \\cite{Beltran:2008xg},\n\\begin{subequations}\n\\begin{align}\n\\Omega^\\text{DM}_0 h^2 \\approx 1.04\\cdot10^9 \\,\\GeV^{-1} \\frac{x_f}\n{m_\\text{Pl} \\sqrt{g_*(x_f)} (a + 3b\/x_f)}\\;. \\label{eqn:Omega0}\n\\end{align}\nHere $m_{\\mathrm{Pl}}$ is the Planck mass. $x_f=M_\\chi\/T_f$ is the\ninverse freeze--out temperature, $T_f$, rescaled by the \\textsc{Wimp} mass,\n$M_\\chi$. It is implicitly given by the equation,\n\\begin{align}\nx_f = \\text{ln} \\left[ c (c+2) \\sqrt{\\frac{45}{8}} \\frac{1}{2 \\pi^3} \n\\frac{g\\ m_\\text{pl} M_\\chi (a + 6b\/x_f)}{\\sqrt{x_f} \\sqrt{g_*(x_f)}} \n\\right]. \\label{eqn:x0}\n\\end{align}\n\\end{subequations}\n$g_*(x_f)$ denotes the relativistic degrees of freedom in equilibrium\nat freeze-out and is given in Ref.~\\cite{Coleman:2003hs}. $a$ and $b$\nare the first two coefficients of the non-relativistic\nexpansion of the thermally averaged annihilation cross section,\n\\begin{equation}\n \\langle\\sigma v \\rangle \\approx a + b v^{2} + O(v^{4}),\n\\end{equation}\nwhere $v$ is the relative velocity of the colliding particles. Here the center-of-mass energy squared is approximated by \\cite{Zheng:2010js,Yu:2011by},\n\\begin{equation}\n s \\approx 4 M_{\\chi}^2+ M_{\\chi}^2 v^2 +3\/4 ~ M_{\\chi}^2 v^4.\n\\end{equation}\n$g$ are the internal degrees of freedom of the \\textsc{Wimp}. $c$ is an order unity parameter which is determined numerically in the solution of the Boltzmann equation and we set this parameter to 0.5.\n\nInstead of testing all the models presented in\nTable~\\ref{tbl:allmodels}, we shall focus on a few exemplary cases.\nFirst, the relic density depends on the possible Standard Model\nparticles, $f$, the \\textsc{Wimp}s can annihilate into $\\chi \\overline{\\chi}\n\\rightarrow f \\overline{f}$. We shall consider two cases for the set of particles $f$:\n(i) all leptons, (ii) all SM fermions.\nSecond, two variants of couplings are tested. In one scenario all SM\nparticles couple via the mediator to the \\textsc{Wimp} with the same strength;\nthis is called \\textit{universal coupling}. In the other they have a\ncoupling proportional to their mass, which we call \\textit{Yukawa-like\ncoupling}. In the cases where we have the same effective operator our\nresults agree with Refs.~\\cite{Zheng:2010js,Yu:2011by}, up to the\nnormalisation (see Appendix \\ref{sec:sigmarelic}).\n\n\nIn order to set constraints, we must determine the total relic\ndensity, which is the sum of the relic density of the particle and the\nanti-particle (if the latter exists). This means the relic density for\na complex particle-pair is two times the density of a real particle.\nIf we consider the \\textsc{Wmap} result as an upper bound on the relic density,\ni.e.\\ allowing for other dark matter, then this corresponds to\na lower bound on the effective coupling of the \\textsc{Wimp} to the SM\nparticles. If we require our \\textsc{Wimp} to be the only dark matter, we shall also obtain an\nupper bound on the effective coupling.\n\nThe strict interpretation that our model only contains a heavy\nmediator and a single \\textsc{Wimp} ensures that there are no\nresonances or co-annihilations. However we also note that in many\nfull theories that contain dark matter, a `co-annihilation' regime can\nexist that can significantly alter the relic density in the\nuniverse. Whilst the co-annihilation mechanism cannot be incorporated\ninto the strict definition of our model, it may actually have no\nobservable effect on the collider based phenomenology. An example of\nsuch a feature could be stau co-annihilation in \\textsc{Susy} that would not\nchange the \\textsc{Ilc} production process of the lightest supersymmetric\nparticle. Another example is that a more complicated model may contain\nresonant annihilations. Both of these examples can significantly\nweaken the relic abundance bounds.\n\n\n\\subsection{Direct Detection}\nWe shall also impose bounds on our operators from the direct detection\nsearches for \\textsc{Wimp} dark matter. The experiments are designed to measure\nthe recoil energy from the scattering between a (dark matter halo)\n\\textsc{Wimp} and the target nucleus. The interactions are difficult to detect\nsince the energy deposited is quite small, 1 to \\unit{100}{\\keV},\n\\cite{Bertone:2004pz}. These experiments give an upper limit for the \ncross section between the dark matter and the nucleus of the\ntarget. One drawback is that in the cases where the \\textsc{Wimp} does not\ncouple to quarks, the coupling can only occur through loop diagrams.\n\nThe direct detection experiments give a much stronger bound on spin\nindependent (SI) interactions than on spin dependent (SD). The reason\nis that in the SI case the interaction with all nucleons add\ncoherently which enhances the corresponding cross section by the\natomic number squared. However, the spins of the nucleons cancel if\nthey are paired. Thus SD interactions are only enhanced for very\nspecial nuclei.\n\nThe SI interactions are scalar or vector interactions in the\n$s$-channel, the axialvector and tensor interactions in the\n$s$-channel give a SD interaction. Note that due to the low\nkinetic energy of the \\textsc{Wimp}s the cross section should be\ncomputed in the non-relativistic limit. In that case the pseudoscalar\ninteraction, $\\overline{\\psi} \\gamma ^5 \\psi$, vanishes.\n\nThe $t\/u$--channel diagrams are cast into a sum of $s$--channel diagrams\nvia the Fierz identities. From this only the SI parts are employed, since any SD\ncontribution is negligibly small.\nTensor interactions occur only via the Fierz identities, since we do\nnot consider fundamental tensor interactions. However, since Fierz\n identities will always give at least one SI contribution, tensor\n terms can be dropped. \n\nFor the SI interactions we shall consider the limits set by the \\textsc{Xenon100} experiment \\cite{Aprile:2012nq}. These are the most recent and set\nthe strictest limits over a broad parameter range. For the SD\ninteractions we consider the \\textsc{Xenon}10 data \\cite{Angle:2008we} since\n\\textsc{Xenon}100 gives no statement on SD interactions. The smaller data \nset along with the physical reasons mentioned above lead to a bound \nthat is $\\sim 10^6$ times weaker than for the SI interactions. The \ncalculations for the \\textsc{Wimp}--nucleus cross sections follow \nRef.~\\cite{Agrawal:2010fh} and for identical models\nwe find the same results. See Appendix \\ref{sec:directdetect} for the\ncomplete list of cross sections.\n\n\n\\subsection{Indirect Detection}\n\nWe also consider the indirect detection searches for dark\nmatter. These are much more model dependent, as the dark matter is\nseen via an agent, for example neutrinos, which could also be produced\nvia other means. Specifically we shall consider the \\textsc{Pamela} experiment\n\\cite{Adriani:2008zr} which measured an excess of positrons. These\ncould potentially originate from dark matter annihilation. To\nimplement this we need to compute the propagation of the produced\npositrons and electrons from the source to the earth. This is\ndescribed by the diffusion--loss equation \\cite{Baltz:1998xv},\n\\begin{equation}\n\\frac{\\partial \\psi}{\\partial t} - \\nabla [K(\\textbf{x},E) \\nabla \n\\psi ] - \\frac{\\partial}{\\partial E} [b(E) \\psi] = q(\\textbf{x},E).\n\\label{diff_loss}\n\\end{equation}\nHere $\\psi(x,E) = \\mathrm{d} n_{e+}\/ \\mathrm{d}E$ is the positron\ndensity per energy. $K(x,E)$ is the diffusion coefficient which\ndescribes the interaction with the galactic magnetic field. $b(E)$\ndenotes the energy loss due to synchrotron emission and inverse\nCompton scattering. $q(x,E)$ is the source term due to dark matter\nannihilation. We note that convection and re--acceleration terms are\nignored as these do not apply to positrons \\cite{Delahaye:2008ua}.\n\nWe use the conventional formalism\n\\cite{Delahaye:2007fr,Perelstein:2010fq} to derive a solution of\nEq.~(\\ref{diff_loss}). It is also possible to use the so-called\nextended formalism that takes the corrections from sources in the free\npropagation zone into account as well as those from the diffusion\nzone. However, this increases the runtime of the calculation\nconsiderably while only giving a small correction that is less than\nthe measurement error. To perform the numerical comparison we use the\ncored isothermal dark matter density profile \\cite{Bahcall:1980fb} and\nthe galactic propagation model M2 \\cite{Delahaye:2007fr}.\n\n\nThe above choices result in the following positron flux,\n\\begin{align}\n\\Phi_{e^+}(E)&=\\frac{\\beta_{e+}}{4 \\pi} \\psi(r_{\\odot},z_{\\odot},E), \\\\\n\\psi(r,z,E) &=\\frac{\\tau_{E}}{\\epsilon^2} \\int^{\\epsilon_{max}}_{\\epsilon} d \\epsilon_S f(\\epsilon_S) I(r,z,\\epsilon,\\epsilon_S), \\\\\nI(r,z,\\epsilon,\\epsilon_S) &= \\sum_i \\sum_n J_0(\\frac{\\alpha_i r}{R}) \\sin{\\frac{n \\pi (z+L)}{2L}} \\nonumber \\\\\n&\\hspace{1.8cm}\\times \\exp{(-\\omega_{i,n} (t-t_S))}\t R_{i,n}, \\\\\n\\omega_{i,n}&= K_0 [ (\\frac{\\alpha_i}{R})^2+ (\\frac{n \\pi}{2L})^2].\n\\end{align}\n\nHere $\\tau_E,~R,~K_0,~L$ are parameters which describe the\nM2 propagation model. They are set to the standard choices \\cite{Delahaye:2007fr,Perelstein:2010fq}\n$\\tau_E = \\unit{\\power{10}{16}}{\\second}$, $R=\\unit{20}{\\kilo \\text{pc}}$ as well as to the M2 propagation model\n$L=\\unit{1}{\\kilo \\text{pc}}$, $K_0 = \\unit{0.00595}{\\kilo \\text{pc$^2$\/Myr}}$, $\\delta=0.55$. $f(\\epsilon)$ is\nthe energy distribution of the positrons from the annihilation and is\ngenerated with \\textsc{Pythia}8 \\cite{Sjostrand:2007gs}.\n$R_{i,n}$ are the coefficients of the Bessel-Fourier expansion of $R(r,z)$,\n\\begin{align}\nR(r,z)&\\equiv \\eta \\langle\\sigma v \\rangle \\left[\\frac{\\rho(r,z)}{M_{\\chi}}\\right]^2, \\\\\n\\rho(r,z) &= \\rho_{\\odot}(\\frac{r_{\\odot}}{r})^{\\gamma}\\left[\\frac{1+(r_{\\odot}\/r_s)^{\\alpha}}{1+(r_{\\odot}\/r)^{\\alpha}}\\right] ^{(\\beta-\\gamma)\/\\alpha}.\n\\end{align}\nHere $\\langle \\sigma v \\rangle$ is the thermally averaged annihilation\ncross section. We include all possible final states, not just those\nresulting in positrons. Furthermore $\\eta= 1\/2$ for real\nparticles and 1\/4 for complex particles. $ r_{\\odot}=\\unit{8.5}{\\kilo \\text{pc}}$ is the\ndistance of the solar system from the galactic center. $\\rho_{\\odot}=\n\\unit{0.3}{\\GeV\\per\\centi\\meter\\cubed}$ is the local dark matter density and $\\alpha=\\beta=2,\n~\\gamma=0, ~ r_S =\\unit{5}{\\kilo \\text{pc}}$ are chosen according to the cored isothermal\ndark matter density distribution \\cite{Delahaye:2007fr,Perelstein:2010fq}.\n\n\\textsc{Pamela} measures the ratio $\\Phi_{e^+}\/(\\Phi_{e^-}+\\Phi_{e^+})$,\nwhere the fluxes, $\\Phi_{e^\\pm}$, contain the flux from dark matter\nannihilation and from any astrophysical background. The background we\ntake is \\cite{Baltz:1998xv},\n\\begin{subequations}\n\\begin{align}\n\\dfrac{d \\Phi_{e^- bg}}{dE} &=\\left(\\frac{0.16 \\epsilon^{-1.1}}{1+11 \\epsilon^{0.9}+3.2 \\epsilon^{2.15}}\\right. \\nonumber \\\\\n&\\hspace{-1.1cm}+\\left.\\frac{0.7 \\epsilon^{0.7}}{1+110 \\epsilon^{1.5}+600 \\epsilon^{2.9}+580 \\epsilon^{4.2}}\\right)\n\\mathrm{GeV}^{-1} \\mathrm{cm}^{-2} \\mathrm{s}^{-1} \\mathrm{sr}^{-1}, \\\\\n\\dfrac{d \\Phi_{e^+ bg}}{dE} &=\\frac{4.5 \\epsilon^{0.7}}{1+650 \\epsilon^{2.3}+1500 \\epsilon^{4.2}} \n\\mathrm{\\GeV}^{-1} \\mathrm{cm}^{-2} \\mathrm{s}^{-1} \\mathrm{sr}^{-1}, \\\\[3mm]\n\\epsilon &\\equiv E\/ \\mathrm{GeV}. \\nonumber\n\\end{align}\n\\end{subequations}\nThe quantity we compare to \\textsc{Pamela} is,\n\\begin{equation}\n\\frac{\\Phi_{e^+}}{\\Phi_{e^+}+\\Phi_{e^-}} = \\frac{\\Phi_{e^+ \\chi} +\\Phi_{e^+ bg}}{\\Phi_{e^+ bg} + \\Phi_{e^+ \\chi} + \\Phi_{e^- \\chi} +\\Phi_{e^- bg}},\n\\end{equation} \nand we note that $\\Phi_{e^+ \\chi }= \\Phi_{e^- \\chi}$. \n\nWe find an upper bound on the annihilation cross section by assuming that all of the excess comes from dark matter. However, it is possible that other background sources contribute and thus we also allow models that produce a flux smaller than the one seen.\n\nWe also note that for dark matter masses above $\\sim$\\unit{1}{\\TeV}, the \\textsc{Fermi--Lat} \\cite{Atwood:2009ez} experiment may provide competitive bounds from inverse Compton scattering \\cite{Cirelli:2009vg,Bernal:2010ip}. However, since we are only interested in models that can be probed at the \\textsc{Ilc} we ignore them here.\n\nThe \\textsc{IceCube} collaboration also sets limits on heavier dark matter masses via annihilations into neutrino final states \\cite{Abbasi:2012ws,IceCube:2011aj}. In addition these bounds may be competitive for spin dependent interactions but we do not consider the limits in this study.\n\n\n\n\n\\section{Cross Sections for Annihilation} \\label{sec:sigmarelic}\n\\allowdisplaybreaks We give the full cross sections for annihilation\nof a pair of dark matter particles with mass $M_\\chi$ into a\npair of Standard Model fermions with mass $m_f$. To find the\nexpansion coefficients in $\\sigma v \\approx a + b v^2$, we perform the\nnon--relativistic approximation $s \\approx 4 M_{\\chi}^2+ M_{\\chi}^2\nv^2 + \\frac{3}{4} M_{\\chi}^2 v^4$ \\cite{Beltran:2008xg}. Note that in order to\nfind the correct result for the $v^2$ term in $\\sigma v$, it is\nnecessary to expand up to order $v^4$ because of the appearance of\n$\\sqrt{s}$ in the cross section formul{\\ae}.\n\nThe total cross section is then given as the sum of the cross\n sections over all allowed final state fermions. This set is\nrestricted both by kinematics ($m_f \\leq M_\\chi$) and by the assumed\nmodel. The latter also determines whether the coupling $G_f$ is\nuniversal or particle--dependent.\n\nWe define the mass ratio $\\xi\\equiv m_{f}\/M_{\\chi}$ and the velocities of both\nparticles $\\beta_X \\equiv \\sqrt{1-4m_X^2\/s}$ to compactify the following expressions.\n\nSome of our effective operators have been analysed before, for example \\cite{Zheng:2010js,\n Yu:2011by}, and we agree with the respective results for the annihilation\n cross sections.\n\\subsection{Scalar \\textsc{Wimp}}\n\\vspace{-1cm}\n\\begin{flalign}\n\\sigma^{\\text{SS}}_{\\text{Sc}} &= \\frac{G_{f}^{2}}{8 \\pi s} \\frac{\\beta_f}{\\beta_\\chi}(s-4 m_{f}^{2}), &\\\\*\n\\sigma v &\\approx \\frac{G_{f}^{2}}{4 \\pi} \\sqrt{1-\\xi^{2}} \\Big[ (1-\\xi^{2}) + \\dfrac{v^{2}}{8}(5 \\xi^{2}-2)\\Big].&\\\\\n\\sigma^{\\text{SS}}_{\\text{Ps}} &= \\frac{G_{f}^{2}}{8 \\pi } \\frac{\\beta_f}{\\beta_\\chi}, &\\\\*\n\\sigma v &\\approx \\frac{G_{f}^{2}}{4 \\pi} \\Big[ \\sqrt{1-\\xi^{2}} + \\dfrac{v^{2}}{8}\\dfrac{3 \\xi^2 -2}{\\sqrt{1-\\xi^{2}}}\\Big].&\\\\\n\\sigma^{\\text{SV}}_{\\text{Vec}} &= \\frac{G_{f}^{2}}{12 \\pi} \\beta_f \\beta_\\chi (s+2 m_{f}^{2}), &\\\\*\n\\sigma v &\\approx \\frac{G_{f}^{2}}{12 \\pi} \\Big[ M_{\\chi}^{2} v^{2}\\sqrt{1-\\xi^{2}}(\\xi^{2}+2)\\Big].&\\\\\n\\sigma^{\\text{SV}}_{\\text{Ax}} &= \\frac{G_{f}^{2}}{12 \\pi } \\beta_f \\beta_\\chi (s-4 m_{f}^{2}), &\\\\*\n\\sigma v &\\approx \\frac{G_{f}^{2}}{6 \\pi} \\Big[ M_{\\chi}^{2}\nv^{2}(1-\\xi^{2})^{3\/2}\\Big]. &\\\\\n\\sigma^{\\text{SV}}_{\\text{Ch}} &= \\frac{G_{f}^{2}}{24 \\pi } \\beta_f\\beta_\\chi(s-m_f^2), &\\\\\n\\sigma v &\\approx \\frac{G_{f}^{2}M_{\\chi}^2}{48 \\pi} v^2\\sqrt{1-\\xi^{2}} (4-\\xi^2). &\\\\\n\\sigma^{\\text{SF}}_{\\text{Sc\/Ps}} &= \\frac{G_{f}^{2}}{48 \\pi s}\n\\frac{ \\beta_f}{ \\beta_\\chi} \\Big[ 2 s(4 m_{f}^{2}-2\nM_{\\chi}^{2}+3 M_{\\Omega}^{2} \\mp 6 m_{f} M_{\\Omega}) \\nonumber &\\\\ &\\qquad -8\nm_{f}^{2} \\Big(3( M_{\\Omega} \\mp m_{f})^{2}+M_{\\chi}^{2}\\Big) +s^{2} \\Big], &\\\\\n\\sigma v &\\approx \\frac{G_{f}^{2}}{4 \\pi} \\sqrt{1-\\xi^{2}} \\Big[ (1-\\xi^{2})(\\xi M_{\\chi} \\mp M_{\\Omega})^{2}. \\nonumber &\\\\*\n&\\qquad + \\dfrac{v^{2}}{24} \\Big((15 \\xi^{2}\n-6) M_{\\Omega}^{2} \\mp 6 \\xi(5 \\xi^{2}-2) M_{\\chi} M_{\\Omega}\\nonumber &\\\\\n&\\qquad +(15 \\xi^{4} -4 \\xi^{2}+4) M_{\\chi}^{2}\\Big)\\Big],&\\\\\n\\sigma^{\\text{SFr}}_{\\text{Sc\/Ps}} &= \\frac{G_{f}^{2}}{2\\pi s} \\frac{ \\beta_f^3}{ \\beta_\\chi} (m_{f} \\mp M_{\\Omega})^{2} &\\\\*\n\\sigma v &\\approx \\frac{G_{f}^{2}}{\\pi} \\sqrt{1-\\xi^{2}}^3 (\\xi M_{\\chi} \\mp\nM_{\\Omega})^{2} \\nonumber &\\\\ & \\qquad \\times \\Big[ 1\n+ \\dfrac{v^{2}}{8} (5 \\xi^{2}-2)\\Big].\n\\end{flalign}\n\\subsection{Fermion \\textsc{Wimp}}\n\\vspace{-1cm}\n\\begin{flalign}\n\\sigma^{\\text{FS}}_{\\text{Sc}} &= \\frac{G_{f}^{2}}{16 \\pi } \\beta_f \\beta_\\chi(s-4 m_{f}^{2}), &\\\\\n\\sigma v &\\approx \\frac{G_{f}^{2}}{8 \\pi} v^{2} M_{\\chi}^{2} (1-\\xi^{2})^{3\/2}.&\\\\\n\\sigma^{\\text{FS}}_{\\text{Ps}} &= \\frac{G_{f}^{2}}{16 \\pi } \\frac{ \\beta_f}{ \\beta_\\chi}s, &\\\\\n\\sigma v &\\approx \\frac{G_{f}^{2} M_{\\chi}^{2}}{2 \\pi} \\Big[ \\sqrt{1-\\xi^{2}} + \\dfrac{v^{2}}{8}\\dfrac{\\xi^2 }{\\sqrt{1-\\xi^{2}}}\\Big].&\\\\\n\\sigma^{\\text{FV}}_{\\text{Vec}} &= \\frac{G_{f}^{2}}{12 \\pi s} \\dfrac{ \\beta_f}{ \\beta_\\chi}(s+2 M_{\\chi}^{2}) (s+2 m_{f}^{2}), &\\\\\n\\sigma v &\\approx \\frac{G_{f}^{2} M_{\\chi}^{2} }{2 \\pi} \\Big[\n\\sqrt{1-\\xi^{2}}(2+\\xi^{2}) \\nonumber &\\\\ &\\qquad +v^{2}\\dfrac{-4+2 \\xi^{2}+11 \\xi^{4}}{24 \\sqrt{1-\\xi^{2}}}\\Big].&\\\\\n\\sigma^{\\text{FV}}_{\\text{Ax}} &= \\frac{G_{f}^{2}}{12 \\pi s} \\dfrac{\n \\beta_f}{ \\beta_\\chi}\\Big[s\\Big(s-4(m_{f}^{2}+M_{\\chi}^{2})\\Big) \\nonumber &\\\\\n& \\qquad \\qquad +28 m_{f}^{2} M_{\\chi}^{2}\\Big], &\\\\\n\\sigma v &\\approx \\frac{G_{f}^{2} M_{\\chi}^{2} }{2 \\pi} \\Big[\n\\sqrt{1-\\xi^{2}}\\xi^{2} \\nonumber &\\\\\n& \\qquad \\qquad + v^{2}\\dfrac{8-28 \\xi^{2}+23 \\xi^{4}}{24 \\sqrt{1-\\xi^{2}}}\\Big].&\\\\\n\\sigma^{\\text{FV}}_{\\text{Ch}} &= \\frac{G_{f}^{2}}{48 \\pi s} \\dfrac{\n \\beta_f}{ \\beta_\\chi}\\Big(s(s-m_f^2+M_{\\chi}^2) \\nonumber &\\\\ & \\qquad \\qquad +4 m_f^2 M_{\\chi}^2\\Big), &\\\\\n\\sigma v &\\approx \\frac{G_{f}^{2} M_{\\chi}^{2} }{8 \\pi} \\Big[\n\\sqrt{1-\\xi^{2}} \\nonumber &\\\\\n& \\qquad \\qquad + v^{2}\\dfrac{(2 -\\xi^2 +2 \\xi^4)}{24\n \\sqrt{1-\\xi^{2}}}\\Big]. &\\\\\n\\sigma^{\\text{FVr}}_{\\text{Ch}} &= \\frac{G_{f}^{2}}{24 \\pi s} \\frac{ \\beta_f}{\n \\beta_\\chi}\\Big((s-4 M_{\\chi}^{2})(s-m_f^2) \\nonumber &\\\\ & \\qquad + 6 m_f^2 M_{\\chi}^2 \\Big), &\\\\\n\\sigma v &\\approx \\frac{G_{f}^{2} M_{\\chi}^{2} }{4\\pi} \\Big[ \\xi^2\n\\sqrt{1-\\xi^2} \\nonumber &\\\\ & \\qquad + v^{2} \\frac{16-32 \\xi^2 +19 \\xi^4}{24 \\sqrt{1-\\xi^{2}}} \\Big]. &\\\\\n\\sigma^{\\text{FtS}}_{\\text{Sc\/Ps}} &= \\frac{G_{f}^{2}}{48 \\pi s} \\dfrac{\n \\beta_f}{ \\beta_\\chi}\\Big(s(s-M_{\\chi}^{2}) \\mp 6 m_{f} M_{\\chi}s \\nonumber &\\\\ &\n\\qquad \\qquad + m_{f}^{2}(16 M_{\\chi}^{2}-s)\\Big), &\\\\*\n\\sigma v &\\approx \\frac{G_{f}^{2} M_{\\chi}^{2}}{8 \\pi} (1 \\mp \\xi)^{2}\n\\Big[ \\sqrt{1-\\xi^{2}} \\nonumber &\\\\ & \\qquad + v^{2} \\dfrac{2 \\pm 16 \\xi+17\n \\xi^{2}}{24 \\sqrt{1-\\xi^{2}}}\\Big].&\\\\\n\\sigma^{\\text{FtSr}}_{\\text{Sc\/Ps}} &= \\frac{G_{f}^{2}}{96 \\pi s} \\dfrac{\n \\beta_f}{ \\beta_\\chi}\\Big(5 s^{2}+80 m_{f}^{2} M_{\\chi}^{2} \\nonumber &\\\\ &\n\\qquad -2s(7 m_{f}^{2}+7 M_{\\chi}^{2}\\mp 6 m_{f} M_{\\chi})\\Big), &\\\\\n\\sigma v &\\approx \\frac{G_{f}^{2} M_{\\chi}^{2}}{8 \\pi}(1 \\pm \\xi)^{2} \\Big[\n\\sqrt{1-\\xi^{2}} \\nonumber &\\\\ &\\qquad + v^{2} \\dfrac{14\\mp 40 \\xi+29\n \\xi^{2}}{24 \\sqrt{1-\\xi^{2}}}\\Big].&\\\\\n\\sigma^{\\text{FtV}}_{\\text{Vec\/Ax}} &= \\frac{G_{f}^{2}}{24 \\pi s} \\dfrac{\n \\beta_f}{ \\beta_\\chi}\\Big(s(4s-7M_{\\chi}^{2}) \\pm 6 m_{f} M_{\\chi}s \\nonumber &\\\\\n& \\qquad \\qquad - m_{f}^{2}(7s-40 M_{\\chi}^{2})\\Big), &\\\\*\n\\sigma v &\\approx \\frac{G_{f}^{2} M_{\\chi}^{2}}{4 \\pi} \\Big[(3 \\pm 2\n\\xi+\\xi^{2}) \\sqrt{1-\\xi^{2}} \\nonumber &\\\\ & \\qquad + v^{2} \\dfrac{14 \\mp 12\n \\xi -31 \\xi^{2} \\pm 18 \\xi^3+29 \\xi^{4}}{24 \\sqrt{1-\\xi^{2}}}\\Big]. \\hspace{-2cm} &\\\\\n\\sigma^{\\text{FtVr}}_{\\text{Vec\/Ax}} &= \\frac{G_{f}^{2}}{12 \\pi s} \\dfrac{\\beta_f}{\\beta_\\chi}\\Big(7 s^{2}+76 m_{f}^{2} M_{\\chi}^{2} \\nonumber &\\\\\n& \\qquad -4s(4 m_{f}^{2}+4 M_{\\chi}^{2} \\pm 3 m_{f} M_{\\chi})\\Big), &\\\\\n\\sigma v &\\approx \\frac{G_{f}^{2} M_{\\chi}^{2}}{2 \\pi} \\Big[(2 \\mp \\xi)^{2}\n\\sqrt{1-\\xi^{2}} \\nonumber &\\\\ & \\qquad + v^{2} \\dfrac{32 \\pm 24 \\xi-64 \\xi^{2} \\mp 36 \\xi^3 +47\\xi^{4}}{24 \\sqrt{1-\\xi^{2}}}\\Big]. \\hspace{-2cm} &\\\\\n\\sigma^{\\text{FtV}}_{\\text{Ch}} &= \\frac{G_{f}^{2}}{48 \\pi s} \\dfrac{\n \\beta_f}{ \\beta_\\chi}\\Big(4 m_f^2 M_{\\chi}^2 + s(s-m_f^2-M_\\chi^2)\\Big), &\\\\\n\\sigma v &\\approx \\frac{G_{f}^{2} M_{\\chi}^{2} }{8 \\pi} \\Big[\n\\sqrt{1-\\xi^{2}} + v^{2}\\dfrac{(2 -\\xi^2 +2 \\xi^4)}{24\n \\sqrt{1-\\xi^{2}}}\\Big]. \\hspace{-2cm} &\\\\\n\\sigma^{\\text{FtVr}}_{\\text{Ch}} &= \\frac{G_{f}^{2}}{24 \\pi s} \\frac{\\beta_f}{\\beta_\\chi}\\Big((s-4 M_{\\chi}^{2})(s-m_f^2) \\nonumber &\\\\ & \\qquad \\qquad + 6 m_f^2 M_{\\chi}^2\\Big), &\\\\\n\\sigma v &\\approx \\frac{G_{f}^{2} M_{\\chi}^{2} }{4\\pi} \\Big[ \\xi^2\n\\sqrt{1-\\xi^2} \\nonumber &\\\\ & \\qquad \\qquad +v^{2} \\frac{16-32 \\xi^2 +19 \\xi^4}{24 \\sqrt{1-\\xi^{2}}}\n\\Big] .\n\\end{flalign}\n\\subsection{Vector \\textsc{Wimp}}\n\\vspace{-1cm}\n\\begin{flalign}\n\\sigma^{\\text{VS}}_{\\text{Sc}} &= \\frac{G_{f}^{2}}{288 M_{\\chi}^{4} \\pi s}\\frac{ \\beta_f}{ \\beta_\\chi}(s-4 m_{f}^{2}) \\nonumber \\\\\n& \\qquad \\times (12 M_{\\chi}^{4} +s^{2}-4 M_{\\chi}^{2} s), \\\\\n\\sigma v &\\approx \\frac{G_{f}^{2}}{12 \\pi}\\sqrt{1-\\xi^{2}} \\Big[ (1-\\xi^{2})\n\\nonumber \\\\ & \\qquad \\qquad + \\dfrac{v^{2}}{24}(2+7 \\xi^2)\\Big].\\\\\n\\sigma^{\\text{VS}}_{\\text{Ps}} &= \\frac{G_{f}^{2}}{288 M_{\\chi}^{4} \\pi } \\frac{ \\beta_f}{ \\beta_\\chi}(12 M_{\\chi}^{4} +s^{2}-4 M_{\\chi}^{2} s), \\\\\n\\sigma v &\\approx \\frac{G_{f}^{2}}{12 \\pi}\\sqrt{1-\\xi^{2}} \\Big[ 1 + \\dfrac{v^{2}}{24}\\dfrac{2+\\xi^{2}}{1-\\xi^{2}}\\Big].\\\\\n\\sigma^{\\text{VV}}_{\\text{Vec}} &= \\frac{G_{f}^{2}}{432 \\pi M_{\\chi}^{4} }\n\\beta_f \\beta_\\chi (s+2 m_{f}^{2}) \\nonumber \\\\ & \\qquad \\times (s^{2}+20 M_{\\chi}^{2} s +12 M_{\\chi}^{4}), \\\\\n\\sigma v &\\approx \\frac{G_{f}^{2}}{4 \\pi} M_{\\chi}^{2} v^{2}\\sqrt{1-\\xi^{2}}(\\xi^{2}+2).\\\\\n\\sigma^{\\text{VV}}_{\\text{Ax}} &= \\frac{G_{f}^{2}}{432 \\pi M_{\\chi}^{4} }\n\\beta_f \\beta_\\chi (s-4 m_{f}^{2}) \\nonumber \\\\ \n& \\qquad \\times (s^{2}+20 M_{\\chi}^{2} s +12 M_{\\chi}^{4}), \\\\\n\\sigma v &\\approx \\frac{G_{f}^{2}}{2 \\pi} M_{\\chi}^{2} v^{2} (1-\\xi^{2})^{3\/2}.\\\\\n\\sigma^{\\text{VV Ch}} &= \\frac{G_{f}^{2}}{864 \\pi M_{\\chi}^{4}} \\beta_f\n \\beta_\\chi (s-m_{f}^{2})\\nonumber \\\\ & \\qquad \\times (s^{2}+20 M_{\\chi}^{2} s +12 M_{\\chi}^{4}), \\\\\n\\sigma v &\\approx \\frac{G_{f}^{2}}{16 \\pi} M_{\\chi}^{2}\nv^{2}\\sqrt{1-\\xi^{2}}(4-\\xi^4). \\\\\n\\sigma^{\\text{VF}}_{\\text{Vec\/Ax}} &= \\frac{G^2}{4320 \\pi M_{\\chi}^4 s } \\frac{\n \\beta_f}{\\beta_\\chi} \\Big[ 8 m_{f}^{4}(-174 M_{\\chi}^{4}+2M_{\\chi}^{2}s+s^{2}) \\nonumber \\\\\n& \\qquad+4 M_{\\chi}^{2}s (s-20 M_{\\Omega}^{2})+s^{2}(10 M_{\\Omega}^{2} + 7s)\\Big) \\nonumber \\\\ \n&\\qquad -2 m_{f}^{2}\\Big(680 M_{\\chi}^{6}+152 M_{\\chi}^{4} (5 M_{\\Omega}^{2}+s) \\nonumber \\\\ \n& \\qquad+s\\Big(40 M_{\\chi}^{6}+2 M_{\\chi}^{4}(70 M_{\\Omega}^{2}-31s) \\nonumber \\\\\n& \\qquad\\pm 240 m_{f}^{3} M_{\\Omega} M_{\\chi}^2 (10 M_{\\chi}^{2}- s^{2}) \\nonumber \\\\\n&\\qquad \\pm 120 m_{f} M_{\\chi}^2 M_{\\Omega} s (M_{\\chi}^{2}-s) \\nonumber \\\\ \n&\\qquad +M_{\\chi}^{2}(76 s^{2}-40 M_{\\Omega}^{2} s )\\nonumber \\\\ & \\qquad+s^{2}(20 M_{\\Omega}^{2}+3s)\\Big) \\Big], \\\\\n\\sigma v &\\approx \\frac{G^2}{36 \\pi }\\sqrt{1-\\xi^2} \\Big[ (1-\\xi^2)\\Big((5\n\\xi^2+4) M_{\\chi}^2 \\nonumber \\\\ & \\qquad \\mp 6 \\xi M_{\\chi} M_{\\Omega}+5 M_{\\Omega}^2\\Big) \\nonumber \\\\\n& \\qquad +\\frac{v^2}{24} \\Big(\\mp 6 \\xi (19 \\xi^2+2) M_{\\chi} M_{\\Omega}\\nonumber\n\\\\ & \\qquad +3 (25 \\xi^2+6) M_{\\Omega}^2\\nonumber \\\\ & \\qquad+(83 \\xi^4+ 136 \\xi^2+156) M_{\\chi}^2\\Big) \\Big].\\\\\n\\sigma^{\\text{VFr}}_{\\text{Vec\/Ax}} &= \\frac{ G_{f}^{2}}{2160 \\pi M_{\\chi}^{4} s} \\frac{ \\beta_f}{ \\beta_\\chi} \\Big[s^4 + 22 m_f^2 M_{\\chi}^2 + 13 M_{\\chi}^4 \\nonumber \\\\ \n& \\qquad -8 s^2\\Big(8 m_f^4 +15 M_{\\Omega}^2 (m_f^2+M_{\\chi}^2) \\nonumber \\\\ \n& \\qquad \\pm 5 m_f M_{\\Omega} (4 m_f^2 + 5 M_{\\chi}^2)\\Big)\\nonumber \\\\\n&\\qquad-32 m_f^2 M_{\\chi}^4 (37 m_f^2 \\pm 50 m_f M_{\\Omega}\\nonumber \\\\\n&\\qquad +70 M_{\\chi}^2+45 M_{\\Omega}^2) \\nonumber \\\\ \n&\\qquad+ 2 s^3 (6 m_f^2 \\pm 20 m_f M_{\\Omega}+16 M_{\\chi}^2+15 M_{\\Omega}^2) \\nonumber \\\\* \n&\\qquad+8 M_{\\chi}^2 s \\Big(24 m_f^4 +15 M_{\\Omega}^2 (4 m_f^2+3M_{\\chi}^2)\n\\nonumber \\\\\n&\\qquad \\pm 50 m_f M_{\\Omega} (2 m_f^2 + M_{\\chi}^2)\\nonumber \\\\\n&\\qquad+119 m_f^2 M_{\\chi}^2+40 M_{\\chi}^4\\Big), \\\\ \n\\sigma v &\\approx \\frac{ G_{f}^{2}}{ 9 \\pi} \\sqrt{1-\\xi^{2}} \\Big[\n(1-\\xi^{2})\\Big(3 M_{\\Omega}^{2} \\pm 2 \\xi M_{\\chi} M_{\\Omega} + \\nonumber \\\\\n&\\qquad (3 \\xi^{2}+4) M_{\\chi}^{2}\\Big) + \\dfrac{v^{2}}{24} \\Big(3(2+7 \\xi^{2}) M_{\\Omega}^{2} \\nonumber \\\\*\n& \\qquad \\pm 6 \\xi (2+\\xi^{2}) M_{\\chi} M_{\\Omega} \\nonumber\\\\\n&\\qquad +(16+30 \\xi^{2}+29 \\xi^{4}) M_{\\chi}^{2}\\Big)\\Big].\n\\end{flalign}\n\\section{Cross Sections for Direct Detection} \\label{sec:directdetect}\nWe now give results for the dark matter--nucleon scattering cross\nsection at zero momentum transfer, $\\sigma^0$, for all defined\nbenchmark models. In a universal scenario, the effective coupling is\nindependent of the quark ($G_q = G$), whereas it grows proportionally\nto the quark mass in a Yukawa-like model ($G_q = G\\ m_q \/ m_e$). We\nuse the following definitions:\n\\begin{align}\n\\frac{f_{p}}{M_P} &\\equiv \\sum_{q=u,d,s} \\hspace{-0.2cm}f_{q}^{p}\\frac{G_q}{m_{q}} + \\frac{2}{27}(1-\\sum_{q=u,d,s}\\hspace{-0.2cm}f_{q}^{p})\\sum_{q=c,b,t}\\hspace{-0.05cm}\\frac{G_{q}}{m_{q}}, \\\\\nd_p &\\equiv \\sum_{q=u,d,s} G_q \\Delta_q^p, \\\\\nb_{p} &\\equiv 2 G_{u} +G_{d}, \\\\\n\\tilde{b}_p &\\equiv b_p M_\\chi + 2 G_u m_u + G_d m_d. \\\\\n\\intertext{with the numerical values for $f_q^p$ and $\\Delta_q^p$ listed in \\cite{fnumbers, deltanumbers}:}\nf_{u}^{p} &= 0.020 \\pm 0.004, \\\\\nf_{d}^{p} &= 0.026 \\pm 0.005, \\\\\nf_{s}^{p} &= 0.118 \\pm 0.062, \\\\ \n\\Delta_{u}^{p} &= -0.427 \\pm 0.013, \\\\\n\\Delta_{d}^{p} &= 0.842\\pm 0.012, \\\\\n\\Delta_{s}^{p} &= -0.085 \\pm 0.018. \\\\\n\\intertext{Furthermore we define the reduced mass of the \\textsc{Wimp} proton system,}\n\\mu &\\equiv \\frac{M_{\\chi} M_{p}}{M_{\\chi}+M_{p}}.\n\\end{align}\nThe cross sections can be evaluated in a nonrelativistic approximation for the\n\\textsc{Wimp} and by using the quark proton form factors listed above. See\ne.g.\\ \\cite{Agrawal:2010fh}. If a model is not listed, its\nscattering cross section\nequals zero, e.g.\\ for pseudoscalar interactions that always vanish in a\nnonrelativistic model. Again, we agree with the respective results in \\cite{Zheng:2010js,\n Yu:2011by} for comparable operators. \n\nCross sections for real final state particles can\neasily be derived from the following list by setting the vector form factors\n$b_p$ and $\\tilde{b}_p$ to zero and rescaling $f_p$ and $d_p$ by a\nfactor of 2.\n\\subsection{Scalar \\textsc{Wimp}}\n\\vspace{-1cm}\n\\begin{flalign}\n\\sigma^0_{\\text{SS Sc.}} &= \\frac{\\mu^{2}}{ 4 \\pi M_{\\chi}^{2}}f_{p}^{2}, &\\\\\n\\sigma^0_{\\text{SV Vec.}} &= \\frac{\\mu^{2}}{\\pi}b_{p}^{2}, &\\\\\n\\sigma^0_{\\text{SF Sc.}} &= \\frac{\\mu^{2}}{4 \\pi}(+f_{p} + \\frac{\\tilde{b}_{p}}{M_\\Omega})^{2}, &\\\\\n\\sigma^0_{\\text{SF Ps.}} &= \\frac{\\mu^{2}}{4 \\pi}(-f_{p} + \\frac{\\tilde{b}_{p}}{M_\\Omega})^{2}, &\\\\\n\\sigma^0_{\\text{SV Chi.}} &= \\frac{\\mu^{2}}{ 4 \\pi}b_{p}^{2}. &\n\\end{flalign}\n\\subsection{Fermion \\textsc{Wimp}}\n\\vspace{-1cm}\n\\begin{flalign}\n\\sigma^0_{\\text{FS Sc.}} &= \\frac{\\mu^{2}}{\\pi}f_{p}^{2},&\\\\ \n\\sigma^0_{\\text{FV Vec.}} &= \\frac{\\mu^{2}}{\\pi}b_{p}^{2}, &\\\\ \n\\sigma^0_{\\text{FV Ax.}} &= 3 \\frac{\\mu^{2}}{ \\pi} d_p^{2},&\\\\ \n\\sigma^0_{\\text{FV Chi.}} &= \\frac{\\mu^{2}}{ 16 \\pi}b_{p}^{2}, &\\\\\n\\sigma^0_{\\text{FVr Chi.}} &= 3 \\frac{\\mu^{2}}{\\pi} d_p^{2}, &\\\\\n\\sigma^0_{\\text{FtS Sc.}} &= \\frac{\\mu^{2}}{16 \\pi}(b_{p} + f_{p} )^{2},&\\\\ \n\\sigma^0_{\\text{FtS Ps.}} &= \\frac{\\mu^{2}}{16 \\pi}(b_{p} - f_{p} )^{2},&\\\\\n\\sigma^0_{\\text{FtV Vec.}} &= \\frac{\\mu^{2}}{\\pi}(1\/2 \\cdot b_{p} -f_{p})^{2},&\\\\ \n\\sigma^0_{\\text{FtV Ax.}} &= \\frac{\\mu^{2}}{\\pi}(1\/2 \\cdot b_{p} + f_{p})^{2},&\\\\ \n\\sigma^0_{\\text{FtV Chi.}} &= \\frac{\\mu^{2}}{ 16 \\pi}b_{p}^{2}. &\n\\end{flalign}\n\\subsection{Vector \\textsc{Wimp}}\n\\vspace{-1cm}\n\\begin{flalign}\n\\sigma^0_{\\text{VS Sc.}} &= \\frac{\\mu^{2}}{4 \\pi M_{\\chi}^{2}}f_{p}^{2},&\\\\ \n\\sigma^0_{\\text{VF Vec.}} &= \\frac{\\mu^{2}}{4 \\pi}(-f_{p}+\\frac{\\tilde{b}_{p}}{M_\\Omega})^{2},&\\\\ \n\\sigma^0_{\\text{VF Ax.}} &= \\frac{\\mu^{2}}{4 \\pi}(+f_{p}+\\frac{\\tilde{b}_{p}}{M_\\Omega})^{2}, &\\\\\n\\sigma^0_{\\text{VF Chi.}} &= \\frac{\\mu^{2}}{4 \\pi}b_{p}^{2}, &\\\\\n\\sigma^0_{\\text{VV Vec.}} &= \\frac{\\mu^{2}}{\\pi}b_{p}^{2}. & \n\\end{flalign}\n\n\\subsection{Photon Loop} \n\\label{app:PhotonLoop} \n\nIf the \\textsc{Wimp} only couples to leptons, the \\textsc{Wimp}--proton\ninteraction can only happen at the loop level. In that case, a low\nenergy photon that couples to a virtual lepton pair interacts with the\nwhole proton. This only happens for models with s--channel vector\nbilinears $\\bar{\\psi} \\gamma^\\mu \\psi$, i.e.\\ models which\ninclude either a, $b_p$, or a, $\\tilde{b}_p$, term in the low energy\ntree level cross section. Results can therefore be derived as follows,\n\\begin{align}\n\\sigma_0^\\text{Loop} &= \\frac{\\alpha^2_{\\text{em}}}{81 \\pi^2} F^2(q^2)\n\\left. \\sigma_0^\\text{Tree} \\right|_{\\text{reduced}}, \\\\\n\\intertext{where the reduced cross section has to be understood as the tree level\n cross section given above after setting $b_p,\\tilde{b}_p = 1$ and $f_p, d_p\n = 0$. This ensures\n that we only take the vector interaction parts. If the tree level cross section includes\n a $b_p$ term, the loop factor is given as,}\nF(q^2) &\\equiv \\sum_lG_l\\ f(q^2, m_l). \\\\\n\\intertext{For $\\tilde{b}_p$ terms, it reads,}\nF(q^2) &\\equiv \\sum_l \\left(m_l + M_\\chi\\right) G_l\\ f(q^2, m_l). \\\\\n\\intertext{In both cases, the loop function can be evaluated as,}\nf(q^{2},m)&\\equiv\\frac{1}{q^{2}} \\left[ 5 q^{2}+12 m^{2}-6(q^{2}+2\nm^{2})\\beta_q \\text{arcoth}~ \\beta_q \\right. \\nonumber \\\\\n& \\qquad \\qquad \\left. - 3 q^2 \\ln m^2 \/ \\Lambda^2 \\right], \\\\\n\\beta_q &\\equiv \\sqrt{1-4 m^{2} \/ q^{2}}.\n\\end{align}\nWe follow the conservative assumption of a maximum scattering angle to find $q^2 = - 4\n\\mu^2 v^2$ with $\\mu$ describing the reduced mass of the \\textsc{Wimp} nucleus system\nand $v = \\unit{500}{\\kilo\\meter\\per\\second}$ being the typical\nescape velocity of a \\textsc{Wimp} in a dark matter halo. Because of the new\n$q$--dependence of the cross section and the fact that the photon only couples\nto the protons inside the nucleus, the official \\textsc{Xenon} results have to\nbe rescaled according to,\n\\begin{align}\n\\sigma^\\text{Loop} = \\sigma^\\text{Tree} \\left[ \\frac{F(\\tilde{q}^2)}{F(q^2)} \\cdot \\frac{A}{Z}\\right]^2,\n\\end{align}\nwhere $\\tilde{q} = q(M_N = M_P)$ uses the reduced mass $\\mu$ of the \\textsc{Wimp} proton\nsystem instead. This weakens the cross section limits by about a factor\n of 10.\n\n\n\\section{Results}\n\\label{sec:results}\nWe begin by presenting the reach at the \\textsc{Ilc} in terms of the\neffective coupling constant in Sec.~\\ref{sec:ilc_bounds}. We then\ncompare these potential bounds with the couplings\npredicted by the cosmological relic density and the bounds coming from\ndirect and indirect detection experiments. Of course we would also like to discover a dark matter at the \\textsc{Ilc} and the bounds provide an estimate of the potential sensitivity of the collider. \n\n\\subsection{ILC Bounds}\n\\label{sec:ilc_bounds}\nWe determine the \\unit{90}{\\%} exclusion bound for the effective\ncoupling constant in each benchmark model for the best case\nscenario. The integrated luminosity is set to \\unit{500}{\\femto\\reciprocal\\barn} and\nthe systematic polarisation error to $\\Delta P\/P =\n\\unit{0.1}{\\%}$. For each benchmark model we choose the polarisation\nsetting that leads to the best signal to background ratio for the\ncorresponding polarisation behaviour according to\nTables~\\ref{tbl:sigoverbkgestimate} and\n\\ref{tbl:sigoverbkgestimate_1tev}. Results for different polarisation\nsettings can be found by rescaling the bound on the coupling according\nto $G^\\prime = G \\sqrt{r^\\prime \/ r}$ with $r$ denoting the ratio\n$N_\\text{S} \/ \\Delta N_\\text{B}$ given in\nTable~\\ref{tbl:sigoverbkgestimate_1tev}. We choose to present all of\nthe results for an \\textsc{Ilc} with a center of mass energy of\n\\unit{1}{\\TeV} due to the increased range of dark matter masses that\nthis option can probe. In addition, smaller effective couplings can be\nprobed, mainly due to the falling Bhabha background.\n\nIn Fig.~\\ref{img:ilcbounds} we show the derived bounds on the coupling\nconstants for an \\textsc{Ilc} center of mass energy of\n\\unit{1}{\\TeV}. The hashed area denotes the region that either\nviolates the tree level approach with a too large dimensionless\ncoupling constant $g^2 > 4 \\pi$, or by having a too small mediator\nmass $M_\\Omega < \\unit{1}{\\TeV}$, for the effective approach to be\nvalid. Note that the leading order in models with fermionic mediators\nhas a different mass dimension and therefore gives a different\ndefinition for the effective coupling constant $G_\\text{eff}$. If a\nmodel has no separate `pseudoscalar' or `axialvector' results, it is\nidentical to the corresponding `scalar'\/ `vector' line due to\nidentical cross section formulas. For masses away from the threshold,\nthe \\textsc{Ilc} is able to exclude coupling constants down to the order of \\unit{\\power{10}{-7}}{\\GeV \\rpsquared}\nor \\unit{\\power{10}{-4}}{\\reciprocal\\GeV}, depending on the mass\ndimension. This corresponds to a total cross section (for the given\nphase space criteria) of about \\unit{0.3}{\\femto\\barn}. Exceptions\nhowever arise for models with vector dark matter that tend to have\nvery strong exclusion limits for small masses. This is caused by the\n$1\/M_\\chi^4$ dependence in the photon cross section, which leads to\ndivergences for very small vector boson masses. It has been shown\n\\cite{Cornwall:1974km} that only spontaneously broken gauge theories\ncan lead to models with massive vector particles that are not\ndivergent. Therefore, our initial fundamental model cannot be the full\ntheory for all energies. In our effective approach, we restrict the\nenergy to a maximum and in that case one can still receive\nperturbative valid results for mass ranges that do not violate unitary\nbounds. However, the perturbatively allowed mass range cannot be given\nin this model independent approach, since such an analysis needs more\ninformation about the size of the individual couplings and the\nrelation between the mass of the mediator and the dark matter mass\nitself. In summary, a more detailed fundamental theory is needed to\nevaluate the breakdown of perturbation theory in this scenario.\n\n\nWe note that in models with fermionic operators, the sub-leading\norder has a negligible effect, as can be seen from the nearly identical lines\nfor fermionic mediators with different masses. \n\n\\subsection{Combined Results}\nThe combined maximum\nexclusion limits for spin independent DM--proton interaction at \\textsc{Pamela}, \\textsc{Wmap}\nand the \\textsc{Ilc} are shown in\nFigs.~\\ref{img:totalbounds1}-\\ref{img:totalbounds3}. We choose a subset of\nmodels that couple to all Standard Model fermions and give an overview\nof the bounds that we can expect. Other models behave similarly and are\ntherefore not shown again separately. We can give the following statements about\nthe comparison of the \\textsc{Ilc} exclusion bound with the current \\textsc{Xenon} limits:\n\\begin{itemize}\n\\item We have sensitivity to spin independent proton cross sections for, as an example, the\n FV Vector model down to \\unit{\\power{10}{-42}}{\\cm\\rpsquared} or\n equivalently \\unit{\\power{10}{-3}}{\\femto\\barn}, which\n is an improvement of about four orders of magnitude compared to\n current \\textsc{Lep} \\cite{Fox:2011fx} and two orders of magnitude\n compared to current Tevatron \\cite{Bai:2010hh} and \\textsc{Cms}\n \\cite{Chatrchyan:2012pa} results.\n\\item An increased center of mass energy can lead to stronger bounds\n by up to one order of magnitude. It also allows a larger dark matter\n mass range to be probed. \n\\item \\textsc{Ilc} bounds get significantly weakened if the interaction is\n Yukawa--like. At the \\textsc{Ilc} the mediator must couple to\n electrons, which have a suppressed Yukawa coupling. The production cross section\nis thus small, leading to weaker bounds.\n\\item Models with scalar mediators give weaker bounds than models with\n vector interactions. For fermionic dark matter we observe a\n difference of about two orders of magnitude, which is in agreement\n with previously mentioned results from e.g.\\\n \\textsc{Lep}. For scalar and vector dark matter the difference is\n mass--dependent and can increase to up to six orders of magnitude,\n which is due to the different mass dimension of the\n couplings.\n\\item The \\textsc{Wmap} bounds are for many effective models very\n constraining, Figs.\\ref{img:totalbounds1}--\\ref{img:totalbounds4}. However, we would like to point out that these can be\n highly dependent on the full theory whilst not affecting the\n \\textsc{Ilc} or direct detection phenomenology. For example,\n annihilation can occur via some resonance or as in some \\textsc{Susy} models,\n co-annihilation with staus or stops.\n\\end{itemize}\n\nIn Fig.~\\ref{img:totalbounds4} we show some models which allow for\nlepton couplings only. In that case, dark matter can only interact\nwith protons via photons through a fermion loop, \\textit{cf.}\nAppendix.~\\ref{app:PhotonLoop}. The loop factor significantly lowers\nthe cross section and therefore increases the bound in the case of\nvector coupled models. Other models allow quark couplings only at the\ntwo--loop level or theoretically completely forbid them\n\\cite{Fox:2011fx}. In all cases, the \\textsc{Ilc} would give the\nstrongest exclusion bounds for dark matter lepton couplings.\nFor models with fermionic mediators there is an extra subtlety when\ncomparing the bounds. In particular the exclusion limit at the\n\\textsc{Ilc} is mainly given by the leading term in the operator\nexpansion, which is scalar like. Loop couplings can only happen for\nvector currents, which in the case of a fermionic mediator is only\ngiven by the sub-leading order and has an additional factor of\n$1\/M_\\Omega^2$. In that case, when translating any exclusion limits\ninto bounds on the {\\textsc{Wimp}--proton cross section, we need to know\nthe exact mass of the mediator. We show this in\nFig.~\\ref{img:totalbounds4} for the two different chosen suppression\nscales `Low' ($M_{\\Omega} =$\\unit{1}{\\TeV}) and `High' ($M_{\\Omega} =$\\unit{10}{\\TeV}), Table~\\ref{tbl:constraints}.\n\nIn Fig.~\\ref{img:totalbounds5} we show the exclusion limits for\nthe spin--dependent interaction. In our\ncase, only the model with fermionic dark matter, a vector mediator and\nan axial--vector coupling leads to such an interaction. In that case,\nwe compare with data from the previous \\textsc{Xenon} experiment\n(\\textsc{Xenon}10), since no results for the \\textsc{Xenon}100 phase\nwere available when this study was completed. Since in this scenario\ndark matter only couples to a single nucleon on average because of the\nnatural spin anti--alignment in nuclei, the \\textsc{Xenon} bounds are\nnot coherently enhanced by the atomic number and therefore strongly\nlose sensitivity. The \\textsc{Ilc} would also give strongest exclusion\nbounds over the whole accessible mass range here.\n\\onecolumngrid\n\\twocolumngrid\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{withpamela_SSScalarplotter_protonxsect_universal} \\hfill\n\\includegraphics[width=0.45\\textwidth]{withpamela_SSScalarplotter_protonxsect_yukawa}\\\\\n\\includegraphics[width=0.45\\textwidth]{withpamela_SFScalarplotter_protonxsect_universal} \\hfill\n\\includegraphics[width=0.45\\textwidth]{withpamela_SFScalarplotter_protonxsect_yukawa}\\\\\n\\includegraphics[width=0.45\\textwidth]{withpamela_SVVectorplotter_protonxsect_universal} \\hfill\n\\includegraphics[width=0.45\\textwidth]{withpamela_SVVectorplotter_protonxsect_yukawa}\n\\caption{Combined \\unit{90}{\\%} exclusion limits on the spin independent dark matter proton cross\n section from \\textsc{Ilc}, \\textsc{Pamela} and \\textsc{Wmap} for a selection of scalar dark matter models.}\n\\label{img:totalbounds1}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{withpamela_FSScalarplotter_protonxsect_universal} \\hfill\n\\includegraphics[width=0.45\\textwidth]{withpamela_FSScalarplotter_protonxsect_yukawa}\\\\\n\\includegraphics[width=0.45\\textwidth]{withpamela_FVVectorplotter_protonxsect_universal} \\hfill\n\\includegraphics[width=0.45\\textwidth]{withpamela_FVVectorplotter_protonxsect_yukawa}\\\\\n\\includegraphics[width=0.45\\textwidth]{withpamela_FtVRightplotter_protonxsect_universal} \\hfill\n\\includegraphics[width=0.45\\textwidth]{withpamela_FtVRightplotter_protonxsect_yukawa}\n\\caption{Combined \\unit{90}{\\%} exclusion limits on the spin independent dark matter proton cross\n section from \\textsc{Ilc}, \\textsc{Pamela} and \\textsc{Wmap} for a selection of fermionic dark matter models.}\n\\label{img:totalbounds2}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{withpamela_VSScalarplotter_protonxsect_universal} \\hfill\n\\includegraphics[width=0.45\\textwidth]{withpamela_VSScalarplotter_protonxsect_yukawa}\\\\\n\\includegraphics[width=0.45\\textwidth]{withpamela_VFVectorplotter_protonxsect_universal} \\hfill\n\\includegraphics[width=0.45\\textwidth]{withpamela_VFVectorplotter_protonxsect_yukawa}\\\\\n\\includegraphics[width=0.45\\textwidth]{withpamela_VVVectorplotter_protonxsect_universal} \\hfill\n\\includegraphics[width=0.45\\textwidth]{withpamela_VVVectorplotter_protonxsect_yukawa}\n\\caption{Combined limits on the spin independent dark matter proton cross\n section from \\textsc{Ilc}, \\textsc{Pamela} and \\textsc{Wmap} for a selection of vector dark matter models.}\n\\label{img:totalbounds3}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{withpamela_FVVectorplotter_chargedleptons_loopxsect_universal} \\hfill\n\\includegraphics[width=0.45\\textwidth]{withpamela_FVVectorplotter_chargedleptons_loopxsect_yukawa}\\\\\n\\includegraphics[width=0.45\\textwidth]{withpamela_SFScalarplotter_loopxsect_universal} \\hfill\n\\includegraphics[width=0.45\\textwidth]{withpamela_SFLScalarplotter_loopxsect_yukawa}\n\\vspace{-0.5cm}\n\\caption{Combined limits for a selection of models\n with loop--coupling to leptons only. `Low' corresponds to $M_{\\Omega} =$\\unit{1}{\\TeV} and `High' to $M_{\\Omega} =$\\unit{10}{\\TeV}, Table~\\ref{tbl:constraints}}\n\\label{img:totalbounds4}\n\\vspace{0.5cm}\n\\includegraphics[width=0.45\\textwidth]{withpamela_FVAxialvectorplotter_protonxsect_universal} \\hfill\n\\includegraphics[width=0.45\\textwidth]{withpamela_FVAxialvectorplotter_protonxsect_yukawa}\n\\vspace{-0.5cm}\n\\caption{Combined limits on the spin dependent dark matter proton cross\n section.}\n\\label{img:totalbounds5}\n\\end{figure*}\n\n\\clearpage\n\\section{Conclusions}\n\\label{sec:conclusions}\nIn this paper we considered a broad range of effective models for dark\nmatter and investigated the possibility that these models could be\nexplored at the \\textsc{Ilc}. The models considered the possibility\nthat dark matter was a new scalar, fermion or vector particle and\nwould be produced at the \\textsc{Ilc} via a new, heavy intermediate\nstate, the mediator particle. For the mediator we also\n considered spins 0, 1\/2 and 1. We obtained the corresponding\n effective theories by integrating out the mediator field.\n\n To be able to compare the reach of the \\textsc{Ilc} with the other\n experimental searches, certain assumptions have to be made on how the\n mediator and dark matter couples to the Standard Model particles. We\n assume in all models that interactions only occur with the Standard\n Model fermions but the relative strength to different particles is\n varied. In the simplest variant we choose that the coupling is equal\n between all the Standard Model states. Another choice is that the\n interaction scales with the mass of the interacting Standard Model\n fermion, a `Yukawa-like' interaction. The last choice we make is the\n most optimistic for \\textsc{Ilc} phenomenology with only the Standard Model\n leptons interacting with the heavy mediator.\n\\balance\n Since the produced dark matter particles will be invisible to the\n \\textsc{Ilc} detectors, we require a radiated photon to be emitted\n from the initial state that will recoil against missing\n momentum. This topology provides a distinctive signal with which to\n discover dark matter. For the \\textsc{Ilc} study, we included the\n dominant backgrounds and most important detector effects. In addition\n we considered the possibility of using polarised initial states to\n reduce backgrounds and improve the signal strength.\n\n The effective theories that we consider provide an efficient way to\n compare the reach of the \\textsc{Ilc} with other methods to discover\n dark matter. Firstly, we consider the dark matter annihilation cross\n section required for the relic density observed by \\textsc{Wmap}. We\n also look at the direct detection bounds at \\textsc{Xenon} by\n calculating the dark matter-nucleon scattering cross section. In\n addition, we include bounds from dark matter annihilation to\n positrons from the \\textsc{Pamela} experiment.\n\nIn terms of the effective dark matter model, we found that the \\textsc{Ilc}\nshould be able to probe couplings \\unit{\\power{10}{-7}}{\\GeV \\rpsquared}, or\n\\unit{\\power{10}{-4}}{\\reciprocal\\GeV} depending on the mass dimension of the theory. In models that contain vector dark matter, the \\textsc{Ilc} may be able to probe even weaker couplings in the case of low dark matter mass.\n\nTo compare with astrophysical bounds, we found that the\n\\textsc{Ilc} reach is strongly dependent on the exact dark matter\nmodel. If we assume that dark matter is relatively heavy ($>$\n\\unit{100}{\\GeV}) and interacts with a Standard Model particle in\nproportion to its mass, then the \\textsc{Ilc} is\nuncompetitive. However, in the case that dark matter is relatively\nlight ($<$ \\unit{10}{\\GeV}) then the bounds from the \\textsc{Ilc} are\ncompetitive with astrophysical bounds in many models. In addition, if\ndark matter happens to only interact with the Standard Model\nleptons then the \\textsc{Ilc} offers a unique possibility to discover\ndark matter. For this reason, an \\textsc{Ilc} search is complementary\nto those done at the \\textsc{Lhc} thanks to the different initial\nstate.\n\n\n\\section{Dark matter search at the \\textsc{Ilc}}\n\\label{sec:ilc}\n\\subsection{Radiative Production of Dark Matter}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\columnwidth]{chichigamma_1.pdf} \\hfill\n\\includegraphics[width=0.49\\columnwidth]{chichigamma_2.pdf}\n\\caption{Diagrams for radiative pair production of dark\n matter. Terms in which the heavy mediator can emit a photon are neglected.}\n\\label{fig:signalprocess_feynmandiagrams}\n\\end{figure}\n\nFor the \\textsc{Ilc} search, we look at the process $\\Ppositron\n\\Pelectron \\rightarrow \\chi \\chi\\Pphoton$ with a hard photon being the\nonly detected particle in the final state, Fig.~\\ref{fig:signalprocess_feynmandiagrams}. We determine the polarized\ndifferential cross section for this process with respect to the\nrelative photon energy $x \\equiv 2 E_\\gamma \/ \\sqrt{s}$ and its polar\nangle $\\theta$ by integrating over the full phase space of the\nfinal state dark matter particles. The results for this\ncalculation are given in Table~\\ref{tbl:2t3crosssections}, with\nfurther explanation of the abbreviations used given in\nAppendix~\\ref{app:xsectterms}. Previous \\textsc{Ilc} studies,\ne.g.\\ \\cite{Bartels:2012ex, Birkedal:2004xn}, have used the\nWeizs\\\"acker--Williams approximation for soft photons. This formula\nrelates the differential photon cross section to the total pair\nproduction cross section $\\Ppositron\\Pelectron \\rightarrow \\chi \\chi$\nwith a reduced center of mass energy $s \\rightarrow \\hat{s} \\equiv\ns(1-x)$ and multiplied by the kinematical function $F_{x \\theta}$,\n\\begin{align}\n\\frac{\\mathrm{d} \\sigma \\left[\\Ppositron \\Pelectron \\rightarrow \\bar{\\chi} \\chi\n \\gamma \\right] }{\\mathrm{d} x \\ \\mathrm{d}\n \\cos \\theta_\\gamma} \\approx F_{x \\theta}\\ \n \\hat{\\sigma} \\left[\\Ppositron \\Pelectron \\rightarrow \\bar{\\chi} \\chi\n \\right].\n\\label{weiz-will}\n\\end{align}\n\\begin{table*}\n\\begin{tabular}{r@{\\quad}l}\n\\hline\nModel & $ \\displaystyle \\frac{ \\mathrm{d}\\sigma}{ \\mathrm{d} x \\\n \\mathrm{d}\\cos \\theta} $\\\\ [1.5ex]\n\\hline\n\\hline\nSS & $ \\displaystyle \\frac{\\hat{\\beta} F_{x \\theta}}{32 \\pi M_\\Omega^4}\nG_{s+p} g_\\chi^2 C_s $\\\\[1.5ex]\nSF & $\\displaystyle \\frac{ \\hat{\\beta} F_{x \\theta}}{32 \\pi M_\\Omega^2} \n\\left[ G_{s-p}^2 C_s + \\frac{\\hat{\\beta}^2 \\hat{s}}{12 M_\\Omega^2} \n \\boldsymbol{ V_{x \\theta}} \\left[(g_s+g_p)^4 C_R + (g_s-g_p)^4 C_L \\right]\n + \\boldsymbol{A_{\\text{SF}}} \\right] $ \\\\[1.5ex]\nSFr & $\\displaystyle \\frac{ \\hat{\\beta}}{16 \\pi M_\\Omega^2} \\left[ F_{x \\theta}\nG_{s-p}^2 C_s + \\boldsymbol{A_{\\text{SFr}}}\\right]$ \\\\[1.5ex]\nSV & $ \\displaystyle \\frac{\\hat{s} \\hat{\\beta}^3 F_{x \\theta}}{96 \\pi M_\\Omega^4} \\boldsymbol{V_{x \\theta}} \n \\left[ g_l^2C_L + g_r^2C_R \\right] g_\\chi^2 $ \\\\[1.5ex]\n\\hline\n\\hline\nFS & $ \\displaystyle \\frac{\\hat{s} \\hat{\\beta} F_{x \\theta}}{16 \\pi M_\\Omega^4} \nG_{s+p} C_s \\left[ g_{s}^2 \\hat{\\beta^2} + g_{p}^2 \\right] $ \\\\[1.5ex]\nFV & $ \\displaystyle \\frac{ \\hat{\\beta} F_{x \\theta} }{48 \\pi\n M_\\Omega^4} \\boldsymbol{V_{x \\theta}} \\left[\nG_{l+r} \\hat{s} \\hat{\\beta}^2 + 3 \\left(g_l + g_r \\right)^2 M_\\chi^2 \\right] \\left[ g_l^2C_L + g_r^2C_R \\right] \n$ \\\\[1.5ex]\nFVr & $ \\displaystyle \\frac{ \\hat{s} \\hat{\\beta^3} F_{x \\theta} }{48 \\pi\n M_\\Omega^4} \\boldsymbol{V_{x \\theta}} \\left(g_l - g_r \\right)^2 \\left[ g_l^2C_L + g_r^2C_R \\right] $\\\\[1.5ex]\nFtS & $\\displaystyle \\frac{ F_{x \\theta} \\hat{\\beta}}{48 \\pi M_\\Omega^4} G_{s+p}^2\n\\left[\\boldsymbol{V_{x \\theta}}(\\hat{s}-M_\\chi^2) + \\boldsymbol{A_{\\text{FtS}}} \\right]$ \\\\[1.5ex]\nFtSr & $\\displaystyle \\frac{ \\hat{\\beta} F_{x \\theta}}{192 \\pi M_\\Omega^4}\nG_{s+p}^2 \\left[ 3 (\\hat{s}-2 M_\\chi^2) C_P + \\boldsymbol{V_{x \\theta}} 2 (\\hat{s}\n - 4 M_\\chi^2) C_V \\right] $\\\\[1.5ex]\nFtV & $\\displaystyle \\frac{\\hat{\\beta} F_{x \\theta}}{48 \\pi M_\\Omega^4} \\left[ \n 6 G_{lr}^2 C_s\n (\\hat{s} - 2 M_\\chi^2 ) + (\\hat{s} - M_\\chi^2) \\boldsymbol{V_{x \\theta}} (g_l^4 C_L + g_r^4 C_R) \\right] $ \\\\[1.5ex]\nFtVr & $\\displaystyle \\frac{\\hat{\\beta} F_{x \\theta}}{48 \\pi M_\\Omega^4} \\left[ \n 12 G_{lr}^2 C_s (\\hat{s} - 2 M_\\chi^2 ) + (\\hat{s} - 4 M_\\chi^2) \\boldsymbol{V_{x \\theta}} (g_l^4 C_L + g_r^4 C_R) \\right] $\\\\[1.5ex]\n\\hline\n\\hline\nVS & $ \\displaystyle \\frac{ \\hat{\\beta} F_{x \\theta}} {128 \\pi M_\\chi^4 M_\\Omega^4} \nG_{s+p} g_\\chi^2 C_s (12 M_\\chi^4-4M_\\chi^2\\hat{s}+\\hat{s}^2) $ \\\\[1.5ex]\nVF & $\\displaystyle \\frac{\\hat{\\beta}F_{x \\theta}}{3840 \\pi M_\\chi^4 M_\\Omega^2} \n\\Big[ 40 G_{lr}^2 C_s (7 M_\\chi^4 - 2 M_\\chi^2 \\hat{s} + \\hat{s}^2) +\\frac{1}{M_\\Omega^2}\\left(g_l^4C_L+g_r^4C_R\\right)\n (40 M_\\chi^6-22M_\\chi^4\\hat{s}+56 M_\\chi^2 \\hat{s}^2 + 3 \\hat{s}^3) + \\boldsymbol{A_{\\text{VF}}}\\Big]$\\\\[1.5ex]\nVFr & $\\displaystyle \\frac{\\hat{\\beta}F_{x \\theta}}{3840 \\pi M_\\chi^4 M_\\Omega^2} \n\\Big[ 60 G_{lr}^2 C_s (12 M_\\chi^4 - 4 M_\\chi^2 \\hat{s} + \\hat{s}^2) + \\frac{1}{M_\\Omega^2}\\left(g_l^4C_L+g_r^4C_R\\right)\n (320 M_\\chi^6-104^4\\hat{s}+32 M_\\chi^2 \\hat{s}^2 + \\hat{s}^3) + \\boldsymbol{A_{\\text{VFr}}}\\Big]$\\\\\nVV & $ \\displaystyle \\frac{ \\hat{s} \\hat{\\beta}^3 F_{x \\theta} \\boldsymbol{V_{x \\theta}}}{3840 \\pi M_\\chi^4 M_\\Omega^4}\\left[ g_l^2C_L + g_r^2C_R \\right] g_\\chi^2 \n( M_\\chi^4 + 20 M_\\chi^2 \\hat{s} + \\hat{s}^2)$ \\\\[1.5ex]\n\\hline\n\\end{tabular}\n\\caption{Analytical differential cross sections for the process $\\Ppositron\n \\Pelectron \\rightarrow \\chi \\chi \\gamma$ in the various effective\n models. Terms in bold do not appear in the Weizs\\\"acker--Williams\n approach and are given in Appendix~\\ref{app:xsectterms} where we also\n define all used abbreviations. Models with a suffix `r' correspond to the case of real particles. \n Cross sections for SSr, FSr and VSr are twice as large as\n in the complex case while SV and VV vanish completely for real particles.}\n\\label{tbl:2t3crosssections}\n\\end{table*}\nDue to the soft collinear approximation used, we expect that the above equation will perform poorly for large angle and high $p_T$ photons. We compare the analytical result to this approximation to test\nthe reliability. In Table~\\ref{tbl:2t3crosssections} we put terms in\nbold, which are purely caused by our analytical\ntreatment. The corrections are either of the form of an additional\nkinematical factor $V_{x \\theta}$, mostly appearing in models with\nvector mediators, or completely new terms that typically appear in\nt--channel interactions. Since $\\lim_{x \\rightarrow 0} V_{x \\theta} =\n1$ and $\\lim_{x \\rightarrow 0} (A_i) = 0$, the WW--approximation is in\nagreement with our full result for small energies. In\nFig.~\\ref{img:comparison} we show the respective photon energy\ndistributions for different models in both the WW--approximation and\nthe full analytical treatment. The curves behave quite congruently\nwith differences visible in the high energy sector. Since most of the\nsignal events lie in the low energy part, the approximation gives\naccurate results for counting experiments. A shape dependent analysis\nwould need to use the analytical result to estimate the correct\nthreshold behaviour for high energies. Our subsequent analysis\nis performed with the full analytical cross section.\n\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{FtS_1tev.pdf}\n\\put(-45,180){(a)} \n\\hfill\n\\includegraphics[width=0.49\\textwidth]{VF_1tev.pdf} \n\\put(-45,180){(b)} \n\\caption{Comparison of tree level photon energy distributions in the\n WW--approximation and the analytical solution for $M_{\\chi} = \\unit{50}{\\GeV}$,\n $|\\cos \\theta_\\gamma|_\\text{max} = 0.98$ and\n $\\sqrt{s} = \\unit{1}{\\TeV}$. (a) SV, (b) FtS.}\n\\label{img:comparison}\n\\end{figure*}\n\n\nWhen we restrict the various couplings in our model according to the benchmark scenarios, Table~\\ref{tbl:constraints}, most of the cross sections simplify and have only one polarisation dependent term $C_i$. To\ndetermine the polarisation leading to the best signal to\nbackground ratio, we only need to consider cases with different $C_i$.\nWe therefore classify our models as follows:\n\\begin{align}\n\\text{Scalar--like}: \\sigma_{\\text{pol}} &= C_S\n\\sigma_{\\text{unpol}}, \\label{eqn:polclasses} \\\\\n\\text{Vector--like}: \\sigma_{\\text{pol}} &= C_V \\sigma_{\\text{unpol}}, \\nonumber\n\\\\\n\\text{Right--like}: \\sigma_{\\text{pol}} &= C_R \\sigma_{\\text{unpol}}, \\nonumber \\\\\n\\text{Left--like}: \\sigma_{\\text{pol}} &= C_L \\sigma_{\\text{unpol}}. \\nonumber\n\\end{align}\n\nModels with t--channel mediators usually have multiple terms with\ndifferent polarisation behaviour and do not fall into one of the basic\npolarisation classes given in Eq.~(\\ref{eqn:polclasses}). We choose\nthe following polarisation settings for those:\n\\begin{itemize}\n\\item Models with fermionic mediators are classified according to their\n leading term, which is always scalar--like. \n\\item All other models have both scalar--like and vector--like parts of about\n the same size. We analyse them in a vector--like scenario that naturally leads to a\n better background suppression.\n\\end{itemize}\n\n\\subsection{Standard Model Background for Monophotons}\nWe consider the two leading dominant Standard Model background contributions\nafter selection, determined with a full \\textsc{Ild} (International Linear Detector concept) detector simulation\n\\cite{BartelsThesis, Bartels:2012ex}. All numbers here and in the following\nparagraphs refer to the nominal \\textsc{Ilc} center of mass energy of\n\\unit{500}{\\GeV} \\cite{Phinney:2007gp}. We also consider the case of an\nincreased energy of \\unit{1}{\\text{TeV}} and mention the differences later.\n\\begin{itemize}\n\\item Neutrinos from $\\Ppositron \\Pelectron \\rightarrow\n \\Pnu \\APnu \\Pphoton (\\Pphoton)$ form a polarisation dependent background. The leading\n contribution comes from t--channel $\\PW$--exchange, which only couples to\n left--chiral leptons. Additional smaller contributions come from s--channel\n $\\PZ$--diagrams with both left-- and right--chiral couplings. We also\n consider the case of one additional undetected photon, which contributes\n with a size of roughly $\\unit{10}{\\%}$. \n\\item Bhabha scattering of leptons with an additional hard photon, $\\Ppositron \\Pelectron \\rightarrow\n \\Ppositron \\Pelectron \\Pphoton$ has a large cross section but a very small\n selection efficiency, since both final state leptons must be undetected. It has been determined to give a contribution of the same order\n of magnitude as the neutrino background, after application of all selection\n criteria. It is mostly polarisation independent \\cite{BartelsThesis, Bartels:2012ex}.\n\\end{itemize}\n Other background sources contribute with less\nthan \\unit{1}{\\%} compared to the neutrino background and are therefore omitted. \n\n\\subsection{Data Modeling}\nTo evade the use of a full detector simulation, we build on the results of\nRefs.~\\cite{BartelsThesis, Bartels:2012ex}. For the signal and monophoton neutrino\nbackground, we generate the events by ourselves with the given phase space criteria. \nWe then apply the \\textsc{Ild} estimates for the\nenergy resolution as well as the reconstruction and selection\nefficiencies\\footnote{From here on, the expression `efficiency' abbreviates\n `reconstruction and selection efficiencies'.} and compare the final energy\ndistributions. For the diphoton neutrino and Bhabha background, we model the\nfinal distributions directly from the given results performed with a full detector simulation \\cite{BartelsThesis, Bartels:2012ex}. \n\n\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{energy_neutrino_color.pdf} \n\\put(-45,180){(a)}\n\\hfill\n\\includegraphics[width=0.49\\textwidth]{energy_FS_color.pdf}\n\\put(-45,180){(b)}\n\\caption{Photon energy distribution before and after application of beam effects\n (\\textsc{Isr} + beamstrahlung) and detector effects (resolution +\n efficiency) for a) unpolarised neutrino background and b) unpolarised FS\n scalar signal with $M_\\chi = \\unit{150}{\\GeV}$. Distributions are normalised to\n $10^6$ tree level events.}\n\\label{img:photonenergy}\n\\end{figure*}\n\n\\begin{figure}\n\\centering\n\\vspace{-0.55cm}\n\\includegraphics[width=0.49\\textwidth]{backgrounds_new.pdf}\n\\vspace{-0.8cm}\n\\caption{Photon energy distributions of the most dominant background\n contributions (stacked) compared to an example signal (FS Scalar, $M_\\chi =\n \\unit{150}{\\GeV}$) with a total cross section of \\unit{100}{\\femto\\barn}. All spectra are taken after selection for an unpolarised\n initial state.}\n\\label{img:backgroundcontributions}\n\\vspace{0.775cm}\n\\begin{tabular}{r@{\\qquad}crr c@{\\quad} rr c@{\\quad} rr }\n\\hline\n$P^-\/P^+$ && \\multicolumn{2}{c}{$\\Pnu \\Pnu \\Pphoton$} && \\multicolumn{2}{c}{$\\Pnu \\Pnu \\Pphoton \\Pphoton$} &&\\multicolumn{2}{c}{$\\Ppositron \\Pelectron \\Pphoton$} \\\\\n\\hline\n\\hline\n$0\/0$ && 2257 & (2240) && 226 & (228) && 1218 & (1229) \\\\\n$+0.8\/-0.3$ && 493 & (438) && 49 & (43) && 1218 & (1204) \\\\\n$-0.8\/+0.3$ && 5104 & (5116) && 510 & (523) && 1218 & (1227) \\\\\n\\hline\n\\end{tabular}\n\\caption{Total number of events in the different background\n sources after application of all selection criteria. The numbers are\n given for an integrated luminosity of \\unit{1}{\\femto\\reciprocal\\barn} in different\n polarisation settings. Numbers in brackets are taken from\n Ref.~\\cite{BartelsThesis} which employed a proper detector\n simulation.}\n\\label{tbl:neventsperscenario}\n\\end{figure}\n\n\nFor the generation of signal and monophoton neutrino events we use\n\\textsc{C}alc\\textsc{hep} \\cite{Pukhov:1999gg}. We produce signal events for all benchmark\nscenarios with dark matter masses ranging from \\unit{1}{\\GeV} to\n\\unit{240}{\\GeV}. To avoid collinear and infrared divergences,\nwe limit phase space in the event generation to $E_\\gamma \\in \\left[\\unit{8}{\\GeV}, \\unit{250}{\\GeV}\n \\right]$ and $\\cos \\theta_\\gamma \\in \\left[ -0.995, 0.995\n \\right]$. Initial State Radiation (\\textsc{Isr})\n and beamstrahlung significantly change the width and position of the neutrino\n \\PZzero--resonance, Fig.~\\ref{img:photonenergy}a), and are taken into account. We\n set the accessible parameters in \\textsc{C}alc\\textsc{hep} according to the \\textsc{Ild}\n Letter of Intent \\cite{Abe:2010aa} to \\unit{645.7}{\\nm} for the bunch size, \\unit{0.3}{\\mm}\n for the bunch length and a total number of particles per bunch of $2 \\cdot\n 10^{10}$. \n\n\nThe finite resolution of the detector components and the use of \nselection criteria to reduce beam--induced background are taken into account by\napplying the following steps to both signal and background data. First we shift the photon\nenergy, given in \\GeV, according to a Gaussian distribution by taking into account the estimated\n resolution of the \\textsc{Ild} detector components \\cite{Abe:2010aa},\n\\begin{align}\n\\frac{\\Delta E_\\gamma}{E_\\gamma} =\n\\frac{\\unit{16.6}{\\%}}{\\sqrt{E_\\gamma \\text{ in \\GeV}}} \\oplus \\unit{1.1}{\\%}.\n\\end{align}\nAfterwards we further limit the phase space to reduce background processes in\n the \\PZzero resonance peak at \\unit{242}{\\GeV} and additional collinear photons from \\textsc{Isr},\n\\begin{align}\nE_\\gamma &\\in \\left[ \\unit{10}{\\GeV}, \\unit{220}{\\GeV} \\right], \\\\\n\\cos \\theta_\\gamma &\\in \\left[ -0.98, 0.98 \\right]. \\nonumber\n\\end{align}\nThe additional angular cut ensures a good photon reconstruction within\nthe detector. Finally a random elimination of events is used\nto simulate the efficiency factor for reconstruction and selection\ndetermined in Ref.~\\cite{BartelsThesis}. The efficiency consists of an\nenergy dependent part $\\epsilon_1$ and a constant part $\\epsilon_2$\nthat are applied successively,\n\\begin{align}\n\\epsilon_1 &= \\unit{97.22}{\\%} - (E_\\gamma \\text{ in \\GeV}) \\cdot\\unit{0.1336}{\\%}, \\label{eqn:efficiency1}\\\\\n\\epsilon_2 &= \\unit{96.8}{\\%}. \\nonumber\n\\end{align}\nFig.~\\ref{img:photonenergy} shows how these settings affect the signal and\nbackground spectrum and Fig.~\\ref{img:backgroundcontributions} shows a stacked histogram of the dominant background processes along with a example dark matter signal.\n\n\n\n\n\\begin{table*}\n\\centering\n\\begin{tabular}{r@{\\quad}r@{\\qquad}c rr@{\\qquad}c rr@{\\qquad}c rrrr}\n\\hline \n&&&&&&&&&&&& \\\\ [-2.ex] \n$P^-\/P^+$ & $N_{\\text{B}}$ && $\\Delta^\\text{stat}_{50}$ & $\\displaystyle\\tilde{\\Delta}^\\text{stat}_{500}$ && $\\Delta^\\text{sys}_{P}$\n& $\\displaystyle\\Delta^\\text{sys}_{\\tilde{P}}$ && $\\displaystyle \\Delta^\\text{tot}_{50 P}$ &\n$\\Delta^\\text{tot}_{50 \\tilde{P}}$ & $\\displaystyle \\tilde{\\Delta}^\\text{tot}_{500 P}$\n& $\\displaystyle \\tilde{\\Delta}^\\text{tot}_{500 \\tilde{P}}$ \\\\ [1ex]\n\\hline\\hline\n0\/0 & 184998 && & && & && & & & \\\\\n\\hline\n$+0.8$\/$+0.3$ & 97568 && 312 & 99 && 312 & 125 && 441 & 336 & 327 & 159 \\\\\n$+0.8$\/$+0.6$ & 102365 && 320 & 101 && 385 & 154 && 500 & 355 & 398 & 184 \\\\\n\\hline\n$+0.8$\/$-0.3$ & 87974 && 297 & 94 && 169 & 68 && 341 & 304 & 193 & 116 \\\\\n$+0.8$\/$-0.6$ & 83177 && 288 & 91 && 104 & 42 && 307 & 291 & 138 & 100 \\\\\n\\hline\n$-0.8$\/$+0.3$ & 341597 && 584 & 185 && 351 & 140 && 682 & 601 & 396 & 232 \\\\\n$-0.8$\/$+0.6$ & 404970 && 637 & 201 && 501 & 200 && 811 & 668 & 546 & 284 \\\\\n\\hline\n$-0.8$\/$-0.3$ & 212851 && 461 & 156 && 233 & 93 && 517 & 471 & 275 & 173 \\\\\n$-0.8$\/$-0.6$ & 148478 && 385 & 122 && 337 & 135 && 512 & 408 & 359 & 182 \\\\\n\\hline\n\\end{tabular}\n\\caption{Total amount of background events, $N_{\\text{B}}$, with statistical\n error, $\\Delta^\\text{stat}$, systematic error, $\\Delta^\\text{sys}$, and the total error,\n $\\Delta^\\text{tot}$. The subscripts 50 and 500 denote the integrated\n luminosity in inverse femtobarn. In case of a ten times larger luminosity,\n one will get ten times as many events in all channels; to better compare to the error\n of the low luminosity case,\n we give $\\tilde{\\Delta}^\\text{stat}_{500} \\equiv \\Delta^\\text{stat}_{50}\/\\sqrt{10}$. The polarisation uncertainties are set to\n \\unit{0.25}{\\%} ($P$) and \\unit{0.1}{\\%}\n ($\\tilde{P}$).}\n\\label{tbl:bkgevts}\n\\end{table*}\n\n\\subsection{Analysis}\n\nWe are interested in determining the lower bound on the effective coupling constants that the\n\\textsc{Ilc} can find for each individual model under the assumption that no signal\nevents are measured. We perform a counting experiment by using the\n\\textsc{TRolke} \\cite{Rolke:2004mj} statistical test. We determine the total\nnumber of background events along with its statistical and systematic fluctuation\n$\\Delta N_{\\text{B}}$ and exclude coupling constants which would lead to a\nlarger number of signal events than the \\unit{90}{\\%} confidence interval of the background--only\nassumption. \n\n\n\\subsection{Systematic Uncertainties}\n\nSystematic uncertainties play an important role in determining the\ntotal error on the background, $\\Delta N_B$, and for estimating the\nbounds on the effective couplings. There are two dominant\ncontributions, motivated in Ref.~\\cite{BartelsThesis} which we now discuss.\n\nThe experimental efficiency given in Eq.~(\\ref{eqn:efficiency1}) will\nbe determined at the real experiment by measuring the\n$\\PZzero$--resonance peak, which is theoretically known to a very good\naccuracy. Systematic uncertainties on that value are given by the\nfinite statistics of this measurement and further broadening of the\npeak by unknown beam effects. These errors can be extrapolated down to\nthe dark matter signal region at small photon energies and, since the\nsame efficiency factor is used for signal and background, is highly\ncorrelated between those two. This global uncertainty will therefore\napproximately cancel in the determination of the maximum coupling\n$G_\\text{eff}$.\n\nCancellation will not take place for model dependent effects\nhowever. This is due to the fact that the signal energy distribution \ndepends on the unknown mass of the dark matter particle and the\nunderlying interaction model. Therefore, the correct function\n$\\epsilon(E_\\gamma)$ for the signal will be different from the used\nneutrino background efficiency given in\nEq.~(\\ref{eqn:efficiency1}). Since we do not know the model a priori,\nwe use the same value for both and introduce an error on the\ndetermination of the signal events, $N_S$. Compared to Ref.~\\cite{BartelsThesis}, we use a\nconservative value of $\\Delta \\epsilon = \\unit{2}{\\%}$.\n\nSince the neutrino spectrum depends on the incoming lepton's\npolarisation $P^\\pm$, any fluctuation within those parameters will\ngive additional systematic uncertainties on the number of expected\nbackground events. One can not use the information from measuring the\n$\\PZzero$--resonance in this case to infer information in the low\nenergy signal range because of the polarisation dependence of the\nshape itself. Given the assumed accuracy of at least $\\Delta P\/P =\n\\unit{0.25}{\\%}$ \\cite{Abe:2010aa} with a possible improvement to\n\\unit{0.1}{\\%} at the \\textsc{Ilc}, we can derive the corresponding\nerror on the polarised number of background events. As an example we\nshow the left handed background,\n\\begin{align}\nN_{\\text{pol}} &= (1+P^+)(1-P^-)N_{\\text{unpol}}, \\nonumber \\\\\n\\Delta N_{\\text{pol}} &= \\sqrt{\\left[P^- (1+P^+) \\right]^2 + \\left[P^+ (1-P^-)\n \\right]^2} \\ \\frac{\\Delta P}{P} \\ N_{\\text{unpol}}. \\label{eqn:deltap}\n\\end{align}\nFrom the numbers in Table~\\ref{tbl:neventsperscenario}, we assume an identical\npolarisation dependence for $\\Pnu \\Pnu \\Pphoton$ and $\\Pnu \\Pnu \\Pphoton\n\\Pphoton$ events and no dependence for the Bhabha background. \n\n\\begin{table}[b]\n\\centering\n\\begin{tabular}{rr@{\\quad}rrrrrr}\n\\hline\nIA type & $P^-\/P^+$ & $N_{\\text{S}}$ && $r_{50 P}$ & $r_{50 \\tilde{P}}$ & $r_{500 P}$ &\n$r_{500 \\tilde{P}}$ \\\\\n\\hline\nScalar &$+0.8\/+0.3$ & 620 && 1.41 & 1.85 & \\textbf{1.90} & 3.90 \\\\\n &$+0.8\/+0.6$ & 740 && \\textbf{1.48} & \\textbf{2.08} & 1.86 & \\textbf{4.02} \\\\\n\\hline\nVector &$+0.8\/-0.3$ & 620 && 1.82 & 2.04 & 3.21 & 5.34 \\\\\n& $+0.8\/-0.6$ & 740 && \\textbf{2.41} & \\textbf{2.54} & \\textbf{5.36}\n& \\textbf{7.40} \\\\\n\\hline\nLeft & $-0.8\/+0.3$ & 1170 && 1.72 & 1.95 & \\textbf{2.95} & 5.04 \\\\\n& $-0.8\/+0.6$ & 1440 && \\textbf{1.78} & \\textbf{2.16} & 2.64 & \\textbf{5.07} \\\\\n\\hline\nRight &$+0.8\/-0.3$ & 1170 && 3.43 & 3.85 & 6.06 & 10.09 \\\\\n& $+0.8\/-0.6$ & 1440 && \\textbf{4.69} & \\textbf{4.95} & \\textbf{10.43} & \\textbf{14.4} \\\\\n\\hline\n\\end{tabular}\n\\caption{Determination of the best ratio $r \\equiv\n {N_{\\text{S}}}\/{\\Delta N_{\\text{B}}}$ with $\\Delta N_{\\text{B}}$ given by the different\n total errors determined in Table~\\ref{tbl:bkgevts}. $N_{\\text{S}}$ describes the\nnumber of polarised signal events for the different classes described in Sec.~\\ref{sec:models}\nwith a common reference value of 500 unpolarised events for an integrated\nluminosity of $\\unit{50}{\\femto\\reciprocal\\barn}$. We only show the polarisation signs with the largest ratios. We mark the\nnumbers which lead to the best signal to background ratio in bold.}\n\\label{tbl:sigoverbkgestimate}\n\\end{table}\n\n\n\\begin{table*}\n\\begin{tabular}{l@{\\qquad}r@{\\quad}rr@{\\quad}rr@{\\quad}rrrr}\n\\hline\n &&&&&&&&& \\\\ [-2.ex] \n$P^-\/P^+$ & $N_{\\text{B}}$ & $\\Delta^\\text{S}_{50}$ & $\\displaystyle\\tilde{\\Delta}^\\text{S}_{500}$ & $\\delta^\\text{P}_{P}$\n& $\\delta^\\text{P}_{\\tilde{P}}$ & $\\Delta^\\text{tot}_{50 P}$ & $\\Delta^\\text{tot}_{50 \\tilde{P}}$ & $\\displaystyle\\tilde{\\Delta}^\\text{tot}_{500 P}$\n& $\\displaystyle \\tilde{\\Delta}^\\text{tot}_{500 \\tilde{P}}$ \\\\ [1ex]\n\\hline\n\\hline\n$0\/0$ & 162437 & & & & & & & & \\\\\n\\hline\n$+0.8$\/$+0.3$ & 54649 & 234 & 74 & 380 & 152 & 446 & 279 & 387 & 169 \\\\\n$+0.8$\/$+0.6$ & 62791 & 251 & 79 & 469 & 188 & 531 & 314 & 476 & 203 \\\\\n\\hline\n$+0.8$\/$-0.3$ & 38365 & 196 & 62 & 201 & 82 & 281 & 212 & 210 & 102 \\\\\n$+0.8$\/$-0.6$ & 30223 & 174 & 55 & 125 & 50 & 214 & 181 & 137 & 74 \\\\\n\\hline\n$-0.8$\/$+0.3$ & 357173 & 598 & 189 & 428 & 171 & 735 & 622 & 468 & 255 \\\\\n$-0.8$\/$+0.6$ & 435979 & 660 & 209 & 612 & 245 & 900 & 704 & 647 & 322 \\\\\n\\hline\n$-0.8$\/$-0.3$ & 199561 & 447 & 141 & 284 & 114 & 530 & 461 & 317 & 181 \\\\\n$-0.8$\/$-0.6$ & 120755 & 348 & 110 & 411 & 165 & 538 & 385 & 425 & 198 \\\\\n\\hline\n\\end{tabular}\n\\caption{Total amount of background events ($N_{\\text{B}}$) and different\n error sources (see Table~\\ref{tbl:bkgevts}) for $\\sqrt{s} = \\unit{1}{\\TeV}$.}\n\\label{tbl:bkgevts_1tev}\n\\end{table*}\n\n\n\\begin{table}[b]\n\\begin{tabular}{r@{\\qquad}r@{\\quad}r@{\\quad}r@{\\quad}}\n\\hline\n$P^-\/P^+$ & $\\Pnu \\Pnu \\Pphoton$ & $\\Pnu \\Pnu \\Pphoton \\Pphoton$ & $\\Ppositron \\Pelectron$ \\\\\n\\hline\n\\hline\n$0\/0$ & 2677 & 268 & 304 \\\\\n$+0.8\/-0.3$ & 421 & 42 & 304 \\\\\n$-0.8\/+0.3$ & 6217 & 622 & 304 \\\\\n\\hline\n\\end{tabular}\n\\caption{Simulated and modeled number of events in the different background\n sources after application of all selection criteria for $\\sqrt{s} = \\unit{1}{\\TeV}$. The numbers are\n calculated for an integrated luminosity of \\unit{1}{\\femto\\reciprocal\\barn} in different\n polarisation settings.}\n\\label{tbl:neventsperscenario_1tev}\n\\end{table}\n\n\\begin{table}[b]\n\\begin{tabular}{r@{\\qquad}rr@{\\quad}rrrr}\n\\hline\nModel & $P^-\/P^+$ & $N_{\\text{S}}$ & $r_{50 P}$ & $r_{50 \\tilde{P}}$ & $r_{500 P}$ &\n$r_{500 \\tilde{P}}$ \\\\\n\\hline\n\\hline\nScalar &$+0.8$\/$+0.3$ & 620 & 1.39 & 2.22 & \\textbf{1.60} & \\textbf{3.7} \\\\\n &$+0.8$\/$+0.6$ & 740 & \\textbf{1.39} & \\textbf{2.36} & 1.55 & 3.65 \\\\\n\\hline\nVector &$+0.8$\/$-0.3$ & 620 & 2.21 & 2.92 & 2.95 & 6.08 \\\\\n& $+0.8$\/$-0.6$ & 740 & \\textbf{3.46} & \\textbf{4.09} & \\textbf{5.40}\n& \\textbf{10.00} \\\\\n\\hline\nLeft & $-0.8$\/$+0.3$ & 1170 & 1.59 & 1.88 & 2.50 & \\textbf{4.59} \\\\\n& $-0.8$\/$+0.6$ & 1440 & \\textbf{1.60} & \\textbf{2.05} & 2.23 & 4.47 \\\\\n\\hline\nRight &$+0.8$\/$-0.3$ & 1170 & 4.16 & 5.52 & 5.57 & 11.47 \\\\\n&$+0.8$\/$-0.6$ & 1440 & \\textbf{6.73} & \\textbf{7.96} & \\textbf{10.51} & \\textbf{19.46} \\\\\n\\hline\n\\end{tabular}\n\\caption{Determination of the best ratio $r \\equiv\n {N_{\\text{S}}}\/{\\Delta N_{\\text{B}}}$ (see\n Table~\\ref{tbl:sigoverbkgestimate}) for $\\sqrt{s} =\n \\unit{1}{\\TeV}$. }\n\\label{tbl:sigoverbkgestimate_1tev}\n\\end{table}\n\\subsection{Polarisation Settings}\nPolarisation can be used to significantly increase the number of\nsignal events according to Eq.~(\\ref{eqn:polclasses}) but also\nincreases the systematical contribution to the total background error,\n$\\Delta N_\\text{B}$ via Eq.~(\\ref{eqn:deltap}). We are interested in\nthe settings for each individual model that leads to the largest\n$N_\\text{S}\/\\Delta N_\\text{B}$ ratio allowing for the strictest bounds\non $G_\\text{eff}$. In Table~\\ref{tbl:bkgevts} we give the total number\nof background events in different polarisation settings $P^-$ = $\\pm\n0.8$ and $P^+$ = $\\pm 0.3$\/$\\pm 0.6$ that are feasible at the\n\\textsc{Ilc} \\cite{Phinney:2007gp}. We give the statistical\nfluctuation for integrated luminosities of\n\\unit{50}{\\femto\\reciprocal\\barn} as well as for\n\\unit{500}{\\femto\\reciprocal\\barn}. Since the latter will give ten times as much events in\nall channels, we reduce the statistical error accordingly to give a value\ncomparable to the small luminosity case.\nWe also give the systematic error that is dominated by the polarisation uncertainty for two estimates of the\npolarisation error $\\delta P\/P = \\unit{0.25}{\\%} $ and $\n\\unit{0.1}{\\%}$ \\cite{Helebrant:2008qz}. Finally we give the total errors adding all combinations of\nindividual errors in quadrature.\n\nOn the signal side, we look at the different classes derived in Sec.~\\ref{sec:models} with respect to their polarisation dependence. For comparison,\nwe use a common reference value of 500 unpolarised events for an integrated\nluminosity of $\\unit{50}{\\femto\\reciprocal\\barn}$ and derive the corresponding number of\nevents for polarised input.\n\nWe look for the maximum ratio $r \\equiv N_{\\text{S}} \/ \\Delta N_{\\text{B}}$ and the results for the best settings are displayed in Table~\\ref{tbl:sigoverbkgestimate}. In most cases the largest possible polarisation for the incoming leptons enhances the result. For high statistics and a non--reduced polarisation error, the systematic uncertainty\nfrom increased polarisation may outweigh the gain in the number of signal events though. In those\ncases, which appear only in scalar-- and left--coupling models, less polarised\nbeams lead to better results.\n\n\n\\subsection{Increasing \\boldsymbol{$\\sqrt{s}$} to \\unit{1}{\\text{TeV}}}\n\n\nWe also consider the possibility of a doubled center of mass\nenergy. This changes the previous analysis as follows:\n\\begin{itemize}\n\\item We generate events in a larger photon energy range $E_\\gamma \\in\n \\left[\\unit{8}{\\GeV}, \\unit{500}{\\GeV}\\right]$ and reduce it to the interval $\\left[\n \\unit{10}{\\GeV}, \\unit{450}{\\GeV} \\right]$ after performing the energy resolution\n shift $\\Delta E \/ E$. This again reduces background events from the\n $\\PZzero$--resonance, which now is positioned at \\unit{496}{\\GeV}.\n\\item Dark matter signal processes can now be produced with masses up to \\unit{490}{\\GeV}.\n\\item We use our previously modeled distribution for the Bhabha background\n and rescale it by a factor of 1\/4, taking into account that the full cross\n section for that process is approximately proportional to $1\/s$. \n\\item We use, as a rough approximation, the same \\textsc{Isr}-- and beamstrahlung parameters in\n\\textsc{C}alc\\textsc{hep}, efficiency factors and systematic error estimates. \n\\end{itemize}\n\nTables~\\ref{tbl:bkgevts_1tev}-\\ref{tbl:sigoverbkgestimate_1tev}\nsummarise again the number of background events per background\nscenario, the individual error sources and the determination of the\nbest polarisation setting for the increased center of mass energy. In\ncontrast to the Bhabha cross section that falls mainly according to\n$\\sigma \\propto 1\/s$, the neutrino background gets significant\ncontributions from t--channel $\\PW^{\\pm}$s, which give $s \/ m_W^4$\n--terms in the evaluation of the total cross section. The left--handed\nneutrino contribution therefore gets enhanced whereas the Bhabha\nbackground becomes less dominant in some polarisation channels. This\nleads to a larger relative polarisation error and therefore a larger\nimpact on the size of the background fluctuation. In the end, vector--\nand right--coupling models receive stronger enhancement for polarised\ninput than in the $\\sqrt{s} = \\unit{500}{\\GeV}$ case, whereas the\nother models suffer from the larger impact of polarisation on the\ntotal error and prefer smaller polarisation.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{finalplot_scalars_couplingBounds.pdf} \\hfill\n\\includegraphics[width=0.45\\textwidth]{finalplot_all12_couplingBounds.pdf} \\\\\n\\includegraphics[width=0.45\\textwidth]{finalplot_fermions_couplingBounds.pdf} \\hfill\n\\includegraphics[width=0.45\\textwidth]{finalplot_fermions2_couplingBounds.pdf} \\\\\n\\includegraphics[width=0.45\\textwidth]{finalplot_all1_couplingBounds.pdf} \\hfill\n\\includegraphics[width=0.45\\textwidth]{finalplot_all2_couplingBounds.pdf} \\\\\n\\caption{\\unit{90}{\\%} exclusion limits on the effective couplings accessible at the \\textsc{Ilc} with $\\sqrt{s} =\n \\unit{1}{\\TeV}$. We only give effectively allowed regions for models with\n dimensionless fundamental couplings $g$. }\n\\label{img:ilcbounds}\n\\end{figure*}\n\n\\section{Introduction}\n\n\nWeakly interacting massive particles (\\textsc{Wimp}s) are one of the leading candidates to solve the dark matter puzzle \\cite{Bertone:2004pz}. Primarily this is due to the fact that a neutral particle that interacts with roughly the strength of the weak force, naturally gives the correct relic abundance. In addition many theoretical models predict that the masses of these \nstates should exist around the scale of electroweak symmetry breaking, e.g.\\ Supersymmetry (\\textsc{Susy}) \\cite{Martin:1997ns,Drees:2004jm}, Universal Extra Dimensions (\\textsc{Ued}) \\cite{Appelquist:2000nn}, Little Higgs \\cite{ArkaniHamed:2001nc} etc.\n\nCurrently, this \\textsc{Wimp} paradigm is being actively explored in a number of different ways. Perhaps the most well known \nare the direct detection searches that aim to observe interactions between the dark matter and an atomic nucleus \\cite{Goodman:1984dc}. As these are extremely low rate experiments, the detectors are typically placed deep underground to reduce \nbackground. The annihilation of dark matter into Standard Model particles in high density regions of our universe offers another potential method to see a signal e.g.\\ \\cite{Bouquet:1989sr}.\n\nIn particle colliders here on Earth the same interactions may be probed in the production of dark matter. Unfortunately, the fact that \\textsc{Wimp}s are neutral and only weakly interacting means that they cannot be detected directly in these experiments. Therefore collider based searches must rely on particles produced in combination with the dark matter candidates. If dark matter is produced directly, one possibility is to use initial state radiation (\\textsc{Isr}), such as gluon jets, or photons, that will recoil against the \\textsc{Wimp}s.\n\nThis idea was first explored in a model independent approach for the International Linear Collider (\\textsc{Ilc}) using mono-photons in a non-relativistic approximation \\cite{Birkedal:2004xn,Konar:2009ae}. Later, detailed detector studies have been performed to understand the full capabilities of the \\textsc{Ilc} for such a signature \\cite{Bartels:2007cv,Bartels:2009fa,Bartels:2010qv,Bartels:2012ex,Bernal:2008zk}. Furthermore the same signature has been considered in the case of \\textsc{Susy} \\cite{Dreiner:2006sb,Dreiner:2007vm}. At the \\textsc{Lhc} (Large Hadron Collider) and Tevatron similar signals have also been studied but with a mono-jet signal \\cite{Cao:2009uw,Bai:2010hh,Fox:2011pm,Goodman:2010ku,Goodman:2010yf,Beltran:2010ww,Rajaraman:2011wf,Bai:2012he,Cheung:2012gi}. All of these papers used the idea of parameterising the dark matter interactions in the form of effective operators. This has the advantage that the bounds can be compared with those coming from direct detection and also that a non-relativistic approximation is not required to compare with the relic density measurement. These methods have now been used by the \\textsc{Lhc} experiments to set bounds on different effective operators that are competitive with other methods \\cite{Chatrchyan:2012pa,ATLAS-CONF-2012-084}. In addition, \\textsc{Lep} (Large Electron-Positron Collider) data has been re-interpreted to determine corresponding constraints \\cite{Fox:2011fx}.\n\nIn this paper we take the effective field theory approach to dark matter and apply this to an \\textsc{Ilc} search \n\\cite{Kurylov:2003ra,Beltran:2008xg,Agrawal:2010fh}. To apply the effective field theory in a consistent way we \nassume that the dark matter particles can only interact with the Standard Model fields via a heavy mediator. \nThe mediator is always assumed to be too heavy to be produced directly at the \\textsc{Ilc} and thus can be \nintegrated out. For our model choices we consider the possibility that the dark matter candidate could be a scalar, \na Dirac (or Majorana) fermion or a vector particle. The same choices are taken for the heavy mediator \nand all combinations are considered. The collider phenomenology can vary significantly, depending on \nwhether the mediator is exchanged in the $s$- or $t$-channel and consequently we examine both. In addition, we also study the different ways in which the mediator can couple to both the dark matter and Standard Model particles. We note that using the effective field theory approach allows us to move away from the non-relativistic approximation that had previously been used in \\textsc{Ilc} studies. This can be especially important if the dark matter candidate happens to be light.\\footnote{The mass determination of a light neutralino dark matter candidate at the \\textsc{Ilc} has been discussed in Ref.~\\cite{Conley:2010jk}.}\n\nFor all models we compare the reach of the \\textsc{Ilc} with the bounds derived from direct and indirect detection. We also calculate the couplings expected to lead to the correct relic density and see whether the \\textsc{Ilc} can probe these regions of parameter space. We also note that an \\textsc{Ilc} search is complementary to that at the \\textsc{Lhc} thanks to the different initial state.\n\nThe paper is laid out as follows. We begin in Sec.~\\ref{sec:models} by explaining how we derive the effective field theories for the dark matter interactions and we explicitly give the Lagrangian for both the full and effective theory. We also describe the benchmark models that we use throughout the study. In Sec.~\\ref{sec:astro} we describe the various astrophysical constraints on our effective theories. We begin with the calculation of the relic density abundance before moving on to explain the bounds from direct and indirect detection.\n\nSection~\\ref{sec:ilc} describes in detail the potential search for dark matter at the \\textsc{Ilc}. Here we explain the calculation of the signal rate and the dominant backgrounds that were considered. In addition we detail how the \\textsc{Ilc} detectors are modeled to account for relevant experimental effects. We find that the polarisation of incoming beams is particularly important for many models of dark matter to discriminate the signal and background. We also investigate the advantage of a doubling of the \\textsc{Ilc} energy to $\\sqrt{s}=1$~TeV. \n\nIn Sec.~\\ref{sec:results} we present the results of the paper. We begin by examining the potential bounds of the \\textsc{Ilc} on the effective coupling of the dark matter model at the collider. Afterwards, we combine these results with those from direct and indirect detection to understand for which models and mass ranges the \\textsc{Ilc} presents a unique opportunity to discover dark matter. Finally in Sec.~\\ref{sec:conclusions} we conclude and summarise the main results of our work.\n\n\n\\section{Models}\n\\label{sec:models}\n\n\n\\subsection{General Motivation}\n\\noindent\nThe idea of parametrising the interaction of a dark matter particle\nwith Standard Model particles by using effective operators is not new,\nsee for example Refs.~\\cite{Agrawal:2010fh, Beltran:2008xg,\n Zheng:2010js, Yu:2011by, Fox:2011fx, Bai:2010hh}. Many authors\nconstruct a list of effective 4--particle-interactions with Lorentz--invariant\ncombinations of $\\gamma^\\mu$, $\\partial_\\mu$ and\nspinor--\/vector--indices up to mass dimension 5 or 6. In many cases\nthere is no explanation how those operators may arise in an underlying\nfundamental theory. That makes it difficult to judge how exhaustive\nthe lists of operators are, whether interference between different\noperators should be taken into account and how the effective model is\nconnected to realistic fundamental theories and their couplings.\n\nWe follow the effective approach introduced in \\cite{Agrawal:2010fh} by\nstarting from different fundamental theories with given renormalisable\ninteractions between Standard Model fermions and the\n hypothesized dark matter particles that are mediated by a very massive\nparticle. From these theories we deduce effective 4--particle--vertices for\nenergies significantly smaller than the mass of the mediator. Working\nwith these effective operators, one can deduce information about the\neffective coupling and propagate this information to the parameters of\nthe corresponding underlying fundamental theory. The effective \napproach allows us to reduce the dimensionality of the parameter space\nand more easily compare the different experimental searches.\n\n\n\\subsection{Deriving Effective Lagrangians}\nWe start with a list of fundamental Lagrangians taken from\n\\cite{Agrawal:2010fh}. However we do not perform a non--relativistic\napproximation, since we are interested in the phenomenology of\n this Lagrangian at a high energy\nexperiment and therefore the results for our effective operators\ndiffer. We also use a different method to evaluate the effective\nvertices, motivated in Ref.~\\cite{Haba:2011vi}, which uses the path\nintegral formalism.\n\nWe give one explicit example for the derivation of the effective\noperators and only mention specific peculiarities for the other cases,\nwhich are apart from that calculated similarly. Let $\\psi$ be a\nStandard Model fermion and $\\chi$ a complex scalar field representing\nthe dark matter candidate. For our example, we assume the mediator to\nbe a real scalar field, $\\phi$, with mass $M_\\Omega$ (we will keep\nthis notation for the mediator mass throughout). The relevant terms in\nthe UV completed Lagrangian are then given by,\n\\begin{align}\n\\mathscr{L_{\\text{UV}}} &= \\frac{1}{2} \\left[\\partial_\\mu \\phi(x) \\right]^2 -\n\\frac{1}{2} M_{\\Omega}^2 \\phi^2(x) - g_\\chi \\chi^\\dagger(x) \\chi(x) \\phi(x) \\nonumber \\\\\n& \\quad - \\bar{\\psi}(x) \\left( g_s +\n i g_p \\gamma^5 \\right) \\psi(x) \\phi(x)\\;, \\\\\n&\\equiv - \\frac{1}{2} \\phi(x) \\square_x \\phi(x) - \\frac{1}{2}\nM_{\\Omega}^2 \\phi^2(x) - F(x) \\phi(x)\\;. \n\\end{align}\nwhere the function $F(x)$ is given by,\n\\begin{align}\nF(x) & \\equiv g_\\chi \\chi^\\dagger(x) \\chi(x) + \\bar{\\psi}(x) \\left( g_s + i\n g_p \\gamma^5 \\right) \\psi(x) \\,.\n\\end{align}\nWe have not included the kinetic terms for $\\chi,\\,\\psi$, as they are\nnot relevant for the computation of the effective Lagrangian. In this\nparticular example, $g_s$, $g_p$ are dimensionless couplings and\n$g_\\chi$ is a dimension one parameter but these definitions can change\ndepending upon the precise model studied and we shall use this notation throughout. \nWe have included the kinetic term\nfor $\\phi$, the heavy mediator field. After integrating out $\\phi$, we\nobtain the effective Lagrangian,\n\\begin{align}\n\\mathscr{L}_\\text{eff} &= \\frac{1}{2 M_\\Omega^2} F^2 \\supset \\frac{g_\\chi}{M_\\Omega^2} \\chi^\\dagger\n \\chi \\bar{\\psi} \\left(g_s + i g_p \\gamma^5 \\right) \\psi \\;.\n\\end{align}\n\\begin{table*}\n\\centering\n\\renewcommand{\\arraystretch}{1.525}\n\\begin{tabular}{c c c l}\n\\hline\n\\multirow{2}{*}{DM} & \\multirow{2}{*}{Med.} & \\multirow{2}{*}{Diagram} &\n$-\\mathscr{L}_{\\text{UV}}$ \\\\\n& & & $-\\mathscr{L}_{\\text{eff}}$ \\\\\n\\hline \\hline\n\\multirow{2}{*}{S} & \\multirow{2}{*}{S} & \\multirow{2}{*}{\\raisebox{-0\\height}{\\includegraphics[width=0.1\\textwidth]{SS_2.pdf}}} & $g_\\chi \\chi^\\dagger \\chi \\phi + \\bar{\\psi} (g_s + i g_p \\gamma^5) \\psi \\phi$ \\\\\n& & & $\\displaystyle \\frac{g_\\chi}{M_\\Omega^2} \\chi^\\dagger \\chi \\bar{\\psi} (g_s + i g_p \\gamma^5) \\psi$ \\\\\n\\hline\n\\multirow{2}{*}{S} & \\multirow{2}{*}{F} &\n \\multirow{2}{*}{\\raisebox{-0\\height}{\\includegraphics[width=0.1\\textwidth]{SF_2.pdf}}}&\n $ \\bar{\\eta} (g_s + g_p \\gamma^5 ) \\psi \\chi + \\bar{\\psi} (g_s - g_p\n \\gamma^5 ) \\eta \\chi^\\dagger$ \\\\\n& & & $ \\displaystyle \\frac{1}{M_\\Omega} \\left[ (g_s^2 - g_p^2) \\bar{\\psi}\n \\psi \\chi^\\dagger \\chi + \\frac{i}{M_\\Omega} \\chi^\\dagger \\bar{\\psi} \\left(g_s^2 + g_p^2 - 2 g_s g_p \\gamma^5 \\right)\\gamma^\\mu \\partial_\\mu \\left( \\psi \\chi \\right) \\right]$ \\\\\n\\hline \n \\multirow{2}{*}{S} & \\multirow{2}{*}{V} &\n \\multirow{2}{*}{\\raisebox{-0\\height}{\\includegraphics[width=0.1\\textwidth]{SV_2.pdf}}}&$\n g_\\chi (\\chi^\\dagger \\partial_\\mu \\chi - \\chi \\partial_\\mu\n \\chi^\\dagger) Z^\\mu + \\bar{\\psi} \\gamma^\\mu (g_l P_L + g_r P_R) \\psi\n Z_\\mu$ \\\\\n& & & $\\displaystyle \\frac{g_\\chi}{M_\\Omega^2} \\bar{\\psi} \\gamma^\\mu \\left( g_l P_L + g_r P_R \\right) \\psi \\left( \\phi^\\dagger \\partial_\\mu \\phi - \\phi \\partial_\\mu \\phi^\\dagger \\right)$\\\\\n\\hline \\hline\n\\multirow{2}{*}{F} & \\multirow{2}{*}{S} &\n\\multirow{2}{*}{\\raisebox{-0\\height}{\\includegraphics[width=0.1\\textwidth]{FS_2.pdf}}}&\n$\\bar{\\chi} \\left(g_s + g_p \\gamma^5 \\right) \\chi \\phi + \\bar{\\psi} \\left( g_s\n + g_p \\gamma^5 \\right) \\psi \\phi$ \\\\\n& & & $\\displaystyle \\frac{1}{M_\\Omega^2} \\bar{\\chi} \\left(g_s + i g_p \\gamma^5 \\right) \\chi \\bar{\\psi} \\left( g_s + i g_p \\gamma^5 \\right)\\psi$\\\\\n\\hline\n\\multirow{2}{*}{F} & \\multirow{2}{*}{V} &\n \\multirow{2}{*}{\\raisebox{-0\\height}{\\includegraphics[width=0.1\\textwidth]{FV_2.pdf}}}&\n $\\bar{\\psi} \\gamma^\\mu (g_l P_L + g_r P_R) \\psi Z_\\mu + \\bar{\\chi}\n \\gamma^\\mu (g_l P_L + g_r P_R) \\chi Z_\\mu $ \\\\\n& & & $\\displaystyle \\frac{1}{M_\\Omega^2} \\bar{\\psi} \\gamma^\\mu \\left( g_l P_L + g_r P_R \\right) \\psi \\ \\bar{\\chi} \\gamma_\\mu \\left(g_l P_L + g_r P_R \\right) \\chi$\\\\\n\\hline\n\\multirow{2}{*}{F} & \\multirow{2}{*}{tS} &\n \\multirow{2}{*}{\\raisebox{-0\\height}{\\includegraphics[width=0.1\\textwidth]{FtS_2.pdf}}}&$\\bar{\\chi}\n \\left(g_s + g_p \\gamma^5 \\right) \\psi \\phi + \\bar{\\psi} \\left( g_s +\n g_p \\gamma^5 \\right) \\chi \\phi$ \\\\\n&&& $\\displaystyle \\frac{1}{M_\\Omega^2} \\bar{\\psi} \\left(g_s - g_p \\gamma^5 \\right) \\chi \\bar{\\chi} \\left( g_s + g_p \\gamma^5 \\right)\\psi $ \\\\\n\\hline\n\\multirow{2}{*}{F} & \\multirow{2}{*}{tV} &\n \\multirow{2}{*}{\\raisebox{-0\\height}{\\includegraphics[width=0.1\\textwidth]{FtV_2.pdf}}}&$\\bar{\\psi}\n \\gamma^\\mu (g_l P_L + g_r P_R) \\chi Z_\\mu + \\bar{\\chi} \\gamma^\\mu (g_l\n P_L + g_r P_R) \\psi Z_\\mu $ \\\\\n&&& $ \\displaystyle \\frac{1}{M_\\Omega^2} \\bar{\\psi}\n \\gamma^\\mu \\left(g_l P_L + g_r P_R \\right) \\chi \\bar{\\chi} \\gamma_\\mu \\left(g_l P_L + g_r P_R \\right) \\psi $ \\\\\n\\hline \\hline\n\\multirow{2}{*}{V} & \\multirow{2}{*}{S} &\n\\multirow{2}{*}{\\raisebox{-0\\height}{\\includegraphics[width=0.1\\textwidth]{VS_2.pdf}}}\n& $-\\chi^\\mu \\chi_\\mu \\phi + \\bar{\\psi} (g_s + i g_p \\gamma^5)\\psi \\phi$ \\\\\n&&& $\\displaystyle - \\frac{g_\\chi}{M_\\Omega^2} \\chi^\\mu \\chi_\\mu \\bar{\\psi} \\left( g_s + i g_p \\gamma^5 \\right)\\psi$\\\\\n\\hline\n\\multirow{2}{*}{V} & \\multirow{2}{*}{F} &\n \\multirow{2}{*}{\\raisebox{-0\\height}{\\includegraphics[width=0.1\\textwidth]{VF_2.pdf}}}&\n $ - \\bar{\\eta} \\gamma^\\mu (g_l P_L + g_r P_R ) \\chi_\\mu\n + \\bar{\\psi} \\gamma^\\mu (g_l P_L + g_r P_R) \\eta \\chi_\\mu^\\dagger$ \\\\\n&&& $\\displaystyle \\frac{1 }{M_\\Omega} \\left[g_l g_r \\bar{\\psi} \\gamma^\\nu\n \\gamma^\\rho \\psi \\ \\chi^\\dagger_\\nu\\chi_\\rho + \\frac{i}{M_\\Omega} \\chi^\\dagger_\\nu \\bar{\\psi}\n \\gamma^\\nu\\gamma^\\mu \\gamma^\\rho \\left(g_l^2 P_L + g_r^2P_R\n \\right) \\partial_\\mu \\left( \\psi \\chi_\\rho \\right) +\\right]$\\\\\n\\hline\n\\multirow{2}{*}{V} & \\multirow{2}{*}{V} &\n \\multirow{2}{*}{\\raisebox{-0\\height}{\\includegraphics[width=0.1\\textwidth]{VV_2.pdf}}}&\n $i g_\\chi \\left[ Z_\\mu \\chi^\\dagger_\\nu {\\partial \\chi}^{\\mu \\nu} + Z_\\mu\n \\chi_\\nu \\partial \\chi^{\\mu \\nu} + \\chi^\\dagger_\\mu\n \\chi_\\nu \\partial Z^{\\mu \\nu} \\right] + \\bar{\\psi} \\gamma_\\mu (g_l P_L + g_r P_R ) \\psi$ \\\\\n&&& $\\displaystyle \\frac{i g_\\chi}{M_\\Omega^2} \\bar{\\psi} \\gamma^\\mu \\left( g_l P_L + g_r\n P_R \\right) \\psi \\left[\\chi^\\nu \\partial \\chi^\\dagger_{\\mu \\nu} - \\chi^{\\dagger, \\nu} \\partial \\chi_{\\mu \\nu} + \\partial^\\nu \\left(\\chi^\\dagger_\\nu\\chi_\\mu - \\chi^\\dagger_\\mu \\chi_\\nu \\right) \\right]$\\\\\n\\hline\n\\end{tabular}\n\\caption{List of interaction vertices for S(calar), F(ermion) and \nV(ector) dark matter, $\\chi$, before and after integrating out the \nheavy mediator scalar field $\\phi$, spinor field $\\eta$ or vector field\n$Z^\\mu$ with mass $M_{\\Omega}$. $\\psi$ denotes the Standard Model fermion.\n$\\partial X^{\\mu \\nu} \\equiv \\partial^\\mu X^\\nu - \\partial^\\nu X^\\mu$.\ntS and tV denote cases where the mediator is exchanged in the $t$-channel.}\n\\label{tbl:allmodels}\n\\end{table*}\nCases with different spin for the dark matter or the mediator particle\nare evaluated similarly. We only want to give some special remarks:\n\\begin{itemize}\n\\item For spin--$1\/2$ mediators, the Dirac propagator\nhas only one power of $M_\\Omega$ in the denominator,\n\\begin{align}\n\\frac{1}{\\slashed{p} - M_\\Omega} \\approx -\\frac{1}{M_\\Omega} - \\frac{\\slashed{p}}{M_\\Omega^2}. \n\\end{align}\nWe therefore get two effective vertices after expanding the Lagrangian up to order $1\/M_\\Omega^2$.\n\n\\item Some effective operators give derivatives on the Standard Model fermion\n fields. These are not negligible, since they only vanish if the\n Dirac equation $i \\slashed{\\partial} \\psi = m \\psi$ can be used and\n the fermion mass $m$ is small. This is not the case for e.g.\\ heavy\n quark contributions in the annihilation sector and processes with\n off--shell fermions.\n\n\\item We use the same list of effective operators for the cases of real scalar\n ($\\chi = \\chi^\\dagger$), real vector ($\\chi_\\mu = \\chi^\\dagger_\\mu$)\n or Majorana fermion \\cite{Denner:1992vza} dark matter\n fields. However, we would like to mention that for consistency we do\n not introduce additional factors of $\\nicefrac{1}{2}$ in the\n couplings as is often done in the case of real fields.\n\n\\end{itemize}\n\nThe full list of models with their respective fundamental and effective\nLagrangians is given in Table~\\ref{tbl:allmodels}. Note that all Lagrangians are\nhermitian by construction.\n\n\\subsection{Benchmark Models}\n\n\nThe effective operators described above have multiple\nindependent parameters to describe the effective coupling, for example $g_\\chi, g_l, g_r$ and $M_\n\\Omega$ in the scalar dark matter, vector mediator (SV) case or\n$g_s, g_p$ and $M_\\Omega$ in the fermion dark matter, scalar mediator (FS) case in Table~\\ref{tbl:allmodels}. \n\\begin{table}\n\\centering\n\\begin{tabular}{l@{\\quad}l@{\\quad}l}\n\\hline\nOperators & Definition & Name \\\\\n\\hline \\hline\nSS, VS, FS, & $g_p = 0$ & scalar \\\\\n FtS, FtSr: & $g_s = 0$ & pseudoscalar \\\\\n\\hline\nSF, SFr: & $g_p = 0, M_\\Omega = \\unit{1}{\\TeV}$ & scalar\\_low \\\\\n & $g_p = 0, M_\\Omega = \\unit{10}{\\TeV}$ & scalar\\_high \\\\\n & $g_s = 0, M_\\Omega = \\unit{1}{\\TeV}$ & pseudoscalar\\_low \\\\\n & $g_s = 0, M_\\Omega = \\unit{10}{\\TeV}$ & pseudoscalar\\_high \\\\\n\\hline\nSV, FV, FtV, & $g_l = g_r$ & vector \\\\\n FtVr, VV: & $g_l = -g_r$ & axialvector \\\\\n & $g_l = 0$ & right--handed \\\\\n\\hline\nVF, VFr: & $g_l = g_r, M_\\Omega = \\unit{1}{\\TeV}$ & vector\\_low \\\\\n & $g_l = -g_r, M_\\Omega = \\unit{10}{\\TeV}$ & vector\\_high \\\\\n & $g_l = g_r, M_\\Omega = \\unit{1}{\\TeV}$ & axialvector\\_low \\\\\n & $g_l = -g_r, M_\\Omega = \\unit{10}{\\TeV}$ & axialvector\\_high \\\\\n\\hline\nFVr : & $g_l = 0$ & right--handed \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\caption{Benchmark models with specific values for the coupling constants shown in Table \\ref{tbl:allmodels}.}\n\\label{tbl:constraints}\n\\end{table}\nConsidering the full range of parameters would lead to a\nplethora of scenarios, well beyond the scope of this paper. Thus we restrict our \nanalysis to specific benchmark models (see\nTable~\\ref{tbl:constraints}) with constraints on the individual\ncouplings such that only one overall multiplicative factor\nremains. The effective coupling constant $G$ for each model is then\ndefined as $G \\equiv g_ig_j\/ M_\\Omega^2$.\nFor models with fermionic\nmediators, the leading term has only a $1\/M_\\Omega$ dependence, which is\nwhy we define $G \\equiv g_ig_j\/ M_\\Omega$ for these. We also choose two\npossible values for $M_\\Omega$ to represent different\nsuppression scales of the respective second order terms. Models with\nreal fields that are trivially connected to the corresponding complex\ncases by multiplicative prefactors are not taken into account\nseparately. We also omit models with left--handed couplings that are\nrelated to the respective right--coupled cases. Information on these\ncan easily be extracted from the related models by rescaling the\ncorresponding result accordingly.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nRecently, the H1~\\cite{H1} and ZEUS~\\cite{zeus} collaborations at HERA\nannounced an anomaly at high-$Q^2$ in the $e^+p\\to eX$ neutral current\n(NC) channel. Using a combined accumulated luminosity of\n$34.3\\invpb$ in $e^+p\\to eX$ mode at $\\sqrt{s}=300\\gev$, the two\nexperiments have\nobserved 24 events with $Q^2>15000\\gev^2$ against a Standard Model (SM)\nexpectation of $13.4\\pm1.0$, and 6 events with $Q^2>25000\\gev^2$ against\nan expectation of only $1.52\\pm0.18$. Furthermore, the\nhigh-$Q^2$ events are clustered at Bjorken-$x$ values near 0.4 to 0.5.\n\nA number of authors have presented proposals for new physics which\nmight explain the NC anomaly, including: \nnew contact interactions~\\cite{bkmrw,altarelli,contact},\n$s$-channel leptoquark production~\\cite{bkmrw,altarelli,leptoquark},\nand R-parity violating supersymmetry~\\cite{altarelli,rparity}, and\nrelated proposals~\\cite{others}.\nThere exist a variety of criticisms for each of the proposed ideas, from\ntheir highly speculative nature, to very concrete flavor problems which\nthey all share~\\cite{bkmrw}. As such, another somewhat less\nspeculative idea has been proposed~\\cite{tung}\\ in which the anomaly in the\nHERA data is simply further evidence of our inability to calculate, and often \neven reliably fit, the parton probability distribution\nfunctions (PDFs) resulting from the non-perturbative dynamics\ninside the proton. That is, Kuhlmann, \\etal~\\cite{tung},\nhave suggested a way by which the behavior of the PDFs at large-$x$ can be \nmodified to explain part, or all, of the HERA NC data without disrupting\nthe fits of the old low-$Q^2$, low-to-moderate-$x$ data. \n\nIt is actually a simple exercise to show that increasing the parton\ndensities at $Q^2\\sim20,000\\gev^2$ and $x\\sim0.4$ to 0.5 can in fact \nfit the NC data.\nIt is a far more complicated question whether these changes are consistent\nwith all other world data.\nWe will not consider here the validity of the claim that such a fit can be\ndone which is consistent with the low-$Q^2$ data. Instead we will show that\nthere can be dramatic consequences for the HERA data itself in the charged\ncurrent mode at high-$Q^2$. We will find that the HERA data already\nrules out a number of theoretically acceptable scenarios for modifying the\nPDFs to agree with the NC data.\n\nHERA is capable of running in two modes: $e^-p$ and $e^+p$. In the former mode\nH1 and ZEUS have accumulated $1.53\\invpb$ of data but have observed\nno statistically significant deviations from the SM. Further, the experiments\ndifferentiate between final state $eX$ and $\\nu X$, where the neutrino is\nidentified through its missing $p_T$. H1 has also announced its\nfindings in the $e^+p\\to\\nu X$ charged current (CC) channel. They find\n3 events at $Q^2>20000\\gev^2$ with an expectation of $0.74\\pm0.39$, but\nno events with $Q^2>25000\\gev^2$. ZEUS has not announced its CC data as of\nthis date.\nAlthough compared to the NC data, the present CC data is much sparser,\none can still conclude from it that there can be no deviations from the SM\nby more than a factor of 2 or 3 in that channel. Whether there is in fact\nany deviation at all is still too uncertain to say. (Explanations of the\npossible CC excess involving non-SM physics have been considered\nby~\\cite{altbkmrCC}.) However, given the \nsizes of the effects which we are going to find, more data in $e^+p$, and \nespecially $e^-p$, mode will provide very strong constraints on the\nviability of this suggestion.\n\n\n\\section{The proposal}\n\nThe class of proposals considered by Kuhlmann, \\etal~\\cite{tung},\nfor explaining the HERA anomaly can be thought of as\nvariations on the so-called ``intrinsic charm'' scenario\\cite{charm}\nin which it is posited that there is some non-perturbative, valence\ncontribution to the charm quark distributions at $Q^2\\to0$.\n(Current fits assume that there is no valence\ncharm component in the proton and generate non-zero\ncontributions at higher $Q^2$ only through perturbative renormalization\ngroup flow, \\ie,\ngluon splitting. In this scenario one can think of the valence\nstructure of the proton being $uudc\\bar c$ versus the usual $uud$.)\n\nThough the underlying dynamics (and motivation) may be subtle, \nthe proposal itself is straightforward: increase by hand\nthe parton densities inside the proton at $x\\sim0.5$ and large $Q^2$ to\nthe point that the NC cross-section of the Standard Model\nmatches that observed at HERA. Such an effect can arise naturally from \nnon-perturbative dynamics at low $Q^2$ if the dynamics produce a narrow\n``bump'' in the PDFs at low $Q^2$ and very large $x\\sim1$. This bump would\nmigrate down to lower $x$ as one flows (through the renormalization group) \nup to higher $Q^2$. Such a structure is difficult to rule out; the data\nat low $Q^2$ and very high $x$ is limited and extractions of the \nstructure functions are problematic due to non-perturbative and higher-twist\neffects which can be important at large $x$. Fits available now typically\nuse only data with $x\\lsim0.8$.\nThe proposal in \\cite{tung}\\ emphasized enhancements to the\n$u$-quark density rather than that of the $c$-quark, but the difference is\nirrelevant from the point of view of this paper because electroweak\nphysics does not distinguish among the generations.\nEnhancement of any or all of the parton densities\ncould in principle explain the HERA data.\n\nThis proposal has strong advantages\n{\\sl and} disadvantages. In its favor, it requires no new physics beyond the\nStandard Model and thus automatically solves the flavor problem associated\nwith many of the new physics interpretations~\\cite{bkmrw}. \nTo its detriment, it invokes\nnon-perturbative QCD effects which are not calculable and cannot be\npredicted {\\sl ab initio}, and its consistency with the moderate\n$Q^2$ data taken at both HERA and the Tevatron has not been fully studied. \n\n\\section{High-$Q^2$ tests at HERA}\n\nPutting aside any questions of how well such a proposal can really do \nat explaining the HERA NC data while\nremaining consistent with all other world data, one can ask:\nwhat are the consequences of such a suggestion\nfor the NC process in $e^-$ mode, and for CC processes in either mode?\n\nThe NC result is simple: in this scenario, the NC cross-sections in both\n$e^-p$ and $e^+p$ modes scale by approximately the same amount. The $Z$\ncouplings introduce a small helicity dependence that keeps the two modes\nfrom being exactly the same. However, the differences between the two are\nless than 10\\% because the photon contribution typically dominates the \ntotal cross-section.\nThus this proposal predicts that once HERA has\naccumulated enough data in $e^-p$ mode, they will observe a NC anomaly there\nas well, and it should be roughly the same size as that observed in $e^+p$.\n\nIn discussing the CC predictions, a number of complications arise,\nmost stemming from the fact that the $x$ and $Q^2$ dependences of the \nNC and CC cross-sections are somewhat different. Nonetheless,\nwe will present fairly precise heuristic arguments about the sizes of CC \neffects, which we will then check in a full numerical calculation.\n\nOne key simplification is this: in the SM, even at high-$Q^2$,\nthe NC scattering is largely dominated by virtual photon exchange\nin the $t$-channel. This need not have been so, since at $Q^2>>m_Z$ there\nis no additional kinematic suppression of the $Z$ contributions; however,\nthe $Z$ coupling to quarks is generally smaller than that of the\nphoton. For the arguments that follow, we will ignore the $Z$ contributions\nand reintroduce them only when going to the full calculation in the next\nsection. \n\nFor photon exchange alone, the NC cross-section at high-$Q^2$ behaves as:\n\\beq\n\\frac{1}{x}\\,\\frac{d\\sigma_{NC}}{dx}\\propto u(x)+\\bar u(x)\n+\\frac{1}{4}\\left\\{d(x)+\\bar d(x)\\right\\}+\\cdots\n\\eeq\nwhere $u(x)$ is the $u$-quark parton probability distribution function (PDF)\ninside a proton,\n$\\bar u(x)$ is the $u$-antiquark\nPDF, {\\sl etc.}, and the ellipses represent heavier ($s$, $c$, $b$, $t$) \nquarks. The factor of $1\/4$ is the relative charge-squared of \n$u$- and $d$-type quarks.\n\nThere are four orthogonal classes of changes to the PDF's that can be\nconsidered, each corresponding to enhancing the densities of either\n$u$, $\\bar u$, $d$ or $\\bar d$ individually. \nThat we do not have to consider changes to\nthe charm and strange densities is clear, since the physics in question cannot\ndistinguish $u$ from $c$, or $d$ from $s$. (This is not a general statement\nfor all experiments at all $Q^2$. For example, it does not hold even at HERA\nif H1 and\/or ZEUS could tag prompt charm production.)\nEach of these four cases leads to a distinctive CC signature.\n\nWe will parameterize the effect of an ``intrinsic quark'' component on\nthe PDFs by:\n\\beq\nq(x)=q_0(x)+q_{\\rm int}(x)\\equiv q_0(x)\\epsilon_q(x)\n\\eeq\nwhere $q(x)$ is the total parton distribution,\n$q_0(x)$ is the usual fit distribution (we will use the CTEQ3 set whenever\nwe have to make an explicit choice~\\cite{cteq}, similar results would\nfollow from the MRS sets~\\cite{mrs}), \n$q_{\\rm int}(x)$ is the intrinsic component and\n$\\epsilon_q(x)\\geq1$ parameterizes the effects of the enhanced\ncomponent. (The $Q^2$ dependence of $q(x)$ and $\\epsilon(x)$\nis implicit and of little relevance to what follows since we will only \nconsider scattering within a small range of $Q^2$ values; therefore\nthe renormalization group running with $Q^2$ can be ignored.)\nFor simplicity, suppose that $\\epsilon_q(x)$\nis exactly 1 everywhere except in a small range of $x$ centered on $x=x_0$\nfor which it is much greater and roughly constant (\\ie, it is roughly a\ntop hat distribution):\n\\beq\n\\eps_q(x)=1+\\eta_q\\theta(x-x_0+\\delta)\\theta(x_0+\\delta-x)\n\\label{epsq}\n\\eeq\nwhere $\\eta_q>0$, $\\theta(x)$ is the usual step function, and $\\delta$ is\nthe half-width of the top hat.\nThough the numerical arguments do not rely on making this\nassumption, it does make the pedagogy simpler.\n\nTo begin, suppose that the correct fit to the HERA NC data is achieved\nby changing only $u(x)$ such that $x_0$, $\\delta$ and $\\eps_u(x_0)$\nhave certain fit values. This will be the canonical scenario to which we\ncompare all others. For example, suppose that instead of changing the $u(x)$\nPDFs, we would like to do a fit for which only $d(x)$ is changed.\nThen to produce the same NC cross-section that $\\eps_u(x_0)$ provided in\nthe $u$-case, $\\eps_d(x_0)$ must shift by a larger amount, though at the same\n$x=x_0$. That the shift must be larger is clear, because $d$-quark effects\non the NC cross-section are suppressed both by electric charge (the 1\/4)\nand by the smaller $d$-quark content in the proton, $d_0(x_0)\/u_0(x_0)$.\nIn terms of an equation, setting the shift due\nto the $\\eps_u$ in the one case equal to the new shift by $\\eps_d$:\n\\beq\n\\eps_u(x_0)u_0(x_0)+\\frac{1}{4}d_0(x_0)=u_0(x_0)+\\frac{1}{4}\\eps_d(x_0)\nd_0(x_0),\n\\eeq\ngiving\n$\\eps_d=1+4(\\eps_u-1)(u_0\/d_0)$, with all functions evaluated at $x=x_0$.\nTo get the same effect by enhancing $\\bar u(x)$ alone requires $\\eps_{\\bar u}\n=1+(\\eps_u-1)(u_0\/\\bar u_0)$ at $x=x_0$. Similarly, changing $\\bar d(x)$ alone\nis identical to the case of $d(x)$, but with $d\\to\\bar d$ in the expression\nabove.\n\nWhat effects do these changes have on the charged current? \nIn terms of the PDFs, the CC cross-sections scale as:\n\\beq\n\\frac{1}{x}\\,\\frac{d\\sigma^+_{CC}}{dx\\,dy}\\propto d(x)(1-y)^2+\\bar u(x)\n+\\cdots,\n\\eeq\n\\beq\n\\frac{1}{x}\\,\\frac{d\\sigma^-_{CC}}{dx\\,dy}\\propto u(x) + \\bar d(x)(1-y)^2\n+\\cdots\n\\eeq\nwhere the $+[-]$ superscript denotes scattering in $e^+p[e^-p]$ mode, and\n$y$ has its usual definition in deep inelastic scattering. In the\ncenter-of-mass frame, $y=\\sin^2(\\vartheta\/2)$ where $\\vartheta$ is the $e^\\pm$\nscattering angle; its presence in the expressions above follows trivially\nfrom angular momentum conservation in the $(V-A)$ \nscattering process. Note also that\nfor the usual PDFs at large $x$, to a very good approximation\n$\\sigma^+_{CC}\\propto d(x)(1-y)^2$ and $\\sigma^-_{CC}\\propto u(x)$.\n\nFor the moment, in order to examine the effects of enhancing the PDFs\non the CC differential cross-section, we will restrict ourselves\nto the region around $x=x_0$ of half-width $\\delta$. We will\ndenote the differential cross-section in this small region\n$d\\sigma^\\pm_{CC}(x_0)$ as a shorthand. In the first case discussed above,\nin which only $u(x_0)$ is changed in response to the NC data,\nthe CC signal in $e^+$ mode is unchanged, while\n$d\\sigma^-_{CC}(x_0)$ increases by $\\eps_u(x_0)$. Thus the same relative\nchange in the NC data will also occur in the $e^-p$ CC data, at least in the\nneighborhood of $x=x_0$.\nHowever, because the kinematic dependence on $x$ (at high-$Q^2$)\nis roughly the same in $e^+u\\to e^+u$ as in $e^-u\\to\\nu d$, the relative\nscaling of the NC to CC in the $x_0$-region will hold for all $x$.\n(The preceding statement is exact in the limit $Q^2\\gg m_W^2$ and when the\n$Z$ contribution to NC can be ignored. Since these two conditions are not\nsimultaneously satisfied in general, there will be important corrections\nto the results of these heuristic arguments. Nonetheless, the results derived\nhere do not differ greatly from the exact results, as we will see.)\nThus, to have a concrete example, if $\\eps_u(x_0)$ is chosen to double the\nhigh $Q^2$ NC cross-section, it will also (approximately)\ndouble the high-$Q^2$ CC\ncross-section in $e^-$ mode, while having no effect in $e^+$ mode.\n\nThe effect on the CC is more marked in the case of using $d(x)$ to explain the\nNC anomaly. As we said above, to have the same effect on the NC data as\n$\\eps_u$, the $\\eps_d$ must be roughly $4[u_0(x_0)\/d_0(x_0)]$ times bigger\nthan the corresponding $\\eps_u$ would have been. And since to a good\napproximation\n\\beq\nd\\sigma^+_{CC}(x_0)\\propto\\eps_d(x_0)\\simeq4\\frac{u_0(x_0)}{d_0(x_0)}\n\\eps_u(x_0),\n\\eeq\nit will scale by the same amount. Consider again our\nexample, where we demanded that $\\eps_u(x_0)$ double the high-$Q^2$ NC data.\nFor $x_0\\simeq0.5$ one finds $u_0(x_0)\\simeq4d_0(x_0)$. Then to fit the\nsame NC data, $\\eps_d(x_0)$ must be $4\\cdot4=16$ times larger than\n$\\eps_u(x_0)$. Moving this enhancement to the CC one finds that\n$d\\sigma^+_{CC}(x_0)$ increases by a factor of $16\\eps_u(x_0)$,\nwhile $\\sigma^-_{CC}$ is unchanged. Again the explicit kinematic\ndependences on $x$ are (approximately) \nthe same in the CC and NC, so the overall $\\sigma^+_{CC}$ will increase \nby roughly a factor of 32 in the high-$Q^2$ region.\n\nThe same arguments go through for $\\bar u$ and $\\bar d$. For $\\bar u$,\n$d\\sigma^+_{CC}(x_0)$ scales by a factor\n\\beq \nd\\sigma^+_{CC}(x_0)\\propto 1+\\frac{\\eps_u-1}{(y-1)^2}\\,\\frac{u_0}{d_0}\n\\simeq\\frac{\\eps_u}{(1-y)^2}\\,\\frac{u_0}{d_0},\n\\eeq\nwhile $\\sigma^-_{CC}$ is unchanged. For explaining the HERA anomaly we \nwould be interested in $Q^2\\simeq2\\times10^4\\gev^2$ and $x\\simeq\n0.5$, and thus $y=Q^2\/(xs)\\simeq0.5$.\nThen in our recurring example wherein the NC signal is doubled,\n$d\\sigma^+_{CC}(x_0)$ scales by a factor of $16\\eps_u$, leading to an\noverall scaling of $\\sigma^+_{CC}$ again by 32.\n\nIn the final case, with enhanced\n$\\bar d(x)$, $d\\sigma^-_{CC}(x_0)$ scales by a factor \n\\beq\nd\\sigma^-_{CC}(x_0)\\propto 1+4(\\eps_u-1)(1-y)^2\n\\eeq\nwhile $\\sigma^+_{CC}$ is unchanged. In our recurring example,\nwith $y\\simeq0.5$, we find $d\\sigma^-_{CC}(x_0)$ scales by approximately\n$\\eps_u$, so that the full $\\sigma^-_{CC}$ at high-$Q^2$ is expected to be\ndouble the usual SM prediction.\n\nThe various scalings of the CC mode are summarized in Table~\\ref{tungtable}.\n\\begin{table}\n\\centering\n\\begin{tabular}{|c|c|c|c|} \\hline\nParton & NC Factor & CC${}^+$ Factor & CC${}^-$ Factor \\\\ \\hline & & & \\\\\n$u$ & $\\eps_u$ & 1 & $\\eps_u$ \\\\ & & & \\\\\n$d$ & $1+4(\\eps_u-1)\\fract{u_0}{d_0}$ & $1+4(\\eps_u-1)\n\\fract{u_0}{d_0}$ & 1 \\\\ & & & \\\\\n$\\bar u$ & $1+(\\eps_u-1)\\fract{u_0}{\\bar u_0}$ & $1+\\frac{\\eps_u-1}{(1-y)^2}\n\\,\\fract{u_0}{d_0}$\n& 1 \\\\ & & & \\\\\n$\\bar d$ & $1+4(\\eps_u-1)\\fract{u_0}{\\bar d_0}$ & 1 &\n$1+4(\\eps_u-1)(1-y)^2$ \\\\ & & & \\\\ \\hline\n\\end{tabular}\n\\label{tungtable}\n\\caption{For each parton is shown the relative sizes of contributions needed\nto explain the HERA NC data, as well as the predictions for the enhancement\nof the CC signal, $d\\sigma^\\pm_{CC}(x_0)$. All quantities are implicitly\nfunctions of $x$ and are to be evaluated at $x=x_0$. These results are derived\nin the heuristic scenario discussed in the text. Note that for $\\eps_u\\gg1$\nthe scaling factors are independent of $\\eps_u$.}\n\\end{table}\nFor each parton, if we assume that it alone must have its density scaled to\nexplain the HERA NC data, the size of the enhancement needed (relative to\nthe enhancement $\\eps_u$ for the $u$-quark) is given in the second column.\nIn the third and fourth column are then shown the relative enhancements\nof $d\\sigma^\\pm_{CC}(x_0)$ given a fit to the NC.\n\nSome profound results can already be extracted from these simple \nconsiderations. If the PDF's of $(u, d, \\bar u, \\bar d)$ are each \nchanged respectively so as \nto double the high-$Q^2$ NC data, then the $e^+$ CC data will increase by \nfactors of about\n(1, 32, 32, 1) while the $e^-$ CC data will increase by (2, 1, 1, 2).\nSince doubling the NC cross-section at high-$Q^2$ is roughly \nconsistent with the\nHERA data, we can conclude that we either expect extremely large enhancements\nin the $e^+$ CC signal, or none at all. The data from H1 is not consistent\nwith an extremely large enhancement (such as a factor of 32), so we can\nconclude that: (1) changing the PDF's of a $d$-type quark or a $\\bar u$-type\nquark to explain the HERA NC data is inconsistent with the HERA CC data;\n(2) any other explanation\ninvolving modifications to the $u$-type or $\\bar d$-type PDFs will not\nlead to a CC signal in $e^+$ mode, but will have one in $e^-$ mode;\n(3) if HERA sees an enhancement in the $e^+p$ CC channel\ncomparable to the one in the NC channel, more than one\nPDF must be modified. Note that\n(2) above does not preclude a small CC enhancement in $e^+$ mode; but\nany such enhancement will not be enough to also explain the NC data.\n\nUp until now, our arguments have ignored the complications introduced by\nkeeping the $Z$ contributions in the NC, and by the non-zero $W$ mass in the\nCC. To show how these results are modified in a complete calculation, we have\nnumerically evaluated the CC signals which would be induced by fitting to the\nNC anomaly. In Figure~\\ref{figure},\nwe have considered each of the $u$, $d$, $\\bar u$ and $\\bar d$ cases\nseparately as a solution to the HERA NC anomaly. (We are only using the H1\nNC and CC data since as of this writing ZEUS has not announced their CC \nresults.) We have\nchosen in each case for $\\eps_q(x)$ to have a top-hat form as in \nEq.~(\\ref{epsq}) with $x_0=0.45$ and $\\delta=0.05$. This region in $x$\nenvelopes the bulk of the H1 high $Q^2$ events. The overall normalization\n$\\eta_q$ is chosen to provide the best possible $\\chi^2$ fit to the H1 data.\n(The best fit values of $\\eta_q$ for $q=u,d,\\bar u,\\bar d$ were found to be\n7.5, 95, 640, 450 using the CTEQ3M PDFs. $\\eta_q$ for the sea quarks shows\na strong dependence on the PDFs used, because of the uncertainty in the sea\nquark distributions at moderate $x$ and large $Q^2$. However the overall \nenhancement of the CC signal is independent of the choice.)\nThen we have plotted the resulting \n$d\\sigma^\\pm_{CC}\/dQ^2$ for each possibility.\n\\begin{figure}\n\\centering\n\\epsfxsize=5in\n\\hspace*{0in}\n\\epsffile{tung.eps}\n\\caption{Differential cross-section for CC $e^+p$ and $e^-p$ scattering\nas a function of $Q^2$. Dotted lines are the SM prediction with unmodified\nPDFs; solid lines are with modified PDFs.}\n\\label{figure}\n\\end{figure}\n\nThe Figure is divided into two frames for the CC processes $e^+p\\to\\bar\\nu X$\nand $e^-p\\to\\nu X$ separately. For the $e^+p$ mode, the enhancement of\nthe CC cross-sections is marked, consistent with our heuristic derivation of\nan enhancement factor of $\\sim32$. For the $e^-p$ mode, the\nenhancements are much smaller, once again consistent with our expectation\nof factors of $\\sim2$ only. \n\nNotice from the figure that the peak enhancements as a function\nof $Q^2$ can be much larger than those of the total integrated cross-sections,\nwhich we give in Table~\\ref{fittable}. There\nwe have done the $Q^2$ integration in the Figure,\nfor $Q^2>10,000\\gev^2$ and $Q^2>20,000\\gev^2$.\n\\begin{table}\n\\centering\n\\begin{tabular}{|c|c||c|c|} \\hline\nPDF & Mode & $Q^2>10,000\\gev^2$ & $Q^2>20,000\\gev^2$ \\\\ \\hline\n$u$ & $e^-p$ & 1.9 & 2.9 \\\\\n$d$ & $e^+p$ & 10 & 21 \\\\\n$\\bar u$ & $e^+p$ & 5.5 & 25 \\\\\n$\\bar d$ & $e^-p$ & 1.3 & 1.2 \\\\ \\hline\n\\end{tabular}\n\\label{fittable}\n\\caption{Multiplicative enhancements of the CC cross-sections for $Q^2$ above\nthe indicated value, in the mode indicated. The PDFs in the first column\nhave been changed to give a best fit to the H1 NC data.}\n\\end{table}\nThe most important H1 cuts have been included in the calculation:\n$y<0.9$ and $p_{T,{\\rm miss}}>50\\gev$. Our pedagogical derivation of the\nenhancements has been shown to work reasonably well, though not exactly.\nThe two PDFs which affect the $e^+p$ mode both induce very large corrections\nin the CC if they are invoked to explain the NC data. However the two\nPDFs which affect the $e^-p$ mode induce small, but observable, corrections.\nGiven high statistics, HERA should be able to probe even the hardest case,\nthat of changing $\\bar d(x)$ to explain the NC data. With the current data,\nit already appears that altering $d(x)$ or $\\bar u(x)$ cannot be invoked to\nexplain the NC anomaly.\n\nThe possibility exists of probing more complicated combinations of \nmodifications because\nthe cross-sections at HERA are linear in the PDFs. Therefore in a scenario in\nwhich several of the PDFs are modified so that the total modification is\na sum of individual parton modifications, weighted by some $\\alpha_i$, then\nthe resulting CC signals are also linear combinations of those shown, again\nweighted by the $\\alpha_i$. For example, if enhancements to the $u$ and $d$ are\nsuch that each one contributes half of the NC excess, then the CC signal\nwill be an average of the individual signals for $u$ and $d$. In particular,\nthe intrinsic charm scenario, for which $\\eps_c=\n\\eps_{\\bar c}$ is responsible for explaining the HERA data, the\n$d\\sigma^+_{CC}(x_0)$ is scaled by $1+(\\eps_u-1)(u_0\/2d_0)\/(1-y)^2$ and\n$d\\sigma^-_{CC}(x_0)$ is scaled by\n$(1+\\eps_u)\/2$ where $\\eps_u=1+2(\\eps_c-1)(c_0\/u_0)$.\nThis linear behavior provides\na powerful, and simple, way to disentangle the form of the modifications\ngiven the size of the excesses over SM in the two CC modes. Unfortunately,\nwith only 2 data points for 4 unknowns, one cannot deduce the full answer\nusing CC data alone. However, it is already enough to rule out the\nintrinsic charm scenario as an explanation of the NC data, given the\nnumbers in Table~\\ref{fittable}.\n\n\n\\section{Conclusions}\n\nModifying the PDFs to explain the HERA $e^+p$ NC data implies\nan immediate modification (by about the same relative size) of\nthe $e^-p$ NC data,\nand also implies striking patterns of modification in the CC data.\nSuch modifications should be easy to observe given the current size of\nthe NC anomaly. Further, they already rule out the intrinsic charm scenario\n(with equal modifications to $c(x)$ and $\\bar c(x)$),\nor any other attempt to use modifications of $d$-type or $\\bar u$-type\ndistribution functions to account for the anomaly in the HERA NC data.\nOur results for the CC channel are summarized in Table~\\ref{fittable}\nand Figure~\\ref{figure}.\n\nIf an excess in the CC signal is detected at HERA in the $e^+p$ data\nof the size currently suggested by H1, then one must go to a scenario in\nwhich more than one PDF are modified, but such that one dominates the \ncontribution to the $e^+p$ NC excess. This will lead to an interesting \nsignal at HERA in the $e^-p$ CC mode. In particular,\nonce significant luminosity has been collected in $e^-p$ mode at HERA, one \nshould also be able to probe solutions to the NC anomaly which involve\nonly changing the distributions of $u$- and $\\bar d$-type quarks.\n\n\n\n\\section*{Acknowledgments}\n\nWe wish to thank J.~Ellis, E.~Weinberg, F.~Wilczek and C.P.~Yuan\nfor useful conversations.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzcjgx b/data_all_eng_slimpj/shuffled/split2/finalzzcjgx new file mode 100644 index 0000000000000000000000000000000000000000..32f8d5aaec002a32f405b3c1159f80f0c6654001 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzcjgx @@ -0,0 +1,5 @@ +{"text":"\n\\section{Introduction} \\label{sec:intro}\n\nBrown dwarfs (BDs) are sub-stellar objects below the hydrogen burning limit ($\\lesssim$80\\;$\\mbox{$M_{\\rm Jup}$}$) but massive enough to fuse deuterium ($\\gtrsim$13\\;$\\mbox{$M_{\\rm Jup}$}$) \\citep{Spiegel_2011,Dieterich_2014}. After their formation, BDs cool radiatively and follow mass-luminosity-age relationships. The degeneracy in these parameters, especially in mass and age, plus assumptions about the initial conditions of BD formation, have long been a major difficulty in calibrating the evolutionary and atmospheric models for substellar objects \\citep{Burrows_1989,Baraffe_2003,Joergens_2006,Gomes_2012,Helling_2014,Caballero_2018}. BDs for which we can measure these parameters independently can benchmark the evolutionary models. As a result, BDs in multiple systems are especially important. Their ages can be determined from the characteristics of their host stars or associated groups, assuming coevality \\citep[e.g.][]{Seifahrt_2010,Leggett_2017}, and some of these BDs can have their masses measured dynamically \\citep[e.g.][]{Crepp+Johnson+Fischer+etal_2012,Crepp+Gonzales+Bechter+etal_2016,Dupuy+Liu_2017,Dieterich_2018,Brandt_2019,Brandt_2020}. \n\n$\\varepsilon$~Indi~B\\xspace, discovered by \\cite{Scholz_2003_EpsIndiB_discovery}, is a distant companion to the high proper motion ($\\sim$4.7 arcsec\/yr) star $\\varepsilon$~Indi\\xspace. It was later resolved to be a binary brown dwarf system by \\cite{McCaughrean_2004}, who estimated the two components of the binary, $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace, to be T dwarfs with spectral types T1 and T6, respectively. It was the first binary T dwarf to be discovered and remains one of the closest binary brown dwarf systems to our solar system; {\\sl Gaia } EDR3 measured a distance of $3.638 \\pm 0.001$\\,pc to $\\varepsilon$~Indi\\xspace~A \\citep{Lindegren+Klioner+Hernandez+etal_2020}. Their proximity makes $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace bright enough and their projected separation wide enough to obtain high quality, spatially resolved images and spectra. And their relatively short orbital period of $\\approx$10\\,yr allows the entire orbit to be traced in a long-term monitoring campaign. Being near the boundary of the L-T transition, $\\varepsilon$~Indi~Ba\\xspace is especially valuable for understanding the atmospheres of these ultra-cool brown dwarfs \\citep{Apai_2010,Goldman_2008,Rajan_2015}.\n\n\\cite{King_2010} carried out a detailed photometric and spectroscopic study of the system, and derived luminosities of $\\log_{10} L\/L_{\\odot} = -4.699 \\pm 0.017$ and $-5.232 \\pm 0.020$ for Ba\\xspace and Bb\\xspace, respectively. They found that neither a cloud-free nor a dusty atmospheric model can sufficiently explain the brown dwarf spectra, and that a model allowing partially settled clouds produced the best match. The relative orbit monitoring was still ongoing at the time, so a preliminary dynamical system mass of $121 \\pm 1\\,\\mbox{$M_{\\rm Jup}$}$ measured by \\citet{Cardoso_2012} was assumed by the authors to derive mass ranges of 60-73 $\\mbox{$M_{\\rm Jup}$}$ and 47-60 $\\mbox{$M_{\\rm Jup}$}$ for Ba\\xspace and Bb\\xspace based on their photometric and spectroscopic observations.\n\n\\cite{Cardoso_2012} and \\cite{Dieterich_2018} both used a combination of absolute and relative astrometry to obtain individual dynamical masses of\n$\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace.\n\\cite{Cardoso_2012} used NACO \\citep{NACO_ins_paper_1,NACO_ins_paper_2} and FORS2 \\citep{Appenzeller+Fricke+Furtig+etal_1998,FORS2_ADC_1997} imaging to measure $77.8\\pm0.3$\\,$M_{\\rm Jup}$ and $61.9\\pm0.3$\\,$M_{\\rm Jup}$ for $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace, respectively, with a parallax of $263.3\\pm 0.3$\\,mas. This parallax disagreed strongly with the {\\sl Hipparcos } parallax { of $\\varepsilon$~Indi\\xspace~A} \\citep{ESA_1997,vanLeeuwen_2007}. Fixing parallax to the {\\sl Hipparcos } 2007 value of $276.1 \\pm 0.3$\\,mas, \\cite{Cardoso_2012} instead obtained masses of $68.0\\pm0.9$\\,$M_{\\rm Jup}$ and $53.1\\pm0.3$\\,$M_{\\rm Jup}$. \\cite{Dieterich_2018} used a different data set to measure individual masses of $75.0\\pm0.8$\\,$M_{\\rm Jup}$ and $70.1\\pm0.7$\\,$M_{\\rm Jup}$ with a parallax of $276.9\\pm0.8$\\,mas, consistent with the {\\sl Hipparcos } distance. The three dynamical mass measurements---two from \\cite{Cardoso_2012} and one from \\cite{Dieterich_2018}---disagree strongly with one another. The highest masses of $\\gtrsim$75\\,$M_{\\rm Jup}$ are in tension with the predictions of substellar cooling models even at very old ages \\citep{Dieterich_2014}.\n\nIn this paper, we use relative orbit and absolute astrometry monitoring of $\\varepsilon$~Indi~B\\xspace from 2005 to 2016 acquired with the VLT to measure the individual dynamical masses of $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace. Much of this data set overlaps with that used by \\cite{Cardoso_2012}, but we have the advantage of a few more epochs of data, {\\sl Gaia } astrometric references \\citep{Lindegren+Klioner+Hernandez+etal_2020} and a better understanding of the direct imaging system thanks to years of work on the Galactic center \\citep{Gillessen+Eisenhauer+Trippe+etal_2009,Plewa+Gillessen+Eisenhauer+etal_2015,Gillessen+Plewa+Eisenhauer+etal_2017}. We structure the paper as follows. We review the stellar properties and its age in Section \\ref{sec:stellarprop}. Section \\ref{sec:data} presents the VLT data that we use, and Section \\ref{sec:positions} describes our method for measuring calibrating the data and measuring the position of the two BDs. Section \\ref{sec:photvar} presents our search for periodic photometric variations, while in Section \\ref{sec: orbit fit} we fit for the orbit and mass of the pair. In Section \\ref{sec:BDtests} we discuss the implications of our results for models of substellar evolution. We conclude with Section \\ref{sec:conclusions}.\n\n\\section{Stellar Properties} \\label{sec:stellarprop}\n\nThe $\\varepsilon$~Indi~B\\xspace system is bound to $\\varepsilon$~Indi\\xspace A (=HIP 108870, HD 209100, HR 8387), a bright K4V or K5V star \\citep{Adams_1935,Evans_1957,Gray_2006}. $\\varepsilon$~Indi\\xspace A has a $2.7^{+2.2}_{-0.4}\\,\\mbox{$M_{\\rm Jup}$}$ planet on a low eccentricity and wide orbit \\citep{Endl+Kurster+Els+etal_2002,Zechmeister+Kurster+Endl+etal_2013,Feng_2019}. The star appears to be slightly metal poor. Apart from a measurement of ${\\rm [Fe\/H]} = -0.6$\\,dex \\citep{Soto+Jenkins_2018}, literature spectroscopic measurements range from ${\\rm [Fe\/H]} = -0.23$\\,dex \\citep{Abia+Rebolo+Beckman+etal_1988} to $+0.04$\\,dex \\citep{Kollatschny_1980}, with a median of $-0.17$\\,dex \\citep{Soubiran_2016}. \n\nSeveral studies have constrained the age of the $\\varepsilon$~Indi\\xspace system via various methods such as evolutionary models, Ca\\,{\\sc ii}~HK age dating techniques, and kinematics. Using a dynamical system mass of $121 \\pm 1 \\mbox{$M_{\\rm Jup}$}$ and evolutionary models, \\citet{Cardoso_2012} predicted a system age of $3.7$-$4.3$\\,Gyr. This age is older than the age of $0.8$-$2.0$\\,Gyr derived from stellar rotation {of $\\varepsilon$~Indi\\xspace A} and the age of $1$-$2.7$\\,Gyr from the Ca \\RomanNumeralCaps{2} activity {of $\\varepsilon$~Indi\\xspace A}, reported in \\cite{Lachaume_1999} { assuming a stellar rotation of $\\sim$20 days}, but is younger than the kinematic estimate of $>$7.4\\,Gyr quoted in the same study. { \\citet{Feng_2019} inferred a longer rotation period of $\\sim$35 days derived from a relatively large data set of high precision RVs and multiple activity indicators for $\\varepsilon$~Indi\\xspace A, and found an age of $\\sim$4 Gyr}. To date, the age of the star still remains a major source of uncertainty in the evolutionary and atmospheric modeling of the system. \n\nWe perform our own analysis on the age of $\\varepsilon$~Indi\\xspace using a Bayesian activity-based age dating tool devised by \\citet{Brandt_2014} and applied in \\cite{Li+Brandt+Brandt+etal_2021}. To do this, we adopt a Ca\\,{\\sc ii} chromospheric index of $\\log R'_{\\rm HK} = -4.72$ from \\citet{Pace_2013}, an X-ray activity index of $R_{X} = -5.62$ from the ROSAT all-sky survey bright source catalog \\citep{Voges_1999}, and Tycho $B_T V_T$ photometry ($B_T = 6.048\\pm0.014$\\,mag, $V_T = 4.826\\pm0.009$\\,mag) from the Tycho-2 catalog \\citep{Hog+Fabricius+Makarov+etal_2000}. The star lacks a published photometric rotation period. Figure \\ref{fig::age} shows our resulting posterior probability distribution, with an age of $3.48_{-1.03}^{+0.78}$ Gyr. This age is somewhat older than the young ages most literature measurements suggest, but is similar to the system age of $3.7$-$4.3$\\,Gyr used by \\citet{Cardoso_2012} for their analysis { based on the preliminary system mass for $\\varepsilon$~Indi\\xspace{} Ba+Bb compared to evolutionary models} { and to the $\\sim$4\\,Gyr age more recently inferred by \\cite{Feng_2019}}. We use our Bayesian age posterior when analyzing the consistency with our dynamical masses with brown dwarf models (Section \\ref{sec:BDtests}).\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.45\\textwidth]{Figures\/age_epsiIndiA.pdf}\n \\caption{Age posterior of $\\varepsilon$~Indi\\xspace A based on the Bayesian activity-age method of \\citet{Brandt_2014}. Our analysis does not use a directly measured rotation period for $\\varepsilon$~Indi\\xspace. The median and 1$\\sigma$ uncertainties are shown by the grey dotted lines; they correspond to $3.48_{-1.03}^{+0.78}$ Gyr.}\n \\label{fig::age}\n\\end{figure}\n\n\\section{Data} \\label{sec:data}\n\n\n\\subsection{Relative Astrometry}\n\nWe measure the relative positions of $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace using nine years of monitoring by the Nasmyth Adaptive Optics System (NAOS) + Near-Infrared Imager and Spectrograph (CONICA), NACO for short \\citep{NACO_ins_paper_1, NACO_ins_paper_2}. We use images taken by the S13 Camera on NACO in the $J$, $H$ and $K_s$ passbands. \nOur images come from Program IDs 072.C-0689(F), 073.C-0582(A), 074.C-0088(A), 075.C-0376(A), 076.C-0472(A), 077.C-0798(A), 078.C-0308(A), 079.C-0461(A), 380.C-0449(A), 381.C-0417(A), 382.C-0483(A), 383.C-0895(A), 384.C-0657(A), 385.C-0994(A), 386.C-0376(A), 087.C-0532(A), 088.C-0525(A), 089.C-0807(A), and 091.C-0899(A), all PI McCaughrean, and 381.C-0860(A), PI Kasper.\n\nThe S13 camera on NACO has a field of view (FOV) of $14'' \\times 14''$ and a plate scale of $\\approx$13.2$\\,{\\rm mas\\,pix^{-1}}$. Most observing sequences consisted of $\\approx$5 dithered images in each filter. The binary system HD~208371\/2 was usually observed on the same nights and in the same mode to serve as an astrometric calibrator. We use a total of 939 images of $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace, taken over 56 nights of observations from 2004 to 2013 for which we have contemporaneous imaging of HD~208371\/2.\n\nWe perform basic calibrations on all of these images. For each night, we use contemporaneous dark images to identify bad pixels and to remove static backgrounds. We construct and use a single, master flat field for all images. We mask pixels for which the flatfield correction deviates by more than 20\\% from its median or for which the standard deviation of the dark frames is more than five times its median standard deviation. We then subtract the median dark image and divide by the flatfield image.\n\nThe data quality varies depending on the observing conditions and the performance of the adaptive optics (AO) system. Therefore, we apply a selection criterion to exclude poor quality data. We first extract the sources in the images using the DAOPHOT program as implemented in the {\\tt photutils} python package \\citep{Stetson_1987, photutils110}. We obtain estimates of the following parameters for $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace: centroid, sharpness (a DAOPHOT parameter that characterizes the width of the source), roundness (a DAOPHOT parameter that characterizes the symmetry of the source), and signal to noise ratio (SNR). We discard images where one or both of the two targets fall outside the field of view, and for the remaining images we apply the following cut-offs in DAOPHOT detection parameters to exclude highly extended, highly elongated and low signal to noise images: sharpness $\\geqslant 0.3$, $- 0.5 \\leqslant $ roundness $\\leqslant 0.5$ and SNR $\\geqslant 25$. We then visually inspect the remaining images to remove ones with bad pixels (cosmic rays or optical defects) landing on or near the target objects, and ones with AO correction artifacts that survived our DAOPHOT cut. Table \\ref{table:relastrodata} summarizes the final data set selected for relative astrometry measurements.\n\n\\begin{deluxetable}{cccc}\n\\tablewidth{0pt}\n\\tablecaption{Relative astrometry data summary \\label{table:relastrodata} }\n\\tablehead{\nDate & \nFilter(s) &\n\\# Frames &\nTotal integration (s)\n}\n\\startdata\n 2004-09-24 & $J$, $H$, $K_s$ & 15 & 150 \\\\\n 2004-11-14 & $J$, $H$, $K_s$ & 14 & 840 \\\\\n 2004-11-15 & $J$ & 5 & 270 \\\\\n 2004-12-15 & $J$, $H$ & 11 & 220 \\\\\n 2005-06-04 & $J$, $H$, $K_s$ & 13 & 780 \\\\\n 2005-07-06 & $K_s$ & 6 & 310 \\\\\n 2005-08-06 & $J$, $H$, $K_s$ & 13 & 780 \\\\\n 2005-12-17 & $J$, $K_s$ & 7 & 210 \\\\\n 2005-12-30 & $J$, $H$, $K_s$ & 14 & 840 \\\\\n 2005-12-31 & $J$, $H$, $K_s$ & 13 & 780 \\\\\n 2006-07-19 & $H$, $K_s$ & 8 & 80 \\\\\n 2006-08-06 & $J$, $H$ & 10 & 100 \\\\\n 2006-09-22 & $J$, $H$, $K_s$ & 15 & 150 \\\\\n 2006-10-03 & $J$, $H$ & 7 & 420 \\\\\n 2006-10-20 & $J$, $H$ & 5 & 300 \\\\\n 2006-11-12 & $J$ & 5 & 300 \\\\\n 2007-06-18 & $J$, $H$, $K_s$ & 12 & 720 \\\\\n 2007-09-09 & $J$, $H$ & 10 & 450 \\\\\n 2007-09-29 & $J$, $H$ & 15 & 900 \\\\\n 2007-11-07 & $J$, $H$ & 10 & 600 \\\\\n 2008-06-05 & $J$, $H$ & 10 & 600 \\\\\n 2008-06-10 & $J$, $H$ & 7 & 70 \\\\\n 2008-06-21 & $J$, $H$ & 10 & 100 \\\\\n 2008-08-25 & $J$, $H$ & 9 & 540 \\\\\n 2008-12-01 & $J$, $H$ & 12 & 720 \\\\\n 2009-06-17 & $J$, $H$, $K_s$ & 12 & 720 \\\\\n 2010-08-01 & $J$, $H$ & 7 & 105 \\\\\n 2010-11-07 & $J$, $H$ & 10 & 300 \\\\\n 2011-07-18 & $J$, $H$, $K_s$ & 13 & 390 \\\\\n 2012-07-18 & $J$, $H$ & 9 & 540 \\\\\n 2012-09-14 & $J$, $H$ & 9 & 540 \\\\\n 2013-06-07 & $J$, $H$ & 10 & 600\n\\enddata\n\\end{deluxetable}\n\n\\subsection{Absolute Astrometry} \\label{subsec:AbsAstData}\n\nThe long term absolute position of $\\varepsilon$~Indi~B\\xspace was monitored with the FOcal Reducer and low dispersion Spectrograph \\citep[FORS,][]{Appenzeller+Fricke+Furtig+etal_1998} installed on ESO's UT1 telescope at the Very Large Telescope (VLT). The FORS system consists of twin imagers and spectrographs FORS1 and FORS2, collectively covering the visual and near UV wavelength. The absolute astrometry monitoring was done with the FORS2 imager coupled with two mosaic MIT CCDs; the camera has a pixel scale of 0$.\\!\\!''$126\/pixel in its unbinned mode and a field of view (FOV) of $\\approx$8$.\\!'6\\times8.\\!'6$. \n\nThe FORS2 monitoring of $\\varepsilon$~Indi~B\\xspace covers a long temporal baseline beginning in 2005 and ending in 2016. Our images come from Program IDs 072.C-0689(D), 075.C-0376(B), 076.C-0472(B), 077.C-0798(B), 078.C-0308(B), 079.C-0461(B), 380.C-0449(B), 381.C-0417(B), 382.C-0483(B), 383.C-0895(B), 384.C-0657(B), 385.C-0994(B), 386.C-0376(B), 087.C-0532(B), 088.C-0525(B), 089.C-0807(B), and 091.C-0899(B), all PI McCaughrean.\nThe FORS-2 focal plane consists of two CCDs, chip1 and chip2. We only consider the data taken with the chip1 CCD. Over the 12 years of absolute position monitoring, 940 images were taken with chip1 over 88 epochs. For the majority of the epochs, 10 dithered images in $I_{\\rm BESSEL}$ filter were obtained, with a 20 second exposure time for each image. We exclude 36 blank image frames over 4 epochs between 2009-8-21 and 2009-11-3, resulting in a final total of 904 image frames for our analysis. A summary of the FORS2 data is given in Table \\ref{tab:absast_obs_log}.\nThese 904 science frames are bias-corrected and flat-fielded using the normalized master values generated from median combination of the flat and bias frames obtained in the same set of observing programs. \n\n\\begin{deluxetable}{cccc}\n\\tablewidth{0pt}\n\\tablecaption{Absolute astrometry data from FORS2\\tablenotemark{a} \\label{tab:absast_obs_log}}\n\\tablehead{Date & \\# Frames & Band &\nTotal integration (s)}\n\\tablewidth{0pt}\n\\startdata\n2005-05-06 & 10 & $I_{\\rm Bess}$ & 200 \\\\\n2005-05-12 & 10 & $I_{\\rm Bess}$ &200 \\\\\n2005-06-08 & 10 & $I_{\\rm Bess}$ &200 \\\\\n2005-07-06 & 10 & $I_{\\rm Bess}$ &200\n\\enddata\n\\tablenotetext{a}{The full observing log is available as an online table; only the first four rows are shown here for reference.}\n\\end{deluxetable}\n\n\\section{Relative and Absolute Positions} \\label{sec:positions}\n\n\\subsection{Point Spread Function (PSF) Fitting} \\label{subsec:joint psf fit}\n\nTo measure the relative separations of the two brown dwarfs in the NACO data, we need to fit their PSFs. \\cite{Cardoso_2012} has demonstrated that Moffat is the best analytical profile for the NACO data compared to Lorentzian and Gaussian. During the epochs when the projected separations of the two brown dwarfs are small, the two PSFs are only separated by one or two full widths at half maximum (FWHM). As a result, the flux near the center of one source has non-negligible contributions from the wings of the other source. This could introduce significant biases in the measured positions if fitting a PSF profile to each source separately. Therefore, we implement a joint fit of the two PSFs using a sum of two elliptical Moffat profiles:\n\\begin{equation}\n \\label{eqn:joint moffat model}\n{\\rm Counts}(x, y) = f_1\\psi_{1}(x, y) + f_2\\psi_{2}(x, y)\n\\end{equation}\nwith\n\\begin{multline}\n\\label{eqn:elliptical moffat}\n \\psi_{i}(x, y) = f_i (1 + c_1(x - x_i)^2 + 2c_2(x - x_i)(y - y_i) \\\\ + c_3(y - y_i)^2 ) ^{-\\beta}\n\\end{multline}\nwhere $\\psi_i$ is a general elliptical 2D Moffat profile centered at \\{$x_i, y_i$\\} with peak intensity $f_i$. Our model is the sum of two such profiles with different fluxes at different locations, sharing the same morphology, i.e., the same \\{$c_1, c_2, c_3$\\}. Instead of fitting for \\{$c_1, c_2, c_3$\\} directly, we fit for three equivalent parameters: \\{${\\rm fwhm}_x, {\\rm fwhm}_y, \\phi$\\}, which are the FWHMs of the elliptical Moffat profile along the x and y axes, and the counter-clockwise rotation angle of the PSF, respectively. These physical parameters are related to \\{$c_1, c_2, c_3, \\beta$\\} through the following equations:\n\\begin{align}\n c_1 &= \\frac{\\cos^2\\phi}{\\sigma_x^2} + \\frac{\\sin^2\\phi}{\\sigma_y^2}\\\\\n c_2 &= \\frac{\\sin 2\\phi}{2\\sigma_x^2} - \\frac{\\sin 2\\phi}{2\\sigma_y^2}\\\\\n c_3 &= \\frac{\\sin^2\\phi}{\\sigma_x^2} + \\frac{\\cos^2\\phi}{\\sigma_y^2}\\\\\n {\\rm fwhm}_{x, y} &= 2\\sigma_{x, y} \\sqrt{(2^{1\/\\beta} - 1)}\n\\end{align} \n\nFor each background subtracted image, we fit for the sum of two PSFs by minimizing $\\chi^2$ over 10 parameters: \\{$x_1, y_1, x_2, y_2, f_1, f_2, {\\rm fwhm}_x, {\\rm fwhm}_y, \\phi, \\beta$\\}. In this case, $\\chi^2$ is defined by:\n\\begin{align}\n\\label{eqn: moffat chisqr}\n\\chi^2 = \\sum_{i}^{n_{pix}} \\frac{(D_{i} - f_1\\; \\psi_{1, i} - f_2\\; \\psi_{2, i})^2}{\\sigma_{i}^2}\n\\end{align}\nWe use scipy's non-linear optimization routines \\citep{2020SciPy-NMeth} to minimize $\\chi^2$ over the 8 non-linear parameters \\{$x_1, y_1, x_2, y_2, {\\rm fwhm}_x, {\\rm fwhm}_y, \\phi, \\beta$\\}, and for each trial set of non-linear parameters, we solve for the best fit linear parameters \\{$f_1, f_2$\\} analytically and marginalize over them.\n\n\\subsection{Calibrations for Relative Astrometry} \\label{subsec: relast calibrations}\n\nIn order to measure precise relative astrometry, we must measure and correct various instrumental properties and atmospheric effects that can alter the apparent separation and position angle (PA) of $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace. In this section we describe our calibrations for the instrument plate scale and orientation, distortion correction and differential atmospheric refraction.\n\n\\subsubsection{Plate scale, Orientation, and Distortion Correction} \\label{subsubsec: platecal}\n\nWe calibrate the plate scale and the north pointing of the NACO S13 camera using NACO's observations of a nearby wide separation binary, HD 208371\/2, observed concurrently with the science data over the $\\sim$10-year relative orbit monitoring period. We calibrate the separation and PA of the binary in NACO data against the high precision measurements from Gaia EDR3 for HD 208371\/2:\n\\begin{align}\n \\label{eq:EDR3 AB sep}\n \\frac{\\rm sep}{\\rm arcsec} &= 8.90612 + 0.00011 \\left({\\rm Jyear} - 2016.0 \\right) \\\\\n \\label{eq:EDR3 AB PA}\n \\frac{\\rm PA}{\\rm degree} &= 348.10345 - 0.00040 \\left({\\rm Jyear} - 2016.0 \\right)\n\\end{align}\nThe uncertainties on these do depend on the epoch, but with proper motion uncertainties $\\lesssim$40\\,$\\mu$as\\,yr$^{-1}$, positional uncertainties are only $\\approx$0.5\\,mas even extrapolated ten years before Gaia. This represents a fractional uncertainty in separation below 10$^{-4}$ and contributes negligibly to our error budget.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{Figures\/NaCoS13platecal.pdf}\n \\caption{Pixel scale and PA zero point calibrations for the NACO S13 camera, derived using the binary HD~208371\/2 as measured by Gaia EDR3.}\n \\label{fig:naco calibrations}\n\\end{figure}\n\n\nTo measure the separation and PA of the calibration binary, we use the Moffat PSF fitting algorithm described in section \\ref{subsec:joint psf fit}. Since the binary is widely separated, a joint PSF fit in this case is effectively equivalent to fitting a single 2D Moffat profile for each star separately {(albeit with the same structure parameters for each star's Moffat function)}. The calibration results are shown in Fig.\\ref{fig:naco calibrations}. We measure an overall average plate scale of $13.260 \\pm 0.001$, but we also note that the plate scale seems to increase slightly with time from 2004 to 2010. Both the plate scale and the increasing trend agree with other measurements in the literature, \\citep{Chauvin_2010,Cardoso_2012}. In \\cite{Cardoso_2012}, the same calibration binary was used to derive the plate scales but a different reference measurement for the binary was used. Adjusting their results to the more precise Gaia measurement of the binary brings their plate scale into agreement with ours. The PA zero point of the instrument varies from observation to observation, and has a long term trend as well. This is in agreement with the analysis in \\cite{Plewa_2018}.\n\nThe distortion correction was shown to be of little significance for the NACO S13 camera \\citep{Trippe_2008} due to the small field of view. For completeness, we still apply the distortion correction derived by \\citet{Plewa_2018_distortion_correction}.\n\n\\subsubsection{Differential Atmospheric Refraction and Annual Aberration}\nThe dominant atmospheric effect that needs to be corrected for is differential atmospheric refraction \\citep{Gubler_1998}. When a light ray travels from vacuum into Earth's atmosphere, it is refracted along the zenith direction and changes the observed zenith angle of the source, making the apparent zenith angle, $z$, deviate from the true zenith angle in the absence of an atmosphere, $z_0$:\n\\begin{equation}\n z = z_0 + R\n\\end{equation}\nwhere R is the total refraction angle experienced by the light ray. The amount of this refraction depends on atmospheric conditions, the wavelength of the incoming light, and the zenith angle of the object. Therefore, for two objects at different positions in the sky and with different spectral types, the total refraction angles are different and can alter the apparent separation and PA of the objects. We can write this differential refraction, $\\Delta R$, in terms of two components, one due to their difference in color, and one due to the difference of their true zenith angles \\citep{Gubler_1998}:\n\\begin{equation}\n \\Delta R = \\Delta R_{\\rm color} + \\Delta R_{\\Delta z_0}\n\\end{equation}\nFor $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace, the second term is {much smaller} as they are separated by only $< 1''$, {and produced negligible effects on the final results compared to the first term. We included both effects for completeness. The total differential refraction can be calculated with:}\n\\begin{equation}\n \\label{eq: ADR}\n \\Delta R = R_2(n_2, z_2) - R_1(n_1, z_1)\n\\end{equation}\nwhere the $n_i$'s are the effective refractive indices of the Earth's atmosphere for the target sources. $n_i$ depends on the effective central wavelength ($\\lambda_i$) of the target in the observed passband, and on observing conditions, most commonly pressure ($P$), temperature ($T$), humidity ($H$) and altitude ($z$). \\citet{Cardoso_2012} calculated the effective central wavelengths for $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace in the $J$, $H$, and $K_s$ bands by integrating high resolution spectra of the two brown dwarfs. To calculate the refractive index, $n_i(\\lambda_i, P, T, H, z)$, we use the models in \\citet{Mathar_2007} covering a wavelength range of $1.3\\;\\mu$m to $24\\; \\mu$m. Then, the total refraction can be approximately expressed as \\citep{smart_1977}:\n\\begin{equation}\n R(n, z) \\approx (n - 1) \\tan(z)\n\\end{equation}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{Figures\/S13_refrac.pdf}\n \\caption{Residual altitude separation of $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace in each band compared to the mean of all bands, before (top panel) and after (bottom panel) applying a correction for differential atmospheric refraction.}\n \\label{fig:refrac consistency}\n\\end{figure}\n\nA comparison of the separations along the zenith direction of $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace are shown in Figure.\\ref{fig:refrac consistency}. We can clearly see the systematic differences between the $J$, $H$, and $K_s$ bands due to differential refraction before the correction. After applying the correction, the three bands are brought to much better agreement as well as having a smaller total scatter around the mean.\n\n{Annual aberration is a phenomenon that describes a change in the apparent position of a light source caused by the observer's changing reference frame due to the orbital motion of the Earth \\citep{Bradley_aberration_discovery, Phipps_relativity_and_aberration}. We correct for the differential annual aberration, the difference in aberration between $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace, in relative astrometry by transforming the measured positions of $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace to a geocentric reference frame using {\\tt astropy}. The effect is generally a small fraction of the relative astrometry error bars and has negligible impact on the relative orbit fit. For absolute astrometry, the aberration is absorbed by the linear component of the distortion correction.}\n\n\\subsubsection{{PSF Fitting Performance and Systematics}} \\label{subsubsec: error inflation}\nIn order to understand how well our PSF fitting algorithm described in Section \\ref{subsec:joint psf fit} performs, we investigate the systematic errors and potential biases of the algorithm in this section, and adjust the errors of our results accordingly. \n\nTo do this, we {crop out boxes around} the stars in the calibration binary, HD~208371\/2, {and use them as} empirical PSFs. {We build a collection of such PSF stamps from the images of the calibration binary based on AO quality and SNR. We use these PSFs stamps and empty background regions of the NACO data to generate mock data sets containing overlapping PSFs}. For each such mock image, we randomly select one empirical PSF from the collection and place two copies of this PSF onto the background of an $\\varepsilon$~Indi~B\\xspace image. We scale the fluxes of the two PSF copies to be similar to those of $\\varepsilon$ Ba\\xspace and Bb\\xspace in a typical image. We then generate a large sample of these mock images at various separations and PAs. Since the calibration binary stars are widely separated, these empirical PSFs are effectively free of nearby star contamination. We then perform the PSF fitting described in Section \\ref{subsec:joint psf fit} on the mock images and compare the measurements to the true, known separations and PAs.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{Figures\/PSF_systematics.pdf}\n \\caption{Root mean square residuals of the measured separations from the true separations of the PSFs in simulated data. Top panel shows the residuals in the radial direction. Bottom panel shows the residuals in the tangential direction in terms of arclength. Arclength is a better indicator of the fitting algorithm's performance than PA, because we expect arclength residuals to be independent of radial separation, but the PA will go up at smaller separation simply due to geometry.}\n \\label{fig:relast error inflation}\n\\end{figure}\n\nThe results for this test are shown in Figure \\ref{fig:relast error inflation}. Each data point is the root mean square residual from fitting 400 mock images at the same separation but with various PAs. The errors we find from these mock data sets are slightly larger but within the same order of magnitude as the scatter in our $\\varepsilon$~Indi~B\\xspace measurements. We also find that the residuals of these mock data measurements do increase as the PSF overlap becomes significant, but they remain at the milliarcsecond level even at a separation equal to the closest separation in the $\\varepsilon$~Indi~B\\xspace data set. The performance for the $K_s$ band is slightly worse due to the large flux ratio of the system in $K_s$. Overall, {our joint PSF fitting algorithm has sub-milliarcsec errors across all three bands for widely separated sources, and has within a few milliarcseconds errors for overlapping sources}. For our final relative astrometry results for $\\varepsilon$~Indi~B\\xspace, we add the systematic errors shown in Figure \\ref{fig:relast error inflation} to the measurement errors of the relative astrometry in quadrature.\n\n\\subsection{Relative Astrometry Results}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{Figures\/relast_PSF_residual.pdf}\n \\caption{Examples of the joint moffat PSF fits to NACO data. Top panel shows an example of negligible PSF overlap. Bottom panel shows an example from the epoch with maximum overlap in NACO data.}\n \\label{fig:relast psf residual}\n\\end{figure}\n\nThe final relative astrometry results are shown in Table \\ref{table:relastroresults}. These are measured by applying the calibrations described in Section \\ref{subsec: relast calibrations} and jointly fitting for the positions of $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace for every selected image, using the PSF fitting method described in Section \\ref{subsec:joint psf fit}. We take the mean and the error on the mean for every epoch, and {add the systematic errors quadratically to the measurement errors} as described in Section \\ref{subsubsec: error inflation}. In Figure \\ref{fig:relast psf residual}, we show a PSF fit for the simple case where the two PSFs are effectively isolated, as well as a PSF fit from the epoch with the closest projected separation and hence maximum PSF overlap. For each case, we take the fit with the median squared residual from that epoch to demonstrate the typical residual level. \n\n\\begin{deluxetable}{ccccc}\n\\tablecaption{Relative astrometry results \\label{table:relastroresults} } \n\\tablehead{\n\\text{Epoch} & \\text{$\\rho$ (arcsec)} & \\text{$\\sigma_{\\rho}${(arcsec)}} & \\text{$\\theta$ (deg)} & \\text{$\\sigma_{\\theta}$ {(deg)}}\n}\n\\startdata\n2004.730 & 0.88310 & 0.00108 & 140.317 & 0.047 \\\\\n2004.869 & 0.89461 & 0.00110 & 140.853 & 0.047 \\\\\n2004.872 & 0.89560 & 0.00126 & 140.814 & 0.067 \\\\\n2004.954 & 0.90200 & 0.00107 & 141.115 & 0.051 \\\\\n2005.423 & 0.93141 & 0.00112 & 142.648 & 0.045 \\\\\n2005.511 & 0.93351 & 0.00126 & 142.888 & 0.054 \\\\\n2005.595 & 0.93654 & 0.00112 & 143.169 & 0.044 \\\\\n2005.959 & 0.94067 & 0.00118 & 144.222 & 0.050 \\\\\n2005.995 & 0.94079 & 0.00117 & 144.352 & 0.049 \\\\\n2005.997 & 0.93987 & 0.00115 & 144.356 & 0.050 \\\\\n2006.546 & 0.92015 & 0.00111 & 146.044 & 0.053 \\\\\n2006.595 & 0.91721 & 0.00107 & 146.230 & 0.049 \\\\\n2006.724 & 0.90802 & 0.00106 & 146.657 & 0.047 \\\\\n2006.754 & 0.90502 & 0.00110 & 146.745 & 0.047 \\\\\n2006.800 & 0.90222 & 0.00138 & 146.984 & 0.053 \\\\\n2006.863 & 0.89489 & 0.00110 & 147.111 & 0.054 \\\\\n2007.461 & 0.81432 & 0.00109 & 149.295 & 0.050 \\\\\n2007.688 & 0.77183 & 0.00104 & 150.250 & 0.053 \\\\\n2007.743 & 0.76014 & 0.00110 & 150.515 & 0.055 \\\\\n2007.849 & 0.73666 & 0.00109 & 151.017 & 0.063 \\\\\n2008.427 & 0.57619 & 0.00104 & 154.647 & 0.078 \\\\\n2008.441 & 0.57103 & 0.00112 & 154.665 & 0.106 \\\\\n2008.471 & 0.56145 & 0.00110 & 154.978 & 0.086 \\\\\n2008.648 & 0.49830 & 0.00103 & 156.664 & 0.082 \\\\\n2008.915 & 0.39126 & 0.00163 & 160.569 & 0.148 \\\\\n2009.458 & 0.14626 & 0.00273 & 186.175 & 0.562 \\\\\n2010.582 & 0.32838 & 0.00120 & 332.295 & 0.157 \\\\\n2010.849 & 0.30942 & 0.00103 & 339.059 & 0.134 \\\\\n2011.543 & 0.17352 & 0.00243 & 12.950 & 0.433 \\\\\n2012.545 & 0.25518 & 0.00110 & 107.165 & 0.186 \\\\\n2012.703 & 0.29394 & 0.00112 & 112.857 & 0.164 \\\\\n2013.431 & 0.47861 & 0.00104 & 126.845 & 0.088\n\\enddata\n\\end{deluxetable}\n\n\\subsection{Calibrations for Absolute Astrometry}\n\\label{sec: absast calibrations}\n\nWe now seek to measure the position of $\\varepsilon$~Indi~Ba\\xspace relative to a set of reference stars in the FORS2 images with known absolute astrometry. We approach the problem in stages. First, we fit for the pixel positions of all stars in the frame. We then use Gaia EDR3 astrometry of a subsample of these stars to construct a distortion map. Next, we use our fit to the NACO data (Section \\ref{sec: orbit fit}) to fix the relative positions of $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace. Finally, we use the PSFs of nearby reference stars to model the combined PSF of $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace and measure their position in a frame anchored by Gaia EDR3.\n\nWe begin by measuring stellar positions in pixel coordinates and using them to derive a conversion between pixel coordinates ($x,y$) and sky coordinates ($\\alpha, \\delta$), i.e., a distortion correction. We identify 46 Gaia sources in the field of view of the FORS2 images; these will serve as reference stars to calibrate and derive the distortion corrections. \n\nWe fit elliptical Moffat profiles to retrieve each individual reference star's pixel location $(x, y)$ on the detector. These are Gaia sources with known $\\alpha$ and $\\delta$ measurements { propagated backwards} from Gaia EDR3's single star astrometry in epoch 2016.0. We adopt the same module used for relative astrometry described in Section~\\ref{subsec:joint psf fit} to fit for the reference stars' positions. For each star, we fit for three additional parameters: the FWHMs along two directions, and a rotation angle in between. \n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=0.8\\textwidth]{Figures\/fig6_withscale.pdf}\n \\caption{Distortion correction to one of the 904 image frames from the FORS2 long term monitoring of $\\varepsilon$~Indi~B\\xspace's absolute positions. The position of $\\varepsilon$~Indi~B\\xspace is indicated by the yellow star while the blue dots are field stars. The red points are reference stars in the Gaia EDR3 catalog. The green lines indicate the residuals of the measured centroids from their distortion-corrected predictions based on EDR3 astrometry. Red open circles are Gaia EDR3 stars that we discard as outliers. \n }\n \\label{fig::distortion}\n\\end{figure*}\n\nWe assume a polynomial distortion solution of order N for FORS2: \n\\begin{align}\n \\alpha*_{\\rm model} \\equiv \\alpha \\cos \\delta &= \\sum_{i = 0}^{N} \\sum_{j = 0}^{N - i} a_{ij} x^i y^j \\\\\n \\delta_{\\rm model} &= \\sum_{i = 0}^{N} \\sum_{j = 0}^{N - i} b_{ij} x^i y^j .\n\\end{align}\n \nminimizing \n\n\\begin{equation}\n \\chi^2 = \\sum_{k=1}^{n_{\\rm ref}} \\left[\\left( \\frac{{\\alpha*_{k}} - {\\alpha*_{{\\rm model},k}}}{\\sigma_{\\alpha*,k}} \\right)^2 + \\left(\\frac{{\\delta_{k}} - {\\delta_{{\\rm model},k}}}{\\sigma_{\\delta_{k}}} \\right)^2 \\right].\n\\end{equation}\n\nThis defines a linear least-squares problem because each of the $a_{ij}$ appears linearly in the data model. \nTo avoid numerical problems, we define $x=y=0$ at the center of the image and subtract $\\alpha_{\\rm ref} = 181.\\!\\!^\\circ327$, $\\delta_{\\rm ref} = -56.\\!\\!^\\circ789$ from all Gaia coordinates. { To determine the best model for distortion correction, we compare 2nd, 3rd and 4th-order polynomial models. We derive distortion corrections excluding one Gaia reference star at a time. We then measure the excluded star's positions, and use the distortion correction built without using this star to derive its absolute astrometry. The consistency of the best-fit astrometric parameters with the Gaia measurements, and the scatter of the individual astrometric measurements about this best-fit sky path, both act as a cross-validation test of the distortion correction. For most stars, a second-order correction outperforms a third-order correction on both metrics. This also holds true dramatically for $\\varepsilon$~Indi~B\\xspace itself, with a second-order distortion correction providing substantially smaller scatter about the best-fit sky path.} \n\nOnce we have a list of pixel coordinates $(x, y)$ and sky coordinates ($\\alpha_{*}$, $\\delta$) for all of our reference stars, we derive { second} order distortion corrections for each image. \nTo avoid having poorly fit stars drive the results, we clip reference stars that are $\\geq$10$\\sigma$ outliers. \nFigure~\\ref{fig::distortion} shows an example of an image frame indicating the displacement of the distortion-corrected centroids according to Gaia with respect to their original ``uncorrected'' centroid locations on the detector. The empty red circles are Gaia stars that were discarded as outliers. \n\nWe now seek to measure the position of $\\varepsilon$~Indi~Ba\\xspace on the distortion-corrected frame defined by the astrometric reference stars. We cannot fit the brown dwarfs' positions in the same way as the reference stars: their light is blended in most images. Instead, we first fix their relative position using an orbital fit to the relative astrometry (Sections \\ref{subsec: relast calibrations} and \\ref{sec: orbit fit}). We then model the two-dimensional image around $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace as a linear combination of the interpolated PSFs of the five nearest field stars. \n\nWith the relative astrometry fixed, our fit to the image around $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace has nine free parameters: five for the normalization of each reference PSF, one for the background intensity, one for the flux ratio between Ba\\xspace and Bb\\xspace, and two for the position of $\\varepsilon$~Indi~Ba\\xspace. The fit is linear in the first six of these parameters. We solve this linear system for each set of positions and flux ratios, and use nonlinear optimization to find their best-fit values in each image. We then fix the flux ratio to its median best-fit value of 0.195 and perform the fits again, optimizing the position of $\\varepsilon$~Indi~Ba\\xspace in each FORS2 image.\n\nFigure~\\ref{fig:selfcal_epsiIndib} shows two examples of the residuals to this fit. The residual intensity exhibits little structure, whether the two components are strongly blended (bottom panel) or clearly resolved (top panel). \n\nOur fit produces pixel coordinates of $\\varepsilon$~Indi~Ba\\xspace in each frame. Our use of self-calibration ensures that these pixel coordinates are in the same reference system as the astrometric standard stars. We then apply the distortion correction derived from these reference stars to convert from pixel coordinates to absolute positions in right ascension and declination. { Another important calibration for absolute astrometry is the correction for atmospheric dispersion. However, our data were taken with an atmospheric dispersion corrector (ADC) in place, which has not been sufficiently well-characterized to model and remove residual dispersion \\citep{FORS2_ADC_1997}. We therefore use only the azimuthal projection of the absolute astrometry in the orbital fit. The effects and implications of the ADC and residual atmospheric dispersion are discussed in Section~\\ref{sec:adc}. } \n\n\n\\begin{figure}\n \\vskip 0.1 truein\n \\includegraphics[width=1\\linewidth]{Figures\/selfcal_separate.pdf} \\quad\n \\includegraphics[width=1.018\\linewidth]{Figures\/selfcal_blended.pdf}\n \\caption{Example fits and residuals to FORS2 images when the two components ($\\varepsilon$~Indi~Ba\\xspace and $\\varepsilon$~Indi~Bb\\xspace) are completely resolved (top panel) or strongly blended (bottom panel). In each panel, all values are normalized to the peak intensity of the model fits.}\n \\label{fig:selfcal_epsiIndib}\n\\end{figure}\n\n\n\\section{Photometric Variability} \\label{sec:photvar}\n\n{\\cite{Koen_2005}, \\cite{Koen_2005_JHK}, and \\cite{Koen_2013} found potential evidence of variability of the system in the Near-Infrared ($I$, $J$, $H$ and $K_s$) but also stated that the results are inconclusive due to correlation between seeing and variability.\nWith the long-term monitoring data acquired by NACO ($J$, $H$, and $K_s$ bands) and FORS2 ($I$ band), we further investigate the photometric variability of $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace in this section.}\n\nWe apply the generalized Lomb-Scargle method \\citep{generalized_Lomb-Scargle_method_2009} and we use the implementation in the {\\tt astropy} Python package \\citep{astropy:2013, astropy:2018} for this work. For NACO data, there are no other field stars within the FOV to calibrate the photometry. Therefore, we take the best fit flux ratios of Ba\\xspace to Bb\\xspace from PSF fitting and apply the periodogram on the time series of this flux ratio. For FORS2 data, we use the {\\tt photutils} python package to perform differential aperture photometry on the sky-subtracted, flat-fielded and dark-corrected FORS2 images. We first measure the total flux of $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace, and the fluxes of fields stars in the field of view. We then normalize the flux of $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace using the median flux of all the non-variable field stars to obtain the relative flux of the $\\varepsilon$~Indi\\xspace system. We apply the periodogram on this relative flux. We choose a minimum frequency of 0 and a maximum frequency of $1 \\; {\\rm hour}^{-1}$, which is roughly an upper frequency limit associated with rotational activities if either object were rotating at break-up velocity. We choose a frequency grid size of $\\Delta f = 1 \/ n_0T$, where $n_0 = 10$, $T=10 \\; {\\rm yr}$ to sufficiently sample the peaks \\citep{VanderPlas_2018}. \n\n\\begin{figure}\n \\centering\n \\includegraphics[width=1\\linewidth]{Figures\/EpsIndi_lightcurves_and_scatter.pdf}\n \\caption{{Top panel shows the flux ratios of $\\varepsilon$~Indi~Ba\\xspace over Bb\\xspace in J, H and Ks bands measured using the joint PSF fitting method, and the flux ratios of Ba\\xspace + Bb\\xspace over the average of the field stars measured using aperture photometry. The bottom panel shows the flux scatter of $\\varepsilon$~Indi\\xspace compared to the field stars in I band FORS2 data.}}\n \\label{fig:light curves}\n\\end{figure}\n\n{Figure 8 top panel shows the measured flux ratios in both NACO and FORS2 data over all epochs. The bottom panels shows that the $\\varepsilon$~Indi\\xspace system has a typical flux scatter for its brightness in FORS2 data. From our simple analysis, we do not see any significant evidence of photometric variability of the system in our periodograms. However, since the observations are not designed for the purpose of investigating variability, the non-uniform and sparsely sampled window function of the observations resulted in very noisy periodograms. Therefore, we also cannot reach any definitive conclusions regarding whether there is any variability of the system with a physical origin.}\n\n\\section{Orbital Fit} \\label{sec: orbit fit}\n\n\\subsection{Relative Astrometry}\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=\\linewidth]{Figures\/rel_orbit_corner_plot.pdf}\n \\caption{Corner plot for the relative orbit fit MCMC chain. The parameters are angular semi-major axis ($a_{\\rm ang}$) in {mas}, period ($P$) in years, eccentricity ($e$), longitude of ascending node ($\\Omega$) in degrees and inclination ($i$) in degrees. {The posterior mean is used as the estimator for each parameter, and the errors are one standard deviation from the mean.}}\n \\label{fig:rel orbit corner plot}\n\\end{figure*}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{Figures\/rel_orbit.pdf}\n \\caption{Relative orbit fit of $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace. The orbit is plotted as the relative separation of Bb\\xspace from Ba\\xspace, where Ba\\xspace is fixed at the origin. The black dots are the measured relative astrometry, the hollow dots show the beginning of each year, and the solid line is the best fit orbit. The bottom panel shows the residuals of the separation in blue and PA in green.}\n \\label{fig:rel orbit}\n\\end{figure}\n\nWe use the relative astrometry measurements summarized in Table \\ref{table:relastroresults} to fit for a relative orbit and obtain the orbital parameters. For this, we use an adaptation of the open-source orbital fitting python package, \\texttt{orvara} \\citep{orvara_2021}, and fit for 7 orbital parameters: period, the angular extent of the semi-major axis ($a_{\\rm ang}$, { hereby referred to as the angular semi-major axis}), eccentricity ($e$), argument of periastron ($\\omega$), time of periastron ($T_0$), longitude of ascending node ($\\Omega$), and inclination ($i$). The corner plot for the MCMC chain is shown in Figure \\ref{fig:rel orbit corner plot}. The best fit relative orbit is shown in Figure \\ref{fig:rel orbit}, and the best fit orbital parameters are summarized in Table. \\ref{table:orbitfit}. The reduced $\\chi^2$ is 0.77 which suggests that we may be slightly overestimating the errors, especially for the earlier epochs. This is possibly because the earlier epochs in general have higher quality data, while we used empirical PSFs of a wider range of qualities in order to generate a large enough sample for the error inflation estimate described in section \\ref{subsubsec: error inflation}. Nevertheless, we are able to produce an excellent fit and obtain very tight constraints on the orbital parameters thanks to high quality direct imaging data and a long monitoring baseline that almost covers an entire period.\n\n\\subsection{Absolute Astrometry}\n\nWe have derived optical geometric distortion corrections for all the FORS2 images in Section~\\ref{sec: absast calibrations}. We describe here our approach to fit for astrometric models to the reference stars, field stars, and most importantly the $\\varepsilon$~Indi~B\\xspace system. We fit standard five-parameter astrometric models, with position, proper motion, and parallax, to the reference stars and field stars in the field of view of the FORS2 images. The results from the fits for reference stars match within 20$\\%$ from the proper motions and parallaxes Gaia provided.\nFor the binary system $\\varepsilon$~Indi~B\\xspace, we fit a six-parameter astrometric solution, adding an extra parameter which is the ratio between the semi-major axes of the orbits of the two components. { We also review, and ultimately project out, the effects of atmospheric dispersion. The wavelength-dependent index of refraction of air causes an apparent, airmass-dependent displacement between the redder brown dwarfs $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace and the bluer field stars along the zenith direction.}\n\nThe results from absolute astrometry give proper motions, parallax, and a ratio between the semi-major axes which can then be converted into a mass ratio and individual masses. In conjuction with our previous relative astrometry results, full Keplerian solutions can be derived that completely characterize the orbits of both $\\varepsilon$~Indi~Ba\\xspace and $\\varepsilon$~Indi~Bb\\xspace. \n\n\\subsubsection{Astrometric Solution}\n\nThe astrometric solution for a single and isolated reference star or background star is a five-parameter linear model in terms of reference pixel coordinates in RA and Declination, proper motions in RA and Declination, and the parallax. A star's instantaneous position $(\\alpha*, \\delta)$ would be its position $(\\alpha*_{\\rm ref}, \\delta_{\\rm ref})$\nat a reference epoch, plus proper motion $(\\mu_{\\alpha*}, \\mu_\\delta)$ multiplied by the time since { the reference epoch $ t_{\\rm ref}$}, and parallax $\\varpi$ times the so-called parallax factors $\\Delta \\pi_{\\alpha*}$ and $\\Delta \\pi_\\delta$:\n\n\\begin{equation}\n\\begin{bmatrix}\n1 & 0 & t - t_{\\rm ref} & 0 & \\rm \\Delta \\pi_{\\alpha*} \\\\\n0 & 1 & 0 & t - t_{\\rm ref} & \\rm \\Delta \\pi_{\\delta} \\\\\n\\end{bmatrix} \n\\begin{bmatrix}\n{\\alpha*_{\\rm ref}} \\\\ \\delta_{\\rm ref} \\\\\n\\mu_{\\alpha*} \\\\\n\\mu_{\\delta} \\\\\n\\varpi\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n\\alpha* \\\\\n\\delta \\\\\n\\end{bmatrix}.\n\\end{equation}\n\n\nTo test the robustness of the distortion corrections in RA and Declination that we have derived for each image, we `reverse engineer' by excluding a particular reference star from the fit and solve for the astrometric solution of that star based on the discussion above for comparison to the Gaia parameters. In particular, we focus on the reference stars close to $\\varepsilon$~Indi~B\\xspace. \n\nFor the binary system $\\varepsilon$~Indi~B\\xspace, the astrometric solution demands an additional parameter $r_{\\rm Ba}$: the ratio between the semi-major axis of $\\varepsilon$~Indi~Ba\\xspace about the barycenter to the total semi-major axis $a$. The parameter $r_{\\rm Ba}$ is related to the binary mass ratio by \n\\begin{equation}\n r_{\\rm Ba} = \\frac{M_{\\rm Bb}}{M_{\\rm Ba} + M_{\\rm Bb}}.\n\\end{equation}\nThe model becomes\n\\begin{equation}\n\\begin{split}\n\\begin{bmatrix}\n1 & 0 & t - t_{\\rm ref} & 0 & \\rm \\Delta \\pi_{\\alpha*} & a_{\\alpha*}\\\\\n0 & 1 & 0 & t - t_{\\rm ref} & \\rm \\Delta \\pi_{\\delta} & a_{\\delta}\\\\\n\\end{bmatrix} \n\\begin{bmatrix}\n\\alpha*_{\\rm ref} \\\\\n\\delta_{\\rm ref} \\\\\n\\mu_{\\alpha*} \\\\\n\\mu_{\\delta} \\\\\n\\varpi \\\\\nr_{\\rm Ba}\n\\end{bmatrix}\n\\\\\n=\n\\begin{bmatrix}\n\\alpha \\!*_{\\rm Ba}-0.5\\dot{\\mu}_{\\alpha*} (t-t_{\\rm ref})^2 \\\\\n\\delta_{\\rm Ba}- 0.5 \\dot{\\mu}_{\\delta}(t-t_{\\rm ref})^2 \\\\\n\\end{bmatrix}.\n\\label{eq:absast_fit}\n\\end{split}\n\\end{equation}\n\n{ We also take into account the perspective acceleration that occurs when a star passes by the observer and its proper motion gets exchanged into radial velocity. This effect is more significant for $\\varepsilon$~Indi\\xspace than for remote stars. We employ constant perspective accelerations of $\\dot{\\mu}_{\\alpha*}$ = 0.165 mas\\,yr$^{-2}$ in RA and $\\dot{\\mu}_{\\delta}$ = 0.078 mas\\,yr$^{-2}$ in Dec for the $\\varepsilon$~Indi~B\\xspace system based on Gaia EDR3 measurements and assuming the radial velocity measured for $\\varepsilon$~Indi\\xspace\\,A. We adopt a reference epoch $t_{\\rm ref}$=2010. With an astrometric baseline of $\\sim$10 years, this gives a displacement of $0.5\\dot{\\mu}(t-t_{\\rm ref})^2\\approx 2$\\,mas at the edges of the observing window, where $\\dot{\\mu}$ is the acceleration. The perspective acceleration, because it is known, is included in the right hand side of Equation \\eqref{eq:absast_fit}.}\n\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=0.8\\textwidth]{Figures\/res_2nd_order.pdf}\n \\caption{Residuals to the best-fit astrometric model for $\\varepsilon$~Indi~Ba\\xspace. The top panel shows the residuals in altitude, and the bottom panel shows the residuals in azimuth, both plotted as a function of altitude. Empty red circles show the rejected epochs from 3$\\sigma$ clipping of the azimuthal residuals. A histogram of the residuals is shown to the right of each scatter plot. Strong systematics are seen in the altitude residuals but not the azimuth ones, evident from the symmetric, roughly Gaussian distribution for the former and the altitude-dependent nonzero mean for the latter. These systematics are consistent with the magnitude expected for uncorrected ADC residual dispersion \\citep{FORS2_ADC_1997}. \\label{fig:residuals}}\n\\end{figure*}\n\n\n\\subsubsection{ Residual Atmospheric Dispersion}\n\\label{sec:adc}\n\nThe FORS2 imaging covers twelve years, with data taken over a wide range of airmasses. This makes it essential to correct for atmospheric dispersion caused by the differential refraction of light of different colors as it passes through the atmosphere. The degree of dispersion is related to the wavelength of light, the filter used, and the airmass, but is always along the zenith direction. \nMany of the FORS2 images were taken at very high airmass. The typical airmass will vary over the course of the year, because the time of observation will vary depending on what part of night the target is up.\n\nAll of the FORS2 images were taken with an atmospheric dispersion corrector, or ADC. The residual dispersion depends on filter, airmass, and position on the FOV, but is typically tens of mas \\citep{FORS2_ADC_1997}. This is smaller than the system's parallax and { angular} semi-major axis, but only by a factor of $\\approx$10. { Further, the ADC is only intended to provide a full correction to a zenith angle of $50^\\circ$ \\citep{FORS2_ADC_1997}. At lower elevations it is parked at its maximum extent; \\cite{Cardoso_2012} applied an additional correction to these data.} Because the { residual dispersion is} only in the zenith direction, we perform two fits to the absolute astrometry of $\\varepsilon$~Indi~Ba\\xspace. First, we use our measurements in RA and Decl.~directly. Second, we use the parallactic angle $\\theta$ to take only the component of our measurement along the azimuth direction, which is immune to the effects of differential atmospheric refraction.\n\nWe project the data into the altitude-azimuth frame by left-multiplying both sides of Equation \\eqref{eq:absast_fit} by the rotation matrix\n\\begin{equation}\n { R} = \n \\begin{bmatrix}\n \\cos \\theta & -\\sin \\theta \\\\\n \\sin \\theta & \\cos \\theta\n \\end{bmatrix}.\n \\label{eq:rot_altaz}\n\\end{equation}\nThe top row of Equation \\eqref{eq:rot_altaz} corresponds to the azimuth direction, while the bottom row corresponds to the altitude direction.\n\nFitting in both the altitude and azimuth directions produces a parallax of 263~mas, in agreement with \\cite{Cardoso_2012} but much lower than both the Hipparcos and Gaia values for $\\varepsilon$~Indi\\xspace~A. We then perform a fit only in the azimuth direction: we multiply both sides of Equation \\eqref{eq:absast_fit} by the top row of Equation \\eqref{eq:rot_altaz}. \nWe exclude eight $3\\sigma$ outliers and assume a uniform per-epoch uncertainty of 8.01 mas to give a normalization factor that gives a reduced $\\chi^2$ value of 1.00. This procedure results in a parallax of $274.99 \\pm 0.43$\\,mas, in good agreement with both the Hipparcos and Gaia measurements. { We note that the 25-year time baseline between Hipparcos and Gaia causes a small parallax difference. $\\varepsilon$~Indi\\xspace~A has a radial velocity of 40.5 km\/s \\citep{GaiaDR2_2018}, which translates to a fractional change of $3\\times 10^{-4}$ or a decrease in parallax of about 0.08 mas over 25 years. This difference is much smaller than the uncertainties of any of these parallax measurements.}\n\n\n\n\nFigure~\\ref{fig:residuals} shows the residual to the best-fit model using only azimuthal measurements: the top panel shows the residuals in altitude, while the bottom panel shows the residuals in azimuth. A upward trend and nonzero mean are seen in the altitude component of the parallax as a function of altitude, but no dependence on altitude was seen in the azimuth-based parallax. This confirms that the altitude component of the position measurements is corrupted by residual atmospheric dispersion of a magnitude consistent with expectations \\citep{FORS2_ADC_1997}. \n\nThe six-parameter azimuth-component-only astrometric solution gives a mass ratio of 0.4431 $\\pm$ 0.0008 between the binary brown dwarf $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace. This mass ratio is consistent with \\cite{Cardoso_2012}. The only differences in our approaches arise from our usage of Gaia EDR3 to anchor the distortion correction and our account of atmospheric dispersion by only taking the azimuthal projection of the motion of the system. Our parallax of $274.99 \\pm 0.43$ mas agrees with the parallax value from Hipparcos and that of \\citet{Dieterich_2018}. \n\n\\subsection{Individual Dynamical Masses} \n\nThe relative astrometry orbital fit provides a precise period and angular semi-major axis. With a parallax from absolute astrometry, we convert the angular semi-major axis to distance units: $2.4058 \\pm 0.0040$\\,au. We then use Kepler's third law to calculate a total system mass of \\hbox{$120.17\\pm 0.62$}\\,$M_{\\rm Jup}$. Finally, the mass ratio derived from absolute astrometry provides individual dynamical masses of \\hbox{$66.92\\pm 0.36$}\\ and \\hbox{$53.25\\pm 0.29$}\\,$M_{\\rm Jup}$ for $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace, respectively. Table \\ref{table:orbitfit} shows the results of each component of the orbital fit, with the final individual mass measurements in the bottom panel. The uncertainty on these masses is dominated by uncertainty in the parallax: the mass ratio is constrained significantly better than the total mass. In the following section, we use both our individual mass constraints and our measurement of the mass ratio to test models of substellar evolution.\n\n\\begin{deluxetable}{cc}\n\\tablewidth{0pt}\n \\tablecaption{Orbital Fit of the $\\varepsilon$~Indi~B\\xspace system \\label{table:orbitfit}}\n \\tablehead{\n \\colhead{{ Fitted Parameters}} & \\colhead{Posterior {mean} $\\pm$1$\\sigma$}}\n \\startdata\n Period (yr) & $11.0197 \\pm 0.0076\\phn$ \\\\\n { Angular} semi-major axis (mas) & $661.58 \\pm 0.37\\phn\\phn$ \\\\\n Eccentricity & $0.54042 \\pm 0.00063$ \\\\\n $\\omega$ (deg) & $328.27 \\pm 0.12\\phn\\phn$ \\\\\n $\\Omega$ (deg) & $147.959 \\pm 0.023\\phn\\phn$ \\\\\n Inclination (deg) & $77.082 \\pm 0.032\\phn$ \\\\\n $\\mu_{\\alpha*}$ (\\hbox{mas\\,yr$^{-1}$}) & $3987.41 \\pm 0.12 \\phn\\phn\\phn$ \\\\\n $\\mu_{\\delta}$ (\\hbox{mas\\,yr$^{-1}$}) & $-2505.35 \\pm 0.10\\phn\\phn\\phn$\\phs \\\\\n $\\Big(\\frac{M_{\\rm Bb}}{M_{\\rm Ba}+M_{\\rm Bb}}\\Big)$ & $0.4431 \\pm 0.0008$ \\\\\n $\\varpi$ (mas) & $274.99 \\pm 0.43\\phn\\phn$\\\\\n reduced $\\chi^2$ & 1.00\\\\\n \\hline\n { Derived Parameters} & Posterior {mean} $\\pm$1$\\sigma$\\\\\n \\hline\n a (AU) & $2.4058 \\pm 0.0040$ \\\\\n System mass ($\\mbox{$M_{\\rm Jup}$}$) & \\hbox{$120.17\\pm 0.62$}$\\phn\\phn$ \\\\\n ${\\rm Mass}_{\\rm Ba}$ ($\\mbox{$M_{\\rm Jup}$}$) & \\hbox{$66.92\\pm 0.36$}$\\phn$\\\\\n ${\\rm Mass}_{\\rm Bb}$ ($\\mbox{$M_{\\rm Jup}$}$) & \\hbox{$53.25\\pm 0.29$}$\\phn$\n \\enddata\n\\end{deluxetable}\n\n\n\n\\section{Testing Models of Substellar Evolution} \\label{sec:BDtests}\n\nThe evolution of substellar objects is characterized by continuously-changing observable properties over their entire lifetimes. Therefore, the most powerful tests to benchmark evolutionary models utilize dynamical mass measurements of brown dwarfs of known age (usually, from an age-dated stellar companion) or of binary brown dwarfs that can conservatively presumed to be coeval. \nA single brown dwarf of known age and mass can test evolutionary models in an absolute sense, and the strength of the test is limited by both the accuracy of the age and of the mass. Pairs of brown dwarfs of known masses can test the slopes of evolutionary model isochrones, even without absolute ages, because their age difference is known very precisely to be near zero unless they are very young. \n\nThe $\\varepsilon$~Indi~B\\xspace system is an especially rare case where both of these types of tests are possible. In fact, it is the only such system containing T~dwarfs where both the absolute test of substellar cooling with time and coevality test of model isochrones are possible.\n\nIn the following, we consider a collection of evolutionary models applicable to $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace covering a range of input physics. The ATMO-2020 grid \\citep{2020A&A...637A..38P} represents the most up-to-date cloudless evolutionary models from the ``Lyon'' lineage that includes DUSTY \\citep{Chabrier:2000sh}, COND \\citep{2003A&A...402..701B}, and BHAC15 \\citep{2015A&A...577A..42B}. For models that include the effect of clouds, we use the hybrid tracks of \\citet[][hereinafter SM08]{2008ApJ...689.1327S}, which are cloudy at $\\mbox{$T_{\\rm eff}$}>1400$\\,K, cloudless at $\\mbox{$T_{\\rm eff}$}<1200$\\,K, and a hybrid of the two in between 1400\\,K and 1200\\,K. These are the most recent models that include cloud opacity from the ``Tucson'' lineage. We also compare to the earlier cloudless Tucson models \\citep{Burrows1997} given their ubiquity in the literature.\n\nIn order to test these models, we chose pairs of observable parameters from among the fundamental properties of mass, age, and luminosity. Using any two parameters, we computed the third from evolutionary models. When the first two parameters were mass and age, we bilinearly interpolated the evolutionary model grid to compute luminosity. When the first two parameters were luminosity and either mass or age, we used a Monte Carlo rejection sampling approach as in our past work \\citep{Dupuy+Liu_2017,2018AJ....156...57D,Brandt2021_Six_Masses}. Briefly, we randomly drew values for the observed independent variable, according to the measured mass or age posterior distribution, and then drew values for the other from an uninformed prior distribution (either log-flat in mass or linear-flat in age). We then bilinearly interpolated luminosities from the randomly drawn mass and age distributions. For each interpolated luminosity $L_{\\rm bol}^{\\prime}$, we computed $\\chi^2 = (\\mbox{$L_{\\rm bol}$}-L_{\\rm bol}^{\\prime})^2\/\\sigma_{L_{\\rm bol}}^2$. For each trial we drew a random number between zero and one, and we only retained trial sets of mass, age, and luminosity in our output posterior if $e^{-(\\chi^2-\\chi^2_{\\rm min})\/2}$ was greater than the random number.\n\nWe used the luminosities of $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace from \\citet{2010A&A...510A..99K}, accounting for the small difference between the {\\sl Hipparcos }\\ parallax of 276.06\\,mas that they used and our value of 274.99\\,mas, which resulted in $\\log(\\mbox{$L_{\\rm bol}$}\/\\mbox{$L_{\\sun}$}) = -4.691\\pm0.017$\\,dex and $-5.224\\pm0.020$\\,dex. Their luminosity errors were dominated by their measured photometry of $\\varepsilon$~Indi~Ba\\xspace and $\\varepsilon$~Indi~Bb\\xspace and the absolute flux calibration of Vega's spectrum, so our errors are identical to theirs.\n\nOur Monte Carlo approach naturally accounts for the relevant covariances between measured parameters. There are six independently-measured parameters for which we randomly drew Gaussian-distributed values: the orbital period ($P$), the semi-major axis in angular units ($a^{\\prime\\prime}$), the ratio of the mass of $\\varepsilon$~Indi~Bb\\xspace to the total mass of $\\varepsilon$~Indi~B\\xspace ($M_{\\rm Bb}\/M_{\\rm tot}$), the parallax in the same angular units as the semi-major axis ($\\varpi$), and the two bolometric fluxes computed from the luminosities and distance in \\citet{2010A&A...510A..99K}. From these, we computed the total mass, $M_{\\rm tot} = (a^{\\prime\\prime}\/\\varpi)^3 (P\/1{\\rm yr})^{-2}$, and the individual masses and luminosities.\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=0.32\\textwidth]{Figures\/epsIndBC_logl-agyr_TUCSON.pdf}\n \\includegraphics[width=0.32\\textwidth]{Figures\/epsIndBC_logl-agyr_ATMO_CEQ.pdf}\n \\includegraphics[width=0.32\\textwidth]{Figures\/epsIndBC_logl-agyr_SM08_hybrid_solar.pdf}\n \\caption{Substellar cooling curves derived from three independent evolutionary models given our measured masses. The top curve in each panel corresponds to $\\varepsilon$~Indi~Ba\\xspace, and the bottom curve corresponds to $\\varepsilon$~Indi~Bb\\xspace. The darker shaded region of each curve shows the 1$\\sigma$ range in our measured mass, and the lighter shading is the 2$\\sigma$ range. On each curve, the ages corresponding to $\\mbox{$T_{\\rm eff}$} = 1400$\\,K and 1200\\,K are marked, indicating the approximate beginning and ending of the L\/T transition. Over-plotted on each panel are the 1$\\sigma$ and 2$\\sigma$ joint uncertainty contours for the age and luminosities of $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace.}\n \\label{fig:lbol-age}\n\\end{figure*}\n\n\\subsection{Absolute test of $\\mbox{$L_{\\rm bol}$}(t)$}\n\nIn general, tests of substellar luminosity as a function of time are either dominated by the uncertainty in the age or in the mass. In the case of the $\\varepsilon$~Indi~B\\xspace system, with highly precise masses having 0.5\\% errors, the uncertainty in the system age ($t = 3.5^{+0.8}_{-1.0}$\\,Gyr) is by far the dominant source of uncertainty. \n\nFigure~\\ref{fig:lbol-age} shows the measured joint confidence intervals on luminosities and age of $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace compared to evolutionary model predictions given their measured masses. The measured luminosity-age contours overlap all model predictions to within $\\approx$1$\\sigma$ or less for both components. To quantitatively test models and observations, we compared our model-derived substellar cooling ages for $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace to $\\varepsilon$~Indi\\xspace~A's age posterior, finding that they are all statistically consistent with the stellar age.\n\nOur results for $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace are comparable to other relatively massive (50--75\\,\\mbox{$M_{\\rm Jup}$}) brown dwarfs of intermediate age (1--5\\,Gyr) that also broadly agree with evolutionary model predictions of luminosity as a function of age \\citep{Brandt2021_Six_Masses}. These include objects such as HR~7672~B \\citep{Brandt_2019}, HD~4747~B \\citep{Crepp+Principe+Wolff+etal_2018}, HD~72946~B \\citep{Maire+Baudino+Desidera_2020}, and HD~33632~Ab \\citep{2020ApJ...904L..25C}.\n\nHowever, despite agreeing with models in an absolute sense, it is evident in Figure~\\ref{fig:lbol-age} that the ATMO-2020 and \\citet{Burrows1997} models prefer a younger age for $\\varepsilon$~Indi~Ba\\xspace than for $\\varepsilon$~Indi~Bb\\xspace. To examine the statistical significance of this difference in model-derived ages between the two components we now consider only their measured masses and luminosities, excluding the rather uncertain stellar age.\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=0.32\\textwidth]{Figures\/epsIndBC_logl-mass_TUCSON.pdf}\n \\includegraphics[width=0.32\\textwidth]{Figures\/epsIndBC_logl-mass_ATMO_CEQ.pdf}\n \\includegraphics[width=0.32\\textwidth]{Figures\/epsIndBC_logl-mass_SM08_hybrid_solar.pdf}\n \\caption{Isochrones from three different evolutionary models, ranging from 2\\,Gyr to 6\\,Gyr. Black and gray contours show the joint 1$\\sigma$ and 2$\\sigma$ confidence intervals of the masses and luminosities of $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace. Because these two brown dwarfs must be coeval they should lie along a single model isochrone. The only models that pass this test are the \\citet{2008ApJ...689.1327S} hybrid models that predict a distinctly different mass--luminosity relation for brown dwarfs. These models have a much shallower dependence of luminosity on mass as objects cool through the L\/T transition over $\\mbox{$T_{\\rm eff}$} = 1400$\\,K to 1200\\,K, changing from cloudy to cloud-free atmosphere boundary conditions.}\n \\label{fig:lbol-mass}\n\\end{figure*}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.45\\textwidth]{Figures\/epsIndBC-coeval.pdf}\n \\caption{Probability distributions of the difference between the model-derived substellar cooling ages ($t_{\\rm cool}$) of $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace. The dashed line shows the expectation that $t_{\\rm cool, Bb} = t_{\\rm cool, Ba}$. Only the \\citet{2008ApJ...689.1327S} hybrid models predict consistent, coeval ages. This is the highest-precision coevality test of brown dwarf binaries to date, and it supports previous results from brown dwarf binaries with mass errors of $\\approx$5\\% \\citep{2015ApJ...805...56D,Dupuy+Liu_2017}.}\n \\label{fig:benchmark-coeval}\n\\end{figure}\n\n\\subsection{Isochrone test of $M$--$\\mbox{$L_{\\rm bol}$}$ relation for T~dwarfs}\n\nEvolutionary models of brown dwarfs, from some of the earliest theoretical calculations up to modern work \\citep[e.g.,][]{1993RvMP...65..301B,2020A&A...637A..38P}, typically predict a power-law relationship between mass and luminosity with a slope of $\\Delta\\log{L} \/ \\Delta\\log{M} = 2.5$--3.0. This general agreement between models with very different assumptions---and that vary greatly in other predictions such as the mass of the hydrogen-fusion boundary---can be seen in the slopes of isochrones for 40--60\\,\\mbox{$M_{\\rm Jup}$}\\ brown dwarfs in Figure~\\ref{fig:lbol-mass}.\n\nOne set of models, from \\citet{2008ApJ...689.1327S}, that substantially alters the atmospheric boundary condition as objects cool from $\\mbox{$T_{\\rm eff}$} = 1400$\\,K to 1200\\,K predicts a much shallower slope from the $M$--$L$ relation during that phase of evolution (Figure~\\ref{fig:lbol-mass}). These so-called hybrid models provide the best match to the $M$--$L$ relation as measured in binaries composed of late-L dwarf primaries ($\\mbox{$T_{\\rm eff}$} \\approx 1400$\\,K) and early-T dwarf secondaries ($\\mbox{$T_{\\rm eff}$} \\approx 1200$\\,K), objects that straddle this evolutionary phase \\citep{2015ApJ...805...56D,Dupuy+Liu_2017}. A fundamental prediction of these models is that during the L\/T transition, objects of similar luminosity can have wider-ranging masses than in other models. The other chief prediction is that luminosity fades more slowly during the L\/T transition, so that the brown dwarfs emerging from this phase are more luminous than in other models.\n\n$\\varepsilon$~Indi~B\\xspace is the only example of a binary with precise individual masses where one component is an L\/T transition brown dwarf and the other is a cooler T~dwarf. This provides a unique test of the $M$--$L$ relation, where the cooler brown dwarf is well past the L\/T transition and the other is in the middle of it. According to hybrid models, the brown dwarf within the L\/T transition will be experiencing slower cooling, so it would be more luminous than in other models. On the other hand, with the immediate removal of cloud opacity in hybrid models below 1200\\,K, a brown dwarf will cool even faster than predicted by other, non-hybrid models. These two effects predict that a system like $\\varepsilon$~Indi~B\\xspace will, in fact, have an especially steep $M$--$L$ relation.\n\nOur measured masses give a particularly steep slope for the $M$--$L$ relation of $\\Delta\\log{L} \/ \\Delta\\log{M} = 5.37\\pm0.08$ between the L\/T-transition primary $\\varepsilon$~Indi~Ba\\xspace and the cooler secondary $\\varepsilon$~Indi~Bb\\xspace. The only evolutionary models that predict such a steep slope are the hybrid models of \\citet{2008ApJ...689.1327S}.\n\nTo quantitatively test models, we compared the model-derived cooling ages of $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace, given their measured masses and luminosities (Figure~\\ref{fig:benchmark-coeval}). Models like ATMO-2020 that assume a single, cloud-free atmospheric boundary condition are 3.9$\\sigma$ inconsistent with our measurements. At 6.9$\\sigma$, models from \\citet{Burrows1997} are even more inconsistent because the bunching up of isochrones around the end of the main sequence, which has a similar effect as the bunching up of isochrones due to slowed cooling in the hybrid models, occurs at higher masses than in ATMO-2020.\n\nThe $\\varepsilon$~Indi~B\\xspace system therefore provides further validation of hybrid evolutionary models, where the atmosphere boundary condition is changed drastically over the narrow range of \\mbox{$T_{\\rm eff}$}\\ corresponding to late-L and early-T dwarfs. No longer just within the L\/T transition, but affirming the consequences of slowed cooling during the L\/T transition to cooler brown dwarfs ($\\mbox{$T_{\\rm eff}$} < 1000$\\,K).\n\n\\subsection{Testing Model Atmospheres: \\mbox{$T_{\\rm eff}$}\\ and \\mbox{$\\log(g)$}}\n\nBrown dwarfs that have both directly measured masses and individually measured spectra have long been used in another type of benchmark test that tests for consistency between evolutionary models and the atmosphere models that they use as their surface boundary condition. Comparison of model atmospheres to observed spectra allows for determinations of \\mbox{$T_{\\rm eff}$}, \\mbox{$\\log(g)$}, and metallicity. Evolutionary models predict brown dwarf radii as a function of mass, age, and metallicity. Combining these radii with empirically determined luminosities produce mostly independent estimates of $\\mbox{$T_{\\rm eff}$} = (\\mbox{$L_{\\rm bol}$}\/4\\pi R^2 \\sigma_{\\rm SB})^{1\/4}$, and with masses gives estimates of $\\mbox{$\\log(g)$} = \\log(GM\/R^2)$. (Evolutionary model radii have a small dependence on the model atmospheres and thus estimates of \\mbox{$T_{\\rm eff}$}\\ and \\mbox{$\\log(g)$}\\ from their radii are not strictly, completely independent.) There are many examples of such benchmark tests, ranging from late-M dwarfs \\citep[e.g.,][]{2001ApJ...554L..67K,2004ApJ...615..958Z,2010ApJ...721.1725D}, L~dwarfs \\citep[e.g.,][]{2004A&A...423..341B,2009ApJ...692..729D,2010ApJ...711.1087K}, and T~dwarfs \\citep[e.g.,][]{2008ApJ...689..436L,Dupuy+Liu_2017}.\n\nFrom \\citet{2010A&A...510A..99K}, $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace have perhaps the most extensive and detailed spectroscopic observations (0.6--5.1\\,$\\mu$m at up to $R\\sim5000$) of any brown dwarfs with dynamical mass measurements. They found that BT-Settl atmosphere models \\citep{2012RSPTA.370.2765A} with parameters of $\\mbox{$T_{\\rm eff}$} = 1300$--1340\\,K and $\\mbox{$\\log(g)$} = 5.50$\\,dex best matched $\\varepsilon$~Indi~Ba\\xspace. For $\\varepsilon$~Indi~Bb\\xspace, they found $\\mbox{$T_{\\rm eff}$} = 880$--940\\,K and $\\mbox{$\\log(g)$} = 5.25$\\,dex.\n\nWe computed evolutionary model-derived values for \\mbox{$T_{\\rm eff}$}\\ and \\mbox{$\\log(g)$}\\ to compare to the model atmosphere results of \\citet{2010A&A...510A..99K}. The most precise estimates result from using the mass and luminosity to derive a substellar cooling age and then interpolating \\mbox{$T_{\\rm eff}$}\\ and \\mbox{$\\log(g)$}\\ from the same evolutionary model grid using the measured mass and the cooling age. The SM08 hybrid models gave $\\mbox{$T_{\\rm eff}$} = 1312\\pm13$\\,K and $972\\pm13$\\,K for $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace, respectively, and $\\mbox{$\\log(g)$} = 5.365\\pm0.006$\\,dex and $5.288\\pm0.003$\\,dex. These evolutionary model-derived values agree remarkably well with the model atmosphere results, which were based on an atmosphere grid with discrete steps of 20\\,K in \\mbox{$T_{\\rm eff}$}\\ and 0.25\\,dex in \\mbox{$\\log(g)$}. \n\nATMO-2020 models are only strictly appropriate for $\\varepsilon$~Indi~Bb\\xspace, and they give $\\mbox{$T_{\\rm eff}$} = 992\\pm13$\\,K and $\\mbox{$\\log(g)$} = 5.311\\pm0.003$\\,dex. This effective temperature is $\\approx$4$\\sigma$ higher than the BT-Settl model atmosphere temperature. ATMO-2020 models are actually based on this family of model atmospheres (BT-Cond and BT-Settl should be effectively equivalent at this \\mbox{$T_{\\rm eff}$}), so this suggests a genuine $\\approx$50\\,K discrepancy between atmosphere model-derived \\mbox{$T_{\\rm eff}$}\\ (too low) and evolutionary model-derived \\mbox{$T_{\\rm eff}$}\\ (too high). If so, this could be due a to a combination of systematics in atmosphere models (e.g., non-equilibrium chemistry, inaccurate opacities) and\/or ATMO-2020 evolutionary model radii (10--20\\% too high).\n\n\\section{Conclusions} \\label{sec:conclusions}\n\nIn this paper we use $\\sim$12 years of VLT data to infer dynamical masses of \\hbox{$66.92\\pm 0.36$}\\,$\\mbox{$M_{\\rm Jup}$}$ and \\hbox{$53.25\\pm 0.29$}\\,$\\mbox{$M_{\\rm Jup}$}$ for the brown dwarfs $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace, respectively. These masses put the the two objects firmly below the hydrogen burning limit. Our system mass agrees with that in \\cite{Cardoso_2009}, who estimated a system mass of $121 \\pm 1 \\; M_{\\rm Jup}$. With extra data from the completed relative and absolute astrometry monitoring campaign, we are able to derive precise individual masses and improve upon their previous analysis on several fronts. Using Gaia EDR3, we provide a much more precise calibration of both relative and absolute astrometry. In addition, we have shown that our joint PSF fitting method accounts for the effect of overlapping halos reasonably well and adjusted our final errors for the relative astrometry according to our systematics analysis. Lastly, we have investigated and corrected for the systematics due to differential atmospheric refraction and residual atmospheric dispersion. As a result, we are able to obtain very tight constraints on the orbital parameters and final masses, and measure a parallax consistent with both the Hipparcos and Gaia values.\n\nOur results disagree with \\cite{Dieterich_2018}, who used the photocenter's orbit together with three NACO epochs to derive a mass of $75.0 \\pm 0.82 \\; M_{Jup}$ for Ba\\xspace, and a mass of $70.1 \\pm 0.68 \\; M_{Jup}$ for Bb\\xspace. These masses are at the boundaries of the hydrogen burning limit, challenging theories of substellar structure and evolution. We cannot conclusively say why \\cite{Dieterich_2018} derive much higher masses. However, we are able to reproduce their results, and find that rotating their measurements into an azimuth-only frame produces a mass closer to ours. We speculate that highly asymmetric uncertainties in RA\/Decl.~for a few of their measurements had a disproportionate effect on the results. \n\nWe also provide a Fourier analysis of $\\varepsilon$~Indi~B\\xspace's fluxes to investigate its potential variability. We find no definitive evidence of variability with a frequency less than 1 hr$^{-1}$.\n\nOur newly precise masses and mass ratios enable new tests of substellar evolutionary models. We find that $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace are generally consistent with cooling models at the activity age of $3.5^{+0.8}_{-1.0}$\\,Gyr we derive for $\\varepsilon$~Indi\\xspace~A. However, the two brown dwarfs are consistent with coevality only under hybrid models like those of \\cite{2008ApJ...689.1327S}, with a transition to cloud-free atmospheres near the L\/T transition. \n\nOur masses for $\\varepsilon$~Indi~Ba\\xspace and Bb\\xspace, precise to $\\approx$0.5\\%, and our mass ratio, precise to $\\approx$0.2\\%, establish the $\\varepsilon$~Indi~B\\xspace binary brown dwarf as a definitive benchmark for substellar evolutionary models. As one of the nearest brown dwarf binaries, it is also exceptionally well-suited to detailed characterization with future telescopes and instruments including JWST. $\\varepsilon$~Indi~B\\xspace, with its two components straddling the L\/T transition, now provides some of the most definitive evidence for cloud clearing and slowed cooling in these brown dwarfs. \\\\\n\n\\acknowledgements{T.D.B.~gratefully acknowledges support from National Aeronautics and Space Administration (NASA) under grant 80NSSC18K0439 and from the Alfred P.~Sloan Foundation. MJM and CVC would like to thank their collaborators on the original ESO VLT NACO and FORS2 programmes which provided the great majority of the $\\varepsilon$~Indi~B\\xspace astrometry data re-reduced and analysed in this paper: Laird Close, Ralf-Dieter Scholz, Rainer Lenzen, Wolfgang Brandner, Nicolas Lodieu, Hans Zinnecker, Rainer K\u00f6hler, and Quinn Konopacky. We would also like to recognise the tremendous efforts made by the many ESO service mode astronomers in carrying out these observations over many runs between 2004 and 2016, more than a full period of the binary orbit, and to the ESO TAC for continuing to support the programme throughout that time. Based on observations collected at the European Southern Observatory under ESO programs 072.C-0689(F), 073.C-0582(A), 074.C-0088(A), 075.C-0376(A), 076.C-0472(A), 077.C-0798(A), 078.C-0308(A), 079.C-0461(A), 380.C-0449(A), 381.C-0417(A), 382.C-0483(A), 383.C-0895(A), 384.C-0657(A), 385.C-0994(A), 386.C-0376(A), 087.C-0532(A), 088.C-0525(A), 089.C-0807(A), 091.C-0899(A), 381.C-0860(A), 072.C-0689(D), 075.C-0376(B), 076.C-0472(B), 077.C-0798(B), 078.C-0308(B), 079.C-0461(B), 380.C-0449(B), 381.C-0417(B), 382.C-0483(B), 383.C-0895(B), 384.C-0657(B), 385.C-0994(B), 386.C-0376(B), 087.C-0532(B), 088.C-0525(B), 089.C-0807(B), and 091.C-0899(B).}\n\n\\software{photutils (\\citealt{Stetson_1987, photutils110}), astropy (\\citealt{Astropycollab2013aaAstropy}), orvara (\\citealt{orvara_2021}), Scipy (\\citealt{2020SciPy-NMeth}), matplotlib (\\citealt{Hunter:2007}), numpy (\\citealt{harris2020array})}\\\\\n\n\\input{main.bbl}\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\nA new race of insidious threats called Advanced Persistent Threats (APTs) have joined the league of eCrime activities on the Internet, and caught a lot of organizations off guard in the fairly recent times. Critical infrastructures and the governments, corporations, and individuals supporting them are under attack by these increasingly sophisticated cyber threats. The goal of the attackers is to gain access to intellectual property, personally identifiable information, financial data, and targeted strategic information. This is not simple fraud or hacking, but intellectual property theft and infrastructure corruption on a grand scale~\\cite{Daly2009}. APTs use multiple attack techniques and vectors that are conducted by stealth to avoid detection, so that hackers can retain control over target systems unnoticed for long periods of time. Interestingly, no matter how sophisticated these attack vectors may be, the most common ways of getting them inside an organization's network are social engineering attacks like phishing, and targeted spear phishing emails. There have been numerous reports of spear phishing attacks causing losses of millions of dollars in the recent past.~\\footnote{\\url{http:\/\/businesstech.co.za\/news\/internet\/56731\/south-africas-3-billion-phishing-bill\/}}~\\footnote{\\url{http:\/\/www.scmagazine.com\/stolen-certificates-used-to-deliver-trojans-in-spear-phishing-campaign\/article\/345626\/}} Although there exist antivirus, and other similar protection software to mitigate such attacks, it is always better to stop such vectors at the entry level itself~\\cite{kumaraguru2010teaching}. This requires sophisticated techniques to deter spear phishing attacks, and identify malicious emails at a very early stage itself.\n\n\nIn this research paper, we focus on identifying such spear phishing emails, wherein the attacker targets an individual or company, instead of anyone in general. Spear phishing emails ususally contain victim-specific context instead of general content. Since it is targeted, a spear phishing attack looks much more realistic, and thus, harder to detect~\\cite{Jakobsson2006}. A typical spear phishing attack can broadly be broken down into two phases. In the first phase, the attacker tries to gather as much information about the victim as possible, in order to craft a scenario which looks realistic, is believable for the victim, and makes it very easy for the attacker to attain the victim's trust. In the second phase, the attacker makes use of this gained trust, and draws the victim into giving out sensitive \/ confidential information like a user name, password, bank account details, credit card details, etc. The attacker can also exploit the victim's trust by infecting the victim's system, through luring them into downloading and opening malicious attachments~\\cite{Jakobsson2006}. While spear phishing may be a timeworn technique, it continues to be effective even in today's Web 2.0 landscape. A very recent example of such a spear phishing attack was reported by FireEye. Here, attackers exploited the news of the disappearance of Malaysian Airlines Flight MH370, to lure government officials across the world into opening malicious attachments (Figure~\\ref{fig:mh370attach}) sent to them over email~\\cite{fireeye:2014}. In 2011, security firm RSA suffered a breach via a targeted attack; analysis revealed that the compromise began with the opening of a spear phishing email.~\\footnote{\\url{http:\/\/blogs.rsa.com\/rivner\/anatomy-of-an-attack\/}} That same year, email service provider Epsilon also fell prey to a spear phishing attack that caused the organization to lose an estimated US\\$4 billion.~\\footnote{\\url{http:\/\/www.net-security.org\/secworld.php?id=10966}} These examples indicate that spear phishing has been, and continues to be one of the biggest forms of eCrime over the past few years, especially in terms of the monetary losses incurred. \n\n\n\\begin{figure}[!h]\n \\begin{center}\n\\fbox{\\includegraphics[scale=0.61]{mh3703.png}}\n \\end{center}\n \\caption{Example of a malicious PDF attachment sent via a spear phishing email. The PDF attachment was said to contain information about the missing Malaysian Airlines Flight 370.\n }\n \\label{fig:mh370attach}\n\n\\end{figure}\n\n\n\nSpear phishing was first studied as context aware phishing by Jakobsson et al. in 2005~\\cite{Jakobsson2005}. A couple of years later, Jagatic et al. performed a controlled experiment and found that the number of victims who fell for context aware phishing \/ spear phishing is 4.5 times the number of victims who fell for general phishing~\\cite{Jagatic2007}. This work was preliminary proof that spear phishing attacks have a much higher success rate than normal phishing attacks. It also highlighted that, what separates a regular phishing attack from a spear phishing attack is the additional information \/ context. Online social media services like LinkedIn, which provide rich professional information about individuals, can be one such source for extracting contextual information, which can be used against a victim. Recently, the FBI also warned that spear phishing emails typically contain accurate information about victims often obtained from data posted on social networking sites, blogs or other websites.~\\footnote{\\url{http:\/\/www.computerweekly.com\/news\/2240187487\/FBI-warns-of-increased-spear-phishing-attacks}} In this work, we investigate if publicly available information from LinkedIn can help in differentiating a spear phishing from a non spear phishing email received by an individual.\nWe attained a dataset of true positive targeted spear phishing emails, and a dataset of a mixture of non targeted, spam and phishing attack emails from the Symantec's enterprise email scanning service, which is deployed at multiple international organizations around the world. To conduct the analysis at the organizational level, we extracted the most frequently occurring domains from the \\emph{``to\"} fields of these emails, and filtered out 14 most targeted organizations, where the first name and last name could be derived from the email address.\nOur final dataset consisted of 4,742 spear phishing emails sent to 2,434 unique employees, and 9,353 non targeted spam \/ phishing emails to 5,914 unique employees. For a more exhaustive analysis, we also used a random sample of 6,601 benign emails from the Enron email dataset~\\cite{Cohen2009} sent to 1,240 unique employees with LinkedIn profiles.\n\nWe applied 4 classification algorithms, and were able to achieve a maximum accuracy of 97.04\\% for classifying spear phishing, and non spear phishing emails using a combination of \\emph{email} features, and \\emph{social} features. However, without the \\emph{social} features, we were able to achieve a slightly higher accuracy of 98.28\\% for classifying these emails. We then looked at the most informative features, and found that \\emph{email} features performed better than \\emph{social} features at differentiating targeted spear phishing emails from non targeted spam \/ phishing emails, and benign Enron emails. To the best of our knowledge, this is the first attempt at making use of a social media profile of a user to distinguish targeted spear phishing emails from non targeted attack emails, received by her.\nHaving found that \\emph{social} features extracted from LinkedIn profiles do not help in distinguishing spear phishing and non spear phishing emails, our results encourage to explore other social media services like Facebook, and Twitter. Such studies can be particularly helpful in mitigating APTs, and reducing the chances of attacks to an organization at the entry level itself. \n\nThe rest of the paper is arranged as follows. Section~\\ref{sec:bg} discuss the related work, Section~\\ref{sec:dcm} describes our email and LinkedIn profile datasets, and the data collection methodology. The analysis and results are described in Section~\\ref{sec:ar}. We conclude our findings, and discuss the limitations, contributions, and scope for future work in Section~\\ref{sec:conclusion}.\n\n\n\n\\section{Background and Related work} \\label{sec:bg}\n\nThe concept of targeted phishing was first introduced in 2005 as \\emph{social phishing} or \\emph{context-aware phishing}~\\cite{Jakobsson2005}. Authors of this work argued that if the attacker can infer or manipulate the context of the victim before the attack, this context can be used to make the victim volunteer the target information. This theory was followed up with an experiment where Jagatic et al. harvested freely available acquaintance data of a group of Indiana University students, by crawling social networking websites like Facebook, LinkedIn, MySpace, Orkut etc.~\\cite{Jagatic2007}. This contextual information was used to launch an actual (but harmless) phishing attack targeting students between the age group of 18-24 years. Their results indicated about 4.5 times increase in the number of students who fell for the attack which made use of contextual information, than the generic phishing attack. However, authors of this work do not provide details of the kind and amount of information they were able to gather from social media websites about the victims.\n\n\\paragraph{Who falls for phish}\n\nDhamija et al. provided the first empirical evidence about which malicious strategies are successful at deceiving general users~\\cite{Dhamija2006}. Kumaraguru et al. conducted a series of studies and experiments on creating and evaluating techniques for teaching people not to fall for phish~\\cite{kumaraguru2009school,kumaraguru2007protecting,kumaraguru2010teaching}. Lee studied data from Symantec's enterprise email scanning service, and calculated the odds ratio of being attacked for these users, based on their area of work. The results of this work indicated that users with subjects \\emph{``Social studies\"}, and \\emph{``Eastern, Asiatic, African, American and Australasian Languages, Literature and Related Subjects\"} were both positively correlated with targeted attacks at more than 95\\% confidence~\\cite{Lee2012}. Sheng et al. conducted an online survey with 1,001 participants to study who is more susceptible to phishing based on demographics. Their results indicated that women are more susceptible than men to phishing, and participants between the ages 18 and 25 are more susceptible to phishing than other age groups~\\cite{Sheng2010}. In similar work, Halevi et al. found a strong correlation between gender and response to a prize phishing email. They also found that neuroticism is the factor most correlated to responding to the email. Interestingly, authors detected no correlation between the participants' estimate of being vulnerable to phishing attacks and actually being phished. This suggests susceptibility to phishing is not due to lack of awareness of the phishing risks, and that real-time response to phishing is hard to predict in advance by online users~\\cite{Halevi2013}. \n\n\\paragraph{Phishing email detection techniques}\n\nTo keep this work focused, we concentrate only on techniques proposed for detecting phishing emails; we do not cover all the techniques used for detecting phishing URLs or phishing websites in general. Abu-Nimeh et al.~\\cite{Abu-Nimeh2007} studied the performance of different classifiers used in text mining such as Logistic regression, classification and regression trees, Bayesian additive regression trees, Support Vector Machines, Random forests, and Neural networks. Their dataset consisted of a public collection of about 1,700 phishing mails, and 1,700 legitimate mails from private mailboxes. They focused on richness of word to classify phishing email based on 43 keywords. The features represent the frequency of ``bag-of-words\" that appear in phishing and legitimate emails. However, the ever-evolving techniques and language used in phishing emails might make it hard for this approach to be effective over a long period of time.\n\nVarious feature selection approaches have also been recently introduced to assist phishing detection. A lot of previous work~\\cite{Abu-Nimeh2007,Chandrasekaran2006,Fette2007} has focused on email content in order to classify the emails as either benign or malicious. Chandrasekaran et al.~\\cite{Chandrasekaran2006} presented an approach based on natural structural characteristics in emails. The features included number of words in the email, the vocabulary, the structure of the subject line, and the presence of 18 keywords. They tested on 400 data points which were divided into five sets with different type of feature selection. Their results were the best when more features were used to classify phishing emails using Support Vector Machine. Authors of this work proposed a rich set of stylometric features, but the dataset they used was very small as compared to a lot of other similar work. Fette et al.~\\cite{Fette2007} on the other hand, considered 10 features which mostly examine URL and presence of JavaScript to flag emails as phishing. Nine features were extracted from the email and the last feature was obtained from WHOIS query. They followed a similar approach as Chandrasekaran et al. but using larger datasets, about 7,000 normal emails and 860 phishing emails. Their filter scored 97.6\\% F-measure, false positive rate of 0.13\\% and a false negative rate of 3.6\\%. The heavy dependence on URL based features, however, makes this approach ineffective for detecting phishing emails which do not contain a URL, or are attachment based attacks, or ask the user to reply to the phishing email with potentially sensitive information. URL based features were also used by Chhabra et al. to detect phishing using short URLs~\\cite{Chhabra2011}. Their work, however, was limited to only URLs, and did not cover phishing through emails. Islam and Abawajy~\\cite{Islam2013} proposed a multi-tier phishing detection and filtering approach. They also proposed a method for extracting the features of phishing email based on weights of message content and message header. The results of their experiments showed that the proposed algorithm reduces the false positive problems substantially with lower complexity.\n\nBehavior-based approaches have also been proposed by various researchers to determine phishing messages~\\cite{Toolan2010,Zhang2007a}. Zhang et al.~\\cite{Zhang2007a} worked on detecting abnormal mass mailing hosts in network layer by mining the traffic in session layer. Toolan et al.~\\cite{Toolan2010} investigated 40 features that have been used in recent literature, and proposed behavioral features such as number of words in \\emph{sender} field, total number of characters in \\emph{sender} field, difference between sender's domain and reply-to domain, and difference between sender's domains and the email's modal domain, to classify ham, spam, and phishing emails. Ma et al.~\\cite{Ma2009} attempted to identify phishing email based on hybrid features. They derived 7 features categorized into three classes, i.e. content features, orthographic features, and derived features, and applied 5 machine learning algorithms. Their results stated that Decision Trees worked best in identifying phishing emails. Hamid et al.~\\cite{Hamid2013} proposed a hybrid feature selection approach based on combination of content and behaviour. Their approach mined attacker behavior based on email header, and achieved an accuracy of 94\\% on a publicly available test corpus.\n\nAll of the aforementioned work concentrates on distinguishing phishing emails from legitimate ones, using various types of features extracted from email content, URLs, header information etc. To the best of our knowledge, there exists little work which focuses specifically on targeted spear phishing emails. Further, there exists no work which utilizes features from the social media profiles of the victim in order to distinguish an attack email from a legitimate one. In this work, we direct our focus on a very specific problem of distinguishing targeted spear phishing emails from general phishing, spam, and benign emails. Further, we apply \\emph{social} features extracted from the LinkedIn profile of recipients of such emails to judge whether an email is a spear phishing email or not; which has never been attempted before, to the best of our knowledge. We performed our entire analysis on a real-world dataset derived from Symantec's enterprise email scanning service.\n\n\n\n\\section{Data collection methodology} \\label{sec:dcm}\n\nThe dataset we used for the entire analysis, is a combination of two separate datasets, viz. a dataset of emails (combination of targeted attack and non targeted attack emails), and a dataset of LinkedIn profiles. We now explain both these datasets in detail. \n\n\\subsection{Email dataset} \\label{sec:emd}\n\n\n\\begin{table*}[!ht]\n\\begin{center}\n \\begin{tabular}{l|l||l|l}\n \\hline\n Spear phishing Attachment Name & \\% & Spam \/ phishing Attachment Name & \\% \\\\ \\hline\n work.doc & 3.46 & 100A\\_0.txt & 20.74 \\\\\nMore detail Chen Guangcheng.rar & 3.01 & 100\\_5X\\_AB\\_PA1\\_MA-OCTET-STREAM\\_\\_form.html & 9.02 \\\\\n ARMY\\_600\\_8\\_105.zip & 2.54 & .\/attach\/100\\_4X\\_AZ-D\\_PA2\\_\\_FedEx=5FInvoice=5FN56=2D141.exe & 4.19 \\\\\n Strategy\\_Meeting.zip & 1.58 & 100\\_2X\\_PM3\\_EMS\\_MA-OCTET=2DSTREAM\\_\\_apply.html & 2.66 \\\\\n 20120404 H 24 year annual business plan 1 quarterly.zip & 1.33 & 100\\_4X\\_AZ-D\\_PA2\\_\\_My=5Fsummer=5Fphotos=5Fin=5FEgypt=5F2011.exe & 1.87 \\\\\n The extension of the measures against North Korea.zip & 1.30 & .\/attach\/100\\_2X\\_PM2\\_EMS\\_MA-OCTET=2DSTREAM\\_\\_ACC01291731.rtf & 1.40 \\\\\n Strategy\\_Meeting\\_120628.zip & 1.28 & 100\\_5X\\_AB\\_PA1\\_MH\\_\\_NothernrockUpdate.html & 1.28 \\\\\n image.scr & 1.24 & .\/attach\/100\\_2X\\_PM2\\_EMS\\_MA-OCTET=2DSTREAM\\_\\_invoice.rtf & 1.15 \\\\\n Consolidation Schedule.doc & 0.98 & 100\\_6X\\_AZ-D\\_PA4\\_\\_US=2DCERT=20Operations=20Center=20Report=2DJan2012.exe & 1.12 \\\\\n DARPA-BAA-11-65.zip & 0.93 & 100\\_4X\\_AZ-D\\_PA2\\_\\_I=27m=5Fwith=5Fmy=5Ffriends=5Fin=5FEgypt.exe & 1.11 \\\\\n Head Office-Y drive.zip & 0.93 & 100\\_4X\\_AZ-D\\_PA2\\_\\_I=27m=5Fon=5Fthe=5FTurkish=5Fbeach=5F2012.exe & 0.80 \\\\\n page 1-2.doc & 0.90 & 100\\_5X\\_AB\\_PA1\\_MA-OCTET-STREAM\\_\\_Lloyds=R01TSB=R01-=R01Login=R01Form.html & 0.69 \\\\\n Aircraft Procurement Plan.zip & 0.90 & 100\\_6X\\_AZ-D\\_PA4\\_\\_Fidelity=20Investments=20Review=2Dfrom=2DJan2012.exe & 0.68 \\\\\n Overview of Health Reform.doc & 0.74 & 100\\_4X\\_AZ-D\\_PA2\\_\\_FedEx=5FInvoice=20=5FCopy=5FIDN12=2D374.exe & 0.64 \\\\\n page 1-2.pdf & 0.64 & 100\\_4X\\_AZ-D\\_PA2\\_\\_my=5Fphoto=5Fin=5Fthe=5Fdominican=5Frepublic.exe & 0.63 \\\\\n fisa.pdf & 0.58 & 100\\_2X\\_PM4\\_EMQ\\_MH\\_\\_message.htm & 0.60 \\\\\n urs.doc & 0.52 & \/var\/work0\/attach\/100\\_4X\\_AZ-D\\_PA2\\_\\_document.exe & 0.58 \\\\\n script.au3 & 0.50 & 100\\_6X\\_AZ-D\\_PA4\\_\\_Information.exe & 0.58 \\\\\n install\\_reader10\\_en\\_air\\_gtbd\\_aih.zip & 0.48 & \/var\/work0\/attach\/100\\_4X\\_AZ-D\\_PA2\\_\\_Ticket.exe & 0.57 \\\\\n dodi-3100-08.pdf & 0.43 & 100\\_4X\\_AZ-D\\_PA2\\_\\_Ticket.exe & 0.57 \\\\ \\hline\n \\end{tabular}\n\\vspace{10pt}\n\\caption{Top 20 most frequently occurring attachment names, and their corresponding percentage share in our spear phishing and spam \/ phishing datasets. Attachment names in the spear phishing emails dataset look much more realistic and genuine as compared to attachment names in spam \/ phishing emails dataset.}\n\\label{tab:attach_names}\n\\end{center}\n\\end{table*}\n\n\nOur email dataset consisted of a combination of targeted spear phishing emails, non targeted spam and phishing emails, and benign emails. We obtained the targeted spear phishing emails from Symantec's enterprise email scanning service. \\emph{Symantec} collects data regarding targeted attacks that consist of emails with malicious attachments. These emails are identified from the vast majority of non-targeted malware by evidence of there being prior research and selection of the recipient, with the malware being of high sophistication and low copy number. The process by which Symantec's enterprise mail scanning service collects such malware has already been described elsewhere~\\cite{Thonnard2012,Lee2013}. The corpus almost certainly omits some attacks, and most likely also includes some non-targeted attacks, but nevertheless it represents a large number of sophisticated targeted attacks compiled according to a consistent set of criteria which render it a very useful dataset to study. \n\n\n\n\nThe non targeted attack emails were also obtained from Symantec's email scanning service. These emails were marked as \\emph{malicious}, and were a combination of malware, spam, and phishing. Both these datasets contained an enormously large number of emails received at hundreds of organizations around the world, where Symantec's email scanning services are being used. Before selecting a suitable sample for organization level analysis, we present an overview of this entire data. Table~\\ref{tab:attach_names} shows the top 20 most frequently occurring attachment names in the complete spear phishing and spam \/ phishing datasets. We found distinct differences in the type of attachment names in these two datasets. While names in spear phishing emails looked fairly genuine and personalized, attachment names in spam \/ phishing emails were irrelevant, and long. It was also interesting to see that the attachment names associated with spear phishing emails were less repetitive than those associated with spam \/ phishing emails. As visible in Table~\\ref{tab:attach_names}, the most commonly occurring attachment name in spear phishing emails was found in less than 3.5\\% of all spear phishing emails, while in the case of spam \/ phishing emails, the most common attachment name was present in over 20\\% of all spam \/ phishing emails. This behavior reflects that attachments in spear phishing emails are named more carefully, and with more effort to make them look genuine.\n\n\n\nWe also looked at the most frequently spread file types in spear phishing, spam, and phishing emails. Table~\\ref{tab:attach_types} shows the top 15 most frequently occurring file types in both the spear phishing and spam \/ phishing email datasets. Not surprisingly, both these datasets had notable presence of executable (.exe, .bat, .com), and compressed (.rar, .zip, .7z) file types. In fact, most of the file types spread through such emails were among the most frequently used file types in general, too. Microsoft Word, Excel, PowerPoint, and PDF files were also amongst the most frequently spread files. It was, however, interesting to note that lesser percentage of targeted spear phishing emails contained executables than spam \/ phishing emails. This reflects that attackers prepare targeted attacks smartly as compared to spammers \/ phishers, and avoid using executables, which are more prone to suspicion.\n\n\n\\begin{table}[!h]\n\\begin{center}\n \\begin{tabular}{p{2.7cm}|p{0.6cm}||p{3cm}|p{0.8cm}}\n \\hline\n Spear phishing Attachment Type & \\% & Spam \/ phishing Attachment Type & \\% \\\\ \\hline\n Zip archive data (zip) & 19.59 & Windows Executable (exe) & 38.39 \\\\\n PDF document (pdf) & 13.73 & ASCII text (txt) & 21.73 \\\\\n Composite Document File & 13.63 & Hypertext (html) & 18.08 \\\\\n Windows Executable (exe) & 11.20 & Hypertext (htm) & 7.06 \\\\\n Rich Text Format data (rtf) & 10.40 & Rich Text Format data (rtf) & 3.04 \\\\\n RAR archive data (rar) & 9.47 & PDF document (pdf) & 2.04 \\\\\n Screensaver (scr) & 5.06 & Zip archive data (zip) & 1.75 \\\\\n data (dat) & 3.00 & Microsoft Word & 1.27 \\\\\n JPEG image data (jpg) & 1.64 & Screensaver (scr) & 1.14 \\\\\n CLASS & 1.56 & Microsoft Excel (xls) & 0.81 \\\\\n Microsoft Word 2007+ & 1.15 & Program Info. file (pif) & 0.80 \\\\\n 7-zip archive data (7z) &1.12 & Dynamic-link Library (dll) & 0.30 \\\\\n Shortcut (lnk) & 1.08 & Windows Batch file (.bat) & 0.24 \\\\\n ASCII text (txt) & 0.80 & JavaScript (js) & 0.17 \\\\\n Dynamic-link Library (dll) & 0.54 & Microsoft HTML Help (chm) & 0.16 \\\\ \\hline\n \\end{tabular}\n\\vspace{5pt}\n\\caption{Top 15 most frequently occurring attachment types, and their corresponding percentage share in our spear phishing and spam \/ phishing datasets. Only 5 file types were common in the top 15 in these datasets.}\n\\label{tab:attach_types}\n\\end{center}\n\\end{table}\n\n\n\nAll the emails present in our full dataset were collected over a period of slightly under 4 years, from March 2009 to December 2013. Figure~\\ref{fig:timeline} presents a timeline of the ``received time\" of all these emails. The spam \/ phishing emails were collected over a period of 3 years, from March 2009 to March 2012. The targeted spear phishing emails were also collected during a period of about 3 years, but from January 2011, to December 2013. The two datasets, thus, had data for a common time period of about 15 months, from January 2011, to March 2012. It was interesting to observe that during this period, while the number of spam and phishing emails saw a tremendous rise, the number of spear phishing emails did not vary too much. This characteristic was observed for the entire 3-year period for spear phishing emails. The number of spear phishing emails received in the beginning and end of our three year observation period saw a 238\\% rise, as compared to a rise of 35,422\\% percent in the number of spam \/ phishing emails.\n\n\\begin{figure*}[!ht]\n \\begin{center}\n\\fbox{\\includegraphics[scale=0.45]{timeline_2.pdf}}\n \\end{center}\n \\caption{Timeline of the number of spear phishing and spam \/ phishing emails in our dataset. The X axis represents time, and the Y axis represents the number of emails on a logarithmic scale. The period of May 2011 - September 2011 saw an exponential increase in the number of spam \/ phishing emails in our dataset.\n }\n \\label{fig:timeline}\n\\end{figure*}\n\n\n\nIn addition to the attachment information and timeline, we also looked at the ``subject\" fields of all the emails present in our datasets. Table~\\ref{tab:subjects} shows the top 20 most frequently occurring ``subject lines\" in our datasets. Evidently, ``subjects\" in both these datasets were very different in terms of context. Targeted spear phishing email subjects seemed to be very professional, talking about jobs, meetings, \\emph{unclassified} information etc. Spam \/ phishing email subjects, however, were observed to follow a completely different genre. These emails' subjects were found to follow varied themes, out of which, three broad themes were fairly prominent: a) fake email delivery failure error messages, which lure victims into opening these emails to see which of their emails failed, and why; b) arrival of packages or couriers by famous courier delivery services -- victims tend to open such messages out of curiosity, even if they are not expecting a package; and c) personalized messages via third party websites and social networks (Hallmark E-Card, hi5 friend request, and Facebook message in this case). Most of such spam \/ phishing emails have generic subjects, to which most victims can relate easily, as compared to spear phishing email subjects, which would seem irrelevant to most common users.\n\n\nIt is important to note that these statistics are for the complete datasets we obtained from Symantec. The total number of emails present in the complete dataset was of the order of hundred thousands. However, we performed our entire analysis on a sample picked from this dataset. The analysis in the rest of the paper talks about only this sample. To make our analysis more exhaustive, we also used a sample of benign emails from the Enron email dataset for our analysis~\\cite{Cohen2009}. All the three email datasets had duplicates, which we identified and removed by using a combination of 5 fields, viz. \\emph{from ID}, \\emph{to ID}, \\emph{subject}, \\emph{body}, and \\emph{timestamp}. On further investigation, we found that these duplicate email messages were different instances of the same email. This happens when an email is sent to multiple recipients at the same time. A globally unique \\emph{message-id} is generated for each recipient, and thus results in duplication of the message. Elimination of duplicates reduced our email sample dataset by about 50\\%. Our final sample email dataset that we used for all our analysis was, therefore, a mixture of targeted attack emails, non targeted attack emails, and benign emails. We now describe this sample.\n\n\n\n\\begin{table*}[!ht]\n\\begin{center}\n \\begin{tabular}{l|l||l|l}\n \\hline\n Spear phishing subjects & \\% & Spam \/ phishing subjects & \\% \\\\ \\hline\n Job Opportunity & 3.45 & Mail delivery failed: returning message to sender & 10.95 \\\\\n Strategy Meeting & 3.09 & Delivery Status Notification (Failure) & 6.71 \\\\\n What is Chen Guangcheng fighting for? & 3.00 & Re: & 2.59 \\\\\n FW: $[$2$]$ for the extension of the measures against North Korea & 1.70 & Re & 2.56 \\\\\n $[$UNCLASSIFIED$]$ 2012 U.S.Army orders for weapons & 1.27 & Become A Paid Mystery Shopper Today! Join and Shop For Free! & 1.28 \\\\\n FW:$[$UNCLASSIFIED$]$2012 U.S.Army orders for weapons & 1.27 & failure notice & 1.09 \\\\\n $<$blank subject line$>$ & 1.17 & Delivery Status Notification (Delay) & 1.06 \\\\\n FW: results of homemaking 2007 annual business plan (min quarter 1 included) & 1.02 & Returned mail: see transcript for details & 0.95 \\\\\n $[$UNCLASSIFIED$]$DSO-DARPA-BAA-11-65 & 0.93 & Get a job as Paid Mystery Shopper! Shop for free and get Paid! & 0.85 \\\\\n Wage Data 2012 & 0.90 & Application number: AA700003125331 & 0.82 \\\\\n U.S.Air Force Procurement Plan 2012 & 0.90 & Your package is available for pickup & 0.78 \\\\\n About seconded expatriate management in overseas offices & 0.80 & Your statement is ready for your review & 0.75 \\\\\n FW:[CLASSIFIED] 2012 USA Government of the the Health Reform & 0.74 & Unpaid invoice 2913. & 0.71 \\\\\n T.T COPY & 0.62 & Track your parcel & 0.70 \\\\\n USA to Provide Declassified FISA Documents & 0.58 & You have received A Hallmark E-Card! & 0.59 \\\\\n FY2011-12 Annual Merit Compensation Guidelines for Staff & 0.55 & Your Account Opening is completed. & 0.57 \\\\\n Contact List Update & 0.45 & Delivery failure & 0.57 \\\\\n DOD Technical Cooperation Program & 0.43 & Undelivered Mail Returned to Sender & 0.56 \\\\\n DoD Protection of Whistleblowing Spies & 0.43 & Laura would like to be your friend on hi5! & 0.56 \\\\\n FW:UK Non Paper on arrangements for the Arms Trade Treaty (ATT) Secretariat & 0.42 & You have got a new message on Facebook! & 0.55 \\\\ \\hline\n \\end{tabular}\n\\vspace{5pt}\n\\caption{Top 20 most frequently occurring subjects, and their corresponding percentage share in our spear phishing, and spam \/ phishing email datasets. Spear phishing email subjects appear to depict that these emails contain highly confidential data. Spam \/ phishing emails, on the other hand, are mainly themed around email delivery error messages, and courier or package receipts.}\n\\label{tab:subjects}\n\\vspace{-15pt}\n\\end{center}\n\\end{table*}\n\n\n\n\\subsection{Email Sample Dataset Description}\n\nTo focus our analysis at the organization level, we identified and extracted the most attacked organizations (excluding free email providing services like Gmail, Yahoo, Hotmail etc.) from the domain names of the victims' email addresses, and picked 14 most frequently attacked organizations. We were however, restricted to pick only those organizations, where the first names and last names were easily extractable from the email addresses. The first name and last name were required to obtain the corresponding LinkedIn profiles of these victims (this process is discussed in detail in Section~\\ref{sec:linkedin_data}). This restriction, in addition to removal of duplicates, left us with a total of 4,742 targeted spear phishing emails sent to 2,434 unique victims (referred to as \\emph{SPEAR} in the rest of the paper); 9,353 non targeted attack emails sent to 5,912 unique non victims (referred to as \\emph{SPAM} in the rest of the paper), and 6,601 benign emails from the Enron dataset, sent to 1,240 unique Enron employees (referred to as \\emph{BENIGN} in the rest of the paper). Further details of this dataset can be found in Table~\\ref{tab:stats}. Table contains the number of victims, and non victims in each of the 15 companies (including Enron), and the number of emails sent to them. The victim and non victim employee sets are mutually exhaustive, and each employee in these datasets received at least one email, and had at least one LinkedIn profile. To maintain anonymity, we do not include the name of the organizations we picked; we only mention the operation sector of these companies.\n\n\\begin{table}[!ht]\n\\begin{center}\n \\begin{tabular}{p{2.1cm}|p{0.8cm}|p{0.7cm}|p{0.8cm}|p{0.7cm}|p{1.3cm}}\n \\hline\n Sector & \\#Victims & \\#Emails & \\#Non Victims & \\#Emails & No. of Employees \\\\ \n\\hline\n Govt. \\& Diplomatic & 206 & 511 & 572 & 1,103 & 10,001+ \\\\\n Info. \\& Broadcasting & 150 & 326 & 240 & 418 & 10,001+ \\\\\n NGO & 131 & 502 & 218 & 472 & 1001-5000 \\\\\n IT\/Telecom\/Defense & 158 & 406 & 68 & 157 & 1001-5000 \\\\\n Pharmaceuticals & 120 & 216 & 589 & 862 & 10,001+ \\\\\n Engineering & 396 & 553 & 1000 & 1,625 & 10,001+ \\\\\n Automotive & 153 & 601 & 891 & 1,204 & 10,001+ \\\\\n Aviation\/Aerospace & 281 & 355 & 161 & 187 & 1001-5000 \\\\\n Agriculture & 94 & 138 & 173 & 264 & 10,001+ \\\\\n IT \\& Telecom & 11 & 12 & 543 & 943 & 5001-10,000 \\\\\n Defense & 388 & 651 & 123 & 147 & 10,001+ \\\\\n Oil \\& energy & 201 & 212 & 680 & 1,017 & 10,001+ \\\\\n Finance & 89 & 129 & 408 & 608 & 10,001+ \\\\\n Chemicals & 56 & 130 & 248 & 346 & 10,001+ \\\\\n Enron & NA & NA & 1,240 & 6,601 & 10,001+ \\\\ \\hline\n Total & 2,434 & 4,742 & 7,154 & 15,954 & ~ \\\\ \\hline\n \\end{tabular}\n\\vspace{5pt}\n\\caption{Detailed description of our dataset of LinkedIn profiles and emails across 15 organizations including Enron. }\n\\label{tab:stats}\n\\vspace{-20pt}\n\\end{center}\n\\end{table}\n\n\n\n\nFigures~\\ref{fig:spearphish_subject},~\\ref{fig:mixed_subject}, and~\\ref{fig:enron_subject} represent the tag clouds of the 100 most frequently occurring words in the ``subject\" fields of our SPEAR, SPAM, and BENIGN datasets respectively. We noticed considerable differences between subjects from all the three datasets. While all three datasets were observed to have a lot of \\emph{forwarded} emails (represented by ``fw\", and ``fwd'' in the tag clouds), SPAM and BENIGN datasets were found to have much more \\emph{reply} emails (signified by ``re\" in the tag clouds) as compared to SPEAR emails. These characteristics of whether an email is forwarded, or a reply, have previously been used as boolean features by researchers to distinguish between phishing and benign emails~\\cite{Toolan2010}. The difference in vocabularies used across the three email datasets is also notable. The SPEAR dataset (Figure~\\ref{fig:spearphish_subject}) was found to be full of attention-grabbing words like \\emph{strategy, unclassified, warning, weapons, defense, US Army} etc. Artifact~\\ref{tab:egspear} shows an example of the attachment name, subject and body of such an email. We removed the received address and other details to maintain anonymity.\n\n\\renewcommand{\\tablename}{ARTIFACT} \n\n\\begin{table}\n\\begin{center}\n \\begin{tabular}{|p{8.5cm}|}\n\\hline\n {\\bf Attachment}: All information about mobile phone.rar \\\\ \\hline\n {\\bf Subject}: RE: Issues with Phone for help \\\\ \\hline\n {\\bf Body}: $<$name$>$,\\\\Thanks for your replying.I contacted my supplier,but he could not resolved it.Now I was worried, so I take the liberty of writing to you.I collect all information including sim card details,number,order record and letters in the txt file.I hope you can deal with the issues as your promised.\\\\Best,\\\\$<$name$>$\\\\\\\\-----Original Message-----\\\\From: Customer Care [mailto:Customer\\_Care@$<$companyDomain$>$]\\\\Sent: 2011-12-8 0:35\\\\To: $<$name$>$\\\\Cc:\\\\Subject: RE: Issues with Phone for help\\\\\\\\Dear $<$name$>$,\\\\\\\\Thank you for your E-mail. I am sorry to hear of your issues. Please can you send your SIM card details or Mobile number so that we can identify your supplier who can assist you further?\\\\\\\\Thank you\\\\\\\\Kind regards,\\\\\\\\$<$name$>$\\\\Customer Service Executive\\\\\\\\$<$Company Name$>$,\\\\$<$Company Address$>$\\\\United Kingdom\\\\\\\\Tel: $<$telephone number$>$\\\\Fax : $<$Fax number$>$ \\\\$<$company website$>$\\\\\\\\-----Original Message-----\\\\From: $<$name$>$ [mailto:$<$email address$>$]\\\\Sent: 08 December 2011 08:27\\\\To: support@$<$companyDomain$>$\\\\Subject: Issues with Phone for help\\\\\\\\Hello,\\\\I purchased order for your IsatPhone Pro six months ago.Now I have trouble that it can't work normally.It often automatic shuts down.Sometimes it tells some information that i can't understand.How to do?Can you help me?\\\\Best,\\\\$<$name$>$\\\\\\\\\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\\\This e-mail has been scanned for viruses by Verizon Business Internet Managed Scanning Services - powered by MessageLabs. For further information visit http:\/\/www.verizonbusiness.com\/uk \\\\ \\hline\n \\end{tabular}\n\\vspace{5pt}\n\\caption{A spear phishing email from our SPEAR dataset. The email shows a seemingly genuine conversation, where the attacker sent a malicious compressed (.rar) attachment to the victim in the middle of the conversation.}\n\\label{tab:egspear}\n\\end{center}\n\\end{table}\n\n\nSPAM emails in our dataset (Figure~\\ref{fig:mixed_subject}) followed a completely different genre, dominated by words like \\emph{parcel, order, delivery, tracking, notification, shipment} etc. We also found mentions of famous courier service brand names like FedEx and DHL which seem to have been used for targeting victims. Such attacks have been widely talked about in the recent past; users have also been warned about scams, and infected payloads (like spyware or malware), that accompany such emails.~\\footnote{\\url{http:\/\/nakedsecurity.sophos.com\/2013\/03\/20\/dhl-delivery-malware\/}}~\\footnote{\\url{http:\/\/www.spamfighter.com\/News-13360-FedEx-and-DHL-Spam-Attack-with-Greater-Ferocity.htm}} Some examples of attachment names, and subjects of such non targeted SPAM emails are shown in Artifact~\\ref{tab:egspam}. BENIGN subjects comprised of diverse keywords like \\emph{report, program, meeting, migration, energy}, which did not seem specific to a particular theme (Figure~\\ref{fig:enron_subject}). These keywords were fairly representative of the kind of typical internal communication that may have been going on in the company.\n\n\n\n\n\\begin{table}[!h]\n\\begin{center}\n \\begin{tabular}{|p{8.5cm}|}\n \\hline\n {\\bf Attachment}: 100A\\_0.txt \\\\\n {\\bf Subject}: DHL Express Notification for shipment 15238305825550113 \\\\ \\hline\n {\\bf Attachment}: .\/attach\/100\\_4X\\_AZ-D\\_PA2\\_\\_FedEx=5FInvoice=5FN 56=2D141.exe \\\\\n {\\bf Subject}: FEDEX Shipment Status NR-6804 \\\\ \\hline\n \\end{tabular}\n\\vspace{5pt}\n\\caption{Examples of \\emph{subject} and \\emph{attachment} names of two spam emails from our SPAM dataset. The \\emph{body} field of the emails was not available in this dataset.}\n\\label{tab:egspam}\n\\end{center}\n\\end{table}\n\n\\renewcommand{\\tablename}{TABLE} \n\n\n\\begin{figure*}[!ht]\n \\begin{center}\n \\subfigure[SPEAR subjects]{\n \\label{fig:spearphish_subject}\n \\includegraphics[scale=0.17]{spearphish_subject.png}\n }\n \t\\subfigure[SPEAR bodies]{\n \\label{fig:spearphish_body}\n \\includegraphics[scale=0.16]{spearphish_body.png}\n }\n \t\\subfigure[SPAM subjects]{\n \\label{fig:mixed_subject}\n \\includegraphics[scale=0.16]{mixed_subject.png}\n }\\\\\n \t\\subfigure[BENIGN subjects]{\n \\label{fig:enron_subject}\n \\includegraphics[scale=0.2]{enron_subject.png}\n }\n \t\\subfigure[BENIGN bodies]{\n \\label{fig:enron_body}\n \\includegraphics[scale=0.2]{enron_body.png}\n }\n \\end{center}\n \\caption{%\nTag clouds of the 100 most frequently occurring words in the subjects and bodies of our SPEAR, SPAM, and BENIGN datasets. Bodies of SPAM emails were not available in our dataset.\n }\n \\label{fig:tags}\n\\end{figure*}\n\n\nWe also compared the body content of SPEAR and BENIGN emails. Figures~\\ref{fig:spearphish_body} and~\\ref{fig:enron_body} represent the tag clouds of the 100 most frequently occurring words in the body fields of the SPEAR and BENIGN datasets respectively. Contrary to our observations from the subject content in the SPEAR dataset (Figure~\\ref{fig:spearphish_subject}), the body content of the SPEAR emails (Figure~\\ref{fig:spearphish_body}) did not look very attention-grabbing or themed. SPEAR bodies contained words like \\emph{attached, please, email, dear, materials, phone} etc., which commonly occur in routine email communications too. The BENIGN body content did not contain anything peculiar or alarming either (Figure~\\ref{fig:enron_body}). Since Symantec's email dataset of spear phishing, spam and phishing emails isn't publicly available, we believe that this characterization of our dataset can be useful for researchers to get a better idea of state-of-the-art, real world malicious email data which circulates in the corporate environment.\n\n\n\n\n\n\n\n\n\n\\subsection{LinkedIn profile dataset} \\label{sec:linkedin_data}\n\nOur second dataset consisted of LinkedIn profiles of the receivers of all the emails present in our email dataset. In fact, we restricted our email dataset to only those emails which were sent to employees having at least one LinkedIn profile. This was done to have a complete dataset in terms of the availability of social and stylometric features. There were two major challenges with data collection from LinkedIn; a) Strict input requirements, and b) Rate limited API.\n\nFirstly, to fetch the profiles of LinkedIn users who are outside a user's network (3$^{rd}$ degree connections and beyond), the LinkedIn People Search API requires first name, last name, and company name as a mandatory input.~\\footnote{\\url{developer.linkedin.com\/documents\/people-search-api}} Understandably, none of the users we were looking for, were in our network, and thus, as specified in the previous subsection, we were restricted to pick up emails of only those companies which followed the format \\emph{firstName}.\\emph{lastName}$@$\\emph{companyDomain} or \\emph{firstName}\\_\\emph{lastName}$@$\\emph{companyDomain}. Restricting our dataset to such email addresses was the only way we could satisfy the API's input requirements.\n\nSecondly, the rate limit of the people search API posed a major hindrance. Towards the end of 2013, LinkedIn imposed a tight limit of 250 calls per day, per application, on the people search API for existing developers, and completely restricted access for new applications and developers, under their Vetted API access program.~\\footnote{\\url{https:\/\/developer.linkedin.com\/blog\/vetted-api-access}} We\nwere able to get access to the Vetted API for two of our applications. Although the new rate limit allowed 100,000 API calls per day, per application, this was still restricted to 100 calls per user, per day, per application. We then created multiple LinkedIn user accounts to make calls to the API. Even with multiple applications, and user accounts, this data collection process took about 4 months. This happened because a lot of our search queries to the API returned no results. On average, we were able to find a LinkedIn profile for only 1 in 10 users in our dataset. This resulted in about 90\\% of the API calls returning no results, and hence getting wasted. Eventually, we were able to collect a total of 2,434 LinkedIn profiles of victims, 5,914 LinkedIn profiles of non victims, across the 14 organizations; and 1,240 LinkedIn profiles of employees from Enron (Table~\\ref{tab:stats}). To obtain these profiles for the 9,588 employees (2,434 victims, 5,914 non victims, and 1,240 Enron employees), the number of API calls we had to make was approximately 100,000 (approx. 10 times the number of profiles obtained). Figure~\\ref{fig:arch} shows the flow diagram describing our data collection process.\n\n\n\\begin{figure}[!h]\n \\begin{center}\n\\includegraphics[scale=0.36]{arch2.pdf}\n \\end{center}\n \\caption{%\nFlow diagram describing the data collection process we used to collect LinkedIn data, and create our final feature vector containing stylometric features from emails, and social features from LinkedIn profiles.\n }\n \\label{fig:arch}\n\\end{figure}\n\n\n\n\nOur first choices for extracting \\emph{social} features about employees were Facebook, and Twitter. However, we found that identifying an individual on Facebook and Twitter using only the first name, last name, and employer company was a hard task. Unlike LinkedIn, the Facebook and Twitter APIs do not provide endpoints to search people using the work company name. This left us with the option to search for people using first name, and last name only. However, searching for people using only these two fields returned too many results on both Facebook and Twitter, and we had no way to identify the correct user that we were looking for. We then visited the profile pages of some users returned by the API results manually, and discovered that the \\emph{work} field for most users on Facebook was private. On Twitter profiles, there did not exist a \\emph{work} field at all. It thus became very hard to be able to find Facebook or Twitter profiles using the email addresses in our dataset. \n\n\n\n\n\\section{Analysis and results} \\label{sec:ar}\n\nTo distinguish spear phishing emails from non spear phishing emails using \\emph{social} features of the receivers, we used four machine learning algorithms, and a total of 27 features; 18 stylometric, and 9 \\emph{social}. The entire analysis and classification tasks were performed using the Weka data mining software~\\cite{Hall2009}. We applied 10-fold cross validation to validate our classification results. We now describe our feature sets, analysis, and results of the classification.\n\n\n\n\\subsection{Feature set description} \\label{sec:fsd}\n\nWe extracted a set of 18 stylometric features from each email in our email dataset, and a set of 9 \\emph{social} features from each LinkedIn profile present in our LinkedIn profile dataset, features described in Table~\\ref{tab:feats}. Features extracted from our email dataset are further categorized into three categories, viz. \\emph{subject} features, \\emph{attachment} features, and \\emph{body} features. It is important to note that we did not have all the three types of these aforementioned features available for all our datasets. While the SPAM dataset did not have \\emph{body} features, the BENIGN dataset did not have the \\emph{attachment} features. Features marked with $^*$ (in Table~\\ref{tab:feats}) have been previously used by researchers to classify spam and phishing emails~\\cite{Toolan2010}. The \\emph{richness} feature is calculated as the ratio of the number of words to the number of characters present in the text content under consideration. We calculate richness value for the email \\emph{subject}, email \\emph{body}, and LinkedIn profile \\emph{summary}. The \\emph{Body\\_hasAttach} features is a boolean variable which is set to true, if the body of the email contain the word ``attached'' or ``attachment'', indicating that an attachment is enclosed with the email. This feature helped us to capture the presence of attachments for the BENIGN dataset, which did not have attachment information. The \\emph{Body\\_numFunctionWords} feature captures the total number of occurrences of function words present in the email body, from a list of function words which includes the words: ~\\emph{account, access, bank, credit, click, identity, inconvenience, information, limited, log, minutes, password, recently, risk, social, security, service,} and \\emph{suspended}. These features have been previously used by Chandrasekaran~\\cite{Chandrasekaran2006}.\n\n\n\\begin{table}[!ht]\n\\begin{center}\n \\begin{tabular}{l|l|l}\n\\hline\n Feature & Data Type & Source \\\\ \\hline\n Subject\\_IsReply$^*$ & Boolean & Email \\\\\n Subject\\_hasBank$^*$ & Boolean & Email \\\\\n Subject\\_numWords$^*$ & Numeric & Email \\\\\n Subject\\_numChars$^*$ & Numeric & Email \\\\\n Subject\\_richness$^*$ & Decimal (0-1) & Email \\\\\n Subject\\_isForwarded$^*$ & Boolean & Email \\\\\n Subject\\_hasVerify$^*$ & Boolean & Email \\\\ \\hline\n Length of attachment name & Numeric & Email \\\\\n Attachment size (bytes) & Numeric & Email \\\\ \\hline\n Body\\_numUniqueWords$^*$ & Numeric & Email \\\\\n Body\\_numNewlines & Numeric & Email \\\\\n Body\\_numWords$^*$ & Numeric & Email \\\\\n Body\\_numChars$^*$ & Numeric & Email \\\\\n Body\\_richness$^*$ & Decimal (0-1) & Email \\\\\n Body\\_hasAttach & Boolean & Email \\\\\n Body\\_numFunctionWords$^*$ & Numeric & Email \\\\\n Body\\_verifyYourAccount$^*$ & Boolean & Email \\\\\n Body\\_hasSuspension$^*$ & Boolean & Email \\\\ \\hline \n Location & Text (country) & LinkedIn \\\\\n numConnections & Numeric (0-500) & LinkedIn \\\\\n SummaryLength & Numeric & LinkedIn \\\\\n SummaryNumChars & Numeric & LinkedIn \\\\\n SummaryUniqueWords & Numeric & LinkedIn \\\\\n SummaryNumWords & Numeric & LinkedIn \\\\\n SummaryRichness & Decimal (0-1) & LinkedIn \\\\\n jobLevel & Numeric (0-7) & LinkedIn \\\\\n jobType & Numeric (0-9) & LinkedIn \\\\\n \\hline\n \\end{tabular}\n\\vspace{5pt}\n\\caption{List of features used in our analysis. We used a combination of stylometric features in addition to normal features. Features marked with $^*$ have been previously used for detecting spam and phishing emails.}\n\\label{tab:feats}\n\n\\end{center}\n\\end{table}\n\n\n\nThe \\emph{social} features we extracted from the LinkedIn profiles, captured three distinct types of information about an employee, viz. location, connectivity, and profession. The \\emph{Location} was a text field containing the state \/ country level location of an employee, as specified by her on her LinkedIn profile. We extracted and used the country for our analysis. The \\emph{numConnections} was a numeric field, and captured the number of connections that a user has on LinkedIn. If the number of connections for a user is more than 500, the value returned is ``500+\" instead of the actual number of connections. These features captured the location and connectivity respectively. In addition to these two, we extracted 5 features from the \\emph{Summary} field, and 2 features from the \\emph{headline} field returned by LinkedIn's People Search API. The \\emph{Summary} field is a long, free-text field comprising of a summary about a user, as specified by her, and is optional. The features we extracted from this field were similar to the ones we extracted from the subject and body fields in our email dataset. These features were, the \\emph{summary length, number of characters, number of unique words, total number of words}, and \\emph{richness}. We introduced two new features, \\emph{job\\_level} and \\emph{job\\_type}, which are numeric values ranging from 0 to 7, and 0 to 9 respectively, describing the position and area of work of an individual. We looked for presence of certain level and designation specific keywords in the ``headline'' field of a user, as returned by the LinkedIn API. The job levels and job types, and their numeric equivalents are as follows:\n\n\\begin{itemize}\n\\item Job\\_level; maximum of the following:\n\n1 - Support\\\\\n2 - Intern\\\\\n3 - Temporary\\\\\n4 - IC\\\\\n5 - Manager\\\\\n6 - Director\\\\\n7 - Executive\\\\\n0 - Other; if none of the above are found.\n\n\\item Job\\_type; minimum of the following:\n\n1 - Engineering\\\\\n2 - Research\\\\\n3 - QA\\\\\n4 - Information Technology\\\\\n5 - Operations\\\\\n6 - Human Resources\\\\\n7 - Legal\\\\\n8 - Finance\\\\\n9 - Sales \/ Marketing\\\\\n0 - Other; if none of the above are found.\n\\end{itemize}\n\nTo see if information extracted about a victim from online social media helps in identifying a spear phishing email sent to her, we performed classification using a) \\emph{email} features~\\footnote{We further split email features into \\emph{subject}, \\emph{body}, and \\emph{attachment} features for analysis, wherever available.}; b) \\emph{social} features, and c) using a combination of these features. We compared these three accuracy scores across a combination of datasets viz. SPEAR versus SPAM emails from Symantec's email scanning service, SPEAR versus benign emails from BENIGN dataset, and SPEAR versus a mixture of emails from BENIGN, and SPAM from the Symantec dataset. As mentioned earlier, not all \\emph{email} features mentioned in Table~\\ref{tab:feats} were available for all the three email datasets. The BENIGN dataset did not have attachment related features, and the \\emph{body} field was missing in the SPAM email dataset. We thus used only those features for classification, which were available in both the targeted, and non targeted emails.\n\n\\subsection{SPEAR versus SPAM emails from Symantec} \\label{sec:sp_vs_spam}\n\nTable~\\ref{tab:sp_spam} presents the results of our first analysis where we subjected SPEAR and SPAM emails from Symantec, to four machine learning classification algorithms, viz. Random Forest~\\cite{Breiman2001}, J48 Decision Tree~\\cite{Quinlan1993}, Naive Bayesian~\\cite{John1995}, and Decision Table~\\cite{Kohavi1995}. Feature vectors for this analysis were prepared from 4,742 SPEAR emails, and 9,353 SPAM emails, combined with \\emph{social} features extracted from the LinkedIn profiles of receivers of these emails. Using a combination of all \\emph{email} and \\emph{social} features, we were able to achieve a maximum accuracy of 96.47\\% using the Random Forest classifier for classifying SPEAR and SPAM emails. However, it was interesting to notice that two out of the four classifiers performed better \\emph{without} the social features. Although the Decision Table classifier seemed to perform equally well with, and without the social features, it performed much better using only \\emph{email} features, as compared to only \\emph{social} features.~\\footnote{This happened because the Decision Table classifier terminates search after scanning for a certain (fixed) number of non-improving nodes \/ features.} In fact, the Decision Table classifier achieved the maximum accuracy using \\emph{attachment} features, which highlights that the attachments associated with SPEAR and SPAM emails were also substantially different in terms of name and size. We achieved an overall maximum accuracy of 98.28\\% using the Random Forest classifier trained on only email features. This behavior revealed that the public information available on the LinkedIn profile of an employee in our dataset, does not help in determining whether she will be targeted for a spear phishing attack or not.\n\n\n\\begin{table}[!h]\n\\begin{center}\n \\begin{tabular}{l|l|p{0.7cm}|p{1cm}|p{0.9cm}|p{0.7cm}}\n \\hline\n Feature set & Classifier & Random Forest & J48 Decision Tree & Naive Bayesian & Decision Table \\\\ \\hline\n Subject (7) & Accuracy (\\%) & 83.91 & 83.10 & 58.87 & 82.04 \\\\\n ~ & FP rate & 0.208 & 0.227 & 0.371 & 0.227 \\\\ \\hline\n Attachment (2) & Accuracy (\\%) & 97.86 & 96.69 & 69.15 & {\\bf 95.05} \\\\\n ~ & FP rate & 0.035 & 0.046 & 0.218 & 0.056 \\\\ \\hline\n All email (9) & Accuracy (\\%) & {\\bf 98.28} & {\\bf 97.32} & 68.69 & {\\bf 95.05} \\\\\n ~ & FP rate & 0.024 & 0.035 & 0.221 & 0.056 \\\\ \\hline\n Social (9) & Accuracy (\\%) & 81.73 & 76.63 & 65.85 & 70.90 \\\\\n ~ & FP rate & 0.229 & 0.356 & 0.445 & 0.41 \\\\ \\hline\n Email + & Accuracy (\\%) & 96.47 & 95.90 & {\\bf 69.35} & {\\bf 95.05} \\\\\n Social (18) & FP rate & 0.052 & 0.054 & 0.232 & 0.056 \\\\ \\hline\n \\end{tabular}\n\\vspace{5pt}\n\\caption{Accuracy and weighed false positive rates for SPEAR versus SPAM emails. Social features reduce the accuracy when combined with email features.}\n\\label{tab:sp_spam}\n\\end{center}\n\\end{table}\n\n\nTo get a better understanding of the results, we looked at the information gain associated with each feature using the InfoGainAttributeEval Attribute Evaluator package.~\\footnote{\\url{http:\/\/weka.sourceforge.net\/doc.dev\/weka\/attributeSelection\/InfoGainAttributeEval.html}} This package calculates the ~\\emph{information gain}~\\footnote{This value ranges between 0 and 1, where a higher value represents a more discriminating feature.} associated with each feature, and ranks the features in descending order of the information gain value. The ranking revealed that the attachment related features were the most distinguishing features between SPEAR and SPAM emails. This phenomenon was also highlighted by the Decision Table classifier (Table~\\ref{tab:sp_spam}). The attachment size was the most distinguishing feature with an information gain score of 0.631, followed by length of attachment name, with an information gain score of 0.485. As evident from Table~\\ref{tab:rankedfeats1}, attachment sizes associated with SPAM emails have very high standard deviation values, even though the average attachment sizes of SPAM and SPEAR emails are fairly similar. It is also evident that attachments associated with SPAM emails tend to have longer names; on average, twice in size as compared to attachments associated with SPEAR emails. Among subject features, we found no major difference in the length (number of characters, and number of words) of the subject fields across the two email datasets.\n\n\n\\begin{table}[!h]\n\\begin{center}\n \\begin{tabular}{l|l|c|c|c|c}\n \\hline\n \\multirow{2}{*}{Feature} & \\multirow{2}{*}{Info. Gain} & \\multicolumn{2}{c|}{SPEAR} & \\multicolumn{2}{c}{SPAM} \\\\ \\cline{3-6}\n ~ & ~ & Mean & Std Dev. & Mean & Std. Dev. \\\\ \\hline\n Attachment size (Kb) & 0.6312 & 285 & 531 & 262 & 1,419 \\\\\n Len. attachment name & 0.4859 & 25.48 & 16.03 & 51.08 & 23.29 \\\\\n Subject\\_richness & 0.2787 & 0.159 & 0.05 & 0.177 & 0.099 \\\\\n Subject\\_numChars & 0.1650 & 29.61 & 17.77 & 31.82 & 23.85 \\\\\n Location & 0.0728 & - & - & - & - \\\\\n Subject\\_numWords & 0.0645 & 4.74 & 3.28 & 4.59 & 3.97 \\\\\n numConnections & 0.0219 & 158.68 & 164.31 & 183.82 & 171.45 \\\\\n Subject\\_isForwarded & 0.0219 & - & - & - & - \\\\ \n Subject\\_isReply & 0.0154 & - & - & - & - \\\\\n SummaryRichness & 0.0060 & 0.045 & 0.074 & 0.053 & 0.078 \\\\ \\hline\n \\end{tabular}\n\\vspace{5pt}\n\\caption{Information gain, mean and standard deviation of the 10 most informative features from SPEAR and SPAM emails.}\n\\label{tab:rankedfeats1}\n\\end{center}\n\\end{table}\n\n\nIt was interesting to see that apart from the Location, number of LinkedIn connections, and SummaryRichness, none of the other social features were ranked amongst the top 10 informative features. Figure~\\ref{fig:countries_spam_sp} shows the top 25 \\emph{Locations} extracted from the LinkedIn profiles of employees of the 14 companies who received SPAM and SPEAR emails. We found a fairly high correlation of 0.88 between the number of SPAM and SPEAR emails received at these locations, depicting that there is not much difference between the number of SPAM and SPEAR emails received by most locations. This falls in line with the low information gain associated with this feature. Among the top 25, only 3 locations viz. France, Australia, and Afghanistan received more SPEAR emails than SPAM emails.\n\n\\begin{figure}[!h]\n \\begin{center}\n\\includegraphics[scale=0.42]{countries_spam_spearphish.png}\n \\end{center}\n \\caption{%\nNumber of SPEAR and SPAM emails received by employees in the top 25 locations extracted from their LinkedIn profiles. Employees working in France, Australia, and Afghanistan received more SPEAR emails than SPAM emails.\n }\n \\label{fig:countries_spam_sp}\n\\end{figure}\n\n\nThe number of LinkedIn connections of the recipients of SPEAR and SPAM emails in our dataset are presented in Figure~\\ref{fig:linkedin1}. There wasn't much difference between the number of LinkedIn connections of recipients of SPEAR emails, and the number of LinkedIn connections of recipients of SPAM emails. We grouped the number of LinkedIn connections into 11 buckets as represented by the X axis in Figure~\\ref{fig:linkedin1}, and found a strong correlation value of 0.97 across the two classes (SPEAR and SPAM). This confirmed that the number of LinkedIn connections did not vary much between recipients of SPEAR and SPAM emails, and thus, is not an informative feature for distinguishing between SPEAR and SPAM emails.\n\n\n\\begin{figure}[!h]\n \\begin{center}\n\\includegraphics[scale=0.33]{linkedin_conn_1_labeled.png}\n \\end{center}\n \\caption{%\nNumber of LinkedIn connections of the recipients of SPEAR and SPAM emails. The number of connections are plotted on the X axis, and the number of employee profiles are plotted on the Y axis. Maximum number of employee profiles had less than 50 LinkedIn connections.\n }\n \\label{fig:linkedin1}\n\\end{figure}\n\n\n\n\n\\subsection{SPEAR emails versus BENIGN emails}\n\nSimilar to the analysis performed in Section~\\ref{sec:sp_vs_spam}, we applied machine learning algorithms on a different dataset containing SPEAR emails, and BENIGN emails. This dataset contained 4,742 SPEAR emails, and 6,601 benign emails from BENIGN. Since BENIGN mostly contains internal email communication between Enron's employees, we believe that it is safe to assume that none of these emails would be targeted spear phishing emails, and can be marked as benign. Similar to our observations in Section~\\ref{sec:sp_vs_spam}, we found that, in this case too, only \\emph{email} features performed slightly better than a combination of \\emph{email} and \\emph{social} features, at distinguishing spear phishing emails from non spear phishing emails. We were able to achieve a maximum accuracy of 97.04\\% using the Random Forest classifier trained on a set of 25 features; 16 \\emph{email}, and 9 \\emph{social} features. However, the overall maximum accuracy that we were able to achieve for this dataset was 97.39\\%, using only \\emph{email} features. Table~\\ref{tab:sp_enron} shows the results of our analysis in detail. Three out of the four classifiers performed best with \\emph{email} features; two classifiers performed best using a combination of \\emph{subject} and \\emph{body} features, while one classifier performed best using only \\emph{body} features. The Naive Bayes classifier worked best using \\emph{social} features.\n\n\n\\begin{table}[!h]\n\\begin{center}\n \\begin{tabular}{l|l|p{0.8cm}|p{1cm}|p{0.9cm}|p{0.7cm}}\n \\hline\n Feature set & Classifier & Random Forest & J48 Decision Tree & Naive Bayesian & Decision Table \\\\ \\hline\n Subject (7) & Accuracy (\\%) & 81.19 & 81.11 & 61.75 & 79.55 \\\\\n ~ & FP rate & 0.210 & 0.217 & 0.489 & 0.228 \\\\ \\hline\n Body (9) & Accuracy (\\%) & 97.17 & 95.62 & 53.81 & {\\bf 90.85} \\\\\n ~ & FP rate & 0.031 & 0.048 & 0.338 & 0.082 \\\\ \\hline\n All email (16) & Accuracy (\\%) & {\\bf 97.39} & {\\bf 95.84} & 54.14 & 89.80 \\\\\n ~ & FP rate & 0.029 & 0.044 & 0.334 & 0.090 \\\\ \\hline\n Social (9) & Accuracy (\\%) & 94.48 & 91.79 & {\\bf 69.76} & 83.80 \\\\\n ~ & FP rate & 0.067 & 0.103 & 0.278 & 0.198 \\\\ \\hline\n Email + & Accuracy (\\%) & 97.04 & 95.28 & 57.27 & 89.80 \\\\\n Social (25) & FP rate & 0.032 & 0.052 & 0.316 & 0.090 \\\\ \\hline\n \\end{tabular}\n\\vspace{5pt}\n\\caption{Accuracy and weighed false positive rates for SPEAR emails versus BENIGN emails. Similar to SPEAR versus SPAM, social features decrease the accuracy when combined with email features in this case too.}\n\\label{tab:sp_enron}\n\\end{center}\n\\end{table}\n\n\nTable~\\ref{tab:rankedfeats2} presents the 10 most informative features, along with their information gain, mean and standard deviation values from the SPEAR and BENIGN datasets. The \\emph{body} features were found to be the most informative in this analysis, with only 2 \\emph{social} features among the top 10. Emails in the BENIGN dataset were found to be much longer than SPEAR emails in our Symantec dataset in terms of number of words, and number of characters in their ``body\". The ``subject\" lengths, however, were found to be very similar across SPEAR and BENIGN. \n\n\n\\begin{table}[!h]\n\\begin{center}\n \\begin{tabular}{p{2.5cm}|p{0.6cm}|c|c|c|c}\n \\hline\n \\multirow{2}{*}{Feature} & Info. & \\multicolumn{2}{c|}{SPEAR} & \\multicolumn{2}{c}{BENIGN} \\\\ \\cline{3-6}\n ~ & Gain & Mean & Std Dev. & Mean & Std. Dev. \\\\ \\hline\n Body\\_richness & 0.6506 & 0.134 & 0.085 & 0.185 & 0.027 \\\\\n Body\\_numChars & 0.5816 & 313.60 & 650.48 & 1735.5 & 8692.6 \\\\\n Body\\_numWords & 0.4954 & 53.12 & 107.53 & 312.81 & 1572.1 \\\\\n Body\\_numUniqueWords & 0.4766 & 38.08 & 49.70 & 149.93 & 416.40 \\\\\n Location & 0.3013 & - & - & - & - \\\\\n Body\\_numNewlines & 0.2660 & 11.29 & 32.70 & 43.58 & 215.77 \\\\\n Subject\\_richness & 0.2230 & 0.159 & 0.051 & 0.174 & 0.056 \\\\\n numConnections & 0.1537 & 158.68 & 164.31 & 259.89 & 167.14 \\\\\n Subj\\_numChars & 0.1286 & 29.61 & 17.77 & 28.54 & 15.23 \\\\ \n Body\\_numFunctionWords& 0.0673 & 0.375 & 1.034 & 1.536 & 5.773 \\\\\n \\hline\n \\end{tabular}\n\\vspace{5pt}\n\\caption{Information gain, mean and standard deviation of the 10 most informative features from SPEAR and BENIGN emails. The \\emph{body} features performed best at distinguishing SPEAR emails from BENIGN emails.}\n\\label{tab:rankedfeats2}\n\\end{center}\n\\end{table}\n\n\n\nThe Random Forest classifier was also able to achieve an accuracy rate of 94.48\\% using only \\emph{social} features; signifying that there exist distinct differences between the LinkedIn profiles of Enron employees, and the LinkedIn profiles of the employees of the 14 companies in our dataset. The \\emph{location} attribute was found to be the most distinguishing feature among the \\emph{social} features. This was understandable since most of the Enron employees were found to be based in the US (as Enron was an American services company). However, we also found a considerable difference in the average number of LinkedIn connections of Enron employees, and employees of the 14 organizations from our dataset (mean values for \\emph{numConnections} feature in Table~\\ref{tab:rankedfeats2}). \n\n\n\\subsection{SPEAR versus a mixture of BENIGN and SPAM}\n\nWhile analyzing SPEAR with SPAM, and BENIGN emails separately, we found similar results where \\emph{social} features were not found to be very useful in both the cases. So we decided to use a mixture of SPAM and BENIGN emails against SPEAR emails, and perform the classification tasks again. We found that in this case, two out of the four classifiers performed better with a combination of email and social features, while two classifiers performed better with only \\emph{email} features. However, the overall maximum accuracy was achieved using a combination of \\emph{email} and \\emph{social} features (89.86\\% using Random Forest classifier). This result is in contradiction with our analysis of SPEAR versus SPAM, and SPEAR versus BENIGN separately, where \\emph{email} features always performed better independently, than a combination of \\emph{email} and \\emph{social} features. Our overall maximum accuracy, however, dropped to 89.86\\% (from 98.28\\% in SPEAR versus SPAM email classification) because of the absence of \\emph{attachment} features in this dataset. Although the \\emph{attachment} features were available in the SPAM dataset, their unavailability in BENIGN forced us to remove this feature for the current classification task. Eventually, merging the SPAM email dataset with BENIGN reduced our email dataset to only 7 features, all based on the email ``subject\". Table~\\ref{tab:sp_spamenron} presents the detailed results from this analysis.\n\n\n\\begin{table}[!h]\n\\begin{center}\n \\begin{tabular}{l|l|p{0.8cm}|p{1cm}|p{0.9cm}|p{0.7cm}}\n \\hline\n Feature set & Classifier & Random Forest & J48 Decision Tree & Naive Bayesian & Decision Table \\\\ \\hline\n Subject (7) & Accuracy (\\%) & 86.48 & 86.35 & {\\bf 77.99} & {\\bf 85.46} \\\\\n ~ & FP rate & 0.333 & 0.352 & 0.681 & 0.341 \\\\ \\hline\n Social (9) & Accuracy (\\%) & 88.04 & 84.69 & 74.46 & 80.61 \\\\\n ~ & FP rate & 0.241 & 0.371 & 0.454 & 0.432 \\\\ \\hline\n Email + & Accuracy (\\%) & {\\bf 89.86} & {\\bf 88.38} & 73.97 & 84.14 \\\\\n Social (16) & FP rate & 0.202 & 0.248 & 0.381 & 0.250 \\\\ \\hline\n \\end{tabular}\n\\vspace{5pt}\n\n\\caption{Accuracy and weighed false positive rates for SPEAR emails versus mix of SPAM emails and BENIGN emails. Unlike SPEAR versus SPAM, or SPEAR versus BENIGN, \\emph{social} features increased the accuracy when combined with email features in this case.}\n\\label{tab:sp_spamenron}\n\\end{center}\n\\end{table}\n\n\nAs mentioned earlier, combining the SPAM email dataset with BENIGN largely reduced our \\emph{email} feature set. We were left with 7 out of a total of 18 email features described in Table~\\ref{tab:feats}. Understandably, due to this depleted \\emph{email} feature set, we found that the email features did not perform as good as \\emph{social} features in this classification task. Despite being fewer in number, the \\emph{subject} features, viz. \\emph{Subject\\_richness} and \\emph{Subject\\_numChars} were found to be two of the most informative features (Table~\\ref{tab:rankedfeats3}). However, the information gain value associated with both these features was fairly low. This shows that even being the best features, the \\emph{Subject\\_richness} and \\emph{Subject\\_numChars} were not highly distinctive features amongst spear phishing, and non spear phishing emails. Similar mean and standard deviation values for both these features in Table~\\ref{tab:rankedfeats3} confirm these outcomes.\n\n\n\\begin{table}[!h]\n\\begin{center}\n \\begin{tabular}{l|l|c|c|c|c}\n \\hline\n \\multirow{2}{*}{Feature} & \\multirow{2}{*}{Info. Gain} & \\multicolumn{2}{c|}{SPEAR} & \\multicolumn{2}{c}{SPAM + BENIGN} \\\\ \\cline{3-6}\n ~ & ~ & Mean & Std Dev. & Mean & Std. Dev. \\\\ \\hline\n Subject\\_richness & 0.1829 & 0.159 & 0.051 & 0.176 & 0.084 \\\\\n Subject\\_numChars & 0.1050 & 29.61 & 17.77 & 30.46 & 20.79 \\\\\n Location & 0.0933 & - & - & - & - \\\\\n numConnections & 0.0388 & 158.68 & 164.31 & 215.30 & 173.76 \\\\\n Subject\\_numWords & 0.0311 & 4.74 & 3.28 & 4.75 & 3.57 \\\\\n Subject\\_isForwarded & 0.0188 & - & - & - & - \\\\\n Subject\\_isReply & 0.0116 & - & - & - & - \\\\ \n SummaryNumChars & 0.0108 & 140.98 & 308.17 & 198.62 & 367.81 \\\\ \n SummaryRichness &0.0090 & 0.045 & 0.074 & 0.057 & 0.080 \\\\ \n jobLevel &0.0088 & 3.41 & 2.40 & 3.71 & 2.49 \\\\ \\hline\n \\end{tabular}\n\\vspace{5pt}\n\\caption{Information gain, mean and standard deviation of the 10 most informative features from SPEAR and a combination of BENIGN and SPAM emails. The \\emph{subject} features performed best at distinguishing SPEAR emails from non SPEAR emails.}\n\\label{tab:rankedfeats3}\n\\end{center}\n\\end{table}\n\n\nContrary to our observations in SPEAR versus SPAM, and SPEAR versus BENIGN emails, we found five \\emph{social} features among the top 10 features in this analysis. These were the \\emph{Location, numConnections, SummaryNumChars, Richness}, and \\emph{jobLevel} features. Although there was a significant difference between the average number of LinkedIn connections in the two datasets, this feature did not have much information gain associated with it due to the very large standard deviation.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Discussion} \\label{sec:conclusion}\n\nIn this paper, we attempted to utilize \\emph{social} features from LinkedIn profiles of employees from 14 organizations, to distinguish between spear phishing and non spear phishing emails. We extracted LinkedIn profiles of 2,434 employees who received a 4,742 targeted spear phishing emails; 5,914 employees who received 9,353 spam or phishing emails; and 1,240 Enron employees who received 6,601 benign emails.\nWe performed our analysis on a real world dataset from Symantec's enterprise email scanning service, which is one of the biggest email scanning services used in the corporate organizational level. Furthermore, we targeted our analysis completely on corporate employees from 14 multinational organizations instead of random real-world users. The importance of studying spear phishing emails in particular, instead of general phishing emails, has been clearly highlighted by Jagatic et al.~\\cite{Jagatic2007}.\nWe performed three classification tasks viz. spear phishing emails versus spam \/ phishing emails, spear phishing emails versus benign emails from Enron, and spear phishing emails versus a mixture of spam \/ phishing emails and benign Enron emails. We found that in two out of the three cases, social features extracted from LinkedIn profiles of employees did not help in determining whether an email received by them was a spear phishing email or not. Classification results from a combination of spam \/ phishing, and benign emails showed some promise, where \\emph{social} features were found to be slightly helpful. The depleted \\emph{email} feature sets in this case, however, aided the enhancement in classifier performance.\nWe believe that it is safe to conclude that publicly available content on an employee's LinkedIn profile was not used to send her targeted spear phishing emails in our dataset. However, we cannot rule out the possibility of such an attack outside our dataset, or in future. These attacks may be better detected with access to richer \\emph{social} features. This methodology of detecting spear phishing can be helpful for safeguarding soft targets for phishers, i.e. those who have strong social media footprint. Existing phishing email filters and products can also exploit this technique to improve their performance, and provide personalized phishing filters to individuals.\n\nThere can be multiple reasons for our results being non-intuitive. Firstly, the amount of social information we were able to gather from LinkedIn, was very limited. These limitations have been discussed in Section~\\ref{sec:linkedin_data}. It is likely that in a real-world scenario, an attacker may be able to gain much more information about a victim prior to the attack. This could include looking for the victim's profile on other social networks like Facebook, Twitter etc., looking for the victim's presence on the Internet in general, using search engines (Google, Bing etc.), and profiling websites like Pipl~\\footnote{\\url{https:\/\/pipl.com\/}}, Yasni~\\footnote{\\url{http:\/\/www.yasni.com\/}} etc. The process of data collection by automating this behavior was a time consuming process, and we were not able to take this approach due to time constraints. Secondly, it was not clear that which all aspect(s) of a user's social profiles were most likely to be used by attackers against them. We tried to use all the features viz. textual information (summary and headline), connectivity (number of connections), work information (job level, and job type) and location information, which were made available by LinkedIn API, to perform our classification tasks. However, it is possible that none of these features were used by attackers to target their victims. In fact, we have no way to verify that the spear phishing emails in our dataset were even crafted using features from social profiles of the victims. These reasons, however, only help us in better understanding the concept of using social features in spear phishing emails.\n\n\nIn terms of research contributions, this work is based on a rich, true positive, real world dataset of spear phishing, spam, and phishing emails, which is not publicly available. We believe that characterization of this data can be very useful for the entire research community to better understand the state-of-the-art spear phishing emails that have been circulated on the Internet over the past two years. To maintain anonymity and confidentiality, we could not characterize this data further, and had to anonymize the names of the 14 organizations we studied. Also, after multiple reports highlighting and warning about social media features being used in spear phishing, there does not exist much work in the research community which studies this phenomenon. \n\n\nWe would like to emphasize that the aim of this work is not to try and improve the existing state-of-the-art phishing email detection techniques based on their header, and content features, but to see if the introduction of social media profile features can help existing techniques to better detect spear phishing emails. We believe that this work can be a first step towards exploring threats posed by the enormous amount of contextual information about individuals, that is present on online social media. In future, we would like to carry out a similar analysis using the same email dataset, with more social features, which we were not able to collect in this attempt due to time constraints. We would also like to apply more machine learning and classification techniques like Support Vector Machines, Stochastic Gradient boosting techniques etc. on this dataset to get more insights into why social features did not perform well.\n\n\n\n\n\\section{Acknowledgement}\nWe would like to thank the Symantec team for providing us with the email data that we used for this work. We would also like to thank the members of Precog Research Group, and Cybersecurity Education and Research Center at IIIT-D for their support.\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\nAssume that $H$ is a separable Hilbert space with inner product $\\langle \\, \\cdot \\,,\\, \\cdot \\, \\rangle$ and consider a self-adjoint operator~$A$ with simple discrete spectrum acting in~$H$. Our aim is to study spectral properties of the rank one perturbations of~$A$, i.e., of the operators $B$ of the form\n\\\n B=A + \\langle \\cdot, \\varphi \\rangle \\psi,\n\\\nwhere $\\varphi$ and $\\psi$ are nonzero elements of $H$.\n\nRank-one perturbations of operators and matrices have been actively studied in both mathematical and physical literature for the reason that, on the one hand, they are simple enough to allow description of the spectral properties of perturbed operators via closed-form formulae which then can be analysed using various techniques; on the other hand, such perturbations turn out to be general enough to produce various non-trivial effects. \n\nOne of the most general results in a finite-dimensional setting is given by Krupnik~\\cite{Kru92} and states that a rank-one perturbation of an $n\\times n$ matrix~$A$ can possess an arbitrary spectrum counting multiplicity. In other words, given any natural number $k$, any pairwise distinct complex numbers~$z_1, z_2, \\dots, z_k$, and any natural numbers $m_1, m_2, \\dots, m_k$ satisfying $m_1+ m_2 + \\dots + m_k =n$, there is a rank-one perturbation~$B$ of~$A$ whose spectrum consists of the points $z_1, z_2, \\dots, z_k$ of the corresponding algebraic multiplicities $m_1,m_2, \\dots, m_n$. This statement is also specialized to cases when both~$A$ and the perturbed matrix~$B$ belong to the Hermitian, unitary, or normal classes. Savchenko~\\cite{Sav03} studies the effect a generic rank-one perturbation has on the Jordan structure of a matrix~$A$; an interesting observation is that, typically, in each root subspace, only the Jordan chain of the largest length splits; in~\\cite{Sav04}, this is further generalized to low-rank perturbations, cf.\\ also~\\cite{MorDop03}. Similar results in infinite-dimensional Banach spaces were earlier derived by H\\\"ormander and Melin in~\\cite{HorMel94}. Bounds on the number of distinct eigenvalues of~$B$ in terms of some spectral characteristics of~$A$ are established in~\\cite{Far16}. \n\n\nStructured perturbations of matrices and matrix pencils have recently been thoroughly studied in a series of papers by Mehl a.o.~\\cite{MehMehRanRod11, MehMehRanRod12, MehMehRanRod13, MehMehRanRod14, MehMehRanRod16, MehMehWoj17, SosMorMeh20}. Changes in the Jordan structures under perturbation within the classes of complex $J$-Hamiltonian and $H$-symmetric matrices and application in the control theory\nis discussed in~\\cite{MehMehRanRod11}; see~\\cite{MehMehRanRod13, MehMehRanRod16} for further treatment in both the real and complex case. The class of $H$-Hermitian matrices, with (skew-)Hermitian $H$, is studied in~\\cite{MehMehRanRod12, MehMehRanRod14} via the canonical form of the pair $(B, H)$. Rank-one perturbations of matrix pencils are discussed e.g.\\ in~\\cite{MehMehWoj17, GerTru17, BarRoc20}. A general perturbation theory for structured matrices is developed in the recent paper~\\cite{SosMorMeh20}.\n\nThe above results typically exploit essentially matrix methods and thus are not directly applicable to the infinite-dimensional case (see, however, \\cite{HorMel94}). Rank-one perturbations of bounded or unbounded operators in infinite-dimensional Hilbert spaces have been studied within the general operator theory. For instance, a comprehensive spectral theory for rank-one perturbations of unbounded operators in the self-adjoint case is developed in~\\cite{Sim95}, where a detailed characterization of discrete, absolutely continuous, and singlularly continuous components of the spectrum of the perturbed operator is given. A thorough overview of the theory of singular point perturbations of Schr\\\"odinger operators (formally corresponding to additive Dirac delta-functions and their derivatives) is given in the monographs by Albeverio a.o.~\\cite{AGHH, AlbKur00}. There has been much work devoted to the so-called singular and super-singular rank-one perturbations of self-adjoint operators, where the functions $\\varphi$ and $\\psi$ belong to the scales of Hilbert spaces $\\operatorname{dom}(A^\\alpha)$ with negative~$\\alpha$, see e.g.~\\cite{AlbKosKurNiz03, AlbKonKos05, AlbKos99, AlbKuzNiz08, Gol18, Kur04, KurLugNeu19, KuzNiz06, DudVdo16}; in this case, a typical approach is through the Krein extension theory of self-adjoint operators. Rank-one and finite-rank perturbations of self-adjoint operators in Krein spaces have been recently discussed in e.g.~\\cite{BehMoeTru14, BehLebPerMoeTru16}.\n\nDespite the extensive research in the area, there seems to be no complete infinite-dimensional generalization of the results by Krupnik~\\cite{Kru92}. The most pertinent work we are aware of include the papers by H\\\"ormader and Melin~\\cite{HorMel94} and by Behrndt a.o.~\\cite{BehLebPerMoeTru15}, which characterize possible changes in the Jordan structure of root subspaces of linear mappings in infinite-di\\-men\\-si\\-o\\-nal linear vector spaces under general finite-rank perturbations. \n\nOur motivation in this work was to understand how the spectrum of an operator in an infinite-dimensional Hilbert space can change under a rank-one perturbation, both locally, i.e., on the level of root subspaces, and globally, i.e., on the level of eigenvalue asymptotics. This task is quite non-trivial even in the case when the unperturbed operator~$A$ is self-adjoint but has generic spectrum, cf.~\\cite{Sim95}. Therefore, we decided to start with deriving a complete spectral picture in the simplest case where the unperturbed operator~$A$ is self-adjoint and has simple discrete spectrum. Under this assumption, our main result (Theorem~\\ref{thm:main}) shows that the rank-one perturbation~$B$ of~$A$ may get eigenvalues of arbitrary algebraic multiplicity in an arbitrary finite set of points; however, all sufficiently large eigenvalues remain simple and asymptotically close to the eigenvalues of~$A$. In the finite-dimensional case, our analysis leads to an extension of the result by Krupnik~\\cite{Kru92}; Theorem~\\ref{thm:finite-dim} states that one of the vectors~$\\varphi$ or $\\psi$ can be fixed arbitrarily in a ``generic'' set, and then one can find the other vector such that the perturbed matrix $B$ possesses the prescribed spectrum and, moreover, such choice is unique. We also specify this result in Theorem~\\ref{thm:phi-arbitrary} to the case when $\\varphi$ or $\\psi$ is fixed arbitrarily. We also note that a complete characterization of the possible spectra of rank-one perturbations of self-adjoint operators in Hilbert space, including precise asymptotic distribution and the constructive algorithm of finding~$\\varphi$ and $\\psi$, is suggested in a subsequent paper~\\cite{DobHry20}.\n\nThe structure of the paper is as follows. In the next section, we introduce the characteristic function of the perturbed operator~$B$ and discuss how it is related to its spectrum. In Section~\\ref{sec:mult}, the algebraic multiplicities of eigenvalues are discussed, and in Section~\\ref{sec:asympt}, the asymptotic distribution of eigenvalues is established. In Section~\\ref{sec:finite-dim}, we specialize the obtained results to the finite-dimensional case, and in the final section we discuss possible generalizations of the main results to wider classes of the operators~$A$.\n\n\n\n\\section{General spectral properties of $B$}\\label{sec:general}\n\n\n\nThroughout the paper, we make the following standing assumption on the operator~$A$:\n\\begin{itemize}\n \\item[(A1)] the operator~$A$ is self-adjoint and has simple discrete spectrum.\n\\end{itemize}\nThe operator~$A$ is necessarily unbounded above or\/and below; clearly, by considering $-A$ in place of~$A$, we reduce the case when $A$ is bounded above to the case when it is bounded below. Therefore, under assumption~(A1), the spectrum of~$A$ consists of real simple eigenvalues that can be listed in increasing order as $\\lambda_n$, $n\\in I$, with the index set~$I$ equal to~$\\mathbb{N}$ in the case where $A$ is bounded below and to~$\\mathbb{Z}$ otherwise. \n\nThe operator~$B$ is a rank one perturbation of the operator~$A$, i.e.,\n\\begin{equation}\\label{eq:B}\n B=A + \\langle \\cdot, \\varphi \\rangle \\psi\n\\end{equation}\nwith fixed nonzero vectors~$\\varphi$ and $\\psi$ in~$H$. Clearly, the operator~$B$\nis well defined and closed on its natural domain $\\dom (B)$ equal to $\\dom (A)$.\nNext, for $\\lambda$ in the resolvent set $\\rho(A)$ of~$A$, we introduce the \\emph{characteristic function}\n\\begin{equation}\\label{eq:F}\n F(\\lambda) := \\langle (A-\\lambda)^{-1}\\psi, \\varphi \\rangle + 1\n\\end{equation}\nand denote by $\\mathcal{N}_F$ the set of zeros of $F$.\nMany spectral properties of the operator~$B$ of~\\eqref{eq:B} will be derived from the explicit formula for its resolvent known as the Krein formula~(see, e.g., \\cite[Sec. 1.1.1]{AlbKur00}); we include its proof for the sake of completeness and to derive some explicit relations to be used later on.\n\n\\begin{lemma}[The Krein formula]\\label{lm:krein}\nThe set $\\rho(A)\\setminus \\cN_F$ consists of resolvent points of the operator~$B$ and, for every~$\\lambda\\in\\rho(A)\\setminus \\cN_F$,\n\\begin{equation}\\label{eq:Krein}\n (B - \\lambda)^{-1}\n = (A - \\lambda)^{-1} - \\frac{\\langle \\, \\cdot \\,, (A - \\overline{\\lambda})^{-1} \\varphi \\rangle}{F(\\lambda)} \\:\n (A-\\lambda)^{-1} \\psi.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nTo prove that a fixed $\\lambda\\in\\rho(A)\\setminus \\cN_F$ is a resolvent point of~$B$, we need to show that for every~$g\\in H$ the equation\n\\begin{equation}\\label{eq:krein-1}\n g = (B - \\lambda) f\n\\end{equation}\ncan be uniquely solved for $f \\in H$. Assuming such an~$f$ exists, writing the equality \\eqref{eq:krein-1} as\n\\begin{equation}\\label{eq:krein-2}\ng = (A - \\lambda) f + \\langle f, \\varphi \\rangle \\psi,\n\\end{equation}\nand applying the resolvent of the operator $A$ to both sides, we obtain\n\\begin{equation}\\label{eq:krein-3}\n (A - \\lambda)^{-1} g\n = f + \\langle f, \\varphi \\rangle (A - \\lambda)^{-1} \\psi.\n\\end{equation}\nTaking the inner product with $\\varphi$ results in the equality\n\\begin{equation}\\label{eq:krein-4}\n \\langle (A - \\lambda)^{-1} g, \\varphi \\rangle\n = \\langle f, \\varphi \\rangle\n + \\langle f, \\varphi \\rangle \\langle (A - \\lambda) ^ {- 1} \\psi, \\varphi \\rangle\n = \\langle f, \\varphi \\rangle F(\\lambda),\n\\end{equation}\nwhich on account of $F(\\lambda)\\ne0$ leads to\n\\begin{equation}\\label{eq:krein-5}\n \\langle f, \\varphi \\rangle =\n \\frac {\\langle (A - \\lambda) ^ {-1} g, \\varphi \\rangle} {F (\\lambda)}.\n\\end{equation}\nSubstituting now \\eqref{eq:krein-5} in \\eqref{eq:krein-3}, we derive the following formula for~$f$:\n\\begin{equation}\\label{eq:krein-6}\n f = (A - \\lambda)^{-1} g\n - \\frac{\\langle (A - \\lambda)^{-1} g, \\varphi \\rangle}{F(\\lambda)} (A-\\lambda)^{-1}\\psi.\n\\end{equation}\nA direct verification shows that $f$ of~\\eqref{eq:krein-6} belongs to~$\\dom(B)=\\dom(A)$ and is indeed a solution of equation~\\eqref{eq:krein-1}.\n\nTherefore the operator $B - \\lambda$ is surjective. It is also injective since if an $f\\in \\dom(B)$ satisfies~\\eqref{eq:krein-1} with $g = 0$, then~\\eqref{eq:krein-3} on account of~\\eqref{eq:krein-5} gives~$f=0$. Thus the operator $B - \\lambda$ is invertible and its inverse is equal to\n$$\n(B - \\lambda)^{-1} = (A - \\lambda)^{-1} - \\frac{\\langle \\, \\cdot \\,, (A - \\overline {\\lambda})^{-1} \\varphi \\rangle} {F(\\lambda)} \\, (A - \\lambda)^{-1} \\psi\n$$\nas claimed. The proof is complete.\n\\end{proof}\n\nThe Krein formula shows that, for every $\\lambda \\in \\rho(A) \\setminus \\cN_F$, the resolvent~$(B-\\lambda)^{-1}$ is a rank one perturbation of the compact operator~$(A-\\lambda)^{-1}$. Therefore, we get the following\n\n\\begin{corollary}\\label{cr:krein}\nThe resolvent of the operator $B$ is compact, i.e., $B$ is an operator with discrete spectrum.\n\\end{corollary}\n\n\nNext we denote by~$v_n$ a normalized eigenvector of~$A$ corresponding to its eigenvalue~$\\lambda_n$; then the set~$\\{v_n\\}_{n\\in I}$ is an orthonormal basis of~$H$. We also denote by $a_n$ and $b_n$ the Fourier coefficients of the vectors~$\\varphi$ and $\\psi$ with respect to this basis, so that%\n\\begin{footnote}\n{In the case $I=\\mathbb{Z}$, the summation will always be understood in the principal value sense}\n\\end{footnote}\n\\[\n \\varphi = \\sum_{n\\in I} a_n v_n, \\qquad \\psi = \\sum_{n\\in I} b_n v_n.\n\\]\n\n\n\n\n\\begin{lemma}\\label{lem:eig-B}\nThe following relations hold between the spectra of the operators $A$ and $B$:\n\\begin{itemize}\n\\item[a)]\nfor $\\lambda\\in\\rho(A)$, $\\lambda$ belongs to the spectrum of~$B$ if and only if~$\\lambda \\in \\cN_F$;\n\n\\item[b)]\nthe eigenvalue $\\lambda = \\lambda_n$ of the operator $A$ belongs to spectrum of the operator $B$ if and only if $a_nb_n = 0$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\na) Let a point $\\lambda \\in \\rho(A) $ belong to the spectrum of the operator~$B$. By Corollary \\ref{cr:krein}, $\\lambda$ is an eigenvalue of the operator~$B$, and we denote by~$y$ a corresponding eigenvector. Then \\eqref{eq:krein-1} holds with~$g=0$ and with $y$ in place of~$f$, so that equations~\\eqref{eq:krein-3} and~\\eqref{eq:krein-4} can be recast as\n\\[\n y = - \\langle y, \\varphi \\rangle (A - \\lambda)^{-1} \\psi\n\\]\nand\n\\[\n \\langle y ,\\varphi \\rangle F (\\lambda) = 0,\n\\]\nrespectively. Since $y$ is a nonzero vector, we see from the former equality that $\\langle y, \\varphi \\rangle\\ne0$, and then the latter one yields $F(\\lambda)=0$.\n\nConversely, if $F(\\lambda)=0$ for some $\\lambda \\in \\rho(A) $, then $y: = (A - \\lambda)^{-1} \\psi $ is an eigenvector of the operator $B$ for the eigenvalue $\\lambda$, as is seen from the equalities\n\\[\n (A - \\lambda) y + \\langle y, \\varphi \\rangle \\psi\n = [1 + \\langle (A - \\lambda)^{-1} \\psi, \\varphi \\rangle]\\psi\n = F(\\lambda)\\psi= 0.\n\\]\nThis completes the proof of part a).\n\nb) Let $\\lambda = \\lambda_n $ belong to the spectrum of the operator $B$; then there is a vector $y \\in \\dom(B)$ such that $B y = \\lambda_n y$, i.e.,\n\\begin{equation}\\label{eq:1-2}\n (B-\\lambda_n) y\n = (A - \\lambda_n) y + \\langle y, \\varphi \\rangle \\psi = 0.\n\\end{equation}\nTaking the inner product with $v_n$ results in\n\\begin{align*}\n\\langle (A - \\lambda_n) y, v_n \\rangle + \\langle y, \\varphi \\rangle \\langle \\psi, v_n \\rangle\n & = \\langle y, (A - \\lambda_n) v_n \\rangle + \\langle y, \\varphi \\rangle \\langle \\psi, v_n \\rangle \\\\\n & = \\langle y, \\varphi \\rangle \\langle \\psi, v_n \\rangle = 0.\n\\end{align*}\nThus $\\langle y, \\varphi \\rangle = 0$ or $\\langle \\psi, v_n \\rangle = 0$. If $\\langle y, \\varphi \\rangle = 0$, then $y = c v_n$ for some constant $c$ on account of~\\eqref{eq:1-2}, so that $a_n = 0 $. If $\\langle \\psi, v_n \\rangle = 0 $, then $b_n = 0$. Therefore the point $\\lambda = \\lambda_n $ belongs to the spectrum of $B$ only if $a_nb_n = 0$.\n\nConversely, let $a_nb_n = 0$; we need to prove that the point $\\lambda = \\lambda_n$ belongs to the spectrum of $B$. If $a_n = 0$, then\n\\[\n (B-\\lambda_n) v_n = (A-\\lambda_n)v_n + a_n \\psi = 0\n\\]\nso that $y = v_n$ is an eigenvector of~$B$ for the eigenvalue $\\lambda_n$. If $b_n = 0$, then for all $y \\in \\dom(B)$\n$$\n \\langle (B - \\lambda_n) y, v_n \\rangle\n = \\langle (A - \\lambda_n) y, v_n \\rangle\n = \\langle y, (A - \\lambda_n)v_n \\rangle =0,\n$$\nso that $B - \\lambda_n$ is not surjective on $\\dom(B)$ and the point $\\lambda = \\lambda_n$ belongs to the spectrum of the operator $B$. The proof is complete.\n\\end{proof}\n\nWe introduce the sets of indices\n$$\nI_0 \\overset{\\text{def}}{=} \\{n \\in I \\, | \\, a_nb_n = 0 \\}, \\quad\nI_1 \\overset{\\text{def}}{=} \\{n \\in I \\, | \\, a_nb_n \\neq 0 \\}\n$$\nof cardinalities (possibly infinite) $N_0$ and $N_1$ respectively, and split the eigenvalues of~$A$ into the respective subsets\n$$\n \\sigma_0 (A) \\overset {\\text{def}}{=} \\{\\lambda_n \\, | \\, n \\in I_0 \\}\n \\text {\\qquad and \\qquad}\n \\sigma_1 (A) \\overset {\\text{def}}{=} \\{\\lambda_n \\, | \\, n \\in I_1 \\}.\n$$\nAccording to Lemma~\\ref{lem:eig-B}, the spectrum of the operator $B$ consists of two parts: $\\sigma_0(A) = \\sigma(A) \\cap \\sigma(B)$, the common eigenvalues of $A$ and $B$, and the set $\\cN_F$ of zeros of the function~$F$ in $\\rho(A)$. Certainly, the latter part of $\\sigma(B)$ is more interesting.\n\n\n\\section{Eigenvalue multiplicity}\\label{sec:mult}\n\n\nIn this section we discuss multiplicity of eigenvalues of the operator~$B$. \n\nFirst we recall that the \\textit{geometric multiplicity} of an eigenvalue $\\lambda$ of an operator~$T$ is the dimension of the corresponding eigenspace, i.e., the number $\\dim \\ker (T - \\lambda)$ \\cite[Ch.~5.1]{Kat95}, and its \\textit{algebraic multiplicity} is the dimension of the corresponding root subspace, i.e., the rank of the corresponding spectral projector \\cite[Ch.~5.4]{Kat95}. Note that for a selfadjoint operator geometric and algebraic multiplicities of every eigenvalue are equal.\n\nBefore proceeding, we recall that the function~$F$ was initially defined only on the resolvent set of the operator~$A$. However, using the spectral theorem for the operator~$A$, we can write the function $F$ as\n\\begin{equation}\\label{eq:F-new}\n F(\\lambda)\n = \\sum_{n \\in I_1} \\frac {\\overline{a_n} b_n}{\\lambda_n - \\lambda} + 1,\n\\end{equation}\nand this formula gives an analytic continuation of~$F$ onto the set $\\sigma_0(A)$. We shall denote this continuation by the same letter~$F$ but will write $\\cN_F^0$ for the set of zeros of $F$ continued onto $\\mathbb{C} \\setminus \\sigma_1(A)$.\n\n\\begin{lemma}\\label{lem:geom-mult}\nAn eigenvalue $\\lambda$ of~$B$ has geometric multiplicity larger than~$1$ if and only if there exists an integer $n$ such that $\\lambda = \\lambda_n$, $a_n = b_n = 0$, and $F(\\lambda_n) = 0$. In that case, the geometric multiplicity of~$\\lambda$ is equal to~$2$.\n\\end{lemma}\n\n\\begin{proof}\nAssume that $\\lambda\\in\\sigma(B)$ has geometric multiplicity larger than~$1$, and denote by $y$ any of the corresponding eigenvectors. Then\n\\[\n (B-\\lambda) y = (A-\\lambda) y + \\langle y, \\varphi \\rangle \\psi =0,\n\\]\nand if $\\lambda$ is a resolvent point of $A$, then $y$ must be collinear to the vector $(A-\\lambda)^{-1}\\psi$ and thus the geometric multiplicity of~$\\lambda$ is one. Therefore $\\lambda\\in\\sigma(A)$, so that $\\lambda=\\lambda_n$ for some $n\\in I_0$.\nNow, as in the proof of part b) of Lemma~\\ref{lem:eig-B}, we find that\n\\[\n 0=\\langle (B-\\lambda_n)y, v_n \\rangle\n = \\langle y, \\varphi \\rangle \\langle \\psi, v_n \\rangle\n = \\langle y, \\varphi \\rangle b_n,\n\\]\nso that $\\langle y, \\varphi \\rangle =0$ or $b_n=0$.\n\nAssume that $b_n\\ne0$; then $\\langle y, \\varphi \\rangle =0$ and\n$(B-\\lambda_n)y =(A-\\lambda_n) y =0$. Thus $y$ in that case must be collinear to $v_n$, and the geometric multiplicity of~$\\lambda_n$ is then~$1$.\nTherefore, $b_n=0$ and the vector $\\psi$ belongs to the subspace $H_n:=H \\ominus \\langle v_n \\rangle$. Since the nullspace of $B-\\lambda_n$ is of dimension at least~$2$, there is an eigenvector $w$ in $H_n$. We denote by $A_n$ the restriction $A|_{H_n}$ of~$A$ onto its invariant subspace~$H_n$ and see that\n\\[\n (A_n - \\lambda_n) w + \\langle w, \\varphi \\rangle \\psi =0.\n\\]\nNote that $\\lambda_n$ is a resolvent point of the operator~$A_n$, so that\nthe above equality implies that $w= c(A_n-\\lambda_n)^{-1}\\psi$ and that\n\\[\n \\langle (A_n-\\lambda_n)^{-1}\\psi, \\phi\\rangle +1 = 0,\n\\]\ni.e., that $F(\\lambda_n)=0$. Therefore, there is at most one (up to a factor) eigenvector of~$B$ in the space~$H_n$, and thus its second eigenvector must be of the form $v_n + w_n$ with some $w_n \\in H_n$. However, then\n\\[\n (B-\\lambda_n)(v_n + w_n) = (A-\\lambda_n)w_n\n + \\langle v_n + w_n, \\varphi \\rangle \\psi = 0\n\\]\nso that $w_n$ is collinear to the eigenvector $(A_n-\\lambda_n)^{-1}\\psi$ found earlier, and thus $v_n$ must also be an eigenvector of~$B$. As $(B-\\lambda_n)v_n = \\langle v_n , \\varphi \\rangle \\psi$, this requires that $a_n=0$.\n\nSumming up, we see that the assumption that $\\dim \\ker (B-\\lambda) > 1$ implies that $\\lambda=\\lambda_n$ for some~$n\\in I$ and $b_n=0$; moreover, there is an eigenvector $w$ in the subspace~$H_n$ if and only if $F(\\lambda_n)=0$, and then $w$ is collinear to~$(A_n-\\lambda_n)^{-1}\\psi$. The second eigenvector must be $v_n$, which is possible if and only if $a_n=0$. Therefore all the conditions are necessary, and the geometric multiplicity is then equal to~$2$.\n\n\nTo prove that these conditions are also sufficient, we assume that $\\lambda=\\lambda_n$ is such that $a_n=b_n=0$ and $F(\\lambda_n)=0$. Then, as shown above, $v_n$ and $w:= (A_n-\\lambda_n)^{-1}\\psi \\in H_n$ are linearly independent eigenvectors of~$B$ for the eigenvalue~$\\lambda_n$. The proof is complete.\n\\end{proof}\n\n\\begin{example}\\rm \nLet $\\lambda$ and $\\mu$ be distinct eigenvalues of an operator $A$ with corresponding normalized eigenvectors~$v$ and $w$; then for the operator\n\\(\n\tB := A + (\\lambda - \\mu)\\langle \\cdot, w\\rangle w\n\\) \nthe number~$\\lambda$ is an eigenvalue of geometric multiplicity two, $v$ and $w$ being the corresponding eigenvectors. As the above lemma shows, geometric multiplicity cannot be made larger by a rank-one perturbation of~$A$.\t\n\\end{example}\n\n\\begin{remark}\\label{rem:multiplicity}\nAssume that $a_n=b_n=0$, so that $\\lambda_n$ is an eigenvalue of~$B$ with eigenvector~$v_n$. Then $v_n$ is also an eigenvector of the adjoint operator~$B^*$, so that the subspaces $\\langle v_n \\rangle$ and $H_n$ are reducing for $B$. Moreover, the restrictions of $A$ and $B$ onto $\\langle v_n\\rangle$ coincide.\n\nMore generally, we denote by $H^{0}$ the closed linear space of all eigenvectors~$v_k$ of~$A$ for which $a_k=b_k=0$. Then the subspace $H^0$ is reducing for~$B$ and the restrictions of~$A$ and of $B$ onto $H^0$ coincide. Therefore, we can concentrate on the study of the restriction of the operator~$B$ onto its invariant subspace $H^1:=H \\ominus H^0$. Without loss of generality, we shall assume that $H^0 = \\{0\\}$, so that $H=H^1$. Under this assumption, all eigenvalues of the operator~$B$ are geometrically simple.\n\\end{remark}\n\nNext we discuss algebraic multiplicity of the eigenvalues of~$B$ in the resolvent set of~$A$. As every such an eigenvalue $\\lambda$ is geometrically simple by Lemma~\\ref{lem:geom-mult}, its algebraic multiplicity coincides with the largest length of chains of eigen- and associated vectors (also called Jordan chains). We recall that a sequence of vectors~$y_0, y_1, \\dots, y_m$ forms a chain of eigen- and associated vectors of~$B$ for an eigenvalue~$\\lambda$ if every $y_k$ is in the domain of~$B$, $(B-\\lambda)y_0=0$, and $(B-\\lambda)y_k = y_{k-1}$ for $k=1,\\dots,m$.\nChains of eigen- and associated vectors are not defined uniquely; however, for geometrically simple eigenvalues all such chains are closely related, as the next lemma demonstrates.\n\n\n\\begin{lemma}\\label{lem:jordan-chains}\nAssume that $\\lambda$ is a (geometrically simple) eigenvalue of the operator~$B$ and $y_0, y_1,\\dots, y_m$ is a chain of eigen- and associated vectors corresponding to~$\\lambda$.\n\\begin{itemize}\n \\item[(i)] For every sequence of complex numbers $c_1, \\dots, c_m$ introduce the vectors $\\tilde y_0 = y_0$ and\n \\begin{equation}\\label{eq:CEAV}\n \\tilde y_k = y_k + c_1 y_{k-1} + \\cdots + c_k y_0\n \\end{equation}\n for $k=1,\\dots,m$. Then $\\tilde y_0, \\tilde y_1,\\dots,\\tilde y_m$ is a chain of eigen- and associated vectors of~$B$ corresponding to $\\lambda$.\n \\item[(ii)] Vice versa, assume that $\\tilde y_0, \\tilde y_1,\\dots,\\tilde y_m$ is another chain of eigen- and associated vectors of~$B$ corresponding to the eigenvalue $\\lambda$ such that $\\tilde y_0 = y_0$. Then there are constants $c_1, \\dots, c_m$ such that for all $k=1,2,\\dots,m$ relations \\eqref{eq:CEAV} hold.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\nBy definition of~$\\tilde y_k$ and $y_k$, we find that\n\\[\n (B-\\lambda)\\tilde y_k =y_{k-1}+ c_1 y_{k-1} + \\cdots + c_{k-1} y_0\n = \\tilde y_{k-1}\n\\]\nfor $k\\ge1$ thus establishing Part~(i).\n\n\\\nFor part (ii), the proof is by induction. Since $(B-\\lambda)(\\tilde y_1 - y_1) = \\tilde y_0 - y_0 =0$, it follows that there is $c_1 \\in \\mathbb{C}$ such that $\\tilde y_1- y_1 = c_1 y_0$ thus establishing the base of induction. Assume that the claim has already been proved for $k =1, \\dots, l-10$). In particular, $l=0$ is equivalent to $m=1$ (recall that the case $a_n = b_n = F(\\lambda_n)=0$ was excluded), and the equality $m=l+1$ is then satisfied.\n\n Assume therefore that $l>0$ and introduce the vectors\n \\[\n y_k:= - \\frac1{b_n}(A_n - \\lambda_n)^{-k}P_n \\psi, \\qquad k\\ge1.\n \\]\n Then one sees that\n \\[\n (B-\\lambda_n) y_k = (A-\\lambda_n) y_k + \\langle y_k ,\\varphi \\rangle \\psi\n\t= y_{k-1} + \\langle y_k ,\\varphi \\rangle \\psi\n \\]\n and\n \\[\n \\langle y_k ,\\varphi \\rangle = - \\frac1{b_n} \\langle (A_n - \\lambda_n)^{-k}P_n \\psi ,\\varphi \\rangle\n\t= -\\frac1{b_n(k-1)!}F^{(k-1)}(\\lambda_n).\n \\]\n It follows that the vectors $y_1, y_2,\\dots,y_l$ form a chain of vectors associated to the eigenvector~$y_0$, so that the algebraic multiplicity $m$ of the eigenvalue~$\\lambda_n$ is at least $l+1$.\n\n Conversely, as in the proof of Lemma~\\ref{lem:alg-mult} one can show that in any chain $\\tilde y_0, \\tilde y_1, \\dots, \\tilde y_{m-1}$ of eigen- and associated vectors for $B$ the vectors $\\tilde y_1, \\dots, \\tilde y_{m-1}$ are related to the above-constructed vectors $y_1, \\dots, y_{m-1}$ via~\\eqref{eq:tilde-y} and that $F(\\lambda_n) = F'(\\lambda_n) = \\dots = F^{(m-2)}(\\lambda_n)=0$. This shows that $l \\ge m-1$ and completes the proof in the case (a).\n\n \\textbf{Case (b)}: $b_n = 0$. Then $\\psi$ belongs to $H_n = H \\ominus v_n$ and thus the range\n $\\ran(B - \\lambda_n)$ of $B-\\lambda_n$ is contained in $H_n$.\n We look for an eigenvector $y_0$ of~$B$ of the form $\\alpha_0 v_n + z_0$ with $z_0 \\in H_n$. Then $(A-\\lambda_n)y_0 = (A_n - \\lambda_n)z_0$, and $(B-\\lambda_n)y_0=0$ can be written as\n \\[\n (A_n - \\lambda_n) z_0 + \\langle y_0 , \\varphi \\rangle \\psi =0,\n \\]\n so that $z_0 = c(A_n - \\lambda_n)^{-1}\\psi$ with an appropriate constant~$c$. Substituting this $z_0$ into the above equation results in the relation\n \\[\n c \\psi + \\bigl[\\alpha_0 \\overline{a_n} + c \\langle (A_n - \\lambda_n)^{-1}\\psi , \\varphi \\rangle \\bigr] \\psi =0,\n \\]\n yielding the equality\n \\begin{equation}\\label{eq:alpha0}\n c F(\\lambda_n) + \\alpha_0 \\overline{a_n} =0.\n \\end{equation}\n\n In order that for the eigenvector~$y_0$ there could exist an associated vector~$y_1$, it is necessary that $y_0 = (B-\\lambda_n)y_1$ belong to $H_n$ and thus that $\\alpha_0 = 0$ and $y_0=z_0$. Equation~\\eqref{eq:alpha0} then yields $cF(\\lambda_n)=0$, and since $c=0$ would lead to the contradiction that $y_0=z_0=0$, we conclude that necessarily $F(\\lambda_n)=0$. In particular, $l=0$ gives $m=1$ as stated.\n\n Assume therefore that $l>0$, so that $F(\\lambda_n)=0$. As the case $a_n=b_n =0$ was excluded earlier, we have $a_n\\ne0$ and thus $\\alpha_0=0$ by~\\eqref{eq:alpha0} and $y_0= z_0 := (A_n - \\lambda_n)^{-1}\\psi$.\n\n We first show that $m\\ge l+1$ by constructing a chain $y_1, \\dots, y_l$ of vectors associated to this~$y_0$. Namely, take\n $y_k := (A_n - \\lambda_n)^{-(1+k)}\\psi$\n for $k=1,\\dots, l-1$ and\n $y_l := \\alpha_l v_n + (A_n - \\lambda_n)^{-(1+l)}\\psi$\n with an~$\\alpha_l$ to be determined later. As in the proof of Case~(a) we find that\n \\[\n (B-\\lambda_n) y_k\n = (A_n - \\lambda_n) y_k + \\langle y_k,\\varphi\\rangle \\psi\n = y_{k-1} + \\frac{1}{k!} F^{(k)}(\\lambda_n) \\psi\n = y_{k-1}\n \\]\n for $k =1, 2, \\dots, l-1$. For $k=l$ we get\n \\[\n (B-\\lambda_n) y_l\n = (A_n - \\lambda_n) y_l + \\langle y_l,\\varphi\\rangle \\psi\n = y_{l-1}\n + \\bigl[\\alpha_l \\overline{a_n}\n +\\frac{1}{l!} F^{(l)}(\\lambda_n)\\bigr] \\psi,\n \\]\n and the equality $(B-\\lambda_n) y_l = y_{l-1}$ is guaranteed by taking (recall that $a_n \\ne0$)\n \\[\n \\alpha_l := - \\frac{1}{\\overline{a_n}l!} F^{(l)}(\\lambda_n).\n \\]\n\n It remains to show that $l \\ge m-1$. We take a chain of eigen- and associated vectors $\\tilde y_0, \\dots, \\tilde y_{m-1}$ of the maximal possible length~$m>1$. The equalities $(B-\\lambda_n) \\tilde y_k = \\tilde y_{k-1}$ for $k=1, \\dots, m-1$ show that the vectors $\\tilde y_0, \\dots, \\tilde y_{m-2}$ belong to $H_n$. Without loss of generality we may assume that $\\tilde y_0 = y_0$ and then prove by induction that with $c_k:= - \\langle \\tilde y_k, \\varphi \\rangle$ for $k=0,1,\\dots, m-2$ we have\n \\\n \\tilde y_k = y_k + c_1 y_{k-1} + \\dots + c_k y_0\n \\\n with $y_k$ defined above and that $F(\\lambda_n) = F'(\\lambda_n)=\\dots = F^{(k)}(\\lambda_n)=0$.\n\n The base of induction was already set up: $\\tilde y_0 = y_0$ and $F(\\lambda_n) = 0$. Assume therefore that the claim holds for all indices $k$ less than $j$ with $0m$. More precisely, we take\n\\[\n\t\\phi(x) = \\sum_{k=-m}^m e^{ikx} = \\frac{\\sin(m+\\tfrac12)x}{\\sin(\\tfrac12x)}\n\\]\nand \n\\[\n\t\\psi(x) = \\sum_{k=1}^m d_k \\sin(kx)\n\\]\nwith coefficients $d_k$ to be determined. Since $\\langle \\psi, v_0 \\rangle = 0$, the corresponding chain of eigen- and associated vectors can be formed as in Case (b) of the above theorem. Namely, with $A_0$ standing for the restriction of~$A$ onto the space $H_0:= H \\ominus v_0$, we take\n\\[\n\ty_k := A_0^{-(k+1)}\\psi, \\qquad k=0,\\dots, 2m-1,\n\\]\nand \n\\[\n\ty_{2m}:= d_0v_0 + A_0^{-(2m+1)}\\psi \n\\]\nfor a suitable $d_0$. We next show that there is a unique set of $d_0,\\dots,d_m$ for which the above $y_0,\\dots, y_{2m}$ form a chain of eigen- and associated vectors of~$B$ and that there is no longer chains of eigen- and associated vectors corresponding to~$\\lambda_0$. \n\nNotice that \n\\[\n\tA_0^{-2l}\\psi(x) = \\sum_{k=1}^m \\frac{d_k}{k^{2l}}\\sin(kx)\n\\]\nand \n\\[\n\tA_0^{-2l+1} = -i \\sum_{k=1}^m \\frac{d_k}{k^{2l-1}}\\cos(kx).\n\\]\nIt then follows that $y_{2l+1}$ are odd functions for all $l=0,\\dots,m-1$, and as~$\\phi$ is an even function, we find that $By_{2l+1} = A y_{2l+1} = y_{2l}$. On the other hand, the equalities\n$By_{2l} = y_{2l-1}$ for $l=0,\\dots,m$ amount to a non-singular system of $m+1$ linear equations in $m+1$ variables $d_0,d_1,\\dots, d_m$,\n\\begin{equation}\\label{eq:example-system}\n\t\\sum_{k=1}^m \\frac{d_k}{k^{2l+1}} = f_l, \\quad l=0,1,\\dots,m,\n\\end{equation}\nwith $f_0 = -i\/(2\\pi)$, $f_1 = \\dots = f_{m-1} = 0$, and $f_m = -id_0\/\\sqrt{2\\pi}$. \n\nNote that $d_0 \\ne0$ as otherwise the system would be inconsistent, so that $y_{2m}$ does not belong to $H_0$ and thus the chain cannot be extended further. In view of Lemma~\\ref{lem:jordan-chains}, this is true of any other chain of EAV's for the eigenvalue~$\\lambda_0$. As $a_0\\ne0$, geometric multiplicity of $\\lambda_0=0$ is equal to one by Lemma~\\ref{lem:geom-mult}; therefore, $\\lambda_0$ is a geometrically simple eigenvalue of~$B$ of algebraic multiplicity~$2m+1$. \n\nThe explicit form of $\\phi$ and $\\psi$ yields their Fourier coefficients: $a_n = b_n = 0$ if $|n|>m$, $a_n = \\sqrt{2\\pi}$ for $|n| \\le m$, and, finally, $b_n = \\sqrt{2\\pi}d_n\/2i$ for $n=1,\\dots,m$, $b_n = -b_{-n}$ for $n=-m,\\dots,-1$, and $b_0=0$. Then the characteristic function,\n\\[\n\tF(z) = \\sum_{n=-m}^m\\frac{\\overline{a_n}b_n}{n-z} + 1 \n\t\t = \\sqrt{2\\pi}\\sum_{n=1}^m\\frac{2nb_n}{n^2-z^2} + 1 \n\t\t = \\frac{2\\pi}i\\sum_{n=1}^m\\frac{nd_n}{n^2-z^2} + 1 \n\\] \nis a rational function of the form $P(z)\/Q(z)$ with $P$ and $Q$ polynomials of degree at most $2m$. Therefore, $F$ has at most $2m$ zeros counting with multiplicity. On the other hand, it is straightforward to verify that equations~\\eqref{eq:example-system} amount to the relations \n\\[\n\tF(0) = F'(0) = \\dots = F^{(2m-1)}(0) = 0,\n\\]\nso that $z = 0$ is a zero of~$F$ of multiplicity~$2m$. This implies that $F$ has no other zeros. In particular, $F(n)\\ne0$ if $n\\ne0$, and thus $\\lambda_n = n$ is an algebraically simple eigenvalue of the operator~$B$ whenever $|n|>m$. \n\nTo sum up, the operator~$B$ has an eigenvalue~$\\lambda_0 = 0$ of algebraic multiplicity~$2m+1$ and simple eigenvalues $\\lambda_n$ for $|n|>m$. Loosely speaking, the rank one perturbation shifts the eigenvalues $\\lambda_{-m}, \\dots, \\lambda_{-1}$, $\\lambda_1, \\dots, \\lambda_m$ towards $\\lambda_0$ respectively enlarging the multiplicity of the latter. \n\\end{example}\n\n\n\n\n\\section{Spectral localization of the operator~$B$}\\label{sec:asympt}\n\nWe next turn to the question, what spectra the rank-one perturbations~$B$ of a given self-adjoint operator~$A$ can have. Keeping in mind the most important and interesting applications to the differential operators, in addition to~$(A1)$ we assume that\n\\begin{itemize}\n\t\\item[(A2)] the eigenvalues of~$A$ are separated, i.e.,\n\t\\begin{equation}\\label{eq:dist}\n\t\\inf_{n \\in I} |\\lambda_{n+1} - \\lambda_n| =: d > 0.\n\t\\end{equation}\n\\end{itemize}\n\nWe next localize the spectrum of~$B$ by studying its characteristic function\n\\begin{equation*}\\label{eq:F1}\n\tF(z) = \\sum_{k \\in I_1} \\frac {\\overline{a_k} b_k}{\\lambda_k - z} + 1.\n\\end{equation*}\nAs the Fourier coefficients $a_k$ and $b_k$ of the functions~$\\phi$ and $\\psi$ are in $\\ell_2(I)$, the sequence $\\overline{a_k}b_k$ is summable and, due to the Cauchy--Bunyakowsky--Schwarz inequality, its $\\ell_1$-norm is bounded by $\\|\\varphi\\|\\|\\psi\\|$. \n\n\\begin{lemma}\n\tThe spectrum of $B$ lies in the strip \n\t\\[\n\t\t\\Pi := \\{z \\in \\bC \\mid |\\myIm z| \\le \\|\\varphi\\|\\|\\psi\\|\\}.\n\t\\]\n\\end{lemma}\n\n\\begin{proof}\n\tIf $z\\not\\in \\Pi$, then $|\\lambda_k - z| \\ge |\\myIm z| > \\|\\varphi\\|\\|\\psi\\|$, so that \n\t\\[\n\t\t\\sum_{k \\in I_1} \\biggl|\\frac {\\overline{a_k} b_k}{\\lambda_k - z}\\biggr| \n\t\t\t< \\sum_{k \\in I_1} |\\overline{a_k} b_k|\/(\\|\\varphi\\|\\|\\psi\\|) \n\t\t\t< 1\n\t\\]\n\tso that $F(z) \\ne 0$. \n\\end{proof}\n\n\nNext, for an $\\varepsilon>0$ we denote by $C_n(\\varepsilon)$ the open circle\n\\[\n\tC_n(\\varepsilon) := \\{ z \\in \\bC \\mid |z - \\lambda_n| < \\varepsilon\\}\n\\]\nand set \n\\[\n\tR_{N, \\varepsilon}:= \\Bigl\\{z \\in \\bC \\mid |\\myRe z| \\ge N\\} \n\t\t\\setminus \\Bigl(\\bigcup\\nolimits_{n\\in I} C_n(\\varepsilon) \\Bigr)\\Bigr\\}\n\\]\n\n\\begin{lemma}\\label{lem:RN}\nFor every $\\varepsilon>0$ there is $N>0$ such that $R_{N,\\varepsilon}$ belongs to the resolvent set of the operator~$B$.\n\\end{lemma}\t\n\n\\begin{proof}\nFor an $\\varepsilon >0$, we choose $N'\\in\\mathbb{N}$ so that%\n\\begin{footnote}\n\t{Throughout this section, the symbol $\\sum{\\hspace*{-2pt}\\vphantom{\\sum}}^{(1)}$ denotes summation over the index set $I_1$}\n\\end{footnote} \n\t\\[\n\t\t\\sumI_{|k|\\ge N'} |\\overline{a_k} b_k| \\le \\frac{\\varepsilon}4; \n\t\\]\n\tthen, for $z$ outside every circle $C_n(\\varepsilon)$,\n\t\\[\n\t\t\\Bigl|\\sumI_{|k|\\ge N'} \\frac{\\overline{a_k} b_k}{\\lambda_k - z}\\Bigr|\n\t\t\t\\le \\frac1\\varepsilon \\sum_{k\\in I_1, |k|\\ge N'} |a_k b_k| \\le \\frac14.\n\t\\]\n\tWe now take $N''\\in\\mathbb{N}$ such that $N''\\ge N' + 4 \\|\\varphi\\|\\|\\psi\\|\/d$ and choose $N\\in \\mathbb{N}$ such that $N\\ge |\\lambda_{N''}|$ and $N \\ge |\\lambda_{-N''}|$ if $-N'' \\in I$. Due to Assumption~$(A2)$ it holds that $|\\lambda_k - \\lambda_m| \\ge d|k-m|$; therefore, \n\t$|\\lambda_k - z| \\ge d (N'' - N') \\ge 4\\|\\varphi\\|\\|\\psi\\|$ whenever $z \\in R_{N,\\varepsilon}$ and $|k| \\le N'$, so that \n\t\\[\n\t \t\\Bigl|\\sumI_{|k| < N'} \\frac{\\overline{a_k} b_k}{\\lambda_k - z}\\Bigr| \n\t \t\t\\le \\frac14\n\t\\]\n\tfor such $z$. \n\tAs a result, for all $z \\in R_{N,\\varepsilon}$ it holds\n\t\\[\n\t\t\t|F(z)| \\ge 1 - \t\\Bigl|\\sum_{k\\in I_1} \\frac{\\overline{a_k} b_k}{\\lambda_k - z}\\Bigr| \n\t\t\t\t\\ge \\frac12;\n\t\\]\n\tby Lemma~\\ref{lem:eig-B} the set $R_{N,\\varepsilon}$ is in the resolvent set of~$B$, and the proof is complete.\n\\end{proof}\n\nCombining the above two lemmata, we conclude that the spectrum of $B$ is localized in the circles $C_n(\\varepsilon)$ and in the rectangular domain\n\\[\n\t\\{z \\in \\bC \\mid |\\myRe|\\le N, \\ |\\myIm z| \\le \\|\\varphi\\|\\|\\psi\\|\\},\n\\]\nwith $N=N(\\varepsilon)$ from Lemma~\\ref{lem:RN}. \n\n\\begin{lemma}\\label{lem:EVinCn}\n\tFor every $\\varepsilon>0$ there is $K=K(\\varepsilon)$ such that for each $n\\in I$ with $|n| > K(\\varepsilon)$ the circle $C_n(\\varepsilon)$ contains precisely one eigenvalue of~$B$.\n\\end{lemma}\n\n\\begin{proof}\n\tBy Lemma~\\ref{lem:RN}, for all $n$ with large enough $|n|$, the boundary $\\partial C_n(\\varepsilon)$ of~$C_n(\\varepsilon)$ is in the resolvent set of~$B$. We next show that the Riesz spectral projections for $A$ and $B$ corresponding to $C_n(\\varepsilon)$ are of the same rank (and thus of rank~$1$) for large enough~$|n|$.\n\t\n\t For every $n$ with $\\partial C_n(\\varepsilon) \\subset \\rho(B)$, we denote by $P_n$ and $P'_n$ the Riesz spectral projectors for $A$ and $B$ respectively on the root subspaces corresponding to the eigenvalues inside $C_n(\\varepsilon)$,\n\t\\[\n\t\tP_n = \\frac1{2\\pi i} \\int_{C_n(\\varepsilon)} (A - z)^{-1}\\,dz, \\qquad \n\t\tP'_n = \\frac1{2\\pi i} \\int_{C_n(\\varepsilon)} (B - z)^{-1}\\,dz.\n\t\\]\n\tBy the Krein resolvent formula~\\eqref{eq:Krein}, we get \n\t\\[\n\t\tP_n - P'_n = \\frac1{2\\pi i} \\int_{C_n(\\varepsilon)}\\frac{dz}{F(z)} \\langle \\, \\cdot \\,, (A - \\overline{z})^{-1} \\varphi \\rangle\n\t\t(A-z)^{-1} \\psi.\n\t\\]\n\tAs the norm of a rank-one operator $\\langle \\, \\cdot \\, u \\rangle v$ is equal to $\\|u\\|\\|v\\|$ and, as proved in Lemma~\\ref{lem:RN}, $|F(z)|\\ge 1\/2$ on $C_n(\\varepsilon)$ for large enough $|n|$, we conclude that \n\t\\[\n\t\t\\|P_n - P'_n\\| \\le d \\max_{z\\in C_n(\\varepsilon)} \\|(A - \\overline{z})^{-1} \\varphi\\| \\|(A - {z})^{-1} \\psi\\|\n\t\\]\n\tfor such $n$. Observe now that for every vector $u = \\sum c_k v_k$ we have \n\t\\[\n\t\t\\|(A - z)^{-1} u \\|^2 = \\sum_{k\\in I} \\frac{|c_k|^2}{|\\lambda_k - z|^2};\n\t\\]\n\tapplying the Lebesgue dominated convergence theorem, we conclude that\n\t\\[\n\t\t\\max_{z\\in C_n(\\varepsilon)} \\|(A - z)^{-1} u \\|^2 \\to 0\n\t\\]\n\tas $|n| \\to \\infty$. Therefore, $\\|P_n - P'_n\\| \\to 0$ as $|n|\\to\\infty$; as a result~\\cite[\\S IV.2]{Kat95}, the ranks of the Riesz projectors $P_n$ and $P'_n$ coincide for all $n$ with large enough $|n|$, and the proof is complete.\n\\end{proof}\n\nTherefore, the operator~$B$ has at most finitely many nonsimple eigenvalues; we next prove that there are no other restrictions on them.\n\n\\begin{lemma}\\label{lem:nonrealEV}\n\tFix an arbitrary $n\\in\\mathbb{N}$, an arbitrary sequence $z_1, z_2, \\dots, z_n$ of pairwise distinct complex numbers, and an arbitrary sequence $m_1$, $m_2$, $\\dots$, $m_n$ of natural numbers. Then there is a rank-one perturbation~$B$ of the operator~$A$ such that, for every $j=1,2,\\dots, n$, the number $z_j$ is an eigenvalue of~$B$ of algebraic multiplicity~$m_j$. \n\\end{lemma}\n\n\\begin{proof}\n\tFor simplicity, we assume that none of $z_j$ is in the spectrum of~$A$; the changes to be made otherwise are not very significant, cf.~Lemma~\\ref{lem:alg-mult0} and Example~\\ref{ex:multiple-EV}.\n\t\n\tSet $N:= m_1 + m_2 + \\dots + m_n$; we will construct a rank-one perturbation~$B$ of $A$ with \n\t\\[\n\t\t\\varphi = \\sum_{k=1}^N a_k v_k, \\qquad \t\t\n\t\t\\psi = \\sum_{k=1}^N b_k v_k.\n\t\\]\n\tAccording to Lemma~\\ref{lem:eig-B}, it suffices to choose $a_k$ and $b_k$ in such a way that the characteristic function~$F$ of~\\eqref{eq:F-new} has zeros $z_1, z_2, \\dots, z_n$ of multiplicity $m_1, m_2, \\dots, m_n$ respectively. Set $c_k := \\overline{a_k}b_k$, $k=1,2,\\dots,n$; then\n\t\\[\n\t\tF(z) = \\sum_{k=1}^n \\frac{c_k}{\\lambda_k - z} + 1,\n\t\\]\n\tand the equalities $F(z_k) = F'(z_k) = \\dots = F^{(m_k-1)}(z_k) = 0$ lead to an inhomogeneous system of $N$ equations in the variables $c_1, c_2, \\dots, c_N$:\n\t\\begin{equation}\\label{eq:system}\n\t\t\\sum_{k=1}^N \\frac{c_k}{(\\lambda_k - z_j)^m} + \\delta_{m1}= 0, \\quad j = 1,2, \\dots, n, \\quad m = 1, 2, \\dots, m_j,\n\t\\end{equation}\n\twith $\\delta_{m1}$ being the Kronecker delta. By Lemma~\\ref{lem:Cauchy} below, the coefficient matrix of the above system is non-singular; therefore, the system possesses a unique solution~$c_1,c_2, \\dots, c_N$. It remains to take $a_k =1$ and $b_k = c_k$ for $k=1,2,\\dots, N$, and the proof is complete.\n\\end{proof}\n\n\n\n\\begin{lemma}\\label{lem:Cauchy}\n\tThe coefficient matrix of system~\\eqref{eq:system} is non-singular.\n\\end{lemma}\n\n\\begin{proof}\n\tFor pairwise distinct numbers $\\omega_1, \\omega_2, \\dots, \\omega_N$ from the resolvent set of~$A$, we introduce the Cauchy matrix $M$ with entries \n\t\\[\n\t(M)_{jk} = \\frac{1}{\\lambda_k - \\omega_j}.\n\t\\]\n\tIt is non-singular and has determinant equal to \n\t\\begin{equation}\\label{eq:Cauchy}\n\tD(\\omega_1,\\omega_2,\\dots,\\omega_{N}) = \\frac{\\prod\\prod_{j>k}(\\lambda_j - \\lambda_k)(\\omega_j-\\omega_k)} \t\t{\\prod_j\\prod_k(\\lambda_j-\\omega_k)}.\n\t\\end{equation}\n\tWe set $C:= \\prod\\prod_{j>k}(\\lambda_j - \\lambda_k)$ for brevity. \n\t\n\tTaking the derivative of that determinant in $\\omega_2$ and setting $\\omega_2 = \\omega_1 = z_1$, we get the determinant of the matrix~$M_2$, whose first and second rows have entries \n\t\\[\n\t\t\t\\frac1{\\lambda_k - z_1 } \\quad \\text{and} \\quad \\frac1{(\\lambda_k - z_1)^2}, \\qquad k = 1, 2, \\dots, N,\n\t\\]\n\trespectively, and the other rows are as in the matrix~$M$. By~\\eqref{eq:Cauchy}, we have \n\t\\[\n\t\tD(\\omega_1,\\omega_2,\\dots,\\omega_{N}) = (\\omega_2 - \\omega_1) D_2(\\omega_1,\\omega_2,\\dots,\\omega_{N}),\n\t\\] \n\tso that \n\t\\[\n\t\t\\frac{\\partial}{\\partial \\omega_2}D(z_1,\\omega_2,\\dots,\\omega_{N})\\Bigr|_{\\omega_2 = z_1}\n\t\t\t= D_2(z_1, z_1, \\omega_3, \\dots, \\omega_N).\n\t\\]\n\tExplicit calculations give \n\t\\begin{multline*}\n\t\t\\det M_2 = D_2(z_1, z_1, \\omega_3, \\dots, \\omega_N) \\\\\n\t\t\t= C \\prod_{j>2}(\\omega_j-z_1)^2 \\frac{\\prod\\prod_{j>k>2}(\\omega_j-\\omega_k)}\n\t\t\t{\\prod_j (\\lambda_j-z_1)^2\\prod_{k>2}(\\lambda_j-\\omega_k)}\n\t\t\t\\ne 0.\n\t\\end{multline*}\n\t\n\tNext, we take the second derivative of $D_2(z_1, z_1, \\omega_3, \\dots, \\omega_N)$ in $\\omega_3$ and set $\\omega_3 = z_1$; this becomes the determinant $D_3(z_1, z_1, z_1, \\omega_4, \\dots, \\omega_N)$ of the matrix $M_3$ that is $M_2$ with its third row replaced by\n\t\\[\n\t\t\\frac2{(\\lambda_k - z_1)^3}, \\qquad k = 1, 2, \\dots, N.\n\t\\]\n\tOn the other hand, \n\t\\begin{multline*}\n\t\t\\det M_3 = D_3(z_1, z_1, z_1, \\omega_4, \\dots, \\omega_N) \n\t\t\t= \\frac{\\partial^2}{\\partial \\omega^2_3}D(z_1, z_1, \\omega_3, \\dots,\\omega_{N})\\Bigr|_{\\omega_3 = z_1} \\\\\n\t\t\t= 2C \\prod_{j>3}(\\omega_j-z_1)^3 \\frac{\\prod\\prod_{j>k>3}(\\omega_j-\\omega_k)}\n\t\t\t\t{\\prod_j (\\lambda_j-z_1)^3\\prod_{k>3}(\\lambda_j-\\omega_k)}\n\t\t\t\\ne 0.\n\t\\end{multline*}\n\tOn each next step, we repeat a similar procedure with the next row and variable until we reach row number $m_1$. \n\t\n\tAfter that, we set $\\omega_{m_1+1} = z_2$, take the derivative in $\\omega_{m_1+2}$ at $\\omega_{m_1 + 2} = z_2$, and repeat with the subsequent rows until we reach row number $m_1 + m_2$. Clearly, the operations described above can be performed on separate groups of variables $\\omega_l$ with $l=m_1 + \\dots + m_j + 1, m_1 + \\dots + m_j + 2, \\dots, m_1 + m_2 + \\dots + m_{j+1}$ independently. At the end, the determinant of the coefficient matrix of the system~\\eqref{eq:system} is found explicitly to be\n\t\\[\n\t\t\\frac{\\prod_{j=k+1}^N\\prod_{k=1}^N(\\lambda_j - \\lambda_k)\t\\prod_{j=k+1}^n\\prod_{k=1}^n(z_j-z_k)^{m_j + m_k}}\n\t\t{\\prod_{j=1}^N\\prod_{k=1}^n(\\lambda_j-z_k)^{m_j}} \\ne 0,\n\t\\]\n\tand the proof is complete.\n\\end{proof}\n\n\\begin{remark}\n\tIn the paper~\\cite{DobHry20}, it is proved that the operators $A$ and $B$ have the same number of eigenvalues in special increasing rectangles exhausting the whole complex plane~$\\bC$. Combined with the results of Lemmata~\\ref{lem:RN} and \\ref{lem:EVinCn}, this allows an enumeration of the eigenvalues of~$B$ as $\\mu_n$, $n\\in I$, such that each value $\\mu_n$ is repeated according to its multiplicity and $\\mu_n- \\lambda_n \\to0$ as $|n|\\to\\infty$. \n\\end{remark}\n\n\nWe summarize the above results in the following theorem.\n\n\\begin{theorem}\\label{thm:main}\n\tAssume that $A$ is an operator in a Hilbert space~$H$ satisfying assumptions~$(A1)$ and $(A2)$ and $B$ is its rank-one perturbation~\\eqref{eq:B}. Then \n\t\\begin{itemize}\n\t\t\\item[(i)] all eigenvalues of $B$ of sufficiently large absolute value are localized within $\\varepsilon$-neighbourhood of the eigenvalues of~$A$ and thus are simple;\n\t\t\\item[(ii)] the eigenvalues of~$B$ can be enumerated as $\\mu_n$, $n\\in I$, so that $\\mu_n-\\lambda_n \\to 0$ as $|n|\\to\\infty$;\n\t\t\\item[(iii)] geometric multiplicity of every eigenvalue of~$B$ is at most~$2$, and multiplicity~$2$ is only possible when the corresponding eigenspace of~$A$ is reducing for $B$.\n\t\\end{itemize}\n\tMoreover, for every prescribed finite set $z_1, z_2, \\dots, z_n$ of pairwise distinct complex numbers, and an arbitrary sequence $m_1$, $m_2$, $\\dots$, $m_n$ of natural numbers there exists a~$B$ such that each $z_j$, $j=1,2,\\dots, n$, is an eigenvalue of~$B$ of algebraic multiplicity~$m_j$.\n\\end{theorem}\n\n\\section{Finite-dimensional case}\\label{sec:finite-dim}\n\nThe analysis of Section~\\ref{sec:mult} allows to essentially complement the results in the fi\\-ni\\-te-di\\-men\\-si\\-o\\-nal case. Namely, assume that $A$ is a Hermitian matrix in $\\bC^n$ with pairwise distinct eigenvalues $\\lambda_1, \\lambda_2, \\dots, \\lambda_n$ and normalized (column) eigenvectors~$\\mathbf{v}_1, \\mathbf{v}_2, \\dots, \\mathbf{v}_n$ and define the \\emph{generic set} $\\mathcal{G}(A)$ of $A$ as\n\\[\n\t\\mathcal{G}(A) = \\{ \\mathbf{x} \\in \\mathbb{C}^n \\mid \\langle \\mathbf{x}, \\mathbf{v}_k\\rangle_{\\bC^n} \\ne 0, \\quad k = 1, 2, \\dots,n \\}. \n\\]\nThen we have the following generalization of the result of~\\cite{Kru92}.\n\n\\begin{theorem}\\label{thm:finite-dim}\n\tUnder the above assumptions, let $\\bm{\\varphi}$ be a vector from the generic set~$\\mathcal{G}(A)$. Then for any natural number $k$, any pairwise distinct complex numbers~$z_1, z_2, \\dots, z_k$, and any natural numbers $m_1, m_2, \\dots, m_k$ satisfying $m_1+ m_2 + \\dots + m_k =n$, there is a unique vector $\\bm{\\psi} \\in \\bC^n$ such that the rank-one perturbation $B = A + \\bm{\\psi}\\bm{\\varphi}^\\top$ of the matrix~$A$ has eigenvalues $z_1, z_2, \\dots, z_k$ of corresponding multiplicities $m_1, m_2, \\dots, m_k$. \n\t\n\tSimilarly, for every fixed $\\bm{\\psi} \\in \\mathcal{G}(A)$ there is a unique $\\bm{\\varphi}\\in \\bC^n$ such that $B$ has the eigenvalues $z_j$ of prescribed multiplicities $m_j$, $j = 1,2, \\dots, k$. \n\\end{theorem}\n\n\\begin{proof}\nDenote by $\\sigma_0(A)$ the common part of the spectrum $\\sigma(A)$ of $A$ and the set $\\{z_1, z_2, \\dots, z_k\\}$, by $\\sigma_1(A):=\\sigma(A) \\setminus\\sigma_0(A)$ the remaining part of $\\sigma(A)$, and let $I_\\ell := \\{ j \\mid \\lambda_j \\in \\sigma_p(A)\\}$, $\\ell=0,1$, be the corresponding index sets. We update the multiplicities $m_j$ to \n\\begin{equation}\\label{eq:reduce-mult}\n\tm'_j := \\begin{cases}\n\t\tm_j - 1, & \\qquad z_j \\in \\sigma(A); \\\\\n\t\tm_j, & \\qquad z_j \\not\\in \\sigma(A);\t\n\t\\end{cases}\n\\end{equation}\nand set\n\\begin{equation}\\label{eq:F-prod}\n\tF(z):= \\frac{\\prod_{j=1}^k (z - z_j)^{m'_j}}{\\prod_{j \\in I_1} (z - \\lambda_j)}.\n\\end{equation}\nDenoting by $-c_j$ the residue of the function~$F$ at the point $z=\\lambda_j$, $j\\in I_1$, we conclude that $F$ can be written in the form\n\\begin{equation}\\label{eq:F-sum}\n\tF(z) = \\sum_{j \\in I_1} \\frac{c_j}{\\lambda_j-z} + 1. \n\\end{equation}\nDenote by $a_j = \\langle \\bm{\\varphi}, \\mathbf{v}_k\\rangle_{\\bC^n}$, $j= 1,2, \\dots, n$, the coefficients of the vector $\\bm{\\varphi}$ in the basis $\\mathbf{v}_1, \\mathbf{v}_2, \\dots, \\mathbf{v}_n$. By assumption, no $a_j$ vanishes, and we set $b_j:= c_j \/ \\overline{a_j}$ for $j \\in I_1$ and $b_j = 0$ for $j \\in I_0$, and define the vector $\\psi$ via\n\\[\n\t\\bm{\\psi} = \\sum_{j=1}^n b_j \\mathbf{v}_j = \\sum_{j \\in I_1} b_j \\mathbf{v}_j. \n\\] \nIt follows from the results of Section~\\ref{sec:mult} that the characteristic function of the matrix $B = A + \\bm{\\psi}\\bm{\\varphi}^\\top$ coincides with the above function~$F$; therefore, the matrix $B$ has eigenvalues $z_1, z_2, \\dots, z_k$ and the multiplicity of the eigenvalue $z_j$ is $m_j'$ if $z_j \\not \\in \\sigma(A)$ or $m_j'+1$ otherwise. \n\nThe second part is proved in a similar manner, by interchanging the roles of $a_n$ and $b_n$.\n\\end{proof}\n\nIf the vector $\\bm{\\varphi}$ is not in the generic set~$\\mathcal{G}(A)$ of $A$, the above theorem has the following analogue. \n\n\\begin{theorem}\\label{thm:phi-arbitrary}\n\tUnder the above assumptions on the matrix~$A$, take a nonzero vector $\\bm{\\varphi} = \\sum _{j=1}^n a_j \\mathbf{v}_j \\in \\bC^n$ and set $I_0 := \\{j \\mid a_j = 0\\}$ and $\\sigma_0(A):=\\{\\lambda_j \\mid j \\in I_0\\}$. Then for every natural $k$, every set $S=\\{z_1, z_2,\\dots, z_k\\}$ of $k$ pairwise distinct complex numbers obeying $S \\cap \\sigma(A) = \\sigma_0(A)$, and every sequence $m_1, m_2, \\dots, m_k$ of natural numbers with $m_1 + m_2 + \\dots + m_k = n$ there is a vector $\\bm{\\psi}\\in\\bC^n$ such that the matrix $B = A + \\bm{\\psi}\\bm{\\varphi}^\\top$ has eigenvalues $z_1, z_2, \\dots, z_k$ of multiplicities $m_1, m_2 , \\dots, m_k$ respectively. \n\t\n\tA similar statement holds with the r\\^oles of $\\bm{\\varphi}$ and $\\bm{\\psi}$ interchanged.\n\\end{theorem}\n\n\\begin{proof}\n\tThe fact that the set $S$ is in the spectrum of~$B$ is proved in Lemma~\\ref{lem:eig-B}. We denote by~$\\sigma_1(A)$ the spectrum of~$A$ not in $\\sigma_0(A)$ and set $I_1$ to be the corresponding set of indices. Reducing by $1$ the multiplicity of each $z_j$ from $S$ and denoting the resulting multiplicities by $m'_j$ as in~\\eqref{eq:reduce-mult}, we construct the function~$F$ of~\\eqref{eq:F-prod} and observe that it assumes the form~\\eqref{eq:F-sum}, with uniquely determined residues~$-c_j$, $j\\in I_1$. Then we define $b_j$ for such $j$ from the relation~$\\overline{a}_jb_j = c_j$, and fix arbitrarily $b_j$ for $j \\in I_0$. \n\t\n\tBy Lemmata~\\ref{lem:alg-mult} and \\ref{lem:alg-mult0}, the numbers $z_j$ not in $\\sigma_0(A)$ are eigenvalues of the matrix~$B$ of multiplicity~$m_j'$, while those in~$\\sigma_0(A)$ have multiplicity $m_j'+1$. The proof is complete.\n\\end{proof}\t\n\n\\begin{remark}\n\tWe can conclude from the above proof that the coordinates of the vector $\\bm{\\psi}$ in the basis $\\mathbf{v}_1, \\mathbf{v}_2, \\dots, \\mathbf{v}_n$ for $j\\in I_0$ are not fixed; therefore, there is an $|I_0|$-dimensional affine set of such vectors producing the required spectrum. \n\\end{remark}\n\n\\section{Concluding remarks}\n\nIt should be noted that some restrictions imposed on~$A$ can be relaxed. For instance, self-adjointness of~$A$ is not essential; the proof with minor amendments will work for rank-one perturbations of every normal operator with simple discrete spectrum, or even in the case when the eigenvectors of~$A$ can be chosen to form a Riesz basis of~$H$. Simplicity of the eigenvalues of~$A$ can also be dropped; however, this will result in a more complicated Jordan structure of the root subspaces of~$B$, cf.~\\cite{BehMoeTru14}. Also, the operator $A$ may possess, in addition to an infinite discrete spectrum, a non-trivial essential component; the results we proved have natural generalization to this case as well. \n\nFinally, this study has found its continuation in~\\cite{DobHry20}, in which a complete characterization of all possible spectra of rank-one perturbations~\\eqref{eq:B} of self-adjoint operators~$A$ with simple discrete spectrum is given. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Acknowledgements}\nThis work was supported in part by the Intel Neuromorphic Research Community Grant to Lule\\aa~ University of Technology, the Swedish Foundation for international Cooperation in Research and Higher Education (STINT) under Mobility Grant for Internationalisation MG2020-8842, and the Russian Science Foundation during the period of 2020-2021 under grant 20-71-10116.\nSK and DH are recipients of the Centre for Data Analytics and Cognition (CDAC) Ph.D. research scholarships. \nDK has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk\u0142odowska-Curie grant agreement No 839179.\n}\n\\thanks{E. Osipov is with the Department of Computer Science, Electrical and Space Engineering at Lule\u00e5 University of Technology, 97187 Lule\u00e5, Sweden. \\mbox{E-mail}: \\mbox{Evgeny.Osipov@ltu.se}\n}\n\\thanks{S. Kahawala, D. Haputhanthri, T.~Kempitiya, D.~De~Silva and D.~Alahakoon are with the Centre for Data Analytics and Cognition (CDAC) at La Trobe University, Melbourne, Australia. \\mbox{E-mail}: \\{S.Kahawala, D.Haputhanthri, T.Kempitiya,\n D.DeSilva, D.Alahakoon\\} @latrobe.edu.au\n }\n\n\\thanks{D. Kleyko is with the Redwood Center for Theoretical Neuroscience at the University of California, Berkeley, CA 94720, USA and also with the Intelligent Systems Lab at Research Institutes of Sweden, 16440 Kista, Sweden. \\mbox{E-mail}: \\mbox{denis.kleyko@ri.se}\n}\n\n}\n\\newcommand{\\marginpar{FIX}}{\\marginpar{FIX}}\n\\newcommand{\\marginpar{NEW}}{\\marginpar{NEW}}\n\n\n\\maketitle\n\n\\begin{abstract} \n\nMotivated by recent innovations in biologically-inspired neuromorphic hardware, this article presents a novel unsupervised machine learning algorithm named Hyperseed that draws on the principles of Vector Symbolic Architectures (VSA) for fast learning of a topology preserving feature map of unlabelled data. It relies on two major operations of VSA, binding and bundling. The algorithmic part of Hyperseed is expressed within Fourier Holographic Reduced Representations model, which is specifically suited for implementation on spiking neuromorphic hardware. The two primary contributions of the Hyperseed algorithm are, few-shot learning and a learning rule based on single vector operation. These properties are empirically evaluated on synthetic datasets as well as on illustrative benchmark use-cases, IRIS classification, and a language identification task using $n$-gram statistics. The results of these experiments confirm the capabilities of Hyperseed and its applications in neuromorphic hardware. \n\\end{abstract}\n \n \n\\begin{IEEEkeywords}\nself-organizing maps, vector symbolic architectures, hyperseed, neuromorphic hardware \n\\end{IEEEkeywords}\n\n\n\\input{tex\/introduction} \n\\input{tex\/Related} \n\\input{tex\/Method} \n\\input{tex\/Hyperseed} \n\\input{tex\/Experiments} \n\\input{tex\/Discussion} \n\\input{tex\/Conclusion} \n\n\n\n\\bibliographystyle{IEEEtran}\n\n\n\n\n\\section{Method: Holographic Reduced Representations (HRR) model}\n\\label{sect:method}\n\nThe Hyperseed algorithm is designed using the Fourier Holographic Reduced Representations (FHRR) model \\cite{PlateNested1994}. \nFHRR facilitates the mathematical treatment of Hyperseed operations. The potential argument that complex numbers used in FHRR add to the memory requirements of Hyperseed is intuitively true in the case of CPU realization. However, in Section \\ref{sect:discussion}, we rationalise this is not an issue for the neuromorphic hardware. Also, due to the equivalence of FHRR and HRR models, the operations of Hyperseed can be implemented with hypervectors from $\\mathbb{R}^d$. \nIn fact, when evaluating the performance of the bottleneck functionality of Hyperseed on Intel's Loihi, we use HRR model\\footnote{\nThe supplementary code base also contains the HRR implementation of the algorithm.\n}. \nThe atomic FHRR hypervectors are randomly sampled from $\\mathbb{C}^d$. Dimensionality $d$ is a hyperparameter of Hyperseed. In high-dimensional random spaces, all random hypervectors are dissimilar to each other (quasi-orthogonal) with an extremely high probability. VSA defines operations and a similarity measure on hypervectors. In this article, we use the cosine similarity of real parts of hypervectors for characterizing the similarity.\nThe three primary operations for computing with hypervectors are superposition, binding, and permutation. \n\n\\subsection{Binding operation} \nThe binding operation is used to bind two hypervectors together. The result of the binding is another hypervector. For example, for two hypervectors $\\textbf{v}_1$ and $\\textbf{v}_2$ the result of binding of their hypervectors (denoted as $\\textbf{b}$) is calculated as follows: \n\\begin{equation}\n\\label{eq:bind} \n\\textbf{b} = \\textbf{v}_1 \\circ \\textbf{v}_2, \n\\end{equation}\nwhere the notation $\\circ$ is used to denote the binding operation. \nIn HRR, the binding operation is implemented as circular convolution of $\\textbf{v}_1$ and $\\textbf{v}_2$, which can be implemented as the component-wise multiplication in the Fourier domain. This observation inspired FHRR where the representations are already in the Fourier domain in a form of phasors so that the component-wise multiplication, which is equivalent to the addition of phase angles modulo $2\\pi$, plays the role of the binding operation. \nBinding is, essentially, a randomizing operation that moves hypervectors to another (random) part of the high-dimensional space. \nThe role played by the binding operation depends on the algorithmic context. \nIn data structures with roll-filler pairs, the binding operation corresponds to the assignment of a value (filler) to a variable (role). There are two important properties of the binding operation. First, the resultant hypervector $\\textbf{b}$ is dissimilar to the hypervectors being bound, i.e., the similarity between $\\textbf{b}$ and $\\textbf{v}_1$ or $\\textbf{v}_2$ is approximately $0$.\n\nSecond, the binding operation preserves similarity. That is the distribution of the similarity measure between hypervectors from some set $\\mathcal{S}$ is preserved after binding of all hypervectors in $\\mathcal{S}$ with the same random hypervector $\\textbf{v}$. \n\nThe binding operation is reversible. The unbinding, denoted as $\\oslash$, is implemented by the circular correlation in HRR. In the case of FHHR this is equivalent to component-wise multiplication with the complex conjugate. Being the inverse of the binding operation, the unbinding obviously has the same similarity preservation property when performed on all hypervectors in $\\mathcal{S}$ with the same hypervector $\\textbf{v}$: \n\n \\begin{equation}\n\\label{eq:unbind} \n\\textbf{v}_2 \\oslash \\textbf{b} = \\textbf{v}_1. \n\\end{equation}\nThe interpretation of the unbinding operation is a retrieval of a value from the hypervector encoding the assignment. When unbinding is performed from the superposition of bindings (see Section~\\ref{sec:vsa:superposition}), the retrieved hypervector contains noise. In VSA, the noisy vector can be cleaned-up by performing a search for the closest atomic hypervector stored in an associative memory.\n\n\\subsection{Permutation operation}\nThe permutation (rotation) operation $\\textbf{b} = \\rho(\\textbf{v})$ is a unitary operation that is commonly used to represent an order of the symbol in a sequence. As with the binding operation, the resultant hypervector $\\textbf{b}$ is dissimilar to $\\textbf{v}$. In this article, this operation is used for encoding a certain type of input data as further described in Section \\ref{sect:perf}.\n\n\\subsection{Superposition operation}\n\\label{sec:vsa:superposition}\nSuperposition is denoted with $+$ and implemented via component-wise addition. \nThe superposition operation combines several hypervectors into a single hypervector. \nFor example, for hypervectors $\\textbf{v}_1$ and $\\textbf{v}_2$ the result of superposition (denoted as $\\textbf{a}$) is simply: \n\\begin{equation}\n\\label{eq:bindle} \n\\textbf{a} = \\textbf{v}_1 + \\textbf{v}_2.\n\\end{equation}\nIn contrast to the binding operation, the resultant hypervector $\\textbf{a}$ is similar to all superimposed hypervectors, i.e., the cosine similarity between $\\textbf{b}$ and $\\textbf{v}_1$ or $\\textbf{v}_2$ is larger than $0$.\nIf several copies of any hypervector are included (e.g., $\\textbf{a} = 3\\textbf{v}_1 + \\textbf{v}_2$), the resultant hypervector is more similar to the dominating hypervector than to other components. \n\nIf superposition is applied to several bindings it is possible to unbind any hypervector from any binding. In this case, the result of the unbinding operation is a noisy version of the second operand of the particular binding. For example, if $\\textbf{a}=\\textbf{v}_1\\circ \\textbf{v}_2 + \\textbf{u}_1 \\circ \\textbf{u}_2$, then $\\textbf{u}_2 \\oslash \\textbf{a}=\\textbf{u}_1+\\mathrm{noise}= \\textbf{u}_1^*$. Given that noiseless atomic hypervectors ($\\textbf{v}_1,\\textbf{v}_2, \\textbf{u}_1, \\textbf{u}_2$) are kept in the associative memory and so vector $\\textbf{u}_1^*$ is expected to have the highest similarity to $\\textbf{u}_1$. The same property holds for the binding of any atomic hypervector with the superposition of unbindings (which we use below in the description of our approach). That is if $\\textbf{a}=\\textbf{v}_1\\oslash \\textbf{v}_2 + \\textbf{u}_1 \\oslash \\textbf{u}_2$, then $\\textbf{u}_2 \\circ \\textbf{a}=\\textbf{u}_1+\\mathrm{noise}= \\textbf{u}_1^*$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Discussion}\n\\label{sect:discussion}\n\nHaving presented the Hyperseed algorithm and its empirical evaluation, it is now pertinent to discuss the following aspects that require further investigation, Hyperseed on neuromorphic hardware, performance comparison of embeddings and limitations of Hyperseed. \n\n\\subsection{Hyperseed on neuromorphic hardware}\n\nThe Hyperseed algorithm from the start was designed targeting an implementation on the neuromorphic hardware. This target departs from recent developments within the Intel neuromorphic research community, where VSA is promoted as an algebraic framework for the development of algorithms on Intel's Loihi \\cite{Loihi18, TPAM, Frady20_KNN}. \n\nSince the main focus of this article is on the algorithmic aspects of Hyperseed, we resort to making rather high level links to a neuromorphic realization of algorithm's operations and present an evaluation of its computational bottleneck using the existing neuromorphic implementation.\n\nBoth HRR and FHRR representations could be mapped onto activities of spiking neurons. \nIn the case of FHRR, the phase of components is used for phase-to-spike-timing mapping.\nIn this way, FHRR representation does not result in higher memory footprint as in the case of the CPU implementation. \nVSA operations are realized in either Resonate-and-Fire neurons~\\cite{TPAM} or in Leaky Integrate and Fire neurons~\\cite{RennerBinding2022, RennerVisualScene}. \n\n\nIn the case of HRR, real-valued components are used in spike-time latency code, where\nearlier spikes represent larger magnitudes. \nVSA operations can realized by Leaky Integrate and Fire neurons. \nIn this article, we use the realization of the dot product calculation presented in \\cite{ Frady20_KNN} (also based on Leaky Integrate and Fire neurons) in order to demonstrate the feasibility of neuromorphic implementation of computational bottleneck of Hyperseed -- the search for the BMV in HD-map for an unbound noisy hypervector $\\textbf{p}^*$ resulting from~(\\ref{eq:seedbind}).\n\nTo measure the performance of the Hyperseed's search procedure in the language identification task HD-maps of various sizes (starting from $30\\times30$ and incrementing the grid size by $10$ along each axis until $90\\times90$) were generated as in Section~\\ref{sect:hdMap}. The hypervectors of each HD-map were then used for mapping their values onto spiking activity of the $k$NN reference base on Intel's Loihi-based Nahuku-32 neuromorphic system. This operation was performed only once as part of the initialization since HD-map remains unchanged for the life-time of Hyperseed. Therefore, the time to construct the $k$NN reference base was not taken into account in the run-time performance evaluation.\n\nFor the experiment we chose the case of identifying five randomly selected languages with a single seed hypervector $\\mathbf{s}$ as the reference scenario. The original dimensionality of the hypervectors used for the encoding of the input data as well as for all other hypervectors of the Hyperseed algorithm was $d=10000$. \nThe reference scenario accuracy of Hyperseed obtained on a CPU was $0.84$. \n\n$k$NN on Loihi was used to model the search operation during the labeling and testing process. To do this, seed hypervector $\\mathbf{s}$ was pre-trained offline on the CPU as described in Section~\\ref{sect:update}. For the labeling process, the binding of all training and test data with the trained $\\mathbf{s}$ was performed and the results (the noisy versions of the BMVs) were used as queries to the $k$NN reference base storing HD-map on Loihi.\nThe dimensionality of the existing Loihi implementation of $k$NN is $d_{kNN}<512$, which is a platform specific limit. \nNote, that in VSA the dimensionality of hypervectors is connected to the information capacity of the superposition. In Hyperseed, every update of seed hypervector $\\mathbf{s}$ increases the cross-talk noise to previous bindings $\\mathbf{s}\\circ \\mathbf{p}_{\\texttt{target}}$. Fig.~\\ref{fig:single} demonstrated the accuracy degradation for dimensionality $d=5000$ with the increase of the number of updates of $\\mathbf{s}$. For smaller dimensionalities, the number of updates for maintaining the acceptable accuracy is even lower. Therefore, in this experiment the training of Hyperseed was done on higher dimensionalities and then the dimensionality was reduced using principal component analysis to $d=400$ to meet the current limitations of the existing implementation. \n\nTo label BMVs, the training data (after binding with $\\mathbf{s}$) were used as queries to the $k$NN reference base and the top-1 index for each query (the earliest fired output neuron) was recorded. After that, the labeling was performed on the CPU using the procedure described above. To compute the accuracy, the test data hypervectors (also after binding with $\\mathbf{s}$) were used as queries to the $k$NN reference base. The top-1 index was recorded and used to compute the accuracy against the list of the labeled indices. \n\nThe average measured accuracy on Loihi was $0.84$, which matched the reference accuracy on the CPU implementation of Hyperseed. This means that the neuromorphic implementation of HD-map and the calculation of similarity did not introduce sensible errors.\n\nNext, in addition to the accuracy we also measured the query time to the $k$NN reference base for different sizes of HD-map. The main outcome of this experiment is illustrated by Fig. \\ref{fig:qTimeVsNumberOfRefVecs}. It shows the computational benefit of implementing the bottleneck operation of Hyperseed on in the neuromorphic hardware: due to parallel, power-efficient computation of the dot product in Loihi, Hyperseed, as expected, was empowered with constant-time search for different sizes of HD-map. \n\n\n\n\n \n\n \n\n\n\\begin{figure}[t!]\n \\centering\n \\includegraphics[width=8cm]{img\/Loihi\/qTimeVsNumberOfRefVecs.png}\n \\caption{Time of querying a noisy BMV in HD-map implemented as the $k$NN reference base against the size of HD-map in the number of reference hypervectors.}\n \\label{fig:qTimeVsNumberOfRefVecs}\n\\end{figure}\n\n\\subsection{Performance comparison of embeddings and distributed representations}\n While a large body of knowledge on methods for encoding data into hypervectors is accumulated throughout the years, the problem is still considered as of the primary importance in the area of VSA. Hyperseed in this respect offers a playground for comparing different embeddings as the embeddings of high quality lead to higher accuracy in classification tasks. It is particularly important to develop the operations of Hyperseed on sparse representations. \n \n \\subsection{Limitations of Hyperseed and future developments}\nHyperseed when using 2D HD-map as evaluated in this article is limited in its capability to describe complex manifold structures. In this sense the algorithm does not show advantages over Self-Organizing Maps, which have similar limitations. This case was chosen to demonstrate the feasibility of implementing non-trivial learning functionality with straightforward VSA operations, which in itself is an original research contribution. However, Hyperseed is in fact scalable in terms of two other aspects: 1) topologies of higher dimensionalities than two as it does not require updates of the hypervectors in HD-map and 2) topologies of other structures than regular grid due to the generality of FPE encoding. The investigation of this capability is part of an ongoing work on Hyperseed extension. \n\n \n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\\section{Conclusions}\n\\label{sect:conclusions}\n\nThe increasing accumulation of unstructured and unlabelled big data initiated a renewed interest in unsupervised machine learning algorithms that are able to capitalize on computational efficiencies of biologically-inspired neuromorphic hardware. \nIn this article, we presented the Hyperseed algorithm that addresses these challenges through the manifestation of a novel unsupervised learning approach that leverages Vector Symbolic Architectures for fast learning from only few input vectors and single vector operation learning rule implementation. A further novelty is that it implements the entire learning pipeline purely in terms of operations of Vector Symbolic Architectures. Hyperseed has been empirically evaluated across diverse scenarios: synthetic datasets from Fundamental Clustering Problems Suite, benchmark classification using the Iris dataset, and the more practical classification of $21$ European languages using their $n$-gram statistics. \nAs future work, we will work on the adaptation of the Hyperseed algorithm for neuromorphic hardware. \n\n\n\n\n\\section{Hyperseed: Unsupervised Learning with Vector Symbolic Architectures}\n\\label{sect:vsaseed}\n\nThis section presents the main contribution of this article -- the method for unsupervised learning -- Hyperseed. Denote the set of FHRR-represented input data as $\\mathcal{D}\\in\\mathbb{C}^d$. \nData hypervectors for training are generated during the encoding phase (see Section \\ref{sect:encoding} for the details of the encoding). They are kept in a working memory. The input query for testing is also a data hypervector obtained using the same encoding procedure as for the training data. \n\nDenote as $\\mathcal{P}\\in\\mathbb{C}^d$ the\nvector space with known similarity properties. Set $\\mathcal{P}$ is created by encoding points (also referred to as nodes further in the text) of a 2D grid using FPE method~\\cite{PlateNested1994,frady2021computing, FradyFunctionsNICE2022, Komer2019ANR, komer2020biologically}. The cardinality of HD-map $|\\mathcal{P}|=n\\times m$, where $n$ and $m$ are the sizes of the grid along the vertical and the horizontal axes, respectively. For the sake of brevity, further in the text we will refer to set $\\mathcal{P}$ as HD-map. HD-map is computed, as described below, once and is stored in the associative memory. This memory is fixed throughout the life-time of the system. \n\nThe Hyperseed algorithm relies on the similarity preservation property of the (un)binding operation. The goal with Hyperseed is to translate the original data hypervectors $\\mathcal{D}$ (with unknown) internal similarity layout to HD-map $\\mathcal{P}$ by unbinding all of its members from hypervector $\\mathbf{s}$, i.e.:\n\n\\begin{equation}\n \\mathcal{D}\\oslash \\mathbf{s}\\Rightarrow\\mathcal{P}.\n\\label{eq:transf}\n\\end{equation}\n\\noindent\nHypervector $\\mathbf{s}$ is obtained as the result of applying an unsupervised learning rule. In essence, during the learning, some selected hypervectors from $\\mathcal{D}$ will be bound to selected vectors from $\\mathcal{P}$ as described further\\footnote{Due to an analogy of ``seeding'' data hypervectors onto HD-map, hypervector $\\mathbf{s}$ is referred as seed vector throughout the article.}. \n\n\\subsection{Initialization Phase: Generation of HD-map $\\mathcal{P}$ and hypervector $\\mathbf{s}$}\n\\label{sect:hdMap}\nThe hypervectors, which are the members of $\\mathcal{P}$, are generated such that the similarity between them relates to topological proximity of grid nodes. Note, however, that the reference to the topological arrangement of $\\mathcal{P}$ is virtual in a sense that in the associative memory, in which hypervectors of $\\mathcal{P}$ are stored does not have any structure. Topology information is kept on a side to be used for visualization purposes only.\n\nThe generation of HD-map starts with two randomly generated unit hypervectors $\\mathbf{x}_0, \\mathbf{y}_0 \\in \\mathbb{C}^d$ as $\\mathbf{x}_0 \\sim e^{j\\cdot 2\\pi\\cdot U(0,1)}$ and $\\mathbf{y}_0 \\sim e^{j\\cdot 2\\pi\\cdot U(0,1)}$.\nLet us denote the bandwidth parameter regulating the similarity between the adjacent coordinates on the grid by $\\epsilon$. The \\textit{i}-th $x$ and $y$ coordinates of the grid will be created using the FPE method as:\n\\noindent\n\\begin{equation}\n \\mathbf{x}_i=\\mathbf{x}_0^{\\epsilon \\cdot i}, \n \\mathbf{y}_i=\\mathbf{y}_0^{\\epsilon \\cdot i}.\n \\label{eq:x_y}\n\\end{equation}\n\\noindent\nThe hypervector $\\mathbf{p}_{(i,j)}$ representing a node with coordinates $(i,j)$ on the grid is computed as $\\mathbf{p}_{(i,j)}=\\mathbf{x}_i \\circ \\mathbf{y}_j$. \nFig.~\\ref{fig:hdplane} illustrates the landscape of similarity between all hypervectors of HD-map stored in the associative memory and one selected hypervector of the same HD-map encoding coordinates (15,15) on $50\\times50$ 2D grid as a function of coordinates $i$ and $j$.\n\n\\begin{figure}[t!\n\\centerline{\\includegraphics[width=0.7\\columnwidth]{img\/HDplane}}\n\\caption{Similarity distribution on an HD-map. The target node is $(15,15)$, the size of the grid is $50\\times50$, bandwidth $\\epsilon=0.05$.\n}\n\\label{fig:hdplane}\n\\end{figure}\n\nHypervector $\\mathbf{s}$ is also initialized randomly: $\\mathbf{s} \\sim e^{j\\cdot 2\\pi\\cdot U(0,1)}$. It is updated over several iterations during the learning phase as described below. The number of update iterations is a hyperparameter of Hyperseed. \n\n\\subsection{Search procedure in Hyperseed: Finding Best Matching Vector on HD-map}\n\\label{sect:search}\nIn the Hyperseed algorithm, HD-map $\\mathcal{P}$ acts as the auto-associative memory~\\cite{FrolovWillshaw2002,FrolovTime2006, GritsenkoAMSurvey2017}. That is the only operation performed on HD-map is the search for the Best Matching Vector (BMV) given some input hypervector. \nThe BMV is found by computing the cosine similarity between the input hypervector and all hypervectors in $\\mathcal{P}$. The output of this procedure is a valid hypervector in $\\mathcal{P}$ with the highest similarity to the input hypervector. \n\nThe mapping ($\\mathbf{d}_i \\rightarrow \\mathbf{p}_i$) of data hypervectors in $\\mathcal{D}$ to hypervectors of HD-map $\\mathcal{P}$ is done by unbinding $\\mathbf{d}_i$ from the trained hypervector $\\mathbf{s}$: \n\n \\begin{equation}\n \\mathbf{p}_i^*=\\mathbf{d}_i \\oslash \\mathbf{s}.\n \\label{eq:seedbind}\n \\end{equation}\n In (\\ref{eq:seedbind}), $\\mathbf{p}_i^*$ is a noisy version of a hypervector in $\\mathcal{P}$.\n\n\\subsection{Update phase: Unsupervised learning of hypervector $\\mathbf{s}$}\n\\label{sect:update}\n The goal with the update procedure on each iteration is to map input hypervector $\\mathbf{d}_i$ as near as possible to some target hypervector in $\\mathcal{P}$ with respect to the cosine similarity.\n \n Therefore, a single learning iteration consists of three steps: \n \\begin{enumerate}\n \\item Choose a target hypervector $\\mathbf{p}_{\\texttt{target}}$ (see the next subsection); \n \\item Compute a hypervector for the perfect mapping \n $\\mathbf{d}_i \\rightarrow \\mathbf{p}_{\\texttt{target}}$ by binding of $\\mathbf{d}_i$ with $\\mathbf{p}_{\\texttt{target}}$. \n \\item Update hypervector $\\mathbf{s}$ by adding this perfect mapping hypervector to hypervector $\\mathbf{s}$:\n \n \n \\begin{equation}\n \\mathbf{s}=\\mathbf{s} + \\mathbf{d}_i \\circ \\mathbf{p}_{\\texttt{target}}.\n \\label{eq:seedupdate}\n \\end{equation}\n \\end{enumerate}\n \\noindent\n Note that after the update, $\\mathbf{s}$ is not a phasor vector anymore so it might be renormalized if necessary. \n Thus, by the end of the learning phase, hypervector $\\mathbf{s}$ is the superposition of bindings $\\mathbf{d}_i \\circ \\mathbf{p}_j$. As such, the result of unbinding of hypervectors similar to $\\mathbf{d}_i$ with hypervector $\\mathbf{s}$ (\\ref{eq:seedbind}) will resemble hypervectors in $\\mathcal{P}$ (i.e., the hypervectors used in the update phase).\n \n\\subsection{Weakest match search (WMS) phase: Finding a data hypervector for the update in a single iteration}\n\n\\label{sect:observe}\n\n\nTo find a data hypervector for the update of hypervector $\\mathbf{s}$ (\\ref{eq:seedupdate}), Hyperseed uses a heuristic based on the farthest-first traversal rule (FFTR). This principle is widely used for defining heuristics in many important computing applications ranging from approximation of Traveling Salesman Problem \\cite{TravelingSalesman} to $k$-center clustering \\cite{k-center} and fast similarity search \\cite{Rachkovskij2017Cybern}. FFTR has been also used as a weight update rule in SOMs \\cite{FFTSOM}, which resulted in better representation of outliers as well as lower topographic and quantization errors. In FFTR, the first point is selected arbitrarily and each successive point is as far as possible from the set of previously selected points. In the case of Hyperseed, FFTR is straightforwardly implemented by checking the cosine similarity between the noisy vector $\\mathbf{p}^*$ (\\ref{eq:seedbind}) with all noiseless hypervectors of HD-map $\\mathcal{P}$. In order to demonstrate this, we shall consider the properties of the transformation from $\\mathcal{D}$ to $\\mathcal{P}$ with the unbinding operation (\\ref{eq:transf}).\n\n\n\\begin{figure}[t!\n\\centering\n\\includegraphics[width=6.5cm]{img\/gridmapping_hyperseed}\n\\caption{Transformation of space $\\mathcal{D}$ into $\\mathcal{P}$ with the unbinding operation.\n}\n\\label{fig:gridmapping}\n\n\\end{figure}\n\n\nConsider an example of transforming vector space $\\mathcal{D}$ of FPE-encoded two dimensional grid structure to HD-map $\\mathcal{P}$. Graphically, the scenario is illustrated in Fig. \\ref{fig:gridmapping}. The data is transformed via FPE as described in Section~\\ref{sect:hdMap} with initial random bases $\\mathbf{x}^\\mathcal{D}_0$ and $\\mathbf{y}^\\mathcal{D}_0$. Importantly, while the transformed hypervectors in $\\mathcal{D}$ are available for observation, the base hypervectors, which were used in the transformation are unknown. The bandwidth $\\epsilon_{\\mathcal{D}}$ used during the encoding is also unknown. \n\nVector space $\\mathcal{P}$ is represented via FPE, now with initial random bases $\\mathbf{x}^P_0$ and $\\mathbf{y}^P_0$. Importantly, the base hypervectors of $\\mathcal{P}$ as well as FPE bandwidth $\\epsilon_{\\mathcal{P}}$ are known. For brevity of calculations, it is assumed the FPE bandwidth of vector space $\\mathcal{D}$ is the same as that of vector space $\\mathcal{P}$, that is $\\epsilon_{\\mathcal{P}}=\\epsilon_{\\mathcal{D}}=\\epsilon$.\n\n\nWe now pick an arbitrary vector from $\\mathcal{D}$ encoding a pair of values $(k,l)$ and bind it to a hypervector from $\\mathcal{P}$ encoding some predefined pair of values $(p,q)$:\\footnote{\nStrictly speaking, equations below should include modulo $2\\pi$ operations but they are omitted for the sake of readability. \n} \n\\begin{equation}\n\\begin{split}\n \\mathbf{s} &=e^{j\\cdot 2\\pi\\epsilon_{\\mathcal{P}}\\cdot x^P_0\\cdot p+j\\cdot 2\\pi\\epsilon_{\\mathcal{P}}\\cdot y^P_0\\cdot q +j\\cdot 2\\pi\\epsilon_{\\mathcal{D}}\\cdot x^D_0\\cdot k+j\\cdot 2\\pi\\epsilon_{\\mathcal{D}} \\cdot y^D_0\\cdot l}\\\\\n &=e^{j\\cdot 2\\pi \\epsilon_{\\mathcal{P}} \\cdot (x^P_0\\cdot p+y^P_0\\cdot q+\\frac{\\epsilon_{\\mathcal{D}}}{\\epsilon_{\\mathcal{P}}}x^D_0\\cdot k+\\frac{\\epsilon_{\\mathcal{D}}}{\\epsilon_{\\mathcal{P}}}y^D_0\\cdot l)}. \n\\end{split}\n\\label{eq:seedvector}\n\\end{equation}\n\n\nObviously, as the result of unbinding of the hypervector representing the coordinate $(k,l)$ from $\\mathcal{D}$ from hypervector $\\mathbf{s}$, it will be translated to the hypervector for $(p,q)$ from $\\mathcal{P}$ as illustrated in Fig. \\ref{fig:gridmapping}. \n\n\nLet us now unbind a hypervector from $\\mathcal{D}$ encoding a point on a certain offset $(a,b)$ from $(k,l)$, that is hypervector: $e^{j\\cdot 2\\pi\\epsilon_{\\mathcal{D}} \\cdot (x^D_0\\cdot (k+a) + y^D_0\\cdot (l+b))}$, which is the farthest from point$(k,l)$ in this scenario. Recall that unbinding in FHRR is implemented as a component-wise multiplication with the complex conjugate:\n\\noindent\n\\begin{equation}\n\\begin{split}\n \\mathbf{v}^* &=e^{j\\cdot 2\\pi \\epsilon_{\\mathcal{P}} \\cdot (x^P_0\\cdot p+y^P_0\\cdot q+\\frac{\\epsilon_{\\mathcal{D}}}{\\epsilon_{\\mathcal{P}}}x^D_0\\cdot k+\\frac{\\epsilon_{\\mathcal{D}}}{\\epsilon_{\\mathcal{P}}}y^D_0\\cdot l)} \\cdot \\\\\n &\\cdot e^{j\\cdot 2\\pi\\epsilon_{\\mathcal{D}} \\cdot (-x^D_0\\cdot (k+a) - y^D_0\\cdot (l+b))}\\\\\n &=e^{j\\cdot 2\\pi \\epsilon_{\\mathcal{P}} \\cdot(x^P_0\\cdot (p+\\alpha_1\\cdot a)+y^P_0\\cdot (q + \\alpha_2 \\cdot b)}. \n\\end{split}\n\\label{eq:unbindoffset}\n\\end{equation}\n\\noindent\nIn \\ref{eq:unbindoffset}, $\\alpha_1=-\\frac{\\epsilon_{\\mathcal{D}} \\cdot x^D_0}{\\epsilon_{\\mathcal{P}} \\cdot x^P_0}$ and $\\alpha_2=-\\frac{\\epsilon_{\\mathcal{D}} \\cdot y^D_0}{\\epsilon_{\\mathcal{P}} \\cdot y^P_0}$ are coefficients introduced in order to align the result of unbinding with HD-map $\\mathcal{P}$ for ease of the interpretation. \n\nIn $\\alpha_1$ and $\\alpha_2$, the parameter of interest is $\\epsilon_{\\mathcal{P}}$, which is the bandwidth of HD-map $\\mathcal{P}$ and is the hyperparameter of the algorithm. Consider the case where $\\epsilon_{\\mathcal{P}}>\\epsilon_{\\mathcal{D}}$. This means that the fidelity of the similarity between hypervectors in HD-map $\\mathcal{P}$ is much coarser compared to the fidelity of the inter-hypervector similarity in the original vector space $\\mathcal{D}$. In this case, all hypervectors from $\\mathcal{D}$ after unbinding will be similar to the hypervector in $\\mathcal{P}$, which was chosen for the update of seed hypervector $\\mathbf{s}$ (e.g., $\\mathbf{v}^P_{(p,q)}$ in Fig.\n~\\ref{fig:gridmapping}). Essentially, we will observe an effect of collapsing of all hypervectors in space $\\mathcal{D}$ onto a single hypervector from space $\\mathcal{P}$. This is demonstrated by a simulation in which hypervector $\\mathbf{s}$ was created as $\\mathbf{s}=\\mathbf{v}^D_{(1,2)} \\circ \\mathbf{v}^P_{(2,2)}$. The size of HD-map $\\mathcal{P}$ is $5\\times 5$. Two simulations were performed: 1.) With the FPE bandwidth for encoding input data being equal $0.2$, while the FPE bandwidth of HD-map was set to 0.03; and 2.) With the FPE bandwidth for encoding input data being equal 0.2, while the FPE bandwidth of HD-map was set to 0.8.\nFig.~\\ref{fig:fft1} and~\\ref{fig:fft2} show the distribution of cosine similarities to all hypervectors of HD-map for every data hypervector after unbinding with hypervector $\\mathbf{s}$ (\\ref{eq:seedbind}) for the first and second simulation, respectively. Fig. \\ref{fig:fft2} demonstrates the effect of collapsing of all points in $\\mathcal{P}$ onto hypervector $\\mathbf{v}^P_{(2,2)}$. Fig. \\ref{fig:fft3} shows cosine similarities for every data hypervector after unbinding with hypervector $\\mathbf{s}$ to the BMV in the second simulation (i.e., $\\mathbf{v}^P_{(2,2)}$). \nObserve that the lowest similarities are for hypervectors $\\mathbf{v}^D_{(1,3)}$ and $\\mathbf{v}^D_{(3,3)}$, which are the farthest away from the hypervector $\\mathbf{v}^D_{(1,2)}$ used to compute hypervector $\\mathbf{s}$. Therefore, following the FFTR heuristic, one of these hypervectors should be used in (\\ref{eq:seedupdate}) to update hypervector $\\mathbf{s}$, thus, creating a new point of attraction in $\\mathcal{P}$. To summarize, the WMS procedure is as follows:\n\n\\begin{enumerate}\n\n\\item Initialize the lowest similarity variable $D_{min}=1$;\n\\item For each hypervector in $\\mathcal{D}$ compute $p^*$ (\\ref{eq:seedbind}) and search for the BMV in HD-map. Store the similarity to BMV $D_{BMV}$;\n\\item If $D_{BMV}\\epsilon_{\\mathcal{D}}$.}} \n \\label{fig:fft2}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[b]{0.3\\columnwidth} \n \\centering \n \\includegraphics[width=\\textwidth]{img\/hits_bmv.png}\n \\caption[]%\n {{\\small Similarities to $\\mathbf{v}^P_{(2,2)}$, $\\epsilon_{\\mathcal{P}}>\\epsilon_{\\mathcal{D}}$.}} \n \\label{fig:fft3}\n \\end{subfigure}\n\n \\caption[ ]\n {\\small Distribution of cosine similarities to hypervectors in $\\mathcal{P}$ for every data hypervector after unbinding with hypervector $\\mathbf{s}$.} \n \\label{fig:FFT}\n \\end{figure}\n\n\n\n\n\\subsection {Finding the target node on HD-map}\n\\label{sect:find}\n\n\n\n\nThe hypervector found during the WMS procedure must be anchored to a hypervector $\\mathbf{p}_{\\texttt{target}}$, encoding a node on HD-map. Since the WMS procedure finds a hypervector, which unbinds a hypervector of HD-map with the lowest similarity, intuitively, it should be bound to a different hypervector. Using the FFTR heuristic, this new hypervector should be located further away from the current BMV in order to create a new point of attraction for the hypervector found by the WMS procedure and other hypervectors similar to it. \n\nWith the FPE encoding of HD-map's hypervectors the availability of farthest hypervectors to one selected hypervector is different depending on the choice of the bandwidth parameter for the same size of the map. For large values of $\\epsilon_{\\mathcal{P}}$ the similarity between hypervectors encoding neighboring nodes decays faster than for small values of $\\epsilon_{\\mathcal{P}}$, therefore, the number of dissimilar hypervectors is larger in the former case. This is demonstrated in Fig. \\ref{fig:fpeplace1} and \\ref{fig:fpeplace2} for $\\epsilon_{\\mathcal{P}}=0.03$ and Fig. \\ref{fig:fpeplace3} and \\ref{fig:fpeplace4} for $\\epsilon_{\\mathcal{P}}=0.008$ on $200\\times200$ HD-map.\n\n\nThe simplest heuristic for finding the hypervector farthest to the given BMV for HD-maps with large $\\epsilon_{\\mathcal{P}}$ is, therefore, a random selection of $\\mathbf{p}_{\\texttt{target}}$.\nAs we demonstrate in the next section, this heuristic leads to an adequate accuracy performance of Hyperseed in classification tasks. At the same time, obviously, the visualization of such projections is not very informative since it does not adequately display the internal disposition of classes. \n\nIn the case of small values of $\\epsilon_{\\mathcal{P}}$, the number of farthest hypervectors on HD-map is limited and has to be selected according to a certain heuristic. When Hyperseed is used in visualization tasks, $\\epsilon_{\\mathcal{P}}$ is chosen such that most dissimilar hypervectors are located in the corners of HD-map. These corner nodes are then chosen as $\\mathbf{p}_{\\texttt{target}}$ during the update phase.\n\nFig.~\\ref{fig:lang_proj} demonstrates an instance of projections of a dataset containing collection of $n$-gram statistics from texts on seven European languages. The dataset is projected onto a $20\\times20$ HD-map with $\\epsilon_{\\mathcal{P}}=0.008$. The complete experiment description follows in the next section. In the figure, crosses in the corners of HD-map show the choice of target nodes during the update procedure. We observe a semantically meaningful projections of languages. The color of the crosses corresponds to the class of the hypervector selected by the WMS procedure. One most important observation at this point is that the classes that were not used for the update of hypervector $\\mathbf{s}$, e.g., Swedish, French, and Bulgarian languages, emerged automatically and adequately projected. \n\n\n \\begin{figure}[t!]\n \\centering\n \\begin{subfigure}[b]{0.475\\columnwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{img\/FPE_largeband.png}\n \\caption[]%\n {{\\small $\\epsilon_{\\mathcal{P}}=0.03$.}} \n \\label{fig:fpeplace1}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[b]{0.475\\columnwidth} \n \\centering \n \\includegraphics[width=\\textwidth]{img\/FPE_largeband_placement.png}\n \\caption[]%\n {{\\small $\\epsilon_{\\mathcal{P}}=0.03$.}} \n \\label{fig:fpeplace2}\n \\end{subfigure}\n \\vskip\\baselineskip\n \\begin{subfigure}[b]{0.475\\columnwidth} \n \\centering \n \\includegraphics[width=\\textwidth]{img\/FPE_lowband.png}\n \\caption[]%\n {{\\small $\\epsilon_{\\mathcal{P}}=0.008$.}} \n \\label{fig:fpeplace3}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[b]{0.475\\columnwidth} \n \\centering \n \\includegraphics[width=\\textwidth]{img\/FPE_lowband_placement.png}\n \\caption[]%\n {{\\small $\\epsilon_{\\mathcal{P}}=0.008$.}} \n \\label{fig:fpeplace4}\n \\end{subfigure}\n \\caption[ ]\n {\\small Distribution of cosine similarities on HD-map for different values of FPE bandwidth $\\epsilon_{\\mathcal{P}}$.} \n \\label{fig:fpeplace}\n \\end{figure}\n\n\n\n\n\n\n \\subsection{The iterative Hyperseed algorithm}\n \\label{sect:iterative}\n \n\n Fig. \\ref{fig:flowchart} displays all phases of the Hyperseed algorithm in a flowchart. The hyperparameters of the algorithm are dimensionality of hypervectors $d$ and the number of iterations $I$ (i.e., the number of updates of hypervector $\\mathbf{s}$). In the flowchart, function \\texttt{SelectD()} returns an arbitrary hypervector from $\\mathcal{D}$ if its argument is ``any'' or $j$-th hypervector from $\\mathcal{D}$ when it is called with argument $j$. Function \\texttt{SelectP()} returns a hypervector from $\\mathcal{D}$ as described in the previous subsection. Function \\texttt{FindBMV}($\\mathbf{p}^*$,$\\mathcal{P}$) performs a search in the associative memory storing hypervectors of $\\mathcal{P}$ and returns the hypervector with the closest cosine similarity to $\\mathbf{p}^*$. Function \\texttt{Sim($\\mathbf{p}^*$,$\\mathbf{BMV}$)} computes the cosine similarity between the two hypervectors.\n\n \n \\begin{figure}[t!]\n \\centering\n \\includegraphics[width=6cm]{img\/lang_proj.png}\n \\caption{Projection of seven European languages after four updates. The chosen $\\mathbf{p}_{\\texttt{target}}$ hypervectors are four nodes (0,0), (20,0), (20,20) and (0, 20). }\n \\label{fig:lang_proj}\n\\end{figure}\n \n \n\n\n \n\n\n\nThe computational complexity of Hyperseed is $\\mathcal{O}(INd|\\mathcal{P}|)$, where $I$ is the number of iterations (updates) of Hyperseed, $N$ is the number of training data hypervectors, $d$ is the dimensionality of hypervectors, and $|\\mathcal{P}|$ is the size of HD-map. When implemented on neuromorphic hardware the search of the BMV happens in a constant time $\\tau$, therefore, the time complexity of Hyperseed in this case is $O(IN)\\cdot \\tau$. The memory complexity, of course, depends on the size of HD-map which is $d \\times |\\mathcal{P}|$. \n\nThe complexity of the SOM algorithm in the Winner-Takes-All phase and in the weight matrix update procedures is $\\mathcal{O}(INd|SOM|)$. Here, $I$ is the number of iterations of SOM algorithm, $N$ is the number of data samples , $|SOM|$ is the number of nodes in the SOM map (corresponds to $|\\mathcal{P}|$ in Hyperseed) and $d$ is the number of neurons per node (corresponds to the dimensionality of hypervectors in Hyperseed). While the $\\mathcal{O}$ complexities of the two algorithms are matching, Hyperseed uses a single vector operation in the update phase and substantially fewer number of iterations. \n\n \n \\begin{figure}[t!]\n \\centering\n \\includegraphics[width=\\columnwidth]{img\/HyperSeedFlowchart.png}\n \\caption{Flowchart of Hyperseed. }\n \\label{fig:flowchart}\n\\end{figure}\n\n \n \n\\section{Introduction}\n\\label{sect:intro}\nVector Symbolic Architectures (VSA) are increasingly leveraged and adapted in machine learning and robotics algorithms and applications \\cite{HDGestureIEEE, Superposition_Olshausen, NeubertRobotics2019, HerscheCompressingBCI2020, Kleyko_RVFL, NeubertAggregation2021}. In classification tasks, the use of VSA leads to order of magnitude increase in energy efficiency of computations on the one hand and natively enables one-shot and multi-task learning on the other~\\cite{ChangHDTaskProjected2020, ChangHDInformationPreserved2020, KarunaratneInMemory2020,KarunaratneHDAugmented2021, KleykoAugmented2022}. It is prospected that VSA will play a key role in the development of novel neuromorphic computer architectures~\\cite{Loihi18} as an algorithmic abstraction \\cite{RahimiNanoscalable2017, KleykoComputingParadigm2021}. The main contribution of this article is a novel algorithm for unsupervised learning called Hyperseed, which relies on the mathematical properties arising from random high-dimensional representation spaces described through the phenomenon of the concentration of measure \\cite{Gorban2018Blessing} and of the main VSA operations of binding and superposition~\\cite{Kanerva09}. The method's name suggests that data samples are encoded as high-dimensional vectors (also called \\textit{hypervectors}, HVs or ``seeds''), which are then mapped (a.k.a. ``sowed'') onto a specially prepared topologically arranged set of hypervectors for revealing the internal cluster structure in the unlabeled data.\n\nThe Hyperseed algorithm bears conceptual similarities to Kohonen's Self-Organizing Maps (SOM) algorithm \\cite{SOMBook, intSOM}, therefore, selected SOM terminology is adopted for the description of our approach. \nHowever, it is designed using cardinally different computing principles to the SOM algorithm. \nHyperseed implements the entire learning pipeline in terms of VSA operations. To the best of our knowledge, this has not been attempted prior and is reported for the first time. \n\nThe Hyperseed algorithm is presented using Frequency Holographic Reduced Representations (FHRR)~\\cite{PlateNested1994} model of VSAs and the concept of Fractional Power Encoding (FPE) \\cite{PlateNested1994, frady2021computing, FradyFunctionsNICE2022, Komer2019ANR, komer2020biologically}. The usage of the FHRR model makes the proposed solution specifically fit for implementation on spiking neural network architectures including Intel's Loihi \\cite{Loihi18}.\n\nTo this end, we introduce the Hyperseed algorithm and demonstrate its performance on three illustrative non-linear classification problems of varying complexity: Synthetic datasets from the Fundamental Clustering Problems Suite, Iris classification, and language identification using $n$-gram statistics. Across all experiments, Hyperseed convincingly demonstrates its key novelties of learning from a few input vectors and single vector operation learning rule, both of which contribute towards reduced time and computation complexity. \n\nThe article is structured as follows. \nSection~\\ref{sect:related} describes related work to Hyperseed operations. The VSA methods leveraged in Hyperseed are presented in Section \\ref{sect:method}.\nSection~\\ref{sect:vsaseed} presents the main contribution -- the method for unsupervised learning -- Hyperseed. \nSection~\\ref{sect:perf} reports the results of the experimental performance evaluation. \nSection~\\ref{sect:discussion} discusses the suitability of Hyperseed for realization on neuromorphic hardware. \nThe conclusions follow in Section~\\ref{sect:conclusions} .\n\n\n\n\n\n\n\n\\section{Related Work}\n\\label{sect:related}\n \nVSA \\cite{PlateNested1994, Rachkovskij2001, KleykoSDR2016, FradySDR2020} is a computing framework providing methods of representing and manipulating concepts and their meanings in a high-dimensional space. VSA finds its applications in, for example, \ncognitive architectures~\\cite{BuildBrain,RachkovskijAnalogical2004,RachkovskijAnalogy2012}, \nnatural language processing~\\cite{BICA16CT,JonesMeaning2007, RPRSKJ2015, RachkovskijRecursiveBinding2022, RachkovskijEquivariant2021}, communications~\\cite{JakimovskiCollective2012, KleykoMACOM2012, KimHDM2018},\nbiomedical signal processing~\\cite{ACCESS_HRV, HDGestureIEEE}, approximation of conventional data structures~\\cite{HD_FSA, ABF},\nand for classification tasks such as gesture recognition~\\cite{TNNLS18, HDGestureIEEE}, cybersecurity threat detection \\cite{christopher2021minority, moraliyage2022evaluating}, physical activity recognition~\\cite{Rasanen14}, character recognition~\\cite{goltsev2005combination, Rachkovskij2022NCA}, speaker identification~\\cite{HuangSpeaker2022}, fault isolation and diagnostics~\\cite{KussulDiagnostics1998, ACCESS_BIOFAULT, EggimannConfigurableHD2021}. Examples of efforts on using VSA for other than classification learning tasks are using data HVs for clustering~\\cite{ImaniHDCluster2019, BandaragodaTrajectoryTraffic2019, HernandezClustering2021}, semi-supervised learning~\\cite{ImaniSemiHD2019}, collaborative privacy-preserving learning~\\cite{ImaniHDColLearn2019, KhaleghiPriveHD2020}, multi-task learning~\\cite{ChangHDTaskProjected2020, ChangHDInformationPreserved2020}, distributed learning~\\cite{RosatoHDDistributed2021, HsiehFL2021}.\nA comprehensive two-part survey of VSA is available in~\\cite{KleykoSurveyVSA2021Part1,KleykoSurveyVSA2021Part2}. \n\nHypervectors of high (but fixed) dimensionality (denoted as $d$) are the basis for representing information in VSA. \nThe information is distributed across hypervector positions, therefore, hypervectors use distributed representations \\cite{Hinton1986}. There are different VSA models that all offer the same operation primitives but differ slightly in terms of the implementation of these primitives. For example, there are VSA models that compute with binary, bipolar \\cite{Kanerva:Hyper_dym13, MAP}, continuous real, and continuous complex vectors \\cite{PlateNested1994}. Thus, the VSA concept has the flexibility to connect to a multitude of different hardware types, such as binary-valued VSAs for analog in-memory computing architectures~\\cite{KarunaratneInMemory2020} or complex-valued VSAs for spiking neuron architectures~\\cite{TPAM, RennerBinding2022, BentSpike2022}.\n\nThe relevant sub-domain of related work to the proposed Hyperseed algorithm is the application of VSA for solving machine learning tasks. In this context, VSA have been used for: 1) Representing input data and interfacing such representations with conventional machine learning algorithms and 2) Implementing the functionality of neural networks with VSA operations. \n\nThe most illustrative use cases for encoding of input data into hypervectors and interfacing conventional machine learning algorithms are \\cite{BandaragodaTrajectoryTraffic2019, RachkovskijClassifiers2007, RIJHK2015, AlonsoHyperEmbed2020, MirusBehavior2019, KleykoBoostingSOM2019, MirusBalanced2020, ShridharEnd2End2020, Kussul1999IJCNN, Rachkovskij2015Cybern}. For example, works~\\cite{AlonsoHyperEmbed2020, KleykoBoostingSOM2019} proposed encoding $n$-gram statistics into hypervectors and subsequently solving typical natural language processing tasks with either supervised or unsupervised learning using standard artificial neural network architectures. The main distinctive property of VSA represented data is the substantial reduction of the memory footprint and the reduced learning time. In \\cite{BandaragodaTrajectoryTraffic2019}, hypervectors were used to encode sequences of variable lengths in the context of unsupervised learning of traffic patterns in intelligent transportation system application.\nIn the context of visual navigation, hypervectors were used as input to Simultaneous Localization and Mapping (SLAM) algorithms \\cite{NeubertRobotics2019} as well as for ego-motion estimation~\\cite{MitrokhinSensorimotor2019, KleykoCommentariesSR2020, HerscheDVSCDT2020}.\n\n\n\n\n\n\n\n\nA great potential of VSAs was demonstrated when used for the implementation of the entire functionality of some classical neural network architectures. In~\\cite{Kleyko_RVFL, KleykointESN2020, DiaoGLVQHD2021} the functionality of an entire class of randomly connected neural networks (random vector functional link networks~\\cite{IgelnikRVFL1995} and echo state networks~\\cite{RC09}) was implemented purely in terms of VSA operations.\nIt was demonstrated that implementing the algorithm functionality with bipolar VSAs allows reducing energy consumption on the specialized digital hardware by the order of magnitude, while substantially decreasing the operation times. \nMoreover, further flexibility can be achieved~\\cite{KleykoCA2020, EggimannConfigurableHD2021} when considering the ways of generating random connections used in the networks.\n\n\n\n\n\n\nThe main contribution of this article in the context of VSA is a novel approach to learning since the dominating learning approach in the area is on creating a single hypervector for a specific class.\nThis is achieved through encoding input data and then forming associative memory storing the prototypical representations for individual classes. \nOur approach to learning is radically different -- it utilizes the similarity preservation property of the binding operation in combination with the FPE encoding method~\\cite{PlateNested1994}. \nFPE was recently used to simulate and predict dynamical systems~\\cite{VoelkerFPEDynamical2021}, perform integer factorization~\\cite{KleykoPrimes2022}, and represent order in time series~\\cite{SchlegelHDC-MiniROCKET2022}.\nThe associative memory in the proposed approach is created once during the initialization phase and remains fixed during the life-time of the system. The update requires a single vector operation. To the best of our knowledge, this is the first reseacrh article to present the usage of VSA in unsupervised learning tasks.\n\n\n \n\\section{Experiments and Results}\n\\label{sect:perf}\n\nThis section describes the results of the experimental evaluation of the proposed Hyperseed algorithm. \nBefore elaborating on the details of the experimental evaluation, it is important to set realistic expectations in order to correctly interpret the results. \nIn particular, in the classification tasks, it would not be reasonable to expect a performance comparable to, e.g., deep learning-based models.\nThis is because classification is not the primary task for unsupervised learning approaches. As any other unsupervised learning algorithm, Hyperseed does not modify the input representations to make them more separable. We chose to evaluate the performance of Hyperseed on classification tasks mainly because visualization (as one of the main applications of the unsupervised learning) is subjective and it is hard to quantify its quality.\n\nWe report three illustrative cases: 1) unsupervised learning from one-shot demonstrations on six synthetic datasets; 2) classification of the Iris dataset; and 3) identification of $21$ languages using their $n$-gram statistics. In all the experiments, we used dense complex FHRR representations \\cite{plate1995holographic} of varying dimensionality. The Python implementation of the Hyperseed algorithm and the code necessary to reproduce all the experiments reported in this study are publicly available\\footnote{Implementation of Hyperseed and the experiments, 2022. [Online.] Available: \\url{https:\/\/github.com\/eaoltu\/hyperseed}}. \n\\begin{table*}[tbh!]\n \\caption{Comparison of projections and classification accuracy of Hyperseed vs. SOMs on FCPS datasets} \\label{tab:hyperseed_fcps}\n \\centering\n \\begingroup\n \\setlength{\\tabcolsep}{6pt}\n \\renewcommand{\\arraystretch}{2}\n \\begin{tabular}{ |c|c|c|c|c|c|c| } \n \\hline\n & Atom & Chain link & Engy time & Hepta & Two diamonds & Lsun 3D \\\\\n \\cline{2-7}\n \\raisebox{20pt}{\\rotatebox{90}{Dataset}}\n &\n {\\includegraphics[scale=0.25]{img\/Atom_data.PNG}}\n &\n {\\includegraphics[scale=0.26]{img\/Chain_link_data.PNG}}\n &\n \\raisebox{-3pt}{\\includegraphics[scale=0.20]{img\/EngyTime_data.PNG}}\n &\n \\raisebox{-3pt}{\\includegraphics[scale=0.26]{img\/Hepta_data.PNG}}\n &\n {\\includegraphics[scale=0.19]{img\/Two_diamonds_data.PNG}}\n &\n \\raisebox{-3pt}{\\includegraphics[scale=0.26]{img\/Lsun3D_data.PNG}} \\\\\n \\hline \n \\raisebox{10pt}{\\rotatebox{90}{SOM}}\n &\n {\\includegraphics[scale=0.185]{img\/Atom_trad.PNG}}\n &\n {\\includegraphics[scale=0.185]{img\/Chainlink_trad.PNG}}\n &\n {\\includegraphics[scale=0.185]{img\/EngyTime_trad.PNG}}\n &\n {\\includegraphics[scale=0.185]{img\/Hepta_trad.PNG}} \n &\n {\\includegraphics[scale=0.185]{img\/TwoDiamonds_trad.PNG}} \n &\n {\\includegraphics[scale=0.185]{img\/Lsun3D_trad.PNG}} \\\\ \\cline{2-7}\n & Accuracy: 0.8878 & Accuracy: 0.9265 & Accuracy: 0.9528 & Accuracy: 0.9777 & Accuracy: 0.9774 & Accuracy: 0.9715 \\\\\n \\hline\n \\raisebox{5pt}{\\rotatebox{90}{Hyperseed}}\n &\n {\\includegraphics[scale=0.15]{img\/Atom_hrr.png}}\n &\n {\\includegraphics[scale=0.15]{img\/Chainlink_hrr.PNG}}\n &\n {\\includegraphics[scale=0.15]{img\/EngyTime_hrr.png}} \n &\n {\\includegraphics[scale=0.15]{img\/Hepta_hrr.png}} \n &\n {\\includegraphics[scale=0.15]{img\/TwoDiamonds_hrr.png}}\n &\n {\\includegraphics[scale=0.15]{img\/Lsun3D_hrr.PNG}} \\\\ \\cline{2-7}\n \n & Accuracy: 0.9821 & Accuracy: 0.9780 & Accuracy: 0.9071 & Accuracy: 0.9552 & Accuracy: 0.9902 & Accuracy: 0.9005 \\\\\n \\hline\n \\end{tabular}\n \\endgroup\n\\end{table*}\n\n\n\\subsection{Data transformation to high-dimensional space}\n\\label{sect:encoding}\nIn the first two experiments, where input data are in the form of feature vectors of dimensionality $K$, the values of each feature $f_k, k \\in [1,K]$ were normalized to range $[0,1]$. The interval $[0,1]$ was then split into $q$ quantization levels. \nFor each feature a base hypervector of unit length $\\mathbf{b}_k$ was randomly generated.\nThen levels for the particular feature $\\textbf{l}_i^k$ were encoded in FHRR representation using FPE~\\cite{PlateNested1994, komer2020biologically, frady2021computing, FradyFunctionsNICE2022, Komer2019ANR}: $\\textbf{l}^k=\\mathbf{b}_k^{\\epsilon i}$, where $\\epsilon \\in \\mathbb{R}$ represents the bandwidth parameter. The feature vector of a data sample was then represented as a single hypervector with the superposition operation: $\\textbf{v}=\\Sigma_{k \\in K} \\textbf{l}_i^k$. \n\nIn the language identification experiments, $n$-gram statistics was represented as hypervectors following the procedure in \\cite{RIJHK2015, KleykoBoostingSOM2019}. First, a bijection of the alphabet letters $i \\in |\\mathcal{A}|$ to random unitary atomic hypervectors $\\mathbf{b}_i$ was created. To encode the position of character $i$ in the $n$-gram, the permutation operation was used on the corresponding atomic hypervector $\\mathbf{b}_i$. For example, the hypervector for character ``b'' on a third position in a tri-gram is the corresponding atomic hypervector rotated three times: $\\rho^3(\\mathbf{b}_{b})$. An $n$-gram of size $n$ was encoded as binding of position based encoded hypervectors for corresponding characters. For example, a tri-gram ``bdf'' was encoded as: $\\rho(\\mathbf{b}_{b}) \\circ \\rho^2( \\mathbf{b}_{d})\\circ \\rho^3( \\mathbf{b}_{f})$. \nThe $n$-gram statistics for a given text sample was then encoded into a single hypervector through the superposition of hypervectors for all observed $n$-grams.\n\n\n\n\\subsection{Hyperseed for classification tasks}\n\nThe proposed Hyperseed algorithm is by definition an unsupervised learning algorithm, therefore, an extra mechanism is needed to use it in supervised tasks such as the considered Iris classification and language identification task. Once Hyperseed was trained, there is a need to assign labels to the best matching hypervectors in HD-map. \n\nRecall that Hyperseed is trained on a small subset of the available training data as described in Section \\ref{sect:iterative}. \nFor the labeling process in the experiments presented below, the training data were presented to the trained Hyperseed HD-map for one full epoch that did not update the seed hypervector $\\mathbf{s}$. The labels of the training data were used to calculate statistics for the BMV in HD-map. The nodes were assigned labels of the input samples that were prominent in the collected statistics. \n\nAt the classification phase, hypervectors of the nodes with the assigned labels were stored in the memory. During the classification phase, samples of the test data were used to assess the trained Hyperseed. For each sample in the test data, the BMV in HD-map was determined using the search procedure (Section~\\ref{sect:search}). The test sample was then assigned the label of the closest labeled hypervector stored in the memory.\n\nAccuracy was used as the main performance metric for evaluation and comparison of Hyperseed runs with different parameters. It should be re-emphasised that the focus of experiments was not on achieving the highest possible accuracy but on a comparative analysis of the Hyperseed algorithm. \n\n\n\n\n\\subsection{Experiment 1: The performance of Hyperseed on synthetic datasets with one-shot demonstration}\n\nThis experiment serves the purpose of highlighting the major property of the Hyperseed algorithm -- the capability of one-shot learning. When talking about one-shot learning, one has to be careful with its definition. In many practical cases, a single example is obviously not enough for accurate inference, instead, it is reasonable to talk about learning from a limited number of data samples, i.e., few-shot learning. This is what we intend to gradually demonstrate with all the experiments in this section. \n\nIn the first experiment, we fix the number of updates of seed hypervector $\\mathbf{s}$ to one. This, essentially, boils down to randomly picking a sample from the particular training data, running the Hyperseed's update phase, and right after that performing labeling, classification on the corresponding test data, and the visualization. \n\nFor this purpose, we used synthetic datasets from Fundamental Clustering Problems Suite (FCPS)~\\cite{Ultsch05}. FCPS provides several non-linear but simple datasets that can be visualized in two or three dimensions, for elementary benchmarking of clustering and non-linear classification algorithms. \n\n\nWe selected six FCPS datasets that are most representative of the non-linearity, which we aim to learn using Hyperseed, they are: ``Atom'', ``Chain Link'', ``Engy Time'', ``Hepta'', ``Two Diamonds'', and ``Lsun 3D''. We repeated the experiment eight times. In each run, all hypervectors used for data encoding as well as for the operation of the Hyperseed algorithm were generated with a new seed used to initialize a pseudorandom generator. \n\nIn Table~\\ref{tab:hyperseed_fcps}, we present the results of this experiment in the form of the comparative evaluation of the projections by the conventional SOM and the projections produced by the Hyperseed algorithm, both visually and as the classification accuracy for each dataset. The sizes of the SOM grid and HD-map of the Hyperseed algorithm were the same $100\\times 100$.\nThe first row in the table presents the visualization of the six selected datasets in their original two or three dimensional data space. The second row presents HD-map projections and classification accuracy of the conventional SOM, while the third row shows HD-map projections and classification accuracy of the Hyperseed algorithm. \n\nThe primary observation in relation to the classification is that the Hyperseed algorithm provided average accuracy on a par with the conventional SOM ($0.948$ versus $0.943$). \n\n\n\nThe visualization of HD-map projections of Hyperseed are more representative of the topology preservation of the original data space, in comparison to the conventional SOM. In three out of six datasets, Engy Time, Hepta, and Lsun 3D, the topology preservation was directly comparable to the original dataset, where data samples of the same class were tightly clustered, in contrast to the conventional SOM, where these data samples were more scattered. For the remaining three datasets (Atom, Chain Link and Two Diamonds), although the topology preservation was not representative, the classification accuracy was high. This can be rationalized by the fixed 2D structure, which inhibits the complete visualization of the data space. \n\n\n\\subsection{Experiment 2: Iris classification with Hyperseed}\n\n\n\nWe continue the evaluation by exploring the details of Hyperseed's operations during several updates. We used the Iris dataset with 150 samples of labeled data. The dataset contains three classes of Iris flowers described by four real-valued features. Data were encoded into hypervectors as described in Section~\\ref{sect:encoding}. \n\nTo further highlight the capabilities of Hypreseed to learn from few-shots, we decided to split the Iris dataset such that the size of the test data was larger than the size of the training data. The results for Hyperseed in this section were obtained for the $20\\% \/ 80\\%$ split for training and test data, respectively. In order to provide a benchmark performance, we trained the conventional SOM with a $30 \\times 30$ grid on different splits of the Iris dataset and counted the number of iterations it required the SOM to reach $95\\%$ accuracy. The experiments with the conventional SOM were repeated 10 times and the maximum accuracy across runs was recorded. Table~\\ref{tab:num_iter} reports the results. \nOne could see that for the 20\/80 split the conventional SOM failed to achieve the target accuracy. Increasing the size of the training data allowed SOM reaching the target accuracy. The number of required iterations, however, was significant (the lowest was 200 for the 80\/20 split).\nThe maximum number of iterations of Hyperseed was set to 3 and 6. \nThis means that algorithm performed 90 (3 times 30 samples of the training data) and 180 (6 times 30 samples of the training data) searches for the BMV and only 3 (and correspondingly 6) updates of seed hypervector $\\mathbf{s}$. This has to be compared to $200 \\times 120$ of searches and updates of the conventional SOM in the 80\/20 split case. Each experiment (with three and six updates) was run ten times with different seeds.\n\nThe target nodes at each update were pre-selected to be (15,15) - for the first update, (20,20) - for the second update, (10,10) - for the third update, (5,5) - for the fourth update, (25,25) - for the fifth update, and (5,25) - for the sixth update. These nodes are marked by red crosses in the visualizations. This highlights again the importance of the target node selection rule for visually adequate projections. In this experiment, however, the adequate visualization was not important since our focus was on the performance characteristics of the Hyperseed algorithm in the classification task. \nTable~\\ref{tab:hyperseed_comp1} shows the results of the experiment. For the fair comparison with the conventional SOM here we also select the best performance across runs. \n\n\\begin{table}[t!]\n \\caption{Number of iterations of the conventional SOM required to reach $95\\%$ accuracy for different splits of Iris.} \\label{tab:num_iter}\n \\centering\n \n \n \\renewcommand{\\arraystretch}{2}\n \\begin{tabular}{ |p{4cm}|c|c|c|c| } \n \\hline\n Data set split (training\/testing), \\% & 20\/80 & 40\/60 & 60\/40 & 80\/20 \\\\ \\hline\n Number of updates \n & - & 2500 & 600 & 200 \\\\\n \\hline\n\n \\end{tabular}\n \n\\end{table}\n\n\\begin{table}[t]\n \\caption{Comparison of the Hyperseed projections with different random seeds for the Iris dataset.} \\label{tab:hyperseed_comp1}\n \\centering\n \\begingroup\n \\setlength{\\tabcolsep}{6pt}\n \\renewcommand{\\arraystretch}{2}\n \\begin{tabular}{ |c|c|c| } \n \\hline\n \n &\n \\# updates: 3, Acc: 0.92 & \\# updates: 6, Acc: 0.95 \\\\\n\n \\hline\n {\\rotatebox[origin=c]{90}{Train Projection}}\n &\n \\raisebox{-45pt}{\\includegraphics[scale=0.2]{img\/Iris_data\/1_train_3itr.png}}\n &\n \\raisebox{-45pt}{\\includegraphics[scale=0.2]{img\/Iris_data\/1_train.png}}\n \\\\\n \n \\hline\n \\rotatebox[origin=c]{90}{Test Projection}\n &\n \\raisebox{-45pt}{\\includegraphics[scale=0.2]{img\/Iris_data\/1_test_3itr.png}} \n &\n \\raisebox{-45pt}{\\includegraphics[scale=0.2]{img\/Iris_data\/1_test.png}}\\\\\n \n \\hline\n \\end{tabular}\n \\endgroup\n\\end{table} \n\nThe first interesting observation comes in the case of three updates. \nWe selected the best runs, which resulted in the highest accuracy on the test data ($0.93$). The projection of the training data show that out of three updates, in total one update was done for the first class (blue circles) and two updates were done for the second class. \nThus, the cluster for the third class emerged automatically. \n\nAnother important observation comes from the relative placement of the data samples (both from the training and test data) on HD-map. For the Iris dataset, it is known that the second and third classes are very similar to each other, which manifests in misclassification of some of their samples. Here, we see that this was, indeed, the case (orange triangles are very close to green triangles in several nodes of HD-map). \nHowever, in the case of Hyperseed this proximity did not lead to large degradation of the classification accuracy. This is because each point in HD-map attracted similar samples.\nNext, Fig.~\\ref{fig:accVsIterIRIS} shows the accuracy of Hyperseed when increasing the number of updates. On average, the classification accuracy increased with more updates of seed hypervector $\\mathbf{s}$.\nAlso, in Fig.~\\ref{fig:accVSqVSbw} we demonstrate the tradeoff between the choice of hyperparameters for the FPE encoding of input data (bandwidth $\\epsilon$ and the number of quantization levels, $q$) and the classification accuracy. \nWe observed that the best accuracy was achieved when $\\epsilon$ and $q$ were in the inverse relationship, that is when $\\epsilon q = 1$.\n\n\n\\begin{figure}[t!]\n \\centering\n \\includegraphics[width=8cm]{img\/Iris_data\/accVSIterations_iris.png}\n \\caption{Number of iterations against the accuracy ($d=500$) of Hyperseed on the Iris dataset. }\n \\label{fig:accVsIterIRIS}\n\\end{figure}\n\n\\begin{figure}[t!]\n \\centering\n \\includegraphics[width=8cm]{img\/Iris_data\/q_vs_bW_vs_acc.png}\n \\caption{Classification accuracy against the choice of FPE hyperparameters for data encoding: bandwidth and the number of quantization levels. Dimensionality of hypervectors was set to $d=500$.}\n \\label{fig:accVSqVSbw}\n\\end{figure}\n\n\n\n\n\n\n\n\n \\begin{figure*}[t!]\n \\centering\n \\begin{subfigure}[b]{0.475\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{img\/textDataPlots\/accVSnumOfClasses-5000D-1SOM-3gram.png}\n \\caption[]%\n {{\\small Accuracy with one seed hypervector $\\mathbf{s}$ using the original update rule from Section~\\ref{sect:update}, $d=5000$.}} \n \\label{fig:single}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[b]{0.475\\textwidth} \n \\centering \n \\includegraphics[width=\\textwidth]{img\/textDataPlots\/accVSnumOfClasses-5000D-10SOM-3gram.png}\n \\caption[]%\n {{\\small Number of classes vs. accuracy with original and modified learning phases ($d=5000$).}} \n \\label{fig:ten}\n \\end{subfigure}\n \\vskip\\baselineskip\n \\begin{subfigure}[b]{0.475\\textwidth} \n \\centering \n \\includegraphics[width=\\textwidth]{img\/textDataPlots\/1-numOfSomsVsAccuracy-5000-3grams.png}\n \\caption[]%\n {{\\small Number of $\\mathbf{s}$ hypervectors vs. accuracy ($d=5000$).}} \n \\label{fig:num_seed}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[b]{0.475\\textwidth} \n \\centering \n \\includegraphics[width=\\textwidth]{img\/textDataPlots\/5-accVsDIM_numOfSoms-10-3grams.png}\n \\caption[]%\n {{\\small Dimensionality of hypervectors vs. accuracy (Number of $\\mathbf{s}$ hypervectors is 10).}} \n \\label{fig:dimensionality}\n \\end{subfigure}\n \\caption[ TResults of Experiment 3. Performance of Hyperseed in 21 languages classification task. ]\n {\\small The performance of Hyperseed in the task of language identification.} \n \\label{fig:Experiment3}\n \\end{figure*}\n\n\n\n\n\\subsection{Experiment 3: Training Hyperseed on $n$-grams in language identification task}\nWhile the previous two experiments were used to get first insights on the major properties of Hyperseed, with this experiment we intend to demonstrate its performance on a larger scale problem. \nWe will use it for the classification task of 21 European languages using $n$-gram statistics. The list of languages is as follows: Bulgarian, Czech, Danish, German, Greek, English,\nEstonian, Finnish, French, Hungarian, Italian, Latvian, Lithuanian, Dutch, Polish, Portuguese, Romanian, Slovak, Slovene, Spanish, Swedish. \nThe training data is based on the Wortschatz Corpora \\cite{LANGDATA}. \nThe average size of each language corpus in the training data was $1085637.3 \\pm 121904.1$ symbols. \nIn this study, we use the method for encoding $n$-gram statistics into hypervectors from~\\cite{KleykoBoostingSOM2019}, where it was used as an input to the conventional SOM. Each original language corpus was divided into samples where the length of each sample was set to $1000$ symbols. \nThe total number of samples in the training data was $22791$. \n\n\nThe test data also contains samples from the same languages and is based on the Europarl Parallel Corpus\\footnote{Available online at \\url{http:\/\/www.statmt.org\/europarl\/}. \n}.\nThe total number of samples in the test data was $21000$, where each language was represented with $1000$ samples. \nEach sample in the test data corresponded to a single sentence. \nThe average size of a sample in the test data was $150.3 \\pm 89.5$ symbols.\n\nThe data for each language was pre-processed such that the text included only lower case letters and spaces. \nAll punctuation was removed. \nLastly, all text used the 26-letter ISO basic Latin alphabet, i.e., the alphabet for both training and test data was the same and it included $27$ symbols. \nFor each text sample, the $n$-gram statistics transformed to hypervectors was obtained, which was then used as input when training or testing Hyperseed.\nSince each sample was pre-processed to use the alphabet of only $a=27$ symbols, the conventional $n$-gram statistics input was $27^{n}$ dimensional. In the experiment, we used tri-grams, therefore, the conventional representation was $19,683$ dimensional. The dimensionality of the mapped $n$-gram statistics into hypervectors as described in Section~\\ref{sect:encoding} depends on the dimensionality of the hypervectors $d$. The results reported below were obtained for different dimensionalities in the range [400, 10000], which corresponds to the dimensionality reduction of the original representation space from $49$ fold (for $d=400$) to $2$ fold (for $d=10,000$). In this experiment, we used $100\\times100$ HD-map.\n\n\nThe first investigation was performed with dimensionality of hypervectors $d=5000$ ($4$ fold dimensionality reduction). \nIn this experiment, we exposed Hyperseed to a different number of classes from the original dataset. The number of Hyperseed updates in each case was set relative to the number of classes at hand. Importantly, this heuristic for choosing the number of updates was adopted for automating experiments only. It only reflects the desire of keeping the number of iterations low. In the unsupervised learning context, the information about the number of classes is, obviously, unavailable. \n\nThe experiment was repeated eight times for each number of classes, each time selecting a different subset of languages for Hyperseed training. Fig.~\\ref{fig:single} shows the results. The reference accuracy 0.97 (the red line) is the accuracy obtained with the conventional tri-gram statistics representation reported in \\cite{HDenergy, RIJHK2015} using the nearest neighbor classifier. \nThe main observation to make from this investigation is that the performance of Hyperseed dropped as it was exposed to larger number of classes. The explanation to this is connected to the finite capacity of hypervectors in terms of the number of hypervectors one can superimpose together while keeping the sufficient accuracy of retrieving them back~\\cite{FradyCapacity2018,KleykoPerceptron2020}. As we discussed above, each update to seed hypervector $\\mathbf{s}$ introduces noise to the previous updates. That is why the number of updates in Hyperseed needs to be kept low. \n\n\\subsubsection{Modified Hyperseed learning phase}\nTo mitigate the problem of the reduced accuracy on large problems (in terms of the number of distinct classes) the learning phase of Hyperseed was modified. \n\nInstead of having a single seed hypervector, it is proposed to use $N$ vectors $\\mathbf{s}_i, i=1..N$, where $N$ becomes another hyperparameter of Hyperseed. During the update procedure, these hypervectors will be updated in a round-robin manner in order to keep the balanced number of updates per a seed hypervector.\n\nDuring the search of the BMV in either the WMS or test procedures, the binding in (\\ref{eq:bind}) is now computed for the current input hypervector and all seed hypervectors $\\mathbf{s}_i$'s. \nHypervector $\\mathbf{p}$ with the highest cosine similarity across all results of unbinding with all $\\mathbf{s}_i$ is selected as the BMV.\n\nThe number of iterations (and as the result the number of updates) with the new update phase should be scaled such that the number of updates per $\\mathbf{s}_i$ is approximately the same. For example, in the experiments with $N=10$ the number of iterations was configured to $30$ to allow for three updates per $\\mathbf{s}_i$. \nFig.~\\ref{fig:ten} demonstrates significant performance improvement with the modified learning phase. The maximum performance of Hyperseed in $21$ classes case is $0.91$ after $30$ iterations.\n\n\n\n\n\nNext, we are interested in the effect of the number of seed hypervectors on the classification performance of Hyperseed. \nFig.~\\ref{fig:num_seed} shows the classification accuracy in the case of 21 classes for different number of seed hypervectors used in the modified learning phase. The main observation here was that the performance stabilized after a certain value of this parameter. \nIn our case, there was no significant increase in the accuracy after using ten seed hypervectors.\n\n\\subsubsection{Hyperseed performance for different dimensionalities of hypervectors}\n\nFinally, we are interested in the effect of the dimensionality of hypervectors on the performance of Hyperseed. We performed an experiment with ten seed hypervectors and varied the dimensionality of the hypervectors used by the algorithm from 400 until 10000. The results are depicted in Fig.~\\ref{fig:dimensionality}. The main observation to make was that the classification accuracy on small dimensionalities was low, as expected. On the positive side, it was substantially higher than the random choice, which is $0.04$ in this case. The reason for this is rather clear and it is connected to the dimensionality of the non-distributed representations, which is high ($19,683$). With only $400$ dimensions and the adopted encoding procedure of the input data, we operated well above the capacity of the hypervectors, this lead to high inter-class similarity. \nIncreasing the dimensionality allowed addressing this issue and resulted in better accuracy. It turned out that in the case of the language identification $5000$ was the optimal dimensionality for obtaining high-quality performance. It is worth noting that the accuracy of Hyperseed was still lower by $0.07$ compared to the baseline. It was, however, not expected that it would necessarily achieve higher accuracy compared to the supervised methods.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\n\n\nThe Authors stress that for a continuous beam, ``the effect of the detuning impedance is to add an additional tune shift to the bare machine working point\". In other words, the detuning impedance is indistinguishable from the external quadrupole focusing. We agree with the Authors on that; our disagreement with them is about the relationship between beam optics and coupling of transverse collective modes. \n\nFirst, let us clarify the general coupling conditions and the terminology for the transverse modes of a coasting beam, since the Article~\\cite{BiancacciPRAB2020} creates confusion in these respects. It is well-known that for a coasting beam, the transverse spectrum consists of two complex-conjugated series, which may be called {\\it positive-based} and {\\it negative-based}, $\\Omega \\approx (Q_\\beta + n)\\omega_0$ and $\\Omega \\approx (- Q_\\beta + n) \\omega_0$ respectfully; $n=0,\\pm1,\\pm2,...$ is an integer harmonic number. Driving impedance terms, which are relatively small, are dropped here for the moment, and the notations are same as in the Article. Each series, in turn, consists of four types of modes, distinguished by the signs and values of the angular phase velocity $\\Omega\/n$, see e.g. Ref.~\\cite{lee2018accelerator}. For the positive-based modes, these types follow,\n\n\\begin{itemize}\n\\item{ Zero mode, $n=0$, $\\Omega= Q_\\beta \\, \\omega_0 \\equiv \\omega_\\beta$;}\n\\item{ Fast mode, $n >0$, hence, $\\Omega\/n > \\omega_0$;}\n\\item{ Backward mode, $-Q_\\beta < n <0$, hence, $\\Omega\/n < 0$;}\n\\item{ Slow mode, $n<-Q_\\beta$, hence, $0<\\Omega\/n < \\omega_0$.}\n\\end{itemize}\n\nNotation for the negative-based modes follows with complex conjugation, $n \\rightarrow -n\\,$, $Q_\\beta \\rightarrow -Q_\\beta$,\n\n\\begin{itemize}\n\\item{ Zero mode, $n=0$, $\\Omega=-Q_\\beta \\omega_0 \\equiv -\\omega_\\beta$;}\n\\item{ Fast mode, $n<0$, hence, $\\Omega\/n > \\omega_0$;}\n\\item{ Backward mode, $0Q_\\beta$, hence, $0<\\Omega\/n < \\omega_0$.}\n\\end{itemize}\n\nDue to the driving impedance properties, only the slow modes can be unstable. An illustrative sketch of the spectrum is presented in Fig.~\\ref{PlotFastSlowModes}, assuming smooth approximation and focusing detuning impedance, when the modes can cross but not couple. \n\nFor the Article's PS example with lattice tunes $Q_\\beta =\\omega_\\beta\/\\omega_0=6.4$, a slow mode with frequency $\\omega_\\beta -7 \\omega_0 = -0.6 \\omega_0$ has the nearest mode $-\\omega_\\beta +6 \\omega_0 = -0.4 \\omega_0$, the backward one. The Article, including the title, calls the latter mode ``fast,\" which is a terminological mistake: the value of the phase velocity of that allegedly ``fast\" mode is actually smallest among all the modes. \n\n\n\\begin{figure*}[tbh!]\n \\centering\n \\includegraphics*[width=0.7\\textwidth]{PlotFastSlowModes2.pdf}\n \\caption{\\label{PlotFastSlowModes} Sketch of a coasting beam spectrum, assuming smooth approximation and focusing detuning impedance. For graphical reasons, the lattice tune $Q_\\beta=1.4$ is assumed, for this Figure only. Positive-based part is in red; negative-based is in blue. Zero modes are shown by thick solid lines, backward modes -- by dashed lines, slow ones -- by dotted lines, and fast modes -- by normal solid lines. All crossing lines necessarily have the same difference of the harmonic numbers, which is the nearest integer above the doubled tune, $n_1-n_2= \\pm \\mathrm{ceil}(2Q_\\beta)=\\pm 3$ in this example.}\n\\end{figure*}\n\nIn linear systems with time-independent coefficients, modes can couple only when their frequencies coincide. Clearly, positive-based modes can couple only with the negative-based ones, and vice versa; one of the modes must be stable (fast, zero, or backward), and another unstable (slow). For the PS example, the positive-based slow mode with $n=n_1=-7$ might couple with the negative-based backward mode with $n=n_2=+6$. The coupling can happen if the lattice tune difference between the two modes, $0.6-0.4=0.2$, is compensated by the detuning impedance, presumably able to shift the betatron tune up by $0.1$. Note that the difference between the harmonic numbers of the coupled modes, $n_2 - n_1 = 13 = \\mathrm{ceil}(2Q_\\beta)$, is just above the doubled betatron tune; the same is true for any pair of coupled modes. \n\nWe beg pardon for this pedantic textbook explanation, but we feel obliged to make it since in the Article the terms are confused and the harmonic numbers are given without signs, making a false impression that the modes of the neighbor harmonics, 6 and 7, can sometimes be coupled. \n\nLet us now come back to Eqs.~(23). They are derived from Eq.~(22) by an ansatz that the collective oscillation $y(s,t)$ is a linear combination of two harmonics, $n_1$ and $n_2$. In that derivation, the dependence on $s$ was dropped without any explanation. For equations with constant coefficients, such as Eqs.~(23), this omission is a mathematical mistake, since the cross terms in Eqs. (23) depend on the coordinate $s$ in a different way than the direct terms, $y_1 \\propto \\exp(i n_1 s\/R)$, while $y_2 \\propto \\exp(i n_2 s\/R)$, and $n_1 \\neq n_2$. To provide the mode coupling, the cross terms can only be built by harmonic $n_1 - n_2=\\pm \\mathrm{ceil}(2Q_\\beta)$ of the driving impedance weighed with the beta-function, $Z^\\mathrm{driv}(\\Omega,s) \\beta(s)$. In the smooth approximation, where $Z^\\mathrm{driv}(\\Omega,s) \\beta(s) = \\mathrm{const}$, these cross terms are equal to zero. Thus, instead of Eqs.~(23) of the Article, the mode coupling problem should be described by the following equations;\n\\begin{equation}\n\\begin{split}\n& \\Ddot{y}_1 + \\omega_\\beta^2 y_1 = -2\\omega_\\beta \\Delta \\Omega^\\mathrm{tot}\\, y_1 -2\\omega_\\beta \\Delta \\Omega^\\mathrm{driv}_{n_1 - n_2} \\,y_2 \\,;\t\\\\\n& \\Ddot{y}_2 + \\omega_\\beta^2 y_2 = -2\\omega_\\beta \\Delta \\Omega^\\mathrm{driv}_{n_2 - n_1} y_1 - 2\\omega_\\beta \\Delta \\Omega^\\mathrm{tot}\\, y_2 \\,.\t\\\\\n\\end{split}\n\\label{TrueEq}\n\\end{equation}\nHere $\\Delta \\Omega^\\mathrm{tot} \\propto i \\int{ \\dd s [Z^\\mathrm{det}(0,s) + Z^\\mathrm{driv}(\\Omega,s)] \\beta(s)}$ is the conventional uncoupled coherent tune shift at the sought-for frequency $\\Omega \\approx \\pm \\omega_\\beta +n \\omega_0$, and the cross-coefficients can be expressed as \n\\begin{equation}\n\\Delta \\Omega^\\mathrm{driv}_{n_1 - n_2} = \\Delta \\Omega^\\mathrm{driv}_0 \n\\frac{\\int{ \\dd s\\, Z^\\mathrm{driv}(\\Omega,s) \\beta(s) \\exp(i (n_1 - n_2)s\/R)}}{\\int{ \\dd\\, s Z^\\mathrm{driv}(\\Omega,s) \\beta(s)}}\\,,\n\\label{CrossCoeff}\n\\end{equation}\nwith $\\Delta \\Omega^\\mathrm{driv}_0$ as the contribution of the driving impedance into the conventional coherent tune shift, $\\Delta \\Omega^\\mathrm{driv}_0 \\propto i \\int{ \\dd s\\, Z^\\mathrm{driv}(\\Omega,s) \\beta(s)}$. Let us stress again, that within the smooth approximation, apparently presumed by the Article, but contrary to its Eqs.~(23), the cross coefficients can only be zeros, since the integral in the numerator of Eq. (2) is equal to zero for $n_1 \\neq n_2$. It is also worth noting, that the mode coupling described by Eq.~(\\ref{TrueEq}) can hardly be of practical importance: the coupling may show itself only near the half-integer resonance, wherefrom the tunes should be kept out anyway. \n\nThe Authors claim that the proposed instability mechanism is conceptually analogous to the transverse mode coupling instability (TMCI) for bunched beams. Here they overlook an important difference between the azimuthal harmonics of a coasting beam and synchrotron harmonics of a bunch. The former are exact eigenfunctions for the coasting beam for arbitrary impedance, provided that the smooth approximation is justified, which is apparently assumed in the Article. This fact is guaranteed by the translation invariance of the dynamic equations. Contrary to that, the synchrotron harmonics of a bunch can, at best, be only approximations for the eigenfunctions, which accuracy deteriorates when the coherent tune shift becomes comparable with the synchrotron tune. For a bunch, the dynamic equations are not invariant under the synchrotron phase shifts, and thus, strictly speaking, the synchrotron harmonics can constitute the eigenfunctions only at zero wake field. That is why coupling of the azimuthal harmonics is forbidden for homogeneous coasting beams in the linear approximation, while coupling of the synchrotron harmonics of a bunch is possible. Note also that the eigenvectors of equidistant multiple bunches are the same harmonic exponents, for any impedance. \n \nNow we are coming to the last point of this comment, to the claimed excellent agreement between the theory of Eqs.~(23) and pyHEADTAIL simulations for the PS presented in Fig.~6. Were this agreement really there, it could only mean that a mistake occurs also in the code. However, we think that Fig.~6 demonstrates something different. Rather, it demonstrates an illusion of agreement. To show this, let us first have a look on the upper plot of this Figure, which represents the growth time $\\tau$[turns]. Here, we clearly see an agreement for all the points with $\\tau \\gg 1$, where the cross terms of Eqs.~(23) do not make any visible difference. For these points we indeed see agreement between the textbooks and the pyHEADTAIL simulations, while nothing at all can be said about the innovative aspect of the Article. Apart from these conventional points, we also see in this plot a sequence of points and the line at the growth time $\\tau \\approx 0$. These results of the code and Eqs.~(23) might really demonstrate the agreement or disagreement between the theory and the simulations, but for that we have to see these data. Instead, we see something indistinguishable from zero, or from infinity, if to speak in terms of the growth rates. Thus, the data of the upper plot of Fig.~6 consist of two parts: one is irrelevant to the suggested model of the mode coupling, and the other is obscured from judgement about the model validity by means of the way the data are presented. \n\nAs to the bottom plot of Fig.~6, we see there that the mode tunes are locked in the half-integer resonance. For the simulations, something like that has to be expected just on the ground of the sufficiently strong detuning quadrupole for a lattice with nonzero harmonic $13$. For a perfectly smooth lattice, however, the half-integer tune $6.5$ would be as good as any other tune; thus, the result of the simulations must be sensitive to the lattice smoothness. Since Eqs.~(23) are fully insensitive to the phase advance per cell or other smoothness parameters, the agreement between the pyHEADTAIL simulations and theory in the bottom plot of Fig.~6 can be only accidental. \n\nWe'd like also to note that, although the half integer resonance is not presented in the smooth approximation, it plays a significant role in any real machine. Approaching this resonance results in a big variation of beta-functions and, consequently, fast increase of effective impedance $ \\left< Z^\\mathrm{driv}(\\Omega,s) \\beta(s) \\right >$ and its coupling-related harmonic, mentioned above. \n\nWe hope that our disagreement with key issues of the Article is clearly expressed, and we would appreciate a response of the Authors.\n\n\n\nThis manuscript has been authored by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office of High Energy Physics. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzecnm b/data_all_eng_slimpj/shuffled/split2/finalzzecnm new file mode 100644 index 0000000000000000000000000000000000000000..98794f7dc86f57a5d954d631df5106b167cf8c60 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzecnm @@ -0,0 +1,5 @@ +{"text":"\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nTo the best of our knowledge, our {\\small GRETA}\\ approach is the first to aggregate event trends that are matched by nested Kleene patterns without constructing these trends. We achieve this goal by compactly encoding all event trends into the {\\small GRETA}\\ graph and dynamically propagating the aggregates along the edges of the graph during graph construction. We prove that our approach has optimal time complexity. Our experiments demonstrate that {\\small GRETA}\\ achieves up to four orders of magnitude speed-up and requires up to 50--fold less memory than the state-of-the-art solutions.\n\n\n\\section{Discussion}\n\\label{sec:discussion}\n\nIn this section, we sketch how our {\\small GRETA}\\ approach can be extended to support additional language features.\n\n\\textbf{Other Aggregation Functions}.\nWhile Theorem~\\ref{theorem:count} defines event trend count computation, i.e., \\textsf{COUNT}(*), we now sketch how the principles of incremental event trend aggregation proposed by our {\\small GRETA}\\ approach apply to other aggregation functions (Definition~\\ref{def:query}).\n\n\\vspace{-2mm}\n\\begin{theorem}[\\textbf{Event Trend Aggregation Computation}]\nLet $G$ be the {\\small GRETA}\\ graph for a query $q$ and a stream $I$,\n$e,e' \\in I$ be events in $G$ such that \n$e.type=E$, \n$e'.type \\neq E$, \n$attr$ is an attribute of $e$, \n$Pr$ and $Pr'$ be the predecessor events of $e$ and $e'$ respectively in $G$, and\n$End$ be the \\textsf{END} events in $G$.\n\\[\n\\begin{array}{l}\ne.count_E = e.count + \\sum_{p \\in Pr} p.count_E\\\\\ne'.count_E = \\sum_{p' \\in Pr'} p'.count_E\\\\\n\\textsf{COUNT}(E) = \\sum_{end \\in End} end.count_E\n\\vspace*{2mm}\\\\\n\ne.min = \\textsf{min}_{p \\in Pr}(e.attr, p.min)\\\\\ne'.min = \\textsf{min}_{p' \\in Pr'}(p'.min)\\\\\n\\textsf{MIN}(E.attr) = \\textsf{min}_{end \\in End}(end.min)\n\\vspace*{2mm}\\\\\n\ne.max = \\textsf{max}_{p \\in Pr}(e.attr, p.max)\\\\\ne'.max = \\textsf{max}_{p' \\in Pr'}(p'.max)\\\\\n\\textsf{MAX}(E.attr) = \\textsf{max}_{end \\in End}(end.max)\n\\vspace*{2mm}\\\\\n\ne.sum = e.attr*e.count + \\sum_{p \\in Pr} p.sum\\\\\ne'.sum = \\sum_{p' \\in Pr'} p'.sum\\\\\n\\textsf{SUM}(E.attr) = \\sum_{end \\in End} end.sum\n\\vspace*{2mm}\\\\\n\n\\textsf{AVG}(E.attr) = \\textsf{SUM}(E.attr) \/ \\textsf{COUNT}(E)\n\\end{array}\n\\]\n\\label{theorem:aggregate}\n\\end{theorem}\n\\vspace{-4mm}\n\n\\begin{figure}[t]\n\\centering\n\\subfigure[\\small \\textsf{COUNT}(*)=11]{\n\\includegraphics[width=0.22\\columnwidth]{figures\/count_star.png} \n\\label{fig:count_star}\n}\n\\hspace*{2mm}\n\\subfigure[\\small \\textsf{COUNT}(A)=20]{\n\\includegraphics[width=0.22\\columnwidth]{figures\/count_e.png} \n\\label{fig:count_e}\n}\n\\hspace*{2mm}\n\\subfigure[\\small \\textsf{MIN}(A.attr)=4]{\n\\includegraphics[width=0.22\\columnwidth]{figures\/min.png} \n\\label{fig:min}\n}\n\\hspace*{2mm}\n\\subfigure[\\small \\textsf{SUM}(A.attr)=100]{\n\\includegraphics[width=0.22\\columnwidth]{figures\/sum.png} \n\\label{fig:sum}\n}\n\\caption{Aggregation of trends matched by the pattern $P=(\\textsf{SEQ}(A+,B))+$ in the stream \\textit{I = \\{a1, b2, a3, a4, b7\\}} where \\textit{a1.attr=5, a3.attr=6,} and \\textit{a4.attr=4}}\n\\label{fig:aggregates}\n\\end{figure} \n\nAnalogously to Theorem~\\ref{theorem:count}, Theorem~\\ref{theorem:aggregate} can be proven by induction on the number of events in the graph $G$.\n\n\\begin{example}\nIn Figure~\\ref{fig:aggregates}, we compute \\textsf{COUNT}$(*)$, \\textsf{COUNT}$(A)$, \\textsf{MIN}$(A.attr)$, and \\textsf{SUM}$(A.attr)$ based on the {\\small GRETA}\\ graph for the pattern $P$ and the stream $I$. Compare the aggregation results with Example~\\ref{ex:aggregates}. \n\\textsf{MAX}$(A.attr)=6$ is computed analogously to \\textsf{MIN}$(A.attr)$. \n\\textsf{AVG}$(A.attr)$ is computed based on \\textsf{SUM}$(A.attr)$ and \\textsf{COUNT} $(A)$.\n\\end{example}\n\n\\textbf{Disjunction} and \\textbf{Conjunction} can be supported by our {\\small GRETA}\\ approach without changing its complexity because the count for a disjunctive or a conjunctive pattern $P$ can be computed based on the counts for the sub-patterns of $P$ as defined below.\nLet $P_i$ and $P_j$ be patterns (Definition~\\ref{def:pattern}).\nLet $P_{ij}$ be the pattern that detects trends matched by both $P_i$ and $P_j$.\n$P_{ij}$ can be obtained from its DFA representation that corresponds to the intersection of the DFAs for $P_i$ and $P_j$~\\cite{theoretical-info}.\nLet $\\textsf{COUNT}(P)$ denote the number of trends matched by a pattern $P$.\nLet \n$C_{ij} = \\textsf{COUNT}(P_{ij})$,\n$C_i = \\textsf{COUNT}(P_i) - C_{ij}$, and \n$C_j = \\textsf{COUNT}(P_j) - C_{ij}$.\n\nIn contrast to event sequence and Kleene plus (Definition~\\ref{def:pattern}),\ndisjunctive and conjunctive patterns do not impose a time order constraint upon trends matched by their sub-patterns.\n\n\\textbf{\\textit{Disjunction}} $(P_i \\vee P_j)$ matches a trend that is a match of $P_i$ or $P_j$. \n$\\textsf{COUNT}(P_i \\vee P_j) = C_i + C_j - C_{ij}$.\n$C_{ij}$ is subtracted to avoid counting trends matched by $P_{ij}$ twice.\n\n\\textbf{\\textit{Conjunction}} $(P_i \\wedge P_j)$ matches a pair of trends $tr_i$ and $tr_j$ where $tr_i$ is a match of $P_i$ and $tr_j$ is a match of $P_j$. \n$\\textsf{COUNT}(P_i \\wedge P_j) = \nC_i * C_j + \nC_i * C_{ij} +\nC_j * C_{ij} +\n\\binom{C_{ij}}{2}$\nsince each trend detected only by $P_i$ (not by $P_j$) is combined with each trend detected only by $P_j$ (not by $P_i$). In addition, each trend detected by $P_{ij}$ is combined with each other trend detected only by $P_i$, only by $P_j$, or by $P_{ij}$.\n\n\\textbf{Kleene Star} and \\textbf{Optional Sub-patterns} can also be supported without changing the complexity because they are syntactic sugar operators. Indeed,\n$\\textsf{SEQ}(P_i*,P_j) = \\textsf{SEQ}(P_i+,$ $P_j) \\vee P_j$ and \n$\\textsf{SEQ}(P_i?,P_j) = \\textsf{SEQ}(P_i,P_j) \\vee P_j$.\n\n\\textbf{Constraints on Minimal Trend Length}.\nWhile our language does not have an explicit constraint on the minimal length of a trend, one way to model this constraint in {\\small GRETA}\\ is to unroll a pattern to its minimal length. For example, assume we want to detect trends matched by the pattern $A+$ and with minimal length 3. Then, we unroll the pattern $A+$ to length 3 as follows: $\\textsf{SEQ}(A,A,A+)$. \n\nAny correct trend processing strategy must keep all current trends, including those which did not reach the minimal length yet. Thus, these constraints do not change the complexity of trend detection. They could be added to our language as syntactic sugar. \n\n\\begin{table}[t]\n\\centering\n\\begin{tabular}{|l|c|c|}\n\\hline\nSemantics & \\textbf{Skipped events} & \\textbf{\\# of trends} \\\\\n\\hline\n\\hline\n\\textbf{Skip-till-any-match}\n& Any\n& Exponential \\\\\n\\hline\n\\textbf{Skip-till-next-match}\n& Irrelevant\n& \\multirow{2}{*}{Polynomial} \\\\\n\\cline{1-2}\n\\textbf{Contiguous}\n& None\n& \\\\\n\\hline\n\\end{tabular}\n\\vspace*{2mm}\n\\caption{Event selection semantics}\n\\label{tab:ess}\n\\end{table}\n\n\\textbf{Event Selection Semantics} are summarized in Table~\\ref{tab:ess}. As explained in Section~\\ref{sec:model}, we focus on Kleene patterns evaluated under the most flexible semantics returning all matches, called \\textit{skip-till-any-match} in the literature~\\cite{ADGI08, WDR06, ZDI14}. \nOther semantics return certain subsets of matches~\\cite{ADGI08, WDR06, ZDI14}. \\textit{Skip-till-next-match} skips only those \\textit{events that cannot be matched}, while \\textit{contiguous} semantics skips \\textit{no} event. \nTo support these semantics, Definition~\\ref{def:pattern} of adjacent events in a trend must be adjusted. Then, fewer edges would be established in the {\\small GRETA}\\ graph than for skip-till-any-match resulting in fewer trends. Based on this modified graph, Theorem~\\ref{theorem:count} defines the event trend count computation.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.5\\columnwidth]{figures\/kleene5.png} \n\\vspace{-2mm} \n\\caption{Count of trends matched by the pattern $P$ in the stream $I=\\{a1,b2,a3,a4,b5\\}$}\n\\label{fig:kleene5}\n\\end{figure}\n\n\\textbf{Multiple Event Type Occurrences in a Pattern}.\nWhile in Section~\\ref{sec:model} we assumed for simplicity that an event type may occur in a pattern at most once, we now sketch a few modifications of our {\\small GRETA}\\ approach allowing to drop this assumption. \nFirst, we assign a unique identifier to each event type. For example, \n\\textsf{SEQ}(A+,B,A,A+,B+)\nis translated into \nP=\\textsf{SEQ}(A1+,B2,A3,A4+,B5+). \nThen, each state in a GRE\\-TA template has a unique label (Figure~\\ref{fig:kleene5}).\nOur {\\small GRETA}\\ approach still applies with the following modifications. \n(1) Events in the first sub-graph are \\textsf{START} events, while events in the last sub-graph are \\textsf{END} events.\n(2)~An event $e$ may not be its own predecessor event since an event may occur at most once in a trend.\n(3)~An event $e$ may be inserted into several sub-graphs. Namely, $e$ is inserted into a sub-graph for $e.type$ if $e$ is a \\textsf{START} event or $e$ has predecessor events. \nFor example, a4 is inserted into the sub-graphs for A1, A3, and A4 in Figure~\\ref{fig:kleene5}. a4 is a \\textsf{START} event in the sub-graph for A1. b5 is inserted into the sub-graphs for B2 and B5. b5 is an \\textsf{END} event in the sub-graph for B5.\n\nSince an event is compared to each previous event in the graph in the worst case, our {\\small GRETA}\\ approach still has quadratic time complexity $O(n^2 k)$ where $n$ is the number of events per window and $k$ is the number of windows into which an event falls (Theorem~\\ref{theorem:optimality}).\nLet $t$ be the number of occurrences of an event type in a pattern. Then, each event is inserted into $t$ sub-graphs in the worst case. Thus, the space complexity increases by the multiplicative factor $t$, i.e., $O(t n k)$, where $n$ remains the dominating cost factor for high-rate streams and meaningful patterns (Theorem~\\ref{theorem:complexity}).\n\n\n\n\n\\section{Performance Evaluation}\n\\label{sec:evaluation}\n\n\\subsection{Experimental Setup}\n\\label{sec:exp_setup}\n\n\\textbf{Infrastructure}. \nWe have implemented our {\\small GRETA}\\ approach in Java with JRE 1.7.0\\_25 running on Ubuntu 14.04 with 16-core 3.4GHz CPU and 128GB of RAM. We execute each experiment three times and report their average.\n\n\\textbf{Data Sets}. \nWe evaluate the performance of our {\\small GRETA}\\ approach using the following data sets.\n\n$\\bullet$~\\textbf{\\textit{Stock Real Data Set}}. \nWe use the real NYSE data set~\\cite{stockStream} with 225k transaction records of 10 companies. Each event carries volume, price, time stamp in seconds, type (sell or buy), company, sector, and transaction identifiers. We replicate this data set 10 times. \n\n$\\bullet$~\\textbf{\\textit{Linear Road Benchmark Data Set}}. \nWe use the traffic simulator of the Linear Road benchmark~\\cite{linear_road} for streaming systems to generate a stream of position reports from vehicles for 3 hours. Each position report carries a time stamp in seconds, a vehicle identifier, its current position, and speed. Event rate gradually increases during 3 hours until it reaches 4k events per second. \n\n\\begin{table}[t]\n\\centering\n\\begin{tabular}{|l||l|l|}\n\\hline\n\\textbf{Attribute} & \\textbf{Distribution} & \\textbf{min--max} \\\\\n\\hline\n\\hline\nMapper id, job id & Uniform & 0--10 \\\\\n\\hline\nCPU, memory & Uniform & 0--1k \\\\\n\\hline\nLoad & Poisson with $\\lambda=100$ & 0--10k \\\\\n\\hline\n\\end{tabular}\n\\vspace{2mm}\n\\caption{Attribute values}\n\\label{tab:parameters}\n\\end{table}\n\n$\\bullet$~\\textbf{\\textit{Cluster Monitoring Data Set}}. \nOur stream generator creates cluster performance measurements for 3 hours. Each event carries a time stamp in seconds, mapper and job identifiers, CPU, memory, and load measurements. The distribution of attribute values is summarized in Table~\\ref{tab:parameters}. The stream rate is 3k events per second.\n\n\\textbf{Event Queries}. \nUnless stated otherwise, we evaluate query $Q_1$ (Section~\\ref{sec:introduction}) and its nine variations against the stock data set. These query variations differ by the predicate $S.price * X < \\textsf{NEXT}(S).price$ that requires the price to increase (or decrease with $>$) by $X \\in \\{1, 1.05, 1.1, 1.15, 1.2\\}$ percent from one event to the next in a trend. \nSimilarly, we evaluate query $Q_2$ and its nine variations against the cluster data set, and query $Q_3$ and its nine variations against the Linear Road data set. \nWe have chosen these queries because they contain all clauses (Definition~\\ref{def:query}) and allow us to measure the effect of each clause on the number of matched trends. The number of matched trends ranges from few hundreds to trillions.\nIn particular, we vary the number of events per window, presence of negative sub-patterns, predicate selectivity, and number of event trend groups.\n\n\n\\textbf{Methodology}. \nWe compare {\\small GRETA}\\ to CET~\\cite{PLAR17}, SA\\-SE~\\cite{ZDI14}, and Flink~\\cite{flink}.\nTo achieve a fair comparison, we have implemented CET and SASE on top of our platform. We execute Flink on the same hardware as our platform. While Section~\\ref{sec:related_work} is devoted to a detailed discussion of these approaches, we briefly sketch their main ideas below.\n\n$\\bullet$~\\textbf{\\textit{CET}}~\\cite{PLAR17} is the state-of-the-art approach to event trend detection. It stores and reuses partial event trends while constructing the final event trends. Thus, it avoids the re-computation of common sub-trends. While CET does not explicitly support aggregation, we extended this approach to aggregate event trends upon their construction.\n\n$\\bullet$~\\textbf{\\textit{SASE}}~\\cite{ZDI14} supports aggregation, nested Kleene patterns, predicates, and windows. It implements the two-step approach as follows. \n(1)~Each event $e$ is stored in a stack and pointers to $e$'s previous events in a trend are stored. For each window, a DFS-based algorithm traverses these pointers to construct all trends. \n(2)~These trends are aggregated.\n\n$\\bullet$~\\textbf{\\textit{Flink}}~\\cite{flink} is an open-source streaming platform that supports event pattern matching. We express our queries using Flink operators. Like other industrial systems~\\cite{esper, dataflow, streaminsight}, Flink does not explicitly support Kleene closure. Thus, we flatten our queries, i.e., for each Kleene query $q$ we determine the length $l$ of the longest match of $q$. We specify a set of fixed-length event sequence queries that cover all possible lengths from 1 to $l$. Flink is a two-step approach.\n\n\\textbf{Metrics}. \nWe measure common metrics for streaming systems, namely, \\textit{latency, throughput}, and \\textit{memory}. \n\\textit{Latency} measured in milliseconds corresponds to the peak time difference between the time of the aggregation result output and the arrival time of the latest event that contributes to the respective result.\n\\textit{Throughput} corresponds to the average number of events processed by all queries per second.\n\\textit{Memory} consumption measured in bytes is the peak memory for storing\nthe {\\small GRETA}\\ graph for {\\small GRETA},\nthe CET graph and trends for CET,\nevents in stacks, pointers between them, and trends for SASE, and \ntrends for Flink.\n\n\\begin{figure*}[t]\n\t\\centering\n \\subfigure[Latency]{\n \t\\includegraphics[width=0.25\\columnwidth]{experiments-sources\/events\/bars\/latency\/events-latency.png}\n \t \\label{fig:events-latency}\n\t}\n\t\\hspace*{5mm}\n\t\\subfigure[Memory]{\n \t\\includegraphics[width=0.25\\columnwidth]{experiments-sources\/events\/bars\/memory\/events-memory.png}\n \t\\label{fig:events-memory}\n\t}\n\t\\hspace*{5mm}\n\t\\subfigure[Throughput]{\n \t\\includegraphics[width=0.25\\columnwidth]{experiments-sources\/events\/bars\/throughput\/events-throughput.png}\n \t \\label{fig:events-throughput}\n\t}\n\t\\vspace{-3mm}\n\t\\caption{Positive patterns (Stock real data set)}\n\t\\label{fig:exp_positive}\n\t\\subfigure[Latency]{\n \t\\includegraphics[width=0.25\\columnwidth]{experiments-sources\/negation\/bars\/latency\/negation-latency.png}\n \t \\label{fig:negation-latency}\n\t}\n\t\\hspace*{5mm}\n\t\\subfigure[Memory]{\n \t\\includegraphics[width=0.25\\columnwidth]{experiments-sources\/negation\/bars\/memory\/negation-memory.png}\n \t\\label{fig:negation-memory}\n\t}\n\t\\hspace*{5mm}\n\t\\subfigure[Throughput]{\n \t\\includegraphics[width=0.25\\columnwidth]{experiments-sources\/negation\/bars\/throughput\/negation-throughput.png}\n \t \\label{fig:negation-throughput}\n\t}\n\t\\vspace{-3mm}\n\t\\caption{Patterns with negative sub-patterns (Stock real data set)}\n\t\\label{fig:exp_negative}\n\\end{figure*}\n\n\\subsection{Number of Events per Window} \n\\label{exp:window}\n\n\\textbf{Positive Patterns}.\nIn Figure~\\ref{fig:exp_positive}, we evaluate positive patterns against the stock real data set while varying the number of events per window. \n\n\\textit{\\textbf{Flink}} does not terminate within several hours if the number of events exceeds 100k because Flink is a two-step approach that evaluates a set of event sequence queries for each Kleene query. Both the unnecessary event sequence construction and the increased query workload degrade the performance of Flink. For 100k events per window, Flink requires 82 minutes to terminate, while its memory requirement for storing all event sequences is close to 1GB. Thus, Flink is neither real time nor lightweight.\n\n\\textit{\\textbf{SASE}}. The latency of SASE grows exponentially in the number of events until it fails to terminate for more than 500k events. Its throughput degrades exponentially. Delayed responsiveness of SASE is explained by the DFS-based stack traversal which re-computes each sub-trend $tr$ for each longer trend containing $tr$. The memory requirement of SASE exceeds the memory consumption of {\\small GRETA}\\ 50--fold because DFS stores the trend that is currently being constructed. Since the length of a trend is unbounded, the peak memory consumption of SASE is significant.\n\n\\textit{\\textbf{CET}}. Similarly to SASE, the latency of CET grows exponentially in the number of events until it fails to terminate for more than 700k events. Its throughput degrades exponentially until it becomes negligible for over 500k events. In contrast to SASE, CET utilizes the available memory to store and reuse common sub-trends instead of recomputing them. To achieve almost double speed-up compared to SASE, CET requires 3 orders of magnitude more memory than SASE for 500k events.\n\n\\textit{\\textbf{{\\small GRETA}}} consistently outperforms all above two-step approaches regarding all three metrics because it does not waste computational resources to construct and store exponentially many event trends. Instead, {\\small GRETA}\\ incrementally computes event trend aggregation. Thus, it achieves 4 orders of magnitude speed-up compared to all above approaches. \n{\\small GRETA}\\ also requires 4 orders of magnitude less memory than Flink and CET since these approaches store event trends. The memory requirement of {\\small GRETA}\\ is comparable to SASE because SASE stores only one trend at a time. Nevertheless, {\\small GRETA}\\ requires 50--fold less memory than SASE for 500k events.\n\n\\textbf{Patterns with Negative Sub-Patterns}.\nIn Figure~\\ref{fig:exp_negative}, we evaluate the same patterns as in Figure~\\ref{fig:exp_positive} but with negative sub-patterns against the stock real data set while varying the number of events. \nCompared to Figure~\\ref{fig:exp_positive}, the latency and memory consumption of all approaches except Flink significantly decreased, while their throughput increased. \nNegative sub-patterns have no significant effect on the performance of Flink because Flink evaluates multiple event sequence queries instead of one Kleene query and constructs all matched event sequences. \nIn contrast, negation reduces the {\\small GRETA}\\ graph, the CET graph, and the SASE stacks \\textit{before} event trends are constructed and aggregated based on these data structures. Thus, both CPU and memory costs reduce. Despite this reduction, SASE and CET fail to terminate for over 700k events.\n\n\\begin{figure*}[t]\n\t\\centering\n \\subfigure[Latency]{\n \t\\includegraphics[width=0.25\\columnwidth]{experiments-sources\/predicates\/latency\/predicates-latency.png}\n \t \\label{fig:predicates-latency}\n\t}\n\t\\hspace*{5mm}\n\t\\subfigure[Memory]{\n \t\\includegraphics[width=0.25\\columnwidth]{experiments-sources\/predicates\/memory\/predicates-memory.png}\n \t\\label{fig:predicates-memory}\n\t}\n\t\\hspace*{5mm}\n\t\\subfigure[Throughput]{\n \t\\includegraphics[width=0.25\\columnwidth]{experiments-sources\/predicates\/throughput\/predicates-throughput.png}\n \t \\label{fig:predicates-throughput}\n\t}\n\t\\vspace{-3mm}\n\t\\caption{Selectivity of edge predicates (Linear Road benchmark data set)}\n\t\\label{fig:exp_predicates}\n\\end{figure*}\n\\begin{figure*}[t]\n\t\\centering\n \\subfigure[Latency]{\n \t\\includegraphics[width=0.25\\columnwidth]{experiments-sources\/grouping\/latency\/grouping-latency.png}\n \t \\label{fig:grouping-latency}\n\t}\n\t\\hspace*{5mm}\n\t\\subfigure[Memory]{\n \t\\includegraphics[width=0.25\\columnwidth]{experiments-sources\/grouping\/memory\/grouping-memory.png}\n \t\\label{fig:grouping-memory}\n\t}\n\t\\hspace*{5mm}\n\t\\subfigure[Throughput]{\n \t\\includegraphics[width=0.25\\columnwidth]{experiments-sources\/grouping\/throughput\/grouping-throughput.png}\n \t \\label{fig:grouping-throughput}\n\t}\n\t\\vspace{-3mm}\n\t\\caption{Number of event trend groups (Cluster monitoring data set)}\n\t\\label{fig:exp_grouping}\n\\end{figure*}\n\n\\subsection{Selectivity of Edge Predicates} \n\\label{exp:predicate}\n\nIn Figure~\\ref{fig:exp_predicates}, we evaluate positive patterns against the Linear Road benchmark data set while varying the selectivity of edge predicates. We focus on the selectivity of edge predicates because vertex predicates determine the number of trend groups (Section~\\ref{sec:filtering}) that is varied in Section~\\ref{exp:grouping}. To ensure that the two-step approaches terminate in most cases, we set the number of events per window to 100k.\n\nThe latency of Flink, SASE, and CET grows exponentially with the increasing predicate selectivity until they fail to terminate when the predicate selectivity exceeds 50\\%. In contrast, the performance of {\\small GRETA}\\ remains fairly stable regardless of the predicate selectivity. {\\small GRETA}\\ achieves 2 orders of magnitude speed-up and throughput improvement compared to CET for 50\\% predicate selectivity. \n\nThe memory requirement of Flink and CET grows exponentially (these lines coincide in Figure~\\ref{fig:predicates-memory}). The memory requirement of SASE remains fairly stable but almost 22--fold higher than for {\\small GRETA}\\ for 50\\% predicate selectivity. \n\n\\subsection{Number of Event Trend Groups} \n\\label{exp:grouping}\n\nIn Figure~\\ref{fig:exp_grouping}, we evaluate positive patterns against the cluster monitoring data set while varying the number of trend groups. The number of events per window is 100k. \n\nThe latency and memory consumption of Flink, SASE, and CET decrease exponentially with the increasing number of event trend groups, while their throughput increases exponentially. Since trends are constructed per group, their number and length decrease with the growing number of groups. Thus, both CPU and memory costs reduce.\nIn contrast, {\\small GRETA}\\ performs equally well independently from the number of groups since event trends are never constructed. Thus, {\\small GRETA}\\ achieves 4 orders of magnitude speed-up compared to Flink for 10 groups and 2 orders of magnitude speed-up compared to CET and SASE for 5 groups.\n\n\n\n\n\n\n\n\\section{Other Language Clauses}\n\\label{sec:filtering}\n\nWe now expand our {\\small GRETA}\\ approach to handle sliding windows, predicates, and grouping. \n\n\\begin{figure*}[t]\n\\begin{minipage}{0.78\\textwidth}\n\\centering\n\\subfigure[\\small {\\small GRETA}\\ sub-graph replication]{\n\t\\includegraphics[width=0.58\\columnwidth]{figures\/window-baseline.png}\n\t\\label{fig:window-baseline}\t\n}\n\\subfigure[\\small {\\small GRETA}\\ sub-graph sharing]{\n\t\\includegraphics[width=0.33\\columnwidth]{figures\/window.png}\n\t\\hspace*{0.2cm}\n\t\\label{fig:window}\t\n}\n\\vspace{-3mm}\n\\caption{Sliding window \\textsf{WITHIN} 10 seconds \\textsf{SLIDE} 3 seconds}\n\\label{fig:rest}\n\\end{minipage}\n\\begin{minipage}{0.2\\textwidth}\n\\centering\n\t\\includegraphics[width=0.92\\columnwidth]{figures\/predicates.png}\n\t\\caption{Edge predicate $A.attr<\\textsf{NEXT}($ $A).attr$}\n\t\\label{fig:predicates}\n\\end{minipage}\n\\end{figure*}\n\n\\textbf{Sliding Windows}.\nDue to continuous nature of streaming, an event may contribute to the aggregation results in several overlapping windows. Furthermore, events may expire in some windows but remain valid in other windows. \n\n$\\bullet$ \\textbf{\\textit{{\\small GRETA}\\ Sub-Graph Replication}}.\nA naive solution would build a {\\small GRETA}\\ graph for each window independently from other windows. Thus, it would replicate an event $e$ across all windows that $e$ falls into. Worse yet, this solution introduces repeated computations, since an event $p$ may be predecessor event of $e$ in multiple windows. \n\n\\vspace{-2mm}\n\\begin{example}\nIn Figure~\\ref{fig:window-baseline}, we count the number of trends matched by the pattern $(\\textsf{SEQ}(A+,B))+$ within a 10-seconds-long window that slides every 3 seconds. The events $a1$--$b9$ fall into window $W_1$, while the events $a4$--$b9$ fall into window $W_2$. If a {\\small GRETA}\\ graph is constructed for each window, the events $a4$--$b9$ are replicated in both windows and their predecessor events are recomputed for each window.\n\\label{ex:window-baseline}\n\\end{example}\n\\vspace{-2mm}\n\n$\\bullet$ \\textbf{\\textit{{\\small GRETA}\\ Sub-Graph Sharing}}.\nTo avoid these drawbacks, we share a sub-graph $G$ across all windows to which $G$ belongs. Let $e$ be an event that falls into $k$ windows. The event $e$ is stored once and its predecessor events are computed once across all $k$ windows. The event $e$ maintains a count fro each window. To differentiate between $k$ counts maintained by $e$, each window is assigned an identifier $wid$~\\cite{LMTPT05-2}. The count with identifier $wid$ of $e$ ($e.count_{wid}$) is computed based on the counts with identifier $wid$ of $e$'s predecessor events (Line~10 in Algorithm~\\ref{lst:ETA_algorithm}). The final count for a window $wid$ ($final\\_count_{wid}$) is computed based on the counts with identifier $wid$ of the \\textsf{END} events in the graph (Line~12). \nIn Example~\\ref{ex:window-baseline}, the events $a4$--$b9$ fall into two windows and thus maintain two counts in Figure~\\ref{fig:window}. The first count is for $W_1$, the second one for $W_2$. \n\n\\textbf{Predicates} on vertices and edges of the {\\small GRETA}\\ graph are handled differently by the {\\small GRETA}\\ runtime. \n\n$\\bullet$ \\textbf{\\textit{Vertex Predicates}} restrict the vertices in the {\\small GRETA}\\ graph. They are evaluated on single events to either filter or partition the stream~\\cite{QCRR14}. \n\n\\textit{Local predicates} restrict the attribute values of events, for example, \\textit{companyID=IBM}. They purge irrelevant events early. We associate each local predicate with its respective state in the \n{\\small GRETA}\\ template.\n\n\\textit{Equivalence predicates} require all events in a trend to have the same attribute values, for example, \\textit{[company, sector]} in query $Q_1$. They partition the stream by these attribute values. Thereafter, {\\small GRETA}\\ queries are evaluated against each sub-stream in a divide and conquer fashion. \n\n$\\bullet$ \\textbf{\\textit{Edge Predicates}} restrict the edges in the graph (Line~4 of Algorithm~\\ref{lst:ETA_algorithm}). Events connected by an edge must satisfy these predicates. Therefore, edge predicates are evaluated during graph construction. We associate each edge predicate with its respective transition in the\n{\\small GRETA}\\ template.\n\n\\vspace{-2mm}\n\\begin{example}\nThe edge predicate $A.attr < \\textsf{NEXT}(A).attr$ in Figure~\\ref{fig:predicates} requires the value of attribute \\textit{attr} of events of type $A$ to increase from one event to the next in a trend. The attribute value is shown in the bottom left corner of a vertex. Only two dotted edges satisfy this predicate. \n\\label{ex:predicates}\n\\end{example}\n\\vspace{-2mm}\n\n\\textbf{Event Trend Grouping}.\nAs illustrated by our motivating examples in Section~\\ref{sec:introduction}, event trend aggregation often requires event trend grouping. Analogously to A-Seq~\\cite{QCRR14}, our {\\small GRETA}\\ runtime first partitions the event stream into sub-streams by the values of grouping attributes. A {\\small GRETA}\\ graph is then maintained separately for each sub-stream. Final aggregates are output per sub-stream.\n\n\n\\section{GRETA Framework}\n\\label{sec:implementation}\n\nPutting Setions~\\ref{sec:positive}--\\ref{sec:filtering} together, we now describe the {\\small GRETA}\\ runtime data structures and parallel processing.\n\n\\textbf{Data Structure for a Single {\\small GRETA}\\ Graph}.\nEdges logically capture the paths for aggregation propagation in the graph. Each edge is traversed \\textit{exactly once} to compute the aggregate of the event to which the edge connects (Lines~8--10 in Algorithm~\\ref{lst:ETA_algorithm}). Hence, edges are not stored. \n\nVertices must be stored in such a way that the predecessor events of a new event can be efficiently determined (Line~4). To this end, we leverage the following data structures.\nTo quickly locate \\textit{previous} events, we divide the stream into non-overlapping consecutive time intervals, called \\textit{\\textbf{Time Pa\\-nes}}~\\cite{LMTPT05}. Each pane contains all vertices that fall into it based on their time stamps. These panes are stored in a time-stamped array in increasing order by time (Figure~\\ref{fig:data-structure}). The size of a pane depends on the window specifications and stream rate such that each query window is composed of several panes -- allowing panes to be shared between overlapping windows~\\cite{AW04, LMTPT05}.\nTo efficiently find vertices of \\textit{predecessor event types}, each pane contains an \\textit{\\textbf{Event Type Hash Table}} that maps event types to vertices of this type. \n\nTo support \\textit{edge predicates}, we utilize a tree index that enables efficient range queries. The overhead of maintaining \\textit{\\textbf{Vertex Trees}} is reduced by event sorting and a pane purge mechanism. \nAn event is inserted into the Vertex Tree for its respective pane and event type. This sorting by time and event type reduces the number of events in each tree.\nFurthermore, instead of removing single expired events from the Vertex Trees, a whole pane with its associated data structures is deleted after the pane has contributed to all windows to which it belongs. \nTo support \\textit{sliding windows}, each vertex $e$ maintains a \\textit{\\textbf{Window Hash Table}} storing an aggregate per window that $e$ falls into.\nSimilarly, we store final aggregates per window in the \\textit{\\textbf{Results Hash Table}}.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.6\\columnwidth]{figures\/data-structure.png} \n\\vspace{-1mm} \n\\caption{Data structure for a single {\\small GRETA}\\ graph}\n\\label{fig:data-structure}\n\\end{figure}\n\n\\textbf{Data Structure for {\\small GRETA}\\ Graph Dependencies}. \nTo support negative sub-patterns, we maintain a \\textbf{\\textit{Graph Dependencies Hash Table}} that maps a graph identifier $G$ to the identifiers of graphs upon which $G$ depends. \n\n\\textbf{Parallel Processing}.\nThe grouping clause partitions the stream into sub-streams that are processed in parallel \\textit{independently} from each other. Such stream partitioning enables a highly scalable execution as demonstrated in Section~\\ref{exp:grouping}. \n\nIn contrast, negative sub-patterns require concurrent maintenance of \\textit{inter-dependent} {\\small GRETA}\\ graphs. To avoid race conditions, we deploy the time-based transaction model~\\cite{sstore}. \nA \\textit{stream transaction} is a sequence of operations triggered by all events with the same time stamp on the same {\\small GRETA}\\ graph. The application time stamp of a transaction (and all its operations) coincides with the application time stamp of the triggering events. \nFor each time stamp $t$ and each {\\small GRETA}\\ graph $G$, our time-driven scheduler waits till the processing of all transactions with time stamps smaller than $t$ on the graph $G$ and other graphs that $G$ depends upon is completed. Then, the scheduler extracts all events with the time stamp $t$, wraps their processing into transactions, and submits them for execution.\n\n\n\\section{Introduction}\n\\label{sec:introduction}\n\nComplex Event Processing (CEP) is a technology for supporting streaming applications from algorithmic trading to traffic management. CEP systems continuously evaluate event queries against high-rate streams composed of primitive events to detect event trends such as stock market down-trends and aggressive driving. In contrast to traditional event sequences of \\textit{fixed} length~\\cite{LRGGWAM11}, event trends have \\textit{arbitrary} length~\\cite{PLAR17}. They are expressed by Kleene closure. \nAggregation functions are applied to these trends to provide valuable summarized insights about the current situation. CEP applications typically must react to critical changes of these aggregates in real time~\\cite{ADGI08, WDR06, ZDI14}.\n\n\\textbf{Motivating Examples}.\nWe now describe three application scenarios of time-critical event trend aggregation.\n\n$\\bullet$ \\textit{\\textbf{Algorithmic Trading}}.\nStock market analytics platforms evaluate expressive event queries against high-rate streams of financial transactions. They deploy event trend aggregation to identify and then exploit profit opportunities in real time. For example, query $Q_1$ computes the \\textit{count} of down-trends per industrial sector. Since stock trends of companies that belong to the same sector tend to move as a group~\\cite{K02}, the number of down-trends across different companies in the same sector is a strong indicator of an upcoming down trend for the sector. When this indicator exceeds a certain threshold, a sell signal is triggered for the whole sector including companies without down-trends. These aggregation-based insights must be available to an algorithmic trading system in \\textit{near real time} to exploit short-term profit opportunities or avoid pitfalls.\n\nQuery $Q_1$ computes the number of down-trends per sector during a time window of 10 minutes that slides every 10 seconds. These stock trends are expressed by the \nKleene plus operator $S+$. All events in a trend carry the same company and sector identifier as required by the predicate $[company,sector]$. The predicate $S.price>\\textsf{NEXT}(S).price$ expresses that the price continually decreases from one event to the next in a trend. \nThe query ignores local price fluctuations by skipping over increasing price records. \n\n\n\\vspace*{-1mm}\n\\begin{lstlisting}[]\n$Q_1$: RETURN$\\;\\text{sector},\\;\\textsf{COUNT}(*)\\ $PATTERN$\\;\\text{Stock}\\ S+$\n WHERE$\\;\\text{[company,sector]}\\;$AND$\\;S.\\text{price}>\\textsf{NEXT}(S).\\text{price}$ \n GROUP$\\text{-}$BY$\\;\\text{sector}\\;$WITHIN$\\;\\text{10 minutes}\\;$SLIDE$\\;\\text{10 seconds}$\n\\end{lstlisting}\n\\vspace*{-1mm}\n\n$\\bullet$ \\textit{\\textbf{Hadoop Cluster Monitoring}}.\nModern computer cluster monitoring tools gather system measurements regarding CPU and memory utilization at runtime. These measurements combined with workflow-specific logs (such as start, progress, and end of Hadoop jobs) form load distribution trends per job over time. These load trends are aggregated to dynamically detect and then tackle cluster bottlenecks, unbalanced load distributions, and data queuing issues~\\cite{ZDI14}. For example, when a mapper experiences increasing load trends on a cluster, we might measure the \\textit{total CPU cycles} per job of such a mapper. These aggregated measurements over load distribution trends are leveraged in \\textit{near real time} to enable automatic tuning of cluster performance.\n\nQuery $Q_2$ computes the total CPU cycles per job of each mapper experiencing increasing load trends on a cluster during a time window of 1 minute that slides every 30 seconds. A trend matched by the pattern of $Q_2$ is a sequence of a job-start event $S$, any number of mapper performance measurements $M+$, and a job-end event $E$. All events in a trend must carry the same job and mapper identifiers expressed by the predicate $[job, mapper]$. The predicate M.load $<$ \\textsf{NEXT}(M).load requires the load measurements to increase from one event to the next in a load distribution trend. \nThe query may ignore any event to detect all load trends of interest for accurate cluster monitoring.\n\n\n\\vspace*{-1mm}\n\\begin{lstlisting}[]\n$Q_2:\\ \\textsf{RETURN}\\ mapper,\\ \\textsf{SUM}(M.cpu)$\n $\\textsf{PATTERN SEQ}(Start\\ S,\\ Measurement\\ M+,\\ End\\ E)$\n $\\textsf{WHERE}\\ [job,mapper]\\ \\textsf{AND}\\ M.load<\\textsf{NEXT}(M).load$\n $\\textsf{GROUP-BY}\\ mapper\\ \\textsf{WITHIN}\\ 1\\ minute\\ \\textsf{SLIDE}\\ 30\\ seconds$\n\\end{lstlisting}\n\\vspace*{-1mm}\n\n$\\bullet$ \\textbf{\\textit{Traffic Management}} is based on the insights gained during continuous traffic monitoring. For example, leveraging the \\textit{maximal} speed per vehicle that follows certain trajectories on a road, a traffic control system recognizes congestion, speeding, and aggressive driving. Based on this knowledge, the system predicts the traffic flow and computes fast and safe routes in real time to reduce travel time, costs, noise, and environmental pollution.\n\nQuery $Q_3$ detects traffic jams which are not caused by accidents. To this end, the query computes the number and the average speed of cars continually slowing down in a road segment without accidents during 5 minutes time window that slides every minute. A trend matched by $Q_3$ is a sequence of any number of position reports $P+$ without an accident event $A$ preceding them. All events in a trend must carry the same vehicle and road segment identifiers expressed by the predicate $[P.vehicle, segment]$. The speed of each car decreases from one position report to the next in a trend, expressed by the predicate $P.\\text{speed}>\\textsf{NEXT}(P).\\text{speed}$. The query may skip any event to detect all relevant car trajectories for precise traffic statistics.\n\n\\vspace*{-1mm}\n\\begin{lstlisting}[]\n$Q_3$: RETURN $\\text{segment},\\ \\textsf{COUNT}(*),\\ \\textsf{AVG}(P.\\text{speed})$\n PATTERN SEQ(NOT $\\text{Accident A, Position P+)}$\n WHERE $\\text{[\\textit{P}.vehicle,segment]}$ AND $P.\\text{speed}>\\textsf{NEXT}(P).\\text{speed}$\n GROUP$\\text{-}$BY $\\text{segment}$ WITHIN $\\text{5 minutes}$ SLIDE $\\text{1 minute}$ \n\\end{lstlisting}\n\\vspace*{-1mm}\n \n\\textbf{State-of-the-Art Systems} do not support incremental aggregation of event trends. They can be divided into:\n\n$\\bullet$ \\textit{\\textbf{CEP Approaches}} including SASE~\\cite{ADGI08,ZDI14}, Cayuga~\\cite{DGPRSW07}, and ZStream~\\cite{MM09} support Kleene closure to express event trends. While their query languages support aggregation, these approaches do not provide any details on how they handle aggregation on top of nested Kleene patterns. Given no special optimization techniques, these approaches construct all trends prior to their aggregation (Figure~\\ref{fig:overview}). These two-step approaches suffer from high computation costs caused by the exponential number of arbitrarily-long trends. Our experiments in Section~\\ref{sec:evaluation} confirm that such two-step approaches take over two hours to compute event trend aggregation even for moderate stream rates of 500k events per window. Thus, they fail to meet the low-latency requirement of time-critical applications. \nA-Seq~\\cite{QCRR14} is the only system we are aware of that targets incremental aggregation of event sequences. However, it is restricted to the simple case of \\textit{fixed-length} sequences such as \\textsf{SEQ}$(A,B,C)$. It supports neither Kleene closure nor expressive predicates. Therefore, A-Seq does not tackle the exponential complexity of event trends -- which now is the focus of our work.\n\n$\\bullet$ \\textit{\\textbf{Streaming Systems}} support aggregation computation over streams~\\cite{AW04, GHMAE07, KWF06, LMTPT05, THSW15}. However, these approaches evaluate simple Select-Project-Join queries with windows, i.e., their execution paradigm is set-based. They support neither event sequence nor Kleene closure as query operators. Typically, these approaches require the construction of join-results prior to their aggregation. Thus, they define incremental aggregation of \\textit{single raw events} but focus on multi-query optimization techniques~\\cite{KWF06} and sharing aggregation computation between sliding windows~\\cite{LMTPT05}.\n\n\\textbf{Challenges}. We tackle the following open problems:\n\n$\\bullet$ \\textbf{\\textit{Correct Online Event Trend Aggregation}}. \nKleene closure matches an exponential number of arbitrarily-long event trends in the number of events in the worst case~\\cite{ZDI14}. Thus, any practical solution must aim to aggregate event trends without first constructing them to enable real-time in-memory execution. At the same time, correctness must be guaranteed. That is, the same aggregation results must be returned as by the two-step approach.\n\n$\\bullet$ \\textbf{\\textit{Nested Kleene Patterns}}.\nKleene closure detects event trends of arbitrary, statically unknown length. Worse yet, Kleene closure, event sequence, and negation may be arbitra\\-rily-nested in an event pattern, introducing complex inter-dependencies between events in an event trend. Incremental aggregation of such arbitrarily-long and complex event trends is an open problem.\n\n$\\bullet$ \\textbf{\\textit{Expressive Event Trend Filtering}}. \nExpressive predicates may determine event relevance depending on other events in a trend. Since a new event may have to be compared to \\textit{any} previously matched event, all events must be kept. The need to store all matched events conflicts with the instantaneous aggregation requirement. \nFurthermore, due to the continuous nature of streaming, events expire over time -- triggering an update of all affected aggregates. However, recomputing aggregates for each expired event would put \\textit{real-time} system responsiveness at risk.\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.7\\columnwidth]{figures\/overview.png} \n\\vspace{-2mm} \n\\caption{State-of-the-art versus our {\\small GRETA}\\ approach}\n\\label{fig:overview}\n\\end{figure} \n\n\\textbf{Our Proposed {\\small GRETA}\\ Approach}.\nWe are the first to tackle these challenges in our Graph-based Real-time Event Trend Aggregation ({\\small GRETA}) approach (Figure~\\ref{fig:overview}). Given an event trend aggregation query $q$ and a stream $I$, the {\\small GRETA}\\ runtime compactly encodes all event trends matched by the query $q$ in the stream $I$ into a {\\small GRETA}\\ graph. During graph construction, aggregates are propagated from previous events to newly arrived events along the edges of the graph following the dynamic programming principle. This propagation is proven to assure incremental aggregation computation without first constructing the trends. The final aggregate is also computed incrementally such that it can be instantaneously returned at the end of each window of $q$.\n\n\\textbf{Contributions}. \nOur key innovations include:\n\n1)~We translate a nested Kleene pattern $P$ into a \n{\\small GRETA}\\ template. \nBased on this template, we construct the \\textit{{\\small GRETA}\\ graph} that compactly captures all trends matched by pattern $P$ in the stream. During graph construction, the aggregates are dynamically propagated along the edges of the graph. We prove the correctness of the {\\small GRETA}\\ graph and the graph-based aggregation computation.\n\n2)~To handle nested patterns with negative sub-patterns, we split the pattern into positive and negative sub-patterns. We maintain a separate {\\small GRETA}\\ graph for each resulting sub-pattern and invalidate certain events if a match of a negative sub-pattern is found. \n\n3)~To avoid sub-graph replication between overlapping sliding windows, we share one {\\small GRETA}\\ graph between all windows. Each event that falls into $k$ windows maintains $k$ aggregates. Final aggregate is computed per window.\n\n4)~To ensure low-latency lightweight query processing, we design the \\textit{{\\small GRETA}\\ runtime data structure} to support dynamic insertion of newly arriving events, batch-deletion of expired events, incremental propagation of aggregates, and efficient evaluation of expressive predicates.\n\n5)~We prove that our {\\small GRETA}\\ approach reduces the time complexity from exponential to quadratic in the number of events compared to the two-step approach and in fact achieves \\textit{optimal time complexity}. We also prove that the space complexity is reduced from exponential to linear. \n\n6)~Our experiments using synthetic and real data sets demonstrates that {\\small GRETA}\\ achieves up to four orders of magnitude speed-up and consumes up to 50--fold less memory compared to the state-of-the-art strategies~\\cite{flink, PLAR17, ZDI14}.\n\n\\textbf{Outline}. \nWe start with preliminaries in Section~\\ref{sec:model}.\nWe overview our {\\small GRETA}\\ approach in Section~\\ref{sec:overview}. \nSection~\\ref{sec:positive} covers positive patterns,\nwhile negation is tackled in Section~\\ref{sec:negative}.\nWe consider other language clauses in Section~\\ref{sec:filtering}. \nWe describe our data structure in Section~\\ref{sec:implementation} and analyze complexity in Section~\\ref{sec:complexity}. \nSection~\\ref{sec:discussion} discusses how our {\\small GRETA}\\ approach can support additional language features.\nSection~\\ref{sec:evaluation} describes the experiments. \nRelated work is discussed in Section~\\ref{sec:related_work}. \nSection~\\ref{sec:conclusions} concludes the paper. \n\\section{GRETA Data and Query Model}\n\\label{sec:model}\n\n\\textbf{Time}.\nTime is represented by a linearly ordered \\textit{set of time points} $(\\mathbb{T},\\leq)$, where $\\mathbb{T} \\subseteq \\mathbb{Q^+}$ and $\\mathbb{Q^+}$ denotes the set of non-negative rational numbers. \n\n\\textbf{Event Stream}.\nAn \\textit{event} is a message indicating that something of interest happens in the real world. An event $e$ has an \\textit{occurrence time} $e.time \\in \\mathbb{T}$ assigned by the event source. For simplicity, we assume that events arrive in-order by time stamps. Otherwise, an existing approach to handle out-of-order events can be employed~\\cite{LTSPJM08, LLGRC09}.\n\nAn event $e$ belongs to a particular \\textit{event type} $E$, denoted $e.type=E$ and described by a \\textit{schema} which specifies the set of \\textit{event attributes} and the domains of their values. \n\nEvents are sent by event producers (e.g., brokers) on an \\textit{event stream I}. An event consumer (e.g., algorithmic trading system) monitors the stream with \\textit{event queries}. We borrow the query syntax and semantics from SASE~\\cite{ADGI08, ZDI14}. \n\n\\vspace*{-2mm}\n\\begin{definition}(\\textbf{Kleene Pattern}.) \nLet $I$ be an event stream. A \\textbf{\\textit{pattern}} $P$ is recursively defined as follows:\n\n$\\bullet$ An \\textit{\\textbf{event type}} $E$ matches an event $e \\in I$, denoted $e \\in matches(E)$, if $e.type = E$.\n\n$\\bullet$ An \\textit{\\textbf{event sequence operator}} \\textsf{SEQ}$(P_i,P_j)$ matches an \\textbf{\\textit{event sequence}} $s=(e_1,\\dots,e_k)$,\ndenoted $s \\in matches(\\textsf{SEQ}($ $P_i,P_j))$,\nif $\\exists m \\in \\mathbb{N}$, $1 \\leq m \\leq k$, such that\n$(e_1,\\dots,e_m) \\in matches (P_i)$,\n$(e_{m+1},\\dots,e_k) \\in matches (P_j)$, and\n$\\forall l \\in \\mathbb{N},$ $1 \\leq l < k,$ $e_l.time < e_{l+1}.time$.\nTwo events $e_l$ and $e_{l+1}$ are called \\textit{\\textbf{adjacent}} in the sequence $s$.\nFor an event sequence $s$, we define $s.start = e_1$ and $s.end = e_k$.\n\n$\\bullet$ A \\textit{\\textbf{Kleene plus operator}} $P_i+$ matches an \\textit{\\textbf{event trend}} $tr=(s_1,\\dots,s_k)$,\ndenoted $tr \\in matches(P_i+))$,\nif \n$\\forall l \\in \\mathbb{N},$ $1 \\leq l \\leq k,$ \n$s_l \\in matches(P_i)$ and \n$s_l.end.time < s_{l+1}.start.$ $time$.\nTwo events $s_l.end$ and $s_{l+1}.start$ are called \\textit{\\textbf{adjacent}} in the trend $tr$.\nFor an event trend $tr$, we define $tr.start = s_1.start$ and $tr.end = s_k.end$.\n\n$\\bullet$ A \\textit{\\textbf{negation operator}} \\textsf{NOT} $N$ appears within an event sequence operator \\textsf{SEQ}$(P_i, \\textsf{NOT}\\; N,$ $P_j)$ (see below).\nThe pattern \\textsf{SEQ}$(P_i,\\textsf{NOT}\\; N, P_j)$ matches an \\textbf{\\textit{event sequence}} $s=(s_i,s_j)$,\ndenoted $s \\in matches(\\textsf{SEQ}(P_i, \\textsf{NOT}\\; N, P_j))$,\nif \n$s_i \\in matches(P_i)$,\n$s_j \\in matches(P_j)$, and\n$\\nexists s_n \\in matches(N)$ such that\n$s_i.end.time < s_n.start.time$ and \n$s_n.end.time < s_j.start.time$.\nTwo events $s_i.end$ and $s_j.start$ are called \\textit{\\textbf{adjacent}} in the sequence $s$.\n\nA \\textit{\\textbf{Kleene pattern}} is a pattern with at least one Kleene plus operator.\nA pattern is \\textit{\\textbf{positive}} if it contains no negation.\nIf an operator in $P$ is applied to the result of another operator, $P$ is \\textit{\\textbf{nested}}. Otherwise, $P$ is \\textit{\\textbf{flat}}. The \\textit{\\textbf{size}} of $P$ is the number of event types and operators in it. \n\\label{def:pattern}\n\\end{definition}\n\\vspace{-2mm}\n\nAll queries in Section~\\ref{sec:introduction} have Kleene patterns. The patterns of $Q_1$ and $Q_2$ are positive. The pattern of $Q_3$ contains a negative sub-pattern \\textsf{NOT} \\textit{Accident A}. The pattern of $Q_1$ is flat, while the patterns of $Q_2$ and $Q_3$ are nested. \n\nWhile Definition~\\ref{def:pattern} enables construction of arbitrarily-nest\\-ed patterns, nesting a Kleene plus into a negation and vice versa is not useful. Indeed, the patterns \\textsf{NOT} $(P+)$ and (\\textsf{NOT} $P)+$ are both equivalent to \\textsf{NOT} $P$. Thus, we assume that a negation operator appears within an event sequence operator and is applied to an event sequence operator or an event type. \nFurthermore, an event sequence operator applied to consecutive negative sub-patterns \\textsf{SEQ}(\\textsf{NOT} $P_i$, \\textsf{NOT} $P_j$) is equivalent to the pattern \\textsf{NOT SEQ}($P_i, P_j$). Thus, we assume that only a positive sub-pattern may precede and follow a negative sub-pattern. Lastly, negation may not be the outer most operator in a meaningful pattern. \nFor simplicity, we assume that an event type appears at most once in a pattern. In Section~\\ref{sec:discussion}, we describe a straightforward extension of our {\\small GRETA}\\ approach allowing to drop this assumption.\n\n\\begin{figure}[t]\n\\[\n\\begin{array}{lll}\nq & := & $\\textsf{\\small RETURN }$ Attributes\\ \\langle A \\rangle\\ $\\textsf{\\small PATTERN}$\\ \\langle P \\rangle\\\\\n&& $(\\textsf{\\small WHERE}$\\ \\langle \\theta \\rangle)?\\\n$(\\textsf{\\small GROUP-BY}$\\ Attributes)?\\\\\n&& $\\textsf{\\small WITHIN }$ Duration\\ $\\textsf{\\small SLIDE }$ Duration \\\\ \n\nA & := & $\\textsf{\\small COUNT}$(* | EventType)\\ |\\\\\n&& ($\\textsf{\\small MIN}$ | $\\textsf{\\small MAX}$ | $\\textsf{\\small SUM}$ | $\\textsf{\\small AVG}$)\n(EventType.Attribute)\\\\\nP & := & EventType\\ |\\ \\langle P \\rangle $\\textsf{+}$\\ |\\ $\\textsf{\\small NOT}$ \\langle P \\rangle\\ |\\ $\\textsf{\\small SEQ}$ ( \\langle P \\rangle , \\langle P \\rangle ) \\\\\n\\theta & := & Constant\\ |\\ EventType . Attribute\\ |\\\\\n&& $\\textsf{\\small NEXT}($EventType$)$ . Attribute\\ |\\\n\\langle \\theta \\rangle\\ \\langle O \\rangle\\ \\langle \\theta \\rangle \\\\\nO & := & +|-|\/|*|\\%|=|\\neq|>|\\geq|<|\\leq|\\wedge|\\vee \\\\\n\\end{array}\n\\]\n\\vspace{-4mm}\n\\caption{Language grammar}\n\\label{fig:language_grammar}\n\\end{figure}\n\n\\begin{definition}(\\textbf{Event Trend Aggregation Query}.) \nAn \\textit{\\textbf{event trend aggregation query}} $q$ consists of five clauses:\n\n$\\bullet$ Aggregation result specification (\\textsf{RETURN} clause),\n\n$\\bullet$ Kleene pattern $P$ (\\textsf{PATTERN} clause),\n\n$\\bullet$ Predicates $\\theta$ (optional \\textsf{WHERE} clause),\n\n$\\bullet$ Grouping $G$ (optional \\textsf{GROUP-BY} clause), and\n\n$\\bullet$ Window $w$ (\\textsf{WITHIN\/SLIDE} clause).\n\nThe query $q$ requires each event in a trend matched by its pattern $P$ (Definition~\\ref{def:pattern}) to be within the same window $w$, satisfy the predicates $\\theta$, and carry the same values of the grouping attributes $G$. \nThese trends are grouped by the values of $G$. An aggregate is computed per group. We focus on distributive (such as \\textsf{COUNT, MIN, MAX, SUM}) and algebraic aggregation functions (such as \\textsf{AVG}) since they can be computed incrementally~\\cite{Gray97}. \n\nLet $e$ be an event of type $E$ and $attr$ be an attribute of $e$.\n\\textsf{COUNT}$(*)$ returns the number of all trends per group, while\n\\textsf{COUNT}$(E)$ computes the number of all events $e$ in all trends per group.\n\\textsf{MIN}$(E.attr)$ (\\textsf{MAX}$(E.attr)$) computes the minimal (maximal) value of $attr$ for all events $e$ in all trends per group.\n\\textsf{SUM}$(E.attr)$ calculates the summation of the value of $attr$ of all events $e$ in all trends per group.\nLastly, \\textsf{AVG}$(E.attr)$ is computed as \\textsf{SUM}$(E.attr)$ divided by \\textsf{COUNT}$(E)$ per group.\n\\label{def:query}\n\\end{definition}\n\\vspace{-3mm}\n\n\\textbf{Skip-Till-Any-Match Semantics}.\nWe focus on Kleene patterns evaluated under the most flexible semantics, called \\textit{skip-till-any-match} in the literature~\\cite{ADGI08, WDR06, ZDI14}. Skip-till-any-match detects \\textit{all possible trends} by allowing to skip \\textit{any} event in the stream as follows. When an event $e$ arrives, it extends each existing trend $tr$ that can be extended by $e$. In addition, the unchanged trend $tr$ is kept to preserve opportunities for alternative matches. Thus, an event doubles the number of trends in the worst case and the number of trends grows exponentially in the number of events~\\cite{QCRR14, ZDI14}. While the number of all trends is exponential, an application selects a subset of trends of interest using predicates, windows, grouping, and negation (Definition~\\ref{def:query}).\n\nDetecting all trends is necessary in some applications such as algorithmic trading (Section~\\ref{sec:introduction}). For example, given the stream of price records $I=\\{10,2,9,8,7,1,6,5,4,3\\}$, skip-till-any-match is the only semantics that detects the down-trend $(10,9,8,7,6,5,4,3)$ by ignoring local fluctuations 2 and 1. Since longer stock trends are considered to be more reliable~\\cite{K02}, this long trend%\n\\footnote{We sketch how constraints on minimal trend length can be supported by {\\small GRETA}\\ in Section~\\ref{sec:discussion}.}\ncan be more valuable to the algorithmic trading system than three shorter trends $(10,2)$, $(9,8,7,1)$, and $(6,5,4,3)$ detected under the skip-till-next-match semantics that does not skip events that can be matched (Section~\\ref{sec:discussion}).\nOther use cases of skip-till-any-match include financial fraud, health care, logistics, network security, cluster monitoring, and e-commerce~\\cite{ADGI08, WDR06, ZDI14}. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.5\\columnwidth]{figures\/aggregates.png} \n\\vspace{-2mm} \n\\caption{Event trends matched by the pattern $P=(\\textsf{SEQ}(A+,$ $B))+$ in the stream $I=\\{a1,b2,a3,a4,b7\\}$ where a1.attr=5, a3.attr=6, and a4.attr=4}\n\\label{fig:aggregates}\n\\end{figure} \n\n\\vspace{-2mm}\n\\begin{example}\nIn Figure~\\ref{fig:aggregates}, the pattern $P$ detects \n\\textsf{COUNT}(*)=11 event trends in the stream $I$ with five events under the skip-till-any-match semantics. \nThere are \\textsf{COUNT}(A)=20 occurrences of $a$'s in these trends.\nThe minimal value of attribute $attr$ in these trends is \\textsf{MIN}(A.attr)=4, while the maximal value of $attr$ is \\textsf{MAX}(A.attr)=6. \\textsf{MAX}$(A.attr)$ is computed analogously to \\textsf{MIN}$(A.attr)$.\nThe summation of all values of $attr$ is all trends is \\textsf{SUM}(A.attr)=100.\nLastly, the average value of $attr$ in all trends is \\textsf{AVG}(A.attr)=\\textsf{SUM}( A.attr)\/\\textsf{COUNT}(A)=5.\n\\label{ex:aggregates}\n\\end{example}\n\\vspace{-2mm}\n\n\n\\section{Patterns with Nested Negation}\n\\label{sec:negative}\n\nTo handle nested patterns with negation, we split the pattern into positive and negative sub-patterns at compile time (Section~\\ref{sec:split}). At runtime, we then maintain the {\\small GRETA}\\ graph for each of these sub-patterns (Section~\\ref{sec:negative-algorithm}).\n\n\\subsection{Static GRETA Template}\n\\label{sec:split}\n\nAccording to Section~\\ref{sec:model}, negation appears within a sequence preceded and followed by positive sub-patterns.\nFurthermore, negation is always applied to an event sequence or a single event type. Thus, we classify patterns containing a negative sub-pattern $N$ into the following groups:\n\nCase 1.~\\textbf{\\textit{A negative sub-pattern is preceded and followed by positive sub-patterns}}. A pattern of the form $P_1=\\textsf{SEQ}(P_i,\\textsf{NOT} N,P_j)$ means that no trends of $N$ may occur between the trends of $P_i$ and $P_j$. A trend of $N$ disqualifies sub-trends of $P_i$ from contributing to a trend detected by $P_1$. A trend of $N$ marks all events in the graph of the \\textit{previous} event type $end(P_i)$ as \\textit{invalid} to connect to any future event of the \\textit{following} event type $start(P_j)$. Only valid events of type $end(P_i)$ connect to events of type $start(P_j)$.\n\n\\vspace{-2mm}\n\\begin{example}\nThe pattern \\textit{(\\textsf{SEQ}(A+,\\textsf{NOT SEQ}(C,\\textsf{NOT} E, D),B))+} is split into one positive sub-pattern $(\\textsf{SEQ}(A+,B))+$ and two negative sub-patterns $\\textsf{SEQ}(C,D)$ and \\textit{E}. Figure~\\ref{fig:dependencies} illustrates the previous and following connections between the template for the negative sub-pattern and the event types in the template for its parent pattern. \n\\label{ex:pattern_split}\n\\end{example}\n\\vspace{-2mm}\n\n\\begin{figure}[t]\n\\centering\n\\subfigure[\\small \\textit{(\\textsf{SEQ}(A+, \\textsf{NOT SEQ}(C, \\textsf{NOT} E, D), B))+}]{\n\\hspace*{2cm}\n\\includegraphics[width=0.205\\columnwidth]{figures\/dependencies.png} \n\\hspace*{2cm} \n\\label{fig:dependencies}\n}\n\\subfigure[\\small \\textit{\\textsf{SEQ}(A+, \\textsf{NOT} E)}]{\n\\hspace*{0.8cm}\n\\includegraphics[width=0.1\\columnwidth]{figures\/dependencies2.png} \n\\hspace*{0.8cm}\n\\label{fig:dependencies2}\n}\n\\subfigure[\\small \\textit{\\textsf{SEQ}(\\textsf{NOT} E, A+)}]{\n\\hspace*{0.8cm}\n\\includegraphics[width=0.1\\columnwidth]{figures\/dependencies3.png} \n\\hspace*{0.8cm}\n\\label{fig:dependencies3}\n}\n\\vspace*{-2mm}\n\\caption{{\\small GRETA}\\ graph templates}\n\\end{figure} \n\nCase 2.~\\textbf{\\textit{A negative sub-pattern is preceded by a positive sub-pattern}}. A pattern of the form $P_2=\\textsf{SEQ}(P_i,\\textsf{NOT}$ $N)$ means that no trends of $N$ may occur after the trends of $P_i$ till the end of the window (Section~\\ref{sec:filtering}). A trend of $N$ marks all previous events in the graph of $P_i$ as \\textit{invalid}.\n\nCase 3.~\\textbf{\\textit{A negative sub-pattern is followed by a positive sub-pattern}}. A pattern of the form $P_3=\\textsf{SEQ}(\\textsf{NOT} N,$ $P_j)$ means that no trends of $N$ may occur after the start of the window and before the trends of $P_j$. A trend of $N$ marks all future events in the graph of $P_j$ as \\textit{invalid}. The pattern of $Q_3$ in Section~\\ref{sec:introduction} has this form.\n\n\\vspace{-2mm}\n\\begin{example}\nFigures~\\ref{fig:dependencies2} and \\ref{fig:dependencies3} illustrate the templates for the patterns \\textit{\\textsf{SEQ}(A+,\\textsf{NOT} E)} and \\textit{\\textsf{SEQ}(\\textsf{NOT} E,A+)} respectively. The first template has only a previous connection, while the second template has only a following connection between the template for the negative sub-pattern $E$ and the event type $A$. \n\\label{ex:pattern_split_special}\n\\end{example}\n\\vspace{-2mm}\n\n\n\\begin{algorithm}[t]\n\\caption{Pattern split algorithm}\n\\label{lst:split_algorithm}\n\\begin{algorithmic}[1]\n\\Require Pattern $P$ with negative sub-patterns\n\\Ensure Set $S$ of sub-patterns of $P$\n\n\\State $S \\leftarrow \\{P\\}$\n\\State $split(P)\\ \\{$ \n\\Switch {$P$}\n\\Case {$P_i+:$}\n\t$S \\leftarrow S \\cup split(P_i)$ \\EndCase\n\\Case {$SEQ(P_i,P_j):$} \n\t$S \\leftarrow S \\cup split(P_i) \\cup split(P_j)$ \\EndCase\n\\Case {$NOT\\ P_i:$} \n\t\\State $Parent \\leftarrow S.getPatternContaining(P)$\n\t\\State $P_i.previous \\leftarrow Parent.getPrevious(P)$\n\t\\State $P_i.following \\leftarrow Parent.getFollowing(P)$\n\t\\State $S.replace(Parent, Parent-P)$\n\t\\State $S \\leftarrow S \\cup \\{P_i\\} \\cup split(P_i)$ \\EndCase\n\\EndSwitch\n\\State\\Return $S \\ \\}$\n\\end{algorithmic}\n\\end{algorithm}\n\n\\textbf{Pattern Split Algorithm}.\nAlgorithm~\\ref{lst:split_algorithm} consumes a pattern $P$, splits it into positive and negative sub-patterns, and returns the set $S$ of these sub-patterns. At the beginning, $S$ contains the pattern $P$ (Line~1). The algorithm traverses $P$ top-down. If it encounters a negative sub-pattern $P=\\textsf{NOT}\\ P_i$, it finds the sub-pattern containing $P$, called $Parent$ pattern, computes the previous and following event types of $P_i$, and removes $P$ from $Parent$ (Lines~7--10). The pattern $P_i$ is added to $S$ and the algorithm is called recursively on $P_i$ (Line~11).\nSince the algorithm traverses the pattern $P$ top-down once, the time and space complexity are linear in the size of the pattern $s$, i.e., $\\Theta(s)$.\n\n\\vspace{-2mm}\n\\begin{definition}(\\textbf{Dependent {\\small GRETA}\\ Graph}.)\nLet $G_N$ and $G_P$ be {\\small GRETA}\\ graph that are constructed according to templates $\\mathcal{T}_N$ and $\\mathcal{T}_P$ respectively. The {\\small GRETA}\\ graph $G_P$ is \\textit{dependent} on the graph $G_N$ if there is a previous or following connection from $\\mathcal{T}_N$ to an event type in $\\mathcal{T}_P$.\n\\label{def:dependent}\n\\end{definition}\n\\vspace{-2mm}\n\n\\subsection{Runtime GRETA Graphs}\n\\label{sec:negative-algorithm}\n\nIn this section, we describe how patterns with nested negation are processed according to the template.\n\n\\vspace{-2mm}\n\\begin{definition}(\\textbf{Invalid Event}.)\nLet $G_P$ and $G_N$ be {\\small GRETA}\\ graphs such that $G_P$ is dependent on $G_N$.\nLet $tr=(e_1, \\dots, e_n)$ be a \\textit{finished} trend captured by $G_N$, i.e., $e_n$ is an \\textsf{END} event.\nThe trend $tr$ marks all events of the previous event type that arrived before $e_1.time$ as \\textit{invalid} to connect to any event of the following event type that will arrive after $e_n.time$.\n\\label{def:invalidation}\n\\end{definition}\n\\vspace{-4mm}\n\n\\begin{example}\nFigure~\\ref{fig:negation} depicts the graphs for the sub-patterns from Example~\\ref{ex:pattern_split}.\nThe match $e3$ of the negative sub-pattern $E$ marks $c2$ as invalid to connect to any future $d$. Invalid events are highlighted by a darker background. \nAnalogously, the match $(c5,d6)$ of the negative sub-pattern \\textsf{SEQ}$(C,D)$ marks all $a$'s before $c5$ ($a1, a3, a4$) as invalid to connect to any $b$ after $d6$. \n$b7$ has no valid predecessor events and thus cannot be inserted. $a8$ is inserted and all previous $a$'s are connected to it. The marked $a$'s are valid to connect to new $a$'s. $b9$ is inserted and its valid predecessor event $a8$ is connected to it. The marked $a$'s may not connect to $b9$. \n\nFigures~\\ref{fig:negation2} and \\ref{fig:negation3} depict the graphs for the patterns from Example~\\ref{ex:pattern_split_special}. The trend $e3$ of the negative sub-pattern $E$ marks all previous events of type $A$ as invalid in Figure~\\ref{fig:negation2}. In contrast, in Figure~\\ref{fig:negation3} $e3$ invalidates all following $a$'s.\n\\label{ex:invalidation}\n\\end{example}\n\\vspace{-2mm}\n\n\\begin{figure}[t]\n\\centering\n\\subfigure[\\small \\textit{\\textsf{SEQ}(A+,\\textsf{NOT} E)}]{\n\\includegraphics[width=0.22\\columnwidth]{figures\/negation2.png} \n\\label{fig:negation2}\n}\n\\hspace*{2mm}\n\\subfigure[\\small \\textit{\\textsf{SEQ}(\\textsf{NOT} E,A+)}]{\n\\includegraphics[width=0.2\\columnwidth]{figures\/negation3.png} \n\\label{fig:negation3}\n}\n\\vspace*{-2mm}\n\\caption{Count of trends matched by the pattern $P$ in the stream $I=\\{a1,b2,c2,a3,e3,a4,c5,d6,b7,a8,b9\\}$}\n\\end{figure} \n\n\\textbf{Event Pruning}.\nNegation allows us to purge events from the graph to speed-up insertion of new events and aggregation propagation. The following events can be deleted:\n\n$\\bullet$ \\textbf{\\textit{Finished Trend Pruning}}. A finished trend that is matched by a negative sub-pattern can be deleted once it has invalidated all respective events. \n\n$\\bullet$ \\textbf{\\textit{Invalid Event Pruning}}. An invalid event of type $end(P_i)$ will never connect to any new event if events of type $end(P_i)$ may precede only events of type $start(P_j)$. The aggregates of such invalid events will not be propagated. Thus, such events may be safely purged from the graph.\n\n\\vspace{-2mm}\n\\begin{example}\nContinuing Example~\\ref{ex:invalidation} in Figure~\\ref{fig:negation}, the invalid $c2$ will not connect to any new event since $c$'s may connect only to $d$'s. Thus, $c2$ is purged. $e3$ is also deleted.\nOnce $a$'s before $c5$ are marked, $c5$ and $d6$ are purged.\nIn contrast, the marked events $a1,a3,$ and $a4$ may not be removed since they are valid to connect to future $a$'s.\nIn Figures~\\ref{fig:negation2} and \\ref{fig:negation3}, $e3$ and all marked $a$'s are deleted.\n\\label{ex:pruning}\n\\end{example}\n\\vspace{-4mm}\n\n\\begin{theorem}(\\textbf{Correctness of Event Pruning}.)\nLet $G_P$ and $G_N$ be {\\small GRETA}\\ graphs such that $G_P$ is dependent on $G_N$.\nLet $G'_P$ be the same as $G_P$ but without invalid events of type $end(P_i)$ if \n$P.pred(start(P_j)) = \\{ end(P_i) \\}$.\nLet $G'_N$ be the same as $G_N$ but without finished event trends.\nThen, $G'_P$ returns the same aggregation results as $G_P$.\n\\label{theo:pruning}\n\\end{theorem}\n\\vspace{-3mm}\n\n\\begin{proof}\nWe first prove that all invalid events are marked in $G_P$ despite finished trend pruning in $G'_N$. We then prove that $G_P$ and $G'_P$ return the same aggregation result despite invalid event pruning. \n\n\\textbf{\\textit{All invalid events are marked in $G_P$}}.\nLet $tr=(e_1,$ $\\dots, e_n)$ be a finished trend in $G_N$. Let $Inv$ be the set of events that are invalidated by $tr$ in $G_P$. By Definition~\\ref{def:invalidation}, all events in $Inv$ arrive before $e_1.time$. According to Section~\\ref{sec:model}, events arrive in-order by time stamps. Thus, no event with time stamp less than $e_1.time$ will arrive after $e_1$. Hence, even if an event $e_i \\in \\{e_1, \\dots, e_{n-1}\\}$ connects to future events in $G_N$, no event $e \\not\\in Inv$ in $G_P$ can be marked as invalid.\n\n\\textbf{\\textit{$G_P$ and $G'_P$ return the same aggregates}}.\nLet $e$ be an event of type $end(P_i)$ that is marked as invalid to connect to events of type $start(P_j)$ that arrive after $e_n.time$. \nBefore $e_n.time$, $e$ is valid and its count is correct by Theorem~\\ref{theorem:count}. Since events arrive in-order by time stamps, no event with time stamp less than $e_n.time$ will arrive after $e_n$. \nAfter $e_n.time$, $e$ will not connect to any event and the count of $e$ will not be propagated if \n$P.pred(start(P_j))=\\{end(P_i)\\}$.\nHence, deletion of $e$ does not affect the final aggregate of $G_P$.\n\\end{proof}\n\\vspace{-2mm}\n\n\\textbf{{\\small GRETA}\\ Algorithm for Patterns with Negation}.\nAlgorithm~\\ref{lst:ETA_algorithm} is called on each event sub-pattern with the following modifications.\nFirst, only valid predecessor events are returned in Line~4. \nSecond, if the algorithm is called on a negative sub-pattern $N$ and a match is found in Line~12, then all previous events of the previous event type of $N$ are either deleted or marked as incompatible with any future event of the following event type of $N$. Afterwards, the match of $N$ is purged from the graph. {\\small GRETA}\\ concurrency control is described in Section~\\ref{sec:implementation}. \n \n\\section{Optimality of GRETA Approach}\n\\label{sec:complexity}\n\n\nWe now analyze the complexity of {\\small GRETA}. Since a negative sub-pattern is processed analogously to a positive sub-pattern (Section~\\ref{sec:negative}), we focus on positive patterns below.\n\n\\vspace{-2mm}\n\\begin{theorem}[\\textbf{Complexity}]\nLet $q$ be a query with edge predicates,\n$I$ be a stream, \n$G$ be the {\\small GRETA}\\ graph for $q$ and $I$,\n$n$ be the number of events per window, and \n$k$ be the number of windows into which an event falls.\nThe time complexity of {\\small GRETA}\\ is $O(n^2k)$, while its space complexity is $O(nk)$.\n\\label{theorem:complexity}\n\\end{theorem}\n\\vspace{-3mm}\n\n\\begin{proof}\\let\\qed\\relax\n\\textbf{Time Complexity}. Let $e$ be an event of type $E$. The following steps are taken to process $e$.\nSince events arrive in-order by time stamps (Section~\\ref{sec:model}), the Time Pane to which $e$ belongs is always the latest one. It is accessed in constant time.\nThe Vertex Tree in which $e$ will be inserted is found in the Event Type Hash Table mapping the event type $E$ to the tree in constant time.\nDepending on the attribute values of $e$, $e$ is inserted into its Vertex Tree in logarithmic time $O(log_b m)$ where $b$ is the order of the tree and $m$ is the number of elements in the tree, $m \\leq n$.\n\nThe event $e$ has $n$ predecessor events in the worst case, since each vertex connects to each following vertex under the skip-till-any-match semantics. Let $x$ be the number of Vertex Trees storing previous vertices that are of predecessor event types of $E$ and fall into a sliding window $wid \\in windows(e)$, $x \\leq n$. Then, the predecessor events of $e$ are found in $O(log_b m + m)$ time by a range query in one Vertex Tree with $m$ elements. The time complexity of range queries in $x$ Vertex Trees is computed as follows:\n\\[\n\\sum_{i=1}^x O(log_b m_i + m_i) =\n\\sum_{i=1}^x O(m_i) =\nO(n).\n\\]\n\nIf $e$ falls into $k$ windows, a predecessor event of $e$ updates at most $k$ aggregates of $e$.\nIf $e$ is an \\textsf{END} event, it also updates $k$ final aggregates.\nSince these aggregates are maintained in hash tables, updating one aggregate takes constant time.\n{\\small GRETA}\\ concurrency control ensures that all graphs this graph $G$ depends upon finishing processing all events with time stamps less than $t$ before $G$ may process events with time stamp $t$. Therefore, all invalid events are marked or purged before aggregates are updated in $G$ at time $t$. Consequently, an aggregate is updated at most once by the same event.\nPutting it all together, the time complexity is:\n\\[\nO(n (log_b m + nk)) = O(n^2k).\n\\] \n\n\\textbf{Space Complexity}. The space complexity is determined by $x$ Vertex Trees and $k$ counts maintained by each vertex.\n\\begin{flalign*} \n\\sum_{i=1}^x O(m_ik) = O(nk). \\rlap{$\\qquad \\Box$}\n\\end{flalign*} \n\\end{proof} \n\n\\vspace{-3mm}\n\\begin{theorem}[\\textbf{Time Optimality}]\nLet $n$ be the number of events per window and $k$ be the number of windows into which an event falls.\nThen, {\\small GRETA}\\ has optimal worst-case time complexity $O(n^2k)$.\n\\label{theorem:optimality}\n\\end{theorem}\n\\vspace{-3mm}\n\n\\begin{proof} \n\\textit{Any} event trend aggregation algorithm must process $n$ events to guarantee correctness of aggregation results. \nSince \\textit{any} previous event may be compatible with a new event $e$ under the skip-till-any-match semantics~\\cite{WDR06}, the edge predicates of the query $q$ must be evaluated to decide the compatibility of $e$ with $n$ previous events in worst case. While we utilize a tree-based index to sort events by the most selective predicate, other predicates may have to be evaluated in addition. Thus, each new event must be compared to each event in the graph in the worst case.\nLastly, a final aggregate must be computed for each window of $q$. An event that falls into $k$ windows contributes to $k$ aggregates.\nIn summary, the time complexity $O(n^2k)$ is optimal.\n\\end{proof} \n\\vspace{-2mm}\n\\section{GRETA Approach In A Nutshell}\n\\label{sec:overview}\n\nOur \\textbf{\\textit{Event Trend Aggregation Problem}} to compute event trend aggregation results of a query $q$ against an event stream $I$ with \\textit{minimal latency}. \n\nFigure~\\ref{fig:system} provides an overview of our {\\small GRETA}\\ framework. The \\textbf{\\textit{{\\small GRETA}\\ Query Analyzer}} statically encodes the query into a {\\small GRETA}\\ configuration. In particular, the pattern is split into positive and negative sub-patterns (Section~\\ref{sec:split}). Each sub-pattern is translated into a \n{\\small GRETA}\\ template\n(Section~\\ref{sec:template}). Predicates are classified into vertex and edge predicates (Section~\\ref{sec:filtering}). \nGuided by the {\\small GRETA}\\ configuration, the \\textit{\\textbf{{\\small GRETA}\\ Runtime}} first filters and partitions the stream based on the vertex predicates and grouping attributes of the query. Then, the {\\small GRETA}\\ runtime encodes matched event trends into a {\\small GRETA}\\ graph. During the graph construction, aggregates are propagated along the edges of the graph in a dynamic programming fashion. The final aggregate is updated incrementally, and thus is returned immediately at the end of each window (Sections~\\ref{sec:positive-algorithm}, \\ref{sec:negative-algorithm}, \\ref{sec:filtering}). \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.6\\columnwidth]{figures\/system.png} \n\\caption{{\\small GRETA}\\ framework}\n\\label{fig:system}\n\\end{figure}\n\n\\section{Positive Nested Patterns}\n\\label{sec:positive}\n\nWe statically translate a positive pattern into a \n{\\small GRETA}\\ template (Section~\\ref{sec:template})\nAs events arrive at runtime, the {\\small GRETA}\\ graph is maintained according to this template (Section~\\ref{sec:positive-algorithm}).\n\n\\subsection{Static GRETA Template}\n\\label{sec:template}\n\nThe {\\small GRETA}\\ query analyzer translates a Kleene pattern $P$ into a Finite State Automaton that is then used as a template during {\\small GRETA}\\ graph construction at runtime. \nFor example, the pattern \\textit{P=(\\textsf{SEQ}(A+,B))+} is translated into the \n{\\small GRETA}\\ template\nin Figure~\\ref{fig:automaton}.\n\n\\textbf{\\textit{States}} correspond to event types in $P$. \nThe initial state is labeled by the first type in $P$, denoted $start(P)$. Events of type $start(P)$ are called \\textsf{START} events. The final state has label $end(P)$, i.e., the last type in $P$. Events of type $end(P)$ are called \\textsf{END} events. All other states are labeled by types $mid(P)$. Events of type $E \\in mid(P)$ are called \\textsf{MID} events. \nIn Figure~\\ref{fig:automaton}, $start(P) = A$, $end(P) = B$, and $mid(P) = \\emptyset$.\n\nSince an event type may appear in a pattern at most once (Section~\\ref{sec:model}), state labels are distinct.\nThere is one $start(P)$ and one $end(P)$ event type per pattern $P$ (Theorem~\\ref{theorem:start-and-end}). There can be any number of event types in the set $mid(P)$. $start(P) \\not\\in mid(P)$ and $end(P) \\not\\in mid(P)$. An event type may be both $start(P)$ and $end(P)$, for example, in the pattern $A+$.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.2\\columnwidth]{figures\/automaton.png} \n\\vspace{-2mm} \n\\caption{{\\small GRETA}\\ template for (\\textsf{SEQ}(A+,B))+}\n\\label{fig:automaton}\n\\end{figure} \n\n\\textbf{Start and End Event Types of a Pattern}.\nLet $trends(P)$ denote the set of event trends matched by a positive event pattern $P$ over any input event stream.\n\n\\vspace*{-2mm}\n\\begin{lemma}\nFor any positive event pattern $P$,\n$trends(P)$ does not contain the empty string.\n\\label{lemma:non-empty}\n\\end{lemma}\n\\vspace*{-2mm}\n\nLet $tr \\in trends(P)$ be a trend and $start(tr)$ and $end(tr)$ be the types of the first and last events in $tr$ respectively.\n\n\\vspace*{-2mm}\n\\begin{theorem}\nFor all $tr_1, tr_2 \\in trends(P)$, $start(tr_1) = start(tr_2)$ and $end(tr_1) = end(tr_2)$. \n\\label{theorem:start-and-end}\n\\end{theorem}\n\\vspace*{-4mm}\n\n\\begin{proof}\nDepending on the structure of $P$ (Definition~\\ref{def:pattern}), the following cases are possible.\n\n\\textbf{\\textit{Case}} $E$. For any $tr \\in trends(E)$, $start(tr) = end(tr) = E$.\n\n\\textbf{\\textit{Case}} $P_i+$. Let $tr_1 = (t_1 t_2 \\dots t_m), tr_2 = (t'_1 t'_2 \\dots t'_n) \\in trends(P_i+)$ where $t_x,t'_y \\in trends(P_i), 1 \\leq x \\leq m, 1 \\leq y \\leq n$. \nBy Lemma~\\ref{lemma:non-empty}, $start(t_1)$ and $start(t'_1)$ are not empty.\nAccording to Algorithm~\\ref{lst:preprocessing_algorithm} Lines 10--14, $start(tr_1) = start(t_1)$ and $start(tr_2) = start(t'_1)$.\nSince $P_i$ contains neither disjunction nor star-Kleene, $start(t_1) = start(t'_1)$. Thus, $start(tr_1) = start(tr_2)$. \nThe proof for $end(P_i+)$ is analogous.\n\n\\textbf{\\textit{Case}} \\textsf{SEQ}$(P_i, P_j)$. Let $tr_1 = (t_1 t_2), tr_2 = (t'_1 t'_2) \\in trends($ \\textsf{SEQ}$(P_i, P_j))$ where $t_1, t'_1 \\in trends(P_i)$ and $t_2, t'_2 \\in trends(P_j)$. \nBy Lemma~\\ref{lemma:non-empty}, $start(t_1)$ and $start(t'_1)$ are not empty.\nAccording to Algorithm~\\ref{lst:preprocessing_algorithm} Lines~10--14, $start(tr_1) = start(t_1)$ and $start(tr_2) = start(t'_1)$. \nSince $P_i$ contains neither disjunction nor star-Kleene, $start(t_1) = start(t'_1)$. Thus, $start(tr_1) = start(tr_2)$. \nThe proof for $end(\\textsf{SEQ}(P_i, P_j))$ is analogous.\n\\end{proof}\n\\vspace*{-2mm}\n\n\\textbf{\\textit{Transitions}} correspond to operators in $P$. They connect types of events that may be adjacent in a trend matched by $P$. \nIf a transition connects an event type $E_i$ with an event type $E_j$, then $E_i$ is a \\textit{predecessor event type} of $E_j$, denoted $E_i \\in P.predTypes(E_j)$.\nIn Figure~\\ref{fig:automaton}, $P.predTypes(A) = \\{A,B\\}$ and $P.predTypes(B) = \\{A\\}$.\n\n\\begin{algorithm}[t]\n\\caption{{\\small GRETA}\\ template construction algorithm}\n\\label{lst:preprocessing_algorithm}\n\\begin{algorithmic}[1]\n\\Require Positive pattern $P$\n\\Ensure GRETA template $\\mathcal{T}$\n\n\\State $generate(P)\\ \\{$\n\\State $S \\leftarrow \\text{event types in } P,\\ T \\leftarrow \\emptyset,\\ \\mathcal{T}=(S,T)$\n\n\\ForAll {$\\textsf{SEQ}(P_i,P_j)$ in $P$}\n\t\\State $t \\leftarrow (end(P_i),start(P_j)),\\ t.label \\leftarrow ``\\textsf{SEQ}\"$\n\t\\State $T \\leftarrow T \\cup \\{t\\}$\n\\EndFor\n\n\\ForAll {$P_i+$ in $P$}\n\t\\State $t \\leftarrow (end(P_i),start(P_i)),\\ t.label \\leftarrow ``+\"$\n\t\\State $T \\leftarrow T \\cup \\{t\\}$\n\\EndFor\n\\State\\Return $\\mathcal{T}\\ \\}$\n\n\\State $start(P)\\ \\{$ \n\\Switch {$P$}\n\\Case {$E$} \\Return $E$ \\EndCase\n\\Case {$P_i+$} \\Return $start(P_i)$ \\EndCase\n\\Case {$\\textsf{SEQ}(P_i,P_j)$} \\Return $start(P_i)\\ \\}$ \\EndCase\n\\EndSwitch\n\n\\State $end(P)\\ \\{$\n\\Switch {$P$}\n\\Case {$E$} \\Return $E$ \\EndCase\n\\Case {$P_i+$} \\Return $end(P_i)$ \\EndCase\n\\Case {$\\textsf{SEQ}(P_i,P_j)$} \\Return $end(P_j)\\ \\}$ \\EndCase\n\\EndSwitch\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\\textbf{{\\small GRETA}\\ Template Construction Algorithm}.\nAlgorithm~\\ref{lst:preprocessing_algorithm} consumes a positive pattern $P$ and returns the automaton-based representation of $P$, called \\textit{GRETA template} $\\mathcal{T}=(S,T)$. The states $S$ correspond to the event types in $P$ (Line~2), while the transitions $T$ correspond to the operators in $P$. Initially, the set $T$ is empty (Line~2). \nFor each event sequence \\textsf{SEQ}$(P_i,P_j)$ in $P$, there is a transition from $end(P_i)$ to $start(P_j)$ with label ``\\textsf{SEQ}\" (Lines~3--5). \nAnalogously, for each Kleene plus $P_i+$ in $P$, there is a transition from $end(P_i)$ to $start(P_i)$ with label ``+\" (Lines~6--8). Start and end event types of a pattern are computed by the auxiliary methods in Lines~10--19. \n\n\\textbf{Complexity Analysis}.\nLet $P$ be a pattern of size $s$ (Definition~\\ref{def:pattern}). To extract all event types and operators from $P$, $P$ is parsed once in $\\Theta(s)$ time. For each operator, we determine its start and event event types in $O(s)$ time. Thus, the time complexity is quadratic $O(s^2)$. \nThe space complexity is linear in the size of the template $\\Theta(|S|+|T|)=\\Theta(s)$.\n\n\n\\begin{figure*}[t]\n\\centering\n\\subfigure[\\small $A+$]{\n\t\\includegraphics[width=0.105\\columnwidth]{figures\/kleene1.png}\n\t\\label{fig:kleene1}\n}\n\\hspace*{2mm}\n\\subfigure[\\small $\\textsf{SEQ}(A+,B)$]{\n\t\\includegraphics[width=0.17\\columnwidth]{figures\/kleene2.png}\n\t\\label{fig:kleene2}\t\n}\n\\hspace*{2mm}\n\\subfigure[\\small $(\\textsf{SEQ}(A+,B))+$]{\n\t\\includegraphics[width=0.17\\columnwidth]{figures\/kleene3.png}\n\t\\label{fig:kleene3}\n}\n\\hspace*{2mm}\n\\subfigure[\\small $(\\textsf{SEQ}(A+, \\textsf{NOT SEQ}(C, \\textsf{NOT} E, D), B))+$]{\n\t\\includegraphics[width=0.41\\columnwidth]{figures\/negation.png}\n\t\\label{fig:negation}\n}\n\\vspace{-3mm}\n\\caption{Count of trends matched by the pattern $P$ in the stream $I=\\{a1,b2,c2,a3,e3,a4,c5,d6,b7,a8,b9\\}$}\n\\label{fig:pattern}\n\\end{figure*}\n\n\\subsection{Runtime GRETA Graph}\n\\label{sec:positive-algorithm}\n\nThe {\\small GRETA}\\ graph is a runtime instantiation of the \n{\\small GRETA}\\ template. \nThe graph is constructed on-the-fly as events arrive \n(Algorithm~\\ref{lst:ETA_algorithm}).\nThe graph compactly captures all matched trends and enables their incremental aggregation.\n\n\\textbf{Compact Event Trend Encoding}.\nThe graph encodes all trends and thus avoids their construction. \n\n\\textbf{\\textit{Vertices}} represent events in the stream $I$ matched by the pattern $P$. Each state with label $E$ in the template is associated with the sub-graph of events of type $E$ in the graph. We highlight each sub-graph by a rectangle frame. If $E$ is an end state, the frame is depicted as a double rectangle. Otherwise, the frame is a single rectangle. An event is labeled by its event type, time stamp, and intermediate aggregate (see below). Each event is stored once.\nFigure~\\ref{fig:kleene3} illustrates the template and the graph for the stream $I$. \n\n\\textbf{\\textit{Edges}} connect adjacent events in a trend matched by the pattern $P$ in a stream $I$ (Definition~\\ref{def:pattern}).\nWhile transitions in the template express predecessor relationships between event types in the pattern, edges in the graph capture predecessor relationships between events in a trend. \nIn Figure~\\ref{fig:kleene3}, we depict a transition in the template and its respective edges in the graph in the same way.\nA path from a \\textsf{START} to an \\textsf{END} event in the graph corresponds to a trend. The length of these trends ranges from the shortest $(a1,b2)$ to the longest $(a1,b2,a3,a4,b7,a8,b9)$. \n\nIn summary, the {\\small GRETA}\\ graph in Figure~\\ref{fig:kleene3} compactly captures all 43 event trends matched by the pattern $P$ in the stream $I$. In contrast to the two-step approach, the graph avoids repeated computations and replicated storage of common sub-trends such as $(a1,b2)$.\n\n\\textbf{Dynamic Aggregation Propagation}.\nIntermediate aggregates are propagated through the graph from previous events to new events in dynamic programming fashion. Final aggregate is incrementally computed based on intermediate aggregates. In the examples below, we compute event trend count \\textsf{COUNT}(*) as defined in Section~\\ref{sec:model}. Same principles apply to other aggregation functions (Section~\\ref{sec:discussion}).\n\n\\textbf{\\textit{Intermediate Count}} $e.count$ of an event $e$ corresponds to the number of (sub-)trends in $G$ that begin with a \\textsf{START} event in $G$ and end at $e$.\nWhen $e$ is inserted into the graph, all predecessor events of $e$ connect to $e$. That is, $e$ extends all trends that ended at a predecessor event of $e$. To accumulate the number of trends extended by $e$, $e.count$ is set to the sum of counts of the predecessor events of $e$. In addition, if $e$ is a \\textsf{START} event, it starts a new trend. Thus, $e.count$ is incremented by 1. \nIn Figure~\\ref{fig:kleene3}, the count of the \\textsf{START} event $a4$ is set to 1 plus the sum of the counts of its predecessor events $a1,b2,$ and $a3$. \n\\[\n\\begin{array}{l}\na4.count=1+(a1.count+b2.count+a3.count)=6 \\\\\n\\end{array}\n\\]\n\n$a4.count$ is computed once, stored, and reused to compute the counts of $b7,a8,$ and $b9$ that $a4$ connects to. For example, the count of $b7$ is set to the sum of the counts of all predecessor events of $b7$.\n\\[\n\\begin{array}{l}\nb7.count=a1.count+a3.count+a4.count=10 \\\\\n\\end{array}\n\\]\n\n\\textbf{\\textit{Final Count}} corresponds to the sum of the counts of all \\textsf{END} events in the graph. \n\\[\n\\begin{array}{l}\nfinal\\_count=b2.count+b7.count+b9.count=43 \\\\\n\\end{array}\n\\]\n\nIn summary, the count of a new event is computed based on the counts of previous events in the graph following the dynamic programming principle. Each intermediate count is computed once. The final count is incrementally updated by each \\textsf{END} event and instantaneously returned at the end of each window.\n\n\\vspace{-2mm}\n\\begin{definition}(\\textbf{{\\small GRETA}\\ Graph}.)\nThe \\textit{{\\small GRETA}\\ graph} $G = (V,E,fi\\-nal\\_count)$ for a query $q$ and a stream $I$ is a directed acyclic graph with a set of vertices $V$, a set of edges $E$, and a $final\\_count$.\nEach vertex $v \\in V$ corresponds to an event $e \\in I$ matched by $q$. A vertex $v$ has the label $(e.type\\ e.time : e.count)$ (Theorem~\\ref{theorem:count}). \nFor two vertices $v_i, v_j \\in V$, there is an edge $(v_i,v_j) \\in E$ if their respective events $e_i$ and $e_j$ are adjacent in a trend matched by $q$. Event $v_i$ is called a \\textit{predecessor event} of $v_j$. \n\\label{def:graph}\n\\end{definition}\n\\vspace{-2mm} \n\nThe {\\small GRETA}\\ graph has different shapes depending on the pattern and the stream.\nFigure~\\ref{fig:kleene1} shows the graph for the pattern $A+$. Events of type $B$ are not relevant for it. Events of type $A$ are both \\textsf{START} and \\textsf{END} events.\nFigure~\\ref{fig:kleene2} depicts the {\\small GRETA}\\ graph for the pattern \\textsf{SEQ}$(A+,B)$. There are no dashed edges since $b$'s may not precede $a$'s. \n\nTheorems~\\ref{theorem:correctness-graph} and \\ref{theorem:count} prove the correctness of the event trend count computation based on the {\\small GRETA}\\ graph.\n\n\\vspace{-2mm}\n\\begin{theorem}[\\textbf{Correctness of the GRETA Graph}]\nLet $G$ be the {\\small GRETA}\\ graph for a query $q$ and a stream $I$. \nLet $\\mathcal{P}$ be the set of paths from a \\textsf{START} to an \\textsf{END} event in $G$.\nLet $\\mathcal{T}$ be the set of trends detected by $q$ in $I$.\nThen, the set of paths $\\mathcal{P}$ and the set of trends $\\mathcal{T}$ are equivalent. That is, for each path $p \\in \\mathcal{P}$ there is a trend $tr \\in \\mathcal{T}$ with same events in the same order and vice versa.\n\\label{theorem:correctness-graph}\n\\end{theorem}\n\\vspace{-4mm}\n\n\\begin{proof}\n\\textbf{\\textit{Correctness}}. For each path $p \\in \\mathcal{P}$, there is a trend $tr \\in \\mathcal{T}$ with same events in the same order, i.e., $\\mathcal{P} \\subseteq \\mathcal{T}$.\nLet $p \\in \\mathcal{P}$ be a path. By definition, $p$ has one \\textsf{START}, one \\textsf{END}, and any number of \\textsf{MID} events. Edges between these events are determined by the query $q$ such that a pair of \\textit{adjacent events in a trend} is connected by an edge. Thus, the path $p$ corresponds to a trend $tr \\in \\mathcal{T}$ matched by the query $q$ in the stream $I$.\n\n\\textbf{\\textit{Completeness}}. For each trend $tr \\in \\mathcal{T}$, there is a path $p \\in \\mathcal{P}$ with same events in the same order, i.e., $\\mathcal{T} \\subseteq \\mathcal{P}$.\nLet $tr \\in \\mathcal{T}$ be a trend. We first prove that all events in $tr$ are inserted into the graph $G$. Then we prove that these events are connected by directed edges such that there is a path $p$ that visits these events in the order in which they appear in the trend $tr$. \nA \\textsf{START} event is always inserted, while a \\textsf{MID} or an \\textsf{END} event is inserted if it has predecessor events since otherwise there is no trend to extend. Thus, all events of the trend $tr$ are inserted into the graph $G$. The first statement is proven.\n\\textit{All} previous events that satisfy the query $q$ connect to a new event. Since events are processed in order by time stamps, edges connect previous events with more recent events. The second statement is proven.\n\\end{proof}\n\\vspace{-4mm}\n\n\\begin{theorem}[\\textbf{Event Trend Count Computation}]\nLet $G$ be the {\\small GRETA}\\ graph for a query $q$ and a stream $I$ and \n$e \\in I$ be an event with predecessor events $Pr$ in $G$. \n(1)~The intermediate count $e.count$ is the number of (sub) trends in $G$ that start at a \\textsf{START} event and end at $e$. \n$e.count = \\sum_{p \\in Pr} p.count$. \nIf $e$ is a \\textsf{START} event, $e.count$ is incremented by one.\nLet $End$ be the \\textsf{END} events in $G$.\n(2)~The final count is the number of trends captured by $G$. \n$final\\_count = \\sum_{end \\in End} end.count$.\n\\label{theorem:count}\n\\end{theorem}\n\\vspace*{-2mm}\n\n\\begin{proof}\n(1)~We prove the first statement by induction on the number of events in $G$.\n\n\\textbf{\\textit{Induction Basis}}: $n=1$. If there is only one event $e$ in the graph $G$, $e$ is the only (sub-)trend captured by $G$. Since $e$ is the only event in $G$, $e$ has no predecessor events. The event $e$ can be inserted into the graph only if $e$ is a \\textsf{START} event. Thus, $e.count=1$.\n\n\\textbf{\\textit{Induction Assumption}}: The statement is true for $n$ events in the graph $G$.\n\n\\textbf{\\textit{Induction Step}}: $n \\rightarrow n+1$. Assume a new event $e$ is inserted into the graph $G$ with $n$ events and the predecessor events $Pred$ of $e$ are connected to $e$. According to the induction assumption, each of the predecessor events $p \\in Pred$ has a count that corresponds to the number of sub-trends in $G$ that end at the event $p$. The new event $e$ continues \\textit{all} these trends. Thus, the number of these trends is the sum of counts of all $p \\in Pred$. In addition, each \\textsf{START} event initiates a new trend. Thus, 1 is added to the count of $e$ if $e$ is a \\textsf{START} event. The first statement is proven.\n\n(2)~By definition only \\textsf{END} events may finish trends. We are interested in the number of finished trends only. Since the count of an \\textsf{END} event $end$ corresponds to the number of trends that finish at the event $end$, the total number of trends captured by the graph $G$ is the sum of counts of all \\textsf{END} events in $G$. The second statement is proven.\n\\end{proof}\n\\vspace{-2mm}\n\n\\begin{algorithm}[t]\n\\caption{{\\small GRETA}\\ algorithm for positive patterns}\n\\label{lst:ETA_algorithm}\n\\begin{algorithmic}[1]\n\\Require Positive pattern $P$, stream $I$\n\\Ensure Count of trends matched by $P$ in $I$\n\\State $process\\_pos\\_pattern(P,I)\\ \\{$\n\\State $V \\leftarrow \\emptyset,\\ final\\_count \\leftarrow 0$\n\\ForAll {$e \\in I$ of type $E$} \n\t\t\\State $Pr \\leftarrow V.predEvents(e)$\n \\If {$E = start(P)$ or $Pr \\neq \\emptyset$}\t\t\n \t\t\t\\State $V \\leftarrow V \\cup e,\\;e.count \\leftarrow (E = start(P))\\;?\\;1\\;:\\;0$\n \t\t\t\\ForAll {$p \\in Pr$} $\\switch e.count += p.count$ \\EndFor \t\t\t\n \t\t\t\\If {$E = end(P)$} $\\switch final\\_count += e.count$ \\EndIf \t\t\t\n\t\\EndIf\n\\EndFor\n\\State\\Return $final\\_count\\ \\}$ \n\\end{algorithmic}\n\\end{algorithm}\n\n\\textbf{{\\small GRETA}\\ Algorithm for Positive Patterns} computes the number of trends matched by the pattern $P$ in the stream $I$. The set of vertices $V$ in the {\\small GRETA}\\ graph is initially empty (Line~2 of Algorithm~\\ref{lst:ETA_algorithm}). Since each edge is traversed exactly once, edges are not stored.\nWhen an event $e$ of type $E$ arrives, the method $V.predEvents(e)$ returns the predecessor events of $e$ in the graph (Line~4).\nA \\textsf{START} event is always inserted into the graph since it always starts a new trend, while a \\textsf{MID} or an \\textsf{END} event is inserted only if it has predecessor events (Lines~5--6).\nThe count of $e$ is increased by the counts of its predecessor events (Line~7). \nIf $e$ is a \\textsf{START} event, its count is incremented by 1 (Line~6).\nIf $e$ is an \\textsf{END} event, the final count is increased by the count of $e$ (Line~8). This final count is returned (Line~9).\n\n\\vspace*{-2mm}\n\\begin{theorem}[\\textbf{Correctness of the {\\small GRETA}\\ Algorithm}]\nGiven a positive event pattern $P$ and a stream $I$, Algorithm~\\ref{lst:ETA_algorithm} constructs the {\\small GRETA}\\ graph for $P$ and $I$ (Definition~\\ref{def:graph}) and computes the intermediate and final counts (Theorem~\\ref{theorem:count}).\n\\label{theorem:correctness-algo}\n\\end{theorem}\n\\vspace*{-4mm}\n\n\\begin{proof}\n\\textbf{\\textit{Graph Construction}}.\nEach event $e \\in I$ is processed (Line~3). A \\textsf{START} event is always inserted, while a \\textsf{MID} or an \\textsf{END} event is inserted if it has predecessor events (Lines~4--6). Thus, the set of vertices $V$ of the graph corresponds to events in the stream $I$ that are matched by the pattern $P$. \nEach predecessor event $p$ of a new event $e$ is connected to $e$ (Lines~8--9). Therefore, the edges $E$ of the graph capture adjacency relationships between events in trends matched by the pattern $P$. \n\n\\textbf{\\textit{Count Computation}}.\nInitially, the intermediate count $e.count$ of an event $e$ is either 1 if $e$ is a \\textsf{START} event or 0 otherwise (Line~7). $e.count$ is then incremented by $p.count$ of each predecessor event $p$ of $e$ (Lines~8, 10). Thus, $e.count$ is correct.\nInitially, the final count is 0 (Line~2). Then, it is incremented by $e.count$ of each \\textsf{END} event $e$ in the graph. Since $e.count$ is correct, the final count is correct too.\n\\end{proof}\n\\vspace*{-2mm}\n\nWe analyze complexity of Algorithm~\\ref{lst:ETA_algorithm} in Section~\\ref{sec:complexity}.\n\n\n\n\\section{Related Work}\n\\label{sec:related_work}\n\n\\textbf{Complex Event Processing}.\nCEP approaches like SASE \\cite{ADGI08,ZDI14}, Cayuga~\\cite{DGPRSW07}, ZStream~\\cite{MM09}, and E-Cube~\\cite{LRGGWAM11} support aggregation computation over event streams. \nSASE and Cayuga deploy a Finite State Automaton (FSA)-based query execution paradigm, meaning that each query is translated into an FSA. Each run of an FSA corresponds to an event trend. \nZStream translates an event query into an operator tree that is optimized based on the rewrite rules and the cost model.\nE-Cube employs hierarchical event stacks to share events across different event queries.\n\nHowever, the expressive power of all these approaches is limited. E-Cube does not support Kleene closure, while Cayuga and ZStream do not support the skip-till-any-match semantics nor the \\textsf{GROUP-BY} clause in their event query languages.\nFurthermore, these approaches define no optimization techniques for event trend aggregation. Instead, they handle aggregation as a post-processing step that follows trend construction. This trend construction step delays the system responsiveness as demonstrated in Section~\\ref{sec:evaluation}.\n\nIn contrast to the above approaches, A-Seq~\\cite{QCRR14} proposes \\textit{online} aggregation of \\textit{fixed-length} event sequences. The expressiveness of this approach is rather limited, namely, it supports neither Kleene closure, nor arbitrarily-nested event patterns, nor edge predicates. Therefore, it does not tackle the exponential complexity of event trends.\n\nThe CET approach~\\cite{PLAR17} focuses on optimizing the \\textit{construction of event trends}. It does not support aggregation, grouping, nor negation. In contrast, our {\\small GRETA}\\ approach focuses on \\textit{aggregation of event trends} without trend construction. Due to the exponential time and space complexity of trend construction, the CET approach is neither real-time nor lightweight as confirmed by our experiments. \n\n\\textbf{Data Streaming}.\nStreaming approaches~\\cite{AW04, GHMAE07, KWF06, LMTPT05, LMTPT05-2, THSW15, ZKOS05, ZKOSZ10} support aggregation computation over data streams. Some approaches incrementally aggregate \\textit{raw input events for single-stream queries}~\\cite{LMTPT05, LMTPT05-2}. Others share aggregation results between overlapping sliding windows~\\cite{AW04, LMTPT05}, which is also leveraged in our {\\small GRETA}\\ approach (Section~\\ref{sec:positive-algorithm}). Other approaches share intermediate aggregation results between multiple queries~\\cite{KWF06, ZKOS05, ZKOSZ10}.\nHowever, these approaches evaluate simple Select-Project-Join queries with window semantics. Their execution paradigm is set-based. They do not support CEP-specific operators such as event sequence and Kleene closure that treat the order of events as first-class citizens. Typically, these approaches require the \\textit{construction of join results} prior to their aggregation. Thus, they define incremental aggregation of \\textit{single raw events} but implement a two-step approach for join results.\n\nIndustrial streaming systems including Flink~\\cite{flink}, Esper~\\cite{esper}, Google Dataflow~\\cite{dataflow}, and Microsoft StreamInsight~\\cite{streaminsight} do not explicitly support Kleene closure nor aggregation of Kleene matches. However, Kleene closure computation can be simulated by a set of event sequence queries covering all possible lengths of a trend. This approach is possible only if the maximal length of a trend is known apriori -- which is rarely the case in practice. Furthermore, this approach is highly inefficient for two reasons. First, it runs a set of queries for each Kleene query. This increased workload drastically degrades the system performance. Second, since this approach requires event trend construction prior to their aggregation, it has exponential time complexity and thus fails to compute results within a few seconds.\n\n\\textbf{Static Sequence Databases}.\nThese approaches extend traditional SQL queries by order-aware join operations and support aggregation of their results~\\cite{LS03, LKHLCC08}. However, they do not support Kleene closure. Instead, \\textit{single data items} are aggregated~\\cite{LS03, MZ97, SZZA04, SLR96}. \nFurthermore, these approaches assume that the data is statically stored and indexed prior to processing. Hence, these approaches do not tackle challenges that arise due to dynamically streaming data such as event expiration and real-time execution. \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nHow should social scientists understand and communicate the uncertainty of statistically estimated causal effects? They usually use a specific decision threshold of a \\textit{p}-value or confidence\/credible interval and conclude whether a causal effect is significant or not, meaning it is not (practically) null \\autocite{Gross2015, Kruschke2018a}. While convenient to make categorical decisions, the significance-vs.-insignificance dichotomy leads us to overlook the full nuance of the statistical measure of uncertainty. This is because uncertainty is the degree of confidence and, therefore, a continuous scale. The dichotomy is also associated with problems such as \\textit{p}-hacking, publication bias, and seeing statistical insignificance as evidence for the null hypothesis \\autocite[e.g., see][]{Amrhein2019, Esarey2016, Gerber2008, McShane2016, McShane2017, Simonsohn2014}. While these cited articles discuss these problems stemming from the Frequentist Null Hypothesis Significance Testing, Bayesian inference is equally susceptible to the same issues if a specific decision threshold is used to interpret a posterior distribution.\n\nBehavioral research suggests both researchers and decision makers understand uncertainty more appropriately if it is presented as a numerical, continuous scale rather than as a verbal, discrete scale \\autocite{Budescu2014, Friedman2018, Jenkins2018, McShane2016, McShane2017, Mislavsky2021}. If researchers are interested in estimating a causal effect, the natural quantity of uncertainty is the probability of an effect, i.e., the probability that a causal factor affects the outcome of interest. Probability is an intuitive quantity and used in everyday life, for example, in a weather forecast for the chance of rain. The probability of an effect can be computed by a posterior distribution estimated by Bayesian statistics \\autocite[or, if appropriate, by a pseudo-Bayesian, confidence distribution; see][]{Wood2019}.\n\nThe standard ways to summarize a posterior is the plot of a probability density function, the probability of a one-sided hypothesis, and a credible interval. These standard ways have drawbacks. A probability density plot is not the best for readers to accurately compute a probability mass for a specific range of parameter values \\autocite{Kay2016}. The probability of a one-sided hypothesis needs a decision threshold for the effect size from which on the probability is computed (typically, the probability of an effect being greater\/smaller than zero). A credible interval also needs a decision threshold for the level of probability based on which the interval is computed (typically, the 95\\% level). From a decision-theoretic perspective, the use of a decision threshold demands a justification based on a context-specific utility function \\autocites{Berger1985}[276--78]{Kruschke2018a}[170]{Lakens2018}{Mudge2012}. Yet, social scientific research often deals with too heterogeneous cases (e.g., all democracies) to find a single utility function, and may want to be agnostic about it.\n\nMotivated by these backgrounds, I propose presenting the uncertainty of statistically estimated causal effects, as the probabilities of different effect sizes. More specifically, it is the plot of a complementary cumulative distribution, where the probability is presented for an effect being greater than different effect sizes (here, ``greater'' is meant in absolute terms: a greater positive value than zero or some positive value, or a greater negative value than zero or some negative value). In this way, it is unnecessary for researchers to use any decision threshold for the ``significance,'' ``confidence,'' or ``credible'' level of uncertainty or for the effect size beyond which the effect is considered practically relevant. This means researchers can be agnostic about a decision threshold and a justification for that. In my approach, researchers play a role of an information provider and present different effect sizes and their associated probabilities as such. My approach applies regardless of the types of causal effect estimate (population average, sample average, individual, etc).\n\nThe positive implications of my approach can be summarized as follows. First, my approach could help social scientists avoid the dichotomy of significance vs. insignificance and present statistical uncertainty regarding causal effects as such: a step also recommended by \\textcite{Gelman2017}. This could enable social scientists to better understand and communicate the continuous nature of the uncertainty of statistically estimated causal effects. I demonstrate this point by applying my approach to a previous social scientific study.\n\nSecond, as a result of social scientists using my approach, decision makers could evaluate whether to use a treatment or not, in light of their own utility functions. The conventional thresholds such as $p<5\\%$ could produce statistical \\textit{in}significance for a treatment effect because of a small sample or effect size, even if the true effect were non-zero. Then, decision makers might believe it is evidence for no effect and therefore decide not to use the treatment. This would be a lost opportunity, however, if the probability of the beneficial treatment effect were actually high enough (if not as high as 95\\%) for these decision makers to use the treatment and accept the risk of failure.\n\nFinally, my approach could help mitigate \\textit{p}-hacking and the publication bias, as researchers could feel less need to report only a particular model out of many that produces an uncertainty measure below a certain threshold. It might be argued that my approach will also allow theoretically or methodologically questionable research to claim credibility based on not so high but decent probability (e.g., 70\\%) that some factor affects an outcome. Yet, the threshold of $p<5\\%$ has not prevented questionable research from being published, as the replication crisis suggests. A stricter threshold of a \\textit{p}-value \\autocite[e.g.,][]{Benjamin2018} may reduce, but does not eliminate, the chance of questionable research satisfying the threshold. A stricter threshold also has a side effect: it would increase false negatives and, as a result, the publication bias. In short, the important thing is to evaluate the credibility of research not only based on an uncertainty measure such as a p-value and probability but also on the entire research design and theoretical arguments: ``No single index should substitute for scientific reasoning'' \\autocite[132]{Wasserstein2016a}. While further consideration and research are necessary to understand what the best practice to present and interpret statistical uncertainty is to maximize scientific integrity, I hope my approach contributes to this discussion. In this article, I assume statistically estimated causal effects are not based on questionable research and the model assumptions are plausible.\n\nThe rest of the article further explains the motivation for, and the detail of, my approach, and then applies it to a previous social scientific study. The accompanying R package makes my approach easy to implement (see the section ``Supplemental Materials''). All statistical analyses for this article were done on RStudio \\autocite{RStudioTeam2020} running R version 4.1.0 \\autocite{RCoreTeam2021}. The data visualization was done by the ggplot2 package \\autocite{Wickham2016}.\n\n\n\\section{Motivation}\nBehavioral research suggests the use of probability as a numerical, continuous scale of uncertainty has merits for both decision makers and researchers, compared to a verbal, discrete scale. In communicating the uncertainty of predictions, a numerical scale more accurately conveys the degree of uncertainty researchers intend to communicate \\autocite{Budescu2014, Jenkins2018, Mandel2021}; improves decision makers' forecasting \\autocite{Fernandes2018, Friedman2018}; mitigates the public perception that science is unreliable in case the prediction of unlikely events fails \\autocite{Jenkins2019}; and helps people aggregate the different sources of uncertainty information in a mathematically consistent way \\autocite{Mislavsky2021}. Even researchers familiar with quantitative methods often misinterpret statistical insignificance as evidence for the null effect, while presenting a numerical probability corrects such a dichotomous thinking \\autocite{McShane2016}. My approach builds on these insights into understanding and communicating uncertainty, adding to a toolkit for researchers.\n\nSocial scientists usually turn the continuous measures of the uncertainty of statistically estimated causal effects, such as a \\textit{p}-value and a posterior, into the dichotomy of significance and insignificance using a decision threshold. Such a dichotomy results in the misunderstandings and misuses of uncertainty measures \\autocite[e.g., see][]{Amrhein2019, Esarey2016, Gerber2008, McShane2016, McShane2017, Simonsohn2014}. \\textit{p}-hacking is a practice to search for a model that produces a statistically significant effect to increase the chance of publication, and results in a greater likelihood of false positives being published \\autocite{Simonsohn2014}. If only statistically significant effects are to be published, our knowledge based on published studies is biased -- the publication bias \\autocite[e.g.,][]{Esarey2016, Gerber2008, Simonsohn2014}. Meanwhile, statistical \\textit{in}significance is often mistaken as evidence for no effect \\autocite{McShane2016, McShane2017}. For example, given a decision threshold of 95\\%, both 94\\% probability and 1\\% probability are categorized as statistically insignificant, although the former presents much smaller uncertainty than the latter. If a study failed to find statistical significance for a beneficial causal effect because of the lack of access to a large sample or because of the true effect size being small, it could be a lost opportunity for decision makers. It might be argued that a small effect size is practically irrelevant, but this is not by definition so but depends on contexts. For example, if a treatment had a small but beneficial effect and also were cheap, it could be useful for decision makers.\n\nA decision threshold is not always a bad idea. For example, it may sometimes be necessary for experts to suggest a way for non-experts to make a yes\/no decision \\autocite[271]{Kruschke2018a}. In such a case, a decision threshold may be justified by a utility function tailored to a particular case \\autocites[276]{Kruschke2018a}[170]{Lakens2018}{Mudge2012}.\n\nHowever, social scientific research often examines too heterogeneous cases to identify a single utility function that applies to every case, and may want to be agnostic about it. This may be one of the reasons why researchers usually resort to the conventional threshold such as whether a point estimate has a \\textit{p}-value of less than 5\\%, or whether a 95\\% confidence\/credible interval does not include zero. Yet, the conventional threshold (or any other threshold) is unlikely to be universally optimal for decision making, exactly because how small the uncertainty of a causal effect should be, depends on a utility function, which varies across decision-making contexts \\autocites{Berger1985}[278]{Kruschke2018a}[170]{Lakens2018}{Mudge2012}.\n\nEven if one adopted the view that research should be free from context-specific utility and should use an ``objective'' criterion universally to detect a causal effect, $p<5\\%$ would not be such a criterion. This is because universally requiring a fixed \\textit{p}-value shifts subjectivity from the choice of a decision threshold for a \\textit{p}-value to that for a sample size, meaning that statistical significance can be obtained by choosing a large enough sample size subjectively \\autocite{Mudge2012}. If ``anything that plausibly could have an effect will not have an effect that is exactly zero'' in social science \\autocite[961]{Gelman2011}, a non-zero effect is by definition ``detectable'' as a sample size increases.\n\nWhile some propose, as an alternative to the conventional threshold, that we evaluate whether an interval estimate excludes not only zero but also practically null values \\autocite{Gross2015, Kruschke2018a}, this requires researchers to justify why a certain range of values should be considered practically null, which still depends on a utility function \\autocite[276--78]{Kruschke2018a}. My approach avoids this problem, because it does not require the use of any decision threshold either for an uncertainty measure or for an effect size. Using a posterior distribution, it simply presents the probabilities of different effect sizes as such. My approach allows researchers to play a role of an information provider for the uncertainty of statistically estimated causal effects, and let decision makers use this information in light of their own utility functions.\n\nThe standard ways to summarize a posterior distribution are a probability density plot, the probability of a one-sided hypothesis, and a credible interval. My approach is different from these in the following respects. A probability density plot shows the full distribution of parameter values. Yet, it is difficult for readers to accurately compute a probability mass for a specific range of parameter values, just by looking at a probability density plot \\autocite{Kay2016}.\n\nPerhaps for this reason, researchers sometimes report together with a density plot either (1) the probability mass for parameter values greater\/smaller than a particular value (usually zero), i.e., the probability of a one-sided hypothesis, or (2) a credible interval. However, these two approaches also have drawbacks. In the case of the probability of a one-sided hypothesis, a specific value needs to be defined as the decision threshold for the effect size from which on the probability is computed. Therefore, it brings the aforementioned problem, i.e., demanding a justification for that particular threshold based on a utility function. In the case of a credible interval, we must decide what level of probability is used to define an interval, and what are practically null values \\autocite{Kruschke2018a}. This also brings the problem of demanding a justification for particular thresholds, one for the level of probability and the other for practically null values, based on a utility function. In addition, when a credible interval includes conflicting values (e.g., both significant positive and significant negative values), it can only imply inconclusive information \\autocite{Kruschke2018a}, although the degree of uncertainty actually differs depending on how much portion of the interval includes these conflicting values (e.g., 5\\% of the interval vs. 50\\% of the interval).\n\nExperimental studies by \\textcites{Allen2014, Edwards2012, Fernandes2018} suggest the plot of a complementary cumulative distribution, such as my approach as explained in the next section, is one of the best ways to present uncertainty (although their findings are about the uncertainty of predictions rather than that of causal effects). The plot of a complementary cumulative distribution presents the probability of a random variable taking a value greater (in absolute terms) than some specific value. Formally: $P(X>x)$, where $X$ is a random variable and $x$ is a particular value of $X$. \\textcites{Allen2014, Edwards2012} find the plot of a complementary cumulative distribution is effective both for probability estimation accuracy and for making a correct decision based on the probability estimation, even under time pressure \\autocite{Edwards2012} or cognitive load \\autocite{Allen2014}. Note that \\textcites{Allen2014, Edwards2012} indicate a complementary cumulative distribution is unsuitable to accurately estimating the mean of a distribution. If the quantities of interest included the mean of a posterior distribution, one could report it together. \\textcite{Fernandes2018} find the plot of a complementary cumulative distribution is one of the best formats for optimal decision making over repeated decision-making contexts, while a probability density plot, a one-sided hypothesis, and a credible interval are not.\n\nI am not arguing my proposed approach should be used or ideal to report the uncertainty of statistically estimated causal effects in every case. For example, if one were interested in the Frequentist coverage of a fixed parameter value, she would rather use a confidence interval over repeated sampling. Note that repeated sampling for the same study is usually uncommon in social science; uncertainty is usually specific to a single study.\n\nIt is possible that there is no universally best way to present uncertainty \\autocite{Visschers2009}. It is also possible that ``the presentation format hardly has an impact on people in situations where they have time, motivation, and cognitive capacity to process information systematically'' \\autocites[284]{Visschers2009}[but also see][]{Suzuki2021}. Yet, even those people would be unable to evaluate uncertainty properly, if the presentation provided only limited information because of its focus on a specific decision threshold without giving any justification for that threshold. For example, if only a 95\\% credible interval were presented, it would be difficult to see what the range of the effect sizes would be if a decision maker were interested in a different probability level (e.g., 85\\%). My approach conveys the uncertainty of statistically estimated causal effects without any decision threshold. \n\n\n\\section{The Proposed Approach}\nMy approach utilizes a posterior distribution estimated by Bayesian statistics \\autocite[for Bayesian statistics, see for example][]{Gelman2013BDA, Gill2015, Kruschke2015, McElreath2016}. A posterior, denoted as $p(\\theta|D,\\ M)$, is the probability distribution of a parameter, $\\theta$, given data, $D$, and model assumptions, $M$, such as a functional form, an identification strategy, and a prior distribution of $\\theta$. Here, let us assume $\\theta$ is a parameter for the effect of a causal factor.\n\nUsing a posterior distribution, we can compute the probabilities of different effect sizes. More specifically, we can compute the probability that a causal factor has an effect greater (in absolute terms) than some effect size. Formally: $P(\\theta>\\tilde\\theta^{+}| D, M)$ for the positive values of $\\theta$, where $\\tilde\\theta^{+}$ is zero or some positive value; $P(\\theta<\\tilde\\theta^{-}| D, M)$ for the negative values of $\\theta$, where $\\tilde\\theta^{-}$ is zero or some negative value. If we compute this probability while changing $\\tilde\\theta$ up to theoretical or practical limits (e.g., up to the theoretical bounds of a posterior or up to the minimum\/maximum in posterior samples), it results in a complementary cumulative distribution.\n\nAs often the case, a posterior distribution might include both positive and negative values. In such a case, we should compute two complementary distribution functions, one for the positive values of $\\theta$, i.e., $P(\\theta>\\tilde\\theta^{+}| D, M)$ and the other for the negative values of $\\theta$, i.e., $P(\\theta<\\tilde\\theta^{-}| D, M)$. Whether it is plausible to suspect the effect can be either positive or negative, depends on a theory. If only either direction of the effect is theoretically plausible \\autocite[e.g., see][]{Vanderweele2010}, this should be reflected on the prior of $\\theta$, e.g., by putting a bound on the range of the parameter.\n\nFigure \\ref{ccdfExa} is an example of my approach. I use a distribution of 10,000 draws from a normal distribution with the mean of 1 and the standard deviation of 1. Let us assume these draws are those from a posterior distribution, or ``posterior samples,'' for a coefficient in a linear regression, so that the values represent a change in an outcome variable. The x-axis is different effect sizes, or different values of minimum predicted changes in the outcome; the y-axis is the probability of an effect being greater than a minimum predicted change. The y-axis has the adjective ``near'' on 0\\% and 100\\%, because the normal distribution is unbounded and, therefore, the plot of the posterior samples cannot represent the exact 0\\% and 100\\% probability.\n\n\\begin{figure}[t]\n \\includegraphics[scale=0.15]{ccdfExa.png}\n \\centering\n \\caption{Example of presenting a posterior as a complementary cumulative distribution plot.}\n \\label{ccdfExa}\n\\end{figure}\n\nFrom the figure, it is possible to see what is the probability of an effect being greater (in absolute terms) than a certain effect size. For example, we can say: ``The effect is expected to increase the outcome by greater than zero point with a probability of approximately 84\\%.'' It is also clear that the positive effect is much more probable than the negative effect: 16\\% probability for $\\theta<0$ while 84\\% probability for $\\theta>0$. Finally, with a little more attention, we can also read the probability of a range of the effect size. For example, we can compute the probability that the effect increases the outcome by greater than one point and up to three points, as $P(\\theta>1) - P(\\theta>3) \\approx .49 - .02 = .47$. This computation is useful if there is such a ``too much'' effect for a treatment to become counterproductive (e.g., the effect of a diet on weight loss).\n\nFor comparison, I also present the standard ways to summarize a posterior, using the same 10,000 draws as in Figure \\ref{ccdfExa}: a probability density plot (Figure \\ref{pdfExa}), a credible interval, and one-sided hypotheses (both in Table \\ref{tableExa}). The probability density plot gives an overall impression that positive values are more probable than negative values. However, it is difficult to compute exactly what probability an effect is greater than, say, 1. Unlike my approach, the y-axis is density rather than probability, and density is not an intuitive quantity.\n\nThe credible interval is computed based on the conventional decision threshold of the 95\\% credible level. It includes both negative and positive values and, therefore, the conventional approach leads either to the conclusion that the effect is not statistically significant, or to the one that there is no conclusive evidence for the effect \\autocite{Gross2015, Kruschke2018a}. The one-sided hypotheses are computed based on the decision threshold of the null effect, i.e., $P(\\theta>0)$ and $P(\\theta<0)$, as commonly done. Because of this threshold, the information of the uncertainty is much more limited than what my approach presents in Figure \\ref{ccdfExa}. Most importantly, the use of these decision thresholds require researchers to justify why these thresholds should be used, rather than, say, the 94\\% credible interval or $P(\\theta>0.1)$ and $P(\\theta<-0.1)$. This problem does not apply to my approach. My approach can be considered as a generalized way to use one-sided hypotheses. It graphically summarizes all one-sided hypotheses (up to a practically computable point), using different effect sizes as different thresholds from which on the probability of an effect is computed.\n\n\\begin{figure}[t]\n \\includegraphics[scale=0.15]{pdfExa.png}\n \\centering\n \\caption{Example of presenting a posterior as a probability density plot.}\n \\label{pdfExa}\n\\end{figure}\n\n\\begin{table}[t]\n\\centering\n\\begin{tabular}{c c c c}\n\\hline\nMean & 95\\% Credible Interval & $P(\\theta>0)$ & $P(\\theta<0)$\\\\\n\\hline\n$0.98$ & [$-1.01$, $2.92$] & $0.84$ & $0.16$\\\\\n\\hline\n\\end{tabular}\n\\caption{Example of presenting a posterior as a credible interval or as one-sided hypotheses.}\n\\label{tableExa}\n\\end{table}\n\nI note three caveats about my approach. First, it takes up more space than presenting regression tables, the common presentation style in social science. Therefore, if we wanted to present all regression coefficients using my approach, it would be too cumbersome. Yet, the purpose of quantitative causal research in social science is usually to estimate the effect size of a causal factor or two, and the remaining regressors are controls to enable the identification of the causal effect(s) of interest. Indeed, it is often hard to interpret all regression coefficients causally because of complex causal mechanisms \\autocite{Keele2020}. When researchers use matching instead of regression, they typically report only the effect of a treatment variable \\autocite[1--2]{Keele2020}, although the identification strategy is the same -- selection on observables \\autocite[321--22]{Keele2015}. Thus, if researchers are interested in estimating causal effects, they usually need to report only one or two causal factors. If so, it is not a problem even if my proposed approach takes more space than regression tables.\n\nSecond, if the effect size is scaled in a nonintuitive measure (such as a log odds ratio in logistic regression), researchers will need to convert it to an intuitive scale to make my approach work best \\autocite[for detail, see][]{Sarma2020}. For example, in the case of logistic regression, researchers can express an effect size as a difference in the predicted likelihood of an outcome variable.\n\nThird, as all models are wrong, Bayesian models are also wrong; they are the simplification of reality. A posterior distribution is conditional on data used and model assumptions. It cannot be a reliable estimate, if data used are inadequate for the purpose of research (e.g., a sample being unrepresentative of the target population or collected with measurement errors), and\/or if one or more of the model assumptions are implausible (which is usually the case). Moreover, in practice a posterior usually needs to be computed by a Markov chain Monte Carlo (MCMC) method, and there is no guarantee that the resulting posterior samples precisely mirror the true posterior. Therefore, the estimated probability of an effect should not be considered as the ``perfect'' measure of uncertainty. For example, even if the estimated probability of an effect being greater than zero is 100\\% \\textit{given the model and the computational method}, it should NOT be interpreted as the certainty of the effect \\textit{in practice}.\n\nGiven these limitations, the same principle applies to my proposed approach as to the \\textit{p}-value: ``No single index should substitute for scientific reasoning'' \\autocite[132]{Wasserstein2016a}. What matters is not the ``trueness'' of a model but the ``usefulness'' of a model. My approach makes a model more useful than the conventional approaches to evaluating and communicating the uncertainty of statistically estimated causal effects, in the following respects. First, it uses the probability of an effect as an intuitive quantity of uncertainty, for a better understanding and communication. Second, it does not require any decision thresholds for uncertainty measures or effect sizes. Therefore, it allows researchers to be agnostic about a utility function required to justify such decision thresholds, and to be an information provider presenting the probabilities of different effect sizes as such.\n\n\n\\section{Application}\nI exemplify my proposed approach by applying it to a previous social scientific study \\autocite{Huff2016} and using its dataset \\autocite{Huff2015a}. \\textcite{Huff2016} collected a nationally representative sample of 2,000 Polish adults and experimented whether more violent methods of protest by an opposition group increase or decrease public support for the government negotiating with the group. Specifically, I focus on the two analyses in Figure 4 of \\textcite{Huff2016}, which present (1) the effect of an opposition group using bombing in comparison to occupation, and (2) the effect of an opposition group using occupation in comparison to demonstrations, on the attitude of the experiment participants towards tax policy in favor of the opposition group. The two treatment variables are measured dichotomously, while the attitude towards tax policy as the dependent variable is measured in a 100-point scale. \\textcite{Huff2016} use linear regression per treatment variable to estimate its average effect. The model is $Y=\\beta_0+\\beta_{1}D+\\epsilon$, where $Y$ is the dependent variable, $D$ is the treatment variable, $\\beta_{0}$ is the constant, $\\beta_{1}$ captures the size of the average causal effect, and $\\epsilon$ is the error term.\n\nIn the application, I convert the model to the following equivalent Bayesian linear regression model:\n\n\\begin{align*}\ny_{i} & \\sim Normal(\\mu_{i}, \\sigma),\\\\\n\\mu_{i} & = \\beta_0 + \\beta_1 d_{i},\\\\\n\\beta_{0} & \\sim Normal(\\mu_{\\beta_0}=50,\\sigma_{\\beta_0}=20),\\\\\n\\beta_{1} & \\sim Normal(\\mu_{\\beta_1}=0,\\sigma_{\\beta_1}=5),\\\\\n\\sigma & \\sim Exponential(rate=0.5),\n\\end{align*}\n\n\\noindent\nwhere $y_{i}$ and $d_{i}$ are respectively the outcome $Y$ and the treatment $D$ for an individual $i$; $Normal(\\cdot)$ denotes a normal distribution; $\\mu_{i}$ is the mean for $i$ and $\\sigma$ is the standard deviation in the normal distribution likelihood. For the quantity of interest $\\beta_{1}$, I use a weakly informative prior of $Normal(\\mu_{\\beta_1}=0,\\sigma_{\\beta_1}=5)$. This prior reflects the point that the original study presents no prior belief in favor of the negative or positive average effect of a more violent method by an opposition group on public support, as it hypothesizes both effects as plausible. For $\\beta_{0}$, the constant term, I use a weakly informative prior of $Normal(\\mu_{\\beta_0}=50,\\sigma_{\\beta_0}=20)$; this prior means that, without the treatment, respondents are expected to have a neutral attitude on average, but the baseline attitude of each individual may well vary. For $\\sigma$ in the likelihood, I put a weakly informative prior of the exponential distribution $Exponential(rate=0.5)$, implying any stochastic factor is unlikely to change the predicted outcome value by greater than 10 points. I use four chains of MCMC process; per chain 10,000 iterations are done, the first 1,000 of which are discarded. The MCMC algorithm used is Stan \\autocite{StanDevelopmentTeam2019b}, implemented via the rstanarm package version 2.21.1 \\autocite{Goodrich2020a}. The $\\hat{R}$ was approximately 1.00 for every estimated parameter, suggesting the models did not fail to converge. The effective sample size exceeded at least 15,000 for every estimated parameter.\n\nTable \\ref{tableHK} presents the results in a conventional way: the mean in the posterior of $\\beta_{1}$ and the 95\\% credible interval for the model examining the effect of bombing in comparison to occupation, and those for the model examining the effect of occupation in comparison to demonstrations. While the mean is a negative value in both models, the 95\\% credible interval includes not only negative values but also zero and positive values. This means the typical interval approach \\autocite[e.g.,][]{Gross2015, Kruschke2018a} would lead us to simply conclude there is no conclusive evidence for the average effect.\n\n\\begin{table}[t]\n\\centering\n\\begin{tabular}{c c c c}\n\\hline\n& Mean $\\beta_{1}$ & 95\\% Credible Interval & N\\\\\n\\hline\nbombing vs. occupation & $-2.49$ & [$-5.48$, $0.49$] & 996\\\\\noccupation vs. demonstration & $-0.53$ & [$-3.31$, $2.32$] & 985\\\\\n\\hline\n\\end{tabular}\n\\caption{Results using the 95\\% credible interval. $\\hat{R}\\cong1.00$ for all parameters.}\n\\label{tableHK}\n\\end{table}\n\nFigure \\ref{ccdfBom} uses my approach for the effect of bombing in comparison to occupation. It enables richer inference than the above conventional approach. If we focus on the probability of $\\beta_{1}<0$ for example, the model expects that if an opposition group uses bombing instead of occupation, it should reduce public support for tax policy in favor of the group by greater than 0 point, with a probability of approximately 95\\%. Thus, the negative effect of bombing is much more likely (95\\% probability) than the positive effect (5\\% probability).\n\n\\begin{figure}[t]\n \\includegraphics[scale=0.15]{ccdfBom.png}\n \\centering\n \\caption{Effect of bombing in comparison to occupation.}\n \\label{ccdfBom}\n\\end{figure}\n \nThe original conclusion was that the effect was not statistically significant at $p<5\\%$, the threshold set at the pre-registration stage \\autocite[1794--1795]{Huff2016}. However, they put a \\textit{post hoc} caveat that if the threshold of statistical significance had been set at 10\\%, the effect would have been regarded as statistically significant \\autocite[1795]{Huff2016}. This interpretation is inconsistent either with the Fisherian paradigm of \\textit{p}-values or with the Neyman-Person paradigm of \\textit{p}-values \\autocite{Lew2012}. According to the Fisherian paradigm, the preset threshold of statistical significance is unnecessary, because a \\textit{p}-value in this paradigm is a local measure and not a global false positive rate -- the rate of false positives over repeated sampling from the same data distribution \\autocite[1562--63]{Lew2012}. An exact \\textit{p}-value should be interpreted as such -- although it is difficult to make intuitive sense of because it is not the probability of an effect but the probability of obtaining data as extreme as, or more extreme than, those that are observed, given the null hypothesis being true \\autocites[1560]{Lew2012}[131]{Wasserstein2016a}. According to the Neyman-Person paradigm, no \\textit{post hoc} adjustment to the preset threshold of statistical significance should be made, because a \\textit{p}-value in this paradigm is used as a global false positive rate and not as a local measure \\autocite[1562--63]{Lew2012}. The estimates from my approach are more straightforward to interpret. We need no \\textit{a priori} decision threshold of a posterior to determine significance, and can \\textit{post hoc} evaluate the probability of an effect \\autocite[328]{Kruschke2015}.\n\nFigure \\ref{ccdfOcc} uses my approach for the effect of occupation in comparison to demonstrations. The model expects that if an opposition group uses occupation instead of demonstrations, it should reduce public support for tax policy in favor of the group by greater than 0 point, with a probability of approximately 64\\%. This suggests weak evidence, rather than no evidence, for the negative effect of occupation, meaning that the negative effect is more likely (64\\% probability) than the positive effect (36\\% probability). Meanwhile, the original conclusion was simply that the effect was not statistically significant at $p<5\\%$ \\autocite[1777, 1795]{Huff2016}.\n\n\\begin{figure}[t]\n \\includegraphics[scale=0.15]{ccdfOcc.png}\n \\centering\n \\caption{Effect of occupation in comparison to demonstration.}\n \\label{ccdfOcc}\n\\end{figure}\n\nIn short, my approach presents richer information about the uncertainty of the statistically estimated causal effects, than the significance-vs.-insignificance approach taken by the original study. It helps a better understanding and communication of the uncertainty of the average causal effects of violent protesting methods.\n\n \n\\section{Conclusion}\nI have proposed the alternative approach to social scientists presenting the uncertainty of statistically estimated causal effects: the probabilities of different effect sizes via the plot of a complementary cumulative distribution function. Unlike the conventional significance-vs.-insignificance approach and the standard ways to summarize a posterior distribution, my approach does not require any preset decision threshold for the ``significance,'' ``confidence,'' or ``credible'' level of uncertainty or for the effect size beyond which the effect is considered practically relevant. It therefore allows researchers to be agnostic about a decision threshold and a justification for that. In my approach, researchers play a role of an information provider and present different effect sizes and their associated probabilities as such. I have shown through the application to the previous study that my approach presents richer information about the uncertainty of statistically estimated causal effects than the conventional significance-vs.-insignificance approach, helping a better understanding and communication of the uncertainty.\n\nMy approach has implications for problems in the current (social) scientific practices, such as \\textit{p}-hacking, the publication bias, and seeing statistical insignificance as evidence for the null effect \\autocite{Amrhein2019, Esarey2016, Gerber2008, McShane2016, McShane2017, Simonsohn2014}. First, if the uncertainty of statistically estimated causal effects were reported by my approach, both researchers and non-experts would be able to understand it intuitively, as probability is used as a continuous measure of uncertainty in everyday life (e.g., a weather forecast for the chance of rain). This could help mitigate the dichotomous thinking of there being an effect or no effect, commonly associated with the significance-vs.-insignificance approach. Second, if research outlets such as journals accepted my approach as a way to present the uncertainty of statistically estimated causal effects, researchers could feel less need to report only a model that produces an uncertainty measure below a certain threshold, such as $p<5\\%$. This could help address the problem of \\textit{p}-hacking and the publication bias.\n\nPresenting the uncertainty of statistically estimated causal effects as such has implications for decision makers as well. If decision makers had access to the probabilities of different effect sizes, they could use this information in light of their own utility functions and decide whether to use a treatment or not. It is possible that even when the conventional threshold of the 95\\% credible level produced statistical \\textit{in}significance for a treatment effect, decision makers could have such a utility function to prefer using the treatment over doing nothing, given the level of probability that does not reach the conventional threshold but they see ``high enough.'' It is also possible that even when a causal effect reaches the conventional threshold of the 95\\% credible level, decision makers could have such a utility function to see the probability not high enough (e.g., must be 99\\% rather than 95\\%) and prefer doing nothing over using the treatment.\n\nI acknowledge my proposed approach may not suit everyone's needs. Yet, I hope this article provides a useful insight into how to understand and communicate the uncertainty of statistically estimated causal effects, and the accompanying R package helps other researchers to implement my approach without difficulty.\n\n\n\\section*{Acknowledgments}\nI would like to thank Johan A. Elkink, Jeff Gill, Zbigniew Truchlewski, Alexandru Moise, and participants in the 2019 PSA Political Methodology Group Annual Conference, the 2019 EPSA Annual Conference, and seminars at Dublin City University and University College Dublin, for their helpful comments. I would like to acknowledge the receipt of funding from the Irish Research Council (the grant number: GOIPD\/2018\/328) for the development of this work. The views expressed are my own unless otherwise stated, and do not necessarily represent those of the institutes\/organizations to which I am\/have been related.\n\n\n\\section*{Supplemental Materials}\nThe R code to reproduce the results in this article and the R package to implement my approach (``ccdfpost'') ia available on my website at \\url{https:\/\/akisatosuzuki.github.io}.\n\n\n\\printbibliography\n \n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{INTRODUCTION}\n\\label{sec:intro}\n\nAmorphous carbon (a-C) has been intensively investigated over the\nyears, but its properties are not fully understood. \nSome of them are strongly debated. Tetrahedral a-C (ta-C), containing a \nhigh fraction of $sp^3$ hybrids, is the form of a-C which\nhas drawn most attention because of its\ndiamondlike properties \\cite{Robertson02,Silva02}, including high\nhardness for mechanical purposes, a wide band gap for optical \napplications, and biocompatibility for biomedical coatings. \nTa-C has also promising applications in micro-electromechanical \ndevices (MEMS).\n\nRecently, nanostructured amorphous carbon (na-C) has attracted \nattention \\cite{Sabra98,Banhart99}. It is a hybrid form of carbon in which \nnanocrystallites are embedded in the a-C matrix. These range from \nnanodiamonds \\cite{Lifshitz}, to open graphene structures \nwith negative curvature (schwarzites) \\cite{Vanderbilt,Donadio},\nto carbyne films (composed of $sp^1$ chainlike structures) \n\\cite{Barborini,Ravagnan}. The variety of embedded nanostructures, \nand the great number of possible configurations in the matrix, open\nsome new ways to explore the physics, tailor the mechanical and electronic \nproperties, and extend the applications of a-C.\n\nFrom a fundamental point of view, the most important characteristic\nof na-C is its inhomogeneous nature, characterized by large gradients of \ndensity and coordination through the system. The challenge for the theorist,\ntherefore, is to provide a global description of the composite\nmaterial, extending and\/or generalizing concepts and trends which\napply to the better understood single-phase system. \n\nHere, we present recent work \\cite{Fyta03,Mathiou04} aiming at such\na theoretical description. It is based on tight-binding molecular\ndynamics (TBMD) and Monte Carlo (MC) simulations. We first review\nwork on single-phase a-C pertained to its structure, stress\nstate, and to physical trends followed by the $sp^3$ fraction and\nelastic moduli as a function of density and mean coordination. We then\ndiscuss na-C, focusing on diamond nanocomposite films. Emphasis is given \non their structure, stability, stress state, and hardness. \nOne of the important findings is that the hardness of nanocomposite \nfilms is considerably higher than that of single-phase a-C films.\n\n\\section{METHODOLOGY}\n\\label{sec:method}\n\nIn much of the work presented here, we treat the interatomic \ninteractions in a-C networks using the tight-binding (TB) method. \nThis bridges the gap between classical and first-principles \ncalculations. It is more accurate and transferable\nthan empirical schemes, providing a quantum-mechanical \ndescription of the interactions, and it yields greater\nstatistical precision than {\\it ab initio} methods, allowing the use \nof larger cells.\n\nWe use the environment-dependent tight-binding\n(EDTB) model of Tang, Wang, Chan, and Ho \\cite{TWCH}. This model\ngoes beyond the traditional two-center approximation and allows the\nTB parameters to change according to the bonding environment. In this\nrespect, it is a considerable improvement over the\nprevious two-center model of Xu, Wang, Chan, and Ho \\cite{XWCH}. Both\naccuracy and transferability are improved, as shown from recent\nsuccessful applications \\cite{Galli1}.\n\nThe TBMD simulations are carried out in the canonical ($N,V,T$) ensemble.\n$T$ is controlled {\\it via} a stochastic temperature control \nalgorithm. The a-C networks are generated by quenching from\nthe melt. Although not directly related to the kinetics of the growth \nprocess of a-C films, it produces generic structures associated with \nthe equilibrium state of the films. (See relevant discussion \nin Ref. \\cite{Kel00}.)\nCubic computational cells of 216 and 512 atoms with periodic boundary \nconditions are used. Quenching at different durations and rates \nwas performed to check the effect on the properties. \nThe longer run was for 52 ps and the rate was 226 K\/ps. \nTwo other runs lasted for 26 and 12 ps, at 226 K\/ps and 500 K\/ps, \nrespectively. No significant changes were found.\nDuring quenching, the volume\/density of the \ncells was kept constant. After quenching, the density was\nallowed to relax by changing homogeneously the dimensions of the cells\nwithin small increments and seeking energy minimization. The minimum \nenergy and the corresponding density and bulk modulus of each cell were \ndetermined by fitting the energy-versus-volume data to Murnaghan's \nequation of state \\cite{Murnaghan}.\n\nDiamond nanocomposite cells are generated by melting and subsequent\nquenching a diamond structure, while keeping a certain number of atoms\nin the central portion of the cell frozen in their ideal crystal \npositions. After quenching, which produces amorphization of the\nsurrounding matrix, the cells are thoroughly relaxed with respect to\natom positions and density. Cells with varying coordination (density) \nof the amorphous matrix can be formed by changing the initial starting \ndensity (volume) of the diamond structure. The size (radius) of the \nnanocrystals is controlled by the choice of the number of the shells kept \nfrozen during quenching. Their shape is spherical. The nanocomposite\ncells produced by TBMD contain 512 atoms.\n\nThe TBMD simulations are supplemented by MC simulations using larger\ncells of 4096 atoms to make certain investigations, such as the\nstability of nanocomposite films as a function of the density\nof the amorphous matrix and the stress analysis, tractable. The\nTersoff potential is used \\cite{Terspot}. \n\nIn addition, we examine the properties of the WWW generic\nmodel \\cite{Wooten,Djord1}. This is a hypothetical model of \n``amorphous diamond'', completely tetrahedral,\nconstructed from the diamond lattice by a bond-switching mechanism.\nWe relaxed its topology and density with the\nEDTB model. The WWW model, although hypothetical, is very useful \nbecause it provides an upper bound to the density, $sp^3$ fraction, \nand bulk modulus of single-phase a-C. Its properties can then \nbe compared to the respective ones from diamond nanocomposites.\n\n\\section{RESULTS AND DISCUSSION}\n\n\\subsection{Single-phase a-C}\n\nWe first discuss the pure a-C phase, reviewing past and recent\nwork. To examine its microstructure, let us look at two \nrepresentative examples, among the sequence of structures characterized\nby different density and $sp^3$ fraction. One for the ta-C dense phase, \nand the other for a low-density phase. Their networks are portrayed in \nFig. 1. \n\nThe ta-C network in panel (a) has a density of 2.99\ngcm$^{-3}$, it shows a clear predominance of $sp^3$ bonding (79\\%), \nand it reveals that the $sp^2$ sites are largely \nclustered \\cite{Frau93,Drabold,Marks96}. Clustering is present in the\nform of olefinic, chainlike geometries. The $sp^2$ chains are isolated\nand do not link (percolate) to a single spanning cluster, in agreement\nwith {\\it ab initio} work \\cite{Marks96}. The driving force behind \nthe clustering effect is stress relief. Earlier work \\cite{Kel00},\naddressing the issue of local rigidity in ta-C, showed \nthat clustering contributes stress relief and rigidity to the \nnetwork. This depends on the degree of clustering. The larger the \ncluster, the higher the stress relaxation and the contribution to \nrigidity in the network.\n\nThe low-density network in panel (b) has a density of 1.20 gcm$^{-3}$\nand contains only 1\\% of $sp^3$ sites. It has an open structure with \nlong chains and large rings, and with numerous $sp^1$ sites (33\\%). \nThis network should be typical of cluster-assembled carbon \nfilms \\cite{Barborini,Ravagnan} with an amorphous $sp^2$ character \nand a sizeable carbyne ($sp^1$ chains) component. Such films have\nattracted attention for various applications, including field emission, \ncatalysis, and gas absorption. \n\nAnalysis of the ring statistics in the ta-C networks reveals the\nexistence of three- and four-membered rings. (The shortest-path criterion \nof Franzblau \\cite{Franzblau} was used to define the ring sizes.)\nThis is the first tight-binding model which predicts three-membered\nrings in ta-C, in agreement with {\\it ab initio} MD \nsimulations \\cite{Marks96} using the Car-Parrinello method.\nThe five-membered rings are slightly more numerous than the six-membered\nrings and significantly more numerous than the seven-membered ones.\n\nThe issue of intrinsic stress and its association to $sp^3$ bonding\nin ta-C films has been strongly debated over the\nyears. We have now reached at a rather clear picture of this issue.\nWe summarize here the important points. Work by McKenzie and \nco-workers \\cite{McKenz1} proposed that the compressive stress in ta-C\nis produced by the energy of ion bombardment in the deposition process \nwhich gives rise to local compression, accompanied by the shallow \nimplantation of incoming atoms. Their model considers the compressive \nstress as the causative factor for the formation of sp$^{3}$ sites \nand supports the idea of a transition from an sp$^{2}$-rich to an \nsp$^{3}$-rich phase at a critical value of the average compressive stress \n(about 4-5 GPa) which stabilizes the sp$^{3}$ bonding.\n\nWhile this scenario can not be ruled out for as-grown films, it fails to\ndescribe the stress state of post-growth annealed\/relaxed films. It has\nnow become apparent, after a series of theoretical works by \nKelires \\cite{Kel00,Kel94,Kel01Phy} and thermal annealing experiments\n\\cite{Friedmann,Ferrari,Kalish,Alam}, that the intrinsic stress is\nnot a crucial factor for the stabilization of sp$^{3}$ bonding.\nA critical {\\it average} compressive stress necessary to sustain a high\nfraction of sp$^{3}$ sites, as required by the McKenzie model,\ndoes not exist. This conclusion is borne out of the {\\it local atomic stress}\nmodel of Kelires \\cite{Kel94}, which proposes that the average intrinsic\nstress of relaxed ta-C films can be zero, while stress at the atomic level\ncan be finite and substantial. It further says that the favored stress \nstate of sp$^{3}$ sites is compression, while that of sp$^{2}$ sites\nis tension, the latter playing the role of relieving stress in the\nnetwork. \n\nAccording to the local stress model, the as-grown, highly strained and\nsp$^{3}$-rich ta-C films are in a metastable state with respect to the \nrelaxed, stress-free and still sp$^{3}$-rich ``quasi-equilibrium'' \nta-C structures. (True equilibrium structures are the \ngraphite-like sp$^{2}$-rich films.) The as-grown films\npossess high intrinsic stress because the stressed non-equilibrium local\nstructures are frozen-in during deposition, but the network at the\nlow deposition temperatures does not acquire enough energy, or it is\nvery slow at typical times in the laboratory, in order to overcome\nthe potential barrier between the two states and relax the excessive\nstress. Post-growth thermal annealing at moderate $T$'s has proved to \nbe a very efficient mechanism for providing the necessary\nenergy in ta-C films to reach their quasi-equilibrium, stress-free state.\nThe stress relief can be achieved with minimal structural \nmodifications \\cite{Ferrari,Sullivan}, without reducing the sp$^{3}$\nfraction. However, further annealing above $\\sim$ 1200 K transforms \nta-C into the graphite-like sp$^{2}$-rich phase.\n\nAnother issue which is still unclear regards the variation of $sp^3$ \nfraction or, equivalently, of mean coordination $\\bar{z}$, \nwith density. The basic question underlying this issue is whether there \nis a linear relationship between these two quantities.\nWe have recently carried out an extensive investigation \\cite{Mathiou04}\nof this issue through the entire range of densities relevant to a-C,\nusing the TBMD method. Several networks have been generated, at\nvarious quenching rates, providing sufficient statistics to reach\na definite conclusion. We also compare to the WWW network relaxed with \nthe EDTB model.\n\nThe variation of $sp^3$ fraction with density is shown in Fig. 2. \n(Hybrid fractions are extracted by counting neighbors\nwithin and up to the first minimum of the pair distribution \nfunctions, not shown.) Without any doubt, the variation\nis linear through the entire range of possible densities.\nA linear fit to the points gives\n\\begin{equation}\n\\rho (\\rm{g\/cm}^{3}) = 1.27 + 2.08\\ (sp^{3} \\rm{fraction}).\n\\label{den-sp3}\n\\end{equation}\nEq. (\\ref{den-sp3}) predicts the minimum density required to\nsustain $sp^3$ bonding in a-C to be $\\sim$ 1.3 gcm$^{-3}$. \nThe $sp^3$ sites are needed in such low-density networks as linking \ngeometries between the main $sp^2$ and $sp^1$ components. For 100\\%\n$sp^3$ bonding, the corresponding density is 3.35 gcm$^{-3}$.\nThis is slightly higher than the density of the WWW network, but still \nless than diamond's by $\\sim$ 3\\%. We conclude that this is the upper \nlimit in the possible densities of ta-C. The highest densities for ta-C \nreported until now by experiment are less than 3.3 gcm$^{-3}$.\n\nUnfortunately, experimental results do not provide a\nclear picture of this issue. Different growth and characterization\ntechniques give sets of data which show a linear variation within\nthe respective set, but not when viewed all together. (A thorough discussion\nof this point is given in Ref.\\ \\cite{Mathiou04}.) Very good agreement \nbetween theory and experiment holds for the ta-C region, i.e., for\ndensities higher than $\\sim$ 2.8 gcm$^{-3}$. At lower densities,\nexperimental points scatter from method to method. For example,\ndata extracted from samples prepared by filtered\ncathodic vacuum arc (FCVA) deposition \\cite{Fallon,FerrPRB00}\ndiffer from data extracted from samples prepared by\nmagnetron sputtering (MS) \\cite{Schwan97}. \nA linear fit over the FCVA data was carried out \nby Ferrari {\\it et al.} \\cite{FerrPRB00}, yielding\n$\\rho$ (g\/cm$^{3})$ = 1.92 + 1.37 ($sp^{3}$ fraction). This gives a\ndensity of $\\sim$ 3.3 gcm$^{-3}$ for 100\\% $sp^{3}$ content, in good\nagreement with our upper limiting value, but the lower limit at \n1.92 gcm$^{-3}$ is higher than ours, suggesting that $sp^{3}$ hybrids \nare absent in networks with lower densities. This can not explain reports \nof $sp^{3}$ sites in low-density carbyne films \\cite{Barborini,Ravagnan}.\nNote, however, that there are uncertainties in the measurements, usually \nby EELS, of the $sp^{3}$ content in such films.\n\nAn equally interesting physical trend in a-C is the\nvariation of elastic moduli as a function of mean coordination.\nWe seek to find simple formulas able to predict the \nhardness and related properties for any given network, over the entire\nrange of densities.\n\nThorpe and collaborators \\cite{Thorpe85,Djord2} suggested that\nthe elastic moduli of bond-depleted crystalline diamond lattices\nand of bond-depleted ``amorphous diamond'' networks (WWW model)\nfollow a power-law behavior $c \\sim (\\bar{z} - \\bar{z}_{f})^{\\nu}$,\nwith the exponent taking the value 1.5 $\\pm$ 0.2. This mean-field\nequation is characteristic of percolation theory, and describes\nthe contributions to rigidity from the local components of the\nsystem as they connect to each other. The critical\ncoordination $\\bar{z}_f$ = 2.4, denotes the transition from rigid to\nfloppy behavior, and comes out of the constraint-counting model of\nPhillips \\cite{Philips79} and Thorpe \\cite{Thorpe83}.\n\nWe examined whether more realistic a-C networks can be described \nby the constraint-counting model, and if their moduli exhibit a power-law\nbehavior. For this, we used the cells generated by TBMD simulations\nand the EDTB model. As a representative quantity, we calculated\nthe equilibrium bulk modulus $B_{eq}$. The results for $B_{eq}$ for\nseveral networks as a function of $\\bar{z}$ are given in Fig. 3.\nAlso included in this figure is the computed $B_{eq}$ for diamond (428 GPa) \nand for the WWW model (361 GPa). The latter value coincides with\nthat calculated with the Tersoff potential for \nWWW \\cite{Kel00,Kel94,Kel01Diam}. \nThe computed data can be fitted to the power-law relation\n\\begin{equation}\nB_{eq} = B_{0}\\ \\left(\\frac{\\bar{z} - \\bar{z}_{f}}{\\bar{z}_{0} - \n\\bar{z}_{f}}\\right)^{\\nu},\n\\label{modul1}\n\\end{equation}\nwhere $B_{0}$ is the bulk modulus of the fully tetrahedral \namorphous network, for which $\\bar{z}_{0}$ = 4.0. Letting all \nfitting parameters in Eq. (\\ref{modul1}) free, we obtain $B_{0}$ = 361 GPa, \nwhich is exactly the computed value for WWW, $\\bar{z}_f$ = 2.25, \nand $\\nu$ = 1.6. (For a measure of the quality of the fit: \n$R^2$ = 0.9907). If we fix $\\nu$ to be 1.5 ($R^2$ = 0.9906), \nwe get 2.33 for $\\bar{z}_f$, and if we fix $\\bar{z}_f$ to be 2.4 \n($R^2$ = 0.9904), we get 1.4 for $\\nu$. We thus conclude that the \nvariation confirms the constraint-counting theory of Phillips and Thorpe, \nwith a critical coordination close to 2.4, and it has a power-law behavior \nwith a scaling exponent $\\nu = 1.5 \\pm 0.1$. For convenience, let us use\n$\\nu = 1.5$, so the modulus obeys the relation\n\\begin{equation}\nB_{eq} = 167.3\\ (\\bar{z} - 2.33)^{1.5}.\n\\label{modul2}\n\\end{equation}\nOur theory also predicts that ``amorphous diamond'' is softer\nthan diamond by $\\sim$ 10\\%.\n\nComparison of these results with experimental moduli derived from \nsurface acoustic waves \\cite{Schultrich} (SAW) and surface Brillouin \nscattering \\cite{FerrAPL99} (SBS) measurements is very good,\nespecially in the ta-C region. For example, the computed modulus for \n$\\bar{z} \\simeq$ 3.9 equals $\\sim$ 330 GPa and nearly\ncoincides with the SBS data. The agreement is less good at lower \ncoordinations, where a fit to experimental \npoints \\cite{Robertson02,Mathiou04} extrapolates to $\\bar{z}_f$ = 2.6, \nhigher than the constraint-counting prediction.\n\n\\subsection{Diamond nanocomposite films}\n\nDiamond nanocomposites consist of diamond nanocrystals embedded in\nan a-C matrix \\cite{Lifshitz}. They are produced by chemical\nvapor deposition (CVD) via a multistage process \\cite{Lifshitz},\nand they differ from pure nanodiamond films with no a-C\ncomponent \\cite{Gruen}. All diamond nanocomposite films reported\nuntil now contain a hydrogenated a-C matrix. Recently, nanodiamonds\nin pure a-C have been successfully grown \\cite{Shay}.\nTheir structure, either with or without H, is rather well known\nexperimentally, but their stability and most of their properties,\nincluding mechanical, are not yet understood.\n\nA first step towards a theoretical description of these films\nwas done recently in our group \\cite{Fyta03}. We summarize here the\nmost important findings of this investigation, based on MC \nsimulations with the Tersoff potential, and we also provide\nsupplementary new results from TBMD simulations using the EDTB model.\n\nA representative diamond nanocomposite network, generated by TBMD, is\nportrayed in Fig. 4. It shows a spherical diamond nanocrystal, \nwhose diameter is 12.5 \\AA, positioned in the\nmiddle of the cell and surrounded by the a-C matrix. Part of the\nimage cells, due to the periodic boundary conditions, are also shown.\nThis would correspond to an ideal case with a homogeneous dense dispersion\nof crystallites of equal size in the matrix, at regularly ordered\npositions. The nanodiamond volume fraction is 31\\%. The density of the\na-C matrix $\\rho_{am}$ is 3 gcm$^{-3}$ and its mean coordination \n$\\bar{z}_{am}$ is 3.8. The size of the diamond crystallite is smaller \nthan seen experimentally, but the overall structure captures the \nessential features of CVD grown nanocomposite films, especially the\nnon-hydrogenated ones.\n\nA crucial issue is the stability of the diamonds as a function of\nthe coordination\/density of the embedding medium. This extensive \ninvestigation required the analysis of many composite structures\nand it was done through MC simulations using larger\ncells (4096 atoms) \\cite{Fyta03}. The quantity of interest is the\nformation energy of a nanocrystal $E_{form}$, which can describe\nthe interaction of the embedded configuration with the host. It is\ndefined as \n\\begin{equation}\nE_{form} = E_{total} - N_{a}E_{a} - N_{c}E_{c},\n\\label{form}\n\\end{equation}\nwhere $E_{total}$ is the total cohesive energy of the composite system\n(amorphous matrix plus nanocrystal), calculated directly from the\nsimulation, $E_{c}$ is the cohesive energy per atom of the respective \ncrystalline phase, $N_{c}$ is the number\nof atoms in the nanocrystal, $N_{a}$ is the number of atoms in the\namorphous matrix, and $E_{a}$ is the cohesive energy per atom of the\npure, undistorted amorphous phase (without the nanocrystal) with\ncoordination $\\bar{z}_{am}$. A negative value of $E_{form}$ denotes stability\nof the nanostructure, a positive value indicates metastability or\ninstability.\n\nThe variation of $E_{form}$ as a function of $\\bar{z}_{am}$ for a diamond \nwith a fixed size embedded in several matrices is shown in Fig. 5. \n$E_{a}$ was computed from a series of calculations on pure a-C cells.\n(For details see Ref.\\ \\cite{Fyta03}.) The most striking result of this\nanalysis is that diamonds are stable in matrices with $\\bar{z}_{am}$ higher\nthan 3.6 ($\\rho_{am} \\simeq$ 2.6 gcm$^{-3}$), and unstable, or metastable\ndepending on temperature, in matrices with lower densities.\nThis nicely explains experimental results from different laboratories\nindicating that diamond nanocrystals precipitate in a dense a-C\nmatrix \\cite{Lifshitz,Shay}.\n\nOne way of checking the stability of nanocrystals is to subject them to \nthermal annealing. A stable structure should be sustained in the \namorphous matrix, while a metastable structure should shrink in favor of \nthe host. Indeed, analysis of the structure of diamonds annealed at high\n$T$ (1500 - 2000 K) reveals \\cite{Fyta03} that metastable nanocrystals\nbecome heavily deformed in the outer regions near the interface \nwith the amorphous matrix. Since only a small core remains intact, this\nmeans that the diamonds extensively shrink. On the other hand,\nthe stable nanodiamonds are only slightly deformed and retain their \ntetrahedral geometry.\n\nIn addition, a stable nanodiamond has, in principle, the potential to \nexpand against the surrounding matrix, provided that the barriers for \nthis transformation can be overcome, possibly by further annealing\nor ion irradiation. This means that nucleation\nof diamond cores in a dense matrix might lead, under the appropriate \nexperimental conditions, to a fully developed nanostructured\nmaterial with large grains.\n\nThe observation that diamonds are stable only in dense matrices\nsuggests a quantitative definition of ta-C, vaguely referred to \nas the form of a-C with a high fraction of sp$^3$ bonding. \nWe can define ta-C as the form of a-C with a fraction of sp$^3$ sites \nabove 60\\%, in which diamond nanocrystals are stable (see Fig. 5). \nIn other words, the predominantly tetrahedral amorphous network of ta-C \nis able to sustain crystalline inclusions. Networks with sp$^3$ fractions\nbelow 60\\% do not belong to the class of ta-C materials, because they\ncan not be transformed into a stable nanocrystalline state.\n\nThe intrinsic stress of the diamond nanocrystals and of the whole\ncomposite material is a crucial quantity. As for pure ta-C, the\naverage stress influences the adhesion properties of the films.\nThe stress within the nanodiamonds is indicative of their stability \nin a-C matrices. To examine these issues, we calculated\nthe stress fields in the nanocomposite cells, using as a probe the\ntool of atomic level stresses, as in the case of single-phase amorphous\ncarbon \\cite{Kel00,Kel94,Kel01Diam}. This gives us the ability to\nextract the stress built up in the nanodiamond and separate it from\nthe stress in the matrix, by summing up the atomic stresses over the\ndesired region. \n\nThe first important aspect of this analysis is that,\nin all cases studied, the average intrinsic stress\nin the fully relaxed composite material is less than 1 GPa, practically\nzero, even in the highly tetrahedral cases. This means, as in the case of\npure ta-C, that diamond nanocomposite films are able to eliminate any \ncompressive stress generated during growth, when brought into their \nequilibrium state, perhaps by moderate thermal annealing. \nThe stress in as-grown films has not yet been experimentally \nreported.\n\nThe other important finding is that the stress in the\nnanocrystal is always found to be tensile, while it is compressive\nin the matrix, yielding a net zero stress. This contrast can be\nexplained by noting that the density of the embedding medium is lower\nthan that of the diamond inclusions. As a result, atoms in the latter\nare forced to strech their bonds in order to conform with the lower\ndensity of the environment. \n\nA typical example of the stress state and its variation in a nanodiamond \nis shown in Fig. 6. The nanocrystal has a diameter of 17 \\AA, and is \nembedded in a ta-C matrix with $\\bar{z}_{am}$ = 3.9. \nThe atomic stresses are averaged over spherical shells starting from \nthe center and moving towards the interface. Negative values denote \ntensile stresses \\cite{Kel94}. The stress in the core\nof the diamond is very small, since the effect of the medium is weak, \nbut it rises up as we move outwards, especially near the interface.\nThis is logical. Atoms near the interface strongly feel the\ninfluence of the medium. However, the average tensile stress is low, \n$\\sim$ -1.5 GPa\/atom, because the density gradient between the nanodiamond \nand the matrix is small.\n\nObviously, the larger the density gradient the higher the tension \nfelt by the inclusion. \nFor example, when $\\bar{z}_{am}$ = 3.84, the nanodiamond stress is \n-6.3 GPa\/atom, and when $\\bar{z}_{am}$ = 3.75, it becomes -9 GPa\/atom. \nThis trend explains the lowering of the relative \nstability of diamonds as $\\bar{z}_{am}$ gets \nsmaller (Fig. 5). Tension substantially increases at the outer regions \nof the nanodiamond, leading to deformation and eventually to amorphization \nand shrinking. This is remarkably evident for nanodiamonds in the region \nof metastability, where the intrinsic tensile stress becomes huge. \nFor example, for a matrix with $\\bar{z}_{am}$ = 3.3, the average stress \nin a typical nanodiamond is -30 GPa\/atom.\n\nFinally, we briefly comment on the hardness of these nanocomposite\nmaterials. Their mechanical properties are not yet measured \nexperimentally. We have preliminary results of calculations of the\nbulk modulus of diamond nanocomposites, generated by TBMD\/EDTB\nsimulations, a typical example of which is shown in Fig. 4.\nWe find moduli which are considerably higher than moduli of single-phase \nfilms of the same density, as calculated using Eq. (\\ref{modul2}).\nFor example, for a nanocomposite with a total $\\bar{z} \\simeq$ 3.75 \nand a density of $\\sim$ 2.85 gcm$^{-3}$, the modulus approaches 350 GPa.\nEq. (\\ref{modul2}) predicts $\\sim$ 280 GPa for a pure a-C network\nhaving the same $\\bar{z}$. This represents a drastic 25\\% increase\nin strength, and opens up the possibility for even harder ta-C\nmaterials for coatings and MEMS applications. A comprehensive account \nof the elastic properties of diamond nanocomposites will be given elsewhere.\n\n\\section{CONCLUSIONS}\n\nResults from TBMD and MC simulations of pure a-C and diamond nanocomposite \nnetworks are presented. Definite trends in a-C regarding\nthe variation of the $sp^3$ fraction and the bulk moduli as a function \nof coordination\/density are shown to firmly hold. Nanodiamonds are\nstable only in dense ta-C matrices. The nanocomposite films are harder \nthan pure a-C films of the same density. They possess zero intrinsic\nstress when they are fully relaxed.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \\label{sec:intro}\n\nHumans are connected in numerous ways, and our many types of interactions with each other influence what we believe and how we act. \nTo model how opinions spread between people or other agents, researchers across many disciplines have developed a variety of models of opinion dynamics \\cite{castellano2009, sune-yy2018, noorazar-et-al2020, noorazar2020, peralta2022, galesic2021}.\nHowever, in part because of the difficulty of gathering empirical data on opinions, much of the work on opinion dynamics has focused on theory and model development, with little empirical validation \\cite{castellano2009, galesic2021, peralta2022} \n\\footnote{Chacoma and Zanette \\cite{chacoma2015} and Vande Kerckhove et al.~\\cite{vandeKerckhove2016} conducted multiple-round experiments in which they asked participants for their opinions and confidence levels on various quantitative questions. They examined the evolution of their answers and compared those to the results of opinion-dynamics models.\nThese experiments have several limitations, including the potential sensitivity of the models to measurement errors \\cite{carpentras2021}.}.\nEven with these difficulties, mechanistic modeling is valuable; it forces researchers to clearly define relationships and assumptions when developing a model, \nand it provides a framework to explore and generate testable hypotheses about complex social phenomena \\cite{holme2015}. \n\nIn an agent-based model (ABM) of opinion dynamics, each agent is endowed with an opinion and an underlying network structure governs which agents can interact with each other. \nWe assume that all interactions are dyadic (i.e., between exactly two agents). We suppose that the opinions take continuous values in a closed interval on the real line.\nThis interval represents a continuous spectrum of agreement for a single belief, such as the strength of support for a political candidate or ideology. \nAt each discrete time step of an ABM, a selection procedure determines which agents interact and then an update rule determines if and how their opinions change. \nBounded-confidence models (BCMs) are a popular class of continuous-opinion models \\cite{noorazar-et-al2020}.\nIn a BCM, interacting agents influence each other only when their opinions are sufficiently similar. This mechanism comes from the psychological idea of selective exposure, which asserts that people tend to seek out information or conversations that support their existing views and avoid those that challenge their views~\\cite{selective_exposure_def}. \nUnder this assumption, an agent's views are influenced directly only by agents with sufficiently similar views. For example, online platforms include polarizing posts, but individuals can choose whether or not to engage with such content and do not adopt the views of everything in their social-media feeds.\n\nThe two most popular BCMs are the Hegselmann--Krause (HK) model \\cite{hegselmann_krause2002} and the Deffuant--Weisbuch (DW) model \\cite{deffuant2000}. \nIn each time step, the HK model has synchronous updates of node opinions, whereas the DW model has asynchronous opinion updates, with a single pair of agents (i.e., a dyad) interacting and potentially updating their opinions at each time. \nAn asynchronous mechanism is consistent with empirical studies, which suggest that individuals in social networks have different times and frequencies of activity \\cite{alizadeh2015}. \nIn the present paper, we generalize the DW model to incorporate different activity levels and sociability of nodes. \nAlthough many heterogeneities and modifications have been incorporated into DW models \\cite{noorazar2020}, to the best of our knowledge, few studies have modified the agent-selection procedure.\nThe ones that have done so (see, e.g., Refs.~\\cite{alizadeh2015, zhang2018, sirbu2019, pansanella2022}) focused on specific scenarios, rather than on investigating baseline effects of heterogeneity in agent-selection probabilities.\nBefore describing previous research on heterogeneous selection of agents in the DW model, we first discuss other generalizations of the model.\nSome studies have drawn the initial opinions of nodes from nonuniform distributions \\cite{jacobmeier2006, carro2013, sobkowicz2015} and have thereby considered differential initial conditions from those in the standard DW model. \nOther investigations have incorporated heterogeneous confidence radii \\cite{dw2002, deffuant-amblard2002, lorenz2008, kou2012, sobkowicz2015, chen2020} and heterogeneous compromises \\cite{dw2002, deffuant-amblard2002, zhang2014, huang2018}. Such generalizations affect the opinion updates of interacting agents.\nOther studies of the effect of network structure on opinion dynamics have examined DW models on time-independent graphs \\cite{meng2018}, hypergraphs \\cite{hickok2022}, and coevolving networks \\cite{unchitta2021}.\nThe standard DW model selects pairs of agents to interact uniformly at random, but social interactions are not uniform in real life.\nHowever, few studies of the DW model have modified the selection procedure that determine which agents interact with each other.\nExamples of such studies include Refs.~\\cite{alizadeh2015, zhang2018, sirbu2019, pansanella2022}.\n\nOne can think of agents that are selected not uniformly at random as having different activity levels that encode the number of interactions in a given time interval. Such ideas have also been employed in activity-driven models of temporal networks \\cite{perra2012}.\nThere have also been studies of activity-driven models of opinion dynamics.\nLi et al.~\\cite{li2017} developed an activity-driven model of opinion dynamics using networks with fixed nodes with assigned activity rates (i.e., assigned activation probabilities).\nAt each time step of their model, all existing edges are removed and activated agents randomly form a fixed number of connections. All agents then evaluate the \nmean opinion of their neighbors to determine if and how to update their own opinion \\cite{li2017}.\nBaronchelli et al.~\\cite{baronchelli2011} studied a voter model with edge activity.\nZhang et al. \\cite{zhang2018} incorporated heterogeneous node activities into a DW model to study social-media networks. As we will discuss shortly, our inspiration is similar to theirs, but we make fundamentally different choices in how we generalize the DW model.\n\nIn social networks, some individuals share their ideas and opinions more frequently than others.\nAlizadeh and Cioffi-Revilla \\cite{alizadeh2015} studied a modified DW model that incorporates a repulsion mechanism, which was proposed initially by Huet et al.~\\cite{huet2008}, in which interacting agents with opinions that differ by more than a cognitive-dissonance threshold move farther away from each other in the space of opinions when they interact.\nThey used 2-dimensional (2D) vector-valued opinions and placed their nodes on complete graphs.\nTo model agents with different activity levels, Alizadeh and Cioffi-Revilla \\cite{alizadeh2015} implemented a Poisson node-selection probability, which one can interpret as independent internal ``clocks'' that determine agent activation.\nIn comparison to selecting agent pairs uniformly at random (as in the standard DW model) the Poisson node-selection probability can either lessen or promote the spread of extremist opinions, depending on which opinions are more prevalent in more-active agents.\n\nZhang et al.~\\cite{zhang2018} studied a modified DW model with asymmetric updates on activity-driven networks. In their model, each node has a fixed activity potential, which they assign uniformly at random from a distribution of activity potentials. The activity potential of an agent is its probability of activating. At each discrete time step, each active agent $i$ randomly either (1) creates a message (e.g., a social-media post) or (2) forwards a message that was created by a neighboring agent $j$. If agent $i$ forwards a message from agent $j$, then $i$ updates its opinion using the standard DW update mechanism.\nZhang et al.~\\cite{zhang2018} simulated their model on a social network from Tencent Weibo\n(\\begin{CJK*}{UTF8}{gbsn}\u817e\u8baf\u5fae\u535a\\end{CJK*})\nand found that the distribution of activity potentials influences the location of the transition between opinion consensus and fragmentation.\nThe node-weights in our BCM are similar in spirit to the the activity potentials of Zhang et al.~\\cite{zhang2018}, and both can encode the social activity levels of individuals such as frequency of posting or commenting on social media. \nHowever, the way we incorporate our node weights in our BCM fundamentally differs from Ref.~\\cite{zhang2018}.\nWe consider a time-independent network $G$ and at each time step select a single pair of neighboring agents for interaction. We randomly select a first agent and then a second neighboring agent with probabilities proportional to their node weights. Both selected agents update their opinions using the DW update mechanism.\n\nIn addition to individuals having different activity levels in social networks, some pairwise interactions are also more likely than others. \nSocial-media feeds tend to curate content that is based on the concept of homophily, which is the idea that people have a tendency to connect with people who are similar to themselves or have similar ideas or beliefs \\cite{mcpherson2001}.\nFor example, social-media feeds tend to show content to a user that closely matches their profile and past activity \\cite{spohr2017}. \nTo examine the effect of such algorithmic bias on opinion dynamics, S\\^{i}rbu et al.~\\cite{sirbu2019} studied a modified DW model that includes a homophily-promoting activation mechanism. \nAt each time step, a first agent is selected uniformly at random, and then one of its neighbors is selected with a probability that depends on the magnitude of the opinion difference between that neighbor and the first agent.\nThe simulations by S\\^{i}rbu et al. of this model on complete graphs suggest that more algorithmic bias yields slower convergence times and more opinion fragmentation \\cite{sirbu2019}. \nPansanella et al.~\\cite{pansanella2022} applied the same algorithmic-bias model to a variety of network topologies (specifically Erd\\H{o}s--R\\'{e}nyi, Barab\\'{a}si--Albert, and Lancichinetti--Fortunato--Radicchi (LFR) graphs), and they found similar trends as S\\^{i}rbu et al. did on complete graphs.\n\nFrom the investigations in Refs.~\\cite{alizadeh2015, zhang2018, sirbu2019, pansanella2022}, \nwe know that incorporating heterogeneous node-selection probabilities into a DW model can influence opinion dynamics.\nEach of these papers examined a specific implementation of heterogeneous agent selection; we are not aware of any systematic investigations of the effects of heterogeneous agent selection.\nIn the present paper, we propose a novel BCM with heterogeneous agent-selection probabilities, which we implement using node weights.\nIn general terms, we are studying a dynamical process on node-weighted networks.\nWe use node weights to model agents with different probabilities of interacting. These probabilities can encode heterogeneities in individual behavior, such as in sociability and activity levels.\nWe conduct a methodical investigation of the effects of incorporating heterogeneous node weights, which we draw from various distributions, into our generalization of the DW model. We compare these effects on a variety of types of networks. \nIn our study, we consider fixed node weights that we assign in a way that disregards network structure and node opinions. However, one can readily can adapt the node weights in our BCM to capture a variety of sociological scenarios in which nodes have heterogeneous selection probabilities.\nWe find that introducing heterogeneous node weights into the DW model results in longer convergence times and more opinion fragmentation than selecting nodes uniformly at random.\nOur results illustrate that it is important to consider the baseline influence of assigning node weights uniformly at random in implementations of heterogeneous node-selection patterns before drawing conclusions about more specific mechanisms such as algorithmic bias \\cite{sirbu2019}.\n\nOur paper proceeds as follows. In Sec.~\\ref{sec:model}, we describe the standard DW model and present our generalized DW model with node weights to incorporate heterogeneous agent-selection probabilities.\nIn Sec.~\\ref{sec:methods}, we discuss our implementation of our BCM, the networks and node-weight distributions that we examine, and the quantities that we compute to characterize the behavior of our model.\nIn Sec.~\\ref{sec:results}, we discuss the results from our numerical simulations of our BCM. \nIn Sec.~\\ref{sec:discussion}, we summarize our results and discuss their implications, present some ideas for future work, and discuss the importance of studying networks with node weights.\nOur code is available at \\url{https:\/\/gitlab.com\/gracejli1\/NodeWeightDW}.\n\n\n\n\\section{Model} \\label{sec:model} \n\nIn this section, we first discuss the Deffuant--Weisbuch (DW) \\cite{deffuant2000} bounded-confidence model (BCM) of opinion dynamics, and we then introduce our BCM with heterogeneous node-selection probabilities.\n\n\n\\subsection{The Standard Deffuant--Weisbuch (DW) BCM} \\label{sec:DW}\n\nThe DW model was introduced over two decades ago \\cite{deffuant2000}, and it and generalizations of it have been studied extensively since then \\cite{noorazar-et-al2020, noorazar2020}. It was examined originally on complete graphs and encoded node opinions as scalar values in a closed interval of the real line. \nDeffuant et al.~\\cite{deffuant2000} let each node have an opinion in $[0,1]$, and we follow this convention. The standard DW model has two parameters. The ``confidence radius'' $c \\in [0,1]$ is a thresholding parameter; if the opinions of two agents (i.e., nodes) differ by more than $c$, then they do not interact. \nThe ``compromise parameter'' $m \\in (0, 0.5]$ (which is also sometimes called a convergence parameter \\cite{deffuant2000} or a cautiousness parameter \\cite{meng2018}) parametrizes the amount that an agent changes its opinion to compromise with the opinion of an agent with whom it interacts.\n\n\nIn the standard DW model, the opinions of the agents update in an asynchronous fashion. We endow each agent with an initial opinion. At each discrete time, one selects a pair of agents uniformly at random.\nAt time $t$, suppose that we pick agents $i$ and $j$, whose associated opinions are $x_i$ and $x_j$, respectively. Agents $i$ and $j$ update their opinions through the following equations:\n\\begin{align} \\label{eq:DW}\n\\begin{split}\n x_i(t+1) &= \n \\begin{cases}\n x_i(t) + m \\Delta_{ij}\\,, & \\text{if } |\\Delta_{ij}(t)| < c \\\\\n x_i(t)\\,, & \\text{otherwise} \\,,\n \\end{cases} \\\\\n x_j(t+1) &= \n \\begin{cases}\n x_j(t) + m \\Delta_{ji}\\,, & \\text{if } |\\Delta_{ij}(t)| < c \\\\\n x_j(t)\\,, & \\text{otherwise}\\,,\n \\end{cases}\n\\end{split}\n\\end{align}\nwhere $\\Delta_{ij}(t) = x_i(t) - x_j(t)$. \n\n\nWhen one extends the DW model to consider an underlying network of agents \\cite{weisbuch2001}, only adjacent agents are allowed to interact. Each node in a network represents an agent, and each edge between two agents encodes a social or communication tie between them. \nAt each discrete time, one selects an edge of a given network uniformly at random and the two agents that are attached to the edge interact as in Eq.~\\eqref{eq:DW}. \nFor the DW model, which updates opinions asynchronously, an alternative to an edge-based approach of randomly selecting an interacting edge is to take a node-based approach to selecting an interacting pair. (See {Ref.}~\\cite{kureh2020} for a discussion of node-based updates versus edge-based updates in the context of voter models.)\nIn a node-based approach, one randomly selects a first node and then randomly selects a second node from its neighbors. \nTo capture the effect of some agents having more frequent interactions (such as from greater sociability or a stronger desire to share their opinions), we implement such a node-based agent-selection procedure in our study. \nThe choice between edge-based and node-based agent selection can have substantial effects on the dynamics of voter models of opinion dynamics \\cite{kureh2020}, and we expect that this is also true for other types of opinion models. \nWe are not aware of a comparison of edge-based and node-based agent selection in asynchronous BCMs (and, in particular, in DW models), and it seems both interesting and relevant to explore this issue.\nMost past work on the DW model has considered edge-based selection \\cite{noorazar2020}. \nHowever, Refs.~\\cite{alizadeh2015, sirbu2019, pansanella2022} used a node-based selection procedure to model heterogeneous activities of agents.\n\n\n\\subsection{A BCM with Heterogeneous Node-Selection Probabilities} \\label{sec:BCM} \n\nWe now introduce our BCM with heterogeneous \nnode-selection probabilities.\nConsider an undirected network $G = (V, E)$, where $V$ is the set of nodes and $E$ is the set of edges between them. Suppose that $V$ has $N$ agents and that each agent $i$ holds a time-dependent opinion $x_i(t)$.\nEach agent also has a fixed node weight $w_i$ that encodes sociability, how frequently it engages in conversations, or simply the desire to share its opinions. \nOne can think of a node's weight as a quantification of how frequently it talks to its friends or posts on social media.\nThrough the incorporation of network structure, the standard DW model can capture agents with different numbers of friends (or other social connections).\nHowever, selecting interacting node pairs uniformly at random is unable to capture heterogeneous interaction frequencies of individuals.\nBy introducing node weights, we encode this heterogeneity\nand then examine how it affects opinion dynamics in a BCM.\nAlthough we employ fixed node weights, one can adapt our model to include time-dependent node weights, such as through purposeful strategies (such as posting on social media more frequently as one's opinions become more extreme).\n\n\nIn our node-weighted DW mode, at each discrete time, we first select an agent $i$ with a probability that is proportional to its weight. Agent $i$ then interacts with a neighbor $j$, which we select with a probability that is equal to its weight divided by the sum of the weights of $i$'s neighbors.\nThat is, the probabilities of first selecting agent $i$ and then selecting agent $j$ are\n\\begin{align} \\label{eq:node-probablity}\n P_1(i) = \\frac{w_i}{\\sum\\limits_{k = 1}^N w_k} \\,, \\quad\n P_2(j|i) = \\frac{w_j}{\\sum\\limits_{k \\in \\mathcal{N}(i)} w_k}\\,, \n\\end{align}\nwhere $\\mathcal{N}(i)$ denotes the neighborhood (i.e., the set of neighbors) of node $i$. Once we select the pair of interacting agents, we update their opinions following the DW opinion update rule in Eq.~\\ref{eq:DW}.\n\n\nOur BCM incorporates heterogeneous node-selection probabilities\nwith node weights that model phenomena such as the heterogeneous sociability of individuals. One can also use edge weights to model heterogeneous agent selection.\nThis variant can encode heterogeneous selection probabilities\nof pairwise (i.e., dyadic) interactions, instead of focusing on the selection of individuals.\nFor instance, in the dyadic interactions of a given individual, that individual may discuss their ideological views with a close friend more frequently than with a work colleague.\nOne can use edge weights to determine the probabilities of selecting each dyadic interaction in a BCM.\nAt each discrete time, one can select an edge with a probability that is proportional to its weight. We do not examine edge-based heterogeneous selection probabilities in the present paper, but it is worth exploring in BCMs.\n\n\\section{Methods and Simulation Details} \\label{sec:methods} \n\nIn this section, we discuss the network structures and node-weight distributions that we consider, the specifications of our numerical simulations, and the quantities that we compute to characterize the results of our simulations.\n\n\n\\subsection{Network Structures} \\label{sec:nets}\n\nWe now describe the details of the networks on which we simulate our node-weighted BCM. We summarize these networks in Table~\\ref{tab:networks}. \n\nWe first simulate our BCM on complete graphs as a baseline scenario that will allow us to examine how incorporating heterogeneous node-selection probabilities affects the opinion dynamics.\nAlthough DW models were introduced more than 20 years ago, it is still the case that complete graphs are the most common type of network on which to study them \\cite{noorazar-et-al2020}. \nTo examine finite-size effects from the networks, we consider complete graphs with 100--1000 nodes in increments of 100. For all other synthetic networks, we consider networks of size $N = 500$ nodes. \n\n\n\\newcommand{1.5in}{1.5in}\n\\newcommand{3.3in}{3.3in}\n\\newcommand{1.5in}{1.5in}\n\n\\begin{table*}\n\\centering\n\\caption{\\label{tab:networks} Summary of the networks on which we simulate our node-weighted BCM.}\n\\begin{ruledtabular}\n\\def\\arraystretch{1.1}\n\\begin{tabular}{m{1.5in} m{3.3in} m{1.5in}}\nNetwork & Description & Parameters \\\\\\hline\n$C(N)$ \n& \\begin{tabular}{m{3.3in}} \nComplete graph with $N$ nodes \\end{tabular}\n& \\begin{tabular}[c]{m{1.5in}}\n$N \\in \\{100, 200, \\ldots, 1000\\}$ \\end{tabular} \n\\\\\\hline\n$G(N,p)$ \n& \\begin{tabular}{m{3.3in}} Erd\\H{o}s--R\\'{e}nyi (ER) random-graph model with $N$ nodes and homogeneous, independent edge probability $p$ \\end{tabular}\n& \\begin{tabular}[c]{m{1.5in}} \n$\\, p \\in \\{0.1, 0.3, 0.5, 0.7\\}$ \\end{tabular} \n\\\\\\hline\nTwo-Community SBM\\footnotemark[1] \n& \\begin{tabular}{m{3.3in}} Stochastic block model with 2 $\\times$ 2 blocks. There is a larger probability of edges within the sets A and B than between the two sets; the block probabilities satisfy $P_{BB} > P_{AA} > P_{AB}$. \\end{tabular}\n& \\begin{tabular}[c]{m{1.5in}}$P_{AA} = 49.9\/374$ \\\\ \n$P_{BB} = 49.9\/124$ \\\\ $P_{AB} = 1\/500$ \\end{tabular}\n\\\\\\hline\nCore--Periphery SBM\\footnotemark[1] \n& \\begin{tabular}{m{3.3in}} Stochastic block model with 2 $\\times$ 2 blocks. Set A is a set of core nodes and set B is a set of peripheral nodes. The block probabilities satisfy $P_{AA} > P_{AB} > P_{BB}$. \\end{tabular} \n& \\begin{tabular}[c]{m{1.5in}}$P_{AA} = 147.9\/374$ \\\\ \n$P_{BB} = 1\/174$ \\\\ $P_{AB} = 1\/25$ \\end{tabular} \n\\\\\\hline\nCaltech Network\n& \\begin{tabular}{m{3.3in}} The largest connected component of the Facebook friendship network at Caltech on one day in fall 2005. This network, which is part of the {\\sc Facebook100} data set \\cite{red2011, traud2012}, has 762 nodes and 16,651 edges. \\end{tabular}\n& \n\\end{tabular}\n\\end{ruledtabular}\n\\footnotetext[1]{In our SBM networks, there are $N=500$ nodes. We partition the network into two sets of nodes; set A has 75\\% of the nodes, and set B has 25\\% of the nodes.}\n\\end{table*}\n\n\nWe then consider synthetic networks that we generate using\nthe $G(N, p)$ Erd\\H{o}s--R\\'{e}nyi (ER) random-graph model, where $p$ is the homogeneous, independent probability of an edge between each pair of nodes \\cite{newman2018}. When $p=1$, this yields a complete graph. We examine $G(500, p)$ graphs with $p \\in \\{0.1, 0.3, 0.5, 0.7\\}$.\n\n\nTo determine how a network with an underlying block structure affects the dynamics of our node-weighted BCM, we consider stochastic-block-model (SBM) networks \\cite{newman2018} with $2 \\times 2$ blocks, where each block consists of an ER graph.\nInspired by the choices of Kureh and Porter \\cite{kureh2020}, we consider two types of SBM networks. The first has a two-community structure, in which there is a larger probability of edges within a community than between communities. \nThe second SBM has a core--periphery structure, in which there is a set of core nodes with a large probability of connections within the set, a set of peripheral nodes with a small probability of connections within the set, and an intermediate probability of connections between core nodes and peripheral nodes. \nTo construct our $2 \\times 2$ SBMs, we partition a network into two sets of nodes; set A has 375 nodes (i.e., 75\\% of the network) and set B has 125 nodes (i.e., 25\\% of the network). We define a symmetric edge-probability matrix\n\\begin{equation}\n P = \\begin{bmatrix}\n P_{AA} & P_{AB} \\\\ P_{AB} & P_{BB}\n \\end{bmatrix} \\,,\n\\end{equation}\nwhere $P_{AA}$ and $P_{BB}$ are the probabilities of an edge between two nodes within set A and set B, respectively, and $P_{AB}$ is the probability of an edge between a node in set A and a node in set B.\n\n\nIn a two-community SBM, the probabilities $P_{AA}$ and $P_{BB}$ are larger than $P_{AB}$, so there is a larger probability of edges within a community than between communities.\nFor our two-community SBM, we choose $P_{AA}$ and $P_{BB}$ so that the expected mean degree matches that of the $G(500, 0.1)$ ER model if we only consider edges within set A or edges within set B. A network from the $G(N, p)$ model has an expected mean degree of $p(N-1)$ \\cite{newman2018}, so we want the two communities in these SBM networks to have an expected mean degree of $49.9 = 0.1 \\times 499$. We thus use edge probabilities $P_{AA} = 49.9\/374$ and $P_{BB} = 49.9\/124$. To ensure that there are few edges between the sets A and B, we choose $P_{AB} = 1\/500$.\n\n\nWe want our core--periphery SBM with core set A and periphery set B to satisfy $P_{AA} > P_{AB} > P_{BB}$.\nWe chose $P_{AA}$ so that the expected mean degree matches that of the $G(500, 0.3)$ model (i.e., it is 147.9) if we only consider edges within the set A. We thus choose the edge probability $P_{AA} = 147.9\/374$. To satisfy $P_{AA} > P_{AB} > P_{BB}$, we choose $P_{AB} = 1\/25$ and $P_{BB} = 1\/174$.\n\n\nFinally, we investigate our node-weighted BCM on a real social network from Facebook friendship data. We use the Caltech network from the {\\sc Facebook100} data set; its nodes encode individuals at Caltech, and its edges encode Facebook ``friendships'' on one day in fall 2005 \\cite{red2011, traud2012}. We only consider the network's largest connected component, which has 762 nodes and 16,651 edges.\n\n\n\\subsection{Node-Weight Distributions} \\label{sec:weights}\n\nIn Table~\\ref{tab:distributions}, we give the parameters and probability density functions of the node-weight distributions that we examine in our BCM. In this subsection, we discuss our choices of distributions.\n\n\n\\begin{table*}\n\\caption{\\label{tab:distributions} Names and specifications of our distributions of node weights. We show both the general mathematical expressions for the means and the specific values of the means for our parameter values.}\n\\def\\arraystretch{1.2}\n\\begin{ruledtabular}\n\\begin{tabular}{lclccl\n\\textbf{Distribution} & \n\\textbf{\\begin{tabular}[c]{@{}c@{}}Probability Density \\\\ Function\\end{tabular}} & \n\\textbf{Parameter values} & \\textbf{Domain} & \\multicolumn{2}{c}{\\textbf{Mean}} \\\\ \\colrule\nConstant & $\\delta (x - 1)$ & N\/A & $\\{1\\}$ & 1 & 1 \\\\ \\colrule\nPareto-80-10 &\n \\multirow{3}{*}{$\\dfrac{\\alpha}{x^{\\alpha+1}}$} &\n $\\alpha = \\log_{4.5}(10)$ &\n \\multirow{3}{*}{$[1,\\infty)$} &\n \\multirow{3}{*}{$\\dfrac{\\alpha}{\\alpha-1}$} &\n 2.8836 \\\\\nPareto-80-20 & & $\\alpha = \\log_{4}(5)$ & & & 7.2126 \\\\\nPareto-90-10 & & $\\alpha = \\log_9 (10)$ & & & 21.8543 \\\\ \\colrule\nExp-80-10 &\n \\multirow{3}{*}{$\\frac{1}{\\beta} \\exp\\left(\\frac{-(x-1)}{\\beta}\\right)$} &\n $\\beta = 1.8836$ &\n \\multirow{3}{*}{$[1,\\infty)$} &\n \\multirow{3}{*}{$\\beta + 1$} &\n 2.8836 \\\\\nExp-80-20 & & $\\beta = 6.2125$ & & & 7.2125 \\\\\nExp-90-10 & & $\\beta = 20.8543$ & & & 21.8543 \\\\ \\colrule\nUnif-80-10 & \n \\multirow{3}{*}{$\\dfrac{1}{b-1}$} & \n $b = 4.7672$ & \n \\multirow{3}{*}{$[1, b]$} & \n \\multirow{3}{*}{$\\dfrac{1}{2} (1+b)$} & \n 2.8836 \\\\\nUnif-80-80 & & $b = 13.425$ & & & 7.2125 \\\\\nUnif-90-10 & & $b = 42.7086$ & & & 21.8543 \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table*}\n\n\nTo study the effects of incorporating node weights \nin our BCM, we compare our model to a baseline DW model.\nTo ensure a fair comparison, we implement a baseline DW model that selects interacting agents uniformly at random using a node-based selection process as in our BCM. \nAs we discussed in Sec.~\\ref{sec:intro}, it is much more common to employ an edge-based selection process.\nWe refer to the case in which all nodes weights are equal to $1$ (that is, $w_i = 1$ for all nodes $i$) as the ``constant distribution''. The constant distribution (and any other situation in which all node weights equal the same positive number) results in a uniformly random selection of nodes for interaction. \nThis is what call the ``baseline DW model''; we compare our DW models with heterogeneous node weights to this baseline model.\nWe reserve the term ``standard DW model'' for the DW model with uniformly random edge-based selection of agents.\n\n\nWe use the node weights in our BCM to model heterogeneities in interaction frequencies, such as when posting content online.\nThe majority of online content arises from a minority of user accounts \\cite{90-9-1}. \nThe ``90-9-1 rule'' has been proposed for such participation inequality; \nin this rule of thumb, about 1\\% of the individuals in online discussions (e.g., on social-media platforms) account for most \ncontributions, about 9\\% of the individuals contribute on occasion,\nand the remaining 90\\% of the individuals are present online (e.g., they consume content) but do not contribute to it \\cite{90-9-1}. \nSimilar participation inequality has been documented in the numbers of posts on digital-health social networks \\cite{vanMierlo2014}, \nposts on internet support groups \\cite{carron2014}, and contributions to open-source software-development platforms \\cite{gasparini2020}.\nAdditionally, the number of tweets on particular topics has also been modeled as following a power-law distribution \\cite{xiong2014}. \nIn 2019, a Pew Research Center survey found that about 10\\% of the accounts generate about 80\\% of the tweets on Twitter from the United States \\cite{twitter_stats}.\n\n\nOne can interpret the node weights in our BCM as encoding the participation of individuals in the form of contribution content to an online social network.\nWe model online participation inequality by using a Pareto distribution for the node weights. This choice of distribution is convenient because of its simple power-law form. \nIt has also been used to model inequality in a variety of other contexts, including distributions of wealth, word frequencies, website visits, and numbers of paper citations \\cite{newman2005}.\nWe are interested in modeling individuals who are active in a finite time interval. For example, when representing social-media interactions, we only care about accounts that make posts or comments; we ignore inactive accounts. Therefore, we impose a minimum node weight in our model.\nWe use the Pareto Type-I distribution, which is defined on $[1, \\infty)$, so each node has a minimum weight of $1$. This positive minimum weight yields a reasonable convergence time for our the simulations of our BCM. \nIf nodes had that weights close to $0$, they would have a very small probability of interacting, and this would prolong simulations.\n\n\nLet Pareto-X-Y denote the continuous Pareto distribution in which (in theory) X\\% of the total node weight is distributed among Y\\% of the nodes. \nIn practice, once we determine the $N$ node weights for our simulations from a Pareteo node-weight distribution, it is not true that precisely X\\% of the total weight is held by Y\\% of the $N$ nodes. \nInspired by the results of the aforementioned Pew Research Center survey of Twitter users \\cite{twitter_stats}, we first consider a Pareto-80-10 distribution, in which we expect 80\\% of the total weight to be distributed among 10\\% of nodes.\nThe Pareto principle (which is also known as the 80-20 rule) is a popular rule of thumb that suggests that 20\\% of individuals have 80\\% of available wealth \\cite{newman2005}. Accordingly, we also consider a Pareto-80-20 distribution. \nFinally, as an example of a node-weight distribution with a more extreme inequality, we also consider a Pareto-90-10 distribution. \n\n\nWe also examine uniform and exponential distributions of node weights. To match the parameters of our Pareto distributions, we shift the uniform and exponential distributions so that their minimum node weight is also $1$. \nWe also choose their parameters to match the means of our Pareto distributions. We use Exp-X-Y and Unif-X-Y to denote the exponential and uniform distributions, respectively, that have the same mean as the Pareto-X-Y distribution. \nIn total, we examine three different families of distributions (Pareto, exponential, and uniform) with different heaviness in their tails.\nIn Table~\\ref{tab:distributions}, we show the details of the probability density functions and the parameters of our node-weight distributions.\n\n\n\n\\subsection{Simulation Specifications} \\label{sec:sims} \n\nIn our node-weighted BCM, agents have opinions in the 1-dimensional (1D) opinion space $[0,1]$. Accordingly, we examine values of the confidence radius $c \\in (0, 1)$\n\\footnote{The extreme case $c=0$ is degenerate (because no agents update their opinions), and the case $c=1$ allows all agents to interact with each other. We are not interested in examining these cases.}.\nWe examine values of the compromise parameter $m \\in (0, 0.5]$, which is the typically studied range for the DW model \\cite{noorazar-et-al2020, meng2018}. \nWhen $m = 0.5$, two interacting agents who influence each other fully compromise and thus average their opinions. \nWhen $m < 0.5$, the two agents move towards each other's opinions, but they do not change their opinions to the mean (i.e., they do not fully compromise).\n\n\nIt is computationally intensive to conduct numerical simulations of a DW model. Additionally, as we will show in Sec.~\\ref{sec:results}, our node-weighted DW model with heterogeneous node weights can converge even more slowly than the baseline DW model to a steady state.\nIn our node-weighted BCM, the generation of graphs in a random-graph ensemble, the node-weight profiles, the initial-opinion profiles, and the selection of pairs of agents to interact at each time step are all stochastic. We use Monte Carlo simulations to reduce these sources of noise in our simulation results.\nFor each of our random-graph models (i.e., the ER and SBM graphs), we generate 5 graphs.\nFor each graph and each node-weight distribution, we randomly generate 10 sets of node weights. For each set of node weights, we generate 10 sets of initial opinions that are distributed uniformly at random. \nIn total, we have 100 distinct sets of initial opinions and node weights for the Monte Carlo simulations of each individual graph.\nWhen we compare simulations from\ndifferent distributions of node weights in the same individual graph, we reuse the same 100 sets of initial opinions.\n\n\nIn theory, the standard DW model and our node-weighted DW model can take infinitely long to approach a steady state.\nWe define an ``opinion cluster'' $S_r$ to be a maximal connected set of agents in which the pairwise differences in opinions are all strictly less than the confidence radius $c$; adding any other agent to $S_r$ will cause a violation in the condition on the opinion differences.\nEquivalently, we are defining an effective-receptivity network $G_{\\mathrm{eff}}(t) = (V, E_{\\mathrm{eff}}(t))$ as the time-dependent network that retains only the edges from the original network in which the associated pair of nodes are receptive to each others' opinions. That is,\n\\begin{equation}\n E_{\\mathrm{eff}}(t) = \\{ (i,j) \\in E : |x_i(t) - x_j(t)| < c \\} \\,.\n\\end{equation}\nThe opinion clusters are the connected components of $G_{\\mathrm{eff}}(t)$.\nIf two opinion clusters $S_1$ and $S_2$ are separated by a distance of at least $c$ (i.e.,\n$|x_i - x_j| \\geq c$ for all $i \\in S_1$ and $j \\in S_2$) at some time $\\tilde{T}$, \nthen (because $c$ is fixed) no agents from $S_1$ can influence the opinion of an agent in $S_2$ (and vice versa) for all $t \\geq \\tilde{T}$. \nTherefore, in finite time, we observe the formation of steady-state clusters of distinct opinions.\nIn practice, we specify that one of our simulations has ``converged'' if all opinion clusters are separated by a distance of at least $c$ and each opinion cluster has an opinion spread that is less than a tolerance of $0.02$. That is, for each cluster $S$, we have that\n$\\max_{i, j \\in S} |x_i - x_j| < 0.02$.\n{We use $T$ to denote the convergence time in our simulations; the connected components of $G_{\\mathrm{eff}}(T)$ are the steady-state opinion clusters.} \n\n\nTo reduce the computational burden of checking for convergence, we do not check at each time step and we compute the convergence time to three significant figures. \nTo guarantee that each simulation stops in a reasonable amount of time, we set a bailout time of $10^9$ iterations. \nIn our simulations, the convergence time is always shorter than the bailout time. We thus report the results of these simulations as steady-state results.\n\n\n\n\\subsection{Quantifying Opinion Consensus and Fragmentation} \\label{sec:quantities}\n\nIn our numerical simulations, we investigate which situations yield consensus (specifically, that result in one ``major'' opinion cluster, which will discuss shortly) at steady state and which situations yield opinion fragmentation (when there are at least 2 distinct major clusters) at steady state.\n\\footnote{Some researchers use the term ``polarization'' to refer to the presence of exactly two opinion clusters (or to refer to exactly 2 major opinion clusters) and ``fragmentation'' to refer to the presence of 3 or more opinion clusters (or 3 or more major opinion clusters) \\cite{hegselmann_krause2002, bramson2016}. However, because we are interested in distinguishing between consensus states and any state that is not consensus, we use the term ``fragmentation'' for any state with at least 2 major opinion clusters. We then quantify the extent of opinion fragmentation.}\nWe are also interested both in how long it takes to determine the steady-state behavior of a simulation and in quantifying the extent of opinion fragmentation when it occurs. \nTo investigate these model behaviors, we compute the convergence time and the number of steady-state opinion clusters.\nIt is common to study these quantities in investigations of BCMs \\cite{noorazar-et-al2020, meng2018, peralta2022}.\n\n\nThere are situations when an opinion cluster has very few agents.\nConsider a 500-node network in which 499 agents eventually have the same opinion, but the remaining agent \n(say, Agent 86, despite repeated attempts by Agent 99 and other agents to convince them) retains a distinct opinion at steady state.\nIn applications, it is not appropriate to think of this situation as opinion fragmentation. \nTo handle this, we use the notions of ``major clusters'' and ``minor clusters'' \\cite{laguna2004, lorenz2008}. We characterize major and minor clusters in an ad hoc way.\nWe define a ``minor'' opinion cluster in a network as an opinion cluster with at most 2\\% of the agents. Any opinion cluster that is not a minor cluster is a ``major'' cluster. \nIn our simulations, we calculate the numbers of major and minor opinion clusters at steady state. We only account for the number of major clusters when determining if a simulation reaches consensus (i.e., one major cluster) or fragmentation (i.e., more than one major cluster).\nWe still track the number of minor clusters and use the minor clusters when quantifying opinion fragmentation.\n\n\nQuantifying opinion fragmentation is much less straightforward than determining whether or not there is fragmentation. \nResearchers have proposed a variety of notions of fragmentation and polarization, and they have also proposed several ways to quantify them~\\cite{bramson2016}.\nIn principle, a larger number of opinion clusters is one indication of more opinion fragmentation. However, as we show in Fig.~\\ref{fig:cluster_size}, there can be considerable variation in the sizes (i.e., the number of nodes) of the opinion clusters. \nFor example, suppose that there are two opinion clusters. If the two opinion clusters have the same size, then one can view the opinions in the system as more polarized \nthan if one opinion cluster has a large majority of the nodes and the other opinion cluster has a small minority. \nAdditionally, although we only use major clusters to determine if a system reaches consensus or opinion fragmentation, we seek to distinguish quantitatively between the scenarios of opinion clusters (major or minor) with similar sizes from ones with opinion clusters with a large range of sizes. \nFollowing Han et al.~\\cite{han2020}, we do this by calculating Shannon entropy.\n\n\n\\begin{figure}[b]\n\\includegraphics[width=0.45\\textwidth]{images\/constant_weight_trajectory-d0.1-mu0.1-weight8-op0.png}\n\\caption{\\label{fig:cluster_size} Sample trajectories of {agent opinions versus time in a single simulation of} our node-weighted BCM on a complete graph with $N = 500$ nodes and a constant weight distribution. Therefore, this situation corresponds to our baseline DW model. We color the trajectory of each node by its final opinion cluster. Observe that the final opinion clusters have different sizes. There is a minor cluster (in black); it consists of a single node whose final opinion is about $0.4$. The opinion cluster that converges to the largest opinion value has about twice as many nodes as the other major clusters.\n}\n\\end{figure}\n\n\nSuppose that we have $K$ opinion clusters, which we denote by $S_r$ for $r \\in \\{1, \\ldots, K\\}$.\nWe call the set $\\{S_r\\}_{r=1}^K$ an ``opinion-cluster profile''; such a profile is a partition of a network.\nThe fraction of agents in opinion cluster $S_r$ is $|S_r|\/N$. The Shannon entropy $H$ of the opinion-cluster profile is\n\\begin{equation}\n H = - \\sum_{r = 1}^{K} \\frac{|S_r|}{N} \\ln \\left( \\frac{|S_r|}{N} \\right) \\,.\n\\end{equation}\nThe quantity $H$ describes the information gain of a particular opinion-cluster profile from knowing the cluster membership of a single agent in comparison to knowing no information.\nFor a fixed $K$, the entropy $H$ is larger if the cluster sizes are closer in magnitude than if there is more heterogeneity in the cluster sizes. For cluster profiles with similar cluster sizes, $H$ is larger if there are more clusters.\nAs in Han et al.~\\cite{han2020}, we use $H$ to quantify opinion fragmentation, with larger $H$ corresponding to greater opinion fragmentation. \nWe calculate $H$ using all steady-state opinion clusters (i.e., both major and minor clusters).\n\n\nAnother way to quantify opinion fragmentation is to look at a local level and consider individual agents. As Musco et al.~\\cite{musco2021} pointed out, if an individual agent has many neighbors with a similar opinion to it, then it may be ``unaware'' of other opinions in the network. \nFor example, an agent can observe that a majority of its neighbors hold an opinion that is uncommon in the global network. This phenomenon is sometimes called a ``majority illusion''~\\cite{lerman2016}.\nIf a set of adjacent agents tend to have neighbors with similar opinions as theirs, they may be in an ``echo chamber'' \\cite{flaxman2016}, as it seems that they are largely exposed only to conforming opinions.\nTo quantify the local observations of agent, Musco et al.~\\cite{musco2021} calculated a notion of local agreement that measures the fraction of an agent's neighbors with opinions on the same side of the mean opinion of the agents in a network.\nIn our simulations, we often observe opinion fragmentation with three or more opinion clusters. \nTherefore, we need to look beyond the mean opinion of an entire network. To do this, we introduce the \\emph{local receptiveness} of an agent.\nAt time $t$, a node $i$ with neighborhood $\\mathcal{N}(i)$ has a local receptiveness of\n\\begin{equation}\n L_i(t) = \n \\frac{ \\left| \\{j \\in \\mathcal{N}(i) : |x_i(t) - x_j(t)| < c \\} \\right|}\n {|\\mathcal{N}(i)|} \\, .\n\\end{equation}\nThat is, $L_i(t)$ is the fraction of the neighbors of agent $i$ at time $t$ to which it is receptive to interacting (and thereby having its opinion influenced). \nIn our paper, we only consider networks with no isolated nodes, so each agent $i$ has $|\\mathcal{N}(i)| \\geq 1$ neighbors. If one wants to consider isolated nodes, one can assign them a local receptiveness of $0$ or $1$.\nIn our numerical simulations, we calculate the local receptiveness of each agent at the convergence time $T$. \nWe then calculate the mean $\\langle L_i(T) \\rangle$ of all agents in a network. This is the steady-state mean local receptiveness, as it is based on edges in the steady-state effective-receptivity network $G_{\\mathrm{eff}}(T)$.\nWhen consensus is not reached, a smaller mean local receptiveness is an indication of greater opinion fragmentation. \nAs we will discuss in Sec.~\\ref{sec:results}, in concert with the number of opinion clusters, both the Shannon entropy and the mean local receptiveness can provide insight into the extent of opinion fragmentation.\n\n\n\n\\section{Numerical Simulations and Results} \\label{sec:results}\n\nIn this section, we present results from our numerical simulations of our node-weighted BCM. \nIn our numerical experiments, the compromise parameter takes the values $m \\in \\{0.1, 0.3, 0.5\\}$. For the confidence radius, we first consider the values $c \\in \\{0.1, 0.3, 0.5, 0.7, 0.9\\}$, and we then examine additional values of $c$ near regions with interesting results.\nAs we discussed in Sec.~\\ref{sec:sims}, we simulate a total of 100 distinct sets of initial opinions and node weights in Monte Carlo simulations of our BCM on each individual graph. For each of our random-graph models (i.e., ER and SBM graphs), we generate 5 graphs. \nFor the 500-node complete graphs, we simulate the 10 weight distributions in Table~\\ref{tab:distributions}. \nBecause of computation time, we simulate the distributions with Pareto-90-10 mean only on 500-node complete graphs. For the other networks in Table~\\ref{tab:networks}, we consider the Pareto, exponential, and uniform families of distributions with the Pareto-80-10 and Pareto-80-20 means.\nSee our repository at \\url{https:\/\/gitlab.com\/gracejli1\/NodeWeightDW} for our code and additional plots.\nIn Table~\\ref{tab:trends}, we summarize the trends that we observe in the examined networks. In the following subsections, we discuss details of our results for each type of network. \nThe numbers of major and minor clusters, Shannon entropies, and values of mean local receptiveness are all steady-state values.\n\n\n\\newcommand{0.5in}{0.5in}\n\\newcommand{2.4in}{2.4in}\n\\newcommand{\\trendvspace}{10pt} \n\n\n\\begin{table}\n\\caption{\\label{tab:trends} Summary of the trends in our simulations of our node-weighted BCM. Unless we note otherwise, we observe these trends for each of the networks that we examine (complete graphs, ER and SBM random graphs, and the Caltech network).}\n\\begin{ruledtabular}\n\\begin{tabular}{m{0.5in} m{2.4in}}\nQuantity & Trends \\\\ \\hline\n\\begin{tabular}{m{0.5in}} Convergence Time \\end{tabular}\n& \\begin{tabular}{m{2.4in}}\n- The heterogeneous weight distributions have longer convergence times than the constant-weight distribution. \n\\end{tabular} \\\\ \\hline\n\\begin{tabular}{m{0.5in}}\nOpinion \\\\ Fragmentation\\footnotemark[1]\n\\end{tabular} &\n\\begin{tabular}{m{2.4in}}\n - For confidence radii $c \\in [0.1, 0.4]$, the heterogeneous weight distributions usually have more opinion fragmentation than the constant-weight distribution. \\\\ \\vspace{\\trendvspace}\n - For a fixed distribution mean, there is more opinion fragmentation as the tail of a distribution becomes heavier. \\\\ \\vspace{\\trendvspace}\n - Within a family of distributions, there is more opinion fragmentation when a distribution has a larger mean.\\footnotemark[2]\n \\end{tabular}\n\\\\ \\hline\n\\begin{tabular}{m{0.5in}}\nNumber of Major Clusters \\end{tabular} \n& \\begin{tabular}{m{2.4in}}\n - In comparison to the constant-weight distribution, the heterogeneous weight distributions need to be above a larger threshold $c$ to consistently reach consensus. \\\\ \\vspace{\\trendvspace}\n - For a fixed distribution mean, there are more major clusters as the tail of a distribution becomes heavier. \\\\ \\vspace{\\trendvspace}\n - Within a family of distributions, there are more major clusters when a distribution has a larger mean.\n \\end{tabular} \\\\ \\hline\n\\begin{tabular}{m{0.5in}}\nNumber of Minor Clusters \\end{tabular}\n& \\begin{tabular}{m{2.4in}}\n- For the constant-weight distribution, there tends to be more minor clusters when the compromise parameter $m \\in \\{0.3, 0.5\\}$ than when $m = 0.1$. The heterogeneous weight distributions do not follow this trend.\\footnotemark[3] \\end{tabular}\n\\end{tabular}\n\\end{ruledtabular}\n\\footnotetext[1]{We quantify opinion fragmentation using Shannon entropy and mean local receptiveness. We observe clearer trends for Shannon entropy than for the mean local receptiveness.}\n\\footnotetext[2]{In contrast to this trend, the associated results are inconclusive for 500-node complete graphs for the uniform and exponential distribution families.\n}\n\\footnotetext[3]{For the Caltech network, we usually observe more minor clusters when $m \\in \\{0.3, 0.5\\}$ than when $m = 0.1$ for each of our heterogeneous weight distributions.}\n\\end{table}\n\n\n\n\\subsection{Complete Graphs} \\label{sec:results-complete} \n\nThe simplest underlying network structure on which we run our node-weighted BCM is a complete graph.\nComplete graphs provide a baseline setting for examining how heterogeneous node-selection probabilities affect opinion dynamics.\nIn our numerical simulations on complete graphs, we consider all three means (which we denote by 80-10, 80-20, and 90-10 in Table~\\ref{tab:distributions}) for each of the uniform, exponential, and Pareto node-weight distribution families.\n\nFor the standard DW model on a complete graph with agents with opinions in the interval $[0,1]$, one eventually reaches consensus if the confidence radius $c \\geq 0.5$.\nAs one decreases $c$ from $0.5$, there are progressively more {steady-state} clusters (both minor and major) \\cite{lorenz2008, benaim2003}.\nLorenz \\cite{lorenz2008} showed using numerical simulations that the number of major clusters is approximately\n$\\lfloor \\frac{1}{2c} \\rfloor$ for the standard DW model. \nTherefore, a transition between consensus and opinion fragmentation occurs for $c \\in [0.25, 0.3]$. We zoom in on these confidence radii in our simulations to explore this transition.\nThe location of the transition is different for the Pareto distributions, so we instead simulate additional values of $c \\in [0.3, 0.4]$ for this situation.\nWe also simulate these additional values of $c$ for the constant-weight distribution, which is our baseline DW model with uniformly random node-based selection of agents.\n\n\n\\begin{figure}[b]\n\\includegraphics[width=0.48\\textwidth]{images\/complete\/T_avg--labeled--with_std.png}\n\\caption{\\label{fig:complete_T} Convergence time (in terms of the number of time steps) in simulations of our node-weighted BCM on a 500-node complete graph. If we only consider the time steps in which interacting nodes actually change their opinions, the times are smaller; however, the trends are the same.\nFor this heat map and all subsequent heat maps, the depicted values are the means of our simulations plus and minus one standard deviation.}\n\\end{figure}\n\n\nIn Fig.~\\ref{fig:complete_T}, we show the convergence times (which we measure in terms of the numbers of time steps) of our simulations for various node-weight distributions. In comparison to the constant-weight distribution, all heterogeneous weight distributions have longer convergence times. \nWithin the same family of distributions (uniform, exponential, or Pareto), the convergence time is progressively larger for distributions with progressively larger means. \nFor a given heterogeneous weight distribution, the convergence time is also progressively larger for progressively smaller values of the compromise parameter $m$.\nWhen calculating convergence time, we include the time steps in which two nodes interact but do not change their opinions.\nTo see if the heterogeneous weight distributions have inflated converges times as a result of more of these futile interactions, we also calculate the number of time steps to convergence when we exclude such time steps. \nThat is, we count the total number of opinion changes to reach convergence. On a logarithmic scale, there is little difference between the total number of opinion changes and the total number of time steps to reach convergence. We include the associated plot and values of the numbers of opinion changes in our code repository.\n\n\n\\begin{figure}[ht!]\n\\includegraphics[width=0.48\\textwidth]{images\/complete\/major_clusters_avg--labeled--with_std.png}\n\\caption{\\label{fig:complete_clusters} Numbers of major opinion clusters {at steady state}\nin simulations of our node-weighted BCM on a 500-node complete graph with various distributions of node weights. We consider a cluster to be major cluster if it has more than 2\\% of the nodes in the network. (In this case, a major cluster must have at least 11 nodes.)}\n\\end{figure}\n\n\nIn Fig.~\\ref{fig:complete_clusters}, we show the numbers of major opinion clusters at steady state in our simulations for various node-weight distributions. For all distributions, consensus occurs consistently (i.e, in all of our simulations) when the confidence radius $c \\geq 0.5$. \nFor the constant-weight distribution, consensus occurs consistently when $c \\geq 0.35$. When $c \\in [0.1, 0.4]$, the heterogeneous weight distributions have more steady-state major clusters than the constant-weight distribution.\nWhen we introduce heterogeneous node weights into our BCM, we need a larger threshold confidence radius $c$ to consistently reach consensus. \nIt appears that our BCM with heterogeneous node weights\ntends to have more opinion fragmentation than the baseline DW model.\nBy comparing the columns in Fig.~\\ref{fig:complete_clusters}, we observe for each distribution family (uniform, exponential, and Pareto) that there are more steady-state major clusters\n(when proceeding left to right in the plot from 80-10 to 80-20 and then to 90-10) when the distribution mean is larger.\nAdditionally, for a fixed mean weight, there are more steady-state major clusters as we proceed from a uniform distribution to an exponential distribution and then to a Pareto distribution.\n\n\n\\begin{figure}[ht!]\n\\includegraphics[width=0.48\\textwidth]{images\/complete\/entropy_avg--labeled--with_std.png}\n\\caption{\\label{fig:complete_entropy} Shannon entropies of the steady-state opinion-cluster profiles in simulations of our node-weighted BCM on a 500-node complete graph with various node-weight distributions.}\n\\end{figure}\n\n\nTo investigate how the model parameters affect the amount of opinion fragmentation, we calculate steady-state Shannon entropy and mean local receptiveness (see Sec.~\\ref{sec:quantities}). In Fig.~\\ref{fig:complete_entropy}, we show the steady-state entropy values of our simulations for various node-weight distributions.\nFor all distributions, when there is opinion fragmentation instead of consensus, a progressively smaller confidence radius $c$ yields progressively larger {steady-state} entropies.\nIn line with our observations in Fig.~\\ref{fig:complete_clusters}, when $c \\in [0.1, 0.4]$, simulations of heterogeneous weight distributions usually yield larger entropies than the constant-weight distribution.\nFor a fixed mean weight, we also observe a slightly larger entropy as we proceed from a uniform distribution to an exponential distribution and then to a Pareto distribution. \nFor the Pareto distributions, there is progressively larger entropy for progressively larger distribution means (from left to right in Fig.~\\ref{fig:complete_entropy}). \nFor the exponential and uniform distributions, although we do not conclusively observe the same trend, we do obtain progressively more major clusters for progressively larger distribution means (see Fig.~\\ref{fig:complete_clusters}). \nFor these distributions, a larger mean weight results in more major clusters, but these clusters are smaller, so the Shannon entropy is similar. \nTherefore, if we quantify fragmentation using Shannon entropy, we conclude that increasing the mean weight of a distribution has little effect on the amount of opinion fragmentation.\nBecause the entropy depends on the sizes of the opinion clusters, it provides more information about the opinion fragmentation than only tracking the number of clusters major clusters.\nOur plot of the mean local receptiveness illustrates the same trends as the entropy. (See our code repository for the relevant figure.) This suggests that both the Shannon entropy and the mean local receptiveness are useful for quantifying opinion fragmentation.\n\n\nWe now discuss the numbers of steady-state minor clusters\nthat we obtain in our numerical experiments on complete graphs. (See our code repository for a plot.)\nFor all of the examined parameter values, once we average over our 100 simulations for a given node-weight distribution and specified values of $c$ and $m$, we obtain at most two minor clusters at steady state. \nWe observe the most minor clusters when $c \\in \\{0.1, 0.2\\}$, which are the smallest confidence radii that we examine.\nFor the constant-weight distribution, there tends to be more minor clusters when $m \\in \\{0.3, 0.5\\}$ than when $m = 0.1$. However, we do not observe this trend for the heterogeneous weight distributions.\nFor example, for the Pareto-80-10 distribution, when $c \\in [0.34, 0.4]$, \nsmaller values of $m$ result in more minor clusters. For the Pareto distribution, for smaller values of $m$, we also observe that minor clusters tend to appear at smaller confidence radii. \nSmaller values of $m$ entail smaller opinion compromises for interacting agents; this may give more time for agents to interact before they settle into their final opinion clusters. \nFor the constant-weight distribution, this may reduce the number of minor clusters by giving more opportunities for agents to be assimilated into a major cluster.\nHowever, for our heterogeneous weight distributions, nodes with larger weights have a larger probability of being involved in interactions and we no longer observe fewer minor clusters for smaller values of $m$. \n\n\n\\begin{figure}[h!]\n\\includegraphics[width=0.45\\textwidth]{images\/pareto_trajectory-d0.2-mu0.1-weight4-op0.png}\n\\caption{\\label{fig:pareto_trajectory} Sample trajectories of {agent opinions versus time in a single simulation of} our node-weighted BCM on a complete graph with $N = 500$ nodes and node weights that are distributed according to a Pareto-80-10 distribution. We color the trajectory of each agent by its node weight. The nodes in the two minor clusters are all small-weight nodes; their weights are close to 0 (and are hence in purple).}\n\\end{figure}\n\n\nWe now propose a possible mechanism by which our node-weighted BCM may promote the trends in Table~\\ref{tab:trends}.\nIn Fig.~\\ref{fig:pareto_trajectory}, we show the trajectories of opinions versus time for a single simulation with node weights that we draw from a Pareto-80-10 distribution.\nTo qualitatively describe our observations, we examine the large-weight and small-weight nodes (i.e., the nodes that are near and at the extremes of a set of node weights in a given simulation). Because our node-selection probabilities are proportional to node weights, to compare the weights in a simulation, we normalize them to sum to $1$. \nIn Fig.~\\ref{fig:pareto_trajectory}, the large-weight nodes in a simulation appear to quickly stabilize into an associated major opinion cluster, and some small-weight nodes are left behind to form the two minor clusters.\nIn our numerical simulations on complete graphs, we observed that introducing heterogeneous node weights results in large-weight nodes interacting more frequently and quickly settling into their respective steady-state major opinion clusters.\nSmall-weight nodes that are not selected early in a simulation are left behind to form the smallest clusters in a steady-state opinion-cluster profile; this increases the amount of opinion fragmentation.\nIn comparison to the constant-weight distribution, when we increase the mean node weight or increase the relative proportion of large-weight nodes in the Pareto-80-10 distribution (thereby increasing the heaviness of the tail of a distribution) or decrease the value of the compromise parameter $m$, \nsmall-weight nodes take longer to settle into their respective opinion clusters; this may promote both opinion fragmentation and the formation of minor clusters.\n\n\n\\subsection{Erd\\H{o}s--R\\'{e}nyi (ER) Graphs} \\label{sec:results-ER} \n\n\nWe now examine random graphs that we generate using $G(N,p)$ ER random-graph models, where $p$ is the homogeneous, independent probability of an edge between any pair of nodes \\cite{newman2018}. For $p=1$, these ER graphs are complete graphs (see Sec.~\\ref{sec:results-complete}). In this subsection, we use the probabilities $p \\in \\{0.1, 0.3, 0.5, 0.7\\}$. \n\n\n\\begin{figure*}[th!]\n\\includegraphics[width=0.95\\textwidth]{images\/ER\/All--entropy_avg--labeled--with_std.png}\n\\caption{\\label{fig:er_entropy} Shannon entropies of the steady-state opinion-cluster profiles in simulations of our node-weighted BCM on $G(500, p)$ ER random graphs with various node-weight distributions.}\n\\end{figure*}\n\n\nFor each value of $p$, we observe the trends in Table~\\ref{tab:trends}. We include the plots of our simulation results for convergence times, \nthe steady-state numbers of major and minor clusters, and the steady-state values of mean local receptiveness in our code repository.\nIn Fig.~\\ref{fig:er_entropy}, we show the steady-state Shannon entropies of our simulations for various node-weight distributions and values of $p$. \nThe entropies are comparable to those that we obtained in our simulations on 500-node complete graphs.\nWhen $c \\in [0.1, 0.4]$, for each of the three distribution families that we examine,\nthe larger-mean distribution (with a Pareto-80-20 mean of 7.2126) has larger entropy than the smaller-mean distribution (with a Pareto-80-10 mean of 2.8836). In our 500-node complete-graph simulations, this trend was inconclusive for the uniform and exponential distributions.\n\n\nFor larger $p$, we expect the results of our simulations on $G(500, p)$ networks to be similar to those of our simulations on a 500-node complete graph.\nFor $p \\in \\{ 0.3, 0.5, 0.7\\}$ and $N=500$, the number of major clusters and the mean local receptiveness are comparable to the corresponding results for a 500-node complete graph. \nWhen $p = 0.1$ and there is opinion fragmentation, we observe fewer major opinion clusters than for larger values of $p$. \nFor $p = 0.1$, when $c\\in [0.1, 0.4]$, we also observe that the mean local receptiveness tends to be larger than the corresponding values for larger $p$. One possible contributing factor for this observation may be that smaller values of $p$ yield $G(N,p)$ graphs with more small-degree nodes, which have fewer possible values of local receptiveness.\nFor example, a node with degree $2$ can only have a local receptiveness in the set $\\{0, 0.5, 1\\}$. \nUnless a small-degree node is an isolated node in the steady-state effective-receptivity network $G_{\\mathrm{eff}}(T)$, it may help inflate the value of the steady-state mean local receptiveness. \n\n\nFor progressively smaller values of $p$, the steady-state number of minor clusters becomes progressively larger. \nFor $p \\in \\{0.5, 0.7\\}$, the steady-state numbers of minor clusters are comparable to the numbers that we obtained for a 500-node complete graph. \nIn our mean of our 100 simulations for each distribution and each combination of $c$ and $m$, we obtain at most 2--3 minor clusters at steady state; this occurs when $c \\in \\{0.1, 0.2\\}$.\nHowever, for $p = 0.1$, we observe up to 9 minor clusters at steady state; this occurs when $c \\in \\{0.35, 0.4\\}$.\nIt seems sensible that smaller values of $p$ yield more minor clusters. For small $p$, there are more small-degree nodes than for larger values of $p$. \nSmall-degree nodes have fewer edges $(i,j)$ than large-degree nodes that need to satisfy the inequality $|x_i - x_j| < c$ to become part of a minor cluster in the steady-state effective-receptivity network.\n\n\n\\subsection{Stochastic-Block-Model (SBM) Graphs} \\label{sec:results-SBM} \n\nWe now examine SBM random graphs that we generate using the parameters in Table~\\ref{tab:networks}. For both the two-community and core--periphery SBM graphs, we observe the trends in Table~\\ref{tab:trends}. \nWe include the plots of our simulation results at steady state for convergence times, numbers of major and minor clusters, values of Shannon entropy, and values of mean local receptiveness in our code repository. \n\n\nFor the two-community SBM graphs, the steady-state Shannon entropies and numbers of major clusters are comparable to those in our simulations on a complete graph. \nWhen there is opinion fragmentation, the steady-state values of mean local receptiveness tend to be similar to the corresponding values for $G(500, 0.1)$ graphs and larger than the values for a complete graph.\nThe steady-state numbers of minor clusters are similar to those for the $G(500, 0.1)$ random graphs. We observe up to 9 minor clusters when $c \\in \\{0.35, 0.4\\}$, which is near the transition between consensus and fragmentation in the standard DW model \\cite{lorenz2008}.\nRecall that we selected the edge probabilities of the two-community SBM so that each of the two communities have an expected mean degree that matches that of $G(500, 0.1)$ graphs.\nTherefore, it is reasonable that we obtain similar results for the two-community SBM and the $G(500, 0.1)$ random graphs. \nIn our numerical experiments, we assign the node weights randomly without consideration of the positions of the nodes in a network. In our numerical simulations, it may be that the sparsity of a graph is more important than community structure because we do not use community structure to influence the assignment of weights (e.g., which specific nodes have large weights) in our networks.\n\n\nFor the core--periphery SBM graphs, both the {steady-state} Shannon entropy and mean local receptiveness tend to be larger than the corresponding values for a complete graph. Larger entropy and smaller local receptiveness are both indications of more opinion fragmentation. \nWhen we consider the opinion-cluster profile of an entire network, Shannon entropy reveals that there is more opinion fragmentation in core--periphery SBM graphs than in a complete graph. \nHowever, the steady-state mean local receptiveness indicates that the nodes in a core--periphery SBM graph tend to be receptive to a larger fraction of their neighbors than the nodes in a complete graph.\n\n\nWe believe that Shannon entropy provides a more useful quantification mean local receptiveness of opinion fragmentation in a network. For networks with a large range of degrees, small-degree nodes can inflate the mean value of local receptiveness. \nA similar trend has been observed for clustering coefficients; the mean local clustering coefficient places more weight on small-degree nodes than the global clustering coefficient of a network \\cite{newman2018}. \nIn the context of our node-weighted BCM, consider a node with degree 2 and a node with degree 100, and suppose that both of them have a local receptiveness of 0.5. The larger-degree node having a local receptiveness of 0.5 gives a better indication that there may be opinion fragmentation in a network than the smaller-degree node having the same local receptiveness. \nHowever, we treat both nodes equally when calculating the mean local receptiveness. \nWe believe that local receptiveness is a useful quantity to calculate for individual nodes to determine how they perceive the opinions of their neighbors. However, the mean local receptiveness appears to be less useful than Shannon entropy for quantifying opinion fragmentation in a network.\n\n\nThe steady-state numbers of major clusters that we obtain in the core--periphery SBM graphs are comparable to the corresponding numbers for a complete graph. \nThe steady-state numbers of minor clusters tend to be larger for core--periphery SBM graphs than for two-community SBM graphs (which have more minor clusters than a complete graph). \nWe observe up to 11 minor clusters at steady state; this occurs when $c = 0.1$. One possibility is that the core--periphery structure makes it easier to disconnect peripheral nodes in the effective-receptivity network, causing these nodes to form minor clusters. \nFor the core--periphery SBM graphs, it also seems interesting to investigate the effect of using network structure to assign which nodes have large weights.\nFor example, if we assign all of the large weights to nodes in the core, will that pull more of the peripheral nodes into opinion clusters with core nodes? If we place a large-weight node in the periphery, will it be able to pull core nodes into its opinion cluster?\n\n\n\n\\subsection{Caltech Network} \\label{sec:results-Caltech} \n\n\\begin{figure*}[th!]\n\\includegraphics[width=0.93\\textwidth]{images\/Caltech\/All--minor_clusters_avg--labeled--with_std.png}\n\\caption{\\label{fig:caltech_minor} The steady-state numbers of minor opinion clusters in simulations of our node-weighted BCM on the Caltech Facebook network with various distributions of node weights. We consider a cluster to be minor cluster if it has at most 2\\% of the nodes (i.e., 15 or fewer nodes) in the network.}\n\\end{figure*}\n\n\\begin{figure*}[th!]\n\\includegraphics[width=0.93\\textwidth]{images\/Caltech\/All--entropy_avg--labeled--with_std.png}\n\\caption{\\label{fig:caltech_entropy} Shannon entropies of the steady-state opinion-cluster profile in simulations of our node-weighted BCM on the Caltech Facebook network with various node-weight distributions.}\n\\end{figure*}\n\n\nWe now discuss the Caltech Facebook network, which is an empirical data set in which the nodes are individuals with Caltech e-mail addresses and the edges represent ``friendships'' on Facebook on one day in fall 2005 \\cite{red2011, traud2012}. We consider the network's largest connected component, which has 762 nodes and 16,651 edges. \nThe Caltech network has all but one of the trends that we reported in Table~\\ref{tab:trends}; the only exception is the trend in the number of minor clusters.\nWhen there is opinion fragmentation, the Caltech network has more steady-state minor clusters and larger steady-state Shannon entropies than in our synthetic networks.\n\n\nIn Fig.~\\ref{fig:caltech_minor}, we show the steady-state numbers of minor clusters in simulations of our BCM on the Caltech network. We obtain the most minor clusters when $c = 0.1$, which is the smallest value of $c$ that we examine. \nOnce we average over our 100 simulations on the Caltech network for each distribution and each pair of values of $c$ and $m$, we obtain up to 78 minor clusters, which is much more than the single-digit numbers that we usually observe for our synthetic networks.\nAdditionally, unlike in our synthetic networks, for all distributions (not just the constant-weight distribution), the Caltech network tends to have more minor clusters for $m \\in \\{0.3, 0.5\\}$ than for $m = 0.1$. \nWe include our plot of the steady-state number of major clusters in our code repository. The Caltech network tends to have fewer major clusters than the $G(500, 0.1)$ random graphs, which in turn tends to have fewer major clusters than our other synthetic networks.\n\n\nIn Fig.~\\ref{fig:caltech_entropy}, we show the steady-state Shannon entropies for the Caltech network.\nWhen there is opinion fragmentation, the Caltech network has a larger entropy than the corresponding entropy for our synthetic networks. This aligns with our observation that the Caltech network has many more minor clusters than our synthetic networks. \nWe show a plot of the steady-state values of mean local receptiveness for the Caltech network in our code repository. The mean local receptiveness tends to be larger for the Caltech network than for the 500-node complete graph. We suspect that this arises from the presence of many small-degree nodes in the Caltech network. We discussed the impact of small-degree nodes on the mean local receptiveness in Sec.~\\ref{sec:results-SBM}. \n\n\nThe degree histogram of the Caltech network in Fig.~\\ref{fig:caltech_degree} differs dramatically from those of our synthetic networks.\nUnlike in our synthetic networks, the most common degrees in the Caltech network are among the smallest degrees. In Fig.~\\ref{fig:caltech_degree}, the tallest bar in the histogram is for nodes of degrees 0--9. These abundant small-degree nodes are likely to disconnect from the largest connected component(s) in the effective-receptivity network and form minor clusters. \nBecause we select the initial opinions uniformly at random from $[0,1]$, for $c = 0.1$, it is possible that small-degree nodes are initially isolated nodes in the effective-receptivity network because of their initial opinions. The abundance of small-degree nodes in the Caltech network helps explain its larger steady-state numbers of minor clusters and the correspondingly larger \nentropies than for our synthetic networks.\nDespite the fact that the Caltech network is structurally very different from our synthetic networks, it follows all of the trends in Table~\\ref{tab:trends} other than the one for the number of minor clusters. \nTherefore, it appears that the trends that we observe in our node-weighted BCM when we assign node weights uniformly at random (and hence in a way that is independent of network structure) are fairly robust to the underlying network structure. \n\n\n\\begin{figure}[th!]\n\\includegraphics[width=0.35\\textwidth]{images\/Caltech\/Caltech_degree_histogram.png}\n\\caption{\\label{fig:caltech_degree} Degree histogram for the Caltech Facebook network. The bins have width 10 and originate at the left end point (i.e., the bins indicate degrees 0--9, 10--19, and so on).}\n\\end{figure}\n\n\n\n\\subsection{Finite-Size Effects} \\label{sec:results-finite_size} \n\nWe now investigate finite-size effects in our BCM results for our simulations on a complete graph. \nTo ensure reasonable computation times, we examined synthetic networks with 500 nodes.\nHowever, it is useful to get a sense \\map{for} the trends in Table~\\ref{tab:trends} hold for networks of different sizes. \nTo start to investigate this, we simulate our BCM on complete graphs of sizes $100, 200, \\ldots, 1000$. We examine $m \\in \\{0.3, 0.5\\}$, and $c \\in \\{0.1, 0.3, 0.5\\}$, which give regimes of opinion fragmentation, transition between fragmentation and consensus for the constant-weight distribution, and opinion consensus. \nWe examine the constant-weight distribution and the uniform, exponential, and Pareto distributions with the same `80-10' mean (i.e., with a mean node weight of 2.8836) because the smaller-mean distributions have shorter computation times. \n\n\nIn Fig.~\\ref{fig:size_T}, we show the convergence times of our simulations of our BCM on complete graphs of various sizes. For all distributions, as the graph size becomes progressively larger, the convergence times also become progressively longer. \nFor each graph size, the convergence times for the heterogeneous weight distributions are longer than those for the constant-weight distribution. \nThe convergence times for the different heterogeneous distributions in Fig.~\\ref{fig:size_T} do not follow a clear trend.\n\n\n\\begin{figure*}[th!]\n\\includegraphics[width=0.95\\textwidth]{images\/finite-size\/T.png}\n\\caption{\\label{fig:size_T} Convergence times (in terms of the number of time steps) in simulations of our node-weighted BCM on complete graphs of various sizes. The plots give results for different choices of $c$ and $m$; the marker shape and color indicates the node-weight distribution. The points are means of 100 simulations, and the error bars represent one standard deviation from the mean. The vertical axes of the plots have different scales.}\n\\end{figure*}\n \n \nIn Fig.~\\ref{fig:size_entropy}, we show the steady-state Shannon entropies from our simulations of our BCM on complete graphs of various sizes. For a given value of $c$, we observe similar results for $m = 0.3$ and $m = 0.5$. \nFor $c = 0.5$, we see that for each distribution, we reach consensus (i.e., the steady-state entropy is $0$) fairly consistently for $N \\geq 300$ nodes. As the size of the network becomes progressively larger, the error bars (which indicate one standard deviation from the mean) also become progressively smaller. \nFor $c = 0.3$, the mean steady-state entropies appear to have settled with respect to $N$ for $N \\geq 400$. For $c = 0.1$, the graph size appears to have little effect on the mean entropy. \n\n\n\\begin{figure*}[th!]\n\\includegraphics[width=0.95\\textwidth]{images\/finite-size\/entropy.png}\n\\caption{\\label{fig:size_entropy} Shannon entropies of the steady-state opinion-cluster profiles in simulations of our node-weighted BCM on complete graphs of various sizes. The plots give results for different choices of $c$ and $m$; the marker shape and color indicates the node-weight distribution. The points are means of 100 simulations, and the error bars represent one standard deviation from the mean. The vertical axes of the plots have different scales.}\n\\end{figure*}\n\n\nWhen there is opinion fragmentation, the heterogeneous distributions yield larger steady-state Shannon entropies (and hence more opinion fragmentation, if measuring it using entropy) than the constant-weight distribution for each graph size. \nAdditionally, for a given distribution mean, we obtain larger entropies (and thus more opinion fragmentation) as we increase the heaviness of the tail of a distribution.\nWe have not explored the effect of graph size on the trend of increasing the distribution mean within the same distribution family.\nIn our code repository, we include a plot of the the steady-state mean local receptiveness for complete graphs of various sizes. In that plot, we also observe the trend of more opinion fragmentation (specifically, in the sense of a smaller mean local receptiveness) for heterogeneous distributions with increasingly heavy tails.\n\n\nWe also examine plots of the steady-state numbers of major and minor clusters in simulations of our BCM on complete graphs of various sizes; we include these plots in our code repository.\nThere is not a clear trend in the numbers of major clusters as the graph size becomes progressively larger. \nFor each graph size, when $c = 0.3$, there are more major clusters as we increase the heaviness of the tail of a distribution. For each graph size, when $c = 0.1$, the Unif-80-10 and constant-weight distributions have similar numbers of major clusters. \nFor $c = 0.1$, for each distribution, there are usually \nprogressively more minor clusters as the complete graph becomes progressively larger.\n(See the associated plot in our code repository.)\n\n\nOverall, for graphs with $N = 500$ or more nodes, the mean steady-state entropies for each distribution appear to have settled with respect to $N$; the mean entropies fluctuate less for $N \\geq 500$ than for smaller values of $N$.\nFor each of the distributions that we consider in this discussion and for each of the graph sizes, the heterogeneous distributions have longer convergence times than the constant-weight distribution. Additionally, in all of these cases, there is more opinion fragmentation as we increase the heaviness of the tail of a distribution.\nBecause of computation time, we have not examined distributions with different means. However, because the mean entropies settle with respect to $N$ for graph sizes of $N \\geq 500$, we hypothesize that the trends in opinion fragmentation and convergence time in Table~\\ref{tab:trends} continue to hold for our synthetic networks when there are more than 500 nodes.\n\n\n\n\\section{Conclusions and Discussion} \\label{sec:discussion} \n\nWe developed a novel bounded-confidence model (BCM) with heterogeneous node-selection probabilities, which we modeled by introducing node weights. One can interpret these node weights as encoding phenomena such as heterogeneous agent sociabilities or activity levels.\nWe studied our node-weighted BCM with fixed node weights that we assign in a way that disregards network structure and node opinions.\nWe also demonstrated that our BCM yields longer convergence times and more opinion fragmentation than our baseline Deffuant--Weisbuch (DW) BCM, in which we uniformly randomly select nodes for interaction.\nIt is straightforward to adapt our BCM to assign node weights in a way that depends on network structure and\/or node opinions. See Sec.~\\ref{sec:discussion-networks} and Sec.~\\ref{sec:discussion-opinion} for discussions.\n\n\n\\subsection{Summary of our Results} \\label{sec:discussion-summary} \n\nWe simulated our node-weighted BCM with a variety of node-weight distributions (see Table~\\ref{tab:distributions}) on several random and deterministic networks (see Table~\\ref{tab:networks}). \nFor each of these distributions and networks, we systematically investigated the convergence time and opinion fragmentation as we varied the confidence radius $c$ and the compromise parameter $m$. \nTo determine if the nodes in a network reach consensus or if there is opinion fragmentation, we calculated the steady-state number of major clusters in our simulations. To quantify the amount of opinion fragmentation, we calculated the steady-state Shannon entropy and mean local receptiveness. \nFor a given network, we found that entropy and mean local receptiveness show the same trends for which distributions have more opinion fragmentation (see Table~\\ref{tab:trends}).\nBased on our results, we believe that a network's Shannon entropy is more useful than its mean local receptiveness for quantifying opinion fragmentation in the network.\nHowever, calculating local receptiveness is relevant for examining the opinion dynamics of individual nodes.\n\n\nIn our simulations of our node-weighted BCM, we observed a variety of typical trends (see Table~\\ref{tab:trends}). \nIn particular, we found that introducing heterogeneous distributions of node weights results in longer convergence times and more opinion fragmentation in comparison to the baseline DW model (which we obtain by using a constant-weight distribution).\nOpinion fragmentation further increases if either (1) for a fixed distribution mean, we make the tail of the distribution heavier or (2) for a given distribution family, we increase the mean of the distribution. \nFor a set of heterogeneous node weights, we propose that large-weight nodes are selected early in a simulation with large probabilities and quickly associate with their respective steady-state major opinion cluster.\nSmall-weight nodes that are not selected early in a simulation are left behind to form small opinion clusters, resulting in more opinion fragmentation than in the baseline DW model.\n\n\n\n\\subsection{Relating Node Weights to Network Structure} \\label{sec:discussion-networks} \n\nWe examined deterministic and random graphs with various structures, and we observed that the trends in Table~\\ref{tab:trends} hold for each of our networks. \nFor each of our simulations, we determined node weights using a specified distribution and then assigned these weights to nodes uniformly at random.\nTherefore, our investigation conveys what trends to expect with fixed, heterogeneous node weights that are assigned to nodes without regard for network structure. \nHowever, our model provides a flexible framework to study the effects of node weights when they are correlated to network structure. \nFor example, one can investigate assigning weights to nodes based on measures of centrality (such as degree).\n{For a given set of node weights,} larger-weight nodes have larger probabilities of being selected for interaction; their position in a network likely affects the dynamics of BCMs and other models of opinion dynamics.\nOne can also investigate the effects of homophily in the assignment of node weights. \nFor example, in social-media platforms, very active accounts may engage with each other more frequently by sharing or commenting on one anothers' posts. \nWe can incorporate such features into our BCM by incorporating a positive node-weight assortativity (such that large-weight nodes have an increased likelihood of being adjacent to each other).\n\n\nIn line with the standard DW model, we assign the initial opinions uniformly at random in our BCM. However, in a real social network with community structure, this choice may not be realistic. \nOne can investigate a social network with communities with different mean opinion values and investigate the effect of placing large-weight nodes in different communities. For example, how does placing all large-weight nodes in the same community affect opinion dynamics and steady-state opinion-cluster profiles?\nHow does the presence of a small community of ``outspoken'' (i.e., large-weight) nodes influence the final opinions of nodes in other communities in a network?\nWill the small community quickly engender an echo chamber \\cite{flaxman2016}, will it pull other nodes into its final opinion cluster, or will something else occur? \n\n\n\n\\subsection{Relating Node Weights to Node Opinions} \\label{sec:discussion-opinion} \n\nIn the present paper, we explored fixed node weights that are independent of node opinions. One can readily adapt our BCM to incorporate time-dependent node weights, such as ones that depend on node opinions.\nOne can allow the probability of selecting a node for interaction to depend on how extreme its opinion is \\cite{alizadeh2015} or on the similarity of its opinion to that of another node \\cite{sirbu2019}).\n\n\nS\\^{i}rbu et al.~\\cite{sirbu2019} studied a modified DW model with heterogeneous node-selection probabilities that model algorithmic bias on social media.\nIn their model, one first selects an agent uniformly at random. One then calculates the magnitude of the opinion difference between that agent and each of its neighbors and then selects a neighbor with a probability that is proportional to this difference. \nIn the context of our BCM, one can represent their mechanism using time-dependent node weights.\nTo do this, at each time $t$, one first assigns the same constant weight to all nodes when selecting a first node $i$.\nWhen selecting a second node $j$ to interact with $i$, one then assign weights to neighbors of $i$ that are a function of the opinion difference $|x_i(t) - x_j(t)|$.\nWe assign a weight of $0$ to nodes that are not adjacent to $i$.\nThe simulations by S\\^{i}rbu et al. on complete graphs suggest that greater algorithmic bias results in longer convergence times and more opinion clusters \\cite{sirbu2019}.\nVery recently, Pansanella et al.~\\cite{pansanella2022} observed similar trends in a study of the algorithmic-bias model of S\\^{i}rbu et al. using various random-graph models.\n\n\nWe also observed similar trends of longer convergence times and more opinion clusters (and opinion fragmentation) than the baseline DW model in our simulations of our BCM with heterogeneous node-selection probabilities.\nOur results show that it is important to consider the baseline effect of assigning node weights uniformly at random in the study of BCMs with heterogenous node-selection probabilities before attributing trends such as longer convergence times and more opinion fragmentation to specific mechanisms such as algorithmic bias.\n\n\n\\subsection{Edge-Based Heterogeneous Activities} \\label{sec:discussion-edges} \n\nIn the standard DW model on a network, at each time, one selects an edge of a network uniformly at random and the two agents that are attached to the edge interact with each other \\cite{weisbuch2001}.\nMost past work on the DW model and its generalizations has focused on this edge-based selection mechanism \\cite{noorazar2020}.\nIn our BCM, to incorporate node weights (e.g., to encode heterogeneous sociabilities or activity levels of individuals), we instead used a node-based selection mechanism.\nFor voter models of opinion dynamics, it is known that the choice between edge-based and node-based agent selection can substantially affect a model's dynamics \\cite{kureh2020}. \nWe are not aware of a comparison of edge-based and node-based agent selection in asynchronous BCMs (and, in particular, in DW models), and it seems interesting to investigate this issue.\n\n\nWe developed our BCM to incorporate node weights that encode \nheterogeneous activity levels of individuals.\nOne can also examine heterogeneous dyadic-activity levels to account for the fact that individuals do not interact with each of their social contacts with the same probability.\nTo encode such heterogeneity, one can construct a variant of our BCM that incorporates edge weights.\nAt each time step, one can select a pair of agents to interact with a probability that is proportional to weight of the edge between them. \nAs we discuss in Sec.~\\ref{sec:discussion-weights}, it is more common in network science to study edge weights than node weights. \nWe have not yet examined edge-based heterogeneous activity levels in a BCM, and we expect that it will be interesting to investigate them.\n\n\n\n\\subsection{Importance of Node Weights} \\label{sec:discussion-weights} \n\n\nThe key novelty of our BCM is our incorporation of node weights into opinion dynamics. \nNode weights have been used in activity-driven models of temporal networks \\cite{perra2012}. Activity-driven frameworks has been used to model what agent interactions are allowed in models of opinion dynamics \\cite{li2017,zhang2018}.\nIn our BCM, the node weights determine the probabilities of selecting agents for interaction in a time-independent network.\nAlizadeh and Cioffi-Revilla \\cite{alizadeh2015}, S\\^{i}rbu et al. \\cite{sirbu2019}, and Pansanella et al. \\cite{pansanella2022} each examined specific scenarios of heterogeneous node-selection probabilities in DW models. \nOur node-weighted BCM provides a general framework to study node weights in an asynchronous BCM. One can use our model to study node weights that are assigned uniformly at random to nodes and fixed (i.e., as we investigated in this paper), assigned according to some other probability distribution and fixed, or assigned in a time-dependent way (e.g., as we discussed in Sec.~\\ref{sec:discussion-opinion}).\n\n\nNode weights have been studied far less than edge weights in network science, and even the term ``weighted network'' usually refers specifically to edge-weighted networks by default. \nFor example, it is very common to study centralities in edge-weighted networks \\cite{opsahl2010}, but studies of centralities in node-weighted networks (e.g., see Refs.~\\cite{heitzig2012, singh2020}) are much less common.\nHeitzig et al. \\cite{heitzig2012} developed generalizations of common network statistics that use node weights that represent the ``sizes'' of nodes in a network.\nThey used their framework to study brain networks with node weights that encode the areas of regions of interest, international trade networks with node weights that encode the gross domestic products (GDPs) of countries, and climate networks with node weights that encode areas in a regular grid on the Earth's surface.\nSingh et al.~\\cite{singh2020} developed centrality measures that incorporate both edge weights and node weights and used them to study service-coverage problems and the spread of contagions.\nThese studies demonstrate the usefulness of node weights for incorporating salient information in network analysis in a variety of applications.\n\n\nIn our BCM, we are interested in determining which nodes in a network are (in some sense) more influential than others and thereby exert larger effects on a steady-state opinion-cluster profile. \nRecently, Brooks and Porter \\cite{brooks2020} quantified the influence of media nodes in their BCM by examining how their ideologies influence other nodes in a network.\nAn interesting area of future work is to develop ways to quantify the influence of specific nodes in models of opinion dynamics with node weights.\nFor example, can one determine which weighted nodes to seed with extreme opinions to try and best spread such opinions? Are there nodes that make it particularly easy for communities to reach consensus and remain\nconnected in the steady-state effective-receptivity network $G_{\\mathrm{eff}}(T)$?\nOne can adapt the node weights in our BCM to capture a variety of sociological scenarios in which nodes have heterogeneous activity levels and interaction frequencies.\nMore generally, our model illustrates the importance of incorporating node weights into network analysis, and we encourage researchers to spend more time studying the effects of node weights on network structure and dynamics. \n\n\n\n\n\n\\begin{acknowledgements}\n\nWe thank Andrea Bertozzi, Deanna Needell, Jacob Foster, Jerry Luo, and the participants of UCLA's Networks Journal Club for helpful comments and discussions. We acknowledge financial support from the National Science Foundation (grant number 1922952) through the Algorithms for Threat Detection (ATD) program. GJL was also supported by NSF grant number 1829071.\n\n\\end{acknowledgements}\n\n\n\n\n\\providecommand{\\noopsort}[1]{}\\providecommand{\\singleletter}[1]{#1}%\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nGroups and clusters of galaxies represent important ingredients in the\nUniverse for many purposes, for example, to test the large-scale structure or\nthe underlying cosmological model. The cluster catalogues by Abell\n(\\cite{abell}) and Abell et al.\\ (\\cite{aco}) were constructed by visual\ninspection of Palomar plates. The catalogues of the new generation of galaxy\ngroups were the Las Campanas catalogue of groups by Tucker et al.\n(\\cite{Tucker00}), the catalogues based on SDSS (Sloan Digital Sky Survey) data\nreleases (EDR, DR1, DR2, DR3, DR4, DR5) and the 2dFGRS (2 degree Field Galaxy\nRedshift Survey) data releases (100K, final, Colless et al.\n\\cite{col01}, \\cite{col03}). This inspired numerous research teams to investigate \nmore refined cluster finding algorithms and to compile catalogues of galaxy\nsystems (de Propris et al.\\ \\cite{dep02a}, Merchan \\& Zandivarez \\cite{mer02}, \n\\cite{mer05}, Bahcall et al.\\ \\cite{bac03}, Lee et al.\\ \\cite{lee04}, Eke et al.\n\\cite{eke04}, Yang et al.\\ \\cite{yang05}, Einasto et al.\\ \\cite{ein05}, Goto\n\\cite{goto02}, Weinmann et al.\\ \\cite{wein06}, Tago et al.\\ \\cite{tago06}, Berlind\net al.\\ \\cite{ber06}). \n\nIn our previous paper Tago et al.\\ (\\cite{tago06}, hereafter Paper~1) we have extracted 2dFGRS \ngroups, and we have\ngiven an extensive review of papers dedicated to group search methods and to\npublished group catalogues. In this introduction we present a short review of\nstudies of galaxy groups. \n\nIn recent years a number of new group finding algorithms and modified well\nknown methods have been applied (Goto et al.\\ \\cite{goto02}, Kim et al.\n\\cite{kim02}, Bahcall et al.\\ \\cite{bac03}, review by Nichol \\cite{nic04}, \nKoester et al.\\ \\cite{koe07}). However, the friends-of-friends method (FoF, \nsometimes called percolation method) remains the most frequently applied for\nredshift surveys. \n\n\n\\begin{table*}\n \\caption[]{The SDSS DR5 Main samples used, and the FoF parameters\n for the group catalogue (DR4 is for comparison but not studied)}\n \\label{Tab1}\n \\begin{tabular}{cccccccccc}\n \\hline\\hline\n \\noalign{\\smallskip}\n Sample & $RA, \\lambda$ & $DEC, \\eta$ & $N_{gal}$ &\n $N_{groups}$&$N_{single}$&$\\Delta V_0$ & $\\Delta R_0$ &\n $ z_* $ & $ a $ \\\\\n & deg & deg & & & & km\/s & Mpc\/h & & \\\\\n \\noalign{\\smallskip}\n \\hline\n 1 & 2 & 3 & 4 & 5 &\n 6 & 7 & 8 & 9 & 10 \\\\\n \\hline\n \\noalign{\\smallskip}\n\nSDSS DR4 E & 120... 255 & -1... 16 & 116471 & 16244 & 65016 &\n250 & 0.25 & 0.138 & 1.46 \\\\\n\nSDSS DR4 N & -63... +63 & 6... 39 & 197481 & 25987 & 115488 &\n250 & 0.25 & 0.138 & 1.46 \\\\\n\n\\\\\nSDSS DR5 E & 120... 255 & -1... 16 & 129985 & 17143 & 75788 &\n250 & 0.25 & 0.055 & 0.83 \\\\\n\nSDSS DR5 N & -63... +63 & 6... 39 & 257078 & 33219 & 152234 &\n250 & 0.25 & 0.055 & 0.83 \\\\\n\n\n \\noalign{\\smallskip}\n \\hline\n \\end{tabular}\\\\\n\n\\small\\rm\\noindent Columns:\n\\begin{itemize}\n\\item[1:] the subsample of the SDSS redshift catalogue used, \n\\item[2:] right ascension limits for the equatorial (E) sample, $\\lambda$\n coordinate limits for the northern (N) sample (degrees), \n\\item[3:] declination limits for the E sample, $\\eta$ coordinate limits for\n the N sample (degrees), \n\\item[4:] number of galaxies in a subsample, \n\\item[5:] number of groups in a subsample, \n\\item[6:] number of single galaxies, \n\\item[7:] the FoF linking length in radial velocity, for $z=0$, \n\\item[8:] the FoF linking length in projected distance in the sky\n , for $z=0$, \n\\item[9:] the characteristic scaling distance for the linking length\n , see Eq.~\\ref{lz}, Sec.~5, \n\\item[10:] the scaling amplitude for the linking length, see\n Eq.~\\ref{lz}, Sec.~5. \n\\end{itemize}\n \\end{table*}\n\nRecently several authors have compiled group catalogues using the 2dF\nGalaxy Redshift Survey. One of the largest sample of groups has been compiled\nby Eke et al.\\ (\\cite{eke04}), who compared the real group samples with\nsamples found for simulated 2dF redshift survey galaxies. \nYang et al.\\ (\\cite{yang05}) applied more strict\ncriteria in group selection, and as a result have obtained a 2dF group\ncatalogue that contains mainly compact groups and a larger fraction of\nsingle galaxies. In Paper~1 we applied criteria yielding groups\nof galaxies with statistical properties between these two catalogues. \n\nUsing earlier releases of the SDSS Lee et al.\\ (\\cite{lee04}, EDR), \nMerchan and Zandivarez (\\cite{mer05}, DR3), Goto (\\cite{goto05}, DR2), Weinmann\net al.\\ (\\cite{wein06}, DR2, see for details Yang et al.\\ \\cite{yang05}), \nZandivarez et al.\\ (\\cite{zan06}, DR4), Berlind et al.\\ (\\cite{ber06}, DR3) have\nobtained catalogues of groups (and clusters) of galaxies with rather different\nproperties. In the present paper we have applied a FoF group search method\nfor the recent public release (DR5) of the SDSS. All these group\ncatalogues are constructed on the basis of spectroscopic data of galaxy\ncatalogues using certain selection criteria. The most important data and\nproperties for these catalogues (if available) are presented in\nTable~\\ref{Tab2}. \n\nApart from the other authors Berlind et al.\\ (\\cite{ber06}) have used\nvolume-limited samples of the SDSS. This yielded one of the most detailed\nsearch method and reliable group catalogue(s). Recently Paz et al.\\ \n(\\cite{paz06}) studied shapes and masses of the 2dFGRS groups (2PIGG), Sloan\nSurvey Data Release 3 groups and numerical simulations, and founda strong\ndependence on richness. \n\nPapers dedicated to group and cluster search show a wide range of both sample\nselection as well as cluster search methods and parameters. The choice of\nthese parameters depends on the goals of the group catalogues obtained. In\nPaper~1 we drew a conclusion that in previous group catalogues the\nluminosity\/density relation in groups have not been applied. In this paper we\napply this property of the observed groups to create a group catalogue for an\nextended sample of the SDSS DR5. \n\nSelection effects in data are important factors in choosing galaxy selection\nmethods and understanding group properties. In the present paper we\ninvestigate various selection effects in SDSS (described in details in\nPaper~1) which influence compilation of group catalogues. We applied for the\nSDSS DR5 (the last published data release) the well-known friends-of-friends\n(FoF) algorithm. Considering earlier experiences we selected a series of\nprocedures discussed below. \n\n\nThe data used are described in Section 2. Sect. 3 discusses the\ngroup-finding algorithm. Selection effects, which influence the\nchoice of parameters for the FoF procedure are discussed in Sect. 4. \nTo select an appropriate cluster-finding algorithm we analyse in Sect. 5\nhow the properties of groups change, if they are observed at various\ndistances. Section 6 describes the final procedure\nused to select the groups, and the group catalogue. We also estimate\nluminosities of groups; this is described in Section 7. \nIn the last Section we compare our groups with groups found by\nother investigators, and present our conclusions. \nAs in Paper~1 we use for simplicity the term ``group'' for all objects in our\ncatalogue including also rich clusters of galaxies. \n\n\n\\section{The Data}\n\nIn this paper we have used the data release 5 (DR5) of the SDSS\n(Adelman-McCarthy et al. \\cite{ade07}; see also \\cite{ade06},DR4) \nthat contains overall 674749 galaxies\nwith observed spectra. The spectroscopic survey is complete from \n${\\rm r} =14.5$ up to ${\\rm r} =17.77$ magnitude.\n\nWe have restricted our study with the main galaxy sample obtained from the\nSDSS Data Archive Server (DAS) which reduced our sample down to 488725\ngalaxies. In present status the survey consists of two main contiguous areas\n(northern and equatorial, hereafter N and E samples, respectively), and 3\nnarrow stripes in the southern sky and a short stripe at high declination. We\nhave excluded smaller areas from our group search. For the two areas the\ncoordinate ranges are given in Table~\\ref{Tab1}.\n\nWe put a lower redshift limit $z=0.009$ to our sample with the aim to exclude\ngalaxies of the Local Supercluster. As the SDSS sample becomes very diluted\nat large distances, we restrict our sample by a upper redshift limit $z=0.2$.\nLater we see that for our purposes this SDSS main sample is more or less\nhomogeneous up to $z=0.12$.\n \nWe have found duplicate galaxies due to repeated spectroscopy for a number of\ngalaxies in the DAS Main galaxy sample. We have excluded from our sample those\n duplicate entries which have spectra of lower accuracy. There were two\ntypes of duplicate galaxies. In one case duplicates had exactly identical ID\nnumbers, coordinates and magnitudes; they were simple to find out and to exclude. \nAnother kind of duplicates had slightly different values of coordinates and\nmagnitudes. This kind of duplicates cannot be seen in the sky distribution of\ngalaxies but were discovered as an enhanced number density of galaxy pairs\nafter the FoF procedure. The majority of the second kind of duplicates have\nbeen found at the common boundary of the data releases DR1 and DR2 (at DEC\n$-1.25$ and $+1.25$). We have excluded them as duplicate galaxies due to features\nseen in Figure~\\ref{fig:duplicate} and Figure~\\ref{fig:rvirduplicate}. In\ntotal we have excluded from both samples 6439 identical galaxies and 1480\ngalaxies with slightly different data. \n\n\\begin{figure}[ht]\n\\centering\n\\resizebox{0.5\\textwidth}{!}{\\includegraphics*{duplicate.eps}}\n\\caption{Duplicate galaxies in the sample E appearing as an increased density\nof groups at the boundaries of the data releases 1 and 2. \n}\n\\label{fig:duplicate}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\n\\resizebox{0.5\\textwidth}{!}{\\includegraphics*{rvirduplicate.eps}}\n\\caption{Duplicate galaxies in the sample E appearing as a separated\nmode (due to false pairs at very low value of virial radius) in the \nvirial radius - distance relation of groups. \n}\n\\label{fig:rvirduplicate}\n\\end{figure}\n\nThe total number of galaxies has reduced to 129985 galaxies in the equatorial\nsample and to 257078 galaxies in the northern sample. Resulting data on the\nsamples are presented in Table~\\ref{Tab1}. In the present paper we have studied\nonly the SDSS DR5 release. The redshifts were corrected for the motion\nrelative to the CMB. For linear dimensions we use co-moving distances (see,\ne.g., Mart{\\`\\i}nez \\& Saar \\cite{mar03}), computed with the standard cosmological\nparameters: the Hubble parameter $H_0=100 h$, the matter density $\\Omega_m =\n0.3$, and the dark energy density $\\Omega_{\\Lambda} = 0.7$.\n\n\n\n\\section{Friends-of-friends algorithm}\n\nOne of the most conventional methods to search for groups of galaxies is\ncluster analysis that was introduced in cosmology by Turner and Gott\n(\\cite{tg76}), and successfully nicknamed as the \"friends-of-friends\"\nalgorithm by Press and Davis (\\cite{pd82}). This algorithm along with the\npercolation method started its world-wide use after suggestions by Zeldovich\net al.\\ (\\cite{zes82}) and by Huchra \\& Geller (\\cite{hg82}). In Paper 1 we\nhave explained the FoF method and the role of linking length (or neighbourhood\nradius) in detail. To summarize here in short: galaxies can be attributed to\nsystems using the FoF algorithm with a certain linking length.\n\nOur experience and analysis show that the choice of the FoF parameters depends\non goals of the authors. For example Weinmann et al.\\ \\cite{wein06} searched\nfor compact groups in a SDSS DR2 sample. They applied strict criteria in FoF\nmethod and obtained, as one of the results, a lower fraction of galaxies in\n\nBerlind et al.\\ (\\cite{ber06} applied the FoF method to volume-limited \nsamples of the SDSS (see Table~\\ref{Tab2}). Their goal\nwas to measure the group multiplicity function and to constrain dark halos.\nThe applied uniform group selection has reduced the incompleteness of the\nsample, but it led also a lower number density of galaxies and of groups.\n\nIn this paper our goal is to obtain DR5 groups for a further determination of\nluminosity density field and to derive properties of \nthe network of the galaxy distribution. Groups are mostly density\nenhancements within filaments, and rich clusters are high-density peaks of the\ngalaxy distribution in superclusters (Einasto et al. \\cite{einm03c}, \n\\cite{einm03d}, \\cite{ein07a}, \\cite{ein07b}). Hence, our goal is to find out\nas many groups as possible to track all of the supercluster network. We\nrealize that differences in the purposes of the different papers which gives a fairly\nwide range of group properties. \n\nA Virialisation condition, or a certain density contrast as alternative methods\ndo not work universally for all density ranges of galaxy distribution.\nHowever, the similar problem arises in the case of FoF method. As shown by\nEinasto et al. (\\cite{e84}), it is not easy to find a suitable linking length\neven for a volume-limited sample of galaxies. The same conclusion has been\nrecently reached by Berlind et al.\\ (\\cite{ber06}), based on a much more larger\nsample and a more detailed analysis. The problem arises due to the variable\nmean density of galaxies in different regions of space. Additional\ndifficulties arise in case of flux-limited samples of galaxies if the linking\nlength depends also on the distance from the observer. In the original\nanalysis by Huchra \\& Geller the linking length was chosen as $l\\sim f^{-1\/3}$,\nwhere $f$ is the selection function of galaxies. This scaling corresponds to\nthe hypothesis that with increasing distance the galaxy field, and the groups,\nare randomly diluted. A recent summary of various methods to find clusters in\ngalaxy samples is given by Eke et al. (\\cite{eke04}).\n\n\\begin{figure}[ht]\n\\centering\n\\resizebox{0.45\\textwidth}{!}{\\includegraphics*{dist_vs_lumgr.eps}}\n\\caption{The total estimated luminosities for groups \n as a function of distance from the observer. \n}\n\\label{fig:2}\n\\end{figure}\n\nThere exists a close correlation between luminosities of galaxies in groups\nand their positions within groups: bright galaxies are concentrated close to\nthe center, and companions lie in the outskirts (for an early analysis of this\nrelationship see Einasto et al. \\cite{eskc74}, for a recent discussion see\nPaper~1). In Paper~1 we have found that while constructing group catalogues\nin the 2dFGRS a slightly growing linking length with distance has to be used.\n\nA similar problem arises in the SDSS. As selection effects were\nanalyzed in detail in Paper~1, then we shall discuss only shortly the selection\neffects in the SDSS survey. We perform tests to find an optimal set of\nparameters for the FoF method in this study.\n\n\n\\section{Selection effects}\n\n\\subsection{Selection effects in group catalogues}\n\nMain selection effects in group catalogues are caused by the fixed interval of\napparent magnitudes in galaxy surveys (see for details in Paper~1). This\neffect is shown for SDSS DR5 groups in Fig.~\\ref{fig:2}.\n\n\n\\begin{figure}[ht]\n\\centering\n\\resizebox{0.45\\textwidth}{!}{\\includegraphics*{spadr5b.eps}}\n\\caption{The number density of the SDSS DR5 MAIN E and N samples of \ngroups in log scale as a function of distance from the observer . \n}\n\\label{fig:3}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\n\\resizebox{0.45\\textwidth}{!}{\\includegraphics*{dr562emult.eps}}\n\\caption{The multiplicity of groups of the sample E \n as a function of distance from the observer. \n}\n\\label{fig:Nrich}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\n\\resizebox{0.45\\textwidth}{!}{\\includegraphics*{dr563nmult.eps}}\\\\\n\\caption{The multiplicity of groups of the sample N\n as a function of distance from the observer. \n}\n\\label{fig:NrichN}\n\\end{figure}\n\nThe main consequence of this selection effect is the inhomogeneous spatial\ndistribution of groups: the decrease of the volume density of groups with\nincreasing distance. The mean volume density of groups as a function of\ndistance is plotted in Fig.~\\ref{fig:3}, separately for the northern and the\nequatorial area.\n\nA consequence of this effect is richness (multiplicity) of groups as a\nfunction of redshift. In Figs.~\\ref{fig:Nrich} and \\ref{fig:NrichN} we show the\nmultiplicity of groups (the number of member galaxies) as a function of\ndistance from the observer for the E and N samples, respectively. We see that\nrich groups are seen only up to a distance of about 300~$h^{-1}$\\thinspace Mpc, thereafter the\nmean multiplicity decreases considerably with distance. This selection effect\nmust be accounted for in the multiplicity analysis. \n\n\\subsection{Selection effects in group sizes}\n\nSizes of groups depend directly on the choice of the linking length, or more\ngenerally on its scaling law. Strong selection effects can be observed\nhere, also. As an example, the median sizes of the distant 2PIGG groups (Eke\net al.\\ \\cite{eke04}) are 7 times larger than those for the nearby groups.\n\nUsually the ratio of radial and transversal linking lengths $ \\Delta V_0 \/\n\\Delta R_0$ is a constant in the FoF process of search of groups. As noted by\nEinasto et al.\\ (\\cite{e84}), and Berlind et al.\\ (\\cite{ber06}) it is impossible\nto fulfill all requirements with any combination of these linking lengths. We\ntry to find the ratio $ \\Delta V_0 \/ \\Delta R_0$ which is the best to fulfill\nthe size ratio of observed groups which was determined by other studies. \nFigure~\\ref{fig:VRmeanratio} demonstrates how the mean group size ratio\ndepends on initial linking length (LL) for three different $\\Delta V \/ \\Delta R $ ratio: 6, 10, \nand 12. If we accept from other considerations the initial $\\Delta R_0 =\n0.25$ $h^{-1}$\\thinspace Mpc, then we could find the best ratio $ \\Delta V_0 \/ \\Delta R_0$ to be 10\n( at $\\Delta R_0=0.25$ the curve 10 is the closest to the same value of\nmean size ratio). \n\nOn the other side, if we accept size ratio 10 (for example from detailedd study\nof cluster shape in redshift space) we could conclude the best $\\Delta R_0$ to\nbe 0.25~$h^{-1}$\\thinspace Mpc\\ where the curve $(\\Delta R_0)$ reach the size ratio $\\Delta V\n\/ \\Delta R= 10$ in Figure~\\ref{fig:VRmeanratio}. \n\nIt is difficult to reliably model the galaxy populations in DM-haloes. Here\nwe summarize in short a solution of the problem.\n\nAt large distances from the observer, only the brightest cluster members are\nvisible, and\nthese brightest members form compact cores of clusters, with sizes much less\nthan the true size of the clusters.\nThis effect work in the opposite direction to the increase of the linking\nlength, and it might cancel it out.\nNext we describe the empirical scaling of\nthe linking length by shifting of the observed groups to growing distances.\n\n\\section{Scaling of linking length}\n\nIn the majority of papers dedicated to group search authors, the group finders\nare tuned using mock $N$-body catalogues (e.g. Eke et al.\\ \\cite{eke04}; Yang\net al.\\ \\cite{yang05}). The mock group catalogues are homogeneous and all\nparameters of the mock groups can be easily found and applied for search of\nreal groups. Still mock groups are only an approximation to the real groups\nusing model galaxies in dark matter haloes. As we have noted, it is difficult\nto properly model the luminosity-density correlation found in real groups.\n\nStarting from these considerations we have used observed groups to study the\nscaling of group properties with distance. The group shifting procedure is\ndescribed in detail in Paper~1. As this is an important part of our search\nmethod, then we present here the method i short and present the results for the SDSS\nDR5 groups.\n \nWe created test group catalogues for the sample SDSS DR5 E with constant and\nvariable linking lengths, selected in the nearby volume $d < 100$~$h^{-1}$\\thinspace Mpc all\nrich groups (with multiplicity $N_{gal}\\ge 20$, in total 222 groups).\nAssuming that the group members are all at the mean distance of the group we\ndetermined their absolute magnitudes and peculiar radial velocities. Then we shifted\nthe groups step by step to larger distances (using a $z=0.001$ step in\nredshift), and calculated new $k$-corrections and apparent magnitudes for the\ngroup members. As with increasing distance more and more fainter members of\ngroups fall outside the observational window of apparent magnitudes, the group\nmembership changes. \nWe found new properties of the groups -- their\nmultiplicities, characteristic sizes, velocity dispersions and densities. \nWe also calculated the minimum FoF linking length, necessary to keep the group\ntogether at this distance. \n\n\\begin{figure}[ht]\n\\centering\n\\resizebox{0.45\\textwidth}{!}{\\includegraphics*{VRmeanratio.eps}}\n\\caption{Mean ratio of radial and perpendicular sizes of groups\n in the sample E as a function of starting value of linking length\nfor three values of linking lengths ratios. \n}\n\\label{fig:VRmeanratio}\n\\end{figure}\n\nTo determine that, we built the minimal spanning\ntree for the group (see, e.g., Martinez and Saar \\cite{mar03}), and found the\nmaximum length of the MST links.\n\nAs the original groups had different sizes and initial redshifts we found the\nrelative changes of their properties, with respect to the redshift change.\nThe individual linking length scaling paths have large scatter. Therefore we\nfound the average scaling path from the individual paths. In\nFigure~\\ref{fig:LLLawdr5} we present the main result of group shifting for our\nlinking length scaling law determination.\n\n\\begin{figure*}[ht]\n\\centering\n\\resizebox{0.48\\textwidth}{!}{\\includegraphics*{LLLawdr5e.eps}}\n\\hspace*{2mm}\n\\resizebox{0.48\\textwidth}{!}{\\includegraphics*{LLLawdr5n.eps}}\n\\caption{The scaling of the group FoF linking length with redshift\n for the samples DR5 E (left panel) and DR5 N (right panel). The ordinate is\n the ratio of the minimal linking length $LL$ at a redshift $z$, necessary to\n keep the group together, to the original linking length $LL_0$ that defined\n the group at its initial redshift $z_0$; the abscissa is the redshift\n difference $\\Delta z=z-z_0$. \n}\n\\label{fig:LLLawdr5}\n\\end{figure*}\n\n\nWe fit the mean values of the linking lengths in $\\Delta z=0.001$ redshift\nbins (the step we used for shifting the groups). We find our scaling law for\nthe case $n\\ge 20$. The fitting law is not sensitive to the richness of\ngroups involved in the LL scaling law determination. The scaling law is\nmoderately different from the scaling law found for the 2dFGRS groups in\nPaper~1 but still can be approximated by a slowly increasing arctan law. \nDue to narrow\nmagnitude window in SDSS, at higher values of $z$ only compact cores of\ngroups or binary galaxies have been found by FoF\nmethod. The deviation from the scaling\nlaw corresponds to the redshift limit above which most groups discovered\ncorrespond only to the compact cores of nearby groups. Therefore, the\ndetermination of the scaling law is a test for redshift limit of homogeneity\nof the group catalogue. A good parametrization of the scaling low is\n\\begin{equation}\n\\label{lz}\nLL\/LL_0=1+a\\, \\mbox{arctan}(z\/z_{\\star}),\n\\end{equation}\nwhere $a=0.83$ and $z_{\\star}=0.055$. \n\nThe main difference between the scaling laws of DR5 and 2dF groups is in the \nvalidity range. This is due to different magnitude limits in these \nflux limited samples. We consider this difference in more details below. \nThe selection of initial groups should not influence much the scaling\nof their properties with distance. \nWe tested group search with three different initial scaling laws for \ngroup selection\n: two lengths constant and one varied with distance. \nThe final scaling relation practically does not depend on\nthe initial group selection (i.e. on initial scaling law). \n \n\n\\section{Group catalogue}\n\n\n\\subsection{The group finder}\n\nWe adopt the scaling of the linking length found above, but we have to select\nyet the initial values for the linking length. In practice, only groups with\nthe observed membership $N_{gal} \\geq 2$ are included in group catalogues.\n\n\\begin{figure*}[ht]\n\\centering\n\\resizebox{0.45\\textwidth}{!}{\\includegraphics*{Size_maxe.eps}}\n\\hspace*{2mm}\n\\resizebox{0.45\\textwidth}{!}{\\includegraphics*{sigVe.eps}}\\\\\n\\caption{ Left panel : the (maximum projected) sizes of our SDSS DR5 groups\n in E sample as a function of distance. \n Right panel shows the velocity dispersions in groups as a function of\n distance in the sample E. The FoF parameters are given in Table~\\ref{Tab1}. \n}\n\\label{fig:10}\n\\end{figure*}\n\n\\begin{figure*}[ht]\n\\centering\n\\resizebox{0.45\\textwidth}{!}{\\includegraphics*{Size_maxn.eps}}\n\\hspace*{2mm}\n\\resizebox{0.45\\textwidth}{!}{\\includegraphics*{sigVn.eps}}\\\\\n\\caption{ Left panel : the (maximum projected) sizes of our SDSS DR5 groups\n in N sample as a function of distance. \n Right panel shows the velocity dispersions of groups as a function of\n distance in the sample N. \n The FoF parameters are given in Table~\\ref{Tab1}. \n}\n\\label{fig:sizesig}\n\\end{figure*}\n\n\nIn order to find the best initial linking lengths in the radial direction, we\ntried a number of different parameter values, $\\Delta V = 100-700$ km\/s and\n$\\Delta R = 0.16 - 0.70$ $h^{-1}$\\thinspace Mpc, and we chose finally the values which were\ndiscussed above, and presented in Table~\\ref{Tab1}. Higher values for $\\Delta\nR$ leads to inclusion of galaxies from neighbouring groups and filaments.\nLower values for $\\Delta V$ exclude the fastest members in intermediate\nrichness groups.\n\nHowever, closer inspection show that one rich group has a richness much larger\n($N=569$) than the rest of them. This is the well-known nearby ($d=27$ $h^{-1}$\\thinspace Mpc)\nbinary Abell cluster A2197\/2199. We consider this cluster as an exception,\nand do not use lower LLs. At slightly lower value of LL this cluster\nfall apart and become the cluster with usual properties.\n\nIn Fig.~\\ref{fig:10} we show the sizes of our groups of the final catalogue.\nWe define the size of the group as its maximum projected diameter, the largest\nprojected galaxy pair distance within the group. We see that the sizes of\nlargest groups slightly increase with distance up to $d = 250$~$h^{-1}$\\thinspace Mpc, and\nthereafter slowly decrease. This decrease is expected since in more distant\ngroups only bright galaxies are seen, and they form the compact cores of\ngroups.\nThe numbers of the groups and the FoF parameters (separately\nfor both SDSS DR5 regions) are given in Table~\\ref{Tab1}. \n\n\n\\subsection{The final catalogue}\n\nOur final catalogue (Table~\\ref{Tab1}) includes 17143 groups in equatorial\narea and 33219 groups in high declination area with richness $\\geq 2$. As an\nexample we present here the first lines of our group table (Table~\\ref{Tab3}),\nwhich include the following columns for each group:\n\n\\begin{itemize}\n \\item[1)] group identification number;\n \\item[2)] group richness (number of member galaxies);\n \\item[3)] RA (J2000.0) in degrees (mean of member galaxies);\n \\item[4)] DEC (J2000.0) in degrees (mean of member galaxies);\n \\item[5)] group distance in $h^{-1}$\\thinspace Mpc\\ (mean comoving distance for member galaxies corrected\nfor CMB);\n \\item[6)] the maximum projected size (in $h^{-1}$\\thinspace Mpc);\n \\item[7)] the rms radial velocity ($\\sigma_V$, in km\/s);\n \\item[8)] the virial radius in $h^{-1}$\\thinspace Mpc\\ (the projected harmonic mean);\n \\item[9)] the luminosity of the cluster main galaxy (in units of $10^{10} h^{-2}\n L_{\\sun}$);\n \\item[10)] the total observed luminosity of visible galaxies ($10^{10} h^{-2}\n L_{\\sun}$);\n \\item[11)] the estimated total luminosity of the group ($10^{10} h^{-2} L_{\\sun}$). \n\\end{itemize}\n\n\\begin{table*}\n \\caption[]{First rows as an example of groups in the SDSS DR5 main\n galaxy catalogue \n described in the present paper}\n \\label{Tab3}\n \\begin{tabular}{ccccccccccc}\n \\hline\\hline\n \\noalign{\\smallskip}\n $ ID_{gr}$ & $N_{g}$ & $RA$ & $DEC$ & Dist &\n $Size_{sky}$&$\\sigma_{V}$&$R_{vir} $ & $ L_{main}$ &\n $ L_{obs} $ & $L_{est}$ \\\\\n & & [deg] & [deg] & [Mpc\/h] & [Mpc\/h] & [km\/s] & [Mpc\/h] &\n [$10^{10} h^{-2} \nL_{\\sun}]$& [$10^{10} h^{-2} L_{\\sun} $] & [$ 10^{10} h^{-2} L_{\\sun} $ ] \\\\\n \\noalign{\\smallskip}\n \\hline\n 1 & 2 & 3 & 4 & 5 &\n 6 & 7 & 8 & 9 & 10 & 11 \\\\\n \\hline\n \\noalign{\\smallskip}\n\n 1 & 4 & 146.57633972 & -0.83209175 & 195.056 & 0.6823 & 53.7783 &\n 0.33341 & 0.17353E+01 & 0.40818E+01 & 0.52815E+01 \\\\\n 2 & 2 & 146.91120911 & -0.31007549 & 385.390 & 0.1291 & 25.2219 &\n 0.12908 & 0.21835E+01& 0.41985E+01 & 0.10160E+02 \\\\\n 3 & 3 & 146.88099670 & -0.49802899 & 249.334 & 0.1522 & 101.6915 &\n 0.09505 & 0.27161E+01& 0.36896E+01 & 0.53377E+01 \\\\\n 4 & 2 & 146.78494263 & 0.02115750 & 368.779 & 0.3185 & 173.4426 & \n 0.31840 & 0.37278E+01& 0.56619E+01 & 0.13310E+02 \\\\\n 5 & 4 & 146.74797058 & -0.25555125 & 383.818 & 0.3404 & 191.9961 &\n 0.15149 & 0.37084E+01& 0.99677E+01 & 0.24499E+02 \\\\\n\n \\noalign{\\smallskip}\n \\hline\n \\end{tabular}\\\\\n \\end{table*}\n\n\nThe identification number is attached to groups by the group finder in\nthe order the groups are found. The calculation of luminosities is\ndescribed in the next section. \n\nWe also give (in an electronic form) a catalogue of all individual\ngalaxies along with their group identification number and the group richness, \nordered by the group identification number, to facilitate search. The\ntables of galaxies end with a list of isolated galaxies (small\ngroups with only one bright galaxy within the observational window of\nmagnitudes); their group identification number is 0 and group richness\nis 1. All tables can be found at\n\\texttt{http:\/\/www.obs.ee\/$\\sim$erik\/index.html}. \n\n\\section{Luminosities of groups}\n\nThe limiting apparent magnitude of the complete sample of the SDSS catalog in\n${\\rm r}$ band is 17.77. The faint limit actually fluctuates from field to\nfield, but in the present context we shall ignore that; we shall take these\nfluctuations into account in our paper on the group luminosity function, based\non our 2dFGRS group catalogue (Einasto et al. \\cite{ets07}).\n\nWe regard every galaxy as a visible member of a group or cluster within the\nvisible range of absolute magnitudes, $M_1$ and $M_2$, corresponding to the\nobservational window of apparent magnitudes at the distance of the galaxy. To\ncalculate total luminosities of groups we have to find for all galaxies of the\nsample the estimated total luminosity per one visible galaxy, taking into\naccount galaxies outside of the visibility window. This estimated total\nluminosity was calculated as follows (Einasto et al.\\ \\cite{e03b})\n\\begin{equation}\nL_{tot} = L_{obs} W_L, \n\\label{eq:ldens}\n\\end{equation}\nwhere $L_{obs}=L_{\\odot }10^{0.4\\times (M_{\\odot }-M)}$ is the\nluminosity of a visible galaxy of an absolute magnitude $M$, and\n\\begin{equation}\nW_L = {\\frac{\\int_0^\\infty L \\phi\n(L)dL}{\\int_{L_1}^{L_2} L \\phi (L)dL}}\n\\label{eq:weight2}\n\\end{equation}\nis the luminous-density weight (the ratio of the expected total luminosity to\nthe expected luminosity in the visibility window). In the last equation\n$L_i=L_{\\odot} 10^{0.4\\times (M_{\\odot }-M_i)}$ are the luminosity limits of\nthe observational window, corresponding to the absolute magnitude limits of\nthe window $M_i$, and $M_{\\odot }$ is the absolute magnitude of the Sun. In\ncalculation of weights we assumed that galaxy luminosities are distributed\naccording to a two power-law function used by Christensen\n(\\cite{chr75}), Kiang (\\cite{kiang76}), Abell (\\cite{abell77}) and Mottmann \\&\nAbell (\\cite{ma77})\n\n\n\\begin{equation}\n \\phi(L)dL \\propto (L\/L^*)^\\alpha(1+(L\/L^*)^\\gamma)^{(\\delta \/ \\gamma)}d(L\/L^*) ,\n\\label{eq:twoplaw}\n\\end{equation}\nwhere $\\alpha $, $\\gamma$, $\\delta$ and $L^{*}$ are parameters. We use two\npower-law rather than Schechter function, because it has\nmore freedom and it gives a better fit for the galaxy luminosity function. \n\nWe used two power-law function with parameters: $\\alpha = -1.123$, $\\gamma=\n1.062$, $\\delta = -17.37$, $L^{*} = 19.61$. We have used all galaxies\n(galaxies in groups and isolated galaxies) for finding the luminosity\nfunction. More detailed explanation about two power-law function and how we\nderive the parameters are given in our paper on the 2dFGRS luminosity function\n(Einasto et al. \\cite{ets07}). \n\nWe derived $k$-correction for SDSS galaxies using the KCORRECT algorithm\n(Blanton \\& Roweis \\cite{bla06}). We also accepted $M_{\\odot} = 4.52$ in the\n${\\rm r}$ photometric system. \n\n We calculated for each group the total observed and corrected luminosities, \nand the mean weight\n\\begin{equation}\nW_m = {\\frac{\\sum L_{tot, i}} {{\\sum L_{obs, i}}}}, \n\\label{eq:sum}\n\\end{equation}\nwhere the subscript $i$ denotes values for individual observed galaxies in\nthe group, and the sum includes all member galaxies of the system. \n\n\\begin{figure}[ht]\n\\centering\n\\resizebox{0.45\\textwidth}{!}\n{\\includegraphics*{dist_vs_weight.eps}}\n\\caption{The mean weights of groups of the SDSS DR5 \n versus the distance from the observer. \n}\n\\label{fig:11}\n\\end{figure}\n\nThe mean weights for the groups of the SDSS DR5 are plotted as a function of\nthe distance $d$ from the observer in Fig.~\\ref{fig:11}. We see that the mean\nweight is slightly higher than unity at a distance $d\\sim 175$~$h^{-1}$\\thinspace Mpc, and\nincreases both toward smaller and larger distances. The increase at small\ndistances is due to the absence of very bright members of groups, which lie\noutside the observational window, and at large distances the increase is\ncaused by the absence of faint galaxies. The weights grow fast for very close\ngroups and for groups farther away than about 400~$h^{-1}$\\thinspace Mpc. At these distances\nthe correction factors start to dominate and the luminosities of groups become\nuncertain.\n\nIn Fig.~\\ref{fig:2} we show the estimated total luminosities of groups as a\nfunction of distance. We produced also colour figures that visualise the\nluminosities of groups. These are too detailed to be presented here, and can be\nfound in our web pages. These figures show that the brightest groups have\ncorrected total luminosities, which are, in the mean, independent of distance.\nThis shows that our calculation of total luminosities is correct.\n\n\n{\\scriptsize\n\\begin{table*}\n \\caption[]{Data for group catalogues based on the SDSS}\n \\label{Tab2}\n\\begin{center}\n \\[\n \\begin{tabular}{llcccccccc}\n \\hline\\hline\n \\noalign{\\smallskip}\n Authors & Release, \n Sample & $N_{gal} $ &\n $N_{gr}(n \\geq 2)$ & $N_{gr}(n \\geq 4)$&$z_{lim}$ & \n $\\Delta V_0 $ & $\\Delta R_0$ & \\% ($\\geq$ 2) & \\% ($\\geq 4$)\\\\\n & & & & & & km\/s & Mpc\/h & & \\\\\n \\noalign{\\smallskip}\n \\hline\n 1 & 2 & 3 & 4 & 5 &\n 6 & 7 & 8 & 9 & 10 \\\\\n \\noalign{\\smallskip}\n \\hline\n\n\nMerchan 2005 & DR3 Main & 300000 & & 10864 & 0 - 0.3 & 200 & & & 22 \\\\\n\nGoto 2005 & DR2 SQL & 259497 & 335 & & 0.03- & 1000 & 1.5 & & 6 ($n\n\\geq$ 20) \\\\ \n\nWeinmann 2006 & DR2 Main VAGC & 184425 & 16012 & 3720 & 0.01 - 0.2 &\n0.3$^1$ & 0.05$^1$ & 30 & 15 \\\\\n\nBerlind 2006 & DR3 sam14 VAGC & 298729 & & & & & & & \\\\\n & vol.lim. Mr20 & 57332 & & 4119$^3$ & 0.015-0.1 & 0.75 &\n 0.14 & 56.3 & 37.2$^3$ \\\\\n & vol.lim. Mr19 & 37938 & & 2696$^3$ & 0.015-0.068 & 0.75 &\n 0.14 & 58.9 & 40.7$^3$ \\\\\n & vol.lim. Mr18 & 18959 & & 1362$^3$ & 0.015-0.045 & 0.75 &\n 0.14 & 60.0 & 42.2$^3$ \\\\\n\n\nTago 2007 & DR5 Main DAS & 387063 & 50362 & 9454 & 0.009 - 0.2 & 250 & 0.25\n& 41.1 & 23.4 \\\\\n \\noalign{\\smallskip}\n \\hline\n \\end{tabular}\n \\]\n\\end{center}\n\n\\small\\rm\\noindent Columns:\n\n\\begin{itemize}\n\\item[1:] authors of group catalog,\n\\item[2:] sample and release number, \n\\item[3:] number of galaxies, \n\\item[4:] number of groups ($n \\geq$ 2), \n\\item[5:] number of groups ($n \\geq$ 4), \n\\item[6:] redshift limits for sample galaxies, \n\\item[7:] the FoF linking length in radial velocity, for $z=0$, \n\\item[8:] the FoF linking length in projected distance in the sky\n , for $z=0$, \n\\item[9:] fraction of galaxies in groups ($n \\geq$ 2), \n\\item[10:] fraction of galaxies in groups ($n \\geq$ 4). \n\\end{itemize}\n\n\\small\\rm\\noindent Notes:\n\n$^1$ for Weinmann et al.\\ groups linking lengths are in the units of mean\ngalaxy separation; \n \n$^3$ for Berlind et al.\\ groups richness $n \\geq 3$\n \n* for Berlind et al.\\ apparent magnitude limit was $r \\leq 17.5$ , for the\n rest $r \\leq 17.77$ \n\n* group-finders : \n \n Merchan: FoF + mock catalog + iterative group re-centering + Schechter LF\n for LL scaling \n\n Goto: FoF + group re-centering \n\n Weinmann: FoF + DM halo mock catalog + group re-centering \n\n Berlind: FoF + DM halo mock catalog\n\n Tago: FoF + DM halo mock + Dens\/Lum relation in groups for LL scaling\n\n \\end{table*}\n}\n\n\n\n\\section{Discussion and conclusions}\n\n\\begin{figure}[ht]\n\\centering\n\\resizebox{0.45\\textwidth}{!}{\\includegraphics*{NSgal.eps}}\n\\caption{The number density of galaxies in the 2dF N and S samples, and SDSS\n DR5 E and N samples as a function of distance from the observer. Histograms\n for 2dF are arbitrary shifted along ordinate axis for clarity. }\n\\label{fig:galden}\n\\end{figure}\n\n\\begin{figure}[ht]\n \\centering \\resizebox{0.5\\textwidth}{!} {\\includegraphics*{dr-comp2.eps}}\n\\caption{The number of sample galaxies, groups and isolated galaxies\n involved in FoF procedure versus total number of galaxies in releases of\n SDSS and 2dF surveys. Note well defined proportional grows with \n releases of SDSS and a higher \"yield\" for 2dF. These relations suggest that\n the FoF method has applied homogeneously to the different releases. \n}\n\\label{fig:dr-comp}\n\\end{figure}\n\n\\subsection{Some issues related to the poor de-blending}\n\nVarious potential caveats related to the automatic pipeline data reduction in\nthe SDSS have been discussed and flagged in the NYU-VAGC, which is based on\nthe SDSS DR2 (Blanton et al.\\ \\cite{bla05}). Most of these issues are related to\npoor de-blending of large and\/or of LSB galaxies with complicated morphology\n(e.g. star-forming regions, dust features etc.). At low redshifts a number\nof SDSS galaxies have been found shredded, i.e. a nearby large galaxy image\nis split by target selection algorithm into several sub-images (e.g. Panter\net al.\\ \\cite{panter07}). Therefore, the treatment of nearby galaxies requires special\ncare. This potential bias is largely reduced in our new catalogue by means of\nsetting reasonably high magnitude ($r > 14.5$) and redshift ($z > 0.009$)\nlimits, which exclude most of luminous and\/or nearby galaxies of the Local\nSupercluster. \n\n\nWe have performed eyeball quality checks of a number of groups in the new\ncatalogue using the SDSS Sky Server Visual Tools. We have inspected a) the\nmembers of the 139 nearest ($z < 0.012$) groups -- 42 groups in the equatorial\n(E) sample and 97 groups in the northern (N) sample; b) conspicuously dense\ngroups as evident on the bottom sections of the Figure~\\ref{fig:rvirduplicate}, \nand of the Figures~\\ref{fig:10} and \\ref{fig:sizesig}.\n The results of these checks can be summarized as follows:\n \n1) {\\it De-blending errors.} In the nearest 139 groups with initially 525\nmember galaxies poor de-blending has been noted for 21 (4\\%) galaxies\ndistributed in 9 (6.5\\%) groups. Poor de-blending means either that the bright\ngalaxy is represented in the DR5 spectroscopic sample with a single off-center\nsource of typically reduced brightness, or that the primary galaxy is shredded\ninto multiple (faint) \\ion{H}{ii} regions. \n\nAs an example of poor de-blending we refer to the group number 30644.\nIts luminous member NGC 3995 ($B_T = 12.7$) with\nknotty morphology is represented in the DR5 with 3 entries, i.e. with 3\ndistinct spectra of its \\ion{H}{ii} knots of magnitudes $r$ = 12.6, 15.13, \nand 17.64, respectively. Other three luminous group members NGC 3966\n($m_B$ = 13.60), NGC 3994 ($B_T$ = 13.30), and NGC 3991 ($m_B$ = 13.50) \nare each represented in the DR5 by two knots with magnitudes\n$r$ = 12.49, 16.88, and $r$ = 12.63, 16.60, and $r$ = 14.81, 17.89, respectively.\nAfter excluding the knots with $r < 14.5$ those intrinsically luminous\ngalaxies will be represented in our catalogue by their faint(er) knots and\ntheir true total magnitudes are underestimated by 1.5 - 3.5 magnitudes.\nIt appears to be one of the most severely biased nearby groups.\n\n2) All the 25 very dense E groups with $R_{vir} < 1~ h^{-1}$ kpc, distributed\nin the bottom section of the Figure~\\ref{fig:rvirduplicate},\n are results from duplicates. Among them there\nare 14 ''pairs'' ( i.e. actually a single galaxy with two records in the DR5\nspectroscopic sample), 7 \"triplets\" and 4 \"quartets\". Among the N groups there\nare only two duplicates in the given $R_{vir}$ range. \n \n3) Considering the Figures~\\ref{fig:10} and \n\\ref{fig:sizesig} (left panels) \\\\\n-- all 13 groups with $Size < 1 h^{-1}$ kpc are among\nthose with $R_{vir} < 1~ h^{-1}$ kpc in the Figure~\\ref{fig:rvirduplicate},\n i.e. they are duplicates. \\\\\n-- The conspicuous lower boundary of the tightly populated region (which varies\nnearly proportional to distance) is probably determined by the fiber collision\ndistance $\\sim 55''$ of the survey. The groups distributed in the range\nbetween this lower boundary and that of $Size = 10~ h^{-1}$ kpc are in the majority\nreal pairs, i.e. no duplicates. \nPairs with $Size < 10 ~h^{-1}$ kpc \nare likely mergers, or advanced mergers\n(with $1 < Size < 5 h^{-1}$ kpc).\\\\\n - The upper boundary of the tightly populated region likely results \nfrom the linking-length\nscaling relation (1), since there is no\nsingle pair above this boundary. That means, our sample could be biased\nagainst the wide (i.e. in the majority optical) pairs. \n \n{\\it To summarize:} As a result of our cursory checks we have found relatively\nfew bad de-blends, either in form of mismatches between spectral targets and\noptical centers, or more severe shreddings of large and\/or LSB galaxies. \nAlthough the redshifts are fine, photometric and structural measurements are\noften erroneous in such cases. The fraction of groups checked so far is small, \nhowever it comprises the nearest, i.e. potentially most affected part of the\nfull sample. We estimate that the net effect of de-blending errors will have\nminor effect, when working with large (sub)samples of groups. \n \n\n\\begin{figure}[ht]\n\\centering\n\\resizebox{0.5\\textwidth}{!}\n{\\includegraphics*{fig15.eps}}\n\\caption{The eight nearby ($z < 0.04$) groups ($n \\geq 2$) as identified \n in this work in a relatively sparce filament. The group members are shown\n with circles and four individual groups are encompassed with large circles. \n The field galaxies in the same redshift range are marked with small circles. \n For comparison, the members of the corresponding Merchan et al. (\\cite{mer05})\n groups ($n\n \\geq 4$) are marked with tilted crosses ($\\times$), and those of the Berlind\n et al. (\\cite{ber06}) groups (Mr18 sample, $n \\geq 3$) are shown with crosses. Note\n that in Merchan et al. (\\cite{mer05}) the rich, elongated group is divided into two (NE\n and SW) subgroups, which are nearly projecting to each other along the\n line-of-sight. }\n\\label{fig:gr8344a}\n\\end{figure}\n\nIn Fig.~\\ref{fig:gr8344a} we give an example of how the group-finder\nalgorithm works. The comparison with groups Merchan et al. (\\cite{mer05})\n and Berlind et al. (\\cite{ber06})\nshows that all three slightly different FoF algorithms identify quite similar\ngroups. The criteria used in Merchan et al. (\\cite{mer05}) tend to split the groups along the\nline-of-sight and\/or exclude the galaxies in outskirts of groups more easily.\n\n\n\\begin{figure}[ht]\n\\centering\n\\resizebox{0.50\\textwidth}{!}{\\includegraphics*{compgr.eps}}\n\\caption{Groups by Berlind et al. (\\cite{ber06}) Mr18 sample (crosses)\ncompared to our groups in the same redshift ($0.015 < z < 0.045$) and \nrichness ($N_{gal} \\geq 3$) range (large circles). The pairs\nof galaxies ($N_{gal} = 2$) in our catalogue are shown with small circles. \n}\n\\label{fig:Berlind_dr5} \n\\end{figure}\n\nIn Fig.~\\ref{fig:Berlind_dr5} we compare the groups in the volume limited Mr18 sample \nof Berlind et al. (\\cite{ber06}) to our groups in a similar redshift range. We conclude that we\ncan detect more groups (121 our groups versus 88 groups in Mr18) and slightly richer groups\n(6.1 galaxies per one our group versus 5.5 galaxies in one Mr18 group), mainly due to \ninclusion of fainter ($Mr > -18$) galaxies.\n\n\n\\subsection{Comparison to other studies}\n\nEarlier catalogues of the SDSS groups of galaxies, based on the first SDSS\nreleases, were obtained by Lee et al. (\\cite{lee04}), Einasto et al.\\ \n(\\cite{e03b}).\n\nAt present there are five extensive catalogues of groups of galaxies available\nto us which are obtained on the basis of the SDSS. Although they are based on\ndifferent SDSS releases they have obtained by incremental addition of\nnew data to previous releases and observational method and parameters are the\nsame. We can reasonably compare these group catalogues. Group catalogues\nare different due to different group search parameters and not under-laying\nsamples of galaxies. An important exception are 3 volume limited samples by\nBerlind et al. At the price of smaller galaxy sample they have the advantage that\nthe most serious incompleteness effect of magnitude limited samples is\nabsent, the missing of faint galaxies in distant parts of the survey. Some characteristics of the\ncatalogues are presented in Table~\\ref{Tab2}. An important characteristic to\ncompare the catalogues is the fraction of single (isolated) galaxies or\nequivalently, the fraction of galaxies in groups. Single galaxies can be\nconsidered as belonging to small groups or to haloes represented only by one\nobserved galaxy in the visibility window. \n\nTherefore, we face the problem how to compare catalogues because different\ngroup-finder criteria have been applied: richness and size of groups, linking\nlengths, the ratio of los\/perpendicular linking lengths, etc. These criteria\ndepend on the goals of a particular study. The last two columns in the table\ngive the fraction of galaxies in groups of richness $n \\geq 2$ and $n \\geq 4$. \nThese are 30 and 42 \\% for the groups by Weinmann et al.\\ and for our groups of\nrichness $\\geq 2$, and 22 and 18.3 \\% for the groups by Merchan et al., and for\nour groups of richness $\\geq 4$, respectively. In fact, these values represent\nthe low richness end of the multiplicity function. \n\nWe note that the fraction of galaxies in our 2dF GRS groups is very similar --\n43 \\% (Paper~1). This suggest that the multiplicity distribution is a robust\ncharacteristic being independent of these two surveys and small\ndifferences in initial parameters of FoF chosen. We see that Weinmann's\ngroups which are intended to determine only compact groups, have remarkably\nlower fraction of galaxies in groups (30 \\%) than ours. Comparing\nthese fractions for Merchan's and our groups the results are much closer (for\nrichness $n \\geq$ 4).\n\nSeveral studies have shown (see, e.g., Kim et al.\\ \\cite{kim02}) that different\nmethods give rather different groups for the SDSS sample. The same is true\nfor the 2dFGRS groups (Paper~1). Although catalogues cited in\nTable~\\ref{Tab2} are FoF-based, the results of Goto et al. \n(\\cite{goto05}) have created a\ncluster catalogue applying a very strong criteria for system search with a\npurpose to study cluster galaxy evolution. It is not much useful to compare\ntheir catalogue with ours due to different purposes and the number of clusters.\nHowever, we present for completeness also properties in\nTable~\\ref{Tab2}. Weinmann et al.\\ (\\cite{wein06}) applied a more strict\ncriteria in group selection based on the idea that galaxies in a common dark\nmatter halo belong to one group. As a result, they obtained a group catalogue\nthat contains mainly compact groups and a large fraction of single galaxies.\n\nThe most detailed search method and reliable group catalogue(s) have been\nobtained by Berlind et al.\\ (\\cite{ber06}; SDSS collaboration). Their purpose\nwas to construct groups of galaxies to test the dark matter halo occupation\ndistribution. For this requirement to get highly reliable groups they choosed\na different way --- volume-limited samples of the SDSS. This way has unwanted\nresult --- much smaller sample, but we see also (Table 2) the advantage ---\nless incompleteness problems and a higher fraction of galaxies in groups than\nin the other catalogues. Berlind et al.\\ (\\cite{ber06}) demonstrated that\nthere exists no combination of radial and perpendicular linking lengths\nsatisfying all three important properties of groups (in mock catalogue):\nthe multiplicity function, the projected size and the velocity dispersion.\n\nThis could explain why the properties of group catalogues, presented in\nTable~\\ref{Tab2}, are so different. We consider this fault as one of\njustifications to use observed groups for determination of linking length\nscaling law.\n \n\n\\subsection{Conclusions}\n\nWe have used the Sloan Digital Sky Survey Data Release 5 to create a new\ncatalogue of groups of galaxies. Our main results are the following:\n\n\\begin{itemize}\n \n\\item[1)] We have taken into account selection effects caused by\n magnitude-limited galaxy samples. Two most important effects are the\n decreasing of group volume density and the decreasing of the group richness\n with increasing distance from the observer. We show that at large distances\n from the observer the population of more massive, luminous and greater\n groups\/clusters dominates. This increase of the mean size of groups is\n almost compensated by the absence of faint galaxies in the observed groups\n at large distances. The remaining bright galaxies form a compact core of\n the group, this compensates for the increase of group sizes caused by\n domination of the population of more massive groups. This confirms the\n similar luminosity\/density relation found for 2dFGRS groups earlier.\n \n\\item[2)] We find the scaling of the group properties and that of the FoF\n linking length empirically, shifting the observed groups to larger\n redshifts. As the SDSS Main and 2dFGRS galaxies have similar redshift\n distributions and luminosity functions, then we find that the linking length\n scaling laws for these catalogues are very close, growing only slightly by\n arctan law, but only up to the redshift $z=0.12$. Beyond this redshift\n the scaling law decreases sharply. At higher redshift we detect mainly compact\n cores of the groups due to more narrow magnitude range (visibility window)\n of the SDSS. This scaling law method can be considered as a test to which\n redshift limit group-finder could be applied. \n \n\\item[3)] We present a catalogue of groups of galaxies for the SDSS Data Release\n 5. We applied the FoF method with a slightly increasing linking length;\n the catalogue is available at the web page\n (\\texttt{http:\/\/www.obs.ee\/$\\sim$erik\/index.html}).\n \n\\item[4)]A wide variety of properties as a result of different purposes of the\n catalogues which involve different parametres for group search algorithms,\n and different samples. \nOthers tried to establish parameters of the halo model\nof the galaxy distribution. We provide a catalogue that was intented most\ncomplete and representative for the survey volume. Thereby we best measure\n the large scale galaxy network over the survey volume.\n\n \n\n\\end{itemize}\n\n\\begin{acknowledgements}\n \n Funding for the Sloan Digital Sky Survey (SDSS) and SDSS-II has been\n provided by the Alfred P. Sloan Foundation, the Participating Institutions, \n the National Science Foundation, the U.S. Department of Energy, the National\n Aeronautics and Space Administration, the Japanese Monbukagakusho, and the\n Max Planck Society, and the Higher Education Funding Council for England. \n The SDSS Web site is http:\/\/www.sdss.org\/. \n \n The SDSS is managed by the Astrophysical Research Consortium (ARC) for the\n Participating Institutions. The Participating Institutions are the American\n Museum of Natural History, Astrophysical Institute Potsdam, University of\n Basel, University of Cambridge, Case Western Reserve University, The\n University of Chicago, Drexel University, Fermilab, the Institute for\n Advanced Study, the Japan Participation Group, The Johns Hopkins University, \n the Joint Institute for Nuclear Astrophysics, the Kavli Institute for\n Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese\n Academy of Sciences (LAMOST), Los Alamos National Laboratory, the\n Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for\n Astrophysics (MPA), New Mexico State University, Ohio State University, \n University of Pittsburgh, University of Portsmouth, Princeton University, \n the United States Naval Observatory, and the University of Washington. \n \n We are pleased to thank the SDSS collaboration for the DAS version of the\n fifth data release, special thanks to James Annis. We acknowledge the\n Estonian Science Foundation for support under grants No. 6104, 6106 and 7146,\n and the Estonian Ministry for Education and Science support by grant\n SF0062465s03. This work has also been supported by the University of\n Valencia through a visiting professorship for Enn Saar and by the Spanish\n MCyT project AYA2003-08739-C02-01 (including FEDER). J.E. thanks\n Astrophysikalisches Institut Potsdam (using DFG-grant 436 EST 17\/2\/06), and\n the Aspen Center for Physics for hospitality, where part of this study was\n performed. \n\n\\end{acknowledgements}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzfiwe b/data_all_eng_slimpj/shuffled/split2/finalzzfiwe new file mode 100644 index 0000000000000000000000000000000000000000..d3b62efc521ed61d42bd1124c316dd56dfa91834 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzfiwe @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nIn 2015, Laser Interferometer Gravitational-Wave Observatory (LIGO)\ndetected gravitational waves from a binary black hole (BBH) merger \\cite{GW150914}.\nIn Observation run 1 and 2, ten BBH merger events were confirmed \\cite{GWTC}.\nCurrently, advanced LIGO and advanced Virgo are operating and KAGRA will join this detector network in 2020 \\cite{KAGRA}.\nBesides the improvement of detectors,\nthe improvement of data analysis methods can contribute to accelerate the gravitational wave physics and astronomy.\n\nRecently, the use of deep learning methods is proposed for various purposes,\ne.g., the detection of gravitational waves \\cite{GeorgeHuerta, SNe},\nthe parameter estimation \\cite{Hongyu, Gabbard, Alvin}.\nthe noise subtraction \\cite{Wei, denoiseHongyu},\nand the classification of glitch noises \\cite{HuertaGravitySpy}.\nOur work is devoted to investigating the accuracy of the parameter estimation.\nOur question is {\\it how accurately deep learning methods can estimate physical parameters},\nor {\\it whether deep learning methods can estimate parameters more accurately than the standard method}.\n\nIn this paper, we focus on the analysis of ringdown gravitational waves.\nThe ringdown is the last stage of a BBH merger.\nThe remnant black hole is largely perturbed just after the merger and the perturbation decays as gravitational waves are emitted.\nLate time perturbations of the black hole is dominated by the black hole quasi-normal modes (QNMs).\nThe ringdown gravitational waves can be modeled by the damped sinusoidal waveforms having the complex-valued QNM frequencies predicted by the black hole perturbation theory \\cite{ReggeWheeler, Zerilli, Teukolsky}.\nIn general relativeity (GR), the QNM frequencies are determined by the black hole mass and spin.\nBecause of this property, the ringdown gravitational waves are useful for the test of GR \\cite{Berti2005, LIGOtestGR}.\nOne way to estimate the QNM frequencies is the matched filtering using the inspiral-merger-ringdown gravitational waves \\cite{GhoshA, GhoshB}. \nThe posterior distribution of the binary masses and the spins is estimated and it can be converted into the mass and the spin of the remnant black hole by the fitting formula obtained from numerical relativity simulations \\cite{Healy}.\nThis method relies on GR and the inference of these parameters is mainly governed by the inspiral part.\nIf the effects caused by exotic theories (e.g. modified theories of gravity, black hole mimickers)\nmodify the merger-ringdown part without changing the inspiral part,\nbias would be introduced in the posterior in this method.\nThus, we need a method to estimate QNM frequencies using only the merger-ringdown part.\n\nThere are two possible directions of investigation:\nimproving the matched filtering and implementing alternative methods.\nIn Ref.~\\cite{MDC}, comparison of various methods for the analysis of ringdown was done using test mock data.\nThe result shows that the deep learning method is competitive with the matched filtering.\nThe deep learning method used in this challenge was the one constructed for the point estimation, that is, the neural network returns only a single estimated value for each parameter that we want to estimate.\nDespite of this shortcoming, deep learning methods are still expected to be a useful method complementary to the matched filtering.\n\nRecently, the authors of Ref.~\\cite{Gabbard} proposed the use of the conditional variational auto encoder (CVAE) for gravitational data analysis.\nIn addition to that the computational speed of the CVAE is much faster than that of the matched filtering, the CVAE can estimate the posterior probability distributions of parameters.\nAlthough the purpose of Ref.~\\cite{Gabbard} was the rapid inference,\nwe apply the CVAE for the off-line analysis and assess the accuracy of the inference of the CVAE.\n\nThis paper is organized as follows.\nIn Sec.~\\ref{sec:dataset}, we present the construction of the waveforms.\nIn Sec.~\\ref{sec:MF}, we briefly review the matched filtering.\nIn Sec.~\\ref{sec:cvae}, the idea and the implementation of CVAE are explained.\nIn Sec.~\\ref{sec:cnn}, we introduce convolutional neural networks (CNNs) as another competitors to the CVAE.\nIn Sec.~\\ref{sec:results}, the results obtained by the CVAE are compared with the matched filtering and the CNN.\nWe focus on the accuracy of the maximum posterior estimations and the area of the confidence regions.\nWe also confirm that the confidence regions obtained by the CVAE have the frequentist meaning by making the P-P plot,\nwith evaluation of the magnitude of the error.\nWe summarize our results and future works in Sec.~\\ref{sec:conclusion}.\nThroughout this paper, we set $G=c=1$.\n\n\n\n\n\n\n\n\\section{Preparing mock templates}\n\\label{sec:dataset}\n\n\n\n\nAs explained in Introduction, the situation we consider is that only the merger-ringdown part is modified from that of GR,\nand we compare deep learning methods and the matched filtering in such a situation.\nFor this purpose, we need to generate a test dataset by modifying only the merger-ringdown part of the waveform.\nIn some modified theories of gravity, gravitational waves from inspiraling BBHs can be calculated in the post-Newtonian approximation.\nBut consistent simulations throughout the inspiral-merger-ringdown phases have not been done so far.\nIn addition, it is a highly speculative assumption that only the merger-ringdown part might be modified.\nTherefore, what we can do for generating modified templates is to modify the merger-ringdown phase of GR templates in a phenomenological manner.\nUsing the modified templates, we prepare a mock test data for comparison of the deep learning methods and the matched filtering.\nThese templates are used not only for preparing a test dataset, but also for training neural networks and for constructing the template bank of the matched filtering.\n\nThe precise modeling of the transition from the inspiral phase to the post-merger phase is difficult,\nbut we would be able to roughly assume that the gravitational waves of the merger-ringdown phase have the following properties,\n\\begin{itemize}\n\\item The amplitude after the peak monotonically decreases.\nAt a later time, the amplitude decays exponentially.\n\\item The frequency monotonically increases and converges to a certain QNM frequency at a later time.\n\\end{itemize}\nWe focus on the case where the waveforms are modified only after the time $t_p^\\mathrm{GR}$, at which the amplitude of GR template reaches its peak.\nTherefore, the inspiral part of the modified waveform coincides with GR one.\nIn this work, we focus only on $l=m=2$ mode and ignore overtones\nas they are much weaker especially for nearly equal-mass binaries.\nThe importance of the multi-modes and overtones has been studied in Refs.~\\cite{multimodeBerti, multimodeJulian, overtone}.\n\n\nWe denote the QNM frequencies for GR templates and for modified templates by $\\omega_\\mathrm{R, I}^\\mathrm{GR}$ and $\\omega_\\mathrm{R, I}$, respectively.\nThe modified templates are constructed by modifying the complex-velued templates in GR, $h^\\text{GR}(t)$.\nFirst, we decompose the strain $h^\\text{GR}(t)$ into the amplitude $A^\\text{GR}(t)$ and the frequency $\\omega^\\text{GR}(t)$ as\n\\begin{equation}\n\th^\\text{GR}(t) = A^\\text{GR}(t) e^{i\\phi^\\text{GR}(t)},\\ \\phi^\\text{GR}(t) = \\int^t dt' \\omega^\\text{GR}(t').\n\\end{equation}\nFrom $A^\\text{GR}(t)$ and $\\omega^\\text{GR}(t)$, the modified amplitude and frequency, $A(t)$ and $\\omega(t)$, are generated.\nOur modified templates are characterized by two parameters, $\\delta\\omega_\\text{R}$ and $\\delta\\omega_\\text{I}$.\nThe real and imaginary parts of the QNM frequency, $\\omega_\\mathrm{R}$ and $\\omega_\\mathrm{I}$, are specified by the fractional deviation from the GR values as\n\\begin{equation}\n\t\\omega_\\mathrm{R, I} = \\omega_\\mathrm{R, I}^\\mathrm{GR} ( 1 + \\delta_\\mathrm{R, I}).\n\\end{equation}\nIn our work, the modifications of the frequencies are assumed to be small.\nThe deviations of the real part and the imaginary part of QNM frequencies are assumed to be less than 30\\% and 50\\%, respectively (i.e. $|\\delta_\\mathrm{R}| < 0.30, |\\delta_\\mathrm{I}| < 0.50$).\n\n\n\n\nModified amplitudes are constructed from two parts, before and after the peak.\nAfter the peak, the amplitudes are modified from GR as\n\\begin{equation}\n\tA'(t) = \\frac{A^\\mathrm{GR}(t)}{1+e^{4 M\\omega_\\mathrm{I}^\\mathrm{GR}x}} +\\frac{A^\\mathrm{RD}(t)}{1+e^{-4 M\\omega_\\mathrm{I}^\\mathrm{GR}x}},\n\\end{equation}\nwith \n\\begin{equation}\n\tA^{\\mathrm{RD}}(t)=\\frac{1.18}{1+e^{- M \\omega_\\mathrm{I}^{\\mathrm{GR}} x}+e^{ M \\omega_{\\mathrm{I}} x}}.\n\\end{equation}\nwhere $M$ is the total mass of the binary, $x$ is the normalized time defined as $x := (t - t^\\text{GR}_p) \/ M$, \nand $t^\\text{GR}_p$ is the time when the GR amplitude $A^\\text{GR}(t)$ reaches its peak.\nThe time when the modified amplitude $A'(t)$ reaches its maximum is denoted by $t'_p$ and can differ from $t_p^\\text{GR}$.\nWe connect the GR amplitude before $t_p^\\mathrm{GR}$ and the modified amplitude after $t'_p$ with an appropriate normalization.\nNamely, the modified amplitude $A(t)$ is obtained as\n\\begin{eqnarray}\n\tA(t) = \n\t\\begin{cases}\n\t\tA^\\mathrm{GR} (t) & (t\\leq t_p^\\mathrm{GR}), \\\\\n\t\t\\alpha A'(t+t'_p-t_p^\\mathrm{GR}) & (t>t_p^\\mathrm{GR}),\n\t\\end{cases}\n\\end{eqnarray}\nwith $\\alpha := A^\\mathrm{GR}(t_p^\\mathrm{GR}) \/ A'(t'_p) $.\n\n\n\n\nThe GW frequency $\\omega(t)$ of the modified waveform is specified as\n\\begin{equation}\n\t\\omega(t)=\\frac{\\omega^{\\mathrm{GR}}(t)}{1+e^{4 M \\omega_{\\mathrm{I}}^{\\mathrm{GR}} x}}+\\frac{\\omega^{\\mathrm{RD}}(t)}{1+e^{- 4 M \\omega_{\\mathrm{I}}^{\\mathrm{GR}} x}},\n\\end{equation}\nwith\n\\begin{equation}\n\t\\omega^{\\mathrm{RD}}(t) = \\omega^\\mathrm{GR}_p + (\\omega_\\text{R} - \\omega^\\text{GR}_p)\\tanh (0.85 M\\omega_\\text{I}^\\text{GR} x),\n\\end{equation}\nand $\\omega^\\text{GR}_p := \\omega^\\text{GR}(t^\\text{GR}_p)$.\n\nFinally, we generate the gravitational wave strain, $h(t)$, by\n\\begin{equation}\n\th(t) = A(t)e^{i\\phi(t)},\\ \\phi(t) = \\int^t dt' \\omega(t').\n\\end{equation}\nThe waveform of the modified model having $\\delta_\\text{R} = \\delta_\\text{I}=0$ coincide with that of GR.\n\nAs a seed for modified templates, we use the waveform SXS:0305~\\cite{SXS} and the total mass is fixed to $M=72.158M_\\odot$.\nThe GR values of QNM frequency is calculated from the fitting formula in Ref.~\\cite{Berti2005}.\nExamples of the modified templates are shown in Fig.~\\ref{fig:templates}.\n\\begin{figure}[t]\n\\centering\n\\begin{minipage}{1.0\\hsize}\n\\includegraphics[width=8cm]{figure\/Amplitude_A0.pdf}\n\\end{minipage}\n\\centering\n\\begin{minipage}{1.0\\hsize}\n\\includegraphics[width=8cm]{figure\/Frequency_A0.pdf}\n\\end{minipage}\n\\caption{The amplitudes ({\\it top}) and the frequencies ({\\it bottom}) of the modified templates having various QNM frequencies.\nThe frequency $f_\\text{R}$ is defined as $f_\\text{R} = \\omega_\\text{R} \/ 2\\pi$.\nWhen $\\delta_\\text{R} = \\delta_\\text{I}=0$, they coincide with those of GR.\nThe black vertical line indicates the time at which the amplitude reaches its peak.}\n\\label{fig:templates}\n\\end{figure}\n\nIn the following analysis, the frequency $f$ is used rather than $\\omega$.\nThey are related with each other by $\\omega_\\text{R,I} = 2\\pi f_\\text{R,I}$.\nThe sampling rate is 4096Hz.\n\n\n\n\n\\section{Matched filtering}\n\\label{sec:MF}\nWhen the waveforms can be theoretically modeled and generated rapidly,\nthe matched filtering is a powerful method for the parameter estimation (see \\cite{Creighton} as a standard textbook).\nThe detection statistic is the signal-to-noise ratio (SNR) and it can be calculated by the noise-weighted inner product between the observational data $s(t)$ and a template $h(t)$,\n\\begin{equation}\n\\text{SNR} = 4\\text{Re} \\int_{f_\\text{min}}^{f_\\text{max}} df\\ \\frac{\\tilde{s}(f) \\tilde{h}^\\ast(f)}{S_n(f)},\n\\label{eq:snr}\n\\end{equation}\nwhere $S_n(f)$ is the noise power spectral density,\n$\\tilde{s}(f)$ and $\\tilde{h}(f)$ are the Fourier transforms of $s(t)$ and $h(t)$, respectively.\nWe use the LIGO O1 noise power spectral density,\n\\begin{align}\n\tS_n(f) = &10^{-44} \\times \\left( \\frac{18.0}{0.1+f} \\right)^4 + 0.49\\times 10^{-46} \\notag \\\\\n\t&+ \\left( \\frac{f}{2000.0} \\right)^2 \\times 16.0 \\times 10^{-46} [\\text{strain}^2\/\\text{Hz}],\n\t\\label{eq:LIGOO1}\n\\end{align}\ngiven in Ref.~\\cite{LIGOnoisecurve}.\n\nWe do not optimize the coalescence time in the present matched filtering analysis.\nInstead, we fix it to the value of the injected templates, assuming that it can be easily guessed from the inspiral part of the gravitational wave data.\nTherefore, our templates are parameterized by the deviation of the QNM frequency, $\\{\\delta_\\text{R}, \\delta_\\text{I}\\}$, and the initial phase, $\\phi_0$.\nSince the initial phase can be marginalized analytically,\nthe parameter search is done on the parameter space of $\\{\\delta_\\text{R}, \\delta_\\text{I}\\}$.\nWith the uniform prior, the posterior distribution of the real and imaginary parts of the QNM frequency $\\{f_\\text{R}, f_\\text{I}\\}$ can be obtained by\n\\begin{equation}\np(f_\\text{R}, f_\\text{I}|s) \\propto \\exp\\left[ \\frac{\\text{SNR}^2(\\delta_\\text{R}, \\delta_\\text{I})}{2} \\right].\n\\end{equation}\n\nFor the post-merger analysis, we set the boundaries of the integration range of frequency to $f_\\text{min} = 160$Hz and $f_\\text{max} = 512$Hz.\nThe lower cutoff frequency, $f_\\text{min}$, is the frequency at which the amplitude of the template reaches the maximum.\n\nIn our work, the template bank is constructed to form a uniform grid in the $(\\delta_\\text{R}, \\delta_\\text{I})$ plane.\nThe parameter $\\delta_\\mathrm{R}$ is varied in the range $[0.7, 1.3]$ with the step of $0.006$,\nwhile $\\delta_\\mathrm{I}$ in the range $[0.5, 1.5]$ with the step of $0.01$.\nThe template bank consists of 10,201 templates.\n\n\n\n\n\n\n\\section{Conditional variational auto encoder}\n\\label{sec:cvae}\n\n\\subsection{Idea of CVAE}\n\\label{subsec:ideaCVAE}\n\nIn this subsection, we explain the idea of CVAE \\cite{Gabbard}.\nIn Bayesian inference, the existence of the true posterior $\\hat{p}(y|x)$, the distribution of the physical parameters $y$ under the assumption that a signal $x$ is given, is assumed.\nHere, the parameterized distributions $p_\\theta(y|x)$ are used as an approximation of $\\hat{p}(y|x)$.\nThe parameter $\\theta$ depends on the input signal $x$.\nThe neural network is trained to estimate the relation between $x$ and $\\theta$ using a training dataset,\nthat is, a lot of pairs of input data and the true values of the physical parameters, $\\{ (x_i, y_i) \\}_{i=1\\dots N}$.\nThe Kullback-Leibler (KL) divergence,\n\\begin{equation}\n\tKL[\\hat{p}(y|x) | p_\\theta(y|x)] := \\int dy\\ \\hat{p}(y|x) \\log \\frac{\\hat{p}(y|x)}{p_\\theta(y|x)},\n\\end{equation}\nis one of the natural choices for quantifying the mismatch between two probability distributions.\nHere, we consider the minimization of the expected value of the KL divergence,\n\\begin{equation}\n\t\\mathbb{E}_{\\hat{p}(x)}\\left[ KL[\\hat{p}(y|x) || p_\\theta(y|x)] \\right].\n\t\\label{eq:avgKL}\n\\end{equation}\nBecause only the terms including $p_\\theta(y|x)$ are essential for optimization,\nthe minimization of \\eqref{eq:avgKL} is equivalent to the maximization of the average of the cross entropy:\n\\begin{align}\n\t&\\mathbb{E}_{\\hat{p}(x)}\\left[ H[\\hat{p}(y|x)||p_\\theta(y|x)] \\right] \\notag \\\\\n\t&:= \\int dxdy\\ \\hat{p}(x) \\hat{p}(y|x) \\log p_\\theta(y|x) \\notag \\\\\n\t&= \\int dxdy\\ \\hat{p}(x, y) \\log p_\\theta(y|x).\n\t\\label{eq: avgLogP}\n\\end{align}\nThis can be approximated by the sample mean,\n\\begin{equation}\n\t\\mathbb{E}_{\\hat{p}(x)} \\left[ H[\\hat{p}(y|x)||p_\\theta(y|x)] \\right] \\simeq \\frac{1}{N}\\sum_{i=1}^N \\log p_\\theta(y_i|x_i).\n\t\\label{eq:crossentropy}\n\\end{equation}\n\nFor example, Gaussian distribution can be used as $p_\\theta(y|x)$.\nHowever, it would be too simple to approximate the posterior.\nIn order to enhance the flexibility of the approximant, the hidden variable model is often employed.\nThe approximated distributions are given as a superposition of simple distributions,\n\\begin{equation}\n\tp_\\theta(y|x) = \\int dz\\ p_{\\theta_\\text{D}}(y|x,z) p_{\\theta_\\text{E}}(z|x).\n\t\\label{eq:hvm}\n\\end{equation}\nThe additional variables $z$, so-called \\textit{hidden variables}, inherit compressed information of the data $x$.\nWith the hidden variable model, $\\log p_\\theta(y|x)$ appeared in R.H.S of Eq.~\\eqref{eq:crossentropy} is bounded by the evidence lower bound (ELBO),\n\\begin{eqnarray}\n\t\\log p_\\theta(y|x) &\\geq& \\mathcal{L}_\\text{ELBO} \\notag \\\\\n\t&:=&\\mathbb{E}_{q_\\phi(z|x,y)} \\left[ \\log p_{\\theta_\\text{D}}(y|x,z) \\right] \\notag \\\\\n\t&&- \\text{KL} \\left[ q_\\phi(z|x,y) | p_{\\theta_\\text{E}}(z|x) \\right]\n\t\\label{eq:ELBO}\n\\end{eqnarray}\nfor an arbitrary distribution $q_\\phi(z|x,y)$.\nThe negative ELBO, $-\\mathcal{L}_\\text{ELBO}$, is employed as the loss function to be minimized.\n\nA CVAE estimates the relation between the parameters of distributions and the conditioning variables.\nAs an example, the distribution $p_{\\theta_\\text{E}}(z|x)$ presents the probability of $z$ conditioned by $x$.\nThe neural network corresponding to $p_{\\theta_\\text{E}}(z|x)$ takes $x$ as an input and predicts the plausible value of $\\theta_\\text{E}$.\nIn Eq.~\\eqref{eq:ELBO}, three distributions, $p_{\\theta_\\text{D}}$, $p_{\\theta_\\text{E}}$ and $q_\\phi$, appear.\nTherefore, we need three networks for emulating these distributions.\n\n\n\nFurther simplification of Eq.~\\eqref{eq:ELBO} can be done as follows.\nFirst, the first term of the R.H.S of Eq.~\\eqref{eq:ELBO} can be approximated by the sample average,\n\\begin{equation}\n\t\\mathbb{E}_{q_\\phi(z|x,y)} \\log p_{\\theta_\\text{D}}(y|x,z)\n\t\\simeq \\frac{1}{N_\\text{z}}\\sum_{j=1}^{N_\\text{z}} \\log p_{\\theta_\\text{D}}(y|x, z_j),\n\\end{equation}\nwhere $z_j$ is the $j$-th sample of $z$ following $q_\\phi(z|y,x)$.\nIn this work, we set $N_\\text{z} = 1$.\nSecond, we adopt multivariate Gaussian distributions with diagonal covariance matrices as $p_{\\theta_\\text{D}}$, $p_{\\theta_\\text{E}}$ and $q_\\phi$.\nWe denote the mean and covariance matrix of $p_{\\theta_\\text{E}}(z|x)$ by\n\\begin{subequations}\n\\begin{eqnarray}\n\t\\vec{\\mu}_\\text{E} &=& (\\mu_{\\text{E}, 1}, \\mu_{\\text{E}, 2}, \\dots, \\mu_{\\text{E}, D_\\text{z}}), \\\\\n\t\\Sigma_\\text{E} &=& \\text{diag}(\\sigma^2_{\\text{E}, 1}, \\sigma^2_{\\text{E}, 2}, \\dots, \\sigma^2_{\\text{E}, D_\\text{z}}),\n\\end{eqnarray}\nthose of $p_{\\theta_\\text{D}}(y|x,z)$ by\n\\begin{eqnarray}\n\t\\vec{\\mu}_\\text{D} &=& (\\mu_{\\text{D}, 1}, \\mu_{\\text{D}, 2}, \\dots, \\mu_{\\text{D}, D_\\text{y}}), \\\\\n\t\\Sigma_\\text{D} &=& \\text{diag}(\\sigma^2_{\\text{D}, 1}, \\sigma^2_{\\text{D}, 2}, \\dots, \\sigma^2_{\\text{D}, D_\\text{y}}),\n\\end{eqnarray}\nand those of $q_\\phi(z|x,y)$ by\n\\begin{eqnarray}\n\t\\vec{\\mu} &=& (\\mu_1, \\mu_2, \\dots, \\mu_{D_\\text{z}}), \\\\\n\t\\Sigma &=& \\text{diag}(\\sigma^2_1, \\sigma^2_2, \\dots, \\sigma^2_{D_\\text{z}}),\n\\end{eqnarray}\n\\end{subequations}\nwhere $D_\\text{z}$ and $D_\\text{y}$ are the dimensions of the hidden variable $z$ and the physical parameters $y$, respectively.\nThus, the parameters $\\theta_\\text{E}$, $\\theta_\\text{D}$ and $\\phi$ denoted abstractly so far are $\\theta_\\text{E} = \\{\\vec{\\mu}_\\text{E}, \\Sigma_\\text{E} \\}$, $\\theta_\\text{D} = \\{\\vec{\\mu}_\\text{D}, \\Sigma_\\text{D} \\}$ and $\\phi = \\{ \\vec{\\mu}, \\Sigma \\}$.\nThen, the loss function for one training data is obtained as \n\\begin{widetext}\n\\begin{equation}\n\t\\text{Loss} = \\frac{D_\\text{y}}{2} \\log 2\\pi + \\sum_{l=1}^{D_\\text{y}} \\log \\sigma_{\\text{D},l} + \\frac{1}{2} \\sum_{l=1}^{D_\\text{y}} \\frac{(y_l - \\mu_{\\text{D},l})^2}{\\sigma^2_{\\text{D},l}}\n\t+\\frac{D_\\text{z}}{2} - \\frac{1}{2} \\sum_{k=1}^{D_\\text{z}} \\left\\{ \\log \\frac{\\sigma^2_{\\text{E},k}}{\\sigma^2_{k}} + \\frac{(\\mu_{k} - \\mu_{\\text{E},k})^2}{\\sigma^2_{\\text{E},k}} + \\frac{\\sigma^2_{k}}{\\sigma^2_{\\text{E},k}} \\right\\}.\n\t\\label{eq:loss}\n\\end{equation}\n\\end{widetext}\n\n\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[scale=0.35]{figure\/CVAE.pdf}\n\\caption{\\label{fig:cvae}\nThe schematic picture of the CVAE.\nEncoder1, Encoder2 and Decoder represent neural networks corresponding to the probability distributions $p_{\\theta_\\text{E}}(z|x)$, $q_\\phi(z|x,y)$ and $p_{\\theta_\\text{D}}(y|x,z)$, respectively.\nHere, we adopt multivariate Gaussian distributions for all three distributions.\nThe parameters characterizing these distributions are $\\theta_\\text{D} = \\{\\mu_\\text{D}, \\Sigma_\\text{D}\\}$, $\\theta_\\text{E} = \\{\\mu_\\text{E}, \\Sigma_\\text{E}\\}$ and $\\phi = \\{\\mu, \\Sigma\\}$.\nAt the training ({\\it left}), three networks are optimized so that the loss function is minimized.\nThe Kulback-Leibller divergence are calculated with the output of the Encoder1 and the Encoder 2.\nThe output of the Decoder is used for assessing the negative log posterior term.\nFor test events ({\\it right}), the Encoder 1 and the Decoder are employed for sampling predicted values.}\n\\end{figure*}\n\n\nFigure~\\ref{fig:cvae} shows the schematic picture of the CVAE we use in this work.\nThe neural networks corresponding to $p_{\\theta_\\text{E}}(z|x)$, $q_\\phi(z|x,y)$ and $p_{\\theta_\\text{D}}(y|x,z)$ are called as Encoder1, Encoder2 and Decoder, respectively.\nEach neural network returns the mean and the diagonal elements of the covariance matrices of each distribution.\nAt the training (the left figure of Fig.~\\ref{fig:cvae}), all networks are simultaneously trained with the loss function \\eqref{eq:loss}.\nWhen the trained the CVAE is applied to a test data (the right figure of Fig.~\\ref{fig:cvae}),\nwe use the networks corresponding to $p_{\\theta_\\text{D}}$ and $p_{\\theta_\\text{E}}$ for estimating a posterior.\nEstimating the posterior for a test event is based on the following sampling method.\nFirst we sample one value of $z$ from the distribution $p_{\\theta_\\text{E}}(z|x)$.\nNext, with the sampled $z$, a sample of the parameter $y$ is obtained from $p_{\\theta_\\text{D}}(y|z, x)$.\nRepeating these sampling processes, we finally obtain many samples of predicted values $y$ that follow the estimated posterior $p_\\theta(y|x)$. \n\n\n\n\n\n\\subsection{Implementation}\n\nIn this subsection, the implementation of the CVAE that we use is described.\nWe use \\verb|PyTorch| \\cite{PyTorch} for the implementation.\n\n\n\\subsubsection{Structure}\n\n\nAs explained in the subsection \\ref{subsec:ideaCVAE}, the CVAE consists of three neural networks, that is, two encoders and one decoder.\nEach of them has six layers and each internal layer has 512 units.\nWe put a ReLU layer after each fully-connected layer except for the last layer of each neural network.\nEncoder1 and Encoder2 will output the mean and the diagonal elements of the covariance matrix of the hidden variables.\nWe set the dimension of the hidden variables as $D_\\text{z} = 16$.\nThe input of Decoder is the sampled variables from the multi-variate Gaussian distribution having the mean and covariance matrix estimated by the encoder.\nDecoder returns the mean and the covariance matrix of the distribution $p_{\\theta_D}(y|z,x)$.\nThe entire structure of the CVAE we use in this work is shown in Table \\ref{tab:structureCVAE}.\n\n\\begin{table}[b]\n\\centering\n\\caption{\\label{tab:structureCVAE}\nThe structure of the CAVE that we use in this work.\nAll layers of Encoder1, Encoder2 and Decoder are fully connected layers.\nEach network consists of six fully connected layers.\nThe input of Encoder1 is the segment of the signal.\nThe inputs of Encoder2 are a segment of the signal and the injected values of $\\delta_\\text{R}$ and $\\delta_\\text{I}$.\nDecoder takes the signal and the hidden variables as input.}\n\\begin{ruledtabular}\n\\begin{tabular}{cc}\nNetwork & \\# of units of respective layers\\\\ \\hline\nEncoder1 & [128, 512, 512, 512, 512, 512, 32] \\\\\nEncoder2 & [130, 512, 512, 512, 512, 512, 32] \\\\\nDecoder & [144, 512, 512, 512, 512, 512, 4]\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\n\\subsubsection{Dataset for training}\n\nFor the training, we use the same templates contained in the template bank for the matched filtering.\nEach template is labeled by $\\{ \\delta_\\text{R}, \\delta_\\text{I}\\}$.\nThe input signals as training data are generated as\n\\begin{equation}\n\tx(t) = A h_\\text{whitened}(t) + n(t)\n\\end{equation}\nwhere $h_\\text{whitened}$ is a template whitened with Eq.~\\eqref{eq:LIGOO1},\nthe noise $n(t)$ is generated from the standard normal distribution,\nand the amplitude $A$ is chosen to realize a specified SNR.\nTo prevent overfitting to a specific noise pattern, the noise realizations are generated and the whitened templates are injected into them for each iteration.\nFrom these simulated signals, we pick up 128 points starting from the amplitude peak,\nwhich is used as the input data of the CVAE.\n\n\n\\subsubsection{Training and inference scheme}\n\nThe Adam procedure \\cite{Adam} is used for the optimization algorithm.\nThe learning rate is set to $10^{-5}$ initially and decreased to $10^{-6}$ on the later stage of the training.\nThe scheduled training is employed, i.e., \nthe amplitude of the signal is gradually decreased from a large initial amplitude.\nThe training schedule is shown in Table \\ref{tab:scheduleCVAE}.\nThe batch size is 256.\n\nWhen the trained CVAE is applied to a test data, the sampling process to estimate the distribution is repeated until $4\\times10^6$ samples are collected.\n\n\n\\begin{table}[t]\n\\caption{\\label{tab:scheduleCVAE}\nThe training schedule for the CVAE.\nIn the last stage of training, input signals have SNR varying from 8 to 30.\nAfter 45000 epochs, the training is terminated when decreasing of training loss saturates.}\n\\begin{ruledtabular}\n\\begin{tabular}{ccc}\nepoch & the range of $A$ & learning rate\\\\ \\hline\n1 - 10000 & [8.0, 10.0] & $1.0\\times10^{-5}$ \\\\\n10001 - 15000 & [6.0, 10.0] & $1.0\\times10^{-5}$ \\\\\n15001 - 20000 & [4.0, 10.0] & $1.0\\times10^{-5}$ \\\\\n20001 - 25000 & [3.0, 10.0] & $1.0\\times10^{-5}$ \\\\\n25001 - 45000 & [2.0, 10.0] & $1.0\\times10^{-5}$ \\\\\n45001 - & [2.0, 10.0] & $1.0\\times10^{-6}$\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\n\n\\section{Convolutional neural network}\n\\label{sec:cnn}\n\nIn this work, an ordinary neural network, which returns a single value for each parameter that we want to estimate,\nis also implemented as one of competitors to the CVAE.\nConvolutional neural networks (CNNs) are used for various research of the gravitational wave data analysis (e.g. \\cite{GeorgeHuerta}).\nOur CNN has three convolutional and four fully-connected layers.\nEach of them, except for the last layer, is followed by a ReLU layer.\nThe output of the last layer is the estimated values of $\\{ \\delta_\\text{R}, \\delta_\\text{I}\\}$.\nFor respective convolutional layers, the numbers of filters are 128, 256 and 512, \nand the sizes of filters are 32, 8 and 8.\nAll of fully connected layers have 512 units.\nWe use mean square loss for the loss function.\nAlso for the training of the CNN, scheduled training is employed.\nThe training schedule is shown in Table \\ref{tab:scheduleCNN}.\nThe CNN is also implemented by \\verb|PyTorch|.\nThe training dataset is the same as the CVAE.\n\n\\begin{table}[t]\n\\caption{\\label{tab:scheduleCNN}\nThe training schedule for the CNN.\nWe set the learning rate as $10^{-4}$ for the whole epoch of training.\nAfter the 4001st epoch, the training is terminated once the decrease of training loss saturates.}\n\\begin{ruledtabular}\n\\begin{tabular}{cc}\nepoch & the range of $A$ \\\\ \\hline\n1 - 1000 & [8.0, 10.0] \\\\\n1001 - 2000 & [6.0, 10.0] \\\\\n2001 - 3000 & [4.0, 10.0] \\\\\n3001 - 4000 & [3.0, 10.0] \\\\\n4001 - & [2.0, 10.0] \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\n\n\\section{Results}\n\\label{sec:results}\n\n\\subsection{Dataset for comparison}\nWe prepare the mock test data in the same way as the training data.\nThe real-valued template $h_\\text{inj}$ is generated from a complex-valued modified template $h = h_+ + ih_\\times$ with the randomly sampled phase $\\phi_0$, i.e.,\n\\begin{equation}\n\th_\\text{inj} = h_+ \\cos\\phi_0 + h_\\times \\sin\\phi_0.\n\\end{equation}\nWe use the noise curve of LIGO O1 for generating the Gaussian noise (Eq.~\\eqref{eq:LIGOO1}).\nThree datasets with SNR of the merger-ringdown part 30.0, 15.0 and 8.0 are prepared (the definition of the merger-ringdown SNR is Eq.~\\eqref{eq:snr}).\nEach dataset consists of 500 simulated data whose $\\delta_\\text{R}$ and $\\delta_\\text{I}$ are randomly sampled from the region satisfying our assumptions, i.e., $|\\delta_\\text{R}|<0.3$ and $|\\delta_\\text{I}|<0.5$.\n\n\n\\subsection{Comparison of the point estimation}\nTo quantify the accuracy of the estimates, we define the following two quantities,\n\\begin{eqnarray}\n&\\overline{\\Delta Q} := \\dfrac{1}{N_\\text{data}} {\\displaystyle \\sum_{i=1}^{N_\\text{data}} } \\left( Q_i^\\text{est} - Q_i^\\text{true} \\right), \\label{eq:DeltaQ} \\\\\n&\\sigma(Q) := \\dfrac{1}{N_\\text{data}} \\left[ {\\displaystyle \\sum_{i=1}^{N_\\text{data}} } \\left( Q_i^\\text{est} - Q_i^\\text{true} \\right)^2 \\right]^{1\/2}. \\label{eq:SigmaQ}\n\\end{eqnarray}\nHere, $Q^\\text{est}$ is given by the estimated value that maximizes the posterior distribution for the matched filtering and the CVAE,\nwhile it is given by the output value for the CNN.\nThe comparison of the errors is shown in Table \\ref{tab:comp}.\nFrom this table, we can conclude that\n\\begin{itemize}\n\\item For both $f_\\text{R}$ and $f_\\text{I}$, the means of the errors $\\overline{\\Delta Q}$ are much smaller than the standard deviations $\\sigma(Q)$.\nTherefore, the estimates of both $f_\\text{R}$ and $f_\\text{I}$ are not significantly biased in all methods.\n\\item Because the standard deviations of the CVAE are smaller than those of the matched filtering and the CNN, we can say that the CVAE estimates the QNM frequencies more accurately than the other two methods.\n\\end{itemize}\n\n\n\n\\begin{table}[t]\n\\centering\n\\caption{\\label{tab:comp}\nThe comparison of the estimation errors.\nThe quantities $\\overline{\\Delta Q}$ and $\\sigma(Q)$ are defined in Eqs.~\\eqref{eq:DeltaQ} and \\eqref{eq:SigmaQ}.\nThe estimation by the CVAE has no significant bias for both of $f_\\text{R}$ and $f_\\text{I}$ and for any values of SNR.\nThe matched filtering and the CNN also estimate QNM frequency with small bias for most cases.\nComparing the values of $\\sigma(f_\\text{R,I})$, we find that the CVAE takes the smallest values for all cases,\nexcept for imaginary part of the dataset having SNR=8.\nFor this case, the CNN has a smaller value of $\\sigma(f_\\text{I})$ than the CVAE.\nHowever, the CNN derives a slightly larger value of $\\overline{\\Delta f_\\text{I}}$ than the CVAE.\nThis means that the estimation by the CNN is more biased.}\n\\begin{ruledtabular}\n\\begin{tabular}{llllll}\n$\\text{SNR}_\\text{RD}$ &method & $\\overline{\\Delta f_\\text{R}}$ [Hz] & $\\sigma(f_\\text{R})$ [Hz] & $\\overline{\\Delta f_\\text{I}}$ [Hz] & $\\sigma(f_\\text{I})$ [Hz] \\\\ \\hline\n& MF & -0.1607 & 3.5243 & -0.1865 & 2.7237 \\\\\n30.0 & CNN & 0.9732 & 8.2192 & -1.1812 & 3.0875 \\\\\n& CVAE & 0.0267 & 3.1180 & -0.2528 & 2.4311 \\\\ \\hline\n& MF & -0.4015 & 7.4448 & -0.5448 & 5.4256 \\\\\n15.0 & CNN & -0.0432 & 9.5206 & -0.6411 & 4.9630 \\\\\n& CVAE & -0.4253 & 6.2759 & -0.2109 & 4.8657 \\\\ \\hline\n& MF & -0.1755 & 15.2181 & -1.7824 & 9.6581 \\\\\n8.0 & CNN & 0.9783 & 14.2067 & 1.7371 & 7.7085 \\\\\n& CVAE & -0.2350 & 12.4485 & 0.4289 & 8.9368\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\n\n\\subsection{Reliability of the confidence regions}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=8cm]{figure\/contour_SNR8.0_00000.pdf}\n\\caption{An example of posterior estimations for a test data whose SNR is 8.0.\nBlue and orange contours are confidence regions estimated by the CVAE and the matched filtering, respectively.\nThe contours show (50, 90, 99)\\% confidence regions.\nBlue circle and orange square are the predicted values of the QNM frequency obtained by the CVAE and the matched filtering, respectively.\nBlack cross shows the injected value of the QNM frequency.}\n\\label{fig:posterior}\n\\end{figure}\n\n\n\\begin{figure*}[t]\n\\begin{tabular}{cc}\n\\begin{minipage}{0.5\\hsize}\n\\begin{center}\n\\includegraphics[width=8cm]{figure\/PPplot_MF_8.0.pdf}\n\\end{center}\n\\end{minipage}\n\\begin{minipage}{0.5\\hsize}\n\\begin{center}\n\\includegraphics[width=8cm]{figure\/PPplot_CVAE_8.0.pdf}\n\\end{center}\n\\end{minipage}\n\\end{tabular}\n\\caption{\\label{fig:ppplot}\nThe P-P plots of the matched filtering ({\\it left}) and of the CVAE ({\\it right}). \nThe SNR of the test dataset is 8.0.\nThe horizontal axis shows percentages of the confidence regions.\nThe vertical axis shows the fraction of events whose true values are located within the confidence regions.\nIf estimated confidence region has the frequentist meaning, the plot (blue line) is consistent with the diagonal line (black dotted line).\nThe orange region is 1-$\\sigma$ error of the binomial distribution.\nThe error estimation by the CVAE seems to be slightly biased.\nA similar feature can be seen for the datasets having SNR 15.0 and 30.0.}\n\\end{figure*}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=7cm]{figure\/ConvergencePPplot_8.0.pdf}\n\\caption{The deviation of the P-P plot from the diagonal line.\nThe SNR of the dataset is 8.0.\nBlue circles and orange squares are obtained with 500 and 10,000 test events, respectively.\nThe CVAE estimates the posterior distributions with $<2$\\% systematic error.}\n\\label{fig:devpp}\n\\end{figure}\n\n\nAn example of the predictions of posterior distributions by the CVAE and the matched filtering is shown in Fig.~\\ref{fig:posterior}.\nBefore comparing the posterior estimations by the CVAE and the matched filtering,\nwe assess the reliability of the posterior distributions estimated by the CVAE.\nIf the estimation of posterior distribution is reliable,\nthe fraction of events whose true values are located within the $x$-\\% confidence region should be $x$-\\%.\nFor visualization, a P-P plot is useful.\nIn a P-P plot, we take the confidence level as horizontal axis and the fraction of events as vertical axis.\nIf the posterior distribution is reliable, the P-P plot reduces to the diagonal line.\nWe show the P-P plots obtained by the CVAE and the matched filtering in Fig.~\\ref{fig:ppplot}.\nIt is found that the error estimation by the matched filtering includes no significant bias.\nOn the other hand, the P-P plot for the CVAE seems to deviate from the 45$^\\circ$ line only slightly.\nIn order to quantify the systematic error,\nwe generate additional 9,500 test events for each SNR.\nFigure~\\ref{fig:devpp} shows the deviation from 45$^\\circ$ line for SNR=8.0 events.\nIt is found that the estimation by the CVAE contains the systematic error less than 2\\%.\nA similar feature can be seen for the events having SNR 15.0 and 30.0.\n\n\n\n\\subsection{Comparison of areas of confidence regions}\nTaking into account the existence of bias at a few percent level,\nwe compare the confidence regions obtained by the CVAE and the matched filtering.\nTo compare them quantitatively, we define \n\\begin{eqnarray}\n\\Delta S_i(x) = S^\\text{CVAE}_i(x) - S^\\text{MF}_i(x), \\\\\n\\overline{\\Delta S(x)} = \\frac{1}{N_\\text{data}} \\sum_{i=1}^{N_\\text{data}} \\Delta S_i(x),\\label{eq:S(x)}\n\\end{eqnarray}\nwhere $S^\\text{CVAE\/MF}_i(x)$ is the area of the $x$-\\% confidence region estimated by the CVAE\/the matched filtering for the $i$-th test event.\nWhen $\\Delta S_i(x)$ is negative, the constraint of the CVAE is tighter than that of the matched filtering.\nThe comparison of the area of the confidence region is shown in Table~\\ref{tab:area}.\nFor all datasets, the CVAE leads to more stringent constraint than the matched filtering.\n\n\\begin{table}[t]\n\\centering\n\\caption{\\label{tab:area}\nThe comparison of the areas of confidence regions.\nThe quantity $\\overline{\\Delta S(x)}$ is defined in Eq.~\\eqref{eq:S(x)}.\nFor all datasets having different SNRs, the CVAE gives tighter constraint than the matched filtering.}\n\\begin{ruledtabular}\n\\begin{tabular}{llll}\n$\\text{SNR}_\\text{RD}$ & $\\overline{\\Delta S(99)} [\\text{Hz}^2]$ & $\\overline{\\Delta S(90)} [\\text{Hz}^2]$ & $\\overline{\\Delta S(50)} [\\text{Hz}^2]$ \\\\ \\hline\n30.0 & -10.8893 & -6.6020 & -2.3531 \\\\\n15.0 & -119.521 & -64.5984 & -20.1443 \\\\\n8.0 & -415.235 & -185.065 & -46.8837 \n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nIn this paper, we investigated how accurately a CVAE can estimate the QNM frequencies using only merger-ringdown waveforms.\nTo do this, we generated modified waveforms by changing the merger-ringdown part of the GR template\nand constructed a test dataset by injecting the waveforms into simulated Gaussian noise data.\nWe compared the accuracies of the CVAE and the matched filtering,\nand showed the CVAE can predict the QNM frequencies with a higher accuracy than the matched filtering.\nNext, we evaluated the reliability of the confidence regions estimated by the CVAE, making a P-P plot.\nThe estimated confidence levels have the systematic error less than 2\\%.\nThe areas of 50\\%, 90\\% and 99 \\% confidence regions obtained by the CVAE and the matched filtering were compared\nand it was found that the CVAE can give more stringent constraint to the QNM frequencies than the matched filtering.\n\nIn this work, we only focused on the case of the Gaussian noise.\nTo make the deep learning method applicable to the real event analysis,\nthe case with the noise having non-Gaussianity need to be investigated.\nThe higher modes of the ringdown signal were also neglected.\nThe importance of the multi-mode analysis is indicated by several authors \\cite{multimodeBerti, multimodeJulian}.\nApplication to the black hole spectroscopy is remaining for future work.\n\nCVAE is not the only method for estimating posteriors (e.g. Bayesian neural network \\cite{Hongyu}, NN with reduced order modeling \\cite{Alvin}).\nComparison (or integration) with these methods would be insightful.\n\nIn this work, the merger-ringdown waveforms modified from those of GR were employed for training the CVAE.\nIn this sense, our method is model-dependent.\nAlthough the post-merger templates based on the specific theory of modified gravity are not obtained so far,\nthe result of our work is insightful when they can be constructed.\nOn the other hand, exploring model independent methods is a possible direction of future work.\nEven in non-GR theories, the ringdown gravitational waves would be expected to have the properties that the frequency is constant and the amplitude decays exponentially.\nNeural networks would be useful to detect these features from noisy signals and estimate the QNM frequencies independently of the way of modification.\n\n\n\n\\begin{acknowledgments}\nThis work was supported by JSPS KAKENHI Grant Number JP17H06358 (and also JP17H06357),\n{\\it A01: Testing gravity theories using gravitational waves,} as a part of the innovative research area,\n``Gravitational wave physics and astronomy: Genesis\".\nWe thank the members of the A01 group for useful discussions.\nSome part of calculation has been performed by using GeForce 2080Ti GPU at Nagaoka University of Technology.\n\\end{acknowledgments}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\nEmergent need to achieve better, more precise and sensitive drug detection in medicine and health care recently has been\naddressed by developing biosensors based on two-dimensional materials (2DM)\\cite{Kostarelos2014,Bolotsky2019,Zhu2019,Daus2021,Lee2014,Campuzano2017,Oh2021,Pang2020}. Not only 2D materials offer new response and\/or transduction mechanisms and better performance, they can be used for label-free biosensing. Importantly, 2DMs could be designed and\/or integrated to generate several signals in response to a single analyte, as it will be illustrated below, or to respond by several channels to a group of substances in parallel, thus achieving a multimodal detection. \n\nThe multimodal operation exceeds single-mode biosensing through its higher throughput as well as ability to differentiate the analyte from background signals in a complex media, and potentially allows the multiplexing of biosensing\\cite{Yen2015,Wang2015,Lee2016,Lei2019,Zhang2017}, {\\em i.e.}, determining multiple analytes through a single test. While significant attention has been paid to exploring new 2D materials and demonstrating their biosensing capabilities at the level of single devices \\cite{Zhang2015,Wang2014,ArjmandiTash2016,Sekhon2021}, overall knowledge on what allows successful multimodal detection and what limits biosensing capabilities of 2DM heterostructures is scarce. Atomically thin 2D materials, having an ultimate surface-to-volume ratio, may possess surface non-uniformities at the nanometer scale (atomic impurities\/adsorbates\/defects, wrinkles\/ruptures) that modulate their optical properties, although their importance and explicit role in producing material's variability yet to be studied. To a large extent, the difficulty to determine physical mechanisms that control performance of 2DM devices is due to disparate scales for atomic non-uniformities compared to a micrometer, or larger, size of active elements of a biosensor. Structural characterization with a high spatial resolution, such as electron microscopy, often does not detect materials optical properties, while optical microscopy lacks the required resolution. Thus, in order to reveal such mechanisms, multiple characterization tools should be combined and correlated\\cite{Kolesnichenko2021}. In this work, correlated multidimensional imaging, including Raman and near-field microscopies, scanning probe and electron microscopies, was applied to unveil physical processes behind label-free multimodal detection of doxorubicin (DOX), an anthracycline cancer drug, by 2DM vertical heterostructures.\n\nDoxorubicin is one of the most common drugs against different types of cancer (haematological, thyroid, breast, ovarian, lung and liver cancer)\\cite{Norouzi2020,Chen2021a,Zhong2017,He2020,Yuan2021}. Since DOX is known for certain drug resistance and side effects\\cite{Carvalho2014,HOFMAN2015168,MITRY201617,Umsumarng2015}, an efficient and sensitive detection of the amount of DOX in various types of biological samples, potentially at the point-of-care, has significant value. Recently, DOX has been loaded on graphene oxide and other nanocomposites\\cite{Sun2008,Chekin2019,Hasanzadeh2016,Yang2020,Pei2020}. Regular Raman microscopy, as well as surface enhanced Raman spectroscopy (SERS) were used to detect DOX in various cell lines and real samples\\cite{Zong2018,Gautier2013,Huang2013,Farhane2015,Farhane2017,Litti2016}. Here, optical signaling of the presence of DOX (deposited from solution) is demonstrated via three independent channels: (1) graphene enhanced Raman spectra (GERS) of DOX, (2) Raman shift of monolayer graphene (MLG) and (3) photoluminescence (PL) shift of single layer MoS$_2$ (Fig.\\ref{fig:fig1}).\n\n\n\n\\begin{figure*}[b]\n\t\\centering \n\t\\includegraphics[width=\n\t\\textwidth]{fig01.jpg\n\n\t\\caption{Multiplexed detection of doxorubicin drug. (a) Schematics of multimode detection by the combination of MoS$_2$ photoluminescence, DOX GERS, and Raman shift of monolayer graphene. (b) GERS signal of DOX\/MLG (red), vs. reference Raman spectra of DOX\/DMSO solution (cyan) and MLG (gray); red (cyan) arrows mark DOX (DMSO) lines. (c) Modulation of MoS$_2$ PL spectrum: with DOX (red) and w\/o DOX (cyan); inset shows DOX molecular structure. (d) Fitting of measured PL spectra from (c): A\/B-exciton and trion (X$^-$) lines are shown; modulation of peak position ($\\Delta\\omega$) and intensity ($\\Delta P$) are indicated using A-exciton fit; inset shows the schematics of optical subbands of MoS$_2$. (e-f) Typical Raman spectra of MLG: with DOX (red) and before incubation (blue); G- and 2D-line intensities were normalized to unity. (g-h) Correlation plots and (i-l) partial distribution functions for peak position and width for G- and 2D-lines, measured locally, at diffraction limited spots across the sample; same color code as in (e-f); clear line red-shift and broadening are detected with DOX.}\n\t\\label{fig:fig1}\n\\end{figure*}\n\n\n\nCurrently, two major approaches are implemented in biosensor technology: label-free and label-based sensing. While the latter shows high selectivity limited only by our ability to find a high-optical-contrast receptor with best binding to known analyte, the former is much more versatile, especially in terms of sensing a wide range of analytes, enabling agnostic biosensing, and being capable to detect yet unknown biothreats for which the receptors have not been developed. Though very promising, label-free biosensors require additional calibration due to lower specificity. To solve the problem sensing multiplexing, combined with machine learning, has been applied\\cite{Zhang,Misun2016,Cui2020}.\n\n\n\n\\begin{table*}\n\t\\caption{The PL fit parameters for Fig.\\ref{fig:fig1}(d): upper\/lower row corresponds to PL with\/without DOX}\n\t\\label{table:tablePL}\n\t\\centering\n\\resizebox{\\textwidth}{!}{\t\n\t\\begin{tabular}{|*{9}{c|}}\n\t\t\\hline\n\t\t\\multicolumn{3}{|c}{Trion} & \\multicolumn{3}{|c}{A-exciton} & \\multicolumn{3}{|c|}{B-exciton} \\\\ \\hline \n\t\t$\\omega_c$, eV & $\\gamma$, meV & P, cts. &$\\omega_c$, eV & $\\gamma$, meV & P, cts. &$\\omega_c$, eV & $\\gamma$, meV & P, cts. \\\\ \\hline\n1.739 $\\pm$ 0.002 & 60. $\\pm$ 3. & 32 $\\pm$ 4 & 1.815 $\\pm$ 0.0002 & 82.0 $\\pm$ 0.3 & 793 $\\pm$ 4 & 1.953 $\\pm$ 0.001 & 135.8 $\\pm$ 2. & 203 $\\pm$ 1 \n\t\t\\\\ \\hline\n1.719 $\\pm$ 0.003 & 60. $\\pm$ 9. & 15 $\\pm$ 3 & 1.806 $\\pm$ 0.002 & 90.7 $\\pm$ 0.3 & 586 $\\pm$ 3 & 1.955 $\\pm$ 0.002 & 135.0 $\\pm$ 2. & 197 $\\pm$ 2 \n\\\\ \\hline\n\\end{tabular}\n}\n\\end{table*}\n\n\n\n\nIn order to achieve multiplexed detection, arrays of different sensors could be integrated in one device\\cite{Kalmykov2019}. To avoid unnecessary complexity of integration, mutimodal sensing materials and heterostructures are developed\\cite{Novoselov2016,Jeong2015,Ma2020,AlaguVibisha2020}. Here we demonstrate multiplexed detection of doxorubicin by vertical heterostructure of monolayer graphene\/transition metal dichalcogenide (TMDC) by measuring response of 2D materials in 3 optical channels: MoS$_2$ photoluminescence, graphene Raman shift and graphene enhanced Raman scattering of molecular fingerprint modes of the molecule itself.\n\n\n\n\n\\section{Results and discussion}\n\\subsection*{Label-free detection of Doxorubicin}\n\nMolybdenum disulfide, a typical TMDC 2D material, is known to show strong PL signal\\cite{Mak2013} which can be modulated by adsorbtion of molecular species\\cite{Mouri2013,CatalanGomez2020,Aryeetey2021,Barja2019,Mitterreiter2021,Schuler2020,Thiruraman2018}. Fig.\\ref{fig:fig1}(c) shows a profound change in PL spectrum of MoS$_2$ photoluminescence (PL) after incubation to 172 nM solution of DOX for 15 minutes (the large area integrated PL is presented here; to not be confused with local micro-PL discussed below). In order to understand physical mechanisms resulting in the DOX recognition, the PL band is fitted with individual excitation lines: as shown in the inset of Fig.\\ref{fig:fig1}(d), the MoS$_2$ optical transitions include typical B- and A-exciton subbands, trion (X$^-$) and, often, additional localized modes. Here the shift in mode peak position ($\\Delta\\omega$), peak intensity ($\\Delta P$) and width ($\\Delta\\gamma$) are indicative for analyte absorption, resulted in subsequent charge transfer\/doping and strain imposed in the 2D material. These shifts are specific for an analyte: panel (d) and data in Table \\ref{table:tablePL} provide the values for DOX analyte. While upper B-exciton is barely influenced by the drug molecules (a small intensity difference is detected, see red arrow in panel (d)), both A-exciton and trion are red-shifted, have lower intensity and larger peak width, that all together lead to the spectral differences in panel (c). The ability to detect DOX at a low (sub-nM) concentration (and differentiate it from other components of a complex solution) would depend on amount of signal over the noise for the biosensor. Importantly, the variation of the signal in the pristine biosensing material adds to the total uncertainty and reduces the device performance as we discuss below.\n\n\\begin{figure}[!]\n\t\\centering\n\t\\includegraphics[width=0.\n\t\\textwidth]{fig12.jpg\n\t\\caption{Stability test of MoS$_2$\/graphene vertical heterostructure. SEM (a,e) and sSNOM (b-d,f-h) images of two MoS$_2$ islands, randomly selected, coated with monolayer graphene. The island (a) shows nearly zero degradation after 242 days in ambient -- from (b) to (c), neither after 705 days -- from (b) to (d); the island (e) was selected near a tear in MLG and shows (g) partial oxidation near the central micro-crystallite of molybdenum after 242 days, followed by (h) almost complete oxidation of MoS$_2$ surface after 705 days. All scale bars are 1 $\\mu$m.\n\t\n\t\n\t\n\t}\n\t\\label{fig:fig12}\n\\end{figure}\n\n\\begin{table*}\n\t\\caption{Measured GERS enhancement factors for major fingerprint Raman lines of DOX}\n\t\\label{table:tableGERS}\n\t\\centering\n\t\\begin{tabular}{|*{8}{c|}}\n\t\t\\hline\n\t\tRaman line position,\tcm$^{-1}$ & 1236 &\t1244 &\t1260 &\t1268 &\t1326 &\t1434 &\t1613 \\\\\n\t\t\\hline\n\t\tGERS enhancement factor & 6.4 & 7.0 &\t23.3 &\t23.3 &\t1.8 &\t2.9 &\t2.1 \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table*}\n\n\n \n\nAgnostic detection of a chemical or biothreat requires multiplexing the receptor signal with additional channels, as there is no calibrated negative control for unknown analyte. In order to differentiate the signal from DOX against any other molecule potentially causing PL modulation, we measure the characteristic fingerprint Raman spectrum of DOX. Fig.\\ref{fig:fig1}(b) shows the Raman spectrum of DOX\/DMSO solution (cyan curve). However, DOX Raman lines (highlighted by red arrows) are mixed, superimposed and even obscured with DMSO (background) response (cyan arrows). Furthermore, the line intensity of analyte would be comparable to background even at a relatively high DOX concentration. On contrary, when deposited on graphene surface, most of DOX lines become clearly visible, due to a significant GERS enhancement of the Raman signal of DOX (compare red and cyan curves). Table \\ref{table:tableGERS} summarizes the amount of signal enhancement for particular lines. In our sample with only two substances, the intensity of fingerprint lines of DOX already allows to confirm the analyte structure and determine the presence of analyte (which cannot be found from a PL data channel alone). While in general, for an agnostic biosensor, the whole Raman spectrum should be analyzed by machine learning correlation analysis of the data. Here, GERS, the second data channel, complements the PL detection which can provide information of the concentration of the drug (while the intensity of the GERS signal depends on enhancement factors and cannot be used to measure the amount of analyte). \n\n\n\n\n\n\n\n\n\\begin{figure*}[!]\n\t\\centering\n\t\\includegraphics[width=\n\t\\textwidth]{fig02_v.jpg\n\t\\caption{Local PL characterization of MLG\/MoS$_2$ heterostructure. \n\t\n\t\t(a) Single-point PL spectra of the MoS$_2$ island in (e). (inset) Total PL intensity map; stars show locations for the point spectra of the same color in main panel.\n\t\t(b) Correlation plots and (c-d) partial distribution functions for peak position and width for A-exciton (red) and trion (orange) lines, measured locally; several clusters are visible in trion data, highlighted by ovals in correlation plot and Gaussian envelope curves in distributions.\n\t\t(f-i) Confocal maps of MoS$_2$ PL: (top row) fitted intensity and (bottom row) peak position for (left) trion and (right) A-exciton; arrows show regions of higher PL intensity for trion (lower for A-exciton). \n\t\tAll scale bars are 1 $\\mu$m.}\n\t\\label{fig:fig2}\n\\end{figure*}\n\n\nAs Fig.\\ref{fig:fig1}(b) shows, several DOX lines are superimposed with the Raman spectrum of graphene (gray curve corresponds to MLG reference), specifically with D- and G-lines near 1350 and 1600 cm$^{-1}$. While obscuring some of the DOX modes, Raman spectra of graphene should be analyzed separately, yielding yet another channel, to be multiplexed with the PL and GERS data. Fig.\\ref{fig:fig1}(e-f) shows pronounced red-shift and the width increase for two major lines of graphene, G- and 2D-band, upon interaction with the DOX analyte (red). Panels (g-l) show detailed statistical information on modulation of both line position and width for both modes; in contrast with previous optical data, each data point in this figure corresponds to a small local region on the sample, less than 0.1 $\\mu$m$^2$, diffraction limited. Clearly, the data points aggregate in two separate clusters, though, point-to-point variability due to non-uniformity of the signal is non-negligible for 2D-mode (compare $\\Delta\\gamma\/\\Delta\\omega$ correlation plot in panel (h) and partial distribution functions in panels (k-l)). Statistical distribution of the data from (g-l) contains important information about the material\/sample, which will be elaborated in detail next.\n\n\n\\subsection*{Stability of 2D van der Waals heterostructure materials}\n\nElectron microscopy of MoS$_2$\/graphene vertical heterostructure, fabricated as described in Methods, reveals structural non-uniformities. A few typical images of several randomly selected single layer MoS$_2$ islands, coated with MLG, are shown in Fig.\\ref{fig:fig12}(a,e) and Fig.\\ref{fig:fig2}(e). White nanocrystallites, likely made of insulating molybdenum oxide, charged under e-beam, are seen either in the center of the island (metal nucleation site) or at the edge (metal precipitation site); in some cases those grow to microcrystals of Mo$_2$O$_3$ (see Fig.\\ref{fig:fig12}(e)) of characteristic triangular (or rectangular, not shown here) shape and size up to 1\/2 micrometer. Graphene seems to be conformal to the substrate, making short wrinkles between nanoscale posts (10-20 nm tall). \n\n\n\nWhile the surface of MoS$_2$ islands appears mostly uniform in scanning electron microscopy (SEM) image, optical properties of 2DM demonstrate substantial variation in agreement with Raman and PL statistics from Fig.\\ref{fig:fig1} and Fig.\\ref{fig:fig2}. The variability of PL in pristine material could produce uncertainty in detection of the analyte. In order to find the origin for such a variation, scattering scanning near-field optical microscopy (sSNOM) has been applied. Careful alignment of large area scans of the same heterostructure allows us to correlate different characterization channels (including SEM, scanning probe imaging, as well as PL and Raman microscopy, having a lower resolution though). In Fig.\\ref{fig:fig12}(b-d) the \nsSNOM image (2nd harmonic optical amplitude, see Methods for details) reveals variation of surface impedance of MLG\/MoS$_2$ heterostructure at the sub-micrometer scale, not captured by SEM (or AFM). We argue that a series of bright regions (on the darker background of MoS$_2$) correspond to the local defects of the TMDC material. Indeed, we regularly observe such a contrast at the edge of the island which is known to be prone to partial oxidation. Similar regions in the bulk of the island should correspond to concentrated sulfur vacancies, reactive to oxygen, and formation of oxy-sulfate regions, often appearing as nanoscale posts that ruckle graphene around (a near-field phase image in Fig.\\ref{fig:fig4}(h) has the best contrast which allows to resolve nano-posts). The series of maps in Fig.\\ref{fig:fig12}(b-d) and (f-h) show evolution of such regions protected (or non-protected) by graphene coating: the larger island (a), covered with intact MLG, preserves the same number of partially oxidized regions after nearly 2 years in ambient, except for a small oxide crystal grown in the bottom right corner, where a trench in graphene (dark line) opens an access to the air. On contrary, the small island (e) has the MLG coating cracked; as a result, the surface is slowly oxidized over the course of retention period, almost entirely on the map in panel (h). sSNOM mapping also shows that the large graphene wrinkles (bright diagonal lines in panel (d)) do not lead to alteration of optical properties. On the opposite, the oxy-sulfate regions will be shown to generate non-uniform doping of the MoS$_2$ and (graphene), leading to the PL variability over the sample.\n\n\n\n\n\n\nLocal fluctuations of PL in the pristine material were analyzed in another island of the same 2DM vertical heterostructure mapped by SEM in Fig.\\ref{fig:fig2}(e) and in Fig.\\ref{fig:fig3}(b) and Fig.\\ref{fig:fig4}(h) by sSNOM. Several features are clearly resolved: graphene ruptures (not reaching the island), an oxide crystallite at the edge of the island, a few oxy-sulfate nano-posts and graphene wrinkles around the posts, and several regions of darker SEM contrast (likely, more conductive than bare MLG), potentially indicating doping\/Fermi level variation. Confocal PL image of the same area is presented in Fig.\\ref{fig:fig2}(a), inset. The large non-uniformity of PL intensity is followed by substantial variability of PL line shape (cf. the curves in main panel taken at three locations shown in the inset). Similar to the large area PL data in Fig.\\ref{fig:fig1}(c), the main variability of micro-PL results from the A and X$^-$ states, to be analyzed separately. Panel (b) presents the correlation plot for fitted PL peak position and width for A-exciton (red) and trion (orange) states by the local optical probe on the surface of MLG\/MoS$_2$ heterostructure shown in Fig.\\ref{fig:fig2}(e). MicroPL reveals large non-uniformity in optical signal. Trion partial distribution functions for both $\\Delta\\gamma$ and $\\Delta\\omega$ show 3 major clusters (highlighted by ovals in panel (b) and green curves in (c-d)), that correspond to the regions of heterostructure where materials properties are locally modulated. \n\n\nMaps in panels (f-i) show actual distribution of the peak position, $\\Delta\\omega$, and peak intensity, $\\Delta P$, with diffraction limited resolution. Importantly, the intensity maps show the anti-correlation for PL strength of A-exciton and trion (as indicated by red and orange arrows): the trion PL is the highest where the A-exciton PL is depressed, compare locations for trion-dominated (blue\/black) and exciton-dominated (purple) PL curves in panel (a). Such a correlation may result from non-uniform doping of the MoS$_2$ island. Indeed, in a highly-doped area the neutral excitons are bound to free charges and, thus, converted into trions\\cite{Mouri2013}. \n\n\n\n\n\n\\subsection*{Multidimensional characterization of heterostructure materials}\n\n\nAlthough useful to shed the light on the PL variability, the confocal PL characterization neither has enough spatial resolution nor enables assessing the MoS$_2$ doping level to uncover the mechanisms of non-uniform optical signaling. Instead, we developed a multidimensional imaging combining sSNOM and Kelvin probe force microscopy (KPFM) to be correlated with PL (and Raman) microscopy. In Fig.\\ref{fig:fig3}(a-b) two maps of the same island -- using KPFM (work function) channel and sSNOM (optical surface impedance) channel -- show identical contrast, further detailed in panel (c) where the cross section profiles allow to quantify the variation of the Fermi level of graphene above the MoS$_2$ layer. The profile of work function is schematically shown in Fig.\\ref{fig:fig3}(h). Charge transfer in the vertical heterojunction decreases the carrier density in both graphene and MoS$_2$ underneath, thus, decreasing the magnitude of graphene work function and doping level. The KPFM probe is in contact with the outermost layer of the heterostructure, graphene, thus it measures the work function of MLG. Graphene above the island appears negatively doped by MoS$_2$. The MLG Fermi level, taken with respect to graphene Dirac point, is negative, corresponding to p-doping. Statistical distribution of the Fermi level values of graphene on\/off the island is shown in panel (f) by red\/green histogram. Knowing $E_F$ in bare graphene and in the vertical heterostructure allows us to calculate the MoS$_2$ doping level, Fig.\\ref{fig:fig3}(d). Using median values for $E_F$, it can be estimated to lie in the range $1-25\\times10^{12}$~cm$^{-2}$, which is further corroborated by independent Raman data below.\n\n\n\n\n\\begin{figure}[!]\n\t\\centering\n\t\\includegraphics[width=0.4\n\t\\textwidth]{fig13_v.jpg\n\t\\caption{Correlation of MLG work function data with sSNOM optical surface impedance. Aligned maps for (a) KPFM and (b) sSNOM (4th harmonic) amplitude. (c) Cross section profiles across the MoS$_2$ area (KPFM, red and sSNOM, pink) vs.MLG reference (KPFM, gray), taken along the lines of the same color in (a-b). (d) Calculated electron density in MoS$_2$ heterostructure, log-scale, vs. Fermi levels in bare\/doped graphene off\/on the island. (f-g) Partial distribution functions for measured $E_F$ in bare graphene (off island, green) and graphene doped by the MoS$_2$ (on island, red) from KPFM map in (a). (e) Partial distribution functions for sSNOM signal from (b) to calibrate near-field signal by $E_F$. Note common abscissa axis for panels (d,f), not (e). Inset (h) shows schematics of charge transfer in the vertical heterojunction on SiO$_2$ substrate with negative charge traps. Pink curve outlines the variation of MLG work function.\n\t\tAll scale bars are 1 $\\mu$m.\n\t\n\t\t}\n\t\\label{fig:fig3}\n\\end{figure}\n\n\nComparison of KPFM and sSNOM profiles in Fig.\\ref{fig:fig3}(c), as well as the distribution functions in Fig.\\ref{fig:fig3}(f-e), allows to calibrate the near-field signal in terms of the Fermi level of the heterostructure. Then, one could interpolate the charge transfer\/doping data to the nanometer features, only resolved by sSNOM (such as wrinkles, oxy-sulfate regions, etc.), and thus, determine the origin for PL non-uniformity. \n\nEnhanced resolution of sSNOM allows us to determine 5 sources of non-uniform doping in the vertical van der Waals heterostructures as shown schematically in Fig.\\ref{fig:fig3}(h). (i) The primary doping is defined by conditions of the MoS$_2$ synthesis: it is known that often the stoichiometry of TMDC is slightly off the equilibrium values. Deficiency in sulfur leads to creation of surface vacancies, typically resulting in n-doping\\cite{Komsa2015}. (ii) Filling of the S-vacancy with oxygen or CH-group yields weaker n- or p-doping\\cite{Mouri2013,Zheng2020}, which was shown to be localized near the defect site\\cite{Nan2014}. (iii) In Mo-abundant synthesis, small micro-crystallites of metal molybdenum form, later oxidized to MoO$_x$, or forming MoO$_x$S$_y$ domains. (iv) In the heterostructure, work function and\/or Fermi level difference between the layers results in charge transfer between the layers. Typically p-doped MLG would become an acceptor for electrons transferred from n-doped MoS$_2$. Finally, (v) the Si\/SiO$_2$ substrate supports the heterostructure, which is known to have a high density of traps at the interface. Such traps, if charged, produce a substantial field and shift of the Fermi level in all 2DM layers above it, generating a random Coulomb potential for charge carriers both in MoS$_2$ and graphene. \n\nAdditional evidence for the existence of defects\/vacancies in TMDC lattice is provided by high-angle annular dark-field (HAADF) Scanning Transmission Electron Microscopy (STEM) imaging. Fig.\\ref{fig:fig4}(e) shows atomic resolution map of a typical MoS$_2$ island. The boundary between dark area and the lower contrast area likely reflects the grain boundary which separates regions of different lattice orientation. Such a twin boundary produces strain and may result in localization of electronic states. Furthermore, several (3-fold) individual defects are seen in the STEM image (approximately half a dozen per 200 nm$^2$ which corresponds to ca. $3\\times10^{12}$~cm$^{-2}$).\n\n\n\n\n\n\n\n\\begin{figure*}[!]\n\t\\centering\n\t\\includegraphics[width=\n\t\\textwidth]{fig04-t.jpg\n\t\\caption{Raman mapping of doping and strain non-uniformity in the heterostructure. (a) Typical MoS$_2$ Raman spectrum, fitted with E$^1$ and A-lines. (b) A-line intensity map. (c) Typical Raman spectra for MLG off\/on MoS$_2$ island, fitted by G (orange), D' (pink) and 2D (green) lines; splitting of G- and 2D-lines is shown in the fit. (d) Raman map of 2D-amplitude showing the island location, cf. map in (b).\t\n\t\t(e) HAADF-STEM image of MoS$_2$ lattice: notice grain boundaries and individual defects; scale bar is 2 nm. \n(f,g) Calculated doping and strain for MoS$_2$ layer overlaid with SEM map; (h) sSNOM phase image of the same area; (i-k) MLG doping, hydrostatic and shear strain maps.\n\t\tAll scale bars, except in (e), are 1 $\\mu$m.\n\t\n\t}\n\t\\label{fig:fig4} \n\\end{figure*}\n\n\n\nMultiple sources of optical non-uniformity, stemming from the variation of the doping level, have been further studied with micro-Raman imaging: typical Raman spectra of MLG\/MoS$_2$ heterostructure are shown in Fig.\\ref{fig:fig4}(a,c). \nPanel (a) presents A- and E$^1$-modes of MoS$_2$ layer, A-intensity map is shown in inset (b). Mode frequencies, fitted as in (a), allow to determine the strain and doping\\cite{Rao2019} of the island underneath the graphene, see Methods, generating the maps presented in panels (f,g). Consistent with the KPFM data, Fig.\\ref{fig:fig3}(a), MoS$_2$ doping is lower along the vertical axis of the island, thus, both the amount of charge transfer and graphene $E_F$ should be lower. \nCharge doping and strain in graphene have been calculated using a similar procedure\\cite{Neumann2015,Mueller2017}. \nUpper\/lower curves in panel (c) correspond to MLG Raman lines off\/on TMDC, where the location of the island is clearly seen, {\\em e.g.}, in the map of 2D-amplitude (d). Fig.\\ref{fig:fig4}(i,j) show graphene doping and isotropic\/hydrostatic strain. Furthermore, the splitting of the G- and 2D-doublet modes (see the fitted curves in panel (c)) yields\\cite{Narula2012} the shear (non-isotropic) component of the strain, panel (k). \n\n\nHigh-resolution map reveals that the hole carrier density in graphene increases next to the location of a large MoO$_x$ crystallite, which should indicate additional chemical doping. Besides doping, all nanoscale features of heterostructure morphology make contributions to the uniform and non-uniform components of graphene strain, thus making Raman line width larger than the natural width\\cite{Neumann2015}, due to the statistical broadening.\n\n\n\n\n\n\\subsection{Outline}\n\\label{sec:conclusions} \n\nCumulatively, multidimensional characterization data above revealed existence of non-uniformities in 2D materials at the nanoscale and allowed to identify doping and\/or strain variations as the origin of statistical distribution of the optical signals used in all three recognition channels (PL shift, Raman spectroscopy and GERS). When integrated over the device area, such a variability in local response would translate in a broadening of the biosensing spectral signal, thus, raising device-to-device variability and, ultimately, lowering the sensitivity and the limit-of-detection by increasing background and\/or systematic error. While the variability of individual device response often could be addressed by careful calibration against known analytes, such a fluctuation and spread of the integrated response would affect biosensing accuracy and, certainly, reduce the ability to perform precise biosensing in the agnostic detection mode. Presented study suggests that in order to improve the performance of biosensors based on 2DM heterostructures, non-uniformity of doping and strain -- two major mechanisms for optical signal variation -- must be addressed. Currently, most of 2DM heterostructures are fabricated by transfer methods, that are known to produce both strain and doping\\cite{Leong2019,Bousige2017,Banszerus2017} (especially for wet transfer). New methods of strain-free and doping-free transfer need to be developed\\cite{Leong2019,Seo2021}. Alternatively, such heterostructure materials should be fabricated in-situ, in synthetic facility, to preserve the layer epitaxy and exclude contamination between the layers.\n\n\n\\section*{Methods}\n\n\\subsection{Sample fabrication} \nThe monolayer MoS$_2$ was grown on a Si substrate with 300 nm thick SiO$_2$ by Chemical Vapor Deposition (CVD) method as described in\\cite{Aryeetey2021}. Optimization of synthesis parameters and stoichiometric ratio of molybdenum to sulfur resulted in producing triangular MoS$_2$ islands with low defect density (cf. STEM image in Fig.\\ref{fig:fig4}(e)), predominantly single layers, with low surface coverage. Monolayer graphene was grown by CVD on Cu foil. MLG was transferred onto MoS$_2$ using the conventional PMMA assisted transfer technique\\cite{Gao2012}. The SEM image of resulted heterostructure is shown on Fig.\\ref{fig:fig2}(e).\n\n\n\\subsection{Sample Characterization.}\nSEM sample imaging was performed in a field emission scanning electron microscope Zeiss Auriga FIB\/FESEM. Atomic resolution images of monolayer MoS$_2$ samples transferred onto Quantaifoil TEM grids were recorded using Nion Ultra HAADF-STEM operating at 60 kV with 3rd-generation C3\/C5 aberration corrector and 0.5 nA current in atomic-size probe $\\sim 1.0 - 1.1$\\AA~ (NCATSU).\nConfocal PL and Raman characterization were performed using a Horiba Jobin Yvon LabRAM HR-Evolution Raman system, 488 nm (for Raman) and 532 nm (for PL) laser excitation wavelengths were used; Horiba XploRA Raman system was used for taking Raman spectra at 532 nm of excitation. Analysis of PL and Raman characterization was performed using home-written codes. \n \nsSNOM maps were collected using scattering type scanning near-field optical microscope (custom-built Neaspec system) in pseudo-heterodyne mode (tapping amplitude $\\sim$70 nm, ARROW-NCPt probes by Nanoworld $<$25 nm radius), excitation by CW Quantum Cascade Laser (MIRCat by Daylight) at power $<$ 2 mW in focal aperture at 1577-1579 cm$^{-1}$ (6.333-6.341 $\\mu$m). Amplitude and phase of high order harmonics ($\\ge 2$) are proportional to the local impedance of the sample under the tip.\n\nThe AFM\/KPFM was performed using Dimension Icon AFM in PeakForce Kelvin Probe Force Microscopy in frequency modulated mode (PFKPFM-FM, Bruker Nano Inc., Santa Barbara, CA) utilizing a PFQNE-AL probe (Bruker SPM Probes, Camarillo, CA). Prior to measuring the samples, the KPFM response of the probe was checked against an Au-Si-Al standard and the work function of the Al reference metal layer was calibrated against a freshly cleaved highly oriented pyrolytic graphite (HOPG) reference sample (PFKPFM-SMPL, HOPG-12M, Bruker SPM Probes, Camarillo, CA); 4.6~eV was used for the work function reference value for HOPG.\n\n\n\\subsection{Supporting Information} \\par \nSupporting Information is available from the Wiley Online Library or from the author.\n\n\\section*{Acknowledgments:} \n\t\nAuthors are personally thankful to Drs. T. Tighe, T. Williams and M. Wetherington (MCL, PSU). S.V.R. acknowledges NSF support (CHE-2032582). T.I. and K.S. acknowledges NSF support (CHE-2032601). T.I. acknowledges Sample Grant from The Pennsylvania State University 2DCC-MI\n, which is supported by NSF cooperative agreement (DMR-1539916). Work at PSU sSNOM facility has been partially supported by NSF MRSEC (DMR-2011839). Part of this work was performed at the Joint School of Nanoscience and Nanoengineering (JSNN), a member of the Southeastern Nanotechnology Infrastructure Corridor (SENIC) and National Nanotechnology Coordinated Infrastructure (NNCI), which is supported by the NSF grant (ECCS-1542174). Scanning Transmission Electron Microscope imaging was conducted at the Center for Nanophase Materials Sciences at ONRL, which is a DOE Office of Science User Facility. \n\n\n\n\n\n\n\n\n\n\n\n\\section*{Supplementary Information}\n\n\n\n\\section{Strain and Doping analysis}\nThe background signal of Raman spectra for monolayer graphene (MLG) was fit and subtracted using air PLS \\cite{Zhang2010}. Then peaks were fit for both MLG and TMDC spectra using the non-linear least-squares minimization and curve-fitting library (LMFIT) for python. Peaks were fit using Lorentzian line shapes around the D, G$^+$, G$^-$, 2D$^+$ and 2D$^-$ peaks, as well as around nearby shoulder peaks if they were distinguishable. \n\n\\begin{figure*}[b]\n\t\\centering \n\t\\includegraphics[width=.5 \n\t\\columnwidth]{SI-01.jpg\n\t\\caption{G vs 2D plot showing the split of graphene on the bare SiO$_2$ substrate (light blue) and over MoS$_2$ island (green). Red\/yellow line indicates a characteristic slope for G-2D data correlation caused by pure doping\/strain (isotropic biaxial).}\n\t\\label{fig:SI-01}\n\\end{figure*}\n\nThe initial separation of strain and doping is accomplished by examining the central peak position of the 2D and G line fits. These Raman frequencies are sensitive to both strain and doping because of the change in lattice constants and force fields that effect the phonon frequencies. Lee et al. created a procedure for extracting the strain and doping of graphene through the statistical analysis of the changes in the Raman frequency position \\cite{Lee2012}. By plotting the 2D and G peaks against each other we are able to see trends in the spectra which represent modulation either by strain or by doping, or both. Strain is seen in the MLG Raman data as a cluster in of the 2D\/G correlation plot (Figure \\ref{fig:SI-01}) with a linear slope of approximately 2.2. The linear slope for p-doped MLG is approximately 0.75 which is also seen in Figure \\ref{fig:SI-01}. At very low values of p-doping and n-doping the dependence should be nonlinear, though, due to Fermi velocity (density of states) renormalization. \n\nIt is known that graphene on MoS$_2$ and SiO$_2$ substrate is typically p-doped. Assuming linear correlation with Raman frequencies, we can extract the relative change in strain and doping by solving the linear equation system:\n\\begin{eqnarray}\n\t\\left(\\begin{array}{l}\n\t\t\\omega_G\n\t\t\\\\ \\\\\\omega_{2D}\n\t\\end{array}\\right)\n\t=\n\t\\begin{array}{|ll|}\n\t\ta_{G,\\varepsilon}\\qquad & a_{G,\\rho}\n\t\t\\\\ &\\\\\n\t\ta_{2D,\\varepsilon} & a_{2D,\\rho}\n\t\\end{array}\n\t\\left(\\begin{array}{l}\n\t\t\\varepsilon\n\t\t\\\\ \\\\\n\t\t\\rho\n\t\\end{array}\\right)\n\t\\label{SI-01}\n\\end{eqnarray}\nWhere the vector ($\\omega_{G}$,$\\omega_{2D}$) should be calibrated against unstrained and undoped graphene reference sample. \n\nGraphene has two different polarizations of optical modes that are degenerate at zero strain. Depending on the axial direction of (uniaxial) strain, position of one of the modes shifts with respect to the other one. This generates a Raman doublet for general strain. Knowing a particular strain configuration is only possible with Raman mapping in polarized light, which resolved the polarization of a phonon mode. However, even in the case of non-polarized Raman data, position of individual components of the doublet allows to separate the isotropic and anisotropic components of the strain. The latter corresponds to the shear strain, although in order to determine specific shear direction, a polarized spectroscopy will be required, also on a calibration sample with know lattice orientation.\n\n\n\n\\begin{figure*}[!]\n\t\\centering \n\t\\includegraphics[width=1 \n\t\\columnwidth]{SI-02.jpg\n\t\\caption{Main figure reproduces the data from Fig. 5c: the dots and filled line represent the experimental data and the total fitted curve. The individual components of the doublet are shown with the thin lines. Additional D'-component is needed for fitting the spectrum in vicinity of the G-doublet for bare graphene. (inset) Schematics of G-line splitting with the shear strain. Hydrostatic strain, on contrary, shifts the whole doublet but does not influence the splitting.}\n\t\\label{fig:SI-02}\n\\end{figure*}\n\nMueller et al. developed a formalism to separate the doping and hydrostatic strain and shear strain components\\cite{Mueller2017} . Critically, the shear strain component does not change the strain\/doping correlation, that is, the slope of the curves in Figure \\ref{fig:SI-01}. While the hydrostatic component does not affect the splitting of the 2D or G peaks into doublet, as it shown on Figure \\ref{fig:SI-02}. The amount of the splitting allows us to determine the magnitude of the shear strain, while the magnitude of the hydrostatic strain can be determined by examining the averaged peak position (after splitting). We can then determine the magnitude of the strain components and the doping by examining the shift of the peaks in a \"zero strain\" case or a \"zero doping\" case. Parametrization follows the paper by Das et al.: in the undoped case the 2D peak shifts at a rate of 1.04 cm$^{-1}$ per 10$^{12}$ cm$^{-2}$ hole density \\cite{Das2009}. We use 2D splitting data and the Grueneisen parameter and the shear deformation potential from \\cite{Mueller2017} to determine the strain components from:\n\\begin{equation}\n\t\\omega_{2D}^{\\pm}= \\langle\\omega_{2D}\\rangle\\, \\left(-\\alpha\\,\\varepsilon_h\\pm \\beta\\,\\varepsilon_s\\right)\n\t\\label{SI-02}\n\\end{equation}\nwhere $\\alpha= 1.8$ is the Grueneisen parameter for MLG, and $\\beta= 0.99$ is the shear deformation potential. \n\n\n\n\n\\begin{figure*}[!]\n\t\\centering \n\t\\includegraphics[width=1 \n\t\\columnwidth]{SI-03A.jpg\n\t\\caption{The maps showing the fitted parameters for splitting of 2D peaks.}\n\t\\label{fig:SI-03A}\n\\end{figure*}\n\n\\begin{figure*}[!]\n\t\\centering \n\t\\includegraphics[width=1 \n\t\\columnwidth]{SI-03B.jpg\n\t\\caption{The maps showing the fitted parameters for splitting of G peaks.}\n\t\\label{fig:SI-03B}\n\\end{figure*}\n\nThe strain and doping of MoS$_2$ can also be determined from Raman correlation data (Rao et al.2019). Peaks for MoS$_2$ were fit in the same way as the graphene peaks (Figure \\ref{fig:SI-04}). For the case of MoS$_2$ we compare E peak and A peak that are near 382 and 404 cm$^{-1}$ respectively. The E peak position is more sensitive to strain, similar to the 2D peak of MLG, while the A peak position is more sensitive to doping, like the G peak of MLG. The slope for strain correlation is $\\sim$4; the slope for doping is $\\sim$0.12. We can then use the undoped E peak position, and a Gruneisen parameter for MoS$_2$ of $\\sim$0.86, to obtain the average strain. Then we examine the unstrained A peak position which shifts at a rate of 4 cm$^{-1}$ per 1.8~10$^{13}$ cm$^{-2}$ \\cite{Chakraborty2012} and determine the doping. Unlike graphene, the peak splitting in MoS$_2$ is absent. \n\n\\begin{figure*}[!]\n\t\\centering \n\t\\includegraphics[width=1 \n\t\\columnwidth]{SI-04.jpg\n\t\\caption{The maps showing the fitted parameters for MoS$_2$ peaks.}\n\t\\label{fig:SI-04}\n\\end{figure*}\n\n\n\n\n\\section{Characterization by Peak-Force Kevin probe force microscopy (KPFM)}\nIn general, the work function value $\\Phi_{sample}$ and, consequently, Fermi level variation can be calculated from KPFM measurements using an equation:\n\\begin{equation}\n\t\\Phi_{sample} = e \\, V_{CPD} - \\Phi_{probe}\t\n\t\\label{SI-03}\n\\end{equation}\nwhere $V_{CPD}$ is the contact potential difference between the sample and the AFM probe, $e$ is elemental charge, and $\\Phi_{probe}$ is the work function of the KPFM probe. Prior to measuring the MoS$_2$\/graphene samples, we checked the KPFM probe response against an Au-Si-Al standard. Topography and $V_{CPD}$ maps of standard are shown on Figure \\ref{fig:SI-04}a,b. Next, the work function of aluminum was calibrated against a freshly cleaved highly oriented pyrolytic graphite (HOPG) reference (Figure \\ref{fig:SI-04}c,d) using value of $\\Phi_{HOPG} =4.6$eV (PFKPFM-SMPL, HOPG-12M, Bruker SPM Probes, Camarillo, CA). To eliminate influence of water condensation the AFM chamber was purged with nitrogen gas.\n\n\n\n\\begin{figure*}[!]\n\t\\centering \n\t\\includegraphics[width=1 \n\t\\columnwidth]{SI-05.jpg\n\t\\caption{(a) AFM topography and (b) KPFM signal for the Au-Si-Al standard. (c) AFM topography and (d) KPFM channel for HOPG reference sample.}\n\t\\label{fig:SI-05}\n\\end{figure*}\n\nThe topography of MoS$_2$\/graphene sample (Figure \\ref{fig:SI-05}a) is dominated by roughness of Si\/SiO$_2$ substrate, and since the thickness of MLG and monolayer MoS$_2$ is below RMS, topographical details of heterostructures cannot be resolved by routine AFM imaging. The spatial distribution of $V_{CPD}$ and calculated work function value of the same area are presented on Figure \\ref{fig:SI-05}b,c. It must be noted that the KPFM probe is in contact with the outermost layer of the heterostructure, graphene, thus it measures the work function of MLG, $\\Phi_{MLG}$, either on or off the MoS$_2$ island. The work function value \"off\" the island reveals p-doping of graphene, likely due to transfer procedure. The Fermi level value, taken with respect to graphene Dirac point, is negative. The \"on\" value is shifted towards the Dirac point, showing non-uniform n-doping effect, originated from the charge transfer in the heterostructure.\n\n\\begin{figure*}[!]\n\t\\centering \n\t\\includegraphics[width=1 \n\t\\columnwidth]{SI-06.jpg\n\t\\caption{The maps of a MoS$_2$\/graphene sample: (a) AFM topography, (b) KPFM, and (c) calculated work function distribution.}\n\t\\label{fig:SI-06}\n\\end{figure*}\n\n\n\n\n\n\\section{Calculation of charge density in MLG and in MoS$_2$ monolayer}\n\\label{app:dos-eqs}\n\nFor 2d-materials with parabolic dispersion relation (with massive fermions), like MoS$_2$, the energy is given by: $E=E_c+\\hbar^2k^2\/(2m^*)$. Then, the density of states (DOS) is constant for each band: $=2m^*\/(\\pi\\hbar^2)$. Then, the following integral gives the carrier density dependence on the Fermi level (spin and valley degeneracy included):\n\\begin{equation}\n\tn(F)=\\frac{2m^*}{\\pi\\hbar^2}\\int_{E_c}^\\infty \\frac{dE}{1+\\exp[\\frac{E\n\t\t\tF}{kT}]}=\\frac{2m^* kT}{\\pi\\hbar^2}\\log\\left(1+\\exp\\left[\\frac{F-E_c}{kT}\\right]\\right)=N_c \\, \\log\\left(1+\\exp\\left[\\frac{|E_c|-|F|}{kT}\\right]\\right)\n\n\t\\label{2d-DOS-MoS2}\n\\end{equation}\nwhere we assume that both Fermi level $F$ and $E_c=-4.21$~eV\\cite{Larentis2014} are taken with respect to the vacuum level and, thus, are negative (this definition is consistent with the definition for Dirac point $E_D$, conduction band edge $E_c$ \nand Fermi level $F\n. The conduction band DOS is given by:\n\\begin{equation}\n\tN_c=\\frac{2m^* kT}{\\pi\\hbar^2}=\\frac{2m^*}{m_o}\\frac{kT}{\\pi a_B^2 \\; E_B}\\simeq 7.6\\,10^{12}\\,\\mathrm{cm}^{-2}\n\t\\label{Nc-MoS2}\n\\end{equation}\nwith $m_o$ being the free electron mass, $a_B=0.53$~\\AA, $E_B=27$~eV, and effective mass in MoS$_2$ is taken to be 0.35$m_o$\\cite{Peelaers2012}.\n\n\n\n\nThere are two limits to be noted: for non-degenerate doping ($|F|>|E_c|$, Fermi level lies below the bottom of CB), one can use $\\log(1+x)\\sim~x$ and write:\n\\begin{equation}\n\tn\\simeq N_c \\, \\exp\\left[-\\frac{|F|-|E_c|}{kT}\\right]\n\t\\label{2d-DOS-MoS2-nondeg}\n\\end{equation}\nwhile in the degenerate doping limit ($|E_c|-|F|\\gg kT>0$, Fermi level is within the CB), unity is neglected compared to the large exponential, and we derive linear dependence of the charge density on the Fermi level:\n\\begin{equation}\n\tn\\simeq N_c \\frac{|E_c|-|F|}{kT} \n\t\\label{2d-DOS-MoS2-ndeg}\n\\end{equation}\n\nCorrespondingly for the monolayer graphene, which is gapless with a linear dispersion relation $E=\\hbar v_F k$, we derive:\n\\begin{equation}\n\tn_g(F)=\\frac{(E_D-F)^2}{\\pi\\hbar^2 v_F^2}=N_g\\,(E_D-F)^2\n\n\t\\label{2d-DOS-MLG}\n\\end{equation}\nwhere the Dirac point $E_D=\\chi_{MLG}\\simeq -4.57$~eV\\cite{Yu2009}, and Fermi velocity $v_F\\simeq 1.16\\times10^6$~m\/s\\cite{Knox2008}. We emphasize that $N_g$ is not a density of carriers, neither it is a 2d-DOS in a classical sense: $N_g\\simeq 5.46\\;10^{13}$~cm$^{-2}$~eV$^{-2}$.\n\n\n\n\n\n\\begin{figure*}[!]\n\t\\centering \n\t\\includegraphics[width=0.5 \n\t\\columnwidth]{SI-07.jpg\n\t\\caption{Matching band structure offsets in MoS$_2$\/graphene van der Waals heterojunction: in order to align the Fermi level, the charg transfer between the 2D-materials must happen, resulted in a drop of vacuum level between the layers.}\n\t\\label{fig:SI-07}\n\\end{figure*}\n\n\n\n\n\nSince the 2d materials are electrically isolated from the Si substrate by the oxide layer, they are at floated potential and the charge transfer produces 2D charge densities $\\pm en_1$, equal (by magnitude and opposite by sign) in both layers, and generates $2\\delta V$, a potential difference between TMDC and MLG ($\\phi(z\\pm d\/2)=\\pm\\delta V$). This potential difference is linearly proportional to the surface charge formed at each of the materials, as the result of charge transfer.\n\nThen, the positions of the Fermi levels, both defined with respect to the higher vacuum level in MLG, are:\n\\begin{equation}\n\t|F_g| =|F_g^{(o)}| -\\Delta_F \\qquad\\qquad\n\t|F_{MoS_2}|=|F_{MoS_2}^{(o)}|+2\\delta V +\\Delta_F{}_{MoS_2}\n\t\\label{levels-03}\n\\end{equation}\nwhere the differences: $\\Delta_F=F^{(o)}_g-F>0$ is the Fermi level (up)shift in graphene, which can be measured as work function difference taken on and off the TMDC island, and $\\Delta_F{}_{MoS_2}$, the Fermi level (down)shift in MoS$_2$.\n\nKnowing the expression for TMDC and MLG DOS, one can easily calculate the charge transfer and, then, the potential difference between the layers in the vertical heterostructure. Thus, the relation between the measured MLG work function and the doping level of TMDC can be established, as shown in the Fig. 4d of main text.\n\n\n\n\n\\section{Animation File}\n\\label{app:movie}\n\nSI includes an animation file representing some of the channels of multidimensional characterization of a particular van der Waals vertical heterostructure (single layer MoS$_2$ packaged by graphene monolayer). The same area of the sample has been imaged using different instrumentation. Theoretical results on the charge transfer (doping) and various components of strain are also shown.\n\n\n~\\newpage~\n\\section*{References}\n\\label{app:ref}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction\nEver since the discovery of a class of objects with extremely bright infrared emission called Ultra\/Luminous Infrared Galaxies (LIRG: $L_{\\rm{FIR}}$\\ = 10$^{11-12}$$L_{\\sun}$; ULIRG: $L_{\\rm{FIR}}$\\ $>$10$^{12}$$L_{\\sun}$) with the \\textit{The Infrared Astronomical Satellite} \\citep[$IRAS$; e.g.][]{Houck1984,Houck1985,Soifer1984b,Soifer1984}, these objects have been studied at every wavelength. High-resolution optical imaging show that most of these systems resemble merging\/interacting systems \\citep[e.g.][]{Murphy1996}. \\cite{Larson2016} showed that major mergers play a significant role for all sources with $L_{\\rm{IR}}$\\ $\\geq$ 10$^{11.5}$$L_{\\sun}$. In addition, optical wavelengths show that these systems are highly dust obscured. Various star formation tracers have shown extreme star formation rates \\citep[e.g.][]{U2012} suggesting that local U\/LIRGs are great laboratories to study extreme modes of star formation. Mid-infrared observations are used to reveal the contribution of an active galactic nucleus (AGN), if present, to the far infrared luminosity \\citep[e.g.][]{Genzel1998,Veilleux2009}. Molecular gas observations via carbon monoxide ($^{12}$CO) have revealed rich concentrations of fuel for current and future star formation \\citep[e.g.][]{Downes1998,Bryant1999,Wilson2008} and also massive, energetic molecular outflows on kiloparsec scales in several sources \\citep[e.g.][and references therein]{Cicone2014}. \n \n\\citet{Gao2004b} used HCN~$J$~=~1$-$0\\ observations to show that there exists a tight correlation between infrared luminosity ($L_{\\rm{IR}}$; proxy for star formation rate) and HCN~$J$~=~1$-$0\\ (proxy for amount of dense molecular gas). This tight relation was extended to span over 7-8 orders of magnitude of $L_{\\rm{IR}}$\\ by \\citet{Wu2005}. These results are interpreted to mean that HCN~$J$~=~1$-$0\\ is tracing the molecular gas component that is directly related to the star formation.\n\n\nSingle dish observations of the $^{12}$CO\\ isotopologue, $^{13}$CO\\ \\citep[e.g.][]{Casoli1992,Garay1993,Aalto1991,Aalto1995,Greve2009,Papadopoulos2012a} in U\/LIRGs have shown a trend of extremely weak emission relative to $^{12}$CO. The integrated brightness temperature line ratios of $^{12}$CO\/$^{13}$CO\\ in U\/LIRGs were found to be usually high ($>$20) when compared to normal disk galaxies suggesting different interstellar medium (ISM) environments between the two classes of galaxies. High-resolution observations \\citep[e.g][]{Aalto1997,Casoli1999,Sliwa2012,Sliwa2013,Sliwa2014} show a similar trend with values evening exceeding 50 for some sources \\citep{Sliwa2017c}. Possible explanations for the unusually high ratios include, photodissociation of $^{13}$CO, excitation and\/or optical depth effects, or abundance variations \\citep[e.g][]{Casoli1992,Henkel1993,Taniguchi1999}. While photodissociation is likely ruled out as the dominant process due to the strong C$^{18}$O\\ emission \\citep[see][]{Casoli1992}, recent radiative transfer studies are showing that the [$^{12}$CO]\/[$^{13}$CO]\\footnote{Square brackets denote abundances while no brackets around ratios denote brightness temperature line ratios unless specifically stated} abundance ratio is higher ($\\geq$100) than what we perceive as normal \\citep{Sliwa2013,Sliwa2014,Sliwa2017a,Papadopoulos2014,Henkel2014,Tunnard2015b}. In this paper, we model the molecular gas of the well studied ULIRG Arp 220 to determine whether it follows the [$^{12}$CO]\/[$^{13}$CO]\\ trend seen in the literature.\n\n\n\\textit{Arp 220:} \\object{Arp 220} (IRAS 15327+2340, UGC 9913, VV 540, IC 1127) is the nearest example of a ULIRG and thus well studied at every wavelength. In this advanced merger, the two nuclei are still distinguishable, separated by 1$^{\\prime\\prime}$\\ \\citep[$\\sim$390~pc;][]{Norris1988} and is modelled to finish merging within 6~$\\times$~10$^{8}$~years \\citep{Konig2012}. With one of the most powerful star-forming environments in the local universe, it is a popular starburst template for dusty, star forming high redshift galaxies. Recently, \\citet{Barcos-Munoz2015} observed 33~GHz continuum at very high-resolution ($\\sim$30~pc) revealing that synchrotron radiation is dominant at this frequency. Combining the sizes measured from the 33~GHz continuum and infrared observations, \\citet{Barcos-Munoz2015} derived very high molecular gas surface densities ($>$2~$\\times$~10$^{5}$~$M_{\\sun}$~pc$^{-2}$) and infrared surface luminosities ($>$4~$\\times$~10$^{13}$~$L_{\\sun}$~pc$^{-2}$). Although there is no clear evidence of an AGN with the 33~GHz continuum \\citep{Barcos-Munoz2015}, \\citet{Downes2007} showed that the dust in the western nucleus is hot ($\\sim$ 170~K) and the size of the dust source is small that it implies a large surface luminosity that can only be plausible by an AGN. \\citet{Wilson2014} used 691~GHz continuum to also show that the western nucleus has a very high luminosity surface density that requires either the presence of an AGN or a \"hot starburst\". \\citet{Lockhart2016} were able to resolve previously observed H$\\alpha$+[NII] emission into a bubble-shaped feature that is aligned with the western nucleus. Either an AGN or extreme star formation within the inner $\\sim$100~pc of the nuclei are the likely possibilities for the origin of this bubble. \\cite{Zsch2016} confirm an outflow from the western nucleus by comparing their high-resolution Very Large Array (VLA) data of several molecular species to the ALMA data.\n\n\nOver the last two decades, Arp 220 has been observed in high-resolution CO many times \\citep{Scoville1997,Downes1998,Sakamoto1999,Downes2007,Sakamoto2008,Matsushita2009,Martin2011,Konig2012,Rangwala2015,Scoville2017}. The observations reveal a large concentration of molecular gas ($\\sim$~5~$\\times$~10$^{9}$~$M_{\\sun}$; e.g. Downes $\\&$ Solomon 1998) within two compact nuclei surrounded by a diffuse kiloparsec-scale disk. Observations of rare CO isotopologues (i.e. $^{13}$CO\\ and C$^{18}$O) have been mainly obtained using single dish telescopes resulting in global fluxes \\citep[e.g.][]{Greve2009,Papadopoulos2012a}; however, \\citet{Matsushita2009} published high-resolution Submillimeter Array (SMA) $^{13}$CO\\ and C$^{18}$O $J$=2$-$1\\ ($\\sim$ 3.5\\arcsec) observations showing that the two isotopologues have similar intensities where they suggest that either the lines are optically thick or there is an overabundance of C$^{18}$O\\ compared to $^{13}$CO. \n\nUsing $Herschel$ Fourier Transform Spectrometer (FTS) spectra, \\citet{Rangwala2011} modelled the global $^{12}$CO\\ emission from $J$ =1-0 to $J$ = 13-12 with a 2-component molecular gas: cold, moderately dense (\\tkin\\ = 50~K and $n_{\\rm{H_{2}}}$\\ = 10$^{2.8}$~cm$^{-3}$) and warm, dense (\\tkin\\ = 1350 K and $n_{\\rm{H_{2}}}$\\ = 10$^{3.2}$ cm$^{-3}$) molecular gas components. The dust continuum from the FTS spectrum was also shown to be consistent with a warm dust ($T_{\\rm{dust}}$ = 66~K) with a large optical depth ($\\tau_{\\rm{dust}}$ $\\sim$ 5 at 100~$\\mu$m; Rangwala et al. 2011). \n\nDense gas tracers such as HCN, HCO$^{+}$, HNC and CS have also been observed in Arp 220 \\citep[e.g.][Barcos-Mu\\~noz et al. in preparation]{Aalto2009,Aalto2015b,Greve2009,Sakamoto2009,Imanishi2010,Scoville2015,Martin2011,Martin2016,Tunnard2015a}. \\cite{Sakamoto2009} reported P-Cygni profiles in the HCO$^+$~$J$~=~3$-$2\\ and $J$~=~3-2 suggesting a $\\sim$100~km s$^{-1}$\\ outflow originating from the inner regions of the nuclear disks. \\cite{Aalto2009} found that HNC $J$~=~3-2 is bright and, perhaps, amplified in the western nucleus and weak in the east suggesting very different physical conditions. \\cite{Martin2016} find that HCN and HCO$^+$~$J$~=~4$-$3\\ and $J$~=~3-2 are optically thick and affected by different absorption systems that can hide up to 70$\\%$ of the total intrinsic emission from these lines. \n\n\\textit{NGC 6240:} \\object{NGC 6240} (IRAS 16504+0228, UGC 10592, VV 617, IC 4625) is unusual among the class of LIRGs in having exceptionally strong emission in the near and mid-IR lines of molecular hydrogen. The consensus of the infrared observers is that the strong H$_{\\rm{2}}$ lines are due to shocked gas, not star formation \\citep{Rieke1985,Depoy1986,Lester1988,Herbst1990,Elston1990,vanderWerf1993,Mori2014}. Since the cooling time of this shocked gas is short ($\\sim$10$^{7}$~yr), we may be seeing NGC 6240 at a privileged moment during the merger process \\citep[e.g.][]{Sugai1997}. Recent work of \\citet{Meijerink2013} showed that the $^{12}$CO\\ spectral line energy distribution (SLED), obtained using the $Herschel$ FTS, is consistent with either an X-ray dominated region (XDR) or with shocked molecular gas; however, the lack of the ionic species OH$^{+}$ and H$_{2}$O$^{+}$, normally found in high abundance in gas clouds near elevated X-ray or cosmic ray fluxes, ruled out the XDR models and once again, shocks are the most likely scenario to explain the observations.\n\nThe near and mid-infrared H$_{\\rm{2}}$ emission peaks between the two nuclei (e.g., Lester et al. 1988; Herbst et al. 1990), suggesting several possible scenarios. Two of the more recent ideas are those of \\citet{Ohyama2003} and \\citet{Nakanishi2005}. The preferred model of Ohyama et al. (2003) is that the merger's tidal forces have channeled the molecular gas into the space between the two nuclei, where it is being shocked by a superwind from the southern nucleus, the stronger source in radio continuum, X-rays, and near- and mid-IR. Nakanishi et al. (2005) propose instead that the two nuclei have had a nearly head-on collision, after which the two supermassive black holes, the older bulge stars, and the recent starburst stars all moved onwards to the present sites of the two nuclei. Due to drag and stripping forces, however, the circumnuclear gas was left behind at the original collision site. \nThis resembles the well-known Bullet Cluster scenario, except that collision speeds are lower, and NGC 6240's gas densities are 10$^{7}$ times higher than those of the Bullet Cluster, so the gas left behind in the middle is cool molecular gas rather than hot X-ray-emitting plasma.\n\nNGC 6240 has also been observed in high-resolution $^{12}$CO\\ many times \\citep{Wang1991,Bryant1999,Tacconi1999,Nakanishi2005,Iono2007,Wilson2008,Feruglio2013a,Feruglio2013b}. The CO emission is peaked in between the two nuclei as is observed for the warm H$_{2}$ emission. \\cite{Feruglio2013b} detected blueshifted CO emission between -200~km s$^{-1}$\\ and -500~km s$^{-1}$\\ peaking near the southern AGN position at the same position where an H$_{2}$ outflow was found \\citep{Ohyama2000,Ohyama2003}. A redshifted CO component peaks in between the two nuclei, similar to the CO emission at the systemic velocity, with a large velocity dispersion ($\\sim$500~km s$^{-1}$\\ at the maximum) suggesting highly turbulent gas \\citep{Feruglio2013b}. \n\nIn addition to CO, dense gas tracers have also been observed, however, not to the extent as that for Arp 220. \\cite{Nakanishi2005} observed HCN and HCO$^+$~$J$~=~1$-$0\\ at 2-3\\arcsec, where both lines peak in between the two nuclei, similar to CO. \\cite{Wilson2008} observed part of the HCO$^+$~$J$~=~4$-$3\\ line, where it still peaks in between the two nuclei. \\cite{Tunnard2015b} observed the rarer $^{13}$C isotopologues of HCN and HCO$^+$\\ in NGC 6240 and found line ratios $>$ 30. \n\\begin{table*\n\\caption{Source Summary}\\label{tab:sourcesum}\n\\centering\n\\begin{tabular}{lccccccc}\n\\hline \\hline\n\t\t&\\multicolumn{2}{c}{\\underline{(0,0) Position}} \t\t&& \\multicolumn{2}{c}{\\underline{Center Velocity}} & & \\\\\nSource\t& RA \t& Dec\t&\t$L_{\\rm{FIR}}$\\ &\tcz$_{lsr}$\t& z$_{lsr}$\t&D$_{L}$\t & Linear Scale \\\\\n\t\t& (J2000)\t& (J2000)\t& ($L_{\\sun}$)\t&(km s$^{-1}$)\t\t&\t\t\t& (Mpc)\t& (pc arcsec$^{-1}$) \\\\\n\\hline\nArp 220\t\t&15 34 57.24\t&+23 30 11.2 \t &1.4 $\\times$ 10$^{12}$\t&5434\t&0.018126\t&81.3\t&\t390 \\\\\nNGC 6240\t&16 52 58.89\t&+02 24 03.7\t &5.4 $\\times$ 10$^{11}$\t&7340\t&0.02448\t&108\t\t&\t520 \\\\\n\\hline\n\\end{tabular}\n\\tablefoot{[1] 9-year WMAP parameters \\citep{Hinshaw2012}: $H_{o}$ = 69.3, $\\Omega_{\\rm{matter}}$ = 0.28, $\\Omega_{\\rm{vacuum}}$ = 0.72.[2] $L_{\\rm{FIR}}$\\ reference: \\cite{Sanders2003}. [3] The (0,0) position is the phase center of the observations.}\n\\end{table*}\n\nIn this paper, we present new Plateau de Bure Interferometer (PdBI) observations of HCN\\ and HCO$^+$~$J$~=~1$-$0\\ and C$_{\\rm{2}}$H~$N$~=~1$-$0\\ for both Arp 220 and NGC 6240 (Table \\ref{tab:sourcesum}). In addition, we present new PdBI observations of $^{13}$CO $J$=1$-$0\\ and $J$ = 2-1, CS~$J$~=~2$-$1\\ and $J$ = 5-4, HNCO~$J_{k,k'}$~=~5$_{0,4}$~-~4$_{0,4}$, CH$_{\\rm{3}}$CN(6-5), SiO $J$ = 2-1, and HN$^{13}$C $J$ = 1-0 and Atacama Large Millimeter\/submillimeter Array (ALMA) Science Verification (SV) observations of CS~$J$~=~4$-$3\\ and CH$_{\\rm{3}}$CN(10-9)\\ for Arp 220. These observations will be made available to the public. The paper is broken down into the following sections: In Section 2, we describe the observations and reduction. In Section 3, we present integrated brightness temperature line ratio maps to show the varying conditions of the ISM across these sources. In Section 4, we present a radiative transfer analysis of $^{12}$CO, $^{13}$CO\\ and C$^{18}$O\\ for Arp 220 at $\\sim$700~pc scales. In Section 5, we discuss the molecules detected in both sources, the results of the radiative transfer analysis of Arp 220, the [$^{12}$CO]\/[$^{13}$CO]\\ and [$^{12}$CO]\/[C$^{18}$O]\\ abundance ratios found in Arp 220 and a comparison of the HCN\/HCO$^{+}$ line ratios. In Section 6, we summarize the major results. We end the paper with an appendix describing the release of the data online and the continuum from the observations.\n\n\n\n\\section{Observations}\n\\subsection{PdBI\nFor the line-ratio investigations in this paper, we used\n$^{12}$CO $J$=1$-$0\\ and $J$~=~2--1\nPdBI data sets on Arp~220 and NGC~6240, that were \npartly data from earlier publications (for Arp 220: Downes \\& Solomon 1998;\nDownes \\& Eckart 2007; K\\\"onig et al.\\ 2012; and for NGC~6240: \nTaconni et al.\\ 1999). We also use the SMA observations of $^{12}$CO $J$=3$-$2\\ in Arp~220 published in \\cite{Sakamoto2008} and the PdBI observations of $^{12}$CO $J$=1$-$0\\ in NGC~6240 published in \\cite{Feruglio2013a}.\n\nOther, previously unpublished, data sets are, for Arp~220, \n$^{13}$CO $J$=1$-$0\\ at 3\\,mm and $^{13}$CO $J$=2$-$1\\ at 1.4\\,mm observed simultaneously, \nand also CS~$J$~=~2$-$1\\ at 3\\,mm and CS~$J$~=~5$-$4\\ at 1\\,mm observed simultaneously.\nAdditional data sets included\nHCN, HN$^{13}$C, HCO$^+$~$J$~=~1$-$0, C$_{\\rm{2}}$H~$N$~=~1$-$0,\nand SiO $J$~=~2--1 (v=0) all observed simultaneously.\n\nFor most of these observations, the six 15\\,m antennas\nwere arranged with spacings from 24\\,m to 400\\,m. The longer baselines,\nobserved in winter, had phase errors $\\leq 40^\\circ$ at 1.4\\,mm, and\n$\\leq 15^\\circ$ at 3\\,mm. Short spacings ($\\leq 80$\\,m), observed in\nsummer at 3\\,mm, had r.m.s.\\ phase errors $\\leq 20^\\circ$.\n\nThe SIS receiver noise plus spillover\nand sky noise gave typical equivalent system temperatures outside the\natmosphere of 150\\,K at 3\\,mm (86\\,GHz) in the lower sideband, and 250\nto 400\\,K at 1.3\\,mm in upper and lower sidebands separated by 3\\,GHz,\nwith the upper band typically at 215 to 225\\,GHz. The spectral\ncorrelators covered 1700\\,km s$^{-1}$ \\ at 3\\,mm and 800\\,km s$^{-1}$ \\ at 1.4\\,mm,\nwith instrumental resolutions of 8 and 4\\,km s$^{-1}$ , respectively.\n\nThese raw data were then smoothed to channels of 10, 20, and 40\\,km s$^{-1}$ . \nThe primary amplitude calibrators were 3C273 (a variable source, but\ntypically 18\\,Jy at 3\\,mm and 13\\,Jy at 1.4\\,mm, at the \ntime of the observations), and MWC349\n(a mostly non-varying source, \nwith 1.0 and 1.7\\,Jy at 3 and 1.4\\,mm during the years of\nthe observations). The uncertainties\nin the flux scales are typically $\\pm 5$\\% at 3\\,mm and $\\pm 10$\\% at 1.4\\,mm.\n\nIn some of the observing epochs, \nthe observing program monitored phases every 20\\,min, at 3 and 1.4\\,mm\nsimultaneously, with the same phase calibrators used in earlier\nobservations of these sources (see Table~1 of Downes \\& Solomon 1998). Prior to 2004, the\ndata processing program used the 1.4\\,mm total power to correct\namplitudes and phases at 3 and 1.4\\,mm for short-term changes in\natmospheric water vapor. After 2004, this was done with \nwater-vapour monitoring receivers at 22 GHz on each antenna.\nA post-observation calibration program took the 3\\,mm curve of phase\nversus time, scaled it to 1.4\\,mm, then subtracted it from the observed\n1.4\\,mm calibrator phases, and then fit the phase difference between\nthe two receivers. All visibilities are weighted by the integration\ntime and the inverse square of system temperature. Most maps were\nmade with this ``natural weighting'' of the $uv$ data.\n\nSome sources were observed with improved IRAM receivers in February\n2008, in a $2\\times 1$\\,GHz spectroscopic mode that simultaneously covered the lines of HCN, HCO$^+$~$J$~=~1$-$0\\ and C$_{\\rm{2}}$H~$N$~=~1$-$0, \nin the newer PdBI extended configuration with antenna\nspacings up to 760\\,m. For these later observations, typical receiver \ntemperatures were 50\\,K at 3\\,mm.\n\nAll data reductions were done with the MAPPING program in the standard\nIRAM GILDAS\\footnote{http:\/\/www.iram.fr\/IRAMFR\/GILDAS} package.\n\nThe datacubes were converted to FITS files and imported into \\verb=CASA= \\citep{McMullin2007} v4.7.1 for data analysis. We created integrated intensity maps using a 1$\\sigma$ cutoff for weak lines such as, for example, HN$^{13}$C and a 5$\\sigma$ cutoff for strong emission lines such as, for example, $^{12}$CO, HCN and HCO$^+$. These integrated intensity maps are presented in Figures \\ref{fig:arp220maps} and \\ref{fig:ngc6240maps}. Table \\ref{tab:observations} presents the various molecular lines detected and their observational properties. We also present spectra in Figures \\ref{fig:arp220spec} and \\ref{fig:n6240spec}. \n\n\\subsection{ALMA\nArp~220 was a target for Band 5 SV observations on 16 July 2016. The four spectral windows, of 1.875GHz bandwidth each, were placed on H$_{2}$O (3$_{13}$ - 2$_{20}$), HNC $J$=2-1, CS $J$=4-3 and CH$_{3}$OH (4$_{31}$ - 3$_{30}$). After checking for any obvious calibration flaws, we use the once phase-only self-calibrated delivered visibility dataset. We image the CS~$J$~=~4$-$3\\ and what we have identified as CH$_{\\rm{3}}$CN(10-9)\\ lines. The H$_{2}$O line has been presented in \\cite{Koenig2016b}. Using \\texttt{CASA} v4.7.1, we create datacubes of 20 km s$^{-1}$\\ channel widths with a natural weighting for maximum sensitivity. We created integrated intensity maps using a 2$\\sigma$ cutoff (Figure \\ref{fig:arp220maps}.) \n\n\n\\subsection{Short-Spacings Flux for $^{12}$CO\n\\cite{Greve2009} present single dish fluxes for both Arp~220 and NGC~6240. Comparing the fluxes measured for Arp~220 in \\citet[][;$^{12}$CO $J$=1$-$0\\ and $J$=2-1]{Downes1998} to that of \\cite{Greve2009} shows that the PdBI maps have recovered nearly all flux ($^{12}$CO $J$=1$-$0: 410 Jy km s$^{-1}$\\ vs 420 Jy km s$^{-1}$; $^{12}$CO $J$=2$-$1: 1100 Jy km s$^{-1}$\\ vs 1130 Jy km s$^{-1}$). \\cite{Sakamoto2008} made comparisons to single dish observations for $^{12}$CO $J$=3$-$2\\ and concluded that $\\sim$10\\% of the total flux is missing. Since 10\\% of the total flux is likely spread over the entire source, each point in Arp~220 will have insignificant missing $^{12}$CO $J$=3$-$2\\ flux when compared to the calibration uncertainty ($\\pm$ 15\\%). \n\nFor NGC~6240, \\cite{Feruglio2013a} made a comparison with single dish observations for $^{12}$CO $J$=1$-$0\\ and found an agreement in fluxes. The $^{12}$CO $J$=2$-$1\\ observations of \\cite{Tacconi1999} are missing $\\sim$30\\% of the total flux when compared to the flux measured in \\cite[][1220 Jy km s$^{-1}$\\ vs 1740 Jy km s$^{-1}$]{Greve2009}.\n\n\\begin{table*\n\\caption{Line Observations Summary}\\label{tab:observations}\n\\centering\n\\begin{tabular}{llccccc}\n\\hline \\hline\nSource \t& Line \t& $\\nu_{\\rm{rest}}$\t&Resolution \t& rms \t\t\t&Channel Width\t\t& Flux \\\\\n\t\t&\t\t& (GHz)\t\t\t&(arcsec)\t\t&(mJy beam$^{-1}$)\t&(km s$^{-1}$) \t& (Jy km s$^{-1}$) \\\\\n\\hline \nArp 220\t&$^{13}$CO $J$=1$-$0\\\t &110.201\t&2.0 x 1.3\t & 1.5\t&40\t&8.0 \\\\\n\t\t&$^{13}$CO $J$=2$-$1\\\t &220.399\t&0.9 x 0.7\t & 1.9\t&20\t&47.0 \\\\\n\t\t&HCO$^+$~$J$~=~1$-$0\\\t &89.189\t&1.3 x 0.8\t &0.7\t \t&20\t& 17.8 \\\\\n\t\t&HCN~$J$~=~1$-$0\\\t &88.631\t&1.3 x 0.8\t &0.7\t \t&20\t& 45.0 \\\\\n\t\t&CS~$J$~=~2$-$1\\\t &97.981\t&1.6 x 1.1\t &1.3\t \t&20\t&17.1 \\\\\n\t\t&CS~$J$~=~4$-$3\\\t&195.954\t&0.9 x 0.7 & 1.4\t&20\t&61.0 \\\\\n\t\t&CS~$J$~=~5$-$4\\\t &244.936\t&0.9 x 0.7\t &2.5\t \t&40\t&40.5 \\\\\n\t\t&C$_{\\rm{2}}$H~$N$~=~1$-$0\\\t &87.3\/4\t&1.3 x 0.8\t &0.8\t \t&20 &8.8 \\\\\n\t\t&CH$_{\\rm{3}}$CN(6-5)\\ & 110.328 - 110.385\t&2.0 x 1.3\t& 1.5\t\t&40\t& $>$3.9 \\\\\n\t\t&CH$_{\\rm{3}}$CN(10-9)\\ & 183.674 - 183.964\t&0.9 x 0.8\t& 2.4\t\t&20\t& 19.7 \\\\\t\t\n\t\t&HNCO~$J_{k,k'}$~=~5$_{0,4}$~-~4$_{0,4}$\\ & 109.906\t&2.0 x 1.3\t\t&1.5\t&40\t& $>$2.0\\\\\n\t\t&HN$^{13}$C $J$ = 1-0 &\t87.091\t&1.3 x 0.8& 0.8\t&40\t& 0.7 \\\\\n\t\t&SiO $J$=2-1 & 86.847\t&1.3 x 0.8& 0.8\t&40\t& 5.5 \\\\\n\t\t&\t \t &\t\t\t &\t \t&\t& \\\\\nNGC 6240&HCO$^+$~$J$~=~1$-$0\\\t &89.189\t&1.1 x 1.1\t &1.0\t \t&20& \t9.9 \t\\\\\t\n\t\t&HCN~$J$~=~1$-$0\\\t &88.63\t&1.1 x 1.1\t &1.0\t \t&20& \t6.3\t\\\\\t\n\t\t&C$_{\\rm{2}}$H~$N$~=~1$-$0\\\t &87.3\/4\t&1.1 x 1.1\t &0.3\t \t&40& \t1.3\t\\\\\t\t\n\\hline\n\\end{tabular} \\\\\n\\textbf{NOTE}: [1] Splatalogue (http:\/\/www.splatalogue.net\/) was used to obtain frequencies. [2] CH$_{3}$CN is composed of several lines that span the stated frequency range. [3] Calibration uncertainties are $\\sim$5$\\%$ and $\\sim$10$\\%$ for 3mm and 1mm observations, respectively.\n\\end{table*}\n\\begin{figure*}[!htbp]\n\\centering\n$\\begin{array}{c@{\\hspace{0.1in}}c@{\\hspace{0.1in}}c}\n\\includegraphics[scale=0.2]{fig1-map-13CO10arp220.pdf} &\\includegraphics[scale=0.2]{fig1-map-13CO21arp220.pdf} & \\includegraphics[scale=0.2]{fig1-map-HN13C10arp220.pdf} \\\\\n\\includegraphics[scale=0.2]{fig1-map-HNCO54arp220.pdf} &\\includegraphics[scale=0.2]{fig1-map-SiO21arp220.pdf} &\\includegraphics[scale=0.2]{fig1-map-C2Harp220.pdf} \\\\\n \\includegraphics[scale=0.2]{fig1-map-CS21arp220.pdf} & \\includegraphics[scale=0.2]{fig1-map-CS43arp220.pdf} & \\includegraphics[scale=0.2]{fig1-map-CS54arp220.pdf} \\\\\n \\end{array}$\n $\\begin{array}{c@{\\hspace{0.1in}}c}\n \\includegraphics[ scale=0.2]{fig1-map-HCN10arp220.pdf} & \\includegraphics[scale=0.2]{fig1-map-HCOP10arp220.pdf} \\\\\n\\includegraphics[scale=0.2]{fig1-map-CH3CN65arp220.pdf} &\\includegraphics[scale=0.2]{fig1-map-CH3CN109arp220.pdf} \n\\end{array}$\n\\caption[]{Arp 220: The ellipse in the bottom left corner of each map represents the synthesized beam size. (TOP ROW) $^{12}$CO $J$=1$-$0, $J$=2-1 and HN$^{13}$C $J$=1-0 with contours corresponding to [4, 6, 8, 10] $\\times$ 0.45 Jy beam$^{-1}$ km s$^{-1}$, [4, 8, 12, 16, 20, 24] $\\times$ 0.5 Jy beam$^{-1}$ km s$^{-1}$\\ and [3, 4, 5, 6] $\\times$ 0.135 Jy beam$^{-1}$ km s$^{-1}$, respectively. (2$^{nd}$ ROW) HNCO~$J_{k,k'}$~=~5$_{0,4}$~-~4$_{0,4}$, SiO $J$=2-1 and C$_{\\rm{2}}$H~$N$~=~1$-$0\\ with contours corresponding to [4, 6, 8,10] $\\times$ 0.36Jy beam$^{-1}$ km s$^{-1}$, [4, 6, 8, 10, 15, 20, 25, 30] $\\times$ 0.135 Jy beam$^{-1}$ km s$^{-1}$\\ and [5, 10, 15, 20, 25] $\\times$ 0.135 Jy beam$^{-1}$ km s$^{-1}$, respectively. (3$^{rd}$ ROW) CS~$J$~=~2$-$1, $J$=4-3 (ALMA) and $J$=5-4 with contours corresponding to [4, 8, 12, 16, 20, 24] $\\times$ 0.5, 1.2 and 0.29 Jy beam$^{-1}$ km s$^{-1}$, respectively. (4$^{th}$ ROW) HCN~$J$~=~1$-$0\\ and HCO$^+$~$J$~=~1$-$0\\ with contours corresponding to [5, 10, 15, 20, 25] $\\times$ 0.52 and 0.15 Jy beam$^{-1}$ km s$^{-1}$, respectively. (BOTTOM ROW) CH$_{\\rm{3}}$CN(6-5)\\ and (10-9) with contours corresponding to [4, 6, 8, 10] and [6, 12, 18, 24, 30, 40, 50] $\\times$ 0.33 Jy beam$^{-1}$ km s$^{-1}$, respectively.\n}\n\\label{fig:arp220maps}\n\\end{figure*}\n\\begin{figure*}[htbp]\n\\centering\n$\\begin{array}{c@{\\hspace{0.1in}}c@{\\hspace{0.1in}}c}\n\\includegraphics[ scale=0.2]{fig2-map-HCN10-n6240.pdf} & \\includegraphics[scale=0.2]{fig2-map-HCOP10-n6240.pdf} & \\includegraphics[scale=0.2]{fig2-map-C2H10-n6240.pdf}\\\\ \n\\end{array}$\n\\caption[]{NGC 6240: The ellipse in the bottom left corner of each map represents the synthesized beam size. HCN $J$=1-0 and HCO$^+$~$J$~=~1$-$0\\ with contours corresponding to [3, 6, 9, 12, 15, 18] $\\times$ 0.184 Jy beam$^{-1}$ km s$^{-1}$\\ and C$_{\\rm{2}}$H~$N$~=~1$-$0\\ with contours corresponding to [1, 2, 3, 4, 5] $\\times$ 0.2 Jy beam$^{-1}$ km s$^{-1}$.}\n\\label{fig:ngc6240maps}\n\\end{figure*}\n\\begin{figure*}[!htbp]\n\\centering\n$\\begin{array}{c@{\\hspace{0.1in}}c}\n\\includegraphics[ scale=0.55]{f4a_13co10.pdf} & \\includegraphics[scale=0.55]{f4a_13co21.pdf} \\\\\n \\includegraphics[scale=0.55]{f4a_c2h.pdf} & \\includegraphics[scale=0.55]{f4a_hcn.pdf} \\\\\n\n \\includegraphics[scale=0.55]{f4a_ch3cn.pdf} &\\includegraphics[scale=0.55]{f4a_CS.pdf}\n \\end{array}$\n\\caption[]{\\textbf{($Top$ and $Middle$) Spectra averaged over a 1\\arcsec\\ diameter aperture centered at Arp 220W and Arp 220E (see Table \\ref{tab:lineratios} and Section \\ref{sec:rad}) and ($Bottom$) Normalized spectra of each of the CS lines averaged over a 3\\arcsec\\ diameter aperture. }}\n\\label{fig:arp220spec}\n\\end{figure*}\n\\begin{figure*}[!htbp]\n\\centering\n$\\begin{array}{c@{\\hspace{0.1in}}c}\n\\includegraphics[ scale=0.55]{f4b_c2h.pdf} & \\includegraphics[scale=0.55]{f4b_hcn.pdf} \\\\\n \\end{array}$\n\\caption[]{ \\textbf{Spectra averaged over a 1\\arcsec\\ diameter aperture for NGC 6240 centered on position (0,0).} }\n\\label{fig:n6240spec}\n\\end{figure*}\n\n\n\\section{Line Ratios\nFor Arp 220, we create the following integrated line brightness temperature (T$_{B}$) ratio \nmaps for $^{12}$CO\\ and $^{13}$CO:\n\nr$_{21}$ = T$_{\\rm{B}} ^{^{12}\\rm{CO}(2-1)}$ \/ T$_{\\rm{B}} ^{^{12}\\rm{CO}(1-0)}$, \n\nr$_{32}$ = T$_{\\rm{B}} ^{^{12}\\rm{CO}(3-2)}$ \/ T$_{\\rm{B}} ^{^{12}\\rm{CO}(2-1)}$, \n\n$^{13}$r$_{21}$ = T$_{\\rm{B}} ^{^{13}\\rm{CO}(2-1)}$ \/ T$_{\\rm{B}} ^{^{13}\\rm{CO}(1-0)}$,\n\nR$_{10}$ = T$_{\\rm{B}} ^{^{12}\\rm{CO}(1-0)}$ \/ T$_{\\rm{B}} ^{^{13}\\rm{CO}(1-0)}$,\n\nR$_{21}$ = T$_{\\rm{B}} ^{^{12}\\rm{CO}(2-1)}$ \/ T$_{\\rm{B}} ^{^{13}\\rm{CO}(2-1)}$.\n\nWe also create integrated T$_{B}$ line ratio maps for molecules other than CO as follows:\n\nH$_{10}$ = T$_{\\rm{B}} ^{\\rm{HCN}(1-0)}$ \/ T$_{\\rm{B}} ^{\\rm{HCO^{+}}(1-0)}$ and\n\nCS \/ HNCO = T$_{\\rm{B}}^{\\rm{CS(5-4)}}$ \/ T$_{\\rm{B}}^{\\rm{HNCO(5-4)}}$ (Figure \\ref{fig:arp220lineratios}).\n\nFor NGC 6240, we show maps of the T$_{B}$ line ratios of \nr$_{21}$ and H$_{10}$ (Figure \\ref{fig:ngc6240lineratios}; with both quantities defined as for Arp 220). Table \\ref{tab:lineratios} gives a summary of the observed line ratios. \n\nTo match the physical scales that we analyze, we smooth the data cubes using a Gaussian kernel to match angular resolution. For Arp 220, we smooth the data to the resolution of $^{13}$CO $J$=1$-$0\\ (Table \\ref{tab:observations}) and for NGC 6240, we use a compromised resolution of 1.2\\arcsec, limited by the resolution of the $^{12}$CO $J$=1$-$0\\ observations of \\citet{Feruglio2013a}. We applied a 5$\\sigma$ cut to each map used to produce the the ratio maps. \n\nSince $^{12}$CO\\ is believed to be optically thick, the R$_{10}$ and R$_{21}$ line ratios give a lower limit to the true [$^{12}$CO]\/[$^{13}$CO]\\ abundance ratio. To illustrate this point, we start with the most general equation for the R line ratios,\n\\begin{eqnarray}\\label{eqn:ratio}\n\\begin{aligned}\n\\rm{R} &= \\frac{T_{\\rm{B}}^{^{12}\\rm{CO}}}{T_{\\rm{B}}^{^{13}\\rm{CO}}} \\\\\n\\rm{R} &= \\frac{T_{\\rm{EX}}^{^{12}\\rm{CO}}}{T_{\\rm{EX}}^{^{13}\\rm{CO}}} \\frac{(1-\\rm{e}^{-\\tau_{^{12}CO}})}{(1-\\rm{e}^{-\\tau_{^{13}CO}})}\n\\end{aligned}\n\\end{eqnarray}\nwhere T$_{\\rm{EX}}$ is the excitation temperature, $\\tau_{\\rm{^{12}CO}}$ and $\\tau_{\\rm{^{13}CO}}$ are the optical depths of $^{12}$CO\\ and $^{13}$CO, respectively. If both $^{12}$CO\\ and $^{13}$CO\\ were to be in local thermal equilibrium (LTE) so that their excitation temperatures were equal, then Equation \\ref{eqn:ratio} simplifies to \n\\begin{equation}\\label{eqn:ratio2}\n\\rm{R} = \\frac{(1-\\rm{e}^{-\\tau_{^{12}CO}})}{(1-\\rm{e}^{-\\tau_{^{13}CO}})} \n\\end{equation}.\nIn addition, if the $^{12}$CO\\ emission is sufficiently optically thick, so that $(1-\\rm{e}^{-\\tau_{^{12}CO}})$ $\\rightarrow$ 1 and if $^{13}$CO\\ is sufficiently optically thin, so that $(1-\\rm{e}^{-\\tau_{^{13}CO}})$ $\\rightarrow$ $\\tau_{^{13}CO}$, then Equation \\ref{eqn:ratio2} can be further simplified to \n\\begin{eqnarray}\\label{eqn:ratio}\n\\begin{aligned}\n\\rm{R} &= \\frac{1}{\\tau_{^{13}CO}}\\\\\n\\rm{R} &= \\frac{\\tau_{^{12}CO}}{\\tau_{^{13}CO}} \\frac{1}{\\tau_{^{12}CO}}\\\\\n\\rm{R} &= \\frac{[^{12}CO]}{[^{13}CO]}\\frac{1}{\\tau_{^{12}CO}}\n\\end{aligned}\n\\end{eqnarray}\nwhere the ratio of the optical depths of the isotopologues are equivalent to the abundance ratio ($\\tau$ $\\propto$ column density). In this case, the observed ratio R of line brightness temperatures, will always be less than the true abundance, [$^{12}$CO]\/[$^{13}$CO], because of the attenuating factor of the $^{12}$CO\\ optical depth. \n\nOnly if $^{12}$CO\\ and $^{13}$CO\\ are both optically thin, will the observed ratio of line brightness temperatures directly trace the abundance ratio, providing that the $^{12}$CO\\ and $^{13}$CO\\ excitation temperatures are equal.\n\nAs is well-known, however, because of resonant trapping in the CO lines, a more realistic approach is to use one of the standard non-LTE approximations, which allows for different excitation temperatures in the different CO lines, and this is what we actually do (Section 4).\n\n\\begin{figure*}[!htbp]\n\\centering\n$\\begin{array}{c@{\\hspace{0.5in}}c}\n\\includegraphics[ scale=0.3]{figA_ratio_co2110_a220.pdf} & \\includegraphics[scale=0.3]{figA_ratio_co3221_a220.pdf} \\\\\n\\includegraphics[ scale=0.3]{figA_ratio_co121310_a220.pdf} & \\includegraphics[ scale=0.3]{figA_ratio_co121321_a220.pdf}\\\\\n\\includegraphics[ scale=0.3]{figA_ratio_co13co2110_a220.pdf} & \\includegraphics[scale=0.3]{figA_ratio_hcnhco_a220.pdf} \\\\\n\\includegraphics[ scale=0.3]{figA_ratio_cshnco54_a220.pdf} \\\\\n\\end{array}$\n\\caption{Arp~220 line ratios: (TOP) r$_{21}$ and r$_{32}$, (2nd ROW) R$_{10}$ and R$_{21}$, (3rd ROW) $^{13}$r$_{21}$ and H$_{10}$, (BOTTOM) CS \/ HNCO. The ellipse in the bottom left corner of each map represents the synthesized beam size. The black contours represent the high-resolution $^{12}$CO $J$=2$-$1\\ emission published in \\cite{Downes2007} to guide the eye to the positions of the two nuclei of Arp~220. Note that the resolution of our observations do not spatially resolve the two nuclei. }\n\\label{fig:arp220lineratios}\n\\end{figure*}\n\\begin{figure*}[!htbp]\n\\centering\n$\\begin{array}{c@{\\hspace{0.5in}}c}\n\\includegraphics[ scale=0.3]{fig_ratio_ngc6240_hcnhco.pdf} & \\includegraphics[ scale=0.3]{fig_ratio_ngc6240_co2110.pdf} \\\\\n\\end{array}$\n\\caption[]{NGC 6240 Line ratios: (LEFT) H$_{10}$ and (RIGHT) r$_{21}$. The ellipse in the bottom left corner of each map represents the synthesized beam size. The black contours represent the 86~GHz continuum emission to guide the eye to the positions of the two nuclei of NGC~6240.}\n\\label{fig:ngc6240lineratios}\n\\end{figure*}\n\\begin{table*\n\\caption{Line Ratios}\\label{tab:lineratios}\n\\centering\n\\begin{tabular}{lcccccccc}\n\\hline\\hline\n\t\t\t& \\multicolumn{2}{c}{\\underline{Position}} & & & & & \\\\\n\t\t\t&RA (J2000)&Dec (J2000)&r$_{\\rm{21}}$\t& r$_{\\rm{32}}$\t & R$_{\\rm{10}}$\t&R$_{\\rm{21}}$\t&$^{13}$r$_{\\rm{21}}$ & H$_{10}$\t\t \\\\\n\\hline\nArp 220E\t\t&15 34 57.342 \t&+23 30 11.610\t &0.8 $\\pm$ 0.1\t&0.8 $\\pm$ 0.1\t\t&23 $\\pm$ 3\t&19 $\\pm$ 2\t&0.9 $\\pm$ 0.1 &2.4 $\\pm$ 0.2\t \\\\\nArp 220W\t\t&15 34 57.197\t&+23 30 11.595\t &1.1 $\\pm$ 0.1\t&0.60 $\\pm$ 0.08\t&38 $\\pm$ 4\t&19 $\\pm$ 2\t&2.3 $\\pm$ 0.3&\t2.7 $\\pm$ 0.2\\\\\nArp 220C\t\t&15 34 57.258\t&+23 30 11.592\t &1.2 $\\pm$ 0.1 &0.60 $\\pm$ 0.08\t&31$\\pm$ 3\t&26 $\\pm$ 3\t&1.3 $\\pm$ 0.1&2.5 $\\pm$ 0.2\t\\\\\nNGC 6240\t&16 52 58.900 & +02 24 03.810\t &1.0 $\\pm$ 0.1 &...\t&...\t&...\t&...\t&0.80 $\\pm$ 0.09\t \\\\\n\\hline \n\\end{tabular}\n\\newline\n\\textbf{NOTE}: The positions for Arp 220E\/W do not correspond to the positions of the two nuclei. Our resolution is too poor to distinguish the nuclei spatially, therefore, the Arp 220E\/W positions are $\\sim$ 2\\arcsec\\ apart (greater than the synthesized beam major axis) with Arp 220C as the central position. Position error = $\\pm$0.1\\arcsec\n\\end{table*}\n\n\\section{Radiative Transfer Analysis}\\label{sec:rad\nTo model the $^{12}$CO\\ emission (and that of the rarer CO isotopologues), we use the escape-probability radiative transfer program RADEX \\citep{vanderTak2007}. To find the most likely RADEX solution, given the observed line strengths, and other constraints, we use the Monte Carlo Markov Chain code\\footnote{https:\/\/github.com\/jrka\/pyradexnest} of \\citet{Kamenetzky2014}. This code implements the nested sampling algorithm MultiNest \\citep{Feroz2009} using its Python wrapper PyMultiNest \\citep{Buchner2014} to constrain parameters. As stated in \\citet{Kamenetzky2014}, ``The algorithm `nests inward' to subsequently smaller regions of high-likelihood parameter space. Unlike calculating the likelihood using a grid method, the algorithm can focus on high likelihood regions and thus estimate parameter constraints more efficiently.\" Model points are generated using RADEX with the following input parameters: kinetic temperature (\\tkin), column density of molecular species X per unit line width ($N_{\\rm{X}}$\/$\\Delta$V) and volume density of the collision partner, molecular hydrogen ($n_{\\rm{H_{2}}}$). In addition, the resulting flux is allowed to scale uniformly down by an area filling factor, $\\Phi_{\\rm{A}}$\\ $\\leq$1. We also implement three priors: \n\\begin{enumerate}\n\\item A column length to constrain the diameter of the molecular emission region. This prior assists in constraining the column and volume densities. We estimate the upper limit to the column length to be the diameter of the synthesized beam ($\\sim$~700~pc). \n\\item A dynamical mass ($M_{\\rm{dyn}}$) as an upper limit to the total mass that can be contained within the column density. We use the equation from \\cite{Wilson1990} assuming a diameter of $\\sim$700~pc and velocity FWHMs from literature presented below.\n\\item An optical depth range of 0 to 100. An optical depth below 0 implies maser emission and the upper limit of 100 is recommended by the RADEX documentation.\n\\end{enumerate}\n We refer the reader to Kamenetzky et al. (2014) for more details. \n\nWe model three molecular species simultaneously: $^{12}$CO, $^{13}$CO\\ and C$^{18}$O. The $^{12}$CO\\ lines modelled are the $J$=1-0, 2-1 \\citep{Downes1998} and 3-2 \\citep{Sakamoto2008}. We use a line ratio of $^{13}$CO\/C$^{18}$O\\ = 1 \\citep{Matsushita2009,Greve2009} to estimate the emission of C$^{18}$O\\ at our angular resolution ($\\sim$2\\arcsec). Due to the lack of resolution elements across the $^{13}$CO\\ emission of Arp 220, we model the molecular gas at the central peak position of $^{13}$CO $J$=1$-$0\\ (see Table \\ref{tab:observations}; Arp 220C) since the peak emission falls in between the two nuclei and 1\\arcsec\\ to each side of Arp 220C, which will place Arp 220E\/W more than one resolution beam apart. We stress that Arp 220E\/W are not the positions of the nuclei and are only modelled for completeness. For Arp 220W, we used only the $J$=1-0 and 2-1 lines because the addition of the $J$=3-2 line resulted in poor SLED fits, likely indicating the presence of a second molecular gas component that cannot be fit with our data. We adopt the following linewidths: 120 km s$^{-1}$\\ and 250 km s$^{-1}$\\ for Arp~220W and Arp~220E measured using ALMA $^{12}$CO $J$=1$-$0\\ observations at 0.09\\arcsec\\ (37~pc) \\citep{Scoville2017}, respectively, and 320 km s$^{-1}$\\ for Arp 220C \\citep{Downes1998} measured from PdBI $^{12}$CO $J$=1$-$0\\ observations. The mean, best fit and $\\pm$1$\\sigma$ results of \\tkin, $n_{\\rm{H_{2}}}$, $N_{\\rm{^{12}CO}}$, $\\Phi_{\\rm{A}}$, thermal pressure, molecular gas mass, CO-to-H$_{2}$ conversion factor ($\\alpha_{\\rm{CO}}$) and the relative abundances of $^{13}$CO\\ and C$^{18}$O\\ to $^{12}$CO\\ per model point are presented in Table \\ref{tab:results}. The best fit SLED for Arp 220C is presented in Figure \\ref{fig:sleds}. The (marginal) probability of a single parameter is computed by integrating over all other parameters. We present the marginal probability distributions of \\tkin, $n_{\\rm{H_{2}}}$, $N_{\\rm{^{12}CO}}$\\ and relative abundances of $^{13}$CO\\ and C$^{18}$O\\ as well as the 2D probability distribution of log$_{10}$(\\tkin) versus log$_{10}$($n_{\\rm{H_{2}}}$) in Figure \\ref{fig:arp220prob}. \n\nWe do not model NGC 6240 due to the lack of available high-resolution observations of $^{13}$CO\\ and the significant missing $^{12}$CO\\ flux in the existing high-resolution maps. We refer the reader to \\citet{Tunnard2015b} for an extensive radiative transfer modelling of the molecular gas in NGC 6240. \n\n\\begin{table*\n\\scriptsize\n\\caption{Radiative Transfer Results}\\label{tab:results}\n\\centering\n\\begin{tabular}{clccccccccc}\n\\hline\\hline\n& \t&\\tkin\\\t&log$_{10}$($n_{\\rm{H_{2}}}$)\t&log$_{10}$(P) &log$_{10}$($\\Phi_{\\rm{A}}$)\t&log$_{10}$($<$$N_{\\rm{^{12}CO}}$\\ $>$) &log$_{10}$($M_{\\rm{H_{2}}}$) & $\\alpha_{\\rm{CO}}$\\ &\t[$^{12}$CO]\/[$^{13}$CO]\\\t\t& [$^{12}$CO]\/[C$^{18}$O]\\ \\\\\n&\t&(K)\t&(cm$^{-3}$)\t& (K cm$^{-3}$) &\t&(cm$^{-2}$) &($M_{\\sun}$) &($M_{\\odot}$ (K km s$^{-1}$ pc$^{2}$)$^{-1}$)\t& & \\\\ \\relax\n&[1]\t&[2]&[3]&[4]&[5]&[6]&[7]&[8]&[9]&[10] \\\\\n\\hline\nArp 220C&Mean \t& 130 & 2.99 & 5.2 &-0.30 \t& 19.33 \t&8.81\t&0.4 &125\t & 125\t\t\\\\\n\t&Best Fit \t\t& 38 \t& 3.35 &\t4.9& -0.036 \t&19.20 \t&8.69 \t&0.3 & 90\t &\t91\t\\\\\n\t&-1$\\sigma$ Value\t&37 \t& 2.55 &\t4.9 & -0.53 \t&18.97\t& 8.46 \t&0.2 &58\t &\t56\t\\\\\n\t&+1$\\sigma$ Value\t&690 & 3.40 &\t5.4 & -0.074\t&19.74 \t& 9.23 \t&1.0\t&309 & 323\\\\\n\t\t\\hline\n\t\t\n\nArp 220E &Mean \t\t&240 & 2.54 & 4.9 &-0.22 \t&19.04 \t&8.53\t&0.4 &93\t & 93\t\t\\\\\n\t\t&Best Fit \t\t&34 \t & 3.10 &\t4.6& -0.017 \t&19.24\t&8.73 \t& 0.6 & 159\t &\t159\t\\\\\n\t\t&-1$\\sigma$ Value\t&10 \t & 2.8 &\t4.3 & -0.17\t&19.03\t& 8.52 \t&0.4 &100\t &\t100\t\\\\\n\t\t&+1$\\sigma$ Value\t&110 & 3.4 &\t4.9 & 0.0\t&19.45 \t& 8.94 \t&1.0\t&255 & 255\\\\\n\t\t\\hline\n\nArp 220W&Mean \t& 105 & 3.78 & 5.8 &-0.32 \t& 19.36 \t&8.85\t&0.5 &151\t & 149\t\t\\\\\n\t&Best Fit \t\t& 300 \t& 3.2 &\t5.7& -0.7 \t&19.34 \t&8.83 \t&0.5 & 142& 143\t\\\\\n\t&-1$\\sigma$ Value\t&90 \t& 2.3 &\t5.0 & -0.91\t&18.96\t& 8.45 \t&0.2 &62\t &\t62\t\\\\\n\t&+1$\\sigma$ Value\t&1000 & 4.1 &\t6.3 & -0.48\t&19.71 \t& 9.20\t&1.1\t&330 & 330\\\\\n\t\t\\hline\t\n\t\t\n\\end{tabular}\n\\textbf{NOTES:} Col [1]: Statistic; Col[2] Kinetic Temperature; Col [3] Volume Density; Col[4]: Thermal Pressure Col[5]: Filling Factor; Col[6]: $^{12}$CO\\ Column Density; Col[7]: Molecular Gas (H$_{2}$) mass assuming [$^{12}$CO]\/[H$_{2}$] = 3 $\\times$ 10$^{-4}$; Col[8]: Conversion Factor $\\alpha_{\\rm{CO}}$\\ = 1.36$M_{\\rm{H_{2}}}$\/$L_{\\rm{CO}}$); Factor of 1.36 is to account for He; Col[9]: [$^{12}$CO]\/[$^{13}$CO]\\ abundance ratio; Col[10]: [$^{12}$CO]\/[C$^{18}$O]\\ abundance ratio; The $\\pm\\sigma$ values denote the range of values within 1$\\sigma$. \n\\end{table*}\n\\begin{figure}[!htbp]\n\\centering\n\\includegraphics[ scale=0.6]{fig4-arp220-sled.pdf} \\\\\n\\caption[]{Arp 220C SLED for $^{12}$CO\\ (black x), $^{13}$CO\\ (green left triangle) and C$^{18}$O\\ (orange right triangle). Solid and dashed lines represent the most probable SLED solution for each molecule. }\n\\label{fig:sleds}\n\\end{figure}\n\\begin{figure*}[!htbp]\n\\centering\n$\\begin{array}{c@{\\hspace{0.5in}}c}\n\\includegraphics[ scale=0.4]{fig5a_Tempdist.pdf} & \\includegraphics[scale=0.4]{fig5b_Densdist.pdf} \\\\\n\\includegraphics[ scale=0.4]{fig5a_abundist.pdf} & \\includegraphics[ scale=0.4]{fig5a_bacdist.pdf} \\\\\n\\end{array}$\n\\includegraphics[scale=0.4]{fig5e_contour.pdf} \\\\\n\\caption[]{Probability distributions for Arp 220C. (\\textit{Top Left}) Temperature, (\\textit{Top Right}) Volume density, (\\textit{Middle Left}) Abundance of $^{13}$CO\\ and C$^{18}$O\\ relative to $^{12}$CO. (\\textit{Middle Right}) Average column density of $^{12}$CO. (\\textit{Bottom}) Temperature versus volume density. Contours are 40, 60, 80 and 95$\\%$ of the most probable solution. Dashed lines represent log(pressure).}\n\\label{fig:arp220prob}\n\\end{figure*}\n\n\n\\section{Discussion\n\\subsection{Molecules Detected\n\\textbf{$^{13}$CO\\ (\\textit{Second most common isotopologue of carbon monoxide, after $^{12}$CO.} )} -The $^{13}$CO $J$=2$-$1\\ peak falls near the western nucleus of Arp 220 while the $^{13}$CO $J$=1$-$0\\ peak falls in between the two nuclei. Both $^{13}$CO $J$=1$-$0\\ and $J$ = 2-1 have an east-west morphology instead of a north-east south-west morphology of the $^{12}$CO\\ maps which suggests that very little $^{13}$CO\\ emission originates from the diffuse kiloparsec disk surrounding the nuclei. The $^{13}$CO\\ lines (and other optically thin tracers) are important to truly trace the physical conditions of the molecular gas as they can probe deeper into the molecular gas ensemble than the optically thick $^{12}$CO\\ (i.e. brick wall effect). This is apparent in the line ratio maps (Figure \\ref{fig:arp220lineratios}) where the $^{12}$CO\\ line ratios do not vary greatly across the system while the $^{13}$CO\\ line ratios show an east-west gradient. \n\n\\textbf{HCN (\\textit{Hydrogen cyanide}) and HCO$^+$\\ (\\textit{Formyl ion})} - \\textit{Arp 220:} The HCN\\ and HCO$^+$~$J$~=~1$-$0\\ emission in Arp 220 peaks at the same position near the western nucleus which may suggest more dense gas in the west than found near the eastern nucleus. A 2D Gaussian fit of the emission shows that the size of HCN (1.4\\arcsec\\ $\\times$ 1.0\\arcsec; size deconvolved from the beam \\textbf{assuming a Gaussian structure}) is more compact than HCO$^+$ (1.9\\arcsec\\ $\\times$ 1.4\\arcsec). The spectra (Figure \\ref{fig:arp220spec} show absorption features in both HCN and HCO$^{+}$ similar to that seen in \\cite{Martin2016} and \\cite{Scoville2017}. The absorption appears to have the same depth in both lines, which seem to be absorbing nearly all of the continuum. The main difference between the two lines is that HCN has more emission. \n\n\\textit{NGC 6240:} The integrated line flux from the HCN\\ and HCO$^+$~$J$~=~1$-$0\\ is roughly half the single-dish flux measured by \\citet{Greve2009}. This is evidence of a larger scale HCN and HCO$^+$~$J$~=~1$-$0\\ emission that is filtered out by the interferometer. With our shortest baseline ($\\sim$ 80~m), 50\\% of the flux is at scales greater than 3\\arcsec\\ ($\\sim$1.5~kpc). Future observations need to include shorter spacings to recover all flux.\n\nAs also observed for Arp 220, the HCN emission is more compact than that for HCO$^+$\\ which is evident in the integrated intensity maps (Figure \\ref{fig:ngc6240maps}). A 2D Gaussian fit on the integrated intensity maps shows that HCN and HCO$^+$~$J$~=~1$-$0\\ are unresolved at our angular resolution (1.1\\arcsec) with an upper limit to the source size of 0.5\\arcsec\\ for HCN and 0.9\\arcsec\\ $\\times$ 0.5\\arcsec\\ for HCO$^+$. The upper limits still indicate that HCO$^+$~$J$~=~1$-$0\\ is coming from a more extended component which supports the analysis of \\citet{Greve2009} suggesting that HCN traces a denser gas phase than HCO$^+$.\n\n\n\\textbf{C$_{2}$H (\\textit{Ethynyl})} - The C$_{2}$H $N$ =1-0 transition consists of six hyperfine structure lines ($J$=3\/2 -1\/2 F=1-1, 2-1, 1-0 and $J$=1\/2-1\/2 F=1-1, 0-1, 1-0). Due to the low spectral resolution, the six lines are blended in our observations for both sources.\n\\textit{Arp 220:} The C$_{\\rm{2}}$H\\ emission is peaked near the western nucleus.\n\\textit{NGC 6240:} The C$_{\\rm{2}}$H\\ emission is peaked in between the two nuclei, similar to the HCN and HCO$^+$\\ emission.\n\n\\textbf{SiO (\\textit{Silicon monoxide})} - The SiO $J$=2-1 emission from Arp 220 has been previously observed by \\citet{Tunnard2015a} at a similar spatial resolution. The total flux we measure agrees with \\citet{Tunnard2015a}; however, we note that there may be contamination from H$^{13}$CO$^{+}$ $J$=1-0. \\citet{Tunnard2015a} explore this possible contamination. We also do not see an absorption feature with the SiO $J$=2-1 line profile in Arp 220W, however, for Arp 220E there is a hint of an absorption feature (Figure \\ref{fig:arp220spec}). Since the detection in the Arp 220E is poor, it is difficult to conclude whether this is a real feature. We note that \\cite{Tunnard2015a} do not see an absorption feature in SiO $J$=2-1.\n\n\\textbf{HNCO (\\textit{Isocyanic acid})} and \\textbf{CS (\\textit{Carbon monosulfide})}: Only part of the HNCO line is observed due to bandwidth constraints and may contain contamination from C$^{18}$O\\ due to the broad linewidths of Arp 220. \\citet{Martin2009} observed HNCO~$J_{k,k'}$~=~5$_{0,4}$~-~4$_{0,4}$\\ in Arp 220 using the IRAM 30m and measured a flux of $\\sim$5.5 Jy km s$^{-1}$, more than twice our measured flux. The emission of HNCO has a horizontal elongation in the vicinity of the two nuclei with very little emission in the surrounding disk. The peak of the emission of HNCO is near the position of the western nucleus (Figure \\ref{fig:arp220maps}); however, this may be influenced by the lack of the eastern portion of the line profile. \n\n\\cite{Greve2009} observed CS~$J$~=~2$-$1\\ and $J$=5-4 in Arp 220. Our high-resolution observations of CS~$J$~=~2$-$1\\ agree with the total flux within uncertainties while the CS~$J$~=~5$-$4\\ is missing 37$\\pm$12$\\%$ of the total flux in the single dish map. \\cite{Galametz2016} observed the CS~$J$~=~4$-$3\\ emission from Arp 220 using APEX. The ALMA SV CS~$J$~=~4$-$3\\ observations total flux agrees with the single dish observations. A comparison of the three CS lines (Figure \\ref{fig:arp220spec}), shows small differences in the line profiles. The CS~$J$~=~4$-$3\\ and $J$=5-4 have a double peak profile similar to what \\cite{Greve2009} noted, while the CS~$J$~=~2$-$1\\ line profile lacks the peak at higher velocities. The CS~$J$~=~5$-$4\\ line also has a third peak around an observed frequency of $\\sim$ 240.8GHz, but this may be due to line contamination from perhaps CH$_{3}$OH and\/or CH$_{2}$NH lines. \\cite{Martin2009} observed a similar third peak in the CS~$J$~=~5$-$4\\ line.\n\nCS is a known tracer of photo-dominated regions (PDRs), where the CS is enhanced through S$^{+}$ chemistry \\citep{Sternberg1995}. HNCO, however, is photodissociated easily by UV photons, but is enhanced in regions with shocks \\citep{Zinchenko2000}. \\citet{Martin2008} proposed that the relative abundance of HNCO to CS can be used as a good diagnostic tool to distinguish between the influence of shocks and molecular clouds illuminated by UV radiation. They determined that an offset region in IC 342 (denoted by IC 342* in their analysis) is dominated by shocks with a CS(5-4)\/HNCO(5-4) line ratio of $\\sim$0.3. M82 was determined to be dominated by PDRs with a CS(5-4)\/HNCO(5-4) line ratio $>$64. In the analysis of Martin et al., Arp 220 is a composite system with contributions from PDRs and shocks. Since we only observe part of the HNCO, our line ratios of CS(5-4)\/HNCO(5-4) (Figure \\ref{fig:arp220lineratios}) is strictly an upper limit to the true line ratios; however, despite this handicap, we see significant variations of the line ratio between the region of the two nuclei ($\\sim$7) and the kiloparsec disk ($\\sim$ 12-30). Assuming both lines are optically thin and in LTE, the line ratios trace the relative abundance and this strongly suggests that the ISM in the kiloparsec disk is not strongly influenced by shocks while there is very likely a more significant contribution of shocks, likely from stellar winds, possible outflows as suggested by \\citet{Sakamoto2009} and\/or supernova explosions, near the two nuclei. Higher spatial resolution imaging is required to properly separate these two regions. \n\n\\textbf{HN$^{13}$C (\\textit{Rare isotopologue of hydrogen isocyanide})} - The HN$^{13}$C emission is peaked oddly below the western nucleus by $\\sim$0.5 $\\pm$ 0.1\\arcsec. Nearly the entire emission is originating from the western side of Arp~220 which may be due to sensitivity since the line is weak and the eastern nucleus is the weaker emitter of the two. \\citet{Wang2016} observed HN$^{13}$C $J$=1-0 and 3-2 using the IRAM~30m and APEX single dish telescopes. They find that the emission is only detected in the blue component of Arp~220 (i.e. the western nucleus region) similar to what we see and our flux measurement (Table \\ref{tab:observations}) agrees with theirs. \n\\citet{Tunnard2015a} observed HN$^{13}$C $J$=3-2 in Arp~220 with the PdBI. The peaks of HN$^{13}$C $J$=3-2 are within the positions of the nuclei (both east and west; see Figure 7 of Tunnard et al. 2015a). The flux measured by \\citet{Tunnard2015a} is a factor of $\\sim$2.5 times higher than that of \\citet{Wang2016}; however, the line in the observations of \\citet{Tunnard2015a} is near the bandpass edge and as noted by \\citet{Wang2016}, the observations may have been difficult to continuum subtract properly. If the case is that the line was poorly continuum subtracted, the peak positions of HN$^{13}$C $J$=3-2 in the PdBI map may not be due to the molecular emission but of the 1.2mm continuum contamination. \nUsing the total flux of HNC~$J$~=~1-0 published in \\cite{Greve2009} and our flux measurement for HN$^{13}$C (Table \\ref{tab:observations}), the line ratio of HNC\/HN$^{13}$C is $\\sim$ 52. Since HNC is likely optically thick, this measured line ratio is a lower limit to the [$^{12}$C]\/[$^{13}$C] abundance. \n\n\\textbf{CH$_{3}$CN (\\textit{Methyl cyanide}}) - Due to bandwidth limitations, only part of the CH$_{3}$CN~(6-5) line is observed. The CH$_{3}$CN is partially blended with the $^{13}$CO $J$=1$-$0\\ line and is nearly as strong in emission as $^{13}$CO $J$=1$-$0. The CH$_{3}$CN~(12-11) line, which would be near the $^{13}$CO $J$=2$-$1, is not included in our bandpass; however a very small part of the CH$_{3}$CN~(12-11) line may be blended in the blue side of the $^{13}$CO $J$=2$-$1\\ line profile. The ALMA CH$_{3}$CN (10-9) observations observed the entire line profile (Figure \\ref{fig:arp220spec}). \n\nSince only the emission on the blue side of the CH$_{3}$CN~(6-5) line is observed, it is not surprising that the emission is peaked near the western nucleus; however, when compared to the CH$_{3}$CN (10-9), the peak positions agree. CH$_{3}$CN is believed to be a tracer of hot cores with the relative ease of detecting the line near hot cores \\citep[e.g.][]{Remijan2004,Purcell2006} and is a good ISM thermometer \\citep{Guesten1985} if multiple transitions are observed. \\citet{Gratier2013} find that CH$_{3}$CN is more abundant in UV illuminated gas and thus may be a PDR tracer. Since U\/LIRGs do experience intense massive star formation, CH$_{3}$CN should be relatively bright (and is in Arp~220). Observations of other transitions of CH$_{3}$CN will be relatively easy in the ALMA\/NOEMA era as demonstrated by the ALMA SV and early PdBI observations.\n\n\n\\subsection{Molecular Gas Conditions in Arp 220\nThe radiative transfer modelling of the molecular gas in Arp 220 is consistent with a warm ($\\sim$~40~K), moderately dense (10$^{3.4}$~cm$^{-3}$ molecular gas component. The best fit molecular gas density is a factor of $\\sim$4 higher than what was found by \\citet{Rangwala2011} using $Herschel$ FTS observations; however, the best fit $n_{\\rm{H_{2}}}$\\ and \\tkin\\ of the two sets of models are consistent within the 1$\\sigma$ uncertainties. We note that this result should not be perceived as a lack of very dense molecular gas as been presented in \\citet{Scoville2015}. Our results are averaged over a $\\sim$700~pc scale which may be dominated by a less dense medium, most likely from the diffuse kiloparsec disk. \n\nWithin the 1$\\sigma$ range, a warm, moderately dense gas component is favoured (Table \\ref{tab:results} and Figure \\ref{fig:arp220prob}). This result is very similar to other advanced mergers such as VV~114 \\citep{Sliwa2013} and NGC~2623 \\citep{Sliwa2017a} where, on average over $\\sim$1~kpc scales, a warm, moderately dense molecular gas is the best fit solution. This can be explained by feedback from massive star formation, which these mergers will experience intensely over the course of the merger process and possibly outflows. The feedback in the form of stellar winds and supernova from massive stars and the outflows that have been observed \\citep[e.g.][]{Cicone2014} are able to push back on the molecular gas, relieving the pressure while decreasing the molecular gas volume density. \\citet{Sakamoto2009} have observed a molecular P-Cygni profile suggesting $\\sim$100~km s$^{-1}$\\ outflows from the nuclei, which can explain our picture of a less dense medium traced by CO. Within early\/intermediate stage mergers we should see a denser molecular medium over roughly kiloparsec scales traced by CO and its isotopologues since the molecular gas is likely still inflowing to the central regions of the merging systems thus increasing the pressure and volume density of the molecular gas \\citep{Sliwa2017a}. In Arp~220, the volume density and thus thermal pressure on $\\sim$kiloparsec scales will likely be on the rise as the two nuclei are on the verge of the final coalescence. \n\n\n\\subsection{[$^{12}$CO]\/[$^{13}$CO]\\ and [$^{12}$CO]\/[C$^{18}$O]\\ Abundance Ratios in Arp 220\nThe R$_{10}$ and R$_{21}$ line ratios of Arp 220 (see Table \\ref{tab:lineratios}) on a position by position basis are typical for ULIRGs but abnormally high when compared with normal disk galaxies \\citep[e.g.][]{Davis2014} The R$_{10}$ line ratio is higher (at some positions) than other LIRGs such as Arp~299 and Arp~55 \\citep{Casoli1999,Sliwa2017a}. However, these high R$_{10}$ line ratios are more common for advanced major mergers such as VV~114, IRAS~13120-5453 and NGC~2623 \\citep{Sliwa2013,Sliwa2017a,Sliwa2017c,Saito2015}. The possible explanations for the higher line ratios include optical depth effects, increased [$^{12}$CO]\/[$^{13}$CO]\\ abundance ratio via some mechanism or excitation effects \\citep[e.g][]{Casoli1992,Henkel1993,Taniguchi1999}.\n\nOur radiative transfer analysis allows the [$^{12}$CO]\/[$^{13}$CO]\\ and [$^{12}$CO]\/[C$^{18}$O]\\ abundances to vary as a free parameter. We find that for Arp 220, the most probable abundance ratio is 90-125, roughly 3 times higher than the ISM value around the Galactic center \\citep[e.g.][]{Milam2005}. In our Galaxy, the abundance ratio increases with increasing radius. If Arp 220 had no enhancements in [$^{12}$CO]\/[$^{13}$CO], and had a\nsimilar [$^{12}$CO]\/[$^{13}$CO]\\ abundance gradient as observed in our Galaxy,\nthen we would expect values of [$^{12}$CO]\/[$^{13}$CO]~$\\leq$50 since the molecular gas in Arp~220 is concentrated within $\\sim$2-3~kpc. Even for NGC~6240, the most probable abundance ratio is 98 \\citep{Tunnard2015b}. \n\nOptical depth effects have been ruled out by our analysis as the best fit solutions have optical depths $>$ 1 for $^{12}$CO\\ and $<<$1 for $^{13}$CO.\n\nSelective photo-dissociation was thought to be a possible mechanism to drive the unusually high [$^{12}$CO]\/[$^{13}$CO]\\ abundance ratio we observe; however, this mechanism is ruled out because C$^{18}$O, a molecule that should be destroyed via photo-dissociation as fast or faster if less abundant than $^{13}$CO, is relatively strong in emission. The likely dominant source of the increased [$^{12}$CO]\/[$^{13}$CO]\\ and [$^{12}$CO]\/[C$^{18}$O]\\ abundance ratios is stellar nucleosynthesis. Short-lived ($\\leq$~10~Myr) massive stars produce \\element[][12]{C}\\ and \\element[][18]{O}, but \\element[][13]{C}\\ is produced in the envelopes of long-lived low\/intermediate mass during the red giant phase. If extreme starbursts have a top-heavy initial mass function, then\nmany massive OB stars will be formed in the Arp 220 starbursts,\nwhose supernovae will enrich the ISM, and we would expect to\nsee increased abundances of \\element[][12]{C}\\ and \\element[][18]{O}. \\citet{Matsushita2009} and \\citet{Greve2009} show that the $^{13}$CO\/C$^{18}$O\\ line ratio is $\\sim$~1 for Arp 220. The $^{13}$CO\/C$^{18}$O\\ line ratio in spirals has been shown to be on average $\\sim$6 \\citep{JD2017}, 6 times higher than for Arp 220. This strongly suggests that stellar nucleosynthesis has enriched the ISM of Arp 220. This is a similar result in other advanced mergers such as NCG 2623 and IRAS 13120-5453 \\citep{Sliwa2017c}. \n\nFractionation of the \\element[][12]{C}\\ and \\element[][13]{C}\\ has also been considered as playing a role in the relative carbon isotope abundances that we find in U\/LIRGs. The most important reaction to exchange carbon isotopes is\n\\begin{equation}\\label{eqn:frac}\n^{13}\\rm{C^{+}} + \\rm{^{12}CO} \\rightleftharpoons \\rm{^{12}C^{+}} + \\rm{^{13}CO} + 35\\rm{K}\n\\end{equation}\npredicted by \\citet{Watson1976}. The forward reaction is exothermic and in cold molecular gas can dominate isotope fractionation favouring the formation of more $^{13}$CO. In hotter temperatures, both the forward and reverse reactions are about equally probable and would not affect the overall abundance ratio greatly. Since the molecular gas in Arp 220 is warm ($\\sim$ 40K), we can rule out fractionation as a possible contaminate in our [$^{12}$CO]\/[$^{13}$CO]\\ abundance. \\citet{Langer1984} showed that oxygen fractionation is very small and the oxygen isotope ratios reflect stellar nucleosynthesis processes further supporting the stellar nucleosynthesis enrichment scenario for the observed line ratios. \n\n\\subsection{HCN and HCO$^+$\\ Line Ratios: AGN or Starburst?\nAs stated in the introduction, a linear relation between HCN~$J$~=~1$-$0\\ and infrared luminosity was found and interpreted to be a direct correlation of dense gas and star formation \\citep{Gao2004a}. Within the last decade or so, the nature of HCN~$J$~=~1$-$0\\ as a true dense gas tracer has been questioned. Studies of samples of AGN and starburst systems have found enhancements of HCN~$J$~=~1$-$0\\ emission among the AGN systems. Recently, \\citet{Privon2015} have shown that composite (i.e. AGN and star forming) and star formation dominated systems can also show a similar enhancement in HCN~$J$~=~1$-$0\\ emission. \\cite{Privon2015} found that for AGN, starbursts and composite systems the H$_{10}$ ratios are 1.84$\\pm$0.43, 0.88$\\pm$0.28 and 1.14$\\pm$0.49, respectively. This overlap of values for the different types of systems has complicated the picture of HCN\\ and HCO$^+$\\ emission suggesting multiple processes contributing to the line ratio differences. \n\nNevertheless, we compare the line ratios of Arp~220 and NGC~6240. We find that the line ratios are significantly different between the two sources. Arp~220 is consistent with AGN-like line ratios while NGC~6240 is consistent with a starburst-like ratio, similar to NGC~4039 and the Antennae Overlap region \\citep{Schirm2016}. This result is quite surprising considering the fact that NGC~6240 has two well known AGNs \\citep[e.g.][]{Komossa2003} while the presence of an AGN in Arp 220 is still debatable. The low line ratio found for NGC~6240 may reflect the different area filling factors of the two species as our observations are slightly unresolved with our beam; however, this is not the case for Arp~220 as our observations do resolve the emission and therefore, line ratios within one beam element would be filled with emission from both species. We note that our line ratio does not suggest the presence of an AGN in Arp 220 as our angular resolution does not distinguish the individual nuclei and the other possible mechanism affecting the H$_{10}$ line ratios \\citep[e.g][]{Privon2015}. Radiative transfer analyses including just these dense gas tracers are required to understand what is driving the line ratios. We also note that other effects such as infrared pumping of HCN and recombination of HCO$^+$\\ may be factors that need to be taken into consideration. \n\n\n\\section{Conclusions\nWe present new PdBI observations of several molecular gas tracers for the two nearby major mergers Arp~220 and NGC~6240. The observations show a wealth of chemistry in Arp~220 and different conditions between the two sources. The main results are summarised as follows:\n \\begin{enumerate}\n \\item $^{13}$CO $J$=2$-$1\\ \/ $J$=1-0 line ratio reveal that molecular gas conditions vary across the disk of Arp~220 where the line ratio increases from east to west.\n \\item The best fit molecular gas conditions at the peak position of $^{13}$CO $J$=1$-$0\\ imply a warm (40~K), moderately dense (10$^{3.4}$~cm$^{-3}$) molecular gas component. This solutions differs in volume density by a factor of $\\sim$4 from the analysis of \\cite{Rangwala2011} most likely because of the spatial resolution difference.\n\t\\item The HCN\/HCO$^+$~$J$~=~1$-$0\\ line ratio for Arp~220 is greater than 2, while for NGC~6240 the line ratios are below 1. \n\t\\item The [$^{12}$CO]\/[$^{13}$CO]\\ abundance ratio for Arp~220 is a factor of $\\sim$3 or more higher than the central value in the Milky Way, while the [$^{12}$CO]\/[C$^{18}$O]\\ abundance ratio is quite low indicating the enhancement of C$^{18}$O\\ in the ISM. The likely explanation that can explain both [$^{12}$CO]\/[$^{13}$CO]\\ and [$^{12}$CO]\/[C$^{18}$O]\\ is an enrichment of $^{12}$C and $^{18}$O in the ISM via stellar nucleosynthesis. Higher-resolution and sensitive observations of $^{13}$CO\\ and C$^{18}$O\\ are needed to look for specific spatial variations as seen in IRAS~13120-5453 \\citep{Sliwa2017c}.\n\t\n\tThese data, along with other sources (see Appendix \\ref{sec:release}) will be made available to the public for download with the intention to use for future analyses and comparisons with new observations. \n \\end{enumerate}\n \n\n\n\n\\begin{acknowledgements\nWe thank the referee for their comments and suggestions that improved this manuscript. We thank K. Sakamoto for passing along the SMA $^{12}$CO $J$=3$-$2\\ observations of Arp 220. KS thanks Julia Kamenetzky for her MCMC code made available via github. KS also thanks L. Barcos-M\\~unoz, A.K. Leroy, E. Schinnerer, F. Walter, and L.K. Zschaechner for fruitful discussions. Based on observations carried out with the IRAM Interferometer PdBI. IRAM is supported by INSU\/CNRS (France), MPG (Germany) and IGN (Spain). The Submillimeter Array is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics and is funded by the Smithsonian Institution and the Academia Sinica.\n\nThis paper makes use of the following ALMA data: ADS\/JAO.ALMA$\\#$2011.0.00018.SV. \nALMA is a partnership of ESO (representing its member states), NSF (USA) and \nNINS (Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan), and KASI (Republic of Korea), \nin cooperation with the Republic of Chile. The Joint ALMA Observatory is \noperated by ESO, AUI\/NRAO and NAOJ.\n\\end{acknowledgements}\n\n\\bibliographystyle{aa.bst}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\IEEEPARstart{D}{eep} discriminative models (DDMs) have been extensively studied recently to solve a variety of computer vision problems, with applications in image classification, action recognition, and object detection, to name a few.\nDDMs use deep neural networks (DNNs)~\\cite{chendeepage, Yan2014Age,chen_using_2017,huang2018mixture} to formulate the input-output mapping in discriminative models.\nDue to a shortage of well-labeled training data, \\emph{i.e.}~without noise and imbalanced distribution problems, learning DDMs is particularly challenging. \nConsiderable literature has grown up around the theme of how to learn DDMs effectively~\\cite{rothe2018deep,huang_soft-margin_2017,ruiz2018fine}.\n\nOne typical approach is to learn more discriminative features through rather deep neural networks and cost-sensitive discriminative functions~\\cite{gao_age_2018,parkhi2015deep,niu_ordinal_2016,rothe2018deep,agustsson2017anchored}.\nThe other typical approaches are to reweight training samples according to estimation errors~\\cite{cui2019class,khan2019striking} or gradient directions~\\cite{ren2018learning} (\\emph{i.e.}~meta learning).\nAlthough these approaches can partially alleviate the noise and imbalanced distribution problem of training data, they are not consistent with the cognitive process of human beings; the difference is we learn new things based on previously learned knowledge.\nIn addition, these approaches do not provide a way to distinguish noisy and underrepresented examples, and thus cannot fundamentally avoid noisy and biased solutions.\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[width=0.9\\textwidth]{Figure1_v2.pdf}\n\t\\caption{The Motivation of considering underrepresented examples in SPL. \\textbf{(a):} The histogram shows the number of facial images of different ages, and the average entropy curve indicates the predictive uncertainty. We observe that the \\emph{underrepresented examples} always have high entropy. \\textbf{(b):} Underrepresented examples are selected only in SPUDRFS at an early pace. Because the underrepresented examples tend to have relatively large loss, they would not be selected at an early pace. \\textbf{(c):} A new self-paced learning method: easy and underrepresented examples first. This builds on the intuition that the underrepresented examples are prone to incur unfair treatment since they are the ``minority'' in training data.}\n\t\\label{Figure1}\n\\end{figure*}\n Intuitively, learning DDMs from easy to hard may be more reasonable because already-learned knowledge, in such a process, can be leveraged to learn DDMs. \nIn addition to this, the noisy and underrepresented examples could be recognized by virtue of already-learned knowledge.\nHence, we resort to a gradual learning strategy \\emph{i.e.}, self-paced learning~\\cite{Kumar2010Self, jiang2015self, ma2017self, ijcai2017}.\nSo far, however, there are few studies on learning DDMs in a self-paced manner.\nThen, a natural question arises: \\emph{can SPL guide DDMs to achieve more robust and less biased solutions?}\n\n\nAn underlying problem in SPL, which is firstly shown by this study, is that it assumes the distribution of training data is balanced, and thus the bias of solutions may be further exacerbated when such an assumption fails.\nThe reason is that the underrepresented examples, due to \\emph{large} prediction error, would not be selected for training at early paces, thereby incurring unfair treatment.\nThis means existing SPL methods have fairness issue.\n\n\n\nTo address the fairness issue in existing SPL methods as well as answer the above questions, this paper proposes a new self-paced learning method for learning DDMs.\nIt tackles the fundamental ranking and selection problems in SPL from a new perspective: fairness.\nOn one hand, our new method keeps SPL's advantage in robustness.\nOn the other hand, it prevents seriously biased solutions induced by SPL.\nThe major contributions of this work include:\n\\begin{itemize}\n\t\\item We for the first time, show that existing SPL methods may lead to seriously biased solutions. To address this problem, we propose a new SPL method: easy and underrepresented examples first. This combines with a typical DDM, \\emph{i.e.}, deep regression forests (DRFs), can lead to a new model---deep regression forests with consideration of underrepresented examples (SPUDRFs). The new model shows considerable performance improvement in both accuracy and fairness.\n\n\n\t\\item To further promote robustness, in SPUDRFs, we define capped likelihood function with respect to DRFs' parameters so as to further exclude noisy examples.\n\t\\item To measure regression fairness, we define a fairness metric for regression problems, which can reflect fairness concerning some sensitive attributes. \n\t\\item For verification, we validate SPUDRFs on three computer vision tasks: (\\romannumeral1) facial age estimation, (\\romannumeral2) head pose estimation, and (\\romannumeral3) gaze estimation.\n\tExtensive experimental results on the Morph \\uppercase\\expandafter{\\romannumeral2}~\\cite{ricanek2006morph}, FG-NET~\\cite{panis2016overview}, BIWI~\\cite{fanelli2013random} , BU3DFE~\\cite{pan2016mixture} and MPIIGaze~\\cite{zhang2015appearance} datasets demonstrate the efficacy of our proposed new self-paced method.\n\tOn all the above tasks, SPUDRFs almost achieve state-of-the-art performance.\n\\end{itemize}\n\nA preliminary version of this work was published in~\\cite{pan2020self}.\nThis paper significantly improves~\\cite{pan2020self} in the following aspects. \n(\\romannumeral1) We extend our model to incorporate capped likelihood, which could further promote robustness.\nThe robust SPUDRFs can recognize and exclude examples with labeling noise, whereas the original is unable to do so, and only places more emphasis on reliable examples. \n(\\romannumeral2) We extend our model to combine with various weighting schemes, whereas \\cite{pan2020self} merely adopts a mixture weighting scheme.\n(\\romannumeral3) We define the fairness metric for regression in this work and show obvious fairness improvement of SPUDRFs against the original SP-DRFs. \n(\\romannumeral4) We also evaluate SPUDRFs on a new computer vision task, \\emph{i.e.}, gaze estimation, and demonstrate its validity on the MPIIGaze dataset.\n(\\romannumeral5) Both robustness and fairness evaluation results are added, along with more comprehensive discussions.\n\nThis paper is organized as follows. Sec.~\\ref{sec:related work} introduces related work on SPL, and DDMs for facial age estimation, head pose estimation and gaze estimation. Sec.~\\ref{sec:SPUDRFs} details our proposed SPUDRFs. Sec.~\\ref{sec:robust SPUDRFs} introduces robust SPUDRFs.\nSec.~\\ref{sec:fairness metric} defines a fairness metric for regression problem.\nSec.~\\ref{sec:experiment} exhibits and analyzes the experimental results on three computer vision tasks and five related datasets. Sec.~\\ref{sec:discussion} discusses the strengths and potential weaknesses of this work, followed by Sec.~\\ref{sec:conclusion} concluding this paper with perspectives.\n\n\\section{Related Work}\n\\label{sec:related work}\nThis section reviews SPL methods and DDMs-based facial age estimation, head pose estimation and gaze estimation.\n\n\n\\subsection{Self-Paced Learning}\nSelf-paced learning, as a gradual learning method, builds on the intuition that, rather than considering all training samples simultaneously, the learning should be presented with the training data from easy to difficult~\\cite{Kumar2010Self,meng_theoretical_2017}.\nTo date, a great deal of study on self-paced learning (SPL) has been undertaken, mostly about learning conventional discriminative models, \\emph{e.g.}~SVM, logistic regressor.\nThere also exists some literature in which SPL is employed for clustering problems~\\cite{Huang2021DSMVC,Huang2021NSMVC, Guo2019adaptive}.\n\n\n\nLearning conventional discriminative models in a self-paced manner exhibits superiority over traditional methods.\nIn~\\cite{Kumar2010Self}, Kumar \\emph{et al.}~proposed the original SPL framework in a regime that suggests processing the samples in order of an easy to hard order. \nIn~\\cite{jiang2015self}, Zhao \\emph{et al.}~generalized the hard weighting scheme in SPL to a soft weighting scheme, which promoted discrimination accuracy.\nIn~\\cite{ma2017self}, Ma \\emph{et al.}~proposed self-paced co-training, where SPL is applied for multi-view or multi-modality problems.\nIn~\\cite{ren2018self}, Ren \\emph{et al.}~introduced capped-norm into the objective function of SPL, so as to further exclude the interference of noisy examples.\nIn fact, the above work can be cast as incorporating SPL with conventional classifiers, \\emph{e.g.}, SVM, logistic regressors, \\emph{etc.}.\nTo incorporate SPL with regression models, in~\\cite{han2017self}, Han \\emph{et al.}~learned a mixture of regressors in a self-paced manner, and added an $\\ell_{1,2}$ norm regularizer to avoid poorly conditioned linear sub-regressors.\n\n\n\nIn computer vision community, recently some researchers have realized that learning DDMs in a self-paced manner may lead to more robust solutions.\nOne of the few studies is~\\cite{ijcai2017}, where Li \\emph{et al.}~sought to enhance the learning robustness of CNNs by SPL.\nHowever, they omitted one important problem in learning discriminative models: the imbalanced distribution of training data.\n\n\n\nOur work is inspired by the existing studies~\\cite{yang2019self,jiang2014self} which take the class diversity in sample selection of SPL into consideration.\nJiang \\emph{et al.}~\\cite{jiang2014self} proposed to ensure the class diversity of samples at the early paces in self-paced training.\nYang \\emph{et al.}~\\cite{yang2019self} defined a metric called ``complexity of image category\" to measure the number of samples in each category, as well as jointly classifying difficulty, and used it to select samples.\nThe two methods mentioned above find that a lack of class diversity in sample selection may result in seriously biased solutions.\nHowever, what causes this problem has not been studied. \nThis work shows that the cause is fundamentally the unfairness of sample ranking in SPL, where underrepresented examples may often have large losses and not be selected at early paces.\nOn the other hand, \\cite{yang2019self,jiang2014self} are only suitable for classification, but not for regression in which the output targets are continuous and high dimensional.\nThis paper will go further in this direction, aiming to tackle the fundamental drawback in SPL---ranking unfairness---and to integrate SPL with DDMs.\n\n\n\\subsection{Facial Age Estimation}\nFacial age estimation has been extensively studied for a decade in the computer vision community.\nIn recent years, a large and growing body of literature~\\cite{niu_ordinal_2016,chen_using_2017,gao_age_2018,shen_deep_2018,li2019bridgenet} has been proposed for age estimation with varying degrees of success.\nOrdinal-based approaches~\\cite{niu_ordinal_2016,chen_using_2017} are the best-known and have demonstrated improved results.\nFor example, Niu \\emph{et al.}~proposed to estimating age through a set of sequential binary queries---each query refers to a comparison with a predefined age---to exploit the inter-relationship (ordinal information) among age labels.\nFurthermore, Gao \\emph{et al.}~\\cite{gao_age_2018} proposed DLDL-v2 to explore the underlying age distribution patterns, so as to effectively accommodate age ambiguity.\nShen \\emph{et al.}~\\cite{shen_deep_2018} used VGG-16 to extract facial features, and mapped the extracted features to age by deep regression forests (DRFs).\nLi \\emph{et al.}~\\cite{li2019bridgenet} proposed BridgeNet to predict age by mixing local regression results, where local regressors are learned on partitioned data space. \nDeng \\emph{et al.}~\\cite{deng2021pml} proposed an age classification method to deal with long-tailed distribution, where a variational margin is used to minimize the influence of head classes that misleads the prediction of tailed samples. \nOverall, these DDMs-based approaches have improved age estimation performance significantly. However, they plausibly ignore one problem: the interference with facial images duo to PIE (\\emph{i.e.}~pose, illumination and expression) variation, occlusion, misalignment and so forth.\nMore importantly, these approaches mostly ignore the fact that the training data is not distributed uniformally.\n\n\n\\subsection{Head Pose Estimation}\nHead pose estimation has attracted vast attention from the computer vision community for a long time.\nRecently, an increasing amount of DDMs-based methods have been proposed for head pose estimation and have dramatically boosted estimation accuracy.\nOf these, Riegler \\emph{et al.}~\\cite{riegler2013hough} utilized convolutional neural networks\n(CNNs) to extract patch features within facial images, and more precise results were achieved.\nIn~\\cite{huang2018mixture}, Huang \\emph{et al.}~proposed multi-modal deep regression networks, with multi-layer perceptron (MLP) architecture, to fuse the RGB and depth images for head pose estimation.\nIn~\\cite{wang2019deep}, Wang \\emph{et al.}~proposed a deep coarse-to-fine network to further boost pose estimation accuracy.\nIn~\\cite{ruiz2018fine}, Ruiz \\emph{et al.}~used a large, synthetically expanded head pose dataset to train a rather deep multi-loss convolutional neural networks (CNNs), gaining obvious accuracy improvement.\nIn~\\cite{kuhnke2019deep}, Kuhnke \\emph{et al.}~proposed a domain adaptation method for head pose estimation where shared label spaces across different domains are assumed.\nAlthough accuracy has improved, handling noisy examples in the above models is not straightforward. \nMoreover, the methods mentioned above seldom deal with imbalanced training data, which may widely exist in visual problems.\n\n\n\\subsection{Gaze Estimation}\nAppearance-based gaze estimation learns a mapping from facial or eye image to gaze.\nRecently, a considerable amount of DDMs based approaches have been proposed for person-independent gaze estimation.\nFor example, in~\\cite{zhang2017s}, Zhang \\emph{et al.}~used spatial weights VGG-16 to encode face image to gaze direction. \nThe spatial weights were used to flexibly suppress or enhance features from different facial regions. \nIn the literature~\\cite{krafka2016eye}, Krafka \\emph{et al.}~implemented a CNNs-based gaze tracker by mapping captured images from the left eye, right eye and face to gaze. The location and size of the head in the original image are provided for mapping networks. \nIn~\\cite{fischer2018rt}, Fischer \\emph{et al.}~adopted two backbone networks to extract features from two eye regions for gaze regression, and achieved improved robustness. \nInstead of predicting gaze directly, in~\\cite{park2018deep}, Park \\emph{et al.}~proposed another method which maps the single eye image to an intermediate pictorial representation as additional supervision to simplify 3D gaze direction estimation. \nWang \\emph{et al.}~\\cite{wang2019generalizing} proposed a Bayesian adversarial learning approach to address the appearance, head pose variations in gaze estimation, and overfitting problem.\nXiong \\emph{et al.}~\\cite{xiong2019mixed} proposed mixed effect neural networks to combine global and person specific gaze estimation results together.\nMeanwhile, other studies show that facial images can help gaze estimation.\nIn \\cite{biswas2021appearance, cheng2020gaze, cheng2020coarse}, multi-channel estimation architectures were adopted to utilize facial image and eye images jointly. \nThese methods outperform the ones that only use eye images as input for gaze estimation.\nIn addition to person-independent methods, there is some literature about person-specific gaze adaption methods~\\cite{park2019few, yu2019improving, chen2020offset}, which have boosted performance dramatically when a few annotated examples are used for calibration.\nThese methods indeed improve gaze estimation results; however, whether we should learn DDMs for gaze estimation in a gradual learning manner has not been discussed, nor has there been discussion of how to deal with imbalanced distribution of training data.\n\n\\section{Self-Paced Deep Regression Forests with Consideration on Underrepresented Samples}\n\\label{sec:SPUDRFs}\nThis section first reviews the basic concepts in DRFs; and introduces the objective formulation for SPUDRFs, as well as variant weighting strategies and an underrepresented example augmentation method; and finally details the optimization algorithm.\n\n\\subsection{Preliminaries}\n\nDeep regression forests (DRFs), as a deep regression model, connect deep neural netwoks (DNNs) to regression forests.\nWe start by reviewing the basic concepts in DRFs~\\cite{shen_deep_2018}.\t\n\n\n\\noindent \\textbf{Deep Regression Tree.} DRFs usually consist of a number of deep regression trees, each of which, given input-output pairs $\\left\\{\\mathbf{x}_i, y_i\\right\\}_{i=1}^N$, map extracted features through DNNs to target output by passing a regression tree. Further, a regression tree $\\mathcal{T}$ consists of split (or decision) nodes $\\mathcal{N}$ and leaf (or prediction) nodes $\\mathcal{L}$~\\cite{shen_deep_2018} (see Fig.~\\ref{Figure1}).\nSpecifically, each split node $n \\in \\mathcal{N}$ determines whether a sample $\\mathbf{x}_i$ goes to left or right sub-tree; each leaf node $\\ell \\in \\mathcal{L}$ represents a Gaussian distribution $p_{\\ell}(y_i)$ with mean $\\mu_l$ and variance $\\sigma^2_l$---parameters of output distribution defined for each tree $\\mathcal{T}$.\n\n\\noindent \\textbf{Split Node.}\nSplit node represents a split function $s_{n}(\\mathbf{x}_i ; \\bm{\\Theta}) : \\mathbf{x}_i \\rightarrow[0,1]$, where $\\bm{\\Theta}$ denotes the parameters of DNNs, as in Fig.~\\ref{Figure1}(c).\nConventionally, $s_{n}(\\mathbf{x}_i ; \\bm{\\Theta})$ is formulated as $\\sigma\\left(\\mathbf{f}_{\\varphi(n)}(\\mathbf{x}_i; \\bm{\\Theta})\\right)$, where $\\sigma(\\cdot)$ denotes a sigmoid function while $\\varphi(\\cdot)$ denotes an index function specifying the $\\varphi(n)$-th element of $\\mathbf{f}(\\mathbf{x}_i; \\bm{\\Theta})$ in correspondence with the split node $n$, and $\\mathbf{f}(\\mathbf{x}_i; \\bm{\\Theta})$ denotes the extracted features through DNNs.\nAn example to illustrate the sketch chart of the DRFs is shown in Fig.~\\ref{Figure1}(c), where $\\varphi_1$ and $\\varphi_2$ are two index functions for two trees.\nFinally, the probability of the sample $\\mathbf{x}_i$ falling into the leaf node $\\ell$ is given by:\n\\begin{equation}\n\t\\label{Eq.1}\n\t\\omega_\\ell( \\mathbf{x}_i | \\bm{\\Theta)}=\\prod_{n \\in \\mathcal{N}} s_{n}(\\mathbf{x}_i ; \\bm{\\Theta})^{[\\ell \\in \\mathcal{L}_{n_{\\text{left}}}]}\\left(1-s_{n}(\\mathbf{x}_i ; \\bm{\\Theta})\\right)^{\\left[\\ell \\in \\mathcal{L}_{n_{\\text{right}}}\\right]},\n\\end{equation}\nwhere $[\\cdot]$ denotes an indicator function conditioned on the argument. \nIn addition, $\\mathcal{L}_{n_\\text{left}}$ and $\\mathcal{L}_{n_\\text{right}}$ correspond to the sets of leaf nodes owned by the sub-trees $\\mathcal{T}_{n_\\text{left}}$ and $\\mathcal{T}_{n_\\text{right}}$ rooted at the left and right children ${n}_{l}$ and ${n}_{r}$ of node $n$.\n\n\t\n\\noindent \\textbf{Leaf Node.} \nFor a tree $\\mathcal{T}$, each leaf node $\\ell \\in \\mathcal{L}$ defines a Gaussian distribution over output $y_i$ conditioned on $\\left\\{\\mu_l,\\sigma^2_l\\right\\}$.\nSince each example $\\mathbf{x}_i$ has the probability $\\omega_\\ell(\\mathbf{x}_i | \\bm{\\Theta)}$ to reach leaf node $\\ell$, considering all leaf nodes, the predictive distribution over $y_i$ shall sum all leaf distribution weighted by $\\omega_\\ell(\\mathbf{x}_i | \\bm{\\Theta)}$:\n\\begin{equation}\n\t\\label{Eq.2}\n\tp_{\\mathcal{T}}(y_i | \\mathbf{x}_i ; \\bm{\\Theta}, \\bm{\\pi})=\\sum_{\\ell \\in \\mathcal{L}} \\omega_\\ell( \\mathbf{x}_i | \\bm{\\Theta)} p_{\\ell}(y_i),\n\\end{equation}\nwhere $\\bm{\\pi}$ represents the distribution parameters of all leaf nodes associated with tree $\\mathcal{T}$.\nNote that $\\bm{\\pi}$ varies along with tree $\\mathcal{T}_k$, and thus we rewrite them in terms of $\\bm{\\pi}_k$.\n\n\\noindent \\textbf{Forests of Regression Trees.}\nSince a forest $\\mathcal{F}$ consists of a number of deep regression trees $\\left\\{\\mathcal{T}_1,...,\\mathcal{T}_k\\right\\}$, the predictive output distribution shall consider all trees:\n\\begin{equation}\n\t\\label{Eq.3}\n\tp_{\\mathcal{F}}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta},\\bm{\\Pi} \\right)\n\t=\n\t\\frac{1}{K}\\sum_{k=1}^K p_{\\mathcal{T}_k}\\left(y_i|\\mathbf{x}_i; \\bm{\\Theta}, \\bm{\\pi}_k\\right),\n\\end{equation}\nwhere $K$ is the number of trees and $\\bm{\\Pi}=\\left\\{\\bm{\\pi}_1,...,\\bm{\\pi}_K\\right\\}$.\n$p_{\\mathcal{F}}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta},\\bm{\\Pi} \\right)$ denotes the likelihood that the $i^{th}$ sample has output $y_i$.\n\n\n\\subsection{Objective}\n\\label{Uncertainty}\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.50\\textwidth]{Rank1_v1.pdf}\n\t\\caption{The average rank of each age group in the $1^{st}$ pace. SP-DRFs tend to rank the underrepresented examples at the end. In contrast, SPUDRFs tend to rank the underrepresented examples in the front to ensure they are selected for training from the very beginning.}\n\t\\label{grouprank}\n\\end{figure}\n\n\n\\noindent\\textbf{Underrepresented Examples.} Considering underrepresented examples in SPL is one of the main contributions of this work.\nIn this work, underrepresented examples mean ``minority''.\nThe underrepresented level could be measured by predictive uncertainty (\\emph{i.e.}~entropy).\nIn fact, underrepresented examples may incur unfair treatment in early paces due to imbalanced data distribution.\nOur proposed method tackles the ranking and selection problems in SPL from a new perspective: \\emph{fairness}.\n\n\t\n\\noindent\\textbf{Entropy.} Given a sample $\\mathbf{x}_i$, its predictive uncertainty is formulated by calculating the entropy of the conditional output distribution $p_{\\mathcal{F}}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta},\\bm{\\Pi} \\right)$:\n\\begin{equation}\n\t\\label{Eq.4}\n\tH\\left [p_{\\mathcal{F}}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta},\\bm{\\Pi} \\right)\\right] = \\frac{1}{K}\\sum^K_{k=1}H\\left [p_{\\mathcal{T}_k}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta}, \\bm{\\pi}_k \\right)\\right],\n\\end{equation}\nwhere $H\\left[ \\cdot\\right ]$ denotes entropy, and the entropy which corresponds to the $k^{th}$ tree is:\n\\begin{multline}\n\t\\label{Eq.5}\n\tH\\left [p_{\\mathcal{T}_k}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta}, \\bm{\\pi}_k \\right)\\right] =\\\\ -\\int p_{\\mathcal{T}_k}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta}, \\bm{\\pi}_k \\right)\\ln p_{\\mathcal{T}_k}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta}, \\bm{\\pi}_k \\right) dy_i.\n\\end{multline}\nGiven $\\mathbf{x}_i$, the larger the entropy $H\\left [p_{\\mathcal{F}}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta},\\bm{\\Pi} \\right)\\right]$ is, the more uncertain the prediction should be, \\emph{i.e.}, the more underrepresented the sample $\\mathbf{x}_i$ is.\n\n\nThe predictive distribution $p_{\\mathcal{T}_k}\\left(y_i|\\mathbf{x}_i; \\bm{\\Theta}, \\bm{\\pi}_k\\right)$ takes the form $\\sum_{\\ell \\in \\mathcal{L}} \\omega_\\ell( \\mathbf{x}_i | \\bm{\\Theta)} p_{\\ell}(y_i)$.\nBecause the integral of mixture of Gaussians in Eq.~(\\ref{Eq.5}) is non-trivial, we resort to Monte Carlo sampling.\nHowever, because this has a large computational cost~\\cite{huber2008entropy}, we turn to use the lower bound of this integral to approximate the integration:\n\\begin{equation}\n\t\\label{Eq.6}\n\tH\\left [p_{\\mathcal{T}_k}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta}, \\bm{\\pi}_k \\right)\\right]\\approx\\frac{1}{2}\\sum_{\\ell\\in\\mathcal{L}}\\omega_\\ell(\\mathbf{x}_i|\\mathbf{\\Theta})\\left[\\ln \\left(2\\pi \\sigma_\\ell^2\\right)+1\\right].\n\\end{equation}\n\t\n\n\\noindent\\textbf{Formulation.} Let $v_i$ denote a latent variable indicating whether the $i^{th}$ sample is selected $(v_i = 1)$ or not $(v_i = 0)$ for training.\nOur objective is to jointly maximize the log likelihood function with\nrespect to DRFs' parameters $\\bm{\\Theta}$ and $\\bm{\\Pi}$, and learn the latent variables $\\mathbf{v}=\\left(v_1,...,v_N\\right)^T$.\nFig.~\\ref{grouprank} shows that the original self-paced method may miss the underrepresented examples at an early pace, resulting in ignorance.\nConsidering fairness, we prefer to select the underrepresented examples, which probably have higher predictive uncertainty (\\emph{i.e.}~entropy), particularly at an early pace.\nTherefore, we maximize the likelihood function of DRFs coupled with the sample selection term meanwhile considering ranking fairness,\n\\begin{equation}\n\t\\label{Eq.7}\n\t\\max_{\\bm{\\Theta},\\bm{\\Pi}, \\mathbf{v}} \\sum_{i=1}^{N} v_{i} \\left \\{ \\log p_{\\mathcal{F}}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta},\\bm{\\Pi} \\right) + \\gamma H_i \\right \\} + \\lambda\\sum_{i=1}^N v_i ,\n\\end{equation}\nwhere $\\lambda$ controls learning pace ($\\lambda\\geq0$), $\\gamma$ is a trade-off between loss and uncertainty ($\\lambda\\geq0$), $H_i$ denotes the entropy of output distribution $p_{\\mathcal{F}}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta},\\bm{\\Pi} \\right)$.\nWhen $\\gamma$ decays to $0$, the objective function is equivalent to the log-likelihood function with respect to DRFs' parameters $\\bm{\\Theta}$ and $\\bm{\\Pi}$.\nIn Eq.~(\\ref{Eq.7}), each sample is weighted by $v_i$, and whether $\\log p_{\\mathcal{F}}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta},\\bm{\\Pi} \\right) + \\gamma H_i>-\\lambda$ determines\nthe $i^{th}$ sample is selected.\nThat is, the sample with high likelihood value or entropy may be selected.\nThe optimal $v_i^*$ is:\n\\begin{align}\n\t\\label{Eq.8}\n\tv_i^* = \\left\\{ \\begin{array}{ll}\n\t\t1 & \\textrm{if $\\log p_{\\mathcal{F}i} + \\gamma H_i > -\\lambda$}\\\\\n\t\t0 & \\textrm{otherwise}\n\t\\end{array} \\right.,\n\\end{align}\nwhere $p_{\\mathcal{F}}\\left(y_i|\\mathbf{x}_i,\\bm{\\Theta},\\bm{\\Pi} \\right)$ is denoted by $ p_{\\mathcal{F}i}$ for simplicity, and we use this representation in all following parts.\nIteratively increasing $\\lambda$ and decreasing $\\gamma$, samples are dynamically involved for training DRFs, starting with easy and underrepresented examples and ending up with all samples.\nThus, SPUDRFs are prone to achieve more robust and less biased solutions.\n\n\nOne might argue that the noisy and hard examples tend to have high predictive uncertainty, with the result that they are selected in the early pace.\nIn fact, from Eq.~(\\ref{Eq.8}), we observe that whether a sample is selected is determined by both its predictive uncertainty and the log likelihood of being predicted correctly.\nThe noisy and hard examples probably have relatively large loss \\emph{i.e.}~low log likelihood $\\log p_{\\mathcal{F}i}$, and would not be selected in the early pace.\n\n\n\n\\subsection{Weighting Schemes}\nWeighting each sample according to its importance would be more reasonable in SPUDRFs.\nIn the previous section, we adopted a hard weighting scheme in SPUDRFs, as defined in Eq.~(\\ref{Eq.7}), where one sample selected or not is determined by $v_i$.\nSuch a weighting scheme appears to omit the importance of samples.\nSPUDRFs can easily incorporate other weighting schemes, including mixture weighting and soft weighting~\\cite{jiang2014easy}.\n\n\n\\noindent\\textbf{Mixture Weighting.}\nMixture weighting scheme~\\cite{jiang2014easy} weights selected sample by its importance, \\emph{i.e.}, $0\\leq v_i \\leq 1$.\nThe objective function under mixture weighting scheme is:\n\\begin{equation}\n\t\\label{Eq.9}\n\t\\max_{\\bm{\\Theta},\\bm{\\Pi}, \\mathbf{v}} \\sum_{i=1}^{N} v_{i} \\left \\{ \\log p_{\\mathcal{F}i} + \\gamma H_i\\right \\} + \\zeta \\sum_{i=1}^N \\log\\left(v_i + \\zeta\/\\lambda\\right) ,\n\\end{equation}\nwhere $\\zeta$ is a parameter controlling the learning pace.\nWe set $\\zeta=\\left(\\frac{1}{\\lambda'}-\\frac{1}{\\lambda}\\right)^{-1}$, and $\\lambda>\\lambda'>0$ to construct a reasonable soft weighting formulation.\nThe self-paced regularizer in Eq.~(\\ref{Eq.9}) is convex with respect to $v\\in\\left[0,1\\right]$.\nThen, setting the partial gradient of Eq.~(\\ref{Eq.9}) with respect to $v_i$ to be zero will lead the following equation:\n\\begin{equation}\n\t\\log p_{\\mathcal{F}i} + \\gamma H_i + \\frac{\\zeta}{v_i + \\zeta\/\\lambda} = 0.\n\\end{equation}\nThe optimal solution of $v_i$ is given by:\n\\begin{align}\n\tv_i^* = \\left\\{ \\begin{array}{ll}\n\t\t1 & \\textrm{if $\\log p_{\\mathcal{F}i} + \\gamma H_i \\geq -\\lambda' $}\\\\\n\t\t0 & \\textrm{if $\\log p_{\\mathcal{F}i} + \\gamma H_i \\leq -\\lambda $}\\\\\n\t\t\\frac{-\\zeta}{\\log p_{\\mathcal{F}i} + \\gamma H_i} - \\zeta\/\\lambda & \\textrm{otherwise}\n\t\\end{array} \\right.\n\t\\label{Eq.11}\n\\end{align}\nIf either the log likelihood or the entropy is too large, $v^*_i$ equals 1.\nIn addition, if the likelihood and entropy are both too small, $v^*_i$ equals 0.\nExcept for the above two cases, the soft weighting, \\emph{i.e.}, the last line of Eq.~(\\ref{Eq.11}), is adopted.\n\n\\noindent\\textbf{Soft Weighting.}\nA soft weighting scheme~\\cite{jiang2014easy} weights a selected sample with respect to its output likelihood and entropy. Such a weighting scheme includes: linear weighting scheme and logarithmic weighting scheme.\n\n\\noindent\\textbf{Linear weighting.} This scheme linearly assigns weights to samples with respect to output likelihood and entropy. The objective of SPUDRFs under linear weighting scheme is formulated as:\n\\begin{equation}\n\t\\max_{\\bm{\\Theta},\\bm{\\Pi}, \\mathbf{v}} \\sum_{i=1}^{N} v_{i} \\left \\{ \\log p_{\\mathcal{F}i} + \\gamma H_i\\right \\} - \\frac{1}{2} \\lambda \\sum_{i=1}^N \\left(v_{i}^2-2v_{i}\\right),\n\t\\label{Eq.12}\n\\end{equation}\nwhere $\\lambda>0$, $v_i\\in[0,1]$. We set the partial gradient of Eq.~(\\ref{Eq.12}) with respect to $v_{i}$ to be zero, then the optimal solution for $v_i$ is:\n\\begin{align}\n\tv_i^* = \\left\\{ \\begin{array}{ll}\n\t\t\\frac{\\log p_{\\mathcal{F}i} + \\gamma H_i + \\lambda}{\\lambda} & \\textrm{if $\\log p_{\\mathcal{F}i} + \\gamma H_i \\geq -\\lambda $}\\\\\n\t\t0 & \\textrm{otherwise}\n\t\\end{array} .\\right.\n\t\\label{Eq.13}\n\\end{align}\nThe larger the log likelihood and entropy are, the higher the weight associated with the $i^{th}$ sample should be. \n\n\\noindent\\textbf{Logarithmic weighting.} This scheme penalizes the output likelihood and entropy logarithmically. The objective of SPUDRFs under a logarithmic weighting scheme can be formulated as:\n\\begin{equation}\n\t\\max_{\\bm{\\Theta},\\bm{\\Pi}, \\mathbf{v}} \\sum_{i=1}^{N} v_{i} \\left \\{ \\log p_{\\mathcal{F}i} + \\gamma H_i\\right \\} - \\sum_{i=1}^N \\left( \\zeta v_{i} - \\frac{\\zeta^{v_i}} {\\log \\zeta} \\right),\n\t\\label{Eq.14}\n\\end{equation}\nwhere $\\zeta=1-\\lambda$, $0 < \\lambda < 1$ and $v_i\\in[0,1]$. Similarly, the optimal solution for $v_i$ is\n\\begin{align}\n\tv_i^* = \\left\\{ \\begin{array}{ll}\n\t\t\\frac{\\log \\left( \\zeta - \\log p_{\\mathcal{F}i} - \\gamma H_i \\right)}{\\log \\zeta} & \\textrm{if $\\log p_{\\mathcal{F}i} + \\gamma H_i \\geq -\\lambda $}\\\\\n\t\t0 & \\textrm{otherwise}\n\t\\end{array} .\\right.\n\t\\label{Eq.15}\n\\end{align}\n\n\n\n\\subsection{Underrepresented Example Augmentation}\n\nAs previously explained, the SPUDRFs method places more emphasis on underrepresented examples and may achieve less biased solutions.\nSince the intrinsic reason for SPL's biased solutions is the ignorance of underrepresented examples, we further rebalance training data via distribution reconstruction.\nSpecifically, we distinguish the underrepresented examples whose $H_i$ are larger than $\\beta$, from regular examples at each pace and augment them.\nAs the number of underrepresented examples increases through augmentation, the label distribution imbalance problem is alleviated. \n\n\n\\subsection{Optimization}\n\\label{Learning}\nTo optimize SPUDRFs, as defined in Eq.~(\\ref{Eq.7}), we propose a two-step alternative search strategy (ASS) algorithm: (\\romannumeral1) For sample selection, update $\\mathbf{v}$ with fixed $\\bm{\\Theta}$ and $\\bm{\\Pi}$ (\\romannumeral2) update $\\bm{\\Theta}$ and $\\bm{\\Pi}$ with current fixed sample weights $\\mathbf{v}$.\nIn addition, with fixed $\\mathbf{v}$, our DRFs are learned by alternatively updating $\\bm{\\Theta}$ and $\\bm{\\Pi}$.\nIn \\cite{shen_deep_2018}, the parameters $\\bm{\\Theta}$ for split nodes (\\emph{i.e.}~parameters for VGG) are updated through gradient descent since the loss is differentiable with respect to $\\bm{\\Theta}$.\nIn comparison, the parameters $\\bm{\\Pi}$ for leaf nodes are updated by virtue of variational bounding~\\cite{shen_deep_2018} when fixing $\\bm{\\Theta}$.\n\n\t\n\n\\renewcommand{\\algorithmicrequire}{ \\textbf{Input:}}\n\\renewcommand{\\algorithmicensure}{ \\textbf{Output:}}\n\\begin{algorithm}[t]\n\t\\caption{The training process of SPUDRFs.}\n\t\\label{alg:The}\n\t\\begin{algorithmic}[1]\n\t\t\\REQUIRE\n\t\t$\\mathcal{D}=\\left\\{\\mathbf{x}_i, \\mathbf{y}_i\\right\\}_{i=1}^N$.\n\t\t\\ENSURE\n\t\tModel parameters $\\bm{\\Pi}$, $\\bm{\\Theta}$.\n\t\t\\STATE Initialize $\\bm{\\Pi}^{0}$, $\\bm{\\Theta}^{0}$, $\\lambda^0$, $\\gamma^0$.\n\t\t\\STATE \\textbf{while} not converged \\textbf{do}\n\t\t\\STATE \\quad Update $\\mathbf{v}$ by Eq.~(\\ref{Eq.8}).\n\t\t\\STATE \\quad \\quad \\textbf{while} not converged \\textbf{do}\n\t\t\\STATE \\quad \\quad \\quad \\quad Randomly select a batch from $\\mathcal{D}$.\n\t\t\\STATE \\quad \\quad \\quad \\quad Calculate $H_i$ for each sample by Eq.~(\\ref{Eq.6}).\n\t\t\\STATE \\quad \\quad \\quad \\quad Update $\\bm{\\Theta}$ and $\\bm{\\Pi}$ to maximize Eq.~(\\ref{Eq.7}).\n\t\t\\STATE \\quad \\quad \\textbf{end while}\n\t\t\\STATE \\quad \\quad Increase $\\lambda$ and decrease $\\gamma$.\n\t\n\t\t\\STATE \\textbf{end while}\n\t\\end{algorithmic}\n\t\\label{alg1}\n\\end{algorithm}\n\n\n\\section{Robust Self-Paced Deep Regression Forests with Consideration on underrepresented examples}\n\\label{sec:robust SPUDRFs}\nAs has already been discussed in Sec.~\\ref{sec:SPUDRFs}, SPL tends to place more emphasis on reliable examples to achieve more robust solutions.\nHowever, SPL's intrinsic selection scheme can not exclude noisy examples, especially the examples with labeling noise.\nTo alleviate this drawback, we introduce capped-likelihood function in SPUDRFs, which can render an output likelihood with an especially small value as zero:\n\\begin{equation}\n\t\\text{cap}(p_{\\mathcal{F}i},\\epsilon) = \\frac{\\max(p_{\\mathcal{F}i}-\\epsilon, 0)}{p_{\\mathcal{F}i}-\\epsilon}p_{\\mathcal{F}i}, \n\t\\label{cap_equation}\n\\end{equation}\nwhere $\\epsilon$ denotes the threshold and $\\epsilon>0$. \nGiven the output likelihood $p_{\\mathcal{F}_i}$, the capped likelihood $p^c_{\\mathcal{F}_i}$ is defined as $\\text{cap}\\left(p_{\\mathcal{F}_i},\\epsilon\\right)$. \nBecause the log-likelihood values of the noisy examples are prone to be\nsmall, such an operation sets capped likelihood $p^c_{\\mathcal{F}_i}$ to be negatively infinite. \nThus, noisy examples would not be selected, especially not examples with labeling noise.\nSPUDRFs with capped likelihood are defined as robust SPUDRFs. \n\\begin{equation}\n\t\\max_{\\bm{\\Theta},\\bm{\\Pi}, \\mathbf{v}} \\sum_{i=1}^{N} v_{i} \\left \\{ \\log p_{\\mathcal{F}i}^{c} + \\gamma H_i \\right \\} + \\lambda\\sum_{i=1}^N v_i,\n\t\\label{Eq.17}\n\\end{equation}\nThe robust SPUDRFs can be optimized using the optimization method proposed in Sec.~\\ref{Learning}. \nNote that, if $p_{\\mathcal{F}_i}\\leq\\epsilon$, the capped likelihood $p^c_{\\mathcal{F}_i}=0$, that is, $\\log p^c_{\\mathcal{F}_i}=-\\infty$. \nSince $\\lambda$ is a positive factor, maximizing Eq.~(\\ref{Eq.17}) must yield $v_i=0$, meaning that the $i^{th}$ sample would not be selected.\nBy adjusting $\\epsilon$, we can exclude a certain ratio of noisy examples to obtain more robust solutions.\n\n\n\\section{Fairness Metric}\n\\label{sec:fairness metric}\nIn addition to \\emph{accuracy}, fairness should also be an essential metric to measure the performance of a regression model, particularly when the regression targets are related to sensitive attributes.\nFor example, in age estimation, we expect our model to treat the younger and the elder fairly, \\emph{i.e.}, not to result in large MAEs for the former but small MAEs for the latter.\nThe notion of fairness was originally defined concerning a protected attribute such as gender, race or age.\nHowever, the term fairness has a range of potential definitions. Here, we adopt a notion of fairness that is for sensitive features~\\cite{fitzsimons2019general}.\n\\emph{Defining a fairness metric for regression is one of the contributions of this work.}\n\nThe present studies~\\cite{agarwal2019fair, berk2021fairness, komiyama2018nonconvex, zafar2017fairness} mostly refer to fairness constraints in regressions, for example, statistical parity or bounded group loss~\\cite{agarwal2019fair, komiyama2018nonconvex}, but seldom refers to fairness metrics.\nIn this work, we define a new fairness metric for regression models.\nTo be specific, the test dataset is divided into $K$ subsets, each of which is denoted by $\\mathbf{D}_k=\\left\\{\\mathbf{x}_i, y_i| y_i \\in \\mathcal{G}_k\\right\\}$, and $\\mathcal{G}_k$ denotes the $k^{th}$ group.\nA fair model is expected to have the same performance on all subsets, which can be described mathematically as:\n\\begin{equation}\n\t\\label{Eq1_fairness}\n\t\\mathbb{E}_{k}\\left[L \\left(\\hat{y}, y\\right)\\right]= \\mathbb{E}_{l}\\left[L\\left(\\hat{y}, y\\right)\\right] \\quad \\forall k,l \\in \\left\\{1,2,..., K\\right\\}, k\\neq l.\n\\end{equation}\nwhere $\\mathbb{E}_k\\left[\\cdot\\right]$ denotes the expectation with respect to the loss $L\\left( \\cdot \\right)$ over group $\\mathbf{D}_k$, $\\hat{y}$ is the predicted value of the model and $y$ is the real target value. Motivated by $p\\%-$rule \\cite{zafar2017fairness}, which measure classification fairness, we evaluate fairness between two subsets $\\mathbf{D}_k$ and $\\mathbf{D}_l$ as follows:\n\\begin{equation}\n\t\\label{Eq2_fairness}\n\tf\\left( \\mathbf{D}_k,\\mathbf{D} _l\\right) = \\min\\left(\\frac{\\mathbb{E}_{k}\\left[L \\left(\\hat{y}, y\\right)\\right]}{\\mathbb{E}_{l}\\left[L\\left(\\hat{y}, y\\right)\\right]} , \\frac{\\mathbb{E}_{l}\\left[L \\left(\\hat{y}, y\\right)\\right]}{\\mathbb{E}_{k}\\left[L \\left(\\hat{y}, y\\right)\\right]} \\right).\n\\end{equation}\nThe loss expectation ratio characterizes the model's fairness on every two subsets. A small ratio means the performance on such two subsets is significantly distinct, reflecting model bias against a particular subset. \nOn the contrary, a large ratio indicates the losses on $\\mathbf{D}_k$ and $\\mathbf{D}_l$ are similar, indicating the model is relatively fair for $\\mathbf{D}_k$ and $\\mathbf{D}_l$. In particularly, when the ratio is equal to 1, the model satisfies Eq.~(\\ref{Eq1_fairness}), and $\\mathbf{D}_k$ and $\\mathbf{D}_l$ are treated fairly.\n\n\n\\begin{figure}\n\t\\centering\n\t\\subfloat[]{\n\t\t\\includegraphics[width=0.48\\textwidth]{age_new_compressed.pdf}%\n\t\t\\label{fig:age_samples}}\\\\\n\t\\subfloat[]{\n\t\t\\includegraphics[width=0.48\\textwidth]{pose_new_compressed.pdf}%\n\t\t\\label{fig:head_samples}}\\\\\n\t\\subfloat[]{\n\t\t\\includegraphics[width=0.48\\textwidth]{gaze_new_compressed.pdf}%\n\t\t\\label{fig:gaze_samples}}\\\\\n\t\\caption{Example images in the Morph \\uppercase\\expandafter{\\romannumeral2}, FG-NET, BIWI, BU3DFE and MPIIGaze datasets. \\textbf{(a):} The images from the Morph \\uppercase\\expandafter{\\romannumeral2} and FG-NET dataset. \\textbf{(b):} The images from the BIWI and BU3DFE dataset. \\textbf{(c):} The images from MPIIGaze dataset.}\n\t\\label{dataset_samples}\n\\end{figure}\n\n\nFinally, the regression fairness is defined as the expectation of the fairness between any two subsets,\n\\begin{equation}\n\t\\label{Eq3_fairness}\n\t\\text{FAIR} = \\mathbb{E}\\left[f\\left(\\mathbf{D}_i,\\mathbf{D}_j\\right)\\right].\n\n\\end{equation}\nIn SPUDRFs, when we set $\\gamma \\textgreater 0$, all samples are sorted by both likelihood and entropy. As a result, easy and underrepresented samples are selected first, which means samples from all subsets would be selected at early paces. Therefore, our model can alleviate the model's prejudice against different subsets and improve regression fairness. More extensive evaluation results can be found in Sec.~\\ref{fairness_section}.\n\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\subfloat[Morph \\uppercase\\expandafter{\\romannumeral2}]{\n\t\t\\includegraphics[width=1.0\\textwidth]{SPUDRFs_validation_v3_compressed.pdf}%\n\t\t\\label{fig:spu_validation_morph}}\\\\\n\t\\subfloat[MPIIGaze]{\n\t\t\\includegraphics[width=1.0\\textwidth]{Fig3_mpii_v4_compressed.pdf}%\n\t\t\\label{fig:spu_validation_mpii}}\\\\\n\t\\caption{Visualization of the SPL process on the Morph \\uppercase\\expandafter{\\romannumeral2} and MPIIGaze datasets. \\textbf{Left:} The typical worst cases at each pace. The images in each panel are sorted by the increasing paces. The two numbers below each image are the labels (left) and predicted targets (right). \\textbf{Right:} MAE comparison between SP-DRFs and SPUDRFs at each pace. The MAE bins are sorted by the increasing paces.}\n\t\\label{SPUDRFs_validation}\n\\end{figure*}\n\n\n\n\t\\section{Experiment}\n\\label{sec:experiment} \n\\subsection{Datasets and Experimental Setup}\n\\label{setup}\n\\noindent\\textbf{Dataset.} There are five datasets used in our experiments, namely Morph \\uppercase\\expandafter{\\romannumeral2}, FG-NET, BIWI, BU3DFE and MPIIGaze.\n\n\n\\noindent\\textbf{Morph \\uppercase\\expandafter{\\romannumeral2}.} \tThe Morph \\uppercase\\expandafter{\\romannumeral2}~\\cite{ricanek2006morph} dataset contains $55,13$4 face images of $13618$ individuals with unbalanced gender and ethnicity distributions.\nThese images are near-frontal pose, neutral expression, and uniform illumination.\n\n\n\\noindent\\textbf{FG-NET.} The FG-NET~\\cite{panis2016overview} dataset includes $1,002$ color and grey images of $82$ people, with each subject having more than $10$ photos at different ages. \nSince all images were taken in an uncontrolled environment, there is a significant deviation on the lighting, pose, and expression (\\emph{i.e.}~PIE) of faces inside the dataset.\n\n\\noindent\\textbf{BIWI.} The BIWI dataset~\\cite{fanelli2013random} includes $15678$ images collected by a Kinect sensor device for different persons, and head poses with pitch, yaw, and roll angles mainly ranging within $\\pm 60^{\\circ}$, $\\pm 75^{\\circ}$ and $\\pm 50^{\\circ}$.\nThese images are from $20$ subjects, including ten males and six females, where four males have been captured twice with\/without glasses.\n\n\n\\noindent\\textbf{BU3DFE.} The BU-3DFE dataset \\cite{pan2016mixture} is collected from $100$ subjects, of whom $44$ are male, and $56$ are female. Following the work \\cite{pan2016mixture}, we randomly rotated the 3D face models to produce $6000$ images with pitch and yaw angles ranging within $\\pm 30^{\\circ}$ and $\\pm 90^{\\circ}$. \n\n\n\\noindent\\textbf{MPIIGaze.} The MPIIGaze dataset~\\cite{zhang2015appearance} includes $213659$ images from $15$ persons. The number of images for each person is between $1498$ and $34745$. The normalized gaze angle is in the range of $[-20^\\circ, +5.0^\\circ]$ degrees in vertical and $[-25^\\circ, +25^\\circ]$ degrees in horizontal.\nDue to massive deviation of image number amongst different persons, similar to \\cite{zhang2015appearance}, $1500$ images from the left and right eyes of each person were chosen for final experiment. \n\n\n\\noindent\\textbf{Reprocessing and Data Augmentation.}\nMTCNN~\\cite{zhang_joint_2016} was used for joint face detection and alignment on the Morph \\uppercase\\expandafter{\\romannumeral2} and FG-NET datasets.\nIn addition, following~\\cite{shen_deep_2018}, two methods were adopted for data augmentation (\\romannumeral1) random cropping; and (\\romannumeral2) random horizontal flipping. \nOn the BIWI and BU3DFE datasets, only random cropping was adopted for augmentation. \nOn the MPIIGaze dataset, two normalized eye images (\\emph{i.e.}~left and right) were obtained following the work~\\cite{zhang2015appearance}, and\nonly random cropping was used for data augmentation.\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[width=0.92\\textwidth]{Uncertainty_efficacy.pdf}\n\t\\caption{Visualization of leaf node distribution. The $1^{st}$, $3^{rd}$, and $6^{th}$ paces are chosen for visualization. For SP-DRFs, the Gaussian means of leaf nodes (denoted by red points in the second row) concentrate in a small range, resulting in seriously biased solutions. For SPUDRFs, the Gaussian means of leaf nodes (denoted by orange points in the third row) distribute in a reasonable range, resulting in lower MAEs.}\n\t\\label{Uncertainty_efficacy}\n\\end{figure*}\n\n\n\\noindent\\textbf{Parameter Setting.}\nVGG-16~\\cite{Simonyan2015} was employed as the backbone network of SPUDRFs. \nFor MPIIGaze, following \\cite{fischer2018rt}, two VGG-16 networks were used.\nIn addition, the pre-trained models were VGG-16 for MPIIGaze, and VGG-Face~\\cite{parkhi2015deep} for other datasets.\nWhen training VGG-16, the batch size were $32$ for Morph~\\uppercase\\expandafter{\\romannumeral2}, BIWI and BU3DFE, $8$ for FG-NET, and $128$ for MPIIGaze.\nThe drop out ratio was $0.5$.\nThe maximun iterations in each pace was $80k$ for Morph~\\uppercase\\expandafter{\\romannumeral2}, $10k$ for FG-NET, $20k$ for BIWI and BU3DFE, and $5k$ for MPIIGaze.\nSGD optimizer was used for BIWI and Adam optimizer was used for other datasets.\nThe initial learning rate was $2 \\times 10^{-5}$ for Morph~\\uppercase\\expandafter{\\romannumeral2} and BU3DFE, $1 \\times 10^{-5}$ for FG-NET, $0.2$ for BIWI, $3 \\times 10^{-5}$ for MPIIGaze.\nWe reduced the learning rate ($\u00d70.5$) per $10k$ iterations for Morph~\\uppercase\\expandafter{\\romannumeral2}, BIWI and BU3DFE.\nHere, some hyper-parameter settings are slightly different from the Caffe version~\\cite{pan2020self}.\t\n\nThe hyper-parameters of DRFs were: tree number ($5$), tree depth ($6$), output unit number of feature learning ($128$).\nThe iterations to update leaf node predictions was $20$, and the number of images to update leaf node predictions was the whole training set for MPIIGaze and $50$ times of batch size for other datasets.\nIn the first pace, $50\\%$ samples were selected for training.\nHere, $\\lambda$ was set to guarantee the first $50\\%$ samples with large $\\log p_{\\mathcal{F}i} + \\gamma H_i$ values were involved.\n$\\lambda'$ was set to ensure that $10\\%\\sim20\\%$ selected samples adopt soft weighting under a mixture weighing scheme.\nTo promote efficiency, the samples selected in the previous pace were preserved and the rest were ranked for current sample selection.\nIn addition, $\\gamma$ was initialized to be $15$ for Morph \\uppercase\\expandafter{\\romannumeral2} and BIWI, and $5$ for FG-NET, BU-3DFE, and MPIIGaze.\n$\\beta$ was used to recognize underrepresented examples that need to be augmented twice in each pace ($1000$ for BIWI, $400$ for BU3DFE, $2000$ for MPIIGaze).\nThe number of paces was empirically set to be 10 for Morph \\uppercase\\expandafter{\\romannumeral2}, $6$ for BIWI and MPIIGaze, $3$ for FG-NET and BU-3DFE. \nExcept for the first pace, an equal proportion of the rest data was gradually involved in each pace.\n\n\n\\noindent\\textbf{Evaluation Details.} To evaluate regression accuracy, we used the mean absolute error (MAE).\nMAE is defined as $\\sum_{i=1}^{N}\\left|\\hat{y}_{i}-y_{i}\\right|\/N$, where $\\hat{y_{i}}$ represents the predicted output for the $i^{th}$ sample, and $N$ is the total number of images for testing.\nIn addition, for age estimation, CS calculates the percentage of images sorted in the range of $\\left[y_{i}-L, y_{i}+L\\right]$: $CS(L)=\\sum_{i=1}^{N}\\left[\\vert\\hat{y}_{i}-y_{i}\\vert \\leq L\\right]\/N \\cdot 100 \\%$, where $[ \\cdot ]$ denotes an indicator function and $L$ is the error level. \nFurther, to measure regression fairness, we adopt our proposed fairness metric FAIR given in Sec.\\ref{sec:fairness metric}.\n\nTo calculate the above metrics, for Morph \\uppercase\\expandafter{\\romannumeral2}, BIWI and BU3DFE, each dataset was divided into a training set ($80\\%$) and a testing dataset ($20\\%$). The random division was repeated $5$ times, and the reported MAEs were averaged over $5$ times. \nThe leave-one-person-out protocol was used for FG-NET~\\cite{shen_deep_2018} and MPIIGaze~\\cite{zhang2015appearance}, where one subject was used for testing and the left subjects for training.\n\n\n\n\\subsection{Validity of SP-DRFs and SPUDRFs}\n\\label{valid}\nThis section validates the original SPL method as well as the new proposed SPL method for learning DDMs.\nIn the following, SP-DRFs denotes self-paced deep regression forests, which learn DRFs using the original SPL method.\n\n\n\n\\noindent \\textbf{Self-Paced Learning.}\nRecall that learning DDMs in a gradual learning manner is more consistent with the cognitive process of human beings, and the noisy examples can be distinguished by virtue of learned knowledge.\nThat means the learning can place more emphasis on ``reliable\" examples.\nTo show this, Fig.~\\ref{SPUDRFs_validation}(a) illustrates the representative face images at each learning pace of SP-DRFs on the Morph \\uppercase\\expandafter{\\romannumeral2} dataset, sorted by increasing $\\lambda$ and decreasing $\\gamma$.\nThe two numbers below each image are the actual age (left) and predicted age (right).\nWe observe that the training images involved in each pace become more confusing and noisy step by step, compared to images in early paces.\nSince the current model is initialized by the results obtained at the last pace, the updated model is adaptively calibrated by ``reliable'' examples rather than by confusing and noisy ones.\nThus, SP-DRFs have improved MAE and CS compared to DRFs.\nWe observe that SP-DRFs gain improvement by 0.33 in MAE $\\left(2.17\\rightarrow1.84\\right)$, by $1.55\\%$ in CS $\\left(91.3\\% \\rightarrow92.85\\%\\right)$, as shown in Fig.~\\ref{morph_experiment}(a).\n\n\nSome representative eye images in the gradual learning sequence for MPIIGaze dataset are shown in Fig.~\\ref{SPUDRFs_validation}(b). \nThe images in each panel are sorted by increasing $\\lambda$ and decreasing $\\gamma$. \nThe two numbers below each pair of images are the actual gaze direction (left) and the predicted gaze direction (right). \nThe same phenomena---the easy examples are prone to be selected in the early paces, while the confusing and noisy ones are prone to be selected in the later paces---can be observed. \nSince the updated model at each pace is adaptively calibrated by ``reliable'' examples rather than by confusing and noisy ones, thus, the MAE associated with each pace decreases step by step.\nFinally, the MAE of SP-DRFs decreases to be $4.57$, whereas the MAE of DRFs is $4.62$ (see Table.~\\ref{MPII_table}).\n\n\n\\noindent\\textbf{Considering Ranking Fairness.}\n\\label{ExpUnderSamples}\nAs was mentioned in Sec.~\\ref{sec:SPUDRFs}, the existing SPL methods may exacerbate the bias of solutioins.\nFig.~\\ref{Uncertainty_efficacy} visualizes the leaf node distributions of SP-DRFs in the gradual learning process.\nThe Gaussian means $\\mu_l$ associated with the $160$ leaf nodes, where each $32$ leaf nodes are defined for $1$ tree, are plotted in each sub-figures.\nThree paces, \\emph{i.e.}~the $1^{st}$, $3^{rd}$, and $6^{th}$ pace, are randomly chosen for visualization.\nFor clarity, only pitch and yaw angles are shown.\nMeanwhile, the leaf node distributions of SPUDRFs are also visualized in Fig.~\\ref{Uncertainty_efficacy}.\n\n\nIn Fig.~\\ref{Uncertainty_efficacy}, the comparison between SP-DRFs and SPUDRFs validates our proposed new SPL method.\nIn SP-DRFs, because the ranking fairness is not considered, the leaf nodes (red points in the $2^{rd}$ row) are only concentrated in a small range that most samples are distributed over, thus leading to seriously biased solutions.\nThe poor MAEs of SP-DRFs can serve as evidence for this.\nIn contrast, because the ranking fairness is considered in SPUDRFs, the leaf nodes are distributed over a wide range that could cover underrepresented examples, thus improving performance.\nSPUDRFs, in the pitch and yaw directions, achieve the best performance with MAEs of $0.71$ and $0.69$, compared to SP-DRFs with MAEs of $1.05$ and $1.23$ ($47.9\\%$ and $78.3\\%$ improvements). \n\n\n\n\n\n\\subsection{Comparison with State-of-the-art Methods}\n\\label{sec:Comparison}\n\nWe compare SPUDRFs with other state-of-the-art methods on three vision tasks: age estimation, head pose estimation and gaze estimation.\n\n\n\n\n\\noindent\\textbf{Results on Age Estimation.} The comparison between our method and other baselines on the Morph \\uppercase\\expandafter{\\romannumeral2} and FG-NET datasets are shown in Fig.~\\ref{morph_experiment}(a) and Fig.~\\ref{fgnet_experiment}(b).\nThe baseline methods include: LSVR~\\cite{guo_human_2009}, RCCA \\cite{Huerta2014Facial}, OHRank~\\cite{Chang2011Ordinal}, OR-CNN \\cite{niu_ordinal_2016}, Ranking-CNN \\cite{chen_using_2017}, DRFs~\\cite{shen_deep_2018}, DLDL-v2~\\cite{gao_age_2018}, and PML~\\cite{deng2021pml}.\nThe results show some consistent trends.\nFirst, SPUDRFs have superior performance compared to conventional discriminative models, such as LSVR~\\cite{guo_human_2009} and OHRank~\\cite{Chang2011Ordinal}.\nSecond, SP-DRFs consistently outperform other DDMs.\nCompared to DRFs, our gains in MAE are $0.33$ on Morph \\uppercase\\expandafter{\\romannumeral2} and 0.12 on FG-NET, and in CS are $1.55\\%$ and $2.91\\%$, respectively.\nThe promotions demonstrate that learning DRFs in a self-paced manner is more reasonable.\nThird, SPUDRFs outperform SP-DRFs on both MAE and CS.\n\n\nFig.~\\ref{morph_experiment}(b) shows the CS curves of SPUDRFs, SP-DRFs and other baseline methods on the Morph \\uppercase\\expandafter{\\romannumeral2} dataset, and Fig.~\\ref{fgnet_experiment}(b) shows the CS curves of different methods on the FG-NET dataset. \nOn both datasets above, SPUDRFs consistently outperform other DDMs. \nWe observe, the CS of SPUDRFs reaches $93.34\\%$ at error level $L=5$, significantly outperforming DRFs by $2.04\\%$ on the Morph \\uppercase\\expandafter{\\romannumeral2} dataset. \nWe also observe that SPUDRFs outperform DRFs by $4.09\\%$ in CS, on the FG-NET dataset.\nThe CS increase clearly validates our proposed self-paced learning method.\n\n\n\n\\begin{figure}[t] \n\t\\centering \n\t\\subfloat[]{\n\t\t\\begin{tabular}{@{}l|c|c}\n\t\t\t\\hline\n\t\t\tMethod & MAE$\\downarrow$ & CS$\\uparrow$\\\\\n\t\t\t\\hline\n\t\t\t\\hline\n\t\t\tLSVR \\cite{guo_human_2009} & 4.31 & 66.2\\% \\\\\n\t\t\tRCCA \\cite{Huerta2014Facial} & 4.25 & 71.2\\% \\\\\n\t\t\tOHRank \\cite{Chang2011Ordinal} & 3.82 & N\/A \\\\\n\t\t\tOR-CNN \\cite{niu_ordinal_2016} & 3.27 & 73.0\\% \\\\\n\t\t\tRanking-CNN \\cite{chen_using_2017} & 2.96 & 85.0\\% \\\\\n\t\t\tDLDL-v2 \\cite{gao_age_2018}& 1.97 & N\/A \\\\\n\t\t\tDRFs \\cite{shen_deep_2018} & 2.17 & 91.3\\% \\\\\n\t\t\tPML \\cite{deng2021pml} & 2.15 & N\/A \\\\\n\t\t\tSP-DRFs & 1.84 & 92.85\\% \\\\\n\t\t\tSPUDRFs & \\textbf{1.78} & \\textbf{93.34\\%} \\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\t\\label{tab:morph_mae}\n\t}\\\\\n\t\\subfloat[]{\n\t\t\\includegraphics[width=0.42\\textwidth]{morph_cs5.pdf}\n\t\t\\label{fig:morph_cs}\n\t} \n\t\\caption{Quantitative comparison with state-of-the-art methods on the Morph \\uppercase\\expandafter{\\romannumeral2} dataset. \\textbf{Upper:} MAE comparison results. \\textbf{Lower:} CS comparison results.}\n\t\\label{morph_experiment}\n\\end{figure}\n\n\\begin{figure} \n\t\\centering \n\t\\subfloat[]{\n\t\t\\begin{tabular}{@{}l|c|c}\n\t\t\t\\hline\n\t\t\tMethod & MAE$\\downarrow$ & CS$\\uparrow$\\\\\n\t\t\t\\hline\n\t\t\t\\hline\n\t\t\tIIS-LDL \\cite{xin_geng_facial_2013} & 5.77 & N\/A \\\\\n\t\t\tLARR \\cite{guodong_guo_image-based_2008} & 5.07 & 68.9\\% \\\\\n\t\t\tMTWGP \\cite{Yu2010Multi} & 4.83 & 72.3\\% \\\\\n\t\t\tDIF \\cite{han_demographic_2015} & 4.80 & 74.3\\% \\\\\n\t\t\tOHRank \\cite{Chang2011Ordinal} & 4.48 & 74.4\\% \\\\\n\t\t\tCAM \\cite{Luu2013Contourlet} & 4.12 & 73.5\\% \\\\\n\t\t\tDRFs \\cite{shen_deep_2018} & 2.80 & 84.50\\% \\\\\n\t\t\tSP-DRFs& 2.68 & 87.41\\% \\\\\n\t\t\tSPUDRFs & \\textbf{2.64} & \\textbf{88.59\\%}\\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\t\\label{tab:fgnet_mae}\n\t}\\\\\n\t\\subfloat[]{\n\t\t\t\\includegraphics[width=0.42\\textwidth]{fgnet_cs_new1.pdf}\n\t\t\\label{fig:fgnet_cs}\n\t}\n\t\\caption{Quantitative comparison with state-of-the-art methods on the FG-NET dataset. \\textbf{Upper:} MAE comparison results. \\textbf{Lower:} CS comparison results.}\n\t\\label{fgnet_experiment}\n\\end{figure}\n \n\t\n\n\\noindent\\textbf{Results on Head Pose Estimation.} \nRecall that considering underrepresented examples in SPL is particularly crucial when the training data has an imbalanced distribution problem.\nSec.~\\ref{valid} has shown the effectiveness of SP-DRFs and SPUDRFs for head pose estimation tasks on the BIWI dataset.\nFig.~\\ref{Uncertainty_efficacy} shows the Gaussian means of learned leaf nodes of SP-DRFs and SPUDRFs on the BIWI dataset.\nIn SP-DRFs, because the underrepresented examples are neglected, the learned leaf nodes are distributed over a small range, resulting in seriously biased solutions.\nIn SPUDRFs, owing to consideration of underrepresented examples, the learned leaf nodes are distributed over a wide range, leading to steadily improved MAE results. \nThis section aims to show that considering ranking fairness in SPL can obtain more reasonable results.\n\n\n\nTab.~\\ref{headpose_table} shows the comparison results of SP-DRFs and SPUDRFs with recent head pose estimation approaches. \nBecause SVR~\\cite{drucker1997support}, RRF~\\cite{liaw2002classification} and KPLS~\\cite{al2012partial} are all conventional regression methods, they have inferior performance relative to MoDRN.\nSPUDRFs achieve the best performance with an MAE of $0.74$ on the BIWI dataset and $0.82$ on the BU-3DFE dataset, which is state-of-the-art performance. \nSuch a significant gain in MAE $(1.13\\rightarrow0.74)$ on BIWI demonstrates that considering ranking fairness in early paces when training DRFs can lead to much more reasonable results. \nThe explanation is that, as illustrated in Fig.~\\ref{Uncertainty_efficacy}, the learned leaf nodes of SPUDRFs are distributed over a wide range that can cover underrepresented examples.\n \\vspace{-0.4cm}\n \\begin{table}[h]\n \t\\centering\n \t\\caption{Quantitative comparison with state-of-the-art methods. \\textbf{Left:} Comparison results on the BIWI dataset, \\textbf{Right:} Comparison results on the BU-3DFE dataset.}\n \t\\label{headpose_table}\n \t\\begin{tabular}[h]{cc}\n \t\t\\small\n \t\t\\scalebox{1.0}{\n \t\t\t\\begin{tabular}{@{}l|c}\n \t\t\t\t\\hline\n \t\t\t\tMethod & MAE$\\downarrow$\\\\\n \t\t\t\t\\hline\n \t\t\t\t\\hline\n \t\t\t\n \t\t\t\tSVR~\\cite{drucker1997support} & 3.14 \\\\\n \t\t\t\tRRF~\\cite{liaw2002classification} & 3.06 \\\\\n \t\t\t\tKPLS~\\cite{al2012partial} & 2.88 \\\\\n \t\t\t\tSAE~\\cite{hinton2006reducing} & 1.94 \\\\\n \t\t\t\tMoDRN~\\cite{huang2018mixture} & 1.62 \\\\\n \t\t\t\tDRFs~\\cite{shen_deep_2018} & 1.33\\footnotemark[1]\\\\\n \t\t\t\tSP-DRFs & 1.13 \\\\\n \t\t\t\tSPUDRFs & \\textbf{0.74} \\\\\n \t\t\t\t\\hline\n \t\t\t\\end{tabular}\n \t\t}\n \t\t&\n \t\t\\small\n \t\t\\scalebox{1.0}{\n \t\t\t\\begin{tabular}{@{}l|c}\n \t\t\t\t\\hline\n \t\t\t\tMethod & MAE$\\downarrow$\\\\\n \t\t\t\t\\hline\n \t\t\t\t\\hline\n \t\t\t\tSVR~\\cite{drucker1997support} & 4.21 \\\\\n \t\t\t\tSAE~\\cite{hinton2006reducing} & 4.14 \\\\\n \t\t\t\tKPLS~\\cite{al2012partial} & 4.12 \\\\\n \t\t\t\tRRF~\\cite{liaw2002classification} & 4.09 \\\\\n \t\t\t\tMoDRN~\\cite{huang2018mixture} & 3.86 \\\\\n \t\t\t\tDRFs~\\cite{shen_deep_2018} & 0.99 \\\\\n \t\t\t\tSP-DRFs & 0.89 \\\\\n \t\t\t\tSPUDRFs & \\textbf{0.82} \\\\\n \t\t\t\t\\hline\n \t\t\t\\end{tabular}\n \t\t} \\\\\n \t\\end{tabular}\n \\end{table}\n\\footnotetext[1]{The reported results are better than our previous work~\\cite{pan2020self} because we used floating point labels in our current experiment but integral labels in~\\cite{pan2020self}.}\n\t\n\n\\noindent\\textbf{Results on Gaze Estimation.}\nFor gaze estimation, the accuracy comparisons between our proposed SP-DRFs, SPUDRFs and other baselines on the MPIIGaze dataset are shown in Tab.~\\ref{MPII_table}. \nBoth the MAE and standard deviation across all persons are reported. \nNote that the standard deviation of RT-GENE~\\cite{fischer2018rt}, Pict-Gaze~\\cite{park2018deep}, and Ordinal Loss~\\cite{guo2021order} are not reported because the original studies do not provide this information. \nAs shown in Tab.~\\ref{MPII_table}, we observe that SPUDRFs outperform all baseline methods in MAE, confirming the effectiveness of our proposed method.\nBecause RT-GENE~\\cite{fischer2018rt} directly maps the features extracted by VGG-16 to gaze through multiple FC layers, it has inferior performance relative to DRFs $(4.62\\rightarrow4.80)$. \nFor a fair comparison, we chose RT-GENE with $1$ model.\nFurther, because SPUDRFs method learn DRFs in a self-paced manner and take into account ranking fairness, it has a further gain in MAE over DRFs $(4.62\\rightarrow4.45)$. \nPict-Gaze~\\cite{park2018deep} regresses an input image to an intermediate pictorial representation and then regresses the representation to the gaze direction.\nOrdinal Loss~\\cite{guo2021order} utilizes ordinal loss with order regularization to solve the regression problem.\nThe two methods mentioned above do not take into consideration the differences amongst samples; they train all examples simultaneously and thus have inferior MAEs. \nWe observe that the MAE of Pict-Gaze~\\cite{park2018deep} and Ordinal Loss~\\cite{guo2021order} are $4.56$ and $4.49$ respectively, while ours is $4.45$.\nIt's noteworthy that SP-DRFs, when compared with DRFs, only promotes MAE slightly.\nThis is probably due to the obvious distribution difference between training data and test data in the leave-one-out setting.\n\n\n\n\\begin{table}[h]\n\t\\centering\n\t\\caption{Quantitative comparison with state-of-the-art methods on the MPIIGaze dataset.}\n\t\\label{MPII_table}\n\t\\begin{tabular}[h]{cc}\n\t\t\\small\n\t\t\\scalebox{1.0}{\n\t\t\t\\begin{tabular}{@{}l|c}\n\t\t\t\t\\hline\n\t\t\t\tMethod & MAE$\\downarrow$\\\\\n\t\t\t\t\\hline\n\t\t\t\t\\hline\n\t\t\t\tMPIIGaze~\\cite{zhang2015appearance} & 6.30$\\pm 1.80$ \\\\\n\t\t\t\tiTracker~\\cite{krafka2016eye} & 6.20$\\pm 0.85$ \\\\\n\t\t\t\tGazeNet+~\\cite{zhang2017mpiigaze} & 5.40$\\pm 0.67$\\\\\n\t\t\t\tMeNets~\\cite{xiong2019mixed} & 4.90$\\pm 0.59$\\\\\n\t\t\t\tRT-GENE~\\cite{fischer2018rt} & 4.80$\\pm -- $\\\\\n\t\t\t\tPict-Gaze~\\cite{park2018deep} & 4.56$\\pm -- $\\\\\n\t\t\t\tOrdinal Loss~\\cite{guo2021order} & 4.49$\\pm -- $\\\\\n\t\t\t\tDRFs~\\cite{shen_deep_2018} & 4.62$\\pm 0.89 $ \\\\\n\t\t\t\tSP-DRFs & 4.57$\\pm 0.78 $ \\\\\n\t\t\t\tSPUDRFs & $\\textbf{4.45}\\pm \\textbf{0.84}$ \\\\\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\\end{tabular}\n\\end{table}\n\n\\begin{table*}\n\t\\centering\n\t\\caption{Quantitative Evaluation Results using Different Weighting Schemes. }\n\t\\begin{tabular}{c|c|c|c|c|c|c|c|c|cc}\n\t\t\\hline\\hline\n\t\t\\multirow{2}{*}{Weighting Schemes} & \\multicolumn{2}{c|}{MORPH} & \\multicolumn{2}{c|}{FGNET} & \\multicolumn{2}{c|}{BIWI} & \\multicolumn{2}{c|}{BU-3DFE} & \\multicolumn{2}{c}{MPIIGaze} \\\\ \\cline{2-11}\n\t\t& SP-DRFs & SPUDRFs & SP-DRFs & SPUDRFs & SP-DRFs & SPUDRFs & SP-DRFs & SPUDRFs & \\multicolumn{1}{c|}{SP-DRFs} & SPUDRFs \\\\ \\hline\\hline\n\t\tHard & 1.85 & \\textbf{1.78}& \\textbf{2.68}& 2.66 & 1.24 & 0.76 & 0.94 & 0.84 & \\multicolumn{1}{c|}{4.58} & \\textbf{4.45} \\\\\n\t\tLinear &1.84 & 1.80 & 2.69 & 2.66 & 1.18 & 0.79 & 0.92 & \\textbf{0.82}& \\multicolumn{1}{c|}{\\textbf{4.57}} & 4.46 \\\\\n\t\tLog & 1.85 & 1.81 & 2.68& \\textbf{2.64} & \\textbf{1.13} & 0.77 & \\textbf{0.89} & 0.82& \\multicolumn{1}{c|}{4.60} & 4.48 \\\\\n\t\tMixture & \\textbf{1.84} & 1.80 & 2.79 & 2.70 & 1.26 & \\textbf{0.74}& 0.93 & 0.86 & \\multicolumn{1}{c|}{4.58} & 4.45\\\\ \\hline\\hline\n\t\\end{tabular}\n\t\\label{ablation_experiments}\n\\end{table*}\n\n\t\\subsection{Different Weighting Schemes}\nA potential concern for SP-DRFs and SPUDRFs is that different weighting schemes could affect the estimation performance on a variety of visual tasks.\nTo evaluate this, we compare SP-DRFs\/SPUDRFs under different weighting schemes, including hard, mixture, and soft weighting. \t\nUnder all weighting schemes, $\\lambda$ was set as in Sec.~\\ref{setup}. \nUnder the mixture weighting scheme, $\\lambda'$ was set to ensure that $10\\%$ selected samples adopt soft weighting for Morph \\uppercase\\expandafter{\\romannumeral2}, and 20\\% samples for other datasets.\n\n\n\n\nTable.~\\ref{ablation_experiments} shows the comparison results.\nThe performances of SP-DRFs\/SPUDRFs with different weighting schemes are only slightly different.\nWe observe that the weights for a large proportion of examples are close to 1.\nFor example, under the log weighting scheme, only $2\\%$ of examples have weights below $0.5$.\nUnder the mixture weighting scheme, the proportion of samples whose weights are 1 can be set manually.\nWe chose to set this proportion to be $0.8\\sim0.9$ on different tasks.\nThe MAEs are not guaranteed to be better than other weighting schemes.\n\n\n\t\n\n\\subsection{Robust SPUDRFs}\nThe intuition for SPUDRFs to work better on different visual tasks is its improved robustness, \\emph{i.e.}, emphasizing more on ``reliable\" examples. \nTo further promote the robustness, we propose robust SPUDRFs in Sec.~\\ref{sec:robust SPUDRFs}, which enable SPUDRFs to handle labeling noise.\nWe added noise to labels to test the validity of robust SPUDRFs on the above datasets. \nSpecifically, we chose $10\\%$ samples in Morph \\uppercase\\expandafter{\\romannumeral2}, BIWI or MPIIGaze datasets, and added Gaussian noise $\\mathcal{N}\\left(0, 10\\right)$ to their labels. \nTo control the proportion ($0\\%\\sim20\\%$) of samples whose likelihoods are capped to be 0, \\emph{i.e.}, the portion of samples to be excluded, we set $\\epsilon$ at variant values.\n\n\n\nFig.~\\ref{capped_experiments} shows the MAE curves of SPUDRFs across variant capped proportions. \nWe observe that, when no example's likelihood is capped at 0, due to the presence of noise, the corresponding MAE ais large for each dataset. \nAs the capped proportion grows, the MAE gradually decreases.\nWhen the capped proportion changes to become $10\\%$, the MAE in Fig.~\\ref{capped_experiments} almost achieves minimal values, which demonstrate that robust SPUDRFs are capable of excluding noisy examples.\nWhen the capped proportion grows continuously, the MAE changes to become large. \nOne explanation is that some regular examples may be discarded when the capped proportion becomes over 10\\%.\n\n\\begin{figure}\n\t\\centering\n\t\\subfloat{\n\t\t\\includegraphics[width=0.4\\textwidth]{capped_morph_ori_new.pdf}%\n\t\t\\label{fig:capped_morph}}\\\\\n\t\\subfloat{\n\t\t\\includegraphics[width=0.4\\textwidth]{capped_biwi_ori_new.pdf}%\n\t\t\\label{fig:capped_biwi}}\\\\\n\t\\subfloat{\n\t\t\\includegraphics[width=0.4\\textwidth]{capped_mpii_ori_new.pdf}%\n\t\t\\label{fig:capped_mpii}}\n\t\\caption{The MAEs of SPUDRFs across different capped proportions evaluated on three datasets: Morph \\uppercase\\expandafter{\\romannumeral2}, BIWI, and MPIIGaze.}\n\t\\label{capped_experiments}\n\\end{figure}\n\n\n\\subsection{Fairness Improvement}\n\\label{fairness_section}\n\\begin{figure*}[t]\n\t\\centering\n\t\\subfloat{\\includegraphics[width=0.48\\textwidth]{fair_biwi_pitch_new.pdf}%\n\t\t\\label{fairness_experiments_pitch}}\n\t\\hfil\n\t\\subfloat{\\includegraphics[width=0.48\\textwidth]{fair_biwi_yaw_new.pdf}%\n\t\t\\label{fairness_experiments_yaw}}\n\t\\caption{Comparison of different methods on the BIWI dataset. In pitch and yaw directions, SP-DRFs and SPUDRFs outperform DRFs in MAE on most groups, but the MAEs of SP-DRFs are worse than DRFs on underrepresented groups.}\n\t\\label{fairness_experiments}\n\\end{figure*}\n\n\t\nThis section discusses how SPUDRFs improve regression fairness on different visual tasks. \n\n\n\nTaking into account of underrepresented examples, SPUDRFs show improved accuracy for age, pose and gaze estimation, compared to SP-DRFs.\nIn this section, we show how SPUDRFs further improve regression fairness.\nIn Sec.~\\ref{sec:fairness metric}, we define FAIR as a regression fairness metric.\nTo evaluate the regression fairness of DRFs, SP-DRFs, and SPUDRFs, we divided all the datasets mentioned above into different subsets and calculated FAIR on them.\nFor Morph \\uppercase\\expandafter{\\romannumeral2}, the entire data was divided into $7$ groups, \\emph{i.e.}, $[10, 20], [20,30], \\ldots, [70,80]$. \nA similar division was conducted on the FG-NET dataset. \nFor BIWI and BU-3DFE, the pitch, yaw, and roll directions were regarded as independent directions, and the interval for each regression group is $10^\\circ$. \nFor MPIIGaze, the pitch and yaw directions were regarded as relevant angles.\nWe set the group interval to be $5^\\circ$ in each direction, resulting in a total of 60 groups.\n\n\n\nTable.~\\ref{fairness_rule} shows the FAIR values on all the datasets. \nThe higher the FAIR is, the more fair the regression model is.\nNote that SP-DRFs always have inferior FAIR relative to DRFs.\nFor example, SP-DRFs have a FAIR of $0.43$, while DRFs have a FAIR of $0.46$ for BIWI, and SP-DRFs have a FAIR of $0.37$ while DRFs have a FAIR of $0.42$ for FG-NET. \nThe results demonstrate that SP-DRFs tend to aggravate the bias of solutions. \nOn the other hand, SPUDRFs have significantly improved FAIR compared to SP-DRFs.\nWe observe SPUDFRs gains improvement by $5\\%$ for FGNET and $27\\%$ for BIWI.\nThis is an evidence that SPUDRFs tend to have more fair estimation results on various visual tasks.\t\n\n\n\n\\begin{table}[h]\n\t\\centering\n\t\\caption{Regression Fairness Evaluation on Different Datasets.}\n\t\\begin{tabular}{c|c|c|c|c|c}\n\t\t\\hline\n\t\tMethods & Morph \\uppercase\\expandafter{\\romannumeral2} & FGNET & BIWI & BU-3DFE & MPIIGaze \\\\\n\t\t\\hline \\hline\n\t\tDRFs & 0.46 & 0.42 & 0.46 & 0.74 & 0.67 \\\\\n\t\tSP-DRFs & 0.44 & 0.37 & 0.43 & 0.72 & 0.67 \\\\\n\t\tSPUDRFs & \\textbf{0.48} & \\textbf{0.42} & \\textbf{0.70} & \\textbf{0.76} & \\textbf{0.69} \\\\ \\hline\n\t\\end{tabular}\n\t\\label{fairness_rule}\n\\end{table}\t\n\nSome pose estimation results of DRFs, SP-DRFs, and SPUDRFs for BIWI are shown in Fig.~\\ref{fairness_experiments}. \nThe estimation accuracy for different pose groups reveals some consistent trends.\nFirst, for most groups, SP-DRFs and SPUDRFs have superior MAEs over DRFs. \nIn Sec.~\\ref{Uncertainty}, we discussed that these gains in fairness are due to SPL, which can guide DDMs to achieve more reasonable solutions.\nSecond, SP-DRFs tend to have even worse MAEs for the underrepresented groups than DRFs, for example, $\\left[30^\\circ, 60^\\circ\\right]$ in the pitch direction and $\\left[-70^\\circ, -40^\\circ\\right]$ in the yaw direction.\nThis demonstrates that existing SPL methods have a fatal drawback: the ranking and selection schemes may incur seriously biased estimation results.\nThird, also for underrepresented groups, SPUDRFs gain significant improvement in MAE compared to DRFs. \nFor example, our gains in MAE are $87.71\\%$ and $77.68\\%$ from $[-60^\\circ, -50^\\circ]$ and $[30^\\circ, 60^\\circ]$ in pitch direction, $80.96\\%$ and $89.52\\%$ from $[-70^\\circ, -40^\\circ]$ and $[60^\\circ, 70^\\circ]$ in yaw direction.\nThis also serves as an evidence that our proposed self-paced method alleviates the fairness problem in existing SPL methods. \n\n\n\n\n\\section{Discussion}\n\\label{sec:discussion}\nIn the experiments above, we have evaluated SPUDRFs against other baseline methods on different visual tasks, such as facial age estimation, head pose estimation and gaze estimation (Sec. \\ref{sec:Comparison}).\nWe also evaluated the SPUDRFs method under different weighting schemes, its extension with capped likelihood formulation, and its performance improvement on fairness.\nOn a number of tasks and datasets, SPUDRFs and SP-DRFs outperform other baseline methods.\nThe advantage of considering ranking fairness in SPUDRFs is most obvious for BIWI.\nFor Morph II and BU-3DFE, the performance improvement when considering ranking fairness in SPUDRFs is also observable.\n\n\n\nLearning DDMs in a self-paced manner has some limitations. \nThe most noticeable one is that, in a leave-one-out setting, SP-DRFs perform only slightly better than DRFs. \nWe speculate that this is due to the data distribution difference between training set and test set.\nTherefore, whether SPL can improve the performance of DDMs may largely depend on the distribution divergence between training data and test data. \n\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nThis paper explored how self-paced learning guided deep discriminative models (DDMs) to obtain robust and less biased solutions on different visual tasks, \\emph{e.g.}~facial age estimation, head pose estimation, and gaze estimation.\nA novel self-paced method, which considers ranking fairness, was proposed.\nSpecifically, a new ranking scheme that jointly considers loss and fairness was proposed in SPL.\nSuch a method was combined with a typical DDM---deep regression forests (DRFs)---and led to a new model, namely deep regression forests with consideration on underrepresented examples (SPUDRFs).\nIn addition, SPUDRFs under different weighting schemes, their extension with capped likelihood formulation, and their performance improvement on fairness were discussed.\nExtensive experiments on three well-known computer vision tasks demonstrated the efficacy of our proposed new self-paced method.\nThe future work will include exploring how to incorporate such a method with other DDMs.\n\n\n\n\\section*{Acknowledgment}\nThe authors would like to thank Jiabei Zeng of the Institute of Computing Technology, Chinese Academy on Sciences, for the valuable advice on the gaze estimation experiments.\nThis work is partially supported by the National Key R\\&D Program of China AI2021ZD0112000, National Natural Science Fundation of China Nos.~62171111, 61806043, 61971106 and 61872068, and the Special Science Foundation of Quzhou No.~2020D013.\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{sec_Int}\nBose-Einstein condensates (BECs) confined in toroidal \ntraps have been the subject of many experimental studies recently~\\cite{Ryu07,Ramanathan11,Moulder12,Wright12,Marti12,Beattie13}.\nThis research covers topics such as the observation of persistent current~\\cite{Ryu07}, phase slips across a stationary barrier~\\cite{Ramanathan11}, stochastic~\\cite{Moulder12} and deterministic~\\cite{Wright12} phase slips between vortex states, the use of toroidal condensates in interferometry~\\cite{Marti12}, and the stability of superfluid flow in a spinor condensate~\\cite{Beattie13}. \nThese experiments have given rise to theoretical studies discussing, e.g., the excitation spectrum and critical velocity of a superfluid BEC~\\cite{Dubessy12} and the simulation of the experiment~\\cite{Ramanathan11} using the Gross-Pitaevskii equation~\\cite{Mathey12,Piazza12} and the truncated Wigner approximation~\\cite{Mathey12}. Most of the experimental and theoretical studies concentrate on the properties of persistent currents. \nThe phase of a toroidal BEC changes by $2\\pi k$ as the toroid is encircled, the integer $k$ being the winding number of the vortex. In a singly connected geometry a vortex with $|k|>1$ is typically unstable against splitting into\nvortices with smaller $k$. In a multiply connected geometry \n this process is suppressed for energetic reasons. \nIn Ref.~\\cite{Moulder12} it was shown experimentally that a vortex with winding number three can persist in a toroidal single-component BEC for up to a minute. \nIn other words, toroidal geometry makes it possible to avoid the fast vortex splitting taking place in a singly connected BEC and study the properties of vortices with large winding number. \nInstead of using a toroidal trap, a multiply connected geometry that stabilizes vortices can also be created by applying a Gaussian potential along the vortex core~\\cite{Kuopanportti10}. \n\nIn this paper, we calculate the Bogoliubov spectrum of a toroidal quasi-one-dimensional (1D) spin-1 BEC. Motivated by the experimental results of Refs.~\\cite{Moulder12,Beattie13}, we assume that the splitting of vortices occurs on a very long time scale in a spinor condensate where only one spin component is populated. The dominant instabilities can then be assumed to arise from the spin-spin interaction. For related theoretical studies on toroidal two-component condensates, see, for example, Refs.~\\cite{Smyrnakis09,Anoshkin12}. \nIn our analysis, the population of the $m_F=0$ spin component is taken to be zero initially, making it possible to calculate the excitation spectrum analytically. \nThis type of a state can be prepared straightforwardly experimentally. The proliferation of instabilities can be observed by measuring the densities of the spin components. \n\nThis paper is organized as follows. In Sec.~\\ref{sec_Ham} we define the Hamiltonian, describe briefly the calculation of the excitation spectrum, and show that the spectrum can be divided into magnetization and spin modes. In Sec.~\\ref{sec_Mag} we analyze the properties of the magnetization modes and \nillustrate how the presence of unstable modes can be seen experimentally. \nWe also compare the analytical results with numerical calculations. \nIn Sec.~\\ref{sec_Spin} we study the spin modes and their experimental observability analytically and numerically and show that a rotonlike spectrum can be realized both in rubidium and sodium condensates. \nIn Sec.~\\ref{sec_Exp} we discuss two recent experiments on toroidal BECs \nand show examples of the instabilities than can be realized in these systems. \nFinally, in Sec.~\\ref{sec_Con} we summarize our results. \n\n\\section{Energy and Hamiltonian}\n\\label{sec_Ham}\nThe order parameter of a spin-$1$ Bose-Einstein condensate reads $\\psi=(\\psi_{1},\\psi_{0},\\psi_{-1})^T$, \nwhere $T$ denotes the transpose. It fulfills the identity $\\psi^\\dag \\psi =n_{3D}$, \nwhere $n_{3D}$ is the total particle density. \nWe assume that the system is exposed to a homogeneous magnetic field oriented along the $z$ axis. \nThe energy functional becomes, then, \n\\begin{align}\n\\nonumber\n&E[\\psi]=\\int d\\mathbf{r} \\left(\\psi^\\dag(\\mathbf{r})\\hat{H}_0(\\mathbf{r})\\psi(\\mathbf{r})\\right.\\\\\n&\\left. +\\frac{1}{2}\\left\\{ g_0 n_{3D}^2(\\mathbf{r}) + g_2 [\\psi^\\dag(\\mathbf{r})\\hat{\\mathbf{F}}\\psi(\\mathbf{r})]^2\\right\\}\\right),\n\\label{eq_E}\n\\end{align}\nwhere the single-particle Hamiltonian $\\hat{H}_0$ is defined as\n\\begin{align}\n\\hat{H}_0(\\mathbf{r})=-\\frac{\\hbar^2\\nabla^2}{2m}+U(\\mathbf{r})-\\mu_{3D}-p\\hat{F}_z+q\\hat{F}_z^2, \n\\end{align}\nand $\\hat{\\mathbf{F}}=(\\hat{F}_x,\\hat{F}_y,\\hat{F}_z)$ is the (dimensionless) spin operator of a spin-1 particle, $U$ is the trapping potential, and $\\mu_{3D}$ \nis the chemical potential. The magnetic field introduces the linear and quadratic Zeeman terms, given by $p$ and $q$, respectively. The sign of $q$ can be controlled experimentally by using a linearly polarized microwave field \\cite{Gerbier06}. \nThe strength of the atom-atom interaction is characterized by $g_0=4\\pi \\hbar^2(a_0+2a_2)\/3m$ and $g_2=4\\pi \\hbar^2(a_2-a_0)\/3m$, where $a_F$ is the $s$-wave scattering length for two atoms colliding with total angular momentum $F$. \nThe scattering lengths of ${}^{87}$Rb used here are $a_0=101.8a_B$ and $a_2=100.4 a_B$ \\cite{vanKempen02}, measured in units of the Bohr radius $a_B$.\nFor ${}^{23}$Na the corresponding values are $a_0=50.0a_B$ and $a_2=55.1a_B$ \\cite{Crubellier99}.\n\n\nThe condensate is confined in a toroidal trap given in cylindrical coordinates as $U(r,z,\\varphi)=m \\left[\\omega_r^2 (R-r)^2+\\omega_z^2 z^2 \\right]\/2$,\nwhere $R$ is the radius of the torus and $\\omega_r,\\omega_z$ are the trapping\nfrequencies in the radial and axial directions, respectively. We assume that the condensate is quasi-1D, so that the order parameter factors as \n$\\psi(r,z,\\varphi;t)=\\psi_{r;z}(r,z)\\psi_\\varphi(\\varphi;t)$, \nwhere $\\psi_{r;z}$ is complex valued and time independent.\nThe normalization of $\\psi_{r;z}$ is chosen such that $\\int\\int r dr dz |\\psi_{r;z}(r,z)|^2=N\/2\\pi$, \nwhere $N$ is the total number of particles. This means that \n\\begin{align}\n\\|\\psi_\\varphi(t)\\|\\equiv\n\\sqrt{\\int_{0}^{2\\pi}d\\varphi\\ \\psi_\\varphi^\\dag(\\varphi;t)\\psi_\\varphi(\\varphi;t)}\n\\end{align}\nhas to be equal to $\\sqrt{2\\pi}$ for any $t$.\nBy integrating over $r$ and $z$ in Eq. \\eqref{eq_E} we obtain\n\\begin{align}\n\\nonumber\n& E_{1\\textrm{D}}[\\psi_\\varphi] = \\\\\n\\nonumber\n&\\int_{0}^{2\\pi} d\\varphi\n\\left( \\psi_\\varphi^\\dag(\\varphi)\\left(-\\epsilon \\frac{\\partial^2}{\\partial \\varphi^2} -\\mu-p\\hat{F}_z+q\\hat{F}_z^2\\right)\\psi_\\varphi(\\varphi)\\right.\\\\\n &\\left. +\\frac{n}{2}\\left\\{ g_0 \\left[\\psi_\\varphi^\\dag(\\varphi)\\psi_\\varphi(\\varphi)\\right]^2 + g_2 \\left[\\psi_\\varphi^\\dag(\\varphi)\\hat{\\mathbf{F}}\\psi_\\varphi(\\varphi)\\right]^2\\right\\}\\right),\n\\label{eq_E1D}\n\\end{align}\nwhere \n\\begin{align}\n\\label{eq_epsilon}\n\\epsilon & =\\frac{2\\pi}{N}\\frac{\\hbar^2}{2m}\\int_{0}^{\\infty} rdr\\int_{-\\infty}^{\\infty} dz\\, \\frac{1}{r^2} |\\psi_{r;z}(r,z)|^2\n\\end{align} \nand \n\\begin{align}\n\\label{eq_n}\nn =\\frac{2\\pi}{N}\\int_{0}^{\\infty} rdr\\int_{-\\infty}^{\\infty} dz\\,|\\psi_{r;z}(r,z)|^4.\n\\end{align}\nIn Eq. \\eqref{eq_E1D} we have omitted an overall factor $N\/2\\pi$ multiplying the right-hand side of this equation. \nThe chemical potential $\\mu$ contains the original chemical potential $\\mu_{3D}$ and terms coming from the integration of the kinetic and potential energies. \n The magnetization in the $z$ direction, \n\\begin{align}\nf_z=\\frac{1}{2\\pi} \\int_{0}^{2\\pi} d\\varphi\\,\\psi_\\varphi^\\dag (\\varphi;t)\\hat{F}_z\\psi_\\varphi (\\varphi;t), \n\\end{align} \nis a conserved quantity; the corresponding Lagrange multiplier can be included into $p$. \nIn the following we drop the superscript $\\varphi$ of $\\psi_\\varphi$. \n\n\nWe assume that in the initial state the spin is parallel to the magnetic field.\nIn \\cite{Makela11} it was argued that in a homogeneous system the most unstable states are almost always of this form.\nThis state can be written as \n\\begin{align}\n\\label{psipara}\n\\psi_\\parallel(\\varphi) = \n\\frac{1}{\\sqrt{2}}\n\\begin{pmatrix}\n e^{i k_1\\varphi}\\sqrt{1+f_z}\\\\\n0 \\\\\ne^{i\\theta} e^{i k_{-1}\\varphi}\\sqrt{1-f_z}\n\\end{pmatrix},\n\\end{align}\nwhere $\\theta$ is the relative phase and \nthe integer $k_{\\pm 1}$ is the winding number of the $m_F=\\pm 1$ component. \nThe energy and stability of $\\psi_\\parallel$ are independent of $\\theta$ and therefore we set $\\theta=0$ in the rest of this article. \nIf $k_1=1$ and $k_{-1}=0$, $\\psi_\\parallel$ describes a half-quantum vortex (Alice string), \nsee, e.g., Refs. \\cite{Leonhardt00,Isoshima01,Hoshi08}. \nThe populations of $\\psi_\\parallel$ are time independent and the Hamiltonian giving the time evolution \nof $\\psi_\\parallel$ reads\n\\begin{align}\n\\label{Hparallel}\n\\hat{H}_\\parallel= \\left(g_0 n - \\mu\\right)\\hat{\\mathbb{I}}\n +(g_2 n f_z -p_{\\textrm{eff}})\\hat{F}_z + q_{\\textrm{eff}}\\hat{F}_z^2,\n\\end{align} \nwhere \n\\begin{align}\np_{\\textrm{eff}}=& p-\\frac{\\epsilon}{2} (k_{1}^2-k_{-1}^2),\\\\\nq_{\\textrm{eff}}=& q+\\frac{\\epsilon}{2}(k_1^2+k_{-1}^2). \n\\end{align}\nThe time evolution operator of $\\psi_\\parallel$ is $\\hat{U}_\\parallel(t)=e^{-it \\hat{H}_\\parallel\/\\hbar}$.\n\n\nWe calculate the linear excitation spectrum in a basis where $\\psi_\\parallel$ is stationary \\cite{Makela11,Makela12} using the Bogoliubov approach, that is, we define \n $\\psi(\\varphi;t)=\\psi_\\parallel(\\varphi) +\\delta\\psi(\\varphi;t)$ \n and expand the time evolution equations to first order in $\\delta\\psi$. \nWe write $\\delta\\psi=(\\delta\\psi_1,\\delta\\psi_0,\\delta\\psi_{-1})^T$ as \n\\begin{align}\n\\label{eq_Bogoliubov}\n\\delta\\psi_j(\\varphi;t) \\equiv e^{ik_j\\varphi}\\sum_{s=0}^{\\infty} u_{j;s}(t)\\,e^{i s \\varphi}- v^*_{j;s}(t)\\,e^{-i s\\varphi},\n\\end{align}\nwhere $j=0,\\pm 1$ and $k_0\\equiv 0$. Due to the toroidal geometry, $\\delta\\psi_j(\\varphi+2\\pi;t)=\\delta\\psi_j(\\varphi;t)$ \nhas to hold. As a consequence, $s$ needs to be an integer. \nIn the next two sections we analyze the excitation spectrum in detail; the actual calculation of the spectrum can be found in the appendix. \nThe normalized wave function reads \n\\begin{align}\n\\label{eq_psi}\n\\tilde{\\psi}(\\varphi;t) = c(t)[\\psi_\\parallel(\\varphi)+\\delta\\psi(\\varphi;t)],\n\\end{align}\nwhere $c(t)$ is determined by the condition $\\|\\tilde{\\psi}(t)\\|=\\sqrt{2\\pi}$.\nTo characterize the eigenmodes we define \n\\begin{align}\n\\label{eq_exp_Fz}\n\\langle\\hat{F}_z\\rangle (\\varphi;t) \\equiv \\tilde{\\psi}^\\dag(\\varphi;t)\\hat{F}_z\\tilde{\\psi}(\\varphi;t),\n\\end{align} \nso that $f_z=1\/2\\pi \\int_0^{2\\pi}d\\varphi\\ \\langle\\hat{F}_z\\rangle(\\varphi;t)$ for any $t$. \nFurthermore, we denote the population of the $m_F=0$ spin component by $\\rho_0$, \n$\\rho_0(\\varphi;t)=|\\tilde{\\psi}_0(\\varphi;t)|^2$. Note that here $\\langle\\hat{F}_z\\rangle$ and $\\rho_0$ are calculated in the basis where $\\psi_\\parallel$ is a stationary state. This basis and the original basis are related \nby a basis transformation that only affects the phases of the $m_F=\\pm 1$ components. \nThe densities of the spin components are thus identical in the original and new basis. The numerical calculations \nare done in the original basis. \n\n\nThe excitation spectrum can be divided into spin and magnetization modes.\nThe spin modes keep the value of $\\langle\\hat{F}_z\\rangle$ unchanged in time, \n$\\langle\\hat{F}_z\\rangle(\\varphi;t)=\\langle\\hat{F}_z\\rangle(\\varphi;0)\\approx f_z$, \n but rotate the spin vector by making $\\rho_0$ nonzero. The magnetization modes, on the other hand, lead to $\\varphi$-dependent $\\langle\\hat{F}_z\\rangle(\\varphi;t)$, \nbut leave $\\rho_0$ unaffected. There are in total six eigenmodes. \nWe denote them by $\\hbar\\omega_j$, where $j=1,2,3,4$ \nlabels the magnetization modes and $j=5,6$ the spin modes. \nWe denote the real and imaginary part of $\\omega_{l}$ by $\\omega^{\\textrm{r}}_{l}$ and \n$\\omega^{\\textrm{i}}_{l}$, respectively. \nThe mode labeled by $l$ is unstable if $\\omega^{\\textrm{i}}_l$ is positive. \nWe discuss first the magnetization modes. \n\n\n\n\\section{Magnetization modes}\n\\label{sec_Mag}\n\\subsection{Eigenmodes}\nWe characterize the eigenmodes by the quantities, \n\\begin{align}\nk_\\pm =\\frac{1}{2}\\left(k_{1}\\pm k_{-1}\\right). \n\\end{align}\nNote that the value of $k_\\pm$ can be a half-integer. \nThe magnetization modes are independent of $q$ and can be written as\n\\begin{align}\n\\hbar\\omega_l(s)=2\\epsilon s k_{+}+\\hbar\\tilde{\\omega}_l(s), \n\\end{align} \nwhere $l=1,2,3,4$. \nThe expression for $\\tilde{\\omega}_l$ is too long to be shown here. \nThe value of $\\tilde{\\omega}_l$ depends on $k_{-}$ but is independent of $k_{+}$. \nConsequently, modes with differing $k_{+}$ but equal $k_{-}$ have identical stability. \n\nIf $f_z=0$, the eigenvalues simplify and read \n\\begin{align}\n\\nonumber\n&\\hbar\\omega_{1,2,3,4}(s)\\big|_{f_z=0} = 2\\epsilon s k_{+}\\\\\n&\\pm \\sqrt{\\epsilon s^2\\left[4\\epsilon k_{-}^2+ w \n\\pm \\sqrt{16\\epsilon k_{-}^2w +(g_0 -g_2)^2 n^2}\\right]},\n\\label{o1234fz0} \n\\end{align}\nwhere \n\\begin{align}\nw=\\epsilon s^2+(g_0+ g_2)n. \n\\end{align}\nThe signs are defined such that $++,-+,+-$, and $--$ correspond to $\\omega_1,\\omega_2,\\omega_3$, and $\\omega_4$, respectively. \nUnstable modes appear when the term inside the square brackets becomes negative. \nFor rubidium and sodium $g_0+g_2 > 0$, which guarantees that $\\omega_1$ and $\\omega_2$ are real. Only $\\omega_3$ can have a positive imaginary part. \n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale=.92]{fig_wi.pdf}\n\\end{center}\n\\caption{(Color online) The amplitudes of the unstable spin and magnetization modes for rubidium and sodium. Here $\\epsilon=0.75|g_2|n$, $q=2.5 |g_2|n$, $f_z=0$, and the unit of $\\omega^{\\textrm{i}}_{3,5}$ is $|g_2|n\/\\hbar$. The lines have been drawn by treating $s$ as a continuous parameter; dots indicate the actual allowed nonvanishing values of $\\omega^{\\textrm{i}}_{3,5}$. In (c) and (d) the curves are reflection symmetric with respect to \n$s=k_{+}=(k_{1}+k_{-1})\/2$. \n\\label{fig_wi}}\n\\end{figure}\nAs can be seen from Figs. \\ref{fig_wi}(a) and 1(b), the value of $\\omega^{\\textrm{i}}_3(s)$ grows as $|k_{-}|$ increases. The allowed values of $s$ are non-negative integers. The modes corresponding to $s=0$ are always stable, but unstable modes are present for $s= 1,2,\\ldots, \\lfloor\\sqrt{4k_{-}^2-2 g_2 n\/\\epsilon}\\rfloor$, where $\\lfloor\\cdots\\rfloor$ is the floor function. \nTherefore, if there are $j$ unstable modes, they have to be the ones corresponding to $s=1,2,\\ldots ,j$. \n A lower bound for the value of $\\epsilon$ yielding at least one unstable mode is given by the equation $\\epsilon (4k_{-}^2-1)\\geq 2g_2 n$. In the case of a sodium BEC ($g_2 >0$) this means that the magnetization modes corresponding to $k_{-}=0$ and $|k_{-}|=1\/2$ are always stable. \n This is visualized in Fig. \\ref{fig_wi}(b), where $\\omega^{\\textrm{i}}_3(s)$ corresponding to $(k_1,k_{-1})=(0,0)$ and $(k_1,k_{-1})=(2,1)$ is seen to vanish for every $s$. \nIn a rubidium condensate $(g_2<0)$ with $k_{-}=0$ unstable modes exist if $\\epsilon\\leq 2|g_2|n$; \nif $|k_{-}|>0$, instabilities are present regardless of the value of $\\epsilon$.\nFor both rubidium and sodium the wave number $s$ of the fastest-growing instability is approximately given by the integer closest to $\\sqrt{2\/3}\\sqrt{4k_{-}^2-2 g_2 n\/\\epsilon}$. \n\n\\subsection{Experimental observability}\nThe properties of unstable magnetization modes can be studied experimentally \nby measuring $\\langle\\hat{F}_z\\rangle$. We assume that there is one dominant unstable mode and that $f_z=0$. \nThe initial time evolution of $\\langle\\hat{F}_z\\rangle$ reads, then (see the appendix), \n\\begin{align}\n\\label{eq_FzexpApprox}\n\\nonumber\n&\\langle \\hat{F}_z\\rangle (\\varphi;t) \\approx c^2(t)\\left\\{ A e^{\\omega^{\\textrm{i}}_3 t} \n\\cos\\left[\\theta+s\\left(\\varphi-\\frac{2\\epsilon k_{+} t}{\\hbar}\\right)\\right]\\right.\\\\\n&\\left.+ B e^{2\\omega^{\\textrm{i}}_3 t} \\cos\\left[2\\theta+2s\\left(\\varphi-\\frac{2\\epsilon k_{+} t}{\\hbar}\\right)\\right]\\right\\},\n\\end{align}\nwhere $c$ is the normalization factor appearing in Eq.~\\eqref{eq_psi} and $A,B,$ and $\\theta$ are defined in Eqs.~\\eqref{eq_A}, \\eqref{eq_B}, and \\eqref{eq_theta}, respectively. Because typically \n$B\\ll A$, the first term on the right-hand side \nof Eq.~\\eqref{eq_FzexpApprox} dominates over the second term during the initial time evolution. \nThis leads to $\\langle\\hat{F}_z\\rangle$ having $s$ maxima and minima. \nIf $k_{+}\\not =0$, these maximum and minimum regions rotate around the torus as time evolves, indicating that the behavior of $\\langle\\hat{F}_z\\rangle$ depends on $k_{+}$, even though the growth rate of the instabilities $\\omega^{\\textrm{i}}_3$ is independent of $k_{+}$. We study the validity of Eq.~\\eqref{eq_FzexpApprox} by considering a rubidium condensate with $\\epsilon=0.75|g_2|n$, \n$q=2.5|g_2|n$, $k_{1}=2$, and $k_{-1}=1$, corresponding to the blue dash-dotted line in Figs.~\\ref{fig_wi}(a) and~\\ref{fig_wi}(c). \nAnalytical results predict that the only unstable mode of this system is a magnetization mode corresponding to $s=1$. The numerically calculated time evolution of $\\langle\\hat{F}_z\\rangle$ is shown \nin Figs.~\\ref{fig_num_mag}(a) and \\ref{fig_num_mag}(b). \n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.85,clip]{fig_mag_k1_2_km1_1.pdf}\n\\vspace{6mm}\n\\caption{(Color online) (a) Numerically calculated $\\langle\\hat{F}_z\\rangle$ for the parameters corresponding to the blue dash-dotted line in Fig.~\\ref{fig_wi}(a), that is, a ${}^{87}$Rb condensate \nwith $\\epsilon =0.75 |g_2|n, q=2.5|g_2|n, f_z=0, k_{1}=2,$ and $k_{-1}=1$. (b) Magnification of the region bounded by the dashed vertical lines in (a). Here we plot $|\\langle\\hat{F}_z\\rangle|$ instead of $\\langle\\hat{F}_z\\rangle$ and use a logarithmic scale to make the initial growth of $|\\langle\\hat{F}_z\\rangle|$ visible. (c) Analytically calculated $\\langle\\hat{F}_z\\rangle$, see Eq.~\\eqref{eq_FzexpApprox}. \n\\label{fig_num_mag}\n}\n\\end{figure}\nThe $s=1$ magnetization mode can be seen to be unstable. \nThe rotation of the minimum and maximum of $\\langle\\hat{F}_z\\rangle$ around the torus is clearly visible in Fig.~\\ref{fig_num_mag}. The analytically obtained behavior of $\\langle\\hat{F}_z\\rangle$ is shown in Fig. \\ref{fig_num_mag}(c). \nBy comparing Figs.~\\ref{fig_num_mag}(b) and \\ref{fig_num_mag}(c), we see that Eq.~\\eqref{eq_FzexpApprox} describes the time evolution of $\\langle\\hat{F}_z\\rangle$ very precisely \nup to $t\\approx 10\\hbar\/|g_2|n$. \nThe only parameters in Eq.~\\eqref{eq_FzexpApprox} that are not fixed by the parameters \nused in the numerical calculation are the initial global phase and length \n$\\|\\delta\\psi(t=0)\\|$ of $\\delta\\psi(t=0)$. \nIn Fig.~\\ref{fig_num_mag}(c) we have chosen the values of these variables in such a way that the match between the numerical and analytical results is the best possible. \n\n\n\\section{Spin modes}\n\\label{sec_Spin}\n\\subsection{Eigenmodes} \nWe now turn to the spin modes. As shown in the appendix, the spin modes read\n\\begin{align}\n\\label{o56}\n&\\hbar\\omega_{5,6}(s) =2\\epsilon k_{+}(s-k_{+})\\\\\n\\nonumber \n&\\pm \\sqrt{\\left\\{\\epsilon [(s-k_+)^2-k_{-}^2]+g_2 n-q\\right\\}^2-(1-f_z^2)(g_2 n)^2},\n\\end{align}\nwhere $+$ ($-$) corresponds to $\\omega_5$ ($\\omega_6$). \nIf $k_{+}=0$, the effect of vortices can be taken into account by scaling $q\\rightarrow q+\\epsilon k_{-}^2$, {\\it i.e.}, the spin modes of a system with $(k_1,k_{-1})=(k,-k)$ and $q=\\tilde{q}$ are equal to the spin modes of a vortex-free condensate with $q=\\tilde{q}+\\epsilon k^2$. Spin modes are unstable if and only if the term inside the square root is negative. Now only $\\omega_5$ can have a positive imaginary part. \nThe fastest-growing unstable mode is obtained at $\\epsilon [(s-k_+)^2-k_{-}^2]+g_2 n-q=0$ and has the amplitude $\\hbar\\omega^{\\textrm{i}}_5(s)=|g_2|n\\sqrt{1-f_z^2}$. \nUnlike in the case of the magnetization modes, the maximal amplitude is bounded from above and is independent of the winding numbers [see Figs.~\\ref{fig_wi}(c) and \\ref{fig_wi}(d)]. By adjusting the strength of the magnetic field, the fastest-growing unstable mode can be chosen to be located at a specific value of $s$, showing that it is \neasy to adjust the stability properties experimentally. \nAt $f_z=0$ the width of the region on the $s$-axis giving positive $\\omega^{\\textrm{i}}_5$ is \n$|\\sqrt{k_{-}^2+q\/\\epsilon}-\\sqrt{k_{-}^2+q\/\\epsilon- 2g_2 n\/\\epsilon}|$. \nThis region can thus be made narrower by increasing $\\epsilon,k_{-}$, or $q$. \nSince the magnetization modes are insensitive to the magnetic field, the properties of the spin and magnetization modes can be tuned independently. \nThe winding number dependence of unstable spin modes is illustrated \nin Figs. \\ref{fig_wi}(c) and \\ref{fig_wi}(d). \n\n\\subsection{Rotonlike spectrum}\nInterestingly, by tuning $\\epsilon$ and $q$, a rotonlike spectrum can be realized (see the solid and dotted blue lines in Fig. \\ref{fig_roton}). \n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale=.90]{fig_roton.pdf}\n\\end{center}\n\\caption{(Color online) The real ($\\omega^{\\textrm{r}}_5$) and imaginary ($\\omega^{\\textrm{i}}_5$) component of the spin mode $\\omega_5$ for rubidium and sodium. Here $\\epsilon=0.2|g_2|n, f_z=0, k_{1}=-k_{-1}$, and $k_1$ is an arbitrary integer. For the blue solid and blue dotted lines $q+\\epsilon k_{-}^2 =2.8|g_2|n$ and for the orange dashed line \n$q+\\epsilonk_{-}^2 =-2|g_2|n$. The unit of $\\omega_5^{\\textrm{r},\\textrm{i}}$ is $|g_2|n\/\\hbar$. \nThe lines have been drawn by treating $s$ as a continuous parameter; dots (open circles) indicate the actual allowed nonvanishing values of $\\omega^{\\textrm{r}}_5$ ($\\omega^{\\textrm{i}}_5$). \n\\label{fig_roton}}\n\\end{figure} \nNow the phonon part of the spectrum is missing, but the roton-maxon feature is present. For $f_z=k_{+}=0$, the roton spectrum exists if \n$q\\geq \\max\\{0,2g_2 n\\}$. Because only integer values of $s$ are allowed, \nit may happen that $\\omega^{\\textrm{i}}_{5}$ is nonzero only in some interval of the $s$ axis that does not contain integers [see Figs. \\ref{fig_wi}(c) and \\ref{fig_wi}(d) for examples of this in the context of magnetization modes]. \nIn this case the rotonic excitations are stable. Alternatively, there can be unstable modes close to the roton minimum (see Fig.~\\ref{fig_roton} and Ref.~\\cite{Matuszewski10}). \nAs evidenced by the orange dashed lines in Fig. \\ref{fig_roton}, the roton spectrum can be made to vanish simply by decreasing $q$. Also the values of $s$ leading to unstable modes can be controlled by varying $q$. For example, using the parameter values corresponding to the blue solid line in Fig. \\ref{fig_roton}, we find that by decreasing (increasing) the value of $q+\\epsilonk_{-}^2$ from $2.8|g_2| n$ to $|g_2| n$ ($4|g_2| n$), the $s=3$ ($s=5$) mode can be made unstable in a rubidium condensate. This opens the way for quench experiments of the type described in Refs.~\\cite{Sadler06,Bookjans11}. \nInstead of altering $q$, instabilities can also be induced by making $\\epsilon$ smaller by changing the trapping frequencies. It is known that a rotonlike spectrum can exist in various types of BECs, such as in a dipolar condensate (see, e.g., Refs.~\\cite{Odell03,Santos03,Cherng09}), in a Rydberg-excited condensate~\\cite{Henkel10}, or in a spin-1 sodium condensate prepared in a specific state~\\cite{Matuszewski10}. \nIn the present case the rotonlike spectrum exists both in a sodium and rubidium BEC and the state \n[Eq.~\\eqref{psipara}] giving rise to it is easy to prepare experimentally. \nNote that the roton-maxon feature exists also in a vortex-free condensate and for \nany $|f_z|<1$. These results suggest that the roton-maxon character of the spectrum is rather a rule than an exception in spinor BECs. \n \n\n\n\n\\subsection{Experimental observability}\nThe properties of unstable spin modes can be studied experimentally by measuring $\\rho_0$. \nAssuming that there is one dominant unstable spin mode located at wave number $s$, we find that (see the Appendix)\n\\begin{align}\n&\\delta\\psi_0(\\varphi;t)\\propto e^{i k_+\\varphi + \\omega^{\\textrm{i}}_5 t}\n\\sin\\left[\\left(s-k_{+}\\right)\\left(\\varphi-\\frac{2\\epsilonk_{+} t}{\\hbar}\\right)+\\frac{\\tilde{\\theta}}{2}\\right].\n\\label{eq_deltapsi0}\n\\end{align}\nThe phase $\\tilde{\\theta}$ is defined in Eq.~\\eqref{eq_thetaSpin}. \nThe sign of $\\delta\\psi_0$ changes at every point where the density $\\rho_0\\propto |\\delta\\psi_0|^2$ vanishes. This is similar to the behavior of the phase of a dark soliton \\cite{Frantzeskakis10}. \nThe number of nodes in $\\rho_0$ is $2|s-k_{+}|$, that is, \nif $2k_{+}$ is even (odd), $\\rho_0$ has an even (odd) number of nodes.\nThe density peaks resulting from the instability rotate around the torus if $k_{+}(s-k_{+})$ is nonzero. In the special case $s=k_{+}$ the density $\\rho_0(\\varphi;t)$ is independent of \n$\\varphi$. A numerically obtained example of this is shown in Fig.~\\ref{fig_Ramanathan}(a). \nIn Fig.~\\ref{fig_num_spin} we compare numerical calculations to analytical results. \n\\begin{figure}[ht]\n\\centering\n\\includegraphics[scale=0.85,clip]{fig_rho0.pdf}\n\\vspace{5mm}\n\\caption{(Color online) (a) Numerically calculated $\\rho_0$ for a ${}^{23}$Na condensate \nwith $\\epsilon =0.75 g_2n, q=2.5g_2n, f_z=0, k_{1}=2,$ and $k_{-1}=1$, corresponding to the \nblue dash-dotted line in Fig.~\\ref{fig_wi}(d). (b) A magnification of the region bounded by the dashed vertical lines in (a). (c) Analytically calculated $\\rho_0$. \nIn (b) and (c) a logarithmic scale has been used. \n\\label{fig_num_spin} \n}\n\\end{figure}\nWe consider a sodium condensate with $\\epsilon=0.75 g_2 n,q=2.5g_2n,k_{1}=2$, and $k_{-1}=1$. \nFor these values the $s=3$ spin mode is the only unstable mode [see the blue dash-dotted line in Figs.~\\ref{fig_wi}(b) and \\ref{fig_wi}(d)]. Numerical calculations give the same result. \nBy comparing Figs.~\\ref{fig_num_spin}(b) and \\ref{fig_num_spin}(c) we see that the analytical \nexpression for $\\rho_0$ approximates the actual dynamics very precisely up to $t\\approx 15\\hbar\/g_2n$. \nAs in the case of the magnetization modes, we choose the initial length and overall phase of $\\delta\\psi(t=0)$ in such a way that the agreement between the numerical and analytical results is the best possible. \n\n\n\\section{Experiments}\n\\label{sec_Exp}\nIn this section we calculate the ratio $\\epsilon\/|g_2|n$ corresponding to two recent experiments. \nTo obtain an analytical estimate for $\\epsilon$, we assume that the particle density $|\\psi_{r;z}(r,z)|^2$ is peaked around $R$ and approximate $1\/r^2\\approx 1\/R^2$ in Eq.~\\eqref{eq_epsilon}. This gives $\\epsilon\\approx \\hbar^2\/2mR^2$. \nApproximating $\\psi_{r;z}$ by the Thomas-Fermi (TF) wavefunction yields \n\\begin{align}\nn &\\approx\\sqrt{\\frac{2 m N\\omega_r\\omega_z}{9\\pi^2g_0 R}}.\n\\end{align} \nWe see that $\\epsilon\/|g_2|n \\propto (\\omega_r\\omega_z N R^3)^{-1\/2}$, so that the properties of the excitation spectrum can be controlled by adjusting the trapping frequencies, number of particles, and the radius of the toroid.\n\nUsing the parameter values of the sodium experiment \\cite{Ramanathan11} we get $\\epsilon\\approx 0.04 g_2 n$. We study numerically the cases $(k_{1},k_{-1})=(0,0)$ and $(k_{1},k_{-1})=(1,0)$.\nWith the help of Eqs.~\\eqref{o1234fz0} and \\eqref{o56} we find that magnetization modes are stable, \nbut spin modes are unstable in both cases. If $0< q \\leq 0.04 g_2 n$, $f_z=0$, and $(k_{1},k_{-1})=(0,0)$, the unstable spin mode leads to a position-independent, homogeneous, increase in $\\rho_0$. If $(k_{1},k_{-1})=(1,0)$, we get $\\rho_0(\\varphi;t)\\sim e^{2\\omega^{\\textrm{i}}_5 t}\\sin^2[(\\epsilon t+\\varphi)\/2]$. The $1$D numerical calculations \nshown in Fig. \\ref{fig_Ramanathan} confirm the validity of these analytical predictions. \nThis example illustrates that even a small $\\epsilon$ can lead to a strongly winding number-dependent \nbehavior of $\\rho_0$. \n\\begin{figure}\n\\center\n\\includegraphics[scale=0.85,clip]{fig_Ramanathan.pdf}\n\\vspace{5mm}\n\\caption{(Color online) Numerically calculated $\\rho_0$ for a ${}^{23}$Na condensate with $\\epsilon=q=0.04|g_2|n$ and $f_z=0$. In (a) $k_{1}=k_{-1}=0$ and in \n(b) $k_{1}=1,k_{-1}=0$. The value of $\\epsilon$ corresponds to that of \\cite{Ramanathan11}. \n\\label{fig_Ramanathan}}\n\\end{figure}\n\n\nThe first experimental realization of a toroidal spin-1 \nBEC was reported recently~\\cite{Beattie13}. The stability of a rubidium BEC with a winding number three vortex in the $m_F=1$ and $m_F=0$ components was found to depend strongly on the population difference of the two components, the most unstable situation corresponding to equal population. Although not directly comparable, our analysis agrees qualitatively with this result: The growth rate of unstable spin and magnetization modes increases as the population difference of the $m_F=1$ and $m_F=-1$ components goes to zero. \nThe parameter values of this experiment yield $\\epsilon\\approx 0.20 |g_2|n$. \nThe $s=1,2,$ and $s=3$ magnetization modes are unstable regardless of the values of winding numbers. \n If $k_{+}=0$ and $q+\\epsilonk_{-}^2 = 2.8|g_2|n$, the spin modes have a rotonlike spectrum (see the left panel of Fig.~\\ref{fig_roton}). The $s=4$ mode can be seen to be the only unstable spin mode. \nThis is confirmed by the numerical results shown in Fig.~\\ref{fig_Beattie}(a). In this figure we have chosen \n$k_{1}=-k_{-1}=1$ and $q=2.6|g_2|n$, so that $q+\\epsilonk_{-}^2 =2.8|g_2|n$. \nBecause $k_{+}=0$, Eqs.~\\eqref{eq_FzexpApprox} and \\eqref{eq_deltapsi0} predict that the nodes of $\\rho_0$ and $\\langle\\hat{F}_z\\rangle$ do not rotate around the torus as time evolves. This is clearly the case in Fig.~\\ref{fig_Beattie}.\nThe $s=3$ magnetization mode can be seen to be the fastest growing unstable mode. However, around $t\\approx 12\\hbar\/g_2 n$, the $s=2$ mode becomes the dominant unstable mode. These observations agree with analytical predictions: Using Eq.~\\eqref{o1234fz0} we find that $\\hbar\\omega^{\\textrm{i}}_3(s)\/|g_2|n=0.72, 1.26$, and $1.34$ for $s=1,2$, and $s=3$, respectively. For other values of $s$ we get $\\omega^{\\textrm{i}}_3(s)=0$. \n\\begin{figure}\n\\center\n\\includegraphics[scale=0.85,clip]{fig_Beattie.pdf}\n\\vspace{5mm}\n\\caption{(Color online) Numerically calculated (a) $\\rho_0$ and (b) $\\langle\\hat{F}_z\\rangle$ for\n a ${}^{87}$Rb condensate with $\\epsilon=0.2|g_2|n,q=2.6|g_2|n,f_z=0$, and $k_{1}=-k_{-1}=1$. \n The value of $\\epsilon$ corresponds to that of Ref.~\\cite{Beattie13}.\n\\label{fig_Beattie}}\n\\end{figure}\n\n\n\n\\section{Conclusions}\n\\label{sec_Con}\nWe have calculated analytically the Bogoliubov spectrum of a toroidal spin-1 BEC that has vortices in the $m_F=\\pm 1$ spin components and is subjected to a homogeneous magnetic field. \nWe treated the strength of the magnetic field and the winding numbers of the vortices as free parameters and assumed that the population of the $m_F=0$ component vanishes. We assumed also that the system is quasi-one-dimensional. We found that the spectrum can be divided into spin and magnetization modes. Spin modes \nchange the particle density of the $m_F=0$ component but leave the particle density difference of the $m_F=1$ and $m_F=-1$ components unchanged. The magnetization modes do the opposite. \nAn important parameter characterizing the spectrum is the ratio of the kinetic to interaction energy, $\\epsilon\/|g_2|n$. \nThe properties of magnetization modes can be tuned by adjusting this ratio, whereas in the case of spin modes also the strength of the magnetic field can be used to control the spectrum. For example, a spin mode spectrum with a roton-maxon structure can be realized both in rubidium and sodium condensates by making the magnetic field strong enough. Furthermore, by changing the strength of the magnetic field or the ratio $\\epsilon\/|g_2|n$, an initially stable condensate can be made unstable. We also showed that some unstable spin modes lead to a transient dark solitonlike wave function of the $m_F=0$ spin component. \nFinally, we discussed briefly two recent experiments on toroidal BECs and \n showed examples of the instabilities that can be realized in these systems. \n\nWe studied the validity of the analytical results by numerical one-dimensional simulations, finding that the former give a very good description of the stability of the condensate and the initial time evolution of the instabilities. \n\n\\begin{acknowledgments}\nThis research has been supported by the Alfred \\mbox{Kordelin} Foundation and the Academy of Finland through\nits Centres of Excellence Program (Project No. 251748).\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzgaao b/data_all_eng_slimpj/shuffled/split2/finalzzgaao new file mode 100644 index 0000000000000000000000000000000000000000..1cff645d01219a170e6554e2145d8c804e7184c0 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzgaao @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nThis article concerns embeddability conditions for pairs of groups from the family of \\emph{symmetric R. Thompson groups} \n$\\{V_m(G)\\}$. The group $V_m(G)$ is the group $V_m(G)=\\langle V_m\\cup\nG\\rangle$, where $V_m\\leq \\operatorname{Aut}(\\mathfrak{C}_m)$ is the Higman-Thompson group\ndenoted $G_{m,1}$ by Higman in \\cite{HI}, acting on the Cantor space $\\mathfrak{C}_m:= \\{0,1,\\ldots, m-1\\}^\\omega$, while $G$ is a particular faithful representation of a finite group $\\widetilde{G}\\leq \\operatorname{Sym}(m)$ in $\\operatorname{Aut}(\\mathfrak{C}_m)$. \n\n\nThe groups $\\{V_m(G)\\}$ have developed as groups of interest for a variety of reasons. Firstly, they were singled out as natural groups of interest in \\cite{NekraCuntz} and \\cite{Roever}, and they arise naturally as a fundamental subfamily of Hughes' $\\mathcal{F}\\mathcal{S}\\mathcal{S}$ groups \\cite{Hughes}. The paper \\cite{BDJ} shows that for $n\\geq2$, $V_m\\cong V_m(G)$ if and only if $\\widetilde{G}$ is semiregular (the nontrivial elements of $\\widetilde{G}$ have no fixed points), and also, that for $m>3$ there exists $\\widetilde{G},\\widetilde{H}\\in\\operatorname{Sym}(m)$ with $\\widetilde{G}\\cong\\widetilde{H}$ but where the induced groups $V_m(G)$ and $V_m(H)$ are not isomorphic (the orbit structure of the actions of the elements of the groups $\\widetilde{G}$ and $\\widetilde{H}$ impacts the isomorphism types of the groups $V_m(G)$ and $V_m(H)$). In another direction, in \\cite{FarleyCoCF} Farley shows the symmetric R. Thompson groups are CoCF groups (see \\cite{HRRT} for the definition of CoCF groups). Thus, if one can show that some group in the family $\\{V_m(G)\\}$ fails to embed in $V=V_2$, then Lehnert's conjecture will be shown to be false (see \\cite{LehnertDiss,LehnertSchweitzer, BMN}).\n\n\nWe investigate conditions on $m\\leq n$, $G\\leq \\operatorname{Sym}(m)$, and $H\\leq \\operatorname{Sym}(n)$ that guarantee the existence of embeddings between the groups $V_m(G)$ and $V_n(H)$ (we now drop the ``tilde'' notation on the groups $G$ and $H$ when thinking of them as subgroups of $\\operatorname{Sym}(m)$ and $\\operatorname{Sym}(n)$, respectively). Thus, this note can be thought of as a continuation of the investigations in \\cite{BDJ}, and is partly inspired by the work of Birget (in \\cite{B}, he gives a method to embed $V_2$ into $V_m$ for $2\\leq m$ (embeddings in the other direction have been known since Higman's book \\cite{HI})), and partly by considering some of the questions alluded to in the previous paragraph. In this context, our two embedding results depend on the direction of the embedding ($V_m(G)\\rightarrowtail V_n(H)$ or $V_n(H)\\rightarrowtail V_m(G)$), and our constructed embeddings require in both cases the Higman condition $n\\equiv 1\\mod(m-1)$.\n\nThe embedding $V_m(G)\\rightarrowtail V_n(H)$ is algebraic in nature, inspired by the embedding of Birget from $V_2$ into $V_n$ in \\cite{B}, while the embedding $V_n(H)\\rightarrowtail V_m(G)$ uses a topological conjugacy by rational group elements (see \\cite{GNS}).\n\nWe can now state and discuss our main results.\n\n\\begin{theorem}\\label{thm:MT1}\nLet $n,m \\geq 2$ be natural numbers such that $m1$, let $\\mathfrak{C}_n$ be the $n$-adic Cantor set, which is constructed inductively as follows: $\\mathfrak{C}_n^1$ corresponds to first subdividing $\\mathfrak{C}_n^0= [0,1]$ into $2n-1$ closed intervals of equal length (so, sharing endpoints with neighbours), numbered $1, \\ldots, 2n-1$ from left to right, and then taking the collection of odd-numbered sub-intervals. Next, $\\mathfrak{C}_n^2$ is obtained from $\\mathfrak{C}_n^1$ by applying the same procedure to each of the intervals forming $\\mathfrak{C}_n^1$, and so on. Then, $C_n$ is the limit of this process, so that \n\\[\n\\mathfrak{C}_n=\\cap_i \\mathfrak{C}_n^i.\n\\] Now, let $\\mathcal{A}_n = \\{0,\\dots,n-1\\}$ and give it the discrete topology. It is easy to build a direct homemorphism from the space $\\mathcal{A}_n^\\mathbb{N}$ equipped with the product topology to $\\mathfrak{C}_n$, so every element $\\zeta \\in \\mathfrak{C}_n$ can be expressed as an infinite word $\\zeta = w_1w_2\\dots $, where $w_i \\in \\mathcal{A}_n$. It is a classical result of Brouwer from \\cite{BRO} that all of the spaces in the set $\\{\\mathfrak{C}_n\\}$ are abstractly homeomorphic to each other.\n\nWe denote by $\\mathcal{A}_n^*$ the set of finite words in $\\mathcal{A}_n$. The empty word $\\varepsilon$ is also in $\\mathcal{A}_n^*$.\n\n\\begin{definition}[Concatenation]\nLet $u = u_1u_2\\dots u_k, u_i \\in \\mathcal{A}_n$ be a finite word and $v \\in \\mathcal{A}_n^* \\cup \\mathcal{A}_n^{\\mathbb{N}}$ with $v=v_1v_2\\ldots$ (where for all valid indices $i$ we have $v_i\\in \\mathcal{A}_n$) . The \\textit{concatenation} of $u$ with $v$ is the (finite or infinite) word: $$u \\vert\\!\\vert v = u_1u_2\\dots u_kv_1v_2\\ldots.$$ \n\\end{definition}\n\nWith concatenation being a fundamental operation, we will often just write the concatenation of two strings without the formal concatenation operator, that is, we might write $u\\vert\\!\\vert v$ as simply $uv$, reserving the formal use of ``$\\vert\\!\\vert$'' for situations where we wish to stress that a concatenation is occuring.\n\n\n\\begin{definition}[Prefix order]\nLet $u \\in \\mathcal{A}_n^*$ and $v \\in \\mathcal{A}_n^{*}\\cup \\mathcal{A}_n^{\\mathbb{N}}$. We say that $u$ is a \\textit{prefix} of $v$ ($u \\leq_{pref} v$) if $v = u\\vert\\!\\vert w$, for some $w \\in\\mathcal{A}_n^{*}\\cup \\mathcal{A}_n^{\\mathbb{N}}$.\n\\end{definition}\n\nNote that this property is transitive for finite length words: If $u \\leq_{pref} v$ and $v \\leq_{pref} w$ then $u \\leq_{pref} w$. In addition, $u \\leq_{pref} u$, as $\\varepsilon \\in \\mathcal{A}_n^*$. That is, $\\leq_{pref}$ provides a partial order on $\\mathcal{A}_n^{*}$.\n\n\\begin{definition}[Prefix code]\nLet $S$ be a finite set of words in $\\mathcal{A}_n^*$. Then $S$ is an \\textit{prefix code} of $\\mathfrak{C}_n$ if for every infinite word $\\zeta \\in \\mathcal{A}_n^{\\mathbb{N}}$ there exists one and only one word $s \\in S$ such that $s \\leq_{pref} \\zeta$. (Specifically, a prefix code is a complete anti-chain for the partial order $\\leq_{pref}$.)\n\\end{definition}\n\nFor convenience, we will use the following notation: let $\\sigma \\in \\operatorname{Sym}(n)$ be an element of the symmetric group of $n$ elements. Given any word $\\zeta = z_1z_2z_3 \\dots \\in \\mathcal{A}_n^\\mathbb{*} \\cup \\mathcal{A}_n^\\mathbb{N}$ we define $\\sigma(\\zeta) = \\sigma(z_1)\\sigma(z_2)\\sigma(z_3)\\dots \\in \\mathcal{A}_n^\\mathbb{*} \\cup \\mathcal{A}_n^\\mathbb{N}$. Let $\\sigma_i \\in H \\leq \\operatorname{Sym}(n)$. (Note that we are using left actions here, so if $\\sigma,\\tau\\in \\operatorname{Sym}(n)$ then the product $\\tau\\sigma$ means employ the permutation $\\sigma$ first, and then employ $\\tau$).\n\nWith the above notation, an element of $V_n(H)$ is a homeomorphism of $\\mathfrak{C}_n$ that can be (non-uniquely) described by a \\textit{table} as follows:\n$$v = \\begin{bmatrix}\np_1 & p_2 & \\cdots & p_k \\\\\n\\sigma_1 & \\sigma_2 & \\cdots & \\sigma_k\\\\\nq_1 & q_2 & \\cdots & q_k \\\\\n\\tau_1 & \\tau_2 & \\cdots & \\tau_k\n\\end{bmatrix},$$\nwhere $p_i,q_i \\in \\mathcal{A}_n^*$, $\\sigma_i, \\tau_i \\in H$ and such that the sets $P = \\{p_i\\}_{i=1}^k$ and $Q = \\{q_i\\}_{i=1}^k$ are prefix codes of $\\mathfrak{C}_n$. We say that $k \\geq 1$ is the \\textit{length} of the table. The homeomorphism of $\\mathfrak{C}_n$ induced can be defined as follows: for every infinite word $\\zeta$ such that $p_i\\leq_{pref}\\zeta$, that is $\\zeta = p_i \\vert\\!\\vert u$ for some $u \\in \\mathcal{A}_n^\\mathbb{N}$, we have \n$$v: p_i \\vert\\!\\vert \\sigma_i(u) \\rightarrow q_i \\vert\\!\\vert \\tau_i(u).$$\nThere are infinitely many tables which induce the same homeomorphism of $\\mathfrak{C}_n$. We proceed to define the four basic moves we can perform on a table in order to obtain an equivalent one (the four basic moves naturally split as two essential sorts of moves, together with their inverse (or ``near-inverse'') moves). \n\nThe first basic move is \\textit{expansion}: for a given prefix code $$P = \\{p_1, \\dots, p_i , \\dots , p_k\\},$$ we can consider $$\\widetilde{P} = \\{p_1, \\dots, p_i0, \\dots, p_i(n-1) , \\dots , p_k\\}$$ by expanding the word $p_i$. This expansion not only occurs in $P$, as the image of $p_i$ must be also expanded. So we have $$\\widetilde{Q} = \\{q_1, \\dots, q_i0, \\dots, q_i(n-1) , \\dots ,q_k\\}.$$ It is easy to see that both $\\widetilde{P}$ and $\\widetilde{Q}$ are also prefix codes. Then:\n$$\\begin{bmatrix}\np_1 & \\cdots& p_i & \\cdots & p_k \\\\\n\\sigma_1 & \\cdots& \\sigma_i & \\cdots & \\sigma_k \\\\\nq_1 & \\cdots& q_i & \\cdots & q_k \\\\\n\\tau_1 & \\cdots& \\tau_i & \\cdots & \\tau_k\n\\end{bmatrix} \\equiv \n\\begin{bmatrix}\np_1 & \\cdots& p_i \\sigma_i(0) & \\cdots & p_i \\sigma_i(n-1) & \\cdots & p_k \\\\\n\\sigma_1 & \\cdots& \\sigma_i & \\cdots& \\sigma_i & \\cdots & \\sigma_k \\\\\nq_1 & \\cdots& q_i \\tau_i(0) & \\cdots & q_i \\tau_i(n-1) & \\cdots & q_k \\\\\n\\tau_1 & \\cdots& \\tau_i & \\cdots& \\tau_i & \\cdots & \\tau_k \\\\\n\\end{bmatrix}.$$ One can always perform an expansion, but not all tables look like the result of an expansion. Naturally, the inverse of an expansion (when it is defined) is called a \\emph{reduction}.\n\nThe second move we can perform on a table is \\textit{pushing down} (resp. \\textit{pushing up}) the action of all $\\sigma_i$ such that $\\sigma_i = Id$ for every $i \\in \\{1, \\dots, k\\}$ (resp. $\\tau_i = Id$ for every $i \\in \\{1, \\dots, k\\}$):\n\\begin{align*}\n\\begin{bmatrix}\np_1 & p_2 & \\cdots & p_k \\\\\n\\sigma_1 & \\sigma_2 & \\cdots & \\sigma_k \\\\\nq_1 & q_2 & \\cdots & q_k \\\\\n\\tau_1 & \\tau_2 & \\cdots & \\tau_k \n\\end{bmatrix} &\\equiv \n\\begin{bmatrix}\np_1 & p_2 & \\cdots & p_k \\\\\nId & Id & \\cdots & Id\\\\\nq_1 & q_2 & \\cdots & q_k \\\\[2pt]\n\\tau_1\\sigma_1^{-1} & \\tau_2\\sigma_2^{-1} & \\cdots & \\tau_k\\sigma_k^{-1}\n\\end{bmatrix} \\\\ &\\equiv \n\\begin{bmatrix}\np_1 & p_2 & \\cdots & p_k \\\\[2pt]\n\\sigma_1\\tau_1^{-1} & \\sigma_2\\tau_2^{-1} & \\cdots & \\sigma_k\\tau_k^{-1} \\\\[2pt]\nq_1 & q_2 & \\cdots & q_k \\\\\nId & Id & \\cdots & Id\\\\\n\\end{bmatrix}.\n\\end{align*}\n\nIt is not hard to see that a table gives a well-defined homeomorphism of the appropriate Cantor space, and if two tables are related by a finite sequence of our four moves then they represent the same homeomorphism. The reader can also check that if a homeomorphism of an appropriate Cantor space is represented by two tables, then in fact these tables are in the same equivalence class under our four basic moves on tables. Thus, we can just consider our group elements to be the equivalence classes of tables with the aforementioned relations. \n\nThe {composition} of two different elements $u,v\\in V_n(H)$ is easy to compute using the equivalences. Let $u$, $v \\in V_n(H)$, such that $u$ takes the prefix code $P$ to the prefix code $Q$ (resp. $v$ takes $P'$ to $Q'$). We need to find a prefix code $S$ such that, for every element $s \\in S$, there exists one element $q \\in Q$ and one element $p' \\in P'$ such that $q\\leq_{pref} s$ and $p'\\leq_{pref} s$. This can always be done by expanding $P'$ and $Q$ until we obtain the same prefix code $S$. Thus, without loss of generality:\n$$u = \\begin{bmatrix}\np_1 & p_2 & \\cdots & p_k \\\\\n\\sigma_1 & \\sigma_2 & \\cdots & \\sigma_k \\\\\ns_1 & s_2 & \\cdots & s_k \\\\\n\\tau_1 & \\tau_2 & \\cdots & \\tau_k\n\\end{bmatrix}, \\quad v = \\begin{bmatrix}\ns_1 & s_2 & \\cdots & s_k \\\\\n\\sigma'_1 & \\sigma'_2 & \\cdots & \\sigma'_k \\\\\nq'_1 & q'_2 & \\cdots & q'_k \\\\\n\\tau'_1 & \\tau'_2 & \\cdots & \\tau'_k\n\\end{bmatrix}.$$\n\nFinally, we push up the action of $u$ and push down the action of $v$:\n\\begin{align*}\nu& = \\begin{bmatrix}\np_1 & p_2 & \\cdots & p_k \\\\\n\\sigma_1\\tau_1^{-1} & \\sigma_2\\tau_2^{-1} & \\cdots & \\sigma_k\\tau_k^{-1}\\\\\ns_1 & s_2 & \\cdots & s_k \\\\\nId & Id & \\cdots & Id\n\\end{bmatrix},\\\\ v & = \\begin{bmatrix}\ns_1 & s_2 & \\cdots & s_k \\\\\nId & Id & \\cdots & Id\\\\\nq'_1 & q'_2 & \\cdots & q'_k \\\\[2pt]\n\\tau'_1(\\sigma'_1)^{-1} & \\tau'_2(\\sigma'_2)^{-1} & \\cdots & \\tau'_k(\\sigma'_k)^{-1}\n\\end{bmatrix},\n\\end{align*}\nso \n$$v \\circ u = \\begin{bmatrix}\np_1 & p_2 & \\cdots & p_k \\\\\n\\sigma_1\\tau_1^{-1}& \\sigma_2\\tau_2^{-1} & \\cdots & \\sigma_k\\tau_k^{-1}\\\\\nq'_1 & q'_2 & \\cdots & q'_k \\\\[2pt]\n\\tau'_1(\\sigma'_1)^{-1} & \\tau'_2(\\sigma'_2)^{-1} & \\cdots & \\tau'_k(\\sigma'_k)^{-1}\n\\end{bmatrix}.$$\n\nWe sum up the previous discussion in the following proposition:\n\\begin{proposition}\n$V_n(H)$ is a group with the composition.\n\\end{proposition}\n\n\n\\section{Topological Embeddings}\n\nIn this section, we present topological embeddings between symmetric Thompson's groups. The key idea is, given any group $V_n(H)$, to translate the action of an element $\\sigma \\in H$ into a permutation $\\widetilde{\\sigma}$ of the elements of some prefix code of $\\mathfrak{C}_m$. Therefore, $\\widetilde{\\sigma} \\in V_m(G)$ for some $G$.\n\nOur method will be first to understand when actions on prefix codes over smaller alphabets can represent embeddings of permutations on larger alphabets which commute with our core operations of expansion and contraction of prefix codes. With that understanding in hand, we can then build the desired embedding from a group $V_n(H)$ to a group $V_m(G)$ for $m\\leq n$.\n\nWe first establish some useful definitions.\n\n\\subsection{The Root Group \\texorpdfstring{$\\boldsymbol{\\mathcal{R}_{G}(S)}$}{RGS}}\n\nGiven a linear order $\\leq$ on $\\mathcal{A}_n$ (we choose $0<1<\\ldotsm$, which is our primary case of interest).\n\n\\subsection{Successors} Here, we give the key idea for our algebraic embeddings, which relies on extending an idea of Birget into our context.\n\nThe \\textit{successor} of an element, expressed as a table, was defined in \\cite{B} in order to embed $V_2(Id)$ in $V_n(Id)$, for all $n \\geq 2$. We generalise Birget's definition.\n\n\n\n\\begin{definition}[Set of prefixes]\\label{spref}\\cite{B}\nLet $P \\subset \\mathcal{A}_n^*$ be a prefix code of $\\mathfrak{C}_n$. We define the \\textit{set of prefixes of $P$}, $\\operatorname{spref}(P)$ as follows:\n$$ \\operatorname{spref}(P) = \\{w \\in \\mathcal{A}_n^* : \\exists p \\in P,\\, w <_{pref} p \\}.$$\nIn other words, $\\operatorname{spref}(P)$ is the set of strict prefixes of the elements of $P$.\n\\end{definition}\n\nWe are embedding a symmetric Thompson's group on alphabet $\\mathcal{A}_m$ into a symmetric Thompson's group on alphabet $\\mathcal{A}_n$, where $m\\leq n$. For this, we will assume $\\mathcal{A}_m\\subseteq \\mathcal{A}_n$. And in particular we set $\\mathcal{A}_m=\\{a_0,a_1,\\ldots,a_{m-1}\\}$ and $\\mathcal{A}_n=\\mathcal{A}_m\\cup\\{a_m,a_{m+1},\\ldots,a_{n-1}\\}$ with symbols with distinct indices being distinct (so that $|\\mathcal{A}_n|=n$).\n\n\nIn what follows, we take a prefix code $P\\subset a_{m-1}\\vert\\!\\vert \\mathcal{A}_m^*$ (so each element of $P$ begins with the letter $a_{m-1}$) and transform it to a new prefix code $\\succ{P}\\subset \\mathcal{A}_n^*$ by appending letters from the set $\\{a_{m},a_{m+1},\\ldots,a_n\\}$. \n\n\n\\begin{definition}[Successor]\\label{successor} Let $P \\subset a_{m-1}\\vert\\!\\vert\\{a_0, \\dots, a_{m-1}\\}^*$ be a prefix code (complete, were we to remove the initial prefix letter $a_{m-1}$, so that $|P|\\equiv 1\\mod (m-1)$) with $\\vert P \\vert = l \\geq 1$, and let $\\{p_1, \\dots, p_l\\}$ be the ordered list of all the elements of $P$, using the \\emph{reverse} dictionary order. \n\nWe build a new prefix code $\\succ{P}$ inductively using our ordered list $(p_1,p_2,\\ldots,p_l)$.\n\nLet $k$ be the smallest non-negative integer so that $n-m=k(m-1)$ (this $k$ will exist when $m$ and $n$ satisfy Higman's condition, which we require to build our embeddings).\n\nWe define (inductively) nested sets $P_{s,i}$, where $s$ will grow from $1$ to $l$, and for each value of $s$, we will have $i$ grow from $1$ to $k$.\n\nSet $\\mathcal{A}_{m,n}:= \\{a_m,a_{m+1},\\ldots,a_{n-1}\\}$. For every $p_s \\in P$, and $i\\in \\{1,2,\\ldots, k\\}$ the \\textit{$i$-th successor} $(p_s)'_i$ of $p_s$ is the element of $\\operatorname{spref}(P)\\vert\\!\\vert \\mathcal{A}_{m,n}$ defined as follows, assuming that \n\\[P_{s,i-1} = \\left\\{\\begin{matrix}(p_1)'_1,(p_1)'_{2},\\ldots,(p_1)'_{k},\\\\(p_2)'_1,(p_2)'_{2},\\ldots,(p_2)'_{k},\\\\\n\\vdots\\\\\n(p_s)'_{1},(p_s)'_{2},\\ldots,(p_s)'_{i-1}\n\\end{matrix}\\right\\}\\]\nhas already been defined, we set:\n\n\\begin{align*}\n(p_s)'_i = \\min \\{ & xa_j \\in \\operatorname{spref}(P)\\vert\\!\\vert\\mathcal{A}_{m,n}: p_s <_{dict} xa_j \\ \\mbox{and} \\ xa_j \\not\\in P_{s,i-1}\\},\n\\end{align*}\nwhere $\\min$ uses the dictionary order in $\\{a_0, \\dots, a_{n-1}\\}$.\n\\end{definition}\n\n\n\\begin{example}\nSuppose $m=3$ and $n=5$, so that $k=1$. In the definition above, $a_{m-1}=2$. So, consider the set $P=\\{20,210,211,212,22\\}$. Now, $k=1$ and $\\operatorname{spref}(P)=\\{\\varepsilon,2,21\\}$. We obtain\n\\[\n\\begin{array}{lll}\n p_1=22&\\quad&(p_1)'_1=23\\\\\n p_2=212&\\quad&(p_2)'_1=213\\\\ \n p_3=211&\\quad& (p_3)'_1=214\\\\\n p_4=210&\\quad& (p_4)'_1=24\\\\\n p_5=20&\\quad&(p_5)'_1=3.\n\\end{array}\n\\]\n\n\\end{example}\n\n\\begin{remark}The three constants, $n,m,k$ are not arbitrary, as the system of successors needs to be well defined. If every element has $k$ successors, then: $$n-m = k(m-1), \\, k \\geq 0,$$\nwhich is Higman's condition.\\end{remark}\n\\begin{proof}\n If we expand an element $p_i \\in P$, we need to assign successors to each element $p_ia_j$ for every $0 \\leq j \\leq m-1$. In particular, the number of successors $k$ of every leaf does not vary, and each element $p_ia_r$ for every $m \\leq r \\leq n-1$ needs to be the successor of some element in $\\widetilde{P} = (P \\backslash \\{p_i\\}) \\cup \\{p_ia_0, \\dots,p_ia_{m-1}\\}$. Then $\\widetilde{P}$ has $m-1$ more elements than $P$ and there are $n-m$ new elements $p_ia_r$ for $m \\leq r \\leq n-1$. Thus, we need $m-1$ to evenly divide $n-m$, and $k$ is the factor of this division.\n\\end{proof}\n\nWe proceed to prove the following lemma, essential for the proof of Theorem \\ref{thm:MT2}: \n\n\\begin{lemma}\\label{lem:successor}Suppose $m\\leq n$ are naturals so that there is $k$ natural with $n-m=k(m-1)$. Suppose $l$ is a positive integer congruent to $m$ modulo $m-1$. Let $S= \\{ a_m , \\dots, a_{n-1}\\}$ and let $P \\subset a_{m-1}\\vert\\!\\vert\\{a_0, \\dots, a_{m-1}\\}^*$ be an $l$-element prefix code, ordered as $p_l <_{dict} p_{l-1}<_{dict}\\dots <_{dict} p_1$. Let $i$ with $1\\leq i\\leq l$. Then, the successors $(p_i)'_1$, $(p_i)'_2$, $\\ldots$, $(p_i)'_k$ are well defined, and furthermore, the expansion in which we replace $P$ by $\\widetilde{P} = (P \\backslash \\{p_i\\} ) \\cup p_i\\{a_0, \\dots, a_{m-1}\\}$ has successors $(p_ia_j)'_i$ uniquely determined as follows: \n$$\\begin{array}{lll}\n(p_ia_{m-1})'_1 & = p_ia_{m}\\\\\n& \\vdots \\\\ (p_ia_{m-1})'_k & = p_ia_{m+k-1}\\\\\n(p_ia_{m-2})'_1 & = p_ia_{m+k}\\\\\n& \\vdots \\\\ (p_ia_{m-2})'_k & = p_ia_{m+2k-1}\\\\\n& \\vdots \\\\\n(p_ia_{1})'_1 & = p_ia_{m+(m-2)k}\\\\ \n& \\vdots \\\\\n(p_ia_{1})'_k & = p_ia_{m+(m-1)k-1} = p_ia_n\\\\\n(p_ia_{0})'_1 & = (p_i)'_1\\\\ \n& \\vdots \\\\\n(p_ia_{0})'_k & = p_ia_{m+(m-1)k} = (p_i)'_k.\\\\\n\\end{array}$$\n\\end{lemma}\n\n\\begin{proof}\nWe prove the two statements by induction on $l$. \n\n{\\flushleft {\\it Base Case ($l=1$):}}\\\\\nIf $l=1$ then $P=\\{a_{m-1}\\}$. We have $\\operatorname{spref}{P}=\\{\\varepsilon\\}$. It then follows that the $k$ successors are, the set $\\{a_m,a_{m+1},\\ldots,a_{m+k-1}\\}$, noting that these are given in order and are the results of the inductive definition of the $k$ successors of $a_{m-1}$. Thus we have in the base case that the successors are well defined. We need to verify the existence of well defined successors for an expansion of $P=\\{a_{m-1}\\}$. In this case, $P$ admits only one expansion, which is precisely the set $\\widetilde{P}=\\{a_{m-1}a_{m-1},a_{m-1}a_{m-2},\\ldots,a_{m-1}a_0\\}.$ We have $\\operatorname{spref}({\\widetilde{P}})=\\{\\varepsilon,a_{m-1}\\}$ and we have $$\\begin{array}{lll}\n(a_{m-1}a_{m-1})'_1 & = a_{m-1}a_{m}\\\\\n& \\vdots \\\\ (a_{m-1}a_{m-1})'_k & = a_{m-1}a_{m+k-1}\\\\\n(a_{m-1}a_{m-2})'_1 & = a_{m-1}a_{m+k}\\\\\n& \\vdots \\\\ (a_{m-1}a_{m-2})'_k & = a_{m-1}a_{m+2k-1}\\\\\n& \\vdots \\\\\n(a_{m-1}a_{1})'_1 & = a_{m-1}a_{m+(m-2)k}\\\\ \n& \\vdots \\\\\n(a_{m-1}a_{1})'_k & = a_{m-1}a_{m+(m-1)k-1} = a_{m-1}a_n\\\\\n(a_{m-1}a_{0})'_1 & = (a_{m-1})'_1\\\\ \n& \\vdots \\\\\n(a_{m-1}a_{0})'_k & = a_{m-1}a_{m+(m-1)k} = (a_{m-1})'_k.\\\\\n\\end{array}$$\nWe can directly observe these successors are well defined and distinct. Thus, the statement is true for $l=1$.\n\n\n{\\flushleft {\\it Inductive Case ($l>1$):}}\\\\\nNow let us assume that $\\widetilde{P}$ is a result of $v$ expansions from the one-element prefix code $\\{a_{m-1}\\}$, for some $v\\geq 1$, where for any prefix code resulting from $u$ expansions from $\\{a_{m-1}\\}$ for $0\\leq u_{dict} p_i$, then $pa_t \\in P_{i-1,k} = \\widetilde{P}_{i-1,k}$, which is also a contradiction. Thus $(p_ia_{m-1})'_1 = p_ia_m$. We can use a similar argument for all $(p_ia_{m-1})'_1 \\dots (p_ia_{1})'_k$. \n\nFor $p_ia_0$, all successors of the form $p_ia_s,\\, a_s \\in \\{a_m ,\\dots, a_{n-1}\\}$, have already been assigned. Thus, the remaining $k$ successors are precisely the $k$ successors of $p_i$, taken in order.\n\\end{proof}\n\n\\begin{remark}\\label{rem:successor}\n Birget in \\cite{B} gives a formula for the $i$-th successor of an element, for the case of $m=2$. The statement of Lemma \\ref{lem:successor} above shows the natural generalisation of that formula holds when we have the Higman Condition (as we must for successors to be well defined). The resulting formula is given as follows:\n \nLet $P \\subset a_{m-1}\\vert\\!\\vert\\{a_0, \\dots, a_{m-1}\\}^*$ be a prefix code with $\\vert P \\vert \\geq 2$, such that the elements of $P$ are ordered in reverse dictionary order. Then every element of $w \\in P$ can be written uniquely in the form $ua_ia_0^t$, where $u\\in \\{a_0 , \\dots, a_{m-1}\\}^*$ and $t \\geq 0$. The $i$-th successor of $w$ is:\n$$(w)'_i = (ua_ja_0^t)'_i = ua_{m-1+(m-1-j)k + i}.$$ \nWe stress that this formula is only valid if $P$ is ordered in reverse dictionary order.\n\\end{remark} \n\n\\subsection{The algebraic embedding}\nWe proceed to define the algebraic embedding of $V_m(G)$ in $V_n(H) = V_n(G_{ext})$. Let $g \\in V_m(G)$, given by the following table:\n\\begin{align*}\ng & = \\begin{bmatrix}\np_1 & p_2 & \\cdots & p_l \\\\\n\\sigma_1 & \\sigma_2 & \\cdots & \\sigma_l\\\\\nq_1 & q_2 & \\cdots & q_l \\\\\n\\tau_1 & \\tau_2 & \\cdots & \\tau_l\n\\end{bmatrix}\n\\end{align*}\nWe define the embedding $\\iota(g)$ below. The resulting tables are large, and our notation requires some explanation. The idea of the embedding is to use the identity map initially, and at $a_{m+k}$ and later letters, but under the address $a_{m-1}$ we place the prefix code $p_1$ to $p_l$, and we also require action under the successors. The first row then has entries following the ordered list given here (wrapped at natural locations due to page length constraints):\n\\[\n\\begin{matrix}\na_0,a_1,\\ldots,a_{m-2},\\\\\na_{m-1}p_1,a_{m-1}p_2,\\ldots,a_{m-1}p_l,\\\\\n(a_{m-1}p_1)'_1,(a_{m-1}p_2)'_1, \\ldots, (a_{m-1}p_l)'_1,\\\\\n(a_{m-1}p_1)'_2,(a_{m-1}p_2)'_2, \\ldots, (a_{m-1}p_l)'_2,\\\\\n\\dots\\\\\n(a_{m-1}p_1)'_k,(a_{m-1}p_2)'_k, \\ldots, (a_{m-1}p_l)'_k,\\\\\na_{m+k},a_{m+k+1},\\ldots,a_{n-1}.\n\\end{matrix}\n\\]\nWe use vertical bars ``$\\vert$'' in our table at the same locations that we placed line-wraps in the row detailed above, for clarity of grouping. The element $\\iota(g)$ is now given by the following table:\n\\begin{flushleft}\n$\\begin{array}{r}\n\\left[\\rule{0cm}{1cm} \\setlength\\arraycolsep{1pt} \\begin{array}{ccccccccccccc}\na_0 & \\cdots & a_{m-2} & \\vert & a_{m-1}p_1 & \\cdots & a_{m-1}p_l & \\vert & (a_{m-1}p_1)'_1 & \\cdots &(a_{m-1}p_l)'_1 &\\vert & \\cdots \\\\\nId & \\cdots & Id & \\vert & \\sigma'_1 &\\cdots & \\sigma'_l & \\vert & \\sigma'_1 &\\cdots & \\sigma'_l & \\vert & \\cdots\\\\\na_0 & \\cdots & a_{m-2} & \\vert & a_{m-1}q_1 & \\cdots & a_{m-1}q_l & \\vert & (a_{m-1}q_1)'_1 & \\cdots &(a_{m-1}q_l)'_1 &\\vert & \\cdots \\\\\nId & \\cdots & Id & \\vert & \\tau'_1 &\\cdots & \\tau'_l & \\vert & \\tau'_1 &\\cdots & \\tau'_l & \\vert & \\cdots \\\\\n\\end{array} \\color{white}\\right] \\\\ \\\\\n\n \\color{white} = \\left[\\rule{0cm}{1cm} \\setlength\\arraycolsep{1pt} \\color{black} \\begin{array}{ccccccccccccc}\n\\cdots & \\vert & (a_{m-1}p_1)'_k & \\cdots & (a_{m-1}p_l)'_k & \\vert & a_{m+k} & \\cdots & a_{n-1} \\\\\n\\cdots & \\vert & \\sigma'_1 &\\cdots & \\sigma'_l & \\vert & Id & \\cdots & Id \\\\\n\\cdots & \\vert & (a_{m-1}q_1)'_k & \\cdots & (a_{m-1}q_l)'_k & \\vert & a_{m+k} & \\cdots & a_{n-1} \\\\\n\\cdots & \\vert & \\tau'_1 &\\cdots & \\tau'_l & \\vert & Id & \\cdots & Id \\\\ \n\\end{array} \\right]\n\\end{array}$\n\\end{flushleft}\n\nNote that the set of successors of $P = \\{p_1, \\dots, p_l\\}$ are assigned supposing that $p_l <_{dict} \\dots <_{dict} p_1$. Therefore, the set of successors of $Q = \\{q_1, \\dots, q_l\\}$ is assigned following the order $q_l \\rightarrow \\dots \\rightarrow q_1$, which does not need to follow the dictionary order on $Q$. \n\nIndeed, the first and third rows of $\\iota(g)$ are both prefix codes of $\\mathfrak{C}_n$. On the one hand, suppose that the number of columns of $g$ is $l = m + d(m-1)$ for some $d \\geq 0$. It follows that the number of columns of $\\iota(g)$ whose elements of the first row start with $a_{m-1}$ is $n+ d(n-1)$ (observe that the last $k$ terms from the successor substitution will not begin with $a_{m-1}$). On the other hand, as the number of columns of $g$ is $(m + d(m-1))$, and we assign $k$ successors to every column, we have $(m+d(m-1))(k+1)$ columns on $\\iota(g)$. As we have Higman's Condition, the reader can verify that $(m+d(m-1))(k+1) = n+ d(n-1) +k$. From this, we see firstly that $(n-1)\\vert k$, but more importantly, this embedding\/successor operation does not place any constraints on the number of expansions $d$ that were used to create the original prefix code for the domain of $g$.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:MT2}]\nIf we push down the action of every $\\sigma_i$, we have:\n\\begin{align*}\n\\operatorname{push}(g) & = \\begin{bmatrix}\np_1 & p_2 & \\cdots & p_l \\\\\nId & Id & \\cdots & Id\\\\\nq_1 & q_2 & \\cdots & q_l \\\\\n\\tau_1\\sigma_1^{-1} & \\tau_2\\sigma_2^{-1} & \\cdots & \\tau_l\\sigma_l^{-1} \\\\\n\\end{bmatrix} \n\\end{align*}\nThus the table for $\\iota(\\operatorname{push}(g))$ is:\n\\begin{flushleft}\n$\\begin{array}{r}\n\\left[\\rule{0cm}{1cm} \\setlength\\arraycolsep{1pt} \\begin{array}{ccccccccccccc}\na_0 & \\cdots & a_{m-2} & \\vert & a_{m-1}p_1 & \\cdots & a_{m-1}p_l & \\vert & (a_{m-1}p_1)'_1 & \\cdots &(a_{m-1}p_l)'_1 &\\vert & \\cdots \\\\\nId & \\cdots & Id & \\vert & Id &\\cdots & Id & \\vert & Id &\\cdots & Id & \\vert & \\cdots\\\\\na_0 & \\cdots & a_{m-2} & \\vert & a_{m-1}q_1 & \\cdots & a_{m-1}q_l & \\vert & (a_{m-1}q_1)'_1 & \\cdots &(a_{m-1}q_l)'_1 &\\vert & \\cdots \\\\[2pt]\nId & \\cdots & Id & \\vert & (\\tau_1\\sigma_1^{-1})' &\\cdots & (\\tau_l\\sigma_l^{-1})' & \\vert & (\\tau_1\\sigma_1^{-1})' &\\cdots & (\\tau_l\\sigma_l^{-1})' & \\vert & \\cdots \\\\\n\\end{array} \\color{white}\\right] \\\\ \\\\\n\n \\color{white} = \\left[\\rule{0cm}{1cm} \\setlength\\arraycolsep{1pt} \\color{black} \\begin{array}{ccccccccccccc}\n\\cdots & \\vert & (a_{m-1}p_1)'_k & \\cdots & (a_{m-1}p_l)'_k & \\vert & a_{m+k} & \\cdots & a_{n-1} \\\\\n\\cdots & \\vert & Id &\\cdots & Id & \\vert & Id & \\cdots & Id \\\\\n\\cdots & \\vert & (a_{m-1}q_1)'_k & \\cdots & (a_{m-1}q_l)'_k & \\vert & a_{m+k} & \\cdots & a_{n-1} \\\\[2pt]\n\\cdots & \\vert & (\\tau_1\\sigma_1^{-1})' &\\cdots & (\\tau_l\\sigma_l^{-1})' & \\vert & Id & \\cdots & Id \\\\\n\\end{array} \\right]\n\\end{array}$\n\\end{flushleft}\nOn the other hand the table for $\\operatorname{push}(\\iota(g))$ is:\n\\begin{flushleft}\n$\\begin{array}{r}\n\\left[\\rule{0cm}{1cm} \\setlength\\arraycolsep{1pt} \\begin{array}{ccccccccccccc}\na_0 & \\cdots & a_{m-2} & \\vert & a_{m-1}p_1 & \\cdots & a_{m-1}p_l & \\vert & (a_{m-1}p_1)'_1 & \\cdots &(a_{m-1}p_l)'_1 &\\vert & \\cdots \\\\\nId & \\cdots & Id & \\vert & Id &\\cdots & Id & \\vert & Id &\\cdots & Id & \\vert & \\cdots\\\\\na_0 & \\cdots & a_{m-2} & \\vert & a_{m-1}q_1 & \\cdots & a_{m-1}q_l & \\vert & (a_{m-1}q_1)'_1 & \\cdots &(a_{m-1}q_l)'_1 &\\vert & \\cdots \\\\[2pt]\nId & \\cdots & Id & \\vert & \\tau'_1(\\sigma'_1)^{-1} &\\cdots & \\tau'_l(\\sigma'_l)^{-1} & \\vert & \\tau'_1(\\sigma'_1)^{-1} &\\cdots & \\tau'_l(\\sigma'_l)^{-1} & \\vert & \\cdots \\\\\n\\end{array} \\color{white}\\right] \\\\ \\\\\n\n \\color{white} = \\left[\\rule{0cm}{1cm} \\setlength\\arraycolsep{1pt} \\color{black} \\begin{array}{ccccccccccccc}\n\\cdots & \\vert & (a_{m-1}p_1)'_k & \\cdots & (a_{m-1}p_l)'_k & \\vert & a_{m+k} & \\cdots & a_{n-1} \\\\\n\\cdots & \\vert & Id &\\cdots & Id & \\vert & Id & \\cdots & Id \\\\\n\\cdots & \\vert & (a_{m-1}q_1)'_k & \\cdots & (a_{m-1}q_l)'_k & \\vert & a_{m+k} & \\cdots & a_{n-1} \\\\[2pt]\n\\cdots & \\vert & \\tau'_1(\\sigma'_1)^{-1} &\\cdots & \\tau'_l(\\sigma'_l)^{-1} & \\vert & Id & \\cdots & Id \\\\ \n\\end{array} \\right]\n\\end{array}$\n\\end{flushleft}\n\nRecall from the statement of Theorem \\ref{thm:MT2} that for an element $\\tau\\in\\operatorname{Sym}(m)$, the extended version $\\tau'$ of $\\tau$ in $\\operatorname{Sym}(n)$ is that element of $\\operatorname{Sym}(n)$ which agrees with $\\tau$ on the set $\\mathcal{A}_m$ and acts as the identity on the points of $\\mathcal{A}_{m,n}$ in $\\mathcal{A}_n$. Thus, both tables are equal, as $(\\tau_i\\sigma_i^{-1})' = (\\tau'_i)(\\sigma'_i)^{-1},\\, \\forall i \\in \\{i,\\dots,l\\}$. If we expand $g$ on $p_i$:\n\n$$\\operatorname{exp}(g) = \\begin{bmatrix}\np_1 & p_2 & \\cdots & p_ia_0 & \\cdots & p_ia_{m-1} & \\cdots & p_l \\\\\nId & Id & \\cdots & Id & \\cdots & Id& \\cdots & Id\\\\\nq_1 & q_2 & \\cdots & q_i \\tau_i(a_0) & \\cdots & q_i\\tau_i(a_{m-1}) & \\cdots & q_l \\\\\n\\tau_1 & \\tau_2 & \\cdots & \\tau_i & \\cdots & \\tau_i& \\cdots & \\tau_l\n\\end{bmatrix}$$\nThen the table for $\\iota(\\operatorname{exp}(g))$ is:\n\\begin{flushleft}\n$\\begin{array}{r}\n\\left[\\rule{0cm}{1cm} \\setlength\\arraycolsep{1pt} \\begin{array}{ccccccccccccc}\na_0 & \\cdots & a_{m-2} & \\vert & a_{m-1}p_1 & \\cdots & a_{m-1}p_ia_0 & \\cdots & a_{m-1}p_ia_{m-1} & \\cdots & a_{m-1}p_l &\\vert & \\cdots \\\\\nId & \\cdots & Id & \\vert & Id &\\cdots & Id & \\cdots & Id &\\cdots & Id & \\vert & \\cdots\\\\\na_0 & \\cdots & a_{m-2} & \\vert & a_{m-1}q_1 & \\cdots & a_{m-1}q_i\\tau_i(a_0) & \\cdots & a_{m-1}q_i\\tau_i(a_{m-1}) & \\cdots & a_{m-1}q_l &\\vert & \\cdots \\\\\nId & \\cdots & Id & \\vert & \\tau'_1 &\\cdots & \\tau'_i &\\cdots & \\tau'_i &\\cdots & \\tau'_l & \\vert & \\cdots \\\\\n\\end{array} \\color{white}\\right] \\\\ \\\\\n\\color{white} = \\left[\\rule{0cm}{1cm} \\setlength\\arraycolsep{1pt} \\color{black} \\begin{array}{ccccccccccccc}\n\\cdots & \\vert & \\cdots & (a_{m-1}p_ia_0)'_j & \\cdots & (a_{m-1}p_ia_{m-1})'_j & \\cdots & \\vert & a_{m+k} & \\cdots & a_{n-1} \\\\\n\\cdots & \\vert & \\cdots & Id & \\cdots & Id & \\cdots & \\vert & Id &\\cdots & Id \\\\\n\\cdots & \\vert & \\cdots & (a_{m-1}q_i\\tau_i(a_0))'_j & \\cdots & (a_{m-1}q_i\\tau_i(a_{m-1}))'_j & \\cdots& \\vert & a_{m+k} & \\cdots & a_{n-1} \\\\\n\\cdots & \\vert& \\cdots & \\tau'_i &\\cdots & \\tau'_i & \\cdots & \\vert & Id & \\cdots & Id \\\\ \n\\end{array} \\right]\n\\end{array}$\n\\end{flushleft}\n\nOn the other hand, the table for $\\operatorname{exp}(\\iota(g))$ is:\n\n\\begin{flushleft}\n$\\begin{array}{r}\n\\left[\\rule{0cm}{1cm} \\setlength\\arraycolsep{1pt} \\begin{array}{ccccccccccccc}\na_0 & \\cdots & a_{m-2} & \\vert & a_{m-1}p_1 & \\cdots & a_{m-1}p_ia_0 & \\cdots & a_{m-1}p_ia_{n-1} & \\cdots & a_{m-1}p_l &\\vert & \\cdots \\\\\nId & \\cdots & Id & \\vert & Id &\\cdots & Id & \\cdots & Id &\\cdots & Id & \\vert & \\cdots\\\\\na_0 & \\cdots & a_{m-2} & \\vert & a_{m-1}q_1 & \\cdots & a_{m-1}q_i\\tau'_i(a_0) & \\cdots & a_{m-1}q_i\\tau'_i(a_{n-1}) & \\cdots & a_{m-1}q_l &\\vert & \\cdots \\\\\nId & \\cdots & Id & \\vert & \\tau'_1 &\\cdots & \\tau'_i &\\cdots & \\tau'_i &\\cdots & \\tau'_l & \\vert & \\cdots \\\\\n\\end{array} \\color{white}\\right] \\\\ \\\\\n\\color{white} = \\left[\\rule{0cm}{1cm} \\setlength\\arraycolsep{1pt} \\color{black} \\begin{array}{ccccccccccccc}\n\\cdots & \\vert &\\cdots & (a_{m-1}p_1)'_j & \\cdots & (a_{m-1}p_i)'_j & \\cdots & (a_{m-1}p_l)'_j & \\cdots & \\vert & a_{m+k} & \\cdots & a_{n-1} \\\\\n\\cdots & \\vert &\\cdots & Id &\\cdots & Id &\\cdots & Id &\\cdots & \\vert & Id & \\cdots & Id \\\\\n\\cdots & \\vert & \\cdots & (a_{m-1}q_1)'_j & \\cdots & (a_{m-1}q_i)'_j & \\cdots & (a_{m-1}q_l)'_j & \\cdots & \\vert & a_{m+k} & \\cdots & a_{n-1} \\\\\n\\cdots & \\vert &\\cdots & \\tau'_1 &\\cdots & \\tau'_i &\\cdots & \\tau'_l & \\cdots & \\vert & Id & \\cdots & Id \\\\ \n\\end{array} \\right]\n\\end{array}$\n\\end{flushleft}\nIt is straightforward to check that for both tables the columns starting at $a_{m-1}p_ia_s$ for $0 \\leq s \\leq m-1$ are equal, as $\\tau'_i(s) = \\tau_i(s), \\forall s \\in \\{0,\\dots, m-1\\}$. \n\n\nFor $ m \\leq s \\leq n-1$ by Lemma \\ref{lem:successor}, we know there is a correspondence between the first two rows of $\\iota(\\operatorname{exp}(g))$ and the first two rows of $\\operatorname{exp}(\\iota(g))$. On the other hand, from the description of the algebraic embedding, the order in which successors for every $q_j$ are selected depends only on the order of $\\{p_1, \\dots, p_l\\}$, so we have the following calculations:\n$$\\begin{array}{lllll}\n(a_{m-1}q_i\\tau_i(a_{m-1}))'_1 & = a_{m-1}q_ia_{m} & = a_{m-1}q_i\\tau'_i(a_{m}) \\\\\n& \\vdots & \\vdots &\\\\ \n(a_{m-1}q_i\\tau_i(a_{m-1}))'_k & = a_{m-1}q_ia_{m+k-1} & = a_{m-1}q_i\\tau'_i(a_{m+k-1}) \\\\\n(a_{m-1}q_i\\tau_i(a_{m-2}))'_1 & = a_{m-1}q_ia_{m+k} & = a_{m-1}q_i\\tau'_i(a_{m+k}) \\\\\n& \\vdots &\\vdots & \\\\\n(a_{m-1}q_i\\tau_i(a_{m-2}))'_k & = a_{m-1}q_ia_{m+2k-1} & = a_{m-1}q_i\\tau'_i(a_{m+2k-1})\\\\\n& \\vdots& \\vdots & \\\\\n(a_{m-1}q_i\\tau_i(a_{1}))'_1 & = a_{m-1}q_ia_{m+(m-2)k} & = a_{m-1}q_i\\tau'_i(a_{m+(m-2)k})\\\\ \n& \\vdots &\\vdots & \\\\\n(a_{m-1}q_i\\tau_i(a_{1}))'_k & = a_{m-1}q_ia_n & = a_{m-1}q_i\\tau'_i(a_{m+(m-1)k-1}) \\\\\n(a_{m-1}q_i\\tau_i(a_{0}))'_1 & = (a_{m-1}q_i)'_1\\\\ \n& \\vdots \\\\\n(a_{m-1}q_i\\tau_i(a_{0}))'_k & = (a_{m-1}q_i)'_k.\\\\\n\\end{array}$$\nThat is, e.g., $(a_{m-1}q_i\\tau_i(a_{m-1}))'_1$ comes first in the choice of successor as $(a_{m-1}p_ia_{m-1})'_1$ appears first under the order of the $p_i$, independent of $\\tau_i$. Thus, the latter two rows of these tables are also equivalent.\n\nFinally, it is easy to see that $\\iota(h \\circ g) = \\iota(h) \\circ \\iota(g)$, as $\\iota$ commutes with expansions and pushings. We only need to obtain row equality on the first part of the table as the remaining part depends entirely on $P$ (resp. on $Q$ for the element $h$): \n\\begin{align*}\n\\iota(g) & = \\begin{bmatrix}\n\\cdots & a_{m-1}p_i & \\cdots & (a_{m-1}p_i)'_j & \\cdots \\\\\n\\cdots & Id & \\cdots & Id &\\cdots \\\\\n\\cdots & a_{m-1}q_i& \\cdots & (a_{m-1}q_i)'_j & \\cdots\\\\\n\\cdots &\\tau'_i & \\cdots & \\tau'_i&\\cdots\\\\ \n\\end{bmatrix},\\\\ \n\\end{align*}\n\\begin{align*}\n\\iota(h) & = \\begin{bmatrix}\n \\cdots & a_{m-1}q_i& \\cdots & (a_{m-1}q_i)'_j& \\cdots\\\\\n\\cdots & \\tau'_i & \\cdots& \\tau'_i & \\cdots\\\\\n \\cdots & a_{m-1}r_i & \\cdots & (a_{m-1}r_i)'_j& \\cdots\\\\\n \\cdots & \\tau''_i & \\cdots& \\tau''_i& \\cdots\\\\ \n\\end{bmatrix},\\\\ \n\\iota(h) \\circ \\iota(g) & = \\begin{bmatrix}\n \\cdots & a_{m-1}p_i& \\cdots & (a_{m-1}p_i)'_j& \\cdots\\\\\n\\cdots & Id & \\cdots& Id & \\cdots\\\\\n \\cdots & a_{m-1}r_i & \\cdots & (a_{m-1}r_i)'_j& \\cdots\\\\\n \\cdots & \\tau''_i & \\cdots& \\tau''_i& \\cdots \\\\ \n\\end{bmatrix} = \\iota(h \\circ g).\n\\end{align*} Thus the result follows. \n\\end{proof}\n\\label{Bibliography}\n\\bibliographystyle{abbrv} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction and Main Results}\nWe consider the Cauchy problem of the generalized SQG (Surface Quasi-geostrophic) equation in the plane as follows\n\\begin{equation}\\label{SQG}\\tag{SQG}\n\\left\\{\\ba\n&\\omega_{t}+u\\cdot\\nabla \\omega =0, \\qquad \\qquad(x,t)\\in \\R^{2}\\times\\R^{+},\\\\\n&u=\\nabla^{\\perp}(-\\Delta)^{-1+\\alpha}\\omega, \\\\\n&\\omega(x,0)=\\omega_0 \\ea\\ \\right.\n\\end{equation}\n with $0\\le \\alpha\\le\\frac 12$. Here, according to the second equation in $\\eqref{SQG}$, the unknown scalar function $\\omega=\\omega(x,t)$ and vector field $u=(u_1(x,t),u_2(x,t))$ can be expressed as the singular integral\n\\begin{equation}\\label{Int}\nu(x)=\\int_{\\R^2} \\frac{(x-y)^\\perp}{|x-y|^{2+2\\alpha}}\\omega(y) \\,\\mathrm{d}y.\n\\end{equation}\nThroughout our paper, we omit some constants before the singular integral \\eqref{Int} for conciseness. Meanwhile, the expression \\eqref{Int} implies that $u$ is divergence-free, that is, $\\nabla\\cdot u=\\partial_{x_1}u_1+\\partial_{x_2}u_2=0.$\n\nWhen $\\alpha=0$, it is well-known that \\eqref{SQG} corresponds to the two-dimensional incompressible Euler equations. In this case, the unknown functions $\\omega=\\omega(x,t)$ and $u=u(x,t)$ are the vorticity and the velocity field, respectively. When $\\alpha=\\frac12$, \\eqref{SQG} corresponds to the surface quasi-geostrophic (SQG) equation which describes a famous approximation model of the nonhomogeneous fluid flow in a rapidly rotating 3D half-space (see \\cite{[P87]}). In this case, the unknown functions $\\omega=\\omega(x,t)$ and $u=u(x,t)$ represent potential temperature and velocity field, respectively. When $0<\\alpha<\\frac12$, \\eqref{SQG} is called the generalized (or modified) SQG equation.\n\n\nThe classical SQG and the generalized SQG equations have been widely studied in the past years and much more progress has been made. In \\cite{[KYZ],[KRYZ]}, it is proved that the generalized SQG in half space $\\R^{2+}=\\{x=(x_1,x_2)| x_2>0\\}$ has a unique local solution for vortex-patch initial data and will appear singularity in finite time for some such kind of initial data when $0<\\alpha<\\frac{1}{24}$. This strongly implies that the SQG equation will appear finite-time singularity (even for smooth initial data) since the velocity has less regularity when $\\alpha=\\frac12$. In fact, the singularity or formation of strong fronts has been suggested in \\cite{[CMT]} although the rigorous derivations have not been reached so far. We note that the global well-posedness or blow-up of the SQG equation is an important issue. As pointed out in \\cite{[CMT]}, the singularity of the SQG equation will be similar to that of the three-dimensional Euler equations. Concerning the dissipative SQG equation, which enjoys a fractional dissipation term $-(-\\Delta)^{\\beta}\\omega$ on the right hand side of the second equation of \\eqref{SQG}, the global well-posedness in the critical case $\\beta=\\frac12$ was proved independently by Caffarelli and Vasseur \\cite{[CV]} and by Kiselev, Nazarov and Volberg \\cite{[KNV]} (see \\cite{[CVi],[KN]} for different approaches). The proof of global regularity for the subcritical case $\\beta>\\frac12$ is standard (see e.g. \\cite{[Con-Wu]}), while in the supercritical case $\\beta<\\frac12$ the global regularity of small solutions is obtained (see e.g. \\cite{[CMZ],[HK],[Ju],[Wu]}) and the slightly supercritical case is studied recently in \\cite{[DKSV]}.\n\n\nIn this paper, our target is to show that, for any $T>0$, if $\\{\\omega^{\\alpha_0}, u^{\\alpha_0}\\}$ defined on $[0,T]$ is the unique smooth solution of \\eqref{SQG} for some $0<\\alpha_0\\le \\frac12$, then there exists $\\delta>0$ such that when $0<\\alpha<\\frac12$ and $0<|\\alpha_0-\\alpha|\\le \\delta$, the problem \\eqref{SQG} with same initial data has also a unique smooth solution $\\{\\omega^\\alpha,u^\\alpha\\}$ defined on $[0,T]$, where we denote the solution of problem \\eqref{SQG} corresponding to $0\\le \\alpha\\le \\frac12$ by $\\{\\omega^\\alpha, u^\\alpha\\}$ satisfying $u^\\alpha=\\nabla^{\\perp}(-\\Delta)^{-1+\\alpha}\\omega^\\alpha$. This is motivated by \\cite{[C86]} in which it is shown that if the Cauchy problem to the three-dimensional incompressible Euler equations have a unique smooth solution on $[0,T]$, then the corresponding three-dimensional incompressible Navier-Stokes equations with the same initial data will also have a unique smooth solution defined on $[0,T]$ when the viscosity is suitably small. Furthermore, our result implies that the construction of the possible singularity of the smooth solution of the Cauchy problem to the generalized SQG with $\\alpha>0$ will be subtle (see Corollary \\eqref{Cor1+}), in comparison with the singularity result presented in \\cite{[KRYZ]}. To prove our main results, we consider the behavior of the difference between $u^\\alpha$ and $u^{\\alpha_0}$. Let us denote\n $$\\overline{\\omega}=\\omega^{\\alpha}-\\omega^{\\alpha_{0}}\\quad\\text{and}\\quad\\overline{u}=u^{\\alpha}-u^{\\alpha_{0}},$$\nwe easily find that the couple $(\\overline{\\omega},\\,\\overline{u})$ satisfies\n\\begin{equation*}\n\\overline{\\omega}_{t}+(u^{\\alpha_{0}}\\cdot\\nabla) \\overline{\\omega}+(\\overline{u}\\cdot\\nabla) \\overline{\\omega}+(\\overline{u}\\cdot\\nabla) \\omega^{\\alpha_{0}} =0.\n\\end{equation*}\nWith this equation, we can establish the following $H^s$-estimate of $\\overline{\\omega}(t)$:\n\\begin{align*}\n\\frac12\\frac{\\mathrm{d}}{\\mathrm{d}t}\\|\\overline{\\omega}(t)\\|_{H^s}^2=&-\\int_{\\R^2} J^s(u^{\\alpha_{0}}\\cdot\\nabla \\overline{\\omega})J^s\\overline{\\omega}\\,\\mathrm{d}x-\\int_{\\R^2} J^s(\\overline{u}\\cdot\\nabla \\overline{\\omega})J^s\\overline{\\omega}\\,\\mathrm{d}x\\\\\n&-\\int_{\\R^2} J^s(\\overline{u}\\cdot\\nabla \\omega^{\\alpha_{0}})J^s\\overline{\\omega}\\,\\mathrm{d}x.\n\\end{align*}\nThe difficulty for us is how to use the information of $\\|\\overline{\\omega}(t)\\|_{H^s}$ to control $\\bar u$ in the nonlinear term. This requires us to consider the behavior in different scale in physical space or in different frequency regime in frequency space. Thus, we decompose $\\bar u$ in two parts\n\\begin{equation}\\label{diff-1-1}\n\\begin{split}\n\\overline{u}=&u^{\\alpha}-u^{\\alpha_{0}}\n\\\\=&\\nabla^{\\perp}(-\\Delta)^{-1+\\alpha}\\overline{\\omega}\n+(\\nabla^{\\perp}(-\\Delta)^{-1+\\alpha}-\\nabla^{\\perp}(-\\Delta)^{-1+\\alpha_0})\\omega^{\\alpha_0}\n\\\\:=&\\overline{u}_I+\\overline{u}_{II},\n\\end{split}\\end{equation}\nwhere $\\bar\\omega=\\omega^\\alpha-\\omega^{\\alpha_0}$. In the above equalities \\eqref{diff-1-1}, we see that the part $u_I$ is related to the difference between the solutions, while the part $u_{II}$ corresponds to the difference between the singular integrals. This observation together with the scale analysis enables us to establish some technique propositions, see Proposition \\ref{uni-est}, Proposition \\ref{add-0} and Proposition \\ref{add1} which will play key roles in our proof of Theorem \\ref{th2} and Theorem \\ref{th3} respectively. More precisely, in view of Riesz potential (see \\eqref{Riesz}), the term $\\overline{u}_I$ can be estimated as\n\\begin{equation}\\label{Riesz1}\n\\|\\overline{u}_I\\|_{L^q(\\R^2)}\\le C(\\alpha)\\|\\bar\\omega\\|_{L^p(\\R^2)}, \\ \\frac1q=\\frac1p-\\frac{1-2\\alpha}{2},\n\\end{equation}\nfor $0<\\alpha<\\frac12$. However, when $\\alpha\\to\\frac12$, the constant $C(\\alpha)$ in \\eqref{Riesz1} will be unbounded. To overcome this difficulty, we establish Propositions \\ref{uni-est}- \\ref{add1} to obtain some new uniform estimates as $\\alpha\\to \\frac12$.\n\nOur main results are stated as follows.\n\\begin{theorem}\\label{th1}\nLet $0< \\alpha_{0}<\\frac12.$ Let $\\omega^{\\alpha_0}$ be a solution of \\eqref{SQG} for $0\\leq t\\leq T$ with $u^{\\alpha_0}=\\nabla^{\\perp}(-\\Delta)^{-1+\\alpha_0}\\omega^{\\alpha_0}$ and $\\omega_0\\in H^{s+1},$ $s>2$. Then, there exists $\\delta>0$ depending on $T$ and $\\int_0^T \\|\\omega^{\\alpha_0}\\|_{H^{s+1}} dt$ such that if $0<\\alpha<\\frac12$ and $|\\alpha_0-\\alpha|\\le \\delta$, the solution $\\omega^{\\alpha}$ to \\eqref{SQG} with\n$u^{\\alpha}=\\nabla^{\\perp}(-\\Delta)^{-1+\\alpha}\\omega^{\\alpha}$ and the same initial data is smooth on $[0,T]$. Moreover, it holds that\n\\begin{equation*}\n\\big\\|\\omega^{\\alpha}(t)-\\omega^{\\alpha_0}(t)\\big\\|_{H^{s}}\\leq C\\left(|\\alpha_0-\\alpha|^{1-2\\alpha_0}+|\\alpha_0-\\alpha||\\log|\\alpha_0-\\alpha||\\right),\n\\end{equation*}\nwhere $C>0$ is a constant depending on $T$ and $\\int_0^T\\|\\omega^{\\alpha_0}\\|_{H^{s+1}} dt$.\n\\end{theorem}\n\n\\begin{theorem}\\label{th2}\nLet $0<\\alpha<\\alpha_{0}=\\frac12.$ Let $\\omega^{\\alpha_0}$ be a solution of \\eqref{SQG} for $0\\leq t\\leq T$ with $u^{\\alpha_0}=\\nabla^{\\perp}(-\\Delta)^{-1+\\alpha_0}\\omega^{\\alpha_0}$ and $\\omega_0\\in H^{s+1}\\cap L^1,$ $s>2$. Then, there exists $\\delta>0$ depending on $T$ and $\\int_0^T\\|\\omega^{\\alpha_0}\\|_{H^{s+1}\\cap L^1} dt$ such that if $0<\\alpha_0-\\alpha<\\delta$, the solution $\\omega^{\\alpha}$ to \\eqref{SQG} with\n$u^{\\alpha}=\\nabla^{\\perp}(-\\Delta)^{-1+\\alpha}\\omega^{\\alpha}$ and the same initial data is smooth on $[0,T]$. Moreover, it holds that\n\\begin{equation*}\n\\big\\|\\omega^{\\alpha}(t)-\\omega^{\\alpha_0}(t)\\big\\|_{H^{s}}\\leq C\\left(\\Big(\\frac12-\\alpha\\Big)+\\Big(\\frac12-\\alpha\\Big)\\log^2\\Big(\\frac12-\\alpha\\Big)\\right),\n\\end{equation*}\nwhere $C>0$ is a constant depending on $T$ and $\\int_0^T\\|\\omega^{\\alpha_0}\\|_{H^{s+1}\\cap L^1} dt$.\n\\end{theorem}\n\n\\begin{theorem}\\label{th3}\n\tLet $0<\\alpha<\\alpha_{0}=\\frac12.$ Let $\\omega^{\\alpha_0}$ be a solution of \\eqref{SQG} for $0\\leq t\\leq T$ with $u^{\\alpha_0}=\\nabla^{\\perp}(-\\Delta)^{-1+\\alpha_0}\\omega^{\\alpha_0}$ and $\\omega_0\\in H^{s+2},$ $s>2$. Then, there exists $\\delta>0$ depending on $T$ and $\\int_0^T \\|\\omega^{\\alpha_0}\\|_{H^{s+2}} dt$ such that if $0<\\alpha_0-\\alpha<\\delta$, the solution $\\omega^{\\alpha}$ to \\eqref{SQG} with\n\t$u^{\\alpha}=\\nabla^{\\perp}(-\\Delta)^{-1+\\alpha}\\omega^{\\alpha}$ and the same initial data is smooth on $[0,T]$. Moreover, it holds that\n\t\\begin{equation*}\n\t\t\\big\\|\\omega^{\\alpha}(t)-\\omega^{\\alpha_0}(t)\\big\\|_{H^{s}}\\leq C\\left(\\Big(\\frac12-\\alpha\\Big)+\\Big(\\frac12-\\alpha\\Big)\\log^2\\Big(\\frac12-\\alpha\\Big)\\right),\n\t\\end{equation*}\nwhere $C>0$ is a constant depending on $T$ and $\\int_0^T \\|\\omega^{\\alpha_0}\\|_{H^{s+2}} dt$.\n\\end{theorem}\n\\begin{remark}\\label{Rm1}\nIn Theorem \\ref{th1}, we consider the case $0<\\alpha_0<\\frac12$.\nIn Theorem \\ref{th2} and Theorem \\ref{th3}, we deal with the case $\\alpha_0=\\frac12$.\n It is noted that in the proof of Theorem \\ref{th1}, the Hardy-Littlewood-Sobolev inequality (see Lemma \\ref{Hardy}) will be used. One point is that the expression $\\bar u_I$ in \\eqref{diff-1-1} can be reduced to $I_{1-2\\alpha}\\bar\\omega$, where $I_{1-2\\alpha}$ is a Riesz operator (see \\eqref{Riesz}) which is bounded from $L^p$ to $L^q$ with $\\frac1q=\\frac1p-\\frac{1-2\\alpha}{n}$ satisfying $0<1-2\\alpha2$. Thanks to the incompressible condition $\\nabla\\cdot u=0$, the solution will stay in $L^1$. In Theorem \\ref{th3}, we drop the restriction on the initial data $\\omega_0\\in L^1$, but more regularity of the initial data $\\omega_0\\in H^{s+2}$ with $s>2$ will be needed.\n\\end{remark}\n\n\\begin{remark}\\label{Rm2+}\nAs mentioned above, \\eqref{SQG} becomes the two-dimensional incompressible Euler equations when $\\alpha_0=0$, of which the global existence of smooth solutions has been known (see \\cite{Majda1} and references therein). In comparison with the singularity for the patch solution with $0<\\alpha<\\frac{1}{24}$ in half space obtained in \\cite{[KRYZ]}, whether the patch or smooth solution of the Cauchy problem to \\eqref{SQG} when $\\alpha>0$ appears singularity in finite time remains open. Theorem \\eqref{th1} implies that the possible blow-up time of the smooth solution to the Cauchy problem of \\eqref{SQG} with $\\alpha>0$ can not be uniformly bounded when $\\alpha\\to 0+$. More precisely, as a corollary of Theorem \\ref{th1}, we have\n\\end{remark}\n\n\\begin{corollary}\\label{Cor1+}\nLet $T^*_\\alpha>0$ the maximal existence time (may be $+\\infty$) of the solution $\\omega^\\alpha\\in C([0,T]; H^{s+1}) (s>2)$ to \\eqref{SQG} with $\\alpha>0$. Then $\\displaystyle\\limsup_{\\alpha\\to 0+}T^*_\\alpha=+\\infty$.\n\\end{corollary}\n\n\nThe rest of the paper is organized as follows. In Section \\ref{pre}, we will present some basic facts which will be needed later. In Section \\ref{SI}, we will investigate a singular integral which can be viewed as an approximation of the Riesz transform. In Section \\ref{BesovE}, we will obtain nonlinear terms and commutator estimates related to $u_I=\\nabla^{\\perp}(-\\Delta)^{-1+\\alpha}\\overline{\\omega}$ in \\eqref{diff-1-1}. The proof of the main results will given in Section \\ref{sec5}. In the end of the paper, Appendix \\ref{sec4} on the Littlewood-Paley decomposition, Besov spaces will be given.\n\n\n\n\n\\section{Preliminaries}\\label{pre}\n\\setcounter{section}{2}\\setcounter{equation}{0}\n\n\nIn this section, we present some basic analysis facts. First of all, we introduce\n\\[\\Lambda^s=(-\\Delta)^\\frac{s}{2}\\quad\\text{and}\\quad\nJ^s=(I-\\Delta)^\\frac{s}{2},\\] where\n $$\\widehat{\\Lambda^sf}(\\xi)=|\\xi|^s\\widehat{f}(\\xi)\\quad\\text{and}\\quad\\widehat{J^sf}(\\xi)\n =\\big(1+|\\xi|^2\\big)^{\\frac{s}{2}}\\widehat{f}(\\xi),~~ s\\in \\R.$$\n\n\\begin{definition}[\\cite{[stein]}]\nLet $s\\in \\R$ and $1\\leq p\\leq \\infty.$ We write\n$$\\|f\\|_{W^{s,p}(\\R^n)}:=\\norm{J^s f}_{L^p(\\R^n)}, ~~\\|f\\|_{\\dot{W}^{s,p}(\\R^n)}:=\\norm{\\Lambda^s f}_{L^p(\\R^n)}.$$\nThe nonhomogeneous Sobolev space $W^{s,p}(\\R^n)$ is defined as\n$$\nW^{s,p}(\\R^n)=\\{f\\in \\mathcal{S'}(\\R^n):\\|f\\|_{W^{s,p}(\\R^n)}<\\infty\\}.\n$$\nThe homogeneous Sobolev space $\\dot{W}^{s,p}(\\R^n)$ is defined as\n$$\n\\dot{W}^{s,p}(\\R^n)=\\{f\\in \\mathcal{S'}(\\R^n):\\|f\\|_{\\dot{W}^{s,p}(\\R^n)}<\\infty\\}.\n$$\nHere $\\mathcal{S'}$ is the Schwarz distributional function space.\n\\end{definition}\nWith this definition in hand, we give a commutator estimate and product estimate (see, e.g., Kenig, Ponce and Vega \\cite{[KPV]}).\n\\begin{lemma}\\label{commutator}\nLet $s>0$ and $10,$ $G>0$ and $m>0$ be given constants and let $F(t)$ be a nonnegative continuous function on $[0,T).$ Let $\\nu_0$\nbe defined by $$\\nu_0=\\frac{1}{4m(2m TG)^\\frac1m\\int_0^T F(t)\\,\\mathrm{d}t}.$$ Then, for all $0<\\nu\\leq\\nu_0,$ all nonnegative solution $y(t)$ of\nthe system \\begin{equation}\\label{ODE}\n\\left\\{\\ba\n&\\frac{\\mathrm{d}\\,y(t)}{\\mathrm{d}t}\\leq \\nu F(t)+Gy(t)^{1+m}\\\\\n&y(0)=0\\ea\\ \\right.\n\\end{equation} is uniformly bounded on $[0,T)$ and\n \\begin{equation}\\label{ODE-0}\ny(t)\\leq\\min\\Big\\{\\frac{4^{\\frac1m}-1}{(2m TG)^\\frac1m},\\,\\, 4m\\big(4^\\frac1m-1\\big)\\nu\\int_0^T F(t)\\,\\mathrm{d}t\\Big\\}.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nLet us define $$\\sigma=\\min\\Big\\{\\frac{1}{(2mT)^{1+\\frac1m}G^{\\frac1m}}, \\Big(4m\\nu\\int_0^T F(t)\\,\\mathrm{d}t\\Big)^{1+m}G\\Big\\}.$$\nDividing the first equation of \\eqref{ODE} by $\\big(1+(\\frac{G}{\\sigma})^{\\frac{1}{m+1}}y\\big)^{1+m}$ yields\n\\begin{equation}\n\\frac1m\\Big(\\frac{\\sigma}{G}\\Big)^{\\frac{1}{m+1}}\\frac{\\mathrm{d}}{\\mathrm{d}t}\\Big(\\frac{1}{\\big(1+(\\frac{G}{\\sigma})^{\\frac{1}{m+1}}y\\big)^{m}}\\Big)\\geq -\\nu F(t)-\\sigma.\n\\end{equation}\nBy integrating from $0$ to $t$, we obtain\n\\begin{equation}\\label{ODE-1}\n\\frac1m\\Big(\\frac{\\sigma}{G}\\Big)^{\\frac{1}{m+1}}\\Big(\\frac{1}{(1+(\\frac{G}{\\sigma})^{\\frac{1}{m+1}}y)^{m}}\\Big)\\geq\n \\frac1m\\Big(\\frac{\\sigma}{G}\\Big)^{\\frac{1}{m+1}}-\\nu \\int_0^T F(t)\\,\\mathrm{d}t-\\sigma T.\n\\end{equation}\nThe choice $\\sigma\\leq \\frac{1}{(2mT)^{1+\\frac1m}G^{\\frac1m}}$ implies $\\sigma T\\leq \\frac{1}{2m}(\\frac{\\sigma}{G})^{\\frac{1}{m+1}}.$\n\nFor $\\nu\\leq \\nu_0,$ we have\n$$\\nu \\int_0^T F(t)\\,\\mathrm{d}t \\leq \\frac{1}{4m}\\Big(\\frac{\\sigma}{G}\\Big)^{\\frac{1}{m+1}}.$$\nIndeed, if $\\sigma=\\big(4m\\nu\\int_0^T F(t)\\,\\mathrm{d}t\\big)^{1+m}G,$ the last inequality is indeed an equality and if $\\sigma= \\frac{1}{(2mT)^{1+\\frac1m}G^{\\frac1m}},$\nit follows from $$\\nu \\int_0^T F(t)\\,\\mathrm{d}t \\leq\\nu_0 \\int_0^T F(t)\\,\\mathrm{d}t=\\frac{1}{4m(2m TG)^\\frac1m}=\\frac{1}{4m}\\Big(\\frac{\\sigma}{G}\\Big)^{\\frac{1}{m+1}}.$$\nThus, we get by \\eqref{ODE-1} that\n \\begin{equation*}\n\\frac{1}{\\big(1+(\\frac{G}{\\sigma})^{\\frac{1}{m+1}}y\\big)^{m}}\\geq\\frac14\n\\end{equation*}\nwhich implies \\eqref{ODE-0}.\n\\end{proof}\n\\section{Estimates on A Singular Integral}\\label{SI}\n\\setcounter{section}{3}\\setcounter{equation}{0}\n\nIn this section, we present some new results on a singular integral which will be needed in the proof of Theorem \\ref{th2}. We denote\n\\begin{equation}\\label{T-operator}\nTf(x)=K*f(x)=\\int_{\\R^n} K(x-y)f(y) dy \\quad\\text{with}\\quad K(x)=\\frac{x}{|x|^{n+1-\\beta}},\\quad 0<\\beta0, \\,\\,0<\\beta0, \\,\\,0<\\beta<\\frac n2;\n\\end{equation}\n\\begin{equation}\\label{T2-2}\n\\|T_2f\\|_{\\dot{H}^s(\\R^n)}\\leq C(\\frac{\\beta}{1-\\beta}+\\frac{2^\\beta-1}{\\beta}\\beta^{-\\beta})\\|f\\|_{\\dot{H}^{s-1}(\\R^n)}, \\quad s\\geq1,\\,\\, 0<\\beta<1.\n\\end{equation}\n\\end{proposition}\n\\begin{remark}\\label{Rm2-1}\nWhen $n=2$, the result of Proposition \\ref{uni-est} holds true if $K(x)$ in \\eqref{T-operator} is replaced by $ K(x)=\\frac{x^\\perp}{|x|^{3-\\beta}}$.\n\\end{remark}\n\n\\begin{remark}\nIt is emphasized that the constants $C$ is independent of $\\beta$, and what is more,\n$\\frac{\\beta^{\\frac n2}}{\\sqrt{n-2\\beta}}$ in \\eqref{T2-1} is sufficiently small, $\\frac{\\beta}{1-\\beta}+\\frac{2^\\beta-1}{\\beta}\\beta^{-\\beta}$ is uniformly bounded\nin \\eqref{T2-2} when $\\beta$ tends to zero.\n\\end{remark}\n\\begin{remark}\n When $\\beta=0$, it follows from \\eqref{T-operator} that $Tf=\\mathcal{R} f$ (in the sense that the integral takes principle values), where $\\mathcal{R}$ is a Riesz transformation which is a strong $(p,p)$ type operator with $10$. By Proposition \\ref{uni-est}, it holds\n\\begin{equation}\\label{Riesz2}\n\\|Tf\\|_{L^2}\\le C(\\|f\\|_{L^2}+\\beta\\|f\\|_{L^1}),\n\\end{equation}\nwhere $C>0$ is an absolute constant. This means that the estimate \\eqref{Riesz2} recovers the corresponding one in \\eqref{Riesz0} with $p=2$.\n\\end{remark}\n\\begin{proof}[Proof of Proposition \\ref{uni-est}]\nWe firstly prove \\eqref{T2-1} and \\eqref{T2-2}.\nNote that\n\\begin{equation*}\nT_2f(x)=\\int_{\\R^n}\\frac{x-y}{|x-y|^{n+1-\\beta}}\\big(1-\\chi_\\beta(|x-y|)\\big)f(y)\\,\\mathrm{d}y.\n\\end{equation*}\nFor $0<\\beta<\\frac n2$, we have\n\\begin{equation*}\n\\begin{split}\n\\big\\|T_2f\\big\\|_{L^2(\\R^n)}&\\leq\\Big\\|\\int_{|x-y|\\geq\\frac{1}{\\beta}}\\frac{1}{|x-y|^{n-\\beta}}|f(y)|\\,\\mathrm{d}y\\Big\\|_{L^2(\\R^n)}\n\\\\&\\leq\\|f\\|_{L^1(\\R^n)}\\Big(\\int_{|x-y|\\geq\\frac{1}{\\beta}}\\frac{1}{|x-y|^{2(n-\\beta)}}\\,\\mathrm{d}y\\Big)^{\\frac12}\n\\\\&\\leq\\frac{C}{\\sqrt{n-2\\beta}}\\beta^{\\frac n2-\\beta}\\|f\\|_{L^1(\\R^n)}.\n\\end{split}\n\\end{equation*}\nSince\n$$\\lim_{\\beta\\to0+}\\beta^{-\\beta}=1,$$\nthere exists an absolutely constant $C(n)>0$ such that, for any $0\\leq\\beta<\\frac n2,$\n\\begin{equation*}\\begin{split}\n\\big\\|T_2f\\big\\|_{L^2(\\R^n)}&\\leq C\\frac{\\beta^{\\frac n2}}{\\sqrt{n-2\\beta}}\\big\\|f\\big\\|_{L^1(\\R^n)}.\n\\end{split}\n\\end{equation*}\nThis means \\eqref{T2-1}.\n\nTo prove \\eqref{T2-2}, we note that for $s\\ge 1$ and $i=1, 2, \\cdots, n$,\n\\begin{equation}\\label{T2-E}\n \\begin{split}\n\\partial_i\\Lambda^{s-1}T_2f(x)=&\\int_{\\R^n}\\partial_i\\Big(\\frac{x-y}{|x-y|^{n+1-\\beta}}\\big(1-\\chi_\\beta(|x-y|)\\big)\\Big)\\Lambda^{s-1}_yf(y)\\,\\mathrm{d}y\n\\\\=&\\int_{|x-y|\\geq\\frac{1}{\\beta}}\\partial_i\\Big(\\frac{x-y}{|x-y|^{n+1-\\beta}}\\Big)\\big(1-\\chi_\\beta(|x-y|)\\big)\\Lambda^{s-1}_yf(y)\\,\\mathrm{d}y\\\\&\n+\\int_{\\frac{1}{\\beta}\\leq|x-y|\\leq\\frac{2}{\\beta}}\\frac{x-y}{|x-y|^{n+1-\\beta}}\\partial_i\\chi_\\beta(|x-y|)\\Lambda^{s-1}_yf(y)\\,\\mathrm{d}y\n\\\\:=&J_1+J_2,\n\\end{split}\n\\end{equation}\nwhere $\\partial_i=\\partial_{x_i}, i=1,2,\\ldots, n$.\n\nThen, for $0<\\beta<1$, we obtain\n\\begin{equation}\\label{J1}\n \\begin{split}\n\\big\\|J_1\\big\\|_{L^2(\\R^n)}&\\leq C\\Big\\|\\int_{|x-y|\\geq\\frac{1}{\\beta}}\\frac{1}{|x-y|^{n+1-\\beta}}|\\Lambda^{s-1}_yf(y)|\\,\\mathrm{d}y\\Big\\|_{L^2(\\R^n)}\n\\\\&\\leq C\\big\\|\\Lambda^{s-1}f\\big\\|_{L^2(\\R^n)}\\int_{|x-y|\\geq\\frac{1}{\\beta}}\\frac{1}{|x-y|^{n+1-\\beta}}\\,\\mathrm{d}y\n\\\\&\\leq\\frac{C}{1-\\beta}\\beta^{1-\\beta}\\big\\|\\Lambda^{s-1}f\\big\\|_{L^2(\\R^n)}.\n\\end{split}\n\\end{equation}\nThe term $J_2$ can be bounded as\n\\begin{equation}\\label{J2}\n \\begin{split}\n\\big\\|J_2\\big\\|_{L^2(\\R^n)}&\\leq C\\Big\\|\\int_{\\frac{1}{\\beta}\\leq|x-y|\\leq\\frac{2}{\\beta}}\\frac{1}{|x-y|^{n-\\beta}}|\\Lambda^{s-1}_yf(y)|\\,\\mathrm{d}y\\Big\\|_{L^2(\\R^n)}\n\\\\&\\leq C\\big\\|\\Lambda^{s-1}f\\big\\|_{L^2(\\R^n)}\\int_{\\frac{1}{\\beta}\\leq|x-y|\\leq\\frac{2}{\\beta}}\\frac{1}{|x-y|^{n-\\beta}}\\,\\mathrm{d}y\n\\\\&\\leq C\\frac{2^\\beta-1}{\\beta}\\beta^{-\\beta}\\big\\|\\Lambda^{s-1}f\\big\\|_{L^2(\\R^n)}.\n\\end{split}\n\\end{equation}\nSubstituting \\eqref{J1} and \\eqref{J2} into \\eqref{T2-E} and using the fact that $$\\|\\Lambda^sT_2f\\|_{L^2}\\le \\|\\nabla\\Lambda^{s-1}T_2f\\|_{L^2},$$ we finish the proof of \\eqref{T2-2}.\n\nNow we turn to prove \\eqref{T1-1}. To do this, it suffice to show that there exists an absolute constant $C>0$ independent $\\beta$ such that\n\\begin{equation}\\label{K1-F}\n\\big\\|\\widehat{K_1}(y)\\big\\|_{L^\\infty(\\R^n)}\\leq C, \\, \\, 0<\\beta0$ such that\n\\begin{equation}\\label{K1-F1}\n\\big|\\widehat{K_1}(y)\\big|\\leq C,\\,\\, 0<\\beta0$ such that\n\\begin{equation}\\label{K1-F2}\n\\big|\\widehat{K_1}(y)\\big|\\leq C\\left(\\frac{2^\\beta}{\\beta+1}\\beta^{-\\beta}+\\frac{2^\\beta-1}{\\beta}\\beta^{-\\beta}\\right)\\le C, \\,\\, 0<\\beta\\beta,$ $\\widehat{K_1}(y)$ can be divided into\n\\begin{equation}\\label{K1-F31}\n\\begin{split}\n\\widehat{K_1}(y)&=\\int_{|x|<\\frac{1}{|y|}} e^{2\\pi ix\\cdot y}K_1(x)\\,\\mathrm{d}x+\\int_{\\frac{1}{|y|}\\leq|x|\\leq\\frac{2}{\\beta}} e^{2\\pi ix\\cdot y}K_1(x)\\,\\mathrm{d}x\n\\\\&=\\int_{|x|<\\frac{1}{|y|}} \\big(e^{2\\pi ix\\cdot y}-1\\big)K_1(x)\\,\\mathrm{d}x+\\int_{\\frac{1}{|y|}\\leq|x|\\leq\\frac{2}{\\beta}} e^{2\\pi ix\\cdot y}K_1(x)\\,\\mathrm{d}x.\n\\end{split}\n\\end{equation}\nFor the first term on the right hand of the above equality, we easily find that\n\\begin{equation}\\label{K1-F311}\n\\begin{split}\n\\Big|\\int_{|x|<\\frac{1}{|y|}} \\big(e^{2\\pi ix\\cdot y}-1\\big)K_1(x)\\,\\mathrm{d}x\\Big|&\\leq C|y|\\int_{|x|<\\frac{1}{|y|}}|x|\\frac{1}{|x|^{n-\\beta}}\\,\\mathrm{d}x\n\\\\&\\leq \\frac{1}{\\beta+1}\\frac{1}{|y|^\\beta} \\leq \\frac{1}{\\beta+1}\\beta^{-\\beta}.\n\\end{split}\n\\end{equation}\nFor the second term,\nwe choose $z=\\frac{y}{2|y|^2}$ with $|z|=\\frac{1}{2|y|}<\\frac{1}{2\\beta}$ such that $e^{2\\pi iy\\cdot z}=-1$\nand\n\\begin{equation*}\n\\int_{\\R^n} e^{2\\pi ix\\cdot y}K_1(x)\\,\\mathrm{d}x=\\frac12\\int_{\\R^n} e^{2\\pi ix\\cdot y}\\big(K_1(x)-K_1(x-z)\\big)\\,\\mathrm{d}x,\n\\end{equation*}\nmoreover, we have\n\\begin{equation}\\label{K1-F321}\n\\begin{split}\n\\int_{\\frac{1}{|y|}\\leq|x|\\leq\\frac{2}{\\beta}} e^{2\\pi ix\\cdot y}K_1(x)\\,\\mathrm{d}x=&\\frac12\\int_{\\frac{1}{|y|}\\leq|x|\\leq\\frac{2}{\\beta}} e^{2\\pi ix\\cdot y}\\big(K_1(x)-K_1(x-z)\\big)\\,\\mathrm{d}x\n\\\\&-\\frac12\\int_{\\frac{1}{|y|}\\leq|x+z|,~ |x|\\leq\\frac{1}{|y|}} e^{2\\pi ix\\cdot y}K_1(x)\\,\\mathrm{d}x\n\\\\&+\\frac12\\int_{|x+z|\\leq\\frac{1}{|y|},~ |x|\\geq\\frac{1}{|y|}} e^{2\\pi ix\\cdot y}K_1(x)\\,\\mathrm{d}x\\\\\n&+\\frac12\\int_{|x+z|\\ge\\frac{2}{\\beta}} e^{2\\pi ix\\cdot y}K_1(x) \\,\\mathrm{d}x\n\\\\:= &I+J+K+L.\n\\end{split}\n\\end{equation}\nTo estimate the term $I$, we see that\n\\begin{equation}\\label{F321-I}\n\\begin{split}\nI&=\\int_{\\frac{1}{|y|}\\leq|x|<\\frac{1}{\\beta},~|x-z|\\leq\\frac{1}{\\beta}}\\Big(\\frac{x}{|x|^{n+1-\\beta}}-\\frac{x-z}{|x-z|^{n+1-\\beta}}\\Big)e^{2\\pi ix\\cdot y}\\,\\mathrm{d}x\n\\\\&+\\int_{\\frac{1}{\\beta}\\leq|x|\\leq\\frac{2}{\\beta},~|x-z|\\leq\\frac{1}{\\beta}}\\Big(\\frac{x}{|x|^{n+1-\\beta}}\\chi_\\beta(x)-\\frac{x-z}{|x-z|^{n+1-\\beta}}\\Big)e^{2\\pi ix\\cdot y}\\,\\mathrm{d}x\n\\\\&+\\int_{\\frac{1}{|y|}\\leq|x|<\\frac{1}{\\beta},~|x-z|\\geq\\frac{1}{\\beta}}\\Big(\\frac{x}{|x|^{n+1-\\beta}}-\\frac{x-z}{|x-z|^{n+1-\\beta}}\\chi_\\beta(x-z)\\Big)e^{2\\pi ix\\cdot y}\\,\\mathrm{d}x\n\\\\&+\\int_{\\frac{1}{\\beta}\\leq|x|\\leq\\frac{2}{\\beta},~|x-z|\\geq\\frac{1}{\\beta}}\\Big(\\frac{x}{|x|^{n+1-\\beta}}\\chi_\\beta(x)-\\frac{x-z}{|x-z|^{n+1-\\beta}}\\chi_\\beta(x-z)\\Big)e^{2\\pi ix\\cdot y}\\,\\mathrm{d}x\n\\\\&:=I_1+I_2+I_3+I_4.\n\\end{split}\n\\end{equation}\nWe first estimate $I_2.$ Thanks to $|x-z|\\geq|x|-|z|\\geq\\frac{1}{\\beta}-\\frac{1}{2|y|}\\geq\\frac{1}{2\\beta},$ one has\n\\begin{equation}\\label{I-2}\n\\begin{split}\n|I_2|&\\leq\\int_{\\frac{1}{\\beta}\\leq|x|\\leq\\frac{2}{\\beta}}\\frac{1}{|x|^{n-\\beta}}\\,\\mathrm{d}x\n+\\int_{\\frac{1}{2\\beta}\\leq|x-z|\\leq\\frac{1}{\\beta}}\\frac{1}{|x-z|^{n-\\beta}}\\,\\mathrm{d}x\n\\\\&\\leq C\\frac{2^\\beta-1}{\\beta}\\beta^{-\\beta}+C\\frac{1-2^{-\\beta}}{\\beta}\\beta^{-\\beta}.\n\\end{split}\n\\end{equation}\nThen, thanks to $|x|=|x-z+z|\\geq|x-z|-|z|\\geq\\frac{1}{\\beta}-\\frac{1}{2|y|}\\geq\\frac{1}{2\\beta},$ $I_3$ can be estimated as follows:\n\\begin{equation}\\label{I-3}\n\\begin{split}\n|I_3|&\\leq\\int_{\\frac{1}{2\\beta}\\leq|x|\\leq\\frac{1}{\\beta}}\\frac{1}{|x|^{n-\\beta}}\\,\\mathrm{d}x\n+\\int_{\\frac{1}{\\beta}\\leq|x-z|\\leq\\frac{2}{\\beta}}\\frac{1}{|x-z|^{n-\\beta}}\\,\\mathrm{d}x\n\\\\&\\leq C\\frac{1 -2^{-\\beta}}{\\beta}\\beta^{-\\beta}+C\\frac{2^\\beta-1}{\\beta}\\beta^{-\\beta}.\n\\end{split}\n\\end{equation}\nThe term $I_4$ is directly estimated as\n\\begin{equation}\\label{I-4}\n\\begin{split}\n|I_4|&\\leq\\int_{\\frac{1}{\\beta}\\leq|x|\\leq\\frac{2}{\\beta}}\\frac{1}{|x|^{n-\\beta}}\\,\\mathrm{d}x\n+\\int_{\\frac{1}{\\beta}\\leq|x-z|\\leq\\frac{2}{\\beta}}\\frac{1}{|x-z|^{n-\\beta}}\\,\\mathrm{d}x\n\\\\&\\leq C\\frac{2^\\beta-1}{\\beta}\\beta^{-\\beta}.\n\\end{split}\n\\end{equation}\nNow we deal with $I_1.$ Note that\n\\begin{equation*}\n\\partial_i\\Big(\\frac{x}{|x|^{n+1-\\beta}}\\Big)=\\frac{\\mathbf{e}_i}{|x|^{n+1-\\beta}}\n+(-n-1+\\beta)\\frac{xx_i}{|x|^{n+3-\\beta}},\\quad i=1,2,\\ldots,n.\n\\end{equation*}\nIn this case, since $|x-z|\\geq|x|-|z|\\geq2|z|-|z|\\geq|z|,$ by Taylor's expansion, one has\n\\begin{equation*}\\begin{split}\n&\\Big|\\frac{x-z}{|x-z|^{n+1-\\beta}}-\\frac{x}{|x|^{n+1-\\beta}}\\Big|\\\\ \\le&\n\\Big|\\sum_{i=1}^n\\Big(\\frac{z_i\\mathbf{e}_i}{|x-z|^{n+1-\\beta}}+\n(-n-1+\\beta)\\frac{(x-z)(x_i-z_i)z_i}{|x-z|^{n+3-\\beta}}\\Big)\\Big|+C\\sum_{k=2}^\\infty \\frac{|z|^k}{k!|x-z|^{n+k-\\beta}}.\n\\end{split}\n\\end{equation*}\nConsequently,\n\\begin{equation}\\label{I-1}\n\\begin{split}\n|I_1|\\leq &\\big(n+2-\\beta\\big)|z|\\int_{|z|\\leq|x-z|<\\frac{1}{\\beta}<\\infty}\\frac{1}{|x-z|^{n+1-\\beta}}\\,\\mathrm{d}x\\\\\n&+\nC\\sum_{k=2}^\\infty \\int_{|z|\\leq|x-z|<\\frac{1}{\\beta}<\\infty}\\frac{|z|^k}{k!|x-z|^{n+k-\\beta}}\\,\\mathrm{d}x\n\\\\ \\leq & C\\frac{|z|^\\beta}{1-\\beta}\\leq C\\frac{\\beta^{-\\beta}}{1-\\beta}.\n\\end{split}\n\\end{equation}\nSubstituting \\eqref{I-2}-\\eqref{I-1} into \\eqref{F321-I} yields\n\\begin{equation}\\label{I-E}\n\\begin{split}\n|I|&=\\Big|\\int_{\\frac{1}{|y|}\\leq|x|<\\frac{1}{\\beta},~|x-z|\\leq\\frac{1}{\\beta}}\\Big(\\frac{x}{|x|^{n+1-\\beta}}-\\frac{x-z}{|x-z|^{n+1-\\beta}}\\Big)e^{2\\pi ix\\cdot y}\\,\\mathrm{d}x\\Big|\\\\\n&\\le C\\Big(\\frac{2^\\beta-1}{\\beta}\\beta^{-\\beta}+\\frac{1-2^{-\\beta}}{\\beta}\\beta^{-\\beta}+\\frac{\\beta^{-\\beta}}{1-\\beta}\\Big)\n\\end{split}\n\\end{equation}\nfor some absolute constant $C>0$.\n\nConcerning the term $J$, thanks to $|x|\\geq|x+z|-|z|\\geq 2|z|-|z|\\geq |z|$, one has\n\\begin{equation}\\label{J-E}\n|J|\\leq\\int_{|z|\\leq|x|\\leq2|z|}\\frac{1}{|x|^{n-\\beta}}\\,\\mathrm{d}x\n\\leq\\frac{1-2^{-\\beta}}{\\beta}\\beta^{-\\beta}.\n\\end{equation}\nUtilizing $|x|\\leq|x+z|+|z|\\leq 2|z|+|z|\\leq 3|z|$, the term $K$ can be bounded by\n\\begin{equation}\\label{K-E}\n|K|\\leq\\int_{2|z|\\leq|x|\\leq3|z|}\\frac{1}{|x|^{n-\\beta}}\\,\\mathrm{d}x\n\\leq\\frac{(\\frac32)^{\\beta}-1}{\\beta}\\beta^{-\\beta}.\n\\end{equation}\nAs for the term $L$, the fact that $\\frac{2}{\\beta}\\ge |x|\\ge |x+z|-|z|\\ge \\frac{2}{\\beta}-\\frac{1}{2\\beta}=\\frac{3}{2\\beta}$ enables us to conclude\n\\begin{equation}\\label{L-E}\n\\begin{split}\n|L| \\leq\\int_{\\frac{3}{2\\beta}\\leq|x|\\leq\\frac{2}{\\beta}}\\frac{1}{|x|^{n-\\beta}}\\,\\mathrm{d}x\n \\leq\\frac{1}{\\beta}\\left(\\Big(\\frac{3}{2\\beta}\\Big)^\\beta-\\Big(\\frac{2}{\\beta}\\Big)^\\beta\\right).\n\\end{split}\n\\end{equation}\nSubstituting \\eqref{I-E}-\\eqref{L-E} into \\eqref{K1-F321}, we readily obtain that there exists an absolute constant $C>0$ such that\n\\begin{equation}\\label{K12}\n\\Big|\\int_{\\frac{1}{|y|}\\leq|x|\\leq\\frac{2}{\\beta}} e^{2\\pi ix\\cdot y}K_1(x)\\,\\mathrm{d}x\\Big|\\le C.\n\\end{equation}\nIn view of \\eqref{K1-F311}, \\eqref{K12} and \\eqref{K1-F31}, there exists an absolute constant $C>0$ such that\n\\begin{equation}\\label{K1-F3}\n\\big|\\widehat{K_1}(y)\\big|\\le C, \\,\\ \\ 0<\\beta\\beta.\n\\end{equation}\nCombining \\eqref{K1-F1}, \\eqref{K1-F2} with \\eqref{K1-F3}, we finish the proof of \\eqref{K1-F}. Applying \\eqref{K1-F}, one has\n\\begin{equation*}\\label{T1-1+}\\begin{split}\n&\\|T_1f\\|_{L^2(\\R^n)}=\\big\\|\\widehat{K_1}\\widehat{f}\\big\\|_{L^2(\\R^n)}\\le C\\big\\|\\widehat{f}\\big\\|_{L^2(\\R^n)}=C\\|f\\|_{L^2(\\R^n)},\\\\\n&\\|\\Lambda^sT_1f\\|_{L^2(\\R^n)}=\\big\\|\\widehat{K_1}\\widehat{\\Lambda^sf}\\big\\|_{L^2(\\R^n)}\\le C\\big\\|\\widehat{\\Lambda^sf}\\big\\|_{L^2(\\R^n)}=C\\|\\Lambda^sf\\|_{L^2(\\R^n)}\n\\end{split}\n\\end{equation*}\nfor any $00,$ there exists a constant $C$ depending only on $s$ and $\\alpha$ such that\n\t\\begin{equation*}\n\t\\big\\|\\overline{u}_I\\cdot \\nabla \\omega^{\\alpha_0}\\big\\|_{H^s(\\R^2)}\\leq C\\left(\\|\\overline{\\omega}\\|_{L^2(\\R^2)}\\|\\omega^{\\alpha_0}\\|_{H^{s+2\\alpha+1}(\\R^2)}\n+\\|\\overline{\\omega}\\|_{H^s(\\R^2)}\\|\\omega^{\\alpha_0} \\|_{B^{1+2\\alpha}_{2,1}(\\R^2)}\\right).\n\t\\end{equation*}\n\\end{proposition}\n\\begin{remark}\nLet us point out that the positive constant $C$ is uniformly bounded as parameter $\\alpha$ goes to $\\frac12.$\n\\end{remark}\n\\begin{proof}[Proof of Proposition \\ref{add-0}]\nIn view of the Bony decomposition, one write\n\\begin{equation*}\n\t\\overline{u}_I\\cdot \\nabla \\omega^{\\alpha_0}=\\sum_{i=1}^2\\left(T_{\\partial_{i} \\omega^{\\alpha_0}}\\overline{u}^i_I +T_{\\overline{u}^i_I}\\partial_i \\omega^{\\alpha_0}+R\\big(\\overline{u}^i_I, \\partial_i \\omega^{\\alpha_0}\\big)\\right),\n\\end{equation*} where\n$$T_{\\partial_i \\omega^{\\alpha_0}}\\overline{u}^i_I=\\displaystyle{\\sum_{q>0}} \\Delta_q\\overline{u}^i_IS_{q-1}\\partial_i \\omega^{\\alpha_0}, \\quad\nT_{\\overline{u}^i_I}\\partial_i \\omega^{\\alpha_0}=\\displaystyle{\\sum_{q>0}}S_{q-1} \\overline{u}^i_I\\Delta_q\\partial_i \\omega^{\\alpha_0},$$\n$$R\\big(\\overline{u}^i_I, \\partial_i \\omega^{\\alpha_0}\\big)=\\displaystyle{\\sum_{q\\geq-1}}\\Delta_{q}\\overline{u}^i_I \\widetilde{\\Delta}_{q}\\partial_i \\omega^{\\alpha_0}.$$\nAccording to the H\\\"{o}lder inequality and Lemma \\ref{B}, we obtain that for $q>0,$\n\\begin{equation*}\n\\begin{split}\n2^{qs}\\big\\|\\Delta_q\\overline{u}^i_I S_{q-1}\\partial_i \\omega^{\\alpha_0}\\big\\|_{L^2}&\\leq 2^{qs}\\big\\|S_{q-1}\\nabla \\omega^{\\alpha_0} \\big\\|_{L^\\infty}\\big\\|\\Delta_q\\overline{u}_I\\big\\|_{L^2}\n\\\\&\\leq 2^{qs}\\sum_{-1\\leq k\\leq q-2}\\|\\Delta_{k}\\nabla \\omega^{\\alpha_0} \\|_{L^\\infty}2^{q(-1+2\\alpha)}\\|\\Delta_q\\overline{\\omega}\\|_{L^2}\n\\\\&\\leq2^{qs}\\|\\Delta_q\\overline{\\omega}\\|_{L^2}\\sum_{-1\\leq k\\leq q-2}2^{(q-k)(2\\alpha-1)}2^{k(1+2\\alpha)}\\|\\Delta_{k} \\omega^{\\alpha_0} \\|_{L^2}.\n\\end{split}\n\\end{equation*}\nTherefore, Lemma \\ref{annulus} and the Young inequality for series yields\n\\begin{equation}\\label{p-1}\n\\begin{split}\n\\norm{T_{\\nabla \\omega^{\\alpha_0}}\\overline{u}_I}_{H^s}&\\leq C_s\\norm{\\big\\{2^{qs}\\big\\|S_{q-1}\\partial_i \\omega^{\\alpha_0} \\Delta_q\\overline{u}^i_I\\big\\|_{L^2}\\big\\}_{q>0}}_{\\ell^2}\n\\\\&\\leq C_s2^{2(2\\alpha-1)}\\|\\overline{\\omega}\\|_{H^s}\\|\\omega^{\\alpha_0} \\|_{B^{1+2\\alpha}_{2,1}}.\n\\end{split}\n\\end{equation}\nSimilarly, for $0<\\epsilon<2\\alpha,$\n\\begin{align*}\n2^{qs}\\big\\|S_{q-1} \\overline{u}^i_I\\Delta_q\\partial_i \\omega^{\\alpha_0}\\big\\|_{L^2}&\\leq2^{qs}\\|S_{q-1}\\overline{u}_I \\|_{L^\\infty}\\|\\Delta_q\\nabla \\omega^{\\alpha_0}\\|_{L^2}\n\\\\&\\leq\\sum_{-1\\leq k\\leq q-2}\\|\\Delta_{k} \\overline{u}_I \\|_{L^\\infty}\\|\\Delta_q \\omega^{\\alpha_0}\\|_{L^2}2^{q(s+1)}\n\\\\&\\leq C\\|\\Lambda^{-(1-2\\alpha+\\epsilon)}\\overline{\\omega}\\|_{L^{\\frac{2}{2\\alpha-\\epsilon}}}\\sum_{k\\leq q-2}2^{2k\\alpha}2^{q(s+1)}\\|\\Delta_{q} \\omega^{\\alpha_0} \\|_{L^2}\n\\\\&\\leq C\\|\\overline{\\omega}\\|_{L^2}2^{q(s+1+2\\alpha)}\\|\\Delta_{q} \\omega^{\\alpha_0} \\|_{L^2}\\sum_{k\\leq q-2} 2^{2\\alpha(k-q)},\n\\end{align*}\nwhere Lemma \\ref{Hardy} has been used in the last inequality. In addition, when $\\alpha\\rightarrow\\frac12,$ the constant $C$ is uniformly bounded.\nHence, by Lemma \\ref{annulus}, we get\n\\begin{equation}\\label{p-2}\n\\begin{split}\n\\big\\|T_{\\overline{u}^i_I}\\partial_i \\omega^{\\alpha_0}\\big\\|_{H^s}&\\leq C_s\\norm{\\big\\{2^{qs}\\|S_{q-1}\\overline{u}_I \\Delta_q\\nabla \\omega^{\\alpha_0}\\|_{L^2}\\big\\}_{q>0}}_{\\ell^2}\n\\\\&\\leq C\\|\\overline{\\omega}\\|_{L^2} \\|\\omega^{\\alpha_0} \\|_{H^{s+1+2\\alpha}}.\n\\end{split}\n\\end{equation}\n\nFor the reminder term, we see that\n\\begin{equation*}\n\\begin{split}\n2^{qs}\\big\\|\\Delta_{q}R(\\overline{u}^i_I, \\partial_i \\omega^{\\alpha_0})\\big\\|_{L^2}&\\leq\n\\sum_{q\\leq q'+N_0}2^{qs}\\big\\|\\Delta_{q}(\\Delta_{q'}\\overline{u}^i_I \\widetilde{\\Delta}_{q'}\\partial_i \\omega^{\\alpha_0})\\big\\|_{L^2}\n\\\\&\\leq\\sum_{q\\leq q'+N_0}2^{qs}\\big\\|\\Delta_{q'}\\overline{u}_I\\big\\|_{L^\\infty} \\big\\|\\widetilde{\\Delta}_{q'}\\nabla \\omega^{\\alpha_0}\\big\\|_{L^2}\n\\\\&\\leq C\\big\\|\\Lambda^{-(1-2\\alpha+\\epsilon)}\\overline{\\omega}\\big\\|_{L^{\\frac{2}{2\\alpha-\\epsilon}}}\\sum_{q\\leq q'+N_0}2^{qs}\n2^{(1+2\\alpha)q'}\\big\\|\\widetilde{\\Delta}_{q'}\\omega^{\\alpha_0}\\big\\|_{L^2}\n\\\\&\\leq C\\|\\overline{\\omega}\\|_{L^{2}}\\sum_{q\\leq q'+N_0}2^{(q-q')s}\n2^{(s+1+2\\alpha)q'}\\|\\widetilde{\\Delta}_{q'}\\omega^{\\alpha_0}\\|_{L^2}.\n\\end{split}\n\\end{equation*}\nThis ensures that by Lemma \\ref{B}, for $s>0,$\n\\begin{equation}\\label{p-3}\n\\begin{split}\n\\big\\|R(\\overline{u}^i_I, \\partial_i \\omega^{\\alpha_0})\\big\\|_{H^s}&\\leq\\big\\|\\big\\{2^{qs}\\|\\Delta_{q}R(\\overline{u}^i_I, \\nabla \\omega^{\\alpha_0})\\|_{L^2}\\big\\}_{q\\geq-1}\\big\\|_{\\ell^2}\n\\\\&\\leq C\\|\\overline{\\omega}\\|_{L^2} \\big\\|\\omega^{\\alpha_0}\\big \\|_{H^{s+1+2\\alpha}}.\n\\end{split}\n\\end{equation}\nCollecting \\eqref{p-1}, \\eqref{p-2} and \\eqref{p-3} above gives the proof of this proposition.\n\\end{proof}\n\n\n\n\\begin{proposition}\\label{add1}\nFor any $s>0,$ there exists a constant $C$ depending only on $s$ such that,\n\t\\begin{equation}\\begin{split}\\label{add-2}\n\t&\\|J^s(\\overline{u}_I\\cdot \\nabla \\overline{\\omega})-\\overline{u}_I\\cdot J^s \\nabla \\overline{\\omega}\\|_{L^2(\\R^2)}\\\\ \\leq& C(\\|\\overline{\\omega}\\|_{H^{2\\alpha}(\\R^2)}^2\n+\\|\\overline{\\omega}\\|_{H^s(\\R^2)}\\|\\overline{\\omega}\\|_{H^{2\\alpha+1}(\\R^2)}+\n\\|\\overline{\\omega}\\|_{H^s(\\R^2)}\\|\\overline{\\omega} \\|_{B^{1+2\\alpha}_{2,1}(\\R^2)}).\n\t\\end{split}\\end{equation}\nIn particular, if $s>2,$ then we have\n\\begin{equation}\\label{c-add}\n\\|J^s(\\overline{u}_I\\cdot \\nabla \\overline{\\omega})-\\overline{u}_I\\cdot J^s \\nabla \\overline{\\omega}\\|_{L^2(\\R^2)}\\leq C\\|\\overline{\\omega}\\|_{H^{s}(\\R^2)}^2.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nWith the help of Bony's decomposition, one writes\n\\begin{equation*}\\begin{split}\n&J^s(\\overline{u}_I\\cdot \\nabla \\overline{\\omega})-\\overline{u}_I\\cdot J^s \\nabla \\overline{\\omega}\\\\=&\n\\sum_{i=1}^2\\Big([J^s, T_{\\overline{u}^i_I}\\partial_i]\\overline{\\omega}+J^s(T_{\\partial_i \\overline{\\omega}}\\overline{u}^i_I)-T_{J^s\\partial_i \\overline{\\omega}}\\overline{u}^i_I+J^s\\big(R(\\overline{u}^i_I,\\partial_i \\overline{\\omega})\\big)\n-R(\\overline{u}^i_I,J^s\\partial_i \\overline{\\omega})\\Big).\n\\end{split}\n\\end{equation*}\nThe last two terms can be further decomposed into three parts\n\\begin{equation*}\\begin{split}\n&J^s\\big(R(\\overline{u}^i_I,\\partial_i \\overline{\\omega})\\big)-R(\\overline{u}^i_I,J^s\\partial_i \\overline{\\omega})\n\\\\=&\\sum_{q'\\geq0}J^s(\\Delta_{q'}\\overline{u}^i_I \\widetilde{\\Delta}_{q'}\\partial_i \\overline{\\omega})-\\sum_{q'\\geq0}\\Delta_{q'}\\overline{u}^i_I \\widetilde{\\Delta}_{q'}J^s\\partial_i \\overline{\\omega}+[J^s, {\\Delta}_{-1}\\overline{u}^i_I \\partial_i]\\widetilde{\\Delta}_{-1}\\overline{\\omega}.\n\\end{split}\n\\end{equation*}\nNext, we are going to establish the standard inner $L^2$-norm of the six terms above one by one.\n\n\\textbf{Bounds for the term $[J^s, T_{\\overline{u}^i_I}\\partial_i]\\overline{\\omega}$}.\nBy virtue of Proposition \\ref{Dy}, we can rewrite $[J^s, T_{\\overline{u}^i_I}\\cdot \\partial_i]$ as a convolution operator. Indeed,\n\\begin{equation*}\\begin{split}\n [J^s, T_{\\overline{u}^i_I}\\partial_i]\\overline{\\omega}\n =&\\sum_{q>0}[J^s\\widetilde{\\Delta}_q, S_{q-1}{u}^i_I \\partial_i]\\Delta_q\\overline{\\omega}\n\\\\=&\\sum_{q>0}\\int_{\\R^2} 2^{2q}G_s(2^qy)\\big(S_{q-1}\\overline{u}^i_I(x-y)-S_{q-1}\\overline{u}^i_I(x)\\big)\\Delta_q\\partial_i\\overline{\\omega}(x-y)\\,\\mathrm{d}y\n,\\end{split}\n\\end{equation*}\nwhere $G_s$ is the inverse Fourier transform of $\\xi\\mapsto \\langle2^{q}\\xi\\rangle^{s}\\varphi(\\xi)$.\n\nFrom the first order Taylor formula, we deduce that\n\\begin{equation*}\\begin{split}\n\\big|[J^s, T_{\\overline{u}^i_I} \\partial_i]\\overline{\\omega}\\big|&\\leq\n\\sum_{q>0}\\int_{\\R^2}\\int_{0}^{1} 2^{2q}|G_s(2^qy)y|\n|\\nabla S_{q-1}\\overline{u}^i_I(x-\\tau y)|\\big|\\Delta_q\\partial_i\\overline{\\omega}(x-y)\\big|\\,\\mathrm{d}\\tau\\mathrm{d}y.\n\\end{split}\n\\end{equation*}\nNow, taking the $L^2$ norm of the above inequality, using the fact that $L^2\\sim B^0_{2,2}$, and using Lemma \\ref{annulus}, we get\n\\begin{equation*}\\begin{split}\n&\\big\\|[J^s, T_{\\overline{u}^i_I}\\partial_i]\\overline{\\omega}\\big\\|_{L^2}\n\\\\ \\leq&\\Big(\\sum_{q>0}\\Big\\|\\int_{\\R^2}\\int_{0}^{1}\n2^{2q}|G_s(2^qy)y|\n|\\nabla S_{q-1}\\overline{u}_I(\\cdot-\\tau y)|\n|\\Delta_q\\nabla\\overline{\\omega}(\\cdot-y)|\\,\\mathrm{d}\\tau\\mathrm{d}y\\Big\\|_{L^2}^2\\Big)^{\\frac12}.\n\\end{split}\n\\end{equation*}\nAdopting to the fact that the\nnorm of an integral is less than the integral of the norm and using H\\\"{o}lder's\ninequality yield\n\\begin{equation*}\\begin{split}\n&\\Big\\|\\int_{\\R^2}\\int_{0}^{1}\n2^{2q}|G_s(2^qy)y|\n\\big|\\nabla S_{q-1}\\overline{u}_I(x-\\tau y)\\big|\n\\big|\\Delta_q\\nabla\\overline{\\omega}(x-y)\\big|\\,\\mathrm{d}\\tau\\mathrm{d}y\\Big\\|_{L^2}\\\\ \\leq&\n\\int_{\\R^2}\\int_{0}^{1}\n2^{2q}|G_s(2^qy)y|\n\\|\\nabla S_{q-1}\\overline{u}_I(\\cdot-\\tau y)\\|_{L^\\infty}\n\\|\\Delta_q\\nabla\\overline{\\omega}(\\cdot-y)\\|_{L^2}\\,\\mathrm{d}\\tau\\mathrm{d}y\\\\ \\leq&\n2^{qs}\\|\\nabla S_{q-1}\\overline{u}_I\\|_{L^\\infty}\\|\\Delta_q\\overline{\\omega}\\|_{L^2},\n\\end{split}\n\\end{equation*}where the translation invariance of the Lebesgue measure is used in the last inequality.\n\nHence, the H\\\"{o}lder inequality and Bernstein's inequality enable us to conclude that\n\\begin{equation*}\\begin{split}\n\\big\\|[J^s, T_{\\overline{u}^i_I}\\partial_i]\\overline{\\omega}\\big\\|_{L^2}&\\leq\n\\Big(\\sum_{q>0}2^{2qs}\\|\\nabla S_{q-1}\\overline{u}_I\\|_{L^\\infty}^2\\|\\Delta_q\\overline{\\omega}\\|_{L^2}^2\\Big)^{\\frac12}\n\\\\&\\leq \\sup_{q>0}\\|\\nabla S_{q-1}\\overline{u}_I\\|_{L^\\infty}\\|\\overline{\\omega}\\|_{H^s}\n\\\\&\\leq C\\|\\overline{\\omega}\\|_{B^{1+2\\alpha}_{2,1}}\\|\\overline{\\omega}\\|_{H^s}.\n\\end{split}\n\\end{equation*}\n\n\\textbf{Bounds for $J^s(T_{\\partial_i \\overline{\\omega}}\\overline{u}^i_I).$} By virtue of the H\\\"{o}lder inequality and Bernstein's inequality, we get\n\\begin{equation*}\n\\begin{split}\n2^{qs}\\big\\|\\Delta_q\\overline{u}_I\\cdot S_{q-1}\\partial_i \\overline{\\omega}\\big\\|_{L^2}&\\leq 2^{qs}\\|S_{q-1}\\partial_i \\overline{\\omega} \\|_{L^\\infty}\\|\\Delta_q\\overline{u}^i_I\\|_{L^2}\n\\\\&\\leq 2^{qs}\\sum_{k\\leq q-2}\\|\\Delta_{k}\\nabla \\overline{\\omega} \\|_{L^\\infty}2^{q(-1+2\\alpha)}\\|\\Delta_q\\overline{\\omega}\\|_{L^2}\n\\\\&\\leq2^{qs}\\|\\Delta_q\\overline{\\omega}\\|_{L^2}\\sum_{k\\leq q-2}2^{(q-k)(2\\alpha-1)}2^{k(1+2\\alpha)}\\|\\Delta_{k} \\overline{\\omega}\\|_{L^2}.\n\\end{split}\n\\end{equation*}\nHence, we have by Lemma \\ref{annulus} that\n\\begin{equation*}\\begin{split}\n\\big\\|J^s(T_{\\partial_i \\overline{\\omega}}\\overline{u}^i_I)\\big\\|_{L^2}&\\leq\nC_s\\norm{\\big\\{2^{qs}\\|\\Delta_q\\overline{u}^i_I S_{q-1}\\partial_i \\overline{\\omega}\\|_{L^2}\\big\\}_{q>0}}_{\\ell^2}\n\\\\&\\leq C_s2^{2(2\\alpha-1)}\\|\\overline{\\omega}\\|_{H^s}\\|\\overline{\\omega}\\|_{B^{1+2\\alpha}_{2,1}}.\n\\end{split}\n\\end{equation*}\n\n\\textbf{A similar bound holds for both terms $T_{J^s\\partial_i \\overline{\\omega}}\\overline{u}^i_I$}, $\\displaystyle\\sum_{q'\\geq0}\\Delta_{q'}\\overline{u}^i_I \\cdot\\widetilde{\\Delta}_{q'}J^s\\partial_i \\overline{\\omega}$. By the H\\\"older inequality, one has\n\\begin{equation*}\n\\begin{split}\n\\big\\|\\Delta_q\\overline{u}^i_I S_{q-1}\\partial_i J^s\\overline{\\omega}\\big\\|_{L^2}&\\leq \\|S_{q-1}\\nabla J^s\\overline{\\omega} \\|_{L^\\infty}\\|\\Delta_q\\overline{u}_I\\|_{L^2}\n\\\\&\\leq \\sum_{k\\leq q-2}\\|\\Delta_{k}\\nabla J^s\\overline{\\omega} \\|_{L^\\infty}2^{q(-1+2\\alpha)}\\|\\Delta_q\\overline{\\omega}\\|_{L^2}\n\\\\&\\leq 2^{q(1+2\\alpha)}\\|\\Delta_q\\overline{\\omega}\\|_{L^2}\\|J^s\\overline{\\omega}\\|_{L^2}\\sum_{k\\leq q-2}2^{2(k-q)},\n\\end{split}\n\\end{equation*}\nfrom which it follows that\n\\begin{equation*}\n\\big\\|T_{J^s\\partial_i \\overline{\\omega}}\\overline{u}^i_I\\big\\|_{L^2} \\leq C\\|\\overline{\\omega}\\|_{H^s} \\|\\overline{\\omega}\\|_{B^{1+2\\alpha}_{2,1}}.\n\\end{equation*}\nFor the term $\\displaystyle\\sum_{q'\\geq0}\\Delta_{q'}\\overline{u}^i_I \\widetilde{\\Delta}_{q'}J^s\\partial_i \\overline{\\omega}$, by the H\\\"older inequality, we immediately obtain\n\\begin{equation*}\n\\begin{split}\n\\Big\\|\\sum_{q'\\geq0}\\Delta_{q'}\\overline{u}^i_I \\widetilde{\\Delta}_{q'}J^s\\partial_i \\overline{\\omega}\\Big\\|_{L^2}&\\leq\n\\sum_{q'\\geq 0}\\big\\|\\Delta_{q'}\\overline{u}^i_I \\widetilde{\\Delta}_{q'}\\partial_i J^s\\overline{\\omega}\\big\\|_{L^2}\n\\\\&\\leq\\sum_{q'\\geq 0}\\|\\Delta_{q'}\\overline{u}_I\\|_{L^2}\\big\\|\\widetilde{\\Delta}_{q'}\\nabla J^s\\overline{\\omega}\\big\\|_{L^\\infty}\n\\\\&\\leq\\|J^s\\overline{\\omega}\\|_{L^2}\\sum_{q'\\geq 0}\\|\\Delta_{q'}\\overline{\\omega}\\|_{L^2}\n2^{q'(1+2\\alpha)}\\\\&\\leq\\|\\overline{\\omega}\\|_{H^s}\\|\\overline{\\omega}\\|_{B^{1+2\\alpha}_{2,1}}.\n\\end{split}\n\\end{equation*}\n\n\\textbf{Bounds for the term $\\displaystyle\\sum_{q'\\geq0}J^s\\big(\\Delta_{q'}\\overline{u}^i_I \\widetilde{\\Delta}_{q'}\\partial_i \\overline{\\omega}\\big)$}. Utilizing again the H\\\"{o}lder inequality and Bernstein's inequality gives\n\\begin{align*}\n\\Big\\|\\sum_{q'\\geq0}J^s\\big(\\Delta_{q'}\\overline{u}^i_I \\widetilde{\\Delta}_{q'}\\partial_i \\overline{\\omega}\\big)\\Big\\|_{L^2}&\\leq\n\\sum_{q'\\geq0}\\Big(\\sum_{q\\leq q'+N_0}2^{2qs}\\big\\|\\Delta_{q}(\\Delta_{q'}\\overline{u}^i_I \\widetilde{\\Delta}_{q'}\\partial_i \\overline{\\omega})\\big\\|_{L^2}^2\\Big)^\\frac12\n\\\\&\\leq\\sum_{q'\\geq0}\\Big(\\sum_{q\\leq q'+N_0}2^{2qs}\\Big)^\\frac12\\|\\Delta_{q'}\\overline{u}_I\\|_{L^2} \\big\\|\\widetilde{\\Delta}_{q'}\\nabla \\overline{\\omega}\\big\\|_{L^\\infty}\n\\\\&\\leq\\sum_{q'\\geq0}2^{q's}\\|\\Delta_{q'}\\overline{\\omega}\\|_{L^2}\n2^{q'(1+2\\alpha)}\\|\\widetilde{\\Delta}_{q'}\\overline{\\omega}\\|_{L^2}\n\\\\&\\leq\\|\\overline{\\omega}\\|_{H^s}\\|\\overline{\\omega}\\|_{H^{1+2\\alpha}}.\n\\end{align*}\n\n\n\n\n\\textbf{Bounds for the last term $[J^s, {\\Delta}_{-1}\\overline{u}_I^i\\partial_i]\\widetilde{\\Delta}_{-1}\\overline{\\omega}$}. Adopting to the similar method to estimate $[J^s, T_{\\overline{u}^i_I} \\partial_i]\\overline{\\omega}$, we get\n\\begin{align*}\n&[J^s, {\\Delta}_{-1}\\overline{u}_I^i\\partial_i]\\widetilde{\\Delta}_{-1}\\overline{\\omega}\n\\\\=&\\sum_{|q+1|\\leq2}[J^s\\Delta_q, {\\Delta}_{-1}\\overline{u}_I^i\\partial_i]\\widetilde{\\Delta}_{-1}\\overline{\\omega}\n\\\\=&\\sum_{|q+1|\\leq2}\\int_{\\R^2} 2^{2q}G_s(2^qy)\\left({\\Delta}_{-1}\\overline{u}^i_I(x-y)-{\\Delta}_{-1}\\overline{u}^i_I(x)\\right)\\widetilde{\\Delta}_{-1}\\partial_i\\overline{\\omega}(x-y)\\,\\mathrm{d}y\n\\\\=&\\sum_{|q+1|\\leq2}\\int_{\\R^2}\\int_{0}^{1} 2^{2q}G_s(2^qy)\\big(y\\cdot\\nabla {\\Delta}_{-1}\\overline{u}^i_I(x-\\tau y)\\big)\\widetilde{\\Delta}_{-1}\\partial_i\\overline{\\omega}(x-y)\\,\\mathrm{d}\\tau\\mathrm{d}y.\n\\end{align*}\nBased on this, the Minkowski inequality and the H\\\"older inequality allow us to infer that\n\\begin{equation*}\\begin{split}\n&\\big\\|[J^s, {\\Delta}_{-1}\\overline{u}_I^i\\partial_i]\\widetilde{\\Delta}_{-1}\\overline{\\omega}\\big\\|_{L^2}\n\\\\ \\leq&\\Big(\\sum_{|q+1|\\leq2}\\Big\\|\\int_{\\R^2}\\int_{0}^{1} 2^{2q}G_s(2^qy)\\big(y\\cdot\\nabla {\\Delta}_{-1}\\overline{u}^i_I(x-\\tau y)\\big)\\widetilde{\\Delta}_{-1}\\partial_i\\overline{\\omega}(x-y)\\,\\mathrm{d}\\tau\\mathrm{d}y\\Big\\|_{L^2}^2\\Big)^{\\frac12}\n\\\\ \\leq&C_s\\|{\\Delta}_{-1}\\nabla\\overline{u}_I\\|_{L^\\infty}\n\\big\\|\\widetilde{\\Delta}_{-1}\\nabla\\overline{\\omega}\\big\\|_{L^2}\n \\leq C_s\\big\\|\\Lambda^{2\\alpha}\\overline{\\omega}\\big\\|_{L^2}\\|\\overline{\\omega}\\|_{L^2}\\leq\nC_s\\|\\overline{\\omega}\\|_{H^{2\\alpha}}^2.\n\\end{split}\n\\end{equation*}\nCombining these estimates above yields \\eqref{add-2}. This ends the proof.\n\\end{proof}\n\n\n\n\n\n\n\n\\section{Proof of main Theorems }\\label{sec5}\n\\setcounter{section}{5}\\setcounter{equation}{0}\n\n This section is devoted to showing the main theorems. Let us begin by proving Theorem~\\ref{th1}.\n \\subsection{Proof of Theorem \\ref{th1}}\n\nFirst of all, let us denote\n$$\\overline{\\omega}=\\omega^{\\alpha}-\\omega^{\\alpha_{0}}\\quad\\text{and}\\quad\\overline{u}=u^{\\alpha}-u^{\\alpha_{0}}.$$\nThen, the couple $(\\overline{\\omega},\\,\\overline{u})$ satisfies\n\\begin{equation}\\label{diffrence}\n\\overline{\\omega}_{t}+u^{\\alpha_{0}}\\cdot\\nabla \\overline{\\omega}+\\overline{u}\\cdot\\nabla \\overline{\\omega}+\\overline{u}\\cdot\\nabla \\omega^{\\alpha_{0}} =0.\n\\end{equation}\nOperating $J^s$ on \\eqref{diffrence} and taking the scalar product of the resulting\nequation with $J^s\\overline{\\omega}$ in $L^2,$ we get\n\\begin{equation}\\label{11-1}\\begin{split}\n\\frac12\\frac{\\mathrm{d}}{\\mathrm{d}t}\\|\\overline{\\omega}(t)\\|_{H^s}^2=&-\\int_{\\R^2} J^s(u^{\\alpha_{0}}\\cdot\\nabla \\overline{\\omega})J^s\\overline{\\omega}\\,\\mathrm{d}x-\\int_{\\R^2} J^s(\\overline{u}\\cdot\\nabla \\overline{\\omega})J^s\\overline{\\omega}\\,\\mathrm{d}x\\\\\n&-\\int_{\\R^2} J^s(\\overline{u}\\cdot\\nabla \\omega^{\\alpha_{0}})J^s\\overline{\\omega}\\,\\mathrm{d}x\\\\:=&\nI_1+I_2+I_3.\n\\end{split}\\end{equation}\n\nWe are going to estimate the three terms on the right hand side of \\eqref{11-1} one by one. By the divergence-free condition and \\eqref{com-2}, we have\n\\begin{equation*}\\begin{split}\nI_1=&-\\int_{\\R^2} (J^s(u^{\\alpha_{0}}\\cdot\\nabla \\overline{\\omega})-u^{\\alpha_{0}}\\cdot\\nabla J^s\\overline{\\omega})\nJ^s\\overline{\\omega}\\,\\mathrm{d}x\\\\ \\leq&\n\\big\\|J^s(u^{\\alpha_{0}}\\cdot\\nabla \\overline{\\omega})-u^{\\alpha_{0}}\\cdot\\nabla J^s\\overline{\\omega}\\big\\|_{L^2}\n\\|J^s\\overline{\\omega}\\|_{L^2}\\\\ \\leq&\\Big(\\|J^su^{\\alpha_{0}}\\|_{L^\\frac{1}{\\alpha_0}}\\|\\nabla\\overline{\\omega}\\|_{L^\\frac{2}{1-2\\alpha_0}}+\\|\\nabla u^{\\alpha_{0}}\\|_{L^\\infty}\\|J^{s-1}\\nabla\\overline{\\omega}\\|_{L^2}\\Big)\n\\|J^s\\overline{\\omega}\\|_{L^2}.\n\\end{split}\\end{equation*}\nNote that $0<\\alpha_0<\\frac12$ and\n$$\nu^{\\alpha_0}=\\nabla^{\\perp}(-\\Delta)^{-1+\\alpha_0}\\omega^{\\alpha_0}.\n$$\nUsing Lemma \\ref{Hardy} yields\n$$\\|J^su^{\\alpha_{0}}\\|_{L^\\frac{1}{\\alpha_0}}\\leq C\\norm{\\omega^{\\alpha_0}}_{H^s}.$$\nApplying Lemma \\ref{embedding1} and Lemma \\ref{Hardy} gives\n$$\\|\\nabla\\overline{\\omega}\\|_{L^\\frac{2}{1-2\\alpha_0}}\\leq C\\|\\overline{\\omega}\\|_{H^s},$$\n\\begin{equation*}\\begin{split}\n\\|\\nabla u^{\\alpha_{0}}\\|_{L^\\infty}\\leq \\norm{u^{\\alpha_{0}}}_{W^{s,\\frac{1}{\\alpha_0}}}\\leq C\\norm{\\omega^{\\alpha_0}}_{H^{s}}.\n\\end{split}\n\\end{equation*}\nThus, \\begin{equation}\\label{wsp-1}\\begin{split}\n|I_1|\\leq C\\|\\overline{\\omega}\\|_{H^s}^2\\norm{\\omega^{\\alpha_0}}_{H^{s}}.\n\\end{split}\\end{equation}\nBy the decomposition \\eqref{diff-1-1}, the second term can be written as\n\\begin{equation*}\\begin{split}\n I_2=&-\\int_{\\R^2}(J^s(\\overline{u}\\cdot\\nabla \\overline{\\omega})-\\overline{u}\\cdot\\nabla J^s\\overline{\\omega})\n J^s\\overline{\\omega}\\,\\mathrm{d}x\\\\=&\n-\\int_{\\R^2}(J^s(\\overline{u}_I\\cdot\\nabla \\overline{\\omega})-\\overline{u}_I\\cdot\\nabla J^s\\overline{\\omega})\nJ^s\\overline{\\omega}\\,\\mathrm{d}x-\\int_{\\R^2}\\big( J^s(\\overline{u}_{II}\\cdot\\nabla \\overline{\\omega})-\\overline{u}_{II}\\cdot\\nabla J^s\\overline{\\omega}\\big)\nJ^s\\overline{\\omega}\\,\\mathrm{d}x.\n\\end{split}\n\\end{equation*}\nUsing \\eqref{com-2} and the Sobolev embedding inequalities, we obtain\n\\begin{equation*}\\begin{split}\n&-\\int_{\\R^2}\\big(J^s(\\overline{u}_I\\cdot\\nabla \\overline{\\omega})-\\overline{u}_I\\cdot\\nabla J^s\\overline{\\omega}\\big)J^s\\overline{\\omega}\\,\\mathrm{d}x\\\\ \\leq&\\big(\\|J^s\\overline{u}_I\\|_{L^\\frac1\\alpha}\\|\\nabla\\overline{\\omega}\\|_{L^\\frac{2}{1-2\\alpha}}+\\|\\nabla \\overline{u}_I\\|_{L^\\infty}\\|J^{s-1}\\nabla\\overline{\\omega}\\|_{L^2}\\big)\n\\|J^s\\overline{\\omega}\\|_{L^2}\\\\ \\leq&\\|\\overline{\\omega}\\|_{H^s}^2\\|J^s\\overline{u}_I\\|_{L^\\frac1\\alpha}.\n\\end{split}\n\\end{equation*}\nSimilarly, for $p>\\frac1\\alpha>2,$\n\\begin{equation*}\\begin{split}\n&-\\int_{\\R^2}\\big(J^s(\\overline{u}_{II}\\cdot\\nabla \\overline{\\omega})-\\overline{u}_{II}\\cdot\\nabla J^s\\overline{\\omega}\\big)J^s\\overline{\\omega}\\,\\mathrm{d}x\\\\ \\leq&\\big(\\|J^s\\overline{u}_{II}\\|_{L^p}\\|\\nabla\\overline{\\omega}\\|_{L^\\frac{2p}{p-2}}+\n\\|\\nabla \\overline{u}_{II}\\|_{L^\\infty}\\|J^{s-1}\\nabla\\overline{\\omega}\\|_{L^2}\\big)\n\\|J^s\\overline{\\omega}\\|_{L^2}\\\\ \\leq&\\|\\overline{\\omega}\\|_{H^s}^2\\|J^s\\overline{u}_{II}\\|_{L^p}.\n\\end{split}\n\\end{equation*}\nTherefore,\n\\begin{equation}\\label{wsp-3}\n|I_2|\\leq\n\\|\\overline{\\omega}\\|_{H^s}^2(\\|J^s\\overline{u}_I\\|_{L^\\frac1\\alpha}+\\|J^s\\overline{u}_{II}\\|_{L^p}).\n\\end{equation}\nConcerning the third term, we use \\eqref{com-1} and the Sobolev embedding inequalities to get\n\\begin{equation}\\label{wsp-4}\\begin{split}\n|I_3|=&\\Big|\\int J^s(\\overline{u}\\cdot\\nabla \\omega^{\\alpha_{0}})J^s\\overline{\\omega}\\,\\mathrm{d}x\\Big|\\\\\n=&\n\\Big|\\int J^s(\\overline{u}_{I}\\cdot\\nabla \\omega^{\\alpha_{0}})J^s\\overline{\\omega}\\,\\mathrm{d}x+\\int J^s(\\overline{u}_{II}\\cdot\\nabla \\omega^{\\alpha_{0}})J^s\\overline{\\omega}\\,\\mathrm{d}x\\Big|\n\\\\ \\leq&\n\\|\\overline{\\omega}\\|_{H^s}(\\|J^s\\overline{u}_I\\|_{L^\\frac1\\alpha}+\\|J^s\\overline{u}_{II}\\|_{L^p})\\|\\omega^{\\alpha_{0}}\\|_{H^{s+1}}.\n\\end{split}\\end{equation}\nPlugging these estimates \\eqref{wsp-1}, \\eqref{wsp-3}, \\eqref{wsp-4} into \\eqref{11-1} yields\n\\begin{equation}\\label{Hs-est}\\begin{split}\n \\frac{\\mathrm{d}}{\\mathrm{d}t}\\|\\overline{\\omega}(t)\\|_{H^s} \\leq&\n\\|\\overline{\\omega}\\|_{H^s}\\big(\\|\\omega^{\\alpha_{0}}\\|_{H^{s}}\n+\\|J^s\\overline{u}_I\\|_{L^\\frac1\\alpha}+\\|J^s\\overline{u}_{II}\\|_{L^p}\\big)\\\\\n&+\n\\|\\omega^{\\alpha_{0}}\\|_{H^{s+1}}\\big(\\|J^s\\overline{u}_I\\|_{L^\\frac1\\alpha}+\n\\|J^s\\overline{u}_{II}\\|_{L^p}\\big)\\\\\n:=&\\tilde {I_1}+\\tilde {I_2}.\n\\end{split}\n\\end{equation}\nThe integral form of $\\overline{u}_I$ can be written as\n\\begin{equation*}\nJ^s\\overline{u}_I(x)=\\int_{\\R^2}\\frac{(x-y)^{\\perp}}{|x-y|^{2+2\\alpha}}\nJ^s\\overline{\\omega}(y)\\,\\mathrm{d}y.\n\\end{equation*}\nThen, using Lemma \\ref{Hardy} enables us to get\n\\begin{equation}\\label{4-16-1}\n\\begin{split}\n\\|J^s\\overline{u}_I\\|_{L^{\\frac1\\alpha}}&\\leq\\Big\\|\\int_{\\R^2}\\frac{1}{|x-y|^{2-(1-2\\alpha)}}\n|J^s\\overline{\\omega}(y)|\\,\\mathrm{d}y\\Big\\|_{L^2}\\\\&\\leq\nC(\\alpha)\\|J^s\\overline{\\omega}\\|_{L^{2}},\n\\end{split}\\end{equation}\nwhere $C(\\alpha)$ depends on $\\alpha$ and will be bounded if $0\\le \\alpha<\\alpha_0<\\frac12$ (but will be unbounded if $\\alpha$ tend to $\\frac12$).\nWhen $p>\\frac1\\alpha>2,$ adopting to the similar way to \\eqref{4-16-1} gives\n\\begin{equation}\\label{4-16-2}\\begin{split}\n\\|J^s\\overline{u}_{II}\\|_{L^p}&\\leq C(\\alpha)\\big(\\|J^s\\omega^{\\alpha_0}\\|_{L^{\\frac{2p}{2+p(1-2\\alpha)}}}+\n \\|J^s\\omega^{\\alpha_0}\\|_{L^{\\frac{2p}{2+p(1-2\\alpha_0)}}}\\big)\n \\\\&\\leq C(\\alpha)\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}.\n\\end{split}\\end{equation}\nHence,\n\\begin{equation}\\label{Hs-1}\n\\tilde {I_1} \\leq C\\|\\overline{\\omega}\\|_{H^s}^{2}+C\\|\\overline{\\omega}\\|_{H^{s}}\n\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}.\n\\end{equation}\nOn the other hand, the estimate \\eqref{4-16-2} is not adaptable to $\\tilde {I_2}$ in \\eqref{Hs-est}. We will use a different way to estimate\n$\\|J^s\\overline{u}_{II}\\|_{L^p}$ in \\eqref{Hs-est}.\nFor $0<\\epsilon<1$ to be determined later, we write $\\overline{u}_{II}$ as\n\\begin{equation*}\\begin{split}\n \\overline{u}_{II}=\\int_{\\R^2}\\Big(\\frac{(x-y)^{\\perp}}{|x-y|^{2+2\\alpha}}\n-\\frac{(x-y)^{\\perp}}{|x-y|^{2+2\\alpha_0}}\\Big)\\omega^{\\alpha_0}(y)\\,\\mathrm{d}y.\n\\end{split}\\end{equation*}\nTherefore,\n\\begin{equation}\\label{11-6-0}\\begin{split}\n J^s\\overline{u}_{II}&=\\int_{\\R^2}\\Big(\\frac{(x-y)^{\\perp}}{|x-y|^{2+2\\alpha}}\n-\\frac{(x-y)^{\\perp}}{|x-y|^{2+2\\alpha_0}}\\Big)J^s\\omega^{\\alpha_0}(y)\\,\\mathrm{d}y\n\\\\&=\\Big(\\int_{|x-y|\\leq\\epsilon}+\\int_{1>|x-y|\\geq\\epsilon}+\\int_{|x-y|\\geq1}\\Big)\\Big(\\frac{(x-y)^{\\perp}}{|x-y|^{2+2\\alpha}}\n-\\frac{(x-y)^{\\perp}}{|x-y|^{2+2\\alpha_0}}\\Big)J^s\\omega^{\\alpha_0}(y)\\,\\mathrm{d}y\n\\\\&:=H_{1}+H_{2}+H_{3}.\n\\end{split}\\end{equation}\nFor the first term $H_1,$ using the Young inequality and the Sobolev embedding, we get\n\\begin{equation*}\\begin{split}\n\\|H_1\\|_{L^{p}}&= \\Big\\|\\int_{|x-y|\\leq\\epsilon}\\Big(\\frac{(x-y)^{\\perp}}{|x-y|^{2+2\\alpha}}\n-\\frac{(x-y)^{\\perp}}{|x-y|^{2+2\\alpha_0}}\\Big)J^s\\omega^{\\alpha_0}(y)\\,\\mathrm{d}y\\Big\\|_{L^{p}}\n\\\\&\\leq C\\Big(\\frac{\\epsilon^{1-2\\alpha}}{1-2\\alpha}+\\frac{\\epsilon^{1-2\\alpha_0}}{1-2\\alpha_0}\\Big)\n\\|J^s\\omega^{\\alpha_0}\\|_{L^{p}}\n\\\\&\\leq C\\Big(\\frac{\\epsilon^{1-2\\alpha}}{1-2\\alpha}\n+\\frac{\\epsilon^{1-2\\alpha_0}}{1-2\\alpha_0}\\Big)\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}.\n\\end{split}\n\\end{equation*}\nAs for $H_2$ and $H_3$, it is divided into two cases.\n\n{\\it Case 1: $\\alpha_0>\\alpha.$} By the mean value theorem, we can obtain\n\\begin{equation*}\\begin{split}\nH_2\\leq|\\alpha_0-\\alpha|\\int_{1>|x-y|\\geq\\epsilon}\\frac{|\\log|x-y||}{|x-y|^{1+2\\alpha_0}}\n|J^s\\omega^{\\alpha_0}(y)|\\,\\mathrm{d}y.\n\\end{split}\n\\end{equation*}\nUtilizing the Young inequality yields\n\\begin{equation*}\\begin{split}\n\\|H_2\\|_{L^p}&\\leq|\\alpha_0-\\alpha|\\|J^s\\omega^{\\alpha_0}\\|_{L^p}\n\\int_{1>|x-y|\\geq\\epsilon}\\frac{|\\log|x-y||}{|x-y|^{1+2\\alpha_0}}\\,\\mathrm{d}y,\n\\\\&\\leq\\frac{C}{1-2\\alpha_0}|\\alpha_0-\\alpha||\\log\\epsilon|\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}.\n\\end{split}\n\\end{equation*}\n To deal with $H_3$, we fix a small number $\\sigma>0$ such that $p>\\frac{2}{2\\alpha-\\sigma}>2.$ Thanks to the fact that $\\log |x-y|\\le C|x-y|^\\sigma$ for any $\\sigma>0$ and $|x-y|\\ge 1$, we have\n\\begin{equation*}\\begin{split}\n\\|H_3\\|_{L^p}&\\leq |\\alpha_0-\\alpha|\\Big\\|\\int_{|x-y|\\geq1}\\frac{|\\log|x-y||}{|x-y|^{1+2\\alpha}}\n|J^s\\omega^{\\alpha_0}(y)|\\,\\mathrm{d}y\\Big\\|_{L^p}\\\\&\\leq|\\alpha_0-\\alpha|\\|J^s\\omega^{\\alpha_0}\\|_{L^r}\n\\Big(\\int_{|x-y|\\geq1}\\Big(\\frac{|\\log|x-y||}{|x-y|^{1+2\\alpha}}\\Big)^q\\,\\mathrm{d}y\\Big)^{\\frac1q}\n\\\\\n&\\leq|\\alpha_0-\\alpha|\\|J^s\\omega^{\\alpha_0}\\|_{L^r}\n\\Big(\\int_{|x-y|\\geq1}\\Big(\\frac{1}{|x-y|^{1+2\\alpha-\\sigma}}\\Big)^q\\,\\mathrm{d}y\\Big)^{\\frac1q}\\\\\n&\\leq|\\alpha_0-\\alpha|\\|J^s\\omega^{\\alpha_0}\\|_{L^r}\\Big(\\frac{1}{(1+2\\alpha-\\sigma)q-2}\\Big)^{\\frac1q},\n\\end{split}\n\\end{equation*}where $\\frac1p+1=\\frac1r+\\frac1q,$ $q>\\frac{2}{1+2\\alpha-\\sigma}$, $p>\\frac{2}{2\\alpha-\\sigma}$, then we can choose some $r>2$ such that\n\\begin{equation*}\nH^{s+1}(\\R^2)\\hookrightarrow W^{s,r}(\\R^2)\n\\end{equation*}holds (see Lemma \\ref{embedding1}).\n\n{\\it Case 2: $\\alpha_0<\\alpha<\\frac12.$} It is similar to {\\it Case 1} by exchanging the position of $\\alpha$ and $\\alpha_0$. For instance, by the mean value theorem, we can obtain\n\\begin{equation*}\\begin{split}\nH_2\\leq|\\alpha_0-\\alpha|\\int_{1>|x-y|\\geq\\epsilon}\\frac{|\\log|x-y||}{|x-y|^{1+2\\alpha}}\n|J^s\\omega^{\\alpha_0}(y)|\\,\\mathrm{d}y.\n\\end{split}\n\\end{equation*}\nThen\n\\begin{equation*}\\begin{split}\n\\|H_2\\|_{L^p}&\\leq|\\alpha_0-\\alpha|\\|J^s\\omega^{\\alpha_0}\\|_{L^p}\n\\int_{1>|x-y|\\geq\\epsilon}\\frac{|\\log|x-y||}{|x-y|^{1+2\\alpha_0}}\\,\\mathrm{d}y,\n\\\\&\\leq\\frac{C}{1-2\\alpha}|\\alpha_0-\\alpha||\\log\\epsilon|\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}.\n\\end{split}\n\\end{equation*}\nNow we fix a small number $\\sigma>0$ such that $p>\\frac{2}{2\\alpha_0-\\sigma}>2.$ Similarly, we have\n\\begin{equation*}\\begin{split}\n\\|H_3\\|_{L^p}\n&\\leq |\\alpha_0-\\alpha|\\Big\\|\\int_{|x-y|\\geq1}\\frac{|\\log|x-y||}{|x-y|^{1+2\\alpha_0}}\n|J^s\\omega^{\\alpha_0}(y)|\\,\\mathrm{d}y\\Big\\|_{L^p}\n\\\\\n&\\leq|\\alpha_0-\\alpha|\\|J^s\\omega^{\\alpha_0}\\|_{L^r}\\Big(\\frac{1}{(1+2\\alpha_0-\\sigma)q-2}\\Big)^{\\frac1q},\n\\end{split}\n\\end{equation*}where $\\frac1p+1=\\frac1r+\\frac1q,$ $q>\\frac{2}{1+2\\alpha-\\sigma}$, $p>\\frac{2}{2\\alpha-\\sigma}$, then we can choose some $r>2$ such that\n\\begin{equation*}\nH^{s+1}(\\R^2)\\hookrightarrow W^{s,r}(\\R^2)\n\\end{equation*}holds (see Lemma \\ref{embedding1}).\n\nAs a consequence, we get\n\\begin{equation*}\n\\|J^s\\overline{u}_{II}(t)\\|_{L^p}\\leq C\\left(\\epsilon^{1-2\\alpha}\n+\\epsilon^{1-2\\alpha_0}+|\\alpha_0-\\alpha|+|\\alpha_0-\\alpha||\\log\\epsilon|\\right)\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}.\n\\end{equation*}\nHence,\n\\begin{equation}\\label{Hs-2}\n\\begin{split}\n\\tilde {I_2}=&\\|\\omega^{\\alpha_{0}}\\|_{H^{s+1}}(\\|J^s\\overline{u}_I\\|_{L^\\frac1\\alpha}+\n\\|J^s\\overline{u}_{II}\\|_{L^p})\\\\ \\leq & C\\|\\omega^{\\alpha_{0}}\\|_{H^{s+1}}\\|\\overline{\\omega}\\|_{H^s}\n+\\|\\omega^{\\alpha_{0}}\\|_{H^{s+1}}^2\\left(\\epsilon^{1-2\\alpha}+\\epsilon^{1-2\\alpha_0}\n+|\\alpha_0-\\alpha|+|\\alpha_0-\\alpha||\\log\\epsilon|\\right).\n\\end{split}\\end{equation}\nSet $\\epsilon=\\alpha_0-\\alpha$.\nBy plugging \\eqref{Hs-1} and \\eqref{Hs-2} into \\eqref{Hs-est}, we get\n\\begin{equation}\\label{Hs-3}\\begin{split}\n \\frac{\\mathrm{d}}{\\mathrm{d}t}\\|\\overline{\\omega}(t)\\|_{H^s} \\leq&\nC\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}\\|\\overline{\\omega}\\|_{H^s}\n +C\\|\\overline{\\omega}\\|_{H^s}^{2}\\\\\n&+\\|\\omega^{\\alpha_{0}}\\|_{H^{s+1}}^2\\left(|\\alpha_0-\\alpha|^{1-2\\alpha}+|\\alpha_0-\\alpha|^{1-2\\alpha_0}+|\\alpha_0-\\alpha||\\log |\\alpha_0-\\alpha||\\right).\n\\end{split}\\end{equation}\nMultiply \\eqref{Hs-3} by $\\exp(-C\\int_0^t\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}\\,\\mathrm{d}s)$ and consider the quantity\n$$y(t)=\\|\\overline{\\omega}\\|_{H^s}\\exp\\Big(-C\\int_0^t\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}\\,\\mathrm{d}s\\Big).$$\nWe then get the inequality\n\\begin{equation*}\n\\frac{\\mathrm{d}y(t)}{\\mathrm{d}t}\\leq \\left(|\\alpha_0-\\alpha|^{1-2\\alpha}+|\\alpha_0-\\alpha|^{1-2\\alpha_0}\\right)F(t)+Gy^2(t),\n\\end{equation*}\nwhere $$F(t)=\\|\\omega^{\\alpha_{0}}\\|_{H^{s+1}}^2\\exp\\Big(-C\\int_0^t\\|\\omega^{\\alpha_0}(s)\\|_{H^{s+1}}\\,\\mathrm{d}s\\Big),$$\nand $$G=C\\exp\\Big(C\\int_0^T\\|\\omega^{\\alpha_0}(t)\\|_{H^{s+1}}\\,\\mathrm{d}t\\Big).$$\nBy Proposition \\ref{o}, there exists a $\\delta>0$ depending on $T$ and $\\int_0^T \\|\\omega^{\\alpha_0}\\|_{H^{s+1}} dt$ such that when $0<\\alpha<\\frac12$ and $|\\alpha-\\alpha_0|<\\delta$,\n\\begin{equation*}\ny(t)\\leq C\\left(|\\alpha_0-\\alpha|^{1-2\\alpha}+|\\alpha_0-\\alpha|^{1-2\\alpha_0}\\right)\\int_0^T F(t)\\,\\mathrm{d}t,\n\\end{equation*}\nwhich implies that\n\\begin{equation*}\n\\|\\overline{\\omega}\\|_{H^s}\\leq C\\left(|\\alpha_0-\\alpha|^{1-2\\alpha_0}+|\\alpha_0-\\alpha||\\log|\\alpha_0-\\alpha||\\right).\n\\end{equation*}\nHere $C>0$ is a constant depending on $T$ and $\\int_0^T \\|\\omega^{\\alpha_0}\\|_{H^{s+1}} dt$ as well.\n\nAssume that $\\omega^{\\alpha_{0}}\\in C([0,T_0];H^{s+1})$, $s>2.$ According to the local well-posedness theory, for $\\alpha<\\alpha_0,$ $\\omega^{\\alpha}\\in C([0,T];H^{s+1})$ for some $T>0$.\nIf $T\\geq T_0$, the proof is finished. If $T0$ the maximal existence time satisfying $\\omega^{\\alpha}\\in C([0,T_{\\max});H^{s+1})$. By performing the $(s+1)$-order energy estimate, we get\n\\begin{equation*}\\begin{split}\n\\frac12\\frac{\\mathrm{d}}{\\mathrm{d}t}\\|\\omega^\\alpha(t)\\|_{H^{s+1}}^2\n&=-\\int_{\\R^2}\\Lambda^{s+1}\\big(u^{\\alpha}\\cdot\\nabla \\omega^\\alpha\\big)\\Lambda^{s+1}\\omega^\\alpha\\,\\mathrm{d}x\n\\\\&=-\\int_{\\R^2}\\Lambda^{s+1}\\big(u^{\\alpha}\\cdot\\nabla \\omega^\\alpha-u^{\\alpha}\\cdot\\Lambda^{s+1}\\nabla \\omega^\\alpha\\big)\\Lambda^{s+1}\\omega^\\alpha\\,\\mathrm{d}x\n\\\\&\\leq\\|\\omega^\\alpha\\|_{H^{s+1}}\\left(\\|\\Lambda^{s+1}u^\\alpha\\|_{L^{\\frac1\\alpha}}\\|\\nabla \\omega^\\alpha\\|_{L^{\\frac{2}{1-2\\alpha}}}\n+\\|\\nabla u^\\alpha\\|_{L^\\infty}\\|\\omega^\\alpha\\|_{H^{s+1}}\\right)\n\\\\&\\leq\\|\\omega^\\alpha\\|_{H^{s+1}}^2\\left(\\|\\nabla \\omega^\\alpha\\|_{L^{\\frac{2}{1-2\\alpha}}}+\\|\\nabla u^\\alpha\\|_{L^\\infty}\\right)\n\\\\&\\leq\\|\\omega^\\alpha\\|_{H^{s+1}}^2\\|\\omega^\\alpha\\|_{H^{s}}.\n\\end{split}\\end{equation*}\nBy the Gronwall inequality, we have\n\\begin{equation*}\\begin{split}\n\\|\\omega^\\alpha(t)\\|_{H^{s+1}}&\\leq e^{\\int_0^{T_{\\max}}\\|\\omega^\\alpha(t)\\|_{H^{s}}\\,\\mathrm{d}t}\\|\\omega^\\alpha_0\\|_{H^{s+1}}\n\\\\&\\leq e^{\\int_0^{T_{\\max}}(\\|\\overline{\\omega}(t)\\|_{H^{s}}+\\|\\omega^{\\alpha_0}(t)\\|_{H^{s}})\\,\\mathrm{d}t}\\|\\omega^\\alpha_0\\|_{H^{s+1}}\n\\leq C\n\\end{split}\\end{equation*}\nfor $t\\in [0,{T_{\\max}}]$ and hence $\\omega^\\alpha(T_{\\max})$ is finite. This deduces a contradiction with $T_{\\max}$ is the maximal existence time by using the local well-posedness theory. In consequence, $T=T_0$ as required and the proof of the theorem is finished.\n\n\n\\subsection{Proof of Theorem \\ref{th2}}\n\n\nTaking the scalar product of \\eqref{diffrence} with $\\overline{\\omega}$ in $H^s$ and using Lemma \\ref{commutator} and the Sobolev embedding inequalities enable us to get\n\\begin{equation}\\label{Hs-est-1-1}\\begin{split}\n\\frac{\\mathrm{d}}{\\mathrm{d}t}\\|\\overline{\\omega}(t)\\|_{H^s}&\\leq\n\\|\\overline{\\omega}\\|_{H^s}\\|u^{\\alpha_{0}}\\|_{H^{s}}+\n(\\|\\overline{\\omega}\\|_{H^s}+\\|\\omega^{\\alpha_{0}}\\|_{H^{s+1}})\\|\\overline{u}\\|_{H^{s}}\\\\\n&\\leq\n\\|\\overline{\\omega}\\|_{H^s}\\|\\omega^{\\alpha_{0}}\\|_{H^{s}}+\n(\\|\\overline{\\omega}\\|_{H^s}+\\|\\omega^{\\alpha_{0}}\\|_{H^{s+1}})(\\|\\overline{u}_{I}\\|_{H^{s}}+\\|\\overline{u}_{II}\\|_{H^{s}}).\n\\end{split}\\end{equation}\nHere we have used the decomposition \\eqref{diff-1-1} with\n\\begin{equation*}\n\\overline{u}_{I}=\\nabla^{\\perp}(-\\Delta)^{-1+\\alpha}\\overline{\\omega}, \\quad \\overline{u}_{II}=\\left(\\nabla^{\\perp}(-\\Delta)^{-1+\\alpha}-\\nabla^{\\perp}(-\\Delta)^{-1+\\alpha_0}\\right)\\omega^{\\alpha_0}.\n\\end{equation*}\nBy using Proposition \\ref{uni-est} (Remark \\ref{Rm2-1}) with $\\beta=1-2\\alpha$, we obtain\n\\begin{equation}\\label{u1-est}\n\\|\\overline{u}_{I}(t)\\|_{H^{s}}\\leq\nC\\big(\\|\\overline{\\omega}\\|_{H^s}+(1-2\\alpha)\\|\\overline{\\omega}\\|_{L^1}\\big),\n\\end{equation}where $C=C(\\alpha,s)$ is an absolutely constant when $\\alpha\\rightarrow\\frac12.$\nBy using Proposition \\ref{uni-est} again ($\\beta=1-2\\alpha$ and $\\beta=0$ respectively), there also exists a uniformly bounded constant $C=C(\\alpha,s)$ when $\\alpha\\rightarrow\\frac12$ such that\n \\begin{equation*}\\| \\overline{u}_{II}(t)\\|_{H^s}\\leq C\\big(\\|\\omega^{\\alpha_0}\\|_{H^s}+\\|\\omega^{\\alpha_0}\\|_{L^1}\\big).\n\\end{equation*}\nIt follows that \\begin{equation}\\label{equ-12}\n\\begin{split}\n\\|\\overline{\\omega}\\|_{H^s}(\\|\\overline{u}_{I}\\|_{H^{s}}+\\|\\overline{u}_{II}\\|_{H^{s}})\\leq C\\|\\overline{\\omega}\\|_{H^s}^2+\nC\\|\\overline{\\omega}\\|_{H^s}(\\|\\omega^{\\alpha_0}\\|_{H^s}+\n\\|\\omega^{\\alpha_0}\\|_{L^1}+\\|\\omega^{\\alpha}\\|_{L^1}).\n\\end{split}\\end{equation}\n\nNow we adopt to anther way to estimate $\\|\\overline{u}_{II}\\|_{H^s}$ in order to deal with $\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}\\|\\overline{u}_{II}\\|_{H^s}$ on the right hand side of \\eqref{Hs-est-1-1}. The decomposition \\eqref{11-6-0} will be applied, which is\n\\begin{equation}\\label{11-6}\\begin{split}\n J^s\\overline{u}_{II}&=\\int_{\\R^2}\\Big(\\frac{(x-y)^{\\perp}}{|x-y|^{2+2\\alpha}}\n-\\frac{(x-y)^{\\perp}}{|x-y|^{2+2\\alpha_0}}\\Big)J^s\\omega^{\\alpha_0}(y)\\,\\mathrm{d}y\n\\\\&=\\Big(\\int_{|x-y|\\leq\\epsilon}+\\int_{1>|x-y|\\geq\\epsilon}+\\int_{|x-y|\\geq1}\\Big)\\Big(\\frac{(x-y)^{\\perp}}{|x-y|^{2+2\\alpha}}\n-\\frac{(x-y)^{\\perp}}{|x-y|^{2+2\\alpha_0}}\\Big)J^s\\omega^{\\alpha_0}(y)\\,\\mathrm{d}y\n\\\\&=H_{1}+H_{2}+H_{3},\n\\end{split}\\end{equation}\nwhere $0<\\epsilon<1$ is to be determined later.\n\nPerforming the fact that $$\\int_{|x|=1}\\frac{x^\\perp}{|x|^{2+2\\alpha}}\\,\\mathrm{d}s=\\int_{|x|=1}\\frac{x^\\perp}{|x|^{3}}\\,\\mathrm{d}s=0,$$\nwe get\n\\begin{equation*}\\begin{split}\nH_{1}&=\\int_{|x-y|\\leq\\epsilon}\\Big(\\frac{(x-y)^{\\perp}}{|x-y|^{2+2\\alpha}}-\\frac{(x-y)^{\\perp}}{|x-y|^{3}}\\Big)\n\\Big(J^s\\omega^{\\alpha_0}(y)-J^s\\omega^{\\alpha_0}(x)\\Big)\\,\\mathrm{d}y\n\\\\&=\\int_{|z|\\leq\\epsilon}\\Big(\\frac{z^{\\perp}}{|z|^{2+2\\alpha}}-\\frac{z^{\\perp}}{|z|^{3}}\\Big)\\big(J^s\\omega^{\\alpha_0}(x-z)-J^s\\omega^{\\alpha_0}(x)\\big)\\,\\mathrm{d}z.\n\\end{split}\n\\end{equation*}\nFrom the mean value theorem, we deduce that\n\\begin{equation*}\\begin{split}\n|H_{1}|&\\leq\\int_0^1\\int_{|z|\\leq\\epsilon}\n\\Big(\\frac{1}{|z|^{2\\alpha}}+\\frac{1}{|z|}\\Big)\\big|\\nabla J^s\\omega^{\\alpha_0}(x-\\tau z)\\big|\\,\\mathrm{d}z\\mathrm{d}\\tau.\n\\end{split}\n\\end{equation*}\nNow, taking the $L^2$ norm of the above inequality, and using the fact that the norm of an integral is less that the integral of the norm, we get\n\\begin{equation*}\\begin{split}\n\\|H_{1}\\|_{L^2}&\\leq \\int_0^1\\int_{|z|\\leq\\epsilon}\n\\Big(\\frac{1}{|z|^{2\\alpha}}+\\frac{1}{|z|}\\Big)\\big\\|\\nabla J^s\\omega^{\\alpha_0}(\\cdot-\\tau z)\\big\\|_{L^2}\\,\\mathrm{d}z\\mathrm{d}\\tau.\n\\end{split}\n\\end{equation*}\nThe translation invariance of the Lebesgue measure then ensures that\n\\begin{equation}\\label{11-7}\\begin{split}\n\\|H_{1}\\|_{L^2}&\\leq C\\Big(\\frac{1}{2-2\\alpha}\\epsilon^{2-2\\alpha}+\\epsilon\\Big)\\big\\|J^{s+1}\\omega^{\\alpha_0}\\big\\|_{L^2}.\n\\end{split}\n\\end{equation}\nFor $2+2\\alpha\\leq\\xi\\leq3,$ we estimate $H_{2}$ as follows,\n\\begin{equation}\\label{11-8}\\begin{split}\n\\|H_{2}\\|_{L^2}&= \\Big(\\frac12-\\alpha\\Big)\\Big\\|\\int_{1>|x-y|\\geq\\epsilon}\\frac{(x-y)^{\\perp}\\big(|x-y|^{\\xi}\\log|x-y|\\big)}{|x-y|^{2+2\\alpha}|x-y|^{2+2\\alpha_0}}\nJ^s\\omega^{\\alpha_0}(y)\\,\\mathrm{d}y\\Big\\|_{L^2}\n\\\\&\\leq \\Big(\\frac12-\\alpha\\Big)\\Big\\|\\int_{1>|x-y|\\geq\\epsilon}\\frac{|\\log|x-y||}{|x-y|^{2}}\n|J^s\\omega^{\\alpha_0}(y)|\\,\\mathrm{d}y\\Big\\|_{L^2}\\\\\n&\\leq\\Big(\\frac12-\\alpha\\Big)|\\log\\epsilon|^2\\|J^s\\omega^{\\alpha_0}\\|_{L^2},\n\\end{split}\n\\end{equation}\nwhere the Young inequality and the mean value theorem have been used.\nAdopting to the similar method to estimate $H_{2}$, we get\n\\begin{equation}\\label{11-9}\\begin{split}\n\\|H_{3}\\|_{L^2}&\\leq \\Big(\\frac12-\\alpha\\Big)\\Big\\|\\int_{|x-y|\\geq1}\\frac{|\\log|x-y||}{|x-y|^{1+2\\alpha}}\n|J^s\\omega^{\\alpha_0}(y)|\\,\\mathrm{d}y\\Big\\|_{L^2}\\\\&\\leq(\\frac12-\\alpha)\\|J^s\\omega^{\\alpha_0}\\|_{L^p}\n\\Big(\\int_{|x-y|\\geq1}\\Big(\\frac{|\\log|x-y||}{|x-y|^{1+2\\alpha}}\\Big)^q\\,\\mathrm{d}y\\Big)^{\\frac1q}\n\\\\&\\leq\\Big(\\frac12-\\alpha\\Big)\\|J^s\\omega^{\\alpha_0}\\|_{L^p}\\Big(\\frac{1}{(1+2\\alpha-\\sigma)q-2}\\Big)^{\\frac1q},\n\\end{split}\n\\end{equation} where $\\frac32=\\frac1p+\\frac1q,$ $q>\\frac{2}{1+2\\alpha-\\sigma}$ whence $p<\\frac{2}{2-2\\alpha+\\sigma}<2$.\nBy the Gagliardo-Nirenberg inequality, we have\n\\begin{equation*}\\begin{split}\n\\|\\Lambda^s\\omega^{\\alpha_0}\\|_{L^p}\\leq\\|\\omega^{\\alpha_0}\\|_{L^1}^{1-\\theta} \\|\\Lambda^{s+1}\\omega^{\\alpha_0}\\|_{L^2}^{\\theta},\n\\end{split}\n\\end{equation*}\nwhere $\\theta=1-\\frac{2}{p(s+2)},$ $\\frac1p\\leq\\frac{1-\\theta}{1}+\\frac{\\theta}{2}$. Then, we conclude that $p\\geq \\frac{2(s+1)}{s+2}.$\nThis enables us to choose some $\\frac{2(s+1)}{s+2}\\leq p<\\frac{2}{2-2\\alpha+\\sigma}.$\nCombining the estimates \\eqref{11-7}-\\eqref{11-9} with \\eqref{11-6}\nand choosing $\\epsilon=(\\frac12-\\alpha)$, we get\n\\begin{equation}\\label{u2-est}\\begin{split}\n\\|\\overline{u}_{II}(t)\\|_{H^{s}}\\leq CL(\\alpha)\\big(\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}+\\|\\omega^{\\alpha_0}\\|_{L^1}\\big).\n\\end{split}\n\\end{equation}\nHere and what in follow,\n\\[L(\\alpha):=\\Big(\\frac12-\\alpha\\Big)^{2-2\\alpha}\n+\\Big(\\frac12-\\alpha\\Big)\\Big|\\log\\Big(\\frac12-\\alpha\\Big)\\Big|^2\n+\\Big(\\frac12-\\alpha\\Big). \\]\nHence, combining \\eqref{u1-est} and \\eqref{u2-est} yields\n\\begin{equation}\\label{equ-12-1}\\begin{split}\n&\\|\\omega^{\\alpha_{0}}\\|_{H^{s+1}}\\|\\overline{u}_{I}\\|_{H^{s}}\n+\\|\\omega^{\\alpha_{0}}\\|_{H^{s+1}}\\|\\overline{u}_{II}\\|_{H^{s}}\n\\\\ \\leq&\\|\\omega^{\\alpha_{0}}\\|_{H^{s+1}}\\|\\overline{\\omega}\\|_{H^s}+\n\\Big(\\frac12-\\alpha\\Big)\\|\\overline{\\omega}\\|_{L^1}\\|\\omega^{\\alpha_{0}}\\|_{H^{s+1}}\n +CL(\\alpha)(\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}^2+\\|\\omega^{\\alpha_0}\\|_{L^1}^2)\n\\\\ \\leq&\\|\\omega^{\\alpha_{0}}\\|_{H^{s+1}}\n\\|\\overline{\\omega}\\|_{H^s}+CL(\\alpha)(\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}^2+\n\\|\\omega^{\\alpha_0}\\|_{L^1}^2+\\|\\omega^{\\alpha}\\|_{L^1}^2).\n\\end{split}\n\\end{equation}\nPlugging \\eqref{equ-12} and \\eqref{equ-12-1} into \\eqref{Hs-est-1-1} gives\n\\begin{equation}\\label{Hs-est-s}\\begin{split}\n \\frac{\\mathrm{d}}{\\mathrm{d}t}\\|\\overline{\\omega}(t)\\|_{H^s}\\leq&\n\\|\\overline{\\omega}\\|_{H^s}(\\|\\omega^{\\alpha_{0}}\\|_{H^{s+1}}\n+\\|\\omega^{\\alpha_0}\\|_{L^1}+\\|\\omega^{\\alpha}\\|_{L^1})+\\|\\overline{\\omega}\\|_{H^s}^2\n\\\\&+CL(\\alpha)(\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}^2+\n\\|\\omega^{\\alpha_0}\\|_{L^1}^2+\\|\\omega^{\\alpha}\\|_{L^1}^2).\n\\end{split}\n\\end{equation}\nThanks to the incompressible condition that $\\nabla\\cdot u=0$, it follows that $\\|\\omega^{\\alpha_0}\\|_{L^1}+\\|\\omega^{\\alpha}\\|_{L^1}$ is bounded if the initial data $\\omega_0\\in L^1$. Arguing similarly as the last part in the proof of Theorem \\ref{th1}, we obtain\n\\begin{equation*}\n\\|\\overline{\\omega}(t)\\|_{H^s}\\leq C\\left(\\Big(\\frac12-\\alpha\\Big)+\\Big(\\frac12-\\alpha\\Big)\\log^2\\Big(\\frac12-\\alpha\\Big)\\right).\n\\end{equation*}\nMoreover, we can prove that $\\omega^{\\alpha}\\in C([0,T];H^{s+1})$. The proof of the theorem is finished.\n\n\n\n\n\\subsection{Proof of Theorem \\ref{th3}}\n\nSimilar to the proof Theorem \\ref{th1}, it follows from the difference equation \\eqref{diffrence} that \\eqref{11-1} holds true.\nThe three terms $I_1,\\, I_2,\\, I_3$ on the right side of \\eqref{11-1} will be estimated as follows. Applying the commutator estimates in Lemma \\ref{commutator} and the Sobolev embedding inequalities, we immediately have\n\\begin{equation}\\label{11-2}\\begin{split}\n|I_1|=&\\Big|\\int_{\\R^2} \\big(J^s(u^{\\alpha_{0}}\\cdot\\nabla \\overline{\\omega})-u^{\\alpha_{0}}\\cdot\\nabla J^s\\overline{\\omega}\\big)\nJ^s\\overline{\\omega}\\,\\mathrm{d}x\\Big|\\\\ \\leq&\n\\|J^s(u^{\\alpha_{0}}\\cdot\\nabla \\overline{\\omega})-u^{\\alpha_{0}}\\cdot\\nabla J^s\\overline{\\omega}\\|_{L^2}\n\\|J^s\\overline{\\omega}\\|_{L^2}\\\\ \\leq&(\\|J^su^{\\alpha_{0}}\\|_{L^2}\\|\\nabla\\overline{\\omega}\\|_{L^\\infty}+\\|\\nabla u^{\\alpha_{0}}\\|_{L^\\infty}\\|J^{s-1}\\nabla\\overline{\\omega}\\|_{L^2})\n\\|J^s\\overline{\\omega}\\|_{L^2}\\\\ \\leq&\\|\\overline{\\omega}\\|_{H^s}^2\\|u^{\\alpha_{0}}\\|_{H^{s}}.\n\\end{split}\\end{equation}\nSubstituting the decomposition \\eqref{diff-1-1} into $I_2$ and $I_3$, respectively, we have\n\\begin{equation}\\label{I_2}\\begin{split}\n&I_2=-\\int_{\\R^2} J^s(\\overline{u}_{I}\\cdot\\nabla \\overline{\\omega})J^s\\overline{\\omega}\\,\\mathrm{d}x-\\int_{\\R^2} J^s(\\overline{u}_{II}\\cdot\\nabla \\overline{\\omega})J^s\\overline{\\omega}\\,\\mathrm{d}x:= I_{21}+I_{22};\\\\\n&I_3=-\\int_{\\R^2} J^s(\\overline{u}_{I}\\cdot\\nabla \\omega^{\\alpha_{0}})J^s\\overline{\\omega}\\,\\mathrm{d}x-\\int_{\\R^2} J^s(\\overline{u}_{II}\\cdot\\nabla \\omega^{\\alpha_{0}})J^s\\overline{\\omega}\\,\\mathrm{d}x:= I_{31}+I_{32}.\n\\end{split}\\end{equation}\nChoose some $\\sigma\\in (0,2\\alpha).$ For any $p>\\frac{2}{2\\alpha-\\sigma}>2,$ applying Lemma \\ref{commutator} and the Sobolev embedding inequalities again, we obtain\n\\begin{equation*}\\begin{split}\n|I_{22}|=&\\Big|\\int_{\\R^2}\\big(J^s(\\overline{u}_{II}\\cdot\\nabla \\overline{\\omega})-\\overline{u}_{II}\\cdot\\nabla J^s\\overline{\\omega}\\big)J^s\\overline{\\omega}\\,\\mathrm{d}x\\Big|\\\\ \\leq&\\|J^s(\\overline{u}_{II}\\cdot \\nabla \\overline{\\omega})-\\overline{u}_{II}\\cdot J^s \\nabla \\overline{\\omega}\\|_{L^2}\\|\\overline{\\omega}\\|_{H^s}\n\\\\ \\leq &\\big(\\|J^s\\overline{u}_{II}\\|_{L^p}\\|\\nabla\\overline{\\omega}\\|_{L^\\frac{2p}{p-2}}+\n\\|\\nabla \\overline{u}_{II}\\|_{L^\\infty}\\|J^{s-1}\\nabla\\overline{\\omega}\\|_{L^2}\\big)\n\\|J^s\\overline{\\omega}\\|_{L^2}\\\\ \\leq&\\|\\overline{\\omega}\\|_{H^s}^2\\|J^s\\overline{u}_{II}\\|_{L^p},\n\\end{split}\n\\end{equation*}\nand\n\\begin{equation*}\\begin{split}\n|I_{32}|&\\leq\n\\|J^s(\\overline{u}_{II}\\cdot\\nabla \\omega^{\\alpha_{0}})\\|_{L^2}\\|\\overline{\\omega}\\|_{H^s}\n\\\\&\\leq\\|\\overline{\\omega}\\|_{H^s}\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}\\|J^s\\overline{u}_{II}\\|_{L^p}.\n\\end{split}\\end{equation*}\nApplying Proposition \\ref{add1} gives\n\\begin{equation*}\\begin{split}\n|I_{21}|=&\\Big|\\int_{\\R^2}\\big(J^s(\\overline{u}_I\\cdot\\nabla \\overline{\\omega})-\\overline{u}_I\\cdot\\nabla J^s\\overline{\\omega}\\big)J^s\\overline{\\omega}\\,\\mathrm{d}x\\Big|\\\\ \\leq&\\|J^s(\\overline{u}_I\\cdot \\nabla \\overline{\\omega})-\\overline{u}_I\\cdot J^s \\nabla \\overline{\\omega}\\|_{L^2}\\|\\overline{\\omega}\\|_{H^s}\n \\leq \\|\\overline{\\omega}\\|^3_{H^s}.\n\\end{split}\n\\end{equation*}\nBy Proposition \\ref{add-0}, one has\n\\begin{equation*}\\begin{split}\n|I_{31}|&\\leq\n\\|J^s(\\overline{u}_I\\cdot\\nabla \\omega^{\\alpha_{0}})\\|_{L^2}\\|\\overline{\\omega}\\|_{H^s}\n\\\\&\\leq\\|\\overline{\\omega}\\|_{H^s}^2\\|\\omega^{\\alpha_0}\\|_{H^{s+2}}.\n\\end{split}\\end{equation*}\nInserting the estimates of $I_{21},\\,I_{22}, \\,I_{31}$ and $I_{32}$ into \\eqref{I_2}, we arrive at\n\\begin{equation}\\label{11-3}\\begin{split}\n&|I_2|+|I_3|\\\\\n\\leq& C\\left(\\|\\overline{\\omega}\\|_{H^s}^2\\|J^s\\overline{u}_{II}\\|_{L^p}+\n\\|\\overline{\\omega}\\|_{H^s}\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}\\|J^s\\overline{u}_{II}\\|_{L^p}+\n\\|\\overline{\\omega}\\|^3_{H^s}+\\|\\overline{\\omega}\\|_{H^s}^2\\|\\omega^{\\alpha_0}\\|_{H^{s+2}}\\right).\n\\end{split}\\end{equation}\nIn view of \\eqref{11-1}, \\eqref{11-2} and \\eqref{11-3}, it deduces\n\\begin{equation}\\label{th3-proof}\\begin{split}\n \\frac{\\mathrm{d}}{\\mathrm{d}t}\\|\\overline{\\omega}(t)\\|_{H^s} \\leq&\\|\\overline{\\omega}\\|_{H^s}(\\|u^{\\alpha_{0}}\\|_{H^{s}}+\\|\\omega^{\\alpha_0}\\|_{H^{s+2}})\n+\\|\\overline{\\omega}\\|^2_{H^s}\\\\\n&+\\|\\overline{\\omega}\\|_{H^s}\\|J^s\\overline{u}_{II}\\|_{L^p}\n+\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}\\|J^s\\overline{u}_{II}\\|_{L^p}.\n\\end{split}\\end{equation}\n\nNow we estimate $\\|J^s\\overline{u}_{II}\\|_{L^p}$. Similar to the proof of Theorem \\ref{th2}, we use the integral form \\eqref{11-6}.\nNote that\n\\begin{equation*}\\begin{split}\nH_{1}&=\\int_{|x-y|\\leq\\epsilon}\\Big(\\frac{(x-y)^{\\perp}}{|x-y|^{2+2\\alpha}}-\\frac{(x-y)^{\\perp}}{|x-y|^{3}}\\Big)\\Big(J^s\\omega^{\\alpha_0}(y)-J^s\\omega^{\\alpha_0}(x)\\Big)\\,\\mathrm{d}y.\n\\end{split}\n\\end{equation*}\nBy the mean value formula and the H\\\"{o}lder inequality, we obtain\n\\begin{equation*}\\begin{split}\n\\|H_{1}\\|_{L^p}&\\leq C\\Big(\\frac{1}{2-2\\alpha}\\epsilon^{2-2\\alpha}+\\epsilon\\Big)\\|J^{s+1}\\omega^{\\alpha_0}\\|_{L^p}\n\\\\&\\leq C\\Big(\\frac{1}{2-2\\alpha}\\epsilon^{2-2\\alpha}+\\epsilon\\Big)\\|\\omega^{\\alpha_0}\\|_{H^{s+2}},\n\\end{split}\n\\end{equation*}\nwhere Lemma \\ref{embedding1} is used in the last inequality.\nSimilarly, for $2+2\\alpha\\leq\\gamma\\leq3,$\n\\begin{equation*}\\begin{split}\n\\|H_{2}\\|_{L^p}&= \\Big(\\frac12-\\alpha\\Big)\\Big\\|\\int_{1>|x-y|\\geq\\epsilon}\\frac{(x-y)^{\\perp}(|x-y|^{\\gamma}\\log|x-y|)}{|x-y|^{2+2\\alpha}|x-y|^{2+2\\alpha_0}}\nJ^s\\omega^{\\alpha_0}(y)\\,\\mathrm{d}y\\Big\\|_{L^p}\n\\\\&\\leq \\Big(\\frac12-\\alpha\\Big)\\Big\\|\\int_{1>|x-y|\\geq\\epsilon}\\frac{|\\log|x-y||}{|x-y|^{2}}\n|J^s\\omega^{\\alpha_0}(y)|\\,\\mathrm{d}y\\Big\\|_{L^p}\\\\&\\leq\\Big(\\frac12-\\alpha\\Big)|\\log\\epsilon|^2\\|J^s\\omega^{\\alpha_0}\\|_{L^p}\n\\\\&\\leq\\Big(\\frac12-\\alpha\\Big)|\\log\\epsilon|^2\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}.\n\\end{split}\n\\end{equation*}\nNote that $\\ln |x-y|\\le C|x-y|^\\sigma$ for $|x-y|\\ge 1$ and any $\\sigma>0$, where $C$ may depend on $\\sigma$. We can apply the mean value formula and the Young inequality to obtain\n\\begin{equation*}\\begin{split}\n\\|H_{3}\\|_{L^p}&\\leq\\Big (\\frac12-\\alpha\\Big)\\Big\\|\\int_{|x-y|\\geq1}\\frac{|\\log|x-y||}{|x-y|^{1+2\\alpha}}\n|J^s\\omega^{\\alpha_0}(y)|\\,\\mathrm{d}y\\Big\\|_{L^p}\\\\&\\leq\\Big(\\frac12-\\alpha\\Big)\\|J^s\\omega^{\\alpha_0}\\|_{L^r}\n\\Big(\\int_{|x-y|\\geq1}\\Big(\\frac{|\\log|x-y||}{|x-y|^{1+2\\alpha}}\\Big)^q\\,\\mathrm{d}y\\Big)^{\\frac1q}\n\\\\&\\leq\\Big(\\frac12-\\alpha\\Big)\\|J^s\\omega^{\\alpha_0}\\|_{L^r}\\Big(\\frac{1}{(1+2\\alpha-\\sigma)q-2}\\Big)^{\\frac1q},\n\\end{split}\n\\end{equation*}\nwhere $\\frac1p+1=\\frac1r+\\frac1q,$ $q>\\frac{2}{1+2\\alpha-\\sigma}$, $p>\\frac{2}{2\\alpha-\\sigma}$, and we can choose some $r>2$ such that the following embedding\n\\begin{equation*}\nH^{s+1}(\\R^2)\\hookrightarrow W^{s,r}(\\R^2)\n\\end{equation*}\nholds (see Lemma \\ref{embedding1}).\nConsequently,\n\\begin{equation}\\label{11-5}\\begin{split}\n \\|J^s\\overline{u}_{II}(t)\\|_{L^p}\\leq &\\|H_{1}\\|_{L^p}+\\|H_{2}\\|_{L^p}+\\|H_{3}\\|_{L^p}\n \\\\\\leq & C\\Big(\\frac{1}{2-2\\alpha}\\epsilon^{2-2\\alpha}+\\epsilon\\Big)\\|\\omega^{\\alpha_0}\\|_{H^{s+2}}\n\\\\\n&+\\Big(\\frac12-\\alpha\\Big)|\\log\\epsilon|^2\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}+\\Big(\\frac12-\\alpha\\Big)\\|\\omega^{\\alpha_0}\\|_{H^{s+1}}.\n\\end{split}\n\\end{equation}\nLet $\\epsilon=\\frac12-\\alpha$. The estimate \\eqref{11-5} combined with \\eqref{th3-proof} yields\n\\begin{equation*}\\begin{split}\n\\frac{\\mathrm{d}}{\\mathrm{d}t}\\|\\overline{\\omega}(t)\\|_{H^s} \\leq &\\|\\overline{\\omega}\\|_{H^s}\\big(\\|u^{\\alpha_{0}}\\|_{H^{s}}\n+\\|\\omega^{\\alpha_0}\\|_{H^{s+2}}\\big)\\\\\n&+\\|\\overline{\\omega}\\|^2_{H^s}+\\|\\omega^{\\alpha_0}\\|_{H^{s+2}}^2\\left(\\Big(\\frac12-\\alpha\\Big)+\\Big(\\frac12-\\alpha\\Big)\\log^2\\Big(\\frac12-\\alpha\\Big)\\right).\n\\end{split}\\end{equation*}\nArguing similarly as the last part in the proof of Theorem \\ref{th1}, we obtain\n\\begin{equation*}\n\\|\\overline{\\omega}(t)\\|_{H^s}\\leq C\\left(\\Big(\\frac12-\\alpha\\Big)+\\Big(\\frac12-\\alpha)\\log^2\\Big(\\frac12-\\alpha\\Big)\\right).\n\\end{equation*}\nMoreover, we can prove that $\\omega^{\\alpha}\\in C([0,T]; H^{s+2})$ for any $t\\in [0,T]$. The proof of the theorem is finished.\n\n\\subsection{Proof of Corollary \\ref{Cor1+}}\n\nSuppose that the result is not true. Then there exists a $M>0$ and a $\\delta_0>$ such that $T^*_\\alpha\\le M$ for all $\\alpha\\in (0,\\delta_0)$. But it is known that for any $T>0$, the smooth solution of the Euler equations exists on $[0,T]$. Take $T=M+1$. According to Theorem \\ref{th1}, there exists a $0<\\delta\\le \\delta_0$ depending on $T$ such that the smooth solution of the generalized SQG exists on $[0,T]$ as well. This contradicts with the assumption that the maximal existence time $T^*_\\alpha\\le M$. The proof of the corollary is complete.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{The Galactic Center}\nThe DSO source approaching the super-massive black hole SgrA* at the center of the Milky Way \nhas spawned great activities in observing that region covering the \nentire electromagnetic spectrum from radio, via infrared to X-ray wavelengths using \ntelescopes across the world.\nThe upcoming events underline the importance of the Galactic Center as a laboratory to \ninvestigate and understand phenomena in the immediate environment of super massive black holes\n(Eisenhauer 2010, Ghez 2009).\nGas and stars within the Galactic Center stellar cluster provide the fuel\nfor the central super-massive black hole and hence the reason for most of the\nflux density variability observed from it.\n\n\n\\subsection{The variability of Sagittarius A*}\nProgress has been made in the understanding of the emission process \nassociated with the immediate surroundings of the super-massive black hole counterpart SgrA* \nas well as the three-dimensional dynamics and the population of the central stellar cluster.\nThere is also ample evidence of interactions between the cluster and SgrA*.\n\nSeries of monitoring observations in the near-infrared (NIR), X-ray, and sub-millimeter (sub-mm) regimes\naccumulated over the years allowed us to perform for the first time detailed\nstatistical studies of the variability of SgrA*\n(Witzel et al. 2012; Eckart et al. 2012).\nThe analyses show that the histogram of the near-infrared flux density is a\npure power-law and the emission process is most likely dominated by \nsynchrotron radiation. \nIn Witzel et al. (2012), we present a comprehensive data description for \nNIR measurements of SgrA*. \nWe characterized the statistical properties of the variability of SgrA* in the near-infrared, which we found to be \nconsistent with a single-state process forming a flux density power-law distribution. \nWe discovered a linear rms-flux relation for the flux density range up to 12 mJy on a time scale of 24 minutes. \nThis and the structure function of the flux density variations imply a phenomenological, formally nonlinear statistical \nbehavior that can be modeled. In this way, we can simulate the observed variability and extrapolate its behavior to higher \nflux levels and longer time scales. \nSgrA* is also strongly variable in the X-ray domain\n(Baganoff et al. 2003, Baganoff et al. 2001, Porquet et al. 2008, Porquet et al. 2003, Eckart et al. 2012 \nand references there in, as well as, Nowak et al. 2012, Barriere et al. 2014 for a recent strong flares observed with\nChandra and NuSTAR).\nThe detailed statistical investigation by Witzel et al. (2012) also suggests that the past strong X-ray variations\nthat give rise to the observed X-ray echos can in fact be explained by the NIR variability histogram under the \nassumption of a Synchrotron Self Compton (SSC) process. \nSgrA* is extremely faint in the X-ray bands, \nthough strong activity has been revealed through the detection of flares. \nTherefore, SgrA* is the ideal target to investigate the \nmass accretion and the ejection physics in the case of an extremely low \naccretion rate onto a super-massive black hole. This is actually the phase\nin which super-massive black holes are thought to spend most of their lifetime.\nThe activity phase onset of a magnetar (see section below) at a separation of only \nabout 3 arcseconds from the Galactic Center presented a problem \nfor the SgrA* monitoring program in 2013\n(Mori et al. 2013, Shannon et al. 2013, Rea et al. 2013).\n\n\nEckart, et al. (2012) present simultaneous observations and modeling \nof the millimeter (mm), NIR, and X-ray flare emission of SgrA*.\nThese data allowed us to investigate physical processes giving rise to the variable emission \nof SgrA* from the radio to the X-ray domain.\nIn the radio cm-regime SgrA* is hardy linearly polarized but shows a fractional circular polarization\nof around 0.4\\% (Bower, Falcke \\& Backer 1999, Bower et al. 1999).\nThe circular polarization decreases towards the mm-domain (Bower 2003), where as\nMacquart et al. (2006) report variable linear polarization from SgrA* of a few percent in the mm-wavelength domain.\nThe observations reveal flaring activity in all wavelength bands.\nThe polarization degree and angle in the sub-mm are likely linked to the magnetic field structure \nor the general orientation of the source.\nIn general - the NIR emission is leading the sub-mm with a delay of about one to two hours (see below)\nand the excursions in the NIR and X-ray emission are rather simultaneous.\nAs a result we found that the observations can be modeled as the signal from an \nadiabatically expanding source component\n(Eckart et al. 2008b, Yusef-Zadeh et al. 2006, Eckart et al. 2006b).\nof relativistic electrons emitting via the synchrotron\/SSC process.\nA large fraction of the lower energy mm\/cm- flux density excursions is not necessarily correlated \nwith the NIR\/X-ray variability\n(e.g. Dexter \\& Fragile 2013, Dexter et al. 2013, and details and further references Eckart et al. 2012).\nOne may compute the SSC spectrum produced by up-scattering\nof a power-law distribution of\nsub-mm-wavelength photons into the NIR and X-ray domain by using the\nformalism given by Marscher (1983) and Gould (1979).\nSuch a single SSC component model may be too simplistic,\nalthough it is considered\nas a possibility in most of the recent modeling approaches.\nIt does not take into account possible deviations from the\noverall spectral index of $\\alpha$=1.3\nat any specific wavelength domain like the NIR or X-ray regime.\n\n\nThe number density distribution of the relativistic electrons responsible for the \nsynchrotron spectrum can be described by \n\\begin{equation}\\label{Equation:Gamma}\nN(E) = N_0 E^{-2 \\alpha +1}~~~,\n\\end{equation}\nwith $\\gamma_e$ between $\\gamma_1$ and $\\gamma_2$\nwhich limit the lower and upper bound of the relativistic electron spectrum\n\\begin{equation}\\label{Equation:spectrum}\n\\gamma_1 mc^2 < E= \\gamma_e m c^2 < \\gamma_2 m c^2~~~.\n\\end{equation}\nLorentz factors $\\gamma_e$ for the emitting electrons of the order of\na few thousand are required to produce a sufficient SSC flux in the\nobserved X-ray domain. In addition the relativistic bulk motion of \nthe orbiting or outward traveling component\nis described by the bulk Lorentz factor $\\Gamma$ (see Fig.~\\ref{Fig:models}).\nOn the left of this figure we show a sketch of a typical relativistic electron \ndensity distribution resulting from MHD calculations\n(e.g. Dexter et al. 2010, Dexter \\& Fragile 2013, Moscibrodzka \\& Falcke 2013). We show a cut through only \none side of the three dimensional structure. The central plane and outflow region \nresulting from these calculations can be modeled in different, dedicated approaches (top and bottom right).\n\nIn the central plane modeling the flux variations are assumed to be the result of the motion of an orbiting blob\nor a hot spot (e.g. Eckart et al. 2006a).\nIn the outflow model the flux variations are assumed to be due\nto the ejection of a blob and its motion along the jet \n(with bulk motions close to the speed of light) or \na much slower overall outflow component. \nIn this case - for VLBI observations - the larger outflow extent at increasingly lower radio frequencies would be\nhidden by the decreasingly lower angular resolutions due to interstellar scattering.\nIn Fig.\\ref{Fig:diskmodel} the accretion disk (here assumed to be edge-on) is shown as a vertical thick line \nto the right, the dashed part indicates the disk sections in the back- and foreground.\nExtending to the left we show one side above the disk.\nHere higher energy flare emission (lower part) is assumed to be responsible for the observed\nNIR\/X-ray flare emission.\nLower energy flare emission (upper part) may substantially contribute to long wavelength\ninfrared emission.\nIn addition to the expansion towards and beyond the the mm-source size, radial and azimuthal\nexpansion within the disk may occur.\nHence, long mm\/cm-wavelength variability may originate from different source components of SgrA*\nand may be difficult to be disentangled based on radio data alone.\n\n\\begin{figure}[!ht]\n \\centering\n \\includegraphics[width=0.99\\textwidth]{diskmodel.eps}\n \\caption{\\small\nSketch of a possible source structure for the accretion disk around the SMBH\nassociated with SgrA* following Fig.12 in Eckart et al. (2008a).\n}\n \\label{Fig:diskmodel} \n\\end{figure}\n\nIn Fig.\\ref{Fig:models}\nthe relativistic boosting vectors for the electrons $\\gamma_e$ and the bulk motion $\\Gamma_{bulk}$ are not \ndrawn to a proper relative scale. One can assume $\\gamma_e$ $\\ge$ $\\Gamma_{bulk}$.\nIn the case of relativistically orbiting gas as well as relativistic outflows\none may use modest values for $\\Gamma$.\nBoth dynamical phenomena are likely to be relevant in the case of SgrA*.\nThe size of the central plane synchrotron component is assumed to be of the\norder of or at most a few times the Schwarzschild radius.\nFrom the overall variable radio\/sub-mm spectrum spectrum we can assume a turnover frequency $\\nu_m$ of a few \n100~GHz (see details in Eckart et al. 2012).\nThe motion of the synchrotron emitting cloud can be described via\n\n\\begin{equation}\\label{Equation:delta}\n\\delta=\\Gamma_{bulk}^{-1}(1-\\beta cos \\phi )^{-1}~~,\n\\end{equation}\n\\begin{equation}\\label{Equation:Gamma}\n\\Gamma_{bulk}= (1-\\beta^2)^{-1\/2}~~.\n\\end{equation}\nHere $\\beta= v\/c $ and $v$ is the speed of the bulk motion of the synchrotron cloud,\n$\\delta$ is the Doppler factor and $\\phi$ the angle to the line of sight.\nFor a relativistic bulk motion with $\\Gamma_{bulk}$ around \n 1.7$\\pm$0.3 (i.e. angles $\\phi$ of about $30^{\\circ} \\pm 15^{\\circ}$) the corresponding magnetic field\nstrengths are often assumed to be of the order of a few ten Gauss,\nwhich is also within the range of magnetic fields expected for RIAF models\n(e.g. Yuan et al. 2006, Narayan et al. 1998).\n\n\nModeling of the light curves shows that (at least for the brighter events) \ntypically the sub-mm flux density excursions follow the NIR emission with a delay of \nabout one to two hours with an expansion velocity of about 0.01c-0.001c\n(Eckart et al. 2008b, Yusef-Zadeh et al. 2006, Eckart et al. 2006b).\nWe find source component sizes of around one Schwarzschild radius, flux densities of a few Janskys, \nand steep spectral indices. \nTypical model parameters suggest that either the adiabatically expanding source components have \na bulk motion larger than its expansion velocity \nor the expanding material contributes to a corona or disk, \nconfined to the immediate surroundings of SgrA*. \nFor the bulk of the synchrotron and SSC models, we find synchrotron turnover frequencies in the \nrange of 300-400 GHz. For the pure synchrotron models, this results in densities of relativistic \nparticles in the mid-plane of the assumed accretion flow of the order of 10$^{6.5}$ cm$^{-3}$, \nand for the SSC models the median densities are about \none order of magnitude higher. However, to obtain a realistic description of the frequency-dependent \nvariability amplitude of SgrA*, models with higher turnover frequencies and even higher \ndensities are required. \nThis modeling approach also successfully reproduces the degree of flux density variability across \nthe radio to far-infrared spectrum of SgrA*. \nIn Fig.~\\ref{Fig:spectrum} we show observed flux densities of SgrA* taken from the \nliterature (blue) compared to a combined\nmodel that consists of the fit given by Falcke et al. (2000), Marrone et al. (2008) (black line), \nand Dexter et al. (2010) (black dashed line). We plotted in red the spectra of synchrotron self-absorption frequencies for\nthe range of models. Here we show results for the preferred synchrotron plus SSC (SYN-SSC) model\nthat most closely represents the observed variability of SgrA*.\n\nValencia-S. et al. (2012) present theoretical polarimetric light curves expected in the \ncase of optically thin NIR emission from over-dense regions close to the marginal stable orbit\n(see also Broderick et al. 2005, Eckart et al. 2006a, Zamaninasab et al. 2010, 2011).\nUsing a numerical code the authors track the time evolution of detectable polarization \nproperties produced by synchrotron emission of compact sources in the vicinity of the black hole. \nThey show that the different setups lead to very special patterns in the time-profiles \nof polarized flux and the orientation of the polarization vector. As such, they may be used \nfor determining the geometry of the accretion flow around SgrA* \n(see also Karas et al. 2011, Zamaninasab, et al. 2011).\n\nDuring the 2013 Bad Honnef and the Granada conference (see Acknowledgments) \nefforts to monitor SgrA* during the DSO fly-by and first observational results from 2013 were reported by\nAkiyama, et al. (2013ab), Eckart et al. (2013abc), Jalali et al. (2013), Meyer et al. (2013), Phifer et al. (2013).\nThe NRAO Karl G. Jansky Very Large Array (VLA) is undertaking an ongoing community \nservice observing program to follow the expected encounter of the DSO \ncloud with the black hole SgrA* in 2013\/14 (Chandler \\& Sjouwerman 2013).\nThe NRAO VLA has been observing the Sgr~A region since \nOctober 2012 on roughly a bi-monthly interval, \ncycling through eight observing bands.\nFor monitoring the flux densities and in particular the radio spectral indices \nthe short wavelength observations ($\\lambda$$<$6cm) are most useful. \nFor 2012\/13 no particular flux density variation \nwas detected that could be attributed to the interaction between SgrA* and the DSO. \nThis may be linked with the fact that the newly determined periapse passage \nis now expected to happen in April\/May of 2014 (Phifer et al. 2013), i.e. later than originally anticipated. \nHowever, in the radio-shock\nframe in which variations of up to several Janskys were expected even during the pre-periapse time. \nHence, the lack of strong radio flares \nindicates that the medium is less dense than expected and\/or that the bow-shock size i.e. the\ncross-section of the dust source is much smaller than assumed\n(Narayan, et al. 2013, Crumley, et al. 2013, Sadowski, et al. 2013, Yusef-Zadeh, et al. 2013, Shcherbakov, et al. 2014).\n\n\\begin{figure}[!ht]\n \\centering\n \\includegraphics[width=0.99\\textwidth]{specgraph1.eps}\n \\caption{\\small\nThe variable radio spectrum of SgrA*: Measurements and model results\n(see text and Eckart et al. 2012 for details).\n}\n \\label{Fig:spectrum} \n\\end{figure}\n\n\n\n\\subsection{VLBI imaging of SgrA*}\nThere is also profound progress in imaging and modeling of the central putative accretion disk\nof SgrA* as well as the jet that may be associated with the source\n(Falcke \\& Markoff, 2013, Moscibrodzka \\& Falcke, 2013, Valencia-S, M. et al. 2012).\nIn fact imaging of SgrA* may turn out to be a Rosetta Stone in the attempts of distinguishing between \ndifferent relativity theories of black holes\n(e.g. Boller \\& M\\\"uller, 2013, on astronomical tests of general relativity and the\n pseudo-complex theory).\n\nVLBI (Very Long Baseline Interferometry) observations at very short millimeter \nradio wavelengths can overcome the effects of interstellar scattering \nand allow us to study the source intrinsic structure of SgrA*.\nLarge mm\/sub-mm facilities like the \nVLBA (Very Long Baseline Array), \nVERA (VLBI Exploration of Radio Astrometry), \nALMA (Atacama Large Millimeter Array), \nPdBI (Plateau de Bure Interferometer) and the sensitive mm-telescopes in the\nEVN (European VLBI Network) - such as the IRAM 30m and the 100m Effelsberg telescopes - are participating in this effort,\nwhich will eventually culminate in the project EHT (Event Horizon Telescope), \na VLBI array especially designed to image the structures close to the\nevent horizons of the larges SMBHs in the sky - namely SgrA* and M87,\nwith 1 Schwarzschild radius extending to an angular size of about 10$\\mu$as and 3.7$\\mu$as, respectively.\nMulti-epoch imaging observations will allow to constrain the locations and sizes of the flaring region of SgrA* \nwithin the putative temporal accretion disk of the accretion stream\/flow towards or an accretion wind from SgrA*.\nThese measurements will also constrain the acceleration processes (e.g. magnetic reconnection events or \nnon-axisymmetric standing shocks) that give rise to the population of relativistic electrons and the variable \nemission we see from SgrA*.\nUltimatly, alternative black hole models will be probed and attempts to test the black hole no-hair theorem \nwill be possible with the new VLB mm\/sub-mm facilities\n(Broderick, et al. 2014, Fish et al. 2014, Akiyama, et al. 2013ab, Huang, et al. 2012, Broderick, et al. 2011,\nBroderick, et al. 2011, Fish et al. 2011, Lu, R.-S., et al. 2011).\n\nThese VLBI experiments will eventually enable spatially resolved studies on sub-horizon scales, leading \nto an unprecedented exploration of a putative predicted black-hole shadow \n(e.g. Huang et al. 2007)\nas an evidence for light trapping by the black hole as well as its interaction with \nthe surrounding material. \nIt will be possible to monitor the possible expansion of source components \nduring flare activity.\nFurthermore, when a rotating black hole is\n immersed in a magnetic field of external origin, the gravito-magnetic \ninteraction is capable of triggering the magnetic reconnection, \naccelerating the particles to very high energy \n(Karas et al. 2012, 2013; Morozova et al. 2014). \nThis frame-dragging phenomenon is particularly interesting in the \ncontext of exploring the strong-gravity effects in astrophysical black holes \nbecause the effect does not have a Newtonian counterpart and it operates \non the border of the ergospheric region (Koide \\& Arai 2008), \ni.e. very close to the black hole horizon, and it can be probed with \nthe future EHT. Also, one can investigate if the black hole proximity generates \nconditions favourable to incite the magnetic reconnection that eventually \nleads to plasma heating and particle acceleration. \nThis effect could contribute to the flaring activity.\n\n\n\\subsection{The importance of dusty sources close to the center}\nA major discussion point is if and how the DSO source will be disrupted during\nits peri-bothron\\footnote{Peri- or apo-bothron is the term used for \nperi- or apoapsis - i.e. closest or furthest separation - for an elliptical orbit with a black hole present \nat one of the foci. \nAs already mentioned by Frank and Rees (1976) word 'bothros' was apparently first suggested in the context of \nblack holes by W.R. Stoeger.\nIt originates from the greek word \n\\`o $\\beta$\\'o$\\theta \\rho o \\varsigma$ \nwith the equivalent meaning of 'the sink' or 'the deep dark pit'.\n} passage.\nIt may be only its dusty envelope that will be disrupted\nsince the K$_s$-band identifications\nof the source suggest that it can also be associated with a star (Eckart et al. 2013a).\nIn addition to the VLT NACO and the Keck NIRC detections of the \nDSO NIR continuum emission (Eckart et al. 2013bc),\nhere we show the detection of the DSO continuum at about K$\\sim$19 using SINFONI data (Fig.~\\ref{Fig:DSO}).\nThe detection of the continuum emission in data sets taken with three different \ninstrumental setups over many years strengthens the case for a substantial\ncontinuum emission from that dusty source.\nAs posted in the astronomer's telegram No.6110 on 2 May 2014 (Ghez et al. 2014), the DSO was detected \n 3.8$\\mu$m during its peri-bothron passage around the central black hole SgrA*. Hence, it appears to be intact\nand up to this point not yet heavily affected by tidal effects. \nThis clearly supports our finding (Eckart et al. 2013bc) that it may very well be a dusty star\nrather than a pure gas and dust cloud.\n\nIn contrast to a pure dust and gas nature of the DSO its possible \nstellar (i.e. a dust enshrouded star) nature is discussed and partially favoured in\nMeyer et al. (2013, 2014), Eckart et al. (2013abc), Scoville \\& Burkert (2013)\nBallone et al. (2013), Phifer et al. (2013).\nEckart et al. (2013a) investigate the possible mass transfer across \nLagrange point L1 in a simple Roche model.\nIf the star has a mass of about 1M$_{\\odot}$, the separation of $L1$ from it will be\nabout 0.1~AU. For a Herbig Ae\/Be stars with 2-8~M$_{\\odot}$ ~that distance will be between\n0.2 and 0.5~AU. For a typical S-cluster stellar mass of $\\sim$20-30~M$_{\\odot}$ ~the separation \nwill be closer to one AU. \nThe interferometrically determined inner ring sizes that one typically finds for young Herbig Ae\/Be and\nT~Tauri stars can indeed be as small as 0.1-1~AU (Monnier \\& Millan-Gabet 2002).\nAny stellar disk or shell may already have been stripped \nsubstantially if the DSO has performed more than a single orbit.\nIf the source has a size of about 1~AU \n(as determined from its MIR-luminosity; Gillessen et al. 2012a) \nthen a significant amount of the dusty circumstellar material\nmay pass beyond $L1$ during peri-bothron passage. \nThis material will then start to move into the Roche lobe associated with SgrA*. \n\n\n\nHowever, it is not at all clear what will happen to the transferred material after \nthe peri-bothron passage around May 2014 or beyond. \nThe fact that this dusty object may be a dust enshrouded star rather than a dust \ncloud will have an influence on the expected flux density variations resulting \nfrom the close approach. They may be much weaker than expected.\nSimulations (e.g. Burkert et al. 2012, Schartmann et al. 2012,\nsee also Zajacek, Karas \\& Eckart 2014) that\nhave discussed the feeding rate of SgrA* as a function of radius\nindicate that a portion of the material may fall towards SgrA*.\nIf SgrA* is associated with a significant wind on scale of the peri-bothron separation, \nthen a large part of the material may be blown away again by an out-bound accretion wind.\nShcherbakov \\& Baganoff (2010) have discussed the feeding rate of SgrA* as a function of radius.\nBased on their modeling one may suggest that the bow-shock sources X3 and X7 (Muzic et al. 2010) are \nstill in the regime in which most of the in-flowing mass is blown away again.\nAnother case for comparison is the star S2. During its peri-bothron passage the star has been well \nwithin the zone in which matter of its (weak) stellar wind could have been accreted by SgrA*. \nThe DSO peri-bothron will be at a larger radius than that of S2 (Phifer et al. 2013). \nThis may imply that \nno enhanced accretion effect will result from it during the peri-bothron passage. \nUntil May 2014 no increase in variability and no significant flux density increase well above normal levels \nhas been reported in the radio to X-ray domains.\n\n\nThe fate of the DSO and the cometary sources X3 and X7 underline the importance\nof investigating the wind properties in the vicinity of SgrA* in more detail.\nIRS~8 is a unique possibility to study the bow shock properties and polarization \nfeatures in the dusty environment at the Galactic Center.\nBased on a detailed study of near-infrared emission \nRauch et al. (2013) present interstellar dust properties for \nthe northern arm in the vicinity of the IRS~8 bow shock. \nThis study allowed us for the first time to determine the relative positioning of \nIRS~8 with respect to the northern arm and the \nsuper-massive black hole SgrA*. The result indicates that the\ncentral star of IRS~8 is in fact located closer towards the observer than the northern arm.\nIn Eckart et al. (2013a) we investigated the near-infrared \nproper motions and spectra of infrared excess sources at the Galactic Center. \nThe work concentrated on a small but dense cluster of comoving \nsources (IRS13N) located ~3'' west of SgrA*. Our analysis shows that \nthese stars are spectroscopically and dynamically young and can indeed be \nidentified with continuum emission at 2 microns and shortward, indicating that these mid-infrared \nsources are not only dust sources but young stars.\nThe possibility of ongoing star formation at the Galactic Center is \nsupported through simulations by Jalali et al. 2014 (submitted) and Jalali et al. (2013).\nIn fact the DSO may be a representative of dusty sources similar to those\ndiscussed in Eckart et al. (2006b; see their Fig.14;\ncompare also to the discussion of sources X3 and X7 given by Muzic et al. 2010).\nMeyer et al. (2014) present NIR spectroscopic data of several of these sources.\nThey also show that the DOS does not seem to be unique, since several red emission-line objects \ncan be found in the central arcsecond. \nIn summary, Meyer et al. (2014) conclude that it seems more likely that G2 is ultimately a\nstellar source that is clearly associated with gas and dust (see also Eckart et al. 2013abc).\n\n\n \\begin{figure}[!ht]\n \\centering\n \\includegraphics[width=0.99\\textwidth]{DSOsumme.eps}\n \\caption{\\small\nThe DSO detected in its K-band continuum emission in 2010 SINFONI data.\nLeft: The original image (positive greyscale); \nRight: A LUCY deconvolved image (negative greyscale) shown at an \nangular resolution close to the diffraction limit of the VLT UT4.\n}\n \\label{Fig:DSO} \n\\end{figure}\n\n\n\n\\subsection{Stellar dynamics and tests of relativity}\nSgrA*, the super-massive black hole at the center of the Milky Way, \nis surrounded by a small cluster of high velocity stars, known as the S-stars\n(Eckart \\& Genzel 1997).\nSabha et al. (2012) aimed at constraining the amount and nature of the stellar \nand dark mass that is associated with the cluster in the immediate vicinity of SgrA*. \nThe authors use near-infrared imaging to determine the Ks-band luminosity function \nof the S-star cluster members, the distribution of the diffuse background \nemission and the stellar number density counts around the central black hole. \nThis allows us to determine the stellar light and mass contribution expected \nfrom the faint members of the cluster. \nSabha et al. (2012) then use post-Newtonian N-body simulations to investigate the effect of stellar \nperturbations on the motion of S2, as a means of detecting the number and masses \nof the perturbers. The authors find that the stellar mass derived from the Ks-band \nluminosity extrapolation is much smaller than the amount of mass that might \nbe present considering the uncertainties in the orbital motion of the star S2. \nAlso the amount of light from the fainter S-cluster members is below the \namount of the residual light at the position of the S-star cluster after one removes \nthe bright cluster members. If the distribution of stars and stellar remnants \nis peaked near SgrA* strongly enough, observed changes in the orbital \nelements of S2 can be used to constrain both the masses and the number of objects inside its orbit. \nBased on simulations of the cluster of high velocity stars we find that in\nthe NIR K-band - close to the confusion level for 8 m class telescopes -\nblended stars will occur preferentially near the position of SgrA*\nwhich is the direction towards which we find the highest stellar density.\nThese blended stars consist of several faint, (with the current facilities) \nindividually undetectable stars that get aligned along the line-of-sight, \nproducing the visual effect of a new point source. \nThe proper motion of stars and the corresponding velocity dispersion\nleads to the fact that such a blended star configuration dissolves typically after 3 years.\n\n\nStars that get very close to the super massive black hole are ideal probes to\nanalyse the gravitational field and to search for effects of relativity due \nto the presence of the high mass concentration and its effect on space time.\nThis can be done by tracing the orbit of stars through proper motions and radial velocities.\nAs discussed in Zucker et al. (2006) relativistic effects should express themselves \nspectroscopically. The redshift $z$ of a black hole orbiting star can be written as:\n\\begin{equation}\\label{eq:aa1}\nz = \\Delta\\lambda\/\\lambda = B_0 + B_1\\beta + B_2\\beta^2 + O(\\beta^3)\n\\end{equation}\nwith $B_0$ being an offset,\n$B_1\\beta$ describing the Doppler velocity \nand $B_2\\beta^2$ expressing the relativistic effects.\nHere the value $B_2$ contains equal contributions from the gravitational redshift and the \nspecial relativistic transverse Doppler effect.\nThe combined effect gives a redshift that is about an order of magnitude larger than\nthe currently achieved spectral resolution of $\\delta\\lambda\/\\lambda$$\\sim$10$^{-4}$.\nFor S2 one expects about a 150-200 km\/s signal measurable over a few months \non top of an orbit-depending radial velocity of more than 4000 km\/s.\nExpectations are high that this will be observable during the next peri-bothron\nfor S2 around 2017.9$\\pm$0.35 (Gillessen et al. 2009b, Eisenhauer et al. 2003) \nor S2-102 around 2021.0$\\pm$0.3 (Meyer et al. 2013).\nRealistically, however, one needs several stars on different orbits \nto detect the relativistic effect with certainty (Zucker et al. 2006; see also \nRubilar \\& Eckart 2001 for peri-bothron shift).\nAlternatively, one has to find stars that are (or get) closer than S2 \nand S2-102 (Meyer et al. 2013; see below) to SgrA*.\n\nDetailed imaging and the analysis of proper motions may be another way to trace relativistic effects.\nAn important deviation from Keplerian motion occurs as a result of\nrelativistic corrections to the equations of motion, which to the lowest\norder predict a certain advance of the argument of \nperi-bothron each orbital period.\nChoosing $a = 5.0$~mpc and $e= 0.88$ for the semi-major axis and eccentricity \nof S2, respectively, and assuming a black hole mass of $M_\\bullet=4.0\\times 10^6 $~M$_\\odot$ this advance will be\n\\begin{equation}\\label{Equation:DomegaGR}\n\\left(\\Delta\\omega\\right)_\\mathrm{GR} \n= \\frac{6\\pi GM_\\bullet}{c^2a(1-e^2)} \\approx 10.8^\\prime.\n\\end{equation}\nThe relativistic precession is prograde, and leaves the orientation of the orbital\nplane unchanged.\n\nThe location of the peri-bothron advances for each orbital period\ndue to the spherically-symmetric component of the distributed mass that \nis resolved by the elliptical orbit of the star.\nThe amplitude of this Newtonian ``mass precession'' is\n\\begin{equation}\\label{eq:nuM}\n\\left(\\Delta\\omega\\right)_\\mathrm{M} = -2\\pi G_\\mathrm{M}(e,\\gamma)\\sqrt{1-e^2}\\left[\\frac{M_\\star}{M_\\bullet}\\right].\n\\end{equation}\nHere, \n$M_\\star = M_\\star(r0$ such that its corresponding eigenfunction\n\t$u\\neq0$ satisfies the linearized equation \\begin{equation}\\label{linprob}\n\t\\partial_x\\mathcal{L}u=\\lambda u.\n\t\\end{equation} Here,\n\t$\\mathcal{L}$ denotes the linearized operator around the traveling\n\twave $\\phi$ defined in $L^2$ given by \\begin{equation}\\label{linop}\n\t\\mathcal{L}=-\\partial_x^2+\\omega-g'(\\phi),\\end{equation}\n\twhere $g'$ indicates the derivative of $g$ in terms of $\\phi$. In the affirmative case, $\\phi$ is said to be spectrally unstable. Otherwise, the periodic wave $\\phi$ is said to be spectrally stable\n\tif the spectrum of $\\partial_x\\mathcal{L}$ is entirely contained in the imaginary axis of the complex plane $\\mathbb{C}$. \\\\\n\t\\indent If we restrict to the case of solitary\n\twaves, $J=\\partial_x$ is a one-to-one operator with no bounded inverse. This fact prevents the use of classical methods of spectral instability as in \\cite{grillakis1}. However,\n\tsufficient conditions have been established by some contributors to overcome this difficulty. For instance,\n\tin \\cite{kap} the authors determined results of spectral\n\tstability related to the problem $(\\ref{linprob})$ by using the\n\tKrein-Hamiltonian instability index. Moreover, it is possible to adapt the method to\n\tconclude similar facts for the BBM-type problems \\begin{equation}\\label{BBM}\n\tu_t+u_x-u_{txx}+(g(u))_x=0,\\end{equation} and the fractional models\n\trelated to those equations. In \\cite{lopes} the author presented\n\tsufficient conditions for the linear instability by\n\tusing the semigroup theory. Interesting results were also given by \\cite{A} and \\cite{lin}.\\\\\n\t\\indent In periodic setting we have the work \\cite{DK}, where sufficient conditions for the spectral stability\/instability have been determined.\n\tHowever in such case, since $J$ is not a one-to-one operator, the authors have\n\tconsidered the modified problem \\begin{equation}\\label{modspecp1}\n\tJ\\mathcal{L}\\big|_{H_0}u=\\lambda u, \\end{equation} where $H_0\\subset\n\tH:=L_{per}^2([0,L])$ is the closed subspace given by\n\t$$H_0=\\left\\{f\\in L^2([0,L]);\\ \\int_0^Lf(x)dx=0\\right\\}.$$ The\n\tKrein-Hamiltonian index formula was applied to deduce the spectral\n\tstability of periodic waves for the equation\n\t$(\\ref{gkdv})$ with $g(s)=\\pm s^3$. However, it was necessary to know the behavior of the first\n\tfive eigenvalues of the linear operator $\\mathcal{L}$ in\n\t$(\\ref{linop})$. \\\\\n\t\\indent In our analysis, the previous knowledge of the periodic wave\n\tis not necessary but it can be useful in order to obtain the spectral stability\/instability. We use a different way to compute the Krein-Hamiltonian index formula for some specific examples but the method can be adapted to other models contained in the regime of $(\\ref{gkdv})$ and related equations. Presented here are the critical KdV and Gardner equations.\\\\\n\t\\indent We are going to illustrate our two basic examples. First, if the critical KdV is considered, we prove the spectral stability\/instability results associated with the positive and zero mean periodic waves. It is well known that both periodic waves appear when, in the equation $(\\ref{travkdv})$, $g(\\phi)=\\phi^5$ and $A=0$. Concerning positive and periodic waves, we determine our results using the explicit solution determined in \\cite{AN2}. After that, we solve numerically some auxiliary initial value problems which give us the precise information about the Krein-Hamiltonian index formula in order to obtain the spectral stability. Explicit zero mean periodic waves were unknown in the current literature until now. To fill this gap, we present a cnoidal wave profile.\\\\\n\t\\indent For the case of positive periodic waves (see ($\\ref{dn4kdv}$)), it is expected that $\\ker(\\mathcal{L})=[\\phi']$ and $n(\\mathcal{L})=1$, where $n(\\mathcal{L})$ indicates the number of negative eigenvalues of $\\mathcal{L}$. When periodic waves with the zero mean property are considered (see $(\\ref{cnoid4kdv})$), we have $n(\\mathcal{L})=2$ and $\\ker(\\mathcal{L})=[\\phi']$. In the second case, and using an explicit solution, it has been determined the same spectral property for the case $g(s)= s^3$ and $A=0$ as determined \\cite{AN1} and \\cite{DK}. It is worth mentioning that the authors had in hands the behavior of the first five eigenvalues of the linearized operator $\\mathcal{L}$ to calculate the sign of $\\langle\\mathcal{L}^{-1}1,1\\rangle$ using Fourier series. This quantity plays an important role to deduce spectral stability results for the gKdV equation in the sense that it is possible to identify an eventual existence of points $(\\omega,A)$ contained in the parameter regime satisfying $\\langle\\mathcal{L}^{-1}1,1\\rangle<0$ and $\\langle\\mathcal{L}^{-1}1,1\\rangle>0$. The change of sign establishes a rupture on the spectral stability scenario when cnoidal waves for the modified KdV equation are considered (\\cite{AN1} and \\cite{DK}). Concerning our zero mean periodic waves for the critical KdV we obtain, in the line $(\\omega,0)$, a threshold value $\\omega_1>0$ such that $\\langle\\mathcal{L}^{-1}1,1\\rangle=0$. In addition, $\\langle\\mathcal{L}^{-1}1,1\\rangle<0$, if $\\omega<\\omega_1$ and $\\langle\\mathcal{L}^{-1}1,1\\rangle>0$, if $\\omega>\\omega_1$. In the first case, the wave is spectrally stable and in the second one, spectrally unstable. For dnoidal waves, there is no threshold value $\\omega_1$ for the quantity $\\langle\\mathcal{L}^{-1}1,1\\rangle$ and the periodic waves are spectrally stable. Summarizing our results, we have the following theorem:\n\t\\begin{teo}\\label{teoest} Let $L>0$ be fixed.\\\\\n\t\ta) For all $\\omega>\\frac{\\pi^2}{L^2}$, positive and periodic waves of dnoidal type for the critical KdV equation are spectrally stable.\\\\\n\t\tb) There exists a unique $\\omega_1>\\frac{4\\pi^2}{L^2}$ such that the zero mean periodic waves of cnoidal type for the critical KdV equation are spectrally stable for $\\omega\\in \\left(\\frac{4\\pi^2}{L^2},\\omega_1\\right)$ and spectrally unstable for $\\omega>\\omega_1$.\n\t\\end{teo}\n\\begin{obs}\n\t Theorem $\\ref{teoest}$-a) establishes the spectral stability of the periodic dnoidal waves. This solution first appeared in \\cite{AN2} and the authors established the existence of a unique $\\omega_0>\\frac{\\pi^2}{L^2}$ such that the dnoidal wave is orbitally stable for $\\omega\\in \\left(\\frac{\\pi^2}{L^2},\\omega_0\\right)$ (using the classical argument in \\cite{grillakis1}) and orbitally unstable for $\\omega>\\omega_0$ (employing an adaptation of the arguments in \\cite{bona}). Since the Cauchy problem for the equation $(\\ref{4kdv})$ is not globally well posed in the energy space $H_{per}^1([0,L])$, we are in conformity with the arguments in \\cite{AN2}. In fact, we are attesting for KdV type equations that spectral stability implies the orbital stability provided that the global well posed in the energy space $H_{per}^1([0,L])$ of the associated Cauchy problem is verified.\n\\end{obs}\n\n\t\\indent Next, we shall give few words about the Gardner equation. We construct explicit periodic waves with cnoidal profile by using the modified KdV equation and its corresponding cnoidal solution. In fact, if $\\phi$ is a solution of the equation $(\\ref{travkdv})$ with $g(s)=s^2+s^3$, thus $\\varphi=\\phi+\\frac{1}{3}$ is a periodic solution with cnoidal profile as $\\varphi(x)=d{\\rm cn}(ex,k)$ and for the corresponding modified KdV equation\n\t\\begin{equation}\\label{mkdv12}\n\t-\\varphi''+\\left(\\omega+\\frac{1}{3}\\right)\\varphi-\\varphi^3=0,\n\t\\end{equation}\n\twhere $d$ and $e$ are smooth functions depending on the wave speed $\\omega+\\frac{1}{3}$. In equation $(\\ref{mkdv12})$, $k\\in(0,1)$ is called modulus of the elliptic function.\\\\\n\t\\indent It is well known that equation $(\\ref{mkdv12})$ admits periodic waves with dnoidal and cnoidal profiles. The corresponding solution with dnoidal profile for the Gardner equation and its respective orbital stability have been determined in \\cite{AP2}. Our intention is to determine spectral stability results of the associated cnoidal profile and we also present a threshold value $\\omega_1$ such that $\\langle\\mathcal{L}^{-1}1,1\\rangle=0$ at $\\omega=\\omega_1$. More specifically, we obtain the same threshold value $\\omega_1$ as obtained for the cnoidal waves for the modified KdV equation and the reason for that concerns a connection between modified KdV and Gardner equations using the Galilean invariance $\\varphi=\\phi+\\frac{1}{3}$. This fact produces that the linearized operator associated to both periodic waves $\\varphi$ and $\\phi$ are the same. As a consequence, if $\\mathcal{L}_{\\varphi}$ is the corresponding linearized operator around $\\varphi$ for the modified KdV equation and $\\mathcal{L}$ the linearized operator around $\\phi$, we have $\\langle\\mathcal{L}_{\\varphi}^{-1}1,1\\rangle=\\langle\\mathcal{L}^{-1}1,1\\rangle$, that is, the sign of $\\langle\\mathcal{L}^{-1}1,1\\rangle$ is determined just by analysing $\\langle\\mathcal{L}_{\\varphi}^{-1}1,1\\rangle$. Thus, $\\omega<\\omega_1$ implies that $\\langle\\mathcal{L}^{-1}1,1\\rangle<0$ (spectral stability) while $\\omega>\\omega_1$ gives us $\\langle\\mathcal{L}^{-1}1,1\\rangle>0$ (spectral instability). Summarizing our results, we have:\n\t\\begin{teo}\\label{teoestG} Let $L>0$ be fixed. There exists a unique $\\omega_2>\\frac{4\\pi^2}{L^2}$ such that the zero mean periodic waves of cnoidal type for the Gardner equation are spectrally stable for $\\omega\\in \\left(\\frac{4\\pi^2}{L^2},\\omega_2\\right)$ and spectrally unstable for $\\omega>\\omega_2$.\n\t\\end{teo}\n\t\n\t\\indent This paper is organized as follows. In Section 2 we give the\n\tbasic framework of the spectral stability following the ideas in \\cite{DK}. In Section 3, we study the existence of periodic waves and\n\ttheir dependence with respect to the parameters, as well as the\n\tspectral analysis of the linearized operator. Finally, Section 4 is devoted to our applications.\n\t\n\t\n\t\n\t\n\t\n\t\n\t\\section{Basic Framework of Spectral Stability of Periodic Waves.}\n\t\n\t\\setcounter{equation}{0}\n\t\\setcounter{defi}{0}\n\t\\setcounter{teo}{0}\n\t\\setcounter{lema}{0}\n\t\\setcounter{prop}{0}\n\t\\setcounter{coro}{0}\n\t\n\t\n\tIn this section we present the basic framework established in\n\t\\cite{DK} which provides us a criterion for determining the spectral\n\tstability of periodic waves related to the abstract Hamiltonian\n\tequations of the form\n\t\n\t\\begin{equation}\\label{Hamilt} u_t=J\\mathcal{E}'(u)\\end{equation} defined on a Hilbert space\n\t$H$, where $J:H\\to {\\rm{range}}(H)\\subset H$ is a skew symmetric, and\n\t$\\mathcal{E}:H\\to\\mathbb{R}$ is a $C^2-$functional. We restrict\n\tourselves to the specific case when $J=\\partial_x$ and\n\t$\\mathcal{E}(u)=\\frac{1}{2}\\int_0^Lu_x^2-2G(u)dx$,\n\twhere $G'=g$. In that case, the equation $(\\ref{Hamilt})$ becomes the well known\n\tgeneralized Korteweg-de Vries equation as in $(\\ref{gkdv})$.\\\\\n\t\\indent Let us consider again the spectral problem related to the\n\tgeneralized KdV equation\n\t\n\t\\begin{equation}\\label{specp}\n\t\\partial_x\\mathcal{L}u=\\lambda u,\n\t\\end{equation} where\n\t$\\mathcal{L}=-\\partial_x^2+\\omega-g'(\\omega,A,\\phi)$ is the\n\tlinearized operator around the periodic wave $\\phi$ which is\n\ta periodic traveling wave solution of the equation $(\\ref{travkdv})$. As we have mentioned before, the standard theories\n\tof spectral instability of traveling waves for the abstract\n\tHamiltonian system as in \\cite{grillakis1} and\n\t\\cite{lopes} can not be applied in this context. To overcome this\n\tdifficulty, we are going to give a brief explanation of the results in \\cite{DK}.\n\tIndeed, let us consider the modified spectral problem obtained from\n\t$(\\ref{specp})$\n\t\n\t\\begin{equation}\\label{modspecp} J\\mathcal{L}\\big|_{H_0}u=\\lambda u,\n\t\\end{equation}\n\twhere $H_0\\subset H=L_{per}^2([0,L])$ is the closed subspace given by\n\t\n\t$$H_0=\\left\\{f\\in L^2([0,L]);\\ \\int_0^Lf(x)dx=0\\right\\}.$$\n\tFor a fixed period $L>0$, we need to assume in this whole section that:\\\\\n\t\n\t{\\rm (a1)} There exists a fixed pair $(\\omega_0,A_0)$ and $\\phi:=\\phi_{(\\omega_0,A_0)}$ smooth even periodic solution for the equation $(\\ref{travkdv})$. Moreover, we assume that\n\t$\\phi'$ has only two zeros in the interval $[0,L)$. \\\\\n\t\n\t{\\rm (a2)} $\\ker(\\mathcal{L})=[\\phi']$.\\\\\n\t\n\t\n\t\\indent Assumption {\\rm (a1)} implies, from the classical Floquet theory in \\cite{Magnus} that $n(\\mathcal{L})=1$ or $n(\\mathcal{L})=2$ where\n\t$n(\\mathcal{L})$ indicates the number of negative eigenvalues of the linearized operator $\\mathcal{L}=-\\partial_x^2+\\omega-g'(\\phi)$. In addition, assumption {\\rm (a2)} allows us to deduce the existence of a non-periodic even solution $\\bar{y}$ which satisfies the Hill equation\n\t\\begin{equation}\\label{Hill1}\n\t-\\bar{y}''+\\omega\\bar{y} -g'(\\phi) \\bar{y}=0,\n\t\\end{equation}\n\twhere $\\{\\bar{y},\\phi'\\}$ is a fundamental set of solutions for the linear equation $(\\ref{Hill1})$.\\\\\n\t\\indent According with Theorem \\ref{teo2} determined in Section 3, one can see that assumption {\\rm (a2)} will provide us the existence of a smooth surface of even periodic waves which solves $(\\ref{travkdv})$ and defined in an open subset $\\mathcal{O}\\subset\\mathbb{R}^2$,\n\t$$(\\omega,A)\\in \\mathcal{O}\\mapsto\\phi_{(\\omega,A)}\\in H_{per}^s([0,L]),\\ \\ s\\gg1,$$ all of them with the same period $L>0$. In what follows and in the whole paper, we shall not distinguish the periodic wave $\\phi$ for a fixed pair $(\\omega_0,A_0)$ and $\\phi$ for a pair $(\\omega,A)\\in\\mathcal{O}$ since both have the \\textit{same fixed period $L>0$}. The intention is to simplify our presentation with easier notations.\\\\\n\t\\indent Next, we describe the arguments in \\cite{DK}. For the spectral problem in $(\\ref{modspecp})$ let $k_r$ be the number of real-valued and positive eigenvalues (counting multiplicities). The quantity $k_c$ denotes the number of complex-valued eigenvalues with a positive real part. Since ${\\rm Im}(\\mathcal{L})=0$, where ${\\rm Im}(z)$ indicates the imaginary part of the complex number $z$, we see that $k_c$ is an even integer. For a self-adjoint operator $\\mathcal{A}$, let $n(\\langle w,\\mathcal{A}w\\rangle)$ be the dimension of the maximal subspace for which $\\langle w,\\mathcal Aw\\rangle<0$. Also, let $\\lambda$ be an eigenvalue and $E_{\\lambda}$ its corresponding eigenspace. The eigenvalue is said to have negative Krein signature if\n\t$$k_i^{-}(\\lambda):=n(\\langle w,(\\mathcal{L}\\big|_{H_0})\\big|_{E_{\\lambda}}w\\rangle)\\geq1,$$\n\totherwise, if $k_i^{-}(\\lambda)=0$, then the eigenvalue is said to have a positive Krein signature. If $\\lambda$ is geometrically and algebraically simple with the eigenfunction $\\psi_{\\lambda}$, then\n\t$$k_i^{-}(\\lambda)=\\left\\{\\begin{array}{llll}\n\t\t0,\\ \\langle \\psi_{\\lambda},(\\mathcal{L}\\big|_{H_0})\\psi_{\\lambda}\\rangle>0\\\\\n\t\t1,\\ \\langle \\psi_{\\lambda},(\\mathcal{L}\\big|_{H_0})\\psi_{\\lambda}\\rangle<0.\\end{array}\n\t\\right.$$\n\t\n\tWe define the total Krein signature as\n\t$$k_i^{-}:=\\sum _{\\lambda\\in i\\mathbb{R}\\backslash\\{0\\}}k_{i}^{-}(\\lambda).$$\n\tThe fact ${\\rm Im}(\\mathcal{L})=0$ implies that $k_i^-(\\lambda)=k_i^{-}(\\overline{\\lambda})$ and $k_i^{-}$ is an even integer.\\\\\n\t\n\tLet us consider\n\t\\begin{equation}\\label{I}\n\t\\mathcal{I}=\\langle \\mathcal{L}^{-1}1,1\\rangle.\n\t\\end{equation}\n\tIf $\\mathcal{I}\\neq0$, denote $\\mathcal{D}$ as the $2\\times 2-$matrix given by\n\t\\begin{equation}\\label{Dmatrix}\n\t\\mathcal{D}=\\frac{1}{\\langle\\mathcal{L}^{-1}1,1\\rangle}\\left[\\begin{array}{llll}\n\t\t\\langle\\mathcal{L}^{-1}\\phi,\\phi\\rangle & & \\langle\\mathcal{L}^{-1}\\phi,1\\rangle\\\\\\\\\n\t\t\\langle\\mathcal{L}^{-1}\\phi,1\\rangle & & \\langle\\mathcal{L}^{-1}1,1\\rangle\\end{array}\\right]\n\t\\end{equation}\n\tWe obtain, then, the following results:\n\t\\begin{teo}\\label{krein}\n\t\tSuppose that assumptions {\\rm (a1)-(a2)} hold. If $\\mathcal{I}\\neq0$ and $\\mathcal{D}$ is non-singular (i.e. $\\det(\\mathcal{D})\\neq 0$) we have for the eigenvalue problem $(\\ref{modspecp})$\n\t\t$$k_r+k_c+k_{i}^{-}=n(\\mathcal{L})-n(\\mathcal{I})-n(\\mathcal{D}).$$\n\t\tThe nonpositive integer $K_{{\\rm Ham}}=k_r+k_c+k_{i}^{-}$ is called \\textit{Hamiltonian-Krein index}.\n\t\\end{teo}\n\t\\begin{proof}\n\t\tSee Theorem 1 in \\cite{DK}.\n\t\\end{proof}\n\t\n\t\\begin{coro}\\label{coroest}\n\t\tUnder the assumptions of Theorem $\\ref{krein}$, if $k_c=k_r=k_{i}^{-}=0$ the periodic wave $\\phi$ is spectrally stable. In addition, if $K_{{\\rm Ham}}=1$ the refereed periodic wave is spectrally unstable.\n\t\\end{coro}\n\t\\begin{proof}\n\t\tSince $k_c=k_r=0$, there is no eigenvalues with positive real part for the problem $(\\ref{modspecp})$ and $\\phi$ is spectrally stable since the total Krein signature is zero. Now, if $K_{{\\rm Ham}}=1$ we deduce that $k_r=1$ since $k_c$ and $k_{i}^{-}$ are even nonnegative integers. So, operator $J\\mathcal{L}$ presented in the spectral problem $(\\ref{modspecp})$ has a positive eigenvalue which enable us to deduce the spectral instability of the periodic wave.\n\t\\end{proof}\n\t\n\tWe shall present some considerations concerning the result determined in Corollary $\\ref{coroest}$ applied to the case of the generalized KdV equation in $(\\ref{gkdv})$. In fact, by assuming that assumption {\\rm (a2)} is verified one has\n\t$$\\langle\\mathcal{L}^{-1}\\phi,\\phi\\rangle=-\\frac{1}{2}\\frac{d}{d\\omega}\\int_0^L\\phi^2dx,\\ \\ \\langle\\mathcal{L}^{-1}\\phi,1\\rangle=-\\frac{d}{d\\omega}\\int_0^L\\phi dx,$$\n\tand\n\t$$\\mathcal{I}=\\langle\\mathcal{L}^{-1}1,1\\rangle=\\frac{d}{dA}\\int_0^L\\phi dx.$$\n\tSo, we have\n\t$$n(\\mathcal{I})=\\left\\{\\begin{array}{llll}\n\t\t0,\\ \\frac{d}{dA}\\int_0^L\\phi dx\\geq0\\\\\\\\\n\t\t1,\\ \\frac{d}{dA}\\int_0^L\\phi dx<0.\\end{array} \\right.$$\n\t\n\tOn the other hand, in order to determine $n(\\mathcal{D})$ it is necessary\n\tto analyze the quantity $\\mathcal{D}$.\n\tIn fact, if $\\mathcal{D}<0$ we have that the associated matrix has a\n\tpositive and a negative eigenvalue and therefore $n(\\mathcal{D})=1$.\n\tHowever, if $\\mathcal{D}>0$, it is not possible to directly decide\n\tabout the quantity $n(\\mathcal{D})$ since we could have\n\t$n(\\mathcal{D})=0$ (two positive eigenvalues for the associated matrix) or $n(\\mathcal{D})=2$\n\t(two negative eigenvalues). In the next section,\n\twe determine sufficient conditions to obtain assumptions {\\rm\n\t\t(a1)-(a2)} for a general class of second order differential equations. In addition, we present two useful initial value problems used to determine a precise way to calculate $\\mathcal{I}$ and $\\mathcal{D}$.\n\t\n\t\n\t\n\t\\section{Basic Framework on Spectral Analysis.}\n\t\\setcounter{equation}{0}\n\t\\setcounter{defi}{0}\n\t\\setcounter{teo}{0}\n\t\\setcounter{lema}{0}\n\t\\setcounter{prop}{0}\n\t\\setcounter{coro}{0}\n\t\n\t\n\t\\indent In a general setting (without considering the arguments in the last section for a while), let us suppose that $\\phi$ is an even $L-$periodic solution of the general equation\n\t\\begin{equation}\\label{ode}\n\t\t-\\phi''+f(\\omega,A,\\phi)=0,\n\t\\end{equation}\n\twhere $f$ is a smooth function depending on $(\\omega,A,\\phi)$ and $(\\omega,A)$ is an element of an admissible set $\\mathcal{P}\\subset\\mathbb{R}^2$. This means that $\\mathcal{P}$ contains all the pairs $(\\omega,A)$ where $\\phi$ is a periodic solution of $(\\ref{ode})$.\\\\\n\t\\indent Let $\\mathcal{L}$ be the linearized equation around $\\phi$, where $\\phi$ is a periodic\n\tsolution of (\\ref{ode}) of period $L$. The linearized operator \\begin{equation}\n\t\\mathcal{L}(y) = - y'' + f'(\\omega, A, \\phi)\\, y,\n\t\\;\\;\\; (\\omega, A) \\in \\mathcal{P} \\label{hill} \\end{equation} is a Hill\n\toperator and $f'$ is the derivative in terms of $\\phi$. According to \\cite{Haupt} and \\cite{Magnus},\n\tthe spectrum of $\\mathcal{L}$ is formed by an unbounded\n\tsequence of real numbers\n\t\\[\n\t\\lambda_0 < \\lambda_1 \\leq \\lambda_2 < \\lambda_3 \\leq \\lambda_4\\;\\;\n\t...\\; < \\lambda_{2n-1} \\leq \\lambda_{2n}\\; \\cdots,\n\t\\]\n\twhere equality means that $\\lambda_{2n-1} = \\lambda_{2n}$ is a\n\tdouble eigenvalue. The spectrum of $\\mathcal{L}$ is\n\tcharacterized by the number of zeros of the eigenfunctions, if $\\Psi$\n\tis an eigenfunction for the eigenvalue $\\lambda_{2n-1}$ or\n\t$\\lambda_{2n}$, then $\\Psi$ has exactly $2n$ zeros in the half-open\n\tinterval $[0,L)$.\n\t\n\tIn order to apply the general theory of orbital stability,\n\t\\cite{bona}, \\cite{grillakis1} and \\cite{W1}, the spectrum of\n\t$\\mathcal{L}$ is of main importance and also of the major\n\tdifficulty in the applications. It is necessary to know exactly the\n\tnon-positive spectrum; more precisely, it is necessary to know the\n\tinertial index $in(\\mathcal{L})$ of\n\t$\\mathcal{L}$, where $in(\\mathcal{L})$ is\n\ta pair of integers $(n,z)$, where $n$ is the dimension of the\n\tnegative subspace of $\\mathcal{L}$ and $z$ is the\n\tdimension of the null subspace of $\\mathcal{L}$.\n\t\n\tThe results of this section are based on \\cite{natali2}, \\cite{neves} and \\cite{neves1} and the first\n\tone concerns the invariance of the index with respect to the parameters. Since the derivative $\\phi'$ is an eigenfunction related to $\\lambda =0$ for every $(\\omega, A)\n\t\\in \\mathcal{P}$, we can state the following result.\n\t\n\t\\begin{teo}\n\t\tLet $\\phi$ a smooth $L-$periodic solution of the equation $(\\ref{ode})$.\n\t\tThen the\n\t\tfamily of operators $\\mathcal{L}(y) = - y'' +\n\t\tf'(\\omega, A, \\phi)\\, y $ is isoinertial with respect to $(\\omega,A)$ in the parameter regime. \\label{teo0}\n\t\\end{teo}\n\t\\begin{proof}\n\t\tSee \\cite{natali2} and \\cite{neves1}.\n\t\\end{proof}\n\t\n\tIn order to calculate the inertial index of $\\mathcal{L}$ for a fixed value of $(\\omega_0,A_0)$, we shall consider the\n\tauxiliary function $\\bar{y}$ the unique solution of the problem \\begin{equation}\n\t\\left\\{\n\t\\begin{array}{l}\n\t\t- \\bar{y}'' + f'(\\omega_0, A_0, \\phi) \\bar{y} = 0 \\\\\n\t\t\\bar{y}(0) = - \\frac{1}{\\phi''(0)} \\\\\n\t\t\\bar{y}'(0)=0,\n\t\\end{array} \\right.\n\t\\label{y} \\end{equation} and also the constant $\\theta$ given by \\begin{equation} \\theta=\n\t\\frac{ \\bar{y}'(L)}{\\phi''(0)}, \\label{theta} \\end{equation} where $L$ is\n\tthe period of $\\phi=\\phi_{(\\omega_0,A_0)}$.\n\t\n\tWe know that the derivative $\\phi'$ is an eigenfunction\n\tfor the eigenvalue $\\lambda = 0$, and also that $\\phi'(x)$\n\thas exactly two zeros in the half-open interval $[0, L)$.\n\tTherefore we have three possibilities: \\begin{itemize}\n\t\t\\item[i)] $\\lambda_1 = \\lambda_2 = 0 \\Rightarrow in(\\mathcal{L}) = (1,2)$,\\\\\n\t\t\\item[ii)] $\\lambda_1 = 0 < \\lambda_2 \\Rightarrow in(\\mathcal{L}) = (1,1)$,\\\\\n\t\t\\item[iii)] $\\lambda_1 < \\lambda_2 = 0 \\Rightarrow in(\\mathcal{L}) = (2,1)$,\n\t\\end{itemize}\n\t\n\tThe method we use to decide and calculate the inertial index is\n\tbased on Lemma 2.1 and Theorems 2.2 and 3.1 of \\cite{neves}. This\n\tresult can be stated as follows.\n\t\n\t\n\t\\begin{teo}\n\t\tLet $\\theta$ be the constant given by (\\ref{theta}), then the\n\t\teigenvalue $\\lambda=0$ is simple if and only if $ \\theta \\neq 0$.\n\t\tMoreover, if $\\theta \\neq 0$, then $ \\lambda_{1}=0$ if $\\theta <\n\t\t0$, and $ \\lambda_{2}=0$ if $\\theta > 0$. \\label{teo1}\n\t\\end{teo}\n\t\\begin{flushright}\n\t\t$\\square$\n\t\\end{flushright}\n\t\n\t\n\t\n\t\n\tLet $L>0$ be fixed. In order to show our spectral stability results, it is convenient to show the existence\n\tof a family $\\phi$ of $L$-periodic solutions for the\n\tequation (\\ref{ode}) that smoothly depends on the parameters\n\t$(\\omega,A)$, for $(\\omega,A)$ in an open set $\\mathcal{O} \\subset\n\t\\mathcal{P}$.\n\t\n\t\n\t\\begin{teo}\n\t\tLet $\\phi_{(\\omega_0,A_0)}$ be an\n\t\teven periodic solution of $(\\ref{ode})$ defined in a fixed pair $(\\omega_0,A_0)$ in the parameter regime. If $\\theta \\neq 0$, where\n\t\t$\\theta$ is the constant given in Theorem \\ref{teo1}, and $L$ is\n\t\tthe period of $\\phi_{(\\omega_0,A_0)}$, then there is an open\n\t\tneighborhood $\\mathcal{O}$ of $(\\omega_0,A_0)$,\n\t\tand a family $\\phi_{(\\omega,A)} \\in H_{per,e}^2([0,L])$ of\n\t\t$L$-periodic solutions of $(\\ref{ode})$, which smoothly depends on\n\t\t$(\\omega,A) \\in \\mathcal{O}$ in a $C^1$ manner. \\label{teo2}\n\t\\end{teo}\n\t\\begin{proof}\n\t\tLet $\\mathcal{P}$ the set of parameters and $\\mathcal{F}$ be the operator given by the equation (\\ref{ode}) restrict\n\t\tto the even functions, precisely, $\\mathcal{F}:\\mathcal{P}\\times\n\t\tH_{per,e}^2([0,L]) \\rightarrow L_{per,e}^2([0,L]) $, \\begin{equation}\n\t\t\\mathcal{F}(\\omega,A,\\phi) = -\\phi''+ f(\\omega,A,\\phi). \\label{eq31}\n\t\t\\end{equation} Then $\\mathcal{F}(\\omega_0,A_0,\\phi_{(\\omega_0,A_0)}) = 0$,\n\t\tsince $\\phi_{(\\omega_0,A_0)}$ is an even periodic solution of the\n\t\tequation (\\ref{ode}). If $\\theta \\neq 0 $, Theorem \\ref{teo1}\n\t\timplies that $ \\mathcal{L}_{(\\omega_0, A_0)}(y) = - y'' +\n\t\tf'(\\omega_0, A_0, \\phi_{(\\omega_0,A_0)})\\, y$, has an one-dimensional nullspace;\n\t\tand from the invariance, this nullspace is spanned by\n\t\t$\\phi'_{(\\omega_0, A_0)}$. Since $\\phi'_{(\\omega_0, A_0)}$ is odd,\n\t\tit is not an element of $H_{per,e}^2([0,L])$, it follows that $\n\t\t\\mathcal{F}_{\\phi}(\\omega_0,A_0,\\phi_{(\\omega_0,A_0)}) =\n\t\t\\mathcal{L}_{(\\omega_0, A_0)}:H_{per,e}^2([0,L]) \\subset L_{per,e}^2([0,L])\\rightarrow\n\t\tL_{per,e}^2([0,L]) $ is invertible and its inverse is bounded.\n\t\tTherefore, the results of the Theorem \\ref{teo2} follows from the\n\t\timplicit function theorem. See Theorem 15.1 and Corollary 15.1 of\n\t\t\\cite{Deimling}.\n\t\\end{proof}\n\t\n\tNext, we turn back to the setting contained in Section 2 by considering $(\\ref{ode})$ as\n\t\n\t\\begin{equation}\\label{odekdv}\n\t\t-\\phi''+\\omega\\phi-g(\\phi)-A=0.\n\t\\end{equation}\n\tWe assume that $\\theta\\neq0$ in a single point $(\\omega_0,A_0)$ in the parameter regime. By Theorem $\\ref{teo2}$ we can define\n\t\\[\n\t\\psi = \\frac{\\partial \\phi}{\\partial \\omega} \\qquad\n\t\\mbox{and} \\qquad \\eta= \\frac{\\partial \\phi}{\\partial\n\t\tA}.\n\t\\]\n\t\n\t\n\tAgain by Theorem \\ref{teo2}, it is easy to see that $\\psi$\n\tabove is an even periodic smooth function which satisfies, for the case of the equation $(\\ref{gkdv})$\n\t\\begin{equation} -\n\t\\psi'' + \\omega \\psi -g'(\\phi)\\psi= -\n\t\\phi. \\label{psi1} \\end{equation} In addition, $\\eta$ is also an\n\teven periodic function satisfying\n\t\n\t\\begin{equation} - \\eta'' + \\omega \\eta\n\t-g'(\\phi)\\eta= 1. \\label{eta1} \\end{equation}\n\t\n\t\\begin{obs}\\label{obsiso}\n\t\tTheorem $\\ref{teo0}$ gives us an important property concerning the quantity and multiplicity of the first two eigenvalues associated to the linearized operator $\\mathcal{L}$ defined in $(\\ref{linop})$. Indeed, if $\\theta\\neq0$ in a certain point $(\\omega_0,A_0)$ in the parameter regime $\\mathcal{P}$, we can conclude that the kernel of $\\mathcal{L}$ is simple and $n(\\mathcal{L})$ is constant for all $(\\omega,A)$ in an open subset contained in $\\mathcal{P}$, that is, the value $in(\\mathcal{L})$ is constant in this subset.\n\t\t\n\t\\end{obs}\n\t\n\tNext result gives us an immediate converse of Theorem $\\ref{teo2}$ for the case $g(s)=s^{p+1}$.\n\t\\begin{prop}\\label{propsimp}\n\t\tLet $\\widetilde{\\mathcal{O}}\\subset\\mathbb{R}^2$ be an open subset. Suppose that $(\\omega,A)\\in\\widetilde{\\mathcal{O}}\\mapsto\\phi_{(\\omega,A)}$ is a smooth surface of even (odd) periodic traveling wave solutions which solves $(\\ref{odekdv})$ with $g(s)=s^{p+1}$ all of them with the same fixed period $L>0$. Then, $\\ker(\\mathcal{L})=[\\phi']$ and the value $n(\\mathcal{L})$ is constant for all $(\\omega,A)\\in\\widetilde{\\mathcal{O}}$. The same result remains valid for the case $A\\equiv0$, by considering $\\widetilde{I}\\subset\\mathbb{R}$ an open subset and $\\omega\\in \\widetilde{I}\\mapsto\\phi_{\\omega}$ a smooth curve of even periodic waves.\n\t\\end{prop}\n\t\\begin{proof}\n\t\tTo simplify the notation, let us denote $\\phi=\\phi_{(\\omega,A)}$ and consider $\\{\\phi',\\bar{y}\\}$ the fundamental set of solutions related to the equation $-y''+\\omega y - (p+1)y^p=0$. By contradiction, assume that $\\bar{y}$ is $L-$periodic. Since $\\phi'$ is odd, the arguments in \\cite{Magnus} give us that $\\bar{y}$ can be considered even. The Wronskian of the set $\\{\\phi',\\bar{y}\\}$ and denoted by $\\mathcal{W}(\\phi',\\bar{y})$ satisfies $\\mathcal{W}(\\phi',\\bar{y})=1$ over $[0,L]$ (see \\cite{Magnus}). Moreover, since $\\bar{y}$ and $\\phi'$ are both periodic functions, we obtain from $(\\ref{travkdv})$ that\n\t\t\\begin{equation}\\label{wronsk}\\begin{array}{lllll}\n\t\t\t\tL&=&\\displaystyle\\int_0^{L}\\mathcal{W}(\\phi',\\bar{y})dx=-2\\int_0^{L} \\bar{y}\\phi''dx=-2\\int_0^{L}\\bar{y}\\left(\\omega\\phi-\\phi^{p+1}-A\\right)dx\\\\\\\\\n\t\t\t\t&=&\\displaystyle-2\\omega\\int_0^{L} \\bar{y}\\phi dx+2\\int_0^{L} \\bar{y}\\phi^{p+1}dx+2A\\int_0^{L}\\bar{y}dx.\\end{array}\n\t\t\\end{equation}\n\t\tSince $\\mathcal{L}\\phi=A-p\\phi^{p+1}$, by $(\\ref{psi1})$ and $(\\ref{eta1})$ we obtain from $(\\ref{wronsk})$\n\t\t\t\\begin{equation}\\label{wronsk1}\n\t\t\t\tL=2\\omega \\langle \\mathcal{L}\\psi,\\bar{y}\\rangle-\\frac{2}{p}\\langle\\mathcal{L}\\phi,\\bar{y}\\rangle+2A\\left(\\frac{1}{p}+1\\right)\\langle\\mathcal{L}\\eta,\\bar{y}\\rangle.\n\t\t\t\\end{equation}\n\\indent The fact that $\\mathcal{L}\\bar{y}=0$ allows us to deduce from the self-adjointness of $\\mathcal{L}$ and $(\\ref{wronsk1})$ that $L=0$. This contradiction shows that $\\ker(\\mathcal{L})=[\\phi']$.\n\t\\end{proof}\n\t\n\t\n\t\n\t\n\tLet us suppose that $\\theta\\neq0$ in a single point $(\\omega_0,A_0)$ in the parameter regime. By Theorem $\\ref{teo2}$ we are able to determine the initial condition\n\t$\\psi(0)$ at the point $(\\omega_0,A_0)$. To do so, we multiply equation (\\ref{psi1}) by $\\bar{y}$, where\n\t$\\bar{y}$ is given in (\\ref{y}), and integrate the\n\tfirst term twice. We get\n\t$$\n\t- \\int_0^{L} \\phi_{(\\omega_0,A_0)} \\;\n\t\\bar{y}\\; dx = \\psi(L) \\bar{y}'(L)=\\psi(0) \\bar{y}'(L).\n\t$$\n\tSimilarly, from $(\\ref{eta1})$ one has\n\t$$\n\t\\int_0^{L}\\bar{y}\\; dx = \\eta(L) \\bar{y}'(L)=\\eta(0) \\bar{y}'(L).\n\t$$\n\t\n\t\n\t\n\tSince $\\theta \\neq 0$ we conclude that $\\bar{y}'(L) \\neq 0$ and then\n\t$\\psi(x)$ and $\\eta(x)$ are obtained by solving, respectively, the\n\tfollowing initial value problems \\begin{equation} \\left\\{\n\t\\begin{array}{lllllllllllll}\n\t\t- \\psi'' + \\omega_0 \\psi -g'(\\omega_0,A_0,\\phi_{(\\omega_0,A_0)})\\psi= -\n\t\t\\phi_{(\\omega_0,A_0)} \\\\\n\t\t\\psi(0) = - \\frac{1}{\\bar{y}'(L)} \\int_0^{L} \\phi_{(\\omega_0,A_0)} \\; \\bar{y}\\; dx \\\\\n\t\t\\psi'(0)=0,\n\t\\end{array} \\right.\\\n\t\\left\\{\n\t\\begin{array}{l}- \\eta'' + \\omega_0 \\eta -g'(\\omega_0,A_0,\\phi_{(\\omega_0,A_0)})\\eta=\n\t\t1 \\\\\n\t\t\\eta(0) = \\frac{1}{\\bar{y}'(L)} \\int_0^{L} \\bar{y}\\; dx \\\\\n\t\t\\eta'(0)=0.\n\t\\end{array} \\right.\n\t\\label{psi2}\n\t\\end{equation}\n\tBoth initial value problems are very useful to determine $\\mathcal{I}$ and $\\mathcal{D}$ given in Section 2. \n\t\\section{Applications - Spectral stability of periodic waves}\n\t\n\t\\setcounter{equation}{0}\n\t\\setcounter{defi}{0}\n\t\\setcounter{teo}{0}\n\t\\setcounter{lema}{0}\n\t\\setcounter{prop}{0}\n\t\\setcounter{coro}{0}\n\t\n\t\n\t\\subsection{Case $g(s)=s^{p+1}$ - Existence of periodic waves using variational methods.} Using a variational method, we establish the existence of periodic waves for the equation $(\\ref{odekdv})$. The main advantage of the approach presented here is that the quantity of negative eigenvalues of $\\mathcal{L}$ in $(\\ref{linop})$ defined for periodic waves $\\phi$ in a single point $(\\omega_0,A_0)\\in\\mathcal{P}$ is precisely determined. Thus, in this specific case, Remark $\\ref{obsiso}$ can be used to deduce the quantity and multiplicity of negative eigenvalues for all $(\\omega,A)$.\n\t\n\t\n\t\\indent Let $L>0$ be fixed. For each $\\gamma>0$, we define the set\n\t$$Y_{\\gamma}=\\left\\{u\\in H_{per}^1([0,L]);\\ \\int_0^{L}u^{p+2}dx=\\gamma\\right\\},$$\n\twhere $p$ is an even integer. Our first goal is to find a minimizer of the constrained minimization problem\n\t\\begin{equation}\n\t\t\\label{infB}\n\t\tm=m_{\\omega}=\\inf_{ u\\in Y_{\\gamma}}\\mathcal{B}_{\\omega}(u),\n\t\\end{equation}\n\twhere for each $\\omega>0$, $\\mathcal{B}_{\\omega}$ is given by\n\t\\begin{equation}\\label{Bfunctional}\n\t\t\\mathcal{B}_{\\omega}(u)=\\frac{1}{2}\\int_{0}^{L}u'^2+\\omega u^2dx.\n\t\\end{equation}\n\tWe observe that $\\mathcal{B}_{\\omega}$ is a smooth functional on $H_{per}^1([0,L])$.\n\t\\medskip\n\t\n\t\\begin{lema}\\label{minlem}\n\t\tThe minimization problem \\eqref{infB} has at least one nontrivial solution, that is, there exists $\\phi\\in Y_{\\gamma}$ satisfying\n\t\t\\begin{equation}\\label{minBfunc}\n\t\t\t\\mathcal{B}_{\\omega}(\\phi)=\\inf_{ u\\in Y_{\\gamma}}\\mathcal{B}_{\\omega}(u).\n\t\t\\end{equation}\n\t\\end{lema}\n\t\\begin{proof}\n\t\tSince $m\\geq0$ and $\\mathcal{B}_{\\omega}$ is a smooth functional, we are enabled to consider $\\{u_n\\}=\\{u_{n,\\omega}\\}$ as a minimizing sequence for \\eqref{infB}, that is, a sequence in $Y_\\gamma$ satisfying\n\t\t$\\displaystyle \\mathcal{B}_{\\omega}(u_n)\\rightarrow\\inf_{u\\in Y_{\\gamma}} \\mathcal{B}_{\\omega}(u)=m, \\ \\mbox{as} \\ n\\rightarrow \\infty.$\n\t\t\n\t\t\\indent The fact that $\\omega>0$ enables us to conclude $\\{u_{n}\\}$ as a bounded set in $H_{per}^1([0,L])$. Thus, modulus a subsequence, there exists\n\t\t$\\phi=\\phi_{\\omega}\\in H_{per}^1([0,L])$ such that\n\t\t$u_n\\rightharpoonup \\phi \\ \\mbox{weakly in} \\ H_{per}^1([0,L]), \\ \\ \\mbox{as} \\ n\\rightarrow \\infty.$\n\t\t\n\t\t\n\t\tNow, since the energy space $H_{per}^1([0,L])$ is compactly embedded in $L_{per}^{p+2}([0,L])\\hookrightarrow L_{per}^2([0,L])$, we have for $n\\rightarrow +\\infty$ that\n\t\t$u_n\\rightarrow \\phi \\ \\mbox{in} \\ L^{p+2}_{per}([0,L]),$\n\t\tthat is, $\\int_0^{L}\\phi^{p+2} dx=\\gamma$.\n\t\t\n\t\tMoreover, the weak lower semi-continuity of $\\mathcal{B}_{\\omega}$ gives us that\n\t\t$\n\t\t\\mathcal{B}_{\\omega}(\\phi)\\leq\\liminf_{n\\rightarrow \\infty} \\mathcal{B}_{\\omega}(u_n)=m.\n\t\t$\n\t\tThe lemma is now proved.\n\t\\end{proof}\n\t\n\t\n\tBy Lemma \\ref{minlem} and Lagrange's Multiplier Theorem, we guarantee the existence of $C_1$ such that\n\t\\begin{equation}\\label{lagrange}\n\t\t-\\phi''+\\omega\\phi=C_1\\phi^{p+1}.\n\t\\end{equation}\n\tWe note that $\\phi$ is nontrivial because $\\gamma>0$ and a standard rescaling argument enables us to deduce that the Lagrange Multiplier $C_1$ can be chosen as $C_1=1$. Now, let $L>0$ be fixed as before. Since the minimization problem $(\\ref{infB})$ can be solved for any $\\omega>0$, we guarantee by arguments of smooth dependence in terms of the parameters for standard ODE (see for instance, \\cite[Chapter I, Theorem 3.3]{hale}), the existence of a convenient open interval $I$ and a smooth curve $\\omega\\in I\\mapsto\\phi\\in H_{per}^n([0,L])$, $n\\in\\mathbb{N}$, satisfying the equation\n\t\\begin{equation}\\label{ode-wave13}\n\t\t-\\phi''+\\omega\\phi-\\phi^{p+1}=0.\n\t\\end{equation}\n\t\n\tIn this setting, the existence of a smooth curve of periodic waves depending on $\\omega$ enables us to conclude by Proposition $\\ref{propsimp}$ that $\\ker(\\mathcal{L})$ is simple. Concerning $n(\\mathcal{L})$, we see that $\\phi$ is a minimizer of $\\mathcal{B}_{\\omega}$ with one constraint. Since $\\langle\\mathcal{L}\\phi,\\phi\\rangle<0$, we obtain by Courant's Min-Max Principle that $n(\\mathcal{L})=1$.\\\\\n\t\\indent Analysis above gives us the following sentence: for $\\phi$ solution of $(\\ref{ode-wave13})$ with $\\omega_0>0$ and the corresponding single point $(\\omega_0,0)$ in the parameter regime, we have that $n(\\mathcal{L}_{(\\omega_0,0)})=1$ and $\\ker(\\mathcal{L}_{(\\omega_0,0)})=[\\phi']$. Therefore, by Theorem $\\ref{teo1}$ and Remark $\\ref{obsiso}$ one has that $\\theta\\neq0$ for all $(\\omega,A)$ in an open subset $\\mathcal{O}\\subset\\mathcal{P}$. This means that the solution $\\phi$ of $(\\ref{odekdv})$ satisfies $n(\\mathcal{L})=1$ and $\\ker(\\mathcal{L})=[\\phi']$ for all $(\\omega,A)\\in \\mathcal{O}$.\n\t\n\t\\begin{obs}\\label{remN1}\n\t\tSolutions $\\phi$ which are minimizers of the problem $(\\ref{minBfunc})$ are well determined by the analysis above. In fact, they are rounding the center point(s) in the phase portrait and have the homoclinic as a limit for large periods. The corresponding solution $\\phi$ enjoys the same property. As we have already mentioned before, the parameter regime is the maximal set constituted of pairs $(\\omega,A)$ such that all periodic waves round the center point(s) in the phase portrait. The analysis above gives us that $n(\\mathcal{L})=1$ and $\\ker(\\mathcal{L})=[\\phi']$ in an open subset contained in the parameter regime.\t\n\t\\end{obs}\n\t\n\t\n\t\\indent As before, let us consider a fixed $L>0$. Again, for a fixed $\\gamma>0$, we define \n\t$$Z_{\\gamma}=\\left\\{u\\in H_{per,odd}^1([0,L]);\\ \\int_0^{L}u^{p+2}dx=\\gamma\\right\\},$$\n\twhere $p$ is an even integer. Now, we need to find a minimizer of the constrained minimization problem\n\t\\begin{equation}\n\t\t\\label{infB1}\n\t\tr=r_{\\omega}=\\inf_{ u\\in Z_{\\gamma}}\\mathcal{B}_{\\omega}(u),\n\t\\end{equation}\n\twhere for each $\\omega>0$, $\\mathcal{B}_{\\omega}$ is given by $(\\ref{Bfunctional})$. We have the following result for the existence of odd periodic waves.\n\t\n\t\\begin{lema}\\label{minlem1}\n\t\tThe minimization problem \\eqref{infB1} has at least one odd nontrivial solution, that is, there exists $\\phi\\in Z_{\\gamma}$ satisfying\n\t\t\\begin{equation}\\label{minBfunc1}\n\t\t\t\\mathcal{B}_{\\omega}(\\phi)=\\inf_{ u\\in Z_{\\gamma}}\\mathcal{B}_{\\omega}(u)=r.\n\t\t\\end{equation}\n\t\\end{lema}\n\t\\begin{proof}\n\t\tThe proof of this result is similar to the proof of Lemma $\\ref{minlem}$.\n\t\\end{proof}\n\t\n\tAs before, by Lemma \\ref{minlem1} and Lagrange's Multiplier Theorem, we guarantee the existence of $C_2$ such that\n\t\\begin{equation}\\label{lagrange1}\n\t\t-\\phi''+\\omega\\phi=C_2\\phi^{p+1}.\n\t\\end{equation}\n\tSince $\\phi$ is nontrivial, a standard rescaling argument gives us that the Lagrange Multiplier can be chosen as $C_2=1$. By using similar arguments as determined above, we guarantee the existence of a convenient open interval $I$ and a smooth curve $\\omega\\in I\\mapsto\\phi\\in H_{per}^n([0,L])$, $n\\in\\mathbb{N}$, satisfying the equation\n\t\\begin{equation}\\label{ode-wave134}\n\t\t-\\phi''+\\omega\\phi-\\phi^{p+1}=0.\n\t\\end{equation}\n\t\n\t\\begin{lema}\\label{lemnLodd} Let $\\phi$ be the solution obtained by Lemma $\\ref{minlem1}$. We have that $\\ker(\\mathcal{L})=[\\phi']$ and $n(\\mathcal{L})=2$.\n\t\\end{lema}\n\t\\begin{proof} The existence of a smooth curve of odd periodic waves depending on $\\omega$ gives us by Proposition $\\ref{propsimp}$ that $\\ker(\\mathcal{L})$ is simple. \\\\\n\t\t\\indent We determine $n(\\mathcal{L})$. In fact, we see that $\\phi$ is a minimizer of $\\mathcal{B}_{\\omega}$ with one constraint in the Sobolev space $H_{per,odd}^1([0,L])$ constituted by odd functions. Since $\\langle\\mathcal{L}|_{odd}\\phi,\\phi\\rangle<0$, we obtain by Courant's Min-Max Principle that $n(\\mathcal{L}|_{odd})=1$. Next, solution $\\phi'$ is even having two zeros in the interval $[0,L)$. Using the standard Floquet theory in \\cite{Magnus}, we see that zero is not the first eigenvalue of $\\mathcal{L}|_{even}$, so that $n(\\mathcal{L}|_{even})\\geq1$. Again from the Floquet theory, since $\\phi'$ has only two zeros in the interval $[0,L)$, we see that $n(\\mathcal{L})\\leq2$. Therefore, the only possibility is that $n(\\mathcal{L})=n(\\mathcal{L}|_{odd})+n(\\mathcal{L}|_{even})=2$.\n\t\\end{proof}\n\t\n\t\\indent Analysis above gives us the following sentence: for $\\phi$ solution of $(\\ref{ode-wave13})$ with $\\omega_0>0$ and the corresponding single point $(\\omega_0,0)$ in the parameter regime, we have that $n(\\mathcal{L}_{(\\omega_0,0)})=2$ and $\\ker(\\mathcal{L}_{(\\omega_0,0)})=[\\phi']$. Therefore, by Theorem $\\ref{teo1}$ and Remark $\\ref{obsiso}$ one has that $\\theta\\neq0$ for all $(\\omega,A)$ in an open subset $\\mathcal{O}\\subset\\mathcal{P}$. This means that the solution $\\phi$ of $(\\ref{odekdv})$ satisfies $n(\\mathcal{L})=2$ and $\\ker(\\mathcal{L})=[\\phi']$ for all $(\\omega,A)\\in \\mathcal{O}$.\n\t\\begin{obs} Defining $\\varphi:=\\phi(\\cdot-L\/4)$, we can consider the solution obtained by Lemma $\\ref{minlem1}$ as being even and satisfying the mean zero condition $\\int_0^{L}\\varphi(x)dx=0$. In our paper and in order to avoid dubiety of notation, we keep the notation $\\phi$ instead of $\\varphi$ to indicate an even zero mean periodic wave satisfying equation $(\\ref{ode-wave134})$. \n\t\\end{obs}\n\t\\subsection{Positive periodic waves for the critical KdV - Proof of Theorem $\\ref{teoest}$-a)} We start our examples studying the spectral stability of periodic waves for $(\\ref{odekdv})$ with $A=0$ and $g(s)=s^5$. According with \\cite{AN2}, it is possible to determine a positive periodic wave with dnoidal profile as\n\t\\begin{equation}\\label{dn4kdv}\n\t\t\\phi(x)=\\frac{a{\\rm dn}\\left(\\frac{2K(k)}{L}x,k\\right)}{\\sqrt{1-b{\\rm sn}^2\\left(\\frac{2K(k)}{L}x,k\\right)}},\n\t\\end{equation}\n\twhere $K(k)=\\int_0^1\\frac{dt}{\\sqrt{(1-t^2)(1-k^2t^2)}}$ is the complete elliptic integral of the first kind. Parameters $a$, $b$ in $(\\ref{dn4kdv})$ and the wave speed $\\omega$ in $(\\ref{ode-wave134})$ depend smoothly on the modulus $k\\in(0,1)$ and they are given by\n\t\\begin{equation}\\label{apos}\t\t\n\t\ta=\\frac{[4(2k^2-1+2(1-k^2+k^4)^{1\/2})K(k)^2L^2]^{1\/4}}{L},\n\t\t\\ \\ \\ \n\t\tb=1-k^2-\\sqrt{k^4-k^2+1},\n\t\\end{equation}\n\tand\n\t\\begin{equation}\\label{wkpos}\n\t\t\\omega=\\frac{4K(k)^2\\sqrt{k^4-k^2+1}}{L^2}.\n\t\\end{equation}\n\t\\indent By Remark $\\ref{remN1}$, one has that $n(\\mathcal{L})=1$ and $\\ker(\\mathcal{L})=[\\phi']$. Therefore, by Theorem $\\ref{teo2}$ we guarantee the existence of a smooth \n\tsurface $(\\omega,A)\\in\\mathcal{O}\\mapsto\\phi_{(\\omega,A)}\\in\n\tH_{per,e}^n([0,L]),\\ n\\gg1,$ of periodic waves, all of them\n\twith the same period $L>0$.\\\\\n\t\\indent Let $L>0$ be fixed. According with the tables below (see also Figure 1), we obtain that $\\mathcal{I}>0$ and $\\det(\\mathcal{D})<0$ for all $k\\in(0,1)$. Then, one has the spectral stability of the periodic wave $\\phi$ in an open neighbourhood of $(\\omega,0)$ where $\\omega>\\frac{\\pi^2}{L^2}$ (see Corollary $\\ref{coroest}$).\n\t\n\t\\begin{obs}\n\t\tLet $L>0$ be fixed. Since $\\omega$ in $(\\ref{wkpos})$ is a strictly increasing function depending smoothly on the modulus $k\\in(0,1)$, we see for $k\\rightarrow 0^+$ and the fact $K(0)=\\frac{\\pi}{2}$ that $\\omega\\rightarrow \\frac{\\pi^2}{L^2}^{+}$. Therefore, we have the basic estimate $\\omega>\\frac{\\pi^2}{L^2}$ for the existence of periodic waves with dnoidal profile in $(\\ref{dn4kdv})$. In addition, if $\\omega>\\frac{\\pi^2}{L^2}$, the standard ODE theory enables us to conclude that $\\phi$ in $(\\ref{dn4kdv})$ is the unique positive solution which solves equation $(\\ref{odekdv})$ with $A=0$ and $g(s)=s^5$. Therefore, $\\phi$ in $(\\ref{dn4kdv})$ satisfies the minimization problem $(\\ref{infB})$\n\t\\end{obs}\n\t\n\t\n\t\\begin{obs} Another important fact: for a fixed $L>0$, we see that $\\det(\\mathcal{D})$ goes to zero when $k\\rightarrow 1^{-}$ (in our tables, this fact also occurs for different values of $L$ by taking $k$ closer to $1$). Thus, we ``recover\" the property $\\frac{d}{d\\omega}\\int_{-\\infty}^{\\infty}Q^2dx=0$, where $Q$ is the hyperbolic secant profile for the critical KdV equation with wave speed $\\omega>0$.\n\t\\end{obs}\n\t\\begin{table}[h]\n\t\t\\centering\n\t\t\\begin{tabular}{|c|c|c|}\\hline\n\t\t\t\\multicolumn{3}{|c|}{$L= 2\\pi$} \\\\\\hline\n\t\t\t$\\kappa_0$ & $\\mathcal{I}$ & $\\det(\\mathcal{D})$ \\\\\\hline\n\t\t\n\t\t\n\t\t\n\t\t\t$0.1$ & $ 5.4972$ &$-0.2692$ \\\\\\hline\n\t\t\t$ 0.3$ & $5.4932$ & $-0.2690$ \\\\\\hline\n\t\t\t$ 0.5$ & $ 5.4568$ & $-0.2669$\\\\\\hline\n\t\t\t$0.7$ & $5.2936$ & $-0.2573$\\\\\\hline\n\t\t\t$ 0.9$ & $4.6183 $& $-0.2148$\\\\\\hline\n\t\t\t$ 0.9999$ & $1.4287$& $-0.0269$\\\\\\hline\n\t\t\\end{tabular}\n\t\t\\centering\n\t\t\\begin{tabular}{|c|c|c|}\\hline\n\t\t\\multicolumn{3}{|c|}{$L= 20$} \\\\\\hline\n\t\t$\\kappa_0$ & $\\mathcal{I}$ & $\\det(\\mathcal{D})$ \\\\\\hline\n\t\n\t\t$0.1$ & $177.272 $ & $-2.7286$\\\\\\hline\n\t\t$ 0.3$ & $177.167$& $-2.7257$ \\\\\\hline\n\t\t$ 0.5$ & $175.993$ & $-2.7046$\\\\\\hline\n\t\t$0.7$ &$170.729$ & $-2.6074$\\\\\\hline\n\t\t$0.9$ & $148.949$ & $-2.1768$\\\\\\hline\n\t\t$0.9999$ & $46.0816$ & $-0.2772$\\\\\\hline\n\t\t\\end{tabular}\n\t\t\\end{table}\n\t\\begin{table}[h]\n\t\t\\centering\n\t\t\\begin{tabular}{|c|c|c|}\\hline\n\t\t\t\\multicolumn{3}{|c|}{$L= 50$} \\\\\\hline\n\t\t\t$\\kappa_0$ & $\\mathcal{I}$ & $\\det(\\mathcal{D})$ \\\\\\hline\n\t\t\n\t\t\n\t\t\t$0.1$ & $2770.18$ & $-17.0531$\\\\\\hline\n\t\t\t$ 0.3$ & $2768.24$ & $-17.0361$\\\\\\hline\n\t\t\t$ 0.5$ & $2749.89 $ & $-16.9039$\\\\\\hline\n\t\t\t$0.7$ &$2667.64$ & $-16.2963$ \\\\\\hline\n\t\t\t$ 0.9$ & $2327.33$ & $-13.6055$\\\\\\hline\n\t\t\t$ 0.9999$ & $720.16$ & $-1.7242$\\\\\\hline\n\t\t\\end{tabular}\n\t\t\\centering\n\t\t\\begin{tabular}{|c|c|c|}\\hline\n\t\t\t\\multicolumn{3}{|c|}{$L= 100$} \\\\\\hline\n\t\t\t$\\kappa_0$ & $\\mathcal{I}$ & $\\det(\\mathcal{D})$\\\\\\hline\n\t\t\n\t\t\n\t\t\t$0.1$ &$22159.9 $ & $-68.2148$\\\\\\hline\n\t\t\t$ 0.3$ & $22145.9 $ & $-68.1444$\\\\\\hline\n\t\t\t$ 0.5$ & $21999.1 $ & $-67.6155$\\\\\\hline\n\t\t\t$0.7$ &$21341.1$ & $-65.1854$\\\\\\hline\n\t\t\t$ 0.9$ & $18618.7$ & $-54.4218$\\\\\\hline\n\t\t\t$ 0.9999$ & $5761.48$ & $-6.8758$\\\\\\hline\n\t\t\\end{tabular}\n\t\t\n\t\\end{table}\n\n\t\\indent Using Maple program, we can plot the behavior of $\\langle\\mathcal{L}^{-1}1,1\\rangle$ in terms of the modulus $k\\in(0,1)$ for the case $L=20$. \n\t\n\t\\begin{figure}[!h]\\begin{center}\n\t\t\t\\includegraphics[scale=0.3]{spline-DN.jpg}\\label{Fig2}\n\t\t\t\\caption{\\small Graphic of $\\langle\\mathcal{L}^{-1}1,1\\rangle$ for $L=20$.}\n\t\t\\end{center}\n\t\\end{figure}\n\t\n\t\n\t\\subsection{Zero mean periodic waves for the critical KdV - Proof of Theorem $\\ref{teoest}$-b)} In what follows, we still consider $A=0$ and $g(s)=s^{5}$ in equation $(\\ref{odekdv})$. Our intention is to give a complete scenario for the spectral stability in this case.\\\\\n\t\\indent Let $L>0$ be fixed. An explicit even periodic wave satisfying the minimization problem in $(\\ref{infB1})$ is given by\n\t\\begin{equation}\\label{cnoid4kdv}\n\t\t\\phi(x)=\\frac{a{\\rm cn}\\left(\\frac{4K(k)}{L}x,k\\right)}{\\sqrt{1-b{\\rm sn}^2\\left(\\frac{4K(k)}{L}x,k\\right)}},\n\t\\end{equation}\n\twhere $a$, $b$ and $\\omega$ also depend smoothly on the elliptic modulus $k\\in(0,1)$ as\n\t\\begin{equation}\\label{ak}\n\t\ta=\\frac{2[(2-k^2+2\\sqrt{k^4-k^2+1})K(k)^2L^2]^{\\frac{1}{4}}}{L},\n\t\\end{equation}\n\t\\begin{equation}\\label{bk}\n\t\tb=-1+k^2-\\sqrt{k^4-k^2+1},\n\t\\end{equation}\n\tand\n\t\n\t\\begin{equation}\\label{wk}\n\t\t\\omega=\\frac{16K(k)^2\\sqrt{k^4-k^2+1}}{L^2}.\n\t\\end{equation}\n\n\\begin{obs}\n\tLet $L>0$ be fixed. Similarly as in the case of positive solutions, we also have the basic estimate $\\omega>\\frac{4\\pi^2}{L^2}$ for the existence of periodic waves with cnoidal profile in $(\\ref{cnoid4kdv})$. Moreover, when $\\omega>\\frac{4\\pi^2}{L^2}$, we obtain by the standard ODE theory that $\\phi$ in $(\\ref{cnoid4kdv})$ is the unique mean zero solution which solves equation $(\\ref{odekdv})$ with $A=0$ and $g(s)=s^5$. The uniqueness of solutions gives us that $\\phi$ in $(\\ref{cnoid4kdv})$ satisfies the minimization problem $(\\ref{infB1})$.\n\\end{obs}\n\tFor a fixed $\\omega_0>0$, we have already determined in the last subsection that the associated linearized operator around the periodic wave $\\phi$ given by $(\\ref{cnoid4kdv})$ satisfies $n(\\mathcal{L}_{(\\omega_0,0)})=2$ and $\\ker(\\mathcal{L}_{(\\omega_0,0)})=[\\phi']$ (see Theorem $\\ref{teo1}$ and Proposition $\\ref{propsimp}$). Therefore, we obtain the same spectral properties for all $(\\omega,A)$ in an open subset contained in the parameter regime. From Theorem $\\ref{teo2}$ we guarantee the existence of a smooth\n\tsurface $$(\\omega,A)\\in\\mathcal{O}\\mapsto\\phi_{(\\omega,A)}\\in\n\tH_{per,e}^n([0,L]),\\ \\ \\ \\ n\\gg1,$$ of periodic waves, all of them\n\twith the same period $L>0$.\\\\\n\t\\indent Since we have constructed the smooth surface\n\t$(\\omega,A)\\in\\mathcal{O}\\mapsto\\phi_{(\\omega,A)}\\in\n\tH_{per,e}^2([0,L])$ of even periodic waves which solves the\n\tnonlinear differential equation $(\\ref{travkdv})$ with $p=4$, the\n\tnext step is to calculate $n(\\mathcal{I})$ and $n(\\mathcal{D})$. Table below shows us the behavior of the quantity $\\mathcal{I}$ for some (fixed) values of $L>0$.\n\t\n\t\\begin{table}[h]\n\t\t\\centering\n\t\t\\begin{tabular}{|c|c|}\\hline\n\t\t\t\t\\multicolumn{2}{|c|}{$L= 2\\pi$} \\\\\\hline\n\t\t\t$\\kappa_0$& $\\mathcal{I}$ \\\\\\hline\n\t\t\t$ 0.0001$ & $ -0.47290$\\\\\\hline\n\t\t\t$0.1$ & $ -0.47296$\\\\\\hline\n\t\t\t$ 0.3$ & $ -0.4643$\\\\\\hline\n\t\t\t$ 0.5$ &$ -0.3886$\\\\\\hline\n\t\t\t$0.7$ & $-0.0985$ \\\\\\hline\n\t\t\t$0.739$ & $-0.0024$ \\\\\\hline\n\t\t\t$0.746$ & $0.0001$ \\\\\\hline\n\t\t\t$ 0.9$ & $ 0.4919 $\\\\\\hline\n\t\t\t$ 0.9999$ & $ 0.9958 $\\\\\\hline\n\t\t\t\\end{tabular}\n\t\t\\centering\n\t\t\\begin{tabular}{|c|c|}\\hline\n\t\t\t\t\\multicolumn{2}{|c|}{$L= 20$} \\\\\\hline\n\t\t\t$\\kappa_0$& $ \\mathcal{I}$ \\\\\\hline\n\t\t\t$ 0.0001$ & $-16.2331$\\\\\\hline\n\t\t\t$0.1$ & $-16.2298 $\\\\\\hline\n\t\t\t$ 0.3$ & $-15.9064$\\\\\\hline\n\t\t\t$ 0.5$ & $ -13.3279 $\\\\\\hline\n\t\t\t$ 0.7$ & $-3.6809 $\\\\\\hline\n\t\t\t$ 0.744$ & $-0.070 $\\\\\\hline\n\t\t\t$0.7449$ & $0.0095$\\\\\\hline\n\t\t\t$ 0.9$ & $15.7314$\\\\\\hline\n\t\t\t$ 0.9999$ & $44.3814$\\\\\\hline\t\n\t\t\\end{tabular}\n\t\t\\centering\n\t\t\\begin{tabular}{|c|c|}\\hline\n\t\t\t\\multicolumn{2}{|c|}{$L= 50$} \\\\\\hline\n\t\t\t$\\kappa_0$& $\\mathcal{I}$ \\\\\\hline\n\t\t\t$ 0.0001$ & $ -255.07$\\\\\\hline\n\t\t\t$0.1$ & $-255.01 $\\\\\\hline\n\t\t\t$ 0.3$ & $-249.89$\\\\\\hline\n\t\t\t$ 0.5$ & $ -209.40 $\\\\\\hline\n\t\t\t$0.7$ & $-58.24$\\\\\\hline\n\t\t\t$0.74521$ & $-0.0263$\\\\\\hline\n\t\t\t$0.74523$ & $0.0017$\\\\\\hline\n\t\t\t$ 0.9$ & $ 245.608$\\\\\\hline\n\t\t\t$ 0.9999$ & $ 71.1702$\\\\\\hline\n\t\t\t\n\t\t\\end{tabular}\n\t\t\\centering\n\t\t\\begin{tabular}{|c|c|}\\hline\n\t\t\\multicolumn{2}{|c|}{$L= 100$} \\\\\\hline\n\t\t$\\kappa_0$& $\\mathcal{I}$ \\\\\\hline\n\t\t$ 0.0001$ & $-2042.21$\\\\\\hline\n\t\t$0.1$ & $ -2041.74 $\\\\\\hline\n\t\t$ 0.3$ & $-2000.67 $\\\\\\hline\n\t\t$ 0.5$ & $-1676.52 $\\\\\\hline\n\t\t$0.7$ & $-466.79$\\\\\\hline\n\t\t$0.74528$ & $-0.1066$\\\\\\hline\n\t\t$0.74529$ & $0.0045$\\\\\\hline\n\t\t$ 0.9$ & $ 1964.63$\\\\\\hline\n\t\t$ 0.9999$ & $813.37$\\\\\\hline\n\t\t\t\n\t\t\\end{tabular}\n\t\t\n\t\\end{table}\n\n\t\n\t\n\t\n\tNow, we plot the graphic of $\\langle\\mathcal{L}^{-1}1,1\\rangle$ for the case $L=2\\pi$.\n\t\\newpage\n\t\n\t\\begin{figure}[!h]\\begin{center}\n\t\t\t\\includegraphics[scale=0.3]{spline.jpg}\\label{Fig3}\n\t\t\t\\caption{\\small Graphic of $\\langle\\mathcal{L}^{-1}1,1\\rangle$ for $L=2\\pi$.}\n\t\t\\end{center}\n\t\\end{figure}\n\t\n\t\n\t\n\tIt remains to calculate $\\det(\\mathcal{D})$. For the case $(\\omega,A)=(\\omega,0),$ we see that $\\langle\\mathcal{L}^{-1}\\phi,1\\rangle=-\\frac{d}{d\\omega}\\int_0^L\\phi dx=0$. If $\\mathcal{I}\\neq0$, we can use $(\\ref{Dmatrix})$ to obtain $$\\det(\\mathcal{D})=-\\frac{1}{2}\\ \\frac{d}{d\\omega}\\int_0^L\\phi^2dx.$$\n\t\n\t\\indent Formula (411.03) in \\cite{byrd} gives us that\n\t\\begin{eqnarray}\\label{norm}\\int_0^L\\phi^2 dx&=&\\frac{a^2L}{K(k)}\\int_0^{K(k)}\\frac{\\textrm{cn}^2(u,k)}{1-b\\ \\textrm{sn}^2(u,k)}du\\nonumber\\\\\n\t\t\\nonumber\\\\\n\t\t&=&\\frac{a^2L}{K(k)}\\ \\frac{\\pi(1-b)\\ [1-\\Lambda_0(\\beta,k)]}{2\\sqrt{b(1-b)(b-k^2)}}\\\\\n\t\t\\nonumber\\\\\n\t\t&=&\\frac{2\\pi\\sqrt{(k^2-2b)(1-b)}\\ [1-\\Lambda_0(\\beta,k)]}{\\sqrt{b(b-k^2)}}:=\\tau(k).\\nonumber\\end{eqnarray}\n\tHere, $\\Lambda_0$ indicates de Lambda Heumann function defined by\n\t$$\\Lambda_0(\\beta,k)=\\frac{2}{\\pi}[E(k)F(\\beta,k')+K(k)E(\\beta,k')-K(k)F(\\beta,k')],$$ \n\twhere $\\beta=\\arcsin\\displaystyle\\left(\\frac{1}{\\sqrt{1-b}}\\right)$ and $ k'=\\sqrt{1-k^2}.$\n\t\n\t\\indent Next, for all $k\\in(0,1)$ we have \n\t\\begin{equation}\\label{dwdk}\\frac{d\\omega}{dk}=-\\frac{16 K(k)\\ [K(k)\\ (k^4-3k^2+2)-2E(k)\\ (k^4-k^2+1)]}{L^2k(1-k^2)\\sqrt{k^4-k^2+1}}>0.\\end{equation}\n\tThus, \n\t\n\t\\begin{equation}\\label{dwnorn}\n\t\t\\frac{d}{d\\omega}\\int_0^L\\phi^2 dx=\\displaystyle\\frac{\\frac{d}{dk}\\int_0^L\\phi^2 dx}{\\frac{d\\omega}{dk}}=\\frac{\\tau'(k)}{\\frac{d\\omega}{dk}}.\n\t\\end{equation}\n\t\n\t\\indent We can plot the behavior of $\\tau'$ in terms of the modulus $k\\in(0,1)$ to conclude from $(\\ref{dwdk})$ and $(\\ref{dwnorn})$ that $\\frac{d}{d\\omega}\\int_0^L\\phi^2\\ dx>0.$ \n\t\n\t\\begin{figure}[!h]\\begin{center}\n\t\t\t\\includegraphics[scale=0.35]{Fig1.jpg}\\label{Fig1}\n\t\t\t\\caption{\\small Graphic of $\\tau'(k)$.}\n\t\t\\end{center}\n\t\\end{figure}\n\t\n\t\n\t\n\t\n\t\\indent Let $L>0$ be fixed. One sees that $\\det(\\mathcal{D})<0$ for all $\\omega>\\frac{4\\pi^2}{L^2}$ and $A\\approx 0$. Previous tables give us a threshold value $k_0\\approx 0.745 $ satisfying \n\t$\\mathcal{I}=\\langle\\mathcal{L}^{-1}1,1\\rangle=0$ at $k=k_0$ with $\\mathcal{I}<0$ if $k\\in(0,k_0)$ and $\\mathcal{I}>0$ if $k\\in(k_0,1)$. Therefore, we can apply Corollary $\\ref{coroest}$ to conclude that $\\phi$ is spectrally stable if $k\\in(0,k_0)$ and spectrally unstable if $k\\in(k_0,1)$.\n\t\n\t\n\t\n\t\n\t\\subsection{The Gardner equation - Proof of Theorem $\\ref{teoestG}$.} Now, we apply the arguments in the previous sections to study the spectral stability for the Gardner equation expressed in a general form as\n\t\n\t\\begin{equation}\\label{gardnereq1}\n\t\tu_t+\\alpha_1 (u^2)_x+\\alpha_2 (u^3)_x+u_{xxx}=0,\n\t\\end{equation}\n\twhere $\\alpha_1$ and $\\alpha_2$ are non-negative constants satisfying $\\alpha_1^2+\\alpha_2^2\\neq0$. When $\\alpha_2=0$, equation $(\\ref{gardnereq1})$ reduces to the well KdV equation while $\\alpha_1=0$, the same equation provides us the modified KdV equation. Results of spectral\/orbital stability of periodic waves in both cases have been discussed in \\cite{AN1}, \\cite{AN2}, \\cite{DK}, and references therein. \\\\\n\t\\indent Let $L>0$ be fixed and consider $\\alpha_1=\\alpha_2=1$. Explicit periodic waves $\\phi$ for this equation can be determined as\n\t\\begin{equation}\\label{cngardner}\n\t\t\\phi(x)=-\\frac{1}{3}+b{\\rm cn}\\left(\\frac{4K(k)}{L}x,k\\right),\n\t\\end{equation}\n\twhere $b$, $\\omega$ and $A$ are given by\n\t\\begin{equation}\\label{ak1}\n\t\tb=\\frac{4\\sqrt{2}kK(k)}{L},\n\t\\end{equation}\n\t\n\t\\begin{equation}\\label{wk1}\n\t\t\\omega=-\\frac{1}{3}-\\frac{16K(k)^2(1-2k^2)}{L^2},\n\t\\end{equation}\n\tand\n\t\\begin{equation}\\label{Ak1}\n\t\tA=\\frac{1}{27}+\\frac{144K(k)^2(1-2k^2)}{27L^2}.\n\t\\end{equation}\n\tNow, let us define $\\varphi(x)=\\frac{1}{3}+\\phi(x)=b{\\rm cn}\\left(\\frac{4K(k)}{L}x,k\\right)$. We see that $\\varphi$ solves the equation\n\t\\begin{equation}\\label{mkdv9}\n\t\t-\\varphi''+\\left(\\omega+\\frac{1}{3}\\right)\\varphi-\\varphi^3=0,\n\t\\end{equation}\n\tthat is, $\\varphi$ is a periodic wave with cnoidal profile for the modified KdV equation.\\\\\n\t\\indent Let $\\mathcal{L}_{\\varphi}$ be the linearized operator around $\\varphi$ for the equation $(\\ref{mkdv9})$ with $g(s)=s^3$. It is a surprising fact that\n\t$$\\mathcal{L}_{\\varphi}=-\\partial_x^2+\\omega+\\frac{1}{3}-3\\varphi^2=-\\partial_x^2+\\omega-2\\phi-3\\phi^2=\\mathcal{L},$$\n\twhere $\\mathcal{L}$ is the linearized operator around $\\phi$ for the Gardner equation $(\\ref{gardnereq1})$. Using the arguments in \\cite{AN1} and \\cite{DK}, we see that $n(\\mathcal{L})=n(\\mathcal{L}_{\\varphi})=2$ and $\\ker(\\mathcal{L})=\\ker(\\mathcal{L}_{\\varphi})=[\\phi']$.\\\\\n\t\\indent Now, since $\\mathcal{L}_{\\varphi}=\\mathcal{L}$, one sees that $\\langle\\mathcal{L}_{\\varphi}^{-1}1,1\\rangle=\\langle\\mathcal{L}^{-1}1,1\\rangle$. In addition, we obtain similarly as determined in \\cite{AN1} and \\cite{DK} that $\\langle\\mathcal{L}^{-1}1,1\\rangle<0$ for $k\\in (0,k_0)$ and $\\langle\\mathcal{L}^{-1}1,1\\rangle>0$ for $k\\in(k_0,1)$, where $k_0\\approx 0.909$.\\\\\n\t\\indent It remains to calculate $\\mathcal{D}$ in this specific case. First, we deal with\n\t\\begin{equation}\\label{norma1}\\int_0^L\\phi^2 dx=\\frac{L}{9}-\\frac{2b}{3}\\int_0^L\\textrm{cn}\\displaystyle\\left(\\frac{4K(k)}{L}x,k\\right) dx+b^2\\int_0^L\\textrm{cn}^2\\displaystyle\\left(\\frac{4K(k)}{L}x,k\\right) dx. \\end{equation}\n\t\n\t\\indent The middle term on the right-hand side of $(\\ref{norma1})$ has zero mean since\n\t\\begin{equation}\\label{int01}\\int_0^L\\textrm{cn}\\displaystyle\\left(\\frac{4K(k)}{L}x,k\\right) dx=\\frac{L}{2K(k)}\\int_0^{2K(k)}\\textrm{cn}(u,k)\\ du=0.\\end{equation} \n\tIn addition, the last term containing the quadratic power can be simplified as\n\t\\begin{equation}\\label{int02}\\int_0^L\\textrm{cn}^2\\displaystyle\\left(\\frac{4K(k)}{L}x,k\\right) dx=\\frac{L}{K(k)}\\int_0^{K(k)}\\textrm{cn}^2(u,k)\\ du=\\frac{L[E(k)-(1-k^2)K(k)]}{k^2K(k)}.\\end{equation}\n\t\n\t\\indent Collecting the arguments contained in (\\ref{ak1}), (\\ref{norma1}), (\\ref{int01}), and (\\ref{int02}), we conclude that\n\t\\begin{equation}\\label{norma2}\\int_0^L\\phi^2 dx=\\frac{L}{9}+\\frac{32 K(k)[E(k)-(1-k^2)K(k)]}{L}. \\end{equation}\n\t\n\t\\indent In order to calculate $\\frac{d}{d\\omega}\\int_0^L\\phi^2 dx$, we need to observe first that for all $k\\in(0,1)$ we have\n\t\\indent \\begin{equation}\\label{dwdk2}\\frac{d\\omega}{dk}=-\\frac{32K(k)[(1-2k^2)E(k)-(1-k^2)K(k)]}{k(1-k^2)L^2}>0.\\end{equation}\n\tThus, we are in position to give a convenient expression for $\\frac{d}{d\\omega}\\int_0^L\\phi^2 dx$ using the chain rule as \n\t\\begin{equation}\\label{derphi2}\\frac{d}{d\\omega}\\int_0^L\\phi^2 dx=\\frac{\\frac{d}{dk}\\int_0^L\\phi^2dx}{\\frac{d\\omega}{dk}}.\\end{equation}\n\tBy (\\ref{norma2}), we obtain for all $k\\in (0,1)$ that \\begin{equation}\\label{dnorm2}\\frac{d}{dk}\\int_0^L\\phi^2 dx=-\\frac{32[(1-k^2)K(k)(2E(k)-K(k))-E(k)^2]}{k(1-k^2)L}>0.\\end{equation}\n\t\n\t\n\t\\indent Finally, by (\\ref{dnorm2}) and (\\ref{dwdk2}) we deduce \\begin{equation}\\label{positivephi2}\\frac{1}{2}\\frac{d}{d\\omega}\\int_0^L\\phi^2 dx=\\frac{\\frac{d}{dk}\\int_0^L\\phi^2 dx}{2\\frac{d\\omega}{dk}}>0.\\end{equation}\n\t\n\t\\indent We are in position to calculate $\\mathcal{D}$. We have already determined that $\\langle\\mathcal{L}_{\\varphi}^{-1}1,1\\rangle=\\langle\\mathcal{L}^{-1}1,1\\rangle$. Similarly, since $\\varphi$ has zero mean it follows that $\\langle\\mathcal{L}_{\\varphi}^{-1}\\varphi,1\\rangle=\\langle\\mathcal{L}^{-1}\\phi,1\\rangle=-\\frac{d}{d\\omega}\\int_0^L\\phi dx=0$. In addition, a straightforward calculation also gives us that $\\langle\\mathcal{L}_{\\varphi}^{-1}\\varphi,\\varphi\\rangle=\\langle\\mathcal{L}^{-1}\\phi,\\phi\\rangle=-\\frac{1}{2}\\frac{d}{d\\omega}\\int_0^L\\phi^2 dx$. Thus, for $k\\neq k_0\\approx 0.909$ one has from $(\\ref{positivephi2})$ that $\\det(\\mathcal{D})$ is \n\t$$\\det(\\mathcal{D})=-\\frac{1}{2}\\frac{d}{d\\omega}\\int_0^L\\phi^2 dx<0.$$ Corollary $\\ref{coroest}$ can be applied to deduce the spectral stability of $\\phi$ when $k\\in(0,k_0)$ and the spectral instability when $k\\in (k_0,1)$. \n\t\n\t\n\t\n\t\\section*{Acknowledgments} S.A. was supported by CAPES. F.N. is partially supported by Funda\\c{c}\\~ao Arauc\\'aria 002\/2017, CNPq 304240\/2018-4 and CAPES MathAmSud 88881.520205\/2020-01.\n\t\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nClustering in metal alloys is known as the very early stage of first-order transformations within a bulk crystal, which largely influences the mechanical properties of quenched\/aged materials. Coherent with the matrix, small clusters of solute atoms have a significantly lower nucleation barrier than the terminal second-phase of a completely different crystal structure. Early clusters nucleate and grow in size forming the so-called GP-zones known to be a metastable precursor of the equilibrium phase~\\cite{christian}. The formation and growth mechanisms of the early clusters are poorly understood presently due to the lack of direct atomistic observations of structural changes during the transition process. However, inspired by the observations made on the quenched structures using transmission electron microscopy (TEM)~\\cite{marceau10,marceau10-2,nutyen67,ozawa70}, 3D atom probe~\\cite{sato04,esmaeili07,marceau10,marceau10-2,biswas11}, and positron annihilation~\\cite{somoza02,somoza10,marceau10-2} techniques, the formation of early clusters has been empirically associated with so-called quenched-in defects. Formed within the bulk crystal upon quenching, excess vacancies and\/or dislocations loops have been presumed to decrease the energy barrier for nucleation facilitating cluster formation~\\cite{esmaeili07,ozawa70,somoza10,marceau10,marceau10-2}.\n\nUnderstanding solute clustering mechanisms is of crucial importance to design an effective age-hardening process producing desired mechanical properties in alloys~\\cite{christian}. To our knowledge, no systematic study of the clustering mechanisms has been done using atomic-scale simulation methods such as molecular dynamics (MD) and Monte Carlo (MC) simulations, due to their main restrictions of accessing the relevant time scales of diffusional transformations. Dynamical calculations using classical density functional theory (CDFT) are also inefficient due to the high spatial resolution required to resolve the sharp density spikes in solid phases~\\cite{jaatinen09}.\n\nThe recently developed phase field crystal (PFC) method~\\cite{elder04,elder07,wu10,greenwood10,greenwood11} has shown promise for simulating structural transformations on diffusive time scales. This new formalism carries the essential physics of CDFT without the need to resolve the sharp atomic density peaks. In the most recent PFC formalism developed by Greenwood et al.~\\cite{greenwood10,greenwood11,greenwood11-2}, various crystal symmetries can be easily stabilized by construction of relevant correlation kernels. This approach preserves the numerical efficiency of the original PFC model and is able to dynamically simulate the precipitation of solid phases within a parent phase of different crystal symmetry~\\cite{greenwood10} and\/or chemical composition~\\cite{greenwood11-2}.\n\nThis letter proposes a new approach to study the clustering phenomenon that relies on atomic-scale simulations using the previously developed alloy PFC model of ref.~\\cite{greenwood11-2}. We explore the formation and growth mechanisms of early clusters in a quenched bulk lattice of a supersaturated Al-Cu alloy initially containing quenched-in defects such as dislocations.\n\n\\section{Model Structure}\nWe start with the free energy functional in the binary PFC model~\\cite{greenwood11-2},\n\\begin{align}\n\\frac{\\Delta\\freedm}{kT\\rho^o} = \\int f dr &=\n\\int \\bigg\\{\\frac {n^2}{2}-\\eta\\frac {n^3}{6}+\\chi\\frac {n^4}{12}+\\nline(n+1)\\Delta\\freedm _{mix} &-\\frac 1{2}n \\int dr'C^n_{eff}(|r-r'|)n'+\\alpha|\\vec{\\nabla}c|^2\\bigg\\} dr\n\\label{PFCalEnrgy2}\n\\end{align}\nwhere $n$ and $c$ represent reduced dimensionless atomic number density and solute concentration fields, respectively. $\\eta$ and $\\chi$ are coefficients added to fit the ideal energy to a polynomial expansion ($\\eta=\\chi=1$ describes a Taylor series expansion of the bulk free energy around the reference density) and\n\\begin{align}\n\\Delta\\freedm_{mix}=\\omega\\{c\\ln(\\frac{c}{c_o})+(1-c)\\ln(\\frac{1-c}{1-c_o})\\}\n\\label{Fmix}\n\\end{align}\nrepresents the energy density associated with the entropy of mixing. The coefficient $\\omega$ is introduced to fit the entropic energy away from the reference composition $c_0$. The parameter $\\alpha$ is a coefficient (taken as 1 in this study). These parameters are discussed further in ref.~\\cite{greenwood11-2}.\n\nFor a binary alloy, Greenwood et al.~\\cite{greenwood11-2} introduced the correlation function\n\\begin{align}\nC^n_{eff}=X_1(c)C^{AA}_2 + X_2(c)C^{BB}_2\n\\label{CorrEff}\n\\end{align}\n, where $X_1(c)=1-3c^2+2c^3$ and $X_2(c)=1-3(1-c)^2+2(1-c)^3$. $C^{AA}_2$ and $C^{BB}_2$ are correlation functions representing, respectively, contributions to the excess free energy for the situations where A atoms are in the preferred crystalline network of B atoms and B atoms which are in a structure preferred by A atoms. The correlation functions $\\hat{C}^{ii}_2(\\vec{k})$ are defined to have reciprocal space peaks (i.e. $k_j$, corresponding to the inverse of interplanar spacings) determined by the main families of planes in the equilibrium crystal unit cell structure for the $i^{th}$ component. Each peak is represented by the following Gaussian form of width $\\alpha_j$, modulated for temperature $\u03c3$ by a Debye-Waller prefactor which accounts for an effective transition temperature $\\sigma_{Mj}$~\\cite{greenwood11-2}.\n\\begin{align}\n\\hat{C}^{ii}_{2j}=e^{-\\frac{\\sigma^2}{\\sigma^2_{Mj}}}e^{-\\frac{(k-k_j)^2}{2\\alpha^2_j}} \n\\label{CorrF}\n\\end{align}\n\nThe equations of motion of the total density and concentration fields follow dissipative dynamics~\\cite{archer05}. The total mass density and total reference density per unit volume are defined as $\\rho=\\rho_A+\\rho_B$ and $\\rho^o=\\rho_A^o+\\rho_B^o$, respectively. Thus, the equations of motion can be written for $n$($=\\rho\/\\rho^o-1$) and $c$($=\\rho_B\/\\rho$) as $\\pxpy{n}{t}=\\vec{\\nabla}.\\{M_n\\vec\\nabla(\\frac{\\delta \\Delta F}{\\delta n})\\}+\\eta_n(\\sigma,t)$ and $\\pxpy{c}{t}=\\vec{\\nabla}.\\{M_c\\vec\\nabla(\\frac{\\delta \\Delta F}{\\delta c})\\}+\\eta_c(\\sigma,t)$, respectively~\\cite{greenwood11-2}. $M_n$ and $M_c$ are dimensionless kinetic mobility parameters (equal to 1 in this study). $\\eta_n(\\sigma,t)$ and $\\eta_c(\\sigma,t)$ are stochastic noise variables subsuming the role of fast atomic vibrations in density and concentration fields, respectively.\n\n\\section{Results}\n\\subsection{Phase diagram reconstruction}\nTo examine the equilibrium properties of this binary PFC model for a 2D Al-Cu system, we construct the phase diagram for the coexistence of two square phases. The coexistence lines between the respective phases are obtained by a common tangent construction of the free energy curves of solid and liquid at the reference density ($\\bar{n}=0$). Following Greenwood et al.~\\cite{greenwood11-2}, the free energy curves of the square phases are calculated using the two-mode approximation of the density fields which is defined by \n\\begin{align}\nn_i(\\vec{r})=\\sum_{j=1}^{N_i}A_j\\sum_{l=1}^{N_j}e^{2\\pi\\mathbf{i}\\vec{k}_{l,j}.\\vec{r}\/a_i}\n\\label{Density}\n\\end{align}\n, where the subscript $i$ denotes a particular solid phase with a lattice spacing $a_i$, and the index $j$ counts the $N_i$ modes of the $i$-phase. $A_j$ is the amplitude of mode $j$ and $l$ is the index over the group of reciprocal space peaks corresponding to mode $j$, $N_j$. Accordingly, $\\vec{k}_{l,j}$ is the reciprocal lattice vector normalized to a lattice spacing of 1, corresponding to each index $l$ in the family $j$. The free energy curve for each phase can be calculated as a function of the composition $c$ by substituting the above density field approximation into Eq.~(\\ref{PFCalEnrgy2}) and integrating over the unit cell. The resulting crystal free energy is then minimized for the amplitudes $A_j$. For the liquid phase, the amplitude $A_j$ is set to zero and the density is considered as the reference density ($\\bar{n}=0$). A more detailed description of this methodology is provided in ref.~\\cite{greenwood11-2}.\n\n\\begin{figure}[htbp]\n\\resizebox{3.3in}{!}{\\includegraphics{PhaseDiagram}}\n\\caption{The constructed phase diagram for a square-square system with the inset showing the Al-rich side of the experimental phase diagram of Al-Cu system taken from Ref.~\\cite{baker}. The parameters for ideal free energy contribution were $\\eta=1.4$ and $\\chi=1$, while $\\omega=0.005$ and $c_0=0.5$ for entropy of mixing. Widths of the correlations peaks are $\\alpha_{11Al}=2.4$, $\\alpha_{10Al}=\\sqrt{2}\\alpha_{11Al}$ (the required ratio to introduce isotropic elastic constants in an square phase~\\cite{greenwood11-2}), $\\alpha_{11\\theta}=2.4$ and $\\alpha_{10\\theta}=\\sqrt{2}\\alpha_{11\\theta}$. The peak positions for pure Al correspond to $k_{11Al}=2\\pi$, $k_{10Al}=\\sqrt{2}k_{11Al}$, $k_{11\\theta}=(81\/38)\\pi$ and $k_{10\\theta}=\\sqrt{2}k_{11\\theta}$. The effective transition temperatures are set to $\\sigma_{M11Al}=0.55$, $\\sigma_{M10Al}=0.55$, $\\sigma_{M11\\theta}=0.55$ and $\\sigma_{M10\\theta}=0.55$; The concentration $c$ is rescaled considering the Cu-content in the $\\theta$-phase.}\n\\label{fig:PhaseDiagram}\n\\end{figure}\n\nIn the Al-rich side of the experimental Al-Cu phase diagram, shown in the inset of Fig.~\\ref{fig:PhaseDiagram}, there is a eutectic transition between the Al-rich $\\alpha$-fcc phase and an intermediate phase $\\theta$ (containing $\\approx 32.5at.\\%$ Cu) with a tetragonal crystal structure. For 2D simulations, in order to approximate these equilibrium properties, we reconstruct the binary phase diagram of Al and $\\theta$, both with a square symmetry but differing in Cu-content. The lattice constant (and thus the reciprocal space peaks) of $\\theta$ is approximated by interpolating between those of Pure Al and Cu. The solid phase free energy is calculated with a variable lattice constant weighted by concentration $c$ using the interpolation functions $X_1$ and $X_2$. The polynomial fitting parameters in Eq.~(\\ref{PFCalEnrgy2}) (namely $\\eta$, $\\chi$ and $\\omega$) and width of various peaks ($\\alpha_j$) in the correlation kernel $\\hat{C}^{ii}_{2j}$ are then chosen so as to obtain the same compositions for $\\alpha$-phase solubility limit and eutectic point as those in the experimental phase diagram. \n\n\\subsection{Simulation of clustering}\nWith the equilibrium properties obtained above, simulations of clustering were performed on a rectangular mesh with grid spacing $dx=0.125$ and time step $dt=1$. Considering the lattice parameter of 1, each atomic spacing is resolved by 8 mesh spacings. The dynamical equations were solved semi-implicitly in Fourier space for higher efficiency. The initial conditions were chosen to study the proposed dominant role of quenched-in dislocation-type defects in the bulk crystal during the early stage precipitation in dilute Al-Cu alloys quenched from a solutionizing temperature~\\cite{nutyen67,ozawa70,somoza02,somoza10,desorbo58}. According to this hypothesis, dislocation loops, generated by excess vacancies, are responsible for local lattice distortions facilitating segregation and diffusion of Cu-atoms, while also driving the system towards a more thermodynamically-stable state~\\cite{nutyen67,ozawa70,somoza10,marceau10,marceau10-2}. Therefore, as initial conditions, we use a crystal lattice of uniform composition distorted by introducing dislocations.\n\n\\begin{figure}[htbp]\n\\resizebox{3.3in}{!}{\\includegraphics{Clustering}}\n\\caption{(colour online) PFC simulation of clustering phenomena on a system of 256$\\times$256 atoms after 225,000 time steps containing clusters with various sizes and concentrations; (a) The developed structure of a long-lived cluster; (b) The initially distorted structure; For graphical illustration, the concentration field is superimposed on the density field, and ranges from dark blue to dark red as the Cu-content increases.}\n\\label{fig:Clustering}\n\\end{figure}\n\nPFC simulation is performed for quench\/aging of Al-2at.$\\%$Cu from the solutionizing temperature of $\\sigma=0.17$ to $\\sigma=0.04$ with the initial conditions shown in Fig.~\\ref{fig:Clustering}(b). During the simulation, first, small clusters form with a slightly higher Cu-content than that of the matrix. As time progresses, some of these clusters shrink in size and concentration and a few get stabilized (e.g. the cluster shown in Fig.~\\ref{fig:Clustering}(a)). In contrast, as expected, quenching the same initial structure from the solutionizing temperature of $\\sigma=0.17$ to a temperature within the single-phase $Sq$-$Al$ region, i.e., $\\sigma=0.16$, leads to complete removal of distortion. \n\n\\section{Discussion}\n\n\\subsection{Evolution of clusters}\nThe dislocation-induced cluster structure shown in Fig.~\\ref{fig:Clustering}(a) is consistent with TEM observations in Al-1.7at.$\\%$Cu~\\cite{nutyen67} and Al-1.1at.$\\%$Cu-0.5at.$\\%$Mg alloys~\\cite{marceau10,marceau10-2}, where dislocation loops appear in the bulk lattice of the quenched structures. Using resistometric measurements and TEM techniques for Al-1.2at.$\\%$Si alloy, Ozawa and Kimura~\\cite{ozawa70,ozawa71} have associated the formation of dislocation (or vacancy) loops upon quenching to the coalescence of excess vacancies. They have further suggested that the solute atoms segregate towards the loops stabilizing them into solute clusters. Also, tracing vacancy clusters by positron annihilation, Somoza et al.~\\cite{somoza02,somoza10} have proposed that vacancy-Cu pairs are present at the quenched-state in Al-1.74at.$\\%$Cu alloy. To our knowledge, our PFC simulations are the first atomic-scale simulations to support the above hypothesis of vacancy\/dislocation-mediated solute clustering and nucleation mechanisms of early stage precipitation.\n\n\\subsection{Analysis of work of formation}\nWe further investigated the above mechanisms of cluster formation and growth by analyzing the system energetics for a long-lived cluster. To avoid possible finite size effects, a test with same conditions as those of the above simulation was performed on a larger system, e.g., 512$\\times$512 atoms. The strain field caused by the dislocations displacement fields is evaluated by\n\\begin{align}\n\\epsilon = \\sum_{i=1}^{N_{tri}} \\sum_{j=1}^{3}\\bigg(\\frac{a_{ij}-a_o}{a_o}\\bigg)\n\\label{Strain}\n\\end{align}\n, which is calculated over triangulated density peaks using the Delaunay Triangulation method. $N_{tri}$ is the number of triangles in the field, $a_o$ is the dimensionless equilibrium lattice parameter (the number of grid points resolving one lattice spacing, i.e., 8), $a_{ij}$ is the length of the $j^{th}$ side of the $i^{th}$ triangle. Small clusters, each accompanied by at least one dislocation, appear to be in local equilibrium with the matrix shown in Fig.~\\ref{fig:StrainField}(a)). During the simulation, following Fig.~\\ref{fig:StrainField}(b) and (c), cluster ``a\" continues to grow while, simultaneously, its accompanying dislocation climbs up towards nearby dislocations, creating larger local strain fields (i.e. 0.001, 0.0016 and 0.014 for cluster ``a\" in Fig.~\\ref{fig:StrainField}(a), (b) and (c), respectively). This mechanism of stress relaxation through solute segregation has been shown through phase-field studies by Leonard and Haataja~\\cite{leonard05} to be the main cause of alloy destabilization by structural spinodal decomposition in the presence of dislocations. Also, PFC studies of thin layers deposition by Muralidharan and Haataja~\\cite{muralidharan10} indicated that, due to the above mechanism, some immiscible alloys exhibit miscibility gap around the inter-layer interface in the presence of coherency stresses.\n\n\\begin{figure*}[htbp]\n\\resizebox{5.1in}{!}{\\includegraphics{StrainField}}\n\\caption{(colour online) (a-c) Snapshots taken at 3 different times showing the structural changes during formation of cluster ``a\"; (d) work of formation (evaluated from Eq.(~\\ref{Nucl.En})) vs. $R$ for increasing dislocation strain fields, i.e., increasing $\\Sigma b_i^2$; the dashed curve represents the work of formation for direct homogeneous nucleation of clusters in absence of dislocations, i.e., when $\\Sigma b_i^2=0$; (e) the variation of numerically evaluated total energy, $\\Delta G_{tot}$, and weighted average burger's vectors, $\\Sigma b_i^2$, due to the formation of cluster ``a\" in the above box;}\n\\label{fig:StrainField}\n\\end{figure*}\n\nThe effect of dislocations on the nucleation of clusters is investigated by considering the following form of work of formation:\n\\begin{align}\nW &= 2\\pi R\\gamma + \\pi R^2 (-\\Delta f + \\Delta G_s) - \\Delta G_{sr} + \\Delta G_d\n\\label{Nucl.En}\n\\end{align}\nwhere $R$ is the cluster radius in terms of number of lattice spacings and\n\\begin{align}\n\\gamma = \\frac {\\int_{Area} {\\alpha|\\vec{\\nabla}c|^2 dr}}{L} \n\\label{Surf.En}\n\\end{align}\nis a Cahn-Hilliard type interfacial free energy per unit length of the interface in 2D. $Area$ represents the area of the surface containing the cluster, and $L$ is the circumferential length of a round cluster of radius $R$. Assuming low dislocation density in the system, the interfacial free energy is taken to be solely chemical, neglecting the structural contributions~\\cite{Turnbull}.\n\\begin{align}\n\\Delta f=f^b-\\mu_c^b|_{c^b}(c^b-c^{cl})-f^{cl} \n\\label{DrivingForce}\n\\end{align}\nis the bulk driving force for nucleation of a cluster at a given concentration, where superscripts `$b$' and `$cl$' denote the bulk matrix and cluster ``phase'' quantities, respectively.\n\\begin{align}\n\\Delta G_s = 2 G_A \\delta^2 \\frac{K_B}{K_B+G_A} \n\\label{StrainEnergy}\n\\end{align}\nrepresents the strain energy for a coherent nucleus~\\cite{hoyt}, where $\\delta$ is the misfit strain and $G_A$ and $K_B$ are 2D shear and bulk moduli, respectively, calculated from PFC 2D mode approximation~\\cite{greenwood11-2}. \n\\begin{align}\n\\Delta G_{sr} = \\eta^2\\chi_d E A \\ln(R) \n\\label{StressRelaxE}\n\\end{align}\nis defined as the stress relaxation term due to segregation of solute into dislocations~\\cite{cahn57}, where $A= \\frac{G_A \\Sigma b_i^2}{4 \\pi (1-\\nu)}$, $\\nu = \\frac{E}{2 G_A}-1$, $\\eta=\\frac{1}{a}\\frac{\\partial a}{\\partial c}$ is the linear expansion coefficient with respect to concentration, $E$ is the 2D Young's modulus~\\cite{greenwood11-2}, $\\chi_d=(\\frac{\\partial^2 f}{\\partial c^2})^{-1}$, $\\Sigma b_i^2$ represents a weighted average of the burger's vectors around the dislocations accompanying the cluster and $a$ is the lattice parameter. The prefactor of the logarithm term, $\\eta^2\\chi_d E A$, approximates how strain energy is reduced due to solute segregation around a dislocation~\\cite{larch\u00e985}. \n\\begin{align}\n\\Delta G_d = \\zeta A \n\\label{Disl.E}\n\\end{align}\naccounts for the increase in the total system energy due to presence of dislocations, where $\\zeta$ is a prefactor of order ten giving the average amount of energy per dislocation core~\\cite{Hull}. Fig.~\\ref{fig:StrainField}(d) plots the evaluation of the above form of work of formation (Eq.~(\\ref{Nucl.En})) for cluster ``a\" at different mean concentrations up to that of the largest cluster shown in Fig.~\\ref{fig:StrainField}(c). The mean concentration of each cluster is estimated within a radius of $R$, defined by radially averaging the radius of the concentration field bound by a threshold of $[c^b+\\frac{\\sum^N {c-c^b}}{N}]$. The dashed curve in Fig.~\\ref{fig:StrainField}(d) represents the work of formation for direct homogeneous nucleation of clusters in absence of dislocations, i.e., $\\Sigma b_i^2=0$. The energy barrier for homogeneous nucleation seems to be smaller than that of the dislocation-assisted clustering by a single dislocation, i.e., $\\Sigma b_i^2=1$. However, according to the plots shown in Fig.~\\ref{fig:StrainField}(d), as Cahn~\\cite{cahn57} also pointed out, the barrier for formation of clusters on dislocations can be significantly reduced or even completely eliminated by increasing the magnitude of strain field around the dislocations (i.e. increasing $\\Sigma b_i^2$). Notably, the local minimum also shifts to larger nucleus sizes until it vanishes (i.e., work of formation continuously slops down vs. $R$). \n\nIt is noteworthy that, in the absence of quenched-in defects, nucleation of the second phase requires introduction of a thermally-activated noise to produce fluctuations in both density and concentration fields. Assuming dislocations are present in the bulk matrix of a supersaturated quenched alloy, in this study, we demonstrate how elasticity itself can drive the system into a phase transition. The influence of a thermally-activated noise on the transformation kinetics will be investigated in a future study through use of a well-defined noise algorithm. We have, however, observed in our simulations that in the case of a mismatch between the two species, such as in Al-Cu alloys, introducing a Gaussian noise to both density and concentration fields will not have a major impact on the overall path of the transformation. In other words, the phase transformation is mainly driven by the interactions between the elastic fields of the dislocations and the solute atoms.\n\nThe total work of formation, $\\Delta G_{tot}$, is also estimated numerically by measuring the change in the grand potential within a box engulfing cluster ``a\" during its formation and growth in the bulk matrix, i.e., \n\\begin{align}\n\\Delta G_{tot} = \\int_V\\Omega-\\int_V\\Omega^b =\\nline\\int_V {[f-\\mu_c c - \\mu_n n]} &-\\int_V {[f^b -\\mu_c^b c^b - \\mu_n^b n^b]}\n\\label{TotalE}\n\\end{align}\n. Here, $\\mu_c=\\frac{\\partial f}{\\partial c}$ and $\\mu_n=\\frac{\\partial f}{\\partial n}$ are diffusion potentials of concentration and density fields, respectively, and $V$ is the total volume. The above work of formation has contributions from the interfacial energy and driving force for formation of clusters (i.e., $\\Delta G_{tot}=\\Delta G_{\\gamma}-\\Delta G_v$), both of which include the elastic effects. Since the above box contains only one cluster, the calculated change in the grand potential accounts for the structural and compositional changes during the formation and growth of only cluster ``a\". While the growth of cluster ``a\" raises the local free energy, other parts of the system may undergo a process of annihilation and\/or shrinkage of sub-critical clusters and their accompanying dislocations leading to an overall decrease in the free energy of the system. As can be seen in Fig.~\\ref{fig:StrainField}(e), the total work of formation increases with the growth of cluster ``a\" until a maximum value, after which it starts to decrease. Also, as can be seen in this figure, the estimated values of $\\Sigma b_i^2$ at various sizes of cluster ``a\" closely corresponds to its analytical relationship with the cluster size at the local minima mapped on the energy plots of Fig.~\\ref{fig:StrainField}(d). Likewise the work of formation, during formation and growth of cluster ``a\", the value of $\\Sigma b_i^2$ reaches a maximum at the critical size of the cluster. \n\nAccording to our data, cluster ``a\" continuously grows in presence of dislocations implying that, at each sub-critical cluster size, the system is sitting at a local energy minimum. Since cluster ``a\" at each sub-critical size is in a local equilibrium with the matrix we call it a metastable precursor to the cluster ``a\" with a critical size. This is analogous to previous PFC studies of crystals solidification which show that metastable amorphous precursors emerge first due to their lower nucleation barrier than that of a crystalline solid~\\cite{toth11,tegze09}. In our case, the nucleation barrier is lowered by the effect of locally straining a sub-critical cluster (as a result of local accumulation of dislocations burger's vectors, as illustrated in Fig.~\\ref{fig:Clustering}(a)), making it thermodynamically favorable for the cluster to receive more solute atoms from the matrix and grow in size. \n\n\\begin{figure}[htbp]\n\\resizebox{3.4in}{!}{\\includegraphics{StrainELand}}\n\\caption{(colour online) Common tangent construction using mean-field free energy curves of unstrained (solid curve) and strained solid phases (dashed curves).}\n\\label{fig:StrainELand}\n\\end{figure}\n\n\\subsection{Metastable phase coexistence}\nThe metastable coexistence between a sub-critical cluster ``a\" and the matrix at the quench\/aging temperature is elucidated by evaluating the mean field free energy of a system comprising an unstrained matrix phase and strained solid phases with different magnitudes of distortion, i.e., a uniform strain. The free energy-concentration curve of a strained solid phase, at a given temperature, can be achieved by calculating the peaks of correlation kernel $\\hat{C}_{2j}^{ii}$, at locations slightly off those of the equilibrium density peaks, $k_j$, for a square structure. The introduced amount of strain is defined in Fourier space by\n\\begin{align}\n\\epsilon=|k-k_j|\/k_j\n\\label{FourierStrain}\n\\end{align}\n, where index $j$ denotes one family of planes in reciprocal space. As can be inferred from Fig.~\\ref{fig:StrainELand}, increasing the amount of strain from 0.0016 to 0.014 (corresponding approximately to the average strain within the cluster ``a\" shown in Fig.~\\ref{fig:StrainField}(b) and (c), respectively), raises the free energy in the strained solid. The free energy wells also shift to different concentrations of solute. Such a configuration admits a common tangent between the free energy curves of the unstrained matrix (e.g. the solid curve) and the distorted ones (e.g. dashed curves)~\\cite{larch\u00e985}, leading to a (metastable) multiphase coexistence with a lower free energy (as demonstrated in Fig.~\\ref{fig:StrainELand}). In other words, at each level of local strain, there is a thermodynamic driving force for a transformation from a single-phase structure of a strained matrix to a phase-coexistence between a strained cluster and an unstrained matrix. On the other hand, despite the fact that the above transformation is thermodynamically favorable, the configuration of energy plots in Fig.~\\ref{fig:StrainELand} implies that the driving force for nucleation is lower for the strained cluster (i.e., using the definition of $\\Delta f$ in Eq.~(\\ref{Nucl.En})). However, since the energy curves in this figure are derived from a mean-field PFC approximation, the illustrated phase-coexistence does not carry the effect of interfacial energy and only includes a mean-field sense of the misfit strain. These factors have a significant impact on the thermodynamics of phase-coexistence at cluster sizes smaller than that of the critical nucleus. In fact, the previously described stress relaxation term in the definition of work of formation (Eq.~(\\ref{StressRelaxE})), $\\Delta G_{sr}$, overcompensates for the effect of reduced driving force for formation of strained clusters. \n\n\\subsection{Clustering mechanism}\nBased on our PFC simulations, we propose the following mechanism of clustering: (1) Stress relaxation by segregation of solute atoms into highly-strained areas in the matrix, such as around dislocations, (2) strain-aided nucleation of sub-critical clusters at concentrations higher than that of the matrix and (3) subsequent growth and enrichment of sub-critical clusters into overcritical sizes, only if a sufficient strain field is preserved, to overcome the nucleation barrier. The above mechanism is consistent with the experimentally observed formation and enrichment of highly-strained coherent GP-zones in quenched-aged dilute Al-Cu alloys~\\cite{biswas11}, proposed as the initial step before precipitation of the semi-coherent and incoherent equilibrium $\\theta$-phases~\\cite{christian}. GP-zones in dilute binary Al alloys are normally known as coherent\/semi-coherent particles often with a crystal structure and composition similar to those of the final equilibrium precipitate~\\cite{christian,hoyt}. Our clusters possess the same chemical composition and lattice parameter as those of the equilibrium theta-phase pre-set by the relevant peaks in our correlation functions. Thus, they would represent an early-stage evolution of the so-called GP-zones. An investigation on the transformation of GP-zones into the subsequent metastable and equilibrium precipitates will be followed in a future study in 3D with more complex crystal structures. We expect to observe a gradual loss of coherency as GP-zones grow in size, as dictated by the energy arguments. We also note that we expect our results to hold qualitatively in 3D, since the same type of elastic effects are expected to appear around the dislocations regardless of their dimension and any possible partial splitting of dislocations around the clusters.\n\n\\section{Summary}\nIn summary, we showed that the alloy phase field crystal model of ref.~\\cite{greenwood11-2} which stabilizes different crystal structures can be used to simulate and analyze the mechanisms of clustering phenomenon in bulk lattice of quenched\/aged alloys. In accordance with the existing experimental observations, our simulations suggests that quenched-in defects, such as dislocations, significantly lower the energy barrier for nucleation of clusters. Furthermore, analysis of overall system energy and local energy changes reveal that the formation and growth of sub-critical clusters are thermodynamically favorable in conjunction with quenched-in mobile dislocations. Consistent with existing experiments, our simulations shed significant light on the elusive energetic mechanism of the growth and enrichment of early clusters which are the precursors of bulk precipitation. \n\n\\begin{acknowledgements}\nWe acknowledge the financial support received from National Science and Engineering Research Council of Canada\n(NSERC), Ontario Ministry of Research and Innovation (Early Researcher Award Program) and the Clumeq High Performance Centre.\n\\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzgicc b/data_all_eng_slimpj/shuffled/split2/finalzzgicc new file mode 100644 index 0000000000000000000000000000000000000000..636e7a73770c4ee1429308950b00c1a1c43a8fd8 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzgicc @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nThe gas kinetic temperature $T_{\\rm gas}$ is a central parameter for\nthe stability of the clouds, their evolution, and their chemistry\n\\citep[e.g.][]{Galli02, Keto05}. The properties of pre-stellar cores\nare of particular interest because they have a direct impact on the\nensuing star formation. The understanding of the temperature\nvariations within the cores is thus crucial for the interpretation of\nobservations and for the development of the theory of star formation\n\\citep{Bergin2007}.\n\nThe early models of the cloud cores were spherically symmetric and\nisothermal, in the case of both stable and collapsing cores \n\\citep[e.g.][]{Bonnor56,Shu77}. The spherical symmetry will eventually\nbreak down because of rotational flattening or core fragmentation but,\nin early evolutionary stages, it is often a reasonable first\napproximation \\citep[e.g.][]{Evans05,Alves01}. The observations of the\ndetailed temperatures structure can be more difficult because the\nmeasured intensities are non-linear averages of the emission\noriginating in different regions along the line-of-sight. The dust\ncolour temperature is known to be a biased estimate of the true grain\ntemperature $T_{\\rm dust}$ \\citep[][]{Shetty09,Malinen11}. The\nsituation is potentially even worse for $T_{\\rm gas}$ where, because\nof abundance variations, the observations may represent conditions in\na small fraction of the source. This is particularly true for cold\npre-stellar cores where many molecules are severely depleted\n\\citep[e.g.][]{Bergin97, Belloche04, Whittet10, Ford11, Parise11}.\nThere are measurements suggesting that, in the centre of a starless\ncore, the kinetic temperature can decrease down to $\\sim$6\\,K\n\\citep{Crapsi07,Harju08}. However, our knowledge of the radial\ntemperature variations depends heavily on models that also form the\nbasis for the interpretation of molecular line data.\n\nThe principles of the thermal balance of dense clouds are well\nunderstood, the main factors being the heating by cosmic rays, the\ncooling by line emission, and the energy exchange between gas and dust\n\\citep{Goldsmith78, Goldsmith01}. Despite the apparent simplicity of\nthe problem, reliable predictions of $T_{\\rm gas}$ are not easy to\nobtain because of the uncertainty of the fractional abundances and\ndust grain sizes, and the potentially complex radiative transfer\neffects. The problem of line cooling has been examined separately,\nwith the Large Velocity Gradient (LVG) approximation \\citep{Neufeld95}\nand with Monte Carlo methods \\citep{Juvela01}, the latter also\nenabling the study of the role of an inhomogeneous medium.\n\nIn this paper, we investigate the uncertainties of the modelled\n$T_{\\rm gas}$ profiles of dense cloud cores. The cores are known or\nare suspected to have significant radial variations in the gas phase\nabundances, grain size distributions, and the velocity field. We wish\nestimate the maximum effects on $T_{\\rm gas}$ that could arise from\nthese systematic changes. This is important for the evaluation of the\nuncertainties of chemical model and, more directly, the interpretation\nof any line observations of dense cores.\n\nWe describe our models in Sect.~\\ref{sect:model}. The results are\npresented in Sect.~\\ref{sect:results}, both for homogeneous models\n(Sect.~\\ref{sect:goldsmith}) and for a Bonnor-Ebert type cloud\n(Sect.~\\ref{sect:BE}), and our conclusions are presented in\nSect.~\\ref{sect:conclusions}. \n\n\n\n\n\\section{The Modelling} \\label{sect:model}\n\nWe examine the thermal balance of spherical clouds without small scale\ninhomogeneity. All models are divided to one hundred concentric\nshells, the innermost shell being 6\\% and the outermost shell 0.6\\% of\nthe outer radius. The same discretization is used for in continuum and\nline radiative transfer calculations and for the determination of the\ntemperature profiles. \nTo separate the radiative transfer effects from those\nof the density, we start by examining homogeneous clouds. The dust\ntemperatures are determined with Monte Carlo radiative transfer\ncalculations \\citep{Juvela05}, using the dust model of\n\\citet{Draine03} and the interstellar radiation field (ISRF) given by\n\\citet{Black94}. \nThe rate for the energy exchange between gas and dust\nis calculated from\n\\begin{eqnarray}\n\\Lambda_{g,d} = 2 \\times 10^{-33} [ n(H_2)\/cm^{3} ]^2\n (T_{\\rm gas}-T_{\\rm dust}) \n \\nonumber \\\\\n (T_{gas}\/10.0\\,K)^{0.5} \n \\, erg \\,cm^{-3} \\,s^{-1}.\n \\label{eq:gd}\n\\end{eqnarray}\nand the rate for cosmic ray heating from\n\\begin{equation}\n\\Gamma_{gas, cr} = 10^{-27} [ n(H_2)\/cm^{3} ] \\, erg\\, cm^{-3} \\,\ns^{-1}\n\\end{equation}\n\\citep{Goldsmith01}. Following \\cite{Goldsmith01}, we calculate the\nline cooling $\\Lambda_{\\rm gas}$ by $^{12}$CO, $^{13}$CO, C$^{18}$O,\nC, CS, and o-H$_2$O, multiplying the last two rates by 10 and two,\nrespectively, to take into account the cooling by other species. The\nradiative transfer problem is solved with Monte Carlo methods\n\\citep{Juvela01}, with abundances given in Table 1 in\n\\citet{Goldsmith01}. As part of the Monte Carlo simulation, the\nprogram saves the net radiative cooling rates for each model cell.\n\nAs a more realistic cloud model we examine a one solar mass,\nmarginally stable Bonnor-Ebert sphere with a value of the stability\nparameter $\\xi=6.5$ \\citep{Bonnor56}. The radial density profile is\ncalculated assuming a constant temperature of 10\\,K. The variations of\nkinetic temperature modify the density profile only slightly\n\\citep[][]{Evans01} and this will not affect the conclusions drawn\nfrom the models.\nAn external UV field can affect the cloud temperatures but, in\nprinciple, only in a thin surface layer. We consider this heating\nthrough the photoelectric effect in connection with some Bonnor-Ebert\nmodels. To estimate the photoelectric heating rate $\\Gamma_{\\rm PE}$,\nwe first calculate the energy that dust absorbs in the energy range\n6--13.6\\,eV. The values are obtained from the continuum radiative\ntransfer calculations. Following \\cite{Juvela03}, $\\Gamma_{\\rm PE}$ is\nobtained by multiplying these energies with an constant efficiency of\n$\\epsilon=0.029$. \nThe dust temperatures are calculated assuming the clouds are heated by\nthe normal ISRF, even when UV heating of gas is ignored. \nPhotoelectric heating is almost completely eliminated if the cloud is\nsurrounded by a dust layer of $A_{\\rm V}\\sim 2^{\\rm m}$ (see\nSect.~\\ref{sect:BE_T}). This would reduce the central dust\ntemperature only by $\\sim$0.5 degrees, less than the uncertainty\nassociated with the selection of a dust model.\n\nBelow we modify several model parameters to determine how their\nvariations are reflected on the gas temperature. These include (1) the\nthermal coupling between gas and dust, $\\Lambda_{g,d}$, that depends\non the grain size distribution, (2) the abundances of the cooling\nspecies that depend on the degree of depletion, (3) the cosmic ray\nheating $\\Gamma_{gas, cr}$ that depends on the rate of cosmic rays,\nand finally (4) the large scale infall motion and (5) the small scale\nvelocity field that both affect the radiative cooling $\\Lambda_{\\rm\ng}$. The degree of modification for each parameter is indicated in\nTable~\\ref{table:parameters} and discussed further in the following\nsection.\n\n\n\n\\section{Results} \\label{sect:results}\n\n\n\\subsection{Homogeneous models} \\label{sect:goldsmith}\n\nAs a first test, before modifying any parameters listed in\nTable~\\ref{table:parameters}, we compared our calculations with those of\n\\citet{Goldsmith01} who estimated the line cooling with LVG modelling\nassuming a velocity gradient of 1\\,km\\,s$^{-1}$\\,pc$^{-1}$. The cloud\nis taken to be well shielded from the external UV field so that the\nheating through the photoelectric effect can be neglected.\nCorresponding to the idea of the LVG models, we calculated $T_{\\rm\ngas}$ at the centre of homogeneous, microturbulent spheres with a\nradius of $R=1$\\,pc and the line Doppler width equal to\n1\\,km\\,s$^{-1}$. The correspondence was found to be good, mostly\nwithin $\\sim$10\\% in $T_{\\rm gas}$ (see Fig.~\\ref{fig:goldsmith}a), in\nspite several basic differences in the respective models. Firstly,\n\\cite{Goldsmith01} calculated the dust temperature for a constant\nshielding that, in the absence of gas-dust coupling resulted in a\ntemperature of $T_{\\rm dust}\\sim$6\\,K. In our calculations the dust\ntemperature was solved self-consistently. The model was assumed to be\nilluminated by the full ISRF, and the radiation field inside the cloud\nwas solved with radiative transfer calculations \\citep{Juvela05}. The\ndifference in dust temperature has no effect on $T_{\\rm gas}$ at low\n$n({\\rm H_2})$. At $n({\\rm H_2})=10^5$\\,cm$^{-3}$ $T_{\\rm dust}$ is\nclose to the value used by \\citet{Goldsmith01} and the $T_{\\rm gas}$\nvalues are in very good agreement. Secondly, in our model the\nexcitation temperature $T_{\\rm ex}$ varies as a function of radius so\nthat the LVG assumption of a uniform medium is not valid. This affects\nthe radiative connection between different parts of the model, is\nreflected in the excitation, and can thus affect the cooling rates.\nThirdly, the photon escape probabilities in the LVG model and our\nMonte Carlo model are not identical even when the total optical depths\nare equal. In our calculations, even at high optical depths, some\nphotons can always escape in the line wings. These effects do not\nappear important for the estimated central temperature of the cores.\nThere are also some differences. For example, in our calculations the\ngas-dust coupling has a smaller effect on $T_{\\rm dust}$ at high\ndensities, apparently because our dust cooling rate is higher than\nthat given by \\citet{Goldsmith01} Eq. 13.\n\n\nWe used the homogeneous models to investigate the effect of the grain\nsize distribution. The gas--dust coupling becomes significant around\n$n({\\rm H}_2)\\sim 10^5$\\,cm$^{-3}$ depending, however, on the total\ngrain area. Equation.~\\ref{eq:gd} is valid for a size distribution\n$dn\/da~\\sim a^{-3.5}$ with $a$ in the range 0.01--1.0\\,$\\mu$m. If the\nlower limit is reduced to 10\\AA\\, \\citep[e.g.][]{Li01}, without\nmodifying the gas-to-dust ratio, $\\Lambda_{g,d}$ increases by a factor\nof $\\sim$3. On the other hand, at the centre of dense cores the size\nof the large grains increases through grain coagulation while small\ngrains may disappear entirely \\citep[e.g.][]{Stepnik03, Ormel09,\nSteinacker10}. If the lower limit of grain sizes increases to 500\\AA,\nthe rate $\\Lambda_{g,d}$ is reduced by 60\\%. If the upper limit is\nfurther increased to 2\\,$\\mu$m, the effect is a factor of three. The\nvalue of $\\Lambda_{g,d}$ is similarly increased (decreased) by a\nfactor of three if the powerlaw exponent of the size distribution is\ndecreased (increased) by $\\sim0.75$, without modifying the size\nlimits.\n\nFigure~\\ref{fig:radial} illustrates the consequences for the gas\ntemperature. The solid lines show the radial profiles of $T_{\\rm gas}$\nand $T_{\\rm dust}$ for the model $n=10^5$\\,cm$^{-3}$ of\nFig.~\\ref{fig:goldsmith}. The increased photon escape probability\nalways decreases $T_{\\rm gas}$ at the cloud surface in spite of the\nincreasing $T_{\\rm dust}$ (Eq.~\\ref{eq:gd}). The other curves\ncorrespond to a three times stronger and a three times weaker gas-dust\ncoupling. The dash-dotted line is schematically the expected behaviour\nwhere, for given density, the coupling becomes weaker in the central\npart because of the grain growth. This is not to be taken so much as a\nmodel of an actual core as an illustration of the uncertainty of\n$T_{\\rm gas}$.\n\n\n\n\n\n\n\n\\subsection{Bonnor-Ebert spheres} \\label{sect:BE}\n\n\nAs more realistic models of dense cores, we examine critically stable\n0.5 and 1.0 solar mass Bonnor-Ebert spheres. We will first investigate\ndifferent factors that could affect their radial temperature profiles\nand then have a look at the predicted line profiles.\n\n\n\n\\subsubsection{Temperature profiles} \\label{sect:BE_T}\n\nWe investigate first the one solar mass Bonnor-Ebert sphere. The basic\nmodel has a constant turbulent line width of $\\sigma_{\\rm\nV}=1$\\,km\\,s$^{-1}$ and no large scale gas motions. The central\ndensity rises above $10^5$\\,cm$^{-3}$ but, because of the smaller\ncloud size, the column densities are lower than in the\nSect.~\\ref{sect:goldsmith} models of similar density. The photon\nescape probability increases outwards and the model predicts a\nsignificant decrease of $T_{\\rm gas}$ towards the cloud surface. This\nis in contrast with the \\citet{Galli02} calculations that employed a\nparameterization of LVG results to estimate $\\Lambda_{gas}$ (see their\nFig. 3; note also the difference in $T_{\\rm dust}$ due to a different\ndust model). The actual surface temperature will be sensitive to the\nabundance profiles and will depend on the amount of UV and cooling\nline radiation entering the cloud from the outside, both effects\nignored in the present model.\n\nFigure~\\ref{fig:BE_1.0} shows the quantitative effects resulting from\npossible variations of the abundances, the velocity field, and\n$\\Lambda_{g,d}$. The question of depletion was already examined by\n\\citet{Goldsmith01}. To illustrate the effect in the context of our\nmodel, we decrease the abundance of all cooling species by a factor of\nten in the cloud centre, within a radius of 0.025\\,pc\n(Fig.~\\ref{fig:BE_1.0}a). Because many lines are already optically\nthick, $\\Lambda_{\\rm g}$ is not expected to decrease linearly with the\nabundances. In the model, $T_{\\rm gas}$ increases by $\\sim$2 degrees\nin the inner part, with only a small effect reflected in the outer\ncloud.\n\nIn quiescent cores the line widths are sometimes observed to be close\nto that determined by thermal broadening \\citep[e.g.][]{Harju08}. When\nthe turbulent line width is reduced to $\\sigma_{\\rm\nV}=0.1$\\,km\\,s$^{-1}$ within innermost 0.025\\,pc, the central\ntemperature again increases by about two degrees \n(Fig.~\\ref{fig:BE_1.0}b). The observed linewidths will remain much\nbroader because of the thermal and opacity broadening and because of\nthe emission from the outer cloud layers where the turbulent line width\nis still $\\sigma_{\\rm V}=1.0$\\,km\\,s$^{-1}$.\n\nAlthough the Bonnor-Ebert model is static, we can introduce a large\nscale velocity field to check its importance on the escape of line\nemission. We add an infall velocity that is zero at the cloud surface\nand increases linearly to 1\\,km\\,s$^{-1}$ in the centre. This is a very\nsimplistic model of the velocity field but the magnitude of the\nvelocity gradient is realistic \\citep[e.g.][]{Zhou93} and the model\nshould capture the main effect on the radiative transfer. In the very\ncentral part of the model, the gas temperature by reduced little less\nthan one degree (Fig.~\\ref{fig:BE_1.0}c). Both the large scale and\nsmall scale velocity fields affect $\\Lambda_{\\rm g}$ through the line\noptical depths. \n\nThe density of the one solar mass model ranges from 1.4$\\times 10^4$\nto 2$\\times 10^5$\\,cm$^{-3}$, a region of densities where the coupling\nbetween gas and dust becomes important. As discussed in\nSect.~\\ref{sect:goldsmith}, changes in the grain size distribution are\nreflected on the efficiency of $\\Lambda_{g,d}$ so that it could be\ndecreased by up to a factor of three in the cloud centre and enhanced\nby up to a factor of three at the surface. It may be very improbable\nthat a single source would exhibit the full range of variation.\nHowever, Fig.~\\ref{fig:BE_1.0}d shows this case where the efficiency\nof $\\Lambda_{g,d}$ jumps from one extreme to the other again at\n0.025\\,pc radius. Because the dust temperature is {\\em above} the gas\ntemperature, the weaker coupling reduces $T_{\\rm gas}$ in the centre\nof the cloud. The effect is again of the order of one degree.\n\nFigure~\\ref{fig:BE_0.5} shows the corresponding results for a half\nsolar mass cloud where the central density is four times the value of\nthe previous model. The main differences result from the enhanced\ngas-dust coupling that, together with lower dust temperature, reduces\nthe central gas temperature by $\\sim 1$\\,K in the cases of low\nabundances and low velocity dispersion. On the other hand, the outer\ncloud is warmer by a similar amount, both because of the stronger\n$\\Lambda_{g,d}$ but also because of the general density dependence\nalready seen in Fig.~\\ref{fig:goldsmith}.\n\n\n\nAn external UV field can directly impact the cloud temperature at\nleast at its surface. We examined another set of one solar mass\nBonnor-Ebert spheres where the photoelectric heating was included and\ncalculated as described in Sect.~\\ref{sect:model}. The ISRF impinging\non the cloud is attenuated by $A_{\\rm V}$=0, 1, or 2$^{\\rm m}$,\ncorresponding to a shielding dust layer that is thought to exist\naround the actual model cloud. This applies to the calculation of both\n$T_{\\rm dust}$ and $\\Gamma_{\\rm PE}$. The resulting temperature\nprofiles inside the model are shown in Fig.~\\ref{fig:PEH}. The effect\nof photoelectric heating becomes negligible once the cloud is shielded\nby $A_{\\rm V}\\sim 2^{\\rm m}$. However, in an unshielded cloud the UV\nfield has a small effect, $\\sim$0.4\\,K, even in the cloud centre. This\nis caused not by a direct photoelectric heating but by the change of\nthe excitation in the outer cloud layers (cf.\nSect.~\\ref{sect:nonlocal}, Fig.~\\ref{fig:goldsmith}b). Again, the\nactual effect will depend critically on the molecular abundances in\nthe region heated by $\\Gamma_{\\rm PE}$.\n\n\n\nAs a final source of uncertainty we consider the rate of cosmic rays.\nThe heating term $\\Gamma_{gas, cr}$ is based on the assumption of a\nrate $\\zeta=3\\times 10^{-17}$\\,s$^{-1}$ but, in diffuse clouds, there\nare reports of rates that are up to two orders of magnitude higher \n\\citep[e.g.][]{McCall03, Liszt03, Shaw06}. To reflect this uncertainty\nwe calculated temperature profiles with $\\zeta$ scaled by factors 1,\n2, 5, and 10. Figure~\\ref{fig:CR} shows the resulting temperature\nprofiles, again for the one solar mass model. With the highest rate,\n$\\zeta=3\\times 10^{-16}$\\,s$^{-1}$, the central temperature has risen\nfrom the original $\\sim$7.5\\,K to $\\sim$17\\,K. For comparison, we show\ntemperatures calculated with the $\\Lambda_{gas}$ parameterization given\nby \\citet{Goldsmith01}. The calculation is done shell by shell using\nthe local density and the local dust temperature ($\\Gamma_{cr}$ and\n$\\Lambda_{gas, dust}$ are as in our calculations). These\n$\\Lambda_{gas}$ rates correspond to a model that has much higher\ncolumn density (per velocity interval and for given density) than the\nBonnor-Ebert spheres. Therefore, also the derived $T_{\\rm gas}$ values\nare higher. Furthermore, the parameterization does not catch the\nincreased photon escape probability at the cloud surface that, in our\nMonte Carlo calculations, results in the decrease of temperature in\nthe outer part.\n\n\n\n\n\n\n\n\\subsubsection{Spectral lines}\n\nFigures~\\ref{fig:spectra_1.0} and \\ref{fig:spectra_0.5} show $^{13}$CO\nand C$^{18}$O line profiles that were calculated for the models of\nFigs.~\\ref{fig:BE_1.0} and \\ref{fig:BE_0.5}. The $J=$1--0, $J=$2--1,\nand $J=$3--2 spectra were calculated as observed towards the centre of\nthe cloud with a beam with the FWHM equal to the half of the cloud\nradius. At 100\\,pc distance, this corresponds to $\\sim$52\\arcsec and\n$\\sim$26\\arcsec for the 1.0\\,$M_{\\sun}$ and the 0.5\\,$M_{\\sun}$\nmodels, respectively.\n\nIn the one solar mass model, the $^{13}$CO(1--0) beam averaged optical\ndepth is only $\\tau \\sim$2.5 through the cloud. In the 0.5\\,$M_{\\sun}$\nthe corresponding optical depth is $\\sim$7.5 meaning that there the\n$^{13}$CO transitions and C$^{18}$O lines originate partially in\ndifferent regions with different kinetic temperatures. One example of\nthis are the line profiles of the 0.5\\,$M_{\\sun}$ cloud with the\ninfall velocity. The $^{13}$CO(2-1) line shows the expected infall\nprofile while in the optically thinner $J=$1-0 line ($\\tau$=3.9 vs\n$\\tau$=6.5 for the second transition) the effect is weaker. The beam\naveraged C$^{18}$O(1--0) lines remain symmetric. When observed with a\npencil beam, the C$^{18}$O $J$=1--0 and $J$=2--1 lines would show\nslight asymmetry but with line profiles with stronger emission on the\nred shifted side. This is caused by the $T_{\\rm gas}$ which decreases\ntowards the cloud centre (see Fig.~\\ref{fig:BE_0.5}c). \n\n\nWe carried out LTE analysis of the $^{13}$CO and C$^{18}$O lines to\ncheck how accurate those column density estimates would be. We used\nthe method described by \\citet{Myers83}. The ratio of the $^{13}$CO\nand C$^{18}$O lines is used to calculate the optical depth\n$\\tau_{18}$, and the excitation temperature of C$^{18}$O,\n$T^{18}_{ex}$, is solved from the radiative transfer equation \n\\citep[see][Eqs. 3-4]{Myers83}. Assuming Gaussian line shapes, the\ncolumn density of C$^{18}$O in the $J=1$ state is \n\\begin{equation}\nN_{J=1} = 3.86 \\times 10^{14} \\tau_{18} J(T^{18}_{\\rm ex}) \\Delta v_{18}\n\\; {\\rm cm}^{-2}\n\\end{equation}\nwhere $J(T) = T_0 \/ [exp(T_0\/T) - 1 ]$ and $T_0 = 5.27$ K. The total\nC$^{18}$O column density is obtained by summing all levels, assuming\nthey are populated according to $T^{18}_{ex}$.\nTable~\\ref{table:LTE} summarizes the results when the line\nparameters are taken from Gaussian fits to the modelled spectral\nprofiles. The errors of these estimates are less than 5\\% which shows\nthat the temperature gradients have little impact on the column\ndensity estimates.\n\n\n\n\n\n\\section{The importance of non-local radiative couplings}\n\\label{sect:nonlocal}\n\nThe LVG method is based on the assumption that the excitation is\nconstant within the radiatively coupled volume. This is not a good\napproximation in dense cores where the gas velocities are small\nand the radial gradients of $T_{\\rm ex}$ are large. To illustrate the\npotential problem further in a schematic way, we took from\nFig.~\\ref{fig:goldsmith} the $n=10^5$\\,cm$^{-3}$ model and forced the\ninner part, $r<0.5$\\,pc, to local thermodynamic equilibrium (LTE) at\n5\\,K or 20\\,K. Fig.~\\ref{fig:LTE} shows the resulting $T_{\\rm\ngas}$ values in the outer part of the cloud, $r>0.5$\\,pc. The assumed\nexcitation of the inner cloud has a strong impact on the remaining\ncloud volume. The same effect was seen in Sect.~\\ref{sect:BE}, where\nthe increased surface temperature caused by photoelectric heating was\nreflected all the way to the centre of the optically thick cloud.\n\n\nOur final example of a system with non-trivial radiative couplings\nconsists of two Bonnor-Ebert spheres with properties identical to\nthose in Sect.~\\ref{sect:BE}. The spheres are touching each other and\nthe mutual shielding and exchange of radiative energy modifies the\ntemperature distributions. Figure~\\ref{fig:two_spheres} shows the\n$T_{\\rm gas}$ estimated without and with photoelectric heating. The\ncalculations were carried out with the same continuum and line\nradiative transfer programs as in the case of spherical models but\ndiscretizing the cloud onto a 128$^3$ cartesian grid. When\nphotoelectric heating is not considered, the absorption of line\nradiation from the other core increases the temperatures between the\ncores. The effect on $T_{\\rm gas}$ is $\\sim 1$ degree at the surface\nbetween the two cores. When UV heating is included, the mutual\nshielding becomes important and $T_{\\rm gas}$ is reduced by up to 3\\,K\nbetween the cores. Compared to the gravitational attraction between\nthe spheres, the force exerted by the resulting pressure asymmetry is\nof the order of one percent (assuming the change of $\\Delta T \\sim\n3$\\,K affects a few percent of the surface area). However, if the\nspheres were to partially coalesce, the affected area and the\ntemperature asymmetry would both increase making the effect\npotentially even dynamically important.\n\n\n\n\n\n\n\n\n\\section{Discussion} \\label{sect:conclusions}\n\n\nWe have modelled the gas and dust temperature of dense clouds. The\nresults emphasize the difference between the $T_{\\rm gas}$\ndistributions obtained with consistent radiative transfer calculation\nand those resulting from the blind application of LVG model results.\nIf LVG calculations are used, also the increased photon escape\nprobability near cloud surface must be taken into account. The effect\ncan be several degrees and this will have consequences for chemical\nmodels and the interpretation of observations. The photoelectric\nheating is capable of raising the surface temperature significantly\nbut only if the cloud is shielded by less than $A_{\\rm V}\\sim 1^{\\rm\nm}$ of extinction. However, the indirect impact of this heating is\nfelt well beyond the region directly penetrated by UV photons. The\ndetails of the radial temperature profile will depend on the\nabundances of the outer cloud layers and the external radiation field,\nboth in terms of line radiation and the UV flux. Further studies\ncoupling the chemistry and the modelling of the thermal balance are\nclearly needed.\n\nWe examined the effects on $T_{\\rm gas}$ resulting from such spatial\nvariation of molecular abundances, velocity field, and dust grain size\ndistribution that are expected in dense cores. Each factor alone can\nchange $T_{\\rm gas}$ by $\\sim$1\\,K or more. In the core, a strong\ndepletion of molecules and the reduction of turbulent motions is\ncapable of rising the temperature by several degrees. As pointed out\nby \\cite{Goldsmith01}, the effect of depletion will become less\nimportant at higher densities when $\\Lambda_{g,d}$ dominates over line\ncooling. The same applies to any effect resulting from the velocity\nfield. However, the increase of grain sizes will significantly\ndecrease the coupling between gas and dust. When $T_{\\rm dust}>T_{\\rm\ngas}$ and the density is close to $10^5$\\,cm$^{-3}$, this can\ncompensate some of the temperature increase predicted for the inner\ncore. There is no observational evidence of a temperature increase at\nthe centre of quiescent cores but also this possibility should be\nconsidered when interpreting observations. In more opaque clouds\n(especially in conjunction with dust coagulation) $T_{\\rm dust}$ will\nbe reduced below $T_{\\rm gas}$ and, at high enough densities, will\neventually force gas temperature down at the centre of starless cores.\nAn increase in the grain sizes can shift this transition to densities\nhigher than usually assumed. However, because of the long time scale\nof dust coagulation \\citep[e.g.][]{Ormel09}, the effect is often\nlikely to be smaller than in our model.\n\nThe value of $T_{\\rm gas}$ is particularly uncertain at the cloud\nsurface where the photoelectric heating and the dissociation of CO \nproduce strong temperature gradients. However, for the dense clouds\nthe largest source of uncertainty still appears to be the rate of\ncosmic ray heating. In the case of the one solar mass Bonnor-Ebert\nsphere, a factor of five increase in $\\zeta$ would raise the central\ntemperature by five degrees to $T_{\\rm gas}\\sim$12.5\\,K. This may\nalready be excluded by direct observational evidence of much lower gas\ntemperatures in dense clouds \\citep{Crapsi07,Harju08}. Nevertheless,\nthe theoretical prediction of the temperatures -- and temperature\nprofiles -- of dense cores still contains significant uncertainty.\n\nThe observed temperature gradients will not strongly modify the radial\ndensity distribution of cores nor significantly affect the core\nstability \\citep[e.g.][]{Harju08, Galli02}. However, in the critically\nstable Bonnor-Ebert models of one solar mass, the difference between\n{\\em isothermal} temperatures of 8\\,K and 10\\,K corresponds to a\nfactor of two increase in the central density. This shows that even\nsmall temperature changes are important in theoretical studies.\nIn Figs.~\\ref{fig:BE_1.0} and \\ref{fig:BE_0.5} we examined separately\nthe effect of various parameters on $T_{\\rm gas}$. In more dense\ncores, with $T_{\\rm dust} < T_{\\rm gas}$, the effects of the $\\chi$,\n$\\sigma_{\\rm v}$, and $\\Lambda_{g,d}$ parameters could accumulate,\nmaking the temperature gradients more pronounced. If gas is not\ncoupled to dust, $T_{\\rm gas}$ could in the central parts remain\nseveral degrees above the temperature of the outer cloud. However, in\nthe centre of the 0.5\\,$M_{\\sun}$ model cloud the gas temperature was\nalready largely determined by the gas-dust coupling. In that case the\nquestion of the dust properties becomes important because different\ndust models can, in the cloud centre, lead to $T_{\\rm dust}$ values\ndiffering by more than one degree (e.g. our $T_{\\rm dust}$ vs. the\n\\citep{Galli02} models).\nThe case of the two Bonnor-Ebert spheres in Sect.~\\ref{sect:nonlocal}\nsuggests that even small temperature anisotropies may sometimes play a\nrole in the long term evolution of clouds. Further studies are also\nrequired to find out how the early evolution of spherical cores is\nmodified relative to the isothermal case.\n\nHowever, the main importance of small $T_{\\rm gas}$ variations may\ncome via chemistry. The gas temperature directly affects chemical\nreaction rates and, in particular, the depletion onto dust grains.\nTherefore, the precise value of the gas temperature is relevant for\ninterpretation of both line and continuum data. Because the collision\nrates are only proportional to $\\sqrt{T_{\\rm gas}}$, the direct effect\non the time scales of depletion and grain mantle accumulation is\nsmall, less than 50\\% for the kind of temperature variations observed\nin our models. However, the effect on steady state abundances is more\nnoticeable. \\citet{Aikawa05} studied chemical evolution in collapsing\nclouds with initial conditions close to critical Bonnor-Ebert spheres.\nIf the evolution was slow enough, significant depletion was observed\nin the central parts of the model clouds. In particular,\n\\citet{Aikawa05} included a comparison of identical models (central\ndensity $3\\times 10^6$\\,cm$^{-3}$) with kinetic temperatures of 10\\,K,\n12\\,K, and 15\\,K. The ice composition was found to be very sensitive\nto the temperature and this was reflected in the gas phase abundances.\nA difference of two degrees could modify some abundances by a factor\nof two and, in the centre where the depletion becomes significant, by\nan order of magnitude or even more \\citep[e.g. NH$_{3}$ and\nN$_2$H$^{\\rm +}$, see][Figs. 2 and 6]{Aikawa05}. It is conceivable\nthat in some cases the depletion will be regulated by the temperature\nrise that results from the decreasing line cooling.\n\nWhen observations are analyzed, there is no guarantee that different\nlines (e.g., different isotopomers or different transitions of the\nsame molecule) would originate in identical gas volume. We already\nnoted that, depending on the opacity of the lines, the spatial\nresolution, and the radial $T_{\\rm gas}$ profile, it is possible to\nobserve both blueshifted and redshifted spectra towards a collapsing\ncloud. In the same fashion, the kinetic temperature measured, e.g.\nwith NH$_3$, may not be representative for other lines. This could\nlead to errors that are propagated to the derived column densities.\nThe evaluation of these uncertainties requires simultaneous modelling\nof the thermal balance, radiative transfer and, in particular, of the\nchemistry. Such a full study is beyond the scope of the present\npaper. However, we did carry out LTE analysis of the $^{13}$CO and\nC$^{18}$O lines calculated for the Bonnor-Ebert models. The\ntemperature gradients did not affect the column density estimates by\nmore than 5\\% and, at least in this case, the LTE analysis would\nproduce accurate estimates for the total $^{13}$CO and C$^{18}$O\ncolumn density.\nThus, the temperature variations examined in this paper mainly affect\nour expectations of the radial abundance profiles. These are important\nconsiderations when line data are used to estimate the central density\nor temperature of a dense core. The direct implications on the\nstability or dynamic evolution of the cores are probably of secondary\nimportance.\n\n\n\n\n\\acknowledgments\n\nMJ and NY acknowledge the financial support by the Academy of Finland Grant\n127015.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Calculation of the linear-response coefficients}\n\\label{sec:linear-response}\nIn what follows we calculate all the linear response coefficients entering Eqs.~(\\ref{w}) and~(\\ref{q}) in the weak-coupling limit between the qubit and the reservoir, following Ref.~\\cite{masterkeldysh}.\nWe rely on the adiabatic quantum-master equation approach under small temperature bias $\\Delta T$ to evaluate the coefficients $\\Lambda_{\\mu,\\nu}$, which enable the calculation of the net transferred heat and work defined in Eqs.~(\\ref{w}) and~(\\ref{q}). \n\nThe derivation of the master equation corresponds to solving the non-equilibrium problem of the driven qubit coupled to the two reservoirs \nexactly up to second order in the coupling constants $V_{\\alpha}^2$ and up to linear order in the velocities of the driving parameters $d_t \\vec{B}$.\n\n\\subsection{Reduced density matrix}\\label{appA:reducedDensityMatrix}\nThe Hamiltonian for the qubit can be expressed, after an appropriate unitary transformation $U$, in the instantaneous diagonal basis $|j\\rangle, j=1,2$ as follows:\n\\begin{equation}\n\\label{hs1}\n{\\cal H}_{\\rm qb}(t)=E_{1}(t) |1\\rangle \\langle 1| + E_{2}(t) |2\\rangle \\langle 2|, \n\\end{equation}\nwhere $E_{1,2}(t)=\\mp B_r(t)$ are the eigenvalues of the Hamiltonian of Eq.~(\\ref{qs}). We focus on the reduced density matrix $\\rho(t)$ expressed in this basis, in the slow-driving regime. We split it into a {\\em frozen} plus an {\\em adiabatic} contribution as \n\\begin{equation}\\label{split}\n\\rho(t) = \\rho^f + \\rho^a,\n\\end{equation}\nwhere\nthe first term corresponds to the description with the Hamiltonian frozen at a given time, for which the parameters take values ${\\vec B}$, while the second one corresponds to the correction $\\propto d_t \\vec B$.\n\nThe master equation for the corresponding matrix elements, $\\rho_{ij}(t)$, reads \\cite{masterkeldysh},\n\\begin{align}\\label{eq:masterEq}\n \\frac{d\\rho_{ij}(t)}{dt} = \\frac{i \\epsilon_{ij}(t)}{h} \\rho_{ij}\n +\\sum_{m,n,\\alpha} \n &\\left[ M^{jn}_{mi,\\alpha}(t) \\rho_{mn}\n + M^{in}_{jm,\\alpha}(t) \\rho_{nm} \\right. \\nonumber \\\\\n &\\left. - M^{mn}_{jm,\\alpha}(t) \\rho_{in}\n - M^{mn}_{mi,\\alpha}(t) \\rho_{nj}\n \\right]\n\\end{align}\nwhere we have introduced a shorthand notation for $E_i(t)-E_j(t) =\\epsilon_{ij}(t)$ for the instantaneous energy differences. The transition rates $M^{ju}_{ml,\\alpha}(t)$ for the present problem are given by\n\\begin{equation}\n\\label{eq:me_coefficients}\n M^{ju}_{ml,\\alpha}(t) = \\frac{\\xi_{\\alpha, ml}(t) \\xi_{\\alpha, ju}(t)}{h}\n \\left(\n n_\\alpha(\\epsilon_{ju}) \\Gamma_\\alpha(\\epsilon_{ju}) +\n [1+n_\\alpha(\\epsilon_{uj})] \\Gamma(\\epsilon_{uj})\n \\right),\n\\end{equation}\nbeing $\\alpha=h,c$ the reservoir indices, $n_\\alpha(\\varepsilon)=1\/\\left(e^{\\varepsilon\/(k_BT_{\\alpha})}-1\\right)$ the Bose-Einstein distribution function corresponding to the temperature of the $\\alpha$-reservoir and \n$\\Gamma_\\alpha(\\varepsilon>0) = \\gamma_\\alpha \\varepsilon e^{-\\varepsilon\/\\varepsilon_C}$ the bath spectral function. The functions $\\xi_\\alpha$ are defined for each bath as $\\xi_\\alpha=\\hat U(t) \\hat \\pi_\\alpha \\hat U^\\dagger(t)$, where $\\pi_\\alpha$ is defined in Eq.~\\eqref{qcont}. In the present problem we use $\\hat \\pi_{h,c} = \\hat \\sigma_{x,z}$. For this problem we consider Ohmic baths with a cutoff frequency $\\varepsilon_C$, and $\\Gamma_\\alpha(\\varepsilon\\leq 0)=0$. The value of $\\gamma_\\alpha$ depends on the coupling strength as $|V_{\\alpha}|^2$. This quantity defines the relaxation time between the q-bit and the reservoirs, $\\tau_{\\rm rel}\\propto \\gamma_{\\alpha}^{-1} \\varepsilon \\approx \\gamma_\\alpha^{-1} k_B T$. \nIn Eq. (\\ref{eq:masterEq}) we have neglected a term proportional to\n$|V_{\\alpha}|^2 |d_t \\vec B|$ \\cite{mak3,riwar2010nov,calvo2012dec,masterkeldysh}.\nThis equation can be written in a compact form as\n\\begin{equation}\n \\frac{d \\mathbf{p}(t)}{dt}= \\mathbf{M}(t) \\mathbf{p}(t),\n\\end{equation}\nby defining $\\mathbf{p}(t)=(\\rho_{11}(t),\\rho_{12}(t),\\rho_{21}(t),\\rho_{22}(t))^T$ with the contributions $\\mathbf{p}^f$ and $\\mathbf{p}^a$ as in Eq. \\eqref{split} and $\\mathbf{M}(t)$ accordingly.\n\nThe frozen contribution $\\mathbf{p}^f$ is calculated as a function of the time-dependent parameters $\\vec B$ only (frozen time), and satisfies the stationary (static) limit $\\frac{d \\mathbf{p}^f(t)}{dt} = 0$. Hence, it can be calculated from\n\\begin{equation}\n\\label{eq:pf}\n 0 = \\mathbf{M}({\\vec B})\\cdot \\mathbf{p}^f({\\vec B}),\n\\end{equation}\nwith the normalization condition $\\rho^f_{11}+\\rho^f_{22}=1$. On the other hand, \nthe adiabatic component satisfies\n\\begin{equation}\n\\label{eq:pa}\n \\sum_\\ell \\frac{\\partial \\mathbf{p}^f({\\vec B})}{\\partial B_\\ell} \\dot B_\\ell(t) = \\mathbf{M}({\\vec B})\\cdot \\mathbf{p^a}(t),\n\\end{equation}\nwith the normalization condition $\\rho^a_{11}+\\rho^a_{22}=0$. An important detail for the calculation of the partial derivatives appearing in the left side of Eq. \\eqref{eq:pa} is to take into account the effects of the basis dependence in $\\vec B$. A practical way to perform the derivative is by expressing $\\rho^f$ in the laboratory (fixed) basis first, and then rotate back to the instantaneous diagonal basis.\n\nWe can modify $\\mathbf{M}$ to include the normalization condition for $\\mathbf{p}^a$ in a single equation~\\cite{riwar2010nov,calvo2012dec}. Naming $\\mathbf{\\tilde M}$ the resulting matrix, we can finally invert Eq. (\\ref{eq:pa}) to obtain the closed expression\n\\begin{equation}\n\\label{pa}\n \\mathbf{p}^a(t) = \\sum_n \\mathbf{\\tilde M}^{-1}({\\vec B}) \\cdot \\frac{\\partial \\mathbf{p}^f}{\\partial B_n}\\dot B_n(t).\n\\end{equation}\n\n\\subsection{Linear-response coefficients}\\label{app-lin-res}\nWe can now use the described approach in (\\ref{appA:reducedDensityMatrix}) to obtain explicit expressions for the linear-response coefficients entering in the thermal geometric tensor $\\Lambda_{\\mu,\\nu}$. To this end we calculate the power developed by the ac-driving sources as follows,\n\\begin{equation}\n\\label{pac}\n P_{ac}(t) = \\mbox{Tr}\\left[\n \\dot {\\cal H}_{\\rm qb}(t)\n \\rho(t)\n \\right],\n\\end{equation}\nbeing\n\\begin{equation}\n\\label{Hdot}\n \\dot {\\cal H}_{\\rm qb}(t) = \\sum_\\ell \n \\frac{\\partial {\\cal H}_{\\rm qb}(t)}{\\partial B_\\ell}\n \\dot B_\\ell(t).\n\\end{equation}\nWe now write \n\\begin{equation}\n W=-\\int_0^{\\tau} dt P_{ac}(t), \n\\end{equation}\nand replace Eq.~(\\ref{split}) into Eq.~(\\ref{pac}), to finally use the solutions for $\\mathbf{p}^f$ and $\\mathbf{p}^a$ obtained in the previous subsection. The resulting expression can be directly compared to the formal relation in Eq. \\eqref{w} for $W$, where the matrix elements of the thermal geometric tensor are multiplied by different powers of $\\dot B_k$.\n\nIn order to discriminate the contribution of the developed power $\\propto \\Delta T$, and recalling that we are considering small temperature differences, we introduce the following extra expansion in frozen component:\n${\\rho}^f= {\\rho}^f_T + \\delta_T {\\rho}^f \\Delta T $. Therefore,\n\\begin{equation}\n \\Lambda_{\\ell, 3}(\\vec B) =-\n \\mbox{Tr} \\left[\n \\frac{\\partial {\\cal H}_{\\rm qb}}{\\partial B_\\ell} \\delta_T {\\rho}^f,\n \\right]. \n\\end{equation}\nwhile \n\\begin{equation}\n\\label{eq:lambda-diss-formal}\n \\Lambda_{\\ell, n}(\\vec B) =\n [\\underline{\\Lambda}]_{\\ell, n}(\\vec B) =\n \\mbox{Tr} \\left[\n \\frac{\\partial {\\cal H}_{\\rm qb}}{\\partial B_\\ell} \n \\left( \\mathbf{\\tilde M}^{-1}_T({\\vec B})\n \\frac{\\partial \\mathbf{p^f}_T}{\\partial B_n} \n \\right)\n \\right],\n\\end{equation}\nwhere we highlight with the sub-index $T$ that the quantities are calculated with the reservoirs at the same temperature $T$ and the quantity between parentheses is to be understood as a $2x2$ matrix.\nNotice that the contribution of Eq. (\\ref{pac}) evaluated with ${\\rho}^f_T$ corresponds to an equilibrium quantity. It represents the power developed by the conservative ac forces, and it, thus, leads to a vanishing value when averaged over the cycle. \n\n\n\nIn the same formalism used to derive the reduced density matrix, the heat current entering the reservoir $\\alpha$, calculated at the second order of perturbation theory \nin the coupling to the reservoirs and within the adiabatic regime, reads \\cite{calvo2012dec,masterkeldysh},\n\\begin{equation}\n\\label{Jheat}\n J_\\alpha(t) = \\sum_{m,n,u}\n \\epsilon_{un}(t)\\mbox{Re}\\left[ M^{nu}_{mn,\\alpha}(t) \\rho_{un}(t) \\right].\n\\end{equation}\nWe can follow the same logic as before to calculate this current at the first order in $\\Delta T$ and $\\dot{\\vec B}$. The first one is \n``thermal'' component \nassociated to the\nthe frozen components evaluated with a thermal bias $\\Delta T$, while the second one is the heat current ``pumped'' by the ac driving without temperature bias.\n\nThe linear-response net transported heat between the two reservoirs is\n\\begin{equation}\n Q=\\int_0^{\\tau} dt J_{\\rm c}(t)=-\\int_0^{\\tau} dt J_{\\rm h}(t).\n\\end{equation}\n\nWe follow the convention of Ref. \\cite{adiageo} and consistently with the definition \\eqref{q}, we focus on the current entering the cold reservoir to define the net transported heat.\nAssociating each term of Eq. \\eqref{q} with those arising from Eq. \\eqref{Jheat} upon substituting ${\\rho}$ by its expansion in $\\Delta T$ and $\\dot{\\vec B}$, we identify the linear coefficients,\n\\begin{equation}\n\\label{L3j}\n \\Lambda_{3,\\ell}(\\vec B) =\n \\vec{\\Lambda}_{\\ell}(\\vec B) = \\sum_{m,n,u}\n \\epsilon_{un} \n \\mbox{Re} \\left[ M^{nu}_{mn,{\\rm c}, T}\n \\left( \\mathbf{\\tilde M}_T^{-1}\n \\frac{\\partial \\mathbf{p^f}}{\\partial B_\\ell} \n \\right)_{un}\n \\right]\n\\end{equation}\nand\n\\begin{equation}\n\\label{L33}\n \\Lambda_{3,3}(\\vec B) =\n \\frac{\\kappa(\\vec B)}{T}\n = \\sum_{m,n,u} \\epsilon_{um}\n \\mbox{Re} \\left[ M^{nu}_{mn,{\\rm c}, T}\n \\delta_T \\rho^f \\right]_{un}\n\\end{equation}\nThese coefficients satisfy the following Onsager equations \\cite{ludovico2016feb,adiageo},\n\\begin{equation}\n\\Lambda_{3,\\ell}=-\\Lambda_{\\ell,3},\\;\\;\\;\\Lambda_{1,2}=\\Lambda_{2,1}.\n\\end{equation}\n\n\n\\section{Explicit expressions for \\texorpdfstring{$\\underline{\\Lambda}(\\vec B)$}{L(B)}, \\texorpdfstring{$\\vec{\\Lambda}(\\vec B)$}{L(B)} and \\texorpdfstring{$\\kappa(\\vec B)$}{k(B)} in the case of equal baths coupling}\n\\label{appendix:sameCouplingStrength}\n\nAssuming equal spectral density in the $L$ and $R$ baths, i.e.\n\\begin{equation}\n\\begin{split}\n \\Gamma_L(\\epsilon) &= \\bar \\Gamma_L \\epsilon e^{-\\epsilon\/\\epsilon_C} =\\Gamma(\\epsilon) \\\\\n \\Gamma_R(\\epsilon) &= \\bar \\Gamma_R \\epsilon e^{-\\epsilon\/\\epsilon_C} =\\Gamma(\\epsilon) \n\\end{split}\n\\end{equation}\n\nwith $\\bar \\Gamma_L = \\bar \\Gamma_R$ constants, we get explicit expressions for the complete geometric tensor. Using $(B_x, B_z) = B_r(\\sin \\phi, \\cos \\phi)$ and $\\beta=1\/(k_B T)$ we arrive to the following results.\n\n\\subsection*{Explicit expression for \\texorpdfstring{$\\underline{\\Lambda}$}{L(B)}}\nUsing Eq. \\eqref{hs1} and the expression for the reduced density matrix in the same basis, we can write for the terms with partial derivatives in Eq. \\eqref{eq:lambda-diss-formal}:\n\n\\begin{equation}\n\\frac {\\partial {\\cal H}_{\\rm qb}}{\\partial B_\\ell}=\n\\sum_j \\partial_\\ell E_j(\\vec{B}) |j\\rangle \\langle j|\n+ E_j(\\vec{B}) ( |\\partial_\\ell j\\rangle \\langle j|\n +|j\\rangle \\langle \\partial_\\ell j|)\n\\end{equation}\n\n\\begin{equation}\n \\frac{\\partial \\mathbf{p^f}_T}{\\partial B_n} =\n \\sum_{ij} \\partial_\\ell p^f_{T,ij}(\\vec{B}) |i\\rangle \\langle j|\n+ p^f_{T,ij}(\\vec{B}) ( |\\partial_\\ell i\\rangle \\langle j|\n +|i\\rangle \\langle \\partial_\\ell j|)\n\\end{equation}\n\nwith the notation $\\frac{\\partial}{\\partial B_\\ell} = \\partial_\\ell$. We note that the term $E_j(\\vec{B})$ depends only on the absolute value of the magnetic field $B_r$. In addition, the density matrix $\\mathbf{p}^f_T(\\vec{B})$ is a function of the energy spectrum, since it is computed in thermal equilibrium considering both baths at equal temperature $T$, and thus depends only on $B_r$ as well.\n\nOn the other hand, the operators $|\\partial_\\ell i\\rangle \\langle j|$ and $|i\\rangle \\langle \\partial_\\ell j|$ are nonzero only for variations in the polar coordinate $\\phi$ because the eigenvectors $|i\\rangle$ are associated to the unitary transformation that makes ${\\cal H}_{\\rm qb}$ diagonal, and do not change when $\\vec{B}$ stays in the same direction.\n\nThese facts allows us to separate the linear response coefficient $\\underline{\\Lambda}$ into a radial contribution ${\\lambda}_{r}$ (changes in the energy spectrum) and a polar contribution ${\\lambda}_{\\phi}$ (basis rotation). Inserting the solution to \\eqref{eq:pf} and \\eqref{pa} into Eq. \\eqref{eq:lambda-diss-formal} we get:\n\n\\begin{equation}\n\\label{eq:lambda_diss_radial}\n\\langle r| \\underline{\\Lambda} |r \\rangle =\n{\\lambda}_{r}(\\vec B)=\n\\frac{\\hbar \\beta \\text{sinh}(\\beta B_r)}{\\Gamma (2 B_r) \\text{cosh}^3(\\beta B_r)}\n\\end{equation}\n\n\\begin{equation}\n\\label{eq:lambda_diss_polar}\n\\langle \\phi| \\underline{\\Lambda} |\\phi \\rangle =\n{\\lambda}_{\\phi}(\\vec B)=\n\\frac{\\hbar \\Gamma (2 B_r)}{4 B_r^3}\n\\end{equation}\n\nEq. \\eqref{eq:lambda-diss-formal} finally reads:\n\n\\begin{equation}\n\\underline{\\Lambda} =\n\\lambda_{r}|r\\rangle \\langle r|+\n\\lambda_{\\phi}|\\phi\\rangle \\langle \\phi|.\n\\end{equation}\n\nWe now turn to analyze $L^2$, which quantifies the dissipated energy for a particular protocol, determined by $\\underline{\\Lambda}$ through Eq. \\eqref{eq:geometric_k}. In the upper panel of Fig. \\ref{fig:dissipation_change} we show the maximum eigenvalue, $\\mbox{max}\\left[\\lambda_{r}, \\lambda_{\\phi}\\right]$ as a function of $(B_z,B_x)$.\nThis plot reflects the behavior resulting from the analytical expressions of Eqs. \\eqref{eq:lambda_diss_radial} and \\eqref{eq:lambda_diss_polar}. \nAt every point, and depending on the values of $\\bar \\Gamma$ and $T$, this maximum eigenvalue corresponds to a pure polar or pure radial displacement of $\\vec{B}$.\nWithin the small circle plotted in dashed lines and outside the one in solid lines, the highest eigenvalue is $\\lambda_{\\phi}$, while within the two circles, $\\lambda_{r}$ is the largest one.\nThis means that for small $B_r < B_{r,\\rm low}$ as well as for large $B_r > B_{r, \\rm high}$, protocols leading to the smallest dissipation are those associated to changes in $B_r$, while \nfor $B_{r,\\rm low}< B_r < B_{r, \\rm high}$, protocols associated to rotations are the least dissipative ones.\nThe specific values $B_{r,\\rm low},\\; B_{r,\\rm high}$\ndepend on the temperature and the coupling between the qubit and the reservoirs, as shown in Fig.~\\ref{fig:dissipation} for two values of $\\Gamma$.\n\nThe precise shape of the interval $B_{r,\\rm low},\\; B_{r,\\rm high}$ is shown in the lower panel of Fig.~\\ref{fig:dissipation_change} and depends only on $\\bar \\Gamma$, while the final value has a linear dependence on $k_B T$. We see that for $\\bar\\Gamma \\gtrapprox 0.6$ the rotational dissipation dominates for all values of $|\\vec B|$.\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.45\\textwidth]{diss_change_panel.png}\n \\includegraphics[width=0.45\\textwidth]{diss_change.png}\n \\caption{Upper panel: Maximum eigenvalue $\\mbox{max}\\left[\\lambda_{r}, \\lambda_{\\phi}\\right]$, which ultimately defines the maximum possible dissipation at a given point $\\vec B$.\n Right panel: $B_{r,\\rm low}$ (dash line) and $B_{r,\\rm high}$ (solid line) as function of $\\bar\\Gamma$. The values $B_r$ for which $\\lambda_{r}=\\lambda_{\\phi}$ are determined by $\\bar \\Gamma$ only and scales linearly with $k_B T$.}\n \\label{fig:dissipation_change}\n\\end{figure}\n\n\n\\subsection*{Explicit expression for \\texorpdfstring{$\\vec \\Lambda$}{L} and \\texorpdfstring{$\\kappa$}{k}}\nAgain, plugging the expressions describing $\\mathbf{p}(t)$ given by Eqs. \\eqref{eq:pf} and \\eqref{pa} into Eq. \\eqref{L3j} we get for the qbit:\n\n\\begin{equation}\n \\vec \\Lambda_{1}(\\vec B) = -\\beta B_r \\sin ^3(\\phi ) \\text{sech}^2(\\beta B_r)\n\\end{equation}\n\\begin{equation}\n \\vec \\Lambda_{2}(\\vec B) = -\\beta B_r \\sin ^2(\\phi ) \\cos (\\phi ) \\text{sech}^2(\\beta B_r).\n\\end{equation}\n\nAnd lastly, using Eq. \\eqref{L33}, the thermal conductance is explicitly written as:\n\n\\begin{equation}\n \\kappa(\\vec B) = \\frac{4 B_r^2 \\sin ^2( \\phi ) \\cos ^2( \\phi ) \\Gamma (2 B_r) \\text{csch}\\left(\\frac{2 B_r}{T k_B}\\right)}{\\hbar k_B T}.\n\\end{equation}\n\\section{More discussion on the circular sector}\n\\label{app:pizzaProtocol}\nHere we present a more detailed study on the power and the efficiency of the circular sector defined in section \\ref{sec:results-C}. In Fig. \\ref{fig:pizza_power} we compute the geometrical value $A^2\/\\mathcal{L}^2$ as a function of the two parameters $R$ and $\\Omega$ that define the curves of this class.\n\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.5\\textwidth]{pizza_power.png}\n \\caption{Geometrical values $A^2\/\\mathcal{L}^2$ for the circular-sector protocols. We found a saturation value equal to $0.022 (k_B T)^2\/\\hbar$ which corresponds to the trajectory defined by $\\Omega=\\pi\/2$ and $R \\rightarrow \\infty$.}\n \\label{fig:pizza_power}\n\\end{figure}\n\nWe see from Fig. \\ref{fig:pizza_power} that for big enough $R$ the value of $A^2\/\\mathcal{L}^2$ depends only on $\\Omega$. This fact can be understood by looking at the upper panel of Fig. \\ref{fig:dissipation_change}: for $B_r>B_{r,{\\rm high}}$ we have $\\underline{\\Lambda} \\approx 0$, which implies that the dissipation outside the solid circle is negligible, and only the radial sections contained in the solid circle contributes significantly to $L^2$. Furthermore, since $\\underline{\\Lambda}$ has rotational symmetry, the value of $L^2$ is independent of the direction of the radial parts. These two observations leads us to the conclusion that $\\mathcal{L}^2$ has a saturation value in the circular sectors with $R>>B_{r,{\\rm high}}$. Finally, the dependence on $\\Omega$ is explained by looking at Fig. \\ref{fig:curl_lambda}, where it is clear that $A=A(\\Omega)$ for $R>>B_{r,{\\rm high}}$ as well, and the maximum occurs at $\\Omega=\\pi\/2$. The saturation value of $A^2\/\\mathcal{L}^2$ for the circular sector is found to be $\\max_{\\rm circ-sec} \\frac{A^2}{\\mathcal{L}^2} = 0.022 (k_B T)^2\/\\hbar$, which comparing to Fig. \\ref{fig:optimal_ellipses} is around $30\\%$ smaller than the ellipses case.\n\nIn Fig. \\ref{fig:pizza_efficiency} we show the computed maximum efficiency $\\eta_{\\rm max}$ in Eq. \\eqref{eq:Etamax} of the circular-sector protocol with $\\Omega=\\pi\/2$ as a function of $R$.\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width=0.5\\textwidth]{pizza_efficiency.png}\n \\caption{Computed efficiency for the circular sector with $\\Omega = \\pi\/2$ as a function of the $R$ parameter. At $R \\rightarrow \\infty$ we recover Carnot efficiency for the optimal parametrization.}\n \\label{fig:pizza_efficiency}\n\\end{figure}\n\nRecall from discussion in section~\\ref{sec:max_efficiency} that the limiting case of the circular sectors with the mentioned $\\Omega$ and $R\\rightarrow\\infty$ defines a Carnot cycle when optimized for maximum efficiency. This fact is clearly seen in Fig.~\\ref{fig:pizza_efficiency} as the saturation value of $\\eta_{max}\/\\eta_C$ goes to $1$ as $R >> B_{r,{\\rm high}}$.\n\n\n\\section{Introduction}\n\nThe development and implementation of thermodynamic processes in few-level quantum systems is currently a very active area of research. Thermodynamic cycles conceived for macroscopic working substances (WS), such as the Otto or Carnot cycle, are now realized \nin single atoms~\\cite{pekola2019mar,von2019spin,rossnagel2016apr,ronzani2018oct,deassis2019jun,peterson2019dec}\nand large theoretical efforts are devoted to its characterization and optimization at the microscopic scale \\cite{rezakhani2009aug,Schmiedl2007,sivak2012may,Kosloff2017,paolo,paolo2,abiuso2019non,cavina2021maximum,adiageo}.\nIn these standard thermodynamic cycles, the WS operates in four steps, of which two are in contact to reservoirs at different temperatures connected one at a time, while the other two steps consist in an evolution decoupled from the reservoirs. \nIt is however typically hard to fully isolate a quantum WS from the environment, which is required to emulate ideal classical cycles. This motivates the study of non-equilibrium systems, where the driven WS is permanently in contact with two or more reservoirs. Unlike standard thermodynamic cycles, these microscopic machines operate away from equilibrium during all the cycle. Thermoelectric devices \\cite{Benenti2017} as well as autonomous refrigerators \\cite{Palao2001,Youssef2009,Brunner2012} are seminal examples of this type of operation. \n\nWhen the WS is connected \nat the same time to two or more thermal reservoirs, it is permanently thread by a heat flux. Hence, the very operation as a machine relies on the mechanism of heat--work conversion\nin order to overcome this effect as well as the dissipation generated by the driving sources. The optimal machine is the one leading to the optimal balance between these two processes.\nIn quantum systems, the operation under a small temperature bias and\n``adiabatic driving\" through parameters which slowly vary on time is of paramount relevance, since this is an appealing scenario to control the non-equilibrium mechanisms. \nIn this regime, the period of the cycle is larger than any characteristic time of the quantum system, including the relaxation time between system and reservoirs~\\cite{thouless1983may,brouwer1998oct,zhou1999jan,moskalets2002nov,moskalets2004dec,reckermann2010jun,cavina2017slow}. \n\nRecently, it was proposed that the dissipation and the heat--work conversion mechanisms are respectively described by different components of the thermal geometric tensor. Furthermore,\n the heat--work conversion component can be expressed in terms of a Berry-type phase~\\cite{adiageo}, which has an associated Berry-type curvature~\\cite{berry1984quantal}, and similar ideas were followed in \\cite{Hino2021,Izumida2021}.\nHence, a length and an area in the parameter space can be defined. Besides, it is well known that dissipation and entropy production admit a geometric description in terms of the concept of thermodynamic length~\\cite{weinhold1975sep,salamon1980jul,salamon1983sep,nulton1985jul,schlogl1985dec,andresen1996nov,diosi1996dec,\ncrooks2007sep,campisi2012,VanVu2021}. This geometric approach has proven useful to optimize finite-time thermodynamic processes (examples can be found in ~\\cite{sivak2012may,Zulkowski2012,Zulkowski2013,bonanca2014jun} for classical and~\\cite{zulkowski2015sep,scandi2019oct,paolo2} for quantum systems), including the finite-time Carnot cycle \\cite{paolo,paolo2} and slowly driven engines~\\cite{brandner2020jan,millermoha,millerTUR,frim2021optimal,topical}.\nAs mentioned before, these cycles are characterized by the WS being coupled to a single reservoir or completely decoupled from reservoirs.\n\n\n\n\n \n \nThe aim of the present work is to optimise the performance of thermal machines with cycles in permanent contact to two or more reservoirs at different temperatures by a geometrical approach. To this end we combine the geometrical description of the two competing mechanisms of the non-equilibrium thermal machine (namely heat-work conversion and dissipation) in order to find optimal protocols for maximizing power generation of the heat-engine operation and the efficiency of the heat engine and refrigerator operational modes. We show that the problem of finding such optimal protocols reduces to an \\emph{isoperimetric problem}~\\cite{ros2001isoperimetric} (also studied as \\emph{Cheeger Problem}~\\cite{parini2011introduction,leonardi2015overview}), that is the task of finding the shape which maximizes the ratio between area and length. This is one of the oldest geometric problems in history, and was solved already by the ancient Greeks in the standard 2-dimensional Euclidean plane~\\cite{blaasjo2005isoperimetric}. Nevertheless, when the underlying area density or length metrics are nontrivial~\\cite{howards1999isoperimetric,morgan2005manifolds,rosales2008isoperimetric,carroll2008isoperimetric}, no general solution is known.\n\nWe illustrate these ideas in a prominent quantum system playing the role of the WS: a qubit driven by two parameters slowly changing in time and asymmetrically coupled to two thermal reservoirs at different temperature (see Fig.~\\ref{fig:pic_model}). We show analytically that the limiting value for the area in the parameter space is given by the celebrated\nLandauer bound~\\cite{landauer61,landauer88}, which has been the motivation of many studies including several experiments (see e.g. \\cite{berut2012experimental,jun2014high}). \nWe also find that, operating as a heat engine, the qubit thermal machine offers a very good ratio between generated power and efficiency in a wide range of parameters. \n\n\n\n\nThe paper is structured as follows. In Sec.~\\ref{sec:setup}, we introduce the set-up and define the relevant thermodynamic quantities to characterize the cycle. In Sec.~\\ref{sec:geom-opt}, we describe the underlying geometry of the system. In Sec.~\\ref{sec:timeopt} we describe the heat engine and refrigeration modes of the machine, and perform the optimization with respect to the driving time. In Sec.~\\ref{sec:results}, we develop the full optimization of the machine. We then compute in detail all the relevant quantities in a model of one of the most paradigmatic and simplest quantum engines, namely a driven qubit system (see Refs.~\\cite{karimi2016nov,paolo2,adiageo}). \n\n\n\n\\section{The setup and its thermodynamics}\n\\label{sec:setup}\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.5\\textwidth]{machine_model3.png}\n \\caption{Schematic configuration of the setup. A working substance WS is in contact with two reservoirs at different temperatures, $T_c$ and $T_h$. The state $\\hat \\rho$ of the system changes at slow but finite speed along a closed path defined by the Hamiltonian $\\mathcal{H}(\\vec B(t))$ in a quasistatic process.\n }\n \\label{fig:pic_model}\n\\end{figure}\n\n\nWe focus on the usual configuration where the WS operates in contact to two reservoirs at different temperatures $T_{\\rm h}$ (hot) and $T_{\\rm c}$ (cold), with $T_{\\rm h}=T+\\Delta T$ and $T_{\\rm c} \\equiv T$. A particular example, which will be studied in detail in forthcoming sections is sketched in Fig.~\\ref{fig:pic_model}.\nThe full system is described by the Hamiltonian \n \\begin{equation}\n {\\cal H}(t)= \\sum_{\\alpha={\\rm c,h}} \\left( {\\cal H}_{\\alpha} + {\\cal H}_{\\rm cont,\\alpha} \\right) + {\\cal H}_{\\rm WS}(t).\n \\label{hamtot}\n \\end{equation}\nThe Hamiltonian for the WS depends on time through a set of control parameters $B_j(t),\\; j=1,N$, which we enclose in a vector $\\vec{ B}(t)=\\left(B_1(t), \\ldots, B_N(t)\\right)$. Hence,\n${\\cal H}_{\\rm WS}(t)={\\cal H}_{\\rm WS}(\\vec{ B}(t))$. We are interested in cycles, so that we consider time-dependent protocols satisfying $\\vec{ B}(t+\\tau)= \\vec{B}(t)$, being $\\tau$ the period of the cycle.\nThe reservoirs are represented by the Hamiltonian\n\\begin{equation}\\label{qres}\n{\\cal H}_{\\alpha}=\\sum_{k}\\varepsilon_{k\\alpha}b_{k\\alpha}^\\dagger b_{k\\alpha}, \\;\\;\\; \\alpha={\\rm c, h},\n\\end{equation}\n with $b_{k\\alpha}$ and $b_{k\\alpha}^\\dagger$ being the annihilation and creation operators of a bosonic excitation.\nThe coupling is represented by\n\\begin{equation}\\label{qcont}\n{\\cal H}_{\\rm cont,\\alpha}=\\sum_{k}V_{k\\alpha}\\hat{\\pi}_{\\alpha}\\Big(b_{k\\alpha}+b_{k\\alpha}^\\dagger\\Big),\n\\end{equation}\nwhere $\\hat{\\pi}_{\\alpha}$ is a matrix with the dimension of the Hilbert space of the WS. \n\n\n\nThe crucial concepts that characterize the operation of the thermal machine are the work performed and the net heat exchanged between the two reservoirs during the cycle. The operation of the driven quantum system as a thermal machine in the presence of a temperature bias relies on the mechanism of heat--work conversion. \nIn the present case we make two main assumptions: \\begin{enumerate}[label=(\\roman*)]\n \\item slow driving \\cite{cavina2017slow}, characterized by a small rate of change of the driving parameters with time, $d_t \\vec{B}$ (short for $\\frac{d}{dt}\\vec{B}$), as well as\n \\item a small temperature bias $\\Delta T$ between the two reservoirs.\n\\end{enumerate} \nThis enables us to work in the linear-response regime with respect to $d_t \\vec{B}$ and $\\Delta T$. \n\nA natural theoretical framework in this context is the adiabatic linear response theory \nproposed in Ref.~\\cite{ludovico2016feb} in the geometric perspective of Ref.~\\cite{adiageo}. This formalism applies to the regime where the period of the cycle is much larger than the longest time-scale characterizing the WS coupled to the reservoirs. In most of the cases, such time scale is determined by the relaxation time $\\tau_{\\rm rel}$ of the WS with the reservoirs.\nMore precisely, the dynamical perturbation to the steady state $\\hat{\\rho}_B$ (corresponding to no driving, i.e. ``frozen\" value of $B$), can be estimated to $\\delta\\hat{\\rho}\\sim\\tau_{\\rm rel}(\\partial_B \\hat{\\rho}_{B}) d_t B$ (cf.~\\cite{cavina2017slow,ludovico2016feb,adiageo} or Appendix~\\ref{appA:reducedDensityMatrix}). Hence, this approach is useful when $\\tau\\gg \\tau_{\\rm rel}$. We also consider small temperature bias, such that $\\Delta T\/T\\ll 1$. This description leads to a linear relation between the relevant energy fluxes operating the cycle and the components of the vector \n$d_t{\\bf X}=(d_t \\vec{B},\\Delta T\/T)$. The relevant quantities are the net {\\em output work} \nand {\\em transferred heat} between the hot and cold reservoirs. They are, respectively, defined as the average over one period of the power developed by the driving sources, and the energy flux into the $\\alpha$-reservoir,\n\\begin{eqnarray}\nW &=&\n-\\int_0^{\\tau}dt\\;\\langle \\frac{\\partial {\\cal H}_{\\rm WS}}{\\partial \\vec{B}}\\rangle \\cdot d_t {\\vec{B}}, \\\\\nQ_\\alpha &=&\n-\\frac{i}{\\hbar}\\int_0^{\\tau} dt\\; \\langle \\left[{\\cal H}_{\\alpha},{\\cal H}\\right] \\rangle\\;, \n\\end{eqnarray}\nwhere $\\langle O \\rangle=\\mbox{Tr}\\left[\\rho O\\right]$, being $\\rho$ the global state of system and baths (which in general will be correlated due to the contacts). The corresponding expectations values are evaluated in linear response with respect to $d_t{\\bf X}$. In such regime $Q_{\\rm c}=- Q_{\\rm h}\\equiv Q$~\\footnote{While the heat flux at each reservoir contains\nboth transported and dissipated components, the latter contributes at the second order in $d_t{\\bf X}$~\\cite{adiageo}, as explicitly shown in Eq. (\\ref{w}).}.\nThe result is \n\\begin{align}\n W &=\\frac{\\Delta T}{T}\\int_0^\\tau dt\\;\\Vec{\\Lambda}\\cdot d_t \\vec{B} -\\int_0^\\tau dt\\; d_t\\vec{B} \\cdot \\underline{\\Lambda}\\cdot d_t \\vec{B} \\;,\n \n \\label{w}\\\\\n Q &=\\int_0^\\tau dt\\;\\Vec{\\Lambda}\\cdot d_t \\vec{B}+\\frac{\\Delta T}{T}\\int_0^\\tau\\ dt\\; \\kappa \\;.\n \n \\label{q}\n\\end{align}\nThese expressions can be derived in the adiabatic linear-response regime as from Ref.~\\cite{adiageo} and we defer the reader to that paper for further details.\nFor the moment it is enough to stress that \\{$\\underline{\\Lambda}$, $\\vec{\\Lambda}$, $\\kappa$\\} are all local functions of $\\vec{B}$, while they also depend on the coupling parameters, the density of states of the thermal baths and $T$.\n\nIn Eq.~(\\ref{w}), the first term represents the mechanism of {\\em heat--work} conversion and the second one corresponds to finite-time dissipation developed by the time-dependent controls. \n\n\nMoreover, in Eq.~\\eqref{q}, the transferred heat $Q$ also contains two terms associated to two different physical processes. The first one describes the heat exchange between the reservoirs related to the driving while the second one is the heat transport as a response to the temperature bias. \nNotice that the fundamental component for the thermal machine to operate is the heat--work conversion term $\\int_0^\\tau dt\\;\\Vec{\\Lambda}\\cdot d_t \\vec{B}$. In fact, without this component, the only surviving processes are the dissipation of the energy supplied by the driving forces and the trivial conduction of heat as a response to the thermal bias.\n\nThe different terms in Eqs.~(\\ref{w}) and (\\ref{q}) can be reinterpreted geometrically, as explained in the following Sec.~\\ref{sec:geom-opt}. This allows for the optimization of the thermodynamic protocols in terms of clear geometrical quantities.\n\nIt is important to notice that the second terms of Eqs.~(\\ref{w}) and (\\ref{q}) have a defined sign. \nIn our convention, $\\underline{\\Lambda}$ is positive definite since it is directly related to the entropy production rate~\\cite{adiageo}, which means that it is detrimental for the work output. Similarly, $\\kappa$ can be seen to be positive, as a consequence of the fact that this component of the transferred heat describes the flux from the hottest to the coldest reservoir. These are direct consequences of the second law of thermodynamics. Instead, the line integral $\\int_0^\\tau dt\\;\\Vec{\\Lambda}\\cdot d_t \\vec{B}$ may have any sign, depending on the driving protocol and it is enough to time-reverse the function $\\vec{B}(t)$ to flip the sign. As mentioned before, this term describes the {\\em heat--work} conversion process and its sign defines the type of operation of the machine.\nIn fact, when it is negative, the contribution of the first term of Eq.~(\\ref{q}) may overcome the heat flowing into the coldest reservoir and enable the operation of the machine as a \\emph{refrigerator}. This has an associated cost, described by the first term of Eq.~(\\ref{w}), which must be developed by the driving sources. In the opposite situation where\n$\\int_0^\\tau dt\\;\\Vec{\\Lambda}\\cdot d_t \\vec{B}\\geq 0$, the first term of Eq.~(\\ref{w}) may overcome the second one, enabling the mechanism of work output. This has an associated\nextra heat transfer from the hot to the cold reservoirs, which is accounted for the first term of Eq.~(\\ref{q}). This operation corresponds to a \\emph{heat engine}. \n\n\n\n\\section{Geometry of the problem}\n\\label{sec:geom-opt}\n\nWe now elaborate on the geometrical interpretation of the quantities presented in the previous section.\n\nFirst, we factorize the total duration $\\tau$ in the expressions Eqs.~\\eqref{w} and \\eqref{q}, such to decouple the time-rescaling from the geometrical contribution to the different quantities. \nIndeed by considering an adimensional time unit $\\theta$ such that\n\\begin{align}\n \\vec{ B}(t)=\\vec{ B}(\\theta \\tau)\\;,\\quad \\theta\\in[0,1]\\;,\n\\end{align}\nwe can define, identifying from now on the adimensional time derivative $\\dot{\\vec B} \\equiv{\\partial \\vec B}\/{\\partial\\theta}=\\tau d_t \\vec{B}$,\n\\begin{align}\n\\label{eq:geometric_A}\n A=& \\int_0^1 d\\theta\\; \\Vec{\\Lambda}\\cdot\\dot{\\vec B}\\;,\n\\\\\n\\label{eq:geometric_L}\n L^2=& \\int_0^1 d\\theta\\; \\dot{\\vec B} \\cdot \\underline{\\Lambda}\\cdot\\dot{\\vec B}\\;,\n\\\\\n\\label{eq:geometric_k}\n \\langle\\kappa\\rangle=& \\int_0^1 d\\theta\\; \\kappa\\;.\n\\end{align}\n\nAccordingly, Eqs. \\eqref{w} and \\eqref{q} can be expressed as follows,\n\\begin{eqnarray}\n\\label{eq:def_W}\n W \n &= & \\frac{\\Delta T}{T}A - \\frac{L^2}{\\tau} \\label{w1}\\\\\n Q \n &= & A+\\frac{\\Delta T}{T}\\tau\\langle\\kappa\\rangle. \\label{q1}\n\\end{eqnarray}\n\n\nThe names $A$ and $L^2$ are related the geometrical meaning of the quantities above, as we discuss below. The representation of Eq.~(\\ref{eq:geometric_A}) highlights the fact that $A$ corresponds to a Berry-type phase in the parameter space as discussed in Ref. \\cite{adiageo}. Notice, that, in order to have a non-vanishing value of $A$, at least two time-dependent parameters are necessary. \nThis is basically the same argument widely discussed in the literature of adiabatic charge pumping\n~\\cite{brouwer1998oct,avron2001nov,moskalets2002nov,arrachea2006}.\nIn addition, it is necessary to break some symmetries in the system to have a finite value of this closed integral \\cite{adiageo}, as discussed below.\n\nGiven that $\\vec B(\\theta)$ represents a closed trajectory in space, we can use Stokes' theorem --in a three-dimensional space or its corresponding generalization in higher dimensions-- to re-express the line-integral defining $A$\n\\begin{align}\n\\label{eq:definition:A}\n A=\\int_{\\partial\\Sigma} \\vec{\\Lambda} \\cdot d\\vec{B}\n =\\int_\\Sigma (\\vec{\\nabla}_B\\wedge \\vec{\\Lambda} )\\cdot d\\vec{\\Sigma}\\;,\n\\end{align}\nwhere $\\Sigma$ is a surface in the $\\vec B$ space, with boundary $\\partial\\Sigma$ coinciding with the control trajectory. In the case of having 4 or more parameters, Eq.~\\eqref{eq:definition:A} should be replaced by the Generalized Stokes' Theorem applied to differential forms in the appropriate dimension~\\footnote{Identifying $\\vec \\Lambda \\cdot d\\vec B$ with a 1-form $\\omega$ over $\\mathbf{R}^n$, we can express Eq.~\\eqref{eq:definition:A} as $A = \\int_{\\partial \\Sigma} \\omega = \\int_{\\Sigma} d\\omega$, where $d\\omega$ is the exterior derivative of $\\omega$.}.\nIn this representation, $A$ is the flux of the vector $\\vec{\\nabla}_B\\wedge \\vec{\\Lambda}$ through the area enclosed by the control trajectory, and can be also interpreted as the integral over this area weighted by the Berry curvature \\cite{berry1984quantal}. \nWe can therefore \\emph{think of $A$ as the area of the surface defined by the control trajectory} (with local weight depending on the Berry curvature). Note that this geometrical translation clarifies as well that $A$ depends $\\emph{only}$ on the geometry of the trajectory $\\vec B(\\theta)$: that is, not only $A$ is independent of $\\tau$, but it is also invariant under any reparametrization $\\theta'(\\theta)$ which might change the local speed and time spent on different points of the trajectory.\n\n\nConcerning $L^2$, \\emph{it can be interpreted as a length squared of the control trajectory} $\\vec{B}(\\theta)$, as it is clear from~\\eqref{eq:geometric_L} that it represents the integral of a quadratic form that defines a metric in the $\\vec B$ space. At the same time, given the presence of two time derivatives, $L^2$ can depend in general on reparametrizations $\\theta'(\\theta)$. However, $L^2$ represents losses due to dissipation in the driving -- see Eq.~\\eqref{w} -- and we are therefore interested in its minimum value, which can be obtained through a Cauchy-Schwarz inequality\n\\begin{align}\n\\label{eq:definition:L}\n L^2\\geq \\left(\\int_0^1 d\\theta\\;\\sqrt{\\dot{\\vec B} \\cdot \\underline{\\Lambda}\\cdot\\dot{\\vec B}}\\right)^2=\\left(\\int_{\\partial\\Sigma}\\sqrt{d\\vec{B}\\cdot \\underline{\\Lambda}\\cdot d\\vec{B}}\\right)^2\\equiv\\mathcal{L}^2\\;.\n\\end{align}\nThe lower bound $\\mathcal{L}$ is fully geometric (it depends solely on $\\partial\\Sigma$) and it is always achievable by choosing the time-parametrization $\\theta'$ such that $\\dot{\\vec B} \\cdot \\underline{\\Lambda}\\cdot\\dot{\\vec B}$ is constant. $\\mathcal{L}$ is a natural extension of the standard thermodynamic length \\cite{weinhold1975sep,salamon1980jul,salamon1983sep,nulton1985jul,schlogl1985dec,andresen1996nov,diosi1996dec,crooks2007sep,sivak2012may,deffner2013feb,bonanca2014jun,scandi2019oct} to non-equilibrium set-ups where the WS is simultaneously interacting with several baths. \n\nFinally, it is apparent that $\\langle\\kappa\\rangle$ Eq.\\eqref{eq:geometric_k} represents the simple average of a scalar number (the heat conductance) along the trajectory. In general it clearly also depends on reparametrizations of the adimensional time $\\theta'(\\theta)$, as the average can be arbitrarily close to the maximum value $\\kappa_{\\max}$ of the trajectoy, in case $\\theta'$ is such to spend almost all the time close to $\\kappa_{\\max}$. Similarly $\\langle\\kappa\\rangle$ can be arbitrarily close to the minimum value along the trajectory $\\kappa_{\\min}$.\n\n\\section{Performance of the machine and time-optimization}\n\\label{sec:timeopt}\n\nIn this section we discuss the different operation modes of the thermal machine, and introduce the relevant figures of merit for its characterization. \n\n\\subsection{Heat engine}\n\\label{sec:heat_eng}\nThe system described in the previous sections can be used to extract work from two reservoirs with a temperature bias. This is the \\emph{engine} operating mode of the system.\nWe write the power of the heat engine and its efficiency as\n\\begin{align}\n\\label{eq:pow_def}\nP=&\\frac{W}{\\tau}=\\frac{\\Delta T}{T}\\frac{A(1-\\frac{\\tau_D}{\\tau})}{\\tau}\\;,\\\\\n\\label{eq:eta_def}\n\\eta=&\\frac{W}{Q}=\\eta_C\n\\frac{1-\\frac{\\tau_D}{\\tau}}{1+\\frac{\\tau}{\\tau_k}}\\;,\n\\end{align}\nwhere we substituted Eqs.~\\eqref{w}-\\eqref{q} and we defined the dissipation and heat leak timescales\n\\begin{align}\n\\label{eq:timescales}\n \\tau_D=\\frac{T}{\\Delta T}\\frac{L^2}{A}\\;,\\quad \\tau_\\kappa= \\frac{T}{\\Delta T}\\frac{A}{\\langle\\kappa\\rangle}\\;.\n\\end{align}\nIn the previous expressions $\\eta_C=\\Delta T\/T$ is the Carnot efficiency.\nGiven the expressions above, we can optimize the duration of the cycles in order to maximize the power or the efficiency, obtaining correspondingly\n\\begin{align}\n \\tau_{P}=2\\tau_D,\\quad \\tau_\\eta=\\tau_D+\\sqrt{\\tau_D(\\tau_D+\\tau_\\kappa)}\\;.\n \\label{eq:opt_taus}\n\\end{align}\nWe see that the duration for maximum efficiency is always larger than the duration for maximum power.\nThe corresponding maximum power and efficiency at maximum power are\n\\begin{align}\n\\label{eq:Pmax}\nP_{\\rm max}=\\frac{1}{4}\\frac{(\\Delta T)^2}{T^2}\\frac{A^2}{L^2}\\;,\\quad \\eta_{P_{\\rm max}}=\n\\frac{\\eta_C}{2} \\frac{x-1}{x+1}\n\\end{align}\nwhile the maximum efficiency and power at maximum efficiency\n\\begin{align}\n\\label{eq:Etamax}\n\\eta_{\\rm max}=\n\\eta_C\\left(1-\\frac{2}{\\sqrt{x}+1}\\right)\\;,\\quad P_{\\eta_{\\rm max}}=\\frac{(\\Delta T)^2}{T^2}\\langle \\kappa\\rangle \\frac{(\\sqrt{x}-1)^2}{\\sqrt{x}}\\;,\n\\end{align}\nwith \n\\begin{equation}\\label{eq:x}\nx=1+\\frac{A^2}{L^2\\langle\\kappa\\rangle}.\n\\end{equation}\nSee Fig.~\\ref{fig:pow-eff_engine} for a summary and visual explanation of these results.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.5\\textwidth]{pow-eff}\n \\caption{\\underline{Engine mode: Power and efficiency vs. cycle duration.} \\\\\n The optimal operating region is the gray interval between the two dashed lines: indeed for any point outside the region, there is a point inside with both larger efficiency and larger power.\\\\\n In the limit of big heat leaks $\\langle\\kappa\\rangle$ the corresponding heat leaks timescale $\\tau_\\kappa$ \\eqref{eq:timescales} is small, and the difference between $\\tau_P$ and $\\tau_\\eta$ \\eqref{eq:opt_taus} shrinks. That is, when the heat leak is the dominant loss, power and efficiency maximization tend to coincide, as one could expect (this can be verified by direct inspection of \\eqref{eq:pow_def} and \\eqref{eq:eta_def}); the corresponding maximum efficiency is also small in this limit. \\\\\n In the opposite limit of no leaks $\\langle \\kappa\\rangle\\rightarrow 0$, $\\tau_\\kappa$ tends to infinite, and we recover the standard scenario in which power is maximized for a finite time, while the efficiency is maximum for $\\tau\\rightarrow\\infty$, where it tends to the Carnot efficiency, as the dominant loss is due to finite-time dissipation. For finite values of $\\langle \\kappa\\rangle$, the scenario is intermediate. In the plot $\\tau_D=1$ and $\\tau_\\kappa=2.5$.\n }\n \\label{fig:pow-eff_engine}\n\\end{figure}\n\n\n\n\\subsection{Refrigerator}\n\\label{sec:heat_pump}\nIn the \\emph{heat pump} or \\emph{refrigerating} mode, external work is supplied to the system to extract heat from the cold bath and transfer it to the hot one. Therefore we define the cooling power $P'$ and the coefficient of performance (COP) $\\eta'$ \n\\begin{align}\n\\label{eq:P'ref}\n P'=&\\frac{-Q}{\\tau}=A\\frac{1-\\frac{\\tau}{|\\tau_k|}}{\\tau}\\;,\\\\\n\\label{eq:eta'ref}\n \\eta'=&\\frac{Q}{W}=\\eta'_C \n \n \\frac{1-\\frac{\\tau}{|\\tau_k|}}{1+\\frac{|\\tau_D|}{\\tau}}\\;,\n\\end{align}\nwhere $\\eta'_C =T\/\\Delta T$ is the Carnot COP.\nThe difference with the engine operating mode is that in this case both $Q$ and $W$ are negative (heat is transferred against the thermal bias and work is performed \\emph{on} the system).\nWe have therefore $A<0$ which implies $\\tau_\\kappa<0$ and $\\tau_D<0$ are formally negative as well (which is the reason of the absolute values in the equations). \nBy direct inspection of~\\eqref{eq:P'ref} we see that the maximum power of such mode is unbounded, as in the limit $\\tau\\rightarrow 0$ the power tends to infinity. \nThe slow-driving approximation $\\tau_{\\rm rel}\/\\tau \\ll 1$ prevents us from analyzing the limit of arbitrary small $\\tau$ and a reliable analysis of the cooling power requires a description beyond linear response~\\cite{hajiloo2020detailed,mateos2021thermoelectric}. Thus, we focus only on maximizing the efficiency of this operation, for which we get\n\\begin{align}\n \\tau_{\\eta'}&=\\tau_{\\eta}=\\sqrt{\\tau_D(\\tau_D+\\tau_\\kappa)}-|\\tau_D|\\;,\n\\end{align}\n\\begin{align}\n \\eta'_{\\rm max}=\\frac{T}{\\Delta T}\\left(1-\\frac{2}{\\sqrt{x}+1}\\right)\\;,\n\\end{align}\n\\begin{align}\n P'_{\\eta'_{\\rm max}}=\\frac{\\Delta T}{T}\\langle \\kappa\\rangle \\sqrt{x}\\;,\n\\end{align}\nwith $x$ defined as in Eq.~(\\ref{eq:x}).\n\n\n\n\n\\section{Full optimization and the isoperimetric problem}\n\\label{sec:results}\nIn the previous section we showed how to choose the optimal duration for cycles of two kinds of thermal machines, and we derived formal expressions for the resulting powers and efficiencies. The resulting figures of merit still depend on the particular trajectory chosen for the cycle. Finding the fully-optimal solution is nontrivial, but we show in the following how the geometrical picture of the thermodynamics introduced in Sections~\\ref{sec:geom-opt} and \\ref{sec:timeopt}, helps in finding the most advantageous control trajectories to be exerted on the machine.\n\nAn interesting question in the present problem is whether we can find a protocol that maximizes the output power of the system. We have shown in Section~\\ref{sec:geom-opt} that, given a parametrization $\\vec B(\\theta\\tau)$ defined over $\\partial \\Sigma$ in the parameter space, we can compute the duration $\\tau$ that upper bounds the power for that protocol. The result is expressed in Eq.~\\eqref{eq:Pmax}.\nBesides, we know from the definition in Eq.~\\eqref{eq:definition:A} that the value of $A^2$ does not depend on reparametrizations $\\theta'$, while the value of $L^2$ can be lower-bounded by $\\mathcal{L}^2$ according to Eq.~\\eqref{eq:definition:L}.\nWith all these considerations, we find that the maximum power developed by a protocol moving along a curve $\\partial \\Sigma$ is expressed by\n\\begin{align}\n\\label{eq:PmaxC}\n \n \n \n P_{{\\rm max}} (\\partial \\Sigma)=\n \\frac{1}{4}\\frac{(\\Delta T)^2}{T^2}\n \\frac{A^2}{\\mathcal{L}^2}.\n\\end{align}\n\nEq.~\\eqref{eq:PmaxC} tells us that the problem of finding the maximum output power of the system is equivalent to the problem of maximizing the term~$A^2 \/ \\mathcal{L}^2$ over the set of all closed curves~$\\partial \\Sigma$ in the parameter space (known as isoperimetric or Cheeger problem \\cite{ros2001isoperimetric,parini2011introduction,leonardi2015overview}).\nThe optimization of this geometrical quantity is not a simple task in general, since one must choose a test curve $\\partial \\Sigma$ that maximizes~$A^2$, while keeping~$\\mathcal{L}^2$ small, being those quantities nontrivial functions of~$ \\partial \\Sigma$ when the corresponding metrics are not flat \\cite{howards1999isoperimetric,morgan2005manifolds,rosales2008isoperimetric,carroll2008isoperimetric}.\n\nFor what concerns the efficiencies, $\\eta_{\\rm max}$, $\\eta'_{\\rm max}$, $\\eta_{P_{\\rm max}}$ are all increasing functions of the same parameter $A^2\/(L^2\\langle\\kappa\\rangle)$. Like in Eq.~\\eqref{eq:definition:L} the denominator can be lower bounded with a Cauchy-Schwarz inequality\n\\begin{multline}\n\\label{eq:CS_eff}\n L^2\\langle \\kappa\\rangle=\\left(\\int_0^1 d\\theta\\; \\dot{\\vec{B}} \\cdot \\underline{\\Lambda}\\cdot\\dot{\\vec{B}}\\right)\\left( \\int_0^1 d\\theta\\; \\kappa\\right)\\\\\n \\geq \\left(\\int_0^1 d\\theta\\;\\sqrt{\\kappa} \\sqrt{\\dot{\\vec{B}} \\cdot \\underline{\\Lambda}\\cdot \\dot{\\vec{B}}}\\right)^2\\;.\n\\end{multline}\nIn complete analogy to Eq.~\\eqref{eq:definition:L}, the bound can be always saturated, by choosing a reparametrization $\\theta'(\\theta)$ such that $\\dot{\\vec{B}} \\cdot \\underline{\\Lambda}\\cdot\\dot{\\vec{B}}\/\\kappa$ is constant in time, and can be interpreted again as a length defined by an underlying metric\n\\begin{align}\n \\left(\\int_{\\partial \\Sigma}\\sqrt{d\\vec{B} \\cdot \\underline{\\Lambda_\\kappa} \\cdot d\\vec{B}}\\right)^2\\equiv \\mathcal{L}^2_\\kappa\\;,\n\\qquad\n\\underline{\\Lambda_\\kappa}=\\underline{\\Lambda}\\kappa\\;.\n \\label{eq:LambdaK_def}\n\\end{align}\nThe length $\\mathcal{L}_\\kappa$ is fully geometric, i.e. it depends only on the set of points defined by the trajectory $\\partial\\Sigma$, and the maximization of $\\eta_{\\rm max}$, $\\eta'_{\\rm max}$, $\\eta_{P_{\\rm max}}$ is also mapped to an isoperimetric problem\n\\begin{align}\n\\label{eq:etamaxC}\n \\max \\frac{A^2}{L^2\\langle\\kappa\\rangle}=\\max_{\\partial\\Sigma} \\frac{A^2}{\\mathcal{L}^2_\\kappa}\\;.\n\\end{align}\nThe geometric expressions \\eqref{eq:PmaxC} and \\eqref{eq:etamaxC}, which map the thermodynamic optimization to an isoperimetric (Cheeger) problem, are the main results of this paper.\n\n\\section{A qubit thermal machine}\\label{sec:qubit}\nWe will exemplify these results for the specific case of a driven qubit, in which case, the Hamiltonian for the working substance entering Eq.~(\\ref{hamtot}) is ${\\cal H}_{\\rm WS}(t)={\\cal H}_{\\rm qb}(t)$, where\n\\begin{equation}\\label{qs}\n{\\cal H}_{\\rm qb}(t)= \\vec{ B}(t) \\cdot \\hat{\\vec{\\sigma}}\n\\end{equation}\nwith $\\hat{\\vec{\\sigma}}=(\\hat{\\sigma}_z,\\hat{\\sigma}_x)$ being the Pauli matrices and $\\vec{ B}(t) \\equiv \\left(B_z(t), B_{x}(t) \\right)$,\nbeing periodic with period $\\tau$. \n\nAs already highlighted in Section~\\ref{sec:setup}, a key ingredient to have the heat--work mechanism in the linear response regime, is some protocol leading to $A \\neq 0$. We recall that this quantity represents also the net pumped heat as a consequence of the time-dependent driving. \nIn linear response, $A$ depends on response functions that are evaluated with the two reservoirs at the same temperature $T$ (see \\cite{ludovico2016feb,adiageo} and Appendix \\ref{app-lin-res}). When the two reservoirs are equally coupled, \nany protocol implemented via changing $\\vec{B}$ generates the same energy flow between them and the qubit. This prevents a net energy transfer between the reservoirs and $A=0$. Therefore, it is necessary to introduce some asymmetry in the coupling between the qubit and the reservoirs in order to have $A \\neq 0$.\nFor this reason, we consider the Hamiltonian describing the coupling to the reservoirs introduced in Eq.~\\eqref{qcont} with $\\hat{\\pi}_{\\rm h} \\equiv \\hat{\\sigma}_{x}, ~\\hat{\\pi}_{\\rm c}=\\hat{\\sigma}_z$, which breaks the c $\\leftrightarrow$ h symmetry in the absence of a temperature bias. Any other combination of Pauli matrices with\n$\\hat{\\sigma}_{\\rm h}\\neq \\hat{\\sigma}_{\\rm c}$ would lead to similar results. As mentioned before, the other crucial ingredient is a protocol depending on at least two parameters, which is necessary to define a non-trivial surface $\\Sigma$. In our case, we consider just two parameters: $B_z(t)$ and $B_x(t)$.\n\nWe solve the problem in the limit of weak coupling between the WS and the reservoirs by deriving the adiabatic master equation by means of the non-equilibrium Green's function formalism at second order of perturbation theory in $V_{\\alpha}$ as explained in Ref.~\\cite{masterkeldysh}. Details are shown in Appendices ~\\ref{sec:linear-response} and ~\\ref{appendix:sameCouplingStrength}.\nIn the specific calculations discussed below, we considered the simplest case, where the two reservoirs have the same spectral density, $\\Gamma_{\\rm c}(\\epsilon)=\\Gamma_{\\rm h}(\\epsilon)=\n\\Gamma(\\epsilon)=\\bar \\Gamma \\epsilon e^{-\\epsilon\/\\epsilon_C}$ for $\\epsilon \\geq 0$. \n\n\\subsection{Adiabatic linear-response coefficients}\nThe adiabatic linear response matrix $\\underline{\\Lambda}$ is originally expressed as a function of the coordinates $(B_z,B_x)$. This matrix is positive defined and symmetric. \nWhen diagonalized it is found that the eigenvectors, $|r\\rangle=\\left(\\sin(\\phi), \\cos(\\phi)\\right)^T$, $|\\phi\\rangle=\\left(\\cos(\\phi), -\\sin(\\phi)\\right)^T$ \ncorrespond to radial and tangential directions and thus $\\underline\\Lambda$ can be expressed as follows,\n\\begin{equation}\\label{eq:underlam}\n\\underline{\\Lambda} =\n\\lambda_{r}|r\\rangle \\langle r|+\n\\lambda_{\\phi}|\\phi\\rangle \\langle \\phi|,\n\\end{equation}\nwith $\\lambda_{r}, \\lambda_{\\phi}\\geq 0$.\nThis suggests that it is natural to implement the following change of coordinates $B_z=B_r\\cos\\phi, \\; B_x=B_r\\sin\\phi$. We get\n\\begin{equation}\n\\dot{\\vec B} \\cdot \\underline{\\Lambda} \\cdot \\dot{\\vec B} \\equiv {\\lambda_{r}} {\\dot B}_r^2 + \n\\lambda_{\\phi} B_r^2 {\\dot \\phi}^2\n\\equiv\n\\lambda_{r} |\\dot{\\vec B}_r|^2\n+ \\lambda_{\\phi} |\\dot{\\vec B}_\\phi|^2.\n\\end{equation}\n\n\n The analytical expression for the radial component reads,\n\\begin{equation}\n\\label{eq:lambda_diss_radial-t}\n\\lambda_{r}(\\vec B)=\n\\frac{\\hbar \\beta \\sinh (\\beta B_r) }{\\Gamma (2 B_r)\\cosh^3{(\\beta B_r)}\n}\n\\end{equation}\n\nand for the tangential one\n is\n\\begin{equation}\n\\label{eq:lambda_diss_polar-t}\n\\lambda_{\\phi}(\\vec B)=\n\\frac{\\hbar \\Gamma (2 B_r)}{4 B_r^3}\n,\n\\end{equation}\nbeing $\\beta=1\/k_B T$.\nThe first component is associated to changes in the energy gap between the two states of the qubit, while the second one leaves the spectrum unchanged but introduces a rotation of the eigenstate basis.\n\nRegarding the other coefficients, the components of the vector $\\vec{\\Lambda}({\\vec B})=(\\Lambda_z, \\Lambda_x) = \\Lambda_r \\langle r| + \\Lambda_\\phi \\langle \\phi|$ \nread\n\\begin{equation}\n \\Lambda_{r}(\\vec B) = \\frac{\\beta B_r \\sin ^2(\\phi) }{\\cosh^2 (\\beta B_r) },\\;\\;\\;\n \\Lambda_{\\phi}(\\vec B) = 0\\;,\n\\end{equation}\nwhile the parametric thermal conductance is\n\\begin{equation}\n\\label{eq:qubit_kappa}\n \\kappa(\\vec B) = \\frac{\\beta B_r^2 \\sin^2 (2 \\phi ) \\Gamma (2 B_r)}{\\sinh{(2\\beta B_r)}\\hbar}.\n\\end{equation}\n\n\\subsection{Geometrical quantities and bound for the heat--work conversion}\n\\label{sec:results-C}\nGiven the above coefficients we can now calculate all \n the relevant geometrical quantities for the characterization of the machine, namely, $A$, $L$ and $\\langle\\kappa\\rangle$ defined respectively in Eqs. \\eqref{eq:geometric_A}, \\eqref{eq:geometric_L} and \\eqref{eq:geometric_k}.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.5\\textwidth]{curl_lambda.png}\n \\caption{The Berry-type curvature $\\left[\\vec\\nabla_B \\wedge \\vec\\Lambda(B)\\right]_y$. The integration of this quantity over the area enclosed by the control trajectory defines the $A$ as in Eq. (\\ref{eq:definition:A}). Parameters are $\\epsilon_C=120 k_B T$ and $\\bar\\Gamma=0.2$. Curves (a), (b) and (c) \n are heuristically searched protocols of elliptic shape, centered in $(0,0)$, $(1,1)$ and $(-1.5, -0.45)$ respectively, that maximize the value $A^2\/\\mathcal{L}^2$ (see Sec. \\ref{subsec:maximum_power}). Curve (d) is a protocol with the shape of a circular sector\n centered at $(0,0)$, with radius $R$ and spanning an angle $\\Omega$ symmetrically with respect to the quadrant's bisector.\n \n }\n \\label{fig:curl_lambda}\n\\end{figure}\nAs already mentioned when the representation of Eq.~\\eqref{eq:definition:A} was introduced, the net pumped heat quantified by $A$ is simply the value of the Berry curvature integrated over the area of the $(B_z,B_x)$ plane enclosed by a particular protocol. \n The Berry curvature as a function of $(B_z,B_x)$ is shown in Fig. \\ref{fig:curl_lambda}.\nBecause of the nature of the setup, this quantity changes sign at $B_x=0$ and $B_z=0$. Therefore, \nprotocols with constant $B_r$ lead to $A=0$. For any protocol, the sign can be simply switched by changing the circulation of the boundary curve, hence switching the operation from heat engine to refrigerator or viceversa.\n\nIt is also easy to visualize in Fig. (\\ref{fig:curl_lambda}), that protocols enclosing a large portion of the dark blue or bright yellow areas lead to a large value of $|A|$.\nFocusing on simple curves that do not cross themselves we consider a circular-sector trajectory like the curve $(d)$ depicted in Fig.~(\\ref{fig:curl_lambda}), characterized by a radius $R$ and an aperture angle $\\Omega$ symmetric with respect to the the quadrant's bisector. \nIt is clear from the figure that the protocol leading to the maximum achievable value of $|A|$ in the present setup corresponds to a trajectory fully enclosing a quadrant.\nSuch a trajectory is, for instance, the special case of the circular-sector trajectory with $\\Omega=\\pi\/2$ that:\ni) starts at the origin and goes to infinity along the $B_x$ axis,\nii) rotates $\\pi\/2$ counterclockwise and aligns in the $B_z$ axis,\niii) returns to the origin along the $B_z$ axis.\n\nThis limiting protocol corresponds to a quasistatic Carnot cycle and the resulting value of $A$ is\n\\begin{equation}\\label{landauer}\n A_{\\rm lim}=\\int_{\\text{quadrant}} (\\vec{\\nabla}_B\\wedge \\vec{\\Lambda} )\\cdot d\\hat{y}=\\pm k_B T \\log(2),\n\\end{equation}\nwhere the signs are determined by the enclosed quadrant and the circulation considered. Notice that, according to Eq.~(\\ref{q}), this corresponds to the extreme values for the energy that could be transported \nbetween the two reservoirs at the same temperature $T$ through the qubit, and coincides with the famous bound obtained by Landauer's argument~\\cite{landauer61} according to which the change of Shannon entropy in the process of erasing the information encoded in a bit is $\\pm \\log(2)$. In the present case, it is associated to the transfer of the same amount of entropy between the reservoirs (a similar result was found quantum dots \\cite{janine}). At finite $\\Delta T$, according to Eq.~(\\ref{w}) this quantity also sets the maximum value of the work that can be extracted in the heat-engine operational mode (for $A_{\\rm lim}>0$) in the limit of\n vanishing dissipation. This result is, respectively,\n\\begin{equation}\n\\label{eq:limitW}\n W_{\\rm lim}= k_B (T_{\\rm h}- T_{\\rm c}) \\log(2)= A_{\\rm lim} \\; \\eta_C.\n\\end{equation}\n\n\n\n\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.45\\textwidth]{dissipation.png}\n \\caption{Positive eigenvalues of $\\underline{\\Lambda}$ -see Eq. \\eqref{eq:underlam}- as a function of $|\\vec B|=B_r$. Parameters are $\\epsilon_C=120 k_B T$ and $\\bar\\Gamma=0.2$ (solid lines), $\\bar\\Gamma=0.05$ (dashed lines). Note for $\\bar \\Gamma=0.2$ (solid lines) that most of the relevant region of Fig. \\ref{fig:curl_lambda} lies inside the interval $(B_{r,\\rm low}, B_{r, \\rm high})$ where the radial dissipation is about one order of magnitude bigger than the polar dissipation.}\n \\label{fig:dissipation}\n\\end{figure}\n\nWe now turn to analyze $L^2$, which assesses\nthe dissipated energy for a particular protocol.\nThis quantity is determined by $\\underline{\\Lambda}$ given by Eq.~(\\ref{eq:geometric_L}). For the qubit, \nthis matrix can be decomposed in two contributions, as expressed in Eq.~(\\ref{eq:underlam}) which are associated to the dissipation of energy originated in the radial and polar changes of $\\vec{B}$.\n\nWe see from the analytical expressions of Eqs. (\\ref{eq:lambda_diss_radial-t}) and (\\ref{eq:lambda_diss_polar-t}) that $\\underline{\\Lambda}$ is symmetric along the polar axis, i.e. it only depends on $B_r$. This is illustrated in the upper panel of Fig. \\ref{fig:dissipation_change} of Appendix~\\ref{appendix:sameCouplingStrength}. In Fig. \\ref{fig:dissipation} we show the dependence of the coefficients $\\lambda_{r}$ and $\\lambda_{\\phi}$ on $B_r$ for two different values of the $\\bar \\Gamma$ parameter. For some values of $\\bar\\Gamma$ we find an interval $(B_{r,{\\rm low}}, B_{r,{\\rm high}})$ for which the dissipation is mainly due to changes in the energy spectrum induced by finite $\\dot B_r$.\nThe specific values $B_{r,\\rm low},\\; B_{r,\\rm high}$\ndepend on the working temperature and the coupling constant $\\bar \\Gamma$ between the qubit and the reservoirs. More details on the $\\underline{\\Lambda}$ submatrix and the dissipation structure of the qubit can be found in Appendix~\\ref{appendix:sameCouplingStrength}.\n\nThe final value of $L^2$ for a protocol $\\vec B(\\theta \\tau)$ defined over $\\partial \\Sigma$ in the parameter space still depends on the chosen parametrization $\\theta$.\nOut of all the possible parametrizations, Eq.~\\eqref{eq:definition:L} tells us that there exists a particular one for which\n $L^2=\\mathcal{L}^2$.\nFurthermore, this corresponds to the lower bound for $L^2$ and, importantly, it is a function of $\\partial \\Sigma$ only (it is geometrical).\n\nIn addition, for a given $\\theta$ associated to $\\partial \\Sigma$, we are able to obtain the optimal parametrization $\\bar\\theta (\\theta)$ that saturates the bound, and defines the less dissipative protocol $\\vec{B} (\\bar{\\theta}\\tau)$ around $\\partial \\Sigma$ in time $\\tau$. The new value of the velocity at a given time can be computed using \\eqref{eq:definition:L}, demanding that $\\dot{\\vec B}(\\theta \\tau)\n\\cdot \\underline{\\Lambda} (\\vec B)\n\\cdot \\dot{\\vec B}(\\theta \\tau)$ is constant at each point.\nThe result is\n\\begin{align}\n\\label{eq:optimal_parametrization_velocity}\n \\frac{\\partial \\vec {B} (\\bar{\\theta}\\tau)}{\\partial \\bar{\\theta}}=\n \\dot{\\vec B}(\\theta \\tau)\n \\sqrt{\\frac{\\mathcal{L}^2}\n {\\dot{\\vec B}(\\theta \\tau)\n \\cdot \\underline{\\Lambda} (\\vec B)\n \\cdot \\dot{\\vec B}(\\theta \\tau)}\n }\n\\end{align}\nwhere the dot in $\\dot{\\vec{B}}$ is the derivative with respect to the original parametrization $\\theta$. This driving ensures constant entropy production along the cycle. \n\n\\subsection{Maximum power}\n\\label{subsec:maximum_power}\nAlthough a global maximum for $P_{\\rm max}(\\partial\\Sigma)$ in Eq.~\\eqref{eq:PmaxC} is hard to find, it is still possible to design simple trajectories with useful output power and reasonable efficiency. \nWe perform a numerical search of $\\mbox{max}_{\\partial \\Sigma} A^2\/\\mathcal{L}^2$ using a gradient descent method, restricted to the space of elliptic trajectories centered at a given point $\\vec B$. The trajectories $(a)$, $(b)$ and $(c)$ shown in Fig.~\\ref{fig:curl_lambda} are examples of the resulting curves. We choose this type of curves because elliptical trajectories are easy to implement and flexible enough to perform an extensive optimal search.\nThe advantage of the elliptical protocols is not obvious, taking into account that Fig.~\\ref{fig:curl_lambda} suggests that\nthe circular-sector protocols are better than the ellipses for maximising~$A$. However, this is not the case for~$A^2\/\\mathcal{L}^2$: we show in Appendix~\\ref{app:pizzaProtocol} that suitable chosen ellipses can clearly outperform circular-sector protocols in terms of power output. \n\n\nFocusing on the elliptic protocols, we see that the highest values of power are achieved for test curves that avoid the region of small $|\\vec B|$, where the dissipation coefficient $\\lambda_{\\phi}$ diverges. The curve $(a)$ centered at~$(0,0)$ is an interesting example. It maximizes $A^2$ by enclosing the two lobes in the first and third quadrant of Fig. \\ref{fig:curl_lambda}, and closes the curve near infinity in order to avoid the central region of high dissipation.\n\nIn Fig.~\\ref{fig:optimal_ellipses} we depict the value of $\\mbox{max}_{\\partial \\Sigma} A^2\/\\mathcal{L}^2$ found by the mentioned heuristic method, as a function of the (fixed) central point of the ellipse.\n We distinguish two different regimes leading to the optimal power, as a consequence of the crossover between the two mechanisms of dissipation discussed in the context of Fig. \\ref{fig:dissipation}. For small $B_r$, where the less dissipative protocol is radial, the optimal trajectories are like the case (a) in Fig. \\ref{fig:curl_lambda}, while in the opposite limit where $B_r$ is large, the optimal protocols are like the ones indicated with (b) and (c) in that Fig.\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.45\\textwidth]{optEllipses.png}\n \\caption{max $A^2\/\\mathcal{L}^2$ as a function of $\\vec B$, for an heuristic optimization of elliptic trajectories centered at $\\vec B$. Parameters are $\\epsilon_C=120 k_B T$ and $\\bar\\Gamma=0.2$. Only positive values of $B_x$ and $B_z$ are shown, since this quantity is symmetric with respect to $B_x=0$ and $B_z=0$.} \n\\label{fig:optimal_ellipses}\n\\end{figure}\n\n\n\\subsection{Maximum efficiency}\n\\label{sec:max_efficiency}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.5\\textwidth]{lambda_K.png}\n \\caption{Maximum eigenvalue of $\\underline{\\Lambda}_{\\kappa}$,\n \n indicating the losses due to the combined effect of dissipation and thermal conduction. Parameters are $\\epsilon_C=120 k_B T$ and $\\bar\\Gamma=0.2$. Curves (b) and (c) are heuristically searched protocols of elliptic shape, centered at $(1,1)$ and $(-1.5, -0.45)$ respectively, that maximize the value $A^2\/\\mathcal{L}_\\kappa^2$. Curve (d) (circular sector) is a quarter of circumference centered at $(0,0)$, joined by two radial lines of length $R$ along the axis.}\n \\label{fig:lambdaK}\n\\end{figure}\n\nFollowing the same philosophy of the analysis of $L^2$ presented in Fig.~\\ref{fig:dissipation}, we plot in Fig.~\\ref{fig:lambdaK} the maximum eigenvalue of \n$\\underline{\\Lambda}_{\\kappa}$ (defined in Eq.~\\eqref{eq:LambdaK_def})\nin order to visualize the value of the thermal losses when the system evolves in the direction of maximum dissipation. Note that since $\\kappa(\\vec B)$ is a scalar, hence, an equivalent decomposition to Eq.~\\eqref{eq:underlam} can be done for $\\Lambda_\\kappa^2$ as follows:\n\\begin{equation}\n\\label{eq:underlamKappa}\n\\underline{\\Lambda_k} =\n{\\lambda_k}_{r}|r\\rangle \\langle r|+\n{\\lambda_k}_{\\phi}|\\phi\\rangle \\langle \\phi|.\n\\end{equation}\nFurthermore the analysis presented for $L^2$ in Section~\\ref{sec:results-C}, and particularly the results shown in Fig.~\\ref{fig:dissipation} and Appendix \\ref{appendix:sameCouplingStrength} still hold for the eigenvalues and eigenvectors of $\\underline{\\Lambda}_\\kappa$.\n\nIn the case of the efficiency, an optimal solution is trivially found by looking at Fig.~\\ref{fig:lambdaK} and considering again the circular-sector curve $(d)$ with $\\Omega=\\pi\/2$. From \\eqref{eq:qubit_kappa} we see that along the $B_x$ and $B_z$ axis we have $\\kappa=0$, because in those regions the system is coupled to only one of the reservoirs. \nIt is clear from Eq.~\\eqref{eq:LambdaK_def} that for the limiting circular-sector protocol with $R \\rightarrow \\infty$, enclosing the full quadrant and leading to Eq.~(\\ref{landauer}) we have $\\left< \\kappa \\right>=0$, which implies $x=\\infty$ in Eq.~\\eqref{eq:Etamax}, hence $\\eta_{\\rm max}=\\eta_C$. In fact, as already mentioned, this protocol is an equilibrium Carnot cycle for the qubit, where the changes along the axis are the isothermal compression and expansion. More details of the efficiency of this protocol can be found in Appendix \\ref{app:pizzaProtocol}.\n\nIn addition to this particular solution of special interest, we illustrate the usefulness of the method in a more generic way. The strong equivalence between the geometrical quantities $A^2\/\\mathcal{L}^2$ and $A^2\/\\mathcal{L}_\\kappa^2$ allows us to replicate the analysis done in the previous subsection in a straightforward manner. Once again, for a given a trajectory the value of $A^2$ is computed from Eq.~\\eqref{eq:definition:A} while the lower bound for $L^2\\left<\\kappa\\right>$ and the corresponding optimal parametrization is given by~\\eqref{eq:LambdaK_def} in complete analogy with Eqs.~\\eqref{eq:definition:L} and~\\eqref{eq:optimal_parametrization_velocity} from the maximum power analysis.\n \nWe perform the numerical search of $\\max_{\\partial \\Sigma}\\lbrace A^2\/\\mathcal{L}_\\kappa^2 \\rbrace(\\vec B)$ again for the special case of closed elliptic curves centered at a given point $\\vec B$. The computed result is presented in Fig.~\\ref{fig:optimal_ellipses_lambdaK}.\nIn Fig.~\\ref{fig:lambdaK} we also show some of these trajectories, centered at the points $\\vec B= \\lbrace (1,1),(-1.5,-0.45) \\rbrace$. Note that, while some differences can be spotted between these trajectories and the ones shown in Fig.~\\ref{fig:curl_lambda}, the qualitative intuition is that efficient protocols are the ones with big $A^2$.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.5\\textwidth]{optEllipsesLambdaK_interp.png}\n \\caption{Maximum $A^2\/\\mathcal{L}_\\kappa^2$ as a function of $\\vec B$ for a heuristic optimization of elliptic trajectories centered at $\\vec B$. Parameters are $\\epsilon_C=120 k_B T$ and $\\bar\\Gamma=0.2$.}\n\\label{fig:optimal_ellipses_lambdaK}\n\\end{figure}\n\n\n\n\n\\subsection{The impact of optimizing the driving speed} \nThe aim of this section is to gather further insight on the effect of selecting the optimal protocol, regarding the trajectory $\\partial \\Sigma$ and the optimal speed for the circulation on the resulting power and efficiency of a heat engine. \n\nWe consider an elliptical protocol for which we can define a ``trivial'' circulation with constant angular velocity. \nFor the case of the power, we compare the results of such trivial circulation with the one corresponding to \n the optimal velocity as defined in\nEq. (\\ref{eq:optimal_parametrization_velocity}). For the case of the efficiency we compare the trivial circulation with \n with the one corresponding to the optimal velocity,\nas defined in Eq. (\\ref{eq:optimal_parametrization_velocity}) with the replacements ${\\cal L} \\rightarrow {\\cal L}_{\\kappa}$\nand $\\underline{\\Lambda} \\rightarrow \\underline{\\Lambda}_{\\kappa}$.\n\nFor sake of concreteness we focus on $\\partial \\Sigma$ given by the protocol (b) of Fig.~\\ref{fig:curl_lambda}. Results are shown in Fig.~\\ref{fig:curve_b_performance}, where we show the power and efficiency of the machine as a function of the cycle total duration. Plots in solid and dashed lines correspond, respectively, to the protocols with constant angular velocity and optimal velocity. \n We note from this figure that the optimized parametrization is around two times bigger in power, and around four times more efficient, with respect to the trivial parametrization of the ellipse circulated at a constant angular velocity.\nDashed lines in Fig.~\\ref{fig:curve_b_performance} are akin to those of Fig.~\\ref{fig:pow-eff_engine} where power values are normalized to $P_{\\rm max}$ and efficiencies to $\\eta_{\\rm max}$, and summarizes the performance of the machine. \n\n\n\n\n\n\n\\subsection{Estimates for the performance}\nTo finalize the analysis of the qubit heat engine, it is interesting to analyze concrete values characterizing its performance.\nAs before, we focus on the protocol (b) of Fig. \\ref{fig:curl_lambda}, \nfor which we have\n\\begin{equation}\\label{al-b}\n A^2 = 0.233 k_B^2 T^2 \\quad\n \\mathcal{L}^2 = 7.71 \\hbar.\n\\end{equation}\nFor these values, we find using Eq.~\\eqref{eq:Pmax}:\n\\begin{align*}\nP_{\\rm max}= \\left( 1.364 \\times 10^{-2} \\frac{pW}{K^2}\\right) (\\Delta T)^2,\n\\end{align*}\nwhich for a working temperature of $T=100mK$ and a temperature bias corresponding to $\\Delta T=0.05 T$, as in previous Figures, gives\n$P_{\\rm max}= \n0.341 aW $\nwith efficiency $\\eta_{P_{max}} = 0.23\\eta_C$. The total time $\\tau_P$ for maximum power output per cycle is computed through Eq.~\\eqref{eq:opt_taus}:\n\\begin{align*}\n \\tau_P = 2 \\tau_D = 48.8 ns\n\\end{align*}\nwhich corresponds to an operation frequency in the order of $0.1GHz$. \n\nIt is interesting to compare the value obtained for the maximum power in the protocol under consideration with the power associated to the limiting value for the work given by Eq. (\\ref{landauer}). Such limiting power can be obtained by \nreplacing $\\tau_P$ of Eq. \\eqref{eq:timescales} in Eq. \\eqref{eq:def_W}, where we see that at finite time the net work done by the machine operating at maximum power is $W_{P_{\\rm max}}= A \\;\\eta_C\/2$. Taking into account that for the heat engine\n$A \\leq \\log(2) k_B T$ -- see Eq. (\\ref{landauer})-- we conclude that the bound for the maximum operating power\nin a cycle of duration $\\tau_P$\nis $P_{\\rm lim}=\\log(2) \\; k_B T \\;\\eta_C\/(2\\tau_P)$. For the case of the protocol (b), given the values of Eq. (\\ref{al-b}), we get\n$P_{\\rm max}=0.7 P_{\\rm lim} $.\n\n\n\nIn a similar way using Eq. \\eqref{eq:Etamax}, the optimized parametrization that maximizes the efficiency of the cycle give us the value\n$\\eta_{\\rm max}=0.34 \\eta_C$.\n\nSpecific values for the maximum efficiency of the machine operating under other protocols can be obtained by substituting in Eq.~(\\ref{eq:x}) the values shown in Fig.\\ref{fig:optimal_ellipses_lambdaK}. This calculation shows that this machine can achieve a performance as high as $\\eta_{\\rm max} > 0.55 \\eta_C$. \nThese results are very encouraging regarding the possibility of the experimental implementation of this system.\n\n\n\n\n\n\n\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.5\\textwidth]{machinePerformance.png}\n \\caption{Power (blue) and efficiency (orange) for curve (b) of Fig. \\ref{fig:lambdaK} as a function of the cycle duration $\\tau$. Solid lines: circulating around the curve at constant angular velocity. Dashed lines: Using the optimal velocities given by Eqs. \\eqref{eq:definition:L} (for power) and \\eqref{eq:LambdaK_def} (for efficiency).}\n\\label{fig:curve_b_performance}\n\\end{figure}\n\n\\section{Summary and conclusions}\nWe have followed a geometrical approach to describe the two competing mechanisms of a non-equilibrium adiabatic thermal machine: the dissipation of energy and the heat--work conversion. While the first mechanism is described in terms of a length, the second one can be represented by and area in the parameter space. \n We then showed that the problem of finding optimal protocols reduces to an isoperimetric problem, which consists in finding the optimal ratio between area and length in a space with non-trivial metrics.\n \n We applied this description to a thermal machine which consists of a single qubit asymmetrically coupled to two bosonic reservoirs at small different temperatures and driven by a cyclic protocol controlled by two parameters that vary slowly in time. \n We solved this problem in the limit of weak coupling between the qubit and the reservoirs. \n We analytically show the limiting value of the pumped heat between reservoirs is given by Landauer bound in an ideal Carnot cycle. We analyzed in this problem the type of cycles leading to optimal performance of the machine. Interestingly, \n the qubit machine has a very good ratio between performance and power within a wide set of parameters. \n \n According to our analysis, efficiencies larger \n than $0.55$ of the Carnot cycle can be achieved and values of the corresponding output power of $0.7$ of the limiting power, corresponding to the work done in an ideal Carnot cycle divided by the duration of the cycle at which the maximum power is achieved.\n These estimates are very encouraging for \n the experimental implementation of this machine. In this sense, a very promising platform is a superconducting qubit coupled to\n resonators, in which there are several configurations under study for some years now \\cite{niskanen2003fast,mottonen2008experimental,cottet2017observing,senior2020heat,upadhyay2021robust,guthrie2021cooper}. Other possible platforms are those in which the Otto cycle has been already implemented, like\n AMO systems \\cite{abah2012nov,von2019spin,brantut2013nov}, as well as spin systems in NMR setups \\cite{peterson2019dec}.\n Quantum dots, where electron pumping has been observed \\cite{pothier1992jan,switkes1999adiabatic} are also candidates for implementing the heat engine and refrigerator operations as well as nanomechanical systems \\cite{bachtold,ares}.\n This geometrical optimization can be also very naturally extended to analyze other systems like motors operating under slow driving and a bias voltage \\cite{marun,magnet,ludovico-capone,lucas1}. In the present work we have focused on the linear-response regime, where the geometric description becomes explicit. The weak-coupling calculations of the heat and work presented in Section \\ref{sec:qubit} and Appendix \\ref{appA:reducedDensityMatrix} can be extended to analyze the operation beyond this regime for specific thermal machines, representing an outlook to further works.\n \n \n \n \\section{Acknowledgements}\n \nWe thank Rosario Fazio and Jukka Pekola for stimulating discussions.\nP.A. is supported by ``la Caixa\" Foundation (ID 100010434, Grant No. LCF\/BQ\/DI19\/11730023), and by the Government of Spain (FIS2020-TRANQI and Severo Ochoa CEX2019-000910-S), Fundacio Cellex, Fundacio Mir-Puig, Generalitat de Catalunya (CERCA, AGAUR SGR 1381).\n L.A. and P.T.A. are supported by CONICET, and acknowledge financial support from PIP-2015 and ANPCyT, Argentina, through PICT-2017-2726, PICT-2018-04536. L.A. thanks KITP for the hospitality in the framework of the activity \"Energy and Information Transport in Non-equilibrium Quantum Systems\" and the support by\n the National Science Foundation under Grant No. PHY-1748958, and the Alexander von Humboldt Foundation, Germany. M. P.-L. acknowledges funding from Swiss National Science Foundation through an Ambizione grant PZ00P2-186067.\n\n\n\\input{appendix.tex}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \\label{sec:hajimeni}\nPassword authentication has long been essential for remote access to services that require user authentication \\cite{password-security-a-case-history}, but there remain many problems with it \\cite{quest-to-replace-passwords}. \nPhishing, for example, is a problem caused by asking users to present their passwords every time they use a service \\cite{anti-phishing}.\nIt is also a problem that users use the same password for many services or use short passwords because they don't want to remember many complex passwords \\cite{users-are-not-the-enemy, password-management-strategies-for-online-accounts}.\n\nPublic key authentication is an alternative to password authentication for stronger authentication.\nPublic key authentication assumes that only a user has a private key, and a service has the corresponding public key.\nA service authenticates a user in the following three steps.\nFirst, the service sends a random number called a challenge to the user.\nSecond, she signs the challenge by the private key corresponding to the public key registered with the service.\nThird, the service verifies the signature by any one of the public keys bound to her account.\n\nServices have to manage the binding of public keys to a user's account.\nServices trust this binding based on several models, where trusted third parties ensure the binding (e.g., Individual Number Card in Japan \\cite{my-number} and WebPKI \\cite{rfc5280}) and where they receive keys from users directly via trusted channels (e.g., FIDO \\cite{how-fido-works} and SSH \\cite{rfc4252}).\nThis study will focus on the last model.\n\nUsers have to manage private keys corresponding to registered public keys on their devices.\nWe call these devices authenticators.\nAuthenticators, such as Yubikey \\cite{yubico-product-doc} and Keychain \\cite{apple-keychain}, store key pairs in secure storage where corresponding private keys cannot be exported nor easily accessed by the outside of the authenticator.\nOperations using keys stored in secure storage require local authentication by authenticators, like PIN or biometrics.\nThis study assumes that users use authenticators having secure storage.\n\nAn authenticator has a mechanism called attestation \\cite{fido-attestation} that proves that an operation is done surely by the authenticator.\nAn attestation includes information about the manufacturer of the authenticator which generates the attestation, the model name of the authenticator, and results of the operation done by the authenticator, such as the public key of a generated key pair.\nAttestations are signed by the attestation key embedded in an authenticator by its manufacturer so that services can validate whether received attestations are generated by the authenticator.\nDuring account registration, an authenticator sends a public key, an attestation about the public key, and the certificate of the attestation key.\nServices can determine the trustworthiness of the received public key and the authenticator that stores the corresponding private key by verifying the attestation with the certificates received from the manufacturer of the authenticator.\n\nPublic key authentication (PKA) is stronger than password authentication for the following reason.\nFirst, PKA is resistant to data breaches on services because attackers cannot sign in to services with only public keys.\nSecond, PKA can be phishing resistant when an authenticator verifies whether the requested service is the same as the service accessed previously without interaction with a user.\nLastly, users don't have to use weak private keys for convenience, because authenticators, not users, remember and manage these keys.\nBesides, Malicious services cannot correlate their account using registered public keys because authenticators generate different public keys for each service.\n\nHowever, public key authentication has the problem that users can only use authenticators storing private keys corresponding to registered public keys when accessing services.\nGiven the following two concerns, it is a burden for users to register, update and revoke public keys in many services.\nFirst, users usually have multiple authenticators such as smartphones, PCs, and tablets.\nThey have to register multiple public keys with their authenticators, but simply registering public keys with each of all authenticators annoys users \\cite{fido-usability}.\nSecond, users add, replace, and throw away their authenticators according to the lifecycles of the authenticators.\nOnce such an event occurs, users need to update and revoke registered public keys in many services.\nCurrently, users can manage registered public keys on services via an authenticated session.\nIf an attacker steals an authenticator and revokes the public keys of authenticators held by a legitimate user before the user revokes the public key of the stolen authenticator, the user may become inaccessible.\n\nThe purpose of this study is that users can access services with public key authentication using any owned authenticators without explicitly registering public keys.\nTo realize this purpose, we propose the mechanism where users and services manage public keys based on the owner of authenticators storing the corresponding private keys.\nWe introduce a key pair, called an Ownership Verification Key (OVK).\nA user proves the ownership of authenticators by the private key of an OVK (Ownership Verification Secret Key; OVSK).\nA service verifies the possession of the authenticators by the public key of the OVK (Ownership Verification Public Key; OVPK).\nAll authenticators owned by a user can derive an OVSK from a seed pre-shared among them.\nA service manages the corresponding OVPK by binding it to her account.\nA service binds public keys signed by the OVSK to her account if verification by the OVPK is successful.\nTo protect user privacy while maintaining convenience, authenticators generate a different OVK for each service from the seed independently. \nUsers and services update OVKs according to the lifecycles of users' authenticators.\nWhen a user changes a set of her authenticators, she updates an OVSK, and services update an OVPK bound to her accounts.\n\nThe main contribution of this paper is that users and services can manage public keys based on the owner of the authenticators storing the corresponding private keys to facilitate their key management in public key authentication.\nWe implemented the Proof of Concept and confirmed that key management works as expected for typical use cases.\nWe analyzed the proposed mechanism to find threats with threat modeling and evaluated what measures our proposal takes against the found threats.\nWe confirmed that our proposal achieves some security goals such as that services cannot correlate accounts and can correctly bind public keys to accounts.\nWe discussed how our proposal mitigates threats for which measures are not sufficient. \n\nThe following is the structure of this paper.\nSection \\ref{sec:haikei} describes related work.\nSection \\ref{sec:teian} describes the key management using OVK.\nSection \\ref{sec:poc} describes the implementation of the Proof of Concept and use cases using the Proof of Concept.\nSection \\ref{sec:eval} describes evaluation with threat modeling.\nSection \\ref{sec:kousatsu} discusses the proposal.\nFinally, Section \\ref{sec:conclusion} summarizes this paper.\n\n\\section{Related Work} \\label{sec:haikei}\nNishimura \\cite{ntt-owner-identity} proposes sharing private keys among authenticators that users own.\nAuthenticators verify the owner of other authenticators to determine whether sharing private keys or not.\nTo verify the owner, a trusted third party issues certificates to authenticators.\nHowever, this approach weakens the authentication level of public key authentication because authenticators export private keys from secure storage.\nThis approach also weakens the trustworthiness of a registered public key because services cannot verify attestations.\nBesides, reliance on a trusted third party has a large management cost and the impact like a certificate authority in WebPKI if it becomes untrustworthy.\n\nJames \\cite{conners-19-lets-authenticate} introduces certificate chains like TLS client authentication to FIDO public keys so that FIDO is capable of registering multiple authenticators and recovering accounts.\nWhen a user registers a new public key generated in her authenticator with a service, she requests a certificate authority to issue the certificate binding the public key to her account of the service.\nThe authenticator sends the certificate to the service and the service can verify the owner of the public key by checking the subject of the certificate.\nHowever, this approach has the problem that it is not clear how a certificate authority authenticates users using multiple authenticators in addition to the same problem due to a trusted third party as \\cite{ntt-owner-identity}.\n\nOogami \\cite{fido-multi-registration} proposes the mechanism in which users register a new FIDO public key of an authenticator via authenticated sessions established by the registered public key of other authenticators.\nPublic keys have high assurance because users use registered authenticators every time users register a new public key of an authenticator. \nHowever, users have to keep multiple authenticators at the same time when registering a new public key, so that users cannot register a new public key when they have only unregistered authenticators.\n\nFrymann \\cite{yubico-webauthn-account-recovery} and Lundberg \\cite{yubico-webatuhn-account-recovery-impl} propose a mechanism for account recovery when losing registered authenticators that users use daily.\nIn this mechanism, a user has two authenticators.\nOne is for daily use by users, called the main authenticator.\nThe other is for backup use by users, called the backup authenticator.\nThe user deposits the backup authenticator in a vault.\nThe main authenticator receives the seed for deriving public keys from the backup authenticator in advance.\nOn behalf of the backup authenticator, the main authenticator generates a different public key of the backup authenticator for each service and registers the public key whose corresponding private key the backup authenticator can only derive.\nThe user can access services with the backup authenticator when losing the main authenticator.\nAs a result, this approach prevents services from correlating their account based on the registered public keys.\nHowever, services cannot verify the attestation of the public key of the backup authenticator during registration.\nBesides, when attackers gain control of the backup authenticator, they sign in with the backup authenticator and can revoke the public key of the main authenticator, and the user cannot sign in with the main authenticator.\n\nIdentity Federations using OpenID Connect \\cite{oidc-core} or SAML \\cite{saml-v2} allow users to reduce the number of services where users register public keys.\nHowever, users still register public keys with several services.\nBesides, there are also privacy issues where the service authenticating users, called an Identity Provider, can know what services they are using.\n\n\\section{Key Management with an Ownership Verification Key} \\label{sec:teian}\n\\subsection{Overview}\nWe propose the mechanism where a user and a service manage keys for authentication based on a public key cryptographic key pair called an Ownership Verification Key (OVK).\nAn OVK is derived by all authenticators of a user to prove that the private key corresponding to the public key to be registered is stored in her owned authenticator.\nThe public key of the OVK (Ownership Verification Public Key; OVPK) is registered with the service via the trusted channel established when registering a new account.\nThe service binds the OVPK to her account.\nThe private key of the OVK (Ownership Verification Secret Key; OVSK) is used for signing the public key to be registered.\nThe service binds the public key to her account if verification by the OVPK is successful.\n\nFig.\\ \\ref{fig:ovk-gaiyou-1} illustrates how a user registers a public key when she has two authenticators (A and B).\nShe shares an OVSK among Authenticators A and B in advance.\nWhen she registers a new account using Authenticator A, she attaches a public key for Authenticator A and an OVPK to the service.\nThen she seamlessly registers a new public key for Authenticator B by signing the public key with the OVSK whose corresponding OVPK has been already registered with Authenticator A.\nThe service verifies the signature by the registered OVPK and, if succeeded, binds the public key to her account.\n\n\\begin{figure}[htb]\n \\centering\n \\includegraphics[width=\\columnwidth]{assets\/ovk-gaiyou-1.png}\n \\caption{Registering Public Keys Using An OVK}\n \\label{fig:ovk-gaiyou-1}\n\\end{figure}\n\nUsers and services update OVKs according to the lifecycles of users' authenticators.\nWhen a user changes a set of her authenticators, she updates an OVK in her all authenticators and notifies services of updating the OVPK.\nTo make an updating message, registered authenticators sign the new OVPK to be updated by the previous OVSK whose corresponding OVPK is now registered with services.\nServices update an OVPK bound to her account based on the most trustworthy updating message and re-bind public keys to her account based on the new OVPK.\nA user can still sign in with authenticators that have notified services of the new OVPK.\nTo invalidate the authenticators that are no longer in use, services revoke the public key corresponding to the authenticators that are not bounded to the new OVPK.\n\n\\begin{figure}[htb]\n \\centering\n \\includegraphics[width=\\columnwidth]{assets\/ovk-gaiyou-2.png}\n \\caption{Updating an OVK}\n \\label{fig:ovk-gaiyou-2}\n\\end{figure}\n\nFig,\\ \\ref{fig:ovk-gaiyou-2} illustrates how a user updates an OVK when she has had two authenticators (A and B) and registered them with a service and replaces Authenticator A with Authenticator C because of losing Authenticator A.\nAfter losing Authenticator A, she shares a new OVSK between Authenticators B and C.\nAuthenticator B derives the new OVK ($OVK^2$) and signs the new OVPK ($OVPK^2$) by the previous OVSK ($OVSK^1$) to make an updating message.\nAuthenticator B notifies the service of the new OVPK ($OVPK^2$) by sending the updating message.\nThe service evaluates the message received from Authenticator B as the most trustworthy and binds the new OVPK to her account.\nIt re-binds the public key for Authenticator B to her account because B has sent the most trustworthy message and revokes the public key for Authenticator A because A has sent no messages.\nThen she can seamlessly register a new public key for Authenticator C by signing the public key with the new OVSK ($OVSK^2$) whose corresponding OVPK ($OVPK^2$) has been already registered by Authenticator B.\n\nThe following is the structure of this section.\nSection \\ref{sec:teian:ovk-derive} describes how to derive an OVK among authenticators.\nWe introduce a pre-shared secret called a seed to derive an OVK.\nSection \\ref{sec:teian:seed-sharing} describes how to share a seed among authenticators.\nSection \\ref{sec:teian:ovk-trust} describes how services verify the trustworthiness of an OVK requested to be registered.\nSection \\ref{sec:teian:ovk-update} describes how to update an OVK after sharing a new seed.\n\n\\subsection{Deriving an OVK from a Shared Secret} \\label{sec:teian:ovk-derive}\nIn this section, we describe how to derive an OVK from the pre-shared secret, called the seed.\nWe assume that the seed has been shared among all authenticators owned by the same user.\nWe also explain how to register public keys using an OVK.\n\n\\subsubsection{Requirement} \\label{sec:teian:ovk-derive:youken}\nWe define the requirements in such a way that our proposal does not interfere with what public key authentication described in Section \\ref{sec:hajimeni}, which we call PKA, can achieve during public key registration \\cite{fido-sec-ref, fido-privacy-principles}.\n\nFirst, our proposed method should not rely on trusted third parties for proving the owner of authenticators except for verifying attestations and establishing secure channels.\nIn PKA, users can register public keys via a trusted channel established when registering a new account or established by registered authenticators.\n\nSecond, our proposed method must prevent services from correlating their account by using the proof of the owner of authenticators.\nIn PKA, users can register different public keys with each service to protect user privacy against services seeking to correlate their accounts based on registered public keys.\n\nThird, in our proposed method, services should verify the attestation of the public key requested to be registered.\nServices can calculate the trustworthiness of the public key by verifying the attestation generated by the authenticator that has the corresponding private key.\n\nFinally, our proposed method should minimize the number of times a user operates multiple authenticators at the same time for convenience.\nOperating multiple authenticators at the same time whenever registering a new public key annoys a user.\n\n\\subsubsection{Deriving an OVK} \\label{sec:teian:ovk-derive:process}\nWe explain our proposal using Fig.\\ \\ref{fig:ovk-derive}, which shows a case where a user has two authenticators, A and B.\nA user registers a new account with service $\\alpha$ using Authenticator A at first, and then she accesses the service using Authenticator B.\nNote that we assume that messages between authenticators and the service have protected in terms of service authentication, confidentiality, and integrity (e.g., via TLS).\n\n\\begin{figure}[htb]\n \\centering\n \\includegraphics[width=\\columnwidth]{assets\/ovk-derive.png}\n \\caption{Deriving an OVK from the shared seed}\n \\label{fig:ovk-derive}\n\\end{figure}\n\nThe two authenticators agree in advance on the following parameters and the identifier of service $\\alpha$ ($sid_\\alpha$).\n\n\\begin{itemize}\n \\item \\texttt{s}: The seed value shared among authenticators (\\textcircled{\\scriptsize 1})\n \\item \\texttt{N}: The number of authenticators sharing the same seed (equal to the number of her authenticators) \n \\item The public key cryptographic algorithm for an OVK\n \\item \\texttt{KDF}: The key derivation function that takes a seed and a random value as inputs and outputs pseudorandom numbers of the length required for an OVSK\n \\item \\texttt{MAC}: The message authentication code function that takes an OVSK as a key\n\\end{itemize}\n\nFirst, the user registers a new account with service $\\alpha$ using Authenticator A.\nAuthenticator A generates a new key pair ($sk_A, pk_A$) and an attestation of the public key (\\textcircled{\\scriptsize 2} in Fig.\\ \\ref{fig:ovk-derive}).\nAt the same time, Authenticator A derives an OVPK and the corresponding metadata and registers them in addition to the public key ($pk_A$) with service $\\alpha$.\nThe derivation consists of the following three steps.\n\n\\begin{enumerate}\n \\renewcommand{\\labelenumi}{\\textcircled{\\scriptsize \\theenumi}}\n \\setcounter{enumi}{2}\n \\item Generate a random number ($R_\\alpha$).\n \\item Calculate an OVSK ($OVSK_\\alpha = \\mathtt{KDF}(\\mathtt{s}, R_\\alpha)$) and the corresponding OVPK ($OVPK_\\alpha$).\n If the authenticator cannot derive validate OVSK using the random number ($R_\\alpha$), start over from \\textcircled{\\scriptsize 3}.\n \\item Register $OVPK_\\alpha$ and the corresponding metadata consisting the following three values.\n \\begin{itemize}\n \\item $R_\\alpha$: The generated random number\n \\item $M_\\alpha$: The message authentication code ($\\mathtt{MAC}(OVSK_\\alpha, R_\\alpha + sid_\\alpha)$)\n \\item \\texttt{N}\n \\end{itemize}\n\\end{enumerate}\nNow, the user has registered a new account with service $\\alpha$.\nThe service binds the public key ($pk_A$), $OVPK_\\alpha$, and the corresponding metadata ($R_\\alpha, M_\\alpha, N$) to the new account.\n\nSecond, the user access service $\\alpha$ using unregistered Authenticator B.\nWhen the service returns a challenge for public key authentication by replying to an authentication request, it also returns the metadata ($R_\\alpha$ and $M_\\alpha$).\nAuthenticator B starts on seamless registration of a new public key because B has no public key for signing in to service $\\alpha$.\nAuthenticator B generates a new key pair ($sk_B, pk_B$) and the attestation of the public key (\\textcircled{\\scriptsize 8} in Fig.\\ \\ref{fig:ovk-derive}).\nAuthenticator B signs the public key ($pk_B$) by $OVSK_\\alpha$ so that service $\\alpha$ verifies whether the owner of Authenticator B storing the corresponding private key ($pk_B$) is the same as the owner of registered authenticators.\nTo derive $OVSK_\\alpha$ from the received metadata, the following two steps are performed.\n\n\\begin{enumerate}\n \\renewcommand{\\labelenumi}{\\textcircled{\\scriptsize \\theenumi}}\n \\setcounter{enumi}{5}\n \\item Derive $OVK_\\alpha$ using the received metadata $R_\\alpha$ in the same way as \\textcircled{\\scriptsize 4}.\n \\item Verify the received metadata $M_\\alpha$ using the derived $OVSK_\\alpha$. \n If this verification failed, the derived OVSK or the received metadata is not for service $\\alpha$.\n\\end{enumerate}\n\nService $\\alpha$ registers the public key ($pk_B$) if the attestation (\\textcircled{\\scriptsize 8}) and the signature (\\textcircled{\\scriptsize 9}) is valid and the number of the registered public keys is not more than \\texttt{N}.\n\n\\subsubsection{Different OVKs per Service} \\label{sec:teian:ovk-derive:ovk-per-svc}\nAuthenticators can derive different OVKs per service because of generating different random numbers (\\texttt{R}) per service.\nFig.\\ \\ref{fig:ovk-derive-per-svc} shows a situation where the user registers with Services $\\alpha$ and $\\beta$.\nIt is impossible to determine the seed value of an OVSK because of the properties of a key derivation function (\\texttt{KDF}).\nBy using a different random number for each service ( $R_\\alpha$ for Service $\\alpha$ and $R_\\beta$ for Service $\\beta$), authenticators can register unlinkable OVPKs with different services.\n\n\\begin{figure}[htb]\n \\centering\n \\includegraphics[width=\\columnwidth]{assets\/ovk-derive-per-svc.png}\n \\caption{Deriving different OVKs per services}\n \\label{fig:ovk-derive-per-svc}\n\\end{figure}\n\nAuthenticators only need to remember the value of the seed. This is because authenticators store random numbers \\texttt{R} to services in a verifiable format. \nMoreover, even though the number of registered services increases, a user does not have to operate multiple authenticators to share a new OVSK.\nThis is convenience for a user.\n\n\\subsection{Sharing a Seed among Authenticators} \\label{sec:teian:seed-sharing}\nIn this section, we describe how to share a seed among authenticators of a user.\n\n\\subsubsection{Requirement} \\label{sec:teian:seed-sharing:youken}\nA user operates multiple authenticators and makes them communicate to share a seed.\nThere are various kinds of short-range communication protocols (e.g., Bluetooth, NFC, and generating and reading QR codes), each of which has its different characteristics in the security of the communication channel.\nWe define the following requirement to be independent of specific communication protocols.\n\n\\begin{itemize}\n \\item Assuming no security features of communication channels\n\\end{itemize}\n\nThis requires that attackers cannot calculate a seed using only the information that authenticators send to the channel (resistance to eavesdropping).\nThis also requires that authenticators can validate whether the received information is generated by the legitimate authenticator to be resistant to tampering.\n\n\\subsubsection{Two Authenticators} \\label{sec:teian:seed-sharing:2-party}\nFig.\\ \\ref{fig:seed-sharing-2-party} shows the case where a user has two authenticators.\nTwo authenticators agree on the same seed based on the Diffie-Hellman key agreement algorithm.\nThey encrypt DH public keys using an authenticated encryption based on a password set by the user to ensure the confidentiality of DH public keys and verify the authenticity.\n\n\\begin{figure}[htb]\n \\centering\n \\includegraphics[width=\\columnwidth]{assets\/seed-sharing-2-party.png}\n \\caption{Sharing a seed between two authenticators}\n \\label{fig:seed-sharing-2-party}\n\\end{figure}\n\nThe authenticators agree on the following parameters in advance.\n\n\\begin{itemize}\n \\item \\texttt{pw}: The password set by the user (\\textcircled{\\scriptsize 1})\n \\item \\texttt{DH}: A Diffie-hellman key agreement algorithm\n \\item The list of authenticated encryption algorithms for encrypting DH public keys (Each algorithm has assigned an identifier)\n \\item The list of password-based encryption algortihms (Each algorithm has assigned an identifier)\n\\end{itemize}\n\nFirst, the user operates Authenticator A.\n\n\\begin{enumerate}\n \\renewcommand{\\labelenumi}{\\textcircled{\\scriptsize \\theenumi}}\n \\setcounter{enumi}{1}\n \\item Generate a DH key pair ($SK_A, PK_A$).\n \\item Generate a random number called a Content Encryption Key ($CEK_A$).\n \\item Encrypt the DH public key ($PK_A$) using $CEK_A$. \n Authenticator A determines the authenticated encryption algorithm from the list.\n \\item Encrypt $CEK_A$ using the password (\\texttt{pw}).\n Authenticator A determines the password-based algorithm from the list.\n\\end{enumerate}\nAuthenticator A sends the generated ciphertexts and the algorithm identifiers (at \\textcircled{\\scriptsize 4} and \\textcircled{\\scriptsize 5}) to Authenticator B.\n\nIn the same way, Authenticator B generates a DH key pair ($(SK_B, PK_B)$ at \\textcircled{\\scriptsize 6}) and a random number ($CEK_B$ at \\textcircled{\\scriptsize 7}), encrypts the DH public key ($PK_B$) using $CEK_B$, and encrypts $CEK_B$ using the password (\\texttt{pw}).\nNote that $CEK_B$ is not necessarily the same as $CEK_A$ generated by Authenticator A.\n\nAuthenticator A receives the ciphertexts from Authenticator B.\n\\begin{enumerate}\n \\renewcommand{\\labelenumi}{\\textcircled{\\scriptsize \\theenumi}}\n \\setcounter{enumi}{7}\n \\item Decrypt a received CEK ($CEK_B$) using the password (\\texttt{pw}).\n Authenticator A identifies the password-based algorithm by the received identifier.\n \\item Decrypt a received DH public key ($PK_B$) using the decrypted CEK ($CEK_B$).\n Authenticator A identifies the authenticated encryption algorithm by the received identifier.\n \\item Agree the same seed using the Diffie-hellman key agreement algorithm.\n\\end{enumerate}\n\n\\subsubsection{Three or More Authenticators}\nWhen a user has three or more authenticators, the authenticators share a seed like the situation in Fig.\\ \\ref{fig:seed-sharing-3-party}.\nFig.\\ \\ref{fig:seed-sharing-3-party} uses the algorithm \\cite{multi-party-diffie-hellman}.\nAll authenticators agree on the following parameters in advance in addition to the agreement for two authenticators.\n\n\\begin{itemize}\n \\item Each authenticator identifier (These identifiers are temporary identifiers used only to share a seed)\n \\item The partner authenticator identifier of each authenticator (Each authenticator receives calculated DH public keys from the same authenticator, called the partner authenticator, every step. The user assigns identifiers without overlap.)\n\\end{itemize}\n\n\\begin{figure}[htb]\n \\centering\n \\includegraphics[width=\\columnwidth]{assets\/seed-sharing-3-party.png}\n \\caption{Sharing a seed among three authenticators}\n \\label{fig:seed-sharing-3-party}\n\\end{figure}\n\nThe user operates authenticators according to the following steps.\nIn each step, encryption means that an authenticator generates a CEK, encrypts a DH public key using the CEK, and encrypts the CEK using the password set by a user.\n\n\\begin{description}\n \\item[Step1] Generate a DH key pair on each authenticator.\n \\item[Step2] \\begin{itemize}\n \\item Authenticator A sends the DH public key ($PK_A$) to Authenticator B with encryption.\n \\item B sends the DH public key ($PK_B$) to C with encryption.\n \\item C sends the DH public key ($PK_C$) to A with encryption.\n \\end{itemize}\n \\item[Step3] \\begin{itemize}\n \\item Authenticator A sends the calculated DH public value ($SK_A * PK_C$) using its DH private key ($SK_A$) and received DH public key ($PK_C$) to Authenticator B with encryption.\n \\item B sends the calculated DH public value ($SK_B * PK_A$) using its DH private key ($SK_B$) and received DH public key ($PK_A$) to C with encryption.\n \\item C sends the calculated DH public value ($SK_C * PK_B$) using its DH private key ($SK_C$) and received DH public key ($PK_B$) to A with encryption.\n \\end{itemize}\n \\item[Step4] \\begin{itemize}\n \\item Authenticator A calculates the DH public value ($SK_A * (SK_C * PK_B)$) and agrees the same seed.\n \\item B calculates the DH public value ($SK_B * (SK_A * PK_C)$) and agrees the same seed.\n \\item C calculates the DH public value ($SK_C * (SK_B * PK_A)$) and agrees the same seed.\n \\end{itemize}\n\\end{description}\n\nWhen a user has more than three authenticators ($N$: the number of her authenticators), she processes the above Step1 to Step $N+1$ with repeating the above Step3.\n\n\\subsection{Verifying the Trustworthiness of an OVK} \\label{sec:teian:ovk-trust}\nBecause a service binds public keys to an account by an OVK, the trustworthiness of public keys can never be higher than the trustworthiness of the OVK.\nWe propose how a service verifies the trustworthiness of an OVK.\n\nA service can evaluate the trustworthiness of an OVK using the following two criteria.\n\n\\begin{description}\n \\item[Criterion1] Whether an OVK is derived as described in Section \\ref{sec:teian:ovk-derive}\n \\item[Criterion2] Whether a seed is securely stored in all authenticators\n\\end{description}\n\nThe proposed verification mechanism depends on the attestation mechanism that authenticators already have.\nAuthenticators send an OVPK as well as the attestation of the OVPK at \\textcircled{\\scriptsize 5} on Fig.\\ \\ref{fig:ovk-derive}.\n\n\\begin{figure}[htb]\n \\centering\n \\includegraphics[width=\\columnwidth]{assets\/ovk-attestation.png}\n \\caption{Sending the attestation of an OVPK}\n \\label{fig:ovk-attestation}\n\\end{figure}\n\nFig.\\ \\ref{fig:ovk-attestation} extracts the registration flow of an OVPK from Fig.\\ \\ref{fig:ovk-derive} and describes more details about attestations of public keys.\nAn attestation private key ($AttsKey_A$) is embedded in an authenticator by its manufacturer. \nA certificate for the attestation public key ($Certificate(AttsKey_A)$) is issued by the manufacturer.\nThe authenticator signs the public key generated at \\textcircled{\\scriptsize 2} and the information about the public key by using $AttsKey_A$ and sends them to a service. \n\nThe authenticator also signs a derived OVPK by $AttsKey_A$ to notify the service that the OVPK is derived from the seed stored in the authenticator as described in Section \\ref{sec:teian:ovk-derive:process}.\nThe service can verify the attestation of the OVPK based on the trusted policy about what authenticators comply with Section \\ref{sec:teian:ovk-derive}.\nThe service can validate Criterion1.\n\nMoreover, an attestation of an OVPK contains the OVPK itself and model names of the other authenticators sharing the same seed.\nAn authenticator gets the model names of the other authenticators by receiving the attestation of their DH public key while sharing the seed at \\textcircled{\\scriptsize 2} on Fig.\\ \\ref{fig:seed-sharing-2-party} (Fig.\\ \\ref{fig:seed-sharing-attestation}).\nA service can verify whether they store the seed securely based on the trusted policy about what authenticator model has secure storage and stores the seed in the storage.\nThe service can validate Criterion2.\n\n\\begin{figure}[htb]\n \\centering\n \\includegraphics[width=\\columnwidth]{assets\/seed-sharing-attestation.png}\n \\caption{Attestation when sharing a seed}\n \\label{fig:seed-sharing-attestation}\n\\end{figure}\n\n\\subsection{Re-sharing a New Seed and Updating an OVK} \\label{sec:teian:ovk-update}\nA user updates a set of her authenticators according to lifecycles of the authenticators, such as theft or loss.\nWe propose a mechanism where a user can revoke an OVPK registered with a service and update a new OVPK in the service.\n\n\\subsubsection{Assumption} \\label{sec:teian:ovk-update:assumption}\nWe assume the following for this proposal.\n\n\\begin{enumerate}\n \\item Attackers can operate the seed and the private keys corresponding to registered public keys stored in a stolen authenticator.\n \\item It takes time for attackers to gain control of a stolen authenticator. \n\\end{enumerate}\n\nAssumption 2 is reasonable when authenticators protect the seed and private keys by local authentication like PIN or biometric.\nIn Section \\ref{sec:kousatsu:update}, we consider the case where authenticators have no local authentication, or where local authentication is immediately passed.\n\n\\subsubsection{Overview}\nAuthenticators share a new seed as described in Section \\ref{sec:teian:seed-sharing}.\nThey hold the previous seed along with a new one without erasing the previous one.\nThey notify a service of updating an OVK by sending an updating message described in Section \\ref{sec:teian:ovk-update:update-message} when a user signs in for the first time after re-sharing the new seed.\nThe service that receives updating messages waits for some period (OVK migration period) and accepts the new OVPK from the most trustworthy updating message.\nA service calculates the trustworthiness of each updating message in the way described in Section \\ref{sec:teian:ovk-update:eval-trust}.\nThe service re-binds public keys to the user's account by verifying with the new OVPK and revokes the public key bound only to the previous OVPK.\n\n\\subsubsection{An Updating Message for a New OVK} \\label{sec:teian:ovk-update:update-message}\nWe describe how an authenticator generates an updating message for a new OVK in Fig.\\ \\ref{fig:ovk-update}.\n\n\\begin{figure}[htb]\n \\centering\n \\includegraphics[width=\\columnwidth]{assets\/ovk-update.png}\n \\caption{Generating an updating message for a new OVK}\n \\label{fig:ovk-update}\n\\end{figure}\n\n\\begin{enumerate}\n \\renewcommand{\\labelenumi}{\\textcircled{\\scriptsize \\theenumi}}\n \\item The authenticator receives the metadata from the service to derive OVKs from seeds.\n There are two kinds of received metadata. \n The first one is the metadata ($R^1, M^1$) for the previously registered OVK ($OVK^1$).\n The second one is the metadata ($[(R^2_i, M^2_i)]$) for OVK ($OVK^2$) candidates that other authenticators have registered as new OVKs.\n The authenticator receives the second one as a list because attackers can also generate a malicious updating message derived from a seed by using a stolen authenticator.\n \\item The authenticator derives the previously registered OVK ($OVK^1$) from the received metadata ($R^1, M^1$) and the previous seed ($s^1$) as described in Section \\ref{sec:teian:ovk-derive}.\n The authenticator uses the same key derivation function (\\texttt{KDF}) as the one used in Section \\ref{sec:teian:ovk-derive}.\n The authenticator verifies whether the derived OVK ($OVK^1$) is legitimate by comparing calculated MAC value with $M^1$.\n If the verification fails, the authenticator aborts this update process.\n \\item The authenticator derives the new OVK ($OVK^2$) from the received metadata ($[(R^2_i, M^2_i)]$) and the newly shared seed ($s^2$).\n If the metadata is not an empty list, the authenticator derives the new OVK from the legitimate metadata ($ (R^2, M^2) = (R^2_l, M^2_l)$) with which the authenticator can verify the MAC value ($M^2_l == MAC(s^2, R^2_l + sid)$).\n If the metadata is an empty list or the metadata has no legitimate metadata, the authenticator generates a new random value ($R^2$) and then derives a new OVK.\n Note that the previously registered OVK ($OVK^1$) is not used to derive the new OVK ($OVK^2$), but used to generate an updating message.\n \\item The authenticator signs the new OVPK ($OVPK^2$) by the OVSK ($OVSK^1$) corresponding to the previously registered OVPK ($OVPK^1$).\n The authenticator sends the signature as an updating message when signing in.\n Because the updating message is signed by the private key corresponding to the registered public key, the service can identify the authenticator sending the updating message.\n\\end{enumerate}\n\nIn the above explanation, we assume that authenticators have two shared seeds.\nHowever, authenticators may have more than two seeds because users change their authenticators many times.\nUsers replace their authenticators when purchasing new devices and may lose their authenticators more than once.\nWe explain that authenticators can generate the correct updating message even when they have more than two seeds.\nAuthenticators select the latest seed as the new seed.\nAs the seed corresponds to registered OVPK, authenticators can select the seed successfully by verifying the MAC value of the received metadata.\nFrom the above, authenticators can send a legitimate updating message to services even when they have more than two seeds.\n\n\\subsubsection{Evaluating the Trustworthiness of an Updating Message} \\label{sec:teian:ovk-update:eval-trust}\nWhen a service receives an updating message from a registered authenticator, it enters the OVK migration period.\nIn this migration period, no authenticators can register a new public key by the registered OVK.\nIf the same updating message comes from more than half of the registered authenticators during the period, the service trusts the message.\nOtherwise, the service trusts the updating message sent from the most registered authenticators at the end of the period.\nIf there is more than one message sent by the most registered authenticators, the service trusts the earliest received message.\n\n\\subsubsection{Reducing the Number of Seed Held by Authenticators}\nAs described in Section \\ref{sec:teian:ovk-derive:ovk-per-svc}, an authenticator can derive many different OVKs for different services from a seed.\nAn authenticator can update OVKs multiple times without consuming a lot of storage space.\nHowever, the secure storage space that an authenticator has is limited.\nTherefore, we propose two methods for limiting the number of seeds that an authenticator holds.\n\n\\begin{enumerate}\n \\item Set a limit on the number of seeds that an authenticator holds.\n If the number of seeds exceeds the limit, it deletes the oldest seed with the consent of the user.\n \\item Set an expiration date for a seed.\n If an authenticator has a seed that is about to expire, it prompts a user to share a new seed and update OVKs.\n It deletes any seed that has expired.\n\\end{enumerate}\n\nThe first method is easier to implement because a user decides whether to delete seeds.\nThe first method does not require a user to renew OVKs periodically, even though she continues to use the same authenticators.\nOn the other hand, in the second method, since a seed has an expiration date, OVKs also have the same expiration date.\nThis means that a service can know when to update a seed.\nIn the second method, a service can send security notifications to reduce operational risks such as forgetting to update OVKs.\n\nBoth have their advantages, and both can reduce disadvantages by setting limits to a large value in the first method or a longer expiration date in the second method.\nThe choice of either method depends on a user's preference or the limitations of authenticators.\n\n\\section{Proof of Concept and Use Cases} \\label{sec:poc}\nWe implement the Proof of Concept (PoC) to demonstrate the feasibility that our proposal allows users to access services with multiple authenticators.\nWe implement the PoC using JavaScript.\nIn the PoC implementation, one browser window is treated as one authenticator, so that we can emulate multiple authenticators on the same device.\nNote that the PoC stores seed, private keys, and the attestation key in not secure storage.\nThe source code is available on GitHub \\footnote{\\url{https:\/\/github.com\/hatake5051\/ovk-poc}}.\n\n\\subsection{Implementation Detail}\n\\subsubsection{Implementation of an Authenticator}\nFig.\\ \\ref{fig:impl-authnor} illustrates the implementation of an authenticator based on data flow diagrams \\cite{dfd}.\nIn this figure, an ellipse represents a process, an entity between an upper line and a lower line represents a data store, and an arrow represents data flow.\n\n\\begin{figure}[htb]\n \\centering\n \\includegraphics[width=\\columnwidth]{assets\/impl-authnor.png}\n \\caption{Authenticator Implementation}\n \\label{fig:impl-authnor}\n\\end{figure}\n\n\\texttt{SeedGenerator} implements Section \\ref{sec:teian:ovk-derive}.\n\\texttt{Seed} stores shared seeds.\nThe bit length of each seed is 256.\nWe statically define the following parameters required in Section \\ref{sec:teian:ovk-derive} and a service identifier as the origin of the service URL.\n\n\\begin{itemize}\n \\item The algorithm of an OVK is elliptic curve cryptography \\cite{sec1} with secp256r1 \\cite{sec2}.\n An OVK is calculated using a KDF output as pseudorandomly selected an integer d of Section 3.2.1 in \\cite{sec1}.\n \\item The key derivation function (\\texttt{KDF}) is HMAC \\cite{rfc2104} using SHA-256 \\cite{fips180-4}.\n \\item The MAC function (\\texttt{MAC}) is HMAC \\cite{rfc2104} using SHA-256 \\cite{fips180-4}.\n\\end{itemize}\n\n\\texttt{SeedNegotiator} implements Section \\ref{sec:teian:seed-sharing} except for encrypting and decrypting a DH public key by a CEK and a CEK by a password, and sending and receiving ciphertexts.\n\\texttt{Device} implements these exceptions instead of \\texttt{SeedNegotiator}.\n\\texttt{SeedNegotiator} stores an ephemeral private key for the DH key agreement algorithm in \\texttt{EDH}.\n\n\\texttt{EDH} stores the seed calculated as a result of the key agreement in \\texttt{Seed} and deletes the ephemeral private key.\nWe statically define the following parameters required in Section \\ref{sec:teian:seed-sharing}.\nWe also use JSON Web Encryption Compact Serialization \\cite{rfc7516} to serialize ciphertexts and algorithm identifiers.\n\n\\begin{itemize}\n \\item The key agreement algorithm (\\texttt{DH}) is elliptic curve diffie-hellman based on elliptic curve cryptography \\cite{sec1} with secp256r1 \\cite{sec2}.\n \\item The authenticated encryption algorithm is AES using 128 bit key \\cite{fips197} in Galois\/Counter Mode (GCM) \\cite{nistsp800-38d}.\n \\item The password-based encryption algorithm is Password Based Encryption Scheme 2 \\cite{rfc8018} using AES-KW \\cite{rfc3394} and SHA-256 \\cite{fips180-4}.\n\\end{itemize}\n\n\\texttt{SeedUpdater} implements Section \\ref{sec:teian:ovk-update}.\nOnly this process and \\texttt{SeedGenerator} can access \\texttt{Seed}.\n\n\\texttt{Device} is the process of generating key pairs and attestations and managing them based on an OVK.\n\\texttt{Credential} stores generated key pairs.\n\\texttt{Attestation} stores the attestation private key and the certificate of the corresponding attestation public key.\n\\texttt{Negotiating} stores data used for Section \\ref{sec:teian:seed-sharing} like a password.\n\\texttt{Device} generates an attestation for these type of keys: public keys stored in \\texttt{Credential}, OVPKs generated by \\texttt{SeedGenerator}, and DH public keys calculated by \\texttt{SeedNegotiator}. \n\n\\texttt{UI} is the process of communicating ciphertexts generated by \\texttt{Device} with other authenticators and interacting with a user.\nWe use reading and generating QR codes as the communication channel among authenticators used for Section \\ref{sec:teian:seed-sharing}.\n\\texttt{FetchAPI} communicates with a service via a secure channel established by TLS.\n\n\\subsubsection{Implementation of a Service}\nFig.\\ \\ref{fig:impl-svc} illustrates the implementation of a service based on data flow diagrams.\n\n\\begin{figure}[htb]\n \\centering\n \\includegraphics[width=0.9\\linewidth]{assets\/impl-svc.png}\n \\caption{Service Implementation}\n \\label{fig:impl-svc}\n\\end{figure}\n\n\\texttt{StartAuthn} accepts authentication requests from a user.\nIt receives the account name of the user and responds with a challenge bound to the account and, if registered, public keys, an OVPK, and the metadata of OVPK.\n\n\\texttt{Register} accepts requests for registering a new account and a new public key bound to an account as described in Section \\ref{sec:teian:ovk-derive}.\nWhen registering a new account, a user sends a new public key, an OVPK, and the metadata of the OVPK.\nThe user also sends an attestation of the public key and, if requested, an attestation of the OVPK.\nA service verifies attestations to determine whether it accepts the registration.\n\n\\texttt{Authn} accepts challenge responses for authentication.\nWe use elliptic curve digital signature algorithm \\cite{sec1} with secp256r1 \\cite{sec2} for authentication.\nIt also receives an updating message as described in Section \\ref{sec:teian:ovk-update}.\n\\texttt{Update} processes updating OVKs.\n\n\\texttt{CredManager} manages the bindings of public keys and OVPKs to accounts.\n\\texttt{Creds} stores public keys bound to OVPKs.\n\\texttt{OVKM} stores OVPKs and the corresponding metadata bound to accounts.\n\\texttt{Updating} stores updating messages when processing updating OVPKs.\n\n\\subsection{Use Cases}\nUsing the PoC in the following scenario, we confirm that our proposal allows authenticators to share a seed, derive an OVK, register a new public key with an OVK seamlessly, and update a registered OVPK.\n\n\\begin{enumerate}\n \\item Three Authenticator A, B, and C share a seed.\n \\item With Authenticator A, the user registers a new public key and an OVPK with Service 1 during account registration.\n \\item With Authenticator B and C, she seamlessly registers new public keys with Service 1.\n \\item With Authenticator B, she registers a new public key and an OVPK with Service 2 during account registration.\n \\item With Authenticator C (not A), she seamlessly registers a new public key with Service 2.\n \\item Two Authenticator A and B updates a new seed (Assuming that Authenticator C is lost). \n \\item With Authenticator A, she notifies Service 1 of updating a new OVPK.\n At this time, it is still possible to sign in to Service 1 with Authenticator C.\n \\item With Authenticator B, she notifies Service 1 of updating the same new OVPK as the one notified with Authenticator A.\n At this time, Service 1 updates an OVPK bound to her account, so that it is impossible to sign in to Service 1 with Authenticator C.\n \\item With Authenticator B, she notifies Service 2 of updating a new OVPK.\n At this time, it is still possible to sign in to Service 2 with Authenticator C.\n After finishing the migration period, Service 2 updates an OVPK bound to her account and revokes the public key of Authenticator C.\n Then, with Authenticator A, she can seamlessly register a new public key with Service 2.\n\\end{enumerate}\n\n\\section{Evaluation with Threat Analysis} \\label{sec:eval}\nWe evaluate our proposal by analyzing the PoC described in Section \\ref{sec:poc} based on threat modeling \\cite{threat-modeling, fido-sec-ref}.\n\n\\subsection{Security Requirement}\n\\subsubsection{Assets to be Protected}\nWe enumerate the assets to be protected in this proposal.\n\\begin{enumerate}\n \\item Private keys stored in authenticators\n \\item Public keys managed by services\n \\item The attestation key stored in each authenticator\n \\item The key signing attestation certificates managed by each authenticator manufacturer\n \\item The trusted root certificate and policy for services to validate attestations\n \\item The TLS private key managed by services\n \\item The trusted root certificate and policy for authenticators to validate TLS certificates\n \\item The seed stored in authenticators\n \\item The OVPK and corresponding metadata stored in services\n \\item The ephemeral DH private key generated by authenticators for sharing a seed\n \\item The temporary password stored in authenticators for sharing a seed\n\\end{enumerate}\n\n\\subsubsection{Security Goals} \\label{sec:eval:sec-goals}\nWe enumerate the goals to achive in our proposal.\nWe define the following goals while referring to goals in \\cite{fido-sec-ref}.\n\n\\begin{description}\n \\item[SG-1] Strong User Authentication: Services can authenticate users based on public key authentication.\n \\item[SG-2] Unlinkability: Services cannot correlate their accounts.\n \\item[SG-3] Credential Binding: Services can bind public keys to legitimate accounts.\n \\item[SG-4] Attestable Properties: Services and authenticators can validate public keys by verifying attestations.\n \\item[SG-5] Forgery Resistance: Be resilient from attempting to modify intercepted communications to masquerade as legitimate users.\n\\end{description}\n\n\\subsubsection{Security Assumptions}\nWe enumerate the assumptions about the environment where our proposal works.\n\n\\begin{description}\n \\item[SA-1] The processes and data stores surrounded by the trusted boundary represented by the inner dotted line in Fig.\\ \\ref{fig:impl-authnor} are isolated from other processes on the authenticator.\n The authenticator requires local authentication before accessing these processes and data stores.\n \\item[SA-2] The cryptographic algorithms used can achieve the objectives of each algorithm.\n \\item[SA-3] A service can correctly validate the certificate chain of attestations.\n \\item[SA-4] A service and an authenticator can establish a secure channel for service authentication, confidentiality for message, and integrity for messages (like TLS).\n\\end{description}\n\n\\subsection{Threat Analysis}\nWe enumerate the threats on data flow diagrams described in Section \\ref{sec:poc}, and explain what goals listed in Section \\ref{sec:eval:sec-goals} each threat violates and what measures our proposal takes.\n\n\\subsubsection{Threats on an Authenticator}\nIn Fig.\\ \\ref{fig:impl-authnor}, a dotted line represents a trusted boundary.\nWe focus on data flows across trusted boundaries and enumerate the threats in each of them.\n\nFirst, threats arising between a user and \\texttt{UI} include the following.\n\n\\begin{description}\n \\item[Homograph Mis-Registration] A malicious service pretends a legitimate service to make the user believes it is legitimate.\n It prompts the user to register a new public key seamlessly, and sends metadata stolen from other services.\n The malicious service correlates OVPKs by whether the user requests a new public key registration or not.\n This threat violates SG-2 because the malicious service can correlate different OVPKs generated for different services.\n Our proposal addresses this threat because authenticators verify the MAC value of the received metadata.\n The data protected by the MAC value contains the identifier of the service that the authenticator communicates.\n Because Assumption SA-4 allows authenticators to trust the communicating service identifier, authenticators can detect spoofing of services by malicious actors.\n \\item[User Verification By-pass] An attacker can operate the authenticator, or an attacker can bypass the local authentication of the authenticator to operate it.\n This threat violates SG-1 because the attacker can masquerade as the legitimate user.\n Our proposal addresses this threat by transferring it to SA-1.\n We consider the case where this SA-1 assumption is not satisfied during the OVK update in Section \\ref{sec:kousatsu:update}.\n\\end{description}\n\nThreats arising between a service and \\texttt{FetchAPI} include the following.\n\n\\begin{description}\n \\item[Service Verification Error] The authenticator cannot properly authenticate services, thus it cannot correctly identify services.\n As a result, an attacker can eavesdrop and tamper with the communication channel.\n This threat violates SG-1 and SG-5 because the attacker can hijack the authenticated session.\n This threat also violates SG-2 because the authenticator fails to address the threat Homograph Mis-Registration described above.\n Our proposal addresses this threat by transferring it to SA-4.\n\\end{description}\n\nThreats arising between another authenticator and \\texttt{UI} include the following.\n\n\\begin{description}\n \\item[Malicious Authenticator Linking] An attacker's authenticator participates in sharing a seed.\n The attacker gets the seed value itself.\n The attacker can use this authenticator to register a new public key with any service that a legitimate user has registered.\n This threat violates SG-3.\n Our proposal addresses this threat because a user protects sharing a seed with a password.\n Users are required to use passwords that are not guessable during sharing a seed.\n \\item[Weak Authenticator] A user allows a weak authenticator to participate in sharing a seed.\n The weak authenticator does not securely protect a seed, so, when an attacker compromises the weak authenticator, the seed may be leaked.\n This threat violates SG-3 because the attacker can register a new public key of the attacker's authenticator by generating the OVK with the compromised seed.\n Our proposal addresses this threat because each authenticator validates the security properties of other authenticators through attestations when sharing a seed.\n\\end{description}\n\nThreats arising between \\texttt{Device} and \\texttt{UI} include the following.\n\n\\begin{description}\n \\item[Malicious Authenticator] A user uses a malicious authenticator.\n Because a user cannot rely on the malicious authenticators, this threat violates any goals.\n It is difficult for users to determine whether it is a legitimate authenticator from a trusted manufacturer.\n Our proposal addresses this threat because services maintain a list of attestation certificates of trusted manufacturers.\n\\end{description}\n\nThreats to the authenticator include the following.\n\n\\begin{description}\n \\item[Side Channel Attack] Access to the data store that is not described in Fig.\\ \\ref{fig:impl-authnor} compromises assets to be protected.\n This threat violates SG-1 if key pairs are compromised and SG-2 and SG-3 if seeds are compromised.\n This threat also arises another threat like Malicious Authenticator Linking if a temporary password for sharing a seed is compromised.\n Our proposal addresses this threat by transferring it to SA-1.\n \\item[Bad Cryptography Primitives] An authenticator uses a compromised cryptographic algorithm or a weak pseudo-random number generator in the process.\n This threat violates SG-3 if an attacker derives the OVK.\n This threat also violates SG-2 if an attacker gets the seed.\n Our proposal addresses this threat by transferring it to SA-2.\n\\end{description}\n\n\\subsubsection{Threat on a Service}\nIn Fig.\\ \\ref{fig:impl-svc}, we focus on data flows across trusted boundaries and enumerate the threats in each of them.\n\nThreats arising between an authenticator and \\texttt{StartAuthn} include the following.\n\n\\begin{description}\n \\item[Linking OVPK] An attacker can obtain the OVPK and metadata associated with the account.\n The attacker receives OVPKs and corresponding metadata from many services and attempts to derive corresponding OVSKs.\n The attacker also attempts to correlate collected OVPKs by checking whether OVPKs are derived from the same seed.\n This threat violates SG-2 and SG-3.\n Our proposal addresses this threat by transferring it to SA-2.\n\\end{description}\n\nThreats arising between an authenticator and \\texttt{Register} include the following.\n\n\\begin{description}\n \\item[Malicious Authenticator Registration] An attacker attempts to register a new public key of his authenticator to a legitimate user account.\n This threat violates SG-3.\n Our proposal addresses this threat because an attacker cannot get the OVSK corresponding to the OVPK bound to the account.\n An attacker cannot also get the seed corresponding to the OVPK.\n The service can verify that trusted authenticators store a seed and OVSKs by verifying OVPK attestations.\n Even if a seed is compromised, our proposal mitigates this threat because the number of authenticators that can be registered is restricted.\n After a user registers public keys of her all authenticators an attacker cannot register his public keys.\n\\end{description}\n\nThreats arising between an authenticator and \\texttt{Authn} include the following.\n\n\\begin{description}\n \\item[Updating Malicious OVPK] An attacker attempts to update to an OVPK derived from the seed held by his authenticator.\n This threat violates SG-3 because the attacker can register his public keys.\n This threat also violates SG-1 because the attacker can revoke the user's public keys.\n Our proposal addresses this threat because an attacker cannot know the seed corresponding to the registered OVPK.\n We consider the case where this SA-1 assumption is not satisfied during updating an OVK in Section \\ref{sec:kousatsu:update}.\n We disscuss whether our proposal mitigates this threat even if the seed is compromised in Section \\ref{sec:kousatsu:update}.\n\\end{description}\n\n\\section{Discussion} \\label{sec:kousatsu}\n\\subsection{Deriving an OVK}\nWe confirm that our proposal achieves the requirements of Section \\ref{sec:teian:ovk-derive:youken}.\nFirst, a user registers an OVK during new account registration where a service can trust information received from a user.\nA service can verify the owner of the authenticator storing a private key corresponding to the public key to be registered without a trusted third party.\nNote that attestation and the safety of communications rely on a trusted third party.\nSecond, malicious services cannot correlate their accounts with sharing OVPKs and corresponding metadata because the security property of a key derivation function makes it impossible to derive a seed from an OVPK and the corresponding metadata.\nBesides, a malicious service cannot correlate the user's account by checking whether a user can use the OVPK and the metadata of another service to request a new public key registration.\nThis is because authenticators verify the MAC value of the received metadata to determine whether the received metadata is for the service.\nThird, an authenticator generates an attestation of the public key to be registered.\nA service can verify the attestation to evaluate the trustworthiness of the public key requested to be registered.\nFinally, once authenticators of a user share a seed, they derive an OVK per service independently.\nShe does not have to operate multiple authenticators whenever registering a new public key.\n\n\\subsection{Sharing a Seed}\nWe confirm that our proposal achieves the requirements of Section \\ref{sec:teian:seed-sharing:youken}.\nOnly authenticators having a password can participate in sharing a seed.\nOnly authenticators having a password can decrypt the ciphertexts generated by other authenticators having the same password.\nThey can also verify the integrity of received ciphertexts by a password.\nA user enters a password directly into each authenticator so that the password does not flow on the communication channel where authenticators share a seed.\nBecause a long enough password allows an attacker to take an extremely long time to decrypt ciphertexts and a secure encryption algorithm prevents him from compromising ciphertexts, it is difficult for the attacker to compromise the password and participate in sharing a seed before the sharing is complete.\nBesides, because the assumption that a DH key agreement algorithm is secure against eavesdropping prevents an attacker from deriving a seed from decrypted ciphertexts including DH public keys, an attacker cannot compromise a seed.\nFrom the above, without assuming the security of the communication channel, we can prevent the leakage of the seed by eavesdropping and tampering.\n\n\\subsection{The Trustworthiness of a Key for Authentication}\nA service considers the next two to determine whether it trusts a key for authentication.\nOne is the trustworthiness of the key itself.\nThe other is the trustworthiness of the binding of a public key to an account.\nA service evaluates the former by verifying that a trusted authenticator stores a private key and that a used cryptographic algorithm has not been compromised.\nTo verify that, a service verifies the attestation of a public key.\n\nA service evaluates the latter by verifying whether the private key corresponding to the public key bound to an account is stored in the authenticator owned by the user having the account.\nThis trustworthiness depends on how a user registers the public key.\nFor example, in \\cite{fido-multi-registration}, a user registers a new public key via an authenticated session established by a registered public key.\nIn our proposal (Section \\ref{sec:teian:ovk-derive} and Section \\ref{sec:teian:seed-sharing}), a user registers a new public key with an OVK derived from the seed shared among authenticators.\nIn the former method, a public key has high assurance because a user uses registered authenticators every time she registers a new public key.\nThe method is not convenient because she has to have a registered authenticator for registering a new one.\nThe latter method (our proposal) is convenient because, once she has shared a seed, a user must have only an authenticator to append a new public key to a service.\nA public key does not have high assurance if the seed can be compromised.\n\nTo make a public key higher assurance in our proposal, We propose a method for a service to verify the trustworthiness of an OVPK (Section \\ref{sec:teian:ovk-trust}).\nBy verifying an attestation of an OVPK to be registered, a service can evaluate whether the seed deriving the OVPK is stored securely on the authenticator communicating with the service.\nBy verifying an attestation of a DH public key when sharing a seed, authenticators can evaluate whether the other authenticators store the seed securely.\nA service can also verify whether all authenticators securely store the seed deriving the registered OVPK by verifying an attestation of the OVPK.\nThis is because the attestation includes the model names of all authenticators storing the seed.\nA service can verify whether they store the seed securely based on the trusted policy about what authenticator model has secure storage and stores the seed in the storage.\n\n\\subsection{Updating an OVK} \\label{sec:kousatsu:update}\nA service calculates the trustworthiness of an updating message and selects the OVK of the most trusted message as the new OVK.\nA service considers the number of registered authenticators sending the same updating message as the trustworthiness of the updating message.\nWe consider that the trustworthiness of all registered authenticators before the OVK migration period is equal.\nThis is because it is difficult for a service to determine whether an authenticator is stolen or held by a legitimate user.\nBased on the assumption that it takes time for an attacker to gain control of a stolen authenticator (Assumption 2 in Section \\ref{sec:teian:ovk-update:assumption}), a service selects the earlier sent message when two or more updating messages have the same and most trustworthiness. \n\nWe discuss what attacks the proposed method prevents when an attacker can operate the seed and the private key with a stolen authenticator (Assumption 1 in Section \\ref{sec:teian:ovk-update:assumption}).\nIn our proposal, the number of registered authenticators is limited to $N$, which is in the metadata sent when an OVK registration.\nNote that an attacker can increase the number of registered authenticators by registering public keys in the way described in Section \\ref{sec:teian:ovk-derive} before sending updating messages.\n\nWe disscuss cases based on the following numbers.\n\\begin{itemize}\n \\item $N$: the number of authenticators sent when an OVK registration\n \\item $N_u$: the number of registered authenticators owned by a legitimate user before an OVK migration period\n \\item $N_a$: the number of registered authenticators controlled by an attacker before an OVK migration period\n\\end{itemize}\n\nIn the case of $N=2$, the service trusts the authenticator that sends updating messages earlier.\nTherefore, in the case of $N_a =1$, if the attacker sends an updating message earlier (Assumption 2 in Section \\ref{sec:teian:ovk-update:assumption} is broken), the service trusts the OVK sending from the authenticator stolen by the attacker and revokes the public key whose corresponding private key is held by the authenticator of the legitimate user.\n\nIn the case of $N \\geq 3$, if $N_u \\geq N \/ 2$ or $N_u > N_a$, then a legitimate user can update an OVK and prevent the attacker from updating an OVK because public keys whose corresponding private keys is stored in stolen authenticators are correctly revoked.\n\n\\section{Conclusion} \\label{sec:conclusion}\nWe introduce a key pair called an Ownership Verification Key (OVK) and propose the mechanism where users and services manage public keys based on the owner of authenticators storing the corresponding private keys.\nThe mechanism allows users to access services with any of their authenticators without registering each of their public keys explicitly.\n\nA user can derive the private key of an OVK (OVSK) on her authenticators from the seed sharing among the authenticators.\nA service binds the public key of OVK (OVPK) to a user's account.\nA service binds a public key signed by an OVSK to the user's account bound to the corresponding OVPK.\nWhen a user changes a set of her authenticators, she updates an OVSK, and a service updates an OVPK binding to her accounts based on the most trustworthy updating message.\n\nWe implemented the Proof of Concept and confirmed that key management works as expected for typical use cases.\nWith threat modeling, we evaluated what measures our proposal takes against the threats.\nWe confirmed that our proposal achieves some security goals, such as that services cannot correlate accounts and can correctly bind public keys to accounts.\nWe discussed how our proposal mitigates threats for which measures are not sufficient. \n\nFuture work includes a model where, in updating an OVK, the trustworthiness of each authenticator having the private key corresponding to a registered public key differs.\nFor example, when a service receives a message that some authenticator is not trustworthy from the registered email address, the service reduces the trustworthiness of the authenticator.\n\n\\bibliographystyle{ACM-Reference-Format}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nIn this paper, we introduce and discuss the possibility to obtain\nsteady solutions with power-like tails starting from conservative\nmolecular systems described by the Boltzmann equation with\nMaxwell-type collision kernels. The starting point of our model is\nto consider binary collisions that result in a linear combination\nof an inelastic collision and a random contribution. As we\nshall see, the random addition to the post-collision velocities\ncan only increase the mean of the collisional energy, and, among\nother things, it gives the possibility to construct a binary collision\nthat preserves (in the mean) mass, momentum and energy. Our model\nis closely related to a kinetic model for economics introduced by\nPareschi and two of the present authors \\cite{CPT}. There, the\nrandom contribution to the collision (trade) was introduced to\ntake into account the returns of the market.\n\nInelastic Maxwell models were introduced by Bobylev, Gamba and one\nof the authors in 2000 \\cite{Bobylev-Carrillo-Gamba}; see also\n\\cite{BK00} for the one dimensional case. Maybe the most\ninteresting result (absent in the elastic case) is the existence\nof self-similar solutions in the homogeneous cooling problem and\nthe non-Maxwellian behavior of these solutions, which displays\npower-like decay for large velocities. It was conjectured in\n\\cite{EB2} and later proved in \\cite{Bobylev-Cercignani2,\nBobylev-Cercignani-Toscani, BisiCT2} that such solutions represent\nintermediate asymptotics for a wide class of initial data. Other\nresults concerned with self-similar solutions in the theory of\nthe classical (elastic) Boltzmann equation for Maxwell molecules were\nalso recently published in \\cite{Bobylev-Cercignani,\nBobylev-Cercignani2}. In light of these results, it looks\nclear that in many aspects both elastic and inelastic Maxwell\nmodels must be studied from a unified point of view. As observed\nby Bobylev and Gamba in\n\\cite{Bobylev-Cercignani-Gamba,Bobylev-Cercignani-Gamba2}, an\ninteresting question arises in connection with power-like tails\nfor high velocities. Is it possible to observe a similar effect,\ni.e., an appearance of power-like tails from initial data with\nexponential tails, in a system of particles interacting according\nto laws of classical mechanics without energy loss? In\n\\cite{Bobylev-Gamba} Bobylev and Gamba gave a partial answer to\nthis question by showing that, under a certain limiting procedure,\nsuch behavior can in principle be observed if one considers a\nmixture of classical Maxwell gases. More precisely, self-similar\nsolutions converging towards maxwellian equilibrium were proved to\nhave power-like tails once normalized by the equilibrium.\n\nIn this paper, we will try to elucidate the same question,\nstarting from a somewhat different point of view. Our starting\npoint will be a suitable modification to the homogeneous Boltzmann\nequation for the inelastic Maxwell molecules introduced in Ref.\n\\cite{Bobylev-Carrillo-Gamba}, in such a way that the usual\nconservations of mass, momentum and energy in the binary\ncollisions still continue to hold in the mean sense. The\nscaled-in-time inelastic Boltzmann equation introduced in\n\\cite{Bobylev-Carrillo-Gamba} reads\n \\begin{equation}\n\\frac{\\p f}{\\p t}=Q_e(f,f)\\,. \\label{BE1}\n \\end{equation}\nHere, $f(v,t)$ is the density for the velocity space distribution\nof the molecules at time $t$, while $Q_e(f,f)$ is the inelastic\nBoltzmann collision operator, which contains the effects of binary\ncollisions of grains. As usual in this context, the collision\noperator $Q_e(f,f)$ is more easily treated if expressed in weak\nform. This corresponds to writing, for every suitable test\nfunction~$\\varphi$,\n \\begin{equation}\n(\\varphi,Q_e(f,f))= \\frac{1}{4 \\pi} \\int_{\\Realr^3}\n\\int_{\\Realr^3} \\int_{S^2} f(v) f(w) \\Big[ \\varphi (v^*)- \\varphi\n(v) \\Big] dv\\,dw\\,d\\sigma. \\label{Qweak}\n \\end{equation}\nIn \\fer{Qweak}, $v^*$ is the outgoing velocity corresponding to a\nparticle in the collision defined by the incoming velocities $v,w$\nand the angular parameter $\\sigma \\in S^2$:\n\\begin{equation}\n \\begin{split}\n &\\displaystyle v^*= \\frac{1}{2} (v+w)+ \\frac{1-e}{4}(v-w)+\\frac{1+e}{4}\n|v-w|\\sigma,\n\\\\*[.3cm]\n&\\displaystyle w^*= \\frac{1}{2} (v+w)- \\frac{1-e}{4}(v-w)-\\frac{1+e}{4}\n|v-w|\\sigma\\,.\n\\end{split}\n\\label{post}\n \\end{equation}\n The parameter $0\\leq e \\leq 1$ represents the restitution coefficient.\n\nIn the model we consider, this restitution coefficient will be chosen as a\nrandom variable that can be interpreted from a physical point of view as the\nstochasticity in the microscopic process of collision due to the randomness of\nthe grains' geometry and the mechanical properties of the medium. We will show in\nthe next section that this random behavior in restoring energy leads to a precise\nform of the energy gain term that differs from the usually chosen diffusion\nterm, the so-called \"thermal bath\". This new form of \"thermal bath\" is thus related\nto the process generated by the randomness of the granular media. We prove\nthat this particular thermal bath yields equilibrium states with power law\ntails.\n\nSuch over-populated tails in distributions at equilibrium arise in\nother contexts. We shall present similar results on the large\ntime behavior of collisional kinetic theory applied to economic\nmodelling. In this framework, the kinetic variable represents the\nwealth of agents and the collision operator describes the\nevolution of the wealth distribution through exchanges. We refer\nto \\cite{CPT, MaTo06} and references therein for a mathematical\npresentation of these models closely related to so called\n\"econo-physics\". In such models, the equations between pre- and\npost-collisional values involve some randomness that is related\nto the stochasticity of the market that provides random returns.\n\n\n\n\nIn the remainder of this paper, we will study in detail the large time\nbehavior of the solution of the Boltzmann equation involving such a stochastic\nprocess. We show that the validity (at a macroscopic level) of the classical\ncollision invariants is enough to guarantee convergence towards a steady\nprofile, but not enough to reach a Maxwellian-like profile. In fact, we will\nshow that there is a class of random perturbations of the coefficient of\nrestitution such that the steady state possesses power-like tails.\n\nA crucial role in our analysis is played by the weak norm\nconvergence, which is obtained by further pushing the development\nof a method first used in \\cite{Gabetta-Toscani-Wennberg} to\ncontrol the exponential convergence of Maxwellian molecules in\ncertain weak norms. This will be done by using the fact that the\nnonlinear operator in the Boltzmann equation (see \\fer{BE2})\n can be expressed in Fourier\nvariables in a simple closed form using Bobylev's identity\n\\cite{Bobylev}. Estimates of the evolution of the Wasserstein\ndistance \\cite{Wasserstein69,Vil03,Vil06} between solutions will\nbe presented for the economic and the inelastic model since they\ngive complementary information with respect to the results in\n\\cite{MaTo06}. Concerning this second aspect, we will take\nadvantage of the recent analysis of Bolley and Carrillo\n\\cite{Bolley-Carrillo, CTPE} of the inelastic Boltzmann equation\nfor Maxwell molecules. From this analysis, we will obtain the\nuniqueness and asymptotic stability of stationary states for this\nmodel. Finally, the appearance of power-like tails for the\nasymptotically stable stationary states will be discussed for both\nmodels, giving explicit examples of random variables producing this\nbehavior.\n\n\nThe paper is organized as follows: in section \\ref{Modeling} we\ndetail the collisional models for both granular media and economy\napplications including random coefficients in the relations\nbetween pre- and post-collisional variables. In section\n\\ref{quick}, we recall the main properties of probability metrics. In\nsection \\ref{eco2}, we investigate large time behavior of the solution\nof the kinetic economy model and section \\ref{seccontr} is devoted to\nlarge time behavior of stochastic granular media.\n\nLastly, let us summarize the two main results of this paper :\nfirst, we give some insight into conditions for a collision\noperator to lead to power-law tails (conservatism in mean being\nsome kind of necessary condition); second, we propose a new form\nfor the thermal bath with a physically relevant origin (the\nrestitution coefficient taking into account the randomness of\ngranular media).\n\n\n\\section{Modelling issues and diffusion approximation}\n\\label{Modeling}\n\n\nLet us present the proposed stochastic granular model\n(with a random restitution coefficient) and its diffusion limit and then\nrecall briefly the similar analysis for the economy model following\n\\cite{CPT}.\n\n\n\\subsection{Stochastic granular media}\n\nConsidering the weak formulation \\fer{Qweak},\neasy computations show that $(\\varphi(v),Q_e(f,f))=0 $ whenever\n$\\varphi(v) =1$ and $\\varphi(v) =v$, while\n$(\\varphi(v),Q_e(f,f))<0 $ if $\\varphi(v) =v^2$. This corresponds\nto conservation of mass and momentum, and, respectively, to loss\nof energy for the solution to equation \\fer{BE1}. For this reason,\nif we fix the initial data to be a centered probability density\nfunction, the solution will remain centered at any subsequent time\n$t>0$. The loss of energy in a single collision\nwith a constant restitution coefficient $e$ is given by\n \\begin{equation}\n \\label{loss}\n |v'|^2+ |w'|^2 = |v|^2 +|w|^2 -\\frac{1-e^2}4 \\left( |v-w|^2 -|v-w|(v-w)\\cdot\n \\sigma \\right).\n \\end{equation}\nThe previous formula is the key to our modification of the collisions.\nLet us replace the constant coefficient of restitution $e$ with\na stochastic coefficient of restitution $\\et$, such that for a\ngiven random variable $\\eta$\n \\begin{equation}\\label{re}\n \\et = e +\\eta, \\qquad \\mbox{with} \\quad\\langle \\eta\\rangle =0 \\quad \\mbox{and} \\quad \\langle \\eta^2 \\rangle =\n \\beta^2.\n \\end{equation}\nIn \\fer{re} and in the rest of the paper, $\\langle \\cdot \\rangle$ denotes the\nmathematical expectation of the real-valued random variable $\\eta$, i.e.,\nintegration against a measure $\\mu$. For obvious physical reasons, the random\nvariable $\\eta$ has to be chosen to satisfy $\\eta \\ge -e$, in order to\nguarantee that the (random) coefficient of restitution $\\et\\geq0$. Using $\\et$\ninstead of $e$ in \\fer{post} gives that the momentum is {\\it conserved in\naverage} for a suitable choice of the variance. In fact, since\n \\begin{equation}\\label{1new}\n \\left\\langle |v'|^2+ |w'|^2\\right\\rangle = |v|^2 +|w|^2 -\\frac{1-e^2 -\\beta^2}4 \\left( |v-w|^2 -|v-w|(v-w)\\cdot\n \\sigma \\right),\n \\end{equation}\nby choosing the variance $\\beta^2 = 1-e^2 >0$, we obtain\n \\begin{equation}\\label{cons}\n\\left\\langle |v'|^2+ |w'|^2\\right\\rangle = |v|^2 +|w|^2.\n \\end{equation}\n We will call a collision process (or equivalently a random cross section)\n satisfying \\fer{cons} \\emph{conservative\n in the mean}.\n\n Let us remark that condition \\fer{cons} \\emph{cannot be satisfied\nif $\\et$ takes only values less than $1$}, since in that case\n$\\et^2$ remains also less than 1 and so does its average\n$<\\et^2>=e^2+\\beta^2 <1$.\n The main idea behind\nthis is that particles can even gain energy in collisions even though\nthe total energy is conserved in the mean.\n\n From the physical point of view, this assumption of energy-gain particle\ncollisions may seem strange. We will show in the sequel that this energy input\ncan be interpreted as a sort of thermal bath. Particles are immersed in a medium\nthat produces this random change in the strength of their relative velocity. We\nwill argue, based on a derivation of a Fokker-Planck approximation, that this\nrandom component in the collision operator can be approximated by a\nsecond-order differential operator whose diffusion matrix depends on the second\nmoments of the solution $f$ itself and the random variable $\\eta$ (see\n\\cite{CPT, PaTo06} for a similar approach in one dimension).\n\n\nThis idea allows us to consider a new class of Maxwell-type models,\nfrom now on called \\emph{conservative in the mean}, which are\nobtained from (post-collision) velocities given by\n\\begin{equation}\n \\begin{split}\n &\\displaystyle v'= \\frac{1}{2} (v+w)+ \\frac{1-\\et}{4}(v-w)+\\frac{1+\\et}{4}\n|v-w|\\sigma,\n\\\\*[.3cm]\n&\\displaystyle w'= \\frac{1}{2} (v+w)- \\frac{1-\\et}{4}(v-w)-\\frac{1+\\et}{4}\n|v-w|\\sigma\\,.\n\\end{split}\n\\label{postcon}\n \\end{equation}\nwhere $\\et$ is the random coefficient of restitution defined in \\fer{re}, and\n$\\beta^2 = 1-e^2$. The corresponding Boltzmann equation reads\n \\begin{equation}\n\\frac{\\p f}{\\p t}= \\tilde{Q}_e(f,f) = \\left\\langle Q_{\\et}(f,f)\n\\right\\rangle\\, , \\label{BE2}\n \\end{equation}\nand its corresponding weak form is\n \\begin{equation}\n(\\varphi,\\tilde{Q}_e(f,f))= \\frac{1}{4 \\pi}\\left\\langle\n\\int_{\\Realr^3} \\int_{\\Realr^3} \\int_{S^2} f(v) f(w) \\Big[ \\varphi\n(v')- \\varphi (v) \\Big] dv\\,dw\\,d\\sigma \\right\\rangle.\n\\label{Qweak1}\n \\end{equation}\nIn view of our choice of the random contribution to the\ncoefficient of restitution, we now have\n$(\\varphi(v),\\tilde{Q}_e(f,f))=0 $ whenever $\\varphi(v) =1,v,\n|v|^2$, that is, the classical collision invariants of the elastic\nBoltzmann equation.\n\n\n\\subsection{ Formal diffusive asymptotics }\n\\label{formal}\n\nBefore entering into the study of the large-time behavior of the\nBoltzmann equation \\fer{BE2}, we shall present here some formal\narguments that hopefully clarify the action of the random\nrestitution coefficient in the collision mechanism \\fer{postcon}.\n\n\n\nTo this end, following the same method as in \\cite{To-Diss}, letting\n$(v',w')$ denote the post-collision velocities \\fer{postcon} in our\nrandom collision with $(v^*,w^*)$ as post collision velocities\ndefined by\nthe classic inelastic collision \\fer{post}, we can split the\nvelocities into their deterministic and random parts\n \\begin{equation}\\label{rel2}\n v' = v^* + \\eta \\Delta(u, \\sigma) \\, , \\quad w' = w^* - \\eta \\Delta(u,\n \\sigma),\n \\end{equation}\n where we let $u = v-w$ and\n \\[\n \\Delta(u, \\sigma) = \\frac 14 \\left( |u|\\sigma - u\\right).\n \\]\nLet us consider a Taylor expansion of $\\varphi(v')$ around $\\varphi(v^*)$\nup to second order in $\\eta$.\nThanks to \\fer{rel2} we get\n\\begin{eqnarray}\\label{exp}\n\\varphi(v') = \\varphi(v^*) + \\eta\\,\\left(\\nabla \\varphi(v^*) \\cdot\n\\Delta(u, \\sigma)\\right) + \\frac 12 \\eta^2\n\\sum_{i,j}\\frac{\\partial^2 \\varphi(v^*)}{\\partial v^*_i\\partial\nv^*_j}\\Delta_i\\Delta_j + \\dots\n\\end{eqnarray}\nThus, taking the mean of the expansion \\fer{exp}, and using the property\n$\\langle \\eta \\rangle= 0$, we get\n \\begin{equation}\\label{exp2}\n\\langle\\varphi(v')\\rangle = \\varphi(v^*) + \\frac 12 \\beta^2\n\\sum_{i,j}\\frac{\\partial^2 \\varphi(v^*)}{\\partial v^*_i\\partial\nv^*_j}\\Delta_i\\Delta_j + \\dots.\n \\end{equation}\nTruncating the expansion \\fer{exp2} after the second--order term and\ninserting \\fer{exp2} into \\fer{Qweak1}, we conclude\n \\begin{align}\\label{weak2}\n(\\varphi\\, , \\, \\tilde Q_e(f,f)) \\, &\\simeq (\\varphi\\, , \\,\nQ_e(f,f)) + (\\varphi, D_e(f,f)) \\\\\n &=(\\varphi\\, , \\, Q_e(f,f))+ \\frac {\\beta^2}{8\\pi}\n\\int_{\\Realr^3}\\int_{\\Realr^3}\\int_{S^2}\\sum_{i,j}\\frac{\\partial^2\n\\varphi(v^*)}{\\partial v^*_i\\partial v^*_j}\\Delta_i\\Delta_j\nf(v)f(w)dv\\, dw\\, d\\sigma \\,\\, .\\nonumber\n \\end{align}\nWhile the first term in \\fer{weak2} $Q_e(f,f)$ is the classical inelastic\nBoltzmann collision operator, the second term $D_e(f,f)$ needs to be further\nanalyzed.\n\nDenoting by $(\\null^{*}v, \\null^{*}w)$ the pre-collision\nvelocities in the inelastic collision, and taking into account the\nfact that the Jacobian of the transformation\n$d\\null^*v\\,d\\null^*w$ into $dv\\,dw$ for a constant restitution\ncoefficient is equal to $e^{-1}$, one obtains\n \\begin{align}\\label{weak3}\n(\\varphi\\, , \\,D_e(f,f)) \\,\\, &= \\frac {\\beta^2}{8\\pi}\n\\int_{\\Realr^3}\\int_{\\Realr^3}\\int_{S^2}\\sum_{i,j}\\frac{\\partial^2\n\\varphi(v^*)}{\\partial v^*_i\\partial v^*_j}\\Delta_i\\Delta_j\nf(v)f(w)dv\\, dw\\, d\\sigma\\nonumber \\\\\n&= \\frac {\\beta^2}{8\\pi}\n\\int_{\\Realr^3}\\int_{\\Realr^3}\\int_{S^2}\\frac\n1e\\sum_{i,j}\\frac{\\partial^2 \\varphi(v)}{\\partial v_i\\partial v_j}\n\\null^*\\Delta_i\\null^*\\Delta_j f(\\null^*v)f(\\null^*w)dv\\, dw\\,\nd\\sigma\\nonumber \\\\\n&= \\frac {\\beta^2}{8\\pi} \\int_{\\Realr^3}\\,\\left[\n\\sum_{i,j}\\frac{\\partial^2 \\varphi(v)}{\\partial v_i\\partial v_j}\n\\int_{\\Realr^3}\\int_{S^2}\\frac 1e \\null^*\\Delta_i\\null^*\\Delta_j\nf(\\null^*v)f(\\null^*w)\\, dw\\, d\\sigma\\right]\\, dv\n\\nonumber \\\\\n&= \\int_{\\Realr^3} \\!\\!\\varphi(v) \\!\\left[\\frac\n{\\beta^2}{8\\pi}\\sum_{i,j}\\frac{\\partial^2 }{\\partial v_i\\partial\nv_j}\\int_{\\Realr^3}\\int_{S^2}\\frac 1e\n\\null^*\\Delta_i\\null^*\\Delta_j f(\\null^*v)f(\\null^*w)\\, dw\\,\nd\\sigma\\right] dv.\n \\end{align}\nThis shows that, at least for small inelasticity, the random part of the\ncollision corresponds to a correction given by the nonlinear diffusion operator\n$D_e(f,f)(v)$, where\n \\begin{equation}\\label{corr}\n D_e(f,f)(v) =\n\\frac {\\beta^2}{8\\pi}\\sum_{i,j}\\frac{\\partial^2 }{\\partial\nv_i\\partial v_j}\\int_{\\Realr^3}\\int_{S^2}\\frac 1e\n\\null^*\\Delta_i\\null^*\\Delta_j f(\\null^*v)f(\\null^*w)\\, dw\\,\nd\\sigma.\n \\end{equation}\nDifferent expressions of the operator \\fer{corr} can be recovered\nowing to the definition of $\\Delta$. For the purposes of the\npresent paper, however, we simply remark that, choosing the test\nfunction $\\varphi(v) = |v|^2$, direct computations show that the\ncorrection $D_e(f,f)$ is such that\n \\begin{align}\\label{temp}\n(|v|^2\\, , \\,D_e(f,f)) \\,\\, &= \\frac {\\beta^2}{64\\pi}\n\\int_{\\Realr^3}\\int_{\\Realr^3}\\int_{S^2}\\left||u|\\sigma-u\\right|^2\nf(v)f(w)dv\\, dw\\, d\\sigma\\nonumber \\\\\n&= \\frac {1}{4}\\beta^2\\left[ \\int_{\\Realr^3}|v|^2f(v)\\, dv -\n\\left|\\int_{\\Realr^3}v\\,f(v)\\, dv\\right|^2\\right].\n\\end{align}\nThis reveals the fundamental fact that the diffusion operator produces a\ngrowth of the second moment proportional to the second moment itself. This\naction is clearly different from the action of a linear diffusion operator (a\nthermal bath), which induces a growth of the second moment proportional to the\nmass. This supports the fact that the Boltzmann equation \\fer{BE2} can\nproduce fat tails.\n\n\n\n\n\n\\subsection{Simple economy market modelling}\n\\label{eco1}\n\n\nIn one dimension of the \"velocity\" variable, a similar construction\nleads to kinetic models for wealth redistribution \\cite{CPT,\nMaTo06}. In this case, the variable\n $v \\in \\Realr_+$ represents\nthe wealth of the agents, binary collisions are trades between\nagents, and the (eventual) power-like tails of the steady\ndistribution of wealth are known in the pertinent literature as\nPareto tails. Due to the fact that the variable is in $\\Realr_+$,\nthe possible conserved quantities reduce to mass and momentum. In\n\\cite{CPT} the collision mechanism is given by\n\\begin{equation}\n\\label{mixing}\nv' = (1-\\lambda)v +\\lambda w + \\eta v; \\qquad\n w' = \\lambda v + (1-\\lambda) w + \\eta^* w\n\\end{equation}\nwhere $0\\leq \\lambda\\leq 1$ represents the constant saving rate\nand $\\eta$ and $\\eta^*$ are random variables with law given by a measure\n$\\mu(s)$ of zero mean, variance $\\beta^2$ and support in\n$[-\\lambda,+\\infty)$. In this way, for all realizations of the\nrandom variable we have $\\eta\\geq -\\lambda$ and wealths after\ntrading are well defined\ni.e.,\nremain nonnegative. This is the so-called {\\em no debt} condition.\nIn this\ncontext, the Boltzmann equation \\fer{BE2} is replaced by\n \\begin{equation}\n(\\varphi,\\tQ_\\lambda(f,f))= \\left\\langle \\int_{\\Realr_+}\n\\int_{\\Realr_+} f(v) f(w) \\Big[ \\varphi (v')- \\varphi (v) \\Big]\ndv\\,dw\\ \\right\\rangle. \\label{Qweak2}\n \\end{equation}\n Here, we use the notation\n$$\n\\left< h\\right> :=\\int_{-\\lambda}^\\infty h(s)d\\mu(s).\n$$\nThe unique possible collision invariants of the one-dimensional Boltzmann\nequation are obtained for $\\varphi(v) =1$ and $\\varphi(v) =v$.\n\n\nThe weak formulation of the Boltzmann equation can also be rewritten\n\\begin{equation}\n\\label{Max1d}\n\\int_{\\rr^+} \\varphi(v)\\,\\tQ_\\lambda(f,f)\\, dv \\!=\\! \\frac12\n\\int_{\\rr^+} \\!\\int_{\\rr^+}\\!\\! f(v) f(w) \\left< \\varphi (v') +\n\\varphi (w')- \\varphi (v)- \\varphi (w) \\right> \\,dv\\, dw .\n \\end{equation}\nIn \\eqref{Max1d} the wealth variables $v,w$ are nonnegative quantities, and the\ncollision mechanism is given by \\fer{mixing}.\nA one-dimensional Boltzmann type equation of the form\n\\begin{equation}\\label{ibe1d}\n \\frac{\\p f}{\\p t} = \\tQ_{\\lambda}(f,f)\n\\end{equation}\nbased on the binary interaction given in \\eqref{mixing} has been\nconsidered in \\cite{CPT,MaTo06} and we refer to them for a deeper\ndiscussion of the model. Without loss of generality, we can fix\nthe initial density $f_0(v) \\in {\\cal P}_2(\\rr)$, with the\nnormalization condition\n\\begin{equation}\\label{norm1}\n m(t):=\\int_{\\rr^+} v f(v,t)\\, dv = \\bar{m},\n\\end{equation}\nsince by choosing $\\varphi(v) = v$, \\eqref{Max1d} shows that\n$m(t)=m(0)$ for all $t\\geq0$.\\\\\n\nAs in section \\ref{formal}, one splits the collision mechanisms into\na deterministic inelastic part and the random part :\n$$v' = v^* + \\eta v; \\qquad\nw' = w^* + \\eta^* w\n$\nwhere $v^*, w^*$ are deterministic wealth (corresponding to inelastic\ncollision with constant restitution coefficient $(1-\\lambda)$)\n$$\nv^* = (1-\\lambda)v +\\lambda w ; \\qquad\nw^* = \\lambda v + (1-\\lambda) w .\n$$\n\n\nA formal Taylor expansion similar to \\fer{formal}, in the limit for\n$\\lambda$ and $\\eta$ small, leads to a drift term for the\ndifference between $(v,w)$ and $(v^*,w^*)$ and a diffusion\nterm proportional to the variance $\\beta^2$.\n$$\n\\varphi(v') = \\varphi(v^*) + \\eta v \\partial_v \\varphi(v^*) +\n\\frac 12 \\eta^2 v^2 \\partial_v^2 \\varphi\"(v^*)+ \\dots\n$$\nTaking the average\n$$\n<\\varphi(v')> = \\varphi(v^*) + \\frac 12 \\beta^2 v^2 \\partial_v^2 \\varphi(v^*)+ \\dots,\n$$\nand on the other hand, the deterministic part gives\n$$\n\\varphi(v^*) = \\varphi(v) + \\lambda (w-v) \\partial_v \\varphi(v) +\n\\frac 12 \\lambda^2 (v-w)^2 \\partial_v^2 \\varphi(v)+ \\dots\n$$\nInserting these expansions into the weak formulation\nof the Boltzmann equation \\fer{Qweak2} and rescaling the time\ngives\n$$\n< (\\varphi,\\tQ_\\lambda(f,f))> = \\int_{\\Realr_+^2}\n f(v) f(w) \\Big[ \\lambda (w-v) \\partial_v \\varphi (v)\n+ \\frac 12 ( \\lambda^2 (v-w)^2 + \\beta^2 v^2 ) \\partial_v^2 \\varphi (v) \\Big]\ndv\\, dw.\n$$\n\nMore precisely, the asymptotics of the\none-dimensional Boltzmann equation for\nwealth distribution \\fer{Qweak2} for\n$\\lambda$ sufficiently small,\nand in the limit $\\frac \\lambda{\\beta^2} \\to \\gamma$,\nhas been studied in \\cite{CPT}. In this so-called\n\"continuous trading limit\",\nit is proved that the solution\nto the Boltzmann equation converges toward the solution to the\nFokker-Planck equation\n \\begin{equation}\\label{FP2b}\n \\frac{\\partial f}{\\partial t} = \\frac \\gamma{2}\\frac{\\partial^2 }\n {\\partial v^2}\\left( v^2 f\\right) + \\frac{\\partial }{\\partial v}\\left((v - \\bar m)\n f\\right),\n \\end{equation}\n which admits a unique stationary state of unit mass, given by the\n $\\Gamma$-distribution\n \\begin{equation}\\label{equi}\nM_\\lambda(v)=\\frac{(\\mu-1)^\\mu}{\\Gamma(\\mu)}\\frac{\\exp\\left(-\\frac{\\mu-1}{v}\\right)}{v^{1+\\mu}}\n \\end{equation}\n where\n $$ \\mu = 1 + \\frac{2}{\\lambda} >1.\n$$\nThis stationary distribution exhibits a Pareto power law tail for large\nvelocities. We remark that in \\fer{FP2b} the growth of the second moment\nfollows the same law as the Boltzmann equation \\fer{BE2}.\n\n\n\n\n\\section{Quick overview of probability metrics}\n\\label{quick}\n\nIn this section, we first briefly recall the main definitions and results\nabout probability metrics and, more precisely, on Wasserstein ($W_2$)\nand Fourier ($d_s$) distances between two probability measures.\n\n\n\\subsection{Wasserstein distances}\n\\label{Wass1}\n\nGiven two probability measures $f,g\\in\\Pn$, the Euclidean\nWasserstein Distance is defined as\n\\begin{equation}\\label{w2def1}\nW_2(f, g) = \\inf_{\\Pi\\in\\Gamma} \\left\\{ \\iint_{\\Rn \\times \\Rn}\n\\vert v -x \\vert^2 \\, d\\Pi(v, x) \\right\\}^{1\/2}\n\\end{equation}\nwhere $\\Pi$ runs over the set of transference plans $\\Gamma$, that\nis, the set of joint probability measures on $\\Rn \\times \\Rn$ with\nmarginals $f$ and $g\\in \\Pn$. From a probabilistic point of view,\nthe Wasserstein distance can be alternatively defined as\n\\begin{equation}\\label{w2def2}\nW_2(f, g) = \\inf_{(V,X)\\in\\tilde\\Gamma} \\left\\{ \\expec\\left[ \\vert\nV-X \\vert^2 \\right] \\right\\}^{1\/2}\n\\end{equation}\nwhere $\\tilde\\Gamma$ is the set of all possible couples of random\nvariables $(V,X)$ with $f$ and $g$ as respective laws. Let us\nremark that $W_2$ is finite for any two probability measures with\nfinite second moments $f,g \\in\\Ptwo$.\n\nThe main properties of the Euclidean Wasserstein distance $W_2$\nare summarized in the following proposition. We refer to\n\\cite{Bolley,Vil03,Vil07} for the proofs and further information\non the connections to optimal mass transport theory.\n\n\\begin{proposition}[$W_2$-properties]\\label{w2properties}\nThe space $(\\Ptwo,W_2)$ is a complete metric space. Moreover, the\nfol\\-lowing properties of the distance $W_2$ hold:\n\\begin{enumerate}\n\\item[i)] {\\bf Optimal transference plan:} The infimum in the\ndefinition of the distance $W_2$ is achieved at a joint\nprobability measure $\\Pi_o$ called an optimal transference plan\nsatisfying:\n$$\nW_2^2(f, g) = \\iint_{\\Rn \\times \\Rn} \\vert v -x \\vert^2 \\,\nd\\Pi_o(v, x).\n$$\n\n\\item[ii)] {\\bf Convergence of measures:} Given $\\{f_n\\}_{n\\ge 1}$\nand $f$ in $\\Ptwo$, the following three assertions are equivalent:\n\\begin{itemize}\n\\item[a)] $W_2(f_n, f)$ tends to $0$ as $n$ goes to infinity.\n\n\\item[b)] $f_n$ tends to $f$ weakly-* as a measure and\n$$\n \\int_{\\Rn} \\vert v \\vert^2 \\, f_n(v) \\, dv \\to\n\\int_{\\Rn} \\vert v \\vert^2 \\, f(v) \\, dv \\, \\mbox{ as } \\,\n\\mbox{n} \\to + \\infty.\n$$\n\\end{itemize}\n\n\\item[iii)] {\\bf Convexity:} Given $f_1$, $f_2$, $g_1$ and $g_2$\nin $\\Ptwo$ and $\\alpha$ in $[0,1]$,\n$$\nW_2^2(\\alpha f_1 + (1-\\alpha) f_2,\\alpha g_1 + (1-\\alpha) g_2)\n\\leq \\alpha W_2^2(f_1,g_1) + (1-\\alpha) W_2^2(f_2,g_2).\n$$\nAs a simple consequence, given $f,g$ and $h$ in $\\Ptwo$,\n$$\nW_2(h * f,h * g) \\leq W_2(f,g)\n$$\nwhere $*$ stands for the convolution in $\\Rn$.\n\n\\item[iv)] {\\bf Additivity with respect to convolution:} Given\n$f_1$, $f_2$, $g_1$ and $g_2$ in ${\\cal P}_2(\\Rn)$ with equal mean values,\n$$\nW_2^2(f_1 * f_2, g_1 * g_2) \\leq W_2^2(f_1,g_1) + W_2^2(f_2,g_2).\n$$\n\n\\end{enumerate}\n\\end{proposition}\n\n\n\n\\subsection{Fourier metrics}\n\\label{Four1}\n\nGiven $f\\in\\Pn$, its Fourier transform or characteristic function\nis defined as\n\\[\n\\hat{f}(k) = \\int_{\\Rn} e^{-iv\\cdot k}\\, df(v).\n\\]\nGiven any $s>0$, the Fourier-based metric $d_s$ is defined as\n\\begin{equation}\\label{d2def}\nd_s(f,g)= \\sup_{k \\in \\Rno} \\frac{|\\hat{f}(k)-\\hat{g}(k)|}{|k|^s}\n\\end{equation}\nwhere $\\Rno=\\Rn-\\{0\\}$, for any pair of probability measures\n$f,g\\in\\Pn$. This metric was introduced in\n\\cite{Gabetta-Toscani-Wennberg} and further used in\n\\cite{CGT,CCG1,Toscani-Villani,GJT}. Only recently, various\napplications to the large-time behavior of the dissipative\nBoltzmann equation \\cite{PT03, BisiCT, BisiCT2} have revealed the\nimportance of this distance.\nWe refer to\n\\cite{CTPE} for a complete survey of this metric and the proofs of\nthe statements below.\n\nThe metric $d_s$ with $s>0$ is well-defined and finite for any two\nprobability measures $f,g\\in {\\cal P}_s(\\Rn)$ with\nequal moments up to $[s]$ if $s\\notin\\N$, or equal moments up to\n$s-1$ if $s\\in \\N$. The main properties of the $d_s$ metrics\nrelevant to the ongoing discussion are summarized in the following\nresult:\n\n\\begin{proposition}\\label{dsproperties}\nThe distances $d_s$ with $s>0$ verify the following properties:\n\\begin{enumerate}\n\n\\item[i)] {\\bf Convexity:} Given $f_1$, $f_2$, $g_1$ and $g_2$ in\n${\\cal P}_s(\\Rn)$ with equal moments up to $[s]$ if\n$s\\notin\\N$, or equal moments up to $s-1$ if $s\\in \\N$ and\n$\\alpha$ in $[0,1]$,\n$$\nd_s(\\alpha f_1 + (1-\\alpha) f_2,\\alpha g_1 + (1-\\alpha) g_2) \\leq\n\\alpha d_s(f_1,g_1) + (1-\\alpha) d_s(f_2,g_2).\n$$\n\n\\item[ii)] {\\bf Superadditivity with respect to convolution:}\nGiven $f_1$, $f_2$, $g_1$ and $g_2$ in ${\\cal P}_s(\\Rn)$\nwith equal moments up to $[s]$ if $s\\notin\\N$, or equal\nmoments up to $s-1$ if $s\\in \\N$,\n$$\nd_s(f_1 * f_2, g_1 * g_2) \\leq d_s(f_1,g_1) + d_s(f_2,g_2).\n$$\n\\end{enumerate}\n\\end{proposition}\n\n\n\n\\section{ Large time behavior for economy model}\n\\label{eco2}\n\n\\subsection{Evolution of Wasserstein distance}\n\nThe Boltzmann equation \\eqref{ibe1d} can be rewritten as\n$$\n\\frac{\\p f}{\\p t}= \\left -f ,\n$$\nwhere we use the shorthand $f_p(v) = (1\/p)f(v\/p)$ with\n$p=\\lambda$ and $q=1-\\lambda$. Here, $f$ is extended by 0 to the\nwhole of $\\rr$ in the convolution. The gain operator is defined as\nthe measure given by\n\\[\n (\\varphi,\\tQ_{\\lambda}^+(f,f))= \\left<\\int_{\\rr^+} \\int_{\\rr^+} f(v) \\, f(w) \\, (\\varphi, \\delta_{(p+\\eta)v+qw}) \\, dv \\,\n dw\\right>\n\\]\nwhere $\\delta_{(p+\\eta)v+qw}$ is the Delta Dirac at the\npost-collisional velocity $v'$ and $(\\cdot,\\cdot)$ is the duality\npair between continuous functions and probability measures. In\nprobabilistic terms, the gain operator is defined as an\nexpectation:\n$$\n \\tQ_{\\lambda}^+(f,f)= =\\expec \\left[ \\delta_{(p+\\eta)V+qW}\\right]\n$$\nwhere $V$ and $W$ are independent random variables with law $f$\nand independent with respect to the random variable $\\eta$. Here\nthe expectation is taken with respect to all random variables.\n\nLet us take two independent pairs of random variables $(V,X)$ and\n$(W,Y)$ such that $V$ and $W$ have law $f_1$ and $X$ and $Y$ have\nlaw $f_2$. From the convexity of $W_2^2$ and the independence of\nthe pairs, it follows that\n$$\nW_2^2(\\tQ_{\\lambda}^+(f_1,f_1),\\tQ_{\\lambda}^+(f_2,f_2))\\leq\n\\expec\\left[\nW_2^2(\\delta_{(p+\\eta)V+qW},\\delta_{(p+\\eta)X+qY})\\right]\n$$\nfor any probability densities $f_1,f_2\\in{\\cal P}_2(\\rr)$. Now, the last term\nis directly computed as the Euclidean distance between the two points $(p+\\eta)V+qW$\nand $(p+\\eta)X+qY$, and thus,\n$$\nW_2^2(\\tQ_{\\lambda}^+(f_1,f_1),\\tQ_{\\lambda}^+(f_2,f_2))\\leq\n\\expec\\left[ |(p+\\eta)(V-X)+q(W-Y)|^2 \\right].\n$$\nUsing independence of the pairs and taking the pairs to be optimal\ncouples for the $W_2(f_1,f_2)$ in the probabilistic definition\n\\eqref{w2def2}, we deduce finally the property\n$$\nW_2^2(\\tQ_{\\lambda}^+(f_1,f_1),\\tQ_{\\lambda}^+(f_2,f_2))\\leq\n\\left[<(p+\\eta)^2>+q^2\\right] \\, W_2^2(f_1,f_2).\n$$\nLet us define, for $s \\ge 1$\n \\begin{equation}\\label{key}\n \\mathfrak{S}(s) := \\langle (p +\\eta)^s\\rangle +q^s -1;\n \\end{equation}\nthen $\\mathfrak{S}(2)=<(p+\\eta)^2>+q^2-1= 2\\lambda(\\lambda\n-1)+\\beta^2$. It is not difficult to see that the convexity\nproperty of $W_2^2$ together with the Duhamel formula for\n\\eqref{ibe1d} and the contractive estimate of the gain operator in\n$W_2$ leads to the result:\n\n\\begin{theorem}\\label{cont1d}\nLet $f_1(t)$ and $f_2(t)$ be two solutions of the one dimensional Boltzmann\nequation \\eqref{ibe1d} corresponding to initial values $f_1^0$ and $f_2^0$ in\n${\\cal P}_2(\\rr^+)$, satisfying conditions \\eqref{norm1}. Then, for all times\n$t \\geq 0$,\n\\begin{equation}\\label{decc}\n W_2(f_1(t), f_2(t)) \\leq \\exp\\left\\{ \\mathfrak{S}(2) t\\right\\} W_2(f_1^0,f_2^0).\n\\end{equation}\nIf $\\beta^2 < 2\\lambda(1-\\lambda)$, then $\\mathfrak{S}(2) <0$, and\nthe Wasserstein metric decays exponentially to zero in time.\n\\end{theorem}\n\n\n\\subsection{Evolution of Fourier metrics}\n\nAnalogous results for the evolution of the $d_s$-metric\n\\fer{d2def} have been obtained recently in \\cite{MaTo06} by a\nsuitable generalization of results in \\cite{PaTo06}. For the\ndetailed computations we refer to \\cite{MaTo06}. The study of the\nevolution of the metric \\fer{d2def}, leading to the understanding\nof the large-time behavior of the solution to the kinetic equation\n\\fer{ibe1d}, requires a fine analysis of the quantity \\fer{key}. As\nshown for the Wasserstein metric in the previous subsection, the\nsign of this quantity is in fact related to the contraction\nproperties of the metric. Moreover, as has been noted in\n\\cite{MaTo06}, the sign of \\fer{key} is also related both to the\nnumber of moments of the solution which remain uniformly bounded\nin time, and to the possibility to conclude the existence and\nuniqueness of a steady state. The results in \\cite{MaTo06} can be\nbriefly summarized into the following\n\n\\begin{theorem}\\label{main}\nTake $s>0$ with $\\mathfrak{S}(s)<\\infty$ and let $f_1(t)$ and\n$f_2(t)$ be two solutions of the one dimensional Boltzmann\nequation \\eqref{ibe1d} corresponding to initial values $f_1^0$\nand $f_2^0$ in ${\\cal P}_r(\\rr^+)$, satisfying conditions\n\\eqref{norm1} with $r = \\max\\{ 1,s\\}$. Then the following bound\nholds:\n\\begin{equation}\\label{conv1}\n d_{s}(f_1(t),f_2(t) ) \\leq\n \\exp\\left\\{\\mathfrak{S}(s) t\\right\\}\\,d_{s}(f_1^0,f_2^0),\n \\end{equation}\n where $ \\mathfrak{S}(s)$ is given by \\fer{key}.\n\\end{theorem}\n\nAlso, the temporal behavior of the moments is almost completely\ndetermined by the function $\\mathfrak{S}(s)$.\n\n\\begin{theorem} \\label{mom1}\nLet $s>1$ and $f_0\\in{\\cal P}_s(\\rr^+)$ with\n$0<\\mathfrak{S}(s)<\\infty$ and let us denote\n$$\nM_s^0:=\\int_{\\rr^+} v^s\\,f_0(v)\\,dv.\n$$\nThen, for the weak solution to the Boltzmann equation, the\nfollowing estimates hold:\n \\begin{enumerate}\n \\item If $\\mathfrak{S}(s)>0$, then, as $t\\to\\infty$,\n \\begin{align*}\n { \\int_{\\Realr_+} v^s f(v,t)\\,dv } \\ge M_s^0 \\,{\\exp\\{\\mathfrak{S}(s)t\\}\n } + o(1).\n \\end{align*}\n \\item If $\\mathfrak{S}(s)<0$, then the $s$th moment is bounded for all times.\n Moreover, as $t\\to\\infty$,\n \\begin{align*}\n { \\int_{\\Realr_+} v^s f(v,t)\\,dv } \\leq M_s^0 \\,{\\exp\\{\\mathfrak{S}(s)t\\}\n } + o(1).\n \\end{align*}\n \\end{enumerate}\nHere, the remainder terms $o(1)$ converge to zero exponentially\nfast.\n\\end{theorem}\n\nAnother important conclusion of the analysis of \\cite{MaTo06} is that the\nessential function $\\mathfrak{S}(s)$ does not only decide whether or not the\nsteady state $f_\\infty$ develops a Pareto tail. In fact, the positive zero of\n$\\mathfrak{S}(s)$ actually determines the value of the Pareto index.\n\nA comparison of the contraction results for the Boltzmann equation\n\\eqref{ibe1d} shows that the contraction properties are heavily linked, through\nthe key function \\fer{key}, to the (eventual) formation of tails. While the\nsituation for equation \\eqref{ibe1d} is reasonably well understood, the\ncorresponding analysis for the Boltzmann equation \\fer{BE2} deserves further\ninvestigation. We will discuss equation \\fer{BE2} in detail in the\nfollowing section.\n\n\n\\section{Large time behavior for stochastic granular media}\n\\label{seccontr}\n\nLet us consider here the modification of the Inelastic Maxwell Model introduced\nin \\cite{Bobylev-Carrillo-Gamba}\n\\begin{align}\\label{him}\n\\frac{\\partial f}{\\partial t} = \\tQ_e(f,f),\n\\end{align}\nwhere the collision operator is defined weakly as\n\\begin{align}\\label{wfqm2}\n(\\varphi,\\tQ_e(f,f)) & = \\frac{1}{4 \\pi} \\left< \\int_{\\rr^3} \\int_{\\rr^3}\n\\int_{S^2} f(v) f(w) \\Big[ \\varphi (v')- \\varphi (v) \\Big] d\\sigma \\,dv\\,dw\n\\right> .\n\\end{align}\nAs discussed in the introduction, the collision mechanism relies\non a random coefficient of restitution,\n\\begin{eqnarray}\nv' &=& \\frac12 (v+w) + \\frac{1-\\et}{4}u+\\frac{1+\\et}{4} |u|\\sigma\n\\nonumber\n\\\\[-2mm]\n\\label{colmech2}\n\\\\[-2mm]\nw' &=& \\frac12 (v+w) - \\frac{1-\\et}{4}u-\\frac{1+\\et}{4} |u|\\sigma. \\nonumber\n\\end{eqnarray}\nAs before, we write $u=v-w$, $\\et=e+\\eta$ and $\\eta$ is a\nreal-valued random variable, with zero mean and variance $\\beta^2$,\ngiven by a measure $\\mu(s)$ with support on $[-e,\\infty)$. Here,\n$<\\cdot>$ means the expectation with respect to $\\eta$, i.e., the\nintegral over $\\rr$ with respect to $\\mu$.\n\nIt is quite straightforward to check that conservation of mass and\nmomentum remains and that\n$$\n\\left<|v'|^2 + |w'|^2- |v|^2 - |w|^2 \\right> =0\n$$\nfor the model that is conservative in the mean in which $\\beta^2=1-e^2$. From\n\\eqref{wfqm2}, we deduce that the temperature evolution is\n$$\\frac{d}{d t} \\int_{\\rr^3} |v|^2 \\,f(t,v)\\,dv = 0,$$ and thus we\ndeduce that $\\theta(t)=\\theta(0)$ for all times $t\\geq0$ and we\nwill fix it to one for convenience.\n\n\n\\subsection{Evolution of Wasserstein distance }\n\nGiven a probability measure $f$ on $\\rr^3$, the gain operator is\nin fact a probability measure $\\tQ_e^+(f, f)$ defined by\n\\[\n (\\varphi,\\tQ_e^+(f,f))= \\left<\\int_{\\rr^3} \\int_{\\rr^3} f(v) \\, f(w) \\, (\\varphi, {\\cal U}_{v, w,\\eta}) \\, dv \\,\n dw\\right>\n\\]\nwhere ${\\cal U}_{v, w,\\eta}$ is the uniform probability\ndistribution on the sphere $S_{v, w}$ with center $c_{v, w} =\n\\frac{1}{2} (v+w)+ \\left[\\frac{1-\\et}{4}+\\eta\\right](v-w)$ and\nradius $r_{v, w} = \\frac{1+\\et}{4} |v-w|$ as in\n\\cite{Bolley-Carrillo}. In probabilistic terms, the gain operator\nis defined as an expectation:\n$$\n\\tQ_e^+(f, f)=\\expec \\left[ {\\cal U}_{V, W,\\eta}\\right]\n$$\nwhere $V$ and $W$ are independent random variables with law $f$\nand independent of the law of $\\eta$. As in\n\\cite{Bolley-Carrillo}, we get the following result:\n\n\\begin{theorem}\\label{contrqw2} Given $f$ and $g$ in\n$\\Ptwot$ with equal mean velocity, then\n\\[\nW_2(\\tQ_e^+(f, f) , \\tQ_e^+(g, g) ) \\leq W_2(f, g).\n\\]\n\\end{theorem}\n\n\\par{\\it Proof.-} \\ignorespaces Let us take two independent pairs of random variables\n$(V,X)$ and $(W,Y)$ such that $V$ and $W$ have law $f$ and $X$ and\n$Y$ have law $g$. Also, let us take two independent random\nvariables $\\eta$ and $\\tilde{\\eta}$ with law $\\mu$. Convexity of\n$W_2^2$ implies\n\\begin{equation}\nW_2^2(\\tQ_e^+(f,f), \\tQ_e^+(g,g)) = W_2^2(\\expec \\left[ {\\cal\nU}_{V, W,\\eta}\\right], \\expec \\left[ {\\cal U}_{X,\nY,\\tilde{\\eta}}\\right])\\leq \\expec \\left[ W_2^2({\\cal U}_{V,\nW,\\eta}, {\\cal U}_{X, Y,\\tilde{\\eta}})\\right]\\label{tech8}\n\\end{equation}\nwhere the expectation is taken with respect to the joint\nprobability density in $\\rr^{14}$ of the six random variables.\nHere, the independence of the pairs of random variables has been\nused.\n\nAs proved in \\cite{Bolley-Carrillo}, the $W_2^2$ distance between\nthe uniform distributions on the sphere with center $O$ and radius\n$r$, ${\\mathcal U}_{O,r}$, and on the sphere with center $O'$ and\nradius $r'$, ${\\mathcal U}_{O',r'}$, in $\\rr^3$ is bounded by\n$\\vert O'-O \\vert^2 + (r' - r)^2$.\n\nWe now estimate the right-hand side of \\eqref{tech8} by using the\nformulas for the center and radii of the spheres given in\n\\eqref{colmech2} to deduce\n\\begin{align*}\nW_2^2(\\tQ_e^+(f,f), \\tQ_e^+(g,g)) \\!\\leq \\!& \\; \\left<\\frac{5 - 2\n\\, \\et + \\et^2}{8}\\right> \\expec \\left[ \\vert V-X \\vert^2 \\right]\n+ \\, \\left<\\frac{(1 + \\et)^2}{8}\\right> \\, \\expec \\left[\n\\vert W-Y \\vert^2 \\right] \\\\\n& + \\, \\left<\\frac{1-\\et^2}{4}\\right> \\, \\expec \\left[ (V-X) \\cdot\n(W-Y)\\right]\n\\end{align*}\nwhere the Cauchy-Schwartz inequality has been used.\n\nFinally, we take both pairs $(V,X)$ and $(W,Y)$ as independent\npairs of variables with each of them being an optimal couple for\n$W_2(f,g)$ in the probabilistic definition \\eqref{w2def2}\nto obtain\n\\begin{align*}\nW_2^2(\\tilde{Q}_e^+(f,f), \\tilde{Q}_e^+(g,g)) \\leq & \\; \\frac{3 +\ne^2+\\beta^2}{4} \\, W_2^2(f,g) + \\, \\frac{1-e^2-\\beta^2}{4} \\,\n\\expec \\left[ (V-X) \\cdot (W-Y)\\right],\n\\end{align*}\nwhere the last term is zero because the random variables are independent\nand have equal means. Since $\\beta^2 =1-e^2$\nin the conservative case, the result is proved.\\endproof\n\n\\\n\nAs a consequence of the previous property of the gain operator, we\ndraw the following conclusion about controlling the distance\nbetween any two solutions of \\eqref{him} in the conservative case.\n\n\\begin{theorem}\\label{contrw2eb} If $f_1$ and $f_2$ are two solutions\nto \\eqref{him} with respective initial data $f_1^0$ and $f_2^0$ in\n$\\Ptwot$ with zero mean velocity, then, for all $t \\geq 0$,\n\\begin{align*}\nW_2^2(f_1(t), f_2(t)) \\leq W_2^2(f_1^0, f_2^0).\n\\end{align*}\n\\end{theorem}\n\n\\par{\\it Proof.-} \\ignorespaces Duhamel's formula for \\eqref{him} reads as\n\\[\nf_i(t) = {\\rm e}^{-t} \\, f_i^0 + \\int_0^{t} {\\rm e}^{-(t-s)} \\,\n\\tilde{Q}_e^+(f_i(s), f_i(s)) \\, ds, \\qquad i=1,2.\n\\]\nAs before, the convexity of the squared Wasserstein distance in\nProposition \\ref{w2properties} and the contraction of the gain\noperator in Theorem \\ref{contrqw2} imply\n\\begin{align*}\nW_2^2(f_1(t), f_2(t)) & \\leq {\\rm e}^{-t} \\, W_2^2(f_1^0, f_2^0) +\n\\int_0^{t} \\!\\! {\\rm e}^{-(t-s)} \\, W_2^2 \\big(\n\\tilde{Q}_e^+(f_1(s), f_1(s)), \\tilde{Q}_e^+(f_2(s), f_2(s)) \\big) \\, ds \\\\\n& \\leq {\\rm e}^{-t} \\, W_2^2(f_1^0, f_2^0) + \\, \\int_0^{t} \\!\\!\n{\\rm e}^{-(t-s)} \\, W_2^2 (f_1(s), f_2(s))\\, ds.\n\\end{align*}\nTherefore, the function $y(t) = {\\rm e}^{t} \\, W_2^2(f_1(t),\nf_2(t))$ satisfies the inequality\n \\[\n y(t) \\leq y(0) + \\int_0^{t} y(s) \\, ds\n \\]\nand thus $y(t) \\leq y(0) \\, {\\rm e}^{t}$ by Gronwall's lemma,\nconcluding the argument.\n\\endproof\n\n\n\\subsection{Evolution of Fourier metrics}\n\nWe start by writing a closed form of the Boltzmann equation in\nFourier variables. In fact, it is not difficult using Bobylev's\nidentity in \\cite{Boby75,Bobylevid,Bobylev,Bobylev-Carrillo-Gamba}\nto get\n$$\n\\widehat{\\tQ_e^+(f,f)} = \\frac{1}{4 \\pi} \\left<\\int_{S^2} \\hat{f}\n(t, k_-) \\hat{f} (t, k_+) \\, d\\sigma\\right>\n$$\nwhere\n$$\nk_-=\\frac{1+\\et}{4} k - \\frac{1+\\et}{4}|k|\\sigma \\quad \\mbox{and}\n\\quad k_+=\\frac{3-\\et}{4}\\, k + \\frac{1+\\et}{4}\\, |k|\\sigma\\,.\n$$\nLet us start by analyzing the evolution of the distance $d_2$ that\nin view of the properties in Propositions \\ref{w2properties} and\n\\ref{dsproperties} should verify the same non-strict contraction\nas the transport distance $W_2$.\n\n\\begin{theorem}\\label{contrqd2} Given $f$ and $g$ in $\\P_2(\\rr^3)$\nwith equal mean velocity,\n\\[\nd_2(\\tQ_e^+(f, f),\\tQ_e^+(g, g) ) \\leq \\frac{3+ e^2+\\beta^2}{4} \\,\nd_2(f,g).\n\\]\n\\end{theorem}\n\n\\par{\\it Proof.-} \\ignorespaces Using the Fourier representation formula above, we deduce\n$$\n\\frac{\\widehat{\\tQ_e^+(f, f)}(k) -\n\\widehat{\\tQ_e^+(g,g)(k)}}{|k|^2} = \\frac{1}{4 \\pi}\\!\\left<\n\\int_{S^2} \\left[ \\frac{\\hat{f}(k_-) \\hat{f}(k_+) - \\hat{g}(k_-)\n\\hat{g}(k_+)}{|k|^2} \\right] d\\sigma\\right>\n$$\nfor all $k\\in\\Rto$. We now estimate the integrand as\n\\begin{align*}\n\\left| \\frac{\\hat{f}(k_-) \\hat{f}(k_+)-\\hat{g}(k_-)\n\\hat{g}(k_+)}{|k|^2} \\right| &\\leq \\sup_{k \\in \\Rto} \\left\\{\n \\frac{|\\hat{f}(k)-\\hat{g}(k)|}{|k|^2} \\right\\} \\left( \\frac{|k_-|^2+\n |k_+|^2}{|k|^2} \\right) \\\\\n & = d_2(f,g) \\left( \\frac{|k_-|^2+\n |k_+|^2}{|k|^2} \\right),\n\\end{align*}\nand thus\n$$\nd_2(\\tQ_e^+(f, f) , \\tQ_e^+(g, g) ) \\leq \\frac{1}{4 \\pi}\\!\n\\left<\\int_{S^2} \\left( \\frac{|k_-|^2+\n |k_+|^2}{|k|^2} \\right) d\\sigma \\right> \\, d_2(f,g).\n$$\n\nWe observe that\n \\[\n \\frac{|k_-|^2+|k_+|^2}{|k|^2}\n \\]\n is a function of\nthe angle between the unit vectors $k\/|k|$ and $\\sigma$ and the\nrandom variable $\\eta$, and that\n$$\nI:=\\frac{1}{4 \\pi} \\left<\\int_{S^2} \\frac{|k_-|^2+|k_+|^2}{|k|^2}\n\\,d\\sigma \\right>= \\frac{3+e^2+\\beta^2}{4}.\n$$\nIn fact, we can compute\n\\begin{equation} \\label{modulusk+-}\n\\begin{split}\n&\\displaystyle |k_-|^2= |k|^2 \\left( \\frac{1+\\et}{4} \\right)^2 2\\,\n\\Big( 1- \\cos \\vartheta \\Big)\n\\\\*[.3cm] &\\displaystyle |k_+|^2=\n|k|^2 \\left[ \\left( \\frac{3-\\et}{4} \\right)^2 + \\left(\n\\frac{1+\\et}{4} \\right)^2 + 2 \\left( \\frac{3-\\et}{4} \\right)\n\\left( \\frac{1+\\et}{4} \\right) \\cos \\vartheta \\right]\n\\end{split}\n\\end{equation}\nwhere $\\vartheta$ is the angle between the unit vectors $k\/|k|$\nand $\\sigma$ from which the value of $I$ is obtained. Putting\ntogether previous estimates we get the contraction in $d_2$ with\nthe same constant as $W_2^2$ as desired.\n\\endproof\n\n\\\n\nNow, let us see that we can also control Fourier-based distances\nwith exponent $2+\\alpha$, $\\alpha\\in [0,\\infty)$. Let us set\n\\begin{align}\\label{key2}\n\\mathfrak{A}(\\alpha,e,\\eta) & := \\displaystyle \\frac{1}{2}\\left< \\int_0^\\pi\n\\left\\{ \\left[ \\left( \\frac{1+\\et}{4} \\right)^2 2 (1- \\cos\n\\vartheta ) \\right]^{\\frac{2+\\alpha}{2}}\n\\right.\\right. \\vspace{0.2 cm} \\nonumber\\\\\n& \\,\\,\\,\\,\\,\\,\\left.\\left. +\\, \\left[ \\left( \\frac{3-\\et}{4} \\right)^2 + \\left(\n\\frac{1+\\et}{4} \\right)^2 + 2 \\left( \\frac{3-\\et}{4} \\right) \\left(\n\\frac{1+\\et}{4} \\right) \\cos \\vartheta \\right]^{\\frac{2+\\alpha}{2}} \\right\\}\n\\sin \\vartheta\\, d\\vartheta \\right>\n \\vspace{0.2 cm} \\nonumber\\\\\n& = \\displaystyle \\frac{2}{4+\\alpha} \\left< \\left( \\frac{1+\\et}{2}\n\\right)^{2+\\alpha} + \\frac{1- \\left| \\frac{1-\\et}{2}\n\\right|^{4+\\alpha}}{1-\\left| \\frac{1-\\et}{2} \\right|^2} \\right> .\n\\end{align}\nWhenever there is no confusion, i.e. for $e$ and $\\eta$ fixed, we\nwill denote just by $\\mathfrak{A}(\\alpha)$ the above constant.\n\n\\begin{theorem}\\label{contrqd2alpha} Given $f,g\\in\n\\P_{2+\\alpha}(\\rr^3)$ with equal moments up to order $2+[\\alpha]$, there\nexists an explicit constant $\\mathfrak{A}(\\alpha,e,\\eta)>0$ given by \\fer{key2}\nsuch that\n\\[\nd_{2+\\alpha}(\\tQ_e^+(f, f) , \\tQ_e^+(g, g) ) \\leq \\mathfrak{A}(\\alpha,e,\\eta) \\,\nd_{2+\\alpha}(f,g).\n\\]\n\\end{theorem}\n\n\\par{\\it Proof.-} \\ignorespaces As in the proof of the previous theorem, we compute\n\\begin{align*}\n\\left| \\frac{\\widehat{\\tQ_e^+(f, f)}(k) -\n\\widehat{\\tQ_e^+(g,g)}(k)}{|k|^{2+\\alpha}} \\right| & = \\frac{1}{4\n\\pi} \\left|\\left< \\int_{S^2} \\frac{\\hat{f}(k^+) \\hat{f}(k^-)\n-\\hat{g}(k^+) \\hat{g}(k^-)}{|k|^{2+\\alpha}}\\, d\\sigma \\right>\\right| \\vspace{0.1 cm}\\nonumber\\\\\n & \\leq A \\, \\sup_{k \\in \\Rto} \\frac{|\\hat{f}(k)-\\hat{g}(k)|}{|k|^{2+\\alpha}}\n\\end{align*}\nwhere $A$ is given by\n\\begin{equation}\nA := \\frac{1}{4 \\pi} \\left< \\int_{S^2} \\frac{|k_+|^{2+\\alpha} +\n|k_-|^{2+\\alpha}}{|k|^{2+\\alpha}}\\, d\\sigma \\right>.\\label{Aalpha}\n\\end{equation}\nBy inserting the expressions of $k_-$ and $k_+$ into\n\\eqref{Aalpha} and computing the integral we conclude\n$A=\\mathfrak{A}(\\alpha,e,\\eta)$ and the proof follows.\n\\endproof\n\n\\\n\nAs a consequence, we obtain an estimate on contraction\/expansion\nof the Fourier distances $d_{2+\\alpha}$ between solutions.\n\n\\begin{theorem}\\label{contrdseb} Let $\\alpha>0$ be such that\n$\\mathfrak{A}(\\alpha,e,\\eta) <\\infty$. Let $f_1$ and $f_2$ be two\nsolutions to~\\eqref{him} corresponding to initial values~$f_1^0$,\n$f_2^0$ with equal moments up to $2+[\\alpha]$. Then, for all $t\n\\geq 0$,\n\\begin{equation} \\label{d2alphadecayeb}\nd_{2+\\alpha}(f_1(t),f_2(t)) \\leq d_{2+\\alpha}(f_1^0,f_2^0)\\, {\\rm\ne}^{-C(\\alpha,e,\\eta) t},\n\\end{equation}\nwith $C(\\alpha,e,\\eta)=1-\\mathfrak{A}(\\alpha,e,\\eta)$.\n\\end{theorem}\n\n\\par{\\it Proof.-} \\ignorespaces The Fourier expression of equation \\eqref{him} is given by\n$$\n\\frac{\\p \\hat{f}}{\\p t} = \\frac{1}{4 \\pi} \\int_{S^2} \\hat{f}(k_+)\n\\hat{f}(k_-) d\\sigma - \\hat{f} = \\widehat{\\tilde{Q}_e^+(f,f)} -\n\\hat{f},\n$$\nwhose solution satisfies\n\\begin{equation}\n\\hat{f}(t,k) = {\\rm e}^{-t} \\hat{f}(0,k) + \\int_0^t {\\rm e}^{-\n(t-s)} \\widehat{\\tilde{Q}_e^+(f,f)}(s,k) \\,ds\\,. \\label{finalg}\n\\end{equation}\n\nTaking the expressions of the two solutions $\\hat{f}_1(t)$ and\n$\\hat{f}_2(t)$ in \\eqref{finalg}, subtracting them and dividing by\n$|k|^{2+\\alpha}$ with $k\\in\\Rto$, we get\n$$\n{\\rm e}^{t} \\frac{(\\hat{f}_1-\\hat{f}_2) (t, k)}{|k|^{2+\\alpha}} =\n\\, \\frac{\\hat{f}_1(0,k)-\\hat{f}_2(0,k)}{|k|^{2+\\alpha}}\\, +\n\\int_0^t \\!{\\rm e}^{s} \\frac{\\Big(\n\\widehat{\\tilde{Q}_e^+(f_1,f_1)}- \\widehat{\\tilde{Q}_e^+(f_2,f_2)}\n\\Big) (s,k)}{|k|^{2+\\alpha}}\\, ds.\n$$\nUsing Theorem \\ref{contrqd2alpha} and taking the supremum in\n$k\\in\\Rto$, we obtain\n\\begin{eqnarray*}\n{\\rm e}^{t} d_{2+\\alpha}(\\hat{f}_1,\\hat{f}_2)(t) \\leq d_{2+\\alpha}\n\\big(\\hat{f}_1(0),\\hat{f}_2(0) \\big) + \\mathfrak{A}(\\alpha,e,\\eta) \\int_0^t\n\\!\\!{\\rm e}^{s} d_{2+\\alpha}(\\hat{f}_1,\\hat{f}_2)(s)ds.\n\\end{eqnarray*}\nLet us set $w(\\tau)={\\rm e}^{t} d_{2+\\alpha}\n(\\hat{f}_1,\\hat{f}_2)(t)$. Then\n$$\nw(t) \\leq w(0) + \\mathfrak{A}(\\alpha,e,\\eta)\\int_0^t w(s)\\, ds ,\n$$\nwhich by Gronwall's inequality implies $w(t) \\leq w(0)\\, {\\rm\ne}^{\\mathfrak{A}(\\alpha,e,\\eta) t}$, concluding the proof.\n\\endproof\n\n\\\n\nThe function $\\mathfrak{A}(\\alpha):[0,\\infty)\\longrightarrow\\rr^+$ is convex\nby direct inspection. Taking into account that $\\mathfrak{A}(0) =\n1$, there are only three possible scenarios for the qualitative\nbehavior of $\\mathfrak{A}$. These are characterized by the sign of\n$\\mathfrak{A}'(0)$. In case $\\mathfrak{A}'(0) \\geq 0$, the\nfunction $\\mathfrak{A}(\\alpha)$ has a minimum at $\\alpha=0$ due\nto convexity, and thus $\\mathfrak{A}(\\alpha) > 1$ for all\n$\\alpha>0$. In this case, there does not exist any $\\bar \\alpha \\in\n\\Realr_+$ such that $\\mathfrak{A}(\\bar \\alpha)<1$ and there are no\ncontraction, only expansion, estimates of $d_s$ for $s> 2$.\n\nSuppose that $\\mathfrak{A}'(0) < 0$. In this case, the\ncontraction properties of $d_s$ depend on whether\n \\[\n \\lim_{\\alpha \\to \\infty} \\mathfrak{A}(\\alpha) < 1\n \\]\n or\n \\[\n \\lim_{\\alpha \\to \\infty} \\mathfrak{A}(\\alpha) > 1.\n \\]\nIn the former case, $\\mathfrak{A}(\\alpha)< 1$ for $\\alpha >0 $.\nTheorem \\ref{contrqd2alpha} then implies that the $d_s$-metric is\ncontractive for all values of the parameter $s>2$. In the latter,\nsince $\\mathfrak{A}(0) = 0$,\nthe convex function\n$\\mathfrak{A}(\\alpha)$ has a minimum attained at some point\n$\\tilde \\alpha\n> 0$, and at the same time there exists $\\bar \\alpha > \\tilde \\alpha$ for which\n$\\mathfrak{A}(\\bar \\alpha)= 1$. Thus, $\\mathfrak{A}(\\alpha)< 1$ in\nthe interval $0 < \\alpha < \\bar \\alpha $, and at the same time\n$\\mathfrak{A}(\\alpha)> 1$ for $\\alpha >\\bar \\alpha$. In this case\nTheorem \\ref{contrqd2alpha} implies that the Boltzmann equation is\ncontractive up to but not including order $\\bar \\alpha$.\n\n\\begin{remark}\nIn order to clarify the behavior of $\\mathfrak{A}(\\alpha,e,\\eta)$, we can\nfix the random variable $\\eta$ to assume only two values, while\nrespecting conditions \\fer{re}. This can be done by assuming that\n$\\eta$ only takes the value ${\\sqrt{1-e^2}}\/\\varrho$ with\nprobability ${\\varrho^2}\/(1+\\varrho^2)$ and the value\n${\\sqrt{1-e^2}}\\varrho$ with probability ${1}\/(1+\\varrho^2)$. By\nvarying the parameters $\\varrho$ and $e$ one encounters the whole\nvariety of possible behaviors of the function $\\mathfrak{A}(\\alpha,e,\\eta)$.\nSince\n \\[\n\\mathfrak{A}(\\alpha,e,\\eta) = \\displaystyle \\frac{2}{4+\\alpha} \\left< \\left(\n\\frac{1+\\et}{2} \\right)^{2+\\alpha} + \\frac{1- \\left| \\frac{1-\\et}{2}\n\\right|^{4+\\alpha}}{1-\\left| \\frac{1-\\et}{2} \\right|^2} \\right>,\n \\]\n$\\mathfrak{A}(\\alpha,e,\\eta)$ results in the sum of four contributions, one\nof which is\n \\[\nC(\\alpha,e,\\eta) = \\displaystyle\n\\frac{1}{1+\\varrho^2}\\frac{2}{4+\\alpha}\\left( \\frac{1+e +\n\\sqrt{1-e^2}{\\varrho}}{2} \\right)^{2+\\alpha}.\n \\]\nFor any fixed values of $\\bar\\alpha >0$ and $e$, since the\nnumerator grows like $\\varrho^{2+\\alpha}$, we can choose\n$\\varrho>>1$ in such a way that $C(\\alpha,e,\\eta) >1$, and Theorem\n{\\rm\\ref{contrqd2alpha}} implies that the Boltzmann equation is\ncontractive up to but not including order $\\bar \\alpha$.\n\nOn the other hand, choosing for example $\\alpha= 2$ to simplify computations, one\nobtains easily\n \\begin{equation}\\label{A4}\n\\mathfrak{A}(2,e,\\eta) = \\displaystyle \\frac{1}{3} \\left< \\left( \\frac{1+\\et}{2}\n\\right)^{4} + 1 + \\left( \\frac{1-\\et}{2} \\right)^{2} + \\left( \\frac{1-\\et}{2}\n\\right)^{4}\\right> =\n \\frac{23 - e + \\langle \\et^4 \\rangle}{24}.\n \\end{equation}\nChoosing now $1-e <<1$, and $\\varrho = \\sqrt{1- e^2}\/e$, one\nobtains that $\\et$ assumes the value $0$ with probability $1-\ne^2$ and the value $1\/e$ with probability $e^2$. Therefore\n$\\langle \\et^4 \\rangle = 1\/e^2$, which implies $\\mathfrak{A}(2,e,\\eta) < 1$\nas long as $1\/e^2 -e < 2$. In this second case Theorem\n{\\rm\\ref{contrqd2alpha}} implies that the Boltzmann equation is\ncontractive at least up to order $4$.\n\\end{remark}\n\n\\subsection{Existence and uniqueness of regular isotropic steady states}\n\nExistence and uniqueness of steady states, as well as the size of their\noverpopulated tails, can be derived in full generality (that is, without imposing\nrestrictive conditions on the random coefficient of restitution) by adapting\nto the present situation the methodology of \\cite{BisiCT2}, which refers to the\ninelastic Boltzmann equation for Maxwell molecules. This methodology, in fact,\nis based only on the contractivity properties of the $d_s$-metric, which are\nanalogous to Theorems \\ref{contrqd2alpha} and \\ref{contrdseb}.\n\nIt has to be remarked that the approach in \\cite{BisiCT2} is not suitable to\nrecover the (eventual) regularity of the steady profile. A regularity result\nfor the steady state of the inelastic Boltzmann equation for Maxwell molecules\nhas been obtained in a recent paper by Bobylev and Cercignani\n\\cite{Bobylev-Cercignani2}. In this paper they were concerned with properties\nof the self-similar profiles of the Boltzmann equation for both elastic and\ninelastic collisions, and, in addition to the existence, they obtained results\non the regularity of the steady profiles by showing that the Fourier transform\nof the steady profile satisfies a suitable upper bound. Their method takes\nadvantage of the existence of a super-solution to the rescaled equation in\nFourier variables (BKW-mode). In our collisional setting, the situation is more\ninvolved, and it requires a precise analysis.\n\nIn Fourier variables, the steady state of \\fer{BE2} is a solution of the\nintegral equation\n \\begin{equation}\\label{sta1}\n\\frac{1}{4 \\pi} \\left<\\int_{S^2} \\hat{f} (k_-) \\hat{f} (k_+) \\, d\\sigma\\right>\n= \\hat{f}(k),\n \\end{equation}\nwhere $k_+$ and $k_-$ are given by the relations\n$$\nk_-=\\frac{1+\\et}{4} k - \\frac{1+\\et}{4}|k|\\sigma \\quad \\mbox{and} \\quad\nk_+=\\frac{3-\\et}{4}\\, k + \\frac{1+\\et}{4}\\, |k|\\sigma\\,.\n$$\nSince isotropy is not destroyed by the collision operator, by choosing\nisotropic initial values, one concludes with the isotropy of the (eventual)\nsteady state. Taking this property into account, the following result can be\nobtained as a consequence of Theorem \\ref{contrdseb} (see \\cite{BisiCT2} for\ndetails).\n\n\\begin{corollary}\\label{exissteadyebds}\nEquation~\\eqref{him} has a unique isotropic steady state $f_\\infty$ in the set\nof isotropic probability measures with unit mass, zero mean velocity and unit\ntemperature. Moreover, given any solution $f$ to \\eqref{him} for the initial\ndata $f_0\\in \\P_2(\\rr^3)$ with zero mean velocity and unit pressure tensor,\n$$\nd_{2+\\alpha}(f(t),f_\\infty) \\leq d_{2+\\alpha}(f_0,f_\\infty)\\, {\\rm\ne}^{-C(\\alpha,e,\\eta) t}\n$$\nfor all $t \\geq 0$, $0<\\alpha<1$. Thus, if $\\mathfrak{A}(\\alpha,e,\\eta) < 1$, $f(t)$\nconverges to the stationary state as $t\\to\\infty$ in the $d_{2+\\alpha}$ sense.\n\\end{corollary}\n\n\\begin{remark}\nThe previous result shows that the stationary states attract all solutions with\ninitial data having zero mean velocity and unit pressure tensor. The assumption\nof having unit pressure tensor can be weakened to having initial unit temperature\nby proceeding similarly to the homogeneous cooling state analysis in\n{\\rm\\cite{Bobylev-Cercignani-Toscani,BisiCT2}}.\n\\end{remark}\n\nLet us define\n\\begin{equation} \\label{+-}\n\\begin{split}\n&\\displaystyle a^2(e,\\eta,\\theta)= \\frac{|k_-|^2}{|k|^2}= \\left(\n\\frac{1+\\et}{4} \\right)^2 2\\, \\Big( 1- \\cos \\vartheta \\Big)\n\\\\*[.3cm] &\\displaystyle b^2(e,\\eta,\\theta)= \\frac{|k_+|^2}{|k|^2}=\n \\left[ \\left( \\frac{3-\\et}{4} \\right)^2 + \\left( \\frac{1+\\et}{4}\n\\right)^2 + 2 \\left( \\frac{3-\\et}{4} \\right) \\left( \\frac{1+\\et}{4} \\right)\n\\cos \\vartheta \\right]\n\\end{split}\n\\end{equation}\n Recalling the definition of $k_+$ and $k_-$ given in\n\\fer{modulusk+-}, it is immediate to show that\n \\begin{equation}\\label{cc1}\n a+b \\ge 1; \\qquad \\frac{1}{2} \\left<\\int_{0}^\\pi (a^2 +b^2) \\,\n \\sin \\theta d\\theta\\right> =1\n \\end{equation}\nThe first property in \\fer{cc1} is a direct consequence of the equality $k_+\n+k_- = k$, while the second is the equality $\\mathfrak{A}(0) =1$ in\n\\fer{Aalpha}.\n Let us set $x= |k|$. Then, for any function $\\psi(x)$, the\n left-hand side of \\fer{sta1} can be rewritten\n in the form\n \\begin{equation}\\label{sta2}\n R[\\psi(x)] = \\frac{1}{2} \\left<\\int_{0}^\\pi \\psi(ax) \\psi(bx) \\,\n \\sin \\theta d\\theta\\right>.\n \\end{equation}\nUnder the conditions of Corollary \\ref{exissteadyebds}, the Boltzmann equation\nhas a unique steady state $\\ff_\\infty(x)$, of unit mass, zero mean velocity and\nunit second moment.\n\nLet us remark that $0 \\le R[\\psi] \\le 1$ if $0 \\le \\psi \\le 1$, and $R[\\psi]\n\\le R[\\phi]$ if $0 \\le \\psi \\le \\phi$. Hence the iteration is monotone\nincreasing and converges point-wise if we choose the initial approximation $0\n\\le \\phi_0 \\le 1$ in such a way that $\\phi_0 \\le R[\\phi_0]$.\n As observed in \\cite{Bobylev-Cercignani2}, $\\phi_0(x) = \\exp\\{-x^2\/2\\}$\nallows us to obtain a monotone increasing sequence. In fact, since the\n function $e^{-r}$, $r \\ge 0$ is convex, by Jensen's inequality we obtain\n \\begin{equation}\\label{in5}\n \\left\\langle {\\rm e}^{-\\frac 12(a^2+b^2)x^2}\\right\\rangle \\ge e^{ -\\left\\langle\n\\frac 12(a^2+b^2)x^2 \\right\\rangle } = e^{-{x^2}\/2}.\n \\end{equation}\nThis implies that the limit $\\ff_\\infty(x) \\ge 0$. The trivial limit\n$\\ff_\\infty(x)=1$ will be excluded if there exists a non-zero function\n$\\phi_0(x)$ such that\n \\begin{equation}\\label{in4}\n \\phi_0(x) \\le \\psi_0(x),\n \\end{equation}\nand at the same time $\\psi_0(x)$ generates a monotone decreasing sequence.\n\nInspired by the ideas of Desvillettes et al in \\cite{DFT}, given a fixed\npositive constant $\\R$, we introduce the fixed point operator\n\\begin{align*}\n {\\mathbf R}[\\psi](x) &:= \\left\\{ \\begin{array}{cl}\n \\ff_\\infty(x) & \\mbox{if $x< \\R$} \\\\\n R[\\psi(x)] & \\mbox{if $x \\geq \\R$}\n \\end{array} \\right.\n\\end{align*}\non bounded complex functions $\\psi:{\\mathbb R}\\to{\\mathbb C}$. Notice that ${\\mathbf R}$ is\nclosely related to the Fourier transform of the collision kernel.\n\n\n\n\n\n\\begin{lemma}\\label{bound}\nLet $0 \\le \\ff_\\infty(x)\\le 1$ be the steady state of the Boltzmann equation,\nand let us define\n\\begin{align*}\n \\psi_0(x) &:= \\left\\{ \\begin{array}{cl}\n \\ff_\\infty(x) & \\mbox{if $ x < \\R$} \\\\\n \\exp(-\\mu x)& \\mbox{if $ x \\geq \\R$.}\n \\end{array} \\right.\n\\end{align*}\nThen, if the random variables $a(e, \\eta, \\theta)$ and $b(e, \\eta, \\theta)$ are\nsuch that\n \\begin{equation} \\label{condd}\n P(a < \\delta) + P(b < \\delta) \\to 0 \\qquad {\\rm as} \\,\\,\\, \\delta \\to 0 ,\n \\end{equation}\n there exist positive constants $\\R$ and $\\mu$ such that\n \\[\n{\\mathbf R}[\\psi_0](x) \\le \\psi_0(x).\n \\]\n \\end{lemma}\n\n \\par{\\it Proof.-} \\ignorespaces\nSince the steady state is an isotropic probability density function of unit\nmass, zero mean velocity and unit second moment, there exist positive constants\n$M < 1\/2$ and $\\R$ such that (cfr. \\cite{DFT})\n \\begin{equation} \\label{max}\n 0 \\le \\ff_\\infty(x) \\le e^{-Mx^2} \\qquad {\\rm if}\\,\\,\\, x \\le \\R.\n \\end{equation}\nHence, we can fix $\\R$ and $M$ to obtain\n \\begin{equation}\\label{bbb}\n \\psi_0(x) \\le e^{-Mx^2} \\qquad {\\rm if}\\,\\,\\, x \\le \\R.\n \\end{equation}\nClearly, thanks to the definition of $\\psi_0$, if $x \\le \\R$, there is nothing\nto prove. Therefore, let us consider the possible cases corresponding to $x\n>\\R$. Since $a+b \\ge 1$, if both $ax \\ge \\R$, $bx \\ge \\R$,\n\\[\n \\langle \\psi_0(ax)\\psi_0(bx)e^{\\mu x} \\rangle \\le 1.\n \\]\nIf now both $ax <\\R$ and $bx <\\R$, using bound \\fer{bbb}, we obtain\n \\[\n \\langle \\psi_0(ax)\\psi_0(bx)e^{\\mu x} \\rangle \\le \\langle e^{g(x)} \\rangle,\n \\]\n where\n \\[\n g(x) = \\mu x - M(a^2 + b^2)x^2.\n \\]\n Since $ a +b \\ge 1$, it follows that $a^2 + b^2 \\ge 1\/2$. Thus\n \\begin{equation} \\label{b2}\n g(x) \\le \\mu x - \\frac 12 (a^2 + b^2)x^2 \\le 0 \\qquad {\\rm if} \\,\\,\\, \\mu \\le \\frac\n M{2\\R}.\n \\end{equation}\n Consider now the case in which $ax \\le \\R$, while $bx >\\R$. In this case\n\\[\n \\langle \\psi_0(ax)\\psi_0(bx)e^{\\mu x} \\rangle \\le \\langle e^{h(x)} \\rangle,\n \\]\n where\n \\[\n h(x) = \\mu(1-a) x - M b^2 x^2.\n \\]\nSince $ a +b \\ge 1$, it follows that $b \\ge 1-a$, and\n \\begin{equation}\\label{b3}\n h(x) \\le z(bx) = \\mu bx - M(bx)^2 \\le \\frac{\\mu^2}{4M^2}.\n \\end{equation}\nIn fact, the function $z(r)$ has a maximum at $\\bar r = \\mu\/(2M)$. Moreover,\nsince $z(r)$ decreases for $r >\\bar r$, if $ r \\ge 3 \\bar r$,\n \\begin{equation}\\label{b4}\n z(r) \\le z(3\\bar r) = -3 \\frac{\\mu^2}{4M^2} .\n \\end{equation}\nLet us split the calculation of the mean value into the sets $A = \\{ bx \\ge 3\n\\bar r\\}$ and $A^c= \\{ bx < 3 \\bar r\\}$. Thanks to conditions \\fer{b3} and\n\\fer{b4} one obtains\n \\begin{equation}\\label{mm}\n\\langle e^{h(x)}\\rangle \\le P(A) \\exp\\left\\{-3 \\frac{\\mu^2}{4M^2}\\right\\} +\nP(A^c)\\exp\\left\\{ \\frac{\\mu^2}{4M^2}\\right\\}.\n \\end{equation}\nLet us set $\\delta = 3\\bar r = 3\\mu\/(2M)$. By hypothesis, since $x >\\rho$,\n \\[\n P(A^c) = P(bx < \\delta) \\le P(b\\rho < \\delta) \\to 0 \\qquad {\\rm if} \\,\\,\\, \\delta \\to 0.\n \\]\n Consider that we can rewrite \\fer{mm} as\n \\[\n\\langle e^{h(x)}\\rangle \\le (1- P(A^c) \\exp\\left\\{- \\frac 13 M\\delta^2 \\right\\}\n+ P(A^c)\\exp\\left\\{ \\frac 13 M\\delta^2\\right\\} =\n \\]\n \\[\n 1 - \\frac 13 (1- 4P(A^c)) M \\delta^2 + o(\\delta^2) \\le 1\n \\]\n if $\\delta$ is sufficiently small. Now, this condition on $\\delta$ can be\n satisfied by choosing $\\mu$ sufficiently small. This is not in contrast with\ncondition \\fer{b2}. Since the case in which $ax > \\R$ while $bx \\le \\R$ can\nbe treated likewise, the lemma is proven.\n\\endproof\n \\\n\n\\begin{remark}\nCondition \\fer{condd} excludes some pathological situations related to the\ndefinition of the random variable $\\eta$ that describes the randomness of the\ncoefficient of restitution $e$. For example, condition \\fer{condd} is violated\nif $\\eta$ is concentrated on some particular point,\n \\[\n P( \\eta = 1-e) = p >0 .\n \\]\nIn this case, in fact, $P(b(e,\\eta, 0) = 0) = p$, and condition \\fer{condd} is\nfalse.\n\\end{remark}\n\nLemma \\ref{bound} implies that, starting from $\\psi_0$, the iteration process\nleads to a monotone decreasing sequence. On the other hand, it is clear that,\nfor $\\mu$ sufficiently small,\n \\[\n 0 \\le \\phi_0(x) \\le \\psi_0(x) \\le 1.\n \\]\n\n\nGiven $\\mu>0$, define $K_\\mu$ as the set of functions $\\psi$ with $\\psi(0)=1$,\n$\\psi'(0)=\\ff_\\infty'(0)$, and satisfying the estimates\n\\begin{align}\n \\label{eq.gevrey}\n |\\psi(x)| \\leq \\exp(-\\kappa x^2) \\quad \\mbox{for $ x < \\R$},\n \\qquad\n |\\psi(x)| \\leq \\exp(-\\mu x) \\quad \\mbox{for $x \\geq \\R$}.\n\\end{align}\n\nThe previous inequalities prove the following\n\n\\begin{theorem}\nFor any pair of functions $a$ and $b$ satisfying conditions \\fer{cc1}and\n\\fer{condd}, the integral equation \\fer{sta2} has a nontrivial solution\n$\\ff_\\infty(x)$ such that $\\ff_\\infty(x)$ belongs to the Gevrey class $K_\\mu$\ndefined by \\fer{eq.gevrey}.\n\\end{theorem}\n\n\\begin{remark}\nAn analogous regularity result can be proven for the steady state to the\none-dimensional kinetic model \\fer{ibe1d} \\cite{MaTo06}. In this case, it is\nimportant to know that the mean wealth of the stationary state is equal to one.\n\\end{remark}\n\n\\subsection{Fat tails of stationary states}\n\nIn this work, we will only examine the case of the fourth moment,\npostponing the complete analysis of moment evolution to future\nresearch. Here, we will show that under certain conditions on the\nrandom variable, the fourth moment diverges or is controlled\nuniformly.\n\n\\begin{lemma}\\label{prop-mom4} Let the restitution coefficient $e$ and\nthe random variable $\\eta$ be chosen so that $\\mathfrak{A}(2,e,\\eta) < 1$. If\n$f^0$ is a Borel probability measure on $\\rr^3$ such that\n$$\n\\int_{\\rr^3} \\vert v \\vert^4 \\, f^0(v) \\, dv<\\infty,\n$$\nthen the solution $f$ to \\eqref{him} with initial datum $f^0$\nsatisfies\n$$\n\\sup_{\\tau \\geq 0} \\int_{\\rr^3} \\vert v \\vert^4 \\, f(t, v) \\,\ndv<\\infty.\n$$\n\\end{lemma}\n\n\\par{\\it Proof.-} \\ignorespaces Without loss of generality we can assume that $f^0$, and\nhence $f(t)$ for all $ \\tau \\geq 0$, has zero mean velocity and\nunit temperature. We let\n$$\nm_4(t) = \\int_{\\rr^3} \\vert v \\vert^4 \\, f(t, v) \\, dv\n$$\ndenote the fourth order moment of $f(t)$. Then, using the weak\nformulation of the inelastic Boltzmann equation, we have:\n\\begin{equation}\\label{eq-mom4}\n\\frac{d m_4(t)}{d t} = \\int_{\\rr^3} \\vert v \\vert^4 \\,\n\\tilde{Q}_e(f(t), f(t)) (v) \\, dv\n\\end{equation}\nthat can be computed as in \\cite{Bolley-Carrillo} by\n\\begin{align*}\n\\int_{\\rr^3} \\vert v \\vert^4 \\, \\tilde{Q}_e(f, f) (v) \\, dv = &-\n<\\zeta> \\, \\int_{\\rr^3} \\vert v \\vert^4 \\, f(v) \\, dv + <\\mu_1>\n\\Big( \\int_{\\rr^3} \\vert v \\vert^2 \\, f(v) \\, dv \\Big)^2\n\\\\ &+ <\\mu_2> \\iint_{\\rr^3 \\times \\rr^3} (v \\cdot w)^2 \\, f(v) \\,\nf(w) \\, dv \\, dw\n\\end{align*}\nwhere\n$$\n\\mu_1 = \\frac{1}{8} (\\nu_1 + \\nu_2 - \\nu_3) \\quad {\\textrm{and}}\n\\quad \\mu_2 = \\frac{1}{4} (\\nu_1 - \\nu_2)\n$$\nwith\n$$\n\\nu_1 = (\\epsilon^2 + \\epsilon'^2)^2 -1 + \\frac{4}{3} \\epsilon^2\n\\epsilon'^2, \\quad \\nu_2 = 2 \\big[ \\epsilon^2 + \\epsilon'^2 -1 +\n\\frac{2}{3} \\epsilon'^2 \\big], \\quad \\nu_3 = 4 (\\epsilon^2 - 1),\n$$\nand\n$$\n\\zeta = \\frac{1}{3} ( 1 + 4 \\, \\epsilon - 7 \\, \\epsilon^2 + 4 \\,\n\\epsilon^3 - 2 \\, \\epsilon^4) \\qquad \\mbox{with} \\qquad \\epsilon =\n\\frac{1-\\et}{2} \\qquad \\mbox{and} \\qquad \\epsilon'=1-\\epsilon .\n$$\nNow, \\eqref{eq-mom4} reads\n\\begin{equation}\\label{new}\n\\frac{d m_4(t)}{d t} = -<\\zeta> m_4(t) + m(t)\n\\end{equation}\nwhere $m(t)$ is a combination of second order moments, which are bounded in\ntime since the kinetic energy is preserved by equation \\eqref{him}. Moreover\none can check from the expression of $\\zeta$ in terms of $e$ that $<\\zeta> = 1-\n\\mathfrak{A}(2,e,\\eta)>0$. This ensures that $m_4(t)$ is bounded uniformly in time if\ninitially finite, and concludes the argument.\n\\endproof\n\n\\\n\nThe preceding result also shows the divergence of the fourth moment in case the\nrandom variable $\\eta$ and the restitution coefficient $e$ are chosen to\nsatisfy $\\mathfrak{A}(2,e,\\eta)\n> 1$ but $\\mathfrak{A}(\\alpha,e,\\eta) < 1$ for some $0<\\alpha<2$.\n\n\\begin{corollary}\nLet the restitution coefficient $e$ and the random variable $\\eta$\nbe chosen so that $\\mathfrak{A}(2,e,\\eta)>1$ but $\\mathfrak{A}(\\alpha,e,\\eta)<1$ for\nsome $0<\\alpha<2$. Then, the unique isotropic steady state\n$f_\\infty$ in $\\Ptwot$ of equation~\\eqref{him} with zero mean\nvelocity and unit pressure tensor has unbounded fourth moment.\n\\end{corollary}\n\n\\par{\\it Proof.-} \\ignorespaces With the notation of the previous subsection, the evolution of the\nfourth moment for isotropic densities given in Lemma \\ref{prop-mom4} ensures\nthat\n$$\n\\frac{d m_4(t)}{d t} = -<\\zeta> m_4(t) + m(t),\n$$\nwhere $m(t)$, which is a combination of second order moments, is bounded from\nbelow. Recall that $<\\zeta>=1-\\mathfrak{A}(2,e,\\eta)<0$ to conclude.\n\\endproof\n\n\n\\bigskip\n\n\\noindent {\\bf Acknowledgements:} JAC acknowledges the support from DGI-MEC\n(Spain) FEDER-project MTM2005-08024 and 2005SGR00611. G.T. acknowledges the\nsupport of the Italian MIUR project ``Kinetic and hydrodynamic equations of\ncomplex collisional systems''. JAC and GT acknowledge partial support of the\nAcc. Integ. program HI2006-0111. JAC acknowledges partial support of the Acc.\nInteg. program HF2006-0198.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{sec:introduction}\nQuantum Computers (QCs) are specialized devices that will be able to solve some problems faster than Classic Computers (CCs)~\\cite{bernstein1997quantum, deutsch1985quantum}. This is known as `quantum advantage'. Examples of such problems (originating in various fields of science) are scalable simulations of quantum systems in physics~\\cite{feynman1982simulating}, efficient modelling of chemical reactions~\\cite{aspuru2005simulated}, accurate pricing of financial instruments and credit scores~\\cite{ORUS2019100028}, and fast breaking of encryption codes in cryptography~\\cite{shor1997}.\n\n\\subsection{QC timeline} \\label{sec:timeline}\n\nThe field of quantum computing is young: Feynman introduced the idea of quantum computing in 1982~\\cite{feynman1982simulating}; Shor proposed the first practically relevant algorithm (for breaking encryption protocols based on integer factorization) that can be efficiently computed on a QC in 1994~\\cite{shor1997}.\n\nIt took a while to implement an actual QC. A partnership between academia and IBM created the first working 2-qubit\\footnote{\nAt any point of execution, the state of a CC is given by a vector of bits taking the values of $0$ and $1$. The state of a QC is, however, given by a vector of qubits and bits. Qubit is the basic unit of quantum information, on which QC operate.\nWe will give a more formal definition of a qubit and compare it with a bit in Section~\\ref{sec:quant_comp}.} QC in 1998~\\cite{Chuang1998}, but it took the company 18 years to make a 5-qubit QC accessible to the public in 2016~\\cite{ibm2016}. \n\nAt present, a few QCs are commercially available. D-Wave started selling adiabatic QC in 2011 (although debate about adiabatic QC being a `true' QC is ongoing\\footnote{A hybrid of adiabatic and gate-based QC is promising~\\cite{barends_digitized_2016}, but no commercial implementation is available.}~\\cite{albash2017}) with the current offerings having $>$~5000 qubits~\\cite{DWaveAnn7:online}. \n\nQCs are also available via fully-managed Cloud services. IBM gave access to 20- and 50-qubit gate-based superconducting QCs to academic and industrial partners to explore practical applications in 2017~\\cite{ibm2017} (and the 65-qubit machine was offered in 2020~\\cite{IBMsRoad67:online}). For non-commercial use, IBM offers 5- and 15-qubit QCs via IBM Q Experience online platform based on IBM Cloud (along with local- and Cloud-based simulators)~\\cite{ibm_quantum}. Rigetti offered 8-qubit superconducting QC in 2017~\\cite{RigettiC96:online}. Google built 72-qubit gate-based superconducting QC in 2018~\\cite{google2018}. IonQ introduced ion-trapped 11-qubit QC in 2019~\\cite{Wright2019}. Honeywell created ion-trapped 10-qubit QC in 2020~\\cite{QuantumC77:online}. Xanadu offered 8- and 12-qubit photonic QCs in 2020~\\cite{XanaduRe21:online,CloudPla55:online}. Microsoft provides access to a simulator\\footnote{A QC can be simulated on a CC~\\cite{ibm_quantum,ms_quantum}. A quantum simulator interprets a mathematical function as part of a physical model~\\cite{Johnson2014}; however, it will not yield performance improvement that a QC would provide, as the underlying host system of the simulator is still based on bits rather than qubits (a basic unit of quantum information). Thus, one needs a real QC to reap performance benefits.} of a topological QC via Microsoft Quantum Development Kit~\\cite{ms_quantum} (and is planning to give access to an actual QC in the future).\n\n\nAggregated Cloud services are starting to appear as well. For example, Amazon Web Services started offering access to QC from various vendors via its Braket service in 2019~\\cite{aws_braket_intro}. Currently, it offers D-Wave adiabatic 2048- and 5640-qubit QCs, IonQ trapped-ion-based 11-qubit QC, and Rigetti 32-qubit superconducting QC~\\cite{AmazonBr86:online}. \n\n\n\\subsection{QC performance} When discussing the performance of the abovementioned QCs, we have to be mindful of the fact that the performance of the QCs (which are based on different architectures) cannot be compared merely based on the number of qubits that each QC has. Conceptually, it is similar to the fact that we cannot compare the performance of CC based solely on the number of central processing unit (CPU) cores and the cores' frequency. Standardization of benchmarks for QC is currently in the works by an IEEE Working Group~\\cite{ieee_wg}. \n\n\nOne of the measurements of the QC performance, introduced by IBM, is the Quantum Volume~\\cite{cross2019}, deemed $V_Q$. This metric combines the number of physical qubits, their inter-connectivity, and measurements error rates. For example, while the Honeywell H1 QC has only 10 ion-trapped physical qubits~\\cite{QuantumC77:online}, its $V_Q=128$~\\cite{Achievin40:online}. In comparison, the IBM's 27-qubit QC has $V_Q=64$~\\cite{jurcevic2020demonstration,IBMDeliv10:online}.\n\nRecently, IonQ introduced a QC with 32 ion-trapped perfect qubits with low gate errors, giving it an expected $V_Q > 4 \\times 10^6$~\\cite{IonQPres32:online}. They have also introduces another measure of QC performance, called Algorithmic Qubit, defined as `the largest number of effectively perfect qubits you can deploy for a typical quantum program~\\cite{ScalingI73:online}.'\n\n\n\n\\subsection{QC applicability} \n\\subsubsection{Theoretical perspective} \\label{sec:applicability_theory}\nOnly those problems falling under the bounded error quantum polynomial time ($\\BQP$) class defined in computational complexity theory~\\cite{nielsen_chuang_2010} can benefit from the QC architecture. The time complexity of algorithms, which solve $\\BQP$ class problems, grows polynomially with the increase of the input size on a QC. On the contrary, the time complexity of the algorithms solving the same problems on a CC is not bounded above by a polynomial function and may grow faster (e.g., exponentially) with the increase of the size of the input. \n\nFormally, it was shown that the relations between $\\BQP$ and other popular complexity classes are as follows: $\\P \\subseteq \\BPP \\subseteq \\BQP \\subseteq \\P^{\\#\\P} \\subseteq \\PSPACE$, where $\\P$ is a polynomial time complexity class, $\\BPP$ is a bounded-error probabilistic polynomial time class, $\\P^{\\#\\P}$ is $\\P$ with $\\#\\P$ oracle class ($\\#\\P$ is a set of counting problems and a class of function problems rather than decision problems), and $\\PSPACE$ is a polynomial space class, see~\\cite{vazirani2002survey} for details. \n\nCurrently, the consensus (although not formally proven) is that some of the nondeterministic polynomial time ($\\NP$) problems do belong to the $\\BQP$ set; however, $\\BQP$ and $\\NP$-complete sets of problems do not overlap (see~\\cite{vazirani2002survey,nielsen_chuang_2010} for review). That is, a QC will not be able to solve an $\\NP$-complete problem.\n\n\n\n\\subsubsection{Practical perspective}\\label{sec:practical_perspective}\n\nQuantum advantage was demonstrated on a superconducting QC in 2019~\\cite{arute2019quantum}. Another demonstration of quantum advantage on a photonic QC was done in 2020~\\cite{zhong2020quantum} (although the setup used in the experiment may be difficult to scale up or generalize~\\cite{choi2021}).\n\nBut when will QCs start solving real-world problems? Quantum chemists are already able to improve simulations of small chemical systems~\\cite{kandala2017hardware} and some large ones, albeit with approximations~\\cite{tavernelli2020resource}, using the existing QCs. Quantitative finacists will need a machine with $\\approx$~7.5K logical qubits to price financial instruments~\\cite{chakrabarti2020threshold}. Hackers will need a computer with 20M qubits to break the 2048-bit RSA key in less than a day~\\cite{gidney2019factor}. \n\nAnother promising area for application of QCs is machine learning. Modern QCs are already capable of solving simple machine learning problems~\\cite{johri2020nearest,broughton2020tensorflow}. They will be able to tackle larger problems as computer size increases. However, we have to be mindful that many subroutines required for machine learning (especially related to the linear algebra computations) will achieve polynomial rather than exponential speedup on QC~\\cite{johri2020nearest,DBLP:conf\/stoc\/Tang19}. Thus, quantum machine learning frameworks (such as Tensorflow Quantum~\\cite{TensorFl66:online}) will have to carefully decide\\footnote{The same strategy is currently used by CC machine learning frameworks to distribute the workload between CPUs and graphics processing units (GPUs).} which subroutines should be executed on a QC and which should stay on a CC. \n\n\nTo start addressing practical use-cases within the next decade, IBM stated that they `need to at least double the Quantum Volume of our quantum computing systems every year.~\\cite{chow2020}' So far, IBM is on track, demonstrating the Quantum Volume of 64 on a QC with 27 qubits in 2020~\\cite{IBMsRoad67:online}. By 2023, IBM plans to ship a computer with 1,121 qubits~\\cite{IBMsRoad67:online} (with the expectation of proportional Quantum Volume growth).\n\n\\subsubsection{Effect of QC on software engineering workloads: vision}\\label{sec:vision}\n\nAs discussed in Section~\\ref{sec:practical_perspective}, currently, the programs for QC are targeting problems from the Science, Technology, Engineering and Mathematics (STEM) domain, e.g., factor integers~\\cite{shor1997} or sample boson particles~\\cite{giordani_experimental_2018}. They are not of interest to a mainstream consumer. Thus, QCs, at the current stage of their evolution, conceptually resemble computers from the 1940s and 1950s. For example, the Electronic Numerical Integrator and Computer (ENIAC), completed in 1945, was used for comparable tasks: to compute the highest factor of $2^{18}$ and simulate decay of neutron particles during nuclear fusion~\\cite{haigh2016eniac}. Peculiarly, this parallel is further supported by the fact that languages designed for QC operate at the level of qubits and quantum circuits~\\cite{heim2020quantum}. \n\n\nDoes this mean that history will repeat itself, and a new Software Crisis~\\cite{DBLP:conf\/qce\/MoguelBGM20}, similar to the one that led to the inception of Software Engineering in the 1960s, is upon us? The authors' position is cautiously optimistic: below, we argue that this is not the case and that the history of computing evolves in an upward spiral rather than a circle.\n\nOn the surface, the current situation with programming QC is similar to programming CC in the 1950s: we are dealing with expensive machines which have to be manipulated at the level of registers and gates. A highly-qualified personnel is required for the machine's maintenance. However, the situation is much better than back in the day. We now understand how to deal with large codebases, e.g., by applying Lehman's laws of software evolution~\\cite{lehman1980}. The tools and access to resources have also improved dramatically. Punched cards and fights for machine time have been replaced with powerful integrated development environment (IDEs) and trivial access to computing resources (as a lot of QC development can be done in a simulator). Thus, programming modern QC is a much more pleasant and forgiving experience than that of early generations of CCs.\n\nFor example, we are programming QCs at the gate level, but we are typically doing from higher-level languages. For example, the QisKit library is coded in Python. Thus, we get superior language constructs, such as for-loops, modularity, or classes\\footnote{A reader interested in exploring high-level language constructs may want to examine the source code~\\cite{qiskit:shor:online} of the \\texttt{Shor} class, used in Figure~\\ref{fig:aqua_shor}.}. We can build automatic test cases using test harnesses designed for these high-level languages. We use potent IDEs with integrated source code management, code-completion, and spell-checking. The code can be executed in the simulators and, in some cases, debugged using an interactive debugger (e.g.,~\\cite{ibm_quantum,ms_ide}). Finally, we have amassed decades of knowledge on requirements engineering and design --- a lot of this knowledge can and will be transferable to the QC programming (see~\\cite{DBLP:journals\/corr\/abs-2007-07047} for review of the latest works). We will also show how we can readily transfer some of software engineering (SE) knowledge, especially in the context of System of Systems (SoS), in Sections~\\ref{sec:usage} and \\ref{sec:qa}.\n\nIronically, the ability of future QCs to break modern encryption schemes may trigger a crisis related to the maintenance of the legacy software executed on a CC (rather than a crisis of developing code for QCs). While the breakage of RSA keys discussed in Section~\\ref{sec:practical_perspective} is years away, we need to start protecting ourselves against these future potential attacks right now. This is because malicious entities can harvest sensitive data communications now and~--- when a powerful QC becomes available~--- leverage that computing power to break today's non-quantum-resistant encryption and gain access to such sensitive data. Many QC-resistant encryption methods have been proposed (see~\\cite{zhang2020quantum} for details). However, their implementations will require significant changes to various software, such as web browsers and web servers, mail and hard drive encryptors. We conjecture that the amount of work required to introduce these changes into legacy software may be similar to that of the Y2K problem~\\cite{britannica_y2k}. For details on protection steps, see~\\cite{zhang2020quantum}.\n\n\n\\subsection{Overview of the rest of the paper}\\label{sec:overview}\n\nThe programming languages for QC are mainly low-level, operating at the level of QC register, e.g., Open Quantum Assembly (OpenQASM) language~\\cite{cross2017open}. However, higher-level languages are being developed (e.g., Scaffold~\\cite{JavadiAbhari2014ScaffCC}). \n\nTo enable usage of the QC, libraries with pre-packaged quantum algorithms start to appear. For example, Qiskit Aqua~\\cite{Qiskit} (an open-source library written in Python) implements quantum algorithms for various domains, such as artificial intelligence, chemistry, and finance. Such a library enables a programmer to treat QC as a black-box and leverage quantum algorithms without having a deep understanding of the QC field. We will discuss the usage of quantum components based on such libraries as part of a software solution in Section~\\ref{sec:usage}. We will then cover the implications of quantum components to testing and debugging SoS and standalone QC programs in Section~\\ref{sec:qa}. \n\nThe quantum libraries themselves have to be developed by programmers with expertise in the QC field. These programmers, inevitably, inject defects in their code (uniting CC and QC programming worlds). After that, the code has to be debugged. We will touch on existing debugging tactics and their applicability to quantum programs in Section~\\ref{sec:traditional}. To better understand programmers' challenges, we will review and compare classic and quantum models of computation in Section~\\ref{sec:quant_comp}. Armed with this knowledge, we will then show tricks for analyzing quantum programs during runtime in Section~\\ref{sec:debug_quantum}. We conclude the paper in Section~\\ref{sec:conclusions}.\n\n\\section{Creation and Usage of Quantum Software Components}\\label{sec:usage}\n\nAs discussed in Section~\\ref{sec:applicability_theory}, $\\P \\subseteq \\BQP$. Thus, one may argue that QCs will replace CCs at some point in time. However, we conjecture~\\cite{miranskyy2019testing} that QCs will not replace CCs in the short run. Rather, QCs will be integrated into an SoS, where QC-based components will solve $\\BQP$ problems (that CCs cannot solve), while the solution will be passed to CC components for post-processing. Let us elaborate on this conjecture.\n\nThe reasons for this lie in economics. Modern QCs are expensive: e.g., D-Wave QC is valued at \\$15 million~\\cite{wired2017}. Many require low-temperature cooling and specialized training to use and maintain. The QCs are bulky, taking significant amount of space. The costs, size, and maintenance requirements will probably be reduced over time (as it happened during the transition of mainframes to personal computers). Let us speculate how various QC architectures may evolve in the future.\n\nThe superconducting QCs, such as the ones from IBM, Google, and Rigetti, require cryogenic cooling. This is the fundamental physics requirement and will not change with time. Thus, even though such QCs may become smaller and cheaper, they will require specialized cooling and maintenance personnel. Thus, these machines will have to be hosted in a public or private Cloud and accessed as a service.\n\nThere are at least two QC architectures for Quantum Processing Units (QPU) that may one day be integrated into a CC, similar to a GPU. The first one is ion-trap QC, such as the one from Honeywell and IonQ, which do not require cryogenic cooling (ion trap QC use vacuum and lasers to `slow down' atoms). IonQ plans to have a rack-based QC by 2023 and a desktop unit by 2025~\\cite{IonQplan81:online}.\n\nThe second architecture is photonic, such as the one from Xanadu. The photonic QC chip operates at room temperature~\\cite{arrazola2021quantum}. However, photon detectors in photonic QC do require cooling~\\cite{choi2021}. Potentially, engineers may be able to come up with a sensor that does not require cooling. This will allow photonic QC to be miniaturized and operate at room temperature.\n\nAnother argument for using QC in an SoS comes from the cost of software development. Theoretically, one can port any CC code to a QC code. However, the cost of porting will make it economically infeasible. Modern QC programming languages, such as IBM QisKit Python package~\\cite{ibm_quantum}, Google Cirq framework~\\cite{cirq:developers:2021:4586899}, or Microsoft Q\\# language~\\cite{svore2018}, operate at the level of qubits and quantum circuits. Creation of, e.g., a graphical user interface, in such a language would be very time-consuming and expensive\\footnote{Notwithstanding, these languages integrate nicely into CC domain, simplifying the creation of SoS. As mentioned above, QisKit is implemented as a Python library, running on a CC. Once translated to QC machine language (via OpenQASM), the code is passed to the QC for execution (the complexities of the call are encapsulated in the library's code). Cirq and Q\\# behaviour is similar: the code is developed on a CC and then passed to a QC for execution.}. \nIn the distant future, as the higher-level languages for QC are created, the replacement of CC with QC will become more probable. \n\nFor now, it is easier to keep the existing code on CC and outsource parts that can run efficiently on a quantum machine to a QC. How can this be done? Let us look at an example.\n\n\\begin{exmp}\\label{ex:sos}\nSuppose that we need to create a software-as-a-service for factoring large integers to break the RSA cryptosystem. The time complexity of the best algorithms available for a CC in the family of general number field sieves) is sub-exponential~\\cite{pomerance96atale}. Thus, these algorithms will be ineffective for large integers. Instead, we will build a software component running Shor's algorithm on a QC, which will be more efficient for large integers, because Shor's algorithm computation time (as other $\\BQP$ class algorithms) will grow polynomially with the growth of the input integer $N$ (when executed on a QC). The rest of the components, such as user interface (UI) and application program interface (API) for obtaining input (i.e., the value of $N$) to be passed to the QC component and to return the vector of factors $\\vec{L}$ back to the user will be implemented on a CC, as depicted in Figure~\\ref{fig:arch}. \n\n\nModern libraries that abstract QC computations can already enable this scheme. For example, the QisKit Aqua library (written in Python) already has Shor's algorithm built-in~\\cite{qiskitaq4:online}. Thus, a programmer does not need to know anything about quantum algorithms and the implementation details. Instead, they will simply call the Python class implementing Shor's algorithm. We show a sample implementation of this approach in Figure~\\ref{fig:aqua_shor}.\n\n\\begin{figure*}[tb]\n \\centering\n \\resizebox{0.9\\linewidth}{!}{\n \\begin{tikzpicture}[outer\/.style={draw=gray,dashed,fill=green!1,thick,inner sep=5pt}]\n \\begin{umlseqdiag}\n \\umlactor[scale = 0.5]{user}\n \\umlbasicobject{WebApp's UI}\n \\umlbasicobject{WebApp's Backend}\n \\umlbasicobject{QC's Controller}\n \\umlbasicobject{QC's Core}\n \\begin{umlcall}[op=$N$, type=synchron, return=Return $\\vec{L}$]{user}{WebApp's UI}\n \\begin{umlcall}[op=$N$, type=synchron, return=Return $\\vec{L}$]{WebApp's UI}{WebApp's Backend}\n \\begin{umlcall}[op=QASM, type=synchron, return=Return $\\vec{L}$]{WebApp's Backend}{QC's Controller}\n \\begin{umlcall}[op=Control sequence, type=synchron, return= Return qubits' state]{QC's Controller}{QC's Core}\n \\end{umlcall}\n \\end{umlcall}\n \\end{umlcall}\n \\end{umlcall}\n \\node (text1) [anchor=north] at ([xshift=5.5em, yshift=1.5em]WebApp's UI.north) {WebApp};\n \\node (text2) [anchor=north] at ([xshift=5.5em, yshift=1.5em]QC's Controller.north) {QC};\n \\begin{pgfonlayer}{background}\n \\node[outer,fit=(WebApp's UI) (WebApp's Backend) (text1)] (A) {};\n \\node[outer,fit=(QC's Controller) (QC's Core) (text2)] (B) {};\n \\end{pgfonlayer}\n \\end{umlseqdiag}\n \\end{tikzpicture}\n }\n \\caption{Sequence diagram for Example~\\ref{ex:sos}. A user submits the value of integer $N$ for factorization via UI of a Web App, which passes $N$ to the WebApp's backend. At the backend where the value of $N$ is passed to, Shor's algorithm is implemented using, say QisKit library~\\cite{ibm_quantum}. The library translates QisKit code into OpenQASM and passes QASM listing to the Controller of a QC. The Controller, which initializes the QC Core based on the OpenQASM code, triggers its execution and measures the values of the qubits once execution ends. The Controller converts the measurements into the elements of $\\vec{L}$. These values are then returned to the Backend, UI, and, finally, the user. The sequence is depicted as synchronous, but can be made asynchronous if required by a use case. Note that the WebApp UI and Backend, as well as the QC Controller, are running on CCs. The QC Core represents the `true' QC. However, the QC Controller and the QC Core can be thought of as one QC system from practical perspective. The figure and example are adopted from~\\cite{miranskyy2019testing}.}\n \\label{fig:arch}\n\\end{figure*}\n\n\\begin{figure}[tbh]\n\\centering\n\\begin{minted}[xleftmargin = 4mm, fontsize = \\scriptsize, numbersep = 2mm, linenos = true]{python3}\nfrom qiskit import Aer\nfrom qiskit.aqua.algorithms import Shor\n\ndef factorize_integer(my_int):\n # Specify the backend\/QC,\n # which will be used for computations.\n # here, we pick a simulator rather than a live QC.\n backend = Aer.get_backend('qasm_simulator')\n # Factor the integer N, \n # a is a random integer that satisfies \n # a < N and gcd(a, N) = 1.\n algorithm = Shor(N = my_int, a = 2)\n result = algorithm.run(backend)\n return result['factors']\n\nint_to_factor = 15\nprint(f\"The factors for {int_to_factor} are \" \n f\"{factorize_integer(int_to_factor)}\")\n# Output:\n# The factors for 15 are [[3, 5]]\n\\end{minted}\n\n\\caption{Factor integer $N=15$ using Qiskit Aqua Python package. The code can be implemented as a one-liner, but we split it into multiple code lines to improve code comprehension.}\\label{fig:aqua_shor}\n\\end{figure}\n\n\\end{exmp}\n\n\n\nThe above example can be easily integrated into any existing SoS. For example, a microservice on a CC can be instantiated to make an asynchronous call to a QC backend offered as a managed Cloud service (such as the ones discussed in Section~\\ref{sec:timeline}). \n\nThe same approach will be readily applicable to rack-based or desktop-based QPU units when they become available. In this case, the backend will simply point to a local rather than a remote device.\n\n\n\\section{Quality Assurance}\\label{sec:qa}\nIn this section, we will look at various viewpoints on quality assurance. We start with a comparison of the white- and black-box testing in Section~\\ref{sec:wb_bb}. Then, we explore black-box testing focusing on using QC as a component in an SoS in Section~\\ref{sec:bb_sos}, followed by the black-box testing of the QC component itself, concentrating on verification and validation in Section~\\ref{sec:vv}. Finally, in Section~\\ref{sec:mapping}, we discuss mapping to test phases of the activities covered in Sections~\\ref{sec:wb_bb}--\\ref{sec:vv}.\n\n\\subsection{White- and black-box testing}\\label{sec:wb_bb}\n\nTwo widespread methods of testing are white- and black-box testing. The former method tests internal data structures and program flow. The latter method tests the functionality, ignoring the inner workings of the software, answering the following question: will we get an expected output for a given input?\n\nWe can perform all of the standard white-box activities on the code listing, such as code reviews and code inspections. We can build linting and code inspection tools similar to those available for CC languages and run them automatically in an IDE.\n\nHowever, interactive debugging (another popular white-box activity) is challenging by construction because a QC is a black-box. Based on the classical quantum mechanics, we cannot observe the inner working of a program (executed on a QC) without altering the program's state and the final result, as measuring a qubit destroys superposition~\\cite{kaye2007introduction}. \n\nThis implies that, currently, we cannot perform interactive debugging of a program running on a QC, as we have to stop the program and take the measurements. Having said that, we may be able to debug the code in a CC simulator\\footnote{For example, Microsoft Q\\#~\\cite{svore2018} provides language constructs to define facts and assertions, and take registry measurements that can be visualized in the Microsoft Visual Studio IDE~\\cite{ms_ide}. } if the CC is powerful enough to perform the computations. \n\nMoreover, suppose the QC program can be decomposed into modules, and a given module produces the measurable expected output. In that case, we can write xUnit test cases for this module that can be tested in a simulator~\\cite{svore2018} or on a real QC\\footnote{Given that most of QC architectures are noisy, we have to run the program multiple times and compute an expected value and the confidence interval for these measurements, which may require the creation of probabilistic test cases similar to~\\cite{DBLP:conf\/icse\/Honarvar0N20}. This comes at a cost~\\cite{miranskyy2019testing}, see Section~\\ref{sec:verification} for details.}. \nWe can even estimate the coverage that our test cases provide using input and output coverage criteria~\\cite{ali2021assessing}. Finally, we can apply clever tricks, such as separate qubits into subgroups and measure them individually to perform approximate measurements; we will discuss these tricks further in Section~\\ref{sec:debug_quantum}.\n\nYet, it is often easier to resort to black-box testing when dealing with a program running on an actual QC. Let us explore the black-box testing further.\n\n\\subsection{Black-box testing: component in an SoS}\\label{sec:bb_sos}\n\nAs we discussed in~\\cite{miranskyy2019testing} and reiterated in Section~\\ref{sec:usage}, we believe that for the foreseeable future, the QC will be used as a component in an SoS, where the code running on a CC will request to compute parts of the code on a QC component. Using a REST interface, we pass the request to the QC (residing in a public or private Cloud) and then get the response to our request via the same interface. The interface is abstracted and hidden in a library, e.g., QisKit, so that the programmer does not have to worry about the details of the interface.\n\nLet us revisit Example~\\ref{ex:sos}, which depicts this flow in a UML sequence diagram in Figure~\\ref{fig:arch}, where the app programmer's `visibility' stops at the WebApp's Backend level. An example of such backend code is given in Figure~\\ref{fig:aqua_shor}. The programmer interacts with the QC as with any generic Platform as a Service (PaaS) via an API (hidden in the QisKit library).\n\nIf we operate at this `granularity level', then we have the full power of existing SE tools at our disposal. For example, we can build UML diagrams to understand the relationship between components, interaction sequences, or system states. Note that we do not need to extend the UML notation at this level of granularity\\footnote{For the actual quantum code, UML extensions might be useful~\\cite{DBLP:conf\/icse\/Perez-DelgadoP20}.}: an architect\/designer with no specialized training in quantum computing can create such diagrams. These diagrams, along with the specifications and requirements for the SoS, can help a tester create test plans without any knowledge of the QC.\n\nWhen testers create automatic test cases, they can either use a simulator backend or, which may be even more efficient, use test doubles and replace calls to the QC with mocks or stubs~\\cite{meszaros2007xunit}. For example, going back to the code in Figure~\\ref{fig:aqua_shor}, as QisKit is written in Python, we can use our favourite Python Mock library (e.g., a built-in library unittest.mock~\\cite{unittest43}) to create test-doubles for the calls to the QC on lines 8, 12, and 13. Alternatively, we can generate a test-double for the whole function \\texttt{factorize\\_integer}.\n\nThe difficulty of creating the unit test may depend on the type of output returned by the QC. Let us explore this further.\n\n\n\\subsection{Black-box testing: verification and validation}\\label{sec:vv}\n\nWhen testing the programs, how can we ensure that our code follows the design document and that the QC is doing what it is supposed to do? And even if our code reflects the design, how can we make certain that the output of the program delivers what a user needs? The former will be discussed in Section~\\ref{sec:verification}, the latter --- in Section~\\ref{sec:validation}. We will conclude with examples of code validation in Sections~\\ref{sec:val_p} and~\\ref{sec:val_super-p}.\n\n\\subsubsection{Verification}\\label{sec:verification}\n\nAs discussed in Section~\\ref{sec:wb_bb}, we can apply the full spectra of verification techniques on the code listings, but verification of a running program on an actual QC is more challenging, as we cannot properly use interactive debugger. To verify the correctness, we can try to run and debug our program in a local or online simulator, such as~\\cite{ibm_quantum,ms_quantum}. However, as the simulators run on CC, we will have to limit the complexity of our test cases to obtain results within a reasonable amount of time. This will help us to eliminate some of the defects (a taxonomy of QC bugs is being developed~\\cite{huang2018qdb}), but does not guarantee that no other defects will be encountered while solving production-scale problems. The same issue, conceptually, arises with CCs too, e.g., when dealing with buffer-overrun- and resource-leak-related defects.\n\nThe above test assumes that a simulator correctly and accurately resembles the actual QC, which is not always the case. Thus, a more definitive verification of correctness should be done on the the actual QC. Given the probabilistic nature of QC, we may have to execute the same code multiple times to increase the accuracy of our answer using Chernoff bound~\\cite{nielsen_chuang_2010} (which is similar to probabilistic algorithms in $\\BPP$ class running on CC~\\cite{nielsen_chuang_2010}). This repeating functionality is built into packages like QisKit, but it requires a higher amount of computing resources (proportional to the number of repetitions).\n\n\nThe above approach assumes that the QC hardware, its operating system, and the compiler\/translator of our program are running correctly, which is not always the case. To verify their correctness, we may need to execute the same program on multiple QCs (preferably from different manufacturers) and compare the results. If results differ significantly --- it may be a sign that there is an issue with one or more QCs under test. This is akin to correctness testing of a database engine by running the same query against different database engines~\\cite{cialini2007method}.\n\nAn award-winning protocol, verifying QC computations with the help of a CC has been proposed~\\cite{mahadev_2018}. It requires a significant amount of computational resources and, probably, will not be implemented shortly. However, as the computing power of QCs will increase, this protocol will become implementable in practice.\n\nFinally, even if all of the above tests pass, it does not guarantee that the actual results returned by the QC are correct. This is where validation comes into play.\n\n\\subsubsection{Validation}\\label{sec:validation}\n\nWhen doing the validation, we need to make sure that the output of the algorithm satisfies the conditions provided in the requirements document (assuming that requirements were captured correctly). In other words, validation of a program running on a QC is similar to that of a program executed by a CC. Essentially, the ease of validation will depend on the difficulty of implementing a program for validating the results and the time needed\\footnote{As discussed in Section~\\ref{sec:applicability_theory}, many problems in $\\BQP$ are solved efficiently on a QC, but are challenging to solve on a CC. However, the time needed to solve a problem is independent of the time needed to validate this solution.} to execute the validation. \n\nBefore implementing a program, we need to estimate how long the validation process would take. To do so, we can resort to complexity analysis. Say, if the execution time of the validation\\footnote{In the algorithm-related literature, the term verification rather than validation is used. We will use the term validation for consistency with the name of this section.} program would belong to $O(1)$, the validation process (given that it is easy to code up) would be straightforward. However, if the execution time would belong, say, to $O(n!)$, where $n$ would be proportionate to the length of input into the validation process (and to the length of the solution), then the validation process for a significantly large $n$ would be formidable. \n\nFor simplicity, we can split the complexity of validation into two classes: polynomial time $\\P$ bounded by $O(n^k)$ (i.e., validation time is bounded above asymptotically by $n^k$, where $k>0$) and super-polynomial time $\\P^\\C$, which is complementary to $\\P$, bounded by $\\omega(n^k)$ (i.e., validation time dominates asymptotically the $n^k$). We will look at examples of the algorithms belonging to these classes in subsections below.\n\nWhere should we implement the validation program: on a CC or a QC? In the program belongs to $\\P$ class, it can be implemented on either one, as $\\P \\subseteq \\BQP$. However, as discussed in Section~\\ref{sec:usage}, it is challenging to program a QC as we are dealing with low-level programming language. Moreover, the cost of running a QC in comparison with a CC is high. Thus, it is simpler and more economically feasible to implement a validation program on a CC, if possible.\n\nIn the case of $\\P^\\C$ class, the answer is less straightforward. If the size of the input into validation program is small, we may be able to still leverage a CC (especially if we can parallelize the validation on a CC cluster). However, we may have to resort to a QC for larger problems. If the validation program belongs to classes which are a subset of $\\BQP$, such as $\\BPP$ class, then QC is a good match. However, if the validation belongs to a `harder' class, such as $\\NP$-complete, then QC may also fail to deliver timely results. In this case, we may have to resort to a heuristic that tries to roughly validate the solution, but does not guarantee that solution is correct.\n\nLet us look at examples of algorithms from both classes and ways to run the validation.\n\n\\subsubsection{Validation: Polynomial $\\left( \\P \\right)$}\\label{sec:val_p}\n\n\\begin{exmp}\\label{ex:shor}\nShor's integer factoring algorithm (which we used in Example~\\ref{ex:sos}) takes integer $N$ as input and returns a vector of prime factors $\\vec{L}$ for $N$~\\cite{shor1997}. The solution runs on a QC in polynomial time, $O((\\log N)^2 (\\log \\log N) (\\log\\log\\log N) )$ to be specific~\\cite{shor1997}. The validation complexity is independent of the solution complexity, growing linearly with the number of elements in $\\vec{L}$, deemed $l$, as we simply need to multiply the elements in $\\vec{L}$ to do the validation. That is, the complexity of validation of Shor's algorithm is $O(l)$. Thus, we can easily\\footnote{Although we may have to leverage existing libraries for multiplication of integers with arbitrary precision, such as Java's BigInteger~\\cite{BigInteg48}.} perform validation on a CC. \n\\end{exmp}\n\n\\begin{exmp}\nGrover's algorithm~\\cite{grover1996fast} takes an unsorted list of $M$ items, out of which there is one item with a unique property (e.g., a unique value) that we would like to retrieve. The algorithm returns the index $i$ of this item of interest. Its complexity on a QC is $O(\\sqrt{M})$, while complexity of the fastest solution running on a CC is $O(M)$~\\cite{grover1996fast}. Suppose that the item is retrieved at $i$; then the time complexity of validation of Grover's algorithm is $O(1)$, because we only need to perform one evaluation of the item at $i$. This can be easily done on a CC. \n\\end{exmp}\n\nTechnically, validation of the two algorithms can all be carried out on a QC. But this is economically inferior, as discussed above.\n\n\\subsubsection{Validation: Super-polynomial $\\left( \\P^\\C \\right)$}\\label{sec:val_super-p}\n\n\\begin{exmp}\nBoson sampling is a good example of a problem that is challenging to validate. Yet, the algorithm is crucial\\footnote{It may lead to the implementation of a non-universal QC, which will still be more efficient than CC for some tasks, see~\\cite{aaronson2011} for details.}. Experimentally, the algorithm is typically implemented using photons (belonging to the family of boson particles~\\cite{nielsen_chuang_2010}). To implement the algorithm, we need a linear-optical circuit with $m$ modes that is injected with $h$ individual photons ($m>h$)~\\cite{giordani_experimental_2018}. In this implementation, the boson sampling task reduces to creating a sample from the probability distribution of individual photon measurements at the circuit's output. \n\nThis algorithm cannot be computed on a CC for large values of $m$ and $h$, as it requires computing a permanent of a matrix which is a $\\#\\P$-hard problem~\\cite{aaronson2011,valiant1979}. At best, it requires $O(h 2^h + mh^2)$ operations~\\cite{clifford2018}.\n\nHowever, the problem does fall~\\cite{aaronson2011} into $\\PostBQP$ class ($\\BQP$ class with post-selection), which can be efficiently computed on a QC. Validation of the results on a CC is also a $\\#\\P$-hard problem, as we again need to compute the permanent of a matrix. However, one may adopt a heuristic to estimate goodness of findings (essentially, performing approximate validation) using machine learning approach~\\cite{giordani_experimental_2018}.\n\\end{exmp}\n\n\\begin{exmp}\\label{ex:gauss}\nThe second example is estimating Gauss sums. The classical algorithm for estimating Gauss sums on CCs (for polynomials of degree $\\geq 3$) belong to the $\\#\\P$-hard class~\\cite{cai10}. However, QCs can estimate Gauss sums to polynomial precision in polynomial time~\\cite{van2002efficient}. To validate the Gauss sums on a CC, we need to take the same inputs and compute the Gauss sum, which, as we know, is a $\\#\\P$-hard problem. Hypothetically, one may invent an approximate validation heuristic (like in the boson sampling case discussed above); to the best of our knowledge, none exist at the time of writing. \n\\end{exmp}\n\nIn the above examples, to perform an accurate validation, we need to do it on a QC. Ideally, this should be done on a different QC to simultaneously check the correctness of the computer itself (as was discussed in Section~\\ref{sec:verification}). The code of the validation software would be similar to the one of the solution software. Thus, if resources permit, one may want to create the validation code from scratch (rather than reusing the existing code from the solution) to avoid migration of the defects from the solution code into the validation code.\n\n\\subsection{Mapping of activities to test phases}\\label{sec:mapping}\n\nLet us now discuss which testing phases we can apply to a QC. Techniques for some of the later test phases are readily transferable from the CC domain. For example, the performance quality assurance team can time the execution of the software on a QC to detect performance degradation during the performance testing phase. Another example is users helping to uncover defects by reporting failures during one of the acceptance test phases (say, beta testing).\n\nIf we treat a QC as a black-box component in an SoS, based on the discussion in Section~\\ref{sec:bb_sos}, the situation is straightforward. We can perform integration testing of a QC component and then see how it behaves in the SoS during the execution of a test scenario during system testing. Given that the QC component is black-box, the testing will be no different from the testing incorporation of yet another PaaS component into the SoS.\n\nHowever, dealing with testing the QC component itself is more convoluted. Let us explore three core testing phases: unit testing (UT), functional testing (FT), and system testing (ST). \n\nUnit tests in the CC domain can leverage black- and white-box testing. Here, we can use the techniques discussed in Section~\\ref{sec:wb_bb}. For the white-box testing, we may be able to debug parts of the QC program on a simulator or a QC. For the black-box (or grey-box) unit testing, we can create test cases for the modules that return a measurable value. \n\nWhat about FT? As we discussed in Section~\\ref{sec:vision}, QC machines are targeting STEM-oriented problems with relatively simple inputs and outputs (for concrete cases revisit Examples~\\ref{ex:shor}--\\ref{ex:gauss}). \nThus, the problems, which algorithms on QC are trying to solve, are self-contained. In other words, an algorithm implemented on QC represents a piece of functionality. Unit testing typically focuses on individual units\/modules of code, where a combination of these modules yields a piece of functionality. FT, on the other hand, tests a particular piece of functionality. This implies that when we perform black-box testing (i.e., checking the correctness of output for a given input) on a program that implemented such an algorithm on a QC, we are doing FT rather than UT. That is, if we are to test expected output of function \\texttt{factorize\\_integer} in Figure~\\ref{fig:aqua_shor}, we may classify this test as FT rather than UT.\n\nWhat about ST? There exists a number of (sometimes conflicting) definitions of FT and ST. Let us adopt the definition that FT `verifies a program by checking it against ... design document(s) or specification(s)' while ST `validate[s] a program by checking it against the published user or system requirements'~\\cite[p. 52]{kaner1999testing}. In the case of usage QC component in SoS, QC output can be tested in isolation to check it against design and specs, hence the FT. Then we can perform inegration testing and then execute QC component in a test workload as part of ST efforts.\n\nWhat if running QC component standalone (akin to Figure~\\ref{fig:aqua_shor}) is all that is required? This is not uncommon for STEM use-cases, where we often would like to compute and save a value. In that case, the boundary between FT and ST blurs. We may argue that we are doing ST (of a small system) rather than FT, as we are mainly checking the correctness based on requirements. For example, if we are to implement Shor's integer factorization algorithm on a QC, the requirement would be as follows. `Given a composite number $N$ as input, the algorithm should output a vector of integers $\\vec{L}$ (with the integers strictly between $1$ and $N$), such that the product of these integers is equal to $N$.' To check the correctness of our implementation of Shor's algorithm, we will multiply the returned integers and compare the resulting product against $N$, thus validating our implementation of the algorithm. Note that we resort to a requirements document rather than a design document to check correctness, thus performing the ST rather than the FT. We can also perform verification, hence the FT, by checking the reproducibility of the results.\n\n\n\\section{Debugging Tactics}\\label{sec:traditional}\nDebugging is a process of removing an error, once this error has been exposed and is often a consequence of successful testing practices~\\cite{pressman2014software} (some of which we discussed above). While we hope that one day debugging will become an orderly and automated process (e.g., by automatically mapping bug reports to code where the defect resides, and then issuing a patch for this code~\\cite{TufanoPWBP19}), currently it is an art more than a science~\\cite{pressman2014software}. \n\nThe high-level tactics~\\cite[Chapter 8]{myers2011art} for debugging a software had not changed significantly over the last 42 years (when the first edition of the seminal work~\\cite{myers2011art} was published), although integrated development environments and various automation tools have streamlined a lot of mundane tasks~\\cite{zeller09debugging, MargineanBCH0MM19}. The three common tactics~\\cite{myers2011art,pressman2014software} are backtracking, cause elimination, and brute force, discussed below.\n\n\\textit{Backtracking} debugging centers around examining the execution tree from the point of the error until a perpetrating code block is found. The analysis techniques for a code listing (such as code reviews and inspections) of a CC program can be readily applied to a QC program~\\cite{miranskyy2019testing}. Thus, these tactics are transferable. Anecdotally, based on discussions with practitioners, code reviews and inspections are the most popular debugging techniques of quantum programs nowadays.\n\n\\textit{Cause elimination} debugging formulates a hypothesis (using inductive or deductive reasoning), specifying a root cause for a bug under study. Then, data are devised, and experiments are conducted to refute or prove this hypothesis. This approach can be applied to QC. Given the probabilistic nature of the QC programs~\\cite{nielsen_chuang_2010, miranskyy2019testing}, we will have to execute the program multiple times to obtain a distribution of the results and assess the accuracy of the answer. Thus, we may be able to extend the techniques used for testing probabilistic programs running on CC, such as~\\cite{DuttaLHM18, DuttaZHM19}, to the QC domain. Such techniques already start to appear~\\cite{huang2019statistical,DBLP:journals\/pacmpl\/LiZYDY020,DBLP:conf\/icse\/Honarvar0N20}.\n\n\\textit{Brute force} debugging --- centred around the analysis of runtime traces, memory dumps, and output statements --- focuses on runtime data analysis. Of the three tactics, this is the most common one~\\cite{pressman2014software}. Some of the analyses of the runtime artifacts can be automated; however, a lot of the brute force debugging is still performed manually~\\cite{pressman2014software}. Can we transfer these tactics? \n\nIf we treat a QC program as a black-box, then the short answer is `yes'. As discussed in Sections~\\ref{sec:usage} and \\ref{sec:bb_sos}, if a QC program will be used as part of an SoS, then we can trace the input (passed from the CC component to the QC component) and the output (from the QC component to the CC component). The input and output data can be recorded in a log, and these data can be compared against the expected values. \n\nBut what if we would like to analyze a QC program at runtime using a white-box approach, e.g., to capture the execution trace of a QC program or perform interactive debugging of the code executed on the QC? In such a case, the short answer is `it depends'~\\cite{DBLP:conf\/icse\/Miranskyy0D20}. Before we delve into the answers, let us compare and formally define classic and quantum models of computation, which will help us understand the issues with debugging quantum programs.\n\n\n\\section{Quantum Computation}\\label{sec:quant_comp}\n\n\nIn this section, we review the basic concepts of quantum computation and set up the conventions and notions that are used in the rest of the paper.\n\n\\subsection{The classical model of computation}\n\\label{sec:cls-cmm-model}\n\nClassical computation can be modeled using the language of circuits. Abstractly, a circuit is a network of gates and wires: the wires transmit bits to gates, the gates perform elementary operations on the input bits, and the results of these operations are again bits that are carried by wires. Figure \\ref{fig:cls-crct} shows a schematic example of a classical circuit. \n\n\\begin{figure}[ht]\n \\centering\n \\begin{quantikz}[row sep = {0.8cm,between origins}]\n \\lstick{$x_0$} & \\ctrl{1} & \\gate{G_2} & \\gate[wires = 2]{G_3} & \\qw & \\qw \\\\\n \\lstick{$x_1$} & \\gate[wires = 2]{G_1} & \\qw & & \\gate{G_5} & \\qw \\\\\n \\lstick{$x_2$} & & \\qw & \\gate{G_4} & \\octrl{-1}\n \\end{quantikz}\n \\caption{A classical circuit.}\n \\label{fig:cls-crct}\n\\end{figure}\n\nAn important concept regarding the circuit model of computation is \\textit{universality}. A set $G$ of gate are said to be universal if given any function $f: \\{0, 1\\}^m \\rightarrow \\{0, 1\\}^n$, where $m$ and $n$ are positive integers, a circuit for $f$ can be constructed using only gates from $G$. For example the set of gates $G = \\{\\textsc{and, not}\\}$ and $G = \\{\\textsc{nand, fanout}\\}$ are universal. Classical computation is generally not reversible. A computation is reversible if for every output it is always possible to uniquely recover the corresponding input. Every classical computation, however, can be made reversible. To do this, one needs to replace every gate in a given circuit with its reversible version. This in turn can be done by constructing the reversible version of the gates in a universal set. \n\nFor example, the set $\\{\\textsc{nand, not}\\}$ is universal. The gate $\\textsc{not}$ is already reversible. The gate $\\textsc{nand}$ can be made reversible by adding an additional input and two additional outputs. This is known as Toffoli gate~\\cite{DBLP:conf\/icalp\/Toffoli80}, \\cite[Section 1.4.1]{nielsen_chuang_2010}. The two new outputs keep a copy of the original inputs, see Figure \\ref{fig:rev-and}.\n\n\\begin{figure}[ht]\n \\centering\n \\begin{quantikz}[row sep = {0.4cm,between origins}]\n \\lstick{$x_0$} & \\gate[wires = 2]{\\textsc{nand}} \\\\\n \\lstick{$x_1$} & & \\qw\\rstick{$\\neg (x_0 \\wedge x_1)$} \\\\ [3mm]\n \\lstick{$x_0$} & \\gate[wires = 3]{\\textsc{nand}} & \\qw\\rstick{$x_0$} \\\\\n \\lstick{$x_1$} & & \\qw\\rstick{$x_1$} \\\\\n \\lstick{$1 \\rightarrow x_2$} & & \\qw\\rstick{$1 \\oplus (x_0 \\wedge x_1) = \\neg (x_0 \\wedge x_1)$}\n \\end{quantikz}\n \\caption{Irreversible (top) to reversible (bottom) \\textsc{nand} gate, where $\\neg$ denotes negation, $\\oplus$ denotes Boolean operator \\textsc{xor}, and $\\wedge$ denotes Boolean operator \\textsc{and}. The reversible gate is implemented using a particular setup of the Toffoli gate: the first two bits are control bits; the third one is a target bit, its input is always set to $1$.}\n \\label{fig:rev-and}\n\\end{figure}\n\nNote that any reversible gate has the same number of inputs and output bits~\\cite{DBLP:conf\/icalp\/Toffoli80}. This means that any reversible circuit has the same number of input and output bits. A bit is usually denoted by a two-dimensional column vector: $[1, 0]^T$ for the bit $0$ and $[0, 1]^T$ for the bit $1$. This notation is used so that one can describe the probabilistic model of computation as well. Encoding bits using vectors leads to the natural representation of gates as linear operators. More concretely, any gate can be represented as a matrix. For example, the \\textsc{not} gate is the following matrix\n\\begin{equation}\n \\label{equ:not}\n \\textsc{not} = \\begin{bmatrix} 0 & 1 \\\\ 1 & 0 \\end{bmatrix},\n\\end{equation}\nwhich flips $0$ to $1$ and vice versa:\n\\begin{equation*}\n \\textsc{not} \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix} = \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix}, \\quad \\textsc{not} \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} = \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix}.\n\\end{equation*}\nTherefore, any circuit can be represented as a matrix that operates on the input bits as a vector and produces the output bits as vector. This formulation of the circuit model of computation proves useful for the quantum model of computation as well. \n\n\n\\subsection{The quantum model of computation}\n\n\n\\subsubsection{The Dirac notation}\n\nThe mathematical formulation of quantum computing is based on that of quantum mechanics. The main mathematical objects used in quantum mechanics are complex Euclidean spaces. These are vector spaces over the complex numbers called Hilbert spaces. For quantum computing we only deal with finite dimensional Euclidean spaces. A complex Euclidean space of dimension $n$ is denoted by $\\mathbb{C}^n$ where $\\mathbb{C}$ is the set of complex numbers and the exponent $n$ means Cartesian product $n$ times. \n\nAn element of $\\mathbb{C}^n$ is called a vector and is usually denoted by an arrow over a letter, for example $\\vec{x}$. In quantum mechanics, however, a vector is denoted by $\\ket{x}$, and its adjoint is denoted by $\\bra{x}$. This is called the Dirac notation, which we will also use in this paper. The standard basis of $\\mathbb{C}^n$ is denoted by $\\ket{0}, \\ket{1}, \\dots, \\ket{n - 1}$, every vector is a unique linear combination of these vectors, i.e., $\\ket{a} = a_0\\ket{0} + \\cdots + a_{n - 1}\\ket{n - 1}$, where $a_i \\in \\mathbb{C}$ for all $i$. If $n$ is a power of two, say $n = 2^k$, then basis vectors are written using binary strings of length $k$: $\\ket{00 \\dots 00}, \\ket{00 \\dots 01}, \\dots, \\ket{11 \\dots 10}, \\ket{11 \\dots 11}$. This basis is usually referred to as the \\textit{computational basis}. \n\nOther equivalent notations for a basis vector $\\ket{b_1, \\dots, b_n}$ in the computational basis are $\\ket{b_1} \\ket{b_2} \\cdots \\ket{b_n}$ or $\\ket{b_1} \\otimes \\ket{b_2} \\otimes \\cdots \\otimes \\ket{b_n}$, where $\\otimes$ is the tensor product operation. Tensor product is an operation that combines two spaces together. For example if $\\{ \\ket{a_i} \\}_{1 \\le i \\le m}$ is a basis for the space $\\mathbb{C}^m$ and $\\{ \\ket{b_j}_{1 \\le j \\le n} \\}$ is a basis for the space $\\mathbb{C}^n$ then $\\{ \\ket{a_i} \\otimes \\ket{b_j} \\}_{1 \\le i \\le m, 1 \\le j \\le n}$ is a basis for the space $\\mathbb{C}^m \\otimes \\mathbb{C}^n$.\n\n\n\n\\subsubsection{Qubits and quantum mechanics}\n\nIn the following, we briefly review the four postulates of quantum mechanics that form the conceptual foundations of quantum computing; see~\\cite{kaye2007introduction,nielsen_chuang_2010} for additional details.\n\nWe start with the \\textit{State Space Postulate}, which says that the state space of a 1-qubit quantum system is described by the set of unit vectors in $\\mathbb{C}^2$. Therefore, in the computational basis, a qubit is described by a linear combination $\\ket{\\psi} = \\alpha \\ket{0} + \\beta \\ket{1}$ where $\\abs{\\alpha}^2 + \\abs{\\beta}^2 = 1$. We refer to $\\ket{\\psi}$ as a quantum state, in this case the state of a 1-qubit system. We also say that $\\ket{\\psi}$ is a \\textit{superposition} of the states $\\ket{0}$ and $\\ket{1}$. The \\textit{Composition of Systems Postulate} states that the combined system of two quantum systems is described by the tensor product of the corresponding state spaces. More precisely, if two quantum systems have state spaces $H_1$ and $H_2$ then the composite system has states space $H_1 \\otimes H_2$. This means that, for example, a $2$-qubit system is described by the superposition of the basis states $\\ket{00}, \\ket{01}, \\ket{10}, \\ket{11}$. More generally, the state of an $n$-qubit system can be written as the superposition\n\\begin{equation}\n \\label{equ:superpos}\n \\ket{\\psi} = \\sum_{x \\in \\{ 0, 1 \\}^n} \\alpha_x \\ket{x}\n\\end{equation}\nof basis states, where $\\alpha_x \\in \\mathbb{C}$ and $\\sum_{x \\in \\{ 0, 1 \\}^n} \\abs{\\alpha_x}^2 = 1$. The quantum state of a system can evolve, over time, to another quantum state. \n\nThe \\textit{Evolution Postulate} says that the state of a closed quantum systems evolves according to unitary operators. This means for any evolution of a system from a state $\\ket{\\psi_1}$ to a state $\\ket{\\psi_2}$ there exists a unitary operator $U$ such that $U \\ket{\\psi_1} = \\ket{\\psi_2}$. An operator is called unitary if $U^* \\, U = I$, where $U^*$ is the adjoint of $U$, and $I$ is the identity operators. If we fix a basis for the state space of the quantum system, an operator is represented by a unique matrix. In that case, the adjoint of an operator is the conjugate-transpose of the corresponding matrix. As explained in the previous section, the evolution of classical systems can also be described by matrices. Therefore, loosely speaking, quantum operations can be thought of as a generalization of classical operations that act on continuous state spaces. A $1$-qubit operator is represented by a $2 \\times 2$ matrix acting on the space $\\mathbb{C}^2$. For example, for the computational basis, the unitary operator that takes the qubit $\\alpha\\ket{0} + \\beta\\ket{1}$ to $\\alpha\\ket{1} + \\beta\\ket{0}$ and vice versa, is the \\textsc{not} operation in \\eqref{equ:not}. Another example is given by a set of well-known $1$-qubit operators deemed the Pauli operators:\n\\begin{equation}\n \\label{eq:pauli}\n \\sigma_0 = \\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix}, \\; \\sigma_1 = \\begin{bmatrix} 0 & 1 \\\\ 1 & 0 \\end{bmatrix}, \\; \\sigma_2 = \\begin{bmatrix} 0 & -i \\\\ i & 0 \\end{bmatrix}, \\; \\sigma_3 = \\begin{bmatrix} 1 & 0 \\\\ 0 & -1 \\end{bmatrix}.\n\\end{equation}\nNote that $\\textsc{not} = \\sigma_1$.\n\nAccording to the \\textit{Measurement Postulate}, quantum measurement on a system $A$ is described by a set of measurement operators which act on the state space of A. The state of the system and the probability of being in that state after the measurement depends on the measurement operators. A concrete example of quantum measurement is the von Neumann measurement with respect to the computational basis: given the state $\\ket{\\psi}$ in \\eqref{equ:superpos}, performing a von Neumann measurement with respect to the basis $\\{ \\ket{x} \\}$ outputs $y$ with probability $\\abs{\\alpha_y}^2$, and the state of the system after the measurement is $\\ket{y}$. In general, performing a measure produces some classical information and leaves the system in a (possibly) new quantum state.\n\n\n\n\\subsubsection{The quantum circuit model}\n\nAs explained in Section \\ref{sec:cls-cmm-model}, any classical circuit can be efficiently converted to a reversible circuit. The reversible model of computation can naturally be generalized to a model of quantum computation. The quantum circuit model is similar to the reversible circuit model with bits and gates replaced by qubits and quantum gates: wires carry qubits to quantum gates, the gates perform quantum operations on the input qubits, and the resulting qubits are again carried by wires. Figure \\ref{fig:qtm-crct} shows a schematic example of a quantum circuit.\n\n\\begin{figure}[ht]\n \\centering\n \\begin{quantikz}[row sep = {0.8cm,between origins}]\n \\lstick{$q_0$} & \\gate{G_1} & \\qw & \\qw & \\gate[wires = 2]{G_5} & \\qw \\\\\n \\lstick{$q_1$} & \\ctrl{-1} & \\gate[wires = 2]{G_3} & \\gate{G_4} & & \\qw \\\\\n \\lstick{$q_2$} & \\gate{G_2} & & \\meter{} & \\qw & \\qw\n \\end{quantikz}\n \\caption{A quantum circuit.}\n \\label{fig:qtm-crct}\n\\end{figure}\n\nNote that since quantum gates represent unitary operations, the number of input wires to a quantum circuit is the same as the number of output wires. This is not the case for classical circuits. As mentioned above, a $1$-qubit operator is a unitary operator that acts on the $2$-dimensional $\\mathbb{C}^2$. Such an operator is called a $1$-qubit gate. When used as quantum gates, the Pauli operators $\\sigma_0, \\sigma_1, \\sigma_2, \\sigma_3$, defined in~\\eqref{eq:pauli}, are denoted by $I, X, Y, Z$, respectively. The gate $X$ is quantum \\textsc{not} gate. Just like classical gates, quantum gates can have \\textit{control} inputs. For example, the control-\\textsc{not} (\\textsc{cnot}) gate acts on a $2$-qubits state as $\\ket{a}\\ket{b} \\mapsto \\ket{a}\\ket{a \\oplus b}$, where $\\oplus$ is the \\textsc{xor} operation. Here, the first qubit is the control qubit, based on the value of which a \\textsc{not} gate is applied to the second qubit. Since \\textsc{cnot} acts on $\\mathbb{C}^4$, it is represented by a $4 \\times 4$ matrix which, with respect to the computational basis, is\n\\[\n\\textsc{cnot} =\n\\begin{bmatrix}\n 1 & 0 & 0 & 0 \\\\\n 0 & 1 & 0 & 0 \\\\\n 0 & 0 & 0 & 1 \\\\\n 0 & 0 & 1 & 0\n\\end{bmatrix}.\n\\]\nThe measurement gate is usually depicted by a meter where the input is a quantum state and the output is classical information (and a post-measurement quantum state).\n\\begin{center}\n \\begin{quantikz}\n \\lstick{quantum state} & \\meter{} & \\qw\\rstick{classical information}\n \\end{quantikz}\n\\end{center}\n\nThe concept of universality can be generalized to quantum computation. A fundamental difference with the classical case is that the set of unitary operations is not discrete, hence a discrete set of quantum gates cannot be used to implement arbitrary unitary operations exactly. \n\nA set $G$ of quantum gates is called universal if any unitary operation can be approximated to arbitrary accuracy by quantum circuits involving only gates from $G$. It can be proved~\\cite[Section 4.3]{kaye2007introduction} that the set $\\{H, T\\}$, where $H$ and $T$ are the Hadamard gate and the $\\frac{\\pi}{8}$-gate defined by\n\\[ H = \\frac{1}{\\sqrt{2}}\\begin{bmatrix} 1 & 1 \\\\ 1 & -1 \\end{bmatrix}, \\quad T = \\begin{bmatrix} 1 & 0 \\\\ 0 & e^{i\\pi \/ 8} \\end{bmatrix}, \\]\nis universal for $1$-qubit gates. It means that any $1$-qubit gate can be arbitrarily approximated by circuits involving only $H$ and $T$. If we add \\textsc{cnot} to the above set, we obtain the universal set of gates $G = \\{\\textsc{cnot}, H, T\\}$ for quantum computation~\\cite[Section 4.3]{kaye2007introduction}.\n\n\n\\section{Debugging Quantum Programs}\\label{sec:debug_quantum}\n\nA quantum program executed on a modern gate-based QC leverages a register of qubits for performing quantum operations and a register of classic bits for recording the measurements of qubits' states and conditionally applying quantum operators~\\cite{cross2017open}. Thus, a typical QC program mixes traditional instructions (to alter the state of bits and apply conditional statements) and quantum instructions (to alter the state of qubits and to measure qubit value).\n\nAs mentioned above, a general quantum program consists of blocks of code each containing classical and quantum instructions. \nQuantum operations can be divided into two kinds: unitary and non-unitary. Unitary operations are \nreversible and preserve the norm of the operands. Non-unitary operations are not reversible and have \nprobabilistic implementations. \n\nThe classical parts of a quantum program can be debugged using traditional methods. The quantum parts, \nhowever, can not be treated in the same way because of the properties of a QC~--- such as superposition, \nentanglement, and no-cloning~--- which are governed by the laws of quantum mechanics. The purpose of \ndebugging a program is to present the user with human readable, i.e., classical, information about \nthe runtime state of the system. Extracting classical information from a quantum state is done using \nmeasurement which is non-unitary and results in collapse of the state, and hence \nan unintended behavior of the program. We shall describe, in the following, different scenarios in a \nQC to which classical debugging techniques cannot be applied, and discuss some potential solutions.\n\n\n\\subsection{Superposition}\\label{sec:superposition}\n\nLet $\\ket{\\psi}$ be the state of an $n$-qubit register. Then we can uniquely write $\\ket{\\psi}$ using a superpostion in the computational basis as in \\eqref{equ:superpos}. By \nthe measurement postulate of quantum mechanics, measuring the state $\\ket{\\psi}$ in the \ncomputational basis results in an outcome $x \\in \\{ 0, 1 \\}^n$ with probability $\\abs{\\alpha_x}^2$, \nand the state of the system after the measurement is $\\ket{x}$. For example, consider the initial state\n$\\ket{010}$ and perform the following steps: first apply a Hadamard transform to each qubit (creating superposition), then\na controlled-not \\textsc{cnot} to qubits 2 and 3, and finally measure qubit 3. If the measured qubit is $0$ \n(which happens with probability $1\/2$), then the state collapses to $\\frac{1}{2}(\\ket{00} - \\ket{01} \n+ \\ket{10} - \\ket{11})$. An implementation of this example in OpenQASM 2.0 is shown in Figure \\ref{fig:spp}.\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\begin{subfigure}[b]{0.25\\columnwidth}\n\t\t\\centering\n\t\t\\includegraphics[width = \\textwidth]{sup_coll_circuit.pdf}\n\t\t\\caption{Circuit}\n\t\t\\label{fig:spp-circ}\n\t\\end{subfigure}\n\t%\n\t\\hspace*{1cm}\n\t%\n\t\\begin{subfigure}[b]{0.25\\columnwidth}\n\t\t\\centering\n\t\t\\begin{minted}[fontsize = \\footnotesize, numbersep = 2mm, linenos = true, autogobble]{cpp}\n OPENQASM 2.0;\n include \"qelib1.inc\";\n \n qreg q[3];\n creg c[1];\n \n x q[1];\n h q[0];\n h q[1];\n h q[2];\n cx q[1],q[2];\n measure q[2] -> c[0];\n \\end{minted}\n \\vspace{-4mm}\n\t\t\\caption{Assembly code}\n\t\t\\label{fig:spp-code}\n\t\\end{subfigure}\n\t\\caption{Example of measuring a superposition. In OpenQASM, \\textsc{not} is denoted by $x$, Hadamard by $h$, and \\textsc{cnot} by $cx$. }\n\t\\label{fig:spp}\n\\end{figure}\n\nA natural feature of a debugger for quantum programs would be to check if the state \nof a variable is in superposition. There are two possible scenarios: when the input state is unknown (e.g., when it is generated as an output of another quantum program) and when the input state is known. Let us elaborate on each of these cases.\n\n\\subsubsection{Unknown input state.}\\label{sec:unknown-inp}\nIf the input to the program is an unknown state $\\ket{\\psi}$, then there is no known general \nalgorithm that can efficiently decide if $\\ket{\\psi}$ is in a superposition. \nNot much can be done here in terms of a general method for debugging, \ndifferent approaches should be considered for different problems.\n\nFor example, in the hidden subgroup problem \\cite[Chapter 7]{kaye2007introduction}, if the group is abelian, then it can be efficiently \ndecided if the coset state of a subgroup is in superposition. For non-abelian groups, however, the same problem is often hard. For example, \nthe best known algorithm for the following problem has subexponential runtime \\cite{kuperberg2005subexponential}: let $N$ be a positive \ninteger, and let $\\mathbb{Z}_N$ be the group of integers mod $N$. For a random unknown $x \\in \\mathbb{Z}_N$ and fixed unknown \n$d \\in \\mathbb{Z}_N$, decide whether a given state is of the form $\\ket{b}\\ket{x}$ or $\\frac{1}{\\sqrt{2}}(\\ket{0}\\ket{x} + \\ket{1}\\ket{x + d})$, where $b \\in \\{0, 1\\}$.\n\n\\subsubsection{Known input state.}\\label{sec:known-inp}\nIf a state is the result of applying a known unitary operation to a known initial state, i.e., \n$\\ket{\\psi} = U \\ket{\\psi_0}$ where $\\ket{\\psi_0}$ and $U$ are both known, then $\\ket{\\psi}$ can be regenerated by the \ndebugger. For example, consider the state \n\\[ \\ket{\\psi} = \\frac{1}{\\sqrt{2^n}} \\sum_{x \\in \\{ 0, 1 \\}^n} (-1)^{h(x)} \\ket{x}, \\]\nwhere $h(x)$ is the Hamming weight of $x$, i.e., the number of non-zero bits in $x$. Then $\\ket{\\psi}$ can be generated by applying the Hadamard transform $U = H^{\\otimes n}$ to the $n$-qubit register $\\ket{11 \\dots 1}$. This is an example of the Quantum Fourier Transform (QFT) over the group $\\mathbb{Z}_2^{\\oplus n}$. QFT can be implemented efficiently over the group $\\mathbb{Z}_N$ where $N$ is an integer, although in this case the implementation is more involved. A toy implementation of QFT for $N = 8$ is shown in Figure \\ref{fig:qft}. \n\n\\begin{figure*}[ht]\n\t\\centering\n\t\\begin{subfigure}[b]{0.45\\columnwidth}\n\t\t\\centering\n\t\t\\includegraphics[width = \\textwidth]{qft_8_circuit.pdf}\n\t\t\\caption{Circuit}\n\t\t\\label{fig:qft-circ}\n\t\\end{subfigure}\n\t%\n\t\\hspace*{10mm}\n\t%\n\t\\begin{subfigure}[b]{0.25\\columnwidth}\n\t\t\\centering\n\t\t\\begin{minted}[fontsize = \\footnotesize, numbersep = 2mm, linenos = true, autogobble]{cpp}\n OPENQASM 2.0;\n include \"qelib1.inc\";\n \n qreg q[3];\n \n h q[0];\n cp(pi\/2) q[1],q[0];\n cp(pi\/4) q[2],q[0];\n h q[1];\n cp(pi\/2) q[2],q[1];\n h q[2];\n swap q[0],q[2];\n \\end{minted}\n \\vspace{-4mm}\n\t\t\\caption{Assembly code}\n\t\t\\label{fig:qft-code}\n\t\\end{subfigure}\n\t\\caption{Quantum Fourier Transform over $\\mathbb{Z}_8$. In OpenQASM, $P_{\\pi \/ 2}$ and $P_{\\pi \/ 4}$ phase gates are denoted by $cp(\\cdot)$; \\textsc{swap} gate by $swap$. }\n\t\\label{fig:qft}\n\\end{figure*}\n\n\nIn the cases as above, there are various methods (depending on \nthe problem) to characterize the state $\\ket{\\psi}$. Often, one relies on \\textit{quantum state \ntomography}, which is the process of reconstructing a quantum state through a series of measurements \\cite{d2003quantum, cramer2010efficient}. \n\n\n\\subsection{Entanglement}\n\nIn a QC, a set of memory cells or registers is said to be in an entangled state if it is impossible \nto classically specify the correlations among them. More precisely, let $X_1, \\dots, X_n$ be the \nstate spaces of a set of quantum systems that represent $n$ registers. The state space of the \ncomposite of these systems, that represents an array, is given by the tensor product $X = X_1 \n\\otimes \\cdots \\otimes X_n$. A state $\\ket{\\psi} \\in X$ that can be written in the form \n$\\ket{\\psi} = \\ket{\\psi_1} \\otimes \\cdots \\otimes \\ket{\\psi_n}$, where $\\ket{\\psi_j} \\in \nX_j$ for $j = 1, \\dots, n$, is called separable. A state that is not separable is called entangled. \nWhen debugging a program that operates on an entangled state, the following problems can be \nconsidered.\n\n\\subsubsection{Checking for separability.}\\label{sec:separability}\nGiven a state $\\ket{\\psi} \\in X$, deciding whether $\\ket{\\psi}$ is separable is an NP-hard \nproblem \\cite{gharibian2010strong, gurvits2003classical}. This is called the \\textit{separability \nproblem} in quantum information theory, see \\cite[Chapter 6]{watrous2018theory} for details. There are a variety of methods (see~\\cite{leinaas2006geometrical,guhne2009entanglement}) for separability\/entanglement detection \nthat can be implemented in practice, specially for lower dimensions. For example, if the debugger can generate \nseveral copies of $\\ket{\\psi}$, then one way to detect the nonlinear properties of \n$\\ket{\\psi}$ is via direct measurement. For the sake of brevity, we do not provide technical details here; see \\cite{leinaas2006geometrical} for a numerical method \nfor examining separability and \\cite{guhne2009entanglement} for other interesting methods and their \nimplementations.\n\n\\subsubsection{Extracting classical information.}\\label{sec:classical}\nMeasuring a subsystem of a larger composite system that is in an entangle state will likely alter \nother subsystems. In debugging terms, if a set of variables are in an entangled state, a debugger will not be able to present any classical information about a subset of those variables to the user without disturbing the state of the whole set. \n\nFor example, consider the entangled state $\\frac{1}{\\sqrt{2}}(\\ket{00} + \\ket{11})$ of two qubits.\nMeasuring any of the two qubits alters the result of the subsequent measurement on the other qubit.\nMore precisely, if the first qubit is measured, then state collapses to $\\ket{00}$ or $\\ket{11}$ with probability $|1\/\\sqrt{2}|^2 =\n1\/2$; the outcome of measuring the second qubit is always $0$ if the resulting state is $\\ket{00}$, and it is always $1$ if the resulting state is $\\ket{11}$. Such a state is called maximally entangled. \n\nA composite system, however, often has subsystems that are not entangled with any other subsystem. In \nthis case, we can measure that subsystem without disturbing the whole state. For example, in the following state of a \n3-qubit register\n\\begin{equation}\n\\label{equ:sep-subsys}\n\t\\frac{1}{2} (\\ket{000} - \\ket{001} + \\ket{110} - \\ket{111}),\n\\end{equation}\nthe last qubit is not entangled with the first two while the first and the second qubits are \nentangled, see \\eqref{equ:gen-clas}. The algorithms for separability detection (discussed in Section~\\ref{sec:separability}) could be used to identify separable subsystems. \n\nThings would be much simpler if the debugger could somehow estimate a given state with a state that \nis generated by applying some operation to a basis state (i.e., classical information). For example, \nthe state in \\eqref{equ:sep-subsys} can be generated as\n\\begin{equation}\n \\label{equ:gen-clas}\n \\begin{aligned}\n & (\\textsc{cnot} \\otimes H)(H \\otimes I_4) \\ket{001} \\\\\n & = \\frac{1}{\\sqrt{2}}(\\ket{00} + \\ket{11}) \\otimes \\frac{1}{\\sqrt{2}}(\\ket{0} - \\ket{1}),\n \\end{aligned}\n\\end{equation}\nwhere $I_4$ is the $4 \\times 4$ identity gate. Therefore, the state in \\eqref{equ:sep-subsys} can be described by the debugger using the classical information \n$\\ket{001}$ and the names of the above operators. An implementation of the sequence of operations in \\eqref{equ:gen-clas} is \nshown in Figure \\ref{fig:entg-sep}.\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\begin{subfigure}[b]{0.25\\columnwidth}\n\t\t\\centering\n\t\t\\includegraphics[width = \\textwidth]{entg_sep_circuit.pdf}\n\t\t\\caption{Circuit}\n\t\t\\label{fig:entg-circ}\n\t\\end{subfigure}\n\t%\n\t\\hspace*{1cm}\n\t%\n\t\\begin{subfigure}[b]{0.25\\columnwidth}\n\t\t\\centering\n\t\t\\begin{minted}[fontsize = \\footnotesize, numbersep = 2mm, linenos = true, autogobble]{cpp}\n OPENQASM 2.0;\n include \"qelib1.inc\";\n \n qreg q[3];\n creg c[2];\n \n x q[2];\n h q[0];\n cx q[0], q[1];\n h q[2];\n measure q[0] -> c[0];\n measure q[1] -> c[1];\n \\end{minted}\n \\vspace{-4mm}\n\t\t\\caption{Assembly code}\n\t\t\\label{fig:entg-code}\n\t\\end{subfigure}\n\t\\caption{A circuit for generating state \\eqref{equ:sep-subsys}.}\n\t\\label{fig:entg-sep}\n\\end{figure}\n\n\n\n\\subsection{No-cloning}\\label{sec:no-cloning}\n\nThe most general method of obtaining information about a variable without disturbing its state is \nto make a copy of the variable and work on the copy. In the classical setting, this is often \nstraightforward. In the quantum setting, however, the situation is much more complicated. In fact, \nit is impossible to make a copy of a given general unknown quantum state. More precisely, given an \nunknown state $\\ket{\\psi}$ and an arbitrary state $\\ket{\\phi}$, it can be shown~\\cite[Theorem 10.4.1]{kaye2007introduction} that \nthere is no unitary operator $U$ that can perform the following:\n\\[ \\ket{\\psi} \\otimes \\ket{\\phi} \\overset{U}{\\longmapsto} \\ket{\\psi} \\otimes \n\\ket{\\psi}. \\] \nIn many practical scenarios, however, a debugger will only need to make an \\textit{approximate copy} \nof a state; a state that is `close enough' to the given state but provides useful debugging \ninformation. For example, for a state $\\ket{\\psi}$ that encodes a probability distribution \\cite{grover2002creating}, such as\nthe Gaussian distribution, an approximate clone would provide valuable information about the distribution.\nThe possibility of approximate cloning was first discussed in \\cite{buvzek1996quantum}. \nMuch research has been done on different cloning methods each optimizing particular aspects of a \ncloner that are desired for different situations, see \\cite{scarani2005quantum} for a survey. \n\nAlso, it is possible to perform exact cloning if the given state belongs to a set of known mutually orthogonal states. Recall that two states $\\ket{\\psi_1}$, $\\ket{\\psi_2}$ are orthogonal if $\\braket{\\psi_1}{\\psi_2} = 0$. For example, consider the set of $n$-qubit states $S = \\{ \\ket{\\psi_j} \\}$ where\n\\[ \\ket{\\psi_j} = \\frac{1}{\\sqrt{2^n}} \\sum_{x \\in \\{ 0, 1 \\}^n} (-1)^{\\lrang{j, x}} \\ket{x}, \\]\nfor $j \\in \\{0, 1\\}^n$. The set $S$ is exponentially large, but for any given state $\\ket{\\psi_j} \\in S$ where $j$ is unknown, we can efficiently make a copy of $\\ket{\\psi_j}$. An example of such a cloning procedure, for $n = 2$, is shown in Figure \\ref{fig:clone-orth}.\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\begin{subfigure}[b]{0.25\\columnwidth}\n\t\t\\centering\n\t\t\\includegraphics[width = \\textwidth]{clone_orth_circuit.pdf}\n\t\t\\caption{Circuit}\n\t\t\\label{fig:clone-orth-circ}\n\t\\end{subfigure}\n\t%\n\t\\hspace{1cm}\n\t%\n\t\\begin{subfigure}[b]{0.25\\columnwidth}\n\t\t\\centering\n\t\t\\begin{minted}[fontsize = \\footnotesize, numbersep = 2mm, linenos = true, autogobble]{cpp}\n OPENQASM 2.0;\n include \"qelib1.inc\";\n \n qreg q[4];\n \n h q[0];\n h q[1];\n cx q[0],q[2];\n h q[0];\n cx q[1],q[3];\n h q[1];\n h q[2];\n h q[3];\n \\end{minted}\n \\vspace{-4mm}\n\t\t\\caption{Assembly code}\n\t\t\\label{fig:clone-orth-code}\n\t\\end{subfigure}\n\t\\caption{A circuit for cloning $2$-qubit states in the set $S$. Here, the state of the first two qubits $q_0, q_1$ is copied into the last two qubits $q_2, q_3$.}\n\t\\label{fig:clone-orth}\n\\end{figure}\n\n\n\n\n\\subsection{Discussion}\nIn Sections~\\ref{sec:superposition}--\\ref{sec:no-cloning}, we discussed various issues preventing the application of the classic debugging techniques and identified some potential solutions. \n\nAs discussed in~\\cite{miranskyy2019testing}, if the input size and the amount of required qubits is small, we can run a quantum program in a simulator (running on a CC). However, the increase of the input size and the qubit register length may force us to run the program on a QC. \n\nIf we can generate multiple approximate copies of the state \\cite{buvzek1996quantum}, then we can produce an empirical distribution of the qubit state and compare it against the expected distribution, to detect problems in the code. The generation of the multiple approximate copies can be readily implemented for moderate inputs sizes using universal cloning methods~\\cite{werner1998optimal, buvzek1998universal, fan2001quantum}. More efficient cloning can be achieved using state-dependent (i.e. non-universal) cloning methods~\\cite{niu1999two, scarani2005quantum}. This would address issues related to superposition with known input state (discussed in Section~\\ref{sec:known-inp}), extraction of classical information (discussed in Section~\\ref{sec:classical}), and no-cloning (discussed in Section~\\ref{sec:no-cloning}). A compiler can automatically generate the code for the approximate copying (akin to compilers for CC that can instrument the code to add debugging information), translating higher-level language into quantum assembly~\\cite{huang2019statistical}. The same principle of multiple approximate copies (aggregated using statistics) can be used to generate runtime assertions~\\cite{li2014debugging,zhou2019quantum,zhou2019quantum_extended,DBLP:journals\/pacmpl\/LiZYDY020}.\n\nFor the case of unknown input states, discussed in Section~\\ref{sec:unknown-inp}, no general solution exists and will require a programmer to make decisions on a case-by-case basis.\n\nFinally, separability checking, discussed in Section~\\ref{sec:separability}, demands the implementation of numerical methods that will require changes to the QC and, hopefully, will be implemented in the future. \n\n\\section{Conclusions}\\label{sec:conclusions}\n\nQC field is rapidly evolving, and the SE community should start bringing SE practices into the QC world.\nIn this paper, we focus on analyzing testing and debugging tactics, highlighting classic ones that are readily applicable and showing that new ones have to be created.\nWe believe that this work would be of interest to practitioners, creating quantum programs, as well as researchers, developing the next generations of tooling for QC.\n\n\\section*{Acknowledgments}\nWe express profuse thanks to the anonymous reviewers of this paper and of~\\cite{miranskyy2019testing} and~\\cite{DBLP:conf\/icse\/Miranskyy0D20}. We are also grateful to the ICSE 2020 committee for rewarding~\\cite{DBLP:conf\/icse\/Miranskyy0D20} with the New Ideas and Emerging Results Distinguished Paper Award.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzhifv b/data_all_eng_slimpj/shuffled/split2/finalzzhifv new file mode 100644 index 0000000000000000000000000000000000000000..8766922705b23d5931b55ed2b30069075edaf202 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzhifv @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nThe Casimir effect \\cite{Casimir48,Lamoreaux05,Bordag09,Dalvit11} is a remarkable consequence of vacuum field fluctuations\nwhich, in its simplest manifestation, leads to the attraction of two neutral ideal \nconducting plates. At very short distances quantum fluctuation forces dominate the interaction \nbetween neutral objects making them an essential consideration for micro-electro mechanical\ndevices (MEMS) and atom traps, among others.\nThe comparison between experimental measurements and theory for Casimir\nforces between metallic plates has been a matter of debate in recent\nyears. This debate is of particular importance if this comparison is\nused to derive constraints on hypothetical new short-range\ninteractions appearing in addition to the gravity force in\nunification models\n\\cite{Fischbach98,Adelberger03,Onofrio06,Antoniadis11}.\n\nTwo recent experiments are at the heart of this debate.\nCasimir force measurements by the IUPUI group\n\\cite{Decca05,Decca07}, performed at distances smaller than 750 nm,\nwere interpreted by the authors as excluding the dissipative Drude\nmodel and agreeing with the lossless plasma model. This has led to a\ndiscrepancy between experiments and physically motivated theoretical\nmodels, such as the Drude model, for real conductors which exhibit\ndissipation. In distinction, a recent experiment by the Yale\n\\cite{Sushkov11} was able to measure Casimir forces at distances up\nto 7 $\\mu$m and was interpreted by the authors as being in agreement\nwith the Drude prediction, including quantum as well as thermal\nfluctuations, once an electrostatic patch contribution has been\ntaken into account.\n\nIt is known that patch effects are a source of concern for\nCasimir experiments \\cite{Speake03,Chumak04,Kim10,deMan10,Kim10b}, as well\nas for other precision measurements\n\\cite{Fairbank67,Camp91,Turchette00,Deslauriers06,Robertson06,Epstein07,%\nPollack08,Adelberger09,Everitt11,Reasenberg11}. For the Yale\nexperiment the patches were assumed to be much larger than the gap\n$D$ between the spherical and planar plates used in the measurement. \nUnder\nthese conditions the patch force is found to be\nproportional to $R V^2_\\mathrm{rms} \/ D$ in the proximity force approximation (see below),\nwhere $R$ is the radius of curvature of the spherical plate and $V_\\mathrm{rms}$ is\nthe root-mean-square (rms) voltage of electrostatic patch potentials\n\\cite{Sushkov11}. For the IUPUI experiment, a patch analysis was\nperformed with different assumptions leading to the conclusion that\nthe patch effect had a negligible influence \\cite{Decca05}.\nUnfortunately, it was not possible in any of these experiments to\nmeasure the patches independently. It follows that the conclusions\nof the theory-experiment comparisons heavily rely on the patch\nmodels used in the data analysis.\n\nIn this paper, we revisit electrostatic patch effects and\nanalyze their possible influence in Casimir force measurements. Our\napproach is based on the method pioneered by Speake and Trenkel\n\\cite{Speake03} with the electrostatic patches described in terms of\na power spectral density. However, we will develop a model for\nthe power spectral density differing from the one proposed in\n\\cite{Speake03} and used in \\cite{Decca05,Decca07}.\n\nOur model is based on the observation that bare metallic surfaces\nare composed of crystallites, each of which constitutes a single\npatch, where the local surface voltage is determined by the local\nwork function \\cite{Gaillard06}.\nBy assuming that through the surface preparation process the\ncrystallographic orientation, and hence the corresponding work\nfunction, of each crystallite is determined independently and\nrandomly we can infer that voltage correlations are restricted to\npoints lying on the same patch: We refer to this as {\\it\nquasi-local correlation}.\nOur model with quasi-local correlations can be compared to the case\nof quenched charge disorder in dielectrics \\cite{Naji10,Sarabadani10}, \nand\nalso shows close similarities with models proposed recently to\ndescribe patch correlation functions for atomic or ionic traps\n\\cite{Dubessy09,Carter11}.\n\n\n\n\nWe will show that the voltage correlation function from our\nquasi-local model strongly differs from that initially proposed in\n\\cite{Speake03} and used in \\cite{Decca05,Decca07}. \nAs a result, in contrast to the claims of \\cite{Decca05,Decca07}, \npatches may have a significant contribution to the IUPUI measurements. \nIn addition we will qualitatively address the issue of surface contamination \nwhich is expected to lead to larger correlation lengths and reduced\nvoltage fluctuations \\cite{Rossi92}. Given that the degree of contamination\nis unknown we perform a fit of the patch model we propose to\nthe difference between measurements and the Casimir theoretical prediction based on the\nDrude model and find that it qualitatively explains the residual signal. \nFor the Yale experiment, our results will essentially\nreproduce those obtained in \\cite{Sushkov11}.\n\n\n\n\\section{Electrostatic patch effect}\n\nIn the present section, we recall a few general results of interest,\nassuming that the validity conditions of the proximity force\napproximation (PFA) are satisfied, that is, the radius of the sphere\nused in the experiment is much greater than the sphere-plane\ndistance. In this case, the expression for the force gradient\n$G_{sp}$ (derivative with distance of the force $F_{sp}$) in the\nsphere-plane geometry is written as follows in terms of the pressure\n$P_{pp}$ (the force per unit area) calculated between two planes\n\\begin{eqnarray}\n\\label{GradientPFA}\nG_{sp}(D) \\equiv \\frac{\\partial\nF_{sp}(D)}{\\partial D} = 2 \\pi R P_{pp}(D) .\n\\end{eqnarray}\nThis expression is used throughout the paper for both Casimir and\npatch effects.\n\nThe basic description of the patch effect after\n\\cite{Speake03} is a statistical ensemble of patch potentials\n$V_i({\\bf r})$ on the surfaces of two planar plates labeled $i=1,2$.\nThe potentials are assumed to have zero mean $\\langle V_i({\\bf r})\n\\rangle =0$, and to be described by the two-point potential\ncorrelation functions\n\\begin{eqnarray}\n\\label{defcorrelations} C_{ij}({\\bf r}) = \\langle V_i({\\bf r})\nV_j({\\bf 0}) \\rangle = \\int \\frac{d^2 {\\bf k}}{4 \\pi^2} e^{ i {\\bf\nk} \\cdot {\\bf r} } C_{ij}[\\bf k] .\n\\end{eqnarray}\nIn the plane-plane geometry, points on the planes are denoted in\ncartesian coordinates as ${\\bf r} = (x,y)$ and the point ${\\bf 0}$\nis an arbitrary origin. The correlation functions $C_{ij}({\\bf\nr})$ and therefore the power spectra $C_{ij}[\\bf k]$ are also\nassumed to be isotropic. The relations between these two functions\ncan be written\n\\begin{eqnarray}\n\\label{FourierBessel}\n&& C_{ij}(r) = \\frac{1}{2\\pi} \\int_0^\\infty d k \\ k \\ J_0(k r) \\ C_{ij}[k] , \\nonumber \\\\\n&& C_{ij}[k] = 2 \\pi \\int_0^\\infty dr \\ r \\ J_0(k r) \\ C_{ij}(r) ,\n\\end{eqnarray}\nwhere we have simply denoted $r\\equiv\\vert{\\bf r}\\vert$ and\n$k\\equiv\\vert{\\bf k}\\vert$ and where $J_n(x)$ is the n-th order\nBessel function \\cite{Gradshteyn}.\nThe patch power spectrum $C_{ij}[k]$ corresponds to the notation\n$\\tilde{C}_{ij}(k)$ in \\cite{Speake03}. As usual, the variances and\ncovariances are given by the integrals\n\\begin{eqnarray}\n\\label{covariances} && C_{ij}(0) = \\langle V_i V_j \\rangle =\n\\frac{1}{2\\pi} \\int_0^\\infty d k \\ k \\ C_{ij}[k] .\n\\end{eqnarray}\n\nThe pressure due to electrostatic patches in the plane-plane\ngeometry can be computed exactly \\cite{Speake03} as\n\\begin{eqnarray}\n\\label{pressure} P_{pp}^\\mathrm{patch}(D) &=& \\frac{\n\\varepsilon_o}{4\\pi} \\int_{0}^{ \\infty }\n\\frac{ d k \\ k^3}{ \\sinh^2 (kD)} \\\\\n&& \\times \\left\\{ C_{11}[k]+ C_{22}[k]- 2 C_{12}[k] \\cosh (kD)\n\\right\\} . \\nonumber\n\\end{eqnarray}\nIt is worth emphasizing at this point that the integral is reduced\nto a very simple expression when patch sizes, with a typical value\ndenoted $\\ell_\\mathrm{patch}$, are larger than the distance $D$. In\nthis case, all wavevectors $k$ contributing to the integral\n(\\ref{pressure}) satisfy $k D \\ll 1$, so that the pressure scales\nuniversally as $1\/D^2$, irrespective of the particular details of\nthe power spectrum (Eq. (\\ref{covariances}) is used)\n\\begin{eqnarray}\n\\label{largepatches} P_{pp}^\\mathrm{patch}(D) &=& \\frac{\n\\varepsilon_o}{2D^2} \\int_{0}^{ \\infty }\n\\frac{ d k \\ k}{2\\pi} \\left\\{ C_{11}[k]+ C_{22}[k]- 2 C_{12}[k] \\right\\} \\nonumber \\\\\n&=& \\frac{ \\varepsilon_o}{2D^2} \\left\\langle \\left(V_i -\nV_j\\right)^2 \\right\\rangle\\quad,\\quad D\\ll\\ell_\\mathrm{patch}.\n\\end{eqnarray}\nThe above result is expected from the analogy with a\nparallel plate capacitor with prescribed voltages. In contrast, when\nthe relevant wavevectors no longer satisfy the above inequality,\ndifferent models for the patch power spectrum result in different\npredictions for the patch contribution to the pressure.\n\nIt is also worth mentioning here some conditions for the expression\n(\\ref{pressure}) of the electrostatic patch pressure between two\nplates to be valid. A fundamental assumption in this analysis is\nthat the ergodic hypothesis is satisfied, which means that the\ndistribution of patches within the interaction area is a fair\napproximation of the ensemble-averaged distribution function defined\nby the power spectrum $C_{ij}[k]$. When applied to two plane plates\nof finite area $A$, we expect this assumption to be well satisfied\nif the effective interaction area contains a large number of patch\ncorrelation areas $A\\gg\\ell_\\mathrm{patch}^2$. For the sphere-plane\ngeometry, the effective area of interaction is of the order of\n$\\pi DR$, leading to the validity requirement\n\\begin{eqnarray}\n\\label{validity} \\pi D R \\gg \\ell_\\mathrm{patch}^2 .\n\\end{eqnarray}\n\nIn the following two subsections we recall a model used in\n\\cite{Speake03} and \\cite{Decca05,Decca07}, and introduce\nanother model with quasi-local correlations which we think to be a\nbetter description of sputtered surfaces.\n\n\n\\subsection{The sharp-cutoff model}\n\nWe now discuss the model of patch correlations which was\nproposed as an example in \\cite{Speake03} and then used in\n\\cite{Decca05,Decca07} to assess the contribution of electrostatic patches\nto the Casimir force measurements.\n\nIt is a simple description based upon two assumptions: a) the power\nspectrum of patches is an annulus in $k$-space possessing no other\ndependence than a sharp cutoff at small ($k_\\mathrm{min}$) and large\n($k_\\mathrm{max}$) wavevectors (hence the name {\\it sharp-cutoff\nmodel}); b) there are no cross correlations between the two plates\n($C_{12} = 0$). This model gives the power spectrum for a single\nplate as\n\\begin{equation}\n\\label{STPS} C_{ii}[k] = \\frac{ 4\\pi V_\\mathrm{rms}^2}{\nk_\\mathrm{max}^2 - k_\\mathrm{min}^2 } \\theta(k_\\mathrm{max} - k)\n\\theta( k - k_\\mathrm{min}),\n\\end{equation}\nwhere $V_\\mathrm{rms}^2$ is the variance of the potential on one\nplate and $\\theta$ is the Heaviside step function.\n\nIn order to determine the parameters of this model, the authors of\n\\cite{Decca05} used the further assumptions~: c) based on AFM images\nof the surfaces, the minimum and maximum grain sizes of the samples\nwere determined to be $\\ell_\\mathrm{patch}^\\mathrm{min}=25$ nm and\n$\\ell_\\mathrm{patch}^\\mathrm{max}=300$ nm ~; d) the patch sizes were\nassumed to be the same as the grain sizes and the cutoffs in\n$k$-space were derived from the inverse maximum and minimum grain sizes\n$k_\\mathrm{min}=2\\pi\/\\ell_\\mathrm{patch}^\\mathrm{max}=20.9 \\mu\\mathrm{m}^{-1}$ and\n$k_\\mathrm{max}=2\\pi\/\\ell_\\mathrm{patch}^\\mathrm{min}=251 \\mu\\mathrm{m}^{-1}$ ~; e) the rms voltage was obtained by computing\nthe variance of the work functions over the different\ncrystallographic planes of gold, which led to $V_\\mathrm{rms} \\approx\n80.8$ mV. Using the five assumptions a) to e), it was concluded in\n\\cite{Decca05} that the patch pressure had a negligible influence on\nthe estimation of the Casimir force. A reasonable agreement was then\nobtained between the experimental data and the prediction for the\nCasimir pressure using the lossless plasma model (more discussions\nbelow).\n\nNow we will argue that model (\\ref{STPS}) is not a good\ndescription for the patch power spectrum for\nthe surfaces used in the experiments, and later on, we\nwill also question the relation between patch and grain sizes. In\norder to make the former point clear, let us write the correlation\nfunction $C_{ii}(r)$ of patches in real space which can be obtained\nthrough an inverse Fourier transform (\\ref{FourierBessel}) from the\nspectrum (\\ref{STPS})\n\\begin{eqnarray}\n\\label{ST_Spatial_Correlation} C_{ii}(r) = 2 V_\\mathrm{rms}^2\n\\frac{k_\\mathrm{max} J_1( k_\\mathrm{max} r )-k_\\mathrm{min} J_1(\nk_\\mathrm{min} r )}{\\left(k_\\mathrm{max}^2 -\nk_\\mathrm{min}^2\\right)r} .&&\n\\end{eqnarray}\nAs one moves away from coincidence the correlation function\n$C_{ii}(r)$ oscillates between positive and negative values with a\nperiod of the order of the smallest patch size, and is contained\nwithin an envelope decaying as $r^{-3\/2}$ (see\nFig.\\ref{correlation_functions}). These oscillations imply that the\npatch potential shows {\\it correlations} as well as {\\it\nanti-correlations} in space. Such behavior could be expected for\nsurfaces exhibiting some kind of antiferroelectric ordering (where\nthe configurational energy is minimized when adjacent surface\ndipoles are antiparallel), but will unlikely describe the random\npotentials on sputtered surfaces.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=2.7in]{CorrSpace}\n\\includegraphics[width=2.7in]{CorrMomentum}\n\\caption{Comparison of the sharp-cutoff and quasi-local patch models\ndescribed in subsections II-A (dashed lines) and II-B (solid lines),\nrespectively. Plot (a) shows the voltage correlation functions in\nreal space while plot (b) shows the associated spectrum in Fourier\nspace. All plots correspond to the correlation function $C \\equiv C_{ii}$ divided\nby $V_\\mathrm{rms}^2$. On the lower plot, the sharp-cutoff spectrum\ndiscussed in II-A is multiplied by a factor of $100$ in order for it\nto appear at the scales shown. The parameters used for both models,\ndiscussed in subsections II-A and II-B, are taken from\n\\cite{Decca05}, but however, do not correspond to the same average\npatch size.} \\label{correlation_functions}\n\\end{figure}\n\nAs already stated, the strict relation between patch sizes and grain\nsizes, assumed in the analysis of \\cite{Decca05}, has also to be\nquestioned. The adsorption of contaminants on the surfaces\nalters patch sizes which, as a result, do not necessarily correspond\nto the grain sizes \\cite{Rossi92}. We expect that contamination\nleads to an effective smearing of the patch layout, so that patch\nsizes will be larger than grain sizes while the voltage variance\nwill be less than the value obtained for a clean sample from the\nassumption e) discussed above. \n\n\n\n\\subsection{The quasi-local correlation model}\n\nWe now propose another patch model which we think to be a better\nmotivated description of the patch correlation function for the\nsurfaces used in the experiments.\n\nTo model the layout of cystallites on a plate, we choose a random\npatch layout and afterward assign a random potential to each patch.\nFor a given micro-realization of patches we write the voltage over\nthe whole surface as\n\\begin{equation}\n\\label{voltage}\nV( {\\bf x} ) = \\sum_a v_a \\Theta_a ({\\bf x}).\n\\end{equation}\nThe sum is over all patches, $v_a$ is a random variable describing\nthe voltage on patch $a$, and the function $\\Theta_a( {\\bf x})$ is\ndefined to be 1 for ${\\bf x}$ on the $a$th patch, and 0 otherwise.\n\nWe now obtain the two-point voltage correlation function by\nperforming ensemble averages over all micro-realizations of the\npatch voltages and layouts. Physically, the voltage on each site is\ndetermined by the crystallite face exposed to the surface. As we\nassume that each crystallite is deposited with a random\ncrystallographic orientation and that each deposition is\nstatistically independent we can infer that\n\\begin{equation}\n\\label{variance} \\langle v_a v_b \\rangle_v = \\delta_{ab}\nV^2_\\mathrm{rms},\n\\end{equation}\nwhere the expectation value $\\langle ... \\rangle_v$ averages over\nthe voltage fluctuations only and $\\delta_{ab}$ is the Kronecker delta. \nAlso, note that we are implicitly assuming that there are no cross correlations\nbetween the patches on different plates, $C_{12} = 0$.\nUsing (\\ref{voltage}) and\n(\\ref{variance}) we construct the two-point voltage correlation for\n{\\it a single micro-realization} of the patch layout\n\\begin{equation}\n\\label{QLPatchModel}\n \\left\\langle V({\\bf x} ) V({\\bf x'}) \\right\\rangle_v =\n V^2_\\mathrm{rms} \\sum_a \\Theta_a({\\bf x} ) \\Theta_a({ \\bf x'}).\n\\end{equation}\n\nThe final step in constructing the ensemble-averaged voltage\ncorrelation function is to average over all patch layouts. We carry\nthis out by exploiting several symmetries:\n\n \\begin{enumerate}\n \\item We assume that the patches are distributed uniformly and\n isotropically which implies that the average patch associated\n with any given point on the surface is circular with a radius\n determined from a distribution of patch sizes. In reality no\n patch is circular and this notion of {\\it patch radius} should\n only be taken in a statistical sense.\n\n \\item For any two points on the sample surface, the voltage\n correlation function $C({\\bf x}, {\\bf x}')$ is proportional to\n the number of patches which contain both points (among all micro-realizations).\n By employing the statistical description of patches, as described above in 1,\n the correlation will be computed by summing over all circular\n patch centers and sizes as depicted in Fig. \\ref{PatchCorrelation}.\n\n \\item As a check one can verify that the correlation at\n coincidence is the constant $V_\\mathrm{rms}^2$. Moreover, translational\n and rotational invariance implies that $C({\\bf x}, {\\bf x}')$\n depends only on $r = |{\\bf x} - {\\bf x}'|$.\n\n \\end{enumerate}\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=2in]{PatchCorrelationDiagram}\n\\caption{The voltage correlation function, $C({\\bf x}, {\\bf x}')$,\nis constructed by summing over all circular patches which contain\nboth ${\\bf x}$ and ${\\bf x}'$. This is undertaken by integrating\nover patch centers ${\\bf y}$, and accounting for the distribution in\npatch sizes with the distribution $\\Pi(\\ell)$. }\n\\label{PatchCorrelation}\n\\end{center}\n\\end{figure}\n\nGiven these considerations we find the following form for the\ncorrelation function:\n\\begin{eqnarray}\nC({\\bf x}, {\\bf x}') &&= \\int_0^\\infty d\\ell \\ \\Pi(\\ell) \\\\\n&& \\times \\frac{4 V^2_\\mathrm{rms} }{\\pi \\ell^2 } \\int d^2 y \\\n\\theta( \\ell\/2 - | {\\bf x} - {\\bf y}|) \\theta( \\ell\/2 - | {\\bf x}'\n- {\\bf y}|), \\nonumber\n\\end{eqnarray}\nwhere the integral over $y$, constrained by the $\\theta$-functions,\nsums over all patches of size $\\ell$ which contain both points. The\nfinal integral over $\\ell$ averages over patch sizes where\n$\\Pi(\\ell)$ is the distribution of patch diameters. Note in\nparticular that the translational invariance of the correlation\nfunction is made apparent by the change of variables ${\\bf x}' -\n{\\bf y} \\to {\\bf z}$ . Subsequently performing the integration over\n$y$ reveals the rotational invariance of the final result\n\\begin{eqnarray}\n\\label{quasi-local-correlation} C(r) \\equiv C_{ii}(r) &&= \\frac{ 2\nV^2_\\mathrm{rms}}{\\pi} \\int_r^\\infty d\\ell \\ \\Pi(\\ell)\n\\nonumber \\\\\n&& \\times \\bigg[ \\cos^{-1}\\left(\\frac{r}{\\ell} \\right) -\n\\frac{r}{\\ell}\\sqrt{ 1 -\\left( \\frac{r}{\\ell} \\right)^2 } \\bigg].\n\\end{eqnarray}\n\nThe patch power spectrum can then be obtained through the Fourier\ntransform (\\ref{FourierBessel}). Some interesting properties can be\ngiven at this point. First, the integration of the correlation\nfunction over all space is simply\n\\begin{eqnarray}\n\\label{QLCorrelationK0} C[k=0] = \\frac{1}{4} \\pi \\overline{\\ell^2}\nV^2_\\mathrm{rms},\n\\end{eqnarray}\nwith $\\overline{\\ell^2}$ the variance of the distribution\n$\\Pi(\\ell)$. Second, the integral of $C[k]$ over wavevectors is just\nthe variance of the potential $\\int_0^\\infty dk k C[k] = 2\\pi\nV^2_\\mathrm{rms}$.\n\nOne can derive some universal scaling laws for the patch\ncontribution to the pressure in some limiting cases. When the\npatches are much larger than the gap ($D\\ll \\bar{\\ell}$), the\nexpression (\\ref{largepatches}) is obtained. This $1\/D^2$ scaling\nlaw for the pressure (and the corresponding $1\/D$ for the energy per\nunit area for planar plates) is universal for all patch power\nspectral densities whenever the typical patch sizes are much larger\nthan the gap. In particular, this scaling was used in\n\\cite{Sushkov11} to model the patch effect \\cite{footnote}. In the\nopposite limit, where the typical patch sizes are much smaller than\nthe gap, one can obtain a simple scaling law. In this case the\nspectrum $C[k]$ is approximately constant over the wavevector range\n$k\\stackrel{<}{\\scriptscriptstyle{\\sim}}1\/D$ which provides the most\nsignificant contribution to (\\ref{pressure}). We then find, when\nusing (\\ref{QLCorrelationK0}),\n\\begin{eqnarray}\n\\label{long-distance} P^\\mathrm{patch}(D) & \\simeq &\n\\frac{\\varepsilon_0}{2\\pi}\\, C[0]\\,\n\\int_0^{\\infty} dk\\,\\frac{k^3}{{\\rm sinh}^2(kD)} \\\\\n&\\simeq& \\frac{3\\zeta(3)}{4} \\frac{\\varepsilon_0 V_\\mathrm{rms}^2\n\\overline{\\ell^2}}{D^4} \\approx 0.90 \\frac{\\varepsilon_0\nV_\\mathrm{rms}^2 \\overline{\\ell^2}}{D^4}. \\nonumber\n\\end{eqnarray}\nWe emphasize at this point that this $1\/D^4$ scaling law is generic\nfor all spectra having a finite limit at $k=0$, but does not hold\nwhen $C[k]$ vanishes at $k=0$. In particular, in the model discussed\nin II-A, there is a sharp-cutoff of the power spectral density at\n$k_\\mathrm{min}>0$. In this case, the pressure (\\ref{pressure}) is\nexponentially small when $k_\\mathrm{min} D\\gg 1$, that is also $D\\gg\n\\ell_\\mathrm{patch}^\\mathrm{max}$. The leading order contribution\nindeed comes from the exponential tail of $1\/\\sinh^2(kD)$ and is\nmuch smaller than the result found in the generic case\n(\\ref{long-distance}). This point will play a crucial role in the\ncomparison to experimental data discussed in the next section.\n\nBefore entering this discussion we choose a specific form for the\npatch size distribution $\\Pi(\\ell)$ which is similar in spirit to\nthe sharp-cutoff model discussed in subsection II-A. By assuming the patch\nsizes are distributed uniformly within a finite interval between a\nminimum $\\ell_\\mathrm{patch}^\\mathrm{min}$ and maximum\n$\\ell_\\mathrm{patch}^\\mathrm{max}$ value, the probability\ndistribution is\n\\begin{equation}\n\\label{SM} \\Pi(\\ell) = \\frac{\\theta(\\ell_\\mathrm{patch}^\\mathrm{max}\n- \\ell) \\theta(\\ell - \\ell_\\mathrm{patch}^\\mathrm{min} ) }{\n\\ell_\\mathrm{patch}^\\mathrm{max}- \\ell_\\mathrm{patch}^\\mathrm{min} }\n,\n\\end{equation}\nand has the following moments\n\\begin{eqnarray}\n\\label{SMmoments} &&\\overline{\\ell} =\n\\frac{\\ell_\\mathrm{patch}^\\mathrm{max} +\n\\ell_\\mathrm{patch}^\\mathrm{min} }{2} \\\\\n&&\\overline{\\ell^2} = \\frac{(\\ell_\\mathrm{patch}^\\mathrm{max})^2 +\n(\\ell_\\mathrm{patch}^\\mathrm{min})^2\n+\\ell_\\mathrm{patch}^\\mathrm{max} \\ell_\\mathrm{patch}^\\mathrm{min}\n}{3}. \\nonumber\n\\end{eqnarray}\nAdditionally, we would like to remark that\nwe have also considered other size patch size distributions $\\Pi(\\ell)$ (log-normal,\nGaussian, generalized gamma, etc.) and have found similar results\nfor the pressure in all cases.\n\nWe emphasize that, despite some similarity in the construction of\nthe two models discussed in subsections II-A and II-B, they\ncorrespond to very different correlation properties, the most\nstriking difference resulting from a nonvanishing value for $C[k=0]$\nin the quasi-local model which gives a distinct large distance behavior. \nIn particular, a patch model employing quasi-local correlations was \nrecently adopted to describe heating in ion traps and dissipation \nin cantilevers \\cite{Dubessy09}. There, the observed large distance \n$(D \\gg \\overline{\\ell})$ scaling of electric field noise \n($\\propto D^{-4}$) is linked with a nonvanishing value of $C[0]$. \n\nTo estimate the effects of contamination, we will assume\nthat the patch power spectrum on a dirty surface\ntakes the same form as on a clean surface (i.e., also given by the quasi-local model), with the exception\nthat the parameters of the model are altered by\nthe contaminants. \nLet us stress that quasi-local correlations may\nbe not as accurate for contaminated surfaces as for clean ones.\nWe employ the above assumptions in a preliminary manner to account for the properties\nof contaminated surfaces, to be confirmed by dedicated studies\nto come in the future.\n\n\n\\section{Comparison with experiments}\n\nWe now compare the theory and experiments by calculating the Casimir\nforce from the Drude model, and the patch pressure arising from the\nmodel with quasi-local correlations. To make the comparison we first\ncalculate the plane-plane Casimir pressure $P_{pp}(D)$ at temperature $T$ using the Lifshitz formula \n\\cite{Lifshitz,Lambrecht00,Lambrecht06}.\nWe use tabulated optical data for gold \\cite{Palik}, \nextrapolated to low frequencies with a Drude model to describe the contribution of conduction electrons, \n$\\varepsilon_{\\rm cond}(\\omega) = 1 - \\Omega_P^2 \/ (\\omega (\\omega+i \\gamma))$,\nwhere $\\Omega_P$ is the plasma frequency and $\\gamma$ quantifies the damping rate.\nTo account for \nroughness corrections to\nthe Casimir pressure\nwe adopt the \nsimplest formulation based on an additive scheme (Eq. (33)\nin \\cite{Decca05}).\nWe will call the resulting pressure as the ``Drude model\" \nCasimir pressure $P_{pp}^{\\rm Drude}(D)$. \n\n\nAs already stated, we use the PFA to relate the experimental data\ncorresponding to the sphere-plane geometry to the predictions\ncalculated in the plane-plane geometry, for the Casimir and the\npatch effects. In the IUPUI experiments the sphere-plane force\ngradient $G_{sp}$ is measured, which is related to the equivalent\nplane-plane pressure as in (\\ref{GradientPFA}). In the Yale\nexperiments the sphere-plane force $F_{sp}$ is measured, which is\nrelated similarly to the plane-plane energy per unit area.\n\nAfter subtracting from the experimental data the theoretical\npredictions for the Casimir interaction, we find a residual signal\n\\begin{equation}\n\\label{residuals} \n\\delta{P}^{\\rm Drude} (D) \\equiv P^{\\rm\nexperiment}_{pp}(D) - P^{\\rm Drude}_{pp}(D) .\n\\end{equation}\nThe question we address in the following is whether or not the\nresidual $\\delta{P}^{\\rm Drude}$ can be explained by a reasonable modeling of\npatch effects. The criterium is then to minimize the remaining\ndifference between the residual signal and the patch pressure\n$\\delta{P}^{\\rm Drude} (D) - P^\\mathrm{patch} (D)$. The residual is\ndefined here for the Drude model and may be as well be defined for\nthe plasma model. The patch pressure $P^\\mathrm{patch} (D)$ is then\ndefined for a given patch model, say in particular the sharp-cutoff\n(subsection II-A) or quasi-local (subsection II-B) models.\n\n\n\n\\subsection{Data analysis for the IUPUI experiment}\n\nFor the comparison with the IUPUI experiment we compute the Casimir\nforce at room temperature $T=295$ K using tabulated optical data extrapolated to low frequencies with a Drude model \nwith parameters $\\Omega_P = 8.9$ eV for the plasma frequency and $\\gamma\n= 0.0357$ eV for the damping rate.\n Root mean square roughness heights for the plane and the sphere\n are $3.6$ nm and $1.9$ nm, respectively.\nThese permittivity and roughness parameters are the ones reported in \\cite{Decca07}. \n\nWe collect in Fig.\\ref{RicardoDataDrude} the information needed to\ncompare IUPUI experimental data with predictions from the Drude\nmodel and modelings of the patch effect. We plot the residuals\n$\\delta{P}^{\\rm Drude}$ defined as in (\\ref{residuals}) as points\nwith error bars and the patch pressure $P_\\mathrm{patch}$ for\ndifferent patch models as lines. The error bars represent the total\nexperimental error described in Fig. 2 of \\cite{Decca07} at $67 \\%$ confidence. \nThe theoretical predictions for the Casimir pressure $P_{pp}^{\\rm Drude}$ are calculated for the Drude model as described above and assumed to have no error. There are four different patch\nmodels represented in Fig.\\ref{RicardoDataDrude}~:\n\\begin{enumerate}\n\n\\item The solid curve is the estimation of the patch effect using\nall the assumptions of subsection II-A. The patches are thus\ndescribed by the sharp-cutoff model (\\ref{STPS}) with the parameters\n$k_\\mathrm{max} = 251 \\, \\mu\\mathrm{m}^{-1}$, $k_\\mathrm{min} =\n20.9 \\, \\mu\\mathrm{m}^{-1}$ and $V_\\mathrm{rms} = 80.8$ mV\n(these are the parameters used in \\cite{Decca05}).\n\n\\item The dotted curve is the result of the quasi-local correlation\nmodel (\\ref{quasi-local-correlation}) with the patch size\ndistribution (\\ref{SM}) described in subsection II-B. The\nparameters, $\\ell_\\mathrm{patch}^\\mathrm{min}=25$nm,\n$\\ell_\\mathrm{patch}^\\mathrm{max}=300$nm and\n$V_\\mathrm{rms}=80.8$ mV, correspond to the assumptions that the\npatch sizes are given by the grain sizes and the rms voltage is\ndetermined by the variance of the work function over the different\ncrystallographic planes (these are the same parameters used in item\n1 above).\n\n\\item The long-dashed curve is obtained from a\nleast-squares minimization of the difference $ \\delta{P}^{\\rm Drude}\n(D) - P^\\mathrm{patch} (D)$, using the quasi-local patch correlation\nmodel given by (\\ref{quasi-local-correlation}) and (\\ref{SM}). As\n$\\ell_\\mathrm{patch}^\\mathrm{min}$ is found to have a small influence, we fix it\nto the smallest grain size $\\ell_\\mathrm{patch}^\\mathrm{min}=25$ nm as discussed\nabove. The best fit on the two remaining parameters gives\n$\\ell_\\mathrm{patch} ^\\mathrm{max} \\approx 2476$ nm and\n$V_\\mathrm{rms} \\approx 9.2$ mV and results in qualitative agreement\nbetween the residual and the fitted patch pressure. \nThe associated first moments of the\npatch size distribution are $\\overline{\\ell}= 1251$ nm and\n$\\overline{\\ell^2}=(1437$ nm$)^2$. The reduced-$\\chi^2$ for this fit,\ncalculated using the total error bars from Fig 2. of \\cite{Decca07} at $67 \\%$,\nis $0.814$. It is important to note that the values of the fit parameters and quality of the fit are \nvery sensitive to the sample's optical parameters, in particular to the plasma \nfrequency used in the extrapolation of optical data to low frequencies \\cite{sampledep}.\nHowever,\none should avoid giving too much importance to any of these values of reduced-$\\chi^2$ as a measure with statistical significance of experiment-theory agreement. Indeed, the influence of sample dependency of optical parameters, the use of a very crude description of roughness corrections to the Casimir pressure, and, most importantly, the lack of precise information of the patch correlation\nfunction in actual experimental samples, all imply that the fits obtained with the quasi-local model for patches have a qualitative nature; dedicated patch effects measurements are required to make metrological claims (see the Conclusions for further discussions).\n\n\\item The short-dashed curve (underneath the long-dashed curve)\nis a fit of a phenomenological model proposed by Carter and Martin\n\\cite{Carter11}. The correlation function of this model, based on a\nMonte-Carlo simulation of patch layouts, can be expressed in terms\nof a shifted Gaussian and is specified by the rms voltage and the\naverage patch area $w^2$, related to our patch radius via $w\n\\approx \\sqrt{\\pi} \\, \\, \\overline{\\ell}\/2$. Our best fit values are\n$\\overline{\\ell} \\approx 1229$ nm and $V_\\mathrm{rms}= 8.6$ mV with\nreduced-$\\chi^2$ of 0.812.\n\n\\end{enumerate}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=3.3in]{RicardoDrudeComparison}\n\\caption{ Comparison of the residual $\\delta{P}^{\\rm Drude}$ between\nthe experimental pressure in \\cite{Decca07} and the Drude prediction\n(points with error bars at $67\\%$ confidence taken from Fig. 2 of \\cite{Decca07}) with patch pressure $P_\\mathrm{patch}$ for\nfour different patch models (more details in the main text)~: 1. The\nsolid curve is the result of the sharp-cutoff model (with\nassumptions of subsection II-A)~; 2. The dotted curve corresponds to\nthe quasi-local patch correlation model assuming that the patch\nsizes are given by the grain sizes and that the rms voltage is given\nby the variance of the work function over different crystallographic\nplanes~; 3. The long-dashed curve is the result of a best-fit on\nthe parameters ($\\ell_\\mathrm{max}$ and $V_\\mathrm{rms}$) of the\nquasi-local patch correlation model~; 4. The short-dashed curve\n(underneath the long-dashed curve) is a fit of a phenomenological\nmodel proposed in \\cite{Carter11}. The inset shows the residual signal\nresulting from subtracting the fit of the quasi-local model (long-dashed curve)\nfrom $\\delta P^{\\rm Drude}$.}\n\\label{RicardoDataDrude}\n\\end{figure}\n\nAfter this description of the information gathered on Fig.\n\\ref{RicardoDataDrude}, let us now comment on the significance of\nthe various results:\n\n\\begin{enumerate}\n\\item The solid curve reproduces and confirms the calculations\nwhich were performed to quantify patch effects in\n\\cite{Decca05,Decca07}. With the assumptions described in subsection\nII-A, the calculated patch pressure is indeed far too small to\nexplain the difference between experimental data and theoretical\npredictions using the Drude model.\n\n\\item The dotted curve gives the result of the quasi-local\nmodel of patch correlations (\\ref{SM}) with parameters\ndetermined as was done in \\cite{Decca05,Decca07}, but here for a\ndifferent patch spectrum model. As a striking illustration of the\nimportance of this difference, the calculated patch pressure is now\nlarger than the difference between experimental data and\ntheoretical predictions using the Drude model. This illustrates the\n highly model dependent nature of the computed patch pressure. Thus, \npatches may be an {\\it important systematic effect} for which their contribution\nto the measured signal should ideally be assessed independently of any Casimir\nforce measurement. \n\n\\item The long-dashed curve corresponds to a least squares fit of the\nquasi-local correlation model to the residual $\\delta P^{\\rm\nDrude}$. With the best-fit parameters\n$\\ell_\\mathrm{patch}^\\mathrm{max}$ and $V_\\mathrm{rms}$, this model\nqualitatively fits the difference between experimental data and\ntheoretical predictions using the Drude model. These parameters have\nreasonable values: $\\ell_\\mathrm{patch}^\\mathrm{max}$ is larger than\nthe maximum grain size on the samples, and $V_\\mathrm{rms}$ smaller\nthan the rms voltage for a clean sample \\cite{Decca05,Decca07}. This\nsuggests the presence of contaminants on the sample surfaces\n\\cite{Rossi92}.\n\n\\item The best-fit of the phenomenological model proposed in\n\\cite{Carter11} is essentially indistinguishable from that of the\nquasi-local correlation model (long-dashed curve). The best-fit\nvalues for $\\overline{\\ell}$ and $V_\\mathrm{rms}$ are consistent with\nthe average patch size and rms\nvoltage obtained from the best-fit parameters of the quasi-local\nmodel.\n\\end{enumerate}\n\nAt this point, we also want to comment on the validity requirement\n(\\ref{validity}), which allows one to calculate the patch effect in\nthe sphere-plane geometry within the PFA. This requirement ensures\nthat the effective area of interaction between the sphere and the\nplane, of the order of $\\pi RD$ for a sphere of radius $R$, contains\na large number of elementary patch areas, so that the sum over the\nmicro-realization of patches on a given plate is a good effective\ndescription of the statistical ensemble-average given by the power\nspectral density. With the numbers in \\cite{Decca07}, that is a\nradius of curvature of the sphere $R=151.3\\mu$m and a shortest\ndistance $D_\\mathrm{min}=160$nm, the interaction area is $\\pi RD\n\\approx 76 (\\mu\\text{m})^2$. \nMeanwhile, the average patch area is\n$(\\pi\/4)\\overline{\\ell^2} \\approx 1.6 (\\mu\\text{m})^2$, \nthere is a large number of elementary patch areas ($\\approx 48$) within\nthe effective area of interaction, but it is possible that one could expect \na small correction to the patch pressure at short distances when the ergodic \nhypothesis begins to break down.\n\nFor completeness we have also studied the residual $\\delta P^{\\rm\nplasma}(D)$, as defined in Eq. (\\ref{residuals}), with the exception\nthat we have compute the plane-plane Casimir pressure \n$P_{pp}^{\\rm plasma}(D)$ using the ``plasma model\", instead of the Drude model. \nMore precisely, we have computed the pressure using for the\npermittivity $\\varepsilon(i \\xi)$ the\n``generalized plasma model\":\n\n\\begin{equation}\n\\label{generalizedplasma} \n\\varepsilon^{\\rm g. plasma}(i \\xi) = 1+ \\frac{\\Omega^2_P}{\\xi^2} + \\sum_{j=1}^6 \\frac{f_j}{\\omega_j^2 + \\xi g_j\n+ \\xi^2} ,\n\\end{equation}\nwhere the first two terms correspond to the permittivity for the\nplasma model for conduction electrons (dissipation of conduction electrons is\nset to zero ad hoc without physical justification), and the second sum of\nterms accounts for the interband transitions of gold \\cite{parameters}. \nTo account for roughness corrections to the Casimir pressure we use\nthe same additive scheme employed above. Computing $\\delta P^{\\rm\nplasma}(D)$ in this way, we have confirmed\nthe findings of \\cite{Decca05,Decca07}, namely that a negligible\ncontribution of the patch effect leads to an agreement of data with\ntheoretical predictions using the plasma model. We note, however,\nthat the patch pressure calculated from the quasi-local model, with\nsizes and voltages used in \\cite{Decca05}, is much larger than the\ndifference between the measurements and the plasma prediction, as\nshown in Fig. \\ref{RicardoDataPlasma} \\cite{footnote2}. We think\nthat this result constitutes a {\\it serious warning} against the claims\naccording to which the plasma model would be confirmed with a high\nconfidence level by Casimir experiments performed with real metals\n\\cite{Klimchitskaya09}.\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=3.3in]{RicardoPlasmaComparison}\n\\caption{ Comparison of the residual $\\delta{P}^{\\rm plasma}$ (points with error bars at $67 \\%$ confidence taken from Fig. 2 of \\cite{Decca07}) with\npatch pressures given by the sharp-cutoff model and the quasi-local\nmodel: 1. The solid curve is the result of the sharp-cutoff model\n(with assumptions of subsection II-A).\nWe find, consistently with the analysis in \\cite{Decca05,Decca07},\nthat the patch pressure from this model gives a negligible\ncontribution to the measured signal.\n2. The dotted curve corresponds to the quasi-local patch correlation\nmodel adopting the same parameters used in \\cite{Decca05,Decca07}\nfor the patch size and the rms voltage.\nIn distinction to the sharp-cutoff model, we find that the\nquasi-local model gives a large signal as compared to the residual\n$\\delta{P}^{\\rm plasma}$. } \\label{RicardoDataPlasma}\n\\end{figure}\n\n\\subsection{Data analysis for the Yale experiment}\n\nIn addition to analyzing the IUPUI experiment we now apply the same\nmodels to the recent experiment by the Yale group \\cite{Sushkov11}.\nA patch analysis was already carried out in \\cite{Sushkov11} and it\nled to a good agreement between experimental data and the Drude\nmodel. This analysis only considered the asymptotic form $\\propto\n1\/D$ of the plane-plane energy due to patches (\\ref{largepatches}).\nHere we extend the analysis by using the more general expression\n(\\ref{pressure}) for the patch pressure with the quasi-local patch\ncorrelation function described in subsection II-B. Because we have\nno information regarding grain or patch sizes in the Yale\nexperiment, we will focus our attention on best-fit estimations of\nthe parameters $\\ell^{\\rm max}_{\\rm patch}$, $\\ell^{\\rm min}_{\\rm\npatch}$, and $V_{\\rm rms}$ characterizing the quasi-local patch\ncorrelation function (\\ref{quasi-local-correlation},\\ref{SM}).\n\nTo analyze Yale experimental data we first compute the Casimir force \nusing tabulated optical data extrapolated to low frequencies with the Drude\nmodel using the plasma frequency $\\Omega_P = 7.54$ eV and \nthe dissipation rate $\\gamma = 0.052$ eV employed in \\cite{Sushkov11}.\nWe set the temperature to be $T=295$K.\nThe roughness correction to the Casimir force is ignored as it gives a negligible\ncorrection to the force at the distances considered in the Yale experiment.\nFig.\\ref{SteveFit} shows the difference of the Yale experimental\nforce data and the Casimir force prediction using the Drude model,\n$\\delta F^{\\rm Drude}$ (defined by analogy with (\\ref{residuals})),\ndepicted by points with error bars (we assume no error for the\ntheory). The solid curve shows the resulting patch force for\nparameters arising from a least-squares minimization of the quantity\n$\\delta F^{\\rm Drude} - F^\\mathrm{patch}$ using the quasi-local\npatch correlation model (\\ref{SM}) described in subsection II-B. The\nbest fit parameters are given by $\\ell_\\mathrm{patch} ^\\mathrm{max}\n\\approx 614 \\mu$m, $\\ell_\\mathrm{patch} ^\\mathrm{min} \\approx 566\n\\mu$m, (corresponding with $\\overline{\\ell}= 590 \\mu$m) and\n$V_\\mathrm{rms} \\approx 3.9$ mV. We should point out, however, that\nthe result of the best-fit is essentially insensitive to the\ndetails of the patch power spectrum. Indeed, since the residual $\\delta P^{\\rm Drude}$ in the\nYale experiment has an approximate $1\/D$ power law we can infer\nusing (\\ref{largepatches}) that the typical patch size is much\nlarger than $D$ for the whole range of distances explored in the\nexperiment (0.7 $\\mu$m - 7$\\mu$m). Performing a constrained fitting\nby requiring that $\\ell^{\\rm max}_{\\rm patch}$ be less than some\npredetermined value (e.g. 500 $\\mu$m), yet still satisfying the\nconstraint $\\bar{\\ell} \\gg D$, we were able to verify that a good\nfit can still be achieved over a large range of patch sizes. In\nsummary, we point out the result of our fitting using the more\ndetailed quasi-local patch model confirms the patch treatment in\n\\cite{Sushkov11}.\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=3in]{SteveFit}\n\\caption{ Comparison of the residual $\\delta F^{\\rm Drude}$ between\nthe data of \\cite{Sushkov11} and the computed sphere-plane force\nwith the associated patch pressure $ F^\\mathrm{patch}$. The dots\ncorrespond to $\\delta F^{\\rm Drude}$, with the errors bars including\nonly the experimental error in the force determination. The solid\nline is a best-fit of the patch force within the quasi-local model\nof subsection II-B. The inset shows the corresponding residual\n$\\delta F^{\\rm plasma}$ with the same convention employed in the\nmain figure. } \\label{SteveFit}\n\\end{center}\n\\end{figure}\n\nFinally, we also report for the sake of completeness some\nsupplementary test we performed for comparing the data in\n\\cite{Sushkov11} with the predictions of the plasma model described by Eq.(\\ref{generalizedplasma})\n(see the inset of Fig. \\ref{SteveFit}). The good agreement obtained for the\nDrude model is dramatically degraded. Therefore, we confirm the\nresult obtained in \\cite{Sushkov11} that patches cannot explain the\ndifference between the experimental data and the plasma model in\nYale data.\n\n\n\n\\section{Concluding Remarks}\n\n\nIn this paper, we have analyzed the patch contribution to Casimir\nexperiments with a model featuring quasi-local voltage correlations.\nOur model is derived from well-motivated physical principles and shares\nkey features with experimentally verified patch models used to describe\nion trap heating and cantilever damping \\cite{Dubessy09}.\nThus, for the description of the surfaces used in the experiments discussed in this paper, we\nbelieve that this model is more appropriate than the sharp-cutoff\nmodel which has been used to the same aim in previous publications\n\\cite{Decca05,Decca07}. \n\nDue to the large difference in the patch\npower spectrum, in particular for small wavevectors, the quasi-local\nmodel gives a larger contribution than the sharp-cutoff model. As a\nstriking consequence, when the patch sizes are deduced from the\ngrain sizes (as was done in \\cite{Decca05,Decca07}), the quasi-local\nmodel produces a patch pressure larger than the difference between\nthe experimental data and the Drude (and plasma) Casimir prediction,\nwhereas the sharp-cutoff model produces a negligible patch pressure.\nTherefore, it is important to emphasize that because of the combination\nof: a) the highly model dependent nature of the computed patch pressure \nand b) the potentially large patch contribution to the \nmeasured signal, patches may lead to nonnegligible systematic effects. \nThis necessitates an independent measurement of patch effects in \norder to meet metrological standards for Casimir force measurements. \n\nWe have also used the new quasi-local patch model to fit the difference between experimental data of the \nIUPUI experiment \\cite{Decca05,Decca07} and the theoretical prediction for the Casimir pressure. The latter\nwas computed taking into a) tabulated optical data extrapolated to low frequencies by means of the \nDrude model, and b) roughness effects modeled by a simple additive technique.\nWe have\nfound best-fit parameters for the\naverage patch size and for the rms voltage that are consistent with\na contamination of the metallic surfaces, which is expected to\nenlarge the patch sizes (with respect to grain sizes) and smear the\npatch voltage (with respect to those of a surface of bare\ncrystallites) \\cite{Rossi92}. \nIndeed, surface contamination is expected, and we believe that preferential adsorption \\cite{Rossi92} and saturation of contaminants \nmay be compatible with the observation of reproducible results in experiments repeated several times with different samples \\cite{DeccaPC}.\n\nTaken together, our results constitute a {\\it strong warning} against\nthe previously published claims of an agreement of Casimir\nexperiments with the plasma model, and an elimination of the Drude\nmodel \\cite{Klimchitskaya09}. However, we want to emphasize that\nthey do not constitute yet a proof of agreement of experimental data\nwith the new model. The parameters of the patch model have been\nfitted and it is still possible that the qualitative agreement thus\nobtained is a fortunate output of the fitting procedure rather than\nan explanation of the experimental data. \n\nIn this paper we have focused our attention\non only the IUPUI and Yale experiments, but of course the analysis can be\nrepeated for other Casimir measurements between metallic plates as\nwell \\cite{Chan01,Bressi02,Chen04,Lisanti05,Svetovoy08,Onofrio08,Jourdan09,deMan09,Masuda09,Antonini09}.\n\nA better characterization of the surfaces used in the experiments is\nnow key to reaching firmer conclusions. The patch distributions can\nbe measured with appropriate technologies such as Kelvin probe force\nmicroscopy which can achieve the necessary size and voltage\nresolutions \\cite{Liscio08,Liscio11}. In addition, the study of cold\natoms and cold ions trapped in the vicinity of metallic surfaces\n\\cite{Epstein07} or the role of patch effects in other precision\nmeasurements \\cite{Adelberger09,Everitt11,Reasenberg11} are other\nways for accessing information of interest for our problem. Let us\nrepeat at this point that our new quasi-local model is similar to\nrecent proposals for patch physics used to achieve a better\nunderstanding of atomic and ionic traps \\cite{Dubessy09,Carter11}.\n\nThe challenges of forthcoming studies may be stated as follows.\nFirst, it is important to confirm the hypothesis that the patch\nvoltages show quasi-local correlations, and to better specify the\npower spectrum which quantitatively describes these correlations.\nSecond, it would also be interesting to study how the patch power\nspectrum depends on contamination, in particular, fabrication,\ntreatment, history of the samples, and on temperature. Finally, an\nindependent determination of the patch power spectrum could lead\neither to a confirmation of the best-fit analysis presented in this\npaper or to new questions. This study is important not only for the\ntest of the Casimir effect, a central prediction of quantum field\ntheory, but also for the searches of the hypothetical new\nshort-range forces predicted by unification models\n\\cite{Fischbach98,Adelberger03,Onofrio06,Antoniadis11}.\n\n\\acknowledgments\n\nWe are grateful to Ricardo Decca, Steve Lamoreaux, Alex Sushkov, and\nWoo-Joong Kim for having kindly provided experimental data and\ninformation needed to analyze them, and for many insightful\ndiscussions. We also acknowledge discussions with Astrid Lambrecht,\nAntoine Canaguier-Durand, Giovanni Carugno, Jo\\\"el Chevrier, Thomas Coudreau, Thomas\nEbbesen, Cyriaque Genet, Romain Gu\\'erout, Harald Haakh, Carsten Henkel, Galina Klimchitskaya, Johann\nLussange, Sven de Man, Umar Mohideen, Vladimir Mostepanenko, Roberto Onofrio, Giuseppe Ruoso, Paolo Samori, Signe\nSeidelin, and Clive Speake.\n\nThis work was supported by the US Department of Energy through contract\nDE-AC52-06NA25396 and was partially funded by LANL LDRD program and by\nDARPA\/MTO's Casimir Effect Enhancement program under DOE\/NNSA\nContract DE-AC52-06NA25396. P. A. M. N. thanks CNPq and FAPERJ-CNE\nfor partial financial support. The authors are thankful for the ESF\nResearch Networking Programme CASIMIR (www.casimirnetwork. com) for\nproviding excellent opportunities for discussions on the Casimir\neffect and related topics.\n\n\n\n\\newcommand{\\Review}[1]{{\\em #1}}\n\\newcommand{\\Volume}[1]{\\textbf{#1}}\n\\newcommand{\\Book}[1]{\\textit{#1}}\n\\newcommand{\\Eprint}[1]{\\textsf{#1}}\n\\def\\textit{et al}{\\textit{et al}}\n\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nOpen-source toolkits have played a critical role in the development of speech processing technology \\cite{Young02, sphinx, julius, rwth, Povey}.\nKaldi \\cite{Povey}, for instance, is an established speech recognition framework, which is implemented in {C\\nolinebreak[4]\\hspace{-.05em}\\raisebox{.4ex}{\\tiny\\bf ++}}{} with recipes built on top of Bash, Perl, and Python scripts. Despite being efficient, its use of {C\\nolinebreak[4]\\hspace{-.05em}\\raisebox{.4ex}{\\tiny\\bf ++}}{} can make prototyping of new deep learning methods difficult.\nWith the advent of general-purpose deep learning libraries like TensorFlow \\cite{tensorflow} and PyTorch \\cite{pytorch}, more flexible speech recognition frameworks have quickly appeared, e.g., DeepSpeech \\cite{deepspeech}, RETURNN \\cite{returnn}, PyTorch-Kaldi \\cite{pytorch_kaldi}, Espresso \\cite{espresso}, Lingvo \\cite{lingvo}, Fairseq \\cite{fairseq}, ESPnet \\cite{espnet}, and NeMo \\cite{nemo}.\n\nRecently, task-specific libraries have also been released. Examples are Asteroid \\cite{asteroid} for speech separation, pyannote \\cite{pyannote} and sidekit \\cite{sidekit} for speaker diarization, and s3prl \\cite{superb} for self-supervised speech representations. \nWhile excelling at specific tasks, these frameworks have different coding styles, standards, and programming languages,\nmaking it challenging and time-consuming to migrate from one codebase to another.\nMoreover, their combination in complex speech processing pipelines poses a challenge for interoperability, as connecting different frameworks might be unnatural and their codebases can interact in unpredictable ways. \n\nOur experience suggests that having a \\textit{single}, \\textit{flexible}, \\textit{multi-task} toolkit can significantly speed up the development of speech technologies. \nDue to growing interest in end-to-end spoken dialog systems (e.g., virtual assistants), implementing composite pipelines within an integrated toolkit offers many advantages.\nA single toolkit, for instance, encourages the exploration of transfer learning and joint training techniques across different tasks \\cite{jointraining, Chen2015sep, GaoDDL15, ravanelli_SLT} and enables the creation of fully differentiable graphs where multiple technologies are trained jointly and learn to interact\n\nInspired by this vision, we have developed SpeechBrain\\footnote{The toolkit website can be found at \\url{speechbrain.github.io\/}.}, an all-in-one PyTorch-based toolkit designed to facilitate the development, portability, and ease of use of speech processing technologies. \nThe name \\textit{SpeechBrain} highlights the need for a holistic system capable of performing multiple tasks at once, for example, recognize speech, understanding its content, language, emotions, and speakers. \nOur toolkit is not only intended for speech researchers, but also for the broader machine learning community, enabling users to easily integrate their models into different speech pipelines and compare them with state-of-the-art (SotA) baselines. Our main contributions in this paper are:\n\n\\begin{itemize}\n \\item The presentation of \\textit{SpeechBrain}, with an emphasis on how we designed it to support multiple tasks without sacrificing simplicity, modularity, or flexibility.\n \\item The implementation and experimental validation of both recent and long-established speech processing models with SotA or competitive performance on a variety of tasks (cf. Table \\ref{tab:tasks}).\n\\end{itemize}\nMore broadly, we believe the SpeechBrain toolkit has the potential to significantly accelerate research and innovation in the field of speech processing and deep learning.\n\n\\begin{table}[t!]\n\\caption{List of speech tasks and corpora that are currently supported by SpeechBrain.}\n \\centering\n \\small\n \\begin{tabular}{p{27mm}p{36mm}p{35mm}p{25mm}} \\toprule\n \\textbf{Task} & \\textbf{Description} & \\textbf{Techniques} & \\textbf{Datasets} \\\\ \\midrule\n Speech recognition & \\textit{Speech-to-text.} & CTC \\cite{CTC} \\newline Transducers \\cite{graves_transducer} \\newline CTC+Attention \\cite{hybrid_ctc_att} \\newline Shallow fusion \\cite{shallow_fusion} & LibriSpeech \\cite{librispeech} \\newline Common Voice \\cite{commonvoice:2020} \\newline AISHELL \\cite{aishell_2017} \\newline TIMIT \\cite{timit}\\\\ \\midrule \n Speaker recognition & \\textit{Speaker verification\/ID.} & X-vectors \\cite{xvector} \\newline ECAPA-TDNN \\cite{ecapa} & VoxCeleb1 \\cite{voxceleb} \\newline VoxCeleb2 \\cite{voxceleb2}\\\\\n\\hline\nSpeaker diarization & \\textit{Detect who spoke when.} & Spectral Clustering \\cite{spec_tutorial} \\newline Neural embeddings \\cite{ecapa_diarization} & AMI corpus \\cite{ami-corpus} \\\\\n\\hline\nSpeech enhancement & \\textit{Noisy to clean speech.} & MetricGAN+ \\cite{fu2021metricgan+} \\newline Mimic Loss \\cite{bagchi2018spectral} & VoiceBank \\cite{voicebank} \\newline DNS \\cite{dns_challenge} \\\\\n\\hline\nSpeech separation & \\textit{Separate overlapped speech.} & ConvTasNet\\cite{convtasnet} \\newline DualPath RNNs \\cite{luo2020dualpath} \\newline SepFormer \\cite{sepformer} & WSJ-mix \\cite{deepclustering} \\newline WHAM \\cite{wham} \\newline WHAMR \\cite{whamr} \\newline\nLibriMix \\cite{librimix}\\\\\n\\hline\nSpoken language \\newline understanding & \\textit{Speech to intent\/slots.} & Decoupled \\cite{timers-and-such} \\newline Multistage \\cite{Haghani2018} \\newline Direct \\cite{Serdyuk2018} & TAS \\cite{timers-and-such} \\newline SLURP \\cite{slurp} \\newline FSC \\cite{fluent}\\\\\n\\hline\nMulti-microphone \\newline processing & \\textit{Combining input signals.} & Delay-and-sum \\newline\nMVDR~\\cite{habets2009new} \\newline GEV~\\cite{heymann2016neural} \\newline\nGCC-PHAT~\\cite{KnappCarter} \\newline SRP-PHAT~\\cite{cobos2010modified} \\newline MUSIC~\\cite{schmidt1986multiple} & Dataset-Independent \\\\ \\bottomrule\n\\end{tabular}\n\\label{tab:tasks}\n\\end{table}\n\n\\section{Related Work}\nA few other toolkits support multiple speech tasks.\nOf these, the ones we consider most related to SpeechBrain are Fairseq \\cite{fairseq}, NeMo \\cite{nemo}, and ESPnet \\cite{espnet}.\nFairseq is developed by Facebook to support sequence-to-sequence processing. It includes models such as ConvS2S \\cite{convs2s}, transformers \\cite{transformers}, and wav2vec \\cite{wav2vect}.\nHowever, speech processing encompasses several paradigms outside of sequence-to-sequence modeling.\nSpeechBrain also supports regression tasks (e.g., speech enhancement, separation), classification tasks (e.g., speaker recognition), clustering (e.g., diarization), and even signal processing techniques (e.g., multi-microphone combination). \n\nNeMo is a toolkit for conversational AI developed by NVIDIA, which provides useful neural modules for many speech processing tasks, including speech recognition, speaker diarization, voice-activity detection and text-to-speech. Due to its industrial orientation, NeMo offers efficient ready-to-use models, such as Jasper \\cite{jasper}, QuartzNet \\cite{quartznet}, and Citrinet \\cite{citrinet}.\nSpeechBrain also provides several ready-to-use models, but focuses more heavily on research and education by providing a wide variety of baselines, models, and recipes that users can easily inspect and modify in the experiments.\n\nESPnet, in its current form, is the closest toolkit to SpeechBrain. Both are academically driven and support numerous speech tasks. \nESPnet started as an end-to-end speech recognition library and progressively grew to support different tasks.\nBy contrast, we designed SpeechBrain to address a wide variety of tasks from the outset. This means that combining technologies and developing recipes for new tasks is extremely simple.\n\n\\section{Design Principles}\n\\label{sec:principles}\nBeyond the multi-task vision highlighted in the introduction, we developed SpeechBrain with the following design principles in mind:\n \n\\textbf{Accessibility}: \nSpeechBrain is designed to be easily understandable by a large user base, including early students and practitioners.\nTherefore, we devoted considerable effort to develop intuitive modules that are easy to interconnect with each other.\nOne remarkable peculiarity of SpeechBrain is that it serves educational purposes as well. \nWe thus have written extensive documentation and tutorials with Google Colab to help newcomers become more familiar with speech technologies.\nPrior work has shown code snippets aid in adopting a codebase~\\cite{fairbanks-2006-design-fragments}. Motivated by this, SpeechBrain provides runnable code snippets in docstrings (documenting interaction at the granular level), tutorial notebooks (explaining single topics), and template files (describing full experiments on different tasks).\nTo make our toolkit as accessible as possible, we have released it under a very permissive license (Apache 2.0).\n \n\\textbf{Ease of use}: \nSpeechBrain employs a simple software stack (i.e., Python $\\rightarrow$ PyTorch $\\rightarrow$ SpeechBrain) to avoid dealing with too many levels of abstractions.\nIt is developed on top of PyTorch directly, without an external API. PyTorch-compatible code works in our toolkit without any further modification.\nSpeechBrain has a minimal list of external dependencies that are all installable via PyPI. The installation process simply requires running the command \\colorbox{backcolour}{\\lstinline{pip install speechbrain}} and is done within a few minutes. The code is Pythonic and maximizes the use of PyTorch routines.\n\n\\textbf{Replicability}: SpeechBrain promotes open and transparent science. We trained most of our models with publicly available data. This way, our results can be easily replicated by the community. \nSeveral pre-trained models, which only require a few lines of code to use, are distributed via Hugging Face \\cite{hugging}.\nBesides sharing the code and the trained models, we also share the whole experiment folder, which contains all the needed details (e.g., logs) to reproduce our results. \n\n\\begin{figure}[t!]\n\\begin{floatrow}\n\\ffigbox{%\n\\centering\n\\includegraphics[trim={0cm 0 0 0.0},clip,width=6.0cm]{figures\/train-script.pdf}\n}{%\n \\caption{An overview of a basic training script.}\n \\label{fig:arch}\n}\n\\ffigbox{%\n \\includegraphics[width=6cm]{figures\/brain-class-light.pdf}\n}{%\n \\caption{Illustration of \\lstinline{Brain.fit()}.}\n \\label{fig:brain_illustration}\n}\n\\end{floatrow}\n\\end{figure}\n\n\\section{Architecture}\n\\label{sec:sb_arch}\nFrom an architectural standpoint, SpeechBrain sits in between a library and a framework. Where libraries require users to manage dataflow by calling library-defined functionality, frameworks primarily define a custom lifecycle in which user-defined functionalities are invoked in specific places (\\textit{inversion of control}).\nMost code in SpeechBrain follows a library-style collection of modular and standalone building blocks, including practical routines for data loading, decoding, signal processing, and other convenient utilities.\nHowever the central \\colorbox{backcolour}{\\lstinline{Brain}} class (see \\S~\\ref{sec:brain_api}), uses inversion of control to define a general training loop.\nTherefore, SpeechBrain is most accurately described as a \\textit{toolkit}.\nAs shown in Figure~\\ref{fig:arch}, the code for training a model is contained within a single Python script. Training begins by calling the script with a set of hyperparameters: \\colorbox{backcolour}{\\lstinline{python train.py hparams.yaml}}. These hyperparameters, declared in human-readable YAML format, contain the location of one or more data manifest files using either CSV or JSON formats (see Appendix \\ref{app:arch_det}).\nUnlike many other toolkits, SpeechBrain orchestrates experiments in Python directly, without relying on external Bash scripts. This allows code for data loading, modeling, optimization, and evaluation to interact naturally. Moreover, the training script exposes the computations likely to be changed most frequently (e.g., forward computations, data transformations, etc.), making them easy to access and modify.\nSpeechBrain treats the user's code as a first-class citizen: all PyTorch-compatible code written by the user is treated the same as SpeechBrain code. In the following sub-sections, we explore the anatomy of a training script in more detail.\n\n\n\\subsection{Hyperparameters}\nThe model hyperparameters, in conjunction with the training script, regulate various properties of the pipeline such as model architecture, training, and decoding. \nSpeechBrain relies on an extended version of YAML called \\textit{HyperPyYAML},\nas shown in the following excerpt:\n\n\\begin{lstlisting}[language=json, firstnumber=1, basicstyle=\\footnotesize\\ttfamily, frame=lines, caption=An excerpt of a YAML file for hyperparameter specification.]\ndropout: 0.2\nfeatures: !new:speechbrain.lobes.features.MFCC\n n_mels: 40\n left_frames: 5\n right_frames: 5\n\nmodel: !new:torch.nn.LSTM\n input_size: 440\n hidden_size: 256\n num_layers: 4\n dropout: !ref \n bidirectional: True\n\\end{lstlisting}\n\nHyperPyYAML is not just an ordinary list of hyperparameters, but allows a complex hyperparameter specification that defines objects along with their corresponding arguments. There is always an explicit reference between the hyperparameter declarations and any object using them, making the code more interpretable and simpler to debug. Overriding the contents of the YAML file (e.g., for hyperparameter search) can also be done easily by passing command-line arguments:\n\n\\begin{lstlisting}[language=bash, basicstyle=\\footnotesize\\ttfamily]\n$ python train.py hparams.yaml --learning_rate=0.1 --dropout=0.5\n\\end{lstlisting}\n\nSpeechBrain initializes the classes automatically when reading the YAML file, thus eliminating boilerplate initialization code from the training script. HyperPyYAML is a general tool for specifying hyperparameters. To enable modular reusuability, we have released it as a separate repository on PyPI\\footnote{\\url{github.com\/speechbrain\/HyperPyYAML}}. \n\n\n\\subsection{Data loading}\\label{sec:data_loading}\nSpeechBrain complements standard PyTorch data loading by addressing the typical challenges that occur when working with speech, such as\nhandling variable-length sequences, large datasets, and complex data transformation pipelines.\nOur \\colorbox{backcolour}{\\lstinline{DynamicItemDataset}} inherits from \\colorbox{backcolour}{\\lstinline{torch.utils.data.Dataset}} and \ncreates a dataset-interface based on a data-manifest file.\nThe data-manifest file contains \\textit{static items}, such as filepaths or speaker labels. Then, \\textit{dynamic items} \nprovide transformations based on the existing items (static or dynamic), as shown in the following example:\n\n\\begin{lstlisting}[language=Python, frame=lines, basicstyle=\\footnotesize\\ttfamily, caption={An example of a custom data pipeline.}, label={lst:data_pip}]\n@speechbrain.utils.data_pipeline.takes(\"file_path\")\n@speechbrain.utils.data_pipeline.provides(\"signal\")\ndef audio_pipeline(file_path):\n return speechbrain.dataio.read_audio(file_path)\n\\end{lstlisting}\nThis function takes an audio file path (a static item) and\nreads it as a tensor called \"signal\" (a dynamic item). Any library for reading audio file can be used here, including torch.audio\\footnote{\\url{https:\/\/github.com\/pytorch\/audio}}.\nThe evaluation order of the items is determined by a dependency graph.\nUsers can define operations such as reading and augmenting an audio file, encoding a text label into an integer, basic text processing, etc. \nThe dynamic items are defined in the training script and are thus directly customizable by the users.\nMoreover, by leveraging the PyTorch \\colorbox{backcolour}{\\lstinline{DataLoader}} class, these data pipelines are automatically applied in parallel across different workers. \n\n\\subsection{Batching}\nSpeech sequences for a given dataset typically vary in length and require zero-padding to create equal-length batches. This tends to add some complication during the training process. \nFirst, the length of each sentence within each batch must be tracked so we can later remove zero-padded elements from computations like normalization, statistical pooling, losses, etc.\nAnother issue that arises is how to avoid wasting computational resources processing zero-padded elements.\n \nOne approach to mitigate this issue is to sort data by sequence length before batching, which minimizes zero-padding but sacrifices randomness in the batch creation process. A more sophisticated approach is to apply dynamic batching \\cite{batching,morishita-etal-2017-empirical}, where sentences are clustered by length and sampled within the same cluster, a trade-off between random and sorted batching. This allows the batch size to be dynamically changed according to sentence length, leading to improved efficiency and better management of available GPU memory.\nAll the aforementioned batching strategies are supported by SpeechBrain, allowing users to choose the approach that meets their specific needs.\n\n\\subsection{The \\colorbox{backcolour}{\\lstinline{Brain}} class}\\label{sec:brain_api}\nSpeechBrain implements a general training loop in\nthe \\colorbox{backcolour}{\\lstinline{Brain}} class. The \\colorbox{backcolour}{\\lstinline{Brain.fit()}} method is inspired by similar methods in libraries such as Scikit-learn \\cite{pedregosa2011scikit}, Scipy~\\cite{virtanen2020scipy}, Keras~\\cite{gulli2017deep}, fastai~\\cite{howard2020fastai}, and PyTorch Lightning~\\cite{falcon2019pytorch}. Figure~\\ref{fig:brain_illustration} illustrates the basic components of the \\colorbox{backcolour}{\\lstinline{Brain.fit()}} method. The following is a simple demonstration:\n\n\n\\begin{lstlisting}[language=Python, frame=lines, basicstyle=\\footnotesize\\ttfamily, caption=Training a simple model with SpeechBrain using the Brain class.]\nimport torch, speechbrain\n\nclass SimpleBrain(speechbrain.Brain):\n def compute_forward(self, batch, stage):\n return self.modules.model(batch[\"input\"])\n def compute_objectives(self, predictions, batch, stage):\n return torch.nn.functional.l1_loss(predictions, batch[\"target\"])\n\nmodules = {\"model\": torch.nn.Linear(in_features=10, out_features=10)}\nbrain = SimpleBrain(modules, lambda x: torch.optim.SGD(x, 0.1))\ndata = [{\"input\": rand(10, 10), \"target\": rand(10, 10)}]\nbrain.fit(epoch_counter=range(15), train_set=data)\n\\end{lstlisting}\n\nWith only about ten lines of code, we can train a neural model. Repetitive boilerplate, such as setting \\colorbox{backcolour}{\\lstinline{train()}} and \\colorbox{backcolour}{\\lstinline{eval()}} flags, putting the models on the specified device, and computing gradients are handled by the \\colorbox{backcolour}{\\lstinline{Brain}} class. Users can override any step of the process, allowing the definition of more complicated (e.g., GAN \\cite{gan}) training procedures. The \\colorbox{backcolour}{\\lstinline{Brain}} class also handles validation, learning rate scheduling, and fault-tolerant model checkpointing, so that training can resume where it left off if execution is interrupted (e.g., by preemption on a cluster). Further details about the \\colorbox{backcolour}{\\lstinline{Brain}} API are provided in \\S~\\ref{sec:brain_api_details}.\n\n\\subsection{Other features}\nBeyond the functionalities mentioned in the previous sections, additional features include:\n\n\\textbf{Multi-GPU training}: SpeechBrain supports both \\colorbox{backcolour}{\\lstinline{DataParallel}} and \\colorbox{backcolour}{\\lstinline{DistributedDataParallel}} modules, allowing the use of GPUs on the same and different machines. Automatic mixed-precision can be enabled by setting a single flag to reduce the memory footprint of the models. Moreover, the library supports PyTorch's Just-In-Time (JIT) compiler for native compilation.\n\n\\textbf{Large-scale experiments}:\nSpeechBrain extends WebDataset\\footnote{\\url{https:\/\/github.com\/webdataset\/webdataset}} with on-the-fly dynamic batching and bucketing. This enables efficient batching in sequential shard-based data reading, which is necessary for processing large corpora on network filesystems.\n\n\n\\textbf{On-the-fly feature generation}: Rather than serializing intermediate features to disk, SpeechBrain loads raw waveforms and supports a wide variety of efficient streaming operations for audio processing. Standard features like the Short-Term Fourier Transform (STFT) and Mel-filterbanks are computed at training time, allowing differentiation and waveform-level augmentation \\cite{Park2019}. Many recipes include on-the-fly augmentations such as adding noise, time warping, or feature masking. \n\n\n\\begin{figure}[t!]\n\\CenterFloatBoxes\n\\begin{floatrow}\n\\capbtabbox{%\n \\begin{tabular}{p{18mm}c c c} \\toprule\n \\textbf{Technique} & \\textbf{\\# Params} & \\textbf{Dev} & \\textbf{Test} \\\\ \\midrule\n CTC & 10 M & 12.34 & 14.15\\\\\n Transducer & 10 M & 12.66 & 14.12\\\\\n CTC+Att & 10 M & 12.74 & 13.83\\\\\n \n CTC+Att+SSL & 318 M & \\textbf{7.11} & \\textbf{8.04}\\\\ \\bottomrule\n \\end{tabular}\n \n}{%\n \\caption{Phoneme Error Rate (PER\\%) achieved with SpeechBrain on TIMIT using different speech recognizers.}%\n \\label{tab:timit}\n}\n\\ffigbox{%\n \\includegraphics[trim={0.7cm 0.7cm 1.6cm 1.4cm},clip,scale=0.43]{soa_TIMIT.pdf}\n}{%\n \\caption{Evolution of the SotA performance for TIMIT. Entries marked with * use extra unlabelled data from the Libri-Light dataset. Source: \\url{https:\/\/paperswithcode.com}.}\n \\label{fig:timit_soa}\n}\n\\end{floatrow}\n\\end{figure}\n\n\\section{Results}\n\\label{sec:results}\nThis section describes use cases of SpeechBrain, highlighting the techniques implemented and the corresponding performance. For more details on datasets, models, and experimental settings, please refer to the appendix (\\S~\\ref{sec:details}). \n\n\\subsection{Speech recognition}\nThe toolkit supports common techniques for end-to-end speech recognition with different levels of complexity. The simplest system employs an encoder trained with Connectionist Temporal Classification (CTC) \\cite{graves_ctc}. An alternative model is the Transducer \\cite{graves_transducer}, which augment CTC with an autoregressive component and a prediction network. The toolkit supports attention-based encoder-decoder architectures as well \\cite{hybrid_ctc_att}. In particular, CTC+Att systems rely on an encoder-decoder architecture with an additional CTC loss applied on the top of the encoder. SpeechBrain is designed such that users can easily plug in any encoder and decoder modules into the speech recognition pipeline. For instance, we implemented an effective CRDNN encoder, which combines convolutional, recurrent (e.g., LSTM \\cite{lstm}, GRU \\cite{gru2}, Light GRU \\cite{li_gru}), and fully connected neural networks. As an alternative, users can plug in one of the transformers that we have made available.\nPre-training based on self-supervised learning (SSL) with wav2vec 2.0 \\cite{wav2vect} is supported.\n\nWe also implemented an efficient GPU-based beam search that combines the acoustic and the language information to retrieve the final sequence of words. The training scripts for language models and tokenizers (using SentencePiece \\cite{sentencepiece}) are provided as well.\nIn the following, we report the performance achieved with SpeechBrain recipes on some popular speech benchmarks.\n\n\\subsubsection{TIMIT}\nTIMIT \\cite{timit} is a small speech dataset with expert-labeled phone sequences. Table \\ref{tab:timit} reports the Phone Error Rate (PER) achieved with the aforementioned techniques. All systems use a CRDNN encoder, except for the CTC+Att+SSL one which uses a pre-trained wav2vec 2.0 encoder \\cite{wav2vect}.\nWe report the average performance out of five runs with different random seeds. The standard deviation ranges between 0.15\\% and 0.2\\% in all the models.\n\nCTC and Transducers provide similar results, while the combination of CTC and attention (CTC+Att) reaches the best performance. The results achieved by our best model (PER 13.8\\%) is SotA for TIMIT performance with no extra data.\nA considerable improvement in PER is observed when Light-GRUs \\cite{li_gru} are used instead of GRUs \\cite{gru2} or LSTMs \\cite{lstm} in the CRDNN encoder.\nWe also observe a performance boost when using self-supervised pre-training with the wav2vec model trained on unlabelled data from the Libri-Light dataset (CTC+Att+SSL) \\cite{librilight}. Our result with this Libri-Light self-supervised pre-training (PER of 8.04\\%) slightly outperforms the previous SotA performance with the same pre-training data (PER of 8.30\\%), as shown in Figure~\\ref{fig:timit_soa}.\n\n\\subsubsection{LibriSpeech}\\label{librispeech_section}\nLibriSpeech \\cite{librispeech} is a popular speech recognition benchmark derived from audiobooks.\nTable \\ref{tab:librispeech_asr} reports the results achieved with different SpeechBrain recipes on this dataset. \n\\begin{table}[h!]\n \\centering\n \\caption{Word Error Rate (WER$\\%$) achieved on LibriSpeech with SpeechBrain.}%\n \\begin{tabular}{p{20mm}p{16mm}p{14mm}p{14mm} c c} \\toprule\n \\textbf{Technique} & \\textbf{Encoder} & \\textbf{Decoder} & \\textbf{\\# Params} & \\textbf{\\lstinline{test-clean}} & \\textbf{\\lstinline{test-other}} \\\\ \\midrule\n \n CTC+Att & CRDNN & GRU & 230 M & 2.91 & 8.07 \\\\ \n \n CTC+Att & Transformer & GRU & 161 M & 2.46 & 5.77 \\\\ \n \n \n \\bottomrule \n \\end{tabular} \n \\label{tab:librispeech_asr}\n\\end{table}\n\nOur best system is a transformer \\cite{transformers} combined with a convolutional front-end based on ContextNet \\cite{contextnet}. The autoregressive decoder estimates 5k subword tokens derived from running byte-pair encoding on top of the training transcriptions \\cite{sentencepiece}. A transformer-based LM is trained on the LibriSpeech text corpus and used within the beam search to rescore partial hypotheses. The best WER that we have achieved on the \\lstinline{test-clean} dataset is 2.46\\%. This performance is comparable with the results reached in the literature when using transformers without additional data \\cite{transformers_asr}. As one can note, the LibriSpeech task is almost perfectly solved by modern speech recognizers.\nWe thus focus on more realistic tasks as well, as suggested in some recent works \\cite{szymanski-etal-2020-wer, rethinking}. See the appendix (\\S~\\ref{app:comp_toolkits}) for a more detailed comparison with other toolkits on LibriSpeech and other tasks.\n\n\n\\begin{table}[t!]\n\\begin{floatrow}\n\\capbtabbox{%\n \\begin{tabular}{p{45mm} c} \\toprule\n \\textbf{Technique} & \\textbf{EER(\\%)} \\\\ \\midrule\n VoxCeleb2 baseline \\cite{voxceleb2} & 3.95 \\\\ \n \n Kaldi x-vector \\cite{xvector} & 3.10 \\\\ \n \n ResNET-50 \\cite{FITPUB12224} & 1.19 \\\\\n ECAPA (original paper) \\cite{ecapa} & 0.87 \\\\ \n \n \n \\midrule\n SpeechBrain x-vector + PLDA & 3.20 \\\\\n SpeechBrain ECAPA & 0.81 \\\\ \n \n SpeechBrain ECAPA (vox1+2) & \\textbf{0.69} \\\\ \\bottomrule\n \\end{tabular}\n \n}{%\n \\caption{Equal Error Rate (EER $\\%$) achieved on VoxCeleb1 - Cleaned dataset.}%\n \\label{tab:speaker_verification}\n}\n\\capbtabbox{%\n\\resizebox{.45\\textwidth}{!}{\n \\begin{tabular}{p{33mm}p{10mm}p{10mm}} \\toprule\n \\textbf{Technique} & \\textbf{Known \\# spks} & \\textbf{Estim. \\# spks} \\\\ \\midrule\n\n MCGAN~\\cite{pal21-meta} & 4.49 & 5.38 \\\\\n \n ClusterGAN~\\cite{pal21-meta} & 3.91 & 8.16 \\\\\n \n xvector+MCGAN~\\cite{pal21-meta} & 4.23 & 4.92 \\\\\n \n xvector+ClusterGAN~\\cite{pal21-meta} & 3.60 & \\textbf{2.87} \\\\\n \n VBx (ResNet101) \\cite{landini2020VBX} & --- & 4.58 \\\\\n \n \\midrule\n SpeechBrain ECAPA & \\textbf{2.82} & 3.01 \\\\ \\bottomrule \n \\end{tabular}\n }\n}{%\n \\caption{Diarization Error Rate (DER$\\%$) on the eval set of the AMI corpus.}%\n \\label{tab:ami_der}\n}\n\\end{floatrow}\n\\end{table}\n\n\\subsubsection{Common Voice}\nThe Common Voice corpus \\cite{commonvoice:2020} is a multilingual open-source collection of transcribed speech based on crowdsourcing data collection. \nCommonVoice is challenging due to significant accented speech, hesitations, presence of foreign words, noise, reverberation, and other recording artifacts.\n\nTable \\ref{tab:commonvoice_asr} reports the results obtained on four different languages.\nNo language models are trained for this task. The best results are obtained with a wav2vec 2.0 encoder pre-trained with 100k hours of multilingual data from the VoxPopuli dataset \\cite{wang2021voxpopuli}. Except for English, the best systems use a GRU decoder on the top of the pre-trained transformer.\nCommonVoice is a newer dataset, and there have been relatively few systems evaluated on it. To the best of our knowledge, however, our results are SotA for these languages.\n\n\\begin{table}[h!]\n\\caption{Word Error Rate (WER$\\%$) achieved with Common Voice Corpus 6.1 using SpeechBrain on the English (En), French (Fr), Italian (It), and Kinyarwanda (Kw) subsets.}\n \\centering\n \\begin{tabular}{p{25mm} c c c c c c c c} \\toprule\n \\textbf{Technique} & \\textbf{Encoder} &\\textbf{Decoder} & \\textbf{\\# Params} & \\textbf{En} & \\textbf{Fr} & \\textbf{It} & \\textbf{Kw}\\\\ \\midrule\n CTC+Att & CRDNN & GRU & 148M & 24.89 & 17.70 & 16.61 & 24.27 \\\\\n CTC+SSL & Transformer & - & 320M & \\textbf{15.58} & 14.44 & 10.93 & 23.12 \\\\ \n \n CTC+Att+SSL & Transformer & GRU & 330M & 15.69 & \\textbf{13.34} & \\textbf{9.86} & \\textbf{18.91} \\\\ \n \\bottomrule\n \\end{tabular}\n \\label{tab:commonvoice_asr}\n\\end{table}\n\n\n\n\\subsection{Speaker recognition and diarization}\nSpeechBrain implements the functionalities needed to support speaker recognition and speaker diarization. It supports popular embeddings \nderived from Time Delay Neural Networks (TDNNs) \\cite{Lang+Hinton88,waibel}, such as x-vectors \\cite{xvector} and the recent ECAPA-TDNN embeddings \\cite{ecapa}. \nFurthermore, SpeechBrain provides traditional Probabilistic Linear Discriminant Analysis (PLDA) for speaker discrimination \\cite{Kenny-plda,GarciaRomero2011AnalysisOI}. \n\nTable \\ref{tab:speaker_verification} reports the performance achieved on a speaker verification task with models trained on VoxCeleb2 \\cite{voxceleb2} and tested on VoxCeleb1-clean \\cite{voxceleb}.\nThe best model for speaker embeddings available in SpeechBrain is the ECAPA-TDNN, which matches the performance achieved in the original paper \\cite{ecapa}. This model outperforms both the x-vectors \\cite{xvector} and the ResNet-34 \\cite{FITPUB12224} by a large margin. To the best of our knowledge, the EER reached so far by SpeechBrain on VoxCeleb is the best so far reached by an open-source toolkit. \n\n\n\n\nTable \\ref{tab:ami_der} reports the performance achieved on speaker diarization with the AMI meeting corpus \\cite{ami-corpus} when using the embeddings available in SpeechBrain.\nIn this case, the embeddings are clustered with spectral clustering to assign a relative speaker label to each segment of the recording \\cite{ecapa_diarization}.\nThe results shown are obtained on the official Full-ASR split of the AMI corpus while keeping 0.25 sec of forgiveness collar.\nThe best diarization system available in SpeechBrain outperforms recent approaches based on meta-learning (MCGAN\/ClusterGAN) \\cite{pal21-meta}, and Variational Bayes (VBx) \\cite{landini2020VBX} when the number of speakers is known (e.g., in a meeting). We have also obtained competitive results when the number of speakers is unknown.\n\n\n\\subsection{Speech enhancement and separation}\n\nSpeechBrain supports speech enhancement models with different input features (e.g., spectral and waveform domain) and training losses (e.g., L1, MSE, and STOI). In addition, it supports a variety of more sophisticated multi-model training techniques such as Mimic Loss~\\cite{bagchi2018spectral} and MetricGAN+~\\cite{fu2021metricgan+}.\n\\begin{figure}\n\\CenterFloatBoxes\n\\begin{floatrow}\n\\capbtabbox{%\n \\setlength{\\tabcolsep}{4pt}\n \n \\begin{tabular}{lccc} \\toprule\n \\textbf{Technique} & \\textbf{\\# Params} & \\textbf{PESQ} & \\textbf{COVL} \\\\ \\midrule\n \\begin{tabular}{@{}l@{}}Facebook \\\\ DEMUCS~\\cite{defossez2020real}\\end{tabular} & 60.8 M & 3.07 & 3.63 \\\\\n \\midrule\n \\begin{tabular}{@{}l@{}}SpeechBrain \\\\ Mimic Loss\\end{tabular} & 22.3 M & 3.05 & \\textbf{3.74} \\\\[10pt]\n \\begin{tabular}{@{}l@{}}SpeechBrain \\\\ MetricGAN+\\end{tabular} & 1.9 M & \\textbf{3.15} & 3.62 \\\\\n \\bottomrule\n \\end{tabular}\n}{%\n \\caption{Speech enhancement performance on VoiceBank-DEMAND.}%\n \\label{tab:voicebank-compare}%\n}\n\\ffigbox{%\n \\includegraphics[trim={0.4cm 0.7cm 1.6cm 1.0cm},clip,scale=0.43]{soa_voicebank.pdf}\n}{%\n \\caption{Evolution of the speech enhancement performance (PESQ) for Voicebank-DEMAND.}%\n \\label{fig:voicebank-evolution}%\n}\n\\end{floatrow}\n\\end{figure}\n\nIn Table~\\ref{tab:voicebank-compare} we compare the best enhancement systems available in SpeechBrain against the SotA DEMUCS model~\\cite{defossez2020real} on the Voicebank-DEMAND corpus~\\cite{valentini2017noisy}. The mimic loss system uses a speech recognition model to provide a perceptual loss, achieving SotA performance on the COVL metric. Combining models for different tasks (as done here) is natural to implement in SpeechBrain. We also re-implemented the recently proposed MetricGAN+, which performs speech enhancement with an adversarially trained metric network \\cite{gan}. Figure~\\ref{fig:voicebank-evolution} shows the evolution of the PESQ performance on this corpus over the last few years. The SpeechBrain implementation of MetricGAN+ achieves the SotA PESQ performance when no extra data are used.\n\nSpeechBrain implements popular models for speech separation as well, namely ConvTasnet \\cite{convtasnet} and Dual-path RNN \\cite{luo2020dualpath}. Moreover, it supports the recently proposed SepFormer \\cite{sepformer}, which uses a pipeline of two transformers within a dual-path framework. Table \\ref{tab:separation} reports the results achieved on the standard WSJ0-2mix and WSJ0-3mix datasets \\cite{deepclustering}, which contain mixtures composed of two or three overlapped speakers, respectively. The last row compares performance achieved with dynamic mixing, in which the training data are generated dynamically on-the-fly instead of using a frozen dataset. \n\\begin{figure}\n\\CenterFloatBoxes\n\\begin{floatrow}\n\\capbtabbox{%\n \\begin{tabular}{p{26mm}p{12mm}p{12mm}} \\toprule\n \\textbf{Technique} & \\textbf{2-mix} & \\textbf{3-mix} \\\\ \\midrule\n ConvTasnet & 15.3 & 12.7 \\\\% \\midrule \n DualPath-RNN & 18.8 & 14.7 \\\\% \\midrule \n SepFormer & 20.4 & 17.6 \\\\% \\midrule \n SepFormer+DM & \\textbf{22.3} & \\textbf{19.5}\\\\ \\bottomrule \n \\end{tabular}\n}{%\n \\caption{Scale-invariant signal-to-noise ratio improvement (SI-SNRi) in dB achieved with SpeechBrain on WSJ2mix and WSJ3mix.}\n \\label{tab:separation}\n}\n\\ffigbox{%\n \\includegraphics[trim={0.6cm 0.6cm 1.6cm 1.4cm},clip,scale=0.43]{soawsj2mix.pdf}\n}{%\n \\caption{Evolution of the SotA performance (SI-SNRi) on the wsj2mix dataset. Source: \\\\\\url{https:\/\/paperswithcode.com}.\n }\n \\label{fig:wsj_soa}\n}\n\\end{floatrow}\n\\end{figure}\nAs shown in Figure~\\ref{fig:wsj_soa}, SpeechBrain's SepFormer implementation achieves SotA on both datasets.\n\n\n\n\\section{Limitations and Future Work}\n\nThe current version of SpeechBrain supports many other tasks, including spoken language understanding, keyword spotting, multi-microphone signal processing, and language modeling. The toolkit also supports complex \\cite{complex} and quaternion neural networks~\\cite{qrnn}. Please refer to~\\ref{sec:other_tasks} for further details. \nIt does not currently support text-to-speech, which will be added shortly (pending pull-requests under review). \nIn the future, we plan to support decoding with Finite State Transducers (FSTs) \\cite{fst} and are considering to adopt the FST implementation of the ongoing k2 project \\cite{k2} once stable. \nWe plan to devote further effort to real-time speech processing, which was not the main focus of the first release. Finally, our goal is to add support for additional languages and further expand the set of recipes to open-source datasets not yet available in the toolkit (e.g., TED-LIUM \\cite{tedlium}).\n\n\n\\section{Conclusion}\nThis paper described SpeechBrain, a novel, open-source, all-in-one speech processing toolkit. Our work illustrated the main design principles behind this toolkit and remarked on the design principles that led us to support multiple tasks without sacrificing simplicity, modularity, or flexibility.\nFinally, we showed several use cases where the technology developed in SpeechBrain reaches SotA or competitive performance.\nThe main contribution to the scientific community is the development of a novel toolkit that can significantly accelerate future research in the fields of speech processing and deep learning. SpeechBrain is a coordinated effort towards making speech processing technology accessible, and are eager to see where its rapidly growing community of users takes the project in the future.\n\n\n\n\\begin{ack}\nWe would like to sincerely thank our generous sponsors: Samsung, Dolby, Nvidia, Nuance, ViaDialog. Special thanks to our institutional partners: Mila, LIA (Avignon University), CaMLSys (University of Cambridge), Sherbrooke University, and Bio-ASP (Academia Sinica). \nWe also would like to acknowledge Breandan Considine, Olexa Bilaniuk, Frederic Osterrath, Mirko Bronzi, Anthony Larcher, Ralf Leibold, Salima Mdhaffar, Yannick Est\u00e8ve, Yu Tsao, Abdelmoumene Boumadane for helpful comments and discussions. We would like to express our gratitude to all the pre-release beta-testers and to the whole community that we are building around this project.\nThanks to Compute Canada for providing computational resources and support.\nSpeechBrain was also granted access to the HPC resources of IDRIS under the allocation 2021-AD011012633 made by GENCI.\n\\end{ack}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \n\\label{sec_intro} \nAccretion of gas from the intergalactic medium (IGM) to the interstellar medium (ISM) and large-scale galactic outflows are the two principal components of current theoretical models of galaxy formation and evolution \\citep[]{Springel03,Keres05,Dekel09,Oppenheimer10,Dave11a,Dave11b,Dave12}. Gas accretion is thought to be bimodal: (a) the ``hot mode\" in which infalling gas onto a massive halo ($M_{\\rm h} \\gtrsim 10^{12} M_{\\odot}$) gets shock-heated near the halo virial radius ($R_{\\rm vir}$) to the halo virial temperature ($T_{\\rm vir} \\sim 10^{6}$~K) and (b) the ``cold mode\" in which gas enters well inside $R_{\\rm vir}$ in a less massive halo with $T \\sim 10^{4} - 10^{5}$~K, without any subsequent shock heating \\citep[]{Keres05,Dekel06,Dekel09,Keres09}. Large-scale wind\/outflows, on the other hand, are the primary mechanism by which energy, baryons, and metals are recycled in galaxies and transported to the IGM \\citep[see][for a review]{Veilleux05}. Such mechanical and chemical feedback is essential to regulate star-formation rates (SFRs) and stellar masses ($M_{\\ast}$) in theoretical models \\citep[]{Springel03,Dave11b}. However, it has been very challenging to directly detect these processes and thereby test the theoretical models. \n\n\nThe infall of metal-poor gas and the outflow of metal-enriched gas both take place through the circumgalactic medium (CGM). Understanding the physical conditions and chemical abundances of the CGM, thus, is of utmost importance in the context of how galaxies form and evolve with cosmic time. In particular, the amount of diffuse baryons and metals, their geometrical distributions, thermal\/ionization states, and the kinematics of the gas in the CGM provide important clues on how galaxies acquire their gas and how they recycle it in the form of feedback. Ultraviolet (UV) absorption lines originating from the CGM of a foreground galaxy in the spectrum of a nearby background quasar (QSO) act as sensitive probes of the physical and chemical conditions of the CGM gas. A significant amount of work has been done, in the recent past, to better understand the CGM gas \\citep[e.g.][]{Chen01,Chen09,Prochaska11,Kacprzak08,Kacprzak10a,Kacprzak12a,Tumlinson11sci,Thom12,Nielsen13b,Nielsen13a,Werk13,Stocke13,Mathes14,Werk14,Bordoloi14b,Borthakur15}. \n\n\nThe primary goals of these above studies are to understand the covering fractions of different phases of the CGM gas and their distributions around different galaxy types, i.e., star-forming (SF) and non-SF. For example, in the recent COS-Halos survey it has been seen that the covering fraction of \\mbox{O\\,{\\sc vi}}\\ absorption, with rest-frame equivalent width $W_{r}>$~0.1~\\AA, is significantly higher around SF galaxies as compared to non-SF galaxies \\citep[]{Tumlinson11sci}. The covering fractions of low-ions (e.g. \\MgII, \\mbox{Si\\,{\\sc iii}}) and \\mbox{H\\,{\\sc i}}, on the other hand, do not show any preference on the galaxy types \\citep[see for example Figure~12 of][]{Werk13}. Nonetheless, \\MgII\\ absorbers are known to exhibit a strong dependence on galaxy orientations \\citep[]{Bordoloi11,Kacprzak11,Kacprzak12a,Bouche12,Bordoloi14a}. For example, \\cite{Kacprzak12a} have reported a bimodality in the azimuthal angle distribution of halo gas traced by \\MgII\\ absorption, such that it prefers to exist near the projected major and minor axes around SF galaxies. Non-SF galaxies, on the contrary, do not show any such preference. The kinematics of \\MgII\\ absorbers are further found to depend on galaxy orientation. For example, in a recent work, \\cite{Nielsen15} have shown that face-on star forming galaxies viewed along their projected minor-axis have the largest velocity dispersion, suggesting that some fraction of \\MgII\\ absorbers directly traces bi-conical outflows. Therefore, connecting absorption properties with both star formation rate of host-galaxy and its orientation relative to the QSO sight-line are essential in order to comprehend the complete picture of the mechanisms by which baryons are processed through galaxies, i.e., the baron cycle. \n\n\n\n\nOne of the important findings of the COS-Halos survey is that \\mbox{O\\,{\\sc vi}}\\ is omnipresent in the halos of SF galaxies which harbor a major reservoir of galactic metals \\citep[i.e.][]{Tumlinson11sci}. Outflows from the SF galaxies are thought to be the origin of highly ionized oxygen in the CGM. However, the outflows need not be active at the time of observations. Active outflows that are detected primarily via low-ions (e.g. \\NaI, \\FeII, \\MgII) absorption in the spectra of host-galaxies (i.e., the ``down-the-barrel\" outflows) are also found to be ubiquitous at both high and low redshifts \\citep[]{Shapley03,Veilleux05,Martin05,Rupke05,Rubin10,Rubin14}. Despite the ubiquity of winds, several basic questions that are key to quantifying galaxy feedback remain unanswered: How far do they propagate? What is the baryon mass that are processed through them? What is the mass-flow rate through winds? What are the mass loading factors and kinetic power of the winds? To what degree are metals processed through them? This is essentially because the location of the outflow with respect to host-galaxy is an unknown in ``down-the-barrel\" outflows. Galaxy outflow probed by a background QSO has the advantage of having the minimum distance between host-galaxy and outflowing material. For example, by analyzing a post-starburst outflow from a galaxy at $z \\sim 0.927$ detected in the spectra of QSO PG~1206+459 at an impact parameter of $\\sim68$~kpc, \\cite{Tripp11} have shown that the entrained gas mass could be as large as $\\sim$ few $100\\times10^{8}M_{\\odot}$. Numerous strong and large velocity spread ($\\Delta v \\gtrsim 400$ km~s$^{-1}$) metal absorption lines along with solar to super-solar metallicities in different absorption components suggest that the absorber is tracing an active outflow. Such wide velocity spread \\mbox{O\\,{\\sc vi}}\\ absorbers at low-$z$ are also reported by \\cite{Tumlinson11} and \\cite{Muzahid14}. In both these cases metallicity of the \\mbox{O\\,{\\sc vi}}\\ bearing gas is low (e.g. $\\rm [X\/H] \\sim -1.0$) which suggest that \\mbox{O\\,{\\sc vi}}\\ is probably tracing an ``ancient outflow\" \\citep[]{Ford14} rather than an active wind. Interestingly, strong \\NV\\ is detected in PG~1206+459 which is not present in the latter two systems. We note that the location of the absorbing gas with respect to the host-galaxy's projected major and minor axes are not known for any of these strong \\mbox{O\\,{\\sc vi}}\\ absorbers. \n\n\n\nHere we present a detailed study of an active outflow in the CGM of a star-forming (SFR~$\\sim6.9 M_{\\odot}$ yr$^{-1}$ ), sub-$L_{\\ast}$ ($\\sim 0.5 L_{B}^{\\ast}$), Sbc galaxy at redshift $z = 0.39853$. The CGM of the galaxy produces several low- and high-ionization absorption lines in the $\\it VLT$(UVES) and $\\it HST$(COS, FOS) spectra of the background quasar Q~$0122-003$ at an impact parameter of 163~kpc. The observed inclination angle, $i = 63\\ensuremath{^\\circ}$ and azimuthal angle, $\\Phi = 73\\ensuremath{^\\circ}$ indicates that the QSO sight-line is at ideal location to probe a large-scale bipolar outflow from the host-galaxy \\citep[]{Kacprzak12a,Bordoloi14a,Fox15}. This article is organized as follows: In Section~\\ref{sec_obs} we present observations of the background QSO and the host-galaxy. Analysis of the absorption system, the host-galaxy, and overall photoionization (PI) models, based on the observed total column densities, are presented in Section~\\ref{sec_ana}. Detailed component-by-component ionization models are presented in Appendix~\\ref{app_models} for the interested readers. In Section~\\ref{sec_diss} we discuss the implications of our findings. The results of our study are summarized in Section~\\ref{sec_summ}. Throughout this article, we adopt a flat $\\Lambda$CDM cosmology with $H_{0} = 71$~km~s$^{-1}$ Mpc$^{-1}$, $\\Omega_{\\rm M} = 0.3$, and $\\Omega_{\\Lambda} = 0.7$. Abundances are given in the notation $\\rm [X\/Y] = \\log (X\/Y) - log (X\/Y)_{\\odot}$ with solar abundances taken from \\cite{Asplund09}. Atomic data for various elements are taken from \\citet{Morton03}. All the distances given are proper (physical) distances. \n\n\n\n\n\\section{Observations} \n\\label{sec_obs} \n\n\\subsection{Absorption Data} \n\\subsubsection{$HST\/$\\rm COS} \n\nUV spectra of the background QSO Q~0122--003 were obtained using $HST\/$COS Cycle-21 observations, under program ID: GO-13398 as a part of our ``Multiphase Galaxy Halos\" survey. The properties of COS and its in-flight operations are discussed by \\cite{Osterman11} and \\cite{Green12}. The observations consist of G160M far-UV (FUV) grating integrations with a wavelength coverage of 1410--1780 \\AA\\ at a medium resolution of $R \\sim 20,000$ (FWHM $\\sim$~18 km~s$^{-1}$). The data were retrieved from the $HST$ archive and reduced using the STScI {\\sc calcos} v2.21 pipeline software. The reduced data were flux calibrated. To increase the spectral $S\/N$, individual G160M integrations were aligned and coadded using the IDL code ``$coadd\\_x1d$\" developed by \\citet{Danforth10}\\footnote{http:\/\/casa.colorado.edu\/\u223cdanforth\/science\/cos\/costools.html}. The combined spectrum has a $S\/N \\sim$ 5 -- 10 per resolution element. As the COS FUV spectra are significantly oversampled (i.e. six raw pixels per resolution element), we binned the data by three pixels. This improves $S\/N$ per pixel by a factor of $\\sqrt3$. All our measurements and analyses were performed on the binned data. Continuum normalization was done by fitting the line-free regions with smooth low-order polynomials. \n \n\n\\subsubsection{$VLT\/$\\rm UVES} \n\nThe optical spectrum of Q~0122--003 was obtained with the Ultraviolet and Visible Echelle Spectrograph \\citep[UVES;][]{Dekker00} mounted on the European Southern Observatory Kueyen 8.2-m telescope at the Paranal Observatory during 28--31 July, 2005 under program ID: 075.A-0841. The UVES observations covered 3290--9466 \\AA\\ at spectral resolution of $R\\sim$~45,000 (FHWM $\\sim6.6$ km~s$^{-1}$; 3 pixels per resolution element). The spectrum has the highest $S\/N$ of $\\sim20$ per pixel roughly between 5000--7000 \\AA. The $S\/N$ drops considerably to $\\sim5$ per pixel in both the blue and red ends of the spectrum. We refer the reader to \\citet{Kacprzak11} and Evans (2011) for the full details of data reduction procedure. \n\n\n\n\\subsection{Galaxy Data} \n\nA 2100 second $HST\/$WFPC2 F702W image of the Q~0122--003 field was obtained under program ID: 6619. The reduced and calibrated image was obtained from the WFPC-2 Associations Science Products Pipeline (WASPP). The image has a limiting magnitude of 26, which translates to an $M_B=-14.5$ and $L=0.002L_{\\ast}$ at $z=0.4$. \n\n\nSpectra of seven galaxies in the Q~0122--003 field were obtained using the Keck Echelle Spectrograph and Imager \\citep[ESI;][]{Sheinis02} on 2014 December 13. The mean seeing was $0.8''$ (${\\rm FWHM}$) with clear skies. Exposure times range between 1800--3600 seconds. The slit is $20''$ long and $1''$ wide and we used $2\\times2$ on-chip CCD binning. Binning by two in the spatial directions results in pixel sizes of $0.27-0.34''$ over the echelle orders of interest. The wavelength coverage of ESI is $\\sim$~4000--10,000~{\\AA}, which provides coverages of multiple emission lines such as {\\mbox{O\\,{\\sc ii}}} doublets, $\\rm{H}\\beta$, {\\mbox{O\\,{\\sc iii}}} doublets, $\\rm{H}\\alpha$, [\\NII] doublets with a velocity dispersion of $22$~km~s$^{-1}$~pixel$^{-1}$ when binning by two in the spectral direction (${\\rm FWHM}\\sim90$~km~s$^{-1}$). The spectra were reduced using the standard echelle package in IRAF along with standard calibrations and were vacuum and heliocentric velocity corrected. Spectra were flux calibrated using the standard star Feige34 taken during the night of the observation. We have made no corrections for slit loss or Galactic reddening. \n\n\n\n\n\n\\section{Analysis} \n\\label{sec_ana} \n\n\n\\begin{figure*}\n\\vskip-0.7cm \n\\centerline{\\vbox{\n\\centerline{\\hbox{ \n\\includegraphics[width=0.9\\textwidth,angle=00]{plots\/0.3989.ps} \n}}\n}} \n\\caption{Velocity plots of different transitions covered in the $\\it HST\/$COS and $\\it VLT\/$UVES spectra. \nThe zero velocity corresponds to the redshift of the host-galaxy, i.e. $z_{\\rm gal}$ = 0.39853. The best \nfitting Voigt profiles for high- (low-) ions are shown in smooth red (blue) curves on top of data. \nIn the Ly$\\alpha$\\ and Ly$\\beta$\\ panels the blue and red curves indicate the contributions of low- and \nhigh-ionization gas phases, respectively (Appendix~\\ref{app_NHIestimate}). The magenta \ncurves represent synthetic profiles corresponding to our adopted PI models for the low-ions \n(Appendix~\\ref{app_photlow}), since the COS spectrum is noisy and does not have enough resolution to resolve \nthe narrow lines. Error in the normalized flux is shown in green histogram. The vertical blue dotted lines \nrepresent the line centroids of the \\MgII\\ components. The small red ticks in each panel show the line \ncentroids of the \\mbox{O\\,{\\sc vi}}\\ components. Unrelated absorption are marked by `x' symbols. Note that \n\\mbox{O\\,{\\sc vi}}~$\\lambda$1037 model subtracted spectrum is shown in the \\mbox{C\\,{\\sc ii}}~$\\lambda$1036 panel.} \n\\label{vplot} \n\\end{figure*} \n\n\n\n\\subsection{Absorption Analysis} \n\\label{sec_absana} \n\n\nThe Ly$\\alpha$, Ly$\\beta$\\ and different ionic transitions originating from the $z_{\\rm abs}$\\ = 0.39853 absorber are shown in Figure~\\ref{vplot}. \\MgII\\ absorption, covered by the high resolution UVES spectrum, is detected in three velocity components. We refer to them as L--1 ($v \\sim-10$~ km~s$^{-1}$), L--2 ($v \\sim+180$~ km~s$^{-1}$), and L--3 ($v \\sim+200$~ km~s$^{-1}$). \\mbox{Mg\\,{\\sc i}}\\ absorption is detected only in the two strong \\MgII\\ components at $v \\sim +200$ km~s$^{-1}$\\ (i.e. in L--2 and L--3). A weak \\FeII\\ absorption is also detected in these two components. Using the {\\sc vpfit}\\footnote{http:\/\/www.ast.cam.ac.uk\/ rfc\/vpfit.html} software we fit \\mbox{Mg\\,{\\sc i}}, \\MgII, and \\FeII\\ lines simultaneously assuming the $b$-parameters of magnesium and iron are connected via pure thermal broadening. The Voigt profile fit parameters are summarized in Table~\\ref{tab_lowion}. In the cases of non-detections, we present standard $3\\sigma$ upper limits estimated from the error spectrum. \n \n\n\nAs we use only the G160M grating data, any transitions with rest-frame wavelength below 1010~\\AA\\ and above 1270~\\AA\\ are not covered by the COS spectrum. The detected \\mbox{C\\,{\\sc ii}}~$\\lambda 1036$ and \\NII~$\\lambda 1083$ absorption clearly trace the \\MgII\\ line kinematics seen in the UVES spectrum. However, the component structure at $v \\sim +200$ km~s$^{-1}$\\ is not resolved due to the lower spectral resolution of COS. Note that the \\NII~$\\lambda1083$ absorption at $v \\sim 0$ km~s$^{-1}$\\ is blended with \\mbox{Si\\,{\\sc iii}}~$\\lambda1206$ absorption from another known intervening system at $z_{\\rm abs}$\\ = 0.25647. The \\mbox{Si\\,{\\sc ii}}~$\\lambda1193$ is blended with the Galactic \\AlII~$\\lambda1670$ absorption. The \\mbox{Si\\,{\\sc ii}}~$\\lambda1190$, $\\lambda1260$ and \\mbox{Si\\,{\\sc iii}}~$\\lambda1206$ transitions are too noisy to get reasonable fitting parameters. Moreover, the latter two transitions are partially blended by unrelated absorption. As the component structure is not resolved and the data are noisy, except for the weak \\mbox{C\\,{\\sc ii}}\\ component at $\\sim 0$ km~s$^{-1}$, we do not attempt to fit the low-ions detected in the COS spectrum. Instead, we generate synthetic profiles corresponding to our adopted PI model (Appendix~\\ref{app_photlow}) and compare with the observed data. \n\n\nWe note here that the line spread function (LSF) of the COS spectrograph is not a Gaussian. A characterization of the non-Gaussian LSF for COS can be found in \\citet{Ghavamian09} and \\citet{Kriss11}. For our Voigt profile fit analysis we adopt the latest LSF given by \\citet{Kriss11}. The LSF was obtained by interpolating the LSF tables at the observed central wavelength for each absorption line and was convolved with the model Voigt profile while fitting absorption lines or generating synthetic profiles using {\\sc vpfit}. Since there is no Ly$\\alpha$\\ forest crowding at low-$z$ and the interested absorption lines do not fall on top of any emission line, we do not account for the continuum fitting error in the absorption line fit parameters. \n\n\nThe \\mbox{O\\,{\\sc vi}}~$\\lambda\\lambda1031,1037$ absorption is detected over a velocity spread of $>500$ km~s$^{-1}$. A formal measurement gives $\\Delta v_{90} = 419$ km~s$^{-1}$\\ with an optical depth-weighted absorption redshift of $\\bar z = 0.39909$ \\citep[see e.g. Figure~3 of][]{Muzahid12a}. As discussed in Section~\\ref{sec_diss}, this is among the four largest velocity spread intervening \\mbox{O\\,{\\sc vi}}\\ absorbers at $z< 1$ known to date. A minimum of five components are required to fit the \\mbox{O\\,{\\sc vi}}\\ doublets satisfactorily. We refer to them as H--1 through H--5 from lowest velocities to highest velocities. Note that the \\mbox{O\\,{\\sc vi}}~$\\lambda1037$ absorption in the H--1 and H--2 components is blended with the \\mbox{C\\,{\\sc ii}}~$\\lambda1036$ absorption. The apparent column density profiles do not suggest any hidden saturation in the three redward \\mbox{O\\,{\\sc vi}}\\ components (i.e. H--3, H--4, and H--5) which are not affected by the \\mbox{C\\,{\\sc ii}}\\ contamination. It is intriguing to note that, unlike most low-$z$ \\mbox{O\\,{\\sc vi}}\\ absorbers, strong \\NV\\ is detected in these three components. However, the \\NV\\ profiles appear noisy and the \\NV~$\\lambda1242$ line is partially blended. To obtain reasonable fitting parameters we fit both \\mbox{O\\,{\\sc vi}}\\ and \\NV\\ simultaneously assuming pure thermal broadening. A pure non-thermal assumption also provides similar component column densities. Voigt profile fit parameters for these components are given in Table~\\ref{tab_highion}. In the cases of non-detections we provide standard $3\\sigma$ upper limits. \n\n\n\nIt is apparent from the Figure~\\ref{vplot} that the absorption kinematics of \\MgII\\ and \\mbox{O\\,{\\sc vi}}\\ are significantly different. \\MgII\\ shows two distinct absorption clumps at $v \\sim -10$~km~s$^{-1}$\\ and at $v \\sim+200$~km~s$^{-1}$\\ with $\\Delta v_{90}$ of $\\sim25$ and $\\sim28$ km~s$^{-1}$, respectively. \\mbox{O\\,{\\sc vi}}, on the contrary, shows a contiguous absorption with a $\\Delta v_{90}$ of 419 km~s$^{-1}$. No low-ions are detected in the H--1, H--3, and H--5 components. \\MgII\\ is detected at velocities similar to those of the H--2 and \\linebreak H--4 components. However, the \\mbox{O\\,{\\sc vi}}\\ components show much larger $b$-parameters as compared to the \\MgII\\ components. The data, thus, clearly indicate that the absorbing gas has multiple phases with different densities (and\/or different temperatures) giving rise to different absorption kinematics. \n\n\nHere, we refer the reader to Appendix~\\ref{app_NHIestimate} for a detailed description of how we estimate $N(\\mbox{H\\,{\\sc i}})$ in the different high- and low-ionization absorption components. In brief, we estimate $\\log N(\\mbox{H\\,{\\sc i}})$ in the range 18.3--18.6 and 14.6--15.3 for the low- and high-ionization components, respectively. \n\n\n\\begin{table} \n\\caption{Low-ionization metal line fit parameters.} \n\\begin{tabular}{cccccc} \n\\hline \\hline \nIon & $z_{\\rm abs}$\\ & ID & $b$ (km~s$^{-1}$) & $\\log N$~(\\sqcm) \\\\ \n\\hline \\\\ \n\\MgII\\ & 0.398479$\\pm$0.000003 & L--1 & 9.1$\\pm$0.8 & 12.44$\\pm$0.03 \\\\ \n\\mbox{C\\,{\\sc ii}} & & & 9.1 & 13.90$\\pm$0.14 \\\\ \n\\FeII & & & 9.1 & $<$12.5 \\\\ \n\\CaII & & & 9.1 & $<$11.3 \\\\ \n\\mbox{Mg\\,{\\sc i}}\\ & & & 9.1 & $<$11.2 \\\\ \n\\NaI\\ & & & 9.1 & $<$11.6 \\\\ \n\\mbox{H\\,{\\sc i}}\\ & & & 9.1$^{a}$ & $<$18.6 \\\\ \\\\ \n\\MgII\\ & 0.399384$\\pm$0.000002 & L--2 & 4.8$\\pm$0.5 & 13.24$\\pm$0.08 \\\\ \n\\FeII\\ & & & 3.1$\\pm$0.5 & 12.66$\\pm$0.10 \\\\ \n\\CaII & & & 4.8 & $<$11.2 \\\\ \n\\mbox{Mg\\,{\\sc i}}\\ & & & 4.8$\\pm$0.5 & 11.62$\\pm$0.05 \\\\ \n\\NaI\\ & & & 4.8 & $<$11.6 \\\\ \n\\mbox{H\\,{\\sc i}}\\ & & & 4.8$^{a}$ & $<$18.3 \\\\ \\\\ \n\\MgII\\ & 0.399449$\\pm$0.000003 & L--3 & 6.7$\\pm$0.6 & 13.11$\\pm$0.04 \\\\ \n\\FeII\\ & & & 4.4$\\pm$0.6 & 12.56$\\pm$0.10 \\\\ \n\\CaII & & & 6.7 & $<$11.3 \\\\ \n\\mbox{Mg\\,{\\sc i}}\\ & & & 6.7$\\pm$0.6 & 11.53$\\pm$0.06 \\\\ \n\\NaI & & & 6.7 & $<$11.4 \\\\ \n\\mbox{H\\,{\\sc i}}\\ & & & 6.7$^{a}$ & $<$18.3 \\\\ \\\\ \n\\hline \n\\end{tabular} \n\\label{tab_lowion} \n\\vskip0.2cm \nNotes -- All column density upper limits are quoted at $3\\sigma$ as estimated from the error \nspectrum for the adopted $b$-parameter. $^{a}$The $b$-values are too low for the gas temperature \nwe derived under PI equilibrium conditions (i.e. $T \\sim 10^{4}$~K). However, we found \nthat the $N(\\mbox{H\\,{\\sc i}})$ limits remain unchanged for $b$-parameters of 14--20 km~s$^{-1}$\\ as suggested by \nour PI models in Appendix~\\ref{app_models}. \n\\end{table} \n\n\n\n\\subsection{Galaxy Analysis} \n\\label{sec_galana} \n\nThe $HST\/$WFPC2 F702W image of the Q~0122--003 field is shown in the left panel of Figure~\\ref{fig_gal}. There are seven galaxies with confirmed spectroscopic redshifts (see Table~\\ref{tab:zfield} for details). Galaxy apparent Vega-magnitudes were determined using 1.5$\\sigma$ isophotes from Source Extractor \\citep{Bertin96}. The sky orientation parameters, such as inclination angles ($i$) and azimuthal angles ($\\Phi$), for all these galaxies were measured using GIM2D \\citep{Simard02} following the methods of \\citet{Kacprzak11}. Here we define the azimuthal angle such that at $\\Phi = 0\\ensuremath{^\\circ}$ the quasar line-of-sight lies along the galaxy projected major axis, and at $\\Phi= 90\\ensuremath{^\\circ}$ it lies along the projected minor axis. \n\n\n\nWe use our own fitting program \\citep[FITTER: see][]{Churchill00a}, which computes best fit Gaussian amplitudes, line centers, and widths, in order to obtain emission-line redshifts and line fluxes. The galaxy redshifts as listed in Table~\\ref{tab:zfield} have accuracy ranges from 6 -- 15~km~s$^{-1}$. Here we focus our study on the $z_{\\rm gal}$~$=0.39853$ galaxy and will present detailed analysis on the other galaxies in future works. We show them here to demonstrate the spectroscopic completeness of the field. Note that no other galaxy redshift matches the absorption redshift within $\\Delta v \\gtrsim \\pm 1000$~ km~s$^{-1}$, suggesting that the host-galaxy is isolated. The host-galaxy has an impact parameter $D = 163$~kpc, $i=63\\ensuremath{^\\circ}$ and $\\Phi = 73\\ensuremath{^\\circ}$. \n\n\n\nThe SFR of the host-galaxy is estimated from the H$\\alpha$ luminosity using the relation of \\cite{Hao11}. The measured H$\\alpha$ flux of $2.69\\times10^{-15}$~erg~cm$^{-2}$~s$^{-1}$, corresponding to a luminosity of $1.50\\times10^{42}$~ erg~s$^{-1}$, leads to a SFR of 6.9~$M_{\\odot}$~yr$^{-1}$. Using the effective radius (3.5 kpc) and ellipticity (0.56) obtained from the GIM2D model we derive a star-formation rate density of $\\Sigma_{\\ast}=$~0.4~$M_{\\odot}$~yr$^{-1}$~kpc$^{-2}$. This is a factor of~4 higher than the known $\\Sigma_{\\ast}$ threshold for driving galactic-scale outflows in local starbursts \\citep{Heckman03}. The observed \\mbox{O\\,{\\sc iii}}$\/$H$\\beta$ and \\NII$\/$H$\\alpha$ emission line ratios (i.e. $-0.323$ and $-0.391$ dex respectively) suggest that while most of the emission is due to star formation, some AGN contamination could also be present \\citep[]{Baldwin81}. Thus, strictly speaking, the measured SFR and $\\Sigma_{\\ast}$ should be considered as upper limit. We compute a gas-phase oxygen abundance for the host-galaxy using the N2 relation of \\citet{Pettini04} where 12+${\\rm \\log(O\/H)}$=8.90+0.57$\\times$N2 (N2~$\\equiv$~log(\\NII\/{H$\\alpha$})). \n\n \nThe galaxy halo mass $M_{\\rm\\,h}$ was obtained by halo abundance matching. We used the methods described in \\citet{Churchill13a}. Systematics and uncertainties in the method are elaborated in \\citet{Churchill13b}. The halo mass library is drawn from the Bolshoi $N$-body cosmological simulation \\citep{Klypin11} and is matched to the observed $r$-band luminosity function from the COMBO-17 survey \\citep{Wolf03}. The galaxy $r$-band and $B$-band absolute magnitudes, $M_r$ and $M_B$ (respectively), were determined by $K$-correcting \\citep[e.g.,][]{Kim96} the $HST\/$WFPC2 F702W observed magnitude using the \\citet{Coleman80} spectral energy distributions (SED) from \\citet{Bolzonella00}. The $K$-correction is fully described in \\citet{Nielsen13a}. The galaxy $B$-band luminosity, i.e. $L_B\/L_B^{\\ast} = 0.5$, was calculated by using the linear fit to $M_B^{\\ast}$ with redshift from \\cite{Faber07}. We have no color information for the galaxy. Given the late-type appearance of the galaxy morphology in the WFPC2 image (see the top-right panel of Figure~\\ref{fig_gal}), we adopt the Sbc SED for the $K$-correction. The adopted halo mass is $\\log M_{\\rm\\,h}\/M_{\\odot} = 12.48 \\pm 0.16$. The $K$-correction varies by no more than $0.46$ from an Irr to an E SED, which translates to a halo mass range of $\\log M_{\\rm\\,h}\/M_{\\odot} = 12.4$--$12.7$, respectively, across the range of ``normal'' galaxy SEDs. \n\n\nThe galaxy virial radius, $R_{\\rm vir}$, is obtained using the formalism of \\citet{Bryan98}. We obtained $R_{\\rm vir} = 278_{-32}^{+38}$ kpc, where the uncertainty accounts for the uncertainty in the adopted $M_{\\rm\\,h}$. For the line-of-sight impact parameter, we probe the CGM of this galaxy with respect to the virial radius at the projected location of $D\/R_{\\rm vir} = 0.6$. \n \n\n\n\n\n\\begin{table} \n\\caption{High-ionization metal line fit parameters.} \n\\begin{tabular}{cccccc} \n\\hline \\hline \nIon & $z_{\\rm abs}$\\ & ID & $b$ (km~s$^{-1}$) & $\\log N$~(\\sqcm) \\\\ \n\\hline \\\\ \n\\mbox{O\\,{\\sc vi}} & 0.398153$\\pm$ 0.000023 & H--1 & 28.3$\\pm$ 6.3 & 14.26$\\pm$ 0.09 \\\\ \n\\NV & & & 28.3 & $<$13.7 \\\\ \n\\mbox{S\\,{\\sc iv}} & & & 28.3 & $<$14.0 \\\\ \n\\mbox{H\\,{\\sc i}} & & & 28.3 & $<$14.9 (15.04$\\pm$0.09) \\\\ \\\\ \n\\mbox{O\\,{\\sc vi}} & 0.398481$\\pm$ 0.000026 & H--2 & 35.4$\\pm$10.5 & 14.28$\\pm$ 0.09 \\\\ \n\\NV & & & 35.4 & $<$13.7 \\\\ \n\\mbox{S\\,{\\sc iv}} & & & 35.4 & $<$14.0 \\\\ \n\\mbox{H\\,{\\sc i}} & & & 35.4 & $<$15.2 (15.36$\\pm$0.11) \\\\ \\\\ \n\\mbox{O\\,{\\sc vi}} & 0.398913$\\pm$ 0.000009 & H--3 & 32.5$\\pm$ 3.2 & 14.62$\\pm$ 0.03 \\\\ \n\\NV & & & 34.7$\\pm$ 0.0$^{a}$ & 14.22$\\pm$ 0.08 \\\\ \n\\mbox{S\\,{\\sc iv}} & & & 32.5 & $<$14.1 \\\\ \n\\mbox{Si\\,{\\sc iii}} & & & 32.5 & $<$12.8 \\\\ \n\\mbox{H\\,{\\sc i}} & & & 32.5 & $<$14.8 (14.83$\\pm$0.05) \\\\ \\\\ \n\\mbox{O\\,{\\sc vi}} & 0.399443$\\pm$ 0.000009 & H--4 & 42.4$\\pm$ 3.6 & 14.61$\\pm$ 0.02 \\\\ \n\\NV & & & 45.3$\\pm$ 0.0$^{a}$ & 14.28$\\pm$ 0.06 \\\\ \n\\mbox{S\\,{\\sc iv}} & & & 42.4 & $<$14.2 \\\\ \n\\mbox{H\\,{\\sc i}} & & & 42.4 & $<$15.3 (15.35$\\pm$0.05) \\\\ \\\\ \n\\mbox{O\\,{\\sc vi}} & 0.399964$\\pm$ 0.000010 & H--5 & 32.5$\\pm$ 3.2 & 14.38$\\pm$ 0.03 \\\\ \n\\NV & & & 34.8$\\pm$ 0.0$^{a}$ & 14.12$\\pm$ 0.08 \\\\ \n\\mbox{S\\,{\\sc iv}} & & & 32.5 & $<$14.1 \\\\ \n\\mbox{Si\\,{\\sc iii}} & & & 32.5 & $<$12.8 \\\\ \n\\mbox{H\\,{\\sc i}} & & & 32.5 & $<$14.6 (14.64$\\pm$0.04) \\\\ \\\\ \n\\mbox{C\\,{\\sc iv}}$^{b}$ & (AOD measurements) & & 191 & $>$14.7 \\\\ \n\\hline \n\\end{tabular} \n\\label{tab_highion} \n\\vskip0.2cm \nNotes -- All column density upper limits are quoted at $3\\sigma$ as estimated from the error spectrum \nfor the adopted $b$-parameter. The $N(\\mbox{H\\,{\\sc i}})$ values in the parenthesis are obtained assuming that the \nLy$\\beta$\\ profile is entirely due to the high-ionization absorption components \n(see Appendix~\\ref{app_NHIestimate}) with $b$-parameters tied with $b(\\mbox{O\\,{\\sc vi}})$. $^{a}$$b(\\NV)$ are tied \nwith $b(\\mbox{O\\,{\\sc vi}})$ via thermal broadening. $^{b}$Apparent Optical Depth (AOD) measurements using FOS \nspectrum. \n\\end{table} \n\n\n\n\\begin{figure*} \n\\centerline{\\vbox{\n\\centerline{\\hbox{ \n\\includegraphics[width=0.98\\textwidth]{plots\/q0122.eps}\n}}\n}} \n\\caption{The $HST\/$WFPC2 F702W image of the Q~0122--003 field on the left, with spectroscopic redshifts \nof the various galaxies labelled. The top right is a $5\\times5''$ zoomed in view of the galaxy of \ninterest, at $z_{\\rm gal} = 0.39853$, which is classified as an Sbc galaxy. A part of the galaxy \nspectrum, obtained with the $\\it KECK\/$ESI, covering H$\\alpha$ and \\NII\\ emission lines, is shown \nin the lower right panel. The data are shown in black, a continuum fit is shown in red and the error \nor sigma spectrum is shown in green. \n} \n\\label{fig_gal} \n\\end{figure*} \n\n\\begin{table*} \n\\begin{center} \n\\caption{Redshifts and properties of the galaxies in the Q~0122--003 field.} \n\\begin{tabular}{ccccccccc} \n\\hline \\hline \nGalaxy ID & RA & Dec & $m_{\\rm F702W}$ & $D$ ($''$) & $D$ (kpc) & $z_{\\rm {gal}}$ & $i$ (degrees) & $\\Phi$ (degrees) \\\\ \n\\hline \nQ~0122--003G1 & 01:25:29.29 & $-$00:05:43.13 & 21.5 & 14.44 & 57.7$\\pm$0.3 & 0.25739$\\pm$0.000050 & $85.0_{-1.4}^{+0.0}$ & $ 3.0_{-1.4}^{+1.2}$ \\\\\nQ~0122--003G2 & 01:25:28.21 & $-$00:05:54.64 & 22.2 & 9.66 & 76.5$\\pm$1.3 & 0.95410$\\pm$0.000095 & $60.5_{-26.7}^{+18.2}$ &$ 13.8_{-19.8}^{+45.2}$ \\\\\nQ~0122--003G3 & 01:25:28.26 & $-$00:06:08.20 & 20.4 & 15.18 & 78.9$\\pm$0.3 & 0.37874$\\pm$0.000022 & $68.9_{-1.8}^{+2.0}$ & $ 57.5_{-3.3}^{+2.4}$ \\\\\nQ~0122--003G4 & 01:25:27.91 & $-$00:05:53:17 & 22.3 & 14.39 & 117.2$\\pm$0.4 & 1.07642$\\pm$0.000069 & $85.0_{-6.3}^{+0.0}$ & $ 7.4_{-11.4}^{+12.9}$ \\\\\nQ~0122--003G5 & 01:25:28.27 & $-$00:06:23.03 & 19.5 & 28.59 & 122.2$\\pm$0.8 & 0.28244$\\pm$0.000021 & $84.3_{-0.6}^{+0.7}$ & $84.9_{-0.4}^{+1.0}$ \\\\ \nQ~0122--003G6 & 01:25:27.67 & $-$00:05:31.39 & 19.4 & 30.39 & 163.0$\\pm$0.1 & 0.39853$\\pm$0.000027 & $63.2_{-2.6}^{+1.7}$ & $ 73.4_{-4.6}^{+4.7}$ \\\\\nQ~0122--003G7 & 01:25:30.68 & $-$00:06:32.43 & 20.3 & 45.98 & 239.2$\\pm$0.1 & 0.37923$\\pm$0.000029 & $56.1_{-8.2}^{+10.4}$ & $ 44.7_{-1.9}^{+2.6}$ \\\\ \n\\hline \n\\label{tab:zfield} \n\\end{tabular} \n\\end{center} \n\\end{table*} \n\n\n\n\n\n\n\n\\subsection{Photoionization modeling} \n\\label{sec_avgmodels} \n\nIn order to understand the physical conditions and the chemical abundances in the absorbing gas we run grids of PI models using {\\sc cloudy} \\citep[v13.03, last described by][]{Ferland13}. In these models absorbing gas is assumed to be a plane parallel slab, exposed to the extra-galactic UV background radiation at $z = 0.39$ \\citep[as computed by][]{Haardt12} from one side. Here we do not consider the effect of a galaxy\/stellar radiation field, as it is negligible at this redshift at a large separation ($\\gtrsim$~100 kpc) from bright ($\\sim L_{\\ast}$) galaxies \\citep[e.g.][]{Narayanan10,Werk14}. We note that the host-galaxy for the present system has $L = 0.5 L_{B}^{\\ast}$ with an impact parameter of 163 kpc. In our models, the relative abundances of heavy elements are assumed to be solar as in \\citet{Asplund09}. \n\n\nNote that here we estimate the average ionization conditions and abundances for the high- and low-ionization gas using the total column densities of relevant ions and \\mbox{H\\,{\\sc i}}. Total column density is derived by summing the component column densities presented in Table~\\ref{tab_lowion}~and~\\ref{tab_highion}. We refer the interested readers to Appendix~\\ref{app_models} for a detailed component-by-component ionization modeling for both the high- and low-ionization phases. In Appendix~\\ref{app_models} we have explored both PI and collisional ionization equilibrium (CIE) and non-equilibrium (non-CIE) models for the high-ionization phase. Moreover, we ruled out both the CIE and non-CIE scenarios for the \\mbox{O\\,{\\sc vi}}\\ bearing high-ionization gas using several observational constraints such as: (i) the Ly$\\beta$\\ profile is too narrow to explain the required gas temperature, (ii) the presence of strong \\mbox{C\\,{\\sc iv}}\\ in the FOS spectrum and (iii) the non-detection of \\mbox{S\\,{\\sc iv}}\\ and \\mbox{Si\\,{\\sc iii}}\\ absorption. The average chemical\/ionization conditions that we present in this section are in good agreement with those from component-by-component analyses. The average model parameters thus provide an adequate description of the high- and low-ionization gas phases. \n\n\nAs illustrated in the left panel of Figure~\\ref{fig_models}, the density of the high-ionization phase, derived using the $N(\\NV)$ to $N(\\mbox{O\\,{\\sc vi}})$ ratio, is $-4.15 \\leqslant \\log n_{\\rm H} \\leqslant -4.00$ corresponding to an ionization parameter of $-1.70 \\geqslant \\log U \\geqslant -1.85$. For this given density, a super-solar metallicity, i.e. $\\rm [X\/H] \\gtrsim 0.3$, is required to reproduce the observed column densities of \\mbox{O\\,{\\sc vi}}\\ and \\NV. This PI solution produced considerable amounts of \\mbox{C\\,{\\sc iii}}\\ ($\\log N = 14.9$) and \\mbox{C\\,{\\sc iv}}\\ ($\\log N = 15.3$) but did not produce any other detectable low-ions (e.g. \\mbox{C\\,{\\sc ii}}, \\mbox{Si\\,{\\sc iii}}). \n\n \nPI models for the low-ionization gas are presented in the right panel of Figure~\\ref{fig_models}. The density is constrained to be $-2.75 \\leqslant \\log n_{\\rm H} \\leqslant -2.40$ (i.e. $-3.10~\\geqslant~ \\log~U~\\geqslant~-~3.45$) from the $N(\\mbox{Mg\\,{\\sc i}})$ to $N(\\MgII)$ ratio. A metallicity of \\linebreak $\\rm [X\/H]\\gtrsim-1.4$ is needed to explain the observed column densities of \\mbox{Mg\\,{\\sc i}}\\ and \\MgII\\ simultaneously. Such a solution does not produce a significant amount of \\NV\\ and\/or \\mbox{O\\,{\\sc vi}}. \n\n\nIn Table~\\ref{tab_models} we summarize the PI model parameters. The large total hydrogen column densities (i.e. $\\log N_{\\rm H} > 19.0$) suggest that a significant amount of baryons is associated with both phases. The thicknesses ($L_{\\rm los}$) of the two phases are tens of kpc and differ from each other by only a factor of $\\sim3$. But, the density of the low-ionization phase is $\\sim 30$ times higher than that of the high-ionization phase. Most interestingly, the metallicity of the high-ionization phase is over an order of magnitude higher than the low-ionization gas phase. The data clearly indicate different origins of the high- and low-ionization gas phases. \n\n\n\n\n\n\\begin{figure*} \n\\centerline{\\vbox{\n\\centerline{\\hbox{ \n\\includegraphics[width=0.45\\textwidth,angle=00]{plots\/Htot_phot.ps} \n\\includegraphics[width=0.45\\textwidth,angle=00]{plots\/Ltot_phot.ps} \n}}\n}} \n\\vskip-0.3cm \n\\caption{Average PI models for the high- (left) and low- (right) ionization gas. \n{\\sl Bottom panels:} The estimation of density\/ionization parameters using ionic ratios. \nThe horizontal dashed and dotted lines represent the observed column density ratios and their \n$1\\sigma$ uncertainties, respectively. The shaded regions indicate the density ranges \nover which model column density ratios (magenta curves) match with the observed values. \nThe ionization parameters corresponding to different $n_{\\rm H}$ values are labelled at \nthe top of each panel. \n{\\sl Top panels:} \nMetallicity required to reproduce the observed column densities of high- (\\mbox{O\\,{\\sc vi}}\\ and \\NV) and \nlow- (\\mbox{Mg\\,{\\sc i}}\\ and \\MgII) ions are plotted in thick curves as a function of gas density. The \ndotted curves correspond to $1\\sigma$ uncertainties in respective column density measurements. \nThe $N(\\mbox{H\\,{\\sc i}})$ values assumed by the models are mentioned. The horizontal dotted lines represent \nthe average metallicities. Over an order of magnitude difference in metallicity between the \nhigh- and low-ionization gas phases is apparent. \n} \n\\label{fig_models} \n\\end{figure*} \n\n\n\n\\begin{table} \n\\begin{center} \n\\caption{Summary of PI model parameters.} \n\\begin{tabular}{cccrcc} \n\\hline \\hline \nPhase & $\\log n_{\\rm H}$ & $\\log U$ & $>\\rm [X\/H]$ & $>\\log N_{\\rm H}$ & $>L_{\\rm los}$ \\\\ \n & (cm$^{-3}$) & & & (cm$^{-2}$) & (kpc) \\\\ \n\\hline \n~\\\\ \nHigh & $-4.1$ & $-1.75$ & 0.3 & 19.1 & 53 \\\\ \nLow & $-2.6$ & $-3.25$ & $-1.4$ & 20.2 & 19 \\\\ \n~\\\\ \n\\hline \n\\label{tab_models} \n\\end{tabular} \n\\end{center} \n\\end{table} \n\n\n\n\n\n\\section{Discussion} \n\\label{sec_diss} \n\n\\subsection{The strong and large velocity spread {\\rm \\mbox{O\\,{\\sc vi}}} absorption} \n\nWe have presented a detailed analysis of an ultra-strong intervening \\mbox{O\\,{\\sc vi}}\\ absorber at $z_{\\rm abs}$\\ = 0.39853 detected in the $HST\/$COS spectrum of Q~0122--003. The absorber exhibits $\\log N(\\mbox{O\\,{\\sc vi}}) = 15.16\\pm0.04$ with a kinematic spread of $\\Delta v_{90} = 419$~ km~s$^{-1}$. Such a strong (i.e. $\\log N > 15.0$) and large velocity spread (i.e. $>400$~ km~s$^{-1}$) intervening \\mbox{O\\,{\\sc vi}}\\ systems are very rare in the Milky Way halo \\citep[]{Wakker03,Savage03}, in high-velocity clouds \\citep[HVCs;][]{Sembach03}, in the CGM \\citep[]{Werk14} and in the IGM \\citep[]{Tripp08,Muzahid12a,Savage14}. For example, only $2\/42$ systems in the COS-Halos sample show $\\log N(\\mbox{O\\,{\\sc vi}}) > 15.0$ \\citep[]{Tumlinson11sci}. In a blind survey of intergalactic \\mbox{O\\,{\\sc vi}}\\ absorbers at high-$z$ ($1.9 < z < 3.1$) \\citet{Muzahid12a} have found only $3\/85$ systems with $\\Delta v_{90} > 200$~ km~s$^{-1}$\\ with none of them having $\\Delta v_{90} > 400$~ km~s$^{-1}$. The systems with large $\\Delta v_{90}$, in their sample, are thought to be associated with outflows from Lyman Break Galaxies \\citep[LBGs; see also][]{Adelberger03,Adelberger05}. The only known \\mbox{O\\,{\\sc vi}}\\ absorbers at low-$z$ with $\\Delta v_{90}$ in excess of 400 km~s$^{-1}$\\ are the $z_{\\rm abs}$\\ = 0.3558 system towards J~1009+0713 \\citep[]{Tumlinson11}, the $z_{\\rm abs}$\\ = 0.927 system towards PG~1206+459 \\citep[]{Fox13} and the $z_{\\rm abs}$\\ = 0.227 systems towards the QSO pair Q~0107--025A and Q~0107--025B \\citep[]{Muzahid14}. \\citet{Tripp11} have argued that a post-starburst outflow is responsible for the wide velocity spread absorption seen towards PG~1206+459 \\citep[but see][for an alternate explanation]{Ding03b}. \\cite{Tumlinson11} have suggested that \\mbox{O\\,{\\sc vi}}\\ arises from interfaces between ``cool\" and ``hot\" gas phases. Moreover, \\cite{Muzahid14} has conjectured that the large velocity spread \\mbox{O\\,{\\sc vi}}\\ absorption seen through the pair of lines-of-sight is originating from an ``ancient outflow\" \\citep[see e.g.][]{Ford14}. \n\n\nRecently, in 15 of 20 Lyman limit systems (LLS) with $\\log N(\\mbox{H\\,{\\sc i}}) > 17.3$ at $2 < z < 3.5$, \\cite{Lehner14} have reported presence of strong (e.g. average $\\log N(\\mbox{O\\,{\\sc vi}}) = 14.9\\pm0.3$) and wide velocity spread i.e. $200 < \\Delta v < 400$ km~s$^{-1}$\\ \\mbox{O\\,{\\sc vi}}\\ absorption in the ISM\/CGM of high-$z$ galaxies and hypothesized that these strong CGM \\mbox{O\\,{\\sc vi}}\\ absorbers probe outflows of star-forming galaxies \\citep[see also][]{Fox07b}. We note that, a vast majority of local starburst galaxies show ultra-strong ``down-the-barrel\" \\mbox{O\\,{\\sc vi}}\\ absorption with an average column density of $\\log N(\\mbox{O\\,{\\sc vi}}) = 15.0\\pm0.3$ \\citep[]{Grimes09}. Additionally, these \\mbox{O\\,{\\sc vi}}\\ absorption lines are broad with a FHWM of $\\sim 367 \\pm 95$ km~s$^{-1}$. \n\n\nIn summary, intervening \\mbox{O\\,{\\sc vi}}\\ absorbers with $\\log N > 15.0$ and $\\Delta v_{90} > 400$ km~s$^{-1}$, as observed in the present system, are extremely rare. Large-scale galaxy outflows are thought to be the most likely origin of such \\mbox{O\\,{\\sc vi}}\\ absorbers whenever detected. The frequent occurrence of strong and large velocity spread \\mbox{O\\,{\\sc vi}}\\ absorption in local starburst galaxies further strengthens this idea. \n \n\n\\subsection{The presence of strong {\\rm \\NV} absorption} \n\nUnlike most low-$z$ intervening \\mbox{O\\,{\\sc vi}}\\ absorbers, the present system shows an ultra-strong ($\\log N(\\NV)=$~ $14.69\\pm0.07$) and wide-spread ($\\Delta v_{90} = 285$ km~s$^{-1}$) \\NV\\ absorption with $\\log N(\\NV)\/N(\\mbox{O\\,{\\sc vi}})$~ $= -0.47 \\pm 0.08$. As the creation and destruction ionization potentials of \\NV\\ are 77.5~eV and 97.9~eV respectively, a hard radiation field at energies above 4~Ryd is required to ionize nitrogen to its {$\\rm N^{+4}$} ionization state. Though no systematic survey exists, weak \\NV\\ absorption ($<10^{14}$ \\sqcm) is detected in only a handful of \\mbox{O\\,{\\sc vi}}\\ systems at low-$z$ (e.g. $z_{\\rm abs}$\\ = 0.20260 system towards PKS~0312--77 \\citep[]{Lehner09}, $z_{\\rm abs}$\\ = 0.16716 system towards PKS~0405--123 \\citep[]{Savage10}, $z_{\\rm abs}$\\ = 0.227 towards Q~0107--025A \\citep[]{Muzahid14}). In all of them \\mbox{O\\,{\\sc vi}}\\ is found to be fairly strong ($>10^{14.7}$ \\sqcm). Here, we note that \\cite{Werk13} have not found any \\NV\\ absorption with $\\log N >14$ in the COS-Halos sample. \n\n\nTo our knowledge, there are only 6 other intervening systems that show $\\log N(\\NV) > 14.0$ and only one of them is at $z<1$. These are the $z_{\\rm abs}$\\ = 0.927 system towards PG~1206+459 \\citep{Tripp11}, $z_{\\rm abs}$\\ = 2.811 system towards Q~0528--250 \\citep{Fox07b}, $z_{\\rm abs}$\\ = 2.83437 system towards J~1343+5721 and $z_{\\rm abs}$\\ = 2.47958 system towards Q~1603+3820 and $z_{\\rm abs}$\\ = 2.18076 system towards Q~1217+499 \\citep{Lehner14}, and the $z_{\\rm abs}$\\ = 1.5965 system towards PKS~0237--23 \\citep{Fechner09}. Interestingly, all these systems show strong \\mbox{O\\,{\\sc vi}}\\ ($\\log N > 14.8$) absorption kinematically spread over $>200$~ km~s$^{-1}$\\ similar to the present system\\footnote{Note that \\mbox{O\\,{\\sc vi}}\\ information is not available for the $z_{\\rm abs}$\\ = 1.5965 system towards PKS~0237--23. However it shows a strong \\mbox{C\\,{\\sc iv}}\\ absorption spread over $\\sim300$ km~s$^{-1}$. The strong correlation between the line spreads of \\mbox{C\\,{\\sc iv}}\\ and \\mbox{O\\,{\\sc vi}}\\ absorption found by \\cite{Muzahid12a} suggests that this system should have a strong and wide-spread \\mbox{O\\,{\\sc vi}}\\ absorption.}, and all of them are LLS. \n\n\nA population of photoionized intervening \\NV\\ absorbers, with $d{\\mathcal N}\/dz\\sim 0.9$ down to $\\log N = 12.7$, at high-$z$ ($1.5 < z < 2.5$) is reported by \\citet{Fechner09}. These absorbers are predominantly weak and only one out of 21 systems shows $\\log N(\\NV) > 14.0$. As the intensity of the extra-galactic UV background radiation is considerably higher as compared to low-$z$ due to the enhanced QSO activity at $z\\sim2$ \\citep[]{Haardt96,Khaire15}, detection of weak \\NV\\ absorption at high-$z$ is consistent with expectations for PI models. However, the presence of \\NV\\ with $\\log N > 14.0$ is very rare at any redshift. \n\n\n \nApart from the present system, the only other \\mbox{O\\,{\\sc vi}}\\ absorber with $\\log N(\\NV) > 14.0$ at $z < 1.0$ is the well known system at $z_{\\rm abs}$\\ = 0.927 towards PG~1206+459 \\citep{Churchill99b,Ding03b,Tripp11}. These two systems have remarkable resemblance in the absorption properties of their high-ions and show $\\log N(\\NV)\/N(\\mbox{O\\,{\\sc vi}}) \\sim -0.5$ (Rosenwasser et al., in preparation). Unlike \\cite{Tripp11}, who favored a collisional ionization scenario for \\NV\\ and could not estimate a high-ionization phase metallicity, Rosenwasser et al. have found that both \\mbox{O\\,{\\sc vi}}\\ and \\NV\\ can be explained under PI equilibrium with solar to super-solar metallicities and with a density of $\\sim 10^{-4}$ cm$^{-3}$. As we discuss next, all these properties are strikingly similar to what we deduce for the present system. \n\n\n\nVery strong \\NV\\ absorption is characteristic of many intrinsic narrow and mini-broad absorption line systems, that are known to be related to the AGN winds\/host-galaxy because of a velocity within $\\sim$~5000 km~s$^{-1}$\\ of the QSO redshift, and evidence of partial coverage of the QSO continuum\/broad emission line source \\citep[]{Hamann00,Srianand02,Misawa07,Wu10}. For example, \\cite{Wu10} reported three $2.6 < z < 3.0$ intrinsic \\NV\\ absorbers with metallicities greater than 10 times the solar value. Strong \\mbox{O\\,{\\sc vi}}\\ and \\mbox{C\\,{\\sc iv}}\\ absorption are common in these systems as well. The present system is clearly not an intrinsic absorber, but the common properties are likely evidence of an origin in the high metallicity, central region of a galaxy. Both the material that feeds an AGN wind, and that which is ejected in a starburst outflow share a similar chemical origin in a high metallicity environment \\citep[]{Hamann99}.\n\n\n\n\n\\subsection{Origins of different gas phases} \n\nIn Section~\\ref{sec_avgmodels} we have demonstrated that the \\mbox{O\\,{\\sc vi}}\\ bearing gas in this absorber can be explained with PI equilibrium models that require a density of $\\sim 10^{-4.1}$~cm$^{-3}$, a super-solar metallicity (i.e. $\\rm [X\/H] \\gtrsim 0.3$), and a solar $\\rm [N\/O]$ ratio. The synthesis of N can arise from primary and secondary production via the CNO cycle. Primary production is synthesis of N from C produced in the core of the same star via helium burning and secondary production is the synthesis of N from C and O produced in previous generations of stars. Secondary N enrichment of the ISM occurs well after massive stars have gone Type-II supernovae and seeded the ISM with oxygen. For enrichment from primary synthesis, the $\\rm [N\/O]$ ratio is roughly $-0.6$ for $\\rm [O\/H] \\lesssim -0.3$. For enrichment from secondary synthesis, the $\\rm [N\/O]$ ratio increases with increasing metallicity for $\\rm [O\/H] \\gtrsim -0.3$, such that by $\\rm [O\/H] \\simeq +0.3$, $\\rm [N\/O] \\simeq 0.0$ i.e. the solar value \\citep[e.g. see Figure~9 of][]{Pettini08a}. Therefore, the assumption of a solar $\\rm [N\/O]$ ratio in this system is reasonable. Solar\/super-solar metallicity with solar $\\rm [N\/O]$ ratio suggests that nitrogen in the high-ionization absorbing gas is predominantly produced via the secondary channel. Thus, the high-ionization, low density gas phase is likely to be originating from a region of the galaxy with a prolonged and high star formation rate \\citep[see also][]{Hussain15}. This is consistent with the host-galaxy properties as discussed in the next section. \n\n\nIt is often claimed that strong \\mbox{O\\,{\\sc vi}}\\ absorbers are collisionally ionized \\citep[e.g.][]{Lehner14} as they require unreasonably large sizes under PI equilibrium \\citep[but see][]{Muzahid14}. However, from the observed narrowness of the Ly$\\beta$\\ profile, the non-detection of \\mbox{S\\,{\\sc iv}}\\ absorption, and the presence of strong \\mbox{C\\,{\\sc iv}}\\ absorption in the low-resolution FOS spectrum, we have ruled out the possibility of \\mbox{O\\,{\\sc vi}}\\ bearing gas in this system being collisionally ionized (Appendix~\\ref{app_CIE} and \\ref{app_nCIE}). \n\n\n\nFor the low-ionization phase, both the overall (Section~\\ref{sec_avgmodels}) and the component-by-component (Appendix~\\ref{app_photlow}) approach of PI modeling suggest a metallicity of $\\rm [X\/H] \\gtrsim -2.0$~to~$-1.3$. The high-ionization phase has more than a factor of $\\sim10$ higher metallicity. This is in contrast to the $z_{\\rm abs}$\\ = 0.927 system towards PG~1206+459 for which Rosenwasser et al. (in preparation) have found that the solar\/super-solar metallicity of the low-ionization gas is indeed similar to the high-ionization gas. We note that in the present case there is no one-to-one correspondence between the high- (e.g. \\mbox{O\\,{\\sc vi}}) and low- (e.g. \\MgII) ionization absorption kinematics. But in the case of PG~1206+459 kinematic similarity between \\MgII\\ and \\mbox{O\\,{\\sc vi}}\\ absorption is quite remarkable. Therefore, while for PG~1206+459 both the high- and low-ionization gas phases are consistent with being part of same multiphase outflow \\citep[e.g.][]{Tripp11}, the low-ionization phase in the present system likely has a very different origin from the high-ionization phase. \n\n\nUsing the methods of \\cite{Kacprzak10a}, we measured the rotation curve of the host-galaxy from the H$\\alpha$ emission line, which extends $\\pm 1''$ (corresponding to 3.5 kpc) from the galaxy center. The projected rotation velocity is $\\simeq 200$~km~s$^{-1}$. We then employed the rotation model of \\cite{Steidel02} to examine the range of allowed velocities along the line-of-sight that are consistent with extended galaxy rotation given the galaxy inclination, azimuthal angle, the quasar impact parameter, and the gas scale height, $h_\\nu$. Following \\cite{Kacprzak10a}, we use $h_\\nu=1$ Mpc (for a non lagging halo above the galaxy plane). The range of velocities along the line-of-sight that could be due to extended rotation from the galaxy are $+25 \\leq v \\leq +150$~{km~s$^{-1}$}. The components of the low-ionization gas are at $v ~\\simeq 0$ and $v \\simeq 200$~ {km~s$^{-1}$}, as probed with {\\MgII} absorption. None of the \\MgII\\ components is consistent with co-rotation of the galaxy. The low metallicity and kinematics of this phase perhaps suggest that it stems from recycled gas as seen in numerical simulation of \\cite{Oppenheimer10}. Such an interpretation is also supported by the recent study of ``bimodal\" metallicity distribution of low-ions in low-$z$ LLS by \\cite{Lehner13}. The metal-poor branch, possibly tracing cold accretion streams, in their sample peaks at $\\rm [X\/H] = -1.6$ which is consistent with what we find for this system. \n\n\n\n\\subsection{The host-galaxy} \n\nAt the epoch of our measurement, the host galaxy has a high SFR of $6.9~M_\\odot$ yr$^{-1}$ and a $\\Sigma_{\\ast}$ of $\\sim0.4~M_{\\odot}$ yr$^{-1}$ kpc$^{-2}$, the latter being a factor of $\\sim 4$ above the well known threshold of 0.1 $M_{\\odot}$ yr$^{-1}$ kpc$^{-2}$ for driving galactic-scale winds \\citep[]{Heckman03}. The observed galaxy orientation parameters, i.e. $i = 63\\ensuremath{^\\circ}$ and $\\Phi = 73\\ensuremath{^\\circ}$, as given in Table~\\ref{tab:zfield} suggest that the QSO line-of-sight is at ideal position to pierce one of the bicones of a bi-conical outflow propagating along the minor-axis of the galaxy \\citep[e.g.][]{Bouche12,Kacprzak12a,Kacprzak14,Bordoloi14a,Schroetter15}. The host-galaxy has a metallicity of $\\rm 12+ \\log (O\/H) =$~ $8.68\\pm0.02$, i.e. $\\rm [O\/H] \\sim 0.0$, which is roughly consistent with the metallicity of high-ionization gas phase. All these observational evidences strongly indicate that the high-ionization gas, as probed by the ultra-strong \\mbox{O\\,{\\sc vi}}\\ and \\NV\\ absorption, is tracing a powerful (see the next section), active, metal-enriched, large-scale galactic outflow. \n\n \n\nHere we emphasize that the absorber is unlikely from the ISM of a dwarf galaxy that happens to be coincident on the quasar. \\cite{vanZee06} found that the H~{\\sc ii} regions of dwarf galaxies have sub-solar metallicity and $\\rm [N\/O]$ ratios consistent with primary N synthesis. On the other hand, \\cite{vanZee98} showed that the H~{\\sc ii} regions of massive spiral galaxies have solar to super-solar metallicity and $\\rm [N\/O]$ ratios consistent with secondary N synthesis. As such, we favor the scenario in which the high-ionization absorbing gas was ejected from the host spiral galaxy we have identified at the absorption redshift. \n\n\n\n\\subsection{The outflow properties} \n\nFrom the halo mass ($M_{h} \\sim 10^{12.5} M_{\\odot}$) and the virial radius ($R_{\\rm vir} \\sim 280$~ kpc) of the host-galaxy we estimate a circular velocity of $v_{c} \\sim 220$~ km~s$^{-1}$. Using the relationship between SFR, $v_c$, and the terminal outflow speed ($v_w$) by \\citet{Sharma12} we estimate $v_w \\sim 230$ km~s$^{-1}$. Here we note that $v_c$ is calculated at $R_{\\rm 200}$ in their models whereas we compute $v_c$ at the virial radius. Nevertheless, this does not change our estimated $v_w$ by more that a few percent. An outflow with a constant speed of 230~ km~s$^{-1}$\\ would require 0.7 Gyr time to travel the minimum projected distance of 163 kpc. This is on the order of the sound crossing time for the individual cool ($T\\sim10^{4}$~K) photoionized clouds with size $\\sim 4-10$~ kpc (see Appendix~\\ref{app_photlow}). However, in practice the outflow could have been ejected with a significantly higher speed as compared to the terminal speed we estimate here. In that case the flow time could be significantly lower as compared to the sound crossing time so that the clouds would have enough time to be stable against any mechanical disturbances before they traverse a distance of 163~kpc. \n\n\nUnder the thin-shell approximation the mass of the outflowing gas can be written as: $M_{\\rm out} = 4 \\pi \\mu m_p C_{\\Omega} C_{f} N_{\\rm H} r^{2}$, where $m_p$ is the proton mass, $\\mu = 1.4$ accounts for the mass of helium, $C_{\\Omega}$ $(C_{f})$ is the global (local) covering factor, and $r$ is the thin-shell radius \\citep[e.g.][]{Rupke05}. The global covering factor is related to the wind's opening angle. Here we assume $C_{\\Omega} = 0.4$ as observed in local starbursts \\citep[e.g.][]{Veilleux05}. The local covering factor, on the other hand, is related to the wind's clumpiness. For simplicity we assume it to be unity for the high-ionization, low density, diffuse gas. Using $r=$~163 kpc and $N_{\\rm H} = 10^{19.1}$~ \\sqcm\\ (see Table~\\ref{tab_models}) we estimate $M_{\\rm out} \\sim 2\\times10^{10} M_{\\odot}$ corresponding to an oxygen mass of $M_{\\rm O} \\sim 10^{7}~ M_{\\odot}$. These are consistent with the masses as estimated in the CGM of $\\sim L_{\\ast}$ galaxies at $z \\lesssim 0.2$ by \\cite{Tumlinson11sci}. Using the wind speed of 230 km~s$^{-1}$, we further estimate the mass-flow rate of $\\dot{M}_{\\rm out} \\sim 54~ M_{\\odot}$~ yr$^{-1}$ and the kinetic luminosity of $\\dot{E_{k}} \\sim 9\\times10^{41}$ erg~s$^{-1}$ (see Appendix~\\ref{app_equations}). These values are typical of what have been seen in ``down-the-barrel\" outflows from infrared-luminous starbursts at $z < 0.5$ \\citep[i.e.][]{Rupke05}. \n\n\nTo better understand how the mass outflow rate is related to the SFR of the host galaxy, it is customary to define the mass loading factor, $\\eta \\equiv \\dot{M}_{\\rm out}\/\\rm SFR$, which is a critical ingredient for numerical simulations of structure formation \\citep[e.g.][]{Oppenheimer10,Dave11b}. We estimate a $\\eta$ of $\\sim 8$ for the outflow studied here. In the infrared-luminous starbursts sample of \\cite{Rupke05}, $\\eta$ is found to range from 0.001 to 10 but is $\\sim0.1$ on an average. However, we note that a vast majority of these starburst galaxies have SFR $> 100~M_{\\odot}~ \\rm yr^{-1}$ which is significantly higher as compared to the present galaxy with SFR of $6.9 ~ M_{\\odot} ~ \\rm yr^{-1}$. Therefore, the galaxy in our study is extremely efficient in entraining mass, in the form of outflow, for its SFR. This probably because of the high $\\Sigma_{\\ast}$. However, it might also be possible that the host-galaxy had much higher SFR at the epoch of ejection compared to what we observe today. \n \n\\section{Summary} \n\\label{sec_summ} \n\nWe have examined the physical conditions, chemical abundances, and energetics of a large-scale galactic outflow in the CGM of a star-forming (SFR~$6.9~M_{\\odot}$~yr$^{-1}$), sub-$L_{\\ast}$ ($0.5 L_{B}^{\\ast}$) galaxy at $z = 0.39853$. Using the halo abundance matching technique we estimate a halo mass of $M_{h} \\sim 10^{12.5 \\pm 0.2} M_{\\odot}$ and a virial radius of $R_{\\rm vir} \\sim 280$ kpc for the galaxy. Along with Ly$\\alpha$\\ and Ly$\\beta$, several low- (i.e. \\mbox{Mg\\,{\\sc i}}, \\MgII, \\mbox{C\\,{\\sc ii}}, \\NII, \\mbox{Si\\,{\\sc ii}}, \\mbox{Si\\,{\\sc iii}}) and high- (i.e. \\mbox{C\\,{\\sc iv}}, \\NV, \\mbox{O\\,{\\sc vi}}) ionization metal absorption lines, originating from the CGM of the galaxy, have been detected in the $HST$(COS, FOS) and $VLT$(UVES) spectra of the background quasar Q~0122--003. The QSO is located at an impact parameter of $D = 163$ kpc ($D\/R_{\\rm vir} = 0.6$) with an azimuthal angle of 73$\\ensuremath{^\\circ}$ with respect to the galaxy major-axis. Our main findings are as follows: \n\n\n\\begin{enumerate}\n \n\\item The low-ionization absorption components trace photoionized gas with a density of $n_{\\rm H} \\sim 10^{-2.6}$~cm$^{-3}$ ($\\log U \\sim -3.2$) and a metallicity of $\\rm [X\/H] \\gtrsim -1.4$. Such a low metallicity gas is consistent with being recycled material in the galaxy halo as predicted in numerical simulations. \n\n\\item The high-ionization gas phase shows ultra-strong \\mbox{O\\,{\\sc vi}}\\ with $\\log N(\\mbox{O\\,{\\sc vi}}) = 15.16\\pm0.04$ kinematically spread over $\\Delta v_{90} = 419$~ km~s$^{-1}$. Such a strong and large velocity spread \\mbox{O\\,{\\sc vi}}\\ absorber is very rare at any redshift. Among the known astrophysical environments in the local universe, only starburst galaxies show such strong \\mbox{O\\,{\\sc vi}}\\ absorption \\citep[]{Grimes09}. \n\n\\item Unlike most intervening \\mbox{O\\,{\\sc vi}}\\ systems, this absorber shows ultra-strong and wide velocity spread \\NV\\ absorption with $\\log N(\\NV) = 14.69\\pm0.07$ and $\\Delta v_{90} = 285$~ km~s$^{-1}$. There is only one system at $z < 1$ \\citep[i.e. $z_{\\rm abs}$\\ = 0.927 towards PG~1206+459,][]{Tripp11} that shows \\NV\\ as strong as this. \n\n\\item It is often claimed that strong \\mbox{O\\,{\\sc vi}}\\ absorbers are collisionally ionized as they commonly require large sizes ($>$ Mpc) under PI equilibrium \\citep[]{Lehner14}. However, for this system our PI models suggest line-of-sight thicknesses for individual components of $\\sim10$~kpc. In fact, based on the $b$-parameter constraints from the Ly$\\beta$\\ profile, the non-detection of \\mbox{S\\,{\\sc iv}}, and the presence of strong \\mbox{C\\,{\\sc iv}}\\ ($W_{r} = 1.7$ \\AA) in the FOS spectrum, we have ruled out CIE and non-CIE models. \n\n\\item The high-ions, \\mbox{O\\,{\\sc vi}}\\ and \\NV, can be well explained as arising in a low-density ($n_{\\rm H} \\sim 10^{-4.2}$~cm$^{-3}$, $\\log U \\sim -1.6$) photoionized gas with super-solar metallicity ($\\rm [X\/H] \\gtrsim 0.3$) and with solar $\\rm [N\/O]$ ratio. The super-solar metallicity with a solar $\\rm [N\/O]$ ratio implies that nitrogen in this system is predominantly produced via secondary synthesis. Thus the high-ionization, low-density gas is presumably stemming from regions of the host-galaxy that have sustained a high star-formation rate for a prolonged period. \n\n\\item The measured $\\Sigma_{\\ast}$ of $\\sim 0.4$ $M_{\\odot}~\\rm yr^{-1}~ \\rm kpc^{-2}$ for the host galaxy is a factor of $\\sim4$ higher than the threshold of $\\sim 0.1~ M_{\\odot}~ \\rm yr^{-1}~ kpc^{-2}$ for driving galactic-scale wind as seen in local starbursts. The azimuthal angle of $73\\ensuremath{^\\circ}$ suggests that the QSO line-of-sight is in ideal position to pierce the bi-conical outflow propagating along the minor-axis of the host-galaxy. \n\n\\item Assuming the high-ionization gas is in outflow, we estimate a outflow mass of $M_{\\rm out} \\sim 2\\times10^{10}~ M_{\\odot}$, mass outflow rate of $\\dot{M}_{\\rm out} \\sim 54~M_{\\odot} \\rm yr^{-1}$, and kinetic luminosity of $\\dot{E}_{k} \\sim 9\\times10^{41}$ erg~s$^{-1}$ using a thin-shell model. These values are consistent with that of ``down-the-barrel\" outflows from infrared-luminous starbursts at $z < 0.5$ \\citep[]{Rupke05}. \n\n\\item We estimate a mass loading factor of $\\eta \\sim 8$. This is among the highest values as seen in infrared-luminous starbursts at low-$z$. However, we note that the SFR of the host-galaxy is significantly lower (by a factor of ten or more) than the starburst galaxies \\citep{Rupke05}. This indicates that the host-galaxy is highly efficient in entraining mass onto the CGM in the form of outflow. \n\n\\end{enumerate} \n\n\\vskip-0.2cm \nFinally, we emphasize that such powerful, large-scale, metal-rich outflows are the essential component by means of which a galaxy imparts sufficient mechanical and chemical feedbacks that regulates star formation and hence evolution of a galaxy. Finding and analyzing more such unique absorption systems is crucial for a comprehensive understanding of galaxy feedback mechanisms and reinforce galaxy evolution theory with useful observational constraints. \n\n\n\\vskip0.4cm \nSupport for this research was provided by NASA through grants HST GO-13398 from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. GGK acknowledges the support of the Australian Research Council through the award of a Future Fellowship (FT140100933). Some of the data presented here were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation. Observations were supported by Swinburne Keck programs 2014A\\_W178E, 2014B\\_W018E, and 2015A\\_W018E. \n\n\n\\vskip0.2cm \n\\noindent \n{\\it Facilities:}~$\\it HST$(COS,~WFPC2),~$\\it KECK$(ESI),~$\\it VLT$(UVES) \n\n\n\\vskip-0.2cm \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{\\label{sec:intro}Introduction}\n\n\\hskip 1.0truecm \nThe experiments with solar, \natmospheric, reactor and accelerator \nneutrinos~\\cite{sol,SKsolaratm,SNO123,KamLAND,K2K}\nhave provided during the last several years \ncompelling evidence for the existence \nof non-trivial\n3-neutrino\nmixing in the weak charged-lepton current \n(see, e.g.,~\\cite{STPNu04}):\n\\begin{equation}\n\\nu_{l \\mathrm{L}} = \\sum_{j=1}^{3} U_{l j} \\, \\nu_{j \\mathrm{L}},~~\nl = e,\\mu,\\tau,\n\\label{3numix}\n\\end{equation}\n\\noindent where \n$\\nu_{lL}$ are the flavour neutrino fields, $\\nu_{j \\mathrm{L}}$ is\nthe field of neutrino $\\nu_j$ having a mass $m_j$ and $U$ is the\nPontecorvo--Maki--Nakagawa--Sakata (PMNS) mixing \nmatrix~\\cite{BPont57}, $U \\equiv \\mbox{$ U_{\\rm PMNS}$}$. \nThe existing data, including the data from the \n$\\mbox{}^3 {\\rm H}$ $\\beta$-decay experiments~\\cite{MoscowH3Mainz} \nimply that the massive neutrinos $\\nu_j$ \nare significantly lighter than the\ncharged leptons and quarks: $m_{j} < 2.3$ eV \n(95\\% C.L.)~\\footnote{More stringent upper limit on $m_j$ follows from the\n constraints on the sum of neutrino masses obtained from\n cosmological\/astrophysical observations, namely, the CMB data of the\n WMAP experiment combined with data from large scale structure\n surveys (2dFGRS, SDSS)~\\cite{WMAPnu}: $\\sum_{j} m_j < (0.7 - 2.0)$\n eV (95\\% C.L.), where we have included a conservative estimate of\n the uncertainty in the upper limit (see, $\\hbox{\\it e.g.}{}$,~\\cite{Hanne03}). }.\n\n The existence of the flavour neutrino mixing, eq.~(\\ref{3numix}),\nimplies that the individual lepton charges, $L_l$, $l =e,\\mu,\\tau$,\nare not conserved (see, $\\hbox{\\it e.g.}{}$,~\\cite{BiPet87}),\nand processes like $\\mu^- \\rightarrow e^- + \\gamma$,\n$\\mu^{-} \\rightarrow e^{-} + e^{+} + e^{-}$, $\\tau^- \\rightarrow e^- +\n\\gamma$, $\\tau^- \\rightarrow \\mu^- + \\gamma$, $\\mu^{-} + (A,Z)\n\\rightarrow e^{-} + (A,Z)$, $\\hbox{\\it etc}{}$. should take place.\nStringent experimental upper limits on\nthe branching ratios and relative cross-sections\nof the indicated $|\\Delta L_l| = 1$ decays and reactions\nhave been obtained\n~\\cite{mega,PDG04,BaBar05} (90\\% C.L.):\n\\begin{equation}\n\\begin{array}{c}\n\\text{BR}(\\mu \\rightarrow e+\\gamma) < 1.2\\times 10^{-11},~~\n\\text{BR}(\\mu \\rightarrow 3e) < 1.2\\times 10^{-12}~,~\\\\[0.24cm]\n\\text{BR}(\\tau \\rightarrow \\mu +\\gamma) < 6.8\\times 10^{-8}~,~~\n\\text{R}(\\mu^{-} + \\text{Ti} \\rightarrow e^{-} + \\text{Ti}) <\n4.3\\times 10^{-12}. \\end{array} \\end{equation}\n\\noindent Future experiments with increased \nsensitivity can reduce the current bounds on $\\text{BR}(\\mu\\rightarrow\ne+\\gamma)$, $\\text{BR}(\\tau\\rightarrow \\mu +\\gamma)$ and on $\\text{R}(\\mu^{-}\n+ (A,Z) \\rightarrow e^{-} + (A,Z))$ by a few orders of magnitude (see,\n$\\hbox{\\it e.g.}{}$,~\\cite{Kuno99}). In the experiment MEG under preparation at \nPSI~\\cite{psi} it is planned to reach a sensitivity to\n\\begin{equation}\n \\text{BR}(\\mu \\rightarrow e+\\gamma)\\sim (10^{-13} - 10^{-14})\\,. \n\\end{equation}\n\\indent In the minimal extension of the Standard Theory\nwith massive neutrinos and neutrino mixing,\nthe rates and cross sections of the \nLFV processes are suppressed by the factor~\\cite{SP76} \n(see also~\\cite{BPP77}) $(m_{j}\/M_W)^4 < 6.7\\times 10^{-43}$, $M_W$ being the\n$W^{\\pm}$ mass, which renders them unobservable.\nIt was shown in ~\\cite{BorzMas86} that\nin SUSY theories with see-saw mechanism of neutrino \nmass generation\n\\footnote{An integral part of the see-saw mechanism are the \nright-handed (heavy) Majorana neutrinos \\cite{Pont67}.}\n\\cite{seesaw} and soft SUSY breaking \nwith universal boundary conditions at a scale $M_X$\nabove the right-handed (RH) Majorana neutrino mass scale $M_R$, $M_{X}>M_R$ \nthe rates and cross sections of the LFV processes can be strongly \nenhanced and can be within the sensitivity \nof presently operating and future planned experiments\n(see also, e.g.,\n\\cite{Hisano96,Iba01,JohnE,Saclay0105,PPY03,PPR3,PPTY03,Eiichi05,PShinYasu05,SPWRTSYasu05}).\nAs is well-known, the see-saw mechanism \nof neutrino mass generation~\\cite{seesaw},\nprovides a very attractive explanation of the\nsmallness of the neutrino masses \nand - through the leptogenesis theory \n\\cite{LeptoG}, of the observed baryon asymmetry \nof the Universe.\n\n One of the basic ingredients of the see-saw mechanism is the matrix of\nneutrino Yukawa couplings, $\\mathbf{Y_{\\nu}}$. Leptogenesis depends\non $\\mathbf{Y_{\\nu}}$ as well. In the large class of SUSY models with\nsee-saw mechanism and SUSY breaking mediated by flavour-universal \nsoft terms at a scale $M_{X}>M_R$ we will consider,\nthe probabilities of LFV processes also depend strongly on\n$\\mathbf{Y_{\\nu}}$ (see, $\\hbox{\\it e.g.}{}$,~\\cite{Iba01,JohnE}). \nThe matrix $\\mathbf{Y_{\\nu}}$ can be expressed \nin terms of the light neutrino and heavy RH neutrino \nmasses, the neutrino mixing matrix $\\mbox{$ U_{\\rm PMNS}$}$, and an \northogonal matrix $\\mathbf{R}$~\\cite{Iba01}.\nObviously, $\\mathbf{Y_{\\nu}}$ \ndepends on the Majorana CP-violation\n(CPV) phases in the PMNS matrix $\\mbox{$ U_{\\rm PMNS}$}$ ~\\cite{BHP80}. \nIn the case of negligible flavour effects in leptogenesis\n\\cite{flavorLG1}, \nsuccessful leptogenesis is possible only if \n$\\mathbf{R}$ is complex (see, e.g., \\cite{LeptoG1}). \nFor $M_R \\ltap 10^{12}$ GeV, \nthe flavour effects in leptogenesis \ncan be substantial \\cite{flavorLG1} \n(see also \\cite{flavorLG2,flavorLG3}). \nDue to the latter, leptogenesis can take place \neven for real $\\mathbf{R}$ \\cite{flavorLG1,PPRi106}, \nbut only if $\\mathbf{R} \\neq {\\bf 1}$. \nIn this case the Dirac and\/or the Majorana \nCPV phases in the PMNS matrix play the role \nof the CPV parameters responsible for \nthe generation of the baryon asymmetry \nof the Universe. Thus, in this case \nthere is a direct link between the \nlow-energy leptonic CP-violation and the \ngeneration of the baryon asymmetry of the \nUniverse \\cite{PPRi106}.\n\n It was shown in~\\cite{PPY03} that if both \nthe light and the heavy Majorana neutrino \nmass spectra are quasi-degenerate (QD),\nthe rates of LFV decays $\\mu\\rightarrow e + \\gamma$, \n$\\tau \\rightarrow e + \\gamma$ and\n$\\tau \\rightarrow \\mu + \\gamma$, \npredicted in the class of SUSY theories \nof interest, can be strongly enhanced by \nthe leptogenesis CP-violating (CPV) parameters\nin the complex matrix $\\mathbf{R}$,\nwith respect to the rates predicted for \nreal $\\mathbf{R} \\neq \\mathbf{1}$ or for\n$\\mathbf{R} = \\mathbf{1}$.\nThe indicated LFV decay rates were also noticed\nin \\cite{PPY03} to depend for complex \n$\\mathbf{R} \\neq \\mathbf{1}$\non the Majorana CPV phases in $\\mbox{$ U_{\\rm PMNS}$}$.\nThis dependence was investigated recently\nin \\cite{PShinYasu05}\nby taking into account the effects of the \nphases in the renormalisation \ngroup (RG) running of the light \n$\\nu$-masses $m_j$ and of the mixing angles in $\\mbox{$ U_{\\rm PMNS}$}$.\nIt was found \\cite{PShinYasu05} that\nthe Majorana phases can affect significantly \nthe predictions for the\n$\\mu\\rightarrow e+\\gamma$ and $\\tau\\rightarrow e+\\gamma$ decay rates.\n\n In the present article we extend the analyses \nperformed in \\cite{PPY03,PShinYasu05} \nto the cases of normal hierarchical and \ninverted hierarchical light neutrino mass spectra. \nWe investigate also in greater detail the case of QD spectrum. \nMore specifically, working in the framework of \nthe class of SUSY theories with see-saw\nmechanism and soft SUSY breaking with flavour-universal \nboundary conditions at a scale $M_X>M_R$,\nwe study in detail the \ndependence of the rates of charged lepton flavour\nviolating (LFV) radiative decays $\\mu\\rightarrow e + \\gamma$, \n$\\tau \\rightarrow e + \\gamma$ and\n$\\tau \\rightarrow \\mu + \\gamma$,\non the Majorana CPV phases in $\\mbox{$ U_{\\rm PMNS}$}$ \nand on the leptogenesis CPV parameters \nin the complex orthogonal matrix $\\mathbf{R}$. \nThe case of quasi-degenerate (QD) in mass heavy \nRH Majorana neutrinos is considered.\nIt is well-known that in the case of \nheavy Majorana neutrinos with QD mass spectrum\n(i.e., negligible splitting between the masses),\nthe rates of LFV radiative decays\nof interest do not depend on the \nmatrix $\\mathbf{R} \\neq 1$ if $\\mathbf{R}$ is \na real matrix (see, e.g., \\cite{PPY03}).\nOur analysis is performed under the condition \nof negligible RG effects for the light neutrino \nmasses $m_j$ and the mixing angles \nand CP-violation phases in $\\mbox{$ U_{\\rm PMNS}$}$.\nThe RG effects in question \n(see, e.g., \\cite{RGrunU,PShinYasu05} \nand the references quoted therein)\nare negligibly small\nin the class of SUSY theories we are considering\nin the case of hierarchical (normal or inverted) \n$\\nu_j$ mass spectrum. \nThe same is valid for QD\n$\\nu_j$ mass spectrum provided the SUSY \nparameter $\\tan\\beta$ is relatively small,\n$\\tan\\beta < 10$, $\\tan\\beta$ being the ratio of the\nvacuum expectation values of the up- and down- type\nHiggs doublet fields in SUSY extensions of the \nStandard Theory. For the three types of \nlight neutrino mass spectrum,\nwe investigate also the predictions\nfor the ratios of the rates of \n$\\mu\\rightarrow e + \\gamma$ and $\\tau \\rightarrow e + \\gamma$,\nand of $\\mu\\rightarrow e + \\gamma$ and\n$\\tau \\rightarrow \\mu + \\gamma$, decays.\nIn a large region of the relevant SUSY parameter space\nthese two ratios are independent of the SUSY parameters \nand are determined completely \nby the neutrino mixing angles,\nMajorana and Dirac CPV phases, leptogenesis\nCPV parameter(s) and, depending on the type of\nthe neutrino mass spectrum - hierarchical or quasi-degenerate,\nby the neutrino mass squared\ndifferences $\\mbox{$ \\Delta m^2_{21}$}$ and $\\mbox{$\\Delta m^2_{31}$}$ or \nthe absolute neutrino mass. A study of the predictions \nfor the two LFV decay rate ratios was performed recently in \nref. \\cite{GIsid05}. In \\cite{GIsid05}, however,\nonly the case of zero leptogenesis CPV \nparameters (i.e., of real matrix $\\mathbf{R}$) \nand of zero Majorana CPV \nphases in $\\mbox{$ U_{\\rm PMNS}$}$ was investigated. \n\\section{\\large{Neutrino Mixing Parameters from Neutrino Oscillation Data}}\n\\hskip 1.0truecm\nWe will use the standard parametrisation of the\nPMNS matrix $\\mbox{$ U_{\\rm PMNS}$}$ (see, $\\hbox{\\it e.g.}{}$,~\\cite{BPP1}): \n\\begin{equation} \\begin{array}{c} \n\\label{eq:Upara}\n\\mbox{$ U_{\\rm PMNS}$} = \\left( \\begin{array}{ccc} \n c_{12} c_{13} & s_{12} c_{13} & s_{13} e^{-i \\delta} \\\\[0.2cm] \n -s_{12} c_{23} - c_{12} s_{23} s_{13} e^{i \\delta} \n & c_{12} c_{23} - s_{12} s_{23} s_{13} e^{i \\delta} & s_{23} c_{13} \n\\\\[0.2cm] \n s_{12} s_{23} - c_{12} c_{23} s_{13} e^{i \\delta} & \n - c_{12} s_{23} - s_{12} c_{23} s_{13} e^{i \\delta} & c_{23} c_{13} \n\\\\ \n \\end{array} \\right) \n{\\rm diag}(1, e^{i \\frac{\\alpha}{2}}, e^{i \\frac{\\beta_M}{2}}) \\, ,\n \\end{array} \\end{equation}\n\\noindent where \n$c_{ij} = \\cos\\theta_{ij}$, $s_{ij} = \\sin\\theta_{ij}$, the angles\n$\\theta_{ij} = [0,\\pi\/2]$, $\\delta = [0,2\\pi]$ is the Dirac\nCP-violating phase and $\\alpha$ and $\\beta_M$ are two Majorana\nCP-violation phases~\\cite{BHP80,SchValle80D81}. One can identify the\nneutrino mass squared difference responsible for solar neutrino\noscillations, $\\mbox{$\\Delta m^2_{\\odot}$}$, with $\\Delta m^2_{21} \\equiv m^2_2 - m^2_1$,\n$\\mbox{$\\Delta m^2_{\\odot}$} = \\Delta m^2_{21} > 0$. The neutrino mass squared difference\ndriving the dominant $\\nu_{\\mu} \\rightarrow \\nu_{\\tau}$\n($\\bar{\\nu}_{\\mu} \\rightarrow \\bar{\\nu}_{\\tau}$) oscillations of\natmospheric $\\nu_{\\mu}$ ($\\bar{\\nu}_{\\mu}$) is then given by\n$|\\mbox{$\\Delta m^2_{\\rm A}$}|=|\\Delta m^2_{31}|\\cong |\\Delta m^2_{32}| \\gg \\Delta m^2_{21}$.\nThe corresponding solar and atmospheric neutrino mixing angles,\n$\\theta_{\\odot}$ and $\\theta_{\\rm A}$, coincide with $\\theta_{12}$ and\n$\\theta_{23}$, respectively. The angle $\\theta_{13}$ is limited by\nthe data from the CHOOZ and Palo Verde experiments~\\cite{CHOOZPV}.\n\n The existing neutrino oscillation data allow us to determine $\\Delta\nm^2_{21}$, $|\\Delta m^2_{31}|$, $\\sin^2\\theta_{12}$ and\n$\\sin^22\\theta_{23}$ with a relatively good precision and to obtain\nrather stringent limits on $\\sin^2\\theta_{13}$ (see,\n$\\hbox{\\it e.g.}{}$,~\\cite{BCGPRKL2,Schwatm05}). The best fit values and the 95\\%\nC.L. allowed ranges of $\\Delta m^2_{21}$, $\\sin^2\\theta_{12}$,\n$|\\Delta m^2_{31}|$ and $\\sin^22\\theta_{23}$ read\n~\\footnote{The data imply, in particular, that \nmaximal solar neutrino mixing is ruled out at $\\sim 6\\sigma$; at 95\\% C.L.\\ \n one finds $\\cos 2\\theta_\\odot \\geq 0.26$~\\cite{BCGPRKL2}, which has\n important implications~\\cite{PPSNO2bb}.}:\n\\begin{equation}\n\\label{bfvsol}\n\\begin{array}{c}\n\\mbox{$ \\Delta m^2_{21}$} = 8.0\\times 10^{-5}~{\\rm eV^2},~~\n\\sin^2\\theta_{21} = 0.31~, \\\\[0.25cm]\n\\mbox{$ \\Delta m^2_{21}$} = (7.3 - 8.5) \\times 10^{-5}~{\\rm eV^2},~~\n\\sin^2 \\theta_{12} = (0.26 - 0.36)~,\n\\end{array}\n\\end{equation}\n\\begin{equation} \n\\label{eq:atmrange}\n\\begin{array}{c}\n|\\mbox{$\\Delta m^2_{31}$}| =2.2\\times 10^{-3}~{\\rm eV^2}~,~~\\sin^22\\theta_{23} = 1.0\n~, \\\\ [0.25cm]\n|\\mbox{$\\Delta m^2_{31}$}| = (1.7 - 2.9)\\times 10^{-3}~{\\rm eV^2}~,~~\n\\sin^22\\theta_{23} \\geq 0.90. \n\\end{array}\n\\end{equation}\n\\noindent\nA combined\n\\footnote{Using the recently\nannounced (but still unpublished) \ndata from the MINOS experiment \\cite{MINOS0306} \nin the analysis leads to somewhat different best fit\nvalue and 95\\% allowed range of $|\\mbox{$\\Delta m^2_{31}$}|$\n\\cite{TSchw0406}:\n$|\\mbox{$\\Delta m^2_{31}$}| = 2.5\\times 10^{-3}~{\\rm eV^2}$\nand $|\\mbox{$\\Delta m^2_{31}$}| = (2.2 - 2.9)\\times 10^{-3}~{\\rm eV^2}$.}\n3-$\\nu$ oscillation analysis of the solar \nneutrino, KL and CHOOZ data gives~\\cite{BCGPRKL2}\n\\begin{equation}\n\\sin^2\\theta_{13} < 0.027~(0.044),~~~~\\mbox{at}~95\\%~(99.73\\%)~{\\rm C.L.}\n\\label{th13}\n\\end{equation}\nThe neutrino oscillation parameters discussed above can (and very\nlikely will) be measured with much higher accuracy in the future (see,\n$\\hbox{\\it e.g.}{}$,~\\cite{STPNu04}).\n\n The sign of $\\mbox{$\\Delta m^2_{\\rm A}$} = \\mbox{$\\Delta m^2_{31}$} $, as it is well known, cannot be\ndetermined from the present (SK atmospheric neutrino and K2K) data.\nThe two possibilities, $\\Delta m^2_{31(32)} > 0$ or $\\Delta\nm^2_{31(32)} < 0$ correspond to two different\ntypes of $\\nu$-mass spectrum:\\\\\n-- {\\it with normal ordering (hierarchy)} \n$m_1 < m_2 < m_3$, $\\mbox{$\\Delta m^2_{\\rm A}$}=\\Delta m^2_{31} >0$, and \\\\\n-- {\\it with inverted ordering (hierarchy)} \n$m_3 < m_1 < m_2$, $\\mbox{$\\Delta m^2_{\\rm A}$} =\\Delta m^2_{32}< 0$. \\\\\n\\noindent Depending on the sign of \\mbox{$\\Delta m^2_{\\rm A}$}, ${\\rm sgn}(\\mbox{$\\Delta m^2_{\\rm A}$})$, and \nthe value of the lightest neutrino mass,\n${\\rm min}(m_j)$, the $\\nu$-mass spectrum can be\\\\\n-- {\\it Normal Hierarchical}: $m_1{\\small \\ll m_2 \\ll }m_3$,\n$m_2{\\small \\cong }(\\mbox{$\\Delta m^2_{\\odot}$})^ {1\\over{2}}{\\small \\sim}$ 0.009 eV,\n$m_3{\\small \\cong }|\\mbox{$\\Delta m^2_{\\rm A}$}|^{1\\over{2}}{\\small \\sim}$ 0.05 eV;\\\\\n-- {\\it Inverted Hierarchical}: $m_3 \\ll m_1 < m_2$,\nwith $m_{1,2} \\cong |\\mbox{$\\Delta m^2_{\\rm A}$}|^{1\\over{2}}\\sim$ 0.05 eV; \\\\\n-- {\\it Quasi-Degenerate (QD)}: $m_1 \\cong m_2 \\cong m_3 \\cong m$,\n$m_j^2 \\gg |\\mbox{$\\Delta m^2_{\\rm A}$}|$, $m \\gtap 0.10$~eV.\n\n The sign of $\\Delta m^2_{31} \\cong \\Delta m^2_{32}$, \nwhich drives the dominant atmospheric neutrino \noscillations, can be \ndetermined by studying \noscillations of neutrinos and\nantineutrinos, say, \n$\\nu_{\\mu} \\rightarrow \\nu_e$\nand $\\bar{\\nu}_{\\mu} \\rightarrow \\bar{\\nu}_e$,\nin which matter effects are sufficiently large.\nThis can be done, e.g., in long-baseline \n$\\nu$-oscillation experiments \n(see, e.g.,~\\cite{AMMS99}).\nInformation about ${\\rm sgn}(\\Delta m^2_{31})$\ncan be obtained also in atmospheric neutrino \nexperiments by studying \nthe oscillations of the\natmospheric $\\nu_{\\mu}$ and $\\bar{\\nu}_{\\mu}$ which\ntraverse the Earth \\cite{JBSP203}.\n\n As is well-known, the theories employing \nthe see-saw mechanism of neutrino mass \ngeneration~\\cite{seesaw} of interest for our \ndiscussion, predict the massive neutrinos \n$\\nu_j$ to be Majorana particles. Determining \nthe nature of massive neutrinos is one of the\nmost formidable and pressing problems in today's neutrino physics\n(see, $\\hbox{\\it e.g.}{}$,~\\cite{STPNu04,APSbb0nu}). \nIf it is established that the massive neutrinos $\\nu_j$ \nare indeed Majorana fermions, getting information about \nthe Majorana CP-violation phases in $\\mbox{$ U_{\\rm PMNS}$}$, \nwould be a very difficult problem.\nThe oscillations of flavour neutrinos, \n$\\nu_{l} \\rightarrow \\nu_{l'}$ and \n$\\bar{\\nu}_{l} \\rightarrow \\bar{\\nu}_{l'}$,\n$l,l'=e,\\mu,\\tau$, are insensitive to the Majorana CP-violation phases\n$\\alpha$ and $\\beta_M$~\\cite{BHP80,Lang87}.\nThe only feasible experiments that at present have the potential of\nestablishing the Majorana nature of light neutrinos $\\nu_j$ and of\nproviding information on the Majorana CP-violation phases in $\\mbox{$ U_{\\rm PMNS}$}$\nare the experiments searching for the neutrinoless double beta\n($\\mbox{$\\beta \\beta_{0 \\nu}$}$-) decay, $(A,Z) \\rightarrow (A,Z+2) + e^- + e^-$ (see,\n$\\hbox{\\it e.g.}{}$,~\\cite{BiPet87,APSbb0nu,STPFocusNu04}). \nThe $\\mbox{$\\beta \\beta_{0 \\nu}$}$-decay effective Majorana mass, $\\mbox{$ \\langle m \\rangle $}$ \n(see, e.g., \\cite{BiPet87}), which contains all the \ndependence of the $\\mbox{$\\beta \\beta_{0 \\nu}$}$-decay amplitude on \nthe neutrino mixing parameters, \nis given by the following expressions \nfor the normal hierarchical (NH), \ninverted hierarchical (IH) \nand quasi-degenerate (QD) neutrino mass spectra\n(see, e.g., \\cite{STPFocusNu04}): \n\\begin{eqnarray}\n\\mbox{$\\left| \\langle m \\rangle\\right|$}&\\cong&\\left|\\sqrt{\\mbox{$ \\Delta m^2_{21}$}}\\sin^2 \\theta_{12}e^{i\\alpha}\n+\\sqrt{\\mbox{$\\Delta m^2_{31}$}}\\sin^2\\theta_{13}\ne^{i\\beta_{M}} \\right|\\;,~m_1\\ll m_2 \\ll m_3~{\\rm (NH)},\n\\label{meffNH2}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\mbox{$\\left| \\langle m \\rangle\\right|$} &\\cong& \\sqrt{\\Delta m^2_{13}}\n\\left|\\cos^2\\theta_{12} + \ne^{i\\alpha}~\\sin^2 \\theta_{12} \\right|\\;,~~m_3 \\ll m_1< m_2~{\\rm (IH)}, \n\\label{meffIH1}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\mbox{$\\left| \\langle m \\rangle\\right|$} &\\cong& m \\left|\\cos^2\\theta_{12}\n + e^{i \\alpha}~\\sin^2 \\theta_{12} \n \\right|\\;,~~m_{1,2,3} \\cong m \\gtap 0.10~{\\rm eV}\\;~{\\rm (QD)}. \n\\label{meffQD0} \n\\end{eqnarray}\nObviously, $\\mbox{$\\left| \\langle m \\rangle\\right|$}$ depends strongly \non the Majorana CP-violation phase(s)\n\\footnote{We assume that the fields of the\nMajorana neutrinos\n$\\nu_j$ satisfy the Majorana conditions:\n$C(\\bar{\\nu}_{1,2})^{T} = \\nu_{1,2}$, and\n$C(\\bar{\\nu}_{3})^{T} = e^{-i 2\\delta} \\nu_{3}$,\nwhere $C$ is the charge conjugation matrix.\nWith the parametrisation we are employing for\n$\\mbox{$ U_{\\rm PMNS}$}$, eq. (\\ref{eq:Upara}), the effective Majorana \nmass $\\mbox{$\\left| \\langle m \\rangle\\right|$}$ does not depend on the Dirac CP-violation\nphase $\\delta$ as a consequence of\nthe presence of the phase factor\n$e^{-i 2\\delta}$ in the Majorana condition\nfor the field $\\nu_{3}$.};\nthe CP-conserving values of \n$(\\alpha - \\beta_M) = 0,\\pm \\pi$\n($\\alpha =0,\\pm\\pi$) \\cite{LW81}, \nin particular, determine the range of \npossible values of $\\mbox{$\\left| \\langle m \\rangle\\right|$}$ in the \ncase of NH (IH, QD) spectrum.\nIf the $\\mbox{$\\beta \\beta_{0 \\nu}$}$-decay is observed, \nthe measurement of the $\\mbox{$\\beta \\beta_{0 \\nu}$}$-decay half-life\ncombined with information on the absolute scale of neutrino masses (or\non ${\\min}(m_j)$), might allow to significantly \nconstrain the Majorana phase\n$\\alpha$ \\cite{BPP1,BGKP96,PPW}, for instance. \n\\section{\\large{The See-Saw Mechanism, Neutrino Yukawa Couplings, \nand LFV Decays $l_i \\to l_j +\\gamma$}}\n\\hskip 1.0truecm In the \nminimal supersymmetric standard model\nwith RH neutrinos $N_j$ and see-saw mechanism of \nneutrino mass generation (MSSMRN) we consider\nit is always possible to choose a basis in which both\nthe matrix of charged lepton Yukawa couplings, \n$\\mathbf{Y_{\\rm E}}$, and the Majorana mass matrix of the \nheavy RH neutrinos, $\\mathbf{M_{\\rm N}}$, are real and diagonal.\nWe will work in that basis and will denote by \n$\\mathbf{D_{\\rm N}}$ the corresponding diagonal \nRH neutrino mass matrix, $\\mathbf{D_{\\rm N}} =\n{\\rm diag}(M_1,M_2,M_3)$, with $M_j > 0$. \nWe will consider in what follows the case of\nQD heavy Majorana neutrinos:\n$M_1 \\cong M_2 \\cong M_3 = M_R$.\nIt will be assumed that \nthe splittings between the masses of the heavy \nMajorana neutrinos are sufficiently small, e.g.,\nthat they are of the order of those considered \nin \\cite{PPY03}. The existence of \nsufficiently small (but nonzero) \nsplittings between the masses \nof the heavy Majorana neutrinos $N_j$ is indeed a \nnecessary condition for the successful\n(resonant) leptogenesis to take place.\nThe requisite small mass splittings\ncan be generated, e.g., by renormalisation group\neffects \\cite{GonzF03}. \nHowever, the mass splittings under discussion,\n$|M_i - M_j| \\ll M_i,M_j$, $i\\neq j =1,2,3$, \ndo not play any significant role in the \npredictions for the rates of the\ndecays $l_i \\rightarrow l_j + \\gamma$,\nwhich is the main subject of the present study.\nThe heavy Majorana neutrino mass $M_R$ \nwill standardly be assumed to be \nsmaller than the GUT scale \n$M_{\\rm GUT} \\simeq 2\\times 10^{16}$ GeV.\n\n In the class of theories of interest,\nthe branching ratio of the $l_i\\to l_j + \\gamma$ decay \nhas the following form (in the ``mass insertion'' \nand leading-log approximations,\nsee, e.g.,~\\cite{Hisano96,JohnE,PPTY03}):\n\\begin{align}\n\\text{BR}(l_i\\to l_j\\gamma)\\cong &\n\\frac{\\Gamma(l_i\\to e\\nu\\bar{\\nu})}{\\Gamma_{\\text{total}}(l_i)}\n\\frac{\\alpha_{\\text{em}}^3}{G_F^2m_S^8}\n\\left|\\frac{(3 + a_0^2)m_0^2}{8\\pi^2}\\right|^2\n\\left|\\sum_k (Y_{\\nu}^{\\dagger})^{ik}~\\ln\\frac{M_{X}}{M_k}~Y_{\\nu}^{kj}\n\\right|^2\\tan^2\\beta\\;,\n\\label{eq_ijg}\n\\end{align}\nwhere $i\\neq j=1,2,3$, $l_1,l_2,l_3\\equiv e,\\mu,\\tau$,\n$m_0$ and $A_0 = a_0m_0$ are the universal SUSY breaking\nscalar masses and trilinear scalar couplings at \n$M_X > M_R$, $m_S$ represents SUSY particle mass (see further),\n$\\tan\\beta$ is the ratio of the \nvacuum expectation values of up-type and down-type\nHiggs fields and $\\mathbf{Y}_{\\nu} = \\mathbf{Y}_{\\nu}(M_R)$ \nis the matrix of neutrino Yukawa couplings evaluated at $M_R$.\nThe matrix $\\mathbf{Y}_{\\nu}$ can be parametrised as~\\cite{Iba01}\n\\begin{align}\n \\mathbf{Y}_{\\nu}(M_R) = \\frac{1}{v_u}\n \\sqrt{\\mathbf{D}_N}~\\mathbf{R}~\n \\sqrt{\\mathbf{D}_{\\nu}}~\\mathbf{U}^{\\dagger}\n\\cong \n\\frac{1}{v_u} \\sqrt{M_R}~\\mathbf{R}~\n \\sqrt{\\mathbf{D}_{\\nu}}~\\mathbf{U}^{\\dagger} \\;.\n\\label{eq_para_yn}\n\\end{align}\nHere $v_u = v \\sin\\beta$, where $v = 174$ GeV,\n$\\mathbf{R}$ is a complex orthogonal matrix \n\\footnote{Equation (\\ref{eq_para_yn}) represents the so-called\n``orthogonal'' parametrisation of $\\mathbf{Y}_{\\nu}$.\nIn certain cases it is more convenient to use the\n``bi-unitary'' parametrisation \\cite{PPR3}\n$\\mathbf{Y}_{\\nu} = \n\\mathbf{U}^{\\dagger}_{R} \\mathbf{Y}^{\\rm diag}_{\\nu}~\\mathbf{U}_{L}$,\nwhere $\\mathbf{U}_{\\rm L,R}$ are unitary matrices and\n$\\mathbf{Y}^{\\rm diag}_{\\nu}$ is a real diagonal matrix.\nThe orthogonal parametrisation is better adapted\nfor our analysis and we will employ it in what follows.}\n$\\mathbf{R}^T\\mathbf{R}= \\mathbf{1}$,\n$\\mathbf{D}_{\\nu} = \\mathrm{diag}(m_1,m_2,m_3)$,\n$m_{1,2,3} > 0$ being the light neutrino masses\n\\footnote{To be more precise, \nwe can have ${\\rm min}(m_j)=0$.} \nand $\\mathbf{U}$ is the PMNS matrix. \n\n In what follows we will consider the case when\nthe RG running of $m_j$ and of the parameters in\n$\\mbox{$ U_{\\rm PMNS}$}$ from approximately $M_Z \\sim 100$ GeV,\nwhere they are measured, to $M_R$ is relatively small \nand can be neglected. This possibility is realised \nin the class of theories under discussion for \nsufficiently small values of $\\tan\\beta$ \nand\/or of the lightest neutrino mass ${\\rm min}(m_j)$,\ne.g., for $\\tan\\beta < 10$ and\/or ${\\rm min}(m_j) \\ltap 0.05$ eV \n(see, e.g., \\cite{RGrunU,PShinYasu05}).\nUnder the indicated condition, \n$\\mathbf{D}_{\\nu}$ and $\\mathbf{U}$\nin eq. (\\ref{eq_para_yn}) should be taken \nat the scale $\\sim M_Z$, at which the neutrino \nmixing parameters are measured.\n\n It was shown in~\\cite{PPTY03} that in \na large region of the relevant soft \nSUSY breaking parameter space, the expression\n\\begin{align}\nm_S^8\\simeq 0.5~m_0^2~m_{1\/2}^2~(m_0^2 + 0.6 ~m_{1\/2}^2)^2\\;,\n\\label{eq_ms}\n\\end{align}\n$m_{1\/2}$ being the universal gaugino \nmass at $M_X$, gives an excellent approximation \nto the results obtained in a full\nrenormalisation group analysis, i.e., \nwithout using the leading-log and \nthe mass insertion approximations. \nFor values of the soft SUSY breaking parameters \nimplying SUSY particle masses in the range of \nfew to several hundred GeV, say,\n$m_0 = m_{1\/2} = 250$ GeV, \n$A_0 = a_0m_0 = -100$ GeV, we have:\n\\begin{equation} \\label{eq:bench}\nBR(l_i \\rightarrow l_j \\gamma) \\cong 9.1 \\times 10^{-10} \n\\left| (\\mathbf{Y_{\\nu}^\\dagger} L \\mathbf{Y_{\\nu}})_{ij} \\right|^2 \n\\, \\tan^2 \\beta ~,\n\\end{equation}\nwhere $L \\cong \\ln(M_{X}\/M_R)$.\nSince $\\tan^2 \\beta \\gtap 10$,\neq. (\\ref{eq:bench}) implies that if indeed the SUSY particle masses\ndo not exceed several hundred GeV, \nthe quantity $|(\\mathbf{Y_{\\nu}^\\dagger} L \\mathbf{Y_{\\nu}})_{21}|$\nhas to be relatively small. \nThis is realised for, e.g., $M_R \\ltap 10^{12}$ GeV.\n\n As follows from eqs. (\\ref{eq_ijg}) and (\\ref{eq:bench})\nand was widely discussed , in the case of soft SUSY\nbreaking mediated by soft \nflavour-universal terms at $M_X>M_R$, the\npredicted rates of LFV processes \nsuch as $\\mu\\to e + \\gamma$ decay are\nvery sensitive to the off-diagonal elements of\n\\begin{align}\n\\mathbf{Y}_{\\nu}^{\\dagger}(M_R)\\mathbf{Y}_{\\nu}(M_R)\n= \\frac{1}{v_u^2}~\n\\mathbf{U}\\sqrt{\\mathbf{D}_{\\nu}}~\\mathbf{R}^{\\dagger}~\n\\mathbf{D}_N~\\mathbf{R}~\\sqrt{\\mathbf{D}_{\\nu}}\n\\mathbf{U}^{\\dagger}\n\\cong \n\\frac{M_R}{v_u^2}~\n\\mathbf{U}\\sqrt{\\mathbf{D}_{\\nu}}~\\mathbf{R}^{\\dagger}\n~\\mathbf{R}~\\sqrt{\\mathbf{D}_{\\nu}}\n\\mathbf{U}^{\\dagger}\n\\;.\n\\label{YnudYnu}\n\\end{align}\n\n It is well-known that in the theories with \nsee-saw mechanism, leptogenesis depends \non $\\mathbf{Y}_{\\nu}(M_R)$ and thus on $\\mathbf{R}$. \nIn the case of negligible flavour \neffects \\cite{flavorLG1},\nthe dependence of interest is realised\nthrough the product \\cite{LeptoG1}\n\\begin{align}\n\\mathbf{Y}_{\\nu}(M_R)\\mathbf{Y}_{\\nu}^{\\dagger}(M_R)\n= \\frac{1}{v_u^2}~\n\\sqrt{\\mathbf{D}_{\\rm N}}~\\mathbf{R}~\n\\mathbf{D}_{\\nu}~\\mathbf{R}^{\\dagger}~\\sqrt{\\mathbf{D}_{\\rm N}}\n\\cong \n\\frac{M_R}{v_u^2}~\\mathbf{R}~\n\\mathbf{D}_{\\nu}~\\mathbf{R}^{\\dagger}\\;.\n\\label{YnuYnud}\n\\end{align}\nIn this case successful leptogenesis \ncan take place only if $\\mathbf{R} \\neq {\\bf 1}$ \nis complex. If $M_R \\ltap 10^{12}$ GeV,\nflavour effects in leptogenesis can be significant\nand leptogenesis can proceed successfully \neven for real $\\mathbf{R} \\neq {\\bf 1}$\n(see, e.g., \\cite{PPRi106}). It follows from eqs.\n(\\ref{eq_ijg}) and (\\ref{YnudYnu}), however, \nthat in the case of QD in mass heavy RH Majorana \nneutrinos of interest, the predicted rates of LFV decays \n$\\mu\\to e + \\gamma$, etc. are independent of the \northogonal matrix $\\mathbf{R}$ if $\\mathbf{R}$ is real.\n\n In what follows we will consider \n$(\\mathbf{R})^{*}\\neq \\mathbf{R}$ \nand will use the parameterizations \nof $\\mathbf{R}$ proposed in \\cite{PPY03}:\n\\begin{align}\n\\mathbf{R} = \\mathbf{O}~e^{i\\mathbf{A}}\\;.\n\\label{RPPY}\n\\end{align}\nHere $\\mathbf{O}$ is a {\\it real orthogonal} \nmatrix \nand $\\mathbf{A}$ is a {\\it real antisymmetric} matrix, \n$(\\mathbf{A})^T = - \\mathbf{A}$,\n\\begin{align}\nA=\n\\begin{pmatrix}\n0&a&b\\\\\n-a&0&c\\\\\n-b&-c&0\n\\end{pmatrix}\\;,\n\\label{Aabc}\n\\end{align}\n$a,b,c$ being real parameters. The following \nrepresentation of $e^{i\\mathbf{A}}$ \nproves useful ~\\cite{PPY03}:\n\\begin{align}\ne^{i\\mathbf{A}} = \n\\mathbf{1}-\\frac{\\cosh r-1}{r^2}\\mathbf{A}^2+i\\frac{\\sinh r}{r}\\mathbf{A}\\,,\n\\label{exp}\n\\end{align}\nwith $r=\\sqrt{a^2+b^2+c^2}$. \nThe requirement of successful\nleptogenesis in the case of QD light and \nheavy Majorana neutrino mass\nspectra and negligible flavour \neffects \\cite{flavorLG1} implies~\\cite{PPY03} \nthat $abc \\neq 0$ and that \nnone of the parameters $|a|$, $|b|$ and $|c|$\ncan be exceedingly small: \n$|abc| \\sim (10^{-6} - 10^{-4})$. \nOne also finds from the condition that \nYukawa couplings should have \nmoduli which do not exceed $\\sim 1$\nthat typically $r\\ltap 1$ \\cite{PPY03}.\n\n The parametrisation given in eq. (\\ref{RPPY})\nis particularly convenient in the analysis \nof the case of QD heavy Majorana neutrinos.\nWe will consider\na range of values of the parameters $a,b,c$\ndetermined by $10^{-4} \\ltap |a|,|b|,|c| \\ltap 0.10$. \nEquations (\\ref{eq_ijg}) and (\\ref{YnudYnu}) imply that\nfor QD heavy Majorana neutrinos \nwe can set $\\mathbf{O} = \\mathbf{1}$ \nand use $\\mathbf{R} = e^{i\\mathbf{A}}$\nin the calculation of $\\text{BR}(l_i\\to l_j+\\gamma)$\nwithout loss of generality. Results for\n$\\text{BR}(l_i\\to l_j+\\gamma)$ in the case of \nreal $\\mathbf{R} \\neq {\\bf 1}$ can be obtained \nby formally replacing $i\\mathbf{A}$ by $\\mathbf{0}$\nin the expressions for \n$\\text{BR}(l_i\\to l_j+\\gamma)$ derived \nusing $\\mathbf{R} = e^{i\\mathbf{A}}$.\n\n\\section{\n\\label{sec:main1}\n\\large{The LFV Decays $l_i\\to l_j + \\gamma$ and Majorana Phases}}\n\\hskip 1.0truecm In the case under discussion\n$M_1 = M_2 = M_3 \\equiv M_R$ and\nthe matrix of neutrino Yukawa couplings \nhas the form $\\mathbf{Y}_{\\nu}=\\frac{\\sqrt{M_R}}{v_u}\n\\mathbf{O}e^{i\\mathbf{A}}\\sqrt{\\mathbf{D}_{\\nu}}\\mathbf{U}^{\\dagger}$. The\noff-diagonal elements of $\\mathbf{Y}_{\\nu}^{\\dagger}\\mathbf{Y}_{\\nu}$\nof interest do not depend on $\\mathbf{O}$ and satisfy \n$(\\mathbf{Y}_{\\nu}^{\\dagger}\\mathbf{Y}_{\\nu})^{*}_{ij}\n= (\\mathbf{Y}_{\\nu}^{\\dagger}\\mathbf{Y}_{\\nu})_{ji}$.\nTo leading order in small quantities they are given by\n(see also \\cite{PPY03,PShinYasu05})\n\\begin{align}\n\\label{12R}\n(Y_{\\nu}^{\\dagger}Y_{\\nu})_{12}&= \\Delta_{21}c_{23}c_{12}s_{12}\n+\\Delta_{31}s_{23}s_{13}e^{-i\\delta}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\\nonumber\\\\\n&+2\\frac{M_R}{v_u^2} i\\biggl[ a\\sqrt{m_1m_2} \\left (\n c_{23}(c_{12}^2e^{-i\\frac{\\alpha}{2}}+\n s_{12}^2e^{i\\frac{\\alpha}{2}})\n+ 2i~s_{13}c_{12}s_{12}s_{23}~e^{-i\\delta}\\sin\\frac{\\alpha}{2} \\right )\n\\nonumber\\\\\n&\\phantom{Space}\n+b\\sqrt{m_1m_3}s_{23} \\left ( c_{12}e^{-i\\frac{\\beta_M}{2}}\n -s_{13}s_{12}e^{i(\\frac{\\beta_M}{2}-\\delta)} \\right )\n\\nonumber\\\\\n&\\phantom{Space}\n+ c\\sqrt{m_2m_3} \\left(s_{23}s_{12}e^{i\\frac{\\alpha-\\beta_M}{2}}\n-c_{23} s_{13}c_{12}e^{-i(\\frac{\\alpha-\\beta_M}{2} + \\delta)} \\right )\n+\\mathcal{O}(s^2_{13})\n\\biggr]+\\mathcal{O}(r^2,s_{13}^2)\\;\n\\end{align}\n\\begin{align}\n\\label{13R}\n(Y_{\\nu}^{\\dagger}Y_{\\nu})_{13}&= -\\Delta_{21}s_{23}c_{12}s_{12}\n+\\Delta_{31}c_{23}s_{13}e^{-i\\delta}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\\nonumber\\\\\n&+2\n\\frac{M_R}{v_u^2} i\\biggl[\na\\sqrt{m_1m_2} \\left ( -s_{23}(c_{12}^2e^{-i\\frac{\\alpha}{2}}+\ns_{12}^2e^{i\\frac{\\alpha}{2}}) \n+ 2i~s_{13}c_{12}s_{12}c_{23}~e^{-i\\delta}\\sin\\frac{\\alpha}{2}\n\\right )\n\\nonumber\\\\\n&\\phantom{Space}\n+b\\sqrt{m_1m_3} \\left ( c_{12}c_{23}e^{-i\\frac{\\beta_M}{2}}\n -s_{13}s_{12}s_{23}e^{i(\\frac{\\beta_M}{2}-\\delta)} \\right )\n\\nonumber\\\\\n&\\phantom{Space}\n+ c\\sqrt{m_2m_3}\\left(s_{12}c_{23}e^{i\\frac{\\alpha-\\beta_M}{2}}\n+ s_{13}c_{12}s_{23} e^{-i(\\frac{\\alpha-\\beta_M}{2} + \\delta)} \\right )\n+\\mathcal{O}(s^2_{13})\n\\biggr]+\\mathcal{O}(r^2,s_{13}^2)\\;\n\\end{align}\n\\begin{align}\n\\label{23R}\n(Y_{\\nu}^{\\dagger}Y_{\\nu})_{23}&= \\Delta_{31}s_{23}c_{23}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\\nonumber\\\\\n&+2\\frac{M_R}{v_u^2}\ni\\biggl[\na\\sqrt{m_1m_2} \\left (-2i c_{12}s_{12}c_{23}s_{23}\\sin\\frac{\\alpha}{2}\n+ s_{13} \\left [ c_{12}^2(s_{23}^2e^{-i(\\frac{\\alpha}{2}-\\delta)} \n+ c_{23}^2e^{i(\\frac{\\alpha}{2} -\\delta)}) \\right. \\right.\n\\nonumber\\\\\n&\\phantom{Space}\n\\left. \\left.\n+ s_{12}^2(s_{23}^2e^{i(\\frac{\\alpha}{2} +\\delta)} \n+ c_{23}^2e^{-i(\\frac{\\alpha}{2} +\\delta)})\\right ]\n\\right )\n\\nonumber\\\\\n&\\phantom{Space}\n+b\\sqrt{m_1m_3} \\left (-s_{12}(s_{23}^2e^{i\\frac{\\beta_M}{2}}\n+c_{23}^2e^{-i\\frac{\\beta_M}{2}})\n+s_{13}c_{12}c_{23}s_{23}~2i\\sin(\\frac{\\beta_M}{2}-\\delta)\n\\right )\n\\nonumber\\\\\n&\\phantom{Space}\n+ c\\sqrt{m_2m_3}\\left ( c_{12}\n(c_{23}^2e^{i\\frac{\\alpha-\\beta_M}{2}}+\ns_{23}^2e^{-i\\frac{\\alpha-\\beta_M}{2}})\n-s_{13}s_{12}c_{23}s_{23}~2i\\sin(\\frac{\\alpha-\\beta_M}{2} + \\delta) \n\\right )\n\\nonumber\\\\\n&\\phantom{Space}\n+\\mathcal{O}(s^2_{13}) \\biggr]+\\mathcal{O}(r^2,s_{13}^2)\\;,\n\\end{align}\nwhere \n\\begin{align}\n\\Delta_{ij} \\equiv\n\\frac{M_R}{v_u^2} \\left(m_i -m_j \\right) = \n\\frac{M_R}{v_u^2}~\\frac{\\Delta m^2_{ij}}{m_i + m_j}\\;.\n\\label{Dij}\n\\end{align}\nEquations (\\ref{12R})-(\\ref{23R})\nare valid for any of the possible \ntypes of light neutrino mass spectrum.\nThe corresponding expressions \nfor real $\\mathbf{R} \\neq {\\bf 1}$\ncan be obtained by formally \nsetting the three leptogenesis CPV parameters\n$a$, $b$ and $c$ to 0 in eqs. (\\ref{12R})-(\\ref{23R}).\n\n In what follows we will concentrate \non the case of complex $\\mathbf{R} \\neq {\\bf 1}$\nand will call the real quantities\n$a,b,c$ ``leptogenesis CP-violation (CPV) \nparameters''. The results in \neqs. (\\ref{12R})-(\\ref{23R})\nimply that in the absence of significant \nRG effects, the ``double'' ratios\n\\begin{align}\n\\text{R}(21\/31) \\equiv \\frac{\\text{BR}(\\mu \\to e + \\gamma)}\n{\\text{BR}(\\tau \\to e + \\gamma)}~\\text{BR}(\\tau \\to e\\nu_{\\tau}\\bar{\\nu}_e)\\;,~ \\text{R}(21\/32) \\equiv \\frac{\\text{BR}(\\mu \\to e + \\gamma)}\n{\\text{BR}(\\tau \\to \\mu + \\gamma)}\n~\\text{BR}(\\tau \\to e\\nu_{\\tau}\\bar{\\nu}_e)\\;,\n\\label{DoubleR}\n\\end{align}\ndepend in the region of validity of eqs. (\\ref{eq_ijg}) \nand (\\ref{eq_ms}) in the relevant SUSY \nparameter space, on the neutrino masses $m_j$,\nmixing angles $\\theta_{12}$, \n$\\theta_{23}$, $\\theta_{13}$ and\nMajorana and Dirac CP-violation phases \n$\\alpha$, $\\beta_M$ and $\\delta$\nat $\\sim M_Z$, \nas well as on the leptogenesis \nCP-violating (CPV) parameters \n$a$, $b$ and $c$. The dependence of \nthe Dirac phase $\\delta$\ncan be significant only if the CHOOZ angle\n$\\theta_{13}$ is sufficiently large.\nThe case of real matrix $\\mathbf{R} \\neq {\\bf 1}$ \ncorresponds to \n$a=0$, $b=0$ and $c=0$. \nAlthough the general expressions for\n$(Y_{\\nu}^{\\dagger}Y_{\\nu})_{ij}$, $i\\neq j$,\neqs. (\\ref{12R})-(\\ref{23R}),\ninclude several terms, there are few\nphysically interesting cases in which \nthe expressions simplify and\nthe dependence on the Majorana\nCP-violation phase(s) and\/or \non the leptogenesis CPV parameters is prominent. \n\n\\subsection{\\label{sec:NH}\\large{Normal Hierarchical Neutrino Mass Spectrum}}\n\n\\hskip 1.0truecm If the neutrino mass spectrum\nis of the {\\it normal hierarchical} (NH) type\nand $m_1$ is negligibly small, i.e., \n$|a|\\sqrt{m_1m_2},|b|\\sqrt{m_1m_3} \\ll |c|\\sqrt{m_2m_3}$,\nthe quantities \n$(Y_{\\nu}^{\\dagger}Y_{\\nu})_{ij}$, $i\\neq j$, depend, \nin particular, on the same Majorana phase difference\n$(\\alpha - \\beta_M)$ on which the effective Majorana mass,\neq. (\\ref{meffNH2}), depends, on the Dirac CPV phase \n$\\delta$ and one leptogenesis CPV parameter, $c$. \nThe terms $\\propto c\\sqrt{m_2m_3}s_{13}e^{-i\\delta}$ \ngive always subdominant contributions \nin $|(Y_{\\nu}^{\\dagger}Y_{\\nu})_{12,13}|$.\nFor $s_{13} \\ll \\tan\\theta_{12} \\sim 0.65$ \nthey are negligible. In this case the expressions for \n$|(Y_{\\nu}^{\\dagger}Y_{\\nu})_{12,13}|$ simplify:\n\\begin{align}\n\\label{12RNH}\n|(Y_{\\nu}^{\\dagger}Y_{\\nu})^{\\rm NH}_{12}|^2&\\cong\n\\frac{M_R^2}{v_u^4}~\n\\left |c_{23}~P^{\\rm NH} + s_{23}~Q^{\\rm NH}\\right |^2 \\;,\\\\%~~{\\rm NH}\\;,\\\\\n\\label{13RNH}\n|(Y_{\\nu}^{\\dagger}Y_{\\nu})^{\\rm NH}_{13}|^2& \\cong\n\\frac{M_R^2}{v_u^4}~\n\\left |-s_{23}~P^{\\rm NH} + c_{23}~Q^{\\rm NH}\\right |^2\\;\n\\end{align}\nwhere \n\\begin{align}\n\\label{NHP}\nP^{\\rm NH} & = (\\mbox{$ \\Delta m^2_{21}$})^{\\frac{1}{2}}c_{12}s_{12}\\;,\\\\%~~~\n\\label{NHQ}\nQ^{\\rm NH} & = (\\mbox{$\\Delta m^2_{31}$})^{\\frac{1}{2}}s_{13}e^{-i\\delta} + \ni~2c~(\\mbox{$ \\Delta m^2_{21}$} \\mbox{$\\Delta m^2_{31}$})^{\\frac{1}{4}}~\ns_{12}~e^{i\\frac{\\alpha-\\beta_M}{2}}\\;. \n\\end{align}\nThe double ratio $\\text{R}(21\/31)$\nis determined completely by the solar and atmospheric \nneutrino oscillation parameters \n$\\mbox{$ \\Delta m^2_{21}$}$, $\\theta_{12}$ and $\\mbox{$\\Delta m^2_{31}$}$ and $\\theta_{23}$,\nby the CHOOZ angle $\\theta_{13}$ and by the\nMajorana and Dirac\nCPV phases $(\\alpha - \\beta_M)$ and $\\delta$ and \nby the leptogenesis CPV parameter $c$:\n\\begin{align}\n\\text{R}(21\/31)\\cong \n\\frac{\\left |c_{23}~P^{\\rm NH} + s_{23}~Q^{\\rm NH}\\right |^2}\n{\\left |-s_{23}~P^{\\rm NH} + c_{23}~Q^{\\rm NH}\\right |^2}\n\\;.\n\\label{R2131NH0} \n\\end{align}\nwhere $P^{\\rm NH}$ and $Q^{\\rm NH}$\nare given by eqs. (\\ref{NHP}) and (\\ref{NHQ}).\nIt follows from eqs. (\\ref{NHP}), (\\ref{NHQ}) and\n(\\ref{R2131NH0}) that if $(\\alpha - \\beta_M) = 0$ and\n$\\delta = \\pm \\pi\/2,\\pm 3\\pi\/2$,\nwe would have\n\\begin{align}\n\\text{R}(21\/31)\\cong \n\\frac{c_{23}^2 \\left |P^{\\rm NH}\\right |^2 + \ns_{23}^2\\left |Q^{\\rm NH}\\right |^2}\n{s_{23}^2 \\left |P^{\\rm NH}\\right |^2 + \nc_{23}^2\\left |Q^{\\rm NH}\\right |^2}\\;.\n\\label{R2131NH1} \n\\end{align}\n\\noindent For the best fit value\n$\\sin^22\\theta_{23} = 1$ we get $\\text{R}(21\/31) = 1$ \nindependently of the value of \nthe leptogenesis CPV parameter\n$c$, although the corresponding branching\nratios $\\text{BR}(\\mu \\to e + \\gamma)$ and\n$\\text{BR}(\\tau \\to e + \\gamma)$ can exhibit strong \ndependence on $c$. If $\\theta_{23}$ differs \nsomewhat from $\\pi\/4$, the dependence of\n$\\text{R}(21\/31)$ on $c$ will, in general,\nbe relatively mild. For \n$|P^{\\rm NH}|^2 \\gg |Q^{\\rm NH}|^2$ \n($|P^{\\rm NH}|^2 \\ll |Q^{\\rm NH}|^2$), however, \nthe dependence of $\\text{R}(21\/31)$ on $c$ \nwill be negligible even if $\\theta_{23} \\neq \\pi\/4$\nand we would have \n$\\text{R}(21\/31) \\cong \\cot^2\\theta_{23}~(\\tan^2\\theta_{23})$.\n\n For $|c| \\ltap 0.1$ and $s_{13}$ having a value close to the \nexisting (3$\\sigma$) upper limit $s_{13} \\cong 0.2$, \nthe term $\\propto c$ in $Q^{\\rm NH}$, eq. (\\ref{NHQ}), \ngives practically negligible\ncontributions in $|(Y_{\\nu}^{\\dagger}Y_{\\nu})_{12,13}|$,\nwhile $P^{\\rm NH}$, eq. (\\ref{NHP}), gives a subleading\nbut non-negligible\ncontribution. Correspondingly, the double ratio\n$\\text{R}(21\/31)$ exhibits in this case a significant dependence\non the Dirac phase $\\delta$ and is essentially independent \nof the leptogenesis CPV parameter $c$ and\nthe Majorana phase $(\\alpha - \\beta_M)$ (Fig. 1). \n\n If $s_{13} \\ltap 0.1$, but $s_{13}$ is \nnot much smaller than \n$\\sqrt{\\mbox{$ \\Delta m^2_{21}$}}c_{12}s_{12}\/\\sqrt{\\mbox{$\\Delta m^2_{31}$}}\\sim 0.07$,\nthe branching ratios $\\text{BR}(\\mu \\to e + \\gamma)$\nand $\\text{BR}(\\tau \\to e + \\gamma)$ \nstill depend on the CHOOZ mixing angle $\\theta_{13}$\nand the phase $\\delta$. For $ 0.01 \\ltap |c| \\ltap 0.10$,\n$\\text{BR}(\\mu \\to e + \\gamma)$\nand $\\text{BR}(\\tau \\to e + \\gamma)$ can \nexhibit significant dependence also\non the Majorana phase $(\\alpha - \\beta_M)$. \nThe dependence of the double ratio \n$\\text{R}(21\/31)$ on $(\\alpha - \\beta_M)$ \nand $\\delta$ can be very strong due to possible\nmutual compensation between $P^{\\rm NH}$ and $Q^{\\rm NH}$\n(see eq. (\\ref{R2131NH0})). For \n$(\\alpha - \\beta_M) \\cong 0$, $s_{13} = 0.10$ and \nsufficiently small $|c|$,\nfor instance, we can have $\\text{R}(21\/31) \\sim 10^{-2}$ or\n$\\text{R}(21\/31) \\sim 10^{2}$ depending on \nwhether $\\delta \\cong \\pi$ or $\\delta \\cong 0$;\nfor $(\\alpha - \\beta_M) \\cong \\pi$ and\n$|c|\\sim 0.03$, $\\text{R}(21\/31)$ can have a value \n$\\text{R}(21\/31) \\sim 10^{-3}$ or\n$\\text{R}(21\/31) \\sim 10^{3}$, respectively (Fig. 1).\n\n For rather small values of $s_{13}$, namely, \n$s_{13}\\ll \\sqrt{\\mbox{$ \\Delta m^2_{21}$}}c_{12}s_{12}\/\\sqrt{\\mbox{$\\Delta m^2_{31}$}}$, \nthe dependence of $|(Y_{\\nu}^{\\dagger}Y_{\\nu})_{12,13}|$ on\n$s_{13}e^{-i\\delta}$ is insignificant and can be neglected.\nUnder the latter condition we also have\n$\\sqrt{\\mbox{$\\Delta m^2_{31}$}}s^2_{13}\\ll \\sqrt{\\mbox{$ \\Delta m^2_{21}$}}s^2_{12}$.\nThe effective Majorana mass in $\\mbox{$\\beta \\beta_{0 \\nu}$}$-decay is given\ncorrespondingly by $\\mbox{$\\left| \\langle m \\rangle\\right|$}_{\\rm NH} \\cong \\sqrt{\\mbox{$ \\Delta m^2_{21}$}}s_{12}^2$.\nThe quantities $P^{\\rm NH}$ and $Q^{\\rm NH}$\ncan be written as:\n\\begin{align}\nP^{\\rm NH}\\cong (\\mbox{$\\left| \\langle m \\rangle\\right|$}_{\\rm NH})^{\\frac{1}{2}}\n(\\mbox{$ \\Delta m^2_{21}$})^{\\frac{1}{4}}c_{12}\\;,\n~Q^{\\rm NH}\\cong i~2c~(\\mbox{$\\left| \\langle m \\rangle\\right|$}_{\\rm NH})^{\\frac{1}{2}}\n(\\mbox{$\\Delta m^2_{31}$})^{\\frac{1}{4}}~\ne^{i\\frac{\\alpha-\\beta_M}{2}}\\;. \n\\label{NHPQ1}\n\\end{align}\nThus, in this case $\\text{BR}(\\mu \\to e + \\gamma)\\propto \\mbox{$\\left| \\langle m \\rangle\\right|$}_{\\rm NH}$\nand $\\text{BR}(\\tau \\to e + \\gamma)\\propto \\mbox{$\\left| \\langle m \\rangle\\right|$}_{\\rm NH}$. \nGiven $\\mbox{$ \\Delta m^2_{21}$}$, $\\mbox{$\\Delta m^2_{31}$}$,\n$\\theta_{12}$ and $\\theta_{23}$,\nthe ratio $\\text{R}(21\/31)$ is \ndetermined by the Majorana phase difference \n$(\\alpha - \\beta_M)$ and the leptogenesis CPV \nparameter $c$: \n\\begin{align}\n\\label{R2131NH1}\n\\text{R}(21\/31) \\cong \\frac{\n\\left |(\\mbox{$ \\Delta m^2_{21}$})^{\\frac{1}{4}}c_{12}\\cot\\theta_{23} + \ni~2c~(\\mbox{$\\Delta m^2_{31}$})^{\\frac{1}{4}}~e^{i\\frac{\\alpha-\\beta_M}{2}}\n\\right |^2}\n{\\left |(\\mbox{$ \\Delta m^2_{21}$})^{\\frac{1}{4}}c_{12} - \ni~2c~(\\mbox{$\\Delta m^2_{31}$})^{\\frac{1}{4}}~\ne^{i\\frac{\\alpha-\\beta_M}{2}}\\cot\\theta_{23}~\n\\right |^2} \\;.\n\\end{align}\nObviously, for $(\\alpha-\\beta_M) \\cong 0$, we have\n$\\text{R}(21\/31) \\cong 1$. If, however, \n$(\\alpha-\\beta_M) \\cong \\pm \\pi$,\nthe double ratio $\\text{R}(21\/31)$\ncan depend strongly on the value of $|c|$, provided\n$|c| \\gtap 0.05$. For \n$2|c| \\sim (\\mbox{$ \\Delta m^2_{21}$}\/\\mbox{$\\Delta m^2_{31}$})^{\\frac{1}{4}} \\cong 0.42$,\nthe two terms in the numerator (denominator) \nof the expression for $\\text{R}(21\/31)$ \ncan compensate (partially) each other and one can have \n$\\text{R}(21\/31) \\sim (10^{-3} - 10^{-2})$ or\n$\\text{R}(21\/31) \\sim (10^{3} - 10^{2})$\ndepending on the sign of $c$ (Fig. 1).\n \n If $|c|$ is relatively small, \n$2|c| \\ll (\\mbox{$ \\Delta m^2_{21}$}\/\\mbox{$\\Delta m^2_{31}$})^{\\frac{1}{4}}\\cong 0.42$,\n$\\text{BR}(\\mu \\to e + \\gamma)$ and \n$\\text{BR}(\\tau \\to e + \\gamma)$ \nare practically independent of $c$ and $(\\alpha - \\beta_M)$.\nThis case was analised recently in \\cite{PShinYasu05}.\nIf, for instance, \n$s_{13}\\ll \\sqrt{\\mbox{$ \\Delta m^2_{21}$}}c_{12}s_{12}\/\\sqrt{\\mbox{$\\Delta m^2_{31}$}}$,\nwe find from eq. (\\ref{R2131NH1}): \n\\begin{align}\n\\text{R}(21\/31) \\cong \\cot^{2}\\theta_{23}\\;.\n\\label{R2131NH}\n\\end{align}\n\\hskip 1.0truecm The results for the double \nratio $\\text{R}(21\/31)$ discussed above are\nillustrated in Fig.~\\ref{NH-R2131-alp}, where the dependence of \n$\\text{R}(21\/31)$ \non the leptogenesis CPV parameter $c$\nfor $s_{13} = 0;~0.10;~0.20$ and few characteristic \nvalues of the Majorana and Dirac CPV phases \n$(\\alpha-\\beta_M) = 0;\\pi\/2;\\pm \\pi$ and\n$\\delta =0;\\pm \\pi\/2;\\pm \\pi$ are shown.\nThe figure was obtained using the best fit values \nof the solar and atmospheric \nneutrino oscillation parameters $\\theta_{12}$,\n$\\mbox{$ \\Delta m^2_{21}$}$, $\\theta_{23}$ and $\\mbox{$\\Delta m^2_{31}$}$. \nThe lightest neutrino mass $m_1$ was set to 0.\nThe quantities $|(Y_{\\nu}^{\\dagger}Y_{\\nu})^{\\rm NH}_{12,13}|^2$\nwere calculated using eqs. (\\ref{YnudYnu}) and \n(\\ref{RPPY}) and not the approximate\nexpressions given in eqs. (\\ref{12R}) and\n(\\ref{13R}). The leptogenesis CPV violating \nparameters $a$ and $b$ can contribute only \nto the higher order corrections \n$\\mathcal{O}(r^2,s_{13}^2)$ in \n$|(Y_{\\nu}^{\\dagger}Y_{\\nu})^{\\rm NH}_{12,13}|^2$. \nThese corrections can be relevant for the evaluation of\n$\\text{R}(21\/31)$ in the case of cancellations between\nthe terms in $|(Y_{\\nu}^{\\dagger}Y_{\\nu})^{\\rm NH}_{12,13}|^2$,\nwhich provide the leading order contributions.\nWe have allowed $a$ and $b$ to vary in the same \ninterval as the parameter $c$ in the calculations.\nThe effects of the higher order corrections \ndue to $a$ and $b$ is reflected\nin the widths of the lines in Fig. 1.\n\n We shall perform next similar analysis for the\ndouble ratio $\\text{R}(21\/32) = \n\\text{BR}(\\mu \\to e + \\gamma)\/\\text{BR}(\\tau \\to \\mu + \\gamma)$.\nAs can be easily verified using eqs. (\\ref{12R}) and (\\ref{23R}) and \nthe known values of the neutrino oscillation parameters, \nfor $|c|\\leq 0.3$ we always have \n\\begin{align}\n\\text{R}(21\/32) < 1\\;.\n\\label{R2132NH00} \n\\end{align}\n\\noindent Typically \nthe stronger inequality $\\text{R}(21\/32) \\ll 1$ holds \n\\footnote{This is in contrast to the case of \nnormal hierarchical heavy Majorana neutrino\nmass spectrum, in which one typically has \n$\\text{R}(21\/32) \\sim 1$ \\cite{PShinYasu05}.}\n(see further).\n\n It is not difficult to convince oneself also that \nthe term $\\propto \\Delta_{31}s_{23}c_{23}$\ndominates in $|(Y_{\\nu}^{\\dagger}Y_{\\nu})_{23}|^2$.\nIndeed, we have $(\\mbox{$ \\Delta m^2_{21}$}\/\\mbox{$\\Delta m^2_{31}$})^{\\frac{1}{2}} \\cong 0.18$,\n$c_{12}|\\cos\\theta_{23}| \\ltap 0.24$,\n$s_{13}s_{12}\\sin2\\theta_{23} \\ltap 0.12$, and \nfor $|c|\\leq 0.2~(0.3)$, the terms $\\propto c$ \nin eq. (\\ref{23R}) give contributions which do not exceed\napproximately 8\\% (18\\%). Keeping only the largest\nof these contributions we have:\n$|(Y_{\\nu}^{\\dagger}Y_{\\nu})_{23}|^2\n\\cong M^2_R v_u^{-4}~|\\sqrt{\\mbox{$\\Delta m^2_{31}$}}s_{23}c_{23}\n+ 2ic~(\\mbox{$ \\Delta m^2_{21}$} \\mbox{$\\Delta m^2_{31}$})^{\\frac{1}{4}}c_{12}\n\\cos(\\alpha -\\beta_M)\/2|^2$. Thus, \nthe branching ratio $\\text{BR}(\\tau \\to \\mu + \\gamma)$\nexhibits very weak dependence on $c$ and $(\\alpha -\\beta_M)$.\nUp to the indicated corrections which for $|c| \\leq 0.3$\ncan increase $|(Y_{\\nu}^{\\dagger}Y_{\\nu})_{23}|^2$ \nby not more than 18\\%, we have:\n\\begin{align}\n\\label{23RNH}\n|(Y_{\\nu}^{\\dagger}Y_{\\nu})_{23}|^2 \n\\cong \\frac{M^2_R}{v^4_u}~\\mbox{$\\Delta m^2_{31}$} s^2_{23} c^2_{23}\n\\;.\n\\end{align}\nThus, for $|c| \\ltap 0.3$ in the case under discussion,\n$\\text{BR}(\\tau \\to \\mu + \\gamma)$ \ndepends essentially only on the atmospheric \nneutrino oscillation parameters \n$\\mbox{$\\Delta m^2_{31}$}$ and $\\theta_{23}$\n(and not on the Dirac and Majorana CPV phases,\nleptogenesis CPV parameters or solar \nneutrino oscillation parameters \n$\\mbox{$ \\Delta m^2_{21}$}$ and $\\theta_{12}$)\nand has a relatively simple form:\n$\\text{BR}(\\tau \\to \\mu + \\gamma) \\cong \n\\text{F} \\times (\\mbox{$\\Delta m^2_{31}$}\/(4v^2_u))~\\sin^22\\theta_{23}$,\nwhere the factor $\\text{F} \\propto M_R^2\/v^2_u$ contains all the\ndependence on $M_R$, $\\tan\\beta$ and \nthe SUSY breaking parameters (see eq. (\\ref{eq_ijg})).\nThe double ratio $\\text{R}(21\/32)$, \nhowever, depends in the case under discussion both on \n$c~e^{i\\frac{\\alpha-\\beta_M}{2}}$ and $s_{13}e^{-i\\delta}$: \n\\begin{align}\n\\text{R}(21\/32)\\cong \n\\frac{\\left |c_{23}~P^{\\rm NH} + s_{23}~Q^{\\rm NH}\\right |^2}\n{\\mbox{$\\Delta m^2_{31}$} s^2_{23}c^2_{23}}\n\\;,\n\\label{R2132NH0} \n\\end{align}\nwhere $P^{\\rm NH}$ and $Q^{\\rm NH}$ are given by \neqs. (\\ref{NHP}) and (\\ref{NHQ}).\n\n For $s_{13} \\cong 0.2$ and $|c| \\ltap 0.25$, \nwe have $\\sqrt{\\mbox{$\\Delta m^2_{31}$}}s_{13} \\cong 2.3\\sqrt{\\mbox{$ \\Delta m^2_{21}$}}c_{12}s_{12}$\nand $\\sqrt{\\mbox{$\\Delta m^2_{31}$}}s_{13}\n\\gtap 1.7~(2c (\\mbox{$\\Delta m^2_{31}$} \\mbox{$ \\Delta m^2_{21}$})^{\\frac{1}{4}}s_{12})$.\nThe double ratio $\\text{R}(21\/32)$ exhibits noticeable dependence\non the CPV phases $(\\alpha - \\beta_{M})$ and $\\delta$.\nFor $(\\alpha - \\beta_{M})=0$ and $\\delta=0$, the term \n$\\propto c$ in $Q^{\\rm NH}$ gives a subdominant contribution\nand $\\text{R}(21\/32)$ is practically independent of $c$.\nIf $\\delta = \\pi$, however, the term \n$\\propto \\sqrt{\\mbox{$\\Delta m^2_{31}$}}s_{13}$ in $Q^{\\rm NH}$\ncan be compensated partially by $P^{\\rm NH}$ and for \nsufficiently large values of $|c|$ the term\n$\\propto c$ in $Q^{\\rm NH}$ can be non-negligible.\nFor $|c| \\ltap 0.1$ in this case we can have \n$\\text{R}(21\/32)\\sim {\\rm few}\\times 10^{-2}$, while if\n$(\\alpha - \\beta_{M})= \\pi$ and $|c| \\cong 0.2$,\n$\\text{R}(21\/32)$ can be as small as \n$\\text{R}(21\/32)\\sim {\\rm few}\\times 10^{-3}$. \n\n In the case of \n$s_{13}\\sim \\sqrt{\\mbox{$ \\Delta m^2_{21}$}}c_{12}s_{12}\/\\sqrt{\\mbox{$\\Delta m^2_{31}$}}\\sim 0.07$,\npartial compensation between the three \nterms in the numerator of the double ratio $\\text{R}(21\/32)$\ncan take place. The double ratio \n$\\text{R}(21\/32)$ \ncan be particularly strongly suppressed for \n$\\delta \\cong \\pi$, when values of $\\text{R}(21\/32) \n\\sim (10^{-3} - 10^{-4})$ for $|c| \\sim 0.05$ are possible.\nSimilar mutual compensations between the terms in the \nnumerator of $\\text{R}(21\/32)$ can be realised if \n$s_{13}\\ll \\sqrt{\\mbox{$ \\Delta m^2_{21}$}}c_{12}s_{12}\/\\sqrt{\\mbox{$\\Delta m^2_{31}$}}$\nand $|c| \\sim (0.15 - 0.20)$. One can have\n$\\text{R}(21\/32) \\sim (10^{-3} - 10^{-4})$ in this case as well.\nFor sufficiently small $s_{13}$ the dependence on the phase \n$\\delta$ is obviously insignificant and we have:\n\\begin{align}\n\\text{R}(21\/32)\\cong \n\\frac{\\mbox{$\\left| \\langle m \\rangle\\right|$}_{\\rm NH}}{\\sqrt{\\mbox{$\\Delta m^2_{31}$}}s_{23}^2c^2_{23}}\n\\left | \\left (\\frac{\\mbox{$ \\Delta m^2_{21}$}}{\\mbox{$\\Delta m^2_{31}$}} \\right )^{\\frac{1}{4}}c_{23}c_{12} \n+ i~2c~s_{23}e^{i\\frac{\\alpha-\\beta_M}{2}}\\right |^2 \n\\;.\n\\label{R2132NHs130} \n\\end{align}\nIf in addition \n$2|c| \\ll (\\mbox{$ \\Delta m^2_{21}$}\/\\mbox{$\\Delta m^2_{31}$})^{\\frac{1}{4}}\\cong 0.42$, \n$\\text{R}(21\/32)$ is also practically independent of\n$c$ and $(\\alpha - \\beta_M)$. It is determined completely \nby the solar and atmospheric neutrino oscillation\nparameters:\n\\begin{align}\n\\text{R}(21\/32)\\cong \n\\mbox{$\\left| \\langle m \\rangle\\right|$}_{\\rm NH}~\n\\frac{(\\mbox{$ \\Delta m^2_{21}$})^{\\frac{1}{2}}}{\\mbox{$\\Delta m^2_{31}$} s_{23}^{2}}~c^2_{12} \n\\cong \\frac{\\mbox{$ \\Delta m^2_{21}$}}{\\mbox{$\\Delta m^2_{31}$} s_{23}^2}~c^2_{12}s^2_{12} \n\\simeq 1.3\\times 10^{-2} \n\\;.\n\\label{R2132NH}\n\\end{align}\n\n\\hskip 1.0truecm \nThe specific features of the double ratio $\\text{R}(21\/32)$\ndiscussed above are evident in Fig. 2, where the dependence of \n$\\text{R}(21\/32)$ on the leptogenesis CPV parameter $c$, \n$|c| \\leq 0.25$, for three values of $s_{13} = 0;~0.1;~0.2$, \nand several characteristic values of the \nMajorana and Dirac CPV phases \n$(\\alpha - \\beta_M)$ and $\\delta$ is shown.\nFigure 2 was obtained using the same method \nand the same best fit values of the oscillation parameters\n$\\mbox{$ \\Delta m^2_{21}$}$, $\\sin^2\\theta_{12}$,\n$\\mbox{$\\Delta m^2_{31}$}$ and $\\sin^22\\theta_{23}$, as Fig. 1.\n\n\\subsection{\\label{sec:NH}\\large{Inverted Hierarchical Spectrum}}\n\n\\hskip 1.0truecm If neutrino mass spectrum is \n{\\it inverted hierarchical} (IH) one has $m_3 \\ll m_{1,2}$,\nand we shall assume that the terms $\\propto \\sqrt{m_3}$ \nin eqs. (\\ref{12R})-(\\ref{23R}) can be neglected. \nFor $\\mbox{$ \\langle m \\rangle $}_{\\rm IH}$ we have\n$\\mbox{$ \\langle m \\rangle $}_{\\rm IH} \\cong \\sqrt{|\\mbox{$\\Delta m^2_{31}$}|}(c_{12}^2 + s_{12}^2e^{i\\alpha})$\n(see eq. (\\ref{meffIH1})).\nNow $|(Y_{\\nu}^{\\dagger}Y_{\\nu})_{ij}|$, $i\\neq j$,\ndepend on the Majorana phase $\\alpha$, on the\nleptogenesis CPV parameter $a$ and, if $s_{13}$\nhas a value close to the current upper limit \n- on the Dirac phase $\\delta$:\n\\begin{align}\n\\label{12RIH}\n|(Y_{\\nu}^{\\dagger}Y_{\\nu})^{\\rm IH}_{12}| & \\cong\n~\\frac{M_R}{v_u^2}~\n\\left |c_{23}~P^{\\rm IH} + s_{23}~Q^{\\rm IH}\\right | \\;,\\\\%~~{\\rm NH}\\;,\\\\\n\\label{13RIH}\n|(Y_{\\nu}^{\\dagger}Y_{\\nu})^{\\rm IH}_{13}| & \\cong\n\\frac{M_R}{v_u^2}~\n\\left |-s_{23}~P^{\\rm IH} + c_{23}~Q^{\\rm IH}\\right | \\;,\\\\\n|(Y_{\\nu}^{\\dagger}Y_{\\nu})^{\\rm IH}_{23}| & \\cong\n\\frac{M_R}{v_u^2}\\sqrt{|\\mbox{$\\Delta m^2_{31}$}|}~c_{23}s_{23}\n\\left |-1 + 4ac_{12}s_{12}\\sin\\frac{\\alpha}{2}\\right | \\;,\n\\label{23RIHa}\n\\end{align}\nwhere \n\\begin{align}\n\\label{IHP}\nP^{\\rm IH} = \\frac{1}{2}~\\frac{\\mbox{$ \\Delta m^2_{21}$}}{\\sqrt{|\\mbox{$\\Delta m^2_{31}$}|}}~c_{12}s_{12} \n+ i~2a~\\mbox{$ \\langle m \\rangle $}_{\\rm IH}~e^{-i\\frac{\\alpha}{2}}\\;, \\\\% ~~~\nQ^{\\rm IH} =\n- \\sqrt{|\\mbox{$\\Delta m^2_{31}$}|}~s_{13}~e^{-i\\delta}\n\\left (1 + 4ac_{12}s_{12}~\\sin\\frac{\\alpha}{2}\\right )\\;. \n\\label{IHPQ}\n\\end{align}\nFor $s_{13}$ satisfying\n\\begin{align}\n\\sin\\theta_{13}(1 + 2|a|\\sin 2\\theta_{12}) \\ll \n{\\rm min}\\left (2|a|\\cos2\\theta_{12}, \n\\frac{\\mbox{$ \\Delta m^2_{21}$}}{4|\\mbox{$\\Delta m^2_{31}$}|}\\sin 2\\theta_{12} \\right )\n\\label{s13IH}\n\\end{align}\nthe dependence of $|(Y_{\\nu}^{\\dagger}Y_{\\nu})^{\\rm IH}_{12,13}|$\non the Dirac phase $\\delta$ would be insignificant.\nThe terms $\\propto Q^{\\rm IH}$ in\neqs. (\\ref{12RIH}) and (\\ref{13RIH}) are negligible,\nand the ratio of $\\text{BR}(\\mu \\to e + \\gamma)$\nand $\\text{BR}(\\tau \\to e + \\gamma)$ is given by\n\\begin{align}\n\\text{R}(21\/31) \\cong \\cot^2\\theta_{23}\\;,\n\\label{R2131IHa}\n\\end{align}\nindependently of the values of the\nMajorana CPV phase $\\alpha$, leptogenesis \nCPV parameter $a$, etc. (Fig. 3). If in addition \n$|a| \\ll (\\mbox{$ \\Delta m^2_{21}$}\/(8|\\mbox{$\\Delta m^2_{31}$}|))\\sin 2\\theta_{12}\\cong 3.6\\times 10^{-3}$,\n$\\text{BR}(\\mu \\to e + \\gamma)$ and $\\text{BR}(\\tau \\to e + \\gamma)$\nalso will not depend on $\\alpha$ and $a$:\\\\\n$|(Y_{\\nu}^{\\dagger}Y_{\\nu})^{\\rm IH}_{12(13)}|^2 \\cong C^2_{12(13)}~\n(M^2_R \\mbox{$ \\Delta m^2_{21}$}\/v_u^4)(\\mbox{$ \\Delta m^2_{21}$}\/(16|\\mbox{$\\Delta m^2_{31}$}|))\\sin^22\\theta_{12}$,\nwhere $C_{12(13)} \\equiv c_{23}~(s_{23})$.\nIn the case of $|a|\\cos2\\theta_{12} \\gg (\\mbox{$ \\Delta m^2_{21}$}\/(8|\\mbox{$\\Delta m^2_{31}$}|))\n\\sin 2\\theta_{12}\\cong 4\\times 10^{-3}$, however, we have:\n\\begin{align}\n|(Y_{\\nu}^{\\dagger}Y_{\\nu})^{\\rm IH}_{12(13)}| \\cong\n2|a|~\\left | \\mbox{$ \\langle m \\rangle $}_{\\rm IH} \\right |\n\\frac{M_R}{v_u^{2}}~C_{12(13)}\\;.\n\\label{Y1213IH}\n\\end{align}\nThus, both $\\text{BR}(\\mu \\to e + \\gamma)$ \nand $\\text{BR}(\\tau \\to e + \\gamma)$ are proportional to \n$|a|^2~|\\mbox{$ \\langle m \\rangle $}_{\\rm IH}|^2$. \n\n We get $\\text{R}(21\/31) \\sim 1$ also when \n$s_{13} \\gg (\\mbox{$ \\Delta m^2_{21}$}\/(4|\\mbox{$\\Delta m^2_{31}$}|))\n\\sin 2\\theta_{12}\\cong 8\\times 10^{-3}$, provided\n$\\alpha \\cong 0$ and $\\delta \\cong 0;~\\pi$ (Fig. 3).\nIn this case $|\\mbox{$ \\langle m \\rangle $}_{\\rm IH}|^2 \\cong |\\mbox{$\\Delta m^2_{31}$}|$,\n$|(Y_{\\nu}^{\\dagger}Y_{\\nu})^{\\rm IH}_{12}| \\cong\n(4a^2c^2_{23} + s^2_{13}s^2_{23})|\\mbox{$\\Delta m^2_{31}$}|M^2_R\/v_u^4$,\n$|(Y_{\\nu}^{\\dagger}Y_{\\nu})^{\\rm IH}_{13}| \\cong\n(4a^2s^2_{23} + s^2_{13}c^2_{23})|\\mbox{$\\Delta m^2_{31}$}|M^2_R\/v_u^4$ and\n\\begin{align}\n\\text{R}(21\/31)\\cong \n\\frac{4a^2c^2_{23} + s^2_{13}s^2_{23}}\n{4a^2s^2_{23} + s^2_{13}c^2_{23}}\\;.\n\\label{R2131IHb}\n\\end{align}\nIf, however, $\\alpha$ is significantly \ndifferent from zero, say $\\alpha \\cong \\pm \\pi\/2;~\\pm \\pi$,\nand $|a|$ is sufficiently large, being comparable \nin magnitude to $s_{13}$,\nthe terms $\\propto P^{\\rm IH}$ and $\\propto Q^{\\rm IH}$ \nin $|(Y_{\\nu}^{\\dagger}Y_{\\nu})^{\\rm IH}_{12}|$\nor $|(Y_{\\nu}^{\\dagger}Y_{\\nu})^{\\rm IH}_{13}|$\ncan partially compensate each other and we can have \n$\\text{R}(21\/31)\\sim (10^{-3} - 10^{-2})$ or\n$\\text{R}(21\/31)\\sim (10^{2} - 10^{3})$ (Fig. 3).\nFor given $|a|$ and $s_{13}$, the degree of \ncompensation depends on the values of $\\alpha$ \nand $\\delta$ and on the ${\\rm sgn}(a)$.\nIt is maximal in $|(Y_{\\nu}^{\\dagger}Y_{\\nu})^{\\rm IH}_{12(13)}|$,\nfor, e.g., $\\alpha = \\pi$ and $\\delta = 0$, \nor $\\alpha = -\\pi$ and $\\delta = \\pi$,\nand $a >0$ ($a <0$) (Fig. 3).\n\n The ratio of $\\text{BR}(\\mu \\to e + \\gamma)$\nand $\\text{BR}(\\tau \\to \\mu + \\gamma)$\ndepends both on $a$ and $\\alpha$. For $|a| \\leq 0.3$\nwe have $\\text{R}(21\/32)\\ltap 1$; \nif $|a| \\leq 0.1$, the stronger inequality \n$\\text{R}(21\/32)\\ll 1$ typically\nholds. For $|a| \\ll (\\mbox{$ \\Delta m^2_{21}$}\/(8|\\mbox{$\\Delta m^2_{31}$}|))\\sin2\\theta_{12} \n\\cong 4\\times 10^{-3}$ and negligibly small $s_{13}$, \nfor instance, one finds \\cite{PShinYasu05} \n$\\text{R}(21\/32) \\cong 10^{-4}$.\nIf, however, the term $\\propto a$ dominates in \n$|P^{\\rm IH}|$, i.e., if $|a|\\cos 2\\theta_{12} \n\\gg 4\\times 10^{-3}$, we get (for $s_{13}\\sim 0$) \n$\\text{BR}(\\mu \\to e + \\gamma) \\propto |a|^2~|\\mbox{$ \\langle m \\rangle $}_{\\rm IH}|^2$\nand correspondingly,\n\\begin{align}\n\\text{R}(21\/32)\\cong \n4~|a|^2 s^{-2}_{23}~r_{\\rm IH}\n\\left | -1 + 2a\\eta \\left (1 - r_{\\rm IH}\n \\right )^{\\frac{1}{2}} \\right|^{-2}\\;,\n\\label{R2132IHa}\n\\end{align}\nwhere $\\eta \\equiv {\\rm sgn}(\\sin2\\theta_{12}\\sin\\frac{\\alpha}{2})$ and \n\\begin{align}\nr_{\\rm IH} \\equiv \\frac{(\\mbox{$\\left| \\langle m \\rangle\\right|$}_{\\rm IH})^2}{|\\mbox{$\\Delta m^2_{31}$}|} =\n1 - \\sin^22\\theta_{12}\\sin^2\\frac{\\alpha}{2}\\;.\n\\label{R2132IHar}\n\\end{align}\nNow $\\text{R}(21\/32)$ can be considerably larger:\nfor $\\alpha$ varying between 0 and $\\pi$\nand $|a|$ having a value, e.g., in the interval\n(0.04 - 0.10), the ratio of interest\nsatisfies $1.9\\times 10^{-3} \\ltap \n\\text{R}(21\/32) \\ltap 8.0\\times 10^{-2}$,\nthe maximal value corresponding to\n$|a|= 0.1$ and $\\alpha = 0$.\n\n The predictions for the double ratios $\\text{R}(21\/31)$ \nand $\\text{R}(21\/32)$, corresponding to \nIH light neutrino mass spectrum \nare illustrated in Figs. 3 and 4, respectively.\nAs in the case of Figs. 1 and 2,\nthe quantities $|(Y_{\\nu}^{\\dagger}Y_{\\nu})^{\\rm IH}_{ij}|^2$, $i\\neq j$,\nhave been calculated using eqs. (\\ref{YnudYnu}) and \n(\\ref{RPPY}) rather than the approximate\nexpressions given in eqs. (\\ref{12R}) and\n(\\ref{13R}). The lightest neutrino mass $m_3$ set to 0.\nThe leptogenesis CPV parameters\n$b$ and $c$, which can contribute only\nto the higher order corrections in \n$|(Y_{\\nu}^{\\dagger}Y_{\\nu})^{\\rm IH}_{ij}|^2$ of interest,\nwere varied in the same interval as the parameter \n$a$ in the calculations.\nThe effects of the higher order corrections \ndue to $b$ and $c$ is reflected\nin the widths of the lines in Figs. 3 and 4.\n\n\\subsection{\\label{sec:NH}\\large{Quasi-Degenerate Neutrinos}}\n\\hskip 1.0truecm For QD light neutrino mass spectrum, \n$m_{1,2,3} \\cong m\\gtap 0.1$ eV,\none has $\\mbox{$ \\langle m \\rangle $}_{\\rm QD} \\cong m(c_{12}^2 + s_{12}^2e^{i\\alpha})$,\nand $\\sqrt{m_im_j} \\cong m$ in eqs. (\\ref{12R}) - (\\ref{23R}). \nBarring ``accidental'' cancellations, we always have \n$|(Y_{\\nu}^{\\dagger}Y_{\\nu})_{12}| \\sim \n|(Y_{\\nu}^{\\dagger}Y_{\\nu})_{13}|$,\nand correspondingly $\\text{BR}(\\mu \\to e + \\gamma)\\sim\n\\text{BR}(\\tau \\to e + \\gamma)$, in this case (Fig. 5).\nThe expressions for $|(Y_{\\nu}^{\\dagger}Y_{\\nu})_{12(13)}|$\nof interest simplify if ${\\rm max}(|a|,|b|,|c|)\\gg \n{\\rm max}(\\mbox{$ \\Delta m^2_{21}$}\/(4m^2),|\\mbox{$\\Delta m^2_{31}$}|s_{13}\/(4m^2))$ and \n$s_{13} \\ltap 0.1$:\n\\begin{align}\n\\label{12RQD}\n|(Y_{\\nu}^{\\dagger}Y_{\\nu})^{\\rm QD}_{12}|&\\cong\n2~\\frac{M_R}{v_u^2}~\n\\left |c_{23}~P^{\\rm QD} + s_{23}~Q^{\\rm QD}\\right | \\;,\\\\%~~{\\rm QD}\\;,\\\\\n\\label{13RQD}\n|(Y_{\\nu}^{\\dagger}Y_{\\nu})^{\\rm QD}_{13}|& \\cong\n2~\\frac{M_R}{v_u^2}~\n\\left |-s_{23}~P^{\\rm QD} + c_{23}~Q^{\\rm QD}\\right |\\;,\n\\end{align}\nwhere \n\\begin{align}\n\\label{QDP}\nP^{\\rm QD} &= a~\\mbox{$ \\langle m \\rangle $}_{\\rm QD}~e^{-i\\frac{\\alpha}{2}}\\;,\\\\\n\\label{QDQ}\nQ^{\\rm QD} &= m \\left [ \\left (bc_{12} + cs_{12}e^{i\\frac{\\alpha}{2}}\\right )\ne^{-i\\frac{\\beta_M}{2}} \n+ ias_{13}\\sin2\\theta_{12}~e^{-i\\delta}\\sin \\frac{\\alpha}{2} \\right ] \n\\;. \n\\end{align}\nThe condition specified above is compatible with the\nleptogenesis constraints on the product $|abc|$ \\cite{PPY03}.\nFor $\\alpha \\cong 0$ and $\\beta_M \\cong \\pm \\pi$ we get:\n\\begin{align}\n\\text{R}(21\/31)\\cong \n\\frac{c_{23}^2 \\left |P^{\\rm QD}\\right |^2 + \ns_{23}^2\\left |Q^{\\rm QD}\\right |^2}\n{s_{23}^2 \\left |P^{\\rm QD}\\right |^2 + \nc_{23}^2\\left |Q^{\\rm QD}\\right |^2}\\;.\n\\label{R2131QD1} \n\\end{align}\nObviously, in this case either $\\text{R}(21\/31)\\cong 1$ \nindependently of the value of $\\theta_{23}$, or \n$\\text{R}(21\/31)\\cong \\tan^2\\theta_{23}~{\\rm or}~\\cot^2\\theta_{23}$.\n\n Under the condition leading to eqs. \n(\\ref{12RQD}) - (\\ref{QDQ}), the quantity\n$|(Y_{\\nu}^{\\dagger}Y_{\\nu})_{23}|$, eq. (\\ref{23R}),\ncannot be simplified. The term \n$\\propto \\Delta_{31}$ in eq. (\\ref{23R}) \nwill be the dominant one if \\\\ \n${\\rm max}(|a\\sin(\\alpha\/2)|,2|a|s_{13},|a|^2),|b|,|c| \\ll \n|\\mbox{$\\Delta m^2_{31}$}|\/(4m^2)$.\nGiven the leptogenesis constraint on $|abc|$, \nthis is realised, e.g., for $m = 0.1$ eV if\n$|a|\\sim |b|\\sim |c|\\cong 10^{-2}$, \nor if $\\sin(\\alpha\/2)\\cong 0$, $s_{13} \\cong 0$ and\n$|a| \\gg |b|,|c|$ but $|a|^2 \\ll |\\mbox{$\\Delta m^2_{31}$}|\/(4m^2)$. \nIn both cases we have\n\\begin{align}\n\\text{R}(21\/32)\\cong \n\\frac{16~m^4~|a|^2}{(\\mbox{$\\Delta m^2_{31}$})^2}\n\\left | \\frac{\\mbox{$ \\langle m \\rangle $}_{\\rm QD}}{m~s_{23}}\n+ \\frac{bc_{12} + cs_{12}e^{i\\frac{\\alpha}{2}}}{a~c_{23}}\ne^{-i\\frac{\\beta_M - \\alpha}{2}}\\; \\right |^2 \\ll 1\\;.\n\\label{R2132QD0}\n\\end{align}\nFor $m = 0.10$ eV and $|a| = |b| = |c| = 10^{-2}$,\nthe ratio $\\text{R}(21\/32)$ given by eq. (\\ref{R2132QD0})\ndepends on $\\alpha$, $\\beta_M$,\n${\\rm sgn}(b\/a)$ and ${\\rm sgn}(c\/a)$ and\nsatisfies $2\\times 10^{-4} \\ltap \\text{R}(21\/32) \n\\ltap 3\\times 10^{-1}$. If, however, \n$\\alpha \\cong 0$, $s_{13} \\cong 0$ and \n$|a|\\cong 0.2$ with $|abc| \\cong 10^{-5}$, the ``corrections''\n$\\propto |a|^2$ in eq. (\\ref{23R}) will be non-negligible\nsince $|a|^2 \\sim |\\mbox{$\\Delta m^2_{31}$}|\/(4m^2)$.\nIn this case we can have even $\\text{R}(21\/32) \\cong 200$ \nas a consequence of rather strong partial \ncancellation between the different terms in \nthe expression for $|(Y_{\\nu}^{\\dagger}Y_{\\nu})_{23}|$ (Fig. 6).\n\n The term $\\propto \\Delta_{31}$ in eq. (\\ref{23R})\ncan be neglected if, e.g., at least one of the \nCPV parameters $|a~\\sin(\\alpha\/2)|$, \n$|b~\\sin(\\beta_M\/2)\\cos2\\theta_{23}|$ \n($|b~\\cos(\\beta_M\/2)|^2$) and \n$|c~\\sin((\\alpha-\\beta_M)\/2)\\cos2\\theta_{23}|$ \n($|c~\\cos((\\alpha-\\beta_M)\/2)|^2$)\nis much bigger than $|\\mbox{$\\Delta m^2_{31}$}|\/(4m^2)$\n($|\\mbox{$\\Delta m^2_{31}$}|^2\/(4m^2)^2$). \nIn this case eqs. (\\ref{12RQD}) - (\\ref{QDQ})\nare also valid. We get particularly simple expressions for\n$|(Y_{\\nu}^{\\dagger}Y_{\\nu})_{ij}^{\\rm QD}|$, $i \\neq j$,\nif the terms $\\propto a$ in eqs. (\\ref{12R}) - (\\ref{23R}) \ndominate:\n\\begin{align}\n\\label{12RQDa}\n|(Y_{\\nu}^{\\dagger}Y_{\\nu})^{\\rm QD}_{12}|&\\cong\n2~|a|~\\frac{M_R}{v_u^2}~\\mbox{$\\left| \\langle m \\rangle\\right|$}_{\\rm QD}~c_{23} \\;,\\\\ \n\\label{13RQDa}\n|(Y_{\\nu}^{\\dagger}Y_{\\nu})^{\\rm QD}_{13}|&\\cong\n|(Y_{\\nu}^{\\dagger}Y_{\\nu})^{\\rm QD}_{12}|~\\tan\\theta_{23}\\;,\\\\\n\\label{23RQDa}\n|(Y_{\\nu}^{\\dagger}Y_{\\nu})^{\\rm QD}_{23}|&\\cong\n2~|a|~\\frac{M_R}{v_u^2}~\n\\sqrt{m^2 - \\mbox{$\\left| \\langle m \\rangle\\right|$}_{\\rm QD}^2}~c_{23}s_{23}\\;.\n\\end{align}\nEquations (\\ref{12RQDa}) - (\\ref{13RQDa}) are valid provided \n$|a| \\gg {\\rm max}(|b|,|c|,|\\mbox{$\\Delta m^2_{31}$}|\/(4m^2))$,\nwhile eq. (\\ref{23RQDa}) holds if \n$|a~\\sin(\\alpha\/2)|\\gg {\\rm max}(|b|,|c|,|\\mbox{$\\Delta m^2_{31}$}|\/(4m^2))$.\nFor $|a| < 1$ and, e.g., $m \\cong 0.1$ eV, \nthe latter condition requires \n$|\\sin(\\alpha\/2)|\\cong 1$.\nIn these cases both \n$\\text{BR}(\\mu \\to e + \\gamma) \\sim |a|^2\\mbox{$\\left| \\langle m \\rangle\\right|$}_{\\rm QD}^2$ \nand $\\text{BR}(\\tau \\to e + \\gamma)\\sim |a|^2\\mbox{$\\left| \\langle m \\rangle\\right|$}_{\\rm QD}^2$,\nwhile $\\text{BR}(\\tau \\to \\mu + \\gamma)\\sim \n|a|^2~(m^2 - \\mbox{$\\left| \\langle m \\rangle\\right|$}_{\\rm QD}^2) \\cong \n|a|^2~m^2\\sin^22\\theta_{12}\\sin^2(\\alpha\/2)$. For the ratio of the\nfirst two we get \n\\begin{align}\n\\text{R}(21\/31) \\cong \\cot^{2}\\theta_{23}\\;,\n\\label{R2131QD}\n\\end{align}\nwhich should be compared with eqs. (\\ref{R2131NH})\nand (\\ref{R2131IHa}).\nThe ratio of $\\text{BR}(\\mu \\to e + \\gamma)$\nand $\\text{BR}(\\tau \\to \\mu + \\gamma)$\nis independent of the leptogenesis CPV parameter $a$.\nGiven $\\theta_{12}$ and $\\theta_{23}$,\nit is determined by the Majorana phase $\\alpha$: \n\\begin{align}\n\\text{R}(21\/32) \\cong \n\\frac{\\mbox{$\\left| \\langle m \\rangle\\right|$}_{\\rm QD}^2}{(m^2 - \\mbox{$\\left| \\langle m \\rangle\\right|$}_{\\rm QD}^2)~s_{23}^2}\n\\cong \n\\frac{1 - \\sin^22\\theta_{12}\\sin^2(\\alpha\/2)}\n{s_{23}^2~\\sin^22\\theta_{12}\\sin^2(\\alpha\/2)}\n\\;.\n\\label{R2132QDa}\n\\end{align}\nWe get similar results if the terms $\\propto b$ \n($\\propto c$) dominate\nin eqs. (\\ref{12R}) - (\\ref{23R}), which in the case of\neq. (\\ref{23R}) would be possible only if\nthe Majorana phase $\\beta_M$\n(phase difference $\\alpha - \\beta_M$) \ndeviates significantly from $\\pi$.\nNow $|(Y_{\\nu}^{\\dagger}Y_{\\nu})^{\\rm QD}_{23}|$ depends on \n$\\beta_M$ ($\\alpha - \\beta_M$). If, e.g, the terms \n$\\propto c$ dominate we get:\n$\\text{R}(21\/32) \\cong s_{23}^2\\tan^2\\theta_{12}~\n(1 - \\sin^22\\theta_{23}\\sin^2((\\alpha - \\beta_M)\/2))^{-1}$.\n\n Our results for the double ratios $\\text{R}(21\/31)$ and \n$\\text{R}(21\/32)$ are illustrated in Fig. 5.\n\\vspace{-0.2cm}\n\\section{\\large{Conclusions}}\n\n\\vspace{-0.3cm}\n\\hskip 1.0cm Working in the framework of the class of \nSUSY theories with see-saw\nmechanism and soft SUSY breaking with flavour-universal \nboundary conditions at a scale $M_X>M_R$,\nwe have analysed the dependence of the rates of \nlepton flavour violating (LFV) decays \n$\\mu\\rightarrow e + \\gamma$, \n$\\tau \\rightarrow e + \\gamma$ \n$\\tau \\rightarrow \\mu + \\gamma$ ($l_i \\to l_j + \\gamma$) \nand of their ratios, on the Majorana and Dirac \nCP-violation (CPV) phases in the PMNS matrix $\\mbox{$ U_{\\rm PMNS}$}$,\n$\\alpha$, $\\beta_M$ and $\\delta$, and \non the leptogenesis CP-violating (CPV) parameters. \nThe case of quasi-degenerate in mass \nheavy RH neutrinos was investigated,\n$M_1 \\cong M_2\\cong M_3 \\equiv M_R$,\nassuming that splitting between the masses\nof the heavy neutrinos is sufficiently small, \nso that it has practically no effect on the\npredictions for the $l_i \\to l_j + \\gamma$\ndecay rates. Results for the \nnormal hierarchical (NH), \ninverted hierarchical (IH) and quasi-degenerate (QD)\nlight neutrino mass spectra\nhave been derived. The analysis was\nperformed under the condition of negligible \nrenormalization group (RG) effects for the \nlight neutrino masses $m_j$ and the mixing angles \nand CPV phases in $\\mbox{$ U_{\\rm PMNS}$}$ \n\\footnote{It is well-known that \nin the class of SUSY theories considered,\nthis condition is satisfied in the cases of \nNH and IH light neutrino mass spectra;\nit is fulfilled for the QD spectrum \nprovided the SUSY parameter $\\tan\\beta$ \nis relatively small, $\\tan\\beta < 10$.}.\nIn the wide region of validity of eqs. (\\ref{eq_ijg}) \nand (\\ref{eq_ms}) in the relevant SUSY \nparameter space, \nthe ratios of rates of the decays \n$\\mu\\rightarrow e + \\gamma$ and \n$\\tau \\rightarrow e + \\gamma$,\nand $\\mu\\rightarrow e + \\gamma$ and\n$\\tau \\rightarrow \\mu + \\gamma$, \nare independent of the SUSY parameters -\nthey are determined by the neutrino masses ($m_j$) and \nmixing angles, Majorana and Dirac CPV phases and by the\nleptogenesis CPV parameter(s). \n For the matrix of neutrino Yukawa couplings, $\\mathbf{Y_{\\nu}}$ -\na basic quantity in the analysis performed,\nwe have used the orthogonal parametrisation \\cite{Iba01}.\nThe latter proved to be the most convenient for the\npurposes of our study \\cite{PPY03}. \nIn this parametrisation $\\mathbf{Y_{\\nu}}$ is expressed \nin terms of the light and heavy Majorana neutrino masses, \n$\\mbox{$ U_{\\rm PMNS}$}$, and an orthogonal matrix $\\mathbf{R}$~\\cite{Iba01}.\nLeptogenesis can take place only if \n$\\mathbf{R} \\neq {\\bf 1}$ \n(see, e.g., \\cite{LeptoG1,PPRi106}).\nIn the case of quasi-degenerate in mass \nheavy Majorana neutrinos \nconsidered in the present \narticle, the rates of the LFV decays \n$l_i \\to l_j + \\gamma$ of interest \ndo not depend on the matrix \n$\\mathbf{R} \\neq {\\bf 1}$ if \n$\\mathbf{R}$ is real.\nFor complex matrix $\\mathbf{R}$,\nonly the three leptogenesis CPV\nreal dimensionless \nparameters of $\\mathbf{R}$,\n$a$, $b$ and $c$ (eqs. (\\ref{RPPY}) and (\\ref{Aabc})),\nenter into the expressions for\nthe $l_i \\to l_j + \\gamma$ \ndecay branching ratios of interest \\cite{PPY03},\n$\\text{BR}(l_i \\to l_j + \\gamma)$.\nIn our analysis we have assumed that\n$|a|,|b|,|c| < 1$, as is suggested by \nthe leptogenesis constraint derived \nfor QD light neutrinos \nand negligible favour effects\n\\cite{PPY03}. In various\nestimates we have considered values of \n$|a|,|b|,|c| \\leq 0.3$. \nThe case of real matrix $\\mathbf{R}$\ncorresponds effectively to \n$a,b,c=0$.\n\n We have found that for NH (IH) spectrum\nand negligible lightest neutrino mass $m_1$ ($m_3$),\nthe branching ratios \n$\\text{BR}(l_i \\to l_j + \\gamma)$\ndepend, in general, on one Majorana \nand the Dirac CPV phases, $\\alpha - \\beta_M$\n($\\alpha$) and $\\delta$, one leptogenesis CPV \nparameter, $c$ ($a$), on the CHOOZ angle\n$\\theta_{13}$ and on the mixing angles\nand mass squared differences associated with solar and \natmospheric neutrino oscillations,\n$\\theta_{12}$, $\\mbox{$ \\Delta m^2_{21}$}$, and\n$\\theta_{23}$, $\\mbox{$\\Delta m^2_{31}$}$.\nThe double ratios \n$\\text{R}(21\/31) \\propto \\text{BR}(\\mu \\to e + \\gamma)\/\n\\text{BR}(\\tau \\to e + \\gamma)$ and \n$\\text{R}(21\/32) \\propto \\text{BR}(\\mu \\to e + \\gamma)\/\n\\text{BR}(\\tau \\to \\mu + \\gamma)$ (see eq. (\\ref{DoubleR}))\nare determined by these parameters.\nThe same Majorana phase $\\alpha - \\beta_M$ ($\\alpha$) \nenters also into the NH (IH) expression for the \neffective Majorana mass in neutrinoless double beta \n($\\mbox{$\\beta \\beta_{0 \\nu}$}$-) decay, $\\mbox{$ \\langle m \\rangle $}$\n(eqs. (\\ref{meffNH2}) and (\\ref{meffIH1})). \nFor the QD spectrum, \n$\\text{BR}(l_i \\to l_j + \\gamma)$ depend, in general,\non the absolute neutrino mass $m$,\nthe three leptogenesis CPV parameters, $a$, $b$, $c$\nand on the two Majorana phases $\\alpha$ and $\\beta_M$. \nFor the IH and QD spectra, \nthe phase $\\alpha$ enters into the expressions for \n$\\text{BR}(\\mu(\\tau) \\to e + \\gamma)$,\nin particular, through the effective \nMajorana mass $\\mbox{$ \\langle m \\rangle $}$ \n(see eqs. (\\ref{IHP}) and (\\ref{QDP})).\nOur results for the double ratios show that \nwe can have \n$\\text{R}(21\/31) \\sim 1$ or $\n\\text{R}(21\/31) \\ll 1$,\nor else $\\text{R}(21\/31) \\gg 1$\nin the cases of NH and IH spectra, \nwhile for the QD spectrum\ntypically $\\text{R}(21\/31) \\sim 1$.\nIn contrast, for the NH and IH spectra one always \ngets $\\text{R}(21\/32) < 1$; in most of the \nrelevant parameter space \n$\\text{R}(21\/32) \\ll 1$ holds.\nFor the QD spectrum, however, $\\text{R}(21\/32) \\gtap 1$ \nis also possible.\n\n More specifically, we find that for the NH (IH) spectrum, \n$\\text{BR}(\\mu (\\tau) \\to e + \\gamma)$\nexhibit significant dependence \non the leptogenesis CPV parameter $c$ ($a$)\nand on the Majorana CPV phase $\\alpha - \\beta_M$\n($\\alpha$) for $|c| \\gtap 0.02$\n($|a|\\gtap 0.02$) and for any \n$s_{13}\\ltap 0.1$ ($s_{13}\\ltap 0.2$). \nIn certain cases the dependence \nof $\\text{BR}(\\mu (\\tau) \\to e + \\gamma)$\non the phase $\\alpha - \\beta_M$ ($\\alpha$)\nand\/or the parameter $c$ ($a$) is dramatic.\nMore generally, the dependence of \n$\\text{BR}(\\mu (\\tau) \\to e + \\gamma)$ \non the Majorana phase can \nbe noticeable only if the corresponding \nleptogenesis parameter is sufficiently large:\nfor $|c| \\ll {\\rm max} ((\\mbox{$\\Delta m^2_{31}$}\/\\mbox{$ \\Delta m^2_{21}$})^{\\frac{1}{4}}s_{13},\n0.5(\\mbox{$ \\Delta m^2_{21}$}\/\\mbox{$\\Delta m^2_{31}$})^{\\frac{1}{4}})$ \nin the NH case, and $|a| \\ll \n{\\rm max}(s_{13}\/(2\\cos2\\theta_{12}),\n\\mbox{$ \\Delta m^2_{21}$} \\tan2\\theta_{12}\/(8|\\mbox{$\\Delta m^2_{31}$}|)$ in the IH one,\nboth $c~(a)$ and the Majorana phase have practically \nno effect on $\\text{BR}(\\mu (\\tau) \\to e + \\gamma)$.\nSimilarly, the CHOOZ angle $\\theta_{13}$ \nand the Dirac phase $\\delta$ can be relevant \nin the evaluation of $\\text{BR}(\\mu (\\tau) \\to e + \\gamma)$ \nin the cases of NH and IH spectra\nonly if $s_{13}$ is large enough, i.e., \nif respectively, $s_{13}\\gtap \\sqrt{\\mbox{$ \\Delta m^2_{21}$}}\n\\sin2\\theta_{12}\/(2\\sqrt{\\mbox{$\\Delta m^2_{31}$}|}) \\cong 0.07$, \nand $s_{13}\\gtap \\mbox{$ \\Delta m^2_{21}$}\n\\sin2\\theta_{12}\/(2|\\mbox{$\\Delta m^2_{31}$}|) \\cong 8\\times 10^{-3}$.\nIn the case of NH (IH) spectrum,\n$\\text{BR}(\\tau \\to \\mu + \\gamma)$\nis practically independent of $s_{13} \\ltap 0.2$;\nthe dependence of $\\text{BR}(\\tau \\to \\mu + \\gamma)$ on \nthe leptogenesis parameter $c$ ($a$) and the Majorana \nphase $\\alpha - \\beta_M$ ($\\alpha$)\nis relatively weak for $|c| \\ltap 0.3$ ($|a| \\ltap 0.1$).\nFor this wide range of values of $|c|$ ($|a|$)\nwe have $\\text{BR}(\\tau \\to \\mu + \\gamma) \\cong \n\\text{F} \\times (|\\mbox{$\\Delta m^2_{31}$}|\/(4v^2_u))~\n\\sin^22\\theta_{23}$, where the factor \n$\\text{F} \\propto M_R^2\/v_u^2$ \ncontains all the dependence on $M_R$, $\\tan\\beta$ and \nthe SUSY breaking parameters (see eq. (\\ref{eq_ijg})).\n\n The double ratios $\\text{R}(21\/31)$ and $\\text{R}(21\/32)$\n(Figs. 1 - 4) can exhibit in the cases of NH and IH spectra\nstrong dependence on the Dirac and\/or Majorana\nphases if $s_{13} \\sim 0.1 - 0.2$ \nand\/or if the relevant leptogenesis \nparameter exceeds approximately \n$10^{-2}$. Under the indicated conditions values of \n$\\text{R}(21\/31) \\sim (10^{-3} - 10^{-2}) \\ll 1$ or\n$\\text{R}(21\/31) \\sim (10^{3} - 10^{2}) \\gg 1$,\nare possible. For, e.g., $s_{13} \\sim 0.1$, \nthe sign of the inequality is \ndetermined by the sign of \nthe leptogenesis parameter,\nthe value of the Majorana phase\nand\/or the value of the Dirac phase (Figs. 1 and 3).\nIf for the NH (IH) spectrum,\n$\\alpha - \\beta_M\\cong 0$ ($\\alpha \\cong \\pi$) and\n$\\delta \\cong \\pm \\pi\/2,\\pm 3\\pi\/2$,\n$\\text{R}(21\/31)$ takes one of the following three values\n$\\text{R}(21\/31) \\cong 1;\\tan^2\\theta_{23};\\cot^2\\theta_{23}$.\nFor $|a| \\gg \\mbox{$ \\Delta m^2_{21}$}\\tan2\\theta_{12}\/(8|\\mbox{$\\Delta m^2_{31}$}|)$\nin the IH case, we find $\\text{BR}(\\mu(\\tau) \\to e + \\gamma)\\cong\nF^{\\rm IH}\\times|a|^2~|\\mbox{$ \\langle m \\rangle $}_{\\rm IH}|^2\/v_u^2$, and thus\n$\\text{R}(21\/31) \\cong 1$, where $\\mbox{$ \\langle m \\rangle $}_{\\rm IH}$ is the \neffective Majorana mass in $\\mbox{$\\beta \\beta_{0 \\nu}$}$-decay \nand the factor $F^{\\rm IH} \\propto M_R^2\/v_u^2$ \nincludes the dependence on $M_R$ and on the SUSY parameters. \nFor sufficiently small $s_{13}$ and $|c|$ \n($s_{13} \\ll 0.07$, $|c| \\ll 0.2)$) \nin the case of NH spectrum, we get: $\\text{R}(21\/32)\\cong \n\\mbox{$ \\Delta m^2_{21}$}\/(\\mbox{$\\Delta m^2_{31}$} s_{23}^2)~c^2_{12}s^2_{12} \n\\cong 10^{-2}$. Smaller values of \n$\\text{R}(21\/32)$ are possible, e.g., for\n$s_{13}\\cong (0.1 - 0.2)$, if \n$|c| \\sim 0.05$ and if for given ${\\rm sgn}(c)$,\nthe Majorana and Dirac phases\n$(\\alpha - \\beta_M)$ and $\\delta$\nhave specific values (Fig. 2).\nFor the IH spectrum we \ntypically have $\\text{R}(21\/32) \\ll 1$ for\n$|a|\\leq 0.1$.\nIf $2|a|\\cos2\\theta_{12},\\sin\\theta_{13} \n\\ll 0.5\\mbox{$ \\Delta m^2_{21}$}~c_{12}s_{12}\/|\\mbox{$\\Delta m^2_{31}$}|$,\n$\\text{R}(21\/32)$ is completely determined \nby the solar and atmospheric neutrino oscillation parameters \n$\\mbox{$ \\Delta m^2_{21}$}$, $\\theta_{12}$, $\\mbox{$\\Delta m^2_{31}$}$ and $\\theta_{23}$, \nand $\\text{R}(21\/32) \\cong 10^{-4}$.\n\n In the case of QD light neutrino mass spectrum, \nthe leptogenesis constraint\nimplies \\cite{PPY03} $10^{-6} \\ltap |abc| \\ltap 10^{-4}$.\nThe expressions for $\\text{BR}(l_i \\to l_j + \\gamma)$\nand for the double ratios \n$\\text{R}(21\/31)$ and $\\text{R}(21\/32)$ simplify\nconsiderably if the terms including\none given leptogenesis parameter dominate.\nWe get, e.g., $\\text{R}(21\/31) \\cong \\tan^2\\theta_{23}$\nand $\\text{R}(21\/32) \\cong 1$ if the terms \n$\\propto b$ ($\\propto c$) are the dominant one. \nThis requires relatively large values of\n$|a|$ or $|b|$ or $|c|$. If, however,\n$|a| \\sim |b| \\sim |c| \\simeq 10^{-2}$,\n$\\text{R}(21\/32)$ lies in the interval\n$\\sim (10^{-4} - 10^{-1})$. \n\n\\vspace{0.1cm}\n{\\bf Acknowledgments.} \nWe would like to thank Y. Takanishi and W. Rodejohann\nfor useful discussions and T. Schwetz for \ninforming us about results of analysis\nincluding the MINOS data prior publication.\nThis work was supported in part by the Italian MIUR and INFN\nprograms on ``Fisica Astroparticellare''\nand by the the European Network of Theoretical Astroparticle Physics \nILIAS\/N6 under the contract RII3-CT-2004-506222\n(S.T.P.).\n\n\\vspace{0.1cm}\n{\\bf Note Added.} During the completion of the present study\nwe became aware \\cite{WurzQD06} that an analysis along seemingly \nsimilar lines is being performed by R. R\\\"uckl et al. \n\n\\vspace{-0.3cm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Introduction} \\label{sec:intro}\n\nType Ia supernovae (SNe) are unique among the various types of SNe because they result from a similar physical process, the thermonuclear explosion of a white dwarf that has acquired enough mass to approach the Chandrasekhar limit ($\\sim$1.4 M\\textsubscript{\\(\\odot\\)}) through either a merger with or mass transfer from a binary companion. Thus, the population of Type Ia SNe are more homogeneous than core-collapse SNe types, which have a wide range of progenitor masses. However, the intrinsic luminosities and colors of Type Ia SNe still vary \\citep{jha06,maeda11}. Asymmetries in the explosion may contribute to their observed differences. In these cases, the viewing angle may be the cause of peculiar features observed from some SNe but not from others \\citep{maeda11}. A direct method for investigating the explosion asymmetry of an SN is to measure the distribution of the ejecta in the resulting supernova remnant (SNR). The ejecta in SNRs that are only a few hundred years old have yet to be significantly disturbed by interactions with the surrounding interstellar medium (ISM). Thus, the explosion details of the SN may be preserved in the ejecta distribution. Since SNe are inherently 3--D events, the structure of an SNR should ultimately be studied in 3--D. \n\n\n\nTycho's SNR (Tycho, hereafter) is the remnant of the Galactic historical supernova SN 1572. Due to its young age, its well-documented SN light curve and its close proximity to Earth ($\\sim$ 2 -- 4 kpc, see \\citet{hayato10}) , Tycho is an ideal choice for studying the structure of a Type Ia SNR (e.g., \\citet{warren05}). Both the X-ray spectrum of the SNR \\citep{badenes06} and the optical spectrum of the light echo \\citep{krause08b} indicate that Tycho is the remnant of a normal Type Ia supernova, neither subluminous nor overluminous. Tycho appears generally circular in shape, with a diameter of $\\sim$ 8\\arcmin{} in radio and X-ray wavelengths. XMM-Newton observations of Tycho showed an overall uniform distribution of X-ray emitting knots and filaments of shocked ejecta gas \\citep{decourchelle01}. The X-ray spectrum shows bright emission from the shocked metal-rich ejecta, indicating that Si, S, Ar, Ca, and Fe abundances are several times greater than solar values \\citep{hwang98}. Suzaku data showed Doppler broadening of X-ray emission lines over large areas of the SNR that suggest a generally spherical expanding ejecta shell \\citep{hayato10}. \\citet{williams17} measured the speeds of blueshifted and redshifted ejecta knots with the Chandra Advanced CCD Imaging Spectrometer (ACIS) data and found no clear evidence for significant asymmetry in the ejecta distribution in Tycho. In many respects, Tycho appears to be the remnant of a standard Type Ia SN explosion. In fact, Type Ia SNe in general show a low degree of continuum polarization, implying that large deviations from spherical symmetry are not common \\citep{wang08}.\n\nAlthough Tycho may be regarded as the remnant of a close approximation of a standard Type Ia SN, it does contain aspherical features whose origin is not fully understood. A prominent example is in the southeast region of the SNR, where a group of metal-rich X-ray emitting ejecta clumps appears to have overtaken the forward shock \\citep{vancura95,decourchelle01,wang01,fang18,williams20,sato20}. These ejecta clumps protruding from the southeastern boundary of the SNR include Fe-rich ejecta gas that can be used to pinpoint specific nucleosynthesis models, e.g., an incomplete Si burning or an $\\alpha$-rich freeze-out regime \\citep{yamaguchi17}. This is in contrast to the western side of the remnant, where there is obvious separation between the ejecta and forward shock \\citep{warren05}. A Chandra study of the proper motions of reverse-shocked gas showed large azimuthal variations on the order of 50\\%, while a Spitzer study suggested an ambient density enhancement by a factor of $\\sim$ 3 -- 10 in the northeastern regions compared to the southwest portion of the remnant \\citep{williams13}. \\citet{sato19} hydrodynamically simulated Tycho's clumpy structure assuming initially clumped ejecta, as well as perfectly smooth ejecta, in all cases evolving through a uniform ambient medium. Even for the perfectly smooth case, a clumpy structure appears in the ejecta due to Rayleigh-Taylor and Kelvin-Helmholtz instabilities. However, the observed structure in Tycho is more consistent with an initial clumped ejecta structure from the SN rather than instabilities arising from the ejecta interaction with the ambient medium. The optical light echo spectrum of Tycho shows an uncommon high-velocity \\ion{Ca}{2} absorption feature \\citep{krause08}, and thus an asymmetry in the ejecta distribution may have developed early in the evolution of the SNR, or in the SN itself. \\citet{sato17tych} performed detailed spectral fits on 27 individual X-ray emitting ejecta clumps across Tycho. They found a disparity in the maximum velocities of redshifted and blueshifted features, $\\lesssim$ 7800 km s \\textsuperscript{-1} and $\\lesssim$ 5000 km s\\textsuperscript{-1}, respectively. The authors also noted large-scale He-like Si K$\\alpha$ line centroid shifts across the SNR, on the order of arcminutes. They suggested that the apparent shifts may be due to differences in the intrinsic intensity of the approaching and receding sides of Tycho.\n\n\n\nHere, we investigate the line--of--sight velocity distributions of the clumpy metal-rich ejecta in Tycho based on our deep 450 ks Chandra High Energy Transmission Grating (HETG) observation. The high resolution HETG spectroscopy has significant advantages over the ACIS spectroscopy in several aspects. For example, the gain energies of the ACIS detectors vary by up to $0.3\\%$ of the laboratory values\\footnote{https:\/\/cxc.harvard.edu\/proposer\/POG\/html\/chap6.html\\#tth\\_sEc6.8}. For the He-like Si K$\\alpha$ energy, this corresponds to an estimated uncertainty in the line--of--sight (radial) velocity up to 900 km s\\textsuperscript{-1}. The type of CCD array, either ACIS-I or ACIS-S, also adds uncertainties to the line-center energies \\citep{sato17tych}. ACIS data show considerable systematic uncertainties on the emission line-center energy depending on background subtraction regions. These effects contribute to overall systematic uncertainties of up to 2000 km s\\textsuperscript{-1} in ACIS radial velocity ($v_r$) measurements \\citep{sato17tych}. The dispersed HETG spectroscopy can avoid such systematic uncertainties associated with the ACIS spectroscopy. Absolute wavelength uncertainties in line-center measurements with the HETG are generally $\\lesssim$ 100 km s\\textsuperscript{-1} \\citep{marshall04,ishibashi06}\\footnote{https:\/\/cxc.harvard.edu\/proposer\/POG\/html\/chap8.html}. Thus, HETG line-center energy measurements are dominated by statistical uncertainties, while the ACIS line-center measurements are dominated by systematic uncertainties. In this work, we combine our radial velocity measurements of clumpy ejecta knots using our deep HETG data with the proper motion measurements of those knots using archival ACIS imaging data--sets to build a 3--D picture of the overall ejecta structure. In Section \\ref{sec:obs}, we present the details of our deep Chandra HETG observation. In Section \\ref{sec:data}, we report our analysis and results, and in Section \\ref{sec:discuss} we discuss our interpretations. We conclude our study in Section \\ref{sec:conclusions}.\n\n\\clearpage\n\n\\section{Observations} \\label{sec:obs}\n\n\nWe performed our Chandra HETG observations of Tycho from 2017 October 17 to 2017 November 19. The aimpoint was set at R.A.(J2000) = 00\\textsuperscript{h}25\\textsuperscript{m}19\\textsuperscript{s}.0, decl.(J2000) = +64$^{\\circ}$08\\arcmin 10\\farcs{}0, which is close to the geometric center of the roughly circular SNR. The date and exposure time of each observation are listed in Table \\ref{tab:obs_hetg}. The total effective exposure time is 443 ks. We processed the raw event files using Chandra Interactive Analysis of Observations (CIAO) \\citep{fruscione06} version 4.11 and the Chandra Calibration Database (CALDB) version 4.8.2 to create a new level=2 event file using the CIAO command, {\\tt\\textbf{chandra\\_repro}}. We extracted the 1st-order dispersed spectra from a number of small regions across the SNR (Section \\ref{subsec:radv}) using the TGCat scripts \\citep{huenemoerder11} {\\tt\\textbf{tg\\_create\\_mask}}\\footnote{http:\/\/cxc.harvard.edu\/ciao\/ahelp\/tg\\_create\\_mask.html}, {\\tt\\textbf{tg\\_resolve\\_events}}\\footnote{http:\/\/cxc.harvard.edu\/ciao\/ahelp\/tg\\_resolve\\_events.html}, and {\\tt\\textbf{tgextract}}\\footnote{http:\/\/cxc.harvard.edu\/ciao\/ahelp\/tg\\_extract.html}, and created the full set of corresponding detector response files using the script, {\\tt\\textbf{make\\_responses}}, which accounts for the zeroth-order position and dispersed region size and orientation. The HETG-dispersed image of Tycho is shown in Figure \\ref{fig:3color}. \n\n\n\n\n\n\nWe also use archival ACIS-I observations of Tycho (Table \\ref{tab:obs_acis}) to supplement the HETG data analysis. For the ObsIDs taken in 2009, we combined all 9 individual ObsIDs, re-projecting them onto ObsID 10095 which had the longest exposure. The main purpose of our archival ACIS data analysis is to measure proper motions of small ejecta knots in Tycho. Thus, we re-align these ACIS images taken at three different epochs using the {\\tt\\textbf{reproject\\_aspect}} command in CIAO, accounting for the astrometric correction based on 5 to 16 background point sources (depending on the ObsID) with their sky positions identified in the NASA\/IPAC Extragalactic Database (NED)\\footnote{https:\/\/ned.ipac.caltech.edu\/}.\n\n\n\\begin{figure}\n\\plotone{hetg_image_draw_v2.pdf}\n\\caption{Chandra HETG 3-color dispersed image of Tycho. Red: 0.7-1.2 keV, Green: 1.7-2.0 keV and Blue: 4.0-8.0 keV. Our color codes are selected to represent the Fe L line complex (red), He-like Si K$\\alpha$ lines (green), and the continuum-dominated band (blue), respectively. The white arrows show the dispersion directions of the Medium and High Energy Gratings.\n\\label{fig:3color}}\n\\end{figure}\n\n\n\n\\section{Data Analysis and Results} \\label{sec:data}\n\n\n\n\\subsection{Region Selection} \\label{subsec:hetgs}\n\n\nBased on the archival Chandra ACIS observation of Tycho, we identified clumpy emission features that are bright in the Si K$\\alpha$ (1.7 - 2.0 keV) band as candidate targets for HETG spectral extraction. We extracted their individual 1st-order Medium Energy Grating (MEG) spectra from our Chandra HETG observation. In Figure \\ref{fig:3color}, it is clear that the zeroth-order and 1st-order images overlap. In some regions, this overlap may cause the 1st-order HETG spectrum to be contaminated by the overlapping zeroth-order emission. To avoid these regions, we compared the counts in the +1 and --1 order spectra. The majority of the knots in our sample have a +1 to --1 order counts ratio that is within a factor of $\\sim$ 2. This difference in counts is consistent with the variations in effective area across the +1 and --1 order legs in the Si K$\\alpha$ band\\footnote{See footnote 2}, and thus their dispersed spectra are unlikely to be significantly affected by the zeroth order emission. We note that two knots in our sample (SW3 and SW7) have a larger disparity in counts between the +1 and -1 orders, up to a factor of 5. The $v_r$ estimates for these knots may be less reliable due to possible contamination from the zeroth order emission. Applying similar methods to those developed by \\citet{millard20}, we select 59 small candidate ejecta regions (angular sizes of $\\sim$ 3\\arcsec{} -- 10\\arcsec{}) to measure the He-like Si K$\\alpha$ line-center energy ($\\sim$ 1.86 keV) for each region. We selected target regions such that similarly bright emission features are at least $\\sim$ 20\\arcsec{} away along the dispersion direction to avoid spectral contamination. We generally avoided choosing knots for our sample that are asymmetrically extended in the dispersion direction. Each region has at least 200 1st order MEG counts in the Si K$\\alpha$ band.\n\n\n\n\n\n\n\n\n\\startlongtable\n\\begin{deluxetable*}{ccC}\n\\tablecaption{Chandra HETG Observations of Tycho's SNR\\label{tab:obs_hetg}}\n\\tablecolumns{3}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{Observation ID} &\n\\colhead{Start Date} &\n\\colhead{Exposure Time (ks) }}\n\\startdata\n19293\t & \t2017-10-17\t & \t49.7\t\\\\\n20813\t & \t2017-10-21\t & \t47.8\t\\\\\n20822\t & \t2017-10-23\t & \t13.9\t\\\\\n19292\t & \t2017-10-26\t & \t19.8\t\\\\\n20820\t & \t2017-10-27\t & \t30.5\t\\\\\n20819\t & \t2017-10-29\t & \t44.5\t\\\\\n19291\t & \t2017-10-30\t & \t40.0\t\\\\\n20832\t & \t2017-11-01\t & \t50.1\t\\\\\n20833\t & \t2017-11-03\t & \t34.6\t\\\\\n20834\t & \t2017-11-04\t & \t35.9\t\\\\\n20835\t & \t2017-11-06\t & \t27.6\t\\\\\n20799\t & \t2017-11-17\t & \t22.2\t\\\\\n20821\t & \t2017-11-19\t & \t25.6\t\\\\\n\\enddata\n\\end{deluxetable*}\n\n\n\\startlongtable\n\\begin{deluxetable*}{ccC}\n\\tablecaption{Archival Chandra ACIS-I Observations of Tycho's SNR\\label{tab:obs_acis}}\n\\tablecolumns{3}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{Observation ID} &\n\\colhead{Start Date} &\n\\colhead{Exposure Time (ks) }}\n\\startdata\n3837\t & \t2003-04-29\t & \t144.6\t\\\\\n10093\t & \t2009-04-13\t & \t117.6\t\\\\\n10094\t & \t2009-04-18\t & \t89.9\t\\\\\n10095\t & \t2009-04-23 \t & \t173.4\t\\\\\n10096\t & \t2009-04-27\t & \t104.9\t\\\\\n10097\t & \t2009-04-11 \t & \t106.9\t\\\\\n10902\t & \t2009-04-15 \t & \t39.3\t\\\\\n10903\t & \t2009-04-17 \t & \t23.9\t\\\\\n10904\t & \t2009-04-13\t & \t34.7\t\\\\\n10906\t & \t2009-05-03 \t & \t40.9\t\\\\\n15998\t & \t2015-04-22\t & \t146.7\t\\\\\n\\enddata\n\\end{deluxetable*}\n\n\n\n\\subsection{Ejecta Identification} \\label{subsec:specmod}\n To identify the overabundant nature of the ejecta-dominated regions (out of our 59 selected candidate regions), we performed spectral model fits for each individual regional spectrum based on the combined 2009 archival Chandra ACIS data (merging all ObsIDs taken in 2009, to achieve the total of $\\sim$ 731 ks). We fitted the observed 1.6 -- 4.5 keV band ACIS spectrum extracted from each region with an absorbed {\\tt\\textbf{vpshock}} model \\citep{borkowski01} using the XSPEC software package version 12.10.1 \\citep{arnaud96}. We estimated the background spectrum using an annulus region encircling the entire remnant. Then, we subtracted the background spectrum from the regional source spectra before fitting them with the spectral model. We fixed the absorption column at $N_H$ = 8 $\\times$ 10\\textsuperscript{21} cm\\textsuperscript{-2} for Tycho \\citep{foight16}. We allowed the electron temperature, $kT$, and ionization timescale, $\\tau$ ($\\tau$ = {\\it n\\textsubscript{e}t}, where {\\it n\\textsubscript{e}} is the electron density, and {\\it t} is the time since being shocked), to vary. We also varied the redshift, normalization, and abundances of Si, S, Ar, and Ca. Since contributions from other elements are negligible in the 1.6 -- 4.5 keV band, we fixed all other elemental abundances at solar values \\citep{wilms00}. The model gave satisfactory fits to the data, with reduced chi-squared values ranging from $\\chi^{2}\/dof$ = 51\/73 -- 119\/60. We confirm that the best--fit abundances are several times solar values, indicating that all knots in our sample are ejecta-dominated. The best-fit electron temperatures of the ejecta knots in our sample are {\\it kT\\textsubscript{e}} $\\sim$ 1 -- 5 keV, with ionization timescales $\\tau$ $\\sim$ 2 -- 50 $\\times$ 10\\textsuperscript{10} cm\\textsuperscript{-3} s, generally consistent with the typical ejecta values reported in \\citet{williams17}. \n\n\n\n\\subsection{Radial Velocities of Ejecta} \\label{subsec:radv}\n\nTo measure the radial velocity of each X-ray emission feature in our sample, we adopt the method used in \\citet{millard20}, using the Interactive Spectral Interpretation System (ISIS) software package version 1.6.2 \\citep{houck00}. The spatially integrated broadband ACIS spectrum of Tycho shows bright Si, S, Ar, Ca, and Fe emission lines. However, we are interested in the HETG spectra of small features that are only a few arcseconds across. The small region sizes and the low HETG detection efficiency greatly reduces the prominence of most of these lines. Ultimately, our HETG spectra of these small emission features are dominated by the He-like Si-K$\\alpha$ line. For each of our regional spectra, we measure the line-center energies in the Si K$\\alpha$ band by fitting six Gaussian curves to the spectrum to account for the three He--like Si K$\\alpha$ lines (6.648 \\AA{} for resonance, 6.688 \\AA{} for intercombination, 6.740 \\AA{} for the forbidden line), and two Li-like Si XII lines at 6.717 \\AA{} and 6.782 \\AA{} \\citep{drake88} and one for the diffuse background emission of the SNR. We jointly fit the model to the MEG +\/- 1 order spectra, tying the line-center wavelengths between spectra dispersed along the positive and negative arms. To account for the extent of the emission along the dispersion direction, the widths of the Gaussian curves are allowed to vary. The widths of the five source Gaussians are tied, while the width of the background Gaussian component varies independently and is generally much broader. Since the individual spectral lines may not be clearly resolved due to the extended nature of the ejecta knots, we fix the flux ratios among the triplet He-like Si K$\\alpha$ lines and Li-like Si lines at those corresponding to the best-fit electron temperature and ionization timescale for each region (Section \\ref{subsec:specmod}). We compared our measured line-center wavelength with the rest value for the resonance line at 6.648 \\AA{}, which is generally the strongest among the five lines in our model. The difference between the rest and observed values gives the Doppler shift, which we use to estimate the {\\it v\\textsubscript{r}} for each knot. The location of each knot is marked in Figure \\ref{fig:knotloc}a, and our results are summarized in Table \\ref{tab:mathmode}. Example HETG spectra and best-fit models for the +1 and -1 arms are shown in Figures \\ref{fig:knotloc}b and \\ref{fig:knotloc}c.\n\n\n\\defcitealias{sato17tych}{SH17a}\n\\defcitealias{williams17}{W17}\n\nIn Figure \\ref{fig:knotloc}a, we show regions for which we measure Doppler shifts of Si lines. Our measured $v_r$ ranges from $\\sim$ --5200 to +5300 km s\\textsuperscript{-1}. We note that our sample partially overlaps with those studied by \\citet{sato17tych} and \\citet{williams17} (\\citetalias{sato17tych} and \\citetalias{williams17}, hereafter), who measured the $v_r$ of small ejecta regions in Tycho based on the lower-resolution ACIS spectroscopy: i.e., 15 and 19 regions of our sample are also included in \\citetalias{sato17tych} and \\citetalias{williams17}, respectively. We find general agreement between our measured values and those from ACIS data, as shown in Figure \\ref{fig:acis_vs_hetg}. We found a few exceptions where our measured radial velocities are smaller than those in \\citetalias{sato17tych} by a few $10^3$ km s\\textsuperscript{-1} (e.g., regions C6 and SW3). The origin of the discrepancy is unclear, but may be due in part to confusion from neighboring emission. Contributions from the diffuse expanding hemispheres of the remnant may be present even in small extraction regions of only a few arcseconds in diameter, and could influence the ACIS velocity estimates. \n\nIt is remarkable that we measure a highly significant radial velocity of $v_r$ = $-1860$ $\\pm$ 170 km s\\textsuperscript{-1} for the SE protrusion (region SE3 in Figure \\ref{fig:knotloc}a). This $v_r$ has been suggested based on the ACIS spectroscopy, but was not constrained due to large uncertainties of a few $10^3$ km s\\textsuperscript{-1} \\citepalias{sato17tych,williams17}. Based on our high resolution HETG spectroscopy, we accurately measure (within $\\sim$ 10\\% uncertainties) this intriguing $v_r$ for an ejecta feature projected beyond the main shell of the SNR with an order of magnitude improved accuracy. \n\n\n\n\n \n\\startlongtable\n\\begin{deluxetable*}{ccCcccccccc}\n\\tablecaption{Radial Velocity and Proper Motion Measurements of Ejecta Features in Tycho's SNR \\label{tab:mathmode}}\n\\tablecolumns{12}\n\\tablewidth{0pt}\n\\tabletypesize{\\scriptsize}\n\\tablehead{\n\\colhead{Region} &\n\\colhead{R.A.\\tablenotemark{a}} &\n\\colhead{Decl.\\tablenotemark{a}} &\n\\colhead{{\\it D}\\tablenotemark{b}} &\n\\colhead{{\\it v\\textsubscript{r}}} &\n\\colhead{$\\mu_{RA}$\\tablenotemark{c}} & \\colhead{$\\mu_{Dec}$\\tablenotemark{c}} & \\colhead{$\\mu_{Tot}$\\tablenotemark{d}} & \\colhead{$\\eta$\\tablenotemark{e}} &\n\\colhead{{\\it v\\textsubscript{s}}\\tablenotemark{f}} &\n \\\\ \n\\colhead{} & \\colhead{(degree)} &\n\\colhead{(degree)} & \\colhead{(arcmin)} & \\colhead{(km s\\textsuperscript{-1})} & \\colhead{(arcsec yr\\textsuperscript{-1})} & \\colhead{(arcsec yr\\textsuperscript{-1})} & \\colhead{(arcsec yr\\textsuperscript{-1})} & & \\colhead{(km s\\textsuperscript{-1})}\n}\n\\startdata\nSE1 & 6.48336 & 64.130282 & 4.07 & -2470$_{-420}^{+410}$ & -0.323 $\\pm$ 0.064 & -0.002\\tablenotemark{*} & 0.323 $\\pm$ 0.064 & 0.6 $\\pm$ 0.12 & 5910 $\\pm$ 980 \\\\\nSE2 & 6.47022 & 64.128169 & 3.74 & -670 $\\pm$ 560 & -0.09 $\\pm$ 0.064 & -0.057 $\\pm$ 0.04 & 0.106 $\\pm$ 0.066 & 0.21 $\\pm$ 0.13 & 1890 $\\pm$ 1040 \\\\\nSE3 & 6.49035 & 64.125542 & 4.28 & -1860 $\\pm$ 170 & -0.299 $\\pm$ 0.063 & -0.096 $\\pm$ 0.04 & 0.314 $\\pm$ 0.064 & 0.56 $\\pm$ 0.11 & 5530 $\\pm$ 1000 \\\\\nSE4 & 6.46192 & 64.12032 & 3.6 & 1650$_{-710}^{+690}$ & - & - & - & - & - \\\\\nSE5 & 6.40962 & 64.125008 & 2.2 & 3840 $\\pm$ 820 & - & - & - & - & - \\\\\nSE6 & 6.47617 & 64.106979 & 4.21 & 1660 $\\pm$ 660 & -0.249 $\\pm$ 0.062 & -0.061 $\\pm$ 0.038 & 0.257 $\\pm$ 0.063 & 0.45 $\\pm$ 0.11 & 4580 $\\pm$ 1000 \\\\\nSE7 & 6.46718 & 64.10879 & 3.95 & 1380$_{-620}^{+610}$ & -0.314 $\\pm$ 0.063 & 0.06 $\\pm$ 0.035 & 0.32 $\\pm$ 0.063 & 0.6 $\\pm$ 0.12 & 5490 $\\pm$ 1020 \\\\\nSE8 & 6.41552 & 64.118707 & 2.47 & 2780$_{-710}^{+690}$ & -0.238 $\\pm$ 0.061 & -0.086 $\\pm$ 0.033 & 0.253 $\\pm$ 0.061 & 0.76 $\\pm$ 0.18 & 5040 $\\pm$ 930 \\\\\nSE9 & 6.3844 & 64.120199 & 1.69 & 4210$_{-1030}^{+980}$ & - & - & - & - & - \\\\\nSE10 & 6.42579 & 64.099514 & 3.29 & 2780 $\\pm$ 680 & - & - & - & - & - \\\\\nSE11 & 6.4111 & 64.088671 & 3.49 & 1070$_{-740}^{+730}$ & - & - & - & - & - \\\\\nSE12 & 6.36942 & 64.113216 & 1.65 & -920 $\\pm$ 490 & - & - & - & - & - \\\\\nSE13 & 6.39798 & 64.086552 & 3.39 & 2100 $\\pm$ 430 & - & - & - & - & - \\\\\nSE14 & 6.36005 & 64.090842 & 2.72 & 5180$_{-740}^{+770}$ & -0.012\\tablenotemark{*} & -0.171 $\\pm$ 0.04 & 0.172 $\\pm$ 0.07 & 0.44 $\\pm$ 0.19 & 5920$_{-860}^{+880}$ \\\\\nSE15 & 6.34372 & 64.085777 & 2.93 & -20\\tablenotemark{*} & -0.053\\tablenotemark{*} & -0.326 $\\pm$ 0.037 & 0.33 $\\pm$ 0.067 & 0.8 $\\pm$ 0.17 & 5480 $\\pm$ 1110 \\\\\nNE1 & 6.36031 & 64.198296 & 3.95 & 960 $\\pm$ 600 & 0.001\\tablenotemark{*} & 0.354 $\\pm$ 0.038 & 0.354 $\\pm$ 0.068 & 0.71 $\\pm$ 0.13 & 5950 $\\pm$ 1120 \\\\\nNE2 & 6.34541 & 64.162038 & 1.74 & 2460$_{-1070}^{+1010}$ & -0.001\\tablenotemark{*} & 0.158 $\\pm$ 0.041 & 0.158 $\\pm$ 0.071 & 0.77 $\\pm$ 0.3 & 3600$_{-1130}^{+1100}$ \\\\\nNE3 & 6.37907 & 64.182406 & 3.19 & -2290$_{-600}^{+590}$ & -0.029\\tablenotemark{*}& 0.243 $\\pm$ 0.038 & 0.245 $\\pm$ 0.068 & 0.61 $\\pm$ 0.16 & 4670$_{-1030}^{+1020}$ \\\\\nNE4 & 6.35505 & 64.156519 & 1.52 & -3590$_{-500}^{+490}$ & -0.062\\tablenotemark{*} & 0.077 $\\pm$ 0.038 & 0.099 $\\pm$ 0.067 & 0.54 $\\pm$ 0.33 & 3950$_{-650}^{+640}$ \\\\\nNE5 & 6.38373 & 64.178972 & 3.06 & -4210$_{-620}^{+600}$ & - & - & - & - & - \\\\\nNE6 & 6.40231 & 64.193381 & 4.06 & 910$_{-470}^{+460}$ & -0.125 $\\pm$ 0.063 & 0.195 $\\pm$ 0.038 & 0.231 $\\pm$ 0.067 & 0.45 $\\pm$ 0.12 & 3940 $\\pm$ 1090 \\\\\nNE7 & 6.40808 & 64.189185 & 3.92 & -400\\tablenotemark{*} & -0.16 $\\pm$ 0.063 & 0.207 $\\pm$ 0.04 & 0.262 $\\pm$ 0.068 & 0.53 $\\pm$ 0.13 & 4360 $\\pm$ 1130 \\\\\nNE8 & 6.40062 & 64.165145 & 2.66 & -1800 $\\pm$ 440 & -0.126 $\\pm$ 0.061 & 0.158 $\\pm$ 0.034 & 0.202 $\\pm$ 0.063 & 0.61 $\\pm$ 0.18 & 3800$_{-940}^{+950}$ \\\\\nNE9 & 6.45171 & 64.179368 & 4.23 & 490\\tablenotemark{*} & -0.133 $\\pm$ 0.062 & 0.159 $\\pm$ 0.035 & 0.207 $\\pm$ 0.064 & 0.38 $\\pm$ 0.11 & 3460 $\\pm$ 1050 \\\\\nNE10 & 6.43524 & 64.161668 & 3.26 & -1660$_{-480}^{+490}$ & -0.204 $\\pm$ 0.062 & 0.049 $\\pm$ 0.035 & 0.21 $\\pm$ 0.062 & 0.51 $\\pm$ 0.14 & 3860 $\\pm$ 950 \\\\\nNE11 & 6.46282 & 64.153989 & 3.72 & 100\\tablenotemark{*} & -0.173 $\\pm$ 0.062 & 0.021\\tablenotemark{*} & 0.174 $\\pm$ 0.062 & 0.36 $\\pm$ 0.12 & 2900 $\\pm$ 1030 \\\\\nNW1 & 6.20057 & 64.139647 & 3.35 & 1900 $\\pm$ 570 & 0.231 $\\pm$ 0.064 & -0.077 $\\pm$ 0.038 & 0.244 $\\pm$ 0.064 & 0.53 $\\pm$ 0.14 & 4470 $\\pm$ 990 \\\\\nNW2 & 6.1934 & 64.153122 & 3.7 & -800 $\\pm$ 670 & 0.264 $\\pm$ 0.064 & 0.103 $\\pm$ 0.039 & 0.284 $\\pm$ 0.065 & 0.57 $\\pm$ 0.13 & 4780 $\\pm$ 1070 \\\\\nNW3 & 6.22578 & 64.149849 & 2.84 & -310\\tablenotemark{*} & 0.17 $\\pm$ 0.064 & 0.071 $\\pm$ 0.036 & 0.185 $\\pm$ 0.064 & 0.48 $\\pm$ 0.17 & 3080 $\\pm$ 1060 \\\\\nNW4 & 6.21326 & 64.154398 & 3.24 & 1180 $\\pm$ 580 & 0.227 $\\pm$ 0.064 & 0.083 $\\pm$ 0.038 & 0.242 $\\pm$ 0.065 & 0.56 $\\pm$ 0.15 & 4190 $\\pm$ 1050 \\\\\nNW5 & 6.17947 & 64.160641 & 4.2 & 220\\tablenotemark{*} & 0.289 $\\pm$ 0.065 & 0.155 $\\pm$ 0.037 & 0.328 $\\pm$ 0.066 & 0.6 $\\pm$ 0.12 & 5450 $\\pm$ 1100 \\\\\nNW6 & 6.25986 & 64.154143 & 2.15 & -3980$_{-660}^{+650}$ & 0.162 $\\pm$ 0.063 & 0.039 $\\pm$ 0.037 & 0.167 $\\pm$ 0.064 & 0.6 $\\pm$ 0.22 & 4850$_{-820}^{+810}$ \\\\\nNW7 & 6.25252 & 64.162735 & 2.62 & -1000 $\\pm$ 600 & 0.126 $\\pm$ 0.064 & 0.14 $\\pm$ 0.04 & 0.189 $\\pm$ 0.068 & 0.55 $\\pm$ 0.19 & 3290 $\\pm$ 1090 \\\\\nNW8 & 6.26811 & 64.162003 & 2.3 & -3430 $\\pm$ 500 & 0.116 $\\pm$ 0.063 & 0.081 $\\pm$ 0.037 & 0.141 $\\pm$ 0.065 & 0.47 $\\pm$ 0.21 & 4150$_{-730}^{+740}$ \\\\\nNW9 & 6.29317 & 64.153676 & 1.49 & 5320$_{-950}^{+920}$ & - & - & - & - & - \\\\\nNW10 & 6.2808 & 64.16993 & 2.48 & -2620$_{-540}^{+560}$ & 0.029\\tablenotemark{*} & 0.102 $\\pm$ 0.034 & 0.106 $\\pm$ 0.064 & 0.34 $\\pm$ 0.19 & 3160$_{-740}^{+750}$ \\\\\nNW11 & 6.30679 & 64.162212 & 1.78 & -1120 $\\pm$ 640 & - & - & - & - & - \\\\\nNW12 & 6.30327 & 64.166397 & 2.05 & -5220 $\\pm$ 890 & - & - & - & - & - \\\\\nNW13 & 6.29901 & 64.192441 & 3.58 & -80* & -0.030\\tablenotemark{*} & 0.147 $\\pm$ 0.04 & 0.15 $\\pm$ 0.07 & 0.33 $\\pm$ 0.14 & 2490 $\\pm$ 1160 \\\\\nNW14 & 6.31804 & 64.168952 & 2.11 & 5280$_{-2070}^{+980}$ & 0.031\\tablenotemark{*}& 0.149 $\\pm$ 0.036 & 0.152 $\\pm$ 0.065 & 0.58 $\\pm$ 0.23 & 5860$_{-1920}^{+1000}$ \\\\\nSW1 & 6.28838 & 64.07014 & 3.97 & 160\\tablenotemark{*} & - & - & - & - & - \\\\\nSW2 & 6.29334 & 64.094728 & 2.53 & 1140 $\\pm$ 520 & 0.076 $\\pm$ 0.063 & -0.194 $\\pm$ 0.039 & 0.208 $\\pm$ 0.068 & 0.57 $\\pm$ 0.2 & 3640 $\\pm$ 1080 \\\\\nSW3 & 6.30962 & 64.113309 & 1.33 & 3860$_{-900}^{+870}$$^{\\dagger{}}$ & - & - & - & - & - \\\\\nSW4 & 6.2801 & 64.091581 & 2.84 & 2060$_{-560}^{+580}$ & 0.061\\tablenotemark{*} & -0.214 $\\pm$ 0.036 & 0.223 $\\pm$ 0.065 & 0.55 $\\pm$ 0.17 & 4240 $\\pm$ 980 \\\\\nSW5 & 6.27296 & 64.096338 & 2.68 & 840$_{-480}^{+470}$ & 0.111 $\\pm$ 0.063 & -0.106 $\\pm$ 0.035 & 0.153 $\\pm$ 0.064 & 0.4 $\\pm$ 0.18 & 2680 $\\pm$ 1020 \\\\\nSW6 & 6.24856 & 64.084384 & 3.64 & 860$_{-630}^{+680}$ & 0.021\\tablenotemark{*} & -0.208 $\\pm$ 0.035 & 0.209 $\\pm$ 0.065 & 0.41 $\\pm$ 0.13 & 3570 $\\pm$ 1060 \\\\\nSW7 & 6.28942 & 64.111983 & 1.67 & 3010$_{-580}^{+590}$$^{\\dagger{}}$ & - & - & - & - & - \\\\\nSW8 & 6.25281 & 64.108776 & 2.49 & 2370$_{-720}^{+730}$ & 0.018\\tablenotemark{*} & -0.073 $\\pm$ 0.038 & 0.076 $\\pm$ 0.067 & 0.21 $\\pm$ 0.2 & 2680$_{-820}^{+830}$ \\\\\nSW9 & 6.22021 & 64.099009 & 3.52 & 2540 $\\pm$ 700 & 0.153 $\\pm$ 0.063 & -0.068 $\\pm$ 0.037 & 0.168 $\\pm$ 0.064 & 0.34 $\\pm$ 0.13 & 3760 $\\pm$ 920 \\\\\nSW10 & 6.22228 & 64.105266 & 3.26 & -1630 $\\pm$ 340 & 0.21 $\\pm$ 0.063 & -0.154 $\\pm$ 0.038 & 0.26 $\\pm$ 0.065 & 0.57 $\\pm$ 0.15 & 4620 $\\pm$ 1020 \\\\\nSW11 & 6.27209 & 64.12145 & 1.65 & 2830$_{-680}^{+700}$ & - & - & - & - & - \\\\\nSW12 & 6.22676 & 64.120373 & 2.78 & 740$_{-630}^{+640}$ & 0.118 $\\pm$ 0.063 & -0.081 $\\pm$ 0.036 & 0.143 $\\pm$ 0.064 & 0.37 $\\pm$ 0.17 & 2490 $\\pm$ 1030 \\\\\nC1 & 6.3738 & 64.143253 & 1.32 & -1700 $\\pm$ 590 & - & - & - & - & - \\\\\nC2 & 6.36353 & 64.132469 & 0.93 & -2860$_{-570}^{+580}$ & - & - & - & - & - \\\\\nC3 & 6.34361 & 64.139439 & 0.52 & 4460$_{-770}^{+780}$ & - & - & - & - & - \\\\\nC4 & 6.34296 & 64.130289 & 0.45 & 4060 $\\pm$ 880 & - & - & - & - & - \\\\\nC5 & 6.33638 & 64.13981 & 0.41 & 3680$_{-950}^{+870}$ & - & - & - & - & - \\\\\nC6 & 6.31697 & 64.132757 & 0.3 & 3700$_{-730}^{+850}$ & - & - & - & - & - \\\\\nC7 & 6.31072 & 64.14762 & 0.93 & 4390$_{-1210}^{+1040}$ & - & - & - & - & - \\\\\n\\enddata\n\\tablenotetext{}{Errors represent a 90\\% confidence interval unless otherwise noted.}\n\\tablenotetext{a}{Position in 2015 (J2000).}\n\\tablenotetext{b}{Projected angular distance from our estimated kinematic center (see Section \\ref{sec:rev_shoc_center})}\n\\tablenotetext{c}{Includes systematic uncertainties (see Section \\ref{subsec:propmot}).}\n\\tablenotetext{d}{$\\mu_{Tot} = \\sqrt[]{\\mu_{R.A.}^{2}+\\mu_{decl.}^{2}}$.}\n\\tablenotetext{e}{Expansion index (see Section \\ref{subsec:propmot}).}\n\\tablenotetext{f}{Estimated space velocity for a distance of 3.5 kpc.}\n\\tablenotetext{*}{The error interval includes 0, and thus the direction of motion is uncertain. We show only our best-fit value.}\n\\tablenotetext{\\dagger}{This estimate may be affected due to spectral contamination from zeroth order emission (Section \\ref{subsec:hetgs}).}\n\\label{table:all}\n\\end{deluxetable*}\n\n\n\n\n\n\n\n\n\n\\subsection{Ejecta Proper Motions} \\label{subsec:propmot}\n\nBased on the archival Chandra ACIS-I data from 2003, 2009, and 2015 (Table \\ref{tab:obs_acis}), we estimate the proper motions of the ejecta regions in our sample. To measure the proper motions, we apply the methods described in \\citet{sato18}. To find the position of each knot at different epochs, we took the image from the long observation in 2009 (ObsID 10095) as the reference image and compared it to the images from the 2003 and 2015 epochs filtered to the 1.6 -- 4.5 keV band, which is dominated by the Si K$\\alpha$ line. We incrementally shifted the 2003 and 2015 images in R.A. and decl. until a statistically good match with the reference image was obtained, i.e., the Cash statistic \\citep{cash79} was minimized. To estimate the systematic uncertainties, we applied this image fitting method to five background point sources. We find the systematic uncertainties of our method to be $\\sigma_{\\mu_{RA}}= 0\\farcs{}06$ yr\\textsuperscript{-1} and $\\sigma_{\\mu_{decl.}}= 0\\farcs{}03$ yr\\textsuperscript{-1}, in reasonable agreement with the uncertainties estimated in \\citet{katsuda10}. We were able to successfully measure proper motions for 37 of the 59 knots in our sample. For other knots, it was difficult to measure proper motions because they were faint or contaminated by complex emission features in the immediate surroundings. Regions projected close to the center of the SNR do not show measurable proper motions (as perhaps expected), and thus their space velocity is dominated by their radial velocity. \n\n\n\n\\begin{figure}\n\\plotone{updated_knot_locations_v9_sep_orders.pdf}\n\\caption{(a): An exposure-corrected Chandra ACIS image of Tycho's SNR in the Si K$\\alpha$ band (1.7 - 2.0 keV) based on the archival Chandra data taken in 2009. The fifty-nine ejecta knots analyzed in this work are marked with circles. The white arrow in the upper left indicates the dispersion direction. Cyan and red circles indicate blue- and red-shifted features, respectively, while green represents statistically negligible {\\it v\\textsubscript{r}} at the 90\\% confidence interval. The image cutouts along the periphery show zoom-in views of example ejecta features. The scale bar in each cutout is 10\\arcsec{} across. (b): An example of our line-center energy fit for region NW9. The MEG +1 spectrum is overlaid with our best-fit Gaussian model (Gray: data; Red: model fit). The dashed lines show individual Gaussian components of our best-fit model. The vertical green lines show the locations of the rest frame He-like Si K$\\alpha$ and Li-like Si XII line-center wavelengths. (c): The same as (b), however the data and model are from the MEG -1 spectrum. \n\\label{fig:knotloc}}\n\\end{figure}\n\n\\begin{figure}\n\\plotone{ACIS_HETG_vr_Sato_Williams_separate_v3.pdf}\n\\caption{Comparison of Chandra HETG vs. ACIS measurements of radial velocity for the common samples of ejecta knots in Tycho between this work and (a) \\citetalias{williams17} and (b) \\citetalias{sato17tych}. The error bars (blue) in the ACIS measurements include systematic uncertainties. \n\\label{fig:acis_vs_hetg}}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\plotone{pm_and_explosion_site_v4.pdf}\n\\caption{The subset of our regions where we measure the proper motion. Each white arrow shows the direction and relative magnitude of the proper motion for each knot. The length of the white arrow at the top right indicates a speed of 5000 km s\\textsuperscript{-1}. The orange cross and ellipse indicate the position and uncertainty of our estimated kinematic center based on our proper motion measurements of ejecta knots (see Section \\ref{sec:rev_shoc_center}). The magenta ``X'' indicates the geometric center \\citep{warren05}. The image is the same as in Figure \\ref{fig:knotloc}, scaled to make the arrows more visible.\n\\label{fig:pm_arrows}}\n\\end{figure}\n\nThe results of our proper motion measurements are summarized in Table \\ref{tab:mathmode}. Our measured values range from $-$0\\farcs{}32 yr\\textsuperscript{-1} to +0\\farcs{}29 yr\\textsuperscript{-1} in R.A. and $-$0\\farcs{}33 yr\\textsuperscript{-1} to +0\\farcs{}35 yr\\textsuperscript{-1} in decl. We define the expansion index, $\\eta$, as $\\mu_{Tot}\/(D\/t_{age})$, where $t_{age}$ is the age of Tycho (445 years in this study), and $D$ is the estimated angular distance from the geometric center. Our proper motion measurements suggest that all of the knots in the sample have undergone some significant deceleration, ranging from $\\eta$ = 0.21 to 0.80, with an average $\\eta = 0.51$. The proper motion directions are shown in Figure \\ref{fig:pm_arrows}.\n\n\n\nWe combine the radial velocity and proper motion measurements to estimate the 3--D space velocity of regions in our sample. We adopt a distance of 3.5 kpc to Tycho \\citep{williams13}. At this distance, the transverse velocities of knots at the boundary and the radial velocities of knots projected near the center of the SNR generally agree, $v_r \\sim 5500$ km s\\textsuperscript{-1}, which is consistent with the maximum range of space velocities for ejecta regions in our sample. Combining the radial velocity and proper motions we estimate the space velocities of $\\sim$ 1900 -- 6000 km s\\textsuperscript{-1}, with an average $v_s \\sim$ 4200 km s\\textsuperscript{-1}. These velocity ranges are in plausible agreement with those estimated by \\citetalias{sato17tych} and \\citetalias{williams17}, but with uncertainties smaller by a factor of $\\sim$ 3 on average.\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Discussion} \\label{sec:discuss}\n\n\n\\subsection{Azimuthal Variations in Ejecta Velocity}\nIn Figure \\ref{fig:space_vel_az_angle}, we plot our estimated space velocity for each knot against its azimuthal angle (position angles measured counter-clockwise from north). For knots projected closer to the center of the remnant, their true location along the periphery of the SNR is more uncertain. Thus, we only included knots with projected positions offset from the center where we have firmly estimated their proper motions. The ejecta knots in the southeast (SE) quadrant of Tycho have $v_s \\sim$ 6000 km s\\textsuperscript{-1}, and thus appear to be among the fastest-moving knots ($\\sim$ 40\\% faster than the average space velocity of our sample). This high space velocity is in plausible agreement with the presence of ejecta bullet-like features (protrusions extending beyond the main SNR shell, \\citep{wang01}) where high-speed overdense clumps overtake the forward shock. However, the protruding knots are not individual ejecta features like in \\citet{wang01}, but are part of a large-scale portion of the ejecta that was propelled from the explosion more energetically than elsewhere in the remnant. The densities along the SE rim are larger by a factor of a few than those in the southwest (SW) \\citep{williams13}. However, the ejecta space velocities in the SE are faster by a factor of $\\sim 2$ than those in the SW. Thus, the ejecta velocities in SE regions may not be directly related to a rarefied ISM in that direction, but probably due to their intrinsically energetic nature. It is interesting to note that there is a prominent high-speed ejecta knot (NW5) approaching $v_{space} \\sim$ 6000 km s\\textsuperscript{-1} projected at the northwest (NW) boundary, in a nearly opposite direction from the protruding SE knots. While it is tempting to speculate strong ejecta outflows along the SE-NW axis connecting these particularly fast-moving knots, we find no additional substantial evidence to support such a bi-polar ejecta outflow along this axis.\n\nIn the northeast (NE), from position angles $10\\degr{}$ to 100$\\degr{}$, the ejecta space velocities appear to decrease, from $\\sim$ 6000 km s\\textsuperscript{-1} to $\\sim$ 2000 km s\\textsuperscript{-1}. The space velocity then sharply rises back up to $\\sim$ 6000 km s\\textsuperscript{-1} for ejecta knots in the SE from position angles $\\sim$ 100\\degr{} -- 170\\degr{}, before decreasing again to 2000--4000 km s\\textsuperscript{-1} in the SW from 200\\degr{} to 250\\degr{}. Some decreases in ejecta velocity with azimuthal angle are coincident with increasing ambient density, suggesting an origin from the SNR's interaction with a dense surrounding medium. A Spitzer study of the ratio of the 70 to 24 $\\mu$m fluxes in Tycho revealed an increase of post-shock densities at the rim from azimuthal angles of roughly 10 -- 80\\degr{} and 300 -- 330\\degr{} \\citep{williams13}, similar to the angle ranges of decreasing velocity (see Figure \\ref{fig:space_vel_az_angle}). The ejecta in these regions may have been slowed either by direct interaction with the higher-density ISM gas or by an enhanced reverse shock that developed due to the shock-ISM interaction, or a combination of both. \n\n\n\n\n\n\n\\begin{figure}\n\\plotone{vspace_vs_azimuthal_angle_v6.pdf}\n\\caption{The upper panel shows the azimuthal distribution of our estimated ejecta space velocities in Tycho. The bottom panel shows the estimated post-shock densities along the rim from \\citet{williams13}.\n\\label{fig:space_vel_az_angle}}\n\\end{figure}\n\n\n\\subsection{3--D Ejecta Structure}\\label{sec:3d_ejec_struc}\nThe X-ray emitting knots and filaments of the shocked ejecta gas in Tycho are distributed, in general, uniformly across the face of the SNR (Figures \\ref{fig:knotloc} and \\ref{fig:3d_view}a). Our kinematic study of these ejecta knots shows that the overall spatial and velocity distributions of ejecta in Tycho are relatively smooth, in contrast to the case of Kepler's SNR where significant deviations from a spherical distribution, such as the ``Ears'' and nearly freely-expanding ejecta knots are present \\citep{sato17kep,millard20}. Our 3--D reconstruction of the ejecta distribution based on our radial velocity and proper motion measurements for a number of clumpy ejecta features indicates a relatively similar ejecta distribution between the eastern and western shells (Figures \\ref{fig:3d_view}d). On the other hand, we find that the southern shell is dominated by redshifted ejecta (23 redshifted vs 6 blueshifted), while the majority of clumpy ejecta features in the northern shell are blueshifted (13 blueshifted vs 8 redshifted, see Figures \\ref{fig:3d_view}b and \\ref{fig:3d_view}c). The Chandra ACIS study by \\citetalias{sato17tych} similarly revealed more blueshifted Si He-like and S He-like line-center energies in the north than in the south. The authors suggested that the observed discrepancy may be caused by a density enhancement of $\\lesssim \\sqrt{3}$ on the near side of the SNR compared with the far side. In this scenario, the density enhancement causes a stronger reverse shock on the northern near side. Thus, more reverse--shocked ejecta is observed in the north than in the south. A similar scenario could account for the north--south (N--S) differential in ejecta knots on the far side of the SNR. This N--S asymmetry of ejecta due to ambient density variation may be supported by the interacting density variations as reported by \\citet{williams13} and \\citet{katsuda10}.\n\n\n\n\\begin{figure}\n\\gridline{\\fig{test_3D_08-18-2022_X-Y_view.pdf}{0.49\\textwidth}{(a)}\n \\fig{test_3D_08-18-2022_Z-Y_view_distorted.pdf}{0.49\\textwidth}{(b)}\n \n }\n\\gridline{\n \\fig{test_3D_08-18-2022_Z-Y_view_all_same_m.pdf}{0.49\\textwidth}{(c)}\n \\fig{test_3D_08-18-2022_X-Z_view_all_same_m.pdf}{0.49\\textwidth}{(d)}\n }\n\\caption{3--D perspectives of the ejecta knots in Tycho. The red markers represent redshifted ejecta and blue are blueshifted ejecta. In (a) -- (c), we also overlay the ACIS measurements of ejecta velocities by \\citetalias{sato17tych} and \\citetalias{williams17}. The circles, squares, and triangles show velocity measurements from our HETG sample, \\citetalias{sato17tych}, and \\citetalias{williams17}, respectively. For shared regions, we plot only our $v_r$ values. For the ACIS data, we include only those high-velocity regions with $v_r >$ 900 km s\\textsuperscript{-1} (the ACIS systematic gain shift uncertainty). For knots where the proper motion was measured, the arrows point in the direction of the estimated 3D velocity. For the rest of the sample, the arrows point from the SNR center to the position of the knot. The length of the arrow represents the magnitude of the space velocity for each knot. The pale shaded circle shows the approximate location of the main X-ray shell of Tycho. The axis along the line of sight increases into the page. In (a), the X and Y components represent the current locations based on each knot's R.A. and decl. In (b), the Z component of each knot is the measured $v_r$ multiplied by the age of the remnant. The Z component in (b) is likely underestimated due to the deceleration of the knots. In (c), the Z component from (b) is divided by the maximum forward shock expansion index, $\\eta = 0.65$ \\citep{katsuda10} to show a general approximation of the current physical positions of knots along the line of sight, accounting for their deceleration. In (d), we show a ``top--down'' view of Tycho with the ejecta in the same positions as in (c). The Y-axis increases out of the page.\n\\label{fig:3d_view}}\n\\end{figure}\n\n \n\n\n\n\\begin{figure*}[!h]\n\\centering\n\\gridline{\n \\fig{vr_vs_r_plot_v4.pdf}{0.32\\textwidth}{(a)}\n \\fig{vr_vs_r_plot_North_only_v4.pdf}{0.32\\textwidth}{(b)}\n \\fig{vr_vs_r_plot_South_only_v4.pdf}{0.32\\textwidth}{(c)} }\n\\gridline{\n \\fig{vr_vs_r_plot_East_only_v4.pdf}{0.32\\textwidth}{(d)}\n \\fig{vr_vs_r_plot_West_only_v4.pdf}{0.32\\textwidth}{(e)} }\n\\caption{In panel (a), the positions of ejecta knots in {\\it v\\textsubscript{r}} vs. {\\it r} (projected angular distance from the center of the SNR) space. The black and gray dashed loci are the approximate locations of the outermost boundary of the main SNR shell and the reverse shock, respectively. The blue locus is a new potential reverse-shock location. A proportionality constant of 0\\farcs{}041 (km s\\textsuperscript{-1})\\textsuperscript{-1} is applied to the loci based off the maximum expansion rate ($\\sim$ 0.15 \\% yr\\textsuperscript{-1}) estimated by \\citet{katsuda10}. Panels (b), (c), (d), and (e) show those knots located only in the northern, southern, eastern, and western hemispheres, respectively. \\label{fig:rev_shock}} \n\\end{figure*}\n\n\nAlthough a variation in the ambient gas density surrounding Tycho is a plausible origin for the N--S ejecta differential, we may also consider that it could be due to an asymmetry in the early ejecta distribution immediately after the explosion. \\citet{seitenzahl13} simulated a range of Type Ia explosion scenarios and found that delayed detonation (DDT) models with fewer ignition points resulted in more asymmetric explosions. \\citet{ferrand19} propagated a fully 3D N100 DDT model of \\citet{seitenzahl13} into the SNR stage. They found that asymmetries in the explosion were required to explain the large-scale structures in X-ray maps of Tycho (specifically, the power spectrum of radius fluctuations around the rim, \\citet{warren05}). \\citet{ferrand21} further explored the early--stage evolutions of SNRs using the N5 (small number of ignition points) and N100 (large number of ignition points) DDT and pure deflagration models of \\citet{seitenzahl13} and \\citet{fink14}. The authors found that the N5ddt models produce a more asymmetric, dipolar remnant whose imprint lasts up to a few hundred years. It is not straightforward to directly compare the results of these simulations with the non-uniform ejecta distribution inferred from our velocity measurements. However, our work suggests the presence of an aspherical ejecta velocity distribution in Tycho.\n\n\n\n\n\n\n\n\n\n\\subsection{Explosion Center and Reverse Shock}\\label{sec:rev_shoc_center}\nWe estimate the kinematic center of Tycho from our proper motion measurements of ejecta knots. We choose knots which have both $\\mu_{R.A.}$ and $\\mu_{decl.}$ greater than the systematic uncertainty. We generally follow the technique employed in \\citet{sato17kep}. Initially, we assume that each knot has moved at its current proper motion speed since the explosion (i.e., $\\eta$ = 1) to estimate its 2-D starting position. We average the starting positions of all individual knots to calculate the tentative ``initial'' kinematic center. Then, we calculate the new expansion index for each knot based on this tentative kinematic center, and trace its motion back to a new starting point, this time dividing the distance traveled by the expansion index to account for its decelerated motion. We repeat this process until the average kinematic center converged on a single value (after about 25 iterations). Our estimated kinematic center is R.A.(J2000) = 00\\textsuperscript{h}25\\textsuperscript{m}18\\textsuperscript{s}.725 $\\pm$ 1\\textsuperscript{s}.157 and decl.(J2000) = +64$^{\\circ}$08\\arcmin 02\\farcs5 $\\pm$ 11\\farcs{}2. This position is $\\sim$ 13\\arcsec{} southwest of the geometric center estimated by \\citet{warren05}. The previously suggested candidates for the companion of Tycho's progenitor, Tycho G \\citep{ruiz-lapuente04}, Tycho E \\citep{ihara07}, and Tycho B \\citep{kerzendorf13,kerzendorf18}, are located approximately 33\\arcsec{} E, 14\\arcsec{} NE, and 17\\arcsec{} NW from our estimated center, respectively. Assuming a distance of 3.5 kpc, their transverse velocities since the explosion would be $\\sim$ 1200 km s\\textsuperscript{-1}, 500 km s\\textsuperscript{-1}, and 600 km s\\textsuperscript{-1}, respectively. This is in contrast to their recently measured proper motions \\citep{kerzendorf13}, which would imply transverse velocities of 100 -- 200 km s\\textsuperscript{-1}. However, since the positions of Tycho E and Tycho B are within a few arcseconds of the error ellipse of our estimated center (see Figure \\ref{fig:pm_arrows}), they may have travelled a shorter distance if our kinematic center is representative of the explosion site. Thus, their transverse velocities since the explosion could be significantly slower, in line with the current values. Tycho G is located several arcseconds outside of the error ellipse, and therefore its current proper motion is still too low to account for the angular distance it would have travelled from our kinematic center since the explosion. \n\n\n\n\n\n\nWe plot our measured radial velocity for each knot against its angular distance from the center of the SNR in Figure \\ref{fig:rev_shock}a. Figures \\ref{fig:rev_shock}b and \\ref{fig:rev_shock}c show the north--south asymmetry discussed in Section \\ref{sec:3d_ejec_struc}, and Figures \\ref{fig:rev_shock}d and \\ref{fig:rev_shock}e show ejecta features projected in the eastern and western hemispheres, respectively. We plot the main shell (the forward shock) and the reverse shock position from \\citet{yamaguchi14} estimated from the location of Fe K$\\beta$ emission generally in the NW quadrant of Tycho. The bulk of the ejecta knots in our sample are positioned between the forward and reverse shocks, as expected. However, there are several knots positioned closer to the SNR center beyond the reverse shock. The locus of these inner ejecta knots appears to form a smaller reverse shock at $\\sim$ 2.0\\arcmin{} from the SNR center, or 75\\% of the 2.6\\arcmin{} radius for the reverse shock estimated by \\citet{yamaguchi14}. According to models of dynamical evolution of SNRs \\citep{truelove99} with an explosion energy of $1.2 \\times 10^{51}$ ergs \\citep{badenes06} and an ejected mass of 1.4 M\\textsubscript{\\(\\odot\\)}, ambient density variation by a factor of $\\sim$ 4 (similar to that reported by \\citet{williams13} for Tycho) may produce $\\sim$ 30\\% deeper-reaching reverse shock. These knots beyond the reach of the previously-known reverse shock are blueshifted, and therefore are positioned on the near side of the SNR. Thus, these inner ejecta knots may represent deviated parts of the reverse shock due to the shock interaction with denser medium on the near side of the SNR. Recently, X-ray proper motion measurements of Tycho's forward shock showed that its expansion has significantly decelerated from 2003 to 2015 (down to 40\\% of its initial value, \\citet{tanaka21}). The authors suggested that the forward shock may be encountering a non-uniform wall of dense gas, possibly created from the winds of the progenitor system. Our results may support the presence of a similar density variation along the line of sight.\n\n\n\n \n\n\\section{Conclusions} \\label{sec:conclusions}\n\nWe have measured the radial velocities of 59 small ejecta features in Tycho's SNR using our deep 450 ks Chandra HETG observation. Based on these measurements, our 3--D reconstruction of Tycho shows a large-scale asymmetry where most knots in the northern half are blueshifted and thus on the near side, and most knots in the southern half are redshifted and therefore are located on the far side. Ambient density variations across the near and far sides of the remnant might have caused non-uniformity in the formation of reverse shock, and thus resulted in the differences in the frequency of detected ejecta knots. Alternatively, the identified asymmetry could be caused by a non-spherical explosion of the progenitor. \n\n\n\nFor 37 of the 59 ejecta features in our sample, we measured their proper motions using archival Chandra ACIS data. We estimate an expansion center based on our measured proper motions. Combining the radial velocities and proper motions, we find space velocities up to 6000 km s\\textsuperscript{-1}. The azimuthal distribution of our measured space velocities shows generally higher speeds of ejecta towards the SE. Regions with low velocity coincide with higher ambient density at the rim. However, the high-velocity SE regions do not coincide with comparatively lower ambient densities, and therefore probably resulted from higher kinetic energy being deposited in that direction from the explosion. Based on our detection of relatively lower radial velocities for several ejecta knots projected near the center of the SNR, we postulate a considerable ambient density variation along the line of sight (e.g., a higher density on the near side of Tycho). \n\n\n\n\n\n\\bigskip\nThis work has been supported in part by NASA Chandra Grant GO7-18061X. TS was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI grant No. JP19K14749. JPH acknowledges support from NASA grant number NNX15AK71G to Rutgers University. We thank the anonymous referee for their useful recommendations that improved the explication of this work.\n \n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzhtys b/data_all_eng_slimpj/shuffled/split2/finalzzhtys new file mode 100644 index 0000000000000000000000000000000000000000..ae33eddd6dfed177627ed55c9dd0c39609fbf448 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzhtys @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\\label{sec:intro}\n\nThe class of force-directed graph drawing algorithms is large\nboth in terms of objectives and optimization algorithms~\\cite{k-fdda-13,b-fdgd-14}. \nExperimental~\\cite{bp-esdbgd-09} and anecdotal evidence suggest\nthat a most desirable objective is\nthe stress function of distance-based multidimensional scaling~\\cite{m-maed-66}.\nGiven a simple undirected graph $G=(V,E)$,\nthe layout $x=(\\mathbb{R}^2)^V$ of a straight-line drawing is considered suitable,\nif the weighted deviation\n\\begin{equation}\n \\mathop{stress}(x)=\\sum_{i0$, is a multiplicative bijection. Nevertheless, $\\mathcal{R} (\\varphi) =X$, so each point is standard for $\\varphi$, but we cannot find a continuous map $p$ as given in Theorem~\\ref{derr-laredo}.\n\\end{rem}\n\n We next state the theorem characterizing all spaces on which can be defined a multiplicative bijection that is not standard. Recall that a topological space $Z$ is said to be {\\em pseudocompact} if every real-valued continuous map on $Z$ is bounded.\n \n \\begin{thm}\\label{sete}\n There exists a bijective and multiplicative map $\\varphi: \\dx \\ra \\dy$ \n that is not standard if and only if $X$ and $Y$ are homeomorphic and there exists a \n subset $Z$ of $Y$, not pseudocompact, such that $Y = \\beta Z$.\n \\end{thm}\n \n Obviously Theorem~\\ref{sete} gives an answer in the negative to Marovt's conjecture. It is enough now to take any completely regular space $Z$ (thus ensuring that its Stone-\\v{C}ech compactification exists) that is not pseudocompact (as for instance $\\mathbb{N}$, $\\mathbb{R}$, or any unbounded subset of a normed space), and we have that there are always multiplicative bijections on $C (\\beta Z, I)$ that\n are not standard. \n\n\n\n\n\n\\section{Some other results and proofs}\n\n The following is a key lemma to prove Theorem~\\ref{derr-laredo}.\n\n\\begin{lem}\\label{azero}\nSuppose that\n$u: \\mathscr{A}_1 \\longrightarrow \\mathscr{A}_2$ is an order preserving multiplicative bijection, \nwhere $\\mathscr{A}_1$ and $ \\mathscr{A}_2$ are semigroups contained in $(0,1)$. Then there exists \n$p \\in (0 , + \\infty)$ such that\n$u(\\gamma) = \\gamma^p$ for every $\\gamma \\in \\mathscr{A}_1$.\n\\end{lem}\n\n\\begin{proof}\nSuppose on the contrary that there exist $\\alpha, \\beta \\in \\mathscr{A}_1$ such that $u( \\alpha) = \\alpha^p$ and $u(\\beta ) =\\beta^q$, where $0 \\beta^{mq}$, that is, $u(\\alpha^n) > u (\\beta^m)$, against the fact that $u$ is order preserving.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Theorem~\\ref{derr-laredo}]\n(\\ref{astamarte})\nFor each $x \\in X$, let $\\mathscr{U}_x$ be the set of all $f \\in \\mathcal{A}$ for which there exists a neighborhood of $x$ where $f \\equiv 1 $. \n\nLet us see that $\\mathscr{C}_x := \\{y \\in Y: \\varphi (f) (y) =1 \\hspace{.03in} \\forall f \\in \\mathscr{U}_x\\}$ is nonempty.\nFirst notice that, given any $f \\in \\mathscr{U}_x$, the set $\\varphi (f)^{-1} (\\{1\\})$ is compact.\nOn the other hand, given $f_1, \\ldots, f_n \\in \\mathscr{C}_x$, we can find $f_0 \\in \\mathcal{A}$, $f_0 \\neq 0$, such that\n$\\mathrm{supp} \\hspace{.02in} f_0 \\subset \\{z \\in X: \\prod_{i=1}^n f_i (z) =1\\}$. This obviously implies that\n$f_0 f_i =f_0$ for each $i$, so $\\varphi(f_0) \\varphi(f_i) = \\varphi (f_0) \\neq 0$. We deduce that \n$\\bigcap_{i=1}^n \\varphi (f_i)^{-1} (\\{1\\}) \\neq \\emptyset$. Then $\\{ \\varphi (f)^{-1} (\\{1\\}) : f \\in \\mathscr{U}_x \\}$ satisfies\nthe finite intersection property, and we conclude that\n$\\mathscr{C}_x $ is nonempty.\n\n\n\nOn the other hand, $\\varphi^{-1}$ is also multiplicative, so given any $y \\in \\mathscr{C}_x$, we have that the set $\\mathscr{C}_y$ (defined in a similar way as $\\mathscr{C}_x$) is nonempty. Let us see that $\\mathscr{C}_y =\\{x\\}$. \nSuppose that there exists $z \\in \\mathscr{C}_y$, $z \\neq x$. Then we can find $f_z \\in \\mathscr{U}_x$ such that $z \\notin \\mathrm{supp} \\hspace{.02in} f_z $. Since $z \\in \\mathscr{C}_y$\n and $\\varphi (f_z) (y) =1$,\nthen we can find $k \\in \\mathcal{B}$ with $\\mathrm{supp} \\hspace{.02in} k \\subset \\mathrm{coz} \\hspace{.02in} \\varphi (f_z)$\nand $\\varphi^{-1} (k) (z) \\neq 0$. Clearly, \nif we now take $g \\in \\mathscr{U}_z$ with $f_z g=0$, then $g \\varphi^{-1} (k) \\neq 0$, but $\\varphi (g) k =0$, which is impossible.\n\n\n\n\n\nThe above process lets us define a map $\\mu : Y \\longrightarrow X$, which turns out to be bijective, such that $\\mu (y)$ is the only point in $\\mathscr{C}_y$, and $y$ is the only point\nin $\\mathscr{C}_{\\mu (y)}$, for each $y \\in Y$.\n\n\n\nWe prove next that $\\mu$ is continuous at every point of $Y$ (and is consequently a homeomorphism). Take any $y \\in Y$, and let $U$ be an open \nneighborhood of $\\mu (y)$. We will see that, if $f \\in \\mathscr{U}_{\\mu (y)}$ and $\\mathrm{supp} \\hspace{.02in} f \\subset U $, \nthen $\\mu (\\mathrm{coz} \\hspace{.02in} \\varphi (f))$ is contained in $U$. Otherwise\nthere exists $z \\in Y$ such that $\\varphi (f) (z) \\neq 0$ and $\\mu (z) \\notin \\mathrm{supp} \\hspace{.02in} f$,\nso we can take $k \\in \\mathscr{U}_{\\mu (z)}$ such that $kf =0$. Obviously \n$\\varphi (k) (z) \\varphi (f) (z) \\neq 0$, which \nis absurd.\n\nFinally, we have that, by definition, if $y \\in \\mathcal{R} (\\varphi)$, then there exist $p(y) \\in (0, + \\infty)$\nand $x \\in X$ such that $\\varphi(f) (y) = f (x)^{p(y)}$ for every $f \\in \\mathcal{A}$. It is easy to check that $x = \\mu (y)$.\nAs for the map $p: \\mathcal{R} (\\varphi) \\longrightarrow (0, + \\infty)$, we have that for each $y \\in \\mathcal{R} (\\varphi)$, \n $$p(y) = \\frac{\\log \\varphi (f) (y)}{\\log f \\left( \\mu \\left( y \\right) \\right)}$$for every $f \\in \\mathcal{A}$ with \n$f \\left( \\mu \\left( y \\right) \\right) \\neq 0,1$. This implies that $p$ is continuous at $y$, and consequently on $\\mathcal{R} (\\varphi)$.\n\n\\medskip\n\n(\\ref{santamarta}) \nFor each $y \\in Y$, let $\\mathcal{B}_y := \\{g (y) : g \\in \\mathcal{B}\\} \\cap (0,1)$, and define $\\mathcal{A}_{x}$ for each $x \\in X$ in a similar way. Consider also \n the set $\\mathcal{R}_1 (\\varphi)$ of all $y \\in Y$ such that\n$\\varphi (f) (y) \\neq 0, 1$ whenever $f \\in \\mathcal{A}$ satisfies $f (\\mu (y)) \\neq 0, 1$. We need the following claim.\n\n\\smallskip\n\n{\\bf Claim.} {\\em Let $y \\in \\mathcal{R}_1 (\\varphi)$. If $f, g \\in \\mathcal{A}$ satisfy \n$g(\\mu (y)) \\le f(\\mu (y))$, then $\\varphi (g) (y) \\le \\varphi (f) (y) $. Moreover $\\mu (y) $ belongs to $\\mathcal{R}_1 \\left( \\varphi^{-1} \\right)$.}\n\\smallskip\n\nSuppose first that $g(\\mu (y)) < f(\\mu (y))$, and take a neighborhood $U$ of $\\mu (y)$ with $ g(x) < f(x)$ for every $x \\in U$. We pick $f_0, g_0 \\in \\mathscr{U}_{\\mu (y)}$\nsuch that $\\mathrm{supp} \\hspace{.02in} f_0 \\subset U$, and \nsuch that $\\mathrm{supp} \\hspace{.02in} g_0 \\subset \\{ x \\in X : f_0 (x) =1\\}$, respectively. Since $\\mathcal{A}$ \nsatisfies Property 2, then \nthere exists $k \\in \\mathcal{A}$ such that $(ff_0)k= gg_0$. Also $k(\\mu(y)) \\in (0,1)$, and consequently $$\\varphi (g) (y) = \\varphi (gg_0) (y) = \\varphi (ff_0) (y) \\varphi (k) (y) < \\varphi (ff_0) (y) = \\varphi (f) (y).$$\nWe now prove that $\\mu (y)$ belongs to $\\mathcal{R}_1 (\\varphi^{-1})$. Let $h \\in \\mathcal{B}$ be such that $ h (y) \\neq 0,1$.\nSuppose that $\\varphi^{-1} (h) (\\mu (y)) = 0$ and take any $l \\in \\mathcal{A}$ with $l (\\mu (y)) \\neq 0, 1$, and \n$n \\in \\mathbb{N}$ such that $\\left( \\varphi(l) (y) \\right)^n < h(y)$. We then have that $ \\varphi^{-1} (h)(\\mu (y)) < l^n (\\mu (y))$ and\n$h (y) > \\varphi(l^n) (y)$, what goes against what we have proved above. We deduce that $\\varphi^{-1} (h) (\\mu (y)) \\neq 0$. In a similar way we can deduce that $\\varphi^{-1} (h) (\\mu (y)) \\neq 1$. Thus, $\\mu (y)$ belongs\nto $\\mathcal{R}_1 (\\varphi)$.\n\nNow, working with $\\varphi^{-1}$, it is clear that if $g(\\mu (y)) \\le f(\\mu (y))$, then we cannot get $\\varphi (g) (y) > \\varphi (f) (y) $.\nThe claim is proved.\n\n\n\\smallskip\n\n\n\n\nFor any $y \\in \\mathcal{R}_1 (\\varphi)$, we may define a map $\\varphi_y : \\mathcal{A}_{\\mu(y)} \\longrightarrow \\mathcal{B}_y$ in\n the following way. Given $\\alpha \\in \\mathcal{A}_{\\mu (y)}$, there exists $f \\in \\mathcal{A}$ such that\n $f(\\mu(y)) = \\alpha$. Then define $\\varphi_y (\\alpha) := \\varphi (f) (y)$. It is clear by the above claim that \n $\\varphi_y$ is well defined, and obviously it is multiplicative, order preserving, and bijective. \nAlso, we have that $\\varphi (f) (y) =1 $ whenever $f(\\mu (y))= 1$, and $\\varphi (f) (y) =0 $ whenever $f(\\mu (y))= 0$,\n$f \\in \\mathcal{A}$.\nConsequently, by \nLemma~\\ref{azero}, we have that $\\mathcal{R}_1 (\\varphi ) \\subset \\mathcal{R} (\\varphi)$. The other inclusion is immediate, so\n$\\mathcal{R}_1 (\\varphi) = \\mathcal{R} (\\varphi) $.\n\n\n\n\n\n\\medskip\n\n \nWe prove that $\\mathcal{R} (\\varphi) $\nis dense in $Y$.\nLet $W_0 \\subset Y$ be open (and nonempty). Pick \n$y \\in W_0$ and assume that $y \\notin \\mathcal{R} (\\varphi) = \\mathcal{R}_1 (\\varphi)$, so \n there exists\n$f_0 \\in \\mathcal{A}$ such that $f_0 (\\mu (y)) \\neq 0, 1$ and $ \\varphi (f_0) (y) \\in \\{0, 1\\}$. Let $V:= \\{x \\in X : 0$0.3 Gyr.\n\nThe MMT observations allowed us to obtain a good determination of the O\/H gradient (together with \nNe\/H, S\/H, and Ar\/H) both from H~{\\sc ii}\\ regions and PNe. In the case of H~{\\sc ii}\\ regions, our cumulative sample includes: \ni) H~{\\sc ii}\\ regions by Magrini et al. \\cite{magrini07b} which comprises their own determinations and all previous abundance determinations with available $t_{e}$, recomputed uniformly; ii) the sample by Rosolowsky \\& Simon \n(2008); iii) the new MMT sample. For PNe, we use the sample of PNe presented by M09. \nThe resulting gradients are the following, for PNe and H~{\\sc ii}\\ regions respectively, 12 + {\\rm log(O\/H)} = \n$$ -0.031 (\\pm 0.013) ~ {\\rm R_{GC}} + 8.44 (\\pm\n0.06),\\eqno(1)$$ \n$$ -0.032 (\\pm 0.009) ~ {\\rm R_{GC}} + 8.42 (\\pm\n0.04),\\eqno(2)$$ where R$_{GC}$ is the galactocentric distance in kpc. \nThe two gradient are identical within the errors, both in the their slopes and central values. \n\\section{The metallicity in M33 and its evolution}\n\\label{sect_distr}\n\nThe large amount of chemical abundance data from H~{\\sc ii}\\ regions and PNe collected \nto date in M33 allow us to reconstruct its metallicity map.\nIn this section, both the 2D and the radial distribution are analyzed\ntaking advantages also of all previous abundance determinations with measured $t_{e}$\\ and of our new\nresults. \n\n\n\\subsection{The 2D distribution of metals}\n\nThe usual way to study the metallicity distribution in disk galaxies is \nto average it azimuthally, assuming that: i) the centre of the galaxy coincides\nwith the peak of the metallicity distribution; ii) at a given radius, the metallicity is the \nsame in each side of the galaxy. \nThe large number of metallicity measurements in M33, both from H~{\\sc ii}\\ regions and from PNe, \nallow us to reconstruct not only their radial gradient, but also their spatial distribution \nprojected onto the disk. \nIn Figure 1, the two-dimensional metallicity distributions for M33 from H~{\\sc ii}\\ regions and \nfrom PNe are shown. \nThe highest metallicity H~{\\sc ii}\\ regions and PNe are not located at the center of the\ngalaxy, but rather lie at a radius of 1-2 kpc, in the southern arm. \nSimon \\& Rosolowsky (2008) noticed this behavior for H~{\\sc ii}\\ regions and suggested as \nexplanation that the material enriched by the most recent generation\nof star formation in the arm has not yet been azimuthally mixed through the\ngalaxy. Whilst it might be true for H~{\\sc ii}\\ regions, it cannot be the reason for the off-center \nmetallicity distribution of PNe since they belong to an older population. \nColin \\& Athanassoula \\cite{colin81} noticed several evidences of asymmetries in the inner regions \nof M33, such as the distribution of HI atomic gas, of H~{\\sc ii}\\ regions, of high luminosity stars. They\nproposed a kinematical model with a displaced bulge having a retrograde motion around the center. \nAlso detailed analysis of the innermost regions of M33 by Corbelli \\& Walterbos \\cite{corbelli07} \nfound possible asymmetries in the stellar and gas velocity pattern which might be related \nto the displacement of a small bulge.\nThe off-center metallicity distribution might thus be related to the lack of a dominant gravitational \nsource in the center of this galaxy with a consequent motion at different epochs of the peak of the highest star \nformation region around the M33 visual center. \n \n \\begin{figure}\n \\resizebox{\\hsize}{!}{\\includegraphics[clip=true,angle=270]{metmap_hii_bw1.ps}}\n \\resizebox{\\hsize}{!}{\\includegraphics[clip=true,angle=270]{metmap_pn_bw1.ps}}\n \\caption{\\footnotesize{The metallicity maps: PNe (top) and H~{\\sc ii}\\ regions (bottom). \n The 12 + log(O\/H) scale is shown on the y axe. The centre of M33 corresponds to the 0,0 position. }}\n \\label{oxy}%\n \\end{figure}\n\n\\subsection{The time-evolution of the abundance gradient}\n\\label{sect_model}\n\nM07 built a chemical evolution model of M33, called {\\em accretion} model, \nable to reproduce its main features, \nincluding the radial trends at present-time of molecular gas, atomic gas, stars, SFR, and the \ntime evolution of the metallicity gradient using the available constraints at that time. \nIn that model, the disk of M33 was formed by continuous accretion of primordial \ngas from the intergalactic medium. \nHowever, new observations have been made recently available, rendering necessary \na revision of the model. \nIn particular, the global slope of the radial O\/H gradient has been confirmed much \nshallower than retained in the past (cf. Rosolowsky \\& Simon 2008) \nand its evolution much slower (cf. M09).\n\nThe general assumption of multiphase chemical evolution models, \nsuch as the M33 one (M07), \nis to describe the formation and disruption of diffuse gas, clouds, and stars, \nby means of physical processes, e.g., Ferrini et al. \\cite{ferrini92}. \nIn particular, the SF is represented \nwith two processes: the interaction of molecular clouds with the \nradiation field of massive stars and the collisions between two molecular clouds.\nThe dominant process is due to cloud collisions. \nIn this kind of model the relationship between the star formation rate (SFR) and the surface density of gas (molecular or total) is, thus, a by-product of the model, and cannot be assumed 'a priori'.\n\nHowever, the law relating the surface densities of SF and cold gas is one of the most fundamental laws \ndescribing the galaxy behaviour. Virtually the entire range of global star formation rates in galaxies \ncan be reproduced by a Schmidt power law relation. \nIn the particular case of M33, the relation between the SFR, measured from the FUV emission, \nand molecular gas has a well-defined slope (Verley et al., in prep.) corresponding to\n\\begin{equation}\n\\Sigma_{SFR} = A \\Sigma^{1.2}_{mol gas}.\n\\end{equation}\nA pure cloud-cloud collision process for the star formation is not able to reproduce it. \nFor this reason, we have taken into account other parameterizations of the SF process. \nIn particular, a process which is dominated by cloud collisions close to the center, while in the \nintermediate and peripheral regions is proportional to the fraction of clouds with a 1.5 exponent, \nis able to reproduce the observations. It reproduces the \nhigher cloud surface density in the inner regions, rendering the conversion into star less effective, \nwhile in the outer regions it takes into account a more efficacious SF. \nThe resulting Schmidt law would have an average exponent all over the radial range (or the molecular gas surface density range) of 1.1, thus consistent with the observations. \n\nIn addition, the introduction of the Schmidt law allowed us to better reproduce \nthe O\/H gradient and its evolution \n(see Figure 2): a gradient almost flat both at present-time both at the epoch of the formation of the PNe progenitors, \nwith a little evolution of its absolute value and slope. \nNote from Figure 2 the steeper gradient predicted by the previous model, where the SF process \nwas dominated by cloud-cloud collision all over the radial range. \n \n \\begin{figure}\n \\centering\n \\resizebox{\\hsize}{!}{\\includegraphics[clip=true]{M33_oxy_sl.ps}} \n \\caption{\\footnotesize{The time-evolution of the O\/H radial gradient. Filled symbols (blue circles and magenta squares) are \n the oxygen abundances of H~{\\sc ii}\\ regions and PNe, respectively, averaged in bins 1 kpc wide.\n Model with the Schmidt law: present time (red continuous line), 1 Gyr ago (long-short dashed line), \n 5 Gyr ago (dot-dashed line). Model with the cloud-cloud collision process (M07): present time (green dashed line)}. \n }\n \\label{sl}%\n \\end{figure}\n\n\n\\section{Summary and conclusions}\n\\label{sect_conclu}\nThe chemical evolution of M33 is studied by means of new spectroscopic observations \nof PNe and H~{\\sc ii}\\ regions.\nTheir 2D metallicity maps have been found both off-centered, with a peak in the southern\narm, at 1-2 kpc from the center. This might be related to the absence of a dominant gravitational \nsource in the center. \nThe slow evolution of the metallicity gradient from the present time to the \nbirth of the PNe progenitors is explained with an {\\em accretion} model \nwhere the SF process is driven by the Schmidt law. \n\n\n\n \n\\begin{acknowledgements}\nI warmly thank Edvige Corbelli, Daniele Galli, \nLetizia Stanghellini, and Eva Villaver for their collaboration in this work. \n\\end{acknowledgements}\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nField perturbations of a curved background spacetime obey a wave equation which dictates that they propagate ``mainly\" along null geodesics. However, in general, there is also a part of the field, called the {\\it tail}, which propagates slower than light (even in the case of a massless field). In other words, the strong Huygens principle is generally violated in curved spacetimes~\\cite{McLenaghan:1974,CzaporMcLenaghan}.\nThis tail is important for different reasons. For example, obtaining the tail term is useful for calculating the self-force~\\cite{Poisson:2011nh} on a point particle via the method of matched expansions~\\cite{Anderson:Wiseman:2005}, where the retarded Green function of the wave equation needs to be regularized~\\cite{CDOW13,PhysRevD.89.084021}.\nAlso, the tail may give rise to an interesting relevant contribution to the communication between quantum particle detectors~\\cite{blasco2015violation,Jonsson:2020npo}.\n\nMathematically, the tail term may be defined in local neighbourhoods of spacetime points via the Hadamard form~\\cite{Hadamard} of the retarded Green function. Specifically, and henceforth focusing on the case of a scalar field for simplicity, the tail is the bitensor $V(x,x')$ in the term in the Hadamard form which has support only {\\it inside} the (past) light cone of the field point. This tail bitensor satisfies the {\\it homogeneous} wave equation, constrained by its value on the light cone, which satisfies a transport equation (along null geodesics) with a given initial condition~\\cite{Poisson:2011nh}.\n\nThe Hadamard tail $V(x,x')$ has been obtained in closed form in only a very few settings of high symmetry - specifically, and to the best of our knowledge, only when the background spacetime is flat~\\cite{M&F} or conformally-flat\\footnote{In these cases, the field considered is non-conformal, so that there exists a nontrivial tail.}, such as simple (spatially-flat) cosmological model spacetimes~\\cite{Burko:2002ge,haas2005mass}, including (a patch of) de Sitter~\\cite{Friedlander}.\nA simplifying feature of flat and conformally-flat spacetimes is that null geodesics emanating from a point do not cross.\nFurthermore, the maximal symmetry of both flat and de Sitter spacetimes, in particular, means that it is possible to rewrite the partial differential wave equations in these spacetimes as {\\it ordinary} differential equations (where the independent variable is the geodesic distance), which are much easier to solve.\nHowever, in other spacetimes which are less symmetric and where null geodesics emanating from a point do cross, such as black hole spacetimes, no closed form expression for $V(x,x')$ is known and, instead, one needs to resort to numerical or approximation analytical techniques. \nFocusing on black hole spacetimes, some of these techniques have been fairly successful in calculating $V(x,x')$ in Schwarzschild~\\cite{CDOWb,Ottewill:2009uj,Decanini:Folacci:2005a,CDOW13,PhysRevD.89.084021} but face much more significant difficulties in the case of Kerr. It is thus important to develop alternative methods for calculating the tail term.\n\nIn this paper we pursue the endeavour of calculating $V(x,x')$ by directly (numerically) integrating the (homogeneous) wave equation with given Characteristic Initial Data (CID) on the light cone.\nSpecifically, we apply this method to the case of a scalar field propagating on Pleba{\\'n}ski-Hacyan spacetime (PH), $\\mathbb{M}_2\\times{\\mathbb{S}^2}$~\\cite{Griffiths&Podolsky}.\nFrom a physical point of view, this spacetime serves as a black hole toy-model and it captures the important feature that null geodesics emanating from a point do cross (in fact, similarly to Schwarzschild, there exist caustics where an ${\\mathbb{S}^2}$-envelope of null geodesics focus).\nIn its turn, from a technical point of view, the fact that PH is not a maximally-symmetric spacetime but is the direct product of two (two-dimensional) maximally-symmetric spacetimes (namely, $\\mathbb{M}_2$ and ${\\mathbb{S}^2}$) means that its wave equation, while it is not an {\\it ordinary} differential equation as in flat or de Sitter spacetimes, it is reduced to a {\\it two}-dimensional PDE; furthermore, the value of $V(x,x')$ on the light cone is known in closed form~\\cite{Casals:2012px}. Our calculation of $V(x,x')$ in the specific case of a massles field with a coupling constant value of $\\xi=1\/8$ and $x$ and $x'$ on a static path, agrees with~\\cite{Casals:2012px}, where $V$ is calculated via a completely different method which involves infinite sums and integrals.\nThis provides a check of our method and calculation and serves as a proof-of-concept for this method.\nWe also obtain new results for $V$ in PH: for {\\it any} pair of spacetime points where it is defined for $\\xi=1\/8$ as well as for $\\xi=0$, $1\/6$, $1\/4$ and $1\/2$.\n\nFor solving the homogeneous \n(two-dimensional)\nwave equation in PH by evolving CID we use a finite-difference scheme. Refs.~\\cite{mark2017recipe,Lousto:1997wf} proposed and implemented a CID scheme for obtaining the multipolar modes of the field (or retarded Green function) in Schwarzschild spacetime. We adapted this scheme to calculate the full $V(x,x')$ in PH spacetime. We note that some peculiarities of the PDE satisfied by $V$ in PH will also be present in the PDE satisfied by $V$ in Schwarzschild, so that our adaptation of the scheme will probably be useful for any future investigation in the latter spacetime. Furthermore, we developed the scheme to higher order than in~\\cite{mark2017recipe,Lousto:1997wf} by providing additional data on the light cone. \n\nThe rest of this paper is organized as follows. In Sec.~\\ref{sec:Hadamard} we introduce the Hadamard form and give the explicit forms of the scalar wave equation and some Hadamard quantities in PH.\nIn Sec.~\\ref{sec:CID} we present the Characteristifc Initial Value problem in PH, our finite difference scheme for solving it and the results of our calculations.\nWe conclude in Sec.~\\ref{sec:Discussion} with a brief discussion.\n\nWe choose units such that $G=c=1$.\n\n\n\\section{Wave equation, Hadamard Form and Tail}\\label{sec:Hadamard}\n\n\n\\subsection{A general spacetime}\n\nA scalar field perturbation of a background spacetime satisfies a wave equation.\nSpecifically, its retarded Green function satisfies:\n\\begin{equation}\\label{eq:GF eq}\n\\left(\\Box-m^2-\\xi R\\right)G_{\\textrm{ret}}(x,{x^\\prime})=-4\\pi\\delta_4(x,{x^\\prime}),\n\\end{equation}\nwhere $m$ is the mass of the field, $R$ is the Ricci scalar, $\\xi$ is a coupling constant and \n$x$ and ${x^\\prime}$ are, respectively, the field and base spacetime points.\n\nThe Hadamard form provides an analytic expression for the singularities of the retarded Green function when $x$ is in a local (normal\\footnote{A normal\nneighbourhood of ${x^\\prime}$ is a region containing ${x^\\prime}$ such that\nevery $x$ in that region is connected to ${x^\\prime}$ by a unique\ngeodesic which lies within the region.}) neighbourhood of ${x^\\prime}$~\\cite{DeWitt:1960,Friedlander,Poisson:2011nh}:\n\\begin{align} & G_{\\textrm{ret}}(x,{x^\\prime})=\n\\label{grhad} \\\\ & [U(x,{x^\\prime})\\delta(\\sigma(x,{x^\\prime}))+V(x,{x^\\prime})\\theta(-\\sigma(x,{x^\\prime}))]\\theta_+(x,{x^\\prime}),\n\\nonumber\n\\end{align}\nwhere $\\delta$ and $\\theta$ are, respectively, the Dirac delta and Heaviside distributions,\n\\[ \\theta_+(x,{x^\\prime})\\equiv \\left\\{\\begin{array}{ll} 1 & \\hbox{if $x$ lies to the future of ${x^\\prime}$};\\\\ 0 & \\hbox{otherwise},\\end{array}\\right. \\]\nand $U$ and $V$ are biscalars which are smooth in that local neighbourhood. \nHere, $\\sigma$ is Synge's world-function, i.e., one-half of the\nsquare of the geodesic distance along the unique geodesic\nconnecting ${x^\\prime}$ and $x$.\nThus, clearly, the term with $U$ in Eq.~\\eqref{grhad} has support only on the light cone whereas the term with $V$ has support {\\it inside} the light cone: this is the tail term which is the focus of this paper.\n\nThe Hadamard tail $V$ satisfies the {\\it homogeneous} wave equation:\n\\begin{equation}\\label{eq:V wave eq}\n\\left(\\Box-m^2-\\xi R\\right)V(x,{x^\\prime})=0,\n\\end{equation}\nconstrained by its value on the light cone:\n\\begin{equation}\\label{eq:transp eq V}\n\\hat{V}_{,\\alpha}\\sigma^{\\alpha}+\\frac{1}{2}\\left(\\sigma^{\\alpha}{}_{\\alpha}-2\\right)\\hat{V}=\\frac{1}{2}\\left(\\Box-m^2-\\xi R\\right)\\left.U\\right|_{\\sigma=0},\n\\end{equation}\nwhere $\\hat{V} \\equiv V|_{\\sigma=0}$ and $\\sigma^{\\alpha}{}_{\\alpha}\\equiv\\nabla_\\alpha\\nabla^\\alpha\\sigma$.\nEq.\\eqref{eq:transp eq V} is in fact a transport equation along a light cone-generating null geodesic. It is to be solved together with the initial condition corresponding to the value of $V$ at coincidence (i.e., at ${x^\\prime}=x$):\n\\begin{equation}\\label{eq:IC V}\nV(x,x)=\\frac{1}{12}\\left(1-6\\xi\\right)R(x)-\\frac{1}{2}m^2,\n\\end{equation}\nwhich follows partly from the fact that $V$ is smooth at coincidence.\n\nOne method for trying to calculate $V(x,{x^\\prime})$ is to express it as an asymptotic series:\n\\begin{equation} \nV(x,x')=\\sum_{n=0}^\\infty \\nu_n(x,x')\\sigma^n\\label{eqn:VsigmaExpansion},\n\\end{equation}\nwhere the coefficients $\\nu_n(x,x')$ satisfy\ncertain recurrence relations~\\cite{DeWitt:1960,Decanini:Folacci:2005a}.\nRef.\\cite{Ottewill:2009uj} provided a complete procedure for calculating $\\nu_n$ by solving transport equations. Unfortunately, however, as the coefficient order $n$ increases, these transport equations become increasingly hard to solve (even numerically and for low $n$).\nFurthermore, although the series in Eq.~\\eqref{eq:IC V} converges uniformly in subregions of normal neighbourhoods~\\cite{DeWitt:1960,Friedlander}, it is not actually guaranteed to converge in the whole maximal normal neighbourhood of a point.\nA more practical method for calculating $V(x,x')$ in spherically-symmetric spacetimes is to expand this bitensor in small {\\it coordinate} distance between $x$ and $x'$~\\cite{CDOWb}.\nAlthough this method has proven to be very useful in Schwarzschild spacetime~\\cite{CDOW13,PhysRevD.89.084021}, it is naturally adapted to spherical symmetry and so it still needs to be developed in Kerr spacetime.\n\nIn this paper, we shall use the first two orders in the series in Eq.~\\eqref{eq:IC V} in order to calculate the characteristic initial data for a numerical scheme for solving the full wave Eq.~\\eqref{eq:V wave eq} for $V(x,x')$ for {\\it any} pair of points in PH spacetime.\n\n\n\\subsection{PH spacetime}\\label{sec:PH}\nPH spacetime is the direct product of two-dimensional Minkowski spacetime $\\mathbb{M}_2$ and the two-sphere ${\\mathbb{S}^2}$~\\cite{Griffiths&Podolsky}. This becomes manifest when writing its line element as\\footnote{We make the units choice that the radius of the two-spheres is equal to one.}\n\\begin{equation} ds^2=-dt^2+dy^2+d\\Omega^2,\\label{PHlel}\\end{equation} where $$d\\Omega^2=d\\theta^2+\\sin^2\\theta d\\varphi^2,$$\nwith $(t,y)\\in \\mathbb{R}^2$ global inertial coordinates in $\\mathbb{M}_2$, and $\\theta\\in [0,\\pi]$ and $\\varphi\\in (-\\pi,\\pi]$\nthe standard angular coordinates in ${\\mathbb{S}^2}$. \nThe Ricci scalar is $R=2$.\n\nPH being the direct product $\\mathbb{M}_2\\times {\\mathbb{S}^2}$, its world function $\\sigma$ is readily given~\\cite{Casals:2012px} as the sum of the world functions in $\\mathbb{M}_2$ and ${\\mathbb{S}^2}$, respectively $\\sigma_{\\mathbb{M}_2}$ and $\\sigma_{{\\mathbb{S}^2}}$: $\\sigma(x,{x^\\prime}) = \\sigma_{\\mathbb{M}_2}+\\sigma_{{\\mathbb{S}^2}}$.\nIn their turn, these world functions are given, in normal neighbourhoods, by\n\\begin{equation} \\sigma_{\\mathbb{M}_2}=-\\frac12\\eta^2\\equiv -\\frac12(t-{t^\\prime})^2+\\frac12(y-{y^\\prime})^2 \\label{sigbardef}\\end{equation}\nand\n\\begin{align} \n\\sigma_{{\\mathbb{S}^2}}=\\,&\\frac{\\gamma^2}{2},\\\\\n\\cos\\gamma\\equiv\\,& \\cos\\theta\\cos{\\theta^\\prime}+\\sin\\theta\\sin{\\theta^\\prime}\\cos(\\varphi-{\\varphi^\\prime}).\\nonumber \\label{gamdef}\\end{align}\nWe thus have\n\\begin{equation} \n\\sigma=\n-\\frac12\\eta^2 + \\frac12\\gamma^2=-\\frac12(t-{t^\\prime})^2+\\frac12(y-{y^\\prime})^2+ \\frac12\\gamma^2.\n\\end{equation}\nClearly, $\\eta\\in \\mathbb{R}$ is the geodesic distance in the whole of $\\mathbb{M}_2$. In its turn, $\\gamma\\in [0,\\pi]$ is the geodesic (or angle) {\\it separation} in ${\\mathbb{S}^2}$, while it also is the geodesic distance in normal neighbourhoods of ${\\mathbb{S}^2}$ (see~\\cite{Casals:2019heg,casals2016global} for this subtle but important distinction between geodesic separation and geodesic distance in the context of Schwarzschild spacetime).\nNull geodesics (for which $\\sigma=0$ in normal neighbourhoods) focus at the first caustic points: $\\eta=\\gamma=\\pi$.\nAfter crossing the first caustic, the envelope of null geodesics emanating from a base point forms the (future) boundary of the maximal normal neighbourhood of the base point; this boundary is given by $\\eta=2\\pi-\\gamma\\in [\\pi,2\\pi]$ (see the left panel of Fig.1 in \\cite{Casals:2012px}).\nSince $V(x,x')$ is only defined in normal neighbourhoods, this will also be part of the boundary of the grid in our numerical scheme.\nThe other part is given by the null hypersurface corresponding to the envelope of future-directed {\\it direct} null geodesics, i.e., by $\\eta=\\gamma\\in [0,\\pi)$ (so that it is $\\sigma=0$ with $\\eta\\geq 0$).\nThat is, the future boundary $\\eta=2\\pi-\\gamma\\in [\\pi,2\\pi]$ of the maximal normal neighourhood of the base point together with the boundary $\\eta=\\gamma\\in [0,\\pi)$ of the causal future of the base point form the boundary of our numerical grid.\n\nIt is straight-forward to obtain the d'Alembertian in PH in the above coordinates:\n\\begin{align}\n&\n\\Box=\\Box_{\\mathbb{M}_2}+\\Box_{{\\mathbb{S}^2}},\\\\\n&\n\\Box_{\\mathbb{M}_2}=-\\frac{\\partial^2}{\\partial t^2}+\\frac{\\partial^2}{\\partial y^2},\n\\nonumber\\\\\n&\n\\Box_{{\\mathbb{S}^2}}=\\frac{1}{\\sin\\theta}\\frac{\\partial}{\\partial \\theta}\\sin\\theta\\frac{\\partial}{\\partial \\theta}+\\frac{1}{\\sin^2\\theta}\\frac{\\partial^2}{\\partial\\varphi^2}.\\nonumber\n\\end{align}\n\nNow, given that both $\\mathbb{M}_2$ and ${\\mathbb{S}^2}$ are maximally-symmetric manifolds, it is easy to see that the above operators $\\Box_{\\mathbb{M}_2}$ and $\\Box_{\\mathbb{S}^2}$ become {\\it ordinary} differential operators when rewritten in terms of the corresponding geodesic distances.\nExplicitly,\n\\begin{equation}\n\\Box_{\\mathbb{M}_2}=-\\frac{\\partial^2}{\\partial \\eta^2}-\\frac{1}{\\eta}\\frac{\\partial^2}{\\partial \\eta}\n\\end{equation}\nand\n\\begin{align}\n\\Box_{\\mathbb{S}^2}=\\frac{\\partial^2}{\\partial \\gamma^2}+\\cot\\gamma \\frac{\\partial}{\\partial \\gamma}=\n\\frac{1}{\\sin\\gamma}\\frac{\\partial}{\\partial \\gamma}\\left(\\sin\\gamma\\frac{\\partial}{\\partial \\gamma}\\right).\n\\end{align}\n\nThe wave equation~\\eqref{eq:V wave eq} thus becomes\n\\begin{equation}\\label{eq:wave eq PH}\n\\left[\n\\frac{\\partial^2}{\\partial \\gamma^2}+\\cot\\gamma \\frac{\\partial}{\\partial \\gamma}\n-\\frac{\\partial^2}{\\partial \\eta^2}-\\frac{1}{\\eta}\\frac{\\partial}{\\partial \\eta}\n-\\zeta\\right]\\! \\! V(x,{x^\\prime})\\!=0\n\\end{equation}\nwhere $\\zeta\\equiv m^2+\\xi R=m^2+2\\xi$.\nThus, we have reduced the wave equation, which is generally a four-dimensional PDE, to a two-dimensional PDE in PH.\n\nIn~\\cite{Casals:2012px}, it was found that, in PH, it is\n$V=V(\\eta,\\gamma)$ and $\\nu_k=\\nu_k(\\gamma)$ and \nclosed form expressions for some Hadamard quantities were obtained. Specifically, it was found that\n\\begin{align} \\label{udef}\n&\nU(x,x')= U(\\gamma)=\n\\left|\\frac{\\gamma}{\\sin\\gamma}\\right|^{1\/2},\n\\end{align}\nand, by solving Eqs.~\\eqref{eq:transp eq V}\nand \\eqref{eq:IC V}, that\n\\begin{align}\n\\hat{V} =\\nu_0(\\gamma)=\\frac18U(\\gamma)\\left(1-4\\zeta\n+\\frac{1}{\\gamma^2}-\\frac{\\cot\\gamma}{\\gamma}\\right).\\label{v0def}\n\\end{align}\nThe higher orders $\\nu_k$, $k>0$, can in principle be obtained from $\\nu_{k-1}$ via a recurrence relation.\nFor the specific case $\\zeta=1\/4$ it was obtained that\n\\begin{align} \\label{eq:hat V1 M2xS2}\n& \\nu_1=\n\\\\ &\nU(\\gamma)\\frac{2\\gamma^2-3 \\csc ^2(\\gamma ) \\left[6 \\gamma ^2+2 \\gamma \\sin (2 \\gamma )+5 \\cos (2 \\gamma )-5\\right]}{256 \\gamma ^4},\n\\nonumber\n\\end{align}\nfor which regularity of $\\nu_1$ at $\\gamma=0$ was required.\n\nWe note that $\\hat{V}$ is regular for all $\\gamma\\in [0,\\pi)$ but it diverges (like $(\\pi-\\gamma)^{-3\/2}$) at the antipodal points $\\gamma=\\pi$. These antipodal points, however, lie outside maximal normal neighbourhoods: \nAs is manifest, ${\\mathbb{S}^2}$-envelopes of null geodesics focus along $\\gamma=\\pi$ (a line of caustics), as in Schwarzschild spacetime. It has been observed~\\cite{Dolan:2011fh,Zenginoglu:2012xe,casals2016global,Ori1,Harte:2012uw,Casals:2012px,CDOWa} that, when this happens, the retarded Green function diverges when ${x^\\prime}$ and $x$ are connected by a null geodesic (even beyond normal neighbourhoods) displaying the following {\\it global} fourfold (leading) singularity structure\\footnote{Here, $\\sigma$ refers to a well-defined extension of the world\nfunction outside normal neighbourhoods~\\cite{casals2016global,Casals:2019heg}. We also note that this structure does not hold at caustics~\\cite{casals2016global} and that the subleading order (in Schwarzschild and outside caustics) is given in~\\cite{casals2016global}.}: $\\delta(\\sigma) \\to \\text{PV}\\left(1\/\\sigma\\right)\\to -\\delta(\\sigma) \\to -\\text{PV}\\left(1\/\\sigma\\right)\\to \\delta(\\sigma)\\dots$, where $\\text{PV}$ denotes the principal value distribution.\nSince $V$ is equal to $G_{\\textrm{ret}}$ in a region of causal separation which lies inside a normal neighbourhood,\n$V$ must diverge like $G_{\\textrm{ret}}$, i.e. as $\\text{PV}\\left(1\/\\sigma\\right)$, when approaching the end of the normal neighbourhood in this direction.\nAs we shall see in Sec.~\\ref{sec:results V}, the divergence of $\\hat{V}$ at $\\gamma=\\pi$ propagates in this manner throughout the end of the maximal normal neighbourhood of the base point.\n\nThe upshot is that, in PH, we have reduced \nthe original wave equation to a two-dimensional PDE (see Eq.~\\eqref{eq:wave eq PH}), and that the Hadamard tail $\\hat{V}$ on the light cone is known in closed form (see Eq.~\\eqref{v0def}).\nIn the next section we will use these advantageous features to help us numerically solve the wave equation \\eqref{eq:wave eq PH} and thus to calculate $V$ inside the light cone.\n\n\n\\section{Solving the Characteristic Initial Value Problem for the Hadamard Tail}\\label{sec:CID}\n\nIn this work, we will directly solve the wave equation \\eqref{eq:wave eq PH} as a characteristic initial value problem.\nMore concretely, we shall develop a numerical scheme which will evolve initial data on the light cone, i.e., on\n$$\n \\sigma=-\\frac{1}{2}\\eta^2+\\frac{1}{2}\\gamma^2=0.\n$$\nThis CID is given by Eqs.\\eqref{eqn:VsigmaExpansion}--\\eqref{eq:hat V1 M2xS2}.\nThis section provides the details of the scheme and the results. We split this section into three subsections: we first rewrite the wave equation in variables suitable to the CID problem; we then describe the numerical scheme; finally, we show our results for $V$.\n\n\\subsection{Wave equation as a Characteristic Initial Value problem} \n\nLet us introduce the variables \n\\begin{equation}\\label{eq:uv}\nu\\equiv \\eta-\\gamma, \\quad v\\equiv \\eta+\\gamma,\n\\end{equation}\nwhich are naturally adapted to the Characteristic Initial Value problem, since $\\sigma=-uv\/2$.\nIn these variables, the d'Alembertian in PH (see Eq.~\\eqref{eq:wave eq PH}) becomes\n\\begin{equation}\n \\Box=-4\\frac{\\partial^2}{\\partial u\\partial v}-Q\\frac{\\partial}{\\partial v}-S\\frac{\\partial}{\\partial u},\n\\end{equation}\nwhere\n\\begin{align*}\n Q\\equiv \\,&\\frac{2}{v+u}-\\cot\\frac{v-u}{2},\\\\\n S\\equiv\\,&\\frac{2}{v+u}+\\cot\\frac{v-u}{2}.\n\\end{align*}\n\nTherefore, Eq.~\\eqref{eq:wave eq PH} turns into\n\\begin{equation}\n \\left(4\\frac{\\partial^2}{\\partial u\\partial v}+Q\\frac{\\partial}{\\partial v}+S\\frac{\\partial}{\\partial u}+\n \\zeta\n \\right)V(x,{x^\\prime})=0.\\label{eqn:VUVEqn}\n\\end{equation}\nWe note the appearance of {\\it first}-order derivatives with respect to $u$ and $v$, arising from the first-order derivatives with respect to $\\eta$ and $\\gamma$ in Eq.~\\eqref{eq:wave eq PH}.\nWe also note that, even though $u$ and $v$ range over the reals to cover the whole spacetime, the domain over which we solve Eq.~\\eqref{eqn:VUVEqn} is limited by the range of $\\gamma\\in [0,\\pi]$ and the region where $V$ is defined.\nIn the next subsection we detail how these constraints are reflected on the domains for $u$ and $v$.\n\nLet us now turn to the CID. On the $u-v$ plane, the light cone is located along the $u=0$ and $v=0$ lines. Thus, $\\hat{V}$ in terms of $u$ and $v$, is given by\n\\begin{equation}\\label{eqn:VCIDBC}\n \\begin{aligned}\n \\left.V\\right|_{u=0}=\\,&\\nu_0\\left(\\frac{v}{2}\\right),\\\\\n \\left.V\\right|_{v=0}=\\,&\\nu_0\\left(-\\frac{u}{2}\\right),\n \\end{aligned}\n\\end{equation}\nwhere $\\nu_0=\\nu_0(\\gamma)$ is given in Eq.~\\eqref{v0def}.\n\n\nThe wave Eq.~\\eqref{eqn:VUVEqn}, together with the CID \\eqref{eqn:VCIDBC} constitutes our Characteristic Initial Value problem.\nWe next present \n the finite difference method that\n we shall use to solve it.\n\n\\subsection{Numerical Scheme}\n\n\\begin{figure}\n \\centering\n \\includegraphics{Figures\/CIDGridM2xS2.pdf}\n \\caption{\n Grid distribution for a finite difference scheme for solving a two-dimensional PDE where $u$ and $v$ denote the independent variables and $2h$ is the stepsize.\n }\n \\label{fig:CIDGridM2xS2}\n\\end{figure}\n\nThere have already been some implementations of CID schemes for solving the two-dimensional PDE (not containing first order derivatives, unlike \\eqref{eqn:VUVEqn}) which is obeyed by the (smooth factor\\footnote{The other factor contains non-smooth Heaviside distributions.} in the) $\\ell$-multipolar modes of the retarded Green function in Schwarzschild spacetime (see, e.g., Eq.~(C2) in~\\cite{Jonsson:2020npo}).\nLet us briefly discuss these schemes.\nIn Ref.~\\cite{mark2017recipe}, the authors implemented a scheme (previously proposed by Lousto and Price \\cite{Lousto:1997wf}) of order $h^4$, where $2h$ is the stepsize of the grid (see Fig.~\\ref{fig:CIDGridM2xS2}). In order to calculate the value of the field at a point, their scheme required field data on the immediately ``previous\" grid points -- e.g., on the points S, E and W in order to obtain the value of the field at the point $N$ in Fig.~\\ref{fig:CIDGridM2xS2}.\nIn~\\cite{Jonsson:2020npo}, together with collaborators, we extended the scheme in~\\cite{mark2017recipe} to order $h^6$ at the expense of requiring the value of the field and of its first-order derivatives at the same grid points as in the lower-order scheme of~\\cite{mark2017recipe}.\nMore recently, Ref.~\\cite{PhysRevD.103.124022} came up with another scheme which they implemented to order $h^6$ (although in principle can be generalized to any higher order) at the expense of requiring the value of the field at more points than in Refs.~\\cite{mark2017recipe,Jonsson:2020npo}.\n\nIn our current work in PH, the PDE \\eqref{eqn:VUVEqn} is also two-dimensional but, unlike in the case of Schwarzschild just reviewed, and as we have emphasized, it contains first order derivatives and it is satisfied by the full field.\nWe note that first order derivatives also appear in Schwarzschild spacetime in the PDE satisfied by the full field as well as in the Teukolsky PDE satisfied by multipolar field modes for fields of higher spin~\\cite{Teukolsky:1973ha}.\nFor solving Eq.~\\eqref{eqn:VUVEqn}, we choose to essentially follow the fourth order scheme of Ref.~\\cite{Jonsson:2020npo} and adapt it to our specific PDE.\n\nAnother difference between our setup and that in Refs.~\\cite{mark2017recipe,Jonsson:2020npo,PhysRevD.103.124022} is that, since Eq.~\\eqref{eqn:VUVEqn} is obeyed by the Hadamard tail $V$, we only solve it inside the maximal normal neighbourhod. Therefore, the independent variables should only range over the finite intervals $v\\in [0,2\\pi)$ and $u\\in [0,v]$ (dictated by the range $\\gamma\\in [0,\\pi)$ inside the maximal normal neighbourhood). This is unlike the problem in Refs.~\\cite{mark2017recipe,Jonsson:2020npo,PhysRevD.103.124022}, which is for the retarded Green function, and so with independent variables, which are null coordinates, in principle ranging over the whole real line.\nIn order to be able to map better our problem to that in Refs.~\\cite{mark2017recipe,Jonsson:2020npo,PhysRevD.103.124022} and the CID problem in Schwarzschild in general, we shall henceforth consider, without loss of generality, that the spacetime points $x$ and ${x^\\prime}$ have $\\theta=\\theta'=\\pi\/2$ and $\\varphi'=0$ and, further, that $\\gamma\\equiv \\varphi\\in(-\\pi,+\\pi)$ denotes the azimuthal angle of $x$ (instead of the angular separation, which is in $[0,\\pi]$, as until now).\nThe variables $u$ and $v$ continue to be defined as in \\eqref{eq:uv} but now with $\\gamma\\in (-\\pi,+\\pi)$ being an azimuthal angle. This means that the region of interest for calculating $V(x,x')$, which is the part of the region of causal separation which lies inside the maximal normal neighbourhood, is bounded by\n$\\eta=2\\pi- \\gamma$ together with $\\eta=\\gamma$ if $\\gamma\\in [0,\\pi]$\\footnote{\\label{ftn:gamma}Here we allow $\\gamma$ to take on the value $\\pi$ (respectively $-\\pi$) so as to refer to the {\\it boundary} of the region of interest.}, as explained in Sec.~\\ref{sec:PH}, and by\n$\\eta=2\\pi+ \\gamma$ together with $\\eta=-\\gamma$ if $\\gamma\\in [-\\pi,0]$\\textsuperscript{\\ref{ftn:gamma}}\nafter extending the range of $\\gamma$ \ninto the negative line.\nIn the CID variables, this means that the region of interest covered by the numerical grid, is given by $u,v\\in [0,2\\pi)$.\n\n\nWe next describe our scheme and its implementation for obtaining $V$ for various values of $\\zeta$.\nWe start by describing a lower order, $\\order{h^4}$ of accuracy, version of the scheme. We do so in order to more clearly highlight the key distinction in solving the PDE \\eqref{eqn:VUVEqn}, containing first-order derivatives, as opposed to the PDE for multipolar modes of Refs.~\\cite{mark2017recipe,Jonsson:2020npo,PhysRevD.103.124022}, not containing any first-order derivatives.\nWe then describe the method at the higher order $\\order{h^6}$ of accuracy. \n\n\\subsubsection{CID scheme setup}\n\nIn order to solve the wave Eq.~\\eqref{eqn:VUVEqn} with the CID in Eq.~\\eqref{eqn:VCIDBC}, we first establish a uniform grid of points on the $u-v$ plane, which we show in Fig.~\\ref{fig:CIDGridM2xS2}. The spacing between grid points is $2h$. The next step is to integrate Eq.~\\eqref{eqn:VUVEqn} on each of the squares in the grid. For the $S$-$E$-$N$-$W$ square in Fig.~\\ref{fig:CIDGridM2xS2} we have\n\n\\begin{widetext}\n \\begin{align}\n 4\\int\\limits_{SENW} \\frac{\\partial^2V}{\\partial v\\partial u} \\,\\textrm{d}v\\,\\textrm{d}u+\\int\\limits_{SENW} Q\\frac{\\partial V}{\\partial v} \\,\\textrm{d}v\\,\\textrm{d}u\n +\\int\\limits_{SENW} S\\frac{\\partial V}{\\partial u} \\,\\textrm{d}v\\,\\textrm{d}u+\n \n \\zeta\n \\int\\limits_{SENW}V \\,\\textrm{d}v\\,\\textrm{d}u=0.\\label{eqn:squareIntegralV}\n \\end{align}\n\\end{widetext}\nThe first integral in the left hand side of Eq.~\\eqref{eqn:squareIntegralV} can be readily evaluated exactly as\n\\begin{equation}\n \\int\\limits_{SENW} \\frac{\\partial^2V}{\\partial v\\partial u} \\,\\textrm{d}v\\,\\textrm{d}u=V_N-V_E-V_W+V_S,\n\\end{equation}\nwhere the subindices $N$, $E$, $W$ and $S$ in $V$ refer to the point on the grid where $V$ is to be evaluated. For the remaining three integrals, we Taylor expand the integrands about the central point $O=(u_O,v_O)$ in the square. The Taylor series expansion of the product of two smooth functions $F(v,u)$ and $G(v,u)$ about $O$ is\n\\begin{widetext}\n \\begin{equation}\n F(v,u)G(v,u)=\\sum_{\\substack{0\\leq m,n\\leq K\\\\m+n\\leq K}}\\frac{1}{m!\\, n!}\\left(\\frac{\\partial^{m+n}}{\\partial v^m\\partial u^n}(F\\,G)\\right)_O(v-v_0)^m(u-u_0)^n+\\mathcal{O}(h^{K+1}),\n \\label{eqn:FGTaylorSeries}\n \\end{equation}\n\\end{widetext}\nwhere $K$ determines the order in the expansion. We expand in this manner the last three integrands in Eq.~\\eqref{eqn:squareIntegralV} (replacing $F$ and $G$ in Eq.~\\eqref{eqn:FGTaylorSeries} by the appropriate functions) to the desired order.\n\n\n\\subsubsection{CID scheme to $\\mathcal{O}(h^4)$}\n\nUsing \\eqref{eqn:FGTaylorSeries} to order $h^2$, the last three\nintegrals in \\eqref{eqn:squareIntegralV} are given by\n\\begin{align}\\label{eqn:squareInts}\n \\int\\limits_{SENW} Q\\frac{\\partial V}{\\partial v} \\,\\textrm{d}v\\,\\textrm{d}u=\\,&4h^2Q_O\\left(\\frac{\\partial V}{\\partial v}\\right)_O+\\mathcal{O}(h^4)\\\\\n \\int\\limits_{SENW} S\\frac{\\partial V}{\\partial u} \\,\\textrm{d}v\\,\\textrm{d}u=\\,&4h^2S_O\\left(\\frac{\\partial V}{\\partial u}\\right)_O+\\mathcal{O}(h^4),\\\\\n \\int\\limits_{SENW}V \\,\\textrm{d}v\\,\\textrm{d}u=\\,&4h^2V_O+\\mathcal{O}(h^4),\n\\end{align}\nwhere, again, the subindex $O$ indicates that the corresponding quantity is evaluated at the point $O$. In this result we only considered the first two leading orders in the Taylor series to obtain the integrals to order $h^3$. However, it can be shown that the contribution to the integral from the next-to-leading order term in the Taylor series vanishes.\n\nIn order to calculate $V$ and its derivatives at the point $O$, we evaluate its Taylor expansion at the points $E,W$ and $S$ to order $h$. This allows us to construct a system of three equations where the unknown variables are $V_O$, $\\left(\\pd{V}{u}\\right)_O$ and $\\left(\\pd{V}{v}\\right)_O$. We find:\n\\begin{align}\n V_O=\\,&\\frac{V_E+V_W}{2}+\\mathcal{O}(h^2),\\\\\n \\left(\\frac{\\partial V}{\\partial u}\\right)_O=\\,&\\frac{V_W-V_S}{2h}+\\mathcal{O}(h),\\\\\n \\left(\\frac{\\partial V}{\\partial v}\\right)_O=&\\,\\frac{V_E-V_S}{2h}+\\mathcal{O}(h).\\label{eqn:dVdvO}\n\\end{align}\n\nIn this way, by using Eqs.~\\eqref{eqn:squareInts}-\\eqref{eqn:dVdvO} and Eq.~\\eqref{eqn:squareIntegralV}, we find that the sought-after value of $V$ at the point $N$ is given by\n\\begin{widetext}\n \\begin{equation}\\label{eqn:VNExpression}\n V_N=-V_S-\\left(\\frac{V_E+V_W-2V_S}{u_O+v_O}+\\frac{1}{2}(V_E-V_W)\\cot{\\frac{v_O-u_O}{2}}\\right)h+\\left(1\n \n -\\frac{\\zeta}{2}\n h^2\\right)(V_E+V_W)+\\order{h^4},\n \\end{equation}\n\\end{widetext}\nIf we compare this expression for $V_N$ with its equivalent in Refs.~\\cite{mark2017recipe,Jonsson:2020npo,PhysRevD.103.124022}, we immediately note the additional term linear in $h$ in Eq.~\\eqref{eqn:VNExpression}. This term is related with the two {\\it first} order derivatives in Eq.~\\eqref{eqn:VUVEqn}, absent in~\\cite{mark2017recipe,Jonsson:2020npo,PhysRevD.103.124022}.\n\nAnother key difference appears when $u_O=v_O$ (i.e., along $\\gamma=0$). The spherical symmetry of $\\mathbb{M}_2\\times{\\mathbb{S}^2}$ implies that $V(\\eta,\\gamma)=V(\\eta,-\\gamma)$ (or $V_E=V_W$) for any square with a central point such that $u_O=v_O$. More precisely, for all squares with $u_O=v_O$, $V_N$ is calculated as\n\\begin{align}\\label{eqn:VNDiag}\n V_N=\\,&-V_S-2h\\left(\\frac{V_E-V_S}{u_O+v_O}\\right)+ \\nonumber\\\\\n &\n \\left(2-\\zeta h^2\n \\right)V_E+\\order{h^4}.\n\\end{align}\nThis symmetry reduces by a half the amount of data to calculate. \n\nWith the above expression for $V_N$, we can reformulate the CID problem in the following way. We first split the grid in Fig.~\\ref{fig:CIDGridM2xS2} into two triangles, one on each side of the $u=v$ line (i.e., $\\gamma=0$). For the bottom triangle, $V_N$ in any square with $u_O=v_O$ only depends on $V_E$ and $V_S$ (see Eq.~\\eqref{eqn:VNDiag}). This implies that, for the bottom triangle, and after having imposed the symmetry along $\\gamma=0$, we only require $V|_{u=0}$ as initial data. As already pointed out, the values at points in the top triangle can just be obtained using the $V(\\eta,\\gamma)=V(\\eta,-\\gamma)$ symmetry.\nAlternatively, one could choose to apply the scheme to points in the top triangle, in which case we would have the opposite situation: the only required initial data in that case would be $V|_{v=0}$.\n\n\nBy knowing the first term, $\\nu_0(\\gamma)$, in Eq.~\\eqref{eqn:VsigmaExpansion}, we were able to calculate $V$ to order $\\order{h^4}$ using this scheme. \nWe next develop a higher order scheme by considering additional initial data on the light cone with the help of the next term, $\\nu_1(\\gamma)$, in Eq.~\\eqref{eqn:VsigmaExpansion}.\n\n\\subsubsection{CID scheme to $\\mathcal{O}(h^6)$}\\label{eq:Oh6}\nIn order to implement a higher order CID scheme, we simply have to calculate the integrals in Eq.~\\eqref{eqn:squareIntegralV} (except the first one) to the desired order. We achieve this by increasing the order in the Taylor expansion of each integrand. In the previous subsection, in order to obtain $V$ to $\\order{h^4}$, we Taylor expanded the integrands in Eq.~\\eqref{eqn:squareIntegralV} to $\\order{h^2}$. Thus, for a CID scheme to $\\order{h^6}$ we Taylor expand it to $\\order{h^4}$. This increase in the order requires additional data since there are more Taylor coefficients to be calculated. These additional data are easily obtained by differentiating Eq.~\\eqref{eqn:VsigmaExpansion} once with respect to $u$ and $v$ and evaluating it at $\\sigma=-\\frac{1}{2}uv=0$. This procedure leads to the following initial data:\n\\begin{align}\\label{eqn:addVCIDBC}\n \\left.\\frac{\\partial V}{\\partial u}\\right|_{u=0}=\\,&-\\frac{1}{2}{\\nu_0}'\\left(\\frac{v}{2}\\right)-\\frac{1}{2}\\nu_1\\left(\\frac{v}{2}\\right)\\,v,\\\\\n \\left.\\frac{\\partial V}{\\partial u}\\right|_{v=0}=\\,&-\\frac{1}{2}{\\nu_0}'\\left(-\\frac{u}{2}\\right),\\\\\n \\left.\\frac{\\partial V}{\\partial v}\\right|_{u=0}=\\,&\\frac{1}{2}{\\nu_0}'\\left(\\frac{v}{2}\\right),\\\\\\label{eqn:addVCIDBCEnd}\n \\left.\\frac{\\partial V}{\\partial v}\\right|_{v=0}=\\,&\\frac{1}{2}{\\nu_0}'\\left(-\\frac{u}{2}\\right)-\\frac{1}{2}\\nu_1\\left(-\\frac{u}{2}\\right)\\,u.\n\\end{align}\nTogether with Eq.~\\eqref{eqn:VCIDBC}, we have just established how to obtain the necessary initial data to calculate $V_N$ using the scheme here. \n\nWe now proceed to calculate the integrals in Eq.~\\eqref{eqn:squareIntegralV}, with the last three integrands now expanded to $\\order{h^4}$. This yields\n\\begin{widetext}\n \\begin{align}\\notag\n V_N=\\,&V_E+V_W-V_S-\\left[Q_O\\evalO{\\pd{V}{v}}+S_O\\evalO{\\pd{V}{u}}+\n \\zeta\\,\n V_O\\right]h^2-\\\\\\notag\n &\\frac{1}{6}\\left[\n \\zeta\\,\n \\evalO{\\pd{{}^2V}{u^2}+\\pd{{}^2V}{v^2}}+2\\evalO{\\pd{Q}{v}}\\evalO{\\pd{^2V}{v^2}}+2\\evalO{\\pd{S}{u}}\\evalO{\\pd{^2V}{u^2}}\\right.+\\\\\\notag\n &\\evalO{\\pd{V}{v}}\\evalO{\\pd{^2Q}{u^2}+\\pd{^2Q}{v^2}}+2\\evalO{\\pd{^2V}{u\\partial v}}\\evalO{\\pd{Q}{u}+\\pd{S}{v}}+\\evalO{\\pd{V}{u}}\\evalO{\\pd{^2S}{u^2}+\\pd{^2S}{v^2}}+\\\\\\label{eqn:higherVNExpresion}\n &\\left.S_O\\evalO{\\pd{{}^3V}{u\\partial v^2}+\\pd{^3V}{u^3}}+Q_O\\evalO{\\pd{{}^3V}{u^2\\partial v}+\\pd{^3V}{v^3}}\\right]h^4+\\order{h^6}.\n \\end{align}\n\\end{widetext}\nIn order to calculate $V$ and its corresponding derivatives at the point $O$, we construct a system of 12 equations by evaluating Eq.~\\eqref{eqn:FGTaylorSeries} (with $F\\cdot G=V$) and its first order derivatives at the points $N$, $E$, $W$ and $S$. Ten of these 12 equations are then used to express $V_O$ and its 9 derivatives appearing in Eq.~\\eqref{eqn:higherVNExpresion} in terms of $V_E$, $V_W$, $V_S$ and their derivatives:\n\\begin{widetext}\n \\begin{align}\n 4V_O&=2V_E+2V_W+h\\left(\\frac{\\partial V}{\\partial u}-\\frac{\\partial V}{\\partial v}\\right)_E-h\\left(\\frac{\\partial V}{\\partial u}-\\frac{\\partial V}{\\partial v}\\right)_W+\\order{h^4},\\\\\n 8h\\left(\\frac{\\partial V}{\\partial u}\\right)_O&=-5V_S-V_E+5V_W+V_N\n -2h\\left(\\frac{\\partial V}{\\partial u}\n +\\frac{\\partial V}{\\partial v}\\right)_S\n -2h\\left(\\frac{\\partial V}{\\partial u}\n -\\frac{\\partial V}{\\partial v}\\right)_W+\\order{h^4},\\\\\n 8h\\left(\\frac{\\partial V}{\\partial v}\\right)_O&=-5V_S+5V_E+V_W-V_N\n -2h\\left(\\frac{\\partial V}{\\partial u}\n +\\frac{\\partial V}{\\partial v}\\right)_S\n +2h\\left(\\frac{\\partial V}{\\partial u}\n -\\frac{\\partial V}{\\partial v}\\right)_E+\\order{h^4},\\\\\n 4h^2\\left(\\frac{\\partial^2V}{\\partial u^2}\\right)_O&=V_S-V_E-V_W+V_N+2h\\left(\\frac{\\partial V}{\\partial u}\\right)_E-2h\\left(\\frac{\\partial V}{\\partial u}\\right)_W+\\order{h^4},\\\\\\label{eqn:GdGofEWNSLast}\n 4h^2\\left(\\frac{\\partial^2V}{\\partial v^2}\\right)_O&=V_S-V_E-V_W+V_N-2h\\left(\\frac{\\partial V}{\\partial v}\\right)_E+2h\\left(\\frac{\\partial V}{\\partial v}\\right)_W+\\order{h^4},\\\\\n 4h^2\\left(\\frac{\\partial^2V}{\\partial v\\partial u}\\right)_O&=V_N+V_S-V_E-V_W+\\order{h^4},\\\\\n \\frac{2}{3}h^3\\left(\\frac{\\partial^3V}{\\partial v^3}\\right)_O&=V_S-V_E+h\\left(\\frac{\\partial V}{\\partial v}\\right)_S+h\\left(\\frac{\\partial V}{\\partial v}\\right)_E+\\order{h^4},\\\\ \\frac{2}{3}h^3\\left(\\frac{\\partial^3V}{\\partial u^3}\\right)_O&=V_S-V_W+h\\left(\\frac{\\partial V}{\\partial u}\\right)_S+h\\left(\\frac{\\partial V}{\\partial u}\\right)_W+\\order{h^4},\\\\\n 4h^3\\left(\\frac{\\partial^2V}{\\partial v^2\\partial u}\\right)_O&=V_N+V_S-V_E-V_W+2h\\left(\\frac{\\partial V}{\\partial v}\\right)_S-2h\\left(\\frac{\\partial V}{\\partial v}\\right)_W+\\order{h^4},\\\\\\label{eqn:GdGofEWNSNLast}\n 4h^3\\left(\\frac{\\partial^2V}{\\partial v\\partial u^2}\\right)_O&=V_N+V_S-V_E-V_W+2h\\left(\\frac{\\partial V}{\\partial u}\\right)_S-2h\\left(\\frac{\\partial V}{\\partial u}\\right)_E+\\order{h^4}.\n \\end{align}\n\\end{widetext}\n\nThe remaining two equations are used to calculate the first order derivatives of $V$ at the point $N$:\n\\begin{widetext}\n \\begin{align}\n \\left(\\pd{V}{u}\\right)_N=\\,&\\frac{V_S-V_E-V_W+V_N}{h}-\\left(\\pd{V}{u}\\right)_E+\\left(\\pd{V}{u}\\right)_W+\\left(\\pd{V}{u}\\right)_S+\\order{h^3},\\\\\n \\left(\\pd{V}{v}\\right)_N=\\,&\\frac{V_S-V_E-V_W+V_N}{h}+\\left(\\pd{V}{v}\\right)_E-\\left(\\pd{V}{v}\\right)_W+\\left(\\pd{V}{v}\\right)_S+\\order{h^3}.\\nonumber\n \\end{align}\n\\end{widetext}\nAs can be seen in the above equations, these derivatives do depend on $V_N$.\n\nSimilarly to the $\\order{h^4}$ scheme of the previous subsection, for squares in the grid with $u_O=v_O$, we have to consider the symmetries of $\\mathbb{M}_2\\times{\\mathbb{S}^2}$ again. Besides the $V_E=V_W$ symmetry condition, the higher order scheme will also make use of additional symmetry conditions for the derivatives of $V$. Specifically, for Eq.~\\eqref{eqn:higherVNExpresion} we make use of the identities \n\\begin{align}\\label{eq:ids high O}\n \\left(\\pd{V}{v}-\\pd{V}{u}\\right)_E=\\,&\\left(\\pd{V}{v}-\\pd{V}{u}\\right)_W,\\\\\\label{eqn:Vsym3}\n \\left(\\pd{V}{u}\\right)_E-\\left(\\pd{V}{u}\\right)_S=\\,&\\left(\\pd{V}{v}\\right)_E-\\left(\\pd{V}{v}\\right)_S,\n\\end{align}\nwhich respectively arise from the following conditions:\n\\begin{equation}\\label{eq:symms high O}\n \\pd{^2V}{u\\partial v}=\\pd{^2V}{v\\partial u}\n \\end{equation}\n and\n \\begin{equation}\n \\left(\\pd{V}{\\gamma}\\right)_{\\gamma=0}=0.\n\\end{equation}\n\nSummarizing, in order to use our CID scheme to $\\order{h^6}$, one should: start with CID given in Eqs.~\\eqref{eqn:VCIDBC} and \\eqref{eqn:addVCIDBC}-\\eqref{eqn:addVCIDBCEnd}, then use Eq.~\\eqref{eqn:higherVNExpresion} to obtain $V$. \nIn fact, this would be the procedure for finding numerically the values at both upper and lower triangles using CID data along both $u=0$ and $v=0$. In practise, however, it is more efficient to just use half the CID data in Eqs.~\\eqref{eqn:VCIDBC} and \\eqref{eqn:addVCIDBC}-\\eqref{eqn:addVCIDBCEnd}, either just along $u=0$ or along $v=0$, and then use $V_E=V_W$ together with the identities in Eqs.~\\eqref{eq:ids high O}-\\eqref{eqn:Vsym3} to evolve the data in one triangle only.\n\nAfter having the prescription for the CID scheme to two different orders, we implemented it using the computer algebra software {\\it Mathematica}. In the following subsection we show our results, compare them against previous results obtained using different approaches to calculate $V$ (see \\cite{Casals:2012px}) and provide new results.\n\n\\subsection{Results for $V$}\\label{sec:results V}\n\n\\begin{figure}\n \\centering\n \\includegraphics[scale=0.65]{Plots\/GretComparisonPlot.pdf}\\\\~\\\\\n \\includegraphics[width=0.45\\textwidth]{Plots\/schemesRelErrorPlot.pdf}\n \n \\caption{Top plot: Retarded Green function as an $\\ell$-mode sum (blue) and $V$ using the higher order CID scheme (dashed red), a small coordinate-separation expansion (green) and approximated by $\\nu_0+\\nu_1\\sigma$ (dashed gray), for $\\mathbb{M}_2\\times{\\mathbb{S}^2}$ with \n $\\zeta=1\/4$,\n $\\Delta y=y-y'=0$ and $\\gamma=\\pi\/2$ as functions of $\\eta=\\Delta t$.\n Bottom plot: Relative error between the $\\mathcal{O}(h^4)$ and $\\mathcal{O}(h^6)$ schemes to calculate $V$ with $h=0.00261799$.}\n \\label{fig:GretComparison}\n\\end{figure}\n\nIn this section we show our results for the Hadamard biscalar $V(x,x')$.\nIn the top plot of Fig.~\\ref{fig:GretComparison} we consider the case of $x$ and $x'$ on a static path with $y=y'$ and $\\gamma=\\pi\/2$ for $\\zeta=1\/4$.\nIn it, we compare the following: $V$ obtained using the higher order CID scheme of Sec.~\\ref{eq:Oh6} with $h=0.00261799$ (dashed red); the retarded Green function $G_{\\textrm{ret}}$ calculated with the multipolar $\\ell$-mode sum (up to $\\ell=800$) expression given in Eq.~(134)~\\cite{Casals:2012px}\\footnote{We note that in the last expression in Eq.~(134)~\\cite{Casals:2012px} there is a missing factor $\\theta(-\\sigma_{\\mathbb{M}_2})$.} (blue); the crude approximation $\\nu_0+\\nu_1\\sigma$ to $V$ (dashed gray) from Eq.~\\eqref{eqn:VsigmaExpansion}; $V$ calculated using a small coordinate distance expansion (see Ref.~\\cite{CDOWb}) (green). \nThe first divergence at $\\eta=\\pi\/2$ of $G_{\\textrm{ret}}$ corresponds to the direct null geodesic divergence $\\delta(\\sigma)$ as per the Hadamard form Eq.\\eqref{grhad}.\nThis divergence signals the start of causal separation.\nAs explained in Sec.\\ref{sec:PH}, $V$ and $G_{\\textrm{ret}}$ should agree in the region in-between this divergence, and the next divergence at $\\eta=3\\pi\/2$, which signals the end of the maximal normal neighbourhood and corresponds to a null geodesic having crossed a caustic at $\\gamma=\\pi$. Thus, this latter divergence should be of type $\\text{PV}\\left(1\/\\sigma\\right)$, in agreement with the plot.\n\n\nThe top plot of Fig.~\\ref{fig:GretComparison} also shows that the CID scheme has good agreement with the $\\ell$-mode-sum calculated $G_\\textrm{ret}$. Indeed, in the bottom plot of Fig.~\\ref{fig:GretComparison} we show that the relative error between the two CID schemes, to $\\order{h^4}$ and to $\\order{h^6}$, with the same stepsize $h=0.00261799$, is at least of order $10^{-4}$. Let us check this value for consistency. Let $e_2$ and $e_4$ be the cumulative errors for the schemes of $\\mathcal{O}(h^4)$ and $\\mathcal{O}(h^6)$, respectively. For the $n$th evolved point in the grid, these errors are given by $e_2=\\order{n(2h)^4}$ and $e_4=\\order{n(2h)^6}$. In the case of Fig.~\\ref{fig:GretComparison}, for a point close to the end of the normal neighbourhood we have $n=\\order{10^5}$. This gives $e_2=\\order{10^{-4}}$ and $e_4=\\order{10^{-9}}$, which can be taken as relative errors since $V=\\order{1}$ close the end of the normal neighbourhood. As expected based on this, $e_2$ does agree with the relative error between the two schemes shown in the bottom plot of Fig.~\\ref{fig:GretComparison}.\n\nIn Fig.~\\ref{fig:V3D} we show the plot of $V$ (to $\\order{h^6}$) for $\\zeta=1\/4$\nfor {\\it any} pair of spacetime\npoints (as long as they lie in normal neighbourhoods, so that $V$ is defined). The red line corresponds to the static worldline ($y=y'$ and $\\gamma=\\pi\/2$) of Fig.~\\ref{fig:GretComparison}. Evolving CID has allowed us to calculate $V$ everywhere where it is defined.\n\n\\begin{figure}\n \\centering\n \\includegraphics[scale=0.5]{Plots\/V3DPlot.pdf}\n \\caption{Plot of $V$ for\n all pairs of points in normal neighbourhoods with\n $\\zeta=1\/4$. The $u=2\\pi$ and $v=2\\pi$ lines correspond to the end of the normal neighbourhood where the (leading) singularity of $G_\\textrm{ret}$, and so of $V$, is of type $\\text{PV}\\left(1\/\\sigma\\right)$ (when away from caustics). The red line is along the static worldline considered in Fig.~\\ref{fig:GretComparison}.}\n \\label{fig:V3D}\n\\end{figure}\n\nWe also calculated $V$ to $\\order{h^4}$ for various values of $\\zeta\\neq 1\/4$. In the top plot of Fig.~\\ref{fig:VXi0468} we show these results for the same worldline as in Fig.~\\ref{fig:GretComparison}. For this particular worldline (which has $\\gamma=\\pi\/2$), as $\\zeta$ increases, the magnitude of $V$ decreases. \nIn the bottom plot of Fig.~\\ref{fig:VXi0468} we again plot $V$ for all possible pairs of spacetime points, but now for $\\zeta=1$. \nWe can see in it that there is a more marked change in the form, with respect to Fig.~\\ref{fig:V3D} for $\\zeta=1\/4$, near the caustic $\\gamma=\\pm\\pi$.\nWe note that we also calculated $V$ for all pairs of points for the other values of $\\zeta$ that we used in the top plot of Fig.\\ref{fig:VXi0468} but we do not display the full results since their behaviour was not so different from Fig.~\\ref{fig:V3D} for $\\zeta=1\/4$. \n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.45\\textwidth]{Plots\/VXi068Plot.pdf}\\\\~\\\\\n \\includegraphics[width=0.45\\textwidth]{Plots\/VXi068Plot3D2.pdf}\n \\caption{Top: Plot of $V$ for different values of $\\zeta$ and for the same scenario (and stepsize) as in Fig.~\\ref{fig:GretComparison}. Bottom: 3D plot of $V$ for $\\zeta=1$.}\n \\label{fig:VXi0468}\n\\end{figure}\n\n\n\\section{Discussion}\\label{sec:Discussion}\n\nIn this work we have presented and implemented a new method for calculating the tail term $V(x,x')$ in wave propagation on curved background spacetimes where null geodesics cross: by integrating the homogeneous wave equation using Characteristic Initial Data on the light cone.\nWe have provided a proof-of-concept for this method by applying it to PH, $\\mathbb{M}_2\\times{\\mathbb{S}^2}$.\nFurthermore, we have calculated $V$ for new cases: at {\\it all} spacetime points where it is defined, for various values of $\\zeta\\equiv m^2+2\\xi$.\nThe calculation of $V$ (and of the retarded Green function) at all pairs of points is useful, in particular, for a potential application to the self-consistent orbital evolution of a particle via the self-force.\n\nThe calculation in $\\mathbb{M}_2\\times{\\mathbb{S}^2}$ is technically easier than in black hole spacetimes: First, the Characteristic Initial Data $\\left.V(x,x')\\right|_{\\sigma=0}$ is known analytically and, second, the wave equation is reduced to a {\\it two}-dimensional PDE. As for the first point, we note that $V(x,x')$ was numerically calculated along null geodesics in Schwarzschild in~\\cite{Ottewill:2009uj} by solving transport equations, and thus its Characteristic Initial Data in Schwarzschild is readily available, while~\\cite{Ottewill:2009uj} provides a prescription for its calculation in Kerr. \nAs for the second point, the wave equation in Schwarzschild would acquire an extra dimension, thus becoming a {\\it three}-dimensional PDE, for which there exist numerical techniques. \nFurthermore, the three-dimensional PDE in Schwarzschild contains first-order derivatives (at least through the angular part), which the scheme presented here in the context of PH has dealt with.\nIn Kerr, the PDE would become {\\it four}-dimensional, entailing a greater numerical challenge.\nWe intend to undertake the calculation of $V(x,x')$ in these black hole spacetimes in the future.\n\n\n\n\\section{{Acknowledgments.}}\n\nWe are grateful to Adrian Ottewill and Barry Wardell for useful discussions.\nM.C.\\ acknowledges partial financial support by CNPq (Brazil), process number 314824\/2020-0. D. Q. A. acknowledges support from FAPERJ (process number 200.804\/2019) and CNPq (process number 140951\/2017-2).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\\subsection{Tree-likeness}\n\nTrees are graphs with some very distinctive and fundamental properties and it is legitimate to ask to what degree those properties can be transferred to more general structures that are tree-like in some sense \\cite[p. 253]{Diestel}.\nRoughly speaking, tree-likeness stands for something related to low\n dimensionality, low complexity, efficient information deduction (from local to global), information-lossless decomposition (from global into simple pieces)\n and nice shape for efficient implementation of divide-and-conquer strategy. For the very basic\n interconnection structures like a graph or a hypergraph,\n tree-likeness is\n naturally reflected by the strength of interconnection, namely its\n connectivity\/homotopy type or cyclicity\/acyclicity, or just the degree of derivation\n from\n some characterizing conditions of a tree\/hypertree and its various associated structures and generalizations.\n\nIn vast applications, one finds that the borderline between\ntractable and intractable cases\n may be the tree-like degree of\n the structure to be\ndealt with \\cite{CYY}. A support to this from the fixed-parameter\ncomplexity point of view is the observation that on various\ntree-structures we can design very good algorithms for many purposes\nand these algorithms can somehow be lifted to tree-like structures\n\\cite{ALS,Downey,DMC,KFL}. It is thus very useful to get information\non approximating general structures by tractable structures, namely\ntree-like structures. On the other hand, one not only finds it\nnatural that tree-like structures appear extensively in many fields,\nsay biology \\cite{Dress96}, structured programs \\cite{Thorup} and\ndatabase theory \\cite{Fagin}, as graphical representations of\nvarious types of hierarchical relationships, but also\n notice surprisingly that many practical structures we\nencounter are just tree-like, say the internet\n\\cite{ABKMRT,Kleinberg,ST} and chemical compounds \\cite{YAM}. This\nprompts in many areas the very active study of tree-like structures.\nEspecially, lots of ways to define\/measure a tree-like structure\nhave been proposed in the literature from many different\nconsiderations, just to name a few, say tree-width \\cite{RS,RS86},\ntree-length \\cite{DG,UY}, combinatorial dimension\n\\cite{Dress96,Dress84}, $\\epsilon$-three-points condition\n\\cite{Dinitz},\n $\\epsilon$-four-points condition \\cite{ABKMRT}, asymptotic connectivity\n \\cite{Bahls},\n tree-partition-width \\cite{Bod,Wood}, tree-degree \\cite{CM},\nMcKee-Scheinerman chordality \\cite{McS}, $s$-elimination dimension\n\\cite{DKT}, linkage (degeneracy) \\cite{DKT,KT,LW}, sparsity order\n\\cite{Laurent},\n persistence \\cite{Downey}, cycle rank \\cite{CYY,Lee}, various\ndegrees of acyclicity\/cyclicity \\cite{Duke,Fagin}, boxicity\n\\cite{Roberts}, doubling dimension \\cite{Gupta}, Domino treewidth\n\\cite{Bod}, hypertree-width \\cite{GLS}, coverwidth \\cite{Chen},\nspread-cut-width \\cite{CJG}, Kelly-width \\cite{Hunter},\n and many other width parameters\n \\cite{DMC,HOSG}. It is clear that many relationships among\n these concepts should be expected as they are all formulated in different ways to represent\n different aspects of our vague but intuitive idea of tree-likeness. To clarify these relationship helps to bridge the study\n in different fields focusing on different tree-likeness measures and helps to improve our understanding\n of the universal tree-like world.\n\nAs a small step in pursuing further understanding of tree-likeness,\nwe take up in this paper the modest task of comparing two parameters\nof tree-likeness,\n namely (Gromov) hyperbolicity and chordality of a graph.\n Our main result is that $k$-chordal graphs must be $\\frac{\\lfloor\n\\frac{k}{2}\\rfloor}{2}$-hyperbolic when $k\\geq 4$ (Theorem\n\\ref{main}). Besides that, we determine a complete set of\nunavoidable\n isometric subgraphs of $5$-chordal graphs attaining hyperbolicity $1$ (Theorem \\ref{main1}), as a minor\n attempt to respond to the general question,\n ``what is the structure of graphs with relative small hyperbolicity'' \\cite[p. 62]{BKM01}, and the even more general\n question,\n ``what is the structure of a very tree-like graph''.\n\nThe plan of the paper is as follows. In Sections \\ref{chordal}\n and \\ref{hyper} we introduce the two tree-likeness parameters,\n chordality and hyperbolicity, respectively.\n Section \\ref{theorem} is devoted to a general discussion of the\n relationship between chordality and hyperbolicity, including a\n presentation of our main results (Theorems \\ref{main} and\n \\ref{main1}).\nSome consequences of our main results will be listed in Section\n \\ref{oct}.\n In\nSection \\ref{parameter}, we study the relationship between several\nother tree-likeness parameters and the main objects of this paper,\nthat is to say, chordality and hyperbolicity, and make use of these\nrelationship to connect chordality and hyperbolicity. The\nrelationship between chordality and hyperbolicity thus obtained by\nnow is not as strong as Theorem \\ref{main}. But the discussion may\nbe of some independent interest.\n We present a complete and\nself-contained proof of Theorems\n \\ref{main} and \\ref{main1} in Section \\ref{proofs} in two stages: some preliminary facts are prepared in Section\n \\ref{lemmas} and the final proof appears in Section \\ref{Proof}. Following \\cite{BKM01,KM02}, the key to our work is to\n examine the extremal local configurations as described by\n Assumptions I and II (see Section \\ref{lemmas}).\nSeveral key lemmas in Section \\ref{lemmas} are basically copied\nfrom \\cite{BKM01,KM02}. It is often that these lemmas are to be\nfound as pieces of a long proof of a big statement in\n\\cite{BKM01,KM02} and so the validity of these technical lemmas\nunder some weaker assumptions needs to be carefully checked. We\ninclude the complete proofs of them, more or less as they were\npresented in \\cite{BKM01,KM02}, not only for the convenience of the\nreader but also to convince the reader that they do hold in our\nsetting.\n\n\\subsection{Chordality}\\label{chordal}\n\n\nWe only consider simple, unweighted, connected, but not necessarily\nfinite graphs. Any graph $G$ together with the usual\nshortest-path metric on it, $d_G: \\ V(G)\\times V(G)\\mapsto\n\\{0,1,2,\\ldots\\}$, gives rise to a metric space. We often suppress\nthe subscript and write $d(x,y)$ instead of $d_G(x,y)$ when the\ngraph is known by context.\n Moreover, we\nmay use the shorthand $xy$ for $d(x,y)$ to further simplify the\nnotation. Note that a pair of vertices $x$ and $y$ form an edge if\nand only if $xy=1.$ For $S,T\\subseteq V(G)$, we write\n$d(S,T)$ for $\\min _{x\\in S,y\\in T}d(x,y)$. We often omit the\nbrackets and adopt the convention that $x$ stands for the\nsingleton set $\\{x\\}$ when no confusion can be caused.\n\n\nLet $G$ be a graph. A {\\em walk of length $n$} in $G$ is a\nsequence of vertices\n $x_0,x_1,x_2,\\ldots ,x_n$ such that $x_{i-1}x_{i}=1$ for $i=1,\\ldots ,n$. If these $n+1$ vertices are\n pairwise different, we call the sequence\n a {\\em path of length $n$}. A {\\em pseudo-cycle} of length $n$ in $G$ is a\ncyclic sequence of $n$ vertices $x_1,\\ldots ,x_n\\in V(G)$ such\nthat $x_ix_{j}=1$ whenever $j=i+1 \\ (\\bmod \\ n)$; we will reserve\nthe notation $[x_1x_2\\cdots x_n]$ for this pseudo-cycle. We\ncall this pseudo-cycle an {\\em $n$-cycle}, or a {\\em cycle of length\n$n$}, if $x_1,\\ldots,x_n$ are $n$ different vertices.\n A {\\em chord} of a path or cycle is an\nedge joining nonconsecutive vertices on the path or cycle. An\n{\\em odd chord} of a cycle of even length is a chord connecting\ndifferent vertices the distance between which in the cycle is odd.\n A cycle without chord is called an\n{\\em induced cycle}, or a {\\em chordless cycle}. For any $n\\geq 3,$\nthe {\\em $n$-cycle graph} is the graph with $n$ vertices\n which has a chordless $n$-cycle and we denote this graph by $C_n$. A subgraph $H$ of a\ngraph $G$ is \\textit{isometric} if for any $u,v\\in V(H)$ it holds\n$d_H(u,v)=d_G(u,v)$. A $4$-cycle of a graph $G$ is an\n {\\em isometric $4$-cycle}\n provided the subgraph of $G$ induced by the vertices of this\ncycle is isometric and the subgraph has only those four edges which\nare displayed in the cycle. Indeed, this amounts to saying that this\ncycle is an induced\/chordless cycle; c.f. Lemma \\ref{EASY}.\n\n\n\nWe say that a graph is {\\em $k$-chordal} if it does not contain any\ninduced $n$-cycle for $n>k.$ Clearly, trees are nothing but\n$2$-chordal graphs. A $3$-chordal graph is usually termed as a {\\em\nchordal\n graph} and a $4$-chordal graph is often called a {\\em\nhole-free\n graph}. The class of $k$-chordal graphs is also\ndiscussed under the name $k$-bounded-hole graphs \\cite{Gavril}.\n\n The {\\em chordality} of a graph $G$ is the smallest integer $k\\geq 2$ such that $G$ is $k$-chordal\n \\cite{BT1}. Following \\cite{BT1}, we use the notation $\\mathbbm{l}\\mathbbm{c}(G)$ for this parameter as it is merely the length of\n the longest chordless cycle in $G$ when $G$ is not a tree. Note that our use of the concept of chordality is\n basically the same as that used in \\cite{CLS,CR} but is very different\n from the usage of this term in \\cite{McS}.\n\n\n\n\nThe recognition of $k$-chordal graphs is coNP-complete for\n$k=\\Theta (n^{\\epsilon})$ for any constant $\\epsilon >0$\n\\cite{Uehara}. Especially, to determine the chordality of the\nhypercube is attracting much attention under the name of the\nsnake-in-the-box problem due to its connection with some\nerror-checking codes problem \\cite{Klee}. Just like the famous\nsnake-in-the-box problem, it looks hard to determine the exact value\nof the chordality of general grid graphs -- it is only easy to see\nthat $\\mathbbm{l}\\mathbbm{c}(G_{m,n})$ should be roughly\nproportional to $nm$ when $\\min (n,m)>2.$\n Nevertheless, just like\nmany other tree-likeness parameters, quite a few natural graph\nclasses are known to have small chordality \\cite{BLS}. We review\n some $5$-chordal ($4$-chordal) graphs in the remainder of this subsection.\n\n\n\n\nAn {\\em asteroidal triple} ($AT$) of a graph $G$ is a a set of three\nvertices of $G$ such that for any pair of them there is a path\nconnecting the two vertices whose distance to the remaining vertex\nis at least two. A graph is {\\em AT-free} if no three vertices form\nan $AT$ \\cite[p. 114]{BLS}. Obviously, all $AT$-free graphs are\n$5$-chordal. A graph is an {\\em interval graph} exactly when it is\nboth chordal and $AT$-free \\cite[Theorem 7.2.6]{BLS}. $AT$-free\ngraphs also include {\\em cocomparability graphs} \\cite[Theorem\n7.2.7]{BLS}; moreover, all {\\em bounded\n tolerance graphs}\nare cocomparability\n graphs \\cite{GMT} \\cite[Theorem 2.8]{MA} and a graph is a {\\em permutation graph} if and only if\n itself and its complement are cocomparability graphs \\cite[Theorem 4.7.1]{BLS}.\nAn important subclass of cocomparability graphs is the class of\n {\\em threshold graphs}, which are those graphs without any induced\nsubgraph isomorphic to the $4$-cycle, the complement of the\n$4$-cycle or the path of length $3$ \\cite[p. 23]{MA}.\n\n\n\n\n A graph is {\\em weakly chordal } \\cite{GMT,Hayward} when both itself and\n its complement are $4$-chordal.\n Note that all tolerance graphs \\cite{MA} are domination graphs \\cite{Rusu} and all domination graphs\n are weakly chordal \\cite{DHMO}.\n A graph is {\\em strongly chordal} if it is chordal and if every even\n cycle of length at least $6$ in this graph has an odd chord \\cite[p. 21]{GMT}.\n A\ngraph is {\\em distance-hereditary} if each of its induced paths, and\nhence each of its connected induced subgraphs, is isometric\n\\cite{Howorka}. We call a graph a {\\em cograph} provided it\ndoes not contain any induced path of length $3$ \\cite[Theorem\n11.3.3]{BLS}. It is easy to see that each cograph is\ndistance-hereditary and all distance-hereditary\n graphs form a proper subclass\nof $4$-chordal graphs. It is also known that cocomparability graphs\nare all $4$-chordal \\cite{BT1, Gallai}.\n\n\n\n\n\n\n\\subsection{Hyperbolicity}\\label{hyper}\n\n\n\\subsubsection{Definition and background}\n\nFor any vertices $x,y,u,v$ of a graph $G,$ put $\\delta\n_G(x,y,u,v)$, which we often abbreviate to $\\delta (x,y,u,v)$, to be\n the difference between the largest and the\nsecond largest of the following three terms:\n$$\\frac{uv+xy}{2},\\, \\frac{ux+vy}{2},\\, \\text{and} \\ \\frac{uy+vx}{2}.$$ Clearly, $\\delta\n(x,y,u,v)=0$ if $x,y,u,v$ are not four different vertices. A graph\n$G$, viewed as a metric space as mentioned above, is {\\em\n$\\delta$-hyperbolic} (or tree-like with defect at most $\\delta$)\nprovided for any vertices $x,y,u,v$ in $G$ it holds $\\delta\n(x,y,u,v)\\leq \\delta$ and the (Gromov) {\\em hyperbolicity} of $G$,\ndenoted $\\delta^* (G)$, is the minimum half integer $\\delta$ such\nthat $G$ is $\\delta$-hyperbolic\n\\cite{Bow91,BH,CDEHV,CDEHVX,DD,Gromov}. Note that it may happen\n$\\delta ^*(G)=\\infty$. But for a finite graph $G$, $\\delta^* (G)$\nis clearly finite and polynomial time computable.\n\n\n\n\nNote that in some earlier literature the concept of Gromov\nhyperbolicity is used\n a little bit different from what we adopt here;\n what we call\n$\\delta$-hyperbolic here is called $2\\delta$-hyperbolic in\n\\cite{ABKMRT,BC,BC08,BKM01,CE,DHHKMW,Dress96,GL,KM02,MS} and hence\nthe hyperbolicity of a graph is always an integer according to their\ndefinition. We also refer to \\cite{Alonso,Bow91,BH, Vai} for some\nequivalent and very accessible definitions of Gromov hyperbolicity\nwhich involve some other comparable parameters.\n\n\nThe concept of hyperbolicity comes from the work of Gromov in\ngeometric group theory which encapsulates many of the global\nfeatures of the geometry of complete, simply connected manifolds of\nnegative curvature \\cite[p. 398]{BH}. This concept not only\nturns out to be strikingly useful in coarse geometry but also\nbecomes more and more important in many applied fields like\nnetworking and phylogenetics\n\\cite{CDEHV1,CDEHV,CDEHVX,CE,DraganX,Dress84,DHHKMW,DHM,Dress96,GL,JLB,JLHB,Kleinberg,ST}.\nThe hyperbolicity of a graph is a way to measure the additive\ndistortion with which every four-points sub-metric of the given\ngraph metric embeds into a tree metric \\cite{ABKMRT}. Indeed, it is\nnot hard to check that the hyperbolicity of a tree is zero -- the\ncorresponding condition for this is known as the\n four-point condition (4PC) and is a characterization of\n general tree-like metric spaces\n\\cite{Dress84,Dress96,Imrich}. Moreover, the fact that hyperbolicity\nis a tree-likeness parameter\n is reflected in the easy fact that the hyperbolicity of a graph is the maximum hyperbolicity of its 2-connected components --\n This observation implies the classical result that\n$0$-hyperbolic graphs are exactly block graphs, namely those graphs\nin which every $2$-connected subgraph is complete, which are also\nknown to be those diamond-free chordal graphs\n\\cite{BM,DMS,Howorka}.\n More results on bounding hyperbolicity of graphs and characterizing low hyperbolicity graphs can be found in\n\\cite{BC,BC08,BKM01,CDEHV1,CDEHV,DG,KM02}; we will only report in\nSection \\ref{MR} some work most closely related to ours and refer\nthe readers to corresponding references for many other interesting\nunaddressed work.\n\n\n\n\n\n\n\n\nFor any vertex $u\\in V(G)$, the {\\em Gromov product}, also known as\n the {\\em overlap function}, of any two vertices $x$ and $y$ of $G$ with respect to $u$ is equal\nto $\\frac{1}{2}(xu + yu - xy)$ and is denoted by $(x\\cdot y)_u$\n\\cite[p. 410]{BH}. As an important context in phylogenetics\n\\cite{DHHKMW,DHM,Farris}, for any real number $\\rho$, the {\\em\nFarris transform} based at $u$, denoted $D_{\\rho ,u}$, is the\ntransformation which sends $d_G$ to the map\n$$D_{\\rho , u}(d_G): V(G)\\times V(G)\\rightarrow \\mathbb{R}: \\ (x,y)\\mapsto \\rho -(x\\cdot\ny)_u.$$\n We\nsay that $G$ is {\\em $\\delta$-hyperbolic with respect to $u\\in\nV(G)$} if the following inequality \\begin{equation}(x\\cdot y)_u\\geq\n\\min ((x\\cdot v)_u,(y\\cdot v)_u)-\\delta \\label{EQ}\\end{equation}\nholds for any vertices $x,y,v$ of $G.$ It is easy to check that the\ninequality \\eqref{EQ} can be rewritten as\n$$xy+uv\\leq \\max (xu+yv, xv+yu)+2\\delta$$ and so we see that $G$ is\n$\\delta$-hyperbolic if and only if $G$ is $\\delta$-hyperbolic with\nrespect to every vertex of $G.$\n By a simple but nice argument, Gromov shows that $G$ is $2\\delta$-hyperbolic provided\n it is $\\delta$-hyperbolic with respect to\n any given vertex \\cite[Proposition 2.2]{Alonso} \\cite[1.1B]{Gromov}.\n\n\n\nThe {\\em tree-length} \\cite{Dour, DG, Lo, UY} of a graph $G$,\ndenoted $\\mathbbm{t}\\mathbbm{l}(G)$, is the minimum integer $k$\nsuch that there is a chordal graph $G'$ satisfying $V(G)=V(G')$,\n$E(G)\\subseteq E(G')$ and $\\max (d_G(u,v):\\ d_{G'}(u,v)=1)= k.$ We\nuse the convention that the tree-length of a graph without any edge\nis 1.\n It is straightforward\nfrom the definition that chordal graphs are exactly the graphs of\ntree-length $1$. It is also known that $AT$-free graphs and\ndistance-hereditary graphs have tree-length at most $2$ \\cite[p.\n367]{Dour}; a way to see this is to use the forthcoming result\nrelating chordality and tree-length as well as the fact that\n$AT$-free graphs are $5$-chordal and distance-hereditary graphs are\n$4$-chordal.\n\n\n\n\n\\begin{te}\\cite[Lemma 6]{GKKPP} \\cite[Theorem 3.3]{GKKPP1} If $G$ is a $k$-chordal graph, then $\\mathbbm{t}\\mathbbm{l}(G)\\leq \\lfloor \\frac{k}{2}\\rfloor.$\n\\label{thm9}\n\\end{te}\n\n\n\\begin{proof}[Outline] To obtain a minimal triangulation of $G,$ it suffices to select a maximal set of pairwise parallel\nminimal separators of $G$ and add edges to make each of them a\nclique \\cite[Theorem 4.6]{Parra}. It is easy to check that each such\nnew edge connects two points of distance at most $\\lfloor\n\\frac{k}{2}\\rfloor$ apart in $G.$\n\\end{proof}\n\nThe following is an interesting extension of the classical result\nthat trees are $0$-hyperbolic and its proof can be given in a way\ngeneralizing the well-known proof of the latter fact.\n\n\n\n\\begin{te}\n\\cite[Proposition 13]{CDEHV} A graph $G$ is $k$-hyperbolic\nprovided its tree-length is no greater than $k.$\n \\label{thm10}\n\\end{te}\n\n\n\n\n\n\n\n\n\nIt is noteworthy that a converse of Theorem \\ref{thm10} has also\nbeen established, which means that hyperbolicity and tree-length are\ncomparable parameters of tree-likeness.\n\n\n\\begin{te}\\cite[Proposition 14]{CDEHV} The inequality $\\mathbbm{t}\\mathbbm{l}(G)\\leq 12k+8k\\log_2n + 17$ holds for any\n $k$-hyperbolic graph $G$ with $n$ vertices.\n\\end{te}\n\n\n\n\\subsubsection{Three examples}\n\n\n\n\nLet us try our hand at three examples to get a feeling of the\nconcept of hyperbolicity. The first example says that graphs with\nsmall diameter, hence those so-called small-world networks, must\nhave low hyperbolicity. Note that additionally similar simple\nresults will be reported as Lemmas \\ref{first} and \\ref{lem14}.\n\n\\begin{example} \\cite[p. 683]{KM02} The hyperbolicity of a graph $G$ with diameter $D$ is at most\n$\\lfloor \\frac{D}{2}\\rfloor$. \\label{diam}\n\\end{example}\n\\begin{proof} Take $x,y,u,v\\in V(G)$. Our goal is to show that $\\delta (x,y,u,v)\\leq\n\\frac{D}{2}$. Without loss of generality, assume that\n\\begin{equation} \\label{Leipzig} xy+uv\\geq xu+yv\\geq xv+yu\n\\end{equation} and hence \\begin{equation}\\delta\n(x,y,u,v)=\\frac{1}{2}((xy+uv)-(xu+yv)). \\label{coffee}\n\\end{equation}\n In the first place, we have\n$$\nxu+yu\\geq xy, ux+vx\\geq uv, xv+yv\\geq xy, vy+uy\\geq uv.\n$$\nSumming up these inequalities yields $(xu+yv)+(xv+yu)\\geq xy+uv$,\nwhich, according to Eq. \\eqref{Leipzig}, implies that\n$$xu+yv\\geq \\frac{1}{2}(xy+uv).$$\nThis along with Eq. \\eqref{coffee} gives $\\delta (x,y,u,v)\\leq\n\\frac{1}{4}(xy+uv)\\leq \\frac{D}{2}.$\n Moreover, if $\\delta (x,y,u,v)=\n\\frac{D}{2}$, then we have\n \\begin{equation}xu+yv=D,\n xv+yv=xy=D, xu+xv=uv=D.\n \\label{JS}\n \\end{equation}\nBy adding the equalities in Eq. \\eqref{JS} together, we see that\n$3D=2(xu+xv+yv)$ and so $D$ must be even.\n\\end{proof}\n\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.4pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{20mm}\n\\begin{center}\n\\begin{picture}(10,10)\n\\put(10,10){\\circle{20}}\\put(10,17){\\circle*{1}}\\put(10,18){\\makebox(0,0)[b]{$x$}}\n\\put(17,10){\\circle*{1}}\\put(18,10){\\makebox(0,0)[l]{$v$}}\n\\put(10,3){\\circle*{1}}\\put(10,2){\\makebox(0,0)[t]{$y$}}\n\\put(3,10){\\circle*{1}}\\put(2,10){\\makebox(0,0)[r]{$u$}}\n\\put(16,15){\\makebox(0,0)[l]{$xv$}}\\put(15,3){\\makebox(0,0)[l]{$vy$}}\n\\put(4,4){\\makebox(0,0)[r]{$yu$}}\\put(4,15){\\makebox(0,0)[r]{$ux$}}\n \\end{picture}\n\\end{center}\n\\caption{Four points in an $n$-cycle.}\\label{n-cycle}\n\\end{figure}\n\n\nThe bound asserted by Example \\ref{diam} is clearly not tight when\n$D=1.$ But, as can be seen from the next example, the bound given\nin Example \\ref{diam} in terms of the diameter $D$ is best\npossible for every $D\\geq 2$. Note that this forthcoming example can\nalso be seen directly via Example \\ref{diam}, as indicated in\n\\cite[p. 683]{KM02}.\n\n\\begin{example} \\cite[p. 683]{KM02} For any $n\\geq 3,$ the chordality of the $n$-cycle is $n$ while\nthe hyperbolicity of the $n$-cycle is \\begin{equation}\n\\delta^*(C_n)=\\begin{cases} \\lfloor\n\\frac{n}{4}\\rfloor-\\frac{1}{2},&\\text{if $n\\equiv 1\\ (\\!\\!\\!\\!\\!\\!\\mod 4)$;}\\\\\n\\lfloor \\frac{n}{4}\\rfloor,&\\text{else.}\n\\end{cases}\n\\label{USTC}\n\\end{equation}\nNote that the diameter of $C_n$ is $\\lfloor \\frac{n}{2}\\rfloor$\nand\n$$ \\delta^*(C_n)=\\begin{cases}\n\\frac{\\lfloor \\frac{n}{2}\\rfloor}{2},&\\text{if $n\\equiv 0\\ (\\!\\!\\!\\!\\!\\!\\mod 4)$;}\\\\\n\\frac{\\lfloor \\frac{n}{2}\\rfloor}{2}-\\frac{1}{2},&\\text{else.}\n\\end{cases}$$\n\\label{exam7}\n\\end{example}\n\\begin{proof} To prove Eq. \\eqref{USTC}, we need to estimate $\\delta\n(x,y,u,v)$ for any four vertices $x,y,u,v$ of the $n$-cycle graph\n$C_n.$ If there is a geodesic connecting two vertices and passing\nthrough all the four vertices $x,y,u,v$, we surely have $\\delta\n(x,y,u,v) =0$ just because we know that the hyperbolicity of a path\nis $0.$ So, we can assume that the cycle $C_n$ is $[c_0c_1\\ldots\nc_{n-1}]$ where $c_0=x,c_{xv}=v,c_{xv+vy}=y,c_{xv+vy+yu}=u$ and\n\\begin{equation}xv+vy+yu+ux=n;\n\\label{sum}\n\\end{equation} see Fig. \\ref{n-cycle}. With no loss of generality, we assume that\n\\begin{equation}xu-vy\\geq |xv-uy|.\n\\label{morning}\n\\end{equation} This implies $ xu+uy \\geq xv+vy\n$ and $ vx+xu\\geq vy+yu$. According to the geometric distribution\nof the four points, we then come to\n$$xy=xv+vy\\ \\ \\text{and} \\ \\ vu=vy+yu.\n$$ It follows that\n\\begin{equation}xy+vu=(xv+yu)+2vy\\label{expo}\\end{equation} and\n$$xy+vu=xv+vy+vy+yu=(xv+vy+yu)+vy\\geq xu+vy.$$ At the moment, we\nsee that there are only two possibilities, either $xy+vu\\geq\nxu+vy>xv+yu$ or $xy+vu\\geq xv+yu \\geq xu+vy.$\n\n If the first case\nhappens, we have\n\\begin{equation}\n\\begin{array}{cll}\\delta(x,y,u,v)&=&\\frac{1}{2}(xy+vu- xu-vy)\n\\\\\n&= & \\frac{n}{2}-xu.\\ \\ \\text{(By Eqs. \\eqref{sum} and\n\\eqref{expo})}\n \\end{array}\n \\label{Kool}\n \\end{equation}\nBy\n Eqs. \\eqref{sum} and \\eqref{morning} and $xu+vy>xv+yu$, we see\n that $xu\\geq\n \\begin{cases} \\lfloor \\frac{n}{4}\\rfloor +1,&\\text{if $n\\equiv 0,1,2\\ (\\!\\!\\!\\!\\mod 4)$,}\\\\\n \\lfloor \\frac{n}{4}\\rfloor +2,&\\text{if $n\\equiv 3\\ (\\!\\!\\!\\!\\mod 4)$,}\n\\end{cases}\n$\n and hence Eq. \\eqref{Kool} forces\n \\begin{equation}\\label{cycle1}\\delta(x,y,u,v)= \\frac{n}{2} - xu \\leq \\begin{cases}\n \\lfloor \\frac{n}{4}\\rfloor -1,&\\text{if $n\\equiv 0\\ (\\!\\!\\!\\!\\!\\!\\mod 4)$;}\\\\\n \\lfloor \\frac{n}{4}\\rfloor ,&\\text{if $n\\equiv 2\\ (\\!\\!\\!\\!\\!\\!\\mod 4)$;}\\\\\n \\lfloor \\frac{n}{4}\\rfloor -\\frac{1}{2},&\\text{if $n\\equiv 1,3\\ (\\!\\!\\!\\!\\!\\!\\mod 4)$.}\n\\end{cases}\n\\end{equation}\n\n\n For the second case, we have\n$$\n\\begin{array}{cll}\\delta(x,y,u,v)&=&\\frac{1}{2}(xy+vu-\nxv-yu)\n\\\\\n&= & vy,\\ \\ \\text{(By Eq. \\eqref{expo})}\n\\end{array}$$\nand hence by\n Eqs. \\eqref{sum} and \\eqref{morning} and $xv+yu\\geq vy +xu$, we further obtain\n \\begin{equation}\\label{cycle2} \\delta(x,y,u,v)=vy\\leq \\begin{cases}\n \\lfloor \\frac{n}{4}\\rfloor ,&\\text{if $n\\equiv 0,2,3\\ (\\!\\!\\!\\!\\!\\mod 4)$;}\\\\\n \\lfloor \\frac{n}{4}\\rfloor -1,&\\text{if $n\\equiv 1\\ (\\!\\!\\!\\!\\!\\mod 4)$.}\n\\end{cases}\n\\end{equation}\n\n\nCombining Eqs. \\eqref{cycle1} and \\eqref{cycle2} yields\n\\begin{equation}\\label{quartet}\n\\delta (x,y,u,v)\\leq \\begin{cases} \\lfloor\n\\frac{n}{4}\\rfloor,&\\text{if $n\\equiv 0,2,3\\ (\\!\\!\\!\\!\\!\\mod 4)$;}\\\\\n\\lfloor\n\\frac{n}{4}\\rfloor-\\frac{1}{2},&\\text{if $n\\equiv 1\\ (\\!\\!\\!\\!\\!\\!\\mod 4)$.}\\\\\n\\end{cases}\n\\end{equation}\nTaking\n\\begin{equation*}\n (x,v,y,u)= \\begin{cases} (c_0,c_k,c_{2k},c_{3k}),&\\text{if $n\\equiv 0\\ (\\!\\!\\!\\!\\!\\!\\mod 4)$,}\\\\\n(c_0,c_k,c_{2k},c_{3k}),&\\text{if $n\\equiv 1\\ (\\!\\!\\!\\!\\!\\!\\mod 4)$,}\\\\\n(c_0,c_k,c_{2k},c_{3k+1}),&\\text{if $n\\equiv 2\\ (\\!\\!\\!\\!\\!\\!\\mod 4)$,}\\\\\n(c_0,c_{k+1},c_{2k+1},c_{3k+2}),&\\text{if $n\\equiv 3\\\n(\\!\\!\\!\\!\\!\\!\\mod 4)$,}\n\\end{cases}\n\\end{equation*}\nwe see that Eq. \\eqref{quartet} is tight and hence Eq. \\eqref{USTC}\nis established.\n\\end{proof}\n\n\nFor any two graphs $G_1$ and $G_2,$ we define its {\\em Cartesian\nproduct } $G_1\\Box G_2$ to be the graph satisfying $V(G_1\\Box\nG_2)=V(G_1)\\times V(G_2)$ and $d_{G_1\\Box\nG_2}((u_1,u_2),(v_1,v_2))=d_{G_1}(u_1,v_1)+d_{G_2}(u_2,v_2)$\n\\cite[\\S 1.4]{IK}.\n\n\\begin{example} \\label{exam3} Let $G_1$ and $G_2$ be two graphs satisfying $\\delta ^*(G_1)=\\delta^*\n(G_2)=0.$ Then $\\delta^*(G_1\\Box G_2)=\\min (D_1,D_2)$ where $D_1$\nand $D_2$ are the diameters of $G_1$ and $G_2$, respectively.\n\\end{example}\n\n\\begin{proof} For any $v\\in V(G_1\\Box G_2)$, we often use\nthe convention that $v=(v_1,v_2)$ for $v_1\\in V(G_1)$ and $v_2\\in\nV(G_2)$. For any $u,v\\in V(G_1\\Box G_2)$, we write $uv$ for\n$d_{G_1\\Box G_2}(u,v)$, $(uv)_1$ for $d_{G_1}(u_1,v_1)$, $(uv)_2$\nfor $d_{G_2}(u_2,v_2)$ and we use $\\delta$ for $\\delta_{G_1\\Box\nG_2}$.\n\nTake $a,b\\in V(G_1)$ such that $d_{G_1}(a,b)=D_1$ and take $c,d\\in\nV(G_2)$ such that $d_{G_2}(c,d)=D_2$. Set\n$x=(a,c),y=(a,d),u=(b,c),v=(b,d)$. It is straightforward that\n$\\delta (x,y,u,v)=\\min (D_1, D_2)$.\n\nTo complete the proof, we pick any four vertices $x,y,u,v$ of\n$G_1\\Box G_2$ and aim to show that \\begin{equation}\\delta\n(x,y,u,v)\\leq \\min (D_1, D_2). \\label{sunny}\\end{equation} Let\n$A=xy+uv,$ $A_1=(xy)_1+(uv)_1$, $A_2=(xy)_2+(uv)_2$, $B=xu+yv,$\n$B_1=(xu)_1+(yv)_1$, $B_2=(xu)_2+(yv)_2$, $C=xv+yu,$\n $C_1=(xv)_1+(yu)_1$, $C_2=(xv)_2+(yu)_2.$ Because\n $\\delta^*(G_1)=\\delta^*(G_2)=0$, we can suppose $A_1=\\max (B_1, C_1)$ and\n $A_2=\\max (B_2,C_2)$.\n\n If it happens either $(A_1,A_2)=(B_1,B_2)$ or\n $(A_1,A_2)=(C_1,C_2)$, we can immediately conclude that $\\delta\n (x,y,u,v)=0.$ By symmetry between $B$ and $C$ and between $G_1$ and $G_2,$ it thus remains to\n deduce Eq. \\eqref{sunny} under the condition that\n\\begin{equation*}B\\geq C, A_1=B_1>C_1, \\ \\text{and}\\ A_2=C_2>B_2.\n\\label{KAIST}\n\\end{equation*}\n\n\n\nSince $A_1=B_1,$ we have $\\delta\n(x,y,u,v)=\\frac{A-B}{2}=\\frac{A_2-B_2}{2}$. We proceed with a direct\ncomputation and find \\begin{equation}\\label{eq14}\\delta (x,y,u,v)\n=\\frac{((xy)_2-(xu)_2)+((uv)_2-(yv)_2 )}{2}\\leq (yu)_2\\leq D_2.\n\\end{equation}\nMaking use of $A_2=C_2$ and $B\\geq C,$ we can obtain instead\n\\begin{equation}\\label{eq15}\n\\begin{array}{cll}\\delta(x,y,u,v)&\\leq&\\frac{A_2-B_2+B-C}{2}=\\frac{A_2-C_2+B_1-C_1}{2}=\\frac{B_1-C_1}{2}\n\\\\\n&= & \\frac{((xu)_1-(xv)_1)+((yv)_1-(yu)_1)}{2}\\leq (uv)_1\\leq D_1\n\\end{array}\n \\end{equation} Combining Eqs. \\eqref{eq14} and \\eqref{eq15} we now get Eq. \\eqref{sunny}, as desired.\n\\end{proof}\n\n\n\\begin{remark}\\label{rem8}\nFor any $t$ natural numbers $m_1,\\ldots , m_t$, the {\\em\n$t$-dimensional\n grid graph} $G_{m_1,\\ldots , m_t}$ is the graph with vertex set $\\{\n1,2,\\ldots,m_1\\}\\times \\cdots \\times \\{ 1,2,\\ldots,m_t\\}$ and\n$(i_1,\\ldots ,i_t)$ and $(j_1,\\ldots, j_t)$ are adjacent in\n$G_{m_1\\ldots,m_t}$ if any only if $\\sum_{p=1}^t(i_p-j_p)^2=1.$\nExample \\ref{exam3} implies that $\\delta^*(G_{m_1,m_2})=\\min\n(m_1,m_2)-1$ and hence $G_{m,m}$ provides another example that the\nbound reported in Example \\ref{diam} is tight. It might be\ninteresting to determine the hyperbolicity of $t$-dimensional\n grid graphs for $t\\geq 3.$\n\\end{remark}\n\n\n\n\n\\begin{remark}\\label{grid}\n Dourisboure and Gavoille show that the tree-length of $G_{n,m}$ is $ \\min (n,m)$ if $n\\not= m$ or $n=m$ is even and is $n-1$ if\n$n=m$ is odd \\cite[Theorem 3]{DG}. Remark \\ref{rem8} tells us that\n$\\delta^*(G_{n,m})=\\min(m,n)-1$. This says that Theorem \\ref{thm10}\nis tight.\n\\end{remark}\n\n\n\n\n\n\n\n\n\n\\section{Chordality vs. hyperbolicity}\\label{MR}\n\n\\subsection{Main results}\\label{theorem}\n\n\nFirstly, we point out that\n a graph with low hyperbolicity may have large\nchordality. Indeed, take any graph $G$ and form the new graph $G'$\nby adding an additional vertex and connecting this new vertex with\nevery vertex of $G$. It is obvious that $\\delta^*(G')\\leq 1$ while\n$\\mathbbm{l}\\mathbbm{c}(G')= \\mathbbm{l}\\mathbbm{c}(G)$ if $G$ is\nnot a tree. Moreover, it is equally easy to see that $G'$ is even\n$\\frac{1}{2}$-hyperbolic if $G$ does not have any induced $4$-cycle\n\\cite[p. 695]{KM02}. Surely, this example does not preclude the\npossibility that for many important graph classes we can bound their\nchordality in terms of their hyperbolicity.\n\nOne of our main results says that hyperbolicity can be bounded\nfrom above in terms of chordality.\n\n\n\n\n\n\\begin{te} \\label{main} For each $k\\geq 4,$ all $k$-chordal graphs are $\\frac{\\lfloor\n\\frac{k}{2}\\rfloor}{2}$-hyperbolic.\n\\end{te}\n\n\n\n\\begin{remark}\n A graph is {\\em bridged} \\cite{AF,LS} if it\ndoes not contain any finite isometric cycles of length at least\nfour, or equivalently, if it is cop-win and has no chordless\ncycle of length $4$ or $5$. In contrast to Theorem \\ref{main}, it\nis interesting to note that the hyperbolicity of bridged graphs can\nbe arbitrarily high \\cite[p. 684]{KM02}.\n\\end{remark}\n\n\n\\begin{remark} Bandelt and Chepoi \\cite[\\S 5.2]{BC08} make the remark that\n ``a characterization of\n all $1$-hyperbolic graphs by forbidden isometric subgraphs\nis not\n in sight, in as much as isometric cycles of lengths up to $7$ may occur, thus\ncomplicating the picture''. Note that our Theorem \\ref{main}\nsays that all $5$-chordal graphs are $1$-hyperbolic and hence the\nappearance of those chordless $6$-cycles and chordless $7$-cycles\nmay be a real headache to deal with in pursuing a characterization\nof all $1$-hyperbolic graphs.\n\\end{remark}\n\n\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.4pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(150,30)\n\n\n\\put(60,25){\\circle*{1}}\\put(60,28){\\makebox(0,0)[t]{$v_8$}}\n\n\\put(55,15){\\circle*{1}}\\put(51,15){\\makebox(0,0)[l]{$v_7$}}\n\n\\put(50,5){\\circle*{1}}\\put(50,1){\\makebox(0,0)[b]{$v_6$}}\n \\put(60,5){\\circle*{1}}\\put(60,1){\\makebox(0,0)[b]{$v_5$}}\n\\put(70,25){\\circle*{1}}\\put(70,29){\\makebox(0,0)[t]{$v_1$}}\n\\put(80,25){\\circle*{1}}\\put(80,28){\\makebox(0,0)[t]{$v_2$}}\n\\put(75,15){\\circle*{1}}\\put(79,15){\\makebox(0,0)[r]{$v_3$}}\n\n\\put(70,5){\\circle*{1}}\\put(70,1){\\makebox(0,0)[b]{$v_4$}}\n\n\\qbezier(60,25)(55,15)(50,5)\\qbezier(80,25)(75,15)(70,5)\n\\qbezier(55,15)(58,9)(60,5)\\qbezier(70,25)(73,19)(75,15)\\qbezier(50,5)(60,5)(70,5)\n\\qbezier(60,25)(70,25)(80,25)\n\\end{picture}\n\\end{center}\n\n\\caption{The outerplanar graph $F_2$ has chordality $6$,\nhyperbolicity $\\frac{3}{2}$, and tree-length $2$.}\\label{counter}\n\\end{figure}\n\n\n\n\n\\begin{example}\\label{exam8} For any $t\\geq 2$ we set\n $F_t$ to be the graph obtained from the $4t$-cycle $[v_1v_2\\cdots v_{4t}]$ by adding the two\n edges $\\{v_1,v_3\\}$ and $ \\{v_{2t+1},v_{2t+3}\\}$; see Fig. \\ref{counter} for an illustartion of $F_2$. Clearly,\n $\\delta(v_2, v_{t+2},v_{2t+2}, v_{3t+2})=t-\\frac{1}{2}$. Furthermore, we\n can check that\n $\\mathbbm{l}\\mathbbm{c}(F_t)=4t-2$ and $\\delta^*(F_t)=t-\\frac{1}{2}=\\delta(v_2, v_{t+2},v_{2t+2},\n v_{3t+2})=\\frac{\\mathbbm{l}\\mathbbm{c}(F_t)}{4}.$ $F_t$ is clearly an outerplanar graph. Thus, applying the result that $\\mathbbm{t}\\mathbbm{l}(G)=\\lceil \\frac{\\mathbbm{l}\\mathbbm{c}(G)}{3}\\rceil$ for\nevery outerplanar graph $G$ \\cite[Theorem 1]{DG}, we even know\nthat $\\mathbbm{t}\\mathbbm{l}(F_t)=\\lceil \\frac{4t-2}{3}\\rceil$.\n\\end{example}\n\n\n\n\n\nIt is clear that if the bound claimed by Theorem \\ref{main} is\ntight for $k=4t$ ($k=4t-2$) then it is tight for $k=4t+1$\n($k=4t-1$). Consequently, Examples \\ref{exam7} and \\ref{exam8}\nindeed mean that the bound reported in Theorem \\ref{main} is tight\nfor every $k\\geq 4.$\n Surely, the logical next step would be to characterize all those extremal\n graphs $G$\nsatisfying\n\\begin{equation}\n\\delta^*(G)=\\frac{\\lfloor\n\\frac{\\mathbbm{l}\\mathbbm{c}(G)}{2}\\rfloor}{2}. \\label{extremal}\n\\end{equation}\n However, there seems to be still a\nlong haul ahead in this direction.\n\n\\begin{remark} \\label{rem13}\nFor any graph $G$ and any positive number $t$, we put $S^t(G)$ to be\na {\\em subdivision graph} of $G$, which is obtained from $G$ by\nreplacing each edge $\\{u,v\\}$ of $G$ by a path $u,n_{u,v}^1,\\ldots,\n n_{u,v}^{t-1}, v$ of length $t$ connecting $u$ and $v$ through a sequence of new vertices $n_{u,v}^1,\\ldots,\n n_{u,v}^{t-1}$ (we surly require that $n_{v,u}^q=n_{u,v}^{t-q}$). For any four vertices $x,y,u,v\\in V(G)$,\nwe obviously have $\\delta_{S^t(G)}(x,y,u,v)=t\\delta_G(x,y,u,v)$ and\nso $\\delta^*(S^t(G))\\geq t\\delta^*(G)$.\n Instead of the trivial fact $\\mathbbm{l}\\mathbbm{c} (S^t(G))\\geq\n t \\mathbbm{l}\\mathbbm{c}(G)$, if the good shape of $G$ permits us to deduce a good upper bound\n of $\\mathbbm{l}\\mathbbm{c} (S^t(G))$ in terms of\n $\\mathbbm{l}\\mathbbm{c}(G)$, we will see that $\\delta^*(S^t(G))$\n is high relative to $\\mathbbm{l}\\mathbbm{c} (S^t(G))$\n provided so is $G.$ Recall that the cycles whose lengths are divisible by $4$ as discussed in Example\n \\ref{exam7} are used to demonstrate the tightness of the bound given in Theorem\n \\ref{main}; also observe that the graphs suggested by Example \\ref{exam8} is nothing but a slight\n ``perturbation''\nof cycles of length divisible by $4.$ Since $C_{4t}=S^t(C_{4})$,\nthese examples can be said to be generated by the ``seed'' $C_4.$\nIt might\n deserve to look for some other good ``seeds\" from which we can use\n the above subdivision operation or its variant to produce graphs\n satisfying Eq. \\eqref{extremal}.\n\\end{remark}\n\n\n\n\n\n\n\n\n\n\nLet $C_4$, $H_1$, $H_2$, $H_3$, $H_4$ and $H_5$ be the graphs\ndisplayed in Fig. \\ref{fig0}. It is simple to check that each of\nthem has hyperbolicity $1$ and is $5$-chordal. Besides Theorem\n\\ref{main}, another main contribution of this paper is the\nfollowing, which says that $5$-chordal graphs will be\n$\\frac{1}{2}$-hyperbolic as soon as these six obvious obstructions\ndo not occur.\n\n\n\n\n\n\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.4pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(150,100)\n\n\\put(30,60){\\circle*{1}}\\put(33,57){\\makebox(0,0)[br]{$y$}}\n\\put(0,60){\\circle*{1}}\\put(-3,57){\\makebox(0,0)[bl]{$u$}}\n\\put(30,90){\\circle*{1}}\\put(33,93){\\makebox(0,0)[tr]{$v$}}\n\\put(0,90){\\circle*{1}}\\put(-3,93){\\makebox(0,0)[tl]{$x$}}\n\\put(15,50){\\makebox(0,0)[l]{$C_{4}$}}\n\n\n\n\\qbezier(30,90)(20,90)(0,90)\\qbezier(0,90)(0,80)(0,60)\\qbezier(0,60)(20,60)(30,60)\\qbezier(30,60)(30,80)(30,90)\n\n\\put(70,60){\\circle*{1}}\\put(70,56){\\makebox(0,0)[b]{$y$}}\n\n\\put(50,80){\\circle*{1}}\\put(47,80){\\makebox(0,0)[l]{$u$}}\n\\put(90,80){\\circle*{1}}\\put(93,80){\\makebox(0,0)[r]{$v$}}\n\n\\put(70,100){\\circle*{1}}\\put(70,103){\\makebox(0,0)[t]{$x$}}\n\n\\put(60,90){\\circle*{1}}\\put(57,90){\\makebox(0,0)[l]{$a$}}\n\\put(80,90){\\circle*{1}}\\put(83,90){\\makebox(0,0)[r]{$b$}}\n\\put(60,70){\\circle*{1}}\\put(57,70){\\makebox(0,0)[l]{$c$}}\n\\put(80,70){\\circle*{1}}\\put(83,70){\\makebox(0,0)[r]{$d$}}\n\\put(70,50){\\makebox(0,0)[l]{$H_{1}$}}\n\n\n\\qbezier(50,80)(60,70)(70,60)\\qbezier(70,60)(80,70)(90,80)\\qbezier(90,80)(80,90)(70,100)\\qbezier(70,100)(60,90)(50,80)\n\\qbezier(60,90)(70,80)(80,70)\\qbezier(60,90)(60,80)(60,70)\\qbezier(60,70)(70,70)(80,70)\\qbezier(80,70)(80,80)(80,90)\n\\qbezier(60,90)(70,90)(80,90)\n\n\n\\put(120,60){\\circle*{1}}\\put(120,56){\\makebox(0,0)[b]{$y$}}\n\n\\put(100,80){\\circle*{1}}\\put(97,80){\\makebox(0,0)[l]{$u$}}\n\\put(140,80){\\circle*{1}}\\put(143,80){\\makebox(0,0)[r]{$v$}}\n\n\\put(120,100){\\circle*{1}}\\put(120,103){\\makebox(0,0)[t]{$x$}}\n\n\\put(110,90){\\circle*{1}}\\put(107,90){\\makebox(0,0)[l]{$a$}}\n\\put(130,90){\\circle*{1}}\\put(133,90){\\makebox(0,0)[r]{$b$}}\n\\put(110,70){\\circle*{1}}\\put(107,70){\\makebox(0,0)[l]{$c$}}\n\\put(130,70){\\circle*{1}}\\put(133,70){\\makebox(0,0)[r]{$d$}}\n\\put(120,50){\\makebox(0,0)[l]{$H_{2}$}}\n\n\n\\qbezier(100,80)(110,70)(120,60)\\qbezier(120,60)(130,70)(140,80)\\qbezier(140,80)(130,90)(120,100)\\qbezier(120,100)(110,90)(100,80)\n\\qbezier(110,90)(120,80)(130,70)\\qbezier(110,70)(120,80)(130,90)\\qbezier(130,90)(130,80)(130,70)\\qbezier(110,90)(120,90)(130,90)\n\\qbezier(110,90)(110,80)(110,70)\\qbezier(110,70)(120,70)(130,70)\n\n\n\n\\put(20,0){\\circle*{1}}\\put(20,-4){\\makebox(0,0)[b]{$y$}}\n\n\\put(0,20){\\circle*{1}}\\put(-3,20){\\makebox(0,0)[l]{$u$}}\n\\put(40,20){\\circle*{1}}\\put(43,20){\\makebox(0,0)[r]{$v$}}\n\n\\put(20,30){\\circle*{1}}\\put(20,33){\\makebox(0,0)[t]{$x$}}\n\n\n\\put(10,10){\\circle*{1}}\\put(7,10){\\makebox(0,0)[l]{$c$}}\n\\put(30,10){\\circle*{1}}\\put(33,10){\\makebox(0,0)[r]{$d$}}\n\\put(22,-6){\\makebox(0,0)[l]{$H_{3}$}}\n\n\n\\qbezier(0,20)(10,10)(20,0)\\qbezier(20,0)(30,10)(40,20)\\qbezier(40,20)(30,25)(20,30)\\qbezier(20,30)(10,25)(0,20)\n \\qbezier(10,10)(20,10)(30,10)\n\n\\put(70,0){\\circle*{1}}\\put(70,-4){\\makebox(0,0)[b]{$y$}}\n\n\\put(50,20){\\circle*{1}}\\put(47,20){\\makebox(0,0)[l]{$u$}}\n\\put(90,20){\\circle*{1}}\\put(93,20){\\makebox(0,0)[r]{$v$}}\n\n\\put(70,40){\\circle*{1}}\\put(70,43){\\makebox(0,0)[t]{$x$}}\n\n\\put(60,30){\\circle*{1}}\\put(57,30){\\makebox(0,0)[l]{$a$}}\n\\put(80,30){\\circle*{1}}\\put(83,30){\\makebox(0,0)[r]{$b$}}\n\\put(60,10){\\circle*{1}}\\put(57,10){\\makebox(0,0)[l]{$c$}}\n\\put(80,10){\\circle*{1}}\\put(83,10){\\makebox(0,0)[r]{$d$}}\n\\put(72,-6){\\makebox(0,0)[l]{$H_{4}$}}\n\n\\qbezier(50,20)(60,10)(70,0)\\qbezier(70,0)(80,10)(90,20)\\qbezier(90,20)(80,30)(70,40)\\qbezier(70,40)(60,30)(50,20)\n\\qbezier(60,30)(70,20)(80,10) \\qbezier(80,30)(70,20)(60,10)\n\n\n\\put(120,0){\\circle*{1}}\\put(120,-4){\\makebox(0,0)[b]{$y$}}\n\\put(110,10){\\circle*{1}}\\put(107,10){\\makebox(0,0)[l]{$c$}}\n\\put(110,30){\\circle*{1}}\\put(107,30){\\makebox(0,0)[l]{$a$}}\n\\put(130,30){\\circle*{1}}\\put(133,30){\\makebox(0,0)[r]{$b$}}\n\\put(130,10){\\circle*{1}}\\put(133,10){\\makebox(0,0)[r]{$d$}}\n\\put(100,20){\\circle*{1}}\\put(96,20){\\makebox(0,0)[l]{$u$}}\n\\put(140,20){\\circle*{1}}\\put(143,20){\\makebox(0,0)[r]{$v$}}\n\\put(120,40){\\circle*{1}}\\put(120,43){\\makebox(0,0)[t]{$x$}}\n\\qbezier(100,20)(110,10)(120,0)\\qbezier(140,20)(130,10)(120,0)\\qbezier(110,30)(120,20)(130,10)\n\\qbezier(120,40)(110,30)(100,20)\\qbezier(120,40)(130,30)(140,20)\n\\put(122,-6){$H_5$}\n \\end{picture}\n\n\\end{center}\n\\caption{Six $5$-chordal graphs with hyperbolicity\n$1$.}\\label{fig0}\n\\end{figure}\n\n\n\n\n\n\n\n\n\\begin{te}\\label{main1}\n A $5$-chordal graph has hyperbolicity one if and only if one\nof $C_4,H_1,H_2,H_3,H_4, H_5$ appears as an isometric subgraph of\nit.\n\\end{te}\n\n\n\n\n\nReturning to Remark \\ref{rem13}, it is natural to investigate if\nsome graphs mentioned in Theorem \\ref{main1} besides $C_4$ can be\nused as ``good seeds''. The next example comes from Gavoille\n\\cite{CGa}.\n\n\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.6pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(50,40)\n\n\\put(20,0){\\circle*{1}}\\put(20,-4){\\makebox(0,0)[b]{$y$}}\n\\put(0,20){\\circle*{1}}\\put(-1,20){\\makebox(0,0)[r]{$u$}}\n\\put(40,20){\\circle*{1}}\\put(41,20){\\makebox(0,0)[l]{$v$}}\n\\put(20,40){\\circle*{1}}\\put(20,44){\\makebox(0,0)[t]{$x$}}\n\\put(10,30){\\circle*{1}}\\put(7,30){\\makebox(0,0)[l]{$a$}}\n\\put(10,10){\\circle*{1}}\\put(9,10){\\makebox(0,0)[r]{$c$}}\n\\put(30,10){\\circle*{1}}\\put(31,10){\\makebox(0,0)[l]{$d$}}\n\\put(30,30){\\circle*{1}}\\put(31,30){\\makebox(0,0)[l]{$b$}}\n\\put(5,25){\\circle*{1}}\\put(4,25){\\makebox(0,0)[r]{$u_a$}}\n\\put(5,15){\\circle*{1}}\\put(4,15){\\makebox(0,0)[r]{$u_c$}}\n\\put(15,5){\\circle*{1}}\\put(14,5){\\makebox(0,0)[r]{$y_c$}}\n\\put(25,5){\\circle*{1}}\\put(26,5){\\makebox(0,0)[l]{$y_d$}}\n\\put(35,15){\\circle*{1}}\\put(36,15){\\makebox(0,0)[l]{$v_d$}}\n\\put(35,25){\\circle*{1}}\\put(36,25){\\makebox(0,0)[l]{$v_b$}}\n\\put(15,35){\\circle*{1}}\\put(14,35){\\makebox(0,0)[r]{$x_a$}}\n\\put(25,35){\\circle*{1}}\\put(26,35){\\makebox(0,0)[l]{$x_b$}}\n\\qbezier[30](0,20)(10,30)(20,40)\\qbezier[30](20,40)(30,30)(40,20)\\qbezier[30](20,0)(30,10)(40,20)\\qbezier[30](20,0)(10,10)(0,20)\n\\qbezier[20](10,30)(10,30)(10,10)\n\\qbezier[20](10,30)(20,30)(30,30)\\qbezier[20](30,10)(30,10)(30,30)\n\\qbezier[20](10,10)(30,10)(30,10)\n\\qbezier(5,15)(5,20)(5,25)\\qbezier(15,5)(20,5)(25,5)\\qbezier(35,15)(35,20)(35,25)\\qbezier(15,35)(20,35)(25,35)\n\\qbezier[20](10,30)(20,20)(30,10)\\qbezier[20](10,10)(20,20)(30,30)\n \\end{picture}\n\\end{center}\n\\caption{$\\mathbb{G}_{4t}^q$.}\\label{example13}\n\\end{figure}\n\n\n\n\\begin{example}\\label{ep} \\cite{CGa} Let $t,q$ be two positive integer with\n$qq$ be two positive integers. We\nconstruct an outerplanar graph $\\mathbb{G}_{6(2t+1)}^q$ by adding\ntwo new edges $\\{v_{21}, v_{23}\\}$ and $\\{v_{65},\n v_{67}\\}$\n to the graph $S^{2t+1}(F_2)$\n where\n $v_{21}=n_{v_2,v_1}^{q}$, $v_{23}=n_{v_3,v_2}^{q-1}$,\n$v_{65}=n_{v_6,v_5}^{q}$, $v_{67}=n_{v_7,v_6}^{q-1}$; see Fig.\n \\ref{expo} for an illustration. It is not hard to check that $\\mathbbm{l}\\mathbbm{c}(\\mathbb{G}_{6(2t+1)}^q)\n=6(2t+1)$ and $\\delta^*(\\mathbb{G}_{6(2t+1)}^q)=3t+\\frac{3}{2}$.\nMoreover, if we replace the edge $\\{v_{21}, v_{23}\\}$\n by the edge $\\{v_{21}, n_{v_3,v_2}^{q}\\} $, then we obtain from $\\mathbb{G}_{6(2t+1)}^q$\n another outerplanar graph $\\mathbb{G}_{6(2t+1)+1}^q$ for which we\n have\n$\\mathbbm{l}\\mathbbm{c}(\\mathbb{G}_{6(2t+1)+1}^q) =6(2t+1)+1$ and\n$\\delta^*(\\mathbb{G}_{6(2t+1)+1}^q)=3t+\\frac{3}{2}$.\n\\end{example}\n\n\n\n\n\n\n\n\nLet $C_6,G_1,G_2,G_3$ be the graphs depicted in Fig.\n \\ref{figconjecture}. It is clear that\n $G_1,G_2,G_3,C_4,C_6,H_i,i=1,\\ldots,5,$ are $6$-chordal graphs with\n hyperbolicity $1$.\n\n\n\n\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.4pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(120,60)\n\n\\put(0,25){\\circle*{1}}\n\n\\put(10,15){\\circle*{1}} \\put(20,25){\\circle*{1}}\n\n\\put(20,5){\\circle*{1}}\n\n\\put(30,5){\\circle*{1}} \\put(30,25){\\circle*{1}}\n\\put(50,25){\\circle*{1}}\\put(40,15){\\circle*{1}}\n\n\\put(25,15){\\circle*{1}}\n\n\n\\qbezier(0,25)(10,15)(20,5)\\qbezier(20,5)(25,5)(30,5)\\qbezier(30,5)(40,15)(50,25)\\qbezier(50,25)(20,25)(0,25)\n\\qbezier(20,5)(25,15)(30,25)\n\\qbezier(20,25)(25,15)(30,5)\\qbezier(10,15)(25,15)(40,15)\\qbezier(20,25)(15,20)(10,15)\\qbezier(40,15)(35,20)(30,25)\n\n\\put(25,-5){\\makebox(0,0){$G_1$}}\n\n\n\n\\put(60,25){\\circle*{1}}\n\n\\put(55,15){\\circle*{1}} \\put(65,15){\\circle*{1}}\n\n\\put(50,5){\\circle*{1}}\n\n\\put(60,5){\\circle*{1}} \\put(70,25){\\circle*{1}}\n\\put(80,25){\\circle*{1}}\\put(75,15){\\circle*{1}}\n\n\\put(70,5){\\circle*{1}}\n\n\n\\qbezier(60,25)(55,15)(50,5)\\qbezier(60,5)(65,15)(70,25)\\qbezier(80,25)(75,15)(70,5)\\qbezier(55,15)(65,15)(75,15)\n\\qbezier(60,25)(65,15)(70,5)\n\\qbezier(70,25)(65,15)(60,5)\\qbezier(55,15)(58,9)(60,5)\\qbezier(70,25)(73,19)(75,15)\\qbezier(50,5)(60,5)(70,5)\n\\qbezier(60,25)(70,25)(80,25)\n\n\\put(60,-5){\\makebox(0,0){$G_2$}}\n\n\\put(55,45){\\circle*{1}}\\put(75,55){\\circle*{1}}\\put(75,35){\\circle*{1}}\\put(85,45){\\circle*{1}}\n\n\\put(65,55){\\circle*{1}}\\put(70,45){\\circle*{1}}\n\n\\put(65,35){\\circle*{1}}\n\\qbezier(55,45)(60,50)(65,55)\\qbezier(55,45)(60,40)(65,35)\\qbezier(65,35)(70,35)(75,35)\n\\qbezier(75,35)(80,40)(85,45)\\qbezier(75,55)(80,50)(85,45)\\qbezier(65,55)(70,55)(75,55)\n\\qbezier(65,55)(70,45)(75,35)\n\n\\put(70,30){\\makebox(0,0){$G_3$}}\n\n\\put(5,45){\\circle*{1}}\\put(15,35){\\circle*{1}}\\put(25,35){\\circle*{1}}\n\\put(35,45){\\circle*{1}}\\put(15,55){\\circle*{1}}\\put(25,55){\\circle*{1}}\n\\qbezier(15,35)(10,40)(5,45)\\qbezier(15,35)(20,35)(25,35)\\qbezier(25,35)(30,40)(35,45)\n\\qbezier(25,55)(30,50)(35,45)\\qbezier(15,55)(20,55)(25,55)\\qbezier(5,45)(10,50)(15,55)\n\\put(20,30){\\makebox(0,0){$C_6$}}\n \\end{picture}\n\\end{center}\n\n\\caption{Four graphs with hyperbolicity $1$ and chordality\n$6$.}\\label{figconjecture}\n\\end{figure}\n\n\n\\begin{conjecture} \\label{conj14} A $6$-chordal graph is\n$\\frac{1}{2}$-hyperbolic if and only if it does not contain any of a\nlist of ten special graphs $G_1,G_2,G_3,C_4,C_6,H_i,i=1,\\ldots,5,$\nas an isometric subgraph.\n\\end{conjecture}\n\n\n\n\n\n\n\n\n\n\nLet $E_1$ and $E_2$ be the graphs depicted in Fig. \\ref{fig\nbridged}. In comparison with Conjecture \\ref{conj14}, when we\nremove the $6$-chordal restriction, we can present the following\ncharacterization of all $\\frac{1}{2}$-hyperbolic graphs obtained\nby Bandelt and Chepoi \\cite{BC}. We refer to \\cite[Fact 1]{BC} for\ntwo other characterizations; also see \\cite{FJ,SC}.\n\n\n \\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.4pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(80,40)\n\\put(10,25){\\circle*{1}}\n\n\\put(5,15){\\circle*{1}} \\put(20,30){\\circle*{1}}\n\n\\put(0,5){\\circle*{1}}\n\n\\put(10,5){\\circle*{1}} \\put(20,15){\\circle*{1}}\n\\put(30,25){\\circle*{1}}\n\\put(20,5){\\circle*{1}}\\put(30,5){\\circle*{1}}\n\\put(30,35){\\circle*{1}}\\put(30,15){\\circle*{1}}\n\n\\qbezier(10,25)(5,15)(0,5)\\qbezier(10,5)(20,15)(30,25)\\qbezier(30,35)(30,15)(30,5)\\qbezier(5,15)(20,15)(30,15)\n\\qbezier(10,25)(15,20)(20,15)\n\\qbezier(20,30)(20,15)(20,5)\\qbezier(5,15)(8,9)(10,5)\\qbezier(10,25)(20,30)(30,35)\\qbezier(0,5)(10,5)(30,5)\n\\qbezier(20,5)(25,10)(30,15)\\qbezier(20,30)(26,27)(30,25)\n\n\\put(15,-5){\\makebox(0,0){$E_1$}}\n\n\\put(40,5){\\circle*{1}}\\put(50,5){\\circle*{1}}\\put(60,5){\\circle*{1}}\\put(70,5){\\circle*{1}}\n\n\\put(40,15){\\circle*{1}}\\put(70,15){\\circle*{1}}\\put(40,25){\\circle*{1}}\n\n\\put(70,25){\\circle*{1}}\\put(40,35){\\circle*{1}}\\put(50,35){\\circle*{1}}\\put(60,35){\\circle*{1}}\n\\put(70,35){\\circle*{1}}\\put(55,20){\\circle*{1}}\n\n\\qbezier(40,5)(60,5)(70,5)\\qbezier(40,5)(40,10)(40,35)\\qbezier(40,35)(50,35)(70,35)\n\\qbezier(70,35)(70,15)(70,5)\\qbezier(40,15)(55,20)(70,25)\\qbezier(40,25)(55,20)(70,15)\n\\qbezier(60,35)(55,20)(50,5)\\qbezier(50,35)(55,20)(60,5)\\qbezier(50,35)(45,30)(40,25)\n\\qbezier(40,15)(45,10)(50,5)\\qbezier(60,35)(65,30)(70,25)\n\n\\qbezier(60,5)(65,10)(70,15)\n\n\\put(55,-5){\\makebox(0,0){$E_2$}}\n \\end{picture}\n\\end{center}\n\n\\caption{Two bridged graphs with hyperbolicity $1$.}\\label{fig\nbridged}\n\\end{figure}\n\n\n\n\\begin{te} \\cite[p. 325]{BC} A graph $G$ is $\\frac{1}{2}$-hyperbolic if and only if\n$G$ does not contain isometric $n$-cycles for any $n>5$, for any\ntwo vertices $x$ and $y$ of $G$ one cannot find two non-adjacent\nneighbors of $x$ which are both closer to $y$ in $G$ than $x$,\nand none of the six graphs $H_1,H_2,G_1,G_2,E_1,E_2$ occurs as an\nisometric subgraph of $G.$ \\label{thmBC}\n\\end{te}\n\n\n\n\\begin{remark} Instead of Theorem \\ref{thmBC}, it would be interesting to determine, if possible, a finite\n list of graphs such that that a graph is $\\frac{1}{2}$-hyperbolic if and only if\nit does not include any graph from that list as an isometric\nsubgraph. Koolen and Moulton point out a possible approach to\ndeduce such kind of a characterization in \\cite[p. 696]{KM02}.\n\\end{remark}\n\n\n\n\n\nNote that a $5$-chordal graph cannot contain any isometric\n$n$-cycle for $n>5$. It is also easy to see that\n$\\mathbbm{l}\\mathbbm{c}(G_1)=\\mathbbm{l}\\mathbbm{c}(G_2)=6,\\mathbbm{l}\\mathbbm{c}(E_1)=7,\n\\mathbbm{l}\\mathbbm{c}(E_2)=8$. Therefore, we obtain the following\neasy consequence of Theorem \\ref{thmBC}. It is interesting to\ncompare it with Theorems \\ref{main} and \\ref{main1}.\n\n\n\\begin{corollary} A $5$-chordal graph $G$ is $\\frac{1}{2}$-hyperbolic if and only if it does not\ncontain the graph $H_1$ and $H_2$ as isometric subgraphs and for\nany two vertices $x$ and $y$ of $G$ the neighbors of $x$ which are\ncloser to $y$ than $x$ are pairwise adjacent.\n\\end{corollary}\n\n\n\n\n\n\n\\subsection{Some consequences}\\label{oct}\n\nNote that $\\mathbbm{l}\\mathbbm{c}(C_4)=4,\n\\mathbbm{l}\\mathbbm{c}(H_1)=\\mathbbm{l}\\mathbbm{c}(H_2)=3,\n\\mathbbm{l}\\mathbbm{c}(H_3)=\\mathbbm{l}\\mathbbm{c}(H_4)=\\mathbbm{l}\\mathbbm{c}(H_5)=5.$\nThe next two results follow immediately from Theorem \\ref{main1}.\n\n\n\\begin{corollary} Every $4$-chordal graph must be $1$-hyperbolic and it has\nhyperbolicity one if and only if it contains one of $C_4$, $H_1$\nand $H_2$ as an isometric subgraph. \\label{cor7}\n\\end{corollary}\n\n\n\n\\begin{corollary} \\cite[Theorem 1.1]{BKM01} Every chordal\ngraph is $1$-hyperbolic and it has hyperbolicity one if and only if\nit contains either $H_1$ or $H_2$ as an isometric\nsubgraph.\\label{BKM}\n\\end{corollary}\n\nWe remark that as long as every $4$-chordal graph is 1-hyperbolic is\nknown, Corollary \\ref{cor7} also immediately follows from Corollary\n\\ref{BKM}. In addition, it is noteworthy that the first part of\nCorollary \\ref{BKM}, namely every chordal graph is $1$-hyperbolic is\nimmediate from Theorem \\ref{thm10} as chordal graphs have\ntree-length $1$.\n\n\n\\begin{corollary} Each weakly chordal graph is $1$-hyperbolic and has\nhyperbolicty one if and only if it contains one of $C_4,H_1,H_2$ as\nan isometric subgraph.\n\\end{corollary}\n\n\\begin{proof} By definition, each weakly chordal graph is $4$-chordal. It is also easy to check that that $C_4,H_1$ and $H_2$\nare all weakly chordal. Hence, the result follows from Corollary\n\\ref{cor7}.\n\\end{proof}\n\n\n\n\\begin{corollary} All strongly chordal graphs are\n$\\frac{1}{2}$-hyperbolic.\n\\end{corollary}\n\\begin{proof} Note that the cycle $C=[x,a,u,c,y,d,v,b]$ in $H_1$\nand $H_2$ does not have any odd chord and hence neither $H_1$ nor\n$H_2$ can appear as an isometric subgraph of a strongly chordal\ngraph. Since strongly chordal graphs must be chordal graphs, this\nresult holds by Corollary \\ref{BKM}.\n\\end{proof}\n\n\n\n\n\n\\begin{corollary} All threshold graphs are $\\frac{1}{2}$-hyperbolic.\n\\end{corollary}\n\\begin{proof} It is obvious that threshold graphs are chordal as they contain neither $4$-cycle\nnor path of length $3$ as induced subgraph. Since the subgraph\ninduced by $x,u,b,c$ in either $H_1$ or $H_2$ is just the\ncomplement of $C_4$, the result follows from Corollary \\ref{BKM}\nand the definition of a threshold graph.\n\\end{proof}\n\n\\begin{corollary} Every $AT$-free graph is $1$-hyperbolic and it has hyperbolicity one if and only if it contains $C_4$\nas an isometric subgraph. \\label{AT-free}\n\\end{corollary}\n\\begin{proof} First observe that an\n $AT$-free graph must be $5$-chordal. Further notice that\nthe triple $u,y,v$ is an $AT$ in any of the graphs $H_1,\\ldots,H_5.$\nNow, an application of Theorem \\ref{main1} concludes the proof.\n\\end{proof}\n\n\\begin{corollary} \\label{com} A cocomparability graph is $1$-hyperbolic and has hyperbolicity one if and only if it contains $C_4$\nas an isometric subgraph.\n\\end{corollary}\n\\begin{proof} We know that cocomparability graphs are $AT$-free and\n $C_4$ is a cocomparability graph. Thus the result\ncomes directly from Corollary \\ref{AT-free}. The deduction of\nthis result can also be made via Corollary \\ref{cor7} and the fact\nthat cocomparability graphs are $4$-chordal \\cite{BT1, Gallai}.\n\\end{proof}\n\n\n\n\n\n\\begin{corollary} A permutation graph is $1$-hyperbolic and has hyperbolicity one if and only\nif it contains $C_4$ as an isometric subgraph.\n\\end{corollary}\n\n\\begin{proof} Every permutation graph is a cocomparability graph and $C_4$ is a permutation\ngraph. So, the result follows from Corollary \\ref{com}.\n\\end{proof}\n\n\n\n\n\n\\begin{corollary} \\cite[p. 16]{BC08}\nA distance-hereditary graph\n is always $1$-hyperbolic and is $\\frac{1}{2}$-hyperbolic exactly when it is\n chordal, or equivalently, when it contains no induced $4$-cycle.\n \\label{distance}\n \\end{corollary}\n\n \\begin{proof} It is easy to see that distance-hereditary graphs\n must be $4$-chordal and can contain neither $H_1$ nor $H_2$ as an\nisometric subgraph. The result now follows from Corollary\n\\ref{cor7}.\n\\end{proof}\n\n\n\n\n\n\n\\begin{corollary} A cograph is $1$-hyperbolic and has hyperbolicity one if and only if it contains\n$C_4$ as an isometric subgraph.\n\\end{corollary}\n\n\\begin{proof} We know that $C_4$ is a cograph and every cograph is ditance-hereditary. Applying Corollary\n\\ref{distance} yields the required result.\n\\end{proof}\n\n\n\n\n\n\n\n\n\\section{Relevant tree-likeness parameters}\\label{parameter}\n\n\\subsection{Tree-length}\n\n\n\n\n\n\nIt turns out that tree-length is a very useful concept for\nconnecting chordality and hyperbolicity.\n Indeed, the following theorem, which can be read from Theorem \\ref{main} (Corollary \\ref{BKM}), comes directly from\n Theorems\n\\ref{thm9} and \\ref{thm10}. This result is firstly notified to us\nby Dragan \\cite{Dragan} and is presumably in the folklore.\n\n\\begin{te} For any $k\\geq 3,$ every $k$-chordal graph is $\\lfloor \\frac{k}{2}\\rfloor $-hyperbolic.\\label{June}\n\\end{te}\n\n\n\nIn view of Remark \\ref{grid}, to get better estimate than Theorem\n\\ref{June} along the same approach\n one may try to beef\nup Theorem \\ref{thm9}. We point out that Dourisboure and Gavoille\n\\cite[Question 1]{DG} posed as an open problem that whether or not\n\\begin{equation} \\label{eq22} \\mathbbm{t}\\mathbbm{l}(G)\\leq \\lceil \\frac{\\mathbbm{l}\\mathbbm{c}(G)}{3}\\rceil\n\\end{equation} is true.\nThe {\\em $k$th-power} of a graph $G$, denoted $G^k,$ is the graph\nwith $V(G)$ as vertex set and there is an edge connecting two\nvertices $u$ and $v$ if and only if $d_G(u,v)\\leq k.$ Let us\ninterpret the problem of Dourisboure and Gavoille as a Chordal\nGraph Sandwich Problem:\n\n\\begin{question}\nFor any graph $G$, is there always a chordal graph $H$ such\nthat $V(H)=V(G)=V(G^{\\lceil\n\\frac{\\mathbbm{l}\\mathbbm{c}(G)}{3}\\rceil})$ and $E(G)\\subseteq E(H)\n\\subseteq G^{\\lceil \\frac{\\mathbbm{l}\\mathbbm{c}(G)}{3}\\rceil}$?\n\\end{question}\n\n\nIf \\eqref{eq22} can be established, it will be the best we can\nexpect in the sense that $\\mathbbm{t}\\mathbbm{l}(G)=\\lceil\n\\frac{\\mathbbm{l}\\mathbbm{c}(G)}{3}\\rceil$ for every outerplanar\ngraph $G$ \\cite[Theorem 1]{DG}.\n\n\n\n\n\n\n\n\n\n\\subsection{Approximating trees, slimness and thinness}\n\n\n\n\nWe introduce in this subsection two general approaches to connect\nchordality with hyperbolicity. A result is given together with a\nproof only when that proof is short and when we do not find it\nappear very explicitly elsewhere. This section also aims to provide\nthe reader a warm-up before entering the longer proof in the main\npart of this paper.\n\n\nA\n result weaker than Theorem \\ref{main} (Theorem \\ref{June}) and reported in\n\\cite[p. 64]{CE} as well as \\cite[p. 3]{CDEHVX} is that each\n$k$-chordal graph is $k$-hyperbolic. The two approaches to be\nreported below by far basically only lead to\n this weaker result.\n Despite of this, it might be interesting to see different ways of bounding\n hyperbolicity in terms of chordality via the use of some other intermediate tree-likeness parameters.\n\nThe first approach is to look at distance approximating trees. A\ntree $T$ is a {\\em distance $t$-approximating tree} of a graph $G$\nprovided $V(T)=V(G)$ and $|d_G(u,v)-d_T(u,v)|\\leq t$ for any $u,v\\in\nV(G)$ \\cite{Balint,BCD,CD,DY}. It is well-known that a graph with a\ngood distance approximating tree will have low hyperbolicity,\nwhich is briefly mentioned in \\cite[p. 3]{CDEHVX} and \\cite[p.\n64]{CE} and is in the same spirit of a general result on\nhyperbolic geodesic metric spaces \\cite[p. 402, Theorem 1.9]{BH}. We\nmake this point clear in the following simple lemma.\n\n\n\n\n\n\\begin{lemma}\\label{lem1} Let $G$ be a graph and $t$ be a nonnegative integer. If $G$ has a distance $t$-approximating tree\n$T$, then $G$ is $2t$-hyperbolic.\n\\end{lemma}\n\\begin{proof}\nFor any $x,y,u,v\\in V(G)$, our aim is to show that\n$\\delta_G(x,y,u,v)\\leq 2t.$ Assume, as we may, that\n$d_G(x,y)+d_G(u,v)\\geq d_G(x,u)+d_G(y,v)\\geq d_G(x,v)+d_G(y,u).$\nSince the tree metric $d_T$ is a four-point inequality metric\n(or additive metric) \\cite{DD}, we know that $\\delta^*(T)=0$ and\nso the following three cases are exhaustive.\n\n\n\n\\paragraph {\\sc Case 1:}\n$d_T(x,y)+d_T(u,v)= d_T(x,u)+d_T(y,v)\\geq d_T(x,v)+d_T(y,u).$\n\n\n\n\n\n\n\n$\\delta_G(x,y,u,v)=\\frac{1}{2}(d_G(x,y)+d_G(u,v))\n-\\frac{1}{2}(d_G(x,u)+d_G(y,v))\\leq \\frac{1}{2}(d_T(x,y)+d_T(u,v)+\n2t ) -\\frac{1}{2}(d_T(x,u)+d_T(y,v)- 2t)=2t. $\n\n\n\\paragraph {\\sc Case 2:}\n$d_T(x,y)+d_T(u,v)= d_T(x,v)+d_T(y,u)\\geq d_T(x,u)+d_T(y,v) $\n\n\n\n\n\n$\\delta_G(x,y,u,v)=\\frac{1}{2}(d_G(x,y)+d_G(u,v))\n-\\frac{1}{2}(d_G(x,u)+d_G(y,v))\\leq \\frac{1}{2}(d_G(x,y)+d_G(u,v))\n-\\frac{1}{2}(d_G(x,v)+d_G(y,u))\\leq \\frac{1}{2} (d_T(x,y)+d_T(u,v)+\n2t ) -\\frac{1}{2}(d_T(x,v)+d_T(y,u)- 2t) = 2t. $\n\n\n\n\n\n\n\\paragraph {\\sc Case 3:} $ d_T(x,v)+d_T(y,u)= d_T(x,u)+d_T(y,v) \\geq d_T(x,y)+d_T(u,v).$\n\n\n\n\n$\\delta_G(x,y,u,v)=\\frac{1}{2}(d_G(x,y)+d_G(u,v))\n-\\frac{1}{2}(d_G(x,u)+d_G(y,v))\\leq \\frac{1}{2}(d_T(x,y)+d_T(u,v)+\n2t) -\\frac{1}{2}(d_T(x,u)+d_T(y,v)- 2t)\\leq\n\\frac{1}{2}(d_T(x,u)+d_T(y,v)+ 2t ) -\\frac{1}{2}(d_T(x,u)+d_T(y,v)-\n2t)= 2t. $\n\\end{proof}\n\n\nAfter showing that the existence of good distance approximating tree\n guarantees low hyperbolicity, in order to connect chordality\nwith hyperbolicity, we need to make sure that low chordality\ngraphs have good distance approximating trees \\cite{BCD,CD} . Here\nis an exact result.\n\n\n\n\n\\begin{te} \\cite{CD} Let $G$ be a $k$-chordal graph. Then, there is a tree\n$T$ with $V(T)=V(G)$ such that for any $u,v\\in V(G)$ it holds\n\\begin{equation*} \\label{eq:1} |d_G(u,v)-d_T(u,v)|\\leq \\left\\{\n\\begin{aligned}\n \\lfloor\n\\frac{k}{2 }\\rfloor +2, & \\ \\ \\ \\ \\ \\text{if}\\ \\ k=4,5, \\\\\n \\lfloor \\frac{k}{2 }\\rfloor +1, & \\ \\ \\ \\ \\ \\text{else.}\n \\end{aligned} \\right.\n \\end{equation*}\n \\label{Chepoi}\n\\end{te}\n\n\n\nThe other possible approach to connect hyperbolicity and chordality\nis via the concept of the thinness\/slimness of geodesic triangles.\nThis approach also consists of two parts, one is to show that a\ngraph with low thinness\/slimness has low hyperbolicity, as\nsummarized in \\cite[Proposition 1]{CDEHV}, and the other part is to\nshow that low chordality implies low thinness\/slimness.\n\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.4pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(30,20)\n\\put(10,5.5){\\circle{11}}\n\\put(0,0){\\circle*{1}}\\put(0,-5){\\makebox(0,0)[b]{$x$}}\n\\put(20,0){\\circle*{1}}\\put(20,-5){\\makebox(0,0)[b]{$y$}}\n\\put(0,0){\\line(1,0){20}}\n\\put(10,17.3){\\circle*{1}}\\put(10,22){\\makebox(0,0)[t]{$z$}}\n\\put(5,8.5){\\circle*{1}}\\put(-5,8.5){\\makebox(0,0)[l]{$m_y^{z,x}$}}\n\\put(15,8.5){\\circle*{1}}\\put(25,8.5){\\makebox(0,0)[r]{$m_x^{y,z}$}}\n\\put(10,0){\\circle*{1}}\\put(10,-5){\\makebox(0,0)[b]{$m_z^{x,y}$}}\n\\qbezier(0,0)(5,8.5)(10,17.3)\\qbezier(20,0)(15,8.5)(10,17.3)\n \\end{picture}\n\\end{center}\n\\caption{An illustration of Gromov product.}\\label{fig-1}\n\\end{figure}\n\n\n\nGiven a graph $G,$ we can put an orientation on it by choosing two\nmaps $\\partial_0$ and $\\partial_1$ from $E(G)$ to $V(G)$ such that\neach edge $e$ just have $\\partial_0(e)$ and $\\partial_1(e)$ as its\ntwo endpoints. The discrete metric space $(V(G),d_G)$ can then be\nnaturally embedded into the {\\em metric graph} \\cite[p. 7]{BH}\n$(X_G, \\widetilde{d_G})$, where $X_G$ is the quotient space of\n$E(G)\\times [0,1]$ under the identification of $(e,i)$ and $(e',i')$\nwhenever $\\partial _i(e)=\\partial_{i'}(e')$ for any $e,e'\\in E(G)$\nand $i,i'\\in \\{0,1\\},$ and $\\widetilde{d_G}$ is the metric on\n$X_G$ satisfying $\\widetilde{d_G}((e,t),(e,t'))=\n |t-t'|$ if $ e=e'$ and $\\widetilde{d_G}((e,t),(e,t'))=\n\\min (d_G(\\partial_0(e),\\partial_0(e'))+t+t',\nd_G(\\partial_0(e),\\partial_1(e'))+t+1-t',\nd_G(\\partial_1(e),\\partial_0(e'))+1-t+t',d_G(\\partial_1(e),\\partial_1(e'))+2-t-t'\n)$ else. It is easy to see that the definition of $(X_G,\n \\widetilde{d_G})$ is indeed independent of the orientation of\n $G.$ Also, any cycle of $G$ naturally corresponds to a\n circle, namely one-dimensional sphere, embedded in $X_G.$\nFor any two points $x,y\\in X_G$, there is a not necessarily unique\ngeodesic connecting them in $(X_G, \\widetilde{d_G})$, which we will\nuse the notation $[x,y]$ if no ambiguity arises.\n We say that $(x,y,z)$ is {\\em $(\\delta_1,\\delta_2)$-thin} provided for any choice\n of the geodesics $[x,y], [y,z], [z,x]$ and $m_x^{y,z}\\in [y,z],m_y^{z,x}\\in [z,x],\n m_z^{x,y}\\in [x,y]$ satisfying \\begin{equation}\\left\\{ \\begin{array}{l }\n \\widetilde{d_G}(m_z^{x,y},x)=\\widetilde{d_G}(m_y^{z,x},x)=(y\\cdot z)_x,\\\\\n\\widetilde{d_G}(m_x^{y,z},y)=\\widetilde{d_G}(m_z^{x,y},y)=(z\\cdot x)_y,\\\\\n \\widetilde{d_G}(m_y^{z,x},z)=\\widetilde{d_G}(m_x^{y,z},z)=(x\\cdot\n y)_z,\n\\end{array}\\right.\\label{E3}\n\\end{equation}\n(Figure \\ref{fig-1} is an illustration of \\eqref{E3} as well as a\nwidely-used geometric interpretation of the Gromov product.) the\nfollowing two conditions hold:\n \\begin{itemize} \\item[(A)]\n$\\delta_1 \\geq \\min (\\widetilde{d_G}(m_x^{y,z},m_y^{z,x}),\n\\widetilde{d_G}(m_z^{x,y}, m_y^{z,x}))$;\n\\item[(B)] $\\{p\\in X_G:\\ \\widetilde{d_G}(p,[y,z]\\cup [y,x])\\leq \\delta_2 \\}\\supseteq [x,z].$\n\\end{itemize}\n Modifying the original definition of Gromov slightly \\cite[p. 8, Definition 1.5]{Alonso} \\cite[p.\n408]{BH} \\cite{Gromov}, we say that a graph $G$ is {\\em\n$(\\delta_1,\\delta_2)$-thin} provided every triple $(x,y,z)$ of its\nvertices is $(\\delta_1,\\delta_2)$-thin.\n\n\n\n\n\n\n\\begin{lemma} Let $G$ be a graph. If $G$ is\n$(\\delta_1,\\delta_2)$-thin, then it is\n$(\\delta_1+\\delta_2)$-hyperbolic. \\label{lemma1}\n\\end{lemma}\n\\begin{proof} The proof is taken from\n \\cite[p. 15, (2) implies (5)]{Alonso}.\n It suffices to establish \\eqref{EQ} for any $x,y,u,v\\in V(G)$.\nBy Eq. \\eqref{E3} and Condition (A) for the\n$(\\delta_1,\\delta_2)$-thinness of $(x,u,y)$, we have\n\\begin{equation}\\label{4}(x\\cdot y)_u+\\delta_1 \\geq \\min ( \\widetilde{d_G}(u,\nm_x^{u,y})+ \\widetilde{d_G}(m_x^{u,y}, m_u^{y,x}),\n \\widetilde{d_G}(u,\nm_y^{x,u})+ \\widetilde{d_G}(m_y^{x,u}, m_u^{y,x}) ) \\geq\n\\widetilde{d_G}(u, m_u^{y,x}).\\end{equation} By Condition (B) for\nthe $(\\delta_1,\\delta_2)$-thinness of $(x,v,y)$, we can suppose,\nwithout loss of generality, that there is $q\\in [y,v]$ such that\n\\begin{equation}\\label{5} \\delta_2 \\geq\n\\widetilde{d_G}(m_u^{y,x},q).\\end{equation} It follows from\n$\\widetilde{d_G}(q,v)+ \\widetilde{d_G}(q,y)=\\widetilde{d_G}(y,v)$,\n$\\widetilde{d_G}(u,q)+\\widetilde{d_G}(q,v)\\geq\n\\widetilde{d_G}(u,v)$, and\n$\\widetilde{d_G}(u,q)+\\widetilde{d_G}(q,y)\\geq \\widetilde{d_G}(u,y)$\nthat\n\\begin{equation}\\label{6}\\widetilde{d_G}(u,q)\\geq (y\\cdot v)_u.\\end{equation}\nWe surely have\n\\begin{equation}\\label{7} \\widetilde{d_G}(u,\nm_u^{y,x})+ \\widetilde{d_G}( m_u^{y,x},q)\\geq\n\\widetilde{d_G}(u,q).\n\\end{equation}\nTo complete the proof, we just need to add together \\eqref{4},\n\\eqref{5}, \\eqref{6}, and \\eqref{7}.\n\\end{proof}\n\nAccording to Gromov \\cite{Gromov}, Rips invents the concept of\nslimness: For any real number $\\delta,$ we say that a graph $G$ is\n{\\em $\\delta$-slim} if for every triple $(x,y,z)$ of vertices of $G,\n$ we have\n\\begin{equation}\\label{eq7} \\{p\\in X_G:\\ \\widetilde{d_G}(p,[y,z]\\cup\n[y,x])\\leq \\delta \\}\\supseteq [x,z].\n\\end{equation}\nAn easy observation is that a $(\\delta_1,\\delta_2)$-thin graph is\n$\\delta_2$-slim.\n It is mentioned in \\cite[Proposition 1]{CDEHV} that every $\\delta$-slim graph is\n$8\\delta$-hyperbolic. The next lemma gives a better bound.\n\n\\begin{lemma} If a graph is $\\delta$-slim, it must be $(2\\delta,\n\\delta)$-thin and hence $3\\delta$-hyperbolic.\n\\end{lemma}\n\\begin{proof} By Lemma \\ref{lemma1}, our task is to prove that any $\\delta$-slim graph $G$ is $(2\\delta,\n\\delta)$-thin. For this purpose, it suffices to deduce $2\\delta \\geq\n\\min (\\widetilde{d_G}(m_x^{y,z},m_y^{z,x}),\n\\widetilde{d_G}(m_z^{x,y}, m_y^{z,x}))$ for any triple $(x,y,z)$ of\nvertices of $G.$ The following argument is almost word-for-word the\nsame as that given in \\cite[p. 13, (1) implies (3)]{Alonso}. By\n\\eqref{eq7}, we suppose, as we may, that there is $w\\in [y,x]$ such\nthat $\\widetilde{d_G}(m_y^{z,x},w)\\leq \\delta.$ Observe that\n$$\\widetilde{d_G}(x,w)\\geq\n \\widetilde{d_G}(m_y^{z,x},x)-\\widetilde{d_G}(m_y^{z,x},w)\\geq\n \\widetilde{d_G}(m_y^{z,x},x)-\\delta=\n \\widetilde{d_G}(m_z^{x,y},x)-\\delta$$\n and that\n$$\\widetilde{d_G}(x,w)\\leq\n \\widetilde{d_G}(m_y^{z,x},x)+\\widetilde{d_G}(m_y^{z,x},w)\\leq\n \\widetilde{d_G}(m_y^{z,x},x)+\\delta=\n \\widetilde{d_G}(m_z^{x,y},x)+\\delta.$$\nIt then follows $\\widetilde{d_G}(m_z^{x,y},w)\\leq \\delta$ and\nhenceforth $\\widetilde{d_G}(m_z^{x,y},m_y^{z,x})\\leq\n \\widetilde{d_G}(m_z^{x,y},w)+ \\widetilde{d_G}(w,m_y^{z,x})\\leq\n 2\\delta,$ as desired.\n\\end{proof}\n\n\n\\begin{lemma}\n Every $k$-chordal graph is\n$ (\\frac{k}{2}, \\frac{k}{2})$-thin.\\label{lemma2}\n\\end{lemma}\n\\begin{proof}\nConsider any triple $(x,y,z)$ of vertices of $G.$ By an abuse of\nnotation as usual, denote by $[x,y],[y,z]$ and $[z,x]$ three\ngeodesic segments joining the corresponding endpoints and put\n$m_x^{y,z}\\in [y,z],$ $ m_y^{z,x}\\in [z,x]$ and $m_z^{x,y}\\in [x,y]$\nbe three points of $X_G$ satisfying Eq. \\eqref{E3}. For any\nnonnegative number $t\\leq (y\\cdot z)_x$, there is a unique point $u$\nlying in $ [x,y]$ such that $\\widetilde{d_G}(u,x)=t$; we use the\nnotation $(z;x)_t$ for this point $u.$ Similarly, we define\n$(y;x)_t$ for any $0\\leq t\\leq (y\\cdot z)_x$ and so on.\n By symmetry, it\nsuffices to show that $\\widetilde{d_G}((z;x)_t,(y;x)_t)\\leq\n\\frac{k}{2} $ for any $0\\leq t\\leq (y\\cdot z)_x$. Take the maximum\n$t'\\leq t$ such that $(z;x)_{t'}=(y;x)_{t'}$. The case of $t'=t$ is\ntrivial and so we assume that $t't$. We can assume that\n$(z;x)_{t'''}\\not=(y;x)_{t'''}$ for any $t't$.\n\nLet $\\Lambda = \\{(z;x)_s, (y;x)_s:\\ 0\\leq s\\leq (y\\cdot x)_x\\}$\n and $\\Upsilon=\\{(z;y)_s:\\ 0\\leq s< (z\\cdot x)_y\\}\\cup \\{(y;z)_s:\\ 0\\leq s<\n(x\\cdot y)_z\\}.$ For any $y\\in \\Upsilon,$\n$\\widetilde{d_G}(x,y)>(y\\cdot z)_x$ holds and for any $y\\in\n\\Lambda$, $\\widetilde{d_G}(x,y)\\leq (y\\cdot z)_x$ holds. This\nsays that\n\\begin{equation}\\label{eq88}\\Lambda \\cap \\Upsilon=\\emptyset . \\end{equation}\nAnalogously, by considering both the\ndistance to $y$ and the distance to $z$, we\n have \\begin{equation}\\label{eq99} \\Lambda\\cap [y,z]=\\emptyset\n.\\end{equation} Combining Eqs. \\eqref{eq88} and \\eqref{eq99}, we get\nthat there is a geodesic $P$ connecting $(z;x)_t$ and $(y;x)_t$\nwhose internal points fall inside $ [y,z]\\cup \\Upsilon$. We\nproduce a circle in $X_G$ as follows: Walk along $P$ from\n$(z;x)_t$ to $(y;x)_t$ and then go along $[x,z]$ from $(y;x)_t$ to\n$(y;x)_{t'}$ and finally return to $(z;x)_t$ by following $[x,y]$.\nThis circle naturally corresponds to a cycle of $G.$ This cycle\nmight have chords. But for each chord which splits the circle into\ntwo smaller circles, our assumption guarantees that the two vertices\n$(z;x)_t$ and $(y;x)_t$ will still appear in one of them\nsimultaneously. This means that there is a circle of $X_G$\ncorresponding to a chordless cycle of $G$ and passing from both\n$(z;x)_t$ and $(y;x)_t$. As $G$ is $k$-chordal,\n$\\widetilde{d_G}((z;x)_t,(y;x)_t)\\leq \\frac{k}{2}$ follows,\n as expected.\n\\end{proof}\n\n\n\n\n\\section{Proofs}\\label{proofs}\n\n\\subsection{Lemmas}\\label{lemmas}\n\nThe proof of our main results, namely Theorems \\ref{main} and\n\\ref{main1}, is divided into a sequence of lemmas\/corollaries.\n\nIn the course of our proof, we will frequently make use of the\ntriangle inequality for the shortest-path metric, namely $ab+bc\\geq\nac$, without any claim. Besides this, we will also freely apply\nthe ensuing simple observation, which is so simple that we need not\nbother to give any proof here.\n\n\\begin{lemma} \\label{EASY} Let $H$ be a vertex induced subgraph of a graph $G.$\nThen $H$ is an isometric subgraph of $G$ if and only if\n$d_{H}(u,v)=d_G(u,v)$ for each pair of vertices $(u,v)\\in V(G)\\times\nV(G)$ satisfying $d_H(u,v)\\geq 3.$ In particular, $H$ must be\nisometric if its diameter is at most $2$.\n\\end{lemma}\n\n\n\n\n\nOne small matter of convention here and in what follows. When we\nrefer to a graph, say a graph depicted in Fig. \\ref{fig0}, we\nsometimes indeed mean that graph together with the special labeling\nof its vertices as indicated when it is introduced and sometimes we\nmean a graph which is isomorphic to it. We just leave it to\nreaders to decide from the context which usage it is. Two immediate\ncorollaries of Lemma \\ref{EASY} are given subsequently. We state\nthem with the above convention and omit their routine proofs.\n\n\n\\begin{corollary}\\label{lem29}\nLet $G$ be a graph. Let $H\\in \\{H_1,H_2,H_4\\}$ be an induced\nsubgraph of $G$ such that $d_G(x,y)=d_G(u,v)=3$. Then $H$ is an\nisometric subgraph of $G$.\n\\end{corollary}\n\n\n\n\n\n\\begin{corollary}\\label{lem30}\nLet $G$ be a graph and $H_3$ be an induced subgraph of $G$. If\n$d_G(x,y)=3$, then $H_3$ is an isometric subgraph of $G$.\n\\end{corollary}\n\nIt is time to deliver some formal proofs.\n\n\\begin{corollary}\\label{lem31}\nLet $G$ be a graph and $H_5$ be an induced subgraph of $G$. If\n$d_G(x,y)=d_G(u,v)=3$ and $d_G(b,c)=4$. Then $H_5$ is an isometric\nsubgraph of $G$.\n\\end{corollary}\n\n\\begin{proof}\nBased on the fact that $d_G(b,c)=4$, we can derive from the triangle\ninequality that $d_G(u,b)=d_G(y,b)=d_G(c,x)=d_G(c,v)=3$. The result\nthen follows from Lemma \\ref{EASY} as $$\\{x,y\\}, \\{ u,v\\}, \\{\nu,b\\}, \\{ y,b\\}, \\{ c,x\\}, \\{ c,v\\}, \\{ b,c\\}$$ are all pairs\ninside $V(H_5)\\choose 2$ which are of distance at least $3$ apart in\n$H_5$.\n\\end{proof}\n\n\n\n\n\n\\begin{lemma}\\label{first} Let $G$ be a graph and let $x,y,u,v\\in V(G)$. Then $\\delta_G\n(x,y,u,v)\\leq \\min (uv,xy,ux,yv,uy,xv)$. \\end{lemma}\n\n\n\n\n\n\\begin{proof} Suppose that\n $d_G(x,S)=d_1, d_G(y,S)=d_2$, where $S=\\{ u,v\\}$. We can check the\n following:\n$$\\left\\{ \\begin{array}{l }\n xy+uv\\leq (d_1+d_2+uv)+uv=d_1+d_2+2uv,\\\\\n d_1+d_2\\leq xu+yv\\leq (d_1+uv)+(d_2+uv)= d_1+d_2+2uv,\\\\\n d_1+d_2\\leq xv+yu\\leq (d_1+uv)+(d_2+uv)=d_1+d_2+2uv,\n\\end{array}\\right.$$\n from which we get $\\delta_G\n(x,y,u,v)\\leq uv$ and hence our claim follows by symmetry.\n\\end{proof}\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.4pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(50,40)\n\n\\put(30,0){\\circle*{1}}\\put(30,-4){\\makebox(0,0)[b]{$y$}}\n\\put(10,20){\\circle*{1}}\\put(7,20){\\makebox(0,0)[l]{$u$}}\n\\put(50,20){\\circle*{1}}\\put(53,20){\\makebox(0,0)[r]{$v$}}\n\\put(30,40){\\circle*{1}}\\put(30,43){\\makebox(0,0)[t]{$x$}}\n\\put(20,30){\\circle*{1}}\\put(17,33){\\makebox(0,0)[l]{$a$}}\n\\put(40,30){\\circle*{1}}\\put(43,33){\\makebox(0,0)[r]{$b$}}\n\\put(20,10){\\circle*{1}}\\put(15,10){\\makebox(0,0)[l]{$c$}}\n\\put(40,10){\\circle*{1}}\\put(45,10){\\makebox(0,0)[r]{$d$}}\n\\put(35,-5){\\makebox(0,0)[l]{$H_{6}$}}\n\\qbezier(10,20)(20,10)(30,0)\\qbezier(30,0)(40,10)(50,20)\\qbezier(10,20)(20,30)(30,40)\\qbezier(30,40)(40,30)(50,20)\n\\qbezier(20,30)(20,20)(20,10)\n\\qbezier(20,30)(30,20)(40,10)\\qbezier(20,10)(30,10)(40,10)\n \\end{picture}\n\\end{center}\n\\caption{A $5$-chordal graph with hyperbolicity $1$.}\\label{fig8}\n\\end{figure}\n\n\nThe next two simple lemmas concern the graph $H_6$ as given in Fig.\n\\ref{fig8}, which is obviously a $5$-chordal graph with\nhyperbolicity $1.$\n\n\\begin{lemma}\\label{lead} Let $H$ be a graph satisfying\n $V(H)=V(H_2)=V(H_5)=\\{x,y,u,v,a,b,c,d\\}$ and $E(H_5)\\subseteq E(H)\\subseteq E(H_2)$.\nLet $t$ be the size of $E(H)\\cap \\{\\{a,b\\},\\{b,d\\}, \\{d,c\\},\n\\{c,a\\}\\}.$ If $t\\in \\{1,2,3\\}$, then either $H$ contains an\ninduced $C_4$ or there is an isomorphism from $H$ to $H_6$.\n\\end{lemma}\n\n\n\n\\begin{proof} For any $v_1,v_2\\in V(H)$, $v_1v_2$ always refers to $d_H(v_1,v_2)$ in the following.\n\n\n\\paragraph {\\sc Case 1:} $bc=1.$\n\nLet us show that $H$ contains an induced $C_4$ in this case.\nSince $t\\in \\{1,2,3\\}$, by symmetry, we can assume that either\n$ac=1, cd\\not=1$ or $cd=1,bd\\not= 1.$\n In the former case,\n$[acyd]$ is an induced $4$-cycle of $H$ and in the latter case\n$[cdvb]$\n is an induced $4$-cycle of $H$.\n\n\n\\paragraph {\\sc Case 2:} $bc\\not= 1.$\n\n\n\nFirst observe that replacing the two edges $\\{ a,c\\}$ and $\\{d,c\\}$\nby the two new edges $\\{ a,b\\}$ and $\\{d,b\\}$ will transform $H_6$\ninto another graph which is isomorphic to $H_6.$ Thus, by symmetry,\nthe condition that $t\\in \\{1,2,3\\}$ means it is sufficient to\nconsider the case that $ac=1,cd>1$ and the case that $ac=cd=1,\nab=bd=2.$ For the first case, $[acyd]$ is an induced $4$-cycle of\n$H$; for the second case, $H$ itself is exactly $H_6$ after\nidentifying vertices of the same labels.\n\\end{proof}\n\n\n\\begin{lemma} \\label{lem10}\nSuppose that $G$ is a $5$-chordal graph which has $H_6$ as\nan induced subgraph. If $d_G(x,y)=d_G(u,v)=3$, then $G$ contains\neither $C_4$ or $H_2$ or $H_3$ as an isometric subgraph.\n\\end{lemma}\n\n\\begin{proof} We can check that the subgraph of $H_6$, and hence of $G,$ induced by $x,a,c,d,v,b$ is\nisomorphic to $H_3.$ If $d_G(b,c)=3$, Corollary \\ref{lem30}\nshows that $G$ contains $H_3$ as an isometric subgraph. Thus, in\nthe remaining discussions we will assume that\n\\begin{equation} \\label{eq2+} d_G(b,c)=2.\n\\end{equation}\n\n\n\n\n\\paragraph {\\sc Case 1:} $\\min(d_G(b,u),d_G(b,y))=2$.\n\n\nAssume, as we may, that $d_G(b,u)=2.$ Take, accordingly, $w\\in\nV(G)$ satisfying $d_G(b,w)=d_G(w,u)=1.$ As $d_{H_6}(b,u)=3,$ we see\nthat $w\\notin V(H_6)$. Observe that\n$$2=3-1=d_G(u,v)-d_G(u,w)\\leq d_G(v,w)\\leq d_G(v,b)+d_G(b,w)=2,$$\nwhich gives \\begin{equation}d_G(v,w)=2. \\label{China}\n\\end{equation}\n\n\n\n\\paragraph {\\sc Case 1.1:} $d_G(w,d)=1$.\n\n\nIn this case, it follows from Eq. \\eqref{China} that $[wbvd]$ is\nan isometric $C_4$ of $G.$\n\n\n\\paragraph {\\sc Case 1.2:} $d_G(w,d)\\geq 2.$\n\n\n\nSince $G$ is $5$-chordal, we know that the $6$-cycle $[wbvdau]$\ncannot be chordless in $G.$ By Eq. \\eqref{China} and the current\nassumption that $d_G(w,d)\\geq 2,$ we can draw the conclusion\nthat $d_G(w,a)=1 $ and hence find that the subgraph of $G$ induced\nby $w,u,a,d,v,b$ is isomorphic to $H_3. $ As we already\n assumed that $d_G(u,v)=3$, this induced $H_3$ is even an isometric subgraph of\n$G,$ taking into account Corollary \\ref{lem30}.\n\n\n\\paragraph {\\sc Case 2:} $d_G(b,u)=d_G(b,y)=3$.\n\nBy Eq. \\eqref{eq2+}, we can choose $w\\in V(G)$ such that\n$d_G(b,w)=d_G(w,c)=1.$ Since $d_{H_6}(b,c)=3,$ we know that\n$w\\notin V(H).$ In addition, we have \\begin{equation}d_G(u,w)\\geq\nd_G(u,b)-d_G(b,w)=3-1=2, \\ \\text{and}\\ \\ d_G(y,w)\\geq\nd_G(y,b)-d_G(b,w)=3-1=2. \\label{full-moon}\n\\end{equation}\nClearly, the map which swaps $u$ and $y$, $a$ and $d$ and $x$\nand $v$ is an automorphism of $H_6$ and the requirement to specify\nour Case 2 will not be affected after applying this automorphism of\n$H_6.$ Therefore, noting that $d_G(w,v), d_G(w,d), d_G(w,x),\nd_G(w,a) \\in \\{1,2\\},$ we may take advantage of this symmetry\nof $H_6$ and merely consider\n the following situations.\n\n\n\n\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.5pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(220,50)\n\n\n\n\n\n\\put(100,0){\\circle*{1}}\\put(100,-4){\\makebox(0,0)[b]{$y$}}\n\n\\put(80,20){\\circle*{1}}\\put(77,20){\\makebox(0,0)[l]{$u$}}\n\\put(120,20){\\circle*{1}}\\put(123,20){\\makebox(0,0)[r]{$v$}}\n\n\\put(100,40){\\circle*{1}}\\put(100,43){\\makebox(0,0)[t]{$x$}}\n\n\\put(90,30){\\circle*{1}}\\put(87,33){\\makebox(0,0)[l]{$a$}}\n\\put(110,30){\\circle*{1}}\\put(113,33){\\makebox(0,0)[r]{$w$}}\n\\put(90,10){\\circle*{1}}\\put(85,10){\\makebox(0,0)[l]{$c$}}\n\\put(110,10){\\circle*{1}}\\put(115,10){\\makebox(0,0)[r]{$d$}}\n\n\\qbezier(80,20)(90,10)(100,0)\\qbezier(100,0)(110,10)(120,20)\\qbezier(80,20)(90,30)(100,40)\\qbezier(100,40)(110,30)(120,20)\n\\qbezier(90,30)(90,20)(90,10)\n\\qbezier(90,30)(100,20)(110,10)\\qbezier(90,10)(100,10)(110,10)\\qbezier(90,30)(100,30)(110,30)\\qbezier(110,10)(110,20)(110,30)\\qbezier(90,10)(100,20)(110,30)\n \\end{picture}\n\\end{center}\n\n\\caption{Case 2.1 in the proof of Lemma \\ref{lem10}.}\\label{fig3}\n\\end{figure}\n\n\n\n\\paragraph {\\sc Case 2.1:} $d_G(w,v)=d_G(w,d)=d_G(w,x)=d_G(w,a)=1.$\n\nFrom Eq. \\eqref{full-moon} and our assumption it follows that\n the subgraph of $G$ induced by\n$x,a,u,c,y,d,v,w$ is isomorphic to $H_2$; see Fig. \\ref{fig3}.\nBecause $d_G(x,y)=d_G(u,v)=3$, Corollary \\ref{lem29} now tells us\nthat $G$ contains $H_2$ as an isometric subgraph.\n\n\n\n\n\\paragraph {\\sc Case 2.2:} $d_G(w,v)=d_G(w,d)=2$.\n\nIt is not difficult to check that the subgraph of $G$ induced\nby $w,b,v,d,c,y$ is isomorphic to $H_3$. The condition that\n$d_G(b,y)=3$ then enables us to appeal to Corollary \\ref{lem30} and\nconclude that $G$ contains the graph $H_3$ as an isometric subgraph.\n\n\n\\paragraph {\\sc Case 2.3:}\n $d_G(w,v)=2$ and $ d_G(w,d)=1$.\n\n\n$G$ contains the induced $4$-cycle $[wbvd]$.\n\n\\paragraph {\\sc Case 2.4:}\n $d_G(w,v)=1$ and $ d_G(w,d)=2$.\n\n $[wvdc]$ is a required induced $C_4$ of $G.$\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\\begin{lemma}\\label{lem27}\nLet $G$ be a $5$-chordal graph which has $H_5$ as an induced\nsubgraph. If $d_G(x,y)=d_G(u,v)=3$, then $G$ contains at least one\nof the subgraphs $C_4,H_2,H_3$ and $H_5$ as an isometric subgraph.\n\\end{lemma}\n\n\n\n\n\n\\begin{proof} Because $H_5$ is an induced subgraph\n of $G,$ it is clear that $d_G(b,u),d_G(b,y),d_G(c,x),d_G(c,v)\\in\n \\{2,3\\}$.\nThere are thus two cases to consider.\n\n\n\n\\paragraph {\\sc Case 1:} $\\min (d_G(b,y),d_G(b,u),d_G(c,x),d_G(c,v))=2$.\n\n\nWithout loss of generality, let us assume that $d_G(b,y)=2.$ There\nis then a vertex $w$ of $G$ such that $d_G(b,w)=d_G(w,y)=1.$\nObserve that\n\\begin{equation}d_G(w,x)\\geq d_G(x,y)-d_G(w,y)=3-1=2.\n\\label{John}\n\\end{equation}\n\n\\paragraph {\\sc Case 1.1:} $d_G(w,a)=1.$\n\nBy Eq. \\eqref{John}, $[wbxa]$ is an induced $4$-cycle of $G.$\n\n\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.5pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(70,50)\n\n\\put(10,10){\\circle*{1}}\\put(7,7){\\makebox(0,0)[l]{$y$}}\n\\put(10,20){\\circle*{1}}\\put(7,20){\\makebox(0,0)[l]{$c$}}\n\\put(20,30){\\circle*{1}}\\put(20,33){\\makebox(0,0)[t]{$a$}}\n\\put(30,20){\\circle*{1}}\\put(33,20){\\makebox(0,0)[r]{$b$}}\n\\put(20,10){\\circle*{1}}\\put(20,7){\\makebox(0,0)[r]{$d$}}\n\\put(10,30){\\circle*{1}}\\put(7,33){\\makebox(0,0)[l]{$u$}}\n\\put(30,10){\\circle*{1}}\\put(33,7){\\makebox(0,0)[r]{$v$}}\n\\put(30,30){\\circle*{1}}\\put(33,33){\\makebox(0,0)[r]{$x$}}\n\\put(20,0){\\circle*{1}}\\put(20,-3){\\makebox(0,0)[t]{$w$}}\n\n\\qbezier(10,30)(10,20)(10,10)\\qbezier(10,30)(20,30)(30,30)\\qbezier(30,30)(30,20)(30,10)\n\\qbezier(10,10)(20,10)(30,10)\\qbezier(20,10)(20,20)(20,30)\\qbezier(20,0)(15,5)(10,10)\\qbezier(20,0)(20,5)(20,10)\\qbezier(20,0)(50,10)(30,20)\n \\end{picture}\n\\end{center}\n\\caption{Case 1.2 in the proof of Lemma \\ref{lem27}.}\\label{fig2.2}\n\\end{figure}\n\n\n\\paragraph {\\sc Case 1.2:} $d_G(w,a)>1.$\n\nConsider the $6$-cycle $[bwydax].$ As $G$ is $5$-chordal, this\ncycle has a chord in $G$. According to Eq. \\eqref{John} and our\nassumption that $d_G(w,a)>1$, the only possibility is that such a\nchord connects $w$ and $d.$ We now examine the subgraph of $G$\ninduced by $w,b,x,a,d,y$ and realize that it is isomorphic with\n$H_3$; see Fig. \\ref{fig2.2}. Armed with Corollary \\ref{lem30}, our\nassumption that $d_G(x,y)=3$ shows that this $H_3$ is even an\nisometric subgraph of $G,$ as wanted.\n\n\n\n\n\n\n\n\n\n\\paragraph {\\sc Case 2:} $d_G(b,u)=d_G(b,y)=d_G(c,x)=d_G(c,v)=3$.\n\n\n\n\n\nBy Corollary \\ref{lem31}, $H_5$ is an isometric subgraph of $G$\nprovided $d_G(b,c)=4$. Thus, we shall\n restrict our attention to the cases that $ d_G(b,c)\\in \\{2, 3\\}.$\n\n\n\\paragraph{\\sc Case 2.1:} $d_G(b,c)=2$.\n\nPick a $w\\in V(G)$ such that $d_G(b,w)=d_G(w,c)=1.$ It is clear\nthat $[bwcydv]$ is a $6$-cycle in the $5$-chordal graph $G$ and\nhence must have a chord. We contend that this chord can be nothing\nbut\n $\\{w, d\\}.$ To see this, one simply needs to notice the following:\n\\begin{equation*}\\left\\{ \\begin{array}{l }\n d_G(w,y)\\geq d_G(b,y)-d_G(b,w)=3-1=2;\\\\\nd_G(w,v)\\geq d_G(c,v)- d_G(c,w)=3-1=2.\n\\end{array}\\right.\n\\end{equation*}\n From the structure of the subgraph of $G$ induced by $b,w,c,y,d,v$ we deduce that both\n $[cwdy]$ and $[bwdv]$ are isometric $4$-cycles in $G$,\n establishing our claim in this case.\n\n\n\\paragraph{\\sc Case 2.2:} $d_G(b,c)=3$.\n\n\n\nWe choose $p,q\\in V(G)$ such that $d_G(b,p)=d_G(p,q)=d_G(q,c)=1$. We\nfirst note that\n$$d_G(q,v)\\geq d_G(c,v)-d_G(c,q)=3-1=2.$$\nDue to the symmetry of $H_5$, it is manifest then that\n\\begin{equation} \\label{Texas} \\min (d_G(q,v),d_G(p,y),d_G(q,x),d_G(p, u))\\geq 2.\n\\frac{}{}\\end{equation}\n We also have $ d_G(q,y)\\in \\{1,2\\}$ as it holds\n$$2=1+1=d_G(q,c)+d_G(c,y) \\geq d_G(q,y)\\geq\nd_G(b,y)-d_G(b,q)=3-2=1.$$ Arguing by analogy, we indeed have\n\\begin{equation} \\label{gang}\n d_G(q,y),d_G(q,u),d_G(p,v),d_G(p,x)\\in \\{1,2\\}.\n\\end{equation}\nEq. \\eqref{Texas} along with Eq. \\eqref{gang} shows that\n\\begin{equation}\\{p,q\\}\\cap V(H_5)=\\emptyset .\n\\label{xuhui}\n\\end{equation}\n\n\n\nAssume, as we may, that\n\\begin{equation}d_G(q,y)+d_G(p,v)\\geq d_G(q,u)+d_G(p,x).\n\\label{eqn3}\n\\end{equation}\n\n\n\n\\paragraph{\\sc Case 2.2.1:} $d_G(q,y)=d_G(p,v)=2$.\n\nWe start with two observations: Thanks to Eq. \\eqref{Texas}, we\nhave $d_G(p,y)\\geq 2, d_G(q,v)\\geq 2$ while as $b,p,q,c$ is a\ngeodesic, we obtain $d_G(p,c)=d_G(q,b)=2.$ Now, let us take a look\nat the $7$-cycle $[bpqcydv]$ of the $5$-chordal graph\n $G$. The cycle must have a chord, which, according to our previous observations and our assumption that $d_G(q,y)=d_G(p,v)=2$,\n can only be the one connecting $d$ to $p$ or to $q.$ Without loss of\ngenerality, let $d_G(q,d)=1$. Then, we can find a $4$-cycle\n$[cqdy]$, as desired.\n\n\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.5pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(40,30)\n\n\\put(10,10){\\circle*{1}}\\put(7,10){\\makebox(0,0)[l]{$y$}}\n\\put(10,20){\\circle*{1}}\\put(7,20){\\makebox(0,0)[l]{$c$}}\n\\put(20,30){\\circle*{1}}\\put(20,33){\\makebox(0,0)[t]{$a$}}\n\\put(30,20){\\circle*{1}}\\put(33,20){\\makebox(0,0)[r]{$b$}}\n\\put(20,10){\\circle*{1}}\\put(20,7){\\makebox(0,0)[r]{$d$}}\n\\put(10,30){\\circle*{1}}\\put(7,33){\\makebox(0,0)[l]{$u$}}\n\\put(30,10){\\circle*{1}}\\put(33,10){\\makebox(0,0)[r]{$v$}}\n\\put(30,30){\\circle*{1}}\\put(33,33){\\makebox(0,0)[r]{$x$}}\n\\put(10,0){\\circle*{1}}\\put(10,-3){\\makebox(0,0)[t]{$q$}}\n\\put(30,0){\\circle*{1}}\\put(30,-3){\\makebox(0,0)[t]{$p$}}\n\\qbezier(10,30)(10,20)(10,10)\\qbezier(10,30)(20,30)(30,30)\\qbezier(30,30)(30,20)(30,0)\n\\qbezier(10,10)(20,10)(30,10)\\qbezier(20,10)(20,20)(20,30)\\qbezier(10,0)(20,0)(30,0)\\qbezier(10,0)(0,10)(10,20)\n\\qbezier(30,0)(25,5)(20,10)\\qbezier(30,0)(40,10)(30,20)\n \\end{picture}\n\\end{center}\n\\caption{Case 2.2.2 in the proof of Lemma\n\\ref{lem27}.}\\label{fig1.2.2}\n\\end{figure}\n\n\\paragraph{\\sc Case 2.2.2:} $d_G(q,y)=2, d_G(p,v)=1$.\n\n\nIn this case, the $5$-chordal graph $G$ possesses the $6$-cycle\n $[pqcydv]$, which must have a chord. We already assume that $d_G(q,y)=2;$ as $b,p,q,c$ is a geodesic, we get $d_G(p,c)=2;$ finally, we have\n\\begin{equation*}\\left\\{ \\begin{array}{l }\n d_G(p,y)\\geq d_G(b,y)-d_G(b,p)=3-1=2;\\\\\nd_G(q,v)\\geq d_G(c,v)- d_G(c,q)=3-1=2.\n\\end{array}\\right.\n\\end{equation*}\nConsequently, it happens either $d_G(q,d)=1$ or $d_G(p,d)=1$.\n If $d_G(q,d)=1$, we will come to an isometric $4$-cycle $[cqdy]$. When $d_G(p,d)=1$ and $d_G(q,d)\\geq 2$, the subgraph of $G$ induced by\n $p,q,c,y,d,v$ is isomorphic to $H_3$ as shown by Fig. \\ref{fig1.2.2}, which is even an isomeric\n subgraph in view of Corollary \\ref{lem30} as well as the assumption that\n $d_G(c,v)=3.$\n\n\n\n\n\n\n\n\n\\paragraph{\\sc Case 2.2.3} $d_G(q,y)=1, d_G(p,v)=2$.\n\n\n\n This case can be disposed of as Case 2.2.2.\n\n\\paragraph{\\sc Case 2.2.4:} $d_G(q,y)=d_G(p,v)=1.$\n\n\n\n\n\nBy Eqs. \\eqref{gang} and \\eqref{eqn3}, we obtain\n$d_G(q,u)=d_G(p,x)=1.$ Noting Eq. \\eqref{xuhui}, we further get\n\\begin{equation}\\left\\{ \\begin{array}{l }\n 1\\leq d_G(p,d)\\leq d_G(p,v)+d_G(v,d)=1+1=2;\\\\\n1\\leq d_G(q,d)\\leq d_G(q,y)+d_G(y,d)=1+1=2;\\\\\n1\\leq d_G(p,a)\\leq d_G(p,x)+d_G(x,a)=1+1=2;\\\\\n1\\leq d_G(q,a)\\leq d_G(q,u)+d_G(u,a)=1+1=2.\\\\\n\\end{array}\\right.\n\\label{picb}\n\\end{equation}\nBecause of Eq. \\eqref{picb}, it is only necessary to consider the\nfollowing three cases, since all others would follow by symmetry.\n\n\n\\paragraph{\\sc Case 2.2.4.1:} $d_G(p,d)=d_G(q,d)=2$.\n\n\n\n\n\n\n\nThe subgraph of $G$ induced by $c,y,q,p,v,d$ is isomorphic to\n$H_3$; see Fig. \\ref{fig1.2.4.1}. But $d_G(c,v)=3$ is among the\nstanding assumptions for Case 2 and hence Corollary\n \\ref{lem30} demonstrates that $G$ has this $H_3$ as an\n isometric subgraph.\n\n\n\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.5pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(40,30)\n\n\\put(10,10){\\circle*{1}}\\put(7,10){\\makebox(0,0)[l]{$y$}}\n\\put(10,20){\\circle*{1}}\\put(7,20){\\makebox(0,0)[l]{$c$}}\n\\put(20,30){\\circle*{1}}\\put(20,33){\\makebox(0,0)[t]{$a$}}\n\\put(30,20){\\circle*{1}}\\put(33,20){\\makebox(0,0)[r]{$b$}}\n\\put(20,10){\\circle*{1}}\\put(20,7){\\makebox(0,0)[r]{$d$}}\n\\put(10,30){\\circle*{1}}\\put(7,33){\\makebox(0,0)[l]{$u$}}\n\\put(30,10){\\circle*{1}}\\put(33,10){\\makebox(0,0)[r]{$v$}}\n\\put(30,30){\\circle*{1}}\\put(33,33){\\makebox(0,0)[r]{$x$}}\n\\put(10,0){\\circle*{1}}\\put(10,-3){\\makebox(0,0)[t]{$q$}}\n\\put(30,0){\\circle*{1}}\\put(30,-3){\\makebox(0,0)[t]{$p$}}\n\\qbezier(10,30)(10,20)(10,0)\\qbezier(10,30)(20,30)(30,30)\\qbezier(30,30)(30,20)(30,0)\n\\qbezier(10,10)(20,10)(30,10)\\qbezier(20,10)(20,20)(20,30)\\qbezier(10,0)(20,0)(30,0)\\qbezier(10,0)(-10,15)(10,30)\\qbezier(10,0)(0,10)(10,20)\n\\qbezier(30,0)(50,15)(30,30)\\qbezier(30,0)(40,10)(30,20)\n \\end{picture}\n\n\\end{center}\n\\caption{Case 2.2.4.1 in the proof of Lemma\n\\ref{lem27}.}\\label{fig1.2.4.1}\n\\end{figure}\n\n\n \\paragraph{\\sc Case 2.2.4.2:}\n$\\{d_G(p,d),d_G(q,d)\\}=\\{ 1, 2\\}$.\n\nThere is no loss of generality in assuming that $d_G(p,d)=1$ and\n$d_G(q,d)=2.$ In such a situation, by recalling from Eq.\n\\eqref{Texas}\n that $d_G(p,y)\\geq 2$, we find that $[pdyq]$ is an induced\n$4$-cycle of $G$, as wanted.\n\n\n\n\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.5pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(50,50)\n\n\n\n\n\n\n\\put(20,0){\\circle*{1}}\\put(20,-4){\\makebox(0,0)[b]{$y$}}\n\n\\put(0,20){\\circle*{1}}\\put(-3,20){\\makebox(0,0)[l]{$u$}}\n\\put(40,20){\\circle*{1}}\\put(43,20){\\makebox(0,0)[r]{$v$}}\n\n\\put(20,40){\\circle*{1}}\\put(20,43){\\makebox(0,0)[t]{$x$}}\n\n\\put(10,30){\\circle*{1}}\\put(7,33){\\makebox(0,0)[l]{$a$}}\n\\put(30,30){\\circle*{1}}\\put(33,33){\\makebox(0,0)[r]{$p$}}\n\\put(10,10){\\circle*{1}}\\put(7,7){\\makebox(0,0)[l]{$q$}}\n\\put(30,10){\\circle*{1}}\\put(33,7){\\makebox(0,0)[r]{$d$}}\n\n\n\\qbezier(0,20)(10,10)(20,0)\\qbezier(20,0)(30,10)(40,20)\\qbezier(40,20)(30,30)(20,40)\\qbezier(20,40)(10,30)(0,20)\n\\qbezier(10,30)(20,20)(30,10)\\qbezier(10,10)(20,20)(30,30)\\qbezier(30,30)(30,20)(30,10)\\qbezier(30,10)(20,10)(10,10)\n\\qbezier(10,10)(10,20)(10,30)\\qbezier(10,30)(20,30)(30,30)\n\n\n\n \\end{picture}\n\n\\end{center}\n\\caption{Case 2.2.4.3 in the proof of Lemma\n\\ref{lem27}.}\\label{fig1.2.4.3}\n\\end{figure}\n\n \\paragraph{\\sc Case 2.2.4.3:} $d_G(p,d)=d_G(q,d)=d_G(p,a)=d_G(q,a)=1$.\n\n\n\n\n\nAfter checking all those existing assumptions on pairs of adjacent\nvertices as well as the fact that $\\{q,v\\},\\{ p,y\\}\\notin E(G)$ as\nguaranteed by Eq. \\eqref{Texas}, we are led to the conclusion that\nthe subgraph of $G$ induced by $x,a,u,q,y,d,v,p$ is just $H_2$; see\nFig. \\ref{fig1.2.4.3}. Noting further our governing assumption in\nthe lemma that\n $d_G(x,y)=d_G(u,v)=3$, Corollary \\ref{lem29} then enables us reach the conclusion that this $H_2$ is indeed an isometric subgraph\n of $G,$ as was to be shown.\n\\end{proof}\n\n\n\n\n\n\n\n\nThe next simple result resembles \\cite[Lemma 2.2]{BKM01} closely.\n\n\n\n\n\\begin{lemma} \\label{lem2.1} Let $G$ be a $k$-chordal graph and let $C=[x_1\\cdots\nx_kx_{k+1}\\cdots x_{k+t}]$ be a cycle of $G$.\n If no chord of $C$ has an endpoint in $\\{ x_2,\\cdots,x_{k-1}\\}$, then $x_1x_{k}=1.$\n \\end{lemma}\n\n\n\\begin{proof} We consider the induced subgraph\n$H=G[x_1,x_{k}, x_{k+1},\\ldots ,x_{k+t}]$. There must exist a\nshortest path in $H$ connecting $x_1$ and $x_{k}$, say $P$. If the\nlength of $P$ is greater than $1$, then we walk along $P$ from\n$x_{k}$ to $x_1$ and then continue with $x_2,x_3,\\ldots,$ and\nfinally get back to $x_{k}$, creating a chordless cycle of length at\nleast $k+1,$ which is absurd as $G$ is $k$-chordal. This proves\nthat $x_1x_k=1,$ as desired.\n\\end{proof}\n\n\n\n\n\n\nLet $G$ be a graph. When studying $\\delta _G(x,y,u,v)$ for some\nvertices $x,y,u,v$ of $G,$ it is natural to look at a {\\em geodesic\nquadrangle} $\\mathcal {Q}(x,u,y,v)$ with {\\em corners} $x,u,y$\nand $v$, which is just the subgraph of $G$ induced by the union\nof all those vertices on four geodesics connecting $x$ and $u,$ $u$\nand $y$, $y$ and $v$, and $v$ and $x$, respectively. Let us fix\nsome notation to be used throughout the paper.\n\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.4pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(60,60)\n\n\\put(20,0){\\circle*{1}}\\put(20,-4){\\makebox(0,0)[b]{$y$}}\n\\put(0,20){\\circle*{1}}\\put(-3,20){\\makebox(0,0)[l]{$u$}}\n\\put(40,20){\\circle*{1}}\\put(42,20){\\makebox(0,0)[l]{$v$}}\n\\put(20,40){\\circle*{1}}\\put(20,43){\\makebox(0,0)[t]{$x$}}\n\n\\put(5,25){\\circle*{1}}\\put(-5,25){\\makebox(0,0)[l]{$a_{xu-1}$}}\n\n\\put(15,35){\\circle*{1}}\\put(10,35){\\makebox(0,0)[l]{$a_{1}$}}\n\\put(5,15){\\circle*{1}}\\put(-5,15){\\makebox(0,0)[l]{$c_{yu-1}$}}\n\n\\put(15,5){\\circle*{1}}\\put(10,5){\\makebox(0,0)[l]{$c_{1}$}}\n\\put(25,5){\\circle*{1}}\\put(30,5){\\makebox(0,0)[r]{$d_{1}$}}\n\n\\put(35,15){\\circle*{1}}\\put(45,15){\\makebox(0,0)[r]{$d_{yv-1}$}}\n\n\\put(25,35){\\circle*{1}}\\put(30,35){\\makebox(0,0)[r]{$b_{1}$}}\n\n\\put(35,25){\\circle*{1}}\\put(45,25){\\makebox(0,0)[r]{$b_{xv-1}$}}\n\n\n\\qbezier[7](5,15)(10,10)(15,5)\\qbezier(0,20)(2,22)(5,25)\\qbezier(0,20)(2,18)(5,15)\n\\qbezier(15,5)(18,2)(20,0)\\qbezier(20,0)(22,2)(25,5)\\qbezier(35,15)(38,18)(40,20)\n\\qbezier[7](25,5)(30,10)(35,15)\\qbezier(15,35)(18,38)(20,40)\\qbezier(20,40)(22,38)(25,35)\n\\qbezier(35,25)(38,22)(40,20)\\qbezier[7](25,35)(30,30)(35,25)\\qbezier[7](5,25)(10,30)(15,35)\n\n \\end{picture}\\end{center}\n\\caption{The geodesic quadrangle $\\mathcal\n{Q}(x,u,y,v)$.}\\label{fig1}\n\\end{figure}\n\n\n\n\\paragraph{\\sc \\bf Assumption I:}\n Let us assume that $x,u,y,v$ are four different vertices of a graph $G$ and\n the four geodesics corresponding to the geodesic quadrangle $\\mathcal\n {Q}(x,u,y,v)$ are\n\\begin{equation*}\\left\\{ \\begin{array}{l }\n P_{a}: x=a_0,a_1,\\ldots,a_{xu}=u;\\\\\n P_b: x=b_0,b_1,\\ldots,b_{xv}=v;\\\\\n P_c: y=c_0,c_1,\\ldots,c_{yu}=u;\\\\\nP_d: y=d_0,d_1,\\ldots,d_{yv}=v.\n\\end{array}\\right.\n\\end{equation*}\nWe call $P_a$, $P_b, P_c$ and $P_d$ the four {\\em sides} of\n$\\mathcal {Q}(x,u,y,v)$ and often just refer to them as vertex\nsubsets of $V(G)$ rather than vertex sequences. We write $\\mathcal\n{P}(x,u,y,v)$ for the pseudo-cycle $$[x, a_1,\\ldots , a_{xu-1}, u,\nc_{yu-1},\\ldots,c_1,y,d_1,\\ldots,d_{yv-1},v,b_{xv-1},\\ldots, b_1].$$\nNote that $\\mathcal {P}(x,u,y,v)$ is not necessarily a cycle as the\nvertices appearing in the sequence may not be all different. Let us\nsay that $x$ is {\\em opposite} to $P_c$ and $P_d$, say that $x$ and\n$y$ are {\\em opposite corners}, say that $x$ and $v$ are {\\em\nadjacent corners}, say that $x$ is the {\\em common peak } of $P_a$\nand $P_b$, say that $P_a$ and $P_b$ are {\\em adjacent} to each\nother, say that $P_a$ and $P_d$ are {\\em opposite} to each other,\n and say that\n those vertices inside $P_a\\setminus \\{x,u\\}$ are {\\em ordinary vertices} of\n $P_a,$ etc.. An edge of $\\mathcal\n {Q}(x,u,y,v)$ which intersects with two adjacent sides but do not lie in any single side is called an $\\mathbb{A}$-edge and an edge of\n $\\mathcal\n {Q}(x,u,y,v)$ which intersect with two opposite sides but does not lie in any single side is called an\n $\\mathbb{H}$-edge.\n\n\n Suppose that $a_i=v_0,v_1,\\ldots, v_{a_id_j}=d_j $ is a geodesic\n connecting $a_i$ and $d_j$ in $G.$ We call the two\n walks $$x,a_1,\\ldots ,a_i,v_1,\\ldots, v_{a_id_j-1}, d_j,d_{j-1},\\ldots ,d_1,y$$\n and\n $$u,a_{xu-1},\\ldots,a_i,v_1,\\ldots, v_{a_id_j-1}, d_j,d_{j+1},\\ldots,d_{yv-1},v$$\n {\\em $\\mathcal\n {Z}$-walks of $\\mathcal\n {Q}(x,u,y,v)$ through $\\{a_i,d_j\\}$} or just {\\em $\\mathcal\n {Z}$-walks of $\\mathcal\n {Q}(x,u,y,v)$ between $P_a$ and $P_d.$} In an apparent way, we\n define similar concepts for {\\em $\\mathcal\n {Z}$-walks of $\\mathcal\n {Q}(x,u,y,v)$ between $P_b$ and $ P_c$.}\n\n\n\n\n\n\n\n\\rz\n\n\n\n\\begin{lemma} \\label{lem11} Let $G$ be a graph and let $\\mathcal\n {Q}(x,u,y,v)$ be one of its geodesic quadrangles for which Assumption I holds.\n Suppose any two adjacent sides of $\\mathcal\n {Q}(x,u,y,v)$ has only one common vertex and that vertex is their common peak.\nThen $\\mathcal\n {Q}(x,u,y,v)$ contains a cycle on which $b_1,x, a_1$ appear in that\norder consecutively. Moreover, if $\\min(d(P_a,P_d),d(P_b,\nP_c))\\geq t$ for some $t$, then we may even require that the length\nof the cycle is no shorter than $4t$.\n\\end{lemma}\n\n\n\\begin{proof}\nIf $\\min(d(P_a,P_d),d(P_b, P_c))\\geq t$ for $t>0$, then $\\mathcal\n {P}(x,u,y,v)$\nitself gives rise to a required cycle. Otherwise, without loss of\ngenerality, assume that $d(P_a, P_d)=0.$ Take the minimum $i$ such\nthat $d(a_i, P_d)=0.$ There is a unique $j>0$ such that $a_i=d_j$.\nIt is plain that $i>0$ and $j0.$ Suppose otherwise, it then follows that\n$b_1,b_2,\\ldots, b_q=a_p, a_{p+1},\\ldots,a_{xu}=u$ is a path\nconnecting $b_1$ and $u$ and so $b_1u0$ and $a_ib_j=1$\nthen\n $i=j$. In the case of $i>j,$ $b_1,b_2,\\ldots, b_j,\na_{i},a_{i+1},\\ldots,a_{xu}=u$ is a path connecting $b_1$ and $u$ of\nlength smaller than $xu ,$ contrary to Lemma \\ref{koolen1}.\nSimilarly, $i0$ and let $\\mathcal\n {Q}(x,u,y,v)$ be a geodesic quadrangle for which Assumptions I and II\n hold. Then $\\mathcal\n {P}(x,u,y,v)$ is a cycle. Moreover, if $\\delta^*(G)>\\frac{1}{2}$,\n then all chords of $\\mathcal\n {P}(x,u,y,v)$ must be either $\\mathbb{A}$-edges or $\\mathbb{H}$-edges.\n\\end{corollary}\n\\begin{proof} This follows from Lemma \\ref{lem14}, Lemma \\ref{lem2.4} (i)\nand Corollary \\ref{cor2.1} in a straightforward fashion.\n\\end{proof}\n\n\nThe next result is very essential to our proof of Theorem\n\\ref{main1} and both its statement and its proof have their origin\nin \\cite[p. 65, Prop. 3.1]{BKM01} and \\cite[p. 691, Claim 2]{KM02}.\n\n\n\n\n\\begin{lemma}\nSuppose that $G$ is a graph for which Assumptions I and II are met\nand $\\mathcal\n {Q}(x,u,y,v)$ has at least one\n$\\mathbb{A}$-edge. Then we have $xu+yv=xv+yu$.\\label{cor12}\n\\end{lemma}\n\n\\begin{proof}\n If the claim were false, without loss of generality, we suppose that\n\\begin{equation}xu+yv>xv+yu.\n\\label{eq5}\n\\end{equation}\nBy symmetry and because of Lemma \\ref{lem2.4} (iii), let us\nwork under the assumption that $a_ib_i=1$. It clearly holds\n\\begin{equation}a_i\\not= x. \\label{clear}\n\\end{equation}\nBefore moving on, let us prove that\n\\begin{equation} a_i\\not= u.\\label{SJTU}\n\\end{equation}\nSuppose for a contradiction that $a_i=u$, we find that $$\n\\begin{array}{cll} yv & \\geq &\nyu+xv-xu+1\\ \\ \\ \\ \\ \\ \\ \\text{(By Eq. \\eqref{eq5})}\\\\ & = &\nyu+xv-xa_i+1\\\\\n& = & yu+xv-i+1\\\\\n& = & yu+b_iv+1.\n\\end{array}\n$$\n But we surely have\n$yv\\leq yu+ub_i+b_iv=yu+b_iv+1$ and so we conclude that we can get\na geodesic $P$ connecting $y$ to $v$ in $G$ by first walking\nalong $P_c$ to go from $y$ to $u$, then moving from $u$ to $b_i$ in\none step and finally traversing along $P_b$ from $b_i$ to $v$.\nSince this geodesic passes through $u$ in the middle, we obtain a\ncontradiction to Lemma \\ref{lem2.4} (ii) and hence establish Eq.\n\\eqref{SJTU}.\n\nTo go one step further, let us check the following:\n\\begin{equation}\\begin{array}{cll} xv+1 & = & xb_i+b_iv+1= xb_i+b_iv+a_ib_i\\\\ & \\geq &\nxb_i+a_iv=xa_i+a_iv \\\\\n& = &xa_1+a_1a_i+a_iv \\ \\ \\ \\ \\ \\ \\ \\text{(By Eq. \\eqref{clear})}\\\\\n& = & 1+a_1a_i+a_iv\\geq\n 1+a_1v\\\\ & \\geq & 1+xv. \\ \\ \\ \\ \\ \\ \\ \\text{(By Lemma\n\\ref{koolen1})}\n\\end{array} \\label{autumn}\\end{equation}\nClearly, equalities hold throughout Eq. \\eqref{autumn}.\n In particular, we have\n\\begin{equation}b_iv+1=a_iv.\n\\label{eqn12}\n\\end{equation}\n From Eq. \\eqref{eqn12} and $xv=b_iv+i$ we deduce that\n\\begin{equation}\\label{eq8} a_iv=xv-(i-1).\\end{equation}\n Here comes the punch line of the proof:\n\\begin{equation}\\begin{array}{cll} a_iy+uv & \\geq &\n(xy-xa_i)+uv\\\\& = & (xy-i)+uv\\\\ & = & \\max ( xu+yv,xv+yu )\n+2\\delta^*(G)-i \\ \\ \\ \\ \\ \\ \\text{(By Eq. \\eqref{key})}\n\\\\ & = &\n\\max ( (xu-i)+yv, xv+yu-(i-1) ) +2\\delta^*(G) \\ \\ \\ \\ \\ \\\n\\text{(By Eq. \\eqref{eq5})}\n\\\\ & = &\n\\max\n ((xu-i)+yv,a_iv+yu) +2\\delta^*(G) \\ \\ \\ \\ \\ \\\n\\text{(By Eq. \\eqref{eq8})}\n\\\\ & = & \\max\n (a_iu+yv,a_iv+yu) +2\\delta^*(G).\n\\end{array} \\label{eq6}\\end{equation}\nAccording to Eqs. \\eqref{clear} and \\eqref{SJTU}, we can apply\nLemma \\ref{lem2.4} (i) to find that $a_i,y,u,v$ are four different\nvertices. We further conclude from the definition of $\\delta^*(G)$\nthat Eq. \\eqref{eq6} should hold equalities throughout, hence that\n$xy+uv\\leq a_iy+uv $ as a result of the minimality of $xy+uv$\n as indicated in our Assumption II, and finally that the first inequality in Eq.\n\\eqref{eq6} must be strict in light of Eq. \\eqref{clear}, getting\na contradiction with the assertion that all equalities in Eq.\n\\eqref{eq6} hold. This is the end of the proof.\n\\end{proof}\n\n\n\\begin{lemma} \\label{lem15} Let $G$ be a graph for which Assumptions I and II\nare required. Suppose that $\\mathcal\n {Q}(x,u,y,v)$ has an $\\mathbb{A}$-edge. If there is $1\\leq i\\leq xu-1$ and\n $0\\leq j\\leq yv$ such that $a_id_j\\leq 1$, then $a_id_j= 1$,\n $a_iu+d_jy=yu$ and $a_ix+d_jv=xv.$\n\\end{lemma}\n\n\n\\begin{proof} We start with an easy observation:\n\\begin{equation}\\label{EWN} \\left\\{ \\begin{array}{l }\n a_iu+d_jy=a_{xu-1}a_i+1+d_jy\\geq a_{xu-1}a_i+a_id_j+d_jy\\geq a_{xu-1}y,\\\\\n a_ix+d_jv=a_{1}a_i+1+d_jv\\geq a_{1}a_i+a_id_j+d_jv\\geq a_{1}v.\n\\end{array}\\right.\n\\end{equation}\n In view of Lemma \\ref{koolen1}, this says that\n\\begin{equation}a_iu+d_jy\\geq uy, \\ \\ a_ix+d_jv\\geq xv.\\label{eq13}\n\\end{equation}\n Adding together the two inequalities in Eq. \\eqref{eq13}, we obtain\n\\begin{equation}\\label{eqn14}xu+yv\\geq xv+yu.\n\\end{equation} But, it follows from\n Lemma \\ref{cor12} and the\nexistence of an $\\mathbb{A}$-edge of $\\mathcal\n {Q}(x,u,y,v)$ that the equality in Eq. \\eqref{eqn14} must occur.\n Consequently, none of the inequalities in Eqs. \\eqref{EWN} and \\eqref{eq13} can be strict, which\n is\n exactly what we want to prove.\n\\end{proof}\n\n\n\n\n\n With a little bit of luck, the forthcoming lemma contributes the number $ \\frac{ \\lfloor \\frac{k}{2}\\rfloor}{2}$, which\n is just the mysterious one we find\nin Theorem\n \\ref{main}. Note that $ \\frac{ \\lfloor\n \\frac{k}{2}\\rfloor}{2}$ is the smallest half integer that is\n greater than $\\frac{k-2}{4}$.\n\n\\begin{lemma} \\label{lemma54} Let $G$ be a $k$-chordal graph for some $k\\geq 4$ and let $\\mathcal\n {Q}(x,u,y,v)$ be a geodesic quadrangle for which Assumptions I and II hold.\n Then we have $\\delta^*(G)\\leq \\frac{ \\lfloor \\frac{k}{2}\\rfloor}{2}$ provided\n\\begin{equation} \\min(d(P_a,P_d),d(P_b,P_c))> 1.\\label{eq00}\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof} Take $i,j,m,\\ell$ as specified in Lemma \\ref{lem49} and follow all the convention made\nin the statement and the proof of Lemma \\ref{lem49}.\n Surely, the result is a direct consequence of Lemma \\ref{lem49} when\n \\begin{equation}\\min (\\pi (a), \\pi (b), \\pi (c), \\pi (d))\\leq \\frac{ \\lfloor \\frac{k}{2}\\rfloor}{2}.\n \\label{xiang}\\end{equation}\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.4pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(60,60)\n\n\\put(20,0){\\circle*{1}}\\put(20,-4){\\makebox(0,0)[b]{$y$}}\n\\put(0,20){\\circle*{1}}\\put(-3,20){\\makebox(0,0)[l]{$u$}}\n\\put(40,20){\\circle*{1}}\\put(42,20){\\makebox(0,0)[l]{$v$}}\n\\put(20,40){\\circle*{1}}\\put(20,43){\\makebox(0,0)[t]{$x$}}\n\n\\put(7,27){\\circle*{1}}\\put(2,27){\\makebox(0,0)[l]{$a_{m}$}}\n\n\n\\put(15,35){\\circle*{1}}\\put(10,35){\\makebox(0,0)[l]{$a_{j}$}}\n\n\\put(7,13){\\circle*{1}}\\put(-10,13){\\makebox(0,0)[l]{$c_{yu-xu+m}$}}\n\n\n\\put(15,5){\\circle*{1}}\\put(10,5){\\makebox(0,0)[l]{$c_{\\ell}$}}\n\\put(25,5){\\circle*{1}}\\put(30,5){\\makebox(0,0)[r]{$d_{\\ell}$}}\n\n\n\\put(33,13){\\circle*{1}}\\put(35,13){\\makebox(0,0)[l]{$d_{yv-xv+i}$}}\n\n\n\\put(25,35){\\circle*{1}}\\put(30,35){\\makebox(0,0)[r]{$b_{j}$}}\n\n\n\\put(33,27){\\circle*{1}}\\put(38,27){\\makebox(0,0)[r]{$b_{i}$}}\n\n\n\n\\qbezier(15,35)(15,35)(25,35)\\qbezier(7,27)(7,13)(7,13)\\qbezier(7,27)(7,27)(9,29)\n\\qbezier[9](9,29)(9,29)(13,33)\\qbezier(13,33)(13,33)(15,35)\\qbezier(7,13)(7,13)(9,11)\n\\qbezier(25,5)(25,5)(27,7)\\qbezier[9](27,7)(27,7)(31,11)\\qbezier[7](0,20)(5,15)(7,13)\\qbezier(33,13)(33,13)(31,11)\n\\qbezier[7](0,20)(2,22)(7,27)\\qbezier(15,5)(15,5)(13,7)\\qbezier[9](13,7)(9,11)(9,11)\\qbezier(15,5)(15,5)(25,5)\n\\qbezier[7](15,5)(18,2)(20,0)\\qbezier[7](20,0)(22,2)(25,5)\\qbezier[7](33,13)(38,18)(40,20)\n\\qbezier(25,5)(25,5)(27,7)\\qbezier[9](27,7)(27,7)(31,11)\\qbezier[7](15,35)(18,38)(20,40)\\qbezier[7](20,40)(22,38)(25,35)\n\\qbezier[7](33,27)(38,22)(40,20)\\qbezier[9](27,33)(27,33)(31,29)\\qbezier(25,35)(25,35)(27,33)\\qbezier(33,27)(33,27)(31,29)\n\n\\qbezier(33,27)(33,20)(33,13)\n\n \\end{picture}\\end{center}\n\\caption{A chordless cycle in $\\mathcal\n{Q}(x,u,y,v)$.}\\label{figlem48}\n\\end{figure}\n\nSuppose, for a contradiction, that the inequality \\eqref{xiang}\ndoes not hold. In this event, as $\\frac{ \\lfloor\n\\frac{k}{2}\\rfloor}{2}\\geq 1$, we know that $\\min ( i-j,\n(yv-xv+i)-\\ell,\n (yu-xu+m)-\\ell, m-j) \\geq \\min (\\pi(a),\\pi(b),\\pi(c),\\pi(d))-1> 0$. By virtue of Lemma \\ref{lem2.4} (i) and Eq. \\eqref{eq00},\n this implies that\n\\begin{equation}C=[a_jb_jb_{j+1}\\cdots b_id_{yv-xv+i}\\cdots\nd_{\\ell-1}d_{\\ell}c_{\\ell}c_{\\ell+1}\\cdots\nc_{yu-xu+m}a_ma_{m-1}\\cdots a_{j+1}] \\label{cycle}\\end{equation} is\na cycle, where the redundant $a_j$ should be deleted from the above\nnotation when $a_j=b_j=x$, the redundant $b_i$ should be deleted\nfrom the above notation when $b_i=d_{yv-xv+i}=v$, etc.; see Fig.\n \\ref{figlem48}. Moreover, by Lemma \\ref{lem2.4} (iii), Eq. \\eqref{eq00}\nand the choice of $i,j,\\ell, m,$ we know that $C$ is even a\nchordless cycle. But the length of $C$ is just $\\pi (a)+\\pi (b)+\\pi\n(c)+\\pi (d)$, which, as the assumption is that \\eqref{xiang} is\nviolated, is no smaller than $4(\\frac{1}{2}+ \\frac{ \\lfloor\n\\frac{k}{2}\\rfloor}{2})$ and hence is at least $k+1.$ This\ncontradicts the assumption that $G$ is $k$-chordal, finishing the\nproof.\n\\end{proof}\n\n\n\\begin{lemma} Let $G$ be a $5$-chordal graph and\nwe demand that\n Assumptions I and II hold.\nTake $i,j,\\ell,m$ to be the numbers as specified in Lemma\n\\ref{lem49}. Suppose that $\\mathcal\n {Q}(x,u,y,v)$ has no $\\mathbb{H}$-edges and $\\min(xu,xv,$ $yu,yv$, $2\\delta^*(G))$ $\\geq\n 2$.\nThen we have\n\\begin{equation}a_jb_j+b_id_{yv-xv+i}+c_{\\ell}d_{\\ell}+a_mc_{yu-xu+m}\\geq\n2. \\label{Nippon}\n\\end{equation} Furthermore, we have the following conclusions: if $a_jb_j=1$, then $(i,m)\\in \\{ (j,j), (j, xu), (xv,j)\\};$\nif $b_id_{yv-xv+i}=1$, then $(j,\\ell)\\in \\{ (i, yv-xv+i), (i,0),\n(0, yv-xv+i) \\}$; if $c_{\\ell}d_{\\ell}=1, $ then $(yu-xu+m,\nyv-xv+i)\\in \\{(\\ell , \\ell), (\\ell ,yv), (yu, \\ell )$; if $a_m\nc_{yu-xu+m}=1,$ then $(j, \\ell)\\in \\{ (m, yu-xu+m), (m, 0), (0,\nyu-xu+m)\\}$. \\label{lem92}\n\\end{lemma}\n\n\n\\begin{proof}\n Since $\\delta^*(G)\\geq 1$, it\nfollows from Lemma \\ref{lem49} that\n\\begin{equation} \\min (i-j,\n(yv-xv+i)-\\ell, (yu-xu+m)-\\ell, m-j)\\geq 0. \\label{mgs}\n\\end{equation}\nUsing Lemma \\ref{lem14} instead, we obtain from $\\delta^*(G)\\geq 1$\n that $\\min (d(P_a,P_d),d(P_b,P_c))\\geq 1$.\nConsidering Lemma \\ref{lem2.4} (i) additionally, this says that\n$G$ has a cycle $C$ as displayed in Eq. \\eqref{cycle} whose length\nis $\\pi (a)+\\pi (b)+\\pi (c)+\\pi (d)$, where $\\pi (a), \\pi (b),\n\\pi (c), \\pi (d)$ stand for the numbers introduced in Eq.\n\\eqref{PI}. As $\\mathcal\n {Q}(x,u,y,v)$ has no $\\mathbb{H}$-edges, by the choice of $i,j,\\ell,m$ and by Lemma \\ref{lem2.4} (iii), we see that $C$ is chordless and are thus led\n to $\\pi (a)+\\pi (b)+\\pi (c)+\\pi (d)\\leq 5.$\n\n\nWe first observe that\n$a_jb_j+b_id_{yv-xv+i}+c_{\\ell}d_{\\ell}+a_mc_{yu-xu+m}>0$; as\notherwise $C$ will be a chordless cycle of length $xv+xu+yv+yu\\geq\n8,$ contradicting $\\mathbbm{l}\\mathbbm{c}(G)\\leq 5.$ Let us proceed\nto consider the case that\n$(a_jb_j,b_id_{yv-xv+i},c_{\\ell}d_{\\ell},a_mc_{yu-xu+m})=(1,0,0,0)$.\nNote that Corollary \\ref{cor2.1} guarantees that $0< j\n<\\min(xu,xv)$. Evoking our assumption $\\min (yv,yu)\\geq 2$, it is\nthen obvious that the cycle $C$ contains at least $6$ different\nvertices $a_j,b_j,v,d_1,y,c_1,u$, which is absurd as $G$ is\n$5$-chordal. By symmetry, Eq. \\eqref{Nippon} is thus established.\n\nAmong the four conclusions, let us now only deal with the one\naccompanied with the assumption that $a_jb_j=1.$ If\n$a_{m}c_{yu-xu+m}=b_id_{yv-xv+i}=0,$ Eq. \\eqref{Nippon}\n implies $c_{\\ell}d_{\\ell}=1$ and so $C$ will have at least $6$ different vertices\n$u,a_j,b_j,v,d_{\\ell},c_{\\ell}$, contrary to\n$\\mathbbm{l}\\mathbbm{c}(G)\\leq 5$. To this point, we can conclude\nthat $\\max (b_id_{yv-xv+i}, a_mc_{yu-xu+m})=1$ and so it suffices to\nprove that $i\\in \\{ j, xv\\}$ and $m\\in \\{ j, xu\\}.$ We only prove\nthe first claim and the second one will follow by symmetry. Since\nwe already have $i\\geq j$ as guaranteed by Eq. \\eqref{mgs}, our task\nis now to get from $i>j$ to $i=xv.$ If this\n is not true, the chordless cycle $C$ will already have four different\n vertices $a_j,b_j,b_i, d_{yv-xv+i}$, which are all outside of $P_c$\n according to Corollary \\ref{cor2.1}. Consequently, due to Corollary \\ref{cor2.1} and $\\mathbbm{l}\\mathbbm{c}(G)\\leq\n 5,$ we find that $C$ must have the fifth vertex $c\\in P_c\\setminus \\{ u,y\\}$ such\n that $ca_j=cd_{yv-xv+i}=1.$ In view of Lemma \\ref{lem2.4}\n (iii), we then see that $$cu=a_ju,cy=\n d_{yv-xv+i}y,a_jx=b_jx,b_iv=d_{yv-xv+1}v.$$ We sum them up and yield\n $xv+yu=xu+yv+b_ib_j>xu+yv$, which is a contradiction with Lemma\n \\ref{cor12}.\n This completes the\nproof of the lemma.\n\\end{proof}\n\n\\begin{lemma}\\label{cor10} Let $G$ be a graph for which we will make Assumptions I and II.\nLet $P$ and $P'$ be two adjacent sides of $\\mathcal\n {Q}(x,u,y,v)$ whose common peak is $w$.\nLet $\\alpha, \\beta\\in P\\setminus \\{w\\}$ and $\\alpha', \\beta'\\in\nP'\\setminus \\{w\\}$ be four vertices of $\\mathcal\n {Q}(x,u,y,v)$ such that $\\alpha\\alpha'=1$ and\n $\\beta w=\\beta 'w<\\alpha w=\\alpha 'w$. Then it holds $\\beta \\beta\n '=1$ in the case that $G$ is $4$-chordal as well as in the case that $G$ is $5$-chordal and $\\beta w=\\beta\n 'w>1$.\n\\end{lemma}\n\n\n\\begin{proof} By symmetry, we only need to show that for any $i\\geq\n3$ ($i\\geq 2$) we can obtain from $a_ib_i=1$ that $a_{i-1}b_{i-1}=1$\nprovided $G$ is $5$-chordal ($4$-chordal). But Lemma \\ref{lem2.4}\n(i) states that $C=[a_{i-1}a_ib_ib_{i-1}\\cdots b_1a_0a_1$ $\\cdots\na_{i-2}]$ is a\n cycle of length at\n least $7$ ($5$). Thus, Lemma\n \\ref{lem2.1} in conjunction with Lemma \\ref{lem2.4} (iii) applies to give\n $a_{i-1}b_{i-1}=1$, as wanted.\n\\end{proof}\n\n\n\\begin{corollary} \\label{cor15} Let $G$ be a $5$-chordal graph without isometric $C_4$ for which we will make Assumptions I and\nII. If there is an $\\mathbb{A}$-edge connecting $\\alpha$ and\n$\\alpha '$ lying in two adjacent sides $P$ and $P'$ with common peak\n$w$, respectively, then this is the only $\\mathbb{A}$-edge\nbetween $P$ and $P'$ and $\\alpha w=\\alpha 'w \\leq 2.$\n\\end{corollary}\n\n\\begin{proof}\nThis follows directly from Lemmas \\ref{lem2.4} (iii) and\n\\ref{cor10}.\n\\end{proof}\n\n\n\n\n\n\\begin{lemma} \\label{lem23} Let $G$ be a $5$-chordal graph with $\\delta^*(G)\\geq 1$ and let Assumptions I and II hold.\nAssume that $\\mathcal {Q}(x,u,y,v)$\n has no $\\mathbb{A}$-edges. (i)\n If there is an $\\mathbb{H}$-edge\n between $P_b$ and $P_c$, then $\\max (xu, yv)\\leq 2.$ (ii) If there is an $\\mathbb{H}$-edge\n between $P_a$ and $P_d$, then $\\max (xv, yu)\\leq 2.$\n\\end{lemma}\n\n\n\n\\begin{proof} We only prove $xu\\leq 2$ under the assumption that there is an $\\mathbb{H}$-edge\n between $P_b$ and $P_c$ and all the other claims follow similarly.\nTake the minimum $i$ such that $b_i$ is incident with an\n$\\mathbb{H}$-edge and then pick the maximum $j$ such that $b_ic_j$\nis an $\\mathbb{H}$-edge. By Corollary \\ref{cor2.1}, we have $\\min\n(i, yu-j)\\geq 1$.\n Since $\\mathcal {Q}(x,u,y,v)$\n has no $\\mathbb{A}$-edges, we find that $$[b_1\\cdots b_ic_jc_{j+1}\\cdots c_{yu-1}ua_{xu-1}\\cdots\n a_1x]$$\n is a chordless cycle of length $xu+1+i+(yu-j)\\geq xu+3$. Finally, because $G$\n is\n $5$-chordal, we conclude that $xu\\leq 2,$ as desired.\n\\end{proof}\n\n\n\n\n\\begin{lemma}\\label{lem19}\nLet $G$ be a $5$-chordal graph with $\\delta^*(G)\\geq 1$. We\nkeep Assumptions I and II. In addition, assume that $\\mathcal\n {Q}(x,u,y,v)$ has a side of length one. Then, $G$ contains at least one\n graph among $C_4,H_3$ and $H_5$ as an isometric subgraph.\n\\end{lemma}\n\n\\begin{proof}\nIt involves no restriction of generality in assuming that $xv=1.$\nOwning to Lemma \\ref{lem2.4} (i), the walk along $P_a, P_c$ and\n$P_d$ will connect $x$ and $v$ without passing through $x$ or $v$\nin the middle and hence there is a shortest path connecting $x$ and\n$v$ in the graph obtained from $\\mathcal\n {Q}(x,u,y,v)$ by deleting the edge $\\{x,v\\}$. This\n says that $\\mathcal\n {Q}(x,u,y,v)$ has an induced cycle passing through $x$ and $v$\n contiguously, say $C=[w_1w_2\\cdots w_n]$, where $w_1=x$ and $w_2=v.$\nFrom Corollary \\ref{cor2.1} we know that $w_3=d_{yv-1}\\not= a_1=w_n$\nand hence $n>3$. Since $G$ is $5$-chordal, our task is to derive\nthat if $n=5$ then $G$ contains an isometric $C_4,$ $H_3$ or\n$H_5$.\n\n\n\n\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.4pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(80,25)\n\n\\put(20,5){\\circle*{1}}\\put(24,1){\\makebox(0,0)[b]{$y=w_3$}}\n\\put(10,5){\\circle*{1}}\\put(10,1){\\makebox(0,0)[b]{$w_4$}}\n\\put(10,15){\\circle*{1}}\\put(5,17){\\makebox(0,0)[l]{$w_5$}}\n\\put(0,5){\\circle*{1}}\\put(-3,3){\\makebox(0,0)[l]{$u$}}\n\\put(30,15){\\circle*{1}}\\put(32,15){\\makebox(0,0)[l]{$v=w_2$}}\n\\put(15,20){\\circle*{1}}\\put(20,23){\\makebox(0,0)[t]{$x=w_1$}}\n\\put(10,-5){\\makebox(0,0)[b]{$H_3$}}\n\\qbezier(0,5)(10,15)(15,20)\\qbezier(0,5)(0,5)(20,5)\\qbezier(10,15)(10,10)(10,5)\n\\qbezier(20,5)(25,10)(30,15)\\qbezier(30,15)(20,18)(15,20)\n\n\n\n\n\\put(70,5){\\circle*{1}}\\put(75,1){\\makebox(0,0)[b]{$y=w_3$}}\n\\put(60,5){\\circle*{1}}\\put(60,1){\\makebox(0,0)[b]{$w_4$}}\n\\put(60,17){\\circle*{1}}\\put(55,19){\\makebox(0,0)[l]{$w_5$}}\n\\put(40,5){\\circle*{1}}\\put(37,3){\\makebox(0,0)[l]{$u$}}\n\\put(80,15){\\circle*{1}}\\put(82,15){\\makebox(0,0)[l]{$v=w_2$}}\n\\put(65,20){\\circle*{1}}\\put(70,23){\\makebox(0,0)[t]{$x=w_1$}}\n\n\\put(50,5){\\circle*{1}}\\put(50,1){\\makebox(0,0)[b]{$c_2$}}\n\\put(50,11){\\circle*{1}}\\put(45,12){\\makebox(0,0)[l]{$a_2$}}\n\n\\qbezier(40,5)(60,17)(65,20)\\qbezier(40,5)(60,5)(70,5)\\qbezier(70,5)(75,10)(80,15)\n\\qbezier(80,15)(70,18)(65,20)\\qbezier(60,5)(60,10)(60,17)\n\\put(55,-5){\\makebox(0,0)[b]{$H_5$}}\n \\end{picture}\n\\end{center}\n \\caption{Case 1 in the proof of Lemma \\ref{lem19}.}\\label{case1}\n\\end{figure}\n\n\\paragraph{\\sc Case 1:} $w_3$ is a\ncorner of $\\mathcal\n {Q}(x,u,y,v)$, namely $yv=1$.\n\nIn light of Corollary \\ref{cor2.1}, we have $w_4=c_1$. If $c_1=u$\nor $w_5=u$ occurs, then $\\mathcal\n {Q}(x,u,y,v) $ turns out to be a $5$-cycle and hence has hyperbolicity $\\frac{1}{2}$. This is impossible as Assumption II means that this hyperbolicity\n can be no smaller than $\\delta^*(G)\\geq 1$. Accordingly, by\n Lemma \\ref{lem2.4} (iii) we know that $w_4u$ and $w_5u$ have a common value, say $m.$\n\n If $m>3 $ or there are two $\\mathbb{A}$-edges between $P_a$ and $P_c$, Corollary \\ref{cor15} says that\n $G$ contains an isometric $C_4.$\n\n\nWhen $m=2$ and there are no two $\\mathbb{A}$-edges between $P_a$\nand $P_c$, the graph $H_5$ as depicted on the right of Fig.\n\\ref{case1} is an induced graph of $G$. Utilizing Eq. \\eqref{key}\nand the assumption that $\\delta^*(G)\\geq 1$, we find that\n$$4=3+1=ux+xv\\geq uv\\geq xv+uy+2\\delta^*(G)-xy=2+2\\delta^*(G)\\geq 4.$$ This\nillustrates that $uv=4$. It follows from Lemma \\ref{koolen1} that\n$a_2y\\geq uy=3$. In addition, we have $a_2y\\leq\na_2a_1+a_1c_1+c_1y=3$ and so we see that $a_2y=3.$ Similarly, we\nhave $c_2x=3.$ Getting that $a_2y=c_2x=3$ and $uv=4,$ we apply\n Corollary \\ref{lem31} and conclude that the above-mentioned $H_5$ must be an isometric\n subgraph of $G.$\n\n\n\nWhen $m=1,$ the graph $H_3$ as depicted on the left of Fig.\n\\ref{case1} is an induced graph of $G$. As in the case of $m=2$,\nwe make use of Eq. \\eqref{key} and $\\delta^*(G)\\geq 1$ to get an\nimportant information: $$3=uw_5+w_5x+xv \\geq uv\\geq xv+uy\n+2\\delta^*(G)-xy=1+2\\delta^*(G)\\geq 3.$$ This implies $uv=3$ and\nhence we deduce from Corollary \\ref{lem30} that this $H_3$ is\neven an isometric subgraph of $G$.\n\n\n\n\n \\paragraph{\\sc Case 2:} $w_5$ is a\ncorner of $\\mathcal\n {Q}(x,u,y,v)$, namely $xu=1$.\n\nThe analysis is symmetric to that of Case 1.\n\n \\paragraph{\\sc Case 3:} Neither $w_3$ nor $w_5$ is a\ncorner. In this case, Corollary \\ref{cor2.1} ensures that $w_4$ is\nnot a corner as well. We proceed to show that this case indeed\ncannot happen.\n\n \\paragraph{\\sc Case 3.1:} $w_4\n \\in P_a$.\n\n Since $P_a$ is a geodesic, we get that $w_4=a_2.$\n It is easy to see that\n $xy\\leq vy+vx=vy+1$ and that\n$uv=uw_2\\leq w_2w_3+w_3w_4+w_4u=2+a_2u=xu.$\n Adding together, we obtain by Assumption II that\n\\begin{equation*} 2\\delta^*(G)=(xy+uv)-\\max(xv+yu,vy+xu)\\leq (xy+uv)-(vy+xu)\\leq\n 1,\n\\end{equation*} violating the assumption that $\\delta^*(G)\\geq 1.$\n\n \\paragraph{\\sc Case 3.2:} $w_4\n \\in P_d$.\n\n\n\n\nReasoning as in Case 3.1 rules out the possibility that this case\nmay happen.\n\n\n \\paragraph{\\sc Case 3.3:} $w_4\n \\in P_c$.\n\n\n\nIn this case, $\\mathcal\n {Q}(x,u,y,v)$ contains $\\mathbb{A}$-edges and hence Lemma\n \\ref{cor12} tells us\n \\begin{equation}\\label{e33}\n xu+yv=xv+yu.\n \\end{equation} But, by\nLemma \\ref{lem2.4} (iii) we have $xu-1=uw_5=uw_4$ and $yv-1=w_3y=\nw_4y$. We therefore get\n that $xu+yv=2+uw_4+w_4y=2+yu=1+xv+yu,$ which contradicts Eq. \\eqref{e33} and finishes the proof.\n\\end{proof}\n\n\n\n\\begin{lemma} Let $G$ be a $5$-chordal graph and let Assumptions I and II hold.\nIf\n $\\min (ux,xv,uy,yv,2\\delta^*(G))\\geq 2$, and $\\mathcal {Q}(x,u,y,v)$\n has no $\\mathbb{A}$-edges, then $\\delta^*(G)= 1$ and either $ux=xv=uy=yv=2$ or $G$ has an isometric $4$-cycle.\n \\label{lem2.6}\\end{lemma}\n\n\\begin{proof} By\nCorollary \\ref{cor45}, $\\mathcal {P}(x,u,y,v)$ is a cycle of length\nat least $8$. As $G$ is $5$-chordal, this cycle must have chords,\nwhich, by Corollary \\ref{cor45} again and by the fact that\n$\\mathcal {Q}(x,u,y,v)$\n has no $\\mathbb{A}$-edges, must\nbe\n $\\mathbb{H}$-edges. So, without loss of generality, suppose that $\\mathcal {Q}(x,u,y,v)$\n has an $\\mathbb{H}$-edge between $P_a$ and $ P_d$.\nOn the one hand, we can thus go to Lemma \\ref{lem23} and get\n\\begin{equation}\\label{eq24} xv=yu=2.\n\\end{equation}\nOn the other hand, this allows us to apply\n Lemmas \\ref{lem14} and \\ref{lemma41} to deduce that $\\delta^*(G)= 1$ and that\n either $G$ has an isometric $C_4$ or has exactly one $\\mathbb{H}$-edge\n between $P_a$ and $P_d.$ If the latter case happens, say we\n have an $\\mathbb{H}$-edge connecting $a_i$ and $d_j,$ we will\n get two chordless cycles of $G$, $[a_ia_{i-1}\\cdots xb_1\\cdots b_{xv-1}vd_{yv-1}\\cdots d_j]$\nand $[a_ia_{i+1}\\cdots uc_{yu-1}\\cdots c_1yd_1\\cdots d_j]$. Since\nneither of these two chordless cycles can be longer than $5$, it\nfollows from Eq. \\eqref{eq24} that $a_ix+d_jv\\leq 2$ and\n$ua_i+yd_j\\leq 2.$ Taking into account additionally that $2\\leq\nux=ua_i+a_ix$ and $2\\leq yv=yd_j+d_jv$, we thus have $ xu=yv=2.$\nThis is the end of the proof.\n\\end{proof}\n\n\n\n\n\n\\begin{lemma} We take a $5$-chordal graph $G$ satisfying $\\delta^*(G)=1$ and require Assumptions I and II. Suppose that $\\mathcal\n {Q}(x,u,y,v)$ has no $\\mathbb{H}$-edge and $[ua_{xu-1}b_{xu-1}d_{yu-1}c_{yu-1}]$ is an induced $5$-cycle of $G$; see Fig. \\ref{fig924}. Then $G$ has at\n least one graph among $C_4$, $H_3$ and $H_5$ as an isometric subgraph.\n \\label{lem54}\n\\end{lemma}\n\n\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.4pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(60,60)\n\n\\put(20,0){\\circle*{1}}\\put(20,-4){\\makebox(0,0)[b]{$y$}}\n\\put(0,20){\\circle*{1}}\\put(-3,20){\\makebox(0,0)[l]{$u$}}\n\\put(40,20){\\circle*{1}}\\put(42,20){\\makebox(0,0)[l]{$v$}}\n\\put(20,40){\\circle*{1}}\\put(20,43){\\makebox(0,0)[t]{$x$}}\n\n\\put(5,25){\\circle*{1}}\\put(-6,25){\\makebox(0,0)[l]{$a_{xu-1}$}}\n\n\\put(15,35){\\circle*{1}}\\put(10,35){\\makebox(0,0)[l]{$a_{1}$}}\n\\put(5,15){\\circle*{1}}\\put(-6,15){\\makebox(0,0)[l]{$c_{yu-1}$}}\n\n\\put(15,5){\\circle*{1}}\\put(10,5){\\makebox(0,0)[l]{$c_{1}$}}\n\\put(25,5){\\circle*{1}}\\put(30,5){\\makebox(0,0)[r]{$d_{1}$}}\n\n\\put(30,10){\\circle*{1}}\\put(42,10){\\makebox(0,0)[r]{$d_{yu-1}$}}\n\n\\put(25,35){\\circle*{1}}\\put(30,35){\\makebox(0,0)[r]{$b_{1}$}}\n\n\\put(30,30){\\circle*{1}}\\put(40,30){\\makebox(0,0)[r]{$b_{xu-1}$}}\n\n\\qbezier[7](5,15)(10,10)(15,5)\n\\qbezier(0,20)(5,25)(5,25)\\qbezier(0,20)(5,15)(5,15)\n\\qbezier(15,5)(18,2)(20,0)\\qbezier(20,0)(22,2)(25,5)\\qbezier[7](30,30)(40,20)(40,20)\\qbezier(35,25)(40,20)(40,20)\n\\qbezier[7](25,5)(30,10)(35,15)\\qbezier(15,35)(18,38)(20,40)\\qbezier(20,40)(22,38)(25,35)\n\\qbezier[7](30,10)(30,10)(40,20)\\qbezier(35,15)(30,10)(40,20)\\qbezier[7](25,35)(30,30)(35,25)\\qbezier[7](5,25)(10,30)(15,35)\n\\qbezier(5,25)(17,27.5)(30,30)\\qbezier(5,15)(15,13)(30,10)\\qbezier(30,30)(30,20)(30,10)\n \\end{picture}\\end{center}\n\\caption{$[ua_{xu-1}b_{xu-1}d_{yu-1}c_{yu-1}]$ is an induced\n$5$-cycle in $\\mathcal\n {Q}(x,u,y,v)$.}\\label{fig924}\n\\end{figure}\n\n\\begin{proof}\nBy Corollary \\ref{cor2.1} and Lemma \\ref{lem2.4} (iii),\n it will be enough to consider the following cases,\n$b_{xu-1}v=d_{yu-1}v>3 $ or $b_{xu-1}v=d_{yu-1}v\\in \\{ 1,2\\}.$\n\n\n\n\\paragraph{\\sc Case 1:} $b_{xu-1}v=d_{yu-1}v>3.$\n\nCorollary \\ref{cor15} implies that $G$ contains an isometric $C_4$.\n\n\n\n\n\n\n\\paragraph{\\sc Case 2:} $b_{xu-1}v=d_{yu-1}v\\in \\{1,2\\}.$\n\n\n\n\nBefore jumping into the analysis of two separate subcases, we make\nsome general observations. Note that\n\\begin{equation}\\begin{array}{cll} xu+yv+2 & = &\n(xa_{xu-1}+ua_{xu-1})+(yd_{yu-1}+ d_{yu-1}v)+2\n\\\\ &=&xb_{xu-1}+ua_{xu-1}+yd_{yu-1}+b_{xu-1}v+(a_{xu-1}b_{xu-1}+b_{xu-1}d_{yu-1})\n\\\\ &=& (xb_{xu-1}+b_{xu-1}d_{yu-1}+yd_{yu-1})+(ua_{xu-1}+a_{xu-1}b_{xu-1}+b_{xu-1}v)\\\\\n &\\geq& xy+(ua_{xu-1}+a_{xu-1}v) \\\\ &\\geq& xy+uv \\\\ &=&\\max(xu+yv,xv+yu)+2\\delta^*(G) \\ \\ \\ \\ \\text{(By Assumption II)}\\\\\n& \\geq & xu+yv+2. \\ \\ \\ \\ \\text{(By $\\delta^*(G)=1$)}\n\\end{array} \\label{eq924} \\end{equation}\nIt follows that all inequalities in Eq. \\eqref{eq924} are best\npossible and hence we have\n\\begin{equation}uv=ua_{xu-1}+a_{xu-1}b_{xu-1}+b_{xu-1}v=2+b_{xu-1}v\\label{925}\n\\end{equation}\nand\n\\begin{equation}a_{xu-1}v=a_{xu-1}b_{xu-1}+b_{xu-1}v=1+b_{xu-1}v.\\label{108}\n\\end{equation}\n\n\\paragraph{\\sc Case 2.1:} $b_{xu-1}v=d_{yu-1}v=1$.\n\nWe derive from Corollary \\ref{cor2.1} that the subgraph of $G$\ninduced by $u,a_{xu-1},b_{xu-1},v,d_{yu-1},c_{yu-1}$ is isomorphic\nto $H_3$ in an obvious way. Thanks to Corollary \\ref{lem30}, in\norder to check that this $H_3$ is isometric, our task is to show\nthat $uv=3.$ But $uv=3$ is an immediate result of\n Eq. \\eqref{925}, proving the claim in this case.\n\n\\paragraph{\\sc Case 2.2:} $b_{xu-1}v=d_{yu-1}v=2$.\n\n\nTo start things off we look at the following:\n\\begin{equation}\\begin{array}{cll} b_{xu-1}v+1 & = & d_{yu-1}v+1 =d_{yu-1}d_{yv-1}+2 \\\\ &=&\nd_{yu-1}d_{yv-1}+ a_{xu-1}b_{xu-1}+b_{xu-1}d_{yu-1}\n\\\\ &\\geq &a_{xu-1}d_{yv-1} \\ \\ \\ \\ \\text{(By the triangle inequality)}\n\\\\ &\\geq & xd_{yv-1}-xa_{xu-1} \\ \\ \\ \\ \\text{(By the triangle inequality)} \\\\\n&\\geq& xv -xa_{xu-1} \\ \\ \\ \\ \\text{(By Lemma \\ref{koolen1})} \\\\ &=& (xb_{xu-1}+b_{xu-1}v)-xa_{xu-1} \\\\\n& = & b_{xu-1}v.\n\\end{array} \\label{KIM} \\end{equation}\nA consequence of Eq. \\eqref{KIM} is that\n\\begin{equation}b_{xu-1}v+1\\geq a_{xu-1}d_{yv-1}\\geq\nb_{xu-1}v.\\label{44}\n\\end{equation}\nBy symmetry, we also have\n\\begin{equation}b_{xu-1}v+1= d_{yu-1}v+1 \\geq c_{yu-1}b_{xv-1}\\geq d_{yu-1}v=b_{xu-1}v.\n\\label{45}\n\\end{equation}\nAs a result of Eqs. \\eqref{44} and \\eqref{45} we\n obtain\n \\begin{equation}3\\geq \\max (a_{xu-1}d_{yv-1}, c_{yu-1}b_{xv-1})\\geq \\min (a_{xu-1}d_{yv-1}, c_{yu-1}b_{xv-1}) \\geq\n 2.\\label{eq47}\n \\end{equation}\nFinally, let us remark that $b_{xu-1}v=2$ implies $xu-1=xv-2$\nand hence\n $b_{xu-1}b_{xv-1}=d_{xu-1}d_{xv-1}=1.$\n\nAccording to Eq. \\eqref{eq47}, the following two subcases are\nexhaustive.\n\n\n\n\n\\paragraph{\\sc Case 2.2.1:}\n$\\min(a_{xu-1}d_{yv-1},c_{yu-1}b_{xv-1})=2$.\n\n\n Without loss of\ngenerality, we suppose that there is a vertex $w\\in V(G)$ such that\n$a_{xu-1}w=wd_{yv-1}=1$. Note that Lemma \\ref{lem2.4} (iii) says\nthat $w\\notin \\{a_{xu-1},b_{xu-1},b_{xv-1},v,d_{yv-1}\\} $ and hence\n $C= [a_{xu-1}b_{xu-1}b_{xv-1}vd_{yv-1}w]$ is a $6$-cycle in $G$.\nBecause $G$ is $5$-chordal, $C$ has at least one chord. Observe\nthat Eq. \\eqref{108} says that\n\\begin{equation} a_{xu-1}v=1+2=3\\label{eq57}\n\\end{equation}\nand so\n$$wv\\geq a_{xu-1}v-a_{xu-1}w =3-1=2.$$\nIn consequence, by virtue of Lemma \\ref{lem2.4} (iii), we have\n $\\min ( wb_{xu-1}, wb_{xv-1}, b_{xv-1}d_{yv-1} )=1.$ There are\n three cases to dwell on.\n\\paragraph{\\sc Case 2.2.1.1:} If $b_{xv-1}d_{yv-1}=1,$ then\n$[b_{xu-1}b_{xv-1}d_{yv-1}d_{xu-1}]$ is a required isometric\n$4$-cycle.\n\n\\paragraph{\\sc Case 2.2.1.2:} If $wb_{xv-1}=1$ and $b_{xv-1}d_{yv-1}>1$, we find that\n$[b_{xv-1}vd_{yv-1}w]$ is an isometric $4$-cycle, as desired.\n\n\n\n\n\\paragraph{\\sc Case 2.2.1.3:}\nIf $\\min (wb_{xv-1}, b_{xv-1}d_{yv-1}) >wb_{xu-1}=1$, as a result\nof Eq. \\eqref{eq57},\n we can make use of Corollary \\ref{lem30}\nto yield that the subgraph induced by\n$a_{xu-1},b_{xu-1},b_{xv-1},v,d_{yv-1},w$ is an isometric $H_3$ in\n$G$; see Fig. \\ref{fig9.24}.\n\n\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.4pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(30,20)\n\\put(0,15){\\circle*{1}}\\put(-5,18){\\makebox(0,0)[l]{$a_{xu-1}$}}\n\\put(0,0){\\circle*{1}}\\put(0,-4){\\makebox(0,0)[b]{$w$}}\n\\put(15,5){\\circle*{1}}\\put(19,5){\\makebox(0,0)[r]{$v$}}\n\\put(10,0){\\circle*{1}}\\put(10,-4){\\makebox(0,0)[b]{$d_{yv-1}$}}\n\n\\put(10,10){\\circle*{1}}\\put(20,10){\\makebox(0,0)[r]{$b_{xv-1}$}}\n\\put(5,15){\\circle*{1}}\\put(15,15){\\makebox(0,0)[r]{$b_{xu-1}$}}\n\n\\qbezier(0,15)(0,15)(0,0)\\qbezier(0,0)(10,0)(10,0)\\qbezier(0,15)(0,15)(5,15)\n\\qbezier(5,15)(5,15)(15,5)\\qbezier(15,5)(15,5)(10,0)\\qbezier(0,0)(5,15)(5,15)\n \\end{picture}\\end{center}\n\\caption{Case 2.2.1.3 in the proof of Lemma\n\\ref{lem54}.}\\label{fig9.24}\n\\end{figure}\n\n\\paragraph{\\sc Case 2.2.2:} $a_{xu-1}d_{yv-1}=c_{yu-1}b_{xv-1}=3$.\n\nFrom Eq. \\eqref{925} we obtain $uv=4.$ This, together with the\nstanding assumption of Case 2.2, enables us to deduce from Corollary\n\\ref{lem31} that the subgraph induced by\n$$u,a_{xu-1},b_{xu-1},b_{xv-1},v,d_{yv-1},d_{yu-1},c_{yu-1}$$\nis an isometric $H_5$.\n\\end{proof}\n\n\n\n\n\n\\subsection{Proofs of Theorems \\ref{main} and\n\\ref{main1}}\\label{Proof}\n\n\nWe now have all necessary tools to prove our main results.\n\n\\rz\n\n\n\n\n\n\n\\par \\noindent \\textbf{Proof of Theorem \\ref{main}: }\nUsing typical compactness argument, it suffices to prove that every\n connected finite induced subgraph of a $k$-chordal graph $G$ is $\\frac{\\lfloor\n\\frac{k}{2}\\rfloor}{2}$-hyperbolic. If $G$ has less than $4$\nvertices, the result is trivial. Thus, we can simply assume that\n$4\\leq |V(G)| <\\infty $ and henceforth there surely exists\n a\ngeodesic quadrangle $\\mathcal\n {Q}(x,u,y,v)$ in $G$ fulfilling\nAssumptions I and II. When\n $\\min(d(P_a,P_d),d(P_b,P_c))\\leq 1$, the result is direct from Lemma\n\\ref{lem14} and the fact that $1\\leq \\frac{\\lfloor\n\\frac{k}{2}\\rfloor}{2}$ while when\n $\\min(d(P_a,P_d),d(P_b,P_c))>\n1$ we are done by Lemma \\ref{lemma54}. {\\QED\\par \\bigskip \\par}\n\n\n\n\n\n\n\\par \\noindent \\textbf{Proof of Theorem \\ref{main1}: }\nConsider a $5$-chordal graph $G$ with $\\delta^*(G)=1$. We surely\ncan get a geodesic quadrangle $\\mathcal\n {Q}(x,u,y,v)$ in $G$ for which Assumption I and Assumption II\n hold. Passing to the proof that $G$ contains one graph from Fig. \\ref{fig0} as an isometric\n subgraph, we have to distinguish four main cases.\n\n\n\\paragraph {\\sc Case 1:}\n $\\min (xu,xv,yu,yv)=1. $\n\n Lemma \\ref{lem19} tells\nus that $G$ has either an isometric $C_4$ or an isometric $H_3$ or\nan isometric $H_5$.\n\n\n\n\\paragraph {\\sc Case 2:}\n $\\min (xu,xv,yu,yv)\\geq 2$ and there exist no\n$\\mathbb{A}$-edges.\n\n\n\n\\paragraph {\\sc Case 2.1:} $\\max (xu,xv,yu,yv)>2$.\n\nBy Lemma \\ref{lem2.6}, $G$ must have an isometric\n $C_4$.\n\n\n\n\n\n\n\\paragraph {\\sc Case 2.2:} $xu=xv=yu=yv=2$.\n\n\n\nBy Corollary \\ref{cor45}, $\\mathcal\n {Q}(x,u,y,v)$ must have an $\\mathbb{H}$-edge. By Corollary\n \\ref{cor2.1}, we may assume, without loss of generality, that\n $a_1d_1=1.$ It then follows from Lemma \\ref{lemma41} (i) that $xy=uv=3.$\n\n\n\n\\paragraph {\\sc Case 2.2.1:}\n $\\mathcal\n {Q}(x,u,y,v)$ has only one $\\mathbb{H}$-edge and hence the subgraph of $G$ induced by its vertices is isomorphic to $H_5$.\n\nBy Lemma \\ref{lem27}, $G$ has one of $C_4,H_2,H_3$ and $H_5$\nas an isometric subgraph.\n\n\\paragraph {\\sc Case 2.2.2:}\n $\\mathcal\n {Q}(x,u,y,v)$ has two $\\mathbb{H}$-edges and hence the subgraph of $G$ induced by its vertices is isomorphic to $H_4$.\n\n\nBy Corollary \\ref{lem29}, $G$ contains $H_4$ as an isometric\nsubgraph.\n\n\n\n\n\n\n\n\\paragraph {\\sc Case 3:} $\\min (xu,xv,yu,yv)\\geq 2$ and there exist no\n$\\mathbb{H}$-edges.\n\n\n\nTake $i,j,\\ell,m$ to be the numbers as specified in Lemma\n\\ref{lem49}. By Lemma \\ref{lem92}, Eq. \\eqref{Nippon} holds. So,\nwithout loss of generality, we can assume that $i=j,$ $a_jb_j=1$ and\n$b_jd_{yv-xv+j}=1.$\n\n\\paragraph {\\sc Case 3.1:} $d_{\\ell}c_{\\ell}=a_mc_{yu-xu+m}=1$.\n\n\n\nBy Lemma \\ref{lem92}, the chordless cycle displayed in Eq.\n\\eqref{cycle} is an isometric $C_4.$\n\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.4pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(60,60)\n\n\\put(20,0){\\circle*{1}}\\put(20,-4){\\makebox(0,0)[b]{$y$}}\n\\put(0,20){\\circle*{1}}\\put(-3,20){\\makebox(0,0)[l]{$u$}}\n\\put(40,20){\\circle*{1}}\\put(41,20){\\makebox(0,0)[l]{$v$}}\n\\put(20,40){\\circle*{1}}\\put(20,43){\\makebox(0,0)[t]{$x$}}\n\\put(12,32){\\circle*{1}}\\put(7,32){\\makebox(0,0)[l]{$a_{j}$}}\n\n\n\\put(28,32){\\circle*{1}}\\put(32,32){\\makebox(0,0)[r]{$b_{j}$}}\n\\put(28,8){\\circle*{1}}\\put(30,8){\\makebox(0,0)[l]{$d_{yv-xv+j}$}}\n\n\\put(12,8){\\circle*{1}}\\put(10,8){\\makebox(0,0)[r]{$c_{yu-xu+j}$}}\n\n\\qbezier(12,32)(12,20)(12,8)\n\\qbezier[7](5,15)(10,10)(15,5)\\qbezier(0,20)(2,22)(5,25)\\qbezier(0,20)(2,18)(5,15)\n\\qbezier(12,8)(18,2)(20,0)\\qbezier(20,0)(22,2)(28,8)\\qbezier(35,15)(38,18)(40,20)\n\\qbezier[7](25,5)(30,10)(35,15)\\qbezier(15,35)(18,38)(20,40)\\qbezier(20,40)(22,38)(25,35)\n\\qbezier(35,25)(38,22)(40,20)\\qbezier[7](25,35)(30,30)(35,25)\\qbezier[7](5,25)(10,30)(15,35)\n\n\\qbezier(28,32)(28,20)(28,8)\\qbezier(12,32)(20,32)(28,32)\n\n\n \\end{picture}\\end{center}\n\\caption{Case 3.2 in the proof of Theorem \\ref{main1}.}\n\\label{case3.2}\n\\end{figure}\n\n\n\n\\paragraph {\\sc Case 3.2:} $(d_{\\ell}c_{\\ell}, a_mc_{yu-xu+m})=\n(0,1)$ or $(1,0)$.\n\n\nWe only consider the case that $(d_{\\ell}c_{\\ell}, a_mc_{yu-xu+m})=\n(0,1)$. For now, the chordless cycle shown in Eq. \\eqref{cycle}\nis just the $5$-cycle $[a_jb_jd_{yv-xv+j}yc_{yu-xu+j}];$ see\n Fig. \\ref{case3.2}. Lemma \\ref{lem54} demonstrates that $G$ contains one\ngraph among $C_4,H_3$ and $H_5$ as an isometric subgraph.\n\n\n\n\\paragraph {\\sc Case 3.3:} $d_{\\ell}c_{\\ell}=a_mc_{yu-xu+m}=0$.\n\nThis case is impossible as the chordless cycle demonstrated in Eq.\n\\eqref{cycle} will contain $6$ different vertices $a_j,b_j,\nd_{yv-xv+j}, y,c_1, u$.\n\n\n\\paragraph {\\sc Case 4:} $\\min (xu,xv,yu,yv)\\geq 2$ and there exist both $\\mathbb{H}$-edges\nand $\\mathbb{A}$-edges.\n\n\n\n\nBefore delving into the case by case analysis, here are some general\nobservations.\n First note that\n Lemma \\ref{cor12} can be applied to give\n \\begin{equation} \\label{eq30} xu+yv=xv+yu.\n \\end{equation}\nSecondly, according to Corollary \\ref{cor2.1},\n we can suppose that\nthere are \\begin{equation}1\\leq i\\leq xu-1\\ \\ \\text{ and} \\ \\ 1\\leq\nj\\leq yv-1 \\label{eq31}\\end{equation} such that $a_id_j=1$ and,\n by Lemma \\ref{lem15}, hence that\n\\begin{equation}a_iu+d_jy=yu\\ \\ \\text{and} \\ \\ a_ix+d_jv=xv.\n\\label{eq23}\n\\end{equation}\n Thirdly, as $\\delta^*(G)=a_id_j=1$, Lemma \\ref{lem14} gives\n \\begin{equation}d(P_a,P_d)=1.\n \\label{JAIST}\n \\end{equation}\nFinally,\n Lemma \\ref{lemma41} (i) says that the $\\mathcal\n {Z}$-walks of $\\mathcal\n {Q}(x,u,y,v)$ through the $\\mathbb{H}$-edge $\\{a_i,d_j\\}$ must\n be\n geodesics. Since any subpath of a geodesic is still a geodesic, we\n come to\n \\begin{equation}\nud_j=ua_i+a_id_j=ua_i+1 \\ \\text{and } \\ \\ a_iy=a_id_j+d_jy=1+d_jy.\n\\label{eqn33}\n\\end{equation}\n\n\n\n\n\\paragraph {\\sc Case 4.1:} $yu=xv=2$.\n\nIn this case, Eq. \\eqref{eq30} forces $xu=xv=yu=yv=2$ and so Eq.\n\\eqref{eq31} tells us that $i=j=1.$ It follows that $\\max\n(xy, uv)\\leq 3$ due to the existence of the path $x,a_1,d_1,y$ and\nthe path $u,a_1,d_1,v.$ For the moment, in view of Eq. \\eqref{key},\nwe can get \\begin{equation}xy=uv=3.\\label{eq33}\n\\end{equation}\n\n\nIdentifying $a_1,b_1,c_1,d_1$ with $a,b,c,d,$ respectively,\nCorollary \\ref{cor2.1} says that $\\mathcal\n {Q}(x,u,y,v)$ is obtained from the graph $H_5$ as depicted in Fig. \\ref{fig0} by adding\n $t$ additional edges among\n$\\{a,b\\},\\{b,d\\}, \\{d,c\\}, \\{c,a\\}$, where $t\\in \\{1,2,3,4\\}$, and\nadding possibly the edge $\\{b,c\\}$.\n\n\n\n\nIf $t=4$ and $bc=1$, we easily infer from Eq. \\eqref{eq33} and\nCorollary \\ref{lem29} that $\\mathcal\n {Q}(x,u,y,v)$ is an isometric\nsubgraph of $G$ which is isomorphic to $H_2$.\n\n\nIf $t=4$ and $bc>1$, we can check that $\\mathcal\n {Q}(x,u,y,v)$ is an induced\nsubgraph of $G$ isomorphic to $H_1$ and then, again by\n Eq. \\eqref{eq33}\nand Corollary\n \\ref{lem29}, $G$ contains an isometric $H_1$.\n\n\n\nIf $t<4$, as a consequence of Lemma \\ref{lead}, either $C_4$\n is an induced subgraph of $G$ or $\\mathcal\n {Q}(x,u,y,v)$ is isomorphic with $H_6$. Accordingly, Eq. \\eqref{eq33} together with Lemma \\ref{lem10} implies that $G$\n has an isometric subgraph which is isomorphic to either $C_4$ or\n $H_2$ or $H_3.$\n\n\n \\paragraph {\\sc Case 4.2:} $\\max (yu, xv) >2$.\n\n\n\n\nWe will show that $G$ contains an isometric subgraph which is\nisomorphic to $H_3$, under the assumption that $G$ has no\nisometric $C_4$. Note that the nonexistence of an isometric $C_4$\nin $G$ together with Eq. \\eqref{JAIST} yields that there exists\nexactly one $\\mathbb{H}$-edge between $P_a$ and $P_d$, namely\n$\\{a_i,d_j\\}$, as a result of Lemma \\ref{lemma41} (i).\n\nIt is no loss of generality in setting\n\\begin{equation} yu>2. \\label{eqn35}\\end{equation}\n By Lemma \\ref{lem2.4} (i) and Eq. \\eqref{JAIST}, the following is a set of pairwise different\nvertices:\n $$y,c_1,\\ldots, c_{yu-1},u,\na_{xu-1},\\ldots,a_i,d_j,d_{j-1},\\ldots ,d_1.$$ In the subgraph $F$\ninduced by these vertices in $G$, $a_i$ and $d_j$ are connected\nby a path disjoint from the edge $\\{a_i,d_j\\}.$ This means that\nthere is a chordless cycle $[w_1w_2\\cdots w_n]$ in $F$ where $n\\geq\n3$ and $w_1 =a_i,w_2=d_j$. Recall that it is already stipulated\nthat the $5$-chordal graph $G$ has no isometric $4$-cycle and hence\n$n$ can only take value either $3$ or $5$.\n\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.4pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(50,50)\n\n\n\n\n\n\\put(30,0){\\circle*{1}}\\put(30,-4){\\makebox(0,0)[b]{$y$}}\n\n\\put(20,0){\\circle*{1}}\\put(20,-4){\\makebox(0,0)[b]{$c_1$}}\n\\put(10,0){\\circle*{1}}\\put(10,-4){\\makebox(0,0)[b]{$c_2$}}\n\n\\put(0,20){\\circle*{1}}\\put(-4,23){\\makebox(0,0)[lt]{$a_i$}}\n\n\\put(30,10){\\circle*{1}}\\put(35,10){\\makebox(0,0)[r]{$d_1$}}\n\\put(30,20){\\circle*{1}}\\put(35,23){\\makebox(0,0)[rt]{$d_2$}}\n\n\n\\qbezier(10,0)(5,10)(0,20)\\qbezier(0,20)(10,20)(30,20)\\qbezier(30,20)(30,10)(30,0)\\qbezier(30,0)(20,0)(10,0)\n\\qbezier(10,0)(20,10)(30,20)\n\n \\end{picture}\n\\end{center}\n\n\\caption{Case 4.2.1 in the proof of Theorem\n\\ref{main1}.}\\label{fig4.2.1}\n\\end{figure}\n\n\n \\paragraph {\\sc Case 4.2.1:} $n=3$.\n\nSince there is exactly one $\\mathbb{H}$-edge between $P_a$ and\n$P_d$, $w_3$ is neither on\n $P_a$ nor on $P_d$. Hence, there is $0< q < yu$ such that $w_3=c_q$. It follows from\n Lemma \\ref{lem2.4} (iii) that $ua_i=uc_q$ and $yc_q=yd_j$.\nFrom Eq. \\eqref{eqn35} we obtain $\\max (yc_q, c_qu) \\geq 2.$\nWithout loss of generality, assume that $yc_q=\\max (yc_q, c_qu) \\geq\n2$. Since $G$ contains no isometric $C_4$, we infer from Corollary\n\\ref{cor15} that $q=j=2$ and $c_1d_1=2$. This then demonstrates that\nthe subgraph induced by the vertices $a_i,d_2,d_1,y,c_1,c_2$ is\n isomorphic to $H_3$; see Fig. \\ref{fig4.2.1}.\nGranting that $a_iy=3$,\n Corollary \\ref{lem30} will bring to us that $G$ contains $H_3$ as an\n isometric subgraph.\n But $a_iy=3$ follows from Eq. \\eqref{eqn33} and $d_jy=j=2.$\n\n\n\n\n\n\n\n \\paragraph {\\sc Case 4.2.2:} $n=5$.\n\n\n\n We aim to prove that this case will never happen by deducing contradictions in all the following subcases.\n\n\n\n \\paragraph {\\sc Case 4.2.2.1:} Both $w_3$ and $w_5$ belong to $P_c.$\n\n\nFirst consider the case that both $w_3$ and $w_5$ are ordinary\n vertices of $P_c.$\n From Lemma \\ref{lem2.4} (iii) we obtain $a_iu=uw_5$ and $d_jy=w_3y.$\n It then follows $uy=uw_5+w_3y$ by means of Eq. \\eqref{eq23}. Since $w_3$ and $w_5$ are on the same geodesic connecting\n $u$ and $y$, this is possible only when\n $w_3=w_5$, yielding a contradiction.\n\n Next the case that at least one of $w_3$ and $w_5$ is a corner. We could assume that\n $w_3$ is a corner, and then, in view of Corollary \\ref{cor2.1},\nit holds $w_3=y$. This implies that $w_5\\not= u$, as otherwise we\nobtain $yu=w_3w_5= 2,$ contradicting Eq. \\eqref{eqn35}.\nAccordingly, it follows from Lemma \\ref{lem2.4} (iii) that\n$a_iu=uw_5$. But, we surely have\n $ 2 = w_5w_3=w_5y$ and $ yd_j=w_3w_2=1.$ Putting together, we get\n$a_iu+yd_j=uw_5+1K$ large enough, the number $x_{jk}=a_{j}+(k-1)210$ is always a composite number. Let $L_{a_{i_0}}$ denote the column or line where the set $\\{x_{i_0 k}\\mid x_{i_0 k}=a_{i_0}+(k-1)210\\}$ is located in Figure \\ref{fig:M7-new}. Choose a prime $p_m$ on the line $L_{a_{i_0}}$ satisfying $p_m>a_{i_0}+210(K-1)$. Then by Proposition \\ref{prop:2}, the first several items in $M_{p_m}$ will not contain the gap (or skip) $2$.\n\nBut every pattern $\\mathcal{P}_{p_m}$ has skip 2. In fact by Theorem \\ref{thm:about-skips}, there are\n\\[\n(3-2)(5-2)(7-2)\\cdots(p_m-2)\n\\]\ngaps of skip $2$ in $\\mathcal{P}_{p_m}$. We can choose some of the corresponding pair of composite numbers in $\\mathcal{P}_{p_m}$ which are located in the adjacent two columns denoted by $L_u$ and $L_v$. Here $u,v\\in M_7^{(0)}$ and $v-u=2$.\n\nNote that there are $15=(3-2)(5-2)(7-2)$ gaps of $2$ in $D_{7}$. And the same number of gap $4$ in $D_{7}$. Hence, we can assume that $L_u$ and $L_v$ are not the column passing through $a_{i_0}$.\n\nSince every two odd composite numbers with gap $2$ are coprime, the elements at the same level of $L_u$ and $L_v$(i.e., they are the form like $u+210(k_s-1)$ and $v+210(k_s-1)$) have different prime divisors.\n\nHence, without loss of generality, we can choose $L_u$ and $L_v$ such that on them there are at least\n\\[\nN:=\\biggl[(3-2)(5-2)(7-2)\\cdots(p_m-2)\\cdot\\frac{1}{15}\\biggr]\n\\]\npairs of composite numbers with gap $2$ in $M_{p_m}^{(0)}$. Each pair of such composite numbers have at least four distinct prime divisors. In the Figure \\ref{fig:main}, $*$ is a central position (not a number) in $M_{p_m}^{(0)}$. For the pairs of red points in the figure, they only can be eliminated by the primes in the set $\\{p_{m+1},p_{m+2},\\ldots,p_{m+k}\\}$. Here $p_{m+k}$ is the largest prime less than $\\sqrt{X\/2}$, and $X=1+L(\\mathcal{P}_{p_n})=1+\\prod_{i=1}^{m}p_i$ is the last number in $M_{p_m}^{(0)}$.\n\nThere are at least $N-\\frac{\\sqrt{X\/2}}{210}$ pair of red points with skip $2$ on the lines $L_u$ and $L_v$. They should be eliminated by these primes $p_{m+1},p_{m+2},\\ldots,p_{m+k}$.\n\n\n\\begin{figure}[htbp]\n \\centering\n\\begin{tikzpicture}[scale=1\n\n\\draw[gray] (0,10) -- (0,0);\n\\draw[gray] (0,0) -- (5,0);\n\\draw[gray] (5,0) -- (5,10);\n\\draw[gray] (5,10) -- (0,10);\n\n\\draw[gray] (0,8) -- (5,8);\n\\draw[gray] (0,4) -- (5,4);\n\\draw[gray,dashed] (0,6) -- (5,6);\n\n\\draw[gray,dashed] (0,7.99) -- (5,7.99);\n\\draw[gray,dashed] (0,4.01) -- (5,4.01);\n\\draw[gray,dashed] (0.01,4.01) -- (0.01,7.99);\n\\draw[gray,dashed] (4.99,4.01) -- (4.99,7.99);\n\n\\draw[gray,dashed] (0,6.75) -- (5,6.75);\n\n\\draw[gray] (2,10) -- (2,0);\n\\draw[gray] (3,10) -- (3,0);\n\\draw[gray] (3.5,10) -- (3.5,0); \n\n\\draw[gray] (3,10.4) -- (3,10.6);\n\\draw[gray] (3.5,10.4) -- (3.5,10.6);\n\\draw[gray][<->] (3,10.5) -- (3.5,10.5);\n\\node at (3.25,10.7) {\\footnotesize $2$};\n\n\\node at (0,7.5)[anchor=east] {\\footnotesize $M_{p_m}$};\n\\node at (2,7.8)[anchor=east] {\\footnotesize $p_{m+1}$};\n\\node at (2,7.6)[anchor=east] {\\footnotesize $p_{m+2}$};\n\\node at (1.75,7.35)[anchor=east] {\\footnotesize $\\vdots$};\n\\node at (2,6.9)[anchor=east] {\\footnotesize $p_{m+k}$};\n\\node at (2.5,6.9) {\\footnotesize $<\\sqrt{X\/2}$};\n\n\\node at (3,10.4)[anchor=north] {\\footnotesize $L_u$};\n\\node at (3.5,10.4)[anchor=north] {\\footnotesize $L_v$};\n\n\\node at (2.5,6) {\\footnotesize $*$};\n\\node at (5,4.2)[anchor=east] {\\footnotesize $X$};\n\n\\fill [black] ($(0.5,9.8)$) circle (1.5pt);\n\\fill [black] ($(0.5,9.3)$) circle (1.5pt);\n\\fill [black] ($(0.5,8.6)$) circle (1.5pt);\n\n\\fill [black] ($(1,9.6)$) circle (1.5pt);\n\\fill [black] ($(1,8.9)$) circle (1.5pt);\n\\fill [black] ($(1,8.3)$) circle (1.5pt);\n\\fill [black] ($(1,8.6)$) circle (1.5pt);\n\n\\fill [black] ($(1.5,9.3)$) circle (1.5pt);\n\\fill [black] ($(1.5,8.8)$) circle (1.5pt);\n\n\\fill [black] ($(2.5,9.8)$) circle (1.5pt);\n\\fill [black] ($(2.5,8.9)$) circle (1.5pt);\n\\fill [black] ($(2.5,8.2)$) circle (1.5pt);\n\n\\fill [black] ($(3,9.5)$) circle (1.5pt);\n\\fill [black] ($(3,8.7)$) circle (1.5pt);\n\\fill [black] ($(3,8.4)$) circle (1.5pt);\n\n\\fill [black] ($(3.5,9.6)$) circle (1.5pt);\n\\fill [black] ($(3.5,9.3)$) circle (1.5pt);\n\\fill [black] ($(3.5,8.5)$) circle (1.5pt);\n\n\\fill [black] ($(4,9.4)$) circle (1.5pt);\n\\fill [black] ($(4,8.6)$) circle (1.5pt);\n\\fill [black] ($(4,8.2)$) circle (1.5pt);\n\n\\fill [black] ($(4.5,9.7)$) circle (1.5pt);\n\\fill [black] ($(4.5,9.2)$) circle (1.5pt);\n\\fill [black] ($(4.5,8.3)$) circle (1.5pt);\n\\fill [black] ($(2,9.8)$) circle (1.5pt);\n\\fill [black] ($(2,9.3)$) circle (1.5pt);\n\\fill [black] ($(2,8.8)$) circle (1.5pt);\n\\fill [black] ($(2,8.3)$) circle (1.5pt);\n\\fill [black] ($(2,7.8)$) circle (1.5pt);\n\\fill [black] ($(2,7.6)$) circle (1.5pt);\n\\fill [black] ($(2,7.2)$) circle (1.5pt);\n\\fill [black] ($(2,6.9)$) circle (1.5pt);\n\\fill [black] ($(2,6.6)$) circle (1.5pt);\n\\fill [black] ($(2,6.3)$) circle (1.5pt);\n\\fill [black] ($(2,5.8)$) circle (1.5pt);\n\\fill [black] ($(2,5.6)$) circle (1.5pt);\n\\fill [black] ($(2,5.3)$) circle (1.5pt);\n\\fill [black] ($(2,4.8)$) circle (1.5pt);\n\\fill [black] ($(2,4.6)$) circle (1.5pt);\n\\fill [black] ($(2,4.3)$) circle (1.5pt);\n\\fill [black] ($(2,3.8)$) circle (1.5pt);\n\\fill [black] ($(2,3.6)$) circle (1.5pt);\n\\fill [black] ($(2,3.3)$) circle (1.5pt);\n\\fill [black] ($(2,2.8)$) circle (1.5pt);\n\\fill [black] ($(2,2.6)$) circle (1.5pt);\n\\fill [black] ($(2,2.3)$) circle (1.5pt);\n\\fill [black] ($(2,1.8)$) circle (1.5pt);\n\\fill [black] ($(2,1.6)$) circle (1.5pt);\n\\fill [black] ($(2,1.3)$) circle (1.5pt);\n\\fill [black] ($(2,0.8)$) circle (1.5pt);\n\\fill [black] ($(2,0.6)$) circle (1.5pt);\n\\fill [black] ($(2,0.3)$) circle (1.5pt);\n\n\\fill [blue] ($(3, 7.8)$) circle (2pt);\n\\fill [blue] ($(3.5,7.8)$) circle (2pt);\n\\fill [blue] ($(3, 7.5)$) circle (2pt);\n\\fill [blue] ($(3.5,7.5)$) circle (2pt);\n\\fill [blue] ($(3, 7.1)$) circle (2pt);\n\\fill [blue] ($(3.5,7.1)$) circle (2pt);\n\\fill [red] ($(3, 6.5)$) circle (2pt);\n\\fill [red] ($(3.5,6.5)$) circle (2pt);\n\\fill [red] ($(3, 6.2)$) circle (2pt);\n\\fill [red] ($(3.5,6.2)$) circle (2pt);\n\\fill [red] ($(3, 5.6)$) circle (2pt);\n\\fill [red] ($(3.5,5.6)$) circle (2pt);\n\\fill [red] ($(3, 5.3)$) circle (2pt);\n\\fill [red] ($(3.5,5.3)$) circle (2pt);\n\\fill [red] ($(3, 5.0)$) circle (2pt);\n\\fill [red] ($(3.5,5.0)$) circle (2pt);\n\\fill [red] ($(3, 4.5)$) circle (2pt);\n\\fill [red] ($(3.5,4.5)$) circle (2pt);\n\\fill [red] ($(3, 4.2)$) circle (2pt);\n\\fill [red] ($(3.5,4.2)$) circle (2pt);\n\n\\end{tikzpicture}\n \\caption{Here $*$ is the center of the $M_{p_m}^{(0)}$}\\label{fig:main}\n\\end{figure}\n\nDuring the process of the elimination by the primes, we should take into account the composite numbers with same prime divisors. Note that the largest value of the red points is greater or equal than\n\\begin{equation}\\label{eqn:main}\n210\\cdot\\biggl(\\frac{1}{15}\\prod_{i=2}^{m}(p_i-2)-\\frac{\\sqrt{X\/2}}{210}\\biggr)+\\sqrt{X\/2}=14\\prod_{i=2}^{m}(p_i-2).\n\\end{equation}\n\n\nThus we should have the following inequality\n\\begin{equation}\\label{eqn:main-inequality}\n14\\prod_{i=2}^{m}(p_i-2)\\cdot(1-\\frac{1}{p_{m+1}})(1-\\frac{1}{p_{m+2}})\\cdots(1-\\frac{1}{p_{m+k}})\\leqslant\\pi(X)-m.\n\\end{equation}\nNote that the left hand side\n\\[\n\\begin{split}\n14\\prod_{i=2}^{m}(p_i-2)\\cdot\\prod_{i=m+1}^{m+k}(1-\\frac{1}{p_{i}})\n&=28\\prod_{i=2}^{m}\\frac{p_i-2}{1-\\frac{1}{p_{i}}}\\cdot\\prod_{i=1}^{m+k}(1-\\frac{1}{p_{i}})\\\\\n&=28\\cdot\\prod_{i=2}^{m}p_i\\cdot\\prod_{i=2}^{m}(1-\\frac{1}{p_i-1})\\cdot\\prod_{i=1}^{m+k}(1-\\frac{1}{p_i})\\\\\n&>14\\cdot\\prod_{i=2}^{m}p_i\\cdot\\prod_{i=2}^{m}(1-\\frac{1}{p_i})\\cdot\\prod_{i=1}^{m+k}(1-\\frac{1}{p_i})\\\\\n&=14\\cdot\\prod_{i=1}^{m}p_i\\cdot\\prod_{i=1}^{m}(1-\\frac{1}{p_i})\\cdot\\prod_{i=1}^{m+k}(1-\\frac{1}{p_i})\\\\\n\\end{split}\n\\]\nBy Merten's estimates\\cite{Hildebrand} in Lemma \\ref{lem:Merten-estimate}, we have\n\\[\n\\begin{split}\n&14\\cdot\\prod_{i=1}^{m}p_i\\cdot\\prod_{i=1}^{m}(1-\\frac{1}{p_i})\\cdot\\prod_{i=1}^{m+k}(1-\\frac{1}{p_i})\\\\\n=&14(X-1)\\cdot\\frac{e^{-\\gamma}}{\\log p_m}\\Bigl(1+O(\\frac{1}{\\log p_m})\\Bigr)\\cdot\\frac{e^{-\\gamma}}{\\log p_{m+k}}\\Bigl(1+O(\\frac{1}{\\log p_{m+k}})\\Bigr)\\\\\n>&13X\\cdot\\frac{e^{-\\gamma}}{\\log p_m}\\cdot\\frac{e^{-\\gamma}}{\\log p_{m+k}}\\\\\n>&\\frac{13}{e^{2\\gamma}\\log p_m}\\cdot\\frac{X}{\\log\\sqrt{X\/2}}\\\\\n>&\\frac{26}{e^{2\\gamma}\\log p_m}\\cdot\\frac{X}{\\log X}.\n\\end{split}\n\\]\nIt will violate the inequality \\eqref{eqn:main-inequality} if the prime number $p_m$ satisfies $\\frac{26}{e^{2\\gamma}\\log p_m}\\geqslant 2\\log 2$ due to the inequality \\eqref{eqn:Erdos}.\n\n Note that $e^{\\gamma}\\approx 1.78107241799$ \\cite{Euler-constant}. Let $p_m\\leqslant e^{5.9}\\approx 365.04$, then we will have\n\\[\n\\frac{26}{e^{2\\gamma}\\log p_m}> 2\\log 2.\n\\]\n\n If $p_m>e^{5.9}$, we change $M_7$ to some $M_{p_n}$ with longer period. Then the coefficient in \\eqref{eqn:main} will be $\\prod_{i=1}^{n}p_i\/\\prod_{i=2}^{n}(p_i-2)$.\n\nNext we follow the same arguments. In detail, equation \\eqref{eqn:main} becomes\n\\[\n\\prod_{i=1}^{n}p_i\\cdot\\biggl(\\frac{1}{\\prod_{i=2}^{n}(p_i-2)}\\prod_{i=2}^{m}(p_i-2)-\\frac{\\sqrt{X\/2}}{\\prod_{i=1}^{n}p_i}\\biggr)+\\sqrt{X\/2}=\\frac{\\prod_{i=1}^{n}p_i}{\\prod_{i=2}^{n}(p_i-2)}\\cdot\\prod_{i=2}^{m}(p_i-2).\n\\]\nAnd \\eqref{eqn:main-inequality} becomes\n\\begin{equation}\\label{eqn:main-inequality2}\n\\frac{\\prod_{i=1}^{n}p_i}{\\prod_{i=2}^{n}(p_i-2)}\\cdot\\prod_{i=2}^{m}(p_i-2)\\cdot(1-\\frac{1}{p_{m+1}})(1-\\frac{1}{p_{m+2}})\\cdots(1-\\frac{1}{p_{m+k}})\\leqslant\\pi(X)-m.\n\\end{equation}\n\nThe left hand side becomes\n\\[\n\\begin{split}\n&\\frac{\\prod_{i=1}^{n}p_i}{\\prod_{i=2}^{n}(p_i-2)}\\cdot\\prod_{i=2}^{m}(p_i-2)\\cdot\\prod_{i=m+1}^{m+k}(1-\\frac{1}{p_{i}})\\\\\n=&2\\cdot\\prod_{i=2}^{n}\\frac{p_i}{p_i-2}\\cdot\\prod_{i=2}^{m}\\frac{p_i-2}{1-\\frac{1}{p_{i}}}\\cdot\\prod_{i=2}^{m+k}(1-\\frac{1}{p_{i}})\\\\\n=&4\\cdot\\prod_{i=2}^{n}\\frac{p_i}{p_i-2}\\cdot\\prod_{i=2}^{m}p_i\\cdot\\prod_{i=2}^{m}(1-\\frac{1}{p_i-1})\\cdot\\prod_{i=1}^{m+k}(1-\\frac{1}{p_i})\\\\\n>&2\\cdot\\prod_{i=2}^{n}\\frac{p_i}{p_i-2}\\cdot\\prod_{i=2}^{m}p_i\\cdot\\prod_{i=2}^{m}(1-\\frac{1}{p_i})\\cdot\\prod_{i=1}^{m+k}(1-\\frac{1}{p_i})\\\\\n=&2\\cdot\\prod_{i=2}^{n}\\frac{p_i}{p_i-2}\\cdot\\prod_{i=1}^{m}p_i\\cdot\\prod_{i=1}^{m}(1-\\frac{1}{p_i})\\cdot\\prod_{i=1}^{m+k}(1-\\frac{1}{p_i}).\\\\\n\\end{split}\n\\]\nBy Merten's estimates, we have\n\\[\n\\begin{split}\n&2\\cdot\\prod_{i=2}^{n}\\frac{p_i}{p_i-2}\\cdot\\prod_{i=1}^{m}p_i\\cdot\\prod_{i=1}^{m}(1-\\frac{1}{p_i})\\cdot\\prod_{i=1}^{m+k}(1-\\frac{1}{p_i})\\\\\n=&2\\cdot\\prod_{i=2}^{n}\\frac{p_i}{p_i-2}\\cdot(X-1)\\cdot\\frac{e^{-\\gamma}}{\\log p_m}\\cdot\\biggl(1+O(\\frac{1}{\\log p_m})\\biggr)\\cdot\\frac{e^{-\\gamma}}{\\log p_{m+k}}\\cdot\\biggl(1+O(\\frac{1}{\\log p_{m+k}})\\biggr)\\\\\n>&(2\\prod_{i=2}^{n}\\frac{p_i}{p_i-2}-1)X\\cdot\\frac{e^{-2\\gamma}}{\\log p_m\\cdot\\log p_{m+k}}\\\\\n>&(2\\prod_{i=2}^{n}\\frac{p_i}{p_i-2}-1)X\\cdot\\frac{1}{e^{2\\gamma}\\log p_m}\\cdot\\frac{1}{\\log\\sqrt{X\/2}}\\\\\n>&\\frac{2(2\\prod_{i=2}^{n}\\frac{p_i}{p_i-2}-1)}{e^{2\\gamma}\\log p_m}\\cdot\\frac{X}{\\log X}=:\\alpha\\cdot\\frac{X}{\\log X}.\n\\end{split}\n\\]\n\n\\noindent{\\bf Claim.} For this $p_m$, there exists a number $n$ with $p_n < p_m$, such that\n\\[\n2(2\\prod_{i=2}^{n}\\frac{p_i}{p_i-2}-1)> (2\\log 2)\\cdot e^{2\\gamma}\\log p_m.\n\\]\nThat is, $\\alpha > 2\\log 2$.\n\nHence, if denote the left hand side of \\eqref{eqn:main-inequality2} as $LHS$, then\n\\[\nLHS>(2\\log 2)\\frac{X}{\\log X}>\\pi(X),\n\\]\nwhich violates the inequality \\eqref{eqn:main-inequality2}.\n\nThus our assumption is wrong. Note that here we work on the table of $M_{p_n}$ similar with Figure \\ref{fig:M7-new} and Figure \\ref{fig:main}. Hence, there exists at least another $a_{j_0}$ such that the set $\\{x_{j_0 k}\\mid x_{j_0 k}=a_{j_0}+(k-1)\\prod_{i=1}^{n}p_i\\}$ contains infinitely many primes.\n\n{\\bf In fact} by above argument we have proved that {\\bf each line of the pair} $L_u$ and $L_v$ with gap $2$ or $4$ in the table about $M_{p_n}$ contains infinitely many primes.\n\nObviously, it infers that {\\bf each line of the pair} $L_u$ and $L_v$ with gap $2$ or $4$ in the table about $M_{7}$ (see Figure \\ref{fig:main}) contains infinitely many primes. And it also true that for the table of $M_3$ and $M_5$. Therefore, since the bigger gap such as $6$ comes from add $2$ and $4$, we conclude that the pair lines $L_u$ and $L_v$ with gap $6$ also both contains infinitely many primes.\n\nTherefore, every column $\\{x_{j k}\\mid x_{j k}=a_{j}+(k-1)\\prod_{i=1}^{n}p_i\\}$ (the $j$-th column) contains infinitely many primes. Here $n=1,2,3,\\ldots$.\n\n\n\nBy using the same idea, we can prove that if fix one of rows of the block $M_7$ in Figure \\ref{fig:M7-new}, then the set\n\\begin{equation}\\label{eqn:2}\n\\{x_{in}\\mid x_{in}=a_{i}+n(k_0-1)\\cdot 210,\\quad n=1,2,3,\\ldots\\}\n\\end{equation}\nalso contains infinitely many primes.\n\nFor the general arithmetic progression $A:=\\{ak+d\\mid k\\in\\mathbb{N}\\}$, $(a,d)=1$. First we can choose suitable $p_k$ such that $d$ or $ah+d$ belongs the set $M_{p_k}$ for some $h$. Then fix this $p_k$, considering the similar rectangle as Figure \\ref{fig:M7-new}. We consider the subset\n\\[\n\\{a(\\prod_{i=1}^{n}p_i)k+d\\mid k=1,2,\\ldots\\}.\n\\]\nThus it contains infinitely many primes. It completes the proof of the theorem of Dirichlet.\n\\end{proof}\n\n\n\\bigskip\n\n\n\n\n\n\n\n\\section{Estimation of the prime counting function in special cases}\n\n\nIn $M_{p_m}^{(0)}$, there are $T_{p_m}$ elements in it. Then after deleting the numbers of the form $p_ih$, $i=m+1,m+2,\\ldots,m+K$, the left are all prime numbers. Here we assume $p_{m+K}$ is the largest prime number which is less than or equal to $X:=L(\\mathcal{P}_{p_m})=1+\\prod_{i=1}^{m}p_i$. Hence we have\n\\[\n\\begin{split}\n\\pi(X)-m&\\geqslant T_{p_m}\\cdot(1-\\frac{1}{p_{m+1}})\\cdot(1-\\frac{1}{p_{m+2}})\\cdots(1-\\frac{1}{p_{m+K}})\\\\\n&=\\prod_{i=1}^{m}(p_i-1)\\cdot\\frac{\\prod_{i=1}^{m+K}(1-p_{i}^{-1})}{\\prod_{i=1}^{m}(1-p_{i}^{-1})}\\\\\n&=\\prod_{i=1}^{m}p_i\\cdot\\prod_{i=1}^{m+K}(1-p_{i}^{-1})\n\\end{split}\n\\]\n\nRecall the Euler's product formula for zeta function\n\\[\n\\zeta(s)=\\sum_{n=1}^{\\infty}\\frac{1}{n^s}=\\prod_{p\\ \\mathrm{prime}}\\frac{1}{1-p^{-s}}.\n\\]\nFor the case $s=1$, we will use the following lemma.\n\\begin{lem}[Merten's estimates\\cite{Hildebrand}]\\label{lem:Merten-estimate}\n\\[\n\\prod_{p\\leqslant x}(1-\\frac{1}{p})=\\frac{e^{-\\gamma}}{\\log x}\\biggl(1+O(\\frac{1}{\\log x})\\biggr),\n\\]\nwhere $\\gamma$ is Euler's constant.\n\\end{lem}\n\n\n\n\\begin{lem}\nFor any positive integer $N$, we have the inequality\n\\[\n\\log(1+N)<\\sum_{n=1}^{N}\\frac{1}{n}<1+\\log N.\n\\]\n\\end{lem}\n\n\nHence, by the above lemmas, we have\n\\[\n\\begin{split}\n\\pi(X)&\\geqslant\\prod_{i=1}^{m}p_i\\cdot\\prod_{i=1}^{m+K}(1-p_{i}^{-1})+m\\\\\n&=(X-1)\\cdot\\frac{e^{-\\gamma}}{\\log X}\\biggl(1+O(\\frac{1}{\\log X})\\biggr)+m.\n\\end{split}\n\\]\nOn the other hand, it is easy to see that\n\\[\n\\begin{split}\n\\pi(X)&\\leqslant T_{p_m}\\cdot 2\\cdot(1-\\frac{1}{p_{m+1}})\\cdot(1-\\frac{1}{p_{m+2}})\\cdots(1-\\frac{1}{p_{m+K}})\\\\\n&=2\\prod_{i=1}^{m}p_i\\cdot\\prod_{i=1}^{m+K}(1-p_{i}^{-1})+m\\\\\n&=2(X-1)\\cdot\\frac{e^{-\\gamma}}{\\log X}\\biggl(1+O(\\frac{1}{\\log X})\\biggr)+m.\n\\end{split}\n\\]\nTherefore, we have\n\\[\n\\frac{1}{e^{\\gamma}}\\cdot\\frac{X-1}{\\log X}\\biggl(1+O(\\frac{1}{\\log X})\\biggr)+m\n\\leqslant\\pi(X)\\leqslant\n\\frac{2}{e^{\\gamma}}\\cdot\\frac{X-1}{\\log X}\\biggl(1+O(\\frac{1}{\\log X})\\biggr)+m.\n\\]\nSince $e^m<1+p_1p_2\\cdots p_m=X$, $m<\\log X$. Thus we have\n\\[\n\\pi(X)\\asymp\\frac{X}{\\log X},\\quad\\text{where}\\ X=1+p_1p_2\\cdots p_m.\n\\]\nTherefore, we get an inequality of prime counting function like the Chebyshev's inequality.\n\n\n\n\n\n\n\n\n\n\\bigskip\n\n\\noindent{\\bf Acknowledgments :}\nWe would like to express our gratitude to Professor Vilmos Komornik who invite the author to visit the Department of Mathematics in University of Strasbourg. This work is done during the stay of the author in Strasbourg.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Conclusions} \n\\vspace{-.1in}\n\nWe have introduced\nthe first method for discovering \ndownward-\\monotonic\\xspace operators\\xspace that is universally applicable.\nPrevious work on automatically detecting {\\po}s\\xspace assumed the existence of a high-quality collection of NPIs, which renders it inapplicable in most languages, where such a resource does not exist. \nWe\novercome this limitation by employing a novel {\\em co-learning} approach, and \ndemonstrate\n its effectiveness on Romanian.\n\n{Also,} we introduce the concept of {\\em pseudo-NPIs}.\nAuxiliary experiments described \\elsewhere show that \n\\mbox{pNPIs\\xspace} are actually more effective \nseeds than a noisy ``true'' NPI list. \n\n\nFinally, we noted some cross-linguistic differences\nin performance, \nand found an interesting connection between these \ndifferences\nand Haspelmath's \n\\citep{Haspelmath:01a} \ncharacterization of cross-linguistic variation in\nthe\n occurrence of indefinite pronouns.\n\n\n\n\n\n\n\n\n\\section{Introduction}\n\n{\\small{\\it\n\\begin{tabular}{ll}\nCristi: & ``Nicio\" ... is that adjective you've mentioned. \\\\\nAnca: & A negative pronominal adjective. \\\\\nCristi: &You mean there are people who analyze that \\\\ & kind of thing? \\\\\nAnca: & The Romanian Academy. \\\\\nCristi: & They're crazy. \\\\\n\\end{tabular}}\n\n\\hfill ---From the movie {\\em Police, adjective}\n}\n\\medskip\t\n\n\nDownward-\\monotonic\\xspace \\shrink{(DE\\xspace)} operators\\xspace\\ {are an interesting and varied class of\nlexical items that change the \ndefault way of dealing with certain types of inferences.\n They thus} play an important role \n\nin \nunderstanding natural language \n\\citep[etc.]{vanBenthem:86a,SanchezValencia:91a,Dowty:94a,vanderWouden:97a}.\n\n\nWe explain what\n{downward \\monotonic\\xspace} \nmeans by first demonstrating the ``default\" behavior, which is {\\em upward \\monotonic\\xspace}. The word \\ex{observed} is an example upward-\\monotonic\\xspaceoperator\\xspace:\nthe statement\n\\newcommand{\\vspace*{-.09in}}{\\vspace*{-.09in}}\n\\vspace*{-.2in}\n\\begin{enumerate}[label=(\\roman*)]\n\\item \\ex{Witnesses \\showupwardop{observed} opium use.} \n\\end{enumerate} \n\\vspace*{-.09in}\nimplies \n\\vspace*{-.09in}\n\\begin{enumerate}[label=(\\roman*),start=2]\n\\item \\ex{Witnesses \\showupwardop{observed} narcotic use.}\n \\end{enumerate}\n but not vice versa (we write i $\\Rightarrow (\\centernot{\\Leftarrow})$ ii). \nThat is, the truth value is preserved if we replace the argument of an upward-\\monotonic\\xspace operator\\xspace by\n a superset (a more general version); in our case,\nthe set \\ex{opium use} was replaced by the superset \\ex{narcotic use}.\n\n{\\em Downward-\\monotonic\\xspace (DE\\xspace)} \n(also known as {\\em downward monotonic} \nor {\\em monotone decreasing}) operators\\xspace\nviolate this default inference rule\\unshrink{:\nwith {\\po}s\\xspace, reasoning instead goes from ``sets to subsets\"}.\nAn example is the word \\ex{bans}:\n\\begin{quote}\\ex{The law \\showop{bans} opium use} \\\\ $\\centernot\\Rightarrow (\\Leftarrow)$\\\\ \\ex{The law \\showop{bans} narcotic use}.\\end{quote}\nAlthough DE\\xspace behavior represents an exception to the \ndefault, \n{\\po}s\\xspace are as a class rather common.\nThey are also quite diverse in sense and even part of speech. \nSome are simple negations, such as \\ex{not},\nbut some other English {\\po}s\\xspace are\n\\ex{without}, \\ex{reluctant to}, \\ex{to doubt}, and \\ex{to allow}.\\footnote{Some examples showing different constructions for analyzing these operators\\xspace: \\ex{The defendant does {not} own a blue car} $\\centernot\\Rightarrow (\\Leftarrow)$ \\ex{The defendant does not own a car}; \\ex{They are reluctant to tango} $\\centernot\\Rightarrow (\\Leftarrow)$ \\ex{They are reluctant to dance}; \\ex{Police doubt Smith threatened Jones} $\\centernot\\Rightarrow (\\Leftarrow)$ \\ex{Police doubt Smith threatened Jones or Brown}; \\ex{You are allowed to use Mastercard} $\\centernot\\Rightarrow (\\Leftarrow)$ \\ex{You are allowed to use any credit card}. }\nThis variety makes\nthem hard to extract\nautomatically.\n\n\n\n\n\n\nBecause {\\po}s\\xspace\n violate the \ndefault\n ``sets to supersets'' inference,\nidentifying them can potentially improve performance in many\nNLP tasks\\unshrink{.\n\nPerhaps the most obvious such tasks are those involving textual\nentailment}, such as \nquestion answering, information extraction, summarization, and the evaluation of machine translation \\citep{Dagan+Glickman+Magnini:06a}.\nResearchers are in fact beginning to build textual-entailment systems that\ncan handle inferences involving {\\polong}s\\xspace other than simple negations,\nalthough these systems almost all rely on small handcrafted lists of {\\po}s\\xspace\n\\citep{Nairn+Condoravdi+Kartunnen:06a,MacCartney+Manning:08a,Christodoulopoulos:08a,Bar-Haim+al:08a,Breck:09a}.\\footnote{The exception \\citep{Breck:09a} employs the list automatically derived by \\citet{Danescu-Niculescu-Mizil+Lee+Ducott:09a}, described later.}\nOther\napplication areas are natural-language generation and human-computer interaction, \nsince downward-entailing inferences\ninduce greater cognitive load than inferences in the opposite\ndirection \\citep{Geurts+vanDerSlik:05a}.\n\n\\unshrink{Most NLP systems for the applications mentioned above have only been deployed for a small subset of languages. \nA \nkey factor is the lack of relevant resources for other languages. While one approach would be to separately develop a method to acquire such resources for each\nlanguage\nindividually, we instead aim to ameliorate the resource-scarcity problem \nin the case of {\\po}s\\xspace\nwholesale:\nwe propose\na single \nunsupervised\nmethod that can\n extract {\\po}s\\xspace\n\nin\nany language\n\nfor which \nraw text\ncorpora exist. \n}\n\n\n{\\paragraph{Overview of our work}\n\n\n\n{\n\\renewcommand{\\ex}[1]{{\\it #1}}\n\\newcommand{\\poex}[1]{\\showop{#1}}%\n\\newcommand{\\npiex}[1]{{\\it #1}}\n\\newcommand{$\\times$\\xspace}{$\\times$\\xspace}\n\\begin{figure*}[t]\n\\begin{center}\n{\\small}\n\\begin{tabular}{r|ll}\n & \\multicolumn{2}{c}{{\\bf NPIs}} \\\\ \n {\\bf {\\po}s\\xspace} & \\multicolumn{1}{c}{$\\npiex{any}^{5}$} &\n\\multicolumn{1}{c}{\\npiex{\\npiex{have a clue}}, idiomatic sense} \\\\ \\cline{2-3}%\n\\poex{not} or \\poex{n't} & $\\checkmark$ We do\\poex{n't} have \\npiex{any} apples & $\\checkmark$ We do\\poex{n't} \\npiex{have a clue} \\\\ \n\\poex{doubt}\t\t\t& \\checkmark I \\poex{doubt} they have \\npiex{any}\napples & $\\checkmark$ I \\poex{doubt} they \\npiex{have a clue} \\\\ %\nno \\po & $\\times$\\xspace They have \\npiex{any} apples & $\\times$\\xspace They \\npiex{have a clue} \\\\\n\\end{tabular}\n\\end{center}\n\\caption{\\label{fig:ladexamples} Examples consistent with Ladusaw's hypothesis that NPIs can only occur within the scope of {\\po}s\\xspace. A $\\checkmark$ denotes an acceptable sentence; a $\\times$ denotes an unacceptable sentence. %\n}\n\\end{figure*}\n}\n\n\nOur approach takes the English-centric work of \\citet{Danescu-Niculescu-Mizil+Lee+Ducott:09a} --- \\dld for short --- as a starting point, as \nthey present\n the first and, until now, only algorithm for automatically extracting {\\po}s\\xspace from data. However, \\unshrink{our work departs significantly from \\dld in the following key respect. \n\n\n}\\dld critically depends on \naccess to\na high-quality, carefully curated collection of \n{\\em\n negative polarity items (NPIs)} --- lexical items such as\n\\ex{any},\n\\ex{ever},\nor \nthe idiom \\ex{have a clue} \nthat tend to occur only in negative\nenvironments\n(see \n\\S \\ref{sec:dld} for more \ndetails).\n\\dld \nuse\nNPIs\nas signals of the occurrence of \n{\\polong}s\\xspace.\nHowever, \n %\n almost every language other than English lacks a high-quality accessible NPI list.\n\n\n\n\n\n\n\n\n\n\n\n\n\n{\nTo circumvent this \\shrink{widespread }problem\\shrink{ of lack of NPI lists}, we introduce a\nknowledge-lean {\\em co-learning} approach\\shrink{ to\nthe unsupervised discovery of {\\polong}s\\xspace}. }\nOur algorithm is initialized with a very small seed set of NPIs\n (which we describe how to \n\ngenerate),\n and then iterates between (a) discovering a set of {\\po}s\\xspace using a collection of {\\em pseudo-NPIs}\n \n \n --- a concept we introduce ---\n and\n(b) using the newly-acquired {\\po}s\\xspace to detect new pseudo-NPIs.\n\n{\\vspace*{-.05in}}\n\n\\paragraph{Why this isn't obvious} \nAlthough the algorithmic idea sketched above \nseems quite simple, it is important to note that {prior experiments in that direction have not proved fruitful.} \n{Preliminary} work on learning\n(German) NPIs using a small list of\nsimple known {\\po}s\\xspace did not yield strong results \n\\citep{Lichte+Soehn:07a}. \n\\citet{Hoeksema:97a}\ndiscusses\nwhy NPIs might be hard to learn from data\n %\n\\footnote{In fact, \nhumans can have trouble agreeing on NPI-hood;\nfor \n \ninstance,\n\\citet{Lichte+Soehn:07a} mention doubts about over half of\n\\citet{Kuerschner:83a}'s 344 manually collected\nGerman NPIs.}\nWe circumvent this problem because we are not interested in learning NPIs per se; rather, for our purposes, pseudo-NPIs suffice.\n{Also}, our preliminary work determined that one of the most famous co-learning algorithms, {\\em hubs and authorities} or {\\em HITS} \\citep{Kleinberg:98a}, is poorly suited to our problem.\\footnote{\n{We} explored three different edge-weighting schemes based on co-occurrence frequencies and seed-set membership, but the results were extremely poor; HITS invariably retrieved very frequent words.\n}\n\n\\paragraph{Contributions}\n\nTo begin with, \nwe apply our algorithm to produce the first large list of {\\po}s\\xspace for a language other than English.\nIn our case study \non\nRomanian\n(\\S \\ref{sec:results}), \n we achieve quite high precisions at $k$\n\n(for example, iteration achieves a precision at 30 of 87\\%).\n\n %\n\nAuxiliary experiments explore the effects of using a large but noisy NPI list, should one be available for the language in question. \n Intriguingly, we find that co-learning new pseudo-NPIs provides better results.\n %\n\n\n\n\n\nFinally (\\S\\ref{sec:multilingual}), we engage in some cross-linguistic analysis \nbased on the results of applying our algorithm to English. \nWe find\nthat\n there are some suggestive connections with \n\nfindings in linguistic typology.\n\n\\paragraph{Appendix available} A more complete account of our work and its implications can be found in a version of this paper containing appendices, available at \n\\elsewhereurl.\n\n\n %\n\n\n\n\n\\section{%\n\\dld: successes and challenges}\\label{sec:dld}\n\n\n\nIn this section, we briefly summarize those aspects of the \\dld\nmethod that are important to understanding how our \nnew co-learning method works.\n\n\\paragraph{{\\po}s\\xspace and NPIs}\nAcquiring {\\po}s\\xspace is challenging because of the \ncomplete\nlack of annotated data.\n\\dld's insight was to make \nuse of {\\em negative polarity items\n(NPIs)},\nwhich are words or phrases that tend to occur only in negative\ncontexts. The reason they \ndid\nso is that Ladusaw's hypothesis \\citep{Fauconnier:75a,Ladusaw:80a} asserts that {\\em NPIs only occur within the scope of {\\po}s\\xspace}.\n Figure \\ref{fig:ladexamples} depicts examples involving the \nEnglish NPIs \\ex{any}\\footnote{The \n{\\em free-choice} sense of \\ex{any}, as in \\ex{I\n can skim any paper in five minutes}, is a known exception.} and \\ex{have a clue} \n(in the idiomatic sense) \n that \nillustrate this relationship. \nSome other English NPIs are \\ex{ever}, \\ex{yet} and \\ex{give a damn}.\n\n\n\n\nThus, NPIs can be treated as clues that a \\po\n might be present (although {\\po}s\\xspace\n may also occur without NPIs).\n %\n\n\\paragraph{\\dld algorithm}\nPotential {\\po}s\\xspace are collected by extracting those words that\n appear in an NPI's \ncontext\n at least once.\\footnote{\\dld policies: (a) \n\n ``NPI context'' was defined as the part of the sentence to the left\nof the NPI\n up to the first comma, \n\n semi-colon or beginning of sentence; (b) to encourage the discovery of new {\\po}s\\xspace, those sentences containing one of a list of 10 well-known {\\po}s\\xspace were discarded. \n\nFor Romanian, we \ntreated\nonly negations (\\ex{nu} and \\ex{n-}) and questions as well-known \nenvironments.\n\n %\n } Then, the potential \n\n {\\po}s\\xspace $x$ are ranked by \n\n{\\small \\vspace*{-.2in}\n$$ f(x) := \\frac\n{\\mbox{fraction of NPI contexts that contain $x$} }\n{\\mbox{relative frequency of $x$ in the corpus}},\n$$\\vspace*{-.2in}}\n\n\\noindent which compares\n$x$'s\nprobability of occurrence conditioned on the appearance of an NPI with \nits\nprobability of occurrence overall.\\footnote{\\dld used an additional {\\em distilled} score, but we found that the distilled score performed worse on Romanian. \\shrink{Our Romanian results for both \\dld and our algorithm are\nbased on the undistilled scores\nso as to make the comparison fair to \\dld.}}\n\nThe method just outlined requires access to a list of NPIs. \\dld's\nsystem used\na subset of\n John Lawler's carefully curated and ``moderately\ncomplete'' \nlist of English \nNPIs.%\n\\footnote{\\url{http:\/\/www-personal.umich.edu\/~jlawler\/aue\/npi.html}} The\nresultant rankings of candidate\nEnglish\n {\\po}s\\xspace were judged to be of high quality.\n\n\\paragraph{The challenge in porting to other languages: cluelessness}\n\nCan\nthe unsupervised approach of\n \\dld be successfully applied to languages other than English? \n\nUnfortunately, for most\nother languages, it does not seem that large, high-quality NPI lists\nare available.\n\nOne might wonder whether one can circumvent the NPI-acquisition problem by simply translating \na known English NPI list into the target language. However,\nNPI-hood need not be preserved under\ntranslation \\citep{ Richter+Rado+Sailer:08a}.\nThus, for most\nlanguages, we\nlack the critical clues that \\dld depends on.\n\n\n\n\n\n\n\\section{Cross-linguistic analysis}\\label{sec:multilingual}\n\n\n\\input{acl-short-english}\n\n\n\\paragraph{Using translation}\nAnother interesting question is whether directly translating {\\po}s\\xspace from English is an alternative to our method. First, \nwe\nemphasize that there exists no complete\nlist of English {\\po}s\\xspace\n (the largest available \n\ncollection\nis the one extracted by \n\\dl\n). Second,\nwe do not know whether\n{\\po}s\\xspace in one language translate into {\\po}s\\xspace in another language.\nEven if that \nwere the case, and we\nsomehow had access to\nideal translations\nof \n\\dld's list,\nthere would still be considerable value in using our method: \n14 (39\\%) of our top\n36 highest-ranked Romanian {\\po}s\\xspace\nfor iteration 9\ndo not, according to \nthe Romanian-speaking author, \nhave English equivalents appearing on \\dld's \n90-item list.\nSome examples are: \n\\ex{ab\\c{t}inut} (abstained), \\ex{criticat} (criticized) and \\ex{reac\\c{t}ionat} (reacted).\n\\noindent Therefore, a significant fraction of the {\\po}s\\xspace derived by our co-learning algorithm would have been missed by the translation alternative even\nunder\n ideal conditions.\n\n\n\n\\section{Getting a clue}\n\n\n\nIn this section, we \ndevelop an iterative co-learning algorithm that\ncan extract {\\po}s\\xspace in the many languages where a high-quality NPI database is not available,\nusing Romanian as a case study.\n\n\\subsection{Data and evaluation paradigm}\nWe used Rada Mihalcea's corpus of $\\approx$1.45 million sentences of raw Romanian newswire articles.\n\n\nNote that we cannot\nevaluate impact on textual inference because, to our knowledge, no publicly available\ntextual-entailment\nsystem \nor evaluation data for Romanian exists. We therefore examine the system outputs\ndirectly\n to determine whether the top-ranked items are actually {\\po}s\\xspace or not. Our evaluation metric is precision at $k$ of a given system's ranked list of candidate {\\po}s\\xspace; \nit is not possible to evaluate recall since no list of Romanian {\\po}s\\xspace \nexists\n (a problem \n\nthat\nis precisely the motivation \n\nfor\n this paper).\n\nTo evaluate the results, two native Romanian speakers labeled the system outputs as being ``DE'', ``not DE'' or ``Hard (to decide)''.\nThe labeling protocol, which was somewhat complex to prevent bias, is described \\elsewhere\n(\\S7.1). \nThe complete system output and\nannotations are\npublicly available at: {\\hypersetup{urlcolor=blue}\\href{http:\/\/www.cs.cornell.edu\/\\%7Ecristian\/acl2010\/}{{http:\/\/www.cs.cornell.edu\/\\string~cristian\/acl2010\/}}\\hypersetup{urlcolor=black}}.\n\n\\subsection{Generating a seed set}\\label{sec:seed}\n\nEven though, as discussed\nabove, \nthe translation of an NPI need not be an NPI, a preliminary review of the literature indicates that in many\nlanguages, there is some NPI that can be translated as \\ex{any} or\nrelated forms like \\ex{anybody}. \nThus, with a small \namount of\neffort, one can\nform a minimal NPI seed\nset\nfor the \\dld method\nby using an appropriate target-language translation of \\ex{any}.\nFor Romanian, we used\n\\ex{vreo} and \\ex{vreun}, which are the feminine and masculine translations of English \\ex{any}.\n\n\\input{acl-short-main-results-fig}\n\n\n\\subsection{\\dld using the Romanian seed set}\nWe first check\nwhether \\dld with the two-item seed set described \nin \n\\S\\ref{sec:seed} \nperforms\n well on Romanian\\shrink{; after all, if the results are good, then there\nis no need \nfor\nnew algorithms}.\nIn fact, the results are fairly poor: for example, the precision at 30 is below 50\\%. (See \nblue\/dark bars in\nfigure\n\\numfigbarsnoiter\\\n\\elsewhere for detailed results.)\n\n\n\n\n\n\nThis relatively unsatisfactory performance\nmay be a consequence \nof the \nvery small\nsize of the NPI list employed,\n and \n\n may\ntherefore indicate that it would be fruitful to investigate automatically\nextending\nour list of clues. \n\n\n\n\n\n\n\\subsection{Main idea: a co-learning approach}\n\nOur main\n insight is that not only \ncan NPIs\nbe used as clues for finding {\\po}s\\xspace, as shown by \\dld, but\nconversely, \n{\\po}s\\xspace (if known)\ncan potentially be used to discover new NPI-like clues,\nwhich we refer to as {\\em pseudo-NPIs} (or {\\em pNPIs\\xspace} for short).\nBy ``NPI-like'' we mean, ``serve as \npossible indicators of the presence of {\\po}s\\xspace, regardless of whether they are actually restricted to \nnegative contexts, \nas true NPIs are''.\nFor example, in English newswire, the words \\ex{allegation} or \\ex{rumor} tend to occur \nmainly \nin DE\\xspace contexts,\nlike \n\\ex{\\showop{denied}} or \n\\ex{\\showop{dismissed}},\n even though they are clearly not true NPIs (the sentence \\ex{I heard a rumor} is fine).\nGiven this insight, we approach the problem using an {\\em iterative co-learning} paradigm\nthat integrates the search for new {\\po}s\\xspace \nwith a search for new pNPIs\\xspace. \n\nFirst, we describe an algorithm that is the ``reverse'' of \\dld (henceforth\n{\\em rDLD\\xspace}), in that it retrieves and ranks pNPIs\\xspace assuming a given list of {\\po}s\\xspace. \nPotential\npNPIs\\xspace are collected by extracting those words that appear in a DE\\xspace context (defined here,\nto avoid the problems of parsing or scope determination,\n as the part of the sentence to the right of a \\po, up to the first comma, semi-colon or end of sentence); these candidates $x$ are then ranked by\n\n{\\small \\vspace*{-.2in} $$\\scorepos_r(x) := \\frac\n{\\mbox{fraction of DE\\xspace contexts that contain $x$} }\n{\\mbox{relative frequency of $x$ in the corpus}}.\n$$\n}\\vspace*{-.2in}\n\n\n\n\nThen, our\n{\\em co-learning} algorithm\n consists of the iteration of the following two steps:\n\n \n\n\\begin{itemize}\n\n\n\\item \\step{DE\\xspace learning} Apply \\dld \nusing a set $\\clues$ of \npseudo-NPIs\nto retrieve a\nlist of candidate {\\po}s\\xspace ranked by $f$ (defined in Section \\ref{sec:dld}).\nLet ${\\cal D}$ be the top $n$ candidates in this list. \n\n\\item \\step{pNPI\\xspace learning} Apply rDLD\\xspace\n using the set ${\\cal D}$ to retrieve a list of\n\n pNPIs\\xspace \n ranked by $\\scorepos_r$; extend $\\clues$ with the top $n_r$ pNPIs\\xspace in this list. Increment $n$.\n\\end{itemize}\nHere, $\\clues$ is initialized with\nthe NPI seed set.\nAt each iteration, we consider the output of the algorithm\nto be the ranked list of {\\po}s\\xspace retrieved in \nthe DE\\xspace-learning step. \nIn our experiments, we initialized $n$ to 10 and set $n_r$ to 1.\n\n\n\n\n\\section{Romanian results}\\label{sec:results}\n\n\n\n\nOur results\n show\nthat there is indeed \nfavorable synergy \nbetween\nDE\\xspace-operator\\xspace and pNPI\\xspace retrieval. \nFigure \\ref{fig:nrops} plots the number of correctly retrieved {\\po}s\\xspace in the top $k$ outputs at each \niteration.\nThe point at iteration $0$ corresponds to a datapoint already discussed above, namely, \\dld applied to the two \\ex{any}-translation NPIs.\nClearly, we see general substantial improvement over \\dld, although the increases level off in later iterations. \n(Determining how to choose the optimal number of iterations is a subject for future research.) \n\n\n\n\n\n\n\n\n\n\n\n\nAdditional experiments, described {\\elsewhere}\n(\\S7.2),\nsuggest \nthat pNPIs\\xspace \ncan\neven be more effective clues \n\nthan a %\n noisy list of NPIs. \n\n (Thus, a larger seed set does not necessarily mean better performance.)\npNPIs\\xspace also have the advantage of being\nderivable automatically, and might be worth investigating from a linguistic perspective in their own right.\n\n\n\\section{Appendices}\\label{sec:appendices}\n\\subsection{Labeling protocol}\nFollowing \\dld, system outputs (i.e., candidate {\\po}s\\xspace) were combined with a set of {\\em distractors} --- words related to but not synonymous with the system outputs, according to WordNet. We chose distractors that were similar to real system outputs so that the distractors would not obviously ``stand out'' as seeming unusual. \n\nThe combined collection was presented in randomized order to \ntwo Romanian native speakers\n(one an author, one not), \nwhose task was to \nlabel each item as either \\mbox{``{DE\\xspace}\"}, ``not {DE\\xspace}\", or ``Hard\", meaning ``hard to decide'',\nand to justify their choice\nby providing\nan inference example of the kind we discussed in the Introduction.\nThe \njudges were\ninformed of the presence of distractors; this served to guard against the judges being biased towards simply defaulting to the ``{DE\\xspace}\" label.\nWe should mention that the annotation process \nwas\nvery time consuming\nbecause generating definitive example inferences can be quite difficult,\nlimiting the number of output items that \nwe could get\nlabeled.\\footnote{Annotators reported a rate of about 10 \nexamples per hour.}\nAlso, because creativity can be needed to construct the requisite supporting inferences, the two judges first worked independently, and then conferred to resolve their disagreements. \nThe complete system output and\nthe annotations are\n publicly available at: {\\small \n \\hypersetup{urlcolor=blue}\\href{http:\/\/www.cs.cornell.edu\/\\%7Ecristian\/acl2010\/}{{http:\/\/www.cs.cornell.edu\/\\string~cristian\/acl2010\/}}\\hypersetup{urlcolor=black}}.\n\n\n\n\\mynumberfig{\\numfigbarsnoiter}\n\\begin{figure}[thb]\n\\includegraphics[width=3.7in,viewport= 100 235 570 557,clip]{esub_figs2\/bars_noiter}\n\\caption{\\label{fig:barsnoiter} \nPrecision at $k= \\{10,20, ... ,50\\}$ using \\dld on Romanian. NPI lists used:\nthe two translations of \\ex{any} (blue\/dark bars); or the relatively large but noisy CoDII-NPI.ro\\xspace list (green\/light bars). \n} \n\\end{figure}\n\n\n\n\n\\mynumbersubsec{2}\n\\subsection{Re-ranking marginal NPIs}\\label{sec:marginal}\nIn this paper we\n proposed a \nDE\\xspace-operator\\xspace discovery algorithm that can be applied to languages for which there is no pre-existing collection of NPIs; this is useful because collecting NPIs is, as discussed in the Introduction, quite difficult\n---\ndetermining NPI-hood can be \nhard\n even for experts. \n\nHowever, there are a few languages wherein the effort has been expended to collect a large {\\em noisy} collection of NPIs, and Romanian is one of them. (This motivated our choice of this language.)\nCoDII-NPI.ro\\xspace\\footnote{\\scriptsize{\\url{http:\/\/www.sfb441.uni-tuebingen.de\/a5\/codii\/}, see Traw\\'nski and Soehn, ``A Multilingual Database of Polarity Items'', {\\it LREC} 2008.}}\ncontains 58 Romanian\nNPIs,\nbut \nin the opinion of the Romanian-speaking author, \nmany\nare {\\em marginal},\nby which we mean\nthat their NPI sense is very infrequent.\\footnote{For example, \nCoDII-NPI.ro\\xspace\ncontains the Romanian verb \\ex{a mi\\c{s}ca} (\\ex{to move}), which is very frequently used in positive contexts. The inclusion of marginal NPIs is presumably due to a design decision to make the list have high recall.}\nThe existence of CoDII-NPI.ro\\xspace allows us to ask: which is more effective, a list including both true and marginal NPIs (presuming one is lucky enough to have one), or pseudo-NPIs? \n\nFirst, it turns out that the CoDII-NPI.ro\\xspace list containing marginal NPIs is much less effective input to the \n\\dld method than are the two \\ex{any}-translation NPIs,\n as shown in Figure \\ref{fig:barsnoiter}. \nGiven that \\ex{vreo} and \\ex{vreun}\nare contained in \nCoDII-NPI.ro\\xspace, we can conclude \nthat the \\dld algorithm is highly sensitive to the \nmarginal items in that list.\n\n\nThere is another way\n to employ the CoDII-NPI.ro\\xspace list:\nuse our co-learning algorithm, but try to \nhave the iterations only add high-quality CoDII-NPI.ro\\xspace items,\nrather than arbitrary pseudo-NPIs; essentially, this amounts to {\\em re-ranking} CoDII-NPI.ro\\xspace.\nWe implement this \nby altering \nthe pNPI\\xspace-learning step\nin our algorithm as follows: extend $\\clues$ with the top $n_r$ \nNPIs on the CoDII-NPI.ro\\xspace list, as opposed to the top $n_r$\npNPIs\n overall. \n However, Figure \\ref{fig:vs} \ndemonstrates\n that this substantially underperforms the algorithm as originally proposed, i.e., based on \\mbox{pNPIs\\xspace}. \n \n\n \\begin{figure}[h!]\n\\includegraphics[width=3.55in,viewport= 100 235 570 557,clip]\n{esub_figs2\/vs2}\n\\caption{\\label{fig:vs} Precision at $k=30$ obtained by \neither re-ranking the CoDII-NPI.ro\\xspace list (dashed line) or by using our co-learning method (solid line).\nResults for other values of $k$ are similar. \n}\n\n\\end{figure}\n \n All\n this suggests that pNPIs\\xspace might be more effective clues \nthan a mixture of non-marginal and marginal NPIs.\nIn addition, pNPIs\\xspace are\nderivable automatically, and might even be worth investigating from a linguistic perspective in their own right.\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\t\t\n\n\n\n\n\\subsection{Co-learning results on English}\n\\mynumberfig{5}\n\\begin{figure}[hbp!]\n\\includegraphics[width=3.55in,viewport= 100 235 570 557,clip]\n{esub_figs2\/nrops_eng_any2}\n\\caption{\\label{fig:english} Number of {\\po}s\\xspace in the top $k$ results returned by the co-learning method at each iteration for English (seed used: `any'). } \\ \n\\end{figure}\n\n\\subsection{Corrigendum} In the proceedings version, we missed the fact that the concept of pseudo-NPIs had been previously discussed by \\citet{Hoeksema:97a}. Hoeksema considers a few examples of what he called ``{\\em pseudo-polarity items}'' in the context of speculating on how polarity sensitivity might arise. \n\n\n\\end{document}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{Introduction}\nIntelligent robots are human companions and assistants with the goal to improve the quality of life. For instance, robots can be designed to guide blind people to navigate by telling them where they are and which direction to go. For this application, there is an urgent need to equip robots with semantic scene understanding ability, e.g. the ability to perceive the surrounding objects and the scene that all objects constitute. Therefore, developing an effective method for indoor scene representation and recognition bears importance in improving the level of robotic perception and intelligence.\n\nIndoor scene representation based recognition has been proposed for more than one decade, and it is still a challenging task in robotics and computer vision due to several issues: (1) conventional networks cannot focus on every object in scenes because they do not extract features at the object granularity \\cite{He2016}; (2) handcrafted object features are sub-optimal to represent scenes \\cite{Quelhas2005}; (3) an effective paradigm is lacking to represent coexisting object relations \\cite{pal2019deduce}.\n\n\n\\begin{figure}[]\n \\centering\n \\includegraphics[width= 0.47\\textwidth]{Fig\/\/1.pdf}\n \\vspace{-1mm}\n \\caption{Comparison of the recognition results of the proposed OTS and ResNet50 on the shown image. The bottom left list shows the objects that the segmentation network in our method detected, and the right charts show the top 5 recognition scores of the two models.}\n \\vspace{-5mm}\n\n \\label{img1}\n\\end{figure}\n\nThis work primarily aims to mitigate the above-mentioned issues and improve the accuracy of indoor scene representation and recognition. To achieve these goals, we propose a novel one-stream method, namely Object-to-Scene (OTS), that constructs object features and calculates object relations to realize indoor scene representation and recognition. In this work, instead of simply using handcrafted features such as \\cite{pal2019deduce} to represent the existence of each object, object features with higher semantic level are extracted using the proposed object feature aggregation module (OFAM) and a segmentation network. Object attention module (OAM) is then used to learn object relations implicitly, the attention mechanism of the module can obtain long-range dependencies, and thus coexisting objects in all situations can be learned. After that, Global relation aggregation module (GRAM) based on a strip depthwise convolution and a pointwise convolution is used to aggregate the object features and convert object features into a scene representation vector. Finally, a fully-connected layer is used for indoor scene recognition, and recognition results are compared with the existing state-of-the-art methods. Fig. \\ref{img1} illustrates the superiority of our method, where the charts on the right show the top 5 recognition scores of our method and ResNet50 on the image. Compared to ResNet50 that focuses on major objects such as stool and table in scenes, the segmentation network in our method can detect tiny but important objects, such as stove and microwave in Fig. \\ref{img1}, and our method can represent object features and relations implicitly. Therefore, ResNet50 hesitates between wet bar and kitchen but OTS, based on object features, can recognize kitchen confidently.\n\n\nIn summary, the major contributions of this paper are as follows:\n\n\\begin{itemize}\n \\vspace{1mm}\n \\item We propose a novel framework OTS that enables using object features and relations for indoor scene representation and recognition, and OTS outperforms existing methods by more than 2\\%.\n \\vspace{1mm}\n \\item We propose OFAM that can extract object features from the segmentation network for indoor scene representation and recognition.\n \\vspace{1mm}\n \\item We propose OAM to learn relations between objects, and it helps to improve the scene representation and recognition ability of our method. Meanwhile, the proposed object attention blocks in OAM are more flexible and effective compared with the well-known self-attention \\cite{Zhang2019} and non-local \\cite{Wang2017}.\n \\vspace{1mm}\n \\item We propose GRAM to fuse the object features and relations into scene representation at a higher semantic level.\n\\end{itemize}\n\nThe remainder of the paper is organized as follows. Section \\ref{Related Works} describes a detailed list of recent works in scene representation and recognition. The details of the proposed OTS are described in Section \\ref{Methodology}. Section \\ref{Experiments and Results} presents the experimental settings and analyzes the results of OTS. Finally, the conclusion and future work are presented in Section \\ref{Conclusions and Future Work}.\n\n\\vspace{2mm}\n\\section{Related Works}\n\\label{Related Works}\n\nUnderstanding surrounding scenes is helpful for robots to make reasonable judgments and behaviours, and a proper scene representation is an important prerequisite \\cite{Liao2016, Ye2017, Yan2019}. Many early approaches used statistics-based and hand-crafted features for scene representation \\cite{Lazebnik2006, Khan2014}. Li et al. proposed codebooks to represent local features and scenes \\cite{Fei-Fei2005}. Quattoni et al. used prototypes that contain object information for different indoor scene representations \\cite{Quattoni2009}. Liu et al. proposed fast adaptive color tags to describe each indoor scene, and used tag matching methods in U-V color space and geometric space for inference \\cite{Liu2009}. However, these statistics-based and hand-crafted features have limited semantic information and are sub-optimal to represent scenes as there are many combinations of objects in the same and different scenes. In that case, an effective scene representation method is urgently needed.\n\nAs computing power increases, many deep learning methods have been proposed for various vision tasks \\cite{Lei2020,He2017mask,Hristov2020,Zhang2018,Herranz2016}. ResNet is one of the most prominent backbone feature extractors since it cleverly avoids the vanishing gradient problem in neural networks \\cite{He2016}. However, ResNet behaves poorly in indoor scene representation and recognition due to the lack of effective expression of coexisting small objects and object relations. To solve this problem, some methods try to combine the backbone features with the object and semantic features from detection networks and segmentation networks for scene representation and recognition. Chen et al. merged backbone features, detection features and segmentation features into an embedding for indoor scene representation \\cite{Chen2018}. Sun et al. used the method of spatial fisher vectors to extract object features from detection network, and combined the object features with contextual and global appearance features for scene recognition \\cite{sun2018fusing}. Alejandro et al. combined the semantic features from segmentation network with backbone features for scene recognition using an attention module \\cite{lopez2020semantic}. Pal et al. used detection network to compute a binary feature vector that represents whether each object existed in the scene, the binary object vector is then combined with backbone features for indoor scene recognition \\cite{pal2019deduce}. Zhou et al. used a Bayesian method to represent the co-occurrence of object pairs for better scene representation and recognition \\cite{Zhou21borm}. Zeng et al. added scene attributes into image features and patch features at multi-scale for scene recognition \\cite{zeng2019}. Although the above mentioned methods improved the scene representation ability and recognition accuracy. These methods are sub-optimal in terms of constructing object features and relations. As shown in Fig. \\ref{img2}, the co-occurrence probability of some coexisting objects, such as bed and lamp, varies significantly from the bedroom to other scenes. Therefore, learning object relations and object features well could further improve the representation ability and recognition accuracy of models. Inspired by the above observations, we propose a novel one-stream method called OTS that enables object features and relations for scene representation and recognition, and our method outperforms existing methods.\n\n\\begin{figure}[]\n \\centering\n \\includegraphics[width= 0.47\\textwidth]{Fig\/\/2.pdf}\n \\vspace{-2mm}\n \\caption{Object pair co-occurrence probability distribution of different scenes in Places365-7classes.}\n \\vspace{-5mm}\n \\label{img2}\n\\end{figure}\n\n\\vspace{-1mm}\n\n\\section{Methodology}\n\\label{Methodology}\n\nWe propose OTS for indoor scene representation and recognition. In OTS, we propose OFAM which enables small but discriminative objects to determine the final recognition results. To ensure effective utilization of object features for scene representation, we propose OAM to learn object relations based on its attention mechanism and a GRAM to fuse object features and relations. Fig. \\ref{img3} illustrates the framework of the proposed OTS. Given an input image $I = (X_{i}, Y_{i})$, where $X_{i}$ is the image data and $Y_{i}$ is the corresponding category, OFAM is used to calculate the object features $X_{obj}$ at first. OAM and GRAM are then used to calculate the scene representation $V_{obj}$. Finally, a fully connected layer and the softmax function are used to calculate the final probability of each scene $P(Y|V_{obj})$.\n\n\n\\begin{figure}[t!]\n \\centering\n \\includegraphics[width= 0.48\\textwidth]{Fig\/\/3.pdf}\n \n \\caption{The proposed OTS includes five parts. (a) Segmentation network: PSPNet is used to calculate the segmentation mask of the input image and to provide its backbone features Conv4\\_6 for the next step. (b) OFAM: the segmentation score map is combined with the Conv4\\_6 feature map to form the object features $X_{obj}$. (c) OAM: cascaded object attention blocks in the OAM are used to calculate object relations. (d) GRAM: a large strip depthwise convolution with the size of the input feature map and a pointwise convolution are used to aggregate object features and form the final scene representation vector. (e) Recognition: a fully connected layer is used to recognize the scene.}\n \\label{img3}\n \\vspace{-4mm}\n\\end{figure}\n\n\\subsection{Object Feature Aggregation Module (OFAM)}\n\\label{OFAM}\n\nThe proposed OFAM can extract object features based on the segmentation network. In this paper, PSPNet that is pretrained on ADE20k dataset is used as the segmentation network \\cite{Zhao2017,Zhou2017}. Fig. \\ref{img4} illustrates the proposed OFAM. OFAM uses Conv4\\_6 feature map $F\\in{R^{C\\times N}}$ and segmentation score map $S\\in{R^{C^{'}\\times N}}$ that obtained from PSPNet to calculate object features $X_{obj}$, where $C$ is the number of channels, $C^{'}$ is the number of objects and $N$ is the number of feature spatial positions. In this paper, $C^{'}$ is set to 150 because of the object number of ADE20K, and $C$ is set to 1024 because Conv4\\_6 is used. For convenience, we call each spatial position in $F$ as a unit, the channel values of each unit form the feature vector $U\\in{R^{C\\times 1}}$ of the unit. The binary mask $M\\in{R^{C^{'}\\times N}}$ of objects is calculated as:\n\\vspace{-1mm}\n$$M_{i,j} = \\begin{cases}\n 1,\\ if\\ max(S_{i,1},\\cdots,S_{i,C^{'}})\\ == \\ S_{i,j} \\\\\n 0,\\ otherwise.\n \\end{cases} \\eqno{(1)}$$\nwhere $i$ is the unit index and $j$ is the object index. Then, the object feature vector $O_{j}\\in{R^{C\\times 1}}$ is calculated as:\n\n$$O_{j} = \\frac{\\sum_{i=1}^{N}{(M_{i,j}*S_{i,j}*U_{i})}}{\\sum_{i=1}^{N}{(M_{i,j}*S_{i,j})}} \\eqno{(2)}$$\nwhere $i$ is the unit index and $j$ is the object index, and $N$ represents the unit number. Finally, the feature vectors stacked together to form the object features $X_{obj}\\in{R^{1024\\times150}}$.\n\n\\subsection{Object Attention Module (OAM)}\n\\label{OAM}\n\nIn addition to object features, relations between objects are also required to improve scene representation ability and recognition accuracy. For example, a model will be more confident about the kitchen label if stove and refrigerator have been detected together. In that case, an attention mechanism is a good choice since it can capture long-range dependencies between all objects \\cite{vaswani2017attention}. However, only global attention mechanism is suitable for our case because object features do not include spatial relations.\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width= 0.48\\textwidth]{Fig\/\/4.pdf}\n \\vspace{-6mm}\n \\caption{Framework of the proposed object feature aggregation module.}\n \\label{img4}\n \\vspace{-4mm}\n\\end{figure}\n\nSelf-attention \\cite{Zhang2019} and non-local \\cite{Wang2017} are one of the most effective global attention mechanisms where three nodes are calculated in parallel with 1$\\times$1 convolutions and the input feature map. We call the three nodes Query ($Q$), Key ($K$), and Value ($V$) for convenience. $Q$ and $K$ are then multiplied together to calculate the correlation matrix of each unit in the feature map, and softmax is used to calculate the activation map. After that, $V$ and the activation map are multiplied together, and an additional 1$\\times$1 convolution is used to form the final attention. Finally, the attention multiplies a learnable scale parameter and adds back to the input feature map to form the output. Fig. \\ref{img5}(b) illustrates the structure of self-attention and non-local. Although self-attention and non-local are effective, they can be further improved. For example, the attention will be more efficient if $Q$ and $K$ are calculated based on $V$, because $V$ has fewer channels compared with input features. Leveraging this insight, we propose an OAM that consists of cascaded object attention blocks as shown in Fig. \\ref{img5}(a). Given an input feature map $F\\in{R^{C\\times N}}$, $V\\in{R^{\\frac{C}{2\\alpha} \\times N}}$ is first obtained using 1$\\times$1 convolutions, where $C$ is the channel number, $N$ is the unit number, $\\alpha$ is a factor to control the compression rate and the output channel number, and $V=W_{v}F$. $Q\\in{R^{\\frac{C}{2\\alpha} \\times N}}$ and $K\\in{R^{\\frac{C}{2\\alpha} \\times N}}$ are then calculated based on $V$, where $Q=W_{q}V$ and $K=W_{k}V$. The object attention is then calculated as follow:\n\n\\begin{tiny}\n\\begin{table}[b!]\n\\vspace{-3mm}\n\\tiny\n\\centering\n\\caption{Comparison of the proposed object attention block, non-local and self-attention.}\\label{tab1}\n\\renewcommand\n\\setlength{\\tabcolsep}{}{\n\\begin{tabular}{lllllllll}\n\\hline\n&$C_{in}$&$C_{out}$&$C_{Q}$&$C_{K}$&$C_{V}$&Parm. (M)&FLOPs (M)\\\\\n\\hline\nSelf-attention&1024&1024&128&128&512&1.3&211.0\\\\\nNon-local&1024&1024&512&512&512&2.1&337.6\\\\\nObject Attention Block&1024&1024&512&512&512&\\textbf{1.1}&\\textbf{180.3}\\\\\n\\hline\n\\end{tabular}}\n\\end{table}\n\\end{tiny}\n\n\\vspace{-1mm}\n$$\\beta_{Q,K} = softmax(Q^{T}K)\\eqno{(3)}$$\n$$Attn = concatenate(\\gamma*V\\beta_{Q,K},\\ V) \\eqno{(4)}$$\nWhere $\\gamma$ is a learnable scale parameter that can be updated gradually. The intuition for why we propose object attention block is straightforward. Firstly, $V$ is used to refine input information with fewer channels (512\/$\\alpha$) in object attention, calculating $Q$ and $K$ based on $V$ enables them to perceive refined input information with half the computational cost. Secondly, $\\alpha$ is used to compress the block and control the output channel number, the connectivity of $Q$, $K$ and $V$ allows them to be automatically compressed proportionally with the change of $\\alpha$. Thirdly, concatenation mechanism in object attention avoids using extra convolution and ensures the later block perceive the refined input information and attention features separately. Table \\ref{tab1} compares the computational cost of three methods. It can be seen that our object attention block is the most efficient of the three methods under the same input and output channel number configuration. Our object attention block is also more flexible because we can adjust $\\alpha$ to reduce the computational cost arbitrarily, and the channel number of $Q$, $K$, $V$, and output will be changed automatically.\n\\vspace{-1mm}\n\n\\begin{figure}[]\n \\centering\n \\includegraphics[width= 0.49\\textwidth]{Fig\/\/5.pdf}\n \\vspace{-5mm}\n \\caption{Structure of the proposed object attention block, non-local and self-attention.}\n \\label{img5}\n \\vspace{-5mm}\n\\end{figure}\n\n\n\\subsection{Global Relation Aggregation Module (GRAM)}\n\\label{GRAM}\n\nAggregation of object features is essential for accurate scene recognition. To solve this problem, we propose a GRAM that consists of a strip depthwise convolution and a pointwise convolution as shown in Fig. \\ref{img6}. Given an input feature map $F_{in}\\in{R^{C\\times N}}$, the strip depthwise convolution first aggregates object features and relations at each channel, and converts the feature map into $F_{mid}\\in{R^{C\\times 1}}$, pointwise convolution is then used to generate scene representation vector $F_{out}\\in{R^{C^{'}\\times 1}}$ that has higher semantic level. Depthwise convolution combined with pointwise convolution is much more efficient compared with conventional convolution kernel \\cite{Francois2017}. The conventional depthwise convolution has a 3x3 kernel to learn local information in each channel. However, we aim to aggregate the object features and relations into a scene representation vector, and learn the global relation at the same time. Therefore, the conventional depthwise convolution is not suitable for our case. To solve this problem, we proposed a strip depthwise convolution, which operates on a list of object features rather than a patch of spatial features. The intuition for why we use this is clear. Firstly, the strip depthwise convolution can convert object feature map into a representation vector, that is more suitable for the final recognition layer. Secondly, the convolution is also able to aggregate the relations between all objects in each channel. Finally, the convolution is more efficient not only because of the depthwise characteristic, but also the avoiding of the use of large amount of fully connected layers.\n\n\\begin{figure}[]\n \\centering\n \\includegraphics[width= 0.47\\textwidth]{Fig\/\/6.pdf}\n \\vspace{-1mm}\n \\caption{Structure of the proposed global relation aggregation module.}\n \\label{img6}\n \\vspace{-4mm}\n\\end{figure}\n\n\\section{Experiments and Results}\n\\label{Experiments and Results}\n\n\\subsection{Implementation Details}\n\\label{Implementation Details}\n\nOur network is implemented in the Pytorch library \\cite{Paszke2019}, and a single RTX 2080Ti GPU is used in the experiments. We use the most common settings to implement the experiments \\cite{He2016}. The batch size for all experiments is set to 256 and cross-entropy loss is used in our method. We adopt Stochastic Gradient Descent (SGD) optimizer with the base learning rate of 0.1 while momentum and weight decay are set to 0.9 and 0.0001, respectively. The learning rate is divided by 10 every 10 epochs. Training is performed for 40 epochs. Meanwhile, object features are calculated off-line to increase the training speed.\n\n\\subsection{Places365 Dataset}\n\\label{Places365 Dataset}\n\nThe reduced Places365 dataset is used in this paper since it is the largest scene recognition dataset with various indoor environment categories \\cite{zhou2017places}. To verify the effectiveness of our method, we used two different class settings of indoor scenes for a fair comparison with other state-of-the-art methods. The first one contains 7 classes: Bathroom, Bedroom, Corridor, Dining room, Kitchen, Living room, and Office. We extract both training data and test data from the target 7 classes in Places365 and form the Places365-7classes, the class setting is totally the same as \\cite{pal2019deduce}. The second one includes 14 indoor scenes in home environment: Balcony, Bedroom, Dining room, Home office, Kitchen, Living room, Staircase, Bathroom, Closet, Garage, Home theater, Laundromat, Playroom, and Wet bar. We also extract both training data and test data from the target 14 classes in Places365 and form the Places365-14classes, the class setting is totally the same as \\cite{Chen2018}.\n\n\\subsection{SUN-RGBD Dataset}\n\\label{SUN-RGBD Dataset}\n\nSUN-RGBD is one of the most challenging datasets for scene understanding \\cite{song2015sun}, which includes images from various sources: 3784 images captured by Kinect v2; 1159 images captured by Intel RealSense cameras; 1449 images from NYU Depth V2 \\cite{Silberman2012indoor} and 554 manually selected realistic scene images from Berkeley B3DO \\cite{janoch2013category} captured by Kinect v1; 3389 selected frames by filtering out significantly blurred frames from the SUN3D videos \\cite{xiao2013sun3d} captured by Asus Xtion. The diversity of categories and sources makes SUN-RGBD more suitable for verifying the generalization ability of methods. We only consider the RGB images in this work, and the 7 classes that chosen in Places365 are used to test our model's generalization ability. Notably, we only used the official test images in SUN-RGBD, and these images are evaluated by the model trained using Places365-7 classes.\n\n\\begin{table}[tp!]\n\\scriptsize\n\\centering\n\\caption{Scene recognition accuracy on the Places365-7classes.}\\label{tab2}\n\\begin{tabular}{lllllll}\n\\hline\n\\multicolumn{1}{c}{} &\n\\multicolumn{2}{c}{\\cite{He2016}} &\n\\multicolumn{3}{c}{\\cite{pal2019deduce}} &\n\\multicolumn{1}{c}{Our} \\\\\n\\cmidrule(lr){2-3}\\cmidrule(lr){4-6}\\cmidrule(lr){7-7}\nCategory&ResNet18 &ResNet50 &Scene &Obj. &Scene+Obj. &OTS \\\\\n\\hline\nBathroom&87&94&92&65&91&92\\\\\nBedroom&82&83&90&74&90&97\\\\\nCorridor&96&93&94&90&96&95\\\\\nDining room&81&71&79&94&79&88\\\\\nKitchen&83&84&87&62&87&92\\\\\nLiving room&55&66&84&25&80&79\\\\\nOffice&79&88&85&29&94&88\\\\\n\\hline\nAvg. Acc. (\\%)&80.4&82.7&87.3&62.6&88.1&\\textbf{90.1}\\\\\n\\hline\n\\end{tabular}\n\\vspace{-6mm}\n\\end{table}\n\n\\begin{tiny}\n\\begin{table}[bp!]\n\\centering\n\\vspace{-3mm}\n\\caption{Scene recognition accuracy on the SUN-RGBD.}\\label{tab3}\n\\begin{tabular}{ll|l|l|l}\n\\hline\nSource&Method&1 Stream&2 Stream&Acc. (\\%)\\\\\n\\hline\n\\multirow{2}{*}{\\cite{He2016}} &ResNet18&\\checkmark&& 63.3\\\\\n&ResNet50&\\checkmark&&67.2\\\\\n\\hline\n\\multirow{3}{*}{\\cite{pal2019deduce}} &Scene&\\checkmark&& 66.8\\\\\n&Obj.&\\checkmark&&53.6\\\\\n&Scene+Obj.&&\\checkmark&70.1\\\\\n\\hline\nOur&OTS&\\checkmark&&\\textbf{70.6}\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\\end{tiny}\n\n\\begin{tiny}\n\\begin{table}[bp!]\n\\centering\n\\vspace{-1mm}\n\\caption{Scene recognition accuracy on the Places365-14classes.}\\label{tab4}\n\\begin{tabular}{ll|l|l|l}\n\\hline\nSource&Method&1 Stream&2 Stream&Acc. (\\%)\\\\\n\\hline\n\\multirow{2}{*}{\\cite{He2016}} &ResNet18&\\checkmark&& 76.0\\\\\n&ResNet50&\\checkmark&& 80.0\\\\\n\\hline\n\\cite{Chen2018} & ResNet50+Word2Vec&&\\checkmark& 83.7\\\\\n\\hline\nOur & OTS &\\checkmark&& \\textbf{85.9}\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\\end{tiny}\n\n\n\\subsection{Main Results}\n\\label{Main Results}\n\nTo evaluate the effectiveness of the proposed OTS, we compare it with other benchmark methods on the Places365-7classes, Places365-14classes and reduced SUN-RGBD datasets. As shown in Table \\ref{tab2}, OTS significantly outperforms the ResNet50 baseline with 7.4\\% on Places365-7classes which shows the effectiveness of OTS. Moreover, OTS has 2\\% higher accuracy than \\cite{pal2019deduce} with only one stream. \\cite{pal2019deduce} used a detection network to calculate one-hot object existence features, and added an additional stream to calculate auxiliary image features.\n\nTo verify the generalization ability of OTS, we further test OTS on SUN-RGBD. It is noteworthy that SUN-RGBD is just used as a test set to verify the generalization ability of models, and all models are trained using Plces365-7classes. As shown in Table \\ref{tab3}, OTS achieves the highest accuracy compared with other methods on SUN-RGBD. The results demonstrate the generalization ability of OTS.\n\nWe further compare OTS with \\cite{Chen2018} on the larger Places365-14classes. \\cite{Chen2018} used a segmentation network to calculate Word2Vec features, and added an additional stream to calculate image features. As shown in Table \\ref{tab4}, our OTS is 2.2\\% higher than \\cite{Chen2018} without any additional streams. The results show the effectiveness of our method. Better segmentation models have the potential to improve the performance of OTS, but they are not the focus of this work. In addition to evaluating the benchmark datasets, we also inference OTS in a real-world office environment, which can be found in our video supplement files.\n\n\\begin{tiny}\n\\begin{table}[tp!]\n\\centering\n\\caption{Ablation studies on each module.}\\label{tab5}\n\\begin{tabular}{lllll}\n\\hline\nResNet50&OFAM&OAM&GRAM&Acc. (\\%)\\\\\n\\hline\n\\checkmark&&&&80.0\\\\\n\\checkmark&\\checkmark&&&77.2\\\\\n\\checkmark&\\checkmark&\\checkmark&&82.0\\\\\n\\checkmark&\\checkmark&\\checkmark&\\checkmark&\\textbf{85.9}\\\\\n\\hline\n\\end{tabular}\n\\vspace{-6mm}\n\\end{table}\n\\end{tiny}\n\n\\begin{figure*}[t!]\n \\centering\n \\includegraphics[width= 0.95\\textwidth]{Fig\/\/7.pdf}\n \\vspace{-1mm}\n \\caption{Failure cases of OTS. Recog. refers to the recognition results of OTS, and Anno. refers to the annotations.}\n \\label{img7}\n \\vspace{-4mm}\n\\end{figure*}\n\n\\subsection{Ablation Experiments}\n\\label{Ablation Experiments}\n\nWe run ablation experiments to analyze the results of our method. Unless specified otherwise, the dataset used in ablation experiments is Places365-14classes since it is larger and contains more categories. We firstly run an ablation experiment to show the effectiveness of each module. As shown in Table \\ref{tab5}, both OAM and GRAM significantly improve the scene recognition accuracy because of the ability to capture the long-range dependencies between all objects. Meanwhile, only adding OFAM degrades the performance of the model because ResNet50 uses the features of Conv5 layer but OFAM extracts the object features from Conv4 layer as shown in Fig. \\ref{img3}, which means the performance gains of our method rely on the object relation construction instead of the backbone features of the pre-trained segmentation model.\n\n\\begin{tiny}\n\\begin{table}[bp!]\n\\centering\n\\vspace{-3mm}\n\\renewcommand\\tabcolsep{4pt}\n\\caption{Ablation studies on the combination methods in the proposed object attention block (OAB).}\\label{tab6}\n\\begin{tabular}{lllllll}\n\\hline\n&Num.&SUM.&CAT.&Acc. (\\%)&Parm. (M)&FLOPs (M)\\\\\n\\hline\nOAB&1&\\checkmark&&85.0&0.4&70.5\\\\\nOAB&2&\\checkmark&&85.2&1.8&321.3\\\\\nOAB&1&&\\checkmark&\\textbf{85.4}&\\textbf{0.4}&\\textbf{70.5}\\\\\nOAB&2&&\\checkmark&\\textbf{85.9}&\\textbf{1.2}&\\textbf{211.5}\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\\end{tiny}\n\n\\vspace{1mm}\n\\subsubsection{Number of object attention blocks}\n\\label{Number of object attention blockss}\n\nAs stated in Section \\ref{OAM}, OAM is built based on one or several cascaded object attention blocks. Therefore, we run ablation experiments to find how many object attention blocks is the optimal choice to balance efficiency and effectiveness. Unless specified otherwise, the $\\alpha$ of the first object attention block is 2 and the $\\alpha$ of the second object attention block is 0.5 to control the output channel number. Before that, we first verify the difference between using summation (SUM.) and using concatenation (CAT.) in the last step of object attention blocks. The experiment results are shown in Table \\ref{tab6}. It can be seen that using concatenation in object attention blocks is both more efficient and effective compared with using summation. This is because simple summation cannot perfectly fuse object features and object relation features together, but concatenation can store both types of features separately. Then, we use concatenation in object attention blocks to find the optimal number of blocks. As shown in Table \\ref{tab7}, we list the results as well as FLOPs obtained with different number of object attention blocks. It can be seen that when two object attention blocks are cascaded, OTS has the best accuracy 85.9\\%. However, adding more object attention blocks exert an adverse impact on the results because these blocks make the model too complicated.\n\n\\begin{tiny}\n\\begin{table}[tp!]\n\\centering\n\\caption{Ablation studies on the number of the proposed object attention block (OAB).}\\label{tab7}\n\\begin{tabular}{lllll}\n\\hline\n&Num.&Acc. (\\%)&Parm. (M)&FLOPs (M)\\\\\n\\hline\nOAB&1&85.4&\\textbf{0.4}&\\textbf{70.5}\\\\\nOAB&2&\\textbf{85.9}&1.2&211.5\\\\\nOAB&3&85.0&1.4&262.3\\\\\nOAB&4&85.0&1.7&313.2\\\\\n\\hline\n\\end{tabular}\n\\vspace{-2mm}\n\\end{table}\n\\end{tiny}\n\n\\vspace{1mm}\n\\subsubsection{Object attention block vs Self-attention and Non-local}\n\\label{Object attention block vs Self-attention and Non-local}\n\nWe have shown the effectiveness of OAM in Table \\ref{tab5}. In addition, We run an ablation experiment to show the effectiveness and efficiency of the proposed object attention blocks in OAM. As shown in Table \\ref{tab8}, the accuracy of the proposed object attention block has an improvement of 1.2\\% compared with self-attention, and an improvement of 3.6\\% compared with non-local. What is more, the proposed object attention block has the least FLOPs compared with others under a similar condition. The object attention block (S) means only one object attention block is used in OAM.\n\n\\vspace{1mm}\n\\subsubsection{Global relation aggregation module (GRAM)}\n\\label{Global relation aggregation module (GRAM)}\n\nWe set two kinds of control groups to show the effectiveness of the proposed GRAM. In the ablation experiment, fully connected layer and Pooling layer are used to replace GRAM, separately. As shown in Table \\ref{tab9}, GRAM obtained the best accuracy with only 2.3M parameters. It is noteworthy that FC in Table \\ref{tab9} is equal to a large conventional convolution kernel with the size of input feature map. FC not only has more computational cost, but also has inferior performance compared with the proposed GRAM. The results indicate that GRAM can balance efficiency and effectiveness well.\n\n\\begin{tiny}\n\\begin{table}[tp!]\n\\centering\n\\caption{Ablation studies on the object attention block (OAB), non-local, and self-attention.}\\label{tab8}\n\\begin{tabular}{llll}\n\\hline\n&Acc. (\\%)&Parm. (M)&FLOPs (M)\\\\\n\\hline\nNon-Local \\cite{Wang2017}&82.3&4.2&675.2\\\\\nSelf-Attention \\cite{Zhang2019}&84.7&2.6&422.0\\\\\nObject Attention Block&\\textbf{85.9}&\\textbf{1.2}&\\textbf{211.5}\\\\\nObject Attention Block (S)&\\textbf{85.4}&\\textbf{0.4}&\\textbf{70.5}\\\\\n\\hline\n\\end{tabular}\n\\vspace{-5mm}\n\\end{table}\n\\end{tiny}\n\n\\begin{tiny}\n\\begin{table}[tp!]\n\\centering\n\\caption{Ablation studies on the GRAM.}\\label{tab9}\n\\begin{tabular}{llll}\n\\hline\n&Acc. (\\%)&Parm. (M)&FLOPs (M)\\\\\n\\hline\nFC&85.0&314.6&314.6\\\\\nMax \\& Avg. Pooling&82.0&\\textbf{0}&\\textbf{0.3}\\\\\nGRAM&\\textbf{85.9}&2.3&2.3\\\\\n\\hline\n\\end{tabular}\n\\vspace{-5mm}\n\\end{table}\n\\end{tiny}\n\n\\vspace{1mm}\n\\subsubsection{Failure cases}\n\\label{Failure cases}\nDuring experiments, we find that ResNet50 misclassified many common images. However, OTS successfully recognized the common images in scenes because it can detect the objects and learn their relations in each scene. Then, we analyze the failure cases of OTS to demonstrate the pros and cons. As shown in Fig. \\ref{img7}, OTS misclassified some images in each scene mainly caused by the missed or wrong detection. For example, kitchen is misclassified as bathroom because the pot is mistaking for a toilet, and office is misclassified as corridor because only wall, painting, ceiling, floor, and light are detected, which are the common coexisting objects in corridor. Therefore, the performance of OTS could be further improved with the help of more accurate segmentation results.\n\n\\section{Conclusions and Future Work}\n\\label{Conclusions and Future Work}\n\nIn this paper, we analyzed the weakness of existing scene representation and recognition methods, and proposed OTS to solve these issues. We further demonstrated that OTS can effectively use object features and relations for scene representation and recognition by comparing OTS with other existing state-of-the-art methods. Based on numerous ablation experiments, we also showed that OAM and GRAM perform well in learning object relations for scene representation. Moreover, the results of our work reflect several interesting conclusions: 1) object features can perform well as long as an appropriate object feature and relation learning method is used; 2) the backbone features in segmentation network can also be used for scene recognition instead of adding an additional stream to calculate; 3) attention mechanism is very suitable for computing object relations. We hope these results could guild future works for scene understanding. What is more, our work can also be promoted in the future. First of all, data augmentation methods can be used to extract more diverse object features during training. Secondly, enriching the number of object categories can offer a better scene representation, and thus improve scene recognition ability of the model. In the future, we plan to extend our methods to other mobile robots and establish more accurate semantic maps. Therefore, they can be better used to improve human's life quality.\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzkxvv b/data_all_eng_slimpj/shuffled/split2/finalzzkxvv new file mode 100644 index 0000000000000000000000000000000000000000..099a68ca237553865f4d80c4c6a0658427e09968 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzkxvv @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\t\n\t\\IEEEPARstart{T}{he} cell voltages in a series-connected battery cell stack become unequal due to manufacturing tolerances, non-uniform aging, and unequal temperature distribution. The differences among the cell voltages increase with the number of charge-discharge cycles, leading to over-charge and over-discharge of some of the cells. A voltage equalizer is essential to improve the charge-capacity and the cycle life of the cell stack by avoiding such a situation. \n\tThe passive voltage equalizers, which are the simplest and cheapest equalizers, dissipate a significant amount of stored energy\\cite{li_diss}.\n\tOn the other hand, the active equalizers transfer charge from the over-charged cells to under-charged cells to equalize all the cell voltages\\cite{14}. The operation of the active equalizers can be of two types: simultaneous equalization of all the cells and serial equalization of the selected cells, as shown in Fig.\\,\\ref{class}. \n\t\n\tThe first type of active equalizer uses multiple power converters or a multi-port power converter so that all the cells can take part in voltage equalization simultaneously. These equalizers can be classified into three categories: adjacent cell\\cite{lee_int,lee_quasi,park_des,cassani_top,ye_zero,hua_cap}, multi-cell to stack\\cite{einhorn,uno_double,chen,uno_single,hua_lifepo4,lim,hua_apwm,hua_rect,shang_mod,li}, and multi-cell to multi-cell\\cite{ling,evzelman,yelaverthi,wang,shang_auto,ye_model,shang_delta,zeltser,ye_star,TPEL} equalizers, among which the multi-cell to multi-cell equalizers offer fastest voltage equalization. Each of these equalizers dedicates one converter port to each cell even though many of the cells in the cell stack may not require voltage equalization at a given point of time. Such dedicated connections result in a higher component count, under-utilization of converter components, and the requirement of many high-frequency isolated gate drivers. \n\t\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\\includegraphics[width=7.0cm]{Drawings\/classification.pdf}\n\t\t\\caption{Classification of active voltage equalization methods.}\n\t\t\\label{class}\n\t\\end{figure}\t\n\t\n\tThe second type of active equalizer uses only one dual-port dc-dc converter shared by all the cells to reduce the component count and several selection switches for cell selection. \t\n\tWhen the number of cells increases, the dc-dc converter's component count remains the same, and only the number of selection switches increases. The charge transfer method in these equalizers is either cell-to-stack or cell-to-cell. The cell-to-stack equalizers \\cite{imtiaz,nazi,lee_double,raber, park_mod,kim_auto,kim_mod2,hannan,zhang_interleaved,lin} selects only one cell is at a time, and charge transfer takes place between that cell and the entire cell stack. On the other hand, cell-to-cell equalizers \\cite{park_c2c,lee_cell_res,lee_cell_tr,yu , xiong,pham,shang_cell} achieve a direct charge transfer from the most over-charged cell to the most under-charged cell. Thus, the cell-to-cell equalizers offer about twice the equalization speed, but they require twice as many selection switches. \n\t\n\tAs the number of selection switches is proportional to the number of cells, any simplification or cost-reduction of the selection switches and their drive circuits is valuable in case of a higher number of cells in the stack. As discussed in Section \\ref{sec_drive_ckt} of this paper, a MOSFET, switched at a high-frequency, requires a complex gate-driver, which is often costlier than the MOSFET itself in case of low-power applications. Hence, the cell-to-stack equalizers in \\cite{park_mod,kim_auto,kim_mod2,hannan,zhang_interleaved,lin} and cell-to-cell equalizers in\\cite{xiong,pham,shang_cell} have been proposed with low-frequency selection switches so that simpler drive circuits can be used. This work aims to propose a Low-Frequency Selection Switch based Cell-to-Cell (LFSSCC) voltage equalizer with high efficiency, simple implementation, and lower switch count. The advantages of the proposed equalizer compared to existing LFSSCC equalizers\\cite{xiong,pham,shang_cell} are discussed below. \t\n\t\n\t\n\tThe dual-port dc-dc converters in \\cite{xiong,pham} use diodes with significant conduction loss and transformer with higher high-frequency losses and size, leading to lower conversion efficiency and increased equalizer size. The dc-dc converter in \\cite{shang_cell} achieves high-efficiency by achieving soft-switched operation with a higher component count. Thus, the dc-dc converters in the existing LFSSCC equalizers suffer from lower efficiency or higher circuit complexity. This work uses a capacitively level-shifted bidirectional Cuk converter, originally proposed for a multi-cell to multi-cell topology in \\cite{ling}. The equalizer in \\cite{ling} uses one closed-loop controlled Cuk converter for each cell, leading to a high component count and control requirements. On the other hand, this work uses only one Cuk converter for a large number of cells, reducing circuit complexity and control effort significantly. Thus, this converter is more suitable for a cell-to-cell equalizer and offers simpler implementation and high efficiency, as no transformer or diode is required in the conduction path. Section \\ref{topology} discusses the converter topology.\n\t\n\tThe LFSSCC equalizers\\cite{xiong,pham,shang_cell} can use either low frequency switched MOSFETs or relays as selection switches. The selection switch networks in these equalizers require $2n$ Double Pole Double Throw (DPDT) switches for $n$ series-connected cells. These switches can be implemented with either $8n$ low-frequency MOSFETs or $2n$ DPDT relays. This work proposes a low-frequency selection switch network with $(n+2)$ DPDT and $2$ SPST switches, leading to a significant reduction in the number of switches and drive circuits.\n\t\n\tThe voltage drop in cell impedance causes cell voltage recovery after equalization current is stopped. This voltage recovery leads to error in detecting end-of-equalization and a large number of switching of the selection switches. For a low-frequency selection switch based equalizer, a higher number of switching leads to longer equalization time and lower reliability. Different methods to reduce the number of selection switching are proposed based on voltage drop estimation\\cite{hannan,pham} and self-learning fuzzy logic based method\\cite{zhang_interleaved}. These methods require additional computation resources and cell charge-discharge characteristics. A simpler method with cell voltage recovery compensation, which does not require charge-discharge characteristics, is proposed to reduce the number of switchings of selection switches significantly. \n\t\n\tThis work in the enhanced version of the work published in \\cite{ecce_volt_eq} with additional theoretical explanations and experimental results.\n\tSection \\ref{selection} explains the proposed low-frequency cell selection network, and Section \\ref{sec_volt_comp} discusses the proposed cell voltage recovery compensation. Sections \\ref{sec_comp} and \\ref{experiment} provides a comparison with existing equalizers and experimental validation of the proposed equalizer respectively.\n\t\n\t\n\n\t\n\t\\section{Drive Circuits for Selection Switches}\\label{sec_drive_ckt}\t\n\tIf a selection switch is operated at high frequency, it is implemented with MOSFETs. In contrast, a low-frequency selection switch is implemented with either MOSFETs or relay with significantly simpler and cheaper driver circuits.\t\n\t\n\t\\subsubsection{ High-Frequency MOSFET Drive Circuit} A high-frequency switched MOSFET for cell selection requires isolated gate driver IC, capable of providing a peak current of a few Ampere to achieve quick turn-on and turn-off, as shown in Fig.\\,\\ref{drivers}(a). Such a driver IC requires an isolated power supply and is often costlier than a low-power MOSFET. \n\t\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\\begin{subfigure}[]{\\includegraphics[width=2.1cm]{Drawings\/mosfet_driver_high_freq.pdf}}\n\t\t\\end{subfigure}\n\t\t\\hspace{0.4cm}\n\t\t\\begin{subfigure}[]{\\includegraphics[width=2.1cm]{Drawings\/mosfet_driver_low_freq.pdf}}\n\t\t\\end{subfigure}\n\t\t\\hspace{0.4cm}\n\t\t\\begin{subfigure}[]{\\includegraphics[width=2.3cm]{Drawings\/relay_driver.pdf}}\n\t\t\\end{subfigure}\t\t\n\t\t\\caption{(a) High-frequency gate driver for MOSFET, (b) low-frequency gate driver for MOSFET, (c) driver for relay.}\n\t\t\\label{drivers}\n\t\\end{figure} \n\t\n\t\\subsubsection{ Low-Frequency MOSFET Drive Circuit} Longer turn-on and turn-off times are acceptable for a low-frequency switched MOSFET. The gate drive resistor $R_g$ is in the order of a few kilo-Ohms, and the driver needs to supply only a few milli-Ampere peak current. Thus, a low-cost digital isolator can replace the costly gate driver IC, as shown in Fig.\\,\\ref{drivers}(b).\n\t\n\t\\subsubsection{ Relay Drive Circuit} The driver circuit for a relay is simpler and cheaper than a MOSFET driver. Fig.\\,\\ref{drivers}(c) shows a relay driver, which requires a signal MOSFET $S$, a gate resistor $R_g$, and a free-whiling diode $D$. These components have to carry only a few tens of milli-Amperes of peak current, leading to a very low-cost implementation compared to high-frequency MOSFET drivers.\n\t\n\tThus, the use of low-frequency selection switches reduces the complexity and cost of the equalizer significantly.\n\t\n\t\n\t\\section{Dc-dc Converter Topology}\\label{topology}\n\tThe capacitively level-shifted Cuk converter allows variable voltage difference between the input and output ports' reference terminals without using any isolation transformer. This converter is used here to transfer charge from the most over-charged cell to the most under-charged cell, as shown in Fig.\\,\\ref{dc_dc_sch}.\n\t\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\\includegraphics[width=5.5cm]{Drawings\/dc_dc_sch.pdf}\n\t\t\\caption{Schematic diagram of the capacitively level shifted Cuk converter connected to two cells of the stack.}\n\t\t\\label{dc_dc_sch}\n\t\\end{figure}\n\t\n\t\n\n\tThis converter has an additional capacitor $C_2$ compared to a conventional bidirectional Cuk converter to block the voltage difference between the grounds of the two ports. Using KVL in Fig.\\,\\ref{dc_dc_sch} and assuming small voltage ripple, the voltage of the capacitors $C_1$ and $C_2$ are given by,\n\t\\begin{eqnarray}\n\t\\label{vc1}\n\tV_{C1}=\\sum_{j=l}^{k}V_{bj} \\qquad \\text{and} \\qquad V_{C2}=\\sum_{j=l+1}^{k-1}V_{bj}\n\n\n\t\\end{eqnarray} \n\tEach inductor connected to the input or output port is split into two coupled inductors to make the circuit symmetric for reducing common mode oscillation. A more detailed discussion on the converter topology is provided in \\cite{ecce_volt_eq}.\n\tA PI controller controls the cell current in one of the converter ports in closed-loop. \n\tThe converter is enabled when at least one cell voltage is out of the acceptable voltage range.\n\t\n\tFor a predetermined tolerance voltage $V_{tol}$ and the average cell voltage $V_{avg}$, the $k^{th}$ cell voltage $V_{bk}$ is in the acceptable voltage range if it satisfies the following condition, \n\t\\begin{eqnarray}\n\tV_{avg}-V_{tol}\\le V_{bk} \\le V_{avg}+V_{tol} \\quad \\text{for all } k\\in [1,n]\n\t\\end{eqnarray}\n\t\n\t\n\t\n\t\n\n\t\n\t\n\t\n\t\n\t\\section{Low-frequency Cell-to-cell Selection Network}\\label{selection}\n\tIn an LFSSCC equalizer topology, it is possible to connect the dc-dc converter between any two cells at a time to achieve power transfer between them. The existing LFSSCC equalizers\\cite{shang_cell,xiong,pham} require $2n$ DPDT switches to implement such a cell selection network for $n$ series-connected cells, as shown in Fig.\\,\\ref{dpdt}(a), where the voltage rails \\textbf{\\textit{a}}, \\textbf{\\textit{b}}, \\textbf{\\textit{c}}, and \\textbf{\\textit{d}} are of fixed polarity. \t\n\tThis work proposes a new low-frequency cell-to-cell selection network with bipolar voltage rails to reduce the selection switch count. \n\t\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\\begin{subfigure}[]{\\includegraphics[width=3.8cm]{Drawings\/sel_net_dpdt_conventional.pdf}}\n\t\t\\end{subfigure}\\hspace{0.4cm}\n\t\t\\begin{subfigure}[]{\\includegraphics[width=3.3cm]{Drawings\/sel_net_dpdt.pdf}}\n\t\t\\end{subfigure}\n\t\n\t\t\\caption{Schematic diagram of the low-frequency cell selection network in (a) existing and (b) proposed cell-to-cell equalizers.}\n\t\t\\label{dpdt}\n\t\\end{figure}\n\t\n\t\n\t\n\t\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\\begin{subfigure}[]{\\includegraphics[width=3.2cm]{Drawings\/sel_net_dpdt_eo.pdf}}\n\t\t\\end{subfigure}\\hspace{0.4cm}\n\t\t\\begin{subfigure}[]{\\includegraphics[width=3.2cm]{Drawings\/sel_net_dpdt_adjacent.pdf}}\n\t\t\\end{subfigure}\n\t\t\\caption{Current flow paths between (a) the $2^{nd}$ and the $7^{th}$ cell, (b) the $4^{th}$ and the $5^{th}$ cell in the proposed low-frequency selection network.}\n\t\t\\label{sel_net_8bat}\n\t\\end{figure}\n\t\n\t\n\t\n\t\n\t\n\t\\subsection{Proposed Cell Selection Network}\n\tFig.\\,\\ref{dpdt}(b) shows the proposed low frequency cell-to-cell selection network, which uses One SPST switch and $n$ DPDT switches to select two cells at a time.\n\tOne DPDT switch is used here to connect each common node between two adjacent cells to a voltage rail. However, each voltage rail is switched to a second rail if an upstream DPDT switch is turned on. Thus, each common node can effectively be connected to two voltage rails. In case the polarities of the voltage rails do not match with the converter port polarities, two DPDT relays $S_{pol1}$ and $S_{pol2}$ are used for polarity reversal. In this case, a polarity reversal is required for each pair of voltage rails when the corresponding cell is even-numbered. \n\tIn the case of two adjacent cells, the common node is connected to both converter ports by an SPST switch $S_{short}$. Thus, the proposed selection network requires ($n+2$) DPDT and $2$ SPST switches, and the selection switch count is reduced to almost half compared to the existing LFSSCC equalizers for a large number of cells.\n\t\n\t\n\tLets consider that the selected cells are the $k^{th}$ and the $l^{th}$ cell, where $k>l$. Then, the $k^{th}$ cell is connected to port 1 and the $l^{th}$ cell is connected to port 2 with following steps, \n\t\\begin{itemize}\n\t\t\\item Turn on $S_{pol1}$ if $k$ is even. \n\t\t\\item Turn on $S_{pol2}$ if $l$ is even. \n\t\t\\item Turn on $S_{short}$ if the $k^{th}$ and the $l^{th}$ cells are adjacent.\n\t\t\\item Turn on $S_k$, $S_{k-1}$, $S_{l}$, and $S_{l-1}$.\n\t\\end{itemize}\n\n\n\t\n\t\n\t\n\t\n\t\n\t\n\tThe selection network's operation is explained here for eight cells in two different equalization situations, as shown in Fig.\\,\\ref{sel_net_8bat}. The red lines indicate the current flow paths.\n\t\n\t\\subsubsection{Two non-adjacent cells} Fig.\\,\\ref{sel_net_8bat}(a) shows the current flow path between two non-adjacent cells, cell 2 and cell 7. The switches $S_{1}$, $S_{2}$, $S_{6}$, and $S_{7}$ are turned on to connect the cells to the converter ports. The polarity reversal switch $S_{pol2}$ is turned on to ensure correct polarity connection.\n\t\n\t\\subsubsection{Two adjacent cells} Fig.\\,\\ref{sel_net_8bat}(b) shows the current flow path for two adjacent cells, cell 4 and cell 5. The switches $S_{3}$, $S_{4}$, $S_{5}$, and $S_{pol2}$ are turned on to connect cell 5 to port 1 and cell 4 to port 2 with the correct polarity. As the cell 4 and cell 5 are adjacent, their common node should be connected to the negative terminal of port 1 and positive terminal of port 2. This is achieved by turning on the shorting switch $S_{short}$.\n\t\n\t\n\t\n\t\\subsection{Implementation of Selection Switches}\n\tThe low-frequency selection switches can be implemented with MOSFETs or relays as discussed below,\n\t\\subsubsection{DPDT switch for cell selection}\n\tFig.\\,\\ref{dpdt_mos}(a) and (b) show the off and on states of the DPDT switches $S_1$ to $S_n$. \t\n\tEach switch blocks the voltages $V_{P1\\_T1b}$, $V_{P2\\_T2b}$ in on-state, and $V_{P2\\_T2a}$ in off-state. Fig.\\,\\ref{dpdt} and Fig.\\,\\ref{sel_net_8bat} show that $V_{P1\\_T1b}$ is bipolar and $V_{P2\\_T2b}$, $V_{P2\\_T2a}$ are unipolar. Hence, the DPDT switch can be implemented with four MOSFETs, as shown in Fig.\\,\\ref{dpdt_mos}(c). Alternatively, it can also be implemented with a DPDT relay.\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\\begin{subfigure}[]{\\includegraphics[width=2.7cm]{Drawings\/dpdt_mos_a.pdf}}\n\t\t\\end{subfigure}\\hspace{0.0cm}\n\t\t\\begin{subfigure}[]{\\includegraphics[width=2.7cm]{Drawings\/dpdt_mos_b.pdf}}\n\t\t\\end{subfigure} \\hspace{0.0cm} \n\t\t\\begin{subfigure}[]{\\includegraphics[width=2.7cm]{Drawings\/dpdt_mos_c.pdf}}\n\t\t\\end{subfigure}\n\t\t\\caption{DPDT switches for cell connection, $S_1$ to $S_n$: (a) off state, (b) on state, (c) a MOSFET based implementation.}\n\t\t\\label{dpdt_mos}\n\t\\end{figure}\n\t\\subsubsection{DPDT switch for polarity reversal}\n\tFig.\\,\\ref{dpdt_pol_mos}(a) and (b) show the off and on state of the polarity reversal DPDT switches $S_{pol1}$ and $S_{pol2}$. It can be observed that the voltage between each pole to each of its throw is positive or zero in off-state. The voltage between a throw and corresponding pole is positive or zero in on-state. The current in the DPDT switch is bi-directional. Hence, the DPDT switch can be implemented with a DPDT relay or four MOSFETs as shown in Fig.\\,\\ref{dpdt_pol_mos}(c).\t\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\\begin{subfigure}[]{\\includegraphics[width=2.7cm]{Drawings\/dpdt_pol_mos_a.pdf}}\n\t\t\\end{subfigure}\\hspace{0.0cm}\n\t\t\\begin{subfigure}[]{\\includegraphics[width=2.7cm]{Drawings\/dpdt_pol_mos_b.pdf}}\n\t\t\\end{subfigure} \\hspace{0.0cm} \n\t\t\\begin{subfigure}[]{\\includegraphics[width=2.7cm]{Drawings\/dpdt_pol_mos_c.pdf}}\n\t\t\\end{subfigure}\n\t\t\\caption{DPDT switch for polarity reversal, $S_{pol1}$ and $S_{pol2}$: (a) off state, (b) on state, (c) a MOSFET based implementation.}\n\t\t\\label{dpdt_pol_mos}\n\t\\end{figure}\n\t\n\t\\subsubsection{SPST switches}\n\tThe SPST switches block unipolar voltages, and each of them can be implemented with an SPST relay or a MOSFET.\t\n\t\n\t\n\tThus, the proposed selection switch network can be implemented with $(n+2)$ DPDT and $2$ SPST relays or with $(4n+10)$ MOSFETs.\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\\section{Cell Voltage Recovery Compensation}\\label{sec_volt_comp}\n\tWhen the discharging of a cell is stopped, the cell voltage $V_b$ recovers to a higher voltage after some time. Similarly, when the cell's charging is stopped, $V_b$ falls and settles to a lower voltage. This recovery in $V_b$ is often modeled as a voltage drop across the cell impedance. Ideally, when the cell's equalization is complete, $V_b$ should settle within the acceptable voltage band. However, if the cell's equalization is stopped when $V_b$ reaches within the acceptable voltage band, $V_b$ settles outside of the band due to voltage recovery. Thus, further rounds of equalization for the same cell are necessary, resulting in a higher number of switching of the selection switches. \n\t\n\t\n\t\n\t\n\tThe control algorithm for a low-frequency cell selection network provides a small amount of time-gap, usually 10s to 30s for Li-ion cell, between two switching transitions for allowing the cell voltages to settle before selecting the next pair of cells for equalization. This time-gap helps to avoid high-frequency switching of the selection switches during transients in cell voltages. The measured cell voltages in low-frequency cell-to-stack\\cite{kim_auto,lin} and cell-to-cell\\cite{shang_cell} equalizers show the voltage recovery effect and resulting high number of charge or discharge rounds of each cell. Due to slow switching transitions and time-gap between transitions, a higher number of switchings leads to a longer equalization time. \n\t\n\tSeveral attempts have been made to consider the effect of the cell voltage recovery within the equalization algorithm to avoid a higher number of switchings of selection switches. Estimation of the impedance drop\\cite{hannan,pham} and self-learning fuzzy logic based method\\cite{zhang_interleaved} have been employed for this purpose. The impedance drop estimation methods are computationally intensive. The self-learning fuzzy logic algorithm requires additional computation resources and prior knowledge of charge-discharge characteristics.\n\n\tA cell voltage recovery compensation is proposed here to reduce the number of switchings of the selection switches. The proposed method is computationally simpler than the impedance drop estimation methods and the fuzzy logic method. It does not require any prior knowledge of the charge-discharge characteristics of the cells.\n\t\n\t\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\\includegraphics[width=8cm]{Drawings\/volt_comp.pdf}\n\t\t\\caption{Cell voltage behavioral response when a step charging of the cell is initiated or terminated.}\n\t\t\\label{volt_comp}\n\t\\end{figure}\n\t\n\t\n\t\n\tThe cell voltage recovery effect is explained here with an example of an under-charged cell. The equalization strategy is to charge the cell so that its voltage becomes equal to the average voltage of all the cells. The equalizer charges the under-charged cell with current $I$ from time $t_1$ to $t_2$. Fig.\\,\\ref{volt_comp} shows the cell voltage and current. \n\tAt the beginning of charging at $t_1$, $V_b$ rises from $V_1$ and mostly settles within a short time $\\triangle t$.\n\tHowever, the cell voltage continues to increase slowly after time $(t_1+\\triangle t)$ due to the charging of the cell. The charging is stopped at $t_2$ when the cell voltage is $(V_2+V_{rcv})$. The cell voltage $V_b$ then settles to voltage $V_2$. Thus, if the cell charging is stopped when $V_b$ reaches the average voltage $V_{avg}$, then $V_b$ settles to ($V_{avg}-V_{rcv}$).\n\t\n\t\n\tHence, the equalizer should charge an under-charged cell until $V_b$ reaches $(V_{avg}+V_{rcv})$, so that $V_b$ settles to $V_{avg}$ after charging is stopped. Similarly, it should should discharge an over-charged cell until $V_b$ reduces to $(V_{avg}-V_{rcv})$. However, the estimation of $V_{rcv}$ with internal battery parameters requires significant computation efforts and often suffers from estimation error. In this work, the change in $V_b$ in time $\\triangle t$ from the start of the charging is measured and stored in memory as $V_{imp}$. The voltage $V_{imp}$ is used as an estimate of $V_{rcv}$. However, $V_{rcv}$ is a function of cell condition and, hence, the use of $V_{imp}$ as an estimate of $V_{rcv}$ will have a lower error when the time duration $(t_2-t_1)$ is small.\n\t\n\tThe cell voltage recovery compensation based algorithm in this work charges an under-charged cell till its voltage reaches $(V_{avg}+V_{imp})$ and discharges an over-charged cell till its voltage reduces to $(V_{avg}-V_{imp})$. \n\tIf the initial cell voltage $V_1$ is not close to the final cell voltage $V_2$, then $(t_2-t_1)$ is large and $V_{imp}$ is not a good estimate of $V_{rcv}$. However, $V_b$ comes close to the acceptable voltage range after the first round of equalization and requires further charging or discharging for a shorter duration. In the second round of equalization, $(t_2-t_1)$ is small and $V_{imp}$ is a good approximation of $V_{rcv}$. Thus, $V_b$ settles within the acceptable voltage range within a few rounds of equalization. \n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\\begin{table*}[h]\n\t\t\\centering\n\t\t\\small\n\t\t\\caption{Comparison of the proposed equalizer with the existing low-frequency selection switch cell-to-cell (LFSSCC) equalizers. \\vspace{0.0cm}}\n\t\n\t\t\\renewcommand{\\arraystretch}{1.2}\n\t\t\\renewcommand{\\tabcolsep}{4pt}\n\t\t\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}\n\t\t\t\\hline\n\t\t\t\\multirow{3}{*}{\\parbox{1.5cm}{\\centering{Topology}}} & \\multirow{3}{*}{\\parbox{2cm}{\\centering{Type of selection switch}}} & \\multicolumn{8}{c|}{\\centering{Number of components}} & \\multirow{3}{*}{\\parbox{1.5cm}{\\centering{Efficiency\\\\ (\\%)}}} \\\\\n\t\t\t\\cline{3-10}\n\t\t\t\n\t\t\t& & \\multicolumn{3}{c|}{Selection switch network} & \\multicolumn{5}{c|}{Dc-dc converter} & \\\\\n\t\t\t\\cline{3-10}\n\t\t\t\n\t\t\t& & MOSFET & DPDT relay & SPST relay & MOSFET & Capacitor & Inductor & Transformer & Diode & \\\\\n\t\t\t\\hline\n\t\t\t\n\t\t\t\\multirow{2}{*}{Ref.\\cite{xiong}} & \\parbox{1.8cm}{\\centering{MOSFET}} & 8$n$ & 0 & 0 & \\multirow{2}{*}{1} & \\multirow{2}{*}{2} & \\multirow{2}{*}{0} & \\multirow{2}{*}{1} & \\multirow{2}{*}{1} & \\multirow{2}{*}{59.4}\\\\\n\t\t\t\\cline{2-5} \n\t\t\t& \\parbox{1.8cm}{\\centering{Relay}} & 0 & 2$n$ & 0 & & & & & & \\\\\n\t\t\t\\hline\n\t\t\t\n\t\t\t\\multirow{2}{*}{Ref.\\cite{pham}} & \\parbox{1.8cm}{\\centering{MOSFET}} & 8$n$ & 0 & 0 & \\multirow{2}{*}{2} & \\multirow{2}{*}{2} & \\multirow{2}{*}{0} & \\multirow{2}{*}{1} & \\multirow{2}{*}{2} & \\multirow{2}{*}{\\parbox{1.5cm}{\\centering{85.3 to 89.5}}}\\\\\n\t\t\t\\cline{2-5} \n\t\t\t& \\parbox{1.8cm}{\\centering{Relay}} & 0 & 2$n$ & 0 & & & & & & \\\\\n\t\t\t\\hline\n\t\t\t\n\t\t\t\\multirow{2}{*}{Ref.\\cite{shang_cell}} & \\parbox{1.8cm}{\\centering{MOSFET}} & 8$n$ & 0 & 0 & \\multirow{2}{*}{5} & \\multirow{2}{*}{2} & \\multirow{2}{*}{2} & \\multirow{2}{*}{0} & \\multirow{2}{*}{5} & \\multirow{2}{*}{\\parbox{1.5cm}{\\centering{98.6 to 99.5}}}\\\\\n\t\t\t\\cline{2-5} \n\t\t\t& \\parbox{1.8cm}{\\centering{Relay}} & 0 & 2$n$ & 0 & & & & & & \\\\\n\t\t\t\\hline\n\t\t\t\n\t\t\t\\multirow{2}{*}{\\parbox{1.5cm}{\\centering{Proposed equalizer}}} & \\parbox{1.8cm}{\\centering{MOSFET}} & 4$n$+10 & 0 & 0 & \\multirow{2}{*}{2} & \\multirow{2}{*}{2} & \\multirow{2}{*}{2} & \\multirow{2}{*}{0} & \\multirow{2}{*}{0} & \\multirow{2}{*}{\\parbox{1.5cm}{\\centering{90.1 to 92.9}}}\\\\\n\t\t\t\\cline{2-5} \n\t\t\t& \\parbox{1.8cm}{\\centering{Relay}} & 0 & $n$+2 & 2 & & & & & & \\\\\n\t\t\t\\hline\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\n\t\t\\end{tabular}\n\t\t\\label{tab_comp_low_freq}\n\t\\end{table*}\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\\begin{table*}[h]\n\t\t\\centering\n\t\t\\small\n\t\t\\caption{Comparison of the proposed low-frequency selection switch cell-to-cell (LFSSCC) equalizer with other types of equalizers. \\vspace{0.0cm}}\n\t\n\t\t\\renewcommand{\\arraystretch}{1.2}\n\t\t\\renewcommand{\\tabcolsep}{2.5pt}\n\t\t\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\n\t\t\t\\hline\n\t\t\t\\multirow{3}{*}{\\parbox{1.2cm}{\\centering{Topology}}} & \\multirow{3}{*}{\\parbox{3.2cm}{\\centering{Type of Equalizer}}} & \\multicolumn{7}{c|}{\\centering{Number of components}} & \\multicolumn{2}{c|}{\\centering{Number of drivers}} & \\multirow{3}{*}{\\parbox{0.7cm}{\\centering{Effici-ency (\\%)}}} & \\multirow{3}{*}{\\parbox{1.0cm}{\\centering{Equaliza- tion speed}}} & \\multirow{3}{*}{\\parbox{1.5cm}{\\centering{Voltage difference dependent}}} \\\\\n\t\t\t\\cline{3-11}\n\t\t\t\n\t\t\t& & \\multicolumn{2}{c|}{Semiconductor} & \\multicolumn{2}{c|}{Relay} & \\multicolumn{3}{c|}{Passives} &\\multirow{2}{*}{\\parbox{1.2cm}{\\centering{High frequency}}} & \\multirow{2}{*}{\\parbox{1.2cm}{\\centering{Low frequency}}} & & &\\\\\n\t\t\t\\cline{3-9}\n\t\t\t\n\t\t\t& & MOSFET & Diode & DPDT & SPST & Cap. & Ind. & Trans. & & & & & \\\\\n\t\t\t\\hline\n\t\t\t\n\t\t\tRef\\cite{ye_zero} & \\parbox{3cm}{\\vspace{0.1 cm}\\centering{Adjacent cell resonant-tank}\\vspace{0.1 cm}} & 2$n$ & 0 & 0 & 0 & 2$n$-1 & $n$-1 & 0 & 2$n$ & 0 & 98.2 & Low & Yes\\\\\t\t\t\t\n\t\t\t\\hline\n\t\t\t\n\t\t\tRef\\cite{li} & \\parbox{3cm}{\\vspace{0.1 cm}\\centering{Multi-cell to stack multi-winding trans.}\\vspace{0.1 cm}} & $n$+1 & 0 & 0 & 0 & $n$ & 0 & \\parbox{0.7cm}{\\centering{$n$+1 wind.}} & $n$+1 & 0 & 84.8 & Moderate & Yes\\\\\t\t\t\t\n\t\t\t\\hline\n\t\t\t\n\t\t\tRef\\cite{ye_star} & \\parbox{3cm}{\\vspace{0.1 cm}\\centering{Multi-cell to multi-cell switched capacitor}\\vspace{0.1 cm}} & 2$n$ & 0 & 0 & 0 & 2$n$ & 0 & 0 & 2$n$ & 0 & - & Good & Yes\\\\\t\t\t\t\n\t\t\t\\hline\n\t\t\t\n\t\t\tRef\\cite{wang} & \\parbox{3cm}{\\vspace{0.1 cm}\\centering{Multi-cell to multi-cell dual-active bridge}\\vspace{0.1 cm}} & 3$n$ & 0 & 0 & 0 & $n$ & 0 & $n$\/2 & 3$n$ & 0 & 84.5 & Excellent & No\\\\\t\t\t\t\n\t\t\t\\hline\n\t\t\t\n\t\t\tRef\\cite{hannan} & \\parbox{3cm}{\\vspace{0.1 cm}\\centering{Cell-to-stack MOSFET based cell-selection}\\vspace{0.1 cm}} & 4$n$+2 & 2 & 0 & 0 & 2 & 0 & 2 & 2 & 4$n$ & 92.0 & Moderate & No\\\\\t\t\t\t\n\t\t\t\\hline\n\t\t\t\n\t\t\tRef\\cite{lin} & \\parbox{3cm}{\\vspace{0.1 cm}\\centering{Cell-to-stack Relay based cell-selection}\\vspace{0.1 cm}} & 1 & 1 & 0 & 2$n$ & 2 & 2 & 0 & 1 & 2$n$ & - & Moderate & No\\\\\t\t\t\t\n\t\t\t\\hline\n\t\t\t\n\t\t\t\\multirow{2}{*}{Proposed} & \\parbox{3.2cm}{\\vspace{0.1 cm}\\centering{MOSFET based cell-to-cell}\\vspace{0.1 cm}} & 4$n$+12 & 0 & 0 & 0 & 2 & 2 & 0 & 2 & 4$n$+10 & \\multirow{2}{*}{\\parbox{0.7cm}{\\centering{90.1-92.9}}} & \\multirow{2}{*}{Good} & \\multirow{2}{*}{No}\\\\\n\t\t\t\\cline{2-11}\n\t\t\t& \\parbox{3.2cm}{\\vspace{0.1 cm}\\centering{Relay based cell-to-cell}\\vspace{0.1 cm}} & 2 & 0 & $n$+2 & 2 & 2 & 2 & 0 & 2 & $n$+2 & & &\\\\\t\t\t\t\n\t\t\t\\hline\n\t\t\t\n\t\t\t\\multicolumn{14}{l}{{$^*Note$: Cap.: Capacitor, Ind.: Inductor, Trans.: Transformer, wind.: winding, $n$ represents the number of cells.}}\n\t\t\t\n\t\t\\end{tabular}\n\t\t\\label{tab_comp_others}\n\t\\end{table*}\n\t\n\t\n\t\n\t\n\t\\section{Comparison of Cell-to-Cell Selection Networks}\\label{sec_comp}\n\tA comparison of the component count and efficiency of the proposed equalizer with the existing LFSSCC equalizers, presented in Table \\ref{tab_comp_low_freq}, shows that it can work with a lower number of selection switches compared to other LFSSCC equalizers. It can also be observed that the proposed equalizer achieves higher efficiency than \\cite{xiong,pham} with similar complexity of the dc-dc converter, which is significantly simpler than \\cite{shang_cell}. Thus, the proposed method achieves above 90\\% efficiency with simple circuit implementation.\n\t\n\tTable \\ref{tab_comp_others} compares the proposed LFSSCC equalizer with different types of existing equalizers. The followings can be observed from the comparison of component counts, driver requirements, efficiency, equalization speed, and voltage difference dependence,\n\t\\begin{enumerate}\n\t\t\\item Although the adjacent cell and multi-cell equalizers can work with a lower number of MOSFETs, they require a large number of passive components and high-frequency isolated gate drivers. \n\t\t\\item The equalization current in the adjacent cell and multi-cell equalizers, except \\cite{wang}, is proportional to the cell voltage differences. Thus, these equalizers become less effective when the voltage difference is not large, especially in Li-ion cells, where even a large difference in SOC results in a small voltage difference. The work in \\cite{wang} achieves voltage difference independence by controlling each of the cell currents simultaneously, leading to higher sensor and computation cost.\n\t\t\\item The proposed equalizer offers similar component counts, driver requirements, and efficiencies compared to low-frequency cell-to-stack equalizers, but achieves twice as fast equalization.\n\t\t\\item A relay-based implementation offers the lowest component count and driver requirements.\n\t\\end{enumerate}\n\t\n\t\n\t\n\t\n\t\n\t\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\n\t\t\\includegraphics[width=8.5cm]{Results\/equalizer_image.pdf}\n\t\t\n\t\t\n\t\t\\caption{Image of the developed 8-cell equalizer prototype.}\n\t\t\\label{setup_image}\n\t\\end{figure}\n\t\n\t\n\t\n\t\n\t\\begin{table}[h]\n\t\t\\centering\n\t\t\\small\n\t\t\\caption{Circuit components and parameters of the proposed equalizer prototypes based on relays. \\vspace{0.0cm}}\n\t\n\t\t\\renewcommand{\\arraystretch}{1.5}\n\t\t\\renewcommand{\\tabcolsep}{3pt}\n\t\t\\begin{tabular}{|c|c|}\n\t\t\t\\hline\n\t\t\tComponent\/parameter & Ratings\/part no. \\\\ \\hline\n\t\t\tSwitching frequency & $30$ kHz\\\\ \\hline\n\t\t\t\\parbox{5cm}{\\centering{\\vspace{0.0cm} Inductances in port 1: $L_1$, $L_{1'}$, $M_1$}} & $62.5$ $\\mu$H, $0.7$ A \\\\ \\hline\n\t\t\t\\parbox{5cm}{\\centering{\\vspace{0.0cm} Inductances in port 2: $L_2$, $L_{2'}$, $M_2$}} & $62.5$ $\\mu$H, $0.7$ A \\\\ \\hline\n\t\t\tCapacitors: $C_1$, $C_2$ & $50$ $\\mu$F, $50$ V \\\\ \\hline\n\t\t\tMOSFETs: $Q_1$, $Q_2$ & BSC009NE2LS5I \\\\ \\hline\n\t\t\tSPST relay & OJE-SH-124LMH \\\\ \\hline\n\t\t\tDPDT relay & RT424024 \\\\ \\hline\n\t\t\tWait time for $V_{imp}$ measurement, $\\triangle t$ & $20$ s\\\\ \\hline \n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\\end{tabular}\n\t\t\\label{tab_comp_ratings}\n\t\\end{table}\t\n\t\n\t\n\t\n\t\\section{Experimental Results}\\label{experiment}\n\tAn 8-cell prototype is developed to validate the proposed low-frequency selection switch cell-to-cell (LFSSCC) equalizer using DPDT relays as the selection switches. Table \\ref{tab_comp_ratings} provides the ratings of the equalizer components, and Fig.\\,\\ref{setup_image} shows the image of the developed prototype. The prototype is tested with eight $3.6$ V, $2.6$ Ah Li-ion cells\\cite{li_cell} to verify the converter operation, equalizer efficiency, and the control algorithm. The current of the cell connected to port $1$ of the converter is controlled to $0.5$ A, and the equalization algorithm decides its direction based on cell voltages.\n\t\n\t\n\t\\subsection{Operation of Cuk Converter} \n\tFig.\\ref{dpdt_wf} shows the measured current and voltage waveforms of the capacitively level-shifted Cuk converter in the prototype to verify the converter operation. The measured current waveforms show that the port 1 current is $0.5$ A, and the peak-to-peak ripple present in each of the port currents is $100$ mA. \n\tFig.\\,\\ref{dpdt_wf}(c) shows the measured capacitor voltages for the case when cell $1$ and cell $4$ are selected for equalization. It can be observed that the capacitor $C_1$ blocks the total voltage of $4$ cells, and $C_2$ blocks that total voltage of $2$ cells, as expected theoretically in (\\ref{vc1}). The measured voltage ripples in $V_{C1}$ and $V_{C2}$ in Fig.\\,\\ref{dpdt_wf}(d) show peak-to-peak ripples of $20$ mV for both of the capacitors. \n\t\n\t\n\t\n\t\n\t\n\t\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\\begin{subfigure}[]{\\includegraphics[width=4cm]{Results\/DPDT_Iout.pdf}}\n\t\t\\end{subfigure}\\hspace{0.0cm}\n\t\t\\begin{subfigure}[]{\\includegraphics[width=4cm]{Results\/DPDT_Iin.pdf}}\n\t\t\\end{subfigure}\n\t\t\\begin{subfigure}[]{\\includegraphics[width=4.25cm]{Results\/DPDT_vc.pdf}}\n\t\t\\end{subfigure}\\hspace{0.0cm}\n\t\t\\begin{subfigure}[]{\\includegraphics[width=4.25cm]{Results\/DPDT_vc_ripple.pdf}}\n\t\t\\end{subfigure}\n\t\t\n\t\t\\caption{Experimental waveforms of (a) port 1 current, (b) port 2 current, (c) capacitor voltages, and (d) capacitor voltage ripples of the Cuk converter.}\n\t\t\\label{dpdt_wf}\n\t\\end{figure}\n\t\n\t\n\t\n\t\n\t\n\t\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\n\t\t\\includegraphics[width=8.5cm]{Plots\/eff_8cell.eps}\n\t\t\n\t\t\n\t\t\\caption{Plot of measured efficiency of the prototype with its output power.}\n\t\t\\label{eff_plot}\n\t\\end{figure}\n\t\n\t\n\tThe efficiency of the prototype is measured for different output powers up to $2$ W and is plotted in Fig.\\,\\ref{eff_plot}. Fig.\\,\\ref{eff_plot} shows that the prototype has a maximum efficiency of $92.9\\%$ and an efficiency of $90.1\\%$ at the rated power. \n\t\n\t\n\t\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\n\t\t\\begin{subfigure}[]{\\includegraphics[width=8.8cm]{Results\/8cell_rest_volt_comp.eps}}\n\t\t\\end{subfigure}\\hspace{0.0cm}\n\t\t\\begin{subfigure}[]{\\includegraphics[width=8.8cm]{Results\/8cell_rest_wo_volt_comp.eps}}\n\t\t\\end{subfigure}\n\t\t\n\t\t\\caption{Plot of measured cell voltages during voltage equalization (a) with, and (b) without cell voltage recovery compensation.}\n\t\t\\label{spst_volt_conv}\n\t\\end{figure}\n\t\n\t\\subsection{Voltage Convergence Test}\n\tThe performance of the equalizer and its control algorithm with the proposed cell voltage recovery compensation is tested on an 8-cell stack under rest condition, and Fig.\\,\\ref{spst_volt_conv}(a) shows the measured cell voltages. The maximum voltage difference is $200$ mV at the beginning. All the cell voltages converge within an acceptable voltage band of $20$ mV in $100$ minutes. \n\t\n\tThe same test is performed without the cell voltage recovery compensation to verify the necessity of this compensation, and Fig.\\,\\ref{spst_volt_conv}(b) shows the measured cell voltages. The initial cell voltages are close to those of the previous test for a proper comparison. The equalizer takes about $130$ minutes in this case to converge the cell voltages within a band of $30$ mV. Thus, the voltage convergence time is $30$\\% longer when the cell voltage recovery compensation is not used. It can also be observed from Fig.\\,\\ref{spst_volt_conv}(a) and (b) that the number of switchings of the relays is significantly lower when cell voltage recovery compensation is employed. For example, the highest switched relay's number of switching transitions is reduced from $166$ to only $18$ using the compensation. The reduction of the number of switchings significantly improves the life and reliability of the relays. \n\t\n\t\n\t\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\n\t\t\\includegraphics[width=8.8cm]{Results\/8cell_ch_dis.pdf}\n\t\t\n\t\t\n\t\t\\caption{Plot of measured cell voltages over a charge-discharge cycle.}\n\t\t\\label{volt_plot_ch_dis}\n\t\\end{figure}\n\t\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\n\t\t\\includegraphics[width=8.8cm]{Results\/8cell_load.pdf}\n\t\t\n\t\t\n\t\t\\caption{Plot of measured cell voltages with time during changes in load current $I_b$ of the cell stack.}\n\t\t\\label{volt_plot_load}\n\t\\end{figure}\n\t\n\t\\subsection{Dynamic Performance of the Equalizer}\n\tThe equalizer's performance is tested during the charge-discharge cycle of the eight-cell Li-ion stack, and Fig.\\,\\ref{volt_plot_ch_dis} shows the cell voltages. The cell voltages are initially unbalanced, with the maximum difference among them being $160$ mV. The cell stack is discharged from this condition at $0.5$ C-rate until the cell voltages reach $3$ V. The cell stack is then subjected to constant current (CC) charging at $0.5$ C-rate and constant voltage (CV) charging at $4$ V. Once, the CV charging is finished the stack is discharged once again at $0.5$ C-rate. It can be observed from the voltage plot that the equalizer equalizes the initially unbalanced cell voltages during the first discharge and maintains these voltages within $20$ mV voltage band. Once the cell voltages are equalized, the equalizer is disabled. However, on the course of charging and discharging, one or two cell voltages occasionally gets out of the $20$ mV voltage band the equalizer is enabled again, as observed near $115$, $170$, $180$, and $190$ minutes of Fig.\\,\\ref{volt_plot_ch_dis}. Fig.\\,\\ref{volt_plot_ch_dis} also shows that a very few switchings of the relays are required to maintain the cell voltages within the $20$ mV band after the initial voltage difference is mitigated. \n\t\n\tThe performance of the equalizer is also verified under variable load conditions. Fig.\\,\\ref{volt_plot_load} shows the load current variation and the corresponding cell voltages. It can be observed that a sudden change in current often produces unbalance in cell voltages, and the equalizer is activated to eliminate this unbalance. The equalizer is also activated when any cell voltage is out of the $20$ mV voltage band during constant current discharging between two step-changes in current, as observed near $65$ and $70$ minutes of Fig.\\,\\ref{volt_plot_load}. Thus, the proposed equalizer meets performance requirements under different practical operating conditions.\n\n\t\n\t\n\t\\section{Conclusion}\n\tA cell-to-cell voltage equalizer with low-frequency selection switches is proposed based on a capacitively level-shifted Cuk converter. The avoidance of an isolation transformer and diodes for the proposed equalizer's operation helps achieve high efficiency. \n\tThe use of low-frequency selection switches with simple drive circuits in the proposed equalizer leads to lower component count and cost.\n\tA low-frequency cell-to-cell selection network is proposed with bipolar voltage buses. This reduces the number of selection switches to almost half compared to the existing low-frequency selection networks for a large number of cells. \n\tA comparison of the proposed equalizer with the existing LFSSCC equalizers shows its advantages in terms of switch count, drive circuit requirements, and efficiency. \n\tAn 8-cell prototype of the proposed equalizer is implemented with relays, and the operation of the capacitively level-shifted Cuk converter is experimentally verified. The developed prototype shows an efficiency above $90$\\% over its entire power range and a peak efficiency of $92.9\\%$. The equalizer is shown to successfully converge the voltages of eight Li-ion cells within a voltage band of $20$ mV from an initial voltage imbalance of $200$ mV. The prototype shows good voltage balancing performance under different conditions such as constant current charging, constant voltage charging, constant current discharge, and varying load. \n\tA cell voltage recovery compensation scheme is proposed that estimates the voltage recovery due to cell impedance to reduce the number of switchings and the total equalization time. Experimental results show that the proposed compensation leads to the voltage convergence in a shorter time with about one order of magnitude reduction in the number of relay switchings. Thus, the proposed equalizer offers cell-to-cell voltage equalization with lower circuit complexity, component count, and good performance.\n\t\n\t\n\t\n\t\n\n\t\n\t\n\n\n\t\n\t\\bibliographystyle{IEEEtranTIE}\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe interaction of the $\\Sigma$ hyperon with nuclear matter may be\nrepresented by the complex single particle (s.p.) optical model\npotential $U_\\Sigma=V_\\Sigma-iW_\\Sigma$. In this paper we present\nour attempts to determine $V_\\Sigma$ and $W_\\Sigma$. We also point\nout the most realistic two-body $\\Sigma N$ interaction among the\navailable OBE models of the baryon-baryon interaction.\n\nIn the present paper we discuss the\nfollowing sources of information on $U_\\Sigma$:\n$\\Sigma N$ scattering data in Sec.2, $\\Sigma^-$ atoms\nin Sec.3, associated production reactions in Sec.4,\nand strangeness exchange reactions in Sec.5. Our conclusions\nare presented in Sec.6.\n\n\\section{$\\Sigma N$ scattering}\n\nThe way from the $\\Sigma N$ scattering data to $U_\\Sigma$ consists\nof two steps: first, we determine the two-body $\\Sigma N$ interaction\n${\\cal V}_{\\Sigma N}$, and second, with this ${\\cal V}_{\\Sigma N}$\nwe calculate $U_\\Sigma$. The scarcity of the two-body $\\Sigma N$\ndata makes the first step\nvery difficult. A way of overcoming these difficulties was followed\nby de Swart and his collaborators in Nijmegen: they assumed the\nmechanism of one-boson exchange (OBE) and the SU(3) symmetry\nwhich enabled them to employ the numerous $NN$ data in determining\nthe parameters of their two-body interaction. In this way they\nproduced a number of the Nijmegen models of the baryon-baryon\ninteraction: models D \\cite{D}, F \\cite{F}, soft core (SC) model\n\\cite{SC}, and the new soft-core (NSC) model \\cite{NSC}.\n\\begin{figure}[h]\n\\vspace{1mm}\n\\begin{minipage}[t]{0.475\\linewidth}\n\\centering \\vspace{-8mm} {\\psfig{file=fig1cc.eps,width=7cm}}\n\\caption{The isoscalar potential $V_\\Sigma$ as a function of the\nnucleon density $\\rho$ at $k_\\Sigma=0$ for the indicated Nijmegen\nmodels of the $\\Sigma N$ interaction.}\n\\end{minipage}\n\\vspace{20mm} \\hspace{1mm}\n\\begin{minipage}[t]{0.475\\linewidth}\n\\centering \\vspace{-8mm}\n\\includegraphics[height=5.5cm,width=8.cm]{fig2cc.eps}\n\\vspace{-6mm} \\caption{The component $W_c, W_\\e$, and $W_t$ of\nthe $\\Sigma$ absorptive potential in nuclear matter of density\n$\\rho_0$ as functions of $k_\\Sigma$.}\n\\end{minipage}\n\\vspace{-16mm}\n\\end{figure}\n\\vspace{-8mm}\n\\subsection{The real potential $V_\\Sigma$}\n\nIn calculating $V_\\Sigma$ we use the real part of the effective\n$\\Sigma N$ interaction YNG \\cite{YM} in nuclear matter. The YNG interaction\nis the configuration space representation of the G matrix calculated\nin the low order Brueckner approximation with the Nijmegen\nmodels of the baryon-baryon interaction. Our results obtained for\n$V_\\Sigma$ as function of the nucleon density\n$\\rho$ are shown in Fig. 1. As the dependence of\n$V_\\Sigma$ on the $\\Sigma$ momentum $k_\\Sigma$ is not very strong\nin the relevant interval of $k_\\Sigma$ \\cite{JJDD}, we use for\n$V_\\Sigma$ its value calculated at $k_\\Sigma=0$.\nWe see that all the Nijmegen interaction models, except for model\nF, lead to pure attractive $V_\\Sigma$ which implies the existence of\nbound states of $\\Sigma$ hyperons in the nuclear core, \\ie,\n$\\Sigma$ hypernuclei. Since no $\\Sigma$ hypernuclei have been\nobserved,\\footnote{The observed bound state of $^4_\\Sigma$He\n\\cite{He} is an exception. In the theoretical description\nof this state, Harada and his collaborators \\cite{Ha} apply\nphenomenological $\\Sigma N$ interactions, in particular, the\ninteraction SAP-F simulating at low energies the Nijmegen\nmodel F interaction. They show that essential for the existence\nof the bound state of $^4_\\Sigma$He is a strong Lane component\n$V_\\tau$ in $V_\\Sigma$, and among the Nijmegen models the\nstrongest $V_\\tau$ is implied by model F.\\cite{JDJD}}\nwe conclude that among the Nijmegen interaction models\nmodel F is the only realistic representation of the $\\Sigma N$\ninteraction.\n\n\\subsection{The absorptive potential $W_\\Sigma$}\n\nAs pointed out in \\cite{YM}, the imaginary part of the YNG\ninteraction is very sensitive to the choice of the intermediate\nstate energies in the $G$ matrix equation. In this situation\nwe decided to use for $W_\\Sigma$ the semi-classical\nexpression in terms of the total cross sections (modified by\nthe exclusion principle) for $\\Sigma N$ scattering,\ndescribed in \\cite{JDPR}. We denote by $W_c$ the contribution to the\nabsorptive potential of\nthe $\\Sigma \\Lambda$ conversion process $\\Sigma N\n\\rightarrow \\Lambda N^\\prime$ and by $W_e$ the contribution\nof the $\\Sigma N$ elastic scattering, and have\n$W_\\Sigma = W_t = W_c + W_e$.\n\\footnote{\nNotice that in the case of the nucleon optical potential\nin nuclear matter (for nucleon energies below the threshold\nfor pion production),\n$V_N-iW_N$, only the elastic $NN$ scattering contributes to\n$W_N$, and the situation is similar as in the case of the\ncontribution $W_e$ to $W_\\Sigma$.}\n\n\nOur results obtained for $W_c, W_e, W_t$ for nuclear matter (with\nN=Z) at equilibrium density $\\rho = \\rho_0 =$ 0.166 fm$^{-3}$ are shown in\nFig. 2.\nWith increasing momentum $k_\\Sigma$ the $\\Sigma\\Lambda$ conversion\ncross section decreases, on the other hand the suppression of $W_c$ by the\nexclusion principle weakens. As the net result\n$W_c$ does not change very much with $k_\\Sigma$. The same two mechanisms\nact in the case of $W_e$. Here, however, the action of the exclusion\nprinciple is much more pronounced: at $k_\\Sigma=0$ the suppression of\n$W_e$ is complete. At higher momenta, where the Pauli blocking is not\nimportant, the total elastic cross section is much bigger than the\nconversion cross section, and we have $W_e>>W_c$, and consequently\n$W_\\Sigma>>W_c$.\n\n\\section{$\\Sigma^-$ atoms}\n\nThe available data on strong interaction effects in $\\Sigma^-$ atoms\nconsist of 23 data points: strong interaction shifts $\\epsilon$ and\nwidths $\\Gamma$ of the observed levels. These shifts and widths can\nbe measured directly only in the lowest $\\Sigma^-$ atomic levels.\nThe widths of the next to the last level can be obtained indirectly\nfrom measurements of the relative yields of X-rays.\n\nIn \\cite{JRA}, we have estimated the 23 values of $\\epsilon$ and\n$\\Gamma$ from the difference between the eigenvalues of the\nSchr\\\"{o}dinger equation of $\\Sigma^-$ in $\\Sigma^-$ atoms with\nthe strong $\\Sigma^-$-atomic nucleus interaction and without\nthis interaction. To obtain this strong interaction, we\napplied the local density approximation, and used our\noptical model of Sec. 2. The agreement of our results,\ncalculated with the optical potentials (obtained with the\n4 Nijmegen $\\Sigma N$ interaction models)\nwith the 23 empirical data points is characterized by the\nfollowing values of $\\chi^2$: $\\chi^2$(model D) $>$ 130,\n$\\chi^2$(model F) = 38.1, $\\chi^2$(model SC) = 55.0,\n $\\chi^2$(model NSC) $>$ 904, and we conclude that the $\\Sigma^-$\natomic data point out at model F as the best representation\nof the $\\Sigma N$ interaction.\\footnote{Notice that the positive\nsign of the measured values of $\\epsilon$ requires an attractive\n$\\Sigma$ potential at the nuclear surface, \\ie at low densities.}\n\n\\section{The associated production reactions}\n\nThe first associated $\\Sigma$ production reaction $(\\pi^-,K^+)$\nwas observed at KEK on $^{28}$Si target at pion momentum of\n1.2 GeV\/c (\\cite{anna1},\\cite{anna3}), and this reaction is\nthe subject of the present analysis. We consider the reaction\n$(\\pi^-,K^+)$ in which the pion $\\pi^-$ with momentum\n${\\bf k}_\\pi$ hits a proton in the $^{28}$Si target in the state $\\psi_P$\nand emerges in the final state as kaon $K^+$ moving in the direction\n$\\hat{k}_K$ with energy $E_K$, whereas the hit proton emerges in the final\nstate as a $\\Sigma^-$ hyperon with momentum $\\bf{k}_\\Sigma$.\nWe apply the simple\nimpulse approximation described in \\cite{I2}, with $K^+$ and $\\pi^-$ plane\nwaves, and obtain:\n\\begin{equation}\n\\label{ia}\nd^3\\sigma\/d\\hat{k}_\\Sigma d\\hat{k}_K dE_K\\sim|\\int d{\\bf r}\\exp(-i{\\bf qr})\n\\psi_{\\Sigma,{\\bf k}_\\Sigma}({\\bf r})^{(-)*}\\psi_P({\\bf r})|^2,\n\\end{equation}\nwhere the momentum transfer ${\\bf q}={\\bf k}_K-{\\bf k}_\\pi$, and\n$\\psi_{\\Sigma,{\\bf k}_\\Sigma}({\\bf r})^{(-)}$ is the $\\Sigma$ scattering\nwave function which is the solution of the s,p. Schr\\\"{o}dinger equation\nwith the s.p. potential\n\\begin{equation}\n\\label{us}\nU_\\Sigma(r)=(V_\\Sigma-iW_\\Sigma)\\theta(R-r),\n\\end{equation}\nwhere for $V_\\Sigma$ and $W_\\Sigma$ we use the nuclear matter results\ndiscussed in Section 2, calculated at $\\rho=n\/[(4\\pi\/3)R^3]$, where\n$n$=27 is the number of nucleons in the final state.\n\nFor the $^{28}$Si target nucleus we assume a simple shell model\nwith a square well s.p, potential $V_P(r)$ (which determines $\\psi_P$)\nwith the radius $R_P$ (and with a spin-orbit term).\nThe parameters of $V_P(r)$ are adjusted to the proton separation\nenergies (in particular $R_P=3.756$fm). For $R$ we make the simple\nand plausible assumption: $R=R_P$.\n\nIn the inclusive KEK experiments \\cite{anna1}-\\cite{anna3} only the energy\nspectrum of kaons at fixed $\\hat{k}_\\Sigma$ was measured.\nTo obtain this\nenergy spectrum, we have to integrate the cross section (\\ref{ia}) over\n$\\hat{k}_\\Sigma$.\n\n\\begin{figure}[h]\n\\vspace{0mm}\n\\begin{minipage}[t]{0.475\\linewidth}\n\\centering \\vspace{-8mm} {\\psfig{file=fig3cc.eps,width=7.cm}}\n\\caption{Kaon spectrum from $(\\pi^-,K^+)$ reaction on $^{28}$Si at\n$\\theta_K=6^\\circ$ at $p_\\pi=1.2$ GeV\/c obtained with $V_\\Sigma$\ndetermined by models F and D of the $\\Sigma N$ interaction. Curves\ndenoted by $c(t)$ were obtained with $W_\\Sigma = W_c(W_t)$. Data\npoints are taken from \\cite{anna3}.}\n\\end{minipage}\n\\vspace{20mm} \\hspace{1mm}\n\\begin{minipage}[t]{0.475\\linewidth}\n\\centering \\vspace{-1mm}\n\\includegraphics[height=4.9cm,width=5.6cm]{fig4cc.eps}\n\\vspace{-3mm} \\caption{Pion spectrum from $(K^-,\\pi^+)$ reaction\non $^9$Be at $\\theta_\\pi=4^\\circ$ at $p_K=0.6$ GeV\/c obtained with\n$V_\\Sigma$ determined by models F and D of the $\\Sigma N$\ninteraction. Curves denoted by $c(t)$ were obtained with $W_\\Sigma\n= W_c(W_t)$. Data points are taken from \\cite{bart}.}\n\\end{minipage}\n\\vspace{-16mm}\n\\end{figure}\nWe present our results for the inclusive cross\nsection as a function of $B_\\Sigma$, the separation (binding)\nenergy of $\\Sigma$ from the hypernuclear system produced.\nOur model F and D results\n\\footnote{The remaining models SC and NSC are\nsimilar to model D: they all lead to attractive $V_\\Sigma$ in\ncontradistinction to model F leading to repulsive $V_\\Sigma$ (at\ndensities inside nulei - see Fig. 1). Consequently, the results\nfor the kaon spectrum for models SC and NSC are expected to be\nsimilar as in case of model D.}\nfor kaon spectrum from $(\\pi^-,K^+)$ reaction on $^{28}$Si\nat $\\theta_K=6^o$ at $p_\\pi= 1.2$ GeV\/c\nare shown in Fig. 3.\nWe see that the best fit to the data points is obtained for\n$V_\\Sigma$ derived from model F and with $W_\\Sigma=W_t=W_c+W_e$.\nThe fit would improve if we considered the distortion of kaon and\nespecially of pion waves\n(it was noticed already in Ref. \\cite{anna1} that this\ndistortion pushes the kaon spectrum down). Inclusion into the\nabsorptive potential of the contribution $W_e$ of the elastic\n$\\Sigma N$ scattering is essential for obtaining this result with\n$V_\\Sigma$(model F) = 17.25 MeV. Earlier estimates of the kaon\nspectrum without this contribution suggested a repulsive\n$V_\\Sigma$ with an unexpected strength of about 100 MeV. Notice\nthat the action of the absorptive potential $W_\\Sigma$ on the\n$\\Sigma$ wave function (decrease of this wave function) is similar\nas the action of a repulsive $V_\\Sigma$. Therefore we achieve with\nstrong absorption the same final effect with a relatively weaker\nrepulsion.\n\n\n\n\\section{The strangeness exchange reactions}\n\nFirst observations of the strangeness exchange $(K^-,\\pi)$ reactions\nwith a reliable accuracy were performed at BNL. Here,\nwe shall discuss the $(K-,\\pi^+)$ reaction observed at BNL on\nBe$^9$ target with 600 MeV\/c kaons.\\cite{bart} Proceeding similarly\nas in the case of the associated production described in Sec.4, we\nget the results shown in Fig. 4. We see that similarly as in Sect. 4\nthe fit to the data points obtained for $V_\\Sigma$\nderived from model F is much better than the fit obtained with\nmodel D.\n\n\\newpage\n\n\\section{Conclusions}\n\n$\\bullet$ The real part $V_\\Sigma$ of the $\\Sigma$ optical potential\nis repulsive inside the nucleus and has a shallow attractive pocket\nat the nuclear surface.\n\n$\\bullet$ Among the Nijmegen models of the baryon-baryon interaction\nonly model F leads to this form of $V_\\Sigma$.\n\n$\\bullet$ The contribution of the elastic $\\Sigma N$ scattering to the\nabsorptive part $W_\\Sigma$ of the $\\Sigma$ optical potential is\nessential in the analysis of $\\Sigma$ production processes.\n\\vspace{0.6cm}\n\nThis research was partly supported by the Polish Ministry of Science\nand Higher Education under Research Project No. N N202 046237.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Introduction} \n\nThis paper is part of a long term project concerned with a \nsystematic approach to unitary representations \nof Banach--Lie groups in terms of conditions \non spectra in the derived representation. A unitary \nrepresentation $\\pi \\: G \\to \\U(\\cH)$ \nis said to be {\\it smooth} if the subspace \n$\\cH^\\infty$ of smooth vectors is dense. \nThis is automatic for continuous \nrepresentations of finite dimensional groups but not in general \n(cf.\\ \\cite{Ne10a}). For any smooth \nunitary representation, the {\\it derived representation} \n\\[ \\dd\\pi \\: \\g = \\L(G)\\to \\End(\\cH^\\infty), \\quad \n\\dd\\pi(x)v := \\derat0 \\pi(\\exp tx)v\\] \ncarries significant information in the sense that the closure of the \noperator $\\dd\\pi(x)$ coincides with the infinitesimal generator of the \nunitary one-parameter group $\\pi(\\exp tx)$. We call $(\\pi, \\cH)$ \n{\\it semibounded} if the function \n\\[ s_\\pi \\: \\g \\to \\R \\cup \\{ \\infty\\},\\ \\ \ns_\\pi(x) \n:= \\sup\\big(\\Spec(i\\dd\\pi(x))\\big) \n= \\sup\\{ \\la i\\dd\\pi(x)v,v\\ra \\: v \\in \\cH^\\infty, \\|v\\|=1\\}\\] \nis bounded on the neighborhood of some point in $\\g$ \n(cf.\\ \\cite[Lemma~5.7]{Ne08}). \nThen the set $W_\\pi$ of all such points \nis an open invariant convex cone in the Lie algebra $\\g$. \nWe call $\\pi$ {\\it bounded} if $s_\\pi$ is bounded on some $0$-neighborhood, \nwhich is equivalent to $\\pi$ being norm continuous \n(cf.\\ Proposition~\\ref{prop:contin}). \nAll finite dimensional unitary representations are bounded \nand many of the unitary representations appearing in physics are semibounded \n(cf. \\cite{Ne10c}). \n\nOne of our goals is a classification of the irreducible semibounded \nrepresentations and the development of \ntools to obtain direct integral decompositions \nof semibounded representations. To this end, realizations \nof unitary representations in spaces of holomorphic sections of \nvector bundles turn out to be extremely helpful. \nClearly, the bundles in question should be allowed to have \nfibers which are infinite dimensional Hilbert spaces, and to \ntreat infinite dimensional groups, we also have to admit \ninfinite dimensional base manifolds. \nThe main point of the present paper is to provide \neffective methods to treat unitary representations \nof Banach--Lie groups in spaces of holomorphic sections of homogeneous \nHilbert bundles. \n\nIn Section~\\ref{sec:1} we explain how to parametrize \nthe holomorphic structures on Banach vector bundles \n$\\bV = G \\times_H V$ over a Banach homogeneous space $M = G\/H$ \nassociated to a norm continuous representation \n$(\\rho, V)$ of the isotropy group~$H$. The main result \nof Section~\\ref{sec:1} is \nTheorem~\\ref{thm:a.2} which generalizes the corresponding \nresults for the finite dimensional case by Tirao and Wolf \n(\\cite{TW70}). As in finite dimensions, the complex bundle \nstructures are specified by \n``extensions'' $\\beta \\: \\fq \\to \\gl(V)$ of the differential \n$\\dd \\rho \\: \\fh \\to \\gl(V)$ to a representation of the complex subalgebra \n$\\fq \\subeq \\g_\\C$ specifying the complex structure on~$M$ in the sense that \n$T_H(M) \\cong \\g_\\C\/\\fq$ \n(cf.~\\cite{Bel05}). The main point in \\cite{TW70} is that the homogeneous space \n$G\/H$ need not be realized as an open $G$-orbit in a complex \nhomogeneous space of a complexification $G_\\C$, which is impossible \nif the subgroup of $G_\\C$ generated by $\\exp \\fq$ is not closed. \nIn the Banach context, two additional difficulties appear: \nThe Lie algebra $\\g_\\C$ may not be integrable in the sense that \nit does not belong to any Banach--Lie group (\\cite{GN03}) and, even if $G_\\C$ \nexists and the subgroup $Q := \\la \\exp \\fq \\ra$ is closed, it \nneed not be a Lie subgroup so that there is no natural construction \nof a manifold structure on the quotient space~$G\/Q$. \nAs a consequence, the strategy of the proof \nin \\cite{TW70} can not be used for Banach Lie groups. Another \ndifficulty of the infinite dimensional context is that there is \nno general existence theory for solutions of $\\oline\\partial$-equations \n(see in particular \\cite{Le99}). \n\nThe next step, carried out in Section~\\ref{sec:2}, \nis to analyze Hilbert subspaces of the space $\\Gamma(\\bV)$ \nof holomorphic sections of $\\bV$ \non which $G$ acts unitarily. In this context \n$(\\rho, V)$ is a bounded unitary representation. The most regular \nHilbert spaces with this property are those that we call \n{\\it holomorphically induced from $(\\rho, \\beta)$}. \nThey contain $(\\rho, V)$ as an $H$-subrepresentation \nsatisfying a compatibility condition with respect to $\\beta$. \nHere we show that if \nthe subspace $V \\subeq \\cH$ is invariant under the commutant \n$B_G(\\cH) = \\pi(G)'$, then restriction to $V$ yields an isomorphism \nof the von Neumann algebra $B_G(\\cH)$ with a suitably defined commutant \n$B_{H,\\fq}(V)$ of $(\\rho,\\beta)$ (Theorem~\\ref{thm:5.5}). \nThis has remarkable \nconsequences. One is that the representation of $G$ on $\\cH$ \nis irreducible (multiplicity free, discrete, type I) if and only \nif the representation of $(H,\\fq)$ on $V$ has this property. \nThe second main result in Section~\\ref{sec:2} is a \ncriterion for a unitary representation \n$(\\pi, \\cH)$ of $G$ to be holomorphically induced \n(Theorem~\\ref{thm:a.3}). \n\nSection~\\ref{sec:3} is devoted to a description \nof environments in which Theorem~\\ref{thm:a.3} applies naturally. \nHere we consider an element $d \\in \\g$ which is {\\it elliptic} in \nthe sense that the one-parameter group $e^{\\R \\ad d}$ of automorphisms \nof $\\g$ is bounded. This is equivalent to the existence of an \ninvariant compatible norm. Suppose that $0$ is isolated in $\\Spec(\\ad d)$. \nThen the subgroup $H := Z_G(d) = \\{ g \\in G \\: \\Ad(g)d = d \\}$ \nis a Lie subgroup and the homogeneous space $G\/H$ carries a natural \ncomplex manifold structure. A smooth unitary representation \n$(\\pi, \\cH)$ of $G$ is said to be of \n{\\it positive energy} if the selfadjoint operator \n$-i \\dd\\pi(d)$ is bounded from below. Note that this is in particular \nthe case if $\\pi$ is semibounded with $d \\in W_\\pi$. \nAny positive energy representation \nis generated as a $G$-representation by the closed subspace \n$V := \\oline{(\\cH^\\infty)^{\\fp^-}}$, where \n$\\fq = \\fp^+ \\rtimes \\fh_\\C \\subeq \\g_\\C$ is the complex subalgebra \ndefining the complex structure on $G\/H$ and \n$\\fp^- := \\oline{\\fp^+}$. \nIf the $H$-representation $(\\rho, V)$ is bounded, then \n$(\\pi, \\cH)$ is holomorphically induced \nby $(\\rho,\\beta)$, where $\\beta$ is determined by $\\beta(\\fp^+) = \\{0\\}$ \n(Theorem~\\ref{thm:6.2}). In Theorem~\\ref{thm:3.15} we further \nshow that, under these assumptions, $\\pi$ is semibounded with $d \\in W_\\pi$. \nThese results are rounded off by Theorem~\\ref{thm:6.2b} which shows \nthat, if $\\pi$ is semibounded with $d \\in W_\\pi$, then \n$\\pi$ is a direct sum of holomorphically induced representations \nand if, in addition, $\\pi$ is irreducible, then $V$ coincides \nwith the minimal eigenspace of $-i\\dd\\pi(d)$ and the representation \nof $H$ on this space is automatically bounded. \nFor all this we use refined analytic tools based on the fact that \nthe space \n$\\cH^\\infty$ of smooth vectors is a Fr\\'echet space on which \n$G$ acts smoothly (\\cite{Ne10a}) and the $\\R$-action on \n$\\cH^\\infty$ defined by $\\pi_d(t) := \\pi(\\exp td)$ is equicontinuous. \nThese properties permit us to use a \nsuitable generalization of Arveson's spectral theory, developed\n in Appendix~\\ref{app:1}. \n\nIn \\cite{Ne11} we shall use the techniques developed \nin the present paper to obtain a complete descriptions of semibounded \nrepresentations for the class of hermitian Lie groups. \nOne would certainly like to extend the tools developed here to \nFr\\'echet--Lie groups such as diffeomorphism groups and groups of \nsmooth maps. Here a serious problem is the construction of \nholomorphic vector bundle structures on associated bundles, \nand we do not know how to extend this beyond the Banach context, \nespecially because of the non-existing solution theory for \n$\\oline\\partial$-equations (cf.\\ \\cite{Le99}). \n\nSeveral of our results have natural predecessors in more restricted\n contexts. In \\cite{BR07} Belti\\c{t}\\u{a} and Ratiu \nstudy holomorphic Hilbert bundles over Banach manifolds $M$ \nand consider the endomorphism bundle $B(\\bV)$ over $M \\times \\oline M$ \n(using a different terminology). They also relate Hilbert subspaces \n$\\cH$ of $\\Gamma(\\bV)$ to \nreproducing kernels, which in this context are \nsections of $B(\\bV)$ satisfying a certain holomorphy condition \nwhich under the assumption of local boundedness of the kernel \nis equivalent to \nholomorphy as a section of $B(\\bV)$ (\\cite[Thm.~4.2]{BR07}; \nsee also \\cite[Thm.~1.4]{BH98} and \\cite{MPW97} for the case of \nfinite dimensional \nbundles over finite dimensional manifolds and \\cite{Od92} for the \ncase of line bundles). \nBelti\\c{t}\\u{a} and Ratiu \nuse this setup to realize certain representations \nof a $C^*$-algebra $\\cA$, which define bounded unitary representations \nof the unitary group $G := \\U(\\cA)$, in spaces of holomorphic sections of a \nbundle over a homogeneous space of the unit group $\\cA^\\times$ on \nwhich $\\U(\\cA)$ acts transitively (\\cite[Thm.~5.4]{BR07}). \nThese homogeneous spaces are of the form \n$G\/H$, where $H$ is the centralizer of some hermitian element \n$a \\in \\cA$ with finite spectrum, so that the realization \nof these representations by holomorphic sections could also be derived \nfrom our Corollary~\\ref{cor:6.2}. This work has been continued by \nBelti\\c{t}\\u{a} with Gal\\'e in a different direction, focusing \non complexifications of real homogeneous spaces \ninstead of invariant complex structures on the real spaces (\\cite{BG08}). \n\nIn \\cite{Bo80} Boyer constructs irreducible unitary representations \nof the Hilbert--Lie group \n$\\U_2(\\cH) = \\U(\\cH) \\cap (\\1 + B_2(\\cH))$ via holomorphic induction \nfrom characters of the diagonal subgroup. They live in spaces of \nholomorphic sections on homogeneous spaces of the complexified \ngroup $\\GL_2(\\cH) = \\GL(\\cH) \\cap (\\1 + B_2(\\cH))$, \nwhich are restricted versions of flag manifolds carrying \nstrong K\\\"ahler structures. \n\n\nIn the present paper we only deal with representations \nassociated to homogeneous vector bundles. To understand branching laws \nfor restrictions of representations to subgroups, one should \nalso study situations where the group $G$ does not act transitively \non the base manifold. For finite dimensional holomorphic vector \nbundles, this has been done extensively by \nT.~Kobayashi who obtained powerful criteria for representations \nin Hilbert spaces of holomorphic sections to be multiplicity \nfree (cf.\\ \\cite{KoT05}, \\cite{KoT06}). It would be interesting to explore \nthe extent to which \nKobayashi's technique of visible actions on complex manifolds \ncan be extended to Banach manifolds. \\\\\n\n\n{\\bf Notation:} For a group $G$ we write $\\1$ for the neutral element \nand $\\lambda_g(x) = gx$, resp., $\\rho_g(x) = xg$ \nfor left multiplications, resp., right multiplications. \nWe write $\\g_\\C$ for the complexification of a real Lie algebra $\\g$ and \n$\\oline{x + iy} := x - iy$ for the complex conjugation on $\\g_\\C$, which is an \nantilinear Lie algebra automorphism. \nFor two Hilbert spaces $\\cH_1, \\cH_2$, we write \n$B(\\cH_1, \\cH_2)$ for the space of continuous (=bounded) linear operators \n$\\cH_1 \\to \\cH_2$ and $B_p(\\cH)$, $1 \\leq p < \\infty$, \nfor the space of Schatten class operators \n$A \\: \\cH \\to \\cH$ of order $p$, i.e., $A$ is compact with \n$\\tr((A^*A)^{p\/2}) < \\infty$. \\\\\n\n{\\bf Acknowledgment:} We thank Daniel Belti\\c{t}\\u{a} for a careful \nreading of earlier versions of this paper, for pointing out references \nand for various interesting discussions on its subject matter. \n\n\\tableofcontents\n\n\n\\section{Holomorphic Banach bundles} \\mlabel{sec:1}\n\nTo realize unitary representations \nof Banach--Lie groups in spaces of holomorphic sections of \nHilbert bundles, we first need a \nparametrization of holomorphic bundle structures on \ngiven homogeneous vector bundles for real Banach--Lie groups. \nAs we shall see in Theorem~\\ref{thm:a.2} below, \nformulated appropriately, the corresponding results\nfrom the finite dimensional case (cf.\\ \\cite{TW70}) \ncan be generalized to the Banach context. \n\nThe following observation provides some information \non the assumptions required on the isotropy representation \nof a Hilbert bundle. It is a slight modification \nof results from \\cite[Sect.~3]{Ne09}. \n\n\\begin{prop}\\label{prop:contin} Let $(\\pi,\\cH)$ be a \nunitary representation of the Banach--Lie group $G$ \nfor which all vectors are smooth. \nThen $\\pi \\: G \\to \\U(\\cH)$ is a morphism of \nLie groups, hence in particular norm continuous, i.e., a \nbounded representation. \n\\end{prop}\n\n\\begin{prf} Our assumption implies that, for each $x\\in \\g$, \nthe infinitesimal generator $\\dd\\pi(x)$ of the unitary \none-parameter group $\\pi_x(t) := \\pi(\\exp_G(tx))$ \nis everywhere defined, hence a bounded operator because its \ngraph is closed. \n\nTherefore the derived representation leads to a morphism of \nBanach--Lie algebras \n$\\dd\\pi \\: \\g \\to B(\\cH).$\nSince the function $s_\\pi$ \nis a sup of a set of continuous linear functionals, it is \nlower semi-continuous. Hence the function \n\\[ x \\mapsto \\|\\dd\\pi(x)\\| \n= \\max(s_\\pi(x), s_\\pi(-x)) \\] \nis a lower semi-continuous seminorm \n and therefore continuous because $\\g$ is barreled \n(cf.\\ \\cite[\\S III.4.1]{Bou07}).\nWe conclude that the set of all linear functional \n$\\la \\dd\\pi(\\cdot)v,v\\ra$, $v \\in \\cH^\\infty$ a unit vector, \n is equicontinuous in $\\g'$, \nso that the assertion follows from \\cite[Thm.~3.1]{Ne09}. \n\\end{prf}\n\nWe now turn to the case where $M$ is a Banach homogeneous space. \nLet $G$ be a Banach--Lie group with Lie algebra $\\g$ and \n$H \\subeq G$ be a split Lie subgroup, i.e., \nthe Lie algebra $\\fh$ of $H$ has a closed complement in $\\g$, \n for which the coset space \n$M := G\/H$ carries the structure of a complex manifold such that \nthe projection $q_M\\: G\\to G\/H$ is a smooth $H$-principal bundle and \n$G$ acts on $M$ by holomorphic maps. \nLet $m_0 = q_M(\\1) \\in M$ be the canonical base point and \n$\\fq \\subeq \\g_\\C$ be the kernel of the complex linear extension \nof the map $\\g \\to T_{m_0}(G\/H)$ to $\\g_\\C$, so that $\\fq$ is a closed \nsubalgebra of~$\\g_\\C$ invariant under $\\Ad(H)$ \n(cf.\\ \\cite[Thm.~15]{Bel05}). We call $\\fq$ the \n{\\it subalgebra defining the complex structure on $M = G\/H$} \nbecause specifying $\\fq$ means to identify \n$T_{m_0}(G\/H)\\cong \\g\/\\fh$ with the complex Banach space $\\g_\\C\/\\fq$ \nand thus specifying the complex structure on~$M$. \n\n\n\\begin{rem} \\mlabel{rem:norm-cont} \nIf the Banach--Lie group $G$ acts smoothly by isometric bundle automorphisms \non the holomorphic Hilbert bundle $\\bV$ over $M = G\/H$, then \nthe action of the stabilizer group $H$ on $V := \\bV_{m_0}$ \nis smooth, so that Proposition~\\ref{prop:contin} shows that it \ndefines a bounded unitary representation \n$\\rho \\: H \\to \\U(V)$. \n\nIf $\\sigma_M \\: G \\times M \\to M$ denotes the corresponding action on \n$M$ and $\\dot\\sigma_M \\: \\g \\to \\cV(M)$ the derived action, \nthen, for the closed subalgebra \n\\[ \\fq := \\{ x \\in \\g_\\C \\: \\dot\\sigma_M(x)(m_0) = 0\\}, \\] \nthe representation $\\beta \\: \\fq \\to B(V)$ \nis given by a continuous bilinear map \n\\[ \\hat\\beta \\: \\fq \\times V \\to V, \\quad \n(x,v) \\mapsto \\beta(x)v.\\] \nThis means that $\\beta$ is a continuous morphism \nof Banach--Lie algebras. \n\\end{rem}\n\nThis observation leads us to the following structures. \n\n\n\\begin{defn}\\mlabel{def:a.1} Let $H \\subeq G$ be a \nLie subgroup and $\\fq \\subeq \\g_\\C$ be a closed subalgebra containing \n$\\fh_\\C$. \nIf $\\rho \\: H \\to \\GL(V)$ is a norm continuous representation \non the Banach space $V$, then a morphism \n$\\beta \\: \\fq \\to \\gl(V)$ of complex Banach--Lie algebras \nis said to be an {\\it extension of $\\rho$} if \n\\begin{equation}\\label{eq:comprel} \n\\dd\\rho = \\beta\\res_\\fh \\quad \\mbox{ and } \\quad \n\\beta(\\Ad(h)x) = \\rho(h)\\beta(x)\\rho(h)^{-1} \\quad \\mbox{ for } \\quad \nh \\in H, x \\in \\fq. \n\\end{equation}\n\\end{defn}\n\n\\begin{defn} (a) \nIf $q \\: \\bV = G \\times_H V \\to M$ is a homogeneous vector bundle \ndefined by the norm continuous representation $\\rho \\: H \\to \\GL(V)$, \nwe associate to each section $s \\: M \\to \\bV$ the function \n$\\hat s \\: G \\to V$ specified by $s(gH) = [g, \\hat s(g)]$. \nA function $f \\: G \\to V$ is of the form $\\hat s$ for a \nsection of $\\bV$ if and only if \n\\begin{equation} \\label{eq:equiv-sec} \nf(gh) = \\rho(h)^{-1} f(g) \\quad \\mbox{ for } \\quad \ng \\in G, h \\in H. \n\\end{equation}\nWe write $C(G,V)_\\rho$, resp., $C^\\infty(G,V)_\\rho$ for the \ncontinuous, resp., smooth functions satisfying \n\\eqref{eq:equiv-sec}. \n\n(b) We associate to each \n$x \\in \\g_\\C$ the left invariant \ndifferential operator on $C^\\infty(G,V)$ defined by \n\\[ (L_x f)(g) := \\derat0 f(g\\exp(tx)) \\quad \\mbox{ for } \\quad \nx \\in \\g. \\]\nBy complex linear extension, we define the operators \n\\[L_{x+ iy} := L_x + i L_y \\quad \\mbox{ for } \\quad z = x + iy \\in \\g_\\C, \nx,y \\in \\g. \\]\n\n(c) For any extension $\\beta$ of $\\rho$, \nwe write $C^\\infty(G,V)_{\\rho,\\beta}$ for the \nsubspace of those elements $f \\in C^\\infty(G,V)_\\rho$ \nsatisfying, in addition, \n\\begin{equation} \\label{eq:inf-equiv}\nL_x f = - \\beta(x) f \\quad \\mbox{ for } \\quad x \\in \\fq. \n\\end{equation}\n\\end{defn}\n\n\\begin{rem}\nFor $x \\in \\g$, $h \\in H$ and any smooth function \n$\\phi$ defined on an open right $H$-invariant subset of \n$G$, we have \n\\begin{eqnarray*}\n(L_x\\phi)(gh) \n&=& \\derat0 \\phi\\big(g\\exp(t\\Ad(h)x)h\\big)=\n (L_{\\Ad(h)x}(\\phi \\circ \\rho_h))(g), \n\\end{eqnarray*}\nso that we obtain for each $x \\in \\g_\\C$ and $h \\in H$ the \nrelation \n\\begin{eqnarray}\\label{eq:liederrel}\n(L_x\\phi)\\circ \\rho_h = L_{\\Ad(h)x}(\\phi \\circ \\rho_h). \n\\end{eqnarray}\n\\end{rem}\n\n\nThe proof of the following theorem is very much inspired by \\cite{TW70}.\n\n\n\\begin{thm} \\mlabel{thm:a.2} Let $M = G\/H$, $V$ be a complex Banach space \nand $\\rho \\: H \\to \\GL(V)$ be a norm continuous representation. \nThen, for any extension $\\beta \\: \\fq \\to \\gl(V)$ of $\\rho$, \nthe associated bundle $\\bV := G \\times_H V$ carries \na unique structure of a holomorphic vector bundle over $M$, \nwhich is determined by the \ncharacterization of the holomorphic sections $s \\: M \\to \\bV$ \nas those for which $\\hat s \\in C^\\infty(G,V)_{\\rho,\\beta}$. \nAny such holomorphic bundle structure is $G$-invariant in the sense that \n$G$ acts on $\\bV$ by holomorphic bundle automorphism. \nConversely, every $G$-invariant holomorphic vector bundle \nstructure on $\\bV$ is obtained from this construction.\n\\end{thm}\n\n\\begin{prf} {\\bf Step 1:} Let $\\beta$ be an extension \nof $\\rho$ and $E \\subeq \\g$ be a closed subspace \ncomplementing $\\fh$. Then $E \\cong \\g\/\\fh \\cong T_{m_0}(M)$ \nimplies the existence \nof a complex structure $I_E$ on $E$ for which $E \\to \\g\/\\fh$ is an isomorphism \nof complex Banach spaces. Therefore \nthe $I_E$-eigenspace decomposition \n\\[ E_\\C = E_+ \\oplus E_-, \\quad E_\\pm = \\ker(I_E \\mp i\\1),\\] \nis a direct decomposition into closed subspaces. The \nquotient map $\\g_\\C \\to \\g_\\C\/\\fq \\cong \\g\/\\fh$ \nis surjective on $E_+$ and annihilates $E_-$. Now \n$\\fr := E_+$ is a closed complex complement of $\\fq \n= \\fh_\\C \\oplus E_-$ in $\\g_\\C$ (\\cite[Thm.~15]{Bel05}). \n\nPick open convex $0$-neighborhoods \n$U_\\fr \\subeq \\fr$ and $U_\\fq \\subeq \\fq$ such that the BCH multiplication \n$*$ defines a biholomorphic map \n$\\mu \\: U_\\fr \\times U_\\fq \\to U, (x,y) \\mapsto x * y$, \nonto the open $0$-neighborhood $U \\subeq \\g_\\C$ \nand that $*$ defines an associative multiplication \ndefined on all triples of elements of $U$. \n\n{\\bf Step 2:} On the $0$-neighborhood $U \\subeq \\g_\\C$, we consider the \nholomorphic function \n$$ F \\: U \\to \\GL(V), \\quad \nF(x * y) := e^{-\\beta(y)}. $$\nLet $U_\\g \\subeq U$ be an open $0$-neighborhood which is \nmapped by $\\exp_G$ diffeomorphically onto an open $\\1$-neighborhood \n$U_G$ of $G$. Then we consider the smooth function \n$$ f \\: U_G \\to \\GL(V), \\quad \\exp_G z \\mapsto F(z). $$ \n\nFor $w \\in \\g$ and $z \\in U_\\g$, the BCH product $z * tw \\in \\g_\\C$ \nis defined if \n$t$ is small enough, and we have \n\\begin{equation}\\label{eq:1.4} \n(L_w f)(\\exp_G z) = \\derat0 F(z * tw) \n= \\dd F(z) \\dd\\lambda_z^*(0)w, \n\\end{equation}\nwhere $\\dd\\lambda_z^*(0) \\: \\g_\\C \\to \\g_\\C$ is the differential \nof the multiplication map $\\lambda_z^*(x) = z * x$ in $0$.\nAs \\eqref{eq:1.4} is complex linear in $w \\in\\g_\\C$, it follows that \n$$ (L_w f)(\\exp_G z) = \\dd F(z) \\dd\\lambda_z^*(0)w $$ \nhold for every $w \\in \\g_\\C$, hence in particular for $w \\in \\fq$. \nFor $w \\in \\fq$ and $z = \\mu(x,y)$, we thus obtain \n\\begin{align*}\n\\dd F(z) \\dd\\lambda_z^*(0)w \n&= \\derat0 F(x * y * tw)\n= \\derat0 e^{-\\beta(y * tw)} \n= \\derat0 e^{-t \\beta(w)} e^{-\\beta(y)}\\\\\n&= -\\beta(w) e^{-\\beta(y)}. \n\\end{align*}\nWe conclude that \n$$ (L_w f)(g) = -\\beta(w) f(g) \\quad \\mbox{ for } \\quad g \\in U_G. $$ \nIn particular, we obtain \n$f(gh) = \\rho(h)^{-1} f(g)$ for $g \\in U_G, h \\in H_0.$ \n\n{\\bf Step 3:} \nSince $H$ is a complemented Lie subgroup, there \nexists a connected submanifold $Z \\subeq G$ containing $\\1$ for which \nthe multiplication map \n$Z\\times H \\to G, (x,h) \\mapsto xh$ is a diffeomorphism onto \nan open subset of $G$. Shrinking $U_G$, we may therefore \nassume that $U_G = U_Z U_H$ holds for a connected open $\\1$-neighborhood \n$U_Z$ in $Z$ and a connected open $\\1$-neighborhood \n$U_H$ in $H$. Then \n\\[ \\tilde f(z h) := \\rho(h)^{-1} f(z) \n\\quad \\mbox{ for } \\quad z \\in U_Z, h \\in H, \\] \ndefines a smooth function $\\tilde f \\: U_Z H \\to \\GL(V)$. \nThat it extends $f$ follows from the fact that \n$u = zh \\in U_G$ with $z \\in U_Z$ and $h \\in U_H$ implies \n$h \\in H_0$, so that \n$$ \\tilde f(u) \n= \\rho(h)^{-1} f(z)\n= f(zh) = f(u). $$\n\nFor $w \\in \\fq$, formula \\eqref{eq:liederrel}\nleads to \n\\begin{align*}\n(L_w \\tilde f)(zh) \n&= (L_{\\Ad(h)w}(\\tilde f \\circ \\rho_h))(z)\n= L_{\\Ad(h)w}(\\rho(h)^{-1}\\tilde f)(z)\n= L_{\\Ad(h)w}(\\rho(h)^{-1}f)(z)\\\\\n&= - \\rho(h)^{-1}\\beta(\\Ad(h)w)f(z) \n= - \\beta(w)\\rho(h)^{-1}f(z) \n= - \\beta(w)\\tilde f(zh) . \n\\end{align*}\nTherefore $\\tilde f$ satisfies \n\\begin{equation}\\label{eq:comprel2}\nL_w \\tilde f = - \\beta(w) \\tilde f \\quad \\mbox{ for } \\quad w \\in \\fq.\n\\end{equation}\n\n{\\bf Step 4:} For $m \\in M$ we choose an element $g_m \\in G$ with $g_m.m_0 = m$ \nand put $U_m := g q_M(U_Z)$, so that \n$$ G^{U_m} := q_M^{-1}(U_m) = g_m U_Z H. $$ \nOn this open subset of $G$, the function \n$$ F_m \\: G^{U_m} \\to \\GL(V), \\quad \nF_m(g) := \\tilde f(g_m^{-1}g) $$ \nsatisfies \n\\begin{description}\n\\item[\\rm(a)] $F_m(gh) = \\rho(h)^{-1} F_m(g)$ for $g \\in G^{U_m}$, $h \\in H$.\n\\item[\\rm(b)] $L_w F_m = -\\beta(w) F_m$ for $w \\in \\fq$. \n\\item[\\rm(c)] $F_m(g_m) = \\1 = \\id_V$. \n\\end{description}\n\nNext we note that the function \n$F_m$ defines a smooth trivialization \n$$ \\phi_m \\: U_m \\times V \\to \\bV\\res_{U_m} = G^{U^m} \\times_H V, \\quad \n(gH, v) \\mapsto [g, F_m(g)v]. $$\nThe corresponding transition functions are given by \n$$ \\phi_{m,n} \\: U_{m,n} := U_m \\cap U_n \\to \\GL(V), \\quad \ngH \\mapsto \\tilde\\phi_{m,n}(g) := F_m(g)^{-1} F_n(g). $$ \n\nTo verify that these transition functions are holomorphic, \nwe have to show that the functions \n$\\tilde\\phi_{m,n} \\: G^{U_{m,n}} \\to B(V)$ \nare annihilated by the differential operators $L_w$, $w \\in \\fq$. \nThis follows easily from the product rule and (b): \n\\begin{align*}\nL_w(F_m^{-1} F_n) \n&= L_w(F_m^{-1}) F_n + F_m^{-1} L_w(F_n)\n= - F_m^{-1} L_w(F_m) F_m^{-1} F_n + F_m^{-1} L_w(F_n)\\\\\n&= F_m^{-1}(\\beta(w) F_m F_m^{-1}- \\beta(w)) F_n \n= F_m^{-1}(\\beta(w) - \\beta(w)) F_n = 0.\n\\end{align*}\nWe conclude that the transition functions $(\\phi_{m,n})_{m,n \\in M}$ \ndefine a holomorphic vector bundle atlas on $\\bV$. \n\n{\\bf Step 5:} To see \nthat this holomorphic structure is determined uniquely \nby $\\beta$, pick an open connected subset $U \\subeq M$ containing \n$m_0$ on which the bundle is holomorphically trivial and \nthe trivialization is specified by a \nsmooth function $F \\: G^U \\to \\GL(V)$. \nThen $L_w F = -\\beta(w)F$ implies that \n$\\beta(w) = - (L_wF)(\\1) F(\\1)^{-1}$ \nis determined uniquely by the holomorphic structure on $\\bV$. \n\n{\\bf Step 6:} The above construction also shows that, for \nany holomorphic vector bundle structure on $\\bV$, for which \n$G$ acts by holomorphic bundle automorphisms, we may consider \n\\begin{equation}\n \\label{eq:beta}\n\\beta \\: \\fq \\to B(V), \\quad \\beta(w) := - (L_wF)(\\1) F(\\1)^{-1} \n\\end{equation}\nfor a local trivialization given by $F \\: G^U \\to \\GL(V)$, where \n$m_0 \\in U$. Then \n$\\beta$ is a continuous linear map. \nTo see that it does not depend on the choice of $F$, we note that, \nfor any other trivialization $\\tilde F \\: G^U \\to \\GL(V)$, \nthe function $F^{-1}\\cdot \\tilde F$ factors through a holomorphic \nfunction on $U$, so that \n$$ 0 = L_w(F^{-1}\\tilde F)(\\1) \n= F(\\1)^{-1}\\Big(- (L_w F)(\\1)F(\\1)^{-1} + (L_w \\tilde F)(\\1)\\tilde F(\\1)^{-1}\\Big)\\tilde F(\\1). $$\nWe conclude that the right hand side of \\eqref{eq:beta} \ndoes not depend on the choice of $F$. \nApplying this to functions of the form \n$F_g(g') := F(gg')$, defined on $g^{-1}G^U$, we obtain in particular \n\\[ -\\beta(w) = -(L_w f)(\\1)F(\\1)^{-1} = (L_w F_g)(\\1) F_g(\\1)^{-1}\n= (L_w F)(g) F(g)^{-1}, \\]\nso that $L_w F = -\\beta(w)F$. \n\nFor $g= h \\in H$, we obtain with \\eqref{eq:liederrel} \n\\begin{align*}\n-\\beta(w) &= (L_w F_h)(\\1) F(h)^{-1}\n= (L_w F)(h) F(\\1)^{-1}\\rho(h)\\\\\n&= \\rho(h)^{-1}(L_{\\Ad(h)w}F)(\\1) F(\\1)^{-1}\\rho(h)\n= -\\rho(h)^{-1}\\beta(\\Ad(h)w)\\rho(h).\n\\end{align*}\nFor $w \\in \\fh$, the relation $\\beta(w) = \\dd\\rho(w)$ \nfollows immediately from the $H$-equivariance of~$F$. \nTo see that $\\beta$ is an extension of $\\rho$, it now remains \nto verify that it is a homomorphism of Lie algebras. \nFor $w_1, w_2 \\in \\fq$, we have \n\\begin{align*}\n\\beta([w_1, w_2])F \n&= - L_{[w_1, w_2]}F\n= L_{w_2} L_{w_1} F - L_{w_1} L_{w_2} F \\\\\n&= -L_{w_2} \\beta(w_1) F + L_{w_1} \\beta(w_2)F \n= \\beta(w_1) \\beta(w_2)F - \\beta(w_2)\\beta(w_1)F,\n\\end{align*}\nwhich shows that $\\beta$ is a homomorphism of Lie algebras. \n\\end{prf} \n\n\\begin{ex} We consider the special case \nwhere $G$ is contained as a Lie subgroup in \na complex Banach--Lie group $G_\\C$ with Lie algebra $\\g_\\C$ \nand where $Q := \\la \\exp \\fq \\ra$ is a Lie subgroup of $G_\\C$ \nwith $Q \\cap G = H$. Then the orbit mapping $G \\to G_{\\C}\/Q, \ng \\mapsto g Q$, induces an open \nembedding of $M = G\/H$ as an open $G$-orbit in the complex manifold\n$G_{\\C}\/Q$.\n\nIn this case every holomorphic representation \n$\\pi \\: Q \\to \\GL(V)$ defines an associated holomorphic Banach \nbundle $\\bV := G_\\C \\times_Q V$ over the complex manifold \n$G_\\C\/Q \\cong G\/H$. \n\nSince the Lie algebra $\\g_\\C$ need not be integrable in the sense \nthat it is the Lie algebra of a Banach--Lie group \n(cf.~\\cite[Sect.~6]{GN03}), \nthe assumption $G \\subeq G_\\C$ is not general enough to \ncover every situation. One therefore needs the general Theorem~\\ref{thm:a.2}.\n\nIn \\cite[Sect.~6]{GN03} one finds examples \nof simply connected Banach--Lie groups \n$G$ for which $\\g_\\C$ is not integrable. Here \n$G$ is a quotient of a simply connected Lie group \n$\\hat G$ by a central subgroup $Z \\cong \\R$, \n$\\hat G$ has a simply connected universal complexification \n$\\eta_{\\hat G} \\: {\\hat G} \\to {\\hat G}_\\C$, and the subgroup \n$\\exp_{{\\hat G}_\\C}(\\fz_\\C)$ is not closed; its closure is a $2$-dimensional \ntorus. \n\nThen the real Banach--Lie group $G$ acts smoothly and faithfully \nby holomorphic maps on a complex Banach \nmanifold $M$ for which $\\g_\\C$ is not integrable. \nIt suffices to pick a suitable tubular neighborhood \n${\\hat G} \\times U \\cong {\\hat G} \\exp(U) \\subeq {\\hat G}_\\C$, \nwhere \n$U \\subeq i\\hat\\fg$ is a convex $0$-neighborhood. \nThen the quotient \n$M := {\\hat G}\/Z \\times (U + i\\z)\/i\\z$ is a complex manifold on which \n$G \\cong {\\hat G}\/Z$ acts faithfully, but $\\g_\\C$ is not integrable. \n\\end{ex}\n\n\\section{Hilbert spaces of holomorphic sections} \\mlabel{sec:2}\n\nIn this section we take a closer look at \nHilbert spaces of holomorphic sections in $\\Gamma(\\bV)$ for \nholomorphic Hilbert bundles $\\bV$ constructed with the methods from \nSection~\\ref{sec:1}. Here we are interested in Hilbert spaces with \ncontinuous point evaluations (to be defined below) \non which $G$ acts unitarily. This leads to the concept of a \nholomorphically induced representation, which for infinite dimensional \nfibers is a little more subtle than in the finite dimensional case. \nThe first key result of this section is \nTheorem~\\ref{thm:5.5} relating the commutant of a \nholomorphically induced unitary $G$-representation \n$(\\pi, \\cH)$ to the commutant of the representation \n$(\\rho,\\beta)$ of $(H,\\fq)$ on the fiber~$V$.\nThis result is complemented by Theorem~\\ref{thm:a.3} which is a \nrecognition devise for holomorphically induced representations. \n \n\n\\begin{defn} Let $q \\: \\bV \\to M$ be a holomorphic Hilbert bundle \non the complex manifold $M$. We write $\\Gamma(\\bV)$ for the space of \nholomorphic sections of $\\bV$. A Hilbert subspace \n$\\cH \\subeq \\Gamma(\\bV)$ is said to have {\\it \ncontinuous point evaluations} if all the evaluation maps \n\\[ \\ev_m \\: \\cH \\to \\bV_m, \\quad s \\mapsto s(m)\\] \nare continuous and the function $m \\mapsto \\|\\ev_m\\|$ is locally bounded. \n\n\nIf $G$ is a group acting on $\\bV$ by holomorphic \nbundle automorphisms, $G$ acts on $\\Gamma(\\bV)$ by \n\\begin{equation}\n \\label{eq:g-act}\n(g.s)(m) := g.s(g^{-1}m). \n\\end{equation}\nWe call a Hilbert subspace $\\cH \\subeq \\Gamma(\\bV)$ with continuous \npoint evaluations {\\it $G$-invariant} if \n$\\cH$ is invariant under the action defined by \\eqref{eq:g-act} \nand the so obtained representation of $G$ on $\\cH$ is \nunitary. \n\\end{defn}\n\n\\begin{rem} \\mlabel{rem:2.2} \nLet $q \\: \\bV \\to M$ be a holomorphic Hilbert bundle \nover $M$. Then we can represent holomorphic sections of this \nbundle by holomorphic functions on the dual bundle $\\bV^*$ \nwhose fiber $(\\bV^*)_m$ is the dual space \n$\\bV_m^*$ of $\\bV_m$. We thus obtain an embedding \n\\begin{equation}\n \\label{eq:funcdual}\n\\Psi \\: \\Gamma(\\bV) \\to \\cO(\\bV^*), \\quad \n\\Psi(s)(\\alpha_m) = \\alpha_m(s(m)), \n\\end{equation}\nwhose image consists of \nholomorphic functions on $\\bV^*$ which are fiberwise linear. \n\nIf $G$ is a group acting on $\\bV$ by holomorphic \nbundle automorphisms, then $G$ also acts naturally \nby holomorphic maps on $\\bV^*$ via \n$(g.\\alpha_m)(v_{g.m}) := \\alpha_m(g^{-1}.v_{g.m})$ \nfor $\\alpha_m \\in \\bV^*_m$. \nTherefore \n\\[ \\Psi(g.s)(\\alpha_m) \n= \\alpha_m(g.s(g^{-1}.m)) \n= (g^{-1}.\\alpha_m)(s(g^{-1}.m)) \n= \\Psi(s)(g^{-1}.\\alpha_m) \\] \nimplies that $\\Psi$ is equivariant with respect to the natural \n$G$-actions on $\\Gamma(\\bV)$ and $\\cO(\\bV^*)$. \n\\end{rem}\n\n\n\\subsection{Existence of analytic vectors} \n\n\n\\begin{lem}\n\\mlabel{lem:3.1} \nIf $M = G\/H$ is a Banach homogeneous space with a $G$-invariant \ncomplex structure and $\\bV = G\\times_H V$ a $G$-equivariant \nholomorphic vector bundle over $M$ defined by \na pair $(\\rho,\\beta)$ as in {\\rm Theorem~\\ref{thm:a.2}}, \nthen the $G$-action on $\\bV$ is analytic. \n\\end{lem}\n\n\\begin{prf} The manifold \n$\\bV$ is a $G$-equivariant quotient of the product \nmanifold $G \\times V$ on which $G$ acts analytically by left \nmultiplications in the left factor. Since the quotient map \n$q \\: G \\times V \\to \\bV$ is a real analytic submersion, the \naction of $G$ on $\\bV$ is also analytic. \n\\end{prf}\n\n\\begin{defn} Let $M$ be a complex manifold (modeled on a locally \nconvex space) and $\\cO(M)$ the space of holomorphic complex-valued \nfunctions on $M$. We write $\\oline M$ for the conjugate complex manifold. \nA holomorphic function \n$Q \\: M \\times \\oline M \\to \\C$\nis said to be a {\\it reproducing kernel} of a Hilbert subspace \n$\\cH \\subeq \\cO(M)$ if for each $w\\in M$ the function \n$Q_w(z) := Q(z,w)$ is contained in $\\cH$ and satisfies \n$$ \\la f, Q_z \\ra = f(z) \\quad \\mbox{ for } \\quad z \\in M, f \\in \\cH. $$\nThen $\\cH$ is called a {\\it reproducing kernel Hilbert space} \nand since it is determined uniquely by the kernel $Q$, it is \ndenoted $\\cH_Q$ (cf.\\ \\cite[Sect.~I.1]{Ne00}). \n\nNow let $G$ be a group and \n$\\sigma \\: G \\times M \\to M, (g,m) \\mapsto g.m$ \nbe a smooth right action of $G$ on $M$ by holomorphic maps. Then \n $(g.f)(m) := f(g^{-1}.m)$ defines a unitary representation \nof $G$ on a reproducing kernel Hilbert space $\\cH_Q \\subeq \\cO(M)$ \nif and only if the kernel $Q$ is {\\it invariant}: \n$$ Q(g.z, g.w) = Q(z,w) \\quad \\mbox{ for } \\quad z,w \\in M, g \\in G $$ \n(\\cite[Rem.~II.4.5]{Ne00}). In this case we call \n${\\cal H}_Q$ a $G$-invariant reproducing kernel Hilbert space. \n\\end{defn} \n\n\n\\begin{lem} \\mlabel{lem:2.2} \nLet $G$ be a Banach--Lie group \nacting analytically via \n\\[\\sigma \\: G \\times M \\to M, \\quad \n(g,m) \\mapsto \\sigma_g(m) \\] \nby holomorphic maps on the complex manifold~$M$. \n\n{\\rm(a)} Let $\\cH \\subeq \\cO(M)$ be a reproducing kernel Hilbert \nspace whose kernel $Q$ is a $G$-invariant holomorphic function on \n$M \\times \\oline M$. \nThen the elements $(Q_m)_{m \\in M}$ representing the evaluation \nfunctionals in $\\cH$ are analytic \nvectors for the representation of $G$, defined by \n$\\pi(g)f := f \\circ \\sigma_g^{-1}$. \n\n{\\rm(b)} Let $\\bV \\to M$ be a holomorphic $G$-homogeneous \nHilbert bundle and $\\cH \\subeq \\Gamma(\\bV)$ be a $G$-invariant \nHilbert space with continuous point evaluations. \nThen every vector of the form \n$\\ev_m^*v$, $m \\in M$, $v \\in \\bV_m$, is analytic for the \n$G$-action on $\\cH$. \n\\end{lem}\n\n\\begin{prf} (a) Since $Q$ is holomorphic on \n$M \\times \\oline M$, it is in particular real analytic. \nFor $m \\in M$, we have \n\\[ \\la \\pi(g)Q_m, Q_m \\ra \n= (\\pi(g)Q_m)(m) = Q_m(g^{-1}m) = Q(g^{-1}m,m), \\]\nand since $Q$ and the $G$-action are real analytic, \nthis function is analytic on $M$. Now \\cite[Thm.~5.2]{Ne10b} \nimplies that $Q_m$ is an analytic vector. \n\n(b) As in Remark~\\ref{rem:2.2}, we \nrealize $\\Gamma(\\bV)$ by holomorphic functions on the dual \nbundle $\\bV^*$. We thus obtain a \nreproducing kernel Hilbert space \n$\\cH_Q := \\Psi(\\cH) \\subeq \\cO(\\bV^*)$, \nand since $\\Psi$ is $G$-equivariant, the reproducing \nkernel $Q$ is $G$-invariant. \nFor $v \\in \\bV_m$, evaluation in the corresponding element \n$\\alpha_m := \\la \\cdot, v \\ra \\in \\bV^*_m$ is given by \n\\[ s \\mapsto \\la s(m), v \\ra = \\la \\ev_m(s), v\\ra \n= \\la s, \\ev_m^*v\\ra.\\] \nFor the corresponding $G$-invariant kernel $Q$ on $\\bV^*$ \nthis means that $Q_{\\alpha_m} = \\ev_m^*v$, so that the assertion \nfollows from (a). \n\\end{prf}\n\n\n\\subsection{The endomorphism bundle and commutants} \n\nThe goal of this section is Theorem~\\ref{thm:5.5} which connects \nthe commutant of the $G$-representation on a Hilbert space $\\cH_V$ \nof a holomorphically induced representation with the \ncorresponding representation $(\\rho, \\beta)$ \nof $(H,\\fq)$ on $V$. The remarkable \npoint is that, under natural assumptions, these commutants are isomorphic, \nso that both representations have the same decomposition theory. \nThis generalizes an important \nresult of S.~Kobayashi concerning irreducibility \ncriteria for the $G$-representation on $\\cH_V$ (\\cite{Ko68}, \n\\cite[Thm.~2.5]{BH98}). \n\nLet $\\bV = G \\times_H V \\to M$ be a $G$-homogeneous holomorphic \nHilbert bundle as in Theorem~\\ref{thm:a.2}. Then \nthe complex manifold $M \\times \\oline M$ is a complex homogeneous \nspace $(G \\times G)\/(H \\times H)$, \nwhere the complex structure is defined by the closed subalgebra \n$\\fq \\oplus \\oline\\fq$ of $\\g_\\C \\oplus \\g_\\C$. \nOn the Banach space $B(V)$ we consider the norm continuous \nrepresentation of $H \\times H$ by \n\\[ \\tilde\\rho(h_1, h_2)A = \\rho(h_1)A\\rho(h_2)^* \\] \nand the the corresponding extension \n$\\tilde\\beta \\: \\fq \\oplus \\oline \\fq \\to \\gl(B(V))$ by \n\\[ \\tilde\\beta(x_1,x_2)A := \\beta(x_1)A + A \\beta(\\oline{x_2})^*.\\]\nWe write ${\\mathbb L} := (G \\times G) \\times_{H \\times H} B(V)$ for the \ncorresponding holomorphic Banach bundle over \n$M \\times \\oline M$ (Theorem~\\ref{thm:a.2}). \n\n\\begin{rem} For every $g \\in G$, we have isomorphisms \n$V \\to \\bV_{gH} = [g,V], v \\mapsto [g,v]$. \nAccordingly, we have for every pair $(g_1, g_2) \\in G \\times G$ an \nisomorphism \n\\[ \\nu \\: B(V) \\to B(\\bV_{g_2 H}, \\bV_{g_1H}), \\quad \n\\nu(A)[g_2,v] \\mapsto [g_1,Av]. \\] \nThis defines a map \n\\[ \\gamma \\: G \\times G \\times B(V) \\to \nB(\\bV) := \\bigcup_{m,n \\in M} B(\\bV_m,\\bV_n), \\quad \n\\gamma(g_1, g_2,A)[g_2,v] = [g_1, Av]. \\] \nFor $h_1, h_2 \\in H$, we then have \n\\begin{align*}\n&\\gamma(g_1h_1, g_2h_2,\\tilde\\rho(h_1, h_2)^{-1}A)[g_2,v] \n= \\gamma(g_1h_1, g_2h_2,\\tilde\\rho(h_1, h_2)^{-1}A)[g_2h_2,\\rho(h_2)^{-1}v] \\\\\n&= [g_1 h_1, \\rho(h_1)^{-1} A v] \n= [g_1, A v] = \\gamma(g_1, g_2, A)[g_2,v], \n\\end{align*}\nso that $\\gamma$ factors through a bijection \n$\\oline\\gamma \\: {\\mathbb L} \\to B(\\bV).$ \nThis provides an interpretation of the bundle ${\\mathbb L}$ as the \n{\\it endomorphism bundle of $\\bV$} (cf.\\ \\cite{BR07}). \n\\end{rem}\n\nSince the group $G$ acts (diagonally) on the bundle \n${\\mathbb L}$, it makes sense to consider $G$-invariant holomorphic \nsections. \n\n\\begin{lem} \\mlabel{lem:2.7} \nThe space $\\Gamma({\\mathbb L})^G$ of $G$-invariant holomorphic \nsections of ${\\mathbb L}$ has the property that the evaluation map \n\\[ \\ev \\: \\Gamma({\\mathbb L})^G \\cong C^\\infty(G \\times G, B(V))_{\\tilde \\rho, \n\\tilde\\beta} \\to B_H(V), \\quad s \\mapsto \\hat s(\\1,\\1) \\] \nis injective. \n\\end{lem}\n\n\\begin{prf} If $\\hat s(\\1,\\1) = 0$, then \nthe corresponding holomorphic section $s \\in \\Gamma({\\mathbb L})$ vanishes on the \ntotally real submanifold \n\\[ \\Delta_M := \\{ (m,m) \\: m \\in M \\} = G(m_0, m_0)\\subeq \nM \\times \\oline M,\\] \nand hence on all of $M \\times \\oline M$. \nFor the function $\\hat s \\: G \\times G \\to B(V)$, we \nhave for $h \\in H$ \n\\[ \\hat s(\\1,\\1) = (h^{-1}.\\hat s)(\\1,\\1) \n= \\hat s(h, h) = \\rho(h)^{-1} \\hat s(\\1,\\1) \\rho(h), \\] \nshowing that $\\hat s(\\1,\\1)$ lies in $B_H(V)$. \n\\end{prf}\n\n\\begin{rem} In general, the holomorphic bundle \n${\\mathbb L}$ does not have any non-zero holomorphic section, although \nits restriction to the diagonal $\\Delta_M$ \nhas the trivial section $R$ given by $R_{(m,m)} = \\id_{\\bV_m}$ \nfor every $m\\in M$. \n\nIf $R \\: M \\times \\oline M \\to {\\mathbb L}$ is a holomorphic section, \nthen we obtain for every $v \\in V$ and $n \\in M$ a holomorphic section \n$R_{n,v} \\: M \\to \\bV, m \\mapsto R(m,n)v$ \nwhich is non-zero in $m$ if $R_{m,m}v \\not=0$. \nTherefore $\\bV$ has nonzero holomorphic sections if ${\\mathbb L}$ does, and this \nis not always the case. \n\\end{rem}\n\n\\begin{ex} \\mlabel{ex:2.11} \n(a) Let $(\\pi, \\cH)$ be a unitary representation of \n$G$ and $\\Psi \\: \\cH \\to \\Gamma(\\bV)$ be $G$-equivariant \nsuch that the evaluation maps $\\ev_m \\circ \\Psi \\: \\cH \\to \\bV_m$ \nare continuous and $m \\mapsto \\|\\ev_m\\|$ is locally bounded. \nThen \n\\[ R(m,n) := (\\ev_m \\Psi) (\\ev_n \\Psi)^* \\in B(\\bV_n, \\bV_m)\\] \ndefines a holomorphic section of $B(\\bV)$. \nSince this assertion is local, it follows from \nthe corresponding assertion for \ntrivial bundles treated in \\cite[Lemma~A.III.9(iii)]{Ne00}. \nThe corresponding smooth function \n\\[ \\hat R \\: G \\times G \\to B(V), \\quad \n\\hat R(g_1, g_2) = (\\ev_{g_1} \\Psi) (\\ev_{g_2} \\Psi)^*\\] \nis a $G$-invariant $B(V)$-valued \nkernel on $G \\times G$ because \n\\[ \\ev_{gh} \\Psi = \\ev_h \\Psi \\circ \\pi(g)^{-1} \n\\quad \\mbox{ for } \\quad g,h \\in G. \\] \nWe conclude that $\\hat R \\in \\Gamma({\\mathbb L})^G$, so that it is completely \ndetermined by \n\\[ \\hat R(\\1,\\1) = (\\ev_\\1 \\Psi) (\\ev_\\1 \\Psi)^*\\] \n(Lemma~\\ref{lem:2.7}). \nIn particular, $\\Psi$ is completely determined by $\\ev_\\1 \\circ \\Psi$. \nThis follows also from its equivariance, which leads to \n$\\Psi(v)(g) = \\ev_\\1(g^{-1}.\\Psi(v)) \n= \\ev_\\1\\Psi(\\pi(g)^{-1}v).$ \n\n(b) For $A, B \\in B(V)$ commuting \nwith $\\rho(H)$ and $\\beta(\\fq)$ and $\\hat R \\in C^\\infty(G \\times G, \nB(V))_{\\tilde\\rho, \\tilde\\beta}$, the function \n\\[ \\hat R^{A,B} \\: G \\times G \\to B(V), \\quad \n\\hat R^{A,B}(g_1, g_2) = A R(g_1, g_2) B\\] \nis also contained in $C^\\infty(G \\times G, B(V))_{\\tilde\\rho, \\tilde\\beta}$, \nhence defines a holomorphic section of the bundle~${\\mathbb L}$. \n\\end{ex} \n\n\\begin{defn} \\mlabel{def:2.10} \n(a) For $(\\rho, \\beta)$ as in Theorem~\\ref{thm:a.2} \nand the corresponding associated bundle $\\bV = G \\times_H V$, \na unitary representation $(\\pi, \\cH)$ of $G$ is said to be \n{\\it holomorphically induced from $(\\rho,\\beta)$} \nif there exists a realization \n$\\Psi \\: \\cH \\to \\Gamma(\\bV)$ as an invariant Hilbert \nspace with continuous point evaluations whose kernel \n$R \\in \\Gamma({\\mathbb L})^G$ satisfies $\\hat R(\\1,\\1) = \\id_V$. \n\n(b) Since this condition determines $R$ by Lemma~\\ref{lem:2.7}, \nit also determines the reproducing kernel space $\\Psi(\\cH)$ \nand its norm. \nWe conclude that, for every pair $(\\rho,\\beta)$, \nthere is at most one holomorphically induced unitary representation \nof $G$ up to unitary equivalence. \nAccordingly, we call $(\\rho, \\beta, V)$ {\\it inducible} \nif there exists a corresponding holomorphically induced unitary \nrepresentation of~$G$. \n\nIf this is the case, then we use the isometric embedding \n$\\ev_\\1^* \\: V \\to \\cH$ to identify $V$ with a subspace of \n$\\cH$ and note that \nthe evaluation map $\\ev_\\1 \\: \\cH \\to V$ corresponds to the\northogonal projection $p_V \\: \\cH \\to V$. \n\\end{defn}\n\n\\begin{rem} \\mlabel{rem:2.13} (a) If $\\cH_V \\into \\Gamma(\\bV)$ \nis a $G$-invariant Hilbert \nspace on which the representation is holomorphically induced, \nthen we obtain a $G$-invariant element $Q \\in B(\\bV)$ by \n\\[ Q(m,n) := \\ev_m \\ev_n^* \\in B(\\bV_n, \\bV_m),\\] \nand the relation $\\ev_g = \\ev_\\1 \\circ \\pi(g)^{-1} \n= p_V \\circ \\pi(g)^{-1}$ yields \n\\[ \\hat Q(g_1, g_2) = \\ev_{g_1} \\ev_{g_2}^* \n= p_V \\pi(g_1)^{-1} \\pi(g_2)p_V = p_V \\pi(g_1^{-1}g_2)p_V \\] \nand in particular $\\hat Q(\\1,\\1) = \\id_V.$\n\n(b) Suppose that $\\cH \\subeq \\Gamma(\\bV)$ is holomorphically \ninduced. Since $\\ev_\\1$ is $H$-equivariant, the closed subspace \n$V \\subeq \\cH$ is $H$-invariant and the $H$-representation \non this space is equivalent to $(\\rho,V)$, hence in particular \nbounded. \n\nMoreover, $V \\subeq \\cH^\\omega$ follows from \nLemma~\\ref{lem:2.2}(b). Since the evaluation maps \\break $\\ev_g \\: \\cH \\to V$ \nseparate the points, the analyticity of the elements \n$\\ev_g^*v$ even implies that $\\cH^\\omega$ is dense in $\\cH$. \n\nFor $x \\in \\fq$ and $s \\in \\cH^\\infty$ we have \n\\[ \\ev_\\1\\dd\\pi(x) s \n= (\\dd\\pi(x)s)\\,\\hat{}(\\1) \n= - L_x \\hat s(\\1) \n= \\beta(x) \\hat s(\\1) \n= \\beta(x) \\ev_\\1 s.\\]\nTherefore $\\dd\\pi(\\fq)$ preserves the subspace \n$\\cH^\\infty \\cap V^\\bot = \\cH^\\infty \\cap \\ker(\\ev_\\1)$. \nFrom $V \\subeq \\cH^\\infty$, we derive \n\\[ \\cH^\\infty = V \\oplus (V^\\bot \\cap \\cH^\\infty), \\] \nso that the density of $\\cH^\\infty$ in $\\cH$ implies that \n$V = (V^\\bot \\cap \\cH^\\infty)^\\bot$, and hence that \nthis space is invariant under the restriction \n$\\dd\\pi(\\oline x)$ of the adjoint $- \\dd\\pi(x)^*$. \nFor $s_j = \\ev_\\1^*v_j \\in V$, $j =1,2$, we further obtain \n\\begin{align*}\n\\la \\dd\\pi(\\oline x)s_1, s_2 \\ra \n&= -\\la \\ev_\\1^* v_1, \\dd\\pi(x) s_2 \\ra \n= -\\la v_1, \\ev_\\1\\dd\\pi(x) s_2 \\ra \n= -\\la v_1, \\beta(x) \\ev_\\1 s_2 \\ra \\\\\n&= -\\la v_1, \\beta(x) v_2 \\ra \n= -\\la \\beta(x)^*v_1,v_2 \\ra,\n\\end{align*}\nso that \n\\begin{equation}\\label{eq:a3} \n\\dd\\pi(\\oline x)\\res_V = - \\beta(x)^*, \\quad x \\in \\fq.\n\\end{equation}\n\nFinally we observe that \n\\[ (\\pi(G)V)^\\bot = \\{ s \\in \\cH \\: (\\forall g \\in G)\\, \\hat s(g) = 0\\}\n= \\{0\\}\\] \n implies that $\\cH = \\oline{\\Spann(\\pi(G)V)}$. \n\\end{rem}\n\nIf $V$ is of the form $\\oline{(\\cH^\\infty)^\\fn}$ for a subalgebra \n$\\fn \\subeq \\g_\\C$, then it is invariant under the commutant $B_G(V)$, \nbut we do not know if this is always true for holomorphically \ninduced representation. \nTo make the following proposition as flexible as possible, we \nassume this naturality condition of~$V$ (cf.\\ Remark~\\ref{rem:2.14} below).\n\n\\begin{thm} \\mlabel{thm:5.5} \nSuppose that $(\\pi, \\cH_V)$ is holomorphically induced from \nthe representation $(\\rho,\\beta)$ of $(H,\\fq)$ on~$V$ \nand that \n\\[ B_{H,\\fq}(V) := \n\\{ A \\in B_H(V) \\: (\\forall x \\in \\fq)\\, \nA \\beta(x) = \\beta(x) A, A^* \\beta(x) = \\beta(x) A^* \\} \\] \nis the involutive commutant of $\\rho(H)$ and $\\beta(\\fq)$. \nIf $V$ is invariant under $B_G(V)$, then the map \n\\[ R \\: B_G(\\cH_V) \\to B_{H,\\fq}(V), \\quad \nA \\mapsto A\\res_V \\]\nis an isomorphism of von Neumann algebras. \n\\end{thm}\n\n\\begin{prf} By assumption, every $A \\in B_G(\\cH_V)$ preserves \n$V$, so that \n$A\\res_V$ can be identified with the operator $p_V A p_V \\in B(V)$, \nwhere $p_V \\: \\cH \\to V$ is the orthogonal projection. \nClearly $A\\res_V$ commutes with each $\\rho(h) = \\pi(h)\\res_V$. \nIt also preserves the subspace $\\cH^\\infty$ on which it satisfies \n$\\pi(x)A = A \\pi(x)$ for every $x \\in \\g_\\C$. Therefore \n$A\\res_V$ commutes with $\\dd\\pi(\\oline\\fq)$ and hence with \n$\\beta(\\fq)$ (cf.~\\eqref{eq:a3} in Remark~\\ref{rem:2.13}). \n\nTherefore $R$ defines a homomorphism of von Neumann algebras. \nIf $R(A) = 0$, then $A V = 0$ implies that \n$A \\pi(G)V = \\{0\\}$, which leads to $A = 0$. \nHence $R$ is injective. \n\nSince each von Neumann algebra is generated by orthogonal projections \n(\\cite[Chap.~1, \\S 1.2]{Dix96}) \nand images of von Neumann algebras under restriction maps \nare von Neumann algebras \n(\\cite[Chap.~1, \\S 2.1, Prop.~1]{Dix96}), \nwe are done if we can show that every orthogonal \nprojection in $B_{H,\\fq}(V)$ is contained in the image of \n$R$. So let $P \\in B_{H,\\fq}(V)$. \nThen $V_1 := P(V)$ and $V_2 := (\\1-P)(V)$ yields an \n$(H,\\fq)$-invariant orthogonal decomposition $V = V_1 \\oplus V_2$. \n\nLet $\\hat Q \\: G \\times G \\to B(V), (g_1, g_2) \\mapsto \np_V \\pi(g_1^{-1}g_2) p_V$ be the natural kernel function \ndefining the inclusion $\\cH_V \\into \\Gamma(\\bV)$ and consider the \n$G$-invariant kernel \n\\[ \\hat R(g_1, g_2) := P \\hat Q(g_1, g_2) (\\1 - P). \\]\nAccording to Example~\\ref{ex:2.11}(b), \nit is contained in $C^\\infty(G \\times G, B(V))_{\\tilde\\rho,\\tilde\\beta}$, \nhence defines an element $R \\in \\Gamma({\\mathbb L})^G$. \nIn view of \n\\[ \\hat R(\\1,\\1) = P Q(\\1,\\1) (\\1 - P) = P(\\1 - P) = 0, \\]\nthis section vanishes in the base point, hence on all of \n$M \\times \\oline M$ (Lemma~\\ref{lem:2.7}). We conclude that \n\\[ 0 = \\hat R(g_1, g_2) \n= P p_V \\pi(g_1^{-1}g_2) p_V (\\1-P) \n= P \\pi(g_1^{-1}g_2) (\\1 - P), \\] \nso that, for every $g \\in G$, we have \n$P \\pi(g)(\\1-P) = 0.$ \nThis leads to $P \\cH_{V_2} = \\{0\\}$, and hence to \n$\\cH_{V_1} \\bot \\cH_{V_2}$. We derive that \n$\\cH_V = \\cH_{V_1} \\oplus \\cH_{V_2}$ is an orthogonal \ndirect sum. Therefore the \northogonal projection $\\tilde P \\in B_G(\\cH_V)$ onto $\\cH_{V_1}$ \nleaves $V$ invariant and satisfies \n$\\tilde P\\res_V = P$. This proves that $R$ is surjective. \n\\end{prf}\n\n\\begin{rem} The preceding proof shows that, under the assumptions of \nTheorem~\\ref{thm:5.5}, the range of the injective map \n\\[ \\ev \\: \\Gamma({\\mathbb L})^G \\to B_H(V), \\quad R \\mapsto \\hat R(\\1,\\1)\\] \ncontains $B_{H,\\fq}(V)$. If $\\beta(\\fq) = \\beta(\\fh_\\C)$, then \n$B_{H,\\fq}(V) = B_H(V)$, so that we obtain a linear isomorphism \n$\\Gamma({\\mathbb L})^G\\cong B_H(V)$. \n\\end{rem}\n\nThe preceding theorem has quite remarkable \nconsequences because it implies that the representations \n$(\\pi, \\cH_V)$ and $(\\rho,V)$ decompose in the same way. \n\n\\begin{cor} \\mlabel{cor:commutant} \nSuppose that $(\\pi, \\cH_V)$ is holomorphically induced from \n$(\\rho,\\beta)$, that $V$ is $B_G(V)$-invariant, \nand that $\\beta(\\fq) = \\beta(\\fh_\\C)$, so that \n$B_H(V) = B_{H,\\fq}(V)$. \nThen the $G$-representation $(\\pi, \\cH_V)$ \nhas any of the following properties if and only if \nthe $H$-representation $(\\rho,V)$ does. \n\\begin{description}\n\\item[\\rm(i)] Irreducibility. \n\\item[\\rm(ii)] Multiplicity freeness. \n\\item[\\rm(iii)] Type $I$, $II$ or $III$. \n\\item[\\rm(iv)] Discreteness, i.e., being a direct \nsum of irreducible representations. \n\\end{description}\n\\end{cor}\n\n\\begin{prf} (i) follows from the fact that, according to Schur's Lemma, \nirreducibility means that the commutant equals $\\C \\1$. \n \n(ii) is clear because multiplicity freeness means that the commutant \nis commutative. \n\n(iii) is clear because the type of a representation is defined as \nthe type of its commutant as a von Neumann algebra. \n\n(iv) That a unitary representation decomposes discretely \nmeans that its commutant is an $\\ell^\\infty$-direct sum of \nfactors of type $I$. Hence the $G$ representation on $\\cH_V$ has\nthis property if only if the $H$-representation on $V$~does.\n\\end{prf}\n\nCorollary~\\ref{cor:commutant}(i) is a version of \nS.~Kobayashi's Theorem in the Banach context \n(cf.\\ \\cite{Ko68}). \n\n\\begin{prob} Theorem~\\ref{thm:5.5} should \nalso be useful to derive direct integral decompositions \nof the $G$-representation on $\\cH_V$ from direct integral \ndecompositions of the $H$-representation on $V$. \n\nSuppose that the bounded unitary representation \n$(\\rho, V)$ of $H$ is of type I and holomorphically inducible. \nThen it is a direct integral of irreducible representations \n$(\\rho_j, V_j)$. Are these irreducible representations of \n$H$ also inducible? \n\\end{prob}\n\n\n\\begin{cor} \\mlabel{cor:2.15} Suppose that the $G$-representations \n$(\\pi_1, \\cH_{V_1})$, resp., $(\\pi_2, \\cH_{V_2})$ \nare holomorphically induced from the $(H,\\fq)$-representations \n$(\\rho_1, \\beta_1, V_1)$, resp., $(\\rho_2, \\beta_2, V_2)$. \nThen any unitary isomorphism \n$\\gamma \\: V_1 \\to V_2$ of $(H,\\fq)$-modules extends uniquely to a \nunitary equivalence $\\tilde\\gamma \\: \\cH_{V_1} \\to \\cH_{V_2}$. \n\\end{cor}\n\n\\begin{prf} Since $\\gamma$ is $(H,\\fq)$-equivariant, we have a \nwell-defined $G$-equivariant bijection \n\\[ \\tilde \\gamma \\: \\Gamma(\\bV_1) \\cong \nC^\\infty(G,V_1)_{\\rho_1, \\beta_1} \\to \nC^\\infty(G,V_2)_{\\rho_2, \\beta_2}, \\quad \nf \\mapsto \\gamma \\circ f \\] \nobtained from a corresponding isomorphism \n$[(g,v)] \\mapsto [(g,\\gamma(v))]$ of holomorphic $G$-bundles. \nTherefore $\\tilde\\gamma(\\cH_{V_1})$ is an invariant Hilbert space \nwith continuous point evaluations. The corresponding \n$G$-invariant kernel $\\hat Q \\in C^\\infty(G \\times G, B(V_2))$ \nsatisfies \n\\[ \\hat Q(\\1,\\1) = \\gamma \\circ \\id_{V_1} \\circ \\gamma^* \n= \\gamma \\gamma^* = \\id_{V_2}.\\] \nThis implies that $\\tilde\\gamma(\\cH_{V_1}) \n= \\cH_{V_2}$ and that the map \n$\\tilde \\gamma \\: \\cH_{V_1} \\to \\cH_{V_2}$ is unitary \nbecause both spaces have the same reproducing kernels \n(cf.\\ Definition~\\ref{def:2.10}). \n\nFor $v \\in V_1$, we further note that \n$(\\tilde\\gamma \\ev_\\1^*v)(\\1) \n= \\gamma \\ev_\\1 \\ev_\\1^*v = \\gamma v$, so that \n$\\tilde\\gamma$ extends $\\gamma$, when considered as a map \non the subspace $V_1 \\cong \\ev_\\1^*V_1 \\subeq \\cH_1$. \n\\end{prf}\n\n\n\\subsection{Realizing unitary representations by holomorphic sections} \n\nWe conclude this section with a result that helps to realize \ncertain subrepresentations of unitary representations in \nspaces of holomorphic sections. We continue in the setting \nof Section~\\ref{sec:1}, where $M = G\/H$ is a Banach homogeneous \nspace with a complex structure defined by the subalgebra \n$\\fq \\subeq \\g_\\C$. \n\nFrom the discussion in Remark~\\ref{rem:2.13}, we know that \nevery holomorphically induced representation \nsatisfies the assumptions (A1\/2) in the theorem below, \nwhich is our main tool to prove that a given unitary representation \nis holomorphically induced. \n\n\\begin{thm}\\mlabel{thm:a.3} \nLet $(\\pi, \\cH)$ be a continuous unitary representation of $G$ \nand $V \\subeq \\cH$ be a closed subspace satisfying the \nfollowing conditions: \n\\begin{description}\n\\item[\\rm(A1)] $V$ is $H$-invariant and the representation \n$\\rho$ of $H$ on $V$ is bounded. In particular, \n$\\dd\\pi \\res_{\\fh}\\: \\fh \\to \\gl(V)$ \ndefines a continuous homomorphism \nof Banach--Lie algebras. \n\\item[\\rm(A2)] The exists a subspace $\\cD_V \\subeq V \\cap \\cH^\\infty$ dense \nin $V$ which is invariant under $\\dd\\pi(\\oline\\fq)$, \nthe operators $\\dd\\pi(\\oline\\fq)\\res_{\\cD_V}$ are bounded, \n and the so obtained representation of \n$\\oline\\fq$ on $V$ defines a continuous morphism \nof Banach--Lie algebras \n$$\\beta \\: \\fq \\to \\gl(V), \\quad x \\mapsto \n-(\\dd\\pi(\\oline x)\\res_V)^*.$$ \n\\end{description}\nThen the following assertions hold: \n\\begin{description}\n\\item[\\rm(i)] $\\beta$ is an extension of $\\rho$ defining on \n$\\bV := G \\times_H V$ the structure of a complex \nHilbert bundle. \n\\item[\\rm(ii)] If $p_V \\: \\cH \\to \\cH$ denotes \nthe orthogonal projection to~$V$, then \n$$ \\Phi \\: \\cH \\to C(G,V)_\\rho, \\quad \\Phi(v)(g) := p_V(\\pi(g)^{-1}v) $$ \nmaps $\\cH$ into $C^\\infty(G,V)_{\\rho,\\beta} \\cong \\Gamma(\\bV)$, \nand we thus obtain a $G$-equivariant unitary isomorphism \nof the closed subspace \n$\\cH_V := \\oline{\\Spann \\pi(G)V}$ \nwith the representation holomorphically induced from $(\\rho, \\beta)$. \n\\item[\\rm(iii)] $V \\subeq \\cH^\\omega$. \n\\end{description}\n\\end{thm}\n\n\\begin{prf} (i) For $x \\in \\fh$, we have \n$\\beta(x) = -\\dd\\rho(x)^*\\ = \\dd\\rho(x),$ \nand it is also easy to see that \n$\\beta(\\Ad(h)x) = \\pi(h) \\beta(x)\\pi(h)^{-1}$ for \n$h \\in H$ and $x \\in \\fq$. Therefore $\\beta$ is an extension \nof $\\rho$, and we can use \nTheorem~\\ref{thm:a.2} to see that \n$\\beta$ defines the structure of a holomorphic Hilbert \nbundle on $\\bV$. \n\n(ii) Clearly, $\\Phi(\\cH^\\infty) \\subeq C^\\infty(G,V)_\\rho$. \nFor $v \\in \\cD_V$, $w \\in \\cH^\\infty$ and $x \\in \\fq$, we further \nderive from (A2) that \n\\begin{align*}\n\\la p_V(\\dd\\pi(x)w), v \\ra \n&= \\la \\dd\\pi(x)w, v \\ra = \\la w, \\dd\\pi(-\\oline x)v \\ra \n= \\la p_V(w), \\dd\\pi(-\\oline x)v \\ra \n= \\la \\beta(x)p_V(w),v \\ra, \n\\end{align*}\nso that the density of $\\cD_V$ in $V$ implies \n$$ p_V \\circ \\dd\\pi(x) = \\beta(x) \\circ p_V\\res_{\\cH^\\infty} \n\\quad \\mbox{ for } \n\\quad x \\in \\fq. $$\nFor $v \\in \\cH^\\infty$ and $x \\in \\fq$, we now obtain \n\\begin{equation}\n \\label{eq:lx-rel}\n\\big(L_x \\Phi(v)\\big)(g) \n= -p_V(\\dd\\pi(x)\\pi(g)^{-1}v) \n= -\\beta(x)p_V(\\pi(g)^{-1}v)\n= -\\beta(x)\\Phi(v)(g). \n\\end{equation}\nThis means that \n$\\Phi(\\cH^\\infty) \\subeq C^\\infty(G,V)_{\\rho,\\beta}$. \nWriting $\\Gamma_c(\\bV)$ for the space of continuous sections of \n$\\bV$, we also obtain a map \n$\\tilde\\Phi \\: \\cH \\to \\Gamma_c(\\bV)$ which is continuous if \n$\\Gamma_c(\\bV)$ is endowed with the compact open topology. \nAs $\\Gamma(\\bV)$ is closed in $\\Gamma_c(\\bV)$ with respect to this topology \n(\\cite[Cor.~III.12]{Ne01}), \n$\\tilde\\Phi(\\oline{\\cH^\\infty}) \\subeq \\Gamma(\\bV)$, \nresp., $\\Phi(\\oline{\\cH^\\infty}) \\subeq C^\\infty(G,V)_{\\rho,\\beta}$. \n\nClearly $\\Phi(v) = 0$ is equivalent to \n$v \\bot \\pi(G)V$, so that $(\\ker \\Phi)^\\bot = \\cH_V$. \nSince (A2) implies that $V \\subeq \\oline{\\cH^\\infty}$, the same \nholds for $\\Spann(\\pi(G)V)$. This shows that \n\\[ \\Phi(\\cH) = \\Phi(\\cH_V) \n= \\Phi(\\oline{\\cH^\\infty}) \\subeq C^\\infty(G,V)_{\\rho,\\beta}.\\] \nThe corresponding kernel $\\hat Q \\: G \\times G \\to B(V)$ is given by \n\\[ \\hat Q(g_1, g_2) = \\ev_{g_1} \\ev_{g_2}^* \n= p_V \\pi(g_1)^{-1} (p_V \\circ \\pi(g_2)^{-1})^* \n= p_V \\pi(g_1^{-1}g_2) p_V,\\] \nso that we have in particular $\\hat Q(\\1,\\1) = \\id_V$. \n\n(iii) To see that $V$ consists of analytic vectors, \nwe may w.l.o.g.\\ assume that $\\cH = \\cH_V$ and hence that \n$\\cH \\subeq \\Gamma(\\bV)$ is holomorphically induced and that \n$\\Phi = \\id_\\cH$. \nLet $\\ev_{\\1 H} \\: \\cH \\to \\bV_{\\1 H} \\cong V$ be the evaluation map. \nThe corresponding map \n\\[ \\ev_\\1 \\: C^\\infty(G,V)_{\\rho,\\beta} \\to V \\] \nis simply given by evaluation in $\\1 \\in G$. \nNow $\\ev_\\1\\res_V \\: V \\to V$ is the identity, so that \n$\\ev_\\1^* \\: V\\to \\cH$ is simply the isometric inclusion. Hence \nthe analyticity of $v = \\ev_\\1^*v = \\ev_{\\1 H}^*v$ \nfollows from Lemma~\\ref{lem:2.2}. \n\\end{prf} \n\n\\begin{rem} \\mlabel{rem:2.14} \nSuppose that there exist subalgebras $\\fp^\\pm \\subeq \\g_\\C$ with \n$\\fq = \\fh_\\C \\rtimes \\fp^+$ and $\\Ad(H)\\fp^\\pm = \\fp^\\pm$. \nThen, for every unitary representation \n$(\\pi, \\cH)$ of $G$, the closed subspace \n$V := \\oline{(\\cH^\\infty)^{\\fp^-}}$ \nis invariant under $H$ and $B_G(\\cH)$. \nIf the representation $(\\rho, H)$ on $V$ is bounded, \nthen the dense subspace $\\cD_V := (\\cH^\\infty)^{\\fp^-}$ \nsatisfies (A2) if we put $\\beta(\\fp^+) = \\{0\\}$. \nTheorem~\\ref{thm:a.3} now implies that \n$V \\subeq \\cH^\\omega$, so that we see in particular that \n$V = (\\cH^\\infty)^{\\fp^-}$ \nis a closed subspace of $\\cH$. \nMoreover, Corollary~\\ref{cor:commutant} applies. \n\\end{rem}\n\n\nThe following remark sheds some extra light on condition (A2). \n\n\\begin{rem}\nLet $\\cH \\subeq \\Gamma(\\bV)$ be a $G$-invariant Hilbert subspace \nwith continuous point evaluations, $m \\in M$ and \n$V := \\oline{\\im(\\ev_{m}^*)} \\subeq \\cH$. \nThen $V$ is a $G_m$-invariant closed subspace of $\\cH$ and \n\\[ V^\\bot = \\im(\\ev_m^*)^\\bot = \\ker(\\ev_m) \n= \\{ s \\in \\cH \\: s(m) =0\\}. \\] \n\nThe action of $G$ on $\\bV$ by holomorphic bundle automorphisms \nleads to a homomorphism $\\dot\\sigma_{\\bV} \\: \\g_\\C \\to \\cV(\\bV)$ \nand each $x \\in \\fq_m$ thus leads to a linear vector field \n$-\\beta_m(x)$ on the fiber $\\bV_m$. Passing to derivatives in the formula \n$(g.s)(m) := g.s(g^{-1}m)$, we obtain for $x \\in \\fq_m$ \n\\[ (x.s)(m) = -\\beta_m(x)\\cdot s(m).\\] \nIn particular, $V^\\bot \\cap \\cH^\\infty$ is invariant under \nthe derived action of $\\fq_m$, so that one can expect that the adjoint \noperators coming from $\\oline{\\fq_m}$ act on $\\bV_m$\n\nAs we have seen in Lemma~\\ref{lem:2.2}(b), the \nsubspace $\\im(\\ev_m^*)$ of $\\cH$ consists of smooth vectors, so that \n$V \\cap \\cH^\\infty$ is dense in $V$. \n\\end{rem}\n\n\n\\begin{ex} \\mlabel{ex:grass} \nWe consider the identical representation of \n$G = \\U(\\cH)$ on the complex Hilbert space $\\cH$. \nLet $\\cK$ be a closed subspace of $\\cH$. Then \nthe subgroup $Q := \\{ g \\in \\GL(\\cH) \\: g\\cK = \\cK\\}$ \nis a complex Lie subgroup of $\\GL(\\cH)$ and the Gra\\ss{}mannian \n$\\Gr_\\cK(\\cH) := \\GL(\\cH)\\cK \\cong \\GL(\\cH)\/Q$ carries the \nstructure of a complex homogeneous space on which the unitary \ngroup $G = \\U(\\cH)$ acts transitively and which is isomorphic \nto $G\/H$ for $H := \\U(\\cH)_\\cK \\cong \\U(\\cK) \\oplus \\U(\\cK^\\bot)$. \n\nWriting elements of $B(\\cH)$ as $(2 \\times 2)$-matrices according to \nthe decomposition $\\cH = \\cK \\oplus \\cK^\\bot$, we have \n\\[ \\fq = \\Big\\{ \\pmat{ a & b \\\\ 0 & d} \\: a \\in B(\\cK), b \\in B(\\cK^\\bot, \\cK), \nd \\in B(\\cK^\\bot)\\Big\\},\\] \nand $\\gl(\\cH) = \\fq \\oplus \\fp^-$ holds for \n$\\fp^- = \\Big\\{ \\pmat{ 0 & 0 \\\\ c & 0} \\: c \\in B(\\cK, \\cK^\\bot)\\Big\\}.$ \n\nThe representation of $\\U(\\cH)$ on $\\cH$ is bounded with \n$V := \\cH^{\\fp^-} = \\cK^\\bot$, and the representation of \n$H \\cong \\U(\\cK) \\oplus \\U(\\cK^\\bot)$ on this space is bounded. \nIn view of Theorem~\\ref{thm:a.3}, the \ncanonical extension $\\beta \\: \\fq \\to \\gl(V), \\beta(x) := (x^*\\res_V)^*$ \nnow leads to a holomorphic vector bundle \n$\\bV := \\GL(\\cH) \\times_Q V \\cong G \\times_H V$ and a $G$-equivariant \nrealization $\\cH \\into \\Gamma(\\bV)$. \n\nIn this sense every Hilbert space can be realized as a space of \nholomorphic sections of a holomorphic vector bundle over any Gra\\ss{}mannian \nassociated to~$\\cH$. Note that $\\U(\\cH)_\\cK = \\U(\\cH)_V$ shows that \n$\\Gr_\\cK(\\cH) \\cong G\/H$ can be identified in a natural way with\n$\\Gr_V(\\cH)$. \n\\end{ex} \n\n\\begin{rem} Let $(\\pi, \\cH)$ be a smooth unitary representation \nof the Lie group $G$ and $V \\subeq \\cH$ be a closed $H$-invariant subspace. \nWe then obtain a natural $G$-equivariant map \n$\\eta \\: G\/H \\to \\Gr_V(\\cH), gH \\mapsto \\pi(g)V$. \nIf this map is holomorphic, then we can pull back the natural \nbundle $\\bV \\to \\Gr_V(\\cH)$ from Example~\\ref{ex:grass} and \nobtain a realization of $\\cH$ in $\\Gamma(\\eta^*\\bV)$. This works \nvery well if the representation \n$(\\pi, \\cH)$ is bounded because in this case $\\pi \\: G \\to \\U(\\cH)$\nis a morphism of Banach--Lie groups, but if \n$\\pi$ is unbounded, then it seems difficult to verify \nthat $\\eta$ is smooth, resp., holomorphic. \n\nIf (A1\/2) in Theorem~\\ref{thm:a.3} are satisfied, then \n$V \\subeq \\cH^\\omega \\subeq \\cH^\\infty$ implies that, \nthe operators $\\dd\\pi(x)$, $x \\in \\g$, are defined on $V$. \nSince they are closable, the graph of these restrictions is closed, \nwhich implies that the restrictions $\\dd\\pi(x)\\res_V \\: V \\to \\cH$ \nare continuous linear operators. We thus obtain a \nnatural candidate for a tangent map \n\\[ T_H(\\eta) \\: T_H(G\/H) \\cong \\g\/\\fh \\to \nT_V(\\Gr_V(\\cH)) \\cong B(V,V^\\bot), \\quad \nx \\mapsto (\\1-p_V)\\dd\\pi(x) p_V.\\] \n\nFor the special case where $\\dim V = 1$, we have \n$V= \\C v_0$ for a smooth vector $v_0$, and since the projective orbit map \n$G \\to \\bP(\\cH) \\cong \\Gr_V(\\cH), g \\mapsto [\\pi(g)v_0]$ is smooth, \nthe induced map $\\eta \\: G\/H \\to \\bP(\\cH)$ is smooth as well.\nThis construction is the key idea \nbehind the theory of coherent state representations \n(\\cite{Od88}, \\cite{Od92}, \\cite{Li91}, \\cite{Ne00}, \\cite{Ne01}), \nwhere one uses a holomorphic map $\\eta \\: G\/H \\to \\bP(\\cH^*)$ of a \ncomplex homogeneous space $G\/H$ to realize a unitary representation \n$(\\pi, \\cH)$ of $G$ in the space of holomorphic sections \nof the line bundle $\\eta^*{\\mathbb L}$, where \n${\\mathbb L} \\to \\bP(\\cH^*)$ is the canonical bundle on the dual projective\nspace with $\\Gamma({\\mathbb L}) \\cong \\cH$. \n\\end{rem}\n\n\n\\section{Realizing positive energy representations} \\mlabel{sec:3}\n\nIn this section we fix an element $d \\in \\g = \\L(G)$ for which \nthe one-parameter group $e^{\\R \\ad d} \\subeq \\Aut(\\g)$ is bounded, \ni.e., preserves an equivalent norm. We call such elements \n{\\it elliptic}. Then $H := Z_G(d)$ is a Lie subgroup and if \n$0$ is isolated in $\\Spec(D)$, then $G\/H$ carries a natural complex \nstructure. The class of representations which one may expect \nto be realized by holomorphic sections of Hilbert bundles \n$\\bV$ over $G\/H$ is the class of {\\it positive energy representations}, \nwhich is defined by the condition that the selfadjoint operator \n$-i\\dd\\pi(d)$ is bounded below. \n\n\\subsection{The splitting condition} \n\nLet $d \\in \\g$ be an elliptic element. Then \n\\[ H = Z_G(\\exp \\R d) = Z_G(d) \n= \\{ g \\in G \\: \\Ad(g)d = d \\} \\] \nis a closed subgroup of $G$, not necessarily connected, \n with Lie algebra $\\fh = \\z_\\g(d) = \\ker(\\ad d)$. \nSince $\\g$ contains arbitrarily small $e^{\\R \\ad d}$-invariant \n$0$-neighborhoods $U$, there exists such an open $0$-neighborhood \nwith $\\exp_G(U) \\cap H = \\exp_G(U \\cap \\L(H)).$ \nTherefore $H$ is a Lie subgroup of $G$, i.e., a Banach--Lie group \nfor which the inclusion $H \\into G$ is a topological embedding. \n\nOur assumption implies that $\\alpha_t := e^{t\\ad d}$ defines an \nequicontinuous one-paramter group of automorphisms of the \ncomplex Banach--Lie algebra $\\g_\\C$. For \n$\\delta > 0$, we consider the Arveson spectral subspace \n$\\fp^+ := \\g_\\C([\\delta,\\infty[)$ \n(cf.\\ Definition~\\ref{def:arv}). \nApplying Proposition~\\ref{prop:spec-add} to the Lie bracket \n$\\g_\\C \\times \\g_\\C \\to \\g_\\C$, we see that \n$\\fp^+$ is a closed complex subalgebra. \nFor $f \\in L^1(\\R)$, $\\alpha(f) := \\int_\\R f(t) \\alpha_t\\, dt$ \nand $x \\in \\g_\\C$, \nthe relations $\\oline{\\alpha(f)x} = \\alpha(\\oline f)\\oline x$ and \n$\\hat{\\oline f}(\\xi) = \\oline{\\hat f(-\\xi)}$ imply that \n$\\fp^- := \\oline{\\fp^+} = \\g_\\C(]-\\infty, -\\delta])$. \nTo make the following constructions work, we assume the \n{\\it splitting condition:} \n\\begin{equation}\n \\label{eq:splitcond}\n\\g_\\C = \\fp^+ \\oplus \\h_\\C \\oplus \\fp^-.\\tag{SC}\n\\end{equation}\nIn view of Lemma~\\ref{lem:a.17}, it is satisfied for some \n$\\delta > 0$ if and only if $0$ is isolated in $\\Spec(\\ad d)$. \n\nSince $\\Ad(H)$ commutes with $e^{\\R \\ad d}$, \nthe closed subalgebras $\\fp^\\pm \\subeq \\g_\\C$ are invariant \nunder $\\Ad(H)$ and $e^{\\R \\ad d}$. \nNow $\\g \\cap (\\fp^+ \\oplus \\fp^-)$ is a closed complement for \n$\\fh$ in $\\g$, so that $M := G\/H$ carries the structure of a \nBanach homogeneous space and \n$\\fq := \\fh_\\C + \\fp^+ \\cong \\fp^+ \\rtimes \\fh_\\C$ \ndefines a $G$-invariant complex manifold \nstructure on $M$ (cf.\\ Section~\\ref{sec:1}). \n\n\\begin{rem} (a) For bounded derivations of compact \n$L^*$-algebras similar splitting conditions have been used \nby Belti\\c{t}\\u{a} in \\cite{Bel03} to obtain K\\\"ahler polarizations \nof coadjoint orbits. In \\cite{Bel04} this is extended to \nbounded normal derivations of a complex Banach Lie algebra. \n\n(b) If $\\g$ is a real Hilbert--Lie algebra, then one can use\n spectral measures to obtain natural \ncomplex structures on $G\/H$ even if the splitting condition is \nnot satisfied, i.e., $0$ need not be isolated in the spectrum \nof $\\ad d$ (\\cite[Prop.~5.4]{BRT07}). \n\\end{rem}\n\n\n\\begin{ex} If $\\alpha$ factors through an action of \nthe circle group $\\T = \\R\/2\\pi\\Z$, then the Peter--Weyl Theorem \nimplies that the sum $\\sum_{n \\in \\Z} \\g_\\C^n$ of the corresponding \neigenspaces $\\g_\\C^n := \\ker(\\ad d - in \\1)$ \nis dense in $\\g_\\C$ with $\\fh_\\C = \\g_\\C^0$. Since the operator \n$\\ad d$ is bounded, only finitely many $\\g_\\C^n$ are non-zero, so that \nwe actually have $\\g_\\C = \\sum_{n \\in \\Z} \\g_\\C^n$. \nFrom $[\\g_\\C^n, \\g_\\C^m] \\subeq \\g_\\C^{n+m}$ it follows \nthat $\\fp^\\pm := {\\sum_{\\pm n > 0} \\g_\\C^n}$ are closed subalgebras \nfor which $\\g_\\C = \\fp^+ \\oplus \\fh_\\C \\oplus \\fp^-$ is direct. \nIn this case the splitting condition is always satisfied \nand $\\Spec(D) \\subeq i\\Z$. \n\nIf, conversely, $d \\in \\g$ is an element for which the complex \nlinear extension of $\\ad d$ to $\\g_\\C$ is diagonalizable \nwith finitely many eigenvalues in $i\\Z$, then $e^{\\R \\ad d} \n\\subeq \\Aut(\\g_\\C)$ is compact, hence preserves a compatible norm. \nAn important special situation, where we have all this \nstructure are hermitian Lie groups (cf.\\ \\cite{Ne11}). \nIn this case we simply have $\\fp^\\pm = \\g_\\C^{\\pm 1}$. \n\\end{ex}\n\n\n\\subsection{Positive energy representations} \n\n\\begin{lem} \\mlabel{lem:c.1} \nLet $\\gamma \\: \\R \\to \\U(\\cH)$ be a strongly continuous \nunitary representation and $A = A^* = -i\\gamma'(0)$ be its selfadjoint \ngenerator, so that $\\gamma(t) = e^{itA}$ in terms of measurable functional \ncalculus. Then the following assertions hold: \n\\begin{description}\n\\item[\\rm(i)] For each $f \\in L^1(\\R)$, we have \n$\\gamma(f) = \\hat f(A),$\nwhere $\\hat f(x) := \\int_\\R e^{ixy} f(y)\\, dy$ is the Fourier transform \nof $f$. \n\\item[\\rm(ii)] Let $P \\: {\\mathfrak B}(\\R) \\to B(\\cH)$ be the unique \nspectral measure with $A = P(\\id_\\R)$. \nThen, for every closed subset $E \\subeq \\R$, the \nrange $P(E)\\cH$ coincides with the Arveson spectral subspace \n$\\cH(E)$. \n\\end{description}\n\\end{lem}\n\n\\begin{prf} Since the unitary representation $(\\gamma,\\cH)$ is a direct sum \nof cyclic representation, it suffices to prove the assertions for \ncyclic representations. Every cyclic representation of \n$\\R$ is equivalent to the representation on \nsome space $\\cH = L^2(\\R,\\mu)$, where $\\mu$ is a Borel probability \nmeasure on $\\R$ and $(\\gamma(t)\\xi)(x) = e^{itx}\\xi(x)$ \n(see \\cite[Thm.~VI.1.11]{Ne00}). \n\n(i) We have $(A\\xi)(x) = x\\xi(x)$, so that \n$(\\hat f(A)\\xi)(x) = \\hat f(x)\\xi(x)$. On the other hand, we have for \n$f \\in L^1(\\R)$ in the space $\\cH = L^2(\\R,\\mu)$ the relation \n\\[ (\\gamma(f)\\xi)(x) \n= \\int_\\R f(t) e^{itx}\\xi(x)\\, dt = \\hat f(x)\\xi(x). \\] \n\n(ii) see \\cite[p.~226]{Ar74}. \n\\end{prf}\n\n\n\\begin{prop} \\mlabel{prop:c.3} \nLet $(\\pi, \\cH)$ be a smooth unitary representation \nof the Banach--Lie group $G$, $d \\in \\g$ be elliptic, \nand $P \\: {\\mathfrak B}(\\R) \\to \\cL(\\cH)$ be the spectral measure \nof the unitary one-parameter group \n$\\pi_d(t) := \\pi(\\exp_G td)$. \nThen the following assertions hold: \n\\begin{description}\n\\item[\\rm(i)] $\\cH^\\infty$ carries a Fr\\'echet structure for which \n$\\pi_d(t)_{t \\in \\R}$ defines a continuous equicontinuous action of \n$\\R$ on $\\cH^\\infty$. In particular, $\\cH^\\infty$ is invariant under \n$\\pi_d(f)$ for every $f \\in L^1(\\R)$. \n\\item[\\rm(ii)] For every closed subset $E \\subeq \\R$, we have \n$\\cH^\\infty(E) = (P(E)\\cH) \\cap \\cH^\\infty$ for the corresponding \nspectral subspace. \n\\item[\\rm(iii)] For every open subset $E \\subeq \\R$, \n$(P(E) \\cH) \\cap \\cH^\\infty$ is dense in $P(E)\\cH^\\infty$. \nMore precisely, there exists a sequence $(f_n)_{n \\in \\N}$ \nin $L^1(\\R)$ for which $\\pi_d(f_n) \\to P(E)$ in the \nstrong operator topology and $\\supp(\\hat f_n) \\subeq E$, so that \n$\\pi_d(f_n)v \\in \\cH^\\infty \\cap P(E)\\cH^\\infty$ for every $v \\in \\cH^\\infty$. \n\\item[\\rm(iv)] \nFor closed subsets $E, F \\subeq \\R$, \nthe Arveson spectral subspaces $\\g_\\C(F)$ satisfies \n\\begin{equation}\n \\label{eq:shift}\n\\dd\\pi(\\g_\\C(F))\\big(\\cH^\\infty \\cap P(E)\\cH\\big) \\subeq P(\\oline{E+ F})\\cH. \n\\end{equation}\n\\end{description}\n\\end{prop} \n\n\\begin{prf} (i) We may w.l.o.g.\\ assume that the norm on \n$\\g$ is invariant under $e^{\\R \\ad d}$. \nOn $\\cH^\\infty$ we consider the Fr\\'echet topology \ndefined by the seminorms \n$$ p_n(v) := \\sup \\{ \\|\\dd\\pi(x_1)\\cdots \\dd\\pi(x_n)v\\| \\: \nx_i \\in \\g, \\|x_i\\| \\leq 1\\} $$\nwith respect to which the action of $G$ on $\\cH^\\infty$ is smooth \n(cf.~\\cite[Thm.~4.4]{Ne10a}). In particular, the bilinear map \n\\begin{equation}\n \\label{eq:applic}\n\\g_\\C \\times \\cH^\\infty \\to \\cH^\\infty, \\quad \n(x,v) \\mapsto \\dd\\pi(x) v\n\\end{equation}\nis continuous because it can be obtained as a \nrestriction of the tangent map of the $G$-action. \n \nIn view of the relation \n$\\pi_d(t) \\dd\\pi(x) \\pi_d(t)^{-1} = \\dd\\pi(e^{t \\ad d}x)$ \nfor $t \\in \\R, x \\in \\g,$ \nthe isometry of $e^{t\\ad d}$ on $\\g$ implies that \nthe seminorms $p_n$ on $\\cH^\\infty$ \nare invariant under $\\pi_d(\\R)$. Since the $\\R$-action \non $\\cH^\\infty$ defined by the operators $\\pi_d(t)$ is smooth, \nhence in particular continuous, we obtain with Definition~\\ref{def:arv} \nan algebra homomorphism \n$$ \\pi_d \\: (L^1(\\R), *) \\to \\End(\\cH^\\infty), \\quad \nf \\mapsto \\int_\\R f(t)\\pi_d(t)\\, dt $$ \n(cf.\\ Definition~\\ref{def:arv} below), and this implies (i). \n\n(ii) Since $\\cH^\\infty(E) = \\cH(E) \\cap \\cH^\\infty$ follows immediately \nfrom the definition of spectral subspaces (Remark~\\ref{rem:a.6}), \nthis assertion is a consequence of Lemma~\\ref{lem:c.1}(ii). \n\n(iii) We write the open set $E$ as an increasing \nunion of compact subsets $E_n$ \nand observe that $\\bigcup_n P(E_n) \\cH$ is dense in $P(E)\\cH$. \nFor every $n$, there exists a compactly supported function \n$h_n \\in C^\\infty_c(\\mathbb R,\\mathbb R)$ such that \n\\[ \\supp(h_n) \\subseteq E, \\quad 0 \\leq h_n \\leq 1, \\quad \n\\mbox{ and } \\quad h_n\\big|_{E_n} = 1.\\] \n Let $f_n \\in \\cS(\\mathbb R)$ with \n$\\hat f_n = h_n$. Then \n$\\pi_d(f_n) = \\hat f_n(-i\\gamma'(0)) = h_n(-i\\gamma'(0))$ \n(Lemma~\\ref{lem:c.1}(i)) and consequently\n\\[ P(E_n)\\cH \\subseteq \\pi_d(f_n)\\cH \\subseteq \nP(E)\\cH. \\]\nTherefore the subspace \n$\\pi_d(f_n)\\cH^\\infty$ of $\\cH^\\infty$ is contained \nin $P(E)\\cH$. \nIf $w = P(E)v$ for some $v \\in \\cH^\\infty$\nthen \n\\[ \\pi_d(f_n)w = \\pi_d(f_n)P(E)v = \\pi_d(f_n)v\n\\in \\cH^\\infty\\] \nand \n\\[ \\|\\pi_d(f_n)w-w\\|^2 \n= \\|h_n(-i\\pi_d'(0))w -w\\|^2 \\leq \\|P(E\\backslash E_n)w\\|^2 \\to 0\\] \nfrom which it follows that $\\pi_d(f_n)w \\to w$. \n\n(iv) This follows from the continuity of \n\\eqref{eq:applic}, Proposition~\\ref{prop:spec-add} and (ii). \n\\end{prf}\n\n\nResults of a similar type as Proposition~\\ref{prop:c.3}(iv) \nand the more universal Proposition~\\ref{prop:spec-add} in the \nappendix are well known in the context of bounded operators \n(cf.\\ \\cite{FV70}, \\cite{Ra85}, \\cite[Prop.~1.1, Cor.~1.2]{Bel04}). \nArveson also obtains variants for automorphism groups of operator algebras \n(\\cite[Thm.~2.3]{Ar74}). \n\n\\begin{rem} Combining Lemma~\\ref{lem:c.1}(i) with \nProposition~\\ref{prop:c.3}(i), we derive that \nthe subspace $\\cH^\\infty$ of $\\cH$ is invariant under all operators \n$P(\\hat f) = \\hat f(-i\\dd\\pi(d))$ for $f \\in L^1(\\R)$. This implies \nin particular to the operators $P(h)$, $h \\in \\cS(\\R)$, but not \nto the spectral projections $P(E)$. If $E \\subeq \\R$ is open, \nProposition~\\ref{prop:c.3}(iii) provides a suitable approximate \ninvariance. \n\\end{rem}\n\nThe following proposition is of key importance for the following. \nIt contains the main consequences of Arveson's spectral theory \nfor the actions on $\\g_\\C$ and $\\cH^\\infty$. \n\n\\begin{prop} \\mlabel{prop:6.2} If $d \\in \\g$ is elliptic with \n$0$ isolated in $\\Spec(\\ad d)$, then for any \nsmooth positive energy representation \n $(\\pi, \\cH)$ of $G$, the $H$-invariant subspace \n$V := \\oline{(\\cH^\\infty)^{\\fp^-}}$ satisfies \n$\\cH = \\oline{\\Spann(\\pi(G)V)}$. \n\\end{prop} \n\n\\begin{prf} First we show that $V \\not=\\{0\\}$ whenever \n$\\cH\\not=\\{0\\}$. Let \n\\[ s := \\inf(\\Spec(-i\\dd\\pi(d))) > -\\infty . \\]\nFor some $\\eps \\in ]0,\\delta[$, we consider the closed subspace \n\\begin{equation}\n \\label{eq:vdef}\nW := P([s,s+ \\eps[) \\cH = P(]s-\\eps, s+ \\eps[) \\cH, \n\\end{equation}\nwhere $P \\: {\\mathfrak B}(\\R) \\to B(\\cH)$ is the spectral measure of~$\\pi_d$. \nThen Proposition~\\ref{prop:c.3} implies that \n$W^\\infty := W\\cap \\cH^\\infty$ is dense in $W$ \nand that \n\\[ \\dd\\pi(\\fp^-) W^\\infty \n\\subeq P(]-\\infty, s+ \\eps - \\delta])\\cH = \\{0\\},\\]\nwhich leads to $\\{0\\}\\not= W \\subeq V$. \n\nApplying the preceding argument to the positive energy \nrepresentation on the orthogonal complement of \n$\\cH_V := \\oline{\\Spann \\pi(G)V}$, the relation \n$V \\cap \\cH_V^\\bot=\\{0\\}$ implies that \n$\\cH_V^\\bot=\\{0\\}$, and hence that $\\cH = \\cH_V$. \n\\end{prf}\n\n\\begin{thm} \\mlabel{thm:6.2} If $d \\in \\g$ is elliptic with \n$0$ isolated in $\\Spec(\\ad d)$ and $(\\pi,\\cH)$ \nis a smooth positive energy representation \nfor which the $H$-representation $\\rho(h) := \\pi(h)\\res_V$ \non $V := \\oline{(\\cH^\\infty)^{\\fp^-}}$ \nis bounded, then $(\\pi, \\cH)$ is holomorphically induced from \nthe representation $(\\rho,\\beta)$ of $(H,\\fq)$ on \n$V$ defined by $\\beta(\\fp^+) = \\{0\\}$. In particular, \n$V$ consists of analytic vectors. \n\\end{thm} \n\n\\begin{prf} Since $\\pi(G)V$ spans a dense subspace of $\\cH$, i.e., \n$\\cH = \\cH_V$ (Proposition~\\ref{prop:6.2}), \nthe assertion follows from Remark~\\ref{rem:2.14}. \n\\end{prf}\n\n\\begin{rem}\nSince $\\cH_V \\cong \\cH_{V'}$ as $G$-representations \nif and only if $V \\cong V'$ as $H$-representations \n(cf.\\ Corollary~\\ref{cor:2.15}), the description of \nall $G$-representations of positive energy \nfor which the $H$-representation $(\\rho, V)$ is bounded \nis equivalent to the determination of all bounded $H$-representations \n$(\\rho, V)$ for which $(\\rho, \\beta, V)$ is \ninducible if we put $\\beta(\\fp^+) = \\{0\\}$. \n\\end{rem}\n\n\\begin{cor} \\mlabel{cor:6.2} If \n$d \\in \\g$ is elliptic with $0$ isolated in $\\Spec(\\ad d)$, \nthen every bounded representation of $G$ is holomorphically induced from \nthe representation $(\\rho,\\beta)$ of $(H,\\fq)$ on \n$V := \\oline{(\\cH^\\infty)^{\\fp^-}}$ defined by $\\beta(\\fp^+) = \\{0\\}$. \n\\end{cor} \n\nFrom Corollary~\\ref{cor:commutant} we obtain in particular:\n\n\\begin{cor} A positive energy representation \n$(\\pi, \\cH)$ of $G$ for which the representation $(\\rho,V)$ of $H$ \nis bounded is a direct sum of irreducible ones if and only \nif $(\\rho, V)$ has this property. \n\\end{cor}\n\n\n\\begin{ex} The complex Banach--Lie algebra \n$\\g$ is called {\\it weakly root graded} if there exists \na finite reduced root system $\\Delta$ such that $\\g$ \ncontains the corresponding \nfinite dimensional semisimple Lie algebra $\\g_\\Delta$ and for some \nCartan subalgebra $\\fh \\subeq \\g_\\Delta$, the Lie algebra $\\g$ is a direct \nsum of finitely many $\\ad \\fh$-eigenspaces. \n\nNow suppose that $\\g$ is a real Banach--Lie algebra for which \n$\\g_\\C$ is weakly root graded, that \n$\\fh$ is invariant under conjugation and, for every \n$x \\in \\fh \\cap i\\g$, the derivation $\\ad x$ has real spectrum.\nThen the realization results for bounded unitary representations \nof $G$ which follows from \n\\cite[Thm.~5.1]{MNS09}, applied to their holomorphic \nextensions $G_\\C \\to \\GL(\\cH)$, can be derived \neasily from Corollary~\\ref{cor:6.2}. \n\\end{ex}\n\nThe following theorem shows that, assuming that $(\\pi, \\cH)$ is semibounded \nwith $d \\in W_\\pi$ permits us to get rid of the quite implicit assumption \nthat the $H$-representation on $V$ is bounded. It is an important \ngeneralization of Corollary~\\ref{cor:6.2} to semibounded representations. \n\n\\begin{thm} \\mlabel{thm:6.2b} Let $(\\pi, \\cH)$ be a semibounded \nunitary representation of the Banach--Lie group $G$ and \n$d \\in W_\\pi$ be an elliptic element for which $0$ \nis isolated in $\\Spec(\\ad d)$. \nWe write $P \\: {\\mathfrak B}(\\R) \\to B(\\cH)$ for the spectral measure of the \nunitary one-parameter group $\\pi_d(t) := \\pi(\\exp(td)$. \nThen the following assertions hold: \n\\begin{description}\n\\item[\\rm(i)] The representation $\\pi\\res_H$ of $H$ is semibounded and, \nfor each bounded measurable subset $B \\subeq \\R$, the \n$H$-representation on $P(B)\\cH$ is bounded. \n\\item[\\rm(ii)] The representation $(\\pi, \\cH)$ is a direct sum of \nrepresentations $(\\pi_j, \\cH_j)$ for which there exist $H$-invariant \nsubspaces $\\cD_j \\subeq (\\cH_j^\\infty)^{\\fp^-}$ for \nwhich the $H$-representation $\\rho_j$ on $V_j := \\oline{\\cD_j}$ is bounded and \n$\\Spann\\big(\\pi_j(G)V_j\\big)$ is dense in $\\cH_j$. \nThen the representations $(\\pi_j, \\cH_j)$ are holomorphically \ninduced from $(\\rho_j, \\beta_j,V)$, where \n$\\beta_j(\\fp^+)= \\{0\\}$. \n\\item[\\rm(iii)] If $(\\pi, \\cH)$ is irreducible and \n$s := \\inf\\Spec(-i\\dd\\pi(d))$, then $P(\\{s\\})\\cH \n= \\oline{(\\cH^\\infty)^{\\fp^-}}$ and \n$(\\pi, \\cH)$ is holomorphically induced \nfrom the bounded $H$-representation $\\rho$ on this space, extended \nby $\\beta(\\fp^+) = \\{0\\}$. \n\\end{description}\n\\end{thm}\n\n\\begin{prf} (i) From $s_{\\pi\\res_H} = s_\\pi\\res_{\\fh}$ it follows that \n$\\pi\\res_H$ is semibounded with $d \\in W_\\pi \\cap \\fh \\subeq W_{\\pi\\res_H}$. \nFor every bounded measurable subset $B \\subeq \\R$, the relation \n$d \\in \\z(\\fh)$ entails that $P(B)\\cH$ is $H$-invariant. Let \n$\\rho_B$ denote the corresponding representation of $H$. \nThen the boundedness of $\\dd\\rho_B(d)$ which commutes with \n$\\dd\\rho(\\fh)$ implies that \n$W_{\\rho_B} + \\R d = W_{\\rho_B}$, so that $d \\in W_{\\pi\\res_H} \n\\subeq W_{\\rho_B}$ leads to $0 \\in W_{\\rho_B}$. \nThis means that $\\rho_B$ is bounded. \n\n(ii) We apply Zorn's Lemma to the ordered set of all \npairwise orthogonal systems of closed $G$-invariant subspaces \nsatisfying the required conditions. \nTherefore it suffices to \nshow that if $\\cH \\not=\\{0\\}$, then there exists a non-zero \n$H$-invariant subspaces $\\cD \\subeq (\\cH^\\infty)^{\\fp^-}$ for \nwhich the $H$-representation $\\rho$ on $\\oline{\\cD}$ is bounded. \n\nFor $s := \\inf\\Spec(-i\\dd\\pi(d))$ and $0 < \\eps < \\delta$, we \nmay take the space $\\cD := \\break \\cH^\\infty \\cap P([s,s+\\eps[)\\cH$ \nfrom the proof of Proposition~\\ref{prop:6.2}. \nAs we have seen there, it is annihilated \nby $\\dd\\pi(\\fp^-)$ and the boundedness of the $H$-representation on its \nclosure follows from~(i). \n\nThat the representations $(\\pi_j, \\cH_j)$ are holomorphically induced\n from the bounded $H$-representations on $V_j$ follows from \nTheorem~\\ref{thm:a.3}. \n\n(iii) For $t > s$, let \n\\[ \\cD_U \n:= (\\cH^\\infty)^{\\fp^-} \\cap P([s,t[)(\\cH^\\infty)^{\\fp^-}\n= (\\cH^\\infty)^{\\fp^-} \\cap P(]s-\\delta, t[)(\\cH^\\infty)^{\\fp^-}. \\] \n\n{\\bf Claim 1:} $\\cD_U$ is dense in \n$U := \\oline{P([s,t[)(\\cH^\\infty)^{\\fp^-}}$. \n\nLet $v \\in (\\cH^\\infty)^{\\fp^-}$ and $w := P(]s-\\delta,t[)v$. \nWith Proposition~\\ref{prop:c.3}(iii), we find a sequence $f_n \\in L^1(\\R)$ \nfor which $\\pi_d(f_n)$ converges strongly to $P(]s-\\delta,t[)$ \nand $\\supp(\\hat f_n) \\subeq ]s-\\delta, t[$, so that $\\pi_d(f_n)v \\to w$ \nand \n\\[ \\pi_d(f_n)v = P(\\hat f_n)v \n= P(]s-\\delta, t[)P(\\hat f_n)v \n\\in P(]s-\\delta, t[)\\cH^\\infty.\\] \nSince $\\fp^-$ is invariant under $e^{\\ad d}$, the closed subspace \n$(\\cH^\\infty)^{\\fp^-}$ is invariant under $\\pi_d(\\R)$, \nso that $\\pi_d(f_n)v\\in (\\cH^\\infty)^{\\fp^-}$. \nThis proves Claim $1$. \n\n{\\bf Claim 2:} $(\\pi, \\cH)$ is holomorphically induced from \nthe bounded $H$-representation $(\\rho, \\beta, U)$, defined by \n$\\beta(\\fp^+) := \\{0\\}$. \n\nFrom (i) we know that the $H$-representation on $U$ is bounded \nand on the dense subspace $\\cD_U$ we have $\\dd\\pi(\\fp^-)\\cD_U = \\{0\\}$. \nTherefore (A1\/2) in Theorem~\\ref{thm:a.3} are satisfied and this proves \nClaim~$2$. \n\n{\\bf Claim 3:} $U = P(\\{s\\})\\cH$ for every $t > s$. \n\nIn view of Claim $2$, Corollary~\\ref{cor:commutant} implies that \nthe $H$-representation on $U$ is irreducible. Since $\\pi_d$ commutes with \n$H$, it follows in particular that $\\rho(\\exp\\R d) \\subeq \\T\\1$. \nThe definition of $s$ now shows that $U \\subeq P(\\{s\\})\\cH$. \n\nFor $0 < \\eps < \\delta$ and $t < \\delta + \\eps$, the proof \nof Proposition~\\ref{prop:6.2} implies that \n$\\big(P([s,t[)\\cH\\big) \\cap \\cH^\\infty$ is dense in \n$P([s,t[)\\cH$ and contained in $(\\cH^\\infty)^{\\fp^-}$, hence \nin $P([s,t[)(\\cH^\\infty)^{\\fp^-}\\subeq U$. \nWe conclude in particular that $P(\\{s\\})\\cH \\subeq U$. \n\n{\\bf Claim 4:} $U = \\oline{(\\cH^\\infty)^{\\fp^-}}$. \n\nFrom the definition of $U$ it is clear that \n$U \\subeq \\oline{(\\cH^\\infty)^{\\fp^-}}$. \nTo see that we actually have equality, we note that Claim $2$ \nshows that $P([s,t[)(\\cH^\\infty)^{\\fp^-} \\subeq P(\\{s\\})\\cH = U$ \nholds for every $t > s$. \nAs $P([s,n]) \\to P([s,\\infty[) = \\id$ holds pointwise, we obtain \n$(\\cH^\\infty)^{\\fp^-} \\subeq U$. \n\nThis completes the proof of (iii). \n\\end{prf}\n\n\\begin{rem} For finite dimensional Lie groups the classification \nof irreducible \nsemibounded unitary representations easily boils down to a situation \nwhere one can apply Theorem~\\ref{thm:6.2b}. Here \n$d \\in \\g$ is a regular element whose centralizer \n$\\fh = \\ft$ is a compactly embedded Cartan subalgebra and \nthe corresponding group $T = H$ is abelian and $V$ is one-dimensional \n(cf.\\ \\cite{Ne00}). In this case $0$ is trivially isolated in \nthe finite set $\\Spec(\\ad d)$. \n\\end{rem}\n\nThe following theorem provides a bridge between the seemingly \nweak positive energy condition and the much stronger semiboundedness \ncondition. \n\n\\begin{thm} \\mlabel{thm:3.15} \nLet $d \\in \\g$ be elliptic with $0$ isolated in $\\Spec(\\ad d)$. \nThen a smooth unitary representation \n$(\\pi, \\cH)$ of $G$ for which the representation $\\rho$ of \n$H$ on $\\oline{(\\cH^\\infty)^{\\fp^-}}$ is bounded \nsatisfies the positive energy condition \n\\begin{equation}\n \\label{eq:posen}\n\\inf\\Spec(-i\\dd\\pi(d)) > -\\infty\n\\end{equation}\nif and only if $\\pi$ is semibounded with $d \\in W_\\pi$. \n\\end{thm}\n\n\\begin{prf} If $\\pi$ semibounded with $d \\in W_\\pi$, then \nwe have in particular \\eqref{eq:posen}. It remains to show the \nconverse if all the assumptions of the theorem are satisfied. \nRecall that the splitting condition \\eqref{eq:splitcond} \nis satisfied because $0$ is isolated in $\\Spec(\\ad d)$. \nWe note that the representation $\\ad_{\\fp^+}$ of \n$\\fh$ on $\\fp^+$ is bounded with $\\Spec(\\ad_{\\fp^+}(-id)) \\subeq ]0, \\infty[$. \nTherefore the invariant cone \n\\begin{equation}\n \\label{eq:cone} \nC := \\{ x \\in \\fh \\: \n\\Spec(\\ad_{\\fp^+}(-ix)) \\subeq ]0,\\infty[ \\} \n\\end{equation}\nis non-empty \nand open because it is the inverse image of the \nopen convex cone \n\\[ \\{ X \\in \\Herm(\\fp^+) \\: \\Spec(X) \\subeq ]0, \\infty[ \\}\\] \n(cf.\\ \\cite[Thm.~14.31]{Up85}) under the continuous linear map \n$\\fh \\to \\gl(\\fp^+), x \\mapsto \\ad_{\\fp^+}(-ix)$. \n\nLet $x \\in C$ and \n\\[ s_\\rho(x) := \\sup(\\Spec(i\\dd\\rho(x))\n= -\\inf(\\Spec(-i\\dd\\rho(x)),\\] so that \n$V := \\oline{(\\cH^\\infty)^{\\fp^-}} \\subeq \\cH^\\infty$ \n(Theorem~\\ref{thm:6.2}) is contained in \nthe spectral subspace $\\cH^\\infty([-s_\\rho(x),\\infty[)$ with respect \nto the one-parameter group $t \\mapsto \\pi(\\exp tx)$. \nSince the map \n\\[ \\g_\\C \\times \\cH^\\infty \\to \\cH^\\infty, \\quad \n(x,v) \\mapsto \\dd\\pi(x)v \\] \nis continuous bilinear and $H$-equivariant, we see with \nProposition~\\ref{prop:spec-add} that \nthe subspace $\\cH^\\infty([-s_\\rho(x),\\infty[)$ of $\\cH^\\infty$ \nis invariant under $\\fp^+$. \n\nFor every $v \\in V \\subeq \\cH^\\omega$, the Poincar\\'e--Birkhoff--Witt \nTheorem shows that it \ncontains the subspace \n\\[ U(\\g_\\C)v = U(\\fp^+)U(\\fh_\\C)U(\\fp^-)v \n= U(\\fp^+)U(\\fh_\\C)v \\subeq U(\\fp^+)V.\\] \nFrom $V \\subeq \\cH^\\omega$ and $\\cH = \\cH_V$ it follows that \n$U(\\g_\\C)V$ is dense in $\\cH$, and hence that \n$\\cH^\\infty([-s_\\rho(x),\\infty[) \\subeq \\cH([-s_\\rho(x),\\infty[)$ \nis dense in $\\cH$. We conclude that \n\\begin{equation}\n \\label{eq:spirho} \ns_{\\pi}(x) = \\sup(\\Spec(i\\dd\\pi(x))) = s_\\rho(x) \n\\quad \\mbox{ for } \\quad x \\in C.\n\\end{equation}\n\nTo see that $C \\subeq W_\\pi$, it now suffices to show that \n$\\Ad(G)C$ has interior points. \nLet $\\fp := (\\fp^+ + \\fp^-)\\cap \\g$ and note that this is a closed \n$H$-invariant complement of $\\fh$ in $\\g$. \nThe map $F \\: \\fh\\times \\fp \\to \\g, F(x,y) := e^{\\ad y}x$ \nis smooth and \n\\[ \\dd F(x,0)(v,w) = [w,x] + v.\\] \nSince the operators $\\ad x$, $x \\in \\h$, \npreserve $\\fp$, the operator \n$\\dd F(x,0)$ is invertible if and only if \n$\\ad x \\: \\fp \\to \\fp$ is invertible, and this is the case \nfor any $x \\in C$ because $\\ad x\\res_{\\fp^\\pm}$ are invertible \noperators. This proves that $C$ is contained in the interior \nof $F(C,\\fp)$, and hence that $C \\subeq W_{\\pi}$. \nTherefore $\\pi$ is semibounded with $d \\in W_{\\pi}$. \n\\end{prf}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\subsection{GPU SM Architecture}\n\\label{sec:sm_arch}\n\\noindent \n\\newedit{Figure~\\ref{fig:gpu1} illustrates a representative SM architecture where shared memory may share a single on-chip memory structure with L1D cache~\\cite{nvidia2009nvidia, nvidia2012nvidia}.\nThe single on-chip memory structure consists of 32 banks with 512 rows, where 128 or 384 contiguous rows can be allocated to shared memory (\\textit{i.e}\\onedot} \\def\\Ie{\\textit{I.e}\\onedot, 16KB or 48KB) based on user configuration and the remaining are allocated as L1D cache~\\cite{gebhart2012unifying}.\nWhile all 32 L1D cache banks operate in tandem for a single contiguous 32$\\times$4-byte (128-byte) L1D cache request, all 32 shared memory banks can be accessed independently and serve upto 32 shared memory requests in parallel.\nL1D cache buffers data from underlying memory and keeps a separate tag array to identify data hit. In such architecture, a L1D cache access is serialized. That is, tag array is accessed before the banks are accessed~\\cite{edmondson1995internal}. \nIn contrast, as shared memory stores intermediate results generated by ALU for each Cooperative Thread Array (CTA) which is explicitly manipulated by programmers, it neither needs tags nor accesses data in underlying memory. \nHence, there is no datapath between shared memory and L2 cache, and no cache write\/eviction policies are applied in shared memory~\\cite{nvidia2009nvidia, nvidia2012nvidia}. In addition, to manage the shared memory space, each SM keeps an independent Shared Memory Management Table (SMMT)~\\cite{yang2012shared}\nwhere each CTA reserves one entry to store the size and base address of allocated shared memory.\n}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1\\linewidth]{figs\/gpu1.eps}\n\\caption{GPU SM architecture.}\n\\label{fig:gpu1}\n\\end{figure}\n\n\\begin{comment}\n\\end{comment}\n\n\n\\begin{figure}[b]\n\\centering\n\\subfloat[]{\\label{fig:l1_contention}\\rotatebox{0}{\\includegraphics[width=0.4\\linewidth]{figs\/l1_contention}}}\n\\hspace{4pt}\n\\subfloat[]{\\label{fig:VTA_mech}\\rotatebox{0}{\\includegraphics[width=0.57\\linewidth]{figs\/VTA_mech}}}\n\\caption{(a) An example of locality and interference and (b) VTA structure.}\n\\end{figure}\n\n\\subsection{Cache Interference}\n\\label{sec:interfere}\n\\noindent \nAs many warps share small L1D cache, they often contend for the same cache line.\nHence, cached data of an active warp are frequently evicted by cache accesses of other active warps.\nThis phenomenon is referred to as \\textit{cache interference} which often changes supposedly a regular memory access pattern into an irregular one. \nFigure~\\ref{fig:l1_contention} depicts an example of how the cache interference worsens data locality in L1D cache, \nwhere warps \\texttt{W0} and \\texttt{W1} send memory requests to get data \\texttt{D0} and \\texttt{D4}, respectively.\nHowever, since \\texttt{D0} and \\texttt{D4} are mapped to the same cache set \\texttt{S0}, \nrepeated memory requests from \\texttt{W0} and \\texttt{W1} to get \\texttt{D0} and \\texttt{D4} keep evicting \\texttt{D4} and \\texttt{D0} at cycles \\texttt{(a)}, \\texttt{(b)}, \\texttt{(e)}, and \\texttt{(f)}.\nUnless the memory requests from \\texttt{W1} and \\texttt{W0} evicted \\texttt{D0} and \\texttt{D4}, respectively, \nthey should have been L1D cache hits.\nSuch a cache hit opportunity is also called \\textit{potential of data locality}, which can be quantified by the frequency of re-referencing the same data unless cache interference occurs.\n\n\n\\subsection{Potential of Data Locality Detection}\n\\label{sec:vta}\n\\noindent\nTo detect the potential of data locality described in Section~\\ref{sec:interfere}, we may leverage a Victim Tag Array (VTA)~\\cite{rogers2012cache} where\n\nwe store a Warp ID (WID) in each cache tag, as shown in Figure~\\ref{fig:VTA_mech}.\nA WID in a cache tag is to track which warp brought current data in a cache line. \nWhen a memory request of a warp evicts data in a cache line, we first take \n(1) the address in the cache tag associated with the evicted data and \n(2) the WID of the warp evicting the data. \nThen we store (1) and (2) in a VTA entry which is indexed by the WID stored in the cache tag (\\textit{i.e}\\onedot} \\def\\Ie{\\textit{I.e}\\onedot, the WID of the warp which brought the evicted data in the cache line).\nWhen memory requests of an active warp repeatedly incur VTA hits, \nthey exhibit potential of data locality.\n\n\n\\subsection{Methodology}\n\\label{sec:method} \n\\noindent\\textbf{GPU architecture.} \nWe use GPGPU-Sim 3.2.2~\\cite{aaamodt2012gpgpu} and configure it to model a GPU similar to NVIDIA GTX 480;\nsee Table \\ref{tab:config} for the detailed GPGPU-Sim configuration parameters~\\cite{nvidia2009nvidia}.\nBesides, we enhance the baseline L1D and L2 caches with a XOR-based set index hashing technique~\\cite{nugteren2014detailed}, making it close to the real GPU device's configuration. \nSubsequently, we implement seven different warp schedulers: \n(1) \\texttt{GTO} (GTO scheduler with set-index hashing \\cite{nugteren2014detailed});\n(2) \\texttt{CCWS};\n(3) \\texttt{Best-SWL} (best static wavefront limiting);\n(4) \\texttt{statPCAL} (representative implementation of bypass scheme\\cite{li2015priority} that performs similar or better than \\cite{li2015locality,tian2015adaptive});\n(5) \\texttt{CIAO-P} (\\texttt{CIAO} with only redirecting memory requests of interfering warp to shared memory); \n(6) \\texttt{CIAO-T} (\\texttt{CIAO} with only selective warp throttling); and\n(7) \\texttt{CIAO-C} (\\texttt{CIAO} with both \\texttt{CIAO-T} and \\texttt{CIAO-P}).\nNote that \\texttt{CCWS}, \\texttt{Best-SWL}, and \\texttt{CIAO-P\/T\/C} leverage \\texttt{GTO} to decide the order of execution of warps. \n\\texttt{CCWS} and \\texttt{CIAO-T\/C} stall a varying number of warps depending on memory access characteristics monitored at runtime.\nIn contrast, \\texttt{Best-SWL} stalls a fixed number of warps throughout execution of a benchmark; we profile each benchmark to determine the number of stalled warps giving the highest performance for each benchmark; see column $\\rm N_{wrp}$ in Table~\\ref{tab:workload_charac}.\n\n\n\n\n\n\n\\noindent \\textbf{Benchmarks.} We evaluate a large collection of benchmarks from \\texttt{PolyBench}~\\cite{grauer2012auto}, \\texttt{Mars}~\\cite{he2008mars} and \\texttt{Rodinia}~\\cite{che2009rodinia}\nwhich are categorized into three classes: \n(1) large-working set (LWS), (2) small-working set (SWS), and (3) compute-intensive (CI). \nTable~\\ref{tab:workload_charac} tabulates chosen benchmarks and their characteristics.\n\n\n\n\n\n\\begin{figure*}\n\\centering\n\\subfloat[IPC]{\\label{fig:ATAX_back_IPC_fig}\\rotatebox{0}{\\includegraphics[width=0.33\\linewidth]{figs\/ATAX_back_IPC_fig}}}\n\\subfloat[Number of active warps]{\\label{fig:ATAX_back_AW_fig}\\rotatebox{0}{\\includegraphics[width=0.33\\linewidth]{figs\/ATAX_back_AW_fig}}}\n\\subfloat[Cache interference]{\\label{fig:ATAX_back_interf_fig}\\rotatebox{0}{\\includegraphics[width=0.33\\linewidth]{figs\/ATAX_back_interf_fig}}}\n\\caption{Comparison between \\texttt{Best-SWL}, \\texttt{CCWS} and \\texttt{CIAO-T} over time: \\texttt{ATAX} and \\texttt{Backprop}}\n\\label{fig:IPCtrace_ATAX_BACK}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\subfloat[IPC]{\\label{fig:SYRK_KMN_IPC}\\rotatebox{0}{\\includegraphics[width=0.33\\linewidth]{figs\/SYRK_KMN_IPC}}}\n\\subfloat[Number of active warps]{\\label{fig:SYRK_KMN_activewarp}\\rotatebox{0}{\\includegraphics[width=0.33\\linewidth]{figs\/SYRK_KMN_activewarp}}}\n\\subfloat[Cache interference]{\\label{fig:SYRK_KMN_interference}\\rotatebox{0}{\\includegraphics[width=0.33\\linewidth]{figs\/SYRK_KMN_interference}}}\n\\caption{Comparison of \\texttt{CIAO-T}, \\texttt{CIAO-P} and \\texttt{CIAO-C} over time: \\texttt{SYRK} and \\texttt{KMN}.}\n\\label{fig:IPCtrace_SYRK_seperate}\n\\end{figure*}\n\n\n\n\\ignore{\n\\begin{figure*}\n\\centering\n\\subfloat[IPC]{\\label{fig:ATAX_back_IPC_fig}\\rotatebox{0}{\\includegraphics[width=1\\linewidth]{figs\/ATAX_back_SYRK_KMN_IPC_fig}}}\n\\subfloat[Number of active warps]{\\label{fig:ATAX_back_AW_fig}\\rotatebox{0}{\\includegraphics[width=1\\linewidth]{figs\/ATAX_back_SYRK_KMN_AW_fig}}}\n\\subfloat[Cache Interference]{\\label{fig:ATAX_back_interf_fig}\\rotatebox{0}{\\includegraphics[width=1\\linewidth]{figs\/ATAX_back_SYRK_KMN_interf_fig}}}\n\\caption{Performance analysis of \\texttt{ATAX}, \\texttt{Backprop}, \\texttt{SYRK}, and \\texttt{KMN}.\n}\n\\label{fig:IPCtrace}\n\\end{figure*}\n}\n\n\\subsection{Performance Analysis}\n\\label{sec:analy}\n\\noindent\nFigure~\\ref{fig:overall_ipc}\nplots the IPC values with the seven warp schedulers and the \\textbf{geometric-mean} IPC values of three benchmark classes (LWS, SWS, and CI), respectively,\nnormalized to those with \\texttt{GTO}. \nOverall, \\texttt{CCWS}, \\texttt{Best-SWL}, \\texttt{statPCAL}, and \\texttt{CIAO-C} provide 2\\%, 16\\%, 24\\% and 56\\% higher performance than \\texttt{GTO}, respectively.\n\n\\texttt{GTO} performs worst among all evaluated schedulers, because, it shuffles only the order of executed warps and does not notably reduce cache thrashing caused by many active warps accessing small L1D cache. \nIn contrast, \\texttt{Best-SWL} outperforms \\texttt{GTO} as it throttles some warps, reducing the number of memory accesses to small L1D and thus cache thrashing. \nNonetheless, as \\texttt{Best-SWL} must decide the number of throttled warps before execution of a given application, \nit cannot effectively capture the optimal number of throttled warps varying within an application compared to warp schedulers that dynamically throttle the number of executed warps such as \\texttt{CCWS} and \\texttt{CIAO}.\nFor example, as \\texttt{ATAX} exhibits very dynamic cache access patterns at runtime, \n\\texttt{CCWS} outperforms \\texttt{Best-SWL} by 49\\%. \nNote that \\texttt{CCWS} gives notably lower performance than \\texttt{Best-SWL}\nespecially for CI benchmarks; considerably affecting its performance.\nThat is because running more active warps achieves higher performance for CI benchmarks, whereas \\texttt{CCWS} unnecessarily stalls some active warps to give a higher priority to a few warps exhibiting high data locality.\n\\texttt{statPCAL} gives up to 37\\% higher performance than \\texttt{Best-SWL} by up to 37\\% because \\texttt{statPCAL} offers higher TLP.\nSpecifically, when \\texttt{statPCAL} detects under-utilization of L2 and\/or main memory bandwidth, it activates throttled warps and makes these warp directly access the underlying memory (\\textit{i.e}\\onedot} \\def\\Ie{\\textit{I.e}\\onedot, bypassing L1D cache).\nDue to the long access latency and limited bandwidth of underlying memory, however,\n\\texttt{statPCAL} cannot significantly improve performance of LWS and SWS workloads such as \\texttt{KMN}, \\texttt{SYRK}, etc.\n\n\\texttt{CIAO-T} provides 32\\% and 34\\% higher performance than \\texttt{CCWS} and \\texttt{GTO}, respectively. \nFurthermore, \\texttt{CIAO-T} offers 22\\% higher performance than \\texttt{Best-SWL} for every benchmark except for \\texttt{SYR2K}, \\texttt{II}, and \\texttt{KMN} exhibiting static cache access patterns at runtime.\nBoth \\texttt{CIAO-T} and \\texttt{CCWS} dynamically stall some active warps at runtime, but\nour evaluation shows that it is often more effective to throttle the warps that considerably interfere with other warps than the warps with low potential of data locality\nas \\texttt{CCWS} does.\nFurthermore, for CI benchmarks, \\texttt{CIAO-T} offers as high performance as \\texttt{GTO} in contrast to \\texttt{CCWS}; \nrefer to our earlier comparison between \\texttt{GTO} and \\texttt{CCWS} for CI benchmarks.\n\n\\texttt{CIAO-P}\ngives 34\\% higher performance than \\texttt{GTO}. \n\\newedit{We observe that \\texttt{CIAO-P} offers the highest TLP among all seven warp schedulers, entailing 28\\% higher performance than \\texttt{CIAO-T} for SWS class benchmarks. This is because \\texttt{CIAO-P} fully utilizes the unused space of shared memory (cf. Figure~\\ref{fig:shmutil_fig}). }\nNonetheless, its benefits can be limited for LWS class benchmarks in which the redirected memory requests of interfering warps are often too intensive and thus thrash the shared memory as well.\nIn such a case, \\texttt{CIAO-T} can perform better than \\texttt{CIAO-P}, giving 48\\% and 66\\% higher performance than \\texttt{CIAO-P} and \\texttt{CCWS}, respectively, as shown in Figure \\ref{fig:IPC_fig}. \nLastly, \\texttt{CIAO-C}, which synergistically integrates \\texttt{CIAO-T} and \\texttt{CIAO-P}, provides 56\\%, 54\\%, 17\\% and 16\\% higher performance than \\texttt{GTO}, \\texttt{CCWS}, \\texttt{CIAO-T}, and \\texttt{CIAO-P}, respectively. \n\n\\begin{figure}[b]\n\\centering\n\\subfloat[Various epoches.]{\\label{fig:epoch}\\rotatebox{0}{\\includegraphics[width=0.48\\linewidth]{figs\/epoch}}}\n\\hspace{2pt}\n\\subfloat[Vairous high cut-off lines.]{\\label{fig:highcutoff}\\rotatebox{0}{\\includegraphics[width=0.48\\linewidth]{figs\/highcutoff}}}\n\\caption{Sensitivity analysis\n}\n\\label{fig:sensi_scheduler}\n\\end{figure}\n\n\\begin{figure*}\n\\centering\n\\subfloat[IPC comparison of varying L1D cache configurations.]{\\label{fig:IPC_fig_sens}\\rotatebox{0}\n{\\includegraphics[width=0.49\\linewidth]{figs\/IPC_fig_sens.eps}}}\n\\subfloat[IPC comparison of vayring DRAM bandwidths.]{\\label{fig:IPC_fig_sens1}\\rotatebox{0}\n{\\includegraphics[width=0.49\\linewidth]{figs\/IPC_fig_sens1.eps}}}\n\\caption{IPC of different L1D cache and DRAM configurations. \n}\n\\label{fig:sensi1_cache}\n\\end{figure*}\n\n\\subsection{Effectiveness of Interference Awareness}\n\\noindent\nFigure~\\ref{fig:IPCtrace_ATAX_BACK}\nshows the IPC, the number of active warps, and cache interference over time of \\texttt{ATAX} as a representative application that exhibits distinct execution phases in a single kernel execution. \nFor example, \\texttt{ATAX} exhibits two distinct execution phases.\nThe first phase comprised of the first 40-million instructions is very memory-intensive, whereas the second phase is very compute-intensive.\nFigure \\ref{fig:ATAX_back_IPC_fig} shows that \\texttt{CIAO-T} outperforms \\texttt{CCWS} and \\texttt{Best-SWL} for the first 40-million instructions executed. \n\\texttt{CIAO-T} exhibits higher performance during this phase because \\texttt{CIAO-T} more effectively reduces cache interference by throttling severely interfering warps, as shown in Figure \\ref{fig:ATAX_back_interf_fig}.\nAfter the first phase, \\texttt{ATAX} starts a compute-intensive phase, performing the computation by fully exploiting data locality on the GPU caches. \nAs \\texttt{Best-SWL} cannot capture this dynamics at runtime, it executes only 2 warps for the second phase execution of \\emph{ATAX}. \nIn contrast, \\texttt{CCWS} and \\texttt{CIAO-C} dynamically reduce the number of stalled warps as they observe fewer cache misses and less cache interference,\ngiving 4$\\times$ higher geometric-mean performance than \\texttt{Best-SWL}.\n\n\n\n\n\nWe choose \\texttt{Backprop} as a representative application that is very compute-intensive but also experiences many cache misses.\nFigure \\ref{fig:IPCtrace_ATAX_BACK} shows the performance change of \\emph{Backprop} over time.\n\\texttt{Best-SWL} and \\texttt{CIAO-T} provide 500 IPC on average. \nHowever, \\texttt{CCWS} notably degrades the performance, ranging from 320 to 150 IPC because \\texttt{CCWS} ends up giving a higher priority to warps with higher data locality and stalling more than 40 warps (or significantly reducing TLP).\nIn contrast, \\texttt{CIAO-T}, which offers performance similar to \\texttt{Best-SWL}, more selectively throttles warps than \\texttt{CCWS} (i.e., only 10$\\sim$20 most interfering warps), better preserving TLP. \n\n\n\n\\subsection{Sensitivity to Working Set Size}\n\\label{sec:tsa}\n\\noindent \\textbf{Small-working set.}\nFigure~\\ref{fig:IPCtrace_SYRK_seperate} shows the performance of three \\texttt{CIAO} schemes for \\texttt{SYRK} over time.\n\\texttt{SYRK} is a representative application with SWS. \nSpecifically, \nFigure~\\ref{fig:IPCtrace_SYRK_seperate}\nillustrates IPC, the number of active warps, and the number of cache conflicts over time of texttt{SYRK} over time with three \\texttt{CIAO} schemes. \nAs shown in Figure \\ref{fig:SYRK_KMN_IPC}, \\texttt{CIAO-P} offers higher IPC than \\texttt{CIAO-T} overall. \nThis is because, it can secure higher TLP (\\textit{cf}\\onedot} \\def\\Cf{\\textit{Cf}\\onedot Figure \\ref{fig:SYRK_KMN_activewarp}), whereas \\texttt{CIAO-T} alone hurts TLP by throttling many active warps. \nUsing the unused shared memory space, \\texttt{CIAO-P} can effectively reduce cache interference without sacrificing TLP in contrast to \\texttt{CIAO-T}. \nAs expected, \\texttt{CIAO-C} selectively stalls very few warps.\n\n\n\n\\noindent \\textbf{Large-working set.}\nFigure~\\ref{fig:IPCtrace_SYRK_seperate} also depicts the performance of three \\texttt{CIAO} schemes \\texttt{KMN}, a representative application with LWS. \nAs shown in Figure~\\ref{fig:SYRK_KMN_IPC}, \\texttt{CIAO-T} provides 50\\% higher IPC than \\texttt{CIAO-P}, and\n\\texttt{CIAO-C} always achieves the highest performance during the entire execution period amongst all three schemes. \nThis is because, as shown in Figure \\ref{fig:SYRK_KMN_interference}, \\texttt{CIAO-P} still suffers from severe shared memory interference \nas the amount of data requested by the partitioned warps exceeds the amount that shared memory can efficiently accommodate. \nIn contrast, \\texttt{CIAO-C} can better utilize shared memory by selectively throttling only the warps that cause severe interference. \n\n\n\n\n\n\n\n\n\n\n\\ignore{\n\\begin{figure*}\n\\centering\n\\subfloat[IPC comparison]{\\label{fig:IPC_fig_16112}\\rotatebox{0}\n{\\includegraphics[width=0.8\\linewidth]{figs\/IPC_fig_16112.eps}}}\n\\subfloat[Geo-mean IPC]{\\label{fig:IPC_fig2_16112}\\rotatebox{0}\n{\\includegraphics[width=0.18\\linewidth]{figs\/IPC_fig2_16112.eps}}}\n\\caption{IPC of 16KB L1D cache and 112KB shared mem. \n}\n\\label{fig:sensi_cache}\n\\end{figure*}\n}\n\n\n\n\n\n\\ignore{the highest reduction in L2 miss rate comes from LRR + CIAO -- compared to LRR and LRR + \\texttt{CCWS} by 52.7\\% and 48.9\\%, respectively. This dramatic decrease in miss rate results from a combined effect of the active warp number throttling and the consideration of cache interference upon warp scheduling. On the other hand, \\texttt{GTO} + CIAO reduces the L2 miss rate by 28.3\\% and 13.8\\% over \\texttt{GTO} and \\texttt{GTO} + \\texttt{CCWS}, respectively. Due to the high number of active warps, even restricting to the oldest warps, \\texttt{GTO} still allows too much data to be contained in L2 cache. Even though, \\texttt{CCWS} can further alleviate the L2-level cache interference by strictly limiting the active warp number, warps with high potential of data locality, which are prioritized in \\texttt{CCWS}, can still contend with each other. The scheduling policy of CIAO successfully exploits this critical observation regarding the potential for further reduction in L2-level interference.\n}\n\n\n\\ignore{\n\\begin{figure}\n\\centering\n\\includegraphics[width=1\\linewidth]{figs\/L1Dynenergy_fig.eps}\n\\caption{L1 total energy analysis. \n}\n\\label{fig:L1Dynenergy_fig}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1\\linewidth]{figs\/L2dynenergy_fig.eps}\n\\caption{L2 total energy analysis. \n}\n\\label{fig:L2dynenergy_fig}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1\\linewidth]{figs\/DRAMdynenergy_fig.eps}\n\\caption{DRAM dynamic energy analysis. \n}\n\\label{fig:DRAMdynenergy_fig}\n\\end{figure}\n}\n\n\n\\subsection{Sensitivity Study}\n\\label{subsubsec:sensi}\n\\noindent \\textbf{Epoch value.} \nFigure~\\ref{fig:epoch} shows the effect of varying \\texttt{high-cutoff} epoch values on the IPC for all the memory-intensive workloads. \nAs we increase the epoch from 1K to 50K instructions, the change in IPC is within 15\\%. \nNote that different workloads can achieve best performance with different epoch values. \nThat is because epoch determines the frequency of checking cache interference for \\texttt{CIAO}.\nA shorter epoch provides fast response to cache interference, while a longer epoch can more accurately detect the warp causing most interference. \nTaking this trade-off into account, we choose 5K instructions as \\texttt{high-cutoff} epoch value. \nAn adaptive scheme can be future work.\n\n\n\n\n\n\\noindent \\textbf{High-cutoff threshold.} \nFigure~\\ref{fig:highcutoff} depicts performance corresponding to different \\texttt{high-cutoff} thresholds, \nwhere the \\texttt{low-cutoff} threshold is fixed to half of it.\nAll benchmarks show steady performance within 5\\% change during the entire execution period.\nThis is because our \\texttt{CIAO} throttles the active warps causing most interference, which can easily exceed the current thresholds we set. 1\\% is chosen in the paper.\n\n\n\\newedit{\n\\noindent \\textbf{L1D cache\/DRAM configurations.} Figure~\\ref{fig:sensi1_cache} illustrates the performance of LWS and SWS workloads by configuring various L1D cache\/DRAM design parameters: \n(1) \\texttt{GTO};\n(2) \\texttt{GTO-cap} (\\texttt{GTO} but increase L1D cache capacity to 48 KB and reduce shared memory size to 16 KB);\n(3) \\texttt{GTO-8way} (\\texttt{GTO} but increase L1D cache associativity to 8 way);\n(4) \\texttt{statPCAL-2X} (\\texttt{statPCAL} but double DRAM bandwidth from 177 GB\/s to 340 GB\/s);\n(5) \\texttt{CIAO-C};\n(6) \\texttt{CIAO-C-2X} (\\texttt{CIAO-C} but double DRAM bandwidth).\nAs shown in Figure \\ref{fig:IPC_fig_sens}, while increasing L1D cache capacity (\\texttt{GTO-cap}) and associativity (\\texttt{GTO-8way}) can effectively improve the overall performance by 108\\% and 51\\% compared to \\texttt{GTO}, \\texttt{CIAO-C} still outperforms \\texttt{GTO-cap} and \\texttt{GTO-8way} by 14\\%, and 57\\%, respectively. This is because, \\texttt{GTO-cap} and \\texttt{GTO-8way} cannot fully eliminate cache interference, as they cannot distinguish the requests between interfering and interfered warps and effectively isolate them. On the other hand, while \\texttt{statPCAL-2X} can benefit from the increased DRAM bandwidth, bypassing requests to underlying DRAM still suffers from long DRAM delay as the latency of DRAM access is much longer than that of L1D cache access. Hence, as shown in Figure \\ref{fig:IPC_fig_sens1}, \\texttt{CIAO-C-2X} outperforms \\texttt{statPCAL-2X} by 16\\%, on average.\n}\n\n\\ignore{\n\\noindent \\textbf{Varying L1D cache sizes.} \nFigure~\\ref{fig:sensi1_cache} \nillustrates the performance of all workloads for the various L1D cache configurations shown in Table~\\ref{tab:config}. \nFor 48KB L1D cache and 16KB shared memory, \\texttt{CIAO-C} gives 29\\%, 27\\%, 13\\%, and 12\\% higher IPC than \\texttt{GTO}, \\texttt{CCWS}, \\texttt{Best-SWL}, and \\texttt{statPCAL}, respectively.\nSince 48KB L1D cache is much larger than the default 16KB size, the performance of \\texttt{GTO} improves greatly and leaves less room for improvement by \\texttt{CIAO}\\xspace. \n}\n\n\\subsection{Overhead Analysis}\n\\noindent \nImplementing the interference detector, \\texttt{CIAO} leverages the VTA structure originally proposed by \\texttt{CCWS}~\\cite{rogers2012cache}, but employs only 8 VTA entries for each warp (\\textit{i.e}\\onedot} \\def\\Ie{\\textit{I.e}\\onedot, half of the VTA entries that \\texttt{CCWS} uses).\nUsing CACTI 6.0~\\cite{muralimanohar2009cacti}, we estimate that the area of one VTA structure is only 0.65 $mm^2$ for 15 SMs, which accounts for only 0.12\\% of the total chip size of NVIDIA GTX480 (529 $mm^2$~\\cite{geforce-gtx-480}).\nIn addition, \\texttt{CIAO} uses 48 registers as VTA-hit counters (one for each warp). \nSince each VTA-hit counter resets at the start of each kernel, a 32-bit counter is sufficient to prevent its overflow. \nThe interference and pair lists are implemented with SRAM arrays indexed by WIDs. \nSince the total number of active warps in a CTA does not exceed 64 (usually, 48 active warps in each SM), we configure the interference and pair lists with 64 entries. \nEach entry of the interference list requires 8 (= 6+2) bits to store one warp index and saturation counter value, while each entry of pair list requires 12 (=6 + 6) bits to store two warp indices.\nUsing CACTI 6.0, we estimate that the combined area of the VTA-hit counters, interference list, and pair lists is 549 $um^{2}$ per SM (8235 $um^{2}$ for 15 SMs).\nOn the other hand, Equation \\ref{eq:irs} is implemented with a few adders, a shifter, and a comparator, which also requires very low cost (2112 gates). \nFor our shared memory modification, the translation unit, multiplexer and MSHR only need 4500 gates and 64B storage per SM.\nWe also track the power consumption of new components employed in \\texttt{CIAO} by leveraging GPUWattch~\\cite{leng2013gpuwattch}, \nwhich reveals the average power is around 79mW. \nOverall, \\texttt{CIAO} improves the performance by more than 50\\% with a negligible area cost (less than 2\\% of the total GTX480 chip area) and power consumption (only 0.3\\% of GTX480 overall power).\n\n\\subsection{Cache Interference Detection}\n\\label{sec:schedule}\n\n\n\\noindent \\textbf{Estimation of cache interference.}\nA level of cache interference experienced by a warp can be quantified by \nan Individual Re-reference Score (IRS)\nwhich can be expressed by: \n\n\n\\begin{equation}\n\\label{eq:irs}\nIRS_i = \\frac{F^i_{VTA-hits}}{N_{executed-inst}\/N_{active-warp}}\n\\end{equation}\n\n\\begin{figure}[b]\n\\centering\n\\includegraphics[width=1\\linewidth]{figs\/vta.eps}\n\\caption{Microarchitecture adaptation for \\texttt{CIAO}\\xspace.}\n\\label{fig:vta}\n\\end{figure}\n\n\n\\noindent where $i$ is active warp number, $F^i_{VTA-hits}$ is the number of VTA hits for warp $i$, $N_{executed-inst}$ is the total number of executed instructions, and $N_{active-warp}$ is the number of active warps running on an SM, respectively. \n$IRS_i$ represents VTA hits per instruction (\\textit{i.e}\\onedot} \\def\\Ie{\\textit{I.e}\\onedot, intensity of VTA hits) for warp $i$.\nHigh $IRS_i$ indicates warp $i$ has experienced severe cache interference in a given epoch.\nBased on $IRS_i$, \\texttt{CIAO}\\xspace (1) decides whether it isolates warps interfering with warp $i$, (2) stalls these interfering warps, or (3) reactivates the stalled warps.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=1\\linewidth]{figs\/shm.eps}\n\\caption{GPU on-chip memory structure adaptation.}\n\\label{fig:shm}\n\\end{figure*}\n\n\n\\noindent \\textbf{Decision thresholds.} \nFor these aforementioned three decisions we introduce two threshold values: (1) \\texttt{high-cutoff} and (2) \\texttt{low-cutoff}. \n$IRS_i$ over \\texttt{high-cutoff} indicates that warp $i$ has experienced severe cache interference.\nSubsequently, \\texttt{CIAO}\\xspace decides to isolate or stall the warp that most recently and severely interfered with warp $i$. \n$IRS_i$ below \\texttt{low-cutoff} often indicates that warp $i$ has experienced light cache interference and\/or completed its execution.\nThen, \\texttt{CIAO}\\xspace decides to reactivate previously stalled warps or\nredirect memory requests of these warps back to L1D cache.\nAs these two thresholds influence the efficacy of \\texttt{CIAO}\\xspace, we sweep these two values, evaluate diverse memory-intensive applications, and determine that \\texttt{high-cutoff} and \\texttt{low-cutoff}, which minimize cache interference and maximize performance, are 0.01 and 0.005, respectively.\nSee Section~\\ref{subsubsec:sensi} for our sensitivity analysis.\n\n\n\\noindent \\textbf{Epochs.} \nAs $IRS_i$ changes over time, \\texttt{CIAO}\\xspace should track the latest $IRS_i$ and compare it against \\texttt{high-cutoff} and \\texttt{low-cutoff} to precisely determine whether a warp needs to be isolated, stalled, or reactivated. However, the update of $IRS_i$ calculation consumes more than 6 cycles, which can be on the critical path of performance.\nTo this end, \\texttt{CIAO}\\xspace divides the execution time into \\texttt{high-cutoff} and \\texttt{low-cutoff} epochs, respectively. \nAt the end of each \\texttt{high-cutoff} (or \\texttt{low-cutoff}) epoch, \\texttt{CIAO}\\xspace updates $IRS_i$ and compares it against \\texttt{high-cutoff} (or \\texttt{low-cutoff}).\nThe \\texttt{low-cutoff} epoch should be shorter than the \\texttt{high-cutoff} epoch because of the following reasons. \nAs preserving high TLP is a key to improve GPU performance, \\texttt{CIAO}\\xspace attempts to minimize a negative effect of stalling warps by reactivating stalled warps as soon as these warps start not to notably interfere with other warps at runtime. \nTo validate this strategy, we sweep \\texttt{high-cutoff} and \\texttt{low-cutoff} epoch values, evaluate diverse memory-intensive applications, and determine that the best \\texttt{high-cutoff} and \\texttt{low-cutoff} epoch values are every 5000 and 100 instructions, respectively.\nSee Section~\\ref{subsubsec:sensi} for our in-depth sensitivity analysis.\n\n\\noindent \\textbf{Microarchitecture support.}\nFigure~\\ref{fig:vta} depicts the necessary hardware, which is built upon the existing VTA organization~\\cite{rogers2012cache}, to implement a cache interference detector.\n\n\nTo capture different levels of cache interference experienced by individual warps, we implement a VTA-hit counter per warp and a total instruction counter per SM (\\texttt{VTACount0-k} and \\texttt{Inst-total} in the figure) atop a VTA.\nEach VTA-hit counter records the number of VTA hits for each warp, and the total instruction counter tracks the total number of instructions executed by a given SM (\\textit{i.e}\\onedot} \\def\\Ie{\\textit{I.e}\\onedot, $N_{executed-inst}$ in Eq.(\\ref{eq:irs})). \nTo compare $IRS_i$ against \\texttt{high-cutoff} and \\texttt{low-cutoff}, we implement the cutoff testing unit which can be implemented by registers, a shifter, and simple comparison logic. \nLastly, we implement the samplers to count the number of executed instructions and determine whether or not the end of a \\texttt{high-cutoff} or \\texttt{low-cutoff} epoch has been reached.\n\n\nTo manage the information related to tracking interfering warps for each warp, we implement the interference list.\nEach entry is indexed by WID of a given warp and stores a 6-bit WID of an interfering warp and a 2-bit saturation counter (\\texttt{C} in the figure). \nWhen a VTA hit occurs, the corresponding entry of interference list is updated, as described in Section~\\ref{sec:interference_detection}.\n\\texttt{CIAO}\\xspace checks the interference list for warp $i$ whenever it needs to isolate or stall an interfering warp based on $IRS_i$.\nTo facilitate this, we also augment a 1-bit active flag (\\texttt{V}) and 1-bit isolation flag (\\texttt{I}) with each ready warp entry in the warp list (\\textit{i.e}\\onedot} \\def\\Ie{\\textit{I.e}\\onedot, a component of warp scheduler). \nUsing \\texttt{V} and \\texttt{I} bits, the warp scheduler can identify whether a given warp is in active (\\texttt{V}=\\texttt{1}, \\texttt{I}=\\texttt{0}), isolated (\\texttt{V}=\\texttt{1}, \\texttt{I}=\\texttt{1}), or stalled state (\\texttt{V}=\\texttt{0}). \n\n\nWe also implement a \\textit{pair list}. \nEach entry is indexed by the WID of a warp at the front of the warp list and composed of two fields \nto record which interfered warp triggered to redirect memory requests of the warp or stall the warp in the past.\nSuppose that warp $i$ is at the front of the warp list.\nBased on WIDs from the first or second field of the entry indexed by warp $i$, \n\\texttt{CIAO}\\xspace checks $IRS_k$ where $k$ is the WID of the interfered warp that previously triggered to either redirect memory requests of warp $i$ or stalling warp $i$. \nThen \\texttt{CIAO}\\xspace decides whether it reactivates warp $i$ or redirects memory requests of warp $i$ back to L1D cache based on $IRS_k$. \nFor example, as \\texttt{W0} is severely interfered by \\texttt{W1}, \\texttt{CIAO}\\xspace decides to redirect memory requests of \\texttt{W1} to unused shared memory space.\nThen \\texttt{W0} is recorded in the first field of the entry indexed by \\texttt{W1} and \\texttt{I} associated with \\texttt{W1} is set, as depicted in Figure~\\ref{fig:vta}. \nSubsequently, \\texttt{W1} begins to send memory requests to the shared memory, but \\texttt{CIAO}\\xspace observes that \\texttt{W1} also severely interferes with \\texttt{W3} that sends its memory requests to the shared memory.\nAs \\texttt{CIAO}\\xspace decides to stall \\texttt{W1}, \\texttt{W3} is recorded in the second field of the entry indexed by \\texttt{W1} and \\texttt{V} associated with \\texttt{W1} is cleared. \nWhen \\texttt{CIAO}\\xspace needs to reactivate \\texttt{W1} later, the second field of the pair list entry and \\texttt{V} corresponding to \\texttt{W1} are cleared to \ninform the warp scheduler of the event that the warp is active. \nWhen \\texttt{CIAO}\\xspace needs to make \\texttt{W1} send its memory request back to L1D cache, the corresponding field in the pair list entry and \\texttt{I} are cleared.\nSee Section~\\ref{sec:pat} for more details on the pair list.\n\n\n\n\\subsection{Shared Memory Architecture}\n\\label{sec:shared_mem_arch}\n\\noindent\nFigure~\\ref{fig:shm}a and b illustrate \\texttt{CIAO}\\xspace on-chip memory architecture and its data placement layout, respectively. \n\n\n\\noindent \\textbf{Determination of unused shared memory space.}\nOne challenge to utilize unused shared memory space is that shared memory is managed by programmers and the used amount of shared memory space varies across implementations of a kernel. \nTo make \\texttt{CIAO}\\xspace on-chip memory architecture transparent to programmers, we leverage the existing SMMT structure to determine the unused shared memory space. \nWhen a CTA is launched, \\texttt{CIAO}\\xspace checks the corresponding SMMT entry to determine the amount of unused shared memory space (\\textit{cf}\\onedot} \\def\\Cf{\\textit{Cf}\\onedot Section~\\ref{sec:sm_arch}). \nThen, \\texttt{CIAO}\\xspace inserts a new entry in the SMMT with the start address and size of unused shared memory to reserve the space for storing 128-byte data blocks and tags. \n\n\n\\noindent \\textbf{Placement of tags and data.}\nIn contrast to L1D cache, shared memory does not have a separate memory array to accommodate tags~\\cite{gebhart2012unifying}. \nIn this work, instead of employing an additional tag array, we propose to place both 128-byte data blocks and their tags into the shared memory.\nThis is to minimize the modification of the current on-chip memory structure architected to be configured as both L1D cache and shared memory. \nAs shown in Figure~\\ref{fig:shm}b, we partition 32 shared memory banks into two bank groups and stripe a 128-byte data block across 16 banks within one bank group. \nEach 128-byte data block can be accessed in parallel since each shared memory bank allows 64-bit accesses~\\cite{nvidia2012nvidia}.\nSince a tag and a WID require only 31 bits (= 25 + 6 bits), two tags can be placed in a single bank which is different from banks storing the corresponding data blocks.\nThen 32 tags can be grouped together to better utilize a row of one bank group (\\textit{i.e}\\onedot} \\def\\Ie{\\textit{I.e}\\onedot, 16 banks). \nThis design strategy, which puts a tag and the corresponding data block into two different bank groups, shuns bank conflicts and thus allows accesses of a tag and a data block in parallel.\nFurthermore, we only use the unused shared memory space as direct-mapped cache so that a pair of a 128-byte data block and the corresponding tag can be accessed with a single shared memory access.\n\n\n\\noindent \\textbf{Address translation unit.}\nAs shown in Figure~\\ref{fig:shm}b, we introduce a hardware address translation unit in front of shared memory to determine where a target 128-byte data block and its tag exist in the shared memory. \nIn practice, a global memory address can be decomposed by cache-related information such as a tag, block index and byte offset. \nHowever, as the usage of shared memory can be varying based on the needs of each CTA, we put an 8-bit mask into the translation unit to decide how many rows will be used for each CTA at runtime. \nFigure~\\ref{fig:shm}c shows how our translation unit determines locations of a target data block and its tag; \nthe data block address (of shared memory) consists of four fields, the byte offset (``\\texttt{F}''), bank index (``\\texttt{B}''), bank group (``\\texttt{G}''), and row index (``\\texttt{R}''), which are presented from LSB to MSB. \nSpecifically, we have 8-byte rows per bank, 16 banks per group, two bank groups and 256 rows (at most), \nwhich in turn 3, 4, 1, and 8 bits for \\texttt{F}, \\texttt{B}, \\texttt{G} and \\texttt{R}, respectively. \nThe remaining bits (16 bits in this example) are used as part of the tag. \nNote that our tags also contain 6-bit WID and 9-bit data block index as the number of cache lines required can be greater than the number of rows. \n\nIn \\texttt{CIAO}\\xspace, one row within a bank group can hold 32 tags since a physical row per bank contains two tags. \nThat is, the actual position of a tag can be indicated by 5 bits (\\textit{i.e}\\onedot} \\def\\Ie{\\textit{I.e}\\onedot, 1 \\texttt{F} and 4 \\texttt{B} bits), \nwhich are also used for the row index of the corresponding data block. \nTo access a data block and the corresponding tag in parallel, \\texttt{G} of the data block will be flipped and assigned to such tag's 5 bits as a significant bit. \nThe remaining \\texttt{R} bits are assigned to the row index of the target tag. \nNote that, as shown in the figure, the start of index for both a data block and a tag can be rearranged by considering the data block and tag offset registers, \nwhich are used to adapt the unused shared memory size allocated for cache.\n\n\n\\noindent \\textbf{Datapath connection.}\nWhen we leverage unused shared memory as cache, we need a datapath between shared memory and L2 cache. \nSince the shared memory is disconnected from the global memory in the conventional GPU, \nwe need to adapt the on-chip memory structure, which is partitioned between L1D cache and shared memory, to share some resources of the L1D cache with the shared memory (\\textit{e.g}\\onedot} \\def\\Eg{\\textit{E.g}\\onedot, datapath to L2 cache, MSHR, etc.). \nAs illustrated in Figure~\\ref{fig:shm}a, \na multiplexer is implemented to connect the write queue (WQ) and response queue (RespQ) to either L1D cache or shared memory. \nThe \\texttt{CIAO}\\xspace cache control logic controls the multiplexer based on the isolation flag bit (\\texttt{I}) and the result of checking cache tags associated with accessing L1D cache or shared memory serving as cache.\nWe also augment an extra field with each MSHR entry to store the shared memory address of a memory request from the aforementioned address translation unit.\nOnce the shared memory issues a fill request after a miss, the request reserves one MSHR entry by filling in its global and translated shared memory addresses. \nIf the response from L2 cache matches the global address recorded in the corresponding MSHR entry, the filling data can be directly stored in the shared memory based on the translated shared memory address.\n\n\n\\noindent \\textbf{Performance optimization and coherence.}\nWhen \\texttt{CIAO}\\xspace redirects memory requests of an interfering warp from L1D cache to shared memory, the shared memory does not have any data. \nThis can incur (1) performance degradation because of cold misses and (2) some coherence issues.\nTo address these two issues,\nwhen \\texttt{CIAO}\\xspace needs to access the shared memory, the cache controller first checks the tag array of L1D cache.\nIf a target data resides in L1D cache (not in shared memory), the L1D cache will evict the data directly to the response queue, \nwhich is used to buffer the fetched data from L2 cache and invalidate the corresponding cache line in L1D cache. \nNote that checking the tag array and accessing L1D cache are serialized as described in Section~\\ref{sec:sm_arch}.\nMeanwhile, the shared memory issues a fill request to MSHR, as the shared memory does not have the data yet. \nDuring this process, the target data will be directly fetch from the response queue to the shared memory (\\textit{cf}\\onedot} \\def\\Cf{\\textit{Cf}\\onedot Figure~\\ref{fig:shm}a) \nIn this way, we naturally migrate data from L1D cache to shared memory, hiding the penalty of cold cache misses and coherence issues.\n\n\\begin{algorithm}[t]\n\\scriptsize\n\\DontPrintSemicolon\ni := getWarpToBeScheduled()\\;\nInstNo := getNumInstructions()\\;\nActiveWarpNo := getNumActiveWarp()\\;\n\\uIf{Warp(i).V == 0 \\textbf{and} end of low cut-off epoch}{\n\t\\tcc{Warp(i) is stalled}\n\tk := Pair\\_List[i][1]\\;\n\t$IRS_k$ := $^{VTAHit[k]}\/_{InstNo\/ActiveWarpNo}$\\;\n \\uIf{$IRS_k$ \\textgreater low-cutoff \\textbf{and} Warp(k) needs executing}{\n \\textbf{continue}\\;\n }\n \\uElse{\n Warp(i).V := 1\\;\n Pair\\_List[i][1] := -1 \\tcp{cleared} } }\n\\uElseIf{Warp(i).I == 1 \\textbf{and} end of low cut-off epoch}{\n\t\\tcc{Warp(i) redirects to access shared memory}\n\tk := Pair\\_List[i][0]\\;\n\t$IRS_k$ := $^{VTAHit[k]}\/_{InstNo\/ActiveWarpNo}$\\;\n \\uIf{$IRS_k$ \\textgreater low-cutoff \\textbf{and} Warp(k) needs executing}{\n \\textbf{continue}\\;\n }\n \\uElse{\n Warp(i).I := 0\\;\n Pair\\_List[i][0] := -1 \\tcp{toggling} } }\n\\uIf{Warp(i).V == 1 \\textbf{and} end of high cut-off epoch }{\n\t\\tcc{Warp(i) is active}\n\t$IRS_i$ := $^{VTAHit[i]}\/_{InstNo\/ActiveWarpNo}$\\;\n\tj := Interference\\_List[i]\\;\n\t\\uIf{$IRS_i$ \\textgreater high-cutoff \\textbf{and} $j$ != $i$ }{\n\t\\uIf{ Warp(j).I == 1}{\n\t\tWarp(j).V := 0\\;\n\t\tPair\\_List[j][1] := i\\;\n\t}\n\t\\uElseIf{ Warp(j).I == 0}{\n\t\tWarp(j).I := 1\\;\n\t\tPair\\_List[j][0] := i\\;\n\t}\n \n \n \n \n\t}\n}\n\\caption{\\texttt{CIAO}\\xspace scheduling algorithm}\n\\label{algo:CIAO}\n\\end{algorithm}\n\n\\subsection{Putting It All Together}\n\\label{sec:pat}\n\\noindent\nAlgorithm~\\ref{algo:CIAO} describes how \\texttt{CIAO}\\xspace schedules warps. \nFor every \\texttt{low-cutoff} epoch, the warp at the front of the warp list (\\textit{e.g}\\onedot} \\def\\Eg{\\textit{E.g}\\onedot, warp $i$), is examined \nto decide whether \\texttt{CIAO}\\xspace redirects memory requests of warp $i$ back to L1D cache or reactivate warp $i$.\nMore specifically, \\texttt{CIAO}\\xspace first checks the first or second field of the \\textit{pair list} entry corresponding to warp $i$. \nOnce \\texttt{CIAO}\\xspace confirms that either \\texttt{CIAO}\\xspace previously redirected memory requests of warp $i$ to shared memory or stalled warp $i$ because warp $i$ severely interfered with another warp (\\textit{e.g}\\onedot} \\def\\Eg{\\textit{E.g}\\onedot, warp $k$),\nit redirects the memory requests of warp $i$ back to L1D cache or reactivate warp $i$,\nunless the following two conditions are satisfied: \n(1) $IRS_k$ is still higher than \\texttt{low-cutoff} and (2) warp $k$ has not completed its execution.\n\nEvery \\texttt{high-cutoff} epoch, \\texttt{CIAO}\\xspace examines $IRS_i$.\nIf warp $i$ is in the active warp list and $IRS_i$ is higher than \\texttt{high-cutoff}, \n\\texttt{CIAO}\\xspace looks up the \\textit{interference} list to determine which warp has most severely interfered with warp $i$. \nOnce \\texttt{CIAO}\\xspace determines the most interfering warp (\\textit{e.g}\\onedot} \\def\\Eg{\\textit{E.g}\\onedot, warp $j$) for warp $i$, \n\\texttt{CIAO}\\xspace checks whether it has redirected memory requests of warp $j$ to shared memory or stalled warp $j$. \nIf \\texttt{CIAO}\\xspace sees that warp $j$ has still sent memory requests to L1D cache, it isolates warp $j$, redirects memory requests of warp $j$ to shared memory,\nand records warp $i$ in the first field of the pair list entry corresponding to warp $j$ to indicate that warp $i$ has triggered to redirect memory requests of warp $j$. \nIf \\texttt{CIAO}\\xspace has already redirected memory requests of warp $j$, then \n\\texttt{CIAO}\\xspace starts to stall warp $j$ and records warp $i$ in second field of the pair list entry corresponding to warp $j$.\nThis record can be referenced when \\texttt{CIAO}\\xspace decides to reactivate warp $i$ in future.\n\n\n\n\n\n\n\n\n\\section{#1}}\n\\newcommand{\\subsect}[1]{\\subsection{#1}}\n\\newcommand{\\subsubsect}[1]{\\subsubsection{#1}}\n\\newcommand{\\mysect}[1]{\\subsect{#1}}\n\n\n\\newtheorem{ourtask}{Task}\n\n\\subsection{Cache Interference Detection}\n\\label{sec:interference_detection}\n\\noindent\nAs introduced in Section~\\ref{sec:interfere}, some warps incur more severe cache interference than other warps (\\textit{i.e}\\onedot} \\def\\Ie{\\textit{I.e}\\onedot, non-uniform cache interference). \nHowever, it is non-trivial to capture such non-uniform interference occurring during the execution of applications \nat compile time \\cite{chenadaptive}.\nThus, we need to determine severely interfering and interfered warps at runtime.\n\n\nAt run time, we may track severely interfered warps, leveraging a VTA structure (\\textit{cf}\\onedot} \\def\\Cf{\\textit{Cf}\\onedot Section~\\ref{sec:vta}).\nA na\\\"ive way to determine severely interfering warps for each warp, however, demands a high storage cost, \nbecause each warp needs to keep track of cache misses incurred by all other $n-1$ warps.\nThis in turn requires a storage structure with $n(n-1)$ entries where $n$ is the number of active warps per SM (\\textit{i.e}\\onedot} \\def\\Ie{\\textit{I.e}\\onedot, 48 warps). \nSearching for a cost-effective way to determine severely interfering warps, we exploit our following observation on an important characteristic of cache interference.\n\n\\begin{figure}\n\\centering\n\\subfloat[]{\\label{fig:unbalance_arrow}\\rotatebox{0}{\\includegraphics[width=0.34\\linewidth]{figs\/unbalance_arrow}}}\n\\subfloat[]{\\label{fig:kmeans_un1}\\rotatebox{0}{\\includegraphics[width=0.61\\linewidth]{figs\/kmeans_un1}}}\n\n\\subfloat[]{\\label{fig:sat_counter}\\rotatebox{0}{\\includegraphics[width=0.92\\linewidth]{figs\/sat_counter}}}\n\\caption{(a) Warps interfering with warp W34 and their interference frequency.\n(b) Min and max interference frequencies experienced by each warp and each evaluated workload. (c) Interference detection example.}\n\\end{figure}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=1\\linewidth]{figs\/toggle_thrott_exam.eps}\n\\caption{\\texttt{CIAO}\\xspace execution flow.}\n\\label{fig:thrott_exam}\n\\end{figure*}\n\n\n\n\\newedit{ Figure~\\ref{fig:unbalance_arrow} shows that \\texttt{W32} interferes with \\texttt{W34}, more than two thousand times, whereas some warps (\\textit{e.g}\\onedot} \\def\\Eg{\\textit{E.g}\\onedot, \\texttt{W2}) do not interfere with \\texttt{W34} at all in \\texttt{KMEANS}~\\cite{che2009rodinia}; \nwe observe a similar trend on cache interference in all other benchmarks that we tested (cf. Figure~\\ref{fig:kmeans_un1}). }\nObserving such an interference characteristic, we propose to track only the most recently and frequently interfering warp for each warp.\nThis significantly reduces the storage cost required to track every interfering warp for each warp.\nSpecifically, \\texttt{CIAO}\\xspace keeps a small memory structure denoted by \\textit{interference list} \nwhere each entry is indexed by the WID of a currently executed warp.\n\nTo track the most recently and frequently interfering warp for a currently executed warp, \nwe may augment each list entry with a 2-bit saturation counter.\nFigure~\\ref{fig:sat_counter} illustrates how \\texttt{CIAO}\\xspace utilizes the counter to track an interfering warp. \nSuppose that a previously executed warp (\\texttt{W32}) interfered with a currently executed warp (\\texttt{W34}), \nThat is, \\texttt{W32} is an interfering WID and \\texttt{W34} is an interfered WID.\nSubsequently, the interfering WID is stored in the list entry indexed by the interfered WID,\nand the counter in the list entry is set to \\texttt{00};\nthe interfering WID is provided by a VTA entry field that tracks which warp incurred the last eviction (\\textit{cf}\\onedot} \\def\\Cf{\\textit{Cf}\\onedot Section~\\ref{sec:vta}).\n\n\nWhenever \\texttt{W32} interferes with \\texttt{W34} (not shown in the figure), the counter is incremented by 1.\nSuppose that the counter has already reached \\texttt{11} (\\redcircled{\\small{1}}) at a given cycle. \nWhen another warp (\\texttt{W42}) interferes with warp \\texttt{W34} in a subsequent cycle, the counter is decremented by 1 (\\redcircled{\\small{2}}). \nThen, if warp \\texttt{W32} interferes with \\texttt{W34} again, the counter is incremented by 1 (\\redcircled{\\small{3}}). \nThe interfering WID in the list entry is replaced with the most recent interfering WID only when its saturation counter is decreased to \\texttt{00}, \nso that the warp with most frequent cache interference can be kept in the interference list. \n\n\n\n\\subsection{CIAO On-Chip Memory Architecture}\n\\label{sec:warp_partitioning}\n\\noindent\nAn effective way to reduce cache interference is to isolate cache accesses of interfering warps from those of interfered warps \nafter partitioning the cache space and allocating separate cache lines to the interfering warps. \nPrior work proposed various techniques to partition the cache space for CPUs (\\textit{e.g}\\onedot} \\def\\Eg{\\textit{E.g}\\onedot, \\cite{qureshi2006utility, srikantaiah2008adaptive}). \nHowever, the size of L1D cache is insufficient to apply such techniques for GPUs, \nas the number of GPU threads sharing L1D cache lines is very large, compared with that of CPU threads. \nFor example, only two or three cache lines can be allocated to each warp, if we apply a CPU-based cache partitioning technique to the L1D cache of GTX480.\nSuch a small number of cache lines per warp can even worsen cache thrashing. \n\n\n\\newedit{Meanwhile, we observe that programmers prefer L1D cache rather than shared memory for programming simplicity and the limited number of running GPU threads constrains the usage of shared memory, leading to \na large fraction of shared memory unused \n(\\textit{cf}\\onedot} \\def\\Cf{\\textit{Cf}\\onedot $F_{smem}$ of Table~\\ref{tab:workload_charac} in Section~\\ref{sec:method}).\nThis agrees to prior work's analysis~\\cite{hayes2014unified, virtualthread}. }\nExploiting such unused shared memory space, \nwe propose to redirect memory requests of severely interfering warps to the unused shared memory space.\n\n\nAs there is no cache interference at the beginning of kernel execution, \nmemory requests of all the warps are directed to L1D cache, as depicted in Figure~\\ref{fig:thrott_exam}a. \nHowever, as the kernel execution progresses, cache accesses begin to compete one another to acquire specific cache lines in L1D cache. \nAs the intensity of cache interference exceeds a threshold, \\texttt{CIAO}\\xspace determines severely interfering warps (\\textit{cf}\\onedot} \\def\\Cf{\\textit{Cf}\\onedot Section~\\ref{sec:interference_detection}).\nSubsequently, \\texttt{CIAO}\\xspace redirects memory requests of these interfering warps to unused shared memory space, \nisolating the interfering warps from the interfered warps in terms of cache accesses, as depicted in Figure~\\ref{fig:thrott_exam}b.\nThis in turn can significantly reduce cache contentions without throttling warps (\\textit{i.e}\\onedot} \\def\\Ie{\\textit{I.e}\\onedot, hurting TLP). \n\\newedit{After the redirection, the memory requests are forwarded from L1D cache to shared memory but the data may already present in the L1D cache (\\textit{cf}\\onedot} \\def\\Cf{\\textit{Cf}\\onedot W3\/D3 in Figure~\\ref{fig:thrott_exam}b). To guarantee cache coherence between L1D cache and shared memory, single data copy needs to be exclusively stored in either shared memory or L1D cache. \nSuch challenge can be addressed by migrating the data copy from L1D cache to shared memory, which may take the steps as follows: 1) a data miss signal would be raised for shared memory, 2) the data copy in L1D cache would be evicted to response queue, and 3) a new entry of MSHR would be filled with the pointer referring to the location of single data copy in the response queue. Later on, to fill the data miss, shared memory fetches data from response queue based on the location information recorded in MSHR.\n}\nWhen \\texttt{CIAO}\\xspace detects significant decrease in cache contentions due to a change in cache access patterns or completion of execution of some warps, \nit redirects the memory requests of these interfering warps from shared memory back to L1D cache (\\textit{cf}\\onedot} \\def\\Cf{\\textit{Cf}\\onedot Figure~\\ref{fig:thrott_exam}c).\n\n\nTo exploit the unused shared memory space for the aforementioned purpose, however, there are two challenges. \nFirst, the shared memory has its own address space separated from the global memory, \nand there is no hardware support that translates a global memory address to a shared memory address. \nSecond, the shared memory does not have a direct datapath to L2 cache and main memory~\\cite{jamshidi2014d}. \nThat is, it always receives and sends data only through the register file.\nTo overcome these limitations, we propose to adapt shared memory architecture as follows. \nFirst, we implement a address translation unit in front of shared memory to translate a given global memory address to a local shared memory address.\nSecond, we slightly adapt the datapath between L1D and L2 caches such that the shared memory can also access L2 cache when the unused shared memory space serves as cache.\n\n\n\\subsection{CIAO Warp Scheduling}\n\\label{sec:warp_throttling}\n\\noindent\nAlthough \\texttt{CIAO}\\xspace on-chip memory architecture\ncan effectively isolate cache accesses of interfering warps from those of interfered warps, its efficacy depends on various run-time factors, such as the number of interfering warps and the amount of unused shared memory space. \nFor example, \nthe interfering warps end up thrashing the shared memory as well when the amount of unused shared memory space\nis insufficient to handle a large number of memory requests from the interfering warps in a short time period (cf. Figure~\\ref{fig:thrott_exam}d).\n\n\nTo efficiently handle such a case, we propose to throttle interfering warps \\textit{only} when it is not effective to redirect memory requests of interfering warps to the shared memory.\nSpecifically, sharing the same cache interference detector used for \\texttt{CIAO}\\xspace on-chip memory architecture,\n\\texttt{CIAO}\\xspace monitors the intensity of interference at the shared memory at runtime.\nOnce the intensity of interference at the shared memory exceeds a threshold, \\texttt{CIAO}\\xspace stalls\nthe most severely interfering warp at the shared memory (\\textit{e.g}\\onedot} \\def\\Eg{\\textit{E.g}\\onedot, \\texttt{W2} in Figure~\\ref{fig:thrott_exam}e). \n\\texttt{CIAO}\\xspace repeats this step until the intensity of interference at the shared memory falls below the threshold. \nAs some warps complete their execution and subsequently the intensity of interference at the shared memory falls below the threshold,\n\\texttt{CIAO}\\xspace starts to reactivate the stalled warp(s) in the reverse order to keep high TLP and maximize the utilization of shared memory (\\textit{cf}\\onedot} \\def\\Cf{\\textit{Cf}\\onedot Figure~\\ref{fig:thrott_exam}f). \n\n\nNote that \\texttt{CIAO}\\xspace warp scheduling shares the same interference detector with \\texttt{CIAO}\\xspace on-chip memory architecture, instead of \nkeeping two separate interference detectors for L1D and shared memory, respectively. \nThis is because isolated interfering warps do not compete L1D cache with warps that exclusively access L1D cache, and memory accesses of isolated interfering warps often interfere with one another. \nIn other words, L1D cache and shared memory interferences do not affect each other. \nHence, L1D cache and shared memory can share the same VTA array to detect interferences. \n\n\\section{Introduction}\n\\label{sec:introduction}\n\\input{introduction}\n\n\\section{Background}\n\\label{sec:background}\n\\input{background}\n\n\\section{Architecture and Scheduling}\n\\label{sec:overview}\n\\input{overview}\n\n\\section{Implementation}\n\\label{sec:implementation}\n\\input{implementation}\n\n\\section{Evaluation}\n\\label{sec:result}\n\\input{evaluation}\n\n\\section{Discussion and Related Work}\n\\label{sec:relatedwork}\n\\input{related}\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\\input{conclusion}\n\n\\section{Acknowledgement}\n\\label{sec:ack}\n\\input{acknowledge}\n\n\n\\bibliographystyle{IEEEtran}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\\subsection{Variance patterns in space-time\\label{mot}}\n\n\nIn many fields of science interest lies on extreme events such as large temperatures or ozone crossing a threshold. Often these processes are observed over space and time and common characteristics are non-normality of observations, presence of outliers or non-constant variance. These characteristics are even more noticeable if data are obtained through long temporal windows, in which case it is often unrealistic to assume that variances are constant for the whole period. In the context of environmental applications, even if seasonality is accounted for, it is rather common to observe changes in variance depending on the influence of air flows or ocean currents. This heterogeneity when not considered in the modelling might lead to poor predictions in out-of-sample locations or future time points.\n\n\nTo illustrate this characteristic of environmental processes, consider the daily maximum ozone data in the United Kingdom observed across 61 locations (Panel (a) of Figure \\ref{figUK1}) from March to November of 2017. This period was chosen because it comprises the highest levels of ozone (Panel (b) of Figure \\ref{figUK1}). Ground level ozone is created by chemical reactions when pollutants emitted by cars, industry, to mention a couple of examples, react with sunlight. Moreover, high levels of ozone can also be found in rural areas due to wind transportation. It is well known that high levels of ozone can be harmful to human health and this problem has motivated several new modelling developments over the last years.\n \nWe start by fitting a multivariate dynamic linear model (MDLM) to this data \\citep{West97}. The mean structure of the MDLM includes time varying effects of latitude, longitude, daily mean temperature and wind speed. In space, we assume a Cauchy correlation function \\citep{Gneit00}, that is, $c(s,s')= \\left[1+ \\left( ||s-s'||\/\\phi \\right)^{\\alpha} \\right]^{-1}$ with \n$s,s'$ any two locations in $D$, $\\phi >0 $ the spatial range parameter and $\\alpha$ the shape parameter.\nPanels (c) to (f) of Figure \\ref{figUK1} show temporal and spatial residuals based on the MDLM fitting. Panel (c) presents the residual temporal precision, whereas panel (d) shows the scatter plot of wind speed versus the residual precision. It is clear that there is some temporal structure left in the residual of this fitted MDLM. Panels (e) and (f), on the other hand, show the spatial residual precision after fitting the MDLM. It is clear that there are smaller residual precisions in the south-eastern portion of the region, and a non-linear relationship of the spatial precision with latitude.\n\nIn this data the heterogeneity is mostly due to volatility in time with peaks of small precision (and large variance) in the months of June and July. This suggests that the proposed model should account for these patterns\nto explain the volatility of ozone observed across the different locations.\nIn what follows we review some attempts to treat the volatility in spatiotemporal applications and present our proposed approach based on modelling the variance laws through a dynamic linear model.\n\n\n\\begin{figure}[H]\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[width=5cm]{MapUKfull.pdf} &\n\\includegraphics[width=5cm]{MeanTimeUKmax.pdf} \\\\\n(a) UK map and spatial locations. & (b) Empirical temporal mean. \\\\\n \\includegraphics[width=5cm]{PrecisionTimeUKmaxResNew.pdf}& \\includegraphics[width=5cm]{VarianceTemporalWindUKmaxRes.pdf}\\\\\n\\\\\n{(c) Residual temporal precision.} & {(d) Residual temporal precision versus wind.} \\\\\n \\includegraphics[width=5cm]{MapVarSpaceUK.pdf}& \\includegraphics[width=5cm]{VarianceLatitudeUKmaxRes.pdf}\\\\\n\\\\\n{(e) Residual spatial precision.} & {(f) Residual spatial precision versus latitude.} \\\\\n\n\\end{tabular}\n\\caption{Data summaries for the ozone data observed over the UK. Panel (a) displays the UK map with the training locations (solid circles) and the testing locations ($\\bf{\\times}$). Panel (b) presents the empirical mean over the year. Panels (c)-(f) present the precision over space and time of the residuals based on the fitting of a multivariate dynamic linear model.\\label{figUK1}\n}\n\\end{figure}\n\n\\clearpage\n\n\\subsection{Related literature}\n\nSeveral papers have investigated the presence of patterns in the variance of spatiotemporal processes and its effects on the predictive performance of the process of interest.\n\\cite{Stein09} discusses the presence of peaks in the temporal variance in the modelling of atmospheric pressure even after including altitude in the mean function. In particular, the author suggests that the observed patterns is possibly due to the passage of weather fronts over the region. Often transformations such as the log or squared root are applied to the data aiming to stabilise the variance \\citep{DeOliveira97,Johns03} or to account for truncated domains \\citep{Allcroft03}. Recently, \\cite{Gent17} proposed to add flexibility to the usual transformed Gaussian fields by considering a large family of possible transformations. However, the transformation approaches will not result in reasonable predictions if changes in variance have a pattern over time. That is, in many applications, even after fitting a Gaussian process to the data, the residuals still present varying variances which might depend on covariates which were already included in the mean function \\citep[see e.g. ][]{Bueno2017}.\nMoreover,\nthe transformation approach may have difficult interpretations and may obscure the relationship between the response and the covariates \\citep[see][for an example]{Bolin15}. In these situations, keeping the observations in their original scale and modelling the variance laws is an appealing alternative. \n\n\n\\cite{Gelf2005} constructed a spatial model based on mixtures via a Dirichlet process which is non-stationary and non-Gaussian. \\cite{Ge07} extend this idea to allow a random surface to be selected in each site based on latent covariates. The approach is non parametric and replications are required for full inference, in which case dynamical models are considered to model temporal dependence.\n\nTo account for outliers \\cite{Bai15} \npropose an estimator to robustify the kriged Kalman filter, extending the spatio-temporal approach of \\cite{Mardia98} which is highly affected by outlying observations. \\cite{bevil20} propose a skew-t model for geostatistical data aiming to accommodate fat tails and asymmetric marginal distributions.\n\nIn the context of variance modelling, \\cite{PSteel06} propose a non-Gaussian process for geostatistical data which accommodates fat tails by scale mixing a Gaussian process; the Gaussian model is a limiting case. This approach was extended by \\cite{Fons11} to account for non-gaussianity in spatio-temporal processes. The proposed model for the variance is the product of two separable mixing processes, one in space by another in time and both are assumed continuous.\n\\cite{Bueno2017} extended \\cite{Fons11} by allowing the use of covariate information\nto explain the spatial patterns observed in the variances, and time is also assumed to vary continuously in $\\mathbb{R}_+$.\n More recently, \\cite{Tadayon2018} propose a modelling approach that considers the use of covariates in the measurement error and can capture the effects of the skewness and heavy tails for datasets with non-Gaussian characteristics. \\cite{Chu18} consider hierarchical modelling of Student-t processes with heterogeneous variance. The dynamic mean and variances depend on the lagged observations in time instead of past states. \n\nNote that, if time is assumed to be continuous in the variance model, correlation matrices will have large dimension and inference becomes too costly for reasonably long temporal windows. Thus, to allow for computational feasibility of real data applications, {different from \\cite{Fons11}}, this paper considers discrete time and dynamic linear models for the spatio-temporal variance process.\nThis proposal modifies the well known multivariate dynamic linear model (MDLM) \\citep{West97} which assumes Gaussianity to account for heterogeneity in spatio-temporal data analysis by modelling the variance laws over space and time. In the context of temporal evolution of variances, \\cite{Uhlig94} extends the usual Gaussian dynamic model by including a sequential evolution for the precision matrix by assuming a Matrix-Beta evolution. An alternative specification considers the Wishart sequential filtering for the variance matrix. \\cite{Liu00b} presents further discussion and model implementations for these proposals. In the context of more flexible state space models, \\cite{Liu00} propose a conditional dynamical model specification which allows for non-Gaussian errors\naccounting for outliers.\nHowever, the model does not consider possible patterns in the variance model and the distributions are the same over time. \n\n\\cite{West97} proposed to model variance laws \nby letting the observational variance to be a function of a known weight. In the context of spatial data, it is reasonable to assume that the weighting depends on Euclidean distances and smoothness in space should be ensured.\nIn the usual multivariate dynamical modelling approach, the variance may vary stochastically as an inverse Gamma (or inverse Wishart) distribution, in which case the resulting sampling distribution for the response is Student-t. However, this extension is not flexible enough to capture spatial heterogeneity as discussed in \\cite{PSteel06} and \\cite{Fons11}. In our proposed solution to this issue, the variance is assumed to vary according to a log-Gaussian process \\citep{PSteel06} and the mixing distribution varies in discrete-time assuming smooth transitions. Besides, the variance laws are allowed to depend on covariates. In this case, recurrence equations for filtering and smoothing are presented for the variance process allowing for feasible computations even for large temporal windows.\n\n\n\n\nThe remaining of the paper is organised as follows. Section \\ref{sec:propmodel} describes the proposed model and its properties. In particular, Sections \\ref{sec:2.2} and \\ref{sec:2.3} describe the inference and prediction procedures for dynamical spatial modelling over time with stochastic variance.\nSections \\ref{sec:real} and \\ref{sec:real2} present the analysis of the maximum temperature in the Spanish Basque Country and the maximum ozone levels in the United Kingdom, respectively. Different models are fitted and these analyses illustrate the effectiveness of our proposal in modelling varying variances over both time and space and the improvement it provides in the precision of predictions. \nSection \\ref{sec:conclusion} concludes with some discussion. Some simulated examples are presented in Appendix \\ref{ApSimD} to verify that our proposed predictive comparison measures indicate the correct data generating models and do not result in overfitting.\n\n\n\n\\section\nNon-Gaussian state-space modelling }\\label{sec:propmodel}\n\nThis section extends the multivariate Gaussian dynamic model by allowing for stochastic variance over space and time.\n\n\\subsection{Spatial Dynamic Linear Models with stochastic variance \n}\\label{sec2.2}\n\n\n\n\n\n\n\n\n\n\nConsider $\\{Z_t(\\mathbf{s}): \\mathbf{s} \\in D\\subseteq \\mathbb{R}^d, \\, \nt\\in T\\subseteq \\mathbb{Z}\\}$ a spatio-temporal random field. We assume $Z_t(\\mathbf{s})$ follows a spatial mixture model, that is,\n\n\n\n\\begin{equation}\\label{model:eq2}\nZ_t(\\bm{s}) = \\mathbf{x}_t(\\mathbf{s})' \\boldsymbol{\\theta}_t + \\sigma \\frac{{\\epsilon}_t(\\mathbf{s})}{\\sqrt{\\lambda_t(\\mathbf{s})}}+\\tau \\rho_t(\\mathbf{s}), \n\\end{equation}\nwhere $\\mathbf{x}_{t}(\\mathbf{s})$ is a vector of observed covariates, $\\bm{\\theta}_t$ is a vector of time varying regression coefficients, $\\sigma$ is a scale parameter and $\\tau$ is a nugget effect. The process ${\\epsilon}_t(\\cdot)$ is a zero mean Gaussian process with correlation function $c(\\cdot,\\cdot)$,\n$\\rho_t(\\cdot)$ is an uncorrelated Gaussian noise with zero mean and unit variance responsible for small scale variation and $\\lambda_t(\\mathbf{s})$ is a mixing process. Conditionally on the mixing process $\\lambda_t(\\cdot)$ and on the coefficients $\\boldsymbol{\\theta}_t$, the process $Z_t(\\cdot)$ has mean function $m_t(\\mathbf{s})=\\mathbf{x}_t(\\mathbf{s})'\\boldsymbol{\\theta}_t$ and covariance function $K(\\mathbf{s},\\mathbf{s}')=\\sigma^2c(\\mathbf{s},\\mathbf{s}')\/\\sqrt{\\lambda_t(\\mathbf{s})\\lambda_t(\\mathbf{s}')}$, $\\mathbf{s},\\mathbf{s}'\\in D$, $t\\in T$. For $\\lambda_t(\\mathbf{s})\\not = 1$ the process $Z(\\mathbf{s})$ has heterogeneous spatiotemporal variance and if $\\lambda_t(s)$ is integrated out the resulting process is non-Gaussian. If $\\lambda_t(\\mathbf{s})=1$ and an evolution state equation is assumed for ${\\boldsymbol{\\theta}_t}$ then the resulting model is the usual Gaussian Dynamic Linear Model \\citep{West97}.\n\n\nIn the sequel, we discuss the mixing process specification which is the crucial part of this spatiotemporal mixture model. Assuming $\\lambda_t(\\mathbf{s}) =\\lambda \\sim {\\rm Gamma}(v\/2, v\/2)$, $\\thinspace \\forall \\thinspace \\mathbf{s} \\in D$ implies that the distribution of $Z_t(\\cdot)$ is a Student-t process \\citep{Omre06} with $v$ degrees of freedom. \\cite{PSteel06} discuss the limitations of assuming a Student-t process for spatial observations. In short, the Student-t process is not able to account for spatial heterogeneity as it inflates the variance of the whole process whenever outliers or spatial heterogeneity is observed. \nOn the other hand, if we assume the Gaussian-log-Gaussian (GLG) model proposed by \\cite{PSteel06}, then ${\\rm ln} \\thinspace[ \\lambda_t(\\cdot)]$ is a Gaussian Process with mean $-\\nu\/2$ and covariance function $\\nu c(\\cdot,\\cdot)$ such that ${\\rm E}[\\lambda_t(\\mathbf{s}) ] = 1$, ${\\rm Var}[\\lambda_t(\\mathbf{s})] = e^{\\nu} -1$ and the kurtosis of the process $Z_t(\\cdot)$ is $3e^{\\nu}$ which is controlled by $\\nu$. This implies that the marginal distribution of $\\lambda_t(\\mathbf{s})$ is concentrated around one for very small value of $\\nu$ (of the order $\\nu= 0.01$) and as $\\nu$ increases, the distribution becomes more spread out and more right-skewed, while the mode becomes zero.\nOur proposed model extends the GLG specification by defining a model for $\\lambda_t(\\mathbf{s})$ through state space equations which assume that, conditionally on state parameters, the variances are independent in time, resulting in computationally efficient estimation algorithms. This approach takes advantage of the recurrence equations of DLM while accounting for more flexible variance laws for spatiotemporal data. \n\n\nNext we specify the dynamical evolution for both the mean and variance states and we discuss the connection between the usual Gaussian dynamic spatial model and the proposed non-Gaussian extension in equation (\\ref{model:eq2}). Let $\\mathbf{Z}_t= (Z_t(\\bm{s}_1), \\ldots, Z_t(\\bm{s}_n))'$ be the data collected at $n$ spatial locations in $D$. Conditional on the latent variables $\\Lambda_t=diag({\\lambda}_t(\\bm{s}_1),\\ldots,{\\lambda}_t(\\bm{s}_n))$, the observation and system equations obtained by integrating $\\epsilon_t(s)$ and $\\rho_t(s)$ out are given by\n\\begin{subequations}\\label{eq5}\n\\begin{equation}\\label{eq5a}\n \\mathbf{Z}_t\\mid \\boldsymbol{\\theta}_t,\\Lambda_t \\sim N\\left ( \\bm{F}_t' \\boldsymbol{\\theta}_t,\\;\\sigma^2 \\Lambda_t^{-1\/2}C_{{\\bm \\psi}}\\Lambda_t^{-1\/2}+\\tau^2I_n\\right ),\n \\end{equation}\n \\begin{equation}\\label{eq5b}\n \\boldsymbol{\\theta}_t\\mid \\boldsymbol{\\theta}_{t-1} \\sim N\\left ( { G}_t \\boldsymbol{\\theta}_{t-1} ,W_t\\right ),\n \\end{equation}\n\\end{subequations}\nwhere $\\bm{F}_t=(\\mathbf{x}_t(\\bm{s}_1),\\ldots,\\mathbf{x}_t(\\bm{s}_n))$ is the $p \\times n$ design matrix with observed $p$ covariates, $\\bm{\\theta}_t$ is the $p-$dimensional state vector, $C_{{\\bm \\psi}}$ represents the correlation matrix with elements computed by $C_{\\bm {\\psi},ij}=c(\\bm{s}_i,\\bm{s}_j)$ that depends on parameters ${\\bm \\psi}$ and the Euclidean distance among locations, $G_t$ represents the evolution matrix and $W_t$ is a $p$-dimensional covariance matrix of the states.\nEquation \\eqref{eq5b} defines the temporal evolution of state variables in the mean function and the smoothness of this evolution is controlled by $W_t$.\n\n\n\n\nWe now focus on the specification of the spatio-temporal mixing process $\\lambda_t(\\bm{s})$, $\\bm{s}\\in D$, $t \\in T$. To keep the model parsimonious,\n we define $\\lambda_t({\\bm s}) = \\lambda_1(\\bm{s})\\lambda_{2t}$ as a separable process.\nThe mixing distributions and the evolution equation for the state space parameters in the variance model are defined as \n\\begin{eqnarray}\\label{eq:lambda1}\n{\\rm ln}(\\mathbf{\\boldsymbol{\\lambda}}_1) & \\sim & N \\left ( -\\frac{\\nu_1}{2}\\bm{1}_n+\\bm{F}_{1}'\\boldsymbol{\\beta},\\nu_1 C_{{\\bm{\\xi}}}\\right ), \n\\end{eqnarray}\n\\begin{subequations}\\label{eqL}\n\\begin{equation}\\label{eq:lambda21}\n{\\rm ln}(\\lambda_{2t}) = \\bm{F}_{2t}'\\boldsymbol{\\eta}_t+v_{2t}, \\; v_{2t}\\sim N \\left ( -\\frac{\\nu_2}{2},\\nu_2 \\right ), \n\\end{equation}\n\\begin{equation}\\label{eq:lambda22}\n\\boldsymbol{\\eta}_t = G_{2t}\\boldsymbol{\\eta}_{t-1}+\\mathbf{\\omega}_{2t}, \\; \\mathbf{\\omega}_{2t}\\sim N \\left ( 0,W_{2t} \\right ),\n\\end{equation}\n\\end{subequations}\nwhere, in equation (\\ref{eq:lambda1}), ${\\rm ln}(\\mathbf{\\boldsymbol{\\lambda}}_1)= ({\\rm ln}(\\lambda(\\bm{s}_1), \\ldots, {\\rm ln}(\\lambda(\\bm{s}_n))'$ and $C_{{\\bm \\xi}}$ the spatial correlation matrix that depends on parameter $\\bm{\\xi}$ and the Euclidean distance between locations. Note that $C_{\\bm{\\xi},ij}=c^*(\\bm{s}_i,\\bm{s}_j)$ which could differ from $c(\\bm{s}_i,\\bm{s}_j)$, that is, in the spatio-temporal context it is possible to estimate a different correlation structure for the process $\\epsilon_t(\\cdot)$ and the process ${\\rm ln}[\\lambda_1(\\cdot)]$. In equation \\eqref{eq:lambda1}, $\\bm{F}_{1}=(\\bm{\\tilde x}(\\bm{s}_1),\\ldots, \\bm{\\tilde x}(\\bm{s}_n))$ is a $p_1\\times n$ design matrix that will allow for the effect of covariates in the spatial variance, and $\\boldsymbol{\\beta}$ is a $p_1$-dimensional vector of coefficients to be estimated. In equation \\eqref{eq:lambda21}, $\\bm{F}_{2t}=\\bm{x}_{t}^*$ is a $p_2$-dimensional vector that will allow for the effect of covariates in the temporal variance. Equation \\eqref{eq:lambda22} defines the temporal evolution of state parameters $\\boldsymbol{\\eta}_t$ in the variance model, with ${W}_{2t}$ controlling the temporal smoothness, and $G_{2t}$ representing the evolution matrix. \n\n\n\n\n\n \n\n\nThe resulting covariance function of $\\left\\{ Z_t(\\bm{s}): \\bm{s} \\in D; t \\in T \\right\\}$\n, defined in \\eqref{model:eq2}, is obtained by integrating out the mixing processes $\\lambda_1(\\bm{s})$ and $\\lambda_{2t}$.\nIf $t_1=t_2=t$ and $\\bm{s}_1=\\bm{s}_2=\\bm{s}$ we obtain the spatio-temporal variance as \\begin{equation}\n Var\\left(Z_t({\\bm s})\\mid \\boldsymbol{\\eta}_{1:T},\\boldsymbol{\\theta}_{1:T}\\right) = \\sigma^2 \\thinspace \\exp\\left\\{\\nu_1 +\\nu_2 -\\bm{F}_1'(\\bm{s})\\bm{\\beta}-\\bm{F}_{2t}'\\boldsymbol{\\eta}_{{t}}\\right\\},\n\\end{equation} \nwith $F_1(\\bm{s})=\\tilde{\\bm{x}}(\\bm{s})$ the vector of spatial covariates at site $\\bm{s}\\in D$. The temporal dependence is carried out by the states $(\\boldsymbol{\\theta}_t,\\boldsymbol{\\eta}_t)$, $t=1,\\ldots,T$ and the conditional spatial correlation is given by\n\\begin{equation}\\label{eq:corr}\nCorr\\left[Z_{t}(\\bm{s}_1), Z_{t}(\\bm{s}_2)\\mid \\boldsymbol{\\eta}_{1:T},\\boldsymbol{\\theta}_{1:T}\\right] = C_{{\\bm \\psi}}(\\mathbf{s}_1,\\mathbf{s}_2)\\exp\\left\\{ \\frac{\\nu_1}{4} \\left( C_{{\\bm \\xi}}(\\mathbf{s}_1,\\mathbf{s}_2) -1 \\right) \\right\\}. \n\\end{equation}\n\nThe kurtosis in each location unconditional on $\\lambda_t({\\bm s})$ is given by\n\\begin{equation}\\label{eq:kurt}\nKurt\\left[Z_t({\\bm s})\\right] = 3 \\thinspace \\exp\\left\\{\\nu_1 + \\nu_2 \\right\\}.\n\\end{equation}\nSee Appendix \\ref{ApB} for the proofs of these results. A particular case of the model proposed in equation \\eqref{model:eq2} is obtained for $\\lambda_t(\\mathbf{s})= 1$ and, consequently, the non-Gaussian distribution converges to the Gaussian distribution for small values of $\\nu_1$ and $\\nu_2$.\n\n\n\\subsection{Resultant posterior distribution and inference procedure}\\label{sec:2.2}\n\nWe follow the Bayesian paradigm to make inference, predictions and model comparisons that are obtained from the joint posterior distribution of the parameters. In particular, we take advantage of the hierarchical structure of our proposal in our iterative estimation algorithm to sample from the joint posterior and to make predictions. In what follows we present the prior, the joint posterior distributions and briefly describe the steps to obtain samples from the posterior distribution.\n\nIn our motivating example, we assume a Cauchy correlation function with range parameter $\\phi>0$ and shape parameter $\\alpha>0$. This function is flexible allowing for long-memory dependence and also correlations at short and intermediate lags.\nWe assume an exponential correlation function for the spatial mixing process $ln[\\lambda_1(\\bm{s})]$ given by $c^*(\\bm{s},\\bm{s}') = \\exp\\left\\{ -||\\bm{s}-\\bm{s}'||\/\\gamma\\right\\}$, where $\\gamma> 0$. \nModel specification is complete after assigning a prior distribution for the static parameters $\\Phi=(\\sigma^2,\\tau^2,\\nu_1,\\nu_2, {\\boldsymbol{\\beta}}, {\\bm \\psi}=(\\alpha,\\phi), {\\bm \\xi}=(\\gamma))$. We assign vague independent priors to the static parameters in $\\Phi$.\n In particular, we assume $\\sigma^{-2} \\sim Gamma(a_{\\sigma^2},b_{\\sigma^2})$ with small values for $a_{\\sigma^2}$ and $b_{\\sigma^2}$.\nFor the range parameter $\\phi$, we take into account that the prior is critically dependent on the scale of the observed distances among locations. For the Cauchy correlation function, we assign a gamma prior $\\phi$, i.e. $\\phi \\sim Gamma\\left(1, c\/med(d) \\right)$, with $med(d)$ representing the median of observed distances and the shape parameter follows a uniform prior, that is, $\\alpha \\sim U(a_{\\alpha};b_{\\alpha})$. For the exponential correlation function parameter, we assume $\\gamma \\sim Gamma(a_{\\gamma}, b_{\\gamma})$. For the mixing parameters $\\nu_i$, $i=1,2$, we assign a $Gamma(a_\\nu,b_{\\nu})$ prior. Notice that very small values of $\\nu_i$ (around 0.01) lead to approximate normality while large values of $\\nu_i$ (of the order of say 3) suggest very thick tails. \n\nFollowing Bayes' theorem, the posterior distribution of model parameters and latent variables given the observed data, $\\mathbf{z}_t = (z_t(\\bm{s}_1), \\ldots, z_t(\\bm{s}_n))'$, $t=1,\\ldots,J$, is proportional to \n\\begin{eqnarray}\\label{eq:post} \\nonumber\n \n p\\left( \\boldsymbol{\\theta}_{1:J}, \\boldsymbol{\\eta}_{1:J} \\boldsymbol{\\lambda}_{1},\\boldsymbol{\\lambda}_{2}, \\Phi \\mid \\mathbf{z}\\right) &\\propto& \\prod_{t=1}^{J} f_{N_{n}}(\\mathbf{z}_t|\\bm{F}'_t\\boldsymbol{\\theta}_t,\\Sigma_t) \\nonumber \\\\ \n \n& \\times & f_{N_{n}}(\\boldsymbol{\\Delta}|{\\bf 0},\\nu_1 C_{\\bm{\\xi}}) \\prod_{t=1}^{J} f_{N_{1}}(L_t|\\bm{F}'_{2t}\\boldsymbol{\\eta}_t,\\nu_2) \\nonumber \\\\ \n \n & \\times & f_{N_{p}}(\\boldsymbol{\\theta}_0\\mid \\bm{m}_0,C_0) \\prod_{t=1}^{J} f_{N_{p}}(\\boldsymbol{\\theta}_t \\mid \\boldsymbol{\\theta}_{t-1},W_t) \\\\\n & \\times & f_{N_{p_{2}}}(\\boldsymbol{\\eta}_0\\mid \\bm{m}_0^*,C_0^*) \\prod_{t=1}^{J} f_{N_{p_{2}}}(\\boldsymbol{\\eta}_t \\mid \\boldsymbol{\\eta}_{t-1},W_{2t}) \\; \\pi(\\Phi), \\nonumber\n\\end{eqnarray}\n\\noindent where $f_{N_{p}}(\\cdot\\mid A,B)$ denotes the density function of a $p$-variate multivariate normal distribution with mean A and covariance matrix B, $\\boldsymbol{\\Delta}= ln (\\boldsymbol{\\lambda}_1) + \\nu_1\/2 \\, {\\bf 1}_n {- \\bm{F}_1'\\boldsymbol{\\beta}}$, ${L}_t= ln \\lambda_{2t} + \\nu_2\/2$, $\\Sigma_t = \\sigma^2 \\Lambda_t^{-1\/2} C_{\\bm{\\psi}} \\Lambda_t^{-1\/2} + \\tau^2 I_n$, $\\Lambda_t = diag(\\lambda_1(\\bm{s}_1), \\ldots, \\lambda_J(\\bm{s}_n))$ and $\\pi(\\cdot)$ the prior distribution of static parameters. Finally, $f_{N_{p}}(\\boldsymbol{\\theta}_0\\mid \\bm{m}_0,C_0)$ and $f_{N_{p_{2}}}(\\boldsymbol{\\eta}_0\\mid \\bm{m}_0^*,C_0^*)$ are the densities for the initial prior information at time $t=0$ for $\\boldsymbol{\\theta}_0$ and $\\boldsymbol{\\eta}_0$, respectively.\n\nThe resultant posterior distribution does not have closed form and we resort to Markov chain Monte Carlo methods \\citep{gamerman} to obtain samples from the posterior. In particular, posterior samples are obtained through a Gibbs sampler algorithm with steps of the Metropolis-Hastings algorithm for $\\phi$, $\\alpha$, $\\gamma$ and $\\nu_i$, $i=1,2$ which are based on random walk proposals.\n\n\n\n\n\n\n\\paragraph{Brief description of the MCMC algorithm} Conditional on the latent variables $\\boldsymbol{\\lambda}_1$ and $\\boldsymbol{\\lambda}_2$, Gaussianity is preserved and samples from the posterior full conditional distributions for the state vectors $\\boldsymbol{\\theta}_t$ in the mean are obtained through the usual forward filtering and backward smoothing recursions (FFBS) proposed by \\cite{Sylvia1994} and \\cite{Carter1994}.\nAnalogously, conditionally on $\\boldsymbol{\\lambda}_2$, the posterior distribution of states $\\boldsymbol{\\eta}_t$ are also obtained through the FFBS algorithm. \nAppendices \\ref{ApA1} and \\ref{ApA2} provide the equations to run the FFBS for $\\boldsymbol{\\theta}_t$ and $\\boldsymbol{\\eta}_t$, respectively. Note that, \ndifferent from \\cite{Bueno2017} and \\cite{Fons11}, as we assume time to be discrete, we do not need to rely on computing the inverse of high dimensional covariance matrices at each iteration of the MCMC. \nConditional on the regression coefficients $\\bm{\\beta}$, the spatial latent mixing variable $\\boldsymbol{\\lambda}_{1}=(\\lambda_{1}(\\bm{s}_1),\\cdots,\\lambda_{1}(\\bm{s}_n))'$ is sampled as part of a Gibbs algorithm using blocks of random walks or the independent sampler proposed in \\cite{PSteel06}. To sample $\\bm{\\beta}$ the Gibbs step is given by\n\\begin{equation}\n \\bm{\\beta}\\mid \\bm{\\lambda}_1,\\nu_1,\\bm{\\xi}\\sim N_{p_1}((\\bm{F}_1'C_{\\bm{\\xi}}^{-1}\\bm{F}_1)^{-1}\\bm{F}_1'C_{\\bm{\\xi}}^{-1}(ln \\bm{\\lambda}_1+\\nu_1\/2\\; \\bm{1}_n),\\nu_1(\\bm{F}_1'C_{\\bm{\\xi}}^{-1}\\bm{F}_1)^{-1}).\n\\end{equation}\nA summary of our proposed sampling algorithm is described in Appendix \\ref{ApPostComp}. The algorithm was coded in {\\tt R} using RStudio \nVersion 1.1.442 \\citep{RCoreTeam} and the source code can be obtained at: https:\/\/github.com\/thaiscofonseca\/DynGLG.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Predictions in space-time}\\label{sec:2.3}\n\n\nFor spatial interpolation {for given observed times}, consider the vector $(\\mathbf{Z}_t^{obs}, \\mathbf{Z}_t^{pred})$, with $\\mathbf{Z}_t^{obs}$ and $\\mathbf{Z}_t^{pred}$ representing, respectively, observed and out-of-sample values of $Z_t(\\bm{s})$, at each time $t= 1, \\ldots, J$. Let $\\Phi= ( \\sigma^2,\\tau^2, \\nu_1,\\nu_2, \\bm{\\beta}, \\bm{\\psi},\\bm{\\xi})$ be the static parameters in the proposed model in equation \\eqref{eq5}.\nIn order to obtain samples from the posterior predictive distribution $p(\\mathbf{Z}_t^{pred} \\mid \\mathbf{Z}_t^{obs})$ we resort to composition sampling; assume that $\\Phi$,\n$\\bm{\\lambda}_{1}^{obs}=(\\lambda_{1}(\\bm{s_1}),\\ldots,\\lambda_{1}(\\bm{s}_n))'$, $\\bm{\\lambda}_{2}^{obs}=(\\lambda_{2,1},\\ldots,\\lambda_{2J})'$, $\\boldsymbol{\\theta}^{obs}=(\\bm{\\theta}_{1},\\ldots,\\bm{\\theta}_J)'$, $\\boldsymbol{\\eta}^{obs}=(\\bm{\\eta}_{1},\\ldots,\\bm{\\eta}_J)'$ were sampled from the joint posterior distribution $p(\\Phi, \\bm{\\lambda}_{1}^{obs},\\bm{\\lambda}_{2}^{obs},\\boldsymbol{\\theta}^{obs},\\boldsymbol{\\eta}^{obs} \\mid \\mathbf{Z}_t^{obs})$.\nThus, samples from $p(\\mathbf{Z}_t^{pred} \\mid \\mathbf{Z}_t^{obs})$ may be obtained by sampling\n\\begin{enumerate}\n \\item[(i)] ${\\rm ln}(\\bm{\\lambda}_{1}^{pred})\\mid \\bm{\\lambda}_{1}^{obs},\\nu_1, \\bm{\\xi} $ and \\item[(ii)] $\\bm{Z}_t^{pred}\\mid \\bm{Z}_t^{obs}, \\bm{\\lambda}_{1}^{pred},\\bm{\\lambda}_2^{obs},\\bm{\\theta}^{obs},\\Phi$. \n\\end{enumerate}\nBoth distributions in (i) and (ii) are Gaussian, the second is the observational model and the first is given by\n\\begin{equation}\\label{eqpredL1}\n{\\rm ln}(\\bm{\\lambda}_1^{pred}) \\mid \\bm{\\lambda}_1^{obs} , \\nu_1, \\bm{\\xi} \\sim N_n \\left[ -\\nu_1\/2 \\; {\\bf 1}_n+{C}_{o,p} {C}_{o,o}^{-1} \\mathbf{a} ; \\nu_1 \\left({C}_{p,p} - {C}_{p,o}{C}_{o,o}^{-1}{C}_{o,p} \\right)\\right] \n\\end{equation}\n\\noindent with $\\mathbf{a} = \\left(ln(\\bm{\\lambda}_1^{obs}) +\\nu_1\/2 \\; {\\bf 1}_n - {\\bm{F}'_1 \\boldsymbol{\\beta}}\\right)$ and\n${C}_{\\bm{\\xi}} = \\begin{pmatrix} \nC_{p,p} & C_{p,o} \\\\\nC_{o,p} & C_{o,o} \n\\end{pmatrix}$. {This result follows from the properties of the partition of the multivariate normal distribution.}\n\n\nSuppose now that interest lies in forecasting future observations at a set of locations given historical data $\\bm{Z}^{obs}=(Z_{1}^{obs},\\ldots,Z_J^{obs})'$. Consider that at time $J$ we want to predict $h$ instants ahead and $h>0$. We define $\\bm{\\lambda}_{2}^{pred}=(\\lambda_{2,J+1},\\ldots,\\lambda_{2,J+h})'$. Thus, samples of $\\bm{Z}_t^{pred}$ may be obtained by sampling from\n\\begin{enumerate}\n \\item[(i)] $\\bm{\\eta}^{pred},\\mid \\bm{\\eta}^{obs},\\bm{\\lambda}_{2}^{obs} $ and $\\bm{\\theta}^{pred}\\mid \\bm{\\theta}^{obs},\\bm{Z}^{obs} $,\n \\item[(ii)] $ln(\\bm{\\lambda}_{2}^{pred})\\mid \\bm{\\eta}^{pred},\\nu_2 $ and \n \\item[(iii)] $\\bm{Z}_t^{pred}\\mid \\bm{Z}_t^{obs}, \\bm{\\lambda}_{1}^{obs},\\bm{\\lambda}_2^{pred},\\bm{\\theta}^{pred},\\Phi$. \n\\end{enumerate}\nIf we are predicting in the future for ungauged locations we replace $\\bm{\\lambda}_1^{obs}$ with $\\bm{\\lambda}_1^{pred}$ obtained using equation (\\ref{eqpredL1}). Steps (ii) and (iii) are performed simulating from the variance and observational models which are all conditionally Gaussian distributions. Step (i) depends on the usual forecast distributions available for the Gaussian Multivariate Dynamical Model \\citep{West97}. \n\n\n\n\\paragraph{Model Comparison}\nTo check the predictive accuracy of competing models, measures based on scoring rules are considered. Scoring rules provide summaries for the evaluation of probabilistic forecasts by comparing the predictive distribution with the actual value which is observed for the process \\citep{GneitRaf07}. In particular, we consider the Interval Score, the Logarithmic Predictive Score and the Variogram Score. Note that the Logarithmic Predictive Score and the Variogram Score are multivariate measures for a $d$-dimensional vector. { We briefly describe how to compute each of these criteria. }\\\\\n\n\\noindent \\textit{Interval Score:} Interval forecast is a crucial special case of quantile prediction \\citep{GneitRaf07}. It compares the predictive credibility interval with the true observed value (validation observation), and it considers the uncertainty in the predictions such that the model is penalised if an interval is too narrow and misses the true value. The Interval Score is given by \n\\begin{equation}\\label{eq:IS}\nIS(u,l;z) = (u-l) + \\frac{2}{\\gamma}(l - z)I_{\\left[z< l\\right]} + \\frac{2}{\\gamma}( z - u)I_{\\left[z> u\\right]},\n\\end{equation}\n\\noindent where $l$ and $u$ represent for the forecaster quoted $\\frac{\\gamma}{2}$ and $1-\\frac{\\gamma}{2}$ quantiles based on the predictive distribution and $z$ is the validation observation. If $\\gamma=0.05$ the resulting interval has 95\\% credibility.\\\\\n\n\\noindent \\textit{Log Predictive Score:} The log predictive score evaluates the predictive density at the observed validation value $\\bm{z}$. It is given by \n\\begin{equation}\\label{eq:LPS}\n LPS(\\bm{z})= -log\\left\\{p(\\bm{z}\\mid \\bm{z}^{obs})\\right\\}.\n\\end{equation}\nThe smaller the log predictive score, the better the model does at forecasting $\\bm{z}^{obs}$.\\\\\n\n\\noindent \\textit{Variogram Score:}\nThe variogram score of order $p$ \\citep{Sche15} was proposed to evaluate forecasts of multivariate quantities. It depends on a matrix $w$ of non-negative weights specified subjectively that allow to emphasize or downweight pairs of observations, for instance, based on Euclidean distances. It is defined as \n\\begin{equation}\n \\mbox{VS-p}(\\bm{z},\\bm{z}^{obs}) = \\sum_{i,j=1}^{d} w_{ij}\\left (|\\bm{z}^{obs}_{i}-\\bm{z}^{obs}_j|^p-\\frac{1}{m}\\sum_{k=1}^M|\\bm{z}_i^{(k)}-\\bm{z}^{(k)}_j|^p\\right )^2,\n\\end{equation}\nwhere $\\{\\bm{z}^{(k)};\\, k=1,\\ldots,M\\}$ are simulated values from the predictive distribution. The smaller the variogram score, the better the model does at forecasting $\\bm{z}^{obs}$. Empirical studies presented in \\cite{Sche15} suggest that $p=0.5$ leads to good model discrimination, however, if the predictive distribution is skewed, then values of $p<0.5$ may lead to better results.\\\\\n\n\n\\section{Data analysis}\\label{sec4}\n\nThis section presents two data analyses relevant in the discussion about extremes in environmental applications: the first application considers the maximum temperature data in the Spanish Basque Country. These data have been previously analysed by \\cite{PSteel06}, \\cite{Fons11} and \\cite{Bueno2017}. As our proposal is able to account for longer temporal windows than \\cite{Fons11}, the analysis shown in Section \\ref{sec:real} considers one year of daily observations instead of one month as in \\cite{Fons11} and \\cite{Bueno2017}. The second application focuses on the maximum ozone data described in Section \\ref{mot}, which illustrates the use of spatial and temporal covariates in the variance model.\nWe define $\\lambda_t(\\bm{s})=\\lambda_1(\\bm{s})\\lambda_{2t}$ and based on equations (\\ref{eq5}) and (\\ref{eqL}) we fit the models described in Table \\ref{tabmodels} which are particular cases of the general model proposed in the previous section.\n\n \n \n \n \n \n \n \n\n\n\n\\begin{table}\n \\caption{Competing models fitted to data applications: Gaussian (G), Student-t (ST), Spatial GLG (GLG), Dynamical (Dyn), Dynamical with covariates (CovDyn), Dynamical GLG (DynGLG), Dynamical GLG with covariates (CovDynGLG) and the complete model (Full). \\label{tabmodels}}\n\\centering\n \\fbox{ \\begin{tabular}{lll}\n \\hline\n \n Model & $\\lambda_1(\\bm{s})$ & $\\lambda_{2t}$\\\\\n \\hline\nG & $1$ & $1$ \\\\\nST & $\\lambda \\sim Gamma(\\nu_1\/2,\\nu_1\/2)$ & $1$\\\\\nGLG & \n $ln (\\bm{\\lambda}_{1})\\sim N\\left (-\\frac{\\nu_1}{2} \\; \\mathbf{1}_n , \\nu_1 C_{\\bm{\\xi}}\\right )$ & $1$\\\\ \nDyn & $1$ & $ln(\\lambda_{2t})\\sim N\\left (-\\frac{\\nu_2}{2} +\\eta_{0t}, \\nu_2\\right )$\\\\\nCovDyn & $1$ & $ln(\\lambda_{2t}) \\sim N\\left (-\\frac{\\nu_2}{2}+\\bm{F}_{2t}'\\bm{\\eta}_t, \\nu_2\\right )$\\\\\nDynGLG & \n $ln (\\bm{\\lambda}_{1})\\sim N\\left (-\\frac{\\nu_1}{2} \\; \\mathbf{1}_n, \\nu_1 C_{\\bm{\\xi}}\\right )$ & $ln(\\lambda_{2t})\\sim N\\left (-\\frac{\\nu_2}{2}+\\eta_{0t}, \\nu_2\\right )$\\\\\nCovDynGLG & $ln (\\bm{\\lambda}_{1})\\sim N\\left (-\\frac{\\nu_1}{2} \\; \\mathbf{1}_n, \\nu_1 C_{\\bm{\\xi}}\\right )$ & $ln(\\lambda_{2t}) \\sim N\\left (-\\frac{\\nu_2}{2}+\\bm{F}_{2t}'\\bm{\\eta}_t, \\nu_2\\right )$\\\\\n Full & $ln (\\bm{\\lambda}_{1})\\sim N\\left (-\\frac{\\nu_1}{2}\\; \\mathbf{1}_n+\\bm{F}_1'\\bm{\\beta} , \\nu_1 C_{\\bm{\\xi}}\\right )$ & $ln(\\lambda_{2t}) \\sim N\\left (-\\frac{\\nu_2}{2}+\\bm{F}_{2t}'\\bm{\\eta}_t, \\nu_2\\right )$\\\\\n\\hline\n \\end{tabular}}\n\\end{table}\n\n\n\n\\subsection{Application to temperature data in the Spanish Basque Country}\\label{sec:real}\n\nThis dataset refers to the maximum temperature recorded in 2006 in the Spanish Basque Country (Figure \\ref{sec6fig1}(a)).\n{Part of these data was analysed by} \\cite{PSteel06}, \\cite{Fons11} and \\cite{Bueno2017} where they only used the maximum temperature recorded in July 2006 at 70 locations.\n\n\\cite{PSteel06} considered only spatial data while \\cite{Fons11} and \\cite{Bueno2017} considered spatio-temporal data. As this region is quite mountainous, with altitude of the monitoring stations lying between 0 and 1188 meters, altitude is included as an explanatory variable in the dynamic mean of the process that also depends on the spatial coordinates, that is, $m_{t}(\\mathbf{s}) = \\theta_{0t} + \\theta_{1t}\\thinspace lat(\\mathbf{s}) + \\theta_{2t}\\thinspace long (\\mathbf{s}) + \\theta_{3t} \\thinspace alt(\\mathbf{s})$, $\\forall \\thinspace t=1, \\ldots J$.\nFor the stations with missing observations, data were inputted using a random forest algorithm \\citep{Stek12}. We considered stations with no more than 5\\% missing data resulting in 68 locations.\n\n\nPanels of Figure \\ref{sec6fig1}(c)--(e) show that the empirical mean, empirical precision over time and space for the residuals of a Gaussian dynamical model is far from constant, suggesting that a spatial model with constant variance might be unsuitable. Panel {(e)} of Figure \\ref{sec6fig1} shows the behaviour of the variability across the region. The diameter of the solid circles is proportional to the value of the residual precision at the respective location. The map suggests that there is a spatial trend left in the variance of the residuals.\n\n\n\n\n\n\\begin{figure\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[width=5.5cm]{MapSpainShape.pdf} \n&\n\\includegraphics[width=4.6cm]{MeanDaysTemp.pdf} \\\\\n(a) Spain Map and spatial locations & (b) Empirical temporal mean \\\\ \n\n \\includegraphics[width=4.6cm]{PrecisionTimeTempmaxResNew.pdf} &\\includegraphics[width=4.6cm]{PrecisionAltitudeTempmaxRes.pdf} \n \\\\\n{(c) Residual temporal precision }& (d) Residual spatial precision versus altitude \\\\\n \\includegraphics[width=5.5cm]{PrecisionSpaceSpainTemp.pdf} &\n\\includegraphics[width=4.6cm]{PrecisionLatitudeTempmaxRes.pdf} \\\\% &\\includegraphics[width=4.6cm]{VarianceLongitudeTemmaxRes.pdf} \\\\\n(e) Residual spatial precision & (f) Residual spatial precision versus latitude \\\\\n\\end{tabular}\n\\caption{Data summaries { for the maximum temperature data observed in the Spanish Basque Country}. Panel (a) displays the map with spatial locations (solid circles) and the crosses are the ones left out from the inference procedure to check the predictive ability of the different models. Panel (b) presents the empirical mean over the year. Panels (c)-(f) represent the empirical precision of the maximum temperature observed data for the residuals of a Gaussian (G) model.\\label{sec6fig1}}\n\\end{figure}\n\nGiven the results from Figure \\ref{sec6fig1}, we move forward and fit models: G, ST, GLG, Dyn, DynGLG and Full as described in Table \\ref{tabmodels} with covariates in the spatial variance. We leave out three locations for predictive comparison (represented by `$\\times$' in Panel (a) of Figure \\ref{sec6fig1}). The parameters to be estimated are the dynamic coefficients ($\\theta_{0t}, \\theta_{1t}, \\theta_{2t},\\theta_{3t}$), $\\boldsymbol{\\eta}_t$, the covariance parameters ($\\sigma^2, \\tau^2, \\phi, \\alpha, \\gamma$), the mixing parameters ($\\nu_1, \\nu_2$), the latent mixing variables ($\\lambda_{1}(\\bm{s})$, $\\lambda_{2t}$) and the variance regression coefficients $\\bm{\\beta}$. As already mentioned in section \\ref{sec2.2} the variances $W_t$ and $W_{2t}$ are estimated through discount factors. We must specify two discount factors referring to the structure of the mean process $\\mathbf{Z}_t$ and the mean of the variance process ${\\lambda}_{2t}$, respectively. In a general context, the value of the discount factor is usually fixed between 0.90 and 0.99, or it is chosen by model selection diagnostics, e.g, looking at the predictive performance of the model for different values of $\\delta = (\\delta_1, \\delta_2)$ using some comparison criteria \\citep{petris2009dynamic}. To illustrate the performance of the competing models, we fixed $\\delta_{1}=0.99$ (for all competing models) and $\\delta_{2}=0.99$ (for the Dyn, the DynGLG and the Full models) for evaluating the behaviour and goodness of fit. \n\n { Following the values of the different model comparison criteria shown in Table \\ref{sec4tab1}, the G model is the one with the worst predictive performance.} As mentioned previously, the G model is not able to accommodate the uncertainty for some observations which presented larger maximum temperature values. { Under LPS the Full and the DynGLG models provide quite similar values, whereas DynGLG results in the smallest value of VS-0.25. Note that the LPS under DynGLG is similar to the one under the Full model. Therefore, in what follows we discuss the main results obtained for the DynGLG model where we do not consider spatial covariates.}\n\n\n\n\\begin{table\n \n \\caption{Model comparison based on the Interval Score (IS), the Log Predictive Score (LPS) and the Variogram Score of order 0.25 (VS-0.25) criteria for the predicted observations at the out-of-sample locations under all fitted models for the maximum temperature dataset. The smallest values are highlighted in boldface. \n\\label{sec4tab1}}\n \\centering\n \\fbox{ \\begin{tabular}{lcccccc}\n \\hline\n \n & G & ST & GLG & Dyn & DynGLG &{Full} \\\\\n \\hline\nIS &6.85 & 4.62 & 4.54 & {\\bf 4.21}& 4.34 & { 4.25}\\\\\nLPS & 1565 & 1355 & 1206 & 1286& { 1097}& {\\bf 1095}\\\\\nVS-0.25 & 1658 & 1304 & 1222 & 1283 & {\\bf 1194} & 1240\\\\\n\\hline\n \\end{tabular}}\n \n\\end{table}\n\nPosterior summaries (limits of the 95\\% posterior credible intervals) of the time varying coefficients (see Figure \\ref{sec6fig2a}) in the mean of the process do not include zero suggesting that latitude, longitude and altitude are important covariates to explain levels of temperature. In particular, and as expected, the coefficient associated with altitude (see panel (d) of Figure \\ref{sec6fig2a}) is negative over time, suggesting that the maximum temperature decreases as the altitude increases.\n\n\\begin{figure\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[width=5cm]{d0glgTemp.pdf}\n&\n \\includegraphics[width=5cm]{d1glgTemp.pdf} \\\\ {(a) Intercept } &{(b) Latitude effect} \\\\ \n \\includegraphics[width=5cm]{d2glgTemp.pdf} &\n \\includegraphics[width=5cm]{d3glgTemp.pdf}\\\\\n{(c) Longitude effect} & {(d) Altitude effect} \\\\\n\\end{tabular}\n\\caption{Temperature data: posterior summaries for the dynamic mean effects $\\theta_t$ under the DynGLG model.\\label{sec6fig2a}}\n\\end{figure}\n\nPanels (a)-(c) of Figure \\ref{sec6fig3} present the dynamic mixing effect indicating that the model captures variability over time and it is able to identify some stations that are potential outliers over space and time. Clearly, the variance is not constant over space-time as previously suggested by Figure \\ref{sec6fig1}.\n\n Figure \\ref{figTempvar1} presents the posterior summaries for the predictive standard deviation of $z_t(s)$, ${ s}=(43.16, -3.28)$ conditional on the latent mixing variables for the DynGLG model compared to the Gaussian model. The posterior predictive standard deviation is obtained numerically by composition sampling that simulates replicated observations from the observational model and computes the empirical standard deviation of these artificial data. Note that the variance is non-constant with some peaks over time. This behaviour cannot be captured by the G model which estimates the standard deviations as almost constant over time. The advantage of our proposed model is clear from panels (a)-(b) of Figure \\ref{figTempvar2}. For this application, the DynGLG model tends to have shorter ranges of the 95\\% posterior predictive intervals whereas uncertainty seems small and it presents larger intervals in periods of more volatile behaviours.\n\n\nAs the Full model provided a similar value of LPS and a smaller value of IS in comparison to the DynGLG we briefly discuss the posterior summaries of the parameters in the model for $\\lambda_1(\\bm{s})$.\nThe Full model includes covariates in spatial variance $\\lambda_1(\\bm{s})$ and the regression coefficients indicate that latitude and longitude do not impact on the variability over space with the 95\\% posterior credible interval for $\\beta_1$ being $(-0.0093 , 0.0132 )$ and, for $\\beta_2$ $(-0.1156, 0.0834 )$, respectively. On the other hand, the effects of altitude $IC(95\\%, \\beta_3)= (-0.0001, 0.0000)$ show that it influences spatial heterogeneity not only in the dynamical mean but also in the variability of the process. Note that the range of $\\beta_3$ is very small and it does not improve the predictive performance substantially.\n\n\n\\begin{figure\n\\begin{center}\n\\begin{tabular}{ccc}\n\\includegraphics[width=4.6cm]{MutTemp.pdf} &\n \\includegraphics[width=4.6cm]{LambdasTemp.pdf} &\n \\includegraphics[width=4.6cm]{LambdatTemp.pdf}\\\\\n (a) $\\eta_{0t}$ &{(b) $\\lambda_{1}(\\bm{s})$} & (c) $\\lambda_{2t}$ \\\\\n\\end{tabular}\n\\caption{Temperature data: posterior summaries for the DynGLG model: (a) dynamic variance mean (solid line), (b) mixing space and (c) mixing temporal.}\n\\label{sec6fig3}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\begin{tabular}{c}\n \\includegraphics[width=11cm]{SeriesEstimatedVarFullTemp2.pdf} \\\\\n \\end{tabular}\n\\caption{Temperature data: Approximated posterior predictive standard deviation over time for the DynGLG model and the Gaussian model for ${\\bf s}=(43.16, -3.28)$. }\n\\label{figTempvar1}\n\\end{center}\n\\end{figure}\n \\begin{figure}[!ht]\n\\begin{center}\n\\begin{tabular}{c}\n\n\\includegraphics[width=12cm]{Site3predTemp.pdf} \\\\\n \\\\ \n (a) ${\\bf s}= (43.18, -2.77)$\\\\\n \\includegraphics[width=12cm]{Site2predTemp.pdf} \\\\\n(b) ${\\bf s}=(43.16, -3.28)$\\\\\n\\end{tabular}\n\\caption{Temperature data: Predictive posterior distribution ($95\\%$ interval) over time for the DynGLG model and the Gaussian model for {${\\bf s}= (43.18,-2.77 )$ and ${\\bf s}=(43.16, -3.28)$.}}\n\\label{figTempvar2}\n\\end{center}\n\\end{figure}\n\n\n\n\n\\clearpage\n\n\n\\subsection{Application to ozone data in the UK}\\label{sec:real2}\n\n\nThis section analyses the ozone data presented in Section \\ref{mot}. The proposed mean function is $m_t(\\mathbf{s}) = \\theta_{0t} + \\theta_{1t}\\thinspace lat(\\mathbf{s}) + \\theta_{2t}\\thinspace long (\\mathbf{s}) + \\theta_{3t} \\thinspace temp_t(\\mathbf{s})+ \\theta_{4t} \\thinspace wind_t(\\mathbf{s})$, $\\forall \\thinspace t=1, \\ldots J$. For the stations with missing observations, data were inputted using a random forest algorithm \\citep{Stek12}. We considered stations with less than $5\\%$ of missing data in all variables resulting in 61 stations, with 56 stations used for model fitting and 5 stations used for prediction comparison. \nThe parameters to be estimated for the complete model are the dynamic coefficients $\\bm{\\theta}_{t}$ and $\\bm{\\eta}_t$, the covariance parameters ($\\sigma^2, \\tau^2, \\phi, \\alpha, \\gamma$), the mixing parameters $\\nu_1$, $\\nu_2$, the latent mixing processes $\\lambda_{1}(\\bm{s})$ and $\\lambda_{2t}$, and the variance regression coefficients $\\bm{\\beta}$. Analogous to the temperature application, smooth evolutions are assumed for the temporal evolution of trend and variance coefficients with discount factors $\\delta_1=0.99$ and $\\delta_2=0.99$, respectively. \nIn what follows we discuss the main results obtained for the best model (CovDynGLG) according to our predictive comparison measures (see Table \\ref{tabozone1}). The different criteria indicate that the Gaussianity is unlikely to hold for this dataset. The most complete models with dynamical effects in the variance have superior predictive performances under all criteria. \n\n\n\n\\begin{table}\n \n \\caption{Model comparison based on the Interval Score (IS), the Log Predictive Score (LPS) and the Variogram Score of order 0.25 (VS-0.25) criteria for the predicted observations at the out-of-sample locations under all fitted models for the maximum ozone dataset.\\label{tabozone1}}\n \\centering\n \\fbox{ \\begin{tabular}{lccccccc}\n \\hline\n \n& G & ST & GLG & CovDyn & DynGLG & CovDynGLG & Full\\\\\n \\hline\n IS & 78 & 75 & 76 & 77 & 70 & {\\bf 68} & 71\\\\\n LPS & 5960 & 5883 & 5696 & 5940 & 5631 & 5563 & {\\bf 5343}\\\\\nVS-25 & 10116 & 9760 & 10081 & 9842 & 9698 & {\\bf 9518} & 9535\\\\\n \\hline\n \\end{tabular}}\n\n\\end{table}\n\nPanels of Figure \\ref{figUK2} present the posterior summaries of the time varying coefficients in the mean \nfor the CovDynGLG model indicating that the maximum ozone mean changes substantially from March to November (Panel (a)). Latitude, longitude and wind are associated with ozone levels resulting in a non-constant behaviour across time, while temperature is mostly not associated with ozone levels as $0$ is within the limits of the 95\\% posterior credible interval.\n\n Panels of Figure \\ref{figUK3} present the time varying coefficients for temperature and wind in the precision model.\nNote that the coefficient for temperature is mostly negative in the precision model while in the mean model the 95\\% posterior credible interval contains zero for most of the instants in time. \nSpecifically, the temperature effect in the precision is negative in July, indicating smaller precision in the exponential scale, when indeed we observe the largest empirical temporal volatility of maximum ozone. For the time-varying coefficients of wind we observe a positive association both in the mean and variance models, however, in the mean model the coefficient has a decreasing pattern (Figure 7 (e)), whereas in the variance model it has an increasing pattern with time (Figure 8 (b)).\n\nFigure \\ref{figUKvar1} presents the posterior summaries for the standard deviation of $z_t(\\mathbf{s})$, $\\mathbf{s}=(50.74,-1.83)$ for the CovDynGLG model compared to the Gaussian model. Note that the variance is non-constant with large peaks in June and July. Differently, the Gaussian dynamic model suggests a nearly constant standard deviation across time.\nThis pattern has a direct effect on the predictive uncertainty of the Gaussian Model which does not capture many extreme observations and tend to have greater variability across the observed period, which is clear from panels of Figure \\ref{figUKvar2}. Note that model CovDynGLG captures the periods of extreme values of ozone while it has shorter ranges of the 95\\% credible intervals for those periods that observations do not change much across time. Regarding the complete model that includes the regression components in the equation for $\\lambda_1(\\bm{s})$, the 95\\% posterior credible interval for latitude is $(-0.0107,-0.0043)$ and for longitude is $(-0.1778,-0.0442)$, suggesting that both variables have a negative association with the precision over space. This results in smaller predictive precision for south-eastern locations. \n\n\n\n\n\n\n\n\n\\begin{figure\n\\begin{center}\n\\begin{tabular}{ccc}\n\\includegraphics[width=4.6cm]{d0UK2a.pdf}\n&\n \\includegraphics[width=4.6cm]{d1UK2a.pdf} &\n \\includegraphics[width=4.6cm]{d2UK2a.pdf}\\\\ {(a) Intercept. } &{(b) Latitude effect} & {(c) Longitude effect.} \\\\ \n \\includegraphics[width=4.6cm]{d3UK2a.pdf} &\n \\includegraphics[width=4.6cm]{d4UK2a.pdf} & \\\\\n{(d) Temperature effect.} & {(e) Wind effect.} & \\\\\n\\end{tabular}\n\\caption{Ozone data: Posterior summaries for the dynamic mean effects, $\\mathbf{\\theta}_t$ in equation (\\ref{eq5b}), under the CovDynGLG model.}\n\\label{figUK2}\n\\end{center}\n\\end{figure}\n\n\\begin{figure\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=5cm]{dd1UK2a.pdf}\n&\n \\includegraphics[width=5cm]{dd2UK2a.pdf} \\\\ \n {(a) Temperature effect. } &{(b) Wind effect} \\\\ \n\n\\end{tabular}\n\\caption{Ozone data: Posterior summaries of the coefficients included the equation for the time-varying variance (CovDynGLG model).\n\\label{figUK3}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\begin{tabular}{c}\n \\includegraphics[width=12cm]{SeriesEstimatedVarFullnew.pdf} \\\\\n \\end{tabular}\n\\caption{Ozone data: Approximated predictive standard deviation over time for the CovDynGLG model and the Gaussian model for ${\\bf s}=(50.74,-1.83)$.}\n\\label{figUKvar1}\n\\end{center}\n\\end{figure}\n\n \\begin{figure}[!ht]\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[width=12cm]{predozonefinals2.pdf} \\\\\n(a) ${\\bf s}=(50.74,-1.83)$\\\\\n\\includegraphics[width=12cm]{predozonefinals5.pdf} \\\\\n(b) ${\\bf s}=(52.95,-1.15)$\\\\\n\\end{tabular}\n\\caption{Ozone data: Predictive distribution ($95\\%$ interval) over time for the CovDynGLG model and the Gaussian model for ${\\bf s}=(50.74,-1.83)$ and ${\\bf s}=(52.95,-1.15)$. }\n\\label{figUKvar2}\n\\end{center}\n\\end{figure}\n\n\n\n\n\\newpage\n\\clearpage\n\n\\section{Conclusions}\\label{sec:conclusion}\nWe have proposed a flexible dynamical non-Gaussian spatio-temporal model that extends the well known multivariate dynamic linear model and accommodates both outliers and regions in space or time with larger observational variance. The dynamic evolution in the variance model proposed in equation (\\ref{eq:lambda21}) is able to account for different regimes of variability over time, which is a desirable feature when modelling environmental data in large temporal windows. For instance, the most complete models with covariates aiding in the representation of uncertainty over space and time presented the best performances in predicting the maximum ozone in the UK. This result indicates that patterns in periods of large variability could be explained by changes in wind and temperature that not only influence the mean but also have an impact on the description of the variance of the process. This results in a better description of the uncertainty associated with temporal predictions and spatial interpolations of interest. As inference is performed under the Bayesian paradigm using MCMC methods, we proposed an efficient sampling algorithm for inference and prediction. It takes advantage of the conditionally Gaussian distributions obtained when we condition the distribution of $Z_t({\\bf s})$ on the mixing latent variables. \n\nAs shown in Section \\ref{sec2.2}, the proposed model allows the resultant variance structure to change across space and time depending on the effect of covariates. Moreover, the kurtosis depends on the mixing scales $\\nu_1$ and $\\nu_2$, which reflect the inflation in the tails when necessary. The correlation structure, on the other hand will not change with the covariates but will have the effect of the correlation structure assumed for the variance model. The model generalizes the well known Gaussian model for spatio-temporal data and adds flexibility to the alternative Student-t model. Although the Student-t model allows for variance inflation, it increases the variance of the process in every location and does not allow for local changes in variability as our proposal does.\n\n\nWe performed extensive simulation studies to investigate the ability of the proposed model to capture different structures of the spatio-temporal process of interest. Our simulated examples in Section D of the Supplementary Material indicate that the correct model is selected with the complete model having worse performance when the data does not have the effect of covariates in the spatial mixing process. The non-Gaussian proposals have equivalent performance when fitted to the Gaussian simulated data. Thus, complexity is not always preferred suggesting that our model does not lead to overfitting. Moreover it seems that the predictive scoring rules used to compare the models are adequate measures of good predictive performance. \n\nWe conclude that allowing for a flexible model for the variance of the process provides coherent posterior predictive credible intervals that accommodates well the structure of the spatio-temporal process under study. A possible drawback of the proposed approach is that prediction of the process to future instants in time depend on covariates that are themselves spatio-temporal processes that need to be predicted. One possible solution is to consider a multivariate spatio-temporal process which is subject for future research.\n\n\n\n\n\n\n\\section*{Acknowledgments}\nSchmidt is grateful for financial support from the Natural Sciences and Engineering Research Council (NSERC) of Canada (Discovery Grant RGPIN-2017-04999).\n\n\n\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzlepc b/data_all_eng_slimpj/shuffled/split2/finalzzlepc new file mode 100644 index 0000000000000000000000000000000000000000..e0673ec3f97337dd00c8f83e5a81d66ca7489f8d --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzlepc @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\\label{Introd}\nDual quaternions, which serve as the core part of Clifford algebra or geometric algebra, were originally introduced by Clifford \\cite{Cl73} in 1873. Mathematically, the dual quaternions are an $8$-dimensional real algebra isomorphic to the tensor product of the quaternions and the dual numbers. In mechanics, the dual quaternions give a neat and succint way to encapsulate both translations and rotaions into a unified representation for rigid transformations in three dimensions. With the rapid developments of artificial intelligence, dual quaternions have been successfully applied to the areas of automatic control, computer vision and bioengineering, such as robotics control \\cite{Da99,MXXLY18,Tho14,WYL12,WZ14} and 3D computer graphics \\cite{BLH19,CKJC16,TRA11}, neuroscience \\cite{LLB13} and biomechanics \\cite{PV10}, to name just a few.\nMore interestingly, we notice that dual quaternion matrices have important application in multi-agent formation control \\cite{QWL22}. In 2011, Wang \\cite{Wa11} raised this issue. In an unpublished manuscript, Wang, Yu and Zheng \\cite{WYZ} introduced three classes of dual quaternion matrices for studying multi-agent formation control, namely relative configuration adjacency matrices, logarithm adjacency matrices and relative twist adjacency matrices. Recently, Qi and Luo \\cite{QL21} showed that dual quaternion Hermitian matrices have very nice spectral properties. They showed that an $n \\times n$ dual quaternion Hermitian matrix has exactly $n$ right eigenvalues, which are all dual numbers and are also the left eigenvalues of this Hermitian matrix. Thus, we may simply call them eigenvalues of that Hermitian matrix. This Hermitian matrix is positive semi-definite or positive definite if and only if these $n$ dual number eigenvalues are nonnegative or positive in the sense of \\cite{QLY21}, respectively. Moreover, the minimax principle and generalized inverses of dual quaternion matrices were studied in \\cite{LQY22}. Then, Qi, Wang and Luo \\cite{QWL22} showed that relative configuration adjacency matrices and logarithm adjacency matrices are Hermitian matrices, and may form positive semi-definite Laplacian matrices with the degree matrix of the mutual visibility graph of the formation control. This linked the study of dual quaternion matrices and multi-agent formation control.\n\n\n\nAs we know, the well-known von Neumann trace inequality \\cite{VN37} (see also \\cite{Mir75}), which bounds the inner product of two matrices via the inner product of their singular vectors, is the key inequality for the analysis of spectral functions and plays a pivot role in the developments of low-rank matrix approximation theory and low-rank optimization, e.g., see \\cite{HJ13}. Since quaternion matrices have wide engineering applications \\cite{CQZ22,CXZ20,Gri17,LM04}, quaternion matrix theory has been received considerable attention in recent years \\cite{WLZZ18,XM15,Zh97}. Particularly, to derive quaternionic proximity operators for trace-norm regularized optimization problems arising from audio separation, Chan and Yang \\cite{CY16} first proved that the von Neumann trace inequality still holds for quaternion matrices. As a combination of dual numbers and quaternions, dual quaternion matrices have been applied to multi-agent formation control \\cite{QWL22}. In theoretical aspects, however, the existence of zero divisors, i.e., infinitesimal dual quaternion numbers makes analysis on dual quaternion matrices difficult. It is still unknown whether the von Neumann trace inequality still holds for dual quaternion matrices. Therefore, to answer such question, we first introduce the spectral norm for dual quaternion matrices in this paper. Then, we present a von Neumann type trace inequality for dual quaternion matrices, which then paves the way to establish a Hoffman-Wielandt type inequality characterizing a simultaneous perturbation bound for all singular values of a general dual quaternion matrix. It is worth pointing out that, when the dual quaternion reduces the quaternion, the von Neumann inequality obtained in this paper is exactly the one presented in \\cite{CY16}, but our proof method is completely different from the way used in \\cite{CY16}, even in the case of quaternions. Moreover, by considering the application of dual quaternion Hermitian matrices in multi-agent formation control, we also discuss the above two inequalities for dual quaternion Hermitian matrices. We believe that our results will enrich the theory of dual quaternion matrices, and they will be of benefit to further study of dual quaternion matrices, algorithmic design, and applications.\n\nThis paper is divided into five sections. In Section \\ref{Prelim}, we present some preliminaries on dual numbers, quaternions, dual quaternions and dual quaternion algebra. In Section \\ref{Spec-Norm}, we introduce the concept of the spectral norm for dual quaternion matrices, which is exactly the largest singular value of the involved dual quaternion matrix. In Section \\ref{VN-TraceIn}, we present a von Neumann type trace inequality for dual quaternion matrices. In Section \\ref{sec_HWineq}, we consider the extension of the well-known Hoffman-Wielandt inequality to dual quaternionic versions. Finally, some concluding remarks are drawn in Section \\ref{Conclusion}.\n\n\\section{Preliminaries}\\label{Prelim}\n\n\\subsection{Dual numbers}\\label{Du-number}\nLet $\\mathbb{R}$ and $\\widehat{\\mathbb R}$ denote the field of the real numbers and the set of the dual numbers, respectively. A dual number $q\\in \\widehat{\\mathbb R}$ has the form $q = q_{\\sf st} + q_{\\sf in}\\epsilon$, where $q_{\\sf st}\\in \\mathbb{R}$ is called the real part or the standard part; $q_{\\sf in}\\in \\mathbb{R}$ represents the dual part or the infinitesimal part of $q$; and $\\epsilon$ is the infinitesimal unit satisfying $\\epsilon^2 = 0$. Particularly, if the standard part $q_{\\sf st}$ of $q$ is nonzero, i.e., $q_{\\sf st} \\not = 0$, we say that $q$ is appreciable; otherwise, we say that $q$ is infinitesimal. Note that the infinitesimal unit $\\epsilon$ is commutative in multiplication with real numbers, complex numbers and\n quaternion numbers (see Section \\ref{Qua_number}). The dual numbers form a commutative algebra of dimension two over the reals.\n\nNow, we recall the recently introduced total order for dual numbers in \\cite{QLY21}. Given two dual numbers $p = p_{\\sf st} + p_{\\sf in}\\epsilon, q = q_{\\sf st} + q_{\\sf in}\\epsilon \\in \\widehat{\\mathbb R}$ with $ p_{\\sf st}, p_{\\sf in},q_{\\sf st}, q_{\\sf in}$ being real numbers, we say that $q < p$ if either $q_{\\sf st} < p_{\\sf st}$, or $q_{\\sf st} = p_{\\sf st}$ and $q_{\\sf in} < p_{\\sf in}$, and we say that $q = p$ if $q_{\\sf st} = p_{\\sf st}$ and $q_{\\sf in} = p_{\\sf in}$. Consequently, if $q > 0$, we say that $q$ is a positive dual number; and if $q \\ge 0$, we say that $q$ is a nonnegative dual number. In what follows, we denote the set of nonnegative and positive dual numbers by $\\widehat{\\mathbb R}_+$ and $\\widehat{\\mathbb R}_{++}$, respectively. For given $p = p_{\\sf st} + p_{\\sf in}\\epsilon, q = q_{\\sf st} + q_{\\sf in}\\epsilon \\in \\widehat{\\mathbb R}$,\nwe have\n\\begin{equation} \\label{e1}\np + q =p_{\\sf st}+q_{\\sf st} +(p_{\\sf in}+q_{\\sf in})\\epsilon,~~~~pq = p_{\\sf st}q_{\\sf st} +(p_{\\sf st}q_{\\sf in}+p_{\\sf in} q_{\\sf st})\\epsilon.\\end{equation}\nFollowing the definition in \\cite{QLY21}, the absolute value of $q=q_{\\sf st}+q_{\\sf in}\\epsilon\\in \\widehat{\\mathbb R}$ is defined by\n\\begin{equation}\\label{e5}\n|q| = \\left\\{ \\begin{array}{ll}|q_{\\sf st}| + {\\rm sgn}(q_{\\sf st})q_{\\sf in}\\epsilon, & {\\rm if~} q_{\\sf st} \\not = 0, \\\\\n|q_{\\sf in}|\\epsilon, & {\\rm otherwise}, \\end{array} \\right.\n\\end{equation}\nwhere `${\\rm sgn}(\\cdot)$' represents the sign function, that is, for any $u \\in \\mathbb R$,\n$${\\rm sgn}(u) = \\left\\{ \\begin{aligned} 1, & \\ {\\rm if}\\ u > 0, \\\\ 0, & \\ {\\rm if}\\ u = 0, \\\\\n-1, & \\ {\\rm if}\\ u < 0. \\end{aligned} \\right. $$\nFor a given dual number $q=q_{\\sf st}+q_{\\sf in}\\epsilon$, if $q$ is appreciable, then $q$ is nonsingular and $q^{-1} = q_{\\sf st}^{-1} - q_{\\sf st}^{-1}q_{\\sf in} q_{\\sf st}^{-1}\\epsilon$. If $q$ is infinitesimal, then $q$ is not nonsingular. If $q$ is nonnegative and appreciable, then the square root of $q$ is still a nonnegative dual number. If $q$ is positive and appreciable, we have\n\\begin{equation} \\label{e4}\n\\sqrt{q} = \\sqrt{q_{\\sf st}} + {q_{\\sf in} \\over 2\\sqrt{q_{\\sf st}}}\\epsilon.\n\\end{equation}\nIn particular, we have $\\sqrt{q} = 0$ when $q=0$.\n\nBelow, we recall a proposition introduced in \\cite{LQY22}.\n\\begin{Prop}\\label{P6.5}\nLet $p,q\\in \\widehat{\\mathbb R}$. Then, we have the following conclusions.\n\\begin{itemize}\n\\item[{\\rm(a).}] If $p, q \\in \\widehat{\\mathbb R}_{+}$, then $pq \\in \\widehat{\\mathbb R}_{+}$.\n\\item[{\\rm(b).}] If $p, q \\in \\widehat{\\mathbb R}_{++}$ and at least one of them is appreciable, then $pq \\in \\widehat{\\mathbb R}_{++}$.\n\\item[{\\rm(c).}] If $p\\geq 0$, then $|p|=p$; otherwise, $|p|>p$.\n\\item[{\\rm(d).}] If $p$ is appreciable, then $|p|=\\sqrt{p^2}$.\n\\item[{\\rm(e).}] If $p, q \\in \\widehat{\\mathbb R}_{++}$ and are both appreciable, then $\\sqrt{pq}=\\sqrt{p}\\sqrt{q}$.\n\\item[{\\rm(f).}] If $q\\in \\widehat{\\mathbb R}_{++}$ and is appreciable, then $p-q \\in \\widehat{\\mathbb R}_{+}$ implies $\\sqrt{p}-\\sqrt{q} \\in \\widehat{\\mathbb R}_{+}$.\n\\end{itemize}\n\\end{Prop}\n\n\n\\subsection{Quaternions}\\label{Qua_number}\nDenote by $\\mathbb Q$ the four-dimensional vector space of the quaternions over $\\mathbb{R}$, with an ordered basis,\ndenoted by $\\ee, \\ii, \\jj$ and $\\kk$. A quaternion $q\\in \\mathbb{Q}$ has the form\n$q = q_0\\ee + q_1\\ii + q_2\\jj + q_3\\kk$, where $q_0, q_1, q_2$ and $q_3$ are real numbers, $\\ii, \\jj$ and $\\kk$ are three imaginary units of quaternions, which satisfy\n$$\\ii^2 = \\jj^2 = \\kk^2 =\\ii\\jj\\kk = -1,\\quad \\ii\\jj = -\\jj\\ii = \\kk, \\quad \\jj\\kk = - \\kk\\jj = \\ii, \\quad \\kk\\ii = -\\ii\\kk = \\jj.$$\nFor notational simplicity, we will omit the real unit $\\ee$ and denote $q \\in \\mathbb Q$ as $q = q_0 + q_1\\ii + q_2\\jj + q_3\\kk$ in the subsequent discussions.\n\nFor a quaternion $q = q_0 + q_1\\ii + q_2\\jj + q_3\\kk$, we call ${\\rm Re}(q):= q_0$ and ${\\rm Im}(q):= q_1\\ii + q_2\\jj +q_3\\kk$ the real and imaginary parts, respectively; the conjugate of $q$ is given by $\\bar q := q_0 -q_1\\ii -q_2\\jj -q_3\\kk$ and the norm of $q$ is defined by $|q|:=\\sqrt{\\bar qq}=\\sqrt{q_0^2+q_1^2+q_2^2+q_3^2}$.\nIn particular, a quaternion is called imaginary if its real part is zero. Given two quaternions $p = p_0 + p_1\\ii + p_2\\jj + p_3\\kk$ and $q = q_0 + q_1\\ii + q_2\\jj + q_3\\kk$, it is easy to verify that\n\\begin{equation}\\label{pqmut}\np\\bar q+q\\bar p=\\bar pq +\\bar qp = 2(p_0q_0+p_1q_1+p_2q_2+p_3q_3).\n\\end{equation}\nHowever, we shall notice that the multiplication of quaternions satisfies the distribution law, but is noncommutative. In fact, $\\mathbb{Q}$ is an associative but non-commutative algebra of four rank over $\\mathbb{R}$, called quaternion skew-field \\cite{WLZZ18}.\n\nThroughout this paper, we denote by $\\mathbb{Q}^{m\\times n}$ the collection of all $m\\times n$ matrices with quaternion entries. Specially, $\\mathbb{Q}^{m\\times 1}$ is abbreviated as $\\mathbb{Q}^m$, which is the collection of quaternion column vectors with $m$ components.\n\n\\subsection{Dual quaternions and dual quaternion algebra}\n\\subsubsection{Dual quaternions}\nDual quaternion is a composite concept, which is the combination of dual numbers and quaternions. Specifically, a dual quaternion $q$ has the form $q = q_{\\sf st} + q_{\\sf in}\\epsilon$,\nwhere $q_{\\sf st}, q_{\\sf in} \\in \\mathbb {Q}$ are the standard part and the infinitesimal part of $q$, respectively. Throughout, we denote by $\\widehat{\\mathbb Q}$ the set of dual quaternions. Recalling the definitions introduced in \\cite{BK20, CKJC16, Ke12}, for any two dual quaterions $p=p_{\\sf st} + p_{\\sf in}\\epsilon$ and $q=q_{\\sf st} + q_{\\sf in}\\epsilon$, the addition and multiplication between them are defined by $$p+q=(p_{\\sf st}+q_{\\sf st})+(p_{\\sf in}+q_{\\sf in})\\epsilon$$\nand\n$$pq=p_{\\sf st}q_{\\sf st} + (p_{\\sf in}q_{\\sf st}+p_{\\sf st}q_{\\sf in})\\epsilon,$$\nrespectively. It is easy to see that $\\widehat{\\mathbb Q}$ is a ring with respect to the two binary algebraic operations defined above. The conjugate of $q$ is $\\bar q = \\bar q_{\\sf st} + \\bar q_{\\sf in}\\epsilon$. Similar to dual numbers, if $q_{\\sf st} \\not = 0$, then we say that $q$ is appreciable, otherwise, we say that $q$ is infinitesimal. We can derive that $q$ is invertible if and only if $q$ is appreciable. In this case, we have $q^{-1} = q_{\\sf st}^{-1} - q_{\\sf st}^{-1}q_{\\sf in} q_{\\sf st}^{-1} \\epsilon$. The magnitude of $q\\in \\widehat{\\mathbb Q}$ is defined as\n\\begin{equation} \\label{e7}\n\\displaystyle|q| := \\left\\{ \\begin{array}{ll} |q_{\\sf st}| +\\displaystyle {(q_{\\sf st}\\bar q_{\\sf in}+q_{\\sf in} \\bar q_{\\sf st}) \\over 2|q_{\\sf st}|}\\epsilon, & \\ {\\rm if}\\ q_{\\sf st} \\not = 0, \\\\\n|q_{\\sf in}|\\epsilon, & \\ {\\rm otherwise},\n\\end{array} \\right.\n\\end{equation}\nwhich is a dual number by (\\ref{pqmut}). Notice that such a definition immediately reduces to the absolute function \\eqref{e5} when $q$ is a dual number, i.e., $q \\in \\widehat{\\mathbb R}$, and it is exactly the magnitude of a quaternion when $q \\in \\mathbb Q$.\n\n\n\\subsubsection{Dual quaternion algebra}\nDenote by $\\widehat{\\mathbb Q}^{m}$ the set of all dual quaternion vectors with $m$ components. For any two $m$-dimensional dual quaternion vectors ${\\bf x}=(x_1,x_2,\\ldots,x_m)^\\top$, ${\\bf y}=(y_1,y_2,\\ldots,y_m)^\\top$ and a dual quaternion $\\alpha\\in \\widehat{\\mathbb Q}$, we define\n$${\\bf x}+{\\bf y}=(x_1+y_1,x_2+y_2,\\ldots,x_m+y_m)^\\top\\quad \\text{and}\\quad {\\bf x}\\alpha=(x_1\\alpha,x_2\\alpha,\\ldots,x_m\\alpha)^\\top,$$\nwhich is called the right multiplication of ${\\bf x}\\in \\widehat{\\mathbb Q}^m$ and $\\alpha\\in \\widehat{\\mathbb Q}$. It is easy to verify that $\\widehat{\\mathbb Q}^m$ is an $m$-dimensional vector space over $\\widehat{\\mathbb Q}$, with respect to the addition and right multiplication defined above.\n\\begin{Def}[\\cite{LQY22}]\\label{Def-DQVecIn}\nLet $\\Xi:=\\{{\\bf u}^{(1)},{\\bf u}^{(2)},\\ldots,{\\bf u}^{(s)}\\}\\subset \\widehat{\\mathbb Q}^m$. We say that $\\Xi$ is right linearly independent, if for any $\\alpha_1,\\alpha_2,\\ldots,\\alpha_s\\in \\widehat{\\mathbb Q}$,\n$$\n{\\bf u}^{(1)}\\alpha_1+{\\bf u}^{(2)}\\alpha_2+\\ldots+{\\bf u}^{(s)}\\alpha_s={\\bf 0}~~~\\Rightarrow~~~\\alpha_1=\\alpha_2=\\ldots=\\alpha_s=0.\n$$\n\\end{Def}\nAs a result of Definition \\ref{Def-DQVecIn}, we can see that, if $\\Xi$ is right linearly independent, then ${\\bf u}^{(i)}$ is appreciable for every $i=1,2,\\ldots,s$. \nFor given ${\\bf u}=(u_1,u_2,\\ldots,u_m)^\\top$ and ${\\bf v}=(v_1,v_2,\\ldots,v_m)^\\top$ in $\\widehat{\\mathbb Q}^m$, denote by $\\langle {\\bf u},{\\bf v}\\rangle$ the dual quaternion-valued inner product, i.e., $\\langle {\\bf u},{\\bf v}\\rangle=\\sum_{i=1}^m\\bar v_iu_i$. It is easy to see that $\\langle {\\bf u}, {\\bf v} \\alpha+{\\bf w} \\beta\\rangle= \\bar\\alpha\\langle {\\bf u}, {\\bf v}\\rangle +\\bar\\beta\\langle{\\bf u},{\\bf w}\\rangle$ and $\\langle {\\bf u},{\\bf v}\\rangle=\\overline{\\langle {\\bf v},{\\bf u}\\rangle}$ for any ${\\bf u},{\\bf v}, {\\bf w}\\in \\widehat{\\mathbb Q}^m$ and $\\alpha,\\beta\\in \\widehat{\\mathbb Q}$. If ${\\bf u}\\in \\widehat{\\mathbb Q}^m$ is appreciable, then $\\langle {\\bf u},{\\bf u}\\rangle$ is an appreciable positive dual number.\n\n\\begin{Def\n\\label{Def-Orthog}\nLet ${\\bf u},{\\bf v}\\in \\widehat{\\mathbb Q}^m$. We say that ${\\bf u},{\\bf v}$ are orthogonal if $\\langle {\\bf u},{\\bf v}\\rangle=0$. An $n$-tuple $\\{{\\bf u}^{(1)},{\\bf u}^{(2)},\\ldots, {\\bf u}^{(s)}\\}$, where ${\\bf u}^{(1)},{\\bf u}^{(2)},\\ldots, {\\bf u}^{(s)}\\in \\widehat{\\mathbb Q}^m$, is said to be orthogonal if $\\langle {\\bf u}^{(i)},{\\bf u}^{(j)}\\rangle=0$ for $i\\neq j$, and orthonormal if it is orthogonal and $\\langle {\\bf u}^{(i)},{\\bf u}^{(i)}\\rangle=1$ for $i=1,2,\\ldots,s$.\n\\end{Def}\n\n\nDenote by $\\widehat{\\mathbb Q}^{m\\times n}$ the set of $m\\times n$ dual quaternion matrices. Then $A=(a_{ij})\\in \\widehat{\\mathbb Q}^{m\\times n}$ can\nbe written as $A = A_{\\sf st} + A_{\\sf in}\\epsilon$, where $A_{\\sf st}, A_{\\sf in}\\in \\mathbb{Q}^{m\\times n}$ are the standard part and the infinitesimal part of $A$, respectively. If $A_{\\sf st}$ is nonzero, i.e., $A_{\\sf st} \\not = O$, we say that $A$ is appreciable, otherwise, we say that $A$ is infinitesimal. For given $A\\in \\widehat{\\mathbb Q}^{m\\times n}$, the transpose of $A$ is denoted as $A^\\top = (a_{ji})$, the conjugate of $A$ is denoted as ${\\bar A} = (\\bar a_{ij})$, and the conjugate transpose of $A$ is denoted as $A^*=(\\bar a_{ji})=\\bar A^\\top$. It is obvious that $A^\\top = A_{\\sf st}^\\top + A_{\\sf in}^\\top\\epsilon$, $\\bar A = \\bar A_{\\sf st}+\\bar A_{\\sf in}\\epsilon$ and $A^* = A_{\\sf st}^* + A_{\\sf in}^*\\epsilon$. In this paper, a square matrix $A\\in \\widehat{\\mathbb Q}^{m\\times m}$ is called nonsingular (invertible) if $AB = BA = I_m$ for some $B\\in \\widehat{\\mathbb Q}^{m\\times m}$. In that situation, we denote $A^{-1} = B$. Moreover, a square matrix $A\\in \\widehat{\\mathbb Q}^{m\\times m}$ is called normal if $AA^*=A^*A$, Hermitian if $A^*=A$, and unitary if $A$ is nonsingular and $A^{-1}=A^*$. We have $(AB)^{-1}=B^{-1}A^{-1}$ if $A$ and $B$ are nonsingular, and $(A^*)^{-1}=(A^{-1})^*$ if $A$ is nonsingular. We say that $A\\in \\widehat{\\mathbb Q}^{m\\times m}$ is unitary, if $A$ satisfies $A^*A=I_m$. It is obvious that $A$ is unitary, if and only if the set consisting of column (row) vectors form an orthonormal basis of $\\widehat{\\mathbb Q}^m$, i.e., it is orthonormal and any vector in $\\widehat{\\mathbb Q}^m$ can be written as a right linear combination of this set. Similarly, we say that $A\\in \\widehat{\\mathbb Q}^{m\\times s} (s\\leq m)$ is partially unitary, if $A$ satisfies $A^*A=I_s$. From Definition \\ref{Def-DQVecIn}, it is easy to see that, the right linear independence of the involved vector set $\\Xi$ is essentially that $A{\\bf x}={\\bf 0}$ has a unique zero solution in $\\widehat{\\mathbb Q}^n$, where $A=[{\\bf u}^{(1)},\\ldots,{\\bf u}^{(s)}]\\in \\widehat{\\mathbb Q}^{m\\times s}$. For given $A=(a_{ij})\\in \\hat{\\mathbb{Q}}^{m\\times m}$, the trace of $A$, named ${\\rm trace}(A)$, is defined as\n\\begin{equation}\\label{TraceDef}\n{\\rm trace}(A)=\\sum_{i=1}^ma_{ii},\n\\end{equation}\nwhich is a dual quaternion.\n\nAs defined in \\cite{QL21}, for a dual quaternion matrix $A\\in \\widehat{\\mathbb Q}^{m\\times m}$, if there exist a $\\lambda\\in \\widehat{\\mathbb Q}$ and an appreciable ${\\bf x}\\in \\widehat{\\mathbb Q}^m$ such that\n\\begin{equation}\\label{RightEig}\nA{\\bf x} = {\\bf x}\\lambda,\n\\end{equation}\nthen we say that $\\lambda$ is a right eigenvalue of $A$, with ${\\bf x}$ as an associated right eigenvector. If $\\lambda$ is a dual number, then we have\n\\begin{equation}\\label{LeftEig}\nA{\\bf x} = \\lambda{\\bf x},\n\\end{equation}\ni.e., $\\lambda$ is also a left eigenvalue of $A$. In this case, $\\lambda$ is simply called an eigenvalue of $A$, and $\\bf x$ an associated eigenvector. In particular, it was shown in \\cite{QL21} that an $m \\times m$ dual quaternion Hermitian matrix has exactly $m$ dual number eigenvalues.\n\n\n\nFor given $A = A_{\\sf st} + A_{\\sf in} \\epsilon = (a_{ij}) \\in {\\widehat{\\mathbb Q}}^{m \\times n}$, the Frobenius norm of $A$, which is a dual number, is defined by\n\\begin{equation}\\label{FNorm-DQM}\n\\|A \\|_F = \\left\\{\\begin{aligned}\\sqrt{\\sum_{i=1}^m \\sum_{j=1}^n |a_{ij}|^2}, & \\quad\\ {\\rm if}\\ A_{\\sf st} \\not = O, \\\\\n\\|A_{\\sf in}\\|_F\\epsilon,\\quad & \\quad \\ {\\rm otherwise.} \\end{aligned}\\right.\n\\end{equation}\nClearly, the Frobenius norm of a matrix is actually the $\\ell_2$-norm of the vectorization of that matrix. Most recently, Ling et al. \\cite{LQY22} proved a dual quaternion version of Cauchy-Schwarz inequality, which can be stated as follows.\n\\begin{Prop}[Cauchy-Schwarz inequality on $\\widehat{\\mathbb Q}^m$]\\label{Ch-SW-In}\n\tFor any ${\\bf u},{\\bf v}\\in \\widehat{\\mathbb Q}^m$, it holds that $$\\|{\\bf u}\\|_2\\|{\\bf v}\\|_2-|\\langle{\\bf u},{\\bf v}\\rangle|\\in \\widehat{\\mathbb R}_+,$$\n\tthat is, $|\\langle{\\bf u},{\\bf v}\\rangle|\\leq\\|{\\bf u}\\|_2\\|{\\bf v}\\|_2$.\n\\end{Prop}\nAs a consequence of (\\ref{FNorm-DQM}) and Proposition \\ref{Ch-SW-In}, for given $A \\in {\\widehat{\\mathbb Q}}^{m \\times n}$ and $\\vx \\in {\\widehat{\\mathbb Q}}^n$, regardless of whether $A\\vx$ is appreciable or not,\nit can be verified that\n$\\|A\\vx\\|_2 \\le \\|A\\|_F \\|\\vx\\|_2.$\n\n\\begin{Prop} \\label{pp3.2}\nSuppose that $U \\in {\\widehat{\\mathbb Q}}^{m \\times n}$ is partially unitary, and $\\vx \\in {\\widehat{\\mathbb Q}}^n$. Then\n\\begin{equation} \\label{eq5}\n\\|U\\vx\\|_2 = \\|\\vx\\|_2.\n\\end{equation}\n\\end{Prop}\n\\begin{proof} Suppose that $\\vx = \\vx_{\\sf st}+\\vx_{\\sf in} \\epsilon$ is appreciable.\n It follows from (\\ref{FNorm-DQM}) that\n$$\\|\\vx\\|_2^2 = \\sum_{i=1}^n |x_i|^2 = \\vx^*\\vx.$$\nOn the other hand, let $U = U_{\\sf st}+U_{\\sf in}\\epsilon$. A direct calculation leads to $U_{\\sf st}^* U_{\\sf st}=I_n$. \nThen, the standard part of $U\\vx$ is $U_{\\sf st}\\vx_{\\sf st} \\not = \\0$, i.e., $U\\vx$ is also appreciable. Consequently, by (\\ref{FNorm-DQM}) again, we have\n$$\\|U\\vx\\|_2^2 = (U\\vx)^*(U\\vx) = \\vx^*U^*U\\vx = \\vx^*\\vx.$$\nHence, $\\|U\\vx\\|_2 = \\|\\vx\\|_2$ in this case.\n\nNow, we assume that $\\vx$ is infinitesimal, i.e., $\\vx = \\vx_{\\sf in}\\epsilon$. Then $U\\vx = U_{\\sf st}\\vx_{\\sf in}\\epsilon$, which means that $U\\vx $ is also infinitesimal. We have $\\|U_{\\sf st}\\vx_{\\sf in}\\|_2 = \\|\\vx_{\\sf in}\\|_2$ since $U_{\\sf st}^* U_{\\sf st}=I_n$. Then by (\\ref{FNorm-DQM}), we still have\n$\\|U\\vx\\|_2 = \\|\\vx\\|_2$ in this case.\n\\end{proof}\n\n\\begin{Prop}[\\cite{QLY21}]\\label{p6.3}\nFor any $\\vu =\\vu_{\\sf st} + \\vu_{\\sf in}\\epsilon\\in{\\widehat{\\mathbb Q}}^m$ with $\\vu_{\\sf st}\\neq \\0$, it holds that\n\\begin{equation}\\label{e12-1}\n\\|\\vu\\|_2=\\|\\vu_{\\sf st}\\|_2+\\displaystyle\\frac{\\langle\\vu_{\\sf st},\\vu_{\\sf in}\\rangle+\\langle\\vu_{\\sf in},\\vu_{\\sf st}\\rangle}{2\\|\\vu_{\\sf st}\\|_2}\\epsilon.\n\\end{equation}\n\\end{Prop}\n\n\\section{Spectral norm of dual quaternion matrices}\\label{Spec-Norm}\nWe begin this section with recalling the following two theorems for dual quaternion matrices.\n\n\\begin{Thm}[\\cite{QL21}]\\label{HUDec}\n\t Let $A=A_{\\sf st}+A_{\\sf in}\\epsilon\\in \\widehat{\\mathbb Q}^{m\\times m}$ be a Hermitian matrix. Then, there exists a unitary matrix $U\\in \\widehat{\\mathbb Q}^{m\\times m}$ and a diagonal matrix $\\Sigma\\in \\widehat{\\mathbb Q}^{m\\times m}$ such that $A=U\\Sigma U^*$, where\n\\begin{equation}\\label{Sigmn}\\Sigma := {\\rm diag} (\\lambda_1(A),\\lambda_2(A),\\ldots,\\lambda_m(A)),\n\\end{equation}\nwhere $\\lambda_1(A)\\geq\\lambda_2(A)\\geq\\ldots\\geq\\lambda_m(A)$ are dual numbers. Counting possible multiplicities $\\lambda_{i,j}$, the form $\\Sigma$ is unique.\n\\end{Thm}\n\nIt is obvious that $\\lambda_i(A)$ is the $i$th largest eigenvalue of $A$, with ${\\bf u}^{(i)}$ as an associated eigenvector, where ${\\bf u}^{(i)}$ is the $i$th column in $U$. When $A$ is Hermitian, since $\\lambda_1(A), \\lambda_2(A), \\ldots,\\lambda_m(A)$ are dual numbers, from (\\ref{Sigmn}) and (\\ref{pqmut}), it is easy to see that\n$$\na_{ii}=\\sum_{j=1}^m\\lambda_j(A)u_{ij}\\bar u_{ij}=\\sum_{j=1}^m\\lambda_j(A)\\bar u_{ij}u_{ij},\n$$\nwhich implies that\n\\begin{equation}\\label{sumEigV}\n\\sum_{i=1}^ma_{ii}=\\sum_{j=1}^m\\lambda_j(A)\\sum_{i=1}^m\\bar u_{ij}u_{ij}=\\sum_{j=1}^m\\lambda_j(A)\\|{\\bf u}^{(j)}\\|_2^2=\\sum_{j=1}^m\\lambda_j(A),\n\\end{equation}\nsince $\\|{\\bf u}^{(j)}\\|_2=1$. Hence, if $A$ is Hermitian, we have ${\\rm trace}(A)={\\rm trace}(UAU^*)$ for any unitary matrix $U\\in \\widehat{\\mathbb{Q}}^{m\\times m}$.\n\n\n\\begin{Thm}[\\cite{QL21}]\\label{SVD-DQM}\nFor a given $A\\in \\widehat{\\mathbb Q}^{m\\times n}$, there exist two dual quaternion unitary matrices $U\\in \\widehat{\\mathbb Q}^{m\\times m}$ and $V\\in \\widehat{\\mathbb Q}^{n\\times n}$, such that\n\\begin{equation}\\label{SVDDQMEQ}\nA=U\\left[\\begin{array}{cc}\\Sigma_t&O\\\\\nO&O\n\\end{array} \\right]_{m\\times n}V^*,\n\\end{equation}\nwhere $\\Sigma_t\\in \\widehat{\\mathbb R}^{t\\times t}$ is a diagonal matrix, taking the form\n$\\Sigma_t={\\rm diag} (\\sigma_1(A),\\ldots, \\sigma_r(A),\\ldots,\\sigma_t(A))$, $r \\leq t\\leq s:={\\min}\\{m, n\\}$, $\\sigma_1(A)\\geq \\sigma_2(A)\\geq\\ldots\\geq\\sigma_r(A)$ are positive appreciable dual numbers, and $\\sigma_{r+1}(A)\\geq \\sigma_{r+2}(A)\\geq\\ldots\\geq\\sigma_t(A)$ are positive infinitesimal dual numbers. Counting possible multiplicities of the diagonal entries, the form $\\Sigma_t$ is unique.\n\\end{Thm}\n\nWe call form \\eqref{SVDDQMEQ} the singular value decomposition (SVD) for dual quaternion matrix $A$, and call $\\sigma_1(A),\\ldots, \\sigma_r(A),\\ldots,\\sigma_t(A)$ and possibly $\\sigma_{t+1}(A)=\\ldots=\\sigma_s(A)=0$ (if $t< s)$ the singular values of\n$A$, where $t$ and $s$ correspond to the rank and the appreciable rank of $A$, respectively.\n\n\nFor given $A\\in {\\widehat{\\mathbb Q}}^{m \\times n}$, the spectral norm $\\|A\\|_2$ is defined by\n\\begin{equation}\\label{SNorm-DQM}\n\\|A \\|_2 = \\max_{{\\bf x}\\in \\widehat{\\mathbb Q}^n,~\\|{\\bf x}\\|_2=1}\\|A{\\bf x}\\|_2.\n\\end{equation}\nNotice that $\\|A\\|_2$ is induced by the $\\ell_2$-norm on dual quaternion vector spaces and hence is a matrix norm. In addition, by (\\ref{SNorm-DQM}), we have\n\\begin{equation}\\label{SpNorm-In}\n\\|A{\\bf x}\\|_2\\leq \\|A\\|_2\\|{\\bf x}\\|_2\n\\end{equation}\nfor any appreciable ${\\bf x}\\in \\widehat{\\mathbb Q}^n$.\nThe following proposition shows that, similar to the common complex matrix situation, $\\|A\\|_2$ defined by \\eqref{SNorm-DQM} is exactly the largest singular value of $A$.\n\n \\begin{Prop}\n Let $A\\in \\widehat{\\mathbb Q}^{m\\times n}$. It holds that $\\|A\\|_2=\\sigma_1(A)$, where $\\sigma_1(A)$ is the largest singular value of $A$.\n \\end{Prop}\n\n \\begin{proof}\nLet $A=U\\Sigma V^*$ be a singular value decomposition of $A$, in which $U$ and $V$ are unitary, $\\Sigma={\\rm diag}(\\sigma_1(A),\\ldots,\\sigma_s(A))$ with $\\sigma_1(A)\\geq\\ldots\\geq\\sigma_s(A)\\geq0$ and $s=\\min\\{m,n\\}$. It follows from (\\ref{SNorm-DQM}) and Proposition \\ref{pp3.2} that\n$$\n\\begin{array}{lll}\n\\|A \\|_2 &=&\\displaystyle \\max_{{\\bf x}\\in \\widehat{\\mathbb Q}^n,~\\|{\\bf x}\\|_2=1}\\|\\Sigma V^*{\\bf x}\\|_2\\\\\n&=&\\displaystyle \\max_{{\\bf y}\\in \\widehat{\\mathbb Q}^n,~\\|V{\\bf y}\\|_2=1}\\|\\Sigma {\\bf y}\\|_2\\\\\n&=&\\displaystyle \\max_{{\\bf y}\\in \\widehat{\\mathbb Q}^n,~\\|{\\bf y}\\|_2=1}\\sqrt{\\sum_{i=1}^n\\sigma^2_i(A) |y_i|^2}\\\\\n&\\leq&\\displaystyle \\max_{{\\bf y}\\in \\widehat{\\mathbb Q}^n,~\\|{\\bf y}\\|_2=1}\\sigma_1(A) \\sqrt{\\sum_{i=1}^n|y_i|^2}\\\\\n&=&\\sigma_1(A),\n\\end{array}\n$$\nwhere the inequality comes from items (a), (c) and (f) of Proposition \\ref{P6.5}. However, $\\|A{\\bf y}\\|_2= \\sigma_1(A)$ for ${\\bf y}={\\bf e}_1$, hence $\\|A \\|_2\\geq \\sigma_1(A)$. Therefore, we conclude that $\\|A \\|_2= \\sigma_1(A)$ and complete the proof.\n\\end{proof}\n\n\n\\section{von Neumann type trace inequality}\\label{VN-TraceIn}\nIn this section, we extend the well-known von Neumann trace inequality to dual quaternionic versions. Because $\\widehat{\\mathbb R}$ is a total order space in the meaning of total order stated in Section \\ref{Du-number}, unless otherwise specified, we have, for $p,q\\in \\widehat{\\mathbb R}$, $p\\leq q$ if and only if $q-p\\in \\widehat{\\mathbb R}_+$. We start this section by introducing the following concept for dual numbers.\n\n\nFor given dual numbers $x_1,x_2,\\ldots, x_m$, we use $\\hat x_1,\\hat x_2,\\ldots, \\hat x_m$ to denote these numbers arranged in non-ascending order of magnitude. If the two sets of dual numbers $x_1,x_2,\\ldots, x_m$ and $y_1,y_2,\\ldots, y_m$ satisfy the\nrelations\n$$\n\\hat x_1+\\hat x_2+\\ldots+ \\hat x_s\\left\\{\\begin{array}{ll}\n\\leq\\hat y_1+\\hat y_2+\\ldots+ \\hat y_s&{\\rm for~}1\\leq s\\leq m-1,\\\\\n=\\hat y_1+\\hat y_2+\\ldots+ \\hat y_s&{\\rm for ~}s=m,\n\\end{array}\\right.\n$$\nwe write $(x_1,x_2, \\ldots,x_m)\\prec (y_1, y_2, \\ldots, y_m)$ for simplicity.\n\n\nThe following lemma is an extension of Lemma in \\cite{Mir59} to the case of dual numbers. Due to the introduction of the total order ``$\\geq$'' in $\\widehat{\\mathbb R}$, it has similar properties to the set of real numbers. For example, if $a,b\\in \\widehat{\\mathbb R}_+$ implies $a+b\\in \\widehat{\\mathbb R}_+$ and $ab\\in \\widehat{\\mathbb R}_+$. Although the proof of this lemma is similar to the way used for the unique lemma in \\cite{Mir59}, we give a proof for the completeness of this paper.\n\\begin{Lem}\\label{xyzIn}\nLet $\\{x_1,x_2,\\ldots, x_m\\}$, $\\{y_1,y_2,\\ldots, y_m\\}$ and $\\{z_1,z_2,\\ldots, z_m\\}\\subset \\widehat{\\mathbb R}$. Suppose $x_1\\geq x_2\\geq \\ldots\\geq x_m$, $y_1\\geq y_2\\geq \\ldots\\geq y_m$ and $(z_1,z_2, \\ldots,z_m)\\prec (y_1, y_2, \\ldots, y_m)$. Then it holds that\n\\begin{equation}\n\\sum_{i=1}^mx_iz_i\\leq \\sum_{i=1}^mx_iy_i.\n\\end{equation}\n\\end{Lem}\n\\begin{proof}\nFor any $1\\leq k\\leq m$, let $X_k=x_1+ x_2+\\ldots+ x_k$ and $Z_k=\\hat z_1+\\hat z_2+\\ldots+ \\hat z_k$. Then, by virtue of hypothesis, we have $Z_k\\leq Y_k$ for $k=1,2,\\ldots,m$. Consequently, it holds that\n$$\n\\begin{array}{lll}\n\\displaystyle\\sum_{k=1}^mx_kz_k&\\leq&\\displaystyle\\sum_{k=1}^mx_k\\hat z_k\\\\\n&=&\\displaystyle x_1Z_1+\\sum_{k=2}^mx_k(Z_k-Z_{k-1})\\\\\n&=&\\displaystyle\\sum_{k=1}^{m-1}(x_k-x_{k+1})Z_k+x_nZ_n\\\\\n&\\leq&\\displaystyle\\sum_{k=1}^{m-1}(x_k-x_{k+1})Y_k+x_nY_n\\\\\n&=&\\displaystyle\\sum_{k=1}^mx_ky_k.\n\\end{array}\n$$\nThe proof is completed.\n\\end{proof}\n\n\n\\begin{Lem}\\label{Lemma2}\nLet $A=(a_{ij})\\in \\widehat{\\mathbb Q}^{m\\times m}$ be a Hermitian matrix, and let the right eigenvalues $\\lambda_i(A)$ of $A$ be arranged so that $\\lambda_1(A)\\geq \\lambda_2(A)\\geq \\ldots\\geq \\lambda_m(A)$. Then, for any given positive integer $k\\leq m$, the sum $\\sum_{i=1}^k\\lambda_i(A)$ is the maximum of $\\sum_{i=1}^k({\\bf x}^{(i)})^*A{\\bf x}^{(i)}$, when $k$ orthonormal vectors ${\\bf x}^{(i)}~(i=1,2,\\ldots,k)$ vary in $\\widehat{\\mathbb Q}^m$. In particular, we have\n\\begin{equation}\\label{TrEigIN}\n\\sum_{i=1}^k a_{ii}\\leq \\sum_{i=1}^k\\lambda_i(A), ~~~~k=1,2,\\ldots,m.\n\\end{equation}\n\\end{Lem}\n\n\\begin{proof}\nBy Theorem \\ref{HUDec}, there exists a unitary matrix $U\\in \\widehat{\\mathbb Q}^{m\\times m}$ such that\n$$U^*AU={\\rm diag}(\\lambda_1(A),\\lambda_2(A),\\ldots,\\lambda_m(A)),$$\nwhich implies $A{\\bf u}^{(i)}={\\bf u}^{(i)}\\lambda_i(A)$ for $i=1,2,\\ldots,m$, where ${\\bf u}^{(i)}$ is the $i$th column of $U$. Since $UU^*=I$, it holds, for any ${\\bf x}^{(j)}\\in \\widehat{\\mathbb Q}^m$, that\n$$\n{\\bf x}^{(j)}=UU^*{\\bf x}^{(j)}=\\sum_{i=1}^m{\\bf u}^{(i)}(({\\bf u}^{(i)})^*{\\bf x}^{(j)})=\\sum_{i=1}^m{\\bf u}^{(i)}\\langle{\\bf x}^{(j)},{\\bf u}^{(i)}\\rangle.\n$$\nSince $A\\in \\widehat{\\mathbb Q}^{m\\times m}$ is a Hermitian matrix, all $\\lambda_i$'s are dual numbers. Consequently, we have\n$$\n\\begin{array}{lll}\n({\\bf x}^{(j)})^*A{\\bf x}^{(j)}&=&\\displaystyle\\sum_{i=1}^m\\lambda_i(A)|\\langle{\\bf x}^{(j)},{\\bf u}^{(i)}\\rangle|^2\\\\\n&=&\\displaystyle\\lambda_k(A)\\sum_{i=1}^m|\\langle{\\bf x}^{(j)},{\\bf u}^{(i)}\\rangle|^2+\\sum_{i=1}^k(\\lambda_i(A)-\\lambda_k(A))|\\langle{\\bf x}^{(j)},{\\bf u}^{(i)}\\rangle|^2\\\\\n&&+\\displaystyle\\sum_{i=k+1}^m(\\lambda_i(A)-\\lambda_k(A))|\\langle{\\bf x}^{(j)},{\\bf u}^{(i)}\\rangle|^2.\n\\end{array}\n$$\nHence, when $\\|{\\bf x}^{(j)}\\|_2=1$, we obtain\n$$\n({\\bf x}^{(j)})^*A{\\bf x}^{(j)}\\leq \\lambda_k(A)+\\sum_{i=1}^k(\\lambda_i(A)-\\lambda_k(A))|\\langle{\\bf x}^{(j)},{\\bf u}^{(i)}\\rangle|^2,\n$$\nsince\n$$\n\\sum_{i=1}^m|\\langle{\\bf x}^{(j)},{\\bf u}^{(i)}\\rangle|^2=({\\bf x}^{(j)})^*UU^*{\\bf x}^{(j)}=({\\bf x}^{(j)})^*{\\bf x}^{(j)}=\\|{\\bf x}^{(j)}\\|_2=1,\n$$\nas well as $\\lambda_i(A)\\leq \\lambda_k(A)$ for $i=k+1,\\ldots, m$. By this, it holds that\n$$\n\\begin{array}{lll}\n\\displaystyle\\sum_{j=1}^k({\\bf x}^{(j)})^*A{\\bf x}^{(j)}&\\leq &\\displaystyle\\sum_{j=1}^k\\lambda_k(A)+\\sum_{j=1}^k\\sum_{i=1}^k(\\lambda_i(A)-\\lambda_k(A))|\\langle{\\bf x}^{(j)},{\\bf u}^{(i)}\\rangle|^2\\\\\n&=&\\displaystyle\\sum_{j=1}^k\\lambda_k(A)+\\sum_{i=1}^k(\\lambda_i(A)-\\lambda_k(A))\\sum_{j=1}^k|\\langle{\\bf x}^{(j)},{\\bf u}^{(i)}\\rangle|^2,\n\\end{array}\n$$\nwhich implies\n$$\n\\begin{array}{lll}\n\\displaystyle\\sum_{i=1}^k\\lambda_i(A)-\\sum_{j=1}^k({\\bf x}^{(j)})^*A{\\bf x}^{(j)}&\\geq &\\displaystyle\\sum_{j=1}^k(\\lambda_j(A)-\\lambda_k(A))-\\sum_{i=1}^k\\sum_{j=1}^k\\left(\\lambda_i(A)-\\lambda_k(A)\\right)|\\langle{\\bf x}^{(j)},{\\bf u}^{(i)}\\rangle|^2\\\\\n&=&\\displaystyle\\sum_{j=1}^k(\\lambda_j(A)-\\lambda_k(A))\\left\\{1-\\sum_{i=1}^k|\\langle{\\bf x}^{(j)},{\\bf u}^{(i)}\\rangle|^2\\right\\}\\\\\n&\\geq &0,\n\\end{array}\n$$\nsince $\\lambda_j(A)\\geq\\lambda_k(A)$ for $j=1,2,\\ldots,k$ and\n$$\n\\sum_{i=1}^k|\\langle{\\bf x}^{(j)},{\\bf u}^{(i)}\\rangle|^2=({\\bf x}^{(j)})^*\\widetilde{U}\\widetilde{U}^*{\\bf x}^{(j)}\\leq \\|\\widetilde{U}\\widetilde{U}^*\\|_2\\|{\\bf x}^{(j)}\\|^2_2\\leq\\|{\\bf x}^{(j)}\\|^2_2=1,\n$$\nwhere $\\widetilde{U}=[{\\bf u}^{(1)},\\ldots,{\\bf u}^{(k)}]$. On the other hand, by taking ${\\bf x}^{(j)}={\\bf u}^{(j)}$ for $j=1,2,\\ldots,k$, it is easy to verify that $\\sum_{i=1}^k\\lambda_i(A)=\\sum_{j=1}^k({\\bf x}^{(j)})^*A{\\bf x}^{(j)}$.\n\nNotice that $a_{ii}\\in \\widehat{\\mathbb R}$ for $i=1,2,\\ldots,m$, since $A$ is Hermitian. In particular, by taking ${\\bf x}^{(j)}={\\bf e}_j$, where ${\\bf e}_j$ is the $j$th column in $I_m$, it is easy to see that (\\ref{TrEigIN}) holds. Therefore, we obtain the desired result and complete the proof.\n\\end{proof}\n\nHereafter, we recall a lemma that can be found in \\cite[Theorem 4.10]{LQY22}.\n\\begin{Lem}[\\cite{LQY22}]\\label{Lemma3}\nLet $A\\in \\widehat{\\mathbb Q}^{m\\times m}$ and $H=(A^*+A)\/2$. Let $\\sigma_1(A)\\geq\\sigma_2(A)\\geq\\ldots\\geq\\sigma_m(A)$ be singular values of $A$, and let $\\lambda_1(H)\\geq\\lambda_2(H)\\geq\\ldots\\geq\\lambda_m(H)$ be the right eigenvalues of $H$. Then, it holds that\n$$\n\\lambda_i(H)\\leq\\sigma_i(A), ~~~i=1,2,\\ldots,m.\n$$\n\\end{Lem}\n\nWith the above preparations, we now state and prove the von Neumann type trace inequality for dual quaternion matrices. Here, we prove the dual quaternionic von Neumann inequality by a unified way, which is applicable to the quaternionic case and is completely different with the proof given in \\cite{CY16}.\n\n\\begin{Thm}\\label{VNINTH}\nFor any $A,B\\in \\widehat{\\mathbb Q}^{m\\times n}$, it holds that\n\\begin{equation}\n{\\rm trace}(A^*B+B^*A)\\leq 2\\sum_{i=1}^s\\sigma_i(A)\\sigma_i(B),\n\\end{equation}\nwhere $s=\\min\\{m,n\\}$, and $\\sigma_1(A)\\geq\\sigma_2(A)\\geq\\ldots\\geq\\sigma_s(A)$ and $\\sigma_1(B)\\geq\\sigma_2(B)\\geq\\ldots\\geq\\sigma_s(B)$ are singular values of $A$ and $B$, respectively.\n\\end{Thm}\n\n\\begin{proof}\nWithout loss of generality, we assume $m\\leq n$. Let $A=U\\Sigma_AV^*$ be the SVD of $A$, where $\\Sigma_A={\\rm diag}(\\sigma_1(A),\\sigma_2(A),\\ldots,\\sigma_m(A))$. It is obvious that $A^*=V\\Sigma_A^\\top U^*$, since all $\\sigma_i(A)$ ($i=1,2,\\ldots,m$) are dual numbers. Consequently, we have\n$$\nV^*(A^*B+B^*A)V=\\Sigma_A^\\top C+C^* \\Sigma_A,\n$$\nwhere $C=U^*BV\\in \\widehat{\\mathbb Q}^{m\\times n}$. It is easy to see that\n$$\n{\\rm trace}(A^*B+B^*A)={\\rm trace}\\left\\{V^*(A^*B+B^*A)V\\right\\}={\\rm trace}(\\Sigma_A^\\top C+C^* \\Sigma_A)=\\sum_{i=1}^m\\sigma_i(A)(c_{ii}+\\bar c_{ii}),\n$$\nwhere $c_{ii}$ is the $i$th diagonal element of $C$ and $\\bar c_{ii}$ is the conjugate of the dual quaternion number $c_{ii}$.\n\nNow, we prove\n$$\n\\sum_{i=1}^m\\sigma_i(A)(c_{ii}+\\bar c_{ii})\\leq2\\sum_{i=1}^{m}\\sigma_i(A)\\sigma_i(B).\n$$\nLet $$D=(d_{ij}):=\\left[\\begin{array}{c}C\\\\\n0_{(n-m)\\times n}\\end{array}\\right]+[C^*,0_{n\\times (n-m)}].$$ It is obvious that $D\\in \\widehat{\\mathbb Q}^{n\\times n}$ satisfies $D^*=D$, i.e., $D$ is a dual quaternion Hermitian matrix. It is obvious that $d_{ii}=c_{ii}+\\bar c_{ii}$ for any $i=1,2,\\ldots,m$. Applying Lemma \\ref{Lemma2} with $A=D$, we obtain\n\\begin{equation}\\label{ccdSinIn}\n\\sum_{i=1}^k(c_{ii}+\\bar c_{ii})=\\sum_{i=1}^kd_{ii}\\leq \\sum_{i=1}^k\\lambda_i(D)\\leq2\\sum_{i=1}^k\\sigma_i(C)=2\\sum_{i=1}^k\\sigma_i(B), ~~k=1,2,\\ldots,m,\n\\end{equation}\nwhere the second inequality comes from Lemma \\ref{Lemma3} and the fact that $\\sigma_i\\left(\\left[\\begin{array}{c}C\\\\0_{(n-m)\\times m}\\end{array}\\right]\\right)=\\sigma_i(C)$ for $i=1,2,\\ldots,m$, and the last equality is due to the fact $\\sigma_i(C)=\\sigma_i(B)$ for $i=1,2,\\ldots,m$ since $U,V$ are unitary. Finally, since $\\sigma_i(A)\\geq $ for $i=1,2,\\ldots, m$, by (\\ref{ccdSinIn}) and Lemma \\ref{xyzIn}, we obtain the desired conclusion and complete the proof.\n\\end{proof}\n\nDue to the application of dual quaternion Hermitian matrices in multi-agent formation control, we below present a variant of the von Neumann inequality expressed by eigenvalues of dual quaternion Hermitian matrices.\n\n\\begin{Thm}\\label{EigFnormIn}\n\tLet $A,B \\in \\widehat{\\mathbb{Q}}^{m\\times m}$. Suppose that both $A$ and $B$ are Hermitian. We have\n\t$$\n\t{\\rm trace}(AB+BA)\\leq 2\\sum_{i=1}^m\\lambda_i(A)\\lambda_i(B),\n\t$$\n\twhere $\\lambda_1(A)\\geq\\lambda_2(A)\\geq\\ldots\\geq\\lambda_m(A)$ and $\\lambda_1(B)\\geq\\lambda_2(B)\\geq\\ldots\\geq\\lambda_{m}(B)$ are the eigenvalues of $A$ and $B$, respectively. In particular, the above inequality holds when $A,B \\in \\mathbb{Q}^{m\\times m}$ are Hermitian.\n\\end{Thm}\n\n\\begin{proof}\n\tSince both $A$ and $B$ are dual quaternion Hermitian matrices, we know that $\\lambda_i(A), \\lambda_i(B)\\in \\widehat{\\mathbb{R}}$ for $i=1,2,\\ldots,m$. By Theorem \\ref{HUDec}, there exists a unitary matrix $U\\in \\widehat{\\mathbb Q}^{m\\times m}$ such that $A=U\\Sigma U^*$ where $\\Sigma={\\rm diag} (\\lambda_1(A),\\lambda_2(A),\\ldots,\\lambda_m(A))$. Let $C:=(c_{ij})=U^*BU$. It is clear that $C$ is a dual quaternion Hermitian matrix, which implies $c_{ii}\\in \\widehat{\\mathbb{R}}$ for $i=1,2,\\ldots,m$. Moreover, we have\n\t\\begin{equation}\\label{trABEq}\n\t{\\rm trace}(AB+BA)={\\rm trace}(U(\\Sigma C+C\\Sigma)U^*)={\\rm trace}(\\Sigma C+C\\Sigma)=2\\sum_{i=1}^m\\lambda_i(A)c_{ii}.\n\t\\end{equation}\n\tSince $C$ is Hermitian, by Lemma \\ref{Lemma2} and {\\color {blue} (\\ref{sumEigV})}, for every $k=1,2,\\ldots m-1$, it holds that\n\t$$\n\t\\sum_{i=1}^kc_{ii}\\leq \\sum_{i=1}^k\\lambda_i(C)=\\sum_{i=1}^k\\lambda_i(B)~~{\\rm and}~~\\sum_{i=1}^mc_{ii}=\\sum_{i=1}^m\\lambda_i(B),\n\t$$\n\tsince $\\lambda_i(C)=\\lambda_i(B)$ for $i=1,2,\\ldots,m$. Consequently, by Lemma \\ref{xyzIn}, we have $$\n\t\\sum_{i=1}^m\\lambda_i(A)c_{ii}\\leq \\sum_{i=1}^m\\lambda_i(A)\\lambda_i(B),\n\t$$\n\twhich implies, together with (\\ref{trABEq}), the desired inequality holds.\n\\end{proof}\n\n\n\n\\section{Hoffman-Wielandt type inequality}\\label{sec_HWineq}\n\nIn this section, we are concerned with the Hoffman-Wielandt type inequality for dual quaternion matrices.\n\nFirst, by Theorem \\ref{VNINTH}, we have the following theorem, which is also a generalization of the well-known Hoffman-Wielandt type inequality \\cite{HW53} in $\\mathbb{C}^{m\\times n}$, and characterizes an upper bound for all singular values simultaneous perturbation of a dual quaternion matrix.\n\n\\begin{Thm}\\label{FNormIn}\n\tLet $A,B\\in \\widehat{\\mathbb Q}^{m\\times n}$. If $A-B$ is appreciable,\n\tthen it holds that\n\t\\begin{equation}\\label{FNormIn-1}\n\t\\|\\sigma(A)-\\sigma(B)\\|_2\\leq \\|A-B\\|_F,\n\t\\end{equation}\n\twhere $s=\\min\\{m,n\\}$, $\\sigma(A)=(\\sigma_1(A), \\ldots,\\sigma_{s}(A))^\\top$ and $\\sigma(B)=(\\sigma_1(B), \\ldots,\\sigma_{s}(B))^\\top$ with $\\sigma_1(A)\\geq\\sigma_2(A)\\geq\\ldots\\geq\\sigma_{s}(A)$ and $\\sigma_1(B)\\geq\\sigma_2(B)\\geq\\ldots\\geq\\sigma_{s}(B)$ being the singular values of $A$ and $B$, respectively.\n\\end{Thm}\n\n\\begin{proof}\n\tLet $A=U\\Sigma_AV^*$ and $B=X\\Sigma_BY^*$ be the SVDs of $A$ and $B$, respectively. Under the condition $A-B$ being appreciable, we divide our proofs into two cases.\n\t\\begin{itemize}\n\t\t\\item If $\\sigma(A)-\\sigma(B)$ is infinitesimal, then $A-B$ being appreciable implies\n\t\t$$\\|A-B\\|_F=\\|A_{\\sf st}-B_{\\sf st}\\|_F+\\delta\\epsilon$$\n\t\tfor some $\\delta\\in \\widehat{\\mathbb R}_+$ by Proposition \\ref{p6.3}. Clearly, it follows from the fact $\\|A_{\\sf st}-B_{\\sf st}\\|_F>0$ that (\\ref{FNormIn-1}) holds.\n\t\t\\item If $\\sigma(A)-\\sigma(B)$ is appreciable, then we only need to prove\n\t\t$$\n\t\t\\sum_{i=1}^{s}|\\sigma_i(A)-\\sigma_i(B)|^2\\leq \\|A-B\\|^2_F,\n\t\t$$\n\t\twhich is equivalent to\n\t\t\\begin{equation}\\label{FNormIn-2}\n\t\t\\sum_{i=1}^{s}\\sigma^2_i(A)-2\\sum_{i=1}^{s}\\sigma_i(A)\\sigma_i(B)+\\sum_{i=1}^{s}\\sigma_i(B)^2\\leq \\|A\\|_F^2-{\\rm trace}(A^*B+B^*A)+\\|B\\|^2_F.\n\t\t\\end{equation}\n\t\tRecalling the fact that $\\sum_{i=1}^{s}\\sigma^2_i(A)=\\|A\\|_F^2$ and $\\sum_{i=1}^{s}\\sigma_i(B)^2=\\|B\\|^2_F$, we immediately prove (\\ref{FNormIn-2}) with the employment of Theorem \\ref{VNINTH}.\n\t\\end{itemize}\n\tTo sum up, we obtain the desired conclusion and complete the proof.\n\\end{proof}\n\n\n\\begin{ReK}\n\tNotice that, if $A,B\\in \\mathbb{Q}^{m\\times n}$, it is obvious that Theorem \\ref{VNINTH} holds. Therefore, if both $A$ and $B$ are either quaternions or infinitesimal, Theorem \\ref{FNormIn} holds. In fact, in the case that both $A$ and $B$ are infinitesimal, we claim that $\\sigma(A)$ and $\\sigma(B)$ are both infinitesimal, and $(\\sigma(A))_{\\sf in}=\\sigma(A_{\\sf in})$ and $(\\sigma(B))_{\\sf in}=\\sigma(B_{\\sf in})$. Consequently, we have\n\t$$\\|\\sigma(A)-\\sigma(B)\\|_2=\\|(\\sigma(A))_{\\sf in}-(\\sigma(B))_{\\sf in}\\|_2\\epsilon=\\|\\sigma(A_{\\sf in})-\\sigma(B_{\\sf in})\\|_2\\epsilon$$ and $$\\|A-B\\|_F=\\|A_{\\sf in}-B_{\\sf in}\\|_F\\epsilon.$$\n\tHence, we only need to prove $\\|\\sigma(A_{\\sf in})-\\sigma(B_{\\sf in})\\|_2\\leq \\|A_{\\sf in}-B_{\\sf in}\\|_F$, which can be proved by applying Theorem \\ref{VNINTH} with $A=A_{\\sf in}$ and $B=B_{\\sf in}$. The conclusion for the case $A, B\\in \\mathbb{Q}^{m\\times n}$ can be proved similarly.\n\\end{ReK}\n\n\\begin{ReK}\\label{remark3}\n\tFrom Theorem \\ref{SVD-DQM}, we know that the standard parts of the singular values of a dual quaternion matrix are exactly the singular values of the standard part of that dual quaternion matrix. Hence, If both $A$ and $B$ are appreciable, but $A-B$ is infinitesimal, i.e., $A_{\\sf st}=B_{\\sf st}\\neq O$, then $\\sigma(A)-\\sigma(B)$ is infinitesimal. In this case, the desired inequality \\eqref{FNormIn-1} becomes $\\|(\\sigma(A)-\\sigma(B))_{\\sf in}\\|_2\\leq \\|A_{\\sf in}-B_{\\sf in}\\|_F$. However, we do not know whether this inequality still holds, and leave it as an open question for one of our future concerns.\n\\end{ReK}\n\nAs mentioned in Remark \\ref{remark3}, the infinitesimal part of dual numbers makes the analysis on dual quaternion matrices is difficult, and Theorem \\ref{FNormIn} holds under the condition that $A-B$ is appreciable. Below, we are concerned with whether the Hoffman-Wielandt inequality still holds for dual quaternion Hermitian matrices when removing the condition $A-B$ being appreciable.\n\nWe first show the quaternionic Hoffman-Wielandt inequality.\n\n\\begin{Prop}\\label{SeProp2}\nLet $A,B \\in \\mathbb{Q}^{m\\times m}$. Suppose that both $A$ and $B$ are Hermitian. We have\n$$\n\\|\\lambda(A)-\\lambda(B)\\|_2\\leq \\|A-B\\|_F,\n$$\nwhere $\\lambda(A)=(\\lambda_1(A), \\ldots,\\lambda_m(A))^\\top$ and $\\lambda(B)=(\\lambda_1(B), \\ldots,\\lambda_m(B))^\\top$ with $\\lambda_1(A)\\geq\\lambda_2(A)\\geq\\ldots\\geq\\lambda_m(A)$ and $\\lambda_1(B)\\geq\\lambda_2(B)\\geq\\ldots\\geq\\lambda_{m}(B)$ being the eigenvalues of $A$ and $B$, respectively.\n\\end{Prop}\n\n\\begin{proof}\nSince ${\\rm trace}(A^2)=\\|A\\|_F^2=\\|\\lambda(A)\\|_2^2$ and ${\\rm trace}(B^2)=\\|B\\|_F^2=\\|\\lambda(B)\\|_2^2$, it follows from Proposition \\ref{EigFnormIn}.\n\\end{proof}\n\nNow, we show that the Hoffman-Wielandt type inequality holds for dual quaternion matrices without the condition $A-B$ being appreciable.\n\n\\begin{Thm}\nLet $A, B\\in \\widehat{\\mathbb{Q}}^{m\\times m}$. If both $A$ and $B$ are Hermitian matrices, then we have\n\\begin{equation}\\label{HW-Ineq}\n\\|\\lambda(A)-\\lambda(B)\\|_2\\leq \\|A-B\\|_F,\n\\end{equation}\nwhere $\\lambda(A)=(\\lambda_1(A), \\ldots,\\lambda_m(A))^\\top$ and $\\lambda(B)=(\\lambda_1(B), \\ldots,\\lambda_m(B))^\\top$ with $\\lambda_1(A)\\geq\\lambda_2(A)\\geq\\ldots\\geq\\lambda_m(A)$ and $\\lambda_1(B)\\geq\\lambda_2(B)\\geq\\ldots\\geq\\lambda_{m}(B)$ being the eigenvalues of $A$ and $B$, respectively.\n\\end{Thm}\n\n\\begin{proof}\nFirstly, since both $A$ and $B$ are Hermitian, we know that $\\lambda(A), \\lambda(B)\\in \\widehat{\\mathbb{R}}^m$. We divide the proofs into two cases: (a) $A-B$ is appreciable, and (b) $A-B$ is infinitesimal.\n\nIn the case (a), since $\\|\\lambda(A)\\|_2^2=\\|A\\|_F^2$ and $\\|\\lambda(B)\\|_2^2=\\|B\\|_F^2$, regardless of whether $\\lambda(A)-\\lambda(B)$ is appreciable or not, by Proposition \\ref{EigFnormIn}, we can prove (\\ref{HW-Ineq}) in a similar way to the proof of Theorem \\ref{FNormIn}.\n\nWe now prove the desired inequality (\\ref{HW-Ineq}) for the case where $A-B$ is infinitesimal. In this case, both $A$ and $B$ must be appreciable or infinitesimal at the same time, since $A_{\\sf st}=B_{\\sf st}$. If both $A$ and $B$ are infinitesimal, i.e., $A_{\\sf st}=B_{\\sf st}=O$, then the inequality (\\ref{HW-Ineq}) becomes\n\\begin{equation}\\label{HW-InQ}\n\\|(\\lambda(A))_{\\sf in}-(\\lambda(B))_{\\sf in}\\|_2\\leq \\|A_{\\sf in}-B_{\\sf in}\\|_F,\n\\end{equation} since $\\lambda(A)=\\lambda(A_{\\sf in})\\epsilon$ and $\\lambda(B)=\\lambda(B_{\\sf in})\\epsilon$. Consequently, since both $A_{\\sf in}$ and $B_{\\sf in}$ are Hermitian, by Proposition \\ref{EigFnormIn} again, we know that the inequality (\\ref{HW-InQ}) holds.\n\nIf $A$ and $B$ are appreciable Hermitian matrices at the same time, i.e., $A_{\\sf st}=B_{\\sf st}$ is a nonzero quaternion Hermitian matrix, then by quaternion matrix theory, there exists a unitary matrix $S\\in \\mathbb{Q}^{m\\times m}$, such that\n$$\nSAS^*=D+C\\epsilon~~~~{\\rm and}~~~~SBS^*=D+G\\epsilon,\n$$\nwhere $C=SA_{\\sf in}S^*$, $G=SB_{\\sf in}S^*$, $D =SA_{\\sf st}S^*={\\rm diag}(\\lambda_1I_{k_1}, \\lambda_2I_{k_2}, \\ldots, \\lambda_rI_{k_r})$. Here, $\\lambda_1 > \\lambda_2 > \\cdots > \\lambda_r$ are real numbers, $I_{k_i}$ is a $k_i \\times k_i$ identity matrix, and $\\sum_{i=1}^r k_i = m$. It is obvious that\n$\\|A_{\\sf in}-B_{\\sf in}\\|_F^2=\\|C-G\\|_F^2$. Notice that both $C$\nand $G$ are quaternion Hermitian matrices. Write\n\\begin{equation*}\nC=\\begin{bmatrix}\nC_{11} & C_{12} & \\cdots & C_{1r} \\\\\n C_{12}^* & C_{22} & \\cdots & C_{2r} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n C_{1r}^* & C_{2r}^* & \\cdots & C_{rr}\n\\end{bmatrix}\n~~~~{\\rm and}~~~~\nG=\\begin{bmatrix}\nG_{11} & G_{12} & \\cdots & G_{1r}\\\\\n G_{12}^*& G_{22} & \\cdots & G_{2r} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n G_{1r}^*& G_{2r}^* & \\cdots & G_{rr}\n\\end{bmatrix},\n\\end{equation*}\nwhere $C_{ij}$ and $G_{ij}$ are quaternion matrices of same adequate dimensions, and $C_{ii}$ and $G_{ii}$ are Hermitian for $i=1,2,\\ldots,r$. Since $C_{ii}$ and $G_{ii}$ are Hermitian, there exist real numbers $\\lambda_{i,1} \\ge \\ldots \\ge \\lambda_{i,k_i}$ and $\\mu_{i,1} \\ge \\ldots \\ge \\mu_{i, k_i}$ for $i=1,2,\\ldots,r$, such that\n\\begin{equation}\\label{blocks}\nC_{ii} = U_{i}{\\rm diag}\\left(\\lambda_{i,1}, \\cdots, \\lambda_{i,k_i}\\right)U_{i}^* ~~{\\rm and}~~G_{ii}= V_{i} {\\rm diag}\\left(\\mu_{i,1}, \\cdots, \\mu_{i,k_i}\\right)V_{i}^* , ~~i=1,\\cdots, r.\n\\end{equation}\nIt is obvious that $\\lambda_{i,j}$ and $\\mu_{i,j}$ are the $j$th largest eigenvalues of $C_{ii}$ and $G_{ii}$ respectively, for $i=1,2,\\ldots,r$. From the proof precess of Theorem 4.1 presented in \\cite{QL21}, we know that\n$$\n\\lambda(A)_{\\sf in}=(\\lambda_{1,1},\\ldots,\\lambda_{1,k_1},\\lambda_{2,1},\\ldots,\\lambda_{2,k_2},\\ldots,\\lambda_{r,1},\\ldots,\\lambda_{r,k_r})^\\top\n$$\nand\n$$\n\\lambda(B)_{\\sf in}=(\\mu_{1,1},\\ldots,\\mu_{1,k_1},\\mu_{2,1},\\ldots,\\mu_{2,k_2},\\ldots,\\mu_{r,1},\\ldots,\\mu_{r,k_r})^\\top.\n$$\nConsequently, it holds that\n$$\n\\begin{array}{lll}\n\\|\\lambda(A)_{\\sf in}-\\lambda(B)_{\\sf in}\\|_2^2&=&\\displaystyle\\sum_{i=1}^r\\sum_{j=1}^{k_i}|\\lambda_{i,j}-\\mu_{i,j}|^2\\\\\n&\\leq &\\displaystyle\\sum_{i=1}^r\\|C_{ii}-G_{ii}\\|^2\\\\\n&\\leq&\\displaystyle\\sum_{i=1}^r\\sum_{j=1}^r\\|C_{ij}-G_{ij}\\|^2\\\\\n&=&\\|C-G\\|_F^2\\\\\n&=&\\|A_{\\sf in}-B_{\\sf in}\\|_F^2,\n\\end{array}\n$$\nwhere the first inequality comes from Proposition \\ref{SeProp2} with $A=C_{ii}$ and $B=G_{ii}$ for $i=1,2,\\ldots,r$. Hence, we obtain $\\|\\lambda(A)_{\\sf in}-\\lambda(B)_{\\sf in}\\|_2\\leq\\|A_{\\sf in}-B_{\\sf in}\\|_F$, which implies $\\|\\lambda(A)-\\lambda(B)\\|_2\\leq\\|A-B\\|_F$ by (\\ref{FNorm-DQM}), since $\\lambda(A)_{\\sf st}=\\lambda(B)_{\\sf st}$ and $A_{\\sf st}=B_{\\sf st}$. We obtain the desired result and complete the proof.\n\\end{proof}\n\n\n\\section{Conclusion}\\label{Conclusion}\nIn this paper, after introducing the concept of spectral norm for dual quaternion matrices, we extended the well-known von Neumann trace inequality for general dual quaternion matrices. Using the proposed trace inequality, we further obtained a Hoffman-Wielandt type inequality, which characterizes the distance between two dual quaternion matrices $A, B\\in \\widehat{\\mathbb Q}^{m\\times n}$ being larger than the distance between their respective singular values. Such an inequality can also be regarded as a simultaneous perturbation bound on all singular values of a general dual quaternion matrix. In particular, we proposed two variants of the above two inequalities expressed by eigenvalues of dual quaternion Hermitian matrices, where the Hoffman-Wielandt type inequality holds without the condition that $A-B$ is appreciable. As a new area of applied mathematics, there are many problems worth exploring to enrich the theory of dual quaternion matrices, such as optimal low-rank approximations and applications of dual quaternion matrices in the fields of data analysis, computer science and intelligent control.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\nOver the last decade, following Nekrasov's~\\cite{Nekrasov:2002qd} and\nPestun's~\\cite{Pestun:2007rz} seminal works, the application of the\nlocalization technique to SUSY gauge theories on various manifolds,\nhas produced an unprecedented amount of exact results (for a\ncomprehensive review see \\cite{Pestun:2016zxk} and references\ntherein).\n\nLocalized partition functions (or vevs of BPS observables) depend\non various parameters such as fugacities for the global\nsymmetries and data specifying the background.\nFor certain backgrounds partition functions do not depend on the gauge coupling \nand can be used to test Seiberg-like dualities and mirror symmetry in various\ndimensions. \n\nThe exact results obtained via localization have also led to the\ndiscovery of AGT-like correspondences which provide dictionaries to\nmap objects in the gauge theories (partition functions, Wilson loops\nvevs etc\\ldots) to objects in different systems such as $2d$ CFTs or\nTQFTs \\cite{Alday:2009aq,Gadde:2009kb}.\n\n\nIt is interesting to study what happens when we take different limits\nof the parameters appearing in the partition functions triggering some\nsort of RG flows. For example, focusing on the global symmetry\nparameters we can explicitly check how certain dualities can be\nobtained by taking massive deformations of other dualities. We can\nalso consider limits involving the data specifying the background. For\nmanifolds of the form $M_{d-1} \\times S^1$ we can explore what happens\nwhen the circle shrinks and in particular gather hints on the fate of\ndualities in $d$ dimensions: do they reduce to dualities in $d-1$\ndimensions? In recent years these questions have been reconsidered\nsystematically in a series of papers\n\\cite{Aharony:2013dha,Aharony:2013kma,Aharony:2016jki,Aharony:2017adm}.\n\nAnother interesting procedure, the so-called Higgsing, involves\nturning on the vev of some operator in a certain UV theory $T'$ which\ntriggers an RG flow to an IR theory $T$ that contains a codimension\ntwo defect, the prototypical case being a surface operator in a $4d$\ntheory. At the level of localized partition functions this procedure\ncan be implemented very efficiently and involves tuning the gauge and\nflavor parameters of the mother theory $T$ to specific values. At\nthese values $T$ typically develops some poles and picking up their\nresidues we obtain the partition function of the theory $T$ with a\ncodimension two defect\n\\cite{Gaiotto:2012xa,Mironov:2009qt,Kozcaz:2010af,Dimofte:2010tz,Dorey:2011pa,Nieri:2013vba,Gaiotto:2014ina}.\n\n\nIn this note we provide a concrete example where all these ideas and\ntechniques come together. We discuss $3d$ mirror symmetry, spectral\nduality and gauge\/CFT correspondences and explore how they behave\nunder dimensional reduction and how they arise via Higgsing.\n\nOur starting point is the $3d$ $T[SU(N)]$ quiver theory introduced in~\\cite{Gaiotto:2008ak} as boundary conditions for the\n$4d$ ${\\cal N}=4$ supersymmetric Yang-Mills theories. $T[G]$ has a\nglobal symmetry group $G\\times G^L$ acting respectively on the Higgs\nand Coulomb branch. The $T[G]$ has the remarkable property of being\ninvariant (or self-mirror) under $3d$ mirror symmetry which acts by\nexchanging the Higgs and the Coulomb branches of the theory.\n\nIn this work we will consider a closely related quiver theory, the\n$FT[SU(N)]$ theory, which contains an additional set of gauge singlet\nfields. The $FT[SU(N)]$ theory is also self-dual under a duality which\nwe call {\\it 3d spectral duality} since it descends from $5d$ spectral\nduality.\n\n\nIn particular we discuss three webs of dualities:\n\\begin{itemize}\n\\item In Duality web I, we relate the $3d$ spectral pair $FT[SU(N)]\\leftrightarrow \\hat{FT}[SU(N)]$ to a pair of spectral dual $q$-CFT blocks via gauge\/CFT correspondence.\n\\item In Duality web II, we view the $3d$ spectral dual pair as the\n result of Higgsing a pair of $5d$ spectral dual theories and the CY\n three-folds which geometrically engineer them.\n\\item In Duality web III, we reduce the $3d$ spectral dual pair to $2d$ and study the corresponding limit of the $q$-CFT blocks.\n\\end{itemize}\n\n\n\\subsection*{Duality web I}\nDuality web I is shown in Fig.~\\ref{fig1}.\n\\begin{figure}[h]\n \\centering\n\\includegraphics{figures\/dual1-crop.pdf} \n\\caption{Duality web I represents the relation between $3d$ $FT[SU(N)]$\n quiver gauge theories and DF representations of the $N+2$-point\n $A_{N-1}$ $q$-Toda conformal blocks with $N$ degenerate\n primaries. $3d$ mirror symmetry of the gauge theories upstairs\n corresponds to the spectral duality of the CFTs downstairs.}\n\\label{fig1}\n\\end{figure}\nIn the top left corner we have~$\\mathcal{B}^{D_2\\times\n S^1}_{FT[SU(N)]}$, the $D_2\\times S^1$ partition function, or\nholomorphic block,\\footnote{The background is actually twisted with\n twisting parameter $q$, i.e.\\ $D_2$ is fibered over $S^1$ so that it\n gets rotated by $\\ln q$ every time one turns around $S^1$. The\n notation $D_2 \\times_q S^1$ would be more proper, however we omit\n the subscript $q$ for the sake of brevity. The name holomorphic\n block is due to the fact that $D_2 \\times_q S^1$ partition functions\n can be used to build partition functions on compact spaces, such as\n $S^3$ or $S^2\\times S^1$\n \\cite{Beem:2012mb,Pasquetti:2016dyl}\\label{fn:1}.} of the\n$FT[SU(N)]$ theory. For this theory we turn real mass deformations\nfor all the flavors and topological symmetries, so that this theory\nhas $N!$ isolated vacua. As we will see the $FT[SU(N)]$ theory is\nself-dual under the action of the spectral duality and correspondingly\nin the top right corner we find the partition function $\\hat{\\cal\n B}^{D_2\\times S^1}_{FT[SU(N)]}$ of the dual theory. This edge of the\nweb is a genuine duality between two theories flowing to the same IR\nSCFT. However, here we are only looking at the map of the \\emph{mass\n deformed} $D_2\\times S^1$ partition functions which can be regarded\nas a refinement of the map between the effective twisted\nsuper-potential evaluated on the Bethe\nvacua~\\cite{Nekrasov:2009uh,Nekrasov:2009ui} of the two\ntheories~\\cite{Gaiotto:2013bwa}. A thorough discussion of this duality\nwill be provided in~\\cite{APZ}.\n\n\n\nIn Section \\ref{secweb1} we discuss in detail the nontrivial map of\nthe $T[SU(N)]$ and $FT[SU(N)]$ holomorphic blocks under mirror symmetry and spectral duality using various\napproaches including direct residue computations, the relation of the\nholomorphic blocks to the integrable Ruijsenaars-Schneider (RS) system as in \\cite{Bullimore:2014awa} \nand, in Sec.~\\ref{sec:higgsing-5d-gauge}, using the relation between holomorphic blocks, $5d$ gauge theories and refined topological strings.\n\n\nThe vertical edges of the web in Fig. \\ref{fig1} represent\ncorrespondences between gauge theories and conformal blocks akin to\nthe AGT\ncorrespondence~\\cite{Alday:2009aq,Wyllard:2009hg,Mironov:2009by}. One\ncan observe that the holomorphic block integrals\n$\\mathcal{B}^{D_2\\times S^1}$ of $3d$ quiver theories can be directly\nidentified with the Dotsenko-Fateev (DF) integral representation of\nthe conformal blocks in $q$-deformed Toda theory. This correspondence\nis part of the so called triality proposed\nin~\\cite{Aganagic:2013tta,Aganagic:2014kja} and generalized\nin~\\cite{Nieri:2015dts,Iqbal:2015fvd,Nedelin:2016gwu,Lodin:2017lrc}.\n\nIn the particular case of the $FT[SU(N)]$ theory we find that the\nholomorphic block ${\\cal B}^{D_2\\times S^1}_{FT[SU(N)]}$ can be mapped\nto the conformal block ${q{\\rm DF}}^{A_{N-1}}_{N+2}$ involving $N$\nfully-degenerate and two generic primaries, and a particular choice of\nscreening charges in the $q$-deformed $A_{N-1}$ Toda theory. The dual\nholomorphic block $\\hat{\\cal B}^{D_2\\times S^1}_{FT[SU(N)]}$ is also\nmapped to a ${q{\\rm DF}}$ integral block $\\hat{q{\\rm\n DF}}^{A_{N-1}}_{N+2}$ which is related to ${q{\\rm\n DF}}^{A_{N-1}}_{N+2}$ by a ``degenerate'' version of spectral\nduality. An exact meaning of this statement should become clear at\nthe end of the discussion of the Duality web II.\n\nThe details of the correspondence between holomorphic blocks of the\n$FT[SU(N)]$ theory and $q$-Toda integral blocks as well as spectral\nduality are presented in the Sec.~\\ref{secweb1}.\n\n\\subsection*{Duality web II}\n\nDuality web I in fig. \\ref{fig1} can actually be understood as a\nconsequence of another web of dualities involving $5d$ $\\mathcal{N}=1$\nquiver theories and correlators of generic (non-degenerate)\n$q$-deformed Toda vertex operators. More precisely, we consider the\nduality web II shown in Fig.~\\ref{fig2} where duality web I\ncorresponds to the bottom face\n(face~{\\color{green!60!black}\\textsf{1}}) of the cube.\n\\begin{figure}[h]\n\\centering\n\\includegraphics{figures\/cube-1-crop}\n\\caption{Duality web II incorporates duality web I (face~{\\color{green!60!black}\\textsf{1}} of the\n cube) in a more general context of $3d$-$5d$-CFT triality.}\n\\label{fig2}\n\\end{figure}\nIn the top left corner we have the $5d$ $\\mathcal{N}=1$ linear quiver\ngauge theory with $(N-1)$ $U(N)$ gauge nodes and $N$\n(anti-)fundamental matter hypermultiplets on each end of the\nquiver. This theory is self-dual under $5d$ spectral duality which\nrelates $U(N)^{M-1}$ to $U(M)^{N-1}$ linear quiver theories\ncompactified on a circle.\n\nThis is a duality between two low energy descriptions of the same\nstrongly interacting UV SCFT which can be conveniently understood\nusing brane setup~\\cite{Aharony:1997bh}. The details of the maps of\nthe parameters of the two theories are nontrivial and have been\nrecently discussed in~\\cite{Bao:2011rc} and~\\cite{Bergman:2013aca}.\nThis duality has been studied also in the context of integrability in\n\\cite{Mironov:2013xva,Mironov:2012uh,Mitev:2014isa,Isachenkov:2014eya,Zenkevich:2014lca}. The\nterm \\emph{spectral} for this duality comes from this interpretation.\n\n\n\nWe will be focusing on the $\\mathbb{R}^4\\times S^1$ instanton\npartition function which can be realised using geometric engineering\nas the refined topological string partition function\n$Z_{\\mathrm{top}}$ associated to the square toric diagram depicted in\nFig.~\\ref{fig:4}~a). Then one can immediately understand invariance of\nthe square quiver theory under spectral duality as the fiber-base\nduality corresponding to the reflection along the diagonal of the\ndiagram.\n\nThe instanton or topological string partition functions are actually\nbased on $U(N)$ quivers, so if we are interested in the $SU(N)$ case,\nwe should strip off the $U(1)$ contribution. This procedure is\ndiscussed for example in~\\cite{Bergman:2013aca}. However, for the\npurpose of this paper, where we discuss instanton partition functions,\nwe can keep the $U(1)$ parts and work with the duality relating\n$U(N)^{M-1}$ to $U(M)^{N-1}$ theories.\n\n \n\n\n\n\n\nIn the other two vertices of face~{\\color{green!60!black}\\textsf{2}}\nwe have an $(N+2)$-point correlator in the the $q$-deformed $A_{N-1}$\nToda theory and its spectral dual\\footnote{In the conformal block\n $\\langle V_1\\cdots V_{N+2}\\rangle_{qA_{N-1}}$ the primaries $V_1$\n and $V_{N+2}$ have generic momenta while all the others have momenta\n proportional to the same fundamental weight and correspond to simple\n punctures in the AGT language.}.\n\n\n\n\nThe $q$-Toda correlators also enjoy the spectral duality relating\n$(K+2)$-point correlators in $A_{N-1}$ $q$-Toda to $(N+2)$-point\ncorrelators in $A_{K-1}$ $q$-Toda theory~\\cite{Zenkevich:2014lca,\n Morozov:2015xya} which is the avatar of the $5d$ spectral duality\nrelating $U(N)^{M-1}$ to $ U(M)^{N-1}$ $5d$ quivers. The\nidentification between $5d$ instanton partition functions and $q$-Toda\ncorrelators is the $5d$ uplift of the AGT correspondence\n\\cite{Awata:2010yy,Awata:2009ur}. More precisely, the AGT map\ncorresponds to the diagonal edges (shown in blue in Fig.~\\ref{fig2}),\nwhile the map along the edges of\nface~{\\color{green!60!black}\\textsf{2}} are from the triality\napproach~\\cite{Aganagic:2013tta,Aganagic:2014kja}.\n \n \nThe vertical arrows going down from the $5d$ web (face\n{\\color{green!60!black}\\textsf{2}}) to the $3d$ web\n(face~{\\color{green!60!black}\\textsf{1}}) indicate a \\emph{tuning\n procedure} where the parameters are fixed to specific discrete\nvalues. On the gauge theory side\n(face~{\\color{green!60!black}\\textsf{3}}) this tuning corresponds to\nthe so called Higgsing procedure~\\cite{Gaiotto:2012xa,\n Mironov:2009qt,Kozcaz:2010af,Dimofte:2010tz,Dorey:2011pa,Nieri:2013vba,Gaiotto:2014ina}. By\ntuning the $5d$ Coulomb branch parameters one can degenerate the $5d$\npartition function into the partition function of a coupled $5d$--$3d$\nsystem describing co-dimension two defect coupled to the remaining\n$5d$ bulk theory. We consider particular tuning of the parameters so\nthat the square $5d$ quiver is Higgsed \\emph{completely,} i.e.\\ it\nreduces to the $3d$ $FT[SU(N)]$ theory coupled to some free $5d$\nhypers\\footnote{The $T[SU(N)]$ vortex partition function has also been\n related to a ramified surface defect in the $5d$ $\\mathcal{N}=2^*$\n theory in~\\cite{Bullimore:2014awa}.}. We demonstrate this in\nSec.~\\ref{sec:higgsing-5d-gauge}. Repeating the Higgsing procedure on\nthe spectral dual side we land on the $3d$ spectral dual $FT[SU(N)]$\ntheory. We then see that $3d$ (self)-duality for $FT[SU(N)]$ follows\nvia Higgsing from the $5d$ spectral duality for the square quiver.\n\nOn the $q$-Toda side (face~{\\color{green!60!black}\\textsf{6}}) the\ntuning procedure corresponds to the tuning of the momenta of the\nvertex operators to special values (corresponding to fully degenerate\nvertex operators) and to a given assignment of screening charges\n(corresponding to conditions on the internal momenta, or Coulomb\nbranch parameters). In this way the $q$-Toda $A_{N-1}$ correlator with\n$N$ semi-degenerate and two full primary operators reduces to the\n$q$-DF representation of the conformal block.\n\nThis explains our previous statement that the integral blocks ${q{\\rm\n DF}}^{A_{N-1}}_{N+2}$ and $\\hat{q{\\rm DF}}^{A_{N-1}}_{N+2}$ are\nrelated by a degenerate version of spectral duality.\n\n\n\n\n\\subsection*{Duality web III}\n\nFinally, starting from duality web I in Fig.~\\ref{fig1} we can obtain\nanother interesting duality web by taking a suitable limit $q\\to 1$ as\nshown in Fig.~\\ref{fig3}, where the duality web I corresponds to face\n{\\color{green!60!black}\\textsf{1}} of the cube.\n\\begin{figure}[h]\n\\centering\n\n \\includegraphics{figures\/cube-2-crop}\n \\caption{Duality web III. Fig.~\\ref{fig1} is the top face (face\n ~{\\color{green!60!black}\\textsf{1}}) of the cube. The arrows going\n downstairs correspond to $q\\to 1$ limits. Notice that the two\n theories related by the spectral duality tend to different theories\n under $q\\to 1$. This asymmetry appears because one needs to choose\n the scaling of the parameters with $q$ and the spectral duality map\n relates two \\emph{different} choices.}\n\\label{fig3}\n\\end{figure}\nLet's consider face~{\\color{green!60!black}\\textsf{3}} in\nFig.~\\ref{fig3}. Here we are performing the reduction of a $3d$\nspectral pair of theories on $D_2\\times S^1$ from $3d$ to $2d$ by\nconsidering the $q\\to 1$ limit, which corresponds to shrinking the\n$S^1$ radius. Taking this limit is subtle, as recently discussed in~\\cite{Aharony:2017adm} (and before in~\\cite{Aganagic:2001uw}), since\nthere exist in fact \\emph{several} meaningful limits. Concretely, one\ncan consider the situation when some of the $3d$ real mass parameters\nare scaled to infinity when going from $3d$ to $2d$ so that $m_{3d}\nR=m_{2d}$ remains finite as $R\\to 0$.\n\nStarting from $\\mathcal{B}^{D_2\\times S^1}_{FT[SU(N)]}$ we take the so\ncalled Higgs limits which reduces it to the $\\mathcal{N}=(2,2)$ gauged\nlinear sigma-model (GLSM) $\\mathcal{B}^{D_2}_{FT[SU(N)]}$. In the\n\\emph{Higgs} limit the real mass scaled to infinity is the FI\nparameter, while the matter remains light, hence the name. This limit\ngenerally reduces a $3d$ gauge theory to a $2d$ gauge theory. However,\nhere we want to lift also the Higgs branch and we turn on all the mass\ndeformations so that the $2d$ gauge theory is massive and has $N!$\nisolated vacua.\n\nSince spectral duality, similarly to mirror symmetry, swaps Higgs and\nCoulomb branch parameters, on the dual side the limit has a very\ndifferent effect. The dual block $\\hat{\\cal B}^{D_2\\times\n S^1}_{FT[SU(N)]}$ in the $q\\to 1$ limit (which is now a\n\\emph{Coulomb} limit) reduces to the partition function of a theory of\ntwisted chiral multiplets with twisted Landau-Ginzburg superpotential\non $D_2$. The horizontal link in\nface~{\\color{green!60!black}\\textsf{2}} of the cube in Fig.~\\ref{fig3}\nis, therefore, a duality of Hori-Vafa type~\\cite{Hori:2000kt} for mass\ndeformed theories.\n\nIn general claiming that a duality for mass deformed theories implies\na duality for massless theories is dangerous. In particular, in this\ncontext the subtleties of inferring a genuine IR $2d$ duality from a\nduality for $2d$ mass deformed theories obtained from the reduction of\npairs of dual theories have been discussed in\n\\cite{Aharony:2016jki,Aharony:2017adm}. Here we are not interested in\nremoving the mass deformations since, as we are about to see, the\nholomorphic blocks for the mass deformed theories can be directly\nmapped to CFT conformal blocks.\n\nIndeed if we look at face~{\\color{green!60!black}\\textsf{4}} of the\nduality web III in Fig.~\\ref{fig3}, we see that we are taking\n\\emph{several} different $q\\to 1$ limits of the $q$-Toda conformal\nblocks in DF representation. Similarly to the gauge theory side there\nare \\emph{several} possible ways to take the limit. The limit when we\nscale the momenta of the vertex operators and keep the insertion\npoints fixed is natural from the CFT point of view and reduces\n$q$-Toda conformal blocks to conformal blocks of the undeformed Toda\nCFT. This is exactly the limit we take when we reduce the spectral\ndual block $q\\hat{{\\rm DF}}^{A_{N-1}}_{N+2}$ down to the undeformed\nconformal block $\\hat{{\\rm DF}}^{A_{N-1}}_{N+2}$ in $2d$ Toda\ntheory. Therefore, we have just discovered that the $2d$ $FT[SU(N)]$\nGLSM holomorphic block $\\mathcal{B}^{D_2}_{FT[SU(N)]}$ is mapped to a\n$2d$ CFT conformal block $\\hat{{\\rm DF}}^{A_{N-1}}_{N+2}$ (red\ndiagonal on the face~{\\color{green!60!black}\\textsf{4}}). In other\nterms we have derived the familiar gauge\/CFT correspondence between\n$S^2$ partition functions and degenerate CFT correlators discussed in\n\\cite{Gomis:2014eya,Gomis:2016ljm,Doroud:2012xw} as a limit of our\n$3d$ spectral duality web.\n\nFinally to complete the picture we study what is the effect of the\n$q\\to 1$ limit on the $q{\\rm DF}^{A_{N-1}}_{N+2}$ conformal block. This is\na less familiar limit which reduces the $q{\\rm DF}^{A_{N-1}}_{N+2}$ to\na block in the channel with the vertex operators of certain bosonized\nalgebra, which we denote by $d$-$W_N$, where $d$ stands for\n\\emph{difference} in the same way as $q$ in $q$-$W_N$ is for\n\\emph{quantum}. The algebra\\footnote{We thank A.~Torrielli for\n pointing out a paper~\\cite{Hou:1996fx} in which a similar algebra\n has appeared earlier in a very different context.} $d$-$W_N$ is a\nparticular limit of the $q$-$W_N$ algebra when $q\\to 1$. We briefly\ndescribe the algebra, correlators and screening charges, leaving a\nmore detailed investigation for the future~\\cite{vertex:future}.\n\nThe Duality web III is discussed in Sec.~\\ref{secweb4}. \n\n\n\n\\section{Duality web I: $3d$ $FT[SU(N)]$ and $q$-Toda blocks}\\label{secweb1}\n\n\n\n\nIn this section we study Duality web I shown in Fig. \\ref{fig1}. We\nfirst introduce the $3d$ holomorphic block ${\\cal B}^{D_2\\times\n S^1}_{T[SU(N)]}(\\vec{\\mu}, \\vec{\\tau}, q, t)$, then we\nshow the effect of adding the flipping fields and discuss the mirror\nand spectral duals of the theory. Finally we introduce the DF\nrepresentation for the $q$-Toda blocks and determine the gauge\/$q$-DF\ndual to ${\\cal B}^{D_2\\times S^1}_{FT[SU(N)]}(\\vec{\\mu},\n\\vec{\\tau}, q, t)$.\n\n\n\\subsection{$3d$ blocks for $T[SU(N)]$, flipping fields, mirror and\n spectral duals}\\label{3dblocks}\n\n\\subsubsection{$3d$ holomorphic blocks}\n\\label{sec:3d-block}\nWe begin by introducing our main character ${\\cal B}^{D_2\\times\n S^1}_{T[SU(N)]}(\\vec{\\mu}, \\vec{\\tau}, q, t)$, the $D_2\\times S^1$\npartition function, or $3d$ holomorphic block integral for the\n$T[SU(N)]$ theory. The $\\mathcal{N}=4$ $T[SU(N)]$ theory is a quiver\ntheory with gauge group $U(1) \\times\nU(2) \\times \\cdots \\times U(N-1)$, with bifundamental hypers\nconnecting the $U(N_a)$ and $U(N_{a+1})$ nodes for $a=1,\\cdots, N-2$\nand $N$ hypermultiplets at the final node. As an example we present\nthe quiver diagram of the $T[SU(4)]$ theory on Fig.~\\ref{fig:2}.\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=8cm]{figures\/3d-quiver-crop}\n \\caption{$3d$ $T[SU(4)]$ gauge theory. $\\tau_a$ are the FI\n parameters. The masses of the chirals are indicated over or under\n the corresponding arrows.}\n \\label{fig:2}\n\\end{figure}\nWe turn on real masses $M^{3d}_a$ in the Cartan of the $SU(N)_H$\nsymmetry rotating the Higgs branch and $T^{3d}_a$ in the Cartan of the\n$SU(N)_C$ symmetry rotating the Coulomb branch. We also turn on an\nextra real axial mass deformation $m^{3d}$ for $U(1)_A$, the\nanti-diagonal combination of $U(1)_C\\times U(1)_H\\in SU(2)_C\\times\nSU(2)_H$, which breaks the super-symmetry down to ${\\cal N}=2^*$. We\ndefine the dimensionless mass parameters $M'_a=R M^{3d}_a$,\n$T'_a=RT^{3d}_a$ and $m'=R m^{3d}$ and the parameter\n$q=e^{\\hbar}=e^{R\\epsilon}$, where $R$ is the $S^1$ circle radius and\n$\\epsilon$ is the equivariant parameter rotating the cigar $D_2$ (see\nfootnote $^{\\ref{fn:1}}$).\n\n\nThe holomorphic block integral for this theory can be constructed as\nexplained in~\\cite{Beem:2012mb} and reads:\n\\begin{multline}\n {\\cal B}^{D_2\\times S^1}_{T[SU(N)]}(\\vec{\\mu}, \\vec{\\tau},\n q, t)= F(q,t,\\vec{\\tau}) \\int_{\\Gamma}\n \\prod\\limits_{a=1}^{N-1}\\prod\\limits_{i=1}^{a} \\left(\n \\frac{dx_i^{(a)}}{x_i^{(a)}} e^{ X_i^{(a)}\n \\left(T_{a}-T_{a+1}\\right)\/\\hbar}\\, t^{-X_i^{(a)}\/\\hbar}\\right) \\times\\\\\n \\prod\\limits_{a=1}^{N-1} \\frac{ \\prod\\limits_{i\\neq j}^{a} \\Big(\n \\frac{x^{(a)}_j}{x^{(a)}_i} ; q\\Big)_\\infty }{\n \\prod\\limits_{i,j=1}^{a} \\Big( t \\frac{x^{(a)}_j}{x^{(a)}_i};\n q\\Big)_\\infty }\n \\prod\\limits_{a=1}^{N-2}\\prod\\limits_{i=1}^{a}\\prod\\limits_{j=1}^{a+1}\n \\frac{ \\Big(t \\frac{x^{(a+1)}_j}{x^{(a)}_i} ; q\\Big)_\\infty }{ \\Big(\n \\frac{x^{(a+1)}_j}{x^{(a)}_i}; q\\Big)_\\infty } \\,\n \\prod\\limits_{p=1}^{N}\\prod\\limits_{i=1}^{N-1} \\frac{ \\Big(t\n \\frac{\\mu_p}{x^{(N-1)}_i} ; q\\Big)_\\infty }{ \\Big(\n \\frac{\\mu_p}{x^{(N-1)}_i}; q\\Big)_\\infty } \\,,\n\\label{TSUN:partition}\n\\end{multline}\nwhere the prefactor $F(q,t,\\vec{\\tau})$ is given by\n\\begin{equation}\n F(q,t,\\vec{\\tau})=e^{-\\frac{2}{3} N(N-1)(2N-1)\\hbar \\beta(1-\\beta)} \n e^{-\\frac{m'^2 N}{4\\hbar}} e^{(1-\\beta) \\sum_{a=1}^{N-1} \\frac{a^2}{2} (T_{a+1} - T_a)}\\,.\n\\end{equation}\nThe integral is performed over the Cartan of the gauge group. For each\ngauge node we have the contribution of vector and adjoint chiral\nmultiplets (first factor in the second line) given by a ratio of\n$q$-Pochhammer symbols defined as\n\\begin{equation}\n(x;\\,q)_\\infty=\\prod\\limits_{k=0}^\\infty \\left( 1- x q^k \\right)\\,.\n\\label{qPoc:symbol}\n\\end{equation}\nThe other factors in the second line are the contributions of the\nbifundamental chirals and of the fundamentals attached to the last\nnode.\n\nMore precisely $(qx^{-1};q)_\\infty$ is the contribution to the block\nintegral of a chiral multiplet of zero $r$-charge and charge $+1$\nunder a flavor symmetry with associated real mass $x$, plus a\n$-\\frac{1}{2}$ Chern-Simons unit. This corresponds to a chiral\nmultiplet with Dirichlet boundary conditions along $\\partial (D_2\n\\times S^1) = T^2$ in \\cite{Yoshida:2014ssa}. A chiral multiplet of\n$r$-charge $+2$, charge $-1$ and $\\frac{1}{2}$ Chern-Simons unit\ncontributes as $(x;q)^{-1}_\\infty$ and corresponds to Neumann boundary\nconditions\\footnote{There is a relation between these two setups:\n\\begin{equation}\n \\label{eq:2}\n (qx^{-1};q)_\\infty = \\frac{\\theta_q(x)}{(q;q)_{\\infty}} (x;q)^{-1}_\\infty,\n\\end{equation}\nwhich can be explained by viewing the $3d$ theory of a single chiral\nmultiplet $\\phi$ as a linear sigma model with target\n$\\mathbb{C}$. Dirichlet boundary conditions on $\\partial (D_2 \\times\nS^1) = T^2$ correspond to a D-brane at a point $\\phi=0$ in the\ntarget. However, one can view the D-brane in a different way, as (an\nequivalence class of) a complex of sheaves\n\\begin{equation}\n \\label{eq:4}\n 0 \\to \\mathcal{O} (\\mathbb{C}) \\stackrel{s}{\\to}\n \\Omega^{1,0}(\\mathbb{C}) \\to 0\n\\end{equation}\nsupported on the whole $\\mathbb{C}$. Here $\\mathcal{O}(\\mathbb{C})$ is\nthe sheaf of functions on $\\mathbb{C}$, and $\\Omega^{1,0}(\\mathbb{C})$\nis that of $(1,0)$ differential forms, e.g.\\ $g(\\phi) \\psi$, where\n$\\psi$ is an anticommuting coordinate on the fiber; the differential\n$s = \\phi \\psi$ is nilpotent because $\\psi^2 = 0$. The relation\nbetween the brane at fixed $\\phi=0$ and the complex is as\nfollows. Mnemonically, one can think that the two terms of the\ncomplex~\\eqref{eq:4} ``cancel'' everywhere outside the point\n$\\phi=0$. More concretely, the space of functions on a point $\\{\n\\phi=0 \\} \\subset \\mathbb{C}$ can be equivalently described by the\ncohomology of the complex~\\eqref{eq:4}:\n\\begin{align}\n \\label{eq:7}\n H^0_s(\\mathbb{C}) &= \\ker s = 0 \\\\\n H^1_s(\\mathbb{C}) &= \\Omega^{1,0}(\\mathbb{C})\/\\textrm{Im}\\, s = \\frac{\\{ \\psi g(\\phi)\n \\}}{ g(\\phi) \\sim g(\\phi) + \\phi f(\\phi)} = \\{ \\mathrm{const} \\}\n = \\mathbb{C},\n\\end{align}\nIn the field theory language $\\mathcal{O}(\\mathbb{C})$ corresponds to\na $3d$ free chiral with \\emph{Neumann} boundary conditions, while to\nget the whole complex $\\Omega^{\\bullet,0}(\\mathbb{C})$\nfrom~\\eqref{eq:4} one needs to add a $2d$ free chiral fermion $\\psi$\nliving on $T^2$ whose partition function is precisely given by\n$\\frac{\\theta_q(x)}{(q;q)_{\\infty}}$. The identity~\\eqref{eq:2} is\ntherefore just the equivalence between two views on the D-brane.}.\n\nIf we assemble the matter contribution to the block integrals taking\nsome chirals with Dirichlet and some with Neumann boundary\nconditions we induce \\emph{mixed} Chern-Simons couplings (because of\nthe attached $\\frac{1}{2}$ units) which we might need to compensate by\nadding extra Chern-Simons terms to the action.\nWith our symmetric choice of boundary conditions the induced dynamical\nChern-Simons couplings vanishes automatically, the induced mixed\ngauge-flavor couplings (the $t^{\\frac{-X^{(a)}_i}{\\hbar}}$ factor in\nthe integrand~\\eqref{TSUN:partition}) renormalize the FI parameters,\nwhile the background mixed couplings contribute as the prefactor\n$F(q,t,\\vec{\\tau})$.\n\n\n\nTo present the block in a form more convenient for the following we\nhave shifted the integration variables and identified a new set of\nexponentiated mass parameters\\footnote{The shifted mass parameters\n satisfy:\n \\begin{equation}\n \\label{eq:57}\n \\sum_{i=1}^N M_i = \\hbar (1-\\beta) \\frac{N^2}{2},\\qquad \\sum_{i=1}^N T_i = \\hbar \\beta \\frac{N^2}{2}.\n\\end{equation}\nThe parameter $t$ introduced in this section will be identified with\nthe parameter of the $5d$ $\\Omega$-background\n$\\mathbb{R}^4_{q,t}\\times S^1$ and with the $(q,t)$-Toda parameter in\nthe next sections. }\n\\begin{gather}\n x^{(a)}_i=e^{X^{(a)}_i}, \\quad\n \\mu_p=e^{M_p}=e^{M'_p}\\left( \\frac{q}{t}\\right)^{N\/2},\\notag \\\\\n \\quad \\tau_p=e^{T_p}=e^{T'_p}t^{N\/2}, \\quad t= q^{\\beta}=-q^{1\/2}\n e^{-m'}.\\label{eq:58}\n\\end{gather}\nAn alternative procedure to write down the block integrand $\\Upsilon$\nis to view it as ``square root'' of the integrand of the partition\nfunction on a compact manifold. Details of this procedure are\npresented in the Appendix~\\ref{appendix:block}. This construction also\nindicates that the contribution of (mixed) Chern-Simons coupling to\nthe partition function should actually be expressed in terms of ratios\nof theta functions rather than exponents\n\\begin{equation}\n\\label{eq:56}\ne^{\\frac{AB}{\\hbar}} \\leadsto\n\\frac{\\theta_q(e^A)\\theta_q(e^B)}{\\theta_q(e^{A+B}) (q;q)_{\\infty}}\\,\n\\end{equation}\nas we discuss in Appendix~\\ref{appendix:block}. However, for the\npurpose of this paper we can avoid introducing theta functions and\nwork with the exponents provided that on the integration contours on\nwhich we are going to evaluate the blocks, the theta functions and the\nexponents have no poles and contribute with the same residue. One can\ncheck that for $T[SU(N)]$ blocks~\\eqref{TSUN:partition} this will\nindeed be the case. As will be shown below, the residues of the\nintegrals~\\eqref{TSUN:partition} come in geometric progressions, i.e.\\\na pole $x_{*}$ is accompanied by a string of poles at $q^k x_{*}$ with\n$k\\in \\mathbb{Z}_{\\geq 0}$. Notice then that both sides of\nEq.~\\eqref{eq:56} transform in the same way under the $\\hbar$-shifts\nof $A$ and $B$ variables, i.e.\\ under $q$-shifts of $e^A$ and\n$e^B$. Thus, their contributions to the residues in the string differ\nonly by an \\emph{overall} constant factor, independent of $k$. This\noverall constant can be factored out of the integral and included in\nthe normalization factors.\n\n\nFinally we need to discuss the integration contour on which we\nevaluate the block integral. The integration in\nEq.~(\\ref{TSUN:partition}) is performed over a basis of integration\ncontours $\\Gamma=\\Gamma_\\alpha$, with $\\alpha=1,\\ldots, N!$ which are\nin one to one correspondence with the SUSY vacua, the critical points\nof the one-loop twisted superpotential\n$\\mathcal{W}^{\\mathbb{R}^2\\times S^1}$. The label of the integration\ncontour $\\alpha$ is essentially an element of the permutation group\n$\\mathfrak{S}_N$. One can understand the origin of the contours\n$\\Gamma_{\\alpha}$ as follows.\n\nThe integrations in Eq.~\\eqref{TSUN:partition} can be done\nstep by step starting from $x_i^{(N-1)}$ and proceeding to\n$x_1^{(1)}$. There are $(N-1)$ integration variables at the first\nstep. The poles of the integrand in $x_i^{(N-1)}$ correspond to zeroes\nof $\\prod_{p=1}^N\\prod_{i=1}^{N-1} ( \\mu_p\/x^{(N-1)}_i;\nq)_\\infty$. Moreover, upon closer examination one can see that each\n$x_i^{(N-1)}$ should be of the form $q^{k_i^{(N-1)}} \\mu_p$ with\ninteger $k_i^{(N-1)}$ and \\emph{distinct} values of $p$, i.e.\\ each of\n$(N-1)$ variables $x_i^{(N-1)}$ settles at a pole close to its own\nmass $\\mu_p$ and no two of the variables can sit near the same\nmass. Therefore, there are $N$ possible configurations with $(N-1)$\nvariables filling $N$ places (the integrand is symmetric in\n$x_i^{(N-1)}$). Evaluating the residues in $x_i^{(N-1)}$ we can\nproceed to the next step of integration. Here the situation is\nrepeated: there are $(N-2)$ integration variables $x_i^{(N-2)}$ and\n$(N-1)$ variables $x_i^{(N-1)}$ from the previous step play the role\nof $\\mu_p$ for them. The poles in $x_i^{(N-2)}$ are located at\n$q^{k_i^{(N-2)} - k_i^{(N-1)}} x_j^{(N-1)}$ with integer $k_i^{(N-2)}\n\\geq k_i^{(N-1)}$ and again no two variables $x_i^{(N-2)}$ can sit\nnear the same $x_j^{(N-1)}$. There are $(N-1)$ possibilities at this\nstep. Proceeding further, one notices the general pattern: the poles\nat each step sit near the poles of the previous step with one free\nplace. Equivalently, there are $(N-1)$ strings of poles with lengths\n$1$, $2$,~\\ldots,~$(N-1)$, in each of which the poles are close\ntogether, e.g.\\ for a string of length $a$ we get\n\\begin{equation}\n \\label{eq:6}\n x^{(a)}_a = q^{k_a^{(a)}-k_a^{(a+1)}} x^{(a+1)}_a = q^{k_a^{(a)}-k_a^{(a+2)}} x^{(a+2)}_a =\\ldots =q^{k_a^{(a)}-k_a^{(N-1)}}\n x^{(N-1)}_a = q^{k_a^{(a)}} \\mu_p\n\\end{equation}\nwhere $k^{(a)}_i$ are all integers and we have used the symmetry of\nthe integrand in $x_1^{(a)}$, \\ldots, $x_a^{(a)}$ to set all the lower\nindices in the string to $a$. Each string terminates at the the free\nplace, not filled by the pole on the next step. Choosing the\nintegration contour is equivalent to specifying which string (of\nlength $a$) sits near which mass $\\mu_p$. Evidently, any choice can be\nobtained from a given one by the unique permutaiton of masses\n$\\mu_p$. There are therefore $N!$ choices in total, each one\ncorresponding to an element of the symmetric group\n$\\mathfrak{S}_N$.\n\nWe will do the calculations for a certain convenient reference choice\nof contour $\\alpha = \\alpha_0$, i.e.\\ in the reference vacuum in which\n\\begin{equation}\n \\label{eq:18}\n x_i^{(a)} = q^{k_i^{(a)}} \\mu_i \\qquad \\text{(in vacuum $\\alpha_0$).}\n\\end{equation}\nIn this vacuum one can expand the vortex partition function as a\ndouble series in $\\frac{\\mu_i}{\\mu_{i+1}}$ and\n$\\frac{\\tau_i}{\\tau_{i+1}}$, i.e.\\ it is implicitly assumed that the\ntheory sits in the chamber of the moduli space where\n$\\frac{\\tau_i}{\\tau_{i+1}} \\ll 1$. Blocks for other vacua can be\nobtained from the block in the reference vacuum by analytic\ncontinuation in $\\frac{\\tau_i}{\\tau_{i+1}}$, taking into account the\nintricate (theta-function) connection coefficients. Let us also notice\nthat since the block is self-dual under mirror symmetry, analytic\ncontinuation in $\\mu_i$ and $\\tau_i$ will give the same results.\n\nThe integration over $\\Gamma_{\\alpha_0}$ yields\n\\begin{equation}\n{\\cal B}^{D_2\\times S^1,\\, (\\alpha_0)}_{T[SU(N)]}=Z^{3d,\\, (\\alpha_0)}_{\\textrm{cl}} \nZ^{3d,\\, (\\alpha_0)}_{1\\mathrm{-loop}} Z^{3d,\\, (\\alpha_0)}_{\\mathrm{vort}}\n\\end{equation}\nwhere $Z^{3d,\\, (\\alpha_0)}_{\\textrm{cl}}$, $Z^{3d,\\,\n (\\alpha_0)}_{1\\mathrm{-loop}}$ and $Z^{3d,\\,\n (\\alpha_0)}_{\\mathrm{vort}}$ denote the classical, perturbative\none-loop and nonperturbative vortex contributions respectively. We\nhave\\footnote{We can trade the theta-functions for exponents using the\n equivalence~\\eqref{eq:56}, but we retain the exact answer for the\n integration for the sake of completeness.}\n\\begin{equation}\n \\label{eq:27}\n Z^{3d,\\, (\\alpha_0)}_{\\textrm{cl}}(\\vec{\\mu}, \\vec{\\tau}, q, t) = F(q,t,\\vec{\\tau})\\,\n (q;q)_{\\infty} ^{-\\frac{N(N-1)}{2}} \\prod_{i=1}^N e^{\\frac{(T_i - T_N)\n M_i}{\\hbar}} t^{-\\frac{(N-i)M_i}{\\hbar}} \\prod_{i0.\n\\end{gather}\nDue to the operator-state correspondence the ket state\n$|\\vec{\\alpha}\\rangle$ can be created by the insertion of the vertex\noperator~(\\ref{vertex:operator:toda}) of weight $\\vec{\\alpha}$ at\npoint $z=0$. Bra state $\\langle \\vec{\\alpha} |$ is created by\ninserting the corresponding operator at $z=\\infty$. We understand the\nweight $\\vec{\\alpha}^{(0)}$ of the vertex operator at zero to be a\nfree parameter of the correlator. Then the weight\n$\\vec{\\alpha}^{(\\infty)}$ is determined uniquely by the momentum\nconservation relation, which needs to be satisfied in order for the\ncorrelator~(\\ref{correlator:toda}) to be nonzero:\n\\begin{equation}\n 2\\sqrt{\\beta}{\\cal Q}_{\\beta}\\vec{\\rho}=\\vec{\\alpha}^{(0)}+\\vec{\\alpha}^{(\\infty)}+\\sum\\limits_{j=1}^l\\vec{\\alpha}^{(j)}+\n \\beta\\sum\\limits_{a=1}^{n}N_a\\vec{e}_{(a)}\\,,\n\\label{charge:conservation}\n\\end{equation}\nwhere $\\vec{\\rho}$ and $\\vec{e}_{(k)}$ are given by\nEqs.~\\eqref{eq:5}. The calculation of the free field\ncorrelator~\\eqref{correlator:toda} is presented in\nAppendix~\\ref{sec:toda-theory} and results in \n\\begin{multline}\n {\\rm DF}^{A_n}_{l+2}(z_1,\\ldots, z_l, \\vec{\\alpha}^{(0)},\n \\vec{\\alpha}^{(1)},\\ldots,\n \\vec{\\alpha}^{(l)}, \\vec{N} , \\beta ) \\sim \\\\\n \\sim \\prod\\limits_{p0}\\frac{1-t^k}{1-q^k}c_k^{(a)}\\frac{x^{-k}}{k}+\\sum\\limits_{k>0}c_{-k}^{(a)}\n\\frac{x^k}{k} \\right)\\times\\\\\n\\times\\exp\\left(\\sum\\limits_{k>0}\\frac{1-t^k}{1-q^k}v^k\\,c_k^{(a+1)}\\frac{x^{-k}}{k}-\n\\sum\\limits_{k>0}v^k\\,c_{-k}^{(a+1)}\\frac{x^k}{k}\\right)\\,\\mathclose{:}\\,\\times\\\\\n\\times e^{\\sqrt{\\beta}Q^{(a)}}\\,x^{\\sqrt{\\beta}P^{(a)}}\\,e^{-\\sqrt{\\beta}Q^{(a+1)}}\\,x^{-\\sqrt{\\beta}P^{(a+1)}}\\,\n\\label{screening:current}\n\\end{multline}\nwhere $t=q^{\\beta}$ and we have introduced $v= \\sqrt{\\frac{q}{t}}$.\nSimilarly to the undeformed case the sector index $a$ runs between $1$\nand $n$. Bosonic operators $c_k^{(a)},\\, Q^{(a)},\\, P^{(a)}$ satisfy\nthe Heisenberg algebra~(\\ref{heisenbeg:algebra})\n\n\n$q$-deformed primary vertex operator is chosen to have the form\n\\begin{eqnarray}\n && V^q_{\\vec{\\alpha}}\\left( z \\right) =\n \\normord{\\exp\\left(\\sum\\limits_{k>0}\\sum\\limits_{a=1}^{n+1}\\frac{q^{k\\alpha_a}-1}{1-q^k}c_k^{(a)}\n v^{-ka}\\frac{z^{-k}}{k}+\n \\right.\\nonumber\\\\&&\n \\hspace{6mm} \\left.\\sum\\limits_{k>0}\\sum\\limits_{a=1}^{n+1}\\frac{\\left(q^{-k\\alpha_a}-v^{2k\\left( N-a-1 \\right)} \\right)}{1-t^k}c_{-k}^{(a)}\n v^{ka} \\frac{z^k}{k}\\right)} \\times\n \n \\textrm{e}^{\\frac{1}{\\sqrt{\\beta}}\\sum\\limits_{a=1}^{n+1}\\alpha_a\\,Q^{(a)}}\\,\n z^{\\frac{1}{\\sqrt{\\beta}}\\sum\\limits_{a=1}^{n+1}\\alpha_a\\,P^{(a)}\n }\\,,\n\\label{vertex:operator}\n\\end{eqnarray} where $\\vec{\\alpha}$ is the weight vector just as in\nEq.~(\\ref{vertex:operator:toda}). Essentially this is the vertex\noperator of the same form\\footnote{For precise matching of\n $\\vec{\\alpha}$-dependent part one also needs to perform shift of\n weights $\\alpha_a\\to \\alpha_a+\\frac{1}{2}a(1-\\beta)$ for the vertex\n operators used in ~\\cite{Aganagic:2013tta,Aganagic:2014kja} } as the\none that can be found\nin~\\cite{Aganagic:2013tta,Aganagic:2014kja}. However in the latter\ncase authors have omitted central part of the operator,\ni.e. $\\vec{\\alpha}$-independent part that commutes with the screening\ncurrent $S(x)$ given in (\\ref{screening:current}). As we will see this part\nappears to be essential for us so we keep it.\n\nAs in the non-deformed case we are interested in the following free field correlator\n\\begin{equation}\n q\\mathrm{DF}_{l+2}^{A_n}(z_1,\\ldots, z_l, \\vec{\\alpha}^{(0)},\n \\vec{\\alpha}^{(1)},\\ldots,\n \\vec{\\alpha}^{(l)}, \\vec{N} , q,t) \\stackrel{\\mathrm{def}}{=} \\langle \\vec{\\alpha}^{(\\infty)}|\\,V^q_{\\vec{\\alpha}^{(1)}}\\left( z_1 \\right)\\dots\n V^q_{\\vec{\\alpha}^{(l)}}\\left( z_l \\right)\\,\n \\prod_{a=1}^{n} Q_{(a)}^{N_a}|\\vec{\\alpha}^{(0)}\\rangle_{\\mathrm{free}}\\,,\n\\label{correlator}\n\\end{equation}\nwhere $Q_{(a)}$ are screening charges related to the screening\ncurrents~(\\ref{screening:current}) in the same way as in non-deformed\ncase~(\\ref{screening:charge}). Initial and final states\n$|\\vec{\\alpha}^{(0)}\\rangle$, $|\\vec{\\alpha}^{(\\infty)}\\rangle$ are\ndefined in Eq.~(\\ref{state}). Conservation\nrelation~(\\ref{charge:conservation}) that constraints weights of the\nvertex operators also holds in the $q$-deformed case.\n\nThe free field calculation in the $q$-Toda conformal block is similar\nto the undeformed case and is presented in\nAppendix~\\ref{sec:q-toad-theory}. The final result is given by the following\nmatrix integral:\n\\begin{multline}\n q{\\rm DF}^{A_n}_{l+2}(z_1,\\ldots, z_l, \\vec{\\alpha}^{(0)},\n \\vec{\\alpha}^{(1)},\\ldots,\n \\vec{\\alpha}^{(l)}, \\vec{N} , q,t)\\sim\n C_{{\\rm vert}}^{q}\\left( \\vec{\\alpha},z \\right)\\prod\\limits_p^lz_p^{\\frac{1}{\\beta}\\left( \\vec{\\alpha}^{(p)},\\vec{\\alpha}^{(0)} \\right)+\n\\sum\\limits_{a=1}^NN_a\\left( \\alpha_a^{(p)}-\\alpha_{a+1}^{(p)} \\right)}\\times\\\\\n \\oint \\prod\\limits_{a=1}^n\\prod\\limits_{i=1}^{N_a} d x^{(a)}_i\n \\prod\\limits_{a=1}^n\\prod\\limits_{i=1}^{N_a}\\left( x_i^{(a)}\n \\right)^{\\beta(N_a-N_{a+1}-1)+(\\alpha_a^{(0)}-\\alpha_{a+1}^{(0)})\n +\\sum\\limits_{p=1}^l\\left( \\alpha_a^{(p)}-\\alpha_{a+1}^{(p)}\\right)}\\times\\\\\n \\times\\prod\\limits_{a=1}^n\\prod\\limits_{i\\neq j}^{N_a}\\frac{\\left(\n \\frac{x_j^{(a)}}{x_i^{(a)}};q \\right)_\\infty}\n {\\left(t\\frac{x_j^{(a)}}{x_i^{(a)}};q \\right)_\\infty}\n \\prod_{a=1}^{n-1}\\prod\\limits_{i=1}^{N_a}\\prod\\limits_{j=1}^{N_{a+1}}\\frac{\\left(\n u \\frac{x_j^{(a+1)}}{x_i^{(a)}}; q \\right)_\\infty} {\\left( v\n \\frac{x_j^{(a+1)}}{x_i^{(a)}};q\\right)_\\infty} \\prod_{p=1}^l\n \\prod_{a=1}^n\\prod_{i=1}^{N_a} \\frac{\\left(\n q^{1-\\alpha^{(p)}_{a}}v^a\\frac{z_p}{x_i^{(a)}}; q\n \\right)_\\infty} {\\left(\n q^{1-\\alpha^{(p)}_{a+1}}v^{a}\\frac{z_p}{x_i^{(a)}};q\n \\right)_\\infty}\\,,\n\\label{correlator1}\n\\end{multline}\nwhere $u=\\sqrt{qt}$ and $C_{ {\\rm vert}}$ is the prefactor coming from\nordering different vertex operators. Precise form of this prefactor is\ngiven in~(\\ref{q:toda:order:vert}). The expression appears to be very\ncomplicated. However, as we will see further, in cases relevant for us,\nin particular when some of the vertices are (semi-)degenerate, this\nexpression simplifies drastically.\n\n\n\n\n\\subsection{Map between $FT[SU(N)]$ and $q$-Toda blocks}\\label{3dqdfmap}\n\n\nThe $q$-Toda blocks in DF representation have been shown to map to the\nholomorphic blocks of the handsaw quiver theory\n\\cite{Aganagic:2013tta,Aganagic:2014kja}. Here we are interested in\nthe simpler case of the $FT[SU(N)]$ holomorphic block which can be\nmapped to a $A_{N-1}$ $q$-Toda block with full primary initial and\nfinal states and $N$ fully degenerate primary vertex operators between\nthem (we again omit the prefactors in front of both integrals in\nholomorphic and conformal blocks):\n\\begin{equation} {\\cal B}^{D_2\\times S^1}_{FT[SU(N)]}\\sim\n q\\mathrm{DF}^{A_{N-1}}_{N+2}\\,.\n\\label{TSUN:map:1}\n\\end{equation}\nwith the identification of parameters which we give momentarily.\n\nWe begin by considering an $(N+2)-$point conformal block with the weights of\nthe vertex operators satisfying the following relation:\n\\begin{equation}\n \\alpha_{a+1}^{(p)}=\\alpha_a^{(p)},\\qquad\n a=1,\\dots, N-2, \\quad p=1,\\dots,N\\,.\n\\label{weights:degeneration}\n\\end{equation}\nThe initial state has generic weight $\\vec{\\alpha}^{(0)}$ and the\nweight $\\vec{\\alpha}^{(\\infty)}$ is fixed by the charge conservation\ncondition~ (\\ref{charge:conservation}). We also specify the number of\nscreening charges to be $N_a = a$ for $a = 1,\\ldots, N-1 $. With this\nchoice of momenta~(\\ref{weights:degeneration}) and screening charges\n$q$-Toda conformal block~(\\ref{correlator1}) reduces to the following\nexpression\n\n\\begin{multline}\n \\langle \\vec\\alpha^{(\\infty)}| V^{q}_{\\vec\\alpha^{(1)}}(z_1)\\cdots\n V^q_{\\vec\\alpha^{(N)}}(z_N)\n \\prod\\limits_{a=1}^{N-1}\\left(Q_{(a)}^q\\right)^a |\\vec\n \\alpha^{(0)}\\rangle\\sim\n \\oint \\prod\\limits_{a=1}^{N-1}\\prod\\limits_{i=1}^{a} d y^{(a)}_i\\times\\\\\n \\times\\prod\\limits_{i=1}^{N-1}\\left( y_i^{(N-1)}\n \\right)^{\\beta\\left( N-2 \\right)+\\alpha_{N-1}^{(0)}-\\alpha_N^{(0)}+\n \\sum\\limits_{p=1}^N\\left( \\alpha_{N-1}^{(p)}-\\alpha_N^{(p)}\n \\right)}\n \\prod\\limits_{a=1}^{N-2}\\prod\\limits_{i=1}^{a}\\left( y_i^{(a)} \\right)^{-2\\beta+\\alpha_a^{(0)}-\\alpha_{a+1}^{(0)}}\\times\\\\\n \\times \\prod\\limits_{a=1}^{N-1}\\prod\\limits_{i\\neq\n j}^{a}\\frac{\\left(\\frac{y_j^{(a)}}{y_i^{(a)}};q \\right)_\\infty}\n {\\left(t\\frac{y_j^{(a)}}{y_i^{(a)}};q \\right)_\\infty}\n \\prod_{a=1}^{N-2}\\prod\\limits_{i=1}^{a}\\prod\\limits_{j=1}^{a+1}\\frac{\\left(\n u \\frac{y_j^{(a+1)}}{y_i^{(a)}}; q \\right)_\\infty}\n {\\left( v \\frac{y_j^{(a+1)}}{y_i^{(a)}}; q\\right)_\\infty} \\prod\\limits_{p=1}^N\\prod\\limits_{i=1}^{N-1}\\frac{\\left(\n q^{1-\\alpha^{(p)}_{N-1}}v^{N-1} \\frac{z_p}{y_i^{(N-1)}};q\n \\right)_\\infty} {\\left(\n q^{1-\\alpha^{(p)}_{N}} v^{N-1} \\frac{z_p}{y_i^{(N-1)}}; q\n \\right)_\\infty}\\,,\n\\label{fdeg}\n\\end{multline}\nwhere we have omitted prefactors coming from the ordering of the vertices to concentrate only on\nthe integral for the moment. Expression on the r.h.s of~(\\ref{fdeg}) is almost of the same form as the integral in ${\\cal\n B}^{D_2\\times S^1}_{FT[SU(N)]}$ block ~(\\ref{TSUN:partition}).\nTo complete the map we need to impose a further restriction on the\n$q$-Toda vertex operator parameters. First of all looking on the\none-loop contribution of the vector and adjoint multiplets in the\nblock integral~(\\ref{TSUN:partition}) we can see that the gauge theory\nparameter $t$ related to the $3d$ axial mass is identified with the\n$t$-parameter of Toda CFT deformation. Then in order to match the\ncontribution of the bifundamental hypers with the corresponding term\nin the correlator~(\\ref{fdeg}) we need to make the following\nidentification between the integration variables $y$ in the $q$-DF\nintegral~(\\ref{fdeg}) and $x$ in the holomorphic block\nintegral~(\\ref{TSUN:partition}):\n\\begin{equation}\ny_i^{(a)}=x_i^{(a)}v^{-a}.\n\\end{equation}\nTo identify the last product in the third line of Eq.~(\\ref{fdeg})\nwith the contribution of the fundamental chiral multiplets we need\n\\begin{gather}\n \\mu_p=q^{1-\\alpha_N^{(p)}}\\,v^{2N-2}z_p\\,,\\notag\\\\\n t\\mu_p=q^{1-\\alpha_{N-1}^{(p)}}v^{2N-2}z_p\\,,\n\\label{mass:map}\n\\end{gather}\nwhich amounts to requiring\n\\begin{equation}\n\\alpha_N^{(p)}-\\alpha_{N-1}^{(p)}=\\beta\\,.\n\\label{eq:10}\n\\end{equation}\nEq.~\\eqref{eq:10} together with the\ncondition~(\\ref{weights:degeneration}) completely fixes all the\ncomponents of the vertex weight vectors in terms of the last\ncomponents so that all weights have the form\n\\begin{equation}\n \\vec{\\alpha}^{(p)}=(g_p-\\beta)\\vec{1}+\\beta\\vec{\\omega}_{N-1}\\,,\n\\end{equation}\nwhere $g_p$ is arbitrary constant and $\\vec{\\omega}_{N-1}$ is the\nhighest weight vector of $A_{N-1}$. The map~(\\ref{mass:map}) give us\nfreedom to choose $g_p$ freely. For example we can absorbe it into the\ndefinition of the insertion points $z'_p=q^{\\beta-g_p}z_p$ and\nconsequently have $\\mu_p=v^{2N}z'_p$. Alternatively we can\nsimultaneously shift of all the components of the vertex operator\nweight. This operation does not affect the $q$-DF\nintegral~(\\ref{correlator1:toda}) as it only contribute an overall\nfactor in front of the integral which we omit anyway. So we choose to\nset $g_p=\\beta$ corresponding to vertices with fully degenerate\nmomenta (corresponding to simple \\emph{degenerate} punctures in the\nAGT setup):\n\\begin{equation}\n \\vec{\\alpha}^{(p)}=\\beta\\vec{\\omega}_{N-1}\\,.\n\\label{full:degenration}\n\\end{equation}\nFinally we need to identify the FI parameters of the $FT[SU(N)]$\ntheory with the components of the initial and final momenta of\n$q$-Toda CFT $\\alpha_a^{(0)}$. This can be done by looking at the\npowers of $y_i^{(a)}$ in the $q$-DF integral~(\\ref{fdeg}) and\n$e^{X_i^{(a)}}$ powers in the block\nintegral~(\\ref{TSUN:partition}). We arrive at the following relation:\n\\begin{equation}\n \\alpha_a^{(0)}-\\alpha_{a+1}^{(0)}+1-2\\beta=\n \\frac{T_a-T_{a+1}}{\\hbar} - \\beta,\n\\end{equation}\nand thus\n\\begin{equation}\nT_a=\\hbar\\left( \\alpha_a^{(0)}+\\left( \\beta-1 \\right)a \\right).\n\\end{equation}\n \nSummarizing, the dictionary between the ${\\cal B}^{D_2\\times\n S^1}_{FT[SU(N)]}$ block~(\\ref{TSUN:partition}) and the\n$q$-Toda block $q{\\rm DF}^{A_{N-1}}_{N+2}\\left( z_1,\\ldots, z_N,\n \\vec{\\alpha}^{(0)}, \\beta \\vec{\\omega}_N,\\ldots, \\beta\n \\vec{\\omega}_N, [1,2,\\ldots,N-1], q, q^{\\beta} \\right)$ is given\nin Table~\\ref{table:map}.\n\\begin{table}[h!]\n\\centering\n\\begin{tabular}{|c|c|c|}\n \\hline\n ${\\cal B}^{D_2\\times S^1}_{FT[SU(N)]}$ & Identification & $q\\mathrm{DF}^{A_{N-1}}_{N+2}$ \\\\\n \\hline\n Integration parameters $x_i^{(a)}$ &$y_i^{(a)}=x_i^{(a)}v^{-a}$ & Screening current positions $y_i^{(a)}$\\\\\n \\hline\n Axial mass $t$ & $t=q^\\beta$ & Central charge parameter $\\beta$ \\\\\n \\hline\n Vector masses $\\mu_p$ &$\\mu_p=v^{2N}z_p$ & Positions of the vertex\n operators $z_p$ \\\\\n \\hline\n FI parameters $T_a$ &$T_a=\\hbar \\left( \\alpha_a^{(0)}+(\\beta-1) a\n \\right)$ & Initial state momentum vector $\\vec{\\alpha}^{(0)}$\\\\\n \\hline\n\\end{tabular}\n\\caption{Map between the parameter of the $FT[SU(N)]$ holomorphic block~(\\ref{TSUN:partition})\nand the conformal block~(\\ref{correlator1}) of the $q$-Toda theory.}\n\\label{table:map}\n\\end{table}\nIt is important to notice here that with the choice (\\ref{full:degenration}) of the vertex operator weights and the map of parameters specified in Table~\\ref{table:map} \n the prefactor $C_{ {\\rm vert}}^q$ in the $q\\mathrm{DF}$ integral (\\ref{correlator1}) simplifies drastically and reduces to:\n \\begin{eqnarray}\n C_{ {\\rm vert}}^q\\rightarrow \\prod\\limits_{p \\; ,\n\\label{feynman}\n\\end{equation}\nwhere ${\\bf J}$ is the nuclear current operator and \\mbox{\\boldmath$\n\\epsilon$}$_{\\lambda}$ the polarization vector of the photon. The choice of\nthe\ncurrent operator is related to the model assumptions which are made\nwith respect to the photoabsorption mechanism. This will be the\nsubject of discussion in sect. 2.3. In the forthcoming section\nwe will address the problem of how to construct appropriate A-body\nwavefunctions $\\mid \\Psi_i>$ and $\\mid \\Psi_f>$\nwhich can be used to calculate\nthe angular cross sections for electromagnetically induced\ntwo-nucleon emission processes. For the time being we will not\nintroduce any SRC corrections to the shell-model wave functions.\nAccordingly, all the nuclear wave functions of this work are Slater\ndeterminants.\n\n\\subsection{Shell-model wave functions with two particles in the\ncontinuum}\n\n\nDespite the fact that the basic electromagnetic interaction of (virtual)\nphotons with the nucleus is relatively well understood,\nthe analysis of coincidence experiments of the type ($\\gamma$,N) and\n(e,e$'$N) on finite nuclei\nis often hampered by the strong interactions between the\nstruck nucleon and the medium in which it is embedded. Only under\nexceptional circumstances it was found that the (A-1) nucleons will\nact as spectators. In the most general case a complex variety of\nstrong interactions between the escaping nucleon and the residual\ncore will affect the cross section. These processes are\ncommonly referred to as the final state interaction (FSI). With\nrespect to the FSI two types of effects can be discriminated. First\nwe have to realize that the escaping nucleon cannot be looked upon as\na free particle. Its wave function will somehow reflect the\ndistortions it undergoes in its way out of the nucleus. Secondly, a\nwhole class of multi-step processes can finally lead to an escaping\nnucleon in a particular channel. In fig. 1 we have sketched some\ndiagrams which are considered to represent the main contributions\nto the FSI in a one nucleon knockout process.\nIn drawing these diagrams we have restricted ourselves\nto the case in which the photon is absorbed on one nucleon.\nAll nucleon lines in the\ndiagrams of fig. 1 have to be associated with mean-field single-particle wave\nfunctions.\nDiagram 1(a) refers to the most simple case. The struck\nnucleon is excited in a continuum state of the real mean-field potential\nin which also the bound-state single-particle wave functions are\ngenerated. The processes of diagram 1(b) will generally lead\nto a reduction of the cross section and can be modeled\nfor through generating the distorted wave for the escaping nucleon in\na complex optical potential. The diagrams of fig. 1(c) are more\ndifficult to implement since they involve a coupled channel\ncalculation of the RPA or continuum shell-model type.\n Depending on the kinematical conditions\nthe diagrams of fig.1(c)\nwill lead to a reduction or increase of the strength in a\nparticular channel.\nThroughout the years a combined effort of several groups has resulted\nin a reasonable understanding of the FSI effects for the case of one\nescaping nucleon.\n\n\\begin{figure}[tbh]\n\\vspace{6.5 cm}\n\\caption{Different classes of diagrams for the FSI in\nelectromagnetically induced one-nucleon emission processes. All nucleon\nlines refer to eigenfunctions of a mean-field potential.\nPhotoabsorption on one nucleon is assumed.}\n\\end{figure}\n\nThe situation in which two nucleons are ejected in coincidence from a\ntarget nucleus represents a complicated three-body problem\nand model\nassumptions regarding the FSI could be expected to be\nindispensable in order to keep the calculations feasible. An equivalent set\nof diagrams for those depicted in fig. 1 for the one-nucleon emission\ncase are shown in fig. 2. For the case of two-nucleon emission we\nhave restricted ourselves to two-body photoabsorption. Diagram 2(a)\nis the equivalent of 1(a) for the one-body case. The processes of\nfig. 2(b) involve an interaction of each of the escaping nucleons\nwith the core and can be accounted for by calculating both\ndistorted waves in an optical potential. The recent calculations by\nthe Pavia group \\cite{Giu92} have been performed along this line.\nIn these calculations, the sole impact of the imaginary part\nof the potential was found to\nbe an overall reduction of the cross section.\nThe processes sketched in fig. 2(c) involve an interaction between\nthe two struck nucleons, whereas for diagram 2(d) both detected\nnucleons are assumed to have undergone RPA-like rescattering effects.\nTo our knowledge, no model is available which has estimated the\ngeneral effect of rescattering effects in ($\\gamma$,NN) or (e,e$'$NN)\nprocesses.\n\nIn what follows we will restrict ourselves to the\ndiagrams of the type sketched in fig. 2(a). It involves direct\ntwo-nucleon knockout following photoabsorption on a two-body current.\nThe bound and the continuum states will be generated in the same\nmean-field potential. In this way we preserve the orthogonality\nbetween the nuclear states.\n\n\\begin{figure}[tbh]\n\\vspace{10. cm}\n\\caption{Different classes of diagrams for the FSI in\nelectromagneticaly induced two-nucleon emission. Two-body photoabsorption\nis assumed.}\n\\end{figure}\n\nDealing with two-nucleon emission processes in a shell-model\npicture, we have to\nconstruct eigenstates $\\mid \\Psi_f>$ of the many-body Hamiltonian, that\nasymptotically for two arbitrary coordinates ${\\bf r}_1$ and ${\\bf r}_2$\ntending to infinity, behave like\n\n\\begin{eqnarray}\n\\left<{\\bf r}_1 \\mbox{\\boldmath$\\sigma$}_1, {\\bf r}_2\n\\mbox{\\boldmath$\\sigma$}_2\n \\mid \\Psi_f\n\\right> & \\stackrel {r_1,r_2 \\gg r_A} {\\longrightarrow} & \\frac {1}\n{\\sqrt{A(A-1)}} {\\cal A}_{2(A-2)} \\nonumber \\\\\n & & \\left[ \\chi_{\\frac{1}{2}m_s} (\\mbox{\\boldmath$\\sigma$}_1)\n\\left(e^{i{\\bf k}_{a} \\cdot\n{\\bf r}_1} + f_{k_{a}}(\\theta_a) \\frac {e^{ik_ar_1}} {r_1} \\right)\n\\right. \\nonumber\n\\\\ & & \\times \\left.\n\\chi_{\\frac{1}{2}m_{s'}} (\\mbox{\\boldmath$\\sigma$}_2) \\left(e^{i{\\bf k}_b {\\bf\n\\\n cdot\nr}_2} + f_{k_{b}}(\\theta_b) \\frac {e^{ik_br_2}} {r_2} \\right) \\mid\n(hh')^{-1} J_R M_R > \\right] \\;,\n\\end{eqnarray}\nwhere ${\\cal A}_{2(A-2)}$ antisymmetrizes the wave functions for the\noutgoing nucleons with respect to themselves and with respect to the\nresidual (A-2) system. In the above wave function the residual (A-2)\nnucleus is described by the two-hole ({\\em 2h}) state $\\mid (hh')^{-1}\nJ_R M_R >$ which is\ndefined according to\n\n\\begin{eqnarray}\n\\mid (hh')^{-1}J_R M_R > = \\sum_{m_hm_{h'}} \\frac{1}{\\sqrt{1+\\delta_{hh'}}}\n \\nonumber \\\\\n\\times (-1)^{j_h+m_h+j_{h'}+m_{h'}} c_{h-m_{h}}\nc_{h'-m_{h'}} \\mid \\Phi_0 >\\;.\n\\end{eqnarray}\nThe {\\em 2h} state $\\mid J_R M_R>$ is a fully antisymmetrized and normalized\n(A-2) wave function and $\\mid \\Phi_0>$ is the uncorrelated\nground-state wave function\nof the target nucleus. The wave function of eq. (4) refers to the situation\nwhere two nucleons are escaping from the system with momentum ${\\bf k}_a\n({\\bf k}_b)$ and spin projection $ m_{s} ( m_{s'})$.\nIn case of an induced one-particle emission process in which a nucleon\nis ejected from the target nucleus with a momentum {\\bf k} and in which\nthe residual nucleus is residing in a pure hole state {\\em h}, an appropriate\nasymptotic wavefunction\n\\begin{equation}\n\\frac{1}{\\sqrt A} {\\cal A} \\left[\n\\chi_{\\frac{1}{2}m_s} (\\mbox{\\boldmath$\\sigma$}_1) \\left(e^{i{\\bf k \\cdot\nr_1}} + f_k(\\theta) \\frac {e^{ikr_1}} {r_1} \\right) (-1)^{j_h+m_h} c_{h-m_h}\n\\mid \\Phi_0> \\right]\n\\end{equation}\nis normally obtained by performing a multipole expansion in terms of\nelementary particle-hole ($ph$) excitations \\cite{Ryc88,Mah} :\n\\begin{eqnarray}\n\\mid \\Psi_f > & = & \\sum_{lm_ljm} \\sum_{JM} 4 \\pi i^l\n\\sqrt{\\frac {\\pi} {2 \\mu k}} \\nonumber \\\\\n & & \\times e^{i(\\delta_l+\\sigma_l)}Y_{lm_l}^{*}(\\Omega_k)\n\\mid p(\\epsilon) h^{-1}(JM)>\\;,\n\\end{eqnarray}\nwhere $\\epsilon \\equiv k^2\/(2 \\mu)$,\n$\\mu$ is the reduced mass of the outgoing nucleon, $\\delta _l$ is the\ncentral phase shift, $\\sigma _l$ is the Coulomb phase shift and $\\Omega _k$\ndetermines the solid angle of the momentum ${\\bf k}$ of the escaping\nnucleon.\nThe above expression has been derived under the following normalization\nconventions\nfor the continuum eigenstates $\\varphi_{lj}$\nof the real mean-field potential~:\n\\begin{equation}\n\\varphi_{lj}(r,E) \\stackrel{r \\gg R} {\\longrightarrow}\n\\sqrt{\\frac{2\\mu} { \\pi k}} \\frac {sin(kr-\\eta ln(2kr) -\n\\frac{\\pi l}{2} + \\delta _l + \\sigma _l)} {r}\\;.\n\\end{equation}\n\nAs a natural extension of the above partial-wave expansion in terms\nof $ph$ excitations for one-particle emission processes, we suggest\nthe following partial-wave expansion in terms of $2h2p$ states for\nthe case of two-nucleon emission processes :\n\\begin{eqnarray}\n\\mid \\Psi_f > & = & \\sum_{lm_ljm}\\sum_{l'm_{l'}j'm'} \\sum_{JMJ_1M_1}\n(4 \\pi)^2 i^{l+l'}\n\\frac {\\pi} {2 \\mu \\sqrt{k_a k_b}}\ne^{i(\\delta_l+\\sigma_l+\\delta_{l'}+\\sigma_{l'\n })}\nY_{lm_l}^{*}(\\Omega_{k_{a}}) Y_{l'm_{l'}}^{*}(\\Omega_{k_{b}}) \\nonumber \\\\\n & & \\times \n \\nonumber \\\\\n& & \\times < j m j' m' \\mid J_1 M_1 >\n\\mid (hh')^{-1} J_R ; (p(\\epsilon _a)p'(\\epsilon _b)) J_1 ; JM> \\;,\n\\label{wave}\n\\end{eqnarray}\nwhere $\\epsilon_a \\equiv k_a^2\/(2 \\mu)$, $\\epsilon_b \\equiv k_b^2\/(2\n\\mu)$ and\nthe ${\\em 2h2p}$ state $\\mid (hh')^{-1} J_R , (pp') J_1 ; JM>$\nis defined according to :\n\\begin{eqnarray}\n\\sum_{mm'}\\sum_{M_1M_R} c_{ljm}^{\\dagger}c_{l'j'm'}^{\\dagger} \\mid (hh')^{-1} ; J_R M_R>\\;.\n\\end{eqnarray}\nSince the wave function of eq. (\\ref{wave}) is a linear combination of ${\\em\n2h2p}$ Slater determinants it obeys the antisymmetry condition. It\ncan be easily shown that the wave function of eq. (\\ref{wave}) has the\nrequired asymptotic behaviour with two escaping particles and a\nresidual (A-2) system remaining in the state $\\mid J_R M_R >$.\n\n\\subsection{Absorption mechanisms and transition matrix elements}\n\nHaving established the proper nuclear wave functions we remain with\ndetermining the dominant photoabsorption mechanism. In this work we\nhave restricted ourselves to the diagrams with one pionic line as\nindicated in fig. 3. Diagrams (a) and (b) correspond to the coupling\nof the electromagnetic field with the pionic current and are commonly\nreferred to as the seagull ($\\pi sea$) and the pion-in-flight ($\\pi\nfli$) diagrams. Next, diagrams (c) are related to intermediate\n$\\bigtriangleup$-isobar creation. In this paper we have employed a\npseudovector $\\pi$NN coupling. Given the charge-exchange nature\nof diagrams 3(a) and 3(b) they will only contribute to the\n($\\gamma$,pn) channel. Of all of the diagrams of fig. 3 the\npion-in-flight term is the most difficult one to calculate, since it\ndoes involve two pion propagators.\n\n\\begin{figure}[tbh]\n\\vspace{10. cm}\n\\caption{Absorption mechanisms which are accounted for in the\npresent calculations.}\n\\end{figure}\n\nIn r-space the currents corresponding to diagrams (a) and (b) read\n\\begin{eqnarray}\n{\\bf J}^{(\\pi sea)}({\\bf r},{\\bf r}_1,{\\bf r}_2) & = & e \\left( \\frac {f_{\\pi\nNN}} {m_{\\pi}} \\right)^2 \\left(\\mbox{\\boldmath$\\tau$}_1 {\\bf \\times}\n\\mbox{\\bold\n math$\\tau$}_2\n\\right)_z \\left\\{\\delta({\\bf r} - {\\bf r}_2) \\mbox{\\boldmath$\\sigma$}_2\n(\\mbox{\\boldmath$\\sigma$}_1 {\\bf \\cdot} \\mbox{\\boldmath$\\nabla$}_1) \\right.\n\\non\n umber \\\\\n& & \\left. - \\delta({\\bf r} - {\\bf r}_1) \\mbox{\\boldmath$\\sigma$}_1\n(\\mbox{\\boldmath$\\sigma$}_2 {\\bf \\cdot} \\mbox{\\boldmath$\\nabla$}_2) \\right\\}\n\\frac {e^{-m_{\\pi} \\mid {\\bf r}_1 - {\\bf r}_2 \\mid}}\n {4 \\pi \\mid {\\bf r}_1 - {\\bf r}_2 \\mid} \\;, \\\\\n{\\bf J}^{(\\pi fli)}({\\bf r},{\\bf r}_1,{\\bf r}_2)\n& = & e \\left( \\frac {f_{\\pi\nNN}} {m_{\\pi}} \\right)^2 \\left(\\mbox{\\boldmath$\\tau$}_1 {\\bf \\times}\n\\mbox{\\boldmath$\\tau$}_2\n\\right)_z \\left\\{ \\mbox{\\boldmath$\\sigma$}_1 {\\bf \\cdot}\n\\mbox{\\boldmath$\\nabla$}_1\n\\mbox{\\boldmath$\\sigma$}_2 {\\bf \\cdot} \\mbox{\\boldmath$\\nabla$}_2\n(\\mbox{\\boldmath$\\nabla$}_2 -\\mbox{\\boldmath$\\nabla$}_1)\n\\right\\} \\nonumber \\\\\n& & \\times\n\\frac {e^{-m_{\\pi} \\mid {\\bf r} - {\\bf r}_1 \\mid}}\n {4 \\pi \\mid {\\bf r} - {\\bf r}_1 \\mid}\n\\frac {e^{-m_{\\pi} \\mid {\\bf r} - {\\bf r}_2 \\mid}}\n {4 \\pi \\mid {\\bf r} - {\\bf r}_2 \\mid} \\;.\n\\end{eqnarray}\nIn the static limit,\nthe diagrams of fig. 3(c) give rise to the following current :\n\\begin{eqnarray}\n{\\bf J}^{(\\pi \\bigtriangleup)}({\\bf r},{\\bf r}_1,{\\bf r}_2) & = &\n\\frac {2 f_{\\gamma N \\bigtriangleup} f_{\\pi N \\bigtriangleup} f_{\\pi\nNN}} {9 m_{\\pi}^3 (M_{\\bigtriangleup} - M_N)}\n\\left\\{ \\left[ \\left(\\mbox{\\boldmath$\\tau$}_1 {\\bf \\times}\n\\mbox{\\boldmath$\\tau$}_2\n\\right)_z \\mbox{\\boldmath$\\sigma$}_2 {\\bf \\cdot} \\mbox{\\boldmath$\\nabla$}_2\n(\\mbox{\\boldmath$\\sigma$}_1 \\times\n\\mbox{\\boldmath$\\nabla$}_2) \\times (\\mbox{\\boldmath$\\nabla$}_1 +\n\\mbox{\\boldmath$\\nabla$}_2) \\right. \\right. \\nonumber \\\\ +\n& & \\left. \\left . 4 (\\mbox{\\boldmath$\\tau$}_2)_z \\mbox{\\boldmath$\\sigma$}_2\n{\\bf \\cdot} \\mbox{\\boldmath$\\nabla$}_2\n(\\mbox{\\boldmath$\\nabla$}_1 {\\bf \\times} \\mbox{\\boldmath$\\nabla$}_2)\n \\delta({\\bf r} - {\\bf r}_1)\n \\right]\n+ 1 \\longleftrightarrow 2 \\right\\}\\frac {e^{-m_{\\pi}\n\\mid {\\bf r}_2 - {\\bf r}_1 \\mid}}\n {4 \\pi \\mid {\\bf r}_2 - {\\bf r}_1 \\mid}\\;.\n\\label{eq.del}\n\\end{eqnarray}\nAt higher photon energies the above ${\\bf J}^{(\\pi \\bigtriangleup)}$\ncurrent has\nto be corrected for the finite lifetime of the $\\bigtriangleup$(1232),\nwhereas also $\\rho$ decay of the resonance could be expected to\nstart playing a role. For the present purposes we have utilized the\nstatic current of eq. (\\ref{eq.del}). A more elaborate treatment of the\n${\\bf J}^{(\\pi \\bigtriangleup)}$ current falls beyond the scope of the\npresent work and will be presented elsewhere \\cite{Mac93}.\nIn order to regularize the above currents at short internucleon\ndistances, hadronic form factors have to be introduced at each\n$\\pi NN$ and $\\pi N \\bigtriangleup$ vertex. As is commonly done we have\nadopted a monopole parametrization for the form factor. After\nintroducing the same monopole form factor at each $\\pi NN$ and $\\pi N\n\\bigtriangleup$ vertex, the pion propagators\n\\begin{eqnarray}\n\\frac {e^{-m_{\\pi} \\mid {\\bf r}_2 - {\\bf r}_1 \\mid}}\n{4 \\pi \\mid {\\bf r}_2 - {\\bf r}_1 \\mid} \\equiv \\frac{1}{(2\\pi)^3}\n\\int d{\\bf p} \\frac{e^{i {\\bf p \\cdot} ( {\\bf r}_1 - {\\bf\nr}_2)}} {p^2+m_{\\pi}^2} \\; ,\n\\end{eqnarray}\nin the above currents will be replaced by :\n\\begin{equation}\n\\frac{1}{(2\\pi)^3}\n\\int d{\\bf p} \\frac{e^{i {\\bf p \\cdot} ( {\\bf r}_1 - {\\bf\nr}_2)}}{p^2+m_{\\pi}^2} \\left(\\frac{\\Lambda_{\\pi}^2 - m_{\\pi}^2}\n{\\Lambda_{\\pi}^2 +p^2} \\right)^{\\beta} \\; ,\n\\end{equation}\nwhere $\\Lambda_{\\pi}$ is the so-called pion cutoff mass and $\\beta$=2\nfor {\\bf J}$^{(\\pi sea)}$ and {\\bf J}$^{(\\pi \\bigtriangleup)}$, whereas\n$\\beta$=1 for {\\bf J}$^{(\\pi fli)}$. Recent\nparametrizations of the Bonn potential lead to $\\Lambda_{\\pi}$=1200\nMeV \\cite{Bonn}. On the other hand, triton binding energy and\nGamow-Teller studies seem to prefer $\\Lambda_{\\pi}$=810\nMeV \\cite{Sas92}. A similar value was found to produce the best\nresults by Wakamatsu and\nMatsumoto in their $^{9}$Be($\\gamma$,pn) and\n$^{9}$Be($\\gamma$,p$\\pi^-$) investigations in a Fermi-gas model\n\\cite{Wak83}.\n\\\\\n\nHaving adopted model assumptions for the nuclear wave functions and\nthe coupling of the electromagnetic photon field to the nucleus, we\nare now in the position to calculate the Feynman amplitude of eq.\n(\\ref{feynman}).\nSince we have performed an angular momentum expansion for the final\nstates we are forced to apply a similar procedure to the current\noperator. This can be easily achieved by performing a multipole\ndecomposition\nin terms of the electric and magnetic transition\noperator. The Feynman amplitude of eq. (\\ref{feynman})\ncan then be written as :\n\\begin{equation}\nm_F^f= - \\sqrt{(2 \\pi)} \\sum _{J \\geq 1} i^J \\hat{J} < \\Psi_f \\mid\nT^{el}_{J \\lambda} (q_{\\gamma}) + \\lambda T^{mag}_{J \\lambda}\n(q_{\\gamma}) \\mid \\ \\Psi_i> \\;,\n\\end{equation}\nwhere the electric and magnetic transition operator are defined\naccording to the conventions of ref. \\cite{For66} and $\\hat{J} \\equiv\n\\sqrt{2J+1}$. Inserting the wave\nfunction of eq. (\\ref{wave}) in the above expression, the Feynman\namplitude for the process in which the absorption of a photon\nwith polarization $\\lambda$ is followed by the emission of a nucleon\npair ({\\bf k}$_a$,m$_{s}$;{\\bf k}$_b$,m$_{s'}$)\nbecomes~:\n\\begin{eqnarray}\nm_F^f & = & - \\sqrt{2 \\pi} \\sum _{J \\geq 1} i^J \\hat{J}\n\\sum_{lm_ljm}\\sum_{l'm_{l'}j'm'} \\sum_{J_1M_1}\n(4 \\pi)^2 (-i)^{l+l'}\n\\frac {\\pi} {2 \\mu \\sqrt{k_a k_b}} e^{-i(\\delta_l+\\sigma_l\n+\\delta_{l'}+\\sigma_{l'})}\n \\nonumber \\\\\n& & \\times Y_{lm_l}(\\Omega_{k_{a}}) Y_{l'm_{l'}}(\\Omega_{k_{b}})\n \\nonumber \\\\\n& & \\times < j m j' m' \\mid J_1 M_1 >\n\\frac{(-1)^{J_R-M_R+1}}{\\hat{J_1}}\n \\nonumber \\\\\n& & \\times _{as} \\;,\n\\label{feynb}\n\\end{eqnarray}\nwhere the antisymmetrized two-body matrix elements are defined according\nto :\n\\begin{eqnarray}\n_{as} & \\equiv &\n \\nonumber \\\\\n& & - (-1)^{j_h+j_{h'}+J_R}\n\\;.\n\\end{eqnarray}\nFor the reduced matrix elements we adopt the conventions of ref.\n\\cite{talmi}.\nReferring to eq. (\\ref{feynb}),\nthe calculation of the ($\\gamma$,NN) cross sections has been\nreduced to determining antisymmetrized two-body matrix elements of the type\n$_{as}$ and\n$_{as}$.\n These matrix elements are a function of the adopted photoabsorption\nmechanism and have\nbeen summarized in Appendix A for the diagrams of fig. 3.\n\nThe contribution from the ($\\gamma$,N$_{a}$N$_{b}$) channel to the total\nphotoabsorption strength $\\sigma$($\\gamma$,N$_{a}$N$_{b}$) can be\ncalculated by integrating the fivefold differential cross section over\nthe respective solid angles and nucleon momentum {\\bf k}$_b$ :\n\\begin{equation}\n\\sigma(\\gamma,N_{a}N_{b}) = \\int d \\Omega_a d \\Omega _b dk_b\n\\frac{d^5 \\sigma^{lab}} {d\\Omega_a d\\Omega_b dk_b} \\;.\n\\label{inte}\n\\end{equation}\nAfter averaging over the intial photon polarization and summing over the\nspin polarizations of the escaping nucleons it can be shown that upon\nsubstituting the Feynman amplitude of eq. (\\ref{feynb}) in the fivefold\ndifferential cross section (\\ref{five}) and integrating over both solid\nangles the following expression is obtained :\n\\begin{eqnarray}\n\\sigma(\\gamma,N_{a}N_{b}) & = & \\int dk_b \\frac{E_a E_{A-2} k_b \\pi}\n{E_{\\gamma} E_A \\mu^2} \\sum_{J\\geq 1} \\sum_{J_1 lj l'j'}\n\\left\\{\n\\left| _{as} \\right|^2\n\\right. \\nonumber \\\\\n& & + \\left. \\left| _{as} \\right|^2 \\right\\} \\;.\n\\label{eq:inte}\n\\end{eqnarray}\nIn deriving this expression we have neglected\nthe effect of nuclear recoil.\n\n\n\\section{Results}\nIn this section we report on the results of the numerical\ncalculations for the $^{16}$O($\\gamma$,pn),\n$^{12}$C($\\gamma$,pn),\n$^{16}$O($\\gamma$,pp) and $^{12}$C($\\gamma$,pp)\ncross sections which we have\nperformed in the theoretical framework discussed above. The\npresented calculations primarily aim at exploring the main sensitivities\nin the calculation of ($\\gamma$,pp) and ($\\gamma$,pn) cross sections.\nGiven that we have assumed that the photon couples exclusively to the\nlightest meson and that apart from the $\\bigtriangleup$ no other\nintermediate nucleon resonances are created, the scope of the\npresent calculations has to be restricted to the photon energy range\nbelow 300 MeV. On the other hand,\ngiven the relatively small momentum transfer involved in real\nphoton reactions, the coupling of the photon to the one-pion exchange\npart of the NN interaction in combination with intermediate\n$\\bigtriangleup$ creation could be expected to set the main trends of\nthe ($\\gamma$,pp) and ($\\gamma$,pn) cross sections in the considered\nenergy range. However, only a detailed confrontation\nof the calculated angular\ncross sections with the data will rule out whether other\neffects, like SRC effects e.g.,\nplay a major role in the photon energy range below the\n$\\bigtriangleup$ production threshold. At present, none of the data\nhave been presented in a model-independent way. In order to\nextrapolate the measurements to the full phase space most analyses\nhave relied on a Monte-Carlo technique based on quasi-deuteron\nkinematics \\cite{Dan88,Gre91}.\nThis procedure hampers the comparison of the data with\nmore fundamental approaches. It is to be hoped that continuous efforts\nto improve on the statistics and resolution\nof the measurements will make this type of\ncomparisons more feasible in the near future.\n\n\\begin{figure}[tbh]\n\\vspace{8.cm}\n\\caption{The fivefold\n$^{16}$O($\\gamma$,pn)$^{14}$N((1p1\/2)$^{-1}_{\\pi}$(1p1\/2)$^{-1}_{\\nu}$;1$^+$)\n angular cross section at E$_{\\gamma}$=200~MeV, T$_p$=88~MeV and $\\theta\n_n$=270$^ \\circ$. The solid line involves all electric and magnetic\nmultipoles up to L=5, the dashed line up to L=3 and the dotted line has\nthe L=1 as the sole contribution.}\n\\label{multi}\n\\end{figure}\n\nAll of the {\\em angular} cross sections\n presented in this work have been obtained in coplanar\nkinematics. This means that the the momentum vectors of the detected\nnucleons ({\\bf k}$_{a}$ and {\\bf k}$_{b}$) and the incoming photon beam\n{\\bf q}$_{\\gamma}$ remain in one fixed plane.\n For all cross sections, we have averaged over\nthe initial photon polarization $\\lambda$ and summed over the spin projections\nm$_{s}$ and m$_{s'}$ of the escaping nucleons. Unless otherwise\nstated a cut-off mass $\\Lambda _{\\pi}$ of 1200 MeV was used. The\nmean-field quantities like the bound state wave functions, the partial waves\nand phase shifts for the escaping particle are selfconsistently\ndetermined through a Hartree-Fock calculation with an effective\ninteraction of the Skyrme type (SkE2) \\cite{Waro}. The calculated cross\nsections were observed to be rather insensitive to the choice of the\nsingle-particle quantities.\n\nThe Feynman amplitude of eq. (17) involves a multipole expansion over\nthe electric and magnetic transition operators in combination with a\npartial-wave expansion for the distorted outgoing nucleon waves. All\nof these expansions were found to converge rather rapidly thus keeping\nthe calculations within reasonable ranges of computer time consumption.\nAt $E_{\\gamma}$=200 MeV the expansion over the electric and magnetic\ntransition operators was found to converge at J=5. This has been\nillustrated in fig. \\ref{multi} where we have plotted the fivefold\ndifferential cross section $d^5 \\sigma \/ d\\Omega _p d\\Omega _n dk_p$ in\nplanar kinematics for the\n$^{16}$O($\\gamma$,pn)$^{14}$N((1p1\/2)$^{-1}_{\\pi}$(1p1\/2)$^{-1}_{\\nu}$;1$^+$)\nprocess. We have assumed that all $2h$ strength is concentrated in the\n1$^+$ ground state in which the residual nucleus $^{14}$N is left.\n For the results of\nfig.\\ref{multi} the photon energy, the proton energy and the neutron\nangle $\\theta _n$ were kept fixed, so that the only remaining functional\ndependence is the proton angle. It is apparent that both the shape and\nthe magnitude of the cross section are determined by the lower\nmultipolarities, thus ensuring a fast convergence of the multipole\nexpansion performed for the electromagnetic interaction hamiltonian.\nRegarding the\nescaping particle wave functions, partial waves of higher order than\n$i\\frac{13}{2}$ were found not to produce any visible change in the\nangular cross sections.\n\nAs a direct application of the framework presented in the previous\nsection we have addressed the question of how important the\ndistortion effects due to the interaction of the outgoing nucleons\nwith the residual (A-2) nucleons are. In order to study this effect we\nhave performed calculations in which a plane wave description for the\ndetected nucleons is adopted. These results have been compared with the\ncross sections obtained after performing a full distorted wave calculation.\nA plane wave description for the ejected particles can be accomplished\nin the scheme presented in sect. 2. It suffices to replace the\nsingle-particle states p and p$'$ in the wave function of eq. (9) by\nspherical Bessel functions to retain an asymptotic behaviour which in\nthe conventions of sect. 2.2 reads~:\n\\begin{eqnarray}\n\\left<{\\bf r}_1 \\mbox{\\boldmath$\\sigma$}_1, {\\bf r}_2\n\\mbox{\\boldmath$\\sigma$}_2\n \\mid \\Psi_f\n\\right> & & \\stackrel {r_1,r_2 \\gg r_A} {\\longrightarrow} \\frac {1}\n{\\sqrt{A(A-1)}} \\nonumber \\\\\n & & \\times {\\cal A}_{2(A-2)} \\left[ \\chi_{\\frac{1}{2}m_s}\n(\\mbox{\\boldmath$\\sig\n ma$}_1)\ne^{i{\\bf k}_a {\\bf \\cdot r}_1}\n\\chi_{\\frac{1}{2}m_{s'}} (\\mbox{\\boldmath$\\sigma$}_2) e^{i{\\bf k}_b {\\bf \\cdot\nr}_2} \\mid\n(hh')^{-1} J_R M_R > \\right]\\;.\n\\end{eqnarray}\nIt is worth mentioning that when adopting a plane-wave description for\nthe escaping particles, the orthogonality condition between the bound\nand the scattering states is lost and spurious contributions can enter\nthe cross sections \\cite{Giutr}. When comparing the plane wave with\nthe full distorted wave calculations we are actually checking in how far\nplane waves are a good description for the escaping particle wave functions.\n The degree of deviation between both approaches will give us a\nhandle on the influence of the distortions in ($\\gamma$,NN) processes.\nWe remind that the plane wave approximation is one of the basic\nassumptions of the factorized approach to ($\\gamma$,NN) processes.\nIn comparison with an unfactorized model, the\nfactorized approach extremely facilitates the interpretation of the data\nand has been observed to give a fair {\\em qualitative} account for the\nrecent $^{12}$C($\\gamma$,pn) \\cite{Dan88}\nand $^{16}$O($\\gamma$,pn) \\cite{Gre91} data.\n\n\\begin{figure}\n\\vspace{13. cm}\n\\caption{The $^{12}$C($\\gamma$,pn) and $^{12}$C($\\gamma$,pp) strength\nfor knockout from the 1s shell versus photon energy. The dashed line is\na plane wave and the solid line a full distorted wave result.}\n\\label{pnppfsi}\n\\end{figure}\n\nIn fig. \\ref{pnppfsi} the calculated $^{12}$C($\\gamma$,pp) and\n$^{12}$C($\\gamma$,pn) strength for photoinduced two-nucleon knockout\nfrom the 1s shell is plotted versus the photon energy. These cross\nsections have been calculated with the expression (\\ref{eq:inte}). In\nthe process of calculating the photoabsorption strength we have summed\nover all angular momentum states $ \\mid J_R M_R >$ of the final nucleus. In\ncomparison with a plane wave calculation, the distortions are noticed to\nyield a reduction of the total strength. Apparently, the reducing\neffect is stronger in the pn than in the pp channel. As will become\nclear in the course of this section, this is attributed to the\ndistortions affecting more the pionic than the isobar contributions.\n\n\n\\begin{figure}[tbh]\n\\vspace{12.4 cm}\n\\caption{The fivefold\n$^{16}$O($\\gamma$,pn)$^{14}$N((1s1\/2)$^{-1}_{\\pi}$(1s1\/2)$^{-1}_{\\nu}$)\n angular cross section at E$_{\\gamma}$=200~MeV and T$_p$=48~MeV.\nFor the right column a full distorted wave calculation for the outgoing\npn pair was performed, whereas the left column results were obtained in\nthe plane wave approximation. The upper cross sections refer to the\nseagull term, the middle to the pion-in-flight and the bottom ones to a\ncoherent sum of both.}\n\\label{five1}\n\\end{figure}\n\n\\begin{figure}[tbh]\n\\vspace{12. cm}\n\\caption{The fivefold\n$^{16}$O($\\gamma$,pn)$^{14}$N((1s1\/2)$^{-1}_{\\pi}$(1s1\/2)$^{-1}_{\\nu}$)\n angular cross section in the\nkinematical conditions of fig. \\protect{\\ref{five1}}. The\nupper cross sections involve only the isobar diagrams ; the bottom ones\ninvolve all pionic and isobar absorption diagrams.}\n\\label{five2}\n\\end{figure}\n\n\n\n\n\n\n\nThe effect of the distortions on the angular cross sections\nhas been investigated in figs. \\ref{five1} and \\ref{five2}. A striking\nfeature of all of the displayed angular cross sections is the dominance\nof the back-to-back emission~: the strength is concentrated in the\nkinematical range 150$^\\circ \\leq \\mid \\theta _p - \\theta_n \\mid \\leq\n210 ^\\circ$. It emerges that the distortions do not affect this picture\nand that their main effect on the angular cross sections is a mere\nreduction.\nThe angular cross sections related to the $\\pi$ absorption are displayed\nin fig. \\ref{five1}. Here, the distortions reduce the strength with\nabout 50~\\%. Remark further the very strong destructive interference\nbetween the seagull and the pion-in-flight diagrams, affecting the\nmagnitude as well as the shape of the angular cross sections. The\neffect of including isobar effects is studied in fig. \\ref{five2}.\nRegarding the distortions, similar trends are noticed as for the pionic\nterms, but the reducing effect is obviously smaller now.\n\n\\begin{figure}[tbh]\n\\vspace{13. cm}\n\\caption{The calculated photoabsorption strength for $^{12}$C($\\gamma$,pp)\nand $^{12}$C($\\gamma$,pn) versus the photon energy. The contributions\nfor knockout to the different {\\em 2h} states are shown : (1s)$^{-2}$\ndotted line ; (1s)$^{-1}$(1p)$^{-1}$ dashed line ; (1p)$^{-2}$\ndot dashed line ; solid line : total cross section.\nAll absorption diagrams of fig. 3 are accounted for.}\n\\label{ic12pn}\n\\end{figure}\n\nThe total photoabsorption strength for $^{12}$C in the ($\\gamma$,pp)\nand the ($\\gamma$,pn) channel is shown in fig. \\ref{ic12pn}. Also shown\nis the contribution for pair emission from the different shells, viz.\n(1s)$^{-2}$, (1s)$^{-1}$(1p)$^{-1}$ and (1p)$^{-2}$. Apparently, the\nrelative strengths of the different {\\em 2h} states in the ($\\gamma$,pn)\nchannel does exhibit some\nphoton energy dependence. From fig. \\ref{ic12pn} it is obvious that\nbelow the pion production threshold, the $^{12}$C($\\gamma$,pn)\nstrength is dominated by (1p)$^{-2}$ and (1s)$^{-1}$(1p)$^{-1}$\nknockout, whereas with increasing\nphoton energies the main contribution is coming from (1s)$^{-1}$(1p)$^{-1}$\nknockout. Remark further that the calculations do not seem to follow the\nnaive picture in which the relative strength in the different {\\em 2h}\nchannels is determined by the number of pn \"quasi-deuteron\" pairs. In\nthis naive model we would expect the (1s)$^{-2}$ to\n(1s)$^{-1}$(1p)$^{-1}$ ratio to be 4\/16, whereas the calculations\npredict that a relatively larger fraction of the ($\\gamma$,pn) strength\ngoes through the\n$^{12}$C($\\gamma$,pn)$^{10}$B((1s1\/2)$^{-2}$) channel. In the pp\nchannel little energy dependence in the ratios between the different\n{\\em 2h} states channels is observed. This has to be attributed to the\nfact that in the presented calculations only one absorption mechanism is\ncontributing to the pp channel. The observed energy dependencies in the\npn channel reflect the sensitivity of the different absorption mechanisms\nto the shell-model structure of the final (A-2) state.\n\n\\begin{figure}\n\\vspace{7. cm}\n\\caption{Ratio of $^{12}$C($\\gamma$,pn)\/$^{12}$C($\\gamma$,pp) absorption\nstrength versus the photon energy. Dotted line : (1s)$^{-2}$ knockout ;\ndashed line : (1s)$^{-1}$(1p)$^{-1}$ ; dot-dashed line : (1p)$^{-2}$ ;\nsolid line : ratio of the total ($\\gamma$,pn) to ($\\gamma$,pp) cross section.\nThe data are from ref. \\protect{\\cite{Gre91}} (circle) and ref.\n\\protect{\\cite{Kana87}} (triangle).}\n\\label{ratio}\n\\end{figure}\n\nFigure \\ref{ratio} shows the photon energy dependence of the\ncalculated ratio of the total\n$^{12}$C($\\gamma$,pn) to the $^{12}$C($\\gamma$,pp) strength. The\nresults clearly illustrate that the ($\\gamma$,pn)\/($\\gamma$,pp) ratio is\nboth photon energy and shell structure dependent.\nDepending on the $2h$ structure of the state in which the residual (A-2)\nnucleus is fed, the ratio can differ significantly from the total\nphotoabsorption value which is determined by a weighted average of all\nof the different $2h$ contributions. Particularly (1s)$^{-2}$ knockout\nis noticed to exhibit a different overall behaviour in\n($\\gamma$,pn)\/($\\gamma$,pp) ratio.\nComparing the calculations with\nthe experimental estimates we observe a general overestimation of the\nratio. This is not surprising given that we assumed a direct knockout\nmechanism. At lower photon energies, the direct ($\\gamma$,pn) strength\nis two orders of magnitude larger than the direct ($\\gamma$,pp)\nstrength. The slightest coupling between both channels, through (n,p)\nrescattering, could yield\nadditional strength in the pp channel and dramatically reduce the\n($\\gamma$,pn)\/($\\gamma$,pp) ratio. In spite of the fact that for\nE$_{\\gamma} \\geq$~250~MeV intermediate isobar creation is the most\ndominant contribution to both channels, the calculated ratio for direct\nknockout seems to converge around 14. On the other hand, the data seem\nto scatter around 10 in the $\\bigtriangleup$(1232) resonance region. It\nremains to be investigated whether this deviation is a mere reflection\nof the coupling between both channels or points towards SRC effects\nputting additional strength in the ($\\gamma$,pp) and ($\\gamma$,pn)\nchannel. The SRC could be expected to decrease the ratio of the\nstrongest to the weakest channel. Other effects which deserve\nto be investigated in the resonance region are off-shell effects of\n$\\bigtriangleup$ propagation, like summarized in the $\\bigtriangleup$-hole\nmodel \\cite{delhole}, and diagrams related to the $\\rho$ meson decay of the\nintermediate isobar.\n\n\\begin{figure}[tbh]\n\\vspace{8. cm}\n\\caption{The $^{16}$O($\\gamma$,pn) strength versus the photon energy. The\nsolid\n line\nis the result of the present calculation. The data are from ref.\n\\protect{\\cite{Carlos}} (squares ($\\gamma$,1n...)), ref.\n\\protect{\\cite{Bonnpn}} (triangle ($\\gamma$,pn)) and ref.\n\\protect{\\cite{Gre91}} (circle ($\\gamma$,pn)).}\n\\label{io16pn}\n\\end{figure}\n\nIn fig. \\ref{io16pn} we compare the calculated $^{16}$O($\\gamma$,pn)\nphotoabsorption strength with the ($\\gamma$,1n...) data of ref.\n\\cite{Carlos} and the ($\\gamma$,pn) of refs. \\cite{Gre91,Bonnpn}. The\nenergy range of comparison is generally considered as the quasi-deuteron\nregion in which the photoabsorption strength is believed to be\ndominated by the ($\\gamma$,pn) channel. Despite the large error bars\nour absolute and parameter-free calculations seem to confirm this\npicture since we exhaust a considerable fraction of the ($\\gamma$,1n...)\nstrength and give a fair account of the ($\\gamma$,pn) data points. In\npassing, it seems that there is very little room for a strong absorptive\neffect in the final state interaction.\nA similar but\nmuch more pronounced effect has been found for the ($\\gamma$,p) and\n($\\gamma$,n) channel for which direct-knockout calculations (DKO) in an\noptical-model approach severely\nunderestimate the data \\cite{gerard}. Part of this discrepancy has been\nattribu\n ted to\nrescattering processes of the long-range type in the final state\ninteraction, which yield a substantial amount of additional strength\n\\cite{Gar81,Ryc88}.\nIt remains to be investigated how big these effects are in the\ntwo-nucleon emission channel.\n\n\\begin{figure}[tbh]\n\\vspace{10. cm}\n\\caption{The fivefold\n$^{12}$C($\\gamma$,pn)$^{10}$B((1p3\/2)$^{-1}_{\\pi}$(1s1\/2)$^{-1}_{\\nu}$)\n angular cross section at E$_{\\gamma}$=200~MeV, T$_p$=78~MeV and\n $\\theta _n$=270$^{\\circ}$ for two values of the pion cut-off mass\n ($\\Lambda _{\\pi}$=1200 MeV\/c$^2$ and $\\Lambda _{\\pi}$=800 MeV\/c$^2$).\nThe dashed line gives the cross section for $\\bf{J} ^{(\\pi sea)}$,\nthe dotted for $\\bf{J}^{(\\pi sea)}$ + $\\bf{J} ^{(\\pi fli)}$ and the\nsolid for $\\bf{J}^{(\\pi sea)}$+$\\bf{J}^{(\\pi fli)}$+$\\bf{J}^{(\\pi\n\\bigtriangleup)}$. In the insert the function F$_{1p1s}$(P=$\\mid \\bf{k}_p\n+ \\bf{k}_n - \\bf{q}_{\\gamma} \\mid$) is plotted\nversus the proton angle.}\n\\label{1s1p}\n\\end{figure}\n\n{}From fig. \\ref{ic12pn} it is obvious that the $^{12}$C($\\gamma$,pn)\nchannel is dominated by (1s)$^{-1}$(1p)$^{-1}$ knockout at higher photon\nenergies. In fig. \\ref{1s1p} we show the fivefold angular cross section\nfor the\n$^{12}$C($\\gamma$,pn)$^{10}$B((1p3\/2)$^{-1}_{\\pi}$(1s1\/2)$^{-1}_{\\nu}$)\nreaction at E$_{\\gamma}$=200 MeV, T$_p$=78 MeV and $\\theta _n$=270$^{\\circ}$.\nThe average excitation energy of the {\\em 2h} state state\n(1p3\/2)$^{-1}_{\\pi}$(1s1\/2)$^{-1}_{\\nu}$ was chosen to be 22 MeV.\nConsequently, the neutron kinetic energy T$_n$ equals almost the\nproton kinetic energy. For the\nresults of fig. \\ref{1s1p} we have summed over the two angular momentum\nstates $J_R=1^-,2^-$ of the residual nucleus.\n The angular cross section is plotted for\ndifferent combinations of the photoabsorption expansion and two values\nof the pion cut-off mass $\\Lambda _{\\pi}$. Apparently, the shape of the\nangular cross section of the type displayed in fig. \\ref{1s1p}\nis roughly independent of the absorption mechanism and the\nvalue of the cut-off mass. On the contrary, the absolute value turns\nout to be sensitive to either of those. Some theoretical ambiguities\nexist with respect to the value of $\\Lambda _{\\pi}$. Obviously, the\ncut-off mass simply affects the magnitude of the cross section. This\ndependence illustrates the sensitivity of the ($\\gamma$,NN) cross\nsections to the short-range effects. In the calculations of refs.\n\\cite{Giu92} the calculated ($\\gamma$,pp) and (e,e$'$pp) cross\nsections were observed to be very sensitive to the choice of the\nshort-range\nJastrow correlation functions which was introduced\nas a correction to the nuclear Slater determinants.\nIt should be stressed, however, that in the calculation of ref.\n\\cite{Giu92} no hadronic form factors were introduced. Therefore, the\nJastrow correlation function acts as the sole regularization of the $\\pi$NN\nvertices at short internucleon distances.\nCombining hadronic form factors with a Jastrow like correlation function\nto correct for the short-range behaviour in nuclear Slater determinants,\ncould be expected to reduce\nthe sensitivity of the calculated cross sections to\nthe Jastrow correlation function.\n\nIn an unfactorized model the angular cross section for knockout from the\nsingle-particle orbits h and h$'$ is given by \\cite{Boa89,Ryc92,Mac93} :\n\\begin{equation}\n\\frac{d^5 \\sigma^{lab}} {d\\Omega_p d\\Omega_n dk_p} = K S_{fi}\nF_{hh'}(P=\\mid \\bf{k}_p\n+ \\bf{k}_n - \\bf{q}_{\\gamma} \\mid) \\;,\n\\end{equation}\nwhere K is a kinematical factor, S$_{fi}$ determines the physics of the\nphotoabsorption and F$_{hh'}$ is proportional to the probability to\nfind a nucleon pair with zero separation and total momentum P in the\nsingle-particle states h and h$'$. In the insert of fig. \\ref{1s1p} we\nshow the function F$_{1p_{3\/2}1s_{1\/2}}$(P) for the kinematical\nconditions of the displayed cross sections. It turns out that even in\nour fully unfactorized model, including FSI effects,\n the shape of the cross section is\nto a large extent determined by the probability F(P). The absorption diagrams\ns\n eem to\nmerely modulate the main trends set by the probability F(P). This could\nbe interpreted as the theoretical justification for the observation that\nthe scaling mechanism of the ($\\gamma$,NN) data in terms of the variable\nP seems to work reasonably well. Therefore, with the eye on future\ncomparisons with the data we would suggest that {\\em absolute} cross sections\nar\n e\ncompared, because any model with a two-body type of absorption mechanism\nwill more or less fit the P dependence of the data, the main trends being set\nby the function F(P). As an alternative to the proposed type of\ncomparions on an absolute\nscale, we suggest to compare angular cross sections at a fixed value of\nthe missing momentum P. In this way the F(P) dependence can be ruled\nout and a much larger sensitivity of the calculated cross sections to the\nabsorption mechanism is reached.\n\nIn contrast to what is noticed in ($\\gamma$,p) and (e,e$'$p) reactions the\ndistorted waves do not yield considerable shifts in the strength\ndistributions which would be expected within a plane wave description.\nA possible explanation starts from considering that the A-2\nsystem will affect the momentum of both escaping nucleons independently.\nIn ($\\gamma$,p) and (e,e$'$p) processes, the effect of the distortions\ncan to a large extent be parametrized\nby introducing an effective momentum {\\bf p}$_{N}^{eff}$\nwith which the nucleon initially escapes from the target nucleus to\narrive at the detectors with momentum {\\bf p}$_{N}$ \\cite{Fin77}. The\nFSI related shifts in the ($\\gamma$,p) cross sections can then be\ncompensated for by plotting the cross sections against the {\\em\neffective} missing momentum {\\bf p}$_{m}^{eff}$={\\bf p}$_{N}^{eff}$-{\\bf\nq}$_{\\gamma}$.\nGiven the back-to-back\nsituation, the concept of the\neffective momentum could be expected to apply to each of the escaping\nnucleon momenta, but the net effect on the total missing momentum ${\\bf P}=\n{\\bf k}_p + {\\bf k}_n - {\\bf q}_{\\gamma}$ will largely cancel out\n({\\bf P}$^{eff} \\approx $ {\\bf P}), which\nmight explain the relatively\nsmall shifts induced by the distortions in two-nucleon\nemission processes.\n\n\n\n\\section{Conclusion and prospects}\nIn conclusion, we have developed a model for the description of\nelectromagnetically induced\ntwo-nucleon emission processes from finite nuclei\nin a general shell-model framework. The method relies on a partial wave\nexpansion for the escaping nucleon waves and a multipole expansion for\nthe coupling of the (virtual) photon field with the nucleus. The basic\nassumptions of the model are that the nucleon pairs escape from the\ntarget in a one-step reaction mechanism following the absorption of the\nphoton on the two-body currents induced by the nucleon-nucleon\ncorrelations in the target system. The currents accounted for in\nthis work are related to one-pion exchange and\nintermediate $\\bigtriangleup$ isobar creation.\n\nWithin this framework we have calculated angular cross sections for the\n($\\gamma$,pn) and the ($\\gamma$,pp) reaction off the target nuclei\n$^{12}$C and $^{16}$O. For these\npurposes the coupling of the photon field to the two-body currents has\nbeen restricted to all diagrams involving at most one intermediate\n$\\bigtriangleup$ line and one pion. It has been\ndemonstrated that the distortions in the outgoing nucleon waves bring\nabout relatively small corrections to the cross sections which would be\nobtained in a plane-wave model. The main effect is that the distorted\nwaves yield a reduction of the cross section. The reduction factor is\ndependent on the choice of the primary photoabsorption diagrams and\nthe photon energy but was found not to be larger than two.\nA striking feature emerging from our investigations\nis that despite of the fact that the distortions can only be treated in\nan unfactorized model, the distortions do not seem to affect drastically\nthe shape of the factorized cross section. Consequently, the\nscaling of the factorized approach in terms of the missing momentum P\nseems to be somehow preserved, even when the plane wave approximation is\ndropped.\n\nIn order to regularize the nucleon-nucleon correlations at short\ndistances, hadronic form factors had to be introduced in the calculations.\nSome theoretical ambiguities exist with respect to the value of the\ncut-off parameter. The\nsensitivity of the calculations to this theoretical ambiguity is\na 50~\\% uncertainty\nin the absolute value of the calculated cross sections.\n\n\nSince angular cross sections for the ($\\gamma$,NN) reaction from finite\nnuclei are as yet not available, we have compared our calculations with\ntotal ($\\gamma$,1n...) and ($\\gamma$,pn) photoabsorption data.\nThe model was found to give a fair\naccount of the data, pointing towards the realistic character of the adopted\nmodel assumptions.\nThe ratio of the ($\\gamma$,pn) strength into the different $2h$ channels\nemerged to be photon energy dependent, reflecting the sensitivity of the\ncalculated cross sections to the nuclear structure of the final state.\nA similar type of dependence was noticed for the\n($\\gamma$,pn)\/($\\gamma$,pp) ratio.\\\\\n\\vspace{2. cm}\n\n\n{\\Large Acknowledgement}\\\\\nThis work has been supported by the National Fund for Scientific\nResearch (NFWO), the Institute for Scientific Research in Industry and\nAgriculture (IWONL) and in part by the NATO through the research grant\nNATO-CRG920171.\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\n\n\nThe observation of oscillations between different neutrino flavors firmly establishes that at least two out of three neutrino mass eigenstates $m_i$ ($i=1,2,3$) are non-zero~\\cite{Fukuda:1998mi,Ahmad:2002jz,Araki:2004mb,Adamson:2008zt,An:2012eh}. Since in the Standard Model of particle physics neutrinos only come with left-handed chirality, it is not possible to generate a mass term for them by the usual Higgs mechanism as for the other leptons. The discovery of neutrino masses then requires new physics beyond the Standard Model,\nwhich may involve the existence of additional right-handed states (see e.g.,~\\cite{Drewes:2013gca} for a review). These states are {\\it sterile}, which means that they do not take part in weak, strong, or electromagnetic interactions, and their number and mass scale depend on the specific model.\nFor phenomenological purposes, in this paper we consider only one light sterile state with a mass at the eV scale~\\cite{Giunti:2019aiy}.\n\nOne way to search for sterile neutrinos is in laboratory experiments. If produced in sufficient numbers in the early universe via oscillations with active states, sterile neutrinos can lead to observable effects in cosmology. As we will discuss in this paper, direct laboratory measurements and cosmological observations are complementary, and both avenues have led to tremendous progress in our understanding of the neutrino sector over the past few years.\n\nNeutrino oscillation experiments have provided high-precision measurements of two mass-squared splittings \\cite{deSalas:2017kay,Capozzi:2018ubv,Esteban:2018azc}:\n\\begin{align}\n\\dmij{21} &= m_2^2-m_1^2 = 7.55\\ex{-5} \\: \\mathrm{eV}^2\\,, \\\\\n|\\dmij{31}| &= |m_3^2-m_1^2| = 2.5\\ex{-3} \\: \\mathrm{eV}^2 \\,.\n\\end{align}\nThese mass splittings are responsible for the oscillations between the three neutrino flavor eigenstates associated with the charged \nleptons of the Standard Model.\nGiven the fact that the sign of $\\dmij{31}$ is not determined, two mass orderings can be possibly realized in nature: the normal ordering, in which the lightest mass state is $m_1$ and $\\dmij{31}>0$, or the inverted ordering, in which $m_3$ is the lightest state and $\\dmij{31}<0$.\nCurrent neutrino oscillation measurements show a mild statistical preference for the normal ordering~\\cite{deSalas:2017kay,Capozzi:2018ubv,Esteban:2018azc,deSalas:2018bym}.\n\nFor many years, however, a number of experimental results that cannot be explained within the context of the three-neutrino oscillation paradigm have been reported.\nSuch anomalies are measured by various so-called short-baseline (SBL) experiments.\nThe first result was found by LSND~\\cite{Aguilar:2001ty} and later partly confirmed by MiniBooNE \\cite{Aguilar-Arevalo:2018gpe}, but anomalies were also detected by the Gallium experiments GALLEX and SAGE\\footnote{GALLEX and SAGE were designed to find solar neutrinos, but the anomalies appear in the short-baseline calibration runs.}~\\cite{Abdurashitov:2005tb,Giunti:2010zu,Kostensalo:2019vmv} and by a number of observations of reactor antineutrino fluxes at short distances~\\cite{Mention:2011rk,Mueller:2011nm,Huber:2011wv}.\nAlthough the situation is not completely clear, (see e.g.~\\cite{Giunti:2019aiy,Boser:2019rta}), \nthe anomalies point to an additional mass splitting, much larger than the other two:\n\\begin{equation}\n\\dmij{\\rm SBL} \\sim \\mathcal O(\\mathrm{eV}^2) \\,.\n\\end{equation}\nThe existence of a new mass splitting implies the presence of a new neutrino mass eigenstate, which\ncorresponds to a fourth flavor eigenstate.\nSince the hypothetical new neutrino has not been found in other weak interaction measurements~\\cite{ALEPH:2005ab}, it cannot take part in Standard Model interactions and is therefore denoted ``sterile'' (see e.g.~\\cite{Gariazzo:2015rra,Giunti:2019aiy}).\nDue to the large gap $\\mathcal O(\\mathrm{eV}^2) \\gg |\\dmij{31}| , \\, \\dmij{21}$, oscillations with the sterile are not expected to affect standard atmospheric and solar neutrino measurements.\nOne typically assumes that the mixing between the three active neutrinos and the fourth mass eigenstate is small, so that the scenario is commonly referred to as the 3+1 model. For recent reviews on experimental searches for eV-scale sterile neutrinos, see also~\\cite{Giunti:2019aiy,Diaz:2019fwt,Boser:2019rta}.\n\nIn case such a new neutrino exists, its presence can affect the evolution of the Universe as well. Even if the fourth neutrino does not interact with the other standard model particles,\nit would be produced in the early Universe via oscillations with the active flavors. Excluding direct detection of relic neutrinos~\\cite{Betti:2019ouf}, the cosmological presence of the sterile state can only be deduced indirectly, for instance from its contribution to the energy density of the Universe~\\cite{Abazajian:2017tcc}.\nIn particular, considering masses around the $\\sim \\mathrm{eV}$ scale as motivated by the SBL anomalies, the sterile neutrino is produced as a relativistic particle. Therefore, it contributes at early times to the number of relativistic degrees of freedom \\ensuremath{N_\\mathrm{eff}}, which quantifies the amount of energy density from relativistic species beyond photons in the early Universe.\nMoreover, the free-streaming length associated to light sterile neutrinos is potentially large, so that they would behave as hot dark matter and leave a distinct imprint on structure formation, similar to the one of active neutrinos~\\cite{Hannestad:2010kz,Wong:2011ip,Lesgourgues:2014zoa,Lattanzi:2017ubx}. Cosmological observations are sensitive to the hot dark matter energy density $\\Omega_\\mathrm{hot} =(\\Omega_\\nu+\\Omega_s) $, neglecting the contribution of other non-standard particles.\nRecent measurements of the temperature and polarisation anisotropies in the cosmic microwave background (CMB) and of large-scale structures (LSS) are able to constrain both $\\ensuremath{N_\\mathrm{eff}}$ and $\\Omega_\\mathrm{hot} h^2$ to high precision, see e.g.~\\cite{Palanque-Delabrouille:2015pga,Cuesta:2015iho,Huang:2015wrx,Giusarma:2016phn,Vagnozzi:2017ovm,Wang:2017htc,Chen:2017ayg,Upadhye:2017hdl,Nunes:2017xon,Giusarma:2018jei,Choudhury:2018byy,Aghanim:2018eyx,Gariazzo:2018meg,RoyChoudhury:2019hls,Gariazzo:2019xhx}. Predictions of big-bang nucleosynthesis (BBN) combined with measurements of primordial element abundances also provide independent, albeit less stringent, constraints on $\\ensuremath{N_\\mathrm{eff}}$ in agreement with CMB+LSS findings~\\cite{Consiglio:2017pot}.\n\n\nThere is another reason to consider additional relativistic particles such as light sterile neutrinos in cosmology, independent of the anomalous results from neutrino experiments. Contributions to the energy density in the early Universe reduce the size of the sound horizon and result in a higher value of the Hubble constant $H_0$ inferred from baryon acoustic oscillations (see e.g. discussions in~\\cite{Bernal:2016gxb,Lemos:2018smw,Poulin:2018cxd,Vagnozzi:2019ezj,Verde:2019ivm}). Currently measurements based on observing the sound horizon in the CMB or in the galaxy distribution at late times are in tension with local distance ladder measurements from supernovae Ia \\cite{Riess:2016jrr,Riess:2018byc,Riess:2019cxk} and lensed quasar time delay measurements~\\cite{Birrer:2018vtm,Chen:2019ejq,Wong:2019kwg}, see e.g.~\\cite{Addison:2017fdm,Aylor:2018drw,Knox:2019rjx} for further discussions. However, a sterile neutrino alone cannot account for the full discrepancy, as discussed in~\\cite{Aghanim:2018eyx}.\n\nIn this work, we study the phenomenology of a light sterile neutrino from various points of view. While the main goal is to compute, for the first time, cosmological bounds on all the mixing angles and the mass splitting between the light sterile state and active neutrinos, we also compare the cosmological results with other probes, such as constraints from direct mass measurements and neutrinoless double-$\\beta$ decay, and limits from neutrino oscillation experiments. We will focus on light sterile neutrinos (i.e., $\\dmij{41}<100\\,\\mathrm{eV^2}$), since we are motivated by anomalies in oscillation experiments that could be explained by the existence of light sterile states.\n\nThe rest of this paper is structured as follows. First, we give a brief overview on the theoretical aspects and the status of sterile neutrino probes in section~\\ref{sec:ster_osc}.\nThe phenomenology of such neutrinos in the early and late Universe is discussed in section~\\ref{sec:sterile_cosmology}. In section~\\ref{sec:data} we discuss the datasets we consider and the method used to obtain the bounds on the sterile neutrino mixing parameters, for which we present upper limits in section~\\ref{sec:results}. We summarize and conclude in section~\\ref{sec:conclusion}.\n\n\\section{Experimental searches for light sterile neutrinos}\n\\label{sec:ster_osc}\n\nIn this section we briefly summarize the theory of active-sterile neutrino oscillations and review the status of light sterile neutrino searches at various experiments. For comprehensive reviews focused on experimental searches for eV-scale sterile neutrinos, see~\\cite{Giunti:2019aiy,Diaz:2019fwt,Boser:2019rta}.\n\n\\subsection{Active-sterile neutrino mixing}\nThe mixing between active and sterile neutrinos can be parameterized by means of the unitary $4\\times4$ lepton mixing matrix $U$. There are several ways to write the mixing matrix, so we will briefly explain our choice of parameterization. In the case of mixing between four neutrino states, $U$ is fully characterized by six mixing angles and three CP-violating Dirac phases, but for the purposes of this work we set the phases to zero and therefore assume that oscillations among neutrinos are identical to those among antineutrinos.\nThree of the angles characterize the oscillations between the three active neutrinos: $\\theta_{13}$, $\\theta_{23}$, and $\\theta_{23}$.\nThe remaining three mixing angles, namely, $\\theta_{14}$, $\\theta_{24}$, and $\\theta_{34}$, describe the mixing with the sterile state.\nWe choose the following parameterization of the mixing matrix~\\cite{Gariazzo:2015rra}:\n\\begin{equation}\\label{eq:mixpar}\nU = R^{34} R^{24} R^{14} R^{23} R^{13} R^{12} \\,,\n\\end{equation}\nwhere each $R^{ij}$ is a real matrix describing a rotation by an angle $\\theta_{ij}$. \n\nBounds on the mixing matrix can be provided in a way that is independent of the parametrization if instead of quoting limits on the individual mixing angles $\\theta_{ij}$ we consider the matrix elements directly. Concerning active-sterile neutrino mixing, the most important elements are those of the fourth column:\n\\begin{align}\n\\Uaj{e4} &= \\sin^2 \\theta_{14} \\,, \\\\\n\\Uaj{\\mu 4} &= \\cos^2 \\theta_{14} \\sin^2 \\theta_{24} \\,, \\\\\n\\Uaj{\\tau 4} &= \\cos^2 \\theta_{14} \\cos^2 \\theta_{24} \\sin^2 \\theta_{34} \\,, \\\\\n\\Uaj{s 4} &= \\cos^2 \\theta_{14} \\cos^2 \\theta_{24} \\cos^2 \\theta_{34} \\,.\n\\end{align}\nWe expect the mixing angles $\\theta_{i4}$ to be small, in order not to substantially alter the phenomenology of three-neutrino mixing beyond what is allowed by current limits. Therefore, the matrix elements \\Uaj{e4}, \\Uaj{\\mu 4} and \\Uaj{\\tau 4} are expected to be small, while \\Uaj{s 4} should in principle be of order unity. For this reason, it is possible to refer to the fourth neutrino mass eigenstate as the ``sterile'' neutrino, even though strictly speaking a sterile neutrino is a flavor eigenstate and has no definite mass. In the rest of the paper, we will often refer to the mass $m_s\\simeq m_4$ as the mass of the sterile neutrino.\n\nIn addition to the mixing angles, we need to specify the three mass-squared splittings \\dmij{21}, \\dmij{31}, and \\dmij{41}. For the active neutrinos, we always assume normal ordering\\footnote{While not yet conclusive, recent cosmological and oscillations probes slightly favor normal ordering over inverted ordering, with varying degree of preference~\\cite{Hannestad:2016fog,Xu:2016ddc,Gerbino:2016ehw,Vagnozzi:2017ovm,Schwetz:2017fey,deSalas:2017kay,Gariazzo:2018pei,deSalas:2018bym,RoyChoudhury:2019hls}.} ($\\dmij{31}>0$) and we fix all the standard mixing parameters to the best-fit values obtained in \\cite{deSalas:2017kay}.\nWhen considering the full oscillation pattern, the complete expressions for the oscillation probabilities are rather involved and depend on all the mass splittings and mixing angles, see e.g.~\\cite{Giunti:2007ry}. When considering SBL oscillations, however, the terms due to the much smaller solar and atmospheric mass-squared differences are suppressed because they correspond to slower oscillations, and only the effect of \\dmij{41} is relevant. As a consequence, the oscillation probabilities can be well approximated by a two-neutrino mixing formula with appropriate mixing matrix elements. This is the appearance probability to populate a flavor state:\n\\begin{equation}\nP^{\\mbox{SBL}}_{\\nu_{\\alpha}\\rightarrow\\nu_{\\beta}} \\simeq \\sin^2 \\left ( 2\\vartheta_{\\alpha\\beta} \\right ) \\sin^2\\left(\\frac{\\dmij{41}L}{4E}\\right)\\,,\n\\label{eq:P_sbl_app}\n\\,\n\\qquad(\\alpha\\neq\\beta)\n\\end{equation}\nwhereas the flux of the initial flavor is modulated by the disappearance probability:\n\\begin{equation}\nP^{\\mbox{SBL}}_{\\nu_{\\alpha}\\rightarrow\\nu_{\\alpha}} \\simeq 1 - \\sin^2 \\left ( 2\\vartheta_{\\alpha\\alpha} \\right )\\sin^2\\left(\\frac{\\dmij{41}L}{4E}\\right)\n\\label{eq:P_sbl_dis}\n\\,,\n\\end{equation}\nwhere $L$ is the distance traveled by the neutrino, $E$ is its energy, and we use Greek letters to refer to the four flavor states $\\alpha, \\beta \\in [e, \\mu, \\tau, s]$. The effective angles $\\vartheta_{\\alpha\\alpha}$ and $\\vartheta_{\\alpha\\beta}$ depend on the elements of the fourth column of the mixing matrix:\n\\begin{eqnarray} \\sin^2 \\left ( 2\\vartheta_{\\alpha\\beta} \\right ) &=& 4\\,\\Uaj{\\alpha4}\\,\\Uaj{\\beta4} \\,,\n\\qquad(\\alpha\\neq\\beta)\n\\label{eq:theta_eff_app}\n\\\\\n\\sin^2 \\left ( 2\\vartheta_{\\alpha\\alpha} \\right ) &=& 4\\,\\Uaj{\\alpha4}\\, (1- \\Uaj{\\alpha4})\n\\label{eq:theta_eff_dis}\n\\,.\n\\end{eqnarray}\nThe last expression is implied by the unitarity of the mixing matrix. From the probabilities in Eqs.~(\\ref{eq:P_sbl_app}) and~(\\ref{eq:P_sbl_dis}) it is clear why the oscillatory behavior manifests when $4E\\sim\\Delta m^2 L$. If $4E\\gg\\Delta m^2 L$, the oscillatory term vanishes; if $4E\\ll\\Delta m^2 L$, the oscillation frequency becomes so high that oscillations cannot be resolved anymore and they are averaged out. The eV scale we focus on arises since various experiments find anomalies pointing towards $(L\/E)^{-1} \\sim \\mathcal O (\\mathrm{eV}^{2})$. We emphasize that the absolute mass scale of the neutrino states does not enter in the equations, so oscillation experiments depend on the differences of the mass-squared values alone. In other words, oscillations are independent of the lightest neutrino mass $m_1$. Both appearance and disappearance channels can be used to measure the effective mixing angles and to constrain the mixing matrix elements.\n\n\\subsection{Current status of oscillation data and global fits}\n\\label{subsec:global_fits}\n\nThere is not a single SBL anomaly, but several different experiments with short baselines find anomalous neutrino fluxes of varying degree. We will briefly summarize the current situation in this section.\n\nFrom the historical point of view, the first anomaly was found by\nLSND \\cite{Aguilar:2001ty},\nwhich reported the unexpected appearance of electron antineutrinos\nin a beam of muon antineutrinos produced from $\\pi^+$ decays.\nYears later, also\nMiniBooNE \\cite{Aguilar-Arevalo:2018gpe} confirmed the excess of $\\bar \\nu_e$ events, with a similar experimental setup, in partial agreement with LSND even though the MiniBooNE excess is too large to be explained by a sterile alone.\n\nAnother anomaly with comparable $L\/E$ was found by $\\nu_e$ disappearance measurements by the Gallium neutrino detectors GALLEX and SAGE \\cite{Abdurashitov:2005tb,Abdurashitov:2009tn,Giunti:2010zu,Kaether:2010ag}. Both experiments measured the electron neutrino flux in proximity to a radioactive source, and found a lack of events at significance of $\\sim 3 \\sigma$. A similar effect is observed in measurements of $\\bar \\nu_e$ disappearance in close proximity to nuclear reactors \\cite{Mention:2011rk}. The lack of electron antineutrinos also reaches a significance of $\\sim 3 \\sigma$ and was noticed after new calculations found a higher expected initial flux \\cite{Mueller:2011nm,Huber:2011wv}.\n\nThe LSND and MiniBooNE $\\bar \\nu_e$ appearance data require a non-zero value of $\\vartheta_{e \\mu}$, while in order to explain the reactor and Gallium disappearance measurements one needs $\\vartheta_{ee} > 0$. Taken together, this also implies a non-zero mixing $\\vartheta_{\\mu \\mu}$ and $\\Uaj{\\mu 4}$.\nHowever, this matrix element can be measured independently by muon (anti) neutrino disappearance experiments, and no corresponding anomaly for the relevant $L\/E$ values is found by either measurements of the atmospheric muon neutrino flux by IceCube \\cite{Aartsen:2017bap,TheIceCube:2016oqi} or by the MINOS+ \\cite{Adamson:2017uda} collaboration using an accelerator beam.\nTherefore, a combination of $\\bar \\nu_e$ appearance data by LSND and MiniBooNE and the disappearance results from electron and muon (anti) neutrinos in a global fit is currently problematic \\cite{Dentler:2018sju,sterile19}.\n\n\n\n\\subsection{Laboratory searches for the absolute neutrino mass scale}\n\\label{sec:lab_numass}\n\nIn addition to neutrino flavor oscillation experiments, laboratory searches for massive neutrinos also include kinematic measurements of $\\beta$-decay and searches for neutrinoless double-$\\beta$ ($\\ensuremath{0\\nu\\beta\\beta}$) decay events. In this section, we briefly summarize the consequences that a light sterile neutrino mixing with the active flavors can have for these searches.\n\nOne way to probe the absolute neutrino mass scale directly is to observe the cutoff of the electron energy spectrum emitted from $\\beta$-decay. It occurs at the effective electron neutrino mass $m_\\beta$, given by the incoherent sum of squared masses and mixing parameters:\n\\begin{equation}\n\\label{eq:m_b}\n m_\\beta^2 = \\sum_{j=1}^{4} |U_{ej}|^2 m_{j}^2\n\\end{equation}\nwhere $j=4$ is the contribution from the additional sterile neutrino. So far, $\\beta$-decay measurements have only been able to set upper limits on the mass scale, with the latest bound $m_\\beta<1.1\\,\\mathrm{eV}$ at 90\\% confidence level (C.L.) recently published by the \nKATRIN collaboration~\\cite{Aker:2019uuj}.\n\nNeutrinoless double-$\\beta$ decay ($0\\nu\\beta\\beta$) searches constrain the half-life $T_{1\/2}$ of the isotope involved in the decay (see e.g.~\\cite{DellOro:2016tmg} for a recent review). Assuming that the mechanism responsible for lepton number violation manifested in $0\\nu\\beta\\beta$ events is the mass mechanism, constraints on the half life can be translated into constraints on the effective Majorana mass $m_{\\beta\\beta}$:\n\n\\begin{equation}\n\\label{eq:halflife}\n T_{1\/2} = \\frac{m_e^2}{G_{0 \\nu} | M_{0 \\nu}|^2 m_{\\beta \\beta}^2},\n\\end{equation}\nwhere $m_e$ is the electron mass, $G_{0 \\nu}$ is the phase space factor and $M_{0\\nu}$ is the nuclear transition matrix element for the decay. The effective mass parameter $m_{\\beta \\beta}$ can be expressed as a coherent sum of mass eigenstates and mixing matrix parameters:\n\n\\begin{equation}\n\\label{eq:m_bb}\n m_{\\beta \\beta} = \\left| \\sum_{j=1}^4 |U_{ej}|^2 \\mathrm{e}^{i \\alpha_j} m_{j} \\right| \\,,\n\\end{equation}\nwhere $\\alpha_j$ are Majorana phases and $j=4$ again contains the contribution from the sterile neutrino. So far, no such event has been detected and only upper limits on the lifetime $T_{1\/2}$ of various isotopes are available, as described in section~\\ref{subsec:decaydata}. These bounds are usually converted into a range of upper limits on the Majorana mass $m_{\\beta \\beta}$ depending on theoretical uncertainty in the calculation of the nuclear matrix elements.\n\n\\section{Light sterile neutrinos in cosmology}\n\\label{sec:sterile_cosmology}\n\nAs explained in section~\\ref{sec:intro}, sterile neutrinos do not take part in weak, strong or electromagnetic interactions. As a consequence, they will not be produced by Standard Model scattering or annihilations in the very early Universe. In this work, we consider a production mechanism via non-resonant oscillations with active states, the so-called Dodelson-Widrow production mechanism~\\cite{Dodelson:1993je,Colombi:1995ze}. \nIn the very early universe, densities are high and weak interactions frequent. This generates an effective matter potential that suppresses neutrino oscillations. Therefore the sterile state is only populated once densities are low enough for oscillations with the active neutrino eigenstates to occur (see e.g.~\\cite{2013neco.book.....L} for a detailed discussion).\n\nThe mass splitting sets the timescale for oscillations with the fourth neutrino and determines the time when flavor oscillations can start to arise. Here we focus on eV-scale sterile neutrinos and consequently consider mass splittings $10^{-2}\\leq\\dmij{41}\/\\text{eV}^2\\leq10^2$. This range corresponds to plasma temperatures between $\\mathcal{O}(100)$ and $\\mathcal{O}(1)$~MeV when the sterile starts to be populated.\n\nThe thermalization process is described by a set of differential equations\nthat encode the evolution of the momentum distribution functions $f_\\alpha(p)$ of the various neutrino flavors $\\alpha \\in [e,\\mu,\\tau,s]$,\nand an additional one for the evolution of the photon temperature, as described in detail in~\\cite{Gariazzo:2019gyi}.\nAs far as cosmological effects of neutrinos are concerned, what matters is the momentum distribution function of all the neutrino states\nafter their decoupling from the thermal plasma and at the end of electron-positron annihilations into photons,\nwhich happens at temperatures around $0.1$~MeV.\n\nThe final momentum distribution function for the active neutrinos is very close, but not exactly equal,\nto a Fermi-Dirac shape with the neutrino temperature $T_\\nu$ \n(see e.g.\\ \\cite{Dolgov:2002wy,Mangano:2005cc,deSalas:2016ztq,Gariazzo:2019gyi,Bennett:2019ewm}). \nThe deviation from thermal equilibrium is mostly due to the fact that neutrino decoupling does not occur instantaneously, so there are small distortions at high momenta that come from the energy transferred\nduring electron-positron annihilations to the few neutrinos still coupled to the plasma.\nFor the sterile neutrino, the momentum distribution function $f_s$ depends on the degree of thermalization that it reaches.\nInitially the sterile is absent, and if the mass splitting and the mixing angles are not large enough, oscillations either start too late or are not efficient enough to bring the fourth neutrino into equilibrium with the active flavors.\nThe degree of thermalization of the sterile can be expressed in terms of the effective number\nof relativistic species ($\\ensuremath{N_\\mathrm{eff}}$), which can be constrained by big-bang nucleosynthesis (BBN)~\\cite{Consiglio:2017pot} \nand more tightly by CMB observations~\\cite{Aghanim:2018oex}. After electron-positron annihilations, this parameter can be expressed as\n\\begin{equation}\n\\ensuremath{N_\\mathrm{eff}}\n=\n\\frac{8}{7}\n\\left(\\frac{11}{4}\\right)^{4\/3}\n\\frac{(\\rho_\\nu+\\rho_s)}{\\rho_\\gamma}\\:,\n\\end{equation}\nwith the energy density in photons $\\rho_\\gamma$ and active plus sterile neutrinos $(\\rho_\\nu+\\rho_s)$. They can be computed from the integrated distribution functions, and for negligible contributions from the sterile we recover the standard value $\\ensuremath{N_\\mathrm{eff}}^{3\\nu}=3.043-3.045$~\\cite{Mangano:2005cc,deSalas:2016ztq,Gariazzo:2019gyi,Bennett:2019ewm,Escudero:2020dfa}.\nOn the other hand, if the mixing parameters are sufficiently large,\nthere is time for neutrino oscillations to fully bring the fourth neutrino\nto equilibrium with the active flavors.\nIn such case, the final distribution function of sterile neutrinos $f_s$ will also be very close to a Fermi-Dirac spectrum,\nwith the same temperature as the active neutrinos,\nand $\\ensuremath{N_\\mathrm{eff}}\\simeq4.05$.\nIntermediate cases correspond to an incomplete thermalization,\nthe sterile neutrino contributes with $\\Delta\\ensuremath{N_\\mathrm{eff}}=\\ensuremath{N_\\mathrm{eff}}-\\ensuremath{N_\\mathrm{eff}}^{3\\nu}<1.01$\nand its distribution function is significantly non-thermal.\nIn such cases, however, it turns out that since the sterile is populated through oscillations from the thermal active states, the distribution function $f_s$ is still proportional to a Fermi-Dirac shape~\\cite{Gariazzo:2019gyi} with temperature $T_\\nu$. Then, we generally have \\cite{Dodelson:1993je,Colombi:1995ze}\n\\begin{equation}\n\\label{eq:f_s}\nf_s(p) = \\frac{\\Delta\\ensuremath{N_\\mathrm{eff}}}{\\exp (p\/T_\\nu) + 1} \\: ,\n\\end{equation}\nwhere $p$ is the neutrino momentum\nand $T_\\nu$ is the temperature of active neutrinos.\nNote that for all cases considered here,\nthe sterile neutrino distribution function reaches its asymptotic values\nwell before primordial BBN at $T\\sim 0.1 \\: \\mathrm{MeV}$ \\cite{Gariazzo:2019gyi}.\nThis means that, under our assumptions,\nthe value of $\\ensuremath{N_\\mathrm{eff}}$ does not change between the time of BBN and CMB decoupling, which would have consequences for the abundance of light elements~\\cite{Consiglio:2017pot} important for the calculation of cosmological perturbations\n(see also \\cite{Schoneberg:2019wmt}). \n\n\n\n\nThe problem of active-sterile oscillations in the early Universe has been studied in many previous papers, some of them published more than 30 years \nago (early references include e.g.\\ \n\\cite{Barbieri:1989ti,Kainulainen:1990ds,Barbieri:1990vx,Enqvist:1990ad,Enqvist:1991qj}, \nsee the review \\cite{Dolgov:2002wy} for an extensive list). Solving the Boltzmann kinetic equations for different neutrino energies is a complex issue, due to the simultaneous presence of neutrino interactions via weak processes and flavor oscillations in an expanding medium. Thus, past analyses \n\\cite{Abazajian:2001nj,Dolgov:2003sg,Cirelli:2004cz,Hannestad:2012ky,Hannestad:2013wwj,Hannestad:2015tea,Mirizzi:2012we,Mirizzi:2013gnd,Saviano:2013ktj,Bridle:2016isd,Knee:2018rvj,Berryman:2019nvr,Adams:2020nue}\nhave considered various approaches that approximated the multi-momentum calculations \nto the evolution of an average momentum and\/or reduced the number of active neutrino states.\nIn particular, the \\texttt{LASAGNA} code \\cite{Hannestad:2013wwj} solves the quantum kinetic equations in the \n1+1 case (assuming one active coupled via oscillations to one sterile neutrino state) with full collision integrals.\nThis code has been used in previous works \n\\cite{Hannestad:2012ky,Bridle:2016isd,Knee:2018rvj,Berryman:2019nvr}\nto map the active-sterile mixing parameters onto two other quantities relevant for cosmology (\\ensuremath{N_\\mathrm{eff}}\\ and the effective sterile neutrino mass).\n\nWhen considering a simplified model with only one active neutrino $\\nu_a$ and one sterile $\\nu_s$, it is possible to relate the degree of thermalization $\\Delta \\ensuremath{N_\\mathrm{eff}}$ directly to the mixing parameters. For a mass splitting $\\delta m^2_{as}$ and mixing angle $\\vartheta_{as}$, this results in \\cite{Dolgov:2003sg}\n\\begin{equation}\\label{eq:deltaneff_mixing_approx}\n\\frac{\\delta m^2_{as}}{\\text{eV}^2}\n\\sin^4\\left(2\\vartheta_{as}\\right)\n\\simeq\n10^{-5}\n\\ln^2\\left(1-\\Delta\\ensuremath{N_\\mathrm{eff}}\\right) \\: ,\n\\end{equation}\nwhere the numerical coefficient is slightly different for electron, muon, or tau flavor neutrinos.\nIf one applies this relation to the 3+1 case,\nwith $\\delta m^2_{as}\\rightarrow\\dmij{41}$\nand\n$\\sin^2\\left(2\\vartheta_{as}\\right)\\simeq4\\Uaj{\\alpha4}\\Uaj{s4}$,\none gets that in order to have a fully thermalized sterile neutrino\nwith a mass splitting around 1~eV$^2$,\na mixing matrix element $\\Uaj{\\alpha4}\\simeq10^{-3}$ is required.\nFrom this relation we can also see that a larger mixing matrix element generally increases \\ensuremath{N_\\mathrm{eff}}\\ towards 4. For larger mass splittings, a smaller mixing is sufficient to generate the same level of thermalization since the oscillations are faster, see Eq.~(\\ref{eq:P_sbl_app})-(\\ref{eq:P_sbl_dis}).\nWhile Eq.~(\\ref{eq:deltaneff_mixing_approx}) is a rough estimate, we will see that it is a quite good approximation of the full calculation,\nas long as one mixing angle is varied at each time.\n\nIncluding also the mixing among the active neutrino states has also been considered before, from the early simplified analyses \\cite{Dolgov:2003sg,Cirelli:2004cz} to more recent multi-angle studies \\cite{Mirizzi:2012we,Mirizzi:2013gnd,Adams:2020nue} performed within the averaged-momentum approximation. Although some authors have studied the multi-angle (2+1 scenario) and multi-momentum problem \\cite{Saviano:2013ktj}, only very recently the evolution in the early Universe of the momentum-dependent kinetic equations for the full $4\\times4$ density matrix of neutrinos was calculated with\na dedicated numerical code, \\texttt{FortEPiaNO}~\\footnote{\\url{https:\/\/bitbucket.org\/ahep_cosmo\/fortepiano_public}}, as\ndescribed in~\\cite{Gariazzo:2019gyi}.\n\nAt late times, the sterile neutrino becomes non-relativistic and it contributes to the matter energy density.\nAt such point in the evolution,\nits contribution to the energy density is~\\cite{2013neco.book.....L,Colombi:1995ze}\n\\begin{equation}\n\\label{eq:Omega_s}\n\\Omega_s h^2 = \\frac{m_s \\Delta \\ensuremath{N_\\mathrm{eff}}}{93.14~\\mathrm{eV}}\\equiv \\frac{m_{s,\\,\\mathrm{eff}}}{93.14~\\mathrm{eV}} \\: ,\n\\end{equation}\nwhere we have introduced the effective mass $m_{s,\\,\\mathrm{eff}} \\equiv m_s \\Delta \\ensuremath{N_\\mathrm{eff}}$. As mentioned in section~\\ref{sec:intro},\nlight sterile neutrinos might behave as a hot dark matter component and affect the evolution of matter perturbations in a similar way as active neutrinos~\\cite{Abazajian:2017tcc}. The form of the distribution function (\\ref{eq:f_s}) implies that the sterile neutrinos have the same average momentum as the active species. Consequently, the maximum free-streaming length of the sterile is equal to the one of an active neutrino with mass $m_\\nu=\nm_s$, corresponding to a comoving wavenumber $k_\\mathrm{fs}$ \\cite{2013neco.book.....L}:\n\\begin{equation}\nk_\\mathrm{fs} = 0.018\\,\\Omega_m^{1\/2} \\left(\\frac{m}{1\\,\\mathrm{eV}}\\right)^{1\/2} h \\,\\mathrm{Mpc}^{-1},\n\\end{equation}\nif neutrinos become nonrelativistic during the matter-dominated era, or\n\\begin{equation}\nk_\\mathrm{fs} = 0.776\\,\\Omega_r^{1\/2} \\left(\\frac{m}{1\\,\\mathrm{eV}}\\right)^{1\/2} h \\,\\mathrm{Mpc}^{-1},\n\\end{equation}\nif they become non-relativistic during the radiation-dominated era. Here $m$ can be either the mass of an active neutrino, or the mass of sterile neutrino produced through non-resonant oscillations. Their large velocities prevent neutrinos from clustering at scales smaller than the free-streaming length, so the collective effect of active and sterile neutrinos is a step-like suppression of the amplitude of matter perturbations below the free-streaming scale~\\cite{Colombi:1995ze,2013neco.book.....L}. \n\\begin{equation}\n \\frac{P_\\nu}{P} \\stackrel{k \\gg k_\\mathrm{fs}}{\\approx} 1 - 8 f_\\mathrm{hot} \\: ,\n\\end{equation}\nwhere $P_\\nu$ and $P$ refer to the matter power spectrum with or without neutrinos respectively. The size of the suppression at small scales depends on the fraction of matter density provided by neutrinos, $f_\\mathrm{hot}=(\\Omega_\\nu + \\Omega_s)\/\\Omega_m$. For sterile neutrinos the onset of the suppression is determined by the free-streaming scale and therefore $m_s$, while the size of the suppression is proportional to $m_{s,\\,\\mathrm{eff}} = m_s \\Delta N_\\mathrm{eff}$. The cosmological effects of the sterile neutrino are therefore completely determined\nonce $m_s$ (or equivalently $\\Omega_s$) and $\\Delta \\ensuremath{N_\\mathrm{eff}}$ are specified.\n\n\nIn figure~\\ref{fig:theoretical_Neff_Omegas} we show the predicted contribution of the sterile neutrino\nto \\ensuremath{N_\\mathrm{eff}}\\ (upper row)\nand\nthe fraction of total dark matter energy density it represents\n(lower row, assuming a total $\\Omega_ch^2 = 0.119$ from the latest Planck measurements~\\cite{Aghanim:2018eyx}), for two different choices for the mixing matrix elements.\nFrom the top row of Fig.~\\ref{fig:theoretical_Neff_Omegas}, it is clear that if at least one of the mixing matrix elements is much larger than $10^{-3}$, one has $\\Delta\\ensuremath{N_\\mathrm{eff}}~\\simeq1$ for $m_s\\sim1\\,\\mathrm{eV}$ in agreement with the expectation from Eq.~(\\ref{eq:deltaneff_mixing_approx}) and the results\nof previous analyses of active-sterile neutrino oscillations in the early Universe (see e.g.\\ \\cite{Hannestad:2012ky,Gariazzo:2019gyi}).\nSuch a value of $\\Delta\\ensuremath{N_\\mathrm{eff}}$ is at odds with what is inferred from Planck measurements of CMB anisotropies, or from astrophysical determinations of the abundances of light elements. In general, we note that in some parts of the parameter space probed by our analysis, corresponding to large mixing angles and\/or large masses, the contribution of sterile neutrinos to both $\\ensuremath{N_\\mathrm{eff}}$ and the dark matter energy density is large. The latter is also a problem, since cosmological observations also constrain the fraction of hot dark matter to be small. As we shall see in the next section, these regions of parameter space will be indeed excluded by data, in agreement with our expectations.\n\n\nAs we will discuss in detail in section~\\ref{sec:results},\nlarge contributions to $\\ensuremath{N_\\mathrm{eff}}$ are in tension with cosmological constraints, so in\norder to allow a sterile neutrino to form all the dark matter\none needs to move to higher masses $m_s \\sim \\mathrm{keV}$ and very small mixing angles. \nIn this mass range, a sterile neutrino produced by oscillations would behave as warm (or even cold) dark matter, so limits on the hot component would not apply. Then even modest contributions to $\\ensuremath{N_\\mathrm{eff}}$\nare sufficient to provide the required dark matter density, although other astrophysical limits apply, as reviewed for instance in~\\cite{Adhikari:2016bei,Abazajian:2017tcc,Boyarsky:2018tvu}.\nThe analysis of the mass range $\\dmij{41}>10^2$ eV$^2$, however, is beyond the scope of this paper, as we are interested in light sterile neutrinos motivated by anomalies in neutrino oscillation experiments. The main question we want to address here is whether such a sterile state is compatible with cosmological bounds.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1\\textwidth]{plots\/theo_Neff_Omegas.pdf}\n\\caption{\\label{fig:theoretical_Neff_Omegas}\n\\textbf{Top:} $\\ensuremath{N_\\mathrm{eff}}$ as a function of the mixing parameters $\\left(\\dmij{41}, \\Uaj{e4}, \\Uaj{\\mu4}, \\Uaj{\\tau4} \\right)$,\nconsidering three equal varying mixing matrix elements ($\\Uaj{e4}=\\Uaj{\\mu4}=\\Uaj{\\tau4}$) in the left panel\nor one varying ($\\Uaj{e4}$) and two fixed ($\\Uaj{\\mu4}=\\Uaj{\\tau4}=10^{-3}$) matrix elements in the right panel.\n\\textbf{Bottom:} Fraction of sterile dark matter compared to the total dark matter density\n$\\Omega_\\mathrm{DM} h^2 = 0.119$ for the same mixing matrix elements assumptions.\n}\n\\end{figure}\n\n\n\\section{Datasets and methods}\n\\label{sec:data}\n\nIn this section, we present the data sets employed in this analysis, and we discuss the method adopted in this work, including the set up adopted for the \\texttt{FortEPianO} code. Since we perform our analysis in a Bayesian framework, we also report the prior choices for the parameters varied in the analysis. \n\n\\subsection{Cosmological data}\n\\label{subsec:cosmodata}\n\nWe consider measurements of the CMB anisotropies in temperature and polarization~\\cite{Aghanim:2019ame},\nas well as determinations of the CMB lensing power spectrum \\cite{Aghanim:2018oex},\nas provided by the latest Planck 2018 data release~\\cite{Akrami:2018vks,Aghanim:2018eyx}. The data sets are incorporated by using the publicly available \\texttt{clik} likelihood code described in \\cite{Aghanim:2019ame}.\n\nTo the CMB dataset, we add late-time distance and expansion rate baryon acoustic oscillation (BAO) measurements\nfrom the 6dFGS~\\cite{Beutler:2011hx}, SDSS-MGS~\\cite{Ross:2014qpa}, and BOSS DR12~\\cite{Alam:2016hwk} surveys. These measurements are fully consistent with the Planck results and greatly help to break parameter degeneracies.\n\n\\subsection{Oscillation experiments}\n\\label{subsec:oscidata}\n\nAs described in section~\\ref{subsec:global_fits}, various oscillation experiments find conflicting results for the active-sterile mixing parameters and it is not possible to combine the measurements in a self-consistent way. Therefore we group the non-conflicting oscillation measurements and present their results separately.\n\nThe $\\bar \\nu_e$ flux from nuclear reactors is measured by Bugey-3 \\cite{Achkar:1996rd},\nDANSS \\cite{Alekseev:2018efk,danilov_epshep19}\nNEOS \\cite{Ko:2016owz},\nand\nPROSPECT \\cite{Ashenfelter:2018iov}. Note that these experiments use differential measurements at a varied distance from the source, so a calibration of the intrinsic neutrino flux is not necessary. A combined frequentist analysis results in a preference for non-zero mixing with a sterile at about $\\sim 2.5 \\sigma$ with two nearly degenerate best-fit points, located at\n($\\dmij{41}\\simeq 0.4~$eV$^2$, $\\Uaj{e4}\\simeq0.01$)\nand\n($\\dmij{41}\\simeq1.3~\\mbox{eV}^2$, $\\Uaj{e4}\\simeq0.01$)\n\\cite{sterile19} (see also \\cite{Gariazzo:2018mwd,Dentler:2017tkw,Dentler:2018sju} for previous analyses).\n\n\nWe also include bounds on the mixing with muon neutrinos from measurements of the atmospheric flux by IceCube \\cite{Aartsen:2017bap,TheIceCube:2016oqi} and from the MINOS+ experiment \\cite{Adamson:2017uda}, which uses an accelerator beam with two detectors, one close ($\\sim 500$~m) and one far ($\\sim 800$~km) from the source, to put strong constraints covering a wide range of $\\dmij{41}$ and $\\Uaj{\\mu 4}$. Both are combined in a frequentist analysis \\cite{sterile19} and we refer to the combination of both data sets as ``$\\nu_\\mu$ disappearance''. Neither IceCube nor MINOS+ find anomalous events and therefore provide upper bounds on the mixing parameters. IceCube also has a limited sensitivity on $\\Uaj{\\tau 4}$ thanks to the low-energy data from DeepCore \\cite{Aartsen:2017bap}.\n\n\\subsection{Decay experiments}\n\\label{subsec:decaydata}\n\nIn this work, we consider the latest measurements of the tritium $\\beta$ decay spectrum released by the KATRIN collaboration~\\cite{Aker:2019uuj} that sets a limit $m_\\beta<1.1\\,\\mathrm{eV}$ at 90\\% C.L. \nWe include the constraint by using the approximated analytical likelihood function presented in Eq.~(B.3) of Ref.~\\cite{Huang:2019tdh}. Since the modification to the decay energy spectrum induced by an extra light sterile species is below the resolution of KATRIN, this likelihood is also valid for the additional sterile neutrino.\n\n\n\nConcerning $\\ensuremath{0\\nu\\beta\\beta}$ searches, we consider the results from the KamLAND-ZEN collaboration phase-I~\\cite{PhysRevLett.110.062502},\nand phase-II~\\cite{PhysRevLett.117.082503},\nfrom the GERDA collaboration~\\cite{Agostini:2017iyd},\nand from the EXO collaboration~\\cite{EXO200}.\nThe constraints are included by using the approximated likelihoods given in \\cite{Caldwell:2017mqu}.\nA summary of all data sets used, the isotopes employed by each collaboration and the respective bounds on the lifetime $T_{1\/2}$ can be found in table~\\ref{tab:0nubb_experiments}, and we show the limits on $m_{\\beta \\beta}$ from the individual experiments in appendix~\\ref{sec_app:individual_0nubb}.\n\nWhile this paper was in preparation, new results from $0\\nu\\beta\\beta$ collaborations have been published from the GERDA collaboration~\\cite{Agostini:2019hzm}, the EXO200 collaboration~\\cite{PhysRevLett.123.161802}, the CUORE collaboration~\\cite{Adams:2018yrj}. We note that the tightest constraints on $m_{\\beta \\beta}$ are still those by KamLAND-ZEN phase-II. As a result, the inclusion of other released data from $0\\nu\\beta\\beta$ searches would not change the conclusions of this work.\n\n\n\\begin{table}[tbp]\n\\centering\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{tabular}{|l c c c c|}\n\\hline\nCollaboration & Isotope &$T_{1\/2}\\,[10^{25}\\,\\mathrm{yr}]$ & $m_{\\beta \\beta} \\, [\\mathrm{eV}]$ & Ref.\\\\\n\\hline\nGERDA & $^{76}\\mathrm{Ge}$ & $>5.3$ & $< 0.25$& \\cite{Agostini:2017iyd}\\\\\nKamLAND ZEN I & $^{136}\\mathrm{Xe}$ & $>1.9$ & $< 0.21$ &\\cite{PhysRevLett.110.062502}\\\\\nKamLAND ZEN II & $^{136}\\mathrm{Xe}$ & $>10.7$ & $< 0.10$ &\\cite{PhysRevLett.117.082503}\\\\\nEXO200 & $^{136}\\mathrm{Xe}$ & $>1.1$ & $<0.24$ &~\\cite{EXO200}\\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{tab:0nubb_experiments} Bounds on the half-life for $0 \\nu \\beta \\beta$ events as measured by the various experiments we consider in our analysis, following~\\cite{Gariazzo:2018pei,Caldwell:2017mqu}. The upper limits on $m_{\\beta \\beta}$ depend on the uncertainty of the nuclear matrix element and are not used directly in our analysis. Details on the individual limits and the effect of the nuclear matrix elements are summarized in appendix~\\ref{sec_app:individual_0nubb}.\n}\n\\end{table}\n\n\\subsection{Method}\n\\label{subsec:method}\n\nIn order to use cosmological data to constrain the sterile neutrino properties, we have to map the fundamental parameters $\\left(\\Delta m_{41}^2, |U_{e4}|^2, |U_{\\mu4}|^2, |U_{\\tau 4}|^2 \\right)$ onto $\\ensuremath{N_\\mathrm{eff}}$ and $m_s$, which determine its impact on cosmological observables.\nAs described in section~\\ref{sec:sterile_cosmology} we use \\texttt{FortEPiaNO} to calculate the evolution of the sterile distribution function including mixing with all three active neutrinos.\nWe pre-compute a $\\ensuremath{N_\\mathrm{eff}}$ table on a four-dimensional grid spanning the parameter space we are interested in.\nFor the mass splitting, we consider nine logarithmically spaced samples over the range $\\Delta m_{41}^2 \\in [10^{-2}, 10^2]$~eV$^2$,\nwhile for each mixing matrix element we take eleven logarithmically spaced samples in the range $\\Uaj{\\alpha 4} \\in [10^{-6}, 10^{-1}]$.\nThe lower boundary chosen for the mass splitting ensures that the hierarchy $\\Delta m_{21}^2 \\ll \\Delta m_{31}^2 \\ll \\Delta m_{41}^2$ always holds, so the sterile does not affect the oscillation patterns among active neutrinos which we keep fixed to their standard values. As a base model we assume a $\\Lambda$CDM cosmology,\nwith its six parameters\\footnote{The $\\Lambda$CDM six parameters are the energy density in baryonic matter $\\Omega_b h^2$ and in cold dark matter $\\Omega_c h^2$, the angular size of the sound horizon at recombination $\\theta$, the reionization optical depth $\\tau$, the amplitude $A_s$ and spectral index $n_s$ of the primordial spectrum of scalar perturbations.},\ncomplemented with the lightest neutrino mass $m_1$, the sterile mass splitting $\\Delta m_{41}^2$\nand the three mixing matrix elements \\Uaj{\\alpha4}, $\\alpha=e,\\,\\mu,\\,\\tau$.\n\nWe calculate the evolution of cosmological perturbations\nusing the numerical Einstein-Boltzmann solver\n\\texttt{CLASS}~\\cite{Lesgourgues:2011re,Blas:2011rf}~\\footnote{\\url{https:\/\/class-code.net\/}}. Since we map the four active-sterile mixing parameters into the two cosmological parameters $\\ensuremath{N_\\mathrm{eff}}$ and $m_s$,\na very large number of samples are necessary to obtain\na well-converged posterior in the full sterile parameter space.\nTo speed the calculations up, we therefore proceed in two steps.\nFirst we use \\texttt{MontePython}~\\cite{Audren:2012wb,Brinckmann:2018cvx} to run a standard Markov Chain Monte Carlo (MCMC) with CMB+BAO data. For this run, we vary all $\\Lambda$CDM parameters and include three massless neutrinos with $\\ensuremath{N_\\mathrm{eff}} = 3.045$ and one additional massive state with varying mass and $\\Delta \\ensuremath{N_\\mathrm{eff}}$.\nNote that with the current sensitivities the data cannot distinguish between various combinations of neutrino parameters as long as they result in the same $\\Delta \\ensuremath{N_\\mathrm{eff}}$ and $\\ensuremath{\\Sigma m_\\nu}$,\nso distributing the mass in different ways over the four available states does not affect the resulting constraints in a significant way (this reflects the fact that cosmology is not yet able to constrain the mass hierarchy, see e.g.~\\cite{Hannestad:2016fog,Giusarma:2016phn,Gerbino:2016sgw,Gerbino:2016ehw,DiValentino:2016foa,Vagnozzi:2017ovm, Mahony:2019fyb}).\nGiven the prior $\\ensuremath{N_\\mathrm{eff}} > 3.045$, the cosmological data sets a $95\\%$-limit of $\\ensuremath{\\Sigma m_\\nu} < 0.15$ eV and $\\ensuremath{N_\\mathrm{eff}} < 3.44$.\nWe then use a Gaussian kernel density estimate (KDE) to interpolate the marginalized $\\ensuremath{N_\\mathrm{eff}} - \\ensuremath{\\Sigma m_\\nu}$ likelihood surface. Evaluating this estimated likelihood is $\\sim 10^5$ faster than running the full CMB likelihood, and we explicitly make sure that sampling from it results in the same constraints as obtained from the MCMC.\n\nIn a second step, we use \\texttt{emcee}~\\cite{ForemanMackey:2012ig} together with the estimated likelihood to sample the neutrino parameter space as follows:\nwe vary the lightest neutrino mass $m_1$, the mass splitting $\\dmij{41}$ and the three mixing matrix elements $\\Uaj{\\alpha 4}$. From the last four, we interpolate $\\Delta \\ensuremath{N_\\mathrm{eff}}$ from the pre-computed \\texttt{FortEPiaNO} grid. Since the mass splittings $\\dmij{21}$ and $\\dmij{31}$ are much smaller than the sensitivity of current constraints on neutrino masses, we assume three degenerate active neutrinos $m_1 = m_2 = m_3$ so we get a total $\\ensuremath{\\Sigma m_\\nu} = 3 m_1 + \\Delta \\ensuremath{N_\\mathrm{eff}} \\, m_4$. We then evaluate the KDE of the cosmological likelihood for $\\ensuremath{N_\\mathrm{eff}} = 3.045 + \\Delta \\ensuremath{N_\\mathrm{eff}}$ and $\\ensuremath{\\Sigma m_\\nu}$. In order to compare the constraints to direct experiments, we also vary the nuclear matrix elements for $^{136}$Xe and $^{76}$Ge defined in Eq.~(\\ref{eq:halflife}) and the three Majorana phases $\\alpha_i$ in Eq.~(\\ref{eq:m_bb}) to derive limits on $m_\\beta$ and $m_{\\beta \\beta}$. A summary over all non-$\\Lambda$CDM parameters varied in the analysis, their prior bounds and shapes is given in table~\\ref{tab:priors}.\n\n\\begin{table}[tbp]\n\\centering\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{tabular}{|c c c c |}\n\\hline\n\\multicolumn{2}{|c}{Parameter} & prior shape & prior bounds \\\\\n\\hline\nlightest neutrino mass & $m_1\/\\mathrm{eV}$ & flat & $[0, 5]$ \\\\\nmass splitting & $\\Delta m_{41}^2\/\\mathrm{eV}^2$ & log & $[10^{-2}, 10^2]$ \\\\\nmixing matrix elements & $|U_{\\alpha 4}|^2$ ($\\alpha \\in [e, \\mu, \\tau]$) & log & $[10^{-6}, 10^{-1}]$ \\\\\nMajorana phases & $\\alpha_j$ ($j \\in [1, 2, 3]$)& flat & $[0, 2 \\pi]$ \\\\\nnuclear matrix elements & $|M_{0 \\nu} ( ^{136}\\mathrm{Xe}) |^2$ & flat & $[2.74, 3.45]$ \\\\\n & $|M_{0 \\nu} ( ^{76}\\mathrm{Ge}) |^2$ & flat & $[4.07, 4.87]$ \\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{tab:priors} Priors for all non-cosmological parameters varied in our analysis. We comment on different prior choices for $\\Delta m_{41}^2$ in detail in section~\\ref{subsec:results_priors}.}\n\\end{table}\n\nFor constraints involving the $\\beta$-decay and $0\\nu\\beta\\beta$ likelihoods described in section~\\ref{subsec:decaydata}, we marginalize over all parameters listed in table~\\ref{tab:priors} as well. The constraints from reactors and $\\nu_\\mu$ appearance measurements from MINOS+ and IceCube (see section~\\ref{subsec:oscidata}), on the other hand, are derived in a frequentist framework and do not depend on the prior choices made here.\n\n\n\\section{Results}\n\\label{sec:results}\n\nIn this section we present the results of our analysis. First we discuss the complementarity of the various data sets either in terms of the fundamental parameters of the sterile mixing matrix, or in terms of the derived parameters $m_\\beta$ and $m_{\\beta \\beta}$ directly constrained by terrestrial experiments. Then we explore the effect of assuming other priors on the sterile mass parameter for the cosmological datasets. In the same context we compare our constraints with the ones previously obtained in a simplified scenario where the sterile is only coupled to one neutrino species.\n\n\\subsection{Constraints from cosmology and direct experiments}\n\\label{subsec:results_compared}\n\nCosmology provides tight limits on the sum of neutrino masses $\\ensuremath{\\Sigma m_\\nu}$ and the effective number of relativistic species $\\ensuremath{N_\\mathrm{eff}}$,\nthat can be translated to tight bounds on the parameter space of the mixing matrix with significant relativistic energy contributions from the sterile neutrino.\nAfter marginalising over all other cosmological and nuisance parameters, we obtain the constraints on the mixing matrix elements in figure~\\ref{fig:cosmology_results} from CMB+BAO. As any mixing reaching a value of $|U_{\\alpha4}|^2 \\approx 10^{-3}$ starts to populate the sterile state and leads to detectable $\\ensuremath{N_\\mathrm{eff}}$ contributions as seen in figure~\\ref{fig:theoretical_Neff_Omegas}, this is where cosmological data becomes constraining.\nThere are only small differences between mixing for the various flavors, so the limits on all of the matrix elements are very similar.\nSince the constraints on both the mass $m_4$ derived from the total sum $\\ensuremath{\\Sigma m_\\nu} = 3 m_1 + \\Delta \\ensuremath{N_\\mathrm{eff}} \\, m_4$ and $\\ensuremath{N_\\mathrm{eff}}$ vanish for small amplitudes of the sterile distribution function set by $\\Delta \\ensuremath{N_\\mathrm{eff}}$, the cosmological data in principle allow large sterile masses as long as the mixing is small and the sterile state is not thermalized or populated to a significant level.\nAs mentioned in section~\\ref{sec:sterile_cosmology}, much higher mass ranges in the keV range and larger are in principle possible if the mixing angles are small enough. In such a case the free-streaming length $k_\\mathrm{fs}$ of the sterile neutrino would be pushed to smaller scales, and it would form a warm or even a cold dark matter component~\\cite{Adhikari:2016bei,Abazajian:2017tcc,Boyarsky:2018tvu}.\n\n\\begin{figure}\n\\centering \n\\includegraphics[width=1.\\textwidth]{plots\/CMB_mixing_triangle_Ui4.pdf}\n\\caption{\\label{fig:cosmology_results}\nCosmological marginalized constraints on the mixing matrix elements $\\Uaj{\\alpha4}$ and mass splitting $\\dmij{41}$ from CMB+BAO. Off-diagonal panels show $68 \\%$ and $95 \\%$ confidence level probability contours. Panels along the diagonal show one-dimensional probability distributions. Cosmological constraints are not very sensitive to the flavor, so bounds on all different matrix elements are similar.\n}\n\\end{figure}\n\n\\begin{table}[tbp]\n\\centering\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{tabular}{|c c c c c|}\n\\hline\n\\multirow{2}{*}{Parameter} & \\multirow{2}{*}{experimental upper limit ($95\\%$)} & \\multicolumn{3}{c|}{cosmological upper limit ($95\\%$)} \\\\\n & & $\\mathcal{P}(\\log \\dmij{14})$ & $\\mathcal{P}(m_4)$ & $\\mathcal{P}(\\dmij{14})$ \\\\\n\\hline\n$m_4$ [eV] & - & $1.6$ & $4.4$ & $6.8$\\\\\n$\\log_{10} \\Uaj{e4}$& - & $-3.04$& $-3.43$& $-4.0$\\\\\n$\\log_{10}\\Uaj{\\mu4}$& $-2.2$ ($\\nu_\\mu$) & $-3.17$ &$-3.55$ & $-4.16$\\\\\n$\\log_{10}\\Uaj{\\tau 4}$& $-0.8$ ($\\nu_\\mu$)& $-3.18$& $-3.55$ &$-4.19$\\\\\n$m_\\beta$ [eV] & $0.9$ (KATRIN)&\\multicolumn{3}{c|}{$0.09$}\\\\\n$m_{\\beta \\beta}$ [eV] & $0.08$ ($0\\nu \\beta \\beta$) &\\multicolumn{3}{c|}{$0.07$}\\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{tab:results} Upper limits ($95 \\%$) for the sterile neutrino mass, the parameters of the mixing matrix, and $m_\\beta$ and $m_{\\beta \\beta}$. For the limits from direct experiments we take a conservative approach and only consider probes that are not in tension with cosmology. For those, we quote the strongest bound on each parameter. We present cosmological limits for the different prior choices for the mass splitting either on $\\log \\dmij{41}$ used in section~\\ref{subsec:results_compared}, or on either $m_4$ or $\\dmij{41}$ discussed in section~\\ref{subsec:results_priors}, but note that the prior choice barely affects the resulting constraints on $m_\\beta$ and $m_{\\beta \\beta}$. \n}\n\\end{table}\n\n\nIn figure~\\ref{fig:0nuBB_CMB} (left panel), we compare the cosmological results with limits obtained from neutrinoless double-$\\beta$ decay and tritium decay measurements by KATRIN \\cite{Aker:2019uuj} in the $\\Delta m_{41}^2 - |U_{e4}|^2$ plane, since the decay experiments are not sensitive to the other matrix elements.\nWhile sensitivity of the neutrinoless double-$\\beta$ decay experiments and $\\beta$-decay measurements from KATRIN on the sterile parameters are comparable, cosmological bounds are orders of magnitude stronger.\nTo address the sterile neutrino interpretation of the reactor anomaly, we also show the parameter space preferred by a combined fit of short-baseline measurements of the reactor antineutrino flux.\nAs explained in section~\\ref{sec:intro},\nsuch experiments observe a preference in favor of a sterile with $\\dmij{41} \\sim \\mathcal O (\\mathrm{eV}^2)$ and $\\Uaj{e4} \\approx 10^{-2}$.\nWhile these parameter values are compatible with current measurements from $0\\nu\\beta\\beta$ and KATRIN, they are in severe tension with the CMB+BAO data. A mixing of the size needed to explain the reactor data would be more than sufficient to bring the sterile in thermal equilibrium in the early universe. On the right of figure~\\ref{fig:0nuBB_CMB} we show a similar comparison for the mixing angle $\\Uaj{\\mu 4}$ and present cosmological constraints together with bounds from $\\nu_\\mu$ disappearance measurements from IceCube and MINOS+ described in section~\\ref{subsec:oscidata}. While the experimental sensitivity is stronger than on $\\Uaj{e 4}$, we still find the CMB+BAO data set to be more constraining.\nWhile the cosmological limits rely on model assumptions and can be slightly relaxed in extended parameter spaces, accommodating $\\ensuremath{N_\\mathrm{eff}} \\approx 4$ is very challenging \\cite{DiValentino:2019dzu, Kreisch:2019yzn}.\n\n\\begin{figure}[tbp]\n\\centering \n\\includegraphics[width=.48\\textwidth]{plots\/CMB_0nuBB_Katrin_14_neosdanss.pdf}\n\\hfill\n\\includegraphics[width=.48\\textwidth]{plots\/CMB_mu_24.pdf}\n\\caption{\\label{fig:0nuBB_CMB}\n\\textbf{Left:} Marginalized $68 \\%$ and $95 \\%$ constraints on the mass splitting \\dmij{41} and\nmixing matrix element \\Uaj{e4} from cosmology (blue),\nfrom the tritium $\\beta$-decay measurements by KATRIN (green)\nand from neutrinoless double-$\\beta$-decay experiments ($0\\nu \\beta \\beta$, red),\ncompared with the preferred frequentist regions by the joint fit \\cite{sterile19} of reactor experiments\ndiscussed in section~\\ref{subsec:oscidata}, which are in strong tension with cosmological bounds.\n\\textbf{Right:} Cosmological $68 \\%$ and $95 \\%$ marginalized constraints on the mixing matrix element \\Uaj{\\mu4}\ncompared to (frequentist) $\\nu_\\mu$ disappearance results \\cite{sterile19}\nfrom IceCube and MINOS+ (grey).\n}\n\\end{figure}\n\nWe also map the cosmological bounds onto the parameter space $m_\\beta$ or $m_{\\beta \\beta}$ directly probed by decay experiments in figure~\\ref{fig:mb_mbb}. On the left side we show limits from CMB+BAO together with the latest results from KATRIN~\\cite{Aker:2019uuj}.\nSince $m_\\beta$ defined in Eq.~(\\ref{eq:m_b}) receives contributions from the sterile, the resulting limit from cosmology in the extended $3+1$ parameter space is slightly higher compared to the expectation $m_\\beta \\approx m_1 \\approx \\ensuremath{\\Sigma m_\\nu} \/ 3$ from standard neutrinos.\nThe CMB+BAO data together with the prior assumption $\\ensuremath{N_\\mathrm{eff}} > 3.045$ lead to an upper bound on the neutrino mass sum of $\\ensuremath{\\Sigma m_\\nu} < 0.15$ eV, corresponding to a limit $m_\\beta^\\mathrm{cosmo} < 0.05$ eV assuming only three active neutrinos, which is slightly degraded to $m_\\beta^\\mathrm{cosmo} < 0.09$ eV if the sterile is included.\nHowever, this constraint is still a factor of $\\sim 10$ stronger than current experimental limits from KATRIN.\n\n\\begin{figure}\n\\centering \n\\includegraphics[width=.9\\textwidth]{plots\/mb_mbb_lims.pdf}\n\\caption{\\label{fig:mb_mbb} Left: One-dimensional probability distribution for the neutrino effective mass $m_\\beta$ from cosmology and $\\beta$-decay measurements from KATRIN. Right: One-dimensional probability distribution for the mass parameter $m_{\\beta \\beta}$ from $0 \\nu \\beta \\beta$ and CMB+BAO. The combined $0\\nu\\beta\\beta$ probability distribution is peaked at slightly non-zero values of $m_{\\beta \\beta}$ due to a modest excess of events observed at EXO200.}\n\\end{figure}\n\nOn the right-hand side of figure~\\ref{fig:mb_mbb} we show a similar comparison of the derived bound on $m_{\\beta \\beta}$ from cosmology and direct $0\\nu \\beta \\beta$ searches. Note that the posterior from the combined $0 \\nu \\beta \\beta$ data has a maximum at slightly non-zero values due to an excess of events observed by EXO200 compared to the background expectation \\cite{PhysRevLett.123.161802}.\nIn table~\\ref{tab:results}, we present a summary of all $95 \\%$ bounds on the sterile mass $m_4$, the mixing matrix elements $\\Uaj{\\alpha4}$ for each flavor, $m_\\beta$ and $m_{\\beta\\beta}$, comparing the cosmological bound to the respective strongest bound from direct searches. Cosmology provides the tightest limits on all parameters, for most of them by orders of magnitude.\n\n\n\n\\subsection{Priors and parameter space volume effects}\n\\label{subsec:results_priors}\n\nSince all cosmological bounds presented in figure~\\ref{fig:cosmology_results} and discussed in the previous section~\\ref{subsec:results_compared} provide only upper bounds on the sterile parameters in a Bayesian framework, the choice of priors affects the limit.\nIn this section we focus on various priors for the mass splitting $\\Delta m_{41}^2$.\nWhile we adopt a logarithmic prior as a standard case, since cosmological data is ignorant of the order of magnitude of the mass splitting, one can also argue that the parameter of interest is either the mass-squared difference itself --- or the sterile mass scale, since cosmological data is approximately sensitive to the energy density parameter $\\Omega_s \\propto m_s \\approx m_4$.\nWe therefore repeat the previous cosmological analysis using a flat prior either on $\\log_{10} \\dmij{41}$, on $\\dmij{41}$, or on $m_4$, always considering the same range specified before in table~\\ref{tab:priors}.\nNote that while sampling over $m_4$ we enforce the additional constraint $\\dmij{41} > 10^{-2} \\, \\mathrm{eV}^2$ to make sure that the neutrino mass hierarchy is unchanged.\n\n\\begin{figure}\n\\centering \n\\includegraphics[width=1.\\textwidth]{plots\/CMB_mixing_triangle_Ui4_priors.pdf}\n\\caption{\\label{fig:priors}\nCosmological marginalized constraints on the mixing matrix elements $\\Uaj{\\alpha4}$ for flat priors on either $\\log \\Delta m_{41}^2$ (blue), $m_4$ (orange) or $\\Delta m_{41}^2$ (green), from CMB+BAO data.\nOff-diagonal panels show $68 \\%$ and $95 \\%$ confidence level probability contours.\nPanels along the diagonal show one-dimensional probability distributions.\nThe more weight the prior gives to higher sterile masses, the lower the allowed $|U_{i4}|^2$ values in order to stay within the $\\ensuremath{N_\\mathrm{eff}}$ range allowed by the cosmological data.\n}\n\\end{figure}\n\nWe present the resulting limits on the mixing matrix elements in figure~\\ref{fig:priors} and in table~\\ref{tab:results}.\nThe different choices indeed affects the results, and bounds on the mixing matrix elements are stronger for the flat priors on $m_4$ or $\\dmij{41}$.\nThis is a consequence of the mapping between parameter spaces: the most important constraint from the data is on $\\ensuremath{N_\\mathrm{eff}}$ (which also contributes to $\\ensuremath{\\Sigma m_\\nu}$ through $\\Delta \\ensuremath{N_\\mathrm{eff}} \\, m_4$), and there is a strong degeneracy between the mass splitting and the mixing matrix elements as explained in section~\\ref{sec:ster_osc} and seen from the upper left plot in figure~\\ref{fig:theoretical_Neff_Omegas}.\nFor a fixed higher sterile mass splitting, the mixing angles have to be smaller to stay within the allowed $\\ensuremath{N_\\mathrm{eff}}$ region. Therefore the more weight the priors give to large masses $m_4$ or $\\dmij{41}$, the smaller the matrix elements $\\Uaj{\\alpha4}$ have to be to fulfill the $\\Delta \\ensuremath{N_\\mathrm{eff}}$ constraint. We want to emphasize that the choice of priors does not affect the degree of tension with reactor and $\\bar \\nu_e$ appearance measurements.\nA different choice of priors changes the weight of the mapping $(\\dmij{41}, \\Uaj{\\alpha 4}) \\rightarrow \\ensuremath{N_\\mathrm{eff}}$, but the region in parameter space necessary to explain the anomalies leads to an almost thermal sterile with $\\Delta \\ensuremath{N_\\mathrm{eff}} \\approx 1$. This is excluded by cosmological data independent of the assumptions made in the analysis.\n\n\n\\begin{figure}[tbp]\n\\centering \n\\includegraphics[width=.48\\textwidth]{plots\/11_11mod_31.pdf}\n\\hfill\n\\includegraphics[width=.48\\textwidth]{plots\/11_e_mu_tau.pdf}\n\\caption{\\label{fig:11vs31} \\textbf{Left:} Cosmological $68 \\%$ and $95 \\%$ marginalized constraints assuming either a $1+1$ case with the sterile only coupled to $\\nu_e$ (purple), compared to the full $3+1$ scenario including the full mixing matrix (blue) from figure~\\ref{fig:cosmology_results}. The difference is mostly a parameter space volume effect, since the limits obtained with a modified coupling only to $\\nu_e$ with an effective mixing $|U^\\mathrm{eff}|^2 = \\Sigma_{\\alpha} |U_{\\alpha 4}|^2$ ($\\alpha \\in [e, \\mu, \\tau]$, red) are almost the same as for the $3+1$ case. \\textbf{Right:} Cosmological $68 \\%$ and $95 \\%$ marginalized constraints on $|U^\\mathrm{eff}|^2$ for a single sterile coupled only to one neutrino species $\\nu_e$ (red, same as left), $\\nu_\\mu$ (green) or $\\nu_\\tau$ (grey). The differences between coupling to the different flavors are very small.}\n\\end{figure}\n\n\nThe resulting limits on the mixing matrix elements summarized in table~\\ref{tab:results} are robustly constrained to be $\\Uaj{\\alpha4} < 10^{-3}$ for all flavors. While mixing with electron neutrinos is the most important channel and the resulting bounds on $\\Uaj{e 4}$ are slightly more stringent, there is overall only a minor difference between mixing with the different active neutrinos.\nIt is therefore interesting to understand the difference between a simplified case where the sterile is only coupled to one active neutrino (often assumed to be $\\nu_e$) and the full mixing with all flavors explored in this paper.\nWe present the comparison between the cosmological constraints on $\\dmij{41}$ and $\\Uaj{e4}$ assuming either a $1+1$ or a $3+1$ scenario on the left side of figure~\\ref{fig:11vs31}.\nThe results clearly differ and the $1+1$ mixing allows for higher mass splittings of the sterile.\nHowever, due to parameter space volume effects it is expected that the limits look different.\nFor every point in the $(\\dmij{41}, \\Uaj{e4})$ plane, in the $3+1$ scenario there are more ways to achieve a higher $\\Delta \\ensuremath{N_\\mathrm{eff}}$ and $\\ensuremath{\\Sigma m_\\nu}$ by increasing the other mixing matrix elements.\nHigher mass splittings with small $\\Uaj{e4}$ that lead to negligible cosmological effects in the $1+1$ model are then still unlikely in the $3+1$ case since the other mixing angles have to be small as well.\nWe explicitly test this explanation by adding another comparison case where the sterile is still only coupled to electron neutrinos, but we consider an effective mixing matrix element\n\\begin{equation}\n\\label{eq:Ueff}\n |U^\\mathrm{eff}|^2 = \\sum_{i}^3 \\Uaj{i4}\n\\end{equation}\nand we sample over three distinct contributions $\\Uaj{i4}$ to make up for the larger parameter space volume in the full $3+1$ mixing scenario.\nAs can be seen on the left side of figure~\\ref{fig:11vs31}, this parameter space volume effect almost completely accounts for the difference between the scenarios. On the right-hand side of figure~\\ref{fig:11vs31} we test this effective mixing scenario by accounting for the parameter space effect in the same way, but coupling the sterile to either one of electron-, muon- or tau-neutrinos respectively. Again, differences between the individual flavors are very small, and an effective coupling to $\\nu_e$ provides a very good approximation to the full dynamics.\n\nAs a consequence we find that the production of light sterile neutrinos via mixing with to three active states can be modelled within a $1+1$ model with a single sterile mixing with $\\nu_e$ as long as the effective mixing matrix element in Eq.~(\\ref{eq:Ueff}) is used. Therefore, the abundance of a single sterile computed in the extended 3+1 model is, to an excellent approximation, very similar to that found in the effective 1+1 scenario, provided that the 1+1 squared mixing matrix element is the sum of the individual squared mixing matrix elements in the 3+1 case.\n\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nSeveral anomalies observed in short-baseline oscillation experiments hint at a new neutrino mass state at the eV scale. In this paper, we have provided a consistent framework to constrain the additional neutrino mass splitting and the active-sterile mixing angles with cosmological data. This also allows us to compare the bounds from cosmological data with results from oscillations, $\\beta$-decay and $\\ensuremath{0\\nu\\beta\\beta}$ measurements in a common parameter space.\n\nFor the first time, we performed this analysis in a full $3+1$ scenario where the sterile state is \nmixed with all three active neutrinos.\nIn order to map the sterile neutrino mass-squared splitting and mixing matrix parameters for each flavor $\\dmij{41}$ and $\\Uaj{\\alpha4}$ onto the cosmological observables, we have solved the evolution of the $4 \\times 4$ neutrino density matrix in the early universe with the \\texttt{FortEPiaNO}~\\cite{Gariazzo:2019gyi} code and find that the resulting distribution function of the fourth neutrino state is well approximated by a Dodelson-Widrow form \\cite{Dodelson:1993je}.\n\nA combination of CMB and BAO data currently provides the strongest available bounds on the sterile mass splitting $\\dmij{41}$ and the mixing matrix elements $\\Uaj{\\alpha4}$. While the limit on the mass scale depends on prior assumptions, we robustly find that once any mixing matrix element reaches a level of $\\Uaj{\\alpha4} \\approx 10^{-3}$, the new state would give rise to a detectable relativistic energy contribution in the early universe not seen in cosmological data. The parameter space needed to explain short baseline anomalies with a sterile neutrino leads to a fully thermalized relativistic species with $\\ensuremath{N_\\mathrm{eff}} \\approx 4$ and is in strong tension with the CMB bounds.\nWe also derive limits on the effective electron neutrino mass $m_\\beta$ and the Majorana mass parameter $m_{\\beta\\beta}$, measurable in $\\beta$- and $\\ensuremath{0\\nu\\beta\\beta}$-decay experiments, from cosmological data, finding $m_\\beta<0.09$ eV and $m_{\\beta \\beta}<0.07$ eV at 95\\% C.L. These constraints are tighter than the ones obtained from the latest direct laboratory measurements. Our main results in terms of limits on the sterile mass scale $m_4$, the mixing matrix parameters for each flavor $\\Uaj{\\alpha4}$, $m_\\beta$, and $m_{\\beta\\beta}$, are summarized in table~\\ref{tab:results}.\n\nWe have explored the effect of prior choices on the cosmological bounds repeating the analysis with different priors on the mass splitting. Since a sizeable sterile contribution to $\\ensuremath{N_\\mathrm{eff}}$ can be produced either by a large mass splittings or large mixing angles, the main effect of priors is to shift the weight between the two quantities. The maximal contribution to $\\ensuremath{N_\\mathrm{eff}}$ allowed by the data is fixed, so the more weight the prior gives to high mass splittings $\\dmij{41}$, the lower the mixing angles have to be to stay within the allowed region.\n\nAlmost all limits on sterile neutrinos from cosmology previously reported in the literature were derived in a simplified $1+1$ scenario where the sterile is only coupled to one active neutrino. Even though the mixing with the different active flavors is almost equivalent, we show that the results differ significantly from the ones obtained with a $3+1$ mixing scheme assumed for this work. This can be largely attributed to parameter space volume effects and can be accounted for by coupling the sterile with one active neutrino with effective mixing as given by Eq.~(\\ref{eq:Ueff}). In any case, our results show that allowing for more than one active-sterile mixing does not relax the tension between the active-sterile neutrino parameters favored by the oscillation anomalies and present cosmological observations.\n\nSince cosmological data currently dominates the constraints, we do not perform a joint analysis with laboratory experiments at this time. However, the framework provided here can be the basis for new global constraints on light sterile neutrino properties once new data from laboratory searches becomes available.\n\n\\acknowledgments\n\nSH, PFdS, and KF acknowledge support from the Vetenskapsr\\r{a}det (Swedish Research Council) through contract No.\\ 638-2013-8993 and the Oskar Klein Centre for Cosmoparticle Physics. SG acknowledges support from the European Union's Horizon 2020 research and innovation program under the Marie Sk\\l odowska-Curie individual Grant Agreement No.\\ 796941. SG and SP acknowledge support from the Spanish grants FPA2017-85216-P (AEI\/FEDER, UE), PROMETEO\/2018\/165 (Generalitat Valenciana) and the Red Consolider MultiDark FPA2017-90566-REDC. MG acknowledges support from Argonne National Laboratory (ANL). ANL's work was supported by the DOE under contract W7405-ENG-36. MG and ML acknowledge support from the ASI grant 2016-24-H.0 COSMOS ``Attivit\\`{a} di studio per la comunit\\`{a} scientifica di cosmologia'' and from INFN through the InDark and Gruppo IV fundings. SV acknowledges support from the Isaac Newton Trust and the Kavli Foundation through a Newton-Kavli Fellowship, and acknowledges a College Research Associateship at Homerton College, University of Cambridge. KF acknowledges support from the Jeff and Gail Kodosky Endowed Chair in\nPhysics, DoE grant DE- SC007859 and the LCTP at the\nUniversity of Michigan.\nThis work is partly based on observations obtained with Planck (http:\/\/www.esa.int\/Planck), an ESA science mission with instruments and contributions directly funded by ESA Member States, NASA, and Canada.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\nThe recent rise of online social media presents an unprecedented opportunity for computational social science:\nuser generated texts provide insight about user's attributes such as employment, education or gender. \nAt the same time the social network structure sheds light on complex real-world relationships between preferences and attributes.\nFor instance people sharing similar\nattributes such as employment background or hobbies have a higher chance of becoming friends.\nUser modeling based on information presented in social networks is an important goal, both for applications such as product recommendation, \ntargeted online advertising, friend recommendation and for helping social scientists and political analysts gain insights into public opinions and user behaviors. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=3in]{1.pdf}\n\\caption{Illustration for the proposed method that learns latent representations for users, attributes and user-generated texts based on social network information.}\\label{brief}\n\\end{figure}\n\nNevertheless, much important information on social networks exists in unstructured data formats.\nImportant social insights are locked away, entangled within a heterogenous combination of social signals \\cite{sun2009ranking} - including text, networks, attributes, relations, preferences, etc.\nWhile recent models have attempted to link one or two aspects of the evidence, how to develop a scalable framework that\n incorporates massive, diverse social signals including user generated texts, tens of thousands of user attributes and network structure in an integrated way, \n\n remains an open problem.\n\n\nIn this paper, we propose a general deep learning framework for jointly analyzing user networks, generated context and attributes.\nWe map users, attributes, and user-generated content to latent vector representations, which are learned in a scalable way from social network data.\nFigure \\ref{brief} gives a brief overview of the mechanism of the proposed model: \nusers are represented by similar vectors if they are friends, share similar attributes or write similar content. Attributes are similarly clustered if \nassociated with similar users.\\footnote{This core idea is similar to collaborative filtering \\cite{kautz1997referral}.} \nIn summary, we incorporate diverse social signals in a unified framework, allowing user embeddings to be jointly optimized \nusing neural networks trained on vast quantities of rich social and linguistic context.\n\n\n\n\nBased on these learned representations, our approach provides with a general \nparadigm \n on a wide range of predictive tasks concerning individual users\nas well as group behavior: user attribute inference (e.g., the city the user lives in), \npersonal interest prediction (e.g, whether a user will like a particular movie),\nand probabilistic logical reasoning over the social network graph. For example, our models infer that:\n \\begin{tightitemize}\n\\item Men in California are 6.8 times more likely to take an engineering occupation than women in California. \n \\item Users who work in the IT industry\\footnote{This information comes from the Standard Occupational Classification (SOC), as will be described later.} \nare 2.5 times more likely to like iPhones than users working in Legal Occupations.\n \\end{tightitemize}\nOur methods also have the potential to seamlessly integrate rich textual context into many social network analysis tasks including: \nlink prediction, community detection, and so on, and the learned user representations can be used as important input features for downstream machine learning models, just as how word embeddings are used in the field of natural language processing. \nThe major contributions of this paper can be summarized as follows:\n\\begin{tightitemize}\n\\item We propose new ways for integrating heterogeneous cues about people's\nrelations or attributes into a single latent representation.\n\\item We present inference algorithms for\nsolving social media inference tasks related to both individual and group behaviors based on the learned user representations.\n\\item Cues that we take advantage of may be noisy and sometimes absent but by combining them via global inference, we can learn latent facts about people.\n\\end{tightitemize}\n\nWe evaluate the proposed model on four diverse tasks: friend-relation prediction, gender identification, occupation identification and user geolocation prediction. \nExperimental results demonstrate improved predictions by our model by incorporating diverse evidence from many sources of social signals.\n\n\\begin{comment}\n\\section{Related Work}\n\\label{related_work}\nMuch work has been devoted to automatic user attribute inference given social signals.\nFor example, \\cite{rao2010classifying,rao2010detecting,ciot2013gender,conover2011political,sadilek2012finding,volkova2014inferring,li2014weakly,hovy2015user}\nfocus on how to infer individual user attributes such as age, gender, political polarity, locations, occupation, educational information (e.g., major, year of matriculation) given user-generated contents or network information. \n\nTaking advantage of large scale user information, recent research has begun exploring logical reasoning \nover the social network (e.g., what's the probability that a New York City resident is a fan of the New York Knicks).\nSome work \\cite{li2014inferring,wang2013programming} relies on logic reasoning paradigms such as Markov Logic Networks (MLNs) \\cite{richardson2006markov} or Probabilistic\nSoft Logic (PSL) \\cite{kimmig2012short}, though there are still many challenges related to scalability.\nSocial network inference usually takes advantage of the fundamental \npropoety of homophily \\cite{mcpherson2001birds}, which states that people sharing\nsimilar attributes have a higher chance of becoming friends\\footnote{Summarized by the proverb ``birds of a feather\nflock together\" \\cite{al2012homophily}.}, and conversely\nfriends (or couples, or people living in the same location) tend to share more attributes. \nSuch properties have been harnessed for applications.\nSomewhat more distantly related to our approach are tasks like community detection \\cite{girvan2002community,yang2013community} and user-link prediction \\cite{perozzi2014deepwalk,tang2009relational,tang2009scalable,yang2013overlapping}. \n\\end{comment}\n\\section{Social Representation Learning}\nOur goal is to learn latent representations\nfrom the \n following three types of online information: (1) user-generated texts (2) friend networks (3) relations and attributes. \n\n\\begin{comment}\n\\subsection{Notation}\nWe first describe the notations used in this paper:\n\\begin{tightitemize1}\n\\item $W=\\{w\\}$: Word list.\n\\item $V=\\{v\\}$: User list.\n\\item $G$: Network graph; each vertex of $G$ represents a user $v\\in V$ in a social network. $g(v_i,v_j)=1$ if $v_i$ and $v_j$ are connected in the social graph and $g(v_i,v_j)=0$ otherwise. \n$G$ can also be rewritten as $G=\\{G_{v_i}\\}$, where \n$G_{v_i}=\\{v| g(v,v_i)=1\\}$ denotes a list of users with whom user $v_i$ is friends\n \n\\item $M=\\{m\\}$: Attribute (entity) list.\n\\item $R=\\{r\\}$ Relation list. A tuple $r(v,m)$ denotes that relation $r$ (e.g., {\\sc StudyAt}) holds between user $v$ and entity $m$ (e.g., Harvard University).\n\\end{tightitemize1}\n\n\nEach entity $m\\in M$, word $w\\in W$, user $v\\in V$ is associated with a K dimensional vector, i.e., $e_m,e_w,e_v \\in \\mathbb{R}^{K\\times 1}$. \nEach relation $r\\in R$ is associated with a matrix $e_r\\in \\mathbb{R}^{K\\times K}$.\nLet $\\Theta$ denote the parameter space of the model.\n \\footnote{Word embeddings $e_w$ are borrowed from external pre-trained embeddings and thus are not treated as parameters to learn.}\n\\begin{equation*}\n\\begin{aligned}\n&\\Theta=[\\Theta_v, \\Theta_m, \\Theta_r] \\\\\n&\\Theta_m=\\{e_m\\}~~~~\\Theta_v=\\{e_v\\}~~~~\\Theta_r=\\{e_r\\}\n\\end{aligned}\n\\end{equation*}\n Each user $v_i$ is associated with the following inventory list: $$I_i=\\{\\text{text}_i, G_i, R_i, M_i\\}$$\nwhere $\\text{text}_i$ denotes user-generated context from user $i$, which corresponds to a sequence of words: $\\text{text}_i=\\{w_1,w_2,...\\}$. \n$R_i=\\{r_{i,j}\\}$ and $M_i=\\{m_{i,j}\\}$ \ndenoting relation $r_{i,j}$ holds between user $v_i$ and entity $m_{i,j}$, i.e., $r_{i,j}(v_i,m_{i,j})=1$. \n\\subsection{Models}\nThe proposed neural model aims at learning latent representations for each user, entity and relation\nobserved from social evidence presented in $I_i$. The framework is optimized through maximizing the probability of predicting all elements within $I_i$ given user $v_i$:\n\\begin{equation*}\n\\begin{aligned}\nP(I_i|v_i:\\Theta)\n&=P(\\text{text}_i|v_i:\\Theta)\\times P(G_i|v_i:\\Theta)\\\\\n&\\times P(R_i,M_i|v_i:\\Theta)\\\\\n&=P(\\text{text}_i|v_i:\\Theta_v)\\times P(G_i|v_i:\\Theta_v)\\\\\n&\\times P(R_i,M_i|v_i:\\Theta_v,\\Theta_r,\\Theta_m)\n\\end{aligned}\n\\label{ref6}\n\\end{equation*}\nWe respectively describe each of the three parts within Equ. \\ref{ref6} in the following sections. \n\\end{comment}\n\n\\subsection{Modeling Text}\nUser generated texts reflect a user's interests, backgrounds, personalities, etc.\nWe thus propose learning user representations based on the text a user generates. \n\nWe represent each user $v$ by a $K$-dimensional vector $e_v$. \nSuppose that $S$ denotes a sequence of tokens $S=\\{w_1, w_2, ..., w_{N_S}\\}$ generated by the current user $v$.\nEach word $w\\in S$ is associated with a $K$-dimensional vector $e_w$.\nLet $C(w)$ denote the list of neighboring words for token $w$. \n$w$\nis generated based on \nnot only \n a general language model shared across all users (namely, a model \n that predicts $w$ given \n $C(w)$), but the \n representation $e_v$ of the\n current user :\n \\begin{equation}\n\\begin{aligned}\n&P(w|C(w), v)= p(w| e_C)\\\\\n&e_C=\\frac{1}{|C_w+1|}[\\sum_{w'\\in C(w)}e_{w'}+e_v]\n\\end{aligned}\n\\label{equ7}\n\\end{equation}\nFrom Eq.\\ref{equ7}, we are predicting the current word given the combination of its neighbors' embeddings and the current user embedding. This is akin to the CBOW model \\cite{mikolov2013efficient} with the only difference that the user embedding is added into contexts. \nSuch an idea also\nresembles \n the {\\it paragraph vector} model \\cite{le2014distributed} and the multimodal language model \\cite{kiros2014multimodal} .\n\nWe use negative sampling, in which we randomly generate negative words $w^*$. \nLet $L_w$ denote a binary variable indicating whether the current word $w$ is generated by the current user. The loss function using negative sampling is given by:\n\\begin{equation*}\n\\text{Loss(text)}=\\log p(L_w=1|v)+\\sum_{w^*}\\log p(L_w=0|v)\n\\end{equation*}\n\n\n\n\\begin{comment}\nLet $L_w$ denote a binary variable indicating whether the current word $w$ is generated by the current user. We have:\n\\begin{equation*}\n\\begin{aligned}\n&\\log p(L_w=1|v)=\\log\\frac{1}{1+\\exp(-g(e_w, e_v))} \\\\\n&\\log p(L_w=0|v)=\\log\\frac{1}{1+\\exp(g(e_w, e_v))}\n\\end{aligned}\n\\end{equation*}\n\\end{comment}\nWord prediction errors are backpropogated to user embeddings, \npushing the representations of users who generate similar texts to be similar. \n\\subsection{Modeling User Networks}\n\\label{graph_objective}\nBy the {\\it homophily} effect, individuals who are friends on social networks tend to share common characteristics. We therefore to encourage users who are friends have similar representations. \n\nWe propose using a strategy similar to skip-gram models, in which we consider users who are friends on social networks analogous to neighboring words, \nthe representations of which we wis to be similar.\nOn the other hand, we want embeddings of individuals who are not friends to be distant, just as words that do not co-appear in context.\nA similar idea of transforming social graph to vector space embeddings has been explored in the recent {\\it deepwalk} model \n\\cite{perozzi2014deepwalk} \nand {\\it Line} \\cite{tang2015line}.\n\nSuppose we have two users $v$ and $v'$. The probability that the two users are and are not friends are respectively given by:\n\\begin{equation}\n\\begin{aligned}\n&\\log p(L(v,v')=1)=\\log\\frac{1}{1+\\exp(-e_v\\cdot e_{v'})}\\\\\n&\\log p(L(v,v')=0)=\\log\\frac{1}{1+\\exp(e_v\\cdot e_{v'})}\n\\end{aligned}\n\\label{eq-graph}\n\\end{equation}\nFrom Eq \\ref{eq-graph}, we can see that the model favors the cases where the dot product of friends' embeddings is large, equivalent to \n their embeddings being similar.\nAgain, we use negative sampling for optimization. For two users $v$ and $v'$ who are friends, we sample $N$ random users $v*$, and we assume friendship does not hold between them. The objective function is therefore given by:\n\\begin{equation*}\n\\begin{aligned}\n&\\text{Loss(graph)} \\\\\n&=\\log p(L(v,v')=1)+\\sum_{v*}\\log p(L(v,v^{*})=0)\n\\end{aligned}\n\\end{equation*}\n\n\n\\subsubsection{Modeling Relations and Attributes}\nIntuitively, users who share similar attributes should also have similar representations and occupy similar position in the vector space. \nSuppose that a specific relation $r$ holds between a user $v$ and an entity $m$.\nWe represent each user and entity by a K-dimensional vector and a relation by a $K\\times K$ matrix. \nFor any tuple $(r, v, m)$, we map it to a scalar within the range of [0,1], indicating the likelihood of relation $r$ holding between user $v$ and entity $m$:\n\\begin{equation*}\n\\begin{aligned}\n&\\log p(L(r, v, m)=1)\\\\\n&~~~~~~~~~~~~~~=\\log\\frac{1}{1+\\exp(-e_{v_i}^T \\cdot R_i\\cdot e_m)}\\\\\n\\end{aligned}\n\\end{equation*}\n\\begin{comment}\n&\\log p(L(r, v, m)=0)\\\\\n&~~~~~~~~~~~~~~=\\log\\frac{1}{1+\\exp(e_{v_i}^T \\cdot R_i\\cdot e_m)}\\\\\n\\end{comment}\nSimilar scoring functions have been applied in a variety of work for relation extraction \\cite{socher2013reasoning,chang2014typed}. \nAgain we turn to negative sampling for optimization. \nThe system randomly samples none-referred-to entities and maximizes the difference between the observed relation tuples and \nrandomly sampled ones.\n\\begin{comment}\n\\begin{equation*}\n\\begin{aligned}\n&\\text{Loss(relation\/attribute)} \\\\\n&=\\log p(L(r,v,m)=1)+\\sum_{v*}\\log p(L(r,v^{*},m)=0)\n\\end{aligned}\n\\end{equation*}\n\\end{comment}\nThrough the model described above, users who share similar attributes will have similar representations. \nAt the same time, entities that are shared by similar users would also\nhave similar representations and thus\n occupy close positions in the vector space. \n\\subsection{Training}\nThe global learning objective is a linear combination of the objectives from the three categories described above. User embeddings are shared across these categories, and each part can communicate with the rest: a user who publishes content about a particular city (text modeling) can have similar embeddings to those who live in that city (relation\/attribution modeling); friends (graph modeling) of a basketball fan (relation\/attribution modeling) are more likely to be basketball fans as well. \nThe final objective function is given as follows:\n\\begin{equation*}\n\\begin{aligned}\nL(\\Theta)=\n&\\text{Loss(text)}+\\lambda_1\\text{Loss(graph)}\\\\\n&+\\lambda_2\\text{Loss(relation\/attribute)}\n\\end{aligned}\n\\end{equation*}\n\\begin{equation*}\n\\begin{aligned}\n&\\Theta=\\argmin_{\\Theta'}L(\\Theta')\n\\end{aligned}\n\\end{equation*}\nwhere $\\lambda_1$ and $\\lambda_2$ denote weights for different constituents. \nWe use \nStochastic gradient decent \\cite{zhang2004solving} to update the parameters.\n\\begin{equation*}\n\\Theta^t:=\\Theta^{t-1}-\\alpha\\frac{\\partial L}{\\partial\\Theta}\n\\label{gradient}\n\\end{equation*}\nThe system jointly learns user embeddings, word embeddings, entity embeddings and relation matrices. \n\\section{Inference on Social Networks}\nIn this section, we describe how to take as input\n a learned user embedding for different inference tasks on social media. \nWe divide inference tasks on social networks\ninto two major categories: inference for individual behaviors and group behaviors. \nThe former focuses on inferring attributes of a specific user such as whether a user likes a specific entity or whether \na specific relation holds between two users, \nwhile the latter focuses on inference over a group of users, e.g., what is the probability of a new yorker being a fan of Knicks.\n\n\\begin{comment}\n\\subsection{Inference for Individual Behaviors}\n Individual Inference is divided into two subcategories:\n{\\it user attribute inference} (e.g., gender, location, etc) where inference is performed for one specific user and {\\it user relation inference}\nwhere inference is performed based on pairwise user embeddings to decide whether a relation holds between two users. \n\nIn NLP, \\newcite{collobert2011natural} described a unified paradigm in which many of NLP inference tasks can be performed based on word representations. \nSimilar ideas can be adapted to inference tasks on social networks as will be described in detail below:\n\\subsubsection{Inferring Individual User Representations}\nWhen we encounter a new user at the test time, we first need to infer user representation given\n information observed for that user, which includes\n his friends, user-generated content, and relations\/entities. \n\nUser representation $e_{v}$ for the current user $v$ is given by:\n\\begin{equation*}\ne_{v_i} =\\argmax_e L(I_i^{\\text{Observed}}|e)\n\\label{eqin}\n\\end{equation*}\nEqu \\ref{eqin} can be intuitively explained as obtaining user embeddings by maximizing the likelihood of observed social signals. \nRepresentations for entities, relations and texts are kept fixed to what have been learned from training. \n$e_{v_i}$ is then optimized through stochastic gradient decent; negative examples are randomly sampled for each component of $L$. \n\\end{comment}\n\n\\subsection{ User Attribute Inference}\nGiven user representation $e_{v_i}$, we wish to infer the label\nfor a specific attribute\n of a specific user. The label can be whether a user likes an entity in a binary classification task, or the state that a user lives in a multi-class classification task. \n\nSuppose that we want to predict an attribute label (denoted by $t_{v} \\in [1,L]$) for\na user $v$.\nWe assume that information for this attribute is embedded in user representations \nand therefore, \nand \nbuild another neural model to expose this information.\nSpecifically, the model takes as input user embedding $e_v$ and outputs the attribute label using a softmax function as follows:\n\\begin{equation*}\n\\begin{aligned}\n&h= \\text{tanh}(W\\cdot e_{v}) \\\\\n&S_l=U_l\\cdot h \\\\\n&p(t_{v_i}=l)=\\frac{\\text{exp}(S_l)}{\\sum_{l'\\in [1,L]} \\text{exp}(S_{l'})}\n\\end{aligned}\n\\end{equation*}\nParameters to learn include $W$ and $U$.\nUser representations are kept fixed during training. The model is optimized \n using AdaGrad \\cite{zeiler2012adadelta}. \n\n\n\\subsection{User Relation Inference}\n{\\it User Relation Inference} specifies whether a particular relationship holds between two users (e.g., whether the two users are friends). \nIt takes \nembeddings for\n both users as inputs.\nGiven a user $v_i$ (associated with embedding $e_{v_i}$) and\na user $v_j$ (associated with embedding $e_{v_j}$), \nwe wish to predict the index of relationship label $t(v_i,v_j)\\in[1,L]$ that holds between the two users.\nA \nneural network prediction model is trained \nthat takes as input the embeddings of the two users.\nThe model considers\n the distances and angle between the two user embeddings. Similar strategies can be found in many existing works, e.g., \\cite{tai2015improved}.\n\nNon-linear composition is first applied to both user representations: \n\\begin{equation*}\n\\begin{aligned}\n&\\hat{h}_{v_1}=\\text{tanh}(W_a\\cdot e_{v_1})\\\\\n&\\hat{h}_{v_2}=\\text{tanh}(W_b\\cdot e_{v_2})\\\\\n\\end{aligned}\n\\end{equation*}\nNext the distance and angle between $\\hat{h}_{v_1}$ and $\\hat{h}_{v_2}$ are computed:\n\\begin{equation*}\n\\begin{aligned}\n&h_{+}=\\hat{h}_{v_1}\\otimes \\hat{h}_{v_2}\\\\\n&h_{\\times}=|\\hat{h}_{v_1}-\\hat{h}_{v_2}| \\\\\n\\end{aligned}\n\\end{equation*}\nThe multiplicative\nmeasure $h_{\\times}$ is the elementwise comparison of the signs of the input representations. \nFinally a softmax function is used to decide the label:\n\\begin{equation*}\n\\begin{aligned}\n&h=\\text{tanh}(W^{\\times}\\cdot h_{\\times}+W^{+}\\cdot h_{+}+W_1\\cdot \\hat{h}_{v_1}+W_2\\cdot \\hat{h}_{v_2})\\\\\n&~~~~p[t(v_i,v_j)=l]=\\text{softmax}(U\\cdot h)\n\\end{aligned}\n\\end{equation*}\nAgain, parameters involved \n are learned using stochastic gradient decent with AdaGrad \\cite{zeiler2012adadelta}. \n\n\\subsection{Inference of Group Behavior}\nWe now return to the example described in Section 1, in which we wish to estimate the probability of a male located in California (Cal for short) having an engineering occupation. \nGiven a list of users, their representations, and gold-standard labels, we first separately train the following neural classifiers:\n\\begin{itemize}\n\\item whether a user is a male, i.e., P(\\text{gender}($e_{v_i}$)=\\text{male}) \n\\item whether a user lives in Cal, i.e., P(\\text{LiveIn}($e_{v_i}$)=\\text{Cal}) \n\\item whether a user takes an engineering occupation,\\\\ i.e., P(\\text{Job}($e_{v_i}$)=\\text{engineering}) \n\\end{itemize}\nNext we estimate the general embedding (denoted by $e_G$) for the group of people that satisfy \nthese premises, namely, they \n are males and live in California at the same time. This can be transformed to the following optimization problem with $e_G$ being the parameters to learn: \\footnote{Note that we assume propositions are independent, so in the example above,\nbeing a male and living in California are independent from each other. We leave relaxing this independence assumption to future work.}\n \\begin{equation}\n \\begin{aligned}\n&e_G=\\argmax_{e} \\log P(\\text{gender}(e)=\\text{male})\\\\\n&~~~~~~~~~~~~~~~~~~~~~~~-\\log P(\\text{LiveIn}(e)=\\text{Cal})\n\\end{aligned}\n\\label{opt}\n\\end{equation}\nEq.\\ref{opt} can be thought of as an optimization problem to find a optimal value of $e_G$.\nThis problem can be solved\n using SGD. \nThe obtained optimal \n $e_G$ is used to represent\n that group embedding for users who satisfy the premises (e..g, males and living in Cal). \n$e_G$ is then used as inputs to classifier \n$P(\\text{Job} (e_G))=\\text{engineering})$ which returns probability of taking an engineering job. \n\nMore formally, given a list of conditions $\\{a_i\\}\\in A$, we want to the compute the probability \nthat another list of conditions (denoted by $B=\\{b_j\\}$) hold, in which $b_j$ can be a user being an engineer.\nThis probability is denoted by $p(B|A)$.\nThe algorithm\nto compute $p(B|A)$ \n for group behavior inference is summarized in Figure \\ref{fig:generative process}.\n\\begin{figure}[!ht]\n\\small\n\\rule{8cm}{0.03cm}\n\\begin{tightitemize}\n\\item For $a_i\\in A$, $b_j\\in B$, train separate classifiers $P(a_i|e), P(b_j|e)$ based on user representations $e$ and labeled datasets.\n\\item Estimate group representation $e_G$ by solving the following optimization problem using SGD:\n$$e_G=\\argmax_{e}\\prod_{a_i\\in A}P(a_i|e)$$\n\\item Infer the probability :\n$$P(B|A)=\\prod_{b_j\\in B}p(b_j|e(\\text{group}))$$\n\\end{tightitemize}\n\\rule{8cm}{0.03cm}\n\\caption{Algorithm for group behavior inference.}\n\\label{fig:generative process}\n\\end{figure}\n\n\\section{Dataset Construction}\nExisting commonly used social prediction datasets (e.g., BlogCatalog and Flickr \\cite{tang2009relational}, YouTube \\cite{tang2009scalable}) are designed with a specific task in mind: classifying whether social links exist between pairs of users.\nThey contain little text or user-attribute information, and are therefore not well suited to evaluate our proposed model. \n\nSocial networks such as Facebook or LinkedIn that support structured profiles would be ideal for the purpose of this work. Unfortunately, they are publicly inaccessible. \nWe advert to Twitter. \nOne downside of relying on Twitter data is that \ngold-standard information is not immediately available.\nEvaluation presented in this paper therefore comes with the flaw that \n it relies on downstream information extraction algorithms or rule-based heuristics for the attainment of ``gold standards\". \nThough not perfect as ``gold-standards\" extraction algorithm can be errorful, such a type of evaluation comes with the advantage that it can be done automatically to compare lots of different systems for development or tuning in relatively large scale.\nMeanwhile\nthe way that our dataset is constructed \n gives important insights about how the proposed framework can be applied when some structured data is missing\\footnote{Facebook and LinkedIn do come with the ideal property of supporting structured user profiles but hardly anyone fills these out.} and how we can address these challenges by directly from unstructured text, making this system applicable to a much wider scenario.\n\n\n\n\n\n\\subsection{User Generated Texts and Graph Networks}\nWe randomly sample a set of Twitter users, discarding non-English tweets and users with less than 100 tweets.\nFor each user, we crawl their published tweets and following \/ follower network\nusing the publicly available Twitter API.\nThis results in a dataset of 75 million tweets.\n\\subsection{User Attributes}\nUnlike social networking websites such as Facebook, Google+ and LinkedIn, Twitter does not support structured user profile attributes such as \ngender, education and employer.\nWe now briefly describe how we enrich our dataset with\nuser attributes (location, education, gender) and user relations (friend, spouse).\nNote that, the goal of this section is to construct a relatively comprehensive dataset for the propose of model evaluation\nrather than developing user attribute extraction algorithms.\n\n\\subsubsection{Location}\\label{sec:location}\nWe first associate one of the 50 US states with each user.\nIn this paper, we employ a rule-based approach for user-location identification.\\footnote{\nWhile there has been a significant work on geolocation inference,\n(e.g., \\cite{cheng2010you,conover2013geospatial,davis2011inferring,onnela2011geographic,sadilek2012finding}), the primary goals of this work are to develop\nuser representations based on heterogenous social signals. We therefore take a simple high-precision, low-recall approach to identifying user locations.}.\nWe select all geo-tagged tweets from a specific user, \nand say an location $e$ corresponds to the location of the current user $i$ if it satisfies\nthe following criteria, designed to ensure high-precision:\n(1) user $i$ published more than 10 tweets from location $e$ .\n(2) user $i$ published from location $e$ in at least three different months of a year.\nWe only consider locations within the United States and entities are matched to state names\nvia Google Geocoding. \n In the end, we are\nable to extract locations for 1.1$\\%$ of the users from our dataset.\n\\subsubsection{Education\/Job}\nWe combine two strategies to harvest gold-standard labels for users' occupation and educational information.\nWe use The Standard Occupational\nClassification (SOC)\\footnote{\\url{http:\/\/www.ons.gov.uk\/ons\/\nguide-method\/classifications\/\ncurrent-standard-classifications\/\nsoc2010\/index.html}} to obtain a list of occupations, a approach similar to \\newcite{preoctiuc2015studying,preotiucanalysis}.\n\\footnote{ SOC is\na UK government system developed by the Office of National Statistics that groups jobs into\n23 major categories (for example: or Engineering Occupations or Legal Occupations), each of which is associated with a set of specific job titles (e.g., mechanical engineer and pediatrist for Professional Occupations).\nWe construct a lookup table from \n job occupations to SOC and apply a rule-based mapping strategy to \nretrieve a users' occupation information based on the free-text user description\nfield from their Twitter profile. Note that this approach introduces some bias: users with high-profile occupations are more likely to self-disclose their occupations than users with less prestigious occupations.}\nA user is assigned an occupation if one of the keywords from the lookup table is identified in his\/her profile.\n$12\\%$ percent of users' occupations are identified using this strategy. \n\\begin{comment}\nThe second method we adopt is based on Google plus. \nFirst, for each user, we obtained their full name and fed it \ninto the Google+ API\\footnote{\\url{https:\/\/developers.google.com\/+\/api\/}}.\nMany Google+ profiles are publicly accessible and \nmany users explicitly list attributes such as their education and employer. \n\nA major challenge in linking Google+ profile information information to our Twitter-based dataset is name \ndisambiguation.\\footnote{A small proportion of Google+ accounts contain direct links to users' Twitter accounts. In these cases, accounts can be directly matched.}\nTo solve this user matching problem across social networking sites, we adopt a \\emph{shared friend} strategy. \nIf more than 10 percent and at least 20 friends are shared by Google+ circles and Twitter followers, we assume the two accounts refer to the same real-world entity.\nUsing this approach $4.8$ percent of users' job or education attributes are identified based on their Google+ accounts.\n\\end{comment}\n\\subsubsection{Gender}\nFollowing a similar strategy as was described for location and occupation, we take implement a simple high-precision approach for obtaining gold-standard \nuser gender information. We leverage the national Social Security Gender Database\n(SSGD)\\footnote{\\url{http:\/\/www.ssa.gov\/oact\/babynames\/names.zip}} to identify users' gender based on their first names.\nSSGD contains first-name records annotated for gender for every US birth since\n1880 A.D\\footnote{Again, we note the large amount of related work on predicting gender of social media users (e.g.,\n\\cite{burger2011discriminating,ciot2013gender,pennacchiotti2011machine,tang2011s}.)\nstudying whether high level tweet features (e.g., link, mention, hashtag\nfrequency) can help in the absence of highly-predictive user name information. As mentioned before, we do not adopted \nmachine learning algorithms for attribute extraction.\n}. \nUsing this database we assign gender to $78\\%$ of users in our dataset.\n\\begin{comment}\n\\subsubsection{User Preferences: Likes and Dislikes}\nIn addition to location, occupation and gender we also gather information on user-preferences.\nThis requires sophisticated information extraction algorithms. \nFor this purpose, we developed an approach that combines {\\em semi-supervised information harvesting} techniques \\cite{davidov2007fully,kozareva2010learning,kozareva2010not,limajor} and \nthe concept of {\\em distant supervision} \\cite{craven1999constructing,go2009twitter,mintz2009distant}.\nAs this subproblem is not the major focus of this paper, we describe relevant details in Appendix A.\nThe approach produces a list of entities that a user likes or dislikes. \n\\end{comment}\n\n\n\n\\section{Experiments}\nWe now turn to experiments on using global inference \nto augment individual local detectors to infer user's attributes,\nrelations and preferences.\nAll experiments are based on datasets described in the previous\nsections. \nWe performed 3 iterations of stochastic gradient descent training over the collected dataset to learn embeddings.\nFor each task, we separate the dataset into 80\\% for training 10\\% development and 10\\% for testing.\n\nFor comparison purposes, neutral models that take into account only part of the training signals presented naturally constitute baselines.\nWe also implement feature-based SVM models as baselines for the purpose of demonstrating strength of neural models. \nFor neural models, we set the latent dimensionality $K$ to $500$.\nPre-trained word vectors are used based on the word2vec package.\\footnote{\\url{https:\/\/code.google.com\/p\/word2vec\/}} Embeddings are trained on a Twitter dataset consisting of roughly 1 billion tokens.\n\n\n\\subsection{Friend-Relation (Graph Link) Prediction}\nTwitter supports two types of following patterns, \\textsc{following} and \\textsc{followed}. \nWe consider two users as friends if they both follow each other.\nThe friendship relation is extracted straightforwardly from the Twitter network. \nModels and baselines we employ include:\n\\begin{tightitemize}\n\\item {\\it All}: The proposed model that considers text, graph structure and user attributes.\n\\item {\\it Only Network}: A simplified version\nof the proposed model \n that only used the social graph structure to learn user representations a\n Note that by making this simplification, the model is similar to DeepWalk \\cite{perozzi2014deepwalk} with the exception that we adopt negative sampling rather than hierarchical softmax.\n\\item {\\it Network+Attribute}: Neural models that consider social graph and relation\/entity information. \n\\item {\\it Network+text}: Neural models that consider social graph and text information. \n\\end{tightitemize}\n\nPerformance for each model is shown in Table \\ref{friend}. \nAs can be seen, taking into account \n different types of social signals yields progressive performance improvement:\n {\\it Graph+Attribute} performs better than {\\it only graph}, and {\\it All}, which consider all different types of social signals is better than {\\it Graph+Attribute} .\n\\begin{table}\n\\centering\n\\small\n\\begin{tabular}{cc}\nModel&Accuracy\\\\\\hline\nAll&0.257\\\\\nOnly Network&0.179\\\\\nNetwork+Attribute&0.198\\\\\nNetwork+Text&0.231\\\\\n\\end{tabular}\n\\caption{Accuracy for different models on friend relationship prediction from social representations.}\n\\label{friend}\n\\end{table}\n\n\\subsection{User Attributes: Job Occupation}\nWe present experimental results for job classification based on user-level representations. \nEvaluation is performed on the subset of users whose job labels are identified by the rule-based approach described in the previous section. \nOur models are trained to classify the top-frequent 10 categories of job occupations\n\nAgain, {\\it all} denotes the model that utilizes all types of information. \nBaselines we consider include:\n\\begin{tightitemize}\n\\item {\\it Text-SVM}: We use SVM-Light package to train a unigram classifier that only considers text-level information. \n\\item {\\it Only Network}: A simplified version of the proposed model that trains user embedding based on network graph and occupation information. \n\\item {\\it Network+Text}: Embeddings are trained from user-generated texts and network information. \n\\end{tightitemize}\n\n\n\nExperimental results are illustrated in Table \\ref{job}. \nAs can be seen, user generated content offers informative evidence about job occupation. We also observe that considering network information yields significant performance improvement \n due to the homophily effect, which has been spotted in earlier work \\cite{li2014weakly}. \nAgain, the best performing model is the one that considers all sorts of evidence. \n\n\\begin{table}\n\\centering\n\\small\n\\begin{tabular}{cc}\nModel&Accuracy\\\\\\hline\nAll&0.402\\\\\nOnly Network&0.259\\\\\nSVM-text&0.330\\\\\nNetwork+Text&0.389\\\\\n\\end{tabular}\n\\caption{Accuracy for different models on 9-class job occupation prediction from social representations.}\n\\label{job}\n\\end{table}\n\n\\subsection{User Attribute: Gender}\n\\begin{table}\n\\centering\n\\small\n\\begin{tabular}{cc}\nModel&Accuracy\\\\\\hline\nAll&0.840\\\\\nOnly Network&0.575\\\\\nOnly Text&0.804\\\\\nSVM-text&0.785\\\\\nAttribute+Text&0.828\\\\\n\\end{tabular}\n\\caption{Accuracy for different models on 9-class job occupation prediction from social representations.}\n\\label{gender}\n\\end{table}\n\nWe evaluate gender based on a dataset of 10,000 users (half male, half female).\nThe subset is \n drawn from the users whose \ngold standard gender labels are assigned by the social-security system described in the previous section.\nBaselines we employ include: {\\it SVM-Text}, in which a SVM binary classification model is trained on unigram features; {\\it Only-Text}, in which user representations are learned only from texts; {\\it Only-Network}, in which user representations are only learned from social graphs; and {\\it Text+Relation}, in which representations are learned from text evidence and relation\/entity information. \n\n\nThe proposed neural model achieves an accuracy value of 0.840. \nwhich is very close to the best performance that we are aware of described in \\newcite{ciot2013gender}, which achieves \nan accuracy of 0.85-0.86 on a different dataset. However, unlike in \\newcite{ciot2013gender}, the proposed model does not require massive efforts in feature engineering, \nwhich involves \n collecting a wide variety of manual features such as entities mentioned, links, wide range of writing style features, psycho-lingsuitic features, etc. \n This demonstrates \n the flexibility and scalability \n of deep learning models\n to utilize and integrate different types of social signals \n on inference tasks over social networks, \n \n \nUser-generated contexts offer significant evidence for gender. \nAgain, we observe that leveraging all sorts of social evidence leads to the best performance. \n\nExperimental results are shown in Figure \\ref{gender}. As can be seen, network information does significantly help the task of gender identification, only achieving slightly better performance than random guess. Such an argument is reinforced by the fact that {\\it Text+Relation} yield almost the same performance as model {\\it all}, which takes Text+ Relation+ network information. \n\n\\begin{table}\n\\centering\n\\small\n\\begin{tabular}{cc}\nModel&Accuracy\\\\\\hline\nAll&0.152\\\\\nOnly Network&0.118\\\\\nOnly Text&0.074\\\\\nNetwork+Text&0.120.\\\\\nAttribute+Text&0.089\\\\\n\\end{tabular}\n\\caption{Accuracy for different models location prediction from social representations.}\n\\label{location}\n\\end{table}\n\n\\subsection{User Attribute: Location}\nThe last task we consider is \n location identification.\n Experiments are conducted \n on users whose locations have been identified using the rule-based approach described in the previous section. \nThe task can be thought of as a 50-class classification problem\nand the goal is to pick \n one from the 50 states (with random-guess accuracy being 0.02). \nWe employ baselines similar to earlier sections: {\\it only-text}, \n{\\it only network},\n{\\it text+attribute} and {\\it text+network}. \n\nResults are presented in Table \\ref{gender}: both text and network evidence provide\ninformative evident about where a user lives, leading to better performances. \nAgain, the best performance is obtained when all types of social signals are jointly considered \n\n\n\\subsection{Examples for Group Behavior Inference}\nGiven the trained classifiers (and additionally trained \\textsc{like-dislike} classifiers with details shown in the Appendix), we are able to perform group behavior inference. \nDue to the lack gold standard labeled dataset, we did not perform evaluations, but rather list a couple of examples to give readers a general sense of the proposed paradigm:\n\\begin{itemize}\n\\item P(isMale$\\Rightarrow$isEngineer)=3.5$\\times$ P(isFemale$\\Rightarrow$isEngineer)\n\\item P(isMale,LiveInCalifornia$\\Rightarrow$isEngineer)=\\\\\n.~~~~~~~~~~~6.8$\\times$$\\cdot$ P(isFemale,LiveInCalifornia$\\Rightarrow$isEngineer)\n\\item P(LiveInColorado$\\Rightarrow$LikeOmelet)=\\\\\n.~~~~~~~~~~~1.4$\\times$P(LiveInCalifornia$\\Rightarrow$LikeOmelet)\n\\item P(LiveInTexas$\\Rightarrow$LikeBarbecue)=\\\\\n.~~~~~~~~~~~1.7$\\times$P(LiveInCalifornia$\\Rightarrow$LikeBarbecue)\n\\end{itemize}\n\\begin{comment}\n\\subsection{Parameter Sensitivity}\nSignificant performance boost is observed by using representations with higher dimensionality. In order to examine how sensitive are performances to dimensionality, we again train models on the previous four tasks using different values of vector dimensionality, i.e., 25, 50, 100, 250, 500 and report performances on development datasets. \n\\begin{figure}\n\\centering\n\\includegraphics[width=2.4in]{7.png}\n\\caption{Performances over dimensionality K on different tasks.}\\label{dimension}\n\\end{figure}\n\nFigure \\ref{dimension} illustrates the effects of increasing the number of latent dimensions in the proposed model for different tasks. As can be seen, \nmodel performance is sensitive to the dimensionality of vector representations:\nlarger dimensionality lead to significantly better performances, yielding around $40\\%$ performance boost when increasing from 25 to 500. \n\\end{comment}\n\n\n\\section{Related Work}\n\\label{related_work}\nMuch work has been devoted to automatic user attribute inference given social signals.\nFor example, \\cite{rao2010classifying,ciot2013gender,conover2011political,sadilek2012finding,hovy2015user}\nfocus on how to infer individual user attributes such as age, gender, political polarity, locations, occupation, educational information (e.g., major, year of matriculation) given user-generated contents or network information. \n\nTaking advantage of large scale user information, recent research has begun exploring logical reasoning \nover the social network (e.g., what's the probability that a New York City resident is a fan of the New York Knicks).\nSome work \\cite{li2014inferring,wang2013programming} relies on logic reasoning paradigms such as Markov Logic Networks (MLNs) \\cite{richardson2006markov}.\n\nSocial network inference usually takes advantage of the fundamental \npropoety of homophily \\cite{mcpherson2001birds}, which states that people sharing\nsimilar attributes have a higher chance of becoming friends\\footnote{Summarized by the proverb ``birds of a feather\nflock together\" \\cite{al2012homophily}.}, and conversely\nfriends (or couples, or people living in the same location) tend to share more attributes. \nSuch properties have been harnessed for applications like community detection \\cite{yang2013community} and user-link prediction \\cite{perozzi2014deepwalk,tang2009relational}. \n\n\nThe proposed framework also focuses on attribute inference, which can be reframed as relation identification \nproblems, i.e., whether a relation holds between a user and an entity. This work is thus related to \na great deal of recent researches on\n relation inference (e.g.,\n\\cite{gu2015traversing,wang2014knowledge,riedel2013relation}). \n \\begin{comment}\n\\cite{riedel2013relation,rocktaschel2015injecting,speer2008analogyspace,socher2013reasoning} treat the problem\n as a matrix completion problem and adopt matrix factorization techniques \nto learn embeddings for entities and relations which are used to fill in missing entries in the matrixes. \nDeep learning methods are incorporated into relation learning models \\cite{socher2013reasoning,chang2014typed} to learn deep representations for relations and entities.\n\\end{comment}\n\\begin{comment}\n \\subsection{Data Harvesting}\nTechniques adopted \nfor data collection \n are related to a strand of work in data harvesting and semantic bootstrapping \\cite{agichtein2000snowball}.\nThis line of work aims to leverage seeds to harvest structured data from text, which is used to learn additional rules or patterns to harvest more data and so on \\cite{davidov2007fully,igo2009corpus,kozareva2010learning,kozareva2010not,riloff1999learning}. \nDistant supervision is a related methodology for data harvesting \\cite{craven1999constructing,hoffmann2011knowledge,mintz2009distant}\nthat leverages large structured databases such as Freebase as an indirect source of supervision for learning to extract information from text.\n\n\\subsection{Multi-Task Learning}\nOur approach is also related to work on Multi-task learning \\cite{thrun1996learning,zhou2011malsar,collobert2008unified}.\nWe learn latent representations that are jointly optimized for a diverse set of predictive tasks (in our case: user\nrelations, text language models and their social relations).\nBy jointly learning from a diverse set of social signals, we are able to demonstrate consistent performance improvements across a range of predictive tasks.\n\\end{comment}\n\nOur work is inspired by classic work on spectral \nlearning for graphs \ne.g., \\cite{kunegis2009learning,estrada2001generalization} and on\nrecent research \\cite{perozzi2014deepwalk,tang2015line}\nthat\n learn embedded representations for a graph's vertices. Our model extends this work\nby modeling not only user-user network graphs, but also incorporating \ndiverse social signals including unstructured text, user attributes, and relations, \nenabling more sophisticated inferences and offering an integrated model of homophily in social relations.\n\n\\section{Conclusions}\nWe have presented a deep learning framework for\nlearning social representations, inferring the latent attributes of people online.\nOur model offers a new way to jointly integrate noisy heterogeneous cues from\npeople's text, social relations, or attributes into a single latent representation.\nThe representation supports an inference algorithm that can\nsolve social media inference tasks related to both individual and group behavior,\nand can scale to the large datasets necessary to provide practical \nsolutions to inferring huge numbers of latent facts about people.\n\nOur model has the ability to incorporate various kinds of information, and it\nincreases in performance as more sources of evidence are added. \nWe demonstrate\n benefits on a range of social media inference tasks, including predicting user gender,\noccupation, location and friendship relations.\n\n\nOur user embeddings naturally capture\nthe notion of homophily---users who are friends, have similar attributes, \nor write similar text are represented by similar vectors.\nThese representations could benefit a wide range of downstream applications, such as\nfriend recommendation, targeted online advertising, and \nfurther applications in the computational social sciences. \nDue to limited \npublicly \naccessible datasets, we only conduct our experiments on Twitter. However, our algorithms hold potentials to yield more benefits by combining different attributes from online social media, such as Facebook, Twitter, LinkedIn, Flickr\\footnote{Images can be similarly represented as vector representations obtained from CovNet \\cite{krizhevsky2012imagenet}, which can be immediately incorporated into the proposed framework.},\n\n\n\\begin{comment}\n\\section{Acknowledgement}\nThe authors \nespecially want to thank David Jurgens for insightful comments and suggestions. \nThe authors also want to thank Will Hamilton, Ignacio Cases Martin, Percy Liang and other members of the\nStanford NLP group for helpful comments. \nJiwei Li is supported by Facebook Fellowship, which we gratefully acknowledge. \n\\end{comment}\n\\bibliographystyle{acl_natbib}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzlumy b/data_all_eng_slimpj/shuffled/split2/finalzzlumy new file mode 100644 index 0000000000000000000000000000000000000000..53511fcde6c917b93bee3c4032bcd79c31be48de --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzlumy @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nThe tensor model was first introduced in \\cite{Ambjorn:1990ge,Sasakura:1990fs,Godfrey:1990dt} \nas an analytical description of simplicial quantum gravity in dimensions higher than \ntwo\\footnote{See, however, \\cite{Fukuma:2015xja,Fukuma:2015haa,Fukuma:2016zea} \nfor a matrix-model-like approach to three-dimensional quantum gravity.} \nby generalizing the matrix model, which successfully describes the two dimensional case.\nWhile the original tensor models are still remaining merely as formal descriptions due to some difficulties, \nthe analyses of the more successful model, the colored tensor model \n\\cite{Gurau:2009tw}, have produced various interesting analytical results concerning the simplicial quantum gravity\nin dimensions higher than two \\cite{Gurau:2011xp}. Among them, it has been shown that the dominant contributions \nof simplicial complexes generated \nfrom the colored tensor model are branched polymers \\cite{Bonzom:2011zz,Gurau:2013cbh}. \nSince the structure of branched polymers is far from the classical spacetime picture of our universe, \nit seems difficult to consider the tensor model as a sensible model of quantum gravity, which\nshould produce wide and smooth spacetimes in certain classical regimes. \n\nOn the other hand, while the models above basically concern the Euclidean case, \nit has been shown that Causal Dynamical Triangulation (CDT),\nwhich is the simplicial quantum gravity with a causal structure, \nsuccessfully produces the 3+1 dimensional world similar to our universe \\cite{Ambjorn:2004qm},\nwhile Dynamical Triangulation, which is the Euclidean version, does not\\footnote{\nWhen coupling many U$(1)$-fields, \nthe authors in \\cite{Horata:2000eg} found a promise of \na phase transition higher than first order, \nwhich, however, is in conflict with the result in \\cite{Ambjorn:1999ix}.\n}.\nThe comparison between the two versions suggests that a causal structure is essentially \nimportant for the emergence of a classical spacetime in quantum gravity. \nThis motivated one of the present authors to formulate\na rank-three tensor model as a totally constrained system in the canonical formalism,\nwhich we call Canonical Tensor Model (CTM) \\cite{Sasakura:2011sq,Sasakura:2012fb}.\nThe constraints of CTM are composed of kinematical symmetry generators and \nthose analogous to the Hamiltonian constraint in the Arnowitt-Deser-Misner (ADM) formalism \n\\cite{Arnowitt:1960es,Arnowitt:1962hi}, \nand form a first-class constraint algebra with a non-linear structure. \nIn fact, the algebraic structure of the constraints is very similar to that of \nthe ADM formalism of general relativity (GR), and \nit can be shown \\cite{Sasakura:2015pxa} that, in a formal continuum limit, the constraint algebra of CTM\nagrees with that of the ADM formalism of GR\\footnote{As well, a certain minisuperspace model of GR can be derived from\nCTM \\cite{Sasakura:2014gia}.}.\nThis is of physical importance, since the algebraic closure of the ADM constraints assures\nthe spacetime covariance of locally defined time evolutions, which is an essence of GR \\cite{Hojman:1976vp}. \n\n\nThe main purpose of this paper is to pursue this correspondence further.\nWe will analyze the classical equation of motion (EOM) of CTM in a formal continuum limit \nthrough a derivative expansion of the tensor of CTM up to the fourth order, \nand will show that it is the same as that of a coupled system of gravity and a scalar field derived from the Hamilton-Jacobi \nequation with an appropriate choice of an action. \nThe action has an exponential potential of the scalar field, and the system is \nclassically invariant under a dilatational symmetry. \nInterestingly, the action is meaningful only \nin spatial dimensions $2\\leq d \\leq 6$, and the system becomes unstable\nin $d>6$ due to the wrong sign of the scalar kinetic term. In the critical dimension $d=6$, \nde Sitter spacetime becomes a solution to the EOM, signaling the emergence of a conformal symmetry.\n\nThe present work may also have some implications to renormalization-group (RG) flow equations of field theories. \nIt has been argued \\cite{Sasakura:2015xxa,Sasakura:2014zwa,Sasakura:2014yoa} \nthat the Hamiltonian constraints of CTM generate the RG flows of statistical systems on random networks \\cite{revnetwork},\nwhich can equivalently be described by randomly connected tensor networks.\nIn addition, it has been shown \\cite{Chen:2016xjx} that classical spaces emerge on boundaries of randomly \nconnected tensor networks by appropriately choosing the tensors. Therefore, it can be expected that \nthe Hamiltonian constraints would generate RG flows of effective field theories on such emergent spaces.\nIf so, the present work would give a hint to the connection between RG flows of field theories and gravity, \nwhich is indeed the subject of the so-called holographic RG (See \\cite{Fukuma:2002sb} for a review.).\n\nWe heavily used a Mathematica package ``xTensor\" \\cite{xtensor} to perform tensorial computations in this paper.\nThe mathematica programs we used can be downloaded from one of the author's \nhomepage \\cite{sasahome}.\n\nThis paper is organized as follows. \nIn Section~\\ref{sec:review}, we review CTM. \nIn Section~\\ref{sec:repP}, we define the fields of CTM in a formal continuum limit in terms of a\nderivative expansion of the tensor of CTM up to the fourth order. There are four fields, a rank-0,2,3,4 tensor\nfield with the weight of negative half-density.\nIn Section~\\ref{sec:gauge}, we study the kinematical symmetry of CTM in the continuum limit. Up to the fourth order, \nwe find two gauge symmetries, the diffeomorphism and a spin-three symmetry.\nIn Section~\\ref{sec:fixed}, by deleting the rank-3 and rank-4 fields by the spin-three gauge symmetry and the EOM, respectively,\nwe write down the EOM of the remaining fields, the rank-0 and rank-2 fields,\nin a static background geometry. \nIn Section~\\ref{sec:backmetric}, we discuss another gauge symmetry which allows us to freely transform the background metric. \nThen, in Section~\\ref{sec:identify}, the background metric is gauge-fixed to a combination of the fields\nso as to remove the odd situation that there exists a static spin-two field, the background metric, other than\nthe rank-2 field of CTM. The EOM with the gauge-fixing condition is written down. \nIn Section~\\ref{sec:delete}, we rewrite the EOM after deleting the weights of the fields. \nIn Section~\\ref{sec:reparametrization}, we \nperform a reparameterization of the fields so that there are no spatial derivative terms of the lapse function in EOM. \nThis is the final form of the EOM of CTM, which is comparable with that of a gravitational system in field theory. \nIn Section~\\ref{sec:conttheory}, we show that the EOM of CTM can be made coincident with\nthat of a coupled system of gravity and a scalar field \nderived from the Hamilton-Jacobi equation by an appropriate choice of an action.\nWe find a critical spatial dimension of the gravitational system, given by six, \nover which the system becomes unstable due to the wrong sign of the kinetic term of the scalar field.\nIn Section~\\ref{sec:mini}, we discuss the time evolution of the scale factor. At the critical dimension, \nde Sitter spacetime is a solution to the EOM, signaling the emergence of a conformal symmetry, \nwhile the time evolution of the scale factor has a power-law behavior below the critical dimension.\nSection~\\ref{sec:summary} is devoted to the summary and future prospects. \n \n\\section{Review of CTM}\n\\label{sec:review}\nIn this section we review the canonical tensor model (CTM) \n\\cite{Sasakura:2011sq, Sasakura:2012fb}, \nexplaining its current status. \n\nWe consider a Hamiltonian system such that \nthe dynamical variables are the real symmetric rank-three tensors, \n$M_{abc}$ and $P_{abc}$ $(a,b,c=1,2,\\cdots, \\mathcal{N})$, \nwhich are canonically conjugate \nin the sense that they satisfy the following Poisson bracket:\n\\[\n\\{ \nM_{abc}, P_{def}\n\\}\n= \n\\sum_{\\sigma}\n\\delta_{a \\sigma_d}\\delta_{b \\sigma_e}\\delta_{c \\sigma_f}, \n\\ \\ \\ \n\\{ \nM_{abc}, M_{def}\n\\}\n= \n\\{ \nP_{abc}, P_{def}\n\\}\n=0, \n\\label{eq:poissopn}\n\\] \nwhere the summation is over all the permutations of $d$, $e$ and $f$, \nreflecting the real symmetric nature of the tensors. \nHere, it would be natural to introduce the O($\\mathcal{N}$) transformation\nas a kinematical symmetry of the system, \n\\[\n\\begin{split}\n&M_{abc} \\to M'_{abc} = L_{aa'}L_{bb'}L_{cc'}M_{a'b'c'}, \\\\\n&P_{abc} \\to P'_{abc} = L_{aa'}L_{bb'}L_{cc'}P_{a'b'c'},\n\\end{split}\n\\label{eq:ontransformation}\n\\]\nwhere the repeated indices are summed over and $L$ is an O($\\mathcal{N}$) matrix, \nsince quantities constructed by the tensors \nwith all indices being contracted are invariant under the O$(\\mathcal{N})$ transformation. \nThe Hamiltonian of CTM is given as follows:\n\\[\nH_{CTM}\n= n_a \\mathcal{H}_a + n_{ab} \\mathcal{J}_{ab},\n\\label{eq:ctmhamiltonian}\n\\] \nwhere $n_a$ and $n_{ab} (=-n_{ba})$ are non-dynamical Lagrange's multipliers, \nand\n\\[\n&\\mathcal{H}_a \n=\\frac{1}{2} \\left(P_{abc}P_{bde}M_{cde} - \\lambda M_{abb} \\right), \n\\label{eq:ctmhamiltonianconstraint}\n\\\\\n&\\mathcal{J}_{ab}\n=-\\mathcal{J}_{ba}\n= \\frac{1}{4} \\left( P_{acd}M_{bcd} - P_{bcd}M_{acd} \\right), \n\\label{eq:ctmmomentumconstraint} \n\\]\nin which $\\lambda$ is a cosntant. \nImitating the nomenclatures in the Arnowitt-Deser-Misner (ADM) formalism of general relativity, \n$\\mathcal{H}_a$ and $\\mathcal{J}_{ab}$ are dubbed as \nHamiltonian constraint and momentum constraint, respectively, \nand they form the following first-class constraint Poisson algebra: \n\\[\n\\begin{split}\n& \\{ \\mathcal{H} (\\xi^1), \\mathcal{H} (\\xi^2) \\} \n= \\mathcal{J} ( [ \\tilde{\\xi}^1, \\tilde{\\xi}^2 ] + 2 \\lambda\\, \\xi^1 \\wedge \\xi^2 ), \\\\\n& \\{ \\mathcal{J} (\\eta), \\mathcal{H} (\\xi) \\} \n= \\mathcal{H} (\\eta \\xi), \\\\\n&\\{ \\mathcal{J} (\\eta^1), \\mathcal{J} (\\eta^2) \\} \n= \\mathcal{J} ([ \\eta^1, \\eta^2 ]), \n\\label{eq:ctmconstraintalgebra}\n\\end{split}\n\\]\nwhere $\\mathcal{H}(\\xi) := \\xi_a \\mathcal{H}_a$, $\\mathcal{J}(\\eta) := \\eta_{ab}\\mathcal{J}_{ab}$, \nand $\\tilde{\\xi}_{ab}:= P_{abc}\\xi_c$. \nIn (\\ref{eq:ctmconstraintalgebra}), the bracket $[\\ , \\ ]$ denotes the matrix commutator, \nand $(\\xi^1\\wedge \\xi^2)_{ab}:= \\xi^1_a \\xi^2_b - \\xi^2_a \\xi^1_b$. \nOne notices that $\\mathcal{J}$ serves as the generators of SO($\\mathcal{N}$),\ninfinitesimally representing the kinematical symmetry of the system.\nThe form of the Hamiltonian constraint has been uniquely fixed \nby the following five assumptions: \nthe Hamiltonian constraint \n(I) carries only one index, \n(II) forms a closed Poisson algebra with $\\mathcal{J}$, \n(III) preserves the time reversal symmetry, $M_{abc} \\to M_{abc}$ and $P_{abc}\\to - P_{abc}$, \n(IV) consists of terms cubic at most, \nand \n(V) allows only ``connected terms,'' \ne.g., \n$P_{abc}P_{bde}M_{cde}$ is allowed but $M_{abb}P_{cde}P_{cde}$ is not allowed \\cite{Sasakura:2012fb}.\nWith the closed Poisson algebra (\\ref{eq:ctmconstraintalgebra}) of the constraints, CTM is a totally constrained system \ngoverned by the Hamiltonian (\\ref{eq:ctmhamiltonian}). \nThe interest of this paper is the classical equation of motion (EOM) of $P$ with $\\lambda=0$, which \nis given by\n\\[\n\\begin{split}\n\\frac{\\text{d}}{\\text{d}t} P_{abc} \n= \\{P_{abc}, H_{CTM}^{\\lambda=0} \\} \n= - \\frac{1}{2} \n\\sum_{\\sigma}\n\\left(\nn_d P_{de \\sigma_{a}} P_{\\sigma_{b}\\sigma_{c}e} \n+ n_{d \\sigma_{a}} P_{\\sigma_{b}\\sigma_{c}d}\n\\right). \n\\label{eq:ctmeom} \n\\end{split}\n\\]\nThe variable $M_{abc}$ will play no roles in this paper.\n\nQuite remarkably, CTM is closely related to general relativity in arbitrary dimensions \nin the following sense. \nFirstly, for $\\mathcal{N}=1$ case, the Hamiltonian (\\ref{eq:ctmhamiltonian}) agrees with that of a certain minisuperspace \nmodel of GR in arbitrary dimensions, \nif we consider \nthe modulus of the tensor, $|M_{111}|$, is proportional to the spatial volume in the minisuperspace model \\cite{Sasakura:2014gia}. \nSecondly, in a formal continuum limit with $\\mathcal{N}\\to \\infty$, \nthe Poisson algebra (\\ref{eq:ctmconstraintalgebra}) coincides with the Dirac algebra in the ADM formalism \\cite{Sasakura:2015pxa}. \nIn this paper we take this argument one step further: \nwe will analyze the EOM (\\ref{eq:ctmeom}) of CTM in a formal continuum limit\nthrough a derivative expansion of $P$ up to the fourth order, and will\nshow that it agrees with the EOM of a coupled system of gravity and a scalar field \nderived from the Hamilton-Jacobi equation with an appropriate choice of an action.\n\n\\section{Representation of the tensor in a derivative expansion}\n\\label{sec:repP}\nIn this paper, we consider CTM in a formal continuum limit.\nWe leave aside for future study the question of dynamics why CTM can be studied in the continuum manner: we \nsimply assume that there exist some regimes where the continuum description is valid. \nThe basic strategy to treat CTM in this limit is the same \nas that in the previous papers \\cite{Sasakura:2015pxa,Chen:2016xjx}. \nWe formally replace the discrete values of the indices to the $d$-dimensional spatial coordinates: \n\\[\na \\rightarrow x \\in R^d.\n\\] \nNamely, the tensor $P_{xyz}$ is a function of three $d$-dimensional coordinates $x,y,z$, symmetric under arbitrary permutations.\nWe further assume a locality: $P_{xyz}$ takes non-vanishing values, only when $x,y,z$ are \nin the neighborhood, $x\\sim y \\sim z$. Mathematically, this can be formulated by that $P_{xyz}$ is a \ndistribution described by delta functions and their \nderivatives\\footnote{In \\cite{Sasakura:2015pxa}, the mathematical formulation is presented\ndifferently as a moment expansion in coordinates. \nThough they are essentially the same from the physical point of view, the present formulation in terms of distributions is superior to \nthe former one in the sense that the covariance can be easily incorporated. }:\n $P_{xyz}\\sim \\delta^d(x-y)\\delta^d (y-z)+\\hbox{derivatives of }\\delta^d(x-y)\\delta^d (y-z)$. \nWe also assume that we can terminate the derivative expansion at a certain order. From the \nphysical point of view, this is an assumption that the scale of the physical process of our interest is \nmuch larger than the fuzziness of the locality of the space. \nIn general, it is more convenient to use test functions to describe distributions \nrather than directly dealing with $\\delta$-functional expressions. \nSo, let us consider a contraction of $P$ with a test function $f$ up to the fourth order of derivatives as follows:\n\\[\n\\begin{split}\nP f^3&:=\n\\int d^dx d^dy d^dz\\, P_{xyz} f(x)f(y)f(z) \\\\\n&=\\int d^dx \\left( \\beta f^3 + \\beta^{\\mu\\nu}f^2 f_{,\\mu\\nu}+ \\beta^{\\mu\\nu\\rho} f^2 f_{,\\mu\\nu\\rho}\n+\\beta^{\\mu\\nu,\\rho\\sigma}f f_{,\\mu\\nu}f_{,\\rho\\sigma}+{\\cal O}(\\nabla^5)\\right),\n\\end{split}\n\\label{eq:pf3}\n\\]\nwhere, for brevity, the arguments $x$ of $\\beta$'s and $f$ are suppressed in the last line, \nand the greek indices represent spatial directions, e.g., $\\mu=1,2,\\cdots,d$.\nHere, the test function $f$ is assumed to have a compact support, \nand the indices of $f$ represent the covariant derivatives associated with a background metric $g_{\\mu\\nu}$, \ni.e., $f_{,\\mu\\nu}:=\\nabla_\\mu \\nabla_{\\nu} f,\\ f_{,\\mu\\nu\\rho}:=\\nabla_\\mu \\nabla_{\\nu} \\nabla_\\rho f$.\nAs will be explained in more detail in Section~\\ref{sec:gauge}, the test function is not a scalar, but \nmust be treated as a scalar half-density. Therefore, the covariant derivatives are defined with \na weight contribution: $\\nabla_\\mu f=(\\partial_\\mu -\\frac{1}{2} \\Gamma_\\mu)f$ with \n$\\Gamma_{\\mu}:=\\Gamma_{\\mu\\nu}^\\nu$,\n$\\nabla_\\mu \\nabla_\\nu f=(\\partial_\\mu -\\frac{1}{2}\\Gamma_\\mu)\\nabla_\\nu f -\\Gamma_{\\mu\\nu}^\\rho \\nabla_\\rho f$, and so on. The tensor fields, $\\beta^{\\mu\\nu}$ and $\\beta^{\\mu\\nu\\rho}$, are symmetric, \nand the field $\\beta^{\\mu\\nu,\\rho\\sigma}$ has the pairwise symmetries,\n\\[\n\\beta^{\\mu\\nu,\\rho\\sigma}=\\beta^{\\nu\\mu,\\rho\\sigma}=\\beta^{\\mu\\nu,\\sigma\\rho}=\n\\beta^{\\rho\\sigma,\\mu\\nu}.\n\\]\nThus, up to the fourth order, the ``components\" of $P$ are represented by the four fields, \n$\\beta(x),\\beta^{\\mu\\nu}(x),\\beta^{\\mu\\nu\\rho}(x)$, and $\\beta^{\\mu\\nu,\\rho\\sigma}(x)$.\nBecause of the weight of $f$ and the invariance of $Pf^3$, \nthese fields are assumed to have the weight of negative half-density\n(the details will be given in Section~\\ref{sec:gauge}):\n\\[\n\\begin{split}\n&[f]=\\frac{1}{2}, \\\\\n&[\\beta]=[\\beta^{\\mu\\nu}]=[\\beta^{\\mu\\nu\\rho}]=[\\beta^{\\mu\\nu,\\rho\\sigma}]=-\\frac{1}{2}.\n\\end{split}\n\\label{eq:weightbeta}\n\\]\nHere, $[X]$ denotes the weight of a quantity $X$, meaning that $X$ has the same weight as $g^\\frac{[X]}{2}$\n($g:=\\hbox{Det}[g_{\\mu\\nu}]$). These weights cancel the weight of the integration measure $d^dx$ to secure the \ninvariance of $Pf^3$. \n\nHere, we will explain more details about the derivative expansion \\eq{eq:pf3}. \nFirstly, as proven in Appendix~\\ref{app:pf3}, a totally symmetric rank-three tensor \ncan be fully characterized by the values of the contraction with an arbitrary vector $\\phi$\\,: \n$P_{abc}\\phi_a \\phi_b \\phi_c \\hbox{ for }^\\forall \\phi$.\nThus, it is enough to know $Pf^3$ for arbitrary $f$ as in \\eq{eq:pf3} for the full characterization of $P$, instead of \nconsidering three different functions for the three indices. \nSecondly, throughout this paper, we will consider the derivative expansion of $P$ up to the fourth order of derivatives, \nas in \\eq{eq:pf3}. \nThe reason is that we are interested in the equations of motion (EOM) of the fields $\\beta,\\beta^{\\mu\\nu}$ up to the \nsecond order of derivatives: as will be discussed later, these fields describe a coupled system of gravity and a scalar field,\nwhich is of physical interest. To correctly describe the EOM of \n$\\beta^{\\mu\\nu}$ (and $\\beta$) up to the second derivatives, \nit is necessary to include the fourth order of derivatives in the expansion of $P$ as in \\eq{eq:pf3}.\nAs one can prove, an independent set of fields describing $P$ up to the fourth order\nare exhausted by the set shown in \\eq{eq:pf3}. \nMore details are given in Appendix~\\ref{app:fourth} and \\ref{app:derivative}.\nLastly, we have introduced a background metric $g_{\\mu\\nu}$, which can be taken arbitrary. \nAs will be explained in detail in Section~\\ref{sec:backmetric},\nthe introduction of the background metric does not change the physical contents,\nbut simply redefines the fields with a linear recombination of them.\nIn fact, we will see that there exists a gauge symmetry which allows one to freely change the background metric\nwith simultaneous change of the fields, and will ultimately\ngauge-fix the background metric to a certain combination of the fields.\n\nIn the analysis of the EOM \\eq{eq:ctmeom} of CTM, \nit is necessary to have an expression corresponding to $3 P_{abc}\\phi_b \\phi_c$. \nIn the continuum limit, one can obtain this by the functional derivative of $Pf^3$ in \\eq{eq:pf3}:\n\\[\n\\begin{split}\nP[f,f]:=&\\frac{\\delta}{\\delta f(x)} Pf^3 \\\\\n=&3 \\beta f^2 + 2 \\beta^{\\mu\\nu} f f_{,\\mu\\nu} + (\\beta^{\\mu\\nu} f^2 )_{,\\mu\\nu} +2 \\beta^{\\mu\\nu\\rho} f f_{,\\mu\\nu\\rho}\n-(\\beta^{\\mu\\nu\\rho}f^2)_{,\\mu\\nu\\rho} +\\beta^{\\mu\\nu,\\rho\\sigma} f_{,\\mu\\nu} f_{,\\rho\\sigma} \\\\\n&+2 (\\beta^{\\mu\\nu,\\rho\\sigma} f f_{,\\mu\\nu})_{,\\rho\\sigma} +{\\cal O}(\\nabla^5)\\\\\n=&(3 \\beta +\\beta^{\\mu\\nu}_{,\\mu\\nu}-\\beta^{\\mu\\nu\\rho}_{,\\mu\\nu\\rho})f^2+(4 \\beta^{\\mu\\nu}_{,\\nu}\n-6 \\beta_{,\\nu\\rho}^{\\mu\\nu\\rho}) f f_{,\\mu}+(2 \\beta^{\\mu\\nu}-6 \\beta^{\\mu\\nu\\rho}_{,\\rho}) f_{,\\mu} f_{,\\nu} \\\\\n&+(4 \\beta^{\\mu\\nu}-6 \\beta_{,\\rho}^{\\mu\\nu\\rho}+\n2\\beta^{\\mu\\nu,\\rho\\sigma}_{,\\rho\\sigma}) f f_{,\\mu\\nu} \n+(-6 \\beta^{\\mu\\nu\\rho}+4\\beta^{\\mu\\sigma,\\nu\\rho}_{,\\sigma})\nf_{,\\mu} f_{,\\nu\\rho} \\\\\n&+4\\beta^{\\mu\\nu,\\rho\\sigma}_{,\\sigma} f f_{,\\rho\\mu\\nu} \n+3\\beta^{\\mu\\nu,\\rho\\sigma}f_{,\\mu\\nu}f_{,\\rho\\sigma}\n+4\\beta^{\\mu\\nu,\\rho\\sigma}f_{,\\mu} f_{,\\nu\\rho\\sigma}\n+2\\beta^{\\mu\\nu,\\rho\\sigma}f f_{,\\mu\\nu\\rho\\sigma}+{\\cal O}(\\nabla^5).\n\\end{split}\n\\label{eq:pff}\n\\]\n\nSimilarly, one can define an expression corresponding to $3 P_{abc}\\phi^1_a \\phi_b^2$ for two different vectors $\\phi^{1,2}$. \nThis is denoted by $P[f,g]$, and is defined by an obvious generalization: \nputting $f,g$ into two $f$'s of each term on the right-hand side of \\eq{eq:pff}, and symmetrizing them.\n\n\\section{Kinematical symmetry in the continuum limit}\n\\label{sec:gauge}\nCTM has the kinematical symmetry generated by the orthogonal group generators ${\\cal J}_{ab}$. \nIn the continuum limit, since the indices represent coordinates, \n${\\cal J}_{xy}$ will become generators of local gauge transformations. \nIn the derivative expansion, the gauge transformations are parameterized by \ntensor fields, like those in \\eq{eq:pf3} for $P$.\nUp to the fourth order, we will find two gauge transformations,\nwhich are the diffeomorphism and a spin-three gauge transformation.\n\nThe orthogonal group transformation of CTM can be characterized by a linear transformation of $f_a$ which preserves\nthe norm square $f_a f_a$. In the continuum limit, this condition is translated to the invariance of \n\\[\n\\Vert f \\Vert^2\\equiv \\int d^dx\\ f(x)f(x),\n\\label{eq:norm}\n\\] \nwhere $f(x)$ is considered to be a scalar half-density, and is assumed to have a compact support.\nIt is easy to show that \\eq{eq:norm} is invariant under the following infinitesimal linear transformations,\n\\[\n\\begin{split}\n\\delta_1 f(x)&=\\frac{1}{2} \n\\left[\\nabla_\\mu( v^\\mu(x) f(x)) + v^\\mu(x) \\nabla_\\mu f(x)\\right]=\\frac{1}{2} v^\\mu_{,\\mu}(x) f(x)+ v^\\mu(x) f_{,\\mu}(x),\\\\\n\\delta_3 f(x)&=\\frac{1}{2} \\left[ \\nabla_\\mu\\nabla_\\nu\\nabla_\\rho( v^{\\mu\\nu\\rho}(x) f(x)) + v^{\\mu\\nu\\rho}(x) \\nabla_\\mu\\nabla_\\nu\n\\nabla_\\rho f(x)\\right] \\\\\n&=\\frac{1}{2} v^{\\mu\\nu\\rho}_{,\\mu\\nu\\rho}(x) f(x)+\\frac{3}{2} v^{\\mu\\nu\\rho}_{,\\mu\\nu}(x) f_{,\\rho}(x)+\\frac{3}{2} v^{\\mu\\nu\\rho}_{,\\mu}(x) f_{,\\nu\\rho}(x)\n+ v^{\\mu\\nu\\rho}(x) f_{,\\mu\\nu\\rho}(x),\n\\end{split}\n\\label{eq:delta13}\n\\]\nwhere $v^\\mu$ and $v^{\\mu\\nu\\rho}$ are a vector field and a symmetric rank-three tensor field, respectively, and \n$\\nabla_\\mu$ is the covariant derivative ($\\nabla_\\mu f=(\\partial_\\mu -\\frac{1}{2} \\Gamma_\\mu)f$ with \n$\\Gamma_\\mu\\equiv \\Gamma_{\\mu\\nu}^\\nu$, etc.).\nHere we use the same simplified notations as in Section~\\ref{sec:repP}, such as $f_{,\\mu\\nu}=\\nabla_\\mu\\nabla_\\nu f$.\nIndeed,\n\\[\n\\delta_1 \\Vert f \\Vert^2 =2 \\int d^dx\\, f(x) \\delta_1 f(x)=\\int d^dx \\, f(x) \\left[ \\nabla_\\mu( v^\\mu(x) f(x)) \n+ v^\\mu(x) \\nabla_\\mu f(x)\\right]=0,\n\\]\nbecause the integrand is a total derivative.\\footnote{Note that $\\Gamma$'s cancel out as \n$\\nabla_\\mu (v^\\mu f^2)=(\\partial_\\mu +\\Gamma_{\\nu\\mu}^\\nu-\\Gamma_\\mu) (v^\\mu f^2)=\\partial_\\mu (v^\\mu f^2)$}\nThe invariance under $\\delta_3$ can also be shown similarly by using partial integrations.\nAs can be seen in \\eq{eq:delta13}, the transformation $\\delta_1$ \nrepresents a diffeomorphism transformation, which transforms $f(x)$ as a scalar-half density, and \n$\\delta_3$ represents a spin-three transformation.\n\nSome comments are in order. Firstly, \n$v^{\\mu\\nu\\rho}$ must be assumed to be symmetric to remove redundancies.\nThe reason is basically the same as that for the symmetry of $\\beta$'s in \\eq{eq:pf3}, which is explained \nin detail in Appendix~\\ref{app:fourth}.\nThe anti-symmetric part of $f_{,\\mu\\nu\\rho}$ in \\eq{eq:delta13} can be rewritten \nin terms of the first derivative of $f$ by using the curvature tensor,\nand therefore the anti-symmetric components of $v^{\\mu\\nu\\rho}$ can be absorbed into $v_\\mu$. \nAnother comment is that one may consider a spin-two transformation with $v^{\\mu\\nu}$\nin a similar manner. However, this is also redundant. \nThe invariance of the norm \\eq{eq:norm} requires that the transformation should be\nin the form, $\\delta_2 f=\\nabla_\\mu\\nabla_\\nu( v^{\\mu\\nu} f) - v^{\\mu\\nu} \\nabla_\\mu\\nabla_\\nu f$, with a minus\nrelative sign in this case. Then, the terms with the second derivative of $f$ cancel, and the transformation \nis equivalent to a diffeomorphism transformation with $v^\\mu=v^{\\mu\\nu}_{,\\nu}$. \nFinally, it is obvious that there exist an infinite tower of spin-odd transformations\nwhich preserve \\eq{eq:norm}. However, \nthe transformations higher than spin-three \nare irrelevant in our treatment up to the fourth order of derivatives.\n \nLet us define the transformations of $\\beta$'s in \\eq{eq:pf3} under $\\delta_{1}$ and $\\delta_{3}$,\nby transferring the transformations of $f$ to $\\beta$'s.\nAs for $\\delta_1$, we obtain\n\\[\n\\begin{split}\n\\delta_1 \\left(Pf^3\\right)&=\\int d^dx \\left[ \n3\\beta f^2 \\delta_1 f \n+\\beta^{\\mu\\nu}\\left( 2 f (\\delta_1 f) f_{,\\mu\\nu} + f^2 (\\delta_1 f)_{,\\mu\\nu} \\right) \\right. \\\\\n&\\hspace{6cm}\n\\left. +\\beta^{\\mu\\nu,\\rho\\sigma}\\left( (\\delta_1f) f_{,\\mu\\nu} f_{,\\rho\\sigma}+2 f f_{,\\mu\\nu}(\\delta_1 f)_{,\\rho\\sigma}\\right)+{\\cal O}(\\nabla^5)\\right]\\\\\n&=\\int d^dx \\left[\n(\\delta_1\\beta) f^3 + (\\delta_1 \\beta^{\\mu\\nu}) f^2 f_{,\\mu\\nu} \n+ (\\delta_1 \\beta^{\\mu\\nu\\rho}) f^2 f_{,\\mu\\nu\\rho}\n+ (\\delta_1 \\beta^{\\mu\\nu,\\rho\\sigma}) f f_{,\\mu\\nu} f_{,\\rho\\sigma}+{\\cal O}(\\nabla^5)\n\\right],\n\\end{split}\n\\label{eq:derdel1}\n\\]\nwhere\n\\[\n\\begin{split}\n&\\delta_1 \\beta=- v^\\mu \\beta_{,\\mu} + \\frac{1}{2} v^\\mu_{,\\mu}\\beta +{\\cal O}(\\nabla^3 ), \\\\\n&\\delta_1 \\beta^{\\mu\\nu} =- v^\\rho \\beta^{\\mu\\nu}_{,\\rho}+\\frac{1}{2}v^\\rho_{,\\rho} \\beta^{\\mu\\nu}+ v^\\mu_{,\\rho} \\beta^{\\rho\\nu}+\nv^\\nu_{,\\rho} \\beta^{\\mu\\rho}+{\\cal O}(\\nabla^3 ), \\\\\n&\\delta_1 \\beta^{\\mu\\nu\\rho}={\\cal O}(\\nabla^2 ),\\\\\n&\\delta_1 \\beta^{\\mu\\nu,\\rho\\sigma} ={\\cal O}(\\nabla ).\n\\end{split}\n\\label{eq:diffeo}\n\\]\nTo derive the result, we have performed some partial integrations to transform the first line of \n\\eq{eq:derdel1} into the form of \\eq{eq:pf3} in the second line.\nWe have assumed $\\beta^{\\mu\\nu\\rho}=0$ initially, which will be discussed later as a gauge condition\nfor the spin-three gauge symmetry.\nThe terms with ${\\cal O}(\\nabla^3)$ in $\\beta$ and $\\beta^{\\mu\\nu}$\ncan also be ignored, because our interest is up to the second derivatives for these fields. \n$\\delta_1 \\beta^{\\mu\\nu\\rho}$ and $\\delta_1 \\beta^{\\mu\\nu,\\rho\\sigma}$ \ncan be ignored, because they are of the fifth order of derivatives in \\eq{eq:derdel1}.\n The result \\eq{eq:diffeo} shows that $\\beta$ transforms as a scalar of negative half-density, \nand $\\beta^{\\mu\\nu}$ as a two-tensor of negative half-density.\nIndeed, \nthis coincides with the weight assignments \\eq{eq:weightbeta},\nwhat is apparently expected from the invariance of \\eq{eq:pf3} under the diffeomorphism.\n\nAs for $\\delta_3$, in a similar manner, we obtain \n\\[\n\\delta_3 (Pf^3)&=\\int d^dx \\left[ 3 \\beta f^2 \\delta_3 f + {\\cal O}(\\nabla^5 ) \\right] \\nonumber \\\\\n&= \\int d^dx \\left[ (\\delta_3 \\beta) f^3 \n+(\\delta_3 \\beta^{\\mu\\nu}) f^2 f_{,\\mu\\nu}+(\\delta_3 \\beta^{\\mu\\nu\\rho}) f^2 f_{,\\mu\\nu\\rho} + \n(\\delta_3 \\beta^{\\mu\\nu,\\rho\\sigma}) f f_{,\\mu\\nu} f_{,\\rho\\sigma}+{\\cal O}(\\nabla^5 )\\right],\n\\]\nwhere \n\\[\n\\begin{split}\n&\\delta_3\\beta = {\\cal O}(\\nabla^3 ),\\\\\n&\\delta_3 \\beta^{\\mu\\nu}=\\frac{9}{2}\\beta v_{,\\rho}^{\\mu\\nu\\rho}, \\\\\n&\\delta_3 \\beta^{\\mu\\nu\\rho} =3 \\beta v^{\\mu\\nu\\rho},\\\\\n&\\delta_3 \\beta^{\\mu\\nu,\\rho\\sigma} = {\\cal O}(\\nabla ).\n\\end{split}\n\\label{eq:delta3}\n\\]\n\nThe equation of motion \\eq{eq:ctmeom} of CTM contains the second term\n$\\sum_\\sigma n_{d \\sigma_{a}} P_{\\sigma_{b}\\sigma_{c}d}$, which \nrepresents the freedom to perform the infinitesimal kinematical transformation along time evolution\nby freely choosing $n_{ab}$ dependent on time. \nWithin our approximation of the continuum limit, the transformations which are relevant\nare $\\delta_1$ and $\\delta_3$. \nThus, we can write \\eq{eq:ctmeom} in a schematic manner as \n\\[\n\\begin{split}\n&\\frac{d}{dt} \\beta =(nPP)+\\delta_1 \\beta, \\\\\n&\\frac{d}{dt} \\beta^{\\mu\\nu}=(nPP)^{\\mu\\nu}+\\frac{9}{2}\\beta v_{,\\rho}^{\\mu\\nu\\rho}+\\delta_1 \\beta^{\\mu\\nu}, \\\\ \n&\\frac{d}{dt} \\beta^{\\mu\\nu\\rho}=(nPP)^{\\mu\\nu\\rho}+3 \\beta v^{\\mu\\nu\\rho},\\\\\n&\\frac{d}{dt} \\beta^{\\mu\\nu,\\rho\\sigma}=(nPP)^{\\mu\\nu,\\rho\\sigma},\\\\\n\\end{split}\n\\label{eq:schemEOM}\n\\]\nwhere we have used \\eq{eq:diffeo} and \\eq{eq:delta3}, and $(nPP),(nPP)^{\\mu\\nu},(nPP)^{\\mu\\nu\\rho},\n(nPP)^{\\mu\\nu,\\rho\\sigma}$ denote \nthe spin-0,2,3,4 components of $\\sum_{\\sigma} n_{d} P_{\\sigma_a de}P_{e\\sigma_b\\sigma_c}$, respectively.\nSince $\\delta_1$ describes the diffeomorphism,\nthe terms with $\\delta_1$ in \\eq{eq:schemEOM} correspond to the freedom to choose the shift-vector in the time-evolution \nin the ADM formalism of general relativity.\nAs for the spin-3 transformation, \nby setting $v^{\\mu\\nu\\rho}=-(nPP)^{\\mu\\nu\\rho}\/3\\beta$ under the assumption $\\beta\\neq0$,\nwe can make a tuning $\\frac{d}{dt} \\beta^{\\mu\\nu\\rho}=0$.\nIn this manner, one can keep the gauge condition $\\beta^{\\mu\\nu\\rho}=0$, which gauges \naway the spin-3 component. As seen in \\eq{eq:schemEOM}, \nby doing this gauge fixing, the time evolution of the spin-2 component will get a contribution by an amount,\n\\[\n-\\frac{3}{2}\\beta \\left(\\frac{(nPP)^{\\mu\\nu\\rho}}{\\beta} \\right)_{,\\rho},\n\\]\nfrom the infinitesimal spin-3 transformation.\nNote that, even if $P$ has no spin-3 component, i.e. $\\beta^{\\mu\\nu\\rho}=0$, $(nPP)^{\\mu\\nu\\rho}$ does not\nvanish in general (This will be seen explicitly later.), \nand the spin-3 infinitesimal transformation must be carried out as above to keep $\\beta^{\\mu\\nu\\rho}=0$\nalong time evolution.\nIn later sections, this and similar procedures will frequently be used to remove the appearance of the spin-3 component.\nIn fact, the spin-3 component can appear not only from the right-hand side of the equation of motion \\eq{eq:ctmeom}, \nbut also from the left-hand side $\\frac{d}{dt}P$, when the background metric has time-dependence as will be discussed in \nSection~\\ref{sec:identify}.\nThis can also be removed by balancing it with the spin-3 transformation on the right-hand side\nin a similar manner as above. \n\n\\section{Equation of motion of CTM in a static background}\n\\label{sec:fixed}\nIn this section, we will study the continuum limit of the equation of motion (EOM) \\eq{eq:ctmeom} of CTM in the case that\nthe background metric $g_{\\mu\\nu}$ is static. Let us take the contractions of both sides of \\eq{eq:ctmeom} \nwith a test function $f$ satisfying $\\dot f=0$.\nThe left-hand side, $\\frac{d}{dt} (Pf^3)$, is simply given by \\eq{eq:pf3} with $\\beta$'s replaced by $\\dot \\beta$'s.\nThe right-hand side is given by \n\\[\n\\delta Pf^3:= \\int d^dx\\, n P[f,P[f,f]],\n\\label{eq:delpf3}\n\\]\nwhere we have left aside the SO($\\mathcal{N}$) rotational part of \\eq{eq:ctmeom} \nfor later discussions, \nhave performed a replacement $n_a\\rightarrow n(x)$, and an overall numerical factor has been \nabsorbed into a constant rescaling of $n(x)$. By rewriting \\eq{eq:delpf3} in the form of \\eq{eq:pf3}, namely,\n\\[\n\\delta P f^3 = \\int d^d x \\left[\n(\\delta\\beta)f^3+(\\delta\\beta^{\\mu\\nu} )f^2 f_{,\\mu\\nu}+(\\delta\\beta^{\\mu\\nu\\rho} )f^2 f_{,\\mu\\nu\\rho}+(\\delta \\beta^{\\mu\\nu,\\rho\\sigma})\nf f_{,\\mu\\nu} f_{,\\rho\\sigma} +{\\cal O}(\\nabla^5)\n\\right],\n\\label{eq:canpf3}\n\\]\none can obtain the explicit expression of the right-hand side of the EOM for the fields $\\beta$'s.\nHere, note that a spin-three component $\\delta \\beta^{\\mu\\nu\\rho}$ of $\\delta P$ may appear\nin general, even though the gauge condition $\\beta^{\\mu\\nu\\rho}=0$ is initially assumed on $P$. \n\nThe symmetric two-tensor field $\\beta^{\\mu\\nu}$ is particularly interesting from \nthe view point of gravity.\nThe lowest order set of fields containing it is given by $\\beta$ and $\\beta^{\\mu\\nu}$.\nTherefore, we want to compute $\\delta \\beta$ and $\\delta \\beta^{\\mu\\nu}$ up to the second order of derivatives,\nwhich would be the minimum for physically interesting dynamics to be expected.\nThe wanted order about the latter field requires that our computations must be correct up to \nthe fourth order in \\eq{eq:canpf3}.\nThis means that $\\delta \\beta^{\\mu\\nu\\rho}$ and $\\delta \\beta^{\\mu\\nu,\\rho\\sigma}$ must be \ncomputed up to the first and the zeroth order of derivatives, respectively.\n\nIt would seem that the fourth order terms\\footnote{There exist no third order terms.} \nin $\\delta \\beta$ must also be included for \nthe consistency of the fourth order computations. However, the order of derivatives of \nthe terms relevant in $\\delta \\beta^{\\mu\\nu},\\delta \\beta^{\\mu\\nu\\rho},\\delta \\beta^{\\mu\\nu,\\rho\\sigma}$ \nare less than four in our computations up to the fourth order. \nThis means that the fourth derivative terms in $\\delta \\beta$ can not affect \n$\\delta \\beta^{\\mu\\nu},\\delta \\beta^{\\mu\\nu\\rho},\\delta \\beta^{\\mu\\nu,\\rho\\sigma}$ even in our later computations,\nwhich more or less mixes $\\delta \\beta,\\delta \\beta^{\\mu\\nu},\\delta \\beta^{\\mu\\nu\\rho},\n\\delta \\beta^{\\mu\\nu,\\rho\\sigma}$. \nTherefore, the fourth derivative terms in $\\delta \\beta$ can be ignored consistently, \nif one is not interested in them: \nour interest \nis up to the second order of derivatives in $\\delta \\beta$. \n\nEven with these upper bounds of our interest on the number of derivatives, the computation of \\eq{eq:delpf3} is \nvery complicated, and we used a Mathematica package ``xTensor\" for the \ntensorial computations.\nThe details of the procedure is explained in Appendix~\\ref{app:explicit}.\nWe have obtained\n\\[\n\\begin{split}\n\\delta \\beta^{\\mu\\nu,\\rho\\sigma}&= 11n \\beta\\beta^{\\mu\\nu,\\rho\\sigma}+4n \\beta \\beta^{\\mu(\\rho,\\sigma)\\nu}\n+4 n \\beta^{\\mu\\nu} \\beta^{\\rho\\sigma}+3 p\\, n \\beta^{(\\mu\\nu}\\beta^{\\rho\\sigma)}+{\\cal O}(\\nabla^2) ,\\\\\n\\delta \\beta^{\\mu\\nu\\rho}&= -14 n \\beta^{(\\mu\\nu,\\rho)\\sigma} \\beta_{,\\sigma}\n-4 p n \\beta^{(\\mu\\nu}\\beta^{\\rho)\\sigma}_{,\\sigma}+4(1-p) n \\beta^{(\\mu\\nu}_{,\\sigma}\\beta^{\\rho)\\sigma}\n-2n \\beta \\beta^{(\\mu\\nu,\\rho)\\sigma}_{,\\sigma}\\\\\n&\\ \\ \\ +4 (1-p) \\beta^{(\\mu\\nu}\\beta^{\\rho)\\sigma} n_{,\\sigma}-8 \\beta \\beta^{(\\mu\\nu,\\rho)\\sigma} n_{,\\sigma}+{\\cal O}(\\nabla^3), \\\\\n\\delta \\beta^{\\mu\\nu} &=15 n \\beta \\beta^{\\mu\\nu}-2(1+p)n \\beta^{\\mu\\nu}_{,\\rho} \\beta^{\\rho\\sigma}_{,\\sigma}\n+2 (1-p) \\beta^{\\mu\\nu} \\beta^{\\rho\\sigma}_{,\\sigma} n_{,\\rho}-2 p n \\beta^{\\mu\\rho}_{,\\sigma}\\beta^{\\nu\\sigma}_{,\\rho} \\\\\n&\\ \\ \\ \n+4 (1-p) \\beta^{\\rho(\\mu} \\beta^{\\nu)\\sigma}_{,\\rho}n_{,\\sigma}\n+4(1-p)\\beta^{\\rho(\\mu}\\beta^{\\nu)\\sigma}_{,\\sigma} n_{,\\rho}+2 (1-p) \\beta^{\\rho\\sigma} \\beta^{\\mu\\nu}_{,\\rho} n_{,\\sigma}\n-2 p n \\beta^{\\mu\\rho}_{,\\rho}\\beta^{\\nu\\sigma}_{,\\sigma}\\\\\n&\\ \\ \\ -10 n \\beta_{,\\rho} \\beta^{\\mu\\nu,\\rho\\sigma}_{,\\sigma}\n -4 \\beta \\beta^{\\mu\\nu,\\rho\\sigma}_{,\\rho} n_{,\\sigma}-8n\\beta_{,\\rho} \\beta^{\\rho(\\mu,\\nu)\\sigma}_{,\\sigma}\n-8 \\beta \\beta^{\\rho(\\mu,\\nu)\\sigma}_{,\\rho} n_{,\\sigma}-4 \\beta_{,\\rho} \\beta^{\\mu\\nu,\\rho\\sigma}n_{,\\sigma}\n\\\\\n&\\ \\ \\ -8 \\beta^{\\rho(\\mu,\\nu)\\sigma}\\beta_{,\\rho} n_{,\\sigma}\n-2n \\beta_{,\\rho\\sigma} \\beta^{\\mu\\nu,\\rho\\sigma}-4n \\beta_{,\\rho\\sigma} \\beta^{\\mu\\rho,\\nu\\sigma}\n+(1-p) n \\beta^{\\mu\\nu} \\beta^{\\rho\\sigma}_{,\\rho\\sigma}\\\\\n&\\ \\ \\ +4 (1-p) n \\beta^{\\rho(\\mu}\\beta^{\\nu)\\sigma}_{,\\rho\\sigma}\n+(2-p) n \\beta^{\\rho\\sigma} \\beta^{\\mu\\nu}_{,\\rho\\sigma}+n \\beta \\beta^{\\mu\\nu,\\rho\\sigma}_{,\\rho\\sigma}\n-4n \\beta \\beta^{\\rho(\\mu,\\nu)\\sigma}_{,\\rho\\sigma}\\\\\n&\\ \\ \\ +(6-p) \\beta^{\\mu\\nu}\\beta^{\\rho\\sigma}n_{,\\rho\\sigma}\n+(4-2p) \\beta^{\\mu\\rho}\\beta^{\\nu\\rho} n_{,\\rho\\sigma}\n+7\\beta \\beta^{\\mu\\nu,\\rho\\sigma}n_{,\\rho\\sigma}-4 \\beta \\beta^{\\mu\\rho,\\nu\\sigma}n_{,\\rho\\sigma}\\\\\n&\\ \\ \\ \n+n\\left(\\frac{4}{3} \\beta^{\\rho\\sigma}\\beta^{\\delta(\\mu}+2 \\beta \\beta^{\\rho\\sigma,\\delta(\\mu}\\right)R^{\\nu)}{}_{\\rho\\sigma\\delta}\n+{\\cal O}(\\nabla^4),\\\\\n\\delta \\beta&= 9 n \\beta^2 - 4n \\beta_{,\\mu}\\beta^{\\mu\\nu}_{,\\nu}+n \\beta^{\\mu\\nu}\\beta_{,\\mu\\nu}\n+n \\beta\\beta^{\\mu\\nu}_{,\\mu\\nu}+5 \\beta \\beta^{\\mu\\nu}n_{,\\mu\\nu}+{\\cal O}(\\nabla^4),\n\\end{split}\n\\label{eq:explicitbetas}\n\\]\nwhere $p=\\frac{4}{3}$ must be taken\\footnote{The parameter $p$ becomes a free parameter\nin the case that the term $\\delta \\beta^{\\mu\\nu\\rho\\sigma} f^2 f_{,\\mu\\nu\\rho\\sigma}$ is also allowed in the expression of \n$\\delta P f^3$. \nAs explained in Appendix~\\ref{app:fourth}, this term can be set to zero by using \\eq{eq:ambiguity}\nfor the unique representation. \nBut, if we leave it, $\\delta \\beta^{\\mu\\nu\\rho\\sigma}=(2-3p\/2)n \\beta^{(\\mu\\nu}\\beta^{\\rho\\sigma)}$, \nand the others will be given by \\eq{eq:explicitbetas} with free $p$.}.\nThe round brackets in the indices represent symmetrization of the indices contained in the pairs of the brackets.\nFor example,\n$\\beta^{\\mu(\\nu,\\rho)\\sigma}=\\frac{1}{2} \\left( \\beta^{\\mu\\nu,\\rho\\sigma}+\\beta^{\\mu\\rho,\\nu\\sigma}\\right)$, and $\\beta^{(\\mu\\nu}\n\\beta^{\\rho\\sigma)}$ represent the total symmetrization.\n\nAs seen in \\eq{eq:explicitbetas}, $\\delta \\beta$'s have complicated expressions with the derivatives of \nboth $\\beta$'s and $n$. \nThe existence of the derivatives of $n$ seems to pose a challenge in comparison with general relativity, \nsince the equation of motion \nof the metric tensor field in the Hamilton-Jacobi formalism of general relativity, \nwritten down in Section~\\ref{sec:conttheory}, contains no derivatives of the lapse function.\nThis absence comes from the fact that the Hamiltonian of the ADM \nformalism $H_{ADM}$ is expressed with no derivatives of the lapse function, and the Poisson brackets\nwith the fields do not produce them either, \nwhere the conjugate momenta to the fields are replaced by some functions of the fields in the Hamilton-Jacobi \nformalism.\n\nThe fundamental reason why we encounter the above difference between CTM and general relativity can intuitively be understood\nby the fact that, in CTM, a space is an emergent object characterized by the tensor $P$. \nAs explained at the beginning of Section~\\ref{sec:repP}, there exists intrinsic fuzziness\nwhich disturbs the exactness of a position specified by the coordinate $x$, where \nthe ambiguity would be in the order of $\\sim\\sqrt{\\beta^{\\mu\\nu}\/\\beta}$ for a dimensional reason.\nThis ambiguity of positions would also make ambiguous the value of a field, here the lapse function, \nas a function of $x$ by an amount in the order of $\\delta n(x)\\sim \\beta^{\\mu\\nu}n_{,\\mu\\nu}\/\\beta$. \nThe real expressions in \\eq{eq:explicitbetas} are much more involved, but this gives an \nintuitive understanding of the reason why the spatial derivatives of the lapse function can appear,\nirrespective of their absence in general relativity. Therefore, to make relations between CTM and general relativity, \nit would be natural to perform some redefinitions of the lapse function and the fields\nby adding some corrections of the spacial derivatives.\nIn fact, we will do so in later sections.\n\nAnother interesting thing to notice in \\eq{eq:explicitbetas}\nis that there appear terms with the background curvature in $\\delta \\beta^{\\mu\\nu}$.\nFor a static background considered in this section, the background curvature appears just as the coefficients \nof the quadratic terms \nof $\\beta$'s, and do not seem to play important roles. \nOn the other hand, as we will discuss in later sections, when the background metric\nbecomes dynamical as a result of the gauge-fixing to a combination of the fields, \nthe curvature terms play essential roles for the consistency of the time evolution. \n\nThe result \\eq{eq:explicitbetas} shows that there appears a spin-three component $\\delta \\beta^{\\mu\\nu\\rho}$,\neven if we assume $\\beta^{\\mu\\nu\\rho}=0$ initially. Therefore, as explained in Section~\\ref{sec:gauge},\nto maintain the gauge condition $\\beta^{\\mu\\nu\\rho}=0$, the spin-three gauge transformation $\\delta_3$ \nin \\eq{eq:delta3} has to be performed simultaneously.\nThis is to bring in the spin-three gauge transformation contained in the SO$({\\cal N})$ \nrotation part of EOM \\eq{eq:ctmeom}.\nBy setting $\\delta \\beta^{\\mu\\nu\\rho}+3 \\beta v^{\\mu\\nu\\rho}=0$,\nwe obtain the EOM for the fields as\n\\[\n\\begin{split}\n&\\dot \\beta=\\delta\\beta, \\\\\n&\\dot \\beta^{\\mu\\nu}=\\delta \\beta^{\\mu\\nu}-\\frac{3}{2} \\beta\\ \\nabla_\\rho \\left(\\frac{1}{\\beta}\\delta\\beta^{\\mu\\nu\\rho}\\right), \\\\\n&\\dot \\beta^{\\mu\\nu,\\rho\\sigma}=\\delta \\beta^{\\mu\\nu,\\rho\\sigma},\n\\end{split}\n\\label{eq:bareeom}\n\\]\nwhere the last term in the second line comes from the second line of \\eq{eq:delta3}, \nthe consequence of maintaining the gauge fixing condition $\\beta^{\\mu\\nu\\rho}=0$.\n\nA physically important consistency check of the EOM \\eq{eq:bareeom}\nis to compute the commutation of two successive \ninfinitesimal time evolutions. This corresponds to the commutation of the Hamiltonian constraints in CTM,\nand, from the first-class nature of the constraint algebra, this should be described by the kinematical \ntransformation ${\\cal J}_{ab}$. \nIn the present context of the continuum limit, the commutation of the time evolutions \nshould be expressed by the gauge transformations discussed in the \npreceding section. Since the spin-three transformation $\\delta_3$ has already been used for the gauge fixing, \none would expect that the commutation should be described by the diffeomorphism transformation $\\delta_1$.\nNote that the lapse function $n(x)$ is a field locally depending on $x$, and the situation is \nthe same as the time evolution in terms of the Hamiltonian constraint in general relativity: \nthe commutation of Hamiltonian constraint being equal to the diffeomorphism is nothing but the assurance of \nthe spacetime covariance of the locally generated time evolution. \nThis is directly connected to the central principle in general relativity, \nand it is highly interesting to check this in the present context.\n\nNow, let us explicitly describe the commutation of two successive infinitesimal time evolutions. \nSuppose we start with a configuration,\n$\\beta,\\beta^{\\mu\\nu},\\beta^{\\mu\\nu,\\rho\\sigma}$.\nAfter an infinitesimal time $\\Delta t$ with lapse $n_1$, the fields evolve to\n\\[\n\\beta_1^i=\\beta^i+\\Delta t \\, \\dot \\beta^i(n_1,\\beta,\\beta^{\\mu\\nu},\\beta^{\\mu\\nu,\\rho\\sigma}),\n\\label{eq:beta1}\n\\]\nwhere $\\beta^i$ represents $\\beta, \\beta^{\\mu\\nu}$, or $\\beta^{\\mu\\nu,\\rho\\sigma}$. \nHere, we have written explicitly the dependence of $\\dot \\beta$'s on $n$ and $\\beta$'s.\nThen, after the second step with lapse $n_2$, \nwe obtain\n\\[\n\\beta^i_{12}=\\beta^i_1+ \\Delta t \\, \\dot \\beta^i(n_2,\\beta_1,\\beta_1^{\\mu\\nu},\\beta_1^{\\mu\\nu,\\rho\\sigma}).\n\\label{eq:beta21}\n\\]\nBy inserting \\eq{eq:beta1} into \\eq{eq:beta21}, expanding in the infinitesimal parameter $\\Delta t$, and \nsubtracting the case that $n_1$ and $n_2$ are interchanged, \none obtains\n\\[\n\\begin{split}\n(\\delta_{n_1} \\delta_{n_2}-\\delta_{n_2}\\delta_{n_1})\\beta^i&=\n\\beta^i_{12}-\\beta^i_{21}\\\\\n&=(\\Delta t)^2 \\int d^d x\\, \\dot \\beta^j(x,n_1,\\beta,\\ldots) \n\\frac{\\delta}{\\delta \\beta^j(x)} \\dot \\beta^i(n_2,\\beta,\\ldots) -(n_1 \\leftrightarrow n_2),\n\\end{split}\n\\label{eq:del12m21}\n\\]\nwhere $j$ is summed over, and we have taken the lowest non-trivial order in $\\Delta t$.\n\nWe have used ``xTensor\" to obtain the following explicit result of \\eq{eq:del12m21}: \n\\[\n(\\delta_{n_1} \\delta_{n_2}-\\delta_{n_2}\\delta_{n_1})\\beta^i=\\delta_1 \\beta^i+{\\cal O}(\\nabla^4),\n\\label{eq:n1n2v}\n\\]\nwhere we have dropped the infinitesimal parameter $\\Delta t$, \n$\\beta^i=\\beta$ or $\\beta^{\\mu\\nu}$, and $\\delta_1$ is the diffeomorphism transformation \\eq{eq:diffeo} with\n\\[\nv^{\\mu}=12 \\beta \\beta^{\\mu\\nu}\\left( n_1 n_{2,\\nu}-n_2 n_{1,\\nu} \\right).\n\\label{eq:vval}\n\\]\nThe case with $\\beta^i=\\beta^{\\mu\\nu,\\rho\\sigma}$ is not considered, because this requires a higher order computation\nthan the fourth.\nIf we make the identification \n\\[\n\\frac{g^{\\mu\\nu}}{\\sqrt{g}}=\\beta \\beta^{\\mu\\nu},\n\\label{eq:geqbeta2}\n\\]\nthe commutation algebra \\eq{eq:n1n2v} with \\eq{eq:vval} agrees with that of the ADM formalism of general relativity\nexcept for a weight factor $1\/\\sqrt{g}$.\nThe weight factor is necessary for the consistency with the weights of $\\beta$ and $\\beta^{\\mu\\nu}$\nshown in \\eq{eq:weightbeta}. \nThe identification \\eq{eq:geqbeta2} was first discussed in \\cite{Sasakura:2015pxa} \nwith a different argument directly taking the formal continuum limit of the constraint algebra,\nand the extra weight factor has been interpreted consistently. \nIn Section~\\ref{sec:identify}, we will use this relation \\eq{eq:geqbeta2} to gauge-fix the background metric,\nand the issue of weights will be treated in Section~\\ref{sec:delete}.\n\nIt is worth mentioning that there exists a scale invariance in the EOM \\eq{eq:bareeom} with \\eq{eq:explicitbetas}.\nThe transformation is given by\n\\[\n\\begin{split}\nt&\\rightarrow Lt,\\ x^\\mu\\rightarrow L x^\\mu,\\\\\n\\beta&\\rightarrow \\frac{\\beta}{L},\\ \\beta^{\\mu\\nu}\\rightarrow L \\beta^{\\mu\\nu},\\ \n\\beta^{\\mu\\nu\\rho}\\rightarrow L^2 \\beta^{\\mu\\nu\\rho},\\ \\beta^{\\mu\\nu,\\rho\\sigma}\\rightarrow L^3 \\beta^{\\mu\\nu,\\rho\\sigma},\n\\end{split}\n\\label{eq:scaletrans}\n\\]\nwhere $L$ is a real free parameter. \nThe lapse function $n$ and the inverse metric $g^{\\mu\\nu}$ do not transform. \nThe transformation is consistent with the identification \\eq{eq:geqbeta2}.\nThis scale invariance will be respected throughout this paper in the other forms of EOM which will appear in due course.\n\nLastly, we will present a solution to the EOM for the highest component $\\beta^{\\mu\\nu,\\rho\\sigma}$. \nLet us assume the following form of a solution,\n\\[\n\\beta^{\\mu\\nu,\\rho\\sigma}=\\frac{a}{\\beta}\\beta^{\\mu\\nu}\\beta^{\\rho\\sigma}+\\frac{b}{\\beta} \\beta^{(\\mu\\nu}\\beta^{\\rho\\sigma)},\n\\label{eq:ansatzbe4}\n\\]\nwhere $a,b$ are real numbers. Note that the form is consistent with the scale transformation \\eq{eq:scaletrans}.\nTo check whether this satisfies the EOM,\nit is enough to compute the time-derivative of the right-hand side of \\eq{eq:ansatzbe4} \nup to non-derivative terms, since we consider $\\beta^{\\mu\\nu,\\rho\\sigma}$ up to the zeroth order. \nSince, from \\eq{eq:bareeom},\n\\[\n\\begin{split}\n\\dot \\beta&=9 n \\beta^2 +\\hbox{derivative terms}, \\\\\n\\dot \\beta^{\\mu\\nu}&=15 n \\beta \\beta^{\\mu\\nu}+\\hbox{derivative terms},\n\\end{split}\n\\label{eq:zeroth}\n\\]\none obtains\n\\[\n\\dot \\beta^{\\mu\\nu,\\rho\\sigma}=21 n \\left(a \\beta^{\\mu\\nu}\\beta^{\\rho\\sigma}+b \\beta^{(\\mu\\nu}\\beta^{\\rho\\sigma)} \\right)\n+\\hbox{derivative terms}\n\\]\nfrom the assumption \\eq{eq:ansatzbe4}. \nOn the other hand, by inserting \\eq{eq:ansatzbe4} into the EOM \n\\eq{eq:bareeom}, one obtains\n\\[\n\\dot \\beta^{\\mu\\nu,\\rho\\sigma}=(9a+4) n \\beta^{\\mu\\nu}\\beta^{\\rho\\sigma}+(6a+15b+4)n\\beta^{(\\mu\\nu}\\beta^{\\rho\\sigma)}\n+\\hbox{derivative terms}.\n\\]\nBy equating the two expressions for $\\dot \\beta^{\\mu\\nu,\\rho\\sigma}$, one obtains\n\\[\na=\\frac{1}{3},\\ b=1.\n\\label{eq:valab}\n\\]\nThe existence of the consistent solution implies that one can \nignore the field $\\beta^{\\mu\\nu,\\rho\\sigma}$ assuming that \nit is given by \\eq{eq:ansatzbe4} with \\eq{eq:valab}. \nThis truncation for simplicity will be assumed in the further analysis in later sections.\n\n\\section{Gauge symmetry of the background metric}\n\\label{sec:backmetric}\nIn the former sections, we considered a static background metric, and this is certainly a consistent treatment. \nHowever, there exist two distinct rank-two symmetric tensors, $g^{\\mu\\nu}$ and $\\beta^{\\mu\\nu}$, and \nthis would be physically awkward from the view point of general relativity, which has a unique symmetric rank-two\ntensor called the metric.\nIn fact, as will be explained below, the background metric can be chosen arbitrarily without changing the physical contents\nof CTM: there exists a gauge symmetry which allows one to freely change the background metric\nwith compensation by the fields.\nIn other words, as illustrated in Figure~\\ref{fig:gauge},\na constant surface of $P$ forms a submanifold in the configuration space of $g_{\\mu\\nu}$ and $\\beta$'s,\nand it is extending in the directions that allow arbitrary infinitesimal changes of the background metric. \nSince the motion of $P$ is determined by $P$ itself as in \\eq{eq:ctmeom} (up to the kinematical gauge symmetry), \nthe motion is actually a time-dependent transition from a constant $P$ submanifold to another. \nSuch transitions can be described by various manners of one's own choice, as illustrated \nfor two examples in Figure~\\ref{fig:gauge}.\nTaking a representative point on each constant $P$ submanifold determines a trajectory of time evolution\nin the configuration space of $g_{\\mu\\nu}$ and $\\beta$'s. This \nis a gauge choice, and, in the former section, we take the gauge that the background metric is static, and the motion is solely \ndescribed by $\\beta$'s.\nThis is illustrated as the dotted arrow in the figure. On the other hand, we may take another choice that $g_{\\mu\\nu}$\nand $\\beta$'s are correlated. This is what we will take for the comparison with general relativity, in which the actual gauge fixing\ncondition will be taken as \\eq{eq:geqbeta2}. This is illustrated as a dashed arrow in the figure. \nNote that the two descriptions are physically equivalent: they are connected by a transformation \nof $g_{\\mu\\nu}$ and $\\beta$'s along a constant $P$ submanifold, while $g_{\\mu\\nu}$ and $\\beta$'s \ntake different values.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=1]{gauge.png}\n\\caption{A schematic illustration of the time evolution in CTM. The horizontal and vertical axes \nrepresent the configurations of the fields and the background metric, respectively. \nThe solid curves represent the submanifolds of constant $P$.\nA time evolution is a transition from a constant $P$ \nsubmanifold to another in the configuration space.\nThe dotted arrow represents a time evolution in the gauge of a static background metric, while the \ndashed arrow represents an evolution which describes time evolution \nin general relativity by the gauge choice \\eq{eq:geqbeta2}.}\n\\label{fig:gauge}\n\\end{center}\n\\end{figure}\n\nLet us describe the submanifold of constant $P$ by considering the infinitesimal changes of $g_{\\mu\\nu}$ and \n$\\beta$'s which keep $P$. This condition is given by $\\delta (Pf^3)=0$ with \nthe test function unchanged $\\delta f=0$, while $g_{\\mu\\nu}$ and $\\beta$'s are allowed to be changed. \nBy taking the infinitesimal of \\eq{eq:pf3}, it is straightforward to derive\n\\[\n\\begin{split}\n\\delta (Pf^3)=\\int d^d x & \\left[\n\\left(\\delta \\beta -\\frac{1}{2} \\beta^{\\mu\\nu} \\delta\\Gamma_{\\mu,\\nu}\n+\\frac{1}{3} \\left( \\beta^{\\mu\\nu}\\delta\\tilde\\Gamma_{\\mu\\nu}^\\rho\\right)_{,\\rho} \\right)f^3 \\right.\\\\\n&\n+\\left(\n\\delta \\beta^{\\mu\\nu} -\\beta^{\\mu\\nu,\\rho\\sigma} \\delta\\Gamma_{\\rho,\\sigma}\n+\\left(\\beta^{\\mu\\nu,\\rho\\sigma} \\delta \\tilde \\Gamma_{\\rho\\sigma}^\\delta \\right)_{,\\delta}\n\\right)f^2 f_{,\\mu\\nu} \n\\\\\n&+\\left.\n\\left(\\delta \\beta^{\\delta \\rho\\sigma}+\\beta^{\\mu\\nu,\\rho\\sigma}\\delta \\tilde \\Gamma_{\\mu\\nu}^\\delta \n\\right)f^2 f_{,\\delta\\rho\\sigma}\n+\\delta \\beta^{\\mu\\nu,\\rho\\sigma}f f_{,\\mu\\nu} f_{,\\rho\\sigma}\n\\right]+{\\cal O}(\\nabla^5),\n\\end{split}\n\\label{eq:delpf3met}\n\\]\nwhere \n\\[\n\\tilde \\Gamma_{\\mu\\nu}^\\rho:= \\Gamma_{\\mu\\nu}^\\rho+ \\delta^{\\rho}_{(\\mu}\\Gamma_{\\nu)},\n\\] \nand \n\\[\n\\delta \\Gamma_{\\mu\\nu}^\\rho=\\frac{1}{2} g^{\\rho\\sigma}\\left(\n\\nabla_\\mu \\delta g_{\\nu\\sigma}+\\nabla_\\nu \\delta g_{\\mu\\sigma}-\\nabla_\\sigma \\delta g_{\\mu\\nu}\\right).\n\\]\nHere, we have assumed the gauge condition $\\beta^{\\mu\\nu\\rho}=0$ as an initial input.\nTo derive the result, we have considered the change of the covariant derivatives under the change of $g_{\\mu\\nu}$, \nnamely,\n\\[\n\\delta f_{,\\mu\\nu}=-\\delta\\Gamma_{\\mu\\nu}^\\rho f_{,\\rho}-\\frac{1}{2} \\delta \\Gamma_\\mu f_{,\\nu}\n+\\nabla_{\\mu} \\left(- \\frac{1}{2}\\delta\\Gamma_\\nu f \\right),\n\\]\nand have performed some partial integrations to obtain \\eq{eq:delpf3met}.\nTo further transform it to the form \\eq{eq:pf3}, we have to symmetrize the third derivative of $f$ \nby using the following equation with the Riemann tensor: \n\\[\n\\begin{split}\n\\beta^{\\mu\\nu,\\rho\\sigma}\\delta \\tilde \\Gamma_{\\mu\\nu}^\\delta f^2 f_{,(\\delta\\rho\\sigma)}\n=&\\frac{1}{3}\\beta^{\\mu\\nu,\\rho\\sigma}\\delta \\tilde \\Gamma_{\\mu\\nu}^\\delta f^2\n\\left(f_{,\\delta\\rho\\sigma}+f_{,\\rho\\delta\\sigma}+f_{,\\rho\\sigma\\delta}\\right)\\\\\n=&\\frac{1}{3}\\beta^{\\mu\\nu,\\rho\\sigma}\\delta \\tilde \\Gamma_{\\mu\\nu}^\\delta f^2 \n\\left(3f_{,\\delta\\rho\\sigma}+2 R_{\\rho\\delta\\sigma}{}^\\kappa f_{,\\kappa}\\right).\n\\end{split}\n\\]\nThe last term in the last line can be transformed to the non-derivative terms of $f$ by a partial integration, because \n$f^2 f_{,\\kappa}=\\frac{1}{3} (f^3)_{,\\kappa}$. Then, the condition $\\delta (Pf^3)=0$ implies\n\\[\n\\begin{split}\n&\\delta \\beta=\\frac{1}{2} \\beta^{\\mu\\nu} \\delta\\Gamma_{\\mu,\\nu}\n-\\frac{1}{3} \\left( \\beta^{\\mu\\nu}\\delta\\tilde\\Gamma_{\\mu\\nu}^\\rho\\right)_{,\\rho} \n-\\frac{2}{9}\\left( \\beta^{\\mu\\nu,\\rho\\sigma}\\delta \\tilde \\Gamma_{\\mu\\nu}^\\delta R_{\\rho\\delta\\sigma}{}^\\kappa\\right)_\\kappa, \\\\\n&\\delta \\beta^{\\mu\\nu}= \\beta^{\\mu\\nu,\\rho\\sigma} \\delta\\Gamma_{\\rho,\\sigma}\n-\\left(\\beta^{\\mu\\nu,\\rho\\sigma} \\delta \\tilde \\Gamma_{\\rho\\sigma}^\\delta \\right)_{,\\delta} ,\\\\\n&\\delta \\beta^{\\mu\\nu\\rho}=-\\beta^{\\sigma\\delta,(\\mu\\nu}\\delta \\tilde \\Gamma_{\\sigma\\delta}^{\\rho)} ,\\\\\n&\\delta \\beta^{\\mu\\nu,\\rho\\sigma}=0.\n\\end{split}\n\\label{eq:deformbeta}\n\\]\nWe have shown that an arbitrary infinitesimal deformation of the background metric can be absorbed by\nthe infinitesimal change of $\\beta$'s shown in \\eq{eq:deformbeta}.\nThe last term in the first line is actually irrelevant, because it is higher order than our range of interest.\nNote that there appear spin-3 components, which must be absorbed in the way discussed in Section~\\ref{sec:gauge}\nto maintain the gauge condition $\\beta^{\\mu\\nu\\rho}=0$.\n\n\\section{Identifying the background geometry with the fields} \n\\label{sec:identify}\nThe background geometry introduced in the preceding sections is arbitrary. \nIn fact, as discussed in Section~\\ref{sec:backmetric},\nan arbitrary change of the background geometry can be absorbed into the change of the fields $\\beta$'s\nwithout changing $P$. This means that there exists a gauge symmetry which changes the background geometry\nwithout changing the dynamical contents of the system.\nThe most reasonable choice of the background geometry is \\eq{eq:geqbeta2}, which \ndetermines the background geometry in terms of $\\beta$ and $\\beta^{\\mu\\nu}$, and \nmakes it a dynamical entity. \n\nIf we impose the identification \\eq{eq:geqbeta2},\nthe diffeomorphism transformation \\eq{eq:diffeo} derived previously for a static background will also be changed.\nThis is because we have to take into account the simultaneous transformation of $g^{\\mu\\nu}$ keeping the \nrelation \\eq{eq:geqbeta2}.\nIt is easy to see that the corrections are given by the minus of \\eq{eq:deformbeta}. \nTherefore, since \\eq{eq:diffeo} is in the first order of derivatives,\nthe corrections are higher than the second order of derivatives.\nThis is out of our range of interest, and the diffeomorphism transformation \nremains in the form \\eq{eq:diffeo}. \nThis is consistent with the naive expectation that\n$\\beta$ and $\\beta^{\\mu\\nu}$ should still behave as a scalar and a two-tensor with the weight of negative half-density,\neven after the identification of the background metric with the fields.\n\nIt is important to see whether the transformation \\eq{eq:diffeo} and the identification \\eq{eq:geqbeta2} \nreproduce the standard diffeomorphism transformation of $g^{\\mu\\nu}$. Let us define\n\\[\n\\tilde g^{\\mu\\nu}\\equiv \\beta\\beta^{\\mu\\nu}=\\frac{g^{\\mu\\nu}}{\\sqrt{g}},\n\\label{eq:deftildeg}\n\\]\nwhere we wrote \\eq{eq:geqbeta2} as well. From \\eq{eq:diffeo}, one obtains\n\\[\n\\delta_1 \\tilde g^{\\mu\\nu}&=(\\delta_1 \\beta)\\beta^{\\mu\\nu}+\\beta (\\delta_1 \\beta^{\\mu\\nu})\\nonumber \\\\\n&=-v^\\rho \\tilde g^{\\mu\\nu}_{,\\rho}+v^\\rho_{,\\rho} \\tilde g^{\\mu\\nu}+v^\\mu_{,\\rho}\\tilde g^{\\rho\\nu}+\nv^\\nu_{,\\rho}\\tilde g^{\\mu\\rho}+{\\cal O}(\\nabla^3).\n\\label{eq:deltagtilde}\n\\]\nThen, by using the second relation in \\eq{eq:deftildeg}, one obtains\n\\[\n\\begin{split}\n\\delta_1 g^{\\mu\\nu}&=\\sqrt{g}\\left( \\delta_1 \\tilde g^{\\mu\\nu}-\\frac{g^{\\mu\\nu}}{d+2} g_{\\rho\\sigma} \\delta_1 \\tilde g^{\\rho\\sigma}\n\\right) \\\\\n&=\\nabla^\\mu v^\\nu+\\nabla^\\nu v^\\mu+{\\cal O}(\\nabla^3 ),\n\\end{split}\n\\]\nwhere we have put \\eq{eq:deltagtilde}.\nThis indeed agrees with the transformation of the metric under the diffeomorphism in general relativity.\n\nOne consequence of the identification \\eq{eq:geqbeta2} is that the expression of $\\delta \\beta$'s \nin \\eq{eq:explicitbetas} is considerably simplified. This comes from $\\nabla_\\mu (\\beta \\beta^{\\nu\\rho})=0$, \nwhich is because the covariant derivative satisfies $\\nabla_\\mu g^{\\nu\\rho}=0$. \nBy substituting \\eq{eq:explicitbetas} with \\eq{eq:geqbeta2}, \\eq{eq:ansatzbe4} and \\eq{eq:valab}, \nwe obtain\n\\[\n\\begin{split}\n\\tilde \\delta \\beta&=9n \\beta^2+\\frac{6n}{\\beta^2} \\tilde g^{\\mu\\nu} \\beta_{,\\mu}\\beta_{,\\nu}+5 \\tilde g^{\\mu\\nu} n_{,\\mu\\nu}\n+{\\cal O}(\\nabla^4),\\\\\n\\tilde \\delta \\beta^{\\mu\\nu}&= 15 n \\tilde g^{\\mu\\nu} -\\frac{20n}{\\beta^4} \\tilde g^{\\mu\\rho}\\tilde g^{\\nu\\sigma} \\beta_{,\\rho}\n\\beta_{,\\sigma}-\\frac{8}{\\beta^3} \\tilde g ^{\\rho(\\mu} \\tilde g^{\\nu)\\sigma}\\beta_{,\\rho} n_{,\\sigma}\n+\\frac{10 n}{\\beta^3} \\tilde g^{\\rho\\mu}\\tilde g^{\\nu\\sigma} \\beta_{,\\rho\\sigma}\n+\\frac{14}{\\beta^2} \\tilde g^{\\mu\\rho}\\tilde g^{\\nu\\sigma} n_{,\\rho\\sigma}\\\\\n&\\ \\ \\ +\\tilde g^{\\mu\\nu}\\left(\\frac{8n}{\\beta^4} \\tilde g^{\\rho\\sigma} \\beta_{,\\rho}\\beta_{,\\sigma} \n-\\frac{4}{\\beta^3} \\tilde g ^{\\rho\\sigma} \\beta_{,\\rho} n_{,\\sigma}\n+\\frac{n}{\\beta^3} \\tilde g^{\\rho\\sigma} \\beta_{,\\rho\\sigma}\n+\\frac{14}{\\beta^2} \\tilde g^{\\rho\\sigma} n_{,\\rho\\sigma}\n\\right) -\\frac{2n}{\\beta^2} \\tilde g^{\\mu\\rho}\\tilde g^{\\nu\\sigma} R_{\\rho\\sigma}+{\\cal O}(\\nabla^4).\n\\end{split}\n\\label{eq:tildebeta}\n\\]\nHere, note that $\\tilde \\delta \\beta^{\\mu\\nu\\rho}$ and $\\tilde \\delta \\beta^{\\mu\\nu,\\rho\\sigma}$ are not considered anymore:\n$\\tilde \\delta \\beta^{\\mu\\nu\\rho}$ has been gauged away to be included in $\\tilde \\delta \\beta^{\\mu\\nu}$ by\nthe spin-three gauge transformation to keep the gauge condition $\\beta^{\\mu\\nu\\rho}=0$,\nand $\\beta^{\\mu\\nu,\\rho\\sigma}$ is assumed to be the solution \\eq{eq:ansatzbe4} with \\eq{eq:valab}.\n\nAs can be seen in \\eq{eq:tildebeta}, while the right-hand sides of the equation of motion (EOM) have considerably been simplified\nin comparison with \\eq{eq:explicitbetas},\nthe left-hand side, $\\frac{d}{dt} (Pf^3)$, \nmust be modified with some additional terms which come from the evolution of the background metric to \nkeep the relation \\eq{eq:geqbeta2}:\nthe left-hand side can not simply be expressed by the time-derivatives of the fields \n$\\dot \\beta,\\dot \\beta^{\\mu\\nu}$, but must also contain some additional terms coming from the time-derivative of $g_{\\mu\\nu}$\ncontained in the covariant derivatives in \\eq{eq:pf3}.\nThe derivation of the explicit expression of the left-hand side is basically the same as that of \\eq{eq:deformbeta}\nthrough \\eq{eq:delpf3met},\nand the additional terms are just the minus of the right-hand sides of \\eq{eq:deformbeta} with the replacement \n$\\delta \\Gamma\\rightarrow \\dot \\Gamma$. In addition, to keep the gauge condition $\\beta^{\\mu\\nu\\rho}=0$, we\nhave to perform the spin-three gauge transformation to transfer \n$\\delta \\beta^{\\mu\\nu\\rho}$ in \\eq{eq:deformbeta} to \n$\\dot \\beta^{\\mu\\nu}$. Then, we obtain the EOM as\n\\[\n\\begin{split}\n&\\dot \\beta-\\frac{1}{2} \\beta^{\\mu\\nu} \\dot \\Gamma_{\\mu,\\nu}+\\frac{1}{3} \\nabla_{\\sigma}\n\\left(\\beta^{\\mu\\nu} \\dot{\\tilde\\Gamma}_{\\mu\\nu}^\\sigma \\right)=\\tilde \\delta \\beta,\\\\\n&\\dot \\beta^{\\mu\\nu}-\\beta^{\\mu\\nu,\\rho\\sigma}\\dot \\Gamma_{\\rho,\\sigma}+\n\\nabla_\\delta \\left(\\beta^{\\mu\\nu,\\rho\\sigma}\\,\\dot{\\tilde \\Gamma}_{\\rho\\sigma}^\\delta \\right)\n-\\frac{3}{2}\\beta \\nabla_\\rho \\left(\\frac{1}{\\beta} \\beta^{\\sigma\\delta,(\\mu\\nu} \\dot{\\tilde \\Gamma}_{\\sigma\\delta}^{\\rho)}\\right)\n=\\tilde \\delta \\beta^{\\mu\\nu},\n\\end{split}\n\\label{eq:eommiddle}\n\\]\nwhere \\eq{eq:geqbeta2}, \\eq{eq:ansatzbe4} and \\eq{eq:valab} are supposed, and the last term on the left-hand side of the \nlast line comes from the spin-three transformation.\nIt would be worth to remind that the time derivative of the Christoffel symbol can be written covariantly as \n\\[\n\\dot \\Gamma_{\\mu\\nu}^\\rho=\\frac{1}{2} g^{\\rho\\sigma}\\left( \\nabla_\\mu \\dot g_{\\nu\\sigma}+\\nabla_\\nu \\dot g_{\\mu\\sigma}-\n\\nabla_\\sigma \\dot g_{\\mu\\nu}\\right),\n\\label{eq:covgam}\n\\]\nand therefore \\eq{eq:eommiddle} is a covariant expression.\n\nLet us simplify \\eq{eq:eommiddle} further.\nIn the zeroth order of derivatives, the equation of motion (EOM) derived from \\eq{eq:eommiddle} is still given by \\eq{eq:zeroth}, \nsince all the corrections in \\eq{eq:eommiddle} are in the second order. Therefore, by using \\eq{eq:geqbeta2}, \nthe EOM of $g^{\\mu\\nu}$ in the zeroth order is given by\n\\[\n\\dot g^{\\mu\\nu}=\\frac{48 n \\beta}{d+2} g^{\\mu\\nu}+{\\cal O}(\\nabla^2).\n\\label{eq:dotg}\n\\]\nHere, the dimensional dependence appears due to\nthe determinant in \\eq{eq:geqbeta2}, while the EOM so far has been independent of it. Then, by\nputting \\eq{eq:dotg} into \\eq{eq:covgam}, one obtains\n\\[\n\\dot\\Gamma_{\\mu\\nu}^\\rho=-\\frac{48}{d+2} \n\\left(\\delta^\\rho_{(\\mu}\\nabla_{\\nu)}(n \\beta)-\\frac{1}{2}g_{\\mu\\nu} \\nabla^\\rho(n\\beta)\\right)+{\\cal O}(\\nabla^3).\n\\label{eq:dotgamma}\n\\]\nThe overall minus sign is from the fact $\\dot g_{\\mu\\nu}=-g_{\\mu\\rho}g_{\\nu\\sigma} \\dot g^{\\rho\\sigma}$.\nThis order of $\\dot \\Gamma$ is enough for our second order computation of \nthe correction terms on the left-hand side of \\eq{eq:eommiddle}. \nBy putting \\eq{eq:dotgamma} into \\eq{eq:eommiddle}, we finally obtain\n\\[\n\\begin{split}\n&\\dot \\beta\n=9 n \\beta^2+\\frac{\\tilde g^{\\mu\\nu}}{d+2}\\left(\\frac{2(3d-2)n}{\\beta^2} \\beta_{,\\mu} \\beta_{,\\nu}\n-\\frac{8(3d-2)}{\\beta} \\beta_{,\\mu}n_{,\\nu}\n-\\frac{4(3d-4)n}{\\beta} \\beta_{,\\mu\\nu}\n-(7d-26) n_{,\\mu\\nu} \\right) \\\\\n&\\ \\ \\ \\ \\ \\ +{\\cal O}(\\nabla^4), \\\\\n&\\dot \\beta^{\\mu\\nu}=15 n \\tilde g^{\\mu\\nu} -\\frac{2n \\tilde g^{\\mu\\rho} \\tilde g^{\\nu\\sigma}}{\\beta^2}R_{\\rho\\sigma} \\\\\n&\\ \\ \\ \\ \\ \\ +\\frac{\\tilde g^{\\rho(\\mu}\\tilde g^{\\nu)\\sigma}}{(d+2)\\beta^2}\\left( -\\frac{4(d-14)n}{\\beta^2}\\beta_{,\\rho}\\beta_{,\\sigma}\n-\\frac{24(d-2)}{\\beta} \\beta_{,\\rho}n_{,\\sigma} \n-\\frac{2(3d-2)n}{\\beta} \\beta_{,\\rho\\sigma}\n-2(d-6) n_{,\\rho\\sigma}\n \\right)\n\\\\\n&\\ \\ \\ \\ \\ \\ +\\frac{\\tilde g^{\\mu\\nu}\\tilde g^{\\rho\\sigma}}{(d+2)\\beta^2}\\left( \\frac{8(5d+8)n}{\\beta^2} \\beta_{,\\rho}\\beta_{,\\sigma} \n-\\frac{4(5d-6)}{\\beta}\\beta_{,\\rho}n_{,\\sigma}\n-\\frac{(23d+6)n}{\\beta} \\beta_{,\\rho\\sigma}\n-10(d-2) n_{,\\rho\\sigma} \n\\right) \\\\\n&\\ \\ \\ \\ \\ \\ +{\\cal O}(\\nabla^4),\n\\end{split}\n\\label{eq:eombetawithg}\n\\]\nwhere \\eq{eq:geqbeta2} is supposed. This is the version of EOM with a dynamical background metric\ndetermined by \\eq{eq:geqbeta2}.\n\nA physically meaningful consistency check of EOM \\eq{eq:eombetawithg} is\ngiven by computing the commutation of two successive infinitesimal time evolutions, as \nthe algebraic structure \\eq{eq:n1n2v} with \\eq{eq:vval} has been obtained for the static background case. \nThe existence of the gauge symmetry discussed in Section \\ref{sec:backmetric}, which \nallows us to freely change the background metric, \nassures the covariance of the time evolution for the evolving background case, too. \nTherefore, we should obtain the same algebraic structure as the static background case. \nHowever, \nthe actual computation for the consistency check is much more complicated and non-trivial than the fixed background case. \nIn the second step of the successive infinitesimal time evolutions, one has to compute the time derivative of \nthe right-hand side of \n\\eq{eq:eombetawithg}.\\footnote{\\eq{eq:del12m21} corresponds to acting the time-derivative on $\\dot \\beta$'s.}\nIn the computation, the main difference from the static background case \nis that we have to take into account the time derivative of the metric as well, which\naffects not only the metric itself but also the covariant derivatives and the curvature tensor.\nTherefore, \nwhile the number of terms in \\eq{eq:eombetawithg} has substantially been reduced from \\eq{eq:explicitbetas}\nby the identification \\eq{eq:geqbeta2}, there appear a number of new terms in the \nsecond step, which someway set back the reduction.\nOne can compute these extra contributions in a similar manner as was done in Section~\\ref{sec:backmetric}.\nFor instance, as for $\\beta$, \n\\[\n\\begin{split}\n\\frac{d}{dt} \\beta_{,\\mu}&={\\dot \\beta}_{,\\mu}+\\frac{1}{2} \\dot \\Gamma_\\mu \\beta,\\\\\n\\frac{d}{dt} \\beta_{,\\mu\\nu}&={\\dot \\beta}_{,\\mu\\nu}-\\dot \\Gamma_{\\mu\\nu}^\\rho \\beta_{,\\rho}+\\frac{1}{2} \n\\dot \\Gamma_\\mu \\beta_{,\\nu}+\\frac{1}{2} \\nabla_\\mu(\\dot \\Gamma_\\nu \\beta),\n\\end{split}\n\\label{eq:dotsecond}\n\\]\nwhere the terms with $\\dot \\Gamma_\\mu$ are due to the weight of $\\beta$'s in \\eq{eq:weightbeta}. Here, \n$\\dot \\Gamma_{\\mu\\nu}^\\rho$ is explicitly given by \\eq{eq:dotgamma}. \nAs for the curvature tensor, since the curvature is in the second order by itself, it is enough to consider the non-derivative part \n\\eq{eq:dotg} of $\\dot g_{\\mu\\nu}$, and we obtain\\footnote{The computation is simplified by noticing that\nthe non-derivative part of \\eq{eq:dotg} is just a conformal transformation.} \n\\[\n\\dot R_{\\mu\\nu}=\\frac{24}{d+2}\\left((d-2)\\nabla_\\mu \\nabla_\\nu (n\\beta)+g_{\\mu\\nu} \\nabla^2(n\\beta)\\right)+{\\cal O}(\\nabla^4).\n\\label{eq:dotR}\n\\]\nBy using these expressions, one can compute the commutation of infinitesimal time evolutions, and obtain \n\\[\n\\begin{split}\n&(\\delta_{n_1}\\delta_{n_2}-\\delta_{n_2}\\delta_{n_1})\\beta\n=-\\tilde g^{\\mu\\nu}v_\\mu \\beta_{,\\nu}+\\frac{1}{2} \\tilde g^{\\mu\\nu}v_{\\mu,\\nu}\\beta +{\\cal O}(\\nabla^4), \\\\\n&(\\delta_{n_1}\\delta_{n_2}-\\delta_{n_2}\\delta_{n_1})\\beta^{\\mu\\nu}\n=\\frac{1}{\\beta^2}\\left(2\\tilde g^{\\rho(\\mu}\\tilde g^{\\nu)\\sigma}v_{\\rho,\\sigma}\n \\beta+\\tilde g^{\\mu\\nu}\\tilde g^{\\rho\\sigma}\\left(\\frac{1}{2} v_{\\rho,\\sigma} \\beta+v_\\rho \\beta_{,\\sigma}\\right)\\right)\n +{\\cal O}(\\nabla^4),\n \\label{eq:hhbeta}\n\\end{split}\n\\]\nwhere \n\\[\nv_\\mu=12 (n_1 n_{2,\\mu}-n_2 n_{1,\\mu}).\n\\]\nOne can easily check that the right-hand sides are the same as \\eq{eq:n1n2v} with \\eq{eq:vval}, when \\eq{eq:geqbeta2} \nis taken into account. \nThus, the right-hand sides of (\\ref{eq:hhbeta}) represent the diffeomorphism transformations, and \nthe consistency of the time evolution in the case of the evolving background with \\eq{eq:geqbeta2} \nhas also been established.\n\n\\section{Deletion of the weights}\n\\label{sec:delete}\nSo far, the field $\\beta$ and the lapse function $n$ have the weights of negative and positive half-densities, respectively.\nWhile these are the natural weights in the framework of CTM, \nscalars with such weights are not standard in general relativity.\nTherefore, we want to transform them into simple scalars with no weights. \nAt first glance, this seems to be a trivial task by doing the replacement,\n$\\beta\\rightarrow g^{-\\frac14} \\beta$ and\n$n\\rightarrow g^\\frac{1}{4} n$, in the equation of motion (EOM) \\eq{eq:eombetawithg}.\nHowever, while the former is obvious, there is a subtle issue in the latter replacement. \n\nWhen we have shown the algebraic relation between the commutation of two infinitesimal time evolutions and \nthe diffeomorphism in the preceding sections,\nit is implicitly assumed that $n_2$ does not change after the first infinitesimal time evolution with $n_1$, and vice versa.\nNamely, the algebraic relation has been shown in the situation that the lapse functions with the weight\nof half-density do not change after the infinitesimal time evolutions.\nOn the other hand, if we do the replacement $n\\rightarrow g^\\frac{1}{4} n$, and assume that the new lapse functions \nwith no weights do not change after a first infinitesimal time evolution, the situation becomes in fact different by the \nevolution of the weight $g^\\frac{1}{4}$ from the original one. This means that the commutation of two infinitesimal time evolutions \nis a sum of a diffeomorphism and an infinitesimal time evolution\nwith the following lapse function:\n\\[\nn_{12}=-\\frac{1}{4} g_{\\mu\\nu}\\dot g^{\\mu\\nu}(n_1)n_2+\\frac{1}{4} g_{\\mu\\nu}\\dot g^{\\mu\\nu}(n_2)n_1.\n\\label{eq:additional}\n\\]\nHere, we have explicitly written the lapse function dependence of $\\dot g^{\\mu\\nu}$, while it depends also on $\\beta$ \nand $g^{\\mu\\nu}$.\nOf course, the appearance of an additional time evolution\nis not a breakdown of the framework, because the algebraic closure of the diffeomorphism and \nthe infinitesimal time evolution anyway holds. But, this deformed algebraic structure is inconvenient, \nif we want to compare CTM with the ADM formalism of general relativity.\n\nTo fix this issue, let us consider the following reparameterization of the lapse function,\n\\[\nn\\rightarrow \\tilde n =n+h(\\beta,g^{\\mu\\nu},n),\n\\label{eq:repn}\n\\] \nwhere $h$ is a scalar function linear in $n$, \nand is assumed to be in the order of second derivatives.\\footnote{A direct way to compensate, \nsuch as $n\\rightarrow g^{-\\frac{1}{4}} n$, cannot be taken, because $n$ is supposed to be a scalar with no weights, and its\nweight should not be changed.}\nThe reason for $h$ to be taken in the second order is that we want to keep the result in the main order,\nnamely, the part expressed by the diffeomorphism. \nThen, the condition to compensate \\eq{eq:additional} is given by\n\\[\n\\begin{split}\n&\\int dx \\left[ \\dot\\beta(x,n_1) \\frac{\\delta}{\\delta \\beta(x)}\n+\\dot g^{\\mu\\nu}(x,n_1)\\frac{\\delta}{\\delta g^{\\mu\\nu}(x)}\\right]h(\\beta,g^{\\mu\\nu},n_2)\n-\\frac{1}{4} g_{\\mu\\nu} \\dot g^{\\mu\\nu}(n_1)n_2 -(n_1 \\leftrightarrow n_2)\\\\\n&\\hspace{13cm}={\\cal O}(\\nabla^4).\n\\end{split}\n\\label{eq:hcond}\n\\]\n\nBefore discussing the solution for $h$ to \\eq{eq:hcond}, \nlet us first discuss the explicit expressions of the EOM\nin the case with no weights.\nSo, let us leave aside the replacement $n\\rightarrow \\tilde n$ for the moment.\nAfter the rescaling by the weight factors, i.e., $\\beta\\rightarrow g^{-\\frac14} \\beta$ and\n$n\\rightarrow g^\\frac{1}{4} n$, the EOM has the form, \n\\[\n\\begin{split}\ng^\\frac{1}{4} \\frac{d}{dt} \\left(g^{-\\frac{1}{4}} \\beta\\right)&=K(\\beta,g^{\\mu\\nu},n), \\\\\ng^\\frac{1}{4}\\frac{d}{dt} \\left( g^{-\\frac{1}{4}}\\beta^{\\mu\\nu}\\right) &=K^{\\mu\\nu}(\\beta,g^{\\mu\\nu},n),\n\\end{split}\n\\label{eq:gK}\n\\]\nwhere $K$ and $K^{\\mu\\nu}$ are given by the right-hand sides of \\eq{eq:eombetawithg} with\nthe formal replacement $\\tilde g^{\\mu\\nu} \\rightarrow g^{\\mu\\nu}$.\nThe left-hand sides of \\eq{eq:gK} can be written in the way,\n\\[\n\\left(\n\\begin{array}{cc}\n1 & \\frac{1}{4} \\beta g_{\\rho\\sigma} \\\\\n-\\frac{1}{\\beta^2}g^{\\mu\\nu} & \\frac{1}{\\beta} I_{\\rho\\sigma}^{\\mu\\nu} +\\frac1{4\\beta} g^{\\mu\\nu}g_{\\rho\\sigma} \n\\end{array}\n\\right)\n\\left(\n\\begin{array}{c}\n\\dot \\beta \\\\\n\\dot g^{\\rho\\sigma}\n\\end{array}\n\\right),\n\\label{eq:dotbdotg}\n\\]\nwhere $I^{\\mu\\nu}_{\\rho\\sigma}= \\delta^{(\\mu}_\\rho \\delta^{\\nu)}_\\sigma$, and \n\\eq{eq:geqbeta2} has been used.\nIt is easy to find the inverse of the matrix in \\eq{eq:dotbdotg}, and we obtain\n\\[\n\\left(\n\\begin{array}{c}\n\\dot \\beta \\\\\n\\dot g^{\\mu\\nu}\n\\end{array}\n\\right)=\n\\left(\n\\begin{array}{cc}\nc_1 & c_2 \\beta^2 g_{\\rho\\sigma} \\\\\n\\frac{c_3}{\\beta}g^{\\mu\\nu} & \\beta I_{\\rho\\sigma}^{\\mu\\nu} +c_4 \\beta g^{\\mu\\nu}g_{\\rho\\sigma} \n\\end{array}\n\\right)\n\\left(\n\\begin{array}{c}\nK(\\beta,g^{\\mu\\nu}, n) \\\\\nK^{\\rho\\sigma}(\\beta,g^{\\mu\\nu},n)\n\\end{array}\n\\right),\n\\label{eq:eomscalar}\n\\]\nwhere\n\\[\nc_1=\\frac{d+4}{2(d+2)},\\ c_2=-\\frac{1}{2(d+2)},\\ c_3=\\frac{2}{d+2},\\ c_4=-\\frac{1}{d+2}.\n\\] \n\nNow let us discuss the replacement $n\\rightarrow \\tilde n$. \nTo solve the condition \\eq{eq:hcond} for $h$, let us assume the following form,\n\\[\nh(\\beta,g^{\\mu\\nu},n)=\\frac{g^{\\mu\\nu}}{\\beta^2}\\left(\nz_1 \\frac{n \\beta_{,\\mu}\\beta_{,\\nu}}{\\beta^2}+z_2 \\frac{\\beta_{,\\mu}n_{,\\nu}}{\\beta}\n+z_3 \\frac{n\\beta_{,\\mu\\nu}}{\\beta}+z_4 n_{,\\mu\\nu}\n\\right),\n\\label{eq:hass}\n\\]\nwhere $z_i$ are parameters. This form is chosen so that the reparameterization \\eq{eq:repn} preserves the\noriginal form of the EOM.\nBy substituting $\\dot g^{\\mu\\nu}$ in \\eq{eq:hcond} with \\eq{eq:eomscalar}, \nwe find that \\eq{eq:hcond} can be solved by\n\\[\n\\begin{split}\n&(d-6) z_3 + 2 (2 + d) z_4 = \\frac{-12 - 44 d + 17 d^2}{6(d+2)},\\\\\n&2 ( d-6) z_1 + (10 + d) z_2 + \n 4 (3 d - 10) z_3 + 8(2 -d) z_4 =\\frac{2 (-12 - 4 d + 11 d^2)}{3 (d + 2)}.\n \\end{split}\n \\label{eq:condz}\n \\]\nThe solutions form a two-parameter family, and any of them can be used for the purpose.\n\nThe final form of the EOM with no weights of the field and the lapse function \nis obtained by \ndoing the replacement $n\\rightarrow \\tilde n$ in \\eq{eq:eomscalar}.\nBecause our concern is up to the second order, the replacement \nis effective only in the zeroth order terms in \\eq{eq:eomscalar}. \nBy explicitly computing \\eq{eq:eomscalar}, we obtain\n\\[\n\\begin{split}\n\\dot \\beta&=-\\frac{3(d-6)}{d+2} \\beta^2 \\left( n+h(\\beta,g^{\\mu\\nu},n)\\right)\n+\\frac{1}{d+2} nR-\\frac{17d^2+20d+36}{(d+2)^2 \\beta^2} n \\beta_{,\\mu}\\beta^{,\\mu}\\\\\n&\\ \\ \\ \\ -\\frac{2(d^2+20d-4)}{(d+2)^2 \\beta} \\beta_{,\\mu} n^{,\\mu} \n+\\frac{11d^2-20 d+60}{2(d+2)^2 \\beta}n \\beta_{,\\mu}^{,\\mu}+\\frac{3 d^2-20 d+92}{2(d+2)^2}n_{,\\mu}^{,\\mu}\n+{\\cal O}(\\nabla^4), \\\\\n\\dot g^{\\mu\\nu}&=\\frac{48 \\beta}{d+2} g^{\\mu\\nu} \\left(n+h(\\beta,g^{\\rho\\sigma},n)\\right)-\\frac{2}{\\beta}nR^{\\mu\\nu}\n+ \\frac{2 }{(d+2)\\beta}nR g^{\\mu\\nu}\\\\\n&\\ \\ \\ \n-\\frac{4(d-14)}{(d+2) \\beta^3} n \\beta^{,\\mu} \\beta^{,\\nu}-\\frac{24(d-2)}{(d+2)\\beta^2} n^{,(\\mu}\\beta^{,\\nu)}\n-\\frac{2(3d-2)}{(d+2)\\beta^2} n \\beta^{,\\mu\\nu}\n-\\frac{2(d-6)}{(d+2)\\beta} n^{,\\mu\\nu}\\\\\n&\\ \\ \\ \n+g^{\\mu\\nu}\\left(\\frac{32(3d+2)}{(d+2)^2\\beta^3}n \\beta_{,\\rho}\\beta^{,\\rho}\n-\\frac{32(2d-1)}{(d+2)^2\\beta^2}n_{,\\rho}\\beta^{,\\rho} -\\frac{16(4d-1)}{(d+2)^2\\beta^2}n\\beta_{,\\rho}^{,\\rho}\n-\\frac{16(2d-5)}{(d+2)^2 \\beta} n_{,\\rho}^{,\\rho} \\right) \\\\\n& \\ \\ \\ +{\\cal O}(\\nabla^4).\n\\end{split}\n\\label{eq:eomfinal}\n\\]\n\nFor a consistency check of this result, one can compute the commutation of two infinitesimal time evolutions,\nas done before.\nThe basic strategy is the same. In the second step of the infinitesimal time evolution, one has to take the \ntime derivative of the right-hand sides of \\eq{eq:eomfinal}. Not only the metric itself, but we also take into account \nthe time derivative of the second covariant derivatives\\footnote{The difference from the previous case \\eq{eq:dotsecond}\nis the absence of weights, namely, $\\dot \\Gamma_\\mu$ is absent.\nBecause of this, the first covariant derivatives have no time-dependencies.} and the curvature.\nSince our concern is up to the second order, the time-derivative of the Christoffel symbol and the curvature can be evaluated by the zeroth order of $\\dot g^{\\mu\\nu}$, as given in \\eq{eq:dotgamma} and \\eq{eq:dotR}, respectively.\nThen, we obtain \n\\[\n\\begin{split}\n&(\\delta_{n_1}\\delta_{n_2} -\\delta_{n_2}\\delta_{n_1})\\beta=-v^\\mu \\beta_{,\\mu}+{\\cal O}(\\nabla^4), \\\\\n&(\\delta_{n_1}\\delta_{n_2} -\\delta_{n_2}\\delta_{n_1})g^{\\mu\\nu}=2v^{(\\mu,\\nu)}+{\\cal O}(\\nabla^4),\n\\end{split}\n\\] \nwhere \n\\[\nv^\\mu=12\\left( n_1 n_2^{,\\mu}-n_2 n_1^{,\\mu}\\right).\n\\label{eq:metricv}\n\\]\nThe right-hand sides certainly agree with the standard diffeomorphism in general relativity for a scalar and a metric.\nIt should be stressed that this result can be obtained, only when the correction $h(\\beta,g^{\\mu\\nu},n)$ with the parameters \nsatisfying \\eq{eq:condz} is included in the equation of motion as in \\eq{eq:eomfinal}.\n\nNow, let us briefly discuss the inclusion of the terms corresponding to those parameterized \nby a shift vector in the Hamiltonian of the ADM formalism of general relativity. \nThe last term of the EOM of CTM in \\eq{eq:ctmeom} represents an arbitrary infinitesimal \nSO$({\\cal N})$ transformation.\nAs discussed in Section~\\ref{sec:gauge}, it contains the diffeomorphism and the spin-three gauge transformation \nin the present context. However, since the latter is used to maintain the gauge-fixing condition $\\beta^{\\mu\\nu\\rho}=0$, \nonly the diffeomorphism can be set arbitrary.\nThe diffeomorphism transformation \\eq{eq:diffeo} and the identification \\eq{eq:geqbeta2} imply that \n$\\beta$ and $g^{\\mu\\nu}$ are transformed in the standard way of general relativity. \nThus, implementing the following replacement in \\eq{eq:eomfinal},\n\\[\n\\begin{split}\n\\dot \\beta&\\rightarrow \\dot\\beta+n^\\mu \\beta_{,\\mu}, \\\\\n\\dot g^{\\mu\\nu}&\\rightarrow \\dot g^{\\mu\\nu}-2n^{(\\mu,\\nu)},\n\\end{split}\n\\]\nwhere $n^\\mu$ is a newly introduced shift vector, \none obtains the EOM with the shift vector.\n\n\\section{Deletion of the derivatives of the lapse function}\n\\label{sec:reparametrization}\nThe equation of motion (EOM) \\eq{eq:eomfinal} contains some terms with the derivatives of $n$.\nAs discussed below \\eq{eq:explicitbetas},\nthis is an obstacle for a general relativistic interpretation of the EOM of CTM.\nIn this section, we will show that, by redefining the fields $\\beta,g^{\\mu\\nu}$ with some \nderivative corrections, one can actually delete all the terms with the derivatives of $n$ from the EOM. \n\nThe reparameterization of the fields we consider is given by adding some correction terms with second order of derivatives:\n\\[\n\\begin{split}\n\\beta&\\rightarrow \\beta+x_1\\, \\frac{\\beta_{,\\mu}\\beta^{,\\mu}}{\\beta^3}+x_2 \\, \\frac{\\beta_{,\\mu}^{,\\mu}}{\\beta^2}\n+x_7\\,\\frac{R}{\\beta}, \\\\\ng^{\\mu\\nu}&\\rightarrow g^{\\mu\\nu} +x_3\\, \\frac{\\beta^{,\\mu}\\beta^{,\\nu}}{\\beta^4}+x_4\\, \\frac{\\beta^{,\\mu\\nu}}{\\beta^3}\n+x_5\\, \\frac{g^{\\mu\\nu} \\beta^{,\\rho}\\beta_{,\\rho}}{\\beta^4}+x_6\\, \\frac{g^{\\mu\\nu}\\beta^{,\\rho}_{,\\rho}}{\\beta^3}\n+x_8\\, \\frac{R^{\\mu\\nu}}{\\beta^2}+x_9\\,\\frac{g^{\\mu\\nu} R}{\\beta^2},\n\\label{eq:replacement}\n\\end{split}\n\\]\nwhere $x_i$'s are parameters. Note that, since the reparameterization is covariant, the algebraic consistency \nbetween the commutation of time evolutions and the diffeomorphism obtained so far should be \nunaltered\\footnote{For sure, we have checked it through explicit computations.}.\n\nThere exist two kinds of effects from this reparameterization.\nThe first one is on the right-hand side of \\eq{eq:eomfinal}. Since the corrections are in the second order of derivatives, \nthe reparameterization is effective only on the zeroth order term, \nand causes some shifts of the coefficients of the non-derivative terms of $n$.\nOn the other hand, the reparameterization affects the left-hand side more importantly for our purpose.\n$\\dot \\beta$ will be replaced by\n\\[\n\\begin{split}\n\\dot \\beta \\rightarrow & \\dot \\beta+x_1 \\left( - \\frac{3\\dot \\beta \\beta_{,\\mu}\\beta^{,\\mu}}{\\beta^4}\n+\\frac{2\\dot\\beta_{,\\mu}\\beta^{,\\mu}+\\dot g^{\\mu\\nu}\\beta_{,\\mu}\\beta_{,\\nu}}{\\beta^3}\\right)\\\\\n&\\ \\ \\ +\nx_2 \\left( -\\frac{2 \\dot \\beta \\beta_{,\\mu}^{,\\mu}}{\\beta^3}+\\frac{\\dot \\beta_{,\\mu}^{,\\mu}+\\dot g^{\\mu\\nu}\\beta_{,\\mu\\nu}\n-g^{\\mu\\nu}\\dot\\Gamma_{\\mu\\nu}^\\rho \\beta_{,\\rho}}{\\beta^2} \\right)\n+x_7\\, \\left( -\\frac{R\\dot \\beta}{\\beta^2}+\\frac{\\dot g^{\\mu\\nu}R_{\\mu\\nu}+g^{\\mu\\nu}\\dot R_{\\mu\\nu}}{\\beta} \\right).\n\\end{split}\n\\label{eq:changedotbeta}\n\\] \nTo evaluate the correction terms in \\eq{eq:changedotbeta} up to the second order of derivatives, \nwe can put the zeroth order expressions of the time-derivative of the fields, \ni.e., the first equation of \\eq{eq:zeroth}, \\eq{eq:dotg}, \\eq{eq:dotgamma}, and \\eq{eq:dotR}, \ninto them. The things are similar for the correction terms in the replacement of $g^{\\mu\\nu}$ in \\eq{eq:replacement}. \nThen, because the zeroth order expressions contain $n$, there emerge a number of terms which contain\nthe derivatives of $n$. In fact, we can delete all the derivative terms of $n$ in the EOM \nby appropriately choosing the $x_i$'s.\nThe condition for the deletion is expressed by six equations, which are explicitly given in Appendix~\\ref{app:deleten}. \nSolving the equations for $x_1,\\cdots, x_6$, and putting the solutions into the EOM, \nwe obtain\n\\[\n\\begin{split}\n\\frac{1}{n}\\dot \\beta&=\n-\\frac{3 (-6 + d) \\beta^2 }{2 + d}+\\frac{(1 + (6 - 9 d) x_7) R}{2 + d}\n-\\frac{16 (-1 + d) (-1 + (-6 + 9 d) x_7) \n\\beta_{,\\mu}^{,\\mu}}{(-6 + d) (2 + d) \\beta}\\\\\n&+\\frac{2 (-8 (11 + 84 x_7) + d^3 (-1 + 360 x_7) + 4 d (43 + 480 x_7) - \n 2 d^2 (19 + 804 x_7)) \\beta_{,\\mu}\\beta^{,\\mu}}{(-6 + d)^2 (2 + d) \\beta^2}\\\\\n&+{\\cal O}(\\nabla^4),\\\\\n\\frac{1}{n} \\dot g^{\\mu\\nu}&=\\frac{48 \\beta g^{\\mu\\nu} }{2 + d}\n+\\frac{2 (1 + 24 x_7 - 3 (2 + d) x_9) g^{\\mu\\nu} R}{(2 + d) \\beta}\n-\\frac{2 (1 + 3 x_8) R^{\\mu\\nu}}{\\beta}\\\\\n&+\\frac{A_1 g^{\\mu\\nu} \\beta_{,\\rho}\\beta^{,\\rho}}{(-6 + d)^2 (2 + d) \\beta^3}\n+\\frac{16 (48 + 84 x_8 + 3 d^2 x_8 - 8 d (1 + 6 x_8)) \\beta^{,\\mu}\\beta^{,\\nu}}{(-6 + d)^2 \\beta^3}\\\\\n&-\\frac{16 (4 + 48 x_7 + 6 x_8 - 12 x_9 + 6 d^2 x_9 + \n d (-1 - 48 x_7 + 3 x_8 + 6 x_9)) g^{\\mu\\nu} \n\\beta_{,\\rho}^{,\\rho}}{(-6 + d) (2 + d) \\beta^2}\\\\\n&-\\frac{8 (-6 + d - 12 x_8 + 6 d x_8) \\beta^{,\\mu\\nu} }{(-6 + d) \\beta^2}+{\\cal O}(\\nabla^4),\n\\end{split}\n\\label{eq:eomCTM}\n\\]\nwhere \n\\[\n\\begin{split}\nA_1&=16 (2 d (13 + 456 x_7 - 6 x_8 - 72 x_9) - 4 (20 + 168 x_7 + 33 x_8 - 42 x_9) + 30 d^3 x_9\\\\\n&\\hspace{3cm}- 3 d^2 (1 + 80 x_7 - 9 x_8 + 18 x_9)).\n\\end{split}\n\\label{eq:A1}\n\\]\nInterestingly, the EOM does not depend on the two-dimensional ambiguity of the solutions of $z_i$'s to \\eq{eq:condz},\nand is parameterized solely by $x_{7,8,9}$.\nIn the following section, we will identify \\eq{eq:eomCTM} with \nthe EOM of general relativity coupled with a scalar field \nbased on the Hamilton-Jacobi approach. \n\nIn the EOM \\eq{eq:eomCTM}, one can see that \nthe scale transformation \\eq{eq:scaletrans} is realized as \n\\[\n\\begin{split}\nt&\\rightarrow L t,\\ x^{\\mu}\\rightarrow L x^\\mu, \\\\\n\\beta&\\rightarrow \\frac{\\beta}{L},\n\\end{split}\n\\label{eq:scaletranseom}\n\\]\nwhile $n,g^{\\mu\\nu}$ are invariant.\n\n\\section{Hamilton-Jacobi equation of general relativity coupled with a scalar field}\n\\label{sec:conttheory}\nIn this section, starting with an action of general relativity coupled with a scalar field, \nand employing the Hamilton-Jacobi approach, \nwe identify the equations of motion (EOM) of this gravitational system with the EOM \\eq{eq:eomCTM} of CTM.\n\nIt is an easy task to guess a possible form of the action for the purpose:\n\\[\nS = \\int_{\\mathcal{M}} d^{d+1}x \\sqrt{- G} \\left( 2 R^{(d+1)} -\\frac{A}{2} G^{ij} \\partial_i \\phi \\partial_i \\phi -\\Lambda e^{2 B \\phi} \\right),\n\\label{eq:contaction}\n\\]\nwhere $G_{ij}$ denotes the $(d+1)$-dimensional metric with $i,j=0,1,2,\\cdots,d$; \n$R^{(d+1)}$ is the $(d+1)$-dimensional Ricci scalar; $\\phi$ is a real scalar field; \n$A,B,\\Lambda$ are real parameters. \nThe scalar field $\\phi$ is assumed to be related to the CTM field $\\beta$ through $\\beta=e^{B\\phi}$. \nThis action would be considered to be an effective action valid up to the second order of derivatives. \nThe classical EOM derived from \\eq{eq:contaction} respects\nthe dilatational symmetry \\eq{eq:scaletranseom}, because $S$ is transformed homogeneously \nby the transformation as $S\\rightarrow L^{d-1} S$. \n\nConsidering that the $(d+1)$-dimensional Lorentzian manifold $\\mathcal{M}$ is globally hyperbolic, \nwe use the following diffeomorphism,\n\\[\n\\varphi : \\ \\Sigma \\times \\mathbb{R} \\to \\mathcal{M}, \n\\label{eq:diff}\n\\] \nwhere $\\Sigma$ is a $d$-dimensional spatial hypersurface, \nto obtain the ADM metric as a pull-back, $\\varphi^*G$:\n\\[\nds^2 = - N^2 dt^2 + g_{\\mu \\nu} (dx^{\\mu} + N^{\\mu}dt )(dx^{\\nu} + N^{\\nu}dt),\n\\]\nwhere $N$, $N^{\\mu}$ and $g_{\\mu \\nu}$ are the lapse function, the shift vector and the $d$-dimensional metric on $\\Sigma$ with $\\mu, \\nu = 1,2, \\cdots, d$. \nHereafter we will turn off the shift vector, i.e., $N^{\\mu}=0$ for simplicity. \nThe terms associated with the non-zero shift vector can be recovered considering the time-dependent spatial diffeomorphism.\n\nBy the diffeomorphism (\\ref{eq:diff}), the action (\\ref{eq:contaction}) becomes\n\\[\nS= \\int dt\\ (K - V),\n\\]\nwhere $K$ is the kinetic term, \n\\[\nK= \\int_{\\Sigma_t} d^dx\\ \n\\left( \n\\frac{1}{2} \\mathcal{G}^{\\mu\\nu,\\rho\\sigma} \\dot g_{\\mu\\nu} \\dot g_{\\rho\\sigma} \n+\\frac{1}{2} \\mathcal{G}^{\\phi,\\phi}\\dot \\phi \\dot \\phi\n\\right)\n\\]\nwith \n\\[\n\\begin{split}\n\\mathcal{G}^{\\mu\\nu,\\rho\\sigma}&=\\frac{\\sqrt{g}}{N} \\left( \\frac{1}{2} (g^{\\mu\\rho}g^{\\nu\\sigma}+g^{\\mu\\sigma}g^{\\nu\\rho})\n-g^{\\mu\\nu}g^{\\rho\\sigma}\\right),\\\\\n\\mathcal{G}^{\\phi,\\phi}&=\\frac{A\\sqrt{g}}{N},\n\\label{eq:quadratic}\n\\end{split}\n\\]\nand $V$ is the potential term, \n\\[\nV \n= \\int_{\\Sigma_t} d^d x\\ N \\sqrt{g} \\left( \\Lambda e^{2B \\phi} -2 R+\\frac{A}{2} (\\nabla \\phi)^2 \\right),\n\\label{eq:potential}\n\\]\nin which $(\\nabla \\phi)^2 := g^{\\mu \\nu} \\nabla_{\\mu} \\phi \\nabla_{\\nu} \\phi$ with $\\nabla_{\\mu}$ being the covariant derivative \nassociated with the metric $g_{\\mu \\nu}$.\n\nTo employ the Hamilton-Jacobi formalism, \nlet us consider the following Hamilton's principal functional: \n\\[\nW =\\int_{\\Sigma_t} d^d x \\sqrt{g} \\left( \\lambda e^{B\\phi} -e^{-B\\phi} \\left( c_1 R+c_2 (\\nabla \\phi)^2 \\right)\\right)+{\\cal O}({\\nabla^4}),\n\\label{eq:W}\n\\]\nwhere $c_1,c_2,\\lambda$ are real parameters. \n$W$ is considered to be expressed as \nperturbative expansions in spatial derivatives up to the second order. \nThe potential in \\eq{eq:potential} and \n$W$ in \\eq{eq:W} must be related by the following Hamilton-Jacobi equation: \n\\[\nV + \\int_{\\Sigma_t} d^dx\\ \n\\frac{1}{2}\\left(\n\\mathcal{G}_{\\mu\\nu,\\rho\\sigma} \\frac{\\delta W}{\\delta g_{\\mu\\nu}} \\frac{\\delta W}{\\delta g_{\\rho\\sigma}}\n+\\mathcal{G}_{\\phi,\\phi} \\frac{\\delta W}{\\delta \\phi} \\frac{\\delta W}{\\delta \\phi} \n\\right) +{\\cal O}(\\nabla^4)\n= 0,\n\\label{eq:VandW}\n\\]\nwhere \n\\[\n\\begin{split}\n\\mathcal{G}_{\\mu\\nu,\\rho\\sigma}&=\\frac{N}{\\sqrt{g}}\\left(\\frac{1}{2} (g_{\\mu\\rho}g_{\\nu\\sigma}+g_{\\mu\\sigma}g_{\\nu\\rho})\n-\\frac{1}{d-1}g_{\\mu\\nu}g_{\\rho\\sigma}\\right),\\\\\n\\mathcal{G}_{\\phi,\\phi}&=\\frac{N}{A\\sqrt{g}},\n\\end{split}\n\\]\nbeing the inverse to \\eq{eq:quadratic}.\nInserting \\eq{eq:W} into \\eq{eq:VandW}, \nwe obtain\n\\[\nV = \n\\int_{\\Sigma_t} d^dx\\ \n\\sqrt{g}N \\,\n\\frac{1}{2}\\left[\n\\frac{1}{d-1} \n\\left( \\frac{\\lambda^2de^{2 B\\phi}}{4} +\\lambda H \\right)\n+\\frac{1}{A} \\left(2 B \\lambda F -B^2 \\lambda^2 e^{2 B \\phi} \\right)\n\\right] +{\\cal O}(\\nabla^4),\n\\label{eq:V}\n\\]\nwhere\n\\[\n\\begin{split}\nH&=\\frac{2-d}{2}\\left( c_1 R+c_2 (\\nabla \\phi)^2\\right)+(d-1)c_1 \\left(B \\nabla^2 \\phi+B^2 (\\nabla\\phi)^2\\right),\\\\\nF&=-B\\left(c_1 R -c_2 (\\nabla \\phi)^2\\right)-2 c_2 \\nabla^2 \\phi.\n\\end{split}\n\\label{eq:HandS}\n\\]\nComparing \\eq{eq:V} with \\eq{eq:potential}, we obtain some conditions for the parameters of $W$ as \n\\[\n\\begin{split}\n\\Lambda&=\\frac{\\lambda^2(-4 B^2 (-1 + d) + A d)}{8 A (-1 + d)},\\\\\n-2&=-\\frac{c_1 \\lambda(A (-2 + d) + 4 B^2 (-1 + d))}{4 A (-1 + d))},\\\\\n\\frac{A}{2}&=\\frac{\\lambda(A (-c_2 (-2 + d) + 2 B^2 c_1 (-1 + d)) + \n 4 B^2 c_2 (-1 + d))}{4 A (-1 + d)},\\\\\n0&=B \\lambda (A c_1 + 4 c_2).\n\\end{split}\n\\label{eq:relpara}\n\\]\nHere, the first equation comes from the comparison of the potential term, the second the curvature, and the third the scalar \nkinetic term. The last equation comes from the absence of $\\nabla^2 \\phi$ term in the potential. \n\nThe flow equations derived from $W$ is given by\\footnote{\nThe flow equations for $\\phi$ and $g_{\\mu \\nu}$ \nare originated with Hamilton's equations for $\\phi$ and $g_{\\mu \\nu}$ \nwith the replacement of conjugate momenta by $\\frac{\\delta W}{\\delta \\phi}$ and $\\frac{\\delta W}{\\delta g_{\\mu \\nu}}$, \nrespectively. \n} \n\\[\n\\begin{split}\n&\\frac{1}{N} \\dot \\phi=\\mathcal{G}_{\\phi,\\phi} \\frac{\\delta W}{\\delta \\phi} \n=\\frac{1}{A} \\left(B \\lambda e^{B\\phi}-e^{-B\\phi} F \\right) + {\\cal O}(\\nabla^4), \\\\\n&\\frac{1}{N}\\dot g_{\\mu\\nu}= \\mathcal{G}_{\\mu\\nu,\\rho\\sigma}\\frac{\\delta W}{\\delta g_{\\rho\\sigma}}\n=\\frac{\\lambda e^{B\\phi}}{2(1-d)}g_{\\mu\\nu}+e^{-B\\phi}\\left(H_{\\mu\\nu}+\\frac{1}{1-d} g_{\\mu\\nu} H\\right)\n+{\\cal O}(\\nabla^4), \n\\end{split}\n\\label{eq:eommetbare}\n\\]\nwhere\n\\[\n\\begin{split}\nH_{\\mu\\nu}=&c_1\\left( R_{\\mu\\nu}-\\frac{1}{2}g_{\\mu\\nu}R+B \\left( \\nabla_\\mu\\nabla_{\\nu}\\phi-g_{\\mu\\nu}\\nabla^2 \\phi \\right)\n-B^2 \\left(\\nabla_\\mu\\phi \\nabla_\\nu\\phi-g_{\\mu\\nu} (\\nabla \\phi)^2 \\right)\\right) \\\\\n&+c_2 \\left(\\nabla_\\mu\\phi \\nabla_\\nu\\phi-\\frac{1}{2}g_{\\mu\\nu} (\\nabla \\phi)^2 \\right).\n\\end{split}\n\\]\nThere is a relation, $H=g^{\\mu\\nu}H_{\\mu\\nu}$. \n\nTo compare \\eq{eq:eommetbare} with the EOM \\eq{eq:eomCTM} from CTM, \nlet us perform \na change of the variable, $\\beta=\\exp[B \\phi]$. Taking into account that \n$\\dot g^{\\mu\\nu}=-g^{\\mu\\mu'}g^{\\nu\\nu'}\\dot g_{\\mu'\\nu'}$, \nthe EOM \\eq{eq:eommetbare} can be rewritten as \n\\[\n\\begin{split}\n\\frac{1}{c_3 n}\\dot \\beta=&\\frac{B^2 \\lambda \\beta^2}{A}+\\frac{B^2 c_1 R}{A}+\\frac{2 c_2 \\beta_{,\\mu}^{,\\mu}}{A \\beta}\n-\\frac{3 c_2 \\beta_{,\\mu}\\beta^{,\\mu}}{A \\beta^2}+ {\\cal O}(\\nabla^4) ,\\\\\n\\frac{1}{c_3 n} \\dot g^{\\mu\\nu}=&\\frac{\\lambda \\beta g^{\\mu\\nu}}{2 (-1 + d)}\n-\\frac{c_1 R^{\\mu\\nu}}{\\beta}\n-\\frac{c_1 \\beta^{,\\mu\\nu}}{\\beta^2}\n+\\frac{(2 B^2 c_1 - c_2) \\beta^{,\\mu}\\beta^{,\\nu}}{B^2 \\beta^3} \\\\\n&+g^{\\mu\\nu}\\left( \\frac{c_1 R}{2 (-1 + d) \\beta}\n+\\frac{c_2 \\beta_{,\\rho} \\beta^{,\\rho}}{2 B^2 (-1 + d) \\beta^3}\n\\right)+ {\\cal O}(\\nabla^4),\n\\end{split}\n\\label{eq:eomcont}\n\\]\nwhere we have introduced possible difference of normalizations between the lapse functions \nof CTM and general relativity as $N=c_3\\, n$ with a constant $c_3$. \nWe want to find the values of the parameters which make \\eq{eq:eomcont} coincident with \\eq{eq:eomCTM}.\nThe number of parameters is smaller than that of the equations to be satisfied (i.e., an overdetermined set of equations), \nbut we can solve the coincidence condition by the following values:\n\\[\n\\begin{split}\n\\lambda c_3 &=\\frac{96(-1+d)}{2+d}, \\\\\nB^2&=\\frac{A(6-d)}{32(d-1)},\\\\\nc_1 c_3&=\\frac{8(2+d)}{-10+7d},\\\\\nc_2 c_3 &=-\\frac{2 A (2 + d)}{-10 + 7 d}, \\\\\nx_7&=\\frac{16 - 88 d + 26 d^2 + d^3}{12 (-20 + 64 d - 65 d^2 + 21 d^3)},\\\\\nx_8&=\\frac{6 - d}{-10 + 7 d}, \\\\\nx_9&=\\frac{14 - 67 d + 17 d^2}{-60 + 192 d - 195 d^2 + 63 d^3}.\n\\end{split}\n\\label{eq:detpara}\n\\]\nThe details of the derivation of the solution are given in Appendix~\\ref{app:solution}.\nThe parameter $c_3$ can be determined by the second (or equivalently the third) \nequation of \\eq{eq:relpara} by putting \\eq{eq:detpara}:\n\\[\nc_3^2=12.\n\\label{eq:valuec3}\n\\] \nThis rather strange value actually normalizes the overall factor\nin the algebraic relation \\eq{eq:metricv} of the CTM to the natural value in GR.\nIt can be checked that the third and fourth equations\nof \\eq{eq:relpara} are also satisfied by \\eq{eq:detpara} and \\eq{eq:valuec3}. \nFrom the first equation of \\eq{eq:relpara}, \\eq{eq:detpara} and \\eq{eq:valuec3}, we obtain\n\\[\n\\Lambda=\\frac{36 (d-1)(3d-2)}{(d+2)^2}.\n\\label{eq:valuelambda}\n\\]\nThe above solution is unique except for the rather obvious ambiguities of the signs of $B$ and $c_3$.\nThese signs are physically irrelevant, because the sign of $B$ can be absorbed by that of $\\phi$, and \nthat of $c_3$ just determines the overall sign of $W$ (or can be absorbed in $n$).\n\nIf we require the positivity of the potential energy from the spatial derivative term of $\\phi$, $A>0$ is required. Then, \nthe second equation of \\eq{eq:detpara} implies that the dimension must be in the range \n$2\\leq d\\leq 6$ (The $d=1$ case is excluded from the beginning in the Hamilton formalism, \nas can be seen at the beginning of this section.).\nIn this range, \\eq{eq:valuelambda} is positive, and one can normalize the value of $\\Lambda$ \nby rescaling the space-time coordinates as $(t,x^\\mu)\\rightarrow L (t,x^\\mu)$ with $L=1\/\\sqrt{\\Lambda}$ and \ndropping an overall factor of the action.\nWe can also rescale the scalar field as $\\phi \\rightarrow \\hbox{sign(}B\\hbox{)}\\phi\/\\sqrt{A}$.\nThen, the action describing CTM is uniquely determined, \nfor a globally hyperbolic $\\mathcal{M}$, \nto be \n\\[\nS_{CTM}=\\int_{\\mathcal{M}} d^{d+1} x\\,\\sqrt{-G} \\left( \n2 R-\\frac{1}{2} G^{ij} \\partial_{i} \\phi \\partial_{j} \\phi \n- e^{\\sqrt{\\frac{6-d}{8(d-1)}}\\phi}\n\\right),\n\\label{eq:CTMaction}\n\\]\nwhich is valid in $2\\leq d \\leq 6$. Thus, \nthe system has a critical dimension $d=6$, over which it becomes unstable due to the \nwrong sign of the scalar kinetic term. \nAt the critical dimension, the scalar is a massless field with no non-derivative couplings.\n\n\\section{Time evolution of the scale factor}\n\\label{sec:mini}\nThe coupled system of gravity and a scalar field described by the action \\eq{eq:CTMaction} \nhas been discussed in the context of models of dark energy \n(See \\cite{Copeland:2006wr} for a comprehensive review.).\nThe exponential potential in \\eq{eq:CTMaction} of the scalar field is known to lead to a power-law \nbehavior (or an exponential behavior in the critical case) of the scale factor. \nLet us see this in our case, analyzing \\eq{eq:eomCTM}.\n\nDiscarding the spatial derivative terms of \\eq{eq:eomCTM} and putting $n=1$, the equation of motion is given by\n\\[\n&\\dot \\beta=d_1 \\beta^2,\n\\label{eq:minibeta} \\\\\n&\\dot g^{\\mu\\nu}=d_2 \\beta g^{\\mu\\nu},\n\\label{eq:minig}\n\\]\nwhere \n\\[\nd_1=\\frac{3(6-d)}{d+2},\\ d_2=\\frac{48}{d+2}.\n\\]\nSubstituting \\eq{eq:minig} with an ansatz $g^{\\mu\\nu}=a(t)^{-2} \\delta^{\\mu\\nu}$ \nwith a scale factor $a(t)$, we obtain\n\\[\n\\frac{2\\dot a}{a}=-d_2 \\beta. \n\\label{eq:minia}\n\\]\nThen, for $d_1\\neq 0\\ (\\hbox{i.e., }d\\neq 6)$, the solution to \\eq{eq:minibeta} and \\eq{eq:minia} is obtained as\n\\[\n\\begin{split}\n&\\beta=\\frac{1}{d_1(t_0-t)},\\\\\n&a=a_0 (t_0-t)^{\\frac{d_2}{2d_1}},\n\\end{split}\n\\]\nwhere $t_0$ and $a_0$ are integration constants.\n\nWhen $d=6$, $d_1$ vanishes. In this case, $\\beta$ is given by a constant, say $\\beta_0$. Then, \n\\eq{eq:minia} gives\n\\[\na=a_0 \\exp \\left[ -\\frac{d_2 \\beta_0}{2} t \\right].\n\\]\nThus, we see that, in the critical case $d=6$, the solution is given by de Sitter spacetime.\n\nAs is well known, de Sitter spacetime has the invariance of a conformal symmetry $SO(d+1,1)$. \nIn statistical physics, the appearance of a conformal symmetry is the sign that a system is on a critical point. \nThis suggests that CTM at $d=6$ is on a critical point in some sense. In fact, as shown in Section~\\ref{sec:conttheory}, \nfor the reality of $B$, the sign of the kinetic term of the scalar field must change its sign at $d=6$.\nIn $d>6$, it gets the wrong sign, and the the scalar field becomes unstable in the direction \nof larger spatial fluctuations.\nThis means that $d=6$ can be thought of as a phase transition point \nbetween a stable phase at $d<6$ and another phase at $d>6$.\nConsidering the instability in the direction of larger spatial fluctuations, \nthe latter phase probably contradicts our assumption of a continuous\nspace. The understanding of the phase transition should be pursued further.\n\n\\section{Summary and future prospects}\n\\label{sec:summary}\nIn this paper, we have analyzed the equation of motion (EOM) of the canonical tensor model (CTM)\nin a formal continuum limit by employing a derivative expansion of its tensor up to the fourth order. \nWe have shown that, up to the order, the EOM of CTM in the continuum limit agrees with \nthat of a coupled system of gravity and a scalar field obtained in the framework of the Hamilton-Jacobi methodology. \nThe action of the gravitational system is composed of the curvature term, the scalar field kinetic term\nand an exponential potential of the scalar field. The system is classically invariant under a dilatational transformation. \nThe action is physically valid in the range of the spatial dimensions, $2\\leq d \\leq 6$,\nand, in $d>6$, the system is unstable due to the wrong sign of the kinetic term of the scalar field.\nAt the critical case $d=6$, de Sitter spacetime is a solution to the EOM, while, in $2\\leq d < 6$,\nthe time evolution of the scale factor of a flat space has a power-law behavior.\n \nThe most significant achievement of this paper is to have concretely shown that CTM indeed derives a \ngeneral relativistic system in a formal continuum limit. This was conjectured in our previous paper \\cite{Sasakura:2015pxa}\nfrom the observation that\nthe constraint algebra of CTM in the continuum limit agrees with that of the ADM formalism, \nbut no concrete correspondences were given. \nOn the other hand, in this paper, we have obtained the one-to-one correspondence of the fields \nbetween CTM in the continuum limit up to the fourth order \nand the gravitational system so that the two systems have a common EOM.\nThe action of the corresponding gravitational system has also been obtained.\n\nAn interesting question arising from our result is what is the meaning of the criticality at $d=6$.\nThe existence of de Sitter spacetime solution implies that the system has a conformal symmetry on this background\nin the dimension.\nOn the other hand, in our previous papers \\cite{Sasakura:2015xxa,Sasakura:2014zwa,Sasakura:2014yoa}, it was shown \nthat the Hamiltonian vector flows of CTM can be regarded as RG flows of statistical systems on random networks.\nThese two aspects of CTM suggest that a statistical system at criticality described by a \nsix-dimensional conformal field theory is associated to CTM \\cite{Strominger:2001pn,Strominger:2001gp}.\nIt would be interesting to identify the conformal field theory in a concrete manner.\n\nAnother interesting direction of study would be to extend the derivative expansion to higher orders,\nwhich includes higher spin fields than two with higher spin gauge symmetries. \nThere are general interests in pursuing higher spin gauge theories (See \\cite{Giombi:2016ejx} for a recent review.).\nSince our approach has a significant difference from the other ones in the sense \nthat we take a formal continuum limit of a consistent \ndiscretized theory in the canonical formalism, we would expect that our model may shed some new lights on the subject. \nFor that purpose, it would be necessary to set up a new efficient methodology for the analysis instead of relying on machine \npowers as in this paper.\n\nThe EOM of CTM is a set of first-order differential equations in time, and has been related \nto a gravitational system through the Hamilton-Jacobi equation.\nWhile the gravitational system contains the phenomena of second-order differential equations like wave propagations,\nit is not clear how to realize such phenomena in the framework of CTM. \nIt would be interesting to improve the canonical formalism of the tensor model in that direction.\n\n\\vspace{1cm}\n\\centerline{\\bf Acknowledgements} \nThe work of N.S. is supported in part by JSPS KAKENHI Grant Number 15K05050. \nThe work of Y.S. is funded under CUniverse research promotion project \nby Chulalongkorn University (grant reference CUAASC).\nN.S. would like to thank the great hospitality and the stimulating discussions with the members\nof Department of Physics of Chulalongkorn University, while he stayed there and part of this work was done. \nY.S. would like to thank the wonderful members in Nagoya University, Japan, \nwhere part of this work was done, for the kind hospitality and fruitful discussions. \nY.S. would like to appreciate Jan Ambj\\o rn for discussions about the dimension in the formal continuum limit. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\\noindent\nLet $G$ be a nontrivial finite group.\nWe consider the minimal number of ramified primes in \nGalois extensions $L\/K$ with Galois group $G$,\nin various situations:\n\nIn the case $K=\\mathbb{Q}$,\nthe primes are as usual equivalence classes of absolute values,\ncorresponding to the prime numbers and the usual absolute value as the prime at infinity.\nLet $r_K(G)$ denote the minimal number of ramified primes\\footnote{the infinite prime is said to ramify in $L\/\\mathbb{Q}$\nif $L$ is not totally real}\n in a Galois extension $L\/K$ with Galois group $G$.\nBased on class field theory, Boston and Markin came to the following conjecture:\n\\begin{conjecture}[Boston--Markin \\cite{BM}]\\label{conj:BM}\nFor every nontrivial finite group $G$,\n$$\n r_\\mathbb{Q}(G) = \\max\\left\\{d(G^{\\rm ab}),1\\right\\},\n$$\nwhere $d(G)$ is the smallest cardinality of a set of \ngenerators of $G$. \n\\end{conjecture}\n\nWhile this \nwas verified by Boston and Markin for abelian groups and for groups of order up to 32,\nand since then was proven for certain solvable groups \\cite{Plans,KisilevskySonn,KisilevskySonn2010,KisilevskyNeftinSonn},\nit is widely open for most other groups.\nFor example, for the symmetric group $G=S_n$, $n>1$, this conjecture predicts realizations\nwith only one ramified prime.\nWhile this can of course be verified for small $n$,\nthe best general bound to date is the recent result of Bary-Soroker and Schlank \\cite{BSS}, who obtain $r_\\mathbb{Q}(S_n)\\leq 4$ for all $n$.\n\nIn the case where $K=k(T)$ is a rational function field over a field $k$, the primes are equivalence classes of ultrametric absolute values on $K$ trivial on $k$,\nwhich correspond to the monic irreducible polynomials $P\\in k[T]$ \nand the degree valuation as the prime at infinity.\nAs before we let $r_K(G)$ be\nthe minimal number of ramified primes\nin Galois extensions $L\/K$ with Galois group $G$,\nwhere in addition we require that $L\/K$ is {\\em geometric}, i.e.~$k$ is algebraically closed in $L$.\nThe geometric situation $k=\\mathbb{C}$ is well understood:\n\n\\begin{theorem}[Riemann Existence Theorem]\\label{thm:RET}\nFor every nontrivial finite group $G$,\n$$\n r_{\\mathbb{C}(T)}(G) = d(G)+1.\n$$\n\\end{theorem}\nIn positive characteristic, wild ramification\nleads to smaller bounds:\n\n\\begin{theorem}[Abhyankar conjecture for the line, \\cite{Ray94,Har94}]\\label{thm:Abhyankar}\nFor every prime number $p$ and every nontrivial finite group $G$,\n$$\n r_{\\overline{\\mathbb{F}}_p(T)}(G) = d(G\/p(G)) + 1,\n$$\nwhere $p(G)$ is the subgroup of $G$ generated by the $p$-Sylow subgroups of $G$.\n\\end{theorem}\n\n\nFor the global field $K=\\mathbb{F}_q(T)$,\nin analogy with Conjecture \\ref{conj:BM},\nwe thus conjecture the following:\n\n\\begin{conjecture}\\label{conj}\\label{bm}\nFor every prime power $q=p^\\nu$ and every nontrivial finite group $G$,\n$$\n r_{\\mathbb{F}_{q}(T)}(G) = \\max\\left\\{d((G\/p(G))^{\\rm ab}),1\\right\\}\n$$\n\\end{conjecture}\n\nWe note that $(G\/p(G))^{\\rm ab}=G^{\\rm ab}\/p(G^{\\rm ab})$,\nand that the close analogy with the case of $K=\\mathbb{Q}$ builds in an essential way on our restriction \nthat \nthe extension $L$ of $K=\\mathbb{F}_q(T)$ with Galois group $G$ is geometric.\nIn Section \\ref{sec:abelian} we prove this conjecture for abelian groups,\nthereby showing that the right hand side is always a lower bound for $r_{\\mathbb{F}_{q}(T)}(G)$.\nThere we also discuss the contribution of De Witt \\cite{DeWitt},\nwho proves Conjecture \\ref{conj} for certain solvable groups.\n\n\\begin{remark}\\label{rem:joachim} \nAnother potential constraint on $r_{\\mathbb{F}_{q}(T)}(G)$ was pointed out to us by Joachim K\\\"onig. \nFor a finite geometric Galois extension $L$ of $K=\\mathbb{F}_q(T)$,\nthe conjugacy classes of inertia groups generate ${\\rm Gal}(L\/K)$,\nand each inertia group is cyclic-by-$p$, i.e.~an extension of a cyclic group by a $p$-group (see \\cite[Corollary IV.2.4]{Ser79}). Thus Conjecture \\ref{conj} is possibly true only if any finite group $G$ is generated by $\\max\\{d((G\/p(G))^{\\rm ab}),1\\}$ conjugacy classes of cyclic-by-$p$ groups. \nWe give the proof of this elementary but nontrivial group-theoretic fact also in Section \\ref{sec:generators}. \nA similar remark applies to Conjecture \\ref{conj:BM}, where Lemma~\\ref{lem:gen_by_cyc} ensures there is no group-theoretic obstruction.\n\\end{remark}\n\nIn the main part of this work (Sections \\ref{sec:tworamified} and \\ref{sec:pleqn}) we prove various results for certain non-solvable groups, both conditional and unconditional,\nwith a focus on symmetric and alternating groups.\nNote that Conjecture \\ref{conj}\npredicts \n$$\n r_{\\mathbb{F}_q(T)}(S_n)=1,\\quad r_{\\mathbb{F}_q(T)}(A_n)=1\n$$\nfor every $n>2$ and every prime power $q=p^\\nu$.\nFor example, we obtain the following for $S_n$ and $A_n$:\n\n\\begin{theorem}\\label{thm:Sn}\nLet $n\\geq 2$ and $q=p^\\nu$ a prime power.\nThen \n\\begin{enumerate}\n\\item $r_{\\mathbb{F}_q(T)}(S_n)\\leq 2$, and \n\\item $r_{\\mathbb{F}_q(T)}(S_n)=1$ in each of the following cases:\n\\begin{enumerate}\n\\item $p(2n-3)^2$\n\\item The function field analogue of Schinzel's hypothesis H (Conjecture \\ref{conj:SchinzelFF}) holds for $\\mathbb{F}_q(T)$.\n\\end{enumerate}\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{theorem}\\label{thm:An}\nLet $n\\geq 3$ and $q=p^\\nu$ a prime power.\n\\begin{enumerate}\n\\item If $p>2$ or $\\mathbb{F}_q\\supseteq\\mathbb{F}_4$, then $r_{\\mathbb{F}_q(T)}(A_n)\\leq 2$, and \n\\item $r_{\\mathbb{F}_q(T)}(A_n)=1$ in each of the following cases:\n\\begin{enumerate}\n\\item $22$ with $n\\neq p+1$ or $\\mathbb{F}_q\\supseteq\\mathbb{F}_{p^2}$ or $p=3$.\n\\item $A_n$ if $p=2$ with either $\\mathbb{F}_q\\supseteq\\mathbb{F}_4$ or $10\\neq n\\ge 8,n\\equiv 0,1,2,6,7\\pmod 8.$\n\\end{enumerate}\nThen $r_{\\mathbb{F}_q(T)}(G)=\\max\\{d((G\/p(G))^{\\rm ab}),1\\}$, i.e. Conjecture \\ref{bm} holds for $G$.\n\\end{theorem}\n\nEn route to proving Theorem~\\ref{thm:main1} we will obtain evidence for the following conjecture of Abhyankar (see \\cite[\\S 16]{Abh01},\\cite[\\S 5]{HOPS}, a weaker form was stated in \\cite[Conjecture 9.2(C)]{Abh95})\nwhich is an analogue over $\\mathbb{F}_q$ of the Abhyankar conjecture for the affine line (which is the case $G=p(G)$ of Theorem~\\ref{thm:Abhyankar}):\n\n\\begin{conjecture}[Abhyankar's arithmetic conjecture for the affine line]\\label{conj:abhyankar} \nLet $G$ be a finite group which is cyclic-by-quasi-$p$, meaning that $G\/p(G)$ is cyclic (see Definition~\\ref{def:quasip} below).\nThen for every power $q$ of $p$ there exists a Galois extension $L\/\\mathbb{F}_q(T)$ (not necessarily geometric)\nramified only over the infinite prime\nsuch that ${\\rm Gal}(L\/\\mathbb{F}_q(T))=G$.\n\\end{conjecture}\n\n\nWe will show that Conjecture \\ref{conj:abhyankar} holds for the alternating group $A_n$ provided $n\\ge p>2,n\\neq p+1$ \n(note that for $n>3$ the group $A_n\/p(A_n)$ is cyclic only if $n\\ge p$).\n\n\\begin{theorem}\\label{thm:abhyankar} Assume $n\\ge p>2$. If $n=p+1$ assume additionally that $\\mathbb{F}_q\\supseteq\\mathbb{F}_{p^2}$. Then there exists a Galois extension $L\/\\mathbb{F}_q(T)$ ramified only over the infinite prime\nwith ${\\rm Gal}(L\/\\mathbb{F}_q(T))=A_n$.\n\\end{theorem}\nTheorem~\\ref{thm:abhyankar} will follow from Theorem~\\ref{thmlist} below. For some values of $q,n$ the theorem above is implicit in the work of Abhyankar \\cite{Abh92,Abh93}, but not in the above generality.\n\n\n\n\nTo obtain these results we apply a variety of tools and recent results\nfrom Galois theory, finite group theory (including the Classification of Finite Simple Groups for some of the results), analytic number theory\nand number theory over function fields,\nand we also employed the help of a computer to eliminate some small exceptional cases.\nSee the summary in Section \\ref{sec:summary} \nfor the proof of Theorems \\ref{thm:Sn} and \\ref{thm:An},\nand the corresponding sections for the full strength and full generality of the results.\nUsing our results for $\\mathbb{F}_q(T)$ we also obtain a conditional result for $\\mathbb{Q}$ (Theorem~\\ref{thm:S_n_over_Q}):\n\n\\begin{theorem}\nSchinzel's hypothesis H (Conjecture \\ref{conj:Schinzel}) implies that\n$r_\\mathbb{Q}(S_n)=1$ for every $n>1$,\ni.e.~Conjecture \\ref{conj:BM} holds for all symmetric groups.\n\\end{theorem}\n\nThis improves upon a result of Plans (see Remark \\ref{remark:plans} below), which shows that under Schinzel's hypothesis H there exists a Galois extension $L\/\\mathbb Q$ with Galois group $S_n$ and ramified over a single finite prime and the infinite prime, whereas our construction has no ramification at the infinite prime.\n\n\n\\section{Preliminaries and notation}\n\\noindent\nWe start by collecting a few definitions and statements that we will use throughout the paper.\n\n\\subsection{Function fields}\n\nFor a thorough introduction to the subject see \\cite{Ros02,Stichtenoth}. Here we only recall some terminology, notation and a few basic facts. \nLet $k$ be a base field.\nA function field (of one variable) over $k$ is a finite extension of the rational function field $k(T)$. \nA \\emph{global function field} is a function field over a finite base field $k=\\mathbb{F}_q$ (this will be our main case of interest).\n\nA function field $K\/k$ is {\\em regular}\nif $K$ is linearly disjoint over $k$ from the algebraic closure $\\bar{k}$ of $k$,\nwhich in case that $k$ is perfect simply means that\n$k$ is relatively algebraically closed in $K$.\nIf $K\/k$ is a regular function field, there exists a unique geometrically integral non-singular projective curve $C\/k$ such that $K$ is the function field of $C$. Conversely the function field of a curve as above is a regular function field over $k$.\n\nLet $K\/k$ be a regular function field and $C$ the corresponding curve. \nA \\emph{prime} $P$ of $K\/k$ is an\nequivalence class of nontrivial absolute values on $K$ trivial on $k$. Each such absolute value corresponds to a discrete valuation $v$,\nand it corresponds to a prime divisor on $C$. \nIf $P$ is a prime of $K$, the residue field $k(P)$ at $P$ is a finite extension of $k$ and $\\deg P=[k(P):k]$ is called the \\emph{degree} of $P$. By a \\emph{point} or \\emph{geometric point} of $C$ we will always mean a closed point of $C\\times_k\\bar k$. \nFor an extension $L\/k$ we denote by $C(L)$ the set of $L$-rational points on $C$.\n\n\\renewcommand{\\div}{\\mathrm{div}}\n\nThe rational function field $k(T)$ corresponds to the projective line $\\P^1_k$ and has primes of two types: the \\emph{finite primes} of the form $\\div(f)_0$ (the zero divisor of a rational function $f$) where $f\\in k[X]$ is an irreducible polynomial (such a prime has degree $\\deg f$) and the \\emph{infinite prime} $\\infty=\\div(1\/T)_0$ which has degree 1.\n\nLet $L\/K$ be a finite extension of function fields over $k$. It is called \\emph{regular}, or \\emph{geometric}, if $L$ is also a regular function field over $k$. Regular extensions correspond to finite covers of the associated curves. \nIf $P$ is a prime of $K$ and $Q$ a prime of $L$ lying over $P$, we denote by $e(Q\/P)$ the ramification index of $Q$ over $P$. \nWe say that $L\/K$ is ramified (resp. wildly ramified) at $Q$ if $e(Q\/P)>1$ (resp.\\ $\\mathrm{char}(k)|e(Q\/P)$). We say that $L\/K$ is ramified (resp. wildly ramified) \\emph{over} $P$ if there exists a prime $Q$ of $L$ lying over $P$ at which the extension is ramified (resp. wildly ramified).\nRamification which is not wild is said to be \\emph{tame}.\nIf $L\/K$ is tamely ramified over at least one prime, then $L\/K$ is separable.\n\nIf $C_L$ and $C_K$ are the underlying curves then the extension $L\/K$ corresponds to a finite morphism $w\\colon C_L\\to C_K$. \nThis correspondence defines an equivalence of categories between regular function fields over $k$ and absolutely irreducible non-singular projective curves defined over $k$. \nIf $P$ is a prime divisor of $C_L$, and $Q=w(P)$ (a prime divisor of $C_K$), we say that $w$ is ramified (resp. tamely ramified) at $P$ whenever the extension $L\/K$ is ramified (resp. tamely ramified) at the corresponding prime of $L$ which we identify with $P$. \nIn this case we also call the geometric points corresponding to $P$ {\\em ramification points} of $w$,\nand the geometric points corresponding to $Q$ {\\em branch points} of $w$.\n\n\\subsection{Ramification theory}\n\nLet $K$ be a field. For a separable polynomial $f\\in K[X]$ of degree $n$ we define its Galois group ${\\rm Gal}(f\/K)$ to be the Galois group of its splitting field over $K$.\nWe will always interpret the Galois group ${\\rm Gal}(f\/K)$ as a subgroup of the symmetric group $S_n$ via its action on the roots of $f$. The embedding into $S_n$ is well-defined up to conjugation.\n\n\\begin{lemma}\\label{lemcycle} \nLet $K$ be a function field over an algebraically closed field $k$\nof characteristic $p\\geq 0$,\nlet $f\\in K[X]$ be a separable irreducible polynomial and let $L=K(\\alpha)$ where $\\alpha$ is a root of $f$. \nLet $P$ be a prime (finite or infinite) of $K$ and let $Q_1,\\ldots,Q_r$ be the primes of $L$ lying over $P$. \nLet $e_i=e(Q_i\/P)$ be the ramification indices.\n\\begin{enumerate}\\item[(i)]\nAssume that $p\\nmid e_i$ for all $i$. \nThen ${\\rm Gal}(f\/K)$ contains a permutation which is a product of $r$ disjoint cycles of lengths $e_1,\\ldots,e_r$.\n\\item[(ii)]\nAssume that\n$p\\nmid e_i$ and $(e_i,e_1)=1$ for $2\\le i\\le r$, \nand either $e_1=p$ or $p\\nmid e_1$. \nThen ${\\rm Gal}(f\/K)$ contains a cycle of length $e_1$.\\end{enumerate}\\end{lemma}\n\n\\begin{proof} See \\cite[\\S 3, Generalized Cycle Lemma]{Abh93}.\\end{proof}\n\n\\begin{lemma}[Abhyankar's Lemma] \\label{lem:abhyankar} \nLet $k$ be perfect, $K\/k$ a function field, and\n$L,M$ finite separable extensions of $K$. \nLet $P$ be a prime of $K$, $Q$ a prime of $L$ lying over $P$, $P'$ a prime of $M$ lying over $P$. \nAssume that \n\\begin{enumerate}\n\\item $Q$ is tamely ramified over $P$, and\n\\item $e(Q\/P)$ divides $e(P'\/P)$.\n\\end{enumerate}\nThen every prime $Q'$ of $LM$ lying over both $Q$ and $P'$ is unramified over $M$, i.e.~$e(Q'\/P')=1$.\n\\end{lemma}\n\n\\begin{proof} \nThis follows from \\cite[Theorem 3.9.1]{Stichtenoth}\nby the multiplicativity of the ramification index.\n\\end{proof}\n\n\\subsection{Group theory}\nWe denote by $C_n$, $D_n$, $S_n$, $A_n$ \nthe cyclic, dihedral, symmetric respectively alternating group of degree $n$.\nWe will need the following group-theoretic result of Jones, which relies on the Classification of Finite Simple Groups (CFSG) for its proof.\n\n\\begin{theorem}[Jones]\\label{thm:jones} Let $G\\leqslant S_n$ be a primitive permutation group containing a cycle of length $l>1$.\n\\begin{enumerate}\\item[(i)]If $l\\le n-3$ then $G\\geqslant A_n$.\n\\item[(ii)]If $l=n-2$ is prime then either $G\\geqslant A_n$ or $l=2^k-1$ is a Mersenne prime and ${PGL}_2(2^k)\\leqslant G\\leqslant{P\\Gamma L}_2(2^k)$ with its standard action on $\\P^1(\\mathbb{F}_{2^k})$.\n\\end{enumerate}\n\\end{theorem} \n\n\\begin{proof} This is a direct consequence of \\cite[Corollary 1.3]{Jon14}.\\end{proof}\n\n\n\\begin{remark}\\label{remark:jordan}\nIn the case that $l$ is prime Theorem~\\ref{thm:jones}(i) is a classical (and elementary) result of Jordan \\cite[Theorem 3.3E]{DiMo96}. Most (though not all) of the applications in Section~\\ref{sec:pleqn} only require this weaker version. In Section~\\ref{sec:tworamified} Jones's theorem is used in its full form.\n\\end{remark}\n\nWe will also make use of the following elementary fact:\n\n\\begin{lemma}\\label{lem:primtrans}\nLet $G\\leqslant S_n$ be a primitive group containing a transposition. Then $G=S_n$.\n\\end{lemma}\n\n\\begin{proof} See \\cite[Theorem 13.3]{Wie64}.\\end{proof}\n\n\n\n\\subsection{Discriminants}\n\nLet $K$ be a field. \nFor a polynomial $f\\in K[X]$ of degree $n$ we denote its leading coefficient by $\\mathrm{lc}(f)$,\nand if $\\alpha_1,\\dots,\\alpha_n\\in\\overline{K}$ are the zeros of $f$ (listed with multiplicity), the discriminant of $f$ is defined by\n\\begin{equation}\\label{eq:discdefinition}\n\\disc(f) \\;=\\; \\mathrm{lc}(f)^{2n-2}\\prod_{i\\deg c$, $f'\\neq0$, and that $c$ is separable. \nThen the finite critical values of $w$ are the roots of $\\disc_X(f-Uc)$, more precisely\n$$\n\\disc_X(f(X)-Uc(X))=a\\cdot\\prod_{(f'c-fc')(\\alpha)=0}(U-w(\\alpha)),\n$$\nwith $a\\in k^\\times$ and the product over the roots of $f'c-fc'$ in $\\bar k$ with multiplicity.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n{\\bf (i).} We identify $U=w(X)$, viewing $U$ as an element of $k(X)$. This gives the extension of function fields $k(X)\/k(U)$ corresponding to the rational map $w$. Let $\\alpha\\in\\bar k$ be such that $c(\\alpha)\\neq 0$ and denote $\\beta=w(\\alpha)$. Then $X-\\alpha$ is a local parameter for $\\P^1_X$ at the point $\\alpha$ and $U-\\beta$ is a local parameter for $\\P^1_U$ at the point $\\beta$. \nWe have $U-\\beta=w(X)-\\beta=(X-\\alpha)^eu(X)$ \nwith $u\\in k(X)$ which has no pole at $\\alpha$. \nTaking derivatives we obtain\n$$\n w'(X)=e(X-\\alpha)^{e-1}u(X)+(X-\\alpha)^eu'(X).\n$$ \nAs $\\alpha$ is neither a pole nor a zero of $u$, and hence also not a pole of $u'$,\nthe multiplicity $\\nu$ of the zero $\\alpha$ of $w'$ \nis $\\nu=e-1$ if ${\\rm char}(k)\\nmid e$, and $\\nu\\geq e$ if ${\\rm char}(k)\\mid e$.\n\n{\\bf (ii).} Denote $h=(f',c')\\in\\bar k[X]$ and write $f'=hu,c'=hv,u,v\\in\\bar k[X]$. \nAs $f'\\neq 0$, also $u\\neq 0$.\nIf $\\deg c=0$ the \nassertion of (ii) is immediate from (i) and (\\ref{eq:disc2}), so let us assume $\\deg c>0$ and then \n$c',v\\neq 0$ by our assumption that $c$ is separable. Using the properties of resultants and \ndiscriminants (\\ref{eq:resdefinition}),(\\ref{eq:disc}),(\\ref{eq:resbimult}),(\\ref{eq:ressym}),(\\ref{eq:resalt}) and using $\\sim$ to denote\nequality up to a non-zero multiplicative constant (in $k$),\nwe calculate:\n\\begin{eqnarray*}\n\\disc_X(f-Uc)&\\stackrel{(\\ref{eq:disc})}{\\sim}&\\mathrm{Res}(f'-Uc',f-Uc)\\\\\n&\\stackrel{(\\ref{eq:resbimult})}=& \\mathrm{Res}(h,f-Uc)\\cdot\\mathrm{Res}(u-Uv,f-Uc)\\\\\n&\\stackrel{(\\ref{eq:resbimult})}=&\\mathrm{Res}(h,f-Uc)\\cdot\\frac{\\mathrm{Res}(u-Uv,fv-Ucv)}{\\mathrm{Res}(u-Uv,v)}\\\\\n&\\stackrel{(\\ref{eq:ressym}),(\\ref{eq:resalt})}\\sim&\\mathrm{Res}(h,f-Uc)\\cdot\\frac{\\mathrm{Res}(u-Uv,fv-uc)}{\\mathrm{Res}(u,v)}\\\\\n&\\stackrel{(\\ref{eq:ressym})}\\sim&\\mathrm{Res}(h,f-Uc)\\cdot\\mathrm{Res}(uc-fv,u-Uv)\\\\\n&\\stackrel{(\\ref{eq:resdefinition})}\\sim&\\prod_{h(\\alpha)=0}\\left(c(\\alpha)U-f(\\alpha)\\right)\\prod_{(uc-fv)(\\alpha)=0}(u(\\alpha)-Uv(\\alpha))\\\\\n&\\sim&\\prod_{h(\\alpha)=0}\\left(U-\\frac{f(\\alpha)}{c(\\alpha)}\\right)\\prod_{(uc-fv)(\\alpha)=0}\\left(U-\\frac{f'(\\alpha)}{c'(\\alpha)}\\right)\\\\\n&\\sim&\\prod_{(f'c-fc')(\\alpha)=0}(U-w(\\alpha)).\n\\end{eqnarray*}\nIn the last line we used the fact that for a root $\\alpha$ of $(f'c-fc')\/h$ we have $f'(\\alpha)\/c'(\\alpha)=w(\\alpha)$.\n\\end{proof}\n\n\\subsection{Monodromy of rational functions}\\label{sec:monodromy}\nThe definitions and results in this subsection will be used only in Section \\ref{sec:two}.\nLet $k$ be a field of characteristic $p\\geq0$. \nThe {\\em (arithmetic) monodromy} $\\mathrm{Mon}(w)$ of a non-constant rational function $w\\in k(X)$ \nfor which the extension $k(X)\/k(w)$ is separable\nis the Galois group of the Galois closure of $k(X)\/k(w)$.\nWhen ambiguity about the base field may arise we write $\\mathrm{Mon}_k(w)$. \nWe usually view $\\mathrm{Mon}(w)$ as a subgroup of $S_n$ with $n=\\deg w$ via its action on the generic fiber of the corresponding morphism $w\\colon \\P^1\\to\\P^1$.\nA rational function $w\\in k(X)$ is called \\emph{indecomposable} if it cannot be written as a composition $w=u\\circ v$ with $u,v\\in\nk(X)$ where $\\deg u,\\deg v>1$. \n\n\n\\begin{lemma}\\label{lem:indecomposable_primitive}\nLet $w\\in k(X)$ with $k(X)\/k(w)$ separable.\nThen $w$ is indecomposable if and only if\n$\\mathrm{Mon}(w)\\leqslant S_n$ is primitive.\n\\end{lemma}\n\n\\begin{proof}\nIf $w=u\\circ v$ with $\\deg u>1$, $\\deg v>1$, \nand $\\alpha_1,\\dots,\\alpha_m\\in\\overline{k(T)}$ are the roots of $u-T$,\nthen $v^{-1}(\\alpha_1),\\dots,v^{-1}(\\alpha_m)$ is a nontrivial partition of the roots of $w-T$,\nwhich is preserved by $\\mathrm{Mon}(w)$.\nThe converse implication is proven for example in \\cite[Lemma 2]{Fried} in the case of polynomials,\nand the case of rational functions can be proven similarly using L\\\"uroth's theorem.\n\\end{proof}\n\n\\begin{definition}\\label{def:ramification type}\nLet $w\\colon \\P^1\\to\\P^1$ be a tamely ramified rational function defined over an algebraically closed field $k$. \nThe \\emph{ramification type} of $w$ (called \\emph{passport} in \\cite{AdZv15}) is the unordered tuple of partitions $(\\lambda_1,\\ldots,\\lambda_r)$, where $b_1,\\ldots,b_r$ are the distinct branch points of $w$ and $\\lambda_i$ is the partition of $n=\\deg w$ given by the ramification indices over $b_i$.\nWe write partitions of $n$ as $n_1^{e_1}\\dots n_r^{e_r}$ with $0\\leq n_1<\\dots1$ over the algebraically closed field $k$\nwith monodromy group $\\mathrm{Mon}(w)\\leqslant S_n$.\nAssume that $w$ has at most two critical values $a,b$,\nand that $w^{-1}(a)$ contains only one critical point $a'$.\nIf the ramification index of $a'$ is prime, and all ramification over $b$ is \ntame, then $\\mathrm{Mon}(w)$ is primitive.\n\\end{lemma}\n\n\\begin{proof}\nLet $L$ be the Galois closure of the extension $K(X)\/K(T)$ induced by $w$.\nLet $K(T)\\subseteq M\\subsetneq K(X)$ be an intermediate extension. \nSince there is only one ramified prime of $K(X)$ over $T=a$ and it has prime ramification index, \nthe extension $M\/K(T)$ has to be unramified over $T=a$. \nHowever, it is also tamely ramified over $T=b$ and unramified elsewhere, \nhence $M=K(T)$ by the Riemann-Hurwitz formula.\nSo $K(X)\/K(T)$ is a minimal proper extension,\nhence by Galois theory ${\\rm Gal}(L\/K(X))$ is maximal in ${\\rm Gal}(L\/K(T))$,\nwhich means that $\\mathrm{Mon}(w)={\\rm Gal}(L\/K(T))\\leqslant S_n$ is primitive.\n\\end{proof}\n\n\\section{Abelian groups}\n\\label{sec:abelian}\n\\label{sec:generators}\n\n\\noindent\nIn this section we first prove the group-theoretic fact mentioned in Remark \\ref{rem:joachim} \nand then verify Conjecture \\ref{conj} for abelian groups. \n\nIf $G$ is a group and $a\\in G$ is an element we denote by $a^G$ the conjugacy class of $a$. For a subset $S\\subseteq G$ we denote $S^G=\\bigcup_{a\\in S}a^G$.\nRecall that for a finite group $G$ we denote by $p(G)$ the subgroup generated by its $p$-Sylow subgroups.\n A \\emph{cyclic-by-$p$} group is an extension of a cyclic group by a $p$-group. \n\n\\begin{lemma}\\label{lem:gen_by_cyc} \nLet $G$ be a finite group. There exist $x_1,\\ldots,x_r\\in G$ with $r=\\max\\{d(G^\\mathrm{ab}),1\\}$ such that $G=\\langle x_1^G,\\ldots,x_r^G\\rangle$.\\end{lemma}\n\n\\begin{proof} \nThis is a special case of the main theorem in \\cite{Kut76},\nsee also \\cite{Bae64}.\n\\end{proof}\n\n\\begin{proposition}\\label{prop:gen_by_cyc_p} \nLet $G$ be a finite group and $p$ a prime number. There exist subgroups $Q_1,\\ldots,Q_r$ of $G$ such that $r=\\max\\{d((G\/p(G))^\\mathrm{ab}),1\\}$, each $Q_i$ is cyclic-by-$p$ and $G=\\langle Q_1^G,\\ldots,Q_r^G\\rangle$.\n\\end{proposition}\n\n\n\\begin{proof} \nLet $H=G\/p(G)$, so that $r=\\max\\{d(H^\\mathrm{ab}),1\\}$. \nBy Lemma~\\ref{lem:gen_by_cyc} there exist $y_1,\\ldots,y_r\\in H$ with $H=\\langle y_1^H,\\ldots,y_r^H\\rangle$. \nLet $P\\leqslant G$ be a $p$-Sylow subgroup. If $P=1$, i.e. $(|G|,p)=1$, then the claim holds with $Q_i=\\langle y_i\\rangle$, so we may assume $P\\neq 1$. By Frattini's argument \\cite[Theorem 3.7]{Gor80} we have $p(G)N_G(P)=G$, so we can choose $x_1,\\ldots,x_r\\in N_G(P)$ such that $y_i = p(G)x_i$.\nDenote $Q_i=\\langle x_i\\rangle P$. Each $Q_i$ is a subgroup of $G$ since $x_i\\in N_G(P)$, it is cyclic-by-$p$ since it is an extension of $\\langle x_i\\rangle\/(\\langle x_i\\rangle\\cap P)$ by $P$, and finally since\n$p(G)=\\langle P^G\\rangle$ and $G=\\langle p(G),x_1^G,\\ldots,x_r^G\\rangle$ we have $G=\\langle P^G,x_1^G,\\ldots,x_r^G\\rangle=\\langle Q_1^G,\\ldots Q_r^G\\rangle$, as required.\n\\end{proof}\n\n\n\\begin{theorem}\\label{thm:abelian}\nConjecture \\ref{conj} holds for $G$ abelian.\n\\end{theorem}\n\n\n\\begin{proof}\n\tLet $q=p^\\nu$ and let $G$ be a nontrivial finite abelian group.\n\tThe claim is that $r_{\\mathbb{F}_q(T)}(G)=1$ if $G$ is a $p$-group and $r_{\\mathbb{F}_q(T)}(G)=d(G\/p(G))$ otherwise.\n\t\n\tWe first prove the lower bound.\n\tLet $L\/\\mathbb{F}_q(T)$ be an abelian Galois extension with Galois group $G$ and $\\mathbb{F}_q$ algebraically closed in $L$.\n\tIf $G$ is a $p$-group, the lower bound $r_{\\mathbb{F}_q(T)}(G)\\geq 1$ follows from the Riemann-Hurwitz formula.\n If $G$ is not a $p$-group, then $G\/p(G)$ is nontrivial, \n and since $r_{\\mathbb{F}_q(T)}(G)\\geqr_{\\mathbb{F}_q(T)}(G\/p(G))$ we can assume that $p(G)=1$. \n Let $P_1,\\ldots, P_k$ be all the finite primes of $\\mathbb{F}_q(T)$ ramified in $L$, and let $I_1,\\ldots, I_k$ be the corresponding inertia groups (note that $I_i$ are well defined since the extension is abelian). Then each $I_i$ is cyclic, as $P_i$ is tamely ramified in $L$. \nAs also the infinite prime of $\\mathbb{F}_q(T)$ is tamely ramified in $L$,\n$G=\\left< I_1,\\ldots, I_k\\right>$, cf.~\\cite[Proposition 4.4.6]{Serre}.\nIn particular, $r_{\\mathbb{F}_q(T)}(G)\\geq k\\geq d(G)$.\n\n\tTo show that the lower bound is tight, we construct a suitable Galois extension. \n\tWrite $G= G\/p(G) \\times p(G)$, let $k$ and $p^\\alpha$ be the exponents of $G\/p(G)$ respectively $p(G)$,\n\tand let $d=\\max\\{d(G\/p(G)),1\\}$. \n\tChoose distinct primes $P_1, \\ldots, P_d$ such that $k(q-1)\\mid q^{\\deg P_i}-1$ (i.e.\\ such that the multiplicative order of $q$ modulo $k(q-1)$ divides $\\deg P_i$). \n\n\tLet $R_i$ be the completion of $\\mathbb{F}_q[T]$ at $P_i$ and $E_i$ its fraction field. \n\tBy \\cite[Proposition~II.5.7]{Neukirch}, $E_i^\\times = \\left<\\pi\\right> \\times(\\mathbb{F}_{q^{\\deg P_i}})^\\times \\times G_i$, where \n\t$\\pi$ is a uniformizer and\n\t\\[\n\t\tG_i = 1+ \\pi\\mathbb{F}_{q^{\\deg P_{i}}}[[\\pi]] \\cong \\mathbb{Z}_p^{\\mathbb{N}}.\n\t\\]\nIn particular, $R_i^\\times = (\\mathbb{F}_{q^{\\deg P_i}})^\\times \\times G_i$. \n\tBy class field theory (take inverse limit in \\cite[Proposition~2.2, Theorem~2.3, and Theorem~3.2]{Hayes}) the maximal abelian extension of $\\mathbb{F}_q(T)$ which is regular over $\\mathbb{F}_q$, tamely ramified at infinity, and unramified outside $\\{P_i,\\infty\\}$ has Galois group $R_i^\\times$ and the tame inertia group at $\\infty$ corresponds to $\\mathbb{F}_{q}^\\times$. Hence $R_i^\\times\/\\mathbb{F}_q^\\times$ is the Galois group of the maximal abelian extension that is regular over $\\mathbb{F}_q$ and unramified outside $\\{P_i\\}$. As $\\mathbb{F}_q(T)$ has no unramified extensions regular over $\\mathbb{F}_q$, those extensions for different $i$ are linearly disjoint, so the Galois group of the maximal abelian extension that is regular over $\\mathbb{F}_q$ and unramified outside $\\{P_1,\\ldots, P_d\\}$ is isomorphic to \n\t\\[\n\t\t\\prod_{i=1}^d R_i^\\times\/\\mathbb{F}_q^\\times \\cong \\prod_{i=1}^d C_{(q^{\\deg P_i}-1)\/(q-1)} \\times \\mathbb{Z}_p^{\\mathbb{N}}.\n\t\\]\n\tBy the assumption that $k\\mid (q^{\\deg P_i}-1)\/(q-1)$, we get that $G\/p(G)$ is a quotient of $ \\prod_i C_{(q^{\\deg P_i}-1)\/(q-1)} $ (this is vacantly true if $G\/p(G)=1$). Obviously, $p(G)$ is a quotient of $\\mathbb{Z}_p^{\\mathbb{N}}$. Hence $G=G\/p(G)\\times p(G)$ is a quotient of $\\prod_i R_i^\\times\/\\mathbb{F}_q^\\times$. This implies we can realize $G$ as the Galois group of a geometric Galois extension of $\\mathbb{F}_q(T)$ unramified outside $\\{P_1,\\ldots, P_d\\}$, as needed.\n\\end{proof}\n\n\\begin{corollary}\\label{cor:lowerbound}\nFor every prime power $q=p^\\nu$ and every nontrivial finite group $G$,\n$$\n r_{\\mathbb{F}_{q}(T)}(G) \\geq \\max\\left\\{d((G\/p(G))^{\\rm ab}),1\\right\\}.\n$$\n\\end{corollary}\n\n\\begin{remark}\nFor $K=\\mathbb{F}_p(T)$, a conjecture \nwas posed in \\cite[Conjecture 1.1]{DeWitt} and claimed to be proven\nfor abelian finite groups,\nnamely that\n$$\n r_{\\mathbb{F}_p(T)}(G) = \\begin{cases} d((G\/p(G))^{\\rm ab})+1,&\\mbox{if }p\\,|\\,|G^{\\rm ab}|\\\\\\max\\{d((G\/p(G))^{\\rm ab}),1\\},\n &\\mbox{otherwise}\\end{cases}.\n$$\nHowever, already \nin the case $G=\\mathbb{Z}\/2p\\mathbb{Z}$, $p>2$,\nthe number obtained there is too big:\nFor example, $G=\\mathbb{Z}\/6\\mathbb{Z}$ can be realized over $\\mathbb{F}_3(T)$ with only one ramified prime of degree $2$,\nfor example by $\\mathbb{F}_3(T,x,y)$ with $x^2=y^3-y=(T^2+1)^{-1}$.\nThe incorrect lower bound in the case $p||G^{\\rm ab}|$ in \\cite{DeWitt} \npresumably comes from the false assumption there that an extension with such group $G$ must be wildly ramified at infinity. \n\n\nIn any case, at least for $q=p$, \\cite[Theorem 2.6]{DeWitt} proves\nConjecture \\ref{conj} for abelian groups of order prime to $p$\n(and thus proves a special case of Theorem~\\ref{thm:abelian}).\nMoreover, \n\\cite[Theorem 3.6]{DeWitt} \nproves Conjecture \\ref{conj}\nfor all semiabelian $\\ell$-groups $G$, with $\\ell$ a prime number different from $p$\n(building on results from \\cite{KisilevskySonn} and \\cite{KisilevskyNeftinSonn}),\nand \\cite[Corollary 6.4]{DeWitt} proves Conjecture \\ref{conj}\nfor all $\\ell$-groups with $\\ell$ a prime number dividing $q-1$\n(both results are stated for $q=p$, but the proofs go through also for general $q$).\n\nWe note however that the proof of \\cite[Theorem 6.9]{DeWitt}, which claims to prove the above conjecture for all nilpotent groups,\nseems to be incomplete regarding both $p$-groups (\\cite[Theorem 2.5]{DeWitt}) \nand $\\ell$-groups with $\\ell$ coprime to $p-1$ (\\cite[Corollary 6.8]{DeWitt}).\n\\end{remark}\n\n\n\n\\section{$S_n$ and $A_n$ for $p>n$}\n\n\\label{sec:tworamified}\n\n\\noindent\nIn this section we obtain Galois extensions of group $S_n$ and $A_n$ with ramification over at most two primes\nin the case where $n$ is smaller than the characteristic $p$.\nWe start with splitting fields of polynomials\nof the form $f(X)-T$ in Section \\ref{sec:Morse}\nand $f(X)-Tc(X)$ in Section \\ref{sec:two},\nwhich produce extensions with one ramified finite prime \nand tame ramification over infinity.\nIn Section \\ref{sec:oneramified}\nwe then explain how to eliminate the tame ramification at infinity\nin many cases.\nFinally, in Section \\ref{sec:twinprimes}\nwe present yet another approach to obtain extensions with two ramified primes,\nby working with trinomials and applying recent results on small gaps between primes in function fields.\n\nWe recall that \nif $f\\in\\mathbb{F}_q[T,X]$ is monic in $X$,\nthen the only primes of $K=\\mathbb{F}_q(T)$ that possibly ramify in \nthe splitting field $L$ of $f$ over $K$\nare the divisors of the discriminant $\\disc_X(f)$, and possibly the infinite prime of $K$ (Lemma~\\ref{lem:Dedekind}).\nIn particular, if $\\disc_X(f)$ is irreducible or, more generally, a prime power, at most two primes of $K$ ramify in $L$.\n\n\\subsection{Two ramified primes via Morse polynomials}\n\\label{sec:Morse}\n\n\nWe start with splitting fields of polynomials of the simple from $f(X)-T$,\nwhere in some cases the classical theory of Morse polynomials\nproduces suitable extensions of group $S_n$ with two ramified primes.\nLet $k$ be a field of characteristic $p\\neq2$,\nand recall that\na polynomial $f\\in k[X]$ of degree $n$ not divisible by $p$ is {\\em Morse} if it has exactly $n-1$ distinct critical values, i.e.\nthe roots $\\alpha_1,\\dots,\\alpha_{n-1}$ of $f'$ in $\\bar{k}$ are simple and\n$f(\\alpha_i)\\neq f(\\alpha_j)$ for $i\\neq j$.\nIf $f$ is Morse, then\n${\\rm Gal}(f(X)-T\/\\overline{k}(T))=S_n$\nby \\cite[Theorem 4.4.5]{Serre},\nso the splitting field of $f(X)-T$ over $k(T)$\nis geometric with Galois group $S_n$.\n\n\\begin{lemma}\\label{lem:f'Morse_disc_irred}\nLet $f\\in k[X]$ be Morse of degree $n$ with $p\\nmid n$. \nThen $f'$ is irreducible if and only if $D(T):=\\disc_X(f(X)-T)\\in k[T]$ is irreducible.\n\\end{lemma}\n\n\\begin{proof}\nLet $A$ be the set of roots of $f'$, and $G={\\rm Gal}(k(A)\/k)$.\nBy (\\ref{eq:disc2}) or Lemma~\\ref{lem:crit}, $D\\sim\\prod_{a\\in A}(T-f(a))$\n(where as before $\\sim$ denotes equality up to a non-zero constant).\nThe fact that $f$ is Morse implies that $f$ is injective on $A$,\nso $f$ induces an isomorphism of $G$-sets $A\\rightarrow f(A)$.\nIn particular, $k(f(a))=k(a)$ for every $a\\in A$,\nhence $D\\sim\\prod_{a\\in A}(T-f(a))$ is irreducible\nif and only if $f'\\sim\\prod_{a\\in A}(X-a)$ is.\n\\end{proof}\n\n\n\n\\begin{proposition}\\label{thm:S_nlargeq}\\label{thm:large_q}\n Let $n\\geq3$ and $q=p^\\nu$ with $p>n$.\n There are $\\frac{q^n}{n-1} + O_n(q^{n-1})$ many \n monic Morse $f\\in\\mathbb{F}_q[X]$ of degree $n$\n such that the splitting field of $f(X)-T$ over $\\mathbb{F}_q(T)$ is geometric with Galois group $S_n$,\n and $\\disc_X(f(X)-T)$ is irreducible.\n\\end{proposition}\n\n\\begin{proof}\nLet $M_n\\cong \\mathbb{A}^n$ be the space of monic polynomials of degree $n$ (identified with the $n$-tuple of non-leading coefficients) considered as a variety over $\\mathbb{F}_q$.\nThe subset $\\mathcal{M}\\subseteq M_n$ of monic Morse polynomials of degree $n$\nis Zariski-open and dense \nwith complement $M_n\\setminus\\mathcal{M}$ the zero set of a polynomial in the coefficients $a_0,\\dots,a_{n-1}$\nof degree $O_n(1)$, see \\cite[Proposition 4.3 and (2) in its proof]{Geyer}.\nThus \nthe elementary estimate \\cite[Lemma 1]{LangWeil} gives that\nthe set $\\mathcal{M}(\\mathbb{F}_q)$ of $f\\in M_n(\\mathbb{F}_q)$ that are Morse\nsatisfies\n$|\\mathcal{M}(\\mathbb{F}_q)|= q^n+O_n(q^{n-1})$.\n\nLet $\\mathcal{P}$ denote the set of $f\\in M_n(\\mathbb{F}_q)$\nfor which $f'$ is irreducible.\nAs $p>n$, the map $M_n(\\mathbb{F}_q)\\rightarrow M_{n-1}(\\mathbb{F}_q)$, $f\\mapsto \\frac{1}{n}f'$\nis surjective with fibers of size $q$,\nso by the Prime Polynomial Theorem,\n$|\\mathcal{P}|=\\frac{q^n}{n-1}+O_n(q^{(n-1)\/2})$.\nIt follows that $|\\mathcal{M}(\\mathbb{F}_q)\\cap\\mathcal{P}|=\\frac{q^n}{n-1} + O_n(q^{n-1})$.\nNow for every $f\\in\\mathcal{M}(\\mathbb{F}_q)$,\nthe splitting field of $f-T$ is geometric with Galois group $S_n$ (see above),\nand\n$\\disc_X(f-T)$ is irreducible\nif and only if $f\\in\\mathcal{P}$ (Lemma~\\ref{lem:f'Morse_disc_irred}),\nso the claim follows.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:f'irred_Morse}\nLet $f\\in\\mathbb{F}_q[X]$ of degree $n$ where $q=p^\\nu$, $p\\nmid n$ and $n-1$ is prime.\nIf $f'$ is irreducible, then $f$ is Morse.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that $f'$ is irreducible\nand that there exist $a\\neq b$ in $\\overline{\\mathbb{F}}_q$ with $f'(a)=f'(b)=0$ and $f(a)=f(b)$. \nAs $f'$ is irreducible of degree $n-1$, we have $a,b\\in\\mathbb{F}_{q^{n-1}}$\nand there exists $1\\neq\\sigma\\in{\\rm Gal}(\\mathbb{F}_{q^{n-1}}\/\\mathbb{F}_q)$ with $b=a^\\sigma$.\nThus $c:=f(a)=f(b)=f(a)^\\sigma$ is in the fixed field of $\\sigma$, which is $\\mathbb{F}_q$ due to the assumption that\n$n-1$ is prime.\nSo $f-c\\in\\mathbb{F}_q[X]$ has root $a$, hence $f'|f-c$.\nDeriving gives $f'|f''$, which contradicts the separability of $f'$.\n\\end{proof}\n\n\\begin{proposition}\\label{prop:Morse2}\nLet $p\\geq3$ and $q=p^\\nu$. Suppose that $n-1$ is prime and $n\\in\\{2,\\dots,p-1\\}\\cup\\{p+1\\}$.\nThen there exists $f\\in\\mathbb{F}_q[X]$ of degree $n$ such that\nthe splitting field of $f(X)-T$ over $\\mathbb{F}_q(T)$ is geometric with Galois group $S_n$,\nand $\\disc_X(f(X)-T)$ is irreducible.\n\\end{proposition}\n\n\\begin{proof}\nThere exists $f\\in\\mathbb{F}_q[X]$ of degree $n$ with $f'$ irreducible:\nIf $n0$.\n\\begin{enumerate}\n\\item[(i)] Assume that $p\\neq 2$, $m\\equiv n\\pmod 2$ and $g=f'c-fc'$ is squarefree. \nThen $\\mathrm{Mon}(w)=S_n$ or one of the following:\n\\begin{itemize}\n\\item $n=6,m=2,\\mathrm{Mon}(w)=PGL_2(5)$ with ram.~type $(1^22^2,1^22^2,2^3,1^24^1)$.\n\\item $n=8,m=2,\\mathrm{Mon}(w)=PGL_2(7)$ with ram.~type $(1^22^3,1^22^3,1^22^3,1^26^1)$.\n\\item $n=9,m=1,\\mathrm{Mon}(w)=AGL_2(3)$ with ram.~type $(1^32^3,1^32^3,1^32^3,1^18^1)$.\n\\item $n=10,m=2,\\mathrm{Mon}(w)=P\\Gamma L_2(9)$ with ram.~type $(1^42^3,1^42^3,2^5,1^28^1)$.\n\\end{itemize}\n\\item[(ii)] Assume that $p\\notin\\{2,3\\}$, $m\\not\\equiv n\\pmod 2$, and $f'c-fc'=g^2$ where $g\\in k[X]$ is squarefree. \nThen $\\mathrm{Mon}(w)=A_n$ or one of the following:\n\\begin{itemize}\n\\item $n=8,m=1,\\mathrm{Mon}(w)=L_2(7)$ or $A\\Gamma L_1(8)$ with ram.~type $(1^23^2,1^23^2,1^17^1)$.\n\\item $n=9,m=0,\\mathrm{Mon}(w)=P\\Gamma L_2(8)$ with ram.~type $(1^33^2,1^33^2,9^1)$.\n\\item $n=12,m=1,\\mathrm{Mon}(w)=M_{12}$ with ram.~type $(1^33^3,1^33^3,1^111^1)$.\n\\item $n=24,m=1,\\mathrm{Mon}(w)=M_{24}$ with ram.~type $(1^63^6,1^63^6,1^123^1)$.\n\\end{itemize}\n\\end{enumerate}\n\\end{proposition}\n\n\n\n\\begin{proof} \n\nThe classification of monodromy groups of indecomposable rational functions with precisely one multiple pole over $\\mathbb{C}$ is given in \\cite[\\S 2.2 and Theorem 12]{AdZv15} (case of 3 critical values) and \\cite[Theorem 1]{Adr17} (case of 4 or more critical values). \nBy Lemma~\\ref{lem:sga} the possible monodromy and ramification type pairs for a tamely ramified rational function over an algebraically closed field in arbitrary characteristic can only be the ones occurring over $\\mathbb{C}$. \nThe lists of possible monodromy groups different from $A_n,S_n,C_n,D_n$ together with the corresponding ramification data appear in \\cite[\\S 3]{AdZv15} and \\cite[Table 1]{Adr17}. We will refer to these henceforth as \\emph{the tables}.\n\nThe case $C_n$ only occurs if $n$ is prime with ram.~type $(n^1,n^1)$,\nand the case $D_n$ occurs only when $n>2$ is prime with ram.~type \n$(1^12^{(n-1)\/2}, 1^12^{(n-1)\/2}, n^1)$, see \\cite[Section 2.2]{AdZv15}\nwhere this can be read off from the corresponding dessins d'enfants. \nIn both cases there is a totally ramified branch point (i.e.\\ after applying a fractional-linear transformation $w$ becomes a polynomial of degree $n$).\n\nConsider the rational map $w=\\frac fc\\colon\\P^1\\to\\P^1$. \nBy our assumptions it has degree $n$, simple poles at the roots of $c$ and a pole of multiplicity $n-m$ at infinity. \nNote that $\\deg(f'c-fc')=n+m-1$ as $m\\not\\equiv n\\pmod p$. \n\\\\\n\n$(i)$.\nAssume that $p\\neq 2$, $m\\equiv n\\pmod 2$ and $g=f'c-fc'$ is squarefree.\nAssume further that $\\mathrm{Mon}(w)\\neq S_n$, in particular $n>2$. \nAs all zeros of $g$ are simple and $p\\neq 2$, \nthe ramification points of $w$ are\nthe zeros of $g$ with ramification index 2 (Lemma~\\ref{lem:crit}) \nand $\\infty$ with ramification index $n-m$. \nIn particular, $w$ is tamely ramified (as $p\\neq 2$ and $p\\nmid n-m$)\nand the ramification type of $w$ has the form \n$$\n \\left(1^{a_1}2^{b_1},\\ldots,1^{a_r}2^{b_r},1^{m}(n-m)^1\\right), \n$$\nwhere $r$ is the number of finite critical values of $w$\nand $a_i+2b_i=n$ for all $i$. \nAs $\\sum_{i=1}^rb_i=\\deg g\\geq n-1>n\/2$, we see that $r\\geq2$.\n\nNote that $\\mathrm{Mon}(w)$ cannot be $A_n$ since the last entry in the ramification type corresponds to an odd permutation, \nit cannot be $C_n$ with $n$ prime since then it would have only one finite critical value and it cannot be $D_n$ with $n$ prime since this case occurs only if $w$ has a totally ramified branch point (i.e. $n^1$ in its ramification type), but if $n>2$ is prime then $m$ must be odd and thus $w$ is not a polynomial. \nTherefore $\\mathrm{Mon}(w)$ must be one of the entries in the tables.\n\nGoing through the tables we see that the only entries with a ramification type as above (with $n-m$ even) are:\n\\begin{itemize}\n\\item {\\bf 4\/6.2} with $\\mathrm{Mon}(w)=PGL_2(5)$ and ramification type $(1^22^2,1^22^2,2^3,1^24^1)$.\n\\item {\\bf 4\/8.4} with $\\mathrm{Mon}(w)=PGL_2(7)$ and ramification type $(1^22^3,1^22^3,1^22^3,1^26^1)$.\n\\item {\\bf 4\/9.1} with $\\mathrm{Mon}(w)=AGL_2(3)$ and ramification type $(1^32^3,1^32^3,1^32^3,1^18^1)$.\n\\item {\\bf 4\/10.1} with $\\mathrm{Mon}(w)=P\\Gamma L_2(9)$ and ramification type $(1^42^3,1^42^3,2^5,1^28^1)$.\n\\end{itemize}\n\n$(ii)$.\nNow assume instead that $p\\notin\\{2,3\\}$, $m\\not\\equiv n\\pmod 2$ and $f'c-fc'=g^2$ with $g$ squarefree.\nAs all zeros of $g^2$ have multiplicity $2$ and $p\\notin\\{2,3\\}$, \nthe ramification points of $w$ are\nthe zeros of $g$ with ramification index 3 (Lemma~\\ref{lem:crit}) \nand $\\infty$ with ramification index $n-m$. \nIn particular, $w$ is tamely ramified (as $p\\neq 3$ and $p\\nmid n-m$) and has a ramification type of the form\n$$\n \\left(1^{a_1}3^{b_1},\\ldots,1^{a_r}3^{b_r},1^{m}(n-m)^1\\right), \n$$\nwhere $r$ is the number of finite critical values of $w$. \nSince now $n-m$ is odd, all entries in the ramification type correspond to even permutations.\nSo since $\\mathrm{Mon}(w)$ is generated by the inertia groups, $\\mathrm{Mon}(w)\\leqslant A_n$, which excludes the case $S_n$. \nIf $n=3$ (and $m=0$), then $r=1$ and $\\mathrm{Mon}(w)=A_3=C_3$.\nIf $n>3$, then similarly to part (i) one can argue that $r\\geq2$\nand that one can exclude the cases $C_n$ and $D_n$ ($D_n$ comes with ramification index 2 at the finite critical points).\n\nNow once again we go over the tables and list the entries with a ramification type of the above form (with $n-m$ odd):\n\\begin{itemize}\n\\item {\\bf 8.1} with $\\mathrm{Mon}(w)=A\\Gamma L_1(8)$ and ramification type $(1^23^2,1^23^2,1^17^1)$.\n\\item {\\bf 8.9} with $\\mathrm{Mon}(w)=L_2(7)$ and ramification type $(1^23^2,1^23^2,1^17^1)$.\n\\item {\\bf 9.7} with $\\mathrm{Mon}(w)=P\\Gamma L_2(8)$ and ramification type $(1^33^2,1^33^2,9^1)$.\n\\item {\\bf 12.10} with $\\mathrm{Mon}(w)=M_{12}$ and ramification type $(1^33^3,1^33^3,1^111^1)$.\n\\item {\\bf 24.5} with $\\mathrm{Mon}(w)=M_{24}$ and ramification type $(1^63^6,1^63^6,1^123^1)$.\\qedhere\n\\end{itemize}\n\\end{proof}\n\n\\begin{remark} \nWhile the above proposition relies on the CFSG in general, \nif we assume that $n-m$ is small, much more elementary group theory is sufficient. \nFor example if $n-m\\le 15$ the result can be derived from the classification by Manning of primitive groups of class $\\le 15$ (see \\cite[\\S II.15]{Wie64} for the list of references).\n\\end{remark}\n\n\n\n\n\n\\begin{lemma}\\label{lem:indec} \nLet $w\\colon\\P^1\\to\\P^1$ be a tamely ramified rational function defined over a field $k$. \nAssume that $w$ has a multiple pole at $\\infty$ and all other poles over $\\overline{k}$ are simple. Assume further that the numerator $g\\in k[X]$ of the derivative $w'$ is irreducible or the square of an irreducible polynomial.\nThen $w$ is indecomposable over $k$.\n\\end{lemma}\n\n\\begin{proof} It will be convenient to introduce two new variable symbols $T,U$ and consider three separate copies of $\\P^1_k$ which we will denote $\\P^1_X,\\P^1_T,\\P^1_U$. \nFor the sake of contradiction, \nassume that $w=v\\circ u$ is the composition of two rational functions\n$u,v\\in k(X)$ with $\\deg u,\\deg v>1$. \nSince $w(\\infty)=\\infty$, assume without loss of generality that $u(\\infty)=\\infty$ and $v(\\infty)=\\infty$.\nConsider the corresponding coverings\n$$\n \\P^1_X\\xrightarrow{u}\\P^1_T\\xrightarrow{v}\\P^1_U.\n$$\nFor a divisor $D=\\sum_P n_PP$ on $\\P^1_X$ we denote \n$$\n D^0=\\sum_{w(P)\\neq\\infty}n_PP,\\quad D^\\infty=\\sum_{w(P)=\\infty}n_PP\n$$ \nand similarly for a divisor $D$ on $\\P^1_T$ we denote\n$$\n D^0=\\sum_{v(P)\\neq\\infty}n_PP,\\quad D^\\infty=\\sum_{v(P)=\\infty}n_PP.\n$$\nConsider the differents $\\mathfrak d_w$, $\\mathfrak{d}_u$, $\\mathfrak{d}_v$ of $w$, $u$ and $v$\nas divisors on $\\P^1_X$, $\\P^1_X$ and $\\P^1_T$, respectively.\nThe Riemann-Hurwitz formula implies that\n$\\deg\\mathfrak{d}_u=2\\deg u-2$ and\n$\\deg\\mathfrak{d}_v=2\\deg v-2$.\nSince all finite poles of $w$ are simple,\nthe same holds for the finite poles of $u$ and $v$.\nMoreover, since $w$ is tamely ramified, so are $u$ and $v$,\nwhich therefore each ramify over at least two geometric primes.\nThus, $\\mathfrak{d}_u^0> 0$ and $\\mathfrak{d}_v^0>0$.\nBy the basic properties of differents we have \n$\\mathfrak d_w=u^*\\mathfrak d_v+\\mathfrak d_u$.\nTherefore, as all divisors in this relation are effective,\n\\begin{equation}\\label{eq:indec1}\n \\mathfrak d_w^0=u^*\\mathfrak d_v^0+\\mathfrak d_u^0.\n\\end{equation}\nNote that $\\mathfrak d_w^0$ is precisely the zero divisor of the numerator of $w'$,\nwhich by assumption is of the form\n$\\mathfrak d_w^0=Q$ or $\\mathfrak d_w^0=2Q$ for some prime divisor $Q$.\nThe first case is already excluded by (\\ref{eq:indec1}), \nso assume that $\\mathfrak d_w^0=2Q$. \nThen (\\ref{eq:indec1}) gives that\n$u^*\\mathfrak d_v^0=\\mathfrak d_u^0=Q$.\nWe have\n$$\n (\\deg u)(\\deg\\mathfrak d_v^0)=\\deg u^*\\mathfrak{d}_v^0=\\deg Q=\\deg \\mathfrak d_u^0\\le\\deg\\mathfrak d_u=2\\deg u-2,\n$$\nwhich implies $(\\deg u)(\\deg\\mathfrak{d}_v^0-2)\\leq -2$,\nhence $\\deg u=2$ and $\\deg\\mathfrak d_v^0=1$. \nSince $\\deg\\mathfrak{d}_v^0+\\deg\\mathfrak{d}_v^\\infty=2\\deg v-2$, the latter shows that $\\mathfrak d_v^\\infty>0$.\nSince $\\deg\\mathfrak{d}_u^0+\\deg\\mathfrak{d}_u^\\infty=2\\deg u-2=2$\nand $\\deg\\mathfrak{d}_u^0=\\deg Q=(\\deg u)(\\deg\\mathfrak{d}_v^0)\\geq 2$ it also follows that $\\mathfrak{d}_u^\\infty=0$.\nThus $v$ ramifies over $\\infty$ but $u$ does not,\nwhich implies that $w=v\\circ u$ has $\\deg u=2$ many\nmultiple poles over $\\overline{k}$, contradicting our assumption.\n\\end{proof}\n\n\\begin{proposition} \\label{prop:monfq}\nLet $p$ be prime, $q$ a power of $p$, and $m\\ge 0$ and $n\\ge m+2$ integers with $n\\not\\equiv m\\pmod p$. \nLet $f,c\\in\\mathbb{F}_q[X]$ be polynomials with ${\\rm deg}(f)=n$, ${\\rm deg}(c)=m$, $(f,c)=1$ and $c$ squarefree.\nLet $w=\\frac{f}{c}$.\n\\begin{enumerate}\n\\item[(i)] Assume $p>2$, $m\\equiv n\\pmod 2$, $m\\neq 2$ and $g=f'c-fc'$ is irreducible. Then $\\mathrm{Mon}_{\\overline{\\F}_q}(w)=S_n$ or \n\\begin{itemize}\n\\item $n=9,m=1$, $\\mathrm{Mon}_{\\overline{\\F}_q}(w)=AGL_2(3)$ with ram.~type $(1^32^3,1^32^3,1^32^3,1^18^1)$\n\\end{itemize}\n\n\\item[(ii)] Assume $p>3$, $m\\not\\equiv n\\pmod 2$, and $f'c-fc'=g^2$ where $g\\in \\mathbb{F}_q[X]$ is irreducible. Then $\\mathrm{Mon}_{\\overline{\\F}_q}(w)=A_n$ or one of the following:\n\\begin{itemize}\n\\item $n=8,m=1,\\mathrm{Mon}_{\\overline{\\F}_q}(w)=A\\Gamma L_1(8)$ or $L_2(7)$ and ram.~type $(1^23^2,1^23^2,1^17^1)$.\n\\item $n=9,m=0,\\mathrm{Mon}_{\\overline{\\F}_q}(w)=P\\Gamma L_2(8)$ and ram.~type $(1^33^2,1^33^2,9^1)$.\n\\item $n=12,m=1,\\mathrm{Mon}_{\\overline{\\F}_q}(w)=M_{12}$ and ram.~type $(1^33^3,1^33^3,1^111^1)$.\n\\item $n=24,m=1,\\mathrm{Mon}_{\\overline{\\F}_q}(w)=M_{24}$ and ram.~type $(1^63^6,1^63^6,1^123^1)$.\n\\end{itemize}\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof} \nNote that $\\mathrm{Mon}_{\\overline{\\F}_q}(w)\\unlhd\\mathrm{Mon}_{\\mathbb{F}_q}(w)$ are transitive\nsubgroups of $S_n$.\nBy Lemma~\\ref{lem:indec}, $w$ is indecomposable over $\\mathbb{F}_q$,\nhence $\\mathrm{Mon}_{\\mathbb{F}_q}(w)$ is primitive (Lemma \\ref{lem:indecomposable_primitive}),\nand Lemma~\\ref{lemcycle}(ii) applied to the infinite prime\nshows that $\\mathrm{Mon}_{\\overline{\\F}_q}(w)$ contains an $(n-m)$-cycle.\nWe treat parts $(i)$ and $(ii)$ simultaneously but distinguish several cases according to the value of $m$:\n\nCase $m=0$: In this case $w$ \nis a polynomial. By \\cite[Theorem 3.5]{FrMa69} a polynomial which is indecomposable over $\\mathbb{F}_q$ of degree coprime with $p$ is also indecomposable over $\\overline{\\F}_q$ and therefore in our case $w$ is indecomposable over $\\overline{\\F}_q$. \nThe conditions of Proposition~\\ref{prop:mon}(i) resp.~(ii) hold and therefore \n$\\mathrm{Mon}_{\\overline{\\F}_q}(w)=S_n$ (none of the exceptional cases has $m=0$) resp.~$\\mathrm{Mon}_{\\overline{\\F}_q}(w)=A_n$ or the exceptional case with $m=0$ appearing in Proposition~\\ref{prop:mon}(ii).\n\nCase $m=1$:\n$\\mathrm{Mon}_{\\overline{\\F}_q}(w)$ is transitive and contains an $(n-1)$-cycle and is therefore primitive, hence $w$ is indecomposable over $\\overline{\\F}_q$ (Lemma \\ref{lem:indecomposable_primitive}). The conditions of Proposition~\\ref{prop:mon}(i) resp.~(ii) hold and therefore $\\mathrm{Mon}_{\\overline{\\F}_q}(w)=S_n$ or the exceptional case with $m=1$ in Proposition~\\ref{prop:mon}(i) resp.~$\\mathrm{Mon}_{\\overline{\\F}_q}(w)=A_n$ or one of the exceptional cases with $m=1$ listed in Proposition~\\ref{prop:mon}(ii).\n\nCase $m=2$: Note that this case is allowed only in part $(ii)$. \nSince $n\\ge 5$ is odd by assumption, $(n-2)$ is coprime with $n$ and so $\\mathrm{Mon}_{\\overline{\\F}_q}(w)$ must be primitive since it is transitive and contains an $(n-2)$-cycle. Hence $w$ is indecomposable over $\\overline{\\F}_q$ (Lemma~\\ref{lem:indecomposable_primitive}). The conditions of Proposition~\\ref{prop:mon}(ii) hold and therefore $\\mathrm{Mon}_{\\overline{\\F}_q}(w)=A_n$.\n\nCase $m>2$: Here $\\mathrm{Mon}_{\\mathbb{F}_q}(w)$ is primitive and contains an $(n-m)$-cycle, so by Theorem~\\ref{thm:jones}(i) it is $S_n$ or $A_n$.\nNote that $n\\geq m+2\\geq 5$, so $A_n$ is simple and $S_n$ has a unique nontrivial cyclic quotient.\nNow $\\mathrm{Mon}_{\\overline{\\F}_q}(w)\\unlhd\\mathrm{Mon}_{\\mathbb{F}_q}(w)$ and $\\mathrm{Mon}_{\\mathbb{F}_q}(w)\/\\mathrm{Mon}_{\\overline{\\F}_q}(w)$ is cyclic,\nhence also $\\mathrm{Mon}_{\\overline{\\F}_q}(w)$ is $S_n$ or $A_n$.\nUnder the assumptions of part $(i)$, the $(n-m)$-cycle is odd, and thus $\\mathrm{Mon}_{\\overline{\\F}_q}(w)=S_n$.\nUnder the assumptions of part $(ii)$, the $(n-m)$-cycle is even, as are all the other inertia subgroups\n(the ramification indices at the finite ramified primes all equal $3$),\nso since $\\mathrm{Mon}_{\\overline{\\F}_q}(w)$ is generated by the inertia subgroups,\n$\\mathrm{Mon}_{\\overline{\\F}_q}(w)=A_n$.\n\\end{proof}\n\nFor a squarefree polynomial $c\\in\\mathbb{F}_q[X]$ we denote \n$$\n \\mathcal{H}_c=\\left\\{R^p: R\\in\\left(\\mathbb{F}_q[X]\/c^2\\mathbb{F}_q[X]\\right)^\\times\\right\\}\\leqslant\\left(\\mathbb{F}_q[X]\/c^2\\mathbb{F}_q[X]\\right)^\\times.\n$$\nDenote also \n\\begin{equation}\\label{eq:def_pi_c}\n \\pi_c=\\prod_{i=1}^m(X-\\alpha_i)^2\\cdot\\sum_{i=1}^m\\frac 1{(X-\\alpha_i)^2}=\\sum_{i=1}^m\\prod_{j\\neq i}(X-\\alpha_j)^2\\in\\mathbb{F}_q[X],\n\\end{equation}\nwhere $\\alpha_1,\\dots,\\alpha_m\\in\\overline{\\F}_q$ are the roots of $c$. Note that $(\\pi_c,c)=1$.\nIn short we write $\\pi_c\\mathcal{H}_c$ for the coset $(\\pi_c+c^2\\mathbb{F}_q[X])\\mathcal{H}_c$ in $(\\mathbb{F}_q[X]\/c^2)^\\times$.\n\n\\begin{lemma} \\label{lem:der} \nLet $c\\in\\mathbb{F}_q[X],\\deg c=m$ be a squarefree polynomial and $g\\in\\mathbb{F}_q[X]$ another polynomial such that $(g,c)=1$ and $\\deg g\\ge 2m$. \nDenote $n=\\deg(g)-m+1$ and assume that $p=\\mathrm{char}(\\mathbb{F}_q)>n-m$. \nAssume further that $g\\bmod c^2\\in\\pi_c\\mathcal H_c$. \nThen there exists $f\\in\\mathbb{F}_q[X],\\deg f=n,(f,c)=1$ such that $g=f'c-fc'$.\n\\end{lemma}\n\n\\begin{proof} \nWithout loss of generality, $c$ is monic.\nWrite $g=uc^2+v,u,v\\in\\mathbb{F}_q[X],\\deg v<\\deg c^2=2m$. Let $\\alpha_1,\\ldots,\\alpha_m\\in\\overline{\\F}_q$ be the \nroots of $c$ (they are pairwise distinct since $c$ is squarefree). By assumption $v\\equiv\\pi_cw\\pmod{c^2}\n$, where $w$ is a $p$-th power modulo each $(X-\\alpha_i)^2$, the latter condition implying $w\\equiv \na_i\\pmod {(X-\\alpha_i)^2}$ for a (uniquely determined) $0\\neq a_i\\in\\overline{\\F}_q$. \nIt follows using (\\ref{eq:def_pi_c}) that $v\\equiv a_i\\prod_{j\\neq i}(X-\\alpha_j)^2\\pmod{(X-\\alpha_i)^2}$ for each $1\\le i\\le m$,\nhence $v\\equiv \\sum_{i=1}^ma_i\\prod_{j\\neq i}(X-\\alpha_j)^2\\pmod{c^2}$. \nSince both sides of this congruence are of degree less than $\\deg c^2$,\nwe get the following\npartial fraction decomposition over $\\overline{\\F}_q$:\n$$\n \\frac v{c^2}=\\sum_{i=1}^m\\frac{a_i}{(X-\\alpha_i)^2}.\n$$ \nWe therefore can write\n$$\n -\\sum_{i=1}^m\\frac{a_i}{X-\\alpha_i}=\\frac\\psi c\\quad\\mbox{ with }\\quad\\psi\\in\\overline{\\F}_q[X],\\deg\\psin-m$. \nLet $c\\in\\mathbb{F}_q[X],\\deg c=m$ be squarefree.\n\\begin{enumerate}\n\\item[(i)] Assume that $q^{\\frac {n-m-1}2}>2m+1$. \nThen there exists $f\\in\\mathbb{F}_q[X]$, $\\deg f=n$, $(f,c)=1$, such that $g=f'c-fc'$ is irreducible in $\\mathbb{F}_q[X]$.\n\\item[(ii)] Assume $n\\not\\equiv m\\pmod 2$ and $q^{\\frac {n-3m-1}4}>2m+1$. Then there exists $f\\in\\mathbb{F}_q[X]$, $\\deg f=n$, $(f,c)=1$, such that $f'c-fc'=g^2$ with $g\\in\\mathbb{F}_q[X]$ irreducible.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof} \nIn case (i) let $N=n+m-1$ and $\\psi_c=\\pi_c$;\nin case (ii) let $N=\\frac{n+m-1}{2}$ and choose $\\psi_c\\in\\mathbb{F}_q[X]$ with $\\psi_c^2\\equiv\\pi_c\\pmod{c^2}$.\nThe latter exists since $\\pi_c$ is a square modulo $c$ (e.g.~$\\pi_c\\equiv\\delta^2\\pmod c$ with\n$\\delta=\\prod_{i=1}^m(X-\\alpha_i)\\cdot\\sum_{i=1}^m\\frac 1{(X-\\alpha_i)}$)\nand the kernel of $(\\mathbb{F}_q[X]\/c^2)^\\times\\rightarrow(\\mathbb{F}_q[X]\/c)^\\times$\nhas order $q^m$, which is odd.\n\nBy Lemma~\\ref{lem:der} it suffices to show that there is a monic irreducible polynomial $g\\in\\mathbb{F}_q[X]\n$ with $\\deg g=N$ such that $g\\bmod c^2\\in\\mathcal \\psi_c\\mathcal{H}_c$ (for case $(ii)$ note that $g\\bmod c^2\\in\n\\psi_c\\mathcal{H}_c$ implies $g^2\\bmod c^2\\in\\pi_c\\mathcal{H}_c$).\nDenote by $\\mathcal H_c^\\perp$ the group of Dirichlet characters modulo $c^2$ that are trivial on $\\mathcal H_c$ (this is the orthogonal group of $\\mathcal H_c$). \nWe have $|\\mathcal H_c^\\perp|=[(\\mathbb{F}_q[X]\/c^2)^\\times:\\mathcal H_c]=q^m$. \nBelow when we sum over $g$ we will always restrict $g$ to be monic,\nand $\\Lambda$ denotes the polynomial von Mangoldt function, \ni.e.~$\\Lambda(g)=\\deg P$ if $g=P^\\nu$ for a monic irreducible $P\\in\\mathbb{F}_q[X]$ and $\\nu\\geq1$, and $\\Lambda(g)=0$ otherwise. \nBy the second orthogonality relation \\cite[Corollary on p. 63]{Ser73} applied to the group $\\left(\\mathbb{F}_q[X]\/c^2\\right)^\\times\/\\mathcal H_c$ (the dual of this group is can be identified with $\\mathcal H_c^\\perp$) and the element $g\/\\psi_c$ we have\n\n$$\\sum_{\\chi\\in\\mathcal H_c^\\perp}\\overline{\\chi(\\psi_c)}\\chi(g)=\\left[\\begin{array}{ll}q^m,&\\mbox{if } g\\bmod c^2\\in\\psi_c\\mathcal H_c,\\\\\n0,&{\\mathrm {otherwise}},\\end{array}\\right.$$ and hence\n\n$$\n \\sum_{\\deg g=N\\atop{g \\bmod c^2\\in\\psi_c\\mathcal H_c}}\\Lambda(g)=\n\\frac 1{q^m}\\sum_{\\chi\\in\\mathcal H_c^\\perp}\\overline{\\chi(\\psi_c)}\\sum_{\\deg g=N}\\chi(g)\\Lambda(g).\n$$\nLet $\\chi_1$ denote the trivial character modulo $c^2$.\nFor all $\\chi\\neq\\chi_1$,\nWeil's Riemann Hypothesis for function fields gives that\\footnote{For example using the notation of \\cite[p.~41-42]{Ros02}:\n$\\sum_{\\deg g=N}\\chi(g)\\Lambda(g)=c_N(\\chi)=-\\sum_{k=1}^{M-1}\\alpha_k(\\chi)^N$\nwhere $M=\\deg c^2$ and $|\\alpha_k(\\chi)|\\leq\\sqrt{q}$.}\n$$\n \\left|\\sum_{\\deg g=N}\\chi(g)\\Lambda(g)\\right|\\le (2m-1)q^{\\frac{N}2}.\n$$\nThus together with the\nelementary relation $\\sum_{\\deg g=N}\\Lambda(g)=q^N$ (see \\cite[Proposition 2.1]{Ros02}) we obtain\n\\begin{multline*}\n\\sum_{\\deg g=N\\atop{g \\bmod c^2\\in\\psi_c\\mathcal H_c}}\\Lambda(g)\\ge\n\\frac{1}{q^m}\\sum_{\\deg g=N\\atop{(g,c)=1}}\\Lambda(g)-\\frac{q^m-1}{q^m}(2m-1)q^{\\frac{N}2}\\ge \n q^{N-m}-1-(2m-1)q^{\\frac{N}2}.\n\\end{multline*}\nBy the same elementary relation,\n$$\n \\sum_{\\deg g=N\\atop{g\\bmod c^2\\in\\mathcal \\psi_cH_c\\atop{g\\,\\mathrm{not\\,irreducible}}}}\\Lambda(g)\\le\\sum_{d|N\\atop{d\\neq N}}q^d\\le 2q^{\\frac{N}2}-1,\n$$ \n and therefore as long as\n$$\n q^{N-m}>(2m+1)q^{\\frac{N}2}\n$$ \nthere must be at least one irreducible $g$ with $\\deg g=N$ and $g\\bmod c^2\\in\\mathcal \\psi_c\\mathcal H_c$.\nTo conclude the proof note that $\\frac{N}{2}-m$ equals $\\frac{n-m-1}{2}$ in case $(i)$\nand $\\frac{n-3m-1}{4}$ in case $(ii)$.\n\\end{proof}\n\n{\n\\allowdisplaybreaks\n\n\\begin{theorem}\\label{thm:tame2} \nLet $p$ be a prime, $q$ a power of $p$, $0\\le m\\le n-2$ integers such that $n-m2$,\n\\item $m\\equiv n\\pmod 2$ and $m\\neq 2$, \n\\item $(n,m)\\neq (9,1)$, and\n\\item $q^{\\frac{n-m-1}2}>2m+1$,\n\\end{enumerate}\nor\n\\item[(ii)] $G=A_n$ and the following conditions hold\n\\begin{enumerate}\n\\item $p>3$,\n\\item $m\\not\\equiv n\\pmod 2$ and $(n,p)=1$,\n\\item $(n,m)\\neq (8,1),(9,0),(12,1),(24,1)$,\n\\item $q^{\\frac{n-3m-1}4}>2m+1$, and\n\\item $m\\ge 2$ or $\\left(\\frac q{n-m}\\right)=1$.\n\\end{enumerate}\n\\end{enumerate}\nThen there exist $f,c\\in\\mathbb{F}_q[X]$ with ${\\rm deg}(f)=n$, ${\\rm deg}(c)=m$, and $c$ squarefree\nsuch that the splitting field of $f(X)-Tc(X)$ over $\\mathbb{F}_q(T)$ is geometric with Galois group $G$\nand is ramified\nover only one finite prime $\\mathcal F$,\nwith $\\disc_X(f-Tc)$ a power of $\\mathcal F$ and ${\\rm deg}(\\disc_X(f-Tc))=n+m-1$.\nIn particular, $r_{\\mathbb{F}_q(T)}(G)\\leq 2$.\n\\end{theorem}\n\n\\begin{proof} \n\n$(i)$. First assume that $G=S_n$, $p>2$, $m\\neq 2$, $n\\equiv m\\pmod 2$, $(n,m)\\neq (9,1)$ and $q^{\\frac{n-m-1}2}>2m+1.$ \nTake any squarefree $c\\in\\mathbb{F}_q[X],\\deg c=m$. \nBy Proposition~\\ref{prop:g}(i) one can find an $f\\in\\mathbb{F}_q[X],\\deg f=n$ such that $g=f'c-fc'$ is irreducible of degree $n+m-1$. By Proposition~\\ref{prop:monfq}(i) the splitting field $K$ of $f(X)-Uc(X)$ over $\\mathbb{F}_q(U)$ is geometric with Galois group $S_n$. \nLet $w=\\frac{f}{c}\\in\\mathbb{F}_q(X)$.\n\nConsider the discriminant $D(U)=\\disc_X(f(X)-Uc(X))$. Let \n$$\n \\alpha_1,\\alpha_2=\\alpha_1^q,\\ldots,\\alpha_{n+m-1}=\\alpha_1^{q^{n+m-2}}\\in\\mathbb{F}_{q^{n+m-1}}\n$$ \nbe the roots of $g$. \nThen by Lemma~\\ref{lem:crit} the roots of $D$ are ${w(\\alpha_i)},1\\le i\\le n+m-1$ (including multiplicity). The \nFrobenius map ${\\rm Fr}_q$ acts cyclically on ${w(\\alpha_i)}$ and hence $D=\\mathcal{F}^r$ is a \npower of some prime $\\mathcal{F}\\in\\mathbb{F}_q[U]$.\nBy Lemma~\\ref{lem:crit}, $w$ is ramified only over $\\mathcal F$ and (possibly) $\\infty$, which concludes the proof in the case $G=S_n$.\n\n$(ii)$.\nNow assume that $G=A_n$, $p>3$, $n\\not\\equiv m\\pmod 2$, $(n,m)\\neq (8,1)$, $(9,0)$, $(12,1)$, $(24,1)$ and $q^{\\frac{n-3m-1}4}>2m+1$.\nLet $c\\in\\mathbb{F}_q[X],\\deg c=m$ be a monic squarefree polynomial. \nBy Proposition~\\ref{prop:g}(ii) there exists a monic irreducible $g\\in\\mathbb{F}_q[X]$ and a monic $f\\in\\mathbb{F}_q[X],\\deg f=n$ such that $f'c-fc'=(n-m)g^2$ (we can always adjust the leading coefficients this way). \nNote that $\\deg f'=n-1$ since $(n,p)=1$.\nLet $w=\\frac{f}{c}\\in\\mathbb{F}_q(X)$ and let $K$ be the splitting field of $f(X)-Uc(X)$ over $\\mathbb{F}_q(U)$.\nBy Proposition~\\ref{prop:monfq}(ii),\n$\\mathrm{Mon}_{\\overline{\\F}_q}(w)=A_n$, hence ${\\rm Gal}(K\/\\mathbb{F}_q(U))=\\mathrm{Mon}_{\\mathbb{F}_q}(w)$ is either $A_n$ or $S_n$ and $K\/\\mathbb{F}_q(U)$ is geometric if it is $A_n$, which happens precisely when $D(U)=\\disc_X(f(X)-Uc(X))$ is a square in $\\mathbb{F}_q(U)$ (Lemma~\\ref{lem:discgalois}).\n\nTo shorten notation, we omit the variable $X$ in the following calculation. All discriminants and resultants are with respect to the variable $X$. Terms which are squares in $\\mathbb{F}_q(U)$ are of no consequence and are simply denoted by $\\square$.\nIf $\\deg c\\ge 1$ then using the properties of resultants and discriminants \nwe calculate:\n\\begin{eqnarray*}\nD(U)&\\stackrel{(\\ref{eq:disc})}=&(-1)^{\\frac{n(n-1)}2}\\mathrm{Res}(f'-Uc',f-Uc)\\\\\n&\\stackrel{(\\ref{eq:resbimult})}=&(-1)^{\\frac{n(n-1)}2}\\mathrm{Res}(c,f-Uc)^{-1}\\mathrm{Res}(f'c-Uc'c,f-Uc)\\\\\n&\\stackrel{(\\ref{eq:resalt})}=&\\square\\cdot(-1)^{\\frac{n(n-1)}2}\\mathrm{Res}(c,f)\\mathrm{Res}(f'c-fc'+c'(f-Uc),f-Uc)\\\\\n&\\stackrel{(\\ref{eq:resalt}),(\\ref{eq:ressym})}=&\\square\\cdot(-1)^{\\frac{n(n-1)}2}\\mathrm{Res}(c,f)\\mathrm{Res}(f'c-fc',f-Uc)\\\\\n&\\stackrel{(\\ref{eq:resbimult}),(\\ref{eq:resdefinition})}=&\\square\\cdot(-1)^{\\frac{n(n-1)}2}\\mathrm{Res}(c,c')\\mathrm{Res}(c,c'f)(n-m)^n\\prod_{g(\\alpha)=0}(f(\\alpha)-Uc(\\alpha))^2\\\\\n&\\stackrel{(\\ref{eq:resalt})}=&\\square\\cdot(-1)^{\\frac{n(n-1)}2}\\mathrm{Res}(c,c')\\mathrm{Res}(c,c'f-cf')(n-m)^n\\\\\n&\\stackrel{(\\ref{eq:resbimult})}=&\\square\\cdot(-1)^{\\frac{n(n-1)}2}\\mathrm{Res}(c,c')\\mathrm{Res}(c,m-n)\\mathrm{Res}(c,g)^2(n-m)^n\\\\\n&\\stackrel{(\\ref{eq:resdefinition})}=&\\square\\cdot(-1)^{\\frac{n(n-1)}2+m}\\mathrm{Res}(c,c')(n-m)^{n+m}\\\\\n&\\stackrel{(\\ref{eq:disc}),(\\ref{eq:ressym})}=&\\square\\cdot(-1)^{\\frac{n(n-1)}2+\\frac{m(m+1)}2}(n-m)\\disc(c).\n\\end{eqnarray*}\nIf $c=1$ a similar calculation gives the same final expression for $D(U)$.\n\nTherefore, $D(U)$ is a square in $\\mathbb{F}_q(U)$ if and only if \n$$\n \\delta:=(-1)^{\\frac{n(n-1)}2+\\frac{m(m+1)}2}(n-m)\\disc(c)\n$$ \nis a square in $\\mathbb{F}_q$.\nIf $m<2$, then $\\disc(c)=1$, and\nby quadratic reciprocity,\n$$\n \\left(\\frac{(-1)^{\\frac{n(n-1)}{2}+m}(n-m)}{p}\\right)\n =(-1)^{\\frac{p-1}{4}(n^2+m-1)}\\left(\\frac{p}{n-m}\\right)\n =\\left(\\frac{p}{n-m}\\right)\n$$\nas $n^2+m-1\\equiv 0\\pmod 4$ if either $m=0$ and $n\\equiv1\\pmod 2$, or $m=1$ and $n\\equiv0\\pmod 2$,\nthus $\\delta$ is a square in $\\mathbb{F}_q$ if and only if $\\left(\\frac q{n-m}\\right)=1$, which is precisely our assumption \n(e) for this case.\nIf $m\\geq2$, then\n$\\disc(c)$ is a square in $\\mathbb{F}_q$ if and only if\n$(-1)^m\\mu(c)=1$ where $\\mu(c)$\nis the polynomial M\\\"obius function, see \\cite[Lemma 4.1]{Conrad}.\nTherefore we can choose $c$ accordingly with an odd or an even number of irreducible factors\nto get the value we need so that $\\delta$ is a square in $\\mathbb{F}_q$. \n\nThis shows that $K\/\\mathbb{F}_q(U)$ is geometric with Galois group $A_n$.\nArguing as in $(i)$ one sees that $D=\\mathcal F^r$ with $\\mathcal F$ prime and that $w$ is ramified only over $\\mathcal F$ and possibly $\\infty$.\n\\end{proof}\n\n}\n\n\\subsection{Eliminating tame ramification at infinity}\n\\label{sec:oneramified}\n\nWe will now explain how to eliminate\nthe tame ramification at infinity of the extensions obtained in the previous subsections,\nconditional on a weak consequence of the function field analogue of Schinzel's hypothesis H\nthat can actually be proven in many cases.\nThe classical Hypothesis H states the following:\n\n\\begin{conjecture}[Schinzel's hypothesis H]\\label{conj:Schinzel}\nLet $f_1,\\dots,f_r\\in\\mathbb{Z}[X]$ irreducible.\nIf $f_1\\cdots f_r$ is not the zero function modulo any prime number $p$,\nthen there exist infinitely many $n\\in\\mathbb{Z}$ with $f_1(n),\\dots,f_r(n)$ simultaneously prime.\n\\end{conjecture}\n\nIn the function field setting, \nthe naive analogue fails due to inseparability issues, \nsee \\cite{CCG}.\nTaking this into account,\none conjectures:\n\n\\begin{conjecture}\\label{conj:SchinzelFF}\nLet $\\mathcal{F}_1,\\dots,\\mathcal{F}_r\\in\\mathbb{F}_q[T][X]$ irreducible and separable in $X$.\nIf $\\mathcal{F}_1\\cdots \\mathcal{F}_r$ is not the zero function modulo any prime polynomial $P\\in\\mathbb{F}_q[T]$,\nthen there exist infinitely many $h\\in\\mathbb{F}_q[T]$ with $\\mathcal{F}_1(h),\\dots,\\mathcal{F}_r(h)$ simultaneously irreducible in $\\mathbb{F}_q[T]$.\n\\end{conjecture}\n\nFor partial results on Conjecture \\ref{conj:SchinzelFF} see for example\n\\cite{Pollack,BS12,EntinBH,Entin,SawinShusterman}.\nWe will work with the following weaker assumption:\n\n\\begin{definition} \nWe say that property $H(q,d,e)$ holds if for any irreducible $\\mathcal{F}\\in\\mathbb{F}_q[X]$ with $\\deg(\\mathcal{F})|d$ there exists $h\\in\\mathbb{F}_q[T]$ with $e|\\deg(h)$ such that \n$\\mathcal{F}(h)$ is irreducible in $\\mathbb{F}_q[T]$. \n\\end{definition}\n\n\nWe write ${\\rm rad}(e)$ for the radical of $e$\nand $\\omega(e)$ for the number of distinct prime divisors of $e$.\nWe define ${\\rm rad}'(e)=\\frac{\\gcd(4,e)}{\\gcd(2,e)}{\\rm rad}(e)$,\ni.e.~${\\rm rad}'(e)$ equals $2{\\rm rad}(e)$ if $4|e$ \nand ${\\rm rad}(e)$ otherwise.\nWe denote by $v_\\ell$ the $\\ell$-adic valuation.\n\n\\begin{proposition}\\label{lem:Hqdm}\n$H(q,d,e)$ holds in each of the following cases:\n\\begin{enumerate}\n\\item Conjecture \\ref{conj:SchinzelFF} holds for $\\mathbb{F}_q$.\n\\item $q$ is sufficiently large with respect to $d$ and $e$.\n\\item ${\\rm rad}'(e)|q-1$ and $\\gcd(d,e)=1$.\n\\item ${\\rm rad}'(e)|q^\\ell-1$ for every prime divisor $\\ell$ of $d$, and $q\\geq(d-1)^2(2^{\\omega(e)}-1)^2$.\n\\item ${\\rm rad}(e)|q-1$ and $q\\geq(2^{\\max\\{v_2(e)-1,0\\}}d-1)^2(2^{\\omega(e)}-1)^2$.\n\\end{enumerate}\nMore precisely, \nfor any irreducible $\\mathcal{F}\\in\\mathbb{F}_q[X]$ with $\\deg(\\mathcal{F})|d$ there exists $h\\in\\mathbb{F}_q[T]$ with $e|\\deg(h)$ such that \n$\\mathcal{F}(h)$ is irreducible in $\\mathbb{F}_q[T]$,\nwhere\n\\begin{enumerate}[(a)]\n\\item there exists such $h$ with ${\\rm deg}(h)=e$ in cases (2), (3), (4) and (5),\n\\item there exists such $h$ that is monic in cases (1), (2), (4) and (5), and\n\\item there exists such $h$ that is monic and satisfies ${\\rm deg}(h)=e$ in cases (4) and (5), as well as in case (2) if $q$ is odd or $e\\geq 4$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nLet $e,d\\geq 1$ and $\\mathcal{F}\\in\\mathbb{F}_q[X]$ irreducible with ${\\rm deg}(\\mathcal{F})=n|d$ be given.\n\n(1): \nWithout loss of generality assume that $q-1|e$.\nThe polynomial\n$$\n g:=\\mathcal{F}(X^e+A_{e-1}X^{e-1}+\\dots+A_0)\\in\\mathbb{F}_q[A_0,\\dots,A_{e-1},X]\n$$ \nis separable in $X$ and irreducible:\nIndeed, $g$ is obtained from $\\mathcal{F}(A_0)$ by the invertible change of variables $A_0\\mapsto X^e+A_{e-1}X^{e-1}+\\dots+A_0$.\nLet $S$ be the set of primes $P\\in\\mathbb{F}_q[T]$ with ${\\rm deg}(P)\\leq\\log_q{\\rm deg}_Xg$\nand let $M=\\prod_{P\\in S}P$.\nBy \\cite[Theorem 1.1]{BaEn19_} there exist $a_0\\in\\mathbb{F}_q+M\\mathbb{F}_q[T]$ and $a_1,\\dots,a_{e-1}\\in\\mathbb{F}_q[T]$ with\n$\\tilde{g}(X):=g(a_0,\\dots,a_{e-1},X)\\in\\mathbb{F}_q[T][X]$ separable in $X$ and irreducible in $\\mathbb{F}_q(T)[X]$,\nhence also irreducible in $\\mathbb{F}_q[T,X]$ (since it is primitive in $X$).\nIn particular, $\\tilde{g}(0)=\\mathcal{F}(a_0)\\not\\equiv 0\\mbox{ mod }P$ for every $P\\in S$,\nand modulo a prime $P\\in\\mathbb{F}_q[T]$ not in $S$, \n$\\tilde{g}$ has at most ${\\rm deg}_X\\tilde{g}1$\nand note that the assumption implies that ${\\rm rad}'(e)|q^n-1$.\nLet $\\alpha\\in\\mathbb{F}_{q^n}$ be a root of $\\mathcal{F}$.\nCohen's result \\cite[Corollary 2.3]{Cohen} gives that\nthe number of $c\\in\\mathbb{F}_q$ for which $\\alpha+c$ is not an $\\ell$-th power in $\\mathbb{F}_{q^n}$ for any prime divisor $\\ell$ of $e$ is bigger than\n$\\frac{\\phi({\\rm rad}(e))}{{\\rm rad}(e)}(q-(n-1)(2^{\\omega(e)}-1)\\sqrt{q})$,\nin particular this number is positive as soon as\n$q\\geq(n-1)^2(2^{\\omega(e)}-1)^2$, which is satisfied by our assumption.\nAs in (3) we conclude that $T^e-(\\alpha+c)\\in\\mathbb{F}_{q^n}[T]$ is irreducible,\nhence $\\mathcal{F}(h)$ is irreducible for $h=T^e-c$.\n\n(5): Let $s=v_2(e)$. If $s\\leq 1$, then ${\\rm rad}'(e)={\\rm rad}(e)$, so the claim follows from (4).\nIf $s\\geq 2$, then $q$ is odd and we first apply (4) with $e=2$ to obtain a monic $h_1\\in\\mathbb{F}_q[T]$ of degree $2$ with $\\mathcal{F}_1(T):=\\mathcal{F}(h_1(T))$ irreducible of degree $2n$. \nWe iterate this until we obtain $\\mathcal{F}_{s-1}=\\mathcal{F}(h_{1}(\\dots h_{s-1}(T)))$\nirreducible of degree $2^{s-1}n$.\nThen ${\\rm rad}'(e\/2^{s-1})={\\rm rad}(e\/2^{s-1})$,\nso we can apply (4)\nto obtain a monic $h_s$ of degree $e\/2^{s-1}$ with $\\mathcal{F}(h_1(\\dots h_s(T)))$\nirreducible.\nThe polynomial $h=h_1\\circ\\dots\\circ h_s$ has degree $e$ and satisfies the claim.\n\\end{proof}\n\n\n\\begin{lemma}\\label{lem:disjoint} \nLet $k$ be a perfect field and let $h\\in k[T]$ be a non-constant polynomial. \nDenote $U=h(T)$ and let $K$ be a Galois extension of $k(U)$ ramified only over $U=\\infty$ and the finite primes $\\mathcal{F}_1,\\dots,\\mathcal{F}_r\\in k[U]$, with tame ramification over $\\infty$. \nAssume that each $\\mathcal{F}_i(h(T))\\in k[T]$ is separable. \nThen the extensions $K,k(T)$ of $k(U)$ are linearly disjoint.\n\\end{lemma}\n\n\\begin{proof} \nDenote $D(U)=\\disc_T(h(T)-U)$. \nBy Lemma~\\ref{lem:crit} the finite branching primes of the extension $k(T)\/k(U)$ are exactly the primes dividing $D(U)$.\nIf $\\mathcal{F}_i|D$ for some $i$, then 0 would be a critical value of $\\mathcal{F}_i\\circ h\\colon\\P^1\\to\\P^1$, which means that $\\mathcal{F}_i(h)$ is not separable, contradicting our assumption. Thus $\\mathcal F_i\\nmid D$ for each $i$,\nso the finite branching loci of $k(T)\/k(U)$ and $K\/k(U)$ are disjoint,\nhence the extension $K\\cap k(T)\/k(U)$ is tamely ramified and ramified only over $\\infty$, and is therefore trivial.\nSince $K\/k(U)$ is Galois, this already implies that $K$ and $k(T)$ are linearly disjoint over $k(U)$. \n\\end{proof}\n\n\n\\begin{lemma}\\label{lem:eliminateinfty}\nLet $k$ be a perfect field of characteristic $p\\geq 0$, and let $f\\in k[U,X]$.\nAssume that the splitting field of $f$ over $k(U)$\nis ramified only over the primes $\\mathcal{F}_1,\\dots,\\mathcal{F}_r\\in k[U]$ \nand over the infinite prime with ramification index $e$ \nnot divisible by $p$.\nIf $h\\in k[T]$ is such that $e|{\\rm deg}(h)$,\nthen the splitting field of $f(h(T),X)$ over $k(T)$\nis ramified at most over the prime factors of $\\mathcal{F}_1(h(T)),\\dots,\\mathcal{F}_r(h(T))$.\n\\end{lemma}\n\n\\begin{proof}\nWe identify $U$ with $h(T)$ and $k(U)=k(h(T))\\subseteq k(T)$. \nLet $K$ be the splitting field of $f(U,X)$ over $k(U)$,\nand consider the splitting field $L$ of $f(h(T),X)$ over $k(T)$, which is the compositum of $K$ and $k(T)$. \nSince $k(T)\/k(U)$ is totally ramified over $\\infty$ of degree ${\\rm deg}(h)$ divisible by $e$,\nby Abhyankar's Lemma (Lemma~\\ref{lem:abhyankar}) the base change from $k(U)$ to $k(T)$ eliminates the ramification over infinity. \nEvery finite prime of $k(T)$ that ramifies in $L$ lies over a \nfinite prime of $k(U)$ that ramifies in $K$,\ni.e.~is a prime factor of some $\\mathcal{F}_i(h)$.\n\\end{proof}\n\n\\begin{theorem}\\label{thm:tame} \nLet $p$ be a prime, $q$ a power of $p$, $0\\le m\\le n-2$ integers such that $n-m2$,\n\\item $m\\equiv n\\pmod 2$ and $m\\neq 2$,\n\\item $(n,m)\\neq (9,1)$,\n\\item $q^{\\frac{n-m-1}2}>2m+1$, and\n\\item $H(q,n+m-1,n-m)$,\n\\end{enumerate}\nor\n\\item[(ii)] $G=A_n$ and the following conditions hold\n\\begin{enumerate}\n\\item $p>3$,\n\\item $m\\not\\equiv n\\pmod 2$ and $(n,p)=1$,\n\\item $(n,m)\\neq (8,1),(9,0),(12,1),(24,1)$,\n\\item $q^{\\frac{n-3m-1}4}>2m+1$, \n\\item $m\\ge 2$ or $\\left(\\frac q{n-m}\\right)=1$, and\n\\item $H\\left (q,\\frac{n+m-1}2,n-m\\right)$.\n\\end{enumerate}\n\\end{enumerate}\nThen there exist $f,c\\in\\mathbb{F}_q[X]$ and $h\\in\\mathbb{F}_q[T]$\nwith ${\\rm deg}(f)=n$, ${\\rm deg}(c)=m$ and $(n-m)|{\\rm deg}(h)$\nsuch that the splitting field of $f-hc$ over $\\mathbb{F}_q(T)$\nis geometric with Galois group $G$\nand is ramified over only one prime of $\\mathbb{F}_q(T)$.\nIn particular, $r_{\\mathbb{F}_q(T)}(G)=1$.\n\\end{theorem}\n\n\\begin{proof}\nIn $(i)$,\nTheorem~\\ref{thm:tame2}$(i)$\nprovides $f,c$ of the right degree such that the splitting field $K$ of \n$f-Uc$ over $\\mathbb{F}_q(U)$ is geometric with Galois group $S_n$\nand is ramified over only one finite prime\n$\\mathcal{F}$ of degree dividing $n+m-1$\nand over the infinite prime, with ramification index $n-m$.\nThus $H(q,n+m-1,n-m)$ provides a suitable $h$ with $\\mathcal{F}(h)$ irreducible,\nhence Lemma~\\ref{lem:disjoint} gives that \nthe splitting field $K\\mathbb{F}_q(T)$ of $f-hc$ over $\\mathbb{F}_q(T)$ is geometric with Galois group $S_n$,\nand Lemma~\\ref{lem:eliminateinfty} gives that it is ramified only over $\\mathcal{F}(h)$.\n\nIn $(ii)$,\nTheorem~\\ref{thm:tame2}$(ii)$ similarly\nprovides $f,c$ of the right degree such that the splitting field of \n$f-Uc$ over $\\mathbb{F}_q(U)$ is geometric with Galois group $A_n$\nand is ramified over only one finite prime\n$\\mathcal{F}$, with $\\disc_X(f-Uc)=\\mathcal{F}^r$ of degree $n+m-1$,\nand over the infinite prime, with ramification index $n-m$.\nAs $G=A_n$ implies that $\\disc_X(f-Uc)$ is a square, $r$ is even,\nand so in particular ${\\rm deg}(\\mathcal{F})$ divides $\\frac{n+m-1}{2}$.\nThus $H(q,\\frac{n+m-1}2,n-m)$ \nprovides a suitable $h$ with $\\mathcal{F}(h)$ irreducible,\nand we conclude again with Lemma~\\ref{lem:disjoint} and Lemma~\\ref{lem:eliminateinfty}.\n\\end{proof}\n\n\\begin{remark}\nNote that in the cases (2)-(5) of Proposition~\\ref{lem:Hqdm},\none can obtain $h$ in Theorem~\\ref{thm:tame} to be of degree {\\em equal} to $n-m$.\n\\end{remark}\n\n\n\\subsection{Two ramified primes via small prime gaps}\n\\label{sec:twinprimes}\nIn this final subsection we present a different approach to produce extensions of group $S_n$ or $A_n$ with two ramified primes.\nFor this we will need the following result on primes with small gaps in $\\mathbb{F}_q[T]$, which follows directly from \\cite[Theorem 1.3]{CHLPT15}, which is a function field adaptation of the method of Maynard \\cite{May15}. \n\n\\begin{proposition}\\label{thm:twinprimes} Assume $q\\ge 107$ and fix $b\\in\\mathbb{F}_q^\\times$. For all sufficiently large $d$ (in terms of $q$) there exist polynomials $h\\in\\mathbb{F}_q[T]$ of degree $d$ (not necessarily monic) such that $h$ and $h-b$ are both irreducible.\n\\end{proposition}\n\n\\begin{proof}\nBy \\cite[Theorem 1.3]{CHLPT15} (applied with $m=2$ and $h_1,\\dots,h_k$ an enumeration of $\\mathbb{F}_q$) and the remark following it, as long as $q>k_0(2)=105$ (using the notation of the cited theorem) and $d\\ge d_0(q)$ is sufficiently large, one can find $f\\in\\mathbb{F}_q[T],\\deg f=d$ such that $f,f-a$ are irreducible for some $a\\in\\mathbb{F}_q^\\times$ (the statement of the cited theorem only asserts the existence of infinitely many such $f$, but the proof shows that such an $f$ exists with any sufficiently large degree). Then $h=\\frac ba f$ satisfies the assertion of the proposition.\n\\end{proof}\n\n\\begin{theorem}\\label{thm:main2} \nLet $p$ be a prime number, $q$ a power of $p$, and $3\\leq n2$ (for both $A_n$ and $S_n$). In all other cases only elementary group theory (mainly Jordan's theorem on primitive permutation groups containing a prime cycle) is used.\\end{remark}\n\n\n\n\\newcommand{L_1^{\\mathrm{gt}}}{L_1^{\\mathrm{gt}}}\n\n\\subsection{$L_1^{\\mathrm{gt}}(q)$-realizable groups}\n\n\\begin{definition}\\label{def:quasip} \nA finite group $G$ is called \\emph{quasi}-$p$ if it is generated by its $p$-Sylow subgroups,\ni.e.~$G=p(G)$. \nA finite group $G$ is called \\emph{cyclic-by-quasi-$p$} if it is an extension of a cyclic group by a quasi-$p$ group,\ni.e.~$d(G\/p(G))\\leq1$.\n\\end{definition}\n\n\\begin{example} \\label{ex:cyclicbyquasip}\nThe groups $S_n$ and $A_n$ are cyclic-by-quasi-$p$ if $n\\ge p$.\n\\end{example}\n\n\\begin{definition} \nWe say that a finite group $G$ is \\emph{$L_1^{\\mathrm{gt}}(q)$-realizable} if there exists a geometric Galois extension $L\/\\mathbb{F}_q(T)$ with ${\\rm Gal}(L\/\\mathbb{F}_q(T))=G$ that is ramified over at most two primes of $\\mathbb{F}_q(T)$, both of degree one, and with wild ramification over at most one of them. \nAn extension $L$ with this property is called an $L_1^{\\mathrm{gt}}(q)$-\\emph{realization} of $G$.\\footnote{Abhyankar introduced the notation $L_1$ for the once-punctured affine line. The letters $\\mathrm{gt}$ remind us that (up to M\\\"obius transformations) we only consider geometric Galois $\\mathbb{F}_q$-covers of $L_1$ which are tamely ramified at $\\infty$.}\\end{definition}\nNote that an $L_1^{\\mathrm{gt}}(q)$-realizable group is automatically $L_1^{\\mathrm{gt}}(q^r)$-realizable for any $r\\ge 1$.\n\n\\begin{lemma}\\label{lemreal}\nLet $G$ be an $L_1^{\\mathrm{gt}}(q)$-realizable group.\nThen $G$ is cyclic-by-quasi-$p$ and satisfies $[G:p(G)]\\mid q-1$,\nand there exists an $L_1^{\\mathrm{gt}}(q)$-realization of $p(G)$ that is ramified over a single prime of degree one.\n\\end{lemma}\n\n\\begin{proof} \nLet $L\/\\mathbb{F}_q(T)$ be an $L_1^{\\mathrm{gt}}(q)$-realization of $G$.\nWe may assume that it is ramified only over $T=0,\\infty$, tamely ramified over $\\infty$. \nLet $H=p(G)\\unlhd G$ and denote $K=L^H$. \nWe have ${\\rm Gal}(K\/\\mathbb{F}_q(T))=C:=G\/H$. Since $(|C|,p)=1$ the extension $K\/\\mathbb{F}_q(T)$ is tamely ramified and by assumption it is\nramified at most over $T=0,\\infty$,\nso it must be of the form $K=\\mathbb{F}_q(T^{1\/n})$ with $n=|C|$ and $n|q-1$\n(e.g.~since $K(T^{1\/n})\/\\mathbb{F}_q(T^{1\/n})$ is geometric, and unramified by Lemma~\\ref{lem:abhyankar}, hence trivial).\nIn particular, $C={\\rm Gal}\\left(\\mathbb{F}_q(T^{1\/n})\/\\mathbb{F}_q(T)\\right)$ is cyclic, so $G$ is cyclic-by-quasi-$p$ and $[G:p(G)]=n\\mid q-1$.\n\nLet $U=T^{1\/n}$ and let $e$ be the ramification index of the prime $U=\\infty$ in $L\/\\mathbb{F}_q(U)$.\nNote that $(e,p)=1$\nand denote $K'=\\mathbb{F}_q(S)$, where $S^e=U$. \nThe extension $K'\\overline{\\F}_q\/\\overline{\\F}_q(U)$ is Galois with ${\\rm Gal}(K'\\overline{\\F}_q\/\\overline{\\F}_q(U))=\\mathbb{Z}\/e\\mathbb{Z}$ and since $L\/\\mathbb{F}_q(U)$ is geometric we have \nthat ${\\rm Gal}(L\\overline{\\F}_q\/\\overline{\\F}_q(U))=p(G)$. \nSince $p(G)$ is quasi-$p$, the groups $p(G)$ and $\\mathbb{Z}\/e\\mathbb{Z}$ have no nontrivial common quotients and therefore the extensions \n$K'\\overline{\\F}_q,L\\overline{\\F}_q$ of $\\overline{\\F}_q(U)$ are linearly disjoint. \nConsequently the geometric extensions $K',L$ of $\\mathbb{F}_q(U)$ are also linearly disjoint and we have ${\\rm Gal}(L\\mathbb{F}_q(S)\/\\mathbb{F}_q(S))={\\rm Gal}(L\/\\mathbb{F}_q(U))=p(G)$. \nBy Lemma~\\ref{lem:eliminateinfty}, $L\\mathbb{F}_q(S)\/\\mathbb{F}_q(S)$ is ramified only over the prime $S$.\n\\end{proof}\n\n\\begin{question}\\label{ques:Abhyankar}\nIs a cyclic-by-quasi-$p$ group $G$ with $[G:p(G)]\\mid q-1$ always $L_1^{\\mathrm{gt}}(q)$-realizable?\n\\end{question}\n\nThis question is a variant of the arithmetic Abhyankar conjecture (Conjecture \\ref{conj:abhyankar}) and is suggested by Abhyankar's general philosophy regarding ramified covers in characteristic $p$. \nWe remark that Harbater and van der Put \\cite[Theorem 5.5]{HavdP02} have given a non-obvious necessary condition for a group to be the Galois group of an extension of $\\mathbb{F}_q(T)$ ramified over two geometric points, but this condition is automatically satisfied for cyclic-by-quasi-$p$ groups, \nas can easily be shown using Frattini's argument. \nThe next theorem answers Question \\ref{ques:Abhyankar} affirmatively for the symmetric and alternating groups (with some mild restrictions on $n,p$).\n\n\\begin{theorem}\\label{thmlist} Let $p$ be a prime, $q$ a power of $p$, and $n\\ge p$. The following groups are $L_1^{\\mathrm{gt}}(q)$-realizable:\n\\begin{enumerate} \n\\item $S_n$ if $n\\neq p+1$ or $p=2$.\n\\item $A_n$ if $p>2$ and either $n\\neq p+1$ or $\\mathbb{F}_q\\supseteq\\mathbb{F}_{p^2}$ or $p=3$.\n\\item $A_n$ if $p=2$ and either $\\mathbb{F}_q\\supseteq\\mathbb{F}_4$ or $10\\neq n\\ge 8$ and $n\\equiv 0,1,2,6,7\\pmod 8$.\n\\end{enumerate}\n\\end{theorem}\n\nTheorem~\\ref{thmlist} will be proved in Section \\ref{secthmlist}. \nNote that in the case of $A_n,n\\ge p>2$ Theorem~\\ref{thmlist} combined with Lemma~\\ref{lemreal}\nand Example \\ref{ex:cyclicbyquasip} immediately imply Theorem~\\ref{thm:abhyankar}.\n\n\\subsection{Application to the minimal ramification problem}\n\\label{sec:thmprod}\n\nWe note that $L_1^{\\mathrm{gt}}(q)$-re\\-al\\-iz\\-abil\\-ity implies\na positive answer to the minimal ramification problem:\n\n\n\\begin{lemma} \\label{removeinf} \\label{lemquasip} \nLet $G$ be an $L_1^{\\mathrm{gt}}(q)$-realizable nontrivial finite group. \nThen for infinitely many primes $h\\in\\mathbb{F}_q[T]$ there exists a geometric Galois extension $K\/\\mathbb{F}_q(T)$ with ${\\rm Gal}(K\/\\mathbb{F}_q(T))=G$ which is ramified only over $h$. \nIn particular, $r_{\\mathbb{F}_q(T)}(G)=1$ and $G$ satisfies Conjecture~\\ref{bm}.\n\\end{lemma}\n\n\\begin{proof} \nLet $K\/\\mathbb{F}_q(U)$ be a geometric realization of $G$ ramified over $U=0$ and (at most) tamely ramified over $U=\\infty$. \nDenote by $e$ the ramification index over $\\infty$,\nwhich by assumption is not divisible by $p$.\nLet $h\\in\\mathbb{F}_q[T]$ irreducible of degree divisible by $e$\nand identify $U=h(T)$.\nBy Lemma~\\ref{lem:disjoint}, $K$ and $\\mathbb{F}_q(T)$ are linearly disjoint over $\\mathbb{F}_q(U)$,\nand $K\\mathbb{F}_q(T)\/\\mathbb{F}_q(T)$ is ramified only over $h$ by Lemma~\\ref{lem:eliminateinfty}.\nTherefore, $K\\mathbb{F}_q(T)\/\\mathbb{F}_q(T)$ is geometric Galois of Galois group $G$ and ramified only over the single prime $h$,\nhence $r_{\\mathbb{F}_q(T)}(G)=1$.\n\\end{proof}\n\nIn this subsection we \nextend this to (certain) products of $L_1^{\\mathrm{gt}}(q)$-realizable groups,\nwhich we then combine with Theorem~\\ref{thmlist} to prove Theorem~\\ref{thm:main1}:\n\n\\begin{proposition}\\label{thmprod}\n Let $G=G_1\\times\\ldots\\times G_m$ be a product with each $G_i$ being $L_1^{\\mathrm{gt}}(q)$-realizable. \n Assume that there is a prime number $\\ell$ such that for each $1\\le i\\le m$ either $G_i$ is quasi-$p$ or $\\ell\\mid [G_i:p(G_i)]$. Then Conjecture \\ref{bm} holds for $G$ over $\\mathbb{F}_q(T)$.\n\\end{proposition}\n\nBy a $k$-{\\em variety} (for a field $k$) we mean a reduced $k$-scheme of finite type.\nIn what follows all maps are morphisms of $\\mathbb{F}_q$-varieties,\nand $\\mathbb{A}^n$ (resp. $\\P^n$) always denotes the affine (resp. projective) $n$-space over $\\mathbb{F}_q$ considered as an $\\mathbb{F}_q$-variety.\nThe function field of an irreducible $\\mathbb{F}_q$-variety $X$ is denoted by $\\mathbb{F}_q(X)$. \nWe will also use the notion of a Galois cover of $\\mathbb{F}_q$-varieties and its Galois group, see \\cite[\\S I.5]{Mil80} for the definition and basic properties. \nIn particular, if $w\\colon Y\\rightarrow X$ is a Galois cover of $\\mathbb{F}_q$-varieties, \nthen $Y$ and $X$ are connected and $w$ is \\'etale, and we denote its Galois group by ${\\rm Gal}(Y\/X)$. \nWe call $w$ a \\emph{geometric} Galois cover if $Y\\times\\mathrm{Spec}\\,\\overline{\\F}_q\\to X\\times\\mathrm{Spec}\\,\\overline{\\F}_q$ is also a Galois cover (if $X$ is geometrically irreducible this is equivalent to $Y$ being geometrically irreducible).\n\n\n\\begin{proposition}\\label{hit}\nLet $U\\subseteq\\mathbb{A}^m$ be an open $\\mathbb{F}_q$-subvariety and $Y\\to U$ a geometric Galois cover of $\\mathbb{F}_q$-varieties with ${\\rm Gal}(Y\/U)=G$.\nThen \nthere exists a morphism \n$\\psi\\colon\\mathbb{A}^1\\to\\mathbb{A}^m$ \n such that $\\psi^{-1}(U)\\neq\\emptyset$ and $Y\\times_U\\psi^{-1}(U)\\to\\psi^{-1}(U)$ is a geometric Galois cover with Galois group $G$.\\end{proposition}\n\n\\begin{proof} \nDenote $\\nu=|G|$ and $Y'=Y\\times_{\\mathbb{F}_q}\\mathrm{Spec}\\,\\mathbb{F}_{q^\\nu},U'=U\\times_{\\mathbb{F}_q}\\mathrm{Spec}\\,\\mathbb{F}_{q^\\nu}$. \nSince $Y\\rightarrow U$ is geometric, $Y'$ is irreducible and the cover $Y'=Y\\times_UU'\\to U$ is Galois with ${\\rm Gal}(Y'\/U)=G\\times {\\rm Gal}(\\mathbb{F}_{q^\\nu}\/\\mathbb{F}_q)$. (Generally, if $Z\\rightarrow X$ and $W\\rightarrow X$ are Galois covers then $Z\\times_X W$ is a union of isomorphic Galois covers with Galois group a subgroup of ${\\rm Gal}(Z\/X)\\times{\\rm Gal}(W\/X)$,\n and if $Z\\times_X W$ is irreducible we have ${\\rm Gal}(Z\\times_X W\/X)\\cong {\\rm Gal}(Z\/X)\\times{\\rm Gal}(W\/X)$. \nThis fact can be deduced from \\cite[Theorem 5.3]{Mil80} or by comparing with the Galois groups of the corresponding function field extensions.)\nWe will show that \none can choose $h_1,\\dots,h_m\\in\\mathbb{F}_q[T]$ such that the corresponding morphism $\\psi\\colon\\mathbb{A}^1\\rightarrow\\mathbb{A}^m$\nsatisfies $\\psi^{-1}(U)\\neq\\emptyset$ and\n$Y'\\times_U\\psi^{-1}(U)\\to\\psi^{-1}(U)$ is a Galois cover with Galois group \n${\\rm Gal}(Y'\/U)=G\\times{\\rm Gal}(\\mathbb{F}_{q^\\nu}\/\\mathbb{F}_q)$,\nwhich will imply that\n$Y\\times_U\\psi^{-1}(U)\\to\\psi^{-1}(U)$ is a {\\em geometric} Galois cover with Galois group $G$,\nsince any extension of the field of constants occurring in this cover must be of degree dividing $\\nu=|G|$.\n\nFirst note that \n$Y'\\times_U\\psi^{-1}(U)\\to\\psi^{-1}(U)$ is again \\'etale\nfor any $\\psi$,\nso it suffices to show that \n$h_1,\\dots,h_m$ can be chosen so that\nit is a Galois cover with Galois group ${\\rm Gal}(Y'\/U)$,\nfor which we are allowed to replace $U$ and $Y'$ by dense open subvarieties.\nSo since an \\'etale morphism is locally standard \\'etale (see e.g.~\\cite[Proposition I.3.19]{Mil80}),\nwe can assume that $U=\\{g(T_1,\\ldots,T_m)\\neq 0\\}$ with $0\\neq g\\in\\mathbb{F}_q[T_1,\\ldots,T_m]$ and\n$$\n Y'=\\{f(T_1,\\ldots,T_m,X)=0,g(T_1,\\ldots,T_m)\\neq 0\\}\\subseteq\\mathbb{A}^{m+1}\n$$ \nwith $f\\in\\mathbb{F}_q[T_1,\\ldots,T_m,X]$,\nand $Y'\\to U$ is the projection to the first $m$ coordinates.\nBy \\cite[Theorem 13.3.5 and Proposition 16.1.5]{FJ} there exist\n$h_1,\\ldots,h_m\\in\\mathbb{F}_q[T]$ with\n$g(h_1,\\ldots,h_m)\\neq 0$, $f(h_1,\\dots,h_m,X)\\in\\mathbb{F}_q(T)[X]$ is irreducible and \n$$\\label{eq:galproduct}\n {\\rm Gal}(f(h_1,\\ldots,h_m,X)\/\\mathbb{F}_q(T))={\\rm Gal}(f(T_1,\\ldots,T_m,X)\/\\mathbb{F}_q(T_1,\\ldots,T_m))={\\rm Gal}(Y'\/U).\n$$ \nThe corresponding $\\psi$ satisfies\n$\\psi^{-1}(U)\\neq\\emptyset$, $Y'\\times_U\\psi^{-1}(U)$ is irreducible, and\n$$\n {\\rm Gal}(Y'\\times_U\\psi^{-1}(U)\/\\psi^{-1}(U))={\\rm Gal}(f(h_1,\\ldots,h_m,X)\/\\mathbb{F}_q(T))={\\rm Gal}(Y'\/U),\n$$ \nconcluding the proof.\n\\end{proof}\n\n\nThe next proposition combined with Lemma $\\ref{lemquasip}$ implies Proposition~\\ref{thmprod} in the special case when all $G_i$ are quasi-$p$.\n\n\\begin{proposition}\\label{propprod} \nLet $G_1,G_2$ be $L_1^{\\mathrm{gt}}(q)$-realizable groups with $G_2$ quasi-$p$. Then $G_1\\times G_2$ is $L_1^{\\mathrm{gt}}(q)$-realizable.\n\\end{proposition}\n\n\\begin{proof} \nFirst assume that both $G_1,G_2$ are quasi-$p$. \nLet $T_1,T_2$ be independent variables and let $L_i\/\\mathbb{F}_q(T_i)$ be an $L_1^{\\mathrm{gt}}(q)$-realization of $G_i$ ramified only over $\\infty$ \n(we may assume this by Lemma~\\ref{lemreal}). \nIt corresponds to a geometric Galois cover $Y_i\\to\\mathbb{A}^1=\\mathrm{Spec}(\\mathbb{F}_q[T_i])$ with ${\\rm Gal}(Y_i\/\\mathbb{A}^1)=G_i$. \nTaking the product of these covers we obtain a geometric Galois cover $Y=Y_1\\times Y_2\\to \\mathbb{A}^2$ with ${\\rm Gal}(Y\/\\mathbb{A}^2)=G_1\\times G_2$.\nBy Proposition~\\ref{hit} there exists a morphism $h\\colon\\mathbb{A}^1\\to \\mathbb{A}^2$ such that $Z=Y\\times_{\\mathbb{A}^2}\\mathbb{A}^1\\to\\mathbb{A}^1$ \nis a geometric Galois cover with ${\\rm Gal}(Z\/\\mathbb{A}^1)={\\rm Gal}(Y\/\\mathbb{A}^2)=G$. \nThe extension $L=\\mathbb{F}_q(Z)$ of $\\mathbb{F}_q(\\mathbb{A}^1)$ now gives a geometric realization of $G$ ramified only over $\\infty$.\n\nNow let $G_1,G_2$ be $L_1^{\\mathrm{gt}}(q)$-realizable with only $G_2$ assumed quasi-$p$. Iterating the above claim we see that an arbitrary product of $L_1^{\\mathrm{gt}}(q)$-realizable quasi-$p$ groups is again $L_1^{\\mathrm{gt}}(q)$-realizable (and of course quasi-$p$ as well). \nIn particular since $G_2$ is quasi-$p$ and $L_1^{\\mathrm{gt}}(q)$-realizable, for every $m$ we can find a geometric Galois extension $L\/\\mathbb{F}_q(T)$ with Galois group $(G_2)^m$ ramified only over $\\infty$, and it contains linearly disjoint subextensions $L_1,\\ldots,L_m$ such that ${\\rm Gal}(L_i\/\\mathbb{F}_q(T))=G_2$. \nLet $L\/\\mathbb{F}_q(T)$ be a geometric realization of $G_1$ ramified over $\\infty$, tamely ramified over 0 and unramified elsewhere. \nFor $m$ sufficiently large, one of the $L_i$ is linearly disjoint from $L$ over $\\mathbb{F}_q(T)$, and then \n$K=LL_i$ is a geometric Galois extension of $\\mathbb{F}_q(T)$ with ${\\rm Gal}(K\/\\mathbb{F}_q(T))=G_1\\times G_2$, ramified only over $0,\\infty$, tamely over $0$.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Proposition~\\ref{thmprod}] \nLet $G=G_1\\times\\dots\\times G_m$ be a product of $L_1^{\\mathrm{gt}}(q)$-realizable groups.\nIf $G$ is quasi-$p$, then by Proposition~\\ref{propprod} the group $G$ is $L_1^{\\mathrm{gt}}(q)$-realizable and by Lemma~\\ref{lemquasip} it satisfies Conjecture \\ref{bm}.\n\nOtherwise we can use Proposition~\\ref{propprod} to absorb all the quasi-$p$ factors into one of the non-quasi-$p$ factors and reduce to the case when none of the $G_i$ are quasi-$p$. Assuming none of the $G_i$ are quasi-$p$ we have $d\\left((G\/p(G))^\\mathrm{ab}\\right)=m$,\nsince by Lemma~\\ref{lemreal} the $G_i$ are cyclic-by-quasi-$p$ and the assumptions of the proposition imply that each $G_i$ has a quotient isomorphic to $\\mathbb{Z}\/\\ell\\mathbb{Z}$. \nBy Lemma~\\ref{removeinf}, for each $G_i$ there exists a geometric Galois extension $L_i\/\\mathbb{F}_q(T)$ with Galois group $G_i$ branched only over a finite prime $h_i\\in\\mathbb{F}_q[T]$ and we may take the $h_i,i=1,\\ldots,m$ to be pairwise distinct. \nThen the extensions $L_i\/\\mathbb{F}_q(T)$ are pairwise linearly disjoint, since their branch loci are disjoint and $\\mathbb{F}_q(T)$ has no unramified geometric extensions. \nTaking $L=L_1\\cdots L_m$ we see that ${\\rm Gal}(L\/\\mathbb{F}_q(T))=G$ and $L\/\\mathbb{F}_q(T)$ is ramified exactly over $h_1,\\ldots,h_m$, so $r_{\\mathbb{F}_q(T)}(G)\\le m$.\nSince by \nCorollary \\ref{cor:lowerbound}\nwe also have $r_{\\mathbb{F}_q(T)}(G)\\geq m$,\nwe conclude that $r_{\\mathbb{F}_q(T)}(G)=m$,\nwhich is what Conjecture \\ref{conj} predicts.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:main1}] \nLet $G=G_1\\times\\ldots\\times G_m$ with each $G_i$ a symmetric or alternating group satisfying the assumptions of Theorem~\\ref{thm:main1}. \nSince Theorem~\\ref{thmlist} has the same assumptions, each $G_i$ is $L_1^{\\mathrm{gt}}(q)$-realizable. \nIf $p>2$ we may take $\\ell=2$ and then each $G_i$ is either an alternating group and thus quasi-$p$ or is a symmetric group and then $\\ell\\mid 2=[G_i:p(G_i)]$. \nThus the conditions of Proposition~\\ref{thmprod} are satisfied and the conclusion follows. \nIf $p=2$ then we take $\\ell=3$ and $G_i$ is quasi-2 unless $G_i=A_3,A_4$ in which case $\\ell\\mid 3=[G_i:p(G_i)]$ and again Proposition~\\ref{thmprod} applies and the conclusion follows.\n\\end{proof}\n\n\n\n\\subsection{Proof of Theorem~\\ref{thmlist}}\\label{secthmlist}\n\nLet $n\\ge p$.\nIn the present subsection we prove for $G\\in\\{S_n,A_n\\}$\nand each of the pairs $(n,q)$ from Theorem~\\ref{thmlist}\nthat the cyclic-by-quasi-$p$ group $G$ is $L_1^{\\mathrm{gt}}(q)$-realizable. \nWe construct realizations for these groups by writing down explicit equations. \nOur constructions are inspired by the work of Abhyankar \\cite{Abh92},\n and in the case $p=2$ we directly use the results of Abhyankar, Ou, Sathaye and Yie \\cite{AOS94,AbYi94}. Some of the Galois group calculations rely on the classification of multiply transitive groups, which in turn relies on the Classification of Finite Simple Groups (CFSG). We will indicate when CFSG and its applications are used. \nIn all cases we will write an explicit polynomial $f\\in\\mathbb{F}_q(T)[X]$ of degree $n$ and show that ${\\rm Gal}(f\/\\mathbb{F}_q(T))=G$ and that the splitting field of $f$ over $\\mathbb{F}_q(T)$ is geometric \nand is ramified over a single prime of degree one if $G$ is quasi-$p$ and over two primes of degree one with tame ramification over at least one of them if $G$ is cyclic-by-quasi-$p$ but not quasi-$p$. Note that we always have the inclusions \n${\\rm Gal}(f\/\\overline{\\F}_q(T))\\leqslant{\\rm Gal}(f\/\\mathbb{F}_q(T))\\leqslant S_n$,\nso for $G=S_n$, proving that ${\\rm Gal}(f\/\\overline{\\F}_q(T))=S_n$ shows both that ${\\rm Gal}(f\/\\mathbb{F}_q(T))=G$ and that the splitting field of $f$ is geometric,\nand for $G=A_n$, proving that ${\\rm Gal}(f\/\\overline{\\F}_q(T))\\geqslant A_n$ and ${\\rm Gal}(f\/\\mathbb{F}_q(T))\\leqslant A_n$ shows\nthat ${\\rm Gal}(f\/\\mathbb{F}_q(T))=G$ and that the splitting field of $f$ is geometric.\nRecall from Lemma~\\ref{lem:discgalois} that if $p>2$ then ${\\rm Gal}(f\/\\mathbb{F}_q(T))\\leqslant A_n$\nif and only if $\\Delta(f)$ is a square in $\\mathbb{F}_q(T)$\n(where we always take the discriminant with respect to $X$).\nWe distinguish four main cases according to $G=S_n$ or $A_n$ and $p=2$ or $\\neq 2$,\nand several subcases according to $n$ and $q$.\n\n\n\n\n\\maincase{$G=S_n,p>2$}\\label{sec:sn}\nIn this case $G$ is cyclic-by-quasi-$p$ but not quasi-$p$, so we are looking for a polynomial $f\\in\\mathbb{F}_q(T)[X],\\deg f=n$ with \n${\\rm Gal}(f\/\\overline{\\F}_q(T))=S_n$ with splitting field ramified exactly over $T=0,\\infty$, tamely over one of these points. \n\n\\case{$n$ odd, $n>p, p\\nmid n+1$} \nLet $g=X^{n+1}-(1+T)X^p+T\\in\\mathbb{F}_p[T,X]$ and\n\\begin{equation}\\label{eq:snodd} \n f=\\frac{g}{X-1}=X^p\\frac{X^{n+1-p}-1}{X-1}-(X-1)^{p-1}T\\in\\mathbb{F}_p[T,X].\n\\end{equation}\nNote that $f$ is a monic polynomial of degree $n$ in $X$. It is irreducible over $\\overline{\\F}_q(T)$ because it is linear in the variable $T$ with coprime coefficients $X^p\\frac{X^{n+1-p}-1}{X-1}$ and $-(X-1)^{p-1}$ (here we use the assumption $p\\nmid n+1$).\nSince $g'=(n+1)X^n$, using (\\ref{eq:discprod}),(\\ref{eq:disc2}) and (\\ref{eq:resdefinition}) we have \n$$\n \\disc(f)=\\disc(g)\/\\mathrm{Res}(X-1,f)^2=\\pm (n+1)^{n+1}T^n\/f(1)^2=\\pm (n+1)^{n-1}T^n.\n$$\nLet $\\alpha$ be a root of $f$ in an algebraic closure of $\\mathbb{F}_q(T)$. \nSince $\\disc(f)=\\pm (n+1)^{n-1}T^n$, the extension $\\overline{\\F}_q(T,\\alpha)\/\\overline{\\F}_q(T)$ (and therefore its normal closure) is ramified at most over $0,\\infty$. We now compute the ramification indices of this extension over 0. To this end observe that\nby (\\ref{eq:snodd}) we have \n\\begin{equation}\\label{eq:snoddt} \n T=\\frac{\\alpha^p(\\frac{\\alpha^{n+1-p}-1}{\\alpha-1})}{(\\alpha-1)^{p-1}}\n\\end{equation} \nand we see that the divisor of zeros of $T$ over $\\overline{\\F}_q(T,\\alpha)=\\overline{\\F}_q(\\alpha)$ is composed of one prime of multiplicity $p$ and other primes of multiplicity 1. Therefore the ramification indices over $T=0$ are $p,1,\\ldots,1$, so by Lemma~\\ref{lemcycle}(ii) $G={\\rm Gal}(f\/\\overline{\\F}_q(T))$ contains a cycle of length $p$. Note also that by (\\ref{eq:snoddt}), the ramification indices over $T=\\infty$ are $p-1,n+1-p$,\nin particular the ramification over $T=\\infty$ is tame.\nTherefore the action of ${\\rm Gal}(f\/\\overline{\\F}_q(T))$ on the roots of $f$ is primitive by Lemma~\\ref{lem:primitive},\nso since it\ncontains a cycle of length $p$ we can apply \nTheorem~\\ref{thm:jones} with $l=p$. \n\nIf $n>p+2$ then Jordan's Theorem (see Remark \n\\ref{remark:jordan}) implies that ${\\rm Gal}(f\/\\overline{\\F}_q(T))\\geqslant A_n$ and since $\\disc(f)$ is not a square \nwe have ${\\rm Gal}(f\/\\mathbb{F}_q(T))={\\rm Gal}(f\/\\overline{\\F}_q(T))=S_n$ and the polynomial $f\n$ satisfies all the required properties. \n\nIf $n=p+2$ then we use Theorem~\\ref{thm:jones}(ii) to \nconclude that either ${\\rm Gal}(f\/\\overline{\\F}_q(T))\\geqslant A_n$ (and then we argue as before) or $p=2^k-1$ is a \nMersenne prime and ${PGL}_2(2^k)\\leqslant {\\rm Gal}(f\/\\overline{\\F}_q(T))\\leqslant{P\\Gamma L}_2(2^k)$ with its action \non the roots being the standard action of such a group on $\\P^1(\\mathbb{F}_{2^k})$. \nHowever,\nthe ramification indices over $\\infty$ are $3,p-1$ and since $p>3$ (as $p\\nmid n+1=p+3$),\n by Lemma \\ref{lemcycle}(i), ${\\rm Gal}(f\/\\overline{\\F}_q(T))$ contains a permutation with cycle structure $3,p-1$. The group \n$P\\Gamma L_2(2^k)$ with its standard action on $\\P^1(\\mathbb{F}_{2^k})$ has no such element (see \n\\cite[Corollary 4.6]{GMPS16}), a contradiction.\n\n\\begin{remark}\nNote that in treating the case $n=p+2$ we made use of Theorem~\\ref{thm:jones}(ii), which makes use of the CFSG.\n\\end{remark}\n\n\\case{$n=p$} Let \n$$\n f=X^p+X^2-T\\in\\mathbb{F}_p[T,X].\n$$ \nWe have $f'=2X$ and therefore by (\\ref{eq:disc2}) we have $\\disc(f)=aT,a\\in\\mathbb{F}_p^\\times$, hence the splitting field of $f$ over $\\overline{\\F}_q(T)$ is ramified at most over 0 and $\\infty$. \nThe polynomial $f$ is irreducible over $\\overline{\\F}_q(T)$ since it is monic (up to sign) and linear in the variable $T$.\n\nLet $\\alpha$ be a root of $f$ in an algebraic closure of $\\mathbb{F}_q(T)$. Then $T=\\alpha^2(\\alpha^{p-2}+1)\n$ and we get (reasoning as in the previous case) that in the extension $\\overline{\\F}_q(\\alpha)=\\overline{\\F}_q(T,\\alpha)\/\n\\overline{\\F}_q(T)$ the prime $T=0$ splits into $p-2$ unramified primes and the prime $\\alpha=0$ with \nramification index 2. \nIn particular, the splitting field of $f$ is tamely ramified over $T=0$\nand ${\\rm Gal}(f\/\\overline{\\F}_q(T))\\leqslant S_n$ contains a transposition by Lemma~\\ref{lemcycle}. \nOn the other hand $T=\\infty$ is totally ramified with ramification index $p$. \nBy Lemma~\\ref{lem:primitive}, ${\\rm Gal}(f\/\\overline{\\F}_q(T))$ is primitive,\nso by Lemma~\\ref{lem:primtrans} we \nhave ${\\rm Gal}(f\/\\overline{\\F}_q(T))=S_n$ and $f$ is as desired.\n\n\\case{$n$ odd, $p|n+1$, $n\\neq 2p-1$}\nNote that these conditions imply $n\\ge 4p-1$. \nChoose $h\\in\\mathbb{F}_p[X],\\deg h=3$ monic irreducible.\nThere exists $u\\in\\mathbb{F}_p[X],\\deg u=3$ with $X^{n+3-p}\\equiv u^p\\pmod h$,\nsince $a\\mapsto a^p$ \nis an automorphism of $\\mathbb{F}_p[X]\/h$ and we can\nassume without loss of generality that $X\\nmid u$ by adding $\\pm h$ if necessary.\nLet $g=X^{n+3}-u(X)^pX^p-h(X)^pT$ and\n\\begin{equation}\\label{eq:snodd1} \n f=\\frac{g}{h}=\\frac{X^{n+3-p}-u(X)^p}{h(X)}X^p-h(X)^{p-1}T\\in\\mathbb{F}_p[T,X].\n\\end{equation}\nThe polynomial $f$ is monic in $X$ with $\\deg_X f=n$.\nLet $v=X^{n+3-p}-u(X)^p$ and note that $v'=(n+3)X^{n+2-p}\\neq 0$, \nin particular $v$ is separable (as $X\\nmid u$),\nwhich also implies that $h^2\\nmid v$ and so $f$ \nis linear in $T$ with coprime coefficients, hence irreducible.\nUsing (\\ref{eq:discprod}),(\\ref{eq:disc2}) and (\\ref{eq:resdefinition}) we compute\n$$\n \\disc(f)=\\disc(g)\/(\\mathrm{Res}(h,f)^2\\disc(h))=aT^{n+2}\n$$ for some $a\\in\\mathbb{F}_p^\\times$,\nas $\\mathrm{Res}(h,f)$ is independent of $T$ and non-zero since $h^2\\nmid v$.\nConsequently the splitting field of $f$\nis ramified only over $T=0,\\infty$. \nSince $n$ is odd, $\\disc(f)$ is not a square in $\\overline{\\F}_q(T)$ and so ${\\rm Gal}(f\/\\overline{\\F}_q(T))\\not\\leqslant A_n$. \n\nLet $\\alpha$ be a root of $f$ in an algebraic closure of $\\mathbb{F}_q(T)$. By (\\ref{eq:snodd1}) we have\n\\begin{equation}\\label{eq:snodd1t} \n T=\\frac{\\left(\\frac{\\alpha^{n+3-p}-u(\\alpha)^p}{h(\\alpha)}\\right)\\alpha^p}{h(\\alpha)^{p-1}}\n \\end{equation}\nand therefore $\\overline{\\F}_q(T,\\alpha)=\\overline{\\F}_q(\\alpha)$. Now since $n\\ge 4p-1$ we see from (\\ref{eq:snodd1t}) that the \nprimes of $\\overline{\\F}_q(\\alpha)$ over $T=0$ are $\\alpha=0$ with ramification index $p$ and the roots of $v\/h$ with ramification index 1,\nso by Lemma~\\ref{lemcycle} the group ${\\rm Gal}(f\/\\overline{\\F}_q(T))$ contains a cycle of length $p$.\nSimilarly, the primes of $\\overline{\\F}_q(\\alpha)$ over $T=\\infty$ are \nthe 3 roots of $h$ with multiplicity $p-1$ each, and $\\alpha=\\infty$ with multiplicity $n-3(p-1)\\equiv 2\\pmod p$,\nin particular the splitting field of $f$ is tamely ramified over $T=\\infty$.\nBy Lemma~\\ref{lem:primitive},\n ${\\rm Gal}(f\/\\overline{\\F}_q(T))$ acts primitively on the roots of $f$. \n By Jordan's Theorem (Theorem~\\ref{thm:jones}(i) in the elementary case $l=p\\le n-3$ prime) we have ${\\rm Gal}(f\/\\overline{\\F}_q(T))\\geqslant A_n$ and therefore ${\\rm Gal}(f\/\\overline{\\F}_q(T))=S_n$.\n\n\\case{\\bf $n=2p-1$} \nLet \n$$\n f=\\frac{X^{2p}-TX^p-X^2+T}{X-1}=\\frac{X^{2p-2}-1}{X-1}X^2-T(X-1)^{p-1}\\in\\mathbb{F}_p[T,X].\n$$ \nThe polynomial $f$ is monic in $X$ and irreducible over $\\overline{\\F}_q(T)$, since it is linear in $T$ with coprime coefficients. \nUsing (\\ref{eq:discprod}),(\\ref{eq:disc2}) and (\\ref{eq:resdefinition}) we compute\n$$\n \\disc(f)=\\disc(X^{2p}-TX^p-X^2+T)\/\\mathrm{Res}(X-1,f)^2=aT\n$$ \nfor some $a\\in\\mathbb{F}_p^\\times$. We see that the splitting field of $f$ over $\\overline{\\F}_q(T)$ is ramified only over $0,\\infty$. Let $\\alpha$ be a root of $f$ in an algebraic closure of $\\mathbb{F}_q(T)$. We have\n$$T=\\frac{\\left(\\frac{\\alpha^{2p-2}-1}{\\alpha-1}\\right)\\alpha^2}{(\\alpha-1)^{p-1}},$$\nfrom which we see that in the extension $\\overline{\\F}_q(\\alpha)=\\overline{\\F}_q(T,\\alpha)$ of $\\overline{\\F}_q(T)$ there are $2p-2$ primes lying over $T=0$ \nwith one of them having ramification index $2$ and the other unramified, \nwhile over $T=\\infty$ we have the primes $\\alpha=\\infty$ with ramification index $p$, and $\\alpha=1$ with ramification index $p-1$.\nTherefore, \nby Lemma~\\ref{lemcycle},\n${\\rm Gal}(f\/\\overline{\\F}_q(T))\\leqslant S_n$ \ncontains both a transposition and cycle of length $p$.\nThe latter implies that it is primitive,\nand so ${\\rm Gal}(f\/\\overline{\\F}_q(T))=S_n$ by Lemma~\\ref{lem:primtrans}. \n\n\\case{$n$ even, $p\\nmid n$, $n>p+1$} \nLet \n$$\n f=X^n+X^p-T\\in\\mathbb{F}_p[T,X].\n$$ \nSince $f$ is monic (up to sign) and linear in $T$, it is irreducible over $\\overline{\\F}_q(T)$. \nSince $f'=nX^{n-1}$, by (\\ref{eq:disc2}) we have $\\disc(f)=aT^{n-1},a\\in\\mathbb{F}_p^\\times$. \nTherefore the splitting field of $f$ is only ramified over $T=0$ and $T=\\infty$, and $\\disc(f)$ is not a square in $\\overline{\\F}_q(T)$ (since $n$ is even) and so ${\\rm Gal}(f\/\\overline{\\F}_q(T))\\not\\leqslant A_n$. \nDenoting by $\\alpha$ a root of $f$ in an algebraic closure of $\\mathbb{F}_q(T)$, we see that $T=\\alpha^p(\\alpha^{n-p}+1)$. \nThe polynomial $X^{n-p}+1$ is separable by the assumption $p\\nmid n$, and so the ramification indices of $\\overline{\\F}_q(T,\\alpha)=\\overline{\\F}(\\alpha)$ over $T=0$ are $p,1,\\ldots,1$, and $T=\\infty$ is totally ramified with ramification index $n$ (in particular tame). \nTherefore, ${\\rm Gal}(f\/\\overline{\\F}_q(T))\\leqslant S_n$ is primitive by Lemma~\\ref{lem:primitive} and contains a cycle of length $p$ by Lemma~\\ref{lemcycle},\nso by Jordan's Theorem (i.e. Theorem~\\ref{thm:jones}(i) in the case $l=p\\le n-3$ prime)\nwe get that ${\\rm Gal}(f\/\\overline{\\F}_q(T))=S_n$.\n\n\n\\case{$n$ even, $p|n$} \nBy our assumptions, $n\\ge 2p$.\nChoose $h(X)\\in\\mathbb{F}_p[X]$ monic irreducible with $\\deg h=2$ and $h'(0)\\neq 0$ (which always exists).\nSince $a\\mapsto a^p$ is an automorphism of $\\mathbb{F}_p[X]\/h$, there exists $u(X)\\in\\mathbb{F}_p[X]$ with $\\deg u2p$ we can add $\\pm h$ to $u$ if necessary, \nwhile if $n=2p$ and we take the unique $u$ with $\\deg u\\le 1$ such that $u^p\\equiv X^{p+2} \\pmod h$ then \nautomatically $u(0)\\neq 0$, since otherwise $u=cX,c\\in\\mathbb{F}_p^\\times$ and then $h=X^2-c$, contradicting our assumption $h'(0)\\neq 0$.\nLet $g=X^{n+2}-u(X)^pX^p-Th(X)^p\\in\\mathbb{F}_p[T,X]$ and\n\\begin{equation}\\label{eq:sneven}\nf=\\frac{g}{h}=\\frac{X^{n+2-p}-u(X)^p}{h(X)}X^p-Th(X)^{p-1}\\in\\mathbb{F}_p[T,X].\n\\end{equation}\nThe polynomial $f$ is monic in $X$ of degree $n$. \nLet $v=X^{n+2-p}-u(X)^p$. \nAs $v'=2X^{n+1-p}$ and $u(0)\\neq 0$, we have \nthat $v$ is separable, in particular $h^2\\nmid v$.\nSo $f$ is linear in $T$ with coprime coefficients, hence irreducible over $\\overline{\\F}_q(T)$. \nUsing (\\ref{eq:discprod}),(\\ref{eq:disc2}) and (\\ref{eq:resdefinition}), we compute\n$$\n \\disc(f)=\\disc(g)\/(\\mathrm{Res}(h(X),f)^2\\Delta(h))=aT^{n+1},a\\in\\mathbb{F}_p^\\times\n$$ \nand conclude that the splitting field of $f$ over $\\overline{\\F}_q(T)$ is ramified at most over $T=0,\\infty$ and that ${\\rm Gal}(f\/\\overline{\\F}_q(T))\\not\\leqslant A_n$ (since $n$ is even).\n\nDenoting by $\\alpha$ a root of $f$ in an algebraic closure of $\\mathbb{F}_q(T)$ we have (by (\\ref{eq:sneven}))\n$$\n T=\\frac{\\alpha^p\\left(\\frac{\\alpha^{n+2-p}-u(\\alpha)^p}{h(\\alpha)}\\right)}{h(\\alpha)^{p-1}}.\n$$\nThe polynomial $v$ is separable, hence the primes of $\\overline{\\F}_q(T,\\alpha)=\\overline{\\F}_q(\\alpha)$ over $T=0$ are $\\alpha=0$ with ramification index $p$ and $n-p$ primes with ramification index 1. \nThe primes over $T=\\infty$ are the two roots of $h$ with ramification index $p-1$ and $\\alpha=\\infty$ with ramification index $n-2(p-1)\\equiv 2\\pmod p$,\nin particular $T=\\infty$ is tamely ramified.\nTherefore again, ${\\rm Gal}(f\/\\overline{\\F}_q(T))\\leqslant S_n$ is primitive by Lemma~\\ref{lem:primitive} and contains a cycle of length $p$ by Lemma~\\ref{lemcycle},\nso by Jordan's Theorem (i.e. Theorem~\\ref{thm:jones}(i) in the case $l=p\\le n-p\\leq n-3$ prime)\nwe get that ${\\rm Gal}(f\/\\overline{\\F}_q(T))=S_n$.\n\n\\maincase{$G=S_n,p=2$}\nIn this case $G$ is quasi-$p$. \n\n\\case{$n$ odd} \nWe have $n\\ge 3$. \nLet\n$$\n f(X)=X^n+TX^{n-2}+1\\in\\mathbb{F}_2[T,X].\n$$ \nBy \\cite[\\S 11.I.5]{Abh92} (with $t=n-2$, $a=1$, $s=1$), \nwe have ${\\rm Gal}(f\/\\overline{\\F}_2(T))=S_n$ and the splitting field of $f$ is ramified only over $T=\\infty$. \n\n\\case{$n$ even} \nLet\n$$\n f(X)=\\left((X+1)^{n-1}+X^{n-1}\\right)(X+1)^2+T^{n-1}X^{n-1}\\in\\mathbb{F}_2[T,X].\n$$ \nBy \\cite[\\S 12.IV.4]{Abh92} (with $t=n-1$, $s=n-1$, $a=1$, $b=1$) \nwe have ${\\rm Gal}(f\/\\overline{\\F}_2(T))=S_n$ and the splitting field of $f$ is ramified only over $T=0$. \n\n\\maincase{$G=A_n, p>2$}~\\\\\n\n\n\\case{$n\\neq p+1$}\nIn this case we have shown that $S_n$ is $L_1^{\\mathrm{gt}}(q)$-realizable, and since $p>2$ we have $p(S_n)=A_n$. Therefore by Lemma~\\ref{lemreal}, the group $A_n=p(S_n)$ is also $L_1^{\\mathrm{gt}}(q)$-realizable.\n\n\\case{$p=3,n=4$} \nLet\n$$\n f=X^4-TX^3+1\\in\\mathbb{F}_3[T,X].\n$$ \nAs $f'=X^3$ and $f(0)=1$ we compute that $\\Delta(f)=1$,\nand thus the splitting field of $f$ is ramified only over $T=\\infty$\nand ${\\rm Gal}(f\/\\mathbb{F}_q(T))\\leqslant A_4$.\nThe group ${\\rm Gal}(f\/\\overline{\\F}_q(T))\\leqslant A_4$ is transitive on the roots (since $f$ is irreducible, being linear in the variable $T$) and has order divisible by $3$ (otherwise the splitting field of $f$ over $\\overline{\\F}_q(T)$ would be tamely ramified and ramified only over infinity, which implies it is trivial). \nThus ${\\rm Gal}(f\/\\mathbb{F}_q(T))={\\rm Gal}(f\/\\overline{\\F}_q(T))=A_4$.\n\n\\case{$n=p+1,p>3, \\mathbb{F}_q\\supseteq\\mathbb{F}_{p^2}$}\\label{sec34} \nLet $s\\ge 1$ and $2\\le a\\le p-1$ be integers and let\n$$\n f=(X+1)\\left(X+\\frac{a}{a-1}\\right)^p-T^sX^a\\in\\mathbb{F}_p[T,X].\n$$ \nBy \\cite[\\S 22]{Abh92} (with $\\tau=a$, $Y=T^s$, $b=\\frac{a}{a-1}$, but note the nonstandard sign convention in the definition of the discriminant),\n\\begin{equation}\\label{eq:disc_mrt}\n \\disc(f)=(-1)^{(p+1)\/2}\\frac{a^{2a-1}}{(a-1)^{2a-3}}T^{s(p+1)}\\in\\mathbb{F}_p[T].\n\\end{equation}\nIt is shown in \\cite[\\S 12.IV.3]{Abh92} (with $t=a$, $b=\\frac{a}{a-1}$) that if $p>5$, $2\\le a\\le \\frac{p-1}2,(a,p+1)=1$ and $a(p+1-a)|s$, then ${\\rm Gal}(f\/\\overline{\\F}_q(T))=A_n$ and the splitting field of $f$ is ramified only over $T=0$. \nIf additionally $\\mathbb{F}_q\\supseteq\\mathbb{F}_{p^2}$, then $\\disc(f)$ is a square in $\\mathbb{F}_q[T]$ and we have ${\\rm Gal}(f\/\\mathbb{F}_q(T))=A_{n}$.\nNote that for all $p>5$, an $a$ with $(a,p+1)=1$ and $a\\not\\equiv\\pm 1\\pmod{p+1}$ can be found \n(as $\\phi(x)>2$ for $x>6$),\nand after replacing $a$ with $p+1-a$ if necessary to assume that $2\\leq a\\leq\\frac{p-1}{2}$,\nwe can set $s=a(p+1-a)$.\n\nIf $p=5,n=6,\\mathbb{F}_q\\supseteq\\mathbb{F}_{25}$ we take the polynomial\n$$\n f=(X+1)(X+2)^5-T^4X^2\\in\\mathbb{F}_p[T,X].\n$$\nThe discriminant of $f$ is computed by (\\ref{eq:disc_mrt}) with $a=2,s=4$ and equals $\\Delta(f)=2T^{24}$, which is a square in $\\mathbb{F}_{25}(T)$ and hence ${\\rm Gal}(f\/\\mathbb{F}_q(T))\\leqslant A_5$. \nOn the other hand, by \\cite[\\S 12.IV.2]{Abh92} \n(with $t=2,b=2$), the splitting field of $f$ over $\\overline{\\F}_5(T)$ is ramified only over $T=0$, and its Galois group is $A_5$, \nhence ${\\rm Gal}(f\/\\mathbb{F}_q(T))={\\rm Gal}(f\/\\overline{\\F}_q(T))=A_5$.\n\n\\begin{remark} If we want to drop the assumption $\\mathbb{F}_q\\supseteq\\mathbb{F}_{p^2}$ in the case $p>5$ we would need to find an $a$ with $2\\le a\\le \\frac{p-1}2,(a,p+1)=1$ such that $(-1)^{(p+1)\/2}a(a-1)$ is a square modulo $p$. It is not hard to show (using a P\\'olya-Vinogradov-type inequality and an elementary sieve argument) that this is possible for all $p$ sufficiently large, and we conjecture that such an $a$ exists for all $p>13$. While proving this should be doable by means of a careful analysis and sufficiently large computer search, we did not pursue this.\\end{remark}\n\n\\maincase{$G=A_n,p=2$}\nThe group $A_n$ is always cyclic-by-quasi-2 and it is quasi-2 iff $n\\neq 3,4$. Most of the required realizations were constructed by Abhyankar, Ou, Sathaye and Yie \\cite{Abh92, Abh93, AOS94,AbYi94}.\n\n\\case{$n=3,4$, $\\mathbb{F}_q\\supseteq\\mathbb{F}_4$} \nFor $A_3\\cong\\mathbb{Z}\/3\\mathbb{Z}$ we can take the extension $K=\\mathbb{F}_q(s)\/F_q(T)$ with $s^3=T$ which is Galois with group $A_3$ if $\\mathbb{F}_q\\supseteq\\mathbb{F}_4$\nand ramified only over $T=0,\\infty$. \nDenote by $\\zeta$ any element of $\\mathbb{F}_4\\setminus\\mathbb{F}_2$, and consider the splitting field $L$ of the polynomial \n$$\n f=(X^2+X+s)(X^2+X+\\zeta s)(X^2+X+\\zeta^2 s)\\in\\mathbb{F}_2(s)[X]\n$$\nover $K$. \nThe extension $L\/\\mathbb{F}_q(T)$ is Galois (since $s,\\zeta s,\\zeta^2 s$ are a Galois orbit over $\\mathbb{F}_q(T)$) and by Artin-Schreier theory ${\\rm Gal}(L\/K)=\\mathbb{Z}\/2\\mathbb{Z}\\times\\mathbb{Z}\/2\\mathbb{Z}$ with ${\\rm Gal}(L\/\\mathbb{F}_q(T))$ acting nontrivially on the order 2 subgroups of ${\\rm Gal}(L\/K)$ by conjugation, in particular ${\\rm Gal}(L\/\\mathbb{F}_q(T))$ is non-abelian.\nThe only non-abelian extension of $\\mathbb{Z}\/3\\mathbb{Z}$ by $\\mathbb{Z}\/2\\mathbb{Z}\\times\\mathbb{Z}\/2\\mathbb{Z}$ is $A_4$. \nFinally observe that $L\/\\mathbb{F}_q(T)$ is ramified only over $0,\\infty$ and is therefore an $L_1^{\\mathrm{gt}}(q)$-realization of $A_4$.\n\n\\begin{remark}\nNote that by Lemma~\\ref{lemreal}, both for $n=3$ and for $n=4$ the condition $\\mathbb{F}_q\\supseteq\\mathbb{F}_4$ is necessary for $A_n$ to be $L_1^{\\mathrm{gt}}(q)$-realizable since $A_3,A_4$ have cyclic quotients of order 3. \n\\end{remark}\n\n\\case{$n=5$, $\\mathbb{F}_q\\supseteq\\mathbb{F}_4$} \nLet\n$$\n f=X^5+TX+1\\in\\mathbb{F}_2[T,X].\n$$ \nIt follows from \\cite[\\S 11.III.1]{Abh92} (with $q=4$, $t=s=1$, $a=-1$, in the notation used there) that ${\\rm Gal}(f\/\\overline{\\F}_2(T))\\cong PSL_2(4)\\cong A_5$ and the splitting field of $f$ is ramified only over $T=\\infty$. \nBy \\cite[2.23]{AOS94} (with $K=\\mathbb{F}_q(T)$, $d=4$, $e=1$, $\\bar{b}_d=T$, $\\bar{b}_n=1$),\n$\\mathbb{F}_q\\supseteq\\mathbb{F}_4$ implies that ${\\rm Gal}(f\/\\mathbb{F}_q(T))\\leqslant A_5$.\nThus ${\\rm Gal}(f\/\\mathbb{F}_q(T))={\\rm Gal}(f\/\\overline{\\F}_q(T))=A_5$.\n\n\n\\case{$n=6,7$, $\\mathbb{F}_q\\supseteq\\mathbb{F}_4$} \nConsider the polynomials\n\\begin{eqnarray*}\nf_6&=&X^6+T^{27}X^5+T^{54}X^4+(T^{18}+T^{36})X^3+T^{108}X^2+(T^{90}+T^{135})X+T^{162},\\\\\nf_7&=&X^7+TX^4+X^2+1.\n\\end{eqnarray*}\n\nThe polynomials $f_6,f_7$ were found by Abhyankar and Yie \\cite[Theorems 2.10 and 2.11]{AbYi94}, who showed that the splitting fields of $f_n,n=6,7$ are ramified only over $\\infty$ and,\nunder the assumption $\\mathbb{F}_q\\supseteq\\mathbb{F}_4$,\nthat ${\\rm Gal}(f_n\/\\mathbb{F}_q(T))={\\rm Gal}(f_n\/\\overline{\\F}_q(T))=A_n$. \n\n\n\n\\case{$n\\ge 9$ odd, $n\\equiv 1,7\\pmod 8$ or $\\mathbb{F}_q\\supseteq\\mathbb{F}_4$} \nLet\n$$\n f=X^n+TX^{n-4}+1\\in\\mathbb{F}_2[T,X].\n$$ \nBy \\cite[Theorem 2]{Abh93} (with $t=n-4$, $q=4$ in the notation of the cited paper), \nthe splitting field of $f$ is ramified only over $\\infty$ and we have ${\\rm Gal}(f\/\\mathbb{F}_q(T))\\geqslant{\\rm Gal}(f\/\\overline{\\F}_q(T))\\geqslant A_n$. \nConversely, by \\cite[(2.27)]{AOS94} (with $t=n-4$, $b_{n-t}^*=T$, $b_n^*=1$) we have ${\\rm Gal}(f\/\\mathbb{F}_q(T))\\leqslant A_n$ if either $\\mathbb{F}_q\\supseteq\\mathbb{F}_4$ or $n\\equiv 1,7\\pmod 8$.\nIn these cases we conclude that ${\\rm Gal}(f\/\\mathbb{F}_q(T))={\\rm Gal}(f\/\\overline{\\F}_2(T))=A_n$.\n\n\\case{$n\\ge 8$ even, $10\\neq n\\equiv 0,2,6\\pmod 8$ or $\\mathbb{F}_q\\supseteq\\mathbb{F}_4$} \nLet $1\\le t\\le n$ with $(t,n)=1$, and\n$$\n f=X^n+X^t+T^t\\in\\mathbb{F}_2[T,X].\n$$ \nBy \\cite[\\S 11.II.5]{Abh92} (with $s=t$, $a=1$), if $2\\le t\\le n-4$ then ${\\rm Gal}(f\/\\mathbb{F}_q(T))\\geqslant{\\rm Gal}(f\/\\overline{\\F}_2(T))\\geqslant A_n$ and the splitting field of $f$ is ramified only over $\\infty$. \nConversely, by \\cite[Theorem 2.27]{AOS94} (with $b_{n-t}^*=1$, $b_n^*=T^t$) we have ${\\rm Gal}(f\/\\mathbb{F}_q(T))\\leqslant A_n$ if either $\\mathbb{F}_q\\supseteq\\mathbb{F}_4$ or $n\\equiv 0\\pmod 8$ or $n\\equiv 2,6\\pmod 8,2t\\equiv n\\pmod 8$. \nIn each of these cases we can choose a suitable $t$:\nFor\n$n\\geq 8$\nthere exists $2\\leq t\\leq n-4$ with $(t,n)=1$\nsince $\\phi(n)>2$,\nand for $n\\equiv 2,6\\pmod 8,n>10$,\nthe choice $t=\\frac n2-4$ satisfies $2\\le t\\le n-4$, $(t,n)=1$ and $2t\\equiv n\\pmod 8$.\n\n\n\n\\section{Summary and application to the minimal ramification problem over $\\mathbb{Q}$}\n\\label{sec:Q}\n\\label{sec:summary}\n\n\\subsection{Summary}\n\\label{subsec:summary}\nWe first summarize some of the results for the groups $S_n$ and $A_n$:\n\n\\begin{theorem}[Main results for $S_n$]\nLet $n\\geq 2$ and $q=p^\\nu$ a prime power.\nThen $r_{\\mathbb{F}_q(T)}(S_n)\\leq 2$,\nand $r_{\\mathbb{F}_q(T)}(S_n)=1$ in each of the following cases:\n\\begin{enumerate}\n\\item $p(2n-3)^2$ \n\\item The function field analogue of Schinzel's hypothesis H (Conjecture \\ref{conj:SchinzelFF}) holds for $\\mathbb{F}_q(T)$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nThe case $n=2$ is very easy and follows for example from Theorem~\\ref{thm:abelian}, so assume from now on that $n\\geq 3$.\nIf $pn$.\n\nFor the rest of this proof, we call an $S_n$-extension of $\\mathbb{F}_q(U)$\na $(q,n,m)$-realization\nif it is the splitting field of $f-Uc$ with $f,c\\in\\mathbb{F}_q[X]$, ${\\rm deg}(f)=n$, ${\\rm deg}(c)=m(2n-7)^{2\/3}$ for all $n\\geq4$).\nFor $n=3,4,6$, Proposition~\\ref{prop:Morse2} gives a $(q,n,0)$-realization.\n\nWe now prove $r_{\\mathbb{F}_q(T)}(S_n)=1$ in cases (2)-(4),\nfor which it suffices by Lemmas~\\ref{lem:disjoint} and \\ref{lem:eliminateinfty} to exhibit a $(q,n,m)$-realization such that $(p,n-m)=1$ and\n$H(q,n+m-1,n-m)$ holds:\n\nIf (4) holds, then $H(q,d,e)$ always holds by Proposition~\\ref{lem:Hqdm}(1),\nso we can take the realizations from above.\n\nIf (3) holds\nand $n\\neq 4$,\nTheorem~\\ref{thm:tame2} with $m=n-2$ gives a $(q,n,n-2)$-realization (as $q>(2n-3)^2=(2m+1)^2$),\nand $H(q,2n-3,2)$ always holds by Proposition~\\ref{lem:Hqdm}(3) (as $q$ is odd).\nIf (3) holds and $n=4$, \nwe take the $(q,4,0)$-realization from above,\nand Proposition~\\ref{lem:Hqdm}(5) gives $H(q,3,4)$ for every $q\\geq (2^1\\cdot 3-1)^2=25=(2n-3)^2$. \n\nIf (2) holds,\nthen $H(q,d,4)$ holds for every odd $d$ by\nProposition~\\ref{lem:Hqdm}(3);\nif $n=5$ or $n\\geq 7$,\nwe obtained a $(q,n,n-4)$-realization above,\nand $H(q,2n-5,4)$ holds;\nif $n=4$ we take the $(q,4,0)$-realization from above,\nand $H(q,3,4)$ holds;\nif $n=3$ or $n=6$ we are in case (3) as soon as $q> 9$ resp.~$q>81$; \nfor $n=6$ and $q\\equiv 1\\mbox{ mod }3$,\n$H(q,5,6)$ holds by Proposition~\\ref{lem:Hqdm}(3)\nso we can take the $(q,6,0)$-realization from above.\nIn each of the remaining cases, the following table provides \n$f,c\\in\\mathbb{F}_p[X]$ with $\\deg f=n$, $\\deg c=m{$}l<{$}}\n\\begin{center}\n\\begin{tabular}{|ll|l|l|l|}\n\\hline\n $n$ & $p$ & $f$ & $c$ & $h$ \\\\\n\\hline\n$3$ &$5$ & $X^3 + 1$ &$X+2$ &$T^2$ \\\\\n$6 $&$5 $&$X^6 + 1 $&$X^2+X $&$T^4+1$ \\\\\n$6 $&$17 $& $X^6 + X^2 + X$ & $1 $ & $T^6+T+2$ \\\\\n$6 $&$29 $&$X^6 + X^2 + X $&$1 $&$T^6+T+6$ \\\\\n$6 $&$41 $&$X^6+X $&$1 $&$T^6+T+1$ \\\\\n$6 $&$53 $&$X^6+X^2+13X $&$1 $&$T^6+T$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{proof}\n\n\\begin{remark}\nNote that instead of $q\\equiv 1\\mbox{ mod }4$ we could also treat other arithmetic progressions:\nFor any prime number $\\ell$, if $q\\equiv 1\\mbox{ mod }2\\ell$, $n\\geq 2\\ell+3$, \nand $2n\\not\\equiv 1\\mbox{ mod }\\ell$,\nlet $m=n-2\\ell$. \nThen, in the language of the previous proof,\nTheorem~\\ref{thm:tame2}\ngives a $(q,n,m)$-realization,\nand $H(q,n+m-1,n-m)=H(q,2n-2\\ell-1,2\\ell)$ holds by Proposition~\\ref{lem:Hqdm}(3).\n\\end{remark}\n\n\\begin{theorem}[Main results for $A_n$]\nLet $n\\geq 3$ and $q=p^\\nu$ a prime power.\nIf $p>2$ \nor \n$\\mathbb{F}_q\\supseteq\\mathbb{F}_{4}$, then $r_{\\mathbb{F}_q(T)}(A_n)\\leq 2$,\nand $r_{\\mathbb{F}_q(T)}(A_n)=1$ in each of the following cases:\n\\begin{enumerate}\n\\item $23$.\n\nIf $p\\le n,p\\neq n-1$ or $p=n-1,\\mathbb{F}_q\\supseteq\\mathbb{F}_{p^2}$, all assertions follow from Theorem~\\ref{thm:main1}. \n\nIf $p=n-1$ and $n\\ge 14$ we may use Theorem~\\ref{thm:tame2} with $m=3$ to obtain $r_{\\mathbb{F}_q(T)}(A_n)\\le 2$ in this case. \nAssuming Conjecture $\\ref{conj:SchinzelFF}$ allows us to apply Theorem~\\ref{thm:tame} combined with Proposition~\\ref{lem:Hqdm}(1) (once again with $m=3$) to conclude $r_{\\mathbb{F}_q(T)}(A_n)=1$ in this case.\n\nIf $p=n-1$ and $n\\in\\{6,8,12\\}$,\nthe following table provides irreducible $f\\in\\mathbb{F}_p[T,X]$ \nwith $\\deg f=n$ and $\\disc_X(f)$ a square in $\\mathbb{F}_p(T)$ with only one prime divisor $T$.\nComputer verification shows that ${\\rm Gal}(f\/\\mathbb{F}_p(T))$\ncontains a $3$-cycle and an $(n-1)$-cycle, hence by Theorem~\\ref{thm:jones} contains $A_n$.\nThus the splitting field of $f$ over $\\mathbb{F}_q(T)$ is ramified at most over $T$ and the infinite prime,\nand ${\\rm Gal}(f\/\\mathbb{F}_q(T))=A_n$ as $A_n$ is simple.\n\\begin{center}\n\\begin{tabular}{|ll|l|l|}\n\\hline\n$n$ &$p$ &$f$ & $\\disc_X(f)$ \\\\\n\\hline\n$6$ &$5$ & $X^6+X^5T - 2X^3T^3 + XT + T^2$ & $4T^{18}$ \\\\\n$8$ & $7$ & $X^8 + 3X^2 + XT - 2$ & $4T^2$ \\\\\n$12$ & $11$ & $X^{12} + 5XT^3 - 5X^2 - 2$ & $4T^6$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\nNow assume that $p>n$.\nIf $n\\ge 13$ or $n=11$ we apply Theorem~\\ref{thm:tame2} \nwith $m=2$ or $m=3$ chosen so that $m\\not\\equiv n\\pmod 2$\nand thus obtain $r_{\\mathbb{F}_q(T)}(A_n)\\le 2$ in this case. \nIf $n=10$ we apply Theorem~\\ref{thm:tame2} with $m=1$ (note that $\\left(\\frac{q}{n-m}\\right)=\\left(\\frac{q}{3}\\right)^2=1$).\nAssuming Conjecture $\\ref{conj:SchinzelFF}$ allows us to apply Theorem~\\ref{thm:tame} combined with Proposition~\\ref{lem:Hqdm}(1) (with $m$ as above) to conclude $r_{\\mathbb{F}_q(T)}(A_n)=1$ in these cases.\nIn all other cases, Theorem~\\ref{thm:main2} gives that \n$r_{\\mathbb{F}_q(T)}(A_n)\\le 2$.\n\\end{proof}\n\n\n\\subsection{Application to the minimal ramification problem over $\\mathbb{Q}$}\nFinally, we apply our results for $S_n$ over $\\mathbb{F}_q(T)$\nto give a conditional proof of Conjecture~\\ref{conj:BM} for $S_n$ over $\\mathbb{Q}$:\n\n\\begin{lemma}\\label{lem:lift_real}\nLet $n\\in\\mathbb{N}$, $h_0\\in\\mathbb{Z}$, and $S$ a finite set of primes numbers,\nand for each $p\\in S$ let $f_p\\in\\mathbb{Z}[X]$ be monic of degree $n$.\nThere exist $f\\in\\mathbb{Z}[X]$ monic of degree $n$ and $c\\in\\mathbb{Z}$ such that $f\\equiv f_p\\pmod p$ and $c\\equiv 1\\pmod p$\nfor every $p\\in S$,\nand $f-c^nh_0$ has $n$ roots in $\\mathbb{R}$.\n\\end{lemma}\n\n\\begin{proof}\nBy the Chinese Remainder Theorem there exists a monic $f_0\\in\\mathbb{Z}[X]$\nwith $f_0\\equiv f_p\\pmod p$ for every finite $p\\in S$.\nLet $\\pi=\\prod_{p\\in S}p$,\nwrite $f_0=\\sum_{i=0}^na_iX^i$ and \nlet $f_\\infty\\in\\mathbb{Z}[X]$ be any monic polynomial of degree $n$\nwith $n$ roots in $\\mathbb{R}$.\nSince fractions of the form $\\frac{x\\pi}{y\\pi+1}$ with $x,y\\in\\mathbb{Z}$ are dense in $\\mathbb{R}$, \nwe can choose\n$x_i,y_i\\in\\mathbb{Z}$ such that\n$\\tilde{f}:=X^n+\\sum_{i=0}^{n-1}(a_i+\\frac{x_i\\pi}{y_i\\pi+1})X^i$\nis arbitrarily close to $f_\\infty+h_0$, \nin particular so close that also $\\tilde{f}-h_0$ has $n$ roots in $\\mathbb{R}$.\nThen with $c=\\prod_{i=0}^{n-1}(y_i\\pi+1)$,\nthe polynomial\n$f(X):=c^n\\tilde{f}(c^{-1}X)\\in\\mathbb{Z}[X]$ satisfies the claim.\n\\end{proof}\n\n\n\\begin{theorem}\\label{thm:S_n_over_Q}\nSchinzel's hypothesis H (Conjecture \\ref{conj:Schinzel}) implies that\n$r_\\mathbb{Q}(S_n)=1$ for every $n\\geq2$, i.e.~Conjecture \\ref{conj:BM} holds for symmetric groups.\n\\end{theorem}\n\n\\begin{proof}\nIn case $n=2$, for example $L=\\mathbb{Q}(\\sqrt{2})$ does the job,\nso assume that $n\\geq 3$.\nLet\n$S_0$ denote the set of prime numbers $\\ell d$,\nso for every such $\\ell$ there exists $a\\in\\mathbb{Z}$ such that\n$g(X,a)\\mbox{ mod }\\ell$ is separable, \nand thus the same argument shows that $\\ell$ is unramified in\n$A^\\phi_{[a:1]}$.\n\nThus in the language of \\cite[Definition 1.10]{BSS},\n$G=S_n$ has a $(U;\\mathbf{d})$-realization with $U=\\emptyset$ and $\\mathbf{d}=(d)$.\nTherefore, \\cite[Proposition 1.11]{BSS} gives a realization of $G$\nover $\\mathbb{Q}$\nwith at most $B(\\mathbf{d})+\\#U$ many ramified primes,\nand under Schinzel's hypothesis H, $B(\\mathbf{d})\\leq 1$, cf.~\\cite[(13) on p.~923]{BSS}.\n\\end{proof}\n\n\n\\begin{remark}\\label{remark:plans}\nPlans \\cite[Remark 3.10]{Plans} sketches how to use Schinzel's hypothesis H (Conjecture \\ref{conj:Schinzel}) to construct $S_n$-extensions $L$ of $\\mathbb{Q}$ ramified in only one finite prime. However, he constructs $L$ as the splitting field of an irreducible trinomial $f=X^n+aX^i+b$, and therefore $L$ is not totally real if $n\\geq5$: \nIndeed, replacing $X$ by $X^{-1}$ if necessary we may assume that $i\\geq\\frac{n}{2}$, \nso since the derivative $f'=X^{i-1}(nX^{n-i}+ia)$ has at most $n-i+1$ distinct roots, in particular at most $n-i+1$ distinct roots in $\\mathbb{R}$,\nRolle's theorem shows that $f$ has at most $n-i+2\\leq\\frac{n}{2}+2\\!0$, they have different set of mathematical equations in the (singular) limit $\\varepsilon=0$ (detailed below). This highlights the fact that these systems are \\textit{singularly} and not \\textit{regularly} perturbed. Resolving these singular limits and reconciling these dual mathematical equations has been the basis of several mathematical and numerical developments within, e.g., \\textit{Catastrophe theory}, \\textit{Singularity theory}, \\textit{Bifurcation theory}, \\textit{Geometrical Singular perturbation theory (Fenichel theory)}, \\textit{geometric desingularization or blow-up method}. In short, these two different limits give rise to the so-called \\textit{slow subsystem} and \\textit{fast subsystem}, respectively. The slow subsystem is a differential-algebraic problem, where the slow variables remain explicitly dynamic through the differential equation $\\dot{y}=g(x,y)$, while the fast variables are enslaved to the slow ones through the algebraic equation $f(x,y)=0$. On the other hand, the fast subsystem is a family of dynamical systems on the fast variables $x$, where the slow variables $y$ have lost their dynamics and have become parameters. The set $\\{f=0\\}$ is referred to as the \\textit{critical manifold} of the system, and it plays a central role in both subsystems. Specifically, it is the phase space of the slow subsystem and it is the set of equilibria of the fast subsystem. Hence the locus of the limiting slow motion corresponds to a subset of the bifurcation diagram of the fast subsystem obtained when varying one or more slow variables. Moreover, varying other system parameters can induce the limiting slow motion to have non trivial trajectories, such as \\textit{canard solutions} that emerge via \\textit{dynamic bifurcations} and that visit both the stable and unstable regions of the critical manifold (see e.g.~\\cite{desroches12} for details). \n\\begin{figure}[!t]\n\\centering\n\\includegraphics{BurstingData.pdf}\n\\caption{Example of electrophysiological recordings of bursting oscillations in four types of neurons: (a) parabolic-type bursting from the CeN neuron from the melibe (a sea slug)~cite{newcomb08}; (b) square-wave-type bursting from a human $\\beta$-cell~\\cite{riz14}; (c) elliptic-type bursting from a dorsal-root-ganglia (DRG) neuron of a rat~\\cite{jian04}; (d) Pseudo-plateau-type bursting from a pituitary cell of a rat~\\cite{tabak11}.}\n\\label{fig.bursting_data}\n\\end{figure}\nThe knowledge of this dissection between the slow and fast subsystems, together with the knowledge of how the trajectory of the full system (i.e. for small $\\varepsilon\\!>\\!0$) evolves along these subsystems is at the basis of the various classification systems for complex slow-fast oscillations that we will subsequently review. Specifically, we will focus on the state-of-the-art mathematical classification systems for \\textit{bursting dynamics}, their limitations and finally propose a novel classification framework. \\red{Our novel classification fundamentally relies upon a certain type of canard configuration, referred to as \\textit{folded node}. Noteworthy, canards are indirectly included in the previous classification frameworks as boundaries between the spiking and the bursting regimes via so-called \\textit{spike-adding transitions} or \\textit{torus canards}, which we will review, thus highlighting the importance of canards in the classification process. We will define the notion of bursting oscillations in the context of neuronal systems} (see examples in Figure~\\ref{fig.bursting_data}) since for historical reasons the notion of bursting emerged within the neuroscientific literature. In particular, \\mdsr{bursting models} appeared in the context of classical single neuron electrophysiological measurements, where \\mdsr{the neuron's voltage time-series \\red{displays} a bursting oscillation either in response to a brief input stimulus or, in absence of any stimuli, in an endogenous manner}. These oscillations are defined as having a periodic succession (sometimes irregular) of two distinct epochs of activity. One epoch features slow and low-amplitude activity, and it is typically referred to as the \\textit{quiescent} (or \\textit{silent}) phase. The other epoch features fast and high-amplitude activations (i.e. several action potentials or spikes) is classically denoted \\textit{active} or \\textit{burst} phase as shown by several examples in Figure~\\ref{fig.bursting_data}. Although a great deal of our discussions will be in the context of neuronal dynamics, the mathematical framework intends to capture complex slow-fast oscillations beyond the scope of neuroscientific applications (e.g. in chemical reactions, genetic switches, material transitions, etc.). Hence some of the mathematical model constructs that we will present here display bursting oscillations not necessarily observed in neural data. Indeed, our \\mdsr{idealized} models will not have direct biophysical interpretation as we aim to be as general as possible in describing the fundamental mathematical mechanisms, which can then be applied to explain complex bursting oscillations in multiple contexts. Moreover, we will focus on the minimal deterministic mathematical setting for bursting oscillations. Specifically, we will cover the case of two-timescale systems with explicit timescale separation (dictated by a single small parameter $0\\!<\\!\\varepsilon\\!\\ll\\!1$), with two fast variables that will enable the description of the active phase of a bursting oscillation (i.e. $x\\in\\mathbb{R}^2$), and one or two slow variables describing the quiescent phase of the bursting dynamics (i.e. $y\\in\\mathbb{R}$ or $y\\in\\mathbb{R}^2$). This minimal setting will inform more complex scenarios involving multi-dimensional systems with multiple timescales.\n\n\\mdsr{This paper is organized as follows. In Section~\\ref{sec:review}, we will review existing classification frameworks for bursting oscillations. \\red{Subsequently}, in Section~\\ref{sec:beyond}, we first introduce the \\red{key} idea of our novel bursting classification based upon the concept of folded-node bursting dynamics. \\red{This is followed by showcasing} several \\red{new} examples of folded-node burster idealized models, first in the case of classical folded node and then in the case of cyclic folded node. Finally, in the conclusion section, we review our findings and propose a number of perspectives and future directions to explore. We \\red{complete the manuscript} by proposing, in Appendix~\\ref{sec:newfast}, few \\red{novel} additional bursting scenarios within the Rinzel-Izhikevich's classification. \\red{These include cases} with transcritical and pitchfork bifurcations of limit cycles, \\textit{isola bursting} and a two-slow-variable bursting scenario with a family of transcritical bifurcations of equilibria.}\n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics{RinzelClassification.pdf}\n\\caption{Rinzel classification of bursting patterns: square-wave bursting, here in the Hindmarsh-Rose model~\\cite{hindmarsh84} (panels (a1)-(b1)); elliptic bursting, here in the FitzHugh-Rinzel model~\\cite{rinzel86,rinzel87} (panels (a2)-(b2)); parabolic bursting, here in Plant's model~\\cite{plant81} (panels (a3)-(b3)).}\n\\label{fig:rinzel}\n\\end{figure}\n\n\\section{Review of the state-of-the-art classification of bursting patterns}\n\\label{sec:review}\n\\subsection{Rinzel's classification (mid 1980s)} Historically, John Rinzel opened the door towards mathematically understanding bursting oscillations. His seminal work on a mathematical analysis and classification of bursting oscillatory patterns, were first published within two companion manuscripts~\\cite{rinzel86,rinzel87}. The fundamental insight behind Rinzel's classification is based on slow-fast dissection and in particular describing the bifurcation structure of the fast subsystem where the slow variables are frozen. Subsequently, the time trajectory of the full system (i.e. for small $\\varepsilon\\!>\\!0$) is superimposed on top of the bifurcation structure of the fast subsystem. This reveals that the quiescent phase of the bursting cycle correspond to trajectory segments where the solution slowly tracks families of stable equilibria, or low-amplitude (subthreshold) limit cycles, of the fast subsystem. Conversely, the burst phase of the full system's cycle correspond to trajectory segments where the solution slowly tracks families of limit cycles of the fast subsystem. The transitions between these two main phases of bursting cycles occur near bifurcation points of the fast subsystem. With this approach, Rinzel proposed three classes of bursting dynamics based on both the bifurcation structure of the fast subsystem and the salient features of the main fast variable's time profile (in the neuronal context this is typically the neuronal membrane potential). These features include spike frequency during the burst, dynamics during the silent phase (oscillatory or not), shape of the burst (on a plateau compared to the silent phase or on the contrary with undershoots). These three features led Rinzel to name three classes as \\textit{square-wave}, \\textit{elliptic} and \\textit{parabolic} bursting. We show an example of each class in Figure~\\ref{fig:rinzel}.\n\n\\subsection{Izhikevich's classification (ca. 2000)} Eugene Izhikevich generalised Rinzel's approach by considering that a bursting pattern is entirely characterised by a pair of bifurcations ($\\mathbf{b_1}$, $\\mathbf{b_2}$) of the fast subsystem. One bifurcation, say $\\mathbf{b_1}$, explains the transition from quiescence to burst, and the other, $\\mathbf{b_2}$, marks the inverse transition, from burst to quiescence. Due to the well established bifurcation theory and indeed knowledge of classes of bifurcation, this led to a systematic identification of at least 120 bursting patterns~\\cite{izhikevich00}. An example of a bursting model that is not within the Rinzel classification is depicted in Fig~\\ref{fig:homhom}. In this example the bursting pattern has a transition from quiescence to burst via a homoclinic bifurcation (involving a small homoclinic connection) and equally, the transition from burst to quiescence is via homoclinic bifurcation (involving a large homoclinic connection). In many ways, Izhikevich's work serves as a key source of reference for classification of complex slow-fast oscillations. This is particularly the case in Neuroscience since some of the assembled examples were motivated by existing conductance-based neuronal models and demonstrated how complex neuronal oscillations could be achieved by adding one slow equation to a spiking system. Indeed, a dedicated book towards Neuroscience was later published, where the derived models where also put into context with neurophysiological processes~\\cite{izhikevich07}. The result of this deeply insightful work is a quasi-complete classification of bursting patterns in terms of pairs of codimension-one bifurcations of the fast subsystem.\n\\begin{figure}[!h]\n\\centering\n\\includegraphics{homhom.pdf}\n\\caption{Small homoclinic \/ big homoclinic bursting, corresponding to Fig.88 of~\\cite{izhikevich00}. Panel (a) shows the slow-fast dissection in the $(u,V)$ phase plane and panel (b) shows the $V$-time series of this bursting solution.}\n\\label{fig:homhom}\n\\end{figure}\n\\subsection{Bertram et al.'s \/ Golubitsky et al.'s classification (mid 1990)} An alternative approach to classification was proposed by Bertram and colleagues in 1995~\\cite{bertram95} and extended mathematically by Golubitsky and collaborators in 2001~\\cite{golubitsky01} using a singularity theory viewpoint. The fundamental idea consists in identifying a codimension-$k$ bifurcation point ($k \\geq 2$) in the fast subsystem and subsequently consider the slow variables of the bursting system as the unfolding parameters of this high codimension bifurcation point. The bursting is then obtained via a slow path made by the slow variables in the unfolding of that point (i.e. within a multidimensional parameter space). The minimum codimension, whose unfolding allows to create a given bursting pattern, defines the class of the associated bursting patterns provided a notion of path equivalence is properly defined. Specifically, two paths are equivalent if one can pass from one to the other via diffeomorphism and a re-parameterization. Recently, a review and a show-case demonstrating the construction of bursting oscillations via this approach, including cases for higher codimension bifurcation points was published in~\\cite{saggio17}. It is worth noting that the Rinzel-Izhikevich approach and the Bertram-Golubitsky approach both focus on the fast subsystem only. Moreover, a way to see a link between the two approaches is that the two bifurcation points ($\\mathbf{b_1}$, $\\mathbf{b_2}$) of the fast subsystem (as characterised by Izhikevich's approach) belong to bifurcation curves in a two-parameter plane, which coalesce at a codimension-two bifurcation point that characterises this particular bursting pattern from the singularity theory viewpoint. This implies that in principle the Rinzel-Izhikevich and the Bertram-Golubitsky approaches both lead to the same number of bursting oscillation cases.\n\n\\mdsr{The bursting patterns covered by these three existing classification schemes have not been exhausted yet, even though a large number (way above one hundred) have already been reported and analysed in previous works. However, we propose a few more cases which we believe have not been considered before and which will be presented with associated idealized models in Appendix~\\ref{sec:newfast}. In particular, we will show bursting scenarios where the burst phase ends due to a transcritical or a pitchfork bifurcation of limit cycles of the fast subsystem. We also propose the concept of \\textit{isola bursting}, where the burst starts and ends through saddle-node bifurcations of limit cycles of the fast subsystem which happen to lie on an isola of limit cycles. Finally, we propose one example (amongst many) of bursting pattern with two slow variables where the burst initiates through a family of transcritical bifurcation of equilibria.}\n\n\n\\section{\\mdsr{Towards a new classification of bursting patterns}}\n\\label{sec:beyond}\n\n\\subsection{\\mdsr{Main idea to go beyond the state-of-the-art}}\nIt is compelling to ask if there are other bursting oscillations beyond the Rinzel-Izhikevich and Bertram-Golubitsky classification approaches, as summarised in Figure~\\ref{fig:sketch} (top panel), and which cannot be explained invoking these state-of-the-art results. If so, could there be an improved classification system that captures a larger class of bursting dynamics beyond the ca. 120 cases captured by the state-of-the-art? Noteworthy, it is reasonable to contemplate that there could be possible extensions for classifying bursting patterns in systems with more than two timescales. In this context, the mathematical analysis would have to deal with nested fast subsystems, which has not yet been achieved (except in very particular cases) and therefore there is substantial work to do in order to extend the state-of-the-art approaches. However, still within the two-timescale framework the question of bursting classification extension remains. This question gains further support since there are electrophysiological recordings of bursting dynamics, which resist the state-of-the-art classification system. A case in point is depicted in Figure~\\ref{fig:MMBO_exp}, where the bursting oscillations has two phases, but the quiescent phase has the peculiarity that it appears to periodically rise close to a threshold, however the neuron does not have a transition to the active phase and instead descends back to its baseline activity and only on the second run the active phase emerges. \n\\begin{figure}[!t]\n\\centering\n\\includegraphics{MMBOexp.pdf}\n\\caption{Electrophysiological recordings of the lateral thalamic nuclei neuron in cat from~\\cite{roy84} show complex bursting oscillations.}\n\\label{fig:MMBO_exp}\n\\end{figure}\nThese observations suggest that there is an underlying complex mechanism for the quiescent phase of the oscillations and therefore point towards a bursting classification framework that has to also incorporate the analysis of the slow subsystem, which is in stark contrast to state-of-the-art approaches. Further motivating this view is our earlier study, which constructed the first example (to the best of our knowledge) of a slow-fast bursting model whereby the initiation of the bursting oscillations could not be explained by the fast subsystem of the underlying slow-fast model~\\cite{desroches13a}. However, therein we did not attempt to derive an improved bursting classification framework. Thus, we herein propose an extension of the state-of-the-art classification system that enforces the importance of considering complex dynamical mechanisms within the slow subsystem as the underlying cause of the initiation or termination of bursting oscillations, which can not be explained by the fast subsystem's analysis solely. To substantiate this novelty we will also show in subsequent sections how to construct a variety of these new cases of bursting oscillations. However, to better guide the reader throughout this manuscript we first replicate the results of our previous work in Figure~\\ref{fig:sketch} (panels (a1)-(c3)) and further summarise the key insights of the proposed novel classification framework. We first considered the minimal setting of systems with two-timescales and that possess at least two slow variables. We then constructed a bursting model whose quiescent phase displays small-amplitude near-threshold oscillations. Mathematically, it turns out that these observations are best explained by the so-called \\textit{folded-node singularities} defined in the slow subsystem ($\\varepsilon=0$) and associated \\textit{canard solutions}, which persist for small enough $\\varepsilon>0$; see Figure~\\ref{fig:sketch} panels (a1)-(c1). Noteworthy, these are not \\emph{per se} bifurcations of the fast subsystem but lie on saddle-node bifurcation curves of the fast subsystem (see Figure~\\ref{fig:sketch} panel (a1)). \n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=14cm]{fnburstRationale.pdf}\n\\caption{(Top part) Rationale behind bursting classifications: that of Rinzel, Izhikevich, Golubitsky et al., respectively, as well as the folded-node bursting approach proposed in this paper. (Main part (ai)-(ci), i=1,2,3) An exemplary folded-node bursting scenario, namely folded-node-homoclinic bursting, is shown in (a2)-(c2) in different 3D phase-space projections; (b2) is a zoomed view of (a2) where the critical manifold $S^0$ has been replaced by attracting $S^a_{\\varepsilon}$ and repelling $S^r_{\\varepsilon}$ slow manifolds in the vicinity of the folded node (dot). The classical analysis using the approaches of Bertram et al. \/ Golubitsky et al. (a3), Rinzel and Izhikevich (b3)-(c3) is based upon the fast subsystem's bifurcation structure. The complementary key approach to fully characterize this bursting pattern, presented in panels (a1)-(c1), uses the slow subsystem and its flow which reveals the presence of the folded node \\textbf{fn}. \\mdsr{Note that the fold curve $\\mathcal{F}^-$ of the critical manifold $S^0$ (panel (a2)) is also a curve of saddle-node bifurcation points (labelled LP in (a1)) of the fast subsystem when both slow variables are considered as parameters, the same curve used in the singularity theory approach on panel (a3).} (Side part) Full system's equations as well as the various subsystems, fast and slow, obtained for $\\varepsilon=0$.}\n\\label{fig:sketch}\n\\end{figure}\nSuch unexpected and non-trivial emergent mathematical objects allow trajectories of the slow subsystem to visit both the attracting ($S^0_a$) and repelling ($S^0_r$) parts of the critical manifold. In the full system (for small $\\varepsilon\\!>\\!0$) the perturbed versions of these manifolds --attracting $S^{\\varepsilon}_a$ and repelling $S^{\\varepsilon}_r$ slow manifolds-- twist and intersect multiple times (see Figure~\\ref{fig:sketch} panels (a2)-(b2)) thereby causing trajectories to non-trivially and robustly oscillate during the quiescent phase. \\mdsr{In essence, if the slow dynamics change direction along a fold then this can create a folded node scenario.} The transition from quiescent to active phase is caused by a repulsion of the trajectory away from the unstable sheet of the critical manifold; this phenomenon is mediated by canard solutions due to the presence of the folded node in the slow subsystem. In this particular example, the fast oscillations of the active phase are due to a nearby family of Hopf bifurcations in the fast subsystem (Figure~\\ref{fig:sketch} panel (c2)). The return back to quiescence is then caused by a family of homoclinic bifurcations of the fast subsystem. The key insight is that the fast subsystem is blind to what is causing these small-amplitude oscillations during the quiescent phase, and thus it is unable to classify the initiation of these oscillations based upon the bifurcations of fast subsystem only. This point is illustrated by the Rinzel-Izhikevich slow-fast dissection and projection of the trajectory of the full system onto the bifurcation diagram of the fast subsystem (see Figure~\\ref{fig:sketch} panels (b3)-(c3)). Note that by employing the Rinzel-Izhikevich classification system, the bursting dynamics would be explained by two bifurcations of the fast subsystem, namely the fold bifurcation LP$_2$ and the homoclinic bifurcation Ho$_2$. In particular, a fold bifurcation (LP$_2$) does not explain an oscillation. Moreover, a similar argument applies to the Golubitsky approach (see Figure~\\ref{fig:sketch} panel (a3)). This panel displays curves of codimension-one bifurcation points of the fast subsystem, which meet at codimension-two e.g. a Bogdanov-Takens BT (within a two-dimensional parameter space). It can then be shown that it is impossible to construct a path for the slow dynamics (within this two-dimensional parameter space), in particular along the homoclinic and saddle-node curves (since these characterise the bursting in the fast subsystem), which could explain folded-node-initiated quiescent phase oscillations. It turns out amongst all possible folded singularities, only folded nodes (and in limiting cases, so-called \\textit{folded saddle-nodes}) can generate such robust small-amplitude oscillations in the full system, and this is due to the twisting of nearby attracting and repelling slow manifolds. The key message is that these folded-node singularities only appear in the slow limit of the underlying two-slow-two-fast bursting systems and are invisible in the fast limit. Therefore one must consider both fast and slow subsystems in order to fully characterise the novel bursting scenarios associated with folded-node singularities, which leads us to a novel bursting classification system (see Figure~\\ref{fig:sketch} top panel in blue for the new framework). We believe these insights will fuel subsequent developments in higher dimensional multi-scale systems.\n\nAs hinted \\mdsr{above}, there are emergent slow dynamical mechanisms (captured by slow variables) that are blind to the state-of-the-art classification and thus this suggests the need to extend the classification of bursting oscillations. An example of such emergent slow dynamics is the so-called \\textit{folded node} and, in a minimal setting, it emerges due to non-trivial interactions between two slow variables. Indeed, herein we focus on new classes of bursting oscillations modelled via slow-fast systems with (at least) two slow variables and (at least) two fast variables and for which some epochs of the oscillatory time-series is explained by folded-node dynamics. This underlying folded-node signature leads us to name the resulting new classes of bursting models, \\textit{folded-node bursters}. Three fundamental cases are envisaged. The first case are bursters characterised by small-amplitude oscillations that occur during the quiescence phase, in which case we will refer to the \\textit{classical folded-node} bursting scenario. The second case involves slow-amplitude modulation of the burst, which we will denote as the \\textit{cyclic folded-node} bursting scenario. The third case, combines \\textit{classical folded-node} and \\textit{cyclic folded-node}. These classes of bursting patterns involve both the fast subsystem and the slow subsystem of the model, unlike traditional bursters. A second key aspect of these new classes is the central role played by \\textit{canards}, namely, \\textit{spike-adding canard} cycles involved in the classical folded-node bursting case, and \\textit{torus canards} in the cyclic folded-node bursting case. In the following subsections, we describe in details these two scenarios.\n\n\\subsection{Classical folded-node case}\n\\label{sec:classicalFN}\nHere we propose several bursting oscillations mediated by a classical folded node and the modelling steps of underlying \\mdsr{idealized} models are given. To guide the reader towards a modelling strategy of these systems, we first recall key concepts and mechanisms.\n\n\\subsubsection{A necessary preliminary step: spike-adding canard explosion}\n\\label{sec:spikeadd}\nFirst, recall that \\textit{canards} are non-trivial trajectories that emerge due timescale separation and unexpectedly, these trajectories contain segments that follow both an attracting and a repelling slow manifold. This phenomenon has been thoroughly studied in planar systems (i.e. with 1 slow variable and 1 fast variable)~\\cite{benoit81,dumortier96,eckhaus83,krupa01,mishchenko94}, as well as, in 3D systems with two slow variables~\\cite{benoit90,desroches12,wechselberger05}. In applications, canards can be associated to complex (bio)physical mechanisms, for example in neuroscience it provides the best approximation to the excitability threshold in certain single-neuron models. This observation was first made by Izhikevich~\\cite{izhikevich07}, who showed that canards organise the transition to the spiking regime of type II neurons. This was later analysed in more details in~\\cite{demaesschalck15,desroches13b,wechselberger13}. Yet another important mechanism is the so-called \\textit{spike-adding canard explosion}. This canard phenomenon arises in bursting oscillations and can be described as a sequence of canard explosions which organise the transition from subthreshold oscillations to bursting solutions with more and more spikes per bursts. This phenomenon was first described and analysed (in the case of chaotic dynamics regime) in~\\cite{terman91} in the context of square-wave bursting. This was revisited more recently in~\\cite{guckenheimer09} from the computational standpoint of saddle-type slow manifolds and further described in~\\cite{nowacki12} in a modeling context to explain transient spikes. These analyses were later refined (from a canard standpoint) in~\\cite{desroches13a} and the canard-mediated spike-adding dynamics was fully analysed in~\\cite{desroches16} in the context of parabolic bursters (with two slow variables), revealing the central role of folded-saddle canards. Noteworthy, bursting oscillations that possess spike-adding mechanism is a limiting (border line) case that already hints to the importance of possibly including the analysis of the slow flow in a bursting classification framework. That is, spike-adding requires a turning point of the slow flow, a so-called \\textit{canard point}, whereby each new added spike (within the bursting phase) is born via a slow (delayed) passage through this turning point. Crucially, the fast subsystem is blind to the underlying canard trajectories occurring near the turning point (well-defined as such only in the slow flow) and instead only sees a fold bifurcation. Therefore the state-of-the-art bursting classification systems does not capture this aspect. Nevertheless, we refrain from declaring this phenomenon as a new bursting mechanism because a spike-adding canard explosion gives rise to canard cycles that exist only within exponentially thin parameter regions. Hence, the robust dynamics is the fold-initiated bursting dynamics, and the fast subsystem analysis still prevail in order to classify it. In contrast, if we consider a fold-initiated bursting scenario undergoing spike-adding canard explosion and if we further add a slow dynamics for the parameter that displays the spike-adding canard explosion (i.e. a second slow variable in the extended model) then we obtain a folded-node bursting system. This has a similar effect to the case in classical (van der Pol type) systems where the canard phenomenon becomes robust if one adds a second slow variable, which has the effect of creating a folded singularity in the resulting two-dimensional slow flow and allows for multiple canard trajectories to exist. The idea here is similar, but with two fast variables, allowing for bursting dynamics in conjunction with folded-node dynamics. A first example of this scenario was termed \\textit{mixed-mode bursting oscillations} in~\\cite{desroches13a} but we prefer to denote it more generally folded-node bursting. Indeed, folded-node bursting is a new form of bursting pattern with two slow variables where the silent phase contains small-amplitude (subthreshold) oscillations due to the presence of a folded node in the slow subsystem. This folded node is responsible for the presence of a funnel region in the full system and trajectories entering this funnel make a number of rotations (which can be controlled by adjusting parameters) before they leave it and start to burst. Hence, the passage through the folded-node funnel organises the transition from quiescence to burst and it can only be understood by suitably analysing the slow subsystem. We subsequently describe a strategy for constructing folded-node bursting systems.\n\\subsubsection{Construction of minimal folded-node bursting systems}\nAs a staring point, we consider the prototypical fold-initiated burster of Hindmarsh-Rose type. By this we mean a three-dimensional slow-fast system with two fast variables and one slow and a cubic-shaped family of equilibria in the fast subsystem, namely the critical manifold $S^0$. We can write the following set of differential equations (using the fast time $\\tau$) to describe the dynamics of such a system\n\\begin{equation}\\label{eq:protoburster}\n\\begin{split}\nx' &= y - f(x) + az,\\\\\ny' &= g(x)-y,\\\\\nz' &= \\varepsilon(\\alpha x + \\gamma\\beta - \\delta z),\n\\end{split}\n\\end{equation}\nwhere $f$ is a cubic polynomial function, $g$ is (at least) quadratic; moreover, $0\\!<\\varepsilon\\!\\ll1$ is a small parameter and $(a,\\alpha,\\beta,\\gamma,\\delta)$ are potential bifurcation parameters; why we use a product of two parameters in the $z$ equation will become clear in the next section. As we shall see in the example section to follow, one can also obtain all fold-initiated scenarios by using an unfolding of a codimension-3 degenerate Bogdanov-Takens (BT) bifurcation; see~\\cite{dumortier91} for details.\n\nA few assumptions are required in order for the system~\\eqref{eq:protoburster} to display fold-initiated bursting. First of all, we assume that $f$ and $g$ are adequately chosen so that the fast subsystem has a cubic-shaped family of equilibria that depends on $z$ as a parameter (for the fast subsystem). This family can be written as a cubic function: \n$$z=\\left(f(x)-g(x)\\right)\/a.$$ \nTherefore, the corresponding bifurcation diagram (of the fast subsystem) in $z$ is S-shaped and will have fold points. \nThe critical manifold is then given by \n\\begin{eqnarray}\\label{eq:critman}\nS^0:=\\left\\{(x,y,z)\\in\\mathbb{R}^3\\;\\big\/\\;\\;y=g(x)\\;,\\;z=\\left(f(x)-y\\right)\/a\\right\\}.\n\\end{eqnarray}\nWe also require bistability in the fast subsystem between equilibria and limit cycles, in an interval of $z$-values. One bound of this interval correspond to a fold bifurcation and, geometrically, to one fold point of the cubic family of equilibria. The other boundary of the region of bistability of the fast subsystem will be a bifurcation of limit cycles and we shall consider three main cases, namely, saddle-homoclinic bifurcation (see Fig.~\\ref{fig:fnhom}), Hopf bifurcation (see Fig.~\\ref{fig:fnhopf}) and fold bifurcation of cycles (see Fig.~\\ref{fig:fnsnp}), but the list is not exhaustive. Now, considering the linear slow dynamics of system~\\eqref{eq:protoburster} for the slow variable $z$, we assume that a variation of one of the two parameters $\\alpha$ and $\\beta$ in the full system induces the linear $z$-nullsurface to cut through the fold point of the critical manifold $S^0$ for a certain value of this parameter. One can show that this creates a Hopf bifurcation in the full system which induces limit cycles to appear. Provided this transversal cut of the $z$-nullsurface with the critical manifold takes place, then a spike-adding canard explosion will emerge, whereby bursting solutions appear from subthreshold (spikeless) periodic solutions along branch of limit cycles undergoing multiple canard explosions; see~\\cite{desroches13a} for an example of this phenomenon in the context of square-wave bursting. As explained in the previous section, one salient feature of the spike-adding canard explosion is the presence of a turning point (a canard point) in the slow flow of system~\\eqref{eq:protoburster}. To compute the slow-flow, we first rescale time in~\\eqref{eq:protoburster} by a factor $\\varepsilon$. That is, we rescale the fast time $\\tau$ (with $x'=dx\/d\\tau$) into the slow time $t$ defined by $t=\\varepsilon\\tau$. This bring the system to the slow-time parametrisation \n\\begin{equation}\\label{eq:protobursterslowtime}\n\\begin{split}\n\\varepsilon\\dot{x} &= y - f(x) + z,\\\\\n\\varepsilon\\dot{y} &= g(x)-y,\\\\\n~~\\dot{z} &= (\\alpha x + \\gamma\\beta - \\delta z),\n\\end{split}\n\\end{equation}\nwhose $\\varepsilon=0$ limit corresponds to the slow subsystem. The slow subsystem is a differential-algebraic equation (DAE), where the dynamics of $z$ is explicitly preserved while $x$ and $y$ are slaved to $z$ by the algebraic constraints that corresponds to the equation of the (here one-dimensional) critical manifold $S^0$. The dynamics of $x$ and $y$ can be revealed by differentiating the algebraic constraint with respect to the slow time, which gives after rearranging the following one-dimensional dynamical system defined on $S^0$\n\\begin{equation}\\label{eq:protobursterslowflow}\n\\begin{split}\n\\dot{x} &= a\\frac{\\alpha x + \\gamma\\beta - \\delta z}{f'(x)-g'(x)}.\n\\end{split}\n\\end{equation}\n\\begin{figure}[!t]\n\\centering\n\\includegraphics{FNHom.pdf}\n\\caption{Folded-node\/Homoclinic bursting. We take $f(x)=x^3-3x^2$ and \\bluebis{$G(x,y,z)=g(x)-y$ with $g(x)=1-5x^2$. The parameter values are: $a=1$, $c=1$, $\\alpha=0.3$, $\\gamma=1$, $\\delta=1.2$, $\\varepsilon=0.002$, $\\mu=0.033$, $\\gamma_y=0.0005$ and $\\gamma_{\\beta}=-0.008$}. Panels (a-b) show the spike-adding transition in system~\\eqref{eq:fnburster}: (a) in the $(z,x)$ plane; (b) associated bifurcation diagram with respect to parameter $\\beta$. Panels (c-d) show a folded-node\/homoclinic bursting orbit: (c) in the $(\\beta,z,x)$ space; (d) $x$-time series. The bottom panels show a comparison between this folded-node bursting orbit from~\\eqref{eq:fnburster} and experimental data from~\\cite{roy84}.}\n\\label{fig:fnhom}\n\\end{figure}\n\\begin{figure}[!t]\n\\centering\n\\includegraphics{FNHopf.pdf}\n\\caption{Folded-node\/Hopf bursting. We take $f(x)=x^3-3x^2$ and \\bluebis{$G(x,y,z)=g(x)-y$ with $g(x)=1-5x^2$. The parameter values are: $a=1$, $c=2$, $\\alpha=0.3$, $\\gamma=1$, $\\delta=1$, $\\varepsilon=0.004$, $\\mu=0.0104$, $\\gamma_y=0.0003$ and $\\gamma_{\\beta}=-0.05$}. Panels (a-b) show the spike-adding transition in system~\\eqref{eq:fnburster}: (a) in the $(z,x)$ plane; (b) associated bifurcation diagram with respect to parameter $\\beta$. Panels (c-d) show a folded-node\/Hopf bursting orbit: (c) in the $(\\beta,z,x)$ space; (d) $x$-time series.}\n\\label{fig:fnhopf}\n\\end{figure}\n\\begin{figure}[!t]\n\\centering\n\\includegraphics{FNSNP.pdf}\n\\caption{Folded-node\/Fold of cycles bursting. We take $f(x)=0$ and \\bluebis{$G(x,y,z)=-x^3+A_1(z)x+A_2(z)-y(A_3(z)-x+x^2)$}, where \\bluebis{$A_1(z)=0.1201z+0.1871$, $A_2(z)=0.0906z-0.0251$, $A_3(z)=0.105z-0.3526$}. \\bluebis{The parameter values are: $a=0$, $c=1$, $\\alpha=0$, $\\gamma=-1$, $\\delta=1$, $\\varepsilon=0.01$, $\\mu=-0.00012$, $\\gamma_y=-0.003$, $\\gamma_{\\beta}=0.0001$}. Panels (a-b) show the spike-adding transition in system~\\eqref{eq:fnburster}: (a) in the $(z,x)$ plane; (b) associated bifurcation diagram with respect to parameter $\\beta$. Panels (c-d) show a folded-node\/fold of cycles bursting orbit: (c) in the $(\\beta,z,x)$ space; (d) $x$-time series.}\n\\label{fig:fnsnp}\n\\end{figure}\nAs is \\red{typical} in slow-fast systems with folded critical manifolds, note that the denominator of the right-hand side of~\\eqref{eq:protobursterslowflow} vanishes at fold points of $S^0$, which makes \\mdsr{generically} the dynamics of $x$ explode \\mdsr{and the corresponding fold point is referred to as a \\textit{jump point}. However, if} the numerator has a zero of the same order \\mdsr{as the denominator, then there can be a cancellation and the dynamics of $x$ does not explode; in this case, the fold point is referred to as a \\textit{canard point} or a \\textit{turning point}. The condition for a canard point to occur \\red{in this system is}} \\mdsr{given} by the following condition: \n\\begin{equation}\\label{eq:protoburstercanardpoint}\n\\begin{split}\nz_f &= (\\alpha x_f + \\gamma\\beta)\/\\delta,\n\\end{split}\n\\end{equation}\nwhere $(x_f,z_f)$ is a fold point of $S^0$. This indeed gives a transversal crossing of the slow nullsurface with the critical manifold at one of its fold points. Even though~\\eqref{eq:protoburstercanardpoint} depends on several parameters, it is a codimension-1 condition, therefore by fixing all parameters but one, then the condition can be satisfied by adjusting the last parameter. We arbitrarily choose to vary $\\beta$, which will become a second slow variable in the full 4D folded-node bursting system that we will construct below. Therefore, the spike-adding transitions leading to bursting in system~\\eqref{eq:protoburster} are obtained as the result of the slow nullsurface moving though one fold point of the critical manifold upon variation of $\\beta$. The same dynamics would be obtained by varying a parameter affecting the critical manifold while maintaining the slow nullsurface fixed, in particular if we were to append an additive parameter $I$ to the $x$ equation of the system. This would mimick the effect of an applied (external) current in a neuron-type model such as the Hindmarsh-Rose model~\\cite{hindmarsh84} or the Morris-Lecar model~\\cite{morris81,terman91}. However, from the pure dynamical viewpoint, varying a parameter in the slow equation results in the same effect and this is the scenario that we chose in order to construct fold-initiated spike-adding transitions in the original 3D burster and folded-node bursting in the extended 4D model.\n\nStarting from a fold-initiated bursting scenario with spike-adding canard explosion (controlled via a static variation of parameter $\\beta$) then a folded-node bursting is obtained by prescribing the dynamics on $\\beta$ by a slow differential equation. That is, we consider the following extended bursting system\n\\bluebis{\n\\begin{equation}\\label{eq:fnburster}\n\\begin{aligned}\nx' &= (y - f(x) + az)\/c,\\\\\ny' &= G(x,y,z),\\\\\nz' &= \\varepsilon(\\alpha z + \\gamma\\beta - \\delta x),\\\\\n\\beta' &= \\varepsilon\\big(\\mu-\\gamma_y(y-y_{\\mathrm{fold}})^2-\\gamma_{\\beta}(\\beta-\\beta_{\\mathrm{fold}})^2\\big).\n\\end{aligned}\n\\end{equation}\n}\n\\bluebis{Note that we will consider prototype systems where either $G$ is directly given as a graph over $x$, described as, $G(x,y,z)=g(x)-y$ (i.e. Folded-node\/homoclinic and folded-node\/Hopf cases), or the level set $\\{G(x,y,z)=0\\}$ is a graph over $(x,z)$, expressed as, $\\{y=g(x,z)\\}$ (i.e. Folded-node\/fold of cycles case). We claim that all folded-node initiated bursting scenarios can be obtained in either of these two ways. In the latter case, our minimal model is inspired by the codimension-3 degenerate Bogdanov-Takens unfolding introduced in~\\cite{dumortier91} and further applied in the context of bursting in~\\cite{saggio17}. In practice (for simulation purposes), $\\mu$, $\\gamma_y$ and $\\gamma_z$ will be taken $O(\\varepsilon)$.} Therefore, system~\\eqref{eq:fnburster} is effectively a three-timescale dynamical systems with dynamics evolving on $O(1)$, $O(\\varepsilon)$ and $O(\\varepsilon^2)$ timescales. For convenience and to ease the folded-node analysis, we will keep the equations written as in~\\eqref{eq:fnburster} with only $\\varepsilon$ has an apparent timescale separation parameter. System~\\eqref{eq:fnburster} is parametrised by the fast time $\\tau$, meaning that the $\\varepsilon=0$ limit of that parametrisation of the system gives the fast subsystem where the slow variables $z$ and $I$ are frozen and become parameters. Introducing the slow time $t=\\varepsilon\\tau$ brings the system into the different parametrisation \n\\bluebis{\n\\begin{equation}\\label{eq:fnburstertau}\n\\begin{aligned}\n\\varepsilon\\dot{x} &= (y - f(x) + az)\/c,\\\\\n\\varepsilon\\dot{y} &= G(x,y,z),\\\\\n\\dot{z} &= \\alpha z + \\gamma\\beta - \\delta x,\\\\\n\\dot{\\beta} &= \\mu-\\gamma_y(y-y_{\\mathrm{fold}})^2-\\gamma_{\\beta}(\\beta-\\beta_{\\mathrm{fold}})^2,\n\\end{aligned}\n\\end{equation}\n}\nwhose $\\varepsilon\\!=\\!0$ limit corresponds to the slow subsystem. We will show that, all other parameters being fixed, the slow subsystem of~\\eqref{eq:fnburstertau} possesses a folded-node singularity, which creates transient subthreshold oscillations that initiate the burst when $0<\\varepsilon\\ll1$, regardless of the values of other parameters. However, simulations require that $\\mu$\\, $\\gamma_y$ and $\\gamma_I$ be $O(\\varepsilon)$ in order for these small subthreshold oscillations to be recurrent, hence entering into a robust periodic bursting attractor which we name folded-node bursting. We provide numerical evidence of this point, based on the strength of the global return mechanism, even though we do not provide a rigorous proof of it.\n\n\\bluebis{Applying the same strategy as in the three-dimensional (bursting) case, and projecting onto the $(x,\\beta)$-plane (the dimension of the slow flow corresponds to the number of slow variables), we obtain the following system for the reduced system (or slow subsystem)\n\\begin{equation}\\label{eq:fnbursterslowflow}\n\\begin{aligned}\n\\dot{x} &= \\frac{(g_z(x,z)+a)(\\alpha z + \\gamma\\beta - \\delta x)}{f'(x)-g_x(x,z)},\\\\\n\\dot{\\beta} &= \\mu-\\gamma_y\\left(g(x,z)-y_{\\mathrm{fold}}\\right)^2-\\gamma_{\\beta}(\\beta-\\beta_{\\mathrm{fold}})^2,\n\\end{aligned}\n\\end{equation}\nafter substituting for $g(x,z)$ for $y$ from the critical manifold condition. Note that in two of the three examples that we will consider, $g(x,z)=g(x)$ depends only on $x$ and hence $g_z(x,z)=0$. The critical manifold of system~\\eqref{eq:fnburster} is not normally hyperbolic everywhere and, hence, the system possesses a (1D here) fold set defined by $$\\mathcal{F}:=\\{(x,y,z)\\in S^0;\\;f'(x)=g_x(x,z)\\}.$$\nThis implies that the slow flow~\\eqref{eq:fnbursterslowflow} of system~\\eqref{eq:fnburster} is not defined along $\\mathcal{F}$. The slow flow can be extended along $\\mathcal{F}$ by preforming an $x$-dependent time rescaling which amounts to multiply the right-hand side of~\\eqref{eq:fnburster} by a factor $f'(x)-g_x(x,z)$, hence yielding the so-called \\textit{desingularised reduced system (DRS)}\n\\begin{equation}\\label{eq:fnbursterDRS}\n\\begin{aligned}\n\\dot{x} &= \\left(g_z(x,z)+a\\right)(\\alpha z + \\gamma\\beta - \\delta x),\\\\\n\\dot{\\beta} &= \\left(f'(x)-g_x(x,z)\\right)\\left(\\mu-\\gamma_y\\left(g(x,z)-y_{\\mathrm{fold}}\\right)^2-\\gamma_{\\beta}(\\beta-\\beta_{\\mathrm{fold}})^2\\right),\n\\end{aligned}\n\\end{equation}\nwith $z=z(x)$ defined by $S^0$, that is, $g(x,z)-f(x)+\\alpha z=0$. In all cases we shall consider (including the general codimension-3 unfolding of a degenerate BT bifurcation from~\\cite{dumortier91}), $z$ can be written as a function of $x$ on $S^0$. \nAs a consequence of this $x$-dependent time rescaling, the DRS~\\eqref{eq:fnbursterDRS} is regular everywhere in $\\mathbb{R}^2$ including on $\\mathcal{F}$, along which it has the possibility for equilibria simply by appearance of the factor $f'(x)-g_x(x,z)$ in the $\\beta$-equation. The equilibrium condition is then that $\\dot{x}=0$ in~\\eqref{eq:fnbursterDRS} together with $f'(x)-g_x(x,z)$, which conveys the idea already seen in the 3D (bursting) case. That is, a singularity of the reduced system at a point on $\\mathcal{F}$ can be resolved if and only if the numerator of the right-hand side of $\\dot{x}$ in that system vanishes at this point and the zeros of the two algebraic expressions to be of the same order. Such points are called \\textit{folded singularities} (or \\textit{folded equilibria}) and they are the equivalent of canard points in the cases with (at least) two slow variables. Folded equilibria are equilibria of the DRS~\\eqref{eq:fnbursterDRS} and, according to their topological type as equilibria of the DRS, one can generically define \\textit{folded nodes}, \\textit{folded saddles} and \\textit{folded foci}. However they are not equilibria of the reduced system~\\eqref{eq:fnbursterslowflow} due to the singular time rescaling performed to pass from one to the other. Indeed, this time rescaling is chosen so that trajectories of the DRS have reversed orientation on the repelling sheet of $S^0$ compared to trajectories of the reduced system (both have the same orientation along the attracting sheet). Hence, in the case of folded nodes and folded saddles, trajectories starting on the attracting sheet of $S^0$ may cross the folded singularity in finite time and with finite speed.\\newline\nThe Jacobian matrix of~\\eqref{eq:fnbursterDRS} evaluated at a folded equilibrium has the form\n\\begin{equation}\\label{eq:JacDRS}\n\\begin{aligned}\n\\mathrm{J}=\\begin{pmatrix}\n(-\\delta+\\alpha z'(x))(g_z(x,z)+a) & \\gamma(g_z(x,z)+a)\\\\\nK_2 & 0\n\\end{pmatrix},\n\\end{aligned}\n\\end{equation}\nwhere $$K_2=(f''(x)-\\partial_xg_x(x,z))\\left(\\mu-\\gamma_y\\left(g(x,z)-y_{\\mathrm{fold}}\\right)^2-\\gamma_{\\beta}(\\beta-\\beta_{\\mathrm{fold}})^2\\right).$$ From~\\eqref{eq:JacDRS}, one can easily write down conditions that enable the emergence of a folded-node singularity ($\\mathrm{tr}(J)<0$, $\\det{J}>0$, $\\mathrm{tr}(J)^2-4\\det{J}>0$) or a folded-saddle singularity ($\\det{J}<0$) in the reduced system. As we will explain below, even though only the folded-node case gives rise to robust bursting patterns, the folded-saddle case is still interesting in the study of 4D bursters with two slow variables. One also can easily verify that our minimal example systems all give rise to a folded-node case. Indeed, in the folded-node\/homoclinic (Figure~\\ref{fig:fnhom}) and folded-node\/Hopf (Figure~\\ref{fig:fnhopf}) bursting cases, system~\\eqref{eq:fnburster} has the form\n\\begin{equation}\\label{eq:fnhomburster}\n\\begin{aligned}\nx' &= (y - x^3 + 3x^2 + z)\/c,\\\\\ny' &= 1 - 5x^2 - y,\\\\\nz' &= \\varepsilon(\\alpha z + \\gamma\\beta - \\delta x),\\\\\n\\beta' &= \\varepsilon\\left(\\mu-\\gamma_y(y-y_{\\mathrm{fold}})^2-\\gamma_{\\beta}(\\beta-\\beta_{\\mathrm{fold}})^2\\right),\n\\end{aligned}\n\\end{equation}\nwhich hence gives the following DRS's Jacobian matrix\n\\begin{equation}\\label{eq:JachomDRS}\n\\begin{aligned}\n\\mathrm{J_{1,2}}=\\begin{pmatrix}\n-\\delta & \\gamma\\\\\nK_2 & 0\n\\end{pmatrix},\n\\end{aligned}\n\\end{equation}\nwith: $K_2=(-6x_{\\mathrm{fs}}-4)\\left(\\mu-\\gamma_y\\left(1-5x_{\\mathrm{fs}}^2-y_{\\mathrm{fold}}\\right)^2-\\gamma_{\\beta}(\\beta-\\beta_{\\mathrm{fold}})^2\\right)$, and $x_{\\mathrm{fs}}=-4\/3$. Given the chosen parameter values corresponding to Figures.~\\ref{fig:fnhom} and~\\ref{fig:fnhopf}, then we immediately conclude that we have indeed a folded node. Likewise, in the folded-node\/fold of cycles case illustrated in Figure~\\ref{fig:fnsnp}, the slow-fast system corresponding to~\\eqref{eq:fnburster} is \n\\begin{equation}\\label{eq:fnsnpburster}\n\\begin{aligned}\nx' &= y,\\\\\ny' &= -x^3+A_1(z)x+A_2(z)-y(A_3(z)-x+x^2),\\\\\nz' &= \\varepsilon(\\alpha z + \\gamma\\beta - \\delta x),\\\\\n\\beta' &= \\varepsilon\\left(\\mu-\\gamma_y(y-y_{\\mathrm{fold}})^2-\\gamma_{\\beta}(\\beta-\\beta_{\\mathrm{fold}})^2\\right),\n\\end{aligned}\n\\end{equation}\nwhere $A_i=a_i z+b_i$ ($i=1,2,3$) are linear functions of $z$. Therefore, we obtain the associated DRS's Jacobian matrix\n\\begin{equation}\\label{eq:JacsnpDRS}\n\\begin{aligned}\n\\mathrm{J_{1,2}}=\\begin{pmatrix}\n\\left(-\\delta+\\alpha\\frac{3x_{\\mathrm{fs}}^2-b_1}{a_1+a_2}\\right)(a_1x_{\\mathrm{fs}}+a_2) & \\gamma(a_1x_{\\mathrm{fs}}+a_2)\\\\\nK_2 & 0\n\\end{pmatrix},\n\\end{aligned}\n\\end{equation}\nwith: $K_2=(6x_{\\mathrm{fs}}-a_1)\\left(\\mu-\\gamma_yy_{\\mathrm{fold}}^2-\\gamma_{\\beta}(\\beta-\\beta_{\\mathrm{fold}})^2\\right)$ and $x_{\\mathrm{fs}}$ solution to $$-3x_{\\mathrm{fs}}^2+a\\frac{x_{\\mathrm{fs}}^3-b_1x_{\\mathrm{fs}}-b_2}{a_1+a_2}+b_1=0.$$ Substituting the parameter values for their chosen numerical value mentioned in the caption of Figure~\\ref{fig:fnsnp} allows to conclude that we are indeed dealing with a folded node.}\\newline\nOne can obtain the general DRS~\\eqref{eq:fnbursterDRS} by applying implicit differentiation to one algebraic equation only (the right-hand side of the $\\dot{x}$ equation in the original system) and substituting $g(x,z)$ for $y$ (coming from the second algebraic equation). This gives the same result as the DRS obtained from both algebraic constraint together. Indeed, in all generality, applying implicit differentiation to the two algebraic equations of the slow subsystem gives \n\\begin{equation}\\label{eq:RS2fast}\n\\begin{aligned}\n\\begin{pmatrix}\n-f'(x) & 1 \\\\\n-g_x(x,z) & 1\n\\end{pmatrix}\n\\begin{pmatrix}\n\\dot{x} \\\\\n\\dot{y}\n\\end{pmatrix}\n&=\n\\begin{pmatrix}\n-a \\\\\ng_z(x,z)\n\\end{pmatrix}\n(\\alpha z + \\gamma\\beta - \\delta x)\\\\\n\\dot{z} &= \\alpha z + \\gamma\\beta - \\delta x,\\\\\n\\dot{\\beta} &= \\mu-\\gamma_y(y-y_{\\mathrm{fold}})^2-\\gamma_{\\beta}(\\beta-\n\\beta_{\\mathrm{fold}})^2,\n\\end{aligned}\n\\end{equation}\nwhich by Kramer's rule is equivalent, after posing $$\\mathrm{J}=\n\\begin{pmatrix}\n-f'(x) & 1 \\\\\n-g_x(x,z) & 1\n\\end{pmatrix}\n,$$ (Jacobian matrix of the original vector field with respect to the fast variables at $\\varepsilon=0$) to \n\\begin{equation}\\label{eq:RS2fast2}\n\\begin{aligned}\n\\det(\\mathrm{J})\n\\begin{pmatrix}\n\\dot{x} \\\\\n\\dot{y}\n\\end{pmatrix}\n&= \\mathrm{Adj(J)}\\begin{pmatrix}\na \\\\\n-g_z(x,z)\n\\end{pmatrix}\n(\\alpha z + \\gamma\\beta - \\delta x)\\\\\n\\dot{z} &= \\alpha z + \\gamma\\beta - \\delta x,\\\\\n\\dot{\\beta} &= \\mu-\\gamma_y(y-y_{\\mathrm{fold}})^2-\\gamma_{\\beta}(\\beta-\n\\beta_{\\mathrm{fold}})^2,\n\\end{aligned}\n\\end{equation}\nwhere $\\det(\\mathrm{J})=g_x(x,z)-f'(x)$ and \n$$\\mathrm{Adj}(\\mathrm{J})=\n\\begin{pmatrix}\n1 & -1\\\\\ng_x(x,z) & -f'(x)\n\\end{pmatrix},$$ denote the determinant and the adjugate matrix of $\\mathrm{J}$, respectively. The previous system is singular when $\\det(\\mathrm{J})$ vanishes, which happens on the fold set. It can be desingularized by rescaling time by a factor $\\det(\\mathrm{J})$, which brings the DRS in its most general form, namely\n\\begin{equation}\\label{eq:DRS2fast}\n\\begin{aligned}\n\\begin{pmatrix}\n\\dot{x} \\\\\n\\dot{y}\n\\end{pmatrix}\n&= \\mathrm{Adj(J)}\\begin{pmatrix}\na \\\\\n-g_z(x,z)\n\\end{pmatrix}\n(\\alpha z + \\gamma\\beta - \\delta x)\\\\\n\\dot{z} &= \\det(\\mathrm{J})(\\alpha z + \\gamma\\beta - \\delta x)\\\\\n\\dot{\\beta} &= \\det(\\mathrm{J})\\big(\\mu-\\gamma_y(y-y_{\\mathrm{fold}})^2-\\gamma_{\\beta}(\\beta-\n\\beta_{\\mathrm{fold}})^2\\big).\n\\end{aligned}\n\\end{equation}\nAfter being projected onto the $(x,\\beta)$-space, system~\\eqref{eq:DRS2fast} then takes the form\n\\begin{equation}\\label{eq:DRS2fast2}\n\\begin{aligned}\n\\dot{x} &=(a+g_z(x,z))(\\alpha z + \\gamma\\beta - \\delta x)\\\\\n\\dot{\\beta} &= (f'(x)-g_x(x,z))\\big(\\mu-\\gamma_y(y-y_{\\mathrm{fold}})^2-\\gamma_{\\beta}(\\beta-\n\\beta_{\\mathrm{fold}})^2\\big),\n\\end{aligned}\n\\end{equation}\nwhich indeed agrees with~\\eqref{eq:fnbursterDRS}.\n\nWith the above analysis, we can construct in principle any folded-node burster of our liking. We showcase three examples. First, a folded-node homoclinic burster is presented in Fig.~\\ref{fig:fnhom} and we also show in the bottom panels a comparison with data from~\\cite{roy84} (also displayed in Fig.~\\ref{fig:MMBO_exp}). Note that our \\mdsr{idealized} model was not initially designed to explain these data, yet the time profiles match remarkably well. The strong similarity between our \\mdsr{idealized} model and these data suggest that folded-node bursting constructions could potentially inform the design of biophysical models. Then, a folded-node Hopf burster is presented in Fig.~\\ref{fig:fnhopf}. Last, a folded-node fold-cycle burster is shown in Fig.~\\ref{fig:fnsnp}. In each of these three figures, we show in panel (a) the classical Rinzel dissection between the bifurcation diagram of the 2D fast subsystem of the underlying 3D fold-initiated bursting model, and several limit cycles of this 3D bursting model undergoing a spike-adding canard explosion. In panel (b), we show the bifurcation diagram of the 3D bursting system upon variation of the parameter which will become the second slow variable of the folded-node burster, and this diagram displays spike-adding canard explosion. Note that a full analysis of these spike-adding canard explosion scenarios is beyond the scope of the present work as it is by-and-large an open research question. In panels (c), we show a folded-node bursting cycle in a 3D phase-space projection together with the critical manifold $S^0$. In panel (d), we plot the time series of this cycle for the fast variable $x$. More examples of folded-node bursting scenarios can be constructed by following the procedure highlighted above and by choosing a different bifurcation of the fast subsystem ending the burst.\n\nFinally, we quickly reflect on why folded-saddle bursting is not robust. The folded-saddle case is simply a different parameter regime in the slow subsystem, however the resulting dynamics is substantially different than that generated by a folded node. In neuron models with (at least) two slow variables, folded saddles and their associated canard solutions play the role of firing threshold. In particular, in the context of bursting system, they have recently been shown to organise the spike-adding transition in parabolic bursters~\\cite{desroches16,desroches18}. Counterintuitively, small-amplitude oscillations can also emerge in the vicinity of a folded saddle; see~\\cite{mitry17} for a rigorous analysis of this phenomenon and also~\\cite{desroches16b,desroches18} for further related work. However, there is no funnel near a folded saddle and the canard dynamics is hence not robust, which applies no matter how many fast variables the system possesses, so in particular in the context of bursting. This is why, in systems with (at least) two fast and two slow variables, only the folded-node case gives rise to a new class of bursting oscillations.\n\n\\subsubsection{Existence of folded-node bursting solutions}\nMixed-mode oscillations have been the subject of intense research in the past few decades, in particular (more recently) in the context of multiple-timescale systems; see~\\cite{desroches12}. However, there are very few results guaranteeing the existence of MMOs. One example is due to Br{\\o}ns, Krupa and Wechselberger in~\\cite{brons06}, where they prove the existence of ``simple'' patterns of MMOs ---with one large-amplitude oscillations and $s$ small-amplitude oscillations, hence denoted a $1^s$ MMO--- using a perturbation argument (in $\\varepsilon$) from a singular orbit that they construct using both slow and fast limits of the original system. Their strategy can be adapted to prove the existence of a folded-node bursting periodic orbit by $\\varepsilon$-perturbation of a singular orbit. The segment of that singular orbit constructed using the slow flow is basically the same as in the MMO case, that is, it lies in the singular funnel or on the strong canard of the folded node. However, in the case of folded-node bursting the fast subsystem is of dimension 2 and displays limit cycles in the fast part of the cycle. Therefore one needs to concatenate the slow-flow segment with a segment along the \\textit{averaged slow flow} of the system; see e.g.~\\cite{roberts15} for details. The averaged slow flow is defined using the fast subsystem in its oscillatory regime and averaging out the fast variables $x$ and $y$ along one cycle to define a slow motion for the slow variables $z$ and $\\beta$; its equations are hence given by:\n\\begin{equation}\\label{eq:avslowflow}\n\\begin{aligned}\nx' &= (y - f(x) + az)\/c,\\\\\ny' &= G(x,y,z),\\\\\nz' &= \\alpha z + \\gamma\\beta - \\delta \\langle x \\rangle,\\\\\n\\beta' &= \\mu-\\frac{\\gamma_y}{T_z}\\int_0^{T(z)}(y(s)-y_{\\mathrm{fold}})^2 -\\gamma_{\\beta}(\\beta(s)-\\beta_{\\mathrm{fold}})^2 ds.\n\\end{aligned}\n\\end{equation}\n\nHence the singular orbit is formed by the following segments (see Figure~\\ref{fig:singorb}):\n\\begin{enumerate}\n\\item A critical fiber connecting the folded node to the landing-up point $\\mathrm{p_u}$;\n\\item A trajectory of the averaged slow flow ending along the line of bifurcation points of the fast subsystem ending the burst;\n\\item A fast fiber connecting that point to landing-down point $\\mathrm{p_d}$;\n\\item A segment of the slow flow connecting $\\mathrm{p_d}$ to the folded node.\n\\end{enumerate}\n\\begin{figure}[!t]\n\\centering\n\\includegraphics{SingPerorb.pdf}\n\\caption{Singular periodic orbit from which an MMBO perturbs for small enough $\\varepsilon>0$. Together with the attracting $S^0_a$ and repelling $S^0_r$ sheets of the critical manifold, the lower fold curve $\\mathcal{F}^-$, and the folded node fn, also shown on this figure are the average slow nullsurface $\\mathcal{A}$, the singular strong $\\gamma_s^0$ and weak $\\gamma_w^0$ canards, the landing-up point $\\mathrm{p_u}$, the landing-down point $\\mathrm{p_d}$ and the singular periodic orbit $\\Gamma_{\\mathrm{sing}}$.}\n\\label{fig:singorb}\n\\end{figure}\n\n\\subsection{Cyclic folded-node case}\n\\begin{figure}[!t]\n\\centering\n\\includegraphics{fnodecyc.pdf}\n\\caption{Cyclic folded-node bursting cases. We use a polar-coordinates formulation in order to construct idealized models. The top panels show the slow-fast dissection for the amplitude $r$ of the underlying bursting model, with three different torus canard scenarios (a), (b) and (c). Adding a slow dynamics on parameter $a$ yields associated cyclic folded-node bursting scenarios for which we show both the slow-fast dissection in the $(a,r)$ plane and the $x$ time series : (a) initiated by a subcritical Hopf bifurcation; (b) terminated by a fold of cycles; (c) initiated by a fold of cycles.}\n\\label{fig:fncy}\n\\end{figure}\n\\mdsr{In the same spirit as in Section~\\ref{sec:classicalFN}, one can construct interesting bursting rhythms where the slow oscillations occur on the envelope of the burst and this is due to what we will \\red{denote} \\textit{cyclic folded node}. Parallel to the construction of a folded-node burster system, one can construct a cyclic-folded-node burster system by considering a three-dimensional slow-fast system which possesses \\textit{torus canard} solutions. Loosely speaking, this corresponds to a canard phenomenon with a fast rotation. Already mentioned by Izhikevich in~\\cite{izhikevich01} in a canonical model, it \\red{was later} found in a biophysical model of Cerebellar Purkinje cell exhibiting fold\/fold cycle bursting~\\cite{kramer08}, and subsequently analysed with more mathematical details in, e.g.,~\\cite{benes11,burke12}. Even though to date not all elements of torus canard transitions have been \\red{mathematically} unravelled, one can summarise this phenomenon by \\red{emphasising that its key feature} corresponds to a canard explosion within a fast oscillatory motion. Instead of following slowly a family of equilibria past a fold bifurcation, the fast-oscillating system follows slowly a family of equilibria past a cyclic fold bifurcation. \\red{Moreover, one can draw} a parallel between classical canards and torus canards in their role of transitional regime in neuronal dynamics: classical canards can explain the rapid transition from rest to the spiking regime, likewise torus canards can explain the rapid transition from the spiking to the bursting regime. \\red{Furthermore}, torus canards are also not robust and only exist within exponentially thin parameter regions. \\red{Thus, the} very same idea that leads from canard point to folded singularities, can lead from torus canard to cyclic folded-node canards, when adding a second slow variable. \\red{In this way}, a cyclic folded-node can be robust even if the torus canards are not robust. This has been proposed very recently by Vo and collaborators~\\cite{vo17,vo16} \\red{via a specific example that links} the resulting dynamics to the amplitude-modulated bursting already mentioned in~\\cite{izhikevich01,kramer08}; see also~\\cite{han18} for other examples of amplitude-modulated bursting. \\red{In summary, we herein} propose a taxonomy of cyclic folded-node bursting patterns, with several numerical examples, which completes our extension of the previous bursting classifications. \\red{We complement this} with few examples of idealized models displaying cyclic-folded-node bursting. \\red{This is achieved by considering systems expressed} in polar form, in which case the condition for cyclic folded node and then for cyclic folded-node bursting reduce to folded-node conditions on $r$; see Figure~\\ref{fig:fncy}. In general, \\red{it is possible to reduce} the system locally near the cyclic fold bifurcation of the fast subsystem \\red{enabling the computation of} normal form coefficients (see~\\cite{roberts15,roberts17,vo16,vo17}) \\red{that effectively characterise} the cyclic folded-node. However, the bursting conditions have not been established in general. \\red{Finally, for sake of completeness, we construct a limiting case of a non-trivial system that displays both classical folded-node bursting and cyclic-folded-node bursting, as depicted in} Figure~\\ref{fig:fnfncy}.}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics{fnfncbursting.pdf}\n\\caption{An example of a folded-node and cyclic folded-node bursting case. The $x$-times series of the folded-node \/ cyclic folded-node bursting solution is shown; panel (b) is a zoom of panel (a) near the folded node.}\n\\label{fig:fnfncy}\n\\end{figure}\n\n\\section{Conclusion and perspectives}\n\\label{sec:conclusion}\n\\mdsr{The research topic of mathematical classification of bursting patterns was initiated with seminal papers published in the mid 1980s with three proposed classes of bursting oscillations~\\cite{rinzel85,rinzel86,rinzel87}.} The key idea of comparing the fast subsystem's bifurcation diagram and the full systems' dynamics may seem natural with \\red{hindsight, but in fact it was a genuine breakthrough, which shaped the way bursting oscillations have been modelled and dissected ever since}. The present manuscript follows \\mdsr{these} footsteps, as well as those of the subsequent contributors on this topic~\\cite{bertram95,golubitsky01,izhikevich00}\\mdsr{, hence it was important to review in detail \\red{these results since they form one pilar of Mathematical Neuroscience but also have impact in other fields. We then take a step forward by proposing an extension of the classification scheme, which allows to cover more types of burster systems, in particular those with two slow variables, namely folded-node bursters}. What we propose is} a conceptual framework which does not pretend to full mathematical rigour \\red{(i.e. theorem derivations since its beyond the focus of the paper)} but rather it aims to \\red{provide key insights that will enable} further theoretical and modelling work on the vast question of bursting. \\mdsr{The extended bursting classification crucially focuses on the dynamics \\red{during the} silent phase where the termination \\red{of the trajectory profile} is not just a simple rise over the fold of the critical manifold but can involve subthreshold oscillations. We emphasize the importance of the slow flow in slow-fast systems with (at least) two slow variables, which was somehow previously overlooked in the context of bursting. In such two-slow-variable bursting systems, the silent phase termination is due to the presence of folded node. This scenario is known to give rise to canard solutions that organise, upon parameter variation but also, transiently, upon change of initial conditions, the number of subthreshold oscillations. This slow cycle-adding phenomenon is indeed entirely due to canards \\red{and it} controls the profile of the underlying bursting oscillations. \\red{Importantly,} it does so in a robust manner in the sense that such bursting patterns with subthreshold oscillations exist over order one ranges of parameter values. Therefore, we have added to the state-of-the-art classifications of bursting patterns the cases of classical and cycle folded-node bursters, which we can summarise in~\\ref{tab:fnburst} below. We also propose a few more cases to the existing classifications (refer to Appendix~\\ref{sec:newfast}), which to the best of our knowledge, have not been reported before.}\n\\begin{table}[!h]\n\\begin{center}\n\\begin{tabular}{|l|c|c|c|}\\hline\n\\diaghead{\\theadfont Diag ColumnmnHead II}%\n {Initiation\\\\of the burst}{Termination\\\\of the burst}&\n\\thead{codim. 1 bif.\\\\of cycles}&\\thead{\\blue{cyclic \\textbf{fn}}}&\\thead{\\blue{classical \\textbf{fn}}}\\\\ \\hline\n\\blue{classical \\textbf{fn}} & \\blue{\\ding{51}},~\\cite{desroches13a} & \\blue{\\ding{51}} & \\cite{vbw}+1 slow var.\\\\ \\hline\ncodim. 1 bif. equilibria & \\cite{izhikevich00,rinzel85} & \\blue{\\ding{51}},~\\cite{vo16} & \\cite{vbw} \\\\ \\hline\ncodim. 1 bif. cycles & \\cite{izhikevich00,rinzel85} & \\blue{\\ding{51}} & \\cite{vbw} \\\\ \\hline\n\\blue{cyclic \\textbf{fn}} & \\blue{\\ding{51}} & \\texttt{modify Fig.~\\ref{fig:fncy} (b)} & \\blue{\\ding{51}} \\\\ \\hline\n\\end{tabular}\n\\vspace*{0.2cm}\n\\caption{Extended classification of bursting patterns.}\n\\label{tab:fnburst}\n\\end{center}\n\\end{table}\n\n\\mdsr{Where do we go from here? Following this initial framework for folded-node bursting, it will be important to develop this approach in the context of biophysical excitable cell models with more than one slow processes. To this extent, a very interesting question for follow-up work is to rethink \\red{about} folded-node bursting dynamics from a \\red{biophysical} modelling viewpoint. In all our \\mdsr{idealized} models of folded-node bursting, we have added feedback terms in the second slow differential equation with both positive and negative coefficients, which tends to indicate that both positive and negative feedback loops are useful to produce the desired output behaviour. \\red{In this context, we hight two interesting aspects associated with the experimental time-series that we attempted to model with our idealized model (folded-node bursting) reproduced in} Figs.~\\ref{fig:MMBO_exp} and~\\ref{fig:fnhom}. First, the subthreshold oscillations \\red{appear to be following} the excitability threshold, which may be harder to obtain in a three-dimensional model, even though some elliptic bursting models --e.g. FitzHugh-Rinzel, Morris-Lecar as well as some MMO models-- could potentially reproduce this aspect. However, \\red{these elliptic bursting models cannot} capture the second aspect. Note that our example of folded-node bursting has 3 time scales; this was done for convenience in the construction and may not be absolutely necessary. Second, the burst phase is located on a plateau (in terms of \\red{neuronal} membrane potential values) compared to the quiescent phase, which is reminiscent of a square-wave type bursting. Indeed our \\mdsr{idealized} folded-node bursting model \\red{reproduces quite well these data and in fact it can effectively be designated as a} folded-node homoclinic bursting model. Three-dimensional elliptic bursting models, or MMO models, would not be able to capture this aspect. \\red{One interesting possibility to find biophysical models with folded-node bursting dynamics is perhaps via existing models} of thalamic bursting, or \\red{alternatively to extend these models to explain the observational data published in}~\\cite{roy84}. In terms of application to neural dynamics, it is legitimate to ask about neural coding and the implications of folded-node dynamics within a bursting regime. There, one would want to compare spike-adding to folded-node cycle-adding. The cycle adding can quantize the slow phase duration which might have significant effect on silent phase (and therefore on active phase) durations. On the other hand, spike-adding has less impact on macroscopic timing and less impact if a spike is added to a burst of several, say, 6 or more, spikes. A single spike added in a 2-4 spike burst might have coding contributions (synaptic transmission) but less so if there are already more than 6 spikes in a burst. These questions go beyond the scope of the present paper but are certainly of direct interest for follow-up work. \\red{On the theoretical side, as aforementioned} we do not claim to have reached mathematical rigour but rather to have introduced a new framework for analysing bursting oscillations with two slow variables. There are clearly several open avenues for \\red{rigorous theorem-driven directions} to be explored. \\red{For instance,} we have \\red{elucidated} how to construct a singular orbit that will perturb to a folded-node bursting orbit, but, proving this perturbation result is not immediate. In general, proving rigorously the existence of canards in this 4D context and how both 3D parts (subthreshold and superthreshold) combine to organise the global dynamics require \\red{non-trivial} mathematical analysis.}\n\n\\mdsr{The question of noise is also a natural one to consider. If small to moderate noise is added to a folded-node bursting systems, \\red{it is likely that noise will not affect significantly the burst phase. However, it is expected that the phase} of spiking oscillations during the burst will be affected, but not the qualitative dynamics. Folded-node dynamics is known to be robust to noise, its time course is parametrically robust and noise-tolerant. The canard phenomenon accounts for subtle dynamic features like cycle-adding however the subthreshold oscillations near a folded node are robust. The noise will affect these subthreshold oscillations by modifying the rotation sector in which the trajectory falls into from one passage to the the next, however the oscillations will remain. To quantify this variability of the \\red{sector of a} folded-node burster with noise, one could use results by Berglund et al.~\\cite{berglund12}. However, here as well the qualitative dynamics and the key role of the slow subsystem and its folded node will remain. A rigorous understanding of the impact of noise on a folded-node burster model is certainly an interesting question that goes beyond the scope of the present work.}\n\n\\mdsr{Finally, the question of bursting dynamics with at least two slow variables and more than two timescales is also of interest and related to the present work. As \\red{aforementioned}, in the limit of folded-saddle-node singularities, small subthreshold oscillations will remain and increase in number and shape. In the context of slow-fast systems with two slow variables, this scenario is well-known to be akin to three-timescale dynamics~\\cite{krupa10}. The associated bifurcation structure is already involved in the three-dimensional setup, with involvement of adding organizing centers such as singular Hopf bifurcation points~\\cite{guckenheimer08}. \\red{Thus,} it is to be expected that the folded-saddle-node bursting profiles will be more rich and \\red{complex} to fully describe than the folded-node bursting cases presented \\red{herein}. Yet, the underlying robust mechanism that gives a bursting pattern and requires the analysis of both slow and fast subsystem will be similar as the one proposed in the present work. A full analysis of this limiting case is a very interesting and natural question for future work. Besides, bursting systems with more than two timescales have recently gained further interest in link with canard solutions~\\cite{desroches18,krupa12,letson17,nan15}, where the additional timescales bring more structure to the system and allow for further geometric singular perturbation analysis. Such approaches would certainly shed further light onto folded-node bursting dynamics as presented here and we regard it as a natural and interesting question for future work.} \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\n\nThe standard setting of \\emph{prediction with expert advice} \\citep{littlestone1994weighted,freund1997boosting,vovk1998mixability,cesabianchi2006plg} aims to provide sound strategies for sequential prediction that combine the forecasts from different sources.\nMore precisely, in the so-called \\emph{Hedge problem} \\citep{freund1997boosting}, at each round the learner has to output a probability distribution on a finite set of \\emph{experts} $\\{1, \\ldots, M\\}$; the losses of the experts are then revealed, and the learner incurs the expected loss from its chosen probability distribution. \nThe goal is then to control the \\emph{regret}, defined as the difference between the cumulative loss of the learner and that of the best expert (with smallest loss).\nThis online prediction problem is typically considered in the \\emph{individual sequences} framework, where the losses may be arbitrary and in fact set by an adversary that seeks to maximize the regret.\nThis leads to regret bounds that hold under virtually no assumption~\\citep{cesabianchi2006plg}.\n\nIn this setting, arguably the simplest and most standard strategy is the \\emph{Hedge algorithm} \\citep{freund1997boosting}, also called the \\emph{exponentially weighted averaged forecaster} \\citep{cesabianchi2006plg}.\nThis algorithm depends on a time-varying parameter $\\eta_t$ called the \\emph{learning rate}, which quantifies by how much the algorithm departs from its initial probability distribution to put more weight on the currently leading experts.\nGiven a known finite time horizon $T$, the standard tuning of the learning rate is fixed and given by $\\eta_t = \\eta \\propto \\sqrt{\\log(M) \/ T}$, which guarantees an optimal worst-case regret of order $O( \\sqrt{T \\log M} )$.\nAlternatively, when $T$ is unknown, one can set $\\eta_t \\propto \\sqrt{\\log (M) \/ t}$ at round $t$, which leads to an \\emph{anytime} $O(\\sqrt{T \\log M})$ regret bound valid for all $T \\geq 1$.\n\n\nWhile worst-case regret bounds are robust and always valid, they turn out to be overly pessimistic in some situations.\nA recent line of research \\citep{cesabianchi2007secondorder,derooij2014followtheleader,gaillard2014secondorder,koolen2014learning,sani2014exploiting,koolen2015squint,luo2015adanormalhedge} designs algorithms that combine $O(\\sqrt{T \\log M})$ worst-case regret guarantees with an improved regret on easier instances of the problem.\nAn interesting example of such an easier instance is the\nstochastic problem, where it is assumed that the losses are stochastic and that at each round the expected loss of a ``best'' expert is smaller than those of the other experts by some gap $\\Delta$.\nSuch algorithms rely either on a more careful, data-dependent tuning of the learning rate $\\eta_t$ \\citep{cesabianchi2007secondorder,derooij2014followtheleader,koolen2014learning,gaillard2014secondorder}, or on more sophisticated strategies \\citep{koolen2015squint,luo2015adanormalhedge}.\nAs shown by~\\citet{gaillard2014secondorder} (see also \\citealt{koolen2016combining}), one particular type of adaptive regret bounds (so-called \\emph{second-order bounds}) \nimplies at the same time a $O (\\sqrt{T\\log M})$ worst-case bound and a better \\emph{constant} $O(\\log (M) \/ \\Delta)$ bound in the stochastic problem with gap $\\Delta$.\nArguably starting with the early work on {second-order bounds} \\citep{cesabianchi2007secondorder}, the design of online learning algorithms that combine robust worst-case guarantees with improved performance on easier instances has been an active research goal in recent years \\citep{derooij2014followtheleader,gaillard2014secondorder,koolen2014learning,sani2014exploiting}.\nHowever, to the best of our knowledge, existing work on the Hedge problem has focused on developing new adaptive algorithms rather than on analyzing the behavior of ``conservative'' algorithms in favorable scenarios. \nOwing to the fact that the standard Hedge algorithm is designed for --- and analyzed in --- the adversarial setting \\citep{littlestone1994weighted,freund1997boosting,cesabianchi2006plg}, and that its parameters are not tuned adaptively to obtain better bounds in easier instances, it may be considered as overly conservative and not adapted to stochastic environments.\n\n\n\\paragraph{Our contribution.}\n\nThis paper fills a gap in the existing literature by providing an analysis of the standard Hedge algorithm in the stochastic setting.\nWe show that the anytime Hedge algorithm with default learning rate $\\eta_t \\propto \\sqrt{\\log (M) \/ t}$ actually \\emph{adapts} to the stochastic setting, in which it achieves an optimal \\emph{constant} $O(\\log (M) \/ \\Delta)$ regret bound \\emph{without any dedicated tuning} for the easier instance, which might be surprising at first sight.\nThis contrasts with previous works, which require the construction of new adaptive (and more involved) algorithms.\nRemarkably, this property is \\emph{not} shared by the variant of Hedge for a known fixed-horizon $T$ with constant learning rate $\\eta \\propto \\sqrt{\\log(M) \/ T}$, since it suffers a $\\Theta (\\sqrt{T \\log M})$ regret even in easier instances.\nThis exhibits a strong difference between the performances of the anytime and the fixed-horizon variants of the Hedge algorithm.\n\nGiven the aforementioned adaptivity of Decreasing Hedge, one may wonder whether there is in fact any benefit in using more sophisticated algorithms in the stochastic regime.\n We answer this question affirmatively, by considering a more refined measure of complexity of a stochastic instance than the gap $\\Delta$.\n Specifically, we show that Decreasing Hedge does not admit improved regret under Bernstein conditions, which are standard low-noise conditions from statistical learning \\citep{mammen1999margin,tsybakov2004aggregation,bartlett2006empirical}.\n By contrast, it was shown by \\citet{koolen2016combining} that algorithms which satisfy some adaptive adversarial regret bound achieve improved regret under Bernstein conditions.\n Finally, we characterize the behavior of Decreasing Hedge in the stochastic regime, by showing that its eventual regret on \\emph{any} stochastic instance is governed\n by the gap $\\Delta$.\n\n\\paragraph{Related work.}\n\nIn the bandit setting, where the feedback only consists of the loss of the selected action, there has also been some interest in ``best-of-both-worlds'' algorithms that combine optimal $O (\\sqrt{M T})$ worst-case regret in the adversarial regime with improved $O (M \\log T)$ regret (up to logarithmic factors) in the stochastic case \\citep{bubeck2012best2worlds,seldin2014practical,auer2016both}.\nIn particular, \\citet{seldin2014practical,seldin2017improved} showed that by augmenting the standard EXP3 algorithm for the adversarial regime (an analogue of Hedge with $\\Theta (1\/\\sqrt{t})$ learning rate)\nwith a special-purpose gap detection mechanism, one can achieve poly-logarithmic regret in the stochastic case.\nThis result is strengthened in some recent follow-up work \\citep{zimmert2019optimal,zimmert2019beating},\nthat appeared since the completion of the first version of the present paper,\nwhich obtains optimal regret in the stochastic and adversarial regimes through a variant of the Follow-The-Regularized-Leader (FTRL) algorithm with $\\Theta (1\/\\sqrt{t})$ learning rate and a proper regularizer choice.\nThis result can be seen as an analogue in the bandit case of our upper bound for Decreasing Hedge.\nNote that, in the bandit setting, the hardness of an instance is essentially characterized by the gap $\\Delta$ \\citep{bubeck2012regret}; in particular, the Bernstein condition, which depends on the correlations between the losses of the experts, cannot be exploited under bandit feedback, where one only observes one arm at each round.\nHence, it appears that the negative part of our results (on the limitations of Hedge) does not have an analogue in the bandit case.\n\nA similar adaptivity result for FTRL with decreasing $\\Theta (1\/\\sqrt{t})$ learning rate has been observed in a different context by \\citet{huang2017curved}. Specifically, it is shown that, in the case of online linear optimization on a Euclidean ball, FTRL with squared norm regularizer and learning rate $\\Theta (1\/\\sqrt{t})$ achieves $O (\\log T)$ regret when the loss vectors are i.i.d.\\@\\xspace\nThis result is an analogue of our upper bound for Hedge, since this algorithm corresponds to FTRL on the simplex with entropic regularizer \\citep{cesabianchi2006plg,hazan2016online}.\nOn the other hand, the simplex lacks the curvature of the Euclidean ball, which is important to achieve small regret; here, the improved regret is ensured by a condition on the distribution, namely the existence of a gap $\\Delta$.\nOur lower bound for Hedge shows that this condition is necessary, thereby characterizing the long-term regret of FTRL on the simplex with entropic regularizer.\nIn the case of the Euclidean ball with squared norm regularizer, the norm of the expected loss vector appears to play a similar role, as shown by the upper bound from \\citet{huang2017curved}.\n\n\\paragraph{Outline.}\n\nWe define the setting of prediction with expert advice and the Hedge algorithm in Section~\\ref{sec:expert-problem-hedge}, and we recall herein its standard worst-case regret bound.\nIn Section~\\ref{sec:hedge-easy}, we consider the behavior of the Hedge algorithm on easier instances, namely the stochastic setting with a gap $\\Delta$ on the best expert. \nUnder an i.i.d assumption on the sequence of losses, we provide in Theorem~\\ref{thm:hedge-stochastic} an upper bound on the regret of order $(\\log M) \/ \\Delta$ for Decreasing Hedge. \nIn Proposition~\\ref{prop:lowerbound-gap}, we prove that the rate $(\\log M) \/ \\Delta$ cannot be improved in this setting.\nIn Theorem~\\ref{thm:adv-gap} and Corollary~\\ref{cor:hedge-martingale}, we extend the regret guarantees to the adversarial with a gap setting, where a leading expert linearly outperforms the others.\nThese results stand for any Hedge algorithm which is worst-case optimal and with any learning rate which is larger than the one of Decreasing Hedge, namely $O(\\sqrt{\\log M\/ t})$.\nIn Proposition~\\ref{prop:lower-bound-hedge-cst}, we prove the sub-optimality of the fixed-horizon Hedge algorithm, and of another version of Hedge based on the so-called ``doubling trick''.\nIn Section~\\ref{sec:advant-second-order}, we discuss the advantages of adaptive Hedge algorithms, and explain what the limitations of Decreasing Hedge are compared to such versions.\nWe include numerical illustrations of our theoretical findings in Section~\\ref{sec:experiments}, conclude in Section~\\ref{sec:conclusion} and provide the proofs in Section~\\ref{sec:proofs}.\n\n\n\\section{The expert problem and the Hedge algorithm}\n\\label{sec:expert-problem-hedge}\n\n\nIn the Hedge setting, also called \\emph{decision-theoretic online learning} \\citep{freund1997boosting}, the learner and its adversary (the Environment) sequentially compete on the following game: at each round $t\\geq 1$,\n\\begin{enumerate}\n\\item the Learner chooses a probability vector $\\bm v_t = (v_{i,t})_{1\\leq i \\leq M}$ on the $M$ experts $1, \\dots, M$;\n\\item the Environment picks a bounded loss vector $\\bm \\ell_t = (\\ell_{i,t})_{1\\leq i \\leq M} \n\\in [0, 1]^M$, where $\\ell_{i, t}$ is the loss of expert $i$ at round $t$, while the Learner suffers loss $\\widehat \\ell_t = \\bm v_t^\\top \\bm \\ell_t$.\n\\end{enumerate}\nThe goal of the Learner is to control its \\emph{regret}\n\\begin{equation}\n \\label{eq:regret}\n R_T = \\sum_{t=1}^T \\widehat \\ell_t - \\min_{1\\leq i \\leq M} \\sum_{t=1}^T \\ell_{i,t}\n\\end{equation}\nfor every $T \\geq 1$, irrespective of the sequence of loss vectors $\\bm \\ell_1, \\bm \\ell_2, \\dots$ chosen by the Environment.\nOne of the most standard algorithms for this setting is the \\emph{Hedge} algorithm.\nThe Hedge algorithm, also called the exponentially weighted averaged forecaster, uses the vector of probabilities $\\bm v_t = (v_{i,t})_{1\\leq i \\leq M}$ given by\n\\begin{equation}\n \\label{eq:hedge-algorithm}\n v_{i, t} = \\frac{e^{-\\eta_t L_{i,t-1}}}{\\sum_{j=1}^M e^{-\\eta_t L_{j,t-1}}}\n\\end{equation}\nat each $t\\geq 1$, where $L_{i,T} = \\sum_{t=1}^T \\ell_{i,t}$ denotes the cumulative loss of \nexpert~$i$ for every $T\\geq 1$.\nLet us also denote $\\widehat L_T := \\sum_{t=1}^T \\widehat \\ell_t$ and $R_{i,T} = \\widehat L_T - L_{i,T}$ the regret with respect to expert $i$.\nWe consider in this paper the following variants of Hedge, where $c_0 > 0$ is a constant.\n\n\\medskip\\noindent\n\\textbf{Decreasing Hedge}~\\citep{auer2002adaptive}.\nThis is Hedge with the sequence of learning rates $\\eta_t = c_0 \\sqrt{\\log(M) \/ t}$.\n\n\\medskip\\noindent\n\\textbf{Constant Hedge}~\\citep{littlestone1994weighted}.\nGiven a finite time horizon $T \\geq 1$, this is Hedge with constant learning rate $\\eta_t = c_0 \\sqrt{\\log(M) \/ T}$.\n\n\\medskip\\noindent\n\\textbf{Hedge with doubling trick}~\\citep{cesabianchi1997doublingtrick, cesabianchi2006plg}.\nThis variant of Hedge uses a constant learning rate on geometrically increasing intervals, restarting the algorithm at the beginning of each interval. Namely, it uses\n\\begin{equation}\n \\label{eq:hedge-doubling-trick}\n v_{i,t} = \\frac{\\exp ( - \\eta_t \\sum_{s = T_k}^{t-1} \\ell_{i,s})}{\\sum_{j=1}^M \\exp ( - \\eta_t \\sum_{s = T_k}^{t-1} \\ell_{j,s})},\n\\end{equation}\nwith $T_l = 2^l$ for $l \\geq 0$, $k \\in \\mathbf N$ such that $T_k \\leq t < T_{k+1}$ and\n$\\eta_t = c_0 \\sqrt{\\log (M) \/ T_k}$.\n\n\\medskip\nLet us recall the following standard regret bound for the Hedge algorithm from \\citet{chernov2010prediction}.\n\\begin{proposition}\n \\label{prop:hedge-adversarial}\n Let $\\eta_1, \\eta_2, \\dots$ be a decreasing sequence of learning rates.\n The Hedge algorithm \\eqref{eq:hedge-algorithm} satisfies the following regret bound:\n \\begin{equation}\n \\label{eq:hedge-adversarial}\n R_T \\leq \\frac{1}{\\eta_T} \\log M + \\frac{1}{8} \\sum_{t=1}^T \\eta_t\n \\, .\n \\end{equation}\n In particular, the choice $\\eta_t = 2 \\sqrt{{\\log (M)}\/{t}}$ yields a regret bound of $\\sqrt{T \\log M}$ for every $T \\geq 1$.\n\\end{proposition}\n\nNote that the regret bound stated in Equation~\\eqref{eq:hedge-adversarial} holds for every sequence of losses $\\bm \\ell_1, \\bm \\ell_2, \\dots$, which makes it valid under no assumption (aside from the boundedness of the losses).\nThe worst-case regret bound in $O(\\sqrt{T \\log M})$ is achieved by Decreasing Hedge, Hedge with doubling trick and Constant Hedge (whenever $T$ is known in advance).\nThe $O(\\sqrt{T \\log M})$ rate cannot be improved either by Hedge or any other algorithm: it is known to be the minimax optimal regret \\citep{cesabianchi2006plg}.\nContrary to Constant Hedge, Decreasing Hedge is anytime, in the sense that it achieves the $O(\\sqrt{T \\log M})$ regret bound simultaneously for each $T \\geq 1$.\nWe note that this worst-case regret analysis fails to exhibit any difference between these three algorithms.\n\n\nIn many cases, this $\\sqrt{T}$ regret bound is pessimistic, and more ``aggressive'' strategies (such as the follow-the-leader algorithm, which plays at each round the uniform distribution on the experts with smallest loss, \\citealp{cesabianchi2006plg}) may achieve constant regret in easier instances, even though they lack regret guarantees in the adversarial regime.\nWe show in Section~\\ref{sec:hedge-easy} below that Decreasing Hedge is actually\nbetter than both Constant Hedge and Hedge with doubling trick in some easier instance of the problem (including in the stochastic setting).\nThis entails that Decreasing Hedge is actually able to adapt, without any modification, to the easiness of the problem considered.\n\n\n\\section{Regret of Hedge variants on easy instances}\n\\label{sec:hedge-easy}\n\n\nIn this section, we depart from the worst-case regret analysis and study the regret of the considered variants of the Hedge algorithm on easier instances of the prediction with expert advice problem.\n\n\\subsection{Optimal regret for Decreasing Hedge in the stochastic regime}\n\\label{sub:optimal-stoch}\n\nWe examine the behavior of Decreasing Hedge in the stochastic regime, where the losses are the realization of some (unknown) stochastic process.\nMore precisely, we consider the standard i.i.d.\\@\\xspace case, where the loss vectors $\\bm \\ell_1, \\bm \\ell_2, \\dots$ are i.i.d.\\@\\xspace (independence holds over rounds, but not necessarily across experts).\nIn this setting, the regret can be much smaller than the worst-case $\\sqrt{T \\log M}$ regret, since the best expert (with smallest expected loss) will dominate the rest after some time.\nFollowing \\citet{gaillard2014secondorder,luo2015adanormalhedge}, the easiness parameter we consider in this case, which governs the time needed for the best expert to have the smallest cumulative loss and hence the incurred regret, is the sub-optimality gap $\\Delta = \\min_{i \\neq i^*} \\mathbb E [\\ell_{i,t} - \\ell_{i^*,t} ]$, where $i^* = \\mathop{\\mathrm{argmin}}_{i} \\mathbb E [ \\ell_{i,t} ]$.\n\nWe show below that, despite the fact that Decreasing Hedge is designed for the worst-case setting described in Section~\\ref{sec:expert-problem-hedge}, it is able to {adapt} to the easier problem considered here, \nIndeed, Theorem~\\ref{thm:hedge-stochastic} shows that Decreasing Hedge achieves a \\emph{constant}, and in fact \\emph{optimal} (by Proposition~\\ref{prop:lowerbound-gap} below) regret bound in this setting, in spite of its ``conservative'' learning rate.\n\nWith the exception of the high-probability bound of Corollary~\\ref{cor:hedge-martingale}, the upper and lower bounds in the stochastic case are stated for the \\emph{pseudo-regret} $\\mathcal{R}_T = \\mathbb E [R_{i^*,T}]$ (similar bounds hold for the the expected regret $\\mathbb E [R_T]$, since $\\mathcal{R}_T \\leq \\mathbb E [R_T]$ and by Remark~\\ref{rem:pseudoregret} in Section~\\ref{sec:proof-theorem-1}).\n\n\\begin{theorem}\n \\label{thm:hedge-stochastic}\n Let $M \\geq 3$.\n Assume that the loss vectors $\\bm \\ell_1, \\bm \\ell_2, \\dots$ are i.i.d.\\@\\xspace random variables, where $\\bm \\ell_t = (\\ell_{i,t})_{1\\leq i \\leq M}$.\n Also, assume that there exists $i^* \\in \\{1, \\dots, M\\}$ and $\\Delta > 0$ such that\n \\begin{equation}\n \\label{eq:gap-condition}\n \\mathbb E [ \\ell_{i,t} - \\ell_{i^*,t} ] \\geq \\Delta\n \\end{equation}\n for every $i \\neq i^*$.\n Then, the Decreasing Hedge algorithm with learning rate $\\eta_t = 2 \\sqrt{(\\log M)\/t}$\n achieves the following pseudo-regret bound\\textup: for every $T \\geq 1$\\textup,\n \\begin{equation}\n \\label{eq:regret-stochastic-exp}\n \\mathcal{R}_T \\leq \\frac{4 \\log M + 25}{\\Delta}\n \\, .\n \\end{equation}\n\\end{theorem}\n\n\nThe proof of Theorem~\\ref{thm:hedge-stochastic} is given in Section~\\ref{sec:proof-theorem-1}.\nTheorem~\\ref{thm:hedge-stochastic} proves that, in the stochastic setting with a gap $\\Delta$, the Decreasing Hedge algorithm achieves a regret $O(\\log (M) \/ \\Delta)$, without any prior knowledge of $\\Delta$.\nThis matches the guarantees of adaptive Hedge algorithms which are explicitly designed to adapt to easier instances \\citep{gaillard2014secondorder,luo2015adanormalhedge}.\nThis result may seem surprising at first: indeed, adaptive exponential weights algorithms\nthat combine optimal regret in the adversarial setting and constant regret in\neasier scenarios, such as Hedge with a second-order tuning \\citep{cesabianchi2007secondorder} or AdaHedge \\citep{derooij2014followtheleader}, typically use a data-dependent learning rate $\\eta_t$ that adapts to the properties of the losses.\nWhile the learning rate $\\eta_t$ chosen by these algorithms may be as low as the worst-case tuning $\\eta_t \\propto \\sqrt{\\log (M) \/ t}$, in the stochastic case those algorithms will use larger, lower-bounded learning rates to ensure constant regret.\nAs Theorem~\\ref{thm:hedge-stochastic} above shows, it turns out that the data-independent, ``safe'' learning rates $\\eta_t \\propto \\sqrt{\\log (M) \/ t}$ used by ``vanilla'' Decreasing Hedge are still large enough to adapt to the stochastic case.\n\n\\paragraph{Idea of the proof.} \n\nThe idea of the proof of Theorem~\\ref{thm:hedge-stochastic} is to divide time in two phases: a short initial phase $\\iint 1{t_1}$,\nwhere $t_1 = O (\\frac{\\log M}{\\Delta^2})$,\nand a second phase $\\iint{t_1}{T}$.\nThe initial phase is dominated by noise, and regret during this period is bounded\nthrough the worst-case regret bound of Proposition~\\ref{prop:hedge-adversarial}, which gives a regret of $O(\\sqrt{t_1 \\log M}) = O (\\frac{\\log M}{\\Delta})$.\nIn the second phase, the best expert dominates the rest, and the weights concentrate on this best expert fast enough that the total regret incurred is small.\nThe control of the regret in the second phase relies on the critical fact that, if $\\eta_t$ is at least as large as $\\sqrt{(\\log M)\/t}$, then the following two things occur simultaneously at $t_1 \\asymp \\frac{\\log M}{\\Delta^2}$, namely at the beginning of the late phase:\n\\begin{enumerate}\n\\item with high probability, the best expert $i^*$ dominates all the others linearly: for every $i \\neq i^*$ and $t \\geq t_1$, $L_{i,t} - L_{i^*,t} \\geq \\frac{\\Delta t}{2}$;\n\\item the total weight of all suboptimal experts is controlled: $\\sum_{i \\neq i^*} v_{i,t_1} \\leq \\frac{1}{2}$. If $\\eta_t \\geq \\sqrt{(\\log M)\/t}$ and the first condition holds, this amounts to $M \\exp (- \\frac{\\Delta}{2} \\sqrt{t \\log M}) \\leq \\frac{1}{2}$, namely $t_1 \\gtrsim \\frac{\\log M}{\\Delta^2}$.\n\\end{enumerate}\nIn other words, the learning rate $\\eta_t \\asymp \\sqrt{(\\log M)\/t}$ ensures that the total weight of suboptimal experts starts vanishing at about the same time as when the best expert starts to dominate the others with a large probability (and remarkably, this property holds for every value of the sub-optimality gap $\\Delta$).\nFinally, the upper bound on the regret in the second phase rests on the two conditions above, together with the bound $\\sum_{t \\geq 1} e^{-c \\sqrt{t}} = O (\\frac{1}{c^2})$ for $c > 0$.\n\n\\begin{remark}\n The fact that $\\sum_{t \\geq 1} e^{-c \\sqrt{t}} = O (1 \/ c^2)$ is also used in the analysis of the EXP3++ bandit algorithm~\\citep[Lemma 10]{seldin2014practical}.\n In the expert setting considered here, summing the contribution of all experts (which suffices in the bandit setting to obtain the correct order of regret) would yield a significantly suboptimal $O(M \/ \\Delta)$ regret bound, with a linear dependence on the number of experts $M$.\n In our case, the decomposition of the regret in two phases, which is explained above, removes the linear dependence on $M$ and allows to obtain the optimal rate $(\\log M) \/ \\Delta$.\n\\end{remark}\n\nWe complement Theorem~\\ref{thm:hedge-stochastic} by showing that the $O((\\log M) \/ \\Delta)$ regret under the gap condition cannot be improved, in the sense that its dependence on both $M$ and $\\Delta$ is optimal.\n\n\\begin{proposition}\n \\label{prop:lowerbound-gap}\n Let $\\Delta \\in (0, \\frac{1}{4})$, $M \\geq 4$ and $T \\geq (\\log M) \/ (16 \\Delta^2)$.\n Then, for any algorithm for the Hedge setting, there exists an i.i.d.\\@\\xspace distribution over the sequence of losses $(\\bm \\ell_{t})_{t \\geq 1}$ such that\\textup:\n \\begin{itemize}\n \\item there exists $i^* \\in \\{ 1, \\dots, M \\}$ such that, for any $i \\neq i^*$, $\\mathbb E [\\ell_{i,t} - \\ell_{i^*, t}] \\geq \\Delta$\\textup;\n \\item the pseudo-regret of the algorithm satisfies\\textup:\n \\begin{equation}\n \\label{eq:lowerbound-gap}\n \\mathcal{R}_T\n \\geq \\frac{\\log M}{256 \\Delta}\n \\, .\n \\end{equation}\n \\end{itemize}\n\\end{proposition}\n\nThe proof of Proposition~\\ref{prop:lowerbound-gap} is given in Section~\\ref{sec:proof-lowerbound-gap}.\nProposition~\\ref{prop:lowerbound-gap} generalizes the well-known minimax lower bound of $\\Theta (\\sqrt{T \\log M})$, which is recovered by taking $\\Delta \\asymp \\sqrt{(\\log M)\/T}$.\n\n\\subsection{Small regret for Decreasing Hedge in the adversarial with a gap problem}\n\\label{sub:adv-gap}\n\nIn this section, we extend the regret guarantee of Decreasing Hedge in the stochastic setting (Theorem~\\ref{thm:hedge-stochastic}), by showing that it holds for more general algorithms and under more general assumptions.\nSpecifically, we consider an ``adversarial with a gap'' regime, similar to the one introduced by \\citet{seldin2014practical} in the bandit case, where the leading expert linearly outperforms the others after some time.\nAs Theorem~\\ref{thm:adv-gap} shows, essentially the same regret guarantee can be obtained in this case, up to an additional $\\log (\\Delta^{-1}) \/ \\Delta$ term.\nTheorem~\\ref{thm:adv-gap} also applies to any Hedge algorithm whose (possibly data-dependent) learning rate $\\eta_t$ is at least as large as that of Decreasing Hedge, and which satisfies a $O(\\sqrt{T \\log M})$ worst-case regret bound;\nthis includes algorithms with \\emph{anytime} first and second-order tuning of the learning rate \\citep{auer2002adaptive,cesabianchi2007secondorder,derooij2014followtheleader}.\nIn what follows, we will assume $M \\geq 3$ for convenience; similar results holds for $M=2$.\n\\begin{theorem}\n \\label{thm:adv-gap}\n Let $M \\geq 3$.\n Assume that there exists $\\tau_0 \\geq 1$\\textup, $\\Delta \\in (0, 1)$ and $i^* \\in \\{1, \\dots, M \\}$ \n such that\\textup, for every $t \\geq \\tau_0$ and $i \\neq i^*$\\textup, one has\n \\begin{equation}\n \\label{eq:condition-gap-adv}\n L_{i, t} - L_{i^*, t} \\geq \\Delta t.\n \\end{equation}\n Consider any Hedge algorithm with \\textup(possibly data-dependent\\textup) learning rate $\\eta_t$ such that \n \\begin{itemize}\n \\item $\\eta_t \\geq c_0 \\sqrt{(\\log M) \/ t}$ for some constant $c_0 > 0$\\textup;\n \\item it admits the following worst-case regret bound: $R_T \\leq c_1 \\sqrt{T \\log M}$ for every $T \\geq 1$\\textup,\n for some $c_1 > 0$.\n \\end{itemize}\n Then, for every $T\\geq 1$, the regret of this algorithm is upper bounded as\n \\begin{equation}\n \\label{eq:regret-gap-adv}\n R_T \n \\leq c_1 \\sqrt{\\tau_0 \\log M} + \\frac{c_2 \\log M + c_3 \\log {\\Delta}^{-1} + c_4}{\\Delta} \n \\end{equation}\n where $c_2 = c_1 + \\frac{\\sqrt{8}}{c_0}$, $c_3 = \\frac{\\sqrt{8}}{c_0}$ and $c_4 = \\frac{16}{c_0^2}$.\n\\end{theorem}\n\nThe idea of the proof of Theorem~\\ref{thm:adv-gap} is the same as that of Theorem~\\ref{thm:hedge-stochastic}, the only difference being the slightly longer initial phase to account for the adversarial nature of the losses.\nAs a consequence of the general bound of Theorem~\\ref{thm:adv-gap}, we can recover the guarantee of Theorem~\\ref{thm:hedge-stochastic} (up to an additional $\\log (\\Delta^{-1}) \/\\Delta$ term), both in expectation and with high probability, under more general stochastic assumptions than i.i.d.\\@\\xspace over time.\nThe proofs of Theorem~\\ref{thm:adv-gap} and Corollary~\\ref{cor:hedge-martingale} are provided in Section~\\ref{sec:proof-thm-adv-gap}.\n\n\\begin{corollary}\n \\label{cor:hedge-martingale}\n Assume that the losses $(\\ell_{i,t})_{1\\leq i \\leq M, t\\geq 1}$ are random variables. \n Also, denoting $\\mathcal{F}_t = \\sigma \\big( (\\ell_{i,s})_{1\\leq i \\leq M, 1\\leq s \\leq t} \\big)$, assume that there exists $i^*$ and $\\Delta >0 $ such that\n \\begin{equation}\n \\label{eq:gap-condition-martingale}\n \\mathbb E \\left[ \\ell_{i,t} - \\ell_{i^*,t} \\,|\\, \\mathcal{F}_{t-1} \\right] \\geq \\Delta\n \\end{equation}\n for every $i\\neq i^*$ and every $t\\geq 1$.\n Then, for any Hedge algorithm satisfying the conditions of Theorem~\\ref{thm:adv-gap}, and every $T \\geq 1$\\textup:\n \\begin{equation}\n \\label{eq:regret-martingale-exp}\n \n \\mathcal{R}_T\n \\leq (5 c_1 + 2 c_2) \\frac{\\log M}{\\Delta} + 2 c_3 \\frac{\\log \\Delta^{-1}}{\\Delta} \n + \\frac{2 c_4}{\\Delta},\n \\end{equation}\n with $c_1, c_2, c_3, c_4$ as in Theorem~\\ref{thm:adv-gap}.\n In addition, for every $\\varepsilon \\in (0, 1)$, we have\n \\begin{equation}\n \\label{eq:regret-martingale-prob}\n R_T\n \\leq \\left( c_1 \\sqrt{8} + 2 c_2 \\right) \\frac{\\log M}{\\Delta} + c_1 \\frac{\\sqrt{8\\log M \\log \\varepsilon^{-1}}}{\\Delta} + 2 c_3 \\frac{\\log \\Delta^{-1}}{\\Delta} + \\frac{2 c_4}{\\Delta}\n \\end{equation}\n with probability at least $1 - \\varepsilon$. \n\\end{corollary}\n\n\n\\subsection{Constant Hedge and Hedge with the doubling trick do not adapt to the stochastic case}\n\\label{sec:negative-results}\n\nNow, we show that the adaptivity of Decreasing Hedge to gaps in the losses, established in Sections~\\ref{sub:optimal-stoch} and~\\ref{sub:adv-gap}, is not shared by the two closely related Constant Hedge and Hedge with the doubling trick, despite the fact that they both achieve the minimax optimal worst-case $O (\\sqrt{T \\log M})$ regret.\nProposition~\\ref{prop:lower-bound-hedge-cst} below shows that both algorithms fail to achieve a constant regret, and in fact to improve over their worst-case $\\Theta (\\sqrt{T \\log M})$ regret guarantee, even in the extreme case of experts with constant losses $0$ (for the leader), and $1$ for the rest (\\ie, $\\Delta = 1$).\n\n\\begin{proposition}\n \\label{prop:lower-bound-hedge-cst}\n Let $T \\geq 1$\\textup, $M \\geq 2$\\textup, and consider the experts $i=1, \\dots, M$ with losses $\\ell_{1, t} = 0$\\textup, \n $\\ell_{i, t} = 1$ $(1 \\leq t \\leq T, 2 \\leq i \\leq M)$.\n Then\\textup, the pseudo-regret of Constant Hedge with learning rate $\\eta_t = c_0 \\sqrt{\\log (M)\/T}$ \n \\textup(where $c_0 > 0$ is a numerical constant\\textup) is lower bounded as follows\\textup:\n \\begin{equation}\n \\label{eq:lower-bound-hedge-cst}\n \\mathcal{R}_T\n \\geq \\min \\Big( \\frac{\\sqrt{T \\log M}}{3 c_0}, \\frac{T}{3} \\Big)\n \\, .\n \\end{equation}\n In addition, Hedge with doubling trick~\\eqref{eq:hedge-doubling-trick} also suffers a pseudo-regret satisfying\n \\begin{equation}\n \\label{eq:lower-bound-hedge-doubling}\n \\mathcal{R}_T \\geq\n \\min \\Big ( \\frac{\\sqrt{T \\log M}}{6 c_0} , \\frac{T}{12} \\Big)\n \\, .\n \\end{equation}\n\\end{proposition}\n\n\nThe proof of Proposition~\\ref{prop:lower-bound-hedge-cst} is given in Section~\\ref{sec:proof-lower-bound-hedge-cst}.\nAlthough Hedge with a doubling trick is typically considered as overly conservative and only suitable for worst-case scenarios \\citealp{cesabianchi2006plg} (especially due to its periodic restarts, after which it discards past observations), to the best of our knowledge Proposition~\\ref{prop:lower-bound-hedge-cst} (together with Theorem~\\ref{thm:hedge-stochastic}) is the first to formally demonstrate the advantage of Decreasing Hedge over the doubling trick version.\nThis implies that Decreasing Hedge should not be seen as merely a substitute for Constant Hedge to achieve anytime regret bounds.\nIndeed, even when the horizon $T$ is fixed, Decreasing Hedge outperforms Constant Hedge in the stochastic setting.\n\n\\section{Limitations of Decreasing Hedge in the stochastic case}\n\\label{sec:advant-second-order}\n\nIn this section, we explore the limitations of the simple Decreasing Hedge algorithm in the stochastic regime, and exhibit situations where it performs worse than more sophisticated algorithms.\nThe starting observation is that the sub-optimality gap $\\Delta$ is a rather brittle measure of ``hardness'' of a stochastic instance, which does not fully reflect the achievable rates.\nWe therefore consider the following fast-rate condition from statistical learning, which refines the sub-optimality gap as a measure of complexity of a stochastic instance.\n\n\\begin{definition}[Bernstein condition]\n \\label{def:bernstein-condition}\n Assume that the losses $\\bm \\ell_1, \\bm \\ell_2, \\dots$ are the realization of a stochastic process.\n Denote $\\mathcal{F}_{t} = \\sigma (\\bm \\ell_1, \\dots, \\bm \\ell_t)$ the $\\sigma$-algebra generated by $\\bm \\ell_1, \\dots, \\bm \\ell_t$.\n For $\\beta \\in [0,1]$ and $B >0$, the losses are said to satisfy the \\emph{$(\\beta,B)$-Bernstein condition} if there exists $i^*$ such that, for every $t \\geq 1$ and $i \\neq i^*$,\n \\begin{equation}\n \\label{eq:bernstein-condition}\n \\mathbb E [(\\ell_{i,t} - \\ell_{i^*,t})^2 \\,|\\, \\mathcal{F}_{t-1}]\n \\leq B \\mathbb E [ \\ell_{i,t} - \\ell_{i^*,t} \\,|\\, \\mathcal{F}_{t-1} ]^\\beta\n \\, . \n \\end{equation}\n\\end{definition}\n\nThe Bernstein condition \\citep{bartlett2006empirical}, a generalization of the Tsybakov margin condition \\citep{tsybakov2004aggregation,mammen1999margin}, is a geometric property on the losses which enables to obtain fast rates (e.g., faster than $O(1\/\\sqrt{n})$ for parametric classes) in statistical learning; we refer to \\citet{vanerven2015fastrates} for a discussion of fast rates conditions.\n The Bernstein condition~\\eqref{eq:bernstein-condition}\n quantifies the \n ``easiness'' of a stochastic instance, and generalizes the gap condition considered in the previous section (see Example~\\ref{ex:bernstein-gap} below).\nRoughly speaking, it states that good experts (with near-optimal expected loss) are highly correlated with the best expert.\nIn the examples below, we assume that the loss vectors $\\bm \\ell_1, \\bm \\ell_2, \\dots$ are i.i.d.\\@\\xspace\n\n\\begin{example}[Gap implies Bernstein]\n \\label{ex:bernstein-gap}\n If $\\Delta_i = \\mathbb E [\\ell_{i,t} - \\ell_{i^*,t}] \\geq \\Delta$ for $i \\neq i^*$, then the $(1, \\frac{1}{\\Delta})$-Bernstein condition holds \\citep[Lemma~4]{koolen2016combining}.\n Furthermore, letting $\\alpha = \\mathbb E [\\ell_{i^*,t}]$ denote the expected loss of the best expert, the $(1, 1 + \\frac{2 \\alpha}{\\Delta})$-Bernstein condition holds.\n Indeed, for any $i \\neq i^*$,\n denoting $\\mu_i := \\mathbb E [\\ell_{i,t}] = \\alpha + \\Delta_i$, we have (since $(u-v)^2 \\leq \\max(u^2,v^2) \\leq u^2+v^2 \\leq u+v$ for $u,v \\in [0, 1]$):\n \\begin{align*}\n \\mathbb E \\big[ (\\ell_{i,t} - \\ell_{i^*,t})^2 \\big]\n &\\leq \\mathbb E \\left[ \\ell_{i,t} + \\ell_{i^*,t} \\right]\n = \\frac{\\mu_i + \\alpha}{\\mu_i - \\alpha} \\mathbb E \\left[ \\ell_{i,t} - \\ell_{i^*,t} \\right]\n = \\Big( 1 + \\frac{2 \\alpha}{\\Delta_i} \\Big) \\, \\mathbb E \\left[ \\ell_{i,t} - \\ell_{i^*,t} \\right]\n \\, ,\n \\end{align*}\n which establishes the claim\n since $\\Delta_i \\geq \\Delta$.\n This provides an improvement when $\\alpha$ is small.\n\\end{example}\n\n\\begin{example}[Bernstein without a gap]\n \\label{ex:bernstein-without-gap}\n Let $P$ be a distribution on $\\mathcal{X} \\times \\{ 0, 1\\}$, where $\\mathcal{X}$ is some measurable space. Assume that $(X_1, Y_1), (X_2, Y_2) \\dots$ are i.i.d.\\@\\xspace samples from $P$, and that the experts $i \\in \\{ 1, \\dots, M\\}$ correspond to classifiers $f_i : \\mathcal{X} \\to \\{ 0, 1 \\}$: $\\ell_{i, t} = \\bm 1 ( f_i (X_t) \\neq Y_t ) $, and that expert $i^*$ is the Bayes classifier: $f_{i^*} (X) = \\bm 1 ( \\eta (X) \\geq 1\/2 ) $, where $\\eta(X) = \\P (Y=1 \\,|\\, X)$.\n Tsybakov's low noise condition \\citep{tsybakov2004aggregation}, namely $\\P ( | 2 \\eta (X) - 1 | \\leq t ) \\leq C t^{\\kappa}$ for some $C > 0$, $\\kappa \\geq 0$ and every $t > 0$, implies the $(\\frac{\\kappa}{\\kappa + 1}, B)$-Bernstein condition for some $B$ (see, e.g., \\citealp{boucheron2005survey}).\n In addition, under the Massart condition \\citep{massart2006risk} that\n $| \\eta (X) - 1\/2 | \\geq c > 0$, the $(1, 1\/(2c))$-Bernstein condition holds.\n Note that these conditions may hold even with an arbitrarily small sub-optimality gap $\\Delta$, since the $f_i$, $i \\neq i^*$, may be arbitrary.\n\\end{example}\n\nTheorem~\\ref{thm:hedge-no-bernstein} below shows that Decreasing Hedge fails to achieve improved rates under Bernstein conditions.\n\n\\begin{theorem}\n \\label{thm:hedge-no-bernstein}\n For every $T \\geq 1$, there exists a $(1, 1)$-Bernstein stochastic instance on which the pseudo-regret of the Decreasing Hedge algorithm with $\\eta_t = c_0 \\sqrt{(\\log M) \/ t}$ satisfies\n $\\mathcal{R}_{T} \\geq \\frac{1}{3} \\min( \\frac{1}{c_0} \\sqrt{T \\log M}, {T})$.\n\\end{theorem}\n\nThe proof of Theorem~\\ref{thm:hedge-no-bernstein} is given in Section~\\ref{sec:proof-lowerbound-no-bernstein}.\nBy contrast, it was shown by \\citet{koolen2016combining}\n(and implicitly used by \\citealp{gaillard2014secondorder}) that\nsome adaptive algorithms with data-dependent regret bounds enjoy improved regret under the Bernstein condition.\nFor the sake of completeness, we state this fact in Proposition~\\ref{prop:second-order-bernstein} below, which corresponds to~\\citet[Theorem~2]{koolen2016combining}, but where the dependence on $B$ is made explicit. We also only provide a bound in expectation, which considerably simplifies the proof.\nThe proof of Proposition~\\ref{prop:second-order-bernstein}, which uses the same ideas as \\citet[Theorem~11]{gaillard2014secondorder}, is provided in Section~\\ref{sec:proof-second-order-bernstein}.\n\n\\begin{proposition}\n \\label{prop:second-order-bernstein}\n Consider an algorithm for the Hedge problem which satisfies the following regret bound: for \n every $i\\in \\{ 1, \\dots, M\\}$\\textup,\n \\begin{equation}\n \\label{eq:second-order-regret}\n R_{i,T} \\leq C_1 \\sqrt{(\\log M) \\sum_{t=1}^T (\\widehat \\ell_t - \\ell_{i,t})^2} + C_2 \\log M\n \\end{equation}\n where\n $C_1, C_2 >0$ are constants.\n Assume that the losses satisfy the $(\\beta, B)$-Bernstein condition.\n Then, the pseudo-regret of the algorithm satisfies\\textup:\n \\begin{equation}\n \\label{eq:regret-bernstein}\n \\mathcal{R}_T\n \\leq C_3 (B \\log M)^{\\frac{1}{2-\\beta}} T^{\\frac{1-\\beta}{2-\\beta}} + C_4 \\log M \n \\end{equation}\n where $C_3 = \\max (1, 4C_1^2)$ and $C_4 = 2 C_2$.\n\\end{proposition}\n\nThe data-dependent regret bound~\\eqref{eq:second-order-regret}, a ``second-order'' bound, is satisfied by adaptive algorithms such as Adapt-ML-Prod \\citep{gaillard2014secondorder} and Squint \\citep{koolen2015squint}.\nA slightly different variant of second-order regret bounds, which depends on some cumulative variance of the losses across experts, has been considered by \\citet{cesabianchi2007secondorder,derooij2014followtheleader}, and is achieved by Hedge algorithms with a data-dependent tuning of the learning rate. Second-order bounds refine so-called \\emph{first-order} bounds \\citep{cesabianchi1997doublingtrick,auer2002adaptive,cesabianchi2006plg}, which are adversarial regret bounds that scale as $O(\\sqrt{L_T^* \\log M} + \\log M)$, where $L_T^*$ denotes the cumulative loss of the best expert.\n While first-order bounds may still scale as the worst-case $O(\\sqrt{T \\log M})$ rate in a typical stochastic instance (where the best expert has a positive expected loss), second-order algorithms are known to achieve constant $O((\\log M) \/ \\Delta)$ regret in the stochastic case with gap $\\Delta$ \\citep{gaillard2014secondorder,koolen2015squint}.\n\nTheorem~\\ref{thm:hedge-no-bernstein}, in light of Proposition~\\ref{prop:second-order-bernstein}, clarifies where the advantage of second-order algorithms compared to Decreasing Hedge lies: unlike the latter, they can exploit Bernstein conditions on the losses.\nThe contrast is most apparent for Bernstein instances with $\\beta = 1$. \nBy Example~\\ref{ex:bernstein-gap}, the existence of a gap $\\Delta$ implies that the $(1,B)$-Bernstein condition holds with $B \\leq \\frac{1}{\\Delta}$.\nHowever, as shown by Example~\\ref{ex:bernstein-without-gap}, $B$ can in fact be much smaller than $\\Delta$, in which case the regret bound~\\eqref{eq:regret-bernstein} satisfied by second-order algorithms, namely $O (B \\log M)$, significantly improves over the upper bound of $O( (\\log M)\/\\Delta)$ of Decreasing Hedge from Theorem~\\ref{thm:hedge-stochastic}.\nTheorem~\\ref{thm:hedge-no-bernstein} provides an instance where the difference does occur, in the most pronounced case where $B =1$, so that second-order algorithms enjoy small $O (\\log M)$ regret, while Decreasing Hedge suffers $\\Theta (\\sqrt{T \\log M})$ regret.\n\n\\begin{remark}\n The advantage of larger learning rates on some stochastic instances may be understood intuitively as follows.\n Consider an instance with $B$ small but small gap $\\Delta$.\n The learning rate of Decreasing Hedge is large enough that it can rule out bad experts (with large enough gap $\\Delta_i$) at the optimal rate (\\ie, at time $(\\log M)\/\\Delta_i^2$).\n However, once these bad experts are ruled out, near-optimal experts (with small gap $\\Delta_i$) are ruled out late (after $(\\log M)\/\\Delta_i^2$ rounds).\n On the other hand, the Bernstein assumption entails that those experts are highly correlated with the best expert, the amount of noise on the relative losses of these near-optimal experts is small, so that a larger learning rate could be safely used and would enable to dismiss near-optimal experts sooner.\n\\end{remark}\n\nSetting the Bernstein condition aside, we conclude by investigating the intrinsic limitations of Decreasing Hedge in the stochastic setting.\nIndeed, it is natural to ask whether Decreasing Hedge can exploit some other regularity of a stochastic instance, apart from the gap $\\Delta$.\nTheorem~\\ref{thm:hedge-characterize-gap} shows that this is in fact not the case.\n\n\\begin{theorem}\n \\label{thm:hedge-characterize-gap} \n For every i.i.d.\\@\\xspace \\textup(over time\\textup) stochastic instance with a unique best expert \n \\begin{equation*}\n i^* = \\mathop{\\mathrm{argmin}}_{1 \\leq i \\leq M} \\mathbb E [\\ell_{i,t}],\n \\end{equation*}\n the pseudo-regret of Decreasing Hedge \\textup(with $c_0 \\geq 1$\\textup) satisfies\n \\begin{equation*}\n \\mathcal{R}_T \\geq \\frac{1}{450 c_0^4 (\\log M)^2 \\Delta}\n \\end{equation*}\n for $T \\geq \\frac{1}{4 \\Delta^2}$\\textup, where $\\Delta := \\inf_{i \\neq i^*} \\mathbb E [\\ell_{i,t} - \\ell_{i^*,t}]$.\n\\end{theorem}\n\nTheorem~\\ref{thm:hedge-characterize-gap} shows (together with the upper bound of Theorem~\\ref{thm:hedge-stochastic}) that the eventual regret of Decreasing Hedge on \\emph{any} stochastic instance is determined by the sub-optimality gap $\\Delta$, and scales (up to a $\\log^3 M$ factor, depending on the number of near-optimal experts) as $\\Theta (\\frac{1}{\\Delta})$.\nThis characterizes the behavior of Decreasing Hedge on any stochastic instance.\n\n\n\\section{Experiments}\n\\label{sec:experiments}\n\n\nIn this section, we illustrate our theoretical results by numerical experiments that compare the behavior of various Hedge algorithms in the stochastic regime.\n\n\\paragraph{Algorithms.} We consider the following algorithms: \\texttt{hedge} is Decreasing Hedge with the default learning rates $\\eta_t = 2\\sqrt{\\log (M) \/ t}$, \\texttt{hedge\\_constant} is Constant Hedge with constant learning rate $\\eta_t = \\sqrt{8 \\log (M) \/ T}$, \\texttt{hedge\\_doubling} is Hedge with doubling trick with $c_0 = \\sqrt{8}$, \\texttt{adahedge} is the AdaHedge algorithm from \\citet{derooij2014followtheleader}, which is a variant of the Hedge algorithm with a data-dependent tuning of the learning rate $\\eta_t$ (based on $\\bm \\ell_1, \\dots, \\bm \\ell_{t-1}$).\nAs shown in the note \\citet{blog}, AdaHedge also benefits from Bernstein conditions.\nA related algorithm, namely Hedge with second-order tuning of the learning rate \\citep{cesabianchi2007secondorder}, performed similarly to AdaHedge on the examples considered below, and was therefore not included. \\texttt{FTL} is Follow-the-Leader \\citep{cesabianchi2006plg} which puts all mass on the expert with the smallest loss (breaking ties randomly).\nWhile FTL serves as a benchmark in the stochastic setting, unlike the other algorithms it lacks any guarantee in the adversarial regime, where its worst-case regret is \\emph{linear} in $T$.\n\n\\paragraph{Results.}\n\nWe report in Figure~\\ref{fig:experiments} the cumulative regrets of the considered algorithms in four examples.\nThe results for the stochastic instances (a), (b) and (c) described below are averaged over $50$ trials.\n\n\n\\medskip\n\\noindent\n\\emph{\\textup(a\\textup) Stochastic instance with a gap.}\nThis is the standard instance considered in this paper.\nThe losses are drawn independently from Bernoulli distributions (one of parameter $0.3$, $2$ of parameter $0.4$ and $7$ of parameter $0.5$, so that $M=10$ and $\\Delta = 0.1$).\nThe results of Figure~\\ref{fig:1a} confirm our theoretical results: Decreasing Hedge achieves a small, constant regret which is close to that of AdaHedge and FTL, while Constant Hedge and Hedge with doubling trick suffer a larger regret of order $\\sqrt{T}$ (note that, although the expected regret of Constant Hedge converges in this case, the value of this limit depends on its learning rate and hence on $T$).\n\n\\medskip\n\\noindent\n\\emph{\\textup(b\\textup) ``Hard'' stochastic instance.}\nThis example has a zero gap $\\Delta = 0$ between the two leading experts and $M=10$, which makes it ``hard'' from the standpoint of Theorem~\\ref{thm:hedge-stochastic} (which no longer applies in this limit case).\nThe losses are drawn from independent Bernoulli distributions, of parameters $0.5$ for the $2$ leading experts, and $0.7$ for the $8$ remaining ones.\nAlthough all algorithms suffer an unavoidable $\\Theta (\\sqrt{T})$ regret due to pure noise, Decreasing Hedge, AdaHedge and FTL achieve better regret than the two conservative Hedge variants (Figure~\\ref{fig:1b}).\nThis is due to the fact that for the former algorithms, the weights of suboptimal experts decrease quickly and only induce a constant regret.\n\n\\medskip\n\\noindent\n\\emph{\\textup(c\\textup) Small loss for the best expert.}\nIn this experiment, we illustrate one advantage of adaptive Hedge algorithms such as AdaHedge over Decreasing Hedge, namely the fact that they admit improved regret bounds when the leading expert has small loss. We considered in this experiment $M = 10$, $\\Delta = 0.04$ and the leading expert is $\\mathsf{Beta}(0.04,0.96)$, then $4$ $\\mathsf{Beta}(0.08,0.92)$, then $5$ $\\mathsf{Beta}(0.5, 0.5)$.\n\n\\medskip\n\\noindent\n\\emph{\\textup(d\\textup) Adversarial with a gap instance.}\nThis simple instance is not random, and satisfies the assumptions of Theorem~\\ref{thm:adv-gap}.\nIt is defined by $M=3$, $\\Delta = 0.04$, $\\ell_{3, t} = \\frac{3}{4}$ for $t \\geq 1$, $(\\ell_{1, t}, \\ell_{2,t}) = (\\frac{1}{2}, 0)$ if $t=1$, $(0, 1)$ if $t \\geq 80$ or if $t$ is even, and $(1, 0)$ otherwise.\nFTL suffers linear regret in the first phase, while Constant Hedge and Hedge with doubling trick suffer $\\Theta (\\sqrt{T})$ during the second phase.\n\n\\begin{figure}\n\\centering\n\\begin{subfigure}[b]{.48\\linewidth}\n \\centering\n \\includegraphics[width=\\linewidth]{stoch_gap_2_legends.pdf}\n \\caption{\n \n }\n \\label{fig:1a}\n\\end{subfigure}%\n\\begin{subfigure}[b]{.48\\linewidth}\n \\centering\n \\includegraphics[width=\\linewidth] {stoch_hard.pdf}\n \\caption{\n \n }\\label{fig:1b}\n\\end{subfigure} \\\\ %\n\\begin{subfigure}[b]{.48\\linewidth}\n \\centering\n \\includegraphics[width=\\linewidth] {small_loss_1000.pdf}\n \\caption{\n \n }\\label{fig:1c}\n\\end{subfigure}%\n\\begin{subfigure}[b]{.48\\linewidth}\n \\includegraphics[width=\\linewidth] {adv_gap_simple.pdf} \n \\caption{\n \n }\\label{fig:1d}\n\\end{subfigure}\n\\caption{Cumulative regret of Hedge algorithms on four examples, see text for a precise description and discussion about the results. (a) Stochastic instance with a gap; (b) ``Hard'' stochastic instance; (c) Small loss for the best expert; (d) Adversarial with a gap instance.}\n\\label{fig:experiments}\n\\end{figure}\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\n\nIn this article, we carried the regret analysis of the standard exponential weights (Hedge) algorithm in the stochastic expert setting, closing a gap in the existing literature.\nOur analysis reveals that, despite being tuned for the worst-case adversarial setting and lacking any adaptive tuning of the learning rate, Decreasing Hedge achieves optimal regret in the stochastic setting.\nThis property also enables one to distinguish it qualitatively from other variants including the one with fixed (horizon-dependent) learning rate or the one with doubling trick, which both fail to adapt to gaps in the losses.\nTo the best of our knowledge, this is the first result that shows the superiority of the decreasing learning rate over the doubling trick.\nIn addition, it suggests that, even for a fixed time horizon $T$, the decreasing learning rate tuning should be favored over the constant one.\n\nFinally, we showed that the regret of Decreasing Hedge on any stochastic instance is essentially characterized by the sub-optimality gap $\\Delta$.\nThis shows that adaptive algorithms, including algorithms achieving second-order regret bounds, can actually outperform Decreasing Hedge on some stochastic instances that exhibit a more refined form of ``easiness''.\n\n\n\\paragraph{A link with stochastic optimization.}\n\\label{par:connection_with_stochastic_optimization}\n\nOur results have a similar flavor to a well-known result \\citep{moulines2011nonasymptotic} about stochastic optimization: stochastic gradient descent (SGD) with learning rate $\\eta_t \\propto 1 \/ \\sqrt{t}$ (which is tuned for the convex case but not for the non-strongly convex case) and Polyak-Ruppert averaging achieves a fast $O(1 \/ (\\mu t))$ excess risk rate for $\\mu$-strongly convex problems, without the knowledge of $\\mu$.\nHowever, this link stops here since the two results are of a significantly different nature: the $O(1 \/ (\\mu t))$ rate is satisfied only by SGD with Polyak-Ruppert averaging, and it does not come from a regret bound;\neven in the $\\mu$-strongly convex case, it can be seen that SGD with step-size $\\eta_t \\propto 1 \/ \\sqrt{t}$ suffers a $\\Theta (\\sqrt{t})$ regret.\nIn fact, the opposite phenomenon occurs: in stochastic optimization, SGD uses a \\emph{larger} $\\Theta(1\/\\sqrt{t})$ step-size than the $\\Theta(1\/(\\mu t))$ step size which exploits the knowledge of strong convexity, but the effect of this larger step-size is balanced by the averaging.\nBy contrast, in the expert setting, Hedge uses a \\emph{smaller} $\\Theta(\\sqrt{(\\log M)\/t})$ learning rate than the constant, large enough learning rate which exploits the knowledge of the stochastic nature of the problem.\n\n\n\\paragraph{Acknowledgments.}\n\nThe authors wish to warmly thank all four Anonymous Reviewers for their helpful feedback and insights on this work.\nIn particular, the proof of Proposition~\\ref{prop:lowerbound-gap} was proposed by an Anonymous Referee, which allowed to shorten our initial proof.\n\n\\section{Proofs}\n\\label{sec:proofs}\n\nWe now provide the proofs of the results from the previous Sections, by order of appearance in the text.\n\n\n\\subsection{Proof of Theorem~\\ref{thm:hedge-stochastic}}\n\\label{sec:proof-theorem-1}\n\nLet $t_0 = \\left\\lceil \\frac{8\\log M}{\\Delta^2} \\right\\rceil$, so that $\\sqrt{t_0} \\leq \\sqrt{1 + \\frac{8 \\log M}{\\Delta^2}} \\leq 1 + \\frac{\\sqrt{8 \\log M}}{\\Delta}$ (since $\\sqrt{a+b} \\leq \\sqrt{a} + \\sqrt{b}$ for $a,b \\geq 0$).\nThe worst-case regret bound of Hedge (Proposition~\\ref{prop:hedge-adversarial}) shows that for $1 \\leq T \\leq t_0$:\n\\begin{equation}\n R_{i^*,T}\n \\leq \\sqrt{T \\log M}\n \\leq \\sqrt{t_0 \\log M}\n \\leq \\sqrt{\\log M} + \\frac{2 \\sqrt{2} \\log M}{\\Delta}\n \\leq \\frac{4 \\log M}{\\Delta}\n \\label{eq:hedge-stochastic-proof-1}\n\\end{equation}\n(since $\\log M \\geq 1$ as $M \\geq 3$, $\\Delta \\leq 1$ and $2 \\sqrt{2} \\leq 3$),\nwhich establishes~\\eqref{eq:regret-stochastic-exp} for $T \\leq t_0$.\nIn order to prove~\\eqref{eq:regret-stochastic-exp} for $T \\geq t_0 +1$, we start by decomposing the regret with respect to $i^*$ as\n\\begin{equation}\n\\label{eq:hedge-stochastic-proof-2}\nR_{i^*,T}\n= \\widehat L_{T} - L_{i^*, T}\n= \\widehat L_{t_0} - L_{i^*, t_0} + \\sum_{t=t_0+1}^T (\\widehat \\ell_t - \\ell_{i^*, t})\n\\, .\n\\end{equation}\nSince $\\widehat L_{t_0} - L_{i^*, t_0} \\leq R_{t_0}$ is controlled by~\\eqref{eq:hedge-stochastic-proof-1}, it remains to upper bound the second term in~\\eqref{eq:hedge-stochastic-proof-2}.\nFirst, for every $t \\geq t_0 +1$,\n\\begin{equation}\n \\label{eq:instant-regret}\n \\widehat \\ell_t - \\ell_{i^*,t}\n = \\sum_{i \\neq i^*} v_{i,t} (\\ell_{i,t} - \\ell_{i^*,t})\n \\, .\n\\end{equation}\nSince $\\bm \\ell_t$ is independent of $\\bm v_t$ (which is $\\sigma (\\bm \\ell_1, \\dots, \\bm \\ell_{t-1})$-measurable),\ntaking the expectation in~\\eqref{eq:instant-regret} yields, denoting $\\Delta_i = \\mathbb E [\\ell_{i,t} - \\ell_{i^*,t}]$,\n\\begin{equation}\n \\label{eq:instant-expected-regret}\n \\mathbb E [\\widehat \\ell_t - \\ell_{i^*,t}]\n = \\sum_{i \\neq i^*} \\Delta_i \\mathbb E [v_{i,t}]\n \\, .\n\\end{equation}\nFirst, for every $i \\neq i^*$,\napplying Hoeffding's inequality to the i.i.d.\\@\\xspace centered variables $Z_{i,t} := - \\ell_{i,t} + \\ell_{i^*,t} + \\Delta_i$, which belong to $[-1 + \\Delta_i, 1 + \\Delta_i]$, yields\n\\begin{align}\n \\label{eq:stochastic-proof-hoeffding}\n \\P \\left( L_{i,t-1} - L_{i^*,t-1} < \\frac{\\Delta_i (t-1)}{2} \\right)\n &= \\P \\left( \\sum_{s=1}^{t-1} Z_{i,s} > \\frac{\\Delta_i (t-1)}{2} \\right) \\nonumber \\\\\n &\\leq e^{ - \\frac{t-1}{2} (\\Delta_i\/2)^2} \\nonumber \\\\\n &= e^{ - {(t-1) \\Delta_i^2}\/{8}}\n \\, .\n\\end{align}\nOn the other hand, if $L_{i,t-1} - L_{i^*,t-1} \\geq \\Delta_i (t-1) \/2$, then \n\\begin{align}\n \\label{eq:hedge-stochastic-proof-3}\n v_{i,t}\n &= \\frac{e^{- \\eta_t (L_{i,t-1} - L_{i^*,t-1})}}{1 + \\sum_{j \\neq i^*} e^{- \\eta_t (L_{j,t-1} - L_{i^*,t-1})}} \\nonumber \\\\\n &\\leq e^{- 2 \\sqrt{(\\log M) \/ t} \\times \\Delta_i (t-1) \/ 2} \\nonumber \\\\\n &\\leq e^{- \\Delta_i \\sqrt{(t-1) (\\log M)\/2}}\n\\end{align}\nsince $t \\leq 2(t-1)$.\nIt follows from~\\eqref{eq:hedge-stochastic-proof-3} and~\\eqref{eq:stochastic-proof-hoeffding} that, for $t \\geq t_0+1 \\geq 2$,\n\\begin{align}\n \\label{eq:hedge-stochastic-two-terms}\n \\mathbb E [v_{i,t}]\n &\\leq \\P \\left( L_{i,t-1} - L_{i^*,t-1} > \\frac{\\Delta_i (t-1)}{2} \\right) + e^{- \\Delta_i \\sqrt{(t-1)(\\log M)\/2}} \\nonumber \\\\\n &\\leq e^{- (t-1) \\Delta_i^2 \/ 8} + e^{- \\Delta_i \\sqrt{(t-1)(\\log M)\/2}}\n \\, .\n\\end{align}\nNow, a simple analysis of functions shows that the functions $f_1 (u) = u e^{-u}$ and $f_2 (u) = u e^{-u^2\/2}$ are decreasing on $[1, + \\infty)$.\nSince $\\Delta_i \\geq \\Delta$, this entails that\n\\begin{equation}\n \\label{eq:hedge-stochastic-proof-5}\n \\Delta_i e^{- (t-1) \\Delta_i^2 \/ 8}\n = \\frac{2}{\\sqrt{t-1}} f_2 \\left( \\frac{\\sqrt{t-1} \\Delta_i}{2} \\right)\n \\leq \\frac{2}{\\sqrt{t-1}} f_2 \\left( \\frac{\\sqrt{t-1} \\Delta}{2} \\right)\n = \\Delta e^{- (t-1) \\Delta^2 \/ 8}\n\\end{equation}\nprovided that $\\frac{\\sqrt{t-1} \\Delta}{2} \\geq 1$, \\ie $t \\geq 1 + \\frac{4}{\\Delta^2}$, which is the case since $t \\geq t_0 +1 \\geq 1 + \\frac{8 \\log M}{\\Delta^2}$.\nLikewise,\n\\begin{equation}\n \\label{eq:hedge-stochastic-proof-6}\n \\Delta_i e^{- \\Delta_i \\sqrt{(t-1) (\\log M) \/2}}\n \\leq \\Delta e^{- \\Delta \\sqrt{(t-1) (\\log M) \/2}}\n\\end{equation}\nif $\\Delta \\sqrt{(t-1) (\\log M) \/2} \\geq 1$, \\ie $t \\geq 1+ \\frac{2}{(\\log M) \\Delta^2}$, which is ensured by $t \\geq t_0 +1$.\nIt follows from~\\eqref{eq:instant-expected-regret}, \\eqref{eq:hedge-stochastic-two-terms}, \\eqref{eq:hedge-stochastic-proof-5} and~\\eqref{eq:hedge-stochastic-proof-6} that for every $t \\geq t_0 + 1$:\n\\begin{align}\n \\label{eq:hedge-stochastic-proof-simult}\n \\mathbb E [\\widehat \\ell_t - \\ell_{i^*,t}]\n &\\leq M \\Delta e^{- (t-1) \\Delta^2 \/ 8} + M \\Delta e^{- \\Delta \\sqrt{(t-1) (\\log M) \/2}} \\nonumber \\\\\n &= \\big( M e^{- t_0 \\Delta^2 \/ 8} \\big) \\big( \\Delta e^{- (t-t_0- 1) \\Delta^2 \/ 8} \\big) + \\big( M e^{- \\Delta \\sqrt{(t-1) (\\log M) \/8}} \\big) \\big( \\Delta e^{- \\Delta \\sqrt{(t-1) (\\log M) \/8}} \\big) \\nonumber \\\\\n &\\leq \\Delta e^{- (t-t_0- 1) \\Delta^2 \/ 8} + \\Delta e^{- \\Delta \\sqrt{(t-1)\/8}}\n\\end{align}\nwhere inequality~\\eqref{eq:hedge-stochastic-proof-simult} comes from the bound $M e^{-t_0 \\Delta^2 \/8} \\leq 1$ (since $t_0 \\geq \\frac{8 \\log M}{\\Delta^2}$) and from the fact that $M e^{- \\Delta \\sqrt{(t-1) (\\log M) \/8}} \\leq 1$ amounts to $t \\geq 1 + \\frac{8 \\log M}{\\Delta^2}$, that is, to $t \\geq t_0 +1$.\nSumming inequality~\\eqref{eq:hedge-stochastic-proof-simult} yields, for every $T \\geq t_0+1$,\n\\begin{align} \n \\mathbb E [ \\sum_{t = t_0 +1}^T (\\ell_{t} - \\ell_{i^*,t}) ]\n &\\leq \\sum_{t=t_0+1}^T \\left\\{ \\Delta e^{- (t-t_0- 1) \\Delta^2 \/ 8} + \\Delta e^{- \\Delta \\sqrt{(t-1)\/8}} \\right\\} \\nonumber \\\\\n &\\leq \\Delta \\sum_{t \\geq 0} e^{-t \\Delta^2\/8} + \\Delta \\sum_{t \\geq 1} e^{- (\\Delta \/ \\sqrt{8}) \\sqrt{t}} \\nonumber \\\\\n &\\leq \\Delta \\left( 1 + \\frac{8}{\\Delta^2} \\right) + \\Delta \\times \\frac{2}{(\\Delta \/ \\sqrt{8})^2} \\label{eq:hedge-stochastic-proof-7} \\\\\n \n \\leq \\frac{25}{\\Delta}\n\\end{align}\nwhere inequality~\\eqref{eq:hedge-stochastic-proof-7} comes from Lemma~\\ref{lem:tail} below.\nFinally, combining inequalities~\\eqref{eq:hedge-stochastic-proof-1} and~\\eqref{eq:hedge-stochastic-proof-7} yields the pseudo-regret bound $\\mathcal{R}_T \\leq \\frac{4 \\log M + 25}{\\Delta}$.\n\n\\begin{lemma}\n \\label{lem:tail}\n For every $\\alpha > 0$,\n \\begin{align}\n \\label{eq:tail-bound}\n \\sum_{t \\geq 1} e^{-\\alpha t} &\\leq \\frac{1}{\\alpha} \\\\\n \\label{eq:tail-bound-sqrt}\n \\sum_{t \\geq 1} e^{- \\alpha \\sqrt{t}} &\\leq \\frac{2}{\\alpha^2} \\, .\n \\end{align}\n\\end{lemma}\n\n\\begin{proof}\n Since the functions $t \\mapsto e^{-\\alpha {t}}$ and $t \\mapsto e^{-\\alpha \\sqrt{t}}$ are decreasing on $\\mathbf R^+$, we have\n \\begin{align*}\n &\\sum_{t \\geq 1} e^{-\\alpha t}\n \\leq \\int_0^{\\infty} e^{-\\alpha t} \\mathrm{d} t\n = \\frac{1}{\\alpha}\n \\, \n \\\\\n &\\sum_{t \\geq 1} e^{-\\alpha \\sqrt{t}}\n \\leq \\int_{0}^{+\\infty} e^{-\\alpha \\sqrt{t}} \\mathrm{d} t\n \\underset{u = \\alpha \\sqrt t}{=} \\frac{2}{\\alpha^2}\n \\int_{0}^{+\\infty} u e^{-u} \\mathrm{d} u\n = \\frac{2}{\\alpha^2}.\n \\end{align*}\n\\end{proof}\n\n\\begin{remark}\n \\label{rem:pseudoregret}\n While the upper bound of Theorem~\\ref{thm:hedge-stochastic} is stated for the pseudo-regret $\\mathcal{R}_T$, a similar upper bound holds for the expected regret $\\mathbb E [R_T]$.\n Indeed, under the assumptions of Theorem~\\ref{thm:hedge-stochastic}, for every $T \\geq \\frac{4 \\log M}{\\Delta^2}$, we have\n $\\mathbb E [R_T] \\leq \\mathcal{R}_T + \\frac{1.1}{\\Delta}$.\n\\end{remark}\n\n\n\\begin{proof}\n Note that $\\mathbb E [R_T] - \\mathcal{R}_T = \\mathbb E [L_{i^*,T} - \\min_{1\\leq i \\leq T} L_{i,T}]$.\n For every $a \\geq 0$, Hoeffding's inequality (applied to the i.i.d.\\@\\xspace centered variables $\\ell_{i^*,t} - \\ell_{i,t} + \\Delta_i \\in [-1 + \\Delta_i, 1 + \\Delta_i]$, $1\\leq t \\leq T$) entails \n \\begin{align}\n \\label{eq:proof-pseudo-regret-1}\n \\P \\left( L_{i^*,T} - \\min_{1\\leq i \\leq T} L_{i,T} \\geq a \\right)\n &\\leq \\sum_{i \\neq i^*} \\P \\left( L_{i^*,T} - L_{i,T} + \\Delta_i T \\geq \\Delta_i T + a \\right) \\nonumber \\\\\n &\\leq \\sum_{i \\neq i^*} e^{- (\\Delta_i T + a)^2 \/ (2T)} \\\\\n &\\leq M e^{- T \\Delta^2\/2} e^{-a^2\/(2T)} \\nonumber \\\\\n &\\leq e^{-T \\Delta^2\/4} e^{-a^2\/(2T)}\n \\label{eq:proof-pseudo-regret-2}\n \\, ,\n \\end{align}\n where inequality~\\eqref{eq:proof-pseudo-regret-2} comes from the fact that $M e^{-T \\Delta^2\/4} \\leq 1$ since $T \\geq \\frac{4 \\log M}{\\Delta^2}$.\n Since the random variable $L_{i^*,T} - \\min_{1\\leq i \\leq T} L_{i,T}$ is nonnegative, this implies that\n \\begin{align} \n \\mathbb E \\left[ L_{i^*,T} - \\min_{1\\leq i \\leq T} L_{i,T} \\right]\n &= \\int_0^{\\infty} \\P \\left( L_{i^*,T} - \\min_{1\\leq i \\leq T} L_{i,T} \\geq a \\right) \\mathrm{d} a \\nonumber \\\\\n &\\leq e^{-T \\Delta^2\/4} \\int_0^{\\infty} e^{-a^2\/(2T)} \\mathrm{d} a \\nonumber \\\\\n &= \\sqrt{\\frac{\\pi}{2}} \\cdot \\sqrt{T} e^{-T \\Delta^2\/4} \\nonumber \\\\\n &= \\frac{\\sqrt{\\pi}}{\\Delta} \\big[ \\Delta \\sqrt{T\/2} \\cdot e^{- (\\Delta \\sqrt{T\/2})^2\/2} \\big] \\nonumber \\\\\n &\\leq \\frac{\\sqrt{\\pi\/e}}{\\Delta}\n \\label{eq:proof-pseudo-regret-3}\n \\end{align}\n where inequality~\\eqref{eq:proof-pseudo-regret-3} comes from the fact that the function $u \\mapsto u e^{-u^2\/2}$ attains its maximum on $\\mathbf R^+$ at $u=1$.\n This concludes the proof, since $\\sqrt{\\pi\/e}\\leq 1.1$.\n\\end{proof}\n\n\n\\subsection{Proof of Proposition~\\ref{prop:lowerbound-gap}}\n\\label{sec:proof-lowerbound-gap}\n\nFix $M$, $\\Delta$ and $T$ as in Proposition~\\ref{prop:lowerbound-gap}.\nFor $i^* \\in \\{ 1, \\dots, M\\}$, denote $\\P_{i^*}$ the following distribution on $[0, 1]^{M \\times T}$: if $(\\ell_{i,t})_{1\\leq i \\leq M, 1\\leq t \\leq T} \\sim \\P_{i^*}$, then the variables $\\ell_{i,t}$ are independent Bernoulli variables, of parameter $\\frac{1}{2} - \\Delta$ if $i = i^*$ and $\\frac{1}{2}$ otherwise; also, denote by $\\mathbb E_{i^*}$ the expectation with respect to $\\P_{i^*}$.\nLet $\\mathcal{A} = (A_t)_{1 \\leq t \\leq T}$ be any Hedging algorithm, where $A_t: [0, 1]^{M \\times (t-1)} \\to \\mathcal{P}_M$ maps past losses $(\\bm \\ell_{1}, \\dots, \\bm \\ell_{t-1})$ to an element of the probability simplex $\\mathcal{P}_M \\subset \\mathbf R^M$ on $\\{ 1, \\dots, M \\}$.\nFor any $i^* \\in \\{ 1, \\dots, M \\}$, let $\\mathcal{R}_{T} (i^*, \\mathcal{A})$ denote the pseudo-regret of algorithm $\\mathcal{A}$ under the distribution $\\P_{i^*}$.\nSince $\\bm \\ell_t$ is independent of $\\bm v_t$ under $\\P_{i^*}$, we have\n\\begin{equation}\n \\label{eq:lowerbound-gap-pseudoregret}\n \\mathcal{R}_T (i^*, \\mathcal{A})\n = \\sum_{t=1}^T \\sum_{i \\neq i^*} \\mathbb E_{i^*} \\big[ v_{i,t} (\\ell_{i,t} - \\ell_{i^*,t}) \\big]\n = \\Delta \\sum_{t=1}^T \\sum_{i \\neq i^*} \\mathbb E_{i^*} [v_{i,t}]\n = \\Delta \\sum_{t=1}^T \\mathbb E_{i^*} [1 - v_{i^*,t}]\n\\end{equation}\nwith $\\bm v_t := A_t (\\bm\\ell_1, \\dots, \\bm\\ell_{t-1})$.\nIt follows from Equation~\\eqref{eq:lowerbound-gap-pseudoregret} that, for every $\\mathcal{A}$ and $i^*$, $\\mathcal{R}_T (i^*, \\mathcal{A})$ increases with $T$.\nHence, without loss of generality we may assume that {$T = \\lfloor (\\log M) \/ (16 \\Delta^2) \\rfloor$}.\nThe maximum pseudo-regret of $\\mathcal{A}$ on the instances $\\P_{i^*}$ is lower-bounded as follows:\n\\begin{equation}\n \\label{eq:lowerbound-gap-1}\n \\sup_{1 \\leq i^* \\leq M} \\mathcal{R}_T (i^*, \\mathcal{A})\n \\geq \\frac{1}{M} \\sum_{1 \\leq i^* \\leq M} \\mathcal{R}_T (i^*, \\mathcal{A})\n = \\frac{1}{M} \\sum_{1 \\leq i^* \\leq M} \\Delta \\sum_{t=1}^T \\mathbb E_{i^*} [ 1 - v_{i^*,t}]\n \\, .\n\\end{equation}\n\nWe now ``randomize'' the algorithm $\\mathcal{A}$, by replacing it with a randomized algorithm which picks expert $i$ at time $t$ with probability $v_{i,t}$.\nFormally, let $\\widetilde P = \\mathcal{U}}%{\\mathsf{Unif} ([0, 1])^{\\otimes T}$ be the distribution of $T$ independent uniform random variables on $[0, 1]$, and denote $\\widetilde \\P_{i^*} = \\P_{i^*} \\otimes \\widetilde P$ for $i^* \\in \\{ 1, \\dots, M \\}$.\nFurthermore, for every $\\bm v \\in \\mathcal{P}_M$, let $I_{\\bm v} : [0, 1] \\to \\{ 1, \\dots, M \\}$ be a measurable map such that $\\P (I_{\\bm v} (U) = i) = v_i$ for every $i \\in \\{ 1, \\dots, M \\}$, where $U \\sim \\mathcal{U}}%{\\mathsf{Unif} ([0 ,1])$.\nFor every sequence of losses $\\bm \\ell_1, \\dots, \\bm \\ell_T$ and random variables $U_1, \\dots, U_T$ and every $1 \\leq t \\leq T$, let $I_t = I_{\\bm v_t} (U_t)$, where $\\bm v_t = A_t (\\bm \\ell_1, \\dots, \\bm \\ell_t)$.\n\nDenote by $\\widetilde \\mathbb E_{i^*}$ the expectation with respect to $\\widetilde \\P_{i^*}$.\nBy definition of $I_{\\bm v}$, we have $\\mathbb E_{i^*} [ v_{i^*, t} ] = \\widetilde \\mathbb E_{i^*} [ \\indic{I_t = i^*} ]$ so that, denoting $N_i = \\sum_{i=1}^T \\indic{I_t = i}$ the number of times expert $i$ is picked,\n\\begin{equation*}\n \\sum_{t=1}^T \\mathbb E_{i^*} [1 - v_{i^*,t}] = \\widetilde \\mathbb E_{i^*} [ T - N_{i^*} ]\n \\geq \\P_{i^*} (N_{i^*} \\leq T\/2) \\cdot \\frac{T}{2}\n \\, .\n\\end{equation*}\nHence, letting $A_i \\subseteq [0, 1]^{M \\times T} \\times [0, 1]^T$ be the event $\\{ N_i > T\/2 \\}$,\nEquation~\\eqref{eq:lowerbound-gap-1} implies that\n\\begin{equation}\n \\label{eq:lowerbound-gap-2}\n \\sup_{1 \\leq i^* \\leq M} \\mathcal{R}_T (i^*, \\mathcal{A})\n \\geq \\frac{\\Delta T}{2} \\times \\frac{1}{M} \\sum_{1\\leq i^* \\leq M} \\big( 1 - \\widetilde \\P_{i^*} (A_{i^*}) \\big)\n \\, .\n\\end{equation}\n\nIt now remains to upper bound $\\frac{1}{M} \\sum_{i^*} \\widetilde \\P_{i^*} (A_{i^*})$.\nTo do this, first note that the events $A_{i^*}$, $1 \\leq i^* \\leq M$, are pairwise disjoint.\nHence, Fano's inequality \\citep[see][p.2]{gerchinovitz2017fano} implies that, for every distribution $\\widetilde \\mathbb Q$ on $[0, 1]^{M \\times T} \\times [0, 1]^T$,\n\\begin{equation}\n \\label{eq:lowerbound-gap-fano}\n \\frac{1}{M} \\sum_{1\\leq i^* \\leq M} \\widetilde \\P_{i^*} ( A_{i^*})\n \\leq \\frac{1}{\\log M} \\bigg\\{ \\frac{1}{M} \\sum_{1 \\leq i^* \\leq M} \\kll{\\widetilde \\P_{i^*}}{\\widetilde \\mathbb Q} + \\log 2 \\bigg\\}\n\\end{equation}\nwhere $\\kll{\\P}{\\mathbb Q}$ denotes the Kullback-Leibler divergence between $\\P$ and $\\mathbb Q$.\nHere, we take $\\widetilde \\mathbb Q = \\mathbb Q \\otimes \\widetilde P$, where $\\mathbb Q$ is the product of Bernoulli distributions $\\mathcal{B} (1\/2)^{\\otimes T}$.\nThis choice leads to\n\\begin{equation*}\n \\kll{\\widetilde \\P_{i^*}}{\\widetilde \\mathbb Q}\n = \\kll{\\P_{i^*}}{\\mathbb Q}\n = T \\cdot \\kll{\\mathcal{B}(1\/2 - \\Delta)}{\\mathcal{B}(1\/2)}\n \\leq 4 T \\Delta^2\n \\leq \\frac{\\log M}{4}\n \\, ,\n\\end{equation*}\nwhere the first bound is obtained by comparing KL and $\\chi^2$ divergences \\citep[Lemma~2.7]{tsybakov2009nonparametric}.\nHence, inequality~\\eqref{eq:lowerbound-gap-fano} becomes (recalling that $M \\geq 4$)\n\\begin{equation*}\n \\frac{1}{M} \\sum_{1\\leq i^* \\leq M} \\widetilde \\P_{i^*} ( A_{i^*})\n \\leq \\frac{(\\log M)\/4}{\\log M} + \\frac{\\log 2}{\\log M}\n \\leq \\frac{3}{4}\n \\, ;\n\\end{equation*}\nplugging this into~\\eqref{eq:lowerbound-gap-2} yields, noting that $T = \\lfloor (\\log M) \/ (16 \\Delta^2) \\rfloor \\geq (\\log M) \/ (32 \\Delta^2)$ since $(\\log M)\/(16 \\Delta^2) \\geq 1$ (as $M \\geq 4$ and $\\Delta \\leq \\frac{1}{4}$),\n\\begin{equation*}\n \\sup_{1 \\leq i^* \\leq M} \\mathcal{R}_T (i^*, \\mathcal{A})\n \\geq \\frac{\\Delta T}{2} \\times \\frac{1}{4}\n \\geq \\frac{\\log M}{256 \\Delta}\n \\, .\n\\end{equation*}\nThis concludes the proof.\n\n\n\\subsection{Proof of Theorem~\\ref{thm:adv-gap} and Corollary~\\ref{cor:hedge-martingale}}\n\\label{sec:proof-thm-adv-gap}\n\n\nLet $t_0 $ be the smallest integer $t\\geq 1$ such that $M e^{- c_0 \\Delta \\sqrt{t \\log (M) \/ 8}} \\leq \\Delta$, namely $t_0 = \\left\\lceil \\frac{8}{c_0^2 \\Delta^2} \\frac{\\log^2 (M \/ \\Delta)}{\\log M} \\right\\rceil$.\n Note that $\\sqrt{t_0} \\leq \\sqrt{1 + \\frac{8}{c_0^2 \\Delta^2} \\frac{\\log^2 (M \/ \\Delta)}{\\log M}} \\leq 1 + \\frac{\\sqrt{8}}{c_0 \\Delta} \\frac{\\log (M \/ \\Delta)}{\\sqrt{\\log M}}$.\n Let $t_1 := t_0 \\vee \\tau_0$.\n For every $T \\leq t_1$, the regret bound in the assumption of Theorem~\\ref{thm:adv-gap} implies\n \\begin{align}\n \\label{eq:proof-adv-1}\n R_T\n &\\leq c_1 \\sqrt{T \\log M} \\nonumber \\\\ \n &\\leq c_1 \\sqrt{\\tau_0 \\log M} + c_1 \\sqrt{t_0 \\log M} \\nonumber \\\\ \n &\\leq c_1 \\sqrt{\\tau_0 \\log M} + c_1 \\sqrt{\\log M} + \\frac{\\sqrt{8} \\log(M\/\\Delta)}{c_0 \\Delta}\n \\end{align}\n which implies~\\eqref{eq:regret-gap-adv} with $c_2 = c_1 + \\frac{\\sqrt{8}}{c_0}$ and $c_3 = \\frac{\\sqrt{8}}{c_0}$ (since $1\\leq \\sqrt{\\log M} \\leq \\frac{\\log M}{\\Delta}$).\n From now on, assume that $T \\geq t_1 + 1$.\n Since $T \\geq \\tau_0$, we have $R_T = \\widehat L_T - L_{i^*,T}$, so that\n \\begin{equation}\n \\label{eq:proof-adv-2}\n R_T\n = \\widehat L_{t_1} - L_{i^*, t_1} + \\sum_{t=t_1 + 1}^T \\big( \\widehat \\ell_t - \\ell_{i^*, t} \\big)\n \\, .\n \\end{equation}\n In addition, we have for $t\\geq t_1 +1$\n \\begin{align} \n \\widehat \\ell_t - \\ell_{i^*, t}\n &= \\sum_{i \\neq i^*} v_{i,t} (\\ell_{i,t} - \\ell_{i^*, t}) \\nonumber \\\\\n &\\leq \\sum_{i \\neq i^*} v_{i,t} \\nonumber \\\\\n &= \\sum_{i\\neq i^*} \\frac{ e^{- \\eta_t (L_{i,t-1} - L_{i^*, t-1})}}{1 + \\sum_{j\\neq i^*} e^{- \\eta_t (L_{j,t-1} - L_{i^*, t-1})}} \\nonumber \\\\\n &\\leq \\sum_{i\\neq i^*} e^{- c_0 \\sqrt{(\\log M)\/t} \\times \\Delta (t-1)} \\label{eq:proof-adv-3} \\\\\n &\\leq M e^{- c_0 \\Delta \\sqrt{(t-1)(\\log M)\/2}} \\nonumber\n \\\\\n &\\leq \\big( M e^{- c_0 \\Delta \\sqrt{t_0 (\\log M)\/8}} \\big) e^{- c_0 \\Delta \\sqrt{(t-1)\/8}} \\label{eq:proof-adv-5} \\\\\n &\\leq \\Delta e^{- c_0 \\Delta \\sqrt{(t-1)\/8}} \\label{eq:proof-adv-6}\n \\end{align}\n where~\\eqref{eq:proof-adv-3} comes from the fact that $\\eta_t \\geq c_0 \\sqrt{(\\log M)\/t}$ and $L_{i,t-1} - L_{i^*,t-1} \\geq \\Delta (t-1)$ (since $t-1\\geq t_1 \\geq \\tau_0$),~\\eqref{eq:proof-adv-5} from the fact that $t-1 \\geq t_0$ and $\\log M \\geq 1$, and~\\eqref{eq:proof-adv-6} from the fact that $M e^{- c_0 \\Delta \\sqrt{t_0 (\\log M)\/8}} \\leq \\Delta$.\n Summing inequality~\\eqref{eq:proof-adv-6}, we obtain \n \\begin{align} \n \\sum_{t=t_1+1}^T (\\widehat \\ell_t - \\ell_{i^*, t}) \\nonumber\n &\\leq \\sum_{t=t_1+1}^T \\Delta e^{- c_0 \\Delta \\sqrt{(t-1)\/8}} \\\\\n &\\leq \\Delta \\sum_{t\\geq 1} e^{- c_0 \\Delta \\sqrt{t\/8}} \\nonumber \\\\\n &\\leq \\Delta \\times \\frac{2}{(c_0 \\Delta \/ \\sqrt{8})^2} \\label{eq:proof-adv-7} \\\\\n &= \\frac{16}{c_0^2 \\Delta}\n \\label{eq:proof-adv-tail}\n \\end{align}\n where~\\eqref{eq:proof-adv-7} follows from Lemma~\\ref{lem:tail}.\n Combining~\\eqref{eq:proof-adv-2},~\\eqref{eq:proof-adv-1} and~\\eqref{eq:proof-adv-tail} proves Theorem~\\ref{thm:adv-gap} with $c_2 = c_1 + \\frac{\\sqrt{8}}{c_0}$, $c_3 = \\frac{\\sqrt{8}}{c_0}$ and $c_4 = \\frac{16}{c_0^2}$.\n\n \\begin{proof}[Proof of Corollary~\\ref{cor:hedge-martingale}]\n Define $\\tau = \\sup \\{ t \\geq 0 , \\exists i \\neq i^* , L_{i,t} - L_{i^*, t} \\leq \\frac{\\Delta t}{2} \\}$.\n By Lemma~\\ref{lem:crossing-time} below, for every $\\varepsilon > 0$ we have, with probability at least $1-\\varepsilon$, $\\tau \\leq {8 ( \\log M + \\log \\varepsilon^{-1})}\/{\\Delta^2}$.\n By Theorem~\\ref{thm:adv-gap}, this implies that, with probability at least $1-\\varepsilon$,\n \\begin{align*}\n \\label{eq:proof-hedge-martingale-1}\n R_T\n &\\leq c_1 \\sqrt{\\tau \\log M} + \\frac{c_2 \\log M + c_3 \\log {\\Delta}^{-1} + c_4}{\\Delta \/ 2} \\\\\n &\\leq \\left( c_1 \\sqrt{8} + 2 c_2 \\right) \\frac{\\log M}{\\Delta} + c_1 \\frac{\\sqrt{8\\log M \\log \\varepsilon^{-1}}}{\\Delta} + 2 c_3 \\frac{\\log \\Delta^{-1}}{\\Delta} + \\frac{2 c_4}{\\Delta} \n \\end{align*}\n where $c_2, c_3, c_4$ are the constants of Theorem~\\ref{thm:adv-gap}.\n The bound~\\eqref{eq:regret-martingale-exp} on the pseudo-regret is obtained similarly from Theorem~\\ref{thm:adv-gap}, by using the fact that $\\mathcal{R}_T \\leq \\mathbb E [R_T]$ and\n \\begin{equation*}\n \\mathbb E [ \\sqrt{\\tau \\log M}] \\leq \\sqrt{\\mathbb E [\\tau] \\log M}\n \\leq \\sqrt{\\log M} \\sqrt{ 1 + \\frac{8 (\\log M + 1)}{\\Delta^2}}\n \\leq \\sqrt{\\log M} \\Big( 1 + \\frac{\\sqrt{8 \\log M} + 1}{\\Delta} \\Big)\n \\end{equation*}\n which is smaller than $(2 + \\sqrt{8}) ({\\log M})\/{\\Delta} \\leq 5 (\\log M)\/\\Delta$ since $M \\geq 3$ and $\\Delta \\leq 1$.\n \\end{proof}\n\n\\begin{lemma}\n \\label{lem:crossing-time}\n Let $(\\ell_{i,t})_{1\\leq i \\leq M, t\\geq 1}$ be as in Theorem~\\ref{thm:hedge-stochastic}.\n Denote $\\tau = \\sup \\{ t \\geq 0 , \\exists i \\neq i^* , L_{i,t} - L_{i^*, t} \\leq \\frac{\\Delta t}{2} \\}$.\n We have\n \\begin{equation}\n \\label{eq:crossing-time-exp}\n \\mathbb E [ \\tau ] \n \\leq 1 + \\frac{8 (\\log M + 1)}{\\Delta^2} \n \\, , \n \\end{equation}\n and for every $\\varepsilon \\in (0, 1)$,\n\\begin{equation}\n \\label{eq:crossing-time-prob}\n \\P \\Big( \\tau \\geq \\frac{8 (\\log M + \\log \\varepsilon^{-1})}{\\Delta^2} \\Big) \\leq \\varepsilon \\, .\n\\end{equation} \n\\end{lemma}\n\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:crossing-time}]\n For every $i \\neq i^*$ and $t \\geq 1$, let $\\Delta_{i,t} := \\mathbb E [ \\ell_{i,t} - \\ell_{i^*,t} \\,|\\, \\mathcal{F}_{t-1}]$.\n Using the Hoeffding-Azuma's maximal inequality to the $(\\mathcal{F}_t)_{t\\geq 1}$-martingale difference sequence $Z_{i,t} = - (L_{i,t} - L_{i^*,t}) + \\Delta_{i,t}$ (such that $\\Delta_{i,t} - 1 \\leq Z_{i,t} \\leq \\Delta_{i,t} +1$), together with the fact that $\\Delta_{i,t} \\geq \\Delta$, implies that\n \\begin{equation}\n \\label{eq:proof-crossing-1}\n \\P \\left( \\exists t \\geq t_0, L_{i,t} - L_{i^*,t} \\leq \\frac{\\Delta t}{2} \\right)\n \\leq \\P \\left( \\sup_{t \\geq t_0} \\frac{1}{t} \\left( \\sum_{s=1}^t Z_{i,s} \\right) \\geq \\frac{\\Delta}{2} \\right)\n \\leq e^{- t_0 \\Delta^2\/8}\n \\, .\n \\end{equation}\n By a union bound, equation~\\eqref{eq:proof-crossing-1} implies that\n \\begin{equation}\n \\label{eq:proof-crossing-2}\n \\P \\left( \\tau \\geq t_0 \\right)\n \\leq M e^{-t_0 \\Delta^2\/8}\n \\, .\n \\end{equation}\n Solving for the probability level in~\\eqref{eq:proof-crossing-2} yields the high probability bound~\\eqref{eq:crossing-time-prob} on $\\tau$.\n The bound on $\\tau$ in expectation~\\eqref{eq:crossing-time-exp} ensues by integrating the high-probability bound over $\\varepsilon$.\n\\end{proof}\n\nWe recall Hoeffding-Azuma's maximal inequality for bounded martingale difference sequences \\citep{hoeffding1963probability,azuma1967weighted}.\nWhile it follows from a standard argument, we provide a short proof for completeness, since the inequality given in Proposition~\\ref{lem:hoeffding-maximal} below differs slightly from the one given in~\\citet{hoeffding1963probability}.\n\n\\begin{proposition}[Hoeffding-Azuma's maximal inequality]\n \\label{lem:hoeffding-maximal}\n Let $(Z_t)_{t \\geq 1}$ be a sequence of random variables adapted to a filtration $(\\mathcal{F}_t)_{t\\geq 1}$.\n Assume that $Z_t$ is a martingale difference sequence: $\\mathbb E [Z_t \\,|\\, \\mathcal{F}_{t-1}] = 0$ for any $t \\geq 1$, and that $A_t - 1 \\leq Z_t \\leq A_t+1$ almost surely, where $A_t$ is $\\mathcal{F}_{t-1}$-measurable.\n Then, denoting $S_n := \\sum_{t=1}^n Z_t$, we have for every $n \\geq 1$ and $a \\geq 0$:\n \\begin{equation}\n \\label{eq:hoeffding-maximal}\n \\P \\left( \\sup_{m \\geq n} \\frac{S_m}{m} \\geq a \\right)\n \\leq e^{- n a^2\/2}\n \\, .\n \\end{equation}\n\\end{proposition}\n\n\\begin{proof}\n Fix $\\lambda > 0$.\n By Hoeffding's inequality, $\\mathbb E [e^{\\lambda Z_t} \\,|\\, \\mathcal{F}_{t-1}] \\leq e^{\\lambda^2\/2}$, so that the sequence $M_t^{\\lambda} := \\exp \\big( \\lambda S_t - \\lambda^2 t \/ 2 \\big)$ is a positive supermartingale.\n Hence, Doob's supermartingale inequality implies that for $\\varepsilon \\in (0, 1]$:\n \\begin{equation}\n \\label{eq:proof-hoeffding-maximal-1}\n \\P \\Big( \\sup_{t \\geq 1} M_t^{\\lambda} \\geq \\frac{1}{\\varepsilon} \\Big)\n \\leq \\frac{\\mathbb E [M_0^\\lambda]}{1\/\\varepsilon} = \\varepsilon\n \\, .\n \\end{equation}\n Rearranging~\\eqref{eq:proof-hoeffding-maximal-1} and letting $\\lambda = \\sqrt{2 \\log (1\/\\varepsilon) \/ n}$ yields: with probability $1 - \\varepsilon$, for every $t \\geq n$,\n \\begin{equation}\n \\label{eq:proof-hoeffding-maximal-2}\n \\frac{S_t}{t}\n \\leq \\frac{\\log \\left( {1}\/{\\varepsilon} \\right)}{\\lambda t} + \\frac{\\lambda}{2}\n = \\sqrt{\\frac{\\log (1\/\\varepsilon)}{2}} \\left( \\frac{\\sqrt{n}}{t} + \\frac{1}{\\sqrt{t}} \\right)\n \\leq \\sqrt{\\frac{2 \\log (1\/\\varepsilon)}{n}}\n \\, .\n \\end{equation}\n Setting $\\varepsilon = e^{- n a^2\/2}$ in~\\eqref{eq:proof-hoeffding-maximal-2} gives the desired bound.\n\\end{proof}\n\n\n\\subsection{Proof of Proposition~\\ref{prop:lower-bound-hedge-cst}}\n\\label{sec:proof-lower-bound-hedge-cst}\n\nNote that, since the loss vectors $\\bm \\ell_t$ are in fact deterministic, $\\mathcal{R}_T = R_T$.\nDenoting $(v_{i,t})_{1 \\leq i \\leq M}$ the weights selected by the Constant Hedge algorithm at time $t$, and letting $c = c_0 \\sqrt{\\log M}$, we have\n\\begin{align}\n R_T\n &= \\sum_{t=1}^T \\sum_{i=2}^M v_{i, t} (\\ell_{i, t} - \\ell_{1, t}) \\nonumber \\\\\n &= \\sum_{t=1}^T \\sum_{i=2}^M \\frac{\\exp \\big( - \\frac{c}{\\sqrt{T}} (L_{i,t-1}-L_{1,t-1}) \\big)}{1 + \\sum_{2 \\leq i' \\leq M} \\exp \\big( - \\frac{c}{\\sqrt{T}} (L_{i',t-1}-L_{1,t-1}) \\big)} \\nonumber \\\\\n &= \\sum_{t=1}^T \\frac{ (M-1) \\exp \\big( - \\frac{c}{\\sqrt{T}} (t-1) \\big)}{1 + (M-1) \\exp \\big( - \\frac{c}{\\sqrt{T}} (t-1) \\big)} \\label{eq:proof-lower-cst1}\n \\, .\n\\end{align}\nNow, let $t_0 \\geq 0$ be the largest integer such that $(M-1) \\exp ( - \\frac{c}{\\sqrt{T}} t ) \\geq 1\/2$, namely\n\\begin{equation*}\n t_0 = \\Big\\lfloor \\frac{\\sqrt{T}}{c} \\log (2(M-1)) \\Big\\rfloor.\n\\end{equation*}\nIt follows from Equation~\\eqref{eq:proof-lower-cst1} that\n\\begin{equation}\n \\label{eq:proof-lower-cst2}\n R_T\n \\geq \\sum_{t=1}^{T \\wedge (t_0 + 1)} \\frac{ (M-1) \\exp \\big( - \\frac{c}{\\sqrt{T}} (t-1) \\big)}{1 + (M-1) \\exp \\big( - \\frac{c}{\\sqrt{T}} (t-1) \\big)} \n \\geq \\frac{1}{3} \\min (T, t_0 + 1)\n\\end{equation}\nwhere the second inequality comes from the fact that $\\frac{x}{1 + x} \\geq \\frac{1}{3}$ for $x \\geq \\frac{1}{2}$, \nwhich we apply to $x = (M - 1) \\exp ( - \\frac{c}{\\sqrt{T}} (t-1)) \\geq \\frac{1}{2}$ for $t \\leq T \\wedge (t_0+1) \\leq t_0 +1$.\nIn order to establish inequality~\\eqref{eq:lower-bound-hedge-cst}, it remains to note that\n\\begin{equation*}\n t_0 + 1\n \\geq \\frac{\\sqrt{T}}{c}\n \\log \\big( 2 (M-1) \\big)\n \\geq \\frac{\\sqrt{T \\log M}}{c_0}\n \\, ,\n\\end{equation*}\nsince $2(M-1) \\geq M$ and $c = \\sqrt{c_0 \\log M}$.\n\nNow, consider the Hedge algorithm with doubling trick. Assume that $T \\geq 2$, and let $k \\geq 1$ such that $T_k \\leq T < T_{k+1}$.\nSince $R_T = \\sum_{t=1}^T \\sum_{2\\leq i \\leq M} v_{i, t} (\\ell_{i, t} - \\ell_{1, t})$ and each of the terms in the sum is nonnegative, $R_T$ is lower bounded by the cumulative regret on the period $\\iint{T_{k-1}}{T_k - 1}$.\nDuring this period of length $T_{k-1}$, the algorithm reduces to the Hedge algorithm with constant learning rate $c_0 \\sqrt{\\log (M) \/ T_{k-1}}$, so that the above bound~\\eqref{eq:lower-bound-hedge-cst} applies; further bounding $T_{k-1} \\geq \\frac{T}{4}$ establishes~\\eqref{eq:lower-bound-hedge-doubling}.\n\n\\subsection{Proof of Proposition~\\ref{prop:second-order-bernstein}}\n\\label{sec:proof-second-order-bernstein}\n\nBy convexity of $x \\mapsto x^2$ and concavity of $x \\mapsto x^{\\beta}$,\nwe have:\n\\begin{align}\n \\label{eq:proof-second-bernstein-1}\n \\mathbb E [ (\\widehat \\ell_{t} - \\ell_{i^*,t})^2 ]\n &\\leq \\mathbb E \\bigg[ \\sum_{i=1}^M v_{i,t} (\\ell_{i,t} - \\ell_{i^*,t})^2 \\bigg] \\\\\n &= \\mathbb E \\bigg[ \\sum_{i=1}^M v_{i,t} \\mathbb E \\left[ (\\ell_{i,t} - \\ell_{i^*,t})^2 \\,|\\, \\mathcal{F}_{t-1} \\right] \\bigg] \\nonumber \\\\\n &\\leq B \\mathbb E \\bigg[ \\sum_{i=1}^M v_{i,t} \\mathbb E \\left[ \\ell_{i,t} - \\ell_{i^*,t} \\,|\\, \\mathcal{F}_{t-1} \\right]^\\beta \\bigg] \\label{eq:proof-second-bernstein-2} \\\\\n &\\leq B \\mathbb E \\bigg[ \\sum_{i=1}^M v_{i,t} \\mathbb E \\left[ \\ell_{i,t} - \\ell_{i^*,t} \\,|\\, \\mathcal{F}_{t-1} \\right] \\bigg]^\\beta \\label{eq:proof-second-bernstein-3} \\\\\n &= B \\mathbb E [\\widehat \\ell_t - \\ell_{i^*,t}]^\\beta\n\\end{align}\nwhere inequalities~\\eqref{eq:proof-second-bernstein-1} and~\\eqref{eq:proof-second-bernstein-3} come from Jensen's inequality, and~\\eqref{eq:proof-second-bernstein-2} from the Bernstein condition~\\eqref{eq:bernstein-condition}.\nTaking the expectation of the regret bound~\\eqref{eq:second-order-regret}, we obtain\n\\begin{align}\n \\label{eq:proof-second-1}\n \\mathbb E [R_{i^*,T}]\n &\\leq \\mathbb E \\Bigg[ C_1 \\sqrt{(\\log M) \\sum_{t=1}^T (\\widehat \\ell_t - \\ell_{i^*,t})^2} + C_2 \\log M \\Bigg] \\nonumber \\\\\n &\\leq C_1 \\sqrt{(\\log M) \\sum_{t=1}^T \\mathbb E \\big[ (\\widehat \\ell_t - \\ell_{i^*,t})^2 \\big]} + C_2 \\log M \\\\\n &\\leq C_1 \\sqrt{(\\log M) B \\sum_{t=1}^T \\mathbb E \\big[ \\widehat \\ell_t - \\ell_{i^*,t} \\big]^\\beta} + C_2 \\log M \\nonumber \\\\\n &= C_1 \\sqrt{B T \\log M} \\bigg( \\frac{1}{T} \\sum_{t=1}^T \\mathbb E \\big[ \\widehat \\ell_t - \\ell_{i^*,t} \\big]^\\beta \\bigg)^{1\/2} + C_2 \\log M \\nonumber \\\\\n &\\leq C_1 \\sqrt{B T \\log M} \\left( \\frac{\\mathbb E [R_{i^*,T}]}{T} \\right)^{\\beta\/2} + C_2 \\log M\n \\label{eq:proof-second-ineq}\n\\end{align}\nwhere inequalities~\\eqref{eq:proof-second-1} and~\\eqref{eq:proof-second-ineq} come from Jensen's inequality.\nLetting $r = {\\mathbb E [R_{i^*,T}]}\/{T}$ and $u = ({\\log M})\/{T}$,\ninequality~\\eqref{eq:proof-second-ineq} writes $r \\leq C_1 \\sqrt{B u} r^{\\beta\/2} + C_2 u$.\nThis implies that (depending on which of these two terms is larger) either $r \\leq 2 C_2 u$, or $r \\leq 2 C_1 \\sqrt{B u} r^{\\beta\/2}$, and the latter condition amounts to $r \\leq (2 C_1)^{2\/(2-\\beta)} (B u)^{1\/(2-\\beta)}$.\nThis entails that\n\\begin{equation*}\n r \\leq (2 C_1)^{\\frac{2}{2-\\beta}} (B u)^{\\frac{1}{2-\\beta}} + 2 C_2 u\n \\, ,\n\\end{equation*}\nwhich amounts to\n\\begin{equation}\n \\label{eq:proof-second-final}\n \\mathbb E [R_{i^*,T}]\n \\leq C_3 (B \\log M)^{\\frac{1}{2-\\beta}} T^{\\frac{1-\\beta}{2-\\beta}} + C_4 \\log M\n\\end{equation}\nwhere $C_3 = (2C_1)^{2\/(2-\\beta)} \\leq \\max (1, 4C_1^2)$ and $C_4 = 2C_2$.\n\n\n\\subsection{Proof of Theorem~\\ref{thm:hedge-no-bernstein}}\n\\label{sec:proof-lowerbound-no-bernstein}\n\nConsider the constant losses $\\ell_{1, t} = 0$, $\\ell_{i,t} = \\Delta$ where $\\Delta = {1} \\wedge c_0^{-1} \\sqrt{(\\log M)\/T}$.\nThese losses satisfy the $(1, 1)$-Bernstein condition since, for every $i>1$,\n$\\mathbb E [ (\\ell_{i,t} - \\ell_{1,t})^2 ] = \\Delta^2 \\leq \\Delta = \\mathbb E [\\ell_{i,t} - \\ell_{1,t}]$.\nOn the other hand, the regret of the Hedge algorithm with learning rate $\\eta_t = c_0 \\sqrt{(\\log M)\/t}$ writes\n\\begin{align}\n \\mathcal{R}_T\n &= \\sum_{t=1}^T \\sum_{i\\neq 1} \\mathbb E [v_{i,t} (\\ell_{i,t} - \\ell_{1,t})] \\nonumber \\\\\n &= \\Delta \\sum_{t=1}^T \\frac{(M-1) e^{-\\eta_t \\Delta (t-1)}}{1 + (M-1) e^{-\\eta_t \\Delta (t-1)}} \\nonumber \\\\\n &\\geq \\frac{\\Delta}{3} \\sum_{t=1}^T \\bm 1 \\Big( (M-1) e^{-\\eta_t \\Delta (t-1)} \\geq \\frac{1}{2} \\Big) \\nonumber \\\\\n &\\geq \\frac{\\Delta}{3} \\sum_{t=1}^T \\bm 1 \\left( M e^{- c_0 \\Delta \\sqrt{(t-1) \\log M}} \\geq 1 \\right) \\label{eq:proof-hedge-no-bernstein-0} \\\\\n &\\geq \\frac{\\Delta}{3} \\times \\min \\left( \\frac{\\log M}{c_0^2 \\Delta^2}, T \\right) \\nonumber \\\\\n &= \\frac{1}{3} \\min \\Big( \\frac{1}{c_0} \\sqrt{T \\log M}, {T} \\Big)\n \\, , \\label{eq:proof-hedge-no-bernstein-0b}\n\\end{align}\nwhere~\\eqref{eq:proof-hedge-no-bernstein-0} relies on the inequalities $2(M-1) \\geq M$ and $({t-1})\/{\\sqrt{t}} \\leq \\sqrt{t-1}$ for $M \\geq 2, t\\geq 1$, while~\\eqref{eq:proof-hedge-no-bernstein-0b}\nis obtained by noting that ${ (\\log M)}\/({c_0^2 \\Delta^2}) \\geq T$ since $\\Delta \\leq c_0^{-1} \\sqrt{(\\log M)\/T}$ and substituting for $\\Delta$.\n\n\\subsection{Proof of Theorem~\\ref{thm:hedge-characterize-gap}}\n\\label{sec:proof-hedge}\n\nAssume that the loss vectors $\\bm \\ell_1, \\bm \\ell_2, \\dots$ are i.i.d.\\@\\xspace, and denote $i^* = \\mathop{\\mathrm{argmin}}_{1\\leq i \\leq M} \\mathbb E [\\ell_{i,t}]$ (which is assumed to be unique), $\\Delta = \\min_{i \\neq i^*} \\Delta_i > 0$ where $\\Delta_i = \\mathbb E [\\ell_{i,t} - \\ell_{i^*,t}]$ and $j \\in \\{1, \\dots, M\\}$ such that $\\Delta_j = \\Delta$.\nThe Decreasing Hedge algorithm with learning rate $\\eta_t = c_0 \\sqrt{(\\log M)\/t}$ satisfies\n\\begin{align} \n \\mathcal{R}_T\n &= \\sum_{t=1}^T \\sum_{i \\neq i^*} \\mathbb E [v_{i,t}] \\Delta_i \\nonumber \\\\\n &\\geq \\Delta \\sum_{t=1}^T \\mathbb E \\left[ \\frac{\\sum_{i \\neq i^*} e^{-\\eta_t (L_{i,t-1} - L_{i^*,t-1})}}{1 + \\sum_{i \\neq i^*} e^{-\\eta_t (L_{i,t-1} - L_{i^*,t-1})}} \\right] \\nonumber \\\\\n &\\geq \\Delta \\sum_{t=1}^T \\mathbb E \\left[ \\frac{e^{-\\eta_t (L_{j,t-1} - L_{i^*,t-1})}}{1 + e^{-\\eta_t (L_{j,t-1} - L_{i^*,t-1})}} \\right] \\label{eq:proof-no-bernstein-b1} \\\\\n &\\geq \\frac{\\Delta}{3} \\sum_{t=1}^T \\mathbb E \\left[ \\bm 1 \\left( e^{-\\eta_t (L_{j,t-1} - L_{i^*,t-1})} \\geq \\frac{1}{2} \\right) \\right] \\nonumber \\\\\n &= \\frac{\\Delta}{3} \\sum_{t=1}^T \\P \\left( \\eta_t (L_{j,t-1} - L_{i^*,t-1}) \\leq \\log 2 \\right) \\label{eq:proof-no-bernstein-b2}\n\\end{align}\nwhere~\\eqref{eq:proof-no-bernstein-b1} relies on the fact that the function $x \\mapsto \\frac{x}{1+x}$ is increasing on $\\mathbf R^+$.\nDenoting $a = (\\log 2)\/(c_0 \\sqrt{\\log M})$, we have for every $1 \\leq t \\leq 1 + \\frac{a^2}{4 \\Delta^2}$:\n\\begin{align} \n \\P \\left( \\eta_t (L_{j,t-1} - L_{i^*,t-1}) > \\log 2 \\right)\n &= \\P \\left( L_{j,t-1} - L_{i^*,t-1} - \\Delta (t-1) > a \\sqrt{t} - \\Delta (t-1) \\right) \\nonumber \\\\\n &\\leq \\P \\left( L_{j,t-1} - L_{i^*,t-1} - \\Delta (t-1) > \\frac{a \\sqrt{t-1}}{2} \\right) \\label{eq:proof-no-bernstein-aDelta} \\\\\n &\\leq e^{- a^2\/8} \\label{eq:proof-no-bernstein-hoeffding}\n\\end{align}\nwhere inequality~\\eqref{eq:proof-no-bernstein-aDelta} stems from the fact that $\\Delta (t-1) \\leq \\frac{a \\sqrt{t-1}}{2}$ (since $t \\leq 1 + \\frac{a^2}{4 \\Delta^2}$), while~\\eqref{eq:proof-no-bernstein-hoeffding} is a consequence of Hoeffding's bound applied to the i.i.d.\\@\\xspace $[-1-\\Delta, 1-\\Delta]$-valued random variables $\\ell_{j,s} - \\ell_{i^*,s} - \\Delta$, $1\\leq s \\leq t-1$.\nAssuming that $c_0 \\geq 1$, we have $a \\leq \\sqrt{\\log 2} \\leq 1$, so that by concavity of the function $x \\mapsto 1- e^{-x\/8}$, $1-e^{-a^2\/8} \\geq (1-e^{-1\/8})a^2$.\nCombining this with inequalities~\\eqref{eq:proof-no-bernstein-b2} and~\\eqref{eq:proof-no-bernstein-hoeffding} and using the fact that $\\left\\lfloor 1+\\frac{a^2}{4 \\Delta^2} \\right\\rfloor \\geq \\frac{a^2}{4 \\Delta^2}$, we obtain for $T \\geq \\frac{1}{4 \\Delta^2} \\geq \\frac{a^2}{4 \\Delta^2}$:\n\\begin{align}\n \\label{eq:proof-no-bernstein-b3}\n \\mathbb E \\left[ R_T \\right]\n \\geq \\frac{\\Delta}{3} \\min \\left( \\frac{a^2}{4 \\Delta^2}, T \\right) (1-e^{-1\/8}) a^2\n = \\frac{(1-e^{-1\/8}) a^4}{12 \\Delta}\n \\geq \\frac{1}{450 c_0^4 (\\log M)^2 \\Delta}\n \\, ,\n\\end{align}\nwhere the last inequality comes from the fact that $(\\log 2)^4 (1-e^{-1\/8})\/12 \\geq \\frac{1}{450}$.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe ground-based gravitational wave (GW) detectors LIGO and Virgo have achieved great success in observing the GWs from the merger of binary neuron stars (BNS), binary black holes (BBH), and neutron star-black hole binaries (NSBH)~\\cite{LIGOScientific:2016aoc,LIGOScientific:2018mvr,LIGOScientific:2020ibl,LIGOScientific:2021usb,LIGOScientific:2021djp}. The third-generation ground-based detectors such as ET~\\footnote{\\url{http:\/\/www.et-gw.eu\/}} and CE~\\footnote{\\url{https:\/\/cosmicexplorer.org\/}} are under design and construction. Together with the space-borne LISA~\\footnote{\\url{https:\/\/www.lisamission.org\/}} and Chinese proposed projects like Taiji~\\cite{Hu:2017mde,Ruan:2018tsw} and Tianqin~\\cite{TianQin:2015yph} , they will be prepared for the detections of GWs around 2035. All of these are laser interferometers (LIs) and they will form a network of GW detectors from ground to space in the future. \n\nA novel type of GW detector called Atom interferometers (AIs) was proposed a decade ago~\\cite{Dimopoulos:2007cj,Dimopoulos:2008sv,Graham:2012sy,Hogan:2015xla}. In the concept AIs, gravitational radiation is sensed through precise measurement of the light flight time between two distantly separated atomic inertial references, each in a satellite in Medium Earth orbit (MEO). Ensembles of ultra-cold atomic Sr atoms at each location serve as precise atomic clocks. Light flight time is measured by comparing the phase of laser beams propagating between the two satellites with the phase of lasers referenced to the Sr optical transitions~\\cite{Graham:2017pmn}. Compared to the LIs, AIs consist of only a single baseline thus the design and building should be easier and cheaper than the traditional LIs. The AI projects such as ground based ZAIGA~\\cite{Zhan:2019quq} in China, AION~\\cite{Badurina:2019hst} in the UK, MIGA~\\cite{Geiger:2015tma} in France, ELGAR~\\cite{Canuel:2019abg} in Europe, and the space-borne MAGIS~\\cite{Graham:2017pmn} and AEDGE~\\cite{AEDGE:2019nxb} have been proposed and in preparation. \n\nAIs are proposed to probe not only the gravitational waves but also the dark matter. For GWs, AIs focus on the Deci-Hz gap between LIGO\/Virgo and LISA.\nIn the mid-frequency range, one can observe the long inspiral period of BNS, BBH, and NSBH. During the long observation, the motion of the space-borne detector around the Sun as well as in Earth orbit would induce large Doppler and reorientation effects, providing a precise angular resolution. Based on the space-borne AEDGE, we compose a series of papers focusing on the GW detections by AIs. \nIn the first paper~\\cite{Cai:2021ooo} (hereafter Paper I) we forecast the bright sirens detected by AEDGE and their applications on cosmology. The specific analysis on the source localization for the dark sirens was conducted in the second paper~\\cite{Yang:2021xox} (hereafter Paper II). The single baseline of AEDGE reorients on a rapid time scale compared to the observation duration. As a detector reorients and\/or moves, the observed waveform and phase are modulated and Doppler-shifted. This allows efficient determination of sky position and polarization information~\\cite{Graham:2017lmg,Graham:2017pmn}. In Paper II, we show AEDGE can even localize the dark sirens in such a small comoving volume that the unique host galaxy can be identified. These dark sirens are called ``golden dark sirens''. The measurements of Hubble constant from the simulated golden dark BNS and BBH are also performed. \n\nMany investigations suggest compact binaries that emit GWs can have non-negligible eccentricities and may contribute observational features in the sensitivity band of ground and space-based detectors~\\cite{Antonini:2012ad,Samsing:2013kua,Thompson:2010dp,East:2012xq}. \nDifferent mechanisms of the dynamic formation of the compact binaries of black holes and neutron stars have been proposed to study their eccentricities~\\cite{Rodriguez:2017pec,Samsing:2017xmd,Samsing:2017oij,Samsing:2018ykz,Wen:2002km,Pratten:2020fqn,OLeary:2008myb,Lee:2009ca}. \nOrbital eccentricity is arguably the most robust discriminator for distinguishing between isolated and dynamical BBH formation scenarios~\\cite{Zevin:2021rtf}. Some studies indicate that a fraction of the binaries possess eccentricities larger than 0.1 at 10 Hz~\\cite{Wen:2002km,Silsbee:2016djf,Antonini:2017ash,Liu:2019gdc}. The source localization improvement by the eccentricity for the ground-based detector networks has been investigated in some detail in~\\cite{Sun:2015bva,Ma:2017bux,Pan:2019anf}. They found that the eccentricity has more distinct effects on localization for higher-mass binaries than for the lower ones. For the case of $100~M_\\odot$ BBH, the improvement factor is about 2 in general when the eccentricity changes from 0.0 to 0.4~\\cite{Pan:2019anf}. Such an improvement is not adequate to considerably shrink the uncertainty of host galaxies (redshift) of dark sirens in the LIGO\/Virgo band, and to provide a conclusive measurement of the expansion of the Universe (e.g. the Hubble constant). While as shown in~\\cite{Yang:2022tig}, the eccentricity can significantly improve the distance estimation and source localization in the mid-band. The multiple harmonics induced by eccentricity can break the degeneracy between parameters in the waveform. In addition, the higher modes can enter the detector band much earlier than the dominant mode, which can provide more angular information. At some specific orientations (inclination angles), the typical compact binaries can achieve $\\mathcal{O}(10^2-10^4)$ improvement for the distance inference and $1.5\\sim{3.5}$ orders of magnitude improvement for the sky localization. Such a huge improvement on the 3-D localization could dramatically shrink the uncertainty of the host galaxies of the dark sirens. Up to now, only GW190521 has been reported to be eccentric in the latest GW catalog GWTC-3~\\cite{Romero-Shaw:2020thy,Gayathri:2020coq}. Considering the fact that the nonvanishing eccentricity is more likely to exist at lower frequency, we expect the dark sirens observed by the mid-band detector have greater potential on probing the cosmic expansion history, dynamics of dark energy, and gravity theory. \n\nIn this paper, we extend our research in Paper II and take the eccentricity effects into account for the dark sirens with AEDGE. In Sec.~\\ref{sec:typical}, we follow the methodology of~\\cite{Yang:2022tig} to check the improvement of distance estimation and source localization by eccentricity for the typical BNS, NSBH, and BBH with AEDGE. In Sec.~\\ref{sec:mock}, we adopt the similar method in Paper II to construct the catalogs of GWs for AEDGE. We first update the construction of catalogs of dark sirens in Paper II in the case of vanishing eccentricity. Comparing to Paper II, we update the waveform and the merger rates of BNS and BBH. We also include the NSBH into the simulation. We refine the fisher matrix calculation to ensure the convergence of the numerical derivatives. These updates and improvements make the simulation more realistic and reliable than that in Paper II. We then take account of eccentricity effects to construct the catalogs of dark sirens. We show how many potential host galaxies would be within the 3-D localization GW sources, with or without eccentricity. We estimate the the population of eccentric dark sirens and pick out the ones whose host galaxies can be best identified. We randomly select the golden dark sirens which AEDGE can track considering the limit of its operational time. The corresponding measurements of the Hubble constant are obtained in Sec.~\\ref{sec:Hubble}. We give the conclusions and discussions in Sec.~\\ref{sec:conclusion}.\n\n\n\n\n\\section{The improvement of distance estimation and localization from eccentricity \\label{sec:typical}}\nBy adopting a similar strategy in~\\cite{Yang:2022tig}, we mock up five types of typical compact binaries in GWTC-3~\\cite{LIGOScientific:2021djp} with component mass ranging from $\\mathcal{O}(1\\sim100)~M_{\\odot}$, i.e., a GW170817-like BNS with $(m_1,m_2)=(1.46,1.27)~M_{\\odot}$, a GW200105-like NSBH with $(9.0,1.91)~M_{\\odot}$, a GW191129-like light-mass BBH with $(10.7,6.7)~M_{\\odot}$, a GW150914-like medium-mass BBH with $(35.6,30.6)~M_{\\odot}$, and a GW190426-like heavy-mass BBH with $(106.9,76.6)~M_{\\odot}$. Note the light, medium, and heavy mass are in the context of the stellar-mass binaries in GWTC-3. The redshifts (distances) are also consistent with the real events in the catalog. We sample 1000 random sets of the angular parameters from the uniform and isotropic distribution for each typical binary and assign six discrete initial eccentricities $e_0=0$, 0.01, 0.05, 0.1, 0.2, and 0.4 at $f_0=0.1$ Hz. Then we have $5\\times 6\\times 1000=3\\times10^4$ cases. For each case, we perform the fisher matrix calculation to infer the errors of distance and sky location. \n\nWe use {\\sc PyCBC}~\\cite{alex_nitz_2021_5347736} to generate the waveform with the non-spinning, inspiral-only EccentricFD waveform approximant available in {\\sc LALSuite}~\\cite{lalsuite}. EccentricFD corresponds to the enhanced post-circular (EPC) model in~\\cite{Huerta:2014eca}. \nTo the zeroth order in the eccentricity, the model recovers the TaylorF2 PN waveform at 3.5 PN order~\\cite{Buonanno:2009zt}. To the zeroth PN order, the model recovers the PC expansion of~\\cite{Yunes:2009yz}, including eccentricity corrections up to order $\\mathcal{O}(e^8)$.\nThe strain can be written as~\\cite{Huerta:2014eca}\n\\begin{equation}\n\\tilde{h}(f)=-\\sqrt{\\frac{5}{384}}\\frac{\\mathcal{M}_c^{5\/6}}{\\pi^{2\/3}d_L}f^{-7\/6}\\sum_{\\ell=1}^{10}\\xi_{\\ell}\\left(\\frac{\\ell}{2}\\right)^{2\/3}e^{-i\\Psi_{\\ell}} \\,.\n\\label{eq:epc}\n\\end{equation} \nThe waveform keeps up to 10 harmonics, which corresponds to a consistent expansion in the eccentricity to $\\mathcal{O}(e^8)$ both in the amplitude and in the phase~\\cite{Yunes:2009yz}. In the vanishing eccentricity case, only the dominant (quadrupole) mode $\\ell=2$ remains, which is identical to the circular TaylorF2 model. With nonvanishing eccentricities, the induced multiple harmonics make the distance and angular parameters nontrivially coupled, enabling us to break the degeneracy among these parameters. In addition, the frequency of each harmonics is $\\ell F$ with $F$ the orbital frequency. Thus the higher harmonics ($\\ell>2$) should enter the detector band much earlier than the dominant mode ($\\ell=2$), which can provide more angular information. The $\\xi_{\\ell}$'s depend on the antenna pattern functions (also called detector response functions) $F_{+,\\times}$. For the space-borne AEDGE, we should consider the motion of the detector thus $F_{+,\\times}$ are functions of time. We give the detailed calculation of the antenna pattern functions in appendix~\\ref{app:F}. \n\nWe have 11 parameters in the waveform, namely the chirp mass $\\mathcal{M}_c$, the symmetric mass ratio $\\eta$, the luminosity distance $d_L$, the inclination angle $\\iota$, the sky location ($\\theta$, $\\phi$), the polarization $\\psi$, the time and phase at coalescence ($t_c$, $\\phi_c$), the initial eccentricity $e_0$ at frequency $f_0$, the azimuthal component of inclination angles (longitude of ascending nodes axis) $\\beta$. To estimate the uncertainty and covariance of the waveform parameters, we adopt the Fisher matrix technique \n\\begin{equation}\n\\Gamma_{ij}=\\left(\\frac{\\partial h}{\\partial P_i},\\frac{\\partial h}{\\partial P_j}\\right)\\,,\n\\end{equation}\nwith $P_i$ one of the 11 waveform parameters.\nThe inner product is defined as\n\\begin{equation}\n(a,b)=4\\int_{f_{\\rm min}}^{f_{\\rm max}}\\frac{\\tilde{a}^*(f)\\tilde{b}(f)+\\tilde{b}^*(f)\\tilde{a}(f)}{2 S_n(f)}df\\,.\n\\label{eq:innerp}\n\\end{equation}\nFor the noise power spectral density (PSD) $S_n(f)$, we adopt the sensitivity curve of AEDGE in the resonant modes (see the envelope in figure 1 of~\\cite{Ellis:2020lxl}).\nThen the covariance matrix of the parameters is $C_{ij}=(\\Gamma^{-1})_{ij}$, from which the uncertainty of each parameter $\\Delta P_i=\\sqrt{C_{ii}}$. The error of the sky localization is~\\cite{Cutler:1997ta}\n\\begin{equation}\n\\Delta \\Omega=2\\pi |\\sin(\\theta)|\\sqrt{C_{\\theta\\theta}C_{\\phi\\phi}-C_{\\theta\\phi}^2}\\,.\n\\end{equation}\nWe calculate the partial derivatives $\\partial \\tilde{h}\/\\partial P_i$ numerically by $[\\tilde{h}(f,P_i+dP_i)-\\tilde{h}(f,P_i)]\/dP_i$, with $dP_i=10^{-n}$. For each parameter, we need to optimize $n$ to make the derivative converged so that the Fisher matrix calculation is reliable. \n\nFor each typical event, the chirp mass $\\mathcal{M}_c$, symmetric mass ratio $\\eta$, and distance $d_L$ are calculated from the component mass and redshift. The angular parameters $P_{\\rm ang}=\\{\\iota,~\\theta,~\\phi,~\\psi,~\\beta\\}$ are sampled from the uniform and isotropic distribution with 1000 sets for each typical event. We use the inclination angle $\\iota$ to represent to angular parameter since we find it is more relevant in terms of the results. Without loss of generality, we fix the coalescence time and phase to be $t_c=\\phi_c=0$. We choose the frequency band of AEDGE to be [0.1, 3] Hz, where the detector is the most sensitive. This range corresponds to lower and upper bounds of frequency in the integral of Eq.~(\\ref{eq:innerp})~\\footnote{This is also different with Paper II in which we naively set the lower bound of frequency to be 0.2 for BNS and 0.05 for BBH.}. However, we should consider the limited operation time of AEDGE for tracking the GWs. We set quadrupole ($\\ell=2$) as the reference mode and its frequency is double of the orbital's, $f_{\\ell=2}=2F$. Then the evolution of the binary orbit can be calculated in terms of the quadrupole frequency. To ensure the observational time of AEDGE for all harmonics is around 400 days ($\\sim1$ year), we set the starting frequency of quadrupole $f_{\\rm start}(\\ell=2)$ to be 0.2, 0.1, 0.059, 0.026, and 0.0105 Hz for the typical BNS, NSBH, light BBH, medium BBH, and heavy BBH, respectively. That is, for the strain Eq.~(\\ref{eq:epc}) we should neglect the contribution of all the harmonics when the quadrupole's frequency is smaller than $f_{\\rm start}(\\ell=2)$~\\footnote{Note in the mid band we can only observe the inspiral phase of these binaries, thus we do not need to care about the upper frequency limit at the innermost-stable circular orbit.}. Thus we multiply the strain by the step function\n\\begin{equation}\n\\tilde{h}_{\\rm AEDGE}(f)=\\tilde{h}(f)\\mathcal{H}(2f-\\ell f_{\\rm start}) \\,,\n\\label{eq:hAEDGE}\n\\end{equation}\nwith the unit step function\n\\begin{equation}\n\\mathcal{H}(x)=\n\\begin{cases}\n1 & {\\rm if}~x\\geq0 \\,, \\\\\n0 & {\\rm otherwise} \\,.\n\\end{cases}\n\\end{equation}\nFor the orbital phase evolution, we numerically solve Eqs. (3.11) and (4.24) in~\\cite{Yunes:2009yz} to obtain the time to coalescence $t(f)$ for a nonvanishing $e_0$. The time to coalescence at a specific frequency is smaller for a larger eccentricity. So for the fixed $f_{\\rm start}(\\ell=2)$, the observational time is shorter with a larger eccentricity.\n\nWe collect all the results of the fisher matrix for the $3\\times10^4$ cases. Same as~\\cite{Yang:2022tig}, for each typical event with a specific orientation, we define the ratios\n\\begin{equation}\nR_{\\Delta d_L}=\\frac{\\Delta d_L|_{e_0={\\rm nonzero}}}{\\Delta d_L|_{e_0=0}}~{\\rm and}~R_{\\Delta \\Omega}=\\frac{\\Delta \\Omega |_{e_0={\\rm nonzero}}}{\\Delta\\Omega|_{e_0=0}} \\,,\n\\end{equation} \nto show the improvement induced by eccentricity in that orientation. If $R<1$, there is an improvement in the relevant parameter. A smaller $R$ indicates a larger improvement. We show the scatter plots of $\\Delta d_L\/d_L$, $R_{\\Delta d_L}$, $\\Delta \\Omega$, and $R_{\\Delta \\Omega}$ against $\\iota$. To give the statistical results, we define the minimum, mean, and maximum value of $x$ in the 1000 orientations as $\\min(x)$, $\\mathbb{E}(x)$, and $\\max(x)$, respectively.\n\nIn figure~\\ref{fig:rep}, we only show the distance inference of GW170817-like BNS and source localization of GW190426-like heavy BBH to represent\nour main results. We just compare the cases with $e_0=$0, 0.1, and 0.4 to give a concise look. The complete results can be found in appendix~\\ref{app:sup}. As shown in left panel of figure~\\ref{fig:rep}, a nonvanishing eccentricity can significantly improve the distance inference in the near face-on orientations (small inclination angle). Among all 1000 orientations, the $\\max(\\Delta d_L\/d_L)$ of GW170817-like BNS is reduced from 27.74 ($e_0=0$) to $0.82$ ($e_0=0.1$) and $0.35$ ($e_0=0.4$). Comparing to $e_0=0$ case, the largest improvement ($\\min(R_{\\Delta d_L})$) corresponds to 47 and 115 times stricter with $e_0=0.1$ and $e_0=0.4$, respectively. The huge improvement of distance inference in the near face-on orientations is true for all the typical events. The binaries with larger component mass and eccentricity can achieve more improvement. As shown in appendix~\\ref{app:sup}, for the heavy BBH with $e_0=0.4$, $\\min(R_{\\Delta d_L})=0.0012$, corresponding to 833 times improvement. We also find that for the heavy BBH case, there is an overall improvement of distance inference in all orientations. Our results indicate that the eccentricity effects are more distinct for the larger mass compact binaries. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{ddL_BNS170817}\n\\includegraphics[width=0.49\\textwidth]{dOmega_BBHheavy} \n\\caption{The distance inference of GW170817-like BNS (left) and source localization of GW190426-like heavy BBH.}\n\\label{fig:rep}\n\\end{figure}\n\nFor the source localization, we find eccentricity can lead to significant improvement for the BBH cases which have larger component mass than BNS and NSBH cases. As shown in the right panel of figure~\\ref{fig:rep}, the localization of heavy BBH is significantly improved by the eccentricities in almost all orientations. The largest improvement $\\min(R_{\\Delta \\Omega})= 8.16\\times 10^{-4}$, corresponding to $1.23\\times10^3$ times tighter. Like the distance inference, the heavier binaries benefits more from the eccentricity for the source localization. \nThe details of the improvement by eccentricity can also be found in the figures summarized in appendix~\\ref{app:sup}.\n\nTo illustrate the improvement of distance inference and localization for these typical binaries with variable eccentricities, we show the largest improvement ($\\min(R)$ in 1000 orientations) of each case in Fig.~\\ref{fig:Rwe}. We can see generally a heavier binary with higher eccentricity can achieve more improvement of distance inference and source localization. With eccentricity $e_0=0.4$, these typical binaries can most achieve \n1.5--3 orders of magnitude\nimprovement for the distance inference (from BNS to heavy BBH). As for the source localization, BNS and NSBH can not benefit much from the eccentricity. While BBHs can most achieve 1.5--3 orders of magnitude improvement (from light BBH to heavy BBH). We should note some anomalies in figure~\\ref{fig:Rwe}. 1) For the distance inference, the typical BNS benefits more from eccentricities than the typical NSBH and light BBH do. 2) For the source localization, BNS behaves similarly with NSBH and both have almost no improvement from eccentricity. 3) The light BBH's tendency is very close to that of medium BBH when $e_0<0.2$ and then they diverge for larger eccentricity. 4) In the BNS, NSBH, and especially for the light BBH cases, the localization achieves largest improvement when $e_0=0.2$, a higher eccentricity ($e_0=0.4$) can even worsen the performance. These anomalies are caused by many factors. On the one hand, eccentricity adds more harmonics in GWs. These harmonics can enlarge the SNR and improve the parameter estimation. The higher modes which enter the detector band much earlier can provide more angular information. On the other hand, eccentricity shrinks the inspiral time within the frequency band, which could lower the SNR and hence worsen the parameter estimation and localization. In addition, due to the different starting frequencies, for each binaries the detector band cover different length of harmonics. For instance, in BNS case, at the starting frequency $f_{\\rm start}(\\ell=2)=0.2$, all harmonics including $\\ell=1$ falls inside the detector band (0.1--3Hz). But in NSBH case, at $f_{\\rm start}(\\ell=2)=0.1$, the $\\ell=1$ mode's frequency is 0.05, falling outside the detector band hence should be truncated. Moreover, we have two more parameters $e_0$ and $\\beta$ in the eccentric waveform, which could degrade the overall performance of the parameter estimation. All above factors compete with each other and make the parameter estimation (distance inference and localization) differ from case to case. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.7\\textwidth]{Rwe}\n\\caption{The largest improvement of distance inference (upper panel) and source localization (lower panel) for the typical compact binaries in 1000 orientations with variable eccentricities.}\n\\label{fig:Rwe}\n\\end{figure}\n\nHere we would like to provide some more explanations for the inverse tendency of the error of the distance and localization versus the inclination angle. We can see in figure~\\ref{fig:rep} that generally the the error of distance is larger in smaller orbital inclination. On the contrary, the source localization is better when inclination is smaller. For distance it is due to the degeneracy between distance and inclination angle. In the amplitude of GW waveform~$h\\sim \\mathcal{A}_++\\mathcal{A}_\\times$, the distance $d_L$ and inclination angle $\\iota$ are tangled in the plus and cross polarization with different form, $\\mathcal{A}_+\\sim\\frac{1}{d_L}\\frac{1+\\cos(\\iota)}{2}$ and $\\mathcal{A}_+\\sim\\frac{1}{d_L}\\cos(\\iota)$. In order to identify the inclination of the binary system using the polarizations of the gravitational wave, we must distinguish the contributions of the plus and cross polarizations. At small $\\iota$, the two amplitudes from plus and cross polarizations have nearly identical contributions to the overall gravitational-wave amplitude. This is the main factor that leads to the strong degeneracy in the measurement of the distance and inclination~\\cite{Usman:2018imj}. So we expect a larger degeneracy between $d_L$ and $\\iota$ in the near face-on orientations and hence larger errors for both distance and inclination angle. As for the source localization, there is no obvious degeneracy between the sky location parameters $(\\theta,\\phi)$ and inclination angle $\\iota$. However, at smaller $\\iota$ the SNR is larger. So the parameter estimation should be better than that at larger $\\iota$. \n\nWe have showed that eccentricity, which is more likely to exist in the mid-band that in LIGO\/Virgo band, can improve the distance inference and source localization of dark sirens with AEDGE significantly. Note GWs are best localized at smallest orbital inclination where the distance are worst determined. But eccentricity happens to improve the distance inference most significantly there. In addition, one of the main targets for the mid-band detector like AEDGE is the intermediate mass black holes (IMBH). While in this paper we showed that the heaviest BBH can benefit most from the eccentricity for the distance inference and source localization. These facts suggests that eccentricity is the perfect ingredient for AEDGE dark sirens as precise probes of the Universe \n\n\\section{The construction of dark sirens catalogs and the host galaxy identification \\label{sec:mock}}\nConsidering the fact that eccentricity plays an important role in the distance inference and source localization of dark sirens with the mid-band detector AEDGE, we should take eccentricity effects into the construction of the dark sirens catalogs. In this section, we first update the construction of the catalogs of dark sirens in Paper II which does not consider the eccentricity effects, i.e., $e_0=0$. We adopt the EccentricFD waveform in which the $e_0=0$ case is equivalent to TaylorF2 at 3.5 PN order. While, in paper II we only expand the waveform to 2 PN order in the phase. We also update the BNS and BBH merger rates from the latest GWTC-3, as well as the BBH population. In addition to BNS and BBH, we add NSBH catalog. More importantly, we refine the numerical derivatives for the fisher matrix calculation, which would make the results more stable and thus robust and reliable. Then we include the eccentricity effects into the construction of the catalogs, to assess the its influence on the population and localization of the binaries.\n\nWe follow Paper II and assume the formation of compact binaries tracks the star formation rate.\nThe merge rate per comoving volume at a specific redshift $R_m(z_m)$ is related to the formation rate of massive binaries and the time delay distribution $P(t_d,\\tau)=\\frac{1}{\\tau}\\exp(-t_d\/\\tau)$ with an e-fold time of $\\tau=100$ Myr~\\cite{Vitale:2018yhm},\n\\begin{equation}\nR_m(z_m)=\\int_{z_m}^{\\infty}dz_f\\frac{dt_f}{dz_f}R_f(z_f)P(t_d) \\,.\n\\label{eq:Rm}\n\\end{equation}\nHere $t_m$ (or the corresponding redshift $z_m$) and $t_f$ are the look-back time when the systems merged and formed. $t_d=t_f-t_m$ is the time delay. $R_f$ is the formation rate of massive binaries and we assume it is proportional to the Madau-Dickinson (MD) star formation rate~\\cite{Madau:2014bja},\n\\begin{equation}\n\\psi_{\\rm MD}=\\psi_0\\frac{(1+z)^{\\alpha}}{1+[(1+z)\/C]^{\\beta}} \\,,\n\\label{eq:psiMD}\n\\end{equation}\nwith parameters $\\alpha=2.7$, $\\beta=5.6$ and $C=2.9$. The normalization factor $\\psi_0$ is determined by the local merger rates. We adopt the local merger rates of BNS, NSBH, and BBH inferred from GWTC-3, with $\\mathcal{R}_{\\rm BNS}=105.5^{+190.2}_{-83.9}~\\rm Gpc^{-3}~\\rm yr^{-1}$, $\\mathcal{R}_{\\rm NSBH}=45^{+75}_{-33}~\\rm Gpc^{-3}~\\rm yr^{-1}$, and $\\mathcal{R}_{\\rm BBH}=23.9^{+14.3}_{-8.6}~\\rm Gpc^{-3}~\\rm yr^{-1}$~\\cite{LIGOScientific:2021psn}. Note we assume the observed NSBH GW200105 and GW200115 are representatives of the population of NSBH. Then we convert the merger rate per comoving volume in the source frame to merger rate density per unit redshift in the observer frame\n\\begin{equation}\nR_z(z)=\\frac{R_m(z)}{1+z}\\frac{dV(z)}{dz} \\,,\n\\label{eq:Rz}\n\\end{equation}\nwhere $dV\/dz$ is the comoving volume element. \n\nHaving the merger rates as redshift, we can sample the redshift distribution of BNS, NSBH, and BBH. Like Paper II, we use the median merger rates to construct the catalogs. We have 11 parameters in the waveform (for vanishing eccentricity there are 9 except $e_0$ and $\\beta$). The luminosity distance $d_L$ is calculated from the sampled redshift by assuming a fiducial cosmological model $\\Lambda$CDM with $H_0=67.72~\\rm km~s^{-1}~Mpc^{-1}$ and $\\Omega_m=0.3104$, corresponding to the mean values obtained from the latest \\textit{Planck} experiment~\\cite{Planck:2018vyg}. The sky localization ($\\theta$, $\\phi$), inclination angle $\\iota$, and polarization $\\psi$ are drawn from isotropic distribution. Without loss of generality we set the time and phase at coalescence to be $t_c=\\phi_c=0$. As for the chirp mass and symmetric mass ration, we consider different strategy for these three binary types. In the BNS case, we assume a uniform distribution of mass in [1, 2.5] $M_{\\odot}$, which is consistent with the assumption for the prediction of the BNS merger rate in GWTC-3~\\cite{LIGOScientific:2021psn}. In the NSBH case, since the merger rate is inferred by assuming the observed NSBH GW200105 and GW200115 are representatives of the population of NSBH, we just randomly choose the component mass of these two events. As for the BBH case, we adopt the same strategy in Paper II with BBH population in GWTC-3. We draw the distribution of component mass of BBH from the histogram of mass distribution of BBH in GWTC-3~\\footnote{We first infer the histograms of primary mass $m_1$ and mass ratio $q$ from GWTC-3. The distribution of $m_1$ and $q$ are sampled accordingly. Then the second mass is just $m_2=m_1q$. We should make sure that $m_2\\ge3~M_{\\odot}$.}. The primary mass and mass ratio peak around 30--40 $M_{\\odot}$ and 0.7. \n\nWe sample the mergers of BNS, NSBH, and BBH in 5 years since the operation time of AEDGE is supposed to be 5--10 years~\\cite{AEDGE:2019nxb}. We set the frequency band and starting frequency to be same as in section~\\ref{sec:typical}. This means the observational time for each event is around 1 year. For each sampled merger, we assume four discrete eccentricities, i.e., $e_0=0$, 01, 0.2, and 0.4 at $f_0=0.1$ Hz. We select the mergers with SNR>8 as the candidate events that could be detected (within the detection range) by AEDGE in 5 years. For each events, we adopt the fisher matrix to derive their distance errors and source localizations. By assigning a uniform eccentricity for each event, we would like to assess the influence of eccentricity on the population and localization of the GWs that could be detected by AEDGE. We will give a discussion about the distribution of eccentricity and the realistic population of eccentric binaries later.\n\nFigure~\\ref{fig:hist} shows the cumulative histogram of events within the detection range of AEDGE in 5 years. The highest redshift AEDGE can reach for BNS and NSBH are around 0.13 and 0.45, respectively. For BBH, the horizon is much larger but we set a cut-off at $z=2$ since the for higher redshift we usually can not obtain the spectroscopic measurement of the redshift. In addition, the large uncertainty of localization makes the BBH at high redshift useless for our purpose in this paper. In the circular case, the total numbers are 106, 1105, and 95369 for BNS, NSBH, and BBH ($z\\leq2$), respectively. The numbers of BNS and BBH are smaller than that in Paper II, which is due to the different choice of the merger rates and the lower limit of frequency band of AEDGE (we adopt $f_{\\rm min}=0.05$ Hz for BBH in Paper II, while in this paper $f_{\\rm min}=0.1$ Hz). We note that a larger eccentricity leads a smaller population of the events. This is due to the fact that eccentricity reduces the inspiral (orbital evolution) time of binaries in the frequency band (0.1--3 Hz). The smaller observational time leads smaller accumulation of SNR, especially for the dominant quadrupole mode. So the GWs whose SNR are just a little above the detection threshold when $e_0=0$ may not be detected if they have nonvanshing eccentricities. In the NSBH case, the largest redshift AEDGE can reach is smaller for eccentric events. Comparing to BNS and BBH, the population of NSBH decrease the most with eccentricity. The reason is that we choose GW200105 and GW200115 as the representatives of NSBH population. The component masses in the NSBH catalog are fixed to be the same as either of these two typical events. So, the high-redshift eccentric events are definitely to be below the SNR threshold. While for BNS and BBH, there may be a larger sampled component mass to compensate for the low SNR. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\textwidth]{hist}\n\\caption{The cumulative histogram of events which are within the detection range of AEDGE in 5 years. Note we set a cut-off at $z=2$ for BBH.}\n\\label{fig:hist}\n\\end{figure}\n\nFigures~\\ref{fig:err_dL} and~\\ref{fig:err_Omega} show the error of distance and localization of the binaries that are within the detection range of AEDGE in 5 years. Eccentricity can significantly improve the overall distance inference of the binaries in the catalogs. For the source localization, BNS and NSBH can not benefit obviously from eccentricity. The localizations of eccentric events are even worsen in some cases. However, the source localization of BNS and NSBH are $\\mathcal{O}(10^{-4})~\\rm deg^{2}$ level even without eccentricity. While BBH's localization is considerably improved by eccentricity. The optimal localization at low redshift is improved to be better than $\\mathcal{O}(10^{-3})~\\rm deg^{2}$. We find that, in some cases, with $e_0=0.2$ the binaries can achieve the most improvements. All of these features can be expected based on the results in section~\\ref{sec:typical}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\textwidth]{err_dL}\n\\caption{The distance error of the events which are within the detection range of AEDGE in 5 years.}\n\\label{fig:err_dL}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\textwidth]{err_Omega}\n\\caption{The localization of the events which are within the detection range of AEDGE in 5 years.}\n\\label{fig:err_Omega}\n\\end{figure}\n\nTo assess the galaxy identification of the binaries in the catalogs, we should calculate their 3-D localization volumes which can be obtained from the errors of distance and localization in figures~\\ref{fig:err_dL} and~\\ref{fig:err_Omega}. We follow the method in~\\cite{Yu:2020vyy} to convert $\\Delta d_L$ and $\\Delta\\Omega$ to the 99\\% confidence ellipsoid of the localization. We use $V_{\\rm loc}$ to denote the 3-D volume of the localization. To estimate the numbers of potential host galaxies in the localization volume, we assume the galaxy is uniformly distributed in the comoving volume and the number density $n_g= 0.01~\\rm Mpc^{-3}$. This number is derived by taking the Schechter function parameters in B-band $\\phi_*=1.6\\times 10^{-2} h^3 {\\rm Mpc^{-3}}, \\alpha=-1.07, L_*=1.2\\times 10^{10} h^{-2} L_{B,\\odot}$ and $h=0.7$, integrating down to 0.12 $L_*$ and comprising 86\\% of the total luminosity~\\cite{Chen:2016tys}. Then the threshold localization volume is $V_{\\rm th}=100~\\rm Mpc^3$. If $V_{\\rm loc}\\leq V_{\\rm th}$, the host galaxy of the dark sirens can be identified uniquely and we call these golden dark sirens.\n\nFigure~\\ref{fig:V_loc} shows the the 99\\% confidence level (C.L.) of the 3-D localization of the events that are within the detection range of AEDGE in 5 years. We can see in the circular case, several BNS and NSBH events at low redshift can be localized within $V_{\\rm th}$. As for BBH, a few events can be localized with only one potential host galaxy. However, through the improvement from eccentricity, the eccentric BBH at low redshift can be well localized to become the golden dark sirens. The number of the golden dark sirens in the catalogs are summarized in table~\\ref{tab:np}. We also show the number of dark sirens whose potential host galaxies' count $n_p$ are less than 10. The result shows BBH can benefit the most from the eccentricity. In the circular case, it is almost impossible to detect the golden dark BBH. While nonvanishing eccentricities significantly increase the possibility to detect the golden BBH at low redshift. Note in the NSBH case, eccentricity would worsen the results compared to the circular case. The $e_0=0.2$ case gives an overall better result than other cases. All of these results are consistent with the expectation in section~\\ref{sec:typical}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\textwidth]{V_loc}\n\\caption{The 3-D localization volumes of the events which are within the detection range of AEDGE in 5 years. The horizontal dashed line corresponds to the threshold volume that the unique host galaxy can be identified.}\n\\label{fig:V_loc}\n\\end{figure}\n\n\\begin{table}\n\\centering\n\\resizebox{\\columnwidth}{!}{\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|} \n\\hline\n & \\multicolumn{2}{c|}{$e_0=0$} & \\multicolumn{2}{c|}{$e_0=0.1$} & \\multicolumn{2}{c|}{$e_0=0.2$} & \\multicolumn{2}{c|}{$e_0=0.4$} \\\\ \n\\hline\nBinary type & Golden~ & $130~\\text{GeV},\\quad |\\eta_{\\mathrm{j}_{1,2}}| <4.5, \\quad \\mathrm{and} \\quad m_{\\mathrm{j}_1\\mathrm{j}_2}>500~\\text{GeV}.\n \\end{align}\n In Fig.~\\ref{fig:colorVSkinemtics}, we show the distribution in the opening angle of the incoming and outgoing quark momenta for the two different pairings. The left plot shows the SM, while the right plot shows a specific NP benchmark point. Depicted in blue is the pairing based on the leading color connection using the color flow variable in the event file, while in red we show the opposite pairing. The plot shows that the momenta of the color connected quarks tend to form a small opening angle and the overlap between the two curves, i.e.~where the interference effects might be sizable, is negligible. This implies that in the experimental analysis the pairing should be done based on this variable. Importantly, the same conclusions can be drawn in the presence of new physics contributions to the contact terms.\n\nThere is a potential caveat to the above argument: the color flow approximation ignores the interference terms that are higher order in $1\/N_C$. \nLet us consider a process with two interfering amplitudes with the final state quarks exchanged, for example in $u u \\to u u h$. The differential cross section receives three contributions proportional to $|F^{f f^\\prime}_L (t_{13},t_{24})|^2$, $|F^{f f^\\prime}_L (t_{13},t_{24}) F^{f f^\\prime}_L (t_{14},t_{23})|$ and $|F^{f f^\\prime}_L (t_{14},t_{23})|^2$, where $t_{ij}=(p_i-p_j)^2=-2E_i E_j (1-\\cos \\theta_{i j})$. For the validity of the momentum expansion it is important that the momentum transfers ($t_{i j}$) remain smaller than the hypothesized scale of new physics. On the other hand, imposing the VBF cuts, the interference terms turn out to depend on one small and one large momentum transfer. However, thanks to the pole structure of the form factors, \nthey give a very small contribution.\n\n\\begin{figure}\n \\begin{center}\n \\includegraphics[width=0.45\\textwidth]{plots\/colorVSkinemtics}\n \\includegraphics[width=0.45\\textwidth]{plots\/colorVSkinemtics-NP}\n \\end{center}\n\\caption{\\small\\label{fig:colorVSkinemtics} Leading order parton level simulation of the Higgs VBF production at $13$~TeV pp c.m. energy. Show in blue is the distribution in the opening angle of the color connected incoming and outgoing quarks $\\measuredangle (\\vec p_3,\\vec p_1)$, while in red is the distribution for the opposite pairing, $ \\angle (\\vec p_3,\\vec p_2)$. The left plot is for the SM, while the plot on the right is for a specific NP benchmark. }\n\\end{figure}\n\n\\begin{figure}\n \\begin{center}\n \\includegraphics[width=0.48\\textwidth]{plots\/qSQvsPT.pdf}\n \\includegraphics[width=0.495\\textwidth]{plots\/qSQvsPT2.pdf}\n \\end{center}\n\\caption{\\small\\label{fig:pTVSq2} Leading order parton level simulation of the Higgs VBF production at $13$~TeV pp c.m. energy. Shown here is the density histogram in two variables; the outgoing quark $\\ensuremath{p_\\mathrm{T}}\\xspace$ and the momentum transfer $\\sqrt{- q^2}$ with the initial ``color-connected'' quark. The left plot is for the SM, while the plot on the right is for a specific NP benchmark. }\n\\end{figure}\n\nEven though in some experimental analyses, after reconstructing the momenta of the two VBF tagged jets and the Higgs boson, one could in principle compute the relevant momentum transfers $q_1$ and $q_2$, adopting the pairing based on the opening angle, in an hadron collider environment like the LHC this is unfeasible.\n Furthermore, for other Higgs decays modes, such as \n$h\\to 2\\ell2\\nu$, it is not possible to reconstruct the Higgs boson momentum.\nTherefore, we want to advocate the use of the $\\ensuremath{p_\\mathrm{T}}\\xspace$ of the VBF jets as a proxy for the momentum transfer $q^2_{1,2}$. \n The quality of this approximation can be understood by explicitly computing the momentum transfer $q^2_{1,2}$ in the VBF limit $|\\ensuremath{p_\\mathrm{T}}\\xspace| \\ll E_{\\mathrm{jet}}$ and for a Higgs produced close to threshold. \nLet us consider the partonic momenta in the c.o.m.~frame for the process: $p_1 = (E, \\vec{0}, E)$, $p_2 = (E, \\vec{0}, -E)$, $p_3 = (E'_1, \\vec{p}_\\mathrm{T,j_1}, \\sqrt{E^{\\prime 2}_1-\\ensuremath{p_\\mathrm{T,j_1}}\\xspace^2})$ and $p_4 = (E'_2, \\vec{p}_\\mathrm{T,j_2}, \\sqrt{E^{\\prime 2}_2 - \\ensuremath{p_\\mathrm{T,j_2}}\\xspace^2})$. Conservation of energy for the whole process dictates $2E = E'_1 + E'_2 + E_h$, where $E_h$ is the Higgs energy, usually of order $m_h$ if the Higgs is not strongly boosted. In this case $E - E'_i = \\Delta E_i \\ll E$ since the process is symmetric in $1 \\leftrightarrow 2$. For each leg, energy and momentum conservation (along the $z$ axis) give\n\\be\n\t\\left\\{ \\begin{array}{l} q^z_i = E - \\sqrt{E^{\\prime 2}_i - p_{\\mathrm{T}i}^2}\\,, \\\\ q^0_i = E - E'_i \\,,\\end{array} \\right.\\ \\quad \\to \\quad\n\t\\left\\{ \\begin{array}{l} q^0_i - q^z_i = \\sqrt{E^{\\prime 2}_i - p_{\\mathrm{T}i}^2} - E'_i \\approx - \\frac{p_{\\mathrm{T}i}^2}{2 E'_i} \\,, \\\\\n\tq^0_i + q^z_i \\approx 2 \\Delta E_i + \\frac{p_{\\mathrm{T}i}^2}{2 E'_i} \\,. \\end{array}\\right. ~.\n\\ee\nPutting together these two relations, one finds\n\\be\n\tq^2_i \\approx - p_{\\mathrm{T}i}^2 - \\frac{p_{\\mathrm{T}i}^2 \\Delta E_i}{2 E'_i} + \\mathcal{O}(p_{\\mathrm{T}i}^4 \/ E'^2) \\approx - p_{\\mathrm{T}i}^2 \\,,\n\\ee\nwhere in the last step we assumed $\\Delta E_i \\ll E'$, i.e. \nthe Higgs being produced near threshold.\n\n\\begin{figure}\n \\begin{center}\n \\includegraphics[width=0.47\\textwidth]{plots\/pTpT-sm}\n \\includegraphics[width=0.47\\textwidth]{plots\/pTpT-np}\n \\end{center}\n\\caption{\\small\\label{fig:smpTpT} Double differential distribution in the two VBF-tagged jet $\\ensuremath{p_\\mathrm{T}}\\xspace$ for VBF Higgs production at 13 TeV LHC. The distribution is normalized such that the total sum of events in all bins is 1. (Left) Prediction in the SM. (Right) Prediction for NP in $\\epsilon_{W u_L} =0.05$.}\n\\end{figure}\n\nIn order to confirm the above conclusion, in Fig.~\\ref{fig:pTVSq2} we show a density histogram in two variables: the (observable) $\\ensuremath{p_\\mathrm{T}}\\xspace$ of the outgoing jet and the\n (unobservable) momentum transfer $\\sqrt{-q^2}$ obtained from the correct color flow pairing (the left and the right plots are for the SM and for a specific NP benchmark, respectively). These plots indicate a very strong correlation of the jet $\\ensuremath{p_\\mathrm{T}}\\xspace$ with the momentum transfer $\\sqrt{-q^2}$ associated with the correct color pairing. \nWe stress that this conclusion holds both within and beyond the SM. \n Therefore, we encourage the experimental collaborations to report the unfolded measurement of the double differential distributions in the two VBF tagged jet $\\ensuremath{p_\\mathrm{T}}\\xspace$'s: $\\tilde F(p_{T j_1}, p_{T j_2})$. This measurable distribution is indeed closely related to the form factor entering the amplitude decomposition, $F_L(q_1^2, q_2^2)$, and encode (in a model-independent way) the dynamical information about the high-energy behavior of the process. \nMoreover, as we will discuss in Section~\\ref{sec:VBF_prospect}, the extraction of the PO in VBF \nmust be done preserving the validity of the momentum expansion: the latter can be checked and enforced setting \nappropriate upper cuts on the $\\ensuremath{p_\\mathrm{T}}\\xspace$ distribution. As an example of the strong sensitivity of the (normalized) $\\tilde F(p_{T j_1}, p_{T j_2})$ distribution to NP effects, in Fig.~\\ref{fig:smpTpT}, we show the corresponding prediction in the SM (left plot) and\n for a specific NP benchmark (right plot).\n\n\n\\subsection{NLO QCD corrections in VBF}\n\\label{sec:vbf_nlo}\n\nInclusive VBF Higgs production in the SM is very stable with respect to higher order QCD corrections \\cite{Han:1992hr,Figy:2003nv,Dittmaier:2011ti,Dittmaier:2012vm}. Employing a fixed renormalization and factorization scale $\\mu_{\\mathrm{R},\\mathrm{F}}=m_W$ inclusive NLO QCD corrections are at the level of $5-10\\%$ with remaining scale uncertainties of a few percent. At the NNLO QCD level these uncertainties on the inclusive cross section are further reduced below $1\\%$~\\cite{Bolzoni:2010xr,Bolzoni:2011cu}. \nHowever, in more exclusive observables, like the $\\ensuremath{p_\\mathrm{T}}\\xspace$ spectra of the VBF jets, or when more exclusive experimental selection cuts are applied, sensitivity to QCD radiation is more severe~\\cite{Figy:2003nv}, yielding non-negligible NLO correction factors while NLO scale uncertainties remain small (mostly well below 10\\%). Recently the dominant NNLO QCD corrections have been calculated fully differentially~\\cite{Cacciari:2015jma} pointing towards a non-trivial phase-space dependence with $5-10\\%$ corrections with respect to NLO. \nBesides higher-order corrections of QCD origin, also EW corrections are relevant for VBF Higgs production~\\cite{Ciccolini:2007jr,Ciccolini:2007ec}. At an inclusive level they amount to about $-5\\%$~\\cite{Ciccolini:2007jr}, while at the differential level due to the presence of large EW Sudakov logarithms they reach for example $-15\\%$ for $\\ensuremath{p_\\mathrm{T,j_1}}\\xspace=400$~GeV and $-10\\%$ for $\\ensuremath{p_\\mathrm{T,j_2}}\\xspace=150$~GeV~\\cite{Ciccolini:2007ec}.\n\n\n\n\nIn the following we will illustrate that the perturbative convergence for exclusive VBF observables can be improved when using a dynamical scale $\\mu_0=\\ensuremath{H_\\mathrm{T}\/2}\\xspace$ (with $H_{\\mathrm{T}}$ being the scalar sum of the $\\ensuremath{p_\\mathrm{T}}\\xspace$ of all final state particles) with respect to a fixed scale $\\mu_0=m_W$. \nIn particular, here we will focus on the $\\ensuremath{p_\\mathrm{T}}\\xspace$ spectra of the VBF jets -- as inputs for a fit of the Higgs PO. To this end we employ the fully automated {\\rmfamily\\scshape Sherpa+OpenLoops}\\xspace framework \\cite{Gleisberg:2007md,Gleisberg:2008ta,Cascioli:2011va,OLhepforge,Ossola:2007ax,vanHameren:2010cp} for the simulation of EW production of $pp\\to hjj$ at LO and NLO QCD in the SM. \nBefore applying the VBF selection cuts defined in Eq.~\\ref{eq:vbf_cuts} we cluster all final state partons into anti-$k_{\\mathrm{T}}$ jets with $R=0.4$ and additionally require a rapidity separation of the two hardest jets of $\\Delta\\eta_{j_1j_2} > 3$. This additional requirement, could slightly reduce the capability of differentiating different tensor structures~\\cite{Maltoni:2013sma}, however, such a cut is, on the one hand, experimentally required in order to suppress QCD backgrounds.\\footnote{~In fact, in most VBF analyses an even tighter selection of $\\Delta\\eta_{j_1j_2} > 4.5$ is imposed.} On the other hand, without such a cut NLO predictions for the $\\ensuremath{p_\\mathrm{T}}\\xspace$ spectra of the jets become highly unstable when the VBF jet selection is just based on the hardness of the jets, i.e.~a bremsstrahlung jet is easily amongst the two hardest jets and spoils the correlation between the \\ensuremath{p_\\mathrm{T}}\\xspace of the jets and the momentum transfer, as discussed in Section~\\ref{sec:vbf_kinematics}.\n\nIn Fig.~\\ref{fig:nlo_kfac_pt_pt} we plot the \\ensuremath{p_\\mathrm{T}}\\xspace distributions of the hardest and the second hardest jet using a dynamical scale $\\mu_0=\\ensuremath{H_\\mathrm{T}\/2}\\xspace$. On the left one-dimensional \\ensuremath{p_\\mathrm{T}}\\xspace spectra are plotted, while on the right we show the corresponding two-dimensional NLO correction factors $K^{\\textrm{NLO}}=\\sigma^{\\textrm{NLO}}\/\\sigma^{\\textrm{LO}}$. \nHere CT10nlo PDFs~\\cite{Lai:2010vv} are used both at LO and NLO and uncertainty bands correspond to 7-point renormalization (only relevant at NLO) and factorization scale variations $\\mu_{\\mathrm{R},\\mathrm{F}}=\\xi_{R,F}\\mu_0$ with $(\\xi_\\mathrm{R},\\xi_\\mathrm{F})=(2,2)$,\n$(2,1)$, $(1,2)$, $(1,1)$, $(1,0.5)$, $(0.5,1)$, $(0.5,0.5)$. \nThanks to the dynamical scale choice NLO corrections to the one-dimensional distributions are almost flat and amount to about $-15\\%$, while the dependence in the two dimensional distribution remains moderate with largest corrections for $\\ensuremath{p_\\mathrm{T,j_1}}\\xspace \\approx \\ensuremath{p_\\mathrm{T,j_2}}\\xspace$.\n\n\nIn the following section we will detail a fit of Higgs PO based on LO predictions of VBF using the scale choice and setup developed in this chapter. Here we already note, that this fit is hardly affected by the overall normalization of the predictions. Thus, with respect to possible small deviations from the SM due to effective form factor contributions we expect a very limited sensitivity to QCD effects assuming a similar stabilization of higher order corrections as observed for the SM employing the scale choice $\\mu_0=\\ensuremath{H_\\mathrm{T}\/2}\\xspace$.\nIn order to verify this assumption and to improve on the Higgs PO fit, we are currently extending the simulations within the Higgs PO framework to the NLO QCD level. To this end, the framework has been implemented in the {\\rmfamily\\scshape OpenLoops}\\xspace one-loop amplitude generator in a process independent way. Here, the $\\mathcal{O}(\\alpha_S)$ rational terms of $R_2$-type required in the numerical calculation of the one-loop amplitudes in {\\rmfamily\\scshape OpenLoops}\\xspace \n have been obtained generalising the corresponding SM expressions~\\cite{Draggiotis:2009yb}.\nThe implementation of the dipole subtraction and parton-shower matching in the {\\rmfamily\\scshape Sherpa}\\xspace Monte Carlo framework is based on the model independent {\\rmfamily\\scshape UFO}\\xspace interface of Sherpa~\\cite{Hoche:2014kca} and is currently being validated.\n \n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.40\\textwidth]{plots\/VBF_hthalf_ptj.pdf}\\quad \n \\includegraphics[width=0.48\\textwidth]{plots\/k-factor-pt-pt}\n\\caption{One- (left) and two-dimensional (right) NLO correction factors and scale uncertainties for EW production of $pp\\to h+2$\\,jets in the SM\nin function of $\\ensuremath{p_\\mathrm{T,j_1}}\\xspace$ and $\\ensuremath{p_\\mathrm{T,j_2}}\\xspace$ employing a central scale $\\mu_0=\\ensuremath{H_\\mathrm{T}\/2}\\xspace$.\n}\n\\label{fig:nlo_kfac_pt_pt}\n\\end{figure}\n\n\n\\subsection{Prospects for the Higgs PO in VBF at the HL-LHC}\n\\label{sec:VBF_prospect}\n\nThe extraction of the PO from the double differential distribution\n$\\tilde F(p_{T j_1}, p_{T j_2})$ has to be done with care. \nHere we make an attempt to perform such analysis. In the following \nwe estimate the sensitivity of the\nHL-LHC, operated at $13$~TeV with $3000$~fb$^{-1}$ of data, on measuring the PO\nassuming maximal flavor symmetry in a seven dimensional fit to $\\kappa_{ZZ}$,\n$\\kappa_{WW}$, $\\epsilon_{Z u_L}$, $\\epsilon_{Z u_R}$, $\\epsilon_{Z d_L}$,\n$\\epsilon_{Z d_R}$ and $\\epsilon_{W u_L}$. The ATLAS search for $h\\to WW^*$\nreported in Ref.~\\cite{ATLAS:2014aga} considers the VBF-enriched category in\nwhich the detection of two jets consistent with VBF kinematics is required. The\nexpected yields in this category are reported in Table VII of\nRef.~\\cite{ATLAS:2014aga}. After the final selection cuts at $8$~TeV with\n$20.3$~fb$^{-1}$ of integrated luminosity, the expected number of Higgs VBF\nevents in the SM is $4.7$ (compared to $5.5$ background events) in the $e\\mu$\nsample. Rescaling the number of expected events with the expected HL-LHC luminosity\n(3000~fb$^{-1}$) and cross section, we expect about $2000$ SM Higgs VBF events \nto be collected by\n each experiment. In the following, we make a brave\napproximation and neglect any background events in the fit and assume that the HL-LHC will observe a total of $2000$ events compatible with the SM expectations.\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.5\\textwidth]{plots\/HPO_HLLHC}\n\\caption{\\small\\label{fig:HPO_HLLHC} Prospects for measuring Higgs PO in electroweak Higgs production at the HL-LHC at 13 TeV with $3000~ \\text{fb}^{-1}$ of integrated luminosity. For VBF and $Zh$ we considered the $h \\to 2\\ell2\\nu$ channel (with $Z\\to 2\\ell$ in $Zh$) while for $Wh$ we considered only the clean $h\\to 4\\ell$, $W \\to \\ell \\nu$ channel. The solid (dashed) intervals represent the $1\\sigma$ ($2\\sigma$) constraints in each PO, where all the others are profiled. The red bounds are from VBF, the blue ones from $Zh$ and the green ones from $Wh$ production. More details can be found in the main text.}\n\\end{figure}\n\nAs anticipated, a key point to be addressed for a consistent extraction of the PO is the\nvalidity of the momentum expansion.\nIn order to control such expansion, we set an upper cut on the $\\ensuremath{p_\\mathrm{T}}\\xspace$ of the leading VBF-tagged jet. \nThe momentum expansion of the form factors in Eq.~\\eqref{eq:FLGL} only makes sense if the higher order terms in $q_{1,2}^2$ are suppressed. \nThis requirement leads to the consistency condition,\n\\be\n\\epsilon_{X_f} ~ |q_{\\rm{max}}^2| \\lesssim m_Z^2 ~ g_X^{f}~,\n\\label{eq:consistency}\n\\ee\nwhere $q_{\\rm{max}}^2$ is the largest momentum transfer in the process. A priori we do not know the size of $\\epsilon_{X_f}$\nor, equivalently, the effective scale of new physics. However, a posteriori we can verify by means of Eq.~(\\ref{eq:consistency}) \nif we are allowed to truncate the momentum expansion to the first non-trivial terms. In practice, \nsetting a cut-off on $\\ensuremath{p_\\mathrm{T}}\\xspace$ we implicitly define a value of $\\sqrt{-q^2_{\\rm{max}}}$. Extracting the $\\epsilon_{X_f}$ for \n $\\ensuremath{p_\\mathrm{T,j}}\\xspace < (\\ensuremath{p_\\mathrm{T,j}}\\xspace)^{\\rm max} \\approx \\sqrt{-q^2_{\\rm{max}}}$ we can \ncheck if Eq.~(\\ref{eq:consistency}) is satisfied. Ideally, the experimental collaborations should perform the extraction of the $\\epsilon_{X_f}$ \nfor different values of $(\\ensuremath{p_\\mathrm{T,j}}\\xspace)^{\\rm max}$ optimizing the range according to the results obtained. \nIn the following exercise we set $(\\ensuremath{p_\\mathrm{T,j}}\\xspace)^{\\rm max} = 600~ \\text{GeV}$ which, a posteriori, will turn out to be a good choice in absence of \nany sizeable deviations from the SM.\n\nIn our analysis we choose the binning in the double differential distributions in the two VBF tagged jet $\\ensuremath{p_\\mathrm{T}}\\xspace$'s as $\\{30-100-200-300-400-600\\}$~\\text{GeV}. We use the \nUFO implementation of the Higgs PO in the Sherpa Monte Carlo generator \\cite{Gleisberg:2008ta,Hoche:2014kca} to simulate VBF Higgs events over the relevant PO parameter space in proton-proton collisions at $13$~TeV c.m. energy. Here we employ the VBF selection cuts as listed in Eq.~(\\ref{eq:vbf_cuts})\nwith the additional requirement $\\Delta\\eta_{j_1j_2} > 3$. We verified that the results of the fit are independent on the precise value of this last cut. Renormalization and factorization scales are set to $\\mu_{\\mathrm{R}\/\\mathrm{F}}=\\ensuremath{H_\\mathrm{T}\/2}\\xspace$, as discussed in Section~\\ref{sec:vbf_nlo}.\n\nAnalyzing the simulation output, we find expressions for the number of expected events in each bin as a quadratic polynomial in the PO:\n\\begin{equation}\n\\label{eq:Nevbin}\n\tN^{\\rm ev}_a = \\kappa^T X^a \\kappa~, \\quad \\mathrm{with} \\quad\n\t\\kappa \\equiv (\\kappa_{ZZ},\\kappa_{WW},\\epsilon_{Z u_L}, \\epsilon_{Z u_R}, \\epsilon_{Z d_L}, \\epsilon_{Z d_R},\\epsilon_{W u_L})^T~,\n\\end{equation}\n where $a$ is a label for each bin.\nAssuming that the HL-LHC ``would-be-measured'' distribution is SM-like and describing the number of events in each bin with a Poisson distribution, we construct a global likelihood $L$ and evaluate the best-fit point from the maximum of the likelihood. We then define the test statistic, $\\Delta \\chi^2 = - 2 \\log (L\/L_{\\mathrm{max}})$, as a function of the seven PO. For more details on the statistical analysis see App.~\\ref{app:stat}.\n\\begin{figure}\n \\begin{center}\n\t\\includegraphics[width=0.47\\textwidth]{plots\/VBF_lead_pTj_hist} \\quad\n \\includegraphics[width=0.48\\textwidth]{plots\/VBF_lead_pTj_hist_norm}\n \\end{center}\n\\caption{\\small\\label{fig:VBF_lead_pTj_hist} Allowed deviations in the distribution of the leading-jet $\\ensuremath{p_\\mathrm{T}}\\xspace$ by varying the PO within the $-2\\log L \/ L_{\\rm max} < 4$ ($2\\sigma$) region obtained after the VBF fit. In the left plot we show the absolute number of events in each bin, while in the right one we show the normalized distribution with respect to the total number of events and the bin width.}\n\\end{figure}\n\nIn Fig.~\\ref{fig:HPO_HLLHC}, we show in red the $1\\sigma$ ($\\Delta \\chi^2 \\le 1$) and $2\\sigma$ ($\\Delta \\chi^2 \\le 4$) bounds for each PO, while profiling over all the others. The expected uncertainty on the $\\kappa_{ZZ,WW}$ is rather large (with a loosely bounded direction: $\\delta \\kappa_{ZZ} \\approx -3 \\delta\\kappa_{WW}$), however in a global fit to all Higgs data, these PO are expected to be much more precisely constrained from $h\\to 4\\ell, 2\\ell 2\\nu$ decays. The most important \nconclusion of this analysis is that at the HL-LHC all five production PO can be constrained at the percent level. \nIn the following we test the robustness of this conclusion.\n\nThe likelihood obtained from the PO fit is highly non-Gaussian, which is mainly due to the fact that Eq.~\\eqref{eq:Nevbin} is quadratic in the PO, and thus the $\\Delta \\chi^2$ is approximately a quartic polynomial. This implies that using the Gaussian approximation to obtain the $1\\sigma$ uncertainties from an expansion around the minimum overestimates these errors (compare with the $1\\sigma$ intervals of Fig.~\\ref{fig:HPO_HLLHC}):\n\\be\\begin{split}\n\t{\\rm VBF:} \\qquad \\sigma^{\\rm Gauss}_{\\rm quad}(\\kappa_{ZZ}, \\kappa_{WW}, \\epsilon_{Z u_L}, \\epsilon_{Z u_R}, \\epsilon_{Z d_L}, \\epsilon_{Z d_R}, \\epsilon_{W u_L}) =& \\\\\n\t= (0.63, 0.18, 0.021, 0.026, 0.032, 0.050, 0.008)~. &\n\t\\label{eq:VBF_GaussFit_Quadr}\n\\end{split}\n\\ee\nIn order to assess if these bounds simply come from the information of the total rate, which in a complete analysis depends \n on the decay parameters and the total Higgs decay width, or it indeed stems from the shape analysis, we introduce a new parameter $\\mu$ as an overall rescaling of the number of events in all bins, $N^{\\rm ev}_a \\to \\mu N^{\\rm ev}_a$. We then perform the same fit as above with this extra parameter and subsequently profile over it.\\footnote{~In order to stabilize the fit we assign a Gaussian distribution for $\\mu$ centered around 1 with $\\sigma = 10$.} As a result, $\\kappa_{ZZ}$ and $\\kappa_{WW}$ become unconstrained but the constraints on the contact terms do not change qualitatively. We thus conclude that their bounds do come from the\n shape information, i.e. the normalized distribution $\\tilde F(p_{T j_1}, p_{T j_2})$.\n\nFurthermore, we have checked that the uncertainties on the entries of the $X^a$ matrices, due to the finite statistics of our Monte Carlo simulations, do not impact the fit results. Details of this analysis are reported in App.~\\ref{app:stat}. The approach sketched there can also be used to estimate the uncertainty of our result caused by missing higher order theory corrections, most notably NLO electroweak effects. As anticipated, \nthe latter can exceed the 10\\% level in VBF~\\cite{Ciccolini:2007jr,Ciccolini:2007ec}; however, the largest contributions are due to factorizable corrections (EW Sudakov logarithms and soft QED radiation) \nthat can be reabsorbed by a redefinition of the PO. From the results in Ref.~\\cite{Denner:2003iy} for the related process $e^+ e^- \\to \\nu \\bar\\nu h$ we estimate non-factorizable NLO electroweak corrections to barely reach $10\\%$ in some dedicated corners of the phase space (being typically well below such values in most of the phase space). To be conservative, we assign uncorrelated relative errors of $10\\%$ in each element of the matrices $X^a$, by introducing appropriate nuisance parameters, and redo the fit. Profiling over these nuisance parameters, in the Gaussian approximation, we find the following $1\\sigma$ uncertainties for the PO: $\\Delta \\kappa_{ZZ} = 0.94$, $\\Delta \\kappa_{WW} = 0.31$, $\\Delta \\epsilon_{Z u_L} = 0.022$, $\\Delta \\epsilon_{Z u_R} = 0.027$, $\\Delta \\epsilon_{Z d_L} = 0.033$, $\\Delta \\epsilon_{Z d_R} = 0.055$ and $\\Delta \\epsilon_{W u_L} = 0.009$. Interestingly, comparing these with the Gaussian errors shown above, we conclude that the estimated sensitivity does not worsen significantly, indicating that statistical errors will still dominate. It is worth noting that the theoretical uncertainties are more relevant for \nthe determination of $\\kappa_{ZZ}$ and $\\kappa_{WW}$ and less relevant for the contact terms PO.\n\nNow that we have obtained the constraint on the PO, we can a posteriori check the consistency condition of the analysis, namely, that we are in the regime of small deviations from the SM prediction. In Fig.~\\ref{fig:VBF_lead_pTj_hist}, we show the envelope of the allowed deviations in the leading-jet $\\ensuremath{p_\\mathrm{T}}\\xspace$ distribution, obtained by varying the PO inside the $2\\sigma$ region. As can be seen, the size of the distribution is well constrained up to $400~\\textrm{GeV}$.\nEquivalently, using $| \\epsilon_{X_f} | \\lesssim 0.01$ to check the consistency condition (\\ref{eq:consistency}), we find \n $0.01 \\times (600~\\textrm{GeV})^2 \/ m_Z^2 \\lesssim 1$, suggesting that we have performed an analysis in a kinematical region \n where the momentum expansion is indeed reliable. \n \n\\section[Higgs PO in VH~production]{Higgs PO in VH~production}\n\\label{sect:VH}\n\n\\subsection[VH kinematics]{VH kinematics}\n\nHiggs production in association with a $W$ or $Z$ boson are respectively the third and fourth most important Higgs production processes in the SM, by total cross section. Combined with VBF studies, they offer complementary handles to limit and disentangle the various Higgs PO. Due the lower cross sections, so far these processes are mainly studied in the highest-rate Higgs decay channels, such as $h\\to b\\bar{b}$ \\cite{Aad:2012gxa,Chatrchyan:2012ww,Chatrchyan:2013zna,Aad:2014xzb} and $h\\to WW^*$ \\cite{Chatrchyan:2012qr,Khachatryan:2014jba,Aad:2015gba,Aad:2015ona}. The drawback of these channels are large backgrounds, which are overwhelming in the $b\\bar{b}$ case and of the same order as the signal in the $WW^*$ channels. In the following we skip over the challenges and the difficulties due to the presence of large backgrounds in these dominant modes, focusing only on $V+h$ decay channels with a good S\/B ratio (that should become accessible at the HL-LHC). In those channels we analyze the prospects for the extraction of the \ncorresponding production PO.\n\n\\begin{figure}\n \\begin{center}\n \\includegraphics[width=0.45\\textwidth]{plots\/pTZ_qSQ_SM} \\quad\n \\includegraphics[width=0.45\\textwidth]{plots\/pTZ_qSQ_BSM}\n \\end{center}\n\\caption{\\small\\label{fig:pTZ_qSQ_SM} The correlation between the $Zh$ invariant mass and the $\\ensuremath{p_\\mathrm{T}}\\xspace$ of the $Z$ boson in $Zh$ associate production at the 13TeV LHC in the SM (left plot) and for a BSM point $\\kappa_{ZZ} = 1$, $\\epsilon_{Z u_L} = 0.1$ (right plot). A very similar correlation is present in the $Wh$ channel.}\n\\end{figure}\n\n\nAn important improvement for future studies of these channels with the much higher luminosity that will be available, can be obtained scrutinizing differential distributions in specific kinematical variables. In Section~\\ref{sec:VHprod_ff} we showed that with this respect the (not always measurable) invariant mass of the $Vh$\\ system is the most important observable in this process, since the form factors directly depend on it.\nIn channels where the invariant mass $m_{Vh}$ can not be reconstructed due to the presence of neutrinos, an accessible kinematical proxy exhibiting a sizable correlation with $q^2$ is given by the transverse momentum of the vector boson $\\ensuremath{p_\\mathrm{T,V}}\\xspace$ or, equivalently, that of the Higgs, as can be seen in the Fig.~\\ref{fig:pTZ_qSQ_SM}. Even though this correlation is not as good as the one between the jet $\\ensuremath{p_\\mathrm{T}}\\xspace$ and the momentum transfer in the VBF Higgs production channel, a measurement of the vector boson (or Higgs) $\\ensuremath{p_\\mathrm{T}}\\xspace$ spectrum\nwould still offer important information on the underlying structure of the form factors appearing in Eq.~\\eqref{eq:FFVh}, namely $F_L^{q_i Z}(q^2)$ or $G_L^{q_{ij} W}(q^2)$, see also Ref.~\\cite{Englert:2015hrx}.\nThe invariant mass of the $Vh$\\ system is given by $m_{Vh}^2 = q^2 = m_V^2 + m_h^2 + 2 p_V\\cdot p_h$.\nIn the c.m. frame, we have $p_V = (E_V, \\vec{p}_{\\mathrm{T}}, p_z)$ and $p_h = (E_h, - \\vec{p}_{\\mathrm{T}}, - p_z)$ and\n\\be\n\tm_{Vh}^2 = m_V^2 + m_h^2 + 2 \\ensuremath{p_\\mathrm{T}}\\xspace^2 + 2 p_z^2 + 2\\sqrt{m_V^2 + \\ensuremath{p_\\mathrm{T}}\\xspace^2 + p_z^2}\\sqrt{m_h^2 + \\ensuremath{p_\\mathrm{T}}\\xspace^2 + p_z^2} \\, \\stackrel{|\\ensuremath{p_\\mathrm{T}}\\xspace| \\to \\infty}{\\longrightarrow} \\, 4 \\ensuremath{p_\\mathrm{T}}\\xspace^2~.\n\\ee\nFor $p_z = 0$ this equation gives the minimum $q^2$ for a given $\\ensuremath{p_\\mathrm{T}}\\xspace$, which can be seen as the left edge of the distributions in Fig.~\\ref{fig:pTZ_qSQ_SM}. This is already a valuable information, especially to address the validity of the momentum expansion.\nFor example the boosted Higgs regime utilized in many $b\\bar{b}$ analyses implies a potentially dangerous lower cut-off on \n$q^2$: here a bin with $\\ensuremath{p_\\mathrm{T}}\\xspace > 300$ GeV implies $\\sqrt{q^2} \\gtrsim 630$ GeV, which might be a problem for the validity of the momentum expansion.\n\nIn the $Wh$ process, for a leptonic $W$ boson decay, the $\\ensuremath{p_\\mathrm{T,W}}\\xspace$ can not be reconstructed independently of the Higgs decay channel. \n It is tempting to consider the $\\ensuremath{p_\\mathrm{T}}\\xspace$ of the charged lepton from the $W$ decay as correlated with the $Wh$ invariant mass. \n However, we checked explicitly that any correlation is washed out by the decay.\n\n\\subsection[NLO QCD corrections in VH]{NLO QCD corrections in VH}\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.45\\textwidth]{plots\/ZH_HT2_pT_Z.pdf}\\quad \n \\includegraphics[width=0.45\\textwidth]{plots\/ZH_HT2_minv_HZ.pdf}\\\\\n\\caption{ NLO correction factors and scale uncertainties for $pp\\to ZH$\\ in the SM\nin function of $\\ensuremath{p_\\mathrm{T,Z}}\\xspace$ (left) and $\\ensuremath{m_\\mathrm{HZ}}\\xspace$ employing a central scale $\\mu_0=\\ensuremath{H_\\mathrm{T}\/2}\\xspace$.\n}\n\\label{fig:nlo_vh_ptZ}\n\\end{figure}\n\nAt the inclusive and exclusive level QCD corrections to VH~processes are well under control~\\cite{Dittmaier:2011ti,Dittmaier:2012vm,Heinemeyer:2013tqa}. The dominant QCD corrections of Drell-Yan-like type are known fully differentially up to NNLO~\\cite{Ferrera:2011bk,Ferrera:2013yga,Ferrera:2014lca} and on the inclusive level amount to about $30\\%$ with respect to the LO predictions for both $Wh$ and $Zh$. Remaining scale uncertainties are at the level of a few percent. \n\nIn Fig.~\\ref{fig:nlo_vh_ptZ} we illustrate the NLO QCD corrections to $Zh$ in the SM looking at differential distributions in $p_{\\textrm{T},Z}$ and $m_{Zh}$, while the qualitative picture is very similar for $Wh$. The employed setup is as detailed already in Section~\\ref{sec:vbf_nlo}, while here we do not apply any phase-space cuts. Although the natural scale choice for VH~clearly is $\\mu_0=Q= \\sqrt{(p_{h}+p_{Z})^2}$, here we employ a scale $\\mu_0=\\ensuremath{H_\\mathrm{T}\/2}\\xspace$. With this scale choice the resulting differential distributions (to be utilized in the Higgs PO fit) are almost free of shape effects due to higher-order QCD corrections. A study of a similar stabilization including deformations in the Higgs PO framework will be performed in the near future.\n\n\nIn the case of $Zh$ besides Drell-Yan-like production there are loop-induced contributions in $g g \\to Z h$ mediated by heavy quark loops, which in particular become important in the boosted regime with $\\ensuremath{p_\\mathrm{T,H}}\\xspace>200$~GeV~\\cite{Englert:2013vua,Goncalves:2015mfa}. \n \n\nBesides QCD corrections also EW corrections give relevant contributions and shape effects to VH~processes due to Sudakov logarithms at large energies. They are known at NLO EW \\cite{Denner:2011id,Denner:2014cla} and decrease the LO predictions by about $10\\%$ for $p_{\\textrm{T},Z}=300$~GeV and by about $15\\%$ for $p_{\\textrm{T},W}=300$~GeV. We stress that, as in the VBF case, the dominant NLO EW effects are \nfactorizable corrections which can be reabsorbed into a redefinition of the PO.\n\n\n\n\\subsection[Prospects for the Higgs PO in ${Zh}$ at the HL-LHC]{Prospects for the Higgs PO in $\\boldsymbol{Zh}$ at the HL-LHC}\n\\label{sec:Zh_prospects}\n\nIn order to estimate the reach of the HL-LHC, at 13 TeV and $3000~ \\text{fb}^{-1}$ of integrated luminosity, for measuring the Higgs PO in $Zh$ production, we consider the all-leptonic channel $Z \\to 2\\ell$, $h \\to 2\\ell2\\nu$. \nThe 8 TeV ATLAS search in this channel \\cite{Aad:2015ona} estimated 0.43 signal events with $20.3~\\text{fb}^{-1}$ (Table X of \\cite{Aad:2015ona}). By rescaling the production cross section and the luminosity up to the HL-LHC we estimate approximately $\\sim 130$ signal events at the SM rate.\n Assuming a sample of this size we perform a fit of the $\\ensuremath{p_\\mathrm{T}}\\xspace$ distribution of the $Z$ boson. In order to control the validity of the momentum expansion we apply an upper cut of $\\ensuremath{p_\\mathrm{T}}\\xspace^{\\rm max} = 280~ \\text{GeV}$, which corresponds approximately to $q^2 \\approx 600~ \\text{GeV}$ (see Fig.~\\ref{fig:pTZ_qSQ_SM}). We bin the \\ensuremath{p_\\mathrm{T,Z}}\\xspace distribution as $\\{0-20-40-60-80-100-120-160-200-240-280\\}~\\text{GeV}$. Using \n the {\\rmfamily\\scshape UFO}\\xspace implementation of the PO within {\\rmfamily\\scshape Sherpa}\\xspace we generate $p p \\to Zh$ events at $13~\\text{TeV}$ of c.o.m. energy. As in the VBF case, in each bin we have obtained the expression of the number of events as a quadratic function in the PO:\n\\be\n\tN_a^{\\rm ev} = \\kappa^T X^a \\kappa~, \\quad \\text{where} \\quad\n\t\\kappa = (\\kappa_{ZZ}, \\epsilon_{Zu_L}, \\epsilon_{Zu_R}, \\epsilon_{Zd_L}, \\epsilon_{Zd_R})~,\n\\ee\nwhere $a$ denotes again the label of each bin. We assume the number of events for each bin to follow a Poisson distribution\n and we build the likelihood $L(\\kappa)$ as a function of the five PO listed above. The best-fit point is defined by $L_{\\textrm max}$ and we determine $\\Delta \\chi^2 = -2\\log L \/ L_{\\rm max}$.\nIn Fig.~\\ref{fig:HPO_HLLHC} we show the resulting $1\\sigma$ ($2\\sigma$) intervals for each PO with solid (dashed) blue lines, when all other PO are profiled. The expected bounds obtained in the $Zh$ channel are comparable in strength with the ones obtained in the VBF channel.\nIn Fig.~\\ref{fig:pTZ_2sigma_range} we illustrate the $2\\sigma$ allowed deviation of the $\\ensuremath{p_\\mathrm{T,Z}}\\xspace$ distribution\n\\begin{figure}\n \\begin{center}\n\t\\includegraphics[width=0.447\\textwidth]{plots\/pTZ_2sigma_range} \\quad\n \\includegraphics[width=0.48\\textwidth]{plots\/pTZ_norm_2sigma_range}\n \\end{center}\n\\caption{\\small\\label{fig:pTZ_2sigma_range} Allowed deviations in the $Z$ boson $\\ensuremath{p_\\mathrm{T}}\\xspace$ distribution by varying the PO within the $-2\\log L \/ L_{\\rm max} < 4$ ($2\\sigma$) region. In the left plot we show the absolute number of events in each bin, while in the right one we show the normalized distribution with respect to the total number of events and the bin width.}\n\\end{figure}\n\n\nA fit based on a binning of the $Zh$ invariant mass spectrum provides very similar errors as those shown in Fig.~\\ref{fig:HPO_HLLHC}.\n Again Gaussian errors obtained by expanding the likelihood as a quadratic function around the minimum overestimates the \n errors compared to the ones shown in Fig.~\\ref{fig:HPO_HLLHC}, although here not as badly as in the VBF case: \n\\be\n\tZh: \\qquad \\sigma^{\\rm Gauss}_{\\rm quad}(\\kappa_{ZZ}, \\epsilon_{Z u_L}, \\epsilon_{Z u_R}, \\epsilon_{Z d_L}, \\epsilon_{Z d_R}) \n\t= (0.085, 0.012, 0.014, 0.013, 0.019)\\, .\n\t\\label{eq:Zh_GaussFit_Quadr}\n\\ee\nBy multiplying the number of events in each bin by an overall rate modifier $\\mu$, as done above for the VBF analysis, and profiling over this parameter, we find $\\kappa_{ZZ}$ being unconstrained but the $1\\sigma$ errors \n on the contact terms, in the Gaussian approximation, are exactly the same as the ones before. \n This clearly implies that the bounds on the contact terms arise from the shape information, and not from the rate.\n\n\n\\subsection[Prospects for the Higgs PO in $Wh$ at the HL-LHC]{Prospects for the Higgs PO in $\\boldsymbol{Wh}$ at the HL-LHC}\n\nIn the case of $Wh$ production, in all the channels used for the Run-1 analysis, the signal manifests itself as a small excess over a \nlarge (dominating) background, see e.g.~Ref.~\\cite{Aad:2015ona}. A detailed analysis for such processes should be performed evaluating carefully the backgrounds, which is beyond the scope of this work. However, given the high luminosity we are looking at, the golden channel $h\\to 4\\ell$, $W\\to \\ell \\nu$ becomes an interesting viable possibility. It has been estimated by ATLAS that 67 signal SM events will be present with 3000~fb$^{-1}$ of integrated luminosity~\\cite{Nisati:2015}.\nWe have thus decided to analyze the prospects of this clean channel only, to constrain the $(\\kappa_{WW}, \\epsilon_{W u_L})$ PO,\nwith an analogous likelihood analysis as those performed for the $Zh$ and VBF channels.\n\nWe have studied in particular the $\\ensuremath{p_\\mathrm{T,H}}\\xspace$ distribution, as reference observable, applying the same binning and upper cut as in the $Zh$ analysis discussed above.\nIn Fig.~\\ref{fig:HPO_HLLHC} we show the resulting $1\\sigma$ ($2\\sigma$) intervals for each PO with solid (dashed) green lines, when the other PO is profiled. In this case the Gaussian approximation works well and provides the following $1\\sigma$ errors:\n\\be\n\tWh: \\qquad \\sigma^{\\rm Gauss}_{\\rm quad}(\\kappa_{WW}, \\epsilon_{W u_L}) = (0.11, 0.0032)~.\n\\ee\n\nUpon introducing a total rate modifier $\\mu$, as done for the previous channels, the bound on $\\kappa_{WW}$ vanishes when $\\mu$ is profiled.\nHowever, the constraint on the contact-term PO $\\epsilon_{W u_L}$ remains unchanged, implying that also in this case the bound \n arises from the shape of the $\\ensuremath{p_\\mathrm{T}}\\xspace$ distribution.\n\n\\medskip\n\nWe conclude the last two phenomenological sections stressing that we have performed simplified estimates of the HL-LHC sensitivity on the \ncontact-terms PO by separately considering a limited set of collider signatures. It is reasonable to expect that, including all possible signatures and performing a global fit, the sensitivity can significantly improve. However, such a global analysis should also consider the effect of backgrounds, neglected in this study.\n\n\\section{Validity of the momentum expansion}\n\\label{sec:mom_exp}\n\\label{sect:expansion}\n\nThe most important check to estimate the validity of the momentum expansion is represented by \nthe consistency condition (\\ref{eq:consistency}), where $q^2_{\\rm{max}}$ is controlled \nby $(\\ensuremath{p_\\mathrm{T,j}}\\xspace)^{\\rm max}$ in VBF and $m_{Vh}$ in VH (or, less efficiently, by $\\ensuremath{p_\\mathrm{T,Z}}\\xspace$ and $\\ensuremath{p_\\mathrm{T,H}}\\xspace$ in VH). \nBesides checking this condition \na further check to assess the validity of the momentum expansion is obtained comparing the fit performed including the full quadratic dependence of $N^{\\rm{ev}}_a$ on the PO, with a fit in which the $N^{\\rm{ev}}_a$ are linearized in $\\delta \\kappa_X \\equiv \\kappa_X-\\kappa_X^{\\rm{SM}}$ and $\\epsilon_X$. The idea behind this procedure is that the quadratic corrections to physical observable in $\\delta \\kappa_X$ and $\\epsilon_X$ are formally of the same order as the interference of the first neglected term in Eq.~\\eqref{eq:FLGL} with the leading SM contribution.\n\nIf the two fits (linear vs.~quadratic) \nprovide similar results, one can safely conclude that the terms neglected in the PO decomposition are indeed subleading.\nIn principle, if the two fits yield significantly different results, the difference might be used to estimate the uncertainty due to the neglected \nhigher-order terms in the momentum expansion. In practice, as will be illustrated below, this estimate turns out to be rather \npessimistic and often an overestimate of the uncertainty on the PO. \n\nTo access the feasibility of this check, we perform a linear fit for the VBF Higgs production closely following the procedure described in Sec.~\\ref{sec:VBF_prospect}.\nThe results obtained in the Gaussian approximation are:\n\\be\\begin{split}\n\t{\\rm VBF:} \\qquad \\sigma^{\\rm Gauss}_{\\rm linear}(\\delta\\kappa_{ZZ}, \\delta \\kappa_{WW}, \\epsilon_{Z u_L}, \\epsilon_{Z u_R}, \\epsilon_{Z d_L}, \\epsilon_{Z d_R}, \\epsilon_{W u_L}) =& \\\\\n\t= (1.7, 0.42, 0.30, 0.57, 0.32, 1.0, 0.038)~. &\n\t\\label{eq:VBF_GaussFit_Lin}\n\\end{split}\n\\ee\nComparing those results with Eq.~\\eqref{eq:VBF_GaussFit_Quadr}, we conclude that the bounds on the contact terms in the linearized case are\nsignificantly weaker (typically one order of magnitude less stringent) than those obtained in the quadratic fit. Similar results are obtained for the $Zh$ analysis, while only in the $Wh$ case the two fits give comparable results:\n\\be\\begin{split}\n\tZh: \\qquad & \\sigma^{\\rm Gauss}_{\\rm linear}(\\delta\\kappa_{ZZ}, \\epsilon_{Z u_L}, \\epsilon_{Z u_R}, \\epsilon_{Z d_L}, \\epsilon_{Z d_R}) \n\t= (0.2, 0.14, 0.32, 0.11, 0.35)~, \\\\\n\tWh: \\qquad &\\sigma^{\\rm Gauss}_{\\rm linear}(\\delta\\kappa_{WW}, \\epsilon_{W u_L}) = (0.11, 0.0033)~.\n\t\\label{eq:Vh_GaussFit_Lin}\n\\end{split}\n\\ee \nGiven the events we have simulated are obtained using SM-like distributions, we cannot attribute this large \ndifference to a possible breakdown of the momentum expansion in the underlying distribution. \nWe dedicate the rest of this section to investigate in more detail the origin of the mismatch and how to address it. \n\nThe most likely explanation for the large difference between linear and quadratic fits reported above is the fact that in the linear fit only a few linear combinations of the PO enter the observables, thus reducing the number of independent constraints one can get. This fact, coupled to the large number \nof free parameters in VBF and $Zh$, could explain the loose constraints obtained in the linear fit. \nIf this was true, we should find that in simple models with less parameters the linear and quadratic fit should agree. \n\nTo check if the constraints obtained on the contact terms can, in fact, be used to bound explicit new physics scenarios, we employ \na simple toy model. To this end, we extent the SM with a new neutral vector boson, $Z'$, coupled to specific fermion currents (to be defined below) and to the Higgs, such \nthat it contributes to VBF and VH (or better $Zh$) production.\nSince the goal of this section is to examine the validity of the momentum expansion with an explicit new physics example, we ignore all other phenomenological constraints on such a model (for example, electroweak precision tests, direct searches, etc).\\footnote{~For recent studies about the validity of the momentum expansion in VBF and $Zh$ using similar toy models see Ref.~\\cite{Brehmer:2015rna,Biekoetter:2014jwa}).}\n\nOne the one hand, we compute the bounds on the mass and couplings of this new state from the analysis of the double differential $\\ensuremath{p_\\mathrm{T}}\\xspace$ distribution in VBF Higgs production (and the $\\ensuremath{p_\\mathrm{T}}\\xspace^Z$ distribution in $Zh$). On the other hand, we integrate out the heavy $Z'$ and match to the Higgs PO framework. Finally, we compare the bounds in the full model with the ones obtained from the Higgs PO fit.\n\nTo be more specific, we consider a $Z'$ which contributes to the form factor $F_L^{f f'}$ of $\\langle J_f^\\mu(q_1) J_{f'}^\\nu(q_2) h \\rangle$ as\n\\beq\n\tF_L(q_1^2, q_2^2)^{f f'} = F_{L, {\\rm SM}}^{f f'}(q_1^2, q_2^2) - \\frac{v}{m_Z} g_H \\left[ \\frac{ g_{Z'}^{f} g_Z^{f'} }{P_{Z'}(q_1^2) P_{Z}(q_2^2)} + \\frac{ g_{Z}^{f} g_{Z'}^{f'} }{P_{Z}(q_1^2) P_{Z'}(q_2^2)} \\right]~,\n\t\\label{eq:ZpFormFactor}\n\\eeq\nSuch a contribution could arise, for example, from the following interaction terms,\n\\beq\n\t\\Lag \\supset - 2 g_H m_Z Z^\\mu Z'_\\mu h + \\sum_{f=f_L,f_R} g_{Z'}^{f} \\bar{f} \\gamma^\\mu f Z'_\\mu~,\n\t\\label{eq:Zprime_int}\n\\eeq\n where all the fields are canonically normalized and in the mass basis. \n Using {\\rmfamily\\scshape FeynRules}\\xspace ~\\cite{Christensen:2008py} (package version 1.6.16) we obtain an {\\rmfamily\\scshape UFO}\\xspace~\\cite{Degrande:2011ua} representation of this $Z'$-model and perform exactly the same analysis previously applied to the PO for VBF and VH~production. This allows us to derive bounds on the combination of couplings $g_f \\equiv g_H g_{Z'}^{f}$ for a set of benchmark $Z^\\prime$ masses, $M_{Z'}$. In this simple model\n the $Z'$ only decays to a pair of fermions as well in $Z+h$. The corresponding partial decay widths, assuming the $Z'$ is much heavier than the daughter particles, are\n\\be\n\\Gamma(Z' \\to \\bar f f) = \\frac{N_c ~M_{Z'}}{24 \\pi} |g_{Z'}^{f}|^2~, \\qquad \\Gamma(Z' \\to Z h) =\\frac{M_{Z'}}{48 \\pi} g_H^2~, \n\\label{eq:Zprime_width}\n\\ee\nwhere $N_c$ is the number of colors. In order to simplify the analysis, we assume that the $Z'$ is a narrow resonance \n($\\Gamma_{Z'} \\ll M_{Z'}$). This allows to interpret bounds from the VBF and VH analyses in terms of the $g_f$ parameters. Using the above relations, we have checked that this condition is satisfied for the benchmark scenarios we consider in the following. \nExpanding the form factor from Eq.~\\eqref{eq:ZpFormFactor} for $q_{1}^2 \\ll M_{Z'}^2$ and $\\Gamma_{Z'} \\ll M_{Z'}$ and keeping only the leading deviation from the SM, we find:\n\\beq\n\t\\epsilon_{Z f} = g_H g_{Z'}^{f} \\frac{v m_Z}{M_{Z'}^2} = g_f \\frac{v m_Z}{M_{Z'}^2} ~.\n\t\\label{eq:ZpHPOmatch}\n\\eeq\n\n\n\\subsection[Effect of the $Z'$ in VBF]{Effect of the $\\boldsymbol{Z'}$ in VBF}\n\n\\begin{figure}[t]\n \\begin{center}\n \\includegraphics[width=0.45\\textwidth]{plots\/vbf_Zprime_dRuR_700_ag} \\quad \n \\includegraphics[width=0.428\\textwidth]{plots\/vbf_Zprime_dRuR_2000_ag}\n \\end{center}\n\\caption{\\small\\label{fig:Zpbound_2d} We show the expected 95 \\% CL bound in the plane $(g_{d_R}, g_{u_R}) \\equiv g_H (g_{Z'}^{d_R}, g_{Z'}^{u_R})$ for $M_{Z'} = $ 700 and 2000 GeV on the left and right plots, respectively. All the bounds are obtained analysing 2000 VBF Higgs production events as discussed in Sec.~\\ref{sec:VBF_prospect}. The solid red line represents the bound obtained in the $Z'$ model, while the solid blue (dotted blue) are the bounds obtained in the Higgs PO fit with quadratic (linear) dependence on the PO. }\n\\end{figure}\n \nWe consider the case where the $Z'$ couples to both the down and up right-handed quarks, with two independent couplings, $g_{Z'}^{d_R}$ and $g_{Z'}^{u_R}$. In addition, we fix the $Z'$ mass to two benchmarks values: (a) $700$~GeV and (b) $2000$ GeV.\nThe main results of the analysis are shown in Fig.~\\ref{fig:Zpbound_2d}. \n\nOn the one hand, we perform a fit to the Higgs PO $\\epsilon_{Z u_R}$ and $\\epsilon_{Z d_R}$, while fixing all other PO to zero, and translate this bound on the relevant parameter space of the $Z'$ model, namely the $\\{g_{d_R}, g_{u_R}\\}$ plane. We report the results of the fit obtained with full quadratic dependence on the PO, as well as the results in which $N_{\\rm{ev}}$ is linearized in $\\delta \\kappa_X$ and $\\epsilon_X$. In both cases, $95\\%$ CL bounds are obtained by requiring $-2\\log L \/ L_{max} \\le 5.99$. \nOn the other hand, using exactly the same binning and statistical treatment, we directly fit the $Z'$ model parameters.\n\n\nComparing the two methods we conclude: (i) for both masses the quadratic PO fit provides a reasonable approximation of the model fit, while the \nlinear fit largely overestimates the errors; (ii) the PO fit performs better for $M_{Z'} = 2000$~GeV than for $M_{Z'} =700$~GeV, as expected from the momentum expansion validity arguments (we recall that we set the cut $\\ensuremath{p_\\mathrm{T,j}}\\xspace < 600$~GeV); however, also for $M_{Z'} = 700$~GeV the quadratic fit does provide a fair approximation to the model fit. In particular, in this case we see that the bound from the PO fit is stronger than in the model, which can be understood by the fact that in VBF the $Z'$ is exchanged in the $t$-channel, and therefore its main effect is to reduce the amplitude for high values of $q^2$.\n\n\n\\subsection[Effect of the ${Z'}$ in $Zh$]{Effect of the $\\boldsymbol{Z'}$ in $Zh$}\n\n\\begin{figure}[t]\n \\begin{center}\n \\includegraphics[width=0.455\\textwidth]{plots\/Zh_partonic_ZprimeLight} \\quad \n \\includegraphics[width=0.47\\textwidth]{plots\/Zh_partonic_ZprimeHeavy}\n \\end{center}\n\\caption{\\small\\label{fig:Zp_Zh_partonic} Partonic cross section $d \\bar{d} \\to Z h$ as a function of the invariant mass $m_{Zh}$ in the SM (dashed gray line) and with a $Z^\\prime$ coupled to right-handed down quarks only. With red lines we show the cross section computed in the full model while the blue ones represent the cross section using the PO decomposition --with matching conditions in\nEq.~\\eqref{eq:ZpHPOmatch}-- using the full dependence (solid line) or only the linear one (dashed line). In the left plot we consider the benchmark light\n $Z^\\prime$ scenario: $M_{Z'} = 700~\\text{GeV}$, $\\Gamma_{Z'} = 100~ \\text{GeV}$ and $g_{d_R} = 0.367$. In the right plot we consider the heavy $Z'$ scenario: $M_{Z'} = 2000~\\text{GeV}$, $\\Gamma_{Z'} = 200~ \\text{GeV}$ and $g_{d_R} = 3$. Both benchmarks give rise to the same contact term: $\\epsilon_{Z d_R} \\simeq 1.68 \\times 10^{-2}$.}\n\\end{figure}\n\nIn order to assess the validity of the momentum expansion in associated production, it is convenient to look first at the underlying partonic cross section. In Fig.~\\ref{fig:Zp_Zh_partonic} we show the partonic cross section $d \\bar{d} \\to Z h$, \nas a function of the $Zh$ invariant mass, for the two benchmark points of $Z'$ model introduced above. \n\nBoth benchmark points have been chosen such that they generate the same contact term when the $Z'$ is integrated out, $\\epsilon_{Z d_R} = 1.68 \\times 10^{-2}$, which is within the $2\\sigma$ bound of our PO fit.\nThe width of the $Z'$ has been fixed to $100~\\text{GeV}$ and $200~\\text{GeV}$ for the light and heavy scenario, respectively. Using Eq.~\\eqref{eq:Zprime_width} and assuming no other decay mode is present, this corresponds to $g_H \\simeq 0.097 ~ (3.0)$ in the light (heavy) scenario. We have checked that our conclusions do no change by varying the total width, as long as the condition $\\Gamma_{Z'} \\ll M_{Z'}$ is satisfied.\n\n\\begin{figure}[t]\n \\begin{center}\n \\includegraphics[width=0.45\\textwidth]{plots\/Zprime700_Zh} \\quad \n \\includegraphics[width=0.43\\textwidth]{plots\/Zprime2000_Zh}\n \\end{center}\n\\caption{\\small\\label{fig:Zpbound_2d_Zh} Expected 95 \\% CL bound in the plane $(g_{d_R}, g_{u_R}) \\equiv g_H (g_{Z'}^{d_R}, g_{Z'}^{u_R})$ for $M_{Z'} = $ 700 and 2000 GeV on the left and right plots, respectively. All the bounds are obtained analysing 130 $Zh$ Higgs production events as discussed in Sec.~\\ref{sec:Zh_prospects}. The solid red line represents the bound obtained in the full model, while the solid blue (dashed blue) are the bounds obtained via the matching in eq.~\\eqref{eq:ZpHPOmatch} from the Higgs PO fit with quadratic (linear) dependence on the PO.}\n\\end{figure}\n\n\nAs expected, in the light scenario the cross section in the full model strongly deviates from the PO one well before the $600~\\text{GeV}$ cutoff imposed in the fit, implying that our PO fit is not reliable in this case.\nOn the other hand, the scenario with a heavy and strongly coupled $Z'$ shows a very good agreement with the full PO analysis up to $\\sim 1~\\text{TeV}$, i.e. well above the UV cutoff of our analysis, implying that the analysis can be safely applied to such scenarios, and that it could be even improved by setting a slighly higher cutoff.\nIn both cases, from Fig.~\\ref{fig:Zp_Zh_partonic} is clear that the linearized dependence on the PO is not sufficient to describe the cross section, even for energies much smaller than the $Z'$ mass.\n\nFrom this analysis we can anticipate the results of a comparison of various fits of $Zh$ data, i.e.~full model fit vs.~PO fits using quadratic and linear dependence, as already done in the VBF case. In Fig.~\\ref{fig:Zpbound_2d_Zh} we show the results of such fits. We stress that in all cases the analysis was exactly the same: we have analyzed the $\\ensuremath{p_\\mathrm{T}}\\xspace^Z$ distribution up to $280~\\text{GeV}$, employing always the same binning \n(as discussed in Sec.~\\ref{sec:Zh_prospects}). The solid red line represents the 95 \\% CL bound in the full model while the solid (dashed) blue line shows the bound obtained from the PO fit with quadratic (linear) dependence. \n\nThe distributions in Fig.~\\ref{fig:Zp_Zh_partonic} allow a straightforward interpretation of these results. In the heavy-$Z^\\prime$ case, the full quadratic expansion in the Higgs PO describes very well the $m_{Zh}$ distribution before the cutoff of $600~\\text{GeV}$, while keeping only the linear dependence underestimates the new physics contribution. It is thus expected that in this case the bound will be much worse.\nIn the light-$Z^\\prime$ case, both expansions with Higgs PO underestimate the cross section, thus providing a worse bound than in the full model. Still, the quadratic dependence does a significantly better job in approximating the complete model than the linear one, as in the VBF case. \n\n\\medskip\n\nFrom this illustrative toy-model example we can draw the following general conclusion with respect to the validity of the PO expansion: \nfor underlying models that respect the momentum expansion, hence for models where the PO extracted from \ndata satisfy, a posteriori, the consistency condition (\\ref{eq:consistency}), the \nquadratic fit provides more reliable and thus more useful constraint on the PO. In such models the difference between quadratic and linear fit represents a large overestimate of the errors.\n\nHowever, the situation is more involved for models with low-scale new physics. The latter should manifest by anomalously \nlarge values of the PO, or sizable differences in the fits performed with different upper $\\ensuremath{p_\\mathrm{T}}\\xspace$ cuts. In such cases \nthe quadratic fit is likely to provide a useful constraint, especially for the class of models with a strong correlation between \nlinear and quadratic terms in the momentum expansion (as the simple $Z^\\prime$ model discussed above). \nStill, for low-scale new-physics we cannot exclude more complicated scenarios where new model parameters appearing \nat higher order in the momentum expansion wash-out an apparent small error on the PO from the quadratic fit.\nIn such cases only the the results of the linear fit (with a properly low $\\ensuremath{p_\\mathrm{T}}\\xspace$ cut) would provide an unbiased constraint on the model. \n\nIn view of these arguments, we encourage the experimental collaborations to report the results of both linear and quadratic fits,\nas well as to perform such fits using different $\\ensuremath{p_\\mathrm{T}}\\xspace$ cuts.\n\n\n\\section{Conclusions }\n\\label{sect:Conc}\n\n\nHiggs physics is entering the era of precision measurements: future\nhigh-statistics data will allow us not only to determine the overall signal\nstrengths of production and decay processes relative to the SM, but also to\nperform detailed kinematical studies. In this perspective, an accurate and\nsufficiently general parameterization of possible NP effects in such\ndistributions is needed. In this paper we have shown how this goal can be\nachieved in the case of VBF and VH production, generalizing the concept of Higgs\nPO already introduced in Higgs decays.\n \nAs summarized in Table~\\ref{tab:POsumm}, the number of additional PO appearing\nin all VBF and VH production amplitudes is manageable.\nIn particular, assuming CP invariance, flavor and custodial symmetry,\nonly 4 new PO should be added to the set of 7 PO appearing in $h\\to 4\\ell, 2\\ell\n2\\nu, 2 \\ell \\gamma, 2\\gamma$ in the same symmetry\nlimit~\\cite{Gonzalez-Alonso:2014eva}. This opens the possibility of precise global determinations of the PO from combined analyses\nof production and decay modes, already starting from the next LHC runs.\n \nAs extensively illustrated in Sections~\\ref{sect:VBF} and \\ref{sect:VH}, the key\naspects of VBF and VH is the possibility of exploring sizable momentum transfers\nin the Green functions of Eq.~(\\ref{eq:corr_func}). On the one hand, this\nmaximizes the sensitivity of such processes to PO that are hardly accessible in\nHiggs decays. On the other hand, it allows us to test the momentum expansion\nthat is intrinsic in the PO decomposition as well as in any EFT approach to\nphysics beyond the SM.\nKey ingredients to reach both of these goals are precise differential measurements of\n ${\\rm d}^2\\sigma\/{\\rm d}\\ensuremath{p_\\mathrm{T,j_1}}\\xspace {\\rm\nd}\\ensuremath{p_\\mathrm{T,j_2}}\\xspace$ in VBF and ${\\rm d}\\sigma\/{\\rm d} m_{Vh}$ in VH\n(or appropriate proxies such as\n$\\ensuremath{p_\\mathrm{T,H}}\\xspace$ and $\\ensuremath{p_\\mathrm{T,Z}}\\xspace$). We thus encourage\nthe experimental collaborations to directly report such differential\ndistributions, especially in the kinematical regions corresponding to high\nmomentum transfer.\n\nAs far as the PO fits in VBF an VH are concerned, we suggest to perform them\nsetting a maximal cut on $\\ensuremath{p_\\mathrm{T,j}}\\xspace$ and $m_{Vh}$, to ensure (and verify a posteriori) the validity of the\nmomentum expansion. As illustrated by matching the PO framework to simplified\ndynamical NP models, it is also important to report the results of fits using\nboth linearized and quadratic expressions for the cross-sections in terms of PO.\nAccording to our preliminary estimates, the production PO could be measured at\nthe percent level at the HL-LHC (in the case of maximal flavor symmetry, without\nthe need of imposing custodial symmetry). This level would be sufficient to\nconstrain (or find evidences) of a wide class of explicit NP models and, among\nother things, to perform non-trivial tests of the relations between electroweak\nobservables and Higgs PO expected in the SMEFT.\n\n\\subsection*{Acknowledgements}\n\nWe would like to thank S. H\\\"oche and S. Kuttimalai for help with the UFO interface of Sherpa. Also we would like thank \nM. Duehrssen-Debling, S. Pozzorini, and A. Tinoco Mendes for useful discussions.\nThis research was supported in\npart by the Swiss National Science Foundation (SNF) under contract 200021-159720.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n Intermetallic binary alloys between the two\n elements of A (Hf\/Zr\/Ti) and B (Fe\/Co\/Ni)\n have many technological applications. They\n have properties like superior strength, corrosion\n resistance, hydrogen storage capacity, ferromagnetism\n and have applications in space craft industry,\n fuel cell etc. The Zr-Ni alloys, particularly,\n have excellent hydrogen storage properties and\n have applications in Ni-metal hydride (MH)\n rechargeable batteries as a negative electrode \\citep{JMJoubert,Ruiz,FCRuiz}. The gaseous hydrogen storage characteristics\n of four intermetallic compounds in the Zr-Ni\n system viz. Zr$_9$Ni$_{11}$, Zr$_7$Ni$_{10}$, Zr$_8$Ni$_{21}$ and\n Zr$_2$Ni$_7$ were compared by Joubert et al. \\citep{JMJoubert}.\n It was found that hydrogen storage\n capacities in hydrogen atoms per metal atom\n (H\/M) for these four compounds are\n 0.93 (10 bar), 1.01 (10 bar), 0.34 (25 bar) and 0.29 (25 bar), respectively,\n at room temperature and the storage capacities for Zr$_8$Ni$_{21}$,\n Zr$_2$Ni$_7$ are completely reversible.\n The hydrogen reversible capacities for\n Zr$_9$Ni$_{11}$ and Zr$_7$Ni$_{10}$, are 50\\% and 77\\%, respectively.\n The electrochemical properties of Zr$_7$Ni$_{10}$,\n Zr$_9$Ni$_{11}$ and Zr$_8$Ni$_{21}$ were studied by Ruiz et al.\n \\citep{Ruiz,FCRuiz} and Nei et al. \\citep{Regmi,S.O.Salley}. It was found \\citep{FCRuiz}\n that Zr$_8$Ni$_{21}$ alloy had a better charge\/discharge\n performance than Zr$_7$Ni$_{10}$ and Zr$_9$Ni$_{11}$. A detailed\n review on the electrochemical properties of various\n compounds of Zr-Ni system for application of these materials\n in Ni-MH battery was reported by Young et al. \\citep{Young}.\n The intermetallic compound ZrNi$_5$ was reported to\n have strong ferromagnetic properties by\n Drulis et al. \\citep{Drulis}. A. Amamou \\citep{Amamou} reported the\n electronic structure of various compounds in\n the Zr-Ni system, namely, Zr$_2$Ni, ZrNi, Zr$_8$Ni$_{21}$\n and ZrNi$_5$. Both magnetic and\n structural properties of the A$_x$B$_y$ compounds\n can be studied experimentally by the nuclear technique of perturbed\n angular correlation (PAC), which uses a suitable\n radioactive isotope (usually $^{181}$Hf) to characterize the materials. Using this technique, \n different intermetallic\n compounds in the Zr-Ni system have recently \n been studied, namely, ZrNi$_5$ \\citep{Dey,SilvaJMMM}, Zr$_2$Ni$_7$ \\citep{CCDeyPhysica} and ZrNi \\citep{CCDeyJPCS}.\n From the studies in ZrNi$_5$ system \\citep{Dey,SilvaJMMM}, however, no magnetic\n interaction was found, in contradiction with the result from previous measurements by Drulis et al. \\citep{Drulis}. \n \n Considering the important applications\n of Zr$_8$Ni$_{21}$, Zr$_9$Ni$_{11}$ and Zr$_7$Ni$_{10}$ as described above,\n we have done detailed PAC measurements in Zr$_8$Ni$_{21}$\n and Hf$_8$Ni$_{21}$ intermetallic compounds. To the best of our knowledge,\n there is no measurement by PAC technique to characterize\n these materials. From previous studies \\citep{Shen}, a pure\n single phase Zr$_8$Ni$_{21}$ was not found to be produced by\n arc melting preparation. It was found \\citep{Shen} that\n Zr$_8$Ni$_{21}$ was not formed congruently from the\n liquid. The Zr$_2$Ni$_7$ was first solidified from the liquid\n and then reacted with the remaining liquid to form\n Zr$_8$Ni$_{21}$ alloy peritectically. The Zr$_7$Ni$_{10}$ was formed eutectically from Zr$_8$Ni$_{21}$. The two other phases\n viz. Zr$_2$Ni$_7$ and Zr$_7$Ni$_{10}$ were produced along with\n Zr$_8$Ni$_{21}$ and were confirmed by scanning electron\n microscopy (SEM)\/X-ray energy dispersive spectroscopy (EDS)\n compositional mapping and transmission electron\n microscopy (TEM) \\citep{Shen}. However,\n in the present report, we have studied both Zr$_8$Ni$_{21}$ and\n Hf$_8$Ni$_{21}$ to\n identify and characterize the different phases produced in these compounds. The secondary\n phases of small fractions that are produced along with the main\n phase can be determined quite accurately by this technique. The structural and compositional stability of Zr$_8$Ni$_{21}$\/Hf$_8$Ni$_{21}$ phases have been \n studied by temperature dependent PAC measurements.\n \n The crystal structure of Zr$_8$Ni$_{21}$ is known to be\n triclinic \\citep{JM} and is isotropic to that of Hf$_8$Ni$_{21}$ \\citep{Bsenko}.\n The lattice parameters of Zr$_8$Ni$_{21}$ are\n $a$=6.476 \\AA, $b$=8.064 \\AA,\n $c$=8.594 \\AA, $\\alpha$=75.15$^\\circ$, $\\beta$=68.07$^\\circ$ and $\\gamma$=75.23$^\\circ$ as\n determined by X-ray diffraction analysis.\n \n In the PAC technique \\citep{Schatz,Catchen,Zacate}, the angular correlation\n of a $\\gamma$-$\\gamma$ cascade in a suitable probe nucleus is perturbed by the interaction of the probe nuclear moments \n with the electric\n field gradients\/magnetic fields generated at\n the probe site due to surrounding charge distribution.\n The crystalline electric field gradient (EFG)\n and the internal magnetic field in a\n magnetic material can be determined by the PAC\n technique if the values of electromagnetic moments\n of the intermediate level of the probe nucleus are known. As the EFG depends on the charge distribution of the probe-nucleus \n environment, the temperature evolution of the \n lattice properties such as crystallographic structure, \n imperfections or defects, can be monitored by applying PAC technique over a wide temperature range. \n The combination of PAC measurements and ab-initio calculations proved to be an excellent method\n to study the structural phase stabilities and the localization of the impurities in the host lattice \\citep{Errico, Wodniecki}.\nIn this paper, results of temperature dependent PAC measurements (77-1073 K) in both Zr$_8$Ni$_{21}$ and Hf$_8$Ni$_{21}$ as well\nas DFT calculations are reported. The calculated EFG values at the $^{181}$Ta impurity sites are compared with experimental results.\n\n\\section{Experimental details}\n\nThe intermetallic compounds Zr$_8$Ni$_{21}$ and Hf$_8$Ni$_{21}$ were prepared by arc melting in an argon atmosphere. Stoichiometric amounts of high purity metals \nprocured from M\/S Alfa Aesar were used to prepare the samples. The purity of Zr (excluding Hf), Hf (excluding Zr) and Ni metals used were 99.2\\%, \n99.95\\% and 99.98\\% respectively. For each sample, the constituent metals were alloyed homogeneously by repeated melting and then\nactivated \nby remelting with a piece of $^{181}$Hf metal. The $^{181}$Hf metal was prepared by thermal neutron capture of natural $^{180}$Hf metal \nin the Dhruba reactor, Mumbai using a flux $\\sim$10$^{13}$\/cm$^2$\/s. In both cases, shiny globule samples were formed and these \nwere then sealed in evacuated quartz tubes for high temperature measurements. Different inactive samples of Zr$_8$Ni$_{21}$ and Hf$_8$Ni$_{21}$ were\nprepared similarly \nfor XRD measurements. The X-ray powder diffraction measurements have been carried out using the Rigaku X-ray diffractometer TTRAX-III and Cu K$_\\alpha$ radiation.\n\nThe TDPAC technique measures the effect of perturbations of $\\gamma$-$\\gamma$ angular correlation of the probe nucleus through the hyperfine \ninteraction. In the present case, the probe $^{181}$Hf substitutes the Zr atom in Zr$_8$Ni$_{21}$ and is a constituent element in Hf$_8$Ni$_{21}$. \nIn the $\\beta^-$ decay of $^{181}$Hf, it\npopulates the 615 keV excited level of $^{181}$Ta which emits two successive $\\gamma$ rays, 133 and 482 keV\npassing through the 482 keV level with a half-life 10.8 ns and a spin angular momentum $I$=5\/2$^+\\hbar$ \\citep{Firestone}. The angular \ncorrelation of the 133-482 keV cascade is perturbed by the extranuclear electric field gradients.\n \nThe perturbation function $G_2(t)$ for the electric quadrupole interaction in a polycrystalline material is given by \\citep{Schatz} \n \\begin{equation}\n G_2(t)=S_{20}(\\eta) + \\sum^{3}_{i=1}S_{2i}(\\eta)cos(\\omega_it)exp(-\\delta\\omega_it)exp\\big[\\frac{-(\\omega_i\\tau_R)^2}{2}\\big]\n \\label{eqn:Stokes}\n \\end{equation}\nThe frequencies $\\omega_i$ correspond to transitions between the sub-levels of the intermediate state that arise due to \nnuclear quadrupole interaction (NQI). \nThe parameter $\\delta$ is the\nfrequency distribution width (Lorentzian damping) which takes care of the chemical inhomogeneities \nin the sample and $\\tau_R$ is the time resolution of the coincidence set up. Due to the presence of \nvarious non-equivalent sites, the perturbation factor $G_2$(t) can generally be expressed as \n \\begin{equation}\n G_2(t)=\\sum_if_iG^{i}_{2}(t),\n \\label{eqn:Bhatnagar}\n \\end{equation}\n where $f_i$ is the site fraction of the\n $i$-th component. \nA fitting to eqn. ($\\ref{eqn:Stokes}$) determines the maximum component $V_{zz}$ of the \nelectric field gradient from the measured quadrupole frequency $\\omega_Q$ given by\n\\begin{equation}\n\\omega_Q= \\frac{eQV_{zz}}{4I(2I-1)\\hbar},\n \\label{eqn:raman}\n\\end{equation}\nwhere $Q$ is the nuclear quadrupole moment of the 482 keV intermediate state (2.36 b). \nFor an axially symmetric EFG ($\\eta=0$), $\\omega_Q$ is related to $\\omega_1$, $\\omega_2$ and $\\omega_3$ by \n\\begin{equation}\n \\omega_Q=\\omega_1\/6=\\omega_2\/12=\\omega_3\/18.\n \\label{eqn:prafulla}\n\\end{equation}\nThe principal EFG components obey the relations\n\\begin{equation}\nV_{xx} + V_{yy} + V_{zz}=0 \\quad \\text{and}\\quad\nV_{zz}\\ge V_{yy}\\ge V_{xx}.\n \\label{eqn:hizenberg}\n\\end{equation} \n The EFG can therefore be designated by two parameters only viz. $V_{zz}$ and $\\eta$. The asymmetry parameter $\\eta$ is defined as \n \\begin{equation}\n \\eta=\\frac{(V_{xx}-V_{yy})}{V_{zz}},\\quad \\text{}\\quad 0\\le\\eta\\le1.\n \\label{eqn:newton}\n \\end{equation} \n \n The TDPAC spectrometer used for present measurements was a four detector\n LaBr$_3$(Ce)-BaF$_2$ set up with crystal sizes 38$\\times$25.4 mm$^2$ for LaBr$_3$(Ce) and 50.8$\\times$50.8 mm$^2$ for \n BaF$_2$. The 133 keV $\\gamma$-rays were selected in LaBr$_3$(Ce) detectors. \n Standard slow-fast\n coincidence assemblies were employed to collect data at 180$^\\circ$ and 90$^\\circ$. \n Typical prompt time resolution (FWHM)$\\sim$800 ps was obtained for the $^{181}$Hf energy window settings. \n The perturbation function $G_2$($t$) is found from the four coincidence spectra at 180$^\\circ$ and 90$^\\circ$.\n Details of the experimental set up and data analysis can be found in reference \\citep{pramana}.\n \n \\begin{table}[t!]\n\\begin{center} \n\\captionof{table}{\\small{ Results of PAC measurements in Zr$_8$Ni$_{21}$}}\n\\scalebox{0.65}{\n\\begin{tabular}{ l l l l l l l l } \n \\hline \\\\ [-0.9ex]\nTemperature (K) &Component & $\\omega_Q$ (Mrad\/s) & $\\eta$ & $\\delta$($\\%$) & $f$($\\%$) & Assignment \\\\ [1.5ex]\n \\hline \\\\ \n \n77 &1 & 77.9(4) & 0.80(1) & 2(1) & 55(3) & Zr$_8$Ni$_{21}^{(1)}$ \\\\ \n &2 & 55.8(7) & 0.68(2) & 0 & 26(3) & Zr$_7$Ni$_{10}$ \\\\ \n &3 & 101.5(9) & 0.73(3) & 0 & 18(3) & Zr$_8$Ni$_{21}^{(2)}$ \\\\ \\\\ \n\n298 &1 & 75.8(2) & 0.77(1) & 1.1(6) & 57(3) & Zr$_8$Ni$_{21}^{(1)}$ \\\\ \n &2 & 54.4(4) & 0.69(1) & 0 & 27(3) & Zr$_7$Ni$_{10}$ \\\\ \n &3 & 100.6(8) & 0.72(3) & 0 & 15(3) & Zr$_8$Ni$_{21}^{(2)}$ \\\\ \\\\ \n\n373 &1 & 74.6(6) & 0.75(2) & 2(1) & 53(3) &Zr$_8$Ni$_{21}^{(1)}$ \\\\ \n &2 & 53(2) & 0.71(3) & 0 & 26(3) & Zr$_7$Ni$_{10}$ \\\\ \n & 3 & 97(2) & 0.76(4) & 0 & 21(3) & Zr$_8$Ni$_{21}^{(2)}$ \\\\ \\\\\n \n473 &1 & 73.1(5) & 0.76(2) & 2.4(8) & 63(3) & Zr$_8$Ni$_{21}^{(1)}$ \\\\ \n &2 & 52.1(7) & 0.72(4) & 0 & 37(3) & Zr$_7$Ni$_{10}$ \\\\ \\\\ \n \n573 &1 & 71.8(3) & 0.77(1) & 2.5(6) & 62(3) & Zr$_8$Ni$_{21}^{(1)}$ \\\\ \n &2 & 52.9(5) & 0.70(2) & 0 & 38(3) & Zr$_7$Ni$_{10}$ \\\\ \\\\ \n \n673 &1 & 70.1(6) & 0.77(2) & 4.5(8) & 71(3) & Zr$_8$Ni$_{21}^{(1)}$ \\\\ \n &2 & 50.8(4) & 0.72(2) & 0 & 29(3) & Zr$_7$Ni$_{10}$ \\\\ \\\\ \n\n773 &1 & 68(1) & 0.73(4) & 4(2) & 64(3) & Zr$_8$Ni$_{21}^{(1)}$ \\\\ \n &2 & 48.4(9) & 0.75(5) & 0 & 36(3) & Zr$_7$Ni$_{10}$ \\\\ \\\\\n \n873 &1 & 67(3) & 0.79(10) & 8(4) & 72(3) & Zr$_8$Ni$_{21}^{(1)}$ \\\\ \n &2 & 48.8(8) & 0.73(5) & 0 & 28(3) & Zr$_7$Ni$_{10}$ \\\\ \\\\ \n \n973 &1 & 65(2) & 0.80(7) & 7(3) & 71(3) &Zr$_8$Ni$_{21}^{(1)}$\\\\ \n &2 & 47.3(6) & 0.74(4) & 0 & 29(3) &Zr$_7$Ni$_{10}$ \\\\ \\\\ \n\n1073 &1 & 52.8(2) & 0 & 1.3(7) & 68(3) & Hf \\\\ \n &2 & 47(1) & 0.71(9) & 5(3) & 32(3) & Zr$_7$Ni$_{10}$ \\\\ \\\\\n \n298$^*$ &1 & 56.8(2) & 0 & 4.0(9) & 70(3) & Hf \\\\ \n &2 & 74.4(8) & 0.79(1) & 0 & 23(3) & Zr$_8$Ni$_{21}^{(1)}$ \\\\ \n &3 & 74(2) & 0.21(10) & 0 & 7(3) & Zr$_2$Ni$_7$ \\\\ \n\n\\hline\n\\end{tabular}}\n\\begin{flushleft}\n \\small{$^*$ after measurement at 1073 K}\\\\\n\\end{flushleft}\n\\label{tab:Zr8Ni21table}\n\\end{center}\n\\end{table}\n\n \\begin{figure*}[t!]\n \\centering\n \\begin{subfigure}[t]{0.45\\textwidth}\n\\centering\n\\includegraphics[scale=.28]{Zr8Ni21_XRD_26Sept2016.eps}\n\\label{XRD_Zr8Ni21}\n\\end{subfigure}\n \\begin{subfigure}[t]{0.45\\textwidth}\n\\centering\n\\includegraphics[scale=.28]{Zr8Ni21_annealed_XRD_26Sept2016.eps}\n\n\\end{subfigure}\n\\caption{ The XRD spectra in Zr$_8$Ni$_{21}$. Figure (a) shows the spectrum in as prepared sample and figure (b) shows the spectrum in a \nsample annealed at 1073 K for two days. The line represents the fit to the measured data, \nthe vertical bars denote the Bragg angles and\nthe bottom line shows the difference between the observed and the fitted pattern. }\n\\label{XRD_Zr8Ni21}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[scale=.65]{Zr8Ni21_PAC_plot_28Sept2018.eps}\n\\end{center}\n\\caption{\\small{TDPAC spectra in Zr$_8$Ni$_{21}$ at different temperatures. Left panel shows the time spectra and\nthe right panel shows the corresponding Fourier transforms. The PAC spectrum designated\nby 298$^*$ K is taken after the measurement at 1073 K. The two sets of arrows in each\nFourier spectrum (up to 373 K) correspond to two non-equivalent $^{181}$Ta sites\nin Zr$_8$Ni$_{21}$. Arrows shown in the Fourier spectra at 1073 and 298$^*$ K correspond to Hf.}}\n\\label{Zr8Ni21TDPAC}\n\\end{figure*}\n\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[scale=.25]{Zr8Ni21_para_plot_30Sept2016_1.eps}\n\\end{center}\n\\caption{Variations of quadrupole frequency ($\\omega_Q$), asymmetry parameter ($\\eta$) and site fraction $f$(\\%)\nwith temperature for the two non-equivalent $^{181}$Ta sites in Zr$_8$Ni$_{21}$. Variation of $\\delta$ is shown for \nthe component Zr$_8$Ni$_{21}^{(1)}$.}\n\\label{Zr8Ni21parameter}\n\\end{figure*}\n\n \\section{PAC results}\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[scale=.40]{Hf8Ni21_XRD_26Sept2016.eps}\n\\end{center}\n\\caption{The background subtracted XRD powder pattern in Hf$_8$Ni$_{21}$. The line represents the fit to the measured data, \nthe vertical bars denote the Bragg angles and\nthe bottom line shows the difference between the observed and the fitted pattern.}\n\\label{XRD_Hf8Ni21}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[scale=.65]{Hf8Ni21_PAC_plot_28Sept2018.eps}\n\\end{center}\n\\caption{\\small{\nTDPAC spectra in Hf$_8$Ni$_{21}$ at different temperatures. Left panel shows the time spectra and the right panel shows the\ncorresponding Fourier transforms. The PAC spectrum designated by 298$^*$ K is taken after the measurement at 1073 K. \nThe two sets of arrows in each Fourier spectrum (up to 773 K) correspond to two\ndifferent $^{181}$Ta sites in Hf$_8$Ni$_{21}$. Two sets of arrows in the Fourier spectrum at 298$^*$ K correspond to HfNi$_3$ and Hf.}}\n\\label{Hf8Ni21TDPAC}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[scale=.25]{Hf8Ni21_para_plot_30Sept2016_1.eps}\n\\end{center}\n\\caption{Variations of quadrupole frequency ($\\omega_Q$), asymmetry parameter ($\\eta$) and site fraction $f$(\\%) with\ntemperature for the two non-equivalent $^{181}$Ta sites in Hf$_8$Ni$_{21}$. \nVariations of $\\delta$ is shown for the site Hf$_8$Ni$_{21}^{(2)}$.}\n\\label{Hf8Ni21parameter}\n\\end{figure*}\n\n\\begin{table}[t!]\n\\begin{center} \n\\captionof{table}{\\small{ Results of PAC measurements in Hf$_8$Ni$_{21}$}}\n\\scalebox{0.65}{\n\\begin{tabular}{ l l l l l l l l } \n \\hline \\\\ [-0.9ex]\nTemperature (K) &Component & $\\omega_Q$ (Mrad\/s) & $\\eta$ & $\\delta$($\\%$) & $f$($\\%$) & Assignment \\\\ [1.5ex]\n \\hline \\\\ \n \n77 &1 & 99.3(3) & 0.62(1) & 2.5(5) & 56(3) & Hf$_8$Ni$_{21}^{(2)}$ \\\\ \n &2 & 80.7(9) & 0.73(3) & 0 & 16(3) & Hf$_8$Ni$_{21}^{(1)}$ \\\\ \n &3 & 32.3(6) & 0 & 0 & 19(3) & HfNi$_3$\\\\ \n &4 & 52(2) & 0 & 0 & 8(3) & Hf \\\\ \\\\ \n\n298 &1 & 97.3(3) & 0.63(1) & 2.3(4) & 63(3) & Hf$_8$Ni$_{21}^{(2)}$ \\\\ \n &2 & 79.1(7) & 0.75(2) & 0 & 21(3) & Hf$_8$Ni$_{21}^{(1)}$ \\\\ \n &3 & 31(1) & 0 & 0 & 10(3) & HfNi$_3$ \\\\ \n &4 & 50(2) & 0 & 0 & 5(3) &Hf \\\\ \\\\ \n\n373 &1 & 95.3(6) & 0.61(2) & 2(1) & 66(3) &Hf$_8$Ni$_{21}^{(2)}$ \\\\ \n &2 & 76(3) & 0.74(1) & 0 & 13(3) & Hf$_8$Ni$_{21}^{(1)}$ \\\\ \n &3 & 31(2) & 0 & 0 & 15(3) & HfNi$_3$ \\\\\n & 4 & 50(4) & 0 & 0 & 7(3) & Hf \\\\ \\\\\n \n473 &1 & 95.5(5) & 0.62(1) & 3(fixed) & 61(3) & Hf$_8$Ni$_{21}^{(2)}$ \\\\ \n &2 & 77(1) & 0.74(3) & 0 & 16(3) & Hf$_8$Ni$_{21}^{(1)}$ \\\\ \n &3 & 28.6(7) & 0 & 0 & 19(3) & HfNi$_3$ \\\\ \n &4 & 45(4) & 0 & 0 & 4(3) & Hf \\\\ \\\\ \n \n573 &1 & 92.1(3) & 0.60(1) & 3.2(5) & 61(3) & Hf$_8$Ni$_{21}^{(2)}$ \\\\ \n &2 & 75(1) & 0.71(3) & 0 & 14(3) & Hf$_8$Ni$_{21}^{(1)}$ \\\\ \n &3 & 27.9(5) & 0 & 0 & 19(3) & HfNi$_3$ \\\\ \n &4 & 46(2) & 0 & 0 & 5(3) & Hf \\\\ \\\\ \n \n673 &1 & 87.7(5) & 0.61(1) & 3(1) & 59(3) & Hf$_8$Ni$_{21}^{(2)}$ \\\\ \n &2 & 73(1) & 0.66(4) & 0 & 17(3) & Hf$_8$Ni$_{21}^{(1)}$ \\\\ \n &3 & 27.5(9) & 0 & 0 & 16(3) & HfNi$_3$ \\\\ \n &4 & 43(2) & 0 & 0 & 8(3) & Hf \\\\ \\\\ \n\n773 &1 & 89.1(6) & 0.56(1) & 6(1) & 74(3) & Hf$_8$Ni$_{21}^{(2)}$ \\\\ \n &2 & 73(2) & 0.69(6) & 0 & 8(3) & Hf$_8$Ni$_{21}^{(1)}$ \\\\ \n &3 & 27.7(5) & 0 & 0 & 18(3) & HfNi$_3$ \\\\ \\\\ \n \n873 &1 & 84.2(5) & 0.58(1) & 7(1) & 80(3) & Hf$_8$Ni$_{21}^{(2)}$ \\\\ \n &2 & 27.0(5) & 0 & 0 & 20(3) & HfNi$_3$ \\\\ \\\\ \n \n973 &1 & 82.7(7) & 0.57(2) & 7(1) & 82(3) & Hf$_8$Ni$_{21}^{(2)}$ \\\\ \n &2 & 26.8(7) & 0 & 0 & 18(3) & HfNi$_3$ \\\\ \\\\\n \n1073 &1 & 26.5(4) & 0 & 5(1) & 70(3) & HfNi$_3$ \\\\ \n &2 & 78.7(6) & 0.57(2) & 0 & 15(3) & Hf$_8$Ni$_{21}^{(2)}$ \\\\ \n &3 & 63.3(9) & 0.23(8) & 0 & 15(3) & Hf$_2$Ni$_7$ \\\\ \\\\ \n \n298$^*$ &1 & 30.2(4) & 0 &9(2) & 62(3) & HfNi$_3$ \\\\ \n &2 & 50.8(4) & 0 & 0 & 25(3) & Hf \\\\ \n &3 & 70.0(8) & 0 & 0 & 12(3) & Hf$_2$Ni$_7$ \\\\ \n\\hline\n\\end{tabular}}\n\\begin{flushleft}\n \\small{$^*$ after measurement at 1073 K}\\\\\n\\end{flushleft}\n\\label{tab:Hf8Ni21table}\n\\end{center}\n\\end{table}\n\n\\subsection{Zr$_8$Ni$_{21}$} \nThe XRD powder pattern of Zr$_8$Ni$_{21}$ sample is shown in figure $\\ref{XRD_Zr8Ni21}$. The X-ray analysis has been\ncarried out using the known crystallographic data of Zr$_8$Ni$_{21}$ \\citep{JM}. \nThe XRD spectrum shows no other peaks except for \nZr$_8$Ni$_{21}$ and this sample is, therefore, found to be produced in an almost pure single component phase. If any small\ncontaminating phases like Zr$_7$Ni$_{10}$ or Zr$_2$Ni$_7$ are produced, it is not observed from the XRD powder pattern.\n\nThe TDPAC spectra of $^{181}$Ta in the as prepared sample of Zr$_8$Ni$_{21}$ are shown in figure \\ref{Zr8Ni21TDPAC}. From PAC\nmeasurements, three frequency components at room temperature have been obtained (Table \\ref{tab:Zr8Ni21table}). \nThe components 1 and 3 have been attributed to Zr$_8$Ni$_{21}$ by \ncomparing with our PAC results in Hf$_8$Ni$_{21}$ and results from ab-initio calculations by density functional theory (discussed later). The component 2 has been\nattributed to Zr$_7$Ni$_{10}$. This follows from the previous X-ray EDS and SEM\/TEM results reported by Shen et al. \\citep{Shen}. \nThe characteristic frequency and $\\eta$ for \nthis component are distinctly different than those found in Zr$_2$Ni$_7$ \\citep{CCDeyPhysica}. Moreover,\nassignment of this component can be supported \nfrom our PAC measurements in Zr$_7$Ni$_{10}$ \\citep{skdeyJAC} where, a component similar to this was found. It is found that\n three frequency components are\n required to fit the spectra in the temperature range 77-373 K with no appreciable\n change in parameters (Table \\ref{tab:Zr8Ni21table}). Variations of $\\omega_Q$, $\\eta$ and site fraction ($f$) with temperature \n for the two components of Zr$_8$Ni$_{21}$ are shown in figure \\ref{Zr8Ni21parameter}.\n \n At 473 K, however, the PAC spectrum gives two components. The component Zr$_8$Ni$_{21}^{(2)}$ does not exist at this temperature. In the temperature\n range 473-973 K, no appreciable changes in results are observed (Table \\ref{tab:Zr8Ni21table}). But at 1073 K, a drastic change in PAC spectrum is found. At\n this temperature, \n the predominant component ($\\sim$68\\%) produces a sharp decrease in quadrupole frequency and asymmetry parameter shows a value equal to zero. This possibly \n indicates a change in local environment of the probe. The component due to Zr$_7$Ni$_{10}$, on the other hand, remains almost unchanged.\n \n To understand the change in PAC spectrum at 1073 K, we have repeated the PAC measurement at room temperature. \n In the re-measured\n spectrum at 298 K, the predominant component ($\\sim$70$\\%$) produces values of $\\omega_Q\\sim$57 Mrad\/s, $\\eta\\sim$0. At the remeasured room temperature, it is found \n that the major component of Zr$_8$Ni$_{21}$ found initially at room temperature reappears with a much smaller fraction (Table \\ref{tab:Zr8Ni21table}). One additional \n new component is observed with a very small fraction which can be assigned to Zr$_2$Ni$_7$ by comparing with the previous result\n in Zr$_2$Ni$_7$ \\citep{CCDeyPhysica}. From SEM\/X-ray EDS measurement \\citep{Shen} also, \nZr$_2$Ni$_7$ was found in a sample of Zr$_8$Ni$_{21}$ annealed at 1233 K.\n\nWe have performed XRD measurement in a sample of Zr$_8$Ni$_{21}$ annealed at 1073 K for two days. The XRD spectrum\n(Fig. \\ref{XRD_Zr8Ni21}) shows peaks mainly due to Zr$_8$Ni$_{21}$. This indicates that no major structural or compositional phase transformation occurs at 1073 K.\n\nPossible explanation for the predominant component observed at 1073 K and subsequently at room temperature is the following. Probably, the probe $^{181}$Hf \nwas not settled well at the position of Zr$_8$Ni$_{21}$ and at 1073 K, these probe atoms got enough energy to go out from the position. The major component is, \ntherefore, observed due to the Hf probe itself. \n \nThe electric field gradients in metal and intermetallic compound are found to vary with temperature following $T$ or $T^{3\/2}$ relationship \\citep{Christiansen}. In Zr$_8$Ni$_{21}$,\nit is found \nthat quadrupole frequencies vary with $T^{3\/2}$ for both components. \nFor the\npredominant site Zr$_8$Ni$_{21}^{(1)}$ (present up to 973 K), the results are fitted by\n \\begin{equation}\n \\omega_Q(T)=\\omega_Q(0)[1-\\beta T^{3\/2}]. \n \\label{eqn:T32}\n \\end{equation}\n A least squares fitting gives results $\\omega_Q$(0)=78.2(1) Mrad\/s and $\\beta$=5.9(1)$\\times$10$^{-6}$ K$^{-3\/2}$. \n \n \\begin{figure*}\n\\begin{center}\n\\includegraphics[scale=.7]{Atomic_position_Hf8Ni21.eps}\n\\end{center}\n\\caption{Models of four types of cells used in this study}\n\\label{Atomic_position_Hf8Ni21}\n\\end{figure*}\n\\begin{table*}[t!]\n\\begin{center} \n\\captionof{table}{\\small{ Calculated EFG values in units of 10$^{21}$ V\/m$^2$ and asymmetry parameters}}\n\\scalebox{0.70}{\n\\begin{tabular}{ l l l l l l l } \n \\hline \\\\ [-0.9ex]\nProbe &Lattice Site & EFG & $\\eta$ & Measured EFG \\\\&&&&extrapolated to 0 K &Measured $\\eta$ at 77 K \\\\ [1.5ex]\n \\hline \\\\ \n \nno probe (pure compound) &Zr(1) 2i 0.0691(4) 0.9052(3) 0.3147(3) & -3.5 & 0.82 \\\\ \n Zr$_8$Ni$_{21}$ &Zr(2) 2i 0.2460(4) 0.4364(3) 0.0344(3) & -4.1 & 0.85 \\\\ \n &Zr(3) 2i 0.2462(4) 0.5604(3) 0.6070(3) & -5.0 & 0.88 \\\\ \n &Zr(4) 2i 0.5685(4) 0.0374(3) 0.2475(3) & -4.7 & 0.74 \\\\ \\\\ \n\n$^{181}$Ta in Zr$_8$Ni$_{21}$ &Zr(1) 2i 0.0691(4) 0.9052(3) 0.3147(3) &8.8 & 0.91 & 8.7(2) & 0.80(1) \\\\ \n &Zr(2) 2i 0.2460(4) 0.4364(3) 0.0344(3) & -11.4 & 0.55 \\\\ \n &Zr(3) 2i 0.2462(4) 0.5604(3) 0.6070(3) & -12.5 & 0.78 &11.4(1) & 0.73(3) \\\\ \n &Zr(4) 2i 0.5685(4) 0.0374(3) 0.2475(3) & -12.3 & 0.56 \\\\ \\\\ \n\nno probe (pure compound) &Hf(1) 2i 0.2487(3) 0.0620(3) 0.1072(3) & -11.5 & 0.75 \\\\ \n Hf$_8$Ni$_{21}$ &Hf(2) 2i 0.0688(3) 0.4040(2) 0.8150(2) &-8.1 & 0.91 \\\\ \n &Hf(3) 2i 0.4312(3) 0.4633(3) 0.2543(3) & -10.8 & 0.69 \\\\ \n &Hf(4) 2i 0.2452(3) 0.9375(3) 0.5340(2) & -10.2 & 0.64 \\\\ \\\\ \n\n$^{181}$Ta in Hf$_8$Ni$_{21}$ &Hf(1) 2i 0.2487(3) 0.0620(3) 0.1072(3) &-13.3 & 0.71 \\\\ \n &Hf(2) 2i 0.0688(3) 0.4040(2) 0.8150(2) & 9.6 & 0.88 &9.1(1) & 0.73(3) \\\\ \n &Hf(3) 2i 0.4312(3) 0.4633(3) 0.2543(3) & -12.5 & 0.60 \\\\ \n &Hf(4) 2i 0.2452(3) 0.9375(3) 0.5340(2) & -11.8 & 0.51 &11.2(2) & 0.62(1) \\\\ \\\\ \n \n\\hline\n\\end{tabular}}\n\\label{tab:DFT_Zr8Ni21_Hf8Ni21}\n\\end{center}\n\\end{table*}\n \\subsection{Hf$_8$Ni$_{21}$}\n The powder XRD pattern of Hf$_8$Ni$_{21}$ is shown in figure \\ref{XRD_Hf8Ni21}. The spectrum is fitted using the known crystallographic \n data of Hf$_8$Ni$_{21}$ \\citep{Bsenko}. The X-ray analysis shows that there are no other peaks except for Hf$_8$Ni$_{21}$ and this sample is also found to be\n produced in an almost pure single \n component phase. No contaminating phase is obtained from the XRD powder pattern.\n \n The TDPAC spectra of $^{181}$Ta in Hf$_8$Ni$_{21}$ are shown in figure \\ref{Hf8Ni21TDPAC}. The spectrum\n at room temperature produces four interaction frequencies.\n The first two components at room temperature with site fractions 63\\% and 21\\%\n (Table \\ref{tab:Hf8Ni21table}) are found to be quite similar to the components found in Zr$_8$Ni$_{21}$. \n The third weak component\n can possibly be attributed to HfNi$_3$ produced along with\n Hf$_8$Ni$_{21}$. L. Bsenko \\citep{LARS BSENKO} reported\n the decomposition of Hf$_8$Ni$_{21}$ to HfNi$_3$ eutectoidally at 1175$\\pm$10$^\\circ$C. This component has been found in the\n whole temperature range. The assignment of HfNi$_3$ in Hf$_8$Ni$_{21}$ can be supported also from our PAC measurements in HfNi$_3$ \\citep{skdeyCCDey}\n where a similar component to this was found. A very weak component ($\\sim$5\\%) found at room temperature can be \n attributed to Hf probe which is not settled in the compound.\n From our temperature dependent PAC measurements, it is found that all\n four components exist in the temperature range 77-673 K. The component 4\n is not observed at 773 K and the minor site of Hf$_8$Ni$_{21}$ (Hf$_8$Ni$_{21}^{(1)}$) disappears\n at 873 K. A drastic change in PAC spectrum is observed at 1073 K\n where the tentatively assigned HfNi$_3$ component suddenly increases at the expense of Hf$_8$Ni$_{21}$ (Table \\ref{tab:Hf8Ni21table}). The component\n due to Hf$_8$Ni$_{21}$ reduces to only 15\\%.\n At this temperature, a new\n frequency component (component 3) is observed which probably can be attributed to Hf$_2$Ni$_7$ by comparing its values with the results reported in the analogous compound Zr$_2$Ni$_7$ \\citep{CCDeyPhysica}. \n \n After measurement at 1073 K, a re-measurement\n at 298 K is carried out. In the re-measurement, HfNi$_3$ is found to be predominant ($\\sim$62\\%) \n which appeared as a minor fraction ($\\sim$10\\%) initially at room temperature. Here, no component due to Hf$_8$Ni$_{21}$ \n is observed. A small component fraction of Hf$_8$Ni$_{21}$ found at 1073 K and absence of this fraction at remeasured room temperature indicates that\n Hf$_8$Ni$_{21}$ is not a stable phase approximately above 1000 K. It is found\n also that, the component due to Hf probe atom re-appears at this temperature with a higher component fraction ($\\sim$25\\%). The quadrupole frequency and asymmetry\n parameter for this component ($\\omega_Q$=50.8 Mrad\/s, $\\eta$=0) are very much similar to the values in a Hf metal \\citep{SKDEYJPCS}.\n \n Variations of $\\omega_Q$, $\\eta$, $\\delta$ and site fraction ($f$) for different\n components observed in Hf$_8$Ni$_{21}$ in the temperature range \n 77-1073 K are shown in figure \\ref{Hf8Ni21parameter}. In Hf$_8$Ni$_{21}$ also,\n the quadrupole frequencies for the two sites show $T^{3\/2}$\n temperature dependent behaviors. A least squares fitting (eqn. \\ref{eqn:T32}) for the predominant site ( present up to 1073 K) \n gives values of $\\omega_Q$(0)=100.1(5) Mrad\/s, $\\beta$=6.0(3)$\\times$10$^{-6}$ K$^{-3\/2}$. For the minor site (present up to 773 K), the fitted results are found \n to be $\\omega_Q$(0)=81.2(4) Mrad\/s, $\\beta$=5.4(3)$\\times$10$^{-6}$ K$^{-3\/2}$. Similar values of $\\beta$ for both components indicate similar temperature \n dependent behaviors for the two sites. Also, the values of $\\beta$ are quite similar to the value of $\\beta$ in Zr$_8$Ni$_{21}$ for the site Zr$_8$Ni$_{21}^{(1)}$.\n\n\\section{DFT calculation}\nThe first-principles density functional theory\n(DFT) calculations were performed to compare with the\nexperimental results and to dispel the doubts existing in\nthe interpretation of the experimental data. All the\ncalculations were done with WIEN2k simulation package \\citep{Blaha}, based on the full potential (linearized) augmented\nplane waves method (FP (L)APW). Electronic exchange-correlation energy was treated with generalized gradient\napproximation (GGA) parametrized by Perdew-Burke-Ernzerhof (PBE) \\citep{Perdew}. In our calculations the muffin-tin\nradii for Zr, Ni and Ta (Hf) were 2.3, 2.2 and 2.4 a. u., \nrespectively. The cut-off parameter $R_{mt}K_{max}$ for limiting\nthe number of plane waves was set to 7.0, where $R_{mt}$ is\nthe smallest value of all atomic sphere radii and $K_{max}$ is\nthe largest reciprocal lattice vector used in the plane\nwave expansion.\n\nThe Brillouin zone integrations within the self-consistency cycles were performed via a tetrahedron\nmethod \\citep{Blochl}, using 18 $k$ points in the irreducible wedge\nof the Brillouin zone (4$\\times$3$\\times$3 mash). The atomic\npositions were relaxed according to Hellmann-Feynman forces calculated at the end of each self-consistent cycle,\nwith the force minimization criterion 2 mRy\/a.u.. In our calculations the self-consistency was\nachieved by demanding the convergence of the\nintegrated charge difference between last two\niterations to be smaller than 10$^{-5}$. Both Zr$_8$Ni$_{21}$ and Hf$_8$Ni$_{21}$ crystallize in the triclinic P1 type structure,\nwhich possesses 15 non-equivalent crystallographic\npositions \\citep{JM,Bsenko}, 4 for Zr (Hf) atoms and 11 for Ni\natoms. \nAll Zr (Hf) non-equivalent positions have the same point group symmetry\n2$i$ and 3 Zr and 12 Ni atoms as nearest neighbors, except Zr(3),\nwhich has 2 Hf and 13 Ni. Each of the four non-equivalent Zr (Hf)\natoms in the unit cell, stated in references \\citep{JM,Bsenko} was replaced by Ta subsequently (figure \\ref{Atomic_position_Hf8Ni21}, \\citep{Kokalj}) preserving the point\ngroup symmetry around original atom and then electric\nfield gradients at thus created Ta positions were\ncalculated using the method developed in reference \\citep{BlahaPRL}.\n\nThe usual convention is to designate the largest\ncomponent of the EFG tensor as $V_{zz}$. The asymmetry\nparameter $\\eta$ is then given by $\\eta=(V_{xx}-V_{yy})\/V_{zz}$, where $V_{zz}\\ge V_{yy}\\ge V_{xx}$. The calculated EFGs in the\npure compounds as well as at Ta probe positions in the\ninvestigated compounds are given in Table \\ref{tab:DFT_Zr8Ni21_Hf8Ni21}.\n\nIt can be observed that there is not much difference in\nthe EFG values for four non-equivalent Zr positions in\nthe pure Zr$_8$Ni$_{21}$ compound. EFG is the smallest at Zr1 and\nthe largest at Zr3. This trend is preserved also for the\nelectric field gradients calculated at the corresponding\nTa positions, but the EFGs are now about 2.5 times\nlarger. In the pure Hf$_8$Ni$_{21}$ compound, EFG values are\nabout doubled, as compared to the corresponding\nones for Zr$_8$Ni$_{21}$, but the $\\eta$ values are similar. Here,\nalso, introduction of Ta atom at one of the non-equivalent Hf sites, leads to increased EFG values.\n\n\\section{Discussion}\nIn the temperature range 77-373 K, Zr$_8$Ni$_{21}$ PAC spectra\nconsist of three frequency components. A uniform conversion\nfrom the measured quadrupole frequencies to the EFGs is\nachieved using the value of 2.36$\\times$10$^{-24}$ cm$^2$ \\citep{Butz} for the\nquadrupole moment of $^{181}$Hf. By comparing the measured\nresults for the EFGs and asymmetry parameters with the\ncalculated ones, components 1 and 3 (Table \\ref{tab:Zr8Ni21table}) are attributed\nto the two non-equivalent Zr sites in Zr$_8$Ni$_{21}$. The measured values of\nEFGs ( 8.7$\\times$10$^{21}$ V\/m$^2$ and 11.3$\\times$10$^{21}$ V\/m$^2$) and $\\eta$ (0.80 and\n0.73) at 77 K are in\nexcellent agreement with the calculated values for the two Zr\nsites in Zr$_8$Ni$_{21}$. However, as Ta doped Zr$_8$Ni$_{21}$ has four non-equivalent crystallographic positions\nwith similar EFG and asymmetry parameter (Table \\ref{tab:DFT_Zr8Ni21_Hf8Ni21}), in order to explain\npreferential site occupation, we performed ab initio total energy calculations\nfor Ta doped Zr$_8$Ni$_{21}$ and found that the configuration obtained when Ta replaces\nZr(3) position has the lowest formation energy, about 0.013 eV lower than the\nstructure when Ta is at Zr(1) postion. The formation energies of the remaining two configurations are about 0.1 eV higher.\n\nAt 1073 K, there is a drastic change of PAC spectrum in Zr$_8$Ni$_{21}$. At this temperature, EFG for the predominant component produces\na zero value of $\\eta$. A similar change in $^{181}$Ta PAC spectra with increasing temperature\nabove 650 K was observed in TiPd$_2$ compound \\citep{BWodniecki} and was explained with the shift\nof Ta atom from Ti to Pd lattice site \\citep{BWodniecki,Cavor}, but in our case, DFT calculations\nexcluded that possibility, as all of the non-equivalent Ni sites in Zr$_8$Ni$_{21}$ has $\\eta$ which differs from zero. \nWe find a resemblance of EFG and $\\eta$ for the component 1 in Zr$_8$Ni$_{21}$\nat re-measured room temperature and component 4 in Hf$_8$Ni$_{21}$ \nwith the calculated values for Ta in pure Hf metal (6.7$\\times$10$^{21}$ V\/m$^2$ and $\\eta$ = 0). \n\nThe PAC spectrum for Hf$_8$Ni$_{21}$ at room temperature consists of four components.\nThe first two components with the EFG values 11.1$\\times$10$^{21}$ V\/m$^2$ and 9.0$\\times$10$^{21}$ V\/m$^2$\nand the corresponding asymmetry parameters 0.62 and 0.73 (at 77 K) obviously correspond\nto the two different Hf positions in Hf$_8$Ni$_{21}$ (Table \\ref{tab:DFT_Zr8Ni21_Hf8Ni21}). The measured results show\nthat quadrupole frequencies for the two corresponding sites in Zr$_8$Ni$_{21}$ and Hf$_8$Ni$_{21}$\nvary in similar manner with temperature. \n\n\\section{Conclusion} \n We have presented the time differential perturbed angular correlations\n measurements and DFT calculations to determine the electric field gradients in Zr$_8$Ni$_{21}$ and Hf$_8$Ni$_{21}$ intermetallic\n compounds. Our results indicate that during the preparation of Zr$_8$Ni$_{21}$ by arc\n melting, other phases like Zr$_7$Ni$_{10}$ can also be formed. The same goes for Hf$_8$Ni$_{21}$,\n in which HfNi$_3$ compound was detected. In both Zr$_8$Ni$_{21}$ and Hf$_8$Ni$_{21}$, EFGs for two\n non-equivalent sites of Zr\/Hf, vary following $T^{3\/2}$ relationship with temperature. Temperature dependent PAC measurement show that Hf$_8$Ni$_{21}$ is \n probably not a stable phase above 1000 K.\n \n \\newpage\n {\\hspace{ -0.4 cm}}{\\bf Acknowledgement}\n \\vspace{0.5cm}\n \nOne of the authors (C.C. Dey) gratefully acknowledges the\nhelp of Prof. Dr. T. Butz, University of Leipzig, Germany in data\nanalysis. The authors acknowledge with thanks A. Karmahapatra and S. Pakhira of Saha Institute of Nuclear Physics, Kolkata for their helps in XRD\nmeasurements and data analysis. The present work is supported by the Department of Atomic Energy, Government of India through the Grant\nno. 12-R\\&D-SIN-5.02-0102. Finally, J. Belo$\\check{\\text{s}}$evi\\'{c}-$\\check{\\text{C}}$avor acknowledges support by The Ministry of Education, \nScience and Technological Department of the Republic of Serbia through the project no. 171001.\n\n\\bibliographystyle{elsarticle-num-names}\n\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nPhysics has long been concerned as a propeller of civilization's evolution in history. The establishment of Newtonian mechanics and thermodynamics drove the ``first technological revolution''. The discovery of the electromagnetic induction phenomenon laid the theoretical foundation for the ``second technological revolution''. Condensed matter physics and quantum physics developed the silicon semiconductor industry for the ``third technological revolution''. With the ongoing ``fourth technological revolution'' currently, physics is also propelling the innovation and development in artificial intelligence, among which the study of complex networks \\cite{strogatz2001exploring, boccaletti2006complex} is a case in point. By using nodes and edges to intuitively describe the nonlinear and heterogeneous interaction patterns of components composing the complex physical, social, or brain systems, its use soon widened to various fields. For decades, scientists have been dedicated to understanding a network's structural and dynamical attributes (like vital node identification \\cite{zhou2019fast, qiu2021identifying}, high-order network structural analysis \\cite{shi2019totally, battiston2020networks, shi2021computing,tang2022optimizing}, and percolation theory \\cite{li2021percolation}) and to utilizing these attributes in specific applications, such as link prediction \\cite{lu2011link}, natural language processing \\cite{pardo2006using}, and recommender systems \\cite{lu2012recommender, xu2020recommending, deng2021recommender}.\n\nRecently, as a pivotal tool to abstract a network's structural and dynamical attributes for utilization in a manner that maps the network or its substructures (like nodes) into a low-dimensional vector space, network representation \\cite{liu2021network, barros2021survey} has intrigued scientists for years, especially in light of ample evidence that network representation has several virtues dear to both academia and industry \\cite{deng2021recommender}: reusable object representations by manual or automated feature engineering, enhanced model precision, and efficient parallel computation based on GPU, among which some are even unprecedented compared to their predecessors. Nevertheless, since the current methods of network representation are mostly based on machine learning \\cite{shalev2014understanding}, almost a black box facing fundamental limits on well-explainable and raising difficulties in tedious hyperparameter tuning attributed to its input-output data fitting rationale, the learned vector space's dimension is generally indeterminable and its elements are not interpreted. Consequently, enormous computing resources are required for searching the suboptimal dimension for the vector space within a range but in most cases researchers still can not interpret why such a dimension works out and what realistic meanings the space's elements may represent. Although recent years have seen massive efforts by computer scientists and mathematics to cope with this issue, the root causes still remain unresolved. For example, although automated machine learning \\cite{yao2018taking} proposed by computer scientists can pave the way for accelerating the search of the space's suboptimal dimension not manually, the search mechanism requiring enormous computing resources still remains. Moreover, although mathematics can conclude an empirical formula used to select the optimal dimension for the space \\cite{gu2021principled}, it is normally built on specific models or data, limiting its interpretability and generalization to other scenarios. Given these inadequacies, determinable and interpretable network representation is still an open and important question.\n\nIn this work, from a physical perspective, this article proposes two methods of determinable and interpretable network representation. Methodologically, the first method is based on the degree, H-index, and Coreness (DHC) theorem \\cite{lu2016h} constructing an operator to generate sequences (with a fixed length) of H-indices for nodes. Regarding their realistic meanings and on the advice of the rich club theory \\cite{colizza2006detecting}, this article utilizes these H-indices to construct node representations, which can represent nodes' local attributes around the neighborhood. To abstract nodes' global attributes in a complex network, the second method is based on the DHC entropy (DHC-E) \\cite{wang2021hyperparameter}, a hyperparameter-free and explainable whole graph embedding algorithm we proposed. If a bipartite network, its $m \\times n$ adjacent matrix can be extended to a $(m+n) \\times (m+n)$ augmented matrix, a simple matrix that can be decomposed to $m+n$ matrices, each of which corresponds to a node and carries the node's global attributes. After implementing the DHC-E algorithm on them, node representations can be generated. Unlike those learned by machine learning-based methods, the node representations generated by the two methods have both a determined dimension and interpretable elements.\n\nTo evaluate the proposed two methods' effectiveness and generalization, this article further proposes Adaptive and Interpretable ProbS (AIProbS), a network-based link prediction model for bipartite networks, which can utilize nodal representations generated by the two methods, as an attempt to enhance the prediction precision. Methodologically, built on a classical network-based framework called ProbS \\cite{zhang2007recommendation}, the AIProbS can control the resource diffusion process of the ProbS framework by setting edge weights quantified with node representations, which can perceive the similarity between nodes. After being equipped with such artificial intelligence as machine learning-based models do, the AIProbS makes the flaw of the classical ProbS framework in self-adaptive perception ability oriented to different scenarios (which is analyzed in Sec.~\\ref{ProbS}). At the same time, compared with machine learning-based link prediction models \\cite{koren2008factorization, rendle2012bpr, chen2019integrating, jiang2020clustering, xu2021topic}, the AIProbS is hyperparameter-free. In addition, implemented on several designed control experiments of diverse recommender systems (a specific application of link prediction in artificial intelligence), experimental results showed that the AIProbS can reach state-of-the-art precision beyond baseline models on some scenarios and can, by and large, make a good trade-off with machine learning-based models on precision, determinacy, and interpretability.\n\n\\section{The model}\n\nIn the first place, this article proposes two novel network representation methods in Sec.~\\ref{generate_node_features}. Then, a classical network-based link prediction framework called ProbS is introduced and its flaws are revealed in Sec.~\\ref{ProbS}. Based on the ProbS framework, this article proposes Adaptive and Interpretable ProbS (AIProbS) in Sec.~\\ref{AIProbS}, a network-based link prediction model for bipartite networks, which can utilize nodal representations generated by the two methods and can enhance the prediction precision of the classical ProbS framework by making up its flaws.\n\n\\subsection{Generate a complex network's nodal representations}\n\\label{generate_node_features}\n\n\\subsubsection{Method one}\n\\label{method_one}\n\nDegree, H-index, and coreness are three measurements used to quantify nodal influence in a complex network. Node's degree measures nodal influence by counting a node's neighbors: the greater a node's degree is, the more neighbors it is connected with, and the higher influence it has. Node's H-index \\cite{hirsch2005index} is the maximum value $h$ such that a node has at least $h$ neighbors with a degree no less than $h$. Furthermore, to take location into account, coreness calculated by $k$-core decomposition \\cite{dorogovtsev2006k} measures a node's centrality: a greater coreness indicates that a node locates more centrally in a complex network and hence has a higher influence.\n\nThe DHC theorem \\cite{lu2016h} reveals that degree, H-index, and coreness are all related. To describe the relationship, the DHC theorem constructs an operator $\\mathcal{H}$, which calculates the maximum value $h$ for each node such that the node has at least $h$ neighbors with H-indices no less than $h$. For each node $i$ in a complex network, taking its degree $k_i$ as the zero-order H-index $h_i^{(0)}$ as the beginning, the first-order H-index $h_{i}^{(1)}$ of node $i$ is calculated by $\\mathcal{H}(h_{j_1}^{(0)}, h_{j_2}^{(0)}, ..., h_{j_{k_i}}^{(0)})$, where $h_{j_1}^{(0)}, h_{j_2}^{(0)}, ..., h_{j_{k_i}}^{(0)}$ are the zero-order H-indices (\\textit{i.e.}, the degree values) of the $k_i$ neighbors of node $i$. By iteratively doing so, $h_i^{(2)} = \\mathcal{H}(h_{j_1}^{(1)}, h_{j_2}^{(1)}, ..., h_{j_{k_i}}^{(1)})$, as well as $h_i^{(3)}, h_i^{(4)},...$, can be calculated. Finally, a sequence $h_i^{(0)}, h_i^{(1)}, h_i^{(2)}, ...$ with a fixed length is generated for node $i$, which is convergent to node $i$'s coreness, as the DHC theorem states:\n\n\\textbf{Theorem 2.1}. \\textit{For each node in a complex network, node $i$'s H-indices sequence $h_i^{(0)}, h_i^{(1)}, h_i^{(2)}, ...$ is convergent to its coreness $c_i$, \\textit{i.e.}, $ \\displaystyle c_i = \\lim_{n \\to \\infty} h_i^{(n)}$.}\n\n\\textit{Proof.} See \\cite{lu2016h}.\n\nAccording to the rich club theory \\cite{colizza2006detecting} (from the field of social network analysis \\cite{scott1988social} and soon widened to interdisciplinary studies like computer science \\cite{zhou2004rich} or cognitive science \\cite{van2011rich}) that a node's influence could reflect its attributes and functions around the neighborhood and the whole network structure, this article proposes the following assumption:\n\n\\textbf{Assumption 2.1}. \\textit{A node's H-indices sequence can abstract the node's multidimensional influence in the neighborhood, where the sequence's convergence steps can reflect the magnitude of the node's influence. The more important role played by the node in the neighborhood, the more slowly its influence decays during the dynamic evolution (\\textit{i.e.}, the convergence process by the DHC theorem), thus the larger its convergence steps are. }\n\nBuilt on assumption 2.1 this article takes a node's H-indices sequence as its node representation. In this way, provided $n$ nodes in a complex network and given that their H-indices sequence converges after up to $s$ steps, this method can map the $n$ nodes to a $s$-dimensional vector space consisting of their H-indices as node representations. This is a determinable and interpretable network representation method, since for an arbitrary complex network the dimension of its nodal representations is determined as $s$ and the elements can be interpreted as nodal multidimensional influence with different magnitudes.\n\n\\subsubsection{Method two}\n\\label{method_two}\n\nFollowing method one, to further abstract a node's global attributes in a complex network, if a bipartite network, its adjacency matrix $A^{m \\times n}$ can be extended to $B^{(m+n)\\times (m+n)}$ constructed by\n$\\displaystyle \\left(\\begin{array}{cc}\nO^{m \\times m} & A^{m \\times n} \\\\\n(A^{m \\times n})^T & O^{n \\times n}\n\\end{array}\\right)$,\nwhere $O$ denotes the null matrix. Based on it, a series of $\\lambda_i$ and $B_i$ can be decomposed by the following theorem.\n\n\\textbf{Theorem 2.2}. \\textit{The adjacency matrix $\\displaystyle B^{(m+n)\\times (m+n)}$ can be decomposed by $\\displaystyle B=\\sum_{i=1}^{m+n}\\lambda_i B_i$, where $\\lambda_i$ is the i-th eigenvalue of $\\displaystyle B^{(m+n)\\times (m+n)}$ and $\\displaystyle B_i$ is the corresponding idempotent matrix.}\n\n\\textit{Proof.} See \\textbf{\\textit{Appendix A}}.\n\nAfter that, this article implements the DHC-E operator $\\mathcal{E}$ \\cite{wang2021hyperparameter} (\\textit{i.e.}, by the DHC theorem to generate a H-index matrix $H^{n \\times s}$ by row containing the H-indices converged after $s$ steps of each of the $n$ nodes in a complex network, the operator $\\mathcal{E}$ calculates the Shannon entropy of each column of $H^{n \\times s}$ and obtains a vector $e^{1 \\times s}$, as the whole graph embedding of the network) on each $B_i$ or $\\lambda_i B_i$ one by one, generating the $m+n$ nodes' representations for the bipartite network correspondingly. Apparently, this method is also a determinable and interpretable network representation method. The characteristics of interpretability and hyperparameter-free of the DHC-E algorithm are thoroughly illuminated in \\cite{wang2021hyperparameter}.\n\n\\subsection{The ProbS framework and its flaws}\n\\label{ProbS}\n\nTo evaluate the two methods' effectiveness and generalization, this article utilizes them in link prediction for bipartite networks. Since network representation can be seen as artificial intelligence that recognizes and abstracts a complex network's underlying structural and dynamical attributes, this article explores how such artificial intelligence (\\textit{i.e.}, nodal representations generated by these methods) can be used to enhance the precision of classical link prediction models.\n\nAmong classical (non-machine learning-based) link prediction models for bipartite networks, the ProbS \\cite{zhang2007recommendation} framework is a typical one. By means of a resource diffusion mechanism inspired by the physical process of Material Diffusion, the ProbS framework can quantify the similarity between nodes after initializing and diffusing resources. Fig.~\\ref{Schematics} includes an example to intuitively illuminate the schematics of the ProbS framework. For instance, when predicting node $B$'s unobserved links with nodes $a$ and $b$, resources are first initialized at nodes $c$ and $d$ (the nodes that are connected with node $B$) with value $1$, then are diffused to nodes $A$, $B$, and $C$ along edges after being equally divided by the degree of each node, finally are diffused back to nodes $a$, $b$, $c$, and $d$ in the same way, which can be used to quantify the similarity between node $B$ and the four nodes, respectively. A larger similarity of two nodes indicates a higher probability of an unobserved link existing between them.\n\nThis article provides a mathematical perspective to describe the ProbS framework, by constructing an operator $T$ to describe its diffusion mechanism. Given a bipartite network consisting of $m+n$ nodes of two different types, respectively, whose adjacency matrix is represented by $A^{m \\times n}$. Let $R^{m \\times n}$ denote the predicted matrix, where $R_{ij}$ represents the similarity (\\textit{i.e.}, the probability of the existence of a link) between nodes $i$ and $j$. Then, through the ProbS framework $R^{m \\times n}$ can be calculated by\n\\begin{equation}\n\\label{Probs}\n R = A \\cdot (D_{I} \\circ A)^T \\cdot (D_{U} \\circ A)\n\\end{equation}\nwhere $\\cdot$ denotes the dot product, and $\\circ$ denotes the Hadamard product. $D_I^{m \\times n} = (a_1, a_2, ..., a_n)$, $\\displaystyle a_i = (\\frac{1}{k_{I_i}}, ... , \\frac{1}{k_{I_i}})^T$ where $k_{I_i}$ is item $i$'s degree. $D_U^{m \\times n} = (a_1, a_2, ..., a_m)^T$, $\\displaystyle a_i = (\\frac{1}{k_{U_i}}, ... ,\\frac{1}{k_{U_i}})$ where $k_{U_i}$ is user $i$'s degree. In Eq.~(\\ref{Probs}) the operator $T=(D_{I} \\circ A)^T \\cdot (D_{U} \\circ A)$.\n\nThe operator $T$ tells why the ProbS framework will converge after deriving $R$ from $A$ and then placing $A$ with the derived $R$ iteratively, stated as the following theorem.\n\n\\textbf{Theorem 2.3}. Let the operator $T=(D_{I} \\circ A)^T \\cdot (D_{U} \\circ A)$ iteratively act on $A$ by $A \\leftarrow A \\cdot T$, the iterative process is convergent.\n\n\\textit{Proof}. See \\textbf{\\textit{Appendix B}}.\n\nSince the difference between the values in $A$ tends to be smoother as the convergent iterative process progresses but link prediction relies for higher precision on the more distinctive differentiation between the predicted values of similarity \\cite{deng2021recommender}, in link prediction the best iteration steps for the ProbS framework is one, and so does the AIProbS proposed in Sec.~\\ref{AIProbS}.\n\nIn addition, from such a mathematical perspective, it is easy to see that the ProbS framework faces fundamental limits on intelligence because its resource diffusion mechanism is just based on equal allocation, shown as $D_I$ and $D_U$ in Eq.~(\\ref{Probs}). In practice like recommender systems (an application of link prediction for bipartite networks in artificial intelligence), such a mechanism raises a key question: if respectively take these nodes of two different types as users and items in recommender systems, the resources diffused between users and items back and forth, to some extent, represent user's preferences for items or item's attractiveness to users, while neither of them should be necessarily equal since user biases \\cite{koren2009matrix, adomavicius2014biasing, manjur2021exploring} and item biases \\cite{koren2009matrix, park2014uncovering} generally exist in reality. Moreover, these biases are usually recommendation scenario-oriented, which means that in different scenarios a user's preferences may differ, and so do an item's attractiveness or popularity. Finally, in practice the ProbS framework fails to take these biases into consideration, let alone adaptively perceive and quantify their differences in various scenarios.\n\n\\subsection{The AIProbS model}\n\\label{AIProbS}\n\nThe essential condition for the ProbS framework to realize that intelligence is to be equipped with self-adaptive perception, an ability to perceive and utilize the attributes of nodes (\\textit{i.e.}, nodal representations) in a complex network toward different scenarios. To utilize the nodal representations generated by the two methods proposed in Sec.~\\ref{generate_node_features} in the ProbS framework, this article proposes Adaptive and Interpretable ProbS (AIProbS).\n\nIn the first step, on the advice that the rich club theory \\cite{colizza2006detecting} gives clues that nodes with high centrality tend to form tightly interconnected communities, this article generalizes this conclusion to the field of link prediction, proposing the following assumption:\n\n\\textbf{Assumption 2.2}. \\textit{The similarity between node pairs having strongly correlated nodal representations (\\textit{i.e.}, similar features or similar influence) is higher than that between weakly correlated ones.}\n\nTo measure the similarity between nodes, the AIProbS uses the cosine similarity metric. Provided two $n$-dimension vectors $x$ and $y$, the cosine similarity between them is calculated by $\\displaystyle \\cos(\\theta)=\\frac{x \\cdot y}{|x| \\cdot |y|}=\\frac{\\sum_{i=1}^n x_i y_i}{\\sqrt{\\sum_{i=1}^n x_i^2} \\cdot \\sqrt{\\sum_{i=1}^n y_i^2}}$. In the same way, provided $m+n$ nodes belonging to two sets $U$ and $I$ of two different types in a bipartite network, respective. Through the network representation methods proposed in Sec.~\\ref{generate_node_features} the representation matrices $F_U^{m \\times s}$ and $F_I^{n \\times s}$ of the two types of nodes are generated, either of which is consist of nodal representations by row. Then, the $m \\times n$ nodal similarity matrix $S^{m \\times n}$ calculated by the cosine similarity metric is\n\\begin{equation}\n\\label{user-item_similarity_matrix}\nS = \\frac{F_{U} \\cdot F_{I}^T}{\\alpha^T \\cdot \\beta},\n\\end{equation}\nwhere vector $\\displaystyle \\alpha = \\big(\\sqrt{\\sum_{j=1}^s{F_U}_{1j}^2}\\,,\\sqrt{\\sum_{j=1}^s{F_U}_{2j}^2}\\,, \\,...\\,, \\sqrt{\\sum_{j=1}^s{F_U}_{mj}^2}\\big)$ and vector $\\displaystyle \\beta = \\big(\\sqrt{\\sum_{j=1}^s{F_I}_{1j}^2}\\,,\\sqrt{\\sum_{j=1}^s{F_I}_{2j}^2}\\,, \\,...\\,, \\sqrt{\\sum_{j=1}^s{F_I}_{nj}^2}\\big)$.\n\nAfter obtaining the nodal similarity matrix $S^{m \\times n}$, utilizing it to control the diffusion process of the classical ProbS framework is the second step. To assign proper weights to every node pair for the diffusion mechanism of the ProbS framework, the AIProbS further complete some normalization and proportioning operations on $S \\circ A$ where $A$ is the adjacency matrix shown in Eq.~(\\ref{Probs}). Since the elements of $S \\circ A$ vary in $[-1,1]$ while the diffused resources are supposed to be positive, the AIProbS normalizes the value range of the elements to $[0,1]$ using the max-min normalization operation, for each row vector $(S\\circ A)_{i*}$ $(i=1,2,...,m)$ of $S \\circ A$, by\n\\begin{equation}\n\\label{max-min_normalization}\n(S \\circ A)_{ij} \\leftarrow \\frac{(S \\circ A)_{ij} - \\min}{ \\max - \\min}, \\,j = 1,2, ..., n,\n\\end{equation}\nwhere the $\\max$ and $\\min$ are the maximum and minimum elements of the row vector $(S \\circ A)_{i*}$, respectively. Based on that, the weight matrix $W_U$ for nodes belonging to set $U$ is calculated by the proportioning operation as\n\\begin{equation}\n\\label{proportioning}\nW_{U_{ij}} = \\frac{1}{(S \\circ A)_{ij}}\\sum_{k=1}^{n} (S \\circ A)_{ik}, \\,i=1,2,...,m, \\, j=1,2,...,n.\n\\end{equation}\nOn the other hand, the same operations are completed on $S^{m \\times n}$ by column, generating the weight matrix $W_I^{m \\times n}$ for nodes belonging to set $I$.\n\nIn the last step, the predicted matrix $R^{m \\times n}$, where $R_{ij}$ represents the prediced similarity between nodes $i$ and $j$, is calculated through the AIProbS by\n\\begin{equation}\n R = A \\cdot W_{I}^T \\cdot W_{U}.\n\\end{equation}\n\nConceivably, there are other metrics for similarity measurement. More combinations were tested in this article (See \\textbf{\\textit{Appendix C}} for details) but none of them performed better than the one proposed in this section. All in all, the whole process of the AIProbS are summarized in the pseudocodes shown in \\textit{Appendix D}. For more intuitive illumination, Fig.~\\ref{Schematics} in \\textbf{\\textit{Appendix D}} presents its schematics.\n\n\\subsection{Performance Evaluation}\n\\label{performance_evaluation}\n\nTo evaluate the AIProbS's precision as well as its pros and cons in link prediction for bipartite networks, which also can be used to reflect the effectiveness of nodal representations generated by the two network representation methods proposed in Sec.~\\ref{generate_node_features}, this article designs control experiments based on recommender systems, an application of link prediction in artificial intelligence.\n\n\\subsubsection{Recommender systems}\n\nBy analyzing observed user-item relations to predict a user's preferred items from millions of candidates, recommender systems \\cite{lu2012recommender, deng2021recommender} are recognized as a pivotal tool to alleviate the information overload problem. Among different user-item relations, implicit user-item interactions (\\textit{e.g.}, user's historical clicks or buys on items) record the existence of a user's interactions with items, defined as a binary state using $1$ and $0$. From the perspective of a complex network, recommendation on implicit user-item interactions can be seen as a process of link prediction for bipartite networks, where users and items correspond to the two types of nodes and implicit user-item interactions represent the edges between nodes. Therefore, the designed experiments in this article are based on the recommendation with implicit user-item interactions, for most current models are based on them.\n\n\\subsubsection{Data sets}\n\nIn light of the no-free-lunch theorem \\cite{adam2019no} that no model can always perform well enough as expected in all different scenarios, this article designs control experiments to evaluate the performance of the AIProbS on diverse real recommendation scenarios, in order to explore not only the AIPobS's pros but also its cons in different scenarios.\n\n\\begin{table}[ht]\n\\caption{\\textbf{Overview of data sets.}}\n\\label{Datasets}\n\\begin{indented}\n\\item[]\\begin{tabular}{ccllcccc}\n\\br\nData sets & \\multicolumn{3}{c}{$|U|$} & $|I|$ & Interactions & Sparsity \\\\\n\\mr\nMovieLens 100K & \\multicolumn{3}{c}{943} & 1680 & 100000 & 93.70\\% \\\\\nMovieLens 1M & \\multicolumn{3}{c}{6040} & 3952 & 1000209 & 95.81\\% \\\\\nLastFM & \\multicolumn{3}{c}{1892} & 17632 & 92834 & 99.72\\% \\\\\n\\br\n\\end{tabular}\n\\end{indented}\n\\end{table}\n\nAs shown in Tab.~\\ref{Datasets}, $|U|$ and $|I|$ represent the number of users and items, respectively, and the interactions between users and items are implicit ones. The sparsity in Tab.~\\ref{Datasets} represents the ratio of the number of unobserved interactions to the maximum number of all possible interactions between users and items (e.g., that between $m$ users and $n$ items is $mn$). As a control group, the MovieLens 100K, MovieLens 1M, and LastFM are three classical data sets from two different recommender systems of movies and music, with distinctive ratios of $|U|$ to $|V|$, data scales, and sparsity, based on which more persuadable results could yield compared with those based on newly published data sets, since these classical data sets have been widely used for evaluation in previous works\n\nIn order to guarantee the reproducibility of experiments, either of the three data sets is obtained from the RecBole public resources (\\url{https:\/\/recbole.io\/dataset_list.html}), organized into tuples (user, item, $0\/1$) without preprocessing. Each of them is randomly split into a ``train\/evaluate\/test'' set by the ratio of ``$80\/10\/10\\%$''. After independently repeating the splitting process $30$ times, $30$ realizations are generated for each data set. One can get the split data used in this article through the hyperlink address posted in Sec.~\\ref{data_and_code_available}.\n\n\\subsubsection{Evaluation metrics}\n\nIn order to quantify the precision of the AIProbS on these data sets, three common-used metrics are chosen in this article. Given a user $u \\in U$ ($U$ is the user set) and the length $N$ of the recommendation list, the set of recommended items for the user is denoted by $\\hat{R}(u)$ and the ground-truth set of items the user interacted with is denoted by $R(u)$. Based on them, the first evaluation metric is the Recall@N \\cite{olson2008advanced}, which calculates the fraction of predicted relevant items out of all ground-truth relevant items by\n\\begin{equation}\n\\mbox{Recall@N}=\\frac{1}{|U|}\\sum_{u \\in U} \\frac{|\\hat{R}(u) \\cap R(u)|}{|R(u)|},\n\\end{equation}\nwhere $|R(u)|$ represents the item count of $R(u)$.\n\nTo calculate the reciprocal rank of the first relevant item recommended to each user, the second evaluation metric MRR@N \\cite{craswell2009mean} is denoted as\n\\begin{equation}\n\\mbox{MRR@N}=\\frac{1}{|U|} \\sum_{u \\in U} \\frac{1}{\\mbox{rank}_u^*},\n\\end{equation}\nwhere $\\mbox{rank}_u^*$ is the rank position of the first relevant item recommended to user $u$.\n\nMoreover, as the third evaluation metric, the NDCG@N \\cite{wang2013theoretical} can further measure the overall ranking quality in a manner that accounts for the position of the hit by assigning higher scores to hits at top ranks as\n\\begin{equation}\n\\footnotesize\n\\mbox{NDCG@N}=\\frac{1}{|U|} \\sum_{u \\in U} \\big( \\frac{1}{\\sum_{i=1}^{\\min(|R(u)|,N)}\\frac{1}{\\log_2(i+1)}} \\sum_{i=1}^N \\delta(i \\in R(u)) \\frac{1}{\\log_2(i+1)}\\big),\n\\end{equation}\nwhere $\\delta(\\cdot)$ is an indicator function and positions are discounted logarithmically.\n\nIn practice, the greater the values of these evaluation metrics are, the higher a model's precision is.\n\n\\subsubsection{Baseline methods}\n\nThis article constructs or chooses nine baseline models as follows, evaluating the pros and cons of the AIProbS compared with its predecessors of both classical and machine learning-based baselines.\n\n\nClassical baselines include two models. As the bedrock, the ProbS \\cite{zhang2007recommendation} is a necessary baseline to evaluate the improvement of the AIProbS. In addition, one might expect to base the recommendation directly on the nodal representations generated by the methods proposed in Sec.~\\ref{generate_node_features}, not the ProbS framework. To test this strategy, this article constructs the Pure-DHC model, used to perform the recommendation by Eq.~(\\ref{user-item_similarity_matrix}) based on the user-item similarity of their H-indices (\\textit{i.e.}, nodal representations).\n\nMachine learning-based baselines include seven models. To avoid the baseline pitfalls that have plagued earlier research on the comprehensive and objective evaluation of proposed models, this article further chooses seven representative machine learning-based models as baselines, among which were based on six different techniques of machine learning frameworks, including NeuMF \\cite{he2017neural} based on deep neural networks, ConvNCF \\cite{he2018outer} and SpectralCF \\cite{zheng2018spectral} based on convolution operations, GCMC \\cite{berg2017graph} based on graph auto-encoder frameworks, LINE \\cite{tang2015line} based on random walking, NGCF \\cite{wang2019neural} based on graph neural networks, and DGCF \\cite{wang2020disentangled} based on attention mechanisms.\n\nTo guarantee the fairness and reproducibility of experiments, the implementation and evaluation of models were hosted to the RecBole \\cite{zhao2021recbole}, a public open pipeline of recommender systems. One can get the codes used in this article through the hyperlink address posted in Sec.~\\ref{data_and_code_available}.\n\n\\section{Results}\n\\label{Experimental_results}\n\nBased on the experimental settings in Sec.~\\ref{performance_evaluation}, this section presents the experimental results on the precision and robustness of the AIProbS and baseline models in Sec.~\\ref{Precision analysis} and Sec.~\\ref{Robustness analysis}, respectively, revealing their pros and cons in different recommendation scenarios.\n\n\n\\subsection{Precision analysis}\n\\label{Precision analysis}\n\nAs shown in Tabs.~\\ref{LastFM}, \\ref{100K} and \\ref{1M}, the results on model precision are presented, where the length $N$ of the recommendation list is set to $10$, and each model's precision is averaged from its independently implementation based on $30$ different realizations. The values in parentheses indicate the percentage of improvement or decline in model precision of the AIProb model compared to the respective baseline models on each specific data set and evaluation metric, where the percentage of improvement is bold.\n\nConceivably, when speaking of the necessity of determinable and interpretable network representation and their utilization in link prediction, one might cast it into doubt: do not nodal representations generated by the two methods in Sec.~\\ref{generate_node_features} be sufficient for link prediction? Can the precision of classical link prediction frameworks really be enhanced by being involved with nodal representations as intelligence? Are the machine learning-based network representation methods not precise enough? As shown in Tabs.~\\ref{LastFM}, \\ref{100K} and \\ref{1M}, on all three data sets the Pure-DHC which directly utilizes the nodal representations generated by the methods in Sec.~\\ref{generate_node_features} for recommendation achieved the worst model precision among the AIProbS and baseline models. That is to say, such generated nodal representations could be nothing with the recommendation if not be utilized in the ProbS or any other recommendation framework. After utilizing these nodal representations in the ProbS framework, as shown in Tabs.~\\ref{LastFM}, \\ref{100K} and \\ref{1M}, the AIProbS outperformed the classical ProbS on all three data sets. When compared to machine learning-based baselines, the AIProbS indeed performed worse than some machine learning models, most obviously on MovieLens 1M. But it still can achieve state-of-the-art performance on model precision on LastFM, suggesting that nodal representations generated by the methods in Sec.~\\ref{generate_node_features} may be able to abstract the underlying attributes of a complex network better than those learned by machine learning methods.\n\n\\begin{table}[ht]\n\\caption{\\textbf{Results of model precision on LastFM.}}\n\\label{LastFM}\n\\begin{indented}\n\\item[]\\begin{tabular}{@{}cccc@{}}\n\\br\n\\multirow{2}{*}{Models} & \\multicolumn{3}{c}{LastFM} \\\\ \\cmidrule(r){2-4}\n & Recall@10 & MRR@10 & NDCG@10 \\\\ \\mr\nPure-DHC & 0.004 & 0.006 & 0.003 \\\\\nProbS & 0.170 (\\textbf{+7.5}\\%) & 0.308 (\\textbf{+9.6}\\%) & 0.166(\\textbf{+9.2}\\%) \\\\\nAIProbS & \\textbf{0.184} & \\textbf{0.340} & \\textbf{0.183} \\\\ \\midrule\nNeuMF & 0.060 (\\textbf{+67.5}\\%) & 0.092 (\\textbf{+73.0}\\%) & 0.050 (\\textbf{+72.5}\\%) \\\\\nConvNCF & 0.056 (\\textbf{+69.7}\\%) & 0.090 (\\textbf{+73.6}\\%) & 0.048 (\\textbf{+73.8}\\%) \\\\\nSpectralCF & 0.066 (\\textbf{+63.9}\\%) & 0.120 (\\textbf{+64.7}\\%) & 0.062 (\\textbf{+66.1}\\%) \\\\\nGCMC & 0.121 (\\textbf{+34.1}\\%) & 0.214 (\\textbf{+37.0}\\%) & 0.116 (\\textbf{+36.5}\\%) \\\\\nLINE & 0.149 (\\textbf{+19.1}\\%) & 0.272 (\\textbf{+20.0}\\%) & 0.145 (\\textbf{+20.5}\\%) \\\\\nNGCF & 0.169 (\\textbf{+8.2}\\%) & 0.301 (\\textbf{+11.4}\\%) & 0.163 (\\textbf{+11.0}\\%) \\\\\nDGCF & 0.177 (\\textbf{+3.7}\\%) & 0.316 (\\textbf{+7.1}\\%) & 0.172 (\\textbf{+6.0}\\%) \\\\ \\br\n\\end{tabular}\n\\end{indented}\n\\end{table}\n\n\\begin{table}[ht]\n\\caption{\\textbf{Results of model precision on MovieLens 100K.}}\n\\label{100K}\n\\begin{indented}\n\\item[]\\begin{tabular}{@{}cccc@{}}\n\\br\n\\multirow{2}{*}{Models} & \\multicolumn{3}{c}{MovieLens 100K} \\\\ \\cmidrule(r){2-4}\n & Recall@10 & MRR@10 & NDCG@10 \\\\ \\mr\nPure-DHC & 0.017 & 0.084 & 0.041 \\\\\nProbS & 0.208 (\\textbf{+3.1}\\%) & 0.413 (\\textbf{+4.8}\\%) & 0.236 (\\textbf{+4.5}\\%) \\\\\nAIProbS & \\textbf{0.215} & \\textbf{0.434} & \\textbf{0.248} \\\\ \\midrule\nNeuMF & 0.070 (\\textbf{+67.2}\\%) & 0.187 (\\textbf{+56.9}\\%) & 0.093 (\\textbf{+62.3}\\%) \\\\\nConvNCF & 0.099 (\\textbf{+53.9}\\%) & 0.245 (\\textbf{+43.5}\\%) & 0.125 (\\textbf{+49.5}\\%) \\\\\nSpectralCF & 0.124 (\\textbf{+42.4}\\%) & 0.293 (\\textbf{+32.5}\\%) & 0.153 (\\textbf{+38.2}\\%) \\\\\nGCMC & 0.196 (\\textbf{+8.9}\\%) & 0.400 (\\textbf{+7.9}\\%) & 0.232 (\\textbf{+6.4}\\%) \\\\\nLINE & 0.190 (\\textbf{+11.3}\\%) & 0.391 (\\textbf{+9.8}\\%) & 0.225 (\\textbf{+9.0}\\%) \\\\\nNGCF & 0.245 (-14.3\\%) & 0.481 (-10.8\\%) & 0.293 (-18.4\\%) \\\\\nDGCF & 0.236 (-9.9\\%) & 0.458 (-5.5\\%) & 0.278 (-12.3\\%) \\\\ \\br\n\\end{tabular}\n\\end{indented}\n\\end{table}\n\n\n\\begin{table}[ht]\n\\caption{\\textbf{Results of model precision on MovieLens 1M.}}\n\\label{1M}\n\\begin{indented}\n\\item[]\\begin{tabular}{@{}cccc@{}}\n\\br\n\\multirow{2}{*}{Models} & \\multicolumn{3}{c}{MovieLens 1M} \\\\ \\cmidrule(r){2-4}\n & Recall@10 & MRR@10 & NDCG@10 \\\\ \\mr\nPure-DHC & 0.002 & 0.031 & 0.014 \\\\\nProbS & 0.108 (\\textbf{+17.6\\%}) & 0.352 (\\textbf{+15.1\\%}) & 0.177 (\\textbf{+15.9\\%}) \\\\\nAIProbS & \\textbf{0.131} & \\textbf{0.414} & \\textbf{0.210\\%} \\\\ \\midrule\nNeuMF & 0.032 (\\textbf{+75.4\\%}) & 0.128 (\\textbf{+69.1\\%}) & 0.053 (\\textbf{+74.6\\%}) \\\\\nConvNCF & 0.073 (\\textbf{+44.2\\%}) & 0.255 (\\textbf{+38.4\\%}) & 0.128 (\\textbf{+38.9\\%}) \\\\\nSpectralCF & 0.147 (-11.5\\%) & 0.416 (-0.5\\%) & 0.236 (-12.2\\%) \\\\\nGCMC & 0.152 (-15.4\\%) & 0.421 (-1.7\\%) & 0.240 (-14.4\\%) \\\\\nLINE & 0.153 (-16.6\\%) & 0.423 (-2.1\\%) & 0.236 (-12.3\\%) \\\\\nNGCF & 0.162 (-23.4\\%) & 0.442 (-6.7\\%) & 0.254 (-21.0\\%) \\\\\nDGCF & 0.172 (-31.0\\%) & 0.460 (-11.1\\%) & 0.266 (-26.8\\%) \\\\ \\br\n\\end{tabular}\n\\end{indented}\n\\end{table}\n\nTo put these results in more general terms, it is definite that designing control experiments to guarantee the comprehensiveness and objectivity of model performance evaluation is indispensable because, as shown in Tabs.~\\ref{LastFM}, \\ref{100K} and \\ref{1M}, the comparative predominance between different models or even that between the classical and machine learning-based models are distinctive. For instance, compared to its predecessor (the ProbS framework), the AIProbS at best improved the Recall@$10$ by $17.6\\%$ on MovieLens 1M and at worst, by $3.1\\%$ on MovieLens 100K. Such a $14.5\\%$ gap shows that the predominance of the AIProbS over the ProbS is not necessarily that significant in all recommendation scenarios. Overall, on MovieLens 1M, although it achieved an appreciable improvement over the ProbS, the AIProbS still performed worse than the other five machine learning-based models, revealing the predominance of the machine learning-based frameworks over the classical ones on this data set. However, that predominance faded on MovieLens 100K because only two machine learning-based models (\\textit{i.e.}, NGCF and DGCF) outperformed the AIProbS. On LastFM, none of the machine learning-based models outperformed the AIProbS, in other words, but the AIProbS achieved state-of-the-art performance on model precision.\n\nFiguring out the determinant factors of model precision in different recommendation scenarios is not easy and intuitive, not to mention accurately predicting a model's performance for one specific scenario. Still, on the advice of the clues given in Tabs.~\\ref{LastFM}, \\ref{100K} and \\ref{1M}, some discoveries could be summarized as follows. (1) The machine learning-based models might have a predominance on data sets with large scales. The recommendation scenario of MovieLens 1M and MovieLens 100K being equal, the machine learning-based models would show a more obvious predominance on the former with a comparative larger data scale than the latter. However, it is hard to assert that the distinctions of ratios of $|U|$ to $|V|$ and the sparsity of the two data sets play a silent role. (2) The classical frameworks might play a large role in the improvement after integration in recommendation scenarios with high sparsity. Since the sparsity of LastFM is the highest among the three data sets, where the machine learning-based models face fundamental limits on lack of enough user-item interactions for training, the AIProbS or the ProbS combined with or of the classical frameworks showed their predominance as a result of their network structure-oriented resolution. Nevertheless, the ratio of $|U|$ to $|V|$ of LastFM, which seems to be a little higher than the other two, could also be a decisive factor.\n\n\\subsection{Robustness analysis}\n\\label{Robustness analysis}\n\nAs revealed in Sec.~\\ref{Precision analysis}, with the increase in data scale the precision of the AIProbS decreased compared with that of machine learning-based baselines, for the mechanism of data fitting (or pattern representation) adopted by machine learning methods can give fully to its play more suitably in scenarios with larger data scale. Nevertheless, it does not mean that the AIProbS is useless in those scenarios. Since hyperparameters having no realistic meanings could make a model's implementing process indeterminable and vague, machine learning-based models lost almost all the interpretability for results, like why a machine learning-based model generates some recommendations for a user. This problem is not confined to recommender systems but still haunts other applications requiring high interpretability for results, such as machine translation or knowledge graph completion. Besides, since tedious hyperparameter tuning is required for up to the optimal performance of a model, generally machine learning methods reach higher precision by sacrificing the model's computing efficiency. Such a strategy brings about heavy financial (\\textit{i.e.}, computing resources) and time costs for the implementation on a huge data. In contrast, the AIProbS is determinable and its results are interpreted, meaning that this model could be more suitable for applications requiring high interpretability and for scenarios with huge data scales but insufficient computing resources.\n\nOn two realizations of ml-100k for instance, Tabs.~\\ref{P1}, \\ref{P2}, \\ref{P3}, and \\ref{P4} present the relations between the different settings of two representative hyperparameters (\\textit{i.e.}, representation dimension and learning rating) and a model's average precision when one hyperparameter is fixed and others are traversed within a specified search range, where the standard deviation of precision is presented by error arrow at a data point and each model's number of hyperparameters is presented in parentheses, reflecting a model's magnitude of performance fluctuation associated with setting disturbance, which actually can quantify the model's robustness. As shown in Tab.~\\ref{P1}, the AIProbS had a stable performance on recall@10, since the representation dimension of its results is determined. However, as a hyperparameter different settings of representation dimension can largely influence machine learning-based baselines' precision. Although the performance on recall@10 of ConvNCF, LightGCN, and NGCF of different representation dimensions was relatively stable among machine learning-based baselines, that of GCMC and SpectralCF largely fluctuated with the change of representation dimension. For example, as for SpectralCF when the representation dimension is set to $48$ its average precision could be around $27\\%$ higher than that when being set to $16$. On top of that, even when the representation dimension of SpectralCF is set to $48$, seemingly the optimal choice, its performance on recall@10 still faces a $125\\%$ gap between the peaks of performance, flowed from the different settings of other hyperparameters. Similar fluctuations on machine learning-based baselines' precision were revealed by the influence of different settings of representation dimension on mrr@10 and by results shown in Tab.~\\ref{P2} when considering learning rate as the controlled hyperparameter. Attributed to such the indeterminable performance of machine learning methods, one may have to repeatedly try different hyperparameter settings for a machine learning-based model to search out the optimal one, which is definitely computing resources-consuming and time costly. If an insufficient searching process turns out improper hyperparameter settings, the model could even end up with its worst performance. In contrast, once implemented the AIProbS can reach its optimal performance.\n\n\\begin{figure}[ht]\n\\includegraphics[scale=0.46]{P1.pdf}\n\\caption{\\textbf{Relation between model precision and representation dimension reflected on ml-100k realization 1}}\n\\label{P1}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\includegraphics[scale=0.46]{P2.pdf}\n\\caption{\\textbf{Relation between model precision and learning rate reflected on ml-100k realization 1}}\n\\label{P2}\n\\end{figure}\n\nIt may be said, though, that virtually determining a machine learning-based model's optimal hyperparameter settings is burdensome but can be once and for all based on one data set. However, it is not a fact. Similar experiments were done on another realization of ml-100k and the results presented in Tabs.~\\ref{P3} and \\ref{P4} revealed that the optimal hyperparameter settings of a machine learning-based model would be changed with the change of data set. For example, as shown in Tab.\\ref{P1} the optimal representation dimension of SpectralCF was $48$ on ml-100k realization 1 but that on ml-100k realization 2 was changed to be $128$, as shown in Tab.~\\ref{P3}. As a result, the searching process for the optimal hyperparameter settings of a machine learning-based model on a new data set appears to be inevitable. In contrast, with the change of data set the representation dimension of the AIProbS is still automatically determined, once implemented.\n\n\\begin{figure}[ht]\n\\includegraphics[scale=0.46]{P3.pdf}\n\\caption{\\textbf{Relation between model precision and representation dimension reflected on ml-100k realization 2}}\n\\label{P3}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\includegraphics[scale=0.46]{P4.pdf}\n\\caption{\\textbf{Relation between model precision and learning rate reflected on ml-100k realization 2}}\n\\label{P4}\n\\end{figure}\n\nAll in all, in the sense that the AIProbS provides a good trade-off with machine learning-based models on precision, interpretability, and determinacy.\n\n\\section{Discussion}\n\nThis article proposes two determinable and interpretable node representation methods. Different from other attempts like automated machine learning methods by computer scientists and computational theory by mathematics to search out and analyze the sub-optimal (or optimal) representation dimension of a machine learning-based network representation model, respectively, and to interpret the implementing process and the results come out of the model, from a perspective of physics the proposed two methods can substantially generate nodal representations with a determined dimension and interpretable elements, reaching its optimal performance once implemented. After utilizing these representations in link prediction for bipartite networks, experimental results showed that the AIProbS can make a good trade-off with machine learning-based models on precision, determinacy, and interpretability, indicating the effectiveness of nodal representations generated by the proposed two representation methods.\n\nImportantly, these methods with good generalization may motivate further research. For example, nodal representations generated by the proposed two methods can also be utilized in machine learning-based models as initial features or network representation, and the AIProbS provides a unified architecture that various nodal representations generated by other methods, like machine learning-based methods, can be involved, which may further improve the precision of link prediction.\n\nNevertheless, like any model under the effect of the no-free-lunch theorem \\cite{adam2019no} that no model can always perform well enough as expected in all different scenarios, the AIProbS has its disadvantages in some scenarios. With the increase in data scales, the AIProbS overall underperformed machine learning-based models on precision. Although the AIProbS can make a good trade-off with machine learning methods on precision and interpretability, in some applications where results' interpretability is unnecessary, like computer vision, machine learning methods seem like a better choice. Besides, since quantum machine learning is usually claimed as the next generation of machine learning, which can exponentially uplift a model's computing efficiency, would costly hyperparameter tuning be no longer an apprehension in the future? In other words, would determinable network representation that could sacrifice some precision but not representation learning-based (\\textit{i.e.}, machine learning-based) methods that are adept in precision still be worthy of quantum computing devices in the future?\n\n\\section{Acknowledgments}\n\nThe author would like to acknowledge Linyuan L\\\"u, Hao Wang, and Fang Zhou for their discussions and suggestions.\n\n\\section{Data availability statement}\n\\label{data_and_code_available}\n\nData and codes are available at \\url{https:\/\/github.com\/pitteryue\/AIProbS}.\n\n\n\n\\section*{References}\n\\bibliographystyle{iopart-num}\n\n\\section{Introduction}\n\nPhysics has long been concerned as a propeller of civilization's evolution in history. The establishment of Newtonian mechanics and thermodynamics drove the ``first technological revolution''. The discovery of the electromagnetic induction phenomenon laid the theoretical foundation for the ``second technological revolution''. Condensed matter physics and quantum physics developed the silicon semiconductor industry for the ``third technological revolution''. With the ongoing ``fourth technological revolution'' currently, physics is also propelling the innovation and development in artificial intelligence, among which the study of complex networks \\cite{strogatz2001exploring, boccaletti2006complex} is a case in point. By using nodes and edges to intuitively describe the nonlinear and heterogeneous interaction patterns of components composing the complex physical, social, or brain systems, its use soon widened to various fields. For decades, scientists have been dedicated to understanding a network's structural and dynamical attributes (like vital node identification \\cite{zhou2019fast, qiu2021identifying}, high-order network structural analysis \\cite{shi2019totally, battiston2020networks, shi2021computing,tang2022optimizing}, and percolation theory \\cite{li2021percolation}) and to utilizing these attributes in specific applications, such as link prediction \\cite{lu2011link}, natural language processing \\cite{pardo2006using}, and recommender systems \\cite{lu2012recommender, xu2020recommending, deng2021recommender}.\n\nRecently, as a pivotal tool to abstract a network's structural and dynamical attributes for utilization in a manner that maps the network or its substructures (like nodes) into a low-dimensional vector space, network representation \\cite{liu2021network, barros2021survey} has intrigued scientists for years, especially in light of ample evidence that network representation has several virtues dear to both academia and industry \\cite{deng2021recommender}: reusable object representations by manual or automated feature engineering, enhanced model precision, and efficient parallel computation based on GPU, among which some are even unprecedented compared to their predecessors. Nevertheless, since the current methods of network representation are mostly based on machine learning \\cite{shalev2014understanding}, almost a black box facing fundamental limits on well-explainable and raising difficulties in tedious hyperparameter tuning attributed to its input-output data fitting rationale, the learned vector space's dimension is generally indeterminable and its elements are not interpreted. Consequently, enormous computing resources are required for searching the suboptimal dimension for the vector space within a range but in most cases researchers still can not interpret why such a dimension works out and what realistic meanings the space's elements may represent. Although recent years have seen massive efforts by computer scientists and mathematics to cope with this issue, the root causes still remain unresolved. For example, although automated machine learning \\cite{yao2018taking} proposed by computer scientists can pave the way for accelerating the search of the space's suboptimal dimension not manually, the search mechanism requiring enormous computing resources still remains. Moreover, although mathematics can conclude an empirical formula used to select the optimal dimension for the space \\cite{gu2021principled}, it is normally built on specific models or data, limiting its interpretability and generalization to other scenarios. Given these inadequacies, determinable and interpretable network representation is still an open and important question.\n\nIn this work, from a physical perspective, this article proposes two methods of determinable and interpretable network representation. Methodologically, the first method is based on the degree, H-index, and Coreness (DHC) theorem \\cite{lu2016h} constructing an operator to generate sequences (with a fixed length) of H-indices for nodes. Regarding their realistic meanings and on the advice of the rich club theory \\cite{colizza2006detecting}, this article utilizes these H-indices to construct node representations, which can represent nodes' local attributes around the neighborhood. To abstract nodes' global attributes in a complex network, the second method is based on the DHC entropy (DHC-E) \\cite{wang2021hyperparameter}, a hyperparameter-free and explainable whole graph embedding algorithm we proposed. If a bipartite network, its $m \\times n$ adjacent matrix can be extended to a $(m+n) \\times (m+n)$ augmented matrix, a simple matrix that can be decomposed to $m+n$ matrices, each of which corresponds to a node and carries the node's global attributes. After implementing the DHC-E algorithm on them, node representations can be generated. Unlike those learned by machine learning-based methods, the node representations generated by the two methods have both a determined dimension and interpretable elements.\n\nTo evaluate the proposed two methods' effectiveness and generalization, this article further proposes Adaptive and Interpretable ProbS (AIProbS), a network-based link prediction model for bipartite networks, which can utilize nodal representations generated by the two methods, as an attempt to enhance the prediction precision. Methodologically, built on a classical network-based framework called ProbS \\cite{zhang2007recommendation}, the AIProbS can control the resource diffusion process of the ProbS framework by setting edge weights quantified with node representations, which can perceive the similarity between nodes. After being equipped with such artificial intelligence as machine learning-based models do, the AIProbS makes the flaw of the classical ProbS framework in self-adaptive perception ability oriented to different scenarios (which is analyzed in Sec.~\\ref{ProbS}). At the same time, compared with machine learning-based link prediction models \\cite{koren2008factorization, rendle2012bpr, chen2019integrating, jiang2020clustering, xu2021topic}, the AIProbS is hyperparameter-free. In addition, implemented on several designed control experiments of diverse recommender systems (a specific application of link prediction in artificial intelligence), experimental results showed that the AIProbS can reach state-of-the-art precision beyond baseline models on some scenarios and can, by and large, make a good trade-off with machine learning-based models on precision, determinacy, and interpretability.\n\n\\section{The model}\n\nIn the first place, this article proposes two novel network representation methods in Sec.~\\ref{generate_node_features}. Then, a classical network-based link prediction framework called ProbS is introduced and its flaws are revealed in Sec.~\\ref{ProbS}. Based on the ProbS framework, this article proposes Adaptive and Interpretable ProbS (AIProbS) in Sec.~\\ref{AIProbS}, a network-based link prediction model for bipartite networks, which can utilize nodal representations generated by the two methods and can enhance the prediction precision of the classical ProbS framework by making up its flaws.\n\n\\subsection{Generate a complex network's nodal representations}\n\\label{generate_node_features}\n\n\\subsubsection{Method one}\n\\label{method_one}\n\nDegree, H-index, and coreness are three measurements used to quantify nodal influence in a complex network. Node's degree measures nodal influence by counting a node's neighbors: the greater a node's degree is, the more neighbors it is connected with, and the higher influence it has. Node's H-index \\cite{hirsch2005index} is the maximum value $h$ such that a node has at least $h$ neighbors with a degree no less than $h$. Furthermore, to take location into account, coreness calculated by $k$-core decomposition \\cite{dorogovtsev2006k} measures a node's centrality: a greater coreness indicates that a node locates more centrally in a complex network and hence has a higher influence.\n\nThe DHC theorem \\cite{lu2016h} reveals that degree, H-index, and coreness are all related. To describe the relationship, the DHC theorem constructs an operator $\\mathcal{H}$, which calculates the maximum value $h$ for each node such that the node has at least $h$ neighbors with H-indices no less than $h$. For each node $i$ in a complex network, taking its degree $k_i$ as the zero-order H-index $h_i^{(0)}$ as the beginning, the first-order H-index $h_{i}^{(1)}$ of node $i$ is calculated by $\\mathcal{H}(h_{j_1}^{(0)}, h_{j_2}^{(0)}, ..., h_{j_{k_i}}^{(0)})$, where $h_{j_1}^{(0)}, h_{j_2}^{(0)}, ..., h_{j_{k_i}}^{(0)}$ are the zero-order H-indices (\\textit{i.e.}, the degree values) of the $k_i$ neighbors of node $i$. By iteratively doing so, $h_i^{(2)} = \\mathcal{H}(h_{j_1}^{(1)}, h_{j_2}^{(1)}, ..., h_{j_{k_i}}^{(1)})$, as well as $h_i^{(3)}, h_i^{(4)},...$, can be calculated. Finally, a sequence $h_i^{(0)}, h_i^{(1)}, h_i^{(2)}, ...$ with a fixed length is generated for node $i$, which is convergent to node $i$'s coreness, as the DHC theorem states:\n\n\\textbf{Theorem 2.1}. \\textit{For each node in a complex network, node $i$'s H-indices sequence $h_i^{(0)}, h_i^{(1)}, h_i^{(2)}, ...$ is convergent to its coreness $c_i$, \\textit{i.e.}, $ \\displaystyle c_i = \\lim_{n \\to \\infty} h_i^{(n)}$.}\n\n\\textit{Proof.} See \\cite{lu2016h}.\n\nAccording to the rich club theory \\cite{colizza2006detecting} (from the field of social network analysis \\cite{scott1988social} and soon widened to interdisciplinary studies like computer science \\cite{zhou2004rich} or cognitive science \\cite{van2011rich}) that a node's influence could reflect its attributes and functions around the neighborhood and the whole network structure, this article proposes the following assumption:\n\n\\textbf{Assumption 2.1}. \\textit{A node's H-indices sequence can abstract the node's multidimensional influence in the neighborhood, where the sequence's convergence steps can reflect the magnitude of the node's influence. The more important role played by the node in the neighborhood, the more slowly its influence decays during the dynamic evolution (\\textit{i.e.}, the convergence process by the DHC theorem), thus the larger its convergence steps are. }\n\nBuilt on assumption 2.1 this article takes a node's H-indices sequence as its node representation. In this way, provided $n$ nodes in a complex network and given that their H-indices sequence converges after up to $s$ steps, this method can map the $n$ nodes to a $s$-dimensional vector space consisting of their H-indices as node representations. This is a determinable and interpretable network representation method, since for an arbitrary complex network the dimension of its nodal representations is determined as $s$ and the elements can be interpreted as nodal multidimensional influence with different magnitudes.\n\n\\subsubsection{Method two}\n\\label{method_two}\n\nFollowing method one, to further abstract a node's global attributes in a complex network, if a bipartite network, its adjacency matrix $A^{m \\times n}$ can be extended to $B^{(m+n)\\times (m+n)}$ constructed by\n$\\displaystyle \\left(\\begin{array}{cc}\nO^{m \\times m} & A^{m \\times n} \\\\\n(A^{m \\times n})^T & O^{n \\times n}\n\\end{array}\\right)$,\nwhere $O$ denotes the null matrix. Based on it, a series of $\\lambda_i$ and $B_i$ can be decomposed by the following theorem.\n\n\\textbf{Theorem 2.2}. \\textit{The adjacency matrix $\\displaystyle B^{(m+n)\\times (m+n)}$ can be decomposed by $\\displaystyle B=\\sum_{i=1}^{m+n}\\lambda_i B_i$, where $\\lambda_i$ is the i-th eigenvalue of $\\displaystyle B^{(m+n)\\times (m+n)}$ and $\\displaystyle B_i$ is the corresponding idempotent matrix.}\n\n\\textit{Proof.} See \\textbf{\\textit{Appendix A}}.\n\nAfter that, this article implements the DHC-E operator $\\mathcal{E}$ \\cite{wang2021hyperparameter} (\\textit{i.e.}, by the DHC theorem to generate a H-index matrix $H^{n \\times s}$ by row containing the H-indices converged after $s$ steps of each of the $n$ nodes in a complex network, the operator $\\mathcal{E}$ calculates the Shannon entropy of each column of $H^{n \\times s}$ and obtains a vector $e^{1 \\times s}$, as the whole graph embedding of the network) on each $B_i$ or $\\lambda_i B_i$ one by one, generating the $m+n$ nodes' representations for the bipartite network correspondingly. Apparently, this method is also a determinable and interpretable network representation method. The characteristics of interpretability and hyperparameter-free of the DHC-E algorithm are thoroughly illuminated in \\cite{wang2021hyperparameter}.\n\n\\subsection{The ProbS framework and its flaws}\n\\label{ProbS}\n\nTo evaluate the two methods' effectiveness and generalization, this article utilizes them in link prediction for bipartite networks. Since network representation can be seen as artificial intelligence that recognizes and abstracts a complex network's underlying structural and dynamical attributes, this article explores how such artificial intelligence (\\textit{i.e.}, nodal representations generated by these methods) can be used to enhance the precision of classical link prediction models.\n\nAmong classical (non-machine learning-based) link prediction models for bipartite networks, the ProbS \\cite{zhang2007recommendation} framework is a typical one. By means of a resource diffusion mechanism inspired by the physical process of Material Diffusion, the ProbS framework can quantify the similarity between nodes after initializing and diffusing resources. Fig.~\\ref{Schematics} includes an example to intuitively illuminate the schematics of the ProbS framework. For instance, when predicting node $B$'s unobserved links with nodes $a$ and $b$, resources are first initialized at nodes $c$ and $d$ (the nodes that are connected with node $B$) with value $1$, then are diffused to nodes $A$, $B$, and $C$ along edges after being equally divided by the degree of each node, finally are diffused back to nodes $a$, $b$, $c$, and $d$ in the same way, which can be used to quantify the similarity between node $B$ and the four nodes, respectively. A larger similarity of two nodes indicates a higher probability of an unobserved link existing between them.\n\nThis article provides a mathematical perspective to describe the ProbS framework, by constructing an operator $T$ to describe its diffusion mechanism. Given a bipartite network consisting of $m+n$ nodes of two different types, respectively, whose adjacency matrix is represented by $A^{m \\times n}$. Let $R^{m \\times n}$ denote the predicted matrix, where $R_{ij}$ represents the similarity (\\textit{i.e.}, the probability of the existence of a link) between nodes $i$ and $j$. Then, through the ProbS framework $R^{m \\times n}$ can be calculated by\n\\begin{equation}\n\\label{Probs}\n R = A \\cdot (D_{I} \\circ A)^T \\cdot (D_{U} \\circ A)\n\\end{equation}\nwhere $\\cdot$ denotes the dot product, and $\\circ$ denotes the Hadamard product. $D_I^{m \\times n} = (a_1, a_2, ..., a_n)$, $\\displaystyle a_i = (\\frac{1}{k_{I_i}}, ... , \\frac{1}{k_{I_i}})^T$ where $k_{I_i}$ is item $i$'s degree. $D_U^{m \\times n} = (a_1, a_2, ..., a_m)^T$, $\\displaystyle a_i = (\\frac{1}{k_{U_i}}, ... ,\\frac{1}{k_{U_i}})$ where $k_{U_i}$ is user $i$'s degree. In Eq.~(\\ref{Probs}) the operator $T=(D_{I} \\circ A)^T \\cdot (D_{U} \\circ A)$.\n\nThe operator $T$ tells why the ProbS framework will converge after deriving $R$ from $A$ and then placing $A$ with the derived $R$ iteratively, stated as the following theorem.\n\n\\textbf{Theorem 2.3}. Let the operator $T=(D_{I} \\circ A)^T \\cdot (D_{U} \\circ A)$ iteratively act on $A$ by $A \\leftarrow A \\cdot T$, the iterative process is convergent.\n\n\\textit{Proof}. See \\textbf{\\textit{Appendix B}}.\n\nSince the difference between the values in $A$ tends to be smoother as the convergent iterative process progresses but link prediction relies for higher precision on the more distinctive differentiation between the predicted values of similarity \\cite{deng2021recommender}, in link prediction the best iteration steps for the ProbS framework is one, and so does the AIProbS proposed in Sec.~\\ref{AIProbS}.\n\nIn addition, from such a mathematical perspective, it is easy to see that the ProbS framework faces fundamental limits on intelligence because its resource diffusion mechanism is just based on equal allocation, shown as $D_I$ and $D_U$ in Eq.~(\\ref{Probs}). In practice like recommender systems (an application of link prediction for bipartite networks in artificial intelligence), such a mechanism raises a key question: if respectively take these nodes of two different types as users and items in recommender systems, the resources diffused between users and items back and forth, to some extent, represent user's preferences for items or item's attractiveness to users, while neither of them should be necessarily equal since user biases \\cite{koren2009matrix, adomavicius2014biasing, manjur2021exploring} and item biases \\cite{koren2009matrix, park2014uncovering} generally exist in reality. Moreover, these biases are usually recommendation scenario-oriented, which means that in different scenarios a user's preferences may differ, and so do an item's attractiveness or popularity. Finally, in practice the ProbS framework fails to take these biases into consideration, let alone adaptively perceive and quantify their differences in various scenarios.\n\n\\subsection{The AIProbS model}\n\\label{AIProbS}\n\nThe essential condition for the ProbS framework to realize that intelligence is to be equipped with self-adaptive perception, an ability to perceive and utilize the attributes of nodes (\\textit{i.e.}, nodal representations) in a complex network toward different scenarios. To utilize the nodal representations generated by the two methods proposed in Sec.~\\ref{generate_node_features} in the ProbS framework, this article proposes Adaptive and Interpretable ProbS (AIProbS).\n\nIn the first step, on the advice that the rich club theory \\cite{colizza2006detecting} gives clues that nodes with high centrality tend to form tightly interconnected communities, this article generalizes this conclusion to the field of link prediction, proposing the following assumption:\n\n\\textbf{Assumption 2.2}. \\textit{The similarity between node pairs having strongly correlated nodal representations (\\textit{i.e.}, similar features or similar influence) is higher than that between weakly correlated ones.}\n\nTo measure the similarity between nodes, the AIProbS uses the cosine similarity metric. Provided two $n$-dimension vectors $x$ and $y$, the cosine similarity between them is calculated by $\\displaystyle \\cos(\\theta)=\\frac{x \\cdot y}{|x| \\cdot |y|}=\\frac{\\sum_{i=1}^n x_i y_i}{\\sqrt{\\sum_{i=1}^n x_i^2} \\cdot \\sqrt{\\sum_{i=1}^n y_i^2}}$. In the same way, provided $m+n$ nodes belonging to two sets $U$ and $I$ of two different types in a bipartite network, respective. Through the network representation methods proposed in Sec.~\\ref{generate_node_features} the representation matrices $F_U^{m \\times s}$ and $F_I^{n \\times s}$ of the two types of nodes are generated, either of which is consist of nodal representations by row. Then, the $m \\times n$ nodal similarity matrix $S^{m \\times n}$ calculated by the cosine similarity metric is\n\\begin{equation}\n\\label{user-item_similarity_matrix}\nS = \\frac{F_{U} \\cdot F_{I}^T}{\\alpha^T \\cdot \\beta},\n\\end{equation}\nwhere vector $\\displaystyle \\alpha = \\big(\\sqrt{\\sum_{j=1}^s{F_U}_{1j}^2}\\,,\\sqrt{\\sum_{j=1}^s{F_U}_{2j}^2}\\,, \\,...\\,, \\sqrt{\\sum_{j=1}^s{F_U}_{mj}^2}\\big)$ and vector $\\displaystyle \\beta = \\big(\\sqrt{\\sum_{j=1}^s{F_I}_{1j}^2}\\,,\\sqrt{\\sum_{j=1}^s{F_I}_{2j}^2}\\,, \\,...\\,, \\sqrt{\\sum_{j=1}^s{F_I}_{nj}^2}\\big)$.\n\nAfter obtaining the nodal similarity matrix $S^{m \\times n}$, utilizing it to control the diffusion process of the classical ProbS framework is the second step. To assign proper weights to every node pair for the diffusion mechanism of the ProbS framework, the AIProbS further complete some normalization and proportioning operations on $S \\circ A$ where $A$ is the adjacency matrix shown in Eq.~(\\ref{Probs}). Since the elements of $S \\circ A$ vary in $[-1,1]$ while the diffused resources are supposed to be positive, the AIProbS normalizes the value range of the elements to $[0,1]$ using the max-min normalization operation, for each row vector $(S\\circ A)_{i*}$ $(i=1,2,...,m)$ of $S \\circ A$, by\n\\begin{equation}\n\\label{max-min_normalization}\n(S \\circ A)_{ij} \\leftarrow \\frac{(S \\circ A)_{ij} - \\min}{ \\max - \\min}, \\,j = 1,2, ..., n,\n\\end{equation}\nwhere the $\\max$ and $\\min$ are the maximum and minimum elements of the row vector $(S \\circ A)_{i*}$, respectively. Based on that, the weight matrix $W_U$ for nodes belonging to set $U$ is calculated by the proportioning operation as\n\\begin{equation}\n\\label{proportioning}\nW_{U_{ij}} = \\frac{1}{(S \\circ A)_{ij}}\\sum_{k=1}^{n} (S \\circ A)_{ik}, \\,i=1,2,...,m, \\, j=1,2,...,n.\n\\end{equation}\nOn the other hand, the same operations are completed on $S^{m \\times n}$ by column, generating the weight matrix $W_I^{m \\times n}$ for nodes belonging to set $I$.\n\nIn the last step, the predicted matrix $R^{m \\times n}$, where $R_{ij}$ represents the prediced similarity between nodes $i$ and $j$, is calculated through the AIProbS by\n\\begin{equation}\n R = A \\cdot W_{I}^T \\cdot W_{U}.\n\\end{equation}\n\nConceivably, there are other metrics for similarity measurement. More combinations were tested in this article (See \\textbf{\\textit{Appendix C}} for details) but none of them performed better than the one proposed in this section. All in all, the whole process of the AIProbS are summarized in the pseudocodes shown in \\textit{Appendix D}. For more intuitive illumination, Fig.~\\ref{Schematics} in \\textbf{\\textit{Appendix D}} presents its schematics.\n\n\\subsection{Performance Evaluation}\n\\label{performance_evaluation}\n\nTo evaluate the AIProbS's precision as well as its pros and cons in link prediction for bipartite networks, which also can be used to reflect the effectiveness of nodal representations generated by the two network representation methods proposed in Sec.~\\ref{generate_node_features}, this article designs control experiments based on recommender systems, an application of link prediction in artificial intelligence.\n\n\\subsubsection{Recommender systems}\n\nBy analyzing observed user-item relations to predict a user's preferred items from millions of candidates, recommender systems \\cite{lu2012recommender, deng2021recommender} are recognized as a pivotal tool to alleviate the information overload problem. Among different user-item relations, implicit user-item interactions (\\textit{e.g.}, user's historical clicks or buys on items) record the existence of a user's interactions with items, defined as a binary state using $1$ and $0$. From the perspective of a complex network, recommendation on implicit user-item interactions can be seen as a process of link prediction for bipartite networks, where users and items correspond to the two types of nodes and implicit user-item interactions represent the edges between nodes. Therefore, the designed experiments in this article are based on the recommendation with implicit user-item interactions, for most current models are based on them.\n\n\\subsubsection{Data sets}\n\nIn light of the no-free-lunch theorem \\cite{adam2019no} that no model can always perform well enough as expected in all different scenarios, this article designs control experiments to evaluate the performance of the AIProbS on diverse real recommendation scenarios, in order to explore not only the AIPobS's pros but also its cons in different scenarios.\n\n\\begin{table}[ht]\n\\caption{\\textbf{Overview of data sets.}}\n\\label{Datasets}\n\\begin{indented}\n\\item[]\\begin{tabular}{ccllcccc}\n\\br\nData sets & \\multicolumn{3}{c}{$|U|$} & $|I|$ & Interactions & Sparsity \\\\\n\\mr\nMovieLens 100K & \\multicolumn{3}{c}{943} & 1680 & 100000 & 93.70\\% \\\\\nMovieLens 1M & \\multicolumn{3}{c}{6040} & 3952 & 1000209 & 95.81\\% \\\\\nLastFM & \\multicolumn{3}{c}{1892} & 17632 & 92834 & 99.72\\% \\\\\n\\br\n\\end{tabular}\n\\end{indented}\n\\end{table}\n\nAs shown in Tab.~\\ref{Datasets}, $|U|$ and $|I|$ represent the number of users and items, respectively, and the interactions between users and items are implicit ones. The sparsity in Tab.~\\ref{Datasets} represents the ratio of the number of unobserved interactions to the maximum number of all possible interactions between users and items (e.g., that between $m$ users and $n$ items is $mn$). As a control group, the MovieLens 100K, MovieLens 1M, and LastFM are three classical data sets from two different recommender systems of movies and music, with distinctive ratios of $|U|$ to $|V|$, data scales, and sparsity, based on which more persuadable results could yield compared with those based on newly published data sets, since these classical data sets have been widely used for evaluation in previous works\n\nIn order to guarantee the reproducibility of experiments, either of the three data sets is obtained from the RecBole public resources (\\url{https:\/\/recbole.io\/dataset_list.html}), organized into tuples (user, item, $0\/1$) without preprocessing. Each of them is randomly split into a ``train\/evaluate\/test'' set by the ratio of ``$80\/10\/10\\%$''. After independently repeating the splitting process $30$ times, $30$ realizations are generated for each data set. One can get the split data used in this article through the hyperlink address posted in Sec.~\\ref{data_and_code_available}.\n\n\\subsubsection{Evaluation metrics}\n\nIn order to quantify the precision of the AIProbS on these data sets, three common-used metrics are chosen in this article. Given a user $u \\in U$ ($U$ is the user set) and the length $N$ of the recommendation list, the set of recommended items for the user is denoted by $\\hat{R}(u)$ and the ground-truth set of items the user interacted with is denoted by $R(u)$. Based on them, the first evaluation metric is the Recall@N \\cite{olson2008advanced}, which calculates the fraction of predicted relevant items out of all ground-truth relevant items by\n\\begin{equation}\n\\mbox{Recall@N}=\\frac{1}{|U|}\\sum_{u \\in U} \\frac{|\\hat{R}(u) \\cap R(u)|}{|R(u)|},\n\\end{equation}\nwhere $|R(u)|$ represents the item count of $R(u)$.\n\nTo calculate the reciprocal rank of the first relevant item recommended to each user, the second evaluation metric MRR@N \\cite{craswell2009mean} is denoted as\n\\begin{equation}\n\\mbox{MRR@N}=\\frac{1}{|U|} \\sum_{u \\in U} \\frac{1}{\\mbox{rank}_u^*},\n\\end{equation}\nwhere $\\mbox{rank}_u^*$ is the rank position of the first relevant item recommended to user $u$.\n\nMoreover, as the third evaluation metric, the NDCG@N \\cite{wang2013theoretical} can further measure the overall ranking quality in a manner that accounts for the position of the hit by assigning higher scores to hits at top ranks as\n\\begin{equation}\n\\footnotesize\n\\mbox{NDCG@N}=\\frac{1}{|U|} \\sum_{u \\in U} \\big( \\frac{1}{\\sum_{i=1}^{\\min(|R(u)|,N)}\\frac{1}{\\log_2(i+1)}} \\sum_{i=1}^N \\delta(i \\in R(u)) \\frac{1}{\\log_2(i+1)}\\big),\n\\end{equation}\nwhere $\\delta(\\cdot)$ is an indicator function and positions are discounted logarithmically.\n\nIn practice, the greater the values of these evaluation metrics are, the higher a model's precision is.\n\n\\subsubsection{Baseline methods}\n\nThis article constructs or chooses nine baseline models as follows, evaluating the pros and cons of the AIProbS compared with its predecessors of both classical and machine learning-based baselines.\n\n\nClassical baselines include two models. As the bedrock, the ProbS \\cite{zhang2007recommendation} is a necessary baseline to evaluate the improvement of the AIProbS. In addition, one might expect to base the recommendation directly on the nodal representations generated by the methods proposed in Sec.~\\ref{generate_node_features}, not the ProbS framework. To test this strategy, this article constructs the Pure-DHC model, used to perform the recommendation by Eq.~(\\ref{user-item_similarity_matrix}) based on the user-item similarity of their H-indices (\\textit{i.e.}, nodal representations).\n\nMachine learning-based baselines include seven models. To avoid the baseline pitfalls that have plagued earlier research on the comprehensive and objective evaluation of proposed models, this article further chooses seven representative machine learning-based models as baselines, among which were based on six different techniques of machine learning frameworks, including NeuMF \\cite{he2017neural} based on deep neural networks, ConvNCF \\cite{he2018outer} and SpectralCF \\cite{zheng2018spectral} based on convolution operations, GCMC \\cite{berg2017graph} based on graph auto-encoder frameworks, LINE \\cite{tang2015line} based on random walking, NGCF \\cite{wang2019neural} based on graph neural networks, and DGCF \\cite{wang2020disentangled} based on attention mechanisms.\n\nTo guarantee the fairness and reproducibility of experiments, the implementation and evaluation of models were hosted to the RecBole \\cite{zhao2021recbole}, a public open pipeline of recommender systems. One can get the codes used in this article through the hyperlink address posted in Sec.~\\ref{data_and_code_available}.\n\n\\section{Results}\n\\label{Experimental_results}\n\nBased on the experimental settings in Sec.~\\ref{performance_evaluation}, this section presents the experimental results on the precision and robustness of the AIProbS and baseline models in Sec.~\\ref{Precision analysis} and Sec.~\\ref{Robustness analysis}, respectively, revealing their pros and cons in different recommendation scenarios.\n\n\n\\subsection{Precision analysis}\n\\label{Precision analysis}\n\nAs shown in Tabs.~\\ref{LastFM}, \\ref{100K} and \\ref{1M}, the results on model precision are presented, where the length $N$ of the recommendation list is set to $10$, and each model's precision is averaged from its independently implementation based on $30$ different realizations. The values in parentheses indicate the percentage of improvement or decline in model precision of the AIProb model compared to the respective baseline models on each specific data set and evaluation metric, where the percentage of improvement is bold.\n\nConceivably, when speaking of the necessity of determinable and interpretable network representation and their utilization in link prediction, one might cast it into doubt: do not nodal representations generated by the two methods in Sec.~\\ref{generate_node_features} be sufficient for link prediction? Can the precision of classical link prediction frameworks really be enhanced by being involved with nodal representations as intelligence? Are the machine learning-based network representation methods not precise enough? As shown in Tabs.~\\ref{LastFM}, \\ref{100K} and \\ref{1M}, on all three data sets the Pure-DHC which directly utilizes the nodal representations generated by the methods in Sec.~\\ref{generate_node_features} for recommendation achieved the worst model precision among the AIProbS and baseline models. That is to say, such generated nodal representations could be nothing with the recommendation if not be utilized in the ProbS or any other recommendation framework. After utilizing these nodal representations in the ProbS framework, as shown in Tabs.~\\ref{LastFM}, \\ref{100K} and \\ref{1M}, the AIProbS outperformed the classical ProbS on all three data sets. When compared to machine learning-based baselines, the AIProbS indeed performed worse than some machine learning models, most obviously on MovieLens 1M. But it still can achieve state-of-the-art performance on model precision on LastFM, suggesting that nodal representations generated by the methods in Sec.~\\ref{generate_node_features} may be able to abstract the underlying attributes of a complex network better than those learned by machine learning methods.\n\n\\begin{table}[ht]\n\\caption{\\textbf{Results of model precision on LastFM.}}\n\\label{LastFM}\n\\begin{indented}\n\\item[]\\begin{tabular}{@{}cccc@{}}\n\\br\n\\multirow{2}{*}{Models} & \\multicolumn{3}{c}{LastFM} \\\\ \\cmidrule(r){2-4}\n & Recall@10 & MRR@10 & NDCG@10 \\\\ \\mr\nPure-DHC & 0.004 & 0.006 & 0.003 \\\\\nProbS & 0.170 (\\textbf{+7.5}\\%) & 0.308 (\\textbf{+9.6}\\%) & 0.166(\\textbf{+9.2}\\%) \\\\\nAIProbS & \\textbf{0.184} & \\textbf{0.340} & \\textbf{0.183} \\\\ \\midrule\nNeuMF & 0.060 (\\textbf{+67.5}\\%) & 0.092 (\\textbf{+73.0}\\%) & 0.050 (\\textbf{+72.5}\\%) \\\\\nConvNCF & 0.056 (\\textbf{+69.7}\\%) & 0.090 (\\textbf{+73.6}\\%) & 0.048 (\\textbf{+73.8}\\%) \\\\\nSpectralCF & 0.066 (\\textbf{+63.9}\\%) & 0.120 (\\textbf{+64.7}\\%) & 0.062 (\\textbf{+66.1}\\%) \\\\\nGCMC & 0.121 (\\textbf{+34.1}\\%) & 0.214 (\\textbf{+37.0}\\%) & 0.116 (\\textbf{+36.5}\\%) \\\\\nLINE & 0.149 (\\textbf{+19.1}\\%) & 0.272 (\\textbf{+20.0}\\%) & 0.145 (\\textbf{+20.5}\\%) \\\\\nNGCF & 0.169 (\\textbf{+8.2}\\%) & 0.301 (\\textbf{+11.4}\\%) & 0.163 (\\textbf{+11.0}\\%) \\\\\nDGCF & 0.177 (\\textbf{+3.7}\\%) & 0.316 (\\textbf{+7.1}\\%) & 0.172 (\\textbf{+6.0}\\%) \\\\ \\br\n\\end{tabular}\n\\end{indented}\n\\end{table}\n\n\\begin{table}[ht]\n\\caption{\\textbf{Results of model precision on MovieLens 100K.}}\n\\label{100K}\n\\begin{indented}\n\\item[]\\begin{tabular}{@{}cccc@{}}\n\\br\n\\multirow{2}{*}{Models} & \\multicolumn{3}{c}{MovieLens 100K} \\\\ \\cmidrule(r){2-4}\n & Recall@10 & MRR@10 & NDCG@10 \\\\ \\mr\nPure-DHC & 0.017 & 0.084 & 0.041 \\\\\nProbS & 0.208 (\\textbf{+3.1}\\%) & 0.413 (\\textbf{+4.8}\\%) & 0.236 (\\textbf{+4.5}\\%) \\\\\nAIProbS & \\textbf{0.215} & \\textbf{0.434} & \\textbf{0.248} \\\\ \\midrule\nNeuMF & 0.070 (\\textbf{+67.2}\\%) & 0.187 (\\textbf{+56.9}\\%) & 0.093 (\\textbf{+62.3}\\%) \\\\\nConvNCF & 0.099 (\\textbf{+53.9}\\%) & 0.245 (\\textbf{+43.5}\\%) & 0.125 (\\textbf{+49.5}\\%) \\\\\nSpectralCF & 0.124 (\\textbf{+42.4}\\%) & 0.293 (\\textbf{+32.5}\\%) & 0.153 (\\textbf{+38.2}\\%) \\\\\nGCMC & 0.196 (\\textbf{+8.9}\\%) & 0.400 (\\textbf{+7.9}\\%) & 0.232 (\\textbf{+6.4}\\%) \\\\\nLINE & 0.190 (\\textbf{+11.3}\\%) & 0.391 (\\textbf{+9.8}\\%) & 0.225 (\\textbf{+9.0}\\%) \\\\\nNGCF & 0.245 (-14.3\\%) & 0.481 (-10.8\\%) & 0.293 (-18.4\\%) \\\\\nDGCF & 0.236 (-9.9\\%) & 0.458 (-5.5\\%) & 0.278 (-12.3\\%) \\\\ \\br\n\\end{tabular}\n\\end{indented}\n\\end{table}\n\n\n\\begin{table}[ht]\n\\caption{\\textbf{Results of model precision on MovieLens 1M.}}\n\\label{1M}\n\\begin{indented}\n\\item[]\\begin{tabular}{@{}cccc@{}}\n\\br\n\\multirow{2}{*}{Models} & \\multicolumn{3}{c}{MovieLens 1M} \\\\ \\cmidrule(r){2-4}\n & Recall@10 & MRR@10 & NDCG@10 \\\\ \\mr\nPure-DHC & 0.002 & 0.031 & 0.014 \\\\\nProbS & 0.108 (\\textbf{+17.6\\%}) & 0.352 (\\textbf{+15.1\\%}) & 0.177 (\\textbf{+15.9\\%}) \\\\\nAIProbS & \\textbf{0.131} & \\textbf{0.414} & \\textbf{0.210\\%} \\\\ \\midrule\nNeuMF & 0.032 (\\textbf{+75.4\\%}) & 0.128 (\\textbf{+69.1\\%}) & 0.053 (\\textbf{+74.6\\%}) \\\\\nConvNCF & 0.073 (\\textbf{+44.2\\%}) & 0.255 (\\textbf{+38.4\\%}) & 0.128 (\\textbf{+38.9\\%}) \\\\\nSpectralCF & 0.147 (-11.5\\%) & 0.416 (-0.5\\%) & 0.236 (-12.2\\%) \\\\\nGCMC & 0.152 (-15.4\\%) & 0.421 (-1.7\\%) & 0.240 (-14.4\\%) \\\\\nLINE & 0.153 (-16.6\\%) & 0.423 (-2.1\\%) & 0.236 (-12.3\\%) \\\\\nNGCF & 0.162 (-23.4\\%) & 0.442 (-6.7\\%) & 0.254 (-21.0\\%) \\\\\nDGCF & 0.172 (-31.0\\%) & 0.460 (-11.1\\%) & 0.266 (-26.8\\%) \\\\ \\br\n\\end{tabular}\n\\end{indented}\n\\end{table}\n\nTo put these results in more general terms, it is definite that designing control experiments to guarantee the comprehensiveness and objectivity of model performance evaluation is indispensable because, as shown in Tabs.~\\ref{LastFM}, \\ref{100K} and \\ref{1M}, the comparative predominance between different models or even that between the classical and machine learning-based models are distinctive. For instance, compared to its predecessor (the ProbS framework), the AIProbS at best improved the Recall@$10$ by $17.6\\%$ on MovieLens 1M and at worst, by $3.1\\%$ on MovieLens 100K. Such a $14.5\\%$ gap shows that the predominance of the AIProbS over the ProbS is not necessarily that significant in all recommendation scenarios. Overall, on MovieLens 1M, although it achieved an appreciable improvement over the ProbS, the AIProbS still performed worse than the other five machine learning-based models, revealing the predominance of the machine learning-based frameworks over the classical ones on this data set. However, that predominance faded on MovieLens 100K because only two machine learning-based models (\\textit{i.e.}, NGCF and DGCF) outperformed the AIProbS. On LastFM, none of the machine learning-based models outperformed the AIProbS, in other words, but the AIProbS achieved state-of-the-art performance on model precision.\n\nFiguring out the determinant factors of model precision in different recommendation scenarios is not easy and intuitive, not to mention accurately predicting a model's performance for one specific scenario. Still, on the advice of the clues given in Tabs.~\\ref{LastFM}, \\ref{100K} and \\ref{1M}, some discoveries could be summarized as follows. (1) The machine learning-based models might have a predominance on data sets with large scales. The recommendation scenario of MovieLens 1M and MovieLens 100K being equal, the machine learning-based models would show a more obvious predominance on the former with a comparative larger data scale than the latter. However, it is hard to assert that the distinctions of ratios of $|U|$ to $|V|$ and the sparsity of the two data sets play a silent role. (2) The classical frameworks might play a large role in the improvement after integration in recommendation scenarios with high sparsity. Since the sparsity of LastFM is the highest among the three data sets, where the machine learning-based models face fundamental limits on lack of enough user-item interactions for training, the AIProbS or the ProbS combined with or of the classical frameworks showed their predominance as a result of their network structure-oriented resolution. Nevertheless, the ratio of $|U|$ to $|V|$ of LastFM, which seems to be a little higher than the other two, could also be a decisive factor.\n\n\\subsection{Robustness analysis}\n\\label{Robustness analysis}\n\nAs revealed in Sec.~\\ref{Precision analysis}, with the increase in data scale the precision of the AIProbS decreased compared with that of machine learning-based baselines, for the mechanism of data fitting (or pattern representation) adopted by machine learning methods can give fully to its play more suitably in scenarios with larger data scale. Nevertheless, it does not mean that the AIProbS is useless in those scenarios. Since hyperparameters having no realistic meanings could make a model's implementing process indeterminable and vague, machine learning-based models lost almost all the interpretability for results, like why a machine learning-based model generates some recommendations for a user. This problem is not confined to recommender systems but still haunts other applications requiring high interpretability for results, such as machine translation or knowledge graph completion. Besides, since tedious hyperparameter tuning is required for up to the optimal performance of a model, generally machine learning methods reach higher precision by sacrificing the model's computing efficiency. Such a strategy brings about heavy financial (\\textit{i.e.}, computing resources) and time costs for the implementation on a huge data. In contrast, the AIProbS is determinable and its results are interpreted, meaning that this model could be more suitable for applications requiring high interpretability and for scenarios with huge data scales but insufficient computing resources.\n\nOn two realizations of ml-100k for instance, Tabs.~\\ref{P1}, \\ref{P2}, \\ref{P3}, and \\ref{P4} present the relations between the different settings of two representative hyperparameters (\\textit{i.e.}, representation dimension and learning rating) and a model's average precision when one hyperparameter is fixed and others are traversed within a specified search range, where the standard deviation of precision is presented by error arrow at a data point and each model's number of hyperparameters is presented in parentheses, reflecting a model's magnitude of performance fluctuation associated with setting disturbance, which actually can quantify the model's robustness. As shown in Tab.~\\ref{P1}, the AIProbS had a stable performance on recall@10, since the representation dimension of its results is determined. However, as a hyperparameter different settings of representation dimension can largely influence machine learning-based baselines' precision. Although the performance on recall@10 of ConvNCF, LightGCN, and NGCF of different representation dimensions was relatively stable among machine learning-based baselines, that of GCMC and SpectralCF largely fluctuated with the change of representation dimension. For example, as for SpectralCF when the representation dimension is set to $48$ its average precision could be around $27\\%$ higher than that when being set to $16$. On top of that, even when the representation dimension of SpectralCF is set to $48$, seemingly the optimal choice, its performance on recall@10 still faces a $125\\%$ gap between the peaks of performance, flowed from the different settings of other hyperparameters. Similar fluctuations on machine learning-based baselines' precision were revealed by the influence of different settings of representation dimension on mrr@10 and by results shown in Tab.~\\ref{P2} when considering learning rate as the controlled hyperparameter. Attributed to such the indeterminable performance of machine learning methods, one may have to repeatedly try different hyperparameter settings for a machine learning-based model to search out the optimal one, which is definitely computing resources-consuming and time costly. If an insufficient searching process turns out improper hyperparameter settings, the model could even end up with its worst performance. In contrast, once implemented the AIProbS can reach its optimal performance.\n\n\\begin{figure}[ht]\n\\includegraphics[scale=0.46]{P1.pdf}\n\\caption{\\textbf{Relation between model precision and representation dimension reflected on ml-100k realization 1}}\n\\label{P1}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\includegraphics[scale=0.46]{P2.pdf}\n\\caption{\\textbf{Relation between model precision and learning rate reflected on ml-100k realization 1}}\n\\label{P2}\n\\end{figure}\n\nIt may be said, though, that virtually determining a machine learning-based model's optimal hyperparameter settings is burdensome but can be once and for all based on one data set. However, it is not a fact. Similar experiments were done on another realization of ml-100k and the results presented in Tabs.~\\ref{P3} and \\ref{P4} revealed that the optimal hyperparameter settings of a machine learning-based model would be changed with the change of data set. For example, as shown in Tab.\\ref{P1} the optimal representation dimension of SpectralCF was $48$ on ml-100k realization 1 but that on ml-100k realization 2 was changed to be $128$, as shown in Tab.~\\ref{P3}. As a result, the searching process for the optimal hyperparameter settings of a machine learning-based model on a new data set appears to be inevitable. In contrast, with the change of data set the representation dimension of the AIProbS is still automatically determined, once implemented.\n\n\\begin{figure}[ht]\n\\includegraphics[scale=0.46]{P3.pdf}\n\\caption{\\textbf{Relation between model precision and representation dimension reflected on ml-100k realization 2}}\n\\label{P3}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\includegraphics[scale=0.46]{P4.pdf}\n\\caption{\\textbf{Relation between model precision and learning rate reflected on ml-100k realization 2}}\n\\label{P4}\n\\end{figure}\n\nAll in all, in the sense that the AIProbS provides a good trade-off with machine learning-based models on precision, interpretability, and determinacy.\n\n\\section{Discussion}\n\nThis article proposes two determinable and interpretable node representation methods. Different from other attempts like automated machine learning methods by computer scientists and computational theory by mathematics to search out and analyze the sub-optimal (or optimal) representation dimension of a machine learning-based network representation model, respectively, and to interpret the implementing process and the results come out of the model, from a perspective of physics the proposed two methods can substantially generate nodal representations with a determined dimension and interpretable elements, reaching its optimal performance once implemented. After utilizing these representations in link prediction for bipartite networks, experimental results showed that the AIProbS can make a good trade-off with machine learning-based models on precision, determinacy, and interpretability, indicating the effectiveness of nodal representations generated by the proposed two representation methods.\n\nImportantly, these methods with good generalization may motivate further research. For example, nodal representations generated by the proposed two methods can also be utilized in machine learning-based models as initial features or network representation, and the AIProbS provides a unified architecture that various nodal representations generated by other methods, like machine learning-based methods, can be involved, which may further improve the precision of link prediction.\n\nNevertheless, like any model under the effect of the no-free-lunch theorem \\cite{adam2019no} that no model can always perform well enough as expected in all different scenarios, the AIProbS has its disadvantages in some scenarios. With the increase in data scales, the AIProbS overall underperformed machine learning-based models on precision. Although the AIProbS can make a good trade-off with machine learning methods on precision and interpretability, in some applications where results' interpretability is unnecessary, like computer vision, machine learning methods seem like a better choice. Besides, since quantum machine learning is usually claimed as the next generation of machine learning, which can exponentially uplift a model's computing efficiency, would costly hyperparameter tuning be no longer an apprehension in the future? In other words, would determinable network representation that could sacrifice some precision but not representation learning-based (\\textit{i.e.}, machine learning-based) methods that are adept in precision still be worthy of quantum computing devices in the future?\n\n\\section{Acknowledgments}\n\nThe author would like to acknowledge Linyuan L\\\"u, Hao Wang, and Fang Zhou for their discussions and suggestions.\n\n\\section{Data availability statement}\n\\label{data_and_code_available}\n\nData and codes are available at \\url{https:\/\/github.com\/pitteryue\/AIProbS}.\n\n\n\n\\section*{References}\n\\bibliographystyle{iopart-num}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe Euler-Poinsot problem is a three degrees-of-freedom (DOF) problem whose super-integrable character limits the solutions to quasi-periodic orbits on two-torus \\citep{Fasso2005}. Because of the symmetry with respect to rotations about the angular momentum vector, the problem is formulated as a 1-DOF Hamiltonian when using \\citet{Andoyer1923} variables. Then, the essentially unique complete reduction that provides the integration of the problem can be performed in different variables, and it is usually done by solving the Hamilton-Jacobi equation.\n\\par\n\nIn the study of the rigid body rotation under external torques the use of suitable variables reveals crucial to the solution by perturbation methods. A common trend is to use action-angle variables \\citep{Sadov1970, SadovRuso1970,Kinoshita1972}, but other variables can be used instead \\citep{HitzlBreakwell1971}.\n\\par\n\nThe Hamilton-Jacobi equation of the Euler-Poinsot problem in Andoyer variables can be solved formally, without need of specifying the new Hamiltonian \\citep{FerrerLara2010}. We show how this new, unspecified Hamiltonian can be cast into a standard form in which the modulus of the elliptic integrals that appear in the solution of the transformation, remains as an undetermined state function of the new momenta. Under certain conditions imposed to the formal transformation, the modulus is determined by solving a system of partial differential equations. The condition we require is ``simplification'' and find a new set of variables that while showing similar performances than action-angles \\citep{Sadov1970,SadovRuso1970}, has the benefit of not requiring the computation of implicit functions. In addition, we demonstrate that Sadov's transformation is also a member of the general family of Euler-Poinsot transformations to Andoyer variables.\n\n\n\\section{Complete Reduction of the Euler-Poinsot Problem}\n\nThe Hamiltonian of the torque-free rotation is \\citep{DepritAJP1967}\n\\begin{equation}\\label{Andoyer}\n\\mathcal{H}=\\left(\\sin^2\\nu\/A+\\cos^2\\nu\/B\\right)(M^2-N^2)\/2+N^2\/2C,\n\\end{equation}\nwhere $A$, $B$, and $C$ are the principal moments of inertia of the body, and the Andoyer variables are defined by three pairs of conjugate variables: the rotation angle on the equatorial plane of the body $\\nu$ and the projection of the angular momentum vector on the body axis of maxima inertia $N$, the precession angle on the invariant plane $\\mu$ and the modulus of the angular momentum vector $M$, and the node angle on the inertial plane $\\lambda$ and the projection of the angular momentum vector on the axis perpendicular to the inertial plane $\\Lambda$. Because $\\lambda$, $\\Lambda$ and $\\mu$ are cyclic $\\lambda=\\lambda_0$, $\\Lambda=\\Lambda_0$, and $M=M_0$ are constant, and Eq. (\\ref{Andoyer}) is a Hamiltonian of 1-DOF.\n\\par\n\nThe integration may be done by complete reduction. To this goal, we look for canonical transformations\n$\\mathcal{T}_{\\mathcal{K}}:(\\lambda,\\mu,\\nu,\\Lambda,M,N)\\rightarrow(\\ell,{g},{h},L,{G},{H})$\nthat convert Eq. (\\ref{Andoyer}) in a new Hamiltonian $\\mathcal{K}$ that depends only on momenta. Because of the two-torus topology of the Euler-Poinsot problem only two momenta are required in $\\mathcal{K}$, and in view of neither $\\lambda$ nor $\\Lambda$ appear in Eq. (\\ref{Andoyer}), we choose $h=\\lambda$, $H=\\Lambda$ and $\\mathcal{K}\\equiv\\mathcal{K}(L,G)$.\n\\par\n\n\\subsection{Formal Solution of the Hamilton-Jacobi Equation}\n\nIn the Hamilton-Jacobi approach, the transformation $\\mathcal{T}_{\\mathcal{K}}$ is derived from a generating function in mixed variables $\\mathcal{S}=\\mathcal{S}(\\mu,\\nu,L,G)$ such that\n\\begin{equation} \\label{transformation0}\n(\\ell,g,M,N)=\\frac{\\partial{S}}{\\partial(L,G,\\mu,\\nu)}\n\\end{equation}\nBecause $\\mu$ is cyclic in Eq. (\\ref{Andoyer}), $\\mathcal{S}$ is chosen in separate variables\n$\\mathcal{S}=G\\,\\mu+W(\\nu,L,G)$. Then, from Eq. (\\ref{Andoyer}) we form the Hamilton-Jacobi equation\n\\begin{equation}\\label{HJsolido}\n\\left(\\frac{\\sin^2\\nu}{2A}+\\frac{\\cos^2\\nu}{2B}\\right)\\left[G^2-\\left(\\frac{\\partial{W}}{\\partial\\nu}\\right)^{\\!\\!2}\\right]+\\frac{1}{2C}\\left(\\frac{\\partial{W}}{\\partial\\nu}\\right)^{\\!\\!2}=\\mathcal{K}\n\\end{equation}\nwhere $W$ may be solved by quadrature. Then, calling $\\beta=L\/G$, the transformation Eqs. (\\ref{transformation0}) are\n\\begin{eqnarray} \n\\label{delta}\n\\ell &=& \\frac{\\mathcal{I}_2}{G^2}\\,\\frac{\\partial\\mathcal{K}}{\\partial\\beta},\n\\\\ \\label{gamma}\ng &=& \\mu+\\mathcal{I}_1-\\frac{\\mathcal{I}_2}{G^2}\\left(2\\mathcal{K}\n+ \\beta\\,\\frac{\\partial\\mathcal{K}}{\\partial\\beta}\\right),\n\\\\ \\label{N}\nN &=& G\\,\\sqrt{Q}, \n\\\\ \\label{M}\nM &=& G,\n\\end{eqnarray}\nwhere\n\\begin{equation}\\label{II12}\n\\mathcal{I}_1=\\int_{\\nu_0}^\\nu\\sqrt{Q}\\,\\mathrm{d}\\nu, \\qquad\n\\mathcal{I}_2=\\int_{\\nu_0}^\\nu\\frac{1}{\\sqrt{Q}}\\,\\frac{\\partial{Q}}{\\partial(1\/\\Delta)}\\,\\mathrm{d}\\nu,\n\\end{equation}\n\\begin{equation}\nQ=\\frac{\\sin^2\\nu\/A+\\cos^2\\nu\/B-1\/\\Delta}{\\sin^2\\nu\/A+\\cos^2\\nu\/B-1\/C},\n\\end{equation}\nand\n\\begin{equation}\\label{K}\n1\/\\Delta=2\\mathcal{K}\/G^2.\n\\end{equation}\n\\par\n\nWe only discuss the general case $A0$ and the function $0\\le{m}\\le1$\n\\begin{eqnarray}\n\\label{efe}\nf=\\frac{C\\,(B-A)}{(C-B)\\,A},\\qquad\nm=\\frac{(C-\\Delta)\\,(B-A)}{(C-B)\\,(\\Delta-A)},\n\\end{eqnarray}\nand the auxiliary variable $\\psi$ defined as\n\\begin{equation}\\label{psi2nu}\n\\cos\\psi=\\frac{\\sqrt{1+f}\\sin\\nu}{\\sqrt{1+f\\sin^2\\nu}},\\qquad\n\\sin\\psi=\\frac{\\cos\\nu}{\\sqrt{1+f\\sin^2\\nu}},\n\\end{equation}\nthen, the quadratures in Eq. (\\ref{II12}) are solved to give\n\\begin{eqnarray}\n\\mathcal{I}_1 &=& \\gamma\\left[\\frac{m}{f+m}\\,F(\\psi|m)-\\Pi(-f,\\psi\\,|\\,m)\\right],\n\\\\[1ex]\n\\mathcal{I}_2 &=& \\gamma\\,\\frac{A\\,C}{C-A}\\,F(\\psi|m),\n\\end{eqnarray}\nwhere\n\\begin{equation}\\label{chi}\n\\gamma=\\sqrt{(1+f)\\,(f+m)\/f}=\\sqrt{\\frac{B\\,\\Delta\\,(C-A)\\,(C-A)}{A\\,C\\,(C-B)\\,(\\Delta-A)}},\n\\end{equation}\n$F(\\psi|m)$ is the elliptic integral of the first kind of modulus $m$ and amplitude $\\psi$, and $\\Pi(-f,\\psi\\,|\\,m)$ is the elliptic integral of the third kind of modulus $m$, amplitude $\\psi$, and characteristic $-f$. It is worth mentioning that for the definition of the elliptic integral of third kind we adhere to the convention in \\citep{ByrdFriedman1971}.\n\\par\n\n\n\\subsection{The Standard Hamiltonian}\nFrom Eqs. (\\ref{K}) and (\\ref{efe}) we note that $\\mathcal{K}$ is characterized by the identity\n\\begin{equation}\\label{SadovHam}\n\\mathcal{K}=\\frac{G^2}{2A}\\left(1-\\frac{C-A}{C}\\frac{f}{f+m}\\right),\n\\end{equation}\nwhich can be taken as a definition by assuming that $m=m(L,G)$ in Eq. (\\ref{SadovHam}). Then, Eqs. (\\ref{delta})--(\\ref{M}) are rewritten\n\\begin{eqnarray}\\label{lg}\n\\ell &=& \n\\frac{1}{2\\gamma}\\,\\frac{1+f}{f+m}\\,\\frac{\\partial{m}}{\\partial\\beta}\\,F(\\psi|m),\\\\ \\label{gg}\ng &=& \n\\mu+\\gamma\\left[\\frac{1}{f+m}\\left(m-\\frac{f}{f+m}\\,\\frac{\\beta}{2}\\,\\frac{\\partial{m}}{\\partial\\beta}\\right)F(\\psi|m)-\\Pi(-f,\\psi|m)\\right]\\!,\\quad\n\\\\ \nN &=& G\\,\\sqrt{\\frac{f}{f+m}}\\,\\sqrt{1-m\\sin^2\\psi}, \\\\ \\label{Mg}\nM &=& G.\n\\end{eqnarray}\n\\par\n\nTransformations in the literature can be obtained as particular cases of the general form Eqs. (\\ref{SadovHam})--(\\ref{Mg}). Thus, the new Hamiltonian selected by \\citet{HitzlBreakwell1971} is the average of the Andoyer Hamiltonian Eq. (\\ref{Andoyer}), which is also the intermediate Hamiltonian of \\citet{Kinoshita1972}, while a previous proposal of ours \\citep{FerrerLara2010} transforms the Andoyer Hamiltonian to the axisymmetric case.\n\\par\n\n\\section{New variables}\n\nSearching for simplification in Eqs. (\\ref{lg}) and (\\ref{gg}) we propose to choose\n\\begin{equation}\\label{simplify}\n\\frac{1}{2\\gamma}\\,\\frac{1+f}{f+m}\\,\\frac{\\partial{m}}{\\partial\\beta}=-1,\\qquad\n\\frac{1}{f+m}\\left(m-\\frac{f}{f+m}\\,\\frac{\\beta}{2}\\,\\frac{\\partial{m}}{\\partial\\beta}\\right)=1.\n\\end{equation}\nEquations (\\ref{simplify}) can be solved for $\\beta=\\beta(m)$ without need of solving any partial differential equation. Furthermore, by squaring $\\beta$ we can express $m$ as a function of $L\/G$\n\\begin{equation}\\label{mmia}\nm=f\\left[(1+f)\\,G^2\/L^2-1\\right]\\!.\n\\end{equation}\nCorrespondingly, the new Hamiltonian in new variables is\n\\begin{equation}\\label{newH}\n\\mathcal{K}=\\frac{G^2}{2A}-\\left(\\frac{1}{B}-\\frac{1}{C}\\right)\\frac{L^2}{2},\n\\end{equation}\nwhose Hessian never vanishes, and which is formally equal to the uniaxial case for a new maximum momentum of inertia, say $P$, such that $1\/P=1\/A+1\/C-1\/B$.\n\\par\n\nThen, the direct transformation is\n\\begin{eqnarray}\\label{ell}\n\\ell &=& -F(\\psi\\,|\\,m) \\\\\ng &=& \\mu+\\sqrt{(1+f)\\,(f+m)\/f}\\,\\left[F(\\psi\\,|\\,m)-\\Pi(-f,\\psi\\,|\\,m)\\right] \\\\\nL &=& N\\,\\sqrt{(1+f)\/(1-m\\sin^2\\psi)} \\\\ \\label{H}\nG &=& M\n\\end{eqnarray}\nwhere $\\psi$ is defined in Eq. (\\ref{psi2nu}) and $m$ is easily computed in Andoyer variables from its definition in Eq. (\\ref{efe}) by noting that $\\Delta=G^2\/(2\\mathcal{K})=M^2\/(2\\mathcal{H})$, where $\\mathcal{H}$ is given in Eq. (\\ref{Andoyer}).\n\\par\n\nThe inverse transformation requires using the Jacobi amplitude $\\mathrm{am}$ to invert Eq. (\\ref{ell})\n\\begin{equation}\\label{psi}\n\\psi=-\\mathrm{am}\\left(\\ell\\,|\\,m\\right),\n\\end{equation}\nwhere $m$ is computed from Eq. (\\ref{mmia}).\nThen, from Eq. (\\ref{psi2nu}) we get\n\\begin{equation}\\label{Deprit2Serretnu}\n\\cos\\nu=-\\frac{\\sqrt{1+f}\\,\\mathrm{sn}(\\ell,m)}{\\sqrt{1+f\\,\\mathrm{sn}^2(\\ell,m)}},\\qquad\n\\sin\\nu=\\frac{\\mathrm{cn}(\\ell,m)}{\\sqrt{1+f\\,\\mathrm{sn}^2(\\ell,m)}}.\n\\end{equation}\nwhere $\\mathrm{sn}$, $\\mathrm{cn}$, $\\mathrm{dn}$, stand for the usual Jacobi elliptic functions.\nFinally, the inverse transformation of Eqs. (\\ref{ell})--(\\ref{H}) is completed with \n\\begin{eqnarray} \\label{mu}\n\\mu &=& g+(1+f)\\,(G\/L)\\left[\\ell+\\Pi(-f,\\mathrm{am}\\left(\\ell\\,|\\,m\\right)\\,|\\,m)\\right], \\\\ \\label{ene}\nN &=& L\\,\\mathrm{dn}(\\ell,m)\/\\sqrt{1+f}, \\\\ \\label{EME}\nM &=& G.\n\\end{eqnarray}\n\n\n\\section{Transformation to action-angle variables}\n\nNote in Eq. (\\ref{psi}) that the variable $\\ell$ is $4\\,K(m)$-periodic, with $K(m)$ the complete elliptic integral of the first kind. With a view on perturbations of the Euler-Poinsot problem, where elliptic functions would be expanded in Fourier series, it could be desired that $\\ell$ be $2\\pi$-periodic (an angle).\n\\par\n\nThe new variable\n\\begin{equation}\\label{ell2}\n\\ell'=-\\frac{\\pi}{2K(m)}\\,F(\\psi|m),\n\\end{equation}\nwill be obtained by requiring to Eq. (\\ref{lg}) that\n\\begin{equation}\\label{aacondition}\n\\frac{1}{2\\gamma}\\,\\frac{1+f}{f+m}\\,\\frac{\\partial{m}}{\\partial\\beta'}=-\\frac{\\pi}{2K(m)},\n\\end{equation}\nwhere $\\beta'=L'\/G'$. Equation (\\ref{aacondition}) is in separate variables and is integrated by quadrature to give, up to an integration constant,\n\\begin{equation}\\label{beta3}\n\\beta'=\\frac{2}{\\pi}\\,\\sqrt{(1+f)\\,(f+m)\/f}\\left[\\Pi(-f,m)-\\frac{m}{f+m}\\,K(m)\\right]\n\\end{equation}\nwhere $\\Pi(-f,m)$ is the complete elliptic integral of the third kind.\nEquation (\\ref{beta3}) defines $m$ as implicit function of $L'\/G'$.\n\\par\n\nNow, we replace $\\beta'$ in Eq. (\\ref{gg}) by its value from Eq. (\\ref{beta3}) to get\n\\begin{equation}\\label{ge2}\ng'=\\mu+\\sqrt{(1+f)\\,(f+m)\/f}\\left[\\frac{\\Pi(-f,m)}{K(m)}\\,F(\\psi|m)-\\Pi(-f,\\psi|m)\\right]\\!.\n\\end{equation}\n\nRemarkably, Eqs. (\\ref{Mg}), (\\ref{ell2}), (\\ref{beta3}) and (\\ref{ge2}) recover the original transformation to action-angle variables \\citep{Sadov1970,SadovRuso1970} without need of relying on their classical definition.\\footnote{%\nIn \\citep{Sadov1970,SadovRuso1970}, $f\\equiv\\kappa^2$ and $m\\equiv\\lambda^2$, Andoyer variables are $(h,\\psi,\\phi,L,G,G_\\zeta)\\equiv(\\lambda,\\mu,\\nu,\\Lambda,M,N) $, and the action-angles are $(f,\\nu,h,I,G,L)\\equiv(\\ell',g',h',L',G',H')$. Besides, Sadov's auxiliary angle is $\\xi=-\\psi$. Note that there is a typo in the definition of $\\lambda^2$ in Eq. (4) of \\citep{Sadov1970}, which should be multiplied by $A\/C$. This typo is easily traced in Eq. (2.20) of \\citep{SadovRuso1970}, but it still remains in \\citep{SadovRuso1984,Kozlov2000}.\n}\n\nFinally, we note that the inverse transformation from action-angles to Andoyer variables requires the computation of $m$ from the implicit function Eq. (\\ref{beta3}).\n\n\n\\section{Conclusions}\n\nAction-angle variables are not necessarily the better option for dealing with perturbed motion. This fact is very well known for perturbed Kep\\-ler\\-i\\-an motion where Delaunay variables are used customarily. The same happens to rotational motion where action-angle variables have the inconvenience of being related to Andoyer variables through implicit relations. But the complete reduction of the Euler-Poinsot problem may be achieved in a variety of canonical variables. Indeed, we demonstrate that when solving the Hamilton-Jacobi equation of the Euler-Poinsot problem in Andoyer variables, the new Hamiltonian can be cast into a standard form as a function of the modulus of the elliptic integrals required in the solution, a quantity that is consubstantial to the problem. Then, the solution of the Hamilton-Hacobi equation can be written formally as a function of the modulus and its partial derivatives with respect to the new momenta. Solving these partial derivatives according to certain criteria provides the desired transformation. In our case, we require ``simplification'' and find a new transformation of variables that, while having similar characteristics than action-angles variables, does not rely on implicit functions. Besides, we show that the transformation to action-angle variables pertains also to this general family.\n\n\n\\section{Acknowledgemnts}\n\nWe thank support from the Spanish Ministry of Science and Innovation, pro\\-jects\nAYA 2009-11896 (M.L.) and MTM 2009-10767 (S.F.), and from Fundaci\\'on S\\'eneca of the autonomous region of Murcia (grant 12006\/PI\/09). We are indebt with Prof.~Sadov, Russian Academy of Sciences, for sending us copies of his preprints in Russian.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nLanguage is a unique symbol of human civilization. Writing system is the development of the language.\nWriting system usually has many different fonts. Unlike the English font library \nthat has only 26 alphabets, \nChinese font library contains tens of thousands of characters. There are also thousands of commonly \nused characters. Making a new style font library is a difficult and time-consuming job due to the \nhuge amount of Chinese characters. Designing a new font of Chinese character \nrequire a lot of human design and adjustment to draw each character.\nIt is necessary to find a automation method to help the font designing \nfor Chinese character.\n\nIn the last few years, the development of deep learning make it possiable in automatic \nimage style transfer. Several researcher have intended to generate Chinese fonts by using\ndifferent deep learning method. Jiang et.al. using a U-net model realize end-to-end\nChinese character mapping, that automatically generate the whole GB2312 font library\n that consists of 6763 Chinese characters from a small number of characters written by the user.\n\\cite{Jiang2017}. Jiang develop an efficient and generalized deep framework W-Net, \nthat is capable of learning and generating any arbitrary characters \nsharing the style similar to the given single font character\\cite{amari_w-net_2017}.\nSun et. al. also propose a variational auto-encoder framework \nto generate Chinese characters\\cite{Sun2017}.\n\nThese methods are based on such an assumption that the latent features\nof a Chinese character can be disentangling \ninto content-related and style-related components. \nCombining different parts content-related and style-related components \ncan construct new style fonts. \nIn this paper, we propose and end-to-end deep Chinese font generation system.\nThis system can generate new style fonts by interpolation of \nlatent style-related embeding variables that could achieve \nsmooth transition between different style. \nOur method is simpler and more effective than other methods,\nwhich will help to improve the font design efficiency\\cite{Sun2017,Lyu2018,Deng,guo_creating_2018,\nchang_rewrite2_nodate,chang_generating_nodate,azadi_multi-content_nodate}.\n\n\n\n\\section{Method Description}\nDue to the complicated structures of Chinese characters, It is not easy transfer \ndeep learning methods widely used in image synthesis to this job.\nThe big problem is that the style and content features of Chinese characters are \ncomplexly entangled. The present deep learning methods, such as cross-domain \ndisentanglement\\cite{Gonzalez-Garcia2018}, Dientangled Representation\\cite{Lee2018},\nU-Net model\\cite{esser_variational_nodate}, are not well work on Chinese characters synthesis.\n\nIn this paper, we using a encoder-decoder model to map Chines character with one type to \nanother type, such as Song fonts to Kai fonts. The mapping just transfer the character style,\nthe characters have the same content. During the mapping training, the encoder-decoder model \nwill extract the fonts style features in the latent space. When we find the right model parameters,\nwe embeding a font style one-hot vector in the latent space. Then we retrain the model with \nstyle embding. This process can be seen as an artificial entanglement process,that entangled \nthe style and content features of the characters.\n\nOur method contains three steps. First, we using a U-Net encoder-decoder model the extract the \nfont feature vectors for about 40 different type fonts. Then, we concatenate an one-hot vector \nto the feature vectors, and retrain the model. The one-hot vector embding about 40 different fonts\nstyle. Finally, we can change the one-hot style vector to arbitrary embedding vectors, and synthesis \nnew style fonts which have mixed styles of different fonts\u3002\n\n\\subsection{U-net encoder-decoder model}\n\nThe U-net encoder-decoder architecture is shown in Figure \\ref{fig1}. The input\nof the network is an Song style Font with 256x256 pixels. The Font throught and\n8-layes convolutional encoder. The output of the encoder is 5x5x512 features \nvectors. This features concatenate a one-hot style \nembedding are input to the decoder. After similar 8-layers deconvolution, the \ndecoder output a new style fonts. \n\n\n\\begin{figure}\n \\centering\n \\includegraphics[scale=0.5]{Unet}\n \\caption{U-Net encoder-decoder architecture}\n \\label{fig1}\n\\end{figure}\n\n\\section{Experiments results}\nFigure \\ref{fig2}\\ref{fig3} is the two different output of the model.\nThe input is all the Hei style fonts. Figure \\ref{fig2} is the Song \nstyle fonts. Song style have a similar style feature of Hei. Figure \\ref{fig3}\nis XingKai style, that has very different characteristics style with the Hei style.\nThis two output examples show that the model has a good ability to generate \nnew fonts. \n\n\\begin{figure}\n \\centering\n \\includegraphics[scale=0.9]{hei}\n \\caption{Hei style fonts generation}\n \\label{fig2}\n\\end{figure}\n\n\\begin{figure}\n \\centering\n \\includegraphics[scale=0.9]{xing}\n \\caption{Xing style fonts generation}\n \\label{fig3}\n\\end{figure}\n\n\nThe style transfer have been show in Figure\\ref{fig4}. The first three \ncolumns are the input fonts. The middle three colums is the output of our \nmodel. The last three colums is the style fonts, which features is embedding \nin the one-hot vectors. This results indicate that our model can control \nthe output fonts through the one-hot embedding vectors.\n\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[scale=0.8]{heitoxing}\n \\caption{Hei style to Xing style transfer}\n \\label{fig4}\n\\end{figure}\n\nFigure \\ref{fig5} show the generate fonts \nof multiple different styles. All\n the generated fonts have no missing strokes, and the font \nfeature details are perfect. The input fonts is all Hei style\nfonts for those multiple generation fonts. These multiple\ngeneration fonts show that the control of \nthe style embedding vector is very good. \n\n\\begin{figure}\n \\centering\n \\includegraphics[scale=0.8]{mfont}\n \\caption{multiple different styles generation}\n \\label{fig5}\n\\end{figure}\n\nSince the above results show the one-hot embedding vector already \nhave stable control ability over the generation fonts, we can explore\nwhether the embedding vector can be used to achieve the style transition\nbetween two fonts.\n\nWhen the input embedding label is [1,0,...], the network will convert \nthe source input font text sample to the target font text sample \nwith label 0 during training, and if the the input embedding label is \n[0,1,0,..], the source font sample is converted into the \ntarget font sample with the label 1. So does the input embedding label\n[0.5,0.5,0,...] mean that the output style is the mixture of the two fonts?\nFigure \\ref{fig6} show the generation of the fonts when the embedding labels are\nassigned different valuse, such as [0.2,0.5,0.7,...]. The \nresults indicate that the font styles can be controlled through assigned \ndifferent valuse in the embedding vector.\n\n\\begin{figure}\n \\centering\n \\includegraphics[scale=0.8]{onehot}\n \\caption{Control effect of one-hot embedding vector}\n \\label{fig6}\n\\end{figure}\n\n\\subsection{Conclusion}\nOur paper proposed a novel and simple method to automatic generate new\nChinese fonts from existing font libraries. The results demonstrated that\nour method is capable of generating new high-quality fonts. \n\n\\subsection{Acknowledgement}\nThis work was supported by National Natural Science Foundation of China (61307080)\n\\bibliographystyle{unsrt} \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzneag b/data_all_eng_slimpj/shuffled/split2/finalzzneag new file mode 100644 index 0000000000000000000000000000000000000000..92a01e40be34a74f868f44c88fb60d9a859f1594 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzneag @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\n\\label{intro}\n\nTexts are hierarchic constructs which consist of several autonomous\nlevels \\cite{hutchins,valgina,hasan}: letters, words, phrases, clauses,\nsentences, paragraphs. If a text is looked at statistically, i.e.\nwithout understanding its meaning (e.g. because it is written in an\nunknown system), how can it be efficiently distinguished from a\nmeaningless collection of words \\cite{baa,orlov,arapov}? Several such\ndistinctions are well-known, e.g. {\\it (i)} meaningful texts have a\nlarge number of words that appear in a text only few times, in\nparticular once (rare words or {\\it hapax legomena}) \\cite{baa}. {\\it\n(ii)} Ranked frequencies of words obey the Zipf's law\n\\cite{estoup,condon,zipf}. {\\it (iii)} Letters and words of a text\ndemonstrate long-range (power-law) correlations\n\\cite{lrc_schenkel,lrc_shnerb,lrc_ebeling,lrc_eckmann,lrc_manin,lrc_altmann}. \n\nHowever, these characteristics can be reproduced by a sufficiently\nsimple stochastic models putting in doubt their direct relation to the\nmeaningfulness of a text. {\\it (i)} Simple stochastic models can\nrecover quite precisely the detailed structure of the hapax legomena\n\\cite{pre}. {\\it (ii)} Zipf's law can be deduced from statistical\napproaches\n\\cite{shrejder,li,simon,zane,kanter,hill,pre,liu,baek,vakarin,dover,latham,mandelbrot,mandel,manin}. The\nfirst model of this type was a random text, where words are generated\nthrough random combinations of letters, i.e. the most primitive\nstochastic process \\cite{mandelbrot,li}. Its drawbacks\n\\cite{howes,seb,cancho} (e.g. many words having the same frequency) are\navoided by more refined models \\cite{simon,zane,kanter,hill,pre}. \nMore generally, it was recently understood that Zipf's law is a statistical regularity\nthat emerges in samples which are informative about the underlying generative process \\cite{cubero}. \n{\\it (iii)} Physics and mathematics of stochastic processes offer a plethora\nof models and approaches for generating long-range correlations\n\\cite{buck} \\footnote{E.g. Ref.~\\cite{lrc_schenkel} points out that\nlong-range correlations are found also in a dictionary, where the\nmeaning of text (as opposed to the meaning of words and phrases) is\nabsent. Ref.~\\cite{lrc_manin} also argued against a direct relation between\nlong-range correlations and semantic structures.}. \n\nHere we contribute to resolving the above question by recalling that\nmeaningful texts evolve sequentially (linearly) from beginning to end.\nThis was taken as one of basic features of language \\cite{sure},\nwhich|together with other design features|allows to distinguish human\nlanguage from other communication systems \\cite{hockett}. Thus we divide\ntexts into two halves, each one containing the same amount of words.\nThereby we neutralize confound variables that are involved in a complex\ntext-producing process (style, genre, subject, the author's motives and\nvocabulary {\\it etc}), because they are the same in both halves. Hence\nby comparing the two halves with each other we hope to see statistical\nregularities that are normally shielded by above variables. Statistical\nregularities hold for the majority of texts, such that it is highly\nunlikely to get this majority for random reasons (as checked by the\n$3\\sigma$ rule). The significance of results will be checked by\nwell-accepted statistical tests; for our purpose this is the W-test\n(Wilcoxon's test) \\cite{wilcoxon}. \n\nIn two sets of several hundred texts we noted the following statistical\nregularities. {\\it (1)} The first half has a larger number of different\nwords, i.e. a larger vocabulary. {\\it (2)} It also has a larger number\nof rare words, i.e. words that appear once or twice. {\\it (3)} The\nfirst half is less compressible than the second half. The\ncompressibility was studied via several different standard approaches,\ne.g. the Lempel-Ziv complexity and the zip algorithm. Lesser\ncompressibility relates to more information in the sense of Shannon\n\\cite{cover}. {\\it (4)} Common words of both halves tend to have a\nlarger overall frequency in the second half. These four features\nsignificantly correlate with each other as quantified by Pearson's\ncorrelation coefficient. {\\it (5)} The words in the first half are\ndistribued less homogeneously, since they have a larger difference\nbetween the frequency and (inverse) spatial period. \n\nOne possible explanation of these result is that the first part of the\ntext normally contains the exposition (which sometimes can be up to 20\n\\% of the text), where the background information about events,\nsettings, and characters is introduced to readers. The first part also\nplots the main conflict (open issue), whose denouement (solution) comes\nin the second half \\footnote{\\label{foo1}Scientific texts contain\nclosely related aspects: introduction, critique of existing approaches,\nstatement of the problem, resolution of the problem, implications of the\nresolution {\\it etc}. The discussion on differences between the halves\napplies also here.}. Hence {\\it (1)-(4)} can be hypothetically explained\nby the fact that the exposition|hence the first half|needs more\ndifferent words {\\it (1)}, more rare words {\\it (2)}, is more\ninformative (in the sense of Shannon) {\\it (3)}, and introduces words that\nare employed in the second half {\\it (4, 5)} (i.e. the second half\nprocesses information introduced in the first half). We emphasize that\nthis explanation is hypothetical, its direct validity is yet to be\nchecked via more refined methods to be developed in future. \n\n{\\it (6)} Many other features|in particular those related to\nhigher-order hierarchic structures of the text\n\\cite{hutchins,valgina,hasan}|do not show any significant difference\nbetween the first and second half: number of sentences, paragraphs,\nrepetitiveness of words (as quantified by Yule's constant), number of\npunctuation signs {\\it etc}. Among such features we especially mention\nthe overall number of letters, since there is a weak statistical evidence\n(the W-test is not always passed) that this quantity is still larger in\nthe first half. \n\nChecking these features does not require any understanding of the text,\ni.e. it is not required that the meaning of words is understood, or\neven their writing system is known. We show that (expectedly) neither\nof them survives if the words of the text are randomly permuted, and only\nafter that the resulting ``text\" is divided into two parts. Hence these\nfeatures are specific for meaningful texts and they can be employed for\ndistinguishing meaningful texts from a random collection of words. \n\n\\comment{Moreover, they do no demand any text parsing, beyond\nconservation of its natural (linear) order. Not even the parsing into\nwords is necessary, since this can be done by detecting the space as the\nmost frequent symbol in the text. }\n\nThis paper is organized as follows. The next section reviews our data\ncollection method and recalls the W-test. Sections \\ref{I}--\\ref{spato}\npresent results from the above points {\\it (1)-(5)}.\nSection \\ref{nego} reviews negative results from point {\\it (6)}. \nThe last section contains the outlook of this research\nand relates it with existing literatures. Our main results\nare briefly summarized in Table~\\ref{tab0}. All other tables are given in\nAppendix \\ref{tablo}. \n\n\n\\begin{table}\n\\centering\n\\begin{tabular}{l|c|c} \\hline\\hline\n& First half & Second half \\\\\n\\hline\nNumber of different words; cf.~(\\ref{udosh}) & + & -- \\\\\n\\hline\nNumber of rare words (absolute and relative); cf.~(\\ref{cov}, \\ref{boris}) & + & -- \\\\\n\\hline\nCompressibility of the size; cf.~(\\ref{compo}, \\ref{sss}) & -- & + \\\\\n\\hline\nThe overall frequency of common words; cf.~(\\ref{cc}) & -- & + \\\\\n\\hline\nDifference between frequency and inverse spatial period; cf.~(\\ref{ort}) & + & -- \\\\\n\\hline\\hline\nNumber of letters; cf.~(\\ref{guppi}) & + & -- \\\\\n\\hline\\hline\nRepetitiveness of words (Yule's constant); see (\\ref{yule}) & $\\emptyset $ & $\\emptyset $ \\\\\n\\hline\nNumber of punctuation signs & $\\emptyset $ & $\\emptyset $ \\\\\n\\hline\nAverage length of words & $\\emptyset $ & $\\emptyset $ \\\\\n\\hline\nNumber of sentences & $\\emptyset $ & $\\emptyset $ \\\\\n\\hline\nAverage length of sentence & $\\emptyset $ & $\\emptyset $ \\\\\n\\hline\nNumber of paragraphs & $\\emptyset $ & $\\emptyset $ \\\\\n\\hline\nSize in bytes & $\\emptyset $ & $\\emptyset $ \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\caption{\\label{tab0}Qualitative comparison of\nvarious features of texts between first and second halves: + (--) means\nthat the feature is larger (smaller) in the corresponding half.\n$\\emptyset $ means that the sought difference does not show up. \nFeatures are divided into two\ngroups by double-lines. The first four features correlate with each other.\nThe number of letters is separated, \nsince there is a weak evidence toward its validity (the \nvalues of test quantities are close to their threshold values). }\n\\end{table}\n\n\n\n\n\\section{Data collection and testing}\n\n\\subsection{Studied texts}\n\nWe selected English texts with a single narrative that are written on relatively few tightly\nconnected subjects, and are sufficiently short for not containing ``texts\ninside of texts'' \\footnote{To ilustrate this point, consider ``War and Peace'' by\nLeo Tolstoy. This big novel is written in two different languages\n(Russian and French), and contains a big amount of heroes and\ncircumstances. It does have several parallel narratives, and describes\nsituations in the course of twenty years. Clearly, this is a text of\ntexts, and it would not be meaningful to focus on dividing it over two\nhalves. But we can divide over two halves one of its (long) chapters. We\nhave not done this so far. }.\nWe divided studied texts into two halves (each half \ncontains equal number of words) \\footnote{We got a\npreliminary evidence that, as expected, dividing texts into more than two\nparts will obscure the text difference effects shown in Table I. Note that\nRef.~\\cite{zano} studies text division into several parts, but that was\ndone for a different purposes. For long texts containing\n$155000-220000$ words, Ref.~\\cite{zano} noted that such texts can be divided into several\nsub-texts of the size of $1000-3000$ words. The criterion of separation\nis qualitatively close to the above concept of ``books inside\nof a book\", because it looked at the spatial clustering of key-words.\nI.e. a group of key-words appearing mostly in one part of the text and\nnot in others, will effectively define a sub-text.} along the flow of the narrative, i.e. from\nthe beginning to end. Several aspects of texts are left unchanged: each half\nis sufficiently large for statistics to apply, they\nhave the same overall number of words, the same author, genre {\\it etc}.\nThe halves are semantically different, since the first half can be understood\nwithout the second half, but the second half generally cannot be\nunderstood alone. Also, the structure of narrative is different: the\nfirst half normally contains the exposition, where actors, situations\nand conflicts are set and defined, while the second half normally\ncontains the denouement; cf.~Footnote~\\ref{foo1}. \n\nWe have chosen to work with two datasets \\cite{online}. The first dataset was taken\nfrom the Gutenberg project at \\cite{gutenberg}. It\nconsists of 156 fiction novels; for each novel the overall number of\nwords is in the range $ [10000, 50000]$, which is sufficiently large for\nstatistics to apply, but sufficiently short to ensure that they do not\namount to ``books inside of a book''. This range of the overall number\nof words $ [10000, 50000]$ is motivated from our own experience as\nreaders. \n\nThe texts within the first dataset are thematically close, since they\nare all fiction novels. The second dataset consists of 350 thematically\ndiverse texts taken from various online sources and collected at\n\\cite{ting}. When collecting those texts we tried to ensure that they\ndo not contain texts that are meaningless to divide into halves, i.e.\nthat they do not contain effectively independent narratives. Hence we\ndid not include in this dataset biographies, poems, collections of short\nstories or essays (in particular, folk stories), lectures, proceedings,\nletters. \n\n\\subsection{Testing the difference between the halves}\n\nAfter processing, the typical form of our data are pairs of numbers for\neach text: $\\{x_{1,i}, x_{2,i}\\}_{i=1}^M$, where $x_{1,i}$ and $x_{2,i}$\nare certain features of (resp.) the first and second half of a text $i$,\nwith $M$ being the overall number of texts in the dataset. E.g. $x_{1,i}$ and $x_{2,i}$\nare the number of different words for (resp.) first and second halves of a text; see below.\n\nTo inquire on\nwhether this data indicate on a difference between two halves, we\nformulate two natural hypotheses: ${\\cal H}_1$ (${\\cal H}_0$) means that\nthe difference $x_{1,i}-x_{2,i}$ does (does not) follow a symmetric\ndistrbution around the zero. Now some understanding on excluding ${\\cal H}_0$ can be gained\nby looking at the percentage of cases, where $x_{1,i}30$ suffices, which always holds in our cases) the law of large numbers works and\n$W$ is a Gaussian random variable, since it is a weigted sum of a large number of uncorrelated random variables. \nIts average is zero, since ${\\rm sgn}[x_{2,i}-x_{i,i}]$ assume values $\\pm 1$ with equal probability (once \n${\\cal H}_0$ is assumed to hold). Its dispersion is calculated directly from (\\ref{gnu2}) \\cite{wilcoxon}:\n\\begin{eqnarray}\n\\label{sigma}\n\\sigma_W^2(M)=\\langle W^2\\rangle=\\sum_{k=1}^M k^2=\\frac{M(M+1)(2M+1)}{6}.\n\\end{eqnarray}\nHence ${\\cal H}_0$ can be excluded via the $3\\sigma$ rule, if \n\\begin{eqnarray}\n\\label{3sigma}\n|W|>3\\sigma_W (M).\n\\end{eqnarray}\nWe accept the $3\\sigma$ rule (\\ref{3sigma}) as the minimal threshold for claiming the statistical\nsignificance of our results. However, we emphasize that the absolute majority of\nour results hold the much stronger $5\\sigma$ rule; see tables in Appendix \\ref{tablo}. \n\n\\section{Words: different, rare, common}\n\\label{I}\n\n\\subsection{Different words (vocabulary)}\n\nThe basic hierarchic level of text is that of words. Neglecting phenomena\nof synonymy and homonymy (which are rare in English, but not at all rare\ne.g. in Chinese \\cite{epjb}), we can say that every word has several\nclosely related meanings (polysemy). Neglecting also the difference\nbetween polysemic meanings, the number of independent meanings in a text\ncan be estimated via the number of different words $n$. Tables\n\\ref{n_156} and \\ref{n_350} show that the first half of a meaningful\ntext has statistically more different words than the second half:\n\\begin{eqnarray}\n\\label{udosh}\nn_1>n_2\n\\end{eqnarray}\nAs expected, this result disappears after random shuffling (random\npermutation of words) of texts that destroys its linear structure; see\nTables \\ref{shuffle_156} and \\ref{shuffle_350}. \n\n\\subsection{Rare words (hapax legomena)}\n\nIn any meaningful text, a sizable number of words appear only very\nfew times ({\\it hapax legomena}). These rare words amount to a finite\nfraction of $n$ (i.e. the number of different words). The existence and\nthe (large) number of rare events is not peculiar for texts, since there\nare statistical distributions that can generate samples with a large\nnumber of rare events \\cite{baa,pre}. One reason\nwhy many rare words should appear in a meaningful text is that a typical\nsentence contains functional words (which come from a small pool), but\nit also has to contain some rare words, which then necessarily have to\ncome from a large pool \\cite{latham} \\footnote{\\label{lato}E.g. this sentence\ncontains rare words {\\it typical} and {\\it pool} that in the present text\nare met only 3 and 2 times, respectively. It also contains frequent\nwords {\\it words}, {\\it since}, {\\it large}.}. \n\n\\comment{\\begin{eqnarray}\n\\label{kusho}\n\\sum_{m=1}^k V_m^{[1]} \\geq \\sum_{m=1}^k V_m^{[2]}, \\qquad k=1,...,5,\n\\end{eqnarray}\nwhere $V_m^{[1]}$ ($V_m^{[2]}$) is the number of words that appear $m$ times in the first (second) half.\nFor the halves (\\ref{kush}) is written as \n\\begin{eqnarray}\n\\label{kusho1}\n&&{\\sum}_{m=1}^{f^{[\\ell]}_1 N\/2}\\, V^{[\\ell]}_m=n_\\ell, \\\\\n&&{\\sum}_{m=1}^{f^{[\\ell ]}_1N\/2} \\, mV^{[\\ell]}_m=N\/2,\n\\qquad \\ell=1,2,\n\\label{kusho2}\n\\end{eqnarray}\nwhere $n_1$ ($n_2$) is the number of different words in the first (second) half, and $f^{[1]}_1$ ($f^{[2]}_1$)\nis the frequency of the most frequent word in the first (second) half. \n}\n\nLet $V_m^{[1]}$ ($V_m^{[2]}$) is the number of words that appear $m$ times in the first (second) half.\nFor defining rare words we focused on \n\\begin{eqnarray}\n\\label{hh}\nh_\\ell(\\kappa) \\equiv \\sum_{m=1}^\\kappa V_m^{[\\ell]}\\qquad \\ell=1,2,\\qquad \\kappa=1,...,5,\n\\end{eqnarray}\ni.e. on words that appear up to $\\kappa$ times. We choose to work with different $\\kappa$'s to ensure that\nour results are robust with respect to varying the definition of ``rare''.\nFor both datasets we observed that in the majority of cases the number\nof rare words in the first half is larger than the number of rare words \nin the second half [see Tables~\\ref{rare_156} and \\ref{rare_350}]:\n\\begin{eqnarray}\n\\label{cov}\nh_1(\\kappa)>h_2(\\kappa),\\qquad \\kappa=1,...,5.\n\\end{eqnarray}\nWe confirmed via the $5\\sigma$ of the W-test that the probability to get (\\ref{cov}) due to random reasons is\nnegligible. \n\nEq.~(\\ref{cov}) suggests that the first half uses more rare words, but\nsuch a conclusion is incomplete, since the two halves have different\nnumbers of distinct words. Denote them as $n_1$ and $n_2$, for the first\nand second half respectively; cf.~(\\ref{udosh}). \nNote that\n\\begin{eqnarray}\n\\label{kusho1}\n{\\sum}_{m=1}^{f^{[\\ell]}_1 N\/2}\\, V^{[\\ell]}_m=n_\\ell, \n\\qquad \\ell=1,2,\n\\label{kusho2}\n\\end{eqnarray}\nwhere $f^{[1]}_1$ ($f^{[2]}_1$)\nis the frequency of the most frequent word in the first (second) half. \nHence in addition to (\\ref{cov}) it is necessary to consider normalized\nquantities, i.e.\n\\begin{eqnarray}\n\\label{boris} \nh_1(\\kappa)\/n_1>h_2(\\kappa)\/n_2, \\qquad \\kappa=1,...,5.\n\\end{eqnarray}\nRelation (\\ref{boris}) does hold statistically; see\nTable~\\ref{rare_156} for the first dataset and Table~\\ref{rare_350} for the\nsecond dataset. \n\nHence the first half has more rare words both in absolute and relative\nterms; see (\\ref{udosh}, \\ref{boris}). These differences between the\nhalves disappear after random shuffling of texts; see Tables\n\\ref{shuffle_156} and \\ref{shuffle_350}. \n\nNote that yet another possibility to define rare words comes from relations\n\\begin{eqnarray}\n\\label{kus}\n{\\sum}_{m=1}^{f^{[\\ell ]}_1N\/2} \\, mV^{[\\ell]}_m=N\/2,\n\\qquad \\ell=1,2.\n\\end{eqnarray}\nEq.~(\\ref{kus}) invites to compare the normalized quantities\n$\\frac{2}{N}\\sum_{m=1}^\\kappa m V_m^{[1]}$ with $\\frac{2}{N}\\sum_{m=1}^\\kappa m V_m^{[2]}$ for $\\kappa=1,...,5$.\nWe carried out this comparison and the results (for percentages and $W$-values) are very similar to (\\ref{cov}), i.e.\nwe obtain\n\\begin{eqnarray}\n\\label{kookoo1}\n\\sum_{m=1}^\\kappa m V_m^{[1]} > \\sum_{m=1}^\\kappa m V_m^{[2]}, \\qquad \\kappa=1,...,5,\n\\end{eqnarray}\nin the same statistical sense as (\\ref{cov}).\n\n\n\\subsection{Common words}\n\nBoth halves of a text have certain common words, e.g. non-common words of the \nsecond half are those that are not met in the first half. Let the number of \ncommon words in a given text is denoted by $C$. Our first result is that for each half\nthe common words are less numerous than non-common ones:\n\\begin{eqnarray}\n\\label{urartu}\nC1\/2,\\qquad c_2>1\/2.\n\\end{eqnarray}\nRelations (\\ref{van}) hold without exclusions for all text we studied.\nInequalities (\\ref{urartu}, \\ref{van}) are well expected, since common\nwords include functional words, which are frequent, but not numerous\n\\cite{pre}. \n\nOur next finding does indicate on a difference between\ntwo halves, and is therefore less expected: the overall frequency of common\nwords is larger in the second half [see Tables \\ref{n_156} and\n\\ref{n_350}]:\n\\begin{eqnarray}\n\\label{cc}\nc_1s_{\\rm inverted}.\n\\end{eqnarray}\nThe percentage of (\\ref{gel}) is remarkably high: it holds for $>97\\%$\ncases in both of our datasets; see Table \\ref{inversion}. \nEq.~(\\ref{gel}) is confirmed via the zip \ncompression method; see Table \\ref{inversion}. \n\nRecall that previous applications of the LZ-complexity in texts\n\\cite{lande,debowski1,debowski2} assumed that the LZ-complexity captures\ncorrelations between different text symbols (letters, words {\\it etc}).\nRelation (\\ref{gel}) can be explained by noting that $C_{\\rm LZ}$ focuses on\nshort range correlations of letters, which may be lost after inversion\nof words. To illustrate this point, let $\\ell_1\\ell_2\\ell_3 $ and\n$\\kappa_1\\kappa_2\\kappa_3 $ are two consecutive 3-letter words. Their\norder in ${\\rm T}$ [in ${\\rm T}_{\\rm inverted}$] is\n$\\ell_1\\ell_2\\ell_3\\,\\,\\kappa_1\\kappa_2\\kappa_3 $\n[$\\kappa_1\\kappa_2\\kappa_3\\,\\,\\ell_1\\ell_2\\ell_3$]. Now if there are\ncorrelations between $\\ell_3$ and $\\kappa_1$ in ${\\rm T}$, and such\ncorrelations are accounted for in $C_{\\rm LZ}$, then in ${\\rm T}_{\\rm\ninverted}$ such correlations will have a longer range, and will not be\nseen in $C_{\\rm LZ}$ \\footnote{\\label{syntagma}Work is in progress to\nunderstand whether such directional correlations are related to\nsyntagmatic correlations between words of the text well-known \nin linguistics \\cite{sure,sahlgren}. Qualitatively, these are correlations\nalong the text determined by co-occurence of words or letters\n\\cite{sure,sahlgren}. They are conceptually different from paradigmatic\nrelations between the words, where two words having such a relation tend to\nappear in the same context, i.e. in the same surrounding of words.}. \n\nNote that instead of inverting texts at the level of words we also\ninverted them at the level of letters: put the last letter as the first\none {\\it etc}. We saw that out of this letter inversion the\ncompressibility does change, i.e. there is more into the LZ-complexity\nthan just short-range correlations of letters. However, no clear\nindications emerged on the analogue of (\\ref{gel}) or on its inverse.\nThe results differ from one dataset to another and from one compression\nmethod to another. \n\n \n\\comment{\nRecall how the lossless compression methods (including zip)\nwork \\cite{cover}: they look for various repeating patterns (within some\nlocal window of consecutive symbols) and code more frequent patterns by\nshorter codewords. Hence sequences that contain more repeating patters\nare more compressible. }\n\n\\subsection{Compressibility of two halves}\n\nTable \\ref{zip} shows that the relative compressibility of the\nfirst half is statistically smaller, i.e. it is compressed less than the second half:\n\\begin{eqnarray}\n\\label{sss}\ns_1\\mu_2,\n\\end{eqnarray}\nand that this effect gets stronger|both in terms of the percentage of\ncases and the value of $W$ in (\\ref{gnu2}), when the the set $\\Omega$ in\n(\\ref{ort}) is restricted to common words of both halves; see Tables\n\\ref{mu_156} and \\ref{mu_350}. \n\n\\section{Features that do not show statistically significant\ndifferences between two halves}\n\\label{nego}\n\nSo far we mostly concentrated on one level of textual hierarchy, i.e.\nwords. Letters are on the hierarchy level below that of words. For the\ntotal number of letters $L$ in each half the statistical evidence we got\nis weaker, since the percentage of cases, where $L_1>L_2$ (i.e. the\nfirst half has a larger overall number of letters than the second half)\nand the W-statistics for $L_1>L_2$ are close to their critical values;\nsee Tables \\ref{n_156} and \\ref{n_350}. Moreover, in one of our datasets\nthe W-test is passed, while in the other it is not. However, there is a\nweak, but a definite evidence for the validity of $L_1>L_2$. First, after\na random shuffling of texts, the percentage of cases where $L_1>L_2$ holds\ndrops down from its value $\\simeq 0.58$ (for original texts) to $\\simeq 0.5$\nfor shuffled texts; see Tables \\ref{shuffle_156} and \\ref{shuffle_350}. Second, \n$L_1>L_2$ shows significant correlations with (\\ref{udosh}) and (\\ref{cov});\nsee Tables \\ref{corr_156} and \\ref{corr_350}. Third, the relation $L_1>L_2$\nholds in average for both datasets: \n\\begin{eqnarray}\n\\label{guppi}\n\\frac{1}{156}\\sum_{k=1}^{156} (L_1^{[k]}-L_2^{[k]})= 146.26,\\qquad\n\\frac{1}{350}\\sum_{k=1}^{350} (L_1^{[k]}-L_2^{[k]})= 340.97.\n\\end{eqnarray}\n\n\\comment{Further distinction between words of the text can be made via\ntheir average length (in letters): content words|which express specific\nmeaning|are normally longer than functional words that mostly serve for\nestablishing grammatic connections \\cite{zipf}. }\n\nFor hierarchy levels higher than that of words, our results are\nnegative, i.e. they do not indicate on a statistically significant\ndifference between the halves. The number of sentences $\\sigma$ does not\nshow significant differences; see Tables \\ref{n_156} and \\ref{n_350}.\nHere we (conventionally) defined the sentence as the shortest sequence\nof words located in between of any of the following symbols: comma, dot,\nsemicolon, question mark, exclamation mark. We also studied the number\nof sentences, when the comma is excluded from the above list (not shown\nin tables). This did not change our conclusion. \n\nWe also calculated the full distribution of sentences over the\nlength (measured in words): $\\kappa_\\alpha$ the fraction of sentences with\nword-length $\\alpha$ ($\\sum_\\alpha\\kappa_\\alpha=1$). Two specific\ncharacteristics of this distribution were looked at: the average\n$\\overline{\\alpha}$, dispersion $\\overline{\\Delta(\\alpha^2)}$ and\nentropy $\\varepsilon$:\n\\begin{gather}\n\\label{deviation}\n\\overline{\\alpha}={\\sum}_\\alpha\\kappa_\\alpha\\alpha, \\qquad\n\\overline{\\Delta(\\alpha^2)}\n={\\sum}_\\alpha\\kappa_\\alpha\n(\\alpha-\\overline{\\alpha})^2.\n\\end{gather}\nNone of these quantities shown a statistically significant difference\nbetween the halves; see Tables \\ref{n_156} and \\ref{n_350}. Another\nlevel of the textual hierarchy is the one containing paragraphs.\nDenoting the number of paragraphs as $\\rho$, we saw that there is no\nstatistical evidence in favor of $\\rho_1>\\rho_2$ or $\\rho_1<\\rho_2$; see\nTables \\ref{n_156} and \\ref{n_350}. Our results on Yule's constant that\ndescribes the repetitiveness of words (see Appendix \\ref{yu} details of\nthe definition) also do not indicate on a significant difference between\nthe halves. \n\n\\section{Outlook}\n\nWe proposed a set of relations between statistical features of the two\nhalves of a meaningful text; see Table~\\ref{tab0} for a summary of our\nresults. The validity of these relations is statistical, i.e. the\nmajority of them holds with $5\\sigma$ significance of the Wilcoxon test;\nsee Appendix \\ref{tablo}. No understanding of the text (or even knowing\nits writing system) is needed for checking these relations. We\nexplicitly confirmed that all these relations disappear after a random\npermutation of words in the text. \n\nWe conjecture that these relations between the halves are connected to a\nspecific, information-carrying structure of the text, where the\ninformation is introduced (defined) in the first half, and then is processed\nin the second half. Such a structure is anticipated in text linguistics,\nwhere the flow of the text narrative is conventionally separated between\nthe exposition and the denouement, which are typically located in the\nfirst and second halves, respectively \\cite{hutchins,valgina,hasan}.\nThis is however a qualitative concept, and hence the connection between\nour results and the exposition-denouement is stated as a\nhypothesis. Work is currently in progress for designing specific tests\nfor checking the hypothesis. \n\nPractically, knowing whether a string of symbols is a meaningful text or\nnot can be useful in cryptography, fraud detection and historical\nanalysis. The latter can refer to inferring whether a given text in\nunknown writing system is meaningful or asemic \\cite{writing_systems}.\nOne interesting application relates to the CETI problem (communication\nwith extra-terrestial intelligence) \\cite{ceti}. Here the code of a\npotential signal is completely unknown, but it can be plausibly\nconjectured that a meaningful signal has similar differences between the\nhalves. (Fractioning into ``words'' is is possible, once the ``space\nsymbol'' is identified as the most frequent symbol in the text\n\\cite{ceti}.)\n\nSome of our results were sporadically observed in literatures.\nRef.~\\cite{minn} emphasized the translation invariance of books, but\nstill noted on a concrete text that its last part has less rare words.\nRef.~\\cite{lrc_manin} noted the following (non-topical) differences\nbetween the first and the second halves of {\\it Moby Dick} (by H.\nMelville): {\\it (1.)} The word {\\it is} is more frequent than {\\it was}\nin the first half, but less frequent in the second half. {\\it (2.)} The\nratio of articles {\\it the} to {\\it a} is larger in the second half,\nwhich may mean that the second half makes more concrete statements.\n\n\\comment{for PRE version:\nAfter the initial version of this paper, one of its unknown referees\ninformed us that he\/she counted the number of rare words in the\nfirst\/second halves of {\\it all} of Project Gutenberg books that have\nbetween 20000 and 40000 words in total (within the lengths of the texts\nchosen by us). For a total of 10364 books, 5649 (54 \\%) had more rare\nwords in the first half compared to the second half. We note that this\nresult has a statistical validity higher than $5\\sigma$, since $0.5 +\n5\/(2 \\sqrt{10364})\\approx 0.5246$. For our situation the corresponding\npercentages of rare words differences are larger by some 10-15 \\%, see\nTables \\ref{rare_156} and \\ref{rare_350}, because we excluded from\nconsideration all those books that are {\\it a priori} meaningless to\ndivide into two halves (hence such texts contribute into the noise):\nbiographies, poems, collections of short stories or essays (in\nparticular, folk stories), lectures, proceedings, letters. }\n\nOur results concerning the compressibility features|in particular, the\nresult in section \\ref{invo} on the compressibility decrease under\ninverting the word order|are especially worth studying in more detail.\nWhile we focused on the Lempel-Ziv complexity and the related zip method\nfor defining the compressibility, it is known that the Lempel-Ziv complexity\nfor long but finite sequences has a drawback of not capturing the real\nrandomness, i.e. not agreeing with the Kolmogorov complexity\n\\cite{cuba}. Hence more refined compression methods are to be studied in\nfuture, e.g. the Huffman coding that is algorithmically slower, but does\ncapture the notion of randomness. It is also important to clarify whether\nthe compressibility difference between the original and inverted text \ncan serve for quantifying syntagmatic correlations between the words; \ncf.~Footnote \\ref{syntagma}.\n\nAnother important open problem relates to modeling the above effects. A\nsuperficial modeling would be possible via altering the existing\nsequential text-generating models (see \\cite{simon,zane,kanter} for\nexample) such that e.g. they generate less rare words towards the end of\nthe text. But we should warn the reader against such quick attempts.\nFirst, the main drawback of sequential models is that they do not\ndescribe sufficiently well the distribution of rare words \\cite{minn},\nwhich was so far possible only via non-sequential statistical models\n\\cite{pre}. So a good model should predicts {\\it both} the distribution\nof rare words and their difference between the halves. Second, the\nexample of the Zipf law|with its numerous models and explanations\n\\cite{shrejder,li,simon,zane,kanter,hill,pre,liu,baek,vakarin,dover,latham,mandelbrot,mandel,manin}|shows\nthat excessive modeling even with a reasonable quantitative agreement is\nnot at all a guarantee for understanding the actual meaning of a complex\ntextual phenomenon. Hence at the present stage of our understanding we\nwant to concentrate on conceptual issues (and novel tests) relating\nstatistics to meaningfulness, more than to develop a purely statistical\nmodel for explaining above results. \n\n\\section*{Acknowledgements}\n\nW. Deng was partially supported by the Fundamental Research Funds for\nthe Central Universities, the Program of Introducing Talents of\nDiscipline to Universities under grant no. B08033, and National Natural\nScience Foundation of China (Grant Nos. 11505071 and 11905163). A.E. Allahverdyan was\nsupported by SCS of Armenia, grants No. 18RF-015 and No. 18T-1C090. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction\\label{sec:intro}}\n\nThe color-magnitude diagram is a powerful diagnostic of galaxy\nevolution and formation. The presence, as early as $z \\sim 1.5$, of a\nprominent and low-scatter, `red-sequence' (RS) in galaxy clusters\nplaces useful constraints on the possible evolutionary pathways in\ngalaxy color and luminosity \\citep*{Mullisetal05, Stanfordetal05,\n Stanfordetal06, Belletal04, Faberetal07}. The red colors of the\nprimarily early-type RS galaxies are due to the observed filters\nspanning the 4000\\AA\\ spectral break. The universality and prominence\nof the RS in appropriately chosen filters have been used to discover\nhigh-redshift clusters \\citep*[e.g.,][]{RCS}. Moreover, the defining\ncharacteristic of galaxy clusters, i.e., the large numbers of galaxies\nall at the same redshift, allows the slope and intrinsic scatter of\nthe RS to be measured with great precision. Based on studies of\ngalaxy clusters at $z < 1.3$, the slope of the RS does not appear to\nevolve and therefore is more likely the by-product of the\nmass-metallicity relation as observed in local galaxy samples\n\\citep*[e.g.,][]{Tremontietal04} rather than the result of a mass-age\ntrend. The scatter, however, is likely due to the fractional age\ndifferences between the RS galaxies \\citep*[e.g.,][]{Blakesleeetal03}.\nBy constructing a set of model galaxies with different star-formation\nhistories and timescales it is possible to fit for the mean epoch of\nlast significant star-formation by matching the intrinsic scatter of\nthe RS. Such studies at $z \\sim 1$ have derived formation redshifts\nof $z_{\\rm form} \\sim 2.0 - 2.5$ \\citep*[e.g.,][]{HalfHubble06,\n vanDokkumvanderMarel07a}. At redshifts beyond $z \\sim 1.5$,\nhowever, the 4000\\AA-break moves into the near-infrared and galaxy\nclusters, and therefore the RS, have not been observed closer to the\ninferred epoch of formation for early-type galaxies. Hence, to\nuncover the younger or forming red-sequence at higher redshifts\nrequires deep near-infrared imaging of suspected (or, preferably,\nconfirmed) protocluster fields.\n\n\\begin{figure*}[t]\n\\plotone{f1.ps}\n\\caption{$J_{110} - H_{160}$ vs. $H_{160}$ color-magnitude diagram for\n the MRC 1138-262 NICMOS field (open black circles). The large\n yellow star is the radio galaxy itself. The blue background points\n are from the NICMOS data of the Ultra Deep Field and Hubble Deep\n Field North. The deep fields cover 2.5x the area of our\n observations. Also shown are the spectroscopically confirmed\n $H\\alpha$ (orange filled circles) and Lyman-$\\alpha$ (green filled\n circles) emitters. The three dot-dash lines show where the CMRs of\n lower redshift clusters would lie under different assumptions. The\n top line is the Coma cluster with no evolution, simply redshifted to\n $z=2.16$ and observed through the NICMOS filters. The next line\n down is the same but for the $z=1.24$ cluster RDCS1252. Finally, if\n we passively de-evolve RDCS1252 to redshift $z=2.16$ (almost exactly\n two Gyr), assuming a median age for the 1252 galaxies of about 3 Gyr\n (or $z_{\\rm form} \\sim 3$), we get the third line.\\label{fig:CMD}}\n\\end{figure*}\n\nWe have undertaken a NICMOS imaging program to study the red galaxy\npopulation in a protocluster at $z=2.16$. Broad and narrow-band\nimaging, both in the optical and near-infrared, of the field\nsurrounding the powerful radio galaxy MRC 1138-262 ($z=2.16$) have\nidentified more than 100 candidate companion galaxies. This target\nserved as the proof-of-concept for the successful VLT Large Program\nsummarized in Venemans et al. (2007)\\nocite{Venemansetal07}. There\nare surface-overdensities of both line-emitting candidates\n(Lyman-$\\alpha$ and H$\\alpha$), X-ray point sources, sub-mm selected\ngalaxies and red optical--near-infrared galaxies\n\\citep*{Pentericcietal02,KurkPhD,Kurketal04,Kurketal04b,Croftetal05,Stevensetal03}.\nFifteen of the Ly$\\alpha$ and 9 of the H$\\alpha$ emitters have now\nbeen spectroscopically confirmed to lie at the same redshift as the\nradio galaxy. The $I-K$-selected Extremely Red Objects (EROs; $I-K >\n4.3$ Vega) seem concentrated around the RG but have no spectroscopic\nredshifts at this time. However, by obtaining deep images through the\nNICMOS $J_{110}$ and $H_{160}$ filters, which effectively span the\n4000\\AA-break at $z=2.16$, accurate and precise colors and basic\nmorphological parameters can be measured for the red galaxy\npopulation. In this paper we present the first results from this\nproject. The article is organized as follows: in Section\n\\S\\ref{sec:obs} we describe the data and their reductions, in Section\n\\S\\ref{sec:overdensity} we present the comparison between the red\ngalaxy counts in this field and in deep field data, in Section\n\\S\\ref{sec:RS} we present the full color-magnitude diagram and our\nfits to the ``red sequence.'' We use a $(\\Omega_{\\Lambda},\\Omega_{M})\n= (0.73,0.27)$, $H_{0} = 71$ ${\\rm km}$ ${\\rm s^{-1}}$ ${\\rm\n Mpc^{-1}}$ cosmology throughout. At $z=2.16$ one arcsecond is\nequivalent to 8.4 kpc. All magnitudes are referenced to the AB system\n\\citep*{AB} unless otherwise noted.\n\n\\section{Observations, Data Reductions and Photometry\\label{sec:obs}}\n\n\\begin{figure}\n\\plotone{f2.ps}\n\\caption{$g_{435} - I_{814}$ vs. $J_{110}-H_{160}$ color-color\n diagram using the ACS and NICMOS data. Arrows represent limits\n where the galaxy is only detected in a single band for that color.\n Filled circles indicate spectroscopically-confirmed Lyman-$\\alpha$\n (green) and H-$\\alpha$ (orange) emitting protocluster members. The\n yellow star is the radio galaxy. The blue, green and red grids\n indicate the regions occupied by galaxies with an\n exponentially-decaying star-formation rate $\\tau = 0.15$ Gyr (red),\n $\\tau = 0.4$ Gyr (green) and a $\\tau = 1000$ Gyr (blue) at $z=2.16$\n for three ages (0.1, 1 and 3 Gyr) and two different extinctions\n ($E(B-V) = 0.0, 1.0$).\\label{fig:colorcolor}}\n\\end{figure}\n\nThe NICMOS instrument on-board {\\it HST} is capable of deep\nnear-infrared imaging more quickly than from the ground but with a\nrelatively small field-of-view ($51\\arcsec \\times 51\\arcsec$). In the\ncase of MRC~1138-262, we know that galaxies are overdense on the scale\nof a few arcminutes \\citep*{Kurketal04, Croftetal05} and are thus\nwell-suited for observations with NICMOS camera 3 on {\\it HST}. We\nused 30 orbits of {\\it HST} time to image 10 of the 24 confirmed\nmembers and $\\sim 70$ of the candidate (narrow-band excess sources and\nEROs) protocluster members in both the $J_{110}$ and $H_{160}$\nfilters. We used seven pointings of NICMOS camera 3 in both filters\nand one additional pointing in $H_{160}$ alone. This single\n`outrigger' $H_{160}$ pointing was included to obtain rest-frame\noptical morphological information for a small concentration of\ncandidate members. These observations reach an AB limiting magnitude\n($m_{10\\sigma}$; 10$\\sigma$, $0\\farcs5$ diameter circular aperture) of\n$m_{10\\sigma}=24.9$~mag in $J_{110}$ and $m_{10\\sigma}=25.1$~mag in\n$H_{160}$. The same field was imaged in the $g_{475}$\n($m_{10\\sigma}=27.5$~mag) and $I_{814}$ ($m_{10\\sigma}=26.8$~mag)\nfilters using the Wide-Field Channel of the Advanced Camera for\nSurveys on {\\it HST} as part of a Guaranteed Time program (\\# 10327;\nMiley et al. 2006\\nocite{Mileyetal06}). These optical data are useful\nfor their higher angular resolution and their coverage of the\nrest-frame far-UV, thus extending the observed SEDs of candidate\nprotocluster members to shorter wavelengths where young stars and\non-going star-formation dominate the emitted spectrum. In particular,\nthe $g_{475}$ and $I_{814}$ data allow us to partially differentiate\nobscured star-formation from evolved stellar populations in the\ncandidate RS galaxies.\n\nThe NICMOS images were reduced using the on-the-fly reductions from\nthe {\\it HST} archive, the IRAF task PEDSKY and the dither\/drizzle\npackage to combine the images in a mosaic. The dither offsets were\ncalculated using image cross-correlation and were refined with one\nfurther iteration of cross-correlation. Alignment of the pointings\nrelative to each other was accomplished using a rebinned version of\nthe ACS $I_{814}$ image as a reference. The final mosaic has a pixel\nscale of $0\\farcs1$. Galaxies were selected using the $H_{160}$-band\nimage for detection within SExtractor \\citep*{SExtractor}. We used a\n$2.2\\sigma$ detection threshold with a minimum connected area of 10\npixels. We also corrected the NICMOS data for the count-rate\ndependent non-linearity \\citep*{CPSNONLINEAR}. Total galaxy\nmagnitudes were estimated by using the MAG\\_AUTO values from\nSExtractor.\n\nThe $J_{110} - H_{160}$ colors were determined by running SExtractor\n\\citep*{SExtractor} in two-image mode using the $H_{160}$ image for\nobject detection and isophotal apertures. The $J_{110}$ image was\nPSF-matched to the $H_{160}$ band. The resulting colors and\nmagnitudes are shown in Figure~\\ref{fig:CMD}. For galaxies which are\nnot detected at $2\\sigma$ significance in the $J_{110}$-band (those to\nthe right of the thick dashed line, representing $J_{110, tot} >\n26.7$, in Fig.~\\ref{fig:CMD}) we consider the color to be a lower\nlimit.\n\nWe also measured similarly PSF-matched, isophotal colors using the two\nACS bands and have used them to construct a $g_{475}-I_{814}$ versus\n$J_{110}-H_{160}$ color-color diagram (Figure~\\ref{fig:colorcolor}).\nWe compared these colors to those of model SEDs for different ages,\nstar-formation histories and dust extinctions. Using the 2007 Charlot\n\\& Bruzual\\nocite{BC} population synthesis models we have constructed\nspectral energy distributions for galaxies with an\nexponentially-decaying star-formation rate with time constants of\n$\\tau = 0.15, 0.4, 1000.0$ Gyr (the red, green and blue grids in\nFig.~\\ref{fig:colorcolor} respectively). Each model's colors are\ncalculated for ages of 0.1, 1 and 3 Gyr and for $E(B-V) = 0.0$ and\n$1.0$. Aging of the population moves primarily the $J_{110}-H_{160}$\ncolor to the red while the dust extinction significantly reddens the\n$g_{475}-I_{814}$ color. From this analysis it appears that the\n$\\tau=0.4$ Gyr model represents well the colors of a majority of the\nred $J_{110}-H_{160}$ galaxies.\n\nTo extend the wavelength coverage for the protocluster galaxies we\nalso incorporated ground-based $U_{n}$-band data from LRIS-B on the\nKeck telescope, $K_{s}$-band imaging from VLT\/ISAAC and three band\nIRAC imaging (the 3.6, 4.5 and 5.8 $\\mu$m bands) from the {\\it Spitzer\n Space Telescope}. The Keck $U$-band data (PI W. van Breugel) were\nobtained in late January and early February of 2003. The ISAAC data\n(PI G. Miley) were taken in Period 73 in service mode. The {\\it\n Spitzer} data are from the IRAC Guaranteed Time program (PI\nG. Fazio, Program \\#17). We have smoothed the imaging data for all\nbands, apart from the IRAC data, to match the resolution of the\n$U_{n}$-band image (approximated by a FWHM~$\\sim 1\\arcsec$ Gaussian).\nWe then used SExtractor to measure galaxy magnitudes within a\n$0\\farcs5$ radius circular aperture for each smoothed image. To\nincorporate the IRAC data, which has much poorer angular resolution,\nwe derived aperture magnitudes which were then corrected to match the\nsmoothed data. These aperture corrections were derived using the\nphotometric curves-of-growth for 20 stars in the field. The resulting\ncatalog was used to generate photometric redshift estimates as\ndescribed below in Section \\S~\\ref{sec:photz}.\n\n\\begin{figure}\n\\plotone{f3.ps}\n\\caption{upper panel: Distribution of high-confidence ($>95\\%$)\n photometric redshifts and their selection function assuming a\n uniform $N(z)$ for our model template galaxies (yellow curve) for\n $1.1 \\leq (J_{110}-H_{160}) \\leq 2.1$ galaxies in the MRC 1138-262\n NICMOS field. The peak between $z=2.1$ and $z=2.4$ is statistically\n highly significant. middle panel: Sum of redshift probability\n distributions for all the galaxies in the upper panel. 38.5\\% of\n the total probability is contained in the redshift interval from 2\n to 2.3. lower 2 panels: two examples of the probability distribution\n function for individual galaxies\\label{fig:photzRS}}\n\\end{figure}\n\n\\section{Photometric Redshifts\\label{sec:photz}}\n\nWe have used the ACS ($g_{475}$, $I_{814}$), NICMOS ($J_{110}$,\n$H_{160}$), ground-based $U_{n}$-band from Keck\/LRIS-B, $K_{S}$-band\nimaging from VLT\/ISAAC and {\\it Spitzer}\/IRAC imaging to estimate\nphotometric redshifts for our $H_{160}$-band selected sample. We\ninput a catalog of aperture galaxy magnitudes, based on the matched,\nsmoothed images described above, into the Bayesian photometric\nredshift code (BPZ) of Ben{\\'{\\i}}tez (2000)\\nocite{BPZ} using a\nuniform prior. We felt that the default prior, based on optical\ngalaxy selection and spectroscopy in the HDF-N, would not necessarily\nrepresent the redshift distribution for our near-infrared selected\ngalaxies. We generated our own extensive set of template spectral\nenergy distributions using the models of Charlot \\& Bruzual\n(2007)\\nocite{CB07}. All these SEDs are $\\tau$ models with values for\n$\\tau = [0.15, 0.4, 1.0, 2.0, 1000.0]$ Gyr and ages $=[0.05, 0.1, 0.5,\n1.0, 2.0, 3.0]$ Gyr. We also included models with internal dust\nextinction ranging from $E(B-V) = [0.0, 0.1, 0.3, 0.5, 0.75, 1.0]$ mag\nand metallicity of $(Z\/Z_{\\odot}) = [0.3, 1.0, 2.5]$.\nWe focused particular attention on the $J_{110}-H_{160}$ selected\nsurface-overdensity. In the upper panel of Figure~\\ref{fig:photzRS}\nwe present the high confidence ($> 95\\%$) photo-$z$ distribution for\nthe NIR-color selected ($1.1 \\leq (J_{110} - H_{160}) \\leq 2.1$)\nsubsample. We ran extensive simulations by redshifting our template\nset, adding appropriate photometric errors and using BPZ to recover\nthe redshifts. The yellow curve represents the redshift selection\nfunction for this color cut, template set and filters assuming that\nthese model galaxies follow a uniform $N(z)$ over this redshift\ninterval. The simulation results were free of significant systematic\nerrors and the random errors are estimated to be $\\delta z\/z \\sim\n0.1$. Based on these SED fits, the approximate luminosity-weighted\nages of the red galaxies lie between 1 and 2.5 Gyrs and their stellar\nmasses are of order a few $\\times 10^{10} M_{\\odot}$. These stellar\nmasses are reasonable as are the absolute magnitudes (see\nFigure~\\ref{fig:CMRfits}). More detailed SED modeling is deferred to\na future paper.\n\nThere is a clear excess of galaxies between $z=2.0$ and $z=2.5$. For\neach galaxy fit by BPZ we have generated the full redshift probability\ndistribution. In the lower panel of Fig.~\\ref{fig:photzRS} we show\nthe $H_{160}$-band weighted-average of these probability\ndistributions. There is a clear peak (containing 38.5\\% of the total\nprobability compared to only 17\\% of the total selection function in\nthe same redshift interval) between $z=2.0$ and $z=2.3$, consistent\nwith the significant peak in the redshift histogram itself.\n\n\\section{NICMOS Galaxy Morphologies\\label{sec:morf}}\n\n\\begin{figure}[b]\n\\plotone{f4.ps}\n\\caption{Distribution of inferred stellar ages (in terms of $\\tau$)\n for both the concentrated ($n\\geq2.5$, red line) and diffuse\n ($n<2.5$, blue line) galaxies which are well-resolved in the NICMOS\n data. Constant star-formation models, for which the $e$-folding\n time is infinite, are placed at the left-hand edge of the plot. The\n blue and red distributions are quite different. Of particular note\n is that the most evolved galaxies generally have high $n$ while the\n low $n$ galaxies dominate the star-forming\n population.\\label{fig:tAn}}\n\\end{figure}\n\nNICMOS camera 3 provides good angular resolution over its entire\nfield-of-view. The FWHM of the PSF in our final mosaic is $\\approx\n0\\farcs27$. To exploit this resolution we have used the GALFIT code\n\\citep*{GALFIT} to fit analytic S{\\'e}rsic surface-brightness profiles\n\\citep*{Sersic} to all the $H_{160} \\leq 24.5$ sources in our\n$H_{160}$-band mosaic. A model point-spread function was created for\neach of these galaxies individually by generating a TinyTim simulated\nPSF \\citep*{TINYTIM} at the galaxies' positions in each exposure and\nthen drizzling these PSFs together in exactly the same fashion as for\nthe data themselves (see Zirm et al. 2007\\nocite{Zirmetal07}). We\nrestricted the S{\\'e}rsic index, $n$, to be between 1 and 5. We will\npresent a full analysis of the morphologies of these galaxies in a\nfuture paper. For the current work, we use these derived sizes and\nprofile shapes to assist us in selecting the morphological\n``early-type'' members of the red galaxy population. \n\n\\begin{figure*}[t]\n\\plotone{f5.ps}\n\\caption{Histogram of the color distributions for the 1138 and deep\n fields (blue). The deep field data has been normalized by total\n area to the 1138 data. Note the clear excess of red galaxies in the\n 1138 field. At $1.1 \\leq (J_{110} - H_{160}) \\leq 2.1$ (horizontal\n dotted lines) for galaxies brighter than the 2$\\sigma$\n $J_{110}$-band limit (dashed line) there is an overdensity of a\n factor of 6.2 in the 1138 field. \\label{fig:hist}}\n\\end{figure*}\n\nIn Figure~\\ref{fig:tAn} we show the distribution of galaxy ages\nderived via these SED fits as parametrized by the $\\tau$ value for\nthe best-fitting model for those galaxies with high and low S{\\'e}rsic\nindex ($n \\geq 2.5$, red line, and $n < 2.5$, blue line). It is clear\nthat while there is substantial overlap between these distributions\nthey are not identical and that they differ in the sense that one\nmight expect, namely, that the concentrated galaxies appear to be\ncomprised of older stellar populations. This trend gives us some\nconfidence in trying to select the ``early-type'' galaxies using these\ndata which is important for our discussion of the color-magnitude\nrelation in Section~\\ref{sec:RS}.\n\t\n\\section{Surface Overdensity of Red Galaxies\\label{sec:overdensity}}\n\n\\begin{figure*}[t]\n\\plotone{f6.ps}\n\\caption{Linear fits to the rest-frame $U-B$ (Vega) color-magnitude\n diagrams for three different sub-samples of the $H_{160}$-band\n selected NICMOS sample. Panel (A) shows the fit (solid line) and\n intrinsic scatter ($1\\sigma$; dotted lines) for a sample selected to\n have $1.1 \\leq (J_{110} - H_{160}) \\leq 2.1$. The crossed-out point\n are those which are rejected as outliers in more than half of the\n realizations (see \\S\\ref{sec:RS}). Both the observed and intrinsic\n scatter are smaller than the initial color cut. Panel (B) shows the\n fit and intrinsic scatter for a photometric redshift selected sample\n with $2.0 < z_{\\rm phot} < 2.5$. The stars indicate galaxies whose\n preferred photometric template has an age $< 4\\tau$, while circles\n represent galaxies with older than this limit. Panel (C) shows the\n fit and scatter for those galaxies which meet the same redshift cut\n but also are well-resolved with a high S{\\'e}rsic index ($n > 2.5$)\n and best-fit by an age $\\geq 4\\tau$ template.\\label{fig:CMRfits}}\n\\end{figure*}\n\nTo compare this protocluster field to more generic `blank' field data\nwe have compiled catalogs for the public NICMOS data in both the\nHubble Deep Field North (HDF-N) and the Ultra Deep Field (UDF).\nFigure~\\ref{fig:CMD} shows the $J_{110} - H_{160}$ color-magnitude\ndiagram (open black circles) and the color distributions for both the\nMRC~1138-262 and the combined HDF-N and UDF galaxy catalogs (blue\ncircles). The deep field data were also $H_{160}$-band selected. The\narea of the two deep fields is roughly 2.5 times the area of our\nprotocluster observations. We have applied no correction to the deep\nfield number counts to account for clustering in those fields. The\ncolor histogram in Figure~\\ref{fig:hist} shows the area-normalized\ngalaxy counts from the two deep fields (blue line) and from the 1138\nfield to the same ($2\\sigma$) limiting magnitude of $J_{110} =\n26.7$~mag (AB). The red dashed line shows the difference between the\ntwo color distributions. It is clear that the radio galaxy field is\noverdense in red galaxies by a large factor. For sources with colors\nbetween $1.1 \\leq (J_{110} - H_{160}) \\leq 2.1$, the\nhorizontal(vertical) dotted lines in\nFig.\\ref{fig:CMD}(\\ref{fig:hist}), and brighter than our\n$J_{110}$-band $2\\sigma$ limit ($26.7$), we calculate an\narea-normalized overdensity of $6.2$ when compared to the deep fields\ndata. We note that the exact value of the measured overdensity is\nrather sensitive to systematic color offsets between the protocluster\nand deep field data. A redward shift of 0.05 for the deep field\ngalaxies would lower the measured overdensity to $5.0$. However, we\nare confident that these systematic offsets remain small ($< 0.05$\nmag) since we have used the same instrument, filters, selection\ntechnique and photometric code with very similar input parameters for\nboth the deep field and 1138 datasets. Looking back at\nFigure~\\ref{fig:colorcolor} we can see that many of the\nspectroscopically-confirmed line emitters (filled blue circles) and\nred galaxies in the overdensity are well-represented by the $\\tau =\n0.4$ Gyr model (green lines) at different ages and extinctions.\n\n\n\nThis current work is not the first to observe red galaxies in this\nfield. Kurk et al. (2004)\\nocite{Kurketal04} identified a small\n($\\sim 1.5\\times$) surface overdensity of extremely red objects (EROs;\n$I-K > 4.3$ Vega magnitudes) using ground-based $I$ and $K$ band data.\nMany of these EROs are also identified as red in the NICMOS\n$J_{110}-H_{160}$ color. More recently Kodama et\nal. (2007)\\nocite{Kodamaetal07} observed this field using the\nwide-field NIR imager, MOIRCS, on the Subaru telescope. These authors\nfound several bright (presumably massive) red galaxies over a wider\nfield-of-view but to shallower depths than the NICMOS data presented\nhere. 24 of their color-selected protocluster candidates are within\nour NICMOS mosaic. 23 of the 24 are identified in our data as being\nred in $J_{110} - H_{160}$. Furthermore, 18 of the 94 galaxies which\nsatisfy our color criteria (and have $J_{110}<26.67$) are also\nidentified by Kodama et al. as protocluster candidates. The much\nlarger number of red galaxies in the NICMOS data is primarily due to\nfainter galaxies detected at high significance in our deeper data.\n\n\\section{An Emergent Red-Sequence?\\label{sec:RS}}\n\nTo study the colors and magnitudes of these galaxies in more detail\nand to possibly identify a red-sequence in the 1138 field we have\nsplit the galaxies into three sub-samples defined by $J_{110}-H_{160}$\ncolor, photometric redshift and morphology (S{\\'e}rsic index). The\nfirst sample (Sample A) comprises all 56 galaxies with $1.1 \\leq\n(J_{110}-H_{160}) \\leq 2.1$ and $H_{160} < 24.5$ and includes the\nradio galaxy itself. Sample B is made up of all 28 galaxies with a\nrobust photometric redshift between 2.0 and 2.5 and $J_{110}-H_{160} >\n0.75$ and $H_{160} < 26.0$. This liberal color cut is included to\nselect galaxies which comprise the large observed surface-overdensity.\nFinally, sample C contains seven galaxies with the same photometric\nredshift cut but which also have well-resolved $H_{160}$-band\nsurface-brightness profiles with S{\\'e}rsic index $n > 2.5$. All of\nthese galaxies' SEDs are also best-fit by models with relatively\nlittle on-going star-formation. We use a limit of (age $\\geq 4 \\times\n\\tau$, cf. Grazian et al 2007 \\nocite{Grazianetal07}). Therefore,\nsample C mimics the color, morphological and photometric redshift\nselection of early-type galaxies in clusters at $z \\mathrel{\\spose{\\lower 3pt\\hbox{$\\mathchar\"218$} 1$. The\nphotometry, photo-$z$s and sizes of the sample C galaxies are listed\nin Table~\\ref{tab:RS}, their rest-frame color-magnitude distributions\nare shown in Figure~\\ref{fig:CMRfits} and the two-dimensional spatial\ndistribution of the Sample A galaxies is plotted in\nFigure~\\ref{fig:spatial}. We note that because the measured\noverdensity is a factor of 6, we statistically expect one of every\nseven sample A galaxies to be a field galaxy. However, this should\nnot effect our results significantly.\n\nFor these three sample selections we have fit a line and measured the\nintrinsic scatter about that best-fit line (see\nFig.~\\ref{fig:CMRfits}). For comparison to lower redshift galaxy\nclusters we have transformed our observed $J_{110}-H_{160}$ color and\n$H_{160}$ magnitudes into rest-frame $U-B$ and $B$ (Vega),\nrespectively, using the following expressions:\n\\begin{equation}\n(U-B)_{\\rm rest} = 0.539 \\times (J_{110}-H_{160})_{\\rm obs} - 0.653\n\\end{equation}\n\n\\begin{equation}\n M_{B, {\\rm rest}} = H_{160,{\\rm obs}} - 0.170 \\times (J_{110}-H_{160})_{\\rm obs} - 43.625\n\\end{equation}\n\nThe small color corrections used in these relations were derived using\na family of $\\tau$-models with a range of ages (0.1-12 Gyr), $\\tau$\n(0.1-5 Gyr) and three metallicities ($0.4, 1$ and $2.5 Z_{\\odot}$).\n\n\\begin{figure*}[t]\n\\epsscale{0.8}\n\\plotone{f7.ps}\n\\caption{Spatial distribution of galaxies (red circles) relative to\n the radio galaxy MRC~1138-262 (yellow star) for the color-selected\n sample defined in the text (A) and shown in the first panel of\n Fig.~\\ref{fig:CMRfits}. The irregular black outline encloses the\n coverage of the 7 NICMOS camera 3 pointing with both $J_{110}$ and\n $H_{160}$ imaging.\\label{fig:spatial}}\n\\end{figure*}\n\nTo fit the ``CMR'' we used a bootstrap re-sampling technique to\nestimate the error on the fitted slope. Then, by assuming that all\nthe red galaxies lie on this fit line, we ran Monte Carlo realizations\nof the contribution of the photometric errors to the observed color\nscatter about the fit line, i.e., by fixing a color-magnitude relation\nwe calculate the measurement scatter with zero intrinsic scatter. We\nthen calculate the intrinsic scatter by subtracting (in quadrature)\nthe estimated measurement scatter from the observed scatter. We show\nthese fits (solid line) and the intrinsic scatters (dotted lines) for\nthe three samples (A, B and C) in Figure~\\ref{fig:CMRfits}. The fits\nto both Sample's A and B have nearly identical rest-frame $U-B$\nslopes, $0.027$ and $0.026$ respectively, and intrinsic scatters\n($0.10$ and $0.12$). While these slopes are comparable to those found\nfor the well-populated lower redshift cluster CMRs, the intrinsic\nscatters are considerably higher. However, the scatter measured for\nthe eight galaxy Sample C is comparable to that of the lower redshift\nsamples but with a much steeper slope ($0.130$). When these scatters\nare compared to model predictions based on lower redshift clusters\n(specifically RDCS 1252.9-2927 at $z=1.24$; Gobat et al. 2008) we find\nthat the 1138 protocluster has lower than predicted scatter. This may\nsuggest that the 1138 protocluster is in a more advanced evolutionary\nstate than RDCS 1252 was at $z=2.2$.\n\nWe have calculated three representative color-magnitude relations for\ncomparison to the colors and magnitudes of the red galaxies (three\ndot-dash lines in Fig.~\\ref{fig:CMD}). We have taken two lower\nredshift clusters, Coma at $z=0.023$ and RDCS~1252.9-2927 at $z=1.24$,\nand transformed them to the observed filters and $z=2.16$ under the\nassumption that the colors do not evolve. In this no evolution case\n(the two dot-dash lines in Fig.~\\ref{fig:CMD}) the CMRs appear at the\nred edge of the observed overdensity. There is almost exactly 2 Gyr\nof cosmic time between $z=2.16$ and $z=1.24$ in our adopted cosmology.\nFrom \\citet{Blakesleeetal03} we know that the median redshift of last\nsignificant star-formation for the RDCS~1252 galaxies is between\n$z=2.7-3.6$. Therefore, if we observe those galaxies at $z=2.16$ they\nwill be significantly younger and hence bluer. In fact, this\npassively de-evolved line (bluest dot-dash line in Fig.~\\ref{fig:CMD},\nlabeled `$z_{\\rm form} \\sim 3$') does fall blueward of the red galaxy\noverdensity. We discuss the implications for these comparisons in\nSection~\\ref{sec:conclusions}.\n\nWe have also translated the Kodama et al. ground-based $J-K$ colors to\nour NICMOS filters assuming all the red galaxies lie at $z=2.16$.\nThese bright galaxies also fall along the passively de-evolved line\nwith the radio galaxy. We have used our suite of SED models to\nestimate the color transformation from their ground-based $J-K_{S}$ to\nour NICMOS $J_{110}-H_{160}$ color. Roughly, the Kodama et al. bright\nred galaxies fall where the RDCS~1252 passive line crosses our color\ncut at $J_{110}-H_{160} = 1.1$. This result hints at a possible\nbi-modality in the red galaxy population of this protocluster.\nNamely, that there are faint red galaxies that are inconsistent with\npassively-evolving cluster members either due to large amounts of\ndust, or due to higher redshifts of formation but that the more\nluminous protocluster members may have already finished forming and\nseem consistent with passive evolution to the present-day.\n\n\n\\section{Discussion\\label{sec:conclusions}}\n\nWe have identified a (6.2$\\times$) surface-overdensity and a\ncorresponding photometric redshift `spike' of red $J_{110}-H_{160}$\ngalaxies which are likely associated with a known protocluster at\n$z=2.16$. The optical-NIR spectral energy distributions of these\nsources suggest that they comprise both evolved galaxies as well as\ndust-obscured star-forming galaxies. Based on our SED fits from the\nphotometric redshift determinations, the approximate\nluminosity-weighted ages of these sources lie between 1 and 2.5 Gyrs\nand their stellar masses are of order a few $\\times 10^{10}\nM_{\\odot}$. Detailed modeling of the SEDs for the protocluster\npopulation, along with their morphologies, is reserved for a future\npaper.\n\nComparison with the CMRs of lower redshift clusters shows that the red\ngalaxy overdensity primarily lies blueward of the no-evolution\npredictions. That the red galaxies in 1138 are also redder than the\n$z_{\\rm form} \\sim 3$ case suggests both that there are galaxies with\nsignificant dust content, an assertion supported by the SED fits, and\nalso that they were perhaps formed at higher redshift than the\nRDCS1252 galaxies. Of course, without a classical, low-scatter\nred-sequence to use as a baseline there remains considerable\nuncertainty in the age of the population as a whole. The results of\nSteidel et al. (2005)\\nocite{Steideletal05} suggest that protocluster\ngalaxies are older than their ``field'' counterparts at $z \\sim 2.3$\nand that these ages and stellar masses were broadly consistent with\nevolution to lower redshift cluster galaxies. However, their\nprotocluster members were all UV-selected and star-forming. With\nfuture spectroscopy of our red galaxy sample it will be possible to\nsee if these differences persist when looking at a more varied galaxy\nsample.\n\nFor three samples of galaxies drawn from the full $H_{160}$-band\nselected dataset we have fit a color-magnitude relation and estimated\nthe intrinsic scatter. The CMR at $z=2.16$ is not as well-defined as\nat $z \\sim 1$ or 0. For sample C, made up of 8 galaxies, the color,\nbest-fit spectral template, morphology and photo-$z$ all point towards\nthem being (proto-)elliptical galaxies within the protocluster. For\nthis small sample, the estimated intrinsic scatter is rather low and\nmay suggest that these galaxies represent the forming red-sequence in\nthis protocluster. The slope of this relation is extremely steep\ncompared to lower redshift clusters. The slope of the CMR is\ngenerally assumed to be a manifestation of the mass-metallicity\nrelation and would therefore flatten at higher redshift. The major\ncaveat regarding the steep slope of Sample C is that none of these\ngalaxies are spectroscopically confirmed protocluster members.\nTherefore, this ``relation'' may just be a random, although somewhat\nunlikely, coincidence rather than a nascent CMR. However, further\ndeep NIR imaging coverage of this field is required to identify\nadditional members of this proto-elliptical galaxy class.\n\n\\acknowledgments\n\nSupport for program \\# 10404 was provided by NASA through a grant\n(GO-10404.01-A) from the Space Telescope Science Institute, which is\noperated by the Association of Universities for Research in Astronomy,\nInc., under NASA contract NAS 5-26555. The work of SAS was performed\nin part under the auspices of the U.S. Department of Energy, National\nNuclear Security Administration by the University of California,\nLawrence Livermore National Laboratory under contract\nNo. W-7405-Eng-48. JK is financially supported by the DFG, grant SFB\n439. WvB acknowledges support for radio galaxy studies at UC Merced,\nincluding the work reported here, with the Hubble Space Telescope and\nthe Spitzer Space Telescope via NASA grants HST \\# 10127, SST \\#\n1264353, SST \\# 1265551, SST \\# 1279182.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nQuantum cohomology, or more general, higher genus Gromov-Witten invariants are defined for symplectic and algebraic manifolds in e.g. \\cite{Ruan96},\\cite{Beh97}, \\cite{LT98}. As a kind of basic varieties, quantum cohomology of the smooth complete intersections in projective spaces has been studied intensively. A kind of (numerical) mirror symmetry for Fano and Calabi-Yau complete intersections has been established (\\cite{Giv96}), which expresses the small J-function in terms of hypergeometric series. From small J-functions one can reconstruct all genus zero Gromov-Witten invariants with only \\emph{ambient cohomology classes} as insertions, i.e. the classes obtained by intersecting linear spaces with the complete intersections. The invariants with \\emph{primitive cohomology classes} as insertions are not well-understood as those with ambient insertions; for 3 point invariants see \\cite{Bea95}. But they are necessary for understanding for example, Dubrovin's conjecture on the relation between quantum cohomology and derived categories. \n\nFrom now on, we call genus zero primary (i.e. without $\\psi$-classes) Gromov-Witten invariants \\emph{correlators}, for brevity. \nIn \\cite{Hu15} we studied the big quantum cohomology of Fano complete intersections, especially the correlators with primitive insertions. One of our basic tool is the Monodromy group of the whole family of complete intersections of a given multidegree. The Zariski closure $\\overline{G}$ of such monodromy groups are either orthogonal groups, symplectic groups, or finite groups. If $\\overline{G}$ is a finite group, we call the complete intersections \\emph{exceptional}, and otherwise \\emph{non-exceptional}. There are only three kinds of exceptional complete intersections: the cubic surfaces, the quadric hypersurfaces, and the even dimensional complete intersections of two quadric hypersurfaces. The first two kinds has been somewhat well-studied, in the sense that all correlators can be effectively computed, and the semisimplicity (of the corresponding Frobenius manifold) is proved. \n\nIn this paper we studied the big quantum cohomology of the even dimensional complete intersections of two quadric hypersurfaces. We will also call them even $(2,2)$-complete intersections for short. To understand our approach and its limitations, let us recall more details in \\cite{Hu15}. The basic idea is still to use WDVV equations\n\\begin{equation}\\label{eq-WDVV-intro}\n\\sum_e\\sum_f\t(\\partial_{t^a}\\partial_{t^b}\\partial_{t^e}F)g^{ef}(\\partial_{t^f}\\partial_{t^c}\\partial_{t^d}F)=\\sum_e\\sum_f(\\partial_{t^a}\\partial_{t^c}\\partial_{t^e}F)g^{ef}(\\partial_{t^f}\\partial_{t^b}\\partial_{t^d}F).\n\\end{equation}\n The most direct way to use WDVV to get recursions is to use the leading terms. Namely, selecting a monomial $t^I$, where $I$ is a multi-index, and extracting the coefficients of $t^I$ on both sides, we get an equation of the form\n \\begin{eqnarray}\n \t&&\\mathrm{Coeff}_{t^I}(\\partial_{t^a}\\partial_{t^b}\\partial_{t^e}F)g^{ef}(\\partial_{t^f}\\partial_{t^c}\\partial_{t^d}F)(0)\n \t+\t(\\partial_{t^a}\\partial_{t^b}\\partial_{t^e}F)(0)g^{ef}\\mathrm{Coeff}_{t^I}(\\partial_{t^f}\\partial_{t^c}\\partial_{t^d}F)\\nonumber\\\\\n \t&&-\\mathrm{Coeff}_{t^I}(\\partial_{t^a}\\partial_{t^c}\\partial_{t^e}F)g^{ef}(\\partial_{t^f}\\partial_{t^b}\\partial_{t^d}F)(0)\n \t-(\\partial_{t^a}\\partial_{t^c}\\partial_{t^e}F)(0)g^{ef}\\mathrm{Coeff}_{t^I}(\\partial_{t^f}\\partial_{t^b}\\partial_{t^d}F)\\nonumber\\\\\n \t&=& \\mbox{combinations of coefficients of lower order terms}.\n \\end{eqnarray}\n Here we have adopted Einstein's summation convention, i.e. omitting the summation notations of the repeated indices $e$ and $f$. We call the resulted recursions \\emph{essentially linear recursions}. Such recursions are effective only when the correlators of length 3 have good properties. For non-exceptional complete intersections and the even $(2,2)$-complete intersections, the correlators with primitive insertions cannot provide a recursion to compute correlators of arbitrary lengths, essentially due to the monodromy reason. For non-exceptional ones, we remedy this by considering the symmetric reduction of the WDVV equations. From the classical theory of polynomial invariants of orthogonal groups or symplectic groups, the dependence of the generating function $F$ on the variables dual to the primitive classes are encoded in only one variable $s$. When the dimension $n$ of the complete intersection $X$ is even, $s$ is defined as\n \\begin{equation}\n \ts=\\frac{1}{2}\\sum (t^i)^2,\n \\end{equation}\n where $t^i$ are dual variables of an orthonormal basis of $H^n_{\\mathrm{prim}}(X)$. The main novelty of \\cite{Hu15} is that we found that the coefficient of $s^k$ in $F$ can be recursively computed by solving quadratic equations of one variable with two equal roots. For a precise conjectural algorithm and partial results on this we refer the reader to \\cite[Section 1]{Hu15}. There is only one case that is excluded in this algorithm: the cubic hypersurfaces. For cubic hypersurfaces we showed in \\cite{Hu15} that the generating function $F$ can be computed by an essentially linear recursion with respect to the variable $s$, using the symmetric-reduced WDVV equations. Note that $s$ is quadratic in $t^i$'s. So this means that we can compute $F$ by a recursion on the \\emph{sub-leading terms} of (\\ref{eq-WDVV-intro}) and the monodromy symmetries. As (even dimensional) cubic hypersurfaces and the even $(2,2)$-complete intersections are the only even dimensional complete intersections of Fano index $n-1$ (recall $n=\\dim X$), it is reasonable to expect an analogy between them.\n\nThe monodromy group of the family of even $n$-dimensional $(2,2)$-complete intersections, or more precisely, it image in $\\mathrm{Aut}(H^n_{\\mathrm{prim}}(X))$, is the Weyl group $D_{n+3}$. Moreover the lattice $H^n_{\\mathrm{prim}}(X)\\cap H^n(X;\\mathbb{Z})$ with the Poincar\u00e9 pairing is isomorphic to a $D_{n+3}$-lattice with $(-1)^{\\frac{n}{2}}$ times its standard inner product. We can choose a basis $\\alpha_1,\\dots,\\alpha_{n+3}$ of the lattice $H^n_{\\mathrm{prim}}(X)\\cap H^n(X;\\mathbb{Z})$ which maps to the $D_{n+3}$ roots under such an isomorphism, and then define a specific orthonormal basis $\\epsilon_1,\\dots,\\epsilon_{n+3}$ of $H^n_{\\mathrm{prim}}(X)$. Let $\\mathsf{h}_i$ be the $i$-th cup product of the hyperplane class $\\mathsf{h}$. Let $t^0,\\dots,t^{n},t^{n+1},\\dots,t^{2n+3}$ be the basis dual to $1,\\mathsf{h},\\dots,\\mathsf{h}_n,\\epsilon_1,\\dots,\\epsilon_{n+3}$, then the generating function of correlators of $X$ is a function of $t^0,\\dots,t^n$ and $s_1,\\dots,s_{n+3}$, where\n\\begin{equation}\\label{eq-invariantsOf-typeD-intro}\n\ts_{i}=\\begin{cases}\n\t\\vspace{0.2cm}\n\t\\frac{1}{(2i)!}\\sum_{j=n+1}^{2n+3}(t^j)^{2i},& \\mbox{for}\\ 1\\leq i\\leq n+2,\\\\\n\t\\prod_{j=n+1}^{2n+3}t^j,& i=n+3.\n\t\\end{cases}\n\\end{equation}\nInstead of doing the reduction of WDVV with the $D_{n+3}$ symmetry, we are going to find recursions based on the sub-leading terms of the WDVV equations, as mentioned above. For this we need first compute all the correlators of length 4. The correlators of length 4 with at most 2 primitive insertions are computed in \\cite{Hu15}. For the correlators of length 4 with 4 primitive insertions, we have a uniform partial result in \\cite[Section 9]{Hu15} which holds for all complete intersections of Fano index $n-1$, which is obtained by an application of Zinger's reduced genus 1 Gromov-Witten invariants \\cite{Zin08}. Combining this result with the above monodromy reason and some integrality, we obtain:\n\n\\begin{theorem}\\label{thm-4points-fanoIndex-even(2,2)-intro}(= Theorem \\ref{thm-4points-fanoIndex-even(2,2)})\nLet $X$ be an even dimensional complete intersection of two quadrics in $\\mathbb{P}^{n+2}$, with $n\\geq 4$. Then\n\\begin{equation}\\label{eq-4points-fanoIndex-even(2,2)-intro}\n\t\\frac{\\partial^2 F}{(\\partial s_1)^2}(0)=1,\\ \\frac{\\partial F}{\\partial s_2}(0)=-2.\n\\end{equation}\nEquivalently, for $1\\leq a,b\\leq n+3$,\n\\begin{equation}\\label{eq-4points-fanoIndex-even(2,2)-ab-intro}\n\t\\langle \\epsilon_a,\\epsilon_a,\\epsilon_b,\\epsilon_b\\rangle_{0,1}=1.\n\\end{equation}\n\\end{theorem}\nUsing Theorem \\ref{thm-4points-fanoIndex-even(2,2)-intro}, a careful study of the sub-leading terms of the WDVV equations leads to the following.\n\\begin{theorem}[Reconstruction]\\label{thm-reconstruction-even(2,2)-intro}(= Theorem \\ref{thm-reconstruction-even(2,2)})\nLet $X$ be an even dimensional complete intersection of two quadrics in $\\mathbb{P}^{n+2}$, with $n\\geq 4$.\nWith the knowledge of the 4-point invariants, all the invariants can be reconstructed from the WDVV, the deformation invariance, and the \\emph{special correlator}\n\\begin{equation}\\label{eq-intro-length(n+3)Invariant-even(2,2)}\n\t\\langle \\epsilon_{1},\\dots,\\epsilon_{n+3}\\rangle_{0,n+3,\\frac{n}{2}}.\n\\end{equation}\n\\end{theorem}\n\nWe tried some more intricate study of WDVV, e.g. some equations that a priori may give quadratic equations of the special correlator (\\ref{eq-intro-length(n+3)Invariant-even(2,2)}). But such equations in examples turn out to be trivial. We conjecture that WDVV does not give any new information on the special correlator; see Conjecture \\ref{conj-specialCorrelator-free} for a precise statement. We can show a weaker result: \n\\begin{lemma}\\label{lem-specialCorrelator-freedomOfSign}\nFor even $(2,2)$-complete intersections, the WDVV equations and the knowledge of correlators of length 4 can at most determine the special correlator with a freedom of signs, unless it vanishes.\n\\end{lemma}\nThe proof is trivial: we change the signs of the basis $\\alpha_1,\\dots,\\alpha_{n+3}$. The resulted basis is still allowable as above. Then each correlator will change by a sign $(-1)^k$, where $k$ is the number of primitive insertions. \n\nWe refer the reader to Remark \\ref{rem:choiceOfBasis}, and Section \\ref{sec:conjecturesOnSpecialCorrelator}, on the choice of the basis. In Section \\ref{sec:explictD-Lattice} we give an explicit construction of the basis, which we take as our standard choice.\n\n\nNevertheless, Theorem \\ref{thm-4points-fanoIndex-even(2,2)-intro} and \\ref{thm-reconstruction-even(2,2)-intro} are enough to imply the following.\n\n\\begin{theorem}[Analyticity and Semisimplicity]\\label{thm-intro-convergence-semisimplicity}\nThere exists an open (in the classical topology) neighborhood of the origin of $\\mathbb{C}^{2n+4}$, on which the generating function $F(t^0,\\dots,t^{2n+3})$ is analytic and defines a semisimple Frobenius manifold.\n\\end{theorem}\nThe analyticity follows from a bound of the correlators, which follows from an induction based on an algorithm given by the proof of Theorem \\ref{thm-reconstruction-even(2,2)-intro}. Let $\\widetilde{E}$ be the matrix of the big quantum multiplication by the Euler field $E$. We show the semisimplicity by showing that \n\\begin{equation}\\label{eq-statement-cutoffEulerField-distinctEigenValues}\n\t\\mbox{the cutoff of $\\widetilde{E}$ at order 2 has pairwise distinct eigenvalues.}\n\\end{equation}\nI would like to provide a prophetic view why this is possible (see also \\cite[Remark 3.2]{Hu15}). If the generating function $F$ has continuous symmetries such as the orthogonal or symplectic groups in the cases of non-exceptional complete intersections, one can construct a family of (normalized) Euler fields for the associated Frobenius manifold $\\mathcal{M}$. But semisimple Frobenius manifold has a unique normalized Euler field \\cite[Theorem I.3.6]{Man99}. So such $\\mathcal{M}$ cannot be generically semisimple. For the same reason, if one wants to show the Frobenius manifold associated to the generating function $F$ of quantum cohomology of an even $(2,2)$-complete intersection $X$ is semisimple, one needs to use the expansion of $F$ to a certain order such that it has no continuous symmetry. This explains why the small quantum cohomology of $X$ is not semisimple: the cutoff of $F$ at order 3 is a function of $t^0,\\dots,t^n$ and \n\\[\ns_1=\\sum_{i=n+1}^{2n+3}(t^i)^2.\n\\]\nSo it has symmetries from the orthogonal group $\\mathrm{O}\\big(H^n_{\\mathrm{prim}}(X)\\big)$. The degree 4 form \n\\[\ns_2=\\sum_{i=n+1}^{2n+3}(t^i)^4\n\\]\nhas only finitely many automorphisms; this is a classical theorem of Jordan \\cite{Jor1880}. So we can expect that the information of correlators of length 4 implies the semisimplicity. To capture the full information of correlators of length 4 we need only the cutoff of the matrix at an order $\\geq 2$.\n\nIn view of Kuznetsov's resulte \\cite[Corollary 5.7]{Kuz08}, the semisimplicity statement above confirms a part of Dubrovin' conjecture \\cite{Dub98} which predicts the equivalence of the existence of full exceptional collections in $D^b(\\mathrm{Coh}(X))$ and the generic semisimplicity of the Frobenius manifold of the quantum cohomology of $X$.\\\\\n\n\n\nBy an analogy to a vanishing conjecture \\cite[Conjecture 10.26]{Hu15} of the coefficient of $s^{n+1}$ in $F$ for cubic hypersurfaces, we make the following conjecture on the special correlator. For details see Section \\ref{sec:conjecturesOnSpecialCorrelator}. \n\\begin{conjecture}\\label{conj-unknownCorrelator-Even(2,2)-intro}\nFor an even $n$ dimensional complete intersection of two quadrics in $\\mathbb{P}^{n+2}$, let $\\varepsilon_1,\\dots,\\varepsilon_{n+3}$ be the basis of $H^n_{\\mathrm{prim}}(X)$ defined in Section \\ref{sec:explictD-Lattice}. Then \n\\begin{equation}\\label{eq-unknownCorrelator-Even(2,2)-intro}\n\t\\langle \\varepsilon_{1},\\dots,\\varepsilon_{n+3}\\rangle_{0,n+3,\\frac{n}{2}}=\\frac{(-1)^{\\frac{n}{2}}}{2}.\n\\end{equation}\n\\end{conjecture}\nHere $\\varepsilon_1,\\dots,\\varepsilon_{n+3}$ is an unnormalized orthogonal basis, see (\\ref{eq-normalizedOrthonormalBasis}).\n\nI must confess that this analogy is not so strong to make us confident about the validity. We can only show Conjecture \\ref{conj-unknownCorrelator-Even(2,2)-intro} in dimension 4.\n\\begin{theorem}\\label{thm-unknownCorrelator-Even(2,2)-4dim-intro}(= Corollary \\ref{cor-unknownCorrelator-Even(2,2)-4dim})\nFor any $4$-dimensional complete intersections of two quadrics in $\\mathbb{P}^{6}$, \n\\begin{equation}\\label{eq-unknownCorrelator-Even(2,2)-4dim-intro}\n\t\\langle \\epsilon_{1},\\dots,\\epsilon_{7}\\rangle_{0,7,2}=\\frac{1}{2}.\n\\end{equation}\n\\end{theorem}\nThe proof is essentially a computation \\emph{from the first principle}. Namely, we find explicit cycles $S_{0,1},S_{1,2},\\dots,S_{5,6},S_{6,0}$ in $X$ such that (\\ref{eq-unknownCorrelator-Even(2,2)-4dim-intro}) is equivalent to\n\\begin{equation}\\label{eq-enumerativeCorrelator-Dim4-intro}\n\t \\langle [S_{0,1}],[S_{1,2}],[S_{2,3}],[S_{3,4}],[S_{4,5}],[S_{5,6}],[S_{6,0}]\\rangle_{0,7,2}=1.\n\\end{equation}\nThen we show (\\ref{eq-enumerativeCorrelator-Dim4-intro}) by the definition of Gromov-Witten invariants: we count curves that intersect with the given cycles, and use deformation theory to compute the intersection multiplicity of the image of the virtual fundamental cycle and cycles in $X^7$ (see Definition \\ref{def-enumerativeCorrelator}). As a byproduct we obtain an enumerative result with somewhat classical flavor.\n\\begin{theorem}\\label{thm-countingConics-4dim-general-intro}(= Theorem \\ref{thm-countingConics-4dim-general})\nFor general 4-dimensional smooth complete intersections $X$ of two quadrics in $\\mathbb{P}^6$, there exists exactly one smooth conic that meets each of the 2-planes $S_{i,i+1}$ in $X$ for $0\\leq i\\leq 6$.\n\\end{theorem}\n\nLet us emphasize the significance of the computation of the special correlator. In view of our previous results and conjectures (see \\cite[Conjecture 1.15 and Table 1]{Hu15}), in the algorithmic sense the special correlator is the only genus 0 Gromov-Witten invariant of complete intersections, at least the even dimensional ones, that we have no definite way to compute. But see Remark \\ref{rem:compute-specialCorrelator-fromHigherGenusInv} for a possible approach via higher genus Gromov-Witten invariants with ambient insertions, as a byproduct of Theorem \\ref{thm-intro-convergence-semisimplicity}.\\\\\n\n\n\nWe are now in a position to make an additional remark on the semisimplicity.\nThere seems to be a folklore conjecture that if a smooth projective variety has generically semisimple quantum cohomology, then there exists finitely many correlators that can determine all correlators by WDVV and the Euler field. One might be temped to make a stronger and, quantitative, conjecture: if the values of the correlators of length $\\leq k$ suffice to imply the generic semisimplicity, then they determine all correlators. Our Theorem \\ref{thm-reconstruction-even(2,2)-intro} confirms the first conjecture in the case of the even $(2,2)$-complete intersections. On the other hand, by the \nstatement (\\ref{eq-statement-cutoffEulerField-distinctEigenValues}), Lemma \\ref{lem-specialCorrelator-freedomOfSign} and Theorem \\ref{thm-unknownCorrelator-Even(2,2)-4dim-intro}, the 4-dimensional $(2,2)$-complete intersections are counterexamples to the latter one. From Example \\ref{example-f(6)} and \\ref{example-f(8)} and integrality, one sees that the special correlator of the 6 dimensional and 8 dimensional $(2,2)$-complete intersections also do not vanish, so they are also counterexamples.\\\\\n\n\n\n\n\n\n\n\n\n\n\nIn the Appendix, we present an algorithm based on the proof of Theorem \\ref{thm-reconstruction-even(2,2)-intro} to compute the primary genus 0 Gromov-Witten invariants of an even $(2,2)$-complete intersection of dimension $\\geq 4$, with the special correlator (\\ref{eq-intro-length(n+3)Invariant-even(2,2)}) as an unknown. The algorithm is implemented in our Macaulay2 package \n\\texttt{QuantumCohomologyFanoCompleteIntersection}.\nMoreover we write also a Macaulay2 package\n \\texttt{ConicsOn4DimIntersectionOfTwoQuadrics} \n for the computations in Section \\ref{sec:EnumerativeGeometry-Even(2,2)}. The reader can find the packages in\n\n\\url{https:\/\/github.com\/huxw06\/Quantum-cohomology-of-Fano-complete-intersections}\n\n\\begin{comment}\nThe paper is organized as follows.\n\nIn Section \\ref{sec:preliminaries} we recall basic properties of Gromov-Witten invariants that we will use in this paper. Then we recall the results on the primitive cohomology lattice of even $(2,2)$-complete intersections.\n\\end{comment}\n\n\\vspace{0.2cm}\n\n\n\\emph{Acknowledgement}\\quad \n I am grateful to Hua-Zhong Ke for enlightening discussions. I also thank Huai-Liang Chang, Weiqiang He, Giosu\u00e8 Muratore, Maxim Smirnov, and Jinxing Xu for discussions on various related topics. \n \n This work is supported by NSFC 11701579.\n\n\n\n\n\\section{Preliminaries}\\label{sec:preliminaries}\n\\subsection{Properties of Gromov-Witten invariants}\nWe recall the definition of the Gromov-Witten invariants, and their properties that we need to use in this paper. Our main reference is\n\\cite[Chapter VI]{Man99}. All schemes in this paper are over $\\mathbb{C}$.\n\nLet $X$ be a smooth projective scheme over $\\mathbb{C}$ of dimension $n$. Let $k\\in \\mathbb{Z}_{\\geq 0}$, and $\\beta\\in H_2(X;\\mathbb{Z})\/\\mathrm{tor}$. The stack $\\overline{\\mathcal{M}}_{g,k}(X,\\beta)$ of stable maps of degree $\\beta$ from $k$-point genus $g$ marked semistable curves to $X$ is a proper Deligne-Mumford stack and carries a virtual fundamental class (\\cite{BF97}, \\cite{LT98}) $[\\overline{\\mathcal{M}}_{g,k}(X,\\beta)]^{\\mathrm{vir}}$ of dimension $(1-g)(n-3)+k+c_1(T_X)\\cdot \\beta$. For each $1\\leq i\\leq k$, the section $\\sigma_i$ pulls back the relative cotangent line bundle of the universal curve to form a line bundle on $\\overline{\\mathcal{M}}_{g,k}(X,\\beta)$, whose first Chern class is denoted by $\\psi_i$; moreover there is an associated \\emph{evaluation map} $\\mathrm{ev}_i=f\\circ \\sigma_i$, where $f$ is the universal stable map. For $\\gamma_1,\\dots,\\gamma_k\\in H^*(X;\\mathbb{Q})$ and $a_1,\\dots,a_k\\in \\mathbb{Z}_{\\geq 0}$, there is an associated \\emph{Gromov-Witten invariant}\n\\begin{equation}\n\t\\langle \\psi_1^{a_1}\\gamma_1,\\dots,\\psi_k^{a_k}\\gamma_k\\rangle_{g,k,\\beta}^X:=\\int_{[\\overline{\\mathcal{M}}_{g,k}(X,\\beta)]^{\\mathrm{vir}}}\\prod_{i=1}^{k}\\psi_i^{a_i}\\mathrm{ev}_i^*\\gamma_i\\in \\mathbb{Q}.\n\\end{equation}\nA term like $\\psi_i^{a_i}\\gamma_i$ in $\\langle \\psi_1^{a_1}\\gamma_1,\\dots,\\psi_k^{a_k}\\gamma_k\\rangle_{g,k,\\beta}$ is called an \\emph{insertion} of this invariant. We say $(g,k,\\beta)$ is in the \\emph{stable range} if either $2g-2+k>0$ or $\\beta$ is a nonzero effective curve class.\nIt is convenient to use simplified notations in the following occasions:\n\\begin{enumerate}\n\t\\item[(i)] The superscript $X$ will be omitted when it is obvious;\n\n\t\\item[(iii)] the subscript $k$ might be dropped when $k$ is obvious from the expression;\n\t\\item[(iv)] the subscript $\\beta$ might be dropped when it can be uniquely determined by the insertions and the following condition (\\ref{eq-Dim}), which is always the case for Fano complete intersections in projective spaces.\n\\end{enumerate}\n\nThe GW invariants $\\langle \\psi_1^{a_1}\\gamma_1,\\dots,\\psi_k^{a_k}\\gamma_k\\rangle_{g,k,\\beta}$ with $a_1=\\dots=a_k=0$ are called \\emph{primary}. In this paper we only deal with primary genus 0 invariants, although some results depends on the computation of certain non-primary invariants and genus 1 invariants in \\cite{Hu15}. For brevity we call the genus 0 primary Gromov-Witten invariants \\emph{correlators}.\n\n\nFor simplicity we assume $H^{\\mathrm{odd}}(X;\\mathbb{Q})=0$, which is the case for even dimensional $(2,2)$-complete intersections. We denote $H^*(X)=H^*(X;\\mathbb{C})$. \nFor two cohomology classes $\\gamma_1$ and $\\gamma_2$, we denote the Poincar\u00e9 pairing by\n\\begin{equation}\n\t(\\gamma_1,\\gamma_2):=\\int_X \\gamma_1\\cup \\gamma_2.\n\\end{equation}\nIf a basis $\\gamma_0,\\dots,\\gamma_N$ of $H^*(X)$ is given, we denote $g_{a,b}=(\\gamma_a,\\gamma_b)$ for $0\\leq a,b\\leq N$, and set $(g^{a,b})_{0\\leq a,b\\leq N}$ to be the inverse matrix of $(g_{a,b})_{0\\leq a,b\\leq N}$.\nLet $T^0,\\dots,T^N$ be dual basis with respect to $\\gamma_0,\\dots,\\gamma^N$, then the genus $g$ generating function is defined as\n \\begin{equation}\\label{eq-def-generatingFunction}\n \t\\mathcal{F}_g(T^0,\\dots,T^{N})=\\sum_{k\\geq 0} \\sum_{\\beta} \\frac{1}{k!}\\big\\langle \\sum_{i=0}^N \\gamma_i T^i,\\dots,\\sum_{i=0}^N \\gamma_i T^i\\big\\rangle_{g,k,\\beta},\n \\end{equation}\nwhere the invariants outside of the stable range are defined to be zero, by convention. Here we have implicitly use that $X$ is Fano, so that with fixed insertions there is finitely many $\\beta$ such that the invariant does not vanish. We denote $F= \\mathcal{F}_0$. \n\nWe record some standard properties of GW invariants as follows.\n\\begin{enumerate}\n\t\\item[(i)] Degree 0 correlators:\n\t\t\\begin{gather}\\label{eq-Deg0}\\tag{Deg0}\n\t\t\\langle \\gamma_1,\\dots,\\gamma_k\\rangle_{g,k,0}=\n\t\t\\begin{cases}\n\t\t\\int_{X}\\gamma_1\\cup \\gamma_2\\cup \\gamma_3,& \\mbox{if}\\ g=0, k=3;\\nonumber\\\\\n\t\t-\\frac{1}{24}\\int_X \\gamma_1\\cup c_{n-1}(T_X), & \\mbox{if}\\ g=1, k=1,\\\\\n\t\t0, & \\mbox{if}\\ 2g-2+k\\geq 2.\n\t\t\\end{cases}\n\t\t\\end{gather}\n\t\\item[(ii)]\tSuppose each $\\gamma_i$ has pure real degree $|\\gamma_i|$, then there is the dimension constraint:\n\t\t\\begin{equation}\\label{eq-Dim}\\tag{Dim}\n\t\t\t\\langle \\gamma_{1},\\dots, \\gamma_{k}\\rangle_{g,k,\\beta}=0\\ \\mbox{unless}\\\n\t\t\t\\sum_{i=1}^k \\frac{1}{2}|\\gamma_{b_k}|=(1-g)(n-3)+k+c_1(T_X)\\cap \\beta.\n\t\t\\end{equation}\n\t\\item[(iii)] The $S_n$-equivariance:\n\t\t\\begin{equation}\\label{eq-Sym}\\tag{Sym}\n\t\t\t\\langle \\gamma_{1},\\dots, \\gamma_{i-1}, \\gamma_{i},\\dots, \\gamma_{b_k}\\rangle_{g,k,\\beta}\n\t\t\t=\\langle \\gamma_{1},\\dots, \\gamma_{i}, \\gamma_{i-1},\\dots,\\gamma_{b_k}\\rangle_{g,k,\\beta}.\n\t\t\\end{equation}\n\t\\item[(iv)]\n\t\t\tThe divisor equation: for $\\gamma\\in H^2(X)$,\n\t\t\t\\begin{gather}\\label{eq-Div}\\tag{Div}\n\t\t\t\t\\langle \\gamma_1,\\dots,\\gamma_k,\\gamma\\rangle_{g,k+1,\\beta}=(\\gamma\\cap \\beta)\\langle \\gamma_1,\\dots,\\gamma_k\\rangle_{g,k,\\beta}.\n\t\t\t\\end{gather}\n\t\t\n\t\\item[(v)] Fundamental class axiom:\n\t\t\t\\begin{gather}\\label{eq-FCA}\\tag{FCA}\n\t\t\t\t\\langle 1, \\gamma_{1},\\dots,\\gamma_{k-1}\\rangle_{g,k,\\beta}=\n\t\t\t\t\\begin{cases}\n\t\t\t\t(\\gamma_1,\\gamma_2),& \\mbox{if}\\ g=0, k=3, \\beta=0;\\\\\n\t\t\t\t0, & \\mbox{if}\\ 3g-3+k\\geq 1\\ \\mbox{or}\\ \\beta\\neq 0.\n\t\t\t\t\\end{cases}\n\t\t\t\\end{gather}\n\t\\item[(vi)] If a basis $\\gamma_0,\\dots,\\gamma_N$ of $H^*(X)$ and $T^0,\\dots,T^N$ is the dual basis, then we have the WDVV equations, for $0\\leq a,b\\leq N$:\n\t\t\\begin{gather}\\label{eq-WDVV}\\tag{WDVV}\n\t\t\\sum_{e=0}^N \\sum_{f=0}^N \\frac{\\partial^3 F}{\\partial T^a \\partial T^b\\partial T^e}g^{ef}\\frac{\\partial^3 F}{\\partial T^f \\partial T^c\\partial T^d}\n\t\t=\\sum_{e=0}^N \\sum_{f=0}^N \\frac{\\partial^3 F}{\\partial T^a \\partial T^c\\partial T^e}g^{ef}\\frac{\\partial^3 F}{\\partial T^f \\partial T^b\\partial T^d}.\n\t\t\\end{gather}\n\n\\end{enumerate}\n\n\n\nNow suppose $\\gamma_0,\\dots,\\gamma_N$ is a basis of $H^*(X)$ such that each $\\gamma_i$ has a pure degree, and let $T^0,\\dots,T^N$ be the dual basis have pure degrees. Let\n\\[\nc_1(T_X)=\\sum_{i=0}^N a_i \\gamma_i.\n\\]\nOf course $a_i=0$ unless $|\\gamma_i|=2$. The Euler vector field is defined as\n\\begin{equation}\\label{eq-EV-0}\n\tE=\\sum_{i=0}^N(1-\\frac{|\\gamma_i|}{2})\\frac{\\partial }{\\partial T^i}+\\sum_{i=0}^N a_i \\frac{\\partial }{\\partial T^i}.\n\\end{equation}\nThen (\\ref{eq-Dim}) and the divisor equation (\\ref{eq-Div}) imply \n\\begin{gather}\\label{eq-EulerVectorField}\\tag{EV}\n\tEF=(3-n)F+ \\sum_{i=0}^N a_i \\frac{\\partial }{\\partial T^i} c,\n\\end{gather}\nwhere $c$ is the classical cubic intersection form \n\\begin{eqnarray}\\label{eq-cubicIntersectionForm}\nc(t_0, \\cdots, t^{n+m})=\\sum_a\\sum_b\\sum_c\\frac{t^{a}t^{b}t^{c}}{6}\\int_{X}\\gamma_a \\gamma_b \\gamma_c.\n\\end{eqnarray}\n\nThe \\emph{big quantum product} is defined as\n\\begin{equation}\n \t\\gamma_a\\star \\gamma_b=\\sum_{e}\\sum_f\\frac{\\partial^3 \\mathsf{F}}{\\partial T^a \\partial T^b\\partial T^e}g^{ef}\\gamma_f,\n \\end{equation} \n and the \\emph{small quantum product} is defined as\n\\begin{equation}\n \t\\gamma_a\\diamond \\gamma_b=\\gamma_a\\star \\gamma_b|_{T^0=\\dots=T^N=0}.\n \\end{equation} \n\n\n\n\\subsection{Monodromy group and the \\texorpdfstring{$D_{n+3}$}{D{n+3}} lattice}\\label{sec:monodromy-lattice}\nLet $n\\geq 4$ be even. \nLet $X$ be a complete intersection of two quadric hypersurfaces in $\\mathbb{P}^{n+2}$. Then $H^*(X)=H^{\\mathrm{even}}(X)$, and the Fano index of $X$ is $n-1$. Denote the hyperplane class of $X$ by $\\mathsf{h}$, and\n\\begin{equation}\n\t\\mathsf{h}_{i}=\\underbrace{\\mathsf{h}\\cup\\dots\\cup\\mathsf{h}}_{i\\ \\mathsf{h}'s}.\n\\end{equation}\n\nLet $V=\\mathbb{R}^{n+3}$ be the Euclidean space with the standard inner product. Let $\\varepsilon_1,\\dots,\\varepsilon_{n+3}$ be an orthonormal basis, and let\n\\begin{equation}\\label{eq-roots-D}\n\t\\begin{cases}\n\t\\alpha_i=\\varepsilon_{i}-\\varepsilon_{i+1}\\ \\mbox{for}\\ 1\\leq i\\leq n+2,\\\\\n\t\\alpha_{n+3}=\\varepsilon_{n+2}+\\varepsilon_{n+3}.\n\t\\end{cases}\n\\end{equation} \nThe Weyl group $D_{n+3}\\subset \\mathrm{GL}(n+3,\\mathbb{R})$ is generated the reflections with respect to the $\\alpha_i$'s. If one writes vectors in $\\mathbb{R}^{n+3}$ in terms of the coordinates according to the basis $\\varepsilon_1,\\dots,\\varepsilon_{n+3}$, i.e. \n\\[\n\\mathbf{v}=(v_1,\\dots,v_{n+3})=\\sum_{i=1}^{n+3}v_i \\varepsilon_i,\n\\]\nthen the group $D_{n+3}$ coincides with the group generated by the permutations of the coordinates, and the change of signs\n\\[\n(v_1,\\dots,v_{n+1},v_{n+2},v_{n+3})\\mapsto (v_1,\\dots,v_{n+1},-v_{n+2},-v_{n+3}).\n\\]\nBy \\cite[\\S 5]{Del73}, the monodromy group of the whole family of smooth complete intersections of two quadrics in $\\mathbb{P}^{n+2}$ is isomorphic to $D_{n+3}$, and in this way the primitive cohomology $H^n_{\\mathrm{prim}}(X)$ is a standard representation of $D_{n+3}$, and the integral lattice $H^n_{\\mathrm{prim}}(X)\\cap H^n(X;\\mathbb{Z})$ is generated by the roots $\\alpha_i$'s of $D_{n+3}$. In other words, there is an isomorphism\n\\begin{equation}\\label{eq-isomorphism-primCoh-even(2,2)}\n\tV\\otimes_{\\mathbb{R}}\\mathbb{C}\\xrightarrow{\\sim} H^n_{\\mathrm{prim}}(X),\n\\end{equation}\nvia which lattice in $V$ generated by $\\alpha_1,\\dots,\\alpha_{n+3}$ is mapped onto $H^n_{\\mathrm{prim}}(X)\\cap H^n(X;\\mathbb{Z})$.\n\\begin{proposition}\\label{prop-lattice-sign}\n We equip $H^n_{\\mathrm{prim}}(X)$ with the inner product $(-1)^{\\frac{n}{2}}(.,.)$, where $(.,.)$ is the Poincar\u00e9 pairing. Then (\\ref{eq-isomorphism-primCoh-even(2,2)}) becomes an isometry. \n\\end{proposition}\n\\begin{proof}\nThe Poincar\u00e9 pairing is invariant under monodromies. As we will recall in the following, the degree 2 polynomial invariant of the standard representation of $D_{n+3}$ is generated by the Poincar\u00e9 pairing. By the Hodge index theorem, $(-1)^{\\frac{n}{2}}(.,.)$ is positive definite. So the inner product on $H^n_{\\mathrm{prim}}(X)$ induced from $V$ by (\\ref{eq-isomorphism-primCoh-even(2,2)}) coincides with $(-1)^{\\frac{n}{2}}(.,.)$ up to a positive constant multiple. Since the discriminant of the lattice $\\alpha_1,\\dots,\\alpha_{n+3}$ equals 4, we are left to show that the discriminant of the lattice $H^n_{\\mathrm{prim}}(X)\\cap H^n(X;\\mathbb{Z})$ is $\\pm 4$. First we note that there exists $\\frac{n}{2}$-planes in $X$ (\\cite[Corollary 3.3]{Rei72}), whose intersection number with $\\mathsf{h}_{n\/2}$ is 1, from which it follows that the sub-lattice $L=H^*_{\\mathrm{amb}}(X)\\cap H^*(X;\\mathbb{Z})$ is generated by $\\mathsf{h}_{n\/2}$. \nSince $H^n(X;\\mathbb{Z})$ is unimodular and $(\\mathsf{h}_{n\/2},\\mathsf{h}_{n\/2})=4$, by \\cite[Prop. 5.3.3]{Kit93} the discriminant of $L^{\\perp}=H^n_{\\mathrm{prim}}(X)\\cap H^n(X;\\mathbb{Z})$ is $\\pm 4$. So we are done.\n\\end{proof}\n\n\\begin{corollary}\n$H^{n}(X;\\mathbb{Z})$ is generated by the classes of $\\frac{n}{2}$-planes in $X$.\n\\end{corollary}\n\\begin{proof}\nBy \\cite[Theorem 3.14]{Rei72}, the discriminant of the lattice of the classes of $\\frac{n}{2}$-planes in $X$ is $\\pm 4$, which coincides with that of the lattice $H^{n}(X;\\mathbb{Z})$ (see the proof of Proposition \\ref{prop-lattice-sign}).\n\\end{proof}\n\n\n Via (\\ref{eq-isomorphism-primCoh-even(2,2)}), we identify the vectors $\\varepsilon_i$ and $\\alpha_i$ with their images in $H^n_{\\mathrm{prim}}(X)$. Moreover we define, for $1\\leq i\\leq n+3$,\n\\begin{equation}\\label{eq-normalizedOrthonormalBasis}\n\t\\epsilon_i=\\begin{cases}\n\t\\varepsilon_i, & \\mbox{if}\\ n\\equiv 0 \\mod 4;\\\\\n\t\\sqrt{-1}\\varepsilon_i, & \\mbox{if}\\ n\\equiv 2 \\mod 4.\n\t\\end{cases}\n\\end{equation}\nThen $\\epsilon_1,\\dots,\\epsilon_{n+3}$ is an \\emph{orthonormal basis} of $H^n_{\\mathrm{prim}}(X)$.\n\n\n Let $t^{n+1},\\dots,t^{2n+3}$ be the basis of $H^*_{\\mathrm{prim}}(X)^{\\vee}$ dual to $\\epsilon_1,\\dots,\\epsilon_{n+3}$. By the invariant theory of Weyl groups \\cite[\\S 3.12]{Hum90}, \nthe polynomial invariants of $D_{n+3}$ are generated by $s_{1},\\dots,s_{n+3}$, where\n\\begin{equation}\\label{eq-invariantsOf-typeD-1}\n\ts_{i}=\\frac{1}{(2i)!}\\sum_{j=n+1}^{2n+3}(t^j)^{2i},\\ \\mbox{for}\\ 1\\leq i\\leq n+2,\n\\end{equation}\nand\n\\begin{equation}\\label{eq-invariantsOf-typeD-2}\n\ts_{n+3}=\\prod_{j=n+1}^{2n+3}t^j.\n\\end{equation}\nMoreover, $s_1,\\dots,s_{n+3}$ are algebraically independent. As a consequence of the deformation invariance of Gromov-Witten invariants, we have:\n\\begin{theorem}\\label{thm-monodromy-evenDim(2,2)}\nThe genus $g$ generating function $\\mathcal{F}_g$ of $X$ can be written in a unique way as a series of $s_1,\\dots,s_{n+3}$.\n\\end{theorem}\nOne can see this by directly quoting the definition of Gromov-Witten invariants via symplectic geometry. For an algebraic proof using \\cite[Theorem 4.2]{LT98}, see \\cite[Corollary 3.2]{Hu15}.\n\nWe introduce some notations that will be used throughout this paper. We denote the genus 0 generating function by $F$. The basis dual to $1,\\mathsf{h},\\dots,\\mathsf{h}_{n},\\epsilon_1,\\dots,\\epsilon_{n+3}$ is denoted by $t^0,\\dots,t^{2n+3}$. Denote the small quantum multiplication by $\\diamond$. Let \n\\begin{equation}\n\t\\tilde{\\mathsf{h}}_{i}=\t\\underbrace{\\mathsf{h}\\diamond\\dots\\diamond\\mathsf{h}}_{i\\ \\mathsf{h}'s}.\n\\end{equation}\nThe basis dual to $1,\\tilde{\\mathsf{h}},\\dots,\\tilde{\\mathsf{h}}_{n},\\epsilon_1,\\dots,\\epsilon_{n+3}$ is denoted by $\\tau^0,\\dots,\\tau^{n+3}$.\n\n\\begin{remark}\\label{rem:choiceOfBasis}\nThe choice of the isomorphism from a $D_{n+3}$-lattice to $\\big(H^n_{\\mathrm{prim}}(X)\\cap H^n(X;\\mathbb{Z}),(-1)^{\\frac{n}{2}}(.,.)\\big)$, or equivalently, the choice of $\\varepsilon_1,\\dots,\\varepsilon_{n+3}$ in \n$H^n_{\\mathrm{prim}}(X)$ as above, is not unique. Different choices differ by automorphisms of the $D_{n+3}$-lattice.\n\nBy \\cite[Theorem 1]{KM83}, the automorphism group, which we denote by $G_{n+3}$, of a $D_{n+3}$-lattice is generated by $D_{n+3}$ and the automorphism group of the $D_{n+3}$ Dykin diagram. Since $n>0$, the latter group is $\\mathbb{Z}\/2 \\mathbb{Z}$. Then $G_{n+3}$ is the semidirect product $D_{n+3}\\rtimes \\mathbb{Z}\/2 \\mathbb{Z}$, and $G_{n+3}$ is generated by $D_{n+3}$ and the map which fixes $\\alpha_i$ for $1\\leq i\\leq n+1$ and interchanges $\\alpha_{n+2}$ and $\\alpha_{n+3}$. Combined this with the description of the $D_{n+3}$ action on $V$ recalled at the beginning of this section, we can also describe $G_{n+3}$ as generated by $D_{n+3}$ and the map $-1$ which sends $\\varepsilon_i$ to $-\\varepsilon_i$ for $1\\leq i\\leq n+3$.\n\nAs a consequence of Theorem \\ref{thm-monodromy-evenDim(2,2)}, two choices of the basis $\\varepsilon_1,\\dots,\\varepsilon_{n+3}$ which differ by an action by $g\\in D_{n+3}$ do not affect the values of the correlators and also the generating functions $\\mathcal{F}_g$. But two choices which differ by an action by the map $-1$, do affect, for example the value of the correlator\n\\[\n\\langle \\varepsilon_1,\\dots,\\varepsilon_{n+3}\\rangle_{0,n+3,\\frac{n}{2}}.\n\\]\nIn Section \\ref{sec:explictD-Lattice} we will give an explicit choice of the $D_{n+3}$-roots $\\alpha_i$ and the basis $\\varepsilon_i$.\n\\end{remark}\n\n\n\\section{Correlators of length at most 4}\\label{sec:correlators-lengt-atMost4}\nIn this section, we fix a smooth complete intersection $X$ of two quadrics in $\\mathbb{P}^{n+2}$, where $n$ is even and $\\geq 4$. \n\n\\subsection{Recap of known results}\\label{sec:knownResults-correlators}\nBy Theorem \\ref{thm-monodromy-evenDim(2,2)}, we expand $F$ as a series of $s_1,\\dots,s_{n+3}$; in particular we denote the constant term of this expansion by $F^{(0)}$, and the coefficient of $s_1$ by $F^{(1)}$. Then $F^{(0)}$ is the generating function of genus 0 primary GW invariants with only \\emph{ambient} insertions, and $F^{(1)}$ is the generating function of genus 0 primary GW invariants with exactly two primitive insertions. By \\cite[Example 6.11]{Hu15},\n\\begin{equation}\\label{eq-tauTot}\n\\begin{cases}\n\\tau^0=t^0-4t^{n-1},\\\\\n\\tau^1=t^1-12t^{n},\\\\\n\\tau^i=t^i\\ \\mbox{for}\\ i\\geq 2.\n\\end{cases}\n\\end{equation}\nThen the information of the correlators of length 3 and length 4 with at most two primitive insertions are encoded in (see \\cite[Lemma 6.5]{Hu15})\n\\begin{eqnarray}\\label{eq-qp1.5}\n\\frac{\\partial F^{(0)}}{\\partial \\tau^a \\partial\\tau^b \\partial\\tau^c}(0)=\\left\\{\n\\begin{array}{cc}\n16^{\\frac{a+b+c-n}{n-1}} \\times 4, & \\mathrm{if}\\ \\frac{a+b+c-n}{n-1}\\in \\mathbb{Z}_{\\geq 0}; \\\\\n0, & \\mathrm{otherwise}.\n\\end{array}\n\\right.\n\\end{eqnarray}\nand (see \\cite[Example 6.11]{Hu15})\n\\begin{equation}\\label{eq-F122-tau}\n\\mathsf{F}^{(1)}(\\tau)\n=\\tau^0-2\\sum_{\\begin{subarray}{c}1\\leq i,j\\leq n\\\\\ni+j=n\\end{subarray}\n}\n\\tau^i \\tau^{j}\n-16\\tau^{n-1}\\tau^{n}\n+O\\big((\\tau)^3\\big),\n\\end{equation}\nor equivalently\n\\begin{eqnarray}\\label{eq-F^122}\n\\mathsf{F}^{(1)}\n=t^0-4t^{n-1}-2\\sum_{i=1}^{n-1}t^{i}t^{n-i}-16t^{n-1}t^{n}+O((t)^3).\n\\end{eqnarray}\n\nAs in \\cite[Section 6.1]{Hu15}, we define two transition matrices $M$ and $W$ by\n\\begin{eqnarray}\\label{eq-transform1}\n\\mathsf{h}_i=\\sum_{j=0}^n M_{i}^{j}\\tilde{\\mathsf{h}}_{j},\\\n\\tilde{\\mathsf{h}}_i=\\sum_{j=0}^n W_{i}^{j}\\mathsf{h}_{j},\\ \\mbox{for } 0\\leq i\\leq n,\n\\label{eq-transform1-2}\n\\end{eqnarray}\nor equivalently\n\\begin{eqnarray}\\label{eq-transform2}\n\\tau^i=\\sum_{j=0}^n M_j^i t^j,\\\nt^i=\\sum_{j=0}^n W_j^i \\tau^j.\n\\end{eqnarray}\nThen by (\\ref{eq-tauTot}), \n\\begin{equation}\\label{eq-tTotau}\n\\begin{cases}\nt^0=\\tau^0+4 \\tau^{n-1},\\\\\nt^1=\\tau^1+12 \\tau^{n},\\\\\nt^i=\\tau^i\\ \\mbox{for}\\ i\\geq 2.\n\\end{cases}\n\\end{equation}\nLet $\\gamma_0,\\dots,\\gamma_{2n+3}$ be the basis $1,\\mathsf{h},\\dots,\\mathsf{h}_n,\\epsilon_1,\\dots,\\epsilon_{n+3}$. For $0\\leq e,f\\leq 2n+3$, we define\n\\[\ng_{ef}=(\\gamma_e,\\gamma_f),\n\\]\nand define $(g^{ef})$ to be the inverse matrix of $(g_{ef})_{0\\leq e,f\\leq 2n+3}$. Let $\\widetilde{\\gamma}_0,\\dots,\\widetilde{\\gamma}_{2n+3}$ be the basis\n\\[\n1,\\tilde{\\mathsf{h}},\\dots,\\tilde{\\mathsf{h}}_n,\\epsilon_1,\\dots,\\epsilon_{n+3}.\n \\] For $0\\leq e,f\\leq 2n+3,$ we define\n\\[\n\\eta_{ef}=(\\widetilde{\\gamma}_e,\\widetilde{\\gamma}_f),\n\\]\nand define $(\\eta^{ef})$ to be the inverse matrix of $(\\eta_{ef})_{0\\leq e,f\\leq 2n+3}$. Then by \\cite[Lemma 6.2]{Hu15},\n\\begin{equation}\\label{eq-etaInversePairing-even(2,2)}\n\t\\eta^{ef}=\\begin{cases}\n\t-4,& \\mbox{if}\\ e+f=1;\\\\\n\t\\frac{1}{4},& \\mbox{if}\\ e+f=n,\\\\\n\t\\delta_{e,f},& \\mbox{if}\\ n+1\\leq e,f\\leq 2n+3,\\\\\n\t0,& \\mbox{otherwise}.\n\t\\end{cases}\n\\end{equation}\n\n\n\\subsection{Some preparatory computations}\\label{sec:preparatoryComputation-even(2,2)}\nIn this section we compute several genus 0 GW invariants of $X$ that are needed in the proof of Theorem \\ref{thm-4points-fanoIndex-even(2,2)}. We follow the notations in Section \\ref{sec:monodromy-lattice}. For convenience in summations we denote the basis $\\gamma_0,\\dots,\\gamma_{2n+3}$ also by $1,\\mathsf{h},\\dots,\\mathsf{h}_{n},\\epsilon_1,\\dots,\\epsilon_{n+3}$, and use Einstein's summation convention, where the summation is from $0$ to $2n+3$.\n\nIn principle one can do the symmetric reduction of the WDVV equation for $X$. But the computation is quite complicated and we have not completed it. The following computations use the deformation invariance and the monodromy group in the same spirit as the symmetric reduction.\n\n \\begin{lemma}\\label{lem-3point-inv-even(2,2)}\n\\begin{equation}\\label{eq-3point-inv-even(2,2)}\n\t\\langle \\mathsf{h}_{n-1},\\mathsf{h}_{n-1},\\mathsf{h}_n\\rangle_{0,2}=192.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nBy (\\ref{eq-tauTot}),\n\\begin{eqnarray*}\n&& \\langle \\mathsf{h}_{n-1},\\mathsf{h}_{n-1},\\mathsf{h}_n\\rangle_{0,2}=\\frac{\\partial^3 F}{\\partial t^{n-1}\\partial t^{n-1}\\partial t^n}(0)\\\\\n&=& \\sum_{0\\leq i,j,k\\leq n}\\frac{\\partial \\tau^i}{\\partial t^{n-1}}\\frac{\\partial \\tau^j}{\\partial t^{n-1}}\\frac{\\partial \\tau^k}{\\partial t^{n}}\\frac{\\partial^3 F}{\\partial \\tau^i \\partial \\tau^j\\partial \\tau^k}(0)\\\\\n&=&\\Big(\\big(-4\\frac{\\partial}{\\partial \\tau^0}+\\frac{\\partial}{\\partial \\tau^{n-1}}\\big)\n\\big(-4\\frac{\\partial}{\\partial \\tau^0}+\\frac{\\partial}{\\partial \\tau^{n-1}}\\big)\n\\big(-12\\frac{\\partial}{\\partial \\tau^1}+\\frac{\\partial}{\\partial \\tau^{n}}\\big)F\\Big)|_{\\tau=0}.\n\\end{eqnarray*}\nThen from (\\ref{eq-qp1.5}) we get (\\ref{eq-3point-inv-even(2,2)}).\n\\end{proof}\n\n\n\\begin{lemma}\\label{lem-5point-withTwoPrim-inv-even(2,2)}\n\\begin{equation}\\label{eq-5point-withTwoPrim-inv-even(2,2)}\n\t\\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_n,\\mathsf{h}_{n-1},\\mathsf{h}_n\\rangle_{0,3}=-192.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nBy (\\ref{eq-WDVV}) for $1\\leq a\\neq b\\leq n+3$,\n\\begin{eqnarray}\\label{lem-5point-withTwoPrim-inv-even(2,2)-1}\n&& \\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_n,\\mathsf{h}_{n-1},\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_b,\\epsilon_b\\rangle_0\n+\\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_n,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\mathsf{h}_{n-1},\\epsilon_b,\\epsilon_b\\rangle_0\\nonumber\\\\\n&&+\\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_{n-1},\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\mathsf{h}_n,\\epsilon_b,\\epsilon_b\\rangle_0\n+\\langle \\epsilon_a,\\epsilon_a,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\mathsf{h}_n,\\mathsf{h}_{n-1},\\epsilon_b,\\epsilon_b\\rangle_0\\nonumber\\\\\n&=& \\langle \\epsilon_a,\\epsilon_b,\\mathsf{h}_n,\\mathsf{h}_{n-1},\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_a,\\epsilon_b\\rangle_0\n+\\langle \\epsilon_a,\\epsilon_b,\\mathsf{h}_n,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\mathsf{h}_{n-1},\\epsilon_a,\\epsilon_b\\rangle_0\\nonumber\\\\\n&&+\\langle \\epsilon_a,\\epsilon_b,\\mathsf{h}_{n-1},\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\mathsf{h}_n,\\epsilon_a,\\epsilon_b\\rangle_0\n+\\langle \\epsilon_a,\\epsilon_b,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\mathsf{h}_n,\\mathsf{h}_{n-1},\\epsilon_a,\\epsilon_b\\rangle_0.\n\\end{eqnarray}\nBy (\\ref{eq-Dim}), (\\ref{eq-FCA}) and Theorem \\ref{thm-monodromy-evenDim(2,2)}, the LHS of (\\ref{lem-5point-withTwoPrim-inv-even(2,2)-1}) equals\n\\begin{eqnarray}\\label{lem-5point-withTwoPrim-inv-even(2,2)-2}\n&& \\frac{1}{4}\\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_n,\\mathsf{h}_{n-1},\\mathsf{h}_n\\rangle_{0,3} \n+\\frac{1}{4}\\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_n,\\mathsf{h}_{n-1},\\mathsf{h}\\rangle_{0,2} \\langle \\mathsf{h}_{n-1},\\epsilon_b,\\epsilon_b\\rangle_{0,1}\\nonumber\\\\\n&&+\\frac{1}{4}\\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_n,\\mathsf{h}_{n-1}\\rangle_{0,2} \\langle \\mathsf{h},\\mathsf{h}_{n-1},\\epsilon_b,\\epsilon_b\\rangle_{0,1}\n+\\frac{1}{4}\\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_{n-1},\\mathsf{h}\\rangle_{0,1} \\langle \\mathsf{h}_{n-1},\\mathsf{h}_n,\\epsilon_b,\\epsilon_b\\rangle_{0,2}\\nonumber\\\\\n&&+\\frac{1}{4}\\langle \\mathsf{h}_n,\\mathsf{h}_n,\\mathsf{h}_{n-1},\\epsilon_b,\\epsilon_b\\rangle_{0,3}\n+\\frac{1}{4}\\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_{n-1}\\rangle_{0,1}\\langle \\mathsf{h},\\mathsf{h}_n,\\mathsf{h}_{n-1},\\epsilon_b,\\epsilon_b\\rangle_{0,2},\n\\end{eqnarray}\nand the RHS of (\\ref{lem-5point-withTwoPrim-inv-even(2,2)-1}) is 0. \nBy (\\ref{eq-F^122}) and (\\ref{eq-Div}),\n\\begin{equation}\\label{lem-5point-withTwoPrim-inv-even(2,2)-3}\n\t\\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_{n-1},\\mathsf{h}\\rangle_{0,1}=\\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_{n-1}\\rangle_{0,1}=-4,\n\\end{equation}\nand\n\\begin{equation}\\label{lem-5point-withTwoPrim-inv-even(2,2)-4}\n\t\\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_n,\\mathsf{h}_{n-1}\\rangle_{0,2}=-16.\n\\end{equation}\nThen by (\\ref{eq-3point-inv-even(2,2)}), (\\ref{lem-5point-withTwoPrim-inv-even(2,2)-3}) and (\\ref{lem-5point-withTwoPrim-inv-even(2,2)-4}), (\\ref{lem-5point-withTwoPrim-inv-even(2,2)-1}) reads\n\\begin{eqnarray*}\n&& \\frac{1}{4}\\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_n,\\mathsf{h}_{n-1},\\mathsf{h}_n\\rangle_{0,3} \n+\\frac{1}{4}\\times 2\\times (-16)\\times (-4)\n+\\frac{1}{4}\\times (-16)\\times (-4)\\\\\n&&+\\frac{1}{4}\\times (-4)\\times(-16)\n+\\frac{1}{4}\\langle \\mathsf{h}_n,\\mathsf{h}_n,\\mathsf{h}_{n-1},\\epsilon_b,\\epsilon_b\\rangle_{0,3}\n+\\frac{1}{4}\\times(-4)\\times 2\\times(-16)=0.\n\\end{eqnarray*}\nBy Theorem \\ref{thm-monodromy-evenDim(2,2)}, \n\\begin{equation}\n\t\\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_n,\\mathsf{h}_{n-1},\\mathsf{h}_n\\rangle_{0,3}=\\langle \\mathsf{h}_n,\\mathsf{h}_n,\\mathsf{h}_{n-1},\\epsilon_b,\\epsilon_b\\rangle_{0,3}.\n\\end{equation}\nSo we obtain (\\ref{eq-5point-withTwoPrim-inv-even(2,2)}).\n\\end{proof}\n\n\n\\begin{lemma}\\label{lem-5point-inv-even(2,2)-1}\n\\begin{equation}\\label{eq-5point-inv-even(2,2)-1}\n\t\\langle \\epsilon_a,\\epsilon_a,\\epsilon_b,\\epsilon_b,\\mathsf{h}_n\\rangle_{0,2}=4-4\\langle \\epsilon_a,\\epsilon_b,\\epsilon_a,\\epsilon_b\\rangle_{0,1}.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nBy (\\ref{eq-WDVV}), for $1\\leq a\\neq b\\leq n+3$,\n\\begin{eqnarray*}\n&& \\langle \\epsilon_a,\\mathsf{h}_n,\\epsilon_a,\\epsilon_b,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_b,\\mathsf{h}_{n-1}\\rangle_0\n+\\langle \\epsilon_a,\\mathsf{h}_n,\\epsilon_a,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_b,\\epsilon_b,\\mathsf{h}_{n-1}\\rangle_0\\\\\n&&+\\langle \\epsilon_a,\\mathsf{h}_n,\\epsilon_b,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_a,\\epsilon_b,\\mathsf{h}_{n-1}\\rangle_0\n+\\langle \\epsilon_a,\\mathsf{h}_n,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_a,\\epsilon_b,\\epsilon_b,\\mathsf{h}_{n-1}\\rangle_0\\\\\n&=& \\langle \\epsilon_a,\\epsilon_b,\\epsilon_a,\\epsilon_b,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\mathsf{h}_n,\\mathsf{h}_{n-1}\\rangle_0\n+\\langle \\epsilon_a,\\epsilon_b,\\epsilon_a,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_b,\\mathsf{h}_n,\\mathsf{h}_{n-1}\\rangle_0\\\\\n&&+\\langle \\epsilon_a,\\epsilon_b,\\epsilon_b,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_a,\\mathsf{h}_n,\\mathsf{h}_{n-1}\\rangle_0\n+\\langle \\epsilon_a,\\epsilon_b,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_a,\\epsilon_b,\\mathsf{h}_n,\\mathsf{h}_{n-1}\\rangle_0.\n\\end{eqnarray*}\nThen (\\ref{eq-Dim}), (\\ref{eq-FCA}) and Theorem \\ref{thm-monodromy-evenDim(2,2)} yield\n\\begin{eqnarray}\\label{eq-lem-5point-inv-even(2,2)-1-1}\n&& \\langle \\epsilon_a,\\mathsf{h}_n,\\epsilon_a,\\epsilon_b,\\epsilon_b\\rangle_{0,2} \\langle \\epsilon_b,\\epsilon_b,\\mathsf{h}_{n-1}\\rangle_{0,1}\n+\\frac{1}{4}\\langle \\epsilon_a,\\mathsf{h}_n,\\epsilon_a,\\mathsf{h}_{n-1}\\rangle_{0,2} \\langle \\mathsf{h},\\epsilon_b,\\epsilon_b,\\mathsf{h}_{n-1}\\rangle_{0,1}\\nonumber\\\\\n&=&\\frac{1}{4} \\langle \\epsilon_a,\\epsilon_b,\\epsilon_a,\\epsilon_b,\\mathsf{h}\\rangle_{0,1} \\langle \\mathsf{h}_{n-1},\\mathsf{h}_n,\\mathsf{h}_{n-1}\\rangle_{0,2}\n+\\langle \\epsilon_a,\\epsilon_b,\\epsilon_a,\\epsilon_b\\rangle_{0,1} \\langle \\epsilon_b,\\epsilon_b,\\mathsf{h}_n,\\mathsf{h}_{n-1}\\rangle_{0,2}\\nonumber\\\\\n&&+\\langle \\epsilon_a,\\epsilon_b,\\epsilon_b,\\epsilon_a\\rangle_{0,1} \\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_n,\\mathsf{h}_{n-1}\\rangle_{0,2}.\n\\end{eqnarray}\nThen by (\\ref{eq-3point-inv-even(2,2)}), (\\ref{lem-5point-withTwoPrim-inv-even(2,2)-3}) and (\\ref{lem-5point-withTwoPrim-inv-even(2,2)-4}), (\\ref{eq-lem-5point-inv-even(2,2)-1-1}) reads\n\\begin{eqnarray*}\n&& -4\\langle \\epsilon_a,\\mathsf{h}_n,\\epsilon_a,\\epsilon_b,\\epsilon_b\\rangle_{0,2}\n+\\frac{1}{4} (-16)(-4)\\\\\n&=&\\frac{1}{4} \\langle \\epsilon_a,\\epsilon_b,\\epsilon_a,\\epsilon_b\\rangle_{0,1} \\times 192\n-16\\langle \\epsilon_a,\\epsilon_b,\\epsilon_a,\\epsilon_b\\rangle_{0,1} -16\\langle \\epsilon_a,\\epsilon_b,\\epsilon_b,\\epsilon_a\\rangle_{0,1}.\n\\end{eqnarray*}\nSo we get (\\ref{eq-5point-inv-even(2,2)-1}).\n\\end{proof}\n\n\\begin{lemma}\\label{lem-5point-inv-even(2,2)-2}\n\\begin{equation}\\label{eq-5point-inv-even(2,2)-2}\n\t \\langle \\epsilon_a,\\mathsf{h}_n,\\epsilon_a,\\epsilon_a,\\epsilon_a\\rangle_{0,2} \n= 12-4\\langle \\epsilon_a,\\epsilon_a,\\epsilon_a,\\epsilon_a\\rangle_{0,1}.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nBy (\\ref{eq-WDVV}), for $1\\leq a\\leq n+3$,\n\\begin{eqnarray}\\label{eq-lem-5point-inv-even(2,2)-1}\n&& \\langle \\epsilon_a,\\mathsf{h}_n,\\epsilon_a,\\epsilon_a,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_a,\\mathsf{h}_{n-1}\\rangle_0\n+2\\langle \\epsilon_a,\\mathsf{h}_n,\\epsilon_a,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_a,\\epsilon_a,\\mathsf{h}_{n-1}\\rangle_0\\nonumber\\\\\n&&+\\langle \\epsilon_a,\\mathsf{h}_n,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_a,\\epsilon_a,\\epsilon_a,\\mathsf{h}_{n-1}\\rangle_0\\nonumber\\\\\n&=& \\langle \\epsilon_a,\\epsilon_a,\\epsilon_a,\\epsilon_a,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\mathsf{h}_n,\\mathsf{h}_{n-1}\\rangle_0\n+2\\langle \\epsilon_a,\\epsilon_a,\\epsilon_a,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_a,\\mathsf{h}_n,\\mathsf{h}_{n-1}\\rangle_0\\nonumber\\\\\n&&+\\langle \\epsilon_a,\\epsilon_a,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_a,\\epsilon_a,\\mathsf{h}_n,\\mathsf{h}_{n-1}\\rangle_0,\n\\end{eqnarray}\nThen (\\ref{eq-Dim}), (\\ref{eq-FCA}) and Theorem \\ref{thm-monodromy-evenDim(2,2)} yield\n\\begin{eqnarray*}\n&& \\langle \\epsilon_a,\\mathsf{h}_n,\\epsilon_a,\\epsilon_a,\\epsilon_a\\rangle_{0,2} \\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_{n-1}\\rangle_{0,1}\n+2\\cdot\\frac{1}{4}\\langle \\epsilon_a,\\mathsf{h}_n,\\epsilon_a,\\mathsf{h}_{n-1}\\rangle_{0,2} \\langle \\mathsf{h},\\epsilon_a,\\epsilon_a,\\mathsf{h}_{n-1}\\rangle_{0,1}\\\\\n&=&\\frac{1}{4} \\langle \\epsilon_a,\\epsilon_a,\\epsilon_a,\\epsilon_a,\\mathsf{h}\\rangle_{0,1}\n\\langle \\mathsf{h}_{n-1},\\mathsf{h}_n,\\mathsf{h}_{n-1}\\rangle_{0,2}\n+2\\langle \\epsilon_a,\\epsilon_a,\\epsilon_a,\\epsilon_a\\rangle_{0,1} \\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_n,\\mathsf{h}_{n-1}\\rangle_{0,2}\\\\\n&&+\\frac{1}{4}\\langle \\epsilon_a,\\epsilon_a,1\\rangle_{0,0} \\langle \\mathsf{h}_n,\\epsilon_a,\\epsilon_a,\\mathsf{h}_n,\\mathsf{h}_{n-1}\\rangle_{0,3}\n+\\frac{1}{4}\\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_{n-1}\\rangle_{0,1} \\langle \\mathsf{h},\\epsilon_a,\\epsilon_a,\\mathsf{h}_n,\\mathsf{h}_{n-1}\\rangle_{0,2}.\n\\end{eqnarray*}\nBy (\\ref{eq-3point-inv-even(2,2)}), (\\ref{lem-5point-withTwoPrim-inv-even(2,2)-3}), (\\ref{lem-5point-withTwoPrim-inv-even(2,2)-4}), (\\ref{eq-5point-withTwoPrim-inv-even(2,2)}) and (\\ref{eq-Div}), (\\ref{eq-lem-5point-inv-even(2,2)-1}) reads\n\\begin{eqnarray*}\n&& -4\\langle \\epsilon_a,\\mathsf{h}_n,\\epsilon_a,\\epsilon_a,\\epsilon_a\\rangle_{0,2} \n+2\\cdot\\frac{1}{4} (-16)(-4)\\\\\n&=& 16 \\langle \\epsilon_a,\\epsilon_a,\\epsilon_a,\\epsilon_a\\rangle_{0,1}\n+\\frac{1}{4}\\cdot (-192)\n+\\frac{1}{4}\\times (-4)\\times 2\\times (-16),\n\\end{eqnarray*}\nso we obtain (\\ref{eq-5point-inv-even(2,2)-2}).\n\\end{proof}\n\n\n\\subsection{Correlators of length 4}\\label{sec:correlators-length4}\nBy Theorem \\ref{thm-monodromy-evenDim(2,2)} and the results recalled in Section \\ref{sec:knownResults-correlators}, all the correlators of length 4 other than the ones with only primitive insertions are computed. To compute the latter ones, we begin to recall a result on the length 4 correlators with only primitive insertions. Let $X$ be a smooth complete intersection of dimension $n$ and of Fano index $n-1$. Then $X$ is a cubic hypersurface i.e. the multidegree $\\mathbf{d}=3$, or a complete intersection of two quadrics i.e. the multidegree $\\mathbf{d}=(2,2)$. Let $m=\\dim H^*_{\\mathrm{prim}}(X)$, and $\\gamma_{n+1},\\dots,\\gamma_{n+m}$ be a basis of $H^*_{\\mathrm{prim}}(X)$. Then by \\cite[Proposition 9.12]{Hu15}, \n\\begin{equation}\\label{eq-4points-sum}\n\\sum_{e=n+1}^{n+m}\\sum_{f=n+1}^{n+m}\\langle \\gamma_b, \\gamma_e, g^{ef}\\gamma_f,\\gamma_c\\rangle_{0,1}\n=g_{bc}\\cdot\\begin{cases}\n\\frac{(-2)^{n+2}+8}{3},&\n \\mbox{if}\\ \\mathbf{d}=3;\\\\\n (-1)^n(n+1)+2,\n &\n \\mbox{if}\\ \\mathbf{d}=(2,2)\n\\end{cases}\n\\end{equation}\nWhen $X$ is a cubic hypersurface, or $X$ is an odd dimensional complete intersection of two quadrics, the Zariski closure of the monodromy group is the orthogonal group or the symplectic group, according to the parity of the dimension. Then as we have seen in \\cite[Theorem 9.13]{Hu15}, (\\ref{eq-4points-sum}) suffices to give all the length 4 correlators of $X$. For the exceptional case that $X$ is an even dimensional complete intersection of two quadrics, we need some ad hoc computations. \n\\begin{proposition}\\label{prop-4points-fanoIndex-even(2,2)}\nLet $X=X_n(2,2)$ of even dimension $n\\geq 4$. Then\n\\begin{equation}\\label{eq-4points-fanoIndex-even(2,2)-0}\n\t(n+5)\\frac{\\partial^2 F}{(\\partial s_1)^2}(0)+\\frac{\\partial F}{\\partial s_2}(0)\n\t=n+3.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nTake $\\gamma_{n+1},\\dots,\\gamma_{2n+3}$ in \\ref{eq-4points-sum} to be the orthonormal basis $\\epsilon_1,\\dots,\\epsilon_{n+3}$ of $H^*_{\\mathrm{prim}}(X)$. Then\n\\[\n\\sum_{e=n+1}^{n+m}\\sum_{f=n+1}^{n+m}\\langle \\gamma_b, \\gamma_e, g^{ef}\\gamma_f,\\gamma_c\\rangle_{0,1}\n=\\sum_{e=1}^{n+3}\\langle \\epsilon_b, \\epsilon_c,\\epsilon_e,\\epsilon_e\\rangle_{0,1}.\n\\]\nBy (\\ref{eq-invariantsOf-typeD-1}) and the definition (\\ref{eq-def-generatingFunction}) of the generating function $F$, one finds\n\\begin{equation}\\label{eq-prop-4points-fanoIndex-even(2,2)-1}\n\\langle \\epsilon_b,\\epsilon_b,\\epsilon_e,\\epsilon_e\\rangle=\\begin{cases}\n\\frac{\\partial^2 F}{(\\partial s_1)^2}(0),& \\mbox{if}\\ 1\\leq e\\leq n+3\\ \\mbox{and}\\ e\\neq b,\\\\\n3\\frac{\\partial^2 F}{(\\partial s_1)^2}(0)+\\frac{\\partial F}{\\partial s_2}(0), & \\mbox{if}\\ e=b.\n\\end{cases}\n\\end{equation}\nSo\n\\begin{equation}\\label{eq-4points-sum-even(2,2)-1}\n\\sum_{e=n+1}^{n+m}\\sum_{f=n+1}^{n+m}\\langle \\gamma_b, \\gamma_e, g^{ef}\\gamma_f,\\gamma_c\\rangle_{0,1}\n=(n+5)\\frac{\\partial^2 F}{(\\partial s_1)^2}(0) \\delta_{b,c}+\\frac{\\partial F}{\\partial s_2}(0)\\delta_{b,c}.\n\\end{equation}\nOn the other hand by (\\ref{eq-4points-sum}) one finds\n\\begin{equation}\\label{eq-4points-sum-even(2,2)-2}\n\\sum_{e=n+1}^{n+m}\\sum_{f=n+1}^{n+m}\\langle \\gamma_b, \\gamma_e, g^{ef}\\gamma_f,\\gamma_c\\rangle_{0,1}\n=(n+3)\\delta_{b,c}.\n\\end{equation}\nComparing (\\ref{eq-4points-sum-even(2,2)-1}) and (\\ref{eq-4points-sum-even(2,2)-2}) we get (\\ref{eq-4points-fanoIndex-even(2,2)-0}).\n\\end{proof}\n\nUsing (\\ref{eq-WDVV}) and Theorem \\ref{thm-monodromy-evenDim(2,2)}, we can get a quadratic equation for $\\frac{\\partial^2 F}{(\\partial s_1)^2}(0)$ and $\\frac{\\partial F}{\\partial s_2}(0)$. Then we determine the correct solution by the integrality of certain invariants.\n\n\\begin{lemma}\\label{lem-4points-even(2,2)-integrality}\nLet $X=X_n(2,2)$. Let $\\alpha_i\\in H^n_{\\mathrm{prim}}(X)$ be the primitive classes corresponding to simple roots of $D_{n+3}$ as (\\ref{eq-roots-D}), for $1\\leq i\\leq n+3$. Then\n\\begin{equation}\\label{eq-4points-even(2,2)-integrality}\n\\langle \\alpha_{i},\\alpha_{j},\\alpha_{k},\\alpha_{l}\\rangle_{0,1}\\in \\mathbb{Z},\\ \\mbox{for}\\ 1\\leq i,j,k,l\\leq n+3.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nBy \\cite[Proposition 13.9]{Lew99}, for a general member $X$ in the family of $n$-dimensional smooth complete intersections of two quadrics, the Fano variety of lines $\\overline{\\mathcal{M}}_{0,0}(X,1)$\nis a smooth scheme of dimension $2n-4$. It follow that $\\overline{\\mathcal{M}}_{0,0}(X,1)$ has the expected dimension. Moreover one directly sees that $\\overline{\\mathcal{M}}_{0,0}(X,1)$ has no stacky point. Since $\\alpha_i\\in H^n(X;\\mathbb{Z})$, we have (\\ref{eq-4points-even(2,2)-integrality}) for such $X$. Then (\\ref{eq-4points-even(2,2)-integrality}) holds for all $n$-dimensional smooth complete intersections of two quadrics, by the deformation invariance. \n\\end{proof}\n\n\\begin{remark}\\label{rem:integreality-symplectic}\nOne can also directly apply the integrality of genus 0 Gromov-Witten invariants of with integral classes as insertions, of semipositive symplectic manifolds (e.g. \\cite[Theorem A]{Ruan96}).\n\\end{remark}\n\n\\begin{theorem}\\label{thm-4points-fanoIndex-even(2,2)}\nLet $X$ be an even dimensional complete intersection of two quadrics in $\\mathbb{P}^{n+2}$, with $n\\geq 4$. Then\n\\begin{equation}\\label{eq-4points-fanoIndex-even(2,2)}\n\t\\frac{\\partial^2 F}{(\\partial s_1)^2}(0)=1,\\ \\frac{\\partial F}{\\partial s_2}(0)=-2.\n\\end{equation}\nEquivalently, for $1\\leq a,b\\leq n+3$,\n\\begin{equation}\\label{eq-4points-fanoIndex-even(2,2)-ab}\n\t\\langle \\epsilon_a,\\epsilon_a,\\epsilon_b,\\epsilon_b\\rangle_{0,1}=1.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nBy (\\ref{eq-WDVV}), for $n+1\\leq a\\neq b\\leq 2n+3$,\n\\begin{eqnarray}\\label{eq-thm-4points-fanoIndex-even(2,2)-1}\n&&\\sum_{e=0}^{2n+3}\\sum_{f=0}^{2n+3}\\big(\\langle \\epsilon_a,\\epsilon_a,\\epsilon_a,\\epsilon_a,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_b,\\epsilon_b\\rangle_0+2\\langle \\epsilon_a,\\epsilon_a,\\epsilon_a,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_a,\\epsilon_b,\\epsilon_b\\rangle_0\n\\nonumber\\\\\n&&+\\langle \\epsilon_a,\\epsilon_a,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_a,\\epsilon_a,\\epsilon_b,\\epsilon_b\\rangle_0\\big)\\nonumber\\\\\n&=&\\sum_{e=0}^{2n+3}\\sum_{f=0}^{2n+3}\\big(2 \\langle \\epsilon_a,\\epsilon_b,\\epsilon_a,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_a,\\epsilon_a,\\epsilon_b\\rangle_0\n+2 \\langle \\epsilon_a,\\epsilon_b,\\epsilon_a,\\epsilon_a,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_a,\\epsilon_b\\rangle_0\\big).\\nonumber\\\\\n\\end{eqnarray}\nBy Theorem \\ref{thm-monodromy-evenDim(2,2)} and (\\ref{eq-prop-4points-fanoIndex-even(2,2)-1}), the RHS of (\\ref{eq-thm-4points-fanoIndex-even(2,2)-1}) equals\n\\begin{eqnarray}\\label{eq-thm-4points-fanoIndex-even(2,2)-2}\n2 \\langle \\epsilon_a,\\epsilon_b,\\epsilon_a,\\epsilon_b\\rangle_{0,1} \\langle \\epsilon_b,\\epsilon_a,\\epsilon_a,\\epsilon_b\\rangle_{0,1}=2\\big(\\frac{\\partial^2 F}{\\partial s_1^2}(0)\n\\big)^2.\n\\end{eqnarray}\nBy (\\ref{eq-Dim}), (\\ref{eq-FCA}) and Theorem \\ref{thm-monodromy-evenDim(2,2)}, the LHS (\\ref{eq-thm-4points-fanoIndex-even(2,2)-1}) equals\n\\begin{eqnarray}\\label{eq-thm-4points-fanoIndex-even(2,2)-3}\n&&\\frac{1}{4}\\langle \\epsilon_a,\\epsilon_a,\\epsilon_a,\\epsilon_a,\\mathsf{h}\\rangle_{0,1} \\langle \\mathsf{h}_{n-1},\\epsilon_b,\\epsilon_b\\rangle_{0,1}\n+\\frac{1}{4}\\langle \\epsilon_a,\\epsilon_a,\\epsilon_a,\\epsilon_a,\\mathsf{h}_n\\rangle_{0,2} \\langle 1,\\epsilon_b,\\epsilon_b\\rangle_{0,0}\\nonumber\\\\\n&&+2\\langle \\epsilon_a,\\epsilon_a,\\epsilon_a,\\epsilon_a\\rangle_{0,1} \\langle \\epsilon_a,\\epsilon_a,\\epsilon_b,\\epsilon_b\\rangle_{0,1}\n+\\frac{1}{4}\\langle \\epsilon_a,\\epsilon_a,1\\rangle_{0,0} \\langle \\mathsf{h}_{n},\\epsilon_a,\\epsilon_a,\\epsilon_b,\\epsilon_b\\rangle_{0,2}\\nonumber\\\\\n&&+\\frac{1}{4}\\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_{n-1}\\rangle_{0,1} \\langle \\mathsf{h},\\epsilon_a,\\epsilon_a,\\epsilon_b,\\epsilon_b\\rangle_{0,1}.\n\\end{eqnarray}\nSince $\\epsilon_a$ and $\\epsilon_b$ are chosen to be orthonormal, we have\n\\begin{equation}\\label{eq-thm-4points-fanoIndex-even(2,2)-4}\n\t1=\\langle 1,\\epsilon_a,\\epsilon_a\\rangle_{0,0}=\\langle 1,\\epsilon_b,\\epsilon_b\\rangle_{0,0}.\n\\end{equation}\nBy (\\ref{eq-F^122}),\n\\begin{equation}\\label{eq-thm-4points-fanoIndex-even(2,2)-5}\n\t\\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_{n-1}\\rangle_{0,1}=\\langle \\epsilon_b,\\epsilon_b,\\mathsf{h}_{n-1}\\rangle_{0,1}=-4.\n\\end{equation}\nBy (\\ref{eq-prop-4points-fanoIndex-even(2,2)-1}),\n\\begin{equation}\\label{eq-thm-4points-fanoIndex-even(2,2)-6}\n\t\\langle \\epsilon_a,\\epsilon_a,\\epsilon_a,\\epsilon_a\\rangle_{0,1}=3\\frac{\\partial^2 F}{\\partial s_1^2}(0)+\\frac{\\partial F}{\\partial s_2}(0).\n\\end{equation}\nUsing (\\ref{eq-Div}), by (\\ref{eq-thm-4points-fanoIndex-even(2,2)-4}), (\\ref{eq-thm-4points-fanoIndex-even(2,2)-5}), (\\ref{eq-thm-4points-fanoIndex-even(2,2)-6}),\n and (\\ref{eq-5point-inv-even(2,2)-1}), (\\ref{eq-5point-inv-even(2,2)-2}) in Section \\ref{sec:preparatoryComputation-even(2,2)}, (\\ref{eq-thm-4points-fanoIndex-even(2,2)-3}) equals\n\\begin{eqnarray*}\n 6\\big(\\frac{\\partial^2 F}{\\partial s_1^2}(0)\\big)^2+2\\frac{\\partial F}{\\partial s_2}(0)\\frac{\\partial^2 F}{\\partial s_1^2}(0)-8\\frac{\\partial^2 F}{\\partial s_1^2}(0)\n-2 \\frac{\\partial F}{\\partial s_2}(0)+4.\n\\end{eqnarray*}\nSo (\\ref{eq-thm-4points-fanoIndex-even(2,2)-1}) yields\n\\begin{equation}\\label{eq-thm-4points-fanoIndex-even(2,2)-7}\n\t4\\big(\\frac{\\partial^2 F}{\\partial s_1^2}(0)\\big)^2+2\\frac{\\partial F}{\\partial s_2}(0)\\frac{\\partial^2 F}{\\partial s_1^2}(0)-8\\frac{\\partial^2 F}{\\partial s_1^2}(0)\n-2 \\frac{\\partial F}{\\partial s_2}(0)+4=0.\n\\end{equation}\nSubstituting (\\ref{eq-4points-fanoIndex-even(2,2)-0}) into (\\ref{eq-thm-4points-fanoIndex-even(2,2)-7}),\nwe obtain\n\\begin{equation*}\n\t\\big((n+3)\\frac{\\partial^2 F}{\\partial s_1^2}(0)-(n+1)\\big)(\\frac{\\partial^2 F}{\\partial s_1^2}(0)-1)=0.\n\\end{equation*}\nThus\n\\begin{equation*}\n\t\\frac{\\partial^2 F}{\\partial s_1^2}(0)=\\frac{n+1}{n+3}\\ \\mbox{or}\\ 1.\n\\end{equation*}\nRecall the simple roots (\\ref{eq-roots-D}). We have\n\\begin{equation*}\n\\langle \\alpha_{1},\\alpha_{1},\\alpha_{n+3},\\alpha_{n+3}\\rangle_{0,1}=\\frac{\\partial^2 F}{\\partial s_1^2}(0)\\cdot (\\alpha_{1},\\alpha_{1})(\\alpha_{n+3},\\alpha_{n+3})=4 \\frac{\\partial^2 F}{\\partial s_1^2}(0).\n\\end{equation*}\nBut by Lemma \\ref{lem-4points-even(2,2)-integrality}, $\\langle \\alpha_{1},\\alpha_{1},\\alpha_{n+3},\\alpha_{n+3}\\rangle_{0,1}\\in \\mathbb{Z}$. Since $n$ is even, $\\frac{4(n+1)}{n+3}$ is never an integer. Hence\n\\begin{equation}\n\t\\frac{\\partial^2 F}{\\partial s_1^2}(0)=1,\n\\end{equation}\nand by (\\ref{eq-4points-fanoIndex-even(2,2)-0}) we get $\\frac{\\partial F}{\\partial s_2}(0)=-2$.\nThe formula (\\ref{eq-4points-fanoIndex-even(2,2)-ab}) follows then from (\\ref{eq-prop-4points-fanoIndex-even(2,2)-1}).\n\\end{proof}\n\n\n\n\\section{A reconstruction theorem}\\label{sec:reconstructionTheorem}\nIn this section, we fix a smooth complete intersection $X$ of two quadrics in $\\mathbb{P}^{n+2}$, where $n$ is even and $\\geq 4$. \nThe aim of this section is to show that, with the results in Section \\ref{sec:correlators-lengt-atMost4}, besides a special GW invariant, we can compute all genus zero GW invariants of $X$ by (\\ref{eq-Dim}), (\\ref{eq-EulerVectorField}), (\\ref{eq-WDVV}) and the deformation invariance. We begin with an easy observation.\nBy (\\ref{eq-Dim}), a length $k$ genus 0 GW invariant of $X$ with only primitive insertions is zero unless\n\\begin{equation}\n\tk\\cdot \\frac{n}{2}=n-3+k+\\beta\\cdot(n-1),\n\\end{equation}\ni.e.\n\\[\n\\beta=\\beta(k):=\\frac{\\frac{k(n-2)}{2}-n+3}{n-1}.\n\\]\nThis is an integer if and only if $n-1$ divides $k-4$. In particular\n\\begin{equation}\n\t \\beta(4)=1,\\ \\beta(n+3)=\\frac{n}{2},\\ \\beta(2n+2)=n-1.\n\\end{equation}\n\nFor the brevity of expressions, we introduce some notations. For $0\\leq j\\leq 2n+3$, we set \n\\[\n\\partial_{t^j}=\\frac{\\partial}{\\partial t^j}.\n\\]\nFor $I=\\{i_0,i_1,\\dots,i_{2n+3}\\}\\in \\mathbb{Z}_{\\geq 0}^{2n+4}$, we define\n\\[\n\\partial_{t^I}=(\\partial_{t^0})^{i_0}\\circ\\dots\\circ (\\partial_{t^{2n+3}})^{i_{2n+3}}.\n\\]\nFor $0\\leq j\\leq 2n+3$, let $e_j$ be the $(j+1)$-th unit vector in $\\mathbb{Z}_{\\geq 0}^{2n+4}$. So\n\\[\n\\partial_{t^{I+e_j}}=\\partial_{t^I}\\circ \\partial_{t^j}.\n\\]\nWe apply similar notations to the coordinates $\\tau^0,\\dots,\\tau^{2n+3}$. \n\nFor $I=(i_0,\\dots,i_{2n+3})$ in $\\mathbb{Z}_{\\geq 0}^{2n+4}$, we set $|I|=\\sum_{k=0}^{2n+3}i_k$, and $I!=\\prod_{k=0}^{2n+3}i_k!$. For $I=(i_0,\\dots,i_{2n+3})$ and $J=(j_0,\\dots,j_{2n+3})$ in $\\mathbb{Z}_{\\geq 0}^{2n+4}$, we say $I\\leq J$ if and only if $i_k\\leq j_k$ for $0\\leq k\\leq 2n+3$. We denote an zero vector $(0,\\dots,0)$ also by $0$ when no confusion arises. Moreover we define\n\\begin{equation}\\label{eq-binomOfLists}\n\\binom{I}{J}=\\prod_{k=0}^{2n+3} \\binom{i_k}{j_k}.\n\\end{equation}\n\n\nIn the following subsections we adopt Einstein's summation convention, where the range of the indices runs over $0,\\dots,2n+3$.\n\n\\subsection{Elimination of ambient classes}\n By \\cite[Theorem 3.1]{KM94}, a correlator with only ambient insertions can be computed from the length 3 correlators with only ambient correlators and (\\ref{eq-WDVV}). In \\cite[Appendix D]{Hu15} we present an explicit algorithm in the $\\tau$-coordinates for any Fano complete intersections in projective spaces.\n The system of $\\tau$-coordinates has the advantage that the linear recursion of the highest order terms in the WDVV equations is quite simple. The cost is that the expression of the Euler field becomes complicated. For the even dimensional intersections of two quadrics in consideration, by (\\ref{eq-tauTot}), the $\\tau$-coordinates are very close to the $t$-coordinates. Nevertheless we still work in $\\tau$-coordinates in this section. \n We will show how to eliminate an ambient class in any correlator by (\\ref{eq-EulerVectorField}) and (\\ref{eq-WDVV}). \n \n\\begin{lemma}\\label{lem-recursion-EulerVecField-even(2,2)}\nLet $I\\in \\mathbb{Z}_{\\geq 0}^{2n+4}$, and suppose $|I|\\geq 4$. Then\n\\begin{eqnarray}\\label{eq-recursion-EulerVecField-even(2,2)}\n\\partial_{\\tau^1}\\partial_{\\tau^I}F(0)\n&=& \\frac{\\sum_{j=0}^n(j-1)i_j+(\\frac{n}{2}-1)\\sum_{j=n+1}^{2n+3}i_j+3-n}{n-1}\\partial_{\\tau^I}F(0)\\nonumber\\\\\n&&-12i_n\\partial_{\\tau^1}\\partial_{\\tau^{I-e_n}}F(0).\n\\end{eqnarray}\n\\end{lemma}\n\\begin{proof}\nBy (\\ref{eq-tauTot}) and (\\ref{eq-tTotau}), we write the Euler vector field (\\ref{eq-EV-0}) in the $\\tau$-coordinates:\n\\begin{eqnarray}\\label{eq-EulerField-tau-Coordinates}\nE&=& \\sum_{i=0}^{n}(1-i)t^{i}\\frac{\\partial}{\\partial t^i}+\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})t^{i}\\frac{\\partial}{\\partial t^i}+(n-1)\\frac{\\partial}{\\partial t^1}\\nonumber\\\\\n\\begin{comment}\n&=&\\sum_{i=0}^{n}\\sum_{j=0}^n\\sum_{k=0}^n\n(1-i)W_j^i M_{i}^k \\tau^j\\frac{\\partial}{\\partial \\tau^k}\n+\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\frac{\\partial}{\\partial \\tau^i}\n+(n-1)\\frac{\\partial}{\\partial \\tau^1}\\nonumber\\\\\n&=&\\sum_{i=0}^{n}(1-i)\\tau^i\\frac{\\partial}{\\partial \\tau^i}+W_{n-1}^0 \\tau^{n-1}\\frac{\\partial}{\\partial \\tau^{0}}+(2-n)M_{n-1}^0\\tau^{n-1}\\frac{\\partial}{\\partial \\tau^{0}}\n+(1-n)M_n^1 \\tau^n\\frac{\\partial}{\\partial t^1}\n\\nonumber\\\\\n&&+\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\frac{\\partial}{\\partial \\tau^i}\n+(n-1)\\frac{\\partial}{\\partial \\tau^1}\\nonumber\\\\\n&=&\\sum_{i=0}^{n}(1-i)\\tau^i\\frac{\\partial}{\\partial \\tau^i}+4 \\tau^{n-1}\\frac{\\partial}{\\partial \\tau^{0}}-4(2-n)\\tau^{n-1}\\frac{\\partial}{\\partial \\tau^{0}}\n+(1-n)(-12) \\tau^n\\frac{\\partial}{\\partial \\tau^1}\n\\nonumber\\\\\n&&+\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\frac{\\partial}{\\partial \\tau^i}\n+(n-1)\\frac{\\partial}{\\partial \\tau^1}\\nonumber\\\\\n\\end{comment}\n&=&\\sum_{i=0}^{n}(1-i)\\tau^i\\frac{\\partial}{\\partial \\tau^i}+(4n-4) \\tau^{n-1}\\frac{\\partial}{\\partial \\tau^{0}}\n\t+(12n-12) \\tau^n\\frac{\\partial}{\\partial \\tau^1}\\nonumber\\\\\n&&+\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\frac{\\partial}{\\partial \\tau^i}\n+(n-1)\\frac{\\partial}{\\partial \\tau^1}.\n\\end{eqnarray}\nRecall (\\ref{eq-EulerVectorField})\n\\[\nE F=(3-n)F+(n-1)\\partial_{t^1}c.\n\\]\nSince $|I|\\geq 4$, we have\n\\begin{eqnarray*}\n&&(n-1)\\partial_{\\tau^1}\\partial_{\\tau^I}F(0)\\\\\n&=&-\\sum_{j=0}^n(1-j)i_j\\partial_{\\tau^I}F(0)-(12n-12)i_n\\partial_{\\tau^1}\\partial_{\\tau^{I-e_n}}F(0)\\\\\n&&-\\sum_{j=n+1}^{2n+3}(1-\\frac{n}{2})i_j \\partial_{\\tau^I}F(0)+(3-n)\\partial_{\\tau^I}F(0)\\\\\n&=& \\big(-\\sum_{j=0}^n(1-j)i_j-\\sum_{j=n+1}^{2n+3}(1-\\frac{n}{2})i_j+3-n\\big)\\partial_{\\tau^I}F(0)\\\\\n&&-(12n-12)i_n\\partial_{\\tau^1}\\partial_{\\tau^{I-e_n}}F(0).\n\\end{eqnarray*}\nSo we obtain (\\ref{eq-recursion-EulerVecField-even(2,2)}).\n\\end{proof}\n\n\\begin{lemma}\\label{lem-recursion-ambient-simplified-even(2,2)}\nLet $I\\in \\mathbb{Z}^{2n+4}_{\\geq 0}$.\nLet $n+1\\leq a,b\\leq 2n+3$, and $2\\leq i\\leq n$. Then\n\\begin{eqnarray}\\label{eq-recursion-ambient-simplified-even(2,2)}\n&&\\partial_{\\tau^i}\\partial_{\\tau^I}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\\nonumber\\\\\n&=& \n4 \\delta_{a,b}\\partial_{\\tau^1}^2\\partial_{\\tau^{i-1}}\\partial_{\\tau^I}F(0)\t\\nonumber\\\\\n&&-\\frac{1}{4} \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|\\end{subarray}}\\sum_{e=0}^n\n\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{i-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\n\t\\partial_{\\tau^{n-e}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\\nonumber\\\\\n&&- \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|\\end{subarray}}\\sum_{e=n+1}^{2n+3}\n\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{i-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\n\t\\partial_{\\tau^{e}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\\nonumber\\\\\t\n&&+\t\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\n\t\\sum_{e=0}^n\n\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\n\t\\partial_{\\tau^{n-e}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{i-1}}\\partial_{\\tau^b}F(0)\\nonumber\\\\\n&&+\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\n\t\\sum_{e=n+1}^{2n+3}\n\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\n\t\\partial_{\\tau^{e}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{i-1}}\\partial_{\\tau^b}F(0).\t\n\\end{eqnarray}\n\\end{lemma}\n\\begin{proof}\nConsider the WDVV equation \n\\begin{equation}\\label{eq-lem-recursion-ambient-simplified-even(2,2)-WDVV-1}\n\t(\\partial_{\\tau^1}\\partial_{\\tau^{i-1}}\\partial_{\\tau^{e}}F)\\eta^{ef}(\\partial_{\\tau^f}\\partial_{\\tau^{a}}\\partial_{\\tau^{b}}F)\n\t=(\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^{e}}F)\\eta^{ef}(\\partial_{\\tau^f}\\partial_{\\tau^{i-1}}\\partial_{\\tau^{b}}F).\n\\end{equation}\nWe apply the differential operator $\\partial_{\\tau^I}$ to (\\ref{eq-lem-recursion-ambient-simplified-even(2,2)-WDVV-1}), and then take the constant terms of both sides. We obtain\n\\begin{eqnarray}\\label{eq-lem-recursion-ambient-simplified-even(2,2)-WDVV-2}\n &&\\sum_{0\\leq J\\leq I}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{i-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\\nonumber\\\\\n&=&\\sum_{0\\leq J\\leq I}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{i-1}}\\partial_{\\tau^b}F(0).\n\\end{eqnarray}\n Note that \n\\[\n\\partial_{\\tau^j}\\partial_{\\tau^{k}}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\\]\nis the structure constant of the small quantum multiplication. By definition of $\\tilde{\\mathsf{h}}_{i}$ we have\n\\[\n\\tilde{\\mathsf{h}}_{1}\\diamond \\tilde{\\mathsf{h}}_{i-1}=\\tilde{\\mathsf{h}}_{i},\n\\]\nand thus\n\\[\n\\partial_{\\tau^1}\\partial_{\\tau^{i-1}}\\partial_{\\tau^e}F(0)\\eta^{ef}=\\delta_{i,f}.\n\\]\nSo the LHS of (\\ref{eq-lem-recursion-ambient-simplified-even(2,2)-WDVV-2}) equals\n\\begin{eqnarray}\\label{eq-lem-recursion-ambient-simplified-even(2,2)-WDVV-3}\n&& \\partial_{\\tau^1}\\partial_{\\tau^{i-1}}\\partial_{\\tau^e}F(0)\\eta^{ef}\\partial_{\\tau^f}\\partial_{\\tau^I}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\\nonumber\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{i-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\\nonumber\\\\\n&=& \\partial_{\\tau^i}\\partial_{\\tau^I}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\n\t+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{i-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0).\n\\end{eqnarray}\nThe RHS of (\\ref{eq-lem-recursion-ambient-simplified-even(2,2)-WDVV-2}) equals\n\\begin{eqnarray*}\n&& \\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^e}F(0)\\eta^{ef}\\partial_{\\tau^f}\\partial_{\\tau^I}\\partial_{\\tau^b}\\partial_{\\tau^{i-1}}F(0)\n\t+\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^I}\\partial_{\\tau^e}F(0)\\eta^{ef}\\partial_{\\tau^f}\\partial_{\\tau^b}\\partial_{\\tau^{i-1}}F(0)\\\\\n&& +\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{i-1}}\\partial_{\\tau^b}F(0)\\\\\n&\\stackrel{\\mbox{\\footnotesize{by} }(\\ref{eq-etaInversePairing-even(2,2)})}{=}&\n\\partial_{\\tau^1}\\partial_{\\tau^{a}}^2F(0)\\partial_{\\tau^I}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^{i-1}}F(0)\n\t+\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^b}\\partial_{\\tau^I}F(0)\\partial_{\\tau^b}^2\\partial_{\\tau^{i-1}}F(0)\\\\\n&& +\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{i-1}}\\partial_{\\tau^b}F(0).\n\\end{eqnarray*}\nBy (\\ref{eq-F122-tau}),\n\\[\n\\partial_{\\tau^1}\\partial_{\\tau^{a}}^2F(0)=0=\\partial_{\\tau^b}^2\\partial_{\\tau^{i-1}}F(0).\n\\]\t\nSo the RHS of (\\ref{eq-lem-recursion-ambient-simplified-even(2,2)-WDVV-2}) equals\n\\begin{equation}\\label{eq-lem-recursion-ambient-simplified-even(2,2)-WDVV-4}\n\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{i-1}}\\partial_{\\tau^b}F(0).\n\\end{equation}\nPutting together (\\ref{eq-lem-recursion-ambient-simplified-even(2,2)-WDVV-2}), (\\ref{eq-lem-recursion-ambient-simplified-even(2,2)-WDVV-3}) and (\\ref{eq-lem-recursion-ambient-simplified-even(2,2)-WDVV-4}) yields\n\\begin{eqnarray}\\label{eq-recursion-ambient-even(2,2)}\n\\partial_{\\tau^i}\\partial_{\\tau^I}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\n&=&- \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{i-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\\nonumber\\\\\n&&+\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{i-1}}\\partial_{\\tau^b}F(0).\n\\end{eqnarray}\nFinally applying (\\ref{eq-etaInversePairing-even(2,2)}) to the RHS of (\\ref{eq-recursion-ambient-even(2,2)}) we obtain (\\ref{eq-recursion-ambient-simplified-even(2,2)}).\n\\end{proof}\n\n\n\n\n\\subsection{Elimination of primitive classes}\n\\begin{comment}\n\\begin{lemma}\\label{lem-recursion-primitive-aabb-even(2,2)}\nLet $I=(i_{n+1},\\dots,i_{2n+3})$ be a list indicating the number of insertions of $\\epsilon_{n+1},\\dots,\\epsilon_{2n+3}$. Suppose $n+1\\leq a\\leq 2n+3$.\nThen\n\\begin{eqnarray}\\label{eq-recursion-primitive-aabb-even(2,2)-3}\n&&\\partial_{t^n}\\partial_{t^I}\\partial_{t^a}^2F(0)=4|I|\\partial_{t^{a}}^2\\partial_{t^I}F(0)-12\\partial_{t^1}\\partial_{t^{a}}^2\\partial_{t^I}F(0)\\nonumber\\\\\n&&-\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{n-1}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^a}^2F(0) \\nonumber\\\\\n&&+\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{a}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^{n-1}}\\partial_{t^a}F(0).\n\\end{eqnarray}\n\\end{lemma}\n\\begin{proof}\nWe make use of the WDVV equation\n\\begin{equation}\\label{eq-WDVV-recursion-primitive-aabb-even(2,2)-3}\n\t(\\partial_{t^1}\\partial_{t^{n-1}}\\partial_{t^e}F)g^{ef}(\\partial_{t^f}\\partial_{t^a}\\partial_{t^a}F)=(\\partial_{t^1}\\partial_{t^a}\\partial_{t^e}F)g^{ef}(\\partial_{t^f}\\partial_{t^{n-1}}\\partial_{t^a}F).\n\\end{equation}\nThe coefficient of $t^I$ of the LHS of (\\ref{eq-WDVV-recursion-primitive-aabb-even(2,2)-3}), after multiplying $I!$, is\n\\begin{eqnarray*}\n&& \\sum_{0\\leq J\\leq I}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{n-1}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^a}^2F(0)\\\\\n&=& \\partial_{t^1}\\partial_{t^{n-1}}\\partial_{t^e}F(0)g^{ef}\\partial_{t^f}\\partial_{t^I}\\partial_{t^a}^2F(0)\n\t+\\sum_{j=n+1}^{2n+3}i_j \\partial_{t^1}\\partial_{t^{n-1}}\\partial_{t^j}^2F(0)\\partial_{t^I}\\partial_{t^a}^2F(0)\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{n-1}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^a}^2F(0)\\\\\n&=& \\partial_{t^n}\\partial_{t^I}\\partial_{t^a}^2F(0)+8\\partial_{t^1}\\partial_{t^I}\\partial_{t^a}^2F(0)-4|I|\\partial_{t^{a}}^2\\partial_{t^I}F(0)\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{n-1}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^a}^2F(0).\n\\end{eqnarray*}\nThe coefficient of $t^I$ of the RHS of (\\ref{eq-WDVV-recursion-primitive-aabb-even(2,2)-3}), after multiplying $I!$, is\n\\begin{eqnarray*}\n&& \\sum_{0\\leq J\\leq I}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{a}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^{n-1}}\\partial_{t^a}F(0)\\\\\n&=& \\partial_{t^1}\\partial_{t^{a}}\\partial_{t^e}F(0)g^{ef}\\partial_{t^f}\\partial_{t^I}\\partial_{t^a}\\partial_{t^{n-1}}F(0)\n\t+\\partial_{t^1}\\partial_{t^{a}}\\partial_{t^I}\\partial_{t^e}F(0)g^{ef}\\partial_{t^f}\\partial_{t^a}\\partial_{t^{n-1}}F(0)\\\\\n&& +\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{a}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^{n-1}}\\partial_{t^a}F(0)\\\\\n&=& \\partial_{t^1}\\partial_{t^{a}}^2F(0)\\partial_{t^I}\\partial_{t^a}^2\\partial_{t^{n-1}}F(0)\n\t+\\partial_{t^1}\\partial_{t^{a}}^2\\partial_{t^I}F(0)\\partial_{t^a}^2\\partial_{t^{n-1}}F(0)\\\\\n&& +\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{a}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^{n-1}}\\partial_{t^a}F(0)\\\\\n&=& -4 \t\\partial_{t^1}\\partial_{t^{a}}^2\\partial_{t^I}F(0)\t\n\t+\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{a}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^{n-1}}\\partial_{t^a}F(0).\n\\end{eqnarray*}\nSo (\\ref{eq-recursion-primitive-aabb-even(2,2)-3}) follows.\n\\end{proof}\n\\end{comment}\n\n\\begin{comment}\n\\begin{lemma}\\label{lem-recursion-primitive-aabb-even(2,2)}\nLet $I=(i_{n+1},\\dots,i_{2n+3})$ be a list indicating the number of insertions of $\\epsilon_{n+1},\\dots,\\epsilon_{2n+3}$. Suppose $n+1\\leq a\\leq 2n+3$.\nThen\n\\begin{eqnarray}\\label{eq-recursion-primitive-aabb-even(2,2)-3}\n&&\\partial_{\\tau^n}\\partial_{\\tau^I}\\partial_{\\tau^a}^2F(0)=4|I|\\partial_{\\tau^{a}}^2\\partial_{\\tau^I}F(0)\\nonumber\\\\\n&&-\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}^2F(0) \\nonumber\\\\\n&&+\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^a}F(0).\n\\end{eqnarray}\n\\end{lemma}\n\\begin{proof}\nWe make use of the WDVV equation\n\\begin{equation}\\label{eq-WDVV-recursion-primitive-aabb-even(2,2)-3}\n\t(\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^e}F)\\eta^{ef}(\\partial_{\\tau^f}\\partial_{\\tau^a}\\partial_{\\tau^a}F)=(\\partial_{\\tau^1}\\partial_{\\tau^a}\\partial_{\\tau^e}F)\\eta^{ef}(\\partial_{\\tau^f}\\partial_{\\tau^{n-1}}\\partial_{\\tau^a}F).\n\\end{equation}\nThe coefficient of $\\tau^I$ of the LHS of (\\ref{eq-WDVV-recursion-primitive-aabb-even(2,2)-3}), after multiplying $I!$, is\n\\begin{eqnarray*}\n&& \\sum_{0\\leq J\\leq I}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}^2F(0)\\\\\n&\\stackrel{\\mbox{by (\\ref{eq-F122-tau}) and (\\ref{eq-etaInversePairing-even(2,2)})} }{=}& \\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^e}F(0)\\eta^{ef}\\partial_{\\tau^f}\\partial_{\\tau^I}\\partial_{\\tau^a}^2F(0)\n\t+\\sum_{k=n+1}^{2n+3}i_k \\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^k}^2F(0)\\partial_{\\tau^I}\\partial_{\\tau^a}^2F(0)\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}^2F(0)\\\\\n&\\stackrel{\\mbox{by (\\ref{eq-F122-tau})}}{=}& \\partial_{\\tau^n}\\partial_{\\tau^I}\\partial_{\\tau^a}^2F(0)-4|I|\\partial_{\\tau^{a}}^2\\partial_{\\tau^I}F(0)\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}^2F(0).\n\\end{eqnarray*}\nThe coefficient of $\\tau^I$ of the RHS of (\\ref{eq-WDVV-recursion-primitive-aabb-even(2,2)-3}), after multiplying $I!$, is\n\\begin{eqnarray*}\n&& \\sum_{0\\leq J\\leq I}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^a}F(0)\\\\\n&=& \\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^e}F(0)\\eta^{ef}\\partial_{\\tau^f}\\partial_{\\tau^I}\\partial_{\\tau^a}\\partial_{\\tau^{n-1}}F(0)\n\t+\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^I}\\partial_{\\tau^e}F(0)\\eta^{ef}\\partial_{\\tau^f}\\partial_{\\tau^a}\\partial_{\\tau^{n-1}}F(0)\\\\\n&& +\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^a}F(0)\\\\\n&\\stackrel{\\mbox{by (\\ref{eq-F122-tau})}}{=}& \\partial_{\\tau^1}\\partial_{\\tau^{a}}^2F(0)\\partial_{\\tau^I}\\partial_{\\tau^a}^2\\partial_{\\tau^{n-1}}F(0)\n\t+\\partial_{\\tau^1}\\partial_{\\tau^{a}}^2\\partial_{\\tau^I}F(0)\\partial_{\\tau^a}^2\\partial_{\\tau^{n-1}}F(0)\\\\\n&& +\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^a}F(0)\\\\\n&\\stackrel{\\mbox{by (\\ref{eq-F122-tau})}}{=}& \t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^a}F(0).\n\\end{eqnarray*}\nSo (\\ref{eq-recursion-primitive-aabb-even(2,2)-3}) follows.\n\\end{proof}\n\\end{comment}\n\n\\begin{comment}\n\\begin{proposition}\\label{proposition-recursion-primitive-aabb-even(2,2)}\nLet $I=(i_{n+1},\\dots,i_{2n+3})$ be a list indicating the number of insertions of $\\epsilon_{n+1},\\dots,\\epsilon_{2n+3}$. Suppose $n+1\\leq a\\neq b\\leq 2n+3$.\nThen\n\\begin{eqnarray}\\label{eq-recursion-primitive-aabb-even(2,2)}\n&&(\\frac{2|I|-4}{n-1}-2i_b)\\partial_{t^{a}}^2\\partial_{t^I}F(0)\n\t+(\\frac{2|I|-4}{n-1}-2i_a)\\partial_{t^{b}}^2\\partial_{t^I}F(0)\\nonumber\\\\\n&=&\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{n-1}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^a}^2F(0) \\nonumber\\\\\n&&- \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{a}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^{n-1}}\\partial_{t^a}F(0)\\nonumber\\\\\t\n&&+\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{n-1}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^b}^2F(0) \\nonumber\\\\\n&&- \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{b}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^{n-1}}\\partial_{t^b}F(0)\\nonumber\\\\\n&&- \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{t^a}^2\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^b}^2F(0)\\nonumber\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{t^a}\\partial_{t^b}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^a}\\partial_{t^b}F(0).\t\n\\end{eqnarray}\t\n\\end{proposition}\n\\begin{proof}\nWe make use of the WDVV equation\n\\begin{equation}\\label{eq-WDVV-aabb}\n\t(\\partial_{t^a}\\partial_{t^a}\\partial_{t^e}F)g^{ef}(\\partial_{t^f}\\partial_{t^b}\\partial_{t^b}F)=(\\partial_{t^a}\\partial_{t^b}\\partial_{t^e}F)g^{ef}(\\partial_{t^f}\\partial_{t^a}\\partial_{t^b}F).\n\\end{equation}\nThe coefficient of $t^I$ of the LHS of (\\ref{eq-WDVV-aabb}), after multiplying $I!$, is\n\\begin{eqnarray*}\n&& \\sum_{0\\leq J\\leq I}\\binom{I}{J}\\partial_{t^a}^2\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^b}^2F(0)\\\\\n&=& \\partial_{t^a}^2\\partial_{t^I}\\partial_{t^1}F(0)\\frac{1}{4}\\partial_{t^{n-1}}\\partial_{t^b}^2F(0)+\n\t\\partial_{t^a}^2\\partial_{t^I}\\partial_{t^n}F(0)\\frac{1}{4}\\partial_{t^{0}}\\partial_{t^b}^2F(0)\\\\\n&&\t+\\partial_{t^a}^2\\partial_{t^0}F(0)\\frac{1}{4}\\partial_{t^{n}}\\partial_{t^I}\\partial_{t^b}^2 F(0)+\n\t\\partial_{t^a}^2\\partial_{t^{n-1}}F(0)\\frac{1}{4}\\partial_{t^{1}}\\partial_{t^I}\\partial_{t^b}^2F(0)\\\\\n&&+ \\sum_{j=n+1}^{2n+3}\ti_j\\partial_{t^a}^2\\partial_{t^I}F(0)\\partial_{t^{j}}^2\\partial_{t^b}^2F(0)\n\t+ \\sum_{j=n+1}^{2n+3}\ti_j\\partial_{t^a}^2\\partial_{t^{j}}^2F(0)\\partial_{t^I}\\partial_{t^b}^2F(0)\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{t^a}^2\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^b}^2F(0)\t\\\\\n&=& -\\partial_{t^a}^2\\partial_{t^I}\\partial_{t^1}F(0)+\n\t\\frac{1}{4}\\partial_{t^a}^2\\partial_{t^I}\\partial_{t^n}F(0)\n\t+\\frac{1}{4}\\partial_{t^{n}}\\partial_{t^I}\\partial_{t^b}^2 F(0)\n\t-\\partial_{t^{1}}\\partial_{t^I}\\partial_{t^b}^2F(0)\\\\\n&&+ |I|\\partial_{t^a}^2\\partial_{t^I}F(0)\n\t+ |I|\\partial_{t^I}\\partial_{t^b}^2F(0)\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{t^a}^2\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^b}^2F(0).\t\n\\end{eqnarray*}\nThe coefficient of $t^I$ of the RHS of (\\ref{eq-WDVV-aabb}), after multiplying $I!$, is\n\\begin{eqnarray*}\n&& \\sum_{0\\leq J\\leq I}\\binom{I}{J}\\partial_{t^a}\\partial_{t^b}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^a}\\partial_{t^b}F(0)\\\\\n&=& i_a\\partial_{t^a}\\partial_{t^b}\\partial_{t^{I-e_a}}\\partial_{t^b}F(0)\\partial_{t^{b}}\\partial_{t^a}\\partial_{t^a}\\partial_{t^b}F(0)\n\t+i_b\\partial_{t^a}\\partial_{t^b}\\partial_{t^{I-e_b}}\\partial_{t^a}F(0)\\partial_{t^{a}}\\partial_{t^b}\\partial_{t^a}\\partial_{t^b}F(0)\\\\\n&&+ i_a\\partial_{t^a}\\partial_{t^b}\\partial_{t^a}\\partial_{t^b}F(0)\\partial_{t^{b}}\\partial_{t^{I-e_a}}\\partial_{t^a}\\partial_{t^b}F(0)\t\n\t+ i_b\\partial_{t^a}\\partial_{t^b}\\partial_{t^b}\\partial_{t^a}F(0)\\partial_{t^{a}}\\partial_{t^{I-e_b}}\\partial_{t^a}\\partial_{t^b}F(0)\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{t^a}\\partial_{t^b}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^a}\\partial_{t^b}F(0)\\\\\t\n&=& i_a\\partial_{t^{I}}\\partial_{t^b}^2F(0)\n\t+i_b\\partial_{t^{I}}\\partial_{t^a}^2F(0)\n\t+ i_a\\partial_{t^{I}}\\partial_{t^b}^2F(0)\t\n\t+ i_b\\partial_{t^{I}}\\partial_{t^a}^2F(0)\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{t^a}\\partial_{t^b}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^a}\\partial_{t^b}F(0)\\\\\t\n&=& 2 i_a\\partial_{t^{I}}\\partial_{t^b}^2F(0)\n\t+2i_b\\partial_{t^{I}}\\partial_{t^a}^2F(0)\n\t+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{t^a}\\partial_{t^b}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^a}\\partial_{t^b}F(0).\n\\end{eqnarray*}\nSo (\\ref{eq-WDVV-aabb}) yields\n\\begin{eqnarray}\\label{proposition-recursion-primitive-aabb-even(2,2)-1}\n&& -\\partial_{t^a}^2\\partial_{t^I}\\partial_{t^1}F(0)+\n\t\\frac{1}{4}\\partial_{t^a}^2\\partial_{t^I}\\partial_{t^n}F(0)\n\t+\\frac{1}{4}\\partial_{t^{n}}\\partial_{t^I}\\partial_{t^b}^2 F(0)\n\t-\\partial_{t^{1}}\\partial_{t^I}\\partial_{t^b}^2F(0)\\nonumber\\\\\n&&+ |I|\\partial_{t^a}^2\\partial_{t^I}F(0)\n\t+ |I|\\partial_{t^I}\\partial_{t^b}^2F(0)\n\t-2 i_a\\partial_{t^{I}}\\partial_{t^b}^2F(0)\n\t-2i_b\\partial_{t^{I}}\\partial_{t^a}^2F(0)\\nonumber\\\\\n&=&- \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{t^a}^2\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^b}^2F(0)\\nonumber\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{t^a}\\partial_{t^b}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^a}\\partial_{t^b}F(0).\n\\end{eqnarray}\nWe make manipulations on the LHS of (\\ref{proposition-recursion-primitive-aabb-even(2,2)-1}). Since\n\\[\n\\frac{n}{2}\\times (2+|I|)-(n-3+2+|I|)=(\\frac{n}{2}-1)|I|+1,\n\\]\nby (Div) we have\n\\begin{equation}\\label{proposition-recursion-primitive-aabb-even(2,2)-2}\n\\partial_{t^a}^2\\partial_{t^I}\\partial_{t^1}F(0)\n=\\frac{(\\frac{n}{2}-1)|I|+1}{n-1}\\partial_{t^a}^2\\partial_{t^I}F(0).\n\\end{equation}\nSo by (\\ref{proposition-recursion-primitive-aabb-even(2,2)-2}) and (\\ref{eq-recursion-primitive-aabb-even(2,2)-3}) we get\n\\begin{eqnarray}\\label{proposition-recursion-primitive-aabb-even(2,2)-4}\n&& -\\partial_{t^a}^2\\partial_{t^I}\\partial_{t^1}F(0)+\n\t\\frac{1}{4}\\partial_{t^a}^2\\partial_{t^I}\\partial_{t^n}F(0)\n\t+\\frac{1}{4}\\partial_{t^{n}}\\partial_{t^I}\\partial_{t^b}^2 F(0)\n\t-\\partial_{t^{1}}\\partial_{t^I}\\partial_{t^b}^2F(0)\\nonumber\\\\\n&&+ |I|\\partial_{t^a}^2\\partial_{t^I}F(0)\n\t+ |I|\\partial_{t^I}\\partial_{t^b}^2F(0)\n\t-2 i_a\\partial_{t^{I}}\\partial_{t^b}^2F(0)\n\t-2i_b\\partial_{t^{I}}\\partial_{t^a}^2F(0)\\nonumber\\\\\n&=&\t-\\partial_{t^a}^2\\partial_{t^I}\\partial_{t^1}F(0)+\n\t|I|\\partial_{t^{a}}^2\\partial_{t^I}F(0)-3\\partial_{t^1}\\partial_{t^{a}}^2\\partial_{t^I}F(0)\\nonumber\\\\\n&&-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{n-1}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^a}^2F(0) \\nonumber\\\\\n&&+ \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{a}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^{n-1}}\\partial_{t^a}F(0)\\nonumber\\\\\t\n&&\t+|I|\\partial_{t^{b}}^2\\partial_{t^I}F(0)-3\\partial_{t^1}\\partial_{t^{b}}^2\\partial_{t^I}F(0)\\nonumber\\\\\n&&-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{n-1}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^b}^2F(0) \\nonumber\\\\\n&&+ \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{b}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^{n-1}}\\partial_{t^b}F(0)\\nonumber\\\\\n&&\t-\\partial_{t^{1}}\\partial_{t^I}\\partial_{t^b}^2F(0)\n\t+ |I|\\partial_{t^a}^2\\partial_{t^I}F(0)\n\t+ |I|\\partial_{t^I}\\partial_{t^b}^2F(0)\n\t-2 i_a\\partial_{t^{I}}\\partial_{t^b}^2F(0)\n\t-2i_b\\partial_{t^{I}}\\partial_{t^a}^2F(0)\\nonumber\\\\\n&=&\t2(|I|-i_b)\\partial_{t^{a}}^2\\partial_{t^I}F(0)-4\\partial_{t^1}\\partial_{t^{a}}^2\\partial_{t^I}F(0)\n\t+2(|I|-i_a)\\partial_{t^{b}}^2\\partial_{t^I}F(0)-4\\partial_{t^1}\\partial_{t^{b}}^2\\partial_{t^I}F(0)\\nonumber\\\\\n&&-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{n-1}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^a}^2F(0) \\nonumber\\\\\n&&+ \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{a}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^{n-1}}\\partial_{t^a}F(0)\\nonumber\\\\\t\n&&-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{n-1}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^b}^2F(0) \\nonumber\\\\\n&&+ \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{b}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^{n-1}}\\partial_{t^b}F(0)\\nonumber\\\\\t\n&=&(2|I|-2i_b-4\\times\\frac{(\\frac{n}{2}-1)|I|+1}{n-1})\\partial_{t^{a}}^2\\partial_{t^I}F(0)\\nonumber\\\\\n&&+(2|I|-2i_a-4\\times\\frac{(\\frac{n}{2}-1)|I|+1}{n-1})\\partial_{t^{b}}^2\\partial_{t^I}F(0)\\nonumber\\\\\n&&-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{n-1}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^a}^2F(0) \\nonumber\\\\\n&&+ \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{a}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^{n-1}}\\partial_{t^a}F(0)\\nonumber\\\\\t\n&&-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{n-1}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^b}^2F(0) \\nonumber\\\\\n&&+ \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{b}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^{n-1}}\\partial_{t^b}F(0)\\nonumber\\\\\t\n&=& (\\frac{2|I|-4}{n-1}-2i_b)\\partial_{t^{a}}^2\\partial_{t^I}F(0)\n\t+(\\frac{2|I|-4}{n-1}-2i_a)\\partial_{t^{b}}^2\\partial_{t^I}F(0)\\nonumber\\\\\n&&-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{n-1}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^a}^2F(0) \\nonumber\\\\\n&&+ \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{a}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^{n-1}}\\partial_{t^a}F(0)\\nonumber\\\\\t\n&&-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{n-1}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^b}^2F(0) \\nonumber\\\\\n&&+ \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{b}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^{n-1}}\\partial_{t^b}F(0).\t\n\\end{eqnarray}\nThen (\\ref{eq-recursion-primitive-aabb-even(2,2)}) follows from (\\ref{proposition-recursion-primitive-aabb-even(2,2)-1}) and (\\ref{proposition-recursion-primitive-aabb-even(2,2)-4}).\n\\end{proof}\n\\end{comment}\n\n\n\\begin{lemma}\\label{lem-recursion-primitive-abcc-even(2,2)}\nLet $I=(i_{n+1},\\dots,i_{2n+3})$ be a list indicating the number of insertions of $\\epsilon_{n+1},\\dots,\\epsilon_{2n+3}$. Suppose $n+1\\leq a,b\\leq 2n+3$.\nThen\n\\begin{eqnarray}\\label{eq-recursion-primitive-abcc-even(2,2)-3}\n&&\\partial_{\\tau^n}\\partial_{\\tau^I}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\n=4|I|\\partial_{\\tau^{a}}\\partial_{\\tau^b}\\partial_{\\tau^I}F(0)\\nonumber\\\\\n&&-\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0) \\nonumber\\\\\n&&+\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^b}F(0).\n\\end{eqnarray}\n\\end{lemma}\n\\begin{proof}\nWe make use of the WDVV equation\n\\begin{equation}\\label{eq-WDVV-recursion-primitive-abcc-even(2,2)-3}\n\t(\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^e}F)\\eta^{ef}(\\partial_{\\tau^f}\\partial_{\\tau^a}\\partial_{\\tau^b}F)=(\\partial_{\\tau^1}\\partial_{\\tau^a}\\partial_{\\tau^e}F)\\eta^{ef}(\\partial_{\\tau^f}\\partial_{\\tau^{n-1}}\\partial_{\\tau^b}F).\n\\end{equation}\nThe coefficient of $\\tau^I$ of the LHS of (\\ref{eq-WDVV-recursion-primitive-abcc-even(2,2)-3}), after multiplying $I!$, is\n\\begin{eqnarray*}\n&& \\sum_{0\\leq J\\leq I}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\\\\\n&\\stackrel{\\mbox{by (\\ref{eq-F122-tau}) and (\\ref{eq-etaInversePairing-even(2,2)})} }{=}& \\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^e}F(0)\\eta^{ef}\\partial_{\\tau^f}\\partial_{\\tau^I}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\n\t+\\sum_{j=n+1}^{2n+3}i_j \\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^j}^2F(0)\\partial_{\\tau^I}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\\\\\n&\\stackrel{\\mbox{by (\\ref{eq-F122-tau}) and $\\tilde{\\mathsf{h}}_1\\diamond \\tilde{\\mathsf{h}}_{n-1}=\\tilde{\\mathsf{h}}_{n}$ }}{=}& \\partial_{\\tau^n}\\partial_{\\tau^I}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)-4|I|\\partial_{\\tau^{a}}\\partial_{\\tau^b}\\partial_{\\tau^I}F(0)\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0).\n\\end{eqnarray*}\nThe coefficient of $\\tau^I$ of the RHS of (\\ref{eq-WDVV-recursion-primitive-abcc-even(2,2)-3}), after multiplying $I!$, is\n\\begin{eqnarray*}\n&& \\sum_{0\\leq J\\leq I}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^b}F(0)\\\\\n&=& \\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^e}F(0)\\eta^{ef}\\partial_{\\tau^f}\\partial_{\\tau^I}\\partial_{\\tau^b}\\partial_{\\tau^{n-1}}F(0)\n\t+\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^I}\\partial_{\\tau^e}F(0)\\eta^{ef}\\partial_{\\tau^f}\\partial_{\\tau^b}\\partial_{\\tau^{n-1}}F(0)\\\\\n&& +\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^b}F(0)\\\\\n&\\stackrel{\\mbox{by (\\ref{eq-F122-tau})}}{=}& \\partial_{\\tau^1}\\partial_{\\tau^{a}}^2F(0)\\partial_{\\tau^I}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^{n-1}}F(0)\n\t+\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^b}\\partial_{\\tau^I}F(0)\\partial_{\\tau^b}^2\\partial_{\\tau^{n-1}}F(0)\\\\\n&& +\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^b}F(0)\\\\\n&\\stackrel{\\mbox{by (\\ref{eq-F122-tau})}}{=}&\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^b}F(0).\n\\end{eqnarray*}\nSo (\\ref{eq-recursion-primitive-abcc-even(2,2)-3}) follows.\n\\end{proof}\n\n\n\\begin{proposition}\\label{proposition-recursion-primitive-aabb-even(2,2)}\nLet $I=(i_{n+1},\\dots,i_{2n+3})$ be a list indicating the number of insertions of $\\epsilon_{n+1},\\dots,\\epsilon_{2n+3}$. Suppose $n+1\\leq a, b\\leq 2n+3$ and $a\\neq b$.\nThen\n\\begin{eqnarray}\\label{eq-recursion-primitive-aabb-even(2,2)}\n&&(\\frac{2|I|-4}{n-1}-2i_b)\\partial_{\\tau^{a}}^2\\partial_{\\tau^I}F(0)\n\t+(\\frac{2|I|-4}{n-1}-2i_a)\\partial_{\\tau^{b}}^2\\partial_{\\tau^I}F(0)\\nonumber\\\\\n&=&\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}^2F(0) \\nonumber\\\\\n&&- \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^a}F(0)\\nonumber\\\\\t\n&&+\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^b}^2F(0) \\nonumber\\\\\n&&- \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{b}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^b}F(0)\\nonumber\\\\\n&&- \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}^2\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^b}^2F(0)\\nonumber\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0).\t\n\\end{eqnarray}\t\n\\end{proposition}\n\\begin{proof}\nWe make use of the WDVV equation\n\\begin{equation}\\label{eq-WDVV-aabb}\n\t(\\partial_{\\tau^a}\\partial_{\\tau^a}\\partial_{\\tau^e}F)\\eta^{ef}(\\partial_{\\tau^f}\\partial_{\\tau^b}\\partial_{\\tau^b}F)=(\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^e}F)\\eta^{ef}(\\partial_{\\tau^f}\\partial_{\\tau^a}\\partial_{\\tau^b}F).\n\\end{equation}\nThe coefficient of $\\tau^I$ of the LHS of (\\ref{eq-WDVV-aabb}), after multiplying $I!$, is\n\\begin{eqnarray*}\n&& \\sum_{0\\leq J\\leq I}\\binom{I}{J}\\partial_{\\tau^a}^2\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^b}^2F(0)\\\\\n&\\stackrel{\\mbox{by (\\ref{eq-F122-tau})}}{=}& \\partial_{\\tau^a}^2\\partial_{\\tau^I}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^b}^2F(0)\n\t+\\partial_{\\tau^a}^2\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I}}\\partial_{\\tau^b}^2F(0)\\\\\n&&+ \\sum_{k=n+1}^{2n+3}\ti_k\\partial_{\\tau^a}^2\\partial_{\\tau^I}F(0)\\partial_{\\tau^{k}}^2\\partial_{\\tau^b}^2F(0)\n\t+ \\sum_{k=n+1}^{2n+3}\ti_k\\partial_{\\tau^a}^2\\partial_{\\tau^{k}}^2F(0)\\partial_{\\tau^I}\\partial_{\\tau^b}^2F(0)\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}^2\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^b}^2F(0)\t\\\\\n&\\stackrel{\\mbox{by (\\ref{eq-F122-tau}), \n\t\t\t\t(\\ref{eq-etaInversePairing-even(2,2)}), \n\t\t\t\tand (\\ref{eq-4points-fanoIndex-even(2,2)-ab})}\n\t\t\t\t}{=}& -4\\partial_{\\tau^a}^2\\partial_{\\tau^I}\\partial_{\\tau^1}F(0)+\n\t\\frac{1}{4}\\partial_{\\tau^a}^2\\partial_{\\tau^I}\\partial_{\\tau^n}F(0)\n\t+\\frac{1}{4}\\partial_{\\tau^{n}}\\partial_{\\tau^I}\\partial_{\\tau^b}^2 F(0)\n\t-4\\partial_{\\tau^{1}}\\partial_{\\tau^I}\\partial_{\\tau^b}^2F(0)\\\\\n&&+ |I|\\partial_{\\tau^a}^2\\partial_{\\tau^I}F(0)\n\t+ |I|\\partial_{\\tau^I}\\partial_{\\tau^b}^2F(0)\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}^2\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^b}^2F(0).\t\n\\end{eqnarray*}\nThe coefficient of $\\tau^I$ of the RHS of (\\ref{eq-WDVV-aabb}), after multiplying $I!$, is\n\\begin{eqnarray*}\n&& \\sum_{0\\leq J\\leq I}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\\\\\n&\\begin{subarray}{c}\\mbox{by (\\ref{eq-F122-tau}) and} \\\\ \\mbox{Theorem \\ref{thm-monodromy-evenDim(2,2)}}\\\\ =\\end{subarray}& i_a\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^{I-e_a}}\\partial_{\\tau^b}F(0)\\partial_{\\tau^{b}}\\partial_{\\tau^a}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\n\t+i_b\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^{I-e_b}}\\partial_{\\tau^a}F(0)\\partial_{\\tau^{a}}\\partial_{\\tau^b}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\\\\\n&&+ i_a\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\\partial_{\\tau^{b}}\\partial_{\\tau^{I-e_a}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\t\n\t+ i_b\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^b}\\partial_{\\tau^a}F(0)\\partial_{\\tau^{a}}\\partial_{\\tau^{I-e_b}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\\\\\t\n&\\stackrel{\\mbox{by (\\ref{eq-4points-fanoIndex-even(2,2)-ab})}}{=}& i_a\\partial_{\\tau^{I}}\\partial_{\\tau^b}^2F(0)\n\t+i_b\\partial_{\\tau^{I}}\\partial_{\\tau^a}^2F(0)\n\t+ i_a\\partial_{\\tau^{I}}\\partial_{\\tau^b}^2F(0)\t\n\t+ i_b\\partial_{\\tau^{I}}\\partial_{\\tau^a}^2F(0)\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\\\\\t\n&=& 2 i_a\\partial_{\\tau^{I}}\\partial_{\\tau^b}^2F(0)\n\t+2i_b\\partial_{\\tau^{I}}\\partial_{\\tau^a}^2F(0)\\\\\n&&\t+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0).\n\\end{eqnarray*}\nSo (\\ref{eq-WDVV-aabb}) yields\n\\begin{eqnarray}\\label{proposition-recursion-primitive-aabb-even(2,2)-1}\n&& -4\\partial_{\\tau^a}^2\\partial_{\\tau^I}\\partial_{\\tau^1}F(0)+\n\t\\frac{1}{4}\\partial_{\\tau^a}^2\\partial_{\\tau^I}\\partial_{\\tau^n}F(0)\n\t+\\frac{1}{4}\\partial_{\\tau^{n}}\\partial_{\\tau^I}\\partial_{\\tau^b}^2 F(0)\n\t-4\\partial_{\\tau^{1}}\\partial_{\\tau^I}\\partial_{\\tau^b}^2F(0)\\nonumber\\\\\n&&+ |I|\\partial_{\\tau^a}^2\\partial_{\\tau^I}F(0)\n\t+ |I|\\partial_{\\tau^I}\\partial_{\\tau^b}^2F(0)\n\t-2 i_a\\partial_{\\tau^{I}}\\partial_{\\tau^b}^2F(0)\n\t-2i_b\\partial_{\\tau^{I}}\\partial_{\\tau^a}^2F(0)\\nonumber\\\\\n&=&- \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}^2\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^b}^2F(0)\\nonumber\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0).\n\\end{eqnarray}\nWe make manipulations on the LHS of (\\ref{proposition-recursion-primitive-aabb-even(2,2)-1}). Since\n\\[\n\\frac{n}{2}\\times (2+|I|)-(n-3+2+|I|)=(\\frac{n}{2}-1)|I|+1,\n\\]\nby (\\ref{eq-tTotau}) and (\\ref{eq-Div}) we have\n\\begin{equation}\\label{proposition-recursion-primitive-aabb-even(2,2)-2}\n\\partial_{\\tau^a}^2\\partial_{\\tau^I}\\partial_{\\tau^1}F(0)\n=\\partial_{t^a}^2\\partial_{t^I}\\partial_{t^1}F(0)\n=\\frac{(\\frac{n}{2}-1)|I|+1}{n-1}\\partial_{\\tau^a}^2\\partial_{\\tau^I}F(0).\n\\end{equation}\nSo from (\\ref{eq-recursion-primitive-abcc-even(2,2)-3}) we obtain\n\\begin{eqnarray}\\label{proposition-recursion-primitive-aabb-even(2,2)-4}\n&& -4\\partial_{\\tau^a}^2\\partial_{\\tau^I}\\partial_{\\tau^1}F(0)+\n\t\\frac{1}{4}\\partial_{\\tau^a}^2\\partial_{\\tau^I}\\partial_{\\tau^n}F(0)\n\t+\\frac{1}{4}\\partial_{\\tau^{n}}\\partial_{\\tau^I}\\partial_{\\tau^b}^2 F(0)\n\t-4\\partial_{\\tau^{1}}\\partial_{\\tau^I}\\partial_{\\tau^b}^2F(0)\\nonumber\\\\\n&&+ |I|\\partial_{\\tau^a}^2\\partial_{\\tau^I}F(0)\n\t+ |I|\\partial_{\\tau^I}\\partial_{\\tau^b}^2F(0)\n\t-2 i_a\\partial_{\\tau^{I}}\\partial_{\\tau^b}^2F(0)\n\t-2i_b\\partial_{\\tau^{I}}\\partial_{\\tau^a}^2F(0)\\nonumber\\\\\n&=&\t-4\\partial_{\\tau^a}^2\\partial_{\\tau^I}\\partial_{\\tau^1}F(0)+\n\t|I|\\partial_{\\tau^{a}}^2\\partial_{\\tau^I}F(0)\\nonumber\\\\\n&&-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}^2F(0) \\nonumber\\\\\n&&+ \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^a}F(0)\\nonumber\\\\\t\n&&\t+|I|\\partial_{\\tau^{b}}^2\\partial_{\\tau^I}F(0)\n-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^b}^2F(0) \\nonumber\\\\\n&&+ \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{b}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^b}F(0)\\nonumber\\\\\n&&\t-4\\partial_{\\tau^{1}}\\partial_{\\tau^I}\\partial_{\\tau^b}^2F(0)\n\t+ |I|\\partial_{\\tau^a}^2\\partial_{\\tau^I}F(0)\n\t+ |I|\\partial_{\\tau^I}\\partial_{\\tau^b}^2F(0)\n\t-2 i_a\\partial_{\\tau^{I}}\\partial_{\\tau^b}^2F(0)\n\t-2i_b\\partial_{\\tau^{I}}\\partial_{\\tau^a}^2F(0)\\nonumber\\\\\n&=&\t2(|I|-i_b)\\partial_{\\tau^{a}}^2\\partial_{\\tau^I}F(0)-4\\partial_{\\tau^1}\\partial_{\\tau^{a}}^2\\partial_{\\tau^I}F(0)\n\t+2(|I|-i_a)\\partial_{\\tau^{b}}^2\\partial_{\\tau^I}F(0)-4\\partial_{\\tau^1}\\partial_{\\tau^{b}}^2\\partial_{\\tau^I}F(0)\\nonumber\\\\\n&&-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}^2F(0) \\nonumber\\\\\n&&+ \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^a}F(0)\\nonumber\\\\\t\n&&-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^b}^2F(0) \\nonumber\\\\\n&&+ \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{b}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^b}F(0)\\nonumber\\\\\t\n&\\stackrel{\\mbox{by (\\ref{proposition-recursion-primitive-aabb-even(2,2)-2})}}{=}&(2|I|-2i_b-4\\times\\frac{(\\frac{n}{2}-1)|I|+1}{n-1})\\partial_{\\tau^{a}}^2\\partial_{\\tau^I}F(0)\\nonumber\\\\\n&&+(2|I|-2i_a-4\\times\\frac{(\\frac{n}{2}-1)|I|+1}{n-1})\\partial_{\\tau^{b}}^2\\partial_{\\tau^I}F(0)\\nonumber\\\\\n&&-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}^2F(0) \\nonumber\\\\\n&&+ \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^a}F(0)\\nonumber\\\\\t\n&&-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^b}^2F(0) \\nonumber\\\\\n&&+ \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{b}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^b}F(0)\\nonumber\\\\\t\n&=& (\\frac{2|I|-4}{n-1}-2i_b)\\partial_{\\tau^{a}}^2\\partial_{\\tau^I}F(0)\n\t+(\\frac{2|I|-4}{n-1}-2i_a)\\partial_{\\tau^{b}}^2\\partial_{\\tau^I}F(0)\\nonumber\\\\\n&&-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}^2F(0) \\nonumber\\\\\n&&+ \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^a}F(0)\\nonumber\\\\\t\n&&-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^b}^2F(0) \\nonumber\\\\\n&&+ \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{b}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^b}F(0).\t\n\\end{eqnarray}\nThen (\\ref{eq-recursion-primitive-aabb-even(2,2)}) follows from (\\ref{proposition-recursion-primitive-aabb-even(2,2)-1}) and (\\ref{proposition-recursion-primitive-aabb-even(2,2)-4}).\n\\end{proof}\n\n\n\n\n\\begin{proposition}\\label{proposition-recursion-primitive-abcc-even(2,2)}\nLet $I=(i_{n+1},\\dots,i_{2n+3})$ be a list indicating the number of insertions of $\\epsilon_{n+1},\\dots,\\epsilon_{2n+3}$. Suppose $n+1\\leq a, b,c\\leq 2n+3$ and $a,b,c$ are pairwise distinct. Then\n\\begin{eqnarray}\\label{eq-recursion-primitive-abcc-even(2,2)}\n &&(\\frac{2|I|-4}{n-1}-2i_c)\\partial_{\\tau^{a}}\\partial_{\\tau^b}\\partial_{\\tau^I}F(0)\\nonumber\\\\\t\n&=&\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0) \\nonumber\\\\\n&&-\\frac{1}{4}\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^b}F(0)\\nonumber\\\\\n&&-\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^c}^2F(0)\\nonumber\\\\\n&& +\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^c}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^b}\\partial_{\\tau^c}F(0).\n\\end{eqnarray}\n\\end{proposition}\n\\begin{proof}\nWe make use of the WDVV equation\n\\begin{equation}\\label{eq-WDVV-abcc}\n\t(\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^e}F)\\eta^{ef}(\\partial_{\\tau^f}\\partial_{\\tau^c}\\partial_{\\tau^c}F)=(\\partial_{\\tau^a}\\partial_{\\tau^c}\\partial_{\\tau^e}F)\\eta^{ef}(\\partial_{\\tau^f}\\partial_{\\tau^b}\\partial_{\\tau^c}F).\n\\end{equation}\nThe coefficient of $\\tau^I$ of the LHS of (\\ref{eq-WDVV-abcc}), after multiplying $I!$, is\n\\begin{eqnarray*}\n&& \\sum_{0\\leq J\\leq I}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^c}^2F(0)\\\\\n&\\begin{subarray}{c}\\mbox{by (\\ref{eq-F122-tau}) and} \\\\ \\mbox{Theorem \\ref{thm-monodromy-evenDim(2,2)}}\\\\ =\\end{subarray}& \\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^I}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^c}^2F(0)\n\t+\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I}}\\partial_{\\tau^c}^2F(0)\\\\\n&&+ \\sum_{k=n+1}^{2n+3}\ti_k\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^I}F(0)\\partial_{\\tau^{k}}^2\\partial_{\\tau^c}^2F(0)\n\t+ i_a\\partial_{\\tau^a}^2\\partial_{\\tau^{b}}^2F(0)\\partial_{\\tau^{I-e_a}}\\partial_{\\tau^b}\\partial_{\\tau^c}^2F(0)\\\\\n&&+ i_b\\partial_{\\tau^a}^2\\partial_{\\tau^{b}}^2F(0)\\partial_{\\tau^{I-e_b}}\\partial_{\\tau^a}\\partial_{\\tau^c}^2F(0)\t\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^c}^2F(0)\t\\\\\t\n&=& \\partial_{t^a}\\partial_{t^b}\\partial_{t^I}\\partial_{t^1}F(0)\\frac{1}{4}\\partial_{t^{n-1}}\\partial_{t^c}^2F(0)+\n\t\\partial_{t^a}\\partial_{t^b}\\partial_{t^I}\\partial_{t^n}F(0)\\frac{1}{4}\\partial_{t^{0}}\\partial_{t^c}^2F(0)\\\\\n&&+ \\sum_{j=n+1}^{2n+3}\ti_j\\partial_{t^a}\\partial_{t^b}\\partial_{t^I}F(0)\\partial_{t^{j}}^2\\partial_{t^c}^2F(0)\n\t+ i_a\\partial_{t^a}^2\\partial_{t^{b}}^2F(0)\\partial_{t^{I-e_a}}\\partial_{t^b}\\partial_{t^c}^2F(0)\\\\\n&&+ i_b\\partial_{t^a}^2\\partial_{t^{b}}^2F(0)\\partial_{t^{I-e_b}}\\partial_{t^a}\\partial_{t^c}^2F(0)\t\\\\\n&& +\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{t^a}\\partial_{t^b}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^c}^2F(0)\\\\\n&\\stackrel{\\mbox{by (\\ref{eq-F122-tau}), \n\t\t\t\t(\\ref{eq-etaInversePairing-even(2,2)}), \n\t\t\t\tand (\\ref{eq-4points-fanoIndex-even(2,2)-ab})}\n\t\t\t\t}{=}& - 4\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^I}\\partial_{\\tau^1}F(0)\n\t+\\frac{1}{4}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^I}\\partial_{\\tau^n}F(0)\\\\\n&&+ |I|\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^I}F(0)\n\t+ i_a\\partial_{\\tau^{I-e_a}}\\partial_{\\tau^b}\\partial_{\\tau^c}^2F(0)\n\t+ i_b\\partial_{\\tau^{I-e_b}}\\partial_{\\tau^a}\\partial_{\\tau^c}^2F(0)\\\\\n&& +\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^c}^2F(0).\t\n\\end{eqnarray*}\nThe coefficient of $\\tau^I$ of the RHS of (\\ref{eq-WDVV-abcc}), after multiplying $I!$, is\n\\begin{eqnarray*}\n&& \\sum_{0\\leq J\\leq I}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^c}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^b}\\partial_{\\tau^c}F(0)\\\\\n&\\begin{subarray}{c}\\mbox{by (\\ref{eq-F122-tau}) and} \\\\ \\mbox{Theorem \\ref{thm-monodromy-evenDim(2,2)}}\\\\ =\\end{subarray}& i_b\\partial_{\\tau^a}\\partial_{\\tau^c}\\partial_{\\tau^{I-e_b}}\\partial_{\\tau^c}F(0)\\partial_{\\tau^{c}}\\partial_{\\tau^b}\\partial_{\\tau^b}\\partial_{\\tau^c}F(0)\n\t+i_c\\partial_{\\tau^a}\\partial_{\\tau^c}\\partial_{\\tau^{I-e_c}}\\partial_{\\tau^b}F(0)\\partial_{\\tau^{b}}\\partial_{\\tau^c}\\partial_{\\tau^b}\\partial_{\\tau^c}F(0)\\\\\n&&+ i_a\\partial_{\\tau^a}\\partial_{\\tau^c}\\partial_{\\tau^a}\\partial_{\\tau^c}F(0)\\partial_{\\tau^{c}}\\partial_{\\tau^{I-e_a}}\\partial_{\\tau^b}\\partial_{\\tau^c}F(0)\t\n\t+ i_c\\partial_{\\tau^a}\\partial_{\\tau^c}\\partial_{\\tau^c}\\partial_{\\tau^a}F(0)\\partial_{\\tau^{a}}\\partial_{\\tau^{I-e_c}}\\partial_{\\tau^b}\\partial_{\\tau^c}F(0)\\\\\n&& +\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^c}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^b}\\partial_{\\tau^c}F(0)\t\\\\\n&\\stackrel{\\mbox{by (\\ref{eq-4points-fanoIndex-even(2,2)-ab})}}{=}& i_b\\partial_{\\tau^a}\\partial_{\\tau^c}^2\\partial_{\\tau^{I-e_b}}F(0)\n\t+2i_c\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^{I}}F(0)\n\t+ i_a\\partial_{\\tau^b}\\partial_{\\tau^{c}}^2\\partial_{\\tau^{I-e_a}}F(0)\\\\\n&& +\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^c}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{t^b}\\partial_{\\tau^c}F(0).\t\n\\end{eqnarray*}\nSo (\\ref{eq-WDVV-abcc}) yields\n\\begin{eqnarray}\\label{proposition-recursion-primitive-abcc-even(2,2)-1}\n&&-4\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^I}\\partial_{\\tau^1}F(0)\n\t+\\frac{1}{4}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^I}\\partial_{\\tau^n}F(0)\n\t+ (|I|-2i_c)\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^I}F(0)\\nonumber\\\\\n&=&-\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^c}^2F(0)\\nonumber\\\\\n&& +\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^c}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^b}\\partial_{\\tau^c}F(0)\n\\end{eqnarray}\nBy (\\ref{eq-tTotau}) and (\\ref{eq-Div}),\n\\begin{equation}\\label{proposition-recursion-primitive-abcc-even(2,2)-2}\n\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^I}\\partial_{\\tau^1}F(0)\n=\\partial_{t^a}\\partial_{t^b}\\partial_{t^I}\\partial_{t^1}F(0)\n=\\frac{(\\frac{n}{2}-1)|I|+1}{n-1}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^I}F(0).\n\\end{equation}\nSo (\\ref{eq-recursion-primitive-abcc-even(2,2)-3}) we obtain\n\\begin{eqnarray}\\label{proposition-recursion-primitive-abcc-even(2,2)-4}\n&&-4\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^I}\\partial_{\\tau^1}F(0)\n\t+\\frac{1}{4}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^I}\\partial_{\\tau^n}F(0)\n\t+ (|I|-2i_c)\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^I}F(0)\\nonumber\\\\\n&=&-4\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^I}\\partial_{\\tau^1}F(0)\n\t+|I|\\partial_{\\tau^{a}}\\partial_{\\tau^b}\\partial_{\\tau^I}F(0)\\nonumber\\\\\n&&-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0) \\nonumber\\\\\n&&+\\frac{1}{4}\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^b}F(0)\\nonumber\\\\\n&& + (|I|-2i_c)\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^I}F(0)\\nonumber\\\\\t\n&\\stackrel{\\mbox{by (\\ref{proposition-recursion-primitive-abcc-even(2,2)-2})}}{=}& (2|I|-2i_c-4\\times\\frac{(\\frac{n}{2}-1)|I|+1}{n-1})\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^I}F(0)\\nonumber\\\\\t\n&&-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0) \\nonumber\\\\\n&&+\\frac{1}{4}\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^b}F(0)\\nonumber\\\\\n&=& (\\frac{2|I|-4}{n-1}-2i_c)\\partial_{\\tau^{a}}\\partial_{\\tau^b}\\partial_{\\tau^I}F(0)\\nonumber\\\\\t\n&&-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0) \\nonumber\\\\\n&&+\\frac{1}{4}\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^b}F(0).\n\\end{eqnarray}\nThen (\\ref{eq-recursion-primitive-abcc-even(2,2)}) follows from (\\ref{proposition-recursion-primitive-abcc-even(2,2)-1}) and (\\ref{proposition-recursion-primitive-abcc-even(2,2)-4}).\n\\end{proof}\n\n\n\n\\subsection{A recursion with an unknown correlator}\n\n\\begin{theorem}\\label{thm-reconstruction-even(2,2)}\nWith the knowledge of the 4-point invariants, all the invariants can be reconstructed from the WDVV, the deformation invariance, and the correlator\n\\begin{equation}\\label{eq-specialLength(n+3)Invariant-even(2,2)}\n\t\\langle \\epsilon_{1},\\dots,\\epsilon_{n+3}\\rangle_{0,n+3,\\frac{n}{2}}.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nBy Theorem \\ref{thm-monodromy-evenDim(2,2)}, a correlator with exactly one primitive insertion vanishes. By (\\ref{eq-FCA}) a correlator of length $\\geq 4$ having an insertion $1$ vanishes. So using Lemma \\ref{lem-recursion-EulerVecField-even(2,2)} and \\ref{lem-recursion-ambient-simplified-even(2,2)}, one can explicitly write a correlator of length $\\geq 5$ with at least one ambient insertion and one primitive insertion as a combination of correlators of smaller lengths.\n\n\nSuppose $I=(i_0,\\dots,i_{2n+3})\\in \\mathbb{Z}_{\\geq 0}^{2n+3}$ and $|I|>4$. What we said above means that if there exists $i_j>0$ for some $j\\leq n$, then $\\partial_{t^I}F(0)$ can be expressed as a combination of correlators with not larger length and less ambient insertions. Repeating this process our task is reduced to compute invariants with only primitive insertions. So we can assume that $I=(0,\\dots,0,i_{n+1},\\dots,i_{2n+3})$, and $|I|>4$. If there exists pairwise distinct $a,b,c$ in $\\{n+1,\\dots,2n+3\\}$ such that \n\\begin{equation}\\label{eq-condition-recursion-even(2,2)}\n\ti_a,i_b>0, \\mbox{and}\\ |I|-4-(n-1)i_c\\neq 0,\n\\end{equation}\nthen using (\\ref{eq-recursion-primitive-abcc-even(2,2)}), $\\partial_{t^I}F(0)$ can be expressed as a combination of invariants with not larger length and less primitive insertions. If there does not exist such $a,b,c$, then one of the following two cases happen:\n\\begin{enumerate}\n \t\\item[(i)] all $i_j$ but one vanish;\n \t\\item[(ii)] all $i_j>0$ and are equal, and $|I|-4-(n-1)i_j=0$ for all $n+1\\leq j\\leq 2n+3$.\n \\end{enumerate} \n In the case (i) one uses (\\ref{eq-recursion-primitive-aabb-even(2,2)}) to reduce to the case that there exists at least two nonvanishing components in $I$. In the case (ii), \n\\[\ni_{j}(n+3)-4=i_{j}(n-1)\n\\]\nfor all $n+1\\leq j\\leq 2n+3$, which implies $i_{n+1}=i_{n+2}=\\dots=i_{2n+3}=1$, and thus corresponds to (\\ref{eq-specialLength(n+3)Invariant-even(2,2)}).\n\\end{proof}\n\nThe following observation will be used in the proof of Theorem \\ref{thm-convergence}.\n\\begin{remark}\\label{rem:recursion-boundOfIndex}\nIn the proof of Theorem \\ref{thm-reconstruction-even(2,2)}, if the cases (i) and (ii) do not happen, we can take $a,b,c$ such that $i_a$ and $i_b$ are the biggest two components in $I$ and $i_c$ is the smallest one in $I$, and thus\n\\begin{equation}\\label{eq-requirementOnic}\n\ti_c\\leq \\frac{|I|}{n+3}. \n\\end{equation}\nIn fact, when $|I|>n+3$, \n\\[\n|I|-4>(n-1)\\cdot \\frac{|I|}{n+3},\n\\]\nso the condition (\\ref{eq-condition-recursion-even(2,2)}) is satisfied. \nWhen $|I|\\leq n+3$, from Theorem \\ref{thm-monodromy-evenDim(2,2)} it follows that the correlator $\\partial_{\\tau^I}F(0)$ has $i_c=0$ unless it is zero or $I=(1,\\dots,1)$. Then the condition (\\ref{eq-condition-recursion-even(2,2)}) is again satisfied. The requirement (\\ref{eq-requirementOnic}) will be used in the proof of Theorem \\ref{thm-convergence}.\n\\end{remark}\n\n\\begin{remark}{}\nWe call (\\ref{eq-specialLength(n+3)Invariant-even(2,2)}) the \\emph{special correlator} of $X$.\nThe proof of Theorem \\ref{thm-reconstruction-even(2,2)} provides an effective algorithm to compute any correlators of $X$, where we regard the special correlator as an indeterminate number. \nWe implement the algorithm as a Macaulay2 package. For further information see Appendix \\ref{sec:algorithm}.\n\\end{remark}\n\n\\subsection{Conjectures on the special correlator}\\label{sec:conjecturesOnSpecialCorrelator}\nWe made some attempts to extract equations on the special correlator of $X$ from the WDVV equation and Theorem \\ref{thm-monodromy-evenDim(2,2)}. All the relations that we found involving the special correlator turn out to be a trivial equation. I guess that this is always true (unfortunately!). More precisely:\n\\begin{conjecture}\\label{conj-specialCorrelator-free}\nSet the special correlator to be an indeterminate $z$. Let $F(t_0,\\dots,t_{2n+3};z)$ be the generating function of primary genus 0 Gromov-Witten invariants of $X$ computed by Algorithm \\ref{algorithm-correlator-even(2,2)} induced by the proof of Theorem \\ref{thm-reconstruction-even(2,2)}. Then $F(t_0,\\dots,t_{2n+3};z)$ satisfies (\\ref{eq-WDVV}) and the conclusion of Theorem \\ref{thm-monodromy-evenDim(2,2)}.\n\\end{conjecture}\nThis means that to compute the special correlator of $X$ one needs to introduce new tools. In the following of this section I try to make a speculation on the value of the special correlators, by making a comparison to the non-exceptional complete intersections of Fano index $n-1$, i.e. the cubic hypersurfaces and the odd dimensional complete intersections of two quadrics as we recalled in the beginning of Section \\ref{sec:correlators-length4}. Recall that for the non-exceptional complete intersections, a correlator of odd length with only primitive insertions vanishes. So to make comparisons we need first find an appropriate correlator of $X$, which has even length, and express it in terms of the special correlator.\n\\begin{conjecture}\\label{conj-unknownCorrelator-Even(2,2)-quadraticEquation}\nFor even $n$ dimensional complete intersections of two quadrics in $\\mathbb{P}^{n+2}$, \n\\begin{equation}\\label{eq-unknownCorrelator-Even(2,2)-quadraticEquation-normalized}\n\\langle \\epsilon_{1},\\epsilon_1,\\dots,\\epsilon_{n+1},\\epsilon_{n+1}\\rangle_{0,2n+2,n-1}=2^{n-3}\\big((\\langle \\epsilon_{1},\\dots,\\epsilon_{n+3}\\rangle_{0,n+3,\\frac{n}{2}})^2-\\frac{1}{4}\\big).\n\\end{equation}\nEquivalently,\n\\begin{equation}\\label{eq-unknownCorrelator-Even(2,2)-quadraticEquation}\n\\langle \\varepsilon_{1},\\varepsilon_1,\\dots,\\varepsilon_{n+1},\\varepsilon_{n+1}\\rangle_{0,2n+2,n-1}=2^{n-3}\\big((\\langle \\varepsilon_{1},\\dots,\\varepsilon_{n+3}\\rangle_{0,n+3,\\frac{n}{2}})^2-\\frac{(-1)^{\\frac{n}{2}}}{4}\\big).\n\\end{equation}\n\\end{conjecture}\nWe verify this conjecture of $n=4,6$ and $8$; see Appendix \\ref{sec:algorithm}. In \\cite[Conjecture 10.26]{Hu15} we conjectured that for cubic hypersurfaces the correlators of the same form as the LHS of (\\ref{eq-unknownCorrelator-Even(2,2)-quadraticEquation-normalized}) (equivalently, the the LHS of (\\ref{eq-unknownCorrelator-Even(2,2)})) vanish. I guess that the same vanishing holds for $X$ when $n\\equiv 0\\mod 4$.\n Note that $\\langle \\varepsilon_{1},\\dots,\\varepsilon_{n+3}\\rangle_{0,n+3,\\frac{n}{2}}\\in \\mathbb{Q}$ and thus the RHS of (\\ref{eq-unknownCorrelator-Even(2,2)-quadraticEquation}) cannot vanish when $n\\equiv 2\\mod 4$.\n This is the reason that we limit the guess to the dimension $n\\equiv 0\\mod 4$, and leads to the following conjecture. \n\nBefore stating the conjecture, we need to recall Remark \\ref{rem:choiceOfBasis} that there are choices of the basis $\\varepsilon_1,\\dots,\\varepsilon_{n+3}$. All the previous statements are independent of such choices, while the following conjecture on the value of the special correlator does depends on. \n\n\\begin{conjecture}\\label{conj-unknownCorrelator-Even(2,2)}\nFor an even $n$ dimensional complete intersection of two quadrics in $\\mathbb{P}^{n+2}$, let $\\varepsilon_1,\\dots,\\varepsilon_{n+3}$ be the basis of $H^n_{\\mathrm{prim}}(X)$ defined in Section \\ref{sec:explictD-Lattice}. Then \n\\begin{equation}\\label{eq-unknownCorrelator-Even(2,2)}\n\t\\langle \\varepsilon_{1},\\dots,\\varepsilon_{n+3}\\rangle_{0,n+3,\\frac{n}{2}}=\\frac{(-1)^{\\frac{n}{2}}}{2},\n\\end{equation}\nand \n\\begin{equation}\n\t\\langle \\varepsilon_{1},\\varepsilon_1,\\dots,\\varepsilon_{n+1},\\varepsilon_{n+1}\\rangle_{0,2n+2,n-1}=\n\t\\begin{cases}\n\t0,& \\mbox{if}\\ n\\equiv 0 \\mod 4,\\\\\n\t2^{n-4}, & \\mbox{if}\\ n\\equiv 2 \\mod 4.\n\t\\end{cases}\n\\end{equation}\n\\end{conjecture}\nWe will prove the $n=4$ case in Section \\ref{sec:EnumerativeGeometry-Even(2,2)}.\nAt this stage we have no further evidence for Conjecture \\ref{conj-unknownCorrelator-Even(2,2)}.\nThe values in (\\ref{eq-unknownCorrelator-Even(2,2)}) is quite speculative, at least when $n\\equiv 2 \\mod 4$. For the reason for our choice of the sign in (\\ref{eq-unknownCorrelator-Even(2,2)}), we refer the reader to Example \\ref{example-f(6)}.\n\n\\section{Convergence of the generating function}\nIn this section, we still fix a smooth complete intersection $X$ of two quadrics in $\\mathbb{P}^{n+2}$, where $n$ is even and $\\geq 4$. \nUsing Algorithm \\ref{algorithm-correlator-even(2,2)} induced by the proof of Theorem \\ref{thm-reconstruction-even(2,2)} we will show that the generating function $F$ has a positive convergence radius. So $F$ is an analytic function rather than only a formal series. The following theorem is a verification of \\cite[Conjecture 1]{Zin14} for $X$.\n\\begin{theorem}\\label{thm-convergence}\nLet $\\gamma_0,\\dots,\\gamma_{n},\\gamma_{n+1},\\dots,\\gamma_{2n+3}$ be a basis of $H^*(X)$. Then there exists a constant $C>0$ such that \n\\begin{equation}\\label{eq-convergence-0}\n\t\\langle \\gamma_{i_1},\\dots,\\gamma_{i_k}\\rangle_{0,k,\\beta}\\leq k! C^{k}\n\\end{equation}\nfor all $k\\geq 0$, and $0\\leq i_1,\\dots,i_k\\leq 2n+3$.\n\\end{theorem}\n\n\n\\begin{lemma}\\label{lem-inequalityOfBinomialOfLists}\nFor any $M\\leq |I|$, \n\\begin{equation}\\label{eq-inequalityOfBinomialOfLists}\n\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ |J|\\leq M\\end{subarray}}\\binom{I}{J}\\leq \\binom{|I|}{M}.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nThis follows from the enumerative meaning of both sides.\n\\end{proof}\n\\begin{lemma}\\label{lem-inequality-1}\nFor $n\\geq 4$,\n\\begin{equation}\\label{eq-inequality-1}\n\t\\sum_{k=2}^{n-2}\\frac{n(n-1)}{k(k-1)(n-k)(n-k-1)}<4.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nWe compute\n\\begin{eqnarray*}\n&&\\sum_{k=2}^{n-2}\\frac{n(n-1)}{k(k-1)(n-k)(n-k-1)}\\\\\n&=& \\sum_{k=2}^{n-2}\\frac{n-1}{(k-1)(n-k)(n-k-1)}+\\sum_{k=2}^{n-2}\\frac{n-1}{k(k-1)(n-k-1)}\\\\\n&=& \\sum_{k=2}^{n-2}\\frac{1}{(n-k)(n-k-1)}+\\sum_{k=2}^{n-2}\\frac{1}{(k-1)(n-k-1)}\\\\\n&&+\\sum_{k=2}^{n-2}\\frac{1}{(k-1)(n-k-1)}+\\sum_{k=2}^{n-2}\\frac{1}{k(k-1)}\\\\\n&=& 2\\sum_{k=2}^{n-2}\\frac{1}{k(k-1)}+2\\sum_{k=2}^{n-2}\\frac{1}{(k-1)(n-k-1)}.\n\\end{eqnarray*}\nThen we estimate the two sums separately. For the first sum,\n\\[\n\\sum_{k=2}^{n-2}\\frac{1}{k(k-1)}=\\sum_{k=2}^{n-2}(\\frac{1}{k-1}-\\frac{1}{k})=1-\\frac{1}{n-2}.\n\\]\nFor the second sum we have\n\\begin{eqnarray*}\n&&\\sum_{k=2}^{n-2}\\frac{1}{(k-1)(n-k-1)}\\\\\n&=& 2\\sum_{k=2}^{\\frac{n}{2}-1}\\frac{1}{(k-1)(n-k-1)}+\\frac{1}{(\\frac{n}{2}-1)^2}\\\\\n&\\leq & 2\\sum_{k=2}^{\\frac{n}{2}-1}\\frac{1}{(k-1)\\frac{n}{2}}+\\frac{1}{(\\frac{n}{2}-1)^2}\\\\\n&\\leq& \\frac{4}{n}\\big(1+\\log(\\frac{n}{2}-2)\\big)+\\frac{1}{(\\frac{n}{2}-1)^2}.\n\\end{eqnarray*}\nIt follows that\n\\begin{eqnarray*}\n\\sum_{k=2}^{n-2}\\frac{n(n-1)}{k(k-1)(n-k)(n-k-1)}\n\\leq 2-\\frac{2}{n-2}+\\frac{8}{n}\\big(1+\\log(\\frac{n}{2}-2)\\big)+\\frac{2}{(\\frac{n}{2}-1)^2}<4.\n\\end{eqnarray*}\n\\end{proof}\n\n\n\\begin{proof}[Proof of Theorem \\ref{thm-convergence}]\nThe statement is independent of the choice of the basis $\\gamma_0,\\dots,\\gamma_{2n+3}$. \nWe take the basis $1,\\tilde{\\mathsf{h}}_1,\\dots,\\tilde{\\mathsf{h}}_n,\\epsilon_1,\\dots,\\epsilon_{n+3}$. Then (\\ref{eq-convergence-0}) is equivalent to the existence of $C>0$ such that\n\\begin{equation}\\label{eq-convergence-01}\n\t|\\partial_{\\tau^I}F(0)|\\leq |I|! C^{|I|}\n\\end{equation}\nfor all $I\\in \\mathbb{Z}_{\\geq 0}^{2n+4}$. \nWithout loss of generality, we can assume that (\\ref{eq-convergence-0}) holds for $k\\leq K$, where $K$ is an arbitrary chosen natural number, and prove (\\ref{eq-convergence-0}) inductively for all $k$. We note that the wanted statement is equivalent to the existence of $C>0$ such that \n\\begin{equation}\\label{eq-convergence-02}\n\t|\\partial_{\\tau^I}F(0)|\\leq (|I|-5)! C^{|I|-5}\n\\end{equation}\n for all $I\\in \\mathbb{Z}_{\\geq 0}^{2n+4}$. By \\cite[Theorem 1]{Zin14}, (\\ref{eq-convergence-0}), equivalently (\\ref{eq-convergence-02}), holds for correlators with only ambient classes; one can also find a simple proof of this fact in \\cite[Remark D.13]{Hu15}. In the following we show (\\ref{eq-convergence-02}) by induction on $|I|$ and the number of primitive insertions. Suppose (\\ref{eq-convergence-02}) holds for $5\\leq |I|\\leq k$. \nBy (\\ref{eq-recursion-EulerVecField-even(2,2)}),\n\\begin{eqnarray*}\n|\\partial_{\\tau^1}\\partial_{\\tau^I}F(0)|\n&=&\\big| \\frac{\\sum_{j=0}^n(j-1)i_j+(\\frac{n}{2}-1)\\sum_{j=n+1}^{2n+3}i_j+3-n}{n-1}\\partial_{\\tau^I}F(0)\\nonumber\\\\\n&&-12i_n\\partial_{\\tau^1}\\partial_{\\tau^{I-e_n}}F(0)\\big|\\\\\n&\\leq & |I|\\cdot (|I|-5)! C^{|I|-5} +12|I|\\cdot (|I|-5)! C^{|I|-5}.\n\\end{eqnarray*}\nThus replacing $C$ by a constant $C>65$ if necessary, we have\n\\begin{eqnarray*}\n|\\partial_{\\tau^1}\\partial_{\\tau^I}F(0)|\\leq (|I|-4)! C^{|I|-4}.\n\\end{eqnarray*}\nIn the following of the proof we use a temporary convention\n\\begin{equation}\n\tk!=1\\ \\mbox{for}\\ k<0.\n\\end{equation}\nThen by (\\ref{eq-recursion-ambient-even(2,2)}) and (\\ref{eq-etaInversePairing-even(2,2)}), for $2\\leq i\\leq n$ and $n+1\\leq a,b\\leq 2n+3$,\n\\begin{eqnarray*}\n&&|\\partial_{\\tau^i}\\partial_{\\tau^I}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)|\\\\\n&\\leq &\\Big|\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{i-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\\Big|\\nonumber\\\\\n&&+\t\\Big|\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{i-1}}\\partial_{\\tau^b}F(0)\\Big|\\\\\n&\\leq & (2n+6)\\times 2\\times\t4\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J} (|J|-2)! C^{|J|-2} (|I|-|J|-2)! C^{|I|-|J|-2}\\\\\n&=& 8(2n+6) (|I|-2)! C^{|I|-4} \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\frac{|J|-2)!(|I|-|J|-2)!}{(|I|-2)!}.\n\\end{eqnarray*}\nBy Lemma \\ref{lem-inequalityOfBinomialOfLists} and Lemma \\ref{lem-inequality-1},\n\\begin{eqnarray*}\n&&\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\frac{|J|-2)!(|I|-|J|-2)!}{(|I|-2)!}\\\\\n&\\leq & \\sum_{M=1}^{|I|}\\binom{|I|}{M}\\frac{(M-2)!(|I|-M-2)!}{(|I|-2)!}\\\\\n&=& 2\\times \\frac{|I|}{|I|-2}+1+\\sum_{M=2}^{|I|-2}\\frac{|I|(|I|-1)}{M(M-1)(|I|-M)(|I|-M-1)}<9.\n\\end{eqnarray*}\nSo if $C^2>72(2n+6)$, we have\n\\begin{eqnarray*}\n&&|\\partial_{\\tau^i}\\partial_{\\tau^I}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)|\\\\\n&<& 72(2n+6) (|I|-2)! C^{|I|-4} <(|I|-2)! C^{|I|-2}.\n\\end{eqnarray*}\n\n\n\n\n\nBy (\\ref{eq-recursion-primitive-abcc-even(2,2)}), \n\\begin{eqnarray*}\n &&\\big|(\\frac{2|I|-4}{n-1}-2i_c)\\partial_{\\tau^{a}}\\partial_{\\tau^b}\\partial_{\\tau^I}F(0)\\big|\\nonumber\\\\\t\n&=&\\Big|\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0) \\nonumber\\\\\n&&-\\frac{1}{4}\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^b}F(0)\\nonumber\\\\\n&&-\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^c}^2F(0)\\nonumber\\\\\n&& +\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^c}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^b}\\partial_{\\tau^c}F(0)\\Big|\\\\\n&\\leq & 4\\times (2n+6)\\times 4 \t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J} (|J|-2)!C^{|J|-2} (|I|-|J|-2)!C^{|I|-|J|-2}\\\\\n&= & 16(2n+6) (|I|-2)! C^{|I|-4}\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J} \\frac{(|J|-2)! (|I|-|J|-2)!}{(|I|-2)!}\\\\\n&<& 144(2n+6) (|I|-2)! C^{|I|-4},\n\\end{eqnarray*}\nBy Remark \\ref{rem:recursion-boundOfIndex}, we can assume\n\\begin{equation}\n\ti_c\\leq \\frac{|I|+2}{n+3}. \n\\end{equation}\nThen \n\\[\n\\frac{2|I|-4}{n-1}-2i_c\\geq \\frac{8(|I|-n-1)}{(n-1)(n+3)},\n\\]\nand thus\n\\begin{eqnarray*}\n &&\\big|\\partial_{\\tau^{a}}\\partial_{\\tau^b}\\partial_{\\tau^I}F(0)\\big|\\nonumber\\\\\t\n&<& 144(2n+6) (|I|-3)! C^{|I|-4}\\cdot (n-1)(n+3) \\frac{|I|-2}{8(|I|-n-1)}.\n\\end{eqnarray*}\nWhen $|I|\\geq 2n+2$, \n\\[\n\\frac{|I|-2}{8(|I|-n-1)}<\\frac{1}{4}.\n\\]\nSo when $|I|\\geq 2n+2$ and \n\\[\nC>\\big(36(2n+6)(n-1)(n+3)\\big)^2,\n\\]\nwe have\n\\begin{equation}\\label{eq-convergence-primitive-abcc}\n\t\\big|\\partial_{\\tau^{a}}\\partial_{\\tau^b}\\partial_{\\tau^I}F(0)\\big|<(|I|-3)! C^{|I|-\\frac{7}{2}}.\n\\end{equation}\n\nBy (\\ref{eq-recursion-primitive-aabb-tau-simplified-even(2,2)}) and (\\ref{eq-convergence-primitive-abcc}), when $i_a=|I|$,\n\\begin{eqnarray*}\n&&(\\frac{2|I|-4}{n-1})\\partial_{\\tau^{a}}^2\\partial_{\\tau^I}F(0)\\nonumber\\\\\n&\\leq &\\Big|(\\frac{2|I|-4}{n-1}-2|I|)\\partial_{\\tau^{b}}^2\\partial_{\\tau^I}F(0)\\Big|\\\\\n&&+\\Big|\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\n\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}^2F(0) \\nonumber\\\\\n&&- \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^a}F(0)\\nonumber\\\\\t\n&&+\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^b}^2F(0) \\nonumber\\\\\n&&- \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{b}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^b}F(0)\\nonumber\\\\\n&&- \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}^2\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^b}^2F(0)\\nonumber\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\\Big|\\\\\n&\\leq & 2|I|\\cdot(|I|-3)! C^{|I|-\\frac{7}{2}}\\\\\n&&\t+ 6\\times(2n+6)\\times 4\\times \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J} (|J|-2)!C^{|J|-2} (|I|-|J|-2)!C^{|I|-|J|-2}\\\\\n&\\leq & 2|I|\\cdot (|I|-3)! C^{|I|-\\frac{7}{2}}\\\\\n&&+ 24(2n+6) (|I|-2)! C^{|I|-4} \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J} \\frac{(|J|-2)! (|I|-|J|-2)!}{(|I|-2)!}\\\\\n&\\leq & (|I|-3)! C^{|I|-3}\\big(\\frac{2|I|}{C^{\\frac{1}{2}}}+\\frac{216(2n+6)(|I|-2)}{C}\\big),\n\\end{eqnarray*}\nso\n\\begin{eqnarray*}\n&&\\partial_{\\tau^{a}}^2\\partial_{\\tau^I}F(0)\n\\leq (|I|-3)! C^{|I|-3}\n\\big(\\frac{|I|(n-1)}{(|I|-2)C^{\\frac{1}{2}}}+\\frac{108(2n+6)(n-1)}{C}\\big)<(|I|-3)! C^{|I|-3}\n\\end{eqnarray*}\nwhen \n\\begin{equation}\\label{eq-convergence-estimateC}\n\\frac{|I|(n-1)}{(|I|-2)C^{\\frac{1}{2}}}+\\frac{108(2n+6)(n-1)}{C}<1.\n\\end{equation}\nWe choose $C$ such that (\\ref{eq-convergence-02}) holds for $|I|<2n+1$, and such that \n\\[\nC>\\max\\{65,\\sqrt{72(2n+6)},\\big(36(2n+6)(n-1)(n+3)\\big)^2\n\\}\n\\]\nand (\\ref{eq-convergence-estimateC}) holds. Then by Algorithm \\ref{algorithm-correlator-even(2,2)} and the above estimates, (\\ref{eq-convergence-02}) holds for all $I$.\n\\end{proof}\n\n\\begin{corollary}\\label{cor-convergence}\nThere exists an open neighborhood of 0 in $\\mathbb{C}^{2n+4}$ on which the generating function $F$ is an analytic function of $t^0,\\dots,t^{2n+3}$.\n\\end{corollary}\n\n\\section{Semisimplicity}\nIn this section, we still fix a smooth complete intersection $X$ of two quadrics in $\\mathbb{P}^{n+2}$, where $n$ is even and $\\geq 4$. By Corollary \\ref{cor-convergence}, there exists an open neighborhood $U$ of $0\\in \\mathbb{C}^{2n+4}$, on which the generating function $F$ defines a Frobenius manifold $\\mathcal{M}_X$. In this section we show\n\\begin{theorem}\\label{thm-semisimplicity}\nThe Frobenius manifold $\\mathcal{M}_X$ is (generically) semisimple.\n\\end{theorem}\n The strategy is to show that at a general point of $U$, the multiplication by the Euler vector field $E$ has only simple eigenvalues. We work in the $\\tau$-coordinates. We use Einstein's summation convention, where the range of the indices runs over $0,\\dots,2n+3$.\n\n\\subsection{Quantum multiplication by the Euler vector field}\nIn $\\tau$-coordinates, using (\\ref{eq-EulerField-tau-Coordinates}), the big quantum multiplication by the Euler vector field $E$ is\n\\begin{eqnarray*}\n E\\star \\partial_{\\tau^j}&=&\\sum_{i=0}^{n}(1-i)\\tau^i(\\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^e}F)\\eta^{ef}\\partial_{\\tau^f}+(4n-4) \\tau^{n-1}\\partial_{\\tau^{j}}\\\\\n&&\t+(12n-12) \\tau^n(\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^e}F)\\eta^{ef}\\partial_{\\tau^f}\n+\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}(\\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^e}F)\\eta^{ef}\\partial_{\\tau^f}\\\\\n&&+(n-1)(\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^e}F)\\eta^{ef}\\partial_{\\tau^f}.\n\\end{eqnarray*}\nDenote by $\\widetilde{\\mathcal{E}}$ the matrix of the big quantum multiplication by $E$ in the basis $\\partial_{\\tau^0},\\dots,\\partial_{\\tau^{2n+3}}$.\nThen by (\\ref{eq-etaInversePairing-even(2,2)}) we get\n\\begin{eqnarray*}\n\\widetilde{\\mathcal{E}}_j^0&=&(4n-4) \\tau^{n-1}\\delta_{j}^0-4\\sum_{i=0}^{n}(1-i)\\tau^i \\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^1}F\n+\\frac{1}{4}\\sum_{i=0}^{n}(1-i)\\tau^i \\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^{n}}F\\\\\n&&\t-48(n-1) \\tau^n\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^1}F\n+3(n-1) \\tau^n\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^n}F\\\\\n&&-4\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^1}F\n+\\frac{1}{4}\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^{n}}F\\\\\n&&-4(n-1)\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^1}F\n+\\frac{1}{4}(n-1)\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^n}F,\n\\end{eqnarray*}\n\\begin{eqnarray*}\n\\widetilde{\\mathcal{E}}_j^1&=&(4n-4) \\tau^{n-1}\\delta_{j}^1-4\\sum_{i=0}^{n}(1-i)\\tau^i \\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^0}F\n+\\frac{1}{4}\\sum_{i=0}^{n}(1-i)\\tau^i \\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^{n-1}}F\\\\\n&&\t-48(n-1) \\tau^n\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^0}F\n+3(n-1) \\tau^n\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^{n-1}}F\\\\\n&&-4\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^0}F\n+\\frac{1}{4}\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^{n-1}}F\\\\\n&&-4(n-1)\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^0}F\n+\\frac{1}{4}(n-1)\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^{n-1}}F,\n\\end{eqnarray*}\nand for $2\\leq k\\leq n$,\n\\begin{eqnarray*}\n\\widetilde{\\mathcal{E}}_j^k&=&(4n-4) \\tau^{n-1}\\delta_{j}^k\n+\\frac{1}{4}\\sum_{i=0}^{n}(1-i)\\tau^i \\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^{n-k}}F\\\\\n&&+3(n-1) \\tau^n\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^{n-k}}F\n+\\frac{1}{4}\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^{n-k}}F\\\\\n&&+\\frac{1}{4}(n-1)\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^{n-k}}F,\n\\end{eqnarray*}\nand for $n+1\\leq k\\leq 2n+3$,\n\\begin{eqnarray*}\n\\widetilde{\\mathcal{E}}_j^k&=&(4n-4) \\tau^{n-1}\\delta_{j}^k\n+\\sum_{i=0}^{n}(1-i)\\tau^i \\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^{k}}F\\\\\n&&+12(n-1) \\tau^n\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^{k}}F\n+\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^{k}}F\\\\\n&&+(n-1)\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^{k}}F.\n\\end{eqnarray*}\n\n\n\n\n\\subsection{2nd order cutoff of the quantum multiplication by Euler vector field}\nTaking the 2nd order cutoff of the matrix $\\widetilde{\\mathcal{E}}$, and taking $\\tau^i=0$ for $0\\leq i\\leq n$, we denote the resulted matrix by $\\mathcal{E}$, i.e.\n\\[\n\\mathcal{E}(\\tau_{n+1},\\dots,\\tau_{2n+3})=\\widetilde{\\mathcal{E}}(0,\\dots,0,\\tau_{n+1},\\dots,\\tau_{2n+3})\n+o(\\tau^2).\n\\]\n Then\n\\begin{eqnarray}\\label{eq-cutoff-EulverVectorField-1}\n\\mathcal{E}_j^0&=&-4\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^1}F\n+\\frac{1}{4}\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^{n}}F\\nonumber\\\\\n&&-4(n-1)\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^1}F\n+\\frac{1}{4}(n-1)\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^n}F\\nonumber\\\\\n&=& \\begin{cases}\n-4(1-\\frac{n}{2})\\cdot (-4)\\sum_{i=n+1}^{2n+3}(\\tau^i)^2\n+\\frac{1}{4}(1-\\frac{n}{2})\\cdot (-64)\\sum_{i=n+1}^{2n+3}(\\tau^i)^2&\\\\\n-4(n-1)\\cdot(-4)\\cdot \\frac{1}{2}\\sum_{i=n+1}^{2n+3}(\\tau^i)^2 &\\\\\n+\\frac{1}{4}(n-1)\\cdot \\big(2\\times (-64)-12\\times (-4)\\big)\\cdot \\frac{1}{2}\\sum_{i=n+1}^{2n+3}(\\tau^i)^2\n,& \\mbox{if}\\ j=n-1,\\\\\n0 & \\mbox{if otherwise},\\\\\n\\end{cases}\\nonumber\\\\\n&=& \\begin{cases}\n-2(n-1)\\sum_{i=n+1}^{2n+3}(\\tau^i)^2,& \\mbox{if}\\ j=n-1,\\\\\n0 & \\mbox{if otherwise}.\n\\end{cases}\n\\end{eqnarray}\n\\begin{eqnarray}\\label{eq-cutoff-EulverVectorField-2}\n\\mathcal{E}_j^1&=&-4\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^0}F\n+\\frac{1}{4}\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^{n-1}}F\\nonumber\\\\\n&&-4(n-1)\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^0}F\n+\\frac{1}{4}(n-1)\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^{n-1}}F\\nonumber\\\\\n&=&\\begin{cases}\nn-1, & \\mbox{if}\\ j=0,\\\\\n\\frac{1}{4}\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\cdot(-4 \\tau^i)\n+\\frac{1}{4}(n-1)\\cdot(-4)\\times \\frac{1}{2}\\sum_{i=n+1}^{2n+3}(\\tau^i)^2,& \\mbox{if}\\ j=1,\\\\\n-16(n-1)+\\frac{1}{4}(n-1)\\times 64,& \\mbox{if}\\ j=n-1,\\\\\n\\frac{1}{4}(1-\\frac{n}{2})\\cdot (-64)\\sum_{i=n+1}^{2n+3}(\\tau^i)^2 &\\\\\n+\\frac{1}{4}(n-1)\\cdot \\big(2\\times (-64)-12\\times (-4)\\big)\\cdot \\frac{1}{2}\\sum_{i=n+1}^{2n+3}(\\tau^i)^2\n& \\mbox{if}\\ j=n,\\\\\n-4(1-\\frac{n}{2})\\tau^j+\\frac{1}{4}(n-1)\\cdot(-4 \\tau^j),&\\mbox{if}\\ n+1\\leq j\\leq 2n+3,\n\\end{cases}\\nonumber\\\\\n&=&\\begin{cases}\nn-1, & \\mbox{if}\\ j=0,\\\\\n-\\frac{1}{2}\\sum_{i=n+1}^{2n+3}(\\tau^{i})^2\n,& \\mbox{if}\\ j=1,\\\\\n0,& \\mbox{if}\\ j=n-1,\\\\\n(-2n-6)\\sum_{i=n+1}^{2n+3}(\\tau^{i})^2,& \\mbox{if}\\ j=n,\\\\\n(n-3)\\tau^j,&\\mbox{if}\\ n+1\\leq j\\leq 2n+3.\n\\end{cases}\n\\end{eqnarray}\nFor $2\\leq k\\leq n-1$,\n\\begin{eqnarray}\\label{eq-cutoff-EulverVectorField-3}\n\\mathcal{E}_j^k&=&\\frac{1}{4}\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^{n-k}}F\n+\\frac{1}{4}(n-1)\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^{n-k}}F\\nonumber\\\\\n&=&\\begin{cases}\nn-1, & \\mbox{if}\\ j=k-1,\\\\\n\\frac{1}{4}\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\cdot(-4 \\tau^i)\n+\\frac{1}{4}(n-1)\\cdot (-4)\\cdot\\frac{1}{2}\\sum_{i=n+1}^{2n+3}(\\tau^{i})^2,& \\mbox{if}\\ j=k,\\\\\n16(n-1), & \\mbox{if}\\ (j,k)=(n,2),\\\\\n0,\\mbox{otherwise},\n\\end{cases}\\nonumber\\\\\n&=&\\begin{cases}\nn-1, & \\mbox{if}\\ j=k-1,\\\\\n-\\frac{1}{2}\\sum_{i=n+1}^{2n+3}(\\tau^{i})^2,& \\mbox{if}\\ j=k,\\\\\n16(n-1), & \\mbox{if}\\ (j,k)=(n,2),\\\\\n0,\\mbox{otherwise}.\n\\end{cases}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\\label{eq-cutoff-EulverVectorField-4}\n\\mathcal{E}_j^n&=&\\frac{1}{4}\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^{0}}F\n+\\frac{1}{4}(n-1)\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^{0}}F\\nonumber\\\\\n&=&\\begin{cases}\nn-1,& \\mbox{if}\\ j=n-1,\\\\\n0, & \\mbox{if}\\ 0\\leq j\\leq n-2\\ \\mbox{of}\\ j=n,\\\\\n\\frac{2-n}{8}\\tau^j,& \\mbox{if}\\ n+1\\leq j\\leq 2n+3.\n\\end{cases}\n\\end{eqnarray}\nFor $n+1\\leq k\\leq 2n+3$,\n\\begin{eqnarray}\\label{eq-cutoff-EulverVectorField-5}\n\\mathcal{E}_j^k&=&\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^{k}}F\n+(n-1)\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^{k}}F\\nonumber\\\\\n&=&\\begin{cases}\n(1-\\frac{n}{2})\\tau^k,& \\mbox{if}\\ j=0,\\\\\n-4(n-1)\\tau^k, & \\mbox{if}\\ j=n-1,\\\\\n0,& \\mbox{if}\\ 1\\leq j\\leq n-2\\ \\mbox{or}\\ j=n,\\\\\n(2-n)\\tau^j \\tau^k+(n-1)\\tau^j \\tau^k, & \\mbox{if}\\ n+1\\leq j\\neq k\\leq 2n+3,\\\\\n(1-\\frac{n}{2})\\sum_{i=n+1}^{2n+3}(\\tau^{i})^2+(n-1)\\cdot\\frac{1}{2}\\sum_{i=n+1}^{2n+3}(\\tau^{i})^2, & \\mbox{if}\\ j=k,\n\\end{cases}\\nonumber\\\\\n&=&\\begin{cases}\n(1-\\frac{n}{2})\\tau^k,& \\mbox{if}\\ j=0,\\\\\n-4(n-1)\\tau^k, & \\mbox{if}\\ j=n-1,\\\\\n0,& \\mbox{if}\\ 0\\leq j\\leq n-2\\ \\mbox{or}\\ j=n,\\\\\n\\tau^j \\tau^k, & \\mbox{if}\\ n+1\\leq j\\neq k\\leq 2n+3,\\\\\n\\frac{1}{2}\\sum_{i=n+1}^{2n+3}(\\tau^{i})^2, & \\mbox{if}\\ j=k.\n\\end{cases}\n\\end{eqnarray}\n\n\\subsection{The characteristic polynomial}\n\n\n\\begin{comment}\n{\\footnotesize\n\\begin{equation*}\n\t\\begin{pmatrix}\n\t\\begin{matrix}\n\t0 & n-1 & 0 & 0 & \\dots & 0 & 0 \\\\\n\t0 & -\\frac{s}{2} & n-1 & 0 & \\dots & 0 & 0 \\\\\n\t0 & 0 & -\\frac{s}{2} & n-1 & \\dots & 0 & 0 \\\\\n\t\\vdots & \\vdots & \\vdots & \\vdots& \\vdots & \\vdots& \\vdots \\\\\n\t0 & 0 & \\dots & 0 & \\dots & n-1 & 0 \\\\\n\t-2(n-1)s & 0 & \\dots & \\dots & \\dots& -\\frac{s}{2} & n-1 \\\\\n\t0 & (-2n-6)s & 16(n-1) & 0 & \\dots& 0 & 0 \n\t\\end{matrix} & \\hspace*{-\\arraycolsep}\\vline\\hspace*{-\\arraycolsep} & \n\t\\begin{matrix}\n\t (1-\\frac{n}{2})\\tau^{n+1} & \\dots & (1-\\frac{n}{2})\\tau^{2n+3} \\\\\n\t 0 & \\dots & 0 \\\\\n\t 0 & \\dots & 0 \\\\\n\t \\vdots& \\vdots& \\vdots\\\\\n\t 0 & \\dots& 0\\\\\n\t (4-4n)\\tau^{n+1} & \\dots & (4-4n)\\tau^{2n+3}\\\\\n\t 0 & \\dots & 0 \n\t \\end{matrix} \\\\\n\t \\hline \n\t \\begin{matrix}\n\t0 & (n-3)\\tau^{n+1} & 0 & 0 & \\dots & 0 & \\frac{2-n}{8}\\tau^{n+1} \\\\\n\t0 & (n-3)\\tau^{n+2} & \\dots & \\dots & \\dots & 0 & \\frac{2-n}{8}\\tau^{n+2} \\\\\n\t\\vdots & \\vdots & \\vdots & \\vdots& \\vdots & \\vdots& \\vdots \\\\\n\t0 & (n-3)\\tau^{2n+3} & 0 & \\dots & \\dots & 0 & \\frac{2-n}{8}\\tau^{2n+3} \n\t\\end{matrix} & \\hspace*{-\\arraycolsep}\\vline\\hspace*{-\\arraycolsep} &\n\t\\begin{matrix}\n\t \\frac{s}{2} & \\dots & \\tau^{n+1}\\tau^{2n+3}\\\\\n\t \\tau^{n+1}\\tau^{n+1} & \\dots & \\tau^{n+2}\\tau^{2n+2} \\\\\n\t\\vdots& \\vdots& \\vdots\\\\\n\t\\tau^{n+1}\\tau^{2n+3} & \\dots & \\frac{s}{2}\n\t\\end{matrix}\n\t\\end{pmatrix}\n\\end{equation*}\n}\n\\end{comment}\n\nLet $\\mathcal{P}_n(z,\\tau^{n+1},\\dots,\\tau^{2n+3})$ be the characteristic polynomial of $\\widetilde{\\mathcal{E}}$ at $(0,\\dots,0,\\tau_{n+1},\\tau_{n+2},\\dots,\\tau_{2n+3})$, and let $P_n(z,\\tau^{n+1},\\dots,\\tau^{2n+3})$ be the characteristic polynomial of $\\mathcal{E}$ at $(\\tau_{n+1},\\tau_{n+2},\\dots,\\tau_{2n+3})$.\nFor brevity of expressions we define (recall (\\ref{eq-invariantsOf-typeD-1}))\n\\[\ns=2s_1=\\sum_{i=n+1}^{2n+3}(\\tau^{i})^2.\n\\]\nDenote by $E_{i,j}$ the elementary matrix whose only nonzero entry is $1$ at the position $(i,j)$.\n\\begin{proposition}\\label{prop-charPoly-EV-2ndOrder}\n\\begin{eqnarray}\\label{eq-charPoly-EV-2ndOrder}\n&&P_n(z,\\tau^{n+1},\\dots,\\tau^{2n+3})\\nonumber\\\\\n&=&\\bigg((n-1)^{n-1}\\big(-\\frac{(n-1)(n-2)^2 s^2}{4}+2(n-1)(n-4)sz\n-4(n-5)z^2\\big)\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\\nonumber\\\\\n&&+4(n-1)^n s z-16(n-1)^{n-1}z^2+z^2(z+\\frac{s}{2})^{n-1}\n -z^2(z+\\frac{s}{2})^{n-1}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2})\\bigg)\\nonumber\\\\\n&&\\cdot \\prod_{i=n+1}^{2n+3}\\big(z-\\frac{s}{2}+(\\tau^i)^2\\big).\n\\end{eqnarray}\n\\end{proposition}\n\\begin{proof}\nBy (\\ref{eq-cutoff-EulverVectorField-1})-(\\ref{eq-cutoff-EulverVectorField-5}), we have\n{\\footnotesize\n\\begin{equation*}\n\\mathcal{E}=\\begin{pNiceArray}{ccccccc|ccc}\n\t0 & n-1 & 0 & 0 & \\dots & 0 & 0 & (1-\\frac{n}{2})\\tau^{n+1} & \\dots & (1-\\frac{n}{2})\\tau^{2n+3} \\\\\n\t0 & -\\frac{s}{2} & n-1 & 0 & \\dots & 0 & 0 & 0 & \\dots & 0 \\\\\n\t0 & 0 & -\\frac{s}{2} & n-1 & \\dots & 0 & 0 & 0 & \\dots & 0 \\\\\n\t\\vdots & \\vdots & \\vdots & \\vdots& \\vdots & \\vdots& \\vdots & \t \\vdots& \\vdots& \\vdots\\\\\n\t0 & 0 & \\dots & 0 & \\dots & n-1 & 0 & \t 0 & \\dots& 0\\\\\n\t-2(n-1)s & 0 & \\dots & \\dots & \\dots& -\\frac{s}{2} & n-1 & (4-4n)\\tau^{n+1} & \\dots & (4-4n)\\tau^{2n+3}\\\\\n\t0 & (-2n-6)s & 16(n-1) & 0 & \\dots& 0 & 0 & 0 & \\dots & 0 \\\\\n\t\\hline\n\t0 & (n-3)\\tau^{n+1} & 0 & 0 & \\dots & 0 & \\frac{2-n}{8}\\tau^{n+1} & \\frac{s}{2} & \\dots & \\tau^{n+1}\\tau^{2n+3}\\\\ \n\t0 & (n-3)\\tau^{n+2} & \\dots & \\dots & \\dots & 0 & \\frac{2-n}{8}\\tau^{n+2} & \\tau^{n+1}\\tau^{n+2} & \\dots & \\tau^{n+2}\\tau^{2n+2} \\\\\n\t\\vdots & \\vdots & \\vdots & \\vdots& \\vdots & \\vdots& \\vdots & \t\\vdots& \\vdots& \\vdots\\\\\n\t0 & (n-3)\\tau^{2n+3} & 0 & \\dots & \\dots & 0 & \\frac{2-n}{8}\\tau^{2n+3} & \t\\tau^{n+1}\\tau^{2n+3} & \\dots & \\frac{s}{2}\n \t\\end{pNiceArray}.\n\\end{equation*}\n}\nIn the following we index the rows and columns by $0\\leq i\\leq 2n+3$. In the above we have blocked the matrix $\\mathcal{E}$ by the first $n+1$ rows and first $n+1$ columns.\nWe perform several similarity transforms of $\\mathcal{E}$.\nLet \n\\[\n\\mathcal{E}_1=\\mathrm{Diag}(\\underbrace{1,\\dots,1}_{n+1},\\tau^{n+1},\\dots,\\tau^{2n+3})\\cdot\n\\mathcal{E}\\cdot \\mathrm{Diag}(\\underbrace{1,\\dots,1}_{n+1},\\frac{1}{\\tau^{n+1}},\\dots,\\frac{1}{\\tau^{2n+3}}). \n\\]\nThe effect is the $i$-th row $\\times \\tau^i$, the $i$-th column $\\times \\frac{1}{\\tau^i}$, for $n+1\\leq i\\leq 2n+3$.\nThen\n{\\footnotesize\n\\begin{equation*}\n\\mathcal{E}_1=\\begin{pNiceArray}{ccccccc|ccc}\n\t0 & n-1 & 0 & 0 & \\dots & 0 & 0 & 1-\\frac{n}{2} & \\dots & 1-\\frac{n}{2} \\\\\n\t0 & -\\frac{s}{2} & n-1 & 0 & \\dots & 0 & 0 & 0 & \\dots & 0 \\\\\n\t0 & 0 & -\\frac{s}{2} & n-1 & \\dots & 0 & 0 & 0 & \\dots & 0 \\\\\n\t\\vdots & \\vdots & \\vdots & \\vdots& \\vdots & \\vdots& \\vdots & \t \\vdots& \\vdots& \\vdots\\\\\n\t0 & 0 & \\dots & 0 & \\dots & n-1 & 0 & \t 0 & \\dots& 0\\\\\n\t-2(n-1)s & 0 & \\dots & \\dots & \\dots& -\\frac{s}{2} & n-1 & 4-4n & \\dots & 4-4n\\\\\n\t0 & (-2n-6)s & 16(n-1) & 0 & \\dots& 0 & 0 & 0 & \\dots & 0 \\\\\n\t\\hline\n\t0 & (n-3)(\\tau^{n+1})^2 & 0 & 0 & \\dots & 0 & \\frac{2-n}{8}(\\tau^{n+1})^2 & \\frac{s}{2} & \\dots & (\\tau^{n+1})^2\\\\ \n\t0 & (n-3)(\\tau^{n+2})^2 & \\dots & \\dots & \\dots & 0 & \\frac{2-n}{8}(\\tau^{n+2})^2 & (\\tau^{n+2})^2 & \\dots & (\\tau^{n+2})^2 \\\\\n\t\\vdots & \\vdots & \\vdots & \\vdots& \\vdots & \\vdots& \\vdots & \t\\vdots& \\vdots& \\vdots\\\\\n\t0 & (n-3)(\\tau^{2n+3})^2 & 0 & \\dots & \\dots & 0 & \\frac{2-n}{8}(\\tau^{2n+3})^2 & \t(\\tau^{2n+3})^2 & \\dots & \\frac{s}{2}\n \t\\end{pNiceArray}.\n\\end{equation*}\n}\n\nLet \n\\[\n\\mathcal{E}_2=(I_{2n+4}-\\frac{8(n-3)}{n-2}E_{n,1})\\cdot \\mathcal{E}_1\\cdot (I_{2n+4}+\\frac{8(n-3)}{n-2}E_{n,1}).\n\\]\nThe effect is\n\\begin{eqnarray*}\n&&\\mbox{1st column}=>\\mbox{1st column}+\\frac{8(n-3)}{n-2}\\times \\mbox{$n$-th column},\\\\\n&&\\mbox{$n$-th row}=>\\mbox{$n$-th row}-\\frac{8(n-3)}{n-2}\\times \\mbox{$1$st row}.\n\\end{eqnarray*}\nThen\n{\\footnotesize\n\\begin{equation*}\n\\mathcal{E}_2=\\begin{pNiceArray}{ccccccc|ccc}\n\t0 & n-1 & 0 & 0 & \\dots & 0 & 0 & 1-\\frac{n}{2} & \\dots & 1-\\frac{n}{2} \\\\\n\t0 & -\\frac{s}{2} & n-1 & 0 & \\dots & 0 & 0 & 0 & \\dots & 0 \\\\\n\t0 & 0 & -\\frac{s}{2} & n-1 & \\dots & 0 & 0 & 0 & \\dots & 0 \\\\\n\t\\vdots & \\vdots & \\vdots & \\vdots& \\vdots & \\vdots& \\vdots & \t \\vdots& \\vdots& \\vdots\\\\\n\t0 & 0 & \\dots & 0 & \\dots & n-1 & 0 & \t 0 & \\dots& 0\\\\\n\t-2(n-1)s & \\frac{8(n-3)(n-1)}{n-2} & 0 & \\dots & \\dots& -\\frac{s}{2} & n-1 & 4-4n & \\dots & 4-4n\\\\\n\t0 & \\frac{-2n^2+2n}{n-2} s & \\frac{8(n-1)^2}{n-2} & 0 & \\dots& 0 & 0 & 0 & \\dots & 0 \\\\\n\t\\hline\n\t0 & 0 & 0 & 0 & \\dots & 0 & \\frac{2-n}{8}(\\tau^{n+1})^2 & \\frac{s}{2} & \\dots & (\\tau^{n+1})^2\\\\ \n\t0 & 0 & \\dots & \\dots & \\dots & 0 & \\frac{2-n}{8}(\\tau^{n+2})^2 & (\\tau^{n+2})^2 & \\dots & (\\tau^{n+2})^2 \\\\\n\t\\vdots & \\vdots & \\vdots & \\vdots& \\vdots & \\vdots& \\vdots & \t\\vdots& \\vdots& \\vdots\\\\\n\t0 & 0 & 0 & \\dots & \\dots & 0 & \\frac{2-n}{8}(\\tau^{2n+3})^2 & \t(\\tau^{2n+3})^2 & \\dots & \\frac{s}{2}\n \t\\end{pNiceArray}.\n\\end{equation*}\n}\nFor $i=n+1,\\dots,2n+2$, we make the following transformation \n\\[\nU=> (I_{2n+4}+E_{2n+3,i})\\cdot U\\cdot (I_{2n+4}-E_{2n+3,i}). \n\\]\nThe effect is, for $i=n+1,\\dots,2n+2$,\n\\begin{eqnarray*}\n&&\\mbox{$i$-th column}=>\\mbox{$i$-th column}- \\mbox{$(2n+3)$-th column},\\\\\n&&\\mbox{$(2n+3)$-th row}=> \\mbox{$(2n+3)$-th row}+\\mbox{$i$-th row}.\n\\end{eqnarray*}\nThen let\n\\[\n\\mathcal{E}_4=(I_{2n+4}-\\frac{8}{n-2}E_{n,2n+3})\\cdot \\mathcal{E}_4\\cdot (I_{2n+4}+\\frac{8}{n-2}E_{n,2n+3}).\n\\]\nThe effect is\n\\begin{eqnarray*}\n&&\\mbox{$(2n+3)$-th column}=>\\mbox{$(2n+3)$-th column}+\\frac{8}{n-2}\\times \\mbox{$n$-th column},\\\\\n&&\\mbox{$n$-th row}=>\\mbox{$n$-th row}-\\frac{8}{n-2}\\times \\mbox{$(2n+3)$-th row}.\n\\end{eqnarray*}\nThe matrices $\\mathcal{E}_3$ and $\\mathcal{E}_4$ are presented in the next page.\n\n\\begin{landscape}\n\\begin{eqnarray*}\n\\mathcal{E}_3=\\begin{pNiceArray}{ccccccc|cccc}\n\t0 & n-1 & 0 & 0 & \\dots & 0 & 0 & 0 & \\dots & 0 & 1-\\frac{n}{2} \\\\\n\t0 & -\\frac{s}{2} & n-1 & 0 & \\dots & 0 & 0 & 0 & \\dots & 0 & 0 \\\\\n\t0 & 0 & -\\frac{s}{2} & n-1 & \\dots & 0 & 0 & 0 & \\dots & 0 & 0 \\\\\n\t\\vdots & \\vdots & \\vdots & \\vdots& \\vdots & \\vdots& \\vdots & \\vdots &\t \\vdots& \\vdots& \\vdots\\\\\n\t0 & 0 & \\dots & 0 & \\dots & n-1 & 0 & \t 0 & \\dots& 0 & 0\\\\\n\t-2(n-1)s & \\frac{8(n-3)(n-1)}{n-2} & \\dots & \\dots & \\dots& -\\frac{s}{2} & n-1 & 0 & \\dots & 0 & 4-4n\\\\\n\t0 & \\frac{-2n^2+2n}{n-2} s & \\frac{8(n-1)^2}{n-2} & 0 & \\dots& 0 & 0 & 0 & \\dots & 0 & 0 \\\\\n\t\\hline\n\t0 & 0 & 0 & 0 & \\dots & 0 & \\frac{2-n}{8}(\\tau^{n+1})^2 & \\frac{s}{2}-(\\tau^{n+1})^2 & \\dots & 0 & (\\tau^{n+1})^2\\\\ \n\t0 & 0 & \\dots & \\dots & \\dots & 0 & \\frac{2-n}{8}(\\tau^{n+2})^2 & 0 & \\ddots & 0 & (\\tau^{n+2})^2 \\\\\n\t\\vdots & \\vdots & \\vdots & \\vdots& \\vdots & \\vdots& \\vdots & \t\\vdots& & \\vdots & \\vdots\\\\\n\t0 & \\dots & \\dots & \\dots & \\dots & 0 & \\frac{2-n}{8}(\\tau^{2n+2})^2 & 0 & \\dots & \\frac{s}{2}-(\\tau^{2n+2})^2 & (\\tau^{2n+2})^2 \\\\\n\t0 & 0 & 0 & \\dots & \\dots & 0 & \\frac{2-n}{8}s & (\\tau^{2n+3})^2-(\\tau^{n+1})^2 & \\dots & (\\tau^{2n+3})^2-(\\tau^{2n+2})^2 & \\frac{3s}{2}-(\\tau^{2n+3})^2 \n \t\\end{pNiceArray}.\n\\end{eqnarray*}\n\n\n\\begin{eqnarray*}\n\\mathcal{E}_4=\\begin{pNiceArray}{ccccccc|cccc}\n\t0 & n-1 & 0 & 0 & \\dots & 0 & 0 & 0 & \\dots & 0 & 1-\\frac{n}{2} \\\\\n\t0 & -\\frac{s}{2} & n-1 & 0 & \\dots & 0 & 0 & 0 & \\dots & 0 & 0 \\\\\n\t0 & 0 & -\\frac{s}{2} & n-1 & \\dots & 0 & 0 & 0 & \\dots & 0 & 0 \\\\\n\t\\vdots & \\vdots & \\vdots & \\vdots& \\vdots & \\vdots& \\vdots & \\vdots &\t \\vdots& \\vdots& \\vdots\\\\\n\t0 & 0 & \\dots & 0 & \\dots & n-1 & 0 & \t 0 & \\dots& 0 & 0\\\\\n\t-2(n-1)s & \\frac{8(n-3)(n-1)}{n-2} & \\dots & \\dots & \\dots& -\\frac{s}{2} & n-1 & 0 & \\dots & 0 & -\\frac{4(n-1)(n-4)}{n-2}\\\\\n\t0 & \\frac{-2n^2+2n}{n-2} s & \\frac{8(n-1)^2}{n-2} & 0 & \\dots& 0 & s & -\\frac{8\\big((\\tau^{2n+3})^2-(\\tau^{n+1})^2\\big)}{n-2} & \\dots & -\\frac{8\\big((\\tau^{2n+3})^2-(\\tau^{2n+2})^2\\big)}{n-2} & -\\frac{8}{n-2}\\big(\\frac{s}{2}-(\\tau^{2n+3})^2\\big) \\\\\n\t\\hline\n\t0 & 0 & 0 & 0 & \\dots & 0 & \\frac{2-n}{8}(\\tau^{n+1})^2 & \\frac{s}{2}-(\\tau^{n+1})^2 & \\dots & 0 & 0 \\\\ \n\t0 & 0 & \\dots & \\dots & \\dots & 0 & \\frac{2-n}{8}(\\tau^{n+2})^2 & 0 & \\ddots & 0 & 0 \\\\\n\t\\vdots & \\vdots & \\vdots & \\vdots& \\vdots & \\vdots& \\vdots & \t\\vdots& & \\vdots & \\vdots\\\\\n\t0 & \\dots & \\dots & \\dots & \\dots & 0 & \\frac{2-n}{8}(\\tau^{2n+2})^2 & 0 & \\dots & \\frac{s}{2}-(\\tau^{2n+2})^2 & 0 \\\\\n\t0 & 0 & 0 & \\dots & \\dots & 0 & \\frac{2-n}{8}s & (\\tau^{2n+3})^2-(\\tau^{n+1})^2 & \\dots & (\\tau^{2n+3})^2-(\\tau^{2n+2})^2 & \\frac{s}{2}-(\\tau^{2n+3})^2 \n \t\\end{pNiceArray}.\n\\end{eqnarray*}\n\\end{landscape}\n\nWe block \n\\[\n\\mathcal{E}_4=\\begin{pmatrix}\nA & B \\\\\nC & D \n\\end{pmatrix}\n\\]\nas indicated above. Compute the characteristic polynomial by the formula\n\\begin{eqnarray*}\n&& P_n(z,\\tau^{n+1},\\dots,\\tau^{2n+3})\\\\\n&=& \\det \\begin{pmatrix}\nz I-A & -B \\\\\n-C & zI- D \n\\end{pmatrix}\n=\\det \\begin{pmatrix}\nz I-A - B(zI-D)^{-1}C & 0 \\\\\n-C & zI- D \n\\end{pmatrix}.\n\\end{eqnarray*}\nSince\n\\[\n(zI-D)^{-1}=\n\\begin{pmatrix}\n\\frac{1}{z-\\frac{s}{2}+(\\tau^{n+1})^2} & 0 & \\dots & 0 \\\\\n0 & \\ddots & & 0\\\\\n0 & & \\ddots & 0\\\\\n\\frac{(\\tau^{2n+3})^2-(\\tau^{n+1})^2}{(z-\\frac{s}{2}+(\\tau^{n+1})^2)(z-\\frac{s}{2}+(\\tau^{2n+3})^2)} & \\dots & \n\\frac{(\\tau^{2n+3})^2-(\\tau^{2n+2})^2}{(z-\\frac{s}{2}+(\\tau^{2n+2})^2)(z-\\frac{s}{2}+(\\tau^{2n+3})^2)} & \\frac{1}{z-\\frac{s}{2}+(\\tau^{2n+3})^2}\n\\end{pmatrix}\n\\]\nwe get\n\\begin{gather*}\nB(zI-D)^{-1}C=\\\\\n\\begin{pNiceArray}{c|c} \n\\Block{6-1}<\\large>{0_{{\\scriptscriptstyle (n+1)\\times n}}} & \n{\\scriptstyle\n\\frac{(n-2)^2}{16}\\big(\\sum_{i=n+1}^{2n+2}\\frac{\\big((\\tau^{2n+3})^2-(\\tau^i)^2\\big)(\\tau^i)^2}{\\big(z-\\frac{s}{2}+(\\tau^{i})^2\\big)\\big(z-\\frac{s}{2}+(\\tau^{2n+3})^2\\big)}+\\frac{s}{z-\\frac{s}{2}+(\\tau^{2n+3})^2}\\big) }\\\\\n \\hspace*{1cm} & {\\scriptstyle\n0} \\\\ \n\\hspace*{1cm} & \\vdots \\\\\n \\hspace*{1cm} & {\\scriptstyle\n0} \\\\ \n \\hspace*{1cm} & \\vspace{1cm} {\\scriptstyle\n \\frac{(n-2)(n-4)}{2}\\big(\\sum_{i=n+1}^{2n+2}\\frac{\\big((\\tau^{2n+3})^2-(\\tau^i)^2\\big)(\\tau^i)^2}{\\big(z-\\frac{s}{2}+(\\tau^{i})^2\\big)\\big(z-\\frac{s}{2}+(\\tau^{2n+3})^2\\big)}+\\frac{s}{z-\\frac{s}{2}+(\\tau^{2n+3})^2}\\big) } \\\\ \n\\hspace*{1cm} & {\\scriptstyle\n\\sum_{i=n+1}^{2n+2}\\frac{\\big((\\tau^{2n+3})^2-(\\tau^i)^2\\big)(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^{i})^2}+\\big(\\frac{s}{2}-(\\tau^{2n+3})^2\\big) \\big(\\sum_{i=n+1}^{2n+2}\\frac{\\big((\\tau^{2n+3})^2-(\\tau^i)^2\\big)(\\tau^i)^2}{(z-\\frac{s}{2}+(\\tau^{i})^2)(z-\\frac{s}{2}+(\\tau^{2n+3})^2)}+\\frac{s}{z-\\frac{s}{2}+(\\tau^{2n+3})^2}\\big) }\n\\end{pNiceArray},\n\\end{gather*}\nwhere $0_{(n+1)\\times n}$ stands for a 0-matrix of size $(n+1)\\times n$.\nUsing \n\\[\n\\sum_{i=n+1}^{2n+2}\\frac{\\big((\\tau^{2n+3})^2-(\\tau^i)^2\\big)(\\tau^i)^2}{\\big(z-\\frac{s}{2}+(\\tau^{i})^2\\big)\\big(z-\\frac{s}{2}+(\\tau^{2n+3})^2\\big)}+\\frac{s}{z-\\frac{s}{2}+(\\tau^{2n+3})^2}\n=\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2},\n\\]\nand\n\\begin{eqnarray*}\n&&\\sum_{i=n+1}^{2n+2}\\frac{\\big((\\tau^{2n+3})^2-(\\tau^i)^2\\big)(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^{i})^2}\\\\\n&&+\\big(\\frac{s}{2}-(\\tau^{2n+3})^2\\big) \\big(\\sum_{i=n+1}^{2n+2}\\frac{\\big((\\tau^{2n+3})^2-(\\tau^i)^2\\big)(\\tau^i)^2}{(z-\\frac{s}{2}+(\\tau^{i})^2)(z-\\frac{s}{2}+(\\tau^{2n+3})^2)}+\\frac{s}{z-\\frac{s}{2}+(\\tau^{2n+3})^2}\\big)\\\\\n&=& z\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2} -s,\n\\end{eqnarray*}\nwe get\n\\begin{eqnarray*}\nB(zI-D)^{-1}C=\\begin{pNiceArray}{c|c} \n\\Block{6-1}<\\large>{0_{{\\scriptscriptstyle\n(n+1)\\times n}}} & \n\\frac{(n-2)^2}{16}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2} \\\\\n \\hspace*{1cm} & 0 \\\\ \n\\hspace*{1cm} & \\vdots \\\\\n \\hspace*{1cm} & 0 \\\\ \n \\hspace*{1cm} & \\frac{(n-2)(n-4)}{2}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2} \\\\ \n\\hspace*{1cm} & z\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2} -s \n\\end{pNiceArray},\n\\end{eqnarray*}\nand thus\n\\begin{gather*}\nzI-A-B(zI-D)^{-1}C=\\\\\n\\begin{pNiceArray}{ccccccc}\n\tz & 1-n & 0 & 0 & \\dots & 0 & -\\frac{(n-2)^2}{16}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2} \\\\\n\t0 & z+\\frac{s}{2} & 1-n & 0 & \\dots & 0 & 0 \\\\\n\t0 & 0 & z+\\frac{s}{2} & 1-n & \\dots & 0 & 0 \\\\\n\t\\vdots & \\vdots & \\vdots & \\vdots& \\vdots & \\vdots& \\vdots \\\\\n\t0 & 0 & \\dots & 0 & \\dots & 1-n & 0 \\\\\n\t2(n-1)s & -\\frac{8(n-3)(n-1)}{n-2} & \\dots & \\dots & \\dots& z+\\frac{s}{2} & 1-n-\\frac{(n-2)(n-4)}{2}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2} \\\\\n\t0 & \\frac{2n^2-2n}{n-2} s & -\\frac{8(n-1)^2}{n-2} & 0 & \\dots& 0 & z\\big(1-\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\\big) \n \t\\end{pNiceArray}.\n\\end{gather*}\nWe compute the determinant of $zI-A-B(zI-D)^{-1}C$ by expanding the determinant according to the last two columns and then the first column.\nSo the characteristic polynomial is\n\\begin{eqnarray*}\n&&\\det \\big(zI-A-B(zI-D)^{-1}C\\big)\\\\\n&=&\\Big(\\frac{(n-2)^2}{16}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\n\\cdot(2n-2)s\\\\\n&& +z\\big(1-n-\\frac{(n-1)(n-4)}{2}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\n\\big)\\Big)\\cdot (1-n)\\\\\n&&\\cdot \\big((-1)^{n-3}\\frac{2n^2-2n}{n-2}s\\cdot (1-n)^{n-3}\n+(-1)^{n-2}(-\\frac{8(n-1)^2}{n-2})\\cdot (1-n)^{n-4}(z+\\frac{s}{2})\\big)\\\\\n&&+z(1-\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2})\\big(\n(-1)^{n-1}(2n-2)s\\cdot (1-n)^{n-1}\\\\\n&&+(-1)^n(-\\frac{8(n-3)(n-1)}{n-2})\\cdot (1-n)^{n-2}z+z(z+\\frac{s}{2})^{n-1}\n\\big).\n\\end{eqnarray*}\nWe simplify its expression as follows, where the change in each step is indicated in \\textcolor{blue}{blue}. Recall that $n$ is even.\n\\begin{eqnarray*}\n&&\\det \\big(zI-A-B(zI-D)^{-1}C\\big)\\\\\n&=&\\Big(-\\frac{(n-2)^2 \\textcolor{blue}{s}}{\\textcolor{blue}{8}}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2} \n+z\\big(1+\\frac{n-4}{2}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\n\\big)\\Big)\\cdot \\textcolor{blue}{(1-n)^2}\\\\\n&&\\cdot \\big(\\frac{2n(n-1)^{\\textcolor{blue}{n-2}}}{n-2}s\n-\\frac{8(n-1)^{\\textcolor{blue}{n-2}}}{n-2})(z+\\frac{s}{2})\\big)\\\\\n&&+z(1-\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2})\\big(\n2(n-1)^{\\textcolor{blue}{n}} s\n-\\frac{8(n-3)(n-1)^{\\textcolor{blue}{n-1}}}{n-2}z+z(z+\\frac{s}{2})^{n-1}\n\\big)\\\\\n&=&(n-1)^2\\Big(-\\frac{(n-2)^2 s}{8}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\n+z\\big(1+\\frac{n-4}{2}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\n\\big)\\Big)\\\\\n&&\\cdot \\big(\\textcolor{blue}{2(n-1)^{n-2}s}-\\frac{8(n-1)^{n-2}}{n-2}z\\big)\\\\\n&&+z(1-\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2})\\big(\n2(n-1)^n s\n-\\frac{8(n-3)(n-1)^{n-1}}{n-2}z+z(z+\\frac{s}{2})^{n-1}\n\\big)\\\\\n&=&(n-1)^{\\textcolor{blue}{n}}\\Big(-\\frac{(n-2)^2 s}{8}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\n+z\\big(1+\\frac{n-4}{2}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\n\\big)\\Big)\\\\\n&&\\cdot \\big(2s-\\frac{8}{n-2}z\\big)\\\\\n&&+z(1-\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2})\\big(\n2(n-1)^n s\n-\\frac{\\textcolor{blue}{8(n-1)}(n-1)^{n-1}}{n-2}z\n\\big)\\\\\n&& +z(1-\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2})\\big(\n\\frac{\\textcolor{blue}{16}(n-1)^{n-1}}{n-2}z+z(z+\\frac{s}{2})^{n-1}\n\\big)\\\\\n&=&(n-1)^n\\Big(-\\frac{(n-2)^2 s}{8}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\n+z\\big(1+\\frac{n-4}{2}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\n\\big)\\Big)\\\\\n&&\\cdot \\big(2s-\\frac{8}{n-2}z\\big)\\\\\n&&+\\textcolor{blue}{(n-1)^n} z(1-\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2})\\big(\n2 s -\\frac{8}{n-2}z\\big)\\\\\n&& +z^2(1-\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2})\\big(\n\\frac{16(n-1)^{n-1}}{n-2}+(z+\\frac{s}{2})^{n-1}\n\\big)\\\\\n&=&(n-1)^n\\big(2s-\\frac{8}{n-2}z\\big)\\Big(-\\frac{(n-2)^2 s}{8}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\n+\\textcolor{blue}{2z+\\frac{n-6}{2}z}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\\Big)\\\\\n&& +z^2(1-\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2})\\big(\n\\frac{16(n-1)^{n-1}}{n-2}+(z+\\frac{s}{2})^{n-1}\n\\big)\\\\\n&=&(n-1)^n\\big(2s-\\frac{8}{n-2}z\\big)\\textcolor{blue}{\\Big(}-\\frac{(n-2)^2 s}{8}\n+\\frac{n-6}{2}z\\textcolor{blue}{\\Big)}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\\\\\n&&+(n-1)^n\\big(2s-\\frac{8}{n-2}z\\big)2z\\\\\n&& +z^2(1-\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2})\\big(\n\\frac{16(n-1)^{n-1}}{n-2}+(z+\\frac{s}{2})^{n-1}\n\\big)\\\\\n&=&(n-1)^n\\big(s-\\frac{\\textcolor{blue}{4}}{n-2}z\\big)\\Big(-\\frac{(n-2)^2 s}{\\textcolor{blue}{4}}\n+(n-6)z\\Big)\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\\\\\n&&+\\textcolor{blue}{4}(n-1)^n\\big(s-\\frac{4}{n-2}z\\big)z+\\textcolor{blue}{z^2\\big(\n\\frac{16(n-1)^{n-1}}{n-2}+(z+\\frac{s}{2})^{n-1}\\big)}\n\\\\\n&& -z^2\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2})\\big(\n\\frac{16(n-1)^{n-1}}{n-2}+(z+\\frac{s}{2})^{n-1}\n\\big)\\\\\n&=&(n-1)^n\\textcolor{blue}{\\big(-\\frac{(n-2)^2 s^2}{4}+(2n-8)sz-\n\\frac{4(n-6)}{n-2}z^2\\big)}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\\\\\n&&+4(n-1)^n s z\\textcolor{blue}{-16(n-1)^{n-1}z^2}+z^2(z+\\frac{s}{2})^{n-1}\n\\\\\n&& -z^2\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2})\\big(\n\\frac{16(n-1)^{n-1}}{n-2}+(z+\\frac{s}{2})^{n-1}\n\\big)\\\\\n&=&(n-1)^{\\textcolor{blue}{n-1}}\\big(-\\frac{(n-1)(n-2)^2 s^2}{4}+2(n-1)(n-4)sz\\\\\n&&\\textcolor{blue}{-\\frac{16}{n-2}z^2}-\\frac{4(n-1)(n-6)}{n-2}z^2\\big)\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\\\\\n&&+4(n-1)^n s z-16(n-1)^{n-1}z^2+z^2(z+\\frac{s}{2})^{n-1}\n-z^2\\textcolor{blue}{(z+\\frac{s}{2})^{n-1}}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\\\\\n\\begin{comment}&=&(n-1)^{n-1}\\big(-\\frac{(n-1)(n-2)^2 s^2}{4}+2(n-1)(n-4)sz\n-\\frac{\\textcolor{blue}{4(n^2-7n+10)}}{n-2}z^2\\big)\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\\\\\n&&+4(n-1)^n s z-16(n-1)^{n-1}z^2+z^2(z+\\frac{s}{2})^{n-1}\n -z^2(z+\\frac{s}{2})^{n-1}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\\\\\n\\end{comment}\n&=&(n-1)^{n-1}\\big(-\\frac{(n-1)(n-2)^2 s^2}{4}+2(n-1)(n-4)sz\n-4\\textcolor{blue}{(n-5)}z^2\\big)\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\\\\\n&&+4(n-1)^n s z-16(n-1)^{n-1}z^2+z^2(z+\\frac{s}{2})^{n-1}\n -z^2(z+\\frac{s}{2})^{n-1}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}.\n\\end{eqnarray*}\nHence we obtain (\\ref{eq-charPoly-EV-2ndOrder}).\n\\end{proof}\n\n\n\\begin{comment}\nWhen $s=0$,\n\\begin{eqnarray*}\n&&P_n(z,\\tau^{n+1},\\dots,\\tau^{2n+3})\\\\\n&=&z^2\\bigg(\n-4(n-5)(n-1)^{n-1}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z+(\\tau^i)^2}\n-16(n-1)^{n-1}+z^{n-1}\\\\\n&&-z^{n-1}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z+(\\tau^i)^2})\\bigg)\n\\cdot \\prod_{i=n+1}^{2n+3}\\big(z+(\\tau^i)^2\\big).\n\\end{eqnarray*}\nIn particular, when $\\tau^i=\\zeta_{n+3}^{i-n}$, \n\\begin{eqnarray*}\nP_n(z,\\tau^{n+1},\\dots,\\tau^{2n+3})\n&=&z^2\\big(z^{2n+2}-16(n-1)^{n-1}z^{n+3}-(n+2)z^{n-1}\\\\\n&&-4(n-1)^{n-1}(n^2-2n-11)\n\\big).\n\\end{eqnarray*}\n\\end{comment}\n\n\\subsection{Proof of semisimplicity}\n\n\\begin{proposition}\\label{prop-charPoly-EV-2ndOrder-simpleRoots}\nFor general $(\\tau^{n+1},\\dots,\\tau^{2n+3})\\in \\mathbb{C}^{n+3}$, the polynomial in $z$\n\\begin{eqnarray}\\label{eq-charPoly-EV-2ndOrder-1}\n&&\\mathcal{P}_n(z,\\tau^{n+1},\\dots,\\tau^{2n+3})=\nP_n(z,\\tau^{n+1},\\dots,\\tau^{2n+3})+o(\\tau^2)\n\\nonumber\\\\\n&=&\\bigg((n-1)^{n-1}\\big(-\\frac{(n-1)(n-2)^2 s^2}{4}+2(n-1)(n-4)sz\n-4(n-5)z^2\\big)\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\\nonumber\\\\\n&&+4(n-1)^n s z-16(n-1)^{n-1}z^2+z^2(z+\\frac{s}{2})^{n-1}\n -z^2(z+\\frac{s}{2})^{n-1}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2})\\bigg)\\nonumber\\\\\n&&\\cdot \\prod_{i=n+1}^{2n+3}\\big(z-\\frac{s}{2}+(\\tau^i)^2\\big)+o(\\tau^2)\n\\end{eqnarray}\nhas only simple roots.\n\\end{proposition}\n\\begin{proof}\nWe have\n\\begin{eqnarray}\nP_n(z,0,\\dots,0)=z^{n+5}(-16(n-1)^{n-1}+z^{n-1}).\n\\end{eqnarray}\nIt has $n-1$ nonzero simple roots and a root of multiplicity $n+5$ at 0. We are going to find a (germ of) line $L$ on the $\\tau$-space starting at 0, parametrized by $\\theta$, and find the branches of solutions on this ray. Let the line $L$ be, say $\\tau^i=\\alpha_i \\theta$, where $\\alpha_i\\in \\mathbb{C}$, for $n+1\\leq i\\leq 2n+3$ are temporarily not specified. By the theory of Puiseux expansion, near the origin of $L$, there are solutions in the form \n\\begin{equation}\n\tz=\\sum_{i=1}^{\\infty}a_{\\frac{i}{M}}\\theta^{\\frac{i}{M}}\n\\end{equation}\nwhere $M$ is a natural number. The index $\\frac{i}{M}$ of the least possible nonzero coefficient $a_{\\frac{i}{M}}$ can be read out from the expression (\\ref{eq-charPoly-EV-2ndOrder}). It is also given by the \\emph{Newton polygon}; indeed, regarding the restriction of $P_n$ on $L$ as a polynomial $P_n(\\theta,z)$ the pair of indices that gives the first slope of the Newton polygon is $(0,n+5)$ and $(2,n+4)$. So the slope is $2$, and we get the expansion \n\\[\nz=a_2 \\theta^2+o(\\theta^{2}).\n\\]\nTo show that $\\mathcal{P}_n$ has $n+5$ distinct branches of solutions of $z$ at 0, we need only to show that the equation \n\\begin{eqnarray}\n\t&&\\big(-\\frac{(n-1)(n-2)^2 s}{4}+2(n-1)(n-4)sz-4(n-5)z^2\\big)\\sum_{i=0}^{n+2}\\frac{\\tau^2_i}{z-\\frac{s}{2}+\\tau^2_i}\\nonumber\\\\\n&&+4(n-1) s z -16z^2=0\n\\end{eqnarray}\nfor $a_2$ has $n+5$ simple roots, for generic choices of $\\tau^{n+1},\\dots,\\tau^{2n+3}$. This follows from the following Proposition \\ref{prop-secondOrderCoeff-distinctRoots}.\n\\end{proof}\n \n\n\n\n\\begin{lemma}\\label{lem-simpleRoots}\nWhen $n\\geq 3$ and $a\\in \\mathbb{Q}$, the equation $z^n-az+1=0$ has only simple roots.\n\\end{lemma}\n\\begin{proof}\n$z^n-az+1$ and $nz^{n-1}-a$ have common factors only if $a=\\frac{n}{n-1}\\cdot (n-1)^{\\frac{1}{n}}$, which is not rational when $n\\geq 3$.\n\\end{proof}\n\n\\begin{proposition}\\label{prop-secondOrderCoeff-distinctRoots}\nLet $n\\geq 4$ be an even natural number.\nLet $\\varsigma=\\varsigma(y_0,\\dots,y_{n+2})=y_0+\\dots+y_{n+2}$. Then for general $(y_0,\\dots,y_{n+2})\\in \\mathbb{C}^{n+3}$, the equation \n\\begin{eqnarray}\\label{eq-secondOrderCoeff-distinctRoots}\n&&\t\\Big(\\big(-\\frac{(n-1)(n-2)^2 \\varsigma^2}{4}+2(n-1)(n-4)\\varsigma z\n-4(n-5)z^2\\big)\\sum_{i=0}^{n+2}\\frac{y_i}{z-\\frac{\\varsigma}{2}+y_i}\\nonumber\\\\\n&&+4(n-1) \\varsigma z-16z^2\\Big)\\cdot \\prod_{i=0}^{n+2}\\big(z-\\frac{\\varsigma}{2}+y_i\\big)=0\n\\end{eqnarray}\nfor $z$ has $n+5$ simple roots.\n\\end{proposition}\n\\begin{proof}\nLet $\\zeta_{n+3}=e^{2\\pi \\sqrt{-1}}$ be a $(n+3)$-th root of unity, and set \n\\begin{equation}\n\ty_i=\\zeta_{n+3}^i+y.\n\\end{equation}\nIt suffices to show that for general $y\\in \\mathbb{C}$, (\\ref{eq-secondOrderCoeff-distinctRoots}) has only simple roots.\nSince\n\\begin{equation*}\n\t\\prod_{i=0}^{n+2}(z+\\zeta_{n+3}^i)=z^{n+3}+1,\n\\end{equation*}\nand \n\\begin{equation*}\n\t\\sum_{i=0}^{n+2}\\frac{1}{z+\\zeta_{n+3}^i}=\\frac{n+3}{z^{n+3}+1},\n\\end{equation*}\nwe have\n\\begin{equation*}\n\t\\prod_{i=0}^{n+2}(z-\\frac{s}{2}+y_i)=(z-\\frac{s}{2}+y)^{n+3}+1,\n\\end{equation*}\nand\n\\begin{equation*}\n\t\\sum_{i=0}^{n+2}\\frac{y_i}{z-\\frac{s}{2}+y_i}\n\t=n+3-(z-\\frac{s}{2})\\sum_{i=0}^{n+2}\\frac{1}{z-\\frac{s}{2}+y_i}\n\t=(n+3)(1-\\frac{z-\\frac{s}{2}}{(z-\\frac{s}{2}+y)^{n+3}+1}).\n\\end{equation*}\nSo (\\ref{eq-secondOrderCoeff-distinctRoots}) reads\n\\begin{eqnarray}\\label{eq-prop-secondOrderCoeff-distinctRoots-1}\n\t&&\\big(-\\frac{(n-1)(n-2)^2(n+3)^2y^2}{4}+2(n-1)(n-4)(n+3)yz\n-4(n-5)z^2\\big)\\nonumber\\\\\n&&\\cdot(n+3)((z-\\frac{n+1}{2}y)^{n+3}+1-z+\\frac{n+3}{2}y) \\nonumber\\\\\n&&+\\big(4(n-1)(n+3)y z-16z^2\\big)\\big((z-\\frac{n+1}{2}y)^{n+3}+1\\big)\n=0.\n\\end{eqnarray}\nWhen $y=0$, this equation is\n\\begin{equation}\n\t4z^2\\big((- n^2+2 n+11) z^{n+3}+( n^2-2 n-15)z+(- n^2 +2 n +11) \\big)=0,\n\\end{equation}\ni.e.\n\\begin{equation}\n\t4(- n^2+2 n+11)z^2\\big( z^{n+3}-\\frac{n^2-2 n-15}{n^2-2 n-11}z+1 \\big)=0.\n\\end{equation}\nBy Lemma \\ref{lem-simpleRoots}, the second factor has only simple roots.\nWe study the branches $z=z(y)$ with $z(0)=0$. They can be written in the form of Puiseux series\n\\[\nz=a_{\\frac{1}{2}}y^{\\frac{1}{2}}+a_1 y+O(y^{\\frac{3}{2}}).\n\\]\nExpanding (\\ref{eq-prop-secondOrderCoeff-distinctRoots-1}), the equation for $a_{\\frac{1}{2}}$ is\n\\[\n0=-4(n-5)(n+3)a_{\\frac{1}{2}}^2-16a_{\\frac{1}{2}}^2=0\n\\]\nso $a_{\\frac{1}{2}}=0$. One can also argue by using Newton polygon.\nThen the equation for $a_1$ is\n\\begin{eqnarray*}\n\t&&(n+3)\\big(-\\frac{(n-1)(n-2)^2(n+3)^2}{4}+2(n-1)(n-4)(n+3)a_1\n-4(n-5)a_1^2\\big)\\\\\n&&+\\big(4(n-1)(n+3)a_1-16a_1^2\\big)=0.\n\\end{eqnarray*}\nThis is a quadratic equation with discriminant equal to\n\\[\n64 (-1 + n) (2 + n) (3 + n)^2.\n\\]\nSo we have two distinct branches with $z(0)=0$.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm-semisimplicity}]\nBy Proposition \\ref{prop-charPoly-EV-2ndOrder} and \\ref{prop-charPoly-EV-2ndOrder-simpleRoots}, the Euler field $E$ has pairwise distinct eigenvalues on a general point in an open neighborhood of 0. This means that $\\mathcal{M}_X$ is semisimple in the sense of \\cite[Definition 3.1]{Dub99}. Then by \\cite[Theorem 3.1]{Dub99}, $\\mathcal{M}_X$ is (generically) semisimple in the usual sense.\n\\end{proof}\n\n\n\\section{Enumerative geometry on even dimensional intersections of two quadrics}\\label{sec:EnumerativeGeometry-Even(2,2)}\nIn this section we study the special correlator \n\\begin{equation}\\label{eq-specialCorrelator}\n\t\\langle \\epsilon_{1},\\dots,\\epsilon_{n+3}\\rangle^X_{0,n+3,\\frac{n}{2}}\n\\end{equation}\nby relating its value to enumerative geometry of $X$. Then we prove the $n=4$ case of Conjecture \\ref{conj-unknownCorrelator-Even(2,2)}. \nWe begin by recalling some facts about the even dimensional intersections of two quadrics from \\cite{Rei72}. Let $\\lambda_0,\\dots,\\lambda_{n+2}\\in\\mathbb{C}$ be pairwise distinct. Let \n\\begin{equation}\\label{eq-definingEquationOfQi}\n\t\\varphi_1(Y_0,\\dots,Y_{n+2})=\\sum_{i=0}^{n+2}Y_i^2,\\\n\t\\varphi_2(Y_0,\\dots,Y_{n+2})=\\sum_{i=0}^{n+2} \\lambda_i Y_i^2,\n\\end{equation}\nand $X=\\{\\varphi_1=\\varphi_2=0\\}\\subset \\mathbb{P}^{n+2}$. By \\cite[Prop. 2.1]{Rei72}, every smooth complete intersections of two quadrics can be obtained in this way by choosing appropriate coordinates on $\\mathbb{P}^{n+2}$.\n\\begin{lemma}\nFor $0\\leq k\\leq \\frac{n}{2}$, let $P_k$ be the point in $\\mathbb{P}^{n+2}$ whose $i$-th homogeneous coordinate is\n\\begin{equation}\n\\frac{\\lambda_i^k}{\\sqrt{\\prod_{\\begin{subarray}{c}0\\leq j\\leq n+2\\\\ j\\neq i\\end{subarray}}\n(\\lambda_i- \\lambda_j)}}. \n\\end{equation}\nThen\n\\begin{equation}\\label{eq-vanish-varphiPk}\n\t\\varphi_1(P_k)=\\varphi_2(P_k)=0,\n\\end{equation}\nand $P_0,\\dots,P_{\\frac{n}{2}}$ span an $\\frac{n}{2}$-plane contained in $X$. Here we choose a root $\\sqrt{\\lambda_i- \\lambda_j}$ uniformly in the expressions of $P_0,\\dots,P_k$, for every pair ${i,j}$, for $0\\leq i\\neq j\\leq n+2$.\n\\end{lemma}\n\\begin{proof}\nDirectly check (\\ref{eq-vanish-varphiPk}) using \n\\[\n\\frac{\\lambda^l}{\\prod_{\\begin{subarray}{c}0\\leq j\\leq n+2\\\\ j\\neq i\\end{subarray}}(\\lambda_i- \\lambda_j)}=0\n\\]\nfor $0\\leq l\\leq n+1$.\n\\end{proof}\n\nWe denote this $\\frac{n}{2}$-plane by $S$. Then $S$ is defined by the following linear equations:\n\\begin{equation}\n\\sum_{i=0}^{n+2}\\frac{\\lambda_i^k}{\\sqrt{\\prod_{\\begin{subarray}{c}0\\leq j\\leq n+2\\\\ j\\neq i\\end{subarray}}\n(\\lambda_i- \\lambda_j)}}Y_i=0,\\ \\mbox{for}\\ 0\\leq k\\leq \\frac{n}{2}+1.\n\\end{equation}\nMake a change of coordinates\n\\begin{equation}\\label{eq-Wcoordinates}\n\tW_i=\\frac{Y_i}{\\sqrt{\\prod_{\\begin{subarray}{c}0\\leq j\\leq n+2\\\\ j\\neq i\\end{subarray}}\n(\\lambda_i- \\lambda_j)}}.\n\\end{equation}\nThen $S$ is defined by\n\\begin{equation}\\label{eq-definingEquations-WCoordinates}\n\t\\sum_{i=0}^{n+2}\\lambda_i^k W_i=0,\\ \\mbox{for}\\ 0\\leq k\\leq \\frac{n}{2}+1.\n\\end{equation}\n\nRecall from \\cite[Lemma 3.10]{Rei72}:\n\\begin{lemma}\\label{lem-intersectionNumber-middelDimension}\n\\begin{enumerate}\n\t\\item[(i)] Let $T$ be a $\\frac{n}{2}$-plane contained in $X$. Then $[T]\\cdot \\mathsf{h}_{n\/2}=1$.\n\t\\item[(ii)] Let $T_1$ and $T_2$ be $\\frac{n}{2}$-planes contained in $X$.\n\tSuppose $\\dim(T_1\\cap T_2)=r$, then \n\t\\begin{equation}\n\t\t\t[T_1]\\cdot [T_2]=(-1)^{r}(\\lfloor\\frac{r}{2}\\rfloor+1).\n\t\\end{equation}\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{notation}\nFor integers $a\\leq b$ let $[a,b]$ be the set of integers $c$ satisfying $a\\leq c\\leq b$. \nFor a subset $I\\subset [0,n+2]$, let $S_I$ be the $\\frac{n}{2}$-plane obtained by reversing the sign of the $i$-th homogeneous coordinate of the points on $S$ for all $i\\in I$. Denote the complement of $I$ by $C(I)$. Then $S_I=S_{C(I)}$. \n\\end{notation}\nBy \\cite[Theorem 3.8]{Rei72}, every $\\frac{n}{2}$-planes of $X$ is equal to $S_I$ for some $I\\subset [0,n+2]$.\nIn particular, let $S_i=S_{\\{i\\}}$. Let $\\varsigma_i=[S_i]$, the homology class of $S_i$. Let $\\varsigma=[S]$. By \\cite[Lemma 3.13]{Rei72},\n\\[\n\\mathsf{h}_{\\frac{n}{2}},\\varsigma_0,\\varsigma_1,\\dots,\\varsigma_{n+2}\n\\]\nis a basis of $H^{\\frac{n}{2}}(X;\\mathbb{Q})$, and \n\\begin{equation}\n\t\\varsigma=\\frac{\\frac{n}{2}+1}{n+1}\\mathsf{h}_{n\/2}-\\frac{1}{n+1}\\sum_{i=0}^{n+2}\\varsigma_i.\n\\end{equation}\nFor $I\\subset [0,n+2]$, let $a(I)=(x_0,\\dots,x_{n+2})\\in \\mathbb{Z}^{n+3}$ such that \n\\[\n x_i=\\begin{cases}\n 1,& \\mbox{if}\\ i\\not\\in I,\\\\\n -1,& \\mbox{if}\\ i\\in I.\n \\end{cases}\n \\] \n For $I$ and $J \\subset [0,n+2]$, suppose $a(I)=(x_0,\\dots,x_{n+2})$ and $a(J)=(y_0,\\dots,y_{n+2})$. Define\n \\[\n a(I,J):=(x_0 y_0,\\dots,x_{n+2}y_{n+2}).\n \\]\n \\begin{lemma}\\label{lem-intersectionDimension}\nLet $m(I,J)$ be the number of $-1$ in the $(n+3)$-tuple $a(I,J)$. Then\n\\begin{equation}\n\t\\dim S_I\\cap S_J=\\begin{cases}\n\t\\frac{n}{2}-m(I,J),& \\mbox{if}\\ m(I,J)\\leq \\frac{n}{2},\\\\\n\t-1,& \\mbox{if}\\ m(I,J)=\\frac{n}{2}+1\\ \\mbox{or}\\ \\frac{n}{2}+2,\\\\\n\tm(I,J)-\\frac{n}{2}-3,& \\mbox{if}\\ m(I,J)\\geq \\frac{n}{2}+3,\n\t\\end{cases}\t\n\\end{equation}\nwhere $\\dim S_I\\cap S_J=-1$ means that the intersection is empty.\n \\end{lemma}\n\\begin{proof}\nOne can show this using directly the defining equations (\\ref{eq-definingEquations-WCoordinates}) of $S$, or from \\cite[Theorem 3.8]{Rei72}.\n\\end{proof} \nBy Lemma \\ref{lem-intersectionNumber-middelDimension} and Lemma \\ref{lem-intersectionDimension}, \n\\begin{equation}\\label{eq-intersectionNumber-Si-Sj}\n\t\\varsigma_i\\cdot \\varsigma_j=\\begin{cases}\n\t(-1)^{\\frac{n}{2}-2}(\\lfloor \\frac{\\frac{n}{2}-2}{2}\\rfloor+1), & \\mbox{if}\\ i\\neq j,\\\\\n\t(-1)^{\\frac{n}{2}}(\\lfloor \\frac{n}{4}\\rfloor+1),& \\mbox{if}\\ i=j.\n\t\\end{cases}\n\\end{equation}\n\n\n\n\\subsection{An explicit orthonormal basis}\\label{sec:explictD-Lattice}\nFor $1\\leq i\\leq n+3$, we define\n \\begin{equation}\n\t\\varepsilon_i=\\varsigma_{i-1}-\\frac{1}{n+1}\\sum_{i=0}^{n+2}\\varsigma_{i}+\\frac{1}{2(n+1)}\\mathsf{h}_{n\/2}.\n\\end{equation}\n\\begin{lemma}\\label{lem-expressingEpsilonbyMiddleDimPlanes}\n\\begin{equation}\n\t\\varsigma_{i-1}=\\varepsilon_i-\\frac{1}{2}\\sum_{i=1}^{n+3}\\varepsilon_i+\\frac{\\mathsf{h}_{n\/2}}{4},\\ \\mbox{for}\\ 1\\leq i\\leq n+3,\n\\end{equation}\n\\begin{equation}\n\t\\varsigma=\\frac{1}{4}\\mathsf{h}_{n\/2}+\\frac{1}{2}\\sum_{i=1}^{n+3}\\varepsilon_i.\n\\end{equation}\n\\end{lemma}\n\\begin{lemma}\\label{lem-intersectionNumber-epsilon}\nFor $1\\leq i\\leq n+3$,\n\\begin{equation}\\label{eq-intersectionNumber-epsilon-h}\n\\varepsilon_i\\cdot \\mathsf{h}_{n\/2}=0,\n\\end{equation}\n\\begin{equation}\\label{eq-intersectionNumber-epsiloni-self}\n\t\\varepsilon_i \\cdot \\varepsilon_i=(-1)^{\\frac{n}{2}},\n\\end{equation}\nfor $i\\neq j$,\n\\begin{equation}\\label{eq-intersectionNumber-epsiloni-epsilonj}\n\t\\varepsilon_i \\cdot \\varepsilon_j=0.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\n(\\ref{eq-intersectionNumber-epsilon-h}) follows from Lemma \\ref{lem-intersectionNumber-middelDimension} (i).\nFor (\\ref{eq-intersectionNumber-epsiloni-self}) and (\\ref{eq-intersectionNumber-epsiloni-epsilonj}) we use Lemma \\ref{lem-intersectionNumber-middelDimension} (i) and (\\ref{eq-intersectionNumber-Si-Sj}): for $1\\leq i\\neq j\\leq n+3$,\n\\begin{eqnarray*}\n&& \\varepsilon_i \\cdot \\varepsilon_j\\\\\n&=& \\big(1-\\frac{2(n+2)}{n+1}+\\frac{(n+3)(n+2)}{(n+1)^2}\\big)\\cdot (-1)^{\\frac{n}{2}-2}(\\lfloor \\frac{\\frac{n}{2}-2}{2}\\rfloor+1)\\\\\n&&+ \\big(-\\frac{2}{n+1}+\\frac{n+3}{(n+1)^2}\\big)\\cdot (-1)^{\\frac{n}{2}}(\\lfloor \\frac{n}{4}\\rfloor+1)\\\\\n&&+ 2(1-\\frac{n+3}{n+1})\\frac{1}{2(n+1)}+\\frac{1}{4(n+1)^2}\\cdot 4\\\\\n&=& \\begin{cases}\n\\frac{n+3}{(n+1)^2}\\cdot \\frac{n}{4}-\\frac{n-1}{(n+1)^2}\\cdot (\\frac{n}{4}+1)-\\frac{1}{(n+1)^2},& \\mbox{if}\\ n\\equiv 0 \\mod 4,\\\\\n-\\frac{n+3}{(n+1)^2}\\cdot \\frac{n-2}{4}+\\frac{n-1}{(n+1)^2}\\cdot (\\frac{n-2}{4}+1)-\\frac{1}{(n+1)^2},& \\mbox{if}\\ n\\equiv 2 \\mod 4,\\\\\n\\end{cases}\\\\\n&=& 0,\n\\end{eqnarray*}\nand\n\\begin{eqnarray*}\n&& \\varepsilon_i \\cdot \\varepsilon_i\\\\\n&=& \\big(-\\frac{2(n+2)}{n+1}+\\frac{(n+3)(n+2)}{(n+1)^2}\\big)\\cdot (-1)^{\\frac{n}{2}-2}(\\lfloor \\frac{\\frac{n}{2}-2}{2}\\rfloor+1)\\\\\n&&+ \\big(1-\\frac{2}{n+1}+\\frac{n+3}{(n+1)^2}\\big)\\cdot (-1)^{\\frac{n}{2}}(\\lfloor \\frac{n}{4}\\rfloor+1)\\\\\n&&+ 2(1-\\frac{n+3}{n+1})\\frac{1}{2(n+1)}+\\frac{1}{4(n+1)^2}\\cdot 4\\\\\n&=& \\begin{cases}\n\\frac{(-n+1)(n+2)}{(n+1)^2}\\cdot \\frac{n}{4}+\\frac{n^2+n+2}{(n+1)^2}\\cdot (\\frac{n}{4}+1)-\\frac{1}{(n+1)^2},& \\mbox{if}\\ n\\equiv 0 \\mod 4,\\\\\n-\\frac{(-n+1)(n+2)}{(n+1)^2}\\cdot \\frac{n-2}{4}-\\frac{n^2+n+2}{(n+1)^2}\\cdot (\\frac{n-2}{4}+1)-\\frac{1}{(n+1)^2},& \\mbox{if}\\ n\\equiv 2 \\mod 4,\\\\\n\\end{cases}\\\\\n&=& (-1)^{\\frac{n}{2}}.\n\\end{eqnarray*}\n\\end{proof}\nWe define \n\\begin{equation}\\label{eq-roots-D-usingExplicitEpsilonClass}\n\t\\begin{cases}\n\t\\alpha_i=\\varepsilon_{i}-\\varepsilon_{i+1}\\ \\mbox{for}\\ 1\\leq i\\leq n+2,\\\\\n\t\\alpha_{n+3}=\\varepsilon_{n+2}+\\varepsilon_{n+3}.\n\t\\end{cases}\n\\end{equation} \nThen from Lemma \\ref{lem-expressingEpsilonbyMiddleDimPlanes} we have\n\\begin{equation}\\label{eq-roots-D-usingExplicitMiddleDimPlanes}\n\t\\begin{cases}\n\t\\alpha_i=\\varsigma_{i-1}-\\varsigma_{i}\\ \\mbox{for}\\ 1\\leq i\\leq n+2,\\\\\n\t\\alpha_{n+3}=\\varsigma_{n+1}+\\varsigma_{n+2}+ 2\\varsigma-\\mathsf{h}_{n\/2}.\n\t\\end{cases}\n\\end{equation} \nSo $\\alpha_i\\in H^*(X;\\mathbb{Z})$. Using Lemma (\\ref{lem-intersectionNumber-epsilon}), and comparing (\\ref{eq-roots-D-usingExplicitEpsilonClass}) and (\\ref{eq-roots-D}), one sees that the group generated by the reflections with respecto $\\alpha_i$'s is the Weyl group $D_{n+3}$. By the Picard-Lefschetz formula, one can take the class $\\alpha_i$ in Section \\ref{sec:monodromy-lattice} to be $\\alpha_i$ defined here. This justifies the notations. Moreover, because of (\\ref{eq-intersectionNumber-epsiloni-self}), we define an orthonormal basis $\\epsilon_i$ of $H^*_{\\mathrm{prim}}(X)$ exactly as (\\ref{eq-normalizedOrthonormalBasis}).\n\n\n\\begin{comment}\nLet $g_i$ be the map that reverses the sign of $Y_i$. Then $g_i$ is an automorphism of $X$, and \n\\[\ng_i(S_{j})=\\begin{cases}\nS_{i,j},& \\mbox{if}\\ i\\neq j,\\\\\nS,& \\mbox{if}\\ i=j.\n\\end{cases}\n\\]\nSo for $i\\neq j-1$,\n\\begin{eqnarray*}\n&& g_{i*}(\\varepsilon_j)=\\varsigma_{j-1,i}-\\frac{1}{n+1}\\sum_{k\\neq i}\\varsigma_{k,i}-\\frac{1}{n+1}\\varsigma+\\frac{1}{2(n+1)}\\mathsf{h}_{n\/2}\\\\\n&=&(\\mathsf{h}_{n\/2}-\\varsigma-\\varsigma_{j-1}-\\varsigma_{i})-\\frac{1}{n+1}\\sum_{k\\neq i}(\\mathsf{h}_{n\/2}-\\varsigma-\\varsigma_{k}-\\varsigma_{i})-\\frac{1}{n+1}\\varsigma+\\frac{1}{2(n+1)}\\mathsf{h}_{n\/2}\\\\\n&=&-\\varsigma_{j-1}+\\frac{1}{n+1}\\varsigma_{i}+\\frac{1}{n+1}\\sum_{k\\neq i}\\varsigma_k-\\frac{1}{2(n+1)}\\mathsf{h}_{n\/2}\\\\\n&=& -\\varepsilon_j,\n\\end{eqnarray*}\nand \n\\begin{eqnarray*}\n&& g_{i*}(\\varepsilon_{i+1})=\\varsigma-\\frac{1}{n+1}\\sum_{k\\neq i}\\varsigma_{k,i}-\\frac{1}{n+1}\\varsigma+\\frac{1}{2(n+1)}\\mathsf{h}_{n\/2}\\\\\n&=&\\varsigma-\\frac{1}{n+1}\\sum_{k\\neq i}(\\mathsf{h}_{n\/2}-\\varsigma-\\varsigma_{k}-\\varsigma_{i})-\\frac{1}{n+1}\\varsigma+\\frac{1}{2(n+1)}\\mathsf{h}_{n\/2}\\\\\n&=&2\\varsigma+\\frac{n+2}{n+1}\\varsigma_{i}+\\frac{1}{n+1}\\sum_{k\\neq i}\\varsigma_k-\\frac{2n+3}{2(n+1)}\\mathsf{h}_{n\/2}\\\\\n&=& \\varsigma_{i}-\\frac{1}{n+1}\\sum_{k\\neq i}\\varsigma_k+\\frac{1}{2(n+1)}\\mathsf{h}_{n\/2}=\\varepsilon_{i+1}.\n\\end{eqnarray*}\n\n\\begin{eqnarray*}\n&& (g_j\\circ g_i)_* (\\varsigma_k)=\\varsigma_{i,j,k}\\\\\n&=& -(\\mathsf{h}_{n\/2}-2\\varsigma-\\varsigma_i-\\varsigma_j-\\varsigma_k)\\\\\n&=& -\\mathsf{h}_{n\/2}+(\\frac{n+2}{n+1}\\mathsf{h}_{n\/2}-\\frac{2}{n+1}\\sum_{l=0}^{n+2}\\varsigma_l)+\\varsigma_i+\\varsigma_j+\\varsigma_k\\\\\n&=& \\varsigma_i+\\varsigma_j+\\varsigma_k-\\frac{2}{n+1}\\sum_{l=0}^{n+2}\\varsigma_l+\\frac{1}{n+1}\\mathsf{h}_{n\/2}\n\\end{eqnarray*}\n\\end{comment}\n\n\n\n\n\\subsection{A potentially enumerative correlator}\\label{sec:enumerativeCorrelator}\nWe begin with giving a working definition of \\emph{enumerative correlators}.\n\\begin{definition}\\label{def-enumerativeCorrelator}\nLet $n\\geq 4$ be an even natural number, and $X$ be an $n$-dimensional smooth complete intersection of two quadrics in $\\mathbb{P}^{n+2}$. Denote the $i$-th projection from $X^{n+3}$ to $X$ by $q_i$. Consider the product of the evaluation morphisms\n\\begin{equation}\n\t\\mathrm{ev}_1\\times\\cdots \\mathrm{ev}_{n+3}: \\overline{\\mathcal{M}}_{0,n+3}(X,\\frac{n}{2})\\rightarrow X^{n+3}.\n\\end{equation}\nLet $I_1,\\dots, I_{n+3}\\subset [0,n+2]$. We say that the correlator \n\\begin{equation}\\label{eq-enumerativeCorrelator-0}\n\t\\langle \\varsigma_{I_1},\\dots,\\varsigma_{I_{n+3}}\\rangle\n\\end{equation}\nis \\emph{enumerative} if there exists an irreducible component $M$ of $\\overline{\\mathcal{M}}_{0,n+3}(X,\\frac{n}{2})$ satisfying the following:\n\\begin{enumerate}\n\t\\item[(i)] $\\dim M$ equals the expected dimension.\n\t\\item[(ii)] The cycles $(\\mathrm{ev}_1\\times\\cdots \\mathrm{ev}_{n+3})(M)$ and $q_1^{-1} S_{I_1}$,\\dots, $q_{n+3}^{-1}S_{I_{n+3}}$ intersect \\emph{properly}, i.e. the dimension of their (scheme theoretic) intersection is 0.\n\t\\item[(iii)] Each irreducible component of $\\overline{\\mathcal{M}}_{0,n+3}(X,\\frac{n}{2})$ other than $M$ has empty intersection with $q_1^{-1} S_{I_1}$,\\dots, $q_{n+3}^{-1} S_{I_n+3}$. \n\\end{enumerate}\n\\end{definition}\n\nThis definition is of course not standard in any sense. It just facilitates the following presentation. \n\nOur strategy to compute the special correlator by the enumerative geometry of $X$ consists of three steps:\n\\begin{enumerate}\n \t\\item Select $I_1,\\dots, I_{n+3}\\subset [0,n+2]$, such that the correlator (\\ref{eq-enumerativeCorrelator-0}) is enumerative.\n\t\\item Express (\\ref{eq-enumerativeCorrelator-0}) in terms of the special correlator, using Lemma \\ref{lem-expressingEpsilonbyMiddleDimPlanes}.\n\t\\item Solve the corresponding enumerative problem by counting curves. More precisely, compute the intersection multiplicities of the intersection \n\t\\begin{equation}\n\t\t(\\mathrm{ev}_1\\times\\cdots \\mathrm{ev}_{n+3})_*[M]\\cap q_1^{*} [S_{I_1}]\\cap\\dots\\cap q_{n+3}^{*} [S_{I_{n+3}}]\n\t\\end{equation}\n\t in the condition (ii) above. \n \\end{enumerate} \n\n \\begin{example}\\label{exp-nonEnumerative-correlator}\n The correlator \n \\begin{equation}\n \t\\langle \\varsigma_{0},\\dots,\\varsigma_{n+3}\\rangle\n \\end{equation}\n should not be enumerative in general. For example, let $n=4$. Then the intersection $S\\cap S_i$ is a line, for $0\\leq i\\leq 6$. The moduli space of conics on $S$ passing through the seven lines has a positive dimension. So there are infinitely many conics passing through $S_0,\\dots,S_6$. Then the conditions (ii) and (iii) in Definition \\ref{def-enumerativeCorrelator} cannot be true simultaneously.\n \\end{example}\n\nStimulated by Example \\ref{exp-nonEnumerative-correlator}, we propose a possibly enumerative correlator.\nLet $S_{[i,i+k-1]}$ be the $\\frac{n}{2}$-plane $S_{i,i+1,\\dots,i+k-1}$ in $X$, where we understand the indices $i$ in the subscript in mod $n+3$ sense. For example, when $n=4$, $S_{[6,6+1]}=S_{6,0}$, $S_{[5,5+3]}=S_{5,6,0,1}$.\n\\begin{lemma}\\label{lem-uniqueS}\n$S$ is the only $\\frac{n}{2}$-plane in $X$ that has non-empty intersections with each of $S_{[i,i+\\frac{n}{2}-1]}$, for $0\\leq i\\leq n+2$. Moreover $S$ meets $S_{[i,i+\\frac{n}{2}-1]}$ at exactly one point.\n\\end{lemma}\n\\begin{proof}\nThe second statement follows from Lemma \\ref{lem-intersectionDimension}.\n\nWe show the first statement.\nLet $I\\subset [0,n+2]$, and suppose $a(I)=(x_0,\\dots,x_{n+2})$. Let $J_i=[i,i+\\frac{n}{2}-1]$ in the mod $n+3$ sense. By Lemma \\ref{lem-intersectionDimension}, the first statement is equivalent to that when $a(I)\\neq (1,1,\\dots,1)$ or $(-1,-1,\\dots,-1)$, there exists $i$ such that $a(I,J_i)=\\frac{n}{2}+1$ or $\\frac{n}{2}+2$, or equivalently the sum of all components of $a(I,J_i)$ is equal to $\\pm 1$. Put\n\\[\np(x_0,\\dots,x_{n+2})=\\sum_{i=0}^{n+2}\\big(a(I,J_i)^2-1\\big).\n\\]\nWe regard $p(x_0,\\dots,x_{n+2})$ as a polynomial of indeterminates $x_0,\\dots,x_{n+2}$.\nThen we are left to show that when $(x_0,\\dots,x_{n+2})\\in \\{1,-1\\}^{n+3}$ and $(x_0,\\dots,x_{n+2})\\neq (1,1,\\dots,1)$ or $(-1,-1,\\dots,-1)$, \n\\begin{equation}\n\tp(x_0,\\dots,x_{n+2})=0.\n\\end{equation}\nPut \n\\[\ny_i=\\sum_{j\\in J_i}x_j.\n\\]\nThen\n\\begin{equation}\n\ta(I,J_i)=\\sum_{i=0}^{n+2}x_i-2y_i.\n\\end{equation}\nSo $p(x_0,\\dots,x_{n+2})$ is manifestly symmetric in $y_i$. It follows that $p(x_0,\\dots,x_{n+2})$ is symmetric in $x_i$, rather than only cyclicly symmetric. So it suffices to show the statement for all $I\\subset[0,n+2]$ of the form $I=[0,k]$ for some $0\\leq k0$ and $\\sum_{k=0}^{2n+3}i_k>3$ then return 0. (By (\\ref{eq-FCA}))\n\t\\item If $i_1>0$ and $\\sum_{k=0}^{2n+3}i_k>3$ then apply (\\ref{eq-recursion-EulerVecField-even(2,2)}).\n\t\\item If $\\sum_{k=2}^{n}i_k>0$ and $\\sum_{k=0}^{2n+3}i_k\\geq 2$ then apply \n\t(\\ref{eq-recursion-ambient-simplified-even(2,2)}).\n\t\\item If $\\sum_{k=n+1}^{2n+3}i_k=1$, return 0. (By Theorem \\ref{thm-monodromy-evenDim(2,2)})\n\n\t\\item If $\\sum_{k=n+1}^{2n+3}i_k=3$, return 0. (By Theorem \\ref{thm-monodromy-evenDim(2,2)})\n\t\\item If $\\sum_{k=0}^{n}i_k=0$ and $\\sum_{k=n+1}^{2n+3}i_k=4$, then if the nonzero components of $(i_{n+1},\\dots,i_{2n+3})$ is a 4, or two 2's, then return 1, otherwise return 0. (By Theorem \\ref{thm-monodromy-evenDim(2,2)} and Theorem \\ref{thm-reconstruction-even(2,2)})\n\t\\item If $\\sum_{k=0}^{n}i_k=0$ and $i_k=1$ for all $n+1\\leq k\\leq 2n+3$, return an indeterminate \\texttt{x}, which stands for the unknown special correlator (\\ref{eq-specialLength(n+3)Invariant-even(2,2)}).\n\t\\item If $\\sum_{k=0}^{n}i_k=0$, $\\sum_{k=n+1}^{2n+3}i_k>4$ and there is only one nonzero component $i_a$ in $(i_{n+1},\\dots,i_{2n+3})$, we rearrange $I$ such that $I=(0,\\dots,0,i_{2n+3})$, and take $b=n+1$ so that $i_b=0$. Then apply (\\ref{eq-recursion-primitive-aabb-tau-simplified-even(2,2)}) to $a=2n+3$, $b=n+1$, and $I=I-2e_a$, and thus return the RHS of (\\ref{eq-recursion-primitive-aabb-tau-simplified-even(2,2)}) divided by a nonzero number.\n\t\\item We define a function \\texttt{indexTriple}. The input is $(i_{n+1},\\dots,i_{2n+3})$. Then we check that neither of the conditions\n\t\\begin{enumerate}\n\t \t\\item[(i)] there is only one nonzero component in $(i_{n+1},\\dots,i_{2n+3})$\n\t \t\\item[(ii)] $i_k=1$ for all $n+1\\leq k\\leq 2n+3$\n\t \\end{enumerate}\n\tis true. When this is the case, the output is the indices $a,b,c$ as given in Remark \\ref{rem:recursion-boundOfIndex}. This produce in a definite way the indices $a,b,c$ in the proof of Theorem \\ref{thm-reconstruction-even(2,2)}.\n\t\\item If $\\sum_{k=0}^{n}i_k=0$, $\\sum_{k=n+1}^{2n+3}i_k>4$, and the neither of the conditions (i) and (ii) in the last step is true, we apply \\texttt{indexTriple} to $(i_{n+1},\\dots,i_{2n+3})$ to get $(a,b,c)$. Then we apply (\\ref{eq-recursion-primitive-abcc-even(2,2)}) to $(a,b,c)$ and $I=I-e_a-e_b$, and thus return the RHS of (\\ref{eq-recursion-primitive-abcc-even(2,2)}) divided by a nonzero number. (As we have shown in the proof of Theorem \\ref{thm-reconstruction-even(2,2)} that the coefficient on the LHS of (\\ref{eq-recursion-primitive-abcc-even(2,2)}) is nonzero in this case.)\n\\end{enumerate}\n\t \t\n\n\n\\end{algorithm}\nWe implemented Algorithm \\ref{algorithm-correlator-even(2,2)} by the command \n\\[\n\\mbox{\\texttt{correlatorEvenIntersecTwoQuadricsRecursionInTauCoord}}\n\\]\n in the Macaulay2 package \\texttt{QuantumCohomologyFanoCompleteIntersection}. The input of this command is a list $\\{n,I\\}$, and the output is $\\partial_{\\tau^I}F(0)$. For example, running\n\\[\n\\mbox{\\texttt{correlatorEvenIntersecTwoQuadricsRecursionInTauCoord} $\\{4,\\{0,0,7,0,0,0,0,0,0,0,0,0\\}\\}$}\n\\]\nreturns 46656, so $\\langle\\mathsf{h}_2,\\mathsf{h}_2,\\mathsf{h}_2,\\mathsf{h}_2,\\mathsf{h}_2,\\mathsf{h}_2,\\mathsf{h}_2\\rangle=46656$. Running\n\\[\n\\mbox{\\texttt{correlatorEvenIntersecTwoQuadricsRecursionInTauCoord} $\\{4,\\{0,0,5,0,0,2,0,0,0,0,0,0\\}\\}$}\n\\]\nreturns $-624$, so $\\langle \\mathsf{h}_2,\\mathsf{h}_2,\\mathsf{h}_2,\\mathsf{h}_2,\\mathsf{h}_2,\\epsilon_1,\\epsilon_1\\rangle=-624$. Similarly one can get the correlators in Lemma \\ref{lem-correlatorsOfLength7-4dim}. The special correlator (\\ref{eq-specialLength(n+3)Invariant-even(2,2)}) is denoted by \\texttt{x} in this package. For example running\n\\begin{multline*}\n\\texttt{correlatorEvenIntersecTwoQuadricsRecursionInTauCoord}\\\\ \\{6,\\{0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,0,0\\}\\}\n\\end{multline*}\nreturns $8\\mbox{\\texttt{x}}^2-2$. One can check Conjecture \\ref{conj-unknownCorrelator-Even(2,2)-quadraticEquation} in this way.\n\n\n\\end{appendices}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction\nInflation theory has claimed that the origin of the large-scale structure of the universe and temperature fluctuations in the cosmic microwave background radiations is quantum fluctuations. Remarkably, the inflation theory also predicts the existence of primordial gravitational waves stemming from the\nquantum fluctuations of the spacetime (relic gravitons). After the discovery\nof gravitational waves from a black hole binary system~\\cite{LIGOScientific:2016aoc},\nthe detection of primordial gravitational waves has been\nthe most important research objective~\\cite{Kawamura:2011zz,Amaro-Seoane:2012aqc}.\n\nThe notable nature of primordial gravitational waves is their quantum origin. If the relic gravitons were found, it would strongly support the inflationary universe. The finding of the relic gravitons would also give a hint of quantum gravity. Hence, it is extremely important to explore\nthe quantum nature of the primordial gravitational waves.\n\nIt is well known that the generation of relic gravitons\ncan be interpreted as the squeezing process of a quantum state during inflation~\\cite{Grishchuk:1989ss,Grishchuk:1990bj,Albrecht:1992kf,Polarski:1995jg}. Since the degree of squeezing is extremely high, the quantum state is highly entangled between two modes with opposite wave number vectors due to conservation of momentum. The squeezed state of the relic gravitons is a key for \nproving the nonclassicality of primordial gravitational waves. \nIn fact, the squeezed gravitons can significantly enhance the quantum noise in interferometers~\\cite{Parikh:2020nrd,Kanno:2020usf,Parikh:2020kfh,Parikh:2020fhy,Kanno:2021gpt}. \nHence, we need to show the degree of the squeezing generated during inflation survives under the decoherence processes in the evolution of the universe. So far, the decoherence process due to short wavelength modes of a field has been investigated~\\cite{Lombardo:2005iz,Martineau:2006ki,Burgess:2006jn,Nelson:2016kjm}. However, it is argued \nthat the decoherence obtained by tracing out the short wavelength modes is false decoherence~\\cite{2011arXiv1110.2199U}. \nThus, it is worth studying different decoherence processes.\n\n \nIn this paper, as a source of the decoherence, we assume the presence of a sizable magnetic field at the beginning of inflation. We then consider conversion process of the squeezed gravitons into photons during inflation in the case of minimal coupling between gravitons and photons~\\cite{Gertsenshtein:1962,Raffelt:1987im,Chen:1994ch,Cillis:1996qy}. \nThe squeezed state of gravitons may turn into the squeezed state of photons due to the graviton-photon conversion. \nHence, it is important to clarify to what extent the squeezing of the relic gravitons survives at present. \nThe purpose of this paper is to compute the \ndegree of squeezing parameters of graviton and photon and cross squeezing parameter between gravitons and photons during inflation.\n\nThe paper is organized as follows: In section 2, we derive basic equations for analyzing the conversion process of gravitons into photons during inflation. \nIn section 3, we explain the perturbative formalism for solving a coupled system between gravitons and photons in order to obtain\nthe time evolution of mode functions.\nIn section 4, we derive Bogoliubov transformations\ndue to the squeezing process in the presence of primordial magnetic fields. \nIn section 5, we deduce formulae for the squeezing parameters and \n reveal the time evolution of the squeezing parameters numerically and analytically. \nWe also discuss the implications of our results.\nThe final section is devoted to the conclusion.\n\n\n\\section{Graviton-photon conversion during inflation\n\n\n\nWe represent the graviton in a spatially flat expanding background by the tensor mode perturbation in the three-dimensional metric, \n\\begin{eqnarray}\nds^2=a^2(\\eta)\\left[-d\\eta^2+\\left(\\delta_{ij}+h_{ij}\\right)dx^idx^j\\right]\\,,\n\\end{eqnarray}\nwhere $\\eta$ is the conformal time and the metric perturbation $h_{ij}$ satisfies the transverse traceless conditions $h_{ij}{}^{,j}=h^i{}_i=0$. The spatial indices $i,j,k,\\cdots$ are raised and lowered by $\\delta^{ij}$ and $\\delta_{k\\ell}$.\n\nThe Einstein-Hilbert action and the action for the electromagnetic field is given by\n\\begin{eqnarray}\nS=S_g+S_A=\\frac{M_{\\rm pl}^2}{2}\\int d^4x \\sqrt{-g}\n\\,\nR-\\frac{1}{4}\\int d^4x \\sqrt{-g}\\,\nF^{\\mu\\nu}\nF_{\\mu\\nu}\n\\label{original action}\\,,\n\\end{eqnarray}\nwhere $M_{\\rm pl}=1\/\\sqrt{8\\pi G}$ is the Planck mass. The gauge field $A_\\mu$ represents the photon and the field strength is defined by $F_{\\mu\\nu}=\\partial_\\mu A_{\\nu}-\\partial_\\nu A_{\\mu}$.\nExpanding the Einstein-Hilbert action up to the second order in perturbations $h_{ij}$, we find\n\\begin{eqnarray}\n\\delta S_g=\\frac{M_{\\rm pl}^2}{8}\\int d^4x\\,a^2\\left[\nh^{ij\\prime}\\,h_{ij}^\\prime-h^{ij,k}h_{ij,k}\n\\right]\\,.\n\\label{action:g}\n\\end{eqnarray}\nHere, a prime denotes the derivative with respect to the conformal time. The action for the photon up to second order in perturbations $A_i$ reads\n\\begin{eqnarray}\n\\delta S_A=\\frac{1}{2}\\int d^4x\\left[A_i^{\\prime\\, 2}-A_{k,i}^2\\right]\\,,\n\\label{action:A}\n\\end{eqnarray}\nwhere the photon field satisfies the Coulomb gauge $A_0=0$ and $A^i{}_{,i}=0$.\nThe action for the interaction between the graviton and the photon up to second order in perturbations $h_{ij}, A^i$ is found to be\n\\begin{eqnarray}\n\\delta S_{\\rm I}=\\int d^4x \\left[\n\\varepsilon_{i\\ell m}B_m h^{ij}\\left(\\partial_j A_\\ell\n-\\partial_\\ell A_j\\right)\n\\right]\\,.\n\\label{action:I}\n\\end{eqnarray}\nNote that $B_m=\\varepsilon_{mj\\ell}\\,\\partial_j A_\\ell$ is a constant background magnetic field that we assumed the presence at the beginning of inflation.\n\nAt quadratic order, it is convenient to expand $h_{ij}(\\eta,x^i)$ and $A_i(\\eta,x^i)$ in the Fourier modes,\n\\begin{align}\n&h_{ij}(\\eta,x^i)=\\frac{2}{M_{\\rm pl}} \\sum_{P}\\frac{1}{(2\\pi)^{3\/2}} \\int d^3 k\\,h^{P}_{\\bm k}(\\eta)\\, e_{ij}^{P}(\\bm{k})\\,e^{i\\bm{k}\\cdot\\bm{x}}\\,,\\\\\n&A_i(\\eta,x^i)=\\sum_{P} \\frac{\\pm i}{(2\\pi)^{3\/2}}\n\\int d^3 k\\,A^{P}_{\\bm k}(\\eta)\\,e_i^{P}(\\bm{k})\\, e^{i\\bm{k}\\cdot\\bm{x}},\n\\end{align}\nwhere three-vectors are denoted by bold math type and $e_{ij}^{P}(\\bm{k})$ and $e_i^{P}(\\bm{k})$ are the polarization tensors and vectors for the ${\\bm k}$ mode respectively normalized as $e^{ijP}(\\bm{k})e_{ij}^{Q}(\\bm{k})=\\delta^{PQ}$ and $e^{iP}(\\bm{k}) e_i^{Q}(\\bm{k})=\\delta^{PQ}$ with $P,Q=+,\\times$.\nUsing the canonical variable $y^P_{\\bm k}(\\eta)=a(\\eta)h_{\\bm k}^{P}(\\eta)$,\nwe can rewrite the quadratic actions~(\\ref{action:g}), (\\ref{action:A}) and (\\ref{action:I}) as\n\\begin{eqnarray}\n\\delta S_g&=&\\frac{1}{2}\\sum_P\\int d^3 k\\,d\\eta\\left[\\,\n|y_{\\bm k}^{P\\,\\prime}|^2\n-k^2|y_{\\bm k}^P|^2\n-\\frac{a^\\prime}{a}y_{\\bm k}^{P}y_{-\\bm k}^{P\\,\\prime}\n-\\frac{a^\\prime}{a}y_{-\\bm k}^{P}y_{\\bm k}^{P\\,\\prime}\n+\\left(\\frac{a^\\prime}{a}\\right)^2|y_{\\bm k}^P|^2\n\\right]\\,,\n\\label{action:g2}\\\\\\\n\\delta S_A&=&\\frac{1}{2}\\sum_P\\int d^3 k\\,d\\eta\\left[\\,\n|A_{\\bm k}^{P\\,\\prime}|^2-k^2|A_{\\bm k}^P|^2\n\\,\\right]\\,,\n\\label{action:A2}\\\\\n\\delta S_I&=&\\frac{2}{M_{\\rm pl}}\\sum_{P,Q}\\int d^3 k\\,d\\eta\\,\\frac{1}{a}\\left[\n\\varepsilon_{i\\ell m}\\,B_m\\,y_{\\bm k}^PA_{-\\bm k}^Q\n\\,e_{ij}^P(\\bm k)\\Bigl\\{ik_\\ell\\,e_{j}^Q(-\\bm k)-ik_j\\,e_{\\ell }^Q(-\\bm k)\\Bigr\\}\\right]\n\\label{action:I2}\\,,\n\\end{eqnarray}\nwhere $k=|\\bm k|$. Polarization vectors $e^{i+}, e^{i\\times}$ and a vector $k^i\/k$ constitute an orthnormal basis.\nWithout loss of generality, we assume the constant background magnetic field is in the ($k^i, e^{i \\times}$)-plane.\n\\begin{figure}[H]\n\\centering\n \\includegraphics[keepaspectratio, scale=0.6]{Configuration.pdf}\n \\renewcommand{\\baselinestretch}{3}\n \\caption{Configuration of the polarization vector ${\\bm e}^P(\\bm k)$, wave number ${\\bm k}$, and background magnetic field ${\\bm B}$.}\n \\label{Configuration}\n \\end{figure}\n\\noindent\nThe polarization tensors can be written in terms of polarization vectors \n$e^{i+}$ and $e^{i\\times}$\n as\n\\begin{align}\n &e_{ij}^+(\\bm{k})\n =\\frac{1}{\\sqrt{2}} \\Bigl\\{\n e^+_i(\\bm{k}) e^+_j(\\bm{k})-e^\\times_i(\\bm{k}) e^\\times_j(\\bm{k})\n \\Bigr\\}\\,,\\\\\n &e_{ij}^\\times(\\bm{k})\n =\\frac{1}{\\sqrt{2}} \n \\Bigl\\{\n e^+_i(\\bm{k}) e^\\times_j(\\bm{k})+e^\\times_i(\\bm{k}) e^+_j(\\bm{k})\n \\Bigr\\}\\, .\n \n\\end{align}\nIn the following, we assume\n\\begin{eqnarray}\ne_i^\\times(-\\bm{k})=-e_i^\\times(\\bm{k})\\,.\n\\end{eqnarray}\nThe action (\\ref{action:I2}) is then reduced into\n\\begin{eqnarray}\n\\delta S_I&=&\\frac{\\sqrt{2}}{M_{\\rm pl}}\\int d^3k\\,d\\eta\\,\\frac{1}{a}\n\\left[\\,\\lambda(\\bm k)\\,y_{\\bm k}^+(\\eta)\\,A_{-\\bm k}^+(\\eta)+\\lambda(\\bm k)\\,y_{\\bm k}^\\times(\\eta)\\,A_{-\\bm k}^\\times(\\eta)\\,\\right]\\,,\n\\label{action:I3}\n\\end{eqnarray}\nwhere we defined the coupling between graviton and photon as \n\\begin{align}\n\\lambda(\\bm{k})\\equiv\\frac{\\sqrt{2}}{M_{\\rm pl}}\n\\varepsilon^{i\\ell m}\\,e_i^+\\,k_\\ell\\,B_m\\,.\n\\label{coupling}\n\\end{align}\nHere, the conditions for the graviton and photon to be real read,\n$h_{-\\bm k}^{+,\\times}(\\eta)=h_{\\bm k}^{*\\,+,\\times}(\\eta)$ and $A_{-\\bm k}^{+,\\times}(\\eta)=-A_{\\bm k}^{*\\,+,\\times}(\\eta)$\\,.\nBelow, we focus on the plus polarization and omit the index $P$ unless there may be any confusion. \n\nIn the case of de Sitter space, the scale factor is given by $a(\\eta)=-1\/(H\\eta)$ where $-\\infty<\\eta<0$.\nThe variation of the actions (\\ref{action:g2}), (\\ref{action:A2}) and (\\ref{action:I3}) with respect to the graviton and the photon fields gives\n\\begin{align}\n &y_{\\bm{k}}''+\\left(k^2-\\frac{2}{\\eta^2}\\right)y_{\\bm{k}}=\\lambda H \\eta A_{\\bm{k}}\\,,\n \\label{eom:graviton}\n \\\\\n &A_{\\bm{k}}''+k^2A_{\\bm{k}}=\\lambda H \\eta\\,y_{\\bm{k}}\\,.\n \\label{eom:photon}\n\\end{align}\nIf we define the Lagrangian in the actions (\\ref{action:g2}) and (\\ref{action:A2}) by $\\delta S_g=\\int d\\eta\\,L_g$ and $\\delta S_A=\\int d\\eta\\,L_A$, the conjugate momenta of graviton $p_{\\bm k}$ and photon $\\pi_{\\bm k}$ are respectively given by\n\\begin{align}\n &p_{\\bm{k}}(\\eta)=\\frac{\\partial L_g}{\\partial y^\\prime_{-\\bm k}}=y_{\\bm{k}}'(\\eta)+\\frac{1}{\\eta}y_{\\bm{k}}(\\eta) \\ , \n \\label{p}\\\\\n &\\pi_{\\bm{k}}(\\eta)=\\frac{\\partial L_A}{\\partial A^\\prime_{-\\bm k}}=A_{\\bm{k}}'(\\eta) \\ .\n \\label{pi}\n\\end{align}\nNow we promote variables $y_{\\bm k}(\\eta), A_{\\bm k}(\\eta)$ and their momenta $p_{\\bm k}(\\eta), \\pi_{\\bm k}(\\eta)$ into operators. The annihilation operator for the graviton is expressed by canonical variables as\n\\begin{eqnarray}\n\\hat{a}_y(\\eta,{\\bm k})=\\sqrt{\\frac{k}{2}}\\hat{y}_{\\bm k}(\\eta)+\\frac{i}{\\sqrt{2k}}\\hat{p}_{\\bm k}(\\eta)\\,.\n\\label{y:annihi}\n\\end{eqnarray}\nIn the same way, the annihilation operator for photon is given by\n\\begin{eqnarray}\n\\hat{a}_A(\\eta,{\\bm k})=\\sqrt{\\frac{k}{2}}\\hat{A}_{\\bm k}(\\eta)+\\frac{i}{\\sqrt{2k}}\\hat{\\pi}_{\\bm k}(\\eta)\\,.\n\\label{A:annihi}\n\\end{eqnarray}\nThe commutation relations $[\\hat{a}_y(\\eta,{\\bm k}),\\hat{a}^\\dag_y(\\eta,-{\\bm k}^\\prime)]=\\delta({\\bm k}+{\\bm k}^\\prime)$ and $[\\hat{a}_A(\\eta,{\\bm k}),\\hat{a}^\\dag_A(\\eta,-{\\bm k}^\\prime)]=\\delta({\\bm k}+{\\bm k}^\\prime)$ guarantee the canonical commutation relations $[y_{\\bm k}(\\eta),p_{{\\bm k}^\\prime}(\\eta)]=i\\delta({\\bm k}-{\\bm k}^\\prime)$ and $[A_{\\bm k}(\\eta),\\pi_{{\\bm k}^\\prime}(\\eta)]=i\\delta({\\bm k}-{\\bm k}^\\prime)$.\nNotice that the annihilation operator becomes time dependent through the time dependence of canonical variables. Thus, the vacuum defined by $\\hat{a}(\\eta,{\\bm k})|0\\rangle =0$ is time dependent as well and the vacuum in this formalism turns out to be defined at every moment.\n\n\n\nIn this paper, we suppose $B_m\/M_{\\rm pl}\\ll 1$ so that the coupling between graviton and photon~(\\ref{coupling}) is weak . Then we solve the Eqs.~(\\ref{eom:graviton}) and (\\ref{eom:photon}) iteratively up to the second order in $y_{\\bm{k}}$ and $A_{\\bm{k}}$ in the next section.\n\n\\section{Time evolution of mode functions\n\nUsing the basic equations presented in the previous section, we perturbatively derive mode functions in this section.\n\n\n\\subsection{Zeroth order}\n\nBy letting $\\lambda=0$ in Eqs.~(\\ref{eom:graviton}) and (\\ref{eom:photon}), the equations of the zeroth order approximation become\n\\begin{align}\n &\\hat{y}_{\\bm{k}}^{(0)\\prime\\prime}+\\left(k^2-\\frac{2}{\\eta^2}\\right)\\hat{y}_{\\bm{k}}^{(0)}=0\\,,\n \\label{GWeqs}\n \\\\\n &\\hat{A}_{\\bm{k}}^{(0)\\prime\\prime}+k^2\\hat{A}_{\\bm{k}}^{(0)}=0 \\, ,\n \\label{EMeqs}\n\\end{align}\nwhere the superscript $(0)$ denotes the zeroth order. The solutions of the above equations are\n\\begin{align}\n &\\hat{y}_{\\bm k}^{(0)}(\\eta)=u_{\\bm k}^{(0)}(\\eta) ~\\hat{c}\n +u_{\\bm k}^{(0)*}(\\eta)~\\hat{c}^\\dagger\\,, \n \\label{0th:graviton}\\\\\n &\\hat{A}_{\\bm k}^{(0)}(\\eta)=v_{\\bm k}^{(0)}(\\eta) ~\\hat{d}\n +v_{\\bm k}^{(0)*}(\\eta)~\\hat{d}^\\dagger,\n \\label{0th:photon}\n\\end{align}\nwhere $\\hat{c}$\\,($\\hat{d}$) and its conjugate \n$\\hat{c}^\\dag$($\\hat{d}^\\dag$) are constant operators of integration. We choose the properly normalized positive frequency mode in the remote past as a basis, which is expressed as\n\\begin{align}\n &u_{\\bm k}^{(0)}(\\eta)=\\frac{1}{\\sqrt{2k}} \\biggl(1-\\frac{i}{k\\eta}\\biggr) e^{-ik\\eta}\n \\,,\\qquad\n v_{\\bm k}^{(0)}(\\eta)=\\frac{1}{\\sqrt{2k}} e^{-ik\\eta}.\n\\end{align}\n\n\n\\subsection{First order}\n\nInserting the solutions of zeroth order approximation (\\ref{0th:graviton}) and (\\ref{0th:photon}) into the r.h.s of Eqs.~(\\ref{eom:graviton}) and (\\ref{eom:photon}) as the source terms, the equations of the first order approximation are written as\n\\begin{align}\n &\\hat{y}_{\\bm{k}}^{(1)\\prime\\prime}+\\left(k^2-\\frac{2}{\\eta^2}\\right)\\hat{y}_{\\bm{k}}^{(1)}\n =\\lambda H \\eta \\hat{A}_{\\bm{k}}^{(0)}\\,,\n \\label{GWeq1}\n \\\\\n &\\hat{A}_{\\bm{k}}^{(1)\\prime\\prime}+k^2\\hat{A}_{\\bm{k}}^{(1)}=\\lambda H \\eta \\hat{y}_{\\bm{k}}^{(0)} \\,.\n \\label{EMeq1}\n\\end{align}\nThe effect of photon comes in Eq.~(\\ref{GWeq1}). Using the Green function\n\\begin{eqnarray}\nG_{\\rm dS}(\\eta,\\eta')=\n\\frac{1}{2ik} \\biggl(1+\\frac{i}{k\\eta'}\\biggr)\n\\biggl(1-\\frac{i}{k\\eta}\\biggr)\ne^{-ik(\\eta-\\eta')}\n-\\frac{1}{2ik} \n\\biggl(1-\\frac{i}{k\\eta'}\\biggr)\n\\biggl(1+\\frac{i}{k\\eta}\\biggr)\ne^{ik(\\eta-\\eta')}\\,,\n\\end{eqnarray}\nwe obtain the solution as\n\\begin{align}\n \\hat{y}^{(1)}_{\\bm k}(\\eta)\n &=-\\int_{\\eta_i}^\\eta d\\eta'\n G_{\\rm dS} (\\eta,\\eta')\n \\lambda H \\eta'\\hat{A}^{(0)}_{\\bm k}(\\eta')\\nonumber\\\\\n &=-\\int_{\\eta_i}^\\eta d\\eta'\n G_{\\rm dS} (\\eta,\\eta')\n \\lambda H \\eta' \n v^{(0)}_{\\bm k}(\\eta')~\\hat{d}\n -\\int_{\\eta_i}^\\eta d\\eta'\n G_{\\rm dS} (\\eta,\\eta')\n \\lambda H\\eta'\n v^{(0)*}_{\\bm k}(\\eta') \n ~\\hat{d}^\\dagger \\nonumber\\\\\n &\\equiv u^{(1)}_{\\bm k}(\\eta)~\\hat{d}+u^{(1)*}_{\\bm k}(\\eta)~\\hat{d}^\\dagger\\,,\n \\label{1st:graviton}\n\\end{align}\nwhere $\\eta_i$ is an initial time. From the first line to the second line we used Eq.~(\\ref{0th:photon}). In the last line, we defined the first order correction due to the source of photon to the positive frequency mode of graviton by\n\\begin{align}\n u^{(1)}_{\\bm k}(\\eta)&\\equiv-\\int_{\\eta_i}^\\eta d\\eta'\n G_{\\rm dS} (\\eta,\\eta')\n \\lambda H \\eta' \n v^{(0)}_{\\bm k}(\\eta') \\ .\n \\label{u1Green}\n\\end{align}\nAfter integration, we have\n\\begin{eqnarray}\nu^{(1)}_{\\bm k}(\\eta)&=&\n\\frac{ \\lambda H }{8 \\sqrt{2} \\eta k^{9\/2}} \n\\left[e^{- i k \\eta } \n\\Bigl\\{2 i \\eta ^3 k^3+\\eta k \n\\Bigl(2 \\eta_i k (2-i \\eta_i k)+3 i\\Bigr)\n-2 \\eta_i k \n(\\eta_i k+2 i)\n+3\\Bigr\\}\n\\right.\\nonumber\\\\\n&&\\left.\n-e^{ i k (\\eta -2\\eta_i ) } \n(\\eta k+i) (2 \\eta_i k-3 i)\n\\right] \\ .\n\\label{u1}\n\\end{eqnarray}\n\nSimilarly, the effect of graviton comes in Eq.~(\\ref{EMeq1}).\nBy using the Green function \n\\begin{eqnarray}\nG_{\\rm M}(\\eta,\\eta')=-\\frac{1}{k} \\sin{k(\\eta-\\eta')}\\,,\n\\end{eqnarray}\nwe have\n\\begin{align}\n \\hat{A}^{(1)}_{\\bm k}(\\eta)\n &=-\\int_{\\eta_i}^\\eta d\\eta'\n G_{\\rm M} (\\eta,\\eta')\n \\lambda H \\eta' \\hat{y}^{(0)}_{\\bm k}(\\eta')\\nonumber\\\\\n &=-\\int_{\\eta_i}^\\eta d\\eta' G_{\\rm M}(\\eta,\\eta')\n \\lambda H \\eta' u_{\\bm k}^{(0)}(\\eta)~\\hat{c}\n -\\int_{\\eta_i}^\\eta d\\eta' G_{\\rm M}(\\eta,\\eta')\n \\lambda H \\eta' u_{\\bm k}^{(0)*}(\\eta)~\\hat{c^\\dagger}\\nonumber\\\\\n &\\equiv v^{(1)}_{\\bm k}(\\eta)~\\hat{c}+v^{(1)*}_{\\bm k}(\\eta)~\\hat{c}^\\dagger \\ ,\n \\label{1st:photon}\n\\end{align}\nwhere we used Eq.~(\\ref{0th:graviton}) from the first line to the second line. We also defined the first order correction due to the source of graviton to the positive frequency mode of photon in the third line by\n\\begin{align}\n v^{(1)}_{\\bm k}(\\eta)&\\equiv-\\int_{\\eta_i}^\\eta d\\eta'\n G_{\\rm M} (\\eta,\\eta')\n \\lambda H \\eta' \n u^{(0)}_{\\bm k}(\\eta') \\ .\n \\label{v1Green}\n\\end{align}\nMore explicitly, the above is written as\n\\begin{eqnarray}\nv^{(1)}_{\\bm k}(\\eta)\n&=&\n\\frac{ \\lambda H }{8 \\sqrt{2} k^{7\/2}}\n\\left[e^{- i k \\eta} \\Bigl\\{2 i k^2 (\\eta^2 -\\eta_i^2) +k (6 \\eta -4 \\eta_i)-3 i\\Bigr\\}\n+e^{ i k(\\eta -2\\eta_i ) } (-2 \\eta_i k+3 i)\\right] \\ .\\nonumber\\\\\n\\label{v1} \n\\end{eqnarray}\n\n\n\n\\subsection{Second order}\n\nBy plugging the solution of the first order approximation (\\ref{1st:graviton}) and (\\ref{1st:photon}) into the r.h.s of Eqs.~(\\ref{eom:graviton}) and (\\ref{eom:photon}) as the source terms, the equations of the second order approximation are\n\\begin{align}\n &y_{\\bm{k}}^{(2)\\prime\\prime}+\\left(k^2-\\frac{2}{\\eta^2}\\right)y_{\\bm{k}}^{(2)}\n =\\lambda H \\eta A_{\\bm{k}}^{(1)}\\,,\n \\label{GWeq2}\n \\\\\n &A_{\\bm{k}}^{(2)\\prime\\prime}+k^2A_{\\bm{k}}^{(2)}=\\lambda H \\eta\\,y_{\\bm{k}}^{(1)} \\,.\n \\label{EMeq2}\n\\end{align}\nAt this order, the effect of graviton itself comes in Eq.~(\\ref{GWeq2}). The solution is written by the Green function $G_{\\rm dS}$ such as\n\\begin{align}\n \\hat{y}^{(2)}_{\\bm k}(\\eta)\n &=-\\int_{\\eta_i}^\\eta d\\eta'\n G_{\\rm dS} (\\eta,\\eta')\n \\lambda H \\eta' \\hat{A}^{(1)}_{\\bm k} (\\eta') \\nonumber\\\\\n &=-\\int_{\\eta_i}^\\eta d\\eta'\n G_{\\rm dS} (\\eta,\\eta')\n \\lambda H \\eta' v^{(1)}_{\\bm k}(\\eta')~\\hat{c}\n -\\int_{\\eta_i}^\\eta d\\eta'\n G_{\\rm dS} (\\eta,\\eta')\n \\lambda H \\eta' v^{(1)*}_{\\bm k}(\\eta')~\\hat{c}^\\dagger\n \\nonumber\\\\\n &=-\\int_{\\eta_i}^{\\eta} d\\eta' G_{\\rm dS} (\\eta,\\eta')\n \\lambda H \\eta' \\Biggl(-\\int_{\\eta_i}^{\\eta'}d\\eta'' G_{\\rm M}(\\eta',\\eta'')\\lambda H \\eta''u^{(0)}_{\\bm k}(\\eta'') \\Biggr)~ \\hat{c} \\nonumber\\\\\n &\\qquad -\\int_{\\eta_i}^{\\eta} d\\eta' G_{\\rm dS} (\\eta,\\eta')\n \\lambda H \\eta' \\Biggl(-\\int_{\\eta_i}^{\\eta'}d\\eta''G_{\\rm M}(\\eta',\\eta'') \\lambda H \\eta''u^{(0)*}_{\\bm k}(\\eta'') \\Biggr)~ \\hat{c}^\\dagger \\nonumber\\\\\n &\\equiv u^{(2)}_{\\bm k}(\\eta)~\\hat{c}+u^{(2)*}_{\\bm k}(\\eta)~\\hat{c}^\\dagger, \n\\end{align}\nwhere we used Eqs.~(\\ref{1st:photon}) and (\\ref{v1Green}) in the second and the third lines respectively. In the last line, we defined\n\\begin{align}\n u^{(2)}_{\\bm k}(\\eta)\n &\\equiv\n -\\int_{\\eta_i}^\\eta d\\eta' G_{\\rm dS} (\\eta,\\eta')\n \\lambda H \\eta'\n v^{(1)}_{\\bm k}(\\eta')\n \\nonumber\\\\\n &=\\int_{\\eta_i}^\\eta d\\eta' G_{\\rm dS} (\\eta,\\eta')\n \\lambda H \\eta'\n \\int_{\\eta_i}^{\\eta'} d\\eta'' G_{\\rm M}(\\eta',\\eta'') \n \\lambda H \\eta'' u^{(0)}_{\\bm k} (\\eta'').\n \\label{u2Green}\n\\end{align}\nBy performing the integration, the explicit form of the $u^{(2)}_{\\bm k}(\\eta)$ is found to be\n\\begin{eqnarray}\nu^{(2)}_{\\bm k}(\\eta)\n&=&\n-\\frac{ \\lambda^2 H^2}{192 \\sqrt{2} \\eta k^{15\/2}}\\nonumber\\\\\n&&\\times\n\\Biggl[\n3 e^{ i k ( \\eta -2\\eta_i)} \n\\biggl\\{2 \\eta^3 k^3 (-3-2 i \\eta_i k)+\\eta k \\biggl(-35+2 \\eta_i k (\\eta_i k (11+2 i \\eta_i k)-23 i)\\bigg)\\nonumber\\\\\n&& \\qquad +2 \\eta_i k (23+\\eta_i k (-2 \\eta_i k+11 i))-35 i\n\\biggr\\}\\nonumber\\\\\n&&\n+e^{- i k \\eta }\n\\Biggl\\{k \n\\Bigl(-105 \\eta +72 \\eta_i+6 \\eta k^4 \\left(\\eta^2-\\eta_i^2\\right)^2-2 i k^3 \\left(17 \\eta^4-12 \\eta^3 \\eta_i-8 \\eta\\eta_i^3+3 \\eta_i^4\\right)\\nonumber\\\\\n&& \\qquad +k^2 \\left(16 \\eta_i^3-52 \\eta^3\\right)+72 i \\eta\\eta_i k\n\\Bigr)\n+105 i\n\\Biggr\\}\n\\Biggr] \\ .\n\\label{u2}\n\\end{eqnarray}\nSimilarly, the effect of photon itself comes in Eq.~(\\ref{EMeq2}) and the solution is given by\n\\begin{align}\n \\hat{A}^{(2)}_{\\bm k}(\\eta)\n &=-\\int_{\\eta_i}^\\eta d\\eta' G_{\\rm M} (\\eta,\\eta')\n \\lambda H \\eta'~\\hat{y}^{(1)}_{\\bm k} (\\eta') \\nonumber\\\\\n &=-\\int_{\\eta_i}^\\eta d\\eta' G_{\\rm M} (\\eta,\\eta')\n \\lambda H \\eta'u^{(1)}_{\\bm k}(\\eta')~\\hat{d} \n -\\int_{\\eta_i}^\\eta d\\eta' G_{\\rm M} (\\eta,\\eta')\n \\lambda H \\eta'~u^{(1)*}_{\\bm k}(\\eta')~\\hat{d}^\\dagger \\nonumber\\\\\n &=-\\int_{\\eta_i}^\\eta d\\eta' G_{\\rm M}(\\eta,\\eta')\n \\lambda H \\eta'~\n \\Biggl( \n -\\int_{\\eta_i}^{\\eta'} d\\eta'' G_{\\rm dS}(\\eta',\\eta'') \n \\lambda H \\eta'' v^{(0)}_{\\bm k} (\\eta'')\n \\Biggr) \\hat{d}\n \\nonumber\\\\\n &\\qquad-\\int_{\\eta_i}^\\eta d\\eta' G_{\\rm M} (\\eta,\\eta')\n \\lambda H \\eta'~\n \\Biggl( \n -\\int_{\\eta_i}^{\\eta'} d\\eta'' G_{\\rm dS}(\\eta',\\eta'') \n \\lambda H \\eta'' v^{(0)*}_{\\bm k} (\\eta'')\n \\Biggr) \\hat{d}^\\dagger\n \\nonumber\\\\\n &=v^{(2)}_{\\bm k}(\\eta)~\\hat{d}+v^{(2)*}_{\\bm k}(\\eta)~\\hat{d}^\\dagger,\n\\end{align}\nwhere we used Eqs.~(\\ref{1st:graviton}) in the second line and (\\ref{u1Green}) in the third line and in the last line. We defined\n\\begin{align}\n v^{(2)}_{\\bm k}(\\eta)\n &\\equiv\n -\\int_{\\eta_i}^\\eta d\\eta' G_{\\rm M} (\\eta,\\eta')\n \\lambda H \\eta'\n u^{(1)}_{\\bm k}(\\eta')\n \\nonumber\\\\&=\\int_{\\eta_i}^\\eta d\\eta' G_{\\rm M}(\\eta,\\eta')\n \\lambda H \\eta'\n \\int_{\\eta_i}^{\\eta'} d\\eta'' G_{\\rm dS}(\\eta',\\eta'') \n \\lambda H \\eta'' v^{(0)}_{\\bm k} (\\eta'')\\,.\n \\label{v2Green}\n\\end{align}\nThe integral of the above reduces to\n\\begin{eqnarray}\nv^{(2)}_{\\bm k}(\\eta)\n&=&\n-\\frac{ \\lambda^2 H^2 }{64 \\sqrt{2} k^{13\/2}}\\nonumber\\\\\n&&\\times\n\\Biggl[\ne^{- i k \\eta } \n\\Biggl(2 k^4 \\left(\\eta^2-\\eta_i^2\\right)^2-4 i \\eta k^3 (\\eta -\\eta_i) (\\eta +3 \\eta_i)\\nonumber\\\\\n&& \\qquad +12 \\eta_i k^2 (\\eta_i-2 \\eta )-12 i k (\\eta -2 \\eta_i)-15\n\\Biggr)\\nonumber\\\\\n&&\\qquad +e^{ i k ( \\eta -2 \\eta_i)} \n\\left(2 k (3+2 i \\eta_i k) \\left(\\eta_i^2 k-\\eta (\\eta k+3 i)\\right)+6 i \\eta_i k+15\n\\right)\n\\Biggr] \\ .\n\\label{v2}\n\\end{eqnarray}\n \n\n\\section{Bogoliubov transformations\n\nBy solving Eqs.(\\ref{eom:graviton}) and (\\ref{eom:photon}) iteratively up to the second order, we can take into account the backreaction of graviton and photon respectively. For the graviton,\nthe field and its conjugate momentum are now given by\n\\begin{eqnarray}\n\\hspace{-1cm}\n\\hat{y}_{\\bm k}(\\eta)&=&\\Bigl(u^{(0)}_{\\bm k}+u^{(2)}_{\\bm k}\\Bigr)\\,\\hat{c}\n+u^{(1)}_{\\bm k}\\,\\hat{d}\n+{\\rm h.c.} \\,,\n\\\\\n\\label{pGWsol}\n\\hat{p}_{\\bm k}(\\eta)&=&\\Bigl(u^{(0)\\,\\prime}_{\\bm k}+u^{(2)\\,\\prime}_{\\bm k}\\Bigr)\\,\\hat{c}\n+u^{(1)\\,\\prime}_{\\bm k}\\,\\hat{d}\n+\\frac{1}{\\eta}\\Bigl\\{\\left(\nu^{(0)}_{\\bm k}+u^{(2)}_{\\bm k}\n\\right)\\hat{c}+u^{(1)}_{\\bm k}\\,\\hat{d}\\Bigr\\}+{\\rm h.c.}\\,,\n\\end{eqnarray}\nwhere we used Eq.~(\\ref{p}) and h.c. represents Hermitian conjugate. For the photon,\nthe field and its conjugate momentum become\n\\begin{eqnarray}\n\\hat{A}_{\\bm k}(\\eta)&=&\\Bigl(v^{(0)}_{\\bm k}+v^{(2)}_{\\bm k}\\Bigr)\\,\\hat{d}\n+v^{(1)}_{\\bm k}\\,\\hat{c}\n+{\\rm h.c.}\\,,\\\\\n\\hat{\\pi}_{\\bm k}(\\eta)&=&\n\\Bigl(v^{(0)\\,\\prime}_{\\bm k}+v^{(2)\\,\\prime}_{\\bm k}\\Bigr)\\,\\hat{d}\n+v^{(1)\\,\\prime}_{\\bm k}\\,\\hat{c}\n+{\\rm h.c.}\\,,\n\\label{pEMsol}\n\\end{eqnarray}\nwhere we used Eq.~(\\ref{pi}).\nThen the annihilation operators for the graviton and photon are obtained by using Eqs.~(\\ref{y:annihi}) and (\\ref{A:annihi}) such as\n\\begin{align}\n \\hat{a}_{y}(\\eta,\\bm{k})\n &= \\Bigl(\n \\psi_{p}^{(0)}\n +\\psi_{p}^{(2)}\n \\Bigr)\\hat{c}\n +\\Bigl(\n \\psi_{m}^{(0)*}\n +\\psi_{m}^{(2)*}\n \\Bigr)\\hat{c}^\\dagger\n +\\psi_{ p}^{(1)} \\hat{d}\n +\\psi_{m}^{(1)*} \\hat{d}^\\dagger\\,,\n \\label{y:annihifull}\n \\\\\n \\hat{a}_{A}(\\eta,\\bm{k})\n &= \\Bigl(\n \\phi_{p}^{(0)}\n +\\phi_{p}^{(2)}\n \\Bigr)\\hat{d}\n +\\Bigl(\n \\phi_{ m}^{(0)*}\n +\\phi_{ m}^{(2)*}\n \\Bigr)\\hat{d}^\\dagger\n +\\phi_{ p}^{(1)} \\hat{c}\n +\\phi_{ m}^{(1)*} \\hat{c}^\\dagger\\,.\n \\label{A:annihifull}\n\\end{align}\nHere, we defined new variables\n\\begin{align}\n&\\psi_{p}^{(j)} \n =\\sqrt{\\frac{k}{2}} u^{(j)}_{\\bm k}(\\eta)\n +\\frac{i}{\\sqrt{2k}}\n \\Bigl(u^{(j)\\prime}_{\\bm k}(\\eta)+\\frac{1}{\\eta}u^{(j)}_{\\bm k} (\\eta)\n \\Bigr) ,\\label{psip}\\\\\n&\\psi_{ m}^{(j)}\n =\\sqrt{\\frac{k}{2}} u^{(j)}_{\\bm k}(\\eta)\n -\\frac{i}{\\sqrt{2k}}\n \\Bigl(u^{(j)\\prime}_{\\bm k}(\\eta)+\\frac{1}{\\eta}u^{(j)}_{\\bm k}(\\eta)\n \\Bigr), \\label{psim}\\\\\n&\\phi_{ p}^{(j)}\n =\\sqrt{\\frac{k}{2}} v^{(j)}_{\\bm k}(\\eta)\n +\\frac{i}{\\sqrt{2k}} v_{\\bm k}^{(j)\\prime}(\\eta),\n \\label{phip}\\\\\n&\\phi_{ m}^{(j)} \n =\\sqrt{\\frac{k}{2}} v^{(j)}_{\\bm k}(\\eta)\n -\\frac{i}{\\sqrt{2k}} v_{\\bm k}^{(j)\\prime}(\\eta),\n \\label{phim}\n\\end{align}\nwhere $j=0,1,2$ denotes the order of perturbations.\n\nWe see that all mode functions other than the zeroth order given in Eqs.~(\\ref{u1Green}), (\\ref{v1Green}) (\\ref{u2Green}) and (\\ref{v2Green}) vanish at the initial time $\\eta_i$. Thus only the zeroth order of the above Eqs.~(\\ref{psip}) $\\sim$ (\\ref{phim}) remains at the initial time. This means that annihilation operators in Eqs.(\\ref{y:annihifull}) and (\\ref{A:annihifull}) at the initial time are expressed by the zeroth order variables\n\\begin{align}\n \\hat{a}_{y}(\\eta_i,\\bm{k})\n &=\\left(1-\\frac{i}{2k\\eta_i}\\right)e^{-ik\\eta_i}\\,\\hat{c}\n +\\frac{i}{2k\\eta_i} e^{ik\\eta_i}\\,\\hat{c}^\\dagger,\n \\label{ycRel}\n \\\\\n \\hat{a}_A(\\eta_i,\\bm{k})\n &=e^{-ik \\eta_i} \\hat{d}.\n\\label{AdRel}\n\\end{align}\nCombining Eqs. (\\ref{ycRel}) and (\\ref{AdRel}) with their complex conjugate, we can express the $\\hat{c}$ and $\\hat{d}$ by the initial creation and annihilation \noperators as\n\\begin{eqnarray}\n \\hat{c} &=& \\left( 1+\\frac{i}{2k\\eta_i}\\right)e^{ik\\eta_i}\\,\\hat{a}_y(\\eta_i,\\bm{k} )\n -\\frac{i}{2k\\eta_i} e^{ik\\eta_i}\\,\\hat{a}_y^\\dagger (\\eta_i,-\\bm{k}) \\ , \\\\\n \\hat{d} &=& e^{ik\\eta_i}\\,\\hat{a}_A(\\eta_i,\\bm{k}) \\ .\n\\end{eqnarray}\nPlugging the above back into Eqs.(\\ref{y:annihifull}) and (\\ref{A:annihifull}), the time evolution of annihilation operator of graviton is described by the Bogoliubov transformation in the form\n\\begin{align}\n &\\hat{a}_y(\\eta,\\bm{k})=\n \\Biggl[\n \\biggl(\n \\psi_{p}^{(0)}\n +\\psi_{p}^{(2)}\n \\biggr)\n \\Bigl( 1+\\frac{i}{2k\\eta_i} \\Bigr)\n e^{ik\\eta_i} \n +\\biggl(\n \\psi_{m}^{(0)*}\n +\\psi_{ m}^{(2)*}\n \\biggr) \\frac{i}{2k\\eta_i} e^{-ik\\eta_i} \n \\Biggr]\n \\hat{a}_y(\\eta_i,\\bm{k})\\nonumber\\\\\n &\\hspace{1.5cm}\n +\\Biggl[\n \\biggl(\n \\psi_{p}^{(0)}\n +\\psi_{p}^{(2)}\n \\biggr)\\Bigl(-\\frac{i}{2k\\eta_i} \\Bigr) e^{ik\\eta_i} \n +\\biggl(\n \\psi_{m}^{(0)*}\n +\\psi_{ m}^{(2)*}\n \\biggr)\n \\Bigl(1-\\frac{i}{2k\\eta_i}\\Bigr)\n e^{-ik\\eta_i} \\Biggr]\\hat{a}_y^\\dagger(\\eta_i,-\\bm{k})\n \\nonumber\\\\\n &\\hspace{1.5cm}\n +\\psi_{p}^{(1)} e^{ik\\eta_i}\n \\hat{a}_A (\\eta_i,\\bm{k})\n +\\psi_{m}^{(1)*} e^{-ik\\eta_i}\n \\hat{a}_A^\\dagger(\\eta_i,-\\bm{k}),\n \\label{y:bogoliubov1}\n\\end{align}\nand the time evolution of annihilation operator of photon is expressed by the Bogoliubov transformation such as\n\\begin{align}\n &\\hat{a}_A(\\eta,\\bm{k})=\n \\Biggl( \n \\phi_{ p}^{(1)}\n \\Bigl(1+\\frac{i}{2k\\eta_i} \\Bigr)\n e^{ik\\eta_i} \n + \\phi_{ m}^{(1)*}\n \\frac{i}{2k\\eta_i}\n e^{-ik\\eta_i} \n \\Biggr)\\hat{a}_y(\\eta_i,\\bm{k})\\nonumber\\\\\n &~~~~~~~~~~~\n +\\Biggl(\n -\\phi_{ p}^{(1)} \\frac{i}{2k\\eta_i}\n e^{ik\\eta_i}\n +\\phi_{ m}^{(1)*}\n \\Bigl(1-\\frac{i}{2k\\eta_i} \\Bigr)\n e^{-ik\\eta_i}\n \\Biggr) \\hat{a}_y^\\dagger (\\eta_i,-\\bm{k})\n \\nonumber\\\\\n &~~~~~~~~~~~\n +\\Bigl(\n \\phi_{p}^{(0)}+\\phi_{p}^{(2)}\n \\Bigr)e^{ik\\eta_i} \\hat{a}_{A}(\\eta_i,\\bm{k})\n +\\Bigl(\n \\phi_{m}^{(0)*}+\\phi_{m}^{(2)*}\n \\Bigr)e^{-ik\\eta_i} \\hat{a}_A^\\dagger(\\eta_i,-\\bm{k}).\n \\label{A:bogoliubov1}\n\\end{align}\nThese Bogoliubov transformations show the particle production during inflation and the mixing between graviton and photon.\n\nIt is useful to use a matrix form for later calculations. \nIn fact, the Bogoliubov transofomation (\\ref{y:bogoliubov1}) and (\\ref{A:bogoliubov1}) and their conjugate can be accommodated into the simple $4\\times 4$ matrix form $M$\n\\begin{eqnarray}\n\\begin{pmatrix}\na_y(\\eta)\\\\\na_y^{\\dagger}(\\eta)\\\\\na_A(\\eta)\\\\\na_A^{\\dagger}(\\eta)\\\\\n\\end{pmatrix}\n=M\n\\begin{pmatrix}\na_y(\\eta_i)\\\\\na_y^{\\dagger}(\\eta_i)\\\\\na_A(\\eta_i)\\\\\na_A^{\\dagger}(\\eta_i)\\\\\n\\end{pmatrix}\n=\n\\left\\{\n\\begin{pmatrix}\nA_{0}& 0\\\\\n0 &D_{0}\\\\\n\\end{pmatrix} \n+\n\\begin{pmatrix}\n0 &B_{1}\\\\\nC_{1}& 0\\\\\n\\end{pmatrix} \n+\n\\begin{pmatrix}\nA_{2}& 0\\\\\n0 & D_{2}\\\\\n\\end{pmatrix} \n\\right\\}\n\\begin{pmatrix}\na_y(\\eta_i)\\\\\na_y^{\\dagger}(\\eta_i)\\\\\na_A(\\eta_i)\\\\\na_A^{\\dagger}(\\eta_i)\\\\\n\\end{pmatrix}\\ .\n\\nonumber\n\\hspace{-6mm}\\\\\n\\label{bogoliubov}\n\\end{eqnarray}\nHere, the zeroth order Bogoliubov transformation consists of $2\\times 2$ matrices $A_0$ and $D_0$ given by \n\\begin{eqnarray}\nA_0=\n\\begin{pmatrix}\nK^* & -L^* \\\\\n-L & K \\\\\n\\end{pmatrix}\n\\ ,\\qquad\nD_0=\n\\begin{pmatrix}\ne^{ik\\,(\\eta-\\eta_i)} & 0\\\\\n0 & e^{-ik\\,(\\eta-\\eta_i)} \\\\\n\\end{pmatrix} \\ ,\n\\end{eqnarray}\nwhere we defined \n\\begin{eqnarray}\nK&=&\\left( 1+\\frac{i}{2k\\eta}\\right)\\left( 1-\\frac{i}{2k\\eta_i}\\right)e^{ik(\\eta-\\eta_i)} \n-\\frac{1}{4k^2\\eta\\eta_i }e^{-ik(\\eta-\\eta_i)}\\ ,\\\\\nL&=&-\\frac{i}{2k\\eta_i }\\left( 1+\\frac{i}{2k\\eta}\\right) e^{ik(\\eta-\\eta_i)} \n+\\frac{i}{2k\\eta }\\left( 1+\\frac{i}{2k\\eta_i}\\right) e^{-ik(\\eta-\\eta_i)} \\ .\n\\end{eqnarray}\nThe first order Bogoliubov transofomation is written by $2\\times 2$ matrices $B_1$ and $C_1$ such as\n\\begin{eqnarray}\nB_1=\n\\begin{pmatrix}\ne^{ik\\eta_i}\\psi_{p}^{(1)} & e^{-ik\\eta_i}\\psi_{m}^{(1)*}\\\\\n e^{ik\\eta_i}\\psi_{m}^{(1)}& e^{-ik\\eta_i}\\psi_{p}^{(1)*} \\\\\n\\end{pmatrix}\n\\end{eqnarray}\nand \n\\begin{eqnarray}\nC_1=\n\\begin{pmatrix}\n\\left( 1+\\frac{i}{2k\\eta_i}\\right)e^{ik\\eta_i}\\phi_{p}^{(1)}+\\frac{i}{2k\\eta_i} e^{-ik\\eta_i}\\phi_{m}^{(1)*} & \\left( 1-\\frac{i}{2k\\eta_i}\\right)e^{-ik\\eta_i}\\phi_{m}^{(1)*}-\\frac{i}{2k\\eta_i} e^{ik\\eta_i}\\phi_{p}^{(1)} \\\\\n \\left( 1+\\frac{i}{2k\\eta_i}\\right)e^{ik\\eta_i}\\phi_{m}^{(1)}+\\frac{i}{2k\\eta_i} e^{-ik\\eta_i}\\phi_{p}^{(1)*}& \\left( 1-\\frac{i}{2k\\eta_i}\\right)e^{-ik\\eta_i}\\phi_{p}^{(1)*}-\\frac{i}{2k\\eta_i} e^{ik\\eta_i}\\phi_{m}^{(1)} \\\\\n\\end{pmatrix}\\ .\n\\end{eqnarray}\nFinally, the second order Bogoliubov transformation $A_2$ and $D_2$ are\n\\begin{eqnarray}\nA_2=\n\\begin{pmatrix}\n\\left( 1+\\frac{i}{2k\\eta_i}\\right)e^{ik\\eta_i}\\psi_{p}^{(2)}+\\frac{i}{2k\\eta_i} e^{-ik\\eta_i}\\psi_{m}^{(2)*} & \\left( 1-\\frac{i}{2k\\eta_i}\\right)e^{-ik\\eta_i}\\psi_{m}^{(2)*}-\\frac{i}{2k\\eta_i} e^{ik\\eta_i}\\psi_{p}^{(2)} \\\\\n \\left( 1+\\frac{i}{2k\\eta_i}\\right)e^{ik\\eta_i}\\psi_{m}^{(2)}+\\frac{i}{2k\\eta_i} e^{-ik\\eta_i}\\psi_{p}^{(2)*} & \\left( 1-\\frac{i}{2k\\eta_i}\\right)e^{-ik\\eta_i}\\psi_{p}^{(2)*}-\\frac{i}{2k\\eta_i} e^{ik\\eta_i}\\psi_{m}^{(2)}\\\\\n\\end{pmatrix}\\ \n\\end{eqnarray}\nand\n\\begin{eqnarray}\nD_2=\n\\begin{pmatrix}\ne^{ik\\eta_i}\\phi_{p}^{(2)} & e^{-ik\\eta_i}\\phi_{m}^{(2)*}\\\\\ne^{ik\\eta_i}\\phi_{m}^{(2)} & e^{-ik\\eta_i}\\phi_{p}^{(2)*} \\\\\n\\end{pmatrix} \\ .\n\\end{eqnarray}\n\n\\section{Time evolution of squeezing parameters\nIn the previous section, we obtained the Bogoliubov transformation that mix the operators $\\hat{a}_y(\\eta)$, $\\hat{a}_A(\\eta)$\nand their Hermitian conjugates $\\hat{a}^\\dagger_y(\\eta)$, $\\hat{a}^\\dagger_A(\\eta)$.\nNote that the initial Bunch-Davies state is defined by\n\\begin{eqnarray}\n\\hat{a}_y(\\eta_i,\\bm{k}) |{\\rm BD}\\rangle= \\hat{a}_A(\\eta_i,\\bm{k}) |{\\rm BD}\\rangle =0\\,.\n\\label{BD}\n\\end{eqnarray}\nIn order to impose these conditions,\nwe need to invert the Bogoliubov transformations (\\ref{y:bogoliubov1}) and (\\ref{A:bogoliubov1})\ninto the form\n\\begin{eqnarray}\n\\hat{a}_y(\\eta_i,\\bm{k})\n&=&\\alpha_y\\,\\hat{a}_y (\\eta,\\bm{k})+ \\beta_y\\,\\hat{a}_y^\\dagger(\\eta,-\\bm{k})\n + \\gamma_A\\,\\hat{a}_A(\\eta,\\bm{k})+ \\delta_A\\,\\hat{a}_A^\\dagger(\\eta,-\\bm{k})\\,,\n \\label{y:invert}\\\\\n\\hat{a}_A(\\eta_i,\\bm{k})\n&=&\\gamma_y\\,\\hat{a}_y(\\eta,\\bm{k})+ \\delta_y\\,\\hat{a}_y^\\dagger(\\eta,-\\bm{k}) \n +\\alpha_A\\,\\hat{a}_A (\\eta,\\bm{k})+ \\beta_A\\,\\hat{a}_A^\\dagger(\\eta,-\\bm{k})\\,,\n \\label{A:invert}\n\\end{eqnarray}\nwhere $\\alpha_y$, $\\beta_y$, $\\gamma_A$, $\\delta_A$, $\\gamma_y$, $\\delta_y$, $\\alpha_A$ and $\\beta_A$ are the Bogoliubov coefficients and we will find these coefficients in the next subsection. \n\n\\subsection{Inversion of the Bogoliubov transformation}\n\nThe matrix $M$ in Eq.~(\\ref{bogoliubov}) can be expanded perturbatively as\n\\begin{eqnarray}\n M = M^{(0)}+M^{(1)}+M^{(2)}\n =M^{(0)}\\left[ 1+ M^{(0)-1}M^{(1)}\n +M^{(0)-1}M^{(2)}\\right] \\ ,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nM^{(0)}=\n\\begin{pmatrix}\nA_{0}& 0\\\\\n0 &D_{0}\\\\\n\\end{pmatrix} \n\\,,\\qquad\nM^{(1)}=\n\\begin{pmatrix}\n0 &B_{1}\\\\\nC_{1}& 0\\\\\n\\end{pmatrix} \n\\,,\\qquad\nM^{(2)}=\n\\begin{pmatrix}\nA_{2}& 0\\\\\n0 & D_{2}\\\\\n\\end{pmatrix}\n\\,.\n\\end{eqnarray}\nThen the inverse of the $M$ is given by\n\\begin{eqnarray}\n M^{-1} \n =\\left[ 1- M^{(0)-1}M^{(1)}\n -M^{(0)-1}M^{(2)}\n + M^{(0)-1}M^{(1)}M^{(0)-1}M^{(1)} \\right]M^{(0)-1}\\ .\n\\end{eqnarray}\nUsing the above general formula, the inverse of the $M$ is obtained in the form\n\\begin{eqnarray}\nM^{-1}=\n\\begin{pmatrix}\nA_0^{-1}-A_0^{-1}A_2 A_0^{-1}\n+A_0^{-1}B_{1}D_0^{-1}C_1A_0^{-1}& -A_0^{-1}B_{1}D_0^{-1}\\\\\n-D_0^{-1}C_1 A_0^{-1}& D_0^{-1}-D_0^{-1}D_{2}D_0^{-1}\n+D_0^{-1}C_{1}A_0^{-1}B_1D_0^{-1}\n\\label{inverseM}\n\\end{pmatrix} \\,.\n\\nonumber\n\\hspace{-5mm}\\\\\n\\end{eqnarray}\nWe see that $A_0^{-1}$ and $D_0^{-1}$ are necessary to calculate the elements of the $M^{-1}$. They are given by\n\\begin{eqnarray}\nA_0^{-1}=\n\\begin{pmatrix}\nK & L^* \\\\\nL & K^* \\\\\n\\end{pmatrix} \n\\ , \\qquad\nD_0^{-1}=\n\\begin{pmatrix}\ne^{-ik\\,(\\eta-\\eta_i)} & 0\\\\\n0 & e^{ik\\,(\\eta-\\eta_i)} \\\\\n\\end{pmatrix} \\ .\n\\end{eqnarray}\nFrom Eqs.~(\\ref{y:invert}) and (\\ref{A:invert}), the $M^{-1}$ is also written as\n\\begin{eqnarray}\nM^{-1}=\n\\begin{pmatrix}\n\\alpha_y & \\beta_y & \\gamma_A & \\delta_A \\\\\n\\beta_y^* & \\alpha_y^* & \\delta_A^* & \\gamma_A^* \\\\\n\\gamma_y & \\delta_y & \\alpha_A & \\beta_A \\\\\n\\delta_y^* & \\gamma_y^* & \\beta_A^* & \\alpha_A^* \\\\\n\\end{pmatrix} \\,,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n && \\alpha_y =\\alpha_y^{(0)} +\\alpha_y^{(2)} \\,,\\qquad\n \\beta_y =\\beta_y^{(0)} +\\beta_y^{(2)} \\,,\\qquad\n \\gamma_A =\\gamma_A^{(1)} \\,,\\qquad\n \\delta_A =\\delta_A^{(1)} \\,,\n \\label{expand1}\\\\\n&& \\alpha_A =\\alpha_A^{(0)} +\\alpha_A^{(2)} \\,,\\qquad\n \\beta_A = \\beta_A^{(2)} \\,,\\qquad\n \\gamma_y =\\gamma_y^{(1)} \\,,\\qquad\n \\delta_y =\\delta_y^{(1)} \\ .\n \\label{expand2}\n\\end{eqnarray}\nThe zeroth order elements are given by\n\\begin{eqnarray}\n&& \\alpha^{(0)}_y= \\left( 1+\\frac{i}{2k\\eta}\\right)\\left( 1-\\frac{i}{2k\\eta_i}\\right)e^{ik(\\eta-\\eta_i)} \n-\\frac{1}{4k^2\\eta\\eta_i }e^{-ik(\\eta-\\eta_i)} \\ ,\\\\\n&& \\beta^{(0)}_y = \\frac{i}{2k\\eta_i }\\left( 1-\\frac{i}{2k\\eta}\\right) e^{-ik(\\eta-\\eta_i)} \n-\\frac{i}{2k\\eta }\\left( 1-\\frac{i}{2k\\eta_i}\\right) e^{ik(\\eta-\\eta_i)} \\ , \\\\\n&& \\alpha^{(0)}_A =e^{ik(\\eta-\\eta_i)} \\ ,\\qquad\n \\beta^{(0)}_A = 0\\ .\n\\end{eqnarray}\nThe first order elements are written as\n\\begin{eqnarray}\n \\gamma^{(1)}_A &=& -\\left( K\\psi_{p}^{(1)}\n+L^*\\psi_{m}^{(1)}\\right)e^{ik\\eta}\\ ,\\\\\n \\delta^{(1)}_A &=& -\\left(K\\psi_{m}^{(1)*}\n+L^*\\psi_{p}^{(1)*}\\right)\ne^{-ik\\eta} \\ , \\\\\n \\gamma^{(1)}_y &=& -K \\left[\\left( 1+\\frac{i}{2k\\eta_i}\\right)e^{ik\\eta}\\phi_{p}^{(1)}\n+\\frac{i}{2k\\eta_i} e^{ik(\\eta-2\\eta_i)}\\phi_{m}^{(1)*} \\right]\\nonumber\\\\\n&&-L \\left[\\left( 1-\\frac{i}{2k\\eta_i}\\right)e^{ik(\\eta-2\\eta_i)}\\phi_{m}^{(1)*}\n-\\frac{i}{2k\\eta_i} e^{ik\\eta}\\phi_{p}^{(1)} \\right]\\ ,\\\\\n \\delta^{(1)}_y &=& -L^* \\left[\\left( 1+\\frac{i}{2k\\eta_i}\\right)e^{ik\\eta}\\phi_{p}^{(1)}\n+\\frac{i}{2k\\eta_i} e^{ik(\\eta-2\\eta_1)}\\phi_{m}^{(1)*} \\right]\\nonumber\\\\\n&&-K^* \\left[\\left( 1-\\frac{i}{2k\\eta_i}\\right)e^{ik(\\eta-2\\eta_i)}\\phi_{m}^{(1)*}\n-\\frac{i}{2k\\eta_i} e^{ik\\eta}\\phi_{p}^{(1)} \\right] \\ .\n\\end{eqnarray}\nThe second order are \n\\begin{eqnarray}\n \\alpha^{(2)}_y &=& -K\\left(KA_{11}+L^* A_{21}\\right)\n-L\\left(KA_{12}+L^* A_{22}\\right) \\nonumber\\\\\n&&+(C_{11} K+C_{12} L)\\left(K\\psi_{p}^{(1)}\n+L^*\\psi_{m}^{(1)}\\right)e^{ik\\eta}\\nonumber\\\\\n&&+(C_{21} K+C_{22} L)\\left(K\\psi_{m}^{(1)*}\n+L^*\\psi_{p}^{(1)*}\\right)e^{-ik\\eta}\\ , \\\\\n \\beta^{(2)}_y &=& -L^*\\left(KA_{11}+L^* A_{21}\\right)\n-K^* \\left(KA_{12}+L^* A_{22}\\right) \\nonumber\\\\\n&&+(C_{11} L^* +C_{12} K^*)\\left(K \\psi_{p}^{(1)}\n+L^*\\psi_{m}^{(1)}\\right)e^{ik\\eta}\\nonumber\\\\\n&&+(C_{21} L^*+C_{22} K^*)\\left(K\\psi_{m}^{(1)*}\n+L^*\\psi_{p}^{(1)*}\\right)e^{-ik\\eta}\\ ,\\\\ \n \\alpha^{(2)}_A &=& - e^{ik(2\\eta-\\eta_i)}\\phi_{p}^{(2)} \\nonumber\\\\\n&& +(C_{11} K+C_{12} L)e^{ik(2\\eta - \\eta_i)}\\psi_{p}^{(1)}\n+(C_{11} L^* +C_{12} K^*)e^{ik(2\\eta - \\eta_i)}\\psi_{m}^{(1)}\n \\ , \\\\\n \\beta^{(2)}_A &=&-e^{-ik\\eta_i}\\phi_{m}^{(2)*}\\nonumber\\\\\n&& +(C_{11} K+C_{12} L) e^{-ik\\eta_i }\\psi_{m}^{(1)*}\n+(C_{11} L^* +C_{12} K^*)e^{-ik\\eta_i}\\psi_{p}^{(1)*}\\ ,\n\\end{eqnarray}\nwhere we have defined\n\\begin{eqnarray}\nA_{11}&=&\\psi_{p}^{(2)}\\left(1+\\frac{i}{2k\\eta_i}\\right)e^{ik\\eta_i}+\\psi_{m}^{(2)*}\\frac{i}{2k\\eta_i}e^{-ik\\eta_i}\\,,\\\\\nA_{12}&=&-\\psi_{p}^{(2)}\\frac{i}{2k\\eta_i}e^{ik\\eta_i}+\\psi_{m}^{(2)*}\\left(1-\\frac{i}{2k\\eta_i}\\right)e^{-ik\\eta_i}\\,,\\\\\nA_{21}&=&\\psi_{p}^{(2)*}\\frac{i}{2k\\eta_i}e^{-ik\\eta_i}+\\psi_{m}^{(2)}\\left(1+\\frac{i}{2k\\eta_i}\\right)e^{ik\\eta_i}\\,,\\\\\nA_{22}&=&\\psi_{p}^{(2)*}\\left(1-\\frac{i}{2k\\eta_i}\\right)e^{-ik\\eta_i}-\\psi_{m}^{(2)}\\frac{i}{2k\\eta_i}e^{ik\\eta_i}\\,,\n\\end{eqnarray}\nand\n\\begin{eqnarray}\nC_{11}&=&\\phi_{p}^{(1)}\\left(1+\\frac{i}{2k\\eta_i}\\right)e^{ik\\eta_i}+\\phi_{m}^{(1)*}\\frac{i}{2k\\eta_i}e^{-ik\\eta_i}\\,,\\\\\nC_{12}&=&\\phi_{m}^{(1)*}\\left(1-\\frac{i}{2k\\eta_i}\\right)e^{-ik\\eta_i}-\\phi_{p}^{(1)}\\frac{i}{2k\\eta_i}e^{ik\\eta_i}\\,,\\\\\nC_{21}&=&\\phi_{p}^{(1)*}\\frac{i}{2k\\eta_i}e^{-ik\\eta_i}+\\phi_{m}^{(1)}\\left(1+\\frac{i}{2k\\eta_i}\\right)e^{ik\\eta_i}\\,,\\\\\nC_{22}&=&\\phi_{p}^{(1)*}\\left(1-\\frac{i}{2k\\eta_i}\\right)e^{-ik\\eta_i}-\\phi_{m}^{(1)}\\frac{i}{2k\\eta_i}e^{ik\\eta_i}\\,.\n\\end{eqnarray}\n\n\n\\subsection{Squeezing operator}\nIn the previous subsection, we obtained the Bogoliubov coefficients of Eqs.~(\\ref{y:invert}) and (\\ref{A:invert}) up to the second order. If we apply the Eqs.~(\\ref{y:invert}) and (\\ref{A:invert}) to the definition of the Bunch-Davies vacuum~(\\ref{BD}) and use the relations $[\\hat{a}_y(\\eta,{\\bm k}),\\hat{a}^\\dag_y(\\eta,-{\\bm k}^\\prime)]=\\delta({\\bm k}+{\\bm k}^\\prime)$\\,, $[\\hat{a}_A(\\eta,{\\bm k}),\\hat{a}^\\dag_A(\\eta,-{\\bm k}^\\prime)]=\\delta({\\bm k}+{\\bm k}^\\prime)$ and $[\\hat{a}_y(\\eta,{\\bm k}),\\hat{a}_A(\\eta,-{\\bm k}^\\prime)]=0$, the Bunch-Davies vacuum can be written by using squeezing parameters $\\Lambda,\\Xi$ and $\\Omega$ such as \n\\begin{eqnarray}\n|{\\rm BD}\\rangle = \\prod_{k=-\\infty}^{\\infty}\\exp\\left[\\frac{\\Lambda}{2}\\,\n\\hat{a}_y^\\dag (\\eta,\\bm{k}) \\hat{a}_y^\\dag (\\eta,-\\bm{k})+\\Xi\\,\\hat{a}_y^\\dag (\\eta,\\bm{k}) \\hat{a}_A^\\dag (\\eta,-\\bm{k})\n+\\frac{\\Omega}{2}\\,\\hat{a}_A^\\dag (\\eta,\\bm{k}) \\hat{a}_A^\\dag (\\eta,-\\bm{k})\\right]|0\\rangle,\\nonumber\n\\hspace{-5mm}\\\\\n\\end{eqnarray}\nwhere $|0\\rangle$ is the instantaneous vacuum defined by\n\\begin{eqnarray}\n\\hat{a}_y(\\eta,{\\bm k}) |0\\rangle=\\hat{a}_A(\\eta,{\\bm k}) |0\\rangle =0 \\,.\n\\end{eqnarray}\nThis describes a four mode squeezed state of pairs of graviton $y$ and photon $A$. In a different context, a four-mode squeezed state of two free massive scalar fields is discussed in~\\cite{Albrecht:2014aga,Kanno:2015ewa}.\nIf we expand the exponential function in Taylor series, we find\n\\begin{eqnarray}\n|{\\rm BD}\\rangle =\n\\prod_{\\bm k} \\sum_{p\\,,q\\,,r=0}^{\\infty}\n \\frac{\\Lambda^p\\,\\Xi^q\\,\\Omega^r}{2^{p+r}p!\\,q!\\,r!} \n |p+q \\rangle_{y,{\\bm k}} \\otimes |p \\rangle_{y,-{\\bm k}} \\otimes |r \\rangle_{A,{\\bm k}} \\otimes |q+r \\rangle_{A,-{\\bm k}}\\,.\n\\end{eqnarray}\nThis is a four-mode squeezed state which consists of an infinite number of entangled particles in the ${\\cal H}_{y,{\\bm k}}\\otimes{\\cal H}_{y,{-\\bm k}}\\otimes{\\cal H}_{A,{\\bm k}}\\otimes{\\cal H}_{A,-{\\bm k}}$ space. \nIn particular, in the highly squeezing limit $\\Lambda\\,,\\Xi\\,,\\Omega\\rightarrow 1$, the Bunch-Davies vacuum becomes the maximally entangled state from the point of view of the instantaneous vacuum. \n\nNow we find the squeezing parameters.\nThe condition $\\hat{a}_y(\\eta_i,{\\bm k})|{\\rm BD}\\rangle=0$ of Eq.~(\\ref{BD}) yields\n\\begin{eqnarray}\n \\alpha_y \\Lambda +\\beta_y +\\gamma_A \\Xi =0 \\ , \\qquad\n \\alpha_y \\Xi +\\gamma_A \\Omega +\\delta_A =0\\,,\n\\end{eqnarray}\nand another condition $\\hat{a}_A(\\eta_i,{\\bm k}) |{\\rm BD}\\rangle=0$ of Eq.~(\\ref{BD}) gives \n\\begin{eqnarray}\n \\alpha_A \\Xi +\\gamma_y \\Lambda +\\delta_y =0 \\ , \\qquad\n \\alpha_A \\Omega +\\beta_A +\\gamma_y \\Xi =0\\,.\n\\end{eqnarray}\nThen, we obtain the three squeezing parameters $\\Lambda,\\Xi$ and $\\Omega$ of the form\n\\begin{eqnarray}\n \\Lambda= \\frac{\\gamma_A\\delta_y -\\beta_y \\alpha_A}{\\alpha_y \\alpha_A -\\gamma_y\\gamma_A} \\ , \\qquad\n \\Xi= \\frac{\\beta_y \\gamma_y - \\alpha_y \\delta_y}{\\alpha_y \\alpha_A -\\gamma_y\\gamma_A} \\ , \\qquad\n \\Omega= \\frac{\\gamma_y\\delta_A -\\beta_A \\alpha_y}{\\alpha_y \\alpha_A -\\gamma_y\\gamma_A}\\,.\n \\label{squeezingparameters}\n\\end{eqnarray}\nWe have four relations for three parameters $\\Lambda,\\Xi$ and $\\Omega$. The remaining relation is turned out to be guaranteed by the commutation relation:\n\\begin{eqnarray}\n [\\hat{a}_y(\\eta,{\\bm k}) \\ , \\hat{a}_A(\\eta,{\\bm k}) ]\n = -\\alpha_A\\delta_A +\\beta_A \\gamma_A\n -\\gamma_y \\beta_y +\\alpha_y \\delta_y =0\\,.\n\\end{eqnarray}\nThus, we find that Eq.~(\\ref{squeezingparameters}) is the unique solution.\nSince the Bogoliubov coefficients are given up to the second order as in Eqs.~(\\ref{expand1}) and (\\ref{expand2}),\nthe squeezing parameters can be expanded up to the second order such as\n\\begin{eqnarray}\n && \\Lambda= -\\frac{\\beta_y^{(0)}}{\\alpha_y^{(0)}} \n \\left[1-\\frac{\\alpha_y^{(2)}}{\\alpha_y^{(0)}}\n + \\frac{\\beta_y^{(2)}}{\\beta_y^{(0)}}\n + \\frac{\\gamma_y^{(1)}\\gamma_A^{(1)}}{\\alpha_y^{(0)}\\alpha_A^{(0)}}\n -\\frac{\\gamma_A^{(1)}\\delta_y^{(1)}}{\\beta_y^{(0)}\\alpha_A^{(0)}}\n \\right]\\ , \\\\\n && \\Xi= \\frac{\\beta_y^{(0)}\\gamma_y^{(1)}}{\\alpha_y^{(0)}\\alpha_A^{(0)}} \n - \\frac{\\delta_y^{(1)}}{\\alpha_A^{(0)}}\\ , \\\\\n && \\Omega= \\frac{\\delta_A^{(1)}\\gamma_y^{(1)}}{\\alpha_y^{(0)}\\alpha_A^{(0)}} \n - \\frac{\\beta_A^{(2)}}{\\alpha_A^{(0)}}\\ .\n\\end{eqnarray}\nIn this way, we obtained the squeezing parameters perturbatively up to the second order. We will discuss the behaviour of the squeezing of graviton $\\Lambda$, the squeezing of mixing between graviton and photon $\\Xi$ and the squeezing of photon $\\Omega$ in the next section.\n\n\n\\subsection{Numerical and analytical results}\n\nThe results of numerical calculations for the amplitude and the phase of the squeezing parameters $\\Lambda$, $\\Xi$, and $\\Omega$ are plotted in FIGs.~\\ref{SqueezingA2}, \\ref{PhaseA}, \\ref{SqueezingB2}, \\ref{PhaseB}, \\ref{SqueezingC2}, and \\ref{PhaseC}, respectively, where we normalized the scale factor at the end of inflation as $a(\\eta_f)=1$.\nThe evolution of the amplitude of $\\Lambda$ in FIG. \\ref{SqueezingA2} shows graviton is squeezed, that is, graviton pair production occurs during inflation ($\\eta<0$).\nWe see that sub-horizon modes oscillates rapidly and no graviton pair production seems to occur before horizon exit. In the presence of coupling with magnetic fields ($\\lambda\\neq 0$), the amplitude of oscillation is relatively small as represented by blue line. After horizon exit, the oscillation ceases and graviton pair production starts to occur and eventually $\\Lambda$ becomes one. This means that almost maximum entangled pair of graviton are produced. This behavior does not change even for $\\lambda\\neq 0$.\nThe evolution of phase of $\\Lambda$ is plotted in FIG. \\ref{PhaseA}, in which we see the phase converges to zero. This is consistent with the result\nin \\cite{Polarski:1995jg}.\nThe time evolution of the amplitude of $\\Xi$ in FIG. \\ref{SqueezingB2} shows that one of pair of gravitons is converted to a photon and graviton-photon pair production occurs. \nWe see that some amount of pair-production occurs when the mode leaves the horizon but the graviton-photon pair production decreases rapidly by the end of inflation.\nThe evolution of the phase of $\\Xi$ plotted in FIG. \\ref{PhaseB} is found to oscillate rapidly but eventually becomes constant after horizon exit. The similar behavior appears in the evolution of phase of $\\Lambda$ in FIG.~{\\ref{PhaseA}}.\nHowever, the final\nphase is found to depend on the initial condition in this case.\nThe time evolution of the amplitude of $\\Omega$ in FIG. \\ref{SqueezingC2} tells us that photon is squeezed, that is, graviton pair production is converged to photon pair production. \nInterestingly, photon pair production occurs rapidly only at the initial time and no more production occurs after that. \nThe behavior of the phase evolution of $\\Omega$ \nin FIG.\\ref{PhaseC} is similar to that of $\\Xi$.\n\\begin{figure}[H]\n\\centering\n \\includegraphics[width=\\textwidth]{SqueezingA2.pdf}\n \\renewcommand{\\baselinestretch}{3}\n \\caption{Squeezing parameter of graviton pair $\\Lambda$ during inflation as a function of the scale factor $a(\\eta)$. We set $\\lambda=5\\times10^{-13}{\\rm GeV}^2$(blue line) and $\\lambda=0~{\\rm GeV}^2$(yellow line). \n Other parameters are set as $k=10^2{\\rm GeV}$, $H = 10^{14}{\\rm GeV}$, $\\eta_i=-2{\\rm GeV}^{-1}$, $\\eta_{f} =-10^{-14}{\\rm GeV}^{-1}$, $a(\\eta_i)=(2\\times 10^{14})^{-1}$, and $a(\\eta_{f})=1$. The red grid line shows the scale factor $a=1.59...\\times10^{-13}$ at the time of horizon exit $\\eta=-2\\pi\/k$.}\n \\label{SqueezingA2}\n \\end{figure}\n \n \n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.55\\textwidth]{PhaseA.pdf}\n\\renewcommand{\\baselinestretch}{1.2}\n\\caption{The phase of the squeezing parameter of graviton pair $\\Lambda(a)$ during inflation as function of the scale factor $a(\\eta)$. We set $\\lambda=5\\times10^{-13}{\\rm GeV}^2$(blue line) and $\\lambda=0~{\\rm GeV}^2$(yellow line). \nOther parameters are set as $k=10^2{\\rm GeV}$, $H=10^{14}{\\rm GeV}$, $\\eta_i=-2{\\rm GeV}^{-1}$, $\\eta_f =-10^{-14}{\\rm GeV}^{-1}$, $a(\\eta_i)=(2\\times 10^{14})^{-1}$, and $a(\\eta_f)=1$. }\n\\label{PhaseA}\n\\end{figure}\n \n\n\n\n\nNow, we investigate the behavior of those squeezing parameters for $k\\eta\\ll 1$ and $k\\eta_i \\gg 1$ analytically. The leading and sub-leading terms of $\\Lambda$ and $\\Xi$ can be calculated as\n\\begin{align}\n\\Lambda=1+ {\\cal O} \\left(\\frac{\\lambda^2 H^2\\eta_i^2 }{k^4 }\\right)\\,,\\qquad\n\\Xi=0+ {\\cal O} \\left(\\frac{\\lambda H \\eta}{k^2 }\\right)\\,.\n\\label{lambdaxi}\n\\end{align}\nWe find that sub-leading terms of $\\Lambda$ and $\\Xi$ are negligibly small near the end of inflation and which is consistent with the numerical results in FIGs.~\\ref{SqueezingA2} and \\ref{SqueezingB2}. This result tells us that the conversion from graviton pair production to graviton-photon pair production is hard to occur.\nFor the squeezing parameter $\\Omega$, we find\n\\begin{align}\n\\Omega= i e^{2ik \\eta_i} \\frac{5\\lambda^2H^2 \\eta_i^3 }{32k^3}\\,.\n\\label{main}\n\\end{align}\nIf we use the numerical values $\\lambda=5\\times10^{-13}\\,{\\rm GeV}^2$, \n $k=10^2\\,{\\rm GeV}$, $H=10^{14}\\,{\\rm GeV}$, and $\\eta_i=-2\\,{\\rm GeV}^{-1}$\n, we find $|\\Omega| \\sim 0.003$ and which agrees with the numerical result in FIG. \\ref{SqueezingC2}. Thus only small amount of conversion from graviton pair production to photon pair production occurs at the end of inflation.\nThese results support the validity of our iterative method to derive squeezing parameters.\n\n\n\n\\begin{figure}[H]\n\\centering\n \\includegraphics[width=0.55\\textwidth]{SqueezingB2.pdf}\n \\renewcommand{\\baselinestretch}{1.2}\n \\caption{Squeezing parameter of graviton-photon pair $\\Xi$ during inflation as a function of the scale factor $a$. We set $\\lambda=5\\times10^{-13}{\\rm GeV}^2$ (blue line). \n Other parameters are set as $k=10^2{\\rm GeV}$, $H=10^{14}{\\rm GeV}$, $\\eta_i=-2{\\rm GeV}^{-1}$, $\\eta_f =-10^{-14}{\\rm GeV}^{-1}$, $a(\\eta_i)=(2\\times 10^{14})^{-1}$, and $a(\\eta_f)=1$. }\n \\label{SqueezingB2}\n \\end{figure}\n \n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=\\textwidth]{PhaseB.pdf}\n\\renewcommand{\\baselinestretch}{1.2}\n\\caption{The phase of the squeezing parameter of photon pair $\\Xi(a)$ during inflation as a function of the scale factor $a(\\eta)$. We set $\\lambda=5\\times10^{-13}{\\rm GeV}^2$, $k=10^2{\\rm GeV}$, $H=10^{14}{\\rm GeV}$, $\\eta_i=-2{\\rm GeV}^{-1}$, $\\eta_f =-10^{-14}{\\rm GeV}^{-1}$, $a(\\eta_i)=(2\\times 10^{14})^{-1}$, and $a(\\eta_f)=1$. }\n\\label{PhaseB}\n\\end{figure}\n \n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.55\\textwidth]{SqueezingC2.pdf}\n\\renewcommand{\\baselinestretch}{1.2}\n\\caption{Squeezing parameter of photon pair $\\Omega$ during inflation as a function of the scale factor $a(\\eta)$. We set $\\lambda=5\\times10^{-13}{\\rm GeV}^2$, $k=10^2{\\rm GeV}$, $H=10^{14}{\\rm GeV}$, $\\eta_i=-2{\\rm GeV}^{-1}$, $\\eta_f =-10^{-14}{\\rm GeV}^{-1}$, $a(\\eta_i)=(2\\times 10^{14})^{-1}$, and $a(\\eta_f)=1$. }\n\\label{SqueezingC2}\n\\end{figure}\n\n\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=\\textwidth]{PhaseC.pdf}\n\\renewcommand{\\baselinestretch}{1.2}\n\\caption{The phase of the squeezing parameter of photon pair $\\Omega$ during inflation as a function of the scale factor $a(\\eta)$. We set $\\lambda=5\\times10^{-13}{\\rm GeV}^2$, $k=10^2{\\rm GeV}$, $H=10^{14}{\\rm GeV}$, $\\eta_i=-2{\\rm GeV}^{-1}$, $\\eta_f =-10^{-14}{\\rm GeV}^{-1}$, $a(\\eta_i)=(2\\times 10^{14})^{-1}$, and $a(\\eta_f)=1$. }\n\\label{PhaseC}\n\\end{figure}\n\n\n\n\n\nLet us discuss implications of our numerical and analytical results. If the squeezing of graviton decreases as time evolves, it implies that the decoherence of graviton occurs. \nHowever, we found that \nthe squeezing parameter of graviton pair increases and becomes $\\Lambda\\rightarrow 1$, so it seems that the decoherence is hard to occur.\nThis behavior can be understood as follows.\nSince the effective coupling $\\lambda H\\eta$ between graviton and photon in Eqs.~(\\ref{eom:graviton}) and (\\ref{eom:photon}) decreases and eventually becomes negligible as $\\eta\\rightarrow 0$ during inflation, practically graviton-photon conversion stops.\nEven after the graviton-photon conversion stops, the squeezing process of graviton pair continues as time evolves during inflation, so the squeezing of graviton pair $\\Lambda$ continues to grow irrespective of the presence of the magnetic field as shown in FIG.~\\ref{SqueezingA2}. \nNext, from FIG.~\\ref{SqueezingB2}, we see the squeezing parameter of graviton-photon pair vanishes $\\Xi\\rightarrow 0$ as time evolves. This is consistent with Eq.~(\\ref{lambdaxi}).\nThis is because the graviton-photon pair production is possible only in the presence of magnetic fields\ndue to spin conservation. In our setup, however, the energy density of the background magnetic field decreases proportional to $a(\\eta)^{-4}$ as the universe expands. \nHence, the rapid decay of magnetic fields lead to the rapid decay of $\\Xi$.\nFinally, we consider the evolution of the squeezing parameter of photon pair $\\Omega$. By using the coupling constant $\\lambda \\simeq Bk \/M_{\\rm pl}$ in Eq.~(\\ref{coupling}) and the scale factor at the initial time $a_i \\equiv -1\/(H\\eta_i)$, the $\\Omega$ reads\n\\begin{eqnarray}\n\\Omega\\simeq \\frac{B^2}{a_i^4 M_{\\rm pl}^2 H^2}\n\\frac{1}{k\\eta_i} \\ .\n\\end{eqnarray}\nThe first factor is the ratio of the energy density of the background magnetic field at the time $\\eta_i$\nto that of the inflaton field. The second factor is the ratio of the mode of graviton to the Hubble radius. In order to have inflation, the energy density of the magnetic field has to be smaller than that of inflaton fields, that is, $B^2\/a_i^4\\ll M_{\\rm pl}^2H^2$. And all modes of graviton is inside horizon initially, that is, $1\/k\\ll \\eta_i$. Hence, the $\\Omega$\nnever exceeds unity after time evolution, which is consistent with FIG.~\\ref{SqueezingC2}. \nMoreover, since graviton-photon conversion stops, \nthe squeezing of photon pair $\\Omega$ converges to a constant value as shown in FIG.~\\ref{SqueezingC2}. \n\n \n\\section{Conclusion\n\nThe relic gravitons are expected to be squeezed during inflation. In that case, quantum noise induced by them can be significantly enhanced in current interferometers. \nHowever, we need to properly take into account the decoherence of the relic gravitons during cosmic history.\nAs a first modest step in this direction, we assumed the presence of a sizable magnetic field at the beginning of inflation. If the squeezing of graviton decreases as time evolves, it implies that the decoherence of graviton occurs.\nSo, we studied the conversion processes of the squeezed gravitons into photons during inflation in the case of minimal coupling between gravitons and photons.\nWe solved the dynamical evolution of a coupled system of graviton and photon \nperturbatively. We numerically plotted\nthe squeezing parameters for the system of graviton and photon. \nFIG.~\\ref{SqueezingA2} showed that magnetic fields do not affect the graviton squeezing parameter. \nIn FIG.~\\ref{SqueezingB2},\nwe numerically checked the parameter of squeezed graviton-photon pair $\\Xi$ and found that the $\\Xi$ rapidly decays at the end of inflation.\nThis fact was confirmed analytically in Eq.~(\\ref{lambdaxi}). \nWe found that the rapid decay of the initial presence of magnetic fields leads to the rapid decay of the $\\Xi$.\nIn FIG.~\\ref{SqueezingC2}, \nwe depicted the squeezing parameter of the photon. It turned out that the amount of squeezed photon produced by the conversion was tiny. \nWe derived an analytic formula for the\nsqueezing parameter of photons $\\Omega$\nand found that the degree of squeezing is at a few percent at most.\n\n\nSince we found that gravitons are robust against the decoherence caused by the cosmological magnetic field, we could expect to find squeezed relic gravitons through quantum noise induced by them in interferometers~\\cite{Parikh:2020nrd,Kanno:2020usf,Parikh:2020kfh,Parikh:2020fhy,Kanno:2021gpt}.\nWe should note that the analysis in our paper can also be applicable to the dark magnetic field models~\\cite{Masaki:2018eut} based on the dark photon scenario~\\cite{Caputo:2021eaa}.\n\nThere are several directions to be pursued. \nIt would be intriguing to follow up the evolution of the squeezed relic gravitons up to the radiation-dominated \nand matter-dominated eras. If we could show the absence of decoherence of the squeezed relic gravitons, the robustness of them would be proved.\nIt would also be interesting to study the case that the primordial magnetic fields persist against the cosmic no-hair theorem during inflation~\\cite{Kanno:2009ei}.\nOn top of gravitons, the squeezing occurs for the light axion dark matter fields~\\cite{Kuss:2021gig,Kanno:2021vwu}.\nThe decoherence of axion fields due to magnetic fields can be discussed in a similar way.\nWe leave these issues for future work.\n\n\n\n\n\n\n\n\n\n\\section*{Acknowledgments}\nS.\\ K. was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Number JP22K03621.\nJ.\\ S. was in part supported by JSPS KAKENHI Grant Numbers JP17H02894, JP17K18778, JP20H01902, JP22H01220.\nK.\\ U. was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Number 20J22946.\n\n\\printbibliography\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:Intro}\nThe well known Lady\\v zenskaja-Babu\\v ska-Brezzi (LBB) condition is a particular instance of the\nso-called\ndiscrete inf--sup condition which is necessary and sufficient for the well-posedness of discrete\nsaddle point problems\narising from discretization via Galerkin methods. If ${\\bf X}_h$ denotes the discrete velocity space and\n$M_h$ the discrete\npressure space, then the LBB condition for the Stokes problem states that there is a constant\n$c$ independent of the discretization parameter $h$ such that\n\\begin{equation}\n c \\| q_h \\|_{L^2} \\leq \\sup_{v_h \\in {\\bf X}_h} \n \\frac{ \\int_\\Omega (\\DIV v_h)\\, q_h}{\\|v_h\\|_{{\\bf H}^1}},\n \\quad \\forall q_h \\in M_h.\n\\tag{LBB}\n\\label{eq:LBB}\n\\end{equation}\nThe reader is referred to \\cite{GR86} for the basic theory on saddle point problems on Banach spaces\nand \ntheir numerical analysis. Simply put, this condition sets a structural restriction on the discrete\nspaces so that\nthe continuous level property that the divergence operator is closed and surjective, see\n\\cite{MR82m:26014,MR1880723}, is preserved uniformly with respect to the discretization parameter.\n\nIn the literature the following condition, which we shall denote the generalized LBB condition, is\nalso assumed\n\\begin{equation}\nc \\| \\GRAD q_h \\|_{{\\bf L}^2} \\leq\n \\sup_{v_h \\in {\\bf X}_h}\n \\frac{ \\int_\\Omega (\\DIV v_h)\\, q_h }{ \\| v_h \\|_{{\\bf L}^2} },\n \\quad \\forall q_h \\in M_h,\n\\tag{GLBB}\n\\label{eq:wLBB}\n\\end{equation}\nhere and throughout we assume $M_h \\subset H^1(\\Omega)$. By properly defining a discrete gradient\noperator, the case of discontinuous pressure spaces can be analyzed with similar arguments to those\nthat we shall present. Condition \\eqref{eq:wLBB}, for example, was used by Guermond\n(\\cite{MR2210084,MR2334774}) to show that approximate solutions to the three-dimensional Navier\nStokes equations constructed using the Faedo-Galerkin method\nconverge to a suitable, in the sense of Scheffer, weak solution.\nOn the basis of \\eqref{eq:wLBB}, the same author has also built (\\cite{MR2520170}) an\n${\\bf H}^s$-approximation theory for the Stokes problem, $0\\leq s \\leq1$.\nOlshanski{\\u\\i}, in \\cite{MR2833487}, under the assumption that the spaces satisfy \\eqref{eq:wLBB}\ncarries out a multigrid analysis for the Stokes problem. Finally, Mardal et al.\\@\\xspace, \n\\cite{schoberlwinther}, use a weighted inf--sup condition to analyze preconditioning techniques\nfor singularly perturbed Stokes problems (see Section \\ref{sec:section5} below).\n\nIt is not difficult to show that, on quasi-uniform meshes, \\eqref{eq:wLBB} implies \\eqref{eq:LBB}, see\n\\cite{MR2210084}. We include the proof of this result below for completeness.\nThe question that naturally arises is whether the converse holds. Recall that a well-known result of\nFortin \\cite{BF91}\nshows that the inf--sup condition \\eqref{eq:LBB} is equivalent to the existence of a so-called Fortin\nprojection that is stable in ${{ \\bf H}^1_0 (\\Omega)}$. In this work, under the assumption that the mesh is \nshape regular and quasi-uniform, we will show that \\eqref{eq:wLBB} is equivalent\nto the existence of a Fortin projection that has ${\\bf L}^2$-approximation properties. Moreover,\nwhen the domain is such that the solution to the Stokes problem \npossesses ${\\bf H}^2$-regularity, we will prove that \n\\eqref{eq:wLBB} is in fact equivalent to \\eqref{eq:LBB}, again on quasi-uniform meshes.\n\nThe work by Girault and Scott (\\cite{MR1961943}) must be mentioned \nwhen dealing with the construction of Fortin projection operators with ${\\bf L}^2$-approximation properties.\nThey have constructed such operators for many commonly \nused inf--sup stable spaces, one notable exception being the lowest order Taylor-Hood element \nin three dimensions.\nHowever, \\eqref{eq:wLBB} has been shown to hold for the lowest order Taylor-Hood\nelement directly \\cite{MR2210084}.\nOur results then can be applied to show that, \\eqref{eq:wLBB} is satisfied \nby almost all inf--sup stable finite element spaces, regardless of the smoothness of the domain.\n\nThis work is organized as follows.\nSection~\\ref{sec:prel} introduces the notation and assumptions we shall work with. Condition\n\\eqref{eq:wLBB} is discussed in Section~\\ref{sec:wLBB}. In Section~\\ref{sec:Equiv} we actually show the\nequivalence of conditions \\eqref{eq:LBB} and \\eqref{eq:wLBB}, provided the domain is smooth enough.\nA weighted inf--sup condition related to uniform preconditioning of the time-dependent Stokes problem\nis presented in Section~\\ref{sec:section5}, where we show that \\eqref{eq:wLBB} implies it.\nSome concluding remarks are provided in Section~\\ref{sec:conclusion}.\n\n\\section{Preliminaries}\n\\label{sec:prel}\nThroughout this work, we will denote by $\\Omega \\subset \\mathbb R^d$ with $d=2$ or $3$ an open bounded\ndomain with Lipschitz boundary. If additional smoothness of the domain is needed, it will be specified\nexplicitly.\n${{ L}^2 (\\Omega)}$, ${{ H}^{1}(\\Omega)}$ and ${{ H}^1_0 (\\Omega)}$ denote, respectively, the usual Lebesgue and Sobolev spaces.\nWe denote by ${L^2_{\\scriptscriptstyle\\!\\int\\!=0} (\\Omega)}$ the set of functions in ${{ L}^2 (\\Omega)}$ with mean zero.\nVector valued spaces will be denoted by bold characters.\n\nWe introduce a conforming triangulation ${\\mathcal T}_h$ of $\\Omega$ which we assume shape-regular and\nquasi-uniform in the sense of \\cite{BF91}. The size of the cells in the triangulation is characterized by \n$h>0$. We introduce finite dimensional spaces ${\\bf X}_h \\subset {{ \\bf H}^1_0 (\\Omega)}$ and \n$M_h \\subset {L^2_{\\scriptscriptstyle\\!\\int\\!=0} (\\Omega)} \\cap {{ H}^{1}(\\Omega)}$ which are\nconstructed, for instance using finite elements,\non the triangulation ${\\mathcal T}_h$. For these spaces, the inverse inequalities\n\\begin{equation}\n \\| v_h \\|_{{\\bf H}^1} \\leq c h^{-1} \\| v_h \\|_{{\\bf L}^2}, \\quad \\forall v_h \\in {\\bf X}_h,\n\\label{eq:invX}\n\\end{equation}\nand\n\\begin{equation}\n \\| q_h \\|_{H^1} \\leq c h^{-1} \\| q_h \\|_{L^2}, \\quad \\forall q_h \\in M_h,\n\\label{eq:invM}\n\\end{equation}\nhold, see \\cite{BF91}. Here and in what follows we denote by $c$ will a constant that is independent of $h$. \n\nWe shall denote by ${\\mathcal C}_h : {{ \\bf H}^1_0 (\\Omega)} \\rightarrow {\\bf X}_h$ the so-called \nScott-Zhang interpolation operator (\\cite{SZ90}) onto the velocity space and we recall that\n\\begin{equation}\n \\| v - {\\mathcal C}_h v \\|_{{\\bf L}^2} + h \\|{\\mathcal C}_h v\\|_{{\\bf H}^1} \\leq c h \\| v \\|_{{\\bf H}^1}, \\quad \\forall v\n\\in {{ \\bf H}^1_0 (\\Omega)}.\n\\label{eq:SZprop}\n\\end{equation}\nand\n\\begin{equation}\n \\| v - {\\mathcal C}_h v\\|_{{\\bf H}^1} \\leq c h \\| v \\|_{{\\bf H}^2}, \\quad \\forall v \\in {{ \\bf H}^1_0 (\\Omega)} \\cap {{ \\bf H}^2 (\\Omega)}\n\\label{eq:SZh1}\n\\end{equation}\nThe Scott-Zhang interpolation operator onto the pressure space\n${\\mathcal I}_h: {L^2_{\\scriptscriptstyle\\!\\int\\!=0} (\\Omega)} \\rightarrow M_h$ can be defined analogously\nand satisfies similar stability and approximation properties.\nWe shall denote by $\\pi_h: {{ \\bf L}^2 (\\Omega)} \\rightarrow {\\bf X}_h$ the ${\\bf L}^2$-projection onto ${\\bf X}_h$ and\nby $\\Pi_0 : {{ L}^2 (\\Omega)} \\rightarrow {{ L}^2 (\\Omega)}$ the $L^2$-projection operator onto the space\nof piecewise constant functions, i.e.,\\@\\xspace\n\\[\n \\Pi_0 q = \\sum_{T \\in {\\mathcal T}_h} \\frac1{|T|}\\left(\\int_T q\\right)\\chi_T, \\quad \\forall q \\in {{ L}^2 (\\Omega)}.\n\\]\n\nFor one result below we shall require full ${\\bf H}^2$-regularity of the solution to the Stokes problem:\n\n\\begin{assumption}\\label{assumption1}\nThe domain $\\Omega$ is such that for any $f\\in {{ \\bf L}^2 (\\Omega)}$, the solution\n$(\\psi,\\theta) \\in {{ \\bf H}^1_0 (\\Omega)} \\times {L^2_{\\scriptscriptstyle\\!\\int\\!=0} (\\Omega)}$ to the Stokes problem\n\\begin{equation}\n\\label{eq:contstokes}\n \\begin{dcases}\n -\\LAP \\psi + \\GRAD \\theta = f, & \\text{in } \\Omega, \\\\\n \\DIV \\psi = 0, & \\text{in } \\Omega, \\\\\n \\psi = 0, & \\text{on } \\partial\\Omega,\n \\end{dcases}\n\\end{equation}\nsatisfies the following estimate:\n\\begin{equation}\n \\| \\psi \\|_{{\\bf H}^2} + \\| \\theta \\|_{H^1} \\leq c \\| f \\|_{{\\bf L}^2}.\n\\label{eq:Cattabriga}\n\\end{equation}\n\\end{assumption}\n\nAssumption~\\ref{assumption1} is known to hold in two and three dimensions ($d=2,3$) whenever\n$\\Omega$ is convex or of class ${\\mathcal C}^{1,1}$, see \\cite[Theorem 6.3]{MR977489}.\n\nBy suitably defining a discrete gradient operator acting on the pressure space, the proofs for \ndiscontinuous pressure spaces can be carried out with similar arguments.\n\nWe introduce the definition of a Fortin projection.\n\\begin{definition}\n\\label{def:Fortin}\nAn operator ${\\mathcal F}_h : {{ \\bf H}^1_0 (\\Omega)} \\rightarrow {\\bf X}_h$ is called a Fortin projection if ${\\mathcal F}_h^2 =\n{\\mathcal F}_h$ and\n\\begin{equation}\n \\int_\\Omega \\DIV(v-{\\mathcal F}_h v)q_h = 0, \\quad \\forall v \\in {{ \\bf H}^1_0 (\\Omega)}, \\quad \\forall q_h \\in M_h.\n\\label{eq:Fortin}\n\\end{equation}\n\\end{definition}\n\nWe shall be interested in Fortin projections ${\\mathcal F}_h$ that satisfy the condition:\n\\begin{equation}\n \\| {\\mathcal F}_h v \\|_{{\\bf H}^1} \\leq c \\| v \\|_{{\\bf H}^1}, \\quad \\forall v \\in {{ \\bf H}^1_0 (\\Omega)},\n\\tag{FH1}\n\\label{eq:fh1}\n\\end{equation}\nor\n\\begin{equation}\n \\| v - {\\mathcal F}_h v \\|_{{\\bf L}^2} \\leq c h \\| v\\|_{{\\bf H}^1}, \\quad \\forall v \\in {{ \\bf H}^1_0 (\\Omega)}.\n\\tag{FL2}\n\\label{eq:fl2}\n\\end{equation}\n\nLet us remark that the approximation property \\eqref{eq:fl2} implies ${\\bf H}^1$-stability.\n\\begin{lem}\n\\label{lem:fl2implfh1}\nIf an operator ${\\mathcal F}_h : {{ \\bf H}^1_0 (\\Omega)} \\rightarrow {\\bf X}_h$ satisfies \\eqref{eq:fl2} then it is ${\\bf H}^1$-stable, i.e., \\eqref{eq:fh1} is satisfied.\n\\end{lem}\n\\begin{proof}\nThe proof relies on the stability and approximation properties \\eqref{eq:SZprop} of the Scott-Zhang operator\nand on the inverse estimate \\eqref{eq:invX}, for if $v \\in {{ \\bf H}^1_0 (\\Omega)}$,\n\\begin{align*}\n \\| {\\mathcal F}_h v \\|_{{\\bf H}^1} &\\leq \\| {\\mathcal F}_h v - {\\mathcal C}_h v \\|_{{\\bf H}^1} + c \\| v \\|_{{\\bf H}^1}\n \\leq c h^{-1} \\| {\\mathcal F}_h v - {\\mathcal C}_h v \\|_{{\\bf L}^2} + c \\| v \\|_{{\\bf H}^1} \\\\\n &\\leq ch^{-1} \\| v - {\\mathcal F}_h v \\|_{{\\bf L}^2} + ch^{-1}\\| v - {\\mathcal C}_h v \\|_{{\\bf L}^2} + c \\|v\\|_{{\\bf H}^1}.\n\\end{align*}\nConclude using the ${\\bf L}^2$-approximation properties of the operators ${\\mathcal F}_h$ and ${\\mathcal C}_h$.\n\\end{proof}\n\n\\begin{rem}\nGirault and Scott, \\cite{MR1961943}, explicitly constructed a Fortin projection \nthat satisfies \\eqref{eq:fh1} and \\eqref{eq:fl2} for many\ncommonly used spaces. In fact, they showed that the approximation is local, i.e.,\\@\\xspace\n\\[\n \\| {\\mathcal F}_h v - v \\|_{{\\bf L}^2(T)}+ h_T\\| {\\mathcal F}_h v -v\\|_{{\\bf H}^1(T)} \\leq c h_T \\| v\n\\|_{{\\bf H}^1({\\mathcal N}(T))}, \\quad \\forall v \\in {{ \\bf H}^1_0 (\\Omega)}\n \\text{ and } \\forall T \\in {\\mathcal T}_h,\n\\]\nwhere ${\\mathcal N}(T)$ is a patch containing $T$.\nIn particular, they have shown the existence of this projection for the Taylor-Hood elements in two \ndimensions. In three dimensions they proved this result for all the Taylor-Hood elements except the \nlowest order case. \n\\end{rem}\n\nIn this work we shall prove the implications\n\\[\n \\xymatrix{\n \\eqref{eq:LBB} \\ar@{<=>}[r] \\ar@{<=}[d]\n &\\exists {\\mathcal F}_h \\mbox{\\, s.t.\\,} \\eqref{eq:Fortin} \\text{ and } \\eqref{eq:fh1} & \\\\\n \\eqref{eq:wLBB} \\ar@{<=>}[r]\n &\\exists {\\mathcal F}_h \\mbox{\\, s.t.\\,} \\eqref{eq:Fortin} \\text{ and } \\eqref{eq:fl2} & \\eqref{eq:LBB} \\text{ and \nAssumption } \\ref{assumption1} \\ar@{=>}[l] \n }\n\\]\nthus showing that, in our setting, all these conditions are indeed equivalent. \nThe top equivalence is\nwell-known, see \\cite{BF91,GR86,MR2050138}. The left implication is also known (see\n\\cite{MR2210084}), for completeness we show\nthis in Theorem~\\ref{thm:wlbbimpllbb}. The bottom implications, although simple to prove, seem to be\nnew. \n\n\\section{The Generalized LBB Condition}\n\\label{sec:wLBB}\nLet us begin by noticing that the generalized LBB condition \\eqref{eq:wLBB} is actually a statement\nabout coercivity of the ${\\bf L}^2$-projection on gradients of functions in the pressure space. Namely,\n\\eqref{eq:wLBB} is equivalent to\n\\begin{equation}\\label{GLBBb}\n \\| \\pi_h \\GRAD q_h \\|_{{\\bf L}^2} \\geq c \\| \\GRAD q_h \\|_{{\\bf L}^2}, \\quad \\forall q_h \\in M_h.\n\\end{equation}\n\nIt is well known that \\eqref{eq:wLBB} implies \\eqref{eq:LBB}. For completeness we present the proof.\nWe\nbegin with a perturbation result.\n\n\\begin{lem}\n\\label{lem:verfurth}\nThere exists a constant $c$ independent of $h$ such that, for all $q_h \\in M_h$, the following holds:\n\\[\n c \\| q_h \\|_{L^2} \\leq \n \\sup_{v_h \\in {\\bf X}_h} \\frac{ \\int_\\Omega (\\DIV v_h)\\, q_h }{\\| \\GRAD v_h \\|_{{\\bf L}^2} }\n + h \\| \\GRAD q_h \\|_{{\\bf L}^2}.\n\\]\n\\end{lem}\n\\begin{proof}\nThe proof relies on the properties \\eqref{eq:SZprop} of the Scott-Zhang interpolation operator ${\\mathcal C}_h$,\n\\begin{align*}\n c \\|q_h\\|_{L^2} &\\leq \\sup_{v \\in {{ \\bf H}^1_0 (\\Omega)}} \\frac{ \\int_\\Omega (\\DIV v)\\, \\, q_h }{\\| \\GRAD v\n\\|_{{\\bf L}^2} }\n \\leq \n \\sup_{ v \\in {{ \\bf H}^1_0 (\\Omega)}} \\frac{ \\int_\\Omega (\\DIV\\,{\\mathcal C}_h v)\\, q_h }{\\| \\GRAD ({\\mathcal C}_h v)\n\\|_{{\\bf L}^2} }\n +\n \\sup_{ v \\in {{ \\bf H}^1_0 (\\Omega)}}\\frac{\\int_\\Omega\\big(\\DIV\\left(v - {\\mathcal C}_h v \\right)\\big)q_h }{\\|\\GRAD\nv\\|_{{\\bf L}^2} }\n \\\\ &\\leq\n \\sup_{ v_h \\in {\\bf X}_h} \\frac{ \\int_\\Omega (\\DIV v_h) \\, q_h }{\\| \\GRAD v_h \\|_{{\\bf L}^2} }\n + \\sup_{ v \\in {{ \\bf H}^1_0 (\\Omega)}} \\frac{ \\int_\\Omega\\left(v - {\\mathcal C}_h v \\right){\\cdot}\\GRAD q_h}{\\|\\GRAD\nv\\|_{{\\bf L}^2}},\n\\end{align*}\nconclude using \\eqref{eq:SZprop}.\n\\end{proof}\nOn the basis of Lemma~\\ref{lem:verfurth} we can readily show that \\eqref{eq:wLBB} implies \\eqref{eq:LBB}. Again,\nthis result is not new and we only include the proof for completeness.\n\\begin{thm}\n\\label{thm:wlbbimpllbb}\n\\eqref{eq:wLBB} implies \\eqref{eq:LBB}.\n\\end{thm}\n\\begin{proof}\nSince we assumed that $M_h \\subset {L^2_{\\scriptscriptstyle\\!\\int\\!=0} (\\Omega)} \\cap {{ H}^{1}(\\Omega)}$, the proof is straightforward:\n\\[\n \\sup_{ v_h \\in {\\bf X}_h} \\frac{ \\int_\\Omega (\\DIV v_h)\\, q_h }{\\|\\GRAD v_h \\|_{{\\bf L}^2} }\n =\n \\sup_{ v_h \\in {\\bf X}_h} \\frac{ \\int_\\Omega v_h {\\cdot} \\GRAD q_h }{\\|\\GRAD v_h \\|_{{\\bf L}^2} }\n \\geq\n \\frac{ \\int_\\Omega \\pi_h \\GRAD q_h {\\cdot}\\GRAD q_h }{\\|\\GRAD \\pi_h \\GRAD q_h \\|_{{\\bf L}^2} }\n =\n \\frac{ \\|\\pi_h \\GRAD q_h \\|_{{\\bf L}^2}^2 }{ \\| \\GRAD \\pi_h \\GRAD q_h \\|_{{\\bf L}^2} }\n \\geq\n c h \\|\\pi_h \\GRAD q_h \\|_{{\\bf L}^2}\n\\]\nwhere, in the last step, we used the inverse inequality \\eqref{eq:invX}.\nThis, in conjunction with Lemma~\\ref{lem:verfurth} and the characterization \\eqref{GLBBb}, implies the result.\n\\end{proof}\n\nLet us now show that the generalized LBB condition \n\\eqref{eq:wLBB} is equivalent to the existence of a Fortin operator satisfying \\eqref{eq:fl2}.\nWe begin with a modification of a classical result.\n\n\\begin{lem}\n\\label{lem:tartarwithh}\nFor all $p\\in {{ H}^{1}(\\Omega)}$ there is $v\\in{{ \\bf H}^1_0 (\\Omega)}$ such that\n\\[\n \\DIV v = p - \\Pi_0 p, \\qquad v|_{\\partial T}=0 \\quad \\forall T\\in {\\mathcal T}_h,\n\\]\nand\n\\[\n \\| v \\|_{{\\bf L}^2} \\leq c \\left( \\sum_{T \\in {\\mathcal T}_h} h_T^4 \\| \\GRAD p \\|_{{\\bf L}^2(T)}^2 \\right)^{1\/2}.\n\\]\n\\end{lem}\n\\begin{proof}\nLet $p\\in {{ H}^{1}(\\Omega)}$ and $T \\in {\\mathcal T}_h$. Clearly,\n\\[\n \\int_T p - \\Pi_0 p = 0.\n\\]\nA classical result (\\cite{MR82m:26014,MR1846644,GR86,MR1880723}) implies that there is a \n$v_T \\in {\\bf H}^1_0(T)$ with\n$ \\DIV v_T = p - \\Pi_0 p$\nin $T$ and\n\\begin{equation}\n \\| \\GRAD v_T \\|_{{\\bf L}^2(T)} \\leq c \\| p - \\Pi_0 p \\|_{L^2(T)}.\n\\label{eq:saves}\n\\end{equation}\nGiven that the mesh is assumed to be shape regular, by mapping to the reference element it is seen\nthat\nthe constant in the last inequality does not depend on $T \\in {\\mathcal T}_h$.\n\nLet $v \\in {{ \\bf H}^1_0 (\\Omega)}$ be defined as $ v|_T = v_T$ for all $T$ in ${\\mathcal T}_h$. By construction,\n\\[\n \\DIV v = p - \\Pi_0 p, \\quad \\text{\\ae in } \\Omega.\n\\]\nMoreover,\n\\[\n \\| v \\|_{{\\bf L}^2}^2 = \\sum_{T \\in {\\mathcal T}_h} \\| v \\|_{{\\bf L}^2(T)}^2\n \\leq c \\sum_{T \\in {\\mathcal T}_h} h_T^2 \\| \\GRAD v \\|_{{\\bf L}^2(T)}^2\n \\leq c \\sum_{T \\in {\\mathcal T}_h} h_T^2 \\| p - \\Pi_0 p \\|_{L^2(T)}^2\n \\leq c \\sum_{T \\in {\\mathcal T}_h} h_T^4 \\| \\GRAD p \\|_{{\\bf L}^2(T)}^2.\n\\]\nThe first equality is by definition; then we applied the Poincar\\'e-Friedrichs inequality (since\n$v|_T = v_T \\in {\\bf H}^1_0(T)$); next we used the properties of the function $v_T$ and the\napproximation properties of the projector $\\Pi_0$.\n\\end{proof}\n\nWith this result at hand we can prove the following.\n\n\\begin{thm}\n\\label{thm:fl2implwlbb}\nIf there exists a Fortin operator ${\\mathcal F}_h$ that satisfies \\eqref{eq:fl2}, then\n\\eqref{eq:wLBB} holds.\n\\end{thm}\n\\begin{proof}\nLet $q_h \\in M_h$. Using the properties of the operator $\\Pi_0$ and \nthe local analogue of\nthe inverse inequality \\eqref{eq:invM}, we get\n\\[\n \\| \\GRAD q_h \\|_{{\\bf L}^2}^2 \n = \\sum_{T \\in {\\mathcal T}_h} \\left\\| \\GRAD \\left(q_h - \\Pi_0 q_h \\right) \\right\\|_{{\\bf L}^2(T)}^2\n \\leq \\sum_{T \\in {\\mathcal T}_h} \\frac1{h_T^2} \\| q_h - \\Pi_0 q_h \\|_{{\\bf L}^2(T)}^2\n \\leq \\frac{c}{h^2} \\| q_h - \\Pi_0 q_h \\|_{{\\bf L}^2}^2.\n\\]\nFrom Lemma~\\ref{lem:tartarwithh} we know there exists $v \\in {{ \\bf H}^1_0 (\\Omega)}$ with $\\DIV v = q_h - \\Pi_0 q_h$\nand\n\\[\n \\| v \\|_{{\\bf L}^2} \\leq c h^2 \\| \\GRAD q_h \\|_{{\\bf L}^2},\n\\]\nhence\n\\[\n \\| \\GRAD q_h \\|_{{\\bf L}^2}^2 \\leq \\frac{c}{h^2} \\| q_h - \\Pi_0 q_h \\|_{L^2}^2\n = \\frac{c}{h^2} \\int_\\Omega (\\DIV v) \\, (q_h-\\Pi_0 q_h)\n = \\frac{c}{h^2} \\int_\\Omega (\\DIV v) \\, q_h,\n\\]\nwhere the last inequality follows from integration by parts over each $T$ and using the fact that $v|_{\\partial T} = 0$ (see Lemma~\\ref{lem:tartarwithh}).\n\nUsing the existence of the operator ${\\mathcal F}_h$,\n\\[\n \\| \\GRAD q_h \\|_{{\\bf L}^2}^2 \\le \\frac{c}{h^2} \\int_\\Omega (\\DIV \\,{\\mathcal F}_h v) q_h\n \\leq \\left( \\sup_{ w_h \\in {\\bf X}_h} \\frac{ \\int_\\Omega (\\DIV w_h) \\, q_h }{ \\| w_h \\|_{{\\bf L}^2}\n}\\right) \n \\frac{c}{h^2}\\| {\\mathcal F}_h v \\|_{{\\bf L}^2}.\n\\]\nIt remains to show that \n\\[\n \\| {\\mathcal F}_h v \\|_{{\\bf L}^2} \\leq c h^2 \\| \\GRAD q_h \\|_{{\\bf L}^2}.\n\\]\nFor this purpose, we use the approximation property \\eqref{eq:fl2} and\nLemma~\\ref{lem:tartarwithh}\n\\[\n \\| {\\mathcal F}_h v \\|_{{\\bf L}^2} \\leq \\| {\\mathcal F}_h v - v\\|_{{\\bf L}^2} + \\|v \\|_{{\\bf L}^2}\n \\leq ch \\| \\GRAD v \\|_{{\\bf L}^2} + c h^2 \\|\\GRAD q_h \\|_{{\\bf L}^2}\n \\leq ch^2 \\| \\GRAD q_h \\|_{{\\bf L}^2},\n\\]\nwhere the last inequality holds because of \\eqref{eq:saves}.\n\\end{proof}\n\nThe converse of Theorem~\\ref{thm:fl2implwlbb} is given in the following.\n\n\\begin{thm}\n\\label{thm:wlbbimplfl2}\nIf \\eqref{eq:wLBB} holds, then there exists a Fortin projector ${\\mathcal F}_h$ that satisfies\n\\eqref{eq:fl2}.\n\\end{thm}\n\\begin{proof}\nLet $v\\in{{ \\bf H}^1_0 (\\Omega)}$. Define $(z_h, p_h) \\in {\\bf X}_h \\times M_h$ as the solution of\n\\begin{equation}\n \\begin{dcases}\n \\int_\\Omega z_h{\\cdot} w_h -\\int_\\Omega p_h \\DIV w_h = \\int_\\Omega v{\\cdot} w_h, & \\forall w_h \\in {\\bf X}_h, \\\\\n \\int_\\Omega q_h \\DIV z_h = \\int_\\Omega q_h \\DIV v, & \\forall q_h \\in M_h.\n \\end{dcases}\n\\label{eq:l2stokes}\n\\end{equation}\nNotice that \\eqref{eq:wLBB} provides precisely necessary and sufficient conditions for this problem\nto have a\nunique solution.\n\nDefine ${\\mathcal F}_h v := z_h$ we claim that this is indeed a Fortin projection that satisfies\n\\eqref{eq:fl2}.\nBy construction, \\eqref{eq:Fortin} holds (see the second equation in \\eqref{eq:l2stokes}). To show\nthat this is indeed a\nprojection, assume that $v=v_h \\in {\\bf X}_h$ in \\eqref{eq:l2stokes}, setting $w_h = z_h - v_h$ we\nreadily obtain that\n\\[\n \\| z_h - v_h \\|_{{\\bf L}^2}^2 =0.\n\\]\nIt remains to show the approximation properties of this operator. We begin by noticing that\n\\eqref{eq:wLBB} implies\n\\begin{equation}\n c \\| \\GRAD p_h \\|_{{\\bf L}^2} \\leq \\sup_{w_h \\in {\\bf X}_h} \\frac{ \\int_\\Omega p_h \\DIV w_h }{ \\| w_h\n\\|_{{\\bf L}^2}}\n \\leq \\sup_{w_h \\in {\\bf X}_h} \\frac{ \\int_\\Omega (v-{\\mathcal F}_h v){\\cdot} w_h }{ \\| w_h \\|_{{\\bf L}^2}}\n \\leq \\| v - {\\mathcal F}_h v \\|_{{\\bf L}^2},\n\\label{eq:boundgradp}\n\\end{equation}\nwhere we used \\eqref{eq:l2stokes}. To obtain the approximation property \\eqref{eq:fl2} we use the\nScott-Zhang interpolation operator ${\\mathcal C}_h$,\n\\begin{align*}\n \\| {\\mathcal F}_h v - v \\|_{{\\bf L}^2}^2 &= \\int_\\Omega ( {\\mathcal C}_h v - v ){\\cdot}( {\\mathcal F}_h v - v )\n + \\int_\\Omega ( {\\mathcal F}_h v - {\\mathcal C}_h v ){\\cdot}( {\\mathcal F}_h v - v ) \\\\\n & \\leq\n \\| {\\mathcal C}_h v - v \\|_{{\\bf L}^2}\\| {\\mathcal F}_h v - v \\|_{{\\bf L}^2}\n + \\int_\\Omega ( {\\mathcal F}_h v - {\\mathcal C}_h v ){\\cdot}( {\\mathcal F}_h v - v ).\n\\end{align*}\nWe bound the first term using the approximation property \\eqref{eq:SZprop} of ${\\mathcal C}_h$. To bound\nthe second\nterm we use problem \\eqref{eq:l2stokes} with $w_h = {\\mathcal F}_h v - {\\mathcal C}_h v$, then\n\\[\n \\int_\\Omega ( {\\mathcal F}_h v - {\\mathcal C}_h v ){\\cdot}( {\\mathcal F}_h v - v )\n = \\int_\\Omega p_h \\DIV( {\\mathcal F}_h v - {\\mathcal C}_h v )\n = \\int_\\Omega p_h \\DIV( v - {\\mathcal C}_h v )\n = -\\int_\\Omega \\GRAD p_h {\\cdot} ( v - {\\mathcal C}_h v ),\n\\]\nwe conclude applying the Cauchy-Schwarz inequality and using \\eqref{eq:boundgradp}.\n\\end{proof}\n\n\\section{Smooth Domains}\n\\label{sec:Equiv}\nHere we show that, provided \\eqref{eq:LBB} holds and, moreover,\nthe domain $\\Omega$ is such that Assumption~\\ref{assumption1} is satisfied, then \\eqref{eq:fl2}\nholds and hence \\eqref{eq:wLBB} holds as well. This is shown in the following.\n\n\\begin{thm}\n\\label{thm:lbbimplwlbb}\nAssume the domain $\\Omega$ is such that the solution to\n\\eqref{eq:contstokes} possesses ${\\bf H}^2$-elliptic regularity, i.e.,\\@\\xspace Assumption \\ref{assumption1}\nholds. Then \\eqref{eq:LBB} implies that there is a Fortin operator\n${\\mathcal F}_h$ that satisfies \\eqref{eq:fl2}.\n\\end{thm}\n\\begin{proof}\nLet $v\\in {{ \\bf H}^1_0 (\\Omega)}$. Define $(z_h,p_h) \\in {\\bf X}_h \\times M_h$ as the solution to the discrete Stokes\nproblem\n\\begin{equation}\n \\begin{dcases}\n \\int_\\Omega \\GRAD z_h {:} \\GRAD w_h -\\int_\\Omega p_h \\DIV w_h = \\int_\\Omega \\GRAD v {:} \\GRAD w_h, & \\forall w_h\n\\in {\\bf X}_h, \\\\\n \\int_\\Omega q_h \\DIV z_h = \\int_\\Omega q_h \\DIV v, & \\forall q_h \\in M_h,\n \\end{dcases}\n\\label{eq:stokes}\n\\end{equation}\nwhere, in \\eqref{eq:stokes}, the colon is used to denote the tensor product of matrices. Notice that \\eqref{eq:LBB} implies that\nthis problem always has a unique solution.\n\nSet ${\\mathcal F}_h v := z_h$. Proceeding as in the proof of Theorem~\\ref{thm:wlbbimplfl2} we see that this is \nindeed a projection. Moreover, \\eqref{eq:Fortin} holds by construction.\nIt remains to\nshow that \\eqref{eq:fl2} is satisfied. To this end, analogously to the proof of\nTheorem~\\ref{thm:wlbbimplfl2},\nwe notice that \\eqref{eq:LBB} implies\n\\[\n \\| p_h \\|_{L^2} \\leq c \\| \\GRAD ({\\mathcal F}_h v - v )\\|_{{\\bf L}^2}.\n\\]\nWe now argue by duality. Let $\\psi$ and $\\phi$ solve\n\\eqref{eq:contstokes} with $f={\\mathcal F}_h v- v$. Assumption \\eqref{eq:Cattabriga} then implies\n\\begin{align*}\n \\| {\\mathcal F}_h v - v \\|_{{\\bf L}^2}^2 &= \\int_\\Omega ({\\mathcal F}_h v - v){\\cdot}(-\\LAP\\psi + \\GRAD \\theta) \\\\\n &= \\int_\\Omega \\GRAD({\\mathcal F}_h v -v ):\\GRAD(\\psi-{\\mathcal C}_h \\psi) -\\int_\\Omega(\\theta - {\\mathcal I}_h\n\\theta)\\,\\DIV({\\mathcal F}_h v -v )\\, \\\\\n & \\phantom{=}+ \\int_\\Omega \\GRAD({\\mathcal F}_h v -v ):\\GRAD({\\mathcal C}_h \\psi) - \\int_\\Omega ({\\mathcal I}_h \\theta)\\,\\DIV({\\mathcal F}_h v -v )\\,\n\\end{align*}\nNotice that since ${\\mathcal I}_h \\theta\\in M_h$, $\\int_\\Omega ({\\mathcal I}_h \\theta)\\,\\DIV({\\mathcal F}_h v -v )=0$.\nSince $\\DIV \\psi = 0$,\nusing \\eqref{eq:stokes}, the estimate for $p_h$, \\eqref{eq:SZh1} and \\eqref{eq:Cattabriga},\n\\[\n \\int_\\Omega \\GRAD({\\mathcal F}_h v -v ):\\GRAD({\\mathcal C}_h \\psi) = \\int_\\Omega p_h \\DIV ({\\mathcal C}_h \\psi-\\psi)\n \\leq ch \\| v - {\\mathcal F}_h v \\|_{{\\bf H}^1} \\| v - {\\mathcal F}_h v \\|_{{\\bf L}^2}.\n\\]\nA direct application of of \\eqref{eq:SZh1}, \\eqref{eq:SZprop} and \\eqref{eq:Cattabriga} allows us to\nobtain the following estimates:\n\\[\n \\int_\\Omega (\\theta - {\\mathcal I}_h \\theta)\\,\\DIV({\\mathcal F}_h v -v ) \n + \\int_\\Omega \\GRAD({\\mathcal F}_h v -v ){:}\\GRAD(\\psi-{\\mathcal C}_h \\psi)\n \\leq\n c h \\| {\\mathcal F}_h v - v \\|_{{\\bf L}^2}\\| v \\|_{{\\bf H}^1}\n\\]\n\nWe conclude using a stability estimate for \\eqref{eq:stokes}\n\\[\n \\| {\\mathcal F}_h v - v \\|_{{\\bf L}^2} \\leq c h \\| {\\mathcal F}_h v - v \\|_{{\\bf H}^1} \\leq c h \\| v \\|_{{\\bf H}^1},\n\\]\nwhich, given \\eqref{eq:LBB}, is uniform in $h$.\n\\end{proof}\n\n\\section{The Weighted LBB condition}\n\\label{sec:section5}\n\nIn relation to the construction of uniform preconditioners for\ndiscretizations of the time dependent Stokes problem, Mardal, Sch\\\"oberl and Winther, \\cite{schoberlwinther}, \nconsider the following inf--sup condition,\n\\begin{equation}\n c\\| q_h \\|_{H^1+\\epsilon^{-1} L^2} \\leq \n \\sup_{v_h \\in {\\bf X}_h} \\frac{ \\int_\\Omega \\DIV v_h q_h}{\\|v_h\\|_{{\\bf L}^2 \\cap \\epsilon {\\bf H}^1}}, \n \\quad \\forall q_h \\in M_h.\n\\label{eq:winf-sup}\n\\end{equation}\nwhere\n\\[\n\\| q \\|_{H^1+\\epsilon^{-1} L^2}^2= \\inf_{q_1+q_2=q} \n \\left( \\|q_1\\|_{H^1}^2+ \\epsilon^{-2} \\|q_2\\|_{L^2}^2 \\right),\n\\]\nand\n\\[\n \\|v\\|_{{\\bf L}^2 \\cap \\epsilon {\\bf H}^1}^2 = \\|v\\|_{{\\bf L}^2}^2 + \\epsilon^2 \\|v\\|_{{\\bf H}^1}^2.\n\\]\n\nBy constructing a Fortin projection operator that is ${\\bf L}^2$-bounded they have showed, on quasi-uniform meshes, \nthat the inf--sup condition \\eqref{eq:winf-sup} holds for the lowest order Taylor-Hood element in two dimension.\nIn addition, they proved the same result, on shape regular meshes, for the mini-element.\nHere, we show that \\eqref{eq:winf-sup} holds if we assume \\eqref{eq:wLBB}. A simple consequence of this \nresult is that, on quasi-uniform meshes, \\eqref{eq:winf-sup} holds for any order Taylor-Hood elements\nin two and three dimensions.\n\n\\begin{thm}\n\\label{thm:wlbbimplweightlbb}\nLet $\\Omega$ be star shaped with respect to ball.\nIf the spaces ${\\bf X}_h$ and $M_h$ are such that \\eqref{eq:wLBB} is satisfied,\nthen the inf--sup condition \\eqref{eq:winf-sup} holds with a constant that \ndoes not depend on $\\epsilon$ or $h$.\n\\end{thm}\n\\begin{proof}\nWe consider two cases: $\\epsilon \\ge h$ and $\\epsilon < h$.\n\nGiven that the domain $\\Omega$ is star shaped with respect to a ball, \nwe can conclude (\\cite{schoberlwinther}) that the following \ncontinuous inf--sup condition holds,\n\\begin{equation}\\label{conweight}\nc \\| q \\|_{H^1+\\epsilon^{-1} L^2} \\leq \\sup_{v \\in {{ \\bf H}^1_0 (\\Omega)}}\n \\frac{ \\int_\\Omega q\\, \\DIV v }{ \\|v\\|_{{\\bf L}^2 \\cap \\epsilon {\\bf H}^1}},\n \\quad \\forall q \\in {L^2_{\\scriptscriptstyle\\!\\int\\!=0} (\\Omega)},\n\\end{equation}\nwith a constant $c$ independent of $\\epsilon$.\n\n\nWe first assume that $\\epsilon \\ge h$. Using \\eqref{conweight} for $q_h \\in M_h$ we have, \n\\begin{align*}\n c \\| q_h \\|_{H^1+\\epsilon^{-1} L^2} &\\leq\n \\sup_{v \\in {{ \\bf H}^1_0 (\\Omega)} } \\frac{ \\int q_h\\,\\DIV v }{ \\| v \\|_{{\\bf L}^2 \\cap \\epsilon {\\bf H}^1 } } =\n \\sup_{v \\in {{ \\bf H}^1_0 (\\Omega)} } \\frac{ \\int_\\Omega q_h\\,\\DIV ({\\mathcal F}_h v) }{\\|{\\mathcal F}_h v\\|_{{\\bf L}^2 \\cap \\epsilon {\\bf H}^1}}\n \\frac{\\|{\\mathcal F}_h v\\|_{{\\bf L}^2 \\cap \\epsilon {\\bf H}^1}}{\\|v\\|_{{\\bf L}^2\\cap \\epsilon {\\bf H}^1} } \\\\\n &\\leq \\sup_{v_h \\in {\\bf X}_h} \\frac{ \\int_\\Omega q_h\\,\\DIV v_h }\n {\\|v_h\\|_{{\\bf L}^2 \\cap \\epsilon {\\bf H}^1} }\n \\sup_{ v \\in {{ \\bf H}^1_0 (\\Omega)}} \\frac{\\|{\\mathcal F}_h v\\|_{{\\bf L}^2 \\cap \\epsilon {\\bf H}^1}}{\\|v\\|_{{\\bf L}^2 \\cap \\epsilon {\\bf H}^1}},\n\\end{align*}\nwhere we used that, since \\eqref{eq:wLBB} holds, Theorem~\\ref{thm:wlbbimplfl2} shows that \nthere exists a Fortin operator ${\\mathcal F}_h$ that satisfies\n\\eqref{eq:Fortin}. By Lemma~\\ref{lem:fl2implfh1} and\nthe approximation properties \\eqref{eq:fl2} of the Fortin operator,\n\\begin{align*}\n \\|{\\mathcal F}_h v \\|_{{\\bf L}^2 \\cap \\epsilon {\\bf H}^1} &\\leq\n c \\left( \\|{\\mathcal F}_h v \\|_{{\\bf L}^2} + \\epsilon \\| {\\mathcal F}_h v \\|_{{\\bf H}^1} \\right) \\leq\n c \\left( \\| v \\|_{{\\bf L}^2} + \\|v - {\\mathcal F}_h v \\|_{{\\bf L}^2} + \\epsilon \\| v \\|_{{\\bf H}^1} \\right) \\\\\n &\\leq c \\left( \\|v\\|_{{\\bf L}^2} + (\\epsilon + h)\\| v \\|_{{\\bf H}^1} \\right)\n \\leq c \\left( \\|v\\|_{{\\bf L}^2} + 2\\epsilon\\| v \\|_{{\\bf H}^1} \\right)\n \\leq c\\| v \\|_{{\\bf L}^2 \\cap \\epsilon {\\bf H}^1},\n\\end{align*}\nwhere we used that $h \\leq \\epsilon$.\n\nOn the other hand, if $\\epsilon < h$ we use $q_1 = q_h$ and $q_2 = 0$ in the definition of the weighted norm for the pressure space.\nCondition \\eqref{eq:wLBB} then implies\n\\[\n \\| q_h \\|_{H^1+\\epsilon^{-1} L^2} \\leq\n c \\| \\GRAD q_h \\|_{{\\bf L}^2} \\leq c \\sup_{v_h \\in {\\bf X}_h} \\frac{ \\int_\\Omega q_h\\,\\DIV v_h }{ \\| v_h \\|_{{\\bf L}^2} }\n \\leq c \\sup_{v_h \\in {\\bf X}_h} \\frac{ \\int_\\Omega q_h\\, \\DIV v_h }{ \\| v_h \\|_{{\\bf L}^2 \\cap \\epsilon {\\bf H}^1} }\n \\sup_{v_h \\in {\\bf X}_h} \\frac{ \\| v_h \\|_{{\\bf L}^2 \\cap \\epsilon {\\bf H}^1} }{ \\| v_h \\|_{{\\bf L}^2} }.\n\\]\nBy the inverse inequality \\eqref{eq:invX},\n\\[\n \\frac{ \\| v_h \\|_{{\\bf L}^2 \\cap \\epsilon {\\bf H}^1} }{ \\| v_h \\|_{{\\bf L}^2} } \\leq\n c\\left( 1 + \\epsilon h^{-1} \\right).\n\\]\nConclude using that $\\epsilon < h $.\n\\end{proof}\n\n\\section{Concluding Remarks}\n\\label{sec:conclusion}\nThere seems to be one main drawback to our methods of proof. Namely, all our results rely heavily on\nthe fact that we have a quasi-uniform mesh. However, at the present moment we do not know whether this\ncondition can be removed. Finally, it will be interesting to see if \\eqref{eq:LBB} is in fact equivalent to\n\\eqref{eq:wLBB} on domains that do not satisfy the regularity assumption \n\\eqref{eq:Cattabriga} (e.g.\\@\\xspace non convex polyhedral domains).\n\nOn the other hand, it seems to us that condition \\eqref{eq:wLBB} must be regarded as the most important one.\nOur results show that, under the sole assumption that the mesh is quasi-uniform, this condition implies\nthe classical condition \\eqref{eq:LBB} (Theorem~\\ref{thm:wlbbimpllbb}). Moreover, as shown in \nTheorem~\\ref{thm:wlbbimplweightlbb}, this condition implies the weighted inf--sup condition \\eqref{eq:winf-sup} on quasi-uniform meshes.\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzpshe b/data_all_eng_slimpj/shuffled/split2/finalzzpshe new file mode 100644 index 0000000000000000000000000000000000000000..3552e225ea554b7b0f368d162cf221c00602401b --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzpshe @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nMulti-armed bandit models were originally introduced in the 1930's as a simple model for a (phase III) clinical trial in which one control treatment is tried against one alternative \\citep{Thompson33}. While those models are nowadays widely studied with completely different applications in mind, like online advertisement \\citep{LiChapelle11}, recommender systems \\citep{LiCLS10News} or cognitive radios \\citep{Anandkumar11}, there has been a surge of interest in the use of bandit algorithms for clinical trials (see \\cite{Villar15BCT}). More broadly, Adaptive Clinical Trials have received an increased attention \\citep{GuidelinesACT18} as the Food and Drug Administration recently updated a draft of guidelines for their actual use \\citep{FDA19}. In this paper, we focus on adaptive designs for phase I and phase I\/II clinical trials for single-agent in oncology, for which adaptations of the original bandit algorithms may be of interest. \n\nPhase I trials are the first stage of testing in human subjects. Their goal is to evaluate the safety (and feasibility) of the treatment and identify its side effects. For non-life-threatening diseases, phase I trials are usually conducted on human volunteers. In life-threatening diseases such as cancer or AIDS, phase I studies are conducted with patients because of the aggressiveness and possible harmfulness of the treatments, possible systemic treatment effects, and the high interest in the new drug's efficacy in those patients directly. The aim of a phase I dose-finding study is to \\emph{determine the most appropriate dose level that should be used in further phases of the clinical trials}. Traditionally, the focus is on determining the highest dose with acceptable toxicity called the Maximum Tolerated Dose (MTD). Once the initial safety of the drug has been confirmed in phase I trials, phase II trials are performed on larger groups and are designed to establish the efficacy of the drug and confirm the safety identified in phase I. In phase II dose-finding studies, the dose-efficacy relationship is modeled in order to estimate the smallest dose to obtain a desired efficacy, called the minimal effective dose (MED). Approaches that use both efficacy and toxicity to find an optimal dose are called phase I\/II designs. If the new potential treatment shows some efficacy in phase II, it is compared to alternative treatments in phase III. We here consider two classes of algorithms for dose-finding in early stage trials:\nalgorithms which consider only toxicity, suited for phase I trials,\nand algorithms which consider both toxicity and efficacy, suited for phase I\/II trials.\n\nUntil recently, cytotoxic agents were the main agent of anti-tumor drug development. A common assumption for these agents is that both toxicity and efficacy of the treatment are monotonically increasing with the dose \\citep{chevret06}. Hence, only toxicity is required to determine the optimal dose which is then the Maximum Tolerated Dose. From a statistical perspective, the MTD is often defined as the dose level closest to an acceptable targeted toxicity probability fixed prior to the trial onset \\citep{faries94,storer89}. However, Molecularly Targeted Agents (MTAs) have emerged as a new treatment option in oncology that have changed the practice of cancer patient care \\citep{Postel-Vinay09,letourneau10,letourneau11,letourneau12}. Previously-common assumptions do not necessarily hold for MTAs. Although toxicity is still assumed to be increasing with the dose, it may be so low that the trial cannot be driven by toxicity occurrence only. Efficacy needs to be studied jointly with toxicity, so that the most appropriate dose is not just the MTD. In particular, for some mechanisms of action, a plateau of efficacy can be observed when increasing the dose \\citep{hoering11}, for instance when the targeted receptors are saturated. In this paper, we aim at providing a unified approach that can be used both for phase I trials involving cytotoxic agents and phase I\/II trials involving MTAs.\n\nPhase I cytotoxic clinical trials in oncology involve several ethical concerns. Therefore, in order to gather information about the dose-toxicity relationship it is not possible to include a large number of patients and randomize them at each different dose level considered in the trial. Patients treated with dose levels over the MTD would be exposed to very high toxicity, and patients treated at low dose levels would be administrated ineffective dose levels. In addition, the total sample size is often very limited. For these reasons, the doses to be allocated should be selected sequentially, taking into account the outcomes of the previous allocated doses, with ideally two objectives in mind: finding the MTD (which is crucial for the next stages of the trial) and treating as many trial participants as possible with this MTD. This trade-off between treatment (curing patients during the study) and experimentation (finding the best treatment) is a common issue in clinical trials. By viewing optimal dose identification as a particular multi-armed bandit problem, this trade-off can be rephrased as a trade-off between rewards and error probability, two performance measures that are well-studied in the bandit literature and that are known to be somewhat antagonistic (see \\cite{Bubeckal11,ESAIM17}). \n\nIn this paper, we investigate the use of Thompson Sampling \\citep{Thompson33} for dose-finding clinical trials. This Bayesian algorithm has gained a lot of popularity in the machine learning community for its successful use for reward maximization in bandit models (see, e.g., \\cite{LiChapelle11}). Interestingly, in the growing literature on Bayesian Adaptive Designs \\citep{Berry06BAD,Berry10BAD}, several designs that may be viewed as variants of Thompson Sampling have been proposed for other types of clinical trials in which different treatments are compared \\citep{Thall07,Berrys16Alzheimer}. However, to the best of our knowledge, the use of Thompson Sampling has not been investigated yet for dose-finding trials, and the present paper aims to fill this gap. We show that, unlike other bandit algorithms that are better suited for phase III trials, Thompson Sampling can indeed be naturally adapted to dose-finding trials. \n\nOur first contribution is a theoretical study in the context of MTD identification showing that the simplest version of Thompson Sampling based on independent prior distributions for each arm asymptotically minimizes the number of sub-optimal allocations during the trial. Albeit asymptotic, this sanity-check for Thompson Sampling with a simple prior motivates our investigation for its use with more realistic prior distributions, where theoretical guarantees are harder to obtain. Our second contribution is to show that Thompson Sampling using more sophisticated prior distributions can compete with state-of-the art dose-finding algorithms. We indeed show that the algorithm can exploit the monotonicity assumption on the toxicity probabilities that are common for MTD identification (Section~\\ref{sec:TSIncreasing}), but also deal with more complex assumptions on both the toxicity and efficacy probabilities that are relevant for trials involving MTAs (Section~\\ref{sec:TSEff}). Through extensive experiments on simulated clinical trials we show that our Thompson Sampling variants typically outperform state-of-the-art dose-finding algorithms. Finally, we propose a discussion revisiting the treatment versus experimentation trade-off through a bandit lens, and explain why an adaptation of existing best arm identification designs \\citep{Bubeck10BestArm,Karnin13} seems currently less promising for dose-finding clinical trials.\n\nThe paper is structured as follows. In Section~\\ref{sec:Bandits}, we present a multi-armed bandit (MAB) model for the MTD identification problem and introduce the Thompson Sampling algorithm. In Section~\\ref{sec:Analysis}, we propose an analysis of Thompson Sampling with independent Beta priors on the toxicity of each dose: We provide finite-time upper-bounds on the number of sub-optimal selections, which match an (asymptotic) lower bound on those quantities. Then in Section \\ref{sec:TS}, we show that Thompson Sampling can leverage the usual monotonicity assumptions in dose-finding clinical trials.\nIn Section~\\ref{sec:Experiments}, we report the results of a large simulation study to assess the quality of the proposed design. Finally in Section~\\ref{sec:Discussion}, we propose a discussion on the use of alternative bandit methods.\n \n\\section{Maximum Tolerated Dose Identification as a Bandit Problem}\\label{sec:Bandits}\n\nIn this section, we propose a simple statistical model for the MTD identification problem in phase I clinical trials and show that it can be viewed as a particular multi-armed bandit problem.\n\nA dose-finding study involves a number $K$ of dose levels that have been chosen by physicians based on preliminary experiments ($K$ is usually a number between $3$ and $10$). Denoting by $p_k$ the (unknown) toxicity probability of dose $k$, the Maximum Tolerated Dose (MTD) is defined as the dose with a toxicity probability closest to a target:\n\\[k^* \\in \\argmin{k \\in \\{1,\\dots,K\\}} |\\theta - p_k|,\\]\nwhere $\\theta$ is the pre-specified targeted toxicity probability (typically between 0.2 and 0.35). For clinical trials in life-threatening diseases, efficacy is often assumed to be increasing with toxicity, hence the MTD is the most appropriate dose to further investigate in the rest of the trial. However, we shall see in Section~\\ref{sec:TS} that under different assumptions the optimal dose may be defined differently. \n\n\\subsection{A (Bandit) Model for MTD Identification}\n\nA MTD identification algorithm proceeds sequentially: at round $t$ a dose $D_t \\in \\{1,\\dots,K\\}$ is selected and administered to a patient for whom a toxicity response is observed. A binary outcome $X_t$ is revealed where $X_t = 1$ indicates that a harmful side-effect occurred and $X_t=0$ indicates than no harmful side-effect occurred. We assume that $X_t$ is drawn from a Bernoulli distribution with mean $p_{D_t}$ and is independent from previous observations. The \\emph{selection rule} for choosing the next dose level to be administered is sequential in that it uses the past toxicity observations to determine the dose to administer to the next patient. More formally, $D_t$ is $\\cF_{t-1}$-measurable where $\\cF_t= \\sigma(U_0,D_1,X_1,U_1,\\dots,D_t,X_t,U_t)$ is the $\\sigma$-field generated by the observations made with the first $t$ patients and the possible exogenous randomness used in each round $t$, $U_{t-1} \\sim \\cU([0,1])$. Along with this selection rule, a ($\\cF_{t}$-measurable) \\emph{recommendation rule} $\\hat{k}_t$ indicates which dose would be recommended as the MTD, if the experiments were to be stopped after $t$ patients. \n\nUsually in clinical trials the total number of patients $n$ is fixed in advance and the first objective is to ensure that the dose $\\hat{k}_n$ recommended at the end of the trial is close to the MTD, $k^*$, but there is also an incentive to treat as many patients as possible with the MTD during the trial. Letting $N_k(t) = \\sum_{s=1}^t\\ind_{(D_s = k)}$ be the number of time dose $k$ has been given to one of the first $t$ patients, this second objective can be formalized as that of minimizing $N_k(n)$ for $k\\neq k^*$. In the clinical trial literature, empirical evaluations of dose-finding designs usually report both the empirical distribution of the recommendation strategy $\\hat{k}_n$ (that should be concentrated on the MTD) and estimates of $\\bE[N_k(n)]\/n$ for all doses $k$ to assess the quality of the selection strategy in terms of allocating MTD as often as possible. \n\n\nThe sequential interaction protocol described above is reminiscent of a stochastic multi-armed bandit (MAB) problem (see \\cite{BanditBook18} for a recent survey). A MAB model refers to a situation in which an agent sequentially chooses arms (here doses) and gets to observe a realization of an underlying probability distribution (here a Bernoulli distribution with mean being the probability that the chosen dose is toxic). Different objectives have been considered in the bandit literature, but most of them are related to \\emph{learning the arm with largest mean}, whereas in the context of clinical trials we are rather concerned with the arm which is the closest to some threshold. \n\n\\subsection{Thompson Sampling for MTD Identification}\n\nEarly works on bandit models \\citep{Robbins52,LaiRobbins85bandits} mostly consider a \\emph{reward maximization} objective: The samples $(X_t)$ are viewed as rewards, and the goal is to maximize the sum of these rewards, which boils down to choosing the arm with largest mean as often as possible. This problem was originally introduced in the 1930s in the context of phase III clinical trials \\citep{Thompson33}. In this context, each arm models the response to a particular treatment, and maximizing rewards amounts to giving the treatment with largest probability of success to as many patients as possible. This suggests a phase III trial is designed for treating as many patients as possible with the best treatment rather than identifying it. The trade-off between treatment and identification is also relevant for MTD identification: besides finding the MTD another objective is to treat as many patients as possible with it during the trial.\n\nReward maximization in a Bernoulli bandit model is a well-studied problem \\citep{Jacko19Binary}. In particular, it is known since \\citep{LaiRobbins85bandits} that any algorithm that performs well on every bandit instance should select each sub-optimal arm $k$ more than $C_k\\log(n)$ times, where $C_k$ is some constant, in a regime of large values of $n$. Algorithms with finite-time upper bounds on the number of sub-optimal selections have been exhibited \\citep{Aueral02,Audibertal09UCBV}, some of which match the aforementioned lower bound on the number of sub-optimal selections \\citep{KLUCBJournal}. In the context of MTD identification, we are also concerned about \\emph{ minimizing the number of sub-optimal selections} but with a different notion of optimal arm: the MTD instead of the arm with largest mean. \n\nAlgorithms for maximizing rewards in a bandit model mostly fall in two categories: frequentist algorithms, based on upper-confidence bounds (UCB) for the unknown means of the arms (popularized by \\cite{KatRob:95Gauss,Aueral02}) and Bayesian algorithms, that exploit a posterior distribution on the means (see, e.g. \\cite{Powell12Book,AISTATS12}). Among those, Thompson Sampling (TS) is a popular approach, known for its practical successes beyond simple bandit problems \\citep{AGContext13,Agrawal17TSRL}. In the context of clinical trials, variants of Thompson Sampling have been notably studied for phase III clinical trials involving two treatments (see \\cite{Thall07} and references therein), or for adaptive trials involving interim analyses \\citep{Berrys16Alzheimer}. Strong theoretical properties have also been established for this algorithm in simple models. In particular, Thompson Sampling was proved to be asymptotically optimal for Bernoulli bandit models \\citep{ALT12,AGAISTAT13}. \n\n\nThompson Sampling, also known as probability matching, implements the following simple Bayesian heuristic. Given a prior distribution over the arms, at each round an arm is selected at random according to its posterior probability of being optimal. In this paper, we advocate the use of Thompson Sampling for dose-finding, using the appropriate notion of optimality. In particular, Thompson Sampling for MTD identification consists of selecting a dose at random according to its posterior probability of being the MTD. Given a prior distribution $\\Pi^0$ on the vector of toxicity probabilities, $\\bm p = (p_1,\\dots,p_K) \\in [0,1]^K$, a posterior distribution $\\Pi^t$ can be computed by taking into account the first $t$ observations. A possible implementation of Thompson Sampling consists of drawing a sample $\\bm \\theta(t) = (\\theta_1(t),\\dots,\\theta_K(t))$ from the posterior distribution $\\Pi^t$ and selecting at round $t+1$ the dose that is the MTD in the sampled model: $D_{t+1} = \\text{argmin}_{k} \\ |\\theta_k(t) - \\theta|$. There are several possible choices for the recommendation rule $\\hat k_t$, which are discussed in the upcoming sections.\n\n\n\\subsection{Why Thompson Sampling?} \n\nThompson Sampling is by far not the only existing bandit algorithm, yet other algorithms may not be as easily adaptable to the MTD identification problem, which justifies our focus on this algorithm.\n\nIndeed, Thompson Sampling only requires defining some notion of \\emph{optimal arm} (or arm to discover), which is naturally defined as the arm with mean closest to the threshold $\\theta$ in the MTD identification problem. Many other popular bandit algorithms instead require a \\emph{value} to be assigned to each sampled arm, and require the optimal arm to be the arm with largest expected value. This is the case for the frequentist \\emph{optimistic} (UCB) algorithms (see, e.g., \\cite{Aueral02,KLUCBJournal}), which construct confidence intervals on the expected value of each arm and select the arm which has the largest statistically plausible expected value (i.e. the largest Upper Confidence Bound). Adapting this optimism in face of uncertainty principle for MTD identification is not straightforward: one can certainly build confidence intervals on the toxicity probability of each dose (several of them may contain the MTD), but there is no natural way to define a ``best plausible value'' for each dose in that case. \n\nIn the literature on Bayesian ranking and selection, value-based approaches have also been proposed. Some algorithms are indeed based on defining some Expected Value of Information \\citep{Chick06}. Among those, knowledge gradient methods \\citep{Powell12Book} are particularly interesting since they permit handling correlations between arms. For example \\cite{XieFrazier16} consider a prior distribution over the arms' means which is a multivariate Gaussian, and \\cite{Wang16KGBinary} consider a Bayesian logistic model (where a Laplace approximation is used for Bayesian inference). However, the proposed algorithms are both tailored to finding an arm $a$ maximizing $\\bE[V(a,D)]$ for some function $V$ that depends on a random variable $D$ under which the expectation is taken (like other algorithms from the Bayesian Optimization (BO) literature \\citep{Brochu10Tuto}). The MTD identification problem cannot naturally be cast in this framework, and adapting, e.g., knowledge gradient methods would require defining an appropriate notion of value of information in this setting. This is why we focused on a Bayesian approach which is easier to adapt to MTD identification, Thompson Sampling. \n\n\\section{Independent Thompson Sampling: an Asymptotically Optimal Algorithm} \\label{sec:Analysis}\n\nInspired by the bandit literature, we introduce the simplest version of Thompson Sampling, that assumes independent uniform prior distributions on the probability of toxicity of each dose. We refer to this algorithm as {Independent Thompson Sampling} and propose some theoretical guarantees for it.\n\n\\subsection{Algorithm Description}\n\nThe prior distribution on $\\bm{p} = (p_1,\\dots,p_K)$ is $\\Pi^0 = \\bigotimes_{i=1}^{K} \\pi_k^0$, where $\\pi_k^0 = \\cU([0,1])$ is a uniform distribution. Letting $\\pi_k^t$ be the posterior distribution of $p_k$ given the observations from the first $t$ patients, the posterior distribution also has a product form, $\\Pi^t =\\bigotimes_{i=1}^{K} \\pi_k^t$. Moreover, each $\\pi_k^t$ can be made explicit: $\\pi_k^t$ is a $\\text{Beta}(S_k(t) + 1, N_k(t) - S_k(t)+1)$ distribution where $S_k(t) = \\sum_{s=1}^t X_s \\ind_{(D_s = k)}$ is the sum of rewards obtained from arm $k$ and $D_s$ is the dose allocated at time $s$. \n\nThe selection rule of Independent Thompson Sampling is simple: a sample from the posterior distribution on the toxicity probability of each dose is generated, and the dose for which the sample is closest to the threshold is selected: \n\\[\\left\\{\\begin{array}{cl}\n& \\forall k \\in \\{1,K\\}, \\ \\theta_k(t) \\sim \\pi_k^t \\\\\n& D_{t+1} = \\text{argmin}_{k} \\ |\\theta_k(t) - \\theta|.\n\\end{array}\\right.\\]\nSeveral recommendation rules may be used for Independent Thompson Sampling. As the randomization induces some exploration, recommending $\\hat{k}_t = D_{t+1}$ is not a good idea. Inspired by what is proposed by \n\\cite{Bubeckal11} for assigning a recommendation rule to rewards maximizing algorithms, a first idea is to recommend $\\hat{k}_t= \\text{argmin}_{k} \\ |\\hat{\\mu}_k(t) - \\theta|$, where $\\hat{\\mu}_k(t)$ is the empirical mean of dose $k$ after the $t$-th patient of the study. Leveraging the fact that TS is supposed to allocate the MTD most of the time, we could also select $\\hat{k}_t= \\text{argmax}_{k} \\ N_k(t)$ or pick $\\hat{k}_t$ uniformly at random among the allocated doses.\n\n\n\\subsection{Upper Bound on the Number of Sub-Optimal Selections}\n\nFor the classical rewards maximization problem, the first finite-time analysis of Thompson Sampling for Bernoulli bandits dates back to \\cite{AGCOLT12} and was further improved by \\cite{ALT12,AGAISTAT13}. \nIn Appendix~\\ref{proof:TS}, building on the analysis of \\cite{AGAISTAT13}, we prove the following for Thompson Sampling applied to MTD identification. \n\n\\begin{theorem}\\label{thm:TS} Introducing for every $k\\neq k^*$ the quantity \n\\[d_k^* := \\argmin{d \\in \\{p_{k^*},2\\theta - p_{k^*}\\}} \\ |p_k - d |,\\]\nIndependent Thompson Sampling satisfies the following. For all $\\varepsilon >0$, there exists a constant $C_{\\varepsilon,\\theta,\\bm p}$ (depending on $\\varepsilon$, the threshold $\\theta$ and the toxicity probabilities) such that for all $k : |p_k - \\theta| \\neq |\\theta - p_{k^*}|$, \n\\[\n\\bE[N_{k}(n)] \\leq \\frac{1 + \\varepsilon}{\\mathrm{kl}(p_k,d_k^*)} \\log(n) + C_{\\varepsilon,\\theta,\\bm p},\n\\]\nwhere $\\mathrm{kl}(x,y) = x\\log(x\/y)+(1-x)\\log((1-x)\/(1-y))$ is the binary Kullback-Leibler divergence.\n\\end{theorem}\n\n\n\nTheorem~\\ref{thm:TS} shows that the total number of allocations to a sub-optimal dose in a trial involving $n$ patients is logarithmic in $n$, which justifies that the MTD is given most of the time, at least in a regime of large values of $n$ (as the second order term can be large). Also, this bounds tells us that in this regime each sub-optimal dose is allocated in inverse proportion of $\\mathrm{kl}(p_k,d_k^*)$, which can be seen as a distance between dose $k$ and an optimal dose with toxicity probability $d_k^*$ which is illustrated in Figure~\\ref{fig:doses}. \n\n\\begin{figure}\\centering\n \\includegraphics[height=6cm,angle=-90]{doses2.pdf}\\hspace{1.2cm}\n \\includegraphics[height=6cm,angle=-90]{doses.pdf}\n \\caption{\\label{fig:doses}\n Optimal dose $d_k^*$ associated with dose $k$.\n In some cases $d_k^*=p_{k^*}$ (left),\n in others $d_k^* = 2\\theta - p_{k^*}$ (right),\n which is symmetric to the MTD with respect to threshold $\\theta$.\n }\n\\end{figure}\n\n\nThe lower bound given in Theorem~\\ref{thm:LB} below furthermore shows that Independent Thompson Sampling actually achieves the \\emph{minimal number of sub-optimal allocations} when $n$ grows large. \n\n\n\\begin{theorem}\\label{thm:LB} We define a uniformly efficient design as a design satisfying for all possible toxicity probabilities $\\bm p$, for all $\\alpha \\in ]0,1[$, for all $k : |\\theta - p_k| \\neq |\\theta - p_{k^*}|$, $\\bE[N_k(n)] = o(n^\\alpha)$ when $n$ goes to infinity. If $p_{k^*} \\neq \\theta$, any uniformly efficient design satisfies, for all $k$: $|\\theta - p_k| \\neq |\\theta - p_{k^*}|$, \n\\begin{align*}\n\\liminf_{n \\rightarrow \\infty} &\\frac{\\bE[N_k(n)]}{\\log(n)}\n\t\\geq \\frac{1}{\\mathrm{kl}(p_k,d_k^*)}.\n\\end{align*}\n\\end{theorem}\n\nTheorem~\\ref{thm:LB} can be viewed as a counterpart of the Lai and Robbins lower bound for classical bandits \\citep{LaiRobbins85bandits}\nand can be easily derived using recent change-of-measure tools (see \\cite{GMS18}). Its proof is given in Appendix~\\ref{proof:LB} for the sake of completeness. \n\n\\subsection{Upper Bound on the Error Probability}\n\nIf the recommendation rule $\\hat{k}_n$ consists of selecting uniformly at random a dose among the doses that were allocated during the trial, $\\{D_1,\\dots,D_n\\}$, it follows from Theorem~\\ref{thm:TS} that \n\\begin{equation}\\bP\\left(\\hat k_n \\neq k^*\\right) = \\sum_{k \\neq k^*} \\frac{\\bE[N_k(n)]}{n} \\leq \\frac{D\\ln(n)}{n},\\label{UpperBoundError}\\end{equation}\nwhere $D$ is a (possibly large) problem-dependent constant. Hence finite-time upper bounds on the number of sub-optimal selection lead to \\emph{non-asymptotic upper bound on the error probability} of the design. Note that for the state-of-the-art dose-finding designs it is not known whether such results can be obtained; the only results available provide conditions for \\emph{consistency}. For example \\cite{ShenOQuigley96,CheungChappell02} exhibit some conditions on the toxicity probabilities under which a classical design called the CRM is such that $\\hat{k}_n$ converges almost surely to $k^*$. \n\nThis being said, the upper bound \\eqref{UpperBoundError} is not very informative, as a very large number of patients is needed for the upper bound to be at least smaller than 1, and one could expect to have an upper bound that is exponentially decreasing with $n$. As we shall see in Section~\\ref{sec:Discussion}, an adaptation of a best arm identification algorithm \\citep{Karnin13} leads to such an upper bound, but may be less desirable for clinical trials from an ethical point of view. This is why we rather chose to investigate in what follows several variants of Thompson Sampling coupled with an appropriate recommendation rule. \n\n\n\nBy using uniform and independent priors on each toxicity probability, Independent Thompson Sampling is the simplest possible implementation of Thompson Sampling. We now explain that using a more sophisticated prior distribution allows the algorithm to leverage some particular constraints of the dose-finding problem, like increasing toxicities or a plateau of efficacy. \n\n\n\n\\section{Exploiting Monotonicity Constraints with Thompson Sampling}\\label{sec:TS}\n\nIndependent Thompson Sampling is an adaptation of a state-of-the-art bandit algorithm for identifying the MTD that does not leverage any prior knowledge on (e.g.) the ordering of the arms' means. While it can be argued that when testing drug combinations no natural ordering between the doses exists (see, e.g., \\cite{Mozgunov17CT}), in most cases monotonicity assumptions can speed up learning. \n\n\\smallskip\n\nA typical assumption in phase I studies is that both efficacy and toxicity are increasing with the dose. We show in Section~\\ref{sec:TSIncreasing} that Thompson Sampling using an appropriate prior is competitive with state-of-the-art phase I approaches leveraging the monotonicity. In Section \\ref{sec:TSEff}, we further show that Thompson Sampling is a flexible method that can be useful in phase I\/II trials, under more complex monotonicity assumptions on both toxicity an efficacy. More specifically, we show it can handle an efficacy ``plateau,'' where efficacy may be non-increasing after a certain dose level. \n\n\\subsection{Thompson Sampling for Increasing Toxicities: A Phase I Design}\\label{sec:TSIncreasing}\n\nIn a phase I study in which both toxicity and efficacy are increasing with the dose, the MTD is the most relevant dose to allocate in further stages. We now focus on algorithms leveraging the extra information that $p_1 \\leq \\dots \\leq p_k$. \nTo exploit this structure, \\emph{escalation procedures} have been developed in the literature, the most famous being the ``3+3'' design \\citep{storer89}. In this design, adjusted for $\\theta = 0.33$, the lowest dose is first given to 3 patients. If no patient experiences toxic effects, one escalates to the next dose and repeats the process. If one patient experiences toxicity, the dose is given to 3 more patients, and if less than two patients among the 6 experience toxicity, one escalates to the next dose. Otherwise the trial is stopped, which is also the case if from the beginning 2 out of the 3 patients experience a toxic effect. Upon stopping, the previous dose is recommended as the MTD, or all doses are deemed too toxic if one stops at the first dose level. Although it is clear that the guarantees in terms of error probability (or sub-optimal selections) are very weak, ``3+3'' is still often used in practice.\n\nAlternative to this first design are variants of the Continuous Reassessment Method (CRM), proposed by \\cite{OQuigley90CRM}. The CRM uses a Bayesian model that combines a parametric dose\/toxicity relationship with a prior on the model parameters. Under this model, CRM appears as a greedy strategy that selects in each round the dose whose expected toxicity under the posterior distribution is closest to the threshold. We propose in this section several variants of Thompson Sampling based on the same Bayesian model, but that favor (slightly) more exploration. \n\n\n\\paragraph{A Bayesian model for increasing toxicities} In the CRM literature, several parametric models that yield an increasing toxicity have been considered. In this paper, we choose a two-parameter logistic model that is among the most popular. Under this model, each dose $k$ is assigned an \\emph{effective dose} $u_k$ (that is usually not related to a true dose expressed in a mass or volume unit) and the toxicity probability of dose $k$ is given by \n\\begin{align*}\np_k(\\beta_0,\\beta_1) &= \\psi(k,\\beta_0,\\beta_1), \\ \\ \\text{ where } \\ \\\n\\psi(k,\\beta_0,\\beta_1) = \\frac{1}{1 + e^{-\\beta_0-\\beta_1u_k}}.\n\\end{align*}\nA typical choice of prior is \n\\[\n\\beta_0 \\sim \\mathcal{N}(0,100) \\ \\ \\text{and} \\ \\ \\beta_1 \\sim \\mathrm{Exp}(1). \\]\nIt is worth noting that this model also heavily relies on the distinct effective dose levels $u_1,\\dots,u_K$ that are usually chosen depending on some \\emph{prior toxicities} set by physicians, $p^0_1 \\leq p^0_2 \\leq \\dots \\leq p^0_K$. Letting $\\overline{\\beta_0}$, $\\overline{\\beta_1}$ be the prior mean of each parameter, the effective doses are calibrated such that for all $k$, $\\psi(k,\\overline{\\beta_0},\\overline{\\beta_1}) = p_k^0$. If there is no medical prior knowledge about the toxicity probabilities, some heuristics for choosing them in a robust way have been developed (see Chapter 9 of \\cite{CRMBook})\n \nUnder this model, given some observations from the different doses one can compute the posterior distribution over the parameters $\\beta_0$ and $\\beta_1$; that is, the conditional distribution of these parameters given the observations. Although there is no closed form for these posterior distributions, they can be easily sampled from using Hamiltonian Monte-Carlo Markov Chain algorithms (HMC) as the log-likelihood under these models is differentiable. In practice, we use the Stan implementation of these Monte-Carlo sampler \\citep{StanManual}, and use (many) samples to approximate integrals under the posterior when needed. \n \n\\subsubsection{Thompson Sampling}\\label{sec:CRMvsTS}\n\nThompson Sampling selects a dose at random according to its posterior probability of being the MTD. Under the two-parameter Bayesian logistic model presented above, letting $\\pi_t$ denote the posterior distribution on $(\\beta_0,\\beta_1)$ after the first $t$ observations, the posterior probability that dose $k$ is the MTD is\n\\begin{align*}\n{q}_k(t) &: = \\bP\\left(\\left. k = \\argmin{\\ell} |\\theta - p_\\ell(\\beta_0,\\beta_1)| \\right| \\cF_t\\right)\n\\\\\n&= \\int_{\\R} \\ind{\\left(k = \\argmin{\\ell} |\\theta - p_\\ell(\\beta_0,\\beta_1)|\\right)} d\\pi_t(\\beta_0,\\beta_1).\n\\end{align*}\n\nA first possible implementation of Thompson Sampling that we use in our experiments consists of computing approximations $\\hat{q}_k(t)$ of the probabilities ${q}_k(t)$ (using posterior samples) and selecting at round $t+1$ a dose $D_{t+1} \\sim \\hat{\\bm q}(t)$, i.e. such that $\\bP\\left(D_{t+1} = k | \\cF_t\\right) = \\hat{q}_k(t)$.\nA second implementation of Thompson Sampling (that may be computationally easier) consists of drawing one sample from the posterior distribution of $(\\beta_0,\\beta_1)$, and selecting the MTD in the sampled model: \n\\begin{align}\n \\left(\\tilde \\beta_0(t), \\tilde \\beta_1(t)\\right) & \\sim \\pi_t, \\nonumber\\\\\n D_{t+1}^{\\text{TS}} & \\in \\argmin{k \\in \\{1,\\dots,K\\}} \\ \\left|\\theta - p_k\\left(\\tilde \\beta_0(t),\\tilde \\beta_1(t)\\right)\\right|.\\label{eq:SampleTS}\n\\end{align}\nIt is easy to see that this algorithm coincides with Thompson Sampling in that $\\bP\\left(D_{t+1}^{\\text{TS}} = k | \\cF_t\\right) = {q}_k(t)$. We will present below a variant of Thompson Sampling based on the first implementation (${\\mathrm{TS}\\_\\mathrm{A}}$) and a variant based on the second implementation (${\\mathrm{TS}(\\varepsilon)}$).\n\n\\paragraph{Recommendation rule} Due to the randomization, Thompson Sampling performs more exploration than the ``greedy'' CRM \\citep{OQuigley90CRM} method, which selects at time $t$ the MTD under the model parameterized by $(\\hat\\beta_0,\\hat\\beta_1)$, the posterior means of the two parameters, given by\n\\begin{equation}\\label{eq:CRMPostMean}\n\\hat{\\beta}_0(t) = \\int_\\R \\beta_0 d\\pi_{t}(\\beta_0,\\beta_1) \\ \\ \\ \\ \\ \\text{and} \\ \\ \\ \\ \\ \\hat{\\beta}_1(t) = \\int_\\R \\beta_1 d\\pi_{t}(\\beta_0,\\beta_1).\n\\end{equation}\nMore precisely, the sampling rule of the CRM is\n\\begin{align*}\nD_{t+1}^{\\text{CRM}} \\in \\argmin{k \\in \\{1,\\dots,K\\}} \\left|\\theta - p_k(\\hat\\beta_0(t),\\hat\\beta_1(t))\\right|.\n\\end{align*}\nThe recommendation rule for CRM after $t$ patients is identical to the next dose that would be sampled under this design, that is $\\hat{k}_t^{\\text{CRM}} = D_{t+1}^{\\text{CRM}}$. For Thompson Sampling, due to the more exploratory nature of the algorithm, we do not want to recommend $\\hat{k}_t^{\\text{TS}} = D_{t+1}^{\\text{TS}}$. Instead, we propose the use of recommendation rule $\\hat{k}_t^{\\text{TS}} = \\underset{k \\in \\{1,\\dots,K\\}}{\\text{argmin}} \\ |\\theta - p_k(\\hat\\beta_0(t),\\hat\\beta_1(t))|$, which coincides with that of the CRM.\n\n\\subsubsection{Two variants of Thompson Sampling}\\label{subsec:ParameterTuning}\n\nThe randomized aspect of Thompson Sampling makes it likely to sample from large or small doses, without respecting some ethical constraints of phase I clinical trials. Indeed, patients should not be exposed to too-high dose levels; overdosing should be controlled. Hence, we also propose two ``regularized'' versions of TS. The first depends on a parameter $\\varepsilon>0$ set by the user that ensures that the expected toxicity of the recommended dose remains within $\\varepsilon$ of the toxicity of the empirical MTD. The second restricts the doses to be tested to a set of \\emph{admissible doses}. These algorithms are formally defined below, and their performance is evaluated in Section~\\ref{sec:Experiments}.\n\n\n\\paragraph{$\\bm{\\mathrm{TS}(\\varepsilon)}$} We first compute the posterior means $\\hat\\beta_0(t),\\hat{\\beta}_1(t)$ from \\eqref{eq:CRMPostMean} and the toxicity of the dose closest to $\\theta$ under the model parameterized by $(\\hat\\beta_0(t),\\hat{\\beta}_1(t))$ (i.e., the toxicity of the dose selected by the CRM): \n\\[\n\\hat{p}(t) = p_{\\hat k_{t}}(\\hat\\beta_0(t),\\hat\\beta_1(t)), \\ \\ \\\n\\text{with} \\ \\ \\ \\hat k_{t} =\\argmin{k \\in \\{1,\\dots,K\\}}\\left|\\theta - p_k(\\hat\\beta_0(t),\\hat\\beta_1(t))\\right| \n\\]\nNext we sample $\\tilde \\beta_0(t), \\tilde{\\beta}_1(t)$ from the posterior distribution $\\pi_t$\nand select a candidate dose level $D_{t+1}$ using \\eqref{eq:SampleTS}.\nIf the predicted toxicity level $p_{D_{t+1}}(\\hat\\beta_0(t),\\hat{\\beta}_1(t))$ is not in the\ninterval $(\\hat{p}(t)-\\varepsilon, \\hat{p}(t)+\\varepsilon)$, then we reject\nour values of $\\tilde\\beta_0(t),\\tilde{\\beta}_1(t)$, draw a new sample from $\\pi_t$ and repeat the process. \nIn order to guarantee that the algorithm terminates, we only reject up to\n50 samples, after which we use the sample that gives the dose with minimum toxicity among all $50$ samples. We choose 50 to limit the computational complexity of the algorithm, but it can also be replaced by a larger value if more computational power is available.\n\n$\\bm{\\mathrm{TS}(\\varepsilon)}$ can be seen as a smooth interpolation between the CRM (which correspond to $\\varepsilon = 0$) and vanilla Thompson Sampling (which corresponds to $\\varepsilon = 1$). Regarding the tuning of the parameter $\\varepsilon$, large values do not reduce much the amount of exploration while too small values lead to a behavior which is indistinguishable from that of the CRM. We did (large scale) experiments with $\\varepsilon \\in \\{0.02, 0.05, 0.1\\}$ and we found that the three values lead to comparable performance across the different scenarios we tried. To ease the presentation, we report results for TS($0.05$) only in Section~\\ref{sec:Experiments}. \n\n\n\\paragraph{$\\bm{\\mathrm{TS}\\_\\mathrm{A}}$} The $\\mathrm{TS}\\_\\mathrm{A}$ algorithm limits exploration by enforcing the selected dose to be in some admissible set $\\cA_t$, by sampling from the modified distribution \n\\[\\bP\\left(D_{t+1} = k | \\cF_{t}\\right) = \\frac{\\hat{q}_k(t) \\ind_{\\left(k \\in \\cA_t\\right)}}{\\sum_{\\ell \\in \\cA_t} \\hat{q}_{\\ell}(t)},\\] \ninstead of sampling directly from $\\bm{\\hat{q}}(t)$ as vanilla Thompson Sampling does. \nThe admissible set $\\cA_t$ is defined as set of doses that meet the following two criteria:\n\\begin{enumerate}\n \\item dose $k$ has either already been tested, or is the next-smallest dose which has not yet been tested \n \\item the posterior probability that the toxicity of dose $k$ exceeds the toxicity of the dose closest to $\\theta$ is smaller than some threshold:\n\\[\n \\bP\\Bigg(\n \\psi(k,\\beta_0,\\beta_1) > \\psi(k',\\beta_0,\\beta_1), \n \\text{ where } k' = \\argmin{k' \\in \\{1,\\dots,K\\}}\\left|\\theta - \\psi(k', \\beta_0,\\beta_1)\\right|\n \\Bigg| \\cF_{t} \\Bigg) \\leq c_1.\n\\]\n \n \n \n \n \n \n \n \n\\end{enumerate}\n$\\cA_t$ is inspired by the admissible set of \\cite{MKR17} described in detail in the next section.\n\nIn our experiments, we tried different values of the parameter $c_1$ and we found that the performance of TS$\\_$A is better with values of $c_1$ that are not too small. In Section~\\ref{sec:Experiments}, we report experiments with $c_1 = 0.8$, but the performance was comparable for the choices $c_1 = 0.6$ or $0.9$. \n\n\n\\subsection{Thompson Sampling for Efficacy Plateau Models: A Phase I\/II Design}\\label{sec:TSEff}\n\n\n\nIn some particular trials, it has been established that efficacy is not always increasing with the dose. Motivated by some concrete examples discussed in their paper, \\cite{MKR17} consider a model in which the dose effectiveness can plateau after some unknown level, while toxicity still increases with dose level. In these models, MTD identification is no longer relevant and the objective is rather to identify the smallest dose with maximal efficacy and with toxicity no more than $\\theta$. More formally, introducing $\\text{eff}_k$ the efficacy probability of dose $k$, the Minimal Effective Dose (MED) is \n\\[k^* = \\min\\left\\{ k : \\text{eff}_k = \\max_{\\ell : p_\\ell \\leq \\theta} \\ \\text{eff}_\\ell\\right\\} \\]\n\nIn a dose-finding study involving efficacy, at each time step $t$ a dose $D_t$ is allocated to the $t$-th patient, and the toxicity $X_t$ is observed, as well as the efficacy $Y_t$. With this two-dimensional observation, assigning a value (or reward) to each sampled arm is even less natural than before. However as one can still define a notion of optimal dose (the MED instead of the MTD), Thompson Sampling can still be applied in this setting. As we shall see, it bears some similarities to the state-of-the-art method developed by \\cite{MKR17}.\n\n\\paragraph{A Bayesian model for toxicity and efficacy} Thompson Sampling requires a Bayesian model for both the dose\/toxicity and the dose\/efficacy relationship that enforces an increasing toxicity and a increasing then plateau efficacy. We use the model proposed by \\cite{MKR17}, that we now describe.\n\nUnder this model, toxicity and efficacy are assumed to be independent. The (increasing) toxicity follows the two-dimensional Bayesian logistic model with effective doses $u_k$: \n\\begin{align*}\np_k &= p_k(\\beta_0,\\beta_1) = \\psi(k,\\beta_0,\\beta_1) \n\\\\\n\\text{and} \\ \\ \\ \\beta_0 &\\sim \\mathcal{N}(0,100), \\ \\ \\ \\beta_1 \\sim \\text{Exp}(1).\n\\end{align*}\nEfficacy also follows a logistic model, with an additional parameter $\\tau$ that indicates the beginning of the plateau of efficacy. The efficacy probability of dose level $k$ is\n\\[\n\\text{eff}_k = \\text{eff}_k(\\gamma_0,\\gamma_1,\\tau) = \\phi(k,\\gamma_0,\\gamma_1,\\tau), \\\n\\text{ where }\n \\ \\phi(k,\\gamma_0,\\gamma_1,\\tau) := \\frac{1}{1 + e^{-\\left[\\gamma_0 + \\gamma_1(\n v_k \\mathds{1}(k<\\tau) + v_{\\tau} \\mathds{1}(k\\ge\\tau) )\\right]}},\n\\]\nwith $v_k$ the \\emph{effective efficacy} of dose $k$. Given $(t_1,\\dots,t_K)$ such that $\\sum_{i=1}^K t_i = 1$, a probability distribution on $\\{1,\\dots,K\\}$, the three parameters $(\\gamma_0,\\gamma_1,\\tau)$ are independent and drawn from the following prior distribution: \n\\[\\gamma_0 \\sim \\mathcal{N}(0,100), \\ \\ \\ \\gamma_1 \\sim \\text{Exp}(1), \\ \\ \\ \\tau \\sim (t_1,\\dots,t_K).\\]\nThe prior on $\\tau$ may be provided by a physician or set to $(1\/K,\\dots,1\/K)$ in case one has no prior information. Just like the effective doses $u_k$ (that we may now call effective toxicities), the effective efficacies $v_k$ are calculated using prior efficacies $\\text{eff}^0_1 \\leq \\dots \\leq \\text{eff}^0_K$:\n\\begin{align*}\nv_k &= \\left( \\log\\left( \\frac{ \\text{eff}^0_k }{1 - \\text{eff}^0_k} \\right)\n - \\overline{\\gamma}_0 \\right) \\bigg\/ \\overline{\\gamma}_1,\n\\end{align*}\nwhere $\\overline{\\gamma}_0 = 0$ and $\\overline{\\gamma}_1 = 1$ are the prior means of the parameters $\\gamma_0$ and $\\gamma_1$. \n\n\n\\paragraph{Posterior sampling} Let\n$\\mathcal{D}^{\\text{eff}}_t = \\{(D_1,Y_1),\\dots,(D_t,Y_t)\\}$\nbe the efficacy data gathered in the first $t$ rounds.\nGenerating samples from the posterior distribution of $(\\gamma_0,\\gamma_1,\\tau)$ given $\\mathcal{D}^{\\text{eff}}_t$ is a bit more involved than generating posterior samples from $(\\beta_0,\\beta_1)$. Indeed, it cannot be handled directly with HMC given that $(\\gamma_0,\\gamma_1)$ are continuous and $\\tau$ is discrete. Thus, we proceed in the following way: we first draw samples from $p(\\gamma_0,\\gamma_1 |\\mathcal{D}^{\\text{eff}}_t)$, which can be performed with HMC (and requires marginalizing out the discrete parameter $\\tau$, following the example of change point models given in the Stan manual \\citep{StanManual}). Then we sample $\\tau$ conditionally to $\\gamma_0,\\gamma_1,\\mathcal{D}^{\\text{eff}}_t$.\n\n\n\\subsubsection{Thompson Sampling}\n\nRecall that the principle of Thompson Sampling is to randomly select doses according to their posterior probability of being optimal. This idea can also be applied in this more complex model, using the corresponding definition of optimality. Given a vector $\\bm\\psi = (\\psi_1,\\dots,\\psi_K)$ of increasing toxicity probabilities and a vector $\\bm\\phi = (\\phi_1,\\dots,\\phi_K)$ of increasing then plateau efficacy probabilities, the optimal dose is \n\\[\n{\\mathrm{MED}}(\\bm{\\psi},\\bm{\\phi}) : = \\min\\left\\{\n k : \\phi_k = \\max_{\\ell : \\psi_\\ell \\leq \\theta} \\phi_\\ell \\right\\}.\n\\]\n\nThe posterior probability of dose $k$ to be optimal in that case is \n\\[{q}_k(t) := \\bP\\left( \\left.k = \\mathrm{MED}\\left( \\psi(\\bm\\cdot, \\beta_0,\\beta_1),\\phi(\\bm\\cdot, \\gamma_0,\\gamma_1,\\tau)\\right) \\right|\\cF_{t}\\right)\\]\nand in our experiments, we implement Thompson Sampling by computing approximations $\\hat{q}_k(t)$ from the quantities $q_k(t)$ (based on posterior samples) and then selecting a dose $D_{t+1}\\sim \\bm{\\hat{q}}(t)$ where $\\bm{\\hat{q}}(t) = (\\hat{q}_1(t),\\dots,\\hat{q}_K(t))$. Just like in the previous model, an alternative implementation of Thompson Sampling would sample parameters from their posterior distributions and select the optimal dose in this sampled model. Letting \n\\[\\tilde{\\beta}_0(t), \\tilde{\\beta}_1(t) \\ \\ \\ \\text{and} \\ \\ \\ \\tilde{\\gamma}_0(t), \\tilde{\\gamma}_1(t), \\tilde{\\tau}(t),\\]\nbe samples from the posterior distributions after $t$ observations of the toxicity and efficacy parameters respectively, one can compute $\\tilde{\\psi}_k(t) = \\psi(k,\\tilde{\\beta}_0(t),\\tilde{\\beta}_1(t))$ and $\\tilde{\\phi}_k(t) = \\phi(k,\\tilde{\\gamma}_0(t),\\tilde{\\gamma}_1(t),\\tilde{\\tau}(t))$ for every dose $k$. Given the toxicity and efficacy vectors \n\\begin{align*}\n\\bm{\\tilde{\\psi}}(t) &= \\left(\\tilde{\\psi}_1(t),\\dots,\\tilde{\\psi}_K(t)\\right) \\\\ \\text{ and} \\ \\ \\bm {\\tilde{\\phi}}(t) &= \\left(\\tilde{\\phi}_1(t),\\dots,\\tilde{\\phi}_K(t)\\right),\n\\end{align*}\nthis implementation of Thompson Sampling selects at round $t+1$ $D_{t+1}^{\\text{TS}} = \\mathrm{MED}\\left(\\bm{\\tilde{\\psi}}(t), \\bm{\\tilde{\\phi}}(t)\\right)$.\n\n\\paragraph{Recommendation rule} Here also we expect Thompson Sampling to be too exploratory for dose recommendation. Hence, we base our recommendation on estimated values. Given the posterior means \n$\\hat{\\beta}_0(t), \\hat{\\beta}_1(t),\\hat{\\gamma}_0(t),\\hat{\\gamma}_1(t)$ (estimated from posterior samples) and $\\hat{\\tau}(t)$ the mode of the posterior distribution of the breakpoint (see the next section for its computation), we compute \n$\\hat{\\psi}_k(t) = \\psi(k,\\hat{\\beta}_0(t),\\hat{\\beta}_1(t))$ and $\\hat{\\phi}_k(t) = \\phi(k,\\hat{\\gamma}_0(t),\\hat{\\gamma}_1(t),\\hat{\\tau}(t))$ and recommend $\\hat{k}_t = \\mathrm{MED}\\left(\\bm{\\hat{\\psi}}(t), \\bm{\\hat{\\phi}}(t)\\right)$.\n\n\n\\subsubsection{A Variant of Thompson Sampling using Adaptive Randomization}\n\nInterestingly, the need for randomization in the context of plateau efficacy has already been observed by \\cite{MKR17}. More precisely, as we explain below, the algorithm $\\mathrm{MTA}$-$\\mathrm{RA}$ described in that work can be viewed as an hybrid approach between Thompson Sampling and a CRM approach. \n\nAdditionally to the use of \\emph{adaptive randomization}, the $\\mathrm{MTA}$-$\\mathrm{RA}$ algorithm also introduces a notion of \\emph{admissible set}. The set of admissible doses after $t$ patients, denoted by $\\cA_t$, is the set of dose levels $k$ meeting all of the following criteria:\n\\begin{enumerate}\n \\item dose $k$ has either already been tested, or is the next-smallest dose which has not yet been tested\n \\item the posterior probability that the toxicity of dose $k$ exceeds $\\theta$ is smaller than some threshold: \n \\begin{equation}\\bP\\left(\\psi(k,\\beta_0,\\beta_1) > \\theta | \\cF_{t}\\right) \\leq c_1\\label{eq:CritTox}\\end{equation}\n \\item if the dose has been tested more than $3$ times, the posterior probability that the efficacy is larger than $\\xi$ is larger than some threshold: \n \\begin{equation}\\bP\\left(\\phi(k,\\gamma_0,\\gamma_1,\\tau) > \\xi | \\cF_{t}\\right) \\geq c_2\\label{eq:CritEff}\\end{equation}\n\\end{enumerate}\nPractical computation of the admissible set can be performed using posterior samples from $(\\beta_0,\\beta_1)$ to check the criterion \\eqref{eq:CritTox} and posterior samples from $(\\gamma_0,\\gamma_1,\\tau)$ to check the criterion \\eqref{eq:CritEff}. \n\nThe $\\mathrm{MTA}$-$\\mathrm{RA}$ algorithm works in two steps. The first step exploits the \\emph{posterior distribution of the breakpoint}, $t_k(t) :=\\bP\\left(\\tau=k | \\cD_{t}^{\\text{eff}}\\right)$, and uses randomization to pick a value $\\hat{\\tau}(t)$ close to the mode of this distribution. More precisely, given $(\\hat{t}_k(t))_{k=1,\\dots,K}$ an estimate of the posterior distribution of $\\tau$, let\n\\[\n\\mathcal{R}_t := \\left\\{\n k : \\left| \\max_{1 \\le \\ell \\le K}(\\hat{t}_\\ell(t)) - \\hat{t}_k(t) \\right| \\le s_1,\n 1 \\le k \\le K\n\\right\\}\n\\]\nbe a set of candidate values for the position of the breakpoint. Then under $\\mathrm{MTA}$-$\\mathrm{RA}$, \n\\[\\bP\\left(\\hat{\\tau}(t) = k |\\cF_t\\right) = \\frac{\\hat{t}_k(t)\\ind_{\\left(k \\in \\cR_t\\right)}}{\\sum_{\\ell \\in \\cR_t} \\hat{t}_\\ell(t)}. \\]\nThe threshold $s_1$ is often adapted such that it is larger in the beginning of the trial when we have high uncertainty about the estimates, but it grows smaller as the trial continues. The second step of $\\mathrm{MTA}$-$\\mathrm{RA}$ doesn't employ randomization. Based on posterior samples from $(\\gamma_0,\\gamma_1)$ conditionally to $\\tau$ being equal to $\\hat{\\tau}(t)$, efficacy estimates $\\hat{\\phi}_k$ are produced (taking the mean of the values of $\\phi(k,\\tilde{\\gamma_0},\\tilde{\\gamma_1},\\hat{\\tau}(t))$ for many samples $\\tilde{\\gamma_0},\\tilde{\\gamma_1}$) and finally the selected dose is \n\\[ D_{t+1}^{\\text{MTA-RA}} = \\inf \\left\\{ k \\in \\mathcal{A}_t : \\hat{\\phi}_k = \\max_{j \\in \\mathcal{A}_t} \\hat{\\phi}_j\\right\\}.\\]\n\nIf $\\hat{\\tau}(t)$ were replaced by a point estimate (e.g. the mode of the breakpoint posterior distribution $\\bm{\\hat t}(t)$), MTA-RA would be close to a CRM approach that computes estimates of all the parameters and acts greedily with respect to those estimated parameters (with the additional constraint that the chosen dose has to remain in the admissible set). However, the first step of MTA-RA bears similarities with the first step of a Thompson Sampling implementation that would sample a parameter $\\tau$ from the $\\bm{\\hat t}(t)$ (and later sample the other parameters conditionally to that value and act greedily in the sampled model). The difference is the use of \\emph{adaptive} randomization, in which the sample is not exactly drawn from $\\bm{\\hat t}(t)$, but is constrained to fall in some set (here $\\cR_t$) that depends on previous observations. \n\n\\paragraph{The $\\bm{\\mathrm{TS}\\_\\mathrm{A}}$ algorithm} We believe that using adaptive randomization is a good idea to control the amount of exploration performed by Thompson Sampling, which leads us to propose the $\\mathrm{TS}\\_\\mathrm{A}$ algorithm, that incorporates the constraint to select a dose that belongs to the admissible set $\\cA_t$. More formally, $\\mathrm{TS}\\_\\mathrm{A}$ selects a dose at random according to \n\\[\\bP\\left(D_{t+1} = k | \\cF_{t}\\right) = \\frac{\\hat{q}_k(t) \\ind_{\\left(k \\in \\cA_t\\right)}}{\\sum_{\\ell \\in \\cA_t} \\hat{q}_{\\ell}(t)},\\]\nwhere we recall that $\\hat{q}_k(t)$ is an estimate of the posterior probability that dose $k$ is optimal. Compared to the variant of $\\mathrm{TS}\\_\\mathrm{A}$ for increasing toxicities that is proposed in Section~\\ref{sec:TSIncreasing}, the difference here is the appropriate definition of the admissible set, that involves both toxicity and efficacy probabilities.\n\n\n\\paragraph{Practical remark} Approximations $\\hat{t}_k(t)$ of the breakpoint distribution can be computed using that\n\\[\nt_k(t) = {t_k \\int \\frac{L(\\mathcal{D}^{eff}_t | \\gamma_0,\\gamma_1,k)}{\\sum_{s=1}^K t_s L(\\mathcal{D}^{eff}_t | \\gamma_0,\\gamma_1,s)} p(\\gamma_0,\\gamma_1 | \\mathcal{D}^{eff}_t) d\\gamma_0 d\\gamma_1},\n\\]\nwhere $L(\\mathcal{D}^{\\text{eff}}_t | \\gamma_0,\\gamma_1,s)$ is the likelihood of the efficacy observations when the efficacy model parameters are $(\\gamma_0,\\gamma_1,s)$ and $p(\\gamma_0,\\gamma_1 | \\mathcal{D}^{\\text{eff}}_t)$ is the density of the distribution of $(\\gamma_0,\\gamma_1)$ given the observations. $\\hat{t}_k(t)$ can be thus be obtained by Monte-Carlo estimation based on samples from $p(\\gamma_0,\\gamma_1 | \\mathcal{D}^{\\text{eff}}_t)$. \n\n\n\n\\section{Experimental Evaluation}\\label{sec:Experiments}\n\nWe now present an empirical evaluation of the variants of Thompson Sampling introduced in the paper first in the context of increasing efficacy and then with the presence of a plateau of efficacy. In both groups of experiments, we adjusted our designs to some common practices in dose-finding trials. We used a start-up phase for all designs (starting from the smallest dose and escalating until the first toxicity is observed) and we also used cohorts of patients of size 3. This means that the same dose is allocated to 3 patients at a time and the model is updated after seeing the outcome for these 3 patients. \n\n\\subsection{Phase I: MTD Identification}\n\nIn this set of experiments, we evaluate the performance of the three algorithms introduced in Section~\\ref{sec:TSIncreasing}, $\\mathrm{TS}$, $\\mathrm{TS}(\\varepsilon)$ and $\\mathrm{TS}\\_\\mathrm{A}$, and compare them to the 3+3 and $\\mathrm{CRM}$ baselines. We report experiments with the value $\\varepsilon = 0.05$ for $\\mathrm{TS}(\\varepsilon)$ and $c_1 = 0.8$ for $\\mathrm{TS}\\_\\mathrm{A}$. We refer the reader to Section~\\ref{subsec:ParameterTuning} for discussions on the choice of these parameters. We also include $\\mathrm{Independent \\ TS}$ as proposed in Section~\\ref{sec:Bandits}, which is agnostic to the increasing structure. \n \nIn Tables~\\ref{tbl-tox} to~\\ref{tbl-tox-c} we provide results for nine different scenarios in which there are $K=6$ doses with a target toxicity $\\theta = 0.30$, budget $n=36$ and prior toxicities\n\\[\\bm{p^{0}} = [0.06 \\ \\ 0.12 \\ \\ 0.20 \\ \\ 0.30 \\ \\ 0.40 \\ \\ 0.50].\\]\nWe choose the same prior toxicities for all scenario, that are sometimes close to actual toxicities (e.g. in Scenario 2) and sometimes quite far, in order to showcase the robustness of Bayesian algorithms. \n\nFor each scenario and algorithm, we report in the first column of these tables the percentage of allocation to each dose, that is, an estimate of $\\bP(\\hat{k}_n = k)$ for each dose $k$, based on $N=2000$ repetitions. In the second column, we report an estimate of the percentage of allocation to each dose during the trial, computed for each dose $k$ as the average value of $100*N_k(n)\/n$ over $N=2000$ repetitions. We add in parenthesis the empirical standard deviation of these allocation percentages, as allocations under bandit algorithms are known to have a large variance. For the 3+3 design, only the recommendation percentages are displayed, as the percentage of allocations would be computed based on a number of patients smaller than 36 (as a 3+3 based trial involves some random stopping). This design is also the only one that would stop and recommend none of the doses if they are all judged too toxic: we add this fraction of no recommendation between brackets in the tables. \n\n\\medskip\n\nFor each scenario, corresponding to different increasing toxicity probabilities, the MTD is underlined\nand we mark in bold the fraction of recommendation or allocation of the MTD that are superior to what is achieved by the CRM. We now comment on the performance of the algorithms on those scenarios.\n\n\\paragraph{Dose recommendation} $\\mathrm{TS}$ outperforms CRM 3 out of 9 times, $\\mathrm{TS}(\\varepsilon)$ does so 5 out of 9 times, and $\\mathrm{TS}\\_\\mathrm{A}$ does so 5 out of 9 times. As expected, $\\mathrm{Independent \\ TS}$, which does not leverage the increasing structure, does not have a remarkable performance. This algorithm would need a larger budget to have a good empirical performance. With $n=36$ in most cases this strategy is not doing much better than selecting the doses uniformly at random. One can also observe that the 3+3 design (that may however require less than 36 patients in the trial) performs very badly in terms of dose recommendation.\n\n\\paragraph{Dose allocation} While $\\mathrm{TS}\\_\\mathrm{A}$ and $\\mathrm{TS}(\\varepsilon)$ do not always have higher allocation percentage at the optimal (underlined) dose compared to CRM, a scan of the dose allocation results in Tables~\\ref{tbl-tox}~to~\\ref{tbl-tox-c} shows that the addition of the admissible set $\\cA$ and $\\varepsilon$ regularity to the Thompson Sampling method consistently reduces the allocation percentage of higher toxicity doses. $\\mathrm{TS}\\_\\mathrm{A}$ performs best in this regard (it is more cautious with allocating higher doses) across all algorithms (e.g. it consistently has superior performance compared to CRM), while $\\mathrm{TS}(\\varepsilon)$ has performance better than or comparable to CRM. We believe this result is of interest in trials where toxicity is an ethical concern.\n\n\n\\subsection{Phase I\/II: MED Identification when Efficacy Plateaus} \\label{subsec:toxOnlyExp}\n\nIn this set of experiments, we evaluate the performance of the two algorithms introduced in Section~\\ref{sec:TSEff}, $\\mathrm{TS}$ and $\\mathrm{TS}\\_\\mathrm{A}$, and compare them to the $\\mathrm{MTA}$-$\\mathrm{RA}$ algorithm. We use the experimental setup of \\cite{MKR17}: several scenarios with $K=6$ doses, budget $n=60$, $\\theta = 0.35$, toxicity and efficacy priors\n\\begin{align*}\n\\bm{p^0} &= [0.02, 0.06, 0.12, 0.20, 0.30, 0.40] \\ \\ \\text{ and } \\ \\ \\bm{\\mathrm{eff}^0} = [0.12, 0.20, 0.30, 0.40, 0.50, 0.59].\n\\end{align*}\nFurthermore, we use the same parameters for the admissible set and the implementation of $\\mathrm{MTA}$-$\\mathrm{RA}$ as those chosen by \\cite{MKR17}: $\\xi=0.2$, $c_1=0.9$, $c_2=0.4$, and $s_1=.2\\left(1-\\frac{I}{n}\\right)$, where $I$ is the number of samples used so far. These parameters are defined above in the main text.\n \n\n\nIn Tables~\\ref{tbl-eff}~to~\\ref{tbl-eff-b} we provide results for several scenarios with increasing toxicities and efficacy, with efficacy which (quasi) plateaus. We report the percentage of allocation to each dose, the percentage of recommendation of each dose when $n=60$, and the percentage of time the trials stopped early (E-Stop), estimated over $N=2000$ repetitions. As before, we also report standard deviations for the percentage of allocations to each dose. \n\nOptimal doses are underlined by a plain line while a dashed line identifies doses whose toxicity is larger than $\\theta$. We mark in bold cases where\nour algorithms makes the optimal decision (in terms of the percentage of recommendations) more often than the\n$\\mathrm{MTA}$-$\\mathrm{RA}$ baseline.\n\n\\paragraph{Dose recommendation} \nRecall that the modeling assumption here is that efficacy increases monotonically in toxicity up to a point and then it plateaus. We present experimental results on several scenarios, some of which are borrowed from \\cite{MKR17}, on which this plateau assumption is not always exactly met. In most of these scenarios, $\\mathrm{TS}\\_\\mathrm{A}$ outperforms the $\\mathrm{MTA}$-$\\mathrm{RA}$ algorithm.\n\nIn scenarios 1 through 4 and in scenarios 12 and 13, there is a plateau of efficacy starting at a reasonable toxicity: in this case the optimal dose corresponds to the plateau breakpoint. Our algorithms make the optimal decision compared to $\\mathrm{MTA}$-$\\mathrm{RA}$ consistently: $\\mathrm{TS}$ 4 out of 6 times and $\\mathrm{TS}\\_\\mathrm{A}$ 5 out of 6 times. In scenarios 5 and 6 the plateau of efficacy starts when the toxicity is already too high, hence the optimal dose is before than the plateau. In scenario 5, $\\mathrm{TS}\\_\\mathrm{A}$ and $\\mathrm{TS}$ both outperform $\\mathrm{MTA}$-$\\mathrm{RA}$, while on scenario 6 $\\mathrm{MTA}$-$\\mathrm{RA}$ has a slight advantage over $\\mathrm{TS}$. \n\n\nIn scenario 7 and 8 there is no true plateau of efficacy, however in both cases there exists a ``breakpoint'' (underlined) after which the efficacy is increasing very slowly while the toxicity is increasing significantly. This breakpoint can thus be argued to be a good trade-off between efficacy and toxicity and should be investigated in further phases. In these two scenarios $\\mathrm{TS}\\_\\mathrm{A}$ identifies this pseudo-optimal dose more often than $\\mathrm{MTA}$-$\\mathrm{RA}$, while $\\mathrm{TS}$ has a slightly worse performance. \n\nLastly, we study the case when there is no clear optimal or near-optimal dose, i.e. scenarios 9-11. In scenario 9 wherein most doses, including the entire quasi-plateau, are too toxic, we would like to stop early or at most recommend dose 1 (the only dose meeting the toxicity constraint but whose efficacy is not very high). Under this interpretation, $\\mathrm{TS}$ and $\\mathrm{TS}\\_\\mathrm{A}$ outperform $\\mathrm{MTA}$-$\\mathrm{RA}$. Note that our algorithms most often either stop early or recommend dose 1, while in comparison $\\mathrm{MTA}$-$\\mathrm{RA}$ recommends the toxic dose 2 a large fraction of the time (33.1 \\%). In scenarios 10 and 11 in which all doses are either too toxic or ineffective a good algorithm would stop early with no recommendation.\n$\\mathrm{TS}\\_\\mathrm{A}$ makes this optimal decision more often than $\\mathrm{MTA}$-$\\mathrm{RA}$ in both scenarios and $\\mathrm{TS}$ in one of the two scenarios. \n\n\n\n\\paragraph{Dose allocation}\nWhile $\\mathrm{TS}$ and $\\mathrm{TS}\\_\\mathrm{A}$ have lower allocation percentage at the optimal (underlined) dose compared to $\\mathrm{MTA}$-$\\mathrm{RA}$, the addition of the admissible set $\\cA$ to the Thompson Sampling method consistently reduces the percentage of dose allocation at doses that are too toxic. Furthermore, $\\mathrm{TS}\\_\\mathrm{A}$ is more cautious in allocating higher doses compared to $\\mathrm{MTA}$-$\\mathrm{RA}$. Our experiments notably reveal that the fraction of allocation to doses whose toxicity is larger than $\\theta$ (that are underlined with a dashed line) is always smaller for $\\mathrm{TS}\\_\\mathrm{A}$ than for $\\mathrm{MTA}$-$\\mathrm{RA}$. Hence, not only is $\\mathrm{TS}\\_\\mathrm{A}$ very good in terms of recommending the right dose, it also manages to avoid too-toxic doses more consistently.\n\n\\input{tbl_toxicity}\n\n\\input{tbl_efficacy}\n\n\\section{Revisiting the Treatment versus Experimentation Trade-off}\\label{sec:Discussion}\n\nIdeally, a good design for MTD identification should be supported by a control of both the error probability $e_n = \\bP(\\hat{k}_n \\neq k^*)$ and the number of sub-optimal selections $\\bE[N_k(n)]$ for $k\\neq k^*$. These two quantities are respectively useful to check whether the design achieves a \\emph{good identification of the optimal dose} and whether \\emph{a large number of patients have been treated with the optimal dose}. \n\nFor classical bandits (in which $k^*$ is the arm with largest mean instead of the MTD), those two performance measures are known to be antagonistic. Indeed, \\cite{Bubeckal11} shows that the smaller the regret (a quantity that can be related to the number of sub-optimal selections), the larger the error probability. Such a trade-off may also exist for the MTD identification problem. However, the precise statement of such a result would be meaningful for large values of the number of patients $n$, which is of little interest for a real clinical trial as it can only involve a small number of patients. In practice, we showed that adaptations of Thompson Sampling, a bandit design aimed at maximizing rewards, achieve good performance in terms of both allocation and recommendation. \n\nStill, another natural avenue of research is to investigate the adaptation of bandit designs aimed at minimizing the error probability. Minimizing the error probability for MTD can be viewed as a variant of the fixed-budget Best Arm Identification (BAI) problem introduced by \\cite{Bubeck10BestArm,Bubeckal11}. In contrast to the standard BAI problem that aims to identify the arm with largest mean (which would correspond here to the most toxic dose), the focus is on identifying the arm whose mean is closest to the threshold $\\theta$. A state-of-the art fixed-budget BAI algorithm is Sequential Halving \\citep{Karnin13}, and we propose in Algorithm~\\ref{alg-ST} a natural adaptation to MTD identification. \n\nSequential Halving for MTD identification proceeds in phases. In each of the $\\log_2(K)$ phases, all the remaining doses are allocated the same amount of times to patients and their empirical toxicity based on these allocations (that is, the average of the toxicity responses) is computed. At the end of each phase the empirical worst half of the doses is eliminated. For MTD identification, rather than the doses with the smallest empirical means (as the vanilla Sequential Halving algorithm would do), the doses whose empirical toxicity are the furthest away from the threshold $\\theta$ are eliminated. Observe that by design of the algorithm, the total number of allocated doses is indeed smaller than the prescribed budget $n$. \n\n\n\\begin{algorithm}[h!]\n\\label{alg-ST}\n\\begin{algorithmic}\n\\State \\textbf{Input:} budget $n$, target toxicity $\\theta$\n\\State \\textbf{Initialization:} Set of dose levels $S_0 \\leftarrow \\{1, \\dots, K\\}$;\n \\For{$r \\leftarrow 0$ to $\\lceil\\log_2(K)\\rceil-1$}\n \\State Allocate each dose $k \\in S_r$ to $t_r = \\left\\lfloor\\frac{n}{|S_r|\\lceil\\log_2(K)\\rceil}\\right\\rfloor$ patients; \n \\State Based on their response compute $\\hat p_k^r$, the empirical toxicity of dose $k$ based on these $t_r$ samples;\n \\State Compute $S_{r+1}$ the set of $\\lceil|S_r|\/2\\rceil$ arms with \n smallest $\\hat d_k^r:=|\\theta - \\hat{p}_k^r|$\n \\EndFor\n\\State \\textbf{Output:} the unique arm in $S_{\\lceil{\\log_2(K)}\\rceil}$\n\\end{algorithmic}\n\\caption{Sequential Halving for MTD Identification}\n\\end{algorithm}\n\n\nBuilding on the analysis of \\cite{Karnin13}, one can establish the following upper bound on the error probability of Sequential Halving for MTD identification. The proof can be found in Appendix~\\ref{sec:SHproof}. \n\n\\begin{theorem}\\label{thm:SH}\nThe error probability of the \\texttt{SH} algorithm is upper bounded as \n\\begin{align*}\n \\bP\\left(\\hat{k}_n \\neq k^*\\right) \\leq 9 \\log_2 K \\cdot \\exp \\left(\n - \\frac{n}{8 H_2 (\\bm p)\\log_2 K}\n \\right),\n\\end{align*}\nwhere $H_2(\\bm p):= \\max_{k \\ne k^*} {k}{\\Delta_{[k]}^{-2}}$ where $\\Delta_k = |p_k - \\theta| - |p_{k^*} - \\theta|$ and $\\Delta_{[1]} \\leq \\Delta_{[2]} \\leq \\dots \\leq \\Delta_{[K]}$.\n\\end{theorem}\nA consequence of Theorem~\\ref{thm:SH} is that in a trial involving more than $n = 8 H_2(\\bm p) \\log_2 K\\log\\left(9\\log_2(K)\/\\delta\\right)$ patients, Sequential Halving is guaranteed to identify the MTD with probability larger than $1-\\delta$. However, this number is typically much larger than the number of patients involved in a clinical trial. Indeed the complexity term $H_2(\\bm p)$ may be quite large, when some doses have a distance to the threshold $\\theta$ which is very close to the smallest distance $|p_{k^*} - \\theta|$. \n\nAn important shortcoming of Sequential Halving is that due to the uniform exploration within each phase each dose is selected at least $n \/ (K\\log_2(K))$ times, even the largest, possibly harmful ones. This is highly unethical in a clinical trial without prior knowledge that too-toxic (or too ineffective) doses have already been eliminated. This problem of allocating too extreme doses is likely to be shared by adaptations of any other BAI algorithm, that are expected to select all the arms a linear number of times. For example the APT algorithm proposed by \\cite{Locatelli16Thres} to identify all arms with mean above a threshold $\\theta$ using a fixed budget $n$ also selects all arms a linear number of times.\n\nTo overcome this problem, an interesting avenue of research would be to try to incorporate monotonicity assumptions in BAI algorithms. \\cite{Garivier17DF} recently proposed such an algorithm, in the fixed confidence setting: given a risk parameter $\\delta$, the goal is to identify a dose $\\hat{k}_\\tau$ such that $\\bP(\\hat{k}_\\tau \\neq k^*) \\leq \\delta$, using as few samples $\\tau$ as possible. Their analysis identifies a minimal sample complexity $\\bE[\\tau]$ that guarantees a $\\delta$-correct identification for any increasing toxicities, which can be obtained under an \\emph{optimal allocation} $w^*$ (where $w_k^*$ indicates the fraction of time dose $k$ is allocated). Interestingly, this optimal allocation is supported only on the neighboring doses of the MTD. The fixed-confidence setting requires allowing for random stopping rules $\\tau$, i.e. for a dose-finding trial based on an adaptively chosen number of patients. This is not always possible in practice, and it would be interesting to investigate optimal allocations in a fixed-budget setting as well. Yet optimality in the fixed-budget setting is a notoriously hard question already for classical bandits \\citep{Locatelli16LBFB}.\n\n\n\n\\section{Conclusion}\\label{sec:Conclusion}\n\nMotivated by the literature on multi-armed bandit models, we advocated the use of the powerful Thompson Sampling principle for dose-finding studies. This Bayesian randomized algorithm can be used in different contexts as it can leverage different prior information about the doses. For increasing toxicities and increasing or plateau efficacies, we proposed variants of Thompson Sampling, notably the $\\mathrm{TS}\\_\\mathrm{A}$ algorithm that often outperforms our baselines in terms of recommendation of the optimal dose, while significantly reducing the allocation to doses with high toxicity. \n\nWe provided theoretical guarantees for the simplest version of Thompson Sampling based on independent uniform priors on each dose toxicity, but advocated the use of more sophisticated priors for practical dose-finding studies. We believe that finding a practical design for which we can also establish non-trivial finite-time performance guarantees is a crucial research question.\n\nAnother interesting direction would be taking contextual information (e.g. a patient's medical history and other medications used) into account for a more ``personalized'' assessment of toxicity and efficacy of a drug. Bayesian methods also seem promising for such an objective, following the success of Thompson Sampling for contextual bandits.\n\n\n\n\n\n\n\n\n\\section*{Acknowledgments} Emilie Kaufmann acknowledges the support of the French Agence Nationale de la Recherche (ANR) under grant ANR-16-CE40-0002 (BADASS project) and ANR-19-CE23-0026-04 (BOLD project). We thank anonymous reviewers of this paper for their helpful suggestions for improvements. \n\n\n\n\n\n\n\\bibliographystyle{biom}\n\n\\section{Analysis of Independent Thompson Sampling: Proof of Theorem~\\ref{thm:TS}} \\label{proof:TS}\n\nFix a sub-optimal arm $k$. Several cases need to be considered depending on the relative position of $p_k$ and $p_{k^*}$ with respect to the threshold. All cases can be treated similarly and to fix the ideas, we consider the case $p_{k^*} \\geq \\theta > p_k$, which is illustrated below. In that case $d^*_k = 2\\theta - p_{k^*}$ satisfies $p_k < d_{k}^* \\leq \\theta$.\n\n\\begin{figure}[h]\\centering\n \\includegraphics[height=7cm,angle=-90]{illustration}\n\\end{figure}\n\nLet $x,y \\in ]0,1[^2$ be such that $p_k < x < y < d_{k^*}$, that will be chosen later. Define $y' = 2\\theta - y > \\theta$ the symmetric of $y$ with respect to the threshold (see the above illustration). We denote by $\\hat{\\mu}_k(t)$ the empirical mean of the toxicity responses gathered from dose $k$ up to the end of round $t$ and recall $\\theta_k(t)$ is the sample from the Beta posterior on $p_k$ after $t$ rounds that is used in the Thompson Sampling algorithm. Inspired by the analysis of \\cite{AGAISTAT13}, we introduce the following two events, that are quite likely to happen when enough samples of arm $k$ have been gathered: \n\\begin{eqnarray*}\n E_k^\\mu (t) & = & \\left(\\hat{\\mu}_k(t) \\leq x\\right) \\ \\ \\ \\text{and} \\ \\ \\ E_k^\\theta (t) = \\left({\\theta}_k(t) \\leq y\\right).\n\\end{eqnarray*}\n\nThe expected number of allocations of dose $k$ is then decomposed in the following way \n\\begin{align*}\n \\bE[N_k(T)] = &\\underbrace{\\sum_{t=0}^{T-1}\\bP\\left(D_{t+1} = k, E_k^\\mu(t),E_k^\\theta(t)\\right)}_{(I)}+ \\underbrace{\\sum_{t=0}^{T-1}\\bP\\left(D_{t+1} = k, E_k^\\mu(t),\\overline{E_k^\\theta(t)}\\right)}_{(II)}\n \\\\ &\n \t+ \\underbrace{\\sum_{t=0}^{T-1}\\bP\\left(D_{t+1} = k, \\overline{E_k^\\mu(t)}\\right)}_{(III)} \n\\end{align*}\nTerms (II) and (III) are easily controlled using some concentration inequalities and the so-called Beta-Binomial trick, that is the fact that the CDF of a Beta distribution with parameters $a$ and $b$, $F^{\\text{Beta}}_{a,b}$, is related to the CDF of a binomial distribution with parameter $n,x$, $F^{B}_{n,x}$, in the following way: \\[F^{\\text{Beta}}_{a,b}(x) = 1 - F^{B}_{a+b-1, x}(a - 1).\\] Term (III) is very small as arm $k$ is unlikely to be drawn often while its empirical mean falls above $x > p_k$ and term (II) grows logarithmically with $T$. More precisely, it can be shown using Lemma 3 and 4 in \\cite{AGAISTAT13} that \n\\[\n (II) \\leq \\frac{\\log(T)}{\\mathrm{kl}(x,y)} + 1 \\ \\ \\ \\text{ and } \\ \\ \\ (III) \\leq \\frac{1}{\\mathrm{kl}(x,y)} + 1.\n\\] \nThe tricky part of the analysis is to control term (I), that is to upper bound the number of selections of dose $k$ when both the empirical mean and the Thompson sample for dose $k$ fall close to the true mean $p_k$. For this purpose, one can prove a counterpart of Lemma 1 in \\cite{AGAISTAT13} that relates the probability of selecting dose $k$ to that of selecting the MTD $k^*$. \n\n\\begin{lemma}\\label{lem:CrucialAG}Define $p_{y}(t) : = \\bP\\left({\\theta}_{k^*}(t) \\in [y,y'] | \\cF_{t}\\right)$, where $\\cF_{s}$ is the filtration generated by the observation up to the end of round $s$. Then \n\\begin{align*}\n&\\bP\\left(D_{t+1} = k | E_{k}^\\theta(t+1),\\cF_{t}\\right)\\leq \\frac{1-p_{y}(t)}{p_y(t)}\\bP\\left(D_{t+1} = k^* | E_k^\\theta(t+1),\\cF_{t}\\right).\n\\end{align*}\n\\end{lemma}\n\n\\begin{proof} The proof is inspired of that of Lemma 1 in \\cite{AGAISTAT13}. We introduce the event in which the Thompson sample for dose $k$ is the closest to the threshold $\\theta$ among all sub-optimal doses:\n\\[M_k(t) = \\{ |\\theta - \\theta_k(t)| \\geq |\\theta - \\theta_\\ell(t)| \\forall \\ell \\neq k^*\\}.\\]\nOn the one hand, one has\n\\begin{align*}\n \\bP\\left(D_{t+1} = k^* | E_k^\\theta(t+1),\\cF_t\\right)& \\geq\n \t\\bP\\left(D_{t+1} = k^*,M_k(t) | E_k^\\theta(t+1),\\cF_t\\right)\n \\\\ & \\geq\n \\bP\\left(\\theta_{k^*}(t) \\in [y,y'],M_k(t) | E_k^\\theta(t+1),\\cF_t\\right) \n\\\\ & =\n\tp_y(t) \\times \\bP\\left(M_k(t) | E_k^\\theta(t+1),\\cF_t\\right).\n\\end{align*}\nOn the other hand, it holds that \n\\begin{align*} \\bP\\left(D_{t+1} = k | E_k^\\theta(t+1),\\cF_t\\right)\n & \\leq \\bP\\left(\\theta_{k^*}(t) \\notin [y, y'], M_k(t) | E_k^\\theta(t+1),\\cF_t\\right)\n\\\\ & = (1 - p_y(t)) \\times \\bP\\left(M_k(t) | E_k^\\theta(t+1),\\cF_t\\right).\n\\end{align*}\nCombining the two inequalities yields Lemma~\\ref{lem:CrucialAG}.\n\\end{proof}\n\nUsing the same steps as \\cite{AGAISTAT13} yields an upper bound on the first term:\n\\[(I) \\leq \\sum_{j=1}^{T-1}\\bE\\left[\\frac{1}{p_y(\\tau_{j})} - 1\\right],\\]\nwhere $\\tau_j$ is the time instant at which dose $k$ is selected for the $j$-th time. The expectation of $1\/p_y(\\tau_{j})$ can be explicitly written \n\\[\\bE\\left[\\frac{1}{p_y(\\tau_{j})}\\right] =\\sum_{s=0}^j \\frac{f^B_{j,p_{k^*}}(s)}{\\bP\\left(y \\leq X_{s+1,j-s+1} \\leq y'\\right)} \\]\nwhere $f^B_{n,x}$ stands for the pdf of a Binomial distribution and $X_{a,b}$ denotes a random variable that has a $\\mathrm{Beta}(a,b)$ distribution. The following lemma is crucial to finish the proof. This original result was specifically obtained for the MTD identification problem and is needed to control the probability that a Beta distributed random variable fall inside an interval, that is $\\bP\\left(y \\leq X_{s+1,j-s+1} \\leq y'\\right)$.\n\n\n\\begin{lemma}\\label{lem:Technical} There exists $j_0$ such that, for all $j \\geq j_0$,\n\\begin{align*}\n&\\forall s \\in \\{0,\\dots, j\\}, \\ \\ \\bP\\left(y \\leq X_{s+1,j-s+1} \\leq y'\\right) \\geq \\frac{1}{2}\\min \\left\\{\\bP\\left(X_{s+1,j+s+1} \\geq y\\right) ; \\bP\\left(X_{s+1,j+s+1} \\leq y'\\right)\\right\\}\n\\end{align*}\n\\end{lemma}\n\nUsing Lemma~\\ref{lem:Technical} and the Beta-Binomial trick, one can write, for $j \\geq j_0$, \n\\begin{align}\n\\nonumber\n\\bE\\left[\\frac{1}{p_y(\\tau_{j})}\\right]\n & \\leq\n\\sum_{s=0}^j \\frac{2f^B_{j,p_{k^*}}(s)}{\\bP\\left(X_{s+1,j+s+1} \\geq y\\right)}+\\sum_{s=0}^j \\frac{2f^B_{j,p_{k^*}}(s)}{\\bP\\left(X_{s+1,j+s+1} \\leq y'\\right)} \n \\\\ \\nonumber & =\n \\sum_{s=0}^j \\frac{2f^B_{j,p_{k^*}}(s)}{F^B_{j+1,y}(s)} + \\sum_{s=0}^j \\frac{2f^B_{j,p_{k^*}}(s)}{1-F^B_{j+1,y'}(s)}\n \\\\ & = \\sum_{s=0}^j \\frac{2f^B_{j,p_{k^*}}(s)}{F^B_{j+1,y}(s)} + \\sum_{s=0}^j \\frac{2f^B_{j,1-p_{k^*}}(s)}{F^B_{j+1,1-y'}(s)},\\label{eq:star}\n\\end{align}\nwhere the last equality relies on the following properties of the Binomial distribution \\[f^B_{n,x}(s) = f^B_{n,1-x}(n-s) \\ \\ \\text{and} \\ \\ F^B_{n,x}(s) = 1 - F_{n,1-x}(n-s-1)\\]\nand a change of variable in the second sum. \n\nNow the following upper bound can be extracted from the proof of Lemma 3 in \\cite{AGAISTAT13}. \n\n\\begin{lemma}\\label{lem:UB} Fix $u$ and $v$ such that $u < v$ and let $\\Delta = v - u$. Then \n\\begin{align*}\n\\sum_{s=0}^j \\frac{f^B_{j,v}(s)}{F^B_{j,u}(s)} \\leq\n \\left\\{\\begin{array}{cl}\n 1 + \\frac{3}{\\Delta} & \\text{if } j < {8}\/\\Delta,\n \\\\\n 1 + \\Theta\\left(e^{-\\Delta^2 j \/2} + \\frac{1}{(j+1)\\Delta^2}e^{-2\\Delta^2 j} + \\frac{1}{e^{\\Delta^2j\/4} - 1}\\right) & \\text{else.} \n\\end{array}\n\\right.\n\\end{align*}\n\\end{lemma}\n\nEach of the two sums in \\eqref{eq:star} can be upper bounded using Lemma~\\ref{lem:UB}. Letting $\\Delta_1 = p_{k^*} - y$ and $\\Delta_2 = y' - p_{k^*}$, one obtains \n\\begin{align*}\n (I) \\leq &\n \t\\sum_{j=1}^{j_0} \\bE\\left[\\frac{1}{p_y(\\tau_{j})} \\right]\n \t- j_0 + \\frac{24}{\\Delta_1^2} + \\frac{24}{\\Delta_2^2} \n\\\\ &\n\t+ C \\sum_{j=0}^{T-1} \\left[e^{-\\Delta_1^2 j \/2} + \\frac{1}{(j+1)\\Delta_1^2}e^{-2\\Delta_1^2 j} + \\frac{1}{e^{\\Delta_1^2j\/4} - 1}\\right]\n\\\\ &\n\t+ C \\sum_{j=0}^{T-1} \\left[e^{-\\Delta_2^2 j \/2} + \\frac{1}{(j+1)\\Delta_2^2}e^{-2\\Delta_2^2 j} + \\frac{1}{e^{\\Delta_2^2j\/4} - 1}\\right], \n\\end{align*}\nwhich is a constant (as the series have a finite sum) that only depends on $y, \\theta$ and $p_{k^*}$ (through $y'$ and the gaps $\\Delta_1$ and $\\Delta_2$ defined above). \n\nPutting things together, we proved that for every $x$ and $y$ satisfying $p_k < x < y < d_{k^*}$, the number of selections of dose $k$ is upper bounded as \n\\[ \\bE[N_k(T)] \\leq \\frac{1}{\\mathrm{kl}(x,y)}\\log(T) + C_{x,y,\\theta,\\bm p}\\]\nfor some constant that depends on the toxicity probabilities, the threshold $\\theta$ and the choice of $x$ and $y$. Now, picking $x$ and $y$ such that $\\mathrm{kl}(x,y) = \\frac{\\mathrm{kl}(p_k,d_{k^*})}{1+\\epsilon}$ yield the result. \n\n\\qed \n\n\\paragraph{Proof of Lemma~\\ref{lem:Technical}.} The proof uses the two equalities below\n\\begin{align}\n \\bP\\left(y \\leq X_{s+1,j-s+1} \\leq y'\\right) & = \\bP\\left(X_{s+1,j-s+1} \\geq y\\right) - \\bP\\left(X_{s+1,j-s+1} \\geq y'\\right)\\label{ForSmallS}\n\\\\ \n\\bP\\left(y \\leq X_{s+1,j-s+1} \\leq y'\\right)\n & = \\bP\\left(X_{s+1,j-s+1} \\leq y'\\right) - \\bP\\left(X_{s+1,j-s+1} \\leq y\\right),\\label{ForLargeS}\n\\end{align}\nas well as the Sanov inequalities: if $S_{n,x}$ is a binomial distribution with parameters $n$ and $x$, then \n\\begin{align}\n\\nonumber\n\\frac{e^{-n\\mathrm{kl}(k\/n,x)}}{n+1}\n& \\leq \\bP\\left(S_{n,x} \\geq k\\right) \n\\\\ & \\leq e^{-n\\mathrm{kl}(k\/n,x)} \\ \\ \\text{if } \\ k > xn \\label{Sanov}\n\\\\ \\nonumber\n\\frac{e^{-n\\mathrm{kl}(k\/n,x)}}{n+1}\n& \\leq \\bP\\left(S_{n,x} \\leq k\\right)\n\\\\ & \\leq\n\te^{-n\\mathrm{kl}(k\/n,x)} \\ \\ \\text{if } \\ k < xn \\label{SanovMin}\n\\end{align}\n\nWe prove the inequality considering 4 cases. We define $y_{\\mathrm{mid}} = \\frac{y+y'}{2}$. \n\n\\paragraph{Case 1: $\\bm{s < (j+1)y}$} Starting from equality \\eqref{ForSmallS} and using the Beta-Binomial trick yields \n\\begin{align*}\n&\\bP\\left(y \\leq X_{s+1,j-s+1} \\leq y'\\right) = \\bP\\left(S_{j+1,y} \\leq s\\right) - \\bP\\left(S_{j+1,y'} \\leq s\\right).\n\\end{align*}\nUsing Sanov inequalities, we shall prove that there exists some $j_1$ such that if $j\\geq j_1$, \n\\[\\forall s \\leq (j+1)y, \\ \\ \\bP\\left(S_{j+1,y'} \\leq s\\right) \\leq \\frac{1}{2}\\bP\\left(S_{j+1,y} \\leq s\\right).\\]\nAs $s$ is smaller than the mean of the two Binomial distributions, by \\eqref{SanovMin} it is sufficient to prove that \n\\begin{align*}\n&\\forall s \\leq (j+1)y,\n\\ \\ e^{-(j+1)\\mathrm{kl}\\left(\\frac{s}{j+1} , y'\\right)} \\leq \\frac{1}{2(j+2)}e^{-(j+1)\\mathrm{kl}\\left(\\frac{s}{j+1} , y\\right)}\n\\end{align*}\nwhich in turn is equivalent to \n\\begin{align*}\n&\\forall s \\leq (j+1)y,\n\\ \\ \\mathrm{kl}\\left(\\frac{s}{j+1} , y'\\right) - \\mathrm{kl}\\left(\\frac{s}{j+1} , y\\right) \\geq \\frac{\\log(2(j+2))}{j+1}.\n\\end{align*}\nAs the function in the left-hand side is non-increasing in $s$, a sufficient condition is that $j$ satisfies \n\\[ \\mathrm{kl}\\left(y , y'\\right)\\geq \\frac{\\log(2(j+2))}{j+1},\\]\nwhich is the case for $j$ superior to some $j_1$. Thus, for $j\\geq j_1$, \n\\begin{align*}\n\\bP\\left(y \\leq X_{s+1,j-s+1} \\leq y'\\right)\\geq \\frac{1}{2}\\bP\\left(S_{j+1,y} \\leq s\\right)= \\frac{1}{2}\\bP\\left(X_{s+1,j-s+1} \\geq y\\right).\n\\end{align*}\n\n\\paragraph{Case 2: $\\bm{(j+1)y \\leq s \\leq (j+1)y_{\\mathrm{mid}}}$} Starting from equality \\eqref{ForSmallS} and using the Beta-Binomial trick and the upper bound in \\eqref{SanovMin} yields \n\\begin{align*}\n\\bP\\left(y \\leq X_{s+1,j-s+1} \\leq y'\\right) \n & \\geq \\bP\\left(S_{j+1,y} \\leq s\\right) - e^{-(j+1)\\mathrm{kl}\\left(\\frac{s}{j+1} , y'\\right)}\n\\\\ & \\geq \\bP\\left(S_{j+1,y} \\leq s\\right) - e^{-(j+1)\\mathrm{kl}\\left(y_{\\mathrm{mid}} , y'\\right)}. \n\\end{align*}\nThe median of $S_{j+1,y}$ is $\\lfloor(j+1)y\\rfloor$ or $\\lceil(j+1)y\\rceil$. As $s \\leq (j+1)y$, it holds that $\\bP\\left(S_{j+1,y} \\leq s\\right) \\geq \\frac{1}{2}$. Therefore, for all $j \\geq j_2 := \\frac{\\ln 4}{\\mathrm{kl}(y_{\\mathrm{mid}},y')}-1$, \n\\[e^{-(j+1)\\mathrm{kl}\\left(y_{\\mathrm{mid}} , y'\\right)} \\leq \\frac{1}{4} \\leq \\frac{1}{2}\\bP\\left(S_{j+1,y} \\leq s\\right).\\]\nTherefore if $j \\geq j_2$, $\\bP\\left(y \\leq X_{s+1,j-s+1} \\leq y'\\right) \\geq \\frac{1}{2}\\bP\\left(X_{s+1,j-s+1} \\geq y\\right)$.\n\n\\paragraph{Case 3: $\\bm{(j+1)y_{\\mathrm{mid}} \\leq s \\leq (j+1)y'}$} Starting from equality \\eqref{ForLargeS} and using the Beta-Binomial trick and the upper bound in \\eqref{Sanov} yields \n\\begin{align*}\n \\bP\\left(y \\leq X_{s+1,j-s+1} \\leq y'\\right) & \\geq \\bP\\left(S_{j+1,y'} \\geq s\\right) - e^{-(j+1)\\mathrm{kl}\\left(\\frac{s}{j+1} , y\\right)}\n\\\\ & \\geq \\bP\\left(S_{j+1,y'} \\geq s\\right) - e^{-(j+1)\\mathrm{kl}\\left(y_{\\mathrm{mid}} , y\\right)}. \n\\end{align*}\nThe median of $S_{j+1,y'}$ is $\\lfloor(j+1)y'\\rfloor$ or $\\lceil(j+1)y'\\rceil$. As $s \\leq (j+1)y'$, it holds that $\\bP\\left(S_{j+1,y'} \\geq s\\right) \\geq \\frac{1}{2}$. Therefore, for all $j \\geq j_3 := \\frac{\\ln 4}{\\mathrm{kl}(y_{\\mathrm{mid}},y)}-1$, \n\\[e^{-(j+1)\\mathrm{kl}\\left(y_{\\mathrm{mid}} , y\\right)} \\leq \\frac{1}{4} \\leq \\frac{1}{2}\\bP\\left(S_{j+1,y'} \\geq s\\right).\\]\nTherefore if $j \\geq j_3$, $\\bP\\left(y \\leq X_{s+1,j-s+1} \\leq y'\\right) \\geq \\frac{1}{2}\\bP\\left(X_{s+1,j-s+1} \\leq y'\\right)$.\n\n\\paragraph{Case 4: $\\bm{s > (j+1)y'}$} Starting from equality \\eqref{ForLargeS} and using the Beta-Binomial trick yields \n\\[\\bP\\left(y \\leq X_{s+1,j-s+1} \\leq y'\\right) = \\bP\\left(S_{j+1,y'} \\geq s\\right) - \\bP\\left(S_{j+1,y} \\geq s\\right).\\]\nUsing Sanov inequalities, we shall prove that there exists some $j_4$ such that if $j\\geq j_4$, \n\\[\\forall s \\geq (j+1)y', \\ \\ \\bP\\left(S_{j+1,y} \\geq s\\right) \\leq \\frac{1}{2}\\bP\\left(S_{j+1,y'} \\geq s\\right).\\]\nAs $s$ is larger than the mean of the two Binomial distributions, by \\eqref{Sanov} it is sufficient to prove that \n\\[\\forall s \\geq (j+1)y', \\ \\ e^{-(j+1)\\mathrm{kl}\\left(\\frac{s}{j+1} , y\\right)} \\leq \\frac{1}{2(j+2)}e^{-(j+1)\\mathrm{kl}\\left(\\frac{s}{j+1} , y'\\right)}\\]\nwhich in turn is equivalent to \n\\[\\forall s \\geq (j+1)y', \\ \\ \\mathrm{kl}\\left(\\frac{s}{j+1} , y\\right) - \\mathrm{kl}\\left(\\frac{s}{j+1} , y'\\right) \\geq \\frac{\\log(2(j+2))}{j+1}.\\]\nAs the function in the left-hand side is non-decreasing in $s$, a sufficient condition is that $j$ satisfies \n\\[ \\mathrm{kl}\\left(y' , y\\right)\\geq \\frac{\\log(2(j+2))}{j+1},\\]\nwhich is the case for $j$ superior to some $j_4$. Thus, for $j\\geq j_4$, \n\\begin{align*}\n\\bP\\left(y \\leq X_{s+1,j-s+1} \\leq y'\\right) \n&\\geq \\frac{1}{2}\\bP\\left(S_{j+1,y'} \\geq s\\right) \n\\\\ &= \\frac{1}{2}\\bP\\left(X_{s+1,j-s+1} \\leq y'\\right).\n\\end{align*}\n\n\\paragraph{Conclusion} Letting $j_0 = \\max(j_1,j_2,j_3,j_4)$, for all $j\\geq j_0$, for every $s \\in \\{0,\\dots,j\\}$,\n\\begin{align*}\n&\\bP\\left(y \\leq X_{s+1,j-s+1} \\leq y'\\right) \\geq \\frac{1}{2}\\min \\left\\{\\bP\\left(X_{s+1,j+s+1} \\geq y\\right) ; \\bP\\left(X_{s+1,j+s+1} \\leq y'\\right)\\right\\}\n\\end{align*}\n\n\n\n\\section{Lower Bound on the Number of Allocation: Proof of Theorem~\\ref{thm:LB}}\\label{proof:LB}\n\nFix a uniformly efficient algorithm and a vector of toxicity probabilities $\\bm p$. We denote by $\\bE_{\\bm p}$ the expectation under the model parameterized by $\\bm p$ when this algorithm is used. Letting $\\bm p'$ be another vector of probabilities, it follows from the change-of-distribution lemma of \\cite{GMS18} that for all random variable $Z_T \\in [0,1]$ which is $\\cF_T$-measurable \n\\begin{equation}\\sum_{\\ell=1}^K \\bE_{\\bm p}[N_{\\ell}(T)]\\mathrm{kl}\\left(p_{\\ell}, p'_{\\ell}\\right) \\geq \\mathrm{kl}\\left(\\bE_{\\bm p}[Z_T], \\bE_{\\bm p'} [Z_T]\\right).\\label{eq:CD}\\end{equation}\nLetting $k^*$ be a MTD in $\\bm p$, we fix $k$ which is not a MTD (i.e. $|p_k - \\theta | > |p_{k^*} - \\theta|$) and we prove that \n\\begin{equation}\\liminf_{T\\rightarrow \\infty}\\frac{\\bE_{\\bm p}[N_{k}(T)]}{\\ln(T)} \\geq \\frac{1}{\\mathrm{kl}(p_k,d_k^*)}\\;.\\label{ToProveLB}\\end{equation} \n\nRecall that we assume $p_{k^*} \\neq \\theta$. Then one can define the alternative model $\\bm p'$ in which for all $\\ell \\neq k$, $p'_{\\ell} = p_{\\ell}$ and $p'_{k} = d_{k}^* + \\epsilon$ if $d_k^* < \\theta$ and $p'_{k} = d_{k}^* - \\epsilon$ if $d_k^* > \\theta$, with $\\epsilon$ small enough such that under $\\bm p'$, dose $k$ is the unique MTD (refer to Figure~\\ref{fig:doses} for an illustration). \n\nFor this particular choice of alternative model $\\bm p'$, \\eqref{eq:CD} becomes\n\\begin{eqnarray*}\\bE_{\\bm p} [N_k(T)] \\mathrm{kl}(p_k,d^*_k \\pm \\epsilon) &\\geq& \\mathrm{kl}\\left(\\bE_{\\bm p}[Z_T], \\bE_{\\bm p'} [Z_T]\\right)\\\\\n & \\geq & \\left(1 -\\bE_{\\bm p}[Z_T]\\right) \\ln \\left(\\frac{1}{1-\\bE_{\\bm p'} [Z_T]}\\right) - \\ln(2)\n\\end{eqnarray*}\nChoosing $Z_T = \\frac{N_{k}(T)}{T}$, exploiting the fact that the algorithm is uniformly efficient we know that \n\\begin{itemize}\n \\item $\\lim_{T\\rightarrow \\infty}\\bE_{\\bm p}[Z_T] = 0$ as $k$ is a sub-optimal dose under $\\bm p$ \n \\item $\\frac{1}{1-\\bE_{\\bm p'} [Z_T]} = \\frac{T}{T - \\bE_{\\bm p'}[N_k(T)]} = \\frac{T}{\\sum_{\\ell \\neq k} \\bE_{\\bm p'}[N_{\\ell}(T)]}$ and $\\sum_{\\ell \\neq k} \\bE_{\\bm p'}[N_{\\ell}(T)] = o(T^\\alpha)$ for all $\\alpha \\in (0,1)$ as $k$ is the only MTD under $\\bm p'$, which yields, for all $\\alpha \\in (0,1)$, \n \\[\\lim_{T\\rightarrow \\infty} \\frac{1}{\\ln(T)} \\ln \\left(\\frac{1}{1-\\bE_{\\bm p'} [Z_T]}\\right) \\geq 1 - \\alpha\\;.\\]\n\\end{itemize}\nLetting $\\alpha$ go to zero, we obtain \n\\[\\liminf_{T\\rightarrow \\infty} \\frac{\\bE_{\\bm p} [N_k(T)] \\mathrm{kl}(p_k,d^*_k \\pm \\epsilon)}{\\ln(T)} \\geq 1\\]\nand \\eqref{ToProveLB} follows by letting $\\epsilon$ go to zero. \n\n\n\n\\section{Analysis of Sequential Halving: Proof of Theorem~\\ref{thm:SH}}\n\\label{sec:SHproof}\n\nRecall $\\hat{d}_k^r = |\\theta - \\hat{p}_k^t|$ is the empirical distance from the toxicity of dose $k$ to the threshold, where $\\hat{p}_k^r$ is the empirical average of the toxicity responses observed for dose $k$ during phase $r$ (based on $t_r$ samples). The central element of the proof is Lemma~\\ref{lemma-inversion} below, that controls the probability that dose $k$ seems to be be closer to the threshold than the MTD $k^*$ in phase $r$. Its proof is more sophisticated than that of Lemma 4.2 in \\cite{Karnin13} as several cases need to be considered. \n\\begin{lemma}\n\\label{lemma-inversion}\nAssume that the arm closest to $\\theta$ was not eliminated\nprior to round $r$.\nThen for any arm $k \\in S_r$,\n\\begin{equation}\\bP(\\hat{d}^r_{k^*} > \\hat{d}^r_k) \\leq 3 \\exp\\left(- \\frac{t_r}{2}\\Delta_k^2\\right).\\label{toproofin4case}\\end{equation}\n\\end{lemma}\n\\begin{proof}\nFor the means $p_{k^*}$ and $p_k$ let $\\hat{p}^r_{k^*}$ and $\\hat{p}^r_k$ denote their\nexpected rewards in round $r$, respectively.\nWe will first derive a probability bound which does not depend on the\nordering of $p_k$ and $p_{k^*}$ w.r.t. $\\theta$, and then we will do a case\nanalysis of the possible orderings to produce our final bound.\n\nThe error event can be decomposed as follows. \n\\begin{align*}\n&\\Set{\\hat{d}^r_{k^*} > \\hat{d}^r_k} =\n\\\\\n\t&~~~\\left( \\Set{\\hat{p}_{{k^*},r} > \\theta}\n\t\t\\cap \\Set{\\hat{p}_{k,r} > \\theta}\n\t\t\\cap \\Set{\\hat{p}_{{k^*},r} - \\theta > \\hat{p}_{k,r} - \\theta}\n\t\t\\right) \\\\\n\t&\\cup \\left(\n\t\t\\Set{\\hat{p}_{{k^*},r} \\le \\theta}\n\t\t\\cap \\Set{\\hat{p}_{k,r} > \\theta}\n\t\t\\cap \\Set{\\theta - \\hat{p}_{{k^*},r} > \\hat{p}_{k,r} - \\theta}\n\t\t\\right) \\\\\n\t&\\cup \\left(\n\t\t\\Set{\\hat{p}_{{k^*},r} > \\theta}\n\t\t\\cap \\Set{\\hat{p}_{k,r} \\le \\theta}\n\t\t\\cap \\Set{\\hat{p}_{{k^*},r} - \\theta > \\theta - \\hat{p}_{k,r}}\n\t\t\\right) \\\\\n\t&\\cup \\left(\n\t\t\\Set{\\hat{p}_{{k^*},r} \\le \\theta}\n\t\t\\cap \\Set{\\hat{p}_{k,r} \\le \\theta}\n\t\t\\cap \\Set{\\theta - \\hat{p}_{{k^*},r} > \\theta - \\hat{p}_{k,r}}\n\t\t\\right)\n\\end{align*}\nFrom there, we distinguish two cases, in which we show the error event is included in a reunion of events whose probability can be controlled using the Hoeffding's inequality. \n\n\\paragraph{Case 1: $\\bm{p_k \\geq \\theta}$.} In that case, it is very unlikely that $\\{\\hat{p}_{k,r} < \\theta\\}$. Hence, we can isolate that event and use the previous decomposition to write\n\\begin{align*}\n&\\Set{\\hat{d}^r_{k^*} > \\hat{d}^r_k} \\subseteq \n\\\\\n&\\Set{\\hat{p}_{k,r} \\le \\theta} \\cup \\Set{\\hat{p}_{{k^*},r} - \\hat{p}_{k,r} > 0}\n\t\t\\cup \\Set{\\hat{p}_{k,r} + \\hat{p}_{{k^*},r} < 2\\theta}\n\t\t.\n\\end{align*}\nWhen $p_k \\geq \\theta$, irrespective of the position of $p_{k^*}$ with respect to $\\theta$, one can justify that $p_k > \\theta$, $p_{k^*} - p_{k} < 0$ and ${p}_{k} + {p}_{{k^*}} > 2\\theta$ (as $p_k \\geq \\max(p_{k^*},2\\theta - p_{k^*})$ because $k$ is a suboptimal arm larger than the threshold). Therefore, the above three events are unlikely. More precisely, using Hoeffding's inequality yields \n\\begin{align*}\n\\bP(\\hat{d}^r_{k^*} > \\hat{d}^r_k) \n &\\leq \n\t\\bP(\\hat{p}_{k,r} \\le \\theta)\n\t+ \\bP(\\hat{p}_{{k^*},r} - \\hat{p}_{k,r} > 0)\n\t+ \\bP(\\hat{p}_{k^*,r} + p_{k,r} < 2\\theta)\n\\\\ &\\leq \n\t\\exp\\left( -2t_r (\\theta - p_k)^2 \\right\\}\n\t+ \\exp\\left\\{ - \\frac{t_r}{2} (p_{k^*} - p_k)^2 \\right\\}\n\\\\\n\t& \\hspace{0.4cm} + \\exp\\left\\{ - \\frac{t_r}{2} (p_{k^*} + p_k - 2\\theta )^2 \\right)\n\\\\ &\\leq \n\t3 \\exp\\left( -\\frac{t_r}{2} \\min\\left\\{(p_{k} - \\theta)^2,\n\t\t(p_{k} - p_{k^*})^2,\n\t\t(p_{k^*}+ p_k - 2\\theta)^2\n\t \\right\\} \\right)\n\\\\ & = \n\t3 \\exp\\left( -\\frac{t_r}{2} \\min\\left\\{\n\t\t(p_{k} - p_{k^*})^2,\n\t\t(p_{k} - (2\\theta - p_{k^*}))^2\n\t \\right\\} \\right)\n\\end{align*}\nEquation~\\eqref{toproofin4case} follows as $\\Delta_k^2 = \\min\\left\\{\n\t\t(p_{k} - p_{k^*})^2,\n\t\t(p_{k} - (2\\theta - p_{k^*}))^2\n\t \\right\\}$. \n\n\\paragraph{Case 2: $\\bm{p_k \\leq \\theta}$.} In that case, the unlikely event is $\\{\\hat{p}_{k,r} > \\theta\\}$ and we write \n\\begin{align*}\n&\\Set{\\hat{d}^r_{k^*} > \\hat{d}^r_k} \\subseteq \n\\Set{\\hat{p}_{k,r} > \\theta} \\cup \\Set{\\hat{p}_{{k},r} - \\hat{p}_{k^*,r} > 0}\\cup \\Set{\\hat{p}_{k,r} + \\hat{p}_{{k^*},r} > 2\\theta}.\n\\end{align*}\nWhen $p_k < \\theta$, irrespective of the position of $p_{k^*}$ with respect to $\\theta$, one can justify that $p_k < \\theta$, $p_{k} - p_{k^*} < 0$ and ${p}_{k} + {p}_{{k^*}} < 2\\theta$ (using the fact that $p_k \\leq \\min(p_{k^*},2\\theta - p_{k^*})$). Then from Hoeffding's inequality, \n\\begin{align*}\n\\bP(\\hat{d}^r_{k^*} > \\hat{d}^r_k) & \\leq \n\t\\bP(\\hat{p}_{k,r} > \\theta)\n\t+ \\bP(\\hat{p}_{{k},r} - \\hat{p}_{k^*,r} > 0)\n\t+ \\bP(\\hat{p}_{k^*,r} + p_{k,r} > 2\\theta)\n\\\\ & \\leq \n\t\\exp\\left( -2t_r (\\theta - p_k)^2 \\right\\}\n\t+ \\exp\\left\\{ - \\frac{t_r}{2} (p_{k^*} - p_k)^2 \\right\\}\n\\\\ & \\hspace{0.4cm}\n\t+ \\exp\\left\\{ - \\frac{t_r}{2} (2\\theta - p_{k^*} - p_k)^2 \\right)\n\\\\ & \\leq \n\t3 \\exp\\left( -\\frac{t_r}{2} \\min\\left\\{(\\theta - p_k)^2,\n\t\t(p_{k^*} - p_k)^2,\n\t\t(2\\theta - p_{k^*} -p_k)^2\n\t \\right\\} \\right)\n\\\\ & = \n\t3 \\exp\\left( -\\frac{t_r}{2} \\min\\left\\{\n\t\t(p_{k^*} - p_{k})^2,\n\t\t((2\\theta - p_{k^*}) - p_k)^2\n\t \\right\\} \\right)\n\\end{align*}\nwhich proves Equation~\\ref{toproofin4case} as $\\Delta_k^2 =\\min\\left\\{\n\t\t(p_{k^*} - p_{k})^2,\n\t\t((2\\theta - p_{k^*}) - p_k)^2\n\t \\right\\}$. \n\\end{proof}\n\n\nBuilding on Lemma~\\ref{lemma-inversion}, the next step is to control the probability that the MTD is eliminated in phase $r$. The proof bears strong similarities with that of Lemma~4.3 in \\cite{Karnin13}. It is given below for the sake of completeness. \n\n\\begin{lemma}\n\\label{lemma-best-survives}\nThe probability that the MTD is eliminated at the end of phase $r$ is at most\n\\begin{align*}\n9 \\exp\\left(\n\t- \\frac{n}{8 \\log_2 K} \\cdot \\frac{\\Delta^2_{k_r}}{k_r}\n\\right)\t\n\\end{align*}\nwhere $k_r = K\/2^{r + 2}$.\n\\end{lemma}\n\nThe end of the proof of Theorem~\\ref{thm:SH} is identical to than of Theorem~4.1 in \\cite{Karnin13}, except that it uses our Lemma~\\ref{lemma-best-survives}. We repeat the argument below with the appropriate modifications. \nObserve that if the algorithm recommends a wrong dose, the MTD must have been eliminated in one of t $\\log_2(K)$ phases. Using Lemma~\\ref{lemma-best-survives} and a union bound yields the upper bound\n\\begin{align*}\n\\bP\\left(\\hat{k}_n \\neq k^*\\right) &\\leq 9 \\sum_{r=1}^{\\log_2 K} \\exp \\left(\n\t- \\frac{n}{8 \\log_2 K} \\cdot \\frac{\\Delta^2_{k_r}}{k_r}\n\t\\right)\t\n\\\\\n\t&\\le 9 \\log_2 K \\cdot \\exp \\left(\n\t\t- \\frac{n}{8 \\log_2 K} \\cdot \\frac{1}{\\max_k k \\Delta^{-2}_k}\n\t\\right)\n\\\\\n\t&\\le 9 \\log_2 K \\cdot \\exp \\left(\n\t\t- \\frac{n}{8 H_2(\\bm p) \\log_2 K}\n\t\\right),\n\\end{align*}\nwhich concludes the proof.\n\n\\paragraph{Proof of Lemma~\\ref{lemma-best-survives}}\nDefine $S_r'$ as the set of arms in $S_r$, excluding the\n$\\frac{1}{4}|S_r| = K\/2^{r+2}$ arms with means closest to $\\theta$.\nIf the MTD $k^*$ is eliminated in round $r$,\nit must be the case that at least half the arms of $S_r$\n(i.e., $\\frac{1}{2}|S_r| = K\/2^{r+1}$ arms)\nhave their empirical average closer to $\\theta$ than its empirical\naverage.\nIn particular, the empirical means of at least\n$\\frac{1}{3}|S_r'| = K\/2^{r+2}$ of the arms in $S_r'$ must be closer to\n$\\theta$ than that of the $k^*$ at the end of round $r$.\nLetting $N_r$ denote the number of arms in $S_r'$ whose empirical average\nis closer to $\\theta$ than that of the optimal arm, we have by\nLemma~\\ref{lemma-inversion}:\n\\begin{align*}\n\t\\mathds{E}[N_r] &= \\sum_{k \\in S_r'}\n\t\t\\P(\\hat d^r_k < \\hat d^r_{k^*})\n\\\\\n\t&\\le \\sum_{k \\in S_r'} 3 \\exp\\left( -\\frac{t_r}{2} \\Delta^2_k \\right)\n\\\\\n\t&\\le 3 \\sum_{k \\in S_r'} \\exp\\left( -\\frac{1}{2} \\Delta^2_k\n\t\t\\cdot \\frac{n}{|S_r| \\log_2 K} \\right)\n\\\\\n\t&\\le 3 |S_r'| \\max_{k \\in S_r'} \\exp\\left( -\\frac{1}{2} \\Delta^2_k\n\t\t\\cdot \\frac{2^r n}{K \\log_2 K} \\right)\n\\\\\n\t&\\le 3 |S_r'| \\exp\\left( -\\frac{n}{8 \\log_2 K}\n\t\t\\cdot \\frac{\\Delta^2_{k_r}}{k_r} \\right)\n\\end{align*}\nWhere the last inequality follows from the fact that there are at least\n$k_r - 1$ arms that are not in $S_r'$ with average reward closer to $\\theta$\nthan that of any arm in $S_r'$.\nWe now apply Markov's inequality to obtain\n\\begin{align*}\n\\P\\left(N_r > \\frac{1}{3}|S_r'|\\right) &\\le 3 \\mathds{E}[N_r] \/ |S_r'|\n\\\\\n\t&\\le 9 \\exp \\left(\n\t\t- \\frac{n}{8 \\log_2 K}\n\t\t\\cdot \\frac{\\Delta^2_{k_r}}{k_r}\n\t\\right),\n\\end{align*}\nand the lemma follows.\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\n\n\\subsection{Policy Abstraction}\\label{sec:mmdp}\n\nIn the context of MARL, a group of $N$ agents interact with each other in a common environment and make decisions influenced by the joint states of all agents. \nAgent decisions can be captured by a joint policy $\\pi: {\\mathcal{X}} \\to \\Delta ({\\mathcal{A}})$,\nwhich is a function mapping the set of joint states ${\\mathcal{X}}=\\{(x^1, \\dots, x^N)\\}$ to a probabilistic distribution over the set of joint actions ${\\mathcal{A}}=\\{(a^1, \\dots, a^N)\\}$, where $x^i$ (\\textit{resp. } $a^i$) denotes the state (\\textit{resp. } action) of agent $i$.\nOnce a policy is trained, agents can act upon it in any given state.\nBut there is a lack of a global view of the entire policy. \nFurther, the size of the policy grows exponentially with the number of agents and state variables. \nTo address these issues, we propose to build an abstract representation of the policy as the basis for generating explanations about agent behavior.\n\nWe use the multi-agent Markov decision process (MMDP) framework to represent MARL policy abstraction. \nFormally, an MMDP is a tuple $({\\mathcal{S}}, {\\mathcal{A}}, {\\mathcal{T}})$, \nwhere ${\\mathcal{S}}=\\{(s^1, \\dots, s^N)\\}$ is the joint (abstract) state space,\n${\\mathcal{A}}$ is the joint action space, and ${\\mathcal{T}}$ is the transition function.\nLet ${\\mathcal{F}}$ be a set of Boolean predicates indicating features of the MARL domain. \nWe denote by $f(x^i)=1$ if an agent state $x^i$ satisfies a feature predicate $f \\in {\\mathcal{F}}$.\nAn abstract state $s^i$ is then given by the satisfaction of all feature predicates $f \\in {\\mathcal{F}}$, where each bit of the binary encoding of $s^i \\in \\mathbb{N}$ corresponds to the satisfaction of a predicate $f(x^i)$.\nThus, the choice of features affects the abstraction level and should include adequate information for explanations. \nIn this work, we assume that users specify a set of feature predicates for a given MARL domain. \n\nOnce an MARL policy is trained, we build an MMDP during the policy evaluation stage.\nFor each sample $(\\vb{x},\\vb{a},\\vb{x}')$, \ndetermine an MMDP transition $\\vb{s} \\xrightarrow{\\vb{a}}\\vb{s}'$ \nby finding the abstract state $\\vb{s}$ (\\textit{resp. } $\\vb{s'}$) corresponding to $\\vb{x}$ (\\textit{resp. } $\\vb{x'}$). \nWhen policy evaluation terminates (e.g., converging to the expected reward),\ncompute the transition probability ${\\mathcal{T}}(\\vb{s},\\vb{a},\\vb{s}')$ via frequency counting.\n\n\\startpara{Properties}\nThe resulting MMDP is a \\emph{sound} abstraction of the MARL policy\nbecause, by construction, every MMDP transition with non-zero probability corresponds to at least one sampled policy decision. \nThe state space size $|{\\mathcal{S}}|$ is bounded by ${\\mathcal{O}}({2^{|{\\mathcal{F}}|}}^N)$, depending on the number of agents $N$ and feature predicates $|{\\mathcal{F}}|$.\nIn practice, a trained MARL policy may only induce a small set of reachable states.\n\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=.75\\columnwidth]{figures\/search_rescue.png}\n \\caption{Example MARL domain of multi-robot search and rescue.}\n \\vspace{-10pt}\n \\label{fig:rs_eg}\n\\end{figure}\n\n\\begin{example} \\label{eg:mmdp}\n\\figref{fig:rs_eg}(a) shows an example MARL domain where three robotic agents cooperate to complete search and rescue tasks. Rescuing the victim requires the cooperation of an unmanned aerial vehicle (UAV) and an unmanned ground vehicle (UGV). Any agent can fight the fire, which is blocked by the wall and obstacle. Removing the obstacle requires the cooperation of two UGVs. Given a trained MARL policy, we build an MMDP abstraction with 6 feature predicates indicating whether each task is detected or completed (e.g., $\\mathsf{victim\\_detect}$, $\\mathsf{victim\\_complete}$).\nAn agent can only detect a task in a neighboring grid (e.g., UAV detects the victim in \\figref{fig:rs_eg}(a)). \nThe resulting MMDP has 63 (reachable) states and 577 transitions.\n\\end{example}\n\n\\subsection{Policy Summarization}\\label{sec:sum}\n\nA policy abstraction containing hundreds of states and transitions is too complex for humans to understand.\nAn alternative way of communicating agent behavior is to show execution traces; however, a lengthy trace may be burdensome for users to review.\nTo overcome these limitations, we develop a method to generate policy summarization, illustrating the agent cooperation and task sequence for the most probable sequence of agent behavior under a given MARL policy.\n\n\\input{3a_ag_sum}\n\n\\agref{ag:sum} shows the proposed method, which takes the input of a policy abstraction ${\\mathcal{M}}$ and a set of predicates ${\\mathcal{F}}_c$ representing the completion of tasks (subgoals) in a given MARL domain. \nThe first step is to compute the most probable path $\\rho = \\vb{s}_0 \\xrightarrow{\\vb{a}_0} \\vb{s}_1 \\xrightarrow{\\vb{a}_1} \\cdots$ from the initial state to a goal state in the MMDP ${\\mathcal{M}}$,\nwhich represents the most probable sequence of agent decisions under the policy. \nThis problem can be solved by converting the MMDP to a directed weighted graph with edge weight $e(\\vb{s},\\vb{a},\\vb{s'}) = -\\log {\\mathcal{T}}(\\vb{s},\\vb{a},\\vb{s'})$ for each transition,\nand then applying the Dijkstra's algorithm~\\cite{dijkstra1959note} to find the shortest path. \n\nNext, the algorithm loops through every joint state $\\vb{s}_t$ in the path $\\rho$ to extract the agent cooperation and task sequence.\nAt each step $t$, the algorithm checks if an agent state $s^i_t$ satisfies any task completion predicate $f \\in {\\mathcal{F}}_c$\nand inserts completed task $f$ into the array element $y[i]$ (line 4-9).\nAn agent only satisfies a task completion predicate at step $t$ when it finishes the task and receives a reward.\nWe assume that if a task is completed via the cooperation of multiple agents, \nthey must satisfy the task predicate $f$ at the same step $t$ and each receive a portion of the reward.\nThus, the agent cooperation is represented as multiple elements of the array $y$ sharing the same task. \nOnly non-empty arrays containing completed tasks are inserted into the summarization ${\\mathcal{Z}}$.\nWhen the algorithm terminates, the generated summarization is visualized as a chart,\nwith each column corresponding to a non-empty $y$-array and each row representing an agent's task sequence.\n\n\\startpara{Properties}\nThe generated policy summarization ${\\mathcal{Z}}$ is \\emph{sound}, because it is derived from the most probable path of a \\emph{sound} policy abstraction (see \\sectref{sec:mmdp}). \nThe complexity of computing the most probable path is bounded by ${\\mathcal{O}}(|{\\mathcal{S}}|^2)$, following the complexity of the Dijkstra's algorithm and depending on the MMDP state space size. \nThe rest of \\agref{ag:sum} is bounded by ${\\mathcal{O}}(|\\rho| \\cdot N \\cdot |{\\mathcal{F}}_c|)$, depending on the path length and the number of agents and tasks. \n\n\n\\begin{example} \\label{eg:sum}\nWe apply \\agref{ag:sum} using the policy abstraction and task predicates from \\egref{eg:mmdp}.\nThere are 8 states in the most probable path from the initial state (i.e., all agents starting in the green grid) to a goal state (i.e., all tasks have been completed). \n\\figref{fig:rs_eg}(b) visualizes the generated summarization, with column names (i.e., T1, T2, T3) indicating the sequence of task completions:\nUGV$_2$ and UAV cooperate to rescue the victim; next, UGV$_1$ and UGV$_2$ cooperate to remove the obstacle; and lastly, UAV fights the fire. \n\\end{example}\n\n\n\\subsection{Explanations for When Query}\\label{sec:when}\n\n\\agref{ag:when} presents both the baseline and proposed methods for answering ``When do agents $G_q$ do actions $A_q$?'', where $G_q$ and $A_q$ are sets of agents and actions, respectively. The text in blue highlights changes about relevancy filters (RF) for the proposed method (called WithRF) compared to the baseline (called NoRF).\n\nWithRF starts the algorithm (line 1-5) by identifying relevant agents $G$, features $F$, and action sets $A$ based on domain knowledge (e.g., agent cooperation requirements). \nFor example, consider a query ``When does UAV rescue the victim?''. \nThe domain knowledge is that rescuing the victim requires the cooperation of a UAV and a UGV (\\egref{eg:mmdp}). \nThus, the relevant agent set $G$ is \\{$\\mathsf{UVA, UGV\\_1, UGV\\_2}$\\}.\nThe relevant feature set $F$ is \\{$\\mathsf{victim\\_detect, victim\\_complete}$\\}, while predicates about the fire and obstacle are irrelevant. \nThe relevant action sets $A$ is an array with each element representing one possible set of agent actions required for cooperation: \n$[\\{\\mathsf{UAV\\_rescue}, \\mathsf{UGV_1\\_rescue}\\}, \\{\\mathsf{UAV\\_rescue}, \\mathsf{UGV_2\\_rescue}\\}]$,\nwhich can be generated based on the aforementioned domain knowledge about agent cooperation requirements. \n\nBoth NoRF and WithRF loop through all the joint states $\\vb{s} \\in {\\mathcal{S}}$ of the policy abstraction MMDP and \ncheck all the \\emph{enabled} (i.e., with non-zero transition probability) joint actions $\\vb{a}$ in state $\\vb{s}$.\nIn line 9 of \\agref{ag:when}, NoRF checks if $\\vb{a}$ is \\emph{compatible} with $A_q$; \nthat is, every agent action $a \\in A_q$ is contained in the joint action $\\vb{a}=(a^1, \\dots, a^N)$.\nBy contrast, WithRF checks if $\\vb{a}$ is compatible with at least one set of relevant actions contained in the array $A$. \nFollowing the previous example, NoRF checks if $\\vb{a}$ contains $\\mathsf{UAV\\_rescue}$,\nwhile WithRF checks if $\\vb{a}$ contains $\\{\\mathsf{UAV\\_rescue}, \\mathsf{UGV_1\\_rescue}\\}$ \\emph{or} $\\{\\mathsf{UAV\\_rescue}, \\mathsf{UGV_2\\_rescue}\\}$. \nSince each element of $A$ is a super-set of $A_q$, the WithRF check is more restrictive.\n\n\\input{4b_ag_when}\n\nIf a state $\\vb{s}$ has at least one enabled action $\\vb{a}$ passing the aforementioned checks, $\\vb{s}$ is inserted to the target states set $V$; and to the non-target states set $\\bar{V}$ otherwise.\nThe intuition is that the generated explanations should describe target states satisfying the query criteria and exclude conditions of non-target states.\nWithRF poses further restrictions that target states need to satisfy criteria captured by relevant actions $A$, such as agent cooperation requirements.\nThus, explanations generated by WithRF can provide information about agent cooperation, which may be missed by NoRF explanations. \n\nNext, the algorithm converts the states set $V$ (\\textit{resp. } $\\bar{V}$) to a list of Boolean formulas $B_1$ (\\textit{resp. } $B_0$) via the function described in line 17-24.\nGiven a joint state $\\vb{s}=(s^1, \\dots, s^N)$, \nNoRF finds valuations of every feature predicates $f \\in {\\mathcal{F}}$ for all agent state $s^i$ and insert them to the list $B$.\nBy contrast, WithRF only inserts to $B$ the valuations of relevant features $f \\in F$ in relevant agent states $s^i$ for all $i \\in G$.\nFollowing the previous example, WithRF only considers Boolean formulas related to relevant feature predicates \n\\{$\\mathsf{victim\\_detect, victim\\_complete}$\\}, filtering out features related to the fire and obstacle.\n\nLastly, the algorithm supplies Boolean formulas $B_1$ and $B_0$ to the Quine-McCluskey algorithm~\\cite{quine1952problem} and obtains a minimized Boolean formula, which can be translated into language explanations following~\\cite{hayes2017improving}.\nThe runtime of Quine-McCluskey grows exponentially with the number of variables. \nThus, WithRF is more efficient than NoRF, due to the decreased number of Boolean variables.\nMoreover, filtering out irrelevant agents and features helps WithRF to prevent redundant information in the generated explanations. \n\n\n\n\\startpara{Properties}\nFollowing the Quine-McCluskey, the complexity of NoRF is bounded by ${\\mathcal{O}}\\big(3^{N \\cdot |{\\mathcal{F}}|} \/ \\ln( N \\cdot |{\\mathcal{F}}|)\\big)$.\nThe complexity of WithRF is reduced to \n${\\mathcal{O}}\\big(3^{|G|\\cdot |F|} \/ \\ln(|G|\\cdot |F|)\\big)$.\n\n\n\\input{4a_table}\n\n\\begin{example} \\label{eg:when}\n\\tabref{tab:qe} (first row) shows the explanations generated by NoRF and WithRF for a when query. \nThe NoRF explanation contains redundant information about the fire and obstacle that are irrelevant to the query. \nThe WithRF explanation completely captures the required agent cooperation for the query, which is missed by the NoRF explanation. \n\\end{example}\n\n\n\\subsection{Explanations for Why Not Query }\\label{sec:why}\n\nThe query ``Why don't agents $G_q$ do actions $A_q$ in the joint State $\\vb{s}_q$?''\ncan be answered by modifying \\agref{ag:when} as follows. \nIn line 10, adding $\\vb{s}$ to $\\bar{V}$ instead of $V$. \nRemove line 11-12 and add a new line for inserting the query state $\\vb{s}_q$ to $V$.\nThe modified algorithm \n\\ifCR\n(see Appendix A in~\\cite{boggess2022marl}) \n\\else\n(see \\appref{app:ag}) \n\\fi\ngenerates an explanation describing the differences between the observed behavior in the target query state $\\vb{s}_q$ and the expected behavior of states with actions compatible with the query actions $A_q$ (NoRF) or relevant action sets $A$ (WithRF). \nThe complexity of the modified algorithm follows \\agref{ag:when}.\n\n\\begin{example} \\label{eg:why}\n\\tabref{tab:qe} (second row) shows the explanations generated by NoRF and WithRF for a why not query about the behavior of two agents UGV$_1$ and UGV$_2$ in the state shown in \\figref{fig:rs_eg}(a).\nThe WithRF explanation captures the required agent cooperation for removing the obstacle, while the NoRF explanation fails to provide such information. \n\\end{example}\n\n\\subsection{Explanations for What Query }\\label{sec:what}\n\nTo answer the query ``What do agents $G_q$ do when satisfying predicates $F_q$?'', \nwe first identify all the satisfying joint states $\\vb{s}=(s^1, \\dots, s^N)$;\nthat is, for all $i \\in G_q$, agent state $s^i$ satisfies predicates $F_q$.\nThe baseline NoRF method is to generate a list of all possible enabled actions for agents $G_q$ in these states. \nThe proposed WithRF method improves the baseline by filtering agent actions that are relevant to predicates $F_q$ and finding the most likely relevant agent actions from the list via frequency counting. \nSince the proposed methods\\footnote{See \n\\ifCR\nAppendix A in~\\cite{boggess2022marl}\n\\else\n\\appref{app:ag} \n\\fi\nfor the pseudos code. } \ndo not need to call the Quine-McCluskey, the complexity of both NoRF and WithRF are only bounded by ${\\mathcal{O}}(|G_q| \\cdot |{\\mathcal{S}}| \\cdot |{\\mathcal{A}}|)$, depending on the number of query agents, joint state space, and joint action space of the policy abstraction. \n\n\n\\begin{example} \\label{eg:what}\n\\tabref{tab:qe} (third row) shows the explanations generated by NoRF and WithRF for a what query. \nThe WithRF explanation is more concise than the NoRF explanation and only contains relevant action for the query predicate. \n\\end{example}\n\n\n\n\n\n\n\n\n\n\n\\subsection{Study Design} \\label{sec:study_design}\n\n\\startpara{Participants} \nWe recruited 116 eligible participants (i.e., fluent English speakers over the age of 18) through university mailing lists. \n62.1\\% of participants self-identified as male, 37.1\\% as female, and 0.8\\% preferred not to say. \nThe age distribution is 76(18-24), 31(25-34), 7(35-49), 2(50-64).\nParticipants were instructed to answer multiple-choice questions about agent behavior for multi-robot search and rescue tasks.\nThey were incentivized with bonus payments to answer questions correctly based on the provided explanations. \nTo ensure data quality, attention checks were injected during the study.\n\n\n\\startpara{Independent variables}\nWe employed a within-subject study design with the explanation generation methods as independent variables. \nParticipants were asked to complete two trials for evaluating policy summarizations.\nThey were presented with charts generated by \\agref{ag:sum} in one trial, \nand GIF animations illustrating the most probable sequence of agent behavior (i.e., visualization of the most probable path in the policy abstraction) in the other trial. \nFor each trial, there were two questions about agent behavior in various environments (i.e., 3$\\times$6 and 6$\\times$6 grid world). Questions used in the two trials are different but had similar difficulty. \nAll participants were presented with the same set of four randomly generated questions for summarization trials. \nTo counterbalance the ordering confound effect, they were randomly assigned to answer the first two questions based on either charts or GIF, and the other two questions based on the remaining method. \nAdditionally, participants were asked to complete two trials for evaluating query-based explanations generated by NoRF and WithRF methods, with 6 questions (2 environments $\\times$ 3 query types) in each trial. \nParticipants answered the same set of 12 randomly generated questions for query-based trials, and were randomly assigned to different groups similarly to summarization trials. \n\n\n\\startpara{Dependent measures} \nWe measured \\emph{user performance} by counting the number of correctly answered questions in each trial. \nIn addition, at the end of each trial, participants were asked to rate in a 5-point Likert scale (1 - strongly disagree, 5 - strongly agree) about \\emph{explanation goodness} metrics (i.e., understanding, satisfaction, detail, completeness, actionability, reliability, trust)~\\cite{hoffman2018metrics}.\n\n\n\n\\startpara{Hypotheses}\nWe make the following hypotheses in this study.\n\\begin{itemize}\n \\item \\textbf{H1:} Chart-based summarizations lead to better user performance than GIF-based.\n \\item \\textbf{H2:} Chart-based summarizations yield higher user ratings on explanation goodness metrics than GIF-based.\n \\item \\textbf{H3:} Query-based explanations generated by WithRF lead to better user performance than those by NoRF. \n \\item \\textbf{H4:} Query-based explanations generated by WithRF yield higher user ratings on explanation goodness metrics than those by NoRF. \n\\end{itemize}\n\n\\subsection{Results Analysis} \\label{sec:study_results}\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=.85\\columnwidth]{figures\/SummaryGoodness.png}\n \\caption{Mean and SD of participant ratings about policy summarizations (``*'' indicates statistically significant difference).}\n \n \\label{fig:sumGood}\n\\end{figure}\n\nWe used a paired t-test to evaluate hypotheses H1 and H3, and used the Wilcoxon Signed-rank test to evaluate hypotheses H2 and H4. We set the significant level as $\\alpha=0.05$.\n\n\\startpara{Evaluating policy summarizations}\nParticipants answered more questions correctly with chart-based summarizations (M=1.8 out of 2, SD=0.6) than GIF-based (M=0.9 out of 2, SD=0.4), \nwith statistical significance ($t$(462)=-15.8, $p\\le$0.01, $d$=1.5).\n\\emph{Thus, the data supports H1.}\n\n\n\n\\figref{fig:sumGood} shows average participant ratings about summarizations.\nChart-based summarizations yield higher ratings on the perceived completeness than GIF-based with statistical significance ($W$=371.5, $Z$=-2.4, $p \\leq$0.02, $r$=-0.2).\nBut no significant difference was found regarding the other metrics.\n\\emph{Thus, the data partially supports H2.}\n\n\n\n\\startpara{Evaluating query-based explanations}\nParticipants answered more questions correctly with explanations generated by WithRF (M=5.2 out of 6, SD=1.7) than NoRF (M=2.3 out of 6, SD=1.0), with statistical significance ($t$(1390)=-21.1, $p\\le$0.01,$d$=2.0).\n\\emph{Thus, the data supports H3.}\n\n\n\\figref{fig:queryGood} shows that participants gave higher average ratings to WithRF explanations than NoRF explanations. \nThe Wilcoxon test found significant differences on all metrics: understanding ($W$=319.5, $Z$=-4.9, $p \\leq$0.01, $r$=-0.3), satisfaction ($W$=266.0, $Z$=-7.0, $p \\leq$0.01, $r$=-0.5), detail ($W$=484.0, $Z$=-3.7, $p \\leq$0.01, $r$=-0.2), completeness ($W$=494.5, $Z$=-6.4, $p \\leq$0.01, $r$=-0.4), actionability ($W$=167.0, $Z$=-6.9, $p \\leq$0.01, $r$=-0.5), reliability ($W$=382.5, $Z$=-3.6, $p \\leq$0.01, $r$=-0.2), and trust ($W$=217.0, $Z$=-3.4, $p \\leq$0.01, $r$=-0.2). \n\\emph{Thus, the data supports H4.}\n\n\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=.85\\columnwidth]{figures\/QueryGoodness.png}\n \\caption{Mean and SD of participant ratings about query-based explanations (``*'' indicates statistically significant difference).}\n \n \\label{fig:queryGood}\n\\end{figure}\n\n\n\n\n\\startpara{Discussion}\nIn summary, the data supports all hypotheses, while H2 is only partially supported because the statistical test found no significant differences between chart-based and GIF-based summarizations on most metrics.\nHowever, \\figref{fig:sumGood} shows that participants rated chart-based summarizations close to 4 (agree) on all metrics, and above GIF-based ratings on all metrics except understanding, reliability, and trust.\nThis may be because users showed a strong preference toward the moving nature of GIF animations and the visualized effects of agents completing tasks.\nBut watching a GIF can be more time-consuming and less informative than a quick glance at the chart. \nThis is supported by the results that participants were able to answer more questions correctly with chart-based summarizations, and they rated this method significantly higher on completeness (i.e., providing needed information). \nMeanwhile, query-based explanations generated by the proposed WithRF method led to significantly better user performance and higher user ratings on all metrics,\nbecause users prefer succinct WithRF explanations with adequate information about agent behavior and cooperation.\nBy contrast, NoRF explanations do not necessarily provide essential information about agent cooperation for correctly answering questions, and may contain redundant information that decreases user satisfaction. \n\n\n\n\n\n\n\\section{Algorithms} \\label{app:ag}\nThe text in blue highlight changes about relevancy filters (RF) \nfor the proposed WithRF method compared to the baseline NoRF method.\n\n\\input{4c_ag_whynot}\n\\input{4d_ag_what}\n\n\\newpage\n\\section{User Study Details} \\label{app:study}\n\n\\startpara{Questionnaire on explanation goodness metrics}\nParticipants were instructed to rate on a 5-point Likert scale (1 - strongly disagree, 5 - strongly agree) about the following statements, which were adapted from~\\cite{hoffman2018metrics}.\n\\begin{itemize}\n \\item The explanations help me \\emph{understand} how the team of robots completes the search and rescue mission.\n \\item The explanations are \\emph{satisfying}.\n \\item The explanations are sufficiently \\emph{detailed}.\n \\item The explanations are sufficiently \\emph{complete}, that is, they provide me with all the needed information to answer the questions.\n \\item The explanations are \\emph{actionable}, that is, they help me know how to answer the questions.\n \\item The explanations let me know how \\emph{reliable} the robot team is for completing the mission.\n \\item The explanations let me know how \\emph{trustworthy} the robot team is for completing the mission.\n\\end{itemize}\n\n\\startpara{User interface and sample questions}\nFigures~\\ref{fig:sample_query_when}-\\ref{fig:sample_summary_gif} show examples of user interface and questions presented to participants during the user study.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=1\\columnwidth]{figures\/sample_query_when.png}\n \\caption{Question based on explanations for a ``when'' query.}\n \\label{fig:sample_query_when}\n\\end{figure}\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=1\\columnwidth]{figures\/sample_query_whynot.png}\n \\caption{Question based on explanations for a ``why not'' query.}\n \\label{fig:sample_query_whynot}\n\\end{figure}\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=1\\columnwidth]{figures\/sample_query_what.png}\n \\caption{Question based on explanations for a ``what'' query.}\n \\label{fig:sample_query_what}\n\\end{figure}\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=1\\columnwidth]{figures\/sample_summary_sch.png}\n \\caption{Question based on policy summarization (sequence chart).}\n \\label{fig:sample_summary_schedule}\n\\end{figure}\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=1\\columnwidth]{figures\/sample_summary_gif.png}\n \\caption{Question based on a policy summarization (GIF animation: \\href{http:\/\/shorturl.at\/uEZ06}{http:\/\/shorturl.at\/uEZ06}).}\n \\label{fig:sample_summary_gif}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction} \\label{sec:intro}\n\\input{1_intro}\n\n\\section{Related Work} \\label{sec:related}\n\\input{2_related}\n\n\\section{Policy Abstraction and Summarization} \\label{sec:policy}\n\\input{3_sum}\n\n\\section{Query-Based Explanations} \\label{sec:query}\n\\input{4_query}\n\n\\section{Computational Experiments} \\label{sec:eval}\n\\input{5_exp}\n\n\\section{User Study} \\label{sec:exp}\n\\input{6_study}\n\n\\section{Conclusion} \\label{sec:conclu}\n\\input{7_conclu}\n\n\\clearpage\n\\section*{Acknowledgments}\nThis work was supported in part by U.S. National Science Foundation grant CCF-1942836, U.S. Office of Naval Research grant N00014-18-1-2829, Israel Science Foundation grant 1958\/20, EU Project TAILOR grant 992215, and by the Data Science Institute at Bar-Ilan\nUniversity.\nAny opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the grant sponsors.\n\n\\bibliographystyle{named}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nLyapunov functionals are widely used in stability analysis of differential and\ndifference equations. However, the extension of utilization of Lyapunov\nfunctionals in dynamical systems on time scales has been lacking behind due to\nthe constrained presented by the particular time scale. For example, in delay\ndifferential equations, a suitable Lyapunov functional will involve a term\nwith double integrals, in which one of the integral's lower limit is of the\nform $t+s$. Such a requirement will restrict the time scale that can be considered.\n\nFor a few references on the study of stability in differential equations,\nusing Lyapunov functionals, we refer the interested reader to \\cite{mayr},\n\\cite{raffoul&adivar}, \\cite{burton 1}-\\cite{yoshizawa}. The reader may\nconsult Yoshizawa \\cite[pp. 183-213]{yoshizawa} (or any book on functional\ndifferential equations and Lyapunov's direct method) for definitions of\nstability and for properties of Lyapunov functionals. For the stability\nanalysis of the delay differential equatio\n\\begin{equation}\nx^{\\prime}(t)=a(t)x(t)+b(t)x(t-h),\\ \\ h>0\\label{delay differential\n\\end{equation}\nwe refer to \\cite{busenberg}-\\cite{hatvani}, and \\cite{wang}. In\n\\cite{adraf2}, the authors improved the results of \\cite{wang} by considering\nthe delay differential equation of the for\n\\begin{equation}\nx^{\\prime}(t)=a(t)x(t)+b(t)x(t-h(t)),\\ \\ 0t\\right\\} $. Hereafter, we denote by $\\mu(t)$ the step size\nfunction $\\mu:\\mathbb{T}\\rightarrow\\mathbb{R}$ defined by $\\mu(t):=\\sigma\n(t)-t$. A point $t\\in\\mathbb{T}$ is said to be right dense (right scattered)\nif $\\mu(t)=0$ ($\\mu(t)>0$). A point is said to be left dense if $\\sup\\left\\{\ns\\in\\mathbb{T}:s0\\mbox{\nfor all }t\\in\\mathbb{T}\\}$.\n\\end{definition}\n\nLet $\\varphi\\in\\mathcal{R}$ and $\\mu(t)>0$ for all $t\\in\\mathbb{T}$. The\n\\emph{exponential function} on $\\mathbb{T}$ is defined by\n\\begin{equation}\ne_{\\varphi}(t,s)=\\exp\\left( \\int_{s}^{t}\\!\\zeta_{\\mu(r)}(\\varphi(r))\\Delta\nr\\right) \\label{exp\n\\end{equation}\nwhere $\\zeta_{\\mu(s)}$ is the cylinder transformation given by\n\\begin{equation}\n\\zeta_{\\mu(r)}(\\varphi(r))\\!:=\\left\\{\n\\begin{array}\n[c]{cc\n\\frac{1}{\\mu(r)}\\mbox{Log}(1+\\mu(r)\\varphi(r)) & if\\text{ }\\mu(r)>0\\\\\n\\varphi(r) & if\\ \\ \\mu(r)=0\n\\end{array}\n\\right. \\,.\\label{cylinder\n\\end{equation}\nIt is well known that if $p\\in\\mathcal{R}^{+}$, then $e_{p}(t,s)>0$ for all\n$t\\in\\mathbb{T}$. Also, the exponential function $y(t)=e_{p}(t,s)$ is the\nsolution to the initial value problem $y^{\\Delta}=p(t)y,\\,y(s)=1$. Other\nproperties of the exponential function are given in the following lemma:\n\n\\begin{lemma}\n\\label{lemma2.3} \\cite[Theorem 2.36]{book} Let $p,q\\in\\mathcal{R}$. Then\n\n\\begin{itemize}\n\\item[i.] $e_{0}(t,s)\\equiv1$ and $e_{p}(t,t)\\equiv1$;\n\n\\item[ii.] $e_{p}(\\sigma(t),s)=(1+\\mu(t)p(t))e_{p}(t,s)$;\n\n\\item[iii.] $\\frac{1}{e_{p}(t,s)}=e_{\\ominus p}(t,s)$ where, $\\ominus\np(t)=-\\frac{p(t)}{1+\\mu(t)p(t)}$;\n\n\\item[iv.] $e_{p}(t,s)=\\frac{1}{e_{p}(s,t)}=e_{\\ominus p}(s,t)$;\n\n\\item[v.] $e_{p}(t,s)e_{p}(s,r)=e_{p}(t,r)$;\n\n\\item[vi.] $\\left( \\frac{1}{e_{p}(\\cdot,s)}\\right) ^{\\Delta}=-\\frac\n{p(t)}{e_{p}^{\\sigma}(\\cdot,s)}$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{theorem}\n\\cite[Theorem 1.117]{book} \\label{theorem 1.117} Let $a\\in\\mathbb{T}^{\\kappa\n$, $b\\in\\mathbb{T}$ and assume that $k:\\mathbb{T}\\times\\mathbb{T}^{\\kappa\n}\\rightarrow\\mathbb{R}$ is continuous at $\\left( t,t\\right) $, where\n$t\\in\\mathbb{T}^{\\kappa}$ with $t>a$. Also assume that $k^{\\Delta}\\left(\nt,.\\right) $ is rd-continuous on $\\left[ a,\\sigma\\left( t\\right) \\right]\n$. Suppose that for each $\\varepsilon>0$ there exists a neighborhood $U$ of\n$t$, independent of $\\tau\\in\\left[ t_{0},\\sigma\\left( t\\right) \\right] $,\nsuch that\n\\[\n\\left\\vert k\\left( \\sigma\\left( t\\right) ,\\tau\\right) -k\\left(\ns,r\\right) -k^{\\Delta}\\left( t,\\tau\\right) \\left( \\sigma\\left( t\\right)\n-s\\right) \\right\\vert \\leq\\varepsilon\\left\\vert \\sigma\\left( t\\right)\n-s\\right\\vert\n\\]\nfor all $s\\in U$, where $k^{\\Delta}$ denotes the derivative of $k$ with\nrespect to the first variable. Then\n\\begin{align*}\ng\\left( t\\right) & :=\\int_{a}^{t}k\\left( t,\\tau\\right) \\Delta\\tau\\text{\nimplies }g^{\\Delta}\\left( t\\right) =\\int_{a}^{t}k^{\\Delta}\\left(\nt,\\tau\\right) \\Delta\\tau+k\\left( \\sigma\\left( t\\right) ,t\\right) \\\\\nh\\left( t\\right) & :=\\int_{t}^{b}k\\left( t,\\tau\\right) \\Delta\\tau\\text{\nimplies }g^{\\Delta}\\left( t\\right) =\\int_{t}^{b}k^{\\Delta}\\left(\nt,\\tau\\right) \\Delta\\tau-k\\left( \\sigma\\left( t\\right) ,t\\right) .\n\\end{align*}\n\n\\end{theorem}\n\n\\section{Shift operators}\n\nNext, we state the generalized shift operators. A limited version of it can be\nfound in \\cite{adivar}.\n\n\\begin{definition}\n\\label{shift} Let $\\mathbb{T}^{\\ast}$ be a non-empty subset of the time scale\n$\\mathbb{T}$ and $t_{0}\\in\\mathbb{T}^{\\ast}$ a fixed number such that there\nexist operators $\\delta_{\\pm}:[t_{0},\\infty)_{\\mathbb{T}}\\times\\mathbb{T\n^{\\ast}\\rightarrow\\mathbb{T}^{\\ast}$ satisfying the following properties:\n\n\\begin{enumerate}\n\\item[P.1] The functions $\\delta_{\\pm}$ are strictly increasing with respect\nto their second arguments, i.e., if\n\\[\n(T_{0},t),(T_{0},u)\\in\\mathcal{D}_{\\pm}:=\\left\\{ (s,t)\\in\\lbrack t_{0\n,\\infty)_{\\mathbb{T}}\\times\\mathbb{T}^{\\ast}:\\delta_{\\pm}(s,t)\\in\n\\mathbb{T}^{\\ast}\\right\\} ,\n\\]\nthen\n\\[\nT_{0}\\leq t\\delta_{-}(T_{2},u),\n\\]\nand if $(T_{1},u),(T_{2},u)\\in\\mathcal{D}_{+}$ with $T_{1}0$,\n\n\\item[ix.] $\\delta_{+}(\\delta_{-}(u,s),\\delta_{-}(s,v))=\\delta_{-}(u,v)$ for\nall $(s,v)\\in\\left( \\lbrack t_{0},\\infty)_{\\mathbb{T}}\\times\\lbrack\ns,\\infty)_{\\mathbb{T}}\\right) \\cap\\mathcal{D}_{-}$ and $(u,s)\\in\\left(\n\\lbrack t_{0},\\infty)_{\\mathbb{T}}\\times\\lbrack u,\\infty)_{\\mathbb{T}}\\right)\n\\cap\\mathcal{D}_{-}$,\n\n\\item[x.] If $(s,t)\\in\\mathcal{D}_{-}$ and $\\delta_{-}(s,t)=t_{0}$, then $s=t$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n(i) is obtained from P.3-5 sinc\n\\[\n\\delta_{-}(t,t)=\\delta_{-}(t,\\delta_{+}(t,t_{0}))=t_{0}\\text{ for all \nt\\in\\mathbb{T}^{\\ast}.\n\\]\n(ii) is obtained from P.3-P.4 sinc\n\\[\n\\delta_{-}(t_{0},t)=\\delta_{-}(t_{0},\\delta_{+}(t_{0},t))=t.\n\\]\nLet $u:=\\delta_{+}(s,t)$. By P.4 we have $(s,u)\\in\\mathcal{D}_{-}$ for all\n$(s,t)\\in\\mathcal{D}_{+}$, and hence\n\\[\n\\delta_{-}(s,u)=\\delta_{-}(s,\\delta_{+}(s,t))=t.\n\\]\nThe latter part of (iii) can be done similarly.. We have (iv) since P.3 and\nP.5 yiel\n\\[\n\\delta_{+}(t,\\delta_{-}(s,t_{0}))=\\delta_{-}(s,\\delta_{+}(t,t_{0}))=\\delta\n_{-}(s,t).\n\\]\nP.3 and P.5 guarantee tha\n\\[\nt=\\delta_{+}(t,t_{0})=\\delta_{+}(t,\\delta_{-}(u,u))=\\delta_{-}(u,\\delta\n_{+}(t,u))\n\\]\nfor all $(u,t)\\in\\left( \\lbrack t_{0},\\infty)_{\\mathbb{T}}\\times\\lbrack\nt_{0},\\infty)_{\\mathbb{T}}\\right) \\cap\\mathcal{D}_{+}$. Using (iii) we hav\n\\[\n\\delta_{+}(u,t)=\\delta_{+}(u,\\delta_{-}(u,\\delta_{+}(t,u)))=\\delta_{+}(t,u).\n\\]\nThis proves (v). To prove (vi) and (vii) we use P.1-2 to ge\n\\[\n\\delta_{+}(s,t)\\geq\\delta_{+}(t_{0},t)=t\\geq t_{0\n\\]\nfor all $(s,t)\\in$ $\\left( [t_{0},\\infty)\\times\\lbrack t_{0},\\infty\n)_{\\mathbb{T}}\\right) \\cap\\mathcal{D}_{+}$ an\n\\[\n\\delta_{-}(s,t)\\geq\\delta_{-}(s,s)=t_{0\n\\]\nfor all $(s,t)\\in\\left( \\lbrack t_{0},\\infty)_{\\mathbb{T}}\\times\\lbrack\ns,\\infty)_{\\mathbb{T}}\\right) \\cap\\mathcal{D}_{-}$. Since $\\delta_{+}(s,t)$\nis strictly increasing in its second variable we have (viii) by\n\\cite[Corollary 1.16]{book2}. (ix) is proven as follows: from P.5 and (v) we\nhav\n\\begin{align*}\n\\delta_{+}(\\delta_{-}(u,s),\\delta_{-}(s,v)) & =\\delta_{-}(s,\\delta\n_{+}(v,\\delta_{-}(u,s)))\\\\\n& =\\delta_{-}(s,\\delta_{-}(u,\\delta_{+}(v,s)))\\\\\n& =\\delta_{-}(s,\\delta_{+}(s,\\delta_{-}(u,v)))\\\\\n& =\\delta_{-}(u,v)\n\\end{align*}\nfor all $(s,v)\\in\\left( \\lbrack t_{0},\\infty)_{\\mathbb{T}}\\times\\lbrack\ns,\\infty)_{\\mathbb{T}}\\right) \\cap\\mathcal{D}_{-}$ and $(u,s)\\in\\left(\n\\lbrack t_{0},\\infty)_{\\mathbb{T}}\\times\\lbrack u,\\infty)_{\\mathbb{T}}\\right)\n\\cap\\mathcal{D}_{-}$. Suppose $(s,t)\\in\\mathcal{D}_{-}$ $=\\left\\{\n(s,t)\\in\\lbrack t_{0},\\infty)_{\\mathbb{T}}\\times\\mathbb{T}^{\\ast}:\\delta\n_{-}(s,t)\\in\\mathbb{T}^{\\ast}\\right\\} $ and $\\delta_{-}(s,t)=t_{0}$. Then by\nP.4 we hav\n\\[\nt=\\delta_{+}(s,\\delta_{-}(s,t))\\in\\delta_{+}(s,t_{0})=s.\n\\]\nThis is (x). The proof is complete.\n\\end{proof}\n\nNotice that the shift operators $\\delta_{\\pm}$ are defined once the initial\npoint $t_{0}\\in\\mathbb{T}^{\\ast}$ is known. For instance, we choose the\ninitial point $t_{0}=0$ to define shift operators $\\delta_{\\pm}(s,t)=t\\pm s$\non $\\mathbb{T}=\\mathbb{R}$. However, if we choose $\\lambda\\in(0,\\infty)$ as\nthe initial point, then the new shift operators associated with $\\lambda$ are\ndefined by $\\widetilde{\\delta}_{\\pm}(s,t)=t\\mp\\lambda\\pm s$. In terms of\n$\\delta_{\\pm}$ the operators $\\widetilde{\\delta}_{\\pm}$ can be given a\n\\[\n\\widetilde{\\delta}_{\\pm}(s,t)=\\delta_{\\mp}(\\lambda,\\delta_{\\pm}(s,t)).\n\\]\n\n\n\\begin{example}\nIn the following, we give some particular time scales to show the change in\nthe formula of shift operators as the initial point changes\n\\\n\\begin{tabular}\n[c]{c||cc|cc|cc}\n& \\multicolumn{2}{||c|}{$\\mathbb{T}=\\mathbb{N}^{1\/2}$} &\n\\multicolumn{2}{|c|}{$\\mathbb{T}=h\\mathbb{Z}$} &\n\\multicolumn{2}{|c}{$\\mathbb{T}=2^{\\mathbb{N}}$}\\\\\\hline\\hline\n$t_{0}$ & $0$ & $\\lambda$ & $0$ & $h\\lambda$ & $1$ & $2^{\\lambda}$\\\\\n$\\delta_{-}(s,t)$ & $\\sqrt{t^{2}-s^{2}}$ & $\\sqrt{t^{2}+\\lambda^{2}-s^{2}}$ &\n$t-s$ & $t+h\\lambda-s$ & $t\/s$ & $2^{\\lambda}ts^{-1}$\\\\\n$\\delta_{+}(s,t)$ & $\\sqrt{t^{2}+s^{2}}$ & $\\sqrt{t^{2}-\\lambda^{2}+s^{2}}$ &\n$t+s$ & $t-h\\lambda+s$ & $ts$ & $2^{-\\lambda}ts\n\\end{tabular}\n\\]\nwhere $\\lambda\\in\\mathbb{Z}_{+}$, $\\mathbb{N}^{1\/2}:=\\{\\sqrt{n}:n\\in\n\\mathbb{N\\}}$, $2^{\\mathbb{N}}:=\\{2^{n}:n\\in\\mathbb{N\\}}$, and $h\\mathbb{Z\n:\\mathbb{=\\{}hn:$ $n\\in\\mathbb{Z\\}}$.\n\\end{example}\n\n\\section{Delay function}\n\nIn this section we introduce the delay function on time scales that will be\nused for the construction of the Lyapunov functional.\n\n\\begin{definition}\nLet $\\mathbb{T}$ be a time scale that is unbounded above\\ and $t_{0\n\\in\\mathbb{T}^{\\ast}$ an element such that there exist the shift operators\n$\\delta_{\\pm}:[t_{0},\\infty)\\times\\mathbb{T}^{\\ast}\\rightarrow\\mathbb{T\n^{\\ast}$ associated with $t_{0}$. Suppose that $h\\in(t_{0},\\infty\n)_{\\mathbb{T}}$ is a constant such that $(h,t)\\in D_{\\pm}$ for all\n$t\\in\\lbrack t_{0},\\infty)_{\\mathbb{T}}$, the function $\\delta_{-}(h,t)$ is\ndifferentiable with an $rd$-continuous derivative, and $\\delta_{-}(h,t)$ maps\n$[t_{0},\\infty)_{\\mathbb{T}}$ onto $[\\delta_{-}(h,t_{0}),\\infty)_{\\mathbb{T}\n$. Then the function $\\delta_{-}(h,t)$ is called the delay function generated\nby the shift $\\delta_{-}$ on the time scale $\\mathbb{T}$.\n\\end{definition}\n\nIt is obvious from P.2 and (iii) of Lemma \\ref{lem pro} tha\n\\begin{equation}\n\\delta_{-}(h,t)<\\delta_{-}(t_{0},t)=t\\text{ for all }t\\in\\lbrack t_{0\n,\\infty)_{\\mathbb{T}}\\text{.} \\label{delay less\n\\end{equation}\nNotice that $\\delta_{-}(h,.)$ is strictly increasing and it is invertible.\nHence, by P.4-5 $\\delta_{-}^{-1}(h,t)=\\delta_{+}(h,t)$.\n\nHereafter, we shall suppose that $\\mathbb{T}$ is a time scale with the delay\nfunction $\\delta_{-}(h,.):[t_{0},\\infty)_{\\mathbb{T}}\\rightarrow\\lbrack\n\\delta_{-}(h,t_{0}),\\infty)_{\\mathbb{T}}$, where $t_{0}\\in\\mathbb{T}$ is\nfixed. Denote by $\\mathbb{T}_{1}$ and $\\mathbb{T}_{2}$ the set\n\\begin{equation}\n\\mathbb{T}_{1}=[t_{0},\\infty)_{\\mathbb{T}}\\text{\\ \\ and }\\mathbb{T}_{2\n=\\delta_{-}(h,\\mathbb{T}_{1}).\\label{T12\n\\end{equation}\nEvidently, $\\mathbb{T}_{1}$ is closed in $\\mathbb{R}$. By definition we have\n$\\mathbb{T}_{2}=[\\delta_{-}(h,t_{0}),\\infty)_{\\mathbb{T}}$. Hence,\n$\\mathbb{T}_{1}$ and $\\mathbb{T}_{2}$ are both time scales. Let $\\sigma_{1}$\nand $\\sigma_{2}$ denote the forward jump operators on the time scales\n$\\mathbb{T}_{1}$ and $\\mathbb{T}_{2}$, respectively. By (\\ref{delay less\n-\\ref{T12})\n\\[\n\\mathbb{T}_{1}\\subset\\mathbb{T}_{2}\\subset\\mathbb{T}.\n\\]\nThus\n\\[\n\\sigma(t)=\\sigma_{2}(t)\\text{ for all }t\\in\\mathbb{T}_{2\n\\]\nan\n\\[\n\\sigma(t)=\\sigma_{1}(t)=\\sigma_{2}(t)\\text{ for all }t\\in\\mathbb{T}_{1}.\n\\]\nThat is, $\\sigma_{1}$ and $\\sigma_{2}$ are the restrictions of the forward\njump operator $\\sigma:\\mathbb{T\\rightarrow T}$ to the time scales\n$\\mathbb{T}_{1}$ and $\\mathbb{T}_{2}$, respectively, i.e.\n\\[\n\\sigma_{1}=\\left. \\sigma\\right\\vert _{\\mathbb{T}_{1}}\\text{ and }\\sigma\n_{2}=\\left. \\sigma\\right\\vert _{\\mathbb{T}_{2}}\\text{.\n\\]\nRecall that the Hilger derivatives $\\Delta$, $\\Delta_{1}$, and $\\Delta_{2}$ on\nthe time scales $\\mathbb{T}$, $\\mathbb{T}_{1}$, and $\\mathbb{T}_{2}$ are\ndefined in terms of the forward jumps $\\sigma$, $\\sigma_{1}$, and $\\sigma_{2\n$, respectively. Hence, if $f$ is a differentiable function at $t\\in\n\\mathbb{T}_{2}$, then we hav\n\\[\nf^{\\Delta_{2}}(t)=f^{\\Delta_{1}}(t)=f^{\\Delta}(t),\\ \\ \\text{\\ for all \nt\\in\\mathbb{T}_{1}.\n\\]\nSimilarly, if $a,b\\in\\mathbb{T}_{2}$ are two points with $a\\widehat{t}\\text{ implies }\\sigma_{2}(\\delta\n_{-}(h,\\widehat{t})>\\delta_{-}(h,\\widehat{t}).\n\\]\n\n\\end{lemma}\n\n\\begin{proof}\nBy definition $\\sigma_{1}(t)\\geq t$ for all $t\\in\\mathbb{T}_{1}$. Thus\n\\[\n\\delta_{-}(h,\\sigma_{1}(t))\\geq\\delta_{-}(h,t).\n\\]\nSince $\\sigma_{2}(\\delta_{-}(h,t))$ is the smallest element satisfyin\n\\[\n\\sigma_{2}(\\delta_{-}(h,t))\\geq\\delta_{-}(h,t),\n\\]\nwe ge\n\\begin{equation}\n\\delta_{-}(h,\\sigma_{1}(t))\\geq\\sigma_{2}(\\delta_{-}(h,t))\\text{ for all \nt\\in\\mathbb{T}_{1}\\text{.} \\label{1\n\\end{equation}\nIf $\\sigma_{1}(\\widehat{t})=\\widehat{t}$, then we hav\n\\[\n\\delta_{-}(h,\\widehat{t})=\\delta_{-}(h,\\sigma_{1}(\\widehat{t}))\\geq\\sigma\n_{2}(\\delta_{-}(h,\\widehat{t})).\n\\]\nThat is\n\\[\n\\delta_{-}(h,\\widehat{t})=\\sigma_{2}(\\delta_{-}(h,\\widehat{t})).\n\\]\nIf $\\sigma_{1}(\\widehat{t})>\\widehat{t}$, the\n\\[\n(\\widehat{t},\\sigma_{1}(\\widehat{t}))_{\\mathbb{T}_{1}}=(\\widehat{t},\\sigma\n_{1}(\\widehat{t}))_{\\mathbb{T}}=\\varnothing\n\\]\nan\n\\[\n\\delta_{-}(h,\\sigma_{1}(\\widehat{t}))>\\delta_{-}(h,\\widehat{t}).\n\\]\nSuppose the contrary. That is $\\delta_{-}(h,\\widehat{t})$ is right dense;\nnamely $\\sigma_{2}(\\delta_{-}(h,\\widehat{t}))=\\delta_{-}(h,\\widehat{t})$. This\nalong with (\\ref{1}) implie\n\\[\n(\\delta_{-}(h,\\widehat{t}),\\delta_{-}(h,\\sigma_{1}(\\widehat{t})))_{\\mathbb{T\n_{2}}\\neq\\varnothing\\text{.\n\\]\nPick one element $s\\in(\\delta_{-}(h,\\widehat{t}),\\delta_{-}(h,\\sigma\n_{1}(\\widehat{t})))_{\\mathbb{T}_{2}}$. Since $\\delta_{-}(h,t)$ is strictly\nincreasing in $t$ and invertible, there should be an element $t\\in(\\widehat\n{t},\\sigma_{1}(\\widehat{t}))_{\\mathbb{T}_{1}}$ such that $\\delta_{-}(h,t)=s$.\nThis leads to a contradiction. Hence, $\\delta_{-}(h,\\widehat{t})$ must be\nright scattered.\n\\end{proof}\n\nUsing the preceding lemma and applying the fact that $\\sigma_{2}(u)=\\sigma(u)$\nfor all $u\\in\\mathbb{T}_{2}$ we arrive at the following result.\n\n\\begin{corollary}\n\\label{Cor 1} We hav\n\\[\n\\delta_{-}(h,\\sigma_{1}(t))=\\sigma_{2}(\\delta_{-}(h,t))\\text{ for all \nt\\in\\mathbb{T}_{1}\\text{.\n\\]\nThus\n\\begin{equation}\n\\delta_{-}(h,\\sigma(t))=\\sigma(\\delta_{-}(h,t))\\text{ for all }t\\in\n\\mathbb{T}_{1}\\text{.} \\label{sigma delta\n\\end{equation}\n\n\\end{corollary}\n\nBy (\\ref{sigma delta}) we hav\n\\[\n\\delta_{-}(h,\\sigma(s))=\\sigma(\\delta_{-}(h,s))\\text{ for all }s\\in\\lbrack\nt_{0},\\infty)_{\\mathbb{T}}\\text{.\n\\]\nSubstituting $s=\\delta_{+}(h,t)$ we obtai\n\\[\n\\delta_{-}(h,\\sigma(\\delta_{+}(h,t)))=\\sigma(\\delta_{-}(h,\\delta\n_{+}(h,t)))=\\sigma(t)\\text{.\n\\]\nThis and (iv) of Lemma \\ref{lem pro} impl\n\\[\n\\sigma(\\delta_{+}(h,t))=\\delta_{+}(h,\\sigma(t))\\text{ for all }t\\in\n\\lbrack\\delta_{-}(h,t_{0}),\\infty)_{\\mathbb{T}}\\text{.\n\\]\n\n\n\\begin{example}\nIn the following, we give some time scales with their shift operators:\n\\\n\\begin{tabular}\n[c]{|c||c|c|c|}\\hline\n$\\mathbb{T}$ & $h$ & $\\delta_{-}(h,t)$ & $\\delta_{+}(h,t)$\\\\\\hline\\hline\n$\\mathbb{R}$ & $\\in\\mathbb{R}_{+}$ & $t-h$ & $t+h$\\\\\\hline\n$\\mathbb{Z}$ & $\\in\\mathbb{Z}_{+}$ & $t-h$ & $t+h$\\\\\\hline\n$q^{\\mathbb{Z}}\\cup\\left\\{ 0\\right\\} $ & $\\in q^{\\mathbb{Z}_{+}}$ &\n$\\frac{t}{h}$ & $ht$\\\\\\hline\n$\\mathbb{N}^{1\/2}$ & $\\in\\mathbb{Z}_{+}$ & $\\sqrt{t^{2}-h^{2}}$ & $\\sqrt\n{t^{2}+h^{2}}$\\\\\\hline\n\\end{tabular}\n\\]\n\n\\end{example}\n\n\\begin{example}\nThere is no delay function $\\delta_{-}(h,.):[0,\\infty)_{\\widetilde{\\mathbb{T\n}}\\rightarrow\\lbrack\\delta_{-}(h,0),\\infty)_{\\mathbb{T}}$ on the time scale\n$\\widetilde{\\mathbb{T}}\\mathbb{=(-\\infty},0]\\cup\\lbrack1,\\infty)$.\\newline\nSuppose the contrary that there exists such a delay function on $\\widetilde\n{\\mathbb{T}}$. Then since $0$ is right scattered in $\\widetilde{\\mathbb{T\n}_{1}:=[0,\\infty)_{\\widetilde{\\mathbb{T}}}$ the point $\\delta_{-}(h,0)$ must\nbe right scattered in $\\widetilde{\\mathbb{T}}_{2}=[\\delta_{-}(h,0),\\infty\n)_{\\mathbb{T}}$, i.e., $\\sigma_{2}(\\delta_{-}(h,0))>\\delta_{-}(h,0)$. Since\n$\\sigma_{2}(t)=\\sigma(t)$ for all $t\\in\\lbrack\\delta_{-}(h,0),0)_{\\mathbb{T}\n$, we hav\n\\[\n\\sigma(\\delta_{-}(h,0))=\\sigma_{2}(\\delta_{-}(h,0))>\\delta_{-}(h,0).\n\\]\nThat is, $\\delta_{-}(h,0)$ must be right scattered in $\\widetilde{\\mathbb{T}\n$. However, in $\\widetilde{\\mathbb{T}}$ we have $\\delta_{-}(h,0)<0$, that is,\n$\\delta_{-}(h,0)$ is right dense. This leads to a contradiction.\n\\end{example}\n\n\\begin{theorem}\n(\\textbf{Substitution}) \\label{thm2.2} \\cite[Theorem 1.98]{book} Assume\n$\\nu:\\mathbb{T}\\rightarrow\\mathbb{R}$ is strictly increasing and\n$\\tilde{\\mathbb{T}}:=\\nu(\\mathbb{T})$ is a time scale. If $f:\\mathbb{T\n\\rightarrow\\mathbb{R}$ is an rd-continuous function and $\\nu$ is\ndifferentiable with rd-continuous derivative, then for $a,b\\in\\mathbb{T}$,\n\\begin{equation}\n\\int_{a}^{b}\\!g(t,s)\\nu^{\\Delta}(s)\\,\\Delta s=\\int_{\\nu(a)}^{\\nu(b)\ng(t,\\nu^{-1}(s))\\,\\tilde{\\Delta}s. \\label{substitute\n\\end{equation}\n\n\\end{theorem}\n\nFirst, since the operator $\\delta:[t_{0},\\infty)_{\\mathbb{T}}\\rightarrow$\n$[\\delta(t_{0}),\\infty)_{\\mathbb{T}}$ is strictly increasing, it is bijection.\nIf we substitute $\\nu(t)=\\delta_{-}(h,t)$ an\n\\[\nf(t,s)=g(t,\\delta_{-}^{-1}(h,s))=g(t,\\delta_{+}(h,s))\n\\]\ninto (\\ref{substitute}), we obtai\n\\begin{equation}\n\\int_{a}^{b}\\!f(t,\\delta_{-}(h,s))\\delta_{-}^{\\Delta_{1}}(h,s)\\,\\Delta\n_{1}s=\\int_{\\delta_{-}(h,a)}^{\\delta_{-}(h,b)}f(t,s)\\,\\Delta_{2}s \\label{5\n\\end{equation}\nfor $a,b\\in\\mathbb{T}_{1}$. For any $t\\in\\mathbb{T}_{1}$, we have $[\\delta\n_{-}(h,t_{0}),t)_{\\mathbb{T}_{1}}\\subset\\mathbb{T}_{2}$. This and (\\ref{5})\nyield\n\\begin{align}\n\\int_{\\delta_{-}(h,t)}^{t}f(t,s)\\Delta s & =\\int_{\\delta_{-}(h,t)\n^{t}f(t,s)\\Delta_{2}s\\nonumber\\\\\n& =\\int_{\\delta_{-}(h,t)}^{\\delta_{-}(h,t_{0})}f(t,s)\\Delta_{2}s+\\int\n_{\\delta_{-}(h,t_{0})}^{t}f(t,s)\\Delta_{2}s\\nonumber\\\\\n& =\\int_{t}^{t_{0}}f(t,\\delta_{-}(h,s))\\delta_{-}^{\\Delta_{1}}(h,s)\\,\\Delta\n_{1}s+\\int_{\\delta_{-}(h,t_{0})}^{t}f(t,s)\\Delta s\\nonumber\\\\\n& =\\int_{t}^{t_{0}}f(t,\\delta_{-}(h,s))\\delta_{-}^{\\Delta}(h,s)\\,\\Delta\ns+\\int_{\\delta_{-}(h,t_{0})}^{t}f(t,s)\\Delta s. \\label{3\n\\end{align}\nThe the formul\n\\begin{align}\n\\left[ \\int_{\\delta_{-}(h,t)}^{t}f(t,s)\\Delta s\\right] ^{\\Delta} &\n=f(\\sigma(t),t)-f(\\sigma(t),\\delta_{-}(h,t))\\delta_{-}^{\\Delta\n(h,t)\\nonumber\\\\\n& +\\int_{\\delta_{-}(h,t)}^{t}f^{\\Delta}(t,s)\\Delta s \\label{4\n\\end{align}\nfollows from (\\ref{3}) and Theorem \\ref{theorem 1.117}.\n\n\\begin{theorem}\n\\label{order of integration} Let $k$ be an $rd$-continuous function. The\n\\begin{equation}\n\\int_{\\delta_{-}(h,t)}^{t}\\Delta s\\int_{s}^{t}k(u)\\Delta u=\\int_{\\delta\n_{-}(h,t)}^{t}\\Delta u\\int_{\\delta_{-}(h,t)}^{\\sigma(u)}k(u)\\Delta s.\\label{8\n\\end{equation}\n\n\\end{theorem}\n\n\\begin{proof}\nSubstitutin\n\\[\nf(s)=s-\\delta_{-}(h,t),\\ \\ \\ \\ g(s)=\\int_{s}^{t}k(u)\\Delta u\n\\]\ninto the formul\n\\[\n\\int_{a}^{z}f(\\sigma(x))g(x)\\Delta x=\\left[ f(x)g(x)\\right] _{a}^{z\n-\\int_{a}^{z}f^{\\Delta}(x)g(x)\\Delta x\n\\]\n(see \\cite[Theorem 1.77]{book}) and using Lemma \\ref{lem pro} we ge\n\\begin{align}\n\\int_{\\delta_{-}(h,t)}^{t}\\Delta s\\int_{s}^{t}k(u)\\Delta u & =\\int\n_{\\delta_{-}(h,t)}^{t}\\left[ \\sigma(s)-\\delta_{-}(h,t)\\right] k(s)\\Delta\ns\\nonumber\\\\\n& =\\int_{\\delta_{-}(h,t)}^{t}\\Delta u\\int_{\\delta_{-}(h,t)}^{\\sigma\n(u)}k(u)\\Delta s.\\label{8a\n\\end{align}\n\n\\end{proof}\n\n\\section{Stability analysis using Lyapunov's method}\n\nLet $\\mathbb{T}$ be a time scale having a delay function $\\delta_{-}(h,t)$\nwhere $h\\geq t_{0}$ and $t_{0}\\in\\mathbb{T}$ is nonnegative and fixed. In this\nsection we consider the equatio\n\\begin{equation}\nx^{\\Delta}(t)=a(t)x(t)+b(t)x(\\delta_{-}(h,t))\\delta_{-}^{\\Delta\n(h,t),\\ \\ \\ t\\in\\lbrack t_{0},\\infty)_{\\mathbb{T}} \\label{9\n\\end{equation}\nand assume tha\n\\begin{equation}\n\\left\\vert \\delta_{-}^{\\Delta}(h,t)\\right\\vert \\leq M<\\infty\\ \\text{ for all\n}t\\in\\lbrack t_{0},\\infty)_{\\mathbb{T}}. \\label{delta turev\n\\end{equation}\nLet $\\psi$: $[\\delta_{-}(h,t_{0}),t_{0}]_{\\mathbb{T}}\\rightarrow\\mathbb{R}$ be\n$rd$-continuous and let $x(t):=x(t,t_{0},\\psi)$ be the solution of Eq.\n(\\ref{9}) on $[t_{0},\\infty)_{\\mathbb{T}}$ with $x(t)=\\psi(t)$ on $[\\delta\n_{-}(h,t_{0}),t_{0}]_{\\mathbb{T}}$. Let $\\left\\Vert \\varphi\\right\\Vert\n=\\sup\\left\\{ |\\varphi(t)|:t\\in\\lbrack\\delta_{-}(h,t_{0}),t_{0})_{\\mathbb{T\n}\\right\\} $.\n\nObserve that using (\\ref{4}) Eq. (\\ref{9}) can be rewritten as follow\n\\begin{equation}\nx^{\\Delta}(t)=Q(t)x(t)-\\left[ \\int_{\\delta_{-}(h,t)}^{t}b(\\delta\n_{+}(h,s))x(s)\\Delta s\\right] ^{\\Delta_{t}}, \\label{10\n\\end{equation}\nwhere\n\\[\nQ(t):=a(t)+b(\\delta_{+}(h,t))\n\\]\nand $\\Delta_{t}$ indicates the delta derivative with respect to $t$.\n\n\\begin{lemma}\n\\label{lem2} Le\n\\begin{equation}\nA(t):=x(t)\n{\\displaystyle\\int\\limits_{\\delta_{-}(h,t)}^{t}}\nb(\\delta_{+}(h,s))x(s)\\Delta s\\label{11a\n\\end{equation}\nand\n\\begin{equation}\n\\beta(t):=t-\\delta_{-}(h,t).\\label{11b\n\\end{equation}\nAssume that there exists a $\\lambda>0$ such tha\n\\begin{equation}\n-\\frac{\\lambda\\delta_{-}^{\\Delta}(h,t)}{\\beta(t)+\\lambda\\left[ \\beta\n(t)+\\mu(t)\\right] }\\leq Q(t)\\leq-\\lambda\\left[ \\beta(t)+\\mu(t)\\right]\nb(\\delta_{+}(h,t))^{2}-\\mu(t)Q^{2}(t)\\label{11\n\\end{equation}\nfor all $t\\in\\lbrack t_{0},\\infty)_{\\mathbb{T}}$. I\n\\begin{equation}\nV(t)=A(t)^{2}+\\lambd\n{\\displaystyle\\int\\limits_{\\delta_{-}(h,t)}^{t}}\n\\Delta \n{\\displaystyle\\int\\limits_{s}^{t}}\nb(\\delta_{+}(h,u))^{2}x(u)^{2}\\Delta u\\label{12\n\\end{equation}\nthen, along the solutions of Eq. (\\ref{9}) we hav\n\\begin{equation}\nV^{\\Delta}(t)\\leq Q(t)V(t)\\text{ for all }t\\in\\lbrack t_{0},\\infty\n)_{\\mathbb{T}}\\text{.}\\label{13\n\\end{equation}\n\n\\end{lemma}\n\n\\begin{proof}\nIt is obvious from (\\ref{10}) and (\\ref{11a}) tha\n\\[\nA^{\\Delta}(t)=Q(t)x(t).\n\\]\nThen by (\\ref{4}) and the formula $A(\\sigma(t))=A(t)+\\mu(t)A(t)$ we hav\n\\begin{align*}\nV^{\\Delta}(t) & =\\left[ A(t)+A(\\sigma(t))\\right] A^{\\Delta}(t)+\\lambda\n\\int_{t}^{\\sigma(t)}b(\\delta_{+}(h,u))^{2}x(u)^{2}\\Delta u\\\\\n& -\\lambda\\delta_{-}^{\\Delta}(h,t)\\int_{\\delta_{-}(h,t)}^{\\sigma(t)\nb(\\delta_{+}(h,u))^{2}x(u)^{2}\\Delta u+\\lambda\\left( t-\\delta_{-\n(h,t)\\right) b(\\delta_{+}(h,t))^{2}x(t)^{2}\\\\\n& =\\left[ 2A(t)+\\mu(t)Q(t)x(t)\\right] Q(t)x(t)-\\lambda\\delta_{-}^{\\Delta\n}(h,t)\\int_{\\delta_{-}(h,t)}^{\\sigma(t)}b(\\delta_{+}(h,u))^{2}x(u)^{2}\\Delta\nu\\\\\n& +\\lambda\\left[ \\beta(t)+\\mu(t)\\right] b(\\delta_{+}(h,t))^{2}x(t)^{2}.\n\\end{align*}\nUsing the identit\n\\begin{equation}\n2A(t)x(t)=x^{2}(t)+A^{2}(t)-\\left(\n{\\displaystyle\\int\\limits_{\\delta_{-}(h,t)}^{t}}\nb(\\delta_{+}(h,s))x(s)\\Delta s\\right) ^{2}\\label{A_Q\n\\end{equation}\nand condition (\\ref{11}) we hav\n\\begin{align}\nV^{\\Delta}(t) & =Q(t)V(t)+R(t)\\nonumber\\\\\n& +x^{2}(t)\\left[ \\lambda\\left( \\beta(t)+\\mu(t)\\right) b(\\delta\n_{+}(h,t))^{2}+Q(t)+\\mu(t)Q^{2}(t)\\right] \\nonumber\\\\\n& \\leq Q(t)V(t)+R(t),\\label{v(t)\n\\end{align}\nwher\n\\begin{align}\nR(t) & =-Q(t)\\left(\n{\\displaystyle\\int\\limits_{\\delta_{-}(h,t)}^{t}}\nb(\\delta_{+}(h,s))x(s)\\Delta s\\right) ^{2}\\nonumber\\\\\n& -\\lambda\\delta_{-}^{\\Delta}(h,t)\\int_{\\delta_{-}(h,t)}^{\\sigma(t)\nb(\\delta_{+}(h,u))^{2}x(u)^{2}\\Delta u\\nonumber\\\\\n& -\\lambda Q(t\n{\\displaystyle\\int\\limits_{\\delta_{-}(h,t)}^{t}}\n\\Delta \n{\\displaystyle\\int\\limits_{s}^{t}}\nb(\\delta_{+}(h,u))^{2}x(u)^{2}\\Delta u.\\label{r(t)\n\\end{align}\nHereafter, we will show that (\\ref{11}) implies $R(t)\\leq0$. This and\n(\\ref{v(t)}) will enable us to derive the desired inequality (\\ref{13}). First\nwe have\n\\begin{equation}\n\\int_{\\delta_{-}(h,t)}^{\\sigma(t)}b(\\delta_{+}(h,u))^{2}x(u)^{2}\\Delta\nu\\geq\\int_{\\delta_{-}(h,t)}^{t}b(\\delta_{+}(h,u))^{2}x(u)^{2}\\Delta\nu.\\label{r1\n\\end{equation}\nFrom H\\\"{o}lder's inequality \\cite[Theorem 6.13]{book} we ge\n\\begin{equation}\n\\left(\n{\\displaystyle\\int\\limits_{\\delta_{-}(h,t)}^{t}}\nb(\\delta_{+}(h,s))x(s)\\Delta s\\right) ^{2}\\leq\\beta(t\n{\\displaystyle\\int\\limits_{\\delta_{-}(h,t)}^{t}}\nb(\\delta_{+}(h,s))^{2}x(s)^{2}\\Delta s.\\label{holder\n\\end{equation}\nOn the other hand, (\\ref{8}) yields\n\\begin{align\n{\\displaystyle\\int\\limits_{\\delta_{-}(h,t)}^{t}}\n\\Delta \n{\\displaystyle\\int\\limits_{s}^{t}}\nb(\\delta_{+}(h,u))^{2}x(u)^{2}\\Delta u & \n{\\displaystyle\\int\\limits_{\\delta_{-}(h,t)}^{t}}\n\\Delta \n{\\displaystyle\\int\\limits_{\\delta_{-}(h,t)}^{\\sigma(u)}}\nb(\\delta_{+}(h,u))^{2}x(u)^{2}\\Delta s\\nonumber\\\\\n& \n{\\displaystyle\\int\\limits_{\\delta_{-}(h,t)}^{t}}\n\\left[ \\sigma(u)-\\delta_{-}(h,t)\\right] b(\\delta_{+}(h,u))^{2}x(u)^{2}\\Delta\nu\\nonumber\\\\\n& \\leq\\left[ \\beta(t)+\\mu(t)\\right]\n{\\displaystyle\\int\\limits_{\\delta_{-}(h,t)}^{t}}\nb(\\delta_{+}(h,u))^{2}x(u)^{2}\\Delta u.\\label{holder1\n\\end{align}\nSubstituting (\\ref{holder}) and (\\ref{holder1}) into (\\ref{r(t)}) and using\n(\\ref{r1}) together with $\\delta_{-}^{\\Delta}(h,t)>0$ we deduc\n\\[\nR(t)\\leq-\\left\\{ \\left( \\beta(t)+\\left[ \\lambda\\beta(t)+\\mu(t)\\right]\n\\right) Q(t)+\\lambda\\delta_{-}^{\\Delta}(h,t)\\right\\}\n{\\displaystyle\\int\\limits_{\\delta_{-}(h,t)}^{t}}\nb(\\delta_{+}(h,s))^{2}x(s)^{2}\\Delta s.\n\\]\nHence, using the left-hand side of (\\ref{11}) we arrive at the inequality\n$R(t)\\leq0$. The proof is complete.\n\\end{proof}\n\nIn preparation for the proof of the next theorem we state the following lemma.\n\n\\begin{lemma}\n\\label{Lemma ep}If $\\varphi\\in\\mathcal{R}^{+}$, the\n\\begin{equation}\n01,\n\\]\nand $\\Lambda(t):=\\delta_{-}(h,t)-\\delta_{-}(h,\\delta_{-}(\\alpha,t)).$\n\\end{theorem}\n\n\\begin{proof}\nSince $t_{0}<\\alpha0$ such that (\\ref{11}) holds for all $t\\in\\lbrack t_{0\n,\\infty)_{\\mathbb{T}}$.\n\n\\begin{enumerate}\n\\item If there exists an $\\alpha\\in(t_{0},h)_{\\mathbb{T}}$ such that\n(\\ref{2.8a}) and (\\ref{2.8}) hold, then any solution $x(t)=x(t,t_{0},\\varphi)$\nof (\\ref{9}) satisfies\n\\[\n\\left\\vert x(t)\\right\\vert \\leq\\sqrt{\\frac{2}{\\left( 1-\\frac{1}{\\xi\n(t)}\\right) }V(t_{0})}e^{-\\frac{1}{2}\\int_{t_{0}}^{\\delta_{-}(\\alpha\n,t)}\\left[ \\lambda\\left( \\beta(s)+\\mu(s)\\right) b(\\delta_{+}(h,s))^{2\n+\\mu(s)Q^{2}(s)\\right] \\Delta s\n\\]\nThus, i\n\\[\n\\lim\\limits_{t\\rightarrow\\infty}\\int_{t_{0}}^{\\delta_{-}(\\alpha,t)}\\left[\n\\lambda\\left( \\beta(s)+\\mu(s)\\right) b(\\delta_{+}(h,s))^{2}+\\mu\n(s)Q^{2}(s)\\right] \\Delta s=\\infty,\n\\]\nthen the zero solution of Eq. (\\ref{9}) is exponentially stable.\n\n\\item If $(t_{0},h)_{\\mathbb{T}}=\\varnothing$, then any solution\n$x(t)=x(t,t_{0},\\varphi)$ of (\\ref{9}) satisfie\n\\[\n\\left\\vert x(t)\\right\\vert \\leq\\sqrt{\\left( 1+\\frac{1}{\\lambda}\\right)\nV(t_{0})}e^{-\\frac{1}{2}\\int_{t_{0}}^{t}\\left[ \\lambda\\left( \\beta\n(s)+\\mu(s)\\right) b(\\sigma(s))^{2}+\\mu(s)Q^{2}(s)\\right] \\Delta s}.\n\\]\nThus, i\n\\[\n\\lim\\limits_{t\\rightarrow\\infty}\\int_{t_{0}}^{t}\\left[ \\lambda\\left(\n\\beta(s)+\\mu(s)\\right) b(\\sigma(s))^{2}+\\mu(s)Q^{2}(s)\\right] \\Delta\ns=\\infty,\n\\]\nthen the zero solution of Eq. (\\ref{9}) is exponentially stable.\n\\end{enumerate}\n\\end{corollary}\n\nLet $q>1$, $\\mathbb{T}=\\overline{q^{\\mathbb{Z}}}=\\left\\{ 0\\right\\}\n\\cup\\left\\{ q^{n}:n\\in\\mathbb{Z}\\right\\} $, $\\delta_{-}(h,t)=q^{-h}t$, and\n$h\\in\\mathbb{Z}_{+}$. Then, Eq. (\\ref{9}) turns into the $q$-difference\nequatio\n\\begin{equation}\nD_{q}x(t)=a(t)x(t)+b(t)x(q^{-h}t)q^{-h},\\ \\ t\\in\\left\\{ 1,q,q^{2\n,...\\right\\} , \\label{qdifference\n\\end{equation}\nwhere $D_{q}x(t)=\\frac{x(qt)-x(t)}{(q-1)t}$. Next, we use Corollary\n\\ref{rem stability} to derive a stability criteria for the $q$-difference\nequation (\\ref{qdifference}).\n\n\\begin{example}\nSuppose that $1+\\mu(t)a(t)>0$, $1+\\mu(t)Q(t)\\neq0$, an\n\\[\n-\\frac{\\lambda q^{-h}}{\\varpi(t)+\\lambda(\\varpi(t)+\\mu(t))}\\leq Q(t)\\leq\n-\\lambda\\left( \\varpi(t)+\\mu(t)\\right) b(\\delta_{+}(h,t))^{2}-\\mu(t)Q^{2}(t)\n\\]\nfor all $t\\in\\left\\{ 1,q,q^{2},...\\right\\} $, where $\\varpi(t):=t\\left(\n1-q^{-h}\\right) $ and $\\mu(t)=t(q-1)$.\n\n\\begin{enumerate}\n\\item If $(1,q^{h})_{q^{\\mathbb{Z}}}\\neq\\varnothing$, then then condition\n(\\ref{2.8}) holds. By Corollary \\ref{rem stability}, we conclude that any\nsolution $x(t)=x(t,t_{0},\\varphi)$ of the $q$-difference equation\n(\\ref{qdifference}) satisfies the exponential inequalitie\n\\[\n\\left\\vert x(t)\\right\\vert \\leq\\sqrt{\\frac{2}{\\left( 1-\\frac{1}{\\xi\n(t)}\\right) }V(t_{0})}\\exp\\left( \\frac{1}{2\n{\\displaystyle\\sum\\limits_{s\\in\\lbrack1,q^{-\\alpha}t)_{q^{Z}}}}\n\\mu(s)Q(s)\\right)\n\\]\nfor all $t\\in\\lbrack q^{\\alpha},\\infty)_{q^{\\mathbb{Z}}}$ and\n\\begin{align*}\n\\left\\vert x(t)\\right\\vert & \\leq\\left\\Vert \\psi\\right\\Vert \\exp\\left(\n{\\displaystyle\\sum\\limits_{s\\in\\lbrack1,t)_{q^{Z}}}}\n\\mu(s)a(s)\\right) \\\\\n& \\times\\left[ 1\n{\\displaystyle\\sum\\limits_{s\\in\\lbrack1,t)_{q^{Z}}}}\nG(s)\\exp\\left( \n{\\displaystyle\\sum\\limits_{u\\in\\lbrack1,s)_{q^{Z}}}}\n\\mu(u)a(u)\\right) \\right]\n\\end{align*}\nfor all $t\\in\\lbrack1,q^{\\alpha})_{q^{\\mathbb{Z}}}$, wher\n\\[\nG(s):=q^{-h}\\mu(s)\\left\\vert \\frac{b(s)}{1+\\mu(s)a(s)}\\right\\vert .\n\\]\nHence, i\n\\[\n\\lim\\limits_{t\\rightarrow\\infty\n{\\displaystyle\\sum\\limits_{s\\in\\lbrack1,q^{-\\alpha}t)_{q^{Z}}}}\ns^{2}\\left[ \\lambda(q-q^{-h})b(q^{h}s)^{2}+(q-1)Q^{2}(s)\\Delta s\\right]\n=\\infty,\n\\]\nthen the zero solution of Eq. (\\ref{qdifference}) is exponentially stable.\n\n\\item If $(1,q^{h})_{q^{\\mathbb{Z}}}=\\varnothing$, then $h=1$ and\n\\[\n\\left\\vert x(t)\\right\\vert \\leq\\sqrt{\\left( 1+\\frac{1}{\\lambda}\\right)\nV(t_{0})}\\exp\\left( \\frac{1}{2\n{\\displaystyle\\sum\\limits_{s\\in\\lbrack1,t)_{q^{Z}}}}\n\\mu(s)Q(s)\\right)\n\\]\nHence, i\n\\[\n\\lim\\limits_{t\\rightarrow\\infty\n{\\displaystyle\\sum\\limits_{s\\in\\lbrack1,t)_{q^{Z}}}}\ns^{2}\\left[ \\lambda(q-q^{-1})b(qs)^{2}+(q-1)Q^{2}(s)\\Delta s\\right]\n=\\infty,\n\\]\nthen the zero solution of Eq. (\\ref{qdifference}) is exponentially stable.\n\\end{enumerate}\n\\end{example}\n\nIn the next result, we will display a Lyapunov functional that involves\n$|x|^{\\Delta}$. Thus, in preparation we have the following. \\newline Using the\nproduct rule $(fg)^{\\Delta}=f^{\\Delta}g^{\\sigma}+fg^{\\Delta}$and\ndifferentiating both sides of $x^{2}(t)=\\left\\vert x(t)\\right\\vert ^{2}$ we\nobtain the derivative $\\left\\vert x(t)\\right\\vert ^{\\Delta}$ as follow\n\\begin{equation}\n\\left\\vert x\\right\\vert ^{\\Delta}=\\frac{x+x^{\\sigma}}{\\left\\vert x\\right\\vert\n+\\left\\vert x^{\\sigma}\\right\\vert }x^{\\Delta}\\text{ for }x\\neq0. \\label{sigma\n\\end{equation}\nSo $\\left\\vert x\\right\\vert ^{\\Delta}$ depends on $\\frac{x(t)}{\\left\\vert\nx(t)\\right\\vert }$ and $\\frac{x^{\\sigma}(t)}{\\left\\vert x^{\\sigma\n}(t)\\right\\vert }$ (i.e., signs of $x$ and $x^{\\sigma}$, respectively). Given\n$x:\\mathbb{T}\\rightarrow\\mathbb{R}$, let the sets $\\mathbb{T}_{x}^{+}$ and\n$\\mathbb{T}_{x}^{-}$ be defined b\n\\begin{align*}\n\\mathbb{T}_{x}^{+} & =\\left\\{ t\\in\\mathbb{T}:x(t)x^{\\sigma}(t)\\geq\n0\\right\\} ,\\\\\n\\mathbb{T}_{x}^{-} & =\\left\\{ t\\in\\mathbb{T}:x(t)x^{\\sigma}(t)<0\\right\\} ,\n\\end{align*}\nrespectively. The set $\\mathbb{T}_{x}^{-}$ consists only of right scattered\npoints of $\\mathbb{T}$. Since the time scale $\\mathbb{T}=\\mathbb{R}$ has no\nany right scattered points, we have $\\mathbb{T}_{x}^{-}=\\varnothing$. Thus for\nall differentiable functions $x:\\mathbb{R}\\rightarrow\\mathbb{R}$, the formula\n(\\ref{sigma}) turns into $\\left\\vert x\\right\\vert ^{\\Delta}=\\frac\n{x}{\\left\\vert x\\right\\vert }x^{\\Delta}$. However, for an arbitrary time scale\n(e.g. $\\mathbb{T}=\\mathbb{Z}$) the set $\\mathbb{T}_{x}^{-}$ may not be empty.\nFor simplicity, we need to have a formula for $\\left\\vert x\\right\\vert\n^{\\Delta}$ which does not include $x^{\\sigma}$. The next result provides a\nrelationship between $\\left\\vert x\\right\\vert ^{\\Delta}$ and $\\frac\n{x}{\\left\\vert x\\right\\vert }x^{\\Delta}$. Its proof can be found in\n\\cite{raffoul&adivar}.\n\n\\begin{lemma}\n\\label{lemma 3.4}\\cite[Lemma 5]{raffoul&adivar} Let $x\\neq0$ be $\\Delta\n$-differentiable. Then\n\\begin{equation}\n\\left\\vert x(t)\\right\\vert ^{\\Delta}=\\left\\{\n\\begin{array}\n[c]{ll\n\\frac{x(t)}{\\left\\vert x(t)\\right\\vert }x^{\\Delta}(t) & \\text{if \nt\\in\\mathbb{T}_{x}^{+}\\\\\n-\\frac{2}{\\mu(t)}\\left\\vert x(t)\\right\\vert -\\frac{x(t)}{\\left\\vert\nx(t)\\right\\vert }x^{\\Delta}(t) & \\text{if }t\\in\\mathbb{T}_{x}^{-\n\\end{array}\n\\right. . \\label{2.2.1\n\\end{equation}\n\n\\end{lemma}\n\n\\begin{theorem}\n\\label{thm2.3}Define a continuous function $\\eta(t)\\geq0$ b\n\\begin{equation}\n\\eta(t):=\\frac{e_{a}(t,t_{0})}{1+\\lambda\\int_{\\delta_{-}(h,t)}^{t}e_{a\n(\\delta_{+}(h,s),t_{0})\\Delta s}. \\label{heta\n\\end{equation}\nSuppose that $a\\in\\mathcal{R}^{+}$ and tha\n\\begin{equation}\n\\left\\vert b(t)\\right\\vert -\\lambda\\eta^{\\sigma}(t)\\delta_{-}^{\\Delta\n}(h,t)\\leq0 \\label{2.28a\n\\end{equation}\nholds for all $t\\in\\lbrack t_{0},\\infty)_{\\mathbb{T}}$. Then any solution of\nEq. (\\ref{9}) satisfies the inequalit\n\\begin{equation}\n\\left\\vert x(t)\\right\\vert \\leq V(t_{0},x_{t_{0}})e_{\\gamma}(t,t_{0})\\text{\nfor all }t\\in\\lbrack t_{0},\\infty)_{\\mathbb{T}}\\text{,} \\label{2.28\n\\end{equation}\nwhere\n\\[\nV(t_{0},x_{t_{0}}):=\\left\\vert x(t_{0})\\right\\vert +\\lambda\\eta(t_{0\n)\\int_{\\delta_{-}(h,t_{0})}^{t_{0}}\\left\\vert x(s)\\right\\vert \\Delta s,\n\\]\n$\\gamma(t):=a(t)+\\lambda\\widetilde{M}\\eta^{\\sigma}(t)$, $\\widetilde{M\n=\\max\\left\\{ 1,M\\right\\} $,and $M$ is as in (\\ref{delta turev}).\n\\end{theorem}\n\n\\begin{proof}\nFor convenience defin\n\\[\n\\zeta(t):=1+\\lambda\\int_{\\delta_{-}(h,t)}^{t}e_{a}(\\delta_{+}(h,s),t_{0\n)\\Delta s.\n\\]\nThen by (\\ref{4})\n\\begin{align}\n\\zeta^{\\Delta}(t) & =\\lambda e_{a}(\\delta_{+}(h,t),t_{0})-e_{a\n(t,t_{0})\\delta_{-}^{\\Delta}(h,t)\\nonumber\\\\\n& =\\lambda e_{a}(t,t_{0})\\left[ e_{a}(\\delta_{+}(h,t),t)-\\delta_{-}^{\\Delta\n}(h,t)\\right] . \\label{zeta derivative\n\\end{align}\nThis and a differentiation of (\\ref{heta}) yiel\n\\begin{align}\n\\eta^{\\Delta}(t) & =\\frac{e_{a}(t,t_{0})}{\\zeta(t)}\\left( \\frac\n{a\\zeta(t)-\\zeta^{\\Delta}(t)}{\\zeta^{\\sigma}(t)}\\right) \\nonumber\\\\\n& =\\eta(t)\\left( \\frac{a\\zeta(t)+a\\mu(t)\\zeta^{\\Delta}(t)-a\\mu\n(t)\\zeta^{\\Delta}(t)-\\zeta^{\\Delta}(t)}{\\zeta(t)+\\mu(t)\\zeta^{\\Delta\n(t)}\\right) \\nonumber\\\\\n& =a(t)\\eta(t)-\\left[ \\left( 1+\\mu(t)a(t)\\right) \\eta(t)\\frac{\\zeta\n(t)}{\\zeta^{\\sigma}(t)}\\right] \\frac{\\zeta^{\\Delta}(t)}{\\zeta(t)}\\nonumber\\\\\n& =a(t)\\eta(t)-\\eta^{\\sigma}(t)\\frac{\\zeta^{\\Delta}(t)}{\\zeta(t)}\\nonumber\\\\\n& =a(t)\\eta(t)+\\lambda\\eta^{\\sigma}(t)\\eta(t)\\delta_{-}^{\\Delta\n(h,t)-\\lambda\\eta^{\\sigma}(t)e_{a}(\\delta_{+}(h,t),t)\\nonumber\\\\\n& \\leq\\eta(t)\\left[ a(t)+\\lambda\\widetilde{M}\\eta^{\\sigma}(t)\\right] ,\n\\label{heta derivative\n\\end{align}\nwhere we also used $\\zeta^{\\sigma}(t)=\\zeta(t)+\\mu(t)\\zeta^{\\Delta}(t)$ and\n\\[\n\\left( 1+\\mu(t)a(t)\\right) \\eta(t)\\frac{\\zeta(t)}{\\zeta^{\\sigma}(t)\n=\\eta^{\\sigma}(t).\n\\]\nDefin\n\\begin{equation}\nV(t,x_{t}):=\\left\\vert x(t)\\right\\vert +\\lambda\\eta(t)\\int_{\\delta_{-\n(h,t)}^{t}\\left\\vert x(s)\\right\\vert \\Delta s. \\label{lyapunov\n\\end{equation}\nLet $t\\in\\mathbb{T}_{x}^{+}\\cap\\lbrack t_{0},\\infty)_{\\mathbb{T}}$. Then by\n(\\ref{2.2.1}) we have $\\left\\vert x(t)\\right\\vert ^{\\Delta}=\\frac\n{x(t)}{\\left\\vert x(t)\\right\\vert }x^{\\Delta}(t)$. Differentiating\n(\\ref{lyapunov}) and utilizing (\\ref{2.28a}) and (\\ref{heta derivative}) we\narrive a\n\\begin{align*}\nV^{\\Delta}(t,x_{t}) & =\\left\\vert x(t)\\right\\vert ^{\\Delta}+\\lambda\n\\eta^{\\Delta}(t)\\int_{\\delta_{-}(h,t)}^{t}\\left\\vert x(s)\\right\\vert \\Delta\ns\\\\\n& +\\lambda\\eta^{\\sigma}(t)\\left[ \\left\\vert x(t)\\right\\vert -\\left\\vert\nx(\\delta_{-}(h,t))\\right\\vert \\delta_{-}^{\\Delta}(h,t)\\right] \\\\\n& \\leq\\frac{x(t)}{\\left\\vert x(t)\\right\\vert }x^{\\Delta}(t)+\\lambda\n\\eta(t)\\left[ a(t)+\\lambda\\widetilde{M}\\eta^{\\sigma}(t)\\right] \\int\n_{\\delta_{-}(h,t)}^{t}\\left\\vert x(s)\\right\\vert \\Delta s\\\\\n& +\\lambda\\eta^{\\sigma}(t)\\left[ \\left\\vert x(t)\\right\\vert -\\left\\vert\nx(\\delta_{-}(h,t))\\right\\vert \\delta_{-}^{\\Delta}(h,t)\\right] \\\\\n& =\\left( a(t)+\\lambda\\widetilde{M}\\eta^{\\sigma}(t)\\right) \\left\\vert\nx(t)\\right\\vert +\\left( \\left\\vert b(t)\\right\\vert -\\lambda\\delta_{-\n^{\\Delta}(h,t)\\eta^{\\sigma}(t)\\right) \\left\\vert x(\\delta_{-\n(h,t))\\right\\vert \\\\\n& +\\lambda\\eta(t)\\left[ a(t)+\\widetilde{M}\\eta^{\\sigma}(t)\\right]\n\\int_{\\delta_{-}(h,t)}^{t}\\left\\vert x(s)\\right\\vert \\Delta s\\\\\n& \\leq\\gamma(t)V(t,x_{t}).\n\\end{align*}\nSimilarly, if $t\\in\\mathbb{T}_{x}^{-}\\cap\\lbrack t_{0},\\infty)_{\\mathbb{T}}$,\nthen $\\left\\vert x(t)\\right\\vert ^{\\Delta}=-\\frac{2}{\\mu(t)}\\left\\vert\nx(t)\\right\\vert -\\frac{x(t)}{\\left\\vert x(t)\\right\\vert }x^{\\Delta}(t)$ by\n(\\ref{2.2.1}). Hence\n\\begin{align*}\nV^{\\Delta}(t,x_{t}) & \\leq\\left\\vert x(t)\\right\\vert ^{\\Delta\n+\\eta(t)\\left[ a(t)+\\lambda\\widetilde{M}\\eta^{\\sigma}(t)\\right] \\int\n_{\\delta_{-}(h,t)}^{t}\\left\\vert x(s)\\right\\vert \\Delta s\\\\\n& +\\lambda\\eta^{\\sigma}(t)\\left[ \\left\\vert x(t)\\right\\vert -\\left\\vert\nx(\\delta_{-}(h,t))\\right\\vert \\delta_{-}^{\\Delta}(h,t)\\right] \\\\\n& \\leq\\left( -\\frac{2}{\\mu(t)}-a(t)+\\lambda\\widetilde{M}\\eta^{\\sigma\n}(t)\\right) \\left\\vert x(t)\\right\\vert \\\\\n& +\\left( \\left\\vert b(t)\\right\\vert -\\lambda\\delta_{-}^{\\Delta\n(h,t)\\eta^{\\sigma}(t)\\right) \\left\\vert x(\\delta_{-}(h,t))\\right\\vert \\\\\n& +\\lambda\\eta(t)\\left[ a(t)+\\widetilde{M}\\eta^{\\sigma}(t)\\right]\n\\int_{\\delta_{-}(h,t)}^{t}\\left\\vert x(s)\\right\\vert \\Delta s\\\\\n& \\leq\\left( a(t)+\\lambda\\widetilde{M}\\eta^{\\sigma}(t)\\right) \\left\\vert\nx(t)\\right\\vert +\\lambda\\eta(t)\\left[ a(t)+\\lambda\\widetilde{M}\\eta^{\\sigma\n}(t)\\right] \\int_{\\delta_{-}(h,t)}^{t}\\left\\vert x(s)\\right\\vert \\Delta s\\\\\n& =\\gamma(t)V(t,x_{t}).\n\\end{align*}\nsince $1+\\mu(t)a(t)>0$ implie\n\\[\n-\\frac{2}{\\mu(t)}-a(t)0$ such that\n\\begin{equation}\n-\\frac{\\lambda\\delta_{-}^{\\Delta}(h,t)}{\\beta(t)+\\lambda\\left[ \\beta\n(t)+\\mu(t)\\right] }\\leq b(\\delta_{+}(h,t))\\leq-b(\\delta_{+}(h,t))^{2}\\left[\n\\lambda\\beta(t)+(1+\\lambda)\\mu(t)\\right] ,\\label{sta5\n\\end{equation}\nholds for all $t\\in\\lbrack t_{0},\\infty)_{\\mathbb{T}}$.\n\n\\begin{enumerate}\n\\item If there exists an $\\alpha\\in(t_{0},h)_{\\mathbb{T}}$ such that\n(\\ref{2.8a}) and (\\ref{2.8}) hold and if\n\\begin{equation}\n\\lim\\limits_{t\\rightarrow\\infty}\\int_{t_{0}}^{\\delta_{-}(\\alpha,t)}\\left[\n\\lambda\\beta(s)+(1+\\lambda)\\mu(s)\\right] b(\\delta_{+}(h,s))^{2}\\Delta\ns=\\infty, \\label{sta 6\n\\end{equation}\nthen the zero solution of Eq. (\\ref{sta3}) is exponentially stable.\n\n\\item If $(t_{0},h)_{\\mathbb{T}}=\\varnothing$ and if\n\\[\n\\lim\\limits_{t\\rightarrow\\infty}\\int_{t_{0}}^{t}\\left[ \\lambda\\beta\n(s)+(1+\\lambda)\\mu(s)\\right] b(\\sigma(s))^{2}\\Delta s=\\infty,\n\\]\nthen the zero solution of Eq. (\\ref{sta3}) is exponentially stable.\n\n\\item Suppose that $a\\in\\mathcal{R}^{+}$ and tha\n\\[\n\\left\\vert b(t)\\right\\vert -\\lambda\\eta^{\\sigma}(t)\\delta_{-}^{\\Delta\n}(h,t)\\leq0\n\\]\nholds for all $t\\in\\lbrack t_{0},\\infty)_{\\mathbb{T}}$, wher\n\\[\n\\eta(t):=\\frac{1}{1+\\lambda\\beta(t)}.\n\\]\nThen any solution of Eq. (\\ref{9}) satisfies the inequalit\n\\[\n\\left\\vert x(t)\\right\\vert \\leq V(t_{0},x_{t_{0}})e^{\\frac{1}{2}\\int_{t_{0\n}^{t}\\gamma(s)\\Delta s}e_{\\gamma}(t,t_{0})\\text{ for all }t\\in\\lbrack\nt_{0},\\infty)_{\\mathbb{T}}\\text{,\n\\]\nwhere\n\\[\nV(t_{0},x_{t_{0}}):=\\left\\vert x(t_{0})\\right\\vert +\\lambda\\eta(t_{0\n)\\int_{\\delta_{-}(h,t_{0})}^{t_{0}}\\left\\vert x(s)\\right\\vert \\Delta s,\n\\]\n$\\gamma(t):=\\lambda\\widetilde{M}\\eta^{\\sigma}(t)$, $\\widetilde{M}=\\max\\left\\{\n1,M\\right\\} $,and $M$ is as in (\\ref{delta turev}).\n\\end{enumerate}\n\\end{remark}\n\nIn \\cite[Theorem 7]{mayr}, the authors utilized fixed point theory and deduced\nthat the conditions\n\\begin{equation}\np(t):=b(\\delta_{+}(h,t))\\neq0\\text{ for all }t\\in\\lbrack t_{0},\\infty\n)_{\\mathbb{T}}, \\label{stap\n\\end{equation\n\\[\n\\lim\\limits_{t\\rightarrow\\infty}e_{p}(t,t_{0})=0,\n\\]\nan\n\\begin{equation}\n\\int_{\\delta_{-}(h,t)}^{t}\\left\\vert p(s)\\right\\vert \\Delta s+\\int_{t_{0}\n^{t}\\left\\vert \\ominus p(s)\\right\\vert e_{p}(t,s)\\left( \\int_{\\delta\n_{-}(h,s)}^{s}\\left\\vert p(u)\\right\\vert \\Delta u\\right) \\Delta s\\leq N<1\n\\label{sta7\n\\end{equation}\nlead to stability of solution $x(t,t_{0};\\psi)$ of Eq. (\\ref{sta3}). Notice\nthat \\cite{mayr} generalizes all the results of \\cite{raffoul}.\n\nMoreover, Wang (see \\cite[Corollary 1]{wang}) proposed the inequalit\n\\begin{equation}\n-\\frac{1}{2h}\\leq a(t)+b(t+h)\\leq-hb^{2}(t+h)\\label{sta7-a\n\\end{equation}\nas sufficient condition for uniform asymptotic stability of the zero solution\nof the delay differential equatio\n\\[\nx^{\\prime}(t)=a(t)+b(t)x(t-h),\\ \\ h>0\\text{.\n\\]\n\n\nIt can be easily seen that the conditions (\\ref{sta7}-\\ref{sta7-a}) are not\nsatisfied for the data given in the following example.\n\n\\begin{example}\nLet $a(t)=0$, $\\mathbb{T}=\\mathbb{R}$, $\\delta_{-}(h,t)=t-h$, and $p<0$ be\nfixed. Then Eq. (\\ref{9}) become\n\\[\nx^{\\prime}(t)=b(t)x(t-h).\n\\]\nWe can simplify condition (\\ref{sta7}) as follow\n\\begin{equation}\nh\\left\\vert p\\right\\vert (2-e^{pt})\\leq N<1. \\label{sta8\n\\end{equation}\nIf $h=\\frac{2}{3}$, and $b(t)=-\\frac{9}{10}$, then (\\ref{stap}) implies\n\\[\nh\\left\\vert p\\right\\vert (2-e^{pt})=\\frac{3}{5}\\left( 2-e^{-\\frac{9}{10\nt}\\right) \\geq1\n\\]\nfor all $t\\geq-\\frac{10}{9}\\ln\\left( \\frac{1}{3}\\right) \\cong1.22$. Thus,\nthe condition (\\ref{sta8}) does not hold. On the other hand, for $h=\\frac\n{2}{3}$ and $\\lambda=\\frac{3}{2}$, condition (\\ref{sta5}) turns int\n\\[\n-\\frac{9}{10}\\leq b(\\delta_{+}(h,t))\\leq-b(\\delta_{+}(h,t))^{2}.\n\\]\nThe last inequality holds for $b(t)=-\\frac{9}{10}$. In addition, setting\n$\\delta_{-}(\\alpha,t)=t-\\frac{1}{3}$ one may easily verify that conditions\n(\\ref{2.8a}), (\\ref{2.8}), and (\\ref{sta 6}) are satisfied. Hence, the first\npart of Remark \\ref{rem sta 2} yields exponential stability while\n\\cite[Theorem 7]{mayr} and \\cite[Corollary 1]{wang} cannot.\n\\end{example}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{\\label{sec:1} Introduction}\nRecently, Mott transitions in multicomponent systems have attracted a lot of attention \\cite{Hofstetter,Gorshkov,Hermele,Gorelik,Inaba}. They give rise to intriguing and rich physics that does not have obvious counterparts in physics of condensed matters. The multicomponent systems are realized by cooling neutral atoms, for example, multicomponent lithium at ultra-low temperature \\cite{Ottenstein,Huckans}. When these multicomponent systems are loaded into optical lattices, they simulate various theoretical models of electron correlations with extensions of degrees of freedom, such as the multicomponent Hubbard and Heisenberg models \\cite{Bloch}. The simulations provide a connection between the theoretical studies and experimental observations. Indeed, experiments already found the theoretically predicted Mott insulating phase in two-component fermion systems which simulate the single-band Hubbard model \\cite{Jordens,Schneider}. In three-component fermion systems\nthe Mott insulating phase still exists at commensurate fillings, and in addition also at incommensurate fillings, depending on the symmetry of the local interactions between the particle components \\cite{Gorelik,Inaba}.\n\nIn a parallel development, experiments have also realized multicomponent atom mixtures that pave the way to the investigation of the Mott insulating states of different particle species \\cite{Tag,Spiegelhalder,Taie,Hara}. This investigation has also attracted a lot of theoretical studies. Various phase transitions have been theoretically discussed for different kind of mixtures, including boson-boson \\cite{Hubener,Sansone}, boson-fermion \\cite{Tit,Lewenstein}, and fermion-fermion mixtures \\cite{Iskin,Bo2013}. In fermion-fermion mixtures, different metal-insulator transitions (MITs), which include collective, species selective, and inverse Mott transitions, were theoretically found \\cite{Bo2013}. The finding is based on\nthe dynamical mean-field theory (DMFT)~\\cite{Metzner,GKKR} for a three-component Falicov-Kimball model (FKM), that describes a fermion-fermion mixture of two-component light atoms and single-component heavy atoms~\\cite{Bo2013}.\nThis model is an extension with odd number of components of the spinless FKM, which exhibits a rich phase diagram and has also attracted a lot of studies \\cite{FKM,FreericksZlatic}. It can also be considered as a mass balanced case of the three-component Hubbard model \\cite{Hofstetter,Gorshkov,Hermele,Gorelik,Inaba}. One may expect\nrich phase transitions in the three-component FKM, that reflect the odd and non-equivalent component multiplicity and do not have obvious counterparts in two-component or three equivalent component systems. Indeed,\nat commensurate fillings both the inter- and intra-species interactions collectively drive the mixture from the metallic to the insulating state, and at the incommensurate half filling, only two-component light atoms are involved in the Mott transition. The species-selective MIT is reminiscent of the orbital-selective one, where the narrow orbital band becomes insulating, while the wide band is still metallic \\cite{Anisimov,Koga,Liebsch2004,Medici}. When the intra-species interaction is weaker than the inter-species one, the correlations between atom components can also drive the system from the insulating to the metallic state \\cite{Bo2013}.\n\nThe aim of the present paper is two-fold. First, in the present paper, the Mott transitions in the three-component Falicov-Kimball model are studied by the Kotliar-Ruckenstein slave boson mean-field approach \\cite{Kotliar,Zwickgnal,Sigrist, anh2012,anh2014}.\nThe previous work has employed the DMFT plus exact diagonalization (ED)~\\cite{Bo2013}. Although the ED is an exact impurity solver, it is limited to a finite number of the energy levels of the dynamical mean field and always admits finite-size effects.\nWithin the DMFT the double occupancy always remains finite even in the Mott insulator state and it cannot be used as an order parameter of the MIT \\cite{GKKR,Tien}.\nIn addition, within the DMFT, the dynamics of the localized heavy fermions is complicated to calculate, because it is completely excluded from the effective mean-field single impurity dynamics \\cite{FreericksZlatic,Brandt1}.\nIn multi-orbital systems, different impurity solvers of the DMFT often capture different aspects of the Mott transitions, and can partially lead to different conclusions \\cite{Anisimov,Koga,Liebsch2004,Medici}.\nIt is thus desirable to investigate the properties of these Mott transitions within a more analytical trackable theory.\nTherefore, we apply the Kotliar-Ruckenstein slave boson approach~\\cite{Kotliar,Zwickgnal,Sigrist,anh2012,anh2014} on the mean-field level to the three-component FKM.\nOur reasonable results not only agree well with the DMFT~\\cite{Bo2013} but also clarify the nature of the Mott transitions in a simple physics picture through the band renormalization factors and the double occupancies. In addition, the slave boson mean field approximation also allows us to determine not only the dynamics of itinerant fermions, but also the dynamics of localized fermions, that is complicated to calculate within the DMFT. The second aim of the present paper is to show that the Kotliar-Ruckenstein slave boson approach at the mean field level already fully captures the description of the MIT in the multicomponent fermion mixtures, and it can even be analyzed in an analytical manner. Since the slave boson mean field approximation is quite simple, and its description of the MIT is physically intuitive, this approach serves an alternative method of investigating multicomponent correlated systems.\n\nThe structure of the paper is as follows. In Sec. \\ref{sec2} we present the model as well as introduce the auxiliary boson representation in the mean-field approximation. In Sec. \\ref{sec3} we present and discuss the numerical results of the ground state in paramagnetic states at both commensurate and incommensurate fillings. Finally, the conclusions are presented in Sec. \\ref{sec4}.\n\n\\section{\\label{sec2} Three-component Falicov-Kimball model and the Kotliar-Ruckenstein slave boson approach}\nWe consider the following Hamiltonian of a three-component FKM that describes a mixture of\nheavy and light fermion atoms loaded in an optical lattice~\\cite{Bo2013}\n\\begin{eqnarray}\nH &=& -J\\sum_{,\\sigma} c^{\\dagger}_{i\\sigma} c_{j\\sigma}-\\mu_c \\sum_{i,\\sigma} c^{\\dagger}_{i\\sigma} c_{i\\sigma} - \\mu_f \\sum_{i} f^{\\dagger}_{i} f_{i}\\nonumber \\\\\n&&+ U_{cc}\\sum_{i}c^{\\dagger}_{i\\uparrow} c_{i\\uparrow} c^{\\dagger}_{i\\downarrow}c_{i\\downarrow} + U_{cf} \\sum_{i,\\sigma} f^{\\dagger}_{i} f_{i} c^{\\dagger}_{i\\sigma} c_{i\\sigma}, \\label{ham}\n\\end{eqnarray}\nwhere $c^{\\dagger}_{i\\sigma}$ ($c_{i\\sigma}$) is the creation\n(annihilation) operator for two-component fermion atoms at lattice site $i$ ($\\sigma\\equiv\\uparrow,\\downarrow$), and\n$f^{\\dagger}_{i}$ ($f_{i}$) is the creation (annihilation)\noperator for single-component fermion atoms at\nlattice site $i$.\n$J$ is the nearest neighbor hopping amplitude of the two-component\nfermion atoms. $\\mu_c$ and $\\mu_f$ are the chemical potentials of the two- and single- component atoms, respectively.\nThey control the\nparticle species fillings $n_{c\\sigma}=\\sum_{i} \\langle c^{\\dagger}_{i\\sigma} c_{i\\sigma}\n\\rangle \/N$ and $n_{f}=\\sum_{i} \\langle f^{\\dagger}_{i} f_{i}\n\\rangle \/N$, where $N$ is the number of lattice sites.\n $U_{cc}$ is the intra-species local repulsive interaction between\nthe two-component atoms, and\n $U_{cf}$ is the inter-species one.\nThe Hamiltonian in Eq.~(\\ref{ham}) describes a fermion-fermion mixture, where only the two-component atoms are able to hop in the lattice, and\nthe single-component atoms are extremely heavy and are always localized. Actually, the hopping amplitude of atoms in optical lattices is tuned by the optical potential and the recoil energy of each atom species of the mixture \\cite{Zwerger}. With sufficiently deep potential, the energy band of atoms in optical lattices can become flat and the hopping amplitude vanishes. The mixture can be considered as an extreme mass imbalance case of the three-component fermion mixtures \\cite{Gorelik,Inaba}. The three-component FKM can also be considered as a multicomponent extension of the spinless FKM~\\cite{FKM} or an asymmetric simplification of the three-component Hubbard model \\cite{Gorelik}. It has two well-known limiting cases. When $U_{cc}=0$, the three-component FKM is reduced to the spinless FKM, which exhibits a Mott-like MIT at half filling \\cite{Dongen92,Dongen}. When $U_{cf}=0$, it is equivalent to the single-band Hubbard model, which also exhibits the Mott MIT at half filling \\cite{Hubbard}. However, when both $U_{cc}$ and $U_{cf}$ are finite, different MIT may occur in the three-component FKM, that reflect the component multiplicity and do not have obvious counterparts of the limiting cases \\cite{Bo2013}.\n\n\\begin{table}\n\\caption{Local states ($|\\Gamma\\rangle$), their energy levels ($E_\\Gamma$) and corresponding slave boson ($\\phi_\\Gamma$) representation.}\n\\begin{tabular}{|c|c|c|c|r|}\n\\hline\n$\\Gamma$ & $\\vert \\Gamma\\rangle$ &\n$E_{\\Gamma}$ &$\\phi^{\\dagger}_{\\Gamma}$& {Slave boson representation}\\\\\n\\hline\n1 & $\\vert 0\\rangle$ & $0$ & $e^{\\dagger}$ &$e^{\\dagger}\\vert 0\\rangle$\\\\\n2 & $c^{\\dagger}_{\\uparrow}\\vert 0\\rangle$ & $0$ & $p^{\\dagger}_{\\uparrow}$ &$\\hat{c}^{\\dagger}_{\\uparrow}p^{\\dagger}_{\\uparrow}\\vert 0\\rangle$\\\\\n3 & $c^{\\dagger}_{\\downarrow}\\vert 0\\rangle$ & $0$ & $p^{\\dagger}_{\\downarrow}$ &$\\hat{c}^{\\dagger}_{\\downarrow}p^{\\dagger}_{\\downarrow}\\vert 0\\rangle$ \\\\\n4 & $f^{\\dagger}\\vert 0\\rangle$ & $0$ & $p^{\\dagger}_{f}$ &$\\hat{f}^{\\dagger}p^{\\dagger}_{f}\\vert 0\\rangle$\\\\\n5 & $c^{\\dagger}_{\\uparrow}c^{\\dagger}_{\\downarrow}\\vert 0\\rangle$ & $U_{cc}$ & $d^{\\dagger}_{\\uparrow\\downarrow}$ &$\\hat{c}^{\\dagger}_{\\uparrow}\\hat{c}^{\\dagger}_{\\downarrow}d^{\\dagger}_{\\uparrow\\downarrow}\\vert 0\\rangle$\\\\\n6 & $c^{\\dagger}_{\\uparrow}f^{\\dagger}\\vert 0\\rangle$ &$U_{cf}$ & $d^{\\dagger}_{\\uparrow f}$ &$\\hat{c}^{\\dagger}_{\\uparrow}\\hat{f}^{\\dagger}d^{\\dagger}_{\\uparrow f}\\vert 0\\rangle$\\\\\n7 & $c^{\\dagger}_{\\downarrow}f^{\\dagger}\\vert 0\\rangle$ & $U_{cf}$ & $d^{\\dagger}_{\\downarrow f}$ &$\\hat{c}^{\\dagger}_{\\downarrow}\\hat{f}^{\\dagger}d^{\\dagger}_{\\downarrow f}\\vert 0\\rangle$\\\\\n8 & $c^{\\dagger}_{\\uparrow}c^{\\dagger}_{\\downarrow}f^{\\dagger}\\vert 0\\rangle$& $U_{cc}+2U_{cf}$ & $t^{\\dagger}_{}$&$\\hat{c}^{\\dagger}_{\\uparrow}\\hat{c}^{\\dagger}_{\\downarrow}\\hat{f}^{\\dagger}t^{\\dagger}\\vert 0\\rangle$ \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nWe generalize the Kotliar-Ruckenstein slave boson representation~\\cite{Kotliar} for the three-component FKM described in Eq. (\\ref{ham}).\nWithin the Kotliar-Ruckenstein slave boson representation, every local state at each lattice site is represented by an auxiliary (slave) boson \\cite{Kotliar,Zwickgnal,Sigrist,anh2012,anh2014}.\nSince each lattice site of the three-component FKM can be empty, singly, doubly, or triply occupied, we use eight slave bosons\n$e^{\\dagger}(i)$, $p^{\\dagger}_{\\sigma}(i)$, $p^{\\dagger}_{f}(i)$, $d^{\\dagger}_{\\uparrow\\downarrow}(i)$, $d^{\\dagger}_{\\uparrow f}(i)$, $d^{\\dagger}_{\\downarrow f}(i)$, $t^{\\dagger}(i)$ to represent these eight different local states.\nThe notations of the slave bosons $e$, $p$, $d$, $t$ denote the empty, singly, doubly and triply occupied states, respectively.\nFor convenient, the eight slave bosons are labeled by numbers as shown in table 1, e.g., $\\phi^{\\dagger}_1(i) = e^{\\dagger}(i)$, $\\phi^{\\dagger}_2(i) = p^{\\dagger}_{\\uparrow}(i)$, etc. In the table 1 for the sake of clarity, we skip the site index.\nFor the three fermion components we use three auxiliary fermions $\\hat{c}_{i\\sigma}$ and $\\hat{f}_{i}$.\nThe original fermion operators are expressed\nin term of auxiliary slave bosons and fermions as follows~\\cite{Kotliar,Zwickgnal,Sigrist,anh2012,anh2014}\n\\begin{eqnarray}\nc_{\\sigma} &=&\n\\hat{R}_{\\sigma}^{\\dagger}[\\Phi]\\hat{c}_{\\sigma}, \\\\\nf &=&\n\\hat{R}_{f}^{\\dagger}[\\Phi]\\hat{f} ,\n\\end{eqnarray}\nwhere the operator $\\hat{R}_{\\alpha}^{\\dagger}[\\Phi]$ is defined as\n\\begin{eqnarray}\n\\hat{R}_{{\\alpha}}^{\\dagger}[\\Phi]\\equiv\n\\frac{\\hat{\\gamma}_{\\alpha}[\\Phi]}{\\sqrt{\\hat{n}_{\\alpha}[\\Phi]\n\\left(1-\\hat{n}_{\\alpha}[\\Phi]\\right)}},\n\\label{eq:QPmatrix}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n\\label{Defgamma}\n\\hat{\\gamma}_{\\alpha}[\\Phi]&\\equiv&\\sum_{\\Gamma\\Gamma'}\n |\\langle \\Gamma|a^{\\dagger}_{\\alpha}|\\Gamma'\\rangle|^2\n\\phi^{\\dagger}_{\\Gamma}\\phi_{\\Gamma'},\\\\\n\\label{Defnjz}\n\\hat{n}_{\\alpha}[\\Phi]\n&\\equiv& \\sum_{\\Gamma}\\phi^\\dagger_{\\Gamma}\\phi_{\\Gamma}\n\\langle \\Gamma |a^\\dagger_{\\alpha} a_{\\alpha}|\\Gamma\\rangle.\n\\end{eqnarray}\nHere we used the notation $a^{\\dagger}_{\\alpha}$ to denote either $c^{\\dagger}_{\\sigma}$ or $f^{\\dagger}$, $\\alpha=\\sigma,f$, and skipped the lattice site index.\nThe introduced slave bosons already enlarge the local state space at each lattice site. Therefore, we should eliminate\nthe unphysical states by imposing the local constraint\n\\begin{equation}\n\\sum_{\\Gamma}\\phi^\\dagger_{\\Gamma}(i)\n\\phi_{\\Gamma}(i)= 1 .\n\\label{constraints1}\n\\end{equation}\nThis constrain is the closure relation of the local states at each lattice site. The other constraints which must be also imposed are\nthe identities of the fermion numbers counted through the auxiliary fermions and original fermions\n\\begin{eqnarray}\n\\sum_{\\Gamma}\\phi^\\dagger_{\\Gamma}(i)\\phi_{\\Gamma}(i)\n\\langle \\Gamma(i) |c^\\dagger_{i\\sigma} c_{i\\sigma}|\\Gamma(i)\\rangle&=&\\hat{c}^\\dagger_{i\\sigma} \\hat{c}_{i\\sigma},\n\\label{constraints2a}\\\\\n\\sum_{\\Gamma}\\phi^\\dagger_{\\Gamma}(i)\\phi_{\\Gamma}(i)\n\\langle \\Gamma(i) |f^\\dagger_{i} f_{i}|\\Gamma(i)\\rangle&=&\\hat{f}^\\dagger_{i} \\hat{f}_{i}.\n\\label{constraints2}\n\\end{eqnarray}\nThe constrains in Eqs. (\\ref{constraints1})-(\\ref{constraints2}) must be taken into account by introducing the Lagrange multipliers.\n\nHamiltonian in Eq.~(\\ref{ham}) without the chemical potential terms now can be rewritten in terms of auxiliary fermions\nand bosons as~\\cite{Zwickgnal,Sigrist,anh2012,anh2014}\n\\begin{eqnarray}\nH &=& -J\\sum_{\\langle i,j\\rangle,\\,\\sigma}\n\\left[\n\\hat{R}_{\\sigma}[\\Phi (i)]\n\\hat{R}_{\\sigma}^\\dagger[\\Phi (j)]\n\\hat{c}^\\dagger_{i \\sigma} \\hat{c}_{j \\sigma}\n+\n{\\rm H. c.}\\right]\\nonumber\\\\\n&& +\n\\sum_{i,\\,\\Gamma} E_{\\Gamma}\n\\phi^\\dagger_{\\Gamma}(i)\n\\phi_{\\Gamma}(i) .\n\\label{eq:auxiliaryHalmintonian}\n\\end{eqnarray}\nActually, the chemical potential terms can be absorbed into the constrain terms (\\ref{constraints2a})-(\\ref{constraints2}). Within the Kotliar-Ruckenstein slave boson representation, the local interaction terms become quadratic of the slave bosons. This feature is a benefit of the slave boson approach on the cost of a complication of the hopping term and additional constraints.\nThe single-component atoms are localized, their effects only enter through the constraint term in Eq. (\\ref{constraints2}) like the chemical potential term. However, the correlation effects still persist with the single-component atoms through the slave bosons.\nAt the mean-field level, the slave bosons are replaced by c-numbers and in the homogeneous phases they can be assumed to be site-independent. This greatly simplifies calculations and allows us to perform a trackable analysis. We use the following notations: $e$, $p_{\\sigma}$, $p_{f}$, $d_{\\uparrow\\downarrow} $, $d_{\\uparrow f}$, $d_{\\downarrow f}$, $t$ for the mean-field value of the slave bosons $e^{\\dagger}(i)$, $p^{\\dagger}_{\\sigma}(i)$, $p^{\\dagger}_{f}(i)$, $d^{\\dagger}_{\\uparrow\\downarrow}(i) $, $d^{\\dagger}_{\\uparrow f}(i)$, $d^{\\dagger}_{\\downarrow f}(i)$, $t^{\\dagger}(i)$, respectively. At zero temperature we obtain the ground-state energy per site~\\cite{Zwickgnal,Sigrist,anh2012,anh2014}\n\\begin{equation}\nE = -\\sum \\limits_{\\sigma}\\frac{W}{2} \\gamma_{\\sigma}[\\Phi]^2 + \\sum_{\\Gamma} E_{\\Gamma}\n\\phi^2_{\\Gamma} .\n\\label{EG}\n\\end{equation}\nHere for simplicity, a constant bare density of states $\\rho_0(\\omega) = \\frac{1}{W} \\theta \\left( {\\frac{W}{2}-\\vert \\omega \\vert}\\right )$ with the bandwidth $W$ has been used.\nIn order to find the mean-field ground state, Eq.~(\\ref{EG}) is minimized with the constraints (\\ref{constraints1})-(\\ref{constraints2}), which are also in the mean field approximation. Solving the minimization equations, we obtain the mean-field values of the slave bosons as well as the particle fillings. After that\nwe calculate the band renormalization factor $Z_\\alpha$, the intra-species double occupancy $D_{cc}$ and the inter-species double occupancy $D_{\\sigma f}$ which are defined as\n\\begin{eqnarray}\nZ_\\alpha = R^2_{\\alpha}[\\Phi]=\\frac{\\gamma^2_{\\alpha}}{n_{\\alpha}\\left( 1 - n_{\\alpha}\\right)},\n\\label{rf} \\\\\nD_{cc}\\equiv \\langle n_{\\uparrow}n_{\\downarrow} \\rangle = d^2_{\\uparrow \\downarrow} + t^2, \\label{docc}\\\\\nD_{\\sigma f} \\equiv \\langle n_{\\sigma}n_{f}\\rangle = d^2_{\\sigma f} + t^2. \\label{docf}\n\\end{eqnarray}\nThe interactions between particles renormalize the effective mass of the particles by the band renormalization factor. When the band renormalization factor vanishes, the effective mass becomes infinite and the particles become localized. This is the standard Brinkman-Rice scenario of the Mott insulating state \\cite{Brinkman}. Although the single-component atoms are localized, the local interactions still renormalize their bare energy level by the band renormalization factor $Z_f$. Within the DMFT the self energy of the localized particles is complicated to calculate, because their dynamics is completely excluded from the effective single impurity dynamics of the dynamical mean field \\cite{FreericksZlatic,Brandt1}. By using the Kotliar-Ruckenstein slave boson approach, the band renormalization factors of both itinerant and localized atoms are determined on the same footing.\nThe band renormalization factors, the intra- and the inter-species occupancies exhibit distinct behaviors in the metallic and the insulating states. Therefore, we can use them to monitor the MIT. Actually, experiments of ultracold atoms loaded in optical lattices have also detected the Mott insulating state by counting the doubly occupied sites \\cite{Jordens}. In this work, we restrict ourself to the paramagnetic phase, however, the calculations for magnetically ordered phases are straightforward. Note that, in the paramagnetic phase, one has $p_{\\uparrow}=p_{\\downarrow}\\equiv p_c$,\n$d_{\\uparrow f}=d_{\\downarrow f}\\equiv d_{cf}$, $n_{c\\uparrow}=n_{c\\downarrow}\\equiv n_c$, $D_{\\uparrow f} = D_{\\downarrow f} = D_{cf} $ and $Z_{\\uparrow}=Z_{\\downarrow}\\equiv Z_c$.\n\\section{\\label{sec3} Numerical results and discussions}\nIn this section we present the numerical results obtained by minimizing the ground-state energy (\\ref{EG}) with the constraints (\\ref{constraints1})-(\\ref{constraints2}). We obtain $8+3$ nonlinear equations.\nThe main difficulty here is how to reach the saddle point\nefficiently in the $8+3$ dimensional parameter space. The simple iterative procedure is very hard to be converged and it is\nalmost useless in practice. Instead, we use a modification of the Powell hybrid method.~\\cite{Powell} This algorithm is a variation of Newton's method, which takes precautions to avoid large step sizes or increasing residuals. In the numerical calculations we take the bare bandwidth $W=1$ as the unit of energy.\n\n\\subsection{Mott transitions and their filling conditions}\nIn this subsection we determine the particle filling conditions, where the Mott transition may occur. We find different kinds of the Mott transition depending on the local interactions and the particle fillings \\cite{Bo2013}.\nFirst, we consider the region of weak inter-species interactions ($U_{cf} < W$).\nIn Fig. \\ref{fig1} we plot the band renormalization factors of both atoms species, as well as the intra- and inter-species double occupancies as a function of two-component atom filling $n_{c}$ for a given weak inter-species local interaction at fixed single-component atom filling $n_f=1\/2$. One can see that for weak intra-species local interactions $U_{cc}$, the band renormalization factor $Z_c$ and the double occupancy $D_{cc}$ of the two-component atoms are finite for all fillings $n_c$. However, at strong intra-species local interactions $U_{cc}$, they together vanish at filling $n_c=1\/2$. This is a signal of the localization of the two-component atoms. However, the single-component atoms exhibit different behaviors.\nFigure \\ref{fig1} also shows that the band renormalization factor $Z_f$ of the single-component atoms\nis always finite for all fillings $n_c$. In contrast to the two-component atoms,\nthe band renormalization factor $Z_f$ of the single-component atoms approaches to $1$ at filling $n_c=1\/2$, when the intra-species local interaction $U_{cc}$ becomes strong.\nThe inter-species double occupancy $D_{cf}$ also equals to the noninteracting value $1\/4$ at filling $n_c=1\/2$.\nThese features indicate that at half filling the single-component atoms behave like free ones at strong intra-species local interactions.\nThe vanishing of the band renormalization factor $Z_c$ and the intra-species double occupancy $D_{cc}$ of the two-component atoms, while the single-component atoms become free at filling $n_c=1\/2$, suggests that the MIT at filling $n_c=1\/2$ deals only with the two-component atoms.\nThis MIT is referred to as a species-selective one \\cite{Bo2013}.\nIn the species-selective MIT, only the two-component atoms are involved in the forming of the Mott insulating state. This is reminiscent of the orbital-selective MIT in the multi-orbital systems, where only the narrow orbital band becomes insulating, while the broad bands still remain metallic \\cite{Anisimov,Koga,Liebsch2004,Medici}. However, in the multi-orbital systems, all bands are always renormalized by interactions, and the orbital-selective MIT requires a finite Hund coupling. In this species-selective MIT, the single-component atoms are free of the interaction renormalization independently of the local interactions and filling $n_f$. Actually, in the insulating state, the particle number fluctuations of the two-component atoms are suppressed; the inter-species interaction acts like a shift of the chemical potential for the single-component atoms. As a result, the single-component atoms become effectively free in the insulating state irrespective of the local interactions and single-component atom filling $n_f$.\nThe freedom of the single-component atoms in the Mott insulator is an unique feature of the species-selective MIT. However, the effect of the inter-species local interaction still persists. It reflects on the dependence of the critical value of the intra-species local interaction, where the MIT occurs, on the inter-species local interaction.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.4\\textwidth]{Z_D_vs_ncUcf0.5.eps}\n\\caption{(Color online) The band renormalization factors $Z_c$ and $Z_f$ and the intra- $D_{cc}$ and the inter-$D_{cf}$\nspecies double occupancies as a function of the two-component atom filling $n_{c}$ in the weak inter-species interaction region ($U_{cf}=0.5$)\nat the single-component atom filling $n_f=1\/2$.}\n\\label{fig1}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.4\\textwidth]{Z_D_vs_ncUcf1.5.eps}\n\\caption{(Color online) The band renormalization factors $Z_c$, $Z_f$ and the intra- $D_{cc}$ and the inter- $D_{cf}$\nspecies double occupancies as a function of the two-component atom filling $n_{c}$ in the strong inter-species interaction region ($U_{cf}=1.5$)\nat the single-component atom filling $n_f=1\/2$.}\n\\label{fig2}\n\\end{center}\n\\end{figure}\n\nNext, we consider the region of strong inter-species interactions ($U_{cf}>W$). In this region, we observe different MITs.\nIn Fig.~\\ref{fig2} we plot the band renormalization factors of both atoms species, as well as the intra- and inter-species double occupancies as a function of two-component atom filling $n_{c}$ for a given strong inter-species local interaction at fixed single-component atom filling $n_f=1\/2$. One can see the vanishing of the band renormalization factor $Z_c$ of the two-component atoms at different fillings $n_c$. First, we still observe the vanishing of $Z_c$ at filling $n_c=1\/2$. However, at filling $n_c=1\/2$ the band renormalization factor $Z_c$ vanishes at both weak and strong intra-species local interactions, for instance, at $U_{cc}=0.5$ and $3.0$ as shown in Fig. \\ref{fig2}. In contrast to the species-selective MIT, for weak intra-species local interactions, the intra-species double occupancy $D_{cc}$ remains finite. The vanishing of the band renormalization factor $Z_c$ indicates the localization of the two-component atoms, but their weak intra-species local interaction does not prevent their double occupation. On the other hand,\nthe band renormalization factor $Z_f$ of the single-component atoms and the inter-species double occupancy $D_{cf}$ vanish at filling $n_c=1\/2$ for weak intra-species local interactions. When the intra-species local interaction is larger than a certain value, both the band renormalization factors $Z_c$ and $Z_f$ stop vanishing. This indicates a transition from insulating to metallic states. This MIT is referred to as an inverse MIT, where particle correlations drive the mixture from insulator to metal \\cite{Bo2013}. For other fillings $n_f$ we still observe the inverse MIT when $n_c+n_f=1$ is fulfilled. This MIT is similar to the Mott-like transition in the spinless Falicov-Kimball model \\cite{FreericksZlatic}, where both band renormalization factors $Z_c$ and $Z_f$, and the double occupancy $D_{cf}$ vanish in the insulating state. \nFigure \\ref{fig2} also shows the vanishing of the band renormalization factor $Z_c$ of the two-component atoms at filling $n_c=1\/2$ for strong intra-species local interactions $U_{cc}$. In this case $D_{cc}=0$ and $Z_f=1$. These features constitute the species-selective MIT, which we have discussed previously.\n\nIn Fig.~\\ref{fig2} we also observe the vanishing of the band renormalization factor $Z_c$ of the two-component atoms at other fillings, $n_c=1\/4$ and $3\/4$, when the local interactions are strong. At filling $n_c=1\/4$, the band renormalization factors of both atom species as well as the inter- and the intra-species double occupancies vanish. This indicate that all atoms are localized and the local interactions prevent any double occupation. In this MIT,\nall atoms of both species equally participate in the transition. Like in the three-component Hubbard model \\cite{Gorelik,Inaba}, the MIT is referred to as the collective MIT \\cite{Bo2013}. The filling $n_c=3\/4$ can be considered as a particle-hole symmetry of filling $n_c=1\/4$ \\cite{Bo2013}. In this case, instead of the particle double occupancies, the hole double occupancies vanish. Since the hole double occupancies\n$D_{cc}^{h}=D_{cc}+1-2 n_c$ and $D_{cf}^h=D_{cf}+ 1-n_f-n_c$, they indeed vanish for strong interactions $U_{cc}$, as one can see in Fig. \\ref{fig2}.\nFor other filings $n_f$ we still observe the collective MIT when\nthe total particle filling is commensurate, i.e. $2 n_c+n_f=1$ or $2 n_c+ n_f=2$.\n\nWe summarize the characteristic features of the MIT, observed in the three-component FKM:\n\\begin{enumerate}\n\\item Collective MIT: $Z_{c}=Z_f=0$, $D_{cc}=D_{cf}=0$ (or $D_{cc}^h=D_{cf}^h=0$ for the hole case ).\n\\item Species-selective MIT: $Z_{c}=0$, $Z_f=1$, $D_{cc}=0$, $D_{cf}=n_c n_f$.\n\\item Inverse MIT: $Z_{c}=Z_f=0$, $D_{cc} \\neq 0$, $D_{cf}=0$.\n\\end{enumerate}\n\nThe filling conditions of these MIT were previously found numerically by the DMFT \\cite{Bo2013}. Within the mean-field slave boson approximation, we can derive them analytically. The constrain (\\ref{constraints1}), and the band renormalized factors of both atom species (\\ref{rf}) in the mean-field approximation read\n\\begin{eqnarray}\ne^2+2 p_c^2 + p_f^2 + 2 d_{cf}^2 + d_{\\uparrow\\downarrow}^2 + t^2 =1 , \\\\\nZ_c = \\frac{(e p_c + p_c d_{\\uparrow\\downarrow} + p_f d_{cf} + d_{cf} t )^2}{n_c(1-n_c)} , \\\\\nZ_f = \\frac{(e p_f + 2 p_c d_{cf} + d_{\\uparrow\\downarrow} t )^2}{n_f(1-n_f)} .\n\\end{eqnarray}\nThe particle fillings of both species can be also calculated analytically in the mean-field approximation\n\\begin{eqnarray}\nn_c = p_c^2 + d_{\\uparrow\\downarrow}^2 + d_{cf}^2 + t^2 , \\\\\nn_f = p_f^2 + 2 d_{cf}^2 + t^2 .\n\\end{eqnarray}\nIn the collective Mott insulator $Z_{c}=Z_f=0$ and $D_{cc}= D_{cf} = 0$. These conditions lead to $e=d_{\\uparrow\\downarrow}=d_{cf}=t=0$. In this case, we obtain\n\\begin{eqnarray}\n2 p_c^2 + p_f^2 =1 , \\\\\nn_c = p_c^2 , \\\\\nn_f = p_f^2.\n\\end{eqnarray}\nThese mean-field equations indeed lead to $2 n_c + n_f=1$. This is the filling condition for the collective Mott transition at the commensurate fillings.\n\nIn the species-selective MIT $Z_{c}=0$, $Z_f=1$ and $D_{cc}=0$, and $D_{cf} \\neq 0$.\nCondition $D_{cc}=0$ leads to $d_{\\uparrow\\downarrow}=t=0$. Then we obtain the mean field equations\n\\begin{eqnarray}\ne^2+2 p_c^2 + p_f^2 + 2 d_{cf}^2 =1 , \\\\\ne p_c + p_f d_{cf} =0 , \\\\\n(e p_f + 2 p_c d_{cf})^2 = n_f(1-n_f) , \\\\\nn_c = p_c^2 + d_{cf}^2 , \\\\\nn_f = p_f^2 + 2 d_{cf}^2.\n\\end{eqnarray}\nWithout difficulty one can show that these mean-field equations lead to $e=p_f=0$, and $n_c=1\/2$.\n\nIn the inverse MIT $Z_c=Z_f=0$, $D_{cf} = 0$, and $D_{cc} \\neq 0$. These conditions lead to $e=p_c=d_{cf}=t=0$. In this case we obtain\n\\begin{eqnarray}\np_f^2 + d_{\\uparrow\\downarrow}^2 =1 , \\\\\nn_c = d_{\\uparrow\\downarrow}^2 , \\\\\nn_f = p_f^2 .\n\\end{eqnarray}\nThese mean-field equations indeed lead to $n_c+n_f=1$. This is the filling condition for the inverse MIT.\n\nSo far we have analytically established the filling conditions for different MIT, which may occur in the three-component FKM.\nFrom these analyses of the slave boson mean-field equations, one can see that in all insulating states there are no empty or triply occupied\nsites. Since the considered mixture has two different double occupancies, there are only three possibilities of their vanishing. These three possibilities lead to three different kinds of the Mott insulator.\nThe collective Mott insulator is similar to the one in the three-component Hubbard model,\nand is quite well studied in literature.~\\cite{Gorelik,Inaba,Bo2013}\nThe species-selective and the inverse MIT exhibit special features in the band renormalization factors and the double occupancies of the atom species, that are not captured by the previous DMFT study~\\cite{Bo2013}. Therefore we will study them in details.\n\n\\subsection{Species-selective and inverse metal-insulator transitions}\n\nSince both the species-selective and the inverse MIT may occur at half filling $n_c=n_f=1\/2$,\nin this subsection we study the half filling case in details. In contrast to the single-band Hubbard model or the spinless FKM, where the half filling is commensurate with the particle component number, the half filling here is incommensurate.\nThe considered three-component FKM has two well-known limiting cases.\nWhen $U_{cf}=0$, the two-component and single-component\natoms are completely decoupled. The two-component atoms just form the single-band\nHubbard model \\cite{Hubbard}. The correlations between the atom components drive the mixture from metal to insulator.\nWithin the Kotliar-Ruckenstein slave boson mean-field approach,\nthe MIT occurs at the critical value\n$U^C_{cc} = 16 \\int\\limits_{0}^{W\/2}\\varepsilon{\\rho _0}\\left( \\varepsilon \\right)d\\varepsilon = 2W$.~\\cite{Kotliar} In the insulating state, both the band renormalization factor $Z_c$ and the\nintra-species double occupancy $D_{cc}$ vanish.\nWhen $U_{cc}=0$, the three-component FKM is equivalent to the spinless FKM \\cite{FKM}. The inter-species local interaction drives the mixture from metal to insulator by splitting the two-component atom band.\nA simple treatment within the Kotliar-Ruckenstein slave boson mean-field approach gives a continuous MIT at $U_{cf}^{C}=W$.\nIn the insulating state the inter-species double occupancy $D_{cf}$ vanishes while the intra-species double occupancy $D_{cc}$ remains finite.\nWhen both local interactions $U_{cc}$ and $U_{cf}$ are finite, these MIT may occur, as we have discussed in the previous subsection.\n\n\\begin{figure}[b]\n\\begin{center}\n\\includegraphics[width=0.4\\textwidth]{ZPD_Ucc_Ucf_small.eps}\n\\caption{(color online) The band renormalization factors $Z_c$ (panel A) and $Z_f$ (panel B), the boson condensation $\\phi^2_{\\Gamma}$ (panel C), and\nthe double occupancies $D_{cc}$ and $D_{cf}$ (panel D) as a function of the intra-species interaction $U_{cc}$ at half filling in the weak inter-species interaction region\n($U_{cf}W$).}\n\\label{Z_Ucc__halffilling_Ucf_large}\n\\end{center}\n\\end{figure}\nIn Figs. \\ref{Z_Ucc__halffilling_Ucf_small} and \\ref{Z_Ucc__halffilling_Ucf_large}\nwe plot the band renormalization factors $Z_c$ and $Z_f$, and the double occupancies $D_{cc}$ and $D_{cf}$, and the boson condensation $\\phi^2_{\\Gamma}$ as a function of $U_{cc}$ for different values of $U_{cf}$.\nIn the region of weak inter-species interactions ($U_{cf}W$, in contrast to the region $U_{cf}W$), there would be a reentrant effect of MIT. With increasing the intra-species interaction $U_{cc}$, the mixture first stays in the insulating state, then it is transformed into the metallic state when\n$U_{cc} > U_{cc}^{\\rm inv}$, and finally, the mixture goes back to the insulating state when $U_{cc} > U_{cc}^{\\rm ss}$. Note that the first and second insulating states are quite different. In the first insulating state, the inter-species double occupancy vanishes, while the intra-species one remains finite. In the second insulating state, the single-component atoms become free, albeit being localized, and the intra-species double occupancy vanishes.\nExperiments would observe this reentrant effect of MITs by measuring the intra- and inter-species double occupancies.\n\n\\section{\\label{sec4} Conclusion}\n\nWe have studied the MIT in the three-component FKM by the Kotliar-Ruckenstein slave boson approach at the mean-field level. Although the slave boson mean field approximation is simple, the obtained results reproduce all important features of the MIT, that are obtained within the more sophisticated DMFT. In particular, the filling conditions and the critical value of the local interaction of the MIT are analytically established. Moreover, the slave boson mean-field approximation allows us to clarify the nature of different MITs, which occur in the three-component FKM, in a simple physics picture. This is an advantage of the slave boson approach. In the collective Mott insulator, all band renormalization factors as well as the double occupancies vanish. The collective MIT occurs only at the commensurate fillings. At the incommensurate fillings, the species-selective or the inverse MIT may occur. The species-selective MIT occurs only at half filling of the two-component atoms irrespective of the filling of the single-component atoms. Actually, in the species-selective MIT only two-component atoms are involved, while the single-component ones become free of interactions. The freedom of the single-component atoms in the species-selective Mott insulator is a special feature that makes its distinction from other Mott insulators.\nThe inverse MIT occurs when the total filling of the single-component atoms and of one component of the two-component atoms reaches a unit. This MIT occurs only for weak intra-species local interactions. The obtained results, which are in good agreement with the DMFT, suggest that the Kotliar-Ruckenstein slave boson at the mean-field level is already adequate to describe the MIT in multicomponent fermion-fermion mixtures of ultracold atoms. In addition, the slave boson mean-field approximation also provides the criteria of the MIT through the band renormalization factors and the double occupancies.\nSince experiments of ultracold atom mixtures detect the Mott insulator by counting the number of doubly occupied sites, it is a challenge to observe the MIT in multicomponent fermion-fermion mixtures of ultracold atoms.\n\n\\begin{acknowledgments}\nWe would like to thank Yoshiro Takahashi for helpful\ndiscussions on experimental realizations of the three-component\nFalicov-Kimball model in optical lattices. This work was financially supported by the National Foundation for Science and Technology Development\n of Vietnam under Grant No. 103.01-2014.23.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\nHistorically, the discovery of the quantum Hall (QH) effect revealed two major concepts that revolutionized our knowledge of quantum transport~\\cite{vonKlitzing:1980,Yoshioka}: the remarkable quantization of the Hall conductivity in terms of topological invariants~\\cite{Thouless:1982,Niu:1985,Haldane1988}, and the simultaneous existence of robust unidirectional (chiral) modes that propagate along the edge of the system. These topological transport properties, which are intimately connected through the bulk-edge correspondence~\\cite{Halperin:1982,Rammel:1983,MacDonald:1984,Hatsugai:1993PRL,Hatsugai:1993PRB,Qi:2006theorem}, recently found their counterparts in a wide family of quantum systems: the topological insulators, superconductors and superfluids~\\cite{Hasan:2010,Qi2011,Bernevig:2013}.\n\nDetecting and analyzing the properties of topological edge excitations constitutes an intense field of research since the early days of the QH effect~\\cite{AllenJr:1983,Glattli:1985,Mast:1985,Wen:1990,Johnson:1991,Ashoori:1992,Meir:1994,Wen:1995,Haldane:1995,Kane:1994,Kane:1995,Milliken:1996,Ji:2003,Bid:2010,Venkatachalam:2012,Gurman:2012,Inoue:2013,Goldstein:2016}. The chiral nature of QH edge modes was first revealed through edge-magnetoplasma experiments~\\cite{Ashoori:1992} (see also the pioneer measurements reported in Refs.~\\cite{AllenJr:1983,Glattli:1985,Mast:1985}), while the topological order associated with fractional quantum Hall (FQH) edge modes~\\cite{Wen:1990,Moon:1993} was first detected by measuring tunneling currents between distinct edges~\\cite{Milliken:1996}. A striking demonstration of the outline trajectory performed by QH edge states was provided by a double-slit-electron-interferometer experiment, which realized a QH-based Mach-Zehnder interferometer~\\cite{Ji:2003}. Interestingly, such geometries have been considered to probe the fractional (anyonic) statistics of FQH excitations~\\cite{Goldstein:2016}. More recent experiments also revealed signatures of exotic counter-propagating (``neutral upstream\") modes in FQH liquids~\\cite{Bid:2010,Gurman:2012,Venkatachalam:2012,Inoue:2013}, in agreement with early theoretical works~\\cite{Haldane:1995,Wen:1995,Kane:1994,Kane:1995}.\n\nSpatially-resolved edge currents were also detected in two-dimensional (2D) topological insulators~\\cite{Nowack2013, Konig2013,Yang2012}, via charge transport measurements and scanning tunneling microscopy, offering an instructive view on the quantum spin Hall effect~\\cite{Hasan:2010, Qi2011}. Similar techniques were also exploited to observe spatially-resolved (non-chiral) edge currents in graphene and graphene nanoribbons \\cite{Tao2011, Allen2015}. Furthermore, topological surface states (``2D Dirac fermions\") were observed in 3D topological insulators using angle-resolved photoemission spectroscopy (ARPES) \\cite{Hsieh2008, Hsieh2009, Zhang2009, Xia2009}.\n\n\n\nToday, basic concepts of QH systems and topological insulators are well established, both in theory and through experimental measurements. However, intriguing and more obscure aspects of these topological phases of matter~\\cite{Hasan:2010, Qi2011} could be further explored and exploited using the controllability of engineered quantum systems. In this context, ultra-cold atoms in optical lattices can offer a promising route towards the realization of (potentially exotic) topological phases, through the implementation of well-designed Hamiltonians leading to distinct topological orders~\\cite{Cooper:2008,Dalibard:2011,Goldman:2014,Goldman:2015BEC}. \n\n Recent experiments successfully achieved to load cold atomic gases into 2D Bloch bands with non-trivial topological properties~\\cite{Aidelsburger2013, Jotzu2014,Aidelsburger2015}, using the notion of Floquet engineering~\\cite{Oka2009,Kitagawa:2010,Lindner:2011,Kolovsky:2011,Bermudez:2011,Cayssol:2013,Goldman2014a, Zheng:2014,Bukov2015}: in this cold-atom context, this consists in trapping a gas in an optical lattice and to subject the system to a (high-frequency) time-periodic modulation. In general, Floquet engineering has been implemented in optical lattices by directly shaking the\nlattice potential~\\cite{Lignier:2007,Sias:2008,Eckardt:2009,Zenesini:2009,Struck:2011,Arimondo:2012,Struck2012,Struck:2013h,Jotzu2014}, or by including additional ``moving\" optical lattices~\\cite{Aidelsburger2013, Miyake2013,Aidelsburger2015,Kennedy:2015}, or time-dependent external fields~\\cite{Jimenez-Garcia2012,Luo:2015,Jotzu:2015}. Such driven 2D optical-lattice settings were used to probe various manifestations of the Berry curvature~\\cite{Duca2015, Flaschner2015}, including the anomalous (transverse) velocity in response to an applied force~\\cite{Jotzu2014,Aidelsburger2015}; recent experiments also reported on the measurement of non-zero Chern numbers~\\cite{Aidelsburger2015,Wu:2015}.\n\n\n\n Bulk QH properties have been revealed in recent cold-atom experiments~\\cite{LeBlanc:2012,Jotzu2014,Aidelsburger2015,Wu:2015}, and preliminary results on the identification of chiral edge modes include the observation of unidirectional motion in ladder geometries (``QH stripes\"), where cold atoms were subjected to a synthetic magnetic flux~\\cite{Atala2014,Mancini2015, Stuhl2015}. These experiments were performed in two-leg ladders created by optical potentials~\\cite{Atala2014}, but also, in three-leg ladders~\\cite{Mancini2015, Stuhl2015} built on the concept of synthetic dimensions (i.e.~the three legs of the ladders were associated with three internal states of an atom~\\cite{Celi:2014}). In addition, a very recent work~\\cite{Leder2016} has reported on the observation of a point-like edge state, situated at the interface between two geometrically distinct regions, in a one-dimensional optical lattice reminiscent of the Su-Schrieffer-Heeger model~\\cite{Su:1979}. Finally, we point out that topological structures were also identified in other engineered systems, such as photonic lattices \\cite{Rechtsman2013, Mittal:2014,Lu2014,Gao2016,Mukherjee:2016,Maczewsky:2016}, superconducting qubits \\cite{Schroer2014, Roushan2014}, mechanical systems~\\cite{Susstrunk:2015} and radio-frequency circuits \\cite{Hu2015, Ningyuan2015}. \n\n\n\n\n\n\n\n\\begin{figure}[h!]\n\\includegraphics[width=9.3cm]{Fig1.png}\n\\vspace{-0.cm} \\caption{Illustration of topological interfaces in a two-band model. \n(a) Abstract two-band model realizing different topological phases. \nDepending on the value of a controllable parameter $\\alpha$, the two bulk bands are either topologically trivial (Chern number $C\\!=\\!0$) or non-trivial ($C\\!=\\!\\pm 1$), where $\\alpha^{\\ast}_{\\mathrm{1}}$ and $\\alpha^{\\ast}_{\\mathrm{2}}$ indicate the transition points (i.e.~gap-closing points). \n(b) Spatially-varying the parameter $\\alpha (x,y)$ in the 2D plane generates different regions, which are associated with distinct (spatially-resolved) topological phases. In this figure, each region is labeled by the Chern number $C$ of the lowest bulk band [see panel (a)], which is evaluated locally in space. The singular spatial regions where $\\alpha (x,y)\\!=\\!\\alpha^{\\ast}_{\\mathrm{1, 2}}$ define the topological interfaces within the 2D plane, where topologically-protected ``edge\" modes are located and propagate.}\n\\label{Fig_topology}\n\\end{figure}\n\n\n\nIn 2D systems, topological interfaces consist of boundary lines separating two distinct topologically-ordered regions, where topologically-protected ``edge\" modes are located and propagate [see Fig.~\\ref{Fig_topology} and Section~\\ref{sect:interfaces_general}]. In this work, we introduce a scheme realizing topological interfaces within a 2D optical lattice, which offers the unique possibility of probing, manipulating and tuning the properties of topological edge modes in ultracold atomic gases. Such controllable properties include the location, the localization length, the chirality, and the trajectory of the propagating topological modes. Our proposal is based on the recent realization of the Haldane model~\\cite{Jotzu2014}, which uses ultracold fermions on a honeycomb optical lattice (see also Refs.~\\cite{Oka2009,Zheng:2014}). As further described below in Section~\\ref{sect:interfaces_general}, this scheme offers an ideal platform to investigate edge-state physics \\emph{within the bulk} of a cold atomic gas~\\cite{GoldmanPRL2010,Tenenbaum:2013,Reichl:2014}, hence limiting the effects of external confinement. In particular, the corresponding topological edge modes appear at genuine topological phase transitions (which, in principle, can be associated with arbitrary changes in the Chern number of the bands), and without the simultaneous action of a potential step. This proposal opens an exciting avenue for the exploration of topological edge modes belonging to various topological classes~\\cite{Hasan:2010,Qi2011,Bernevig:2013}, in a highly controllable environment.\n\n\n\n\n\n\\begin{figure}[h!]\n\\includegraphics[width=9.3cm]{Fig2.png}\n\\vspace{-0.cm} \\caption{Edge-state structures in various external potentials and interfaces, for an abstract two-band model. \\emph{Left of each panel}: Schematics of the bulk energy bands $E_{\\pm}(\\bs x)$, as evaluated locally in space along the $x$ direction. The Chern numbers $C$ associated with the bulk bands are indicated; the Fermi energy $E_{\\text{F}}$ is set within the bulk gap, where edge modes (not shown) are expected whenever the bands reach the topological regimes $C\\!=\\!\\pm1$. The location and localization length $\\lambda_{\\text{loc}}$ of the edge modes is represented on the $x$ axis. The external potential or confinement is indicated by thick black lines. \\emph{Right of each panel}: Illustration of the corresponding topological edge modes in the 2D plane. The chirality and the localization length of the edge modes are indicated, as well as the Chern number of the lowest band (note that vacuum is associated with $C\\!=\\!0$). (a) In an ideal box-like potential $V_{\\mathrm{box}}$, the bulk bands are shifted to higher energies. This naturally generates a topological interface between the inner topological system ($C\\!=\\!1$) and vacuum ($C\\!=\\!0$). In this standard case, chiral edge modes appear at the topological interfaces defined by the edges of the box. (b) In the case of a smoothly varying potential (e.g.~a harmonic trap), the bulk bands are locally deformed, and edge modes potentially survive within the bulk gaps. However, smooth traps significantly increase the localization length and reduce the group velocity of the edge modes. (c) Varying a system parameter in space allows for the creation of tunable topological interfaces within the 2D plane [Fig.~\\ref{Fig_topology}]: topological edge modes are located in the vicinity of the transition point, which can be designed at the center of the (smooth) trap. In the latter case, robust edge modes appear at a genuine topological phase transition (here, $C\\!=\\!0\\!\\leftrightarrow\\!1$), without the simultaneous action of a potential step [panel (a)].}\n\\label{Fig_traps}\\end{figure}\n\n\n\n\n\\subsection{Engineered interfaces and the detection of topologically-protected modes}\\label{sect:interfaces_general}\n\nIn the standard realization of the QH effect in solids, topological edge modes appear at the physical boundary of a two-dimensional electron gas~\\cite{Halperin:1982}, which is typically set by the confining potential created by an external metallic electrode gate~\\cite{Yoshioka,Chklovskii:1992}. While theoretical models generally assume that QH systems display sharp edges, the confining potentials of real QH samples are in fact quite smooth: the electronic density slowly drops to zero in the vicinity of the edge of the electron gas~\\cite{Chklovskii:1992,Dempsey:1993,Chamon:1994,Wen:1995}. From a theoretical point of view, the smooth nature of the confining potential was also shown to generate additional edge-state dispersion branches in the spectrum, as compared to the ideal sharp-edges configuration, see Refs.~\\cite{Chamon:1994,Meir:1994,StanescuPRA,CocksPRA}.\n\nIn cold-atom experiments, the atomic cloud is generally confined by an external optical harmonic (or quartic) potential. As was discussed in Refs.~\\cite{CocksPRA,GoldmanPRL,GoldmanEPJST,GoldmanPNAS}, this smooth confinement can significantly affect the properties of topological edge states. In cold-atom systems realizing the QH effect, chiral propagating modes were shown to survive in the presence of smooth external traps, however, their localization length was found to be largely increased and their velocity significantly reduced~\\cite{GoldmanPRL,GoldmanEPJST,GoldmanPNAS}. Furthermore, the distinction between bulk states and edge states becomes complicated in the limit of a purely harmonic trap~\\cite{CocksPRA}. Altogether, this strongly limits the prospect of probing and analyzing topological-edge-state physics in current cold-atom experiments, suggesting the necessity of developing methods to design sharp (box) confinement for these systems~\\cite{Gaunt:2013}. The distinction between the box-potential and smooth-confinement configurations is illustrated in Fig.~\\ref{Fig_traps}(a)-(b), where real-space spectra and topological edge states are schematically represented.\n\nImportantly, the bulk-edge correspondence emanating from topological band theory is not limited to physical edges, which are defined as the boundary separating a sample (e.g.~an electron gas or a cold-atom gas) from vacuum~\\cite{Hasan:2010}. Indeed, the general bulk-edge correspondence states that any \\emph{interface} separating two topologically-different regions of space necessarily hosts topologically-protected edge modes. While this includes the standard case of a sample surrounded by vacuum (whose topology is trivial), this suggests the intriguing possibility of engineering topological interfaces \\emph{within} a sample~\\cite{GoldmanPRL2010,Tenenbaum:2013,Reichl:2014}, e.g.~in a region of space where the effects of external confinement are strongly limited [Fig.~\\ref{Fig_traps}(c)]. Moreover, in a QH system exhibiting chiral edge states, engineered interfaces would offer a tool to design flexible guides for the propagation of topologically-protected modes. \n\nSince topological interfaces play a central role in this work, let us briefly describe this notion using a simple local-density-approximation (LDA) argument. Let us consider an abstract two-band model depending on a constant parameter $\\alpha$, which is topologically trivial for $\\alpha^{\\ast}_{\\mathrm{1}} \\!<\\!\\alpha\\!<\\! \\alpha^{\\ast}_{\\mathrm{2}}$ and topologically non-trivial otherwise [see Fig. \\ref{Fig_topology}(a)]. In the non-trivial regime, the bulk gap hosts topologically-protected edge states, localized at the physical boundary of the system, in agreement with the bulk-edge correspondence. Now, let us suppose that the parameter $\\alpha(\\bs x)$ can be varied continuously in space. Then, in an LDA approach, one can estimate the band structure locally in space, in a region located around some position $\\bs x^*$, based on the value $\\alpha(\\bs x^*)$ that the spatially-varying parameter takes there. Following the topological phase diagram of the model, one finds that different regions of the lattice can then be associated with different topological phases [Fig.~\\ref{Fig_topology}(b)]: the bulk bands that are evaluated locally can have zero or non-zero topological invariants depending on the value $\\alpha(\\bs x)$. In particular, some singular regions are associated with a (local) gapless band structure: this occurs when $\\alpha(\\bs x)\\!=\\!\\alpha^{\\ast}_{\\mathrm{1,2}}$, which defines the local topological interfaces \\emph{within} the lattice. The band structure being locally gapless at these interfaces, and since the latter are associated with a change in the topology, these regions host topologically-protected modes~\\cite{Qi:2006theorem,Hasan:2010}. These modes share the general properties of the topological edge-states associated with the uniform model, except that they are now localized within the interior of the system [Fig. \\ref{Fig_topology}(b) and Fig.~\\ref{Fig_traps}(c)].\n\nThe heart of our proposal is to create a tunable topological interface at the center of a two-dimensional ultracold gas, through a suitable adjustment of optical-lattice parameters. As schematically represented in Fig.~\\ref{Fig_traps}(c), our scheme creates different topological regions within an optical lattice, and is designed so as to localize topologically-protected modes at the center of the trap. This configuration offers a promising platform for the study of topologically-protected modes in cold-atom experiments, where the effects associated with inter-particle interactions~\\cite{Chin:2010} and disorder~\\cite{Sanchez-Palencia\/Lewenstein} could be analyzed in a clean and controllable way. We focus our study on a two-band system realizing the QH effect, hence exhibiting unidirectional (chiral) topological modes, and discuss possible extensions in Section~\\ref{Section:conclusions}. Importantly, the versatility of our scheme allows one to design the shape of the interface within the 2D optical lattice, to control its location, but also, to tune the localization length of the associated topological propagating states. Finally, we point out that the number of topological modes (dispersion branches) associated with an interface is directly given by $n_{\\text{int}}\\!=\\!\\vert C_1 \\!-\\! C_2 \\vert$, where $C_{1,2}$ are the Chern numbers of the lowest bulk band evaluated in the two spatial regions separated by the interface~\\cite{Hasan:2010}; these topological invariants, and hence the number of modes, could also be tuned in a cold-atom experiment.\n\n\n\n\n\n\\subsection{Outline}\n\nOur paper is structured as follows. In Sec. \\ref{section:Haldanemodel} we summarize and present the Haldane model and its general topological properties. \nWe recapitulate the main features and motivate our choice of the Haldane model that lays out the basis for creating and manipulating topological interfaces. \nIn Sec. \\ref{sect:space_dep} we propose a new and variable method to create and probe topological interfaces in cold-atom experiments. \nWe discuss the general strategy of generating a topological interface in the center of the system, by spatially varying the lattice potential, and study the corresponding edge-state structures. \nIn Sec. \\ref{section:radial_interface} we advance our idea of spatially differing topological phases to a radial geometry, and discuss how we can realize a radial-symmetric topological interface. \nSec. \\ref{Section:dynamics} is dedicated to the actual measurement, and provides numerical calculations for possible observables of the topological edge mode appearing at the interface. \nWe present wave-packet dynamics for our proposed schemes, both for projections onto edge and bulk states, and show how this allows one to probe the motion of chiral edge modes in the presence of a harmonic trap. \nIn Sec. \\ref{Section:disorderb}, we study how the dynamics are affected by the presence of disorder in the lattice. In particular, we show that disorder can be used to improve the detection of the chiral propagating modes, by reducing the dispersion of non-chiral (bulk) states.\nThe spatially differing optical lattice configurations and their possible realization in a realistic experimental setup is described in Sec. \\ref{Section:experiment}. \nWe conclude and summarize our tunable approach of generating topological interfaces in Sec. \\ref{Section:conclusions}, and explore further possible applications and outlooks. \n\n\n\n\n\\section{The Haldane model \\\\ and the topological phase diagram}\\label{section:Haldanemodel}\n\nBefore introducing our proposal for the creation of topological interfaces, let us briefly summarize the general topological properties of the model considered in this work. This will allow us to introduce the relevant parameters of the model and the notations used in the following. The choice of this specific model will be motivated at the end of this section (see \\ref{section:motivations}).\n\n\\subsection{The model}\\label{section:themodel}\n\nWe consider a two-dimensional (2D) honeycomb lattice realizing the two-band Haldane model \\cite{Haldane1988,Jotzu2014}. The actual configuration used for our calculations, the so-called brickwall geometry, is depicted in Fig.~\\ref{Fig_one}. Neglecting the effects of inter-particle interactions, and in the absence of any external trapping potential, the tight-binding Hamiltonian is given by \n\\begin{align}\n\\hat H =& - t_{\\text{NN}} \\sum_{\\langle j,k \\rangle} \\hat a^{\\dagger}_j \\hat a_k + t_{\\text{NNN}} \\sum_{\\langle \\langle j,k \\rangle \\rangle} i^{\\circlearrowleft} \\hat a^{\\dagger}_j \\hat a_k ,\\label{Eq:Haldane}\\\\\n&+(\\Delta_{\\text{AB}}\/2) \\left ( \\sum_{j \\in \\text{A sites}} \\hat a^{\\dagger}_j \\hat a_j - \\sum_{j \\in \\text{B sites}} \\hat a^{\\dagger}_j \\hat a_j \\right ), \\notag\n\\end{align}\nwhere $\\hat a^{\\dagger}_j$ creates a particle at lattice site $j$, $t_{\\text{NN}}$ [resp.~$t_{\\text{NNN}}$] is the tunneling amplitude for hopping processes between nearest-neighboring (NN) [resp.~next-nearest-neighboring (NNN)] sites, and $i^{\\circlearrowleft}\\!=\\!\\pm i$ depending on the orientation of the NNN hopping, e.g. $i^{\\circlearrowleft}\\!=\\!+ i$ for clockwise hopping (see colored arrows in Fig.~\\ref{Fig_one}). We emphasize that the chirality imposed by this orientation-dependent hopping term is responsible for the breaking of time-reversal symmetry (TRS) in the model, in analogy with the Lorentz force induced by an external magnetic field. Similarly to the traditional QH effect, this TRS-breaking term opens a gap in the two-band spectrum, and favors bulk bands with non-trivial Chern numbers $C \\!=\\!\\pm1$, see Ref.~\\cite{Haldane1988} and below. In the second line of Eq.~\\eqref{Eq:Haldane}, we introduced an offset $\\Delta_{\\text{AB}}$ between $A$ and $B$ sites [Fig.~\\ref{Fig_one}], which breaks inversion symmetry in the model: This term also opens a gap in the spectrum, but it favors bulk bands with trivial Chern numbers $C \\!=\\!0$. The competing effects associated with these two terms become evident when writing the Hamiltonian~\\eqref{Eq:Haldane} in momentum representation,\n\\begin{align}\n&\\hat H(\\bs k)= d_x (a \\bs k) \\hat \\sigma_x + d_y (a \\bs k) \\hat \\sigma_y + d_z (a \\bs k) \\hat \\sigma_z, \\\\\n&d_x (\\bs k)=- t_{\\text{NN}} \\left ( \\cos (k_x) + 2\\cos k_y \\right) ; \\, d_y (\\bs k)=- t_{\\text{NN}} \\sin k_x ; \\notag \\\\\n&d_z (\\bs k)= (\\Delta_{\\text{AB}}\/2) + 2 t_{\\text{NNN}} \\left [ \\sin (k_x+k_y) - \\sin (k_x-k_y) \\right ].\\notag\n\\end{align}\nHere, we considered the brickwall geometry depicted in Fig.~\\ref{Fig_one} and neglected purely-vertical NNN hopping, which is small in the experimental realisation of the Haldane model of Ref.~\\cite{Jotzu2014}. In the absence of NNN hopping and offset ($t_{\\text{NNN}}\\!=\\!\\Delta_{\\text{AB}}\\!=\\!0$) the spectrum is gapless and displays conical intersections at the two Dirac points $\\bs K_+=(0,\\pi\/3a)$ and $\\bs K_-=(\\pi\/a,2\\pi\/3a)$. Including these two effects then potentially adds a mass term $M \\hat \\sigma_z$ to the effective Dirac equations associated with the two Dirac points $\\bs K_\\pm$, with masses given by $M_\\pm\\!=\\!d_z (\\bs K_\\pm)$, respectively. On the one hand, the constant offset term $(\\Delta_{\\text{AB}}\/2) \\hat \\sigma_z$ generates the same mass term at the two Dirac points, with effective mass $M_\\pm\\!=\\!(\\Delta_{\\text{AB}}\/2)$: This opens a topologically-trivial bulk gap in the spectrum, since the Chern numbers of the two bulk bands are given by $C\\!=\\!\\pm(1\/2) [\\text{sign}(M_+) - \\text{sign}(M_-)]\\!=\\!0$, see Ref.~\\cite{Haldane1988}. On the other hand, adding the $\\bs k$-dependent term associated with the chiral NNN-hopping generates opposite mass terms at the two Dirac points, with masses $M_{\\pm}\\!=\\!\\pm 2 \\sqrt{3} t_{\\text{NNN}}$: This opens a bulk gap and generates two bands with non-zero Chern numbers $C\\!=\\!\\pm 1$.\n\nThe topology of the system is thus determined by the competition between these two opposite effects. For instance, keeping $t_{\\text{NNN}}$ fixed, a variation of the offset $\\Delta_{\\text{AB}}$ can be exploited to drive topological phase transitions: These are marked by a closing of the bulk gap ($M_{\\pm}\\!=\\!0$) and a change in the Chern number of the bands. Noting that the mass terms at the two Dirac points are given by $M_{\\pm}\\!=\\!(\\Delta_{\\text{AB}}\/2) \\pm 2 \\sqrt{3} t_{\\text{NNN}}$, one finds that the critical offset at which a topological phase transition occurs is given by $\\Delta_{\\text{AB}}\\!=\\!\\pm \\Delta_{\\text{trans}}$, where $\\Delta_{\\text{trans}}\\!\\equiv\\! 4 \\sqrt{3} t_{\\text{NNN}}$; see Fig.~\\ref{Fig_two} for an illustration of the topological phase diagram associated with the model.\n\n\\begin{figure}[h!]\n\\includegraphics[width=9.3cm]{Fig3.png}\n\\vspace{-0.cm} \\caption{The Haldane model defined on the ``brickwall\" lattice used in this work. The lattice spacing between NN sites is denoted $a$, the NN [resp.~NNN] tunneling amplitude is denoted $t_{\\text{NN}}$ [resp.~$t_{\\text{NNN}}$]. The NNN tunneling matrix elements $ \\pm i t_{\\text{NNN}}$ are complex; their sign is positive [resp.~negative] for clockwise [resp.~anti-clockwise] paths. Non-equivalent lattice sites are denoted $A$ and $B$. Note that we neglect purely vertical NNN hoppings, which are small in experimental realizations of the model; this slightly modifies the original Haldane model, but does not alter its topological properties.}\\label{Fig_one}\\end{figure}\n\nThe manifestation of topology, and the related phase transitions, are directly visible when analyzing the model in Eq.~\\eqref{Eq:Haldane} on a cylinder, e.g.~by applying periodic boundary conditions along the $y$ direction only. The corresponding spectrum $E\\!=\\!E(k_y)$, represented as a function of the quasi-momentum $k_y$, is shown in Fig.~\\ref{Fig_two} for $\\Delta_{\\text{AB}}\\!=\\!0$ and $\\Delta_{\\text{AB}}\\!=\\!2 \\Delta_{\\text{trans}}$, keeping $t_{\\text{NNN}}=0.15t_{\\text{NN}}$ fixed. In the absence of offset, the bulk gap hosts two edge-state branches, namely, a single edge-state mode per edge of the cylinder. \nImportantly, the edge mode associated with a given edge has a well-defined chirality, $\\text{sign}(\\partial_{k_y}E)$, which describes the orientation of propagation along this edge [Fig.~\\ref{Fig_two}\\textcircled{1}]. These edge modes are topologically protected, in that they cannot be removed by weak perturbations that preserve the bulk gap. \nCrossing the topological phase transition, i.e.~$\\vert\\Delta_{\\text{AB}}\\vert\\!>\\!\\vert\\Delta_{\\text{trans}}\\vert$, the edge modes disappear from the bulk gap \\footnote{\nIn Fig.~\\ref{Fig_two}\\textcircled{2}, some edge states, which are clearly distinguishable from the bulk states, are still visible in the spectrum. However, as they are located away from the bulk gap, they are not topologically protected (these can be removed without changing the topology of the bulk bands). In addition, we find that their presence depends on the specific details of the boundary, e.g.~the orientation of the boundaries with respect to the lattice.\n} and the system reduces to a trivial band insulator [Fig.~\\ref{Fig_two}\\textcircled{2}]. \n\nIt is worth pointing out that for traditional physical realizations of the model, i.e.~for a finite-size lattice defined on a 2D plane, the system only displays a single edge, and hence, a single topologically-protected edge mode (when parameters are set in the topological regime). We also refer the reader to Ref.~\\cite{Lacki2016}, where a scheme to engineer cylindrical optical lattices has been proposed.\n\n\\begin{figure}[h!]\n\\includegraphics[width=9.3cm]{Fig4.png}\n\\vspace{-0.cm} \\caption{Topological phase diagram and two representative spectra. (top) The boundary lines in the phase diagram indicate gap-closing events in the two-band spectrum, through which the Chern number of the bulk bands is allowed to change. (bottom) Spectrum $E\\!=\\!E(k_y)$ of the Haldane model \\eqref{Eq:Haldane}, set on a cylinder geometry aligned along $x$: \\textcircled{1} $\\Delta_{\\text{AB}}\\!=\\!0$ and \\textcircled{2} $\\Delta_{\\text{AB}}\\!=\\!2 \\Delta_{\\text{trans}}$; here $t_{\\text{NNN}}\\!=\\!0.15 t_{\\text{NN}}$ and $\\Delta_{\\text{trans}}\\!\\equiv\\! 4 \\sqrt{3} t_{\\text{NNN}}$. The situation in \\textcircled{1} corresponds to a topological band configuration, where the bulk bands are associated with Chern numbers $C \\!=\\!\\pm1$, and where the bulk gap hosts a single edge-state mode (per edge); note that the two dispersion branches, with opposite group velocity (chirality), correspond to edge modes located on opposite edges of the cylinder. The case in \\textcircled{2} is a trivial band insulator with $C \\!=\\!0$. These two situations are indicated in the topological phase diagram.}\\label{Fig_two}\\end{figure}\n\n\n\n\\subsection{Motivations behind the choice of the model}\\label{section:motivations}\n\nThe topological interfaces and detection methods that we are about to discuss can be applied to a wide family of physical platforms featuring topological band structures. However, it is worth mentioning that the model considered in this work [Eq.~\\eqref{Eq:Haldane}] does present several advantages. First of all, the Haldane model (and its variants) has been recently implemented in photonics and cold-atom experiments \\cite{Rechtsman2013,Jotzu2014}, where the relevant model parameters ($t_{\\text{NNN}}$, $\\Delta_{\\text{AB}}$) can be finely tuned. Moreover, the minimal topological-band structure associated with the two-band Haldane model, which displays a single topologically-protected edge mode (per edge), significantly simplifies the analysis of edge-state physics: Indeed, a state that is prepared in the vicinity of an edge (or more generally, close to an interface separating topologically-different regions) will necessarily project unto two types of eigenstates: (1) bulk states, and (2) edge states that are associated with a well-defined chirality. This is in contrast with models displaying many bands, e.g.~the Hofstadter model \\cite{Hofstadter1976}, where the different bulk gaps host edge-state modes of different chirality, which can all be potentially populated when preparing the initial state close to an edge\/interface. We finally point out that the efficiency with which edge states are populated depends on various parameters, e.g.~the Fermi energy or the mean quasi-momentum of a prepared wave packet [the edge-mode dispersions shown in Fig.~\\ref{Fig_two}\\textcircled{1} are local in momentum space]. This latter aspect will be illustrated below.\n\n\n\n\n\\section{Creating topological interfaces}\\label{sect:space_dep}\n\n\\subsection{The general strategy}\\label{Section:general_strategy}\n\nThe topological properties of the system [Eq.~\\eqref{Eq:Haldane}] have been discussed above for a homogeneous configuration of the parameters $t_{\\text{NN}}$, $t_{\\text{NNN}}$ and $\\Delta_{\\text{AB}}$, and in the absence of any external trapping potential. In this case, the topological edge modes identified in Fig.~\\ref{Fig_two}\\textcircled{1} are located at the interface between the topological system [associated with non-zero Chern numbers $C \\!=\\!\\pm1$] and vacuum [associated with trivial topology, $C \\!=\\!0$]: The edge states are \\emph{located at the edges of the system}. However, it is possible to engineer interfaces, separating different topologically-ordered regions, \\emph{within the system}, as we now explain. \n\n\nA first proposal in Ref.~\\cite{GoldmanPRL2010} suggested to achieve topological interfaces by locally (but strongly) modifying hopping parameters in a central region of the system. This local change of hopping parameters, which can indeed split the system into topologically distinct regions, is particularly suitable for atom-chip implementations, where these parameters are set by tunable (and local) current-carrying wires; see also Refs.~\\cite{Tenenbaum:2013,Reichl:2014}. \n\nIn this work, we explore a different strategy, more practical for current optical-lattice experiments, which is based on the introduction of a spatially dependent offset $\\Delta_{\\text{AB}}(\\bs x)$ between neighboring sites. We take the offset to be a \\emph{linear} function of one of the spatial coordinates, i.e.~we consider an offset of the form\n\\begin{equation}\n\\Delta_{\\text{AB}}(x)=\\left (\\delta_{\\text{max}}\/L_x \\right ) x + \\Delta_{\\text{trans}} - \\left (\\delta_{\\text{max}}\/2 \\right ) , \\label{space_offset}\n\\end{equation}\nwhere we introduced $L_x$, the system length along the $x$ direction, and the parameter $\\delta_{\\text{max}}$, which determines the slope of the space-dependent offset. \nIn Section \\ref{Section:experiment_lin}, we will show how such a spatial variation can be implemented experimentally, through a direct modification of the lattice potential.\nNote that the function $\\Delta_{\\text{AB}}(x)$ in Eq.~\\eqref{space_offset} is chosen such that the critical value,\n\\begin{equation}\n \\Delta_{\\text{trans}}\\!=\\!\\Delta_{\\text{AB}}(x\\!=\\!L_x\/2),\n \\end{equation} \n is exactly reached \\emph{at the center of system}, i.e.~where the external trapping potential $V_{\\text{trap}}(\\bs x)$ is minimal [Sec.~\\ref{section_trap}]. This critical position will be denoted $x_R\\!=\\!L_x\/2$, as it defines the location of the \\emph{right interface}, where topologically-protected chiral modes are localized. \nNote that the other critical value $-\\Delta_{\\text{trans}}$ can also be reached within the system, at the position\n\\begin{equation}\nx_L=L_x \\left ( \\frac{1}{2} - \\frac{2 \\Delta_{\\text{trans}}}{\\delta_{\\text{max}}} \\right ) = x_R - \\left (\\frac{8 \\sqrt{3} L_x }{\\delta_{\\text{max}}} \\right ) t_{\\text{NNN}}, \\label{left_edge}\n\\end{equation}\nwhich defines the location of the \\emph{left interface}; see Figs.~\\ref{Fig_three}(a)-(b). \n\nAt this stage, it is important to note that we are dealing with a competition between two relevant effects. On the one hand, increasing the ratio $ t_{\\text{NNN}}\/\\delta_{\\text{max}}$ allows to spatially separate the edge modes located on different interfaces [Eq. \\eqref{left_edge}], which is an important feature in order to detect clean chiral edge-state propagation (and potentially, to limit back-scattering processes in the presence of engineered disorder). On the other hand, increasing the slope $\\delta_{\\text{max}}$ allows to improve the localization of the edge states within each interface, as shown below. \n\n\\begin{figure}[h!]\n\\includegraphics[width=9.3cm]{Fig5.png}\n\\vspace{-0.cm} \\caption{(a) Space-dependent offset $\\Delta_{\\text{AB}}(x)$, as defined in Eq.~\\eqref{space_offset}, for $\\delta_{\\text{max}}\\!=\\!20t_{\\text{NN}}$, $t_{\\text{NNN}}\\!=\\!0.4 t_{\\text{NN}}$ and $L_x=100a$. The offset $\\Delta_{\\text{AB}}(x)$ is expressed in units of the critical value $\\Delta_{\\text{trans}}\\!\\equiv\\! 4 \\sqrt{3} t_{\\text{NNN}}$, at which topological transitions occur (see horizontal dotted lines). (b) The corresponding separation between topological and trivial regions, as dictated by the two interfaces located at $x\\!=\\!x_R\\!=\\!L_x\/2$ and at $x\\!=\\!x_L$; see Eq.~\\eqref{left_edge}. The inner region of the lattice corresponds to the topological region ($-\\Delta_{\\text{trans}}\\!<\\!\\Delta_{\\text{AB}}(x)\\!<\\!\\Delta_{\\text{trans}}$), while the regions outside these frontiers are topologically trivial. Chiral topological edge modes are localized and propagate in the vicinity of these two interfaces, as indicated by the colored arrows. }\\label{Fig_three}\\end{figure}\n\n\nWe demonstrate these two competing effects by diagonalizing the system on a cylinder aligned along the $x$ direction, noting that the system still preserves translational symmetry along the $y$ direction. The corresponding spectrum, as well as the amplitude $\\vert \\psi (x) \\vert^2$ of two representative states, are represented in Fig.~\\ref{Fig_4} for two different values of the parameter $ t_{\\text{NNN}}$. First of all, we note that the states represented in Fig.~\\ref{Fig_4}(a) are well localized in the vicinity of the interfaces located at $x\\!=\\!x_{L,R}$ and that the dispersion relation of these modes, $E\\!=\\!E(k_y)$ are reminiscent of those associated with standard topological edge states [Fig.~\\eqref{Fig_two}\\textcircled{1}]: These dispersions are approximately linear and they are well isolated from the bulk bands associated with delocalized states (the size of the corresponding ``bulk gap\" is found to be $\\Delta_{\\text{gap}}\\!\\approx\\!0.9 t_{\\text{NN}}$ in Fig.~\\ref{Fig_4}). Hence, these dispersions describe one-dimensional Dirac fermions, propagating along the $y$ direction, with an approximately constant group velocity $+v_g^y$ [resp.~$-v_g^y$], along the right [resp.~left] interface. Besides, we note that these edge states have a localization length of about five lattice sites for the (realistic) parameters chosen in these calculations [$\\delta_{\\text{max}}\\!=\\!20t_{\\text{NN}}$ and $L_x=100a$]. Figure~\\ref{Fig_4}(b) shows that reducing the parameter $ t_{\\text{NNN}}$ affects the spatial separation between the two localized modes, as predicted by Eq.~\\eqref{left_edge}; however, quite surprisingly, we observe that this change only very slightly modifies the localization length and dispersion (group velocity) of the modes. Importantly, the localization length of the localized mode propagating along the central interface $x\\!=\\!x_R$ is significantly reduced by increasing the slope parameter $\\delta_{\\text{max}}$, as we illustrate in Fig.~\\ref{Fig_5}(a). We further show in Fig.~\\ref{Fig_5}(b) the robustness of the group velocity $v_g^y$ against changes in the slope parameter $\\delta_{\\text{max}}$, which stabilizes around the value $ v_g^y \\approx 1.7 a t_{\\text{NN}}\/\\hbar$. \n\n \\begin{figure}[h!]\n\\includegraphics[width=9.3cm]{Fig6.png}\n\\vspace{-0.cm} \\caption{Energy spectrum $E\\!=\\!E(k_y)$, as a function of quasi-momentum $k_y$, and edge-state amplitudes for two values of the NNN hopping amplitude: (a) $t_{\\text{NNN}}\\!=\\!0.4 t_{\\text{NN}}$, (b) $t_{\\text{NNN}}\\!=\\!0.2 t_{\\text{NN}}$. The system is solved on a cylinder of length $L_x\\!=\\!100 a$, and with a space-dependent offset $\\Delta_{\\text{AB}}(x)$, Eq.~\\eqref{space_offset}, characterized by $\\delta_{\\text{max}}\\!=\\!20 t_{\\text{NN}}$. Each eigenvalue $E(k_y)$ is colored in terms of the mean position $\\langle x \\rangle$ of its corresponding eigenstate (see colorbars). The eigenstates represented on the right correspond to the eigenenergies $E(k_y)$ indicated by blue and green circles in the spectrum, respectively. Vertical dotted lines in the right panels correspond to the interface positions, $x_{L,R}$, as predicted by Eqs.~\\eqref{space_offset}-\\eqref{left_edge}. We note that the group velocity $v_g^y$ associated with the edge-modes remains approximately constant as $t_{\\text{NNN}}$ is increased. Moreover, the bulk bands are only slightly distorted in the vicinity of the bulk gap, whose size is approximatively $\\Delta_{\\text{gap}}\\!\\approx\\!0.9 t_{\\text{NN}}$. We point out that only the states located within the bulk gap, with an approximately linear dispersion relation, are well (spatially) localized. }\\label{Fig_4}\\end{figure}\n\n\nFinally, we diagonalized the full 2D open-boundary lattice, and we present the corresponding spectrum and a representative eigenstate in Fig.~\\ref{Fig_2D_states}. The spectrum is shown in Fig.~\\ref{Fig_2D_states}(a), where the presence of the bulk gap is identified through a severe reduction of the density of states around $E\\!\\approx\\!0$. Note that this spectrum is in agreement with the one presented in Fig.~\\ref{Fig_4}, for the cylinder-geometry case. Fig.~\\ref{Fig_2D_states}(b) shows a representative eigenstate, whose energy $E_{\\text{edge}}\\!\\approx\\!0$ is located within the bulk gap. We find that the states present in the bulk gap are indeed well localized on the topological interfaces located at $x\\!=\\!x_{R,L}$. \n\nThis method of displacing the conducting topological interfaces by tuning $t_{\\text{NNN}}$ raises an interesting possibility for technological applications: \nIn the realization of the Haldane model based on Floquet engineering, the value of $t_{\\text{NNN}}$ is controlled by the frequency, amplitude and polarization of an oscillating external force \\cite{Oka2009,Jotzu2014}. \nIndeed, the original proposal considered circularly polarized light illuminating a sheet of graphene.\nIf such a sheet were to be placed on a substrate with an incommensurate lattice spacing (see Sec.~\\ref{Section:experiment_lin} and Refs.~\\cite{Fain1980,Tang2013,Woods2014}), this may lead to a spatially varying site-offset.\nThen, tuning the properties of the illumination could be used as a method for dynamically displacing conducting channels in a material.\n\n\n\n\n\n\n\n\\begin{figure}[h!]\n\\includegraphics[width=9.3cm]{Fig7.png}\n\\vspace{-0.cm} \\caption{(a) Amplitude of an edge state localized around $x_R$ for $t_{\\text{NNN}}\\!=\\!0.4 t_{\\text{NN}}$, and four different values of the slope parameter $\\delta_{\\text{max}}$, as indicated on the figure. The edge mode propagating along the central interface is more localized as $\\delta_{\\text{max}}$ is increased. (b) Dispersion relation $E\\!=\\!E(k_y)$ of the edge mode at $x_R$, for the same values of the system parameters. The group velocity of this localized mode is found to be stable around the value $ v_g^y \\approx 1.7 a t_{\\text{NN}}\/\\hbar$. Here the system size along the $x$ direction is $L_x\\!=\\!100 a$. }\\label{Fig_5}\\end{figure}\n\n\\begin{figure}[h!]\n\\includegraphics[width=9.3cm]{Fig8.png}\n\\vspace{-0.cm} \\caption{(a) Energy spectrum for a finite-size open-boundary honeycomb lattice [Eq.~\\eqref{Eq:Haldane}], with spatially-dependent offset given by Eq.~\\eqref{space_offset}. Here $\\lambda$ is an integer labeling the eigenvalues by increasing order. The bulk gap has a size $\\Delta_{\\text{gap}}\\!\\approx\\! t_{\\text{NN}}$, as indicated by the green region where the density of states is strongly reduced. (b) Amplitude of a representative eigenstate $\\vert \\psi_{\\text{edge}} (\\bs x)\\vert^2$, with energy $E_{\\text{edge}}\\!\\approx\\!0$ in the bulk gap. This state is localized on the topological interfaces located at $x\\!=\\!x_{R,L}$, with a localization length of about five lattice sites [in agreement with Fig.~\\ref{Fig_4}(a)]. System parameters are $t_{\\text{NNN}}\\!=\\!0.4 t_{\\text{NN}}$, $\\delta_{\\text{max}}\\!=\\!20 t_{\\text{NN}}$, $L_x\\!=\\!100a$ and $L_y\\!=\\!56a$. }\\label{Fig_2D_states}\\end{figure}\n\n\n\\subsection{On the effects of time-reversal-symmetry breaking and spatially-resolved edge states}\\label{Section:TRS}\n\n\nIn the previous section, we demonstrated that increasing the NNN hopping parameter allows one to spatially separate the topological interfaces induced by $\\Delta_{\\text{AB}}(x)$, and hence, resolve the edge modes propagating with opposite chirality in the system. \nWe now address a natural question: ``{\\emph{What happens when $ t_{\\text{NNN}}\\!=\\!0$, namely when time-reversal-symmetry is present in the system?}\". \nIn this situation, we note that $\\Delta_{\\text{AB}}(x)\\!>\\!0$ in the region defined by $x\\!>\\!L_x\/2$: The system is locally gapped, with positive mass terms at both Dirac points (the gap is topologically trivial, as it is only due to local inversion-symmetry breaking). \nIn the other region, $x\\!<\\!L_x\/2$, the system is also locally gapped since $\\Delta_{\\text{AB}}(x)\\!<\\!0$, but now with negative mass terms at both Dirac points. \nWe deduce from this that the full system is topologically trivial, but that the mass terms are locally reversed at the center of the system $x\\!=\\!x_R$. \nConsequently the system is locally gapless at $x\\!=\\!x_R$, where $\\Delta_{\\text{AB}}(x_R)\\!=\\!t_{\\text{NNN}}\\!=\\!0$; see also the phase diagram in Fig.~\\ref{Fig_two}. \nHence, when setting $t_{\\text{NNN}}\\!=\\!0$, there is a single ``trivial interface\" at $x\\!=\\!x_R$, where states are localized (due to the fact that, in the LDA picture, the system is gapless there); see the sketch in Fig.~\\ref{Fig_TRS}(a) and the numerical calculation (spectrum and localized states) shown in Fig.~\\ref{Fig_TRS}(b). \nThis interface, however, is of a different nature than the ones discussed above for $t_{\\text{NNN}}\\!\\ne\\!0$, since time-reversal-symmetry is satisfied when $t_{\\text{NNN}}\\!=\\!0$: \nThe single interface now hosts localized edge modes with both chirality (i.e.~the interface is equivalent to a standard, isolated, 1D lattice) meaning that conduction in these modes is not protected from back-scattering. \nIn Ref. \\cite{Leder2016}, a related type of localized state was created in a \\emph{one-dimensional} lattice, by locally reversing the mass of a 1D Dirac point.\n\nNote that the pair of edge modes shown in Fig.~\\ref{Fig_4} and~\\ref{Fig_TRS}(b) can be resolved in $k$-space for all values of $t_{\\text{NNN}}$, including for the time-reversal-invariant case $t_{\\text{NNN}}\\!=\\!0$. This indicates that edge modes with opposite chirality can still be individually selected on this interface, even in the time-reversal-invariant case, e.g.~by tuning the mean quasi-momentum of a Gaussian wave packet prepared at the center of the system. For instance, in the TRS case represented in Fig.~\\ref{Fig_TRS}(b), tuning the mean quasi-momentum to the value $k_y^0\\!\\approx\\!+2.1\/a$ will mostly project the wave packet unto the localized states with positive group velocity $v_g^y\\!>\\!0$. We demonstrate this selectivity of localized chiral edge modes, for the singular TRS case, in Appendix \\ref{Appendix:TRS}, where the dynamics of Gaussian wave packets are obtained through numerical simulations of the full 2D lattice.\n\nThis analysis has an important corollary:~Showing the uni-directional propagation of states along the central interface is not enough to demonstrate the existence of topologically-non-trivial edge modes in the system. Indeed, it is important to prove that a non-trivial (spatially separated) interface only hosts modes with a given chirality. A possible protocol for verifying this consists in initially preparing a wave packet on an interface (e.g.~around $x\\!=\\!x_R$), and imaging the time-evolution of the wave packet for various values of the mean quasi-momentum (i.e.~scanning the full Brillouin zone): In the non-trivial-topological situation, there should only be a single interval of values $k_y^0$ that gives rise to an unidirectional motion along this specific interface. This would unambiguously demonstrate the chirality associated with this interface. Then one could perform the same analysis on the other interface ($x_L$), and demonstrate that the observed motion has the opposite chirality in that case. In photonic and mechanical systems, adding dislocations to topological edge states has been used as a method to probe their chirality \\cite{Rechtsman2013,Susstrunk:2015}.\n\n \\begin{figure}[h!]\n\\includegraphics[width=9.cm]{Fig9.png}\n\\vspace{-0.cm} \\caption{(a) In the absence of NNN hopping, $t_{\\text{NNN}}\\!=\\!0$, the system is invariant under time reversal: The topological region disappears, which is indicated here by the equality $x_R\\!=\\!x_L$ for $t_{\\text{NNN}}=0$; see Eq.~\\eqref{left_edge}. The trivial ``interface\" at $x\\!=\\!L_x\/2$ hosts localized propagating states, with opposite chirality: In this scheme, the chirality of localized modes can only be spatially resolved when setting $t_{\\text{NNN}}\\!\\ne\\!0$, i.e.~when time-reversal symmetry is broken [see Figs.~\\ref{Fig_three} and~\\ref{Fig_4}]. (b) Energy spectrum $E\\!=\\!E(k_y)$, as a function of quasi-momentum $k_y$, and amplitude of two localized states (green) and one delocalized state (grey) for the time-reversal-invariant case $t_{\\text{NNN}}\\!=\\!0$. The energy of these depicted states is indicated in the spectrum by two green circles and one grey circle, respectively. All other system parameters are the same as in Fig.~\\ref{Fig_4}. We point out that only the states located within the bulk gap are well (spatially) localized [see the bulk state depicted in grey versus the localized states in green]. }\\label{Fig_TRS}\\end{figure}\n\n\n\\subsection{Adding the external harmonic trap}\\label{section_trap}\n\nIn this section, we discuss the effects of a harmonic trapping potential $V_{\\text{trap}} (\\bs x)$ added on top of the lattice system introduced in Sec.~\\ref{Section:general_strategy}. Since this is a typical feature of optical-lattice experiments, such a harmonic trap will be included in the numerical simulations presented in Sec.~\\ref{Section:dynamics}-\\ref{Section:disorderb}. The aim of this section is to identify realistic configurations of the trap for which the propagation of chiral edge modes can still be observed in experiments.\n\nThe robustness of topological edge states against smoothly varying potentials (e.g.~harmonic traps) has already been investigated numerically in Refs.~\\cite{StanescuPRA,GoldmanPRL,GoldmanEPJST,GoldmanPNAS,CocksPRA}. These results can also be understood based on a general LDA argument, which can be summarized as follows: A topological edge mode, spatially localized in some region $R$ in the absence of the external trap, will survive (in this region) in the presence of the trap $V_{\\text{trap}} (\\bs x)$ as long as \\begin{equation}\nV_{\\text{trap}} (\\bs x\\! \\in R) \\!<\\! \\Delta_{\\text{gap}},\n\\end{equation} \nwhere $\\Delta_{\\text{gap}}$ denotes the size of the bulk gap that hosts and protects the topological edge mode. This criterium signifies that the topological gap, locally estimated in the region of interest $R$, should not collapse due to the presence of the trap, in order to probe the propagation of the edge mode in this region.\n\nOne can directly apply this argument to the system introduced in Section~\\ref{Section:general_strategy}, where the regions of interest correspond to the topological interfaces at $x\\!=\\!x_{L,R}$. To analyze this situation, let us write the harmonic potential in the following form\n\\begin{equation}\nV_{\\text{trap}} (\\bs x)= V_x \\left ( \\frac{x-x_R}{x_L\\!-\\!x_R} \\right)^2 + V_y \\left ( \\frac{y-y_0}{L_{\\text{obs}}} \\right)^2 , \\label{xy_trap}\n\\end{equation}\nwhere we introduced an ``observation\" length $L_{\\text{obs}}$, and where $(x_R,y_0)\\!=\\!(L_x\/2,L_y\/2)$ is the center of the trap. First of all, let us focus on the central interface located at $x\\!=\\!x_R$. This region effectively feels a one-dimensional harmonic trap, aligned along the propagation direction ($y$), of the form $V_{\\text{trap}} (y)\\!=\\!V_y [(y-y_0)\/L_{\\text{obs}}]^2$. Hence, following the LDA argument above, we find that the detection of chiral modes, propagating along this interface over a distance $L_{\\text{obs}}$, is possible as long as $V_y\\!<\\!\\Delta_{\\text{gap}}$. A similar criterion can also be introduced if one is interested in the detection of the other topological interface [$x\\!=\\!x_L$], which sets a condition on the other trap parameter $V_x\\!<\\!\\Delta_{\\text{gap}}$. We verified these LDA predictions through a direct numerical diagonalization of the lattice Hamiltonian [Eq.~\\eqref{Eq:Haldane},\\eqref{space_offset}], in the presence of the trap [Eq.~\\eqref{xy_trap}]. Figure \\ref{Fig_harmonic} compares the shape of the harmonic potential $V_{\\text{trap}} (\\bs x)$ in the 2D plane with the amplitude $\\vert \\psi_{\\text{edge}}(\\bs x)\\vert^2$ of a representative eigenstate, whose energy $E\\!\\approx\\!0$ is located within the bulk gap. This figure shows that this ``edge\" state is indeed well localized along the two topological interfaces, but only within regions where $V_{\\text{trap}} (\\bs x)\\!<\\!\\Delta_{\\text{gap}}\\!\\approx\\!0.9 t_{\\text{NN}}$; see the regions encircled by the white dotted ellipses in the right panels of Fig.~\\ref{Fig_harmonic}. \n\nThis analysis, which confirms the general LDA prediction, identifies the trapping-potential configurations for which clear chiral motion can be observed in an experimental realization of our topological-interface model [Eq.~\\eqref{Eq:Haldane},\\eqref{space_offset}]. In particular, it highlights the robustness of the chiral modes propagating \\emph{at the center} of the trap, i.e.~along the engineered interface at $x\\!=\\!x_R$.\n\n\\begin{figure}[h!]\n\\includegraphics[width=9.3cm]{Fig10.png}\n\\vspace{-0.cm} \\caption{Comparison between the shape of the harmonic potential $V_{\\text{trap}} (\\bs x)$ in the 2D plane (left) and the amplitude $\\vert \\psi_{\\text{edge}}(\\bs x)\\vert^2$ of an eigenstate whose energy is located within the bulk gap (right). The trap parameters are given by (a)-(b) $V_{x}\\!=\\!V_{y}\\!=\\!0.2 t_{\\text{NN}}$, (c)-(d) $V_{x}\\!=\\!V_{y}\\!=\\!0.5 t_{\\text{NN}}$, and (e)-(f) $V_{x}\\!=\\!V_{y}\\!=\\!1.0 t_{\\text{NN}}$. In both cases, $L_{\\text{obs}}\\!=\\!20a \\sqrt{V_y\/t_{\\text{NN}}}$, so that the available propagation distance along the central interface is of about $40$ sites. The ``edge\" state is indeed well localized along the topological interfaces, but only within regions where $V_{\\text{trap}} (\\bs x)\\!<\\!\\Delta_{\\text{gap}}\\!\\approx\\!0.9 t_{\\text{NN}}$, see the white dotted ellipses on the right panels. System parameters are $t_{\\text{NNN}}\\!=\\!0.4 t_{\\text{NN}}$, $\\delta_{\\text{max}}\\!=\\!20 t_{\\text{NN}}$, $L_x\\!=\\!100a$ and $L_y\\!=\\!50a$. On the left panels, the trap $V_{\\text{trap}} (\\bs x)$ is measured in units of $t_{\\text{NN}}$.}\\label{Fig_harmonic}\\end{figure}\n\n\n\\section{The radial-symmetric topological interface}\\label{section:radial_interface}\n\nIn this section, we discuss how the space-dependent offset $\\Delta_{\\text{AB}}(x)$ in Eq.~\\eqref{space_offset} can be modified so as to generate a single radial-symmetric topological interface. As will be shown below, this configuration allows one to further reduce the effects of the external harmonic trap, and hence, to probe the physics of topologically-protected modes on potentially larger length scales (and longer time scales). \n\nIn order to achieve such a radial-symmetric interface, we now consider that the space-dependent offset $\\Delta_{\\text{AB}}(\\bs x)$ can be created in the form of a radial-symmetric Gaussian\n\\begin{align}\n\\Delta_{\\text{AB}}(r) &= \\frac{5 \\Delta_{\\text{trans}}}{2} \\left [ 1 - \\exp ( - r^2 \/ 2 R_{\\text{inter}}^2 ) \\right ] ,\\label{space_offset_radial}\\\\\n r &= \\sqrt{(x-L_x\/2)^2+(y-L_y\/2)^2}, \\nonumber\n\\end{align}\nwhere we introduced the radius of the interface $R_{\\text{inter}}$, which separates the (inner) topologically-non-trivial region from the (outer) trivial region; see also Ref.~\\cite{Reichl:2014}. Indeed, the function $\\Delta_{\\text{AB}}(r)$ in Eq.~\\eqref{space_offset_radial} is chosen such that $\\Delta_{\\text{AB}}\\!=\\!0$ at the center of the trap ($r\\!=\\!0$), then increases as a Gaussian function, and reaches the topological transition point in the vicinity of the radius $R_{\\text{inter}}$, i.e.~$\\Delta_{\\text{AB}}(r\\!=\\!R_{\\text{inter}})\\!\\approx\\!\\Delta_{\\text{trans}}$. Note that the location of this radial topological interface corresponds to the width of the Gaussian-shaped offset function $\\Delta_{\\text{AB}}(r)$: in the vicinity of the critical radius $r\\!\\approx\\!R_{\\text{inter}}$, the offset $\\Delta_{\\text{AB}}(r)$ depends approximately linearly on $r$, with a slope given by \n\\begin{equation}\n\\Delta_{\\text{AB}}'(r\\!=\\!R_{\\text{inter}})\\!\\approx\\!(3\/2)\\Delta_{\\text{trans}}\/R_{\\text{inter}}.\n\\end{equation}\nSetting the radius parameter to be $R_{\\text{inter}}\\!\\approx\\! 20a$, one approximately recovers the slope associated with the linear-interface scheme illustrated in Fig.~\\ref{Fig_three}(a). The experimental implementation of such an offset is discussed in Section~\\ref{Section:experiment_circ}.\n\nIn this radial-symmetric configuration, the LDA argument predicts that the chiral edge mode is now localized along the radius $r\\!=\\!R_{\\text{inter}}$, where it performs a circular motion. One estimates the angular velocity of this chiral mode to be given by $\\dot \\theta\\!\\approx\\!v_g^y\/R_{\\text{inter}}$, where $v_g^y$ is the group velocity evaluated above for the linear-interface case; since the microscopic details of the boundary are changed, the velocity $v_g^y$ can potentially slightly differ from the linear case. \n\nWe verified these predictions by performing a direct diagonalization of the full 2D lattice described by Eqs.~\\eqref{Eq:Haldane} and \\eqref{space_offset_radial}. The corresponding energy spectrum, shown in Fig.~\\ref{Fig_2D_states_radial}(a), indicates that the new shape of the interface does not significantly modify the bulk bands obtained in the linear-interface case:~In particular, the edge modes are still protected by a bulk gap of size $\\Delta_{\\text{gap}}\\!\\approx\\! t_{\\text{NN}}$. A representative edge state, whose energy is located within the bulk gap, is shown in Fig.~\\ref{Fig_2D_states_radial}(b). We find that these edge states are indeed well localized around the radius $r\\!=\\!R_{\\text{inter}}$, with a typical localization length of about five lattice sites [similarly to the linear case]. We point out that, when comparing the radial and linear configurations in Fig.~\\ref{Fig_2D_states_radial}(a), we set the corresponding system parameters in such a way that the offset slopes are of the same order in both configurations, i.e.~$\\delta_{\\text{max}}\/L_x\\!\\approx\\!(3\/2)\\left (\\Delta_{\\text{trans}}\/R_{\\text{inter}} \\right )$. \n\n\\begin{figure}[h!]\n\\includegraphics[width=9.3cm]{Fig11.png}\n\\vspace{-0.cm} \\caption{(a) Energy spectrum for the radial-interface (red) configuration [Eq.~\\eqref{space_offset_radial}]; this spectrum is compared with the linear-interface case (blue). Here $\\lambda$ is an integer labeling the eigenvalues by increasing order. In both cases, the bulk gap has a size $\\Delta_{\\text{gap}}\\!\\approx\\! t_{\\text{NN}}$, as indicated by the green region where the density of states is strongly reduced. (b) Amplitude of a representative eigenstate $\\vert \\psi_{\\text{edge}} (\\bs x)\\vert^2$, with energy $E_{\\text{edge}}\\!\\approx\\!0$ within the bulk gap, for the radial-interface configuration. This state is localized around the radial topological interface, defined by the radius $r\\!=\\!R_{\\text{inter}}$. System parameters are $t_{\\text{NNN}}\\!=\\!0.4 t_{\\text{NN}}$, $R_{\\text{inter}}\\!=\\!17 a$, and $L_x\\!=\\!L_y\\!=\\!75a$.}\\label{Fig_2D_states_radial}\\end{figure}\n\n\nFinally, we discuss the effects of the harmonic trap in the radial-interface configuration. Following the discussion of Section~\\ref{section_trap}, we write the trapping potential in the form\n\\begin{equation}\nV_{\\text{trap}}(r)=V_0 (r\/R_{\\text{inter}})^2 . \\label{trap}\n\\end{equation}\nIn this case, the LDA argument predicts that the edge mode propagating along the radial interface $r\\!=\\!R_{\\text{inter}}$ will survive in the presence of the trap, as long as $V_0\\!<\\!\\Delta_{\\text{gap}}$. We verified this statement by diagonalizing the lattice system in the presence of the trap, for various values of the trap parameter $V_0$. As an illustration, we present a perfectly conserved edge state in Fig.~\\ref{Fig_harmonic_radial}, obtained for the parameters $V_0\\!=\\!0.25 t_{\\text{NN}}$ and $R_{\\text{inter}}\\!=\\!17a$. The remarkable robustness of this radial-symmetric edge mode relies on that it is \\emph{entirely} located in a region where $V_{\\text{trap}}(\\bs x)\\!<\\!\\Delta_{\\text{gap}}$; see the white dotted ellipse in Fig.~\\ref{Fig_harmonic_radial}(b). This constitutes a significant advantage, as compared to the linear-interface configuration [Fig.~\\ref{Fig_harmonic}].\n\n\\begin{figure}[h!]\n\\includegraphics[width=11.cm]{Fig12.png}\n\\vspace{-0.cm} \\caption{Comparison between (a) the shape of the harmonic potential $V_{\\text{trap}} (\\bs x)$ in the 2D plane, and (b) the amplitude $\\vert \\psi_{\\text{edge}}(\\bs x)\\vert^2$ of an eigenstate whose energy is located within the bulk gap, for the radial-interface configuration [Eq.~\\eqref{space_offset_radial}]. The radius of the circular interface $R_{\\text{inter}}$ is indicated by a white [resp. blue] dotted circle in (a) [resp.~(b)]. The robustness of this localized edge state relies on that it is entirely located in a region where $V_{\\text{trap}}(\\bs x)\\!<\\!\\Delta_{\\text{gap}}$, see the white dotted ellipse in (b); compare with Fig.~\\ref{Fig_harmonic}. The system parameters are $V_0\\!=\\!0.25 t_{\\text{NN}}$, $t_{\\text{NNN}}\\!=\\!0.4 t_{\\text{NN}}$, $R_{\\text{inter}}\\!=\\!17 a$, and $L_x\\!=\\!L_y\\!=\\!75a$. In panel (a), the trap $V_{\\text{trap}} (\\bs x)$ is measured in units of $t_{\\text{NN}}$.}\\label{Fig_harmonic_radial}\\end{figure}\n\n\n\\section{Wave-packet dynamics}\\label{Section:dynamics}\n\nIn this section, we explore the dynamical properties of our topological-interface scheme, by studying the motion of Gaussian wave packets in various relevant configurations. \nThe wave packets are defined as\n\\begin{align}\n&\\vert \\psi_{\\text{0}} \\rangle = \\sum_{j} G_j \\hat a_j^{\\dagger} \\vert 0 \\rangle, \\\\\n&G_j = (1\/\\mathcal{N}) \\mathrm{e}^{-(x_j - x_0)^2\/2\\sigma_x^2}\\mathrm{e}^{-(y_j - y_0)^2\/2\\sigma_y^2}\\mathrm{e}^{i k_x^0(x_j - x_0)}\\mathrm{e}^{i k_y^0(y_j - y_0)},\\nonumber\n\\end{align}\nThe time-evolution of wave packets presented below was obtained through a numerical implementation of the time-evolution operator $\\exp (-i t \\hat H\/\\hbar)$ associated with the full 2D system, including the effects of the external trap introduced above, and the space-dependent offset $\\Delta_{\\text{AB}}(\\bs x)$. This study aims to highlight the applicability of our topological-interface scheme, in view of detecting the chiral motion of topological modes within an optical-lattice setup.\n\n\\subsection{The linear-interface case}\\label{section:dynamics_linear}\n\nLet us start by considering the linear-interface scheme associated with the space-dependent offset in Eq.~\\eqref{space_offset}. We show in Fig.~\\ref{Fig_dynamics_linear} the time-evolution of a small wave packet moving in the 2D lattice, for four different initial conditions. In the first case [Fig.~\\ref{Fig_dynamics_linear}(a)], the wave packet is initially prepared on the central interface at $x\\!=\\!x_R$, with a mean quasi-momentum $\\bs k^0\\!=\\!(0,2.1\/a)$, which maximizes the projection unto the chiral localized mode [see the dispersion in Fig.~\\ref{Fig_4}(a)]. We find that this initial state, whose small mean-deviation $\\sigma_x\\!=\\!3a$ is of the order of the localization length of the central chiral mode, projects unto this localized mode with about $90\\%$ efficiency. This wave packet undergoes a chiral motion along the topological interface, with positive mean velocity $v_g^y\\!\\approx\\! 1.7 a t_{\\text{NN}}\/\\hbar$, which is in agreement with the approximately-linear dispersion shown in Figs.~\\ref{Fig_4}(a)-\\ref{Fig_5}(b). We then show in Figs.~\\ref{Fig_dynamics_linear}(b)-(c) how shifting the initial position of the wave packet, or changing its mean quasi-momentum, dramatically affects the projection unto chiral modes: In both these cases, the wave packet projects unto delocalized modes [with about $100\\%$ efficiency], and accordingly, it undergoes an irregular (non-chiral) motion within the 2D lattice, and diffuses into the bulk. Figure~\\ref{Fig_dynamics_linear}(d) shows the dynamics of a wave packet initially prepared on the other interface at $x\\!=\\!x_L$, with a mean quasi-momentum $\\bs k^0\\!=\\!(0,-2.1\/a)$ that maximizes the projection unto the other localized mode: In this situation, the wave packet performs a regular motion along the second topological interface, $x\\!=\\!x_L$, with an opposite chirality. The center-of-mass motion of these wave packets, along the ``propagation\" direction $y$, is further illustrated in Fig.~\\ref{Fig_COM_linear}. As a technical remark, we note that the group velocity of the localized modes slightly differ on the two interfaces $x_{R,L}$, which is due to the presence of the trap and microscopic details of the two interfaces.\n\nThis numerical study demonstrates how the topological chiral modes, which are localized on the topological interfaces, can be populated and probed \\emph{in situ}, through a careful preparation of the wave-packet's position and momentum. We verified, through numerical simulations, that this preparation can be achieved by performing a partial Bloch oscillation: This consists in applying a force $\\bs F\\!=\\! F_y \\bs{1}_y$ for a time $\\Delta_t$, in such a way that the mean quasi-momentum $k^0_y(\\Delta_t)\\!=\\! F \\Delta_t$ reaches the desired value. Note that in this scheme, the initial (spatial) position of the wave packet should also be adjusted, so that it exactly reaches the desired topological interface after the duration $\\Delta_t$.\n\n\\begin{figure\n\\includegraphics[width=9.3cm]{Fig13.png}\n\\vspace{-0.cm} \\caption{Wave-packet dynamics in the 2D honeycomb lattice. (a) A small Gaussian wave packet is initially prepared at the center of the system [i.e. on the central interface at $x\\!=\\!x_R$], and its mean quasi-momentum $\\bs k^0\\!=\\!(0,2.1\/a)$ is adjusted so as to maximize the projection unto the chiral localized modes. Due to its small mean-deviation $\\sigma_x\\!=\\!3a$, which is of the order of the localization length of the central chiral mode, this wave packet projects unto this chiral mode with about $90\\%$ efficiency, leading to a clear chiral motion along the central topological interface. (b) Same wave packet, but shifting its initial mean position away from the interface $x^0\\!=\\!x_R\\!+\\!20a$: This wave packet mainly projects unto the (non-chiral) delocalized states and diffuses into the bulk. (c) The wave packet is prepared at the center of the system, but with an opposite initial mean quasi-momentum $\\bs k^0\\!=\\!(0,-2.1\/a)$: As in (b), the wave packet projects unto delocalized states and diffuses into the bulk. (d) Initially preparing the wave packet on the other interface [$x\\!=\\!x_L$], with mean quasi-momentum $\\bs k^0\\!=\\!(0,-2.1\/a)$: This wave packet projects unto the other localized mode and undergoes the opposite chiral motion. In all cases, the system parameters are $t_{\\text{NNN}}\\!=\\!0.4 t_{\\text{NN}}$, $\\delta_{\\text{max}}\\!=\\!20 t_{\\text{NN}}$, $L_x\\!=\\!100a$ and $L_y\\!=\\!150a$, and the trap parameters are $V_{x}\\!=\\!V_{y}\\!=\\!0.2 t_{\\text{NN}}$ and $L_{\\text{obs}}\\!=\\! (L_y\/2) \\sqrt{V_y\/t_{\\text{NN}}}$. In all figures, time is expressed in units of $\\hbar\/t_{\\text{NN}}$.}\n\\label{Fig_dynamics_linear}\n\\end{figure}\n\n\\begin{figure\n\\includegraphics[width=9.3cm]{Fig14.png}\n\\vspace{-0.cm} \\caption{Center-of-mass evolution along the $y$ direction, $\\langle y \\rangle\\!=\\!\\langle y \\rangle(t)$, for the four situations [(a)-(d)] illustrated in Fig.~\\ref{Fig_dynamics_linear}. The cases (a) and (d) correspond to opposite chiral motion along the interfaces $x_R$ and $x_L$, respectively. The cases (b) and (c) illustrate non-chiral motion generated by the bulk (delocalized) states. Time is expressed in units of $\\hbar\/t_{\\text{NN}}$.}\n\\label{Fig_COM_linear}\n\\end{figure}\n\n\n\\subsection{Large clouds and the differential measurement}\n\nIn the previous Section~\\ref{section:dynamics_linear}, we investigated the dynamics of small wave packets, whose size maximizes the projection unto the localized mode; in the realistic system configurations considered here, the localization length is found to be of the order of five lattice sites [see also Figs.~\\ref{Fig_4} and \\ref{Fig_2D_states}(b)]. In this section, we discuss the fate of more experimentally realistic wave packets, whose size typically exceeds the localization length of the topological modes. \n\n\\subsubsection{Dynamics of larger clouds}\\label{Section:large}\n\nThe dynamics shown in Fig.~\\ref{Fig_larger_dynamics}(a) corresponds to the same system configuration as in Fig.~\\ref{Fig_dynamics_linear}(a), but with an initial Gaussian wave packet of mean-deviation $\\sigma_x\\!=\\!20 a$ instead of $\\sigma_x\\!=\\!3 a$ (and a slightly larger system size along the propagation direction $y$). Due to its larger width, this initial state only projects with about $30\\%$ efficiency unto the chiral mode that is localized on the central topological interface. This leads to significant bulk noise in the background of the moving cloud, as compared with the result shown in Fig.~\\ref{Fig_dynamics_linear} (a), which limits the detection of the chiral motion associated with the localized topological mode. We emphasize that, although the larger cloud potentially overlaps with the other topological interface at $x\\!=\\!x_L$, this initial state does not project unto the corresponding mode (of opposite chirality): This is due to the fact that the dispersion relations of the two counter-propagating modes are associated with disconnected regions in $\\bs k$-space [Fig.~\\ref{Fig_4}(a)]. Note that in experiments, in particular when using fermionic atoms, the cloud may be broad in both real- and momentum-space, which is expected to increase the contribution of bulk states.\n\n\\subsubsection{The differential measurement}\\label{section_differential}\n\n\nAs proposed in Ref.~\\cite{GoldmanPNAS}, a differential measurement can be performed to improve the detection of chiral modes in the presence of noisy backgrounds, which are typically associated with the contribution of delocalized states to the particle density. This idea is based on the fact that only chiral modes are severely affected when performing a time-reversal (TR) transformation to the system, as we now briefly recall. For a 2D square lattice in a uniform magnetic field, this TR transformation consists in reversing the sign of the applied magnetic field: This leaves the dispersion of the bulk bands perfectly unchanged, but it reverses their Chern number, and hence, also the chirality of all the topological edge modes present in the bulk gaps \\cite{GoldmanPNAS}: Hence, subtracting the particle density associated with two time-reversal-related configurations potentially annihilates any contribution from the bulk, allowing for clean detection of the chiral modes dynamics. In the context of the Haldane model, Eq.~\\eqref{Eq:Haldane}, this transformation consists in reversing the sign of the TR-breaking term, i.e.~$t_{\\text{NNN}}\\!\\rightarrow\\! -t_{\\text{NNN}}$. As a technical remark, one should note that in the presence of the offset $\\Delta_{\\text{AB}}$, inversion symmetry is also broken: As a consequence, the bulk dispersions are no longer perfectly immune to the TR transformation. This observation is irrelevant when considering a completely filled band, but it does affect the differential measurement when it is applied to wave packets that are localized in $\\bs k$-space, see e.g. Ref. \\cite{Jotzu2014}.\n\nWe now illustrate how this differential measurement can be exploited in the present proposal, by applying it to the situation discussed above in Section~\\ref{Section:large} and depicted in Fig.~\\ref{Fig_larger_dynamics}(a).\nFirst, in Fig.~\\ref{Fig_larger_dynamics}(b) we show the time-reversal counterpart of Fig.~\\ref{Fig_larger_dynamics}(a), which was obtained by reversing the sign of the TR-breaking term, i.e.~$t_{\\text{NNN}}\\!\\rightarrow\\! -t_{\\text{NNN}}$, as well as the sign of the mean quasi-momentum $\\bs k^0\\!\\rightarrow\\! -\\bs k^0$ of the initial Gaussian wave packet. \nEven if the particle density again shows a significant contribution from the delocalized states, this result shows how the TR transformation indeed reverses the general direction of propagation along $y$. Next, we subtract the densities associated with the two TR-counterparts, and we show the corresponding differential measurement in Fig.~\\ref{Fig_larger_dynamics}(c). As anticipated above, we find that the contribution of the delocalized (bulk) states is largely annihilated by the differential measurement, allowing for a clear detection of the chiral modes (including the measurement of their group velocity), even in the regime of large atomic clouds. We note that the residual noise, which is visible in Fig.~\\ref{Fig_larger_dynamics}(c), is due to a weak asymmetry in the bulk bands, which is a direct consequence of the inversion-symmetry-breaking offset $\\Delta_{\\text{AB}}$, as announced above.\n\n\\begin{figure\n\\includegraphics[width=9.3cm]{Fig15.png}\n\\vspace{-0.cm} \\caption{Dynamics in the 2D honeycomb lattice for a large wave packet, of mean deviation $\\sigma_x\\!=\\!20 a$. The system parameters are the same as in Fig.~\\ref{Fig_dynamics_linear}, except $L_y\\!=\\!180a$. (a) The large wave packet is prepared on the central topological interface, with mean quasi-momentum $\\bs k^0\\!=\\!(0,2.1\/a)$, so as to maximize the projection unto the localized modes [as in Fig.~\\ref{Fig_dynamics_linear} (a)]. However, due to its large width, this state only projects unto the localized mode with about $30\\%$ efficiency, which leads to significant bulk noise in the background of the moving cloud [compare with Fig.~\\ref{Fig_dynamics_linear} (a)]. (b) The time-reversal (TR) counterpart of the configuration depicted in (a), namely, when reversing the TR-breaking term $t_{\\text{NNN}}\\!\\rightarrow\\! -t_{\\text{NNN}}$ and the mean quasi-momentum of the initial state $\\bs k^0\\!\\rightarrow\\! -\\bs k^0$. (c) The differential measurement, obtained by subtracting the TR-related situations in (a) and (b), allows one to significantly reduce the noise associated with the bulk contributions, and hence, highlight the motion of topological chiral modes. In all figures, time is expressed in units of $\\hbar\/t_{\\text{NN}}$.}\n\\label{Fig_larger_dynamics}\n\\end{figure}\n\n\\subsection{The radial-interface case}\n\nIn this section, we present the time-evolution of a wave packet initially prepared on the radial topological interface defined in Eq.~\\eqref{space_offset_radial}, and in the presence of the trap [Eq.~\\ref{trap}]. The corresponding time-evolving particle density is shown in Fig.~\\ref{Fig_radial_dynamics}, for a Gaussian wave packet whose mean-deviation and phase have been adjusted so as to maximize projection unto the chiral localized mode. This result highlights the advantage of probing the physics of chiral modes using radial topological interfaces, as the latter is particularly immune to the presence of the harmonic trap [see also Fig.~\\ref{Fig_harmonic_radial}], and hence allows for the analysis of edge-state physics over ``arbitrarily\" long observation times. \n\n\\begin{figure}[h!]\n\\includegraphics[width=9.3cm]{Fig16.png}\n\\vspace{-0.cm} \\caption{Wave-packet dynamics in the 2D honeycomb lattice for the radial-interface configuration [Eq.~\\eqref{space_offset_radial}], with an interface radius $R_{\\text{inter}}\\!=\\!17a$, $t_{\\text{NNN}}\\!=\\!0.4 t_{\\text{NN}}$, and a harmonic potential of strength $V_0\\!=\\!0.25t_{\\text{NN}}$, see also Eq.~\\ref{trap} and Fig.~\\ref{Fig_harmonic_radial}. In all figures, time is expressed in units of $\\hbar\/t_{\\text{NN}}$.}\n\\label{Fig_radial_dynamics}\n\\end{figure}\n\n\n\n\n\n\\section{Studying the topological nature of interfaces using disorder}\\label{Section:disorderb}\n\nThe observation of the quantum Hall effect in solid-state physics relies on the robustness of chiral edge modes against disorder~\\cite{Yoshioka}. Indeed, whereas the bulk states are localized by disorder, the chiral nature of topological edge modes prevents them from any backscattering processes (as long as opposite edges are well spatially separated). In this section, we verify that the topological modes that propagate along engineered topological interfaces within the system are equally robust against disorder. In our study of disorder, we take this perturbation to be in the form of a random (on-site) potential, with energies uniformly distributed between 0 and $D\\!=\\!2 t_{\\text{NN}}$. We point out that such a disordered potential can be engineered in a cold-atom experiment, using optical speckle potentials \\cite{Sanchez-Palencia\/Lewenstein}. \n\nIn Fig.~\\ref{Fig_small_dynamics_disorder} we show how the dynamics of small wave packets is modified by disorder. Comparing these results with the clean situation previously shown in Figs.~\\ref{Fig_dynamics_linear}(a)-(b) clearly reveals the robustness of the edge mode that propagates along the central topological interface [Fig.~\\ref{Fig_small_dynamics_disorder} (a)], as well as the disorder-induced localization of wave packets made of bulk (non-chiral) states [Fig.~\\ref{Fig_small_dynamics_disorder} (b)].\n\nMost importantly, we show in Fig.~\\ref{Fig_larger_dynamics_disorder}(a) how disorder can be \\emph{exploited} to improve the detection of topological modes in cold-atom systems: By annihilating the dispersion of the bulk states, which typically constitute the majority of populated states in realistic situations involving wide atomic clouds, the disorder naturally enhances the signal associated with the propagating chiral modes. Indeed, the time-evolving cloud depicted in Fig.~\\ref{Fig_larger_dynamics_disorder}(a) directly reveals the propagation of the central topological mode, in contrast with the disorder-free situation previously shown in Fig.~\\ref{Fig_larger_dynamics} (a) for the same system parameters. Finally, Fig.~\\ref{Fig_larger_dynamics_disorder}(b) shows how combining disorder and the aforementioned differential measurement [Section~\\ref{section_differential}] allows one to reach a remarkably precise visualization of the edge-mode propagation, in the absence of residual noise associated with the bulk [compare with the disorder-free case in Fig.~\\ref{Fig_larger_dynamics}(c)]. The fact that the differential measurement is improved by disorder relies on that this perturbation dephases the cloud: This smoothes out the residual background noise associated with the asymmetry in the bulk bands under the TR transformation [see the discussion in Section~\\ref{section_differential}].\n\nThe results in Fig.~\\ref{Fig_larger_dynamics_disorder} highlight how engineered disorder, as created by optical speckle potentials \\cite{Sanchez-Palencia\/Lewenstein}, could be used as a powerful tool for the detection and study of topological-edge-state physics in cold-atom systems. In this sense, the important role played by disorder in revealing QH physics~\\cite{Yoshioka} is thus not restricted to solid-state physics.\n\n\n\\begin{figure\n\\includegraphics[width=9.4cm]{Fig17.png}\n\\vspace{-0.cm} \\caption{Wave-packet dynamics in the 2D honeycomb lattice for the same configuration as in Fig.~\\ref{Fig_dynamics_linear}(a)-(b), but including a disordered potential of strength $D\\!=\\!2 t_{\\text{NN}}$, and averaging over ten realizations of disorder. (a) The edge mode localized on the central topological interface is immune to disorder, and freely propagates along the topological interface. (b) A wave packet associated with (non-chiral) bulk states is localized by the disorder [compare with Fig.~\\ref{Fig_dynamics_linear}(b)].}\n\\label{Fig_small_dynamics_disorder}\n\\end{figure}\n\n\n\n\\begin{figure\n\\includegraphics[width=9.3cm]{Fig18.png}\n\\vspace{-0.cm} \\caption{Dynamics of a large wave packet, in the same configuration as in Fig.~\\ref{Fig_larger_dynamics}(a), but including a disordered potential of strength $D\\!=\\!2 t_{\\text{NN}}$, and averaging over thirty realizations. (a) The chiral motion of the localized mode is clearly identified in the presence of disorder, as the latter annihilates the propagation of the bulk states. (b) The differential measurement in the presence of disorder reveals clean edge-mode dynamics at the central topological interface, by removing the contribution of the (dephased) bulk states to the density [compare with Fig.~\\ref{Fig_larger_dynamics}(c)].}\n\\label{Fig_larger_dynamics_disorder}\n\\end{figure}\n\n\n\\section{Experimental implementation}\\label{Section:experiment}\n\\subsection{The linear interface}\\label{Section:experiment_lin}\n\n\\begin{figure\n\\includegraphics[width=8.6cm]{Fig19.png}\n\\vspace{-0.cm} \\caption{Experimental implementation of the linear scheme. \n(a) Beam setup. The retro-reflected laser beams $X_1$ and $Y_1$, with wavelength $\\lambda_1$ interfere with each other at the position of the atomic cloud (red) and are phase stabilized. The additional beam $\\overline{X}$ (blue) has a different wavelength $\\overline{\\lambda}$ and has the same transverse spatial mode as $X_1$. \n(b) Cut through the optical lattice potentials at $y=0$. The standing wave created by $\\overline{X}$ (blue) has a lattice spacing of $\\overline{\\lambda}\/2$, which is slightly more than half of $\\lambda_1$, the $x$-direction spacing of the interference lattice created by $X_1$ and $Y_1$ (red). Therefore, their extrema only coincide at a particular point in the system. The resulting total potential (green) then has a spatially varying site-offset $\\Delta_{\\text{AB}}$.\n(c) Sketch of the resulting potential, with the tight-binding lattice structure (black) superimposed. Minima are dark green, maxima light green. The variation of $\\Delta_{\\text{AB}}$ in the $x$ direction is exaggerated for better visibility.}\n\\label{Fig_Setup1}\n\\end{figure}\n\nThe honeycomb lattices with a spatially dependent site offset discussed in this work can be implemented experimentally using an extension of the tunable-geometry lattice introduced in Ref.~\\cite{Tarruell2012}. \nA linear variation of $\\Delta_{\\text{AB}}$, as introduced in Eq.~\\eqref{space_offset}, can be created as follows:\nFirst, a pair of red-detuned retro-reflected laser beams with identical wavelength $\\lambda_1$ and single-beam lattice depths $V_{X1}$ and $V_{Y1}$ are phase-stabilized with respect to each other and oriented along the $x$ and $y$ direction respectively [see Fig. \\ref{Fig_Setup1}(a)]. \nAt their intersection, where the atomic cloud is placed, the resulting potential experienced by the atoms is given by \n\\begin{align}\nV_1(x,y) = & - V_{X1} \\cos^2(k_1\nx)-V_{Y1} \\cos^2(k_1 y)\\nonumber \\\\\n&- 2\\sqrt{V_{X1}V_{Y1}}\\cos(k_1 x)\\cos(k_1\ny), \\label{eqlattice_xy1}\n\\end{align} \nwhere $k_1=2\\pi\/\\lambda_1$, which corresponds to a checkerboard lattice \n\\footnote{Here and henceforth, the potential is given in the $xy$ plane only. In the $z$ direction, either a weak harmonic trap, as in Ref.~\\cite{Tarruell2012}, or an additional optical lattice, as in Ref.~\\cite{Uehlinger2013PRL}, can be used. }.\nIts unit vectors are oriented at $\\pm 45^{\\circ}$ with respect to the $x$ axis and the site spacing along the $x$ direction is $\\lambda_1$, see Fig.~\\ref{Fig_Setup1}(b).\n\n\n\n\nAn additional laser beam with lattice depth $V_{\\overline{X}}$, operating at wavelength $\\overline{\\lambda}$ and oriented along the $x$ direction, gives rise to an additional standing wave with site spacing $\\overline{\\lambda}\/2$ and potential \n\\begin{equation}\n\\overline{V}(x,y) = -V_{\\overline{X}} \\cos^2(\\overline{k}x+\\theta\/2),\n\\label{eqlattice_xb}\n\\end{equation}\nwhere $\\overline{k}=2\\pi\/\\overline{\\lambda}$.\nWe first consider the case where $\\overline{\\lambda}$ is so close to $\\lambda_1$ \nthat we can assume $\\overline{k}=k_1$ over the size of the atomic cloud. \nHowever, because of the distance between the retro-reflecting mirror and the cloud, the small difference between $\\overline{\\lambda}$ and $\\lambda_1$ (typically on the order of $1$\\,pm or - in terms of frequency - a few hundred MHz) still leads to a shift between the two potentials, which is captured \\textit{via} $\\theta$. \nSetting $\\theta\\approx\\pi$, this gives rise to the honeycomb lattice of Ref. \\cite{Tarruell2012}. \nIts near-constant site-offset $\\Delta_{\\text{AB}}$ is tuned via $\\theta$ and can be calibrated using Bloch-Zener oscillations \\cite{Uehlinger2013EPJ}. It becomes 0 when $\\theta\\!=\\!\\pi$. \n\nIn order to achieve a significant spatial variation of $\\Delta_{\\text{AB}}$ over the size of the cloud, we increase the difference between $\\overline{\\lambda}$ and $\\lambda_1$, such that $\\overline{k}=k_1$ is not a good approximation any more. \nThen, the extrema of $\\overline{V}(x,y)$ and $V_1(x,y)$ will line up differently depending on the position within the atomic cloud, leading to a smooth variation of $\\Delta_{\\text{AB}}$ along the $x$ direction, as illustrated in Fig. \\ref{Fig_Setup1}(b)-(c). \n\nThe tight-binding parameters corresponding to this optical lattice depend on the choice of atomic species and laser wavelength \\cite{Lewenstein2012}.\nFor example, when using $^{40}$K atoms and setting the laser intensity such that $t_{\\mathrm{NN}}=h\\times 200$\\,Hz, a variation of $0.2t_{\\mathrm{NN}}$ per site (i.e. $\\delta_{\\text{max}}\/L_x=0.2t_{\\mathrm{NN}}\/a$ in Eq.~\\eqref{space_offset} is achieved with $\\lambda_1 = 1064.0$\\,nm and $\\overline{\\lambda} = 1064.5$\\,nm\n\\footnote{\nFor comparison, in the case discussed above, the ``near-constant'' $\\Delta_{\\text{AB}}$ as realized in Ref.~\\cite{Tarruell2012}, typically corresponds to changes in $\\Delta_{\\text{AB}}$ of about $5 \\times 10^{-4}t_{\\mathrm{NN}}$ per site.\n}.\nFor these parameters, the spatial variation of $\\Delta_{\\text{AB}}$ deviates from a linear function by less than $0.1\\%$, whilst $t_{\\mathrm{NN}}$ changes by at most 1\\% over a range of 100 sites.\n\n\\subsection{The radial-symmetric scheme}\\label{Section:experiment_circ}\n\n\\begin{figure\n\\includegraphics[width=8.6cm]{Fig20.png}\n\\vspace{-0.cm} \\caption{\nExperimental implementation of the radial-symmetric scheme. \n(a) Beam setup. Lasers $X_1$, $Y_1$ (red) and $\\overline{X}$ (dark red, dashed) are similar as in Fig.~\\ref{Fig_Setup2}, except that now $\\overline{\\lambda} \\approx \\lambda_1$. In addition, a pair of phase-stabilised beams $X_2$ and $Y_2$ with a smaller beam waist are focused onto the center of the atomic cloud.\n(b) Laser beams $X_1$ and $Y_1$, together with $\\overline{X}$ form an imbalanced honeycomb lattice potential, which is nearly uniform over the extent of the atomic cloud (dark blue minima, light blue maxima, tight-binding structure indicated by black lines). \nThe minima of the additional potential (orange) created by $X_2$ and $Y_2$ are aligned with every second site of the honeycomb lattice. Their intensity decreases away from the center.\n(c) The resulting potential has a radially varying site offset (dark green minima, light green maxima). The spatial variation is exaggerated for better visibility.}\n\\label{Fig_Setup2}\n\\end{figure}\n\n\n\nIn order to create a two-dimensional radial-symmetric variation of $\\Delta_{\\text{AB}}$, as introduced in Section \\ref{section:radial_interface}, a different scheme can be used: \nA honeycomb lattice with a near-constant site offset is created by laser beams $X_1, Y_1$ and $\\overline{X}$ (taking $\\overline{\\lambda}$ very close to $\\lambda_1$ \nso that $k_1 = \\overline{k}$ is a good approximation) as outlined above and shown in Fig.~\\ref{Fig_Setup2}(a)-(b). Its lattice structure can be assumed homogeneous over the size of the atomic cloud. \n\nTwo additional phase-stabilised, retro-reflected laser beams $X_2$ and $Y_2$, operating at $\\lambda_2 \\approx \\lambda_1$ have much smaller transverse beam waists, $w$, than the other beams. The checkerboard lattice potential they create is given by \n\\begin{align}\nV_2(x,y) = & \\;\\mathrm{e}^{-2(x^2+y^2)\/w^2}\\times \\nonumber \\\\ \n \\big( &-V_{X2} \\cos^2(k_2\nx+\\theta\/2)-V_{Y2} \\cos^2(k_2 y)\\nonumber \\\\\n&- 2\\sqrt{V_{X2}V_{Y2}}\\cos(k_2 x + \\theta\/2)\\cos(k_2y) \\big). \n\\label{eqlattice_xy2}\n\\end{align} \nThe small detunings between $\\lambda_2$, $\\lambda_1$ and $\\overline{\\lambda}$ as well as the distances between retro-reflecting mirrors and the atomic cloud are chosen such that $\\theta$ is the same as in Eq.~\\eqref{eqlattice_xb} and we can assume $k_2=k_1=\\overline{k}$. Then, the minima of this potential coincide with every other site (i.e. with either the A or B sublattice, see Fig.~\\ref{Fig_one}) of the honeycomb lattice [see Fig.~\\ref{Fig_Setup2}(b)], thereby contributing to the site offset $\\Delta_{\\text{AB}}$. The Gaussian envelope of the beams means that this contribution varies spatially. For example, by choosing $V_{X2}$ and $V_{Y2}$ correctly, we can set $\\Delta_{\\mathrm{AB}}=0$ in the center of the cloud and let it increase as a Gaussian function of the radial distance to the center, as illustrated in Fig.~\\ref{Fig_Setup2}(c).\n\n\\section{Concluding remarks and outlooks}\\label{Section:conclusions}\n\nIn this work, we introduced a novel scheme allowing for the direct detection of topological propagating modes within 2D ultracold atomic gases. Our proposal is based on the engineering of topological interfaces, which localize topologically-protected modes in desired regions of space (e.g.~at the center of a harmonic trap, typically present in cold-atom experiments). This allows for real-space detection of topological transport in a highly controllable and versatile platform. In particular, we stress that the trajectory performed by these topologically-protected modes within the system can be tuned by shaping the form of the topological interfaces, which suggests interesting applications based on topological quantum transport. In particular, this opens an exciting avenue for the manipulation and probing of topological modes in atomic systems, where disorder, inter-particle interactions and external gauge fields can be induced and controlled at will. We note that similar topological interfaces could be created within photonic crystals, where system parameters (e.g.~the hopping amplitudes and on-site potentials entering engineered tight-binding models) can also be precisely addressed in a local manner~\\cite{Longhi:2006,Mukherjee:2016}. \n\n\nThe scheme introduced in this work has been illustrated based on a 2D model~\\cite{Haldane1988}, which exhibits the integer quantum Hall effect; namely, a non-interacting system featuring 2D Bloch bands with non-zero Chern numbers~\\cite{Nagaosa:2010}. However, we point out that our scheme, which consists in varying a model parameter in space in view of creating local topological phase transitions (i.e.~topological interfaces), can be applied to any model exhibiting topological band structures. For instance, similar interfaces could be engineered in the quantum-spin-Hall regime of $Z_2$ (time-reversal-invariant) topological insulators: Considering the 2D Kane-Mele model~\\cite{Kane:2005}, which is a direct extension of the Haldane model~\\cite{Haldane1988} to spin-1\/2 particles, this could be realized either by introducing a spatially-varying offset $\\Delta_{\\mathrm{AB}}$ between the sites of the honeycomb lattice (as proposed in this work), or by engineering a spatially-varying spin-orbit coupling. In this spinful TRS situation, the topological interfaces would host helical topological modes, namely, modes associated with opposite spins and propagating in opposite directions (see Refs.~\\cite{Kane:2005,GoldmanPRL2010} and the recent photonics proposal~\\cite{Barik:2016}). We stress that, in contrast to the spinless TRS configuration of Section~\\ref{Section:TRS}, these helical counter-propagating modes are topologically protected (backscattering processes are forbidden by time-reversal symmetry in this spinful case~\\cite{Kane:2005}). Hence, engineering interfaces in $Z_2$ topological insulators would introduce adjustable guides for topologically-protected spin transport within 2D systems. The same strategy could be applied to higher-dimensional systems, such as 3D topological insulators~\\cite{Fu:2007}: By varying the spin-orbit coupling of such systems in space, one could engineer 2D topological interfaces hosting a single 2D Dirac fermion. In this scheme, these intriguing excitations would be located \\emph{within} the system, instead of at its surfaces, which could also open interesting avenues for spin transport in 3D systems. It is worth pointing out that $Z_2$ topological insulators can be realized in 2D optical lattices~\\cite{GoldmanPRL2010,Beri:2011,Kennedy:2013}; see Ref.~\\cite{Aidelsburger2013} for a first experimental realization of such a model with cold atoms. Furthermore, these setups could be extended in view of creating 3D topological insulators~\\cite{Bermudez:2010}, but also, to reveal the 4D QH effect~\\cite{Price:2015,Price:2016}, in cold-atom experiments. Creating topological interfaces within a 4D QH atomic system~\\cite{Price:2015} offers a unique platform to investigate 3D topological surface modes (i.e.~spatially isolated Weyl fermions) in the laboratory.\n\nFinally, the physics of topological edge modes is certainly not restricted to non-interacting quantum systems. In particular, edge modes play a crucial role in fractional quantum Hall (FQH) liquids, where they present exotic (sometimes counter-intuitive) structures~\\cite{Wen:1995}. For instance, FQH liquids potentially exhibit counter-propagating edge modes (allowing back-scattering on the edge in the presence of impurities), while still presenting a finite and quantized Hall conductivity~\\cite{Haldane:1995,Wen:1995,Kane:1995,Kane:1994,Inoue:2013}. While these counter-propagating modes remained undetectable in standard edge-magnetoplasma experiments~\\cite{Ashoori:1992,Wen:1995}, they were recently revealed through shot noise~\\cite{Bid:2010,Gurman:2012,Inoue:2013} and thermometry~\\cite{Venkatachalam:2012} measurements. Even more recently, it was suggested in Ref.~\\cite{Goldstein:2016} that the presence of such counter-propagating edge modes could have important consequences on the detection of fractional (anyonic) statistics, based on QH Mach-Zehnder interferometers~\\cite{Ji:2003}. The unique possibility of creating topological interfaces in a cold-atom experiment offers a promising platform for the analysis of these topological edge structures, where Bragg spectroscopy~\\cite{Liu:2010,GoldmanPRL} and high-resolution imaging techniques could be exploited in view of revealing their exotic dispersion relations and dynamical properties in the presence of controllable interactions and disorder.\n\n\n\\acknowledgements We acknowledge A. Dauphin, M. Kolodrubetz, and D. T. Tran for helpful discussions. N.G. is financed by the FRS-FNRS Belgium and by\nthe BSPO under PAI Project No. P7\/18 DYGEST. The ETH team acknowledges SNF, NCCR-QSIT, and QUIC (Swiss State Secretary for Education, Research and Innovation contract number 15.0019) for funding. R.D. acknowledges support from ETH Zurich Postodoctoral Program and Marie Curie Actions for People COFUND program.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzrdst b/data_all_eng_slimpj/shuffled/split2/finalzzrdst new file mode 100644 index 0000000000000000000000000000000000000000..25ae5fef23a8e185150c1f67c0d1c74a90a8140e --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzrdst @@ -0,0 +1,5 @@ +{"text":"\n\nCopyright \u00a9 2012, 2009 by The McGraw-Hill Companies, Inc. All rights reserved. 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McGraw-Hill and its licensors do not warrant or guarantee that the functions contained in the work will meet your requirements or that its operation will be uninterrupted or error free. Neither McGraw-Hill nor its licensors shall be liable to you or anyone else for any inaccuracy, error or omission, regardless of cause, in the work or for any damages resulting therefrom. McGraw-Hill has no responsibility for the content of any information accessed through the work. Under no circumstances shall McGraw-Hill and\/or its licensors be liable for any indirect, incidental, special, punitive, consequential or similar damages that result from the use of or inability to use the work, even if any of them has been advised of the possibility of such damages. This limitation of liability shall apply to any claim or cause whatsoever whether such claim or cause arises in contract, tort or otherwise.\n\n## **Contents**\n\nPreface\n\n**1. Introduction: The Paradox of Excess and Deprivation**\n\n**2. Paying for Health Care**\n\n**3. Access to Health Care**\n\n**4. Reimbursing Health Care Providers**\n\n**5. How Health Care Is Organized\u2014I: Primary, Secondary, and Tertiary Care**\n\n**6. How Health Care Is Organized\u2014II: Health Delivery Systems**\n\n**7. The Health Care Workforce and the Education of Health Professionals**\n\n**8. Painful Versus Painless Cost Control**\n\n**9. Mechanisms for Controlling Costs**\n\n**10. Quality of Health Care**\n\n**11. Prevention of Illness**\n\n**12. Long-Term Care**\n\n**13. Medical Ethics and Rationing of Health Care**\n\n**14. Health Care in Four Nations**\n\n**15. Health Care Reform and National Health Insurance**\n\n**16. Conflict and Change in America's Health Care System**\n\n**17. Conclusion: Tensions and Challenges**\n\n**18. Questions and Discussion Topics**\n\nIndex\n\n## **Preface**\n\n_Understanding Health Policy: A Clinical Approach_ is a book about health policy as well as about individual patients and caregivers and how they interact with each other and with the overall health system. We, the authors, are practicing primary care physicians\u2014one in a public hospital and clinic and the other, for many years, in a private practice. We are also analysts of our nation's health care system. In one sense, these two sides of our lives seem quite separate. When treating a patient's illness, it seems that health expenditures as a percentage of gross domestic product or variations in surgical rates between one city and another seem remote if not irrelevant\u2014but they are neither remote nor irrelevant. Health policy affects the patients we see on a daily basis. Managed care referral patterns determine to which specialist we can send a patient, the coverage gaps for outpatient medications in the Medicare benefit package affects how we prescribe medications for our elderly patients, and differences in access to care between families on Medicaid and those with private coverage influences which patients ended up seeing one of us (in the private sector) and which the other (in a public setting). In _Understanding Health Policy_ , we hope to bridge the gap separating the microworld of individual patient visits and the macrouniverse of health policy.\n\n### **THE AUDIENCE**\n\nThe book is primarily written for health science students\u2014medical, nursing, nurse practitioner, physician assistant, pharmacy, social work, public health, and others\u2014who will benefit from understanding the complex environment in which they will work. Physicians feature prominently in the text, but in the actual world of clinical medicine, patients' encounters with other health care givers are an essential part of their health care experience. Physicians would be unable to function without the many other members of the health care team. Patients seldom appreciate the contributions made to their well-being by public health personnel, research scientists, educators, and many other health-related professionals. We hope that the many nonphysician members of the clinical care, public health, and health science education teams as well as students aspiring to join these teams will find the book useful. Nothing can be accomplished without the combined efforts of everyone working in the health care field.\n\n### **THE GOAL OF THE BOOK**\n\n_Understanding Health Policy_ attempts to explain how the health care system works. We focus on basic principles of health policy in hopes that the reader will come away with a clearer, more systematic way of thinking about health care in the United States, its problems, and the alternatives for managing these problems. Most of the principles also apply to understanding health care systems in other nations.\n\nGiven the public's concerns about health care in the United States, the book concentrates on the failures of the system. We spend less time on the successful features because they need less attention. Only by recognizing the difficulties of the system can we begin to fix its problems. The goal of this book, then, is to help all of us understand the health care system so that we can better work in the system and change what needs to be changed.\n\n### **CLINICAL VIGNETTES**\n\nIn our attempt to unify the overlapping spheres of health policy and health care encounters by individuals, we use clinical vignettes as a central feature of the book. These short descriptions of patients, physicians, and other care-givers interacting with the health care system are based on our own experiences as physicians, the experiences of colleagues, or cases reported in the medical literature or popular press. Most of the people and institutions presented in the vignettes have been given fictitious names to protect privacy. Some names used are emblematic of the occupations, health problems, or attitudes portrayed in the vignettes; most do not have special significance.\n\n### **OUR OPINIONS**\n\nIn exploring the many controversial issues of health policy, our own opinions as authors inevitably color and shade the words we use and the conclusions we reach. We present several of our most fundamental values and perspectives here.\n\n### **THE RIGHT TO HEALTH CARE**\n\nWe believe that health care should be a right enjoyed equally by everyone. Certain things in life are considered essential. No one gets excited if someone is turned away from a movie or concert because he or she cannot afford a ticket. But sick people who are turned away from a medical practice can make headlines, and rightly so. A simple statement of the right to health care reads something like this: All people should have equal access to a reasonable level of appropriate health services, regardless of ability to pay.\n\nIn 2009, the United States entered into a fierce debate over whether health care should be a right. The debate focused on President Barack Obama's campaign to enact universal health insurance. Following a year of public ferment, Congress passed the Affordable Care Act, which goes a long way toward guaranteeing health care as a right. Yet, at the time of writing this edition of _Understanding Health Policy,_ the controversy continues with challenges to implementation of the Affordable Care Act.\n\n### **THE IMPERATIVE TO CONTAIN COSTS**\n\nWe believe that limits must be placed on the costs of health care. Cost controls can be imposed in a manner that does relatively little harm to the health of the public. The rapidly rising costs of health care are in part created by scientific advances that spawn new, expensive technologies. Some of these technologies truly improve health care, some are of little value or harmful, and others are of benefit to some patients but are inappropriately used for patients whom they do not benefit. Eliminating medical services that produce no benefit is one path to \"painless\" cost control (see Chapter 8).\n\nReduction in the rapidly rising cost of administering the health care system is another route to painless cost containment. Administrative excess wastes money that could be spent for useful purposes, either within or outside the health care sector. While large bureaucracies do have the advantage of creating jobs, the nation and the health care system have a great need for more socially rewarding and productive jobs (eg, home health aides, drug rehabilitation counselors, childcare workers, and many more) that could be financed from funds currently used for needless administrative tasks.\n\nThere is a growing consensus that health care cost increases are bad for the economy. Employers complain that the high cost of health insurance for employees reduces international competitiveness. If government health expenditures continue their rapid rise, other publicly financed programs essential to the nation's economy (eg, education and transportation) will be curtailed and the unsustainable government budget deficits will strain the future of the nation's well-being.\n\nRising costs are harmful to everyone because they make health services and health insurance unaffordable. Many companies are shifting more health care costs onto their employees. As government health budgets balloon, cutbacks are inevitable, generally hurting the elderly and the poor. Individuals with no health insurance or inadequate coverage have a far harder time paying for care as costs go up. As a general rule, when costs go up, access goes down.\n\nFor these reasons, we believe that health care costs should be contained, using strategies that do the least harm to the health of the population.\n\n### **THE NEED FOR POPULATION-BASED MEDICINE**\n\nMost physicians, nurses, and other health professionals are trained to provide clinical care to individuals. Yet clinical care is not the only determinant of health status; standard of living and public health measures have an even greater influence on the health of a population (see Chapter 3). Health care, then, should have another dimension: concern for the population as a whole. Individual physicians may be first-rate in caring for their patients' heart attacks, but may not worry enough about the prevalence of hypertension, smoking, elevated cholesterol levels, uncontrolled diabetes, and lack of exercise in their city, in their neighborhood, or among the group of patients enrolled in their practices. For years, clinical medicine has divorced itself from the public health community, which does concern itself with the health of the population. We believe that health caregivers should be trained to add a population orientation to their current role of caring for individuals.\n\n### **ACKNOWLEDGMENTS**\n\nWe could not have written this book by ourselves. The circumstances encountered by hundreds of our patients and dozens of our colleagues provided the insights we needed to understand and describe the health care system. Any inaccuracies in the book are entirely our responsibility. Our warmest thanks go to our families, who have provided both encouragement and patience.\n\nEarlier versions of Chapters 2, , , , , and were published serially as articles in the _Journal of the American Medical Association_ (1994;272:634\u2013639, 1994;272:971\u2013977, 1994;272:1458\u20131464, 1995;273:160\u2013167, 1995;274:85\u201390, and 1996;276:1025\u20131031) and are published here with permission (copyright, 1994, 1995, and 1996, American Medical Association).\n\n### **CONCLUSION**\n\nThis is a book about health policy. As such, we will cite technical studies and will make cross-national generalizations. We will take matters of profound personal meaning\u2014sickness, health, providing of care to individuals in need\u2014and discuss them using the detached language of \"inputs and outcomes,\" \"providers and consumers,\" and \"cost-effectiveness analysis.\" As practicing physicians, however, we are daily reminded of the human realities of health policy. _Understanding Health Policy: A Clinical Approach_ is fundamentally about the people we care for: the uninsured janitor enduring the pain of a gallbladder attack because surgery might leave him in financial ruin, or the retired university professor who sustains a stroke and whose life savings are disappearing in nursing home bills uncovered by her Medicare or private insurance plans.\n\nAlmost every person, whether a mother on public assistance, a working father, a well-to-do physician, or a millionaire insurance executive, will someday become ill, and all of us will die. Everyone stands to benefit from a system in which health care for all people is accessible, affordable, appropriate in its use of resources, and of high quality.\n\n_Thomas Bodenheimer \nKevin Grumbach \nSan Francisco, California \nFebruary, 2012_\n\n## **1 Introduction: The Paradox of Excess and Deprivation**\n\n_Louise Brown was an accountant with a 25-year history of diabetes. Her physician taught her to monitor her glucose at home, and her dietician helped her follow a diabetic diet. Her diabetes was brought under good control. Diabetic retinopathy was discovered at yearly eye examinations, and periodic laser treatments of her retina prevented loss of vision. Ms. Brown lived to the age of 88, a success story of the US health care system._\n\n_Angela Martini grew up in an inner-city housing project, never had a chance for a good education, became pregnant as a teenager, and has been on public assistance while caring for her four children. Her Medicaid coverage allows her to see her family physician for yearly physical examinations. A breast examination located a suspicious lesion, which was found to be cancer on biopsy. She was referred to a surgical breast specialist, underwent a mastectomy, was treated with a hormonal medication, and has been healthy for the past 15 years._\n\nFor people with private or public insurance who have access to health care services, the melding of high-quality primary and preventive care with appropriate specialty treatment can produce the best medical care in the world. The United States is blessed with thousands of well-trained physicians, nurses, pharmacists, and other health caregivers who compassionately provide up-to-date medical attention to patients who seek their assistance. This is the face of the health care system in which we can take pride. Success stories, however, are only part of the reality of health care in the United States.\n\n### **EXCESS AND DEPRIVATION**\n\nThe health care system in the United States has been called \"a paradox of excess and deprivation\" (Enthoven and Kronick, 1989). Some persons receive too little care because they are uninsured, inadequately insured, or have Medicaid coverage that many physicians will not accept.\n\n_James Jackson's Medicaid benefits were terminated because of state cutbacks. At age 34, he developed abdominal pain but did not seek care for 10 days because he had no insurance and feared the cost of treatment. He began to vomit, became weak, and was finally taken to an emergency room by his cousin. The physician diagnosed a perforated ulcer with peritonitis and septic shock. The illness had gone on too long; Mr. Jackson died on the operating table. Had he received prompt medical attention, his illness would likely have been cured._\n\n_Betty Yee was a 68-year-old woman with angina, high blood pressure, and diabetes. Her total bill for medications, which were only partly covered under her Medicare plan, came to $200 per month. She was unable to afford the medications, her blood pressure went out of control, and she suffered a stroke. Ms. Yee's final lonely years were spent in a nursing home; she was paralyzed on her right side and unable to speak._\n\n_Mary McCarthy became pregnant but could not find an obstetrician who would accept her Medicaid card. After 7 months, she began to experience severe headaches, went to the emergency room, and was found to have hypertension and preeclampsia. She delivered a stillborn baby._\n\nWhile some people cannot access the care they need, others receive too much care that is costly and may be harmful.\n\n_At age 66, Daniel Taylor noticed that he was getting up to urinate twice each night. It did not bother him much. His family physician sent him to a urologist, who found that his prostate was enlarged (though with no signs of cancer) and recommended surgery. Mr. Taylor did not want surgery. He had a friend with the same symptoms whose urologist had said that surgery was not needed. Since Mr. Taylor never questioned doctors, he went ahead with the procedure anyway. After the surgery he became incontinent of urine._\n\n_Consuelo Gonzalez had a minor pain in her back, which was completely relieved by over-the-counter acetaminophen. She went to the doctor just to make sure the pain was nothing serious, and it was not. The physician gave Ms. Gonzalez a stronger medicine, indomethacin, to take three times a day. The indomethacin caused a bleeding ulcer requiring a 9-day hospital stay at a cost of $27,000 to her health insurer._\n\n#### **Too Little Care**\n\nIn 2009, over 50 million people in the United States had no health insurance. Many are victims of the changing economy, which has shifted from a manufacturing economy based on highly paid full-time jobs with good fringe benefits, toward a service economy with lower-paying jobs that are often part-time and have poor or no benefits (Renner and Navarro, 1989). Three-fourths of uninsured adults are employed. Lack of insurance is not simply a problem of the poor but has also become a middle-class phenomenon, particularly for families of people who are self-employed or work in small establishments. Many people with health insurance have inadequate coverage. In 2007, 45% of adults could not get needed care because they could not afford to pay the bills (Collins et al, 2008).\n\n#### **Too Much Care**\n\nAccording to health services expert Robert Brook (1989):\n\n_...almost every study that has seriously looked for overuse has discovered it, and virtually every time at least double-digit overuse has been found. If one could extrapolate from the available literature, then perhaps one-fourth of hospital days, one-fourth of procedures, and two-fifths of medications could be done without. (Brook, 1989)_\n\nA 1998 report estimated that 20%\u201330% of patients continue to receive care that is not appropriate (Schuster et al, 1998). A 2003 study found that elderly patients in some areas of the country receive 60% more services\u2014hospital days, specialty consultations, and medical procedures\u2014than similar patients in other areas; the patients receiving fewer services had the same mortality rates, quality of care, access to care, and patient satisfaction as those receiving more services (Fisher et al, 2003a and 2003b). In 2009, health care quality expert Brent James estimated that half of all health care dollars are wasted (James, 2009).\n\n### **THE PUBLIC'S VIEW OF THE HEALTH CARE SYSTEM**\n\nHealth care in the United States encompasses a wide spectrum, ranging from the highest-quality, most compassionate treatment of those with complex illnesses, to the turning away of the very ill because of lack of an ability to pay; from well-designed protocols for prevention of illness to inappropriate high-risk surgical procedures performed on uninformed patients. While the past three decades have been witness to major upheavals in health care, one fundamental truth remains: the United States has the least universal, most costly health care system in the industrialized world (Davis et al, 2010).\n\nMany people view the high costs of care and the lack of universal access as indicators of serious failings in the health care system. In 2009, only 15% of people in the United States believed that the system was working well (Blendon et al, 2009). In 2010, 33% of Americans reported not seeing a doctor or not filling a prescription due to costs, a prevalence of access problems considerably higher than that in other developed nations (Schoen et al, 2010).\n\n### **UNDERSTANDING THE CRISIS**\n\nIn order to correct the weaknesses of the health care system while maintaining its strengths, it is necessary to understand how the system works. How is health care financed? What are the causes and consequences of incomplete access to care? How are physicians paid, and what is the effect of their mode of reimbursement on health care costs? How are health care services organized and quality of care enhanced? Is sufficient attention paid to the prevention of illness, and what are different strategies for preventing illness?\n\nHow can the problems of health care be solved? Does the health reform law enacted by Congress in 2010 provide the answer? Can costs be controlled in a manner that does not reduce access? Can access be expanded in a manner that does not increase costs? How have other nations done it\u2014or attempted to do it? How might the health care system in the United States change in the future?\n\n### **REFERENCES**\n\nBlendon RJ et al. The American public and the next phase of the health care debate. _N Engl J Med_. 2009;361:e48.\n\nBrook RH. Practice guidelines and practicing medicine. _JAMA_. 1989;262:3027.\n\nCollins SR et al. _Losing Ground: How the Loss of Adequate Health Insurance Is Burdening Working Families_. New York: Commonwealth Fund; 2008.\n\nDavis K et al. _Mirror, Mirror on the Wall_. New York: Commonwealth Fund; 2010.\n\nEnthoven A, Kronick R. A consumer-choice health plan for the 1990s. _N Engl J Med._ 1989;320:29.\n\nJames BC. A conversation with Brent C James, MD. Health reform debate overlooks physician\u2013patient dynamic. _Managed Care_. December 2009:18(12):31.\n\nFisher ES et al. The implications of regional variations in Medicare spending. Part 1: The content, quality, and accessibility of care. _Ann Intern Med_. 2003a;138:273.\n\nFisher ES et al. The implications of regional variations in Medicare spending. Part 2: Health outcomes and satisfaction with care. _Ann Intern Med._ 2003b;138:288.\n\nRenner C, Navarro V. Why is our population of uninsured and underinsured persons growing? The consequences of the \"deindustrialization\" of the United States. _Int J Health Serv_. 1989;19:433.\n\nSchoen C et al. How health insurance design affects access to care and costs, by income, in eleven countries. _Health Aff (Millwood)_. 2010;29:2323.\n\nSchuster M et al. How good is the quality of health care in the United States? _Milbank Q_. 1998;76:517.\n\n## **2 Paying for Health Care**\n\nHealth care is not free. Someone must pay. But how? Does each person pay when receiving care? Do people contribute regular amounts in advance so that their care will be paid for when they need it? When a person contributes in advance, might the contribution be used for care given to someone else? If so, who should pay how much?\n\nHealth care financing in the United States evolved to its current state through a series of social interventions. Each intervention solved a problem but in turn created its own problems requiring further intervention. This chapter will discuss the historical process of the evolution of health care financing.\n\n### **MODES OF PAYING FOR HEALTH CARE**\n\nThe four basic modes of paying for health care are out-of-pocket payment, individual private insurance, employment-based group private insurance, and government financing (Table 2\u20131). These four modes can be viewed both as a historical progression and as a categorization of current health care financing.\n\n**Table 2\u20131.** Health care financing in 2009 _a_\n\n#### **Out-of-Pocket Payments**\n\n_Fred Farmer broke his leg in 1911. His son ran 4 miles to get the doctor, who came to the farm to splint the leg. Fred gave the doctor a couple of chickens to pay for the visit. His great-grandson, Ted, who is uninsured, broke his leg in 2011. He was driven to the emergency room, where the physician ordered an x-ray and called in an orthopedist who placed a cast on the leg. The cost was $1800._\n\nOne hundred years ago, people like Fred Farmer paid physicians and other health care practitioners in cash or through barter. In the first half of the twentieth century, out-of-pocket cash payment was the most common method of reimbursement. This is the simplest mode of financing\u2014direct purchase by the consumer of goods and services (Figure 2\u20131).\n\n**Figure 2\u20131.** Out-of-pocket payment is made directly from patient to provider.\n\nPeople in the United States purchase most consumer items, from DVD players to haircuts, through direct out-of-pocket payments. This is not the case with health care (Arrow, 1991; Evans, 1984), and one may ask why health care is not considered a typical consumer item.\n\n##### **Need versus Luxury**\n\nWhereas a DVD player is considered a luxury, health care is regarded as a basic human need by most people.\n\n_For 2 weeks, Marina Perez has had vaginal bleeding and has felt dizzy. She has no insurance and is terrified that medical care might eat up her $500 in savings. She scrapes together $100 to see her doctor, who finds that her blood pressure falls to 90\/50 mm Hg upon standing and that her hematocrit is 26%. The doctor calls Marina's sister Juanita to drive her to the hospital. Marina gets into the car and tells Juanita to take her home._\n\nIf health care is a basic human right, then people who are unable to afford health care must have a payment mechanism available that is not reliant on out-of-pocket payments.\n\n##### **Unpredictability of Need and Cost**\n\nWhereas the purchase of a DVD player is a matter of choice and the price is known to the buyer, the need for and cost of health care services are unpredictable. Most people do not know if or when they may become severely ill or injured or what the cost of care will be.\n\n_Jake has a headache and visits the doctor, but he does not know whether the headache will cost $100 for a physician visit plus the price of a bottle of ibuprofen, $1000 for an MRI, or $100,000 for surgery and irradiation for brain cancer._\n\nThe unpredictability of many health care needs makes it difficult to plan for these expenses. The medical costs associated with serious illness or injury usually exceed a middle-class family's savings.\n\n##### **Patients Need to Rely on Physician Recommendations**\n\nUnlike the purchaser of a DVD player, a person in need of health care may have little knowledge of what he or she is buying at the time when care is needed.\n\n_Jenny develops acute abdominal pain and goes to the hospital to purchase a remedy for her pain. The physician tells her that she has acute cholecystitis or a perforated ulcer and recommends hospitalization, an abdominal CT scan, and upper endoscopic studies. Will Jenny, lying on a gurney in the emergency room and clutching her abdomen with one hand, use her other hand to leaf through a textbook of internal medicine to determine whether she really needs these services, and should she have brought along a copy of Consumer Reports to learn where to purchase them at the cheapest price?_\n\nHealth care is the foremost example of asymmetry of information between providers and consumers (Evans, 1984). A patient with abdominal pain is in a poor position to question a physician who is ordering laboratory tests, x-rays, or surgery. When health care is elective, patients can weigh the pros and cons of different treatment options, but even so, recommendations may be filtered through the biases of the physician providing the information. Compared with the voluntary demand for DVD players (the influence of advertising notwithstanding) the demand for health services is partially involuntary and is often physician-rather than consumer-driven.\n\nFor these reasons among others, out-of-pocket payments are flawed as a dominant method of paying for health care services. Because the direct purchase of health services became increasingly difficult for consumers and was not meeting the needs of hospitals and physicians to be reliably paid, health insurance came into being.\n\n#### **Individual Private Insurance**\n\n_Bud Carpenter is self-employed. He recently purchased a health insurance policy from his insurance broker for his family. To pay the $500 monthly premium, he had to work some extra jobs on weekends, and the $2500 deductible meant he would still have to pay quite a bit of his family's medical costs out of pocket. Mr. Carpenter preferred to pay these costs rather than take the risk of spending the money saved for his children's college education on a major illness. When his son became ill with leukemia and the hospital bill reached $80,000, Mr. Carpenter appreciated the value of health insurance. Nonetheless he had to feel disgruntled when he read a newspaper story listing his insurance company among those that paid out on average less than 60 cents for health services for every dollar collected in premiums._\n\nWith private health insurance, a third party, the insurer, is added to the patient and the health care provider, who are the two basic parties of the health care transaction. While the out-of-pocket mode of payment is limited to a single financial transaction, private insurance requires two transactions\u2014a premium payment from the individual to an insurance plan (also called a health plan), and a reimbursement payment from the insurance plan to the provider (Figure 2\u20132). In nineteenth-century Europe, voluntary benefit funds were set up by guilds, industries, and mutual societies. In return for paying a monthly sum, people received assistance in case of illness. This early form of private health insurance was slow to develop in the United States. In the early twentieth century, European immigrants set up some small benevolent societies in US cities to provide sickness benefits for their members. During the same period, two commercial insurance companies, Metropolitan Life and Prudential, collected 10\u201325 cents per week from workers for life insurance policies that also paid for funerals and the expenses of a final illness. The policies were paid for by individuals on a weekly basis, so large numbers of insurance agents had to visit their clients to collect the premiums as soon after payday as possible. Because of the huge administrative costs, individual health insurance never became a dominant method of paying for health care (Starr, 1982). In 2009, individual policies provided health insurance for only 5% of the US population (see Table 2\u20131).\n\n**Figure 2\u20142.** Individual private insurance. A third party, the insurance plan (health plan), is added, dividing payment into a financing component and a reimbursement component.\n\n#### **Employment-Based Private Insurance**\n\n_Betty Lerner and her schoolteacher colleagues each paid $6 per year to Prepaid Hospital in 1929. Ms. Lerner suffered a heart attack and was hospitalized at no cost. The following year Prepaid Hospital built a new wing and raised the teachers' prepayment to $12._\n\n_Rose Riveter retired in 1961. Her health insurance premium for hospital and physician care, formerly paid by her employer, had been $25 per month. When she called the insurance company to obtain individual coverage, she was told that premiums at age 65 cost $70 per month. She could not afford the insurance and wondered what would happen if she became ill._\n\nThe development of private health insurance in the United States was impelled by the increasing effectiveness and rising costs of hospital care. Hospitals became places not only in which to die, but also in which to get well. However, many patients were unable to pay for hospital care, and this meant that hospitals were unable to attract \"customers.\"\n\nIn 1929, Baylor University Hospital agreed to provide up to 21 days of hospital care to 1500 Dallas school-teachers such as Betty Lerner if they paid the hospital $6 per person per year. As the Great Depression deepened and private hospital occupancy in 1931 fell to 62%, similar hospital-centered private insurance plans spread. These plans (anticipating health maintenance organizations [HMOs]) restricted care to a particular hospital. The American Hospital Association built on this prepayment movement and established statewide Blue Cross hospital insurance plans allowing free choice of hospital. By 1940, 39 Blue Cross plans controlled by the private hospital industry had enrolled over 6 million people. The Great Depression reduced the amount patients could pay physicians out of pocket, and in 1939, the California Medical Association set up the first Blue Shield plan to cover physician services. These plans, controlled by state medical societies, followed Blue Cross in spreading across the nation (Starr, 1982; Fein, 1986).\n\nIn contrast to the consumer-driven development of health insurance in European nations, coverage in the United States was initiated by health care providers seeking a steady source of income. Hospital and physician control over the \"Blues,\" a major sector of the health insurance industry, guaranteed that reimbursement would be generous and that cost control would remain on the back burner (Law, 1974; Starr, 1982).\n\nThe rapid growth of employment-based private insurance was spurred by an accident of history. During World War II, wage and price controls prevented companies from granting wage increases, but allowed the growth of fringe benefits. With a labor shortage, companies competing for workers began to offer health insurance to employees such as Rose Riveter as a fringe benefit. After the war, unions picked up on this trend and negotiated for health benefits. The results were dramatic: Enrollment in group hospital insurance plans grew from 12 million in 1940 to 142 million in 1988.\n\nWith employment-based health insurance, employers usually pay most of the premium that purchases health insurance for their employees (Figure 2\u20133). However, this flow of money is not as simple as it looks. The federal government views employer premium payments as a tax-deductible business expense. The government does not treat the health insurance fringe benefit as taxable income to the employee, even though the payment of premiums could be interpreted as a form of employee income. Because each premium dollar of employer-sponsored health insurance results in a reduction in taxes collected, the government is in essence subsidizing employer-sponsored health insurance. This subsidy is enormous, estimated at $260 billion per year (Gruber, 2010).\n\n**Figure 2\u20133.** Employment-based private insurance. In addition to the direct employer subsidy, indirect government subsidies occur through the tax-free status of employer contributions for health insurance benefits.\n\nThe growth of employment-based health insurance attracted commercial insurance companies to the health care field to compete with the Blues for customers. The commercial insurers changed the entire dynamic of health insurance. The new dynamic was called **experience rating.** (The following discussion of experience rating can be applied to individual as well as employment-based private insurance.)\n\n_Healthy Insurance Company insures three groups of people\u2014a young healthy group of bank managers, an older healthy group of truck drivers, and an older group of coal miners with a high rate of chronic illness. Under experience rating, Healthy sets its premiums according to the experience of each group in using health services. Because the bank managers rarely use health care, each pays a premium of $200 per month. Because the truck drivers are older, their risk of illness is higher, and their premium is $400 per month. The miners, who have high rates of black lung disease, are charged a premium of $600 per month. The average premium income to Healthy is $400 per member per month._\n\n_Blue Cross insures the same three groups and needs the same $400 per member per month to cover health care plus administrative costs for these groups. Blue Cross sets its premiums by the principle of community rating. For a given health insurance policy, all subscribers in a community pay the same premium. The bank managers, truck drivers, and mine workers all pay $400 per month._\n\nHealth insurance provides a mechanism to distribute health care more in accordance with human need rather than exclusively on the basis of ability to pay. To achieve this goal, funds are redistributed from the healthy to the sick, a subsidy that helps pay the costs of those unable to purchase services on their own.\n\n**Community rating** achieves this redistribution in two ways:\n\n1. Within each group (bank managers, truck drivers, and mine workers), people who become ill receive benefits in excess of the premiums they pay, while people who remain healthy pay premiums while receiving few or no health benefits.\n\n2. Among the three groups, the bank managers, who use less health care than their premiums are worth, help pay for the miners, who use more health care than their premiums could buy.\n\nExperience rating is far less redistributive than community rating. Within each group, those who become ill are subsidized by those who remain well, but among the different groups, healthier groups (bank managers) do not subsidize high-risk groups (mine workers). Thus the principle of health insurance, which is to distribute health care more in accordance with human need rather than exclusively on the ability to pay, is weakened by experience rating (Light, 1992).\n\nIn the early years, Blue Cross plans set insurance premiums by the principle of community rating, whereas commercial insurers used experience rating as a \"weapon\" to compete with the Blues (Fein, 1986). Commercial insurers such as Healthy Insurance Company could offer cheaper premiums to low-risk groups such as bank managers, who would naturally choose a Healthy commercial plan at $200 over a Blue Cross plan at $400. Experience rating helped commercial insurers overtake the Blues in the private health insurance market. While in 1945 commercial insurers had only 10 million enrollees, compared with 19 million for the Blues, by 1955 the score was commercials 54 million and the Blues 51 million.\n\nMany commercial insurers would not market policies to such high-risk groups as mine workers, leaving Blue Cross with high-risk patients who were paying relatively low premiums. To survive the competition from the commercial insurers, Blue Cross had no choice but to seek younger, healthier groups by abandoning community rating and reducing the premiums for those groups. In this way, many Blue Cross and Blue Shield plans switched to experience rating. Without community rating, older and sicker groups became less and less able to afford health insurance.\n\nFrom the perspective of the elderly and those with chronic illness, experience rating is discriminatory. Healthy persons, however, might have another viewpoint on the situation and might ask why they should voluntarily transfer their wealth to sicker people through the insurance subsidy. The answer lies in the unpredictability of health care needs. When purchasing health insurance, an individual does not know if he or she will suddenly change from a state of good health to one of illness. Thus, _within a group,_ people are willing to risk paying for health insurance, even though they may not use it. _Among different groups,_ however, healthy people have no economic incentive to voluntarily pay for community rating and subsidize another group of sicker people. This is why community rating cannot survive in a market-driven competitive private insurance system (Aaron, 1991).\n\nThe most positive aspect of health insurance\u2014that it assists people with serious illness to pay for their care\u2014has also become one of its main drawbacks\u2014the difficulty in controlling costs in an insurance environment. With direct purchase, the \"invisible hand\" of each individual's ability to pay holds down the price and quantity of health care. However, if a patient is well insured and the cost of care causes no immediate fiscal pain, the patient will use more services than someone who must pay for care out of pocket. In addition, particularly before the advent of fee schedules, health care providers could increase fees more easily if a third party was available to foot the bill.\n\nThus health insurance was originally an attempt by society to solve the problem of unaffordable health care under an out-of-pocket payment system, but its very capacity to make health care more affordable created a new problem. If people no longer had to pay out of their own pockets for health care, they would use more health care; and if health care providers could charge insurers rather than patients, they could more easily raise prices, especially during the era when the major insurers (the Blues) were controlled by hospitals and physicians. The solution of insurance fueled the problem of rising costs. As private insurance became largely experience rated and employment based, persons who had low incomes, who were chronically ill, or who were elderly found it increasingly difficult to afford private insurance.\n\n#### **Government Financing**\n\n_In 1984 at age 74 Rose Riveter developed colon cancer. She was now covered by Medicare, which had been enacted in 1965. Even so, her Medicare premium, hospital deductible expenses, physician copayments, short nursing home stay, and uncovered prescriptions cost her $2700 the year she became ill with cancer._\n\nEmployment-based private health insurance grew rapidly in the 1950s, helping working people and their families to afford health care. But two groups in the population received little or no benefit: the poor and the elderly. The poor were usually unemployed or employed in jobs without the fringe benefit of health insurance; they could not afford insurance premiums. The elderly, who needed health care the most and whose premiums had been partially subsidized by community rating, were hard hit by the trend toward experience rating. In the late 1950s, less than 15% of the elderly had any health insurance (Harris, 1966). Only one program could provide affordable care for the poor and the elderly: tax-financed government health insurance.\n\nGovernment entered the health care financing arena long before the 1960s through such public programs as municipal hospitals and dispensaries to care for the poor and through state-operated mental hospitals. But only with the 1965 enactment of Medicare (for the elderly) and Medicaid (for the poor) did public insurance payments for privately operated health services become a major feature of health care in the United States. Medicare Part A (Table 2\u20132) is a hospital insurance plan for the elderly financed largely through social security taxes from employers and employees. Medicare Part B (Table 2\u20133) insures the elderly for physician services and is paid for by federal taxes and monthly premiums from the beneficiaries. Medicare Part D, enacted in 2003, offers prescription drug coverage and is paid for by federal taxes and monthly premiums from beneficiaries. Medicaid (Table 2\u20134) is a program run by the states that is funded by federal and state taxes, which pays for the care of certain low-income groups. In 2009, Medicare and Medicaid expenditures totaled $502 and $374 billion, respectively (Martin et al, 2011).\n\n**Table 2\u20132.** Summary of Medicare Part A, 2011\n\n**Table 2\u20133.** Summary of Medicare Part B, 2011\n\n**Table 2\u20134.** Summary of Medicaid, 2011\n\nWith its large deductibles, copayments, and gaps in coverage, Medicare paid for only 48% of the average beneficiary's health care expenses in 2006 (Kaiser Family Foundation, 2010a). Most of the 47 million Medicare beneficiaries (2010) have supplemental coverage. In 2010, nearly 30% of beneficiaries had additional coverage from their previous employment, about 20% purchased supplemental private insurance (called \"Medigap\" plans), 24% were enrolled in the Medicare Advantage program, and 19% were enrolled in both Medicare and Medicaid (Kaiser Family Foundation, 2010b).\n\nThe Medicare Modernization Act (MMA) of 2003 made two major changes in the Medicare program: the expansion of the role of private health plans (the Medicare Advantage program, Part C) and the establishment of a prescription drug benefit (Part D). Under the Medicare Advantage program, a beneficiary can elect to enroll in a private health plan contracting with Medicare, with Medicare subsidizing the premium for that private health plan rather than paying hospitals, physicians, and other providers directly as under Medicare Parts A and B. Beneficiaries joining a Medicare Advantage plan sacrifice some degree of freedom of choice of physician and hospital in return for lower out-of-pocket payments and are only allowed to receive care from health care providers who are connected with that plan. Two-thirds of Medicare Advantage plans are health maintenance organizations (HMOs) (see Chapter 6). In order to channel more patients into Medicare Advantage plans, the MMA provided generous payments to those plans, with the result that they cost the federal government 14% more than the government paid for health care services for similar Medicare beneficiaries in the traditional Part A and Part B programs. The 2010 health care reform law passed by the Obama Administration (the Accountable Care Act) reduced payments to Medicare Advantage plans with the goal of saving the Medicare program $136 billion over the following 10 years (Kaiser Family Foundation, 2010c).\n\nMedicare Part D provides partial coverage for prescription drugs. In 2010, 82% of Part D was financed through tax revenues, with 10% coming from beneficiary premiums (Kaiser Family Foundation, 2010a). As of 2010, 59% of Medicare beneficiaries had enrolled in the voluntary Medicare Part D program. Part D has been criticized because (1) there are major gaps in coverage, (2) coverage has been farmed out to private insurance companies rather than administered by the federal Medicare program, and (3) the government is not allowed to negotiate with pharmaceutical companies for lower drug prices. These 3 features of the program have caused confusion for beneficiaries, physicians, and pharmacists and a high cost for the program. Beneficiaries desiring Medicare Part D can enroll in one of 1500 stand-alone private prescription drug plans or receive their Part D coverage through a Medicare Advantage plan. Different plans cover different medications and require different premiums, deductibles, and coinsurance payments. The standard plan in 2010 had a $310 yearly deductible and 25% coinsurance up to an initial coverage limit of $2830 in total drug costs, after which coverage stops until the beneficiary has spent $4550 out of pocket (excluding premiums) for prescription drugs. Above $4550, coverage resumes with 5% coinsurance. The coverage gap, called the \"donut hole,\" becomes a major problem for patients with chronic illness needing several medications. The Accountable Care Act of 2010 gradually reduces the amounts beneficiaries must pay in the donut hole.\n\nIn 2009, the trustees of the Medicare program estimated that the Part A trust fund would be depleted by 2017. The Accountable Care Act, by raising social security payments and reducing expenditures, has extended Medicare's solvency through 2029.\n\nThe Medicaid program (Table 2\u20134) is jointly administered by the federal and state governments. Although designed for low-income Americans, not all poor people are eligible for Medicaid. In addition to being poor, Medicaid has required that people also meet \"categorical\" eligibility criteria such as being a young child, pregnant, elderly, or disabled. Medicaid enrollment is growing dramatically, increasing from 32 million to 58 million people between 2000 and 2010 (9 million of whom are \"dual eligibles\" receiving both Medicare and Medicaid). The Accountable Care Act includes a huge expansion of Medicaid starting in 2014, eliminating the categorical eligibility criteria and offering the program to all citizens and legal residents with family income below 133% of the federal poverty line. The additional 16 million people on Medicaid will be financed largely by the federal government at a cost of over $40 billion per year in new dollars (see Chapter 15).\n\nFrom 2000 to 2010, Medicaid expenditures rose from $200 billion to $374 billion. To slow down this expenditure growth, the federal government ceded to states enhanced control over Medicaid programs through Medicaid waivers, which allow states to reduce the number of people eligible for Medicaid, make alterations in the scope of covered services, require Medicaid recipients to pay part of their costs, and obligate Medicaid recipients to enroll in managed care plans (see Chapter 4). In 2010, over half of Medicaid recipients were enrolled in managed care plans. Because Medicaid pays physicians an average of 72% of Medicare fees, the majority of adult primary care physicians limit the number of Medicaid patients they will see; these patients are increasingly concentrated in academic health centers and community health centers.\n\nIn 1997, the federal government created the State Children's Health Insurance Program (SCHIP), a companion program to Medicaid. SCHIP covers children in families with incomes at or below 200% of the federal poverty level, but above the Medicaid income eligibility level. States legislating a SCHIP program receive generous federal matching funds and can administer SCHIP through Medicaid or by creating a separate program. In 2009, almost 8 million children were enrolled in the program.\n\nGovernment health insurance for the poor and the elderly added a new factor to the health care financing equation: the taxpayer (Figure 2\u20134). With government-financed health plans, the taxpayer can interact with the health care consumer in two distinct ways:\n\n**Figure 2\u20134.** Government-financed insurance. Under the social insurance model (eg, Medicare Part A), only individuals paying taxes into the public plan are eligible for benefits. In other models (eg, Medicaid), an individual's eligibility for benefits may not be directly linked to payment of taxes into the plan.\n\n1. The social insurance model, exemplified by Medicare, allows only those who have paid a certain amount of social security taxes to be eligible for Part A and only those who pay a monthly premium to receive benefits from Part B. As with private insurance, social insurance requires people to make a contribution in order to receive benefits.\n\n2. The contrasting model is the Medicaid public assistance model, in which those who contribute (taxpayers) may not be eligible for benefits (Bodenheimer and Grumbach, 1992).\n\nIt must be remembered that private insurance contains a subsidy: redistribution of funds from the healthy to the sick. Tax-funded insurance has the same subsidy and usually adds another: redistribution of funds from upper- to lower-income groups. Under this double subsidy, exemplified by Medicare and Medicaid, healthy middle-income employees generally pay more in social security payments and other taxes than they receive in health services, whereas unemployed, disabled, and lower-income elderly persons tend to receive more in health services than they contribute in taxes.\n\nThe advent of government financing improved financial access to care for some people, but, in turn, aggravated the problem of rising costs. The federal government and state governments have responded by attempting to limit Medicare and Medicaid payments to physicians and hospitals. At the same time, the rising costs of private insurance continue to place employment-based coverage out of the fiscal reach of many employers and employees.\n\n### **THE BURDEN OF FINANCING HEALTH CARE**\n\nDifferent methods of financing health care place different burdens on the various income levels of society. Payments are classified as **progressive** if they take a rising percentage of income as income increases, **regressive** if they take a falling percentage of income as income increases, and **proportional** if the ratio of payment to income is the same for all income classes (Pechman, 1985).\n\nWhat principle should underlie the choice of revenue source for health care? A central purpose of the health care system is to maintain and improve the health of the nation's population. As discussed in Chapter 3, rates of mortality and disability are far higher for low-income people than for the wealthy. Burdening low-income families with high levels of payments for health care (ie, regressive payments) reduces their disposable income, amplifies the ill effects of poverty, and thereby worsens their health. It makes little sense to finance a health care system\u2014whose purpose is to improve health\u2014with payments that worsen health. Thus, regressive payments could be considered \"unhealthy.\"\n\n_Rita Blue earns $10,000 per year for her family of 4. She develops pneumonia, and her out-of-pocket health costs come to $1000, 10% of her family income._\n\n_Cathy White earns $100,000 per year for her family of 4. She develops pneumonia, and her out-of-pocket health costs come to $1000, 1% of her family income._\n\nOut-of-pocket payments are a regressive mode of financing. According to the 1987 National Medical Care Expenditure Survey, out-of-pocket payments took 12% of the income of families in the nation's lowest-income quintile, compared with 1.2% for families in the wealthiest 5% of the population (Bodenheimer and Sullivan, 1997). This pattern is confirmed by the 2000 Medical Expenditure Panel Survey (MEPS, 2003). Many economists and health policy experts would consider this regressive burden of payment as unfair. Aggravating the regressivity of out-of-pocket payments is the fact that lower-income people tend to be sicker and thus have more out-of-pocket payments than the wealthier and healthier.\n\n_Jim Hale is a young, healthy, self-employed accountant whose monthly income is $6000, with a health insurance premium of $200, or 3% of his income._\n\n_Jack Hurt is a disabled mine worker with black lung disease. His income is $1800 per month, of which $400 (22%) goes for his health insurance._\n\nExperience-rated private health insurance is a regressive method of financing health care because increased risk of illness tends to correlate with reduced income. If Jim Hale and Jack Hurt were enrolled in a community-rated plan, each with a premium of $300, they would respectively pay 5% and 17% of their incomes for health insurance. With community rating, the burden of payment is regressive, but less so than with experience rating.\n\nMost private insurance is not individually purchased but rather obtained through employment. How is the burden of employment-linked health insurance premiums distributed?\n\n_Jill is an assistant hospital administrator. To attract her to the job, the hospital offered her a package of salary plus health insurance of $6500 per month. She chose to take $6200 in salary, leaving the hospital to pay $300 for her health insurance._\n\n_Bill is a nurse's aide, whose union negotiated with the hospital for a total package of $2800 per month; of this amount $2500 is salary and $300 pays his health insurance premium._\n\nDo Jill and Bill pay nothing for their health insurance? Not exactly. Employers generally agree on a total package of wages and fringe benefits; if Jill and Bill did not receive health insurance, their pay would probably go up by nearly $300 per month. That is why employer-paid health insurance premiums are generally considered deductions from wages or salary, and thus paid by the employee (Blumberg et al, 2007). For Jill, health insurance amounts to only 5% of her income, but for Bill it is 12%. The MEPS corroborates the regressivity of employment-based health insurance; in 2001\u20132003, premiums took an average of 10.9% of the income of families in between 100% and 200% of the federal poverty line compared with 2.3% for those above 500% of poverty (Blumberg et al, 2007).\n\n_Larry Lowe earns $10,000 and pays $410 in federal and state income taxes, or 4.1% of his income._\n\n_Harold High earns $100,000 and pays $12,900 in income taxes, or 12.9% of his income._\n\nThe progressive income tax is the largest tax providing money for government-financed health care. Most other taxes are regressive (eg, sales and property taxes), and the combined burden of all taxes that finance health care is roughly proportional (Pechman, 1985).\n\nIn 2009, 46% of health care expenditures were financed through out-of-pocket payments and premiums, which are regressive, while 47% was funded through government revenues (Martin et al, 2011), which are proportional. The sum total of health care financing is regressive. In 1999, the poorest quintile of households spent 18% of income on health care, while the highest-income quintile spent only 3% (Cowan et al, 2002). Overall, the US health care system is financed in a manner that is unhealthy.\n\n### **CONCLUSION**\n\nNeither Fred Farmer nor his great-grandson Ted had health insurance, but the modern-day Mr. Farmer's predicament differs drastically from that of his ancestor. Third-party financing of health care has fueled an expansive health care system that offers treatments unimaginable a century ago, but at tremendous expense.\n\nEach of the four modes of financing health care developed historically as a solution to the inadequacy of the previous modes. Private insurance provided protection to patients against the unpredictable costs of medical care, as well as protection to providers of care against the unpredictable ability of patients to pay. But the private insurance solution created three new, interrelated problems:\n\n1. The opportunity for health care providers to increase fees to insurers caused health services to become increasingly unaffordable for those with inadequate insurance or no insurance.\n\n2. The employment-based nature of group insurance placed people who were unemployed, retired, or working part-time at a disadvantage for the purchase of insurance, and partially masked the true costs of insurance for employees who did receive health benefits at the workplace.\n\n3. Competition inherent in a deregulated private insurance market gave rise to the practice of experience rating, which made insurance premiums unaffordable for many elderly people and other medically needy groups.\n\nTo solve these problems, government financing was required, but government financing fueled an even greater inflation in health care costs.\n\nAs each \"solution\" was introduced, health care financing improved for a time. But rising costs have jeopardized private and public coverage for many people and made services unaffordable for those without a source of third-party payment. The problems of each financing mode, and the problems created by each successive solution, have accumulated into a complex crisis characterized by inadequate access for some and high costs for everyone.\n\n### **REFERENCES**\n\nAaron HJ. _Serious and Unstable Condition: Financing America's Health Care_. Washington, DC: Brookings Institution; 1991.\n\nArrow KJ. Uncertainty and the welfare economics of medical care. _Am Econ Rev_. 1963;53:941.\n\nBlumberg LJ et al. Setting a standard of affordability for health insurance coverage. _Health Affairs_. 2007;26:w463-w473.\n\nBodenheimer T, Grumbach K. Financing universal health insurance: Taxes, premiums, and the lessons of social insurance. _J Health Polit Policy Law_. 1992;17:439.\n\nBodenheimer T, Sullivan K. The logic of tax-based financing for health care. _Int J Health Services_. 1997;27:409.\n\nCowan CA et al. Burden of health care costs: Businesses, households, and governments, 1987\u20132000. _Health Care Financ Rev_. 2002;23:131.\n\nEvans RG. _Strained Mercy_. Toronto, Ontario, Canada: Butterworths; 1984.\n\nFein R. _Medical Care, Medical Costs_. Cambridge, MA: Harvard University Press; 1986.\n\nGruber J. The tax exclusion for employer-sponsored health insurance. _National Bureau of Economic Research_ ; February 2010. www.nber.org\/papers\/w15766. Accessed November 11, 2011.\n\nHarris R. _A Sacred Trust_. New York, NY: New American Library; 1966.\n\nKaiser Family Foundation. Medicare Spending and Financing. 2010a. www.kff.org. Accessed August 3, 2011.\n\nKaiser Family Foundation. Medicare at a Glance. 2010b. www.kff.org. Accessed August 3, 2011.\n\nKaiser Family Foundation. Medicare Advantage 2010 Data Spotlight. 2010c. www.kff.org. Accessed August 3, 2011.\n\nKaiser Family Foundation. The Medicaid Program at a Glance. 2010d. www.kff.org. Accessed August 3, 2011.\n\nLaw SA. _Blue Cross: What Went Wrong?_ New Haven, CT: Yale University Press; 1974.\n\nLight DW. The practice and ethics of risk-rated health insurance. _JAMA_. 1992;267:2503.\n\nMartin A et al. Recession contributes to slowest annual rate of increase in health spending in five decades. _Health Affairs_. 2011;30:11.\n\nMedical Expenditure Panel Survey. Health insurance coverage of the civilian non-institutionalized population, first half of 2002. Agency for Healthcare Research and Quality, June 2003. www.meps.ahrq.gov. Accessed November 11, 2011.\n\nPechman JA. _Who Paid the Taxes, 1966\u20131985_. Washington, DC: Brookings Institution; 1985.\n\nStarr P. _The Social Transformation of American Medicine_. New York, NY: Basic Books; 1982.\n\n## **3 Access to Health Care**\n\nAccess to health care is the ability to obtain health services when needed. Lack of adequate access for millions of people is a crisis in the United States.\n\nAccess to health care has two major components. First and most frequently discussed is ability to pay. Second is the availability of health care personnel and facilities that are close to where people live, accessible by transportation, culturally acceptable, and capable of providing appropriate care in a timely manner and in a language spoken by those who need assistance. The first and longest portion of this chapter dwells on financial barriers to care. The second portion touches on nonfinancial barriers. The final segment explores the influences other than health care (in particular, socioeconomic status and race) that are important determinants of the health status of a population.\n\n### **FINANCIAL BARRIERS TO HEALTH CARE**\n\n#### **Lack of Insurance**\n\n_Ernestine Newsome was born into a low-income working family living in South Central Los Angeles. As a young child, she rarely saw a physician and was behind on her childhood immunizations. When Ernestine was 7 years old, her mother began working for the telephone company, and this provided the family with health insurance. Ernestine went to a neighborhood physician for regular checkups. When she reached 19, she left home and began work as a part-time secretary. She was no longer eligible for her family's health insurance coverage, and her new job did not provide insurance. She has not seen a physician since starting her job._\n\nHealth insurance coverage, whether public or private, is a key factor in making health care accessible. In 1980, 25 million people were uninsured, but by 2009 the number had increased to 51 million (Table 3\u20131 and Figure 3\u20131) (US Census Bureau, 2010). The particular pattern of uninsurance is related to the employment-based nature of health care financing. Most people, like Ernestine Newsome, obtain health insurance when employers voluntarily decide to offer group coverage to employees and their families and their employers help pay for the costs of health insurance. People whose employers choose not to provide health insurance, are self-employed, or are unemployed are left to fend for themselves outside of the employer-sponsored group health insurance market, with the result that many are uninsured. Often people without employment-based insurance are not eligible for public programs such as Medicare and Medicaid, and are unable to purchase individual private coverage because they cannot afford the premiums.\n\n**Table 3\u20131.** Estimated principal source of health insurance, 2009\n\n**Figure 3\u20131.** Number of uninsured persons in the United States, 1980 to 2009 (US Census Bureau, 2010).\n\nBetween the 1930s and mid-1970s, because of the growth of private health insurance and the 1965 passage of Medicare and Medicaid, the number of uninsured persons declined steadily, but since 1976, the number has been growing. The single most important factor explaining the growing number of uninsured is a 25-year trend of decreasing private insurance coverage in the United States. Virtually all people aged 65 and older are covered by Medicare, and the number of people enrolled in Medicaid has increased. However, a dwindling proportion of children and working age adults are covered by private insurance, exposing the limitations of the employment-linked system of private insurance in the United States. If the 2010 health care reform law, the Accountable Care Act, is fully implemented, the number of uninsured people is expected to drop from 51 million to 22 million (Buettgens et al, 2010).\n\n##### **Why People Lack Insurance?**\n\n_Joe Fortuno dropped out of high school and went to work for Car Doctor auto body shop in 2003. His employer paid the full cost of health insurance forJoe and his family. Joe's younger cousin Pete Luckless got a job working at an auto mechanic shop in 2005. The company did not offer health insurance benefits. In 2008, Car Doctor, after experiencing a doubling of health insurance premium rates over the prior few years, began requiring that its employees pay $150 per month for the employer-sponsored health plan. Joe could not afford the monthly payments and lost his health insurance._\n\nWhy has private health insurance coverage decreased over the past decades, creating the uninsurance crisis? There are several explanations:\n\n1. The skyrocketing cost of health insurance has made coverage unaffordable for many businesses and individuals. From 2000 to 2010, employer-sponsored health insurance premiums rose by 114%. In 2010, the average annual cost of health insurance, including employer and employee contributions, was $5049 for individuals and $13,770 for families (Claxton et al, 2010). Some employers responded to rising health insurance costs by dropping insurance policies for their workers. Many employers have shifted more of the cost of health insurance premiums and health services onto their employees, resulting in employees dropping health coverage because of unaffordability. On average, employee contributions represent 19% of the premium for individual employee coverage and 30% for family coverage, though some employees have to pay more than half of the premium for family coverage (Claxton et al, 2010). Low-income workers are hit especially hard by the combination of rising insurance costs and declining employer subsidies.\n\n_Jean Irons worked for Bethlehem Steel as a clerk and her fringe benefits included health insurance. Bethlehem Steel was bought by a global corporation and her plant moved to another country. She found a job as a food service worker in a small restaurant. Her pay decreased by 35%, and the restaurant did not provide health insurance._\n\n2. During the past few decades, the economy in the United States has undergone a major transition. The number of highly paid, largely unionized, full-time manufacturing workers with employer-sponsored health insurance has declined, and the workforce has shifted toward more low-wage, increasingly part-time, nonunionized service, and clerical workers whose employers are less likely to provide insurance (Renner and Navarro, 1989). Between 1980 and 2006, the number of workers in the manufacturing sector decreased by 30% while the number working in the service sector increased by 75%. From 1957 to 2000, the percentage of workers with part-time jobs\u2014generally without health benefits\u2014increased from 12% to 21%.\n\nThese two factors\u2014increasing health care costs and a changing labor force\u2014eroded private insurance coverage. One countervailing trend has been a major expansion of public insurance coverage through the Medicaid and State Children's Health Insurance Program (SCHIP) programs. Without these changes, many more millions of Americans would currently be uninsured.\n\n_Sally Lewis worked as a receptionist in a physician's office. She received health insurance through her husband, who was a construction worker. They got divorced, she lost her health insurance, and her physician employer told her he could not provide her with health insurance because of the cost._\n\n3. The link of private insurance with employment inevitably produces interruptions in coverage because of the unstable nature of employment. People who are laid off from their jobs or who leave jobs because of illness may also lose their insurance. Family members insured through the workplace of a spouse may lose their insurance in cases of divorce, job loss, or death of the working family member. People who leave their employment may be eligible to pay for continued coverage under their group plan for 18 months, as stipulated in the Consolidated Omnibus Budget Reconciliation Act of 1985 (COBRA), with the requirement that they pay the full cost of the premium; however, many people cannot afford the premiums, which may exceed $1000 per month for a family.\n\nThe often transient nature of employment-linked insurance is compounded by difficulties in maintaining eligibility for Medicaid. Small increases in family income can mean that families no longer qualify for Medicaid. The net result is that millions of people cycle in and out of the ranks of the uninsured every month. A total of 87 million people, 29% of the entire US population, went without health insurance for all or part of the 2-year period 2007\u20132008 (Families USA, 2009). Health insurance may be a fleeting benefit.\n\n##### **Who Are the Uninsured?**\n\nIn 2009, 12% of non-Hispanic whites were uninsured, compared with 21% of African Americans, 17% of Asians, and 32% of Latinos (Figure 3\u20132). Twenty-seven percent of individuals with annual household incomes less than $25,000 were uninsured, compared with 9% of individuals with household incomes of $75,000 or more (Figure 3\u20133) (US Census Bureau, 2010).\n\n**Figure 3\u20132.** Percentage of population lacking health insurance by race and ethnicity in 2009 (US Census Bureau, 2010).\n\n**Figure 3\u20133.** Lack of insurance by income in 2009 (US Census Bureau, 2010).\n\n_Morris works for a corner grocery store that employs five people. Morris once asked the owner whether the employees could receive health insurance through their work, but the owner said it was too expensive. Morris, his wife, and their three kids are uninsured._\n\n_Norris, a shipyard worker, was laid off 3 years ago, and at age 60 is unable to get another job. He lives on county general assistance of $400 per month, but is ineligible for Medicaid because he is not a parent, not older than 65, and not disabled. He is uninsured._\n\nThe uninsured can be divided into two major categories: the employed uninsured (Morris) and the unemployed uninsured (Norris). Seventy-five percent of the uninsured are employed or the spouses and children of those who work. Most of the jobs held by the employed uninsured are low paying, in small firms, and may be part time (Figures 3\u20134 and 3\u20135). Twenty-five percent of the uninsured are unemployed, often with incomes below the poverty line, but like Norris are ineligible for Medicaid.\n\n**Figure 3\u20134.** Lack of insurance by employment status in 2009 (US Census Bureau, 2010).\n\n**Figure 3\u20135.** Lack of job-based insurance by size of employer in 2007 (Kaiser Family Foundation, Health Insurance Coverage in America, 2008, www.kff.org).\n\n##### **Does Health Insurance Make a Difference?**\n\n_Two US senators are debating the issue of access to health care. One decries the stigma of uninsurance and claims that people without insurance receive less care and suffer worse health than those with insurance. The other disagrees, claiming that hospitals and physicians deliver large amounts of charity care, which allows uninsured people to receive the services they need._\n\nTo resolve this debate, the US Congress Office of Technology Assessment (1992) conducted a comprehensive review to determine whether health insurance makes a difference in the use of health care and in health outcomes. The findings, corroborated by the Institute of Medicine (2002), proved that people lacking health insurance receive less care and have worse health outcomes.\n\n##### **Health Insurance and Use of Health Services**\n\n_Percy, a child whose parents were both employed but not insured, was refused admission by a private hospital for treatment of an abscess. Outpatient treatment failed, and his mother attempted to admit Percy to other area hospitals, which also refused care. Finally an attorney arranged for the original hospital to admit the child; the parents then owed the hospital $6000._\n\nAccess to health care is most simply measured by the number of times a person uses health care services. Commonly used data are numbers of physician visits, hospital days, and preventive services received. In addition, access can be quantified by surveys in which respondents report whether or not they failed to seek care or delayed care when they felt they needed it. In 2009, 56% of uninsured adults, compared with 10% of those with insurance, had no usual source of care, 32%, compared with 8% of those with insurance, postponed seeking care due to cost, and 26%, compared with 4% for those with insurance, went without needed care due to cost (Kaiser Family Foundation, 2010a).\n\n##### **Health Insurance and Health Outcomes**\n\n_Dan Sugarman noticed that he was urinating a lot and feeling weak. His friend told him that he had diabetes and needed medical care, but lacking health insurance, Mr. Sugarman was afraid of the cost. Eight days later, his friend found him in a coma. He was hospitalized for diabetic ketoacidosis._\n\n_Penny Evans worked in a Nevada casino. She was uninsured and ignored a growing mole on her chest. After many months of delay, she saw a dermatologist and was diagnosed with malignant melanoma, which had metastasized. She died 2 years later at the age of 44._\n\n_Leo Morelli, a hypertensive patient, was doing well until his company relocated to Mexico and he lost his job. Lacking both paycheck and health insurance, he became unable to afford his blood pressure medications. Six months later, he collapsed with a stroke._\n\nThe uninsured suffer worse health outcomes than those with insurance. Compared with insured persons, the uninsured like Mr. Sugarman have more avoidable hospitalizations; like both Mr. Sugarman and Ms. Evans, they tend to be diagnosed at later stages of life-threatening illnesses, and they are on average more seriously ill when hospitalized (American College of Physicians, 2000). Higher rates of hypertension and cervical cancer and lower survival rates for breast cancer among the uninsured, compared with those with insurance, are associated with less frequent blood pressure screenings, Pap smears, and clinical breast examinations (Ayanian et al, 2000). People without insurance have greater rates of uncontrolled hypertension, diabetes, and elevated cholesterol than those with insurance (Wilper et al, 2009). Most significantly, people who lack health insurance suffer a higher overall mortality rate than those with insurance. After adjusting for age, sex, education, poorer initial health status, and smoking, it was found that lack of insurance alone increased the risk of dying by 25% (Franks et al, 1993). The Institute of Medicine estimated that lack of health insurance accounts for 18,000 deaths annually in the United States (Institute of Medicine, 2004).\n\n##### **Does Medicaid Make a Difference?**\n\nMedicaid, the federal and state public insurance plan, has made great strides in improving access to care for two-thirds of people with incomes below the federal poverty level, but Medicaid has its limitations.\n\n##### **Medicaid and Use of Health Services**\n\n_Concepcion Ortiz lived in a town of 25,000 persons. When she became pregnant, her sister told her that she was eligible for Medicaid, which she obtained. She called each obstetrician in town and none would take Medicaid patients. When she reached her sixth month, she became desperate._\n\nFor those people with Medicaid coverage, access to care is by no means guaranteed. Medicaid pays physicians far less than does Medicare or private insurance with the result that many physicians do not accept Medicaid patients.\n\nAs a rule, people with Medicaid have a level of access to medical care that is intermediate between those without insurance and those with private insurance. Compared with uninsured people, those with Medicaid are more likely to have a regular source of medical care and are less likely to report delays in receiving care. But these access measures for Medicaid recipients are not as good as for people with private insurance (Kaiser Family Foundation, 2010a).\n\n##### **Medicaid and Health Outcomes**\n\nHealth outcomes for Medicaid recipients lag behind those for privately insured people. Compared with privately insured people, Medicaid recipients have lower rates of immunizations, screening for breast and cervical cancer, hypertension and diabetes control, and timeliness of prenatal care. (Landon et al, 2007). Medicaid patients with cancer have their disease detected at significantly later stages than privately insured patients, with the delays in diagnosis comparable for uninsured and Medicaid patients (Halpern et al, 2007). Persons with Medicaid are sometimes relegated, with the uninsured, to the lowest tier of the health care system.\n\n#### **Underinsurance**\n\nHealth insurance does not guarantee financial access to care. Many people are underinsured; that is, their health insurance coverage has limitations that restrict access to needed services (Table 3\u20132). An estimated 20% of insured Americans between the ages of 19 and 64 were underinsured in 2007, up from 12% in 2003 (Gabel et al, 2009).\n\n**Table 3\u20132.** Categories of underinsurance\n\n##### **Limits to Insurance Coverage**\n\nIn 2007, 71% of privately insured people with low incomes and substantial medical expenditures were underinsured. This number is rising as health care costs rise and insurance coverage becomes less comprehensive (Gabel et al, 2009). In 2007, 62% of bankruptcies in the United States were caused by inability to pay medical bills; 75% of these individuals had health insurance at the onset of their illness (Himmelstein et al, 2009).\n\n##### **Insurance Deductibles and Copayments**\n\n_Eva Stefanski works as a legal secretary and has a Blue Cross high-deductible health plan policy with a $2500 deductible. Last year, she failed to show up for her mammogram appointment because she did not have $150 to pay for the test. This year, she also decides to forego making an appointment for her periodic pap test._\n\nFor people with low or moderate incomes, insurance deductibles and copayments may represent a substantial financial problem. From 2006 to 2010, the percent of people with employer-sponsored insurance having a deductible of $1000 or more for single (not family) coverage grew from 10% to 27%. In 2010, 13% of insured employees (up from 4% in 2006) had high-deductible insurance plans, with families paying an average deductible of $3500 plus the employee premium contribution and copayments (Claxton et al, 2010).\n\n##### **Gaps in Medicare Coverage**\n\n_Corazon Estacio suffers from angina, congestive heart failure, and high blood pressure, in addition to diabetes. She takes 17 pills per day: four each of glyburide and metformin, three isosor-bide, two carvedilol and two furosemide, and one each of benazepril and aspirin. Because of the deductibles and the \"doughnut hole\" in her Medicare Part D plan, her yearly medication bill comes to $3840._\n\n_Ferdinand Foote was covered by Medicare and had no Medigap, Medicare Advantage, or Medicaid coverage. He was hospitalized for peripheral vascular disease caused by diabetes and a non-healing infected foot ulcer. He spent 4 days in the acute hospital and 1 month in the skilled nursing facility and made weekly physician visits following his discharge. The costs of illness not covered by Medicare included a $1132 deductible for acute hospital care, a $141.50 per day copayment for days 21 to 30 of the skilled nursing facility stay, a $162 physician deductible, and a 20% ($12) physician copayment per visit for 12 visits. The total came to $2853 not including the cost of uncovered outpatient medications._\n\nMedicare paid for only 48% of the average beneficiary's health care expenses in 2006 (Kaiser Family Foundation, 2010b). For the 5% of beneficiaries in poorest health, uncovered costs in 2004 averaged $7646, up 48% from 1992 (Riley, 2008). As discussed in Chapter 2, Medicare Part D requires beneficiaries to continue shouldering large out-of-pocket expenses for their medications, a situation that is expected to improve with the Accountable Care Act of 2010.\n\n##### **Lack of Coverage for Long-Term Care**\n\n_Victoria and Gus Pappas had $80,000 in the bank when Gus had a stroke. After his hospitalization, he was still paralyzed on the right side and unable to speak or swallow. After 18 months in the nursing home, most of the $80,000 was gone. At that point, Medicaid picked up the nursing home costs._\n\nMedicare paid only 20% of the elderly's nursing home bills in 2009, and private insurance policies picked up only an additional 8% (see Chapter 12). Many elderly families spend their life savings on long-term care, qualifying for Medicaid only after becoming impoverished.\n\n##### **The Effects of Underinsurance**\n\nDoes underinsurance represent a serious barrier to the receipt of medical care? The famous Rand Health Insurance Experiment compared nonelderly individuals who had health insurance plans with no out-of-pocket costs and those who had plans with varying amounts of patient cost sharing (deductibles or copayments). The study found that cost sharing reduces the rate of ambulatory care use, especially among the poor, and that patients with cost-sharing plans demonstrate a reduction in both appropriate and inappropriate medical visits. For low-income adults, the cost-sharing groups received Pap smears 65% as often as the free-care group. Hypertensive adults in the cost-sharing groups had higher diastolic pressures, and children had higher rates of anemia and lower rates of immunization (Brook et al, 1983; Lohr et al, 1986; Lurie et al, 1987).\n\nIn 2003, underinsured adults aged 19 to 64 with health problems were much more likely than well-insured adults to skip recommended tests or follow-up, forego seeing a physician when they felt sick, and fail to fill a prescription on account of cost (Schoen et al, 2005). In 2006, 20% of Medicare beneficiaries with Part D coverage did not fill, or delayed filling, a prescription due to inability to pay the uncovered costs (Neuman et al, 2007). In summary, lack of comprehensive insurance reduces access to health care services and may contribute to poorer health outcomes.\n\n### **NONFINANCIAL BARRIERS TO HEALTH CARE**\n\nNonfinancial barriers to health care include inability to access care when needed, language, literacy, and cultural differences between patients and health caregivers, and factors of gender and race. Excellent discussions of these issues can be found in the book \"Medical Management of Vulnerable and Underserved Patients\" (King and Wheeler, 2007).\n\n#### **Lack of Prompt Access**\n\nMedical practices often fail to provide their patients with access at the time when the patient needs care. This problem has worsened with the growing shortage of primary care practitioners. In 2008, 28% of Medicare beneficiaries without a primary care physician reported difficulty finding such a physician, a 17% increase from 2006. Thirty-one percent of privately insured patients had an unwanted delay in obtaining an appointment for routine care in 2008. In 2006, only 27% of adults with a usual source of care could easily contact their physician by phone, obtain care or advice after hours, and experience timely office visits. After Massachusetts passed its health insurance expansion in 2006, demand for primary care increased without an increase in supply, resulting in the average wait time to see a primary care internist increasing from 17 days in 2005 to 31 days in 2008. Fewer primary care physicians are accepting Medicaid patients, and inappropriate emergency department visits are growing, especially for Medicaid patients, due to inability to access timely primary care (Bodenheimer and Pham, 2010).\n\n#### **Gender and Access to Health Care**\n\n_Olga Madden is angry. Her male physician had not listened. He told her that her incontinence was from too many childbirths and that she would have to live with it. She had questions about the hormones he was prescribing, but he always seemed too busy, so she never asked. Ms. Madden calls her HMO and gets the names of two female physicians, a female physician assistant, and a nurse practitioner. She calls them. Their receptionists tell her that none of them is accepting new patients; they are all too busy._\n\nAccess problems for women often begin with finding a physician who communicates effectively. Women are 50% more likely than men to report leaving a physician because of dissatisfaction with their care, and they are more than twice as likely to report that their physician \"talked down\" to them or told them their problems were \"all in their head\" (Leiman et al, 1997). Female physicians have a more patient-centered style of communicating and spend more time with their patients than do male physicians (Roter and Hall, 2004). In a study of patients with insurance coverage for Pap smears and mammo-grams, the patients of female physicians were almost twice as likely to receive a Pap smear and 1.4 times as likely to have a mammogram than the patients of male physicians (Lurie et al, 1993).\n\nPhysicians are less likely to counsel women than men about cardiac prevention\u2014diet, exercise, and weight reduction. After having a heart attack, women are less likely than men to receive recommended diagnostic tests and are less likely to be prescribed recommended aspirin and beta-blockers (Agency for Healthcare Research and Quality, 2005).\n\nBecause women are more likely than men to have a chronic condition, women use more chronic medications and are more likely than men not to fill a prescription because of cost. Because more women than men are Medicaid recipients, they are more likely to be turned away from physicians who do not accept Medicaid. Fewer than one-third of women of reproductive age have received counseling about emergency contraception, sexually transmitted diseases, or domestic violence (Kaiser Family Foundation, 2005b).\n\nFor those women who wish to terminate a pregnancy, access to abortions is limited in many areas of the country. In 2009, 87% of US counties had no identifiable abortion provider. While women have reduced access to certain kinds of care, an equally serious problem may be instances of inappropriate care. A study conducted in a managed-care medical group in California found that 70% of hysterectomies were inappropriate (Broder et al, 2000).\n\n#### **Race and Access to Health Care**\n\n_Jose is suffering. The pain from his fractured femur is excruciating, and the emergency department physician has given him no pain medication. In the next room, Joe is asleep. He has received 10 mg of morphine for his femur fracture._\n\nAt a California emergency department, 55% of Latino patients with extremity fractures received no pain medication compared with 26% of non-Latino whites. This marked difference in treatment was attributable not to insurance status but to ethnicity (Todd et al, 1993). African American patients similarly receive poorer pain control than whites (Todd et al, 2000).\n\nBecause a far higher proportion of minorities than whites is uninsured, has Medicaid coverage, or is poor, access problems are amplified for these groups. African Americans and Latinos in the United States are less likely to have a regular source of care or to have had a physician visit in the past year (King and Wheeler, 2007). Racial and ethnic differences in access to care are not always a matter of differences in financial resources and insurance coverage. Studies have shown that African Americans and Latinos receive fewer services even when compared with non-Hispanic whites who have the same level of health insurance and income (Agency for Healthcare Research and Quality, 2009).\n\nStudies have also detected such disparities in quality of care. Looking at 38 measures of quality for such conditions as diabetes, asthma, HIV\/AIDS, cardiac care, and cancer, African Americans receive poorer quality of care than whites for 66% of these quality measures; American Indians and Alaska Natives and Latinos also have lower quality indicators (King and Wheeler, 2007).\n\nNeighborhoods that have high proportions of African American or Latino residents have far fewer physicians practicing in these communities. African American and Latino primary care physicians are more likely than white physicians to locate their practices in underserved communities (Komaromy et al, 1996).\n\nWhat explains these disparities in access to care across racial and ethnic groups that are not fully accounted for by differences in insurance coverage and socioeconomic status? Several hypotheses have been proposed. Cultural differences may exist in patients' beliefs about the value of medical care and attitudes toward seeking treatment for their symptoms. However, differences in patient preferences do not account for substantial amounts of the racial variations seen in cardiac surgery rates (Mayberry et al, 2000). A related factor may be ineffective communication between patients and caregivers of differing races, cultures, and languages. African Americans are more likely than whites to report that their physicians did not properly explain their illness and its treatment (LaVeist et al, 2000). Access barriers related to communication problems may be particularly acute for the subset of Latino patients for whom Spanish is the primary language. However, language issues do not fully account for access barriers faced by Latinos. In the study of emergency department pain medication cited previously, even Latinos who spoke English as their primary language were much less likely than non-Latino whites to receive pain medication.\n\nBecause many of these hypotheses do not satisfactorily explain the observed racial disparities in access to care, an important consideration is whether racism may also contribute to these patterns (King and Wheeler, 2007). Medicine in the United States has not escaped the nation's legacy of institutionalized racism toward many minority groups. Many hospitals, including institutions in the North, were for much of the twentieth century either completely segregated or had segregated wards, with inferior facilities and services available to nonwhites. Explicit segregation policies persisted in many hospitals until a few decades ago. Racial barriers to entry into the medical profession gave rise to the establishment of black medical schools such as the Howard, Morehouse, and Meharry schools of medicine. Although such overt racism is a diminishing feature of medicine in the United States, more insidious and often unconscious forms of discrimination may continue to color the interactions between patients and their caregivers and influence access to care for minorities (Van Ryn, 2002).\n\n### **THE RELATION BETWEEN HEALTH CARE AND HEALTH STATUS**\n\nAccess to health care does not by itself guarantee good health. A complex array of factors, only one of which is health care, determines whether a person is healthy or not.\n\n_Ace Banks is 48, an executive vice president, with four grandparents who lived past 90 years of ageand parents alive and well in their late 70s. Mr. Banks went to an Ivy League college where he was a star athlete. He has never seen a physician except for a sprained ankle._\n\n_Keith Cole is a coal miner who at age 48 developed pneumonia. He had excellent health insurance through his union and went to see the leading pulmonologist in the state. He was hospitalized but became less and less able to breathe because the pneumonia was severely complicated by black lung disease, which he contracted through his job. He received high-quality care in the intensive care unit at a fully insured cost of $65,000, but he died._\n\n_Bill Downes, an African American man, knew that his father was killed by high blood pressure and his mother died of diabetes. Mr. Downes spent his childhood in poverty living with eight children at his grandmother's house. He had little to eat except what was provided at the school lunch program, a diet heavily laden with cheese and butter. To support the family, he left school at age 15 and got a job. At age 24, he was diagnosed with high blood pressure and diabetes. He did not smoke and was meticulous in following the diet prescribed by his physician. He had private health insurance through his job as a security guard and was cared for by a professor of medicine at the medical school. In spite of excellent medical care, his glucose and cholesterol levels and blood pressure were difficult to control, and he developed retinopathy, kidney failure, and coronary heart disease. At age 48, he collapsed at work and died of a heart attack._\n\n#### **Health Status and Income**\n\nThe gap between the rich and the poor has widened markedly in the United States. Between 1952 and 2005, the proportion of pretax income reported by the wealthiest decile of the population increased from 31% to 44%; the share of income for the richest 1% doubled from 8% in 1980 to 17% in 2005. At the same time, income is decreasing for the great majority of households (Woolf, 2007). As the stories of Ace Banks, Keith Cole, and Bill Downes suggest, the health of an individual or a population is influenced less by medical care than by broad socioeconomic factors such as income and education (Braveman et al, 2010). People in the United States with incomes above four times the poverty level live on average 7 years longer than those with incomes below the poverty level (Table 3\u20133). The mortality rate for heart disease among laborers is more than twice the rate for managers and professionals. The incidence of cancer increases as family income decreases, and survival rates are lower for low-income cancer patients. Higher infant mortality rates are linked to low income and low educational level. Not only does the income level of individuals affect their health and life expectancy, the way in which income is distributed within communities also appears to influence the overall health of the population. In the United States, overall mortality rates are higher in states that have a more unequal distribution of income, with greater concentration of wealth in upper income groups (Lochner et al, 2001). Some social scientists have concluded that the toxic health effects of social inequality in developed nations result from the psychosocial stresses of social hierarchies and social oppression, not simply from material deprivation (Kawachi and Kennedy, 1999).\n\n**Table 3\u20133.** Income, race, and life expectancy in years (at age 25) _a_\n\n#### **Health Status and Race**\n\nAfrican Americans experience dramatically worse health than white Americans. Life expectancy is lower for African Americans than for other racial and ethnic groups in the United States (Table 3\u20134). Infant mortality rates among African Americans are more than double those for whites (Table 3\u20135), and the relative disparity in infant mortality has widened during the past decade. Mortality rates for African Americans exceed those for whites for 7 of the 10 leading causes of death in the United States, including the most common killers in the US population\u2014heart disease, strokes, and cancer (Table 3\u20136) (US Department of Health and Human Services, 2009). African American men younger than 45 years have 10 times the likelihood of dying of hypertension than white men in the same age group. Although the incidence of breast cancer is lower in African American women than in white women, in African American women this disease is diagnosed at a more advanced stage of illness, and thus they are more likely to die of breast cancer (Institute of Medicine, 2003; Halpern et al, 2007).\n\n**Table 3\u20134.** Life expectancy in years\n\n**Table 3\u20135.** Infant mortality, 2006 (per 1000 live births)\n\n**Table 3\u20136.** Age-adjusted death rates per 100,000 population, 2006\n\nNative Americans are another ethnic group with far poorer health than that of whites. Native Americans younger than 45 years have far higher death rates than whites of comparable age, and the Native American infant mortality rate is 50% higher than the rate of whites (US Department of Health and Human Services, 2009).\n\nLatinos and Asians and Pacific Islanders are minority groups characterized by great diversity. Health status varies widely between Cuban Americans, who tend to be more affluent, and poor Mexican American migrant farm workers, as well as between Japanese families, who are more likely to be middle class, and Laotians, who are often indigent. Compared with whites, Latinos have markedly higher death rates for diabetes and the acquired immune deficiency syndrome. Overall, Latinos have lower age-adjusted mortality rates than whites because of less cardiovascular disease and cancer. Asians in the United States have lower death rates than whites for all age groups (US Department of Health and Human Services, 2009).\n\nSome of the differences in mortality rates of African Americans and Native Americans compared with whites are related to the higher rates of poverty among these minority groups. In 2009, the white poverty rate was 12% compared with 26% for African Americans, and 25% for Latinos (US Census Bureau, 2010). However, even compared with whites in the same income class, African Americans as a group have inferior health status. Although mortality rates decline with rising income among both African Americans and whites, at any given income level, the mortality rate for African Americans is consistently higher than the rate for whites (Table 3\u20133). Thus, social factors and stresses related to race itself seem to contribute to the relatively poorer health of African Americans. The inferior health outcomes among African Americans, such as higher mortality rates for heart disease, cancer, and stroke, are in part explained by the lower rate of access to health services among this group.\n\nIf lower income is associated with poorer health, and if Latinos tend to be poorer than non-Latino whites in the United States, then why do Latinos have overall lower mortality rates than non-Latino whites? This is possibly related to the fact that many Latinos are immigrants, and foreign-born people often have lower mortality rates than people born in the United States at the same level of income (Abraido-Lanza et al, 1999; Goel et al, 2004). This phenomenon is often referred to as the \"healthy immigrant\" effect. If this is the case, mortality rates for Latinos may rise as a higher proportion of their population is born in the United States.\n\n### **CONCLUSION**\n\nHealth outcomes are determined by multiple factors. Socioeconomic status appears to be the dominant influence on health status; yet medical care and public health interventions are also extremely important (King and Wheeler, 2007). The advent of the polio vaccine markedly reduced the number of paralytic polio cases. From 1970 to 2004, age-adjusted death rates from stroke decreased by more than 100%\u2014a successful result of hypertension diagnosis and treatment. Early prenatal care can prevent low-birth-weight and infant deaths. Irradiation and chemotherapy have transformed the prognosis of some cancers (eg, Hodgkin disease) from a certain fatal outcome toward complete cure. A 1980 study of mortality rates in 400 counties in the United States found that after controlling for income, education, cigarette consumption, and prevalence of disability, a 10% increase in per capita medical care expenditures was associated with a reduced average mortality rate of 1.57% (Roemer, 1991). Moreover, the health care system provides patients with chronic disease welcome relief from pain and suffering and helps them to cope with their illnesses. Access to health care does not guarantee good health, but without such access health is certain to suffer.\n\n### **REFERENCES**\n\nAbraido-Lanza AF et al. The Latino mortality paradox: A test of the \"salmon bias\" and healthy migrant hypothesis. _Am J Public Health_. 1999;89:1543.\n\nAgency for Healthcare Research and Quality. _Women's Health Care in the United States_. May 2005. www.ahrq.gov.\n\nAgency for Healthcare Research and Quality. National Healthcare Disparities Report, 2009. www.ahrq.gov.\n\nAyanian JZ et al. Unmet needs of uninsured adults in the United States. _JAMA_. 2000;284:2061.\n\nBodenheimer T, Pham HH. Primary care: Current problems and proposed solutions. _Health Aff (Millwood)_. 2010:29:799.\n\nBraveman PA et al. Socioeconomic disparities in health in the United States: What the patterns tell us. _Am J Public Health_. 2010;100:S186\u2013S196.\n\nBroder MS et al. The appropriateness of recommendations for hysterectomy. _Obstet Gynecol_. 2000;95:199.\n\nBrook RH et al. Does free care improve adults' health? _N Engl J Med_. 1983;309:1426.\n\nBuettgens M et al. _Why the Individual Mandate Matters_. Urban Institute Web Site. December 2010. www.urban.org.\n\nClaxton G et al. Health benefits in 2010: Premiums rise modestly, workers pay more toward coverage. _Health Aff (Millwood)_. 2010;29:1942.\n\nFamilies USA. _Americans at Risk_. _One in Three Uninsured. March 2009_. www.familiesusa.org.\n\nFranks P et al. Health insurance and mortality. _JAMA_. 1993;270:737.\n\nGabel JR et al. Trends in underinsurance and the affordability of employer coverage, 2004\u20132007. _Health Aff (Millwood)_. 2009; 28:w595.\n\nGoel MS et al. Obesity among US immigrant subgroups by duration of residence. _JAMA_. 2004;292:2860.\n\nHalpern MT et al. Insurance status and stage of cancer at diagnosis among women with breast cancer. _Cancer_. 2007;110:231.\n\nHimmelstein DU et al. Medical bankruptcy in the United States, 2007. _Am J Med_. 2009;122:741.\n\nInstitute of Medicine. _Care Without Coverage: Too Little, Too Late_. Washington, DC: National Academies Press; 2002.\n\nInstitute of Medicine. _Unequal Treatment_. Washington, DC: National Academies Press; 2003.\n\nInstitute of Medicine. _Insuring America's Health_. Washington, DC: National Academies Press; 2004.\n\nKaiser Family Foundation. The Uninsured and the Difference Health Insurance Makes. September 2010a. www.kff.org.\n\nKaiser Family Foundation. Medicare Spending and Financing. 2010b. www.kff.org.\n\nKaiser Family Foundation: _Women and Health Care_. Menlo Park, CA: Kaiser Family Foundation; July 2005. www.kff.org.\n\nKawachi I, Kennedy BP. Income inequality and health: Pathways and mechanisms. _Health Serv Res._ 1999;34:215.\n\nKing TE, Wheeler MB. _Medical Management of Vulnerable and Underserved Patients_. New York, NY: McGraw-Hill; 2007.\n\nKomaromy M et al. The role of Black and Hispanic physicians in providing health care for underserved populations. _N Engl J Med._ 1996;334:1305.\n\nLandon BE et al. Quality of care in Medicaid managed care and commercial health plans. _JAMA_. 2007;298:1674.\n\nLaVeist TA et al. Attitudes about racism, medical mistrust, and satisfaction with care among African American and white cardiac patients. _Med Care Res Rev_. 2000;57(Suppl 1):146.\n\nLeiman JM et al. _Selected Facts on U.S. Women's Health: A Chart Book._ New York: The Commonwealth Fund; 1997.\n\nLochner K et al. State-level income inequality and individual mortality risk: A prospective, multilevel study. _Am J Public Health_. 2001;91:385.\n\nLohr KN et al. Use of medical care in the Rand Health Insurance Experiment. _Med Care._ 1986;24(Suppl):S1.\n\nLurie N et al. Preventive care: Do we practice what we preach? _Am J Public Health._ 1987;77:801.\n\nLurie N et al. Preventive care for women: Does the sex of the physician matter? _N Engl J Med._ 1993;329:478.\n\nMayberry RM et al. Racial and ethnic differences in access to medical care. _Med Care Res Rev._ 2000;57(Suppl 1):108.\n\nNeuman P et al. Medicare prescription drug benefit progress report. _Health Aff (Millwood)_. 2007;26:w630.\n\nRenner C, Navarro V. Why is our population of uninsured and underinsured persons growing? The consequences of the \"deindustrialization\" of the United States. _Int J Health Serv._ 1989;19:433.\n\nRiley GF. Trends in out-of-pocket healthcare costs among older community-dwelling Medicare beneficiaries. _Am J Manag Care_. 2008;14:692.\n\nRoemer MI. _National Health Systems of the World_. New York: Oxford University Press; 1991.\n\nRoter DL, Hall JA. Physician gender and patient-centered communication. _Annu Rev Public Health._ 2004;25:497.\n\nSchoen et al. Insured but not protected: How many adults are underinsured? _Health Affairs Web Exclusive_. June 14, 2005:w5\u2013289. . Accessed November 11, 2011.\n\nTodd KH et al. Ethnicity and analgesic practice. _Ann Emerg Med._ 2000;35:11.\n\nTodd KH et al. Ethnicity as a risk factor for inadequate emergency department analgesia. _JAMA_. 1993;269: 1537.\n\nUS Census Bureau. Income, Poverty, and Health Insurance Coverage in the United States, 2009; pp. 60\u2013238, September, 2010.\n\nUS Congress, Office of Technology Assessment. _Does Health Insurance Make a Difference_? OTA-BP-H-99. US Government Printing Office; 1992.\n\nUS Department of Health and Human Services. _Health United States 2009_. www.cdc.gov.\n\nVan Ryn M. Research on the provider contribution to race\/ethnicity disparities in medical care. _Med Care._ 2002;40(Suppl):I-140.\n\nWilper et al. Hypertension, diabetes, and elevated cholesterol among insured and uninsured U.S. adults. _Health Aff (Millwood)._ 2009;28:1151.\n\nWoolf SH. Future health consequences of the current decline in US household income. _JAMA_. 2007;298:1931.\n\n## **4 Reimbursing Health Care Providers**\n\nChapter 2 described the different modes of financing health care: out-of-pocket payments, individual health insurance, employment-based health insurance, and government financing. Each of these mechanisms attempted to solve the problem of unaffordable care for certain groups, but each \"solution\" in turn created new problems by stimulating rapid rises in health care costs. One of the factors contributing to this inflation was reimbursement of physicians and hospitals by insurance companies and government programs. Therefore, new methods of reimbursement have been tried as one way of lowering the growth rate in health care costs.\n\n_Dr. Mary Young has recently finished her family medicine residency and joined a small group practice, PrimaryCare. On her first day, she has the following experiences with health care financing: her first patient is insured by Blue Shield; Primary Care is paid a fee for the physical examination and for the electrocardiogram (ECG) performed. Dr. Young's second patient requires the same services, for which PrimaryCare receives no payment but is forwarded $10 for each month that the patient is enrolled in the practice. In the afternoon, a hospital utilization review physician calls Dr. Young, explains the diagnosis-related group (DRG) payment system, and suggests that she send home a patient hospitalized with pneumonia. In the evening, she goes to the emergency department, where she has agreed to work two shifts per week for $85 per hour._\n\nDuring the course of a typical day, some physicians will be involved with four or five distinct types of reimbursement. This chapter will describe the different ways in which physicians and hospitals are paid. Although reimbursement has many facets, from the setting of prices to the processing of claims, this discussion will focus on one of its most basic elements: establishing the unit of payment. This basic principle must be grasped before one can understand the key concept of physician-borne risk.\n\n### **UNITS OF PAYMENT**\n\nMethods of payment can be placed along a continuum that extends from the least to the most aggregated unit. The methods range from the simplest (one fee for one service rendered) to the most complex (one payment for many types of services rendered), with many variations in between (Table 4\u20131).\n\n**Table 4\u20131.** Units of payment\n\n#### **Definitions of Methods of Payment**\n\n##### **Fee-for-Service Payment**\n\nThe unit of payment is the visit or procedure. The physician or hospital is paid a fee for each office visit, ECG, intravenous fluid, or other service or supply provided. This is the only form of payment that is based on individual components of health care. All other reimbursement modes aggregate or group together several services into one unit of payment.\n\n##### **Payment by Episode of Illness**\n\nThe physician or hospital is paid one sum for all services delivered during one illness, as is the case with global surgical fees for physicians and DRGs for hospitals.\n\n##### **Per Diem Payments to Hospitals**\n\nThe hospital is paid for all services delivered to a patient during 1 day.\n\n##### **Capitation Payment**\n\nOne payment is made for each patient's care during a month or year.\n\n##### **Payment for All Services Delivered to All Patients within a Certain Time Period**\n\nThis includes global budget payment of hospitals and salaried payment of physicians.\n\n#### **Managed Care Plans**\n\nTraditionally physicians and hospitals have been paid on a fee-for-service basis. The development of managed care plans introduced changes in the methods by which hospitals and physicians are paid, for the purpose of controlling costs. Managed care is discussed in more detail in Chapter 6; in this chapter, only those aspects needed to understand physician and hospital reimbursement will be considered.\n\nThere are three major forms of managed care: fee-for-service practice with utilization review, preferred provider organizations (PPOs), and health maintenance organizations (HMOs).\n\n##### **Fee-for-Service Reimbursement with Utilization Review**\n\nThis is the traditional type of payment, with the addition that the third-party payer (whether private insurance company or government agency) assumes the power to authorize or deny payment for expensive medical interventions such as hospital admissions, extra hospital days, and surgeries.\n\n##### **Preferred Provider Organizations**\n\nPPOs are loose-knit organizations in which insurers contract with a limited number of physicians and hospitals who agree to care for patients, usually on a discounted fee-for-service basis with utilization review.\n\n##### **Health Maintenance Organizations**\n\nHMOs are organizations whose patients are required (except in emergencies) to receive their care from providers within that HMO. There are several types of HMOs which are discussed in Chapter 6. Some HMOs pay physicians and hospitals by more highly bundled units of payment (eg, per diem, capitation, or salary).\n\n### **METHODS OF PHYSICIAN PAYMENT**\n\n#### **Payment per Procedure: Fee-for-Service**\n\n_Roy Sweet, a patient of Dr. Weisman, is seen for recent onset of diabetes. Dr. Weisman spends 20 minutes performing an examination, fingerstick blood glucose test, urinalysis, and ECG. Each service has a fee set by Dr. Weisman: $92 for a complex visit, $8 for a fingerstick glucose test, $15 for a urinalysis, and $70 for an ECG. Because Mr. Sweet is uninsured, Dr. Weisman reduces the total bill from $185 to $90._\n\n_In 1988, Dr. Lenz, an ophthalmologist, requested that Dr. Weisman do a medical consultation for Gertrude Rales, who developed congestive heart failure and arrhythmias following cataract surgery. Dr. Weisman took 90 minutes to perform the consultation and was paid $100 by Medicare. Dr. Lenz had spent 90 minutes on the surgery plus pre- andpostoperative care and received $1600 from Medicare. In 1998, Dr. Weisman did a similar consultation for Dr. Lenz and received $130; Dr. Lenz was sent $900 for the operation._\n\n_Melissa High, a Medicaid recipient, makes three visits to Dr. Weisman for hypertension. He bills Medicaid $92 for one complex visit and $52 each for two shorter visits. He is paid $26 per visit, 40% of his total charges. Under Medicaid, Dr. Weisman may not bill Ms. High for the balance of his fees._\n\n_Dr. Weisman contracted with Blue Cross to care for its PPO patients at 70% of his normal fee. Rick Payne, a PPO patient, comes in with a severe headache and is found to have left arm weakness and hyperreflexia. Dr. Weisman is paid $84.40 for a complex visit. Before a magnetic resonance imaging (MRI) scan can be ordered, the PPO must be asked for authorization._\n\nTraditionally, private physicians have been reimbursed by patients and insurers through the fee-for-service mechanism. Before the passage of Medicare and Medicaid, physicians often discounted fees for elderly or poor patients, and even afterward many physicians have continued to assist uninsured people in this way.\n\nPrivate insurers, as well as Medicare and Medicaid in the early years, usually reimbursed physicians according to the usual, customary, and reasonable (UCR) system, which allowed physicians a great deal of latitude in setting fees. As cost containment became more of a priority, the UCR approach to fees was largely supplanted by payer-determined fee schedules. An example of this is Melissa High's three visits, which incurred charges of $196 of which Medicaid paid only $78 ($26 per visit).\n\nIn the early 1990s, Medicare moved to a fee schedule determined by a resource-based relative-value scale (RBRVS). With this system, fees (which vary by geographic area) are set for each service by estimating the time, mental effort and judgment, technical skill, physical effort, and stress typically related to that service (Bodenheimer et al, 2007). The RBRVS system made a somewhat feeble attempt to correct the bias of physician payment that has historically paid for surgical and other procedures at a far higher rate than primary care and cognitive services. In 1998, Dr. Weisman was paid nearly 15% of Dr. Lenz's surgery fee, compared with 6% of that fee in 1988, before the advent of RBRVS.\n\nPPO managed care plans often pay contracted physicians on a discounted fee-for-service basis and require prior authorization for expensive procedures.\n\nWith fee-for-service payments, physicians have an economic incentive to perform more services because more services bring in more payments (see Chapter 10). The fee-for-service incentive to provide more services contributed to the rapid rise in health care costs in the United States (Relman, 2007).\n\n#### **Payment per Episode of Illness**\n\n_Dr. Nick Belli removes Tom Stone's gallbladder and is paid $1300 by Blue Cross. Besides performing the cholecystectomy, Dr. Belli sees Mr. Stone three times in the hospital and twice in his office for postoperative visits. Because surgery is paid by means of a global fee, Dr. Belli may not bill separately for the visits, which are included in his $1300 cholecystectomy fee._\n\n_Joan Flemming complains of having had coughing, fever, and green sputum for 1 week. Dr. Violet Gramm analyzes a sputum smear and orders a chest x-ray and makes the diagnosis of pneumonia. She treats Ms. Flemming as an outpatient with azithromycin, checking her once a week for 3 weeks. With the experimental episode-based system, Dr. Gramm is paid one fee for all services and procedures involved in treating Ms. Flemming's pneumonia._\n\nSurgeons usually receive a single payment for several services (the surgery itself and postoperative care) that have been grouped together, and obstetricians are paid in a similar manner for a delivery plus pre- and postnatal care. This bundling together of payments is often referred to as reimbursement at the unit of the case or episode.\n\nWith payment by episode, surgeons have an economic incentive to limit the number of postoperative visits because they do not receive extra payment for extra visits. On the other hand, they continue to have an incentive to perform more surgeries, as with the traditional fee-for-service system. Some health care experts recommend paying physicians through an episode-based system similar to that used by Medicare for hospital reimbursement (Pham et al, 2010). Under such a system one fee would be paid for one episode of illness, no matter how many times the patient visited the physician.\n\nAt this point, it is helpful to introduce the important concept of risk. Risk refers to the potential to lose money, earn less money, or spend more time without additional payment on a reimbursement transaction. With the traditional fee-for-service system, the party paying the bill (insurance company, government agency, or patient) absorbs all the risk; if Dr. Weisman sees Rick Payne ten times rather than five times for his headaches, Blue Cross pays more money and Mr. Payne spends more in copayments. Bundling of services transfers a _portion_ of the risk from the payer to the physician; if Dr. Belli sees Tom Stone ten times rather than five times for follow-up after cholecystectomy, he does not receive any additional money. However, Blue Cross is also partially at risk; if more Blue Cross enrollees require gallbladder surgery, Blue Cross is responsible for more $1300 payments. As a general rule, the more services bundled into one payment, the larger the share of financial risk that is shifted from payer to provider. ( _Payer_ is a general term referring to whomever pays the bill; in Chapter 16, a distinction is made between purchasers of health insurance such as employers, and insurers, who can both be payers.)\n\n#### **Payment per Patient: Capitation**\n\nCapitation payments (per capita payments or payments \"by the head\") are monthly payments made to a physician for each patient signed up to receive care from that physician\u2014generally a primary care physician. The essence of capitation is a shift in financial risk from insurers to providers. Under fee-for-service, patients who require expensive health services cost their health plan more than they pay the plan in insurance premiums; the insurer is at risk and loses money. Physicians and hospitals who provide the care earn more money for treating ill people. In a 180-degree role reversal, capitation frees insurers of risk by transferring risk to providers. An HMO that pays physicians via capitation has little to fear in the short run from patients who become ill. The HMO pays a fixed sum no matter how many services are provided. The providers, in contrast, earn no additional money yet spend a great deal of time and incur large office and hospital expenditures to care for people who are sick. (In the long term, HMOs do want to limit services in order to reduce provider pressure for higher capitation payments.)\n\nCertain methods have been developed to mitigate the financial risk associated with capitation payment. One method involves reintroducing fee-for-service payments for specified services. Such types of services provided but not covered within the capitation payment are called _carve-outs;_ their reimbursement is \"carved out\" of the capitation payment and paid separately. Pap smears, immunizations, office ECGs, and minor surgical procedures may be carved out and paid on a fee-for-service basis.\n\nA common method of managing risk is called \"risk-adjusted capitation.\" For physicians paid by capitation, patients with serious illnesses require a great deal more time without any additional payment, creating an incentive to sign up healthy patients and avoid those who are sick. Risk-adjusted capitation provides higher monthly payments for elderly patients and for those with chronic illnesses. However, risk adjustment poses a major challenge. Researchers have investigated measures for risk-adjusting capitation payments by appraising an individual's state of health or risk of needing health care services (Brown et al, 2010).\n\nCapitation has potential merits as a way to control costs by providing an alternative to the inflationary tendencies of fee-for-service payment. In addition, capitation has been advocated for its potential beneficial influence on the organization of care. Capitation payments require patients to register with a physician or group of physicians. The clear enumeration of the population of patients in a primary care practice offers advantages for monitoring appropriate use of services and planning for these patients' needs. Capitation also potentially allows for more flexibility at the practice level in how to most effectively and efficiently organize and deliver services. For example, fee-for-service typically only pays for an in-person visit with a physician; under capitation payment, a physician could substitute \"virtual visits\" such as e-mail and telephone contacts for in-person visits for following up on blood pressure or diabetes control, or delegate routine preventive care tasks to nurses or medical assistants in the practice, without experiencing a financial disincentive for these alternative ways of delivering care. Capitation also explicitly defines\u2014in advance\u2014the amount of money available to care for an enrolled population of patients, providing a better framework for rational allocation of resources and innovation in developing better modes of delivering services. For a large group of primary care physicians, the sheer size of the aggregated capitation payments provides clout and flexibility over how to best arrange ancillary and specialty services.\n\n##### **Capitation with Two-Tiered Structures**\n\n_Jennifer is a young woman in England who develops an ear infection; her general practitioner, Dr. Walter Liston, sees her and prescribes antibiotics. Jennifer pays no money at the time of the visit and receives no bill. Dr. Liston is paid the British equivalent of $12 per month to care for Jennifer, no matter how many times she requires care. When Jennifer develops appendicitis and requires an x-ray and surgical consultation, Dr. Liston sends her to the local hospital for these services; payment for these referral services is incorporated into the hospital's operating budget paid for separately by the National Health Service._\n\n_**British System**_ \u2014Capitation payments to physicians in the United States are complicated, as will shortly be seen. But in the United Kingdom, they have traditionally been simple (see Chapter 14). Under the traditional British National Health Service, each person enrolls with a general practitioner, who becomes the primary care physician (PCP). For each person on the general practitioner's list, the physician receives a monthly capitation payment. The more patients on the list, the more money the physician earns. Patients are required to route all nonemergency medical needs through the general practitioner \"gatekeeper,\" who when necessary makes referrals for specialist services or hospital care. Patients can freely change from one general practitioner to another. This simple arrangement, illustrated in Figure 4\u20131, is referred to as a two-tiered capitation structure. One tier is the health plan (the government in the case of the UK) and the other tier the individual PCP or a small number of physicians in group practice.\n\n_**United States System**_ \u2014In the United States, capitation payment is associated with HMO plans and not with traditional or PPO insurance. Some HMO plans have two-tiered structures, with HMOs paying capitation fees directly to PCPs (Figure 4\u20131). However, capitation payment in US managed care organizations more often involves a three-tiered structure.\n\n**Figure 4\u20131.** Two-tiered capitated payment structures. The health plan pays the primary care physician by capitation and pays for referral services (eg, x-rays and specialist consultations) through a different reimbursement stream.\n\n##### **Capitation with Three-Tiered Structures**\n\nIn three-tiered structures, HMOs do not pay capitation fees directly to individual physicians or small group practices, but instead rely on an intermediary administrative structure for processing these payments (Robinson and Casalino, 1995). In one variety of such three-tiered structures (Figure 4\u20132A), physicians remain in their own private offices but join together into physician groups called independent practice associations (IPAs).\n\n**Figure 4\u20132.** Three-tiered capitated payment structures. **(A)** The CapCap Associates type of arrangement, in which primary physicians receive a capitation payment plus a bonus from the IPA if there is an end-of-the-year surplus in the pool for paying for referral services. **(B)** The CapFee Associates type of arrangement, in which the IPA receives capitation payments from the health plans, but pays its primary care physicians on a fee-for-service basis.\n\n_George is enrolled through his employer in Smart-Care, an HMO run by Smart Insurance Company. SmartCare has contracted with two IPAs to provide physician services for its enrollees in the area where George lives. George has chosen to receive his care from Dr. Bunch, a PCP affiliated with one of these IPA groups, CapCap Associates IPA. Smart-Care pays CapCap Associates a $60 monthly capitation fee on George's behalf for all physician and related outpatient services. CapCap Associates in turn pays Dr. Bunch a $15 monthly capitation fee to serve as George's primary care physician._\n\n_George develops symptoms of urinary obstruction consistent with benign prostatic hyperplasia. Dr. Bunch orders some laboratory tests and refers George to a urologist for cystoscopy. The laboratory and the urologist bill CapCap Associates on a fee-for-service basis and are paid by the IPA from a pool of money (called a risk pool) that the IPA has set aside for this purpose from the capitation payments CapCap Associates receives from Smart-Care. At the end of the year, CapCap Associates has money left over in this diagnostic and specialist services risk pool. CapCap Associates distributes this surplus revenue to its PCPs as a bonus._\n\nSorting out the flow of payments and nature of risk sharing becomes difficult in this type of three-tiered capitation structure. In most three-tiered HMOs, the financial risk for diagnostic and specialist services is borne by the overall IPA organization and spread among all the participating PCPs in the IPA. In the 1980s and 1990s, the CapCap Associates type of IPA often provided financial incentives to PCPs to limit the use of diagnostic and specialist services by returning to these physicians any surplus funds that remain at the end of the year. This method of reimbursement is known as capitation-plus-bonus payment. The less frequent the use of diagnostic and specialist services, the higher the year-end bonus for IPA physician gatekeepers. This arrangement came under criticism as representing a conflict of interest for PCPs because their personal income was increased by denying diagnostic and specialty services to their patients (Rodwin, 1993). More recently some managed care organizations have begun to tie bonus payments to quality measures\u2014\"pay for performance\"\u2014rather than to cost control (see Chapter 10). A considerable price must be paid for setting up a three-tiered structure because administrative costs are substantial for both the health plan and the IPA.\n\n_George's brother Steve works for the same company as George and also has SmartCare insurance. Steve, however, obtains his primary care from a physician in the other SmartCare IPA plan, CapFee Associates. Like CapCap Associates, CapFee Associates is an IPA that receives $60 per month in capitation fees for every patient enrolled. Unlike CapCap Associates, CapFee Associates pays its PCPs on a fee-for-service basis._\n\nThree-tiered IPA structures become even more confusing when the unit of reimbursement differs across tiers. In the CapCap Associates model, capitation is the basic payment method for both the IPA as a whole and its constituent primary care physicians. However, in the CapFee Associates model the IPA receives capitation payments from the health insurance plan but then reimburses its participating PCPs on a fee-for-service basis (Figure 4\u20132B). Under this arrangement, the fees billed by the IPA physicians may well exceed the amount of money the IPA has received from the insurance plan on a capitated basis to pay for physician and related outpatient services. To reduce this risk, many IPAs of the CapFee Associates type pay their physicians only a portion, perhaps 60%, of a predetermined fee schedule and withhold the other 40%. If money is left over at the end of the year, the physicians receive a portion of the withheld money.\n\nWith the CapFee system, the IPA is the main entity at risk because provision of more services can cause the IPA to lose money. But individual physicians are also partially at risk because if expenditures by the IPA are high, they will not receive the withheld funds. The economic incentive for individual primary care physicians is a mixed one. It is to the physician's financial advantage to schedule as many patient visits as possible because the physician receives a fee for each visit. But a large number of visits overall by IPA patients, as well as high use of laboratory and x-ray studies and specialist services, will deplete the IPA budget, thereby increasing the possibility that the IPA could go bankrupt, leaving its physicians with thousands of unpaid charges.\n\n#### **Payment per Time: Salary**\n\n_Dr. Joyce Parto is employed as an obstetrician-gynecologist by a large staff model HMO. She considers the financial security and lack of business worries in her current work setting an improvement over the stresses she faced as a solo fee-for-service practitioner before joining the HMO. However, she has some concerns that the other obstetricians are allowing the hospital's obstetric house staff to manage most of the deliveries during the night, and wonders if the lack of financial incentives to attend deliveries may be partly to blame. She is also annoyed by the bureaucratic hoops she has to jump through to cancel an afternoon clinic to attend her son's school play._\n\nIn contrast with traditional private physicians, physicians in the public sector (municipal, Veterans Health Administration and military hospitals, state mental hospitals), and in community clinics are usually paid by salary. Salaried practice aggregates payment for all services delivered during a month or year into one lump sum. Managed care has brought salaried practice to the private sector, sometimes with a salary-plus-bonus arrangement, particularly in integrated medical groups and group and staff model HMOs (see Chapter 6). Group and staff model HMOs bring physicians and hospitals under one organizational roof.\n\nThe distinction between staff and group model HMOs is analogous to the difference between the two-and three-tiered IPA model HMOs discussed previously. The staff model HMO is a two-tiered payment structure, with an HMO insurance plan directly employing physicians on a salaried basis (Figure 4\u20133A). In the group model HMO, the HMO insurance plan contracts on a capitated basis with an intermediary physician group, which in turn pays its individual physicians a salary (Figure 4\u20133B).\n\n**Figure 4\u20133.** Salaried payment. **(A)** In the staff model HMO, the plan directly employs physicians. **(B)** In the group model HMO, a \"prepaid group practice\" receives capitation payments from the plan and then reimburses its physicians by salary.\n\nHMO physicians paid purely by salary bear little if any individual financial risk; the HMO or physician group is at risk if expenses are too great. To manage risk, administrators at group and staff model HMOs may place constraints on their physician employees, such as scheduling them for a high volume of patient visits or limiting the number of available specialists. Salaried physicians are at risk of not getting extra pay for extra work hours. For a physician paid an annual salary without allowances for overtime pay, a high volume of complex patient visits may turn an 8-hour day into a 12-hour day with no increase in income. HMOs and medical groups may offer bonuses to salaried physicians if overall expenses are less than the amounts budgeted for these expenses or if the physician performs high quality care (pay for performance).\n\n### **METHODS OF HOSPITAL PAYMENT**\n\n#### **Payment per Procedure: Fee-for-Service**\n\n_Kwin Mock Wong is hospitalized for a bleeding ulcer. At the end of his 4-day stay, the hospital sends a $14,000 seven-page itemized hospital bill to Blue Cross, Mr. Wong's insurer._\n\nIn the past, insurance companies made fee-for-service payments to private hospitals based on the principle of \"reasonable cost,\" a system under which hospitals had a great deal of influence in determining the level of payment. Because the American Hospital Association and Blue Cross played a large role in writing reimbursement regulations for Medicare, that program initially paid hospitals according to a similar reasonable cost formula (Law, 1974). More recently, private and public payers concerned with cost containment have begun to question hospital charges and negotiate lower payments, or to shift financial risk toward the hospitals by using per diem, DRG, or capitation payments.\n\n#### **Payment per Day: Per Diem**\n\n_John Johnson, an HMO patient, with a severe headache is admitted to the hospital. During his 3-day stay, he undergoes MRI scanning, lumbar puncture, and cerebral arteriography, procedures that are all costly to the hospital in terms of personnel and supplies. The hospital receives $4800, or $1600 per day from the HMO; Mr. Johnson's stay costs the hospital $7200._\n\n_Tom Thompson, in the same HMO, is admitted for congestive heart failure. He receives intravenous furosemide for 3 days and his condition improves. Diagnostic testing is limited to a chest x-ray, ECG, and basic blood work. The hospital receives $4800; the cost to the hospital is $4200._\n\nMany insurance companies and Medicaid plans contract with hospitals for per diem payments rather than paying a fee for each itemized service (room charge, MRI, arteriogram, chest x-ray, and ECG). The hospital receives a lump sum for each day the HMO patient is in the hospital. The insurer may send a utilization review nurse to the hospital to review the charts of its patients, and if the nurse decides that a patient is not acutely ill, the HMO may stop paying for additional days.\n\nPer diem payments represent a bundling of all services provided for one patient on a particular day into one payment. With traditional fee-for-service payment, if the hospital performs several expensive diagnostic studies, it makes more money because it charges for each study, whereas with per diem payment the hospital receives no additional money for expensive procedures. Per diem bundling of services into one fee removes the hospital's financial incentive because it loses, rather than profits, by performing expensive studies.\n\nWith per diem payment, the insurer continues to be at risk for the number of days a patient stays in the hospital because it must pay for each additional day. However, the hospital is at risk for the number of services performed on any given day because it incurs more costs without additional payment when it provides more services. It is in the insurer's interest to conduct utilization reviews to reduce the number of hospital days, but the insurer is less concerned about how many services are performed within each day; that fiscal concern has been transferred to the hospital.\n\n#### **Payment per Episode of Hospitalization: Diagnosis-Related Groups**\n\n_Bill is a 67-year-old man who enters the hospital for acute pulmonary edema. He is treated withfurosemide and oxygen in the emergency room, spends 36 hours in the hospital, and is discharged. The cost to the hospital is $5200. The hospital receives a $7000 DRG payment from Medicare._\n\n_Will is an 82-year-old man who enters the hospital for acute pulmonary edema. In spite of repeated treatments with furosemide, captopril, digoxin, and nitrates, he remains in heart failure. He requires telemetry, daily blood tests, several chest x-rays, electrocardiograms, and an echocardio-gram, and is finally discharged on the ninth hospital day. His hospital stay costs $23,000 and the hospital receives $7000 from Medicare._\n\nThe DRG method of payment for Medicare patients started in 1983. Rather than pay hospitals on a fee-for-service basis, Medicare pays a lump sum for each hospital admission, with the size of the payment dependent on the patient's diagnoses. The DRG system has gone one step further than per diem payments in bundling services into one payment. While per diem payment lumps together all services performed during one day, DRG reimbursement lumps together all services performed during one hospital episode. (Although an episode of illness may extend beyond the boundaries of the acute hospitalization [eg, there may be an outpatient evaluation preceding the hospitalization and transfer to a nursing facility for rehabilitation afterward], the term _episode_ under the DRG system refers only to the portion of the illness actually spent in the acute care hospital.)\n\nWith the DRG system, the Medicare program is at risk for the number of admissions, but the hospital is at risk for the length of hospital stay and the resources used during the hospital stay. Medicare has no financial interest in the length of stay, which (except in unusually long \"outlier\" stays) does not affect Medicare's payment. In contrast, the hospital has an acute interest in the length of stay and in the number of expensive procedures performed; a long, costly hospitalization such as Will's produces a financial loss for the hospital, whereas a short stay yields a profit. Hospitals therefore conduct internal utilization review to reduce the costs incurred by Medicare patients.\n\n#### **Payment per Patient: Capitation**\n\n_Jane is enrolled in Blue Cross HMO, which contracts with Upscale Hospital to care for Jane if she requires hospitalization. Upscale receives $60 per month as a capitation fee for each patient enrolled in the HMO. Jane is healthy, and during the 36 months that she is an HMO member, the hospital receives $2160, even though Jane never sets foot in the hospital._\n\n_Wayne is also enrolled in Blue Cross HMO. Twenty-four months following his enrollment, he contracts Pneumocystis carinii pneumonia, and in the following 12 months he spends 6 weeks in Upscale Hospital at a cost of $35,000. Upscale receives a total of $2160 (the $60 capitation fee per month for 36 months) for Wayne's care._\n\nWith capitation payment, hospitals are at risk for admissions, length of stay, and resources used; in other words, hospitals bear all the risk and the insurer, usually an HMO, bears no risk. Capitation payment to hospitals has almost disappeared as a method of payment.\n\n#### **Payment per Institution: Global Budget**\n\n_Don Samuels, a member of the Kaiser Health Plan, suffers a sudden overwhelming headache and is hospitalized for 1 week at Kaiser Hospital in Oakland, California, for an acute cerebral hemorrhage. He goes into a coma and dies. No hospital bill is generated as a result of Mr. Samuels' admission, and no capitation payments are made from any insurance plan to the hospital._\n\nKaiser Health Plan is a large integrated delivery system that in some regions of the United States operates its own hospitals. Kaiser hospitals are paid by the Kaiser Health Plan through a global budget: a fixed payment is made for all hospital services for 1 year. Global budgets are also used in Veterans Health Administration, Department of Defense, and local municipal or county hospitals in the United States, as well as being a standard payment method in Canada and many European nations. In managed care par-lance, one might say that the hospital is entirely at risk because no matter how many patients are admitted and how many expensive services are performed, the hospital must figure out how to stay within its fixed budget. Global budgets represent the most extensive bundling of services: Every service performed on every patient during 1 year is aggregated into one payment.\n\n### **CONCLUSION**\n\nDuring the 1990s, the push for cost containment created a movement to change\u2014in two ways\u2014how physicians and hospitals are paid:\n\n1. Private insurers, Medicare, and Medicaid often replaced fee-for-service payment, which encourages use of more services, with reimbursement mechanisms that place economic pressure on physicians and hospitals to limit the number and cost of services offered. The bundling of services into one payment tends to shift financial risk away from payers toward physicians and hospitals.\n\n2. Whereas levels of payment were formerly set largely by providers themselves (reasonable cost reimbursement for hospitals and usual, customary, and reasonable fees for physicians), payment levels are increasingly determined by negotiation between payers and providers or by fee schedules set by payers.\n\nThe second of these trends appears to be a permanent feature of provider payment. But the first change, the substitution of capitation and other bundled mechanisms in place of fee-for-service, was largely reversed for physician payment, although more bundled forms of payment are still common for hospital reimbursement. Fee-for-service made a comeback. However, with the accelerating health cost crisis, a great deal of discussion has been taking place since 2010 about reintroducing alternatives to fee for service.\n\nOne of the challenges in designing an optimal payment system is striking the right balance between economic incentives for overtreatment and undertreatment (Casalino, 1992). The British National Health Service has traditionally mixed units of payment for general practitioners, paying a global budget for overhead costs (eg, office rent and staff), a capitation payment for each patient enrolled in the practice, and fee-for-service payments selectively for preventive services (eg, vaccinations and Pap tests) and some home visits in order to encourage provision of these items. In the United States, some managed care organizations are following the British example, creating blended payments for physicians that include elements of both capitation and fee for service (Robinson, 1999). This innovation has the potential to balance overtreatment and undertreatment incentives.\n\n### **REFERENCES**\n\nBodenheimer T et al. The primary care-specialty income gap: Why it matters. _Ann Intern Med._ 2007;146:301.\n\nBrown J et al. Does Risk Adjustment Reduce Selection in the Private Health Insurance Market?, 2010. www.wcas.north-western.edu\/csio\/Conferences\/DugganPaper.pdf. Accessed November 23, 2011.\n\nCasalino LP. Balancing incentives: How should physicians be reimbursed? _JAMA_. 1992;267:403.\n\nLaw SA. _Blue Cross: What Went Wrong?_ New Haven, CT: Yale University Press; 1974.\n\nPham HH et al. Episode-Based Payments: Charting a Course for Health Care Payment Reform. Center for Studying Health System Change Policy Analysis, January 2010. www.hschange.com.\n\nRelman A. _Second Opinion: Rescuing America's Health Care_. New York: Public Affairs; 2007.\n\nRobinson JC. Blended payment methods in physician organizations under managed care. _JAMA_. 1999;282:1258.\n\nRobinson JC, Casalino LP. The growth of medical groups paid through capitation in California. _N Engl J Med._ 1995;333:1684.\n\nRodwin MA. _Medicine, Money, and Morals: Physicians' Conflicts of Interest_. New York, NY: Oxford University Press; 1993.\n\n## **5 How Health Care Is Organized\u2014I: Primary, Secondary, and Tertiary Care**\n\n_Frank Hope has walked with a limp since contracting polio in the 1940s. When he watches his daughter run after her young toddler, he feels a sense of gratitude that the era of vaccination has protected his child and grandchild from such a disabling infection. He recalls the excitement that gripped the nation as the Salk polio vaccine was first tested and then adopted into widespread use. In Frank's mind, these types of scientific breakthroughs attest to the wonders of the US health care system._\n\n_Frank's grandson attends a day-care program. Ruby, a 3-year-old girl in the program, was recently hospitalized for a severe asthma attack complicated by pneumococcal pneumonia. She spent 2 weeks in a pediatric intensive care unit, including several days on a respirator. Ruby's mother works full-time as a bus driver while raising three children. She has comprehensive private health insurance through her job but finds it difficult to keep track of all her children's immunization schedules and to find a physician's office that offers convenient appointment times. She takes Ruby to an evening-hours urgent care center when Ruby has some wheezing but never sees the same physician twice. Ruby never received all her pneumococcal vaccinations or consistent prescription of a steroid inhaler to prevent a severe asthma attack. Ruby's mother blames herself for her child's hospitalization._\n\nPeople in the United States rightfully take pride in the technologic accomplishments of their health care system. Innovations in biomedical science have almost eradicated scourges such as polio and measles and have allowed such marvels as organ transplantation, \"knifeless\" gamma-ray surgery for brain tumors, and intensive care technology that saves the lives of children with asthma complicated by pneumonia. Yet for all its successes, the health care system also has its failures. For example, asthma is the most common cause of hospitalization in childhood (Akinbami et al, 2009). Proper medical care can markedly reduce the frequency of severe asthma symptoms and of asthma hospital admissions. In cases such as Ruby's, the failure to prevent severe asthma flare-up is not related to financial barriers, but rather reflects organizational problems, particularly in the delivery of primary care and preventive services.\n\nThe organizational task facing all health care systems is one of \"assuring that the right patient receives the right service at the right time and in the right place\" (Rodwin, 1984). An additional criterion could be \"... and by the right caregiver.\" The fragmented care Ruby received for her asthma is an example of this challenge. Who is responsible for planning and ensuring that every child receives the right service at the right time? Can an urgent care center or an in-store clinic at Wal-Mart designed for episodic needs be held accountable for providing comprehensive care to all patients passing through its doors? Should parents be expected to make appointments for routine visits at medical offices and clinics, or should public health nurses travel to homes and day-care centers to provide preventive services out in the community? What is the proper balance between intensive care units that provide life-saving services to critically ill patients and primary care services geared toward less dramatic medical and preventive needs?\n\nThe previous chapters have emphasized financial transactions in the health care system. In this chapter and the following one, the organization of the health care system will be the main focus. While considerable debate has dwelled on how to improve financial access to care, less emphasis has been given to the question \"access to what?\" In this chapter, organizational systems will be viewed through a wide-angle lens, with emphasis on such broad concepts as the relationship between primary, secondary, and tertiary levels of care, and the influence of the biomedical paradigm and medical professionalism in shaping US health care delivery. In Chapter 6, a zoom lens will be used to focus on specific organizational models that have appeared (often only to disappear) in this country over the past century.\n\n### **MODELS OF ORGANIZING CARE**\n\n#### **Primary, Secondary, and Tertiary Care**\n\nOne concept is essential in understanding the topography of any health care system: the organization of care into primary, secondary, and tertiary levels. In the Lord Dawson Report, an influential British study written in 1920, the author (1975) proposed that each of the three levels of care should correspond with certain unique patient needs.\n\n1. Primary care involves common health problems (eg, sore throats, diabetes, arthritis, depression, or hypertension) and preventive measures (eg, vaccinations or mammograms) that account for 80% to 90% of visits to a physician or other caregiver.\n\n2. Secondary care involves problems that require more specialized clinical expertise such as hospital care for a patient with acute renal failure.\n\n3. Tertiary care, which lies at the apex of the organizational pyramid, involves the management of rare and complex disorders such as pituitary tumors and congenital malformations.\n\nTwo contrasting approaches can be used to organize a health care system around these levels of care: (1) the carefully structured Dawson model of regionalized health care and (2) a more free-flowing model.\n\n1. One approach uses the Dawson model as a scaffold for a highly structured system. This model is based on the concept of regionalization: the organization and coordination of all health resources and services within a defined area (Bodenheimer, 1969). In a regionalized system, different types of personnel and facilities are assigned to distinct tiers in the primary, secondary, and tertiary levels, and the flow of patients across levels occurs in an orderly, regulated fashion. This model emphasizes the primary care base.\n\n2. An alternative model allows for more fluid roles for caregivers, and more free-flowing movement of patients, across all levels of care. This model tends to place a higher value on services at the tertiary care apex than at the primary care base.\n\nAlthough most health care systems embody elements of both models, some gravitate closer to one polarity or the other. The British National Health Service (NHS) and some large integrated delivery systems in the United States resemble the regionalized approach, while US health care as a whole traditionally followed the more dispersed format.\n\n#### **The Regionalized Model: The Traditional British National Health Service**\n\n_Basil, a 60-year-old man living in a London suburb, is registered with Dr. Prime, a general practitioner in his neighborhood. Basil goes to Dr. Prime for most of his health problems, including hay fever, back spasms, and hypertension. One day, he experiences numbness and weakness in his face and arm. By the time Dr. Prime examines him later that day, the symptoms have resolved. Suspecting that Basil has had a transient ischemic attack, Dr. Prime prescribes aspirin and refers him to the neurologist at the local hospital, where a carotid artery sonogram reveals high-grade carotid stenosis. Dr. Prime and the neurologist agree that Basil should make an appointment at a London teaching hospital with a vascular surgeon specializing in head and neck surgery. The surgeon recommends that Basil undergo carotid endarterectomy on an elective basis to prevent a major stroke. Basil returns to Dr. Prime to discuss this recommendation and inquires whether the operation could be performed at a local hospital closer to home. Dr. Prime informs him that only a handful of London hospitals are equipped toperform this type of specialized operation. Basil schedules his operation in London and several months later has an uncomplicated carotid endarterectomy. Following the operation, he returns to Dr. Prime for his ongoing care._\n\nThe British NHS has traditionally typified a relatively regimented primary\u2014secondary\u2014tertiary care structure (Figure 5\u20131).\n\n**Figure 5\u20131.** Organization of services under the traditional National Health Service model in the United Kingdom. Care is organized into distinct levels corresponding to specific functions, roles, administrative units, and population bases.\n\n1. For physician services, the primary care level is virtually the exclusive domain of general practitioners (commonly referred to as GPs), who practice in small- to medium-sized groups and whose main responsibility is ambulatory care. Two-thirds of all physicians in the United Kingdom are GPs.\n\n2. The secondary tier of care is occupied by physicians in such specialties as internal medicine, pediatrics, neurology, psychiatry, obstetrics and gynecology, and general surgery. These physicians are located at hospital-based clinics and serve as consultants for outpatient referrals from GPs, in turn routing most patients back to GPs for ongoing care needs. Secondary-level physicians also provide care to hospitalized patients.\n\n3. Tertiary care subspecialists such as cardiac surgeons, immunologists, and pediatric hematologists are located at a few tertiary care medical centers.\n\nHospital planning follows the same regionalized logic as physician services. District hospitals are local facilities equipped for basic inpatient services. Regional tertiary care medical centers handle highly specialized inpatient care needs.\n\nPlanning of physician and hospital resources within the NHS occurs with a population focus. GP groups provide care to a base population of 5000 to 50,000 persons, depending on the number of GPs in the practice. District hospitals have a catchment area population of 50,000 to 500,000, while tertiary care hospitals serve as referral centers for a population of 500,000 to 5 million (Fry, 1980).\n\nPatient flow moves in a stepwise fashion across the different tiers. Except in emergency situations, all patients are first seen by a GP, who may then steer patients toward more specialized levels of care through a formal process of referral. Patients may not directly refer themselves to a specialist.\n\nWhile nonphysician health professionals, such as nurses, play an integral role in staffing hospitals at the secondary and tertiary care levels, especially noteworthy is the NHS' multidisciplinary approach to primary care. GPs work in close collaboration with practice nurses (similar to nurse practitioners in the United States), home health visitors, public health nurses, and midwives (who attend most deliveries in the United Kingdom). Such teamwork, along with accountability for a defined population of enrolled patients and universal health care coverage, helps to avert such problems as missed childhood vaccinations. Public health nurses visit all homes in the first weeks after a birth to provide education and assist with scheduling of initial GP appointments. A national vaccination tracking system notifies parents about each scheduled vaccination and alerts GPs and public health nurses if a child has not appeared at the appointed time. As a result, more than 85% of British preschool children receive a full series of immunizations. (The British NHS is discussed at greater length in Chapter 14.)\n\nA number of other nations, ranging from industrialized countries in Scandinavia to developing nations in Latin America, have adopted a similar approach to organizing health services. In developing nations, the primary care tier relies more on community health educators and other types of public health personnel than on physicians.\n\n#### **The Dispersed Model: Traditional United States Health Care Organization**\n\n_Polly Seymour, a 55-year-old woman with private health insurance who lives in the United States, sees several different physicians for a variety of problems: a dermatologist for eczema, a gastroenterologist for recurrent heartburn, and an orthopedist for tendinitis in her shoulder. She may ask her gastroenterologist to treat a few general medical problems, such as borderline diabetes. On occasion, she has gone to the nearby hospital emergency department for treatment of urinary tract infections. One day, Polly feels a lump in her breast and consults a gynecologist. She is referred to a surgeon for biopsy, which indicates cancer. After discussing treatment options with Polly, the surgeon performs a lumpectomy and refers her to an oncologist and radiation therapy specialist for further therapy. She receives all these treatments at a local hospital, a short distance from her home._\n\nThe US health care system has had a far less structured approach to levels of care than the British NHS. In contrast to the stepwise flow of patient referrals in the United Kingdom, insured patients in the United States, such as Polly Seymour, have traditionally been able to refer themselves and enter the system directly at any level. While many patients in the United Kingdom have a primary care physician (PCP) to initially evaluate all their problems, many people in the United States have become accustomed to taking their symptoms directly to the specialist of their choice.\n\nOne unique aspect of the US approach to primary care has been to broaden the role of internists and pediatricians. While general internists and general pediatricians in the United Kingdom and most European nations serve principally as referral physicians in the secondary tier, their US counterparts share in providing primary care. Moreover, the overlapping roles among \"generalists\" in the United States (GPs, family physicians, general internists, and general pediatricians) are not limited to the outpatient sector. PCPs in the United States have assumed a number of secondary care functions by providing substantial amounts of inpatient care. Only recently has the United States moved toward the European model that removes inpatient care from the domain of PCPs and assigns this work to \"hospitalists\"\u2014physicians who exclusively practice within the hospital (Wachter and Goldman, 1996).\n\nIncluding general internists and general pediatricians, the total supply of generalists amounts to approximately one-third of all physicians in the United States, a number well below the 50% or more found in Canada and many European nations (Starfield, 1998). To fill in the primary care gap, some physicians at the tertiary care level in the United States have also acted as PCPs for some of their patients. In contrast to physicians, nurse practitioners and physician assistants are more likely to work in primary care settings and are a key component of the nation's clinical workforce.\n\nUS hospitals are not constrained by rigid secondary and tertiary care boundaries. Instead of a pyramidal system featuring a large number of general community hospitals at the base and a limited number of tertiary care referral centers at the apex, hospitals in the United States each aspire to offer the latest in specialized care. In most urban areas, for example, several hospitals compete with each other to perform open heart surgery, organ transplants, radiation therapy, and high-risk obstetric procedures. The resulting structure resembles a diamond more than a pyramid, with a small number of hospitals (mostly rural) that lack specialized units at the base, a small number of elite university medical centers providing highly superspecialized referral services at the apex, and the bulk of hospitals providing a wide range of secondary and tertiary services in the middle.\n\n#### **Which Model Is Right?**\n\nCritics of the US health care system find fault with its \"top-heavy\" specialist and tertiary care orientation and lack of organizational coherence. Analyses of health care in the United States over the past half century abound with such descriptions as \"a nonsystem with millions of independent, uncoordinated, separately motivated moving parts,\" \"fragmentation, chaos, and disarray,\" and \"uncontrolled growth and pluralism verging on anarchy\" (Somers, 1972; Halvorson and Isham, 2003). The high cost of health care has been attributed in part to this organizational disarray. Quality of care may also suffer. For example, when many hospitals each perform small numbers of surgical procedures such as coronary artery bypass grafts, mortality rates are higher than when such procedures are regionalized in a few higher-volume centers (Grumbach et al, 1995).\n\nDefenders of the dispersed model reply that pluralism is a virtue, promoting flexibility and convenience in the availability of facilities and personnel. In this view, the emphasis on specialization and technology is compatible with values and expectations in the United States, with patients placing a high premium on direct access to specialists and tertiary care services, and on autonomy in selecting caregivers of their choice for a particular health care need. Similarly, the desire for the latest in hospital technology available at a convenient distance from home competes with plans to regionalize tertiary care services at a limited number of hospitals.\n\n#### **Balancing the Different Levels of Care**\n\n_Dr. Billie Ruben completed her residency training in internal medicine at a major university medical center. Like most of her fellow residents, she went on to pursue subspecialty training, in her case gastroenterology. Dr. Ruben chose this career after caring for a young woman who developed irreversible liver failure following toxic shock syndrome. After a nerve-racking, touch-and-go effort to secure a donor liver, transplantation was performed and the patient made a complete recovery._\n\n_Upon completion of her training, Dr. Ruben joined a growing subspecialty practice at Atlantic Heights Hospital, a successful private hospital in the city. Even though the metropolitan area of 2 million people already has two liver transplant units, Atlantic Heights has just opened a third such unit, feeling that its reputation for excellence depends on delivering tertiary care services at the cutting edge of biomedical innovation. In her first 6 months at the hospital, Dr. Ruben participates in the care of only two patients requiring liver transplantation. Most of her patients seek care for chronic, often illdefined abdominal pain and digestive problems. As Dr. Ruben begins seeing these patients on a regular basis, she starts to give preventive care and treat nongastrointestinal problems such as hypertension and diabetes. At times she wishes she had experienced more general medicine during her training._\n\nAdvocates of a stronger role for primary care in the United States believe that it is too important to be considered an afterthought in health planning. In this view, overemphasis on the tertiary care apex of the pyramid creates a system in which health care resources are not well matched to the prevalence and incidence of health problems in a community. In an article entitled \"The Ecology of Medical Care\" published more than four decades ago, Kerr White recorded the monthly prevalence of illness for a general population of 1000 adults (White et al, 1961). In this group, 750 experienced one or more illnesses or injuries during the month. Of these patients, 250 visited a physician at least once during the month, nine were admitted to a hospital, and only one was referred to a university medical center. Dr. White voiced concern that the training of health care professionals at tertiary care\u2013oriented academic medical centers gave trainees like Dr. Billie Ruben an unrepresentative view of the health care needs of the community.\n\n_Serious questions can be raised about the nature of the average medical student's experience, and perhaps that of some of this student's clinical teachers, with the substantive problems of health and disease in the community. In general, this experience must be both limited and unusually biased if, in a month, only 0.0013 of the \"sick\" adults.... or 0.004 of the patients . . . . in a community are referred to university medical centers. . . . Medical, nursing, and other students of the health professions cannot fail to receive unrealistic impressions of medicine's task in contemporary Western society. . . . (White et al, 1961)_\n\nUpdating Kerr White's findings, Larry Green found precisely the same patterns four decades later (Green et al, 2001).\n\nAn English GP, John Fry (1980) conducted a related study of the ecology of care, in which he systematically recorded the types of health problems that brought patients to his office in the 1970s. Because of the GP's function as a gatekeeper under the NHS, Dr. Fry's investigation provides a close approximation of the full incidence and prevalence of diseases requiring medical attention among his population of registered patients (Table 5\u20131). The dominant pathology in this unselected population consisted of minor ailments (many of which would have improved without treatment), chronic conditions such as hypertension and arthritis, and gradations of mental illness. The incidence of new cancers was relatively rare, and only a handful of patients manifested complex syndromes such as multiple sclerosis. Although the specific pattern of illnesses differs for a US family physician practicing in the 21st century compared with the pattern for a British GP in the 1970s (eg, human immunodeficiency virus infection and Alzheimer disease do not appear in Table 5\u20131), the general pattern remains true. Dr. Fry's study confirms the adage that \"common disorders commonly occur and rare ones rarely happen.\"\n\n**Table 5\u20131.** Persons per year seeking care in a general practitioner practice with a registered population of 2500, according to problem\n\nAlthough these analyses suggest that most health needs can be met at the primary care level, this observation should not imply that most health care resources should be devoted to primary care. The minority of patients with severe or complicated conditions requiring secondary or tertiary care will command a much larger share of health care resources per capita than the majority of people with less dramatic health care needs. Treating a patient with liver failure costs a great deal more than treating a patient for a sore throat. Even in the United Kingdom, where the 65% of physicians who are GPs provide 60% of all ambulatory care, expenditures on their services account for less than 10% of the overall NHS budget, whereas the cost of inpatient and outpatient hospital care at the secondary and tertiary levels consumes nearly two-thirds of the budget. Thus, the pyramidal shape shown in Figure 5\u20131 better represents the distribution of health care problems in a community than the apportionment of health care expenditures. While almost all industrialized nations devote a dominant share of health care resources to secondary and tertiary care, the ecologic view reminds us that most people have health care needs at the primary care level.\n\n#### **The Functions and Value of Primary Care**\n\n_Dr. O. Titus Wells has cared for all six of Bruce and Wendy Smith's children. As a family physician whose practice includes obstetrics, Dr. Wells attended the births of all but one of the children. The Smiths' 18-month-old daughter Ginny has had many ear infections. Even though this is a common problem, Dr. Wells finds that it presents a real medical challenge. Sometimes examination of Ginny's ears indicates a raging infection and at other times shows the presence of middle ear fluid, which may or may not represent a bona fide bacterial infection. He tries to reserve antibiotics for clear-cut cases of bacterial otitis. He feels it is important that he be the one to examine Ginny's ears because her eardrums never look entirely normal and he knows what degree of change is suspicious for a genuinely new infection._\n\n_When Ginny is 2 years old, Dr. Wells recommends to the Smiths that she see an otolaryngologist andaudiologist to check for hearing loss and language impairment. The audiograms show modest diminution of hearing in one ear. The otolaryngologist informs the Smiths that ear tubes are an option. At Ginny's return visit with Dr. Wells, he discusses the pros and cons of tube placement with the Smiths. He also uses the visit as an opportunity to encourage Mrs. Smith to quit smoking, mentioning that research has shown that exposure to tobacco smoke may predispose children to ear infections._\n\nBarbara Starfield, one of the world's foremost scholars in the field of primary care, conceptualized the key tasks of primary care as (1) first contact care, (2) longitudinality, (3) comprehensiveness, and (4) coordination. Dr. Wells' care of the Smith family illustrates these essential features of primary care. He is the first-contact physician performing the initial evaluation when Ginny or other family members develop symptoms of illness. _Longitudinality_ (or _continuity_ ) refers to sustaining a patient\u2013caregiver relationship over time. Dr. Wells' familiarity with Ginny's condition helps him to better discern an acute infection. Comprehensiveness consists of the ability to manage a wide range of health care needs, in contrast with specialty care, which focuses on a particular organ system or procedural service. Dr. Wells' comprehensive, family-oriented care makes him aware that Mrs. Smith's smoking cessation program is an important part of his treatment plan for Ginny. Coordination builds upon longitudinality. Through referral and follow-up, the primary care provider integrates services delivered by other caregivers. These tasks performed by Dr. Wells meet the definition of primary care as defined by the Institute of Medicine: \"Primary care is the provision of integrated, accessible health care services by clinicians who are accountable for addressing a large majority of personal health care needs, developing sustained partnerships with patients, and practicing in the context of family and community\" (Institute of Medicine, 1996).\n\nA functional approach helps characterize which health care professionals truly fill the primary care niche. Among physicians in the United States, family physicians, general internists, and general pediatricians typically provide first contact, longitudinal, comprehensive, coordinated care. Emergency medicine physicians provide first contact care that may be relatively comprehensive for acute problems, but they do not provide continuity of care or coordinate care for patients on an ongoing basis. Some obstetrician-gynecologists provide first contact and longitudinal care, but usually only for reproductive health conditions; it is the rare obstetrician-gynecologist who is trained and inclined to comprehensively care for the majority of a woman's health needs throughout the lifespan (Rivo et al, 1994). Similarly, a patient with kidney failure or a patient with cancer may have a strong continuity of care relationship with a nephrologist or an oncologist, but these medical subspecialists rarely assume responsibility for comprehensive care of clinical problems outside of their specialty area or coordinate most ancillary and referral services (Rosenblatt et al, 1998).\n\nIn addition to physicians, many generalist nurse practitioners and physician assistants in the United States deliver the four key Starfield functions and serve as primary care providers for their patients. Research performed in a selected number of practices have demonstrated comparable quality of care for patients treated by primary care physicians and nurse practitioners (Horrocks et al, 2002).\n\nStudies have found that the core elements of good primary care advance the \"triple aims\" of health system improvement: better patient experiences, better patient outcomes, and lower costs (Starfield, 1998). For example, continuity of care is associated with greater patient satisfaction, higher use of preventive services, reductions in hospitalizations, and lower costs (Saultz and Albedaiwi, 2004; Saultz and Lochner, 2005). There is evidence that having a regular source of care results in better control of hypertension and less reliance on emergency department services (Shea et al, 1992). A Canadian study found that children undergoing tonsil-lectomy were more likely to have the operation performed for appropriate indications when they were referred to the otolaryngologist by a pediatrician than when care was directly sought from the otolaryngologist (Roos, 1979). Persons whose care meets a primary care\u2013oriented model have better perceived access to care are more likely to receive recommended preventive services, are more likely to adhere to treatment, and are more satisfied with their care (Bindman et al, 1996; Stewart et al, 1997; Safran et al, 1998). International comparisons have indicated that nations with a greater primary care orientation tend to have more satisfied patients and better performance on health indicators such as infant mortality, life expectancy, and total health expenditures (Starfield et al, 2005). Within the United States, states with more PCPs per capita have lower total mortality rates, lower heart disease and cancer mortality rates, and higher life expectancy at birth compared with states having fewer PCPs, adjusting for other factors such as age and per capita income. In contrast, increases in specialist supply are associated with greater costs but not improved quality (Starfield et al, 2005). In an analysis of quality and cost of care across states for Medicare beneficiaries, Baicker and Chandra (2004) found that states with more PCPs per capita had lower per capita Medicare costs and higher quality. States with more specialists per capita had lower quality and higher per capita Medicare expenditures.\n\n#### **Care Coordination and \"Gatekeeping\"**\n\n_Polly Seymour, described earlier in the chapter, feels terrible. Every time she eats, she feels nauseated and vomits frequently. She has lost 8 pounds, and her oncologist is worried that her breast cancer has spread. She undergoes blood tests, an abdominalCT scan, and a bone scan, all of which are normal. She returns to her gastroenterologist, who tells her to stop the ibuprofen she has been taking for tendinitis. Her problem persists, and the gastroenterologist performs an endoscopy, which shows mild gastric irritation. A month has passed, $3000 has been spent, and Polly continues to vomit._\n\n_Polly's friend Martha recommends a nurse practitioner who has been caring for Martha for many years and who, in Martha's view, seems to spend more time talking with patients than do many physicians. Polly makes an appointment with the nurse practitioner, Sara Steward. Ms. Steward takes a complete history, which reveals that Polly is taking tamoxifen for her breast cancer and that she began to take aspirin after stopping the ibuprofen. Ms. Steward explains that either of these medications can cause vomiting and suggests that they be stopped for a week. Polly returns in a week, her nausea and vomiting resolved. Ms. Steward then consults with Polly's oncologist, and together they decide to restart the tamoxifen but not the aspirin. Polly becomes nauseated again, but eventually begins to feel well and gains weight while taking a reduced dose of tamoxifen. In the future, Ms. Steward handles Polly's medical problems, referring her to specialty physicians when needed, and making sure that the advice of one consultant does not interfere with the therapy of another specialist._\n\nA concept that incorporates many of the elements of primary care is that of the primary care provider as gatekeeper. Gatekeeping took on pejorative connotations in the heyday of managed care, when, as described in Chapter 4, some types of financial arrangements with PCPs provided incentives for them to \"shut the gate\" in order to limit specialist referrals, diagnostic tests, and other services (Grumbach et al, 1998). A more accurate designation of the role of the PCP in helping patients navigate the complexities of the health care system is that of coordinator of care (Franks et al, 1992). Stories such as Polly's demonstrate the importance of having a generalist care coordinator who can advocate on behalf of his or her patients and work in partnership with patients to integrate an array of services involving multiple providers to avoid duplication of services, enhance patient safety, and care for the whole person.\n\n#### **The Patient-Centered Medical Home**\n\n_Dr. Retro is counting the days until he can retire from his solo practice of family medicine. He feels overwhelmed most days. The next available appointment in his office is in 10 weeks, and patients call every day frustrated about not being able to get appointments. A health plan just sent him a quality report card indicating that many diabetic patients in his practice have not achieved the targeted levels of control of their blood sugar, blood pressure, and lipids. He is also behind in keeping his patients up to date on their mammograms and colorectal cancer screening. Many days he has trouble finding information in the thick paper medical records about when his patients last received their preventive care services or diabetic tests. He was hoping to recruit a new family medicine residency graduate to take over his practice, but most young physicians in his region are pursuing more highly paid careers in non-primary care specialties._\n\n_Dr. Avantgard has always embraced innovation. When she read a series of articles in the Journal of the American Medical Association about new primary care practice models (Bodenheimer and Grumbach, 2007), she proposed to her 3 physician and 2 nurse practitioner partners that their primary care practice become a Patient Centered Medical Home. Dr. Avantgard starts by identifying a consultant to help the practice completely revamp their scheduling system to a \"same-day\" appointment system, where 50% of appointment slots are to be left unbooked until the day prior so that patients can call and be guaranteed a same day or next day appointment. Despite her partners' concerns about being overrun with patient appointments, the new scheduling system results in the same number of patients being seen each day, but with happier patients who are delighted to be able to get prompt access to care. The practice buys an electronic medical record system and uses the EMR to develop registries of all the patients in the practice due for preventive and chronic care services. Dr. Avantgard and her associates train their medical assistants to use the EMR, along with standing orders, to proactively order mammograms and blood lipid tests whendue and to administer vaccinations and screen for depression during patient intake at medical visits. Now that many of the routine preventive and chronic care tasks are being capably handled by other staff, Dr. Avantgard and her clinician colleagues have more time during office visits to focus on the problems patients want to talk with them about and to work through complex medical problems. With the quality indicators and patient satisfaction scores for the practice rising to the top decile of scores for practitioners in the region, Dr. Avantgard plans to start negotiations with several health plans to add a monthly care coordination payment to the current fee-for-service payments they pay, so that the practice can be compensated for all the work they perform in care coordination outside of office visits._\n\nBy the turn of the 21st century, primary care in the US had reached a critical juncture (Bodenheimer, 2006). In 2006, the American College of Physicians sounded the alarm about an \"impending collapse of primary care medicine\" (American College of Physicians, 2006). Primary care clinicians like Dr. Retro struggled to meet patient demands for accessible, comprehensive, well-coordinated care. Many gaps in quality existed, and care often fell short of being patient centered. PCPs were demoralized by outmoded practice models ill-equipped to meet the demands of modern-day primary care and an ever-widening gap between their take-home pay and the escalating earnings of specialists. In response to this crisis, the 4 major professional organizations representing the nation's primary care physicians\u2014the American Academy of Family Physicians, American College of Physicians, American Academy of Pediatrics, and American Osteopathic Association\u2014came together in 2007 and issued a report on a shared vision for reform of primary care. The _Joint Principles of a Patient-Centered Medical Home_ has served as a rallying point for building a broad movement to revitalize primary care in the US (Grundy et al, 2010).\n\nThe term \"medical home\" dates back to 1967, when it was first used by the American Academy of Pediatrics to describe the notion of a primary care practice that would coordinate care for children with complex needs. While the _Joint Principles_ have several specific elements, Rittenhouse and Shortell (2009) have provided a straightforward conceptualization of the patient-centered medical home as consisting of four basic cornerstones: primary care, patient-centered care, new-model practice, and payment reform. This framework begins by reaffirming the fundamental functions of primary care and the goal of delivering accessible, comprehensive, longitudinal, and coordinated care. The concept then builds on those foundational principles by calling for greater attention to patient-centeredness, such as the type of same-day scheduling methods adopted by Dr. Avantgard; implementation of innovative practice models, such as Dr. Avantgard's development of team-care models that reengineer workflows and tasks; and changes in physician payment, such as blending fee-for-service with partial capitation and quality incentives. Another perspective on the patient-centered medical home is shown in Table 5\u20132.\n\n**Table 5\u20132.** \"Old\" and \"new\" model primary care: some elements of transforming a practice into a patient-centered medical home\n\nThe primary care reform movement in the United States has gathered momentum, with many large employers and consumer groups joining the physician organizations authoring the _Joint Principles_ and other health professional groups to form the Patient Centered Primary Care Collaborative (Grundy et al, 2010). The Collaborative advocates and provides technical assistance for policy reforms to support primary care and transformation of practices into patient-centered medical homes. The push to enact the Affordable Care Act in 2010 focused lawmakers' attention on primary care. President Obama and many members of Congress recognized that expanding insurance coverage requires an adequate primary care workforce to provide first contact care for millions of newly insured people. The Affordable Care Act includes several measures to strengthen primary care, including increases in Medicare fees for primary care and support of patient-centered medical home reforms. Evaluation of the first wave of practices and systems implementing the types of practice innovations called for under patient-centered medical home reforms have demonstrated improvements in patient satisfaction and quality of care and reductions in use of costly emergency department and hospital services (Grumbach and Grundy, 2010). Whether the new-found enthusiasm for reform and renewal of primary care can be sustained and lead to a fundamental reorientation of the health system in the United States remains to be determined.\n\n### **FORCES DRIVING THE ORGANIZATION OF HEALTH CARE IN THE UNITED STATES**\n\n#### **The Biomedical Model**\n\nThe growth of the dispersed mode of health care delivery in the United States was shaped by several forces. One factor was the preeminence of the biomedical model among medical educators and young physicians throughout the 20th century. The combination of stricter state licensing laws and an influential national study, the Flexner report of 1906, led to consolidation of medical training in academically oriented medical schools (Starr, 1982). These academic centers embraced the biomedical paradigm that was the legacy of such renowned 19th-century European microbiologists as Pasteur and Koch. The antimicrobial model engendered the faith that every illness has a discrete, ultimately knowable cause and that \"magic bullets\" can be crafted to eradicate these sources of disease. Physicians were trained to master pathophysiologic changes within a particular organ system, leading to the development of specialization (Luce and Byyny, 1979).\n\nAdvocates of a larger role for generalism and primary care in US health care have not so much rejected the concepts of scientific medicine and professional specialism as they have attempted to broaden the interpretation of these terms. They have called for a more integrated scientific approach to understanding health and illness that incorporates information about the individual's psychosocial experiences and family, cultural, and environmental context as well as physiologic and anatomic constitution (Engel, 1977). The attempt to more rigorously define the scientific and clinical basis of generalism contributed to the emergence of family medicine in the 1970s as a specialty discipline in its own right, and the 1-year general practice internship was replaced by a 3-year residency program and specialty board certification.\n\n#### **Financial Incentives**\n\nA second and related factor influencing the structure of health care was the financial incentive for physician specialization and hospital expansion, which played out in a number of ways.\n\n1. Insurance benefits first offered by Blue Cross covered hospital costs but not physician visits and other outpatient services.\n\n2. As physician services came to be covered later under Blue Shield and other plans, a growing differential in reimbursement between generalist and specialist physicians developed. New technologic and other procedures often required considerable physician time when first introduced, and higher fees were justified for these procedures. But as the procedures became routine, fees remained high, while the time and effort required to perform them declined (Starr, 1982); this resulted in an increasing disparity in income between PCPs and specialists (Bodenheimer et al, 2007). In the mid-1980s, the average PCP's income was 75% of the average specialist's income; by 2006, PCP income had dropped to only 50% of specialists' income (Council of Graduate Medical Education, 2010). As Figure 5\u20132 shows, the percentage of graduating medical students planning to enter careers in primary care tracks the PCP-specialist income gap closely, with the proportion of students entering primary care decreasing as the earnings of PCPs relative to specialists declines.\n\n**Figure 5\u20132.** Proportion of US medical students entering primary care strongly tracks relative incomes of primary care physicians. The figure shows trends over time in the average income of primary care physicians relative to specialist physicians in the United States, and in the percentage of graduating medical students in the United States planning on entering careers in primary care. In 1990, when the average primary care physician income was about half that of specialists, fewer than 20% of graduating students planned to enter primary care fields. By 1997, when primary care physician incomes had risen to more than 60% that of specialists, the proportion of graduating students entering primary care had increased in a parallel direction with 40% of graduates planning to enter primary care fields in 1997. Both relative incomes and intentions to enter primary care decreased after 1997. (From the Council on Graduate Medical Education. _Twentieth Report: Advancing Primary Care,_ December 2010).\n\n3. Federal involvement in health care financing further fueled the expansion of hospital care and specialization. The Hill\u2013Burton Hospital Construction Act of 1946 allocated nearly $4 billion between 1946 and 1971 for expansion of hospital capacity rather than development of ambulatory services (Starr, 1982). The enactment of Medicare and Medicaid in 1965 perpetuated the private insurance tradition of higher reimbursement for procedurally oriented specialists than for generalists. Medicare further encouraged specialization through its policy of extra payments to hospitals to cover costs associated with residency training. Linking Medicare teaching payments to the hospital sector added yet another bias against community-based primary care training.\n\nThe growth of hospitals and medical specialization was intertwined. As medical practice became more specialized and dependent on technology, the site of care increasingly shifted from the patient's home or physician's office to the hospital. The emphasis on acute hospital care had an effect on the nursing profession comparable to that on physicians. World War I was a watershed period in the transition of nursing from a community-based to a hospital-based orientation. During the war, US military hospitals overseas were much heralded for their success in treating acute war injuries. At the war's conclusion, the nation rallied behind a policy of boosting the civilian hospital sector. According to Rosemary Stevens (1989),\n\n_Before the war, public-health nursing was the elite area; nurses had been instrumental in the campaigns against tuberculosis and for infant welfare. In contrast, the war emphasized the supremacy and glamour of hospitals. . . . nurses, like physicians, were trained\u2014and ready\u2014to perform in an increasingly specialized, acute-care medical environment rather than to expand their interests in social medicine and public health (Stevens, 1989)._\n\n#### **Professionalism**\n\nThe final factor accounting for the organizational evolution of US health care delivery was the nature of control over health planning. The United States is unique in its relative laxity of public regulation of health care resources. In most industrialized nations, governments wield considerable control over health planning through measures such as regulation of hospital capacity and technology, allocation of the number of residency training positions in generalist and specialist fields, and coordination of public health with medical care services. In the United States, the government has provided much of the financing for health care, but without an attendant degree of administrative control. The Hill\u2013Burton program, for example, did not make grants for hospital construction contingent upon any rigorous community-wide plan for regionalized hospital services. Medicare funding for physician training did not stipulate any particular distribution of residency positions according to specialty.\n\nWith government controls kept largely at bay, the professional \"sovereignty\" of physicians emerged as the preeminent authority in health care (Starr, 1982). Societies grant certain occupations special status as \"professions\" because of the unique knowledge and skill required of members of the profession, and the expectation that this knowledge and skill will be applied beneficially (Friedson, 1970; Light and Levine, 1988). Professionalism thus involves a social contract; in return for the privilege of autonomy, physicians bear the responsibility for acting as the patient's agent, and the profession must regulate itself to preserve the public trust.\n\nTheir professional status vested physicians with special authority to guide the development of the US health care system. As described in Chapter 2, third-party payment for physician services was established with physician control of the initial Blue Shield insurance plans. Physician judgment about the need for technology and greater inpatient capacity drove the expansion of hospital facilities.\n\nWhat was the nature of the profession that so heavily influenced the development of the US health care organization? It was a profession that, because of the primacy of the biomedical paradigm and the nature of financial incentives, was weighted toward hospital and specialty care. Small wonder that US health care has emphasized its tertiary care apex over its primary care base. In Chapter 16, we discuss the shifting power relationships in health care that are challenging the professional dominance of physicians.\n\n### **CONCLUSION**\n\n_Jeff leaves a town forum at the local medical center feeling confused. It featured two speakers, one of whom criticized the medical center as being out of touch with the community's needs, and the other of whom defended the center's contributions to society. Jeff found the first speaker very convincing about the need to pay more attention to primary care, prevention, and public health. He had never had a regular primary care physician, and the idea of having a family physician appealed to him. He was equally impressed by the second speaker, whose account of how research atthe medical center had led to life-saving treatment of children with a hereditary blood disorder was very moving, and whose description of the hospital's plan for a new imaging center was spellbinding. Jeff felt that if he ever became seriously ill, he would certainly want all the specialized services the medical center had to offer._\n\nThe professional model and the biomedical paradigm are responsible for many of the attractive characteristics of the US health care system. The biomedical model has instilled respect for the scientific method and has helped to curtail medical quackery. Professionalism has directed physicians to serve as agents acting in their patients' best interests and has made the practice of medicine more than just another business. Expansion of hospital facilities has meant that people with health insurance have had convenient access to tertiary care services and new technology. Patients have been able to take advantage of the expertise and availability of a wide variety of specialists. In many circumstances, the system is well organized to deliver the \"right care.\" For a patient in cardiogenic shock, the right place to be is an intensive care unit; for a patient with a detached retina, an ophthalmologist's office is the right place to be.\n\nHowever, there is widespread concern that despite the benefits of biomedical science and medical professionalism, the US health care system is precariously off balance. A model of excellence focused on specialization, technology, and curative medicine has led to relative inattention to basic primary care services, including such needs as disease prevention and supportive care for patients with chronic and incurable ailments. The value placed on individualism and autonomy for health care professionals and institutions has contributed to a pluralistic delivery system in which care is often fragmented and lacking coordination. A system that prizes specialists who focus on organ systems and researchers who concentrate on splitting genes has bred apprehension that health care has somehow lost sight of the whole person and the whole community. The net result is a system structured to perform miraculous feats for individuals who are ill, but at great expense and often without satisfactorily attending to the full spectrum of health care needs of the entire population. During the 2009 debate in Congress leading up to the passage of the Affordable Care Act, one of the harshest critiques of the status quo in US health care came not from a Congressional Democrat, but from Senator Orrin Hatch, the senior Republican Senator from Utah. At a hearing on health reform, Senator Hatch said, \"The US is first in providing rescue care, but this care has little or no impact on the general population. We must put more focus on primary care and preventive medicine. How do we transform the system to do this?\" (Grundy et al, 2010).\n\n### **REFERENCES**\n\nAkinbami LJ et al. Status of childhood asthma in the United States, 1980\u20132007. _Pediatrics_. 2009;123(Suppl 3):S131-45.\n\nAmerican College of Physicians. _The Impending Collapse of Primary Care Medicine and Its Implications for the State of the Nation's Health Care_. January 30, 2006. .\n\nBaicker K, Chandra A. Medicare spending, the physician work-force, and beneficiaries' quality of care. _Health Aff (Millwood)_. 2004:W4-184.\n\nBindman AB et al. Primary care and receipt of preventive services. _J Gen Intern Med_. 1996;11:269.\n\nBodenheimer T. Regional medical programs: No road to regionalization. _Med Care Rev_. 1969;26:1125.\n\nBodenheimer T. Primary care\u2014will it survive? _N Engl J Med_. 2006;355:861.\n\nBodenheimer T et al. The primary care-specialty income gap: Why it matters. _Ann Intern Med_. 2007;146:301.\n\nBodenheimer T, Grumbach K. _Improving Primary Care. Strategies and Tools for a Better Practice_. New York, NY: McGraw-Hill; 2007.\n\nCouncil on Graduate Medical Education. _Twentieth Report: Advancing Primary Care_ , December 2010.\n\nDawson W. Interim report on the future provision of medical and allied services. In: Saward EW, ed. _The Regionalization of Personal Health Services_. London, England: Prodist; 1975.\n\nEngel GL. The need for a new medical model: A challenge for biomedicine. _Science_. 1977;196:129.\n\nFranks P et al. Gatekeeping revisited: Protecting patients from overtreatment. _N Engl J Med_. 1992;327:424.\n\nFriedson E. _Professional Dominance: The Social Structure of Medicine_. Atherton, CA: Atherton Publishing; 1970.\n\nFry J. Primary care. In: Fry J, ed. _Primary Care_. London, England: William Heinemann; 1980.\n\nGreen LA et al. The ecology of medical care revisited. _N Engl J Med_. 2001;344:2021.\n\nGrumbach K, Grundy P. _Outcomes of Implementing Patient Centered Medical Home Interventions: A Review of EvidenceFrom Prospective Evaluation Studies in the United States_. Washington, DC: Patient Centered Primary Care Collaborative, 2010. .\n\nGrumbach K et al. Primary care physicians' experience of financial incentives in managed care systems. _N Engl J Med_. 1998;339:1516.\n\nGrumbach K et al. Regionalization of cardiac surgery in the United States and Canada: Geographic access, choice, and outcomes. _JAMA_. 1995;274:1282.\n\nGrundy P et al. The multi-stakeholder movement for primary care renewal and reform. _Health Affairs_. 2010;29:791.\n\nHalvorson GC, Isham GJ. _Epidemic of Care_. San Francisco, CA: Jossey-Bass; 2003.\n\nHorrocks S et al. Systematic review of whether nurse practitioners working in primary care can provide equivalent care to doctors. _BMJ_. 2002;324:819.\n\nInstitute of Medicine. _Primary Care: America's Health in a New Era._ Washington, DC: National Academies Press; 1996.\n\nLight D, Levine S. The changing character of the medical profession: A theoretical overview. _Milbank Mem Fund Q_. 1988;66:10.\n\nLuce JM, Byyny RL. The evolution of medical specialism. _Perspect Biol Med_. 1979;22:377.\n\nRivo ML et al. Defining the generalist physician's training. _JAMA_. 1994;271:1499.\n\nRodwin VG. _The Health Planning Predicament._ Berkeley, CA: University of California Press; 1984.\n\nRoos N. Who should do the surgery? Tonsillectomy-adenoidectomy in one Canadian province. _Inquiry_. 1979;16:73.\n\nRosenblatt RA et al. The generalist role of specialty physicians: is there a hidden system of primary care? _JAMA_. 1998;279:1364.\n\nSafran DG et al. Linking primary care performance to outcomes of care. _J Fam Pract_. 1998;47:213.\n\nSaultz JW, Albedaiwi W. Interpersonal continuity of care and patient satisfaction: A critical review. _Ann Fam Med_. 2004;2:445.\n\nSaultz JW, Lochner J. Interpersonal continuity of care and care outcomes: a critical review. _Ann Fam Med_. 2005;3:159.\n\nShea S et al. Predisposing factors for severe, uncontrolled hypertension in an inner-city minority population. _N Engl J Med_. 1992;327:776.\n\nSomers AR. Who's in charge here? Alice searches for a king in Mediland. _N Engl J Med_. 1972;287:849.\n\nStarfield B. _Primary Care._ New York, NY: Oxford University Press; 1998.\n\nStarfield B et al. Contribution of primary care to health systems and health. _Milbank Q_. 2005;83:457.\n\nStarr P. _The Social Transformation of American Medicine_. New York, NY: Basic Books; 1982.\n\nStevens R. _In Sickness and in Wealth: American Hospitals in the Twentieth Century._ New York, NY: Basic Books; 1989.\n\nStewart AL et al. Primary care and patient perceptions of access to care. _J Fam Pract_. 1997;44:177.\n\nWachter RM, Goldman L. The emerging role of \"hospitalists\" in the American health care system. _N Engl J Med_. 1996;335:514.\n\nWhite KL et al. The ecology of medical care. _N Engl J Med_. 19; 265:885.\n\n## **6 How Health Care Is Organized\u2014II: Health Delivery Systems**\n\nThe last chapter explored some general principles of health care organization, including levels of care, regionalization, physician and other practitioner roles, and patient flow through the system. This chapter looks more closely at actual structures of medical practice.\n\nThe traditional dispersed model of the US medical practice has been referred to as a \"cottage industry\" of independent private physicians working as solo practitioners or in small groups. A number of alternative organizational forms have existed in the United States, ranging from community health centers to prepaid group practices. The traditional model is in competition with a system of larger practice organizations and networks structured along a more integrated model of health care delivery.\n\n### **THE TRADITIONAL STRUCTURE OF MEDICAL CARE**\n\n#### **Physicians and Hospitals**\n\n_Dr. Harvey Commoner finished his residency in general surgery in 1956. For the next 30 years, he and another surgeon practiced medicine together in a middle-class suburb near St. Peter's Hospital, a nonprofit church-affiliated institution. Dr. Commoner received most of his cases from general practitioners and internists on the St. Peter's medical staff. By 1965, the number of surgeons operating at St. Peter's had grown. Because Dr. Commoner was not getting enough cases, he and his partner joined the medical staff of Top Dollar Hospital, a for-profit facility 3 miles away, and University Hospital downtown. On an average morning, Dr. Commoner drove to all three hospitals to perform operations or to do postoperative rounds on his patients. The afternoon was spent seeing patients in his office. He was on call every other night and weekend._\n\n_Dr. Commoner was active on the St. Peter's medical staff executive committee, where he frequently proposed that the hospital purchase new radiology and operating room equipment needed to keep up with advances in surgery. Because the hospital received hundreds of thousands of dollars each year for providing care to Dr. Commoner's patients, and because Dr. Commoner had the option of admitting his patients to Top Dollar or University, the St. Peter's administration usually purchased the items that Dr. Commoner recommended. The Top Dollar Hospital administrator did likewise._\n\nDuring the period when Dr. Commoner was practicing, most medical care was delivered by fee-for-service private physicians in solo or small group practices. Most hospitals were private nonprofit institutions, sometimes affiliated with a religious organization, occasionally with a medical school, often run by an independent board of trustees composed of prominent people in the community. Most physicians in traditional fee-for-service practice were not employees of any hospital, but joined one or several hospital medical staffs, thereby gaining the privilege of admitting patients to the hospital and at times acquiring the responsibility to assist the hospital through work on medical staff committees or by caring for emergency department patients who have no physician.\n\nFor many years, the physicians were the dominant power in the hospital, because physicians admit the patients, and hospitals without patients have no income. Because physicians were free to admit their patients to more than one hospital, the implicit threat to take their patients elsewhere gave them influence. Under traditional fee-for-service medicine, physicians used informal referral networks, often involving other physicians on the same hospital medical staff. In metropolitan areas with a high ratio of physician specialists to population, referrals could become a critical economic issue. Most surgeons obtained their cases by referral from primary care physicians (PCPs) or medical specialists; surgeons like Dr. Commoner who were not readily available when called soon found their case load drying up.\n\n### **THE SEEDS OF NEW MEDICAL CARE STRUCTURES**\n\nThe dispersed structure of independent fee-for-service private practice was not always the dominant model in the United States. When modern medical care took root in the first half of the twentieth century, a variety of structures blossomed. Among these were multispecialty group practices, community health centers, and prepaid group practices. Some of these flourished but then wilted, while others became the seeds from which the future health care system of the twenty-first century may germinate.\n\n#### **Multispecialty Group Practice**\n\n_In 1905, Dr. Geraldine Giemsa joined the department of pathology at the Mayo Clinic. The clinic, led by the brothers William and Charles Mayo, was becoming a nationally renowned referral center for surgery and was recruiting pathologists, microbiologists, and other specialized diagnosticians to support the work of the clinic's group of surgeons. Dr. Giemsa received a salary and became an employee of the group practice. With time, she became a senior partner and part owner of the Mayo Clinic._\n\nTogether with their father, the Mayo brothers, who were general practitioners skilled at surgical techniques, formed a group practice in the small town of Rochester, MN, in the 1890s. As the brothers' reputation for clinical excellence grew, the practice added several surgeons and physicians in laboratory-oriented specialties. By 1929, the Mayo Clinic had more than 375 physicians and 900 support staff and eventually went on to open its own hospitals (Starr, 1982). Although the clinic paid its physician staff by salary, the clinic itself billed patients, and later third-party insurance plans, on a fee-for-service basis. The Mayo Clinic was the inspiration for other group practices that developed in the United States, such as the Menninger Clinic in Topeka, KS, and the Palo Alto Medical Foundation in California. These clinics were owned and administered by physicians and featured physicians working in various specialties\u2014hence the common use of the term _multispecialty group practice_ to describe this organizational model. As in the case of the Mayo Clinic, these multispecialty group practices were innovative in the manner in which they brought a large number of physicians together under one roof to deliver care.\n\nBy formally integrating specialists into a single clinic structure, group practice attempted to promote a collaborative style of care. Lacking a strong role for the PCP as coordinator of services, the specialty-oriented group practice model attempted to use the structure of the practice organization itself as a means of creating an environment for coordinated care among specialist physicians. Enhancement of quality of care was also expected from the greater opportunity for formal and informal peer review and continuing education when colleagues worked together and shared responsibility for the care of patients. Critics of group practice warned that large practice structures would jeopardize the intimate patient\u2013physician relationship possible in a solo or small group setting, arguing that large groups would subject patients to an impersonal style of care with no single physician clearly accountable for the patient's welfare.\n\nIn 1932, the blue ribbon Committee on the Costs of Medical Care recommended that the delivery of care be organized around large group practices (Starr, 1982). The eight physicians in private practice who were members of the committee dissented from the recommendations, roundly criticizing the sections on group practice. An editorial in the _Journal of the American Medical Association_ was even more scathing in its attack on the committee's majority report:\n\n_The physicians of this country must not be misled by utopian fantasies of a form of medical practice,which would equalize all physicians by placing them in groups under one administration. The public will find to its cost, as it has elsewhere, that such schemes do not answer that hidden desire in each human breast for human kindliness, human forbearance, and human understanding. It is better for the American people that most of their illnesses be treated by their own physicians rather than by industries, corporations, or clinics. (The Committee on the Costs of Medical Care, 1932)_\n\nSeveral multispecialty group practices flourished during the period between the world wars, and to this day remain among the most highly regarded systems of care in the United States. Yet multispecialty group practice did not become the dominant organizational structure. In part, resistance to this model by professional societies blunted the potential for growth. In addition, as hospitals assumed a central role in medical care, group practice lost some of its unique attractions. Hospitals could provide the ancillary services physicians needed for the increasingly specialized and technology-dependent work of medicine. Hospitals also served as an organizational focus for the informal referral networks that developed among private physicians in independent practice.\n\n#### **Community Health Centers**\n\nOne of the most far-reaching alternatives to fee-for-service medical practice is the community health center, emphasizing primary and preventive care and also striving to take responsibility for the health status of the community served by the health center. An early twentieth-century example of such an institution was the Greater Community Association at Creston, IA. The association brought together civic, religious, education, and health care groups in a coordinated system centered on the community hospital serving a six-county area with 100,000 residents. The plan placed its greatest emphasis on preventive care and public health measures administered by public health nurses. In describing the association, Kepford (1919) wrote:\n\n_The motto of the Greater Community Association is \"Service.\" Among the principles of the hospital management are the precept that it shall be a long way from the threshold of the hospital to the operating room. . . . We have a hospital that makes no attempt to pattern after the great city institutions, but is organized to meet the needs of a rural neighborhood. The Greater Community Association has been taught to regard the hospital as a repair shop, necessary only where preventive medicine has failed. (Kepford, 1919)_\n\n_In 1928, Sherry Kidd joined the Frontier Nursing Service in Appalachia as a nurse midwife. For $5 per year, families could enroll in the service and receive pregnancy-related care. Sherry was responsible for all enrolled families within a 100-mile radius. She referred patients with complications to an obstetrician in Lexington, KY, who was the service's physician consultant._\n\nAnother pioneering model, the Frontier Nursing Service was established by Mary Breckinridge, an English-trained midwife, in 1925 (Dye, 1983). Breckin-ridge designed the service to meet the needs of a poor rural area in Kentucky that lacked basic medical and obstetric care and suffered from high rates of maternal and infant mortality. The Frontier Nursing Service shared many of the features of the Creston, IA, model: regionalized services planned on a geographic basis to serve rural populations with an emphasis on primary care and health education. Like the Creston system, the service relied on nurses to provide primary care, with physicians reserved for secondary medical services on a referral basis.\n\nThese rural programs had their urban counterparts in health centers that focused on maternal and child health services during the early 1900s (Rothman, 1978; Stoeckle and Candib, 1969). The clinics primarily served populations in low-income districts in large cities and were often involved with large immigrant populations. As in the rural systems, public health nurses played a central role in an organizational model geared toward health education, nutrition, and sanitation. Both the urban and rural models of community health centers waned during the middle years of this century. Public health nursing declined in prestige as hospitals became the center of activity for nursing education and practice (Stevens, 1989). A team model of nurses working in collaboration with physicians withered under a system of hierarchical professional roles.\n\nThe community health center model was revived in 1965, when the federal Office of Economic Opportunity, the agency created to implement the \"War on Poverty,\" initiated its program of community health centers. The program's goals included the combining of comprehensive medical care and public health to improve the health status of defined low-income communities, the building of multidisciplinary teams to provide health services, and participation in the governance of the health centers by community members.\n\n_Dr. Franklin Jefferson was professor of hematology at a prestigious medical school. His distinguished career was based on laboratory research, teaching, and subspecialty medical practice, with a focus on sickle cell anemia. Dr. Jefferson felt that his work was serving his community, but that he would like to do more. In 1965, with the advent of the federal neighborhood health center program, he left his laboratory in the hands of a well-trained assistant and began to talk with community leaders in the poor neighborhood that surrounded the medical school. After a year, the trust that was developed between Dr. Jefferson and members of the neighborhood bore fruit in a decision to approach the medical school dean about a joint medical school\u2013community application for funds to create a neighborhood health center. Two years later, the center opened its doors, with Dr. Jefferson as its first medical director._\n\nBy the early 1980s, 800 federally funded community health centers were in operation in the United States, administered by governing boards that included patients enrolled in the health center. Many of the centers trained community members as outreach workers, who became members of health care teams that included public health nurses, physicians, mental health workers, and health educators. Some of the health centers made a serious attempt to meld clinical services with public health activities in programs of community-oriented primary care. For example, the rural health center in Mound Bayou, MS, helped organize a cooperative farm to improve nutrition in the county, dig wells to supply safe drinking water, and train community residents to become health care professionals. By improving the care of low-income ambulatory patients, the centers were able to reduce hospitalization and emergency department visits by their patients. Community health centers also had some success in improving community health status, particularly by reducing infant and neonatal mortality rates among African Americans (Geiger, 1984). In the past decade, the federal government invested in a new period of expansion of community health centers, and these health centers are viewed as a critical access point for the reforms enacted in the Affordable Care Act of 2010. In 2008, more than 1000 community health centers at 7500 sites were serving 17 million people, three-quarters of them uninsured or covered by Medicaid (Kaiser Commission, 2010).\n\n#### **Prepaid Group Practice and Health Maintenance Organizations**\n\nHistorically, one alternative to small office-based, fee-for-service practice became the major challenge to that traditional model: prepaid group practice, one of the models upon which the modern HMO is based.\n\nIn 1929, the Ross\u2013Loos Clinic began to provide medical services for employees of the Los Angeles Department of Water and Power on a prepaid basis. By 1935, the clinic had enrolled 37,000 employees and their dependents, who each paid $2 per month for a specified list of services. Also in 1929, an idealistic physician, Dr. Michael Shadid, organized a medical cooperative in Elk City, OK, based on four principles: group practice, prepayment, preventive medicine, and control by the patients, who were members of the cooperative. In the late forties, more than a hundred rural health cooperatives were founded, many in Texas, but they tended to fade away, partly from the stiff opposition of organized medicine. In the 1950s, another version of the consumer-managed prepaid group practice sprang up in Appalachia, where the United Mine Workers established union-run group practice clinics, each receiving a budget from the union-controlled, coal industry\u2013financed medical care fund. Meanwhile, the Group Health Association of Washington, DC, had been organized in 1937 as a prepaid group practice whose board was elected by the cooperative's membership. A few years later in Seattle, Group Health Cooperative of Puget Sound acquired its own hospital, began to grow, and by the mid-1970s had 200,000 subscribers, a fifth of the Seattle-area population. In 1947, the Health Insurance Plan of New York opened its doors, operating 22 group practices; within 10 years, Health Insurance Plan's enrollment approached 500,000 (Starr, 1982).\n\nThe most successful of the prepaid group practices that emerged in the 1930s and 1940s was the Kaiser Health Plan. In 1938, a surgeon named Sidney Garfield began providing prepaid medical services for industrialist Henry J. Kaiser's employees working at the Grand Coulee Dam in Washington State. Rather than receiving a salary from Kaiser, Garfield was prepaid a fixed sum per employee, a precursor to modern capitation payment. Kaiser transported this concept to 200,000 workers in his shipyards and steel mills on the West Coast during World War II (Garfield, 1970; Starr, 1982). In this way, company-sponsored medical care in a remote area gave birth to today's largest alternative to fee-for-service practice. Kaiser opened its doors to the general public after World War II. Kaiser now operates in nine states and Washington, DC, with nearly 9 million patients enrolled.\n\nThe contemporary systems that grew out of the Kaiser and consumer cooperative models share several important features. Rather than preserving a separation between insurance plans and the providers of care, these models attempt to meld the financing and delivery of care into a single organizational structure. Paying a premium for health insurance coverage in this approach does not just mean that a third-party payer will reimburse some or all the costs of care delivered by independent practitioners. Rather, the premium serves to directly purchase, in advance, health services from a particular system of care. This is the notion of \"prepaid\" care that is one component of the prepaid group practice model. (As discussed in Chapter 2, the Baylor Hospital plan in the 1930s was a parallel attempt to develop a model of prepaid hospital care.) The second component is care delivered by a large group of practitioners working under a common administrative structure\u2014the \"group practice\" aspect of prepaid group practice.\n\nSystems such as Kaiser and Group Health Cooperative of Puget Sound were commonly referred to as _prepaid group practices_ until the 1970s, when terminology underwent a transformation as part of a political effort to sell the public and Congress on this model of care as a centerpiece of health care reform under the Nixon administration. Paul Ellwood, a Minnesota physician and advisor to President Nixon, suggested that prepaid group practices be referred to as \"health maintenance organizations\" (Ellwood et al, 1971; Starr, 1982). This change in name was intended in part to break from the political legacy of the prepaid group practice movement, a legacy colored with populist tones from the cooperative plans and tainted by organized medicine's common criticism of prepaid group practice as a socialist threat. The term _health maintenance_ was also designed to suggest that these systems would place more emphasis on preventive care than had the traditional medical model. Although HMOs were initially synonymous with prepaid group practice, by the 1980s, several varieties of HMO plans emerged that departed from the prepaid group practice organizational form. We describe the Kaiser model to fully illustrate the first-generation HMO model, and then proceed to discuss the second-generation HMOs known as independent practice associations (IPAs) or network HMOs.\n\n### **FIRST-GENERATION HEALTH MAINTENANCE ORGANIZATIONS AND VERTICAL INTEGRATION: THE KAISER\u2013PERMANENTE MEDICAL CARE PROGRAM**\n\n_Mario Fuentes was a professor at the University of California. He and his family belonged to the Kaiser Health Plan, and the university paid his family's premium. Professor Fuentes had once fractured his clavicle, for which he went to the urgent care clinic at Kaiser Hospital in Oakland; otherwise, he had not used Kaiser's facilities. Mrs. Fuentes suffered from rheumatoid arthritis; her regular physician was a salaried rheumatologist at the Permanente Medical Clinic, the group practice in which Kaiser physicians work. One of the Fuentes' sons, Juanito, had been in an automobile accident a year earlier near a town 90 miles away from home. He had been taken to a local emergency department and released; Kaiser had paid the bill because no Kaiser facility was available in the town. Three days after returning home, Juanito developed a severe headache and became drowsy; he was taken to the urgent care clinic, received a CT scan, and was found to have a subdural hematoma. He was immediately transported to Kaiser's regional neurosurgery center in Redwood City, CA, where he underwent surgery to evacuate the hematoma._\n\n_Dr. Roberta Short had mixed feelings about working at Kaiser. She liked the hours, the salary, and the paucity of administrative tasks. She particularly liked working in the same building with other general internists and specialists, providingthe opportunity for frequent discussions on diagnostic and therapeutic problems. However, she was not happy about seeing 4 or 5 patients per hour. Such a pace left little time to talk to the patients or to make important phone calls to patients or specialists. It was tough for Dr. Short's patients to get appointments with her, and it was even harder to arrange prompt appointments with specialists, who were as busy as she was. Moreover, the rules for ordering magnetic resonance imaging scans and other expensive tests were strict, though by and large reasonable. Overall, Dr. Short felt that the Kaiser system worked well but needed more physicians per enrolled patient._\n\nThe Kaiser\u2013Permanente Medical Care Program is the largest of the nation's prepaid group practice HMOs, consisting of three interlocking administrative units:\n\n1. The Kaiser Foundation Health Plan, which performs the functions of health insurer, such as administering enrollment and other aspects of the financing of care.\n\n2. The Kaiser Foundation Hospitals Corporation, which owns and administers Kaiser hospitals (the same individuals sit on the boards of directors for the Health Plan and the Hospitals Corporation).\n\n3. Permanente medical groups, the physician organizations that administer the group practices and provide medical services to Kaiser plan members under a capitated contract with the Kaiser plan.\n\nThe organizational model typified in the Kaiser\u2013Permanente HMO has come to be known as vertical integration. _Vertical integration_ refers to consolidating under one organizational roof and common ownership all levels of care, from primary to tertiary care, and the facilities and staff necessary to provide this full spectrum of care (Figure 6\u20131). Although structures differ somewhat across Kaiser's regional health plans, most Kaiser\u2013Permanente regional units own their hospitals and clinics, hire the nurses and other personnel staffing these facilities, and contract with a single large group practice (Permanente) to exclusively serve patients covered by the Kaiser health plan.\n\n**Figure 6\u20131.** Vertical integration consolidates health services under one organizational roof.\n\nThe Kaiser form of HMO differs from traditional fee-for-service models in how it pays physicians (salary) and hospitals (global budget). It also differs in how health services are organized. Most obvious is the prepaid group practice structure that contrasts with the traditional US style of solo, independent private practice. In addition, Kaiser has typically regionalized tertiary care services at a select number of specialized centers. For example, Northern California Kaiser has centralized all neurosurgical care at only two hospitals; patients with spinal cord injuries, brain tumors, and other neurosurgical conditions are referred to these centers from other Northern California Kaiser hospitals. The distribution of specialties within the physician staff in The Permanente Medical Group is approximately half generalists and half specialists. Most regions have also integrated nonphysicians, such as nurse practitioners and physician assistants, into the primary care team.\n\nMany observers consider this ability to coherently plan and regionalize services to be a major strength of vertically integrated systems (Figure 6\u20131). Unlike a public district health authority in the United Kingdom, an HMO such as Kaiser\u2013Permanente is not responsible for the entire population of a region, but these private, vertically integrated systems in the United States do assume responsibility for organizing and delivering services to a population of plan enrollees. The prepaid nature of enrollment in the Kaiser plan permits Kaiser to orient its care more toward a population health model.\n\n### **SECOND-GENERATION HEALTH MAINTENANCE ORGANIZATIONS AND \"VIRTUAL INTEGRATION\": NETWORK MODEL HMOs, INDEPENDENT PRACTICE ASSOCIATIONS, AND INTEGRATED MEDICAL GROUPS**\n\n_As more and more of her patients switched from fee-for-service health plans to the new HMO plans run by commercial insurers that were capturing a growing share of the private health insurance market in California, Dr. Westcoast figured she had no choice but to start contracting with these HMOs if she wanted to retain her patients. She joined the Good Health Independent Practice Association (IPA), an organization that helped solo practitioners like Dr. Westcoast contract with different HMOs. Within 3 years, 30% of the patients in her internal medicine practice were covered by 4 HMO plans that contracted with the Good Health IPA._\n\n_Although having HMO contracts was clearly proving to be important for the viability of her practice, Dr. Westcoast found much of the new arrangements frustrating. Each HMO sent her annual reports on various quality-of-care measures for the diabetic patients that the HMO showed as having Dr. Westcoast as their PCP. The trouble was that, many of the patients were not actually patients in her practice, and it was hard to reconcile the reports for a few diabetic patients from each of the 4 HMOs with all the diabetic patients she saw, including many not enrolled in HMOs, to understand how she really was doing in meeting quality standards for all the diabetic patients in her practice. Good Health IPA sent her its own quality report about the diabetic patients that Good Health thought had Dr. Westcoast as a PCP, and the information in that report didn't match the data sent by the HMOs. To make matters worse, each HMO had a different formulary of the diabetic medications that were covered by the health plan, and Dr. West-coast spent a lot of time helping exasperated patients who needed their prescriptions changed to a different medication. She wondered about giving up her practice to join Kaiser, where she would be less independent but at least she wouldn't have to deal with so many different HMOs, each with their different set of rules._\n\nIn 1954, the medical society in San Joaquin County, CA, fretted about the possibility of Kaiser moving into the county. Private fee-for-service patients might go to the lower cost Kaiser, and physicians' incomes would fall. An idea was born: To compete with Kaiser, the San Joaquin Foundation for Medical Care was set up as a network of physicians in independent private practice to contract as a group with employers for a monthly payment per enrollee; the foundation would then pay the physicians on a discounted fee-for-service basis and conduct utilization review to discourage overtreatment (Starr, 1982). It was hoped that the plan would reduce the costs to employers, who would choose the foundation rather than Kaiser.\n\nWhen the Health Maintenance Organization Act of 1973 was enacted into law as the outcome of President Nixon's health care reform strategy, network model HMOs were included along with prepaid group practice as legitimate HMOs. The HMO law stimulated HMO development by requiring large- and mediumsized businesses that provided health insurance to their employees to offer at least one federally qualified HMO as an alternative to traditional fee-for-service insurance if such an HMO existed in the vicinity (Starr, 1982). Network-model HMOs were far easier to organize than prepaid group practices; a county or state medical society, a hospital, or an insurance company could simply recruit the office-based, fee-for-service physicians practicing in the community into network, and thereby create the basis for an HMO. The physicians could continue to see their non-HMO patients as well. The inclusion of the network form of HMO in the 1973 legislation ensured that the HMO movement would not produce rapid alterations in the traditional mode of delivering medical care.\n\nSome of the initial network-model HMOs were organized on the two-tiered payment model described in Chapter 4. Under this model, an HMO contracts with many individual physicians to care for HMO enrollees. Some network-model HMOs have evolved into models that use a three-tiered payment structure whereby the HMO does not contract directly with individual physicians but rather with a large group of physicians. These groups may take several forms. The San Joaquin Foundation for Medical Care was an early example of the Independent Practice Association (IPA) model, consisting of a network of physicians who agree to participate in an association for purposes of contracting with HMOs and other managed care plans. Physicians maintain ownership of their practices and administer their own offices. The IPA serves as a vehicle for negotiating and administering HMO contracts.\n\nUnlike the \"monogamous\" arrangement between each Kaiser region and its respective Permanente medical group, in network models physicians can establish contractual relationships with numerous HMOs and IPAs. A physician may participate in more than one IPA, and each IPA may in turn have contracts with many HMO and managed care plans. The result of this more open HMO\u2013physician relationship is a series of physician panels in the same community that overlap partially, but not completely, for patients covered by different HMOs. While this more open-ended network approach may have some appeal to physicians and patients in contrast to more tightly integrated HMO models like Kaiser, it can also produce the types of frustrations experienced by Dr. Westcoast. A PCP, who may see patients from several HMOs and participate in more than one IPA, often finds that a specialist or hospital participates in the network for one HMO or IPA but not another, causing disruption and confusion when it comes to figuring out which specialist or hospital is eligible to accept a referral (Bodenheimer, 2000). Patients may find that their PCP is in one IPA but their preferred specialist is not in the same network\u2014with physicians often moving in and out of various networks as contracts are renegotiated.\n\nIPAs initially did little more than to act as brokers between physicians and HMOs, replacing the need for physicians to negotiate contracts on an individual basis. As IPAs took on a larger portion of financial risk for care (see Chapter 4), they became more active in attempting to control costs and assumed responsibility for authorizing utilization of services, profiling physicians' practice patterns, and administering other cost control strategies. Some IPAs have attempted to fashion themselves into more than simply contractual and financial intermediaries by facilitating quality improvement efforts and adoption of electronic medical records among participating practices.\n\nAnother structure related to second-generation HMOs is the integrated medical group. Integrated medical groups have a tighter organizational structure than IPAs, consisting of groups in which physicians no longer own their practices and office assets, but become employees of an organization that owns and manages their practice. Some modern-day integrated groups are survivors of the original breed of multispecialty group practices, such as the Mayo Clinic and Palo Alto Medical Foundation described earlier. Others lack these clinics' historical genesis and consist of new organizations created in the managed care era. Some of these newer organizations were created by large, for-profit companies buying up the practices of formerly independent physicians and hiring these same physicians to work as employees of the medical group (Robinson and Casalino, 1996). Others are owned by hospitals or medical schools or are privately held companies with physician partners as owners. Similar to IPAs, integrated medical groups contract with multiple managed care plans and also typically care for patients in fee-for-service private insurance plans and Medicare.\n\nYet another organizational structure to have emerged is the Physician Hospital Organization (PHO). PHOs developed in the 1980s as an alternative to the IPA model. Instead of creating a physician association to negotiate health plan contracts, physicians partnered with a hospital to jointly contract with health plans for both physician and hospital payment rates. The physicians participating in PHOs often consisted of both private practitioners on the hospital's medical staff and physicians directly employed by the hospital. Formation of PHOs received a setback in the 1990s when the Federal Trade Commission deemed that some PHO arrangements constituted collusion in price setting between physicians and hospitals to an extent that violated anti-trust laws.\n\nThe network model HMO represents an alternative to the vertically integrated HMO. As shown in Figure 6\u20132, managed care relationships involving IPAs and medical groups consist of a network of contractual links between HMOs and autonomous physician groups, hospitals, and other provider units, rather than the \"everything-under-one-roof\" model of vertical integration. Observers have dubbed the network forms of managed care organization \"virtual integration,\" signifying an integration of services based on contractual relationships rather than unitary ownership (Robinson and Casalino, 1996). In these virtually integrated systems, HMOs do not directly provide health services through their own hospitals and physician organizations.\n\n**Figure 6\u20132.** Virtual integration involves contractual links between HMOs and physician groups, hospitals, and other provider units.\n\n### **COMPARING VERTICALLY AND VIRTUALLY INTEGRATED MODELS**\n\nIn 2009, about one in four people in the United States was enrolled in some type of HMO, including Medicare and Medicaid beneficiaries participating in HMOs (Kaiser Family Foundation, 2011) (Figure 6\u20133). There is wide variation across states in HMO enrollment, ranging from more than half of insured people in California to under 10% in many other states. For many years, policy analysts predicted that the organizational efficiency and coherence of vertically integrated, first-generation HMOs would position these systems of care to prevail as health care entered a more competitive era. These predictions have not come true, as enrollment in virtually integrated systems has surpassed that of traditional HMOs. Whereas most vertically integrated HMOs are regionally based, non-profit health plans, national for-profit commercial insurers operate the plans enrolling the majority of patients in virtual HMO models.\n\n**Figure 6\u20133.** Trends in HMO enrollment in the United States. Enrollment includes individuals enrolled through Medicare and Medicaid HMO options, as well as through employment-based and other privately insured HMO plans. (MCOL Managed Care Fact Sheets, 2011; .)\n\nIn response to the reluctance of many patients to be locked into a limited panel of physicians and hospitals in conventional HMO plans, insurers have developed a variety of other products, such as the Preferred Provider Organization (PPO), which allow patients to see physicians not in the insurer's physician network, with the stipulation that patients pay a higher share of the cost out of pocket when they use non-network physicians and hospitals. Physicians joining the PPO network agree to accept discounted fees from the health plan with the hope that being listed as a \"preferred\" provider will attract more patients to their practice. PPO enrollment was about 50 million in 2009, compared with HMO enrollment of 70 million. Although both HMOs and PPOs are considered forms of \"managed care,\" PPOs are essentially a variation on insurance product benefit structure and, unlike HMOs, involve very little in the way of change in the organization of the delivery of care.\n\nVertically integrated HMOs clearly represent a significantly different organizational model than the traditional dispersed cottage industry model. Are IPAs and PHOs and the other organizational forms that predominate under network model HMOs a meaningful break from the dispersed model, offering a better framework for the delivery of health care? Or, are these virtual organizations just that\u2014loose confederacies of providers organized primarily for contracting and business objectives that offer little in the way of tangible gains in organizational coherence in the actual provision of health care?\n\nResearch suggests that more integrated organizational models have their advantages. Vertically integrated HMOs of the traditional prepaid group practice model tend to rank higher than network model HMOs on various measures quality of care, such as evidence-based care of chronic illnesses (Himmelstein et al, 1999). Integrated medical groups have been shown to perform better than IPAs in delivering up-to-date preventive care such as mammograms and Pap tests (Mehrotra et al, 2006). On general health plan satisfaction ratings, patients tend to rate integrated HMOs such as Kaiser ahead of network-model HMOs. Compared with physicians in IPAs or those not affiliated with any network, physicians in prepaid group practices report greater adoption of tools for quality improvement, such as more structured systems for planning and following through on care of patients with diabetes and other chronic illnesses (Rittenhouse et al, 2004). Size also seems to matter when it comes to the ability of medical groups to adopt the infrastructure for creating patient-centered medical homes; larger medical groups are more likely than small groups to have systems in place for care coordination, enhanced patient access, and related processes (Rittenhouse et al, 2008).\n\nThere is also evidence that moving from a completely dispersed model to a somewhat more organized model, even if only of a virtual network variety, can yield improvements in the delivery of care, especially when the network emphasizes linking patients with primary care clinicians and supports better coordination of care. An example is Community Care of North Carolina. Community Care linked North Carolina Medicaid and State Children's Health Insurance Program recipients with a primary care medical home at more than 1000 small private offices and community health centers and provided technical assistance to practices to improve chronic care services including a cadre of nurses to collaborate with practices in care management of high-risk patients. This model resulted in both better patient outcomes, such as reductions in emergency hospitalizations for children with asthma, and reductions in health care costs (Steiner et al, 2008).\n\nThe dispersed model does appear to have one important strength from the patient perspective, which is the satisfaction that comes from receiving care from a small practice where patients have a sense that clinicians and staff know them personally and the patient\u2013clinician relationship is less encumbered by organizational bureaucracy. Studies of patient preferences have found that satisfaction is highest when care is received in small offices rather than larger clinic structures (Rubin et al, 1993). People value having a familiar receptionist at the end of the line when they call about a child with a fever rather than experiencing the frustration of navigating impersonal HMO and clinic switchboard operators and voicemail systems\u2014what has been described as the \"chain store\" persona of some HMOs and large delivery systems (Mechanic, 1976). The computer era may be allowing many large medical groups and vertically integrated HMOs to jump ahead of small practices in providing the means for patients to communicate personally with their clinicians. Many large groups have implemented electronic medical records systems that allow patients to securely e-mail their clinicians and to promptly receive diagnostic test results through web-based \"patient portals\" providing patients direct access to their personal medical record.\n\n### **ACCOUNTABLE CARE ORGANIZATIONS**\n\nAs discussed in Chapter 5, the Affordable Care Act of 2010 is propelling not only an expansion of health insurance coverage, but also reform in how health care is organized and delivered. During the policy debates leading up to passage of the Affordable Care Act, the concept of Accountable Care Organizations (ACOs) emerged as a centerpiece of delivery system reform. ACOs have been defined as \"a provider-led organization whose mission is to manage the full continuum of care and be accountable for the overall costs and quality of care for a defined population\" (Rittenhouse et al, 2009). The Affordable Care Act authorized Medicare to initiate an ACO program beginning in 2012 (Health Policy Brief, 2010)\n\nACOs are envisioned as spanning a spectrum of organizational structures. On one end of this spectrum lie vertically integrated HMOs, with their medical groups and facilities well suited to provide comprehensive care to a defined population of enrolled patients under capitated payment. At the other end of the ACO spectrum lie loosely knit affiliations of providers in the traditional dispersed structure discussed at the beginning of this chapter, consisting of a hospital and the collection of private practice physicians who admit their patients to that hospital and maintain informal referral networks. Proponents of the ACO concept have proposed that Medicare could encourage such an informal hospital-physician network to function in a more cohesive and efficient manner by creating a \"shared savings\" program. Under the shared savings program, Medicare would still pay physicians by fee-for-service and hospitals by DRG, and patients would be allowed free choice of physician and hospital. However, Medicare would compare the costs and quality for the Medicare patients cared for by this virtual network; if the physicians were able to achieve certain targeted quality goals and keep costs below the predicted expenditures for these patients, Medicare would share a portion of the cost savings with the physicians and hospitals in this loosely affiliated ACO network. To qualify for such a shared-savings program, physicians and hospitals would have to create a formal ACO entity to accept shared savings payments. The variety of other organizational models discussed in this chapter, such as IPAs and PHOs, that fall between the poles of the hospital staff model and vertically integrated HMO would be other candidates for ACOs (Shortell et al, 2010).\n\nOne key difference between the proposed ACO program and the existing Medicare Advantage Program, discussed in Chapter 2, is the role of insurance companies. Medicare Advantage involves Medicare contracting with insurance plans that operate as HMOs, including both network and traditional prepaid group practice model HMOs. The Medicare ACO program is envisioned as a more direct financial relationship between Medicare and provider organizations. Prepaid group practice model HMOs could qualify as ACOs because they consist of an integrated delivery organization attached to a single insurance plan. However, network model HMO insurers would not, and, under the ACO program, Medicare would look to establish financial arrangements directly with IPAs, PHOs, and other provider-led organizations.\n\nWhether the Medicare ACO program will succeed in moving the US further along the road to more organized, integrated care structures that can improve quality and \"bend the cost curve\" remains to be determined. Regardless of whether the ACO program as articulated by its proponents will be fully realized, the ACO concept has stimulated considerable attention in the United States on reforming the delivery system to focus more on proactive care of a defined population of patients rather than just reactive care for individual patients, and on holding providers accountable for the quality and costs of care delivered.\n\n### **FROM MEDICAL HOMES TO MEDICAL NEIGHBORHOODS**\n\nIn Chapter 5, we discussed the concept of medical homes. Much of what has been discussed in this chapter on delivery system organization could aptly be described as the attempt to create well-functioning medical neighborhoods. The medical _neighborhood_ is a term coined by Fisher to describe the constellation of services, providers, and organizations in a health system that contributes to the care of a population of patients (Fisher, 2008). The primary care medical home resides in the medical neighborhood, but the medical neighborhood consists of much more than just medical homes and includes the secondary, tertiary, community, and related services needed by different patients at different times to meet their comprehensive health care needs. High-performing health care requires both excellent medical homes and excellent medical neighborhoods (Rittenhouse et al, 2009). The distinguishing feature of a hospitable medical neighborhood is care that is functionally integrated, but not necessarily structurally integrated along the lines of traditional HMOs. According to one definition, \"Integrated health care starts with good primary care and refers to the delivery of comprehensive health care services that are well coordinated with good communication among providers; includes informed and involved patients; and leads to high-quality, cost-effective care. At the center of integrated health care delivery is a high-performing primary care provider who can serve as a medical home for patients\" (Aetna Foundation, 2010).\n\nOrganizations that are structurally integrated have an advantage in being able to provide care that is functionally integrated. These organizations have assets such as multispecialty groups, a unified electronic medical record, interdisciplinary health care teams, and a quality improvement infrastructure equipped to promote care coordination and the free flow of information among all providers involved in a patient's care. One of the ongoing challenges in the United States is whether organizations that are less structurally integrated than traditional prepaid group practice HMOs will be able to achieve the degree of functional integration needed to deliver more effective and efficient care and overcome what we cited in Chapter 5 as the \"fragmentation, chaos, and disarray\" that has long plagued the US health system.\n\nDespite the admonitions of the dissenting _JAMA_ editorialists in 1932 who warned physicians not to be \"misled by utopian fantasies\" of group practice, it appears that in the early part of the twenty-first century a tipping point has occurred in the United States and the health care cottage industry is rapidly giving way to larger organizations for delivering care. From 2002 to 2008, the percentage of medical practices that are doctor-owned fell from 70% to 48% while the percentage owned by hospitals grew from 24% to 50% (Figure 6\u20134). Most new residency graduates are eschewing the tradition of becoming autonomous proprietors of their own private practices and seeking employed positions, seeking what the _New York Times_ has described as \"regular paychecks instead of shopkeeper risks\" (Harris, 2011). Hospitals are merging with one another and with physician organizations and creating large, regional delivery systems.\n\n**Figure 6\u20134.** Ownership of medical practices. (Harris G. More doctors giving up private practices. _New York Times_ , March 25, 2010.)\n\nWill the rapid organizational changes occurring in health care in the United States result in a higher-quality, more affordable health system? Will patients be cared for at the proper level of care\u2014primary, secondary, and tertiary? Will the flow of patients among these levels be constructed in an orderly way within each geographic region\u2014a regionalized structure? Will a sufficient number of primary care providers\u2014generalist physicians, physician assistants, and nurse practitioners\u2014be available so that everyone in the United States can have a regular source of primary care that allows for continuity and coordination of care? Will HMOs, ACOs, and other organizations require their physicians to take responsibility for the health of their enrollee population, or will physicians be content to care only for whoever walks in the door? What is an ideal health delivery system? Different people would have different answers. One vision is a system in which people choose their own primary care clinicians in modest-sized, decentralized, prepaid group practices that would be linked to community hospitals, including specialists' offices providing secondary care. Difficult cases could be referred to the academic tertiary care center in the region. In the primary care practices, teams of health caregivers would endeavor to provide medical care to those people seeking attention, and would also concern themselves with the health status of the entire population served by the practice.\n\n### **REFERENCES**\n\nAetna Foundation. Program Areas: Specifics, 2010. .\n\nBodenheimer T. Selective chaos. _Health Aff (Millwood)_. 2000;19(4):200.\n\nThe Committee on the Costs of Medical Care. Editorial. _JAMA._ 1932;99:1950.\n\nDye NS. Mary Breckinridge, the Frontier Nursing Service and the introduction of nurse-midwifery in the United States. _Bull Hist Med_. 1983;57:485.\n\nEllwood PM et al. Health maintenance strategy. _Med Care._ 1971;9:291.\n\nFisher ES. Building a medical neighborhood for the medical home. _N Engl J Med._ 2008;359:1202.\n\nGarfield SR. The delivery of medical care. _Sci Am_. 1970;222:15.\n\nGeiger HJ. Community health centers: Health care as an instrument of social change. In: Sidel VW, Sidel R, eds. _Reforming Medicine._ New York: Pantheon Books; 1984.\n\nHarris G. More Doctors Giving up Private Practice, and Family Physicians Can't Give Away Solo Practice. _New York Times_. March 25 and April 22, 2011.\n\nHealth Policy Brief: Accountable care organizations. _Health Affairs._ July 27, 2010.\n\nHimmelstein DU et al. Quality of care in investor-owned vs not-for-profit HMOs. _JAMA_. 1999;282:159.\n\nKaiser Commission on Medicaid and the Uninsured. Issue Brief. _Community Health Centers: Opportunities and Challenges of Health Reform_. Washington, DC: The Kaiser Commission on Medicaid and the Uninsured; 2010. .\n\nKaiser Family Foundation. Total HMO Enrollment, July 2009. State Health Facts, 2011. http:\/\/www.statehealthfacts.org\/comparetable.jsp?ind=348&cat=7&sub=85&yr=194&typ=1&sort=a.\n\nKepford AE. The Greater Community Association at Creston, Iowa. _Mod Hosp_. 1919;12:342.\n\nMechanic D. _The Growth of Bureaucratic Medicine._ New York: John Wiley & Sons; 1976.\n\nMehrotra A et al. Do integrated medical groups provide higher-quality medical care than individual practice associations? _Ann Intern Med._ 2006;145:826.\n\nRittenhouse DR et al. Physician organization and care management in California: From cottage to Kaiser. _Health Aff (Millwood)_. 2004;23(6):51.\n\nRittenhouse DR et al. Measuring the medical home infrastructure in large medical groups. _Health Aff (Millwood)._ 2008;27:1246.\n\nRittenhouse DR et al. Primary care and accountable care\u2014two essential elements of delivery-system reform. _N Engl J Med._ 2009;361:2301.\n\nRobinson JC, Casalino LP. Vertical integration and organizational networks in health care. _Health Aff (Millwood)_. 1996;15:7.\n\nRothman SM. _Woman's Proper Place: A History of Changing Ideals and Practices._ New York: Basic Books; 1978.\n\nRubin HR et al. Patients' ratings of outpatient visits in different practice settings. _JAMA_. 1993;270:835.\n\nShortell SM et al. How the Center for Medicare and Medicaid innovation should test accountable care organizations. _Health Aff (Millwood)_. 2010;29:1293.\n\nStarr P. _The Social Transformation of American Medicine._ New York: Basic Books; 1982.\n\nSteiner BD et al. Community care of North Carolina: Improving care through community health networks. _Ann Fam Med_. 2008;6:361.\n\nStevens R. _In Sickness and in Wealth: American Hospitals in the Twentieth Century._ New York: Basic Books; 1989.\n\nStoeckle JD, Candib LM. The neighborhood health center: Reform ideas of yesterday and today. _N Engl J Med_. 1969;280:1385.\n\n## **7 The Health Care Workforce and the Education of Health Professionals**\n\nA health care system is only as good as the people working in it. The most valuable resource in health care is not the latest technology or the most state-of-the-art facility, but the health care professionals and other workers who are the human resources of the health care system.\n\nIn this chapter, we discuss the nation's three largest health professions\u2014nurses, physicians, and pharmacists, as well as a closely linked profession, physician assistants (Table 7\u20131). What are the educational pathways and licensing processes that produce the nation's practicing physicians, nurses (including nurse practitioners), pharmacists, and physician assistants? How many of these health care professionals are working in the United States, and where do they practice? Do we have the right number? Too many? Too few? How would we know if we had too many or too few? Are more women becoming physicians? Are more men becoming nurses? Is the growing racial and ethnic diversity of the nation's population mirrored in the racial and ethnic composition of the health professions? To answer these questions, we begin by providing an overview of each of these professions, describing the overall supply and educational pathways. We then discuss several cross-cutting issues pertinent to all these professions.\n\n**Table 7\u20131.** Number of active practitioners in selected health professions in the United States, by profession and year\n\n### **PHYSICIANS**\n\n_Susan Gasser entered medical school in 1997. During college, she had worked in the laboratory of an anesthesiologist, which made her seriously consider a career in that specialty. During her first year of medical school, the buzz among the fourth-year students was that practice opportunities were drying up fast in anesthesiology. Health maintenance organizations (HMOs) wanted more primary care physicians, not more specialists. Almost none of the fourth-year students applied to anesthesiology residency programs that year. Susan started to think more about becoming a primary care physician. In her third year of school, she had a gratifying experience during her family practice rotation working in a community health center and started to plan to apply for family practice residencies._\n\n_At the beginning of her fourth year of school, Susan spent a month in the office of a suburban family physician, Dr. Woe. Dr. Woe frequently remarked to Susan about the pressures he felt to see more patients and about how his income had fallen because of low reimbursement and higher practice expenses. He mentioned that the local anesthesiology group was having difficulty finding a new anesthesiologist to join the group to help keep up with all the surgery being performed in the area. The group was guaranteeing a first-year salary that was twice what Dr. Woe earned as an experienced family physician. Susan quickly began to reconsider applying to anesthesiology residency programs._\n\nApproximately 873,000 physicians are professionally active in the United States. One-third are in primary care fields, and two-thirds in non\u2013primary care fields. Of physicians who have completed residency training, more than 90% have patient care as their principal activity, with the remainder primarily active in teaching, research, or administration (US Department of Health and Human Services, 2008a). Licensing of all types of health care professionals, including physicians, is a state jurisdiction. State medical boards require that physicians applying for licen-sure document a passing grade on national licensing examinations, certification of graduation from medical school, and (in most states) completion of at least one year of residency training after medical school.\n\n#### **Medical Education**\n\nThe University of Pennsylvania opened the first medical school in the colonies in 1765, promoting a curriculum that emphasized the therapeutic powers of blood letting and intestinal purging. Many other medical sects coexisted in this era, including the botanics, \"natural bonesetters,\" midwives, and homeopaths, without any one group winning dominance. Few regulations impeded entry into a medical career; physicians were as likely to have completed informal apprenticeships as to have graduated from medical schools. Most medical schools operated as small, proprietary establishments profiting their physician owner rather than as university-centered academic institutions (Starr, 1982).\n\nThe modern era of the US medical profession dates to the 1890\u20131910 period. In 1893, the opening of the Johns Hopkins University School of Medicine ushered in a new tradition of medical education. Johns Hopkins University implemented many features that remain the standard of medical education in the United States: a 4-year course of study at the graduate school level, competitive selection of students, emphasis on the scientific paradigms of clinical and laboratory science, close linkage between a medical school and a medical center hospital, and cultivation of academically renowned faculty.\n\nThe second key event in the creation of a reformed twentieth-century medical profession was the publication of the Flexner Report in 1910. At the behest of the American Medical Association, the Carnegie Foundation for the Advancement in Teaching commissioned Abraham Flexner to perform an evaluation of medical education in the United States. Flexner's report indicted conventional medical education as conducted by most proprietary, nonuniversity medical schools. Flexner held up the example of Johns Hopkins as the standard by which the nation's institutions of medical education should be judged. Flexner's report was extremely influential. More than 30 medical schools closed in the decades following the Flexner Report, and academic standards at the surviving schools became much more stringent (Starr, 1982). More vigorous regulatory activities in respect to credentialing of medical schools and licensure for medical practice soon enforced the standards promoted in the Flexner Report, and only schools meeting the standards of the Licensing Council on Medical Education (LCME) were allowed to award MD degrees. Unlike the state boards licensing practice entry, which are government agencies, the LCME was a private agency operating under the authority of medical professional organizations. LCME-accredited schools became known as \"allopathic\" medical schools to distinguish themselves from homeopathic schools and practitioners. Although homeopaths still practice in the United States (there is now a resurgence of homeopathic practitioners), homeopaths are not officially sanctioned as \"physicians\" by licensing agencies in the United States. However, one alternative medical tradition has survived in the United States that carries the official imprimatur of the physician rank\u2014osteopathy. Osteopathy originated as a medical practice developed by a Missouri physician, Andrew Still, in the 1890s, emphasizing mechanical manipulation of the body as a therapeutic maneuver (Starr, 1982). Schools of osteopathy award DO degrees and have their own accrediting organization. Much of the educational content of modern-day osteopathic medical schools has converged with that of allopathic schools. Most state licensing boards grant physicians with MD and DO degrees equivalent scopes of practice, such as prescriptive authority. By the middle of the twentieth century, regulatory restrictions on practice entry, institutionalization of a rigorous standard of academic training, and the rapid growth of medical science and technology solidified the prestige and authority of licensed physicians in the United States.\n\nIn 2010, allopathic schools had 16,838 graduates, and osteopathic schools 3631. The annual number of allopathic school graduates changed little between 1980 and 2008, and only started to increase in 2009 in response to a new surge of medical school expansion starting in the first decade of the twenty-first century. In contrast, the annual number of osteopathic graduates has grown steadily over past decades, increasing threefold between 1980 and 2010.\n\n#### **Postdoctoral Education**\n\nAt least one year of formal education after medical school is required for licensure in most states, and most physicians complete additional training to become certified in a particular specialty. Traditionally, the first year of postdoctoral training was referred to as an \"internship,\" with subsequent years referred to as \"residency.\" Before the advent of specialization, many physicians completed only a single year of a general \"rotating\" internship. Physicians aspiring to full specialty training became residents (with trainees often literally \"residing\" in the hospital because of endless hours of on-call duty). Now, almost all physicians in the United States complete a full residency training experience.\n\nResidency training is much more decentralized than medical school education. Although some residency training programs are integrated into the same large academic medical centers that are home to the nation's allopathic medical schools, many smaller community hospitals sponsor residency-training programs, often in only one or two specialties. The Accreditation Council for Graduate Medical Education (ACGME), a private agency, accredits allopathic residency training programs. Residency training ranges from 3 years for generalist fields, such as family medicine and pediatrics, through 4 to 5 years for specialty training in fields such as surgery and obstetrics\u2013gynecology, to 6 years or longer for physicians pursuing highly subspecialized training. Some osteopathic schools sponsor osteopathic residency programs.\n\nOnce physicians have completed residency training, another private consortium, the American Board of Medical Specialties, certifies physicians for board certification in their particular specialty field. Criteria for board certification usually consist of completion of training in an ACGME-accredited program and passing of an examination administered by the specific specialty board (eg, the American Board of Pediatrics). Board certification is not required for state licensure. Physicians may advertise to patients their status as specialty board-certified to promote their expertise and qualifications, and board certification may be a factor considered by hospitals when deciding whether to allow a physician to have \"privileges\" to care for patients in the hospital or for managed care organizations deciding whether to include a physician in the organization's physician network. Many specialty boards now require periodic reexamination to maintain certification.\n\nEach year, approximately 25% more physicians enter ACGME residency programs than the number of students graduating from US allopathic medical schools. Who fills these extra residency positions? Approximately 7% are filled by graduates of schools of osteopathy; half of DO graduates enter allopathic residencies rather than residency programs sponsored by schools of osteopathy. The remainder of the ACGME residency positions are filled by physicians who graduated from medical schools outside the United States. A complex regulatory structure exists to govern which international medical graduates are eligible to enter residency training in the United States, involving state licensing board sanctioning of the graduate's foreign medical school and graduates completing US medical licensing examinations. There is almost no opportunity for international graduates to become licensed to practice in the United States without first undergoing residency training in the United States, even if the physician has been fully trained abroad and has years of practice experience. Some international medical graduates are US citizens who decided to train abroad, often because they were not admitted to a US medical school. However, the majority are not US residents, and most of these physicians come from India, the Philippines, sub-Saharan Africa, and other developing nations (Mullan, 2005). International medical graduates who are not US citizens receive only a temporary educational visa while in residency training, and in principle there is an expectation that these individuals will return to their nations of origin once they have completed training. However, various visa-waiver programs exist to allow these physicians to remain in the United States after completing training, usually linked to a period of service in a US community with a physician shortage. Controversy exists about this reliance on international medical graduates to meet US physician workforce needs, with critics arguing that the United States fosters a \"brain drain,\" depleting developing nations of vital human resources (Mullan, 2005).\n\n#### **Financing Medical Education**\n\nWho pays the cost of medical education in the United States? Unlike the case in most developed nations, where medical schools levy no or only nominal tuition, students pay high amounts of tuition and fees to attend US medical schools. Approximately half of US medical schools are public state institutions, with state tax revenues helping to subsidize medical school education. The Federal Government plays a minor role in financing medical student education, but is a major source of funds to support residency training. Medicare allocates \"graduate medical education\" funding to hospitals that sponsor residency programs. These funds are considerable, amounting to $9.5 billion annually, and include \"direct\" education payments for resident stipends and faculty salaries plus indirect education payments to defray other costs associated with being a teaching hospital. The joint federal\u2013state Medicaid programs contribute an additional $3 billion annually to residency education (Iglehart, 2010). Although in 1997, Medicare capped the number of residency program slots it would pay for, Medicare gives hospitals considerable latitude in how to spend their Medicare medical education dollars. Hospitals can decide which specialties, and how many slots in each specialty, they wish to sponsor for residency training, and can qualify for Medicare education payments as long as the positions are ACGME accredited. Hospitals may also invest non-Medicare revenues in their residency education programs and are not beholden to a prescriptive national workforce planning policy. Hospitals have tended to preferentially add new residency positions in non\u2013primary care fields, guided more by the value of residents as low-cost labor to staff hospital-based specialty services than by an assessment of regional physician workforce needs and priorities. Between 2002 and 2007, hospitals added nearly 8000 new residency positions despite the cap on Medicare-funded positions, with virtually all the gains occurring in specialist positions and family medicine residency positions losing ground during the same period (Salsberg et al, 2008).\n\n### **PHYSICIAN ASSISTANTS**\n\n_Jillian Boca was a speech therapist at a community hospital. She liked her work but wanted to advance in her career. She was talking to some of her colleagues who were physical therapists and x-ray technicians; they were thinking of going back to school to become physician assistants. One of the registered nurses at the hospital was also planning to go back to school to become a nurse practitioner. A local medical school sponsored a program with physician assistant and nurse practitioner students receiving their training together. Jillian and two of her colleagues were admitted to the program._\n\nAs the name suggests, physician assistants (PAs) are closely linked with physicians. The profession of PA originated in the United States in 1965 with the establishment of the first PA training program at Duke University School of Medicine. The PA profession developed to fill the niche of a broadly skilled clinician who could be trained without the many years of medical school and residency education required to produce a physician, and who would work in close collaboration with physicians to augment the effective medical workforce, especially in primary care fields and under-served communities. The first wave of PAs trained in the United States included many veterans who had acquired considerable clinical skills working as medical corpsmen in the Vietnam War. PA training programs served as an efficient means to allow these veterans to \"retool\" their skills for civilian practice.\n\nThe American Academy of Physician Assistants defines PAs as \"health professionals licensed to practice medicine with physician supervision\" (Jones, 2007). PAs are usually licensed by the same state boards that license physicians, with the requirement that PAs work under the delegated authority of a physician. In practical terms, \"delegated authority\" means that PAs are permitted to perform many of the tasks performed by physicians as long as the tasks are performed under physician supervision. Studies of PAs in primary care settings have found that their scope overlaps with approximately 80% of the scope of work of primary care physicians. To be eligible for licensure in most states, PAs must have graduated from an accredited training program and pass the Physician Assistant National Certifying Examination, administered by the National Commission on Certification of Physician Assistants. Approximately 60,000 PAs are professionally active in the United States. Traditionally, the majority of PAs worked in primary care fields. However, currently only one-third of PAs now practice in primary care, with many finding employment opportunities in surgical and medical specialty fields (Jones, 2007). PAs work in diverse settings, including private physician offices, community clinics, HMOs, and hospitals.\n\n#### **Physician Assistant Education**\n\nPA training has been described as a \"condensed version of medical school\" (Jones, 2007). The duration of training ranges from 20 to 36 months, with an average of 27 months (Hooker, 2006). Many of the initial training programs did not award degrees and accepted applicants with varying levels of prior formal education. Currently, of the 136 accredited PA training programs in the United States, 79% award a master's degree and require applicants to have attained a baccalaureate degree (Jones, 2007). Approximately half of PA training programs are based at academic health centers and are directly affiliated with medical schools. Several PA programs have established postgraduate training programs, typically one year in duration and focused on subspecialty training.\n\nPA programs produce about 5,600 graduates annually, compared with the 20,500 graduates of allopathic and osteopathic medical schools. Enrollment in PA programs has grown steadily over the past decades, with the number of PA graduates more than doubling between 1990 and 2010.\n\n### **REGISTERED NURSES**\n\n_Felicia Comfort has worked for 20 years as a registered nurse on hospital medical\u2013surgical wards. Although the work has always been hard, Felicia has found it gratifying to care for patients when they are acutely ill and need the clinical skills and compassion of a good nurse. But lately the work seems even more difficult. The pressure to get patients in and out of the hospital as soon as possible has meant that the only patients occupying hospital beds are those who are severely ill and require a tremendous amount of nursing care. At age 45, Felicia finds that her back has problems tolerating the physical labor of moving patients around in bed. Making matters worse, the hospital recently decided to \"re-engineer\" its staffing as a cost containment strategy and has hired more nursing aides and fewer registered nurses, adding to Felicia's work responsibilities. Felicia decides that it is time for a change. She takes a job as a visiting nurse with a home health care agency, providing services to patients after their discharge from a hospital. She likes the pace of her new job and finds the greater clinical independence refreshing after her years of dealing with rigid hospital regimentation of nurses and physicians._\n\nRegistered nurses represent the single largest health profession in the United States. In 2008, approximately 3,000,000 registered nurses were licensed in the United States (US Department of Health and Human Services, 2010). Approximately 80% of licensed registered nurses are actively employed in nursing jobs, with most of these nurses working full-time. In 2008, hospitals were the primary employment setting for 62% of nurses. Approximately 25% work in ambulatory care or other community-based settings, and 5% in long-term care facilities. The national licensing examination for registered nurses is administered by the National Council of State Boards of Nursing, a nonprofit organization comprising representatives of each of the state boards of nursing.\n\n#### **Registered Nurse Education**\n\nHistorically, many nurses received their education in vocational programs administered by hospitals not integrated into colleges and universities. These programs awarded diplomas of nursing rather than college degrees and tended to have the least demanding curricula. Over time, nursing education shifted into academic institutions. Most nurses are now educated either in 2- to 3-year associate degree programs administered by community colleges, or in baccalaureate programs administered by 4-year colleges. Of nurses active in 2008, 20% received their basic nursing training in diploma programs, 45% in associate degree programs, and 34% in baccalaureate degree programs (US Department of Health and Human Services, 2010). Many nursing leaders have called for nursing education to move almost completely to baccalaureate-level programs. At least one study has found that patient outcomes are better when hospitals are staffed with baccalaureate, trained nurses (Aiken et al, 2003). Of the nurses sitting for the national licensing examination in 2005, only 4% attended diploma programs. However, associate degree programs have remained a more affordable and accessible option than baccalaureate programs for many students, with nearly twice as many new registered nurses coming from community college programs as from baccalaureate programs.\n\nEnrollment in registered nurse training programs has had a cyclical pattern over recent decades, corresponding to perceptions of surpluses and shortages in the labor market for nurses. The number of US-educated nurses taking the national licensing examination for the first time (a proxy for new nurse graduates) increased by approximately 50% between 1990 and 1995, reaching 96,610 in 1995, and then fell back to 1990 levels by 2000 (National Council of State Boards of Nursing, 2006). Graduation numbers have recently rebounded in response to aggressive advertising campaigns promoting nursing as a career, such as the Campaign for Nursing's Future led by the Johnson & Johnson Company, and large increases in starting salaries for nurses. In 2006, nearly 110,000 nurses graduated from US programs (National Council of State Boards of Nursing, 2006).\n\nHistorically, most registered nurses in the United States were educated at US schools. However, as the numbers of US nursing graduates decreased in the late 1990s and hospital demand for nurse labor increased, growing numbers of foreign-educated nurses began entering the US health workforce. Unlike the situation for physicians, international nursing school graduates do not have to undergo training in the United States to become eligible for licensure. They may sit for the US registered nurse licensing examination, and upon passing the examination may apply for an occupational visa to work as a nurse. According to Dr. Linda Aiken, the United States has now become the \"world's largest importer of nurses,\" with approximately 15,000 internationally trained nurses passing the US licensing examination in 2005 (Aiken, 2007). Approximately one-third of internationally educated nurses in the United States immigrated from a single nation, the Philippines. This recent upswing in nurse immigration has raised the same concerns about a brain drain from developing nations that has been voiced about physician immigration.\n\n### **NURSE PRACTITIONERS**\n\n_Felicia Comfort has now been working as a home care nurse for 2 years. She has taken on growing responsibility as a case manager for many home care patients with chronic, debilitating illnesses, coordinating services among the physicians, physical therapists, social workers, and other personnel involved in caring for each patient. She decides that she would like to become the primary caregiver for these types of patients, and applies to a nurse practitioner training program in her area. After completing her 2 years of nurse practitioner education, she finds a job as a primary care clinician at a geriatric clinic._\n\nEight percent of registered nurses in the United States have obtained advanced practice education in addition to their basic nursing training (US Department of Health and Human Services, 2010). Advanced practice nurses include clinical nurse specialists, nurse anesthetists, clinical nurse midwives, and nurse practitioners. The approximately 140,000 professionally active nurse practitioners represent the largest single group of advanced practice nurses.\n\nNurse practitioner education typically involves a 2-year master's degree program for individuals who previously attained a baccalaureate degree in nursing. Education emphasizes primary care, prevention, and health promotion, preparing nurse practitioners for a broad scope of clinical practice, although some training programs also prepare nurse practitioners for work in non\u2013primary care fields. Approximately 50% to 60% of nurse practitioners work in primary care settings.\n\nMany nurse practitioner programs were established in the 1970s with federal funding as part of the same national effort to boost the number of primary care clinicians that gave rise to PA training programs. Enrollment in nurse practitioner programs grew slowly in the 1980s and exploded in the 1990s, with the number of nurse practitioner training programs more than doubling between 1992 and 1997. Whereas 1500 nurse practitioners graduated in 1992, more than 8000 graduated in 1997 (Hooker and Berlin, 2002). Unlike the trend for PAs, the number of annual nurse practitioner graduates has decreased in recent years, falling to approximately 6500 graduates in 2005 (Hooker, 2006); the number of graduates is projected to decrease further to 4000 annually by 2015 (Robert Graham Center, 2005). The causes of this decrease are multifactorial, including an initial pent-up demand for advanced practice training among the existing pool of registered nurses that was met by the expansion of programs in the 1990s, leaving a lower \"steady state\" demand once the initial demand was met, and increases in salaries for registered nurses that has lessened the additional earnings that may be gained by advanced practice training.\n\nLicensing and related regulations for nurse practitioners are less uniform across states than those for physicians, physician assistants, and registered nurses. Slightly more than half of state nursing boards require nurse practitioners to have attained a master's degree, but other states accept less extensive training (Christian et al, 2007). Rather than a single national licensing examination for all nurse practitioners, certification examinations are administered by different organizations and are specialty-specific, akin to medical specialty board certification. State boards of nursing also vary in the scope of practice they allow nurse practitioners. Most states require that nurse practitioners work in collaboration with a physician, usually with written practice protocols in place. Eleven states have more liberal regulations permitting nurse practitioners to practice with complete independence from physicians, while at the other extreme, 10 states require physicians to directly supervise nurse practitioners (Christian et al, 2007).\n\nSimilar to physician assistants, nurse practitioners working in primary care settings typically perform approximately 80% of the types of tasks performed by physicians. Two meta-analyses provide evidence that nurse practitioners can deliver care of equivalent quality to that delivered by primary care physicians (Brown and Grimes, 1995; Horrocks et al, 2002), with the caveat that most studies reviewed included small numbers of clinicians and few examined long-term outcomes for patients with chronic illness or complex conditions.\n\nMuch of the initial impetus for developing training programs for both nurse practitioners and PAs in the 1960s was to create substitutes for physicians in an era when there was a perceived shortage of physicians, especially in primary care fields. As concerns about a physician shortage waned in subsequent decades and the era of cost containment arrived, substitution came to mean less a matter of filling shortages than of finding a less expensive type of clinician to substitute for physician labor. A different view of nurse practitioners and PAs sees them less as physician substitutes than as complements in a health care team that includes a variety of personnel. In this view, each profession brings its own unique training and skills to create a health care team in which the whole is more than the sum of its parts (Wagner, 2000). For example, care of patients with chronic diseases such as diabetes is enhanced by multidisciplinary teams (Grumbach and Bodenheimer, 2004). In these types of teams, nurse practitioners often play a leading role by providing care management, health promotion, and instruction in patient self-care, while physicians focus more on medication management and treatment of acute complications.\n\nThe boldest effort to promote advance practice nurses as substitutes for physicians comes from proponents of doctoral-level professional degrees for nurses, known as doctor of nursing practice (DrNP) degrees. A few DrNP training programs have been established in the United States, involving a 4-year graduate education experience following the initial baccalaureate nursing training. Leaders of these programs have articulated the vision of producing nursing graduates carrying the title of \"doctor\" who will be able to practice autonomously with a scope equivalent to that of physicians, including independent practice in acute care hospital settings. Whether there will be ample numbers of registered nurses interested in pursuing this level of training, along with sufficient liberalization of state scope of practice regulations, to actualize this vision for DrNPs in the health workforce in the United States remains to be determined.\n\n### **PHARMACISTS**\n\n_Rex Hall has worked for 5 years as a pharmacist at a chain drug store. He is not sure that his extensive professional education and skills as a pharmacist are being fully utilized in his current job. Some of his time is spent discussing possible drug interactions with physicians and suggesting alternative drug regimens, as well as counseling patients about side effects and proper use of their medications. But too much of his time is taken up answering calls from physicians and patients who are ordering prescription refills, counting out pills, filling pill bottles, and figuring out which medications are covered by which health plan. He sees a job posting for a new pharmacist position at a local hospital. The job description states that the pharmacist will review drug use in the hospital and develop strategies to work with physicians, nurses, and other staff to minimize drug errors and inappropriate prescribing practices. Rex decides to apply for the job._\n\nPharmacists constitute the nation's third largest health care profession. About 250,000 pharmacists were actively practicing in 2010 (US Department of Health and Human Services, 2008b). Although historically most pharmacists were educated in baccalaureate degree programs, in 2004 all programs were required to extend the training period by 1 to 2 years and award Doctorate of Pharmacy degrees. Pharmacy education is in a period of expansion, with the number of accredited schools increasing from 82 in 2000 to 119 in 2011, and the number of graduates growing from 7300 in 2000 to 11,500 in 2010 (US Department of Health and Human Services, 2008b). Approximately 60% of pharmacists work in retail pharmacies, mostly as employees rather than as owners. Over the past decades, drug store chains such as Walgreens and Longs have largely displaced the independently owned pharmacy. Hospitals are the second largest employer of pharmacists, with HMOs and other managed care organizations, long-term care facilities, and clinics also offering practice settings for pharmacists. The content of pharmacists' work is changing, as noted in the vignette above and in a further discussion later in this chapter.\n\n### **SOCIAL WORKERS**\n\nSocial work is a growing profession, with the number of social workers projected to increase from 642,000 in 2008 to 745,000 in 2018; 43% of social workers are dedicated to health care, with about half of these in the fields of mental health and substance abuse. Social workers are trained in assessment skills, diagnostic impressions, psychosocial support to patients and families; and assistance with navigation of the health and social service systems including transitions between hospital, extended care facilities, and home. Some specific tasks carried out by social workers include assessing patients' personal, behavioral, and family\/home\/job situation for the health care team, connecting patients to durable medical equipment and in-home services, finding placements for hospital in-patients unable to go home, helping patients to get health insurance and other community services, investigating possible neglect or abuse, and counseling patients on healthy behavior change (Kitchen and Brook, 2005).\n\nThe minimum educational requirement is a bachelor's degree, but most social work positions in the health care field require a masters in social work plus state licensure. Licensed clinical social workers (LCSWs) must have at least a master's degree plus 2 years of academic and practical experience in the field, during which they serve as members of care teams in hospital, primary care, and behavioral health settings. LCSWs may be generalists or be specialized in the management of geriatric patients, children, or persons with developmental disabilities, mental health, and substance abuse diagnoses.\n\n### **SUPPLY, DEMAND, AND NEED**\n\n_Justin Case began his premed studies in college in 1993. He was taken aback one morning to read an article in the newspaper reporting that a prestigious national commission had just issued a report declaring that the United States was training too many physicians and that medical schools should reduce their enrollment by 25%. Nonetheless, hepressed on in his studies, medical schools did not decrease the number of first-year positions, and Justin succeeded in gaining admission to medical school. By the time he finished his residency training in internal medicine in 2004, he was hearing reports that the United States was facing a shortage of physicians and he received many offers to join medical practices as a primary care internist. However, he opted to do a fellowship in cardiology at a prominent cardiac center in Miami, FL. One of his classmates warned him that Miami already had more cardiologists than most cities of comparable size. Justin told his friend, \"I'm not worried about finding a good job in Miami when I finish my fellowship. Everyone tells me that there will always be more than enough work for interventional cardiologists in Florida.\"_\n\nThe supply of health workers in all the professions discussed in this chapter has been growing over past decades (Figures 7\u20131 to 7\u20133). Between 1975 and 2005, the number of active registered nurses per capita in the United States nearly doubled, the number of physicians per capita grew by approximately 75%, and the number of pharmacists per capita increased by approximately 50%. Increases in the supply of PAs and nurse practitioners have been even more dramatic. For physicians, virtually all the growth in supply is accounted for by increasing numbers of non\u2013primary care specialists. Interestingly, although supply has steadily increased during these years, health workforce analysts have alternated between sounding alarms about shortages and surpluses of physicians and nurses. For example, in the 1980s and 1990s, several commissions warned of a surplus of physicians in the United States (Graduate Medical Education National Advisory Committee, 1981; Pew Health Professions Commission, 1995; Council on Graduate Medical Education, 1996). By the early years of the twenty-first century, some policy analysts were declaring a physician shortage (Council on Graduate Medical Education, 2005). Similarly, concerns about an oversupply of nurses in the mid-1990s were supplanted in 1998 by declarations of a nursing shortage (Buerhaus et al, 2000).\n\n**Figure 7\u20131.** Supply of practicing physicians in the United States. Note: Includes patient care physicians who have completed training, and excludes physicians employed by the federal government (Council on Graduate Medical Education (COGME). _Patient Care Physician Supply and Requirements: Testing COGME Recommendations_. US Department of Health and Human Services; 1996 [HRSA-P-DM 95\u20133].)\n\n**Figure 7\u20132.** Supply of active registered nurses per 100,000 population in the United States. (Peter I. Buerhaus, PhD, RN, FAAN, Douglas O. Staiger, PhD, and David I. Auerbach, PhD, _The Future of the Nursing Workforce in the United States: Data, Trends and Implications,_ 2009: Jones & Bartlett Publishers, Sudbury, MA. www.jbpub.com. Reprinted with permission.)\n\n**Figure 7\u20133.** Supply of active pharmacists per 100,000 population in the United States. (Bureau of Health Professions. _The Pharmacist Workforce. A Study of the Supply and Demand for Pharmacists_. Rockville, MD: Health Resources and Services Administration; 2000.)\n\nWhat explains why perceptions turned from surplus to shortage when supply was continuing to increase? One concern was that overall supply trends might present a misleading picture of the actual labor participation of health care professionals. For example, female physicians work on average fewer hours per week than male physicians. Women constitute a growing share of the physician workforce, and therefore head counts of the number of practicing physicians may overstate the full-time equivalent supply of physicians. In nursing, concerns were voiced that overly stressful working conditions on hospital wards were driving licensed nurses out of the workforce. This concern was magnified by the fear that the sudden plummeting of enrollment in nursing schools portended a major downturn in entry of newly trained nurses into the workforce.\n\nHowever, the supply of health care professionals is only one part of the equation for determining the adequacy of the workforce. The other part of the equation is a judgment about how many physicians, nurses, or pharmacists are actually required. Even when the supply of health care professionals per capita is growing, there may be a perception of a workforce shortage if the requirements for these workers are judged to be increasing more rapidly than supply. There are two general schools of thought about how to define health workforce requirements (Grumbach, 2002). One view considers market demand as the arbiter of workforce requirements. According to this view, if there is unmet market demand for, let us say, nurses, as indicated by many vacant nursing positions at hospitals, then a shortage exists. Or, to the contrary, if many nurses are unemployed or underemployed, a surplus exists. An alternative approach defines workforce requirements on the basis of population need rather than market demand. For example, a need-based approach for nursing would attempt to evaluate whether a certain level of nursing supply optimizes patient outcomes, such as by determining whether higher registered nurse staffing levels for a given volume and acuity of hospital inpatients result in fewer medication errors and hospital-acquired infections and better overall patient outcomes.\n\nIn the case of registered nursing, both demand and need perspectives converged to conclude that a shortage existed in the late 1990s (Bureau of Health Professions, 2002). As the intensity of hospital care increased and hospitals sought more highly trained registered nurses to staff their facilities, vacancy rates increased for hospital nurses. In response, hospitals began to increase wages to attract nurses into the workforce. Researchers around this time also began to produce evidence that lower levels of registered nurse staffing in hospitals were associated with worse clinical outcomes for hospitalized patients (Aiken et al, 2002; Needleman et al, 2002), suggesting a true medical need for more registered nurses in hospitals. One state, California, proceeded to codify a need-based approach to nurse supply by enacting legislation requiring a minimum nurse staffing level per occupied hospital bed (Spetz, 2004). In response to concerns about a nurse shortage, comprehensive strategies have been implemented that appear to be succeeding in attracting more applicants to nursing programs, increasing enrollment in these programs, and increasing the proportion of licensed nurses who are working as nurses. These strategies include actions by private entities, such as hospitals increasing wages for nurses and the Johnson & Johnson\u2013sponsored advertising campaign mentioned above, and actions by government agencies, such as appropriating more funds for expansion of community and state college nursing program capacity.\n\nThe case of the physician workforce has been less straightforward. While most nurses work as employees of hospitals or other employers, most physicians are self-employed or part-owners of a medical group that acts as their employer, making vacancy rates or other typical labor market metrics less reliable indicators of the demand for physicians. Moreover, physicians' authority and influence over medical care give them considerable market power and create opportunities for supplier-induced demand (see Chapter 9), particularly when costs are covered by third-party payers. In a health care environment like that in the United States, in which demand for physician labor may be almost limitless, physicians tend to keep busy even as supply continues to rise. Dr. Richard Cooper has been the most vocal advocate of the position that the United States currently faces a physician shortage, based on his view that the public's demand for physician services is increasing rapidly because of an aging population and the expanding national economy, while growth in physician supply per capita in the United States is beginning to level off (Cooper et al, 2002). Countering this view has been research that raises questions about whether the public really needs and benefits from more physicians, particularly more specialists. Studies comparing patient outcomes across regions in the United States have found that while a very low supply of physicians is associated with higher mortality, once supply is even modestly greater, patients derive little further survival benefit (Goodman and Grumbach, 2008). For example, mortality rates for high-risk newborns are worse in regions with a very low supply of neonatologists than in regions with a somewhat greater supply, but above that level, further increases in the supply of neonatologists are not associated with better clinical outcomes for newborns (Goodman et al, 2002). At the other age extreme, Medicare beneficiaries residing in areas with high physician supply do not report better access to physicians or higher satisfaction with care and do not receive better quality of care (Goodman and Grumbach, 2008). One exception to these patterns is when studies focus on primary care physician supply, rather than on overall physician supply or the supply of specialists. These studies tend to find that patient outcomes and quality of care are better in regions with a more primary care-oriented physician workforce (Baicker and Chandra, 2004; Starfield et al, 2005). Proponents of a need-based approach to physician workforce planning argue that because much of physician training is supported by tax dollars, and because there is little true market restraint on demand for medical care, society should plan physician supply based on considerations of quality, affordability, and prioritization of health care services informed by the type of research evidence cited above (Grumbach, 2002).\n\nIn assessing the adequacy of health care professional supply, it is important not just to count the number of workers, but to examine how these workers are deployed. The quest for effective deployment of the workforce has been characterized using the following analogy: \"Before adding another spoonful of sugar to your tea, first stir up the sugar already in your tea cup.\" In other words, does the health system make the most of its existing supply of highly trained health care professionals? The case of the pharmacist workforce highlights this issue. As has been the case for nurses and physicians, concerns have recently been raised about a shortage of pharmacists. One of the factors cited is the steep rise in the prescribing of medications, which may be considered an indicator of the demand for pharmacists. Approximately 3.6 billion prescriptions were dispensed in 2005, 70% more than in 1994 (US Health and Human Services, 2008b). The estimated number of prescriptions filled per pharmacist in retail pharmacies grew from 17,400 in 1992 to 22,900 in 1999 (Bureau of Health Professions, 2000). In response, pharmacies sought to hire more pharmacists, and between 1998 and 2000, the number of unfilled pharmacist positions in chain store pharmacies more than doubled (Bureau of Health Professions, 2000; Cooksey et al, 2002). Partly in response to increased output from pharmacy schools, the percentage of pharmacist employment positions unfilled dropped from 9% to 5% between 2000 and 2004 (US Department of Health and Human Services, 2008b).\n\nAlthough these trends would suggest a shortage of pharmacists based on a traditional demand model, some observers have questioned whether the existing supply of pharmacists is optimally deployed. Many pharmacists still spend a great deal of time performing the basic \"pill counting\" tasks of drug dispensing. Should pharmacists continue to perform most dispensing functions, or would their extensive training be better utilized in more clinically challenging activities\u2014especially now that all newly graduated pharmacists in the United States are required to have doctoral-level training? The occupation of pharmacy technician has been developed in the United States to assist pharmacists with drug dispensing (Cooksey et al, 2002). An estimated 69% of pharmacists' time is spent on activities that properly trained technicians could perform\u2014counting, packaging, and labeling prescriptions, and resolving third-party insurance issues. Greater use of properly supervised pharmacy technicians might increase the productivity of the existing pharmacists. In addition, innovations in automation of pill dispensing could reduce pharmacist workload. Delegating more tasks to pharmacy assistants and automated systems would allow pharmacists to optimize their clinical training and skills for patient counseling about medications, collaborating on patient safety programs to reduce the epidemic of medication errors, monitoring drug use for chronic disease management programs, and participating in multidisciplinary clinical teams in both hospitals and ambulatory settings. These same types of concerns have been raised about whether other health care professionals are being deployed with maximum efficiency and productivity and working at their highest level of skill. For example, new models of primary care are emphasizing that many preventive and chronic care tasks traditionally performed by physicians could be delegated to medical assistants and assisted by electronic technologies (Bodenheimer and Grumbach, 2007), allowing more productive use of the work effort of primary care clinicians.\n\n### **WOMEN IN THE HEALTH PROFESSIONS**\n\n_Dr. Jenny Wong works as a general internist for the Suburbia Medical Group. She never has to check her schedule in advance, because she knows that every appointment is always booked, not to mention the last minute add-ons. As one of only two women in a group of eleven primary care physicians, she is in demand. In particular, female patients in the practice have sought her out to become their primary care physician. While gratified to be responding to this demand, Dr. Wong also finds it a bit daunting. She senses that her patients expect her to spend more time with them to explain diagnoses and treatments and discuss their overall well-being. But Dr. Wong has the same 15-minute appointment times as every other physician in the practice and continually finds herself falling behind in her schedule. Today Dr. Wong is feeling especially stressed. She is scheduled to meet at lunchtime with the director of Suburbia Medical Group to discuss plans for her impending maternity leave. She knows he will not take kindly to her intention of taking 4 months off after the birth of her child._\n\nHistorically, most physicians and pharmacists in the United States have been men, and most nurses women. For physicians and pharmacists, this demographic pattern is in the midst of a dramatic change. In 1970, 13% of pharmacists were women, but by 2010, more than half of pharmacists were women. The proportion of women among physicians increased from 8% in 1970 to more than 30% in 2010 (Figure 7\u20134). The figures are even more dramatic when examining the makeup of current students in training: women constituted 47% of medical students and 61% of pharmacy students in 2010. In contrast, nursing has long been a profession mainly comprising women, and this is changing very slowly. In 2008, only 10% of registered nurses were men, up slightly from 5% in 1996.\n\n**Figure 7\u20134.** Women as a percentage of physicians, nurses, and pharmacists in the United States. (US Department of Health and Human Services. The Physician Workforce: Projections and Research into Current Issues Affecting Supply and Demand. Health Resources and Services Administration, Bureau of Health Professions, 2008a. US Department of Health and Human Services. The Adequacy of Pharmacist Supply, 2004\u20132030. Health Resources and Services Administration, Bureau of Health Professions, 2008b. US Department of Health and Human Services. The Registered Nurse Population. Findings from the 2008 National Sample Survey of Registered Nurses. Health Resources and Services Administration, Bureau of Health Professions, 2010.)\n\nAs noted above, women, on average, work fewer hours per week than men and are more likely to work on a part-time basis. However, the practices of male and female health care professionals differ in ways other than simply the number of hours worked. Female physicians attract more female patients, in part because female patients highly value more time spent and clearer explanations from their physicians than do male patients, and female physicians spend more time with their patients than do male physicians. Several studies have shown that female physicians deliver more preventive services than male physicians, especially for their female patients (Lurie et al, 1993). Female physicians appear to communicate differently with their patients, with both adults and children, being more likely to discuss lifestyle and social concerns, and to give more information and explanations during a visit (Elderkin-Thompson and Waitzkin, 1999; Roter et al, 2002). Female physicians are more likely to involve patients in medical decision-making than male physicians (Cooper-Patrick et al, 1999).\n\n### **UNDERREPRESENTED MINORITIES IN THE HEALTH PROFESSIONS**\n\n_Cynthia Cuidado is the first person in her family to go to college, much less the first to become a health professional. A large contingent of her extended family celebrates her graduation from her master's degree family nurse practitioner training program. Although HMOs in the city where Cynthia trained had several open positions for nurse practitioners, she has decided to take a job at a migrant farm worker clinic in a rural community near where she grew up._\n\nThe United States is a nation of growing racial and ethnic diversity. According to the 2010 US census, African Americans, Latinos, and Native Americans now account for nearly one-third of the population, yet the health professions fail to reflect the rich racial and ethnic diversity of the US population. Only about 10% of pharmacists, 9% of physicians, 8% of physician assistants, 10% of nurses, and 5% of dentists are from these three underrepresented racial and ethnic groups (Grumbach and Mendoza, 2008).\n\nHealth professions have made efforts to increase the number of underrepresented minorities enrolling in their training programs. In nursing, these efforts appear to be paying dividends (Figure 7\u20135). Underrepresented minorities as a proportion of students in baccalaureate nursing programs increased from 12.2% in 1991 to 18.1% in 2005. Medical schools have experienced a different trend. Underrepresented minorities as a percentage of medical students increased in the early 1990s, from 12.2% in 1991 to 15.5% in 1997. However, the percentage of underrepresented minority medical students dropped after 1997, falling to 13.9% in 2005. The decrease in underrepresented minority student enrollment in medical schools beginning in the mid-1990s coincided with the onset of a wave of antiaffirmative action policies, such as Proposition 209 in California and the Hopwood vs. Texas federal court ruling that curtailed the ability of university admissions committees to give special consideration to applicants' race and ethnicity (Grumbach and Mendoza, 2008). Pharmacy schools also showed little net increase in underrepresented minority enrollment, with 11% of pharmacy students in 1990 and 2010 being from underrepresented minority groups.\n\nThe problem of underrepresented minorities in the health professions is an especially compelling policy concern. As discussed in Chapter 3, minority communities experience poorer health and access to health care compared with communities populated primarily by non-Latino whites. Minority health care professionals are more likely to practice in underserved minority communities and serve disadvantaged patients, such as the uninsured and those covered by Medicaid (Moy and Bartman, 1995; Cantor et al, 1996; Komaromy et al, 1996; Mertz and Grumbach, 2001). Research has also found salutary effects of ethnically concordant relationships between minority patients and health care professionals on the use of preventive services, patient satisfaction, and ratings of the physician's participatory decision-making style (Saha et al, 2000; Cooper et al, 2003; US Department of Health and Human Services, 2006). Some studies focusing specifically on language concordance when patients have limited English proficiency have also found that access to language concordant clinicians is associated with better patient experiences and outcomes such as reductions in patient reports of medication errors (Wilson et al, 2005). Thus, the underrepresentation of minorities is not just a matter of equality of opportunity; it has profound implications for racial and ethnic disparities in access to care and in health status.\n\n**Figure 7\u20135.** Underrepresented minorities as a percentage of students in selected health professions in the United States. Note: Medical schools include only allopathic schools. (American Association of Colleges of Nursing, Enrollment and Graduations in Baccalaureate & Graduate Programs in Nursing; Association of American Medical Colleges, Data Warehouse. _Applicant Matriculant File_. Association of Colleges of Pharmacy, Profile of Pharmacy Students Application Trends; 2007.)\n\n### **CONCLUSION**\n\nAn intricate array of educational pathways, accreditation of teaching institutions, and credentialing of individuals to legally practice a healing profession defines the composition of the health workforce. Access, cost, and quality\u2014the three overriding issues in health care\u2014are all inextricably linked to trends in the health care workforce. An inadequate supply of health care professionals may impede patients' access to care or compromise the quality of care. But increases in the supply of health care professionals may fuel intolerable escalation of health care costs. It is not surprising, then, to find disagreement about whether a health system has enough, too few, or too many of a particular class of health care professionals. The recent consensus in the United States about a shortage of registered nurses is one of the rare instances in which analyses based on demand models and on need models arrived at similar conclusions. The current debate over the adequacy of the physician workforce in the United States is more typical of the challenges in coming to agreement about the adequacy of supply, revealing how different frames of reference for judging the nation's requirement for health care professionals lead to different policy conclusions. In addition to the overall supply of health professionals, the demographic composition of the workforce in terms of gender and race\u2013ethnicity also has important policy implications.\n\nAlthough making definitive determinations about the \"right\" number of health care professionals often proves elusive, two conclusions may be made with more confidence. First, all health systems should deploy their workers in a manner that makes the best use of their training and skills, creating practice structures that allow each health care professional to operate at his or her highest level of capability and ensuring that those patients most in need benefit from the clinical expertise of the health care professionals working in the system. Most systems fall short of this goal and have not fully \"stirred the sugar in the cup of tea,\" failing to continually reassess and adapt the roles and responsibilities of the members of the health care team to the changing needs of modern-day health systems. Second, all systems need to ensure that their health professionals are highly qualified and embrace a culture of continuous quality improvement (discussed in Chapter 10). To echo the opening of this chapter, a health care system is only as good as the people working in it.\n\n### **REFERENCES**\n\nAiken LH. U.S. nurse labor market dynamics are key to global nurse sufficiency. _Health Serv Res_. 2007;42:1299.\n\nAiken LH et al. Hospital nurse staffing and patient mortality, nurse burnout, and job dissatisfaction. _JAMA_. 2002;288:1987.\n\nAiken LH et al. Educational levels of hospital nurses and surgical patient mortality. _JAMA_. 2003;290:1617.\n\nBaicker K, Chandra A. 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Nurse-staffing levels and the quality of care in hospitals. _N Engl J Med_. 2002;346:1715.\n\nPew Health Professions Commission. _Critical Challenges. Revitalizing the Health Professions for the Twenty-First Century._ San Francisco: UCSF Center for the Health Professions; December 1995.\n\nRobert Graham Center Policy Studies in Family Medicine and Primary Care. _Physician Assistant and Nurse Practitioner Workforce Trends_. One-pagers, 37, 2005. . Accessed November 14, 2011.\n\nRoter D et al. Physician gender effects in medical communication: a meta-analytic review. _JAMA_. 2002;288:756.\n\nSalsberg E et al. US residency training before and after the 1997 Balanced Budget Act. _JAMA._ 2008;300:1174.\n\nSaha S et al. Do patients choose physicians of their own race? _Health Aff (Millwood)._ 2000;19(4):76.\n\nSpetz J. California's minimum nurse-to-patient ratios: The first few months. _J Nurs Adm_. 2004;34:571.\n\nStarfield B et al. The effects of specialist supply on populations' health: assessing the evidence. _Health Aff Web Exclusives_. 2005;(suppl):W5-97\u2013W5-107.\n\nStarr P. _The Social Transformation of American Medicine._ New York: Basic Books; 1982.\n\nUS Department of Health and Human Services. The Rationale for Diversity in the Health Professions: A Review of the Evidence. Health Resources and Services Administration; 2006. .\n\nUS Department of Health and Human Services. The Physician Workforce: Projections and Research into Current Issues Affecting Supply and Demand. Health Resources and Services Administration, Bureau of Health Professions, 2008a. \n\nUS Department of Health and Human Services. The Adequacy of Pharmacist Supply, 2004\u20132030. Health Resources and Services Administration, Bureau of Health Professions, 2008b. \n\nUS Department of Health and Human Services. The Registered Nurse Population. Findings From the 2008 National Sample Survey of Registered Nurses. Health Resources and Services Administration, Bureau of Health Professions, 2010. \n\nWagner EH. The role of patient care teams in chronic disease management. _Br Med J._ 2000;320:569.\n\nWilson E et al. Effects of limited English proficiency and physician language on health care comprehension. _J Gen Intern Med_. 2005;20(9):800-806.\n\n## **8 Painful Versus Painless Cost Control**\n\n_Dr. Joshua Worthy is chief of neurology at a large staff model health maintenance organization (HMO) and serves as the physician representative to the HMO's executive committee. A national health plan has just been enacted that imposes mandatory cost controls. The HMO's budget for the coming year will be frozen at the current year's level. In past years, the annual growth in the HMO's budget has averaged 12%._\n\n_The health plan CEO begins the committee meeting by groaning, \"These cuts are draconian! To meet these new budget limits we'll have to cut staff and ration life-saving technologies. Patients will suffer.\" A consumer member responds, \"We all know there's fat in the system. Why, in the newspaper just the other day there was an article about how rates of back surgery in our city are twice the national average. And if we're going to talk about cuts, maybe we should start by looking at your salary and the number of administrators working here. I'm not so sure patients have to suffer just because we're adopting the kind of reasonable spending limits that they have in most countries.\"_\n\n_Dr. Worthy remains silent for much of the meeting. He wonders to himself, \"Is the CEO right? Is cost containment inevitably a painful process that will deprive our patients of valuable health services? Or, could we be doing a better job with the resources we're already spending? Is there a way that our HMO could implement these cost controls in a relatively painless fashion as far as our patients' health is concerned?\" Interpreting Dr. Worthy's silence as an indication of great wisdom and judgment, the committee assigns him to chair the HMO's task force charged with developing a cost control strategy to meet the new budgetary realities._\n\nConcerns about the rise of health care costs dominate the health policy agenda in the United States. Another pressing health policy concern\u2014lack of adequate insurance and access to care for tens of millions of people\u2014is in part attributable to the problem of rising costs. Health care inflation has made health insurance and health services unaffordable to many families and employers.\n\nPrivate and public payers in the United States have taken aim at health care inflation and discharged volleys of innovative strategies attempting to curb expenditure growth, such as creating new approaches to utilization review, encouraging HMO enrollment, making patients pay more out-of-pocket for care, and a multitude of other measures. These approaches had little noticeable impact on the rate of growth of health care costs in the United States. National health expenditures per capita increased over sevenfold between 1980 and 2009, rising from $1110 to $8086 per capita (Figure 8\u20131). Viewed as a percentage of gross domestic product (GDP), US health expenditures increased from 9.2% in 1980 to 17.6% in 2009 (Figure 8\u20132). Health expenditures as a percentage of GDP are projected to rise to 19.6% by 2019 (Sisko et al, 2010).\n\n**Figure 8\u20131.** US per capita health care expenditures. (Martin A et al. Recession contributes to slowest annual rate of increase in health spending in five decades. _Health Aff (Millwood)_. 2011;30:11.)\n\n**Figure 8\u20132.** US health care expenditures as a percentage of the gross domestic product. (Martin A et al. Recession contributes to slowest annual rate of increase in health spending in five decades. _Health Aff (Millwood)_. 2011;30:11.)\n\nHealth care providers are discovering that they have to adjust to the prospect of practicing in an era of finite resources. Like Dr. Worthy, physicians and other health caregivers need to deliberate about how constraints on expenditure growth may affect patients' health. Must cost control necessarily be painful, leading to rationing of beneficial services? Or, is there a painless route to containing costs, reached by eliminating unnecessary medical treatments and administrative expenses?\n\nIn this chapter, the painful\u2013painless cost control debate will be explored. First a model will be constructed describing the relationship between health care costs and benefits in terms of improved health outcomes. Then different general approaches to cost containment and their potential for achieving painless cost control will be discussed. Chapter 9 will describe specific cost control measures in more detail.\n\n### **HEALTH CARE COSTS AND HEALTH OUTCOMES**\n\n_Before entering medical school, Dr. Worthy worked in the Peace Corps in a remote area in Central America. At the time he first arrived in the region, the infant mortality rate was quite high, withmany deaths due to infectious gastroenteritis. Dr. Worthy participated in the creation of a sewage treatment system and clean well-water sources for the region, as well as a program for implementing oral rehydration techniques for infants. By the end of Dr. Worthy's 2-year stay, the infant mortality rate had dropped by nearly 25%. The cost for the entire program amounted to 15 cents per capita, paid for by the World Health Organization._\n\n_Conditions have been very different for Dr. Worthy as a practicing neurologist in the United States. In the past 5 years, over a dozen new magnetic resonance imaging (MRI) scanners have been installed in the city in which his HMO is located, an urban area with a population of 800,000. Dr. Worthy has found that MRI scans provide images that are better than those of computed tomography (CT) scans, allowing him to more accurately diagnose conditions such as multiple sclerosis in earlier stages. He is less certain about the extent to which these superior images allow superior health care for his patients._\n\nFrom society's point of view, the value of health care expenditures lies in purchasing better health for the population. The concept of \"better health\" is a broad one, encompassing improved longevity and quality of life, reduced mortality and morbidity rates from specific diseases, relief of pain and suffering, enhanced ability to function independently for those with chronic illnesses, and reduction in fear of illness and death. Thus, it is important to know whether investing more resources in health care buys improved health outcomes for society, and if so, what magnitude of the improvement in outcomes may be relative to the amount of resources invested.\n\nFigure 8\u20133, drawn from the work of Robert Evans (1984), illustrates a theoretic relationship between health care resource input and health care outcomes. Initially, as health care resources increase, these outcomes improve, but above a certain level, the slope of the curve diminishes, signifying that increasing investments in health care yield more marginal benefits. In terms of Dr. Worthy's experiences, the Central American region in which he worked lay on the steep slope of this cost\u2013benefit curve: A small investment of resources to create more sanitary water supplies and to administer inexpensive rehydration therapy yielded dramatic improvements in health. On the other hand, purchasing MRI scanners to supplement CT scanners represents a health care system operating on the flatter portion of the curve: Large investments of resources in new technologies may produce more marginal and difficult-to-measure improvements in the overall health of a population.\n\n**Figure 8\u20133.** A theoretic model of costs and health outcomes. Moving from point A to point B on the curve is associated with both higher costs and better health outcomes.\n\nNaturally, different medical interventions lie on steeper (eg, childhood immunizations) or on flatter (eg, the costly prolongation of life for an anencephalic infant) portions of the curve. The curve in Figure 8\u20133 may be viewed as an aggregate cost\u2013benefit curve for the functioning of a health care system as a whole. The system may be an entire nation or a smaller entity such as an HMO, with its defined population of enrollees.\n\nOverall, the US health care system currently operates somewhere along the flatter portion of the curve. Let us assume that Dr. Worthy's HMO system lies at point A on the curve in Figure 8\u20133, with average total health care expenditures per HMO enrollee being the same as the average overall per capita health care cost in the United States (roughly $8000 in 2009). If stringent new cost containment policies forced the HMO to virtually freeze spending at point A rather than increasing annual expenditures at their usual clip to move to point B, then Figure 8\u20133 implies that the HMO would sacrifice improving the health of its enrollees by an amount equal to the distance between points A and B on the vertical axis.\n\nSuch an analysis would confirm the opinion of those who argue that cost containment requires painful choices that affect the health of the population. Among the most forceful proponents of this view are Aaron and Schwartz (1984 and 1990), who have described cost containment as a \"painful prescription\" requiring rationing of beneficial care. In Figure 8\u20133, the distance between points A and B on the _y_ axis measures how much health \"pain\" accompanies the decision to limit spending at point A instead of advancing to point B. Some degree of pain is inherent in the curve. As Evans (1984) observes, \"if its slope is everywhere positive, then in a world of finite resources, unmet needs are inevitable.\" No matter where we sit on the curve, it will always be true that if we spent more we could do a little better.\n\nIn Figure 8\u20133, the distance between points A and B on the y axis is small, given the relatively flat slope of the curve at these points. But reassurances about relatively mild cost containment pain bring to mind the physician, scalpel in hand, hovering over a patient and declaring that \"it will only hurt a little bit.\" A little pain, necessary as it may be, is not the same as no pain; or as Fuchs (1993) puts it, \"'low yield' medicine is not 'no yield' medicine.\"\n\nBefore allowing ourselves (and Dr. Worthy) to become overly chagrined at the inevitable painfulness of cost containment, let us add the new dimension of efficiency. We can picture a point C (Figure 8\u20134) at which spending is the same as that at point A, but outcomes improve. How does the model account for point C, a point off the curve?\n\n**Figure 8\u20134.** Moving off the curve. Point C represents achievement of better health outcome without increased costs.\n\nThe move to point C requires a shifting of the curve (Figure 8\u20135), signifying a new, more efficient (or productive) relationship between costs and health outcomes (Donabedian, 1988). There are numerous possible routes to greater efficiency. For example, diagnostic radiographic imaging services are a rapidly inflating expenditure in the United States. Research has concluded that 20% to 40% of imaging studies are not clinically necessary, and that radiation exposure from diagnostic x-rays carries a risk of inducing malignant cancers (Brenner and Hricak, 2010). Eliminating unnecessary diagnostic radiographic procedures, such as head CT scans for patients with uncomplicated tension headaches, could simultaneously decrease health care costs and improve health. In the remainder of this chapter, we will examine in greater detail the various possible methods that Dr. Worthy's cost control task force could consider to achieve more health \"bang\" for the health care \"buck.\" Before turning to this discussion, however, it is necessary to make explicit three assumptions about this model of costs and outcomes.\n\n**Figure 8\u20135.** Shifting the curve. The shift of the curve represents moving to a more efficient relationship between costs and health outcomes.\n\n1. Implicit in the model is the notion that the relevant outcome of interest is the overall health of a population rather than of any one individual patient. A number of authors have emphasized the need for physicians to broaden their perspective to encompass the health of a general population, as well as their narrower traditional focus on providing the best possible care for each patient (Eddy, 1991; Greenlick, 1992). The population-oriented model of costs and outcomes depicted in Figures 8\u20133, 8\u20134, and 8\u20135 may not fit easily with many physicians' experiences of caring for a particular patient. At the level of the individual patient, the outcome may be all or nothing (eg, the patient will almost certainly live if he or she receives an operation and die without it) and not easily thought about in terms of curves and slopes. Rather than focusing on any one particular intervention or patient, the curve attempts to represent the overall functioning of a health care system in the aggregate for the population under its care. (The ethical issues of the population health perspective are discussed in Chapter 13.)\n\n2. The model assumes that it is possible to quantify health at a population level. Traditionally, health status at this level has been measured relatively crudely, using vital statistics such as life expectancy and infant mortality rates. While an index such as infant mortality rates may be a sensitive, meaningful way of evaluating the impact of health care and public health programs in rural Central America, many analysts have questioned whether such crude indicators accurately gauge the impact of health care services in wealthier industrialized nations. In these latter nations, much of health care focuses on \"softer\" health outcomes such as enhancement of functional status and quality of life in individuals with chronic diseases\u2014aspects more difficult to monitor at the population level than death rates and related vital statistics. In other words, it may be difficult to conceptualize a scale on the _y_ axis of Figures 8\u20133, 8\u20134, and 8\u20135 that can register both the effects of managing gastroenteritis in a poor nation and the addition of MRI scanners in a US city.\n\n3. When evaluating population health, it is difficult to disentangle the effects of health care on health from the effects of such basic social factors as poverty, education, lifestyle, and social cohesiveness (see Chapter 3). For the purpose of our discussion of cost control, we view the curves depicted in Figures 8\u20133, 8\u20134, and 8\u20135 as representing the workings of the health care system (including public health) per se rather than of the broader economic and social milieu. We therefore use the term _health outcomes_ to describe the _y_ axis, a term intended to suggest that we are evaluating those aspects of health status directly under the influence of health care. The _x_ axis correspondingly represents expenditures for formal health care services.\n\n#### **Prices and Quantities**\n\nWe have shown that painless cost control is theoretically possible. But can efficiency be improved in the real world? What strategies could Dr. Worthy's task force propose to move the HMO from point A to point C on the curve? An answer to these questions requires further scrutiny of resource costs in the health care sector.\n\nCosts may be described by the equation\n\n_Price_ refers to such items as the hospital daily room charge or the physician fee for a routine office visit. _Quantity_ represents the volume and intensity of health service use (eg, the length of stay in an intensive care unit, or the number and types of major diagnostic tests performed during a hospitalization). Lomas and colleagues (1989), noting this distinction between prices (Ps) and quantities (Qs), refer to cost containment as \"minding the Ps and Qs\" of health care costs.\n\nLet us look at an example of the equation:\n\n_Blue Shield pays Dr. Morton $600 for 10 office visits at a fee of $60 per visit. The next year, the insurer pays Dr. Morton $720 for 10 visits at $72 per visit._\n\n_Prudential pays Dr. Norton $600 for 10 office visits, and the next year pays $720 for 12 visits at the same $60 fee. An identical cost increase is a price rise for Dr. Morton but an increase in quantity of care for Dr. Norton._\n\nChanges in prices and quantities have different implications for patients and providers (Reinhardt, 1987). In the preceding example, both physicians increase their income (and both insurance plans increase their expenditures) by $120, though in the case of the price increase, the additional income does not require a higher volume of work. To the patient, however, only the additional $120 spent on a greater number of visits purchases more health care services. (For simplicity's sake, we assume that all visits are identical and that the price rise does not reflect increased quality of service, but simply a higher price for the same product.) A cost increase that merely represents higher prices without additional quantities of health care is an inefficient use of resources from the patient's point of view. Returning to the diagrams in Figures 8\u20133 and 8\u20134, if real costs in a health care system were rising only because medical price inflation was exceeding general price inflation while the quantity of care per capita remained static, then increased health costs would not bring about improved health outcomes, and the overall curve would become absolutely flat.\n\n### **COST CONTROL STRATEGIES**\n\n#### **Controlling Price Inflation**\n\n_After intense deliberation, Dr. Worthy's task force submits a plan for \"painless cost containment\" to the HMO executive committee. The first proposal calls for the HMO to aggressively seek discounts on the prices paid for supplies, equipment, and pharmaceuticals by having the HMO selectively contract with suppliers for bulk purchases and stock a more limited variety of product lines and drugs within the same therapeutic class. The proposal also calls for a 10% reduction in salaries for all HMO employees earning over $150,000 per year, as well as a 10% reduction in the capitation fee paid to the HMO's physician group. The executive committee never gets beyond this part of the plan, as furious argument erupts over the proposed income cuts._\n\nPrice inflation has been a major contributor to the rise of health care costs in recent decades. Between 1947 and 1987, US health care costs rose 2.5% per year faster than the growth in the overall economy. Two-thirds of this higher growth rate, or 1.6%, was due to health care prices rising more rapidly than prices in the overall economy. The remaining 0.9% differential was due to differences in the rate of increase of quantities of health care relative to increases in the overall quantity of goods and services (Fuchs, 1990).\n\nThe rapid rise of health care prices manifests itself in such ways as prices for prescription drugs in the United States often being over 50% higher than prices for the same products sold in other nations. Also, specialist physician incomes have increased rapidly. Higher prices explain much of the higher costs of health care in the United States compared with the costs in other industrialized nations (Peterson and Burton, 2007). Limiting this type of price inflation is one way to restrain expenditures without inflicting \"pain\" on the public's health (Table 8\u20131).\n\n**Table 8\u20131.** Examples of painless cost control\n\n#### **Eliminating Ineffective and Inappropriate Care**\n\n_After a brief hiatus to let the furor subside, the HMO executive committee reconvenes. Dr. Worthy introduces his task force's second recommendation\u2014developing appropriateness of care guidelines\u2014by recounting one of his own clinical experiences. WhenDr. Worthy first came to the HMO, the neurologists were keeping their stroke patients at bed rest for 1 week before initiating physical therapy. Dr. Worthy, in contrast, began physical therapy and discharge planning for stroke patients the moment their neurologic status was stable. The average length of stay in the acute hospital for his stroke patients was 3 days, compared with 9 days for other neurologists. Dr. Worthy gave a grand rounds presentation demonstrating that 4 days of exercise are required to regain the strength lost from each day of bed rest, meaning that stroke patients would have better outcomes and use fewer resources\u2014shorter acute hospital stays and less rehabilitation\u2014under his care than under the care of his colleagues. Dr. Worthy cites this as just one example of how the HMO may be devoting resources to ineffective, or even harmful, care._\n\nIf controlling prices is one approach to painless cost control, are there also ways to contain the \"Q\" (quantity) factor in a manner that does not sacrifice beneficial care? Earlier, we cited unnecessary diagnostic imaging studies as an example of a source of inefficient resource use in terms of quantities of services that add to costs without, in many cases, adding health benefits. A number of researchers have found convincing evidence of substantial amounts of unnecessary care in the United States (Brook and Lohr, 1986; Leape, 1992; Brownlee, 2007; Kilo and Larsen, 2009). Physicians in the United States perform large numbers of inappropriate procedures (Schuster et al, 1998; Deyo et al, 2009), and physicians may inappropriately and harmfully accept new technologies as a result of industry influence rather than proven efficacy (Grimes, 1993; Avorn, 2007).\n\nPersuasive evidence comes from the work of Fisher, Wennberg and colleagues, who found that per capita Medicare costs are over twice as high in some cities (eg, Miami) than in other metropolitan areas (eg, Minneapolis). This difference is explained not by prices or degree of illness but is related to the quantity of services provided, which in turn is associated with the predominance of specialists in the higher-cost areas (Fisher et al, 2003). Moreover, residents of areas with a greater per capita supply of hospital beds are up to 30% more likely to be hospitalized than those in areas with fewer beds, after controlling for socioeconomic characteristics and disease burden (Fisher et al, 2000). As for the value of this spending, quality of care and health outcomes are, if anything, worse in the highest spending regions than in areas with less intensive use of services. These findings suggest that a great deal of unnecessary care is taking place in the high-cost areas.\n\nThe slope of the cost\u2013benefit curve would become more favorable if a system could eliminate those components of rising expenditures that have flat slopes (no medical benefit) or negative slopes (harm exceeding benefit, as in the case of inappropriate surgical procedures or prolonged bed rest after strokes). However, inducing physicians and patients to selectively eliminate unnecessary care is no easy matter.\n\n#### **Administrative Waste**\n\n_The third item on Dr. Worthy's painless cost containment plan targets the HMO's administrative costs. The task force proposes eliminating the HMO's TV and radio advertising budget, laying off 25% of all HMO administrative personnel, and reassigning 25 of the 50 staff members in the department that handles contracts with employers to a new department designed to develop a program to ensure that the HMO provides up-to-date child immunizations and adult preventive care services for 100% of plan enrollees. The HMO's marketing director patiently explains to Dr. Worthy that although he, in principle, agrees with these recommendations, he does not consider it in the HMO's best interest to cut costs in a way that jeopardizes the plan's ability to maintain its market share of enrollees._\n\nNot all quantities in the health care cost equation are clinical in nature. The tremendous administrative overhead of the US health care system has come under increasing scrutiny in recent years as a source of inefficiency in health care expenditures. Woolhandler and colleagues (2003) have estimated that as many as 31 cents of every dollar of US health care spending goes for such quantities of administrative services as insurance marketing, billing and claims processing, and utilization review, rather than for actual clinical services. US administrative costs are over twice as high proportionately as those in nations such as Canada and have been rising more rapidly than the rate of overall national health care inflation. While some level of administrative service is necessary for health care finance management and related activities such as quality assurance, few argue that the burgeoning administrative and marketing activities translate into meaningful improvement in patient health. Reducing administrative services is another route to painless cost containment.\n\nEliminating purely wasteful quantities of health care services, be they ineffective clinical services or unnecessary administrative activities, is a relatively straightforward approach to painless cost control. The motto of this approach is: Stop doing things of no clinical benefit. More complicated are approaches to efficiency that involve not simply ceasing completely unproductive activities, but doing things differently. Examples of this latter approach include innovations that substitute less costly care of equal benefit, preventive care, and redistribution of resources from services with some benefit to services with greater benefit relative to cost. Let us examine each of these examples in turn.\n\n#### **Innovation and Cost Savings**\n\nMuch of the process of innovation in health care involves the search for less costly ways of producing the same or better health outcomes. A new drug is developed that is less expensive but is equally efficacious and well tolerated as a conventional medication. Services provided by highly paid physicians can often be delivered with the same quality by nurses, nurse practitioners, or physician assistants. A clinical trial documents that infusion of chemotherapy for many cancer treatments may be done safely on an outpatient basis, averting the expense of hospitalization. Often new technologies are introduced in hopes that they will ultimately prove to be less costly than existing treatment methods.\n\nHowever, new technologies often fail to live up to cost-saving expectations (Bodenheimer, 2005). A case in point is that of laparoscopic cholecystectomy. Through the use of fiberoptic technology, the gallbladder may be surgically removed using a much smaller abdominal incision than that required for traditional open cholecystectomy, thereby significantly shortening the time required for postoperative recuperation in the hospital. The shorter length of hospital stay reduces the overall cost of the operation, with improved outcomes due to less postoperative pain and disability\u2014seemingly a classic case of \"efficient substitution\" that lowers costs and improves health outcomes. There's a catch, however. The necessity of gallbladder surgery is not always clear-cut for patients with gallstones. Many patients have only occasional, mild symptoms, and prefer to tolerate these symptoms rather than undergo an operation. Rates of cholecystectomy increased dramatically following the advent of the laparoscopic technique, apparently because more patients with milder symptoms were undergoing gallbladder surgery. In one HMO, the cholecystectomy rate increased by 59% between 1988 and 1992 after the introduction of the laparoscopic technique. Even though the average cost per cholecystectomy declined by 25%, the total cost for all cholecystectomies in the HMO rose by 11% because of the increased number of procedures done (Legorreta et al, 1993).\n\n#### **Ounces of Prevention**\n\nIf an ounce of prevention is worth a pound of cure, then replacement of expensive end-stage treatment with low-cost prevention would appear to be an ideal candidate for the \"painless cost controller award.\" Investing in prevention sometimes generates this type of efficiency in health care spending (eg, many childhood vaccinations cost less than caring for children with infections) (Armstrong, 2007). However, the prevention story is not always so simple. In many cases, the cost of implementing a widespread prevention program may exceed the cost of caring for the illness it aims to prevent. For example, screening the general population for elevated blood pressure and providing long-term treatment for those with mild-to-moderate hypertension to prevent strokes and other cardiovascular complications has been found to cost more than the expense of treating the eventual complications themselves (Russell, 2009). For some diseases, this is the case because the complications are rapidly and inexpensively fatal, while successful prevention leads to a long life with high medical costs, perhaps for a different illness, required at some point. Similarly a program of routine mammography screening and biopsy following abnormal test results costs more than it saves by detecting breast cancers at earlier stages. Blood pressure and breast cancer screening programs result in the improved health of the population but require a net investment in additional resources.\n\n#### **Prioritization and Analysis of Cost Effectiveness**\n\n_A fourth recommendation of Dr. Worthy's task force involves the diagnosis and treatment of colon cancer. Many HMO physicians suggest screening colonoscopy for their patients over age 50 for early detection of colon cancer. All the HMO's oncologists strongly recommend chemotherapy for patients who develop metastatic colon cancer. Analysis of cost-effectiveness has demonstrated that screening colonoscopy saves many more years of life per dollar spent than chemotherapy for metastatic colon cancer. Yet chemotherapy allows some patients with metastatic disease to enjoy an extra 6\u201312 months of life. The task force takes the position that the HMO's physicians should do screening colonosco-pies, but that the HMO insurance plan should not cover chemotherapy for metastatic colon cancer._\n\nThe most controversial strategy for making health care more efficient is the redistribution of resources from services with some benefit to services with greater benefit relative to cost. This approach is commonly guided by cost-effectiveness analysis, which as defined by Eisenberg (1989).\n\n_... measures the net cost of providing a service (expenditures minus savings) as well as the outcomes obtained. Outcomes are reported in a single unit of measurement, either a conventional clinical outcome (eg, years of life saved) or a measure that combines several outcomes on a common scale. (Eisenberg, 1989)_\n\nAn example is a cost-effectiveness analysis of different strategies to prevent heart disease, showing that the cost per year of life saved (in 1984 dollars) was approximately $1000 for brief advice about smoking cessation during a routine office visit, $24,000 for treating mild hypertension, and nearly $100,000 for treating elevated cholesterol levels with drugs (Cummings et al, 1989). In order to get the most \"bang\" for the health care \"buck,\" this analysis suggests that a system operating under limited resources would do better by maximizing resources for smoking cessation before investing in cholesterol screening and treatment.\n\nCost-effectiveness analysis must be used with caution. If the data used are inaccurate, the conclusions may be incorrect. Moreover, cost-effectiveness analysis may discriminate against people with disabilities. Researchers are likely to assign less worth to a year of life of a disabled person than does the person himself or herself; thus, analyses using \"quality-adjusted life years\" may have a built-in bias against persons with less capacity to function independently (Menzel, 1992).\n\nDr. David Eddy (1991, 1992, 1993), in a series of provocative articles in the _Journal of the American Medical Association,_ has discussed the practical and ethical challenges of applying cost-effectiveness analysis to medical practice. Two of the essays involve the case of an HMO trying to decide whether to adopt routine use of low-osmolar contrast agents, a type of dye for special x-ray studies that carries a lower risk of provoking allergic reactions than the cheaper conventional dye. With the use of this agent for all x-ray dye studies, 40 nonfatal allergic reactions would be avoided annually and the cost to the HMO would be $3.5 million more per year, compared with costs for use of the older agent in routine cases and use of the low-osmolar dye only for patients at high risk of allergy. The same $3.5 million dollars invested in an expanded cervical cancer screening program in the HMO would prevent approximately 100 deaths from cervical cancer per year.\n\nIn discussing how best to deploy these resources, Eddy highlights several points of particular relevance to clinicians:\n\n1. It must be agreed upon that resources are truly limited. Although the cost-effectiveness of low-osmolar contrast dye and cervical cancer screening is quite different, both programs offer some benefit (ie, they are not flat-of-the-curve medicine). If no constraints on resources existed, the best policy would be to invest in both services.\n\n2. If resources are limited and trade-offs based on cost-effectiveness considerations are to be made, these trade-offs will have professional legitimacy only if it is clear that resources saved from denying services of low cost-effectiveness will be reinvested in services with greater cost-effectiveness, rather than siphoned off for ineffective care or higher profits.\n\n3. Ethical tensions exist between maximizing health outcomes for a group or population as opposed to the individual patient. The radiologist experiences the trauma of patients having severe allergic reactions to the injection of contrast dye. Preventing future deaths from cervical cancer in an unspecified group of patients not directly under the radiologist's care seems an abstract and remote benefit from his or her perspective\u2014one that may be perceived as conflicting with the radiologist's obligation to provide the best care possible to his or her patients.\n\nMany analysts, including those who question the methods of cost-effectiveness analysis, share Eddy's conclusion: Physicians must broaden their perspective to balance the needs of individual patients directly under their care with the overall needs of the population served by the health care system, whether the system is an HMO or the nation's health care system as a whole (see Chapter 13). Professional ethics will have to incorporate social accountability for resource use and population health, as well as clinical responsibility for the care of individual patients (Greenlick, 1992; Hiatt, 1975).\n\n_The final recommendation of Dr. Worthy's task force is for the HMO to hire a consultant to advise the HMO on the relative cost-effectiveness of different services offered by the HMO, in order to prioritize the most cost-effective activities. While waiting for the consultant's report, the task force suggests that the HMO begin implementing this strategy by allocating an extra 5 minutes to every routine medical appointment for patients who smoke, so that the physician, nurse practitioner, or physician assistant has time to counsel patients on smoking cessation, as well as by setting up two dozen new community-based group classes in smoking cessation for HMO members. The costs of these new activities are to be funded from the HMO's existing budget for coronary artery stenting, and the number of these stent procedures is to be restricted to 30 fewer than the number performed during the current year. The day following the executive committee meeting, the HMO's health education director buys Dr. Worthy lunch and compliments him on his \"enlightened\" views. On the way back from lunch, the chief of cardiology accosts Dr. Worthy in the corridor and says, \"Why don't you just take a few dozen of my patients with severe coronary artery disease out and shoot them? Get it over with quickly, instead of denying them the life-saving stents they need.\"_\n\n### **CONCLUSION**\n\nThe relationship between health outcomes and health care costs is not a simple one. The cost\u2013benefit curve has a diminishing slope as increasing investment of resources yields more marginal improvements in the health of the population. The curve itself may shift up or down, depending on the efficiency with which a given level of resources is deployed.\n\nThe ideal cost containment method is one that achieves progress in overall health outcomes through the \"painless\" route of making more efficient use of an existing level of resources. Examples of this approach include restricting price increases, reducing administrative waste, and eliminating inappropriate and ineffective services. \"Painful\" cost containment represents the other extreme, when controls on expenditures are accomplished only by sacrificing quantities of medically beneficial services. Making trade-offs in services based on relative cost-effectiveness may be felt as painless or painful, depending on one's point of view; some individuals may experience the pain of being denied potentially beneficial services, but at a net gain in health for the overall population through more efficient use of the resources at hand.\n\nCost containment in the real world tends to fall somewhere between the entirely painless paragon and the completely painful pariah (Ginzberg, 1983). As the experiences of Dr. Worthy reveal, putting painless cost control into practice may be impeded by political, organizational, and technical obstacles. Price controls may make economic sense but risk intense opposition from providers. Administrative savings may be largely beyond the control of any single HMO or group of providers and require an overhaul of the entire health care system. Identifying and modifying inappropriate clinical practices is a daunting task, as is prioritizing services on the basis of cost-effectiveness. But while painless cost control may be difficult to achieve, few would argue that the US health care system currently operates anywhere near a maximum level of efficiency. Regions in the nation with higher health care spending do not have better health outcomes (Fisher et al, 2003). The nation's lackluster performance on indices such as infant mortality and life expectancy rates suggests that the prolific degree of spending on health care in the United States has not been matched by a commensurate level of excellence in the health of the population (Davis et al, 2010). Making better use of existing resources must be the priority of cost control strategies in the United States.\n\n### **REFERENCES**\n\nAaron H, Schwartz WB. Rationing health care: The choice before us. _Science_. 1990;247:418.\n\nAaron H, Schwartz WB. _The Painful Prescription: Rationing Hospital Care_. Washington, DC: Brookings Institution; 1984.\n\nArmstrong EP. Economic benefits and costs associated with target vaccinations. _J Manag Care Pharm_. 2007;13(Suppl S-b):S12.\n\nAvorn J. Keeping science on top in drug evaluation. _N Engl J Med_. 2007;357:633.\n\nBodenheimer T. High and rising health care costs. Part 2: technologic innovation. _Ann Intern Med_. 2005;142:932.\n\nBrenner DJ, Hricak H. Radiation exposure from medical imaging. _JAMA_. 2010;304:208.\n\nBrook RH, Lohr KN. Will we need to ration effective health care? _Issues Sci Technol_. 1986;3:68.\n\nBrownlee S. _Overtreated_. New York: Bloomsbury; 2007.\n\nCummings SR et al. The cost-effectiveness of counseling smokers to quit. _JAMA_. 1989;261:75.\n\nDavis K et al. _Mirror, Mirror on the Wall_. New York: Commonwealth Fund; 2010.\n\nDeyo RA et al. Overtreating chronic back pain: Time to back off. _J Am Board Fam Med._ 2009;22:62.\n\nDonabedian A. Quality and cost: Choices and responsibilities. _Inquiry_. 1988;25:90.\n\nEddy DM. Applying cost-effectiveness analysis. _JAMA._ 1992;268:2575.\n\nEddy DM. Broadening the responsibilities of practitioners. _JAMA._ 1993;269:1849.\n\nEddy DM. The individual vs. society: Is there a conflict? _JAMA._ 1991;265:1446.\n\nEisenberg JM. Clinical economics. _JAMA._ 1989;262:2879.\n\nEvans RG. _Strained Mercy: The Economics of Canadian Health Care._ Toronto, Ontario, Canada: Butterworths; 1984.\n\nFisher ES et al. The implications of regional variation in medicare spending. _Ann Intern Med._ 2003;138:273.\n\nFisher ES et al. Associations among hospital capacity, utilization, and mortality of US Medicare beneficiaries, controlling for sociodemographic factors. _Health Serv Res._ 2000;34:1351.\n\nFuchs VR. No pain, no gain: Perspectives on cost containment. _JAMA._ 1993;269:631.\n\nFuchs VR. The health sector's share of the gross national product. _Science._ 1990;247:534.\n\nGinzberg E. Cost-containment: Imaginary and real. _N Engl J Med._ 1983;308:1220.\n\nGreenlick MR. Educating physicians for population-based clinical practice. _JAMA._ 1992;267:1645.\n\nGrimes DA. Technology follies: The uncritical acceptance of medical innovation. _JAMA._ 1993;269:3030.\n\nHiatt HH. Protecting the medical commons: Who is responsible? _N Engl J Med._ 1975;293:235.\n\nKilo CM, Larsen EB. Exploring the harmful effects of health care. _JAMA._ 2009;302:89.\n\nLeape LL. Unnecessary surgery. _Annu Rev Public Health._ 1992;13:363.\n\nLegorreta AP et al. Increased cholecystectomy rate after the introduction of laparoscopic cholecystectomy. _JAMA._ 1993;270:1429.\n\nLomas J et al. Paying physicians in Canada: Minding our Ps and Qs. _Health Aff (Millwood)._ 1989;8(1):80.\n\nMartin A et al. Recession contributes to slowest annual rate of increase in health spending in five decades. _Health Aff (Millwood)._ 2011;30:11.\n\nMenzel PT. Oregon's denial: Disabilities and the quality of life. _Hastings Cent Rep_. 1992;22:21.\n\nPeterson CL, Burton R. _U.S. Health Care Spending: Comparison with Other OECD Countries_ (RL34175) [Electronic copy]. Washington, DC: Congressional Research Service; 2007. . Accessed November 14, 2011.\n\nReinhardt UE. Resource allocation in health care: The allocation of lifestyles to providers. _Milbank Mem Fund Q_. 1987;65:153.\n\nRussell LB. Preventing chronic disease: An important investment, but don't count on cost savings. _Health Aff (Millwood)_. 2009;28:42.\n\nSchuster M et al. How good is the quality of health care in the United States? _Milbank Q_. 1998;76:517.\n\nSisko AM et al. National health spending projections: The estimated impact of reform through 2019. _Health Aff (Millwood)_. 2010;29:1933.\n\nWoolhandler S et al. Costs of health care administration in the United States and Canada. _N Engl J Med_. 2003;349:768.\n\n## **9 Mechanisms for Controlling Costs**\n\nIn Chapter 8, we discussed the general relationship between costs and health outcomes and explored the tension between painful and painless approaches to cost containment. In this chapter, we examine specific methods for controlling costs. Our emphasis is on distinguishing among the different types of cost control mechanisms and understanding their intent and rationale. We briefly cite evidence about how these mechanisms may affect cost and health outcomes.\n\nFinancial transactions under private or public health insurance (see Chapter 2, Figures 2\u20132, 2\u20133, and 2\u20134) may be divided into two components:\n\n1. _Financing,_ the flow of dollars (premiums or taxes) from individuals and employers to the health insurance plan (private health insurance or government programs), and\n\n2. _Reimbursement,_ the flow of dollars from insurance plans to physicians, hospitals, and other providers.\n\nCost-control strategies can be divided into those that target the financing side versus those that impact the reimbursement side of the funding stream (Figure 9\u20131 and Table 9\u20131).\n\n**Figure 9\u20131.** Cost-control mechanisms may be applied to both the financing and reimbursement components of health care spending under a system of health insurance.\n\n**Table 9\u20131.** Categories of cost controls\n\n### **FINANCING CONTROLS**\n\nCost controls aimed at the financing of health insurance attempt to limit the flow of funds into health insurance plans, with the expectation that the plans will then be forced to modify the outflow of reimbursement. Financing controls come in two basic flavors\u2014regulatory and competitive.\n\n#### **Regulatory Strategies**\n\n_Dieter Arbeiter, a carpenter in Berlin, Germany, is enrolled in one of his nation's health insurance plans, the \"sick fund\" operated by the Carpenter's Guild. Each month, Dieter pays 7.5% of his wages to the sick fund and his employer contributes an equal 7.5%. The German federal government regulates these payroll tax rates. When the government proposes raising the rate to 8.5%, Dieter and his coworkers march to the parliament building to protest the increase. The government backs down, and the rate remains at 7.5%. As a result, physician fees do not increase that year._\n\nIn nations with tax-financed health insurance, government regulation of taxes serves as a control over public expenditures for health care. This regulatory control is most evident when certain tax funds are earmarked for health insurance, as in the case of the German health insurance plans (see Chapter 14) or Medicare Part A in the United States. Under these types of social insurance systems, an increase in expenditures for health care requires explicit legislation to raise the rate of earmarked health insurance taxes. Public antipathy to tax hikes may serve as a political anchor against health care inflation.\n\nA somewhat different model of financing regulation was offered by President Clinton's 1994 health care proposal (which never passed). This proposal called for government regulation of premiums paid to private health insurance plans.\n\n#### **Competitive Strategies**\n\nAn alternative US proposal for containing health costs attempts to control the financing flow through a competitive strategy rather than through regulation. The basic premise of competitive financing strategies is to make employers, employees, and individuals more cost-conscious in their health insurance purchasing decisions. Health insurance plans would be encouraged to compete on the basis of price, with lower-cost plans being rewarded with a greater number of enrollees. Instead of having a government agency regulate financing, the competitive market would pressure plans to restrain their premium prices and overall costs.\n\n_Giovanni Costa works for General Auto (GA). It is 1985, and he and his family have Blue Cross health insurance that covers most services provided by the health care provider of his choice, with no deductible. Giovanni does not know how much his health plan costs, because GA pays the total premium. Once Giovanni asked his friend in the employee benefits department whether the company was worried about the costs of health insurance. \"It's a problem,\" Giovanni was told, \"but it's not too bad because our health insurance premiums are tax deductible for the company. Also, if we gave you higher wages you'd have to pay taxes on those wages, but if we give you better health care coverage, you don't pay taxes on the value of that coverage. So we're both better off by providing generous health care benefits. When it comes right down to it, the government's paying a portion of those premiums.\"_\n\nWhen considering competitive strategies that attempt to make purchasers more price sensitive, it is important to consider who the purchaser of health insurance really is. For employment-based health insurance, is the purchaser the employer selecting which health plans to offer employees, or is it the individual employee deciding to enroll in a specific plan? As in the case of Giovanni Costa and GA, the answer is often both: GA selects which plans to offer employees and what portion of the premium to subsidize, and Giovanni chooses a particular plan from those offered by GA.\n\nHistorically, several factors have blunted both employers' and employees' consideration of price in the purchase of health insurance (Enthoven, 1993). For employees, the fact that employers who provide health benefits usually pay a large share of the premium for their employees' private health insurance has insulated insured employees from the costs of insurance. Employees view health insurance premiums as an expense to the employer rather than as a cost borne by themselves. In fact, many employees might receive higher wages if the costs of health insurance were lower, but employees do not generally perceive health insurance benefits as foregone wages.\n\nMoreover, the federal policy of treating health care benefits as nontaxable to both employee and employer makes it in the employee's financial interest to receive generous health care benefits and reduces the burden to the employer of paying for such benefits. A dollar contributed directly by the employer to a health plan goes farther toward the purchase of health insurance than a dollar in wages that is first taxed as income and then spent by the employee for health insurance. This dynamic, which cost the federal government about $260 billion in 2010 (Gruber, 2010), has shielded employees from the real price of health insurance and given employees less incentive to be cost-conscious consumers when selecting an insurance plan.\n\nFor employers, inflation of health insurance premiums in the 1950s and 1960s was an acceptable part of doing business when the economy was booming and health insurance costs consumed only a small portion of overall business expenses. However, as health insurance costs continued to spiral upward and economic growth slowed in recent decades, employers became more active in their approach to health insurance costs (see Chapter 16).\n\n_It is 2010, and GA now offers Giovanni Costa three choices of health insurance plans: The health maintenance organization (HMO) plan costs $1000 per month for family coverage, with GA paying 70% and Giovanni paying 30%; the preferred provider organization (PPO) plan is worth $1200 per month; and the fee-for-service plan runs $1400 a month. If Giovanni chooses the HMO plan, GA pays $700 (70%) and Giovanni pays $300 (30%). If Giovanni signs up for the $1200 PPO plan, GA still pays $700 (70% of the lowest-cost plan) and Giovanni must pay the rest\u2014$500. If Giovanni wants to choose the fee-for-service plan, GA pays $700 and Giovanni pays $700. GA negotiated with all three of its health plans that premium levels would be frozen at their 2008 rates for the next 3 years. A fourth plan previously offered by GA refused to agree to this stipulation, and GA dropped this plan from its portfolio of employee benefits. After 2011, however, the three health plans can demand yearly premium increases, increasing health insurance costs for both GA and Giovanni._\n\nThe competitive approach to health insurance financing encourages price-sensitive purchasing by both employer and employee. For employers, the competitive strategy calls for businesses to be more aggressive in their negotiations with health plans over premium rates. Employers bargain actively with health plans and offer employees only plans that keep their rates below a certain level. Moreover, employers make employees more cost-aware when selecting a health plan by limiting the amount of the insurance premium that the employer will pay. Rather than paying all or most of the premium, many employers offer a fixed amount of insurance subsidy\u2014often indexed to the cost of the cheapest health plan\u2014and compel employees selecting more costly plans to pay the extra amount. Economist Alain Enthoven, one of the chief proponents of the competitive approach, has called this strategy \"managed competition\" (Enthoven, 2003). The strategy is also known as the \"defined contribution\" approach.\n\nIs the evolving competitive approach succeeding at controlling costs? From 2000 to 2010, employer-sponsored health insurance premiums rose by 114%, a major cost-control failure (Claxton et al, 2010). However, competition has never been truly instituted in the United States; 94% of metropolitan markets are controlled by one or two large commercial insurance companies that can extract increasing premiums from employers (Arnst, 2009). Moreover, insurance plans find it easier to compete by \"gaming\" the market through selection of low-cost enrollees rather than by disciplining providers to deliver a lower-cost, higher-quality product. Studies have shown that competing Medicare HMOs have utilized precisely that strategy (Mehrotra et al, 2006).\n\nIf competition could succeed at containing costs, would the outcome be painful or painless cost control? A fundamental concern about market-oriented reforms is that whatever pain may be produced would be experienced most acutely by individuals with lower incomes. Under competition, individuals with higher incomes would be the ones most likely to pay the extra premium costs to enroll in more expensive health plans, while individuals of lesser means could not afford the extra premiums and would be relegated to the lower-cost plans. If the differential in premium prices across plans were large, enrollees in low-cost plans might experience inferior quality of care and health outcomes.\n\n#### **The Weaknesses of Financing Controls**\n\nFor cost controls\u2014whether regulatory or competitive\u2014on the financing side of the health care equation to be successful, these strategies ultimately must produce reductions in the flow of funds on the reimbursement side. A government may try to limit the level of taxes earmarked for health care. However, if payments to physicians, hospitals, and other providers continue to grow at a rapid clip, the imbalance between the level of financing and level of reimbursement will produce budget deficits and ultimately force the government to raise taxes. Similarly, under competition, health insurers will attempt to hold down premium increases in order to gain more customers, but if these health plans cannot successfully control what they pay to hospitals, physicians, pharmacies, and other providers, then insurers will be forced to raise their premiums, and competitive relief from health care inflation will prove elusive. It is on the reimbursement side of the equation that the rubber meets the road in health care cost containment. Governments in nations with publicly financed insurance programs do not simply regulate health care financing, but are actively involved in controlling provider reimbursement. Competition would place the onus on private health insurance plans\u2014rather than a public agency\u2014to regulate reimbursement costs. We now turn to an examination of the options available to private insurers or government for controlling the flow of funds in the reimbursement transaction.\n\n### **REIMBURSEMENT CONTROLS**\n\nIn Chapter 8, we distinguished between the \"Ps\" and \"Qs\" of health care costs: prices and quantities. Cost equals price multiplied by quantity\n\nStrategies to control costs on the reimbursement side can primarily target either prices or quantities (see Table 9\u20131).\n\n#### **Price Controls**\n\n_Under California's fee-for-service Medicaid program, Dr. Vincent Lo's reimbursement for a routine office visit has remained below $25 for the past 8 years._\n\n_The Medicare program reduced Dr. Ernesto Ojo's fee for cataract surgery from $1600 to $900._\n\n_Instead of paying all hospitals in the area the going rate for magnetic resonance imaging (MRI) brain scans ($1200), Apple a Day HMO contracts only with those hospitals who agree to perform scans for $800, and will not allow its patients to receive MRIs at any other hospital._\n\n_Metropolitan Hospital wants a contract with Apple a Day HMO at a per diem rate of $1750. Because Apple a Day can hospitalize its patients at Cross-town Hospital for $1400 a day, Metropolitan has no choice but to reduce its per diem rate to Apple a Day to $1400 in order to get the contract. In turn, to make up the $350 per day shortfall, Metropolitan increases its charges to several other private insurers._\n\nIn Canada and most European nations, a public or quasipublic agency regulates a uniform fee schedule for physician and hospital payments. Often, negotiations occur between the payers (payer is a general term that includes both purchasers and insurers\u2014see Chapter 16) and professional organizations in establishing these fee schedules. In the United States, as discussed in Chapter 4, Medicare, Medicaid, and many private insurance plans have replaced \"usual, customary, and reasonable\" physician payment with predetermined prices for particular services. Competitive approaches to controlling prices have also been attempted in the United States. In the 1980s, California initiated competitive bidding among hospitals for Medicaid contracts, with contracts awarded to hospitals offering lower per diem charges. Private insurance plans have also used competitive bidding to bargain for reductions in physician and hospital fees.\n\nControlling prices has produced some limited success at restraining the growth of overall health care expenditures. However, two major problems limit the potency of price controls for containing overall costs, particularly when prices are regulated at the fee-for-service level.\n\n1. The first problem occurs when price controls are implemented in a piecemeal fashion by different payers. Providers, like Metropolitan Hospital, often respond to price controls imposed by one payer by increasing charges to other payers with less restrictive policies on fees\u2014a phenomenon known as _cost shifting._ The cost-shifting problem may be avoided when a uniform fee schedule is used by all payers (as in Germany) or by a single payer (as in Canada).\n\n2. The quantity of services provided often surges when prices are strictly controlled, leading analysts to conclude that providers respond to fee controls by inducing higher use of services in order to maintain earnings (Bodenheimer, 2005).\n\nPrice controls have the appeal of being a relatively painless form of cost control insofar as they do not limit the quantity of services provided. However, variations in fee schedules may compromise access to care for certain populations; Medicaid fee-for-service rates to physicians are far below private insurance rates in most states, making it difficult for Medicaid patients to find private physicians who will accept Medicaid payment. In nations with uniform fee schedules, concerns have been voiced that ratcheting down of fees may result in \"patient churning\" (high volumes of brief visits), with a deterioration in quality of care and patient satisfaction.\n\n#### **Utilization (Quantity) Controls**\n\nBecause the effectiveness of price controls may be limited by increases in quantity, payers need to consider methods for containing the actual use of services. As indicated in Table 9\u20131, there are a variety of methods for attempting to control use. We begin by examining one strategy, changing the unit of payment, that we introduced in Chapter 4. We then describe additional mechanisms that attempt to restrain the quantity of services.\n\n##### **Changing the Unit of Payment**\n\n_Dr. John Wiley is upset when the PPO reduces his fee from $75 to $60 per visit. In order to maintain his income, Dr. Wiley lengthens his day by half an hour so he can schedule more patient visits._\n\n_Dr. Jane Stuckey is angry when the HMO reduces her capitation payment from $20 to $15 per patient per month. She is unable to maintain her income by providing more visits because more patient visits do not bring her more money. She hopes that more HMO patients will enroll in her practice so that she can receive more capitation payments._\n\nOne simple way to get a handle on the quantity factor is by redefining the unit of payment. In Chapter 4, we discussed how services may be bundled into more aggregate units of payment, such as capitated physician payment and diagnosis-related group (DRG) episode-of-care hospital payment. The more bundled the unit of payment, the more predictable the quantity tends to be. For example, in the case of Dr. Wiley receiving fee-for-service payment, there is a great potential for costs to rise due to increases in the number of physician visits, surgical procedures, and diagnostic tests. When the unit of payment is capitation, as in the case of Dr. Stuckey, the quantity factor is not the number of visits but rather the number of individuals enrolled in a practice or plan. From a health plan's perspective, the formula still applies when paying physicians by capitation, but now the P is the capitation fee and the Q is the number of individuals covered. Other than by raising birth rates, physicians have little discretion in inducing a higher volume of \"quantities\" at the capitation level for the health care system as a whole. Similarly, under global budgeting of hospitals, P represents the average global budget per hospital and Q is the number of hospitals.\n\nShifting payment to a more aggregated unit has obvious appeal as a way for payers to counter cost inflation due to the quantity factor. Life is never so simple, however. In Chapter 4, we discussed how more aggregate units of payment shift financial risk to providers of care. Another way of describing this shifting of risk is that one person's solution to the quantity problem becomes another person's new quantity problem. A hospital paid by global budget instead of by fee-for-service now must monitor its own internal quantities of service lest these quantities drive hospital operating costs over budget. To the extent that providers are unsuccessful in managing resources under more global forms of payment, pressures mount to raise the prices paid at these more aggregated payment units.\n\nChanges in policies for units of payment rarely occur independent of other reforms in cost-control strategies, making it difficult to isolate the specific effects of changing the unit of payment. For example, physician capitation usually occurs in the context of other organizational and cost-control features within a managed care plan. For example, group- and staff-model HMOs receiving capitation payments from employers and paying physicians by salary have been shown to reduce costs by reducing the quantity of services provided, in particular by reducing rates of hospitalization (Hellinger, 1996; Bodenheimer, 2005).\n\nFor hospitals, changing Medicare payments from a fee-for-service to an episode-of-care unit under the DRG-based system in 1983 resulted in a modest slowing of the rate of increase in Medicare Part A expenditures. However, hospitals were able to shift costs to private payers to make up for lower DRG revenues, and national health expenditures as a whole were not affected by Medicare's new payment mechanism (Rice, 1996). Global hospital budgeting in Canada has been a key element of that nation's relative success at containing hospital costs (Rice, 1996).\n\nThe health care system in Germany and in some Canadian provinces has countered the open-ended dynamic of fee-for-service payment by introducing global budgeting, called expenditure caps, for physician payment (Bodenheimer, 2005). Under Canadian expenditure caps, a budget is established for all physician services in a province. Although individual physicians continue to bill the provincial health plan on a fee-for-service basis, if increases in the use of services cause overall physician costs to exceed the budget, fees are reduced (or fee increases for the following year are sacrificed) to stay within the expenditure cap. Evidence from Canada suggests that implementation of expenditure caps was associated with stabilization of physician costs in the mid-1990s (Barer et al, 1996). In the United States, the Medicare program adopted a less-stringent version of an expenditure cap for physician fees, known as the \"sustainable growth rate\" (Vladeck, 2010). Expenditure caps for physician payments allow the payer to focus on the aggregate C part of the equation\u2014in this case, the total physician budget. The shared savings program proposed under the new Medicare Accountable Care Organization initiative, discussed in Chapter 6, is a related strategy attempting to provide a global expenditure feedback loop to modulate fee-for-service payments.\n\n##### **Patient Cost Sharing**\n\n_Randy Payton has an insurance policy with a $2000 deductible and 20% copayment for all services; if he incurs medical expenses of $6000, he pays the first $2000 plus 20% of $4000, for a total of $2800._\n\n_Joseph Mednick's health plan requires that he pay $20 each time he fills a prescription for a medication, with the health plan paying the cost above $20; because he suffers from diabetes, hypertension, and coronary artery disease, the copayments for his multiple medications cost him $1200 per year._\n\nCost sharing refers to making patients pay directly out of pocket for some portion of their health care. In managed competition, cost sharing occurs as part of the financing transaction _at the point of purchasing a health insurance plan_. In this section, we discuss the more traditional notion of cost sharing\u2014using deductibles, copayments, and uncovered services as part of the reimbursement transaction to make patients pay a share of costs _at the point of receiving health care services_.\n\nThe primary intent of cost sharing at the point of service is to discourage patient demand for services. (Cost-sharing also shifts some of the overall bill for health care from third party payers to individuals in the form of greater out-of-pocket expenses.) As discussed in Chapter 3, when individuals have insurance coverage, they are more likely to use services than when they have no insurance. While protection against individual financial risk is one of the essential benefits of insurance, insurance coverage removes the market restraint on costs that occurs in a system of out-of-pocket payment.\n\nCost sharing at the point of service has been one of the few cost-containment devices subjected to the rigorous evaluation of a randomized controlled experiment. In the Rand Health Insurance Experiment, individuals were randomly assigned to health insurance plans with varying degrees of cost sharing. Individuals with cost-sharing plans made about one-third fewer visits and were hospitalized one-third less often than individuals randomized to the plan with no cost sharing (Newhouse et al, 1981).\n\nAlthough the randomized controlled trial provides an excellent laboratory for scrutinizing the effect of a single cost-containment mechanism, some observers have cautioned that analyses based on controlled research designs may produce results that cannot be generalized to the real world of health policy. For example, the United States has a greater level of cost sharing than many industrialized nations, but also the highest overall costs. Studies have found that when cost sharing begins to produce lower use of services for a large population of patients rather than for a small number of patients in an experiment, physicians may increase the volume of services provided to patients with better insurance coverage (Beck and Horne, 1980; Fahs, 1992). Moreover, 70% of health care expenditures are incurred by 10% of the population\u2014people who are extremely ill and generate huge costs through lengthy ICU stays and other major expenses. Cost sharing has little influence over this component of care. Compared to the micro-world of one not-very-sick patient deciding whether to spend some money on a physician visit, patient cost sharing in the macro-world may remove only a thin slice from a large, expanding pie (Bodenheimer, 2005).\n\nThe Rand experiment also evaluated the influence of cost sharing on appropriateness of care and health outcomes. Cost sharing did not reduce medically inappropriate use of services selectively, but equally discouraged use of appropriate and inappropriate services. Study patients (especially those with low incomes) with cost sharing received less preventive services and had poorer hypertension control than those without cost sharing (Brook et al, 1983). Patients are less likely to purchase needed medications under cost-sharing policies, for example Medicare Part D's \"donut hole\" (see Chapter 2), leading to worse control of chronic illnesses and more emergency hospitalizations (Hsu et al, 2006; Tamblyn et al, 2001; Goldman et al, 2007; Schneeweiss et al, 2009). These studies suggest that cost sharing is not a painless form of cost control.\n\nCost sharing for emergency department care may reduce inappropriate use of emergency services without adversely affecting appropriate use or patient health outcomes (Goodell et al, 2009). Cost sharing may be a painless form of cost control when used in modest amounts, not applied to low-income patients, and designed to encourage patients to use lower-cost alternative sources of care (eg, clinics instead of emergency departments) rather than to discourage use of services altogether.\n\n##### **Utilization Management**\n\n_Thelma Graves suffers from a severe hyperthyroid condition; she and her physician agree that she will undergo thyroid surgery. Before scheduling the surgery, the physician has to call Ms. Graves' insurance company to obtain preauthorization, without which the insurer will not pay for the surgery._\n\n_Fred Brady is hospitalized for an acute myocardial infarction. The hospital contacts the utilization management firm for Mr. Brady's insurer, which authorizes 5 hospital days. On the fourth day, Mr. Brady develops a heart rate of 36 beats\/min, requiring the insertion of a temporary pacemaker and prolonging the hospital stay for 10 extra days. After the fifth hospital day, Mr. Brady's physician has to call the utilization management (UM) firm every 2 days to justify why the insurer should continue to pay for the hospitalization._\n\n_Derek Jordan has juvenile-onset diabetes and at age 42 becomes eligible for Medicare due to his permanent disability from complications of his diabetes. He is admitted to the hospital for treatment of a gangrenous toe. Under Medicare's DRG method of payment, the hospital receives the same payment for Derek's hospitalization regardless of whether it lasts 2 days or 12 days. Therefore, the hospital wants Derek's physician to discharge Derek as soon as possible. Each day, a hospital UM nurse reviews Derek's chart and suggests to the physician that Derek no longer requires acute hospitalization._\n\nUtilization management involves the surveillance of and intervention in the clinical activities of physicians for the purpose of controlling costs (Grumbach and Bodenheimer, 1990). In contrast to cost sharing, which attempts to restrict health care use by influencing patient behavior, UM seeks to influence physician behavior. The mechanism of influencing physician decisions is simple and direct: denial of payment for services deemed unnecessary.\n\nUM is related to the unit of payment in the following way: Whoever is at financial risk (see Chapter 4) performs UM. Under fee-for-service reimbursement, insurance companies perform UM to reduce their payments to hospitals and physicians. The DRG system induces hospitals, at risk for losing money if their patients stay too long, to perform UM. Under an HMO capitation contract with a primary physician group, the physician group conducts UM so that it does not pay more to physicians than it receives in capitation payments. If an HMO pays a hospital a per diem rate, the HMO may send a UM nurse to the hospital each day to review whether the patient is ready to go home.\n\n_Micromanage, Inc., performs UM for several insurance companies. Each day, Rebecca Hasselbach reviews the charts of each patient hospitalized by these insurers to determine whether the patients might be ready for discharge. In some cases, Ms. Hasselbach discusses the case with her medical director and with the patient's attending physician. Usually, if the attending physician wants the patient to remain in the hospital, his or her opinion is honored. By pushing for early discharges, Ms. Hasselbach, her Micromanage colleagues around the country, and the medical director save their insurers about $1,000,000 each year. The annual cost of the UM operation is $900,000._\n\nAlthough a few case studies of UM have shown some short-term reduction in rates of hospitalization and surgery, there is little evidence that this approach yields substantial savings, particularly when the overhead of administering the UM program itself is taken into account (Wickizer, 1990). If successful at containing costs, UM would appear to be a painless form of cost control because it intends to selectively reduce inappropriate or unnecessary care. However, reviewers often make decisions on a case-by-case basis without explicit guidelines or criteria, with the result that decisions may be inconsistent both between different reviewers for the same case and among the same reviewer for different cases (Light, 1994).\n\nUM has come under fire as a process of micromanagement of clinical decisions that intrudes into the physician\u2013patient relationship and places an unwelcome administrative burden on physicians and other caregivers. Physicians in the United States have been called the most \"second-guessed and paperwork-laden physicians in western industrialized democracies\" (Lee and Etheredge, 1989). Substantial physician time goes into appealing denials and persuading insurers about the appropriateness of services delivered. A physician and public backlash to UM forced many health insurance plans to relax their UM activities in the late 1990s. However, many plans reintroduced UM around 2003 as costs escalated (Mays et al, 2004).\n\nSeveral approaches to UM have been developed that attempt to avoid some of the onerous features of case-by-case utilization review. Practice profiling, rather than focusing on individual cases, uses summary data on practice patterns to identify physicians whose overall use of services significantly deviates from the standards set by other physicians in the community. Physician outliers identified by practice profiling are then subject to various interventions. In Canada and Germany, these interventions consist of educational and monitoring activities performed by regional medical societies. The questionable accuracy of some profile data and the need to account for underlying differences in patients' clinical needs that may in part explain practice variation have limited the utility of practice profiling as a cost-control tool (Bindman, 1999). Perhaps the most blunt form of utilization management is when a health plan simply refuses to cover an entire class of services, such as in vitro fertilization or experimental treatments for cancer. This approach is discussed in more detail in Chapter 13.\n\n##### **Supply Limits**\n\n_Bob is a patient in the Canadian province of Alberta. He develops back pain, and after several visits to his family physician requests an MRI of his spine to rule out disk disease. His physician, who does not suspect a disk herniation, agrees to place him on the waiting list for an MRI, which for non-urgent cases is 5 months long._\n\n_Rob lives in Alberta, and after lifting an 80-pound load at work, experiences severe lower back pain radiating down his right leg. Finding a positivestraight-leg-raising test on the right with loss of the right ankle reflex, his family physician calls the radiologist and obtains an emergency MRI scan within 3 days._\n\nSupply limits are controls on the number of physicians and other caregivers and on material resources such as the number of hospital beds or MRI scanners. Supply limits can take place within an organized delivery system such as an HMO in the United States, or for an entire geographic region such as a Canadian province.\n\nThe number of elective operations and invasive procedures, such as cardiac catheterization, performed per capita increases with the per-capita supply of surgeons and cardiologists, respectively (Bodenheimer, 2005). This phenomenon is sometimes called \"supplier-induced demand\" (Evans, 1984; Rice and Labelle, 1989; Phelps, 2003). Controlling physician supply may reduce the use of physician services and thereby contribute to cost containment.\n\nSupplier-induced demand pertains to material capacity as well as to physician supply. Per-capita spending for fee-for-service Medicare patients is over twice as high in some regions of the United States than in others (Gawande, 2009, www.dartmouthatlas.org). This remarkable cost variation is not explained by differences in demographic characteristics of the population, prices of services, or levels of illness, but is due to the quantity of services provided. Residents of areas with a greater per-capita supply of hospital beds are up to 30% more likely to be hospitalized than those in areas with fewer beds (Fisher et al, 2000). The maxim that \"empty beds tend to become filled\" has been known as Roemer's law (Roemer and Shain, 1959). Conversely, strictly regulating the number of centers allowed to perform heart surgery establishes a limit for the total number of cardiac operations that can be performed. In situations of limited supply, physicians must determine which patients are most in need of the limited supply of services. Ideally, those truly in need gain access to appropriate services, with physicians possessing the wisdom to distinguish those patients truly in need (Rob) from those not requiring the service (Bob).\n\nAlthough there may not always be a directly linear relationship between supply and use of services, there are clear instances in which limitations of capacity restrain use. For example, international comparisons demonstrate large variations in use of coronary revascularization procedures (coronary artery bypass surgery and angioplasty), with a relatively low rate of surgery in the United Kingdom, an intermediate rate in Canada, and the highest rate in the United States. These rates correspond to the degree to which these nations regulate (minimally in the case of the United States) the number of centers performing cardiac surgery. In spite of the large variations in the quantity of care, with the US performing almost four times the number of procedures per capita than the UK, there are minimal differences in heart disease mortality among these countries (OECD, 2009).\n\nA \"natural experiment\" provides an illustration of how restricting the supply of a high cost resource may be implemented in a relatively painless manner for patients' clinical outcomes. A US hospital experiencing a nursing shortage abruptly reduced the number of staffed intensive care unit beds from 18 to 8 (Singer et al, 1983). For patients admitted to the hospital for chest pain, physicians became more selective in admitting to the intensive care unit only those patients who actually suffered heart attacks. Limiting the use of ICU beds did not result in any adverse health outcomes for patients admitted to nonintensive care unit beds, including those few nonintensive care unit patients who actually sustained heart attacks. This study suggests that when faced with supply limits, physicians may be able to prioritize patients on clinical grounds in a manner that selectively reduces unnecessary services. Establishing supply limits that require physicians to prioritize services based on the appropriateness and urgency of patient need represents a very different (and less intrusive) approach to containing costs than UM, which relies on external parties to authorize or deny individual services in a setting of relatively unconstrained capacity.\n\n##### **Controlling the Type of Supply**\n\nA specific form of supply control is regulation of the _types_ (rather than the total number) of providers. Chapter 5 explored the balance between the number of generalist and specialist physicians in a health care system. Increasing the proportion of generalists may yield savings for two reasons. First, generalists earn lower incomes than specialists. Second, and of greater impact for overall costs, generalists appear to practice a less resource-intensive style of medicine and generate lower overall health care expenditures, including less use of hospital and laboratory services (Bodenheimer and Grumbach, 2007).\n\n### **CONCLUSION**\n\nIn the real world, cost containment strategies are applied not as isolated phenomena in a static system, but as an array of policies concerned with modes of financing, organization of health care delivery, and cost control all mixed together. Managed care is a strategy that utilizes mixture of cost control mechanisms: changing the unit of payment, utilization management, price discounts, and in some cases supply controls. The Canadian health care system (see Chapter 14) also relies on regulation of prices, global budgets and supply controls.\n\nThere is no perfect mechanism for controlling health care costs. Strategies must be judged by their relative success at containing costs and doing so in as painless a manner as possible\u2014without compromising health outcomes. In the view of Dr. John Wennberg, the key to cost control in the United States\n\n_is not in the micromanagement of the doctor-patient relationship but the management of capacity and budgets. The American problem is to find the will to set the supply thermostat somewhere within reason. (Wennberg, 1990)_\n\nAlthough US managed care plans and Canadian provincial health plans are often viewed as diametrically opposed paradigms for health care reform, both the Canadian plans and US group and staff model HMOs base their cost control approaches on what Wennberg terms \"the management of capacity and budgets.\" In Canada, this management is under public control through regulation of physician supply, physician and hospital budgets, and technology. In the United States, private group and staff model HMOs adjust their own \"thermostats\" by setting their own budgets and numbers of physicians, hospital beds, and high-cost equipment.\n\nIf there is a lesson to be learned from attempts to control health care costs in the United States over the past decades, it is that cost-containment policies affecting provider reimbursement need to focus more on macromanagement and less on micromanagement. Trying to manage costs at the level of individual patient encounters (ie, regulating fees for each service, reviewing daily practice decisions, or imposing cost sharing for every prescription and visit to the physician) is a cumbersome and largely ineffectual strategy for containing overall expenditures. Moreover, one payer lowering its costs by shifting expenses to another payer does not produce systemwide cost savings. Those systems that have been most successful in moderating the inexorable increase in health care costs have tended to emphasize global cost containment tools, such as paying by capitation or other aggregate units, limiting the size and specialty mix of the physician workforce, and concentrating high-technology services in regional centers. The future debate over cost containment in the United States will center on whether these cost-containment tools are best wielded by private health care plans operating in a price competitive market or by public regulation of health care providers and suppliers.\n\n### **REFERENCES**\n\nArnst C. In most markets, a few health insurers dominate. _Business Week_. July 23, 2009.\n\nBarer ML et al. Re-minding our Ps and Qs: Cost controls in Canada. _Health Aff (Millwood)_. 1996;15(2):216.\n\nBindman AB. Can physician profiles be trusted? _JAMA_. 1999;281:2142.\n\nBodenheimer T. High and rising health care costs. _Ann Intern Med_. 2005;142:847, 932, 996.\n\nBodenheimer T, Grumbach K. Improving primary care. _Strategies and Tools for a Better Practice_. New York: McGraw-Hill; 2007.\n\nBrook RH et al. Does free care improve adults' health? _N Engl J Med_. 1983;309:1426.\n\nClaxton G et al. Health benefits in 2010. _Health Aff (Millwood)_. 2010;29:1942.\n\nEnthoven AC. Employment-based health insurance is failing: now what? _Health Aff (Millwood)_. 2003;(suppl web exclusives):W3\u2013W237.\n\nEnthoven AC. The history and principles of managed competition. _Health Aff (Millwood)_. 1993;12(suppl): 24\u201348.\n\nEvans RG. _Strained Mercy: The Economics of Canadian Health Care._ Toronto, Ontario, Canada: Butterworths; 1984.\n\nFisher ES et al. Associations among hospital capacity, utilization, and mortality of U.S. Medicare beneficiaries, controlling for sociodemographic factors. _Health Serv Res_. 2000; 34:1351.\n\nGawande A. The cost conundrum. _The New Yorker_. June 1, 2009.\n\nGoldman DP et al. Prescription drug cost sharing: associations with medication and medical utilization and spending and health. _JAMA_. 2007;298:61.\n\nGoodell S et al. _Emergency Department Utilization and Capacity_. Robert Wood Johnson Foundation Policy Brief. No. 17, July 2009. www.rwjf.org\/files\/research\/45929.emergencyutilization.brief.pdf. Accessed November 14, 2011.\n\nGruber J. _The Tax Exclusion for Employer-Sponsored Health Insurance_. National Bureau of Economic Research. February 2010. www.nber.org\/papers\/w15766. Accessed November 14, 2011.\n\nGrumbach K, Bodenheimer T. Reins or fences: A physician's view of cost containment. _Health Aff (Millwood)_. 1990;9(3):120.\n\nHellinger FJ. The impact of financial incentives on physician behavior in managed care plans: A review of the evidence. _Med Care Res Rev_. 1996;53:294.\n\nHsu J et al. Unintended consequences of caps on Medicare drug benefits. _N Engl J Med_. 2006;354:2349.\n\nLee PR, Etheredge L. Clinical freedom: Two lessons for the UK from U.S. experience with privatisation of health care. _Lancet_. 1989;1:263.\n\nLight DW. Life, death, and the insurance companies. _N Engl J Med_. 1994;330:498.\n\nMays GP et al. Managed care rebound? Recent changes in health plans' cost containment strategies. _Health Aff (Millwood)_. 2004;(suppl web exclusive):w4\u2013427\u201336.\n\nMehrotra A et al. The relationship between health plan advertising and market incentives: Evidence of risk-selective behavior. _Health Aff (Millwood)_. 2006;25:759.\n\nNewhouse JP et al. Some interim results from a controlled trial of cost sharing in health insurance. _N Engl J Med_. 1981;305:1501.\n\nOECD. Health at a Glance. Organization for Economic Cooperation and Development, 2009. www.oecd.org\/health.\n\nPhelps CE. _Health Economics._ Boston, MA: Addison Wesley; 2003.\n\nRice TH. Containing health care costs. In: Andersen RM, Rice TH, Kominski GF, eds. _Changing the U.S. Health Care System_. San Francisco, CA: Jossey-Bass; 1996.\n\nRice TH, Labelle RJ. Do physicians induce demand for medical services? _J Health Polit Policy Law_. 1989;14:587.\n\nRoemer MI, Shain M. _Hospital Utilization Under Insurance_. Chicago, IL: American Hospital Association; 1959.\n\nSchneeweiss S et al. The effect of Medicare Part D coverage on drug use and cost sharing among seniors without prior drug benefits. _Health Aff (Millwood)_. 2009;28:w305.\n\nSinger DE et al. Rationing intensive care: Physician responses to a resource shortage. _N Engl J Med_. 1983;309:1155.\n\nVladeck BC. Fixing Medicare's physician payment system. _N Engl J Med_. 2010;362:1955.\n\nWennberg JE. Outcomes research, cost containment, and the fear of health care rationing. _N Engl J Med_. 1990;323:1202.\n\nWickizer TM. The effect of utilization review on hospital use and expenditures: A review of the literature and an update on recent findings. _Med Care Rev_. 1990;47:327.\n\n## **10 Quality of Health Care**\n\nEach year in the United States, millions of people visit hospitals, physicians, and other caregivers and receive medical care of superb quality. But that's not the whole story. Some patients' interactions with the health care system fall short (Institute of Medicine, 1999, 2001).\n\nAt the beginning of the twenty-first century, an estimated 32,000 people died in US hospitals each year as a result of preventable medical errors (Zahn and Miller, 2003). In addition, an estimated 57,000 people in the United States died because they were not receiving appropriate health care\u2014in most cases, because common medical conditions such as high blood pressure or elevated cholesterol are not adequately controlled (National Committee for Quality Assurance, 2010). Hospitals vary greatly in their risk-adjusted mortality rates for Medicare patients; for 2000 to 2002, if hospitals with mortality rates higher than expected reduced deaths to the levels that were expected given their patient mix, 17,000 to 21,000 fewer deaths per year would have occurred (Schoen et al, 2006).\n\nFatal medication errors among outpatients doubled between 1983 and 1993 (Phillips et al, 1998). Prescribing errors occur in 7.6% of outpatient prescriptions (Gandhi et al, 2005), which amounts to 228 million errors in 2004. In 2007, about 25% of elderly patient received high-risk medications (Zhang et al, 2010). Diagnostic error rates are around 10% for a variety of medical conditions (Wachter, 2010). In some primary care practices, patients are not informed about abnormal laboratory results over 20% of the time (Casalino et al, 2009).\n\nForty-five percent of adults do not receive recommended chronic and preventive care, and 30% seeking care for acute problems receive treatment that is contra-indicated (Schuster et al, 1998; McGlynn et al, 2003). Only 50% of people with hypertension are adequately treated (Egan, 2010). Sixty-three percent of people with diabetes are inadequately controlled (Saydah et al, 2004). In many studies, racial and ethnic minority patients experience an inferior quality of care compared with white patients (Agency for Healthcare Research and Quality, 2010). The likelihood of patients being harmed by medical negligence is almost three times as great in hospitals serving largely low-income and minority patients than in hospitals with more affluent populations (Burstin et al, 1993a; Ayanian, 1994; Fiscella et al, 2000). A recent study of multiple quality measures found that the US continues to have serious quality problems and lags behind other developed nations (Schoen et al, 2006).\n\nA prominent Institute of Medicine report (2001) concluded that between what we _know_ and what we _do_ lies not just a gap, but a chasm. Quality problems have been categorized as overuse, underuse, and misuse (Chassin et al, 1998). We will first examine the factors contributing to poor quality and then explore what can be done to elevate all health care to the highest possible level.\n\n### **THE COMPONENTS OF HIGH-QUALITY CARE**\n\nWhat is high-quality health care? It is care that assists healthy people to stay healthy, cures acute illnesses, and allows chronically ill people to live as long and fulfilling a life as possible. What are the components of high-quality health care? (Table 10\u20131)\n\n**Table 10\u20131.** Components of high-quality care\n\n#### **Adequate Access to Care**\n\n_Lydia and Laura were friends at a rural high school; both became pregnant. Lydia's middle-class parents took her to a nearby obstetrician, while Laura, from a family on welfare, could not find a physician who would take Medicaid. Lydia became the mother of a healthy infant, but Laura, going without prenatal care, delivered a low-birth-weight baby with severe lung problems._\n\nTo receive quality care, people must have access to care. People with reduced access to care suffer worse health outcomes in comparison to those enjoying full access\u2014the quality problem of underuse (see Chapter 3). Quality requires equality (Schiff et al, 1994).\n\n#### **Adequate Scientific Knowledge**\n\n_Brigitte Levy, a professor of family law, was started on estrogen replacement in 1960 when she reached menopause. Her physician prescribed the hormone pills for 10 years. In 1979, she was diagnosed with invasive cancer of the uterus, which spread to her entire abdominal cavity in spite of surgical treatment and radiation. She died in 1980 at age 68, at the height of her career._\n\nA body of knowledge must exist that informs physicians what to do for the patient's problem. If clear scientific knowledge fails to distinguish between effective and ineffective or harmful care, quality may be compromised. During the 1960s, medical science taught that estrogen replacement, without the administration of progestins, was safe. Sadly, cases of uterine cancer caused by estrogen replacement did not show up until many years later. Brigitte Levy's physician followed the standard of care for his day, but the medical profession as a whole was relying on inadequate scientific knowledge. A great deal of what physicians do has never been evaluated by rigorous scientific experiment (Eddy, 1993), and many therapies have not been adequately tested for side effects. Treatments of uncertain safety and efficacy may cause harm and cost billions of dollars each year.\n\n#### **Competent Health Care Providers**\n\n_Ceci Yu, age 77, was waking up at night with shortness of breath and wheezing. Her physician told her she had asthma and prescribed albuterol, a bronchodilator. Two days later, Ms. Yu was admitted to the coronary care unit with a heart attack. Writing to the chief of medicine, the cardiologist charged that Ms. Yu's physician had misdiagnosed the wheezing of congestive heart failure and had treated Ms. Yu incorrectly for asthma. The cardiologist charged that the treatment might have precipitated the heart attack._\n\nThe provider must have the skills to diagnose problems and choose appropriate treatments. An inadequate level of competence resulted in poor quality care for Ms. Yu.\n\nThe Harvard Medical Practice study reviewed 30,000 medical records in 51 hospitals in New York State in 1984 (Studdert et al, 2004). The study found that in approximately 4% of hospital admissions, the patient experienced a medical injury (ie, a medical problem caused by the management of a disease rather than by the disease itself); this is the quality problem of misuse. A more recent study placed the percent of hospital patients experiencing a medical injury at 13.8% (Meurer et al, 2006). Medical injuries can be classified as negligent or not negligent.\n\n_Jack was given a prescription for a sulfa drug. When he took the first pill, he turned beet red, began to wheeze, and fell to the floor. His friend called 911, and Jack was treated in the emergency department for anaphylactic shock, a potentially fatal allergic reaction. The emergency medicine physician learned that Jack had developed a rash the last time he took sulfa. Jack's physician had never asked him if he was allergic to sulfa, and Jack did not realize that the prescription contained sulfa._\n\n_Mack was prescribed a sulfa drug, following which he developed anaphylactic shock. Before writingthe prescription, Mack's physician asked whether he had a sulfa allergy. Mack had said \"No.\"_\n\nMedical negligence is defined as failure to meet the standard of practice of an average qualified physician practicing in the same specialty. Jack's drug reaction must be considered negligence, while Mack's was not. Of the medical injuries discovered in the Harvard study, 28% were because of negligence. In those injuries that led to death, 51% involved negligence. The most common injuries were drug reactions (19%) and wound infections (14%). Eight percent of injuries involved failure to diagnose a condition, of which 75% were negligent. Seventy percent of patients suffering all forms of medical injury recovered completely in 6 months or less, but 47% of patients in whom a diagnosis was missed suffered serious disabilities (Brennan et al, 1991; Leape et al, 1991).\n\nNegligence cannot be equated with incompetence. Any good health care professional may have a mental lapse, may be overtired after a long night in the intensive care unit, or may have failed to learn an important new research finding.\n\n#### **Money and Quality of Care**\n\n_Nina Brown, a 56-year-old woman with diabetes, arrived at her primary care physician's office complaining of several bouts of chest pain over the past month. Her physician examined Ms. Brown, performed an electrocardiogram (ECG), which showed no abnormalities, diagnosed musculoskeletal pain, and recommended she take some ibuprofen. Five minutes later in the parking lot, Ms. Brown collapsed of a heart attack. The health plan insuring Ms. Brown had an incentive arrangement with primary care physicians whereby the physicians receive a bonus payment if the physicians reduce use of emergency department and referral services below the community average._\n\n_Completely healthy at age 45, Henry Fung reluctantly submitted to a treadmill exercise test at the local YMCA. The study was possibly abnormal, and Mr. Fung, who had fee-for-service insurance, sought the advice of a cardiologist. The cardiologist knew that treadmill tests are sometimes positive in healthy people. He ordered a coronary angiogram, which was perfectly normal. Three hours after the study, a clot formed in the femoral artery at the site of the catheter insertion, and emergency surgery was required to save Mr. Fung's leg._\n\nNo one can know what motivated the physician to send Ms. Brown home instead of to an emergency department when unstable coronary heart disease was one possible diagnosis (underuse); nor can one guess what led the fee-for-service cardiologist to perform an invasive coronary angiogram of questionable appropriateness on Mr. Fung (overuse). One factor that bears close attention is the impact of financial considerations on the quantity (and thus the quality) of medical care (Relman, 2007). As noted in Chapter 4, fee-for-service reimbursement encourages physicians to perform more services, whereas capitation payment rewards those who perform fewer services.\n\nMore than 40 years ago, Bunker (1970) found that the United States performed twice the number of surgical procedures per capita than Great Britain. He postulated that this difference could be accounted for by the greater number of surgeons per capita in the United States and concluded that \"the method of payment appears to play an important, if unmeasured, part.\" Most surgeons in the United States are compensated by fee-for-service, whereas most in Great Britain are paid a salary. From 8% to 86% of surgeries\u2014depending on the type\u2014have been found to be unnecessary and have caused substantial avoidable death and disability (Leape, 1992). As an example, spinal fusion surgery increased by 77% from 1996 to 2001, though little evidence supports this procedure in many cases. Complications are frequent and rates of reoperation (because of failure to relieve pain or worsening pain) are high. Reimbursement for this procedure is greater than that provided for most other procedures performed by orthopedists and neurosurgeons (Deyo et al, 2004).\n\n_It was a nice dinner, hosted by the hospital radiologist and paid for by the company manufacturing magnetic resonance imaging (MRI) scanners. After the meal came the pitch: \"If you physicians invest money, we can get an MRI scanner near our hospital; if the MRI makes money, you all share in the profits.\" One internist explained later, \"After I put in my $10,000, it was hard to resist ordering MRI scans. With headaches, back pain, and knee problems, the indications for MRIs are kind of fuzzy. You might order one or you might not. Now, I do.\"_\n\nRelman (2007) writes about the commercialization of medicine: \"The introduction of new technology in the hands of specialists, expanded insurance coverage, and unregulated fee-for-service payments all combined to rapidly increase the flow of money into the health care system, and thus sowed the seeds of a new, profit-driven industry.\"\n\nDuring the 1980s, many physicians formed partnerships and joint ventures, giving them part ownership in laboratories, MRI scanners, and outpatient surgicenters. Forty percent of practicing physicians in Florida owned services to which they referred patients. Ninety-three percent of diagnostic imaging facilities, 76% of ambulatory surgery centers, and 60% of clinical laboratories in the state were owned wholly or in part by physicians. The rates of use for MRI and CT scans were higher for physician-owned compared with non-physician-owned facilities (Mitchell and Scott, 1992). In a national study, physicians who received payment for performing x-rays and sonograms within their own offices obtained these examinations four times as often as physicians who referred the examinations to radiologists and received no reimbursement for the studies. The patients in the two groups were similar (Hillman et al, 1990).\n\nAfter 2000, profitable diagnostic, imaging, and surgical procedures have rapidly migrated from the hospital to free-standing physician-owned ambulatory surgery centers, endoscopy centers, and imaging centers (Berenson et al, 2006). For example, the number of CT scans performed for Medicare patients increased by 65% from 2000 to 2005; during those years, the number of MRI scans jumped by 94% (Bodenheimer et al, 2007). The number of CT scans is growing by more than 10% per year, increasing patients' risk of radiation-related cancer (Smith-Bindman, 2010). A significant association exists between surgeon ownership of ambulatory surgery centers and a higher volume of surgeries; surgery volume increases immediately following surgeons' acquisition of the surgicenter (Hollingsworth et al, 2010).\n\nMoving to the other side of the overuse\u2013underuse spectrum, payment by capitation, or salaried employment by a for-profit business, may create a climate hostile to the provision of adequate services. In the 1970s, a series of HMOs called prepaid health plans (PHPs) sprang up to provide care to California Medicaid patients. The quality of care in several PHPs became a major scandal in California. At one PHP, administrators wrote a message to health care providers: \"Do as little as you possibly can for the PHP patient,\" and charts audited by the California Health Department revealed many instances of undertreatment. The PHPs received a lump sum for each patient enrolled, meaning that the lower the cost of the services actually provided, the greater the PHP's profits (US Senate, 1975).\n\nThe quantity and quality of medical care are inextricably interrelated. Too much or too little can be injurious. The research of Fisher et al (2003) has shown that similar populations in different geographic areas have widely varying rates of surgeries and days in the hospital, with no consistent difference in clinical outcomes between those in high-use and low-use areas.\n\n#### **Health Care Systems and Quality of Care**\n\n_The personnel cutbacks were terrible; staffing had diminished from four RNs per shift to two, with only two aides to provide assistance. Shelley Rush, RN, was 2 hours behind in administering medications and had five insulin injections to give, with complicated dosing schedules. A family member rushed to the nursing station saying, \"The lady in my mother's room looks bad.\" Shelley ran in and found the patient unconscious. She quickly checked the blood sugar, which was disastrously low at 20 mg_ \/ _dL. Shelley gave 50% glucose, and the patient woke up. Then it hit her\u2014she had injected the insulin into the wrong patient._\n\nHealth care institutions must be well organized, with an adequate, competent staff. Shelley Rush was a superb nurse, but understaffing caused her to make a serious error. The book _Curing Health Care_ by Berwick et al (1990) opens with a heartbreaking case:\n\n_She died, but she didn't have to. The senior resident was sitting, near tears, in the drab office behind the nurses' station in the intensive care unit. It was 2:00 AM, and he had been battling for thirty-two hours to save the life of the 23-year-old graduate student who had just suffered her final cardiac arrest._\n\n_The resident slid a large manila envelope across the desk top. \"Take a look at this,\" he said. \"Routine screening chest x-ray, taken 10 months ago. The tumor is right there, and it was curable\u2014then. Bythe time the second film was taken 8 months later, because she was complaining of pain, it was too late. The tumor had spread everywhere, and the odds were hopelessly against her. Everything we've done since then has really just been wishful thinking. We missed our chance. She missed her chance.\" Exhausted, the resident put his head in his hands and cried._\n\n_Two months later, the Quality Assurance Committee completed its investigation.... \"We find the inpatient care commendable in this tragic case,\" concluded the brief report, \"although the failure to recognize the tumor in a potentially curable stage 10 months earlier was unfortunate....\" Nowhere in this report was it written explicitly why the results of the first chest x-ray had not been translated into action. No one knew._\n\n_One year later.... it was 2:00 AM, and the night custodian was cleaning the radiologist's office. As he moved a filing cabinet aside to sweep behind it, he glimpsed a dusty tan envelope that had been stuck between the cabinet and the wall. The envelope contained a yellow radiology report slip, and the date on the report\u2014nearly two years earlier\u2014convinced the custodian that this was, indeed, garbage... He tossed it in with the other trash, and 4 hours later it was incinerated along with other useless things. (Berwick et al, 1990)_\n\nThis patient may have had perfect access to care for an illness whose treatment is scientifically proved; she may have seen a physician who knew how to make the diagnosis and deliver the appropriate treatment; and yet the quality of her care was disastrously deficient. Dozens of people and hundreds of processes influence the care of one person with one illness. In her case, one person\u2014perhaps a file clerk with a near-perfect record in handling thousands of radiology reports\u2014lost control of one report, and the physician's office had no system to monitor whether or not x-ray reports had been received. The result was the most tragic of quality failures\u2014the unnecessary death of a young person.\n\nHow health care systems and institutions are organized has a major impact on health care outcomes. For example, large multispecialty group practices in 22 metropolitan areas have better-quality measures at lower cost than dispersed physician practices in those areas (Weeks et al, 2009). Studies have shown that hospitals with more RN staffing have lower surgical complication rates (Kovner and Gergen, 1998) and lower mortality rates (Aiken et al, 2002).\n\n_Oliver Hart lived in a city with a population of 80,000. He was admitted to Neighborhood Hospital with congestive heart failure caused by a defective mitral valve. He was told he needed semiurgent heart surgery to replace the valve. The cardiologist said \"You can go to University Hospital 30 miles away or have the surgery done here.\" The cardiologist did not say that Neighborhood Hospital performed only seven cardiac surgeries last year. Mr. Hart elected to remain for the procedure. During the surgery, a key piece of equipment failed, and he died on the operating table._\n\nQuality of care must be viewed in the context of regional systems of care (see Chapter 6), not simply within each health care institution. In one study, 27% of deaths related to coronary artery bypass graft (CABG) surgery at low-volume hospitals might have been prevented by referral of those patients to hospitals performing a higher volume of those surgeries (Dudley et al, 2000). Quality improves with the experience of those providing the care (Kizer, 2003; Peterson et al, 2004). Had Mr. Hart been told the relative surgical mortality rates at University Hospital, which performed 500 cardiac surgeries each year, and at Neighborhood Hospital, he would have chosen to be transferred 30 miles down the road. Not only does the volume of surgeries in a hospital matter; equally important is the volume of surgeries performed by the specific surgeon (Birkmeyer et al, 2003).\n\nIn the late 1980s, Dr. Donald Berwick (1989) and others realized that quality of care is not simply a question of whether or not a physician or other caregiver is competent. If poorly organized, the complex systems within and among medical institutions can thwart the best efforts of professionals to deliver high-quality care.\n\n_There are two approaches to the problem of improving quality... [One is] the Theory of Bad Apples, because those who subscribe to it believe that quality is best achieved by discovering bad apples and removing them from the lot.... The Theory of Bad Apples gives rise readily to what can be called the my-apple-is-just-fine-thank-you response... andseeks not understanding, but escape. [The other is] the Theory of Continuous Improvement . . . . Even when people were at the root of defects,... the problem was generally not one of motivation or effort, but rather of poor job design, failure of leadership, or unclear purpose. Quality can be improved much more when people are assumed to be trying hard already, and are not accused of sloth. Fear of the kind engendered by the disciplinary approach poisons improvement in quality, since it inevitably leads to the loss of the chance to learn._\n\n_Real improvement in quality depends... on continuous improvement throughout the organization through constant effort to reduce waste, rework, and complexity. When one is clear and constant in one's purpose, when fear does not control the atmosphere (and thus the data), when learning is guided by accurate information... and when the hearts and talents of all workers are enlisted in the pursuit of better ways, the potential for improvement in quality is nearly boundless . . . . A test result lost, a specialist who cannot be reached, a missing requisition, a misinterpreted order, duplicate paperwork, a vanished record, a long wait for the CT scan, an unreliable on-call system\u2014these are all-too-familiar examples of waste, rework, complexity, and error in the physician's daily life . . . . For the average physician, quality fails when systems fail. (Berwick et al, 1989)_\n\n#### **The Components of Quality: Summary**\n\nGood-quality care can be compromised at a number of steps along the way.\n\n_Angie Roth has coronary heart disease and may need CABG surgery. (1) If she is uninsured and cannot get to a physician, high-quality care is impossible to obtain. (2) If clear evidence-based guidelines do not exist regarding who benefits from CABG and who does not, Ms. Roth's physician may make the wrong choice. (3) Even if clear guidelines exist, if Angie Roth's physician fails to evaluate her illness correctly or sends her to a surgeon with poor operative skills, quality may suffer. (4) If indications for surgery are not clear in Ms. Roth's case but the surgeon will benefit economically from the procedure, the surgery may be inappropriately performed. (5) Even if the surgery is appropriate and performed by an excellent surgeon, faulty equipment in the operating room or poor teamwork among the operating room surgeons, anesthesiologists, and nurses may lead to a poor outcome._\n\nThe Institute of Medicine, in its influential 2001 report _Crossing the Quality Chasm_ , conceptualized six core dimensions of quality: safe, effective, patient-centered, timely, efficient, and equitable. These dimensions, defined in greater detail in Table 10\u20132, are consistent with the components of quality discussed earlier.\n\n**Table 10\u20132.** Quality aims as defined by the Institute of Medicine\n\n### **PROPOSALS FOR IMPROVING QUALITY**\n\n_Several infants at a hospital received epinephrine in error and suffered serious medical consequences. An analysis revealed that several pharmacists had made the same mistake; the problem was caused by the identical appearance of vitamin E and epinephrine bottles in the pharmacy. This was a system error._\n\n_An epidemic of unexpected deaths on the cardiac ward was investigated. The times of the deaths were correlated with personnel schedules, leading to the conclusion that one nurse was responsible. Itturned out that she was administering lethal doses of digoxin to patients. This was not a system error._\n\nQuality issues must be investigated to determine if they are system errors or problems with a particular caregiver. Traditionally, quality assurance has focused on individual caregivers and institutions in a \"bad apple\" approach that relies heavily on sanctions. More recently, quality has been viewed through the lens of the continuous quality improvement (CQI) model that seeks to enhance the clinical performance of all systems of care, not just the outliers with flagrantly poor quality of care. The move to a CQI model has required development of more formalized standards of care that can be used as benchmarks for measuring quality, and more systematic collection of data to measure overall performance and not just performance in isolated cases (Tables 10\u20132).\n\n#### **Traditional Quality Assurance: Licensure, Accreditation, and Peer Review**\n\nTraditionally, the health care system has placed great reliance on educational institutions and licensing and accrediting agencies to ensure the competence of individuals and institutions in health care. Health care professionals undergo rigorous training and pass special licensing examinations intended to ensure that caregivers have at least a basic level of knowledge and competence. However, not all individuals who have successfully completed their education and passed licensing examinations are competent clinicians. In some cases, this reflects a failure of the educational and licensing systems. In other cases, clinicians may have been competent practitioners at the time they took their examinations, but their skills lapsed or they developed impairment from alcohol or drug use, depression, or other conditions (Leape and Fromson, 2006).\n\nLicensing agencies in the United States do not require periodic reexaminations. In most cases, licensing boards only respond to patient or health care professional complaints about negligent or unprofessional behavior. Many organizations that confer specialty board certification require physicians to pass examinations on a periodic basis to maintain active specialty certification. Some specialties also require physicians to perform and document systematic quality reviews of their own clinical practices for maintenance of certification. However, while some hospitals may require active specialty certification for a physician to be granted privileges to practice in the hospital, certification is not required for medical licensure, diluting some of the consequences of not participating in specialty recertification.\n\nThe traditional approach to quality assurance has also relied heavily on peer pressure within hospitals, HMOs, and the medical community at large. Peer review is the evaluation by health care practitioners of the appropriateness and quality of services performed by other practitioners, usually in the same specialty. Peer review has been a part of medicine for decades (eg, tissue committees study surgical specimens to determine whether appendectomies and hysterectomies have actually removed diseased organs; credentials committees review the qualifications of physicians for hospital staff privileges). But peer review moved to center stage with the passage of the law enacting Medicare in 1965.\n\nMedicare anointed the Joint Commission on Accreditation of Hospitals (now named simply the Joint Commission) with the authority to terminate hospitals from the Medicare program if quality of care was found to be deficient. The Joint Commission requires hospital medical staff to set up peer review committees for the purpose of maintaining quality of care.\n\nThe Joint Commission uses criteria of structure, process, and outcome to assess quality of care. Structural criteria include such factors as whether the emergency department defibrillator works properly. Criteria of process include whether medical records are dictated and signed in a timely manner, or if the credentials committee keeps minutes of its meetings. Outcomes include such measures as mortality rates for surgical procedures, proportions of deaths that are preventable, and rates of adverse drug reactions and wound infections. Medicare also contracts with quality improvement organizations (QIOs) in each state to promote better quality of care among physicians caring for Medicare beneficiaries.\n\n_Angela Lopez, age 57, suffered from metastatic ovarian cancer but was feeling well and prayed she would live 9 months more. Her son was the first family member ever to attend college, and she hoped to see him graduate. It was decided to infuse chemotherapy directly into her peritoneal cavity.As the solution poured into her abdomen, she felt increasing pressure. She asked the nurse to stop the fluid. The nurse called the physician, who said not to worry. Two hours later, Ms. Lopez became short of breath and demanded that the fluid be stopped. The nurse again called the physician, but an hour later Ms. Lopez died. Her abdomen was tense with fluid, which pushed on her lungs and stopped circulation through her inferior vena cava. The quality assurance committee reviewed the case as a preventable death and criticized the physician for giving too much fluid and failing to respond adequately to the nurse's call. The physician replied that he was not at fault; the nurse had not told him how sick the patient was. The case was closed._\n\nThe traditional quality assurance strategies of licensing and peer review have not been particularly effective tools for improving quality. Peer review often adheres to the theory of bad apples, attempting to discipline physicians (to remove them from the apple barrel) for mistakes rather than to improve their practice through education. The physician who caused Ms. Lopez's preventable death responded to peer criticism by blaming the nurse rather than learning from the mistake. With the hundreds of decisions physicians make each day, often in time-constrained situations, serious errors are relatively common in medical practice. Yet 42% of physicians recently surveyed had never disclosed a serious error to a patient (Gallagher et al, 2006). Hiding mistakes rather than correcting them is the legacy of a punitive quality assurance apparatus (Leape, 1994).\n\nEven if sanctions against the truly bad apples had more teeth, these measures would not solve the quality problem. Removing the incontrovertibly bad apples from the barrel does not address all the quality problems that emanate from competent caregivers who are not performing optimally. Health care systems do need to ensure basic clinical competence and to forcefully sanction caregivers who, despite efforts at remediation, cannot operate at a basic standard of acceptable practice. But measures are also needed to \"shift the curve\" of overall clinical practice to a higher level of quality, not just to trim off the poor-quality outliers.\n\nPeer reviewers frequently disagree as to whether the quality of care in particular cases is adequate or not (Laffel and Berwick, 1993). Because of these limitations, efforts are underway to formalize standards of care using clinical practice guidelines and to move from individual case review to more systematic monitoring of overall practice patterns (Table 10\u20133).\n\n**Table 10\u20133.** Proposals for improving quality\n\n#### **Clinical Practice Guidelines**\n\n_Dr. Benjamin Waters was frustrated by patients who came in with urinary incontinence. He never learned about the problem in medical school, so he simply referred these patients to a urologist. In his managed care plan, Dr. Waters was known to over-refer, so he felt stuck. He could not handle the problem, yet he did not want to refer patients elsewhere. He solved his dilemma by prescribing incontinence pads and diapers, but did not feel good about it._\n\n_Dr. Denise Drier learned about urinary incontinence in family medicine residency but did not feel secure about caring for the problem. On the web, she found \"Urinary Incontinence in Adults: Clinical Practice Guideline Update.\" She studied the material and applied it to her incontinence patients. After a few successes, she and the patients were feeling better about themselves._\n\nFor many conditions, there is a better and a worse way to make a diagnosis and prescribe treatment. Physicians may not be aware of the better way because of gaps in training, limited experience, or insufficient time or motivation to learn new techniques. For these problems, clinical practice guidelines can be helpful in improving quality of care. In 1989, Congress established the Agency for Health Care Policy and Research, now called the Agency for Healthcare Research and Quality (AHRQ), to develop practice guidelines, among other tasks. Produced by panels of experts, practice guidelines make specific recommendations to physicians on how to treat clinical conditions such as diabetes, osteoporosis, urinary incontinence, or cataracts. However, some powerful physician interests, displeased by AHRQ practice guidelines that recommended against surgical treatment for most cases of back pain, pressured Congress to reduce AHRQ's budget and bar AHRQ from issuing its own guidelines.\n\nMore than 2000 guidelines exist; written by dozens of organizations, they vary in scientific reliability. Most are developed by societies of medical specialists (Steinbrook, 2007). Ideally, practice guidelines are based on a rigorous and objective review of scientific evidence, with explicit ratings of the quality of the evidence. However, 87% of clinical practice guideline authors in one survey had ties to the pharmaceutical industry, a bias often not disclosed to readers of the guidelines (Shaneyfelt and Centor, 2009). For example, eight of nine authors of widely used guidelines recommending broad use of cholesterol-lowering statin drugs had financial ties to companies making or selling statins (Abramson and Starfield, 2005). Moreover, clinical practice guidelines developed based on research on a narrowly defined population, such as nonelderly patients with a single chronic condition, may not be applicable to different patient populations, such as elderly patients with multiple diseases (Boyd et al, 2005).\n\nPractice guidelines are not appropriate for many clinical situations. Uncertainty pervades clinical medicine, and practice guidelines are applicable only for those cases in which we enjoy \"islands of knowledge in our seas of ignorance.\" Practice guidelines can assist but not replace clinical judgment in the quest for high-quality care.\n\n_Pedro Urrutia, age 59, noticed mild nocturia and urinary frequency. His friend had prostate cancer, and he became concerned. The urologist said that his prostate was only slightly enlarged, his prostate-specific antigen (blood test) was normal, and surgery was not needed. Mr. Urrutia wanted surgery and found another urologist to do it._\n\n_At age 82, James Chin noted nocturia and urinary hesitancy. He had two glasses of wine on his wife's birthday and later that night was unable to urinate. He went to the emergency department, was found to have a large prostate without nodules, and was catheterized. The urologist strongly recommended a transurethral resection of the prostate. Mr. Chin refused, thinking that the urinary retention was caused by the alcohol. Five years later, he was in good health with his prostate intact._\n\nThe difficulty with creating a set of indications for surgery, for example surgery for benign enlargement of the prostate gland, is that patient preferences vary markedly. Some, like Mr. Urrutia, want prostate surgery, even though it is not clearly needed; others, like Mr. Chin, have strong reasons for surgery but do not want it. Practice guidelines must take into account not only scientific data, but also patient preferences (O'Connor et al, 2007).\n\nDo practice guidelines in themselves improve quality of care? Studies reveal that by themselves they are unsuccessful in influencing physicians' practices (Cabana et al, 1999). However, guidelines can be an important foundation for more comprehensive quality improvement strategies, such as computer systems to remind physicians when patients are in need of certain services according to a guideline (eg, a reminder system about women due for a mammogram) or having trusted colleagues (\"opinion leaders\") or visiting experts (\"academic detailing\") conduct small group sessions with clinicians to review and reinforce practice guidelines (Bodenheimer and Grumbach, 2007).\n\n#### **Measuring Practice Patterns**\n\nOne of the central tenets of the CQI approach is the need to systematically monitor how well individual caregivers, institutions, and organizations are performing. There are two basic types of indicators that are used to evaluate clinical performance: process measures and outcome measures. _Process_ of care refers to the types of services delivered by caregivers. Examples are prescribing aspirin to patients with coronary heart disease, or turning immobile patients in hospital beds on a regular schedule to prevent bed sores. _Outcomes_ are death, symptoms, mental health, physical functioning, and related aspects of health status, and are the gold standard for measuring quality. However, outcomes (particularly those dealing with quality of life) may be difficult to measure. More easily counted outcomes such as mortality may be rare events, and therefore uninformative for evaluating quality of care for many conditions that are not immediately life-threatening. Also, outcomes may be heavily influenced by the underlying severity of illness and related patient characteristics, and not just by the quality of health care that patients received (King and Wheeler, 2007) When measuring patient outcomes, it is necessary to \"risk adjust\" these outcome measurements for differences in the underlying characteristics of different groups of patients. Because of these challenges in using outcomes as measures to monitor quality of care, process measures tend to be more commonly used. For process measures to be valid indicators of quality, there must first be solid research demonstrating that the processes do in fact influence patient outcomes.\n\n_Dr. Susan Cutter felt horrible. It was supposed to have been a routine hysterectomy. Somehow she had inadvertently lacerated the large intestine of the patient, a 45-year-old woman with symptomatic fibroids of the uterus but otherwise in good health prior to surgery. Bacteria from the intestine had leaked into the abdomen, and after a protracted battle in the ICU the patient died of septic shock._\n\n_Dr. Cutter met with the Chief of Surgery at her hospital. The Chief reviewed the case with Dr. Cutter, but also pulled out a report showing the statistics on all of Dr. Cutter's surgical cases over the previous 5 years. The report showed that Dr. Cutter's mortality and complication rates were among the lowest of surgeons on the hospital's staff. However, the Chief did note that another surgeon, Dr. Dehisce, had a complication rate that was much higher than that of all the other staff surgeons. The Chief of Surgery asked Dr. Cutter to serve on a departmental committee to review Dr. Dehisce's cases and to meet with Dr. Dehisce to consider ways to address his poor performance._\n\nThe contemporary approach to quality monitoring moves beyond examining a few isolated cases toward measuring processes or outcomes for a large population of patients. For example, a traditional peer review approach is to review every case of a patient who dies during surgery. Reviewing an individual case may help a surgeon and the operating team understand where errors may have occurred\u2014a process known as \"root cause\" analysis. However, it does not indicate whether the case represented an aberrant bad outcome for a surgeon or team that usually has good surgical outcomes, or whether the case is indicative of more widespread problems. To answer these questions requires examining data on all the patients operated on by the surgeon and the operating team to measure the overall rate of surgical complications, and having some benchmark data that indicate whether this rate is higher than expected for similar types of patients.\n\n_Mel Litus was the nurse in charge of diabetes education for a large medical group. After seeing yet another patient return to clinic after having had a foot amputation or suffering a heart attack, Mel wondered how the clinic team could do a better job in preventing diabetic complications. The medical group had recently implemented a new computerized clinical information system. Mel met with the administrator in charge of the computer system and arranged to have a printout made of all the laboratory findings, referrals, and medications for the diabetic patients in the medical group. When Mel reviewed the printout, he noticed that many of the patients didn't attend appointments very regularly and were not receiving important services like regular ophthalmology visits and medications that protect the kidneys from diabetic damage. Mel met with the medical director for quality improvement to discuss a plan for sharing this information with the clinical staff and creating a system for more closely monitoring the care of diabetic patients._\n\nMany practice organizations, from small groups of office-based physicians to huge, vertically integrated HMOs are starting to monitor patterns of care and provide feedback on this care to physicians and other staff in these organizations. The goal of this feedback is to alert caregivers and health care organizations about patterns of care that are not achieving optimal standards, in order to stimulate efforts to improve processes of care. The response may range from individual clinicians systematically reviewing their care of certain types of patients and clinical conditions, to entire organizations redesigning the system of care. A typical example of this practice profiling is measuring the rate at which diabetic patients receive recommended services, such as annual eye examinations, periodic testing of HbA1c levels, and evaluation of kidney function. Process of care profiles alert individual caregivers to specific diabetic patients who need to be called in for certain tests, and point out patterns of care that suggest that the organization should implement systematic reforms, such as developing case management programs for diabetic patients in poor control (Bodenheimer and Grumbach, 2007).\n\n#### **Continuous Quality Improvement**\n\nMaximizing excellence for individual health care professionals is only one ingredient in the recipe for high-quality health care. Improving institutions is the other, through CQI techniques. CQI involves the identification of concrete problems and the formation of interdisciplinary teams to gather data and propose and implement solutions to the problems.\n\n_In LDS Hospital in Salt Lake City, variation in wound infection rates by different physicians was related to the timing of the administration of prophylactic antibiotics. Patients who received antibiotics 2 hours before surgery had the lowest infection rates. The surgery department adopted a policy that all patients receive antibiotics precisely 2 hours before surgery; the rate of postoperative wound infections dropped from 1.8% to 0.9%. (Burke, 2001)_\n\nSuch successes only dot, but do not yet dominate, the health care quality landscape (Solberg, 2007). The Institute for Healthcare Improvement (IHI) has led efforts to spread CQI efforts by sponsoring \"collaboratives\" to assist institutions and groups of institutions to improve health care outcomes and access while ideally reducing costs. Hundreds of health care organizations have participated in collaboratives concerned with such topics as improving the care of chronic illness, reducing waiting times, improving care at the end of life, and reducing adverse drug events. Collaboratives involve learning sessions during which teams from various institutions meet and discuss the application of a rapid change methodology within institutions. Some of IHI's successes have taken place in the area of chronic disease, with a variety of institutions\u2014from large integrated delivery systems to tiny rural community health centers\u2014implementing the chronic care model to improve outcomes for conditions such as diabetes, asthma, and congestive heart failure (Bodenheimer et al, 2002). Collaboratives that assist institutions to implement the chronic care model have shown modest improvement in patient outcomes compared with controls (Vargas et al, 2007). In the area of patient safety, in 2004, IHI launched the 100,000 Lives Campaign (www.ihi.org) to reduce mortality rates in hospitals, followed by a 5 Million Lives Campaign between 2006 and 2008; more than 4000 hospitals in the United States participated in these campaigns. There is evidence that these campaigns have contributed to reductions in hospital mortality, although there is debate about the magnitude of the impact (Berwick et al, 2006; Wachter and Pronovost, 2006).\n\n#### **Computerized Information Systems**\n\nThe advent of computerized information systems has created opportunities to improve care and to monitor the process and outcomes of care for entire populations. Electronic medical records can create lists of patients who are overdue for services needed for preventive care or the management of chronic illness and can generate reminder prompts for physicians and patients (Baron, 2007). In-hospital medical errors related to drug prescribing are reduced with computerized physician order entry (CPOE), systems requiring physicians to enter hospital orders directly into a computer rather than handwriting them. The computer can alert the physician about inappropriate medication doses or medications to which the patient is known to be allergic (Kaushal et al, 2003). However, hospital-based electronic health records have not yet been proven to significantly improve quality (Eslami et al, 2007; DesRoches et al, 2010). By themselves, computerized information systems are unlikely to improve quality; computerization must be accompanied by changes in the organization of informational processes (Bodenheimer and Grumbach, 2007).\n\n#### **Public Reporting of Quality**\n\nThe CQI approach emphasizes systematic monitoring of care to provide internal feedback to clinicians and health organizations to spur improved processes of care. A different approach to monitoring quality of care is to direct this information to the public. This approach views public release of systematic measurements of quality of care\u2014commonly referred to as health care \"report cards\"\u2014as a tool to empower health care consumers to select higher-quality caregivers and institutions. Advocates of this approach argue that armed with this information, patients and health care purchasers will make more informed decisions and preferentially seek out health care organizations with better report card grades.\n\nAn important experiment in individual physician report cards was initiated by the New York State Department of Health in 1990. The department released data on risk-adjusted mortality rates for coronary bypass surgery performed at each hospital in the state, and in 1992, mortality rates were also published for each cardiac surgeon. Each year's list was big news and highly controversial. However, difficulties in measurement were highlighted by the fact that within 1 year, 46% of the surgeons had moved from one-half of the ranked list to the other half.\n\nSeveral fascinating results came of this project: (1) Patients did not switch from hospitals with high mortality rates to those with lower mortality rates. (2) With the release of each report, one in five bottom quartile surgeons relocated or ceased practicing within two years. (3) In 4 years, overall risk-adjusted coronary artery bypass mortality dropped by 41% in New York State. Mortality for this operation also dropped in states without report cards, but not as much. (4) Some surgeons, worried about the report cards, may have elected not to operate on the most risky patients in order to improve their report card ranking. It is possible that the reduction in surgical mortality in part resulted from withholding surgery for the sickest patients. The New York State experiment had less effect on changing the market decisions of patients and purchasers than on motivating quality improvements in hospitals that had poor surgical outcomes (Marshall et al, 2000; Jha and Epstein, 2006).\n\nIn 2011, the federal Centers for Medicare and Medicaid Services (CMS) launched its Physician Compare website, which will report on quality-of-care measures for specific physicians by 2015 (www.medicare.gov\/find-a-doctor\/provider-search.aspx).\n\nThe most important report card program is the Healthcare Effectiveness Data and Information Set (HEDIS). Developed by the National Committee for Quality Assurance (NCQA), a private organization controlled by large HMOs and large employers, HEDIS for 2010 is a list of 71 performance indicators including the percentage of children immunized; the percentage of enrollees of certain ages who have received Pap smears, colorectal screening, mammograms, and glaucoma screening; the percentage of pregnant women who received prenatal care in the first trimester; the percentage of diabetic patients who received retinal examinations; and the percentage of smokers for whom physicians made efforts at smoking cessation; the appropriateness of treatment for asthma, bronchitis, osteoporosis, depression, and others. NCQA chiefly reports on the performance of health plans; some critics believe that reporting on physicians and hospitals would be more helpful. Another problem is that few employers use quality data when selecting health plans for their employees; cost is the driving factor in most employer decisions (Galvin and Delbanco, 2005).\n\nReport cards are based on a philosophy that says \"if you can't count it, you can't improve it.\" Albert Einstein expressed an alternative philosophy that might illuminate the report card enterprise: \"Not everything that can be counted counts, and not everything that counts can be counted.\" Increasingly, the focus on quality is switching to a focus on value, with value referring to quality divided by cost. Thus an increase in a quality measure associated with a growth in cost may not improve value, where improved quality with a stable or reduced cost increases value (Owens et al, 2011).\n\n#### **Pay for Reporting**\n\nIn 2003, the Medicare program initiated public reporting for hospitals, focusing on risk-adjusted quality of care for heart attacks, heart failure, and pneumonia. More recently, surgical care and other measures have been added. Reports on individual hospitals and an explanation of the program are available at www.hospitalcompare.hhs.gov. The program, the Hospital Quality Initiative, is voluntary but nonparticipating hospitals receive a reduction in their Medicare payments. One might say that the program is in essence no-pay for no-reporting. Hospital quality has improved for some measures that are reported (Chassin et al, 2010), but hospitals focus their quality activities on the specific measures prescribed by the program, at times to the detriment of other quality activities (Pham et al, 2006).\n\nIn 2007, Medicare began the Physician Quality Reporting System, under which physicians who report certain quality measures may receive a 2% increase in their Medicare fees. This is not a full-fledged pay for reporting program because the reports for individual physicians or physician practices are not made public (www.cms.hhs.gov\/pqri\/).\n\n#### **Pay for Performance**\n\nBy 2003, a new concept\u2014\"pay for performance\"\u2014was gaining widespread acceptance in health care (Epstein et al, 2004). Pay for performance (P4P) goes one step beyond pay for reporting; physicians or hospitals receive more money if their quality measures exceed certain benchmarks or if the measures improve from year to year.\n\nOne of the largest P4P programs is the Integrated Healthcare Association (IHA) program in California. IHA, representing employers, health plans, health systems and physician groups, launched the program in 2002 with a set of uniform performance measures. In 2010, seven health plans and 221 physician organizations\u2014involving 35,000 physicians and 10 million patients\u2014participated in the IHA program (www.iha.org).\n\nIn 2010, the health plans paid physician organizations $49 million in performance-based bonuses. The physician organizations receive funds for demonstrating improved clinical care (eg, cancer screening, immunizations, and management of asthma, diabetes, and cardiovascular disease), patient satisfaction, and development of information technology. In 2009, IHI added cost containment measures including inpatient utilization, hospital readmissions, and generic drug prescribing. Physician organizations distribute a substantial amount of the money to individual physicians but keep a portion of the bonus for organizationwide quality-enhancing initiatives. Quality measures have improved modestly since the program began, between 5% and 12%, but patient satisfaction did not. A limitation of the program is that the bonuses are small (about 2% of physician group revenues) and practices must spend money to organize and report their data (Damberg et al, 2009). Moreover, health plans are becoming less enthusiastic about P4P as they are not seeing the return on investment hoped for (Integrated Healthcare Organization, 2009).\n\nThe IHA program is unique for two reasons: All major health plans collaborated in choosing the measures upon which performance bonuses are based, and most physicians in California belong to a large medical group or independent practice association (see Chapter 6). If only one health plan sets up a P4P program with physicians, there may not be enough patients from that health plan to accurately measure the physician's quality; with all health plans participating, a substantial portion of a physician's patient panel is included in the measures. If P4P targets individual physicians rather than larger physician organizations, the small numbers of patients may distort the results. The ability of the California experience to aggregate a large number of patients allows for more accurate performance evaluation.\n\nA P4P program initiated by large employers rather than health plans is Bridges to Excellence. This program involves more than 80 employers, large national health plans, and 3000 physicians in about 15 states. Physicians receive bonus payments for implementing computerized office systems and for improving the care of patients with diabetes, asthma, chronic lung disease, heart disease, back pain, and high blood pressure. Physicians practicing high-quality medicine in these areas receive public recognition and may receive bonuses, with the employers financing the program counting on higher-quality translating into lower costs. Performance is measured only for patients who are employees of the employers participating in the program, a small number for many physicians (www.bridgestoexcellence.org).\n\nIn 2003, Medicare launched a P4P program for 268 hospitals, measuring certain quality indicators for heart attack, heart failure, pneumonia, coronary artery bypass surgery, and hip and knee replacements. High-performing hospitals receive bonuses and the lowest performers may be subject to penalties. Performance on ten measures for heart attack, heart failure, and pneumonia in the P4P hospitals improved more than in control hospitals (Lindenauer, 2007). Another study looked at more than 100,000 heart attack patients treated at P4P and control hospitals; between 2003 and 2006, quality measures for these patients improved equally at P4P and control hospitals (Glickman et al, 2007). From 2003 to 2008, quality scores for participating hospitals improved by 18% (CMS, 2010), which was a 2% to 4% greater improvement than in control hospitals (Mehrotra et al, 2009).\n\nA P4P program described as \"an initiative to improve the quality of primary care that is the boldest such proposal attempted anywhere in the world\" was launched in the United Kingdom in 2004 (Roland, 2004). This program is described in Chapter 14.\n\nSome authors urge caution, pointing out that P4P programs could encourage physicians and hospitals to avoid high-risk patients in order to keep their performance scores up (McMahon et al, 2007). Another difficulty is that many patients see a large number of physicians in a given year, making it impossible to determine which physician should receive a performance bonus (Pham et al, 2007). Moreover, P4P programs could increase disparities in quality by preferentially rewarding physicians and hospitals caring for higher-income patients and having greater resources available to invest in quality improvement, and penalizing those institutions and physicians attending to more vulnerable populations in resource-poor environments (Casalino et al, 2007).\n\n#### **Financially Neutral Clinical Decision Making**\n\nThe quest for quality care encompasses a search for a financial structure that does not reward over- or under-treatment and that separates physicians' personal incomes from their clinical decisions. Balanced incentives (see Chapter 4), combining elements of capitation or salary and fee-for-service, may have the best chance of minimizing the payment\u2013treatment nexus (Robinson, 1999), encouraging physicians to do more of what is truly beneficial for patients while not inducing inappropriate and harmful services. Completely financially neutral decision making will always be an ideal and not a reality.\n\n### **WHERE DOES MALPRACTICE REFORM FIT IN?**\n\n_During a coronary angiogram, emboli traveled to the brain of Ivan Romanov, resulting in a serious stroke, with loss of use of his left arm and leg. The angiogram was appropriate and performed without any technical errors. Mr. Romanov had suffered a medical injury (an injury caused by his medical treatment), but the event was not because of negligence._\n\n_During a dilation and curettage (D &C), Judy Morrison's physician unknowingly perforated her uterus and lacerated her colon. Ms. Morrison reported severe pain but was sent home without further evaluation. She returned 1 hour later to the emergency department with persistent pain and internal bleeding. She required a two-stage surgical repair over the following 4 months. This medical injury was found by the legal system to be because of negligence._\n\nA peculiar set of institutions called the malpractice liability system forms an important part of US health care (Sage and Kersh, 2006). The goals of the malpractice system are twofold: To financially compensate people who in the course of seeking medical care have suffered medical injuries and to prevent physicians and other health care personnel from negligently causing harm to their patients.\n\nThe existing malpractice system scores miserably on both counts. According to the Harvard Medical Practice Study, only 2% of patients who suffer adverse events caused by medical negligence file malpractice claims that would allow them to receive compensation, meaning that the malpractice system fails in its first goal. Moreover, the system does not deal with 98% of negligent acts performed by physicians, making it difficult to attain its second goal. More recent research has confirmed the findings of the Harvard study (Sage and Kersh, 2006).\n\nOn the other hand, as many as 40% of malpractice claims do not involve true medical errors (Studdert et al, 2006), with an even smaller proportion representing actual negligence. Nonetheless, one-quarter of these inappropriate claims result in the patient receiving monetary compensation. Overall, for every dollar in compensation received by patients in malpractice awards, legal costs and fees come to 54 cents (Studdert et al, 2006).\n\nThe malpractice system has serious negative side effects on medical practice (Localio et al, 1991).\n\n1. The system assumes that punishment, which usually involves physicians paying large amounts of money to a malpractice insurer plus enduring the overwhelming stress of a malpractice jury trial, is a reasonable method for improving the quality of medical care. Berwick's analysis of the Theory of Bad Apples suggests that fear of a lawsuit closes physicians' minds to improvement and generates an \"I didn't do it\" response. The entire atmosphere created by malpractice litigation clouds a clear analytic assessment of quality.\n\n2. The system is wasteful, with a huge portion of malpractice insurance premiums spent on lawyers, court costs, and insurance overhead almost as costly as payments to patients (Mello et al, 2010). Many claims have no merit but create enormous waste and wreak an unnecessary stress upon physicians. Patients granted malpractice award payments sometimes experienced no negligent care, and patients subjected to negligent care often receive no malpractice payments (Brennan et al, 1996). Total costs of the malpractice system came to $55.6 billion in 2008, 2.4% of total national health spending (Mello et al, 2010).\n\n3. The system is based on the assumption that trial by jury is the best method of determining whether there has been negligence, a highly questionable assumption.\n\n4. People with lower incomes generally receive smaller awards (because wages lost from a medical injury are lower) and are therefore less attractive to lawyers, who are generally paid as a percentage of the award. Accordingly, low-income patients, who suffer more medical injury, are less likely than wealthier people to file malpractice claims (Burstin et al, 1993b).\n\nIn summary, the malpractice system is burdened with expensive, unfounded litigation that harasses physicians who have done nothing wrong, while failing to discipline or educate most physicians committing actual medical negligence and to compensate most true victims of negligence.\n\n_Mei Tagaloa underwent neurosurgery for compression of his spinal cord by a cervical disk. On awakening from the surgery, Mr. Tagaloa was unable to move his legs or arms at all. After 3 months of rehabilitation, he ended up as a wheelchair-bound paraplegic. He sued the neurosurgeon and his family physician. The physicians' malpractice insurer paid for lawyers to defend them. Mr. Tagaloa's lawyer used the system of contingency fees, whereby he would receive one-third of the settlement if Mr. Tagaloa won the case, but would receive nothing if Mr. Tagaloa lost._\n\n_After 18 months, the case went to trial; the physicians left their practices and sat in the courtroom for 3 weeks. Each physician spent many hours going over records and discussing the case with the lawyers. The family physician, who had nothing to do with the surgery, was so upset with the proceedings that he developed an ulcer. The jury found the family physician innocent and the neurosurgeon guilty of negligence. The family physician lost $8000 in income because of absence from his practice. The neurosurgeon's malpractice insurer paid $900,000 to Mr. Tagaloa, who paid $300,000 to the lawyer._\n\nA number of proposals have been made for malpractice reform (Mello and Gallagher, 2010).\n\n#### **Tort Reform**\n\nMedical malpractice fits into the larger legal field of torts (wrongful acts or injuries done willfully or negligently). The California Medical Injury Compensation Reform Act and the Indiana Medical Malpractice Act are examples of tort reform, placing caps on damages awarded to injured parties and limits on lawyers' contingency fees. Tort reform can help physicians by slowing the growth of malpractice insurance premiums. However, caps on awards can be unfair to patients, limiting payments to those with the worst injuries (Mello et al, 2003; Localio, 2010) (Table 10\u20134).\n\n**Table 10\u20134.** Malpractice reform options\n\n#### **Alternative Dispute Resolution**\n\nThese programs would substitute mediation, arbitration, or private negotiated settlements for jury trials in the case of medical injury. Alternatives to the jury trial could bring more compensation to injured parties by reducing legal costs and might shift the dispute settlement to a more scientific, less emotional theater.\n\n#### **No-Fault Malpractice Reform**\n\nProposals have been made to switch compensation for medical injury from the tort system to a no-fault plan (Studdert and Brennan, 2001; Localio, 2010). Under no-fault malpractice, patients suffering medical injury would receive compensation whether or not the injury was caused by negligence. Without costly lawyers' fees and jury trials, overhead costs would drop from more than 50% to approximately 20%. A no-fault system would compensate far more people and would cost approximately the same as the current tort system (Johnson et al, 1992). In addition, the no-fault approach might allow physicians to be more inclined to identify and openly discuss medical errors for the purpose of correcting them (Studdert et al, 2004).\n\n#### **Enterprise Liability**\n\nA relatively new idea for malpractice reform is to make health care institutions\u2014primarily hospitals and HMOs\u2014responsible for compensating medical injuries (Studdert et al, 2004; Sage and Kersh, 2006; Chan, 2010). As with no-fault proposals, patients suffering medical injury would be compensated whether or not the injury is negligent. Enterprise liability improves upon the no-fault concept by making institutions pay higher insurance premiums if they are the site of more medical injuries (whether caused by system failure or physician error). Hospitals and HMOs would have a financial incentive to improve the quality of care.\n\n### **CONCLUSION**\n\nEach year people in the United States make more than 1 billion visits to physicians' offices and spend more than 100 million days in acute care hospitals. While quality of care provided during most of these encounters is excellent, the goal of the health care system should be to deliver high-quality care every day to every patient. This goal presents an unending challenge to each health caregiver and health care institution. Physicians make hundreds of decisions each day, including which questions to ask in the patient history, which parts of the body to examine in the physical examination, which laboratory tests and x-rays to order and how urgently, which diagnoses to entertain, which treatments to offer, when to have the patient return for follow-up, and whether other physicians need to be consulted. Nurse practitioners, physician assistants, nurses, and other caregivers face similar numbers of decisions. It is humanly impossible to make all of these decisions correctly every day. For health care to be of high quality, mistakes should be minimized, mistakes with serious consequences should be avoided, and systems should be in place that reduce, detect, and correct errors to the greatest extent possible. Even when all decisions are technically accurate, if caregivers are insensitive or fail to provide the patient with a full range of informed choices, quality is impaired.\n\nFor the clinician, each decision that influences quality of care may be simple, but the sum total of all decisions of all caregivers impacting on a patient's illness makes the achievement of high-quality care elusive. To safeguard quality of care, our nation needs laws and regulations, including standards for health care professional education, rules for licensure, boards with the authority to discipline clear violators, and measurement to inform institutions, practitioners, and patients about the quality of their care. Improvement of health care quality cannot solely rely on regulators in Washington, DC, in state capitals, or across town; it must come from within each institution, whether a huge academic center, a community hospital, or a small medical office.\n\n### **REFERENCES**\n\nAbramson J, Starfield B. 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Claims, errors, and compensation payments in medical malpractice litigation. _N Engl J Med._ 2006;354:2024.\n\nUS Senate. Hearings Before the Permanent Subcommittee on Investigations, Committee on Government Operations, March 13 and 14, 1975. Prepaid Health Plans. US Government Printing Office; 1975.\n\nVargas RB et al. Can a chronic care model collaborative reduce heart disease risk in patients with diabetes? _J Gen Intern Med._ 2007;22:215.\n\nWachter RM. Why diagnostic errors don't get any respect\u2014and what can be done about them. _Health Aff (Millwood)_. 2010;29:1605.\n\nWachter RM, Pronovost PJ. The 100,000 Lives Campaign: A scientific and policy review. _Jt Comm J Qual Patient Saf_. 2006;32:621.\n\nWeeks WB et al. Higher health care quality and bigger savings found at large multispecialty medical groups. _Health Aff (Millwood)_. 2009;29:991.\n\nZahn C, Miller MR. Excess length of stay, charges, and mortality attributable to medical injuries during hospitalization. _JAMA._ 2003;290:1868.\n\nZhang Y et al. Geographic variation in the quality of prescribing. _N Engl J Med_. 2010;363:1985.\n\n## **11 Prevention of Illness**\n\n### **WHAT IS PREVENTION?**\n\n_In 2009, the United States spent_ $ _2.5 trillion on health care. Only 3% of this total was dedicated to government public health activities designed to prevent illness._\n\nThe renowned medical historian Henry Sigerist, writing in 1941, listed the main items that must be included in a national health program. The first three items were free education, including health education, for all; the best possible working and living conditions; and the best possible means of rest and recreation. Medical care rated only fourth on his list (Terris, 1992a). For Sigerist (1941), medical care was\n\n_A system of health institutions and medical personnel, available to all, responsible for the people's health, ready and able to advise and help them in the maintenance of health and in its restoration when prevention has broken down. (Sigerist, 1941)_\n\nMany people working in the fields of medical care and public health believe that \"prevention has broken down\" too often; sometimes because modern science has insufficient knowledge to prevent disease, but more often because society has dedicated insufficient resources and commitment to prevention.\n\n_Primary prevention_ seeks to avert the occurrence of a disease or injury (eg, immunization against polio; taxes on the sale of cigarettes to reduce their affordability, and thereby their use). _Secondary prevention_ refers to early detection of a disease process and intervention to reverse or retard the condition from progressing (eg, Pap smears to screen for premalignant and malignant lesions of the cervix, and mammograms for early detection of breast cancer).\n\nThe promotion of good health and the prevention of illness encompass three distinct levels or strategies (Terris, 1986; Table 11\u20131):\n\n**Table 11\u20131.** Strategies of prevention\n\n1. The first and broadest level includes measures to address the fundamental social determinants of illness; as evidence presented in Chapter 3 shows, lower income is associated with higher morbidity and mortality rates. Improvement in the standard of living and social equity (eg, through job creation programs to reduce or eliminate unemployment) may have a greater impact on preventing disease than specific public health programs or medical care services.\n\n2. The second level of prevention involves public health interventions to reduce the incidence of illness in the population as a whole. Examples are water purification systems, the banning of cigarette smoking in the workplace, and public health education on human immunodeficiency virus (HIV) prevention in the schools. These strategies generally consist of primary prevention. The 3% figure cited in the opening paragraph represents these public health activities.\n\n3. The third level of prevention involves individual health care providers performing preventive interventions for individual patients; these activities can be either primary or secondary prevention. The US Preventive Services Task Force and other organizations have established regular schedules for preventive medical care services (US Preventive Services Task Force, 2010).\n\n### **THE FIRST EPIDEMIOLOGIC REVOLUTION**\n\nUntil modern times, the conditions that produced the greatest amount of illness and death in the population were infectious diseases. The initial decline of infectious disease mortality rates took place even before the cause of these illnesses was understood. In the eighteenth and nineteenth centuries, food production increased markedly throughout the Western world. By the early nineteenth century, infectious disease mortality rates were dropping in England, Wales, and Scandinavia, probably as a result of improved nutrition that allowed individuals, particularly children, to resist infectious agents. Thus, the initial success of illness prevention took place through the improvement of overall living conditions rather than from specific public health or medical interventions (McKeown, 1990).\n\nIn the nineteenth century, scientists and public health practitioners discovered many of the agents causing infectious diseases. By comprehending the causes (such as bacteria and viruses) and the risk factors (eg, poverty, overcrowding, poor nutrition, and contaminated water) associated with these illnesses, public health measures (such as water purification, sewage disposal, and pasteurization of milk) were implemented that drastically reduced their incidence. This was the first epidemiologic revolution (Terris, 1985).\n\nFrom 1870 to 1930, the death rate from infectious diseases fell rapidly. Medical interventions, whether immunizations or treatment with antibiotics, were introduced only after much of the decline in infectious disease mortality had taken place. The first effective treatment against tuberculosis, the antibiotic streptomycin, was developed in 1947, but its contribution to the decrease in the tuberculosis death rate since the early nineteenth century has been estimated to be a mere 3%. For whooping cough, measles, scarlet fever, bronchitis, and pneumonia, mortality rates had fallen to similarly low levels before immunization or antibiotic therapy became available. Pasteurization and water purification were probably the main reason for the decline in infant mortality rates (McKeown, 1990).\n\nSome illnesses are exceptions to the rule that infectious disease mortality is influenced more by improved living standards and public health measures than by medical interventions. Immunization for smallpox, polio, and tetanus and antimicrobial therapy for syphilis had a substantial impact on mortality rates from those illnesses. Considering infectious diseases as a group, however, medical measures probably account for less than 5% of the decrease in mortality rates for these conditions over the past century (McKinlay et al, 1989; McKeown, 1990).\n\nAs infectious diseases waned in importance during the first half of the twentieth century and as life expectancy increased, rates of noninfectious chronic illness grew rapidly. Eleven major infectious diseases accounted for 40% of total deaths in the United States in 1900, but less than 10% in 1980. In contrast, heart disease, cancer, and stroke (cerebrovascular disease) caused 16% of total deaths in 1900 but 64% by 1980 (McKinlay et al, 1989).\n\n### **THE SECOND EPIDEMIOLOGIC REVOLUTION**\n\nFifty years ago, epidemiologists did not understand the causes of noninfectious chronic diseases.\n\n_Unable to prevent the occurrence of these diseases, we retreated to a second line of defense, namely, early detection and treatment\u2014so-called secondary prevention. But secondary prevention has\u2014with few exceptions\u2014proved disappointing; it cannot compare in effectiveness with measures for primary prevention. The periodic physical examination, the cancer detection center, multiphasic screening, and ahost of variations on these themes have incurred enormous expenditures for relatively modest benefits... Major exceptions are cancer of the cervix, for which early detection has proved dramatically effective, and, to a lesser extent, cancer of the breast._\n\n_Beginning in 1950, dramatic breakthroughs occurred in the epidemiology of the noninfectious diseases. During the next three decades, our epidemiologists forged powerful weapons to combat most of the major causes of death. In doing so, they initiated a second epidemiologic revolution, which, if we act appropriately, will result in an enormous reduction in premature death and disability. (Terris, 1992b)_\n\nDuring the second epidemiologic revolution, it was learned that the major illnesses in the United States have a few central causes and are in large part preventable. In 2007, 2.4 million people died in the United States (Table 11\u20132). A surprisingly small number of risk factors are implicated in 37% of these deaths. It has been estimated that use of tobacco causes 435,000 fatalities, a high-fat diet and inactivity contributes to 365,000 more, and alcohol is responsible for 85,000 deaths annually in the United States (Heron et al, 2009; Xu et al, 2010). By discovering and educating the population about the risk factors of smoking, rich diet, and lack of exercise, the second epidemiologic revolution has already been very successful. From 1980 to 2006, age-adjusted mortality rates for coronary heart disease (CHD) declined by an astonishing 61%. This decline was associated with reduced rates of tobacco use and lowered mean serum cholesterol levels in the population. As with infectious diseases a century earlier, this decline was in substantial part related to public health interventions regarding smoking and diet (US Department of Health and Human Services, 2009). The unfortunate side of this success story is that those in the poorest socioeconomic position and the least education have considerably higher mortality rates than those with higher socioeconomic status (Loucks, 2009).\n\n**Table 11\u20132.** Causes of death in the United States, 2007 _a_\n\n### **INDIVIDUAL OR POPULATION?**\n\nChronic disease prevention may be viewed from two distinct perspectives: that of the individual and that of the population (Rose, 1985). The medical model seeks to identify high-risk individuals and offer them individual protection, often by counseling on such topics as smoking cessation and low-fat diet. The public health approach seeks to reduce disease in the population as a whole, using such methods as mass education campaigns to counter drinking and driving, the taxation of tobacco to drive up its price, and the labeling of foods to indicate fat and cholesterol content. Both approaches have merits but the medical model suffers from some drawbacks.\n\nThe individual-centered approach of the medical model may produce tunnel vision regarding the causation, and thus the prevention, of disease. Let us take the example of cholesterol.\n\nAncel Keys (1970) performed a famous study comparing CHD in different nations. In east Finland, CHD was common, 20% of diet calories came from saturated fat, and 56% of men aged 40 to 59 years had cholesterol levels greater than 250 mg\/dL. In Japan, CHD was rare, 3% of calories were provided by saturated fat, and only 7% of men aged 40 to 59 years had cholesterol levels above 250 mg\/dL. If we compared two individuals in east Finland who eat the same diet, one with a cholesterol level of 200 mg\/dL and the other with a level of 300 mg\/dL, we might conclude that the variation in cholesterol levels among individuals is caused by genetic or other factors, but not diet. If, on the other hand, we remove our individual blinders and look at entire populations, studying the average cholesterol level and the percentage of fat in the diet in east Finland and in Japan, we will conclude that high-fat diets correlate with high levels of cholesterol and with high rates of CHD.\n\nIndividual variations within each country are often of less importance than variations between one nation and another. The clues to the causes of diseases \"must be sought from differences between populations or from changes within populations over time\" (Rose, 1985).\n\nThe medical model may also target its interventions to the wrong individuals. Let us continue with the cholesterol example. In the United States, most people with high cholesterol levels remain healthy for years, and some people with low levels have heart attacks at an early age. Why is this so? Because the risk of CHD for persons with high cholesterol levels or low cholesterol levels is not so different; even for the low-risk individual, CHD is the most likely cause of death. Everyone in the United States is at risk for this disease. A \"low\" cholesterol level of 180 mg\/dL is low by US standards, but high when compared with levels in poor nations. A large number of people at small risk for a disease may give rise to more cases of the disease than the smaller number of people who are at high risk (Brown et al, 1992). This fact limits the utility of the medical model's \"high-risk\" approach to prevention. A public health approach (eg, mass educational campaigns on the health effects of rich diets and the labeling of foods) strives to reduce the mean population cholesterol level. A 10% reduction in the serum cholesterol distribution of the entire population would do far more to reduce the incidence of heart disease than a 30% reduction in the cholesterol levels of those relatively few individuals with counts greater than 300 mg\/dL.\n\nA coherent ideology underlies the medical model of chronic disease prevention\u2014the concept that in the arena of noninfectious chronic disease, individuals play a major role in causing their own illnesses by such behaviors as smoking, drinking alcohol, and eating high-fat foods. The corollary to this view is that chronic disease mortality rates can be reduced by persuading individuals to change their lifestyles. These statements are true, but they do not tell the whole story.\n\nAn alternative ideology, which fits more closely with the public health approach to chronic disease prevention, argues that modern industrial society, rather than the individuals living in that society, creates the conditions leading to heart disease, cancer, stroke, and other major chronic diseases of the developed world. Tobacco advertising; processed high-fat, high-salt foods in \"supersized\" portions; easy availability of alcoholic beverages; societal stress; an urbanized and suburbanized existence that substitutes automobile travel for exercise; and a markedly unequal distribution of wealth are the substrates upon which the modern epidemic of chronic disease has flourished. Such a worldview leads to an emphasis on societal rather than individual strategies for chronic disease prevention (Fee and Krieger, 1993).\n\nBoth the medical and the public health models (seeing responsibility as both individual and societal) must be joined to further implement the second epidemio-logic revolution; medical caregivers must attempt to change high-risk lifestyles of their individual patients, and society must search for ways to reduce the consumption of tobacco, alcohol, and rich foods. One model that bridges the medical and public health approaches is community-oriented primary care. In this model, primary care clinicians systematically define a target population, determine its health needs, and develop community-based interventions to address these needs (Nutting, 1990). The target population could be as simple as the patients enrolled in a primary care practice, or more ambitiously, an entire neighborhood. For example, a pediatrician might review data on her enrolled patients and find that many children are obese. In addition to counseling individual families in her practice, in the Community Oriented Primary Care model the pediatrician would also work with community members and agencies on broader public health interventions, such as advocating for improved school lunch programs and more time for physical education classes in the local schools, or encouraging the local health department to launch a media campaign promoting consumption of water instead of sweetened beverages.\n\n### **MODELS OF PREVENTION**\n\nTo provide examples of different approaches to preventing illness, we have chosen to discuss two serious health problems in the United States: coronary heart disease and breast cancer.\n\n#### **Coronary Heart Disease**\n\nCoronary heart disease (CHD) is associated with four major risk factors: the eating of a rich diet (the principal cause of the CHD epidemic), elevated levels of serum cholesterol, cigarette smoking, and hypertension (Stamler, 1992a).\n\nPrimary prevention strategies are available for CHD because the causes of the disease are well understood. Primary CHD prevention involves risk factor reduction, including cessation of cigarette smoking, replacement of rich diets by low-fat diets, and control of hypertension. These strategies have been largely responsible for the large decrease in CHD death rates (Figure 11\u20131).\n\n**Figure 11\u20131.** Trends in age-adjusted mortality from coronary heart disease in the United States, 1980\u20132006.\n\n#### **Cigarette Smoking**\n\nTobacco has been called the smallpox virus of chronic disease\u2014a harmful agent whose elimination from the planet would benefit humankind (Fee and Krieger, 1993). Since the 1964 release of the first Surgeon General's Report on the Health Consequences of Smoking, the smoking behavior of the US population has changed dramatically. Between 1965 and 2007, the age-adjusted percentage of adult men who were current smokers dropped from 51% to 22%; for adult women, the decline was from 34% to 18% (Figure 11\u20132). These reductions in smoking prevalence avoided an estimated 3 million deaths between 1964 and 2000\u2014a major public health achievement (Warner, 1989). However, rates of smoking are far higher among people with lower educational levels and smoking continues to be the leading cause of death in the United States (US Department of Health and Human Services, 2009).\n\n**Figure 11\u20132.** Cigarette smoking by persons 18 years and older in the United States in 1965 (light blue bars) and 2007 (dark blue bars). Percentages are age adjusted. (US Department of Health and Human Services. _Health United States_. 2009.)\n\nAntismoking campaigns have been relatively successful for well-educated people, but less so for people with less education, who also tend to be poorer. Between 1974 and 2007, cigarette smoking declined 38% among the least educated persons, while it dropped 67% among the most educated. In 2006, 30% of the least educated persons smoked cigarettes, compared with only 9% of the most educated (US Department of Health and Human Services, 2009).\n\nSince the 1969 ban on radio and television cigarette advertising, the tobacco industry has increased its advertising expenditures dramatically in the print media and through sponsorship of community events. In 2005, tobacco advertising expenditures exceeded $13 billion, almost double the 1999 figure (Bayer et al, 2002; Cokkinides et al, 2009). Tobacco industry documents prove that the principal target group for cigarette advertising is young adults (Ling and Glantz, 2002). The antismoking campaign of the past 30 years has merged the medical and public health models of prevention. Physician counseling can influence smokers to quit. In 2006, however, only 34% of low-income smokers had smoking cessation discussions with their health care provider (Cokkinides et al, 2009) and relapse rates for those who quit after receiving active treatment are 77% at 12 months (Mannino, 2009). Public health measures are more effective, including public education, cigarette taxes, and restriction of smoking in public places. A 10% increase in the price of cigarettes reduces cigarette consumption by 3% to 5%. Yet compared with other developed nations, the United States has relatively low taxes on tobacco (Cokkinides et al, 2009; Schroeder and Warner, 2010).\n\n#### **Rich Diet**\n\nA rich diet is a diet high in fat, saturated fat, cholesterol, salt, and often alcohol, and one with a high caloric intake in relation to the amount of energy expended (Stamler, 1992a). The rich diet produces CHD primarily by causing an increase in low-density-lipoprotein cholesterol. Lowering cholesterol levels has been shown to reduce the risk of heart attacks caused by CHD.\n\nIn the late 1980s, a major national campaign was launched by the National Institutes of Health (NIH) to reduce serum cholesterol levels. This National Cholesterol Education Program is based on the medical model, with health care providers screening individuals for elevated cholesterol and aggressively treating hyperlipidemic patients with diet, cholesterol-lowering medications, or both (Grundy et al, 2004).\n\nPublic health analysts have criticized the NIH strategy as relying too heavily on a medical model of prevention that is expensive and of potentially limited effectiveness. The NIH approach targets more than 100 million people who need dietary changes and recommends drug treatment for many of these individuals.\n\nThe use of statin drugs to treat hyperlipidemia in people with known CHD (secondary prevention) and without CHD (primary prevention) has been shown to reduce deaths from CHD and deaths from all causes (Steinberg and Gotto, 1999). However, the effectiveness of drug treatment is far greater if it is used in secondary rather than primary prevention (Hayward et al, 2010). For primary prevention, 53 patients would have to take a statin drug for 5 years to prevent one patient from experiencing a fatal or nonfatal coronary event. For secondary prevention (patients with known CHD), statin drugs can prevent approximately one nonfatal myocardial infarction or death for every 10 patients treated, at a far lower cost for every year of life saved (Pharoah and Hollingworth, 1996; Lloyd-Jones, 2001).\n\nThe NIH cholesterol reduction strategy highlights the paradox of primary prevention: Prevention within a population of healthy individuals may be better (and less expensively) served by broad public health efforts to reduce risk among the majority of people at moderate risk than by concentrating intensive medical interventions on the smaller number of high-risk persons (Rose, 1985). The traditional orientation of physicians toward individual patients (the medical model) has led the medical profession and the NIH to emphasize identification and treatment of high-risk individuals with elevated cholesterol levels. Pharmaceutical manufacturers also have an interest in promoting a medical model of prevention that relies on prescribing medications. Reducing the mean cholesterol level of the US population rather than reducing the individual cholesterol counts of hyperlipidemic patients may have better long-term results for primary prevention.\n\nCurrently, public health efforts to curb the consumption of rich foods are failing; 74% of adults in the United States were classified as overweight or obese in 2008, compared with 46% in the early 1960s (Ogden and Carroll, 2010). The food industry spends billions of dollars on advertising, a substantial portion of which promotes high-fat fast foods. Proposals have been made to copy the strategy used by tobacco prevention campaigns in reducing the availability of high-fat foods; for example, taxing unhealthy foods, changing school lunch programs to reduce their fat content, restricting food advertising directed at children, and eliminating school-based candy and soft-drink vending machines are primary preventive measures that are gaining public acceptance (Frieden et al, 2010). Growing attention is also being paid to the billions of dollars annually in federal government subsidies to agribusinesses for growing corn, which has contributed to the flooding of the nation with low-cost, high fructose corn sweeteners and other high-calorie processed foods. Public health advocates have called for reforms to the federal farm bill to reduce subsidies for obesogenic foods and to provide more support for sustainable farming of healthful fruits and vegetables (Pollan, 2007; Wallinga, 2010).\n\n#### **Hypertension**\n\nRisk factors for hypertension include high salt intake, low potassium intake, high ratio of dietary sodium to potassium, obesity, and excess alcohol intake; other important risk factors likely exist. Prior to the advent of modern agriculture, intake of sodium was low and intake of potassium high, and high levels of physical exertion prevented persons from being overweight.\n\nCHD risk is associated with increased blood pressure, even at relatively moderate levels of blood pressure elevation. Individuals with systolic blood pressures of 130 to 140 mm Hg have almost twice the cardiovascular risk of those with systolic blood pressures less than 110 mm Hg. One quarter of hypertension-related cardiovascular deaths take place among borderline hypertensives, and in the United States, 90% of men aged 35 to 57 years have blood pressure levels that create excess cardiovascular risk. Thus, it can be said that high blood pressure as a risk factor for CHD is a problem for the entire population and not simply a problem for the 20% to 25% of the population with frank hypertension. Similarly to the cholesterol situation, the greatest impact in reducing hypertension-related CHD mortality rates will come from a reduction in the blood pressure of the large number of borderline hypertensives rather than from focusing solely on people with very high blood pressure (Stamler, 1992b).\n\nPrimary prevention of high blood pressure can be accomplished by a reduction in the daily intake of salt by 3 g per person. Currently, the average man in the United States consumes 10.4 g of salt per day, with women eating 7.3 g. Such a change would reduce the number of new CHD cases by 60,000 per year. This public health approach would be as effective as the use of medical treatment to control the blood pressures of the 65 million people in the United States with hyper-tension (Bibbins-Domingo et al, 2010).\n\nPrevention of hypertension has focused on screening and early treatment of elevated blood pressure. These measures are considered secondary prevention (early diagnosis and intervention) with respect to high blood pressure as a disease but are categorized as primary prevention (averting the occurrence) with respect to CHD. American medicine has a poor record in lowering elevated blood pressures; only 50% of hypertensives are adequately controlled (Egan et al, 2010)\n\n#### **Breast Cancer**\n\nWhereas mortality rates for cardiovascular disease declined since the late 1960s, cancer mortality rates continued to increase through 1990. Between 1990 and 2006, cancer mortality rates dropped by 16%, probably as a result of reductions in cigarette smoking. Breast cancer mortality rates have also decreased during those years, but are considerably higher for African American women than for white women (US Department of Health and Human Services, 2009).\n\nThe designing of effective primary prevention for a disease generally depends on an understanding of the epidemiology of that disease. In the case of lung cancer, the discovery of the link with cigarette smoking allowed a widespread primary prevention program to be developed\u2014the antismoking campaign. But the causes of many cancers are still unclear, meaning that preventive strategies must use secondary rather than primary prevention. Pap smears for early detection of cervical cancer, fecal occult blood testing and colonoscopy for early detection of colorectal cancer, and mammography for early detection of breast cancer are examples of secondary prevention.\n\nMultiple risk factors for breast cancer have been uncovered, including age greater than 65 years, a family history of breast cancer, atypical hyperplasia on breast biopsy, birth in North America or northern Europe, and genetic susceptibility related to the BRCA geno-type. Women with more years of ovulatory menstrual cycles have a greater risk, indicating a hormonal influence on the disease (American Cancer Society, 2011).\n\nHowever, only one-fourth of breast cancer cases can be accounted for by these risk factors. The differences between high and low age-adjusted breast cancer risk in the United States are small compared with the differences between such high-incidence nations as the United States and low-incidence (generally underdeveloped) nations. Perhaps unknown agents related to modern industrialization are the primary causes of breast cancer, while such influences as female hormones are secondary promoters of the disease.\n\nThe age-adjusted incidence (new cases) of breast cancer fell sharply in 2003 compared with 2002 and continued to fall slightly through 2006, a phenomenon temporally related to the drop in the use of hormone replacement therapy by women in the United States, occasioned by the widely publicized report from the Women's Health Initiative providing new data on the risks of hormone replacement therapy (Ravdin et al, 2007). This association suggests that estrogen is an important cause or facilitator of breast cancer.\n\nEvidence linking dietary fat to cancer of the breast is inconsistent and weak, and further research is needed on the role of environmental carcinogens (American Cancer Society, 2011). From the 1940s to the 1980s, industrial production of synthetic organic chemicals rose from 1 billion to 400 billion pounds annually, and the volume of hazardous wastes also increased 400-fold during that period (Epstein, 1990, 1994). One study estimated that toxic chemicals encountered at work-places are responsible for 20% of all human cancers (Landrigan, 1992). Estrogens have been used as additives to poultry and cattle feed, and pesticide residues contain estrogen-like compounds that may contribute to breast cancer causation (Davis and Bradlow, 1995). Some studies have linked breast cancer risk to organo-chlorine insecticides, polycyclic aromatic hydrocarbons, and organic solvents, but research on these environmental causes of breast cancer has been inadequate and inconsistent (Brody and Rudel, 2003).\n\nLack of knowledge has forced modern medicine to retreat to secondary prevention (ie, early diagnosis through breast examinations and mammography) to reduce mortality rates in women with the disease. Thankfully, breast cancer, like cervical cancer, lends itself to secondary prevention techniques. Periodic mammograms can reduce breast cancer mortality rates in women aged above 50 years. Yet many breast cancer activists decry the relatively paltry sums going for basic epidemiologic research to determine the causes of breast cancer.\n\n#### **Summary**\n\nThe examples of CHD and breast cancer illustrate different aspects of illness prevention. Primary prevention has been successful in reducing mortality rates for CHD. Both public health and medical approaches have been used, with far greater emphasis given to the latter strategy. Secondary prevention has had some success in reducing breast cancer mortality rates, but the incidence of the disease remains high and primary prevention is badly needed.\n\n### **DOES PREVENTION REDUCE MEDICAL CARE COSTS?**\n\nThe influence of prevention on medical care costs is a complex one. As a rule, primary prevention using public health measures is far more cost-effective than primary prevention through medical care; public health measures do not require many millions of expensive one-to-one interactions with medical care providers.\n\nIn the arena of individual medical care prevention, some measures save money and some do not. Every dollar invested in measles, mumps, and rubella immunizations saves many more dollars in averted medical care costs. Physician counseling on smoking cessation is a low-cost activity that can reduce the multibillion dollar cost of caring for people with tobacco-related illness. These preventive care activities do reduce health care spending in the long run. In contrast, medical care to reduce cholesterol and high blood pressure are unlikely to result in significant savings to the health care system (Cohen et al, 2008).\n\nPrimary prevention through public health action can be enormously effective in reducing the burden of human suffering and the cost of treating disease. From 1900 to 1940, the nation's public health efforts achieved a 97% reduction in the death rate for typhoid fever; 97% for diphtheria; 92% for infectious diarrhea; 91% for measles, scarlet fever, and whooping cough; and 77% for tuberculosis (Winslow, 1944). The imposition of a $2-per-pack increase in the tobacco tax could substantially reduce the $50-plus billion annual cost of tobacco-related disease, while at the same time yielding tens of billions of dollars per year in tax revenues\u2014an ideal preventive measure that actually earns money. If the three primary preventive methods known to reduce the incidence of coronary heart disease, cancer, and stroke (ie, reduction in smoking, cholesterol levels, and blood pressure) were intensified, the medical care costs of these illnesses could be reduced by 50%. These three illnesses account for 20% of personal health care costs in the United States and reducing their incidence could yield a cost savings of billions of dollars per year. However, these savings are overstated because money saved by preventing disease X will ultimately be spent on the treatment of disease Y or Z, which will strike those people spared from disease X.\n\n### **CONCLUSION**\n\nThe goals of disease prevention are to delay disability and death and to maximize illness-free years of life. Improvements in living standards, public health measures, and preventive medical care have made enormous contributions toward the achievement of these goals. Producing further improvements in the overall health of society will likely depend on reducing the growing gap between the rich and the poor and shifting a greater proportion of the health dollar to disease prevention.\n\n### **REFERENCES**\n\nAmerican Cancer Society. What are the risk factors for breast cancer, 2011. www.cancer.org\/cancer\/breastcancer\/detailedguide\/breast-cancer-risk-factors. Accessed November 15, 2011.\n\nBayer R et al. Tobacco advertising in the United States. _JAMA_. 2002;287:2990.\n\nBibbins-Domingo K et al. Projected effect of dietary salt reductions on future cardiovascular disease. _N Engl J Med._ 2010;362:590.\n\nBrody JG, Rudel RA. Environmental pollutants and breast cancer. _Environ Health Perspect_. 2003;111:1007.\n\nBrown EY, Viscoli CM, Horwitz RI. Preventive health strategies and the policy makers? paradox. _Ann Intern Med_. 1992;116:593.\n\nCohen JT et al. Does preventive care save money? _N Engl J Med_. 2008;358:661.\n\nCokkinides V et al. Tobacco control in the United States: recent progress and opportunities. _CA Cancer J Clin_. 2009;59:352.\n\nDavis DL, Bradlow HL. Can environmental estrogens cause breast cancer? _Sci Am_. 1995;273:167.\n\nEgan BM et al. US trends in prevalence, awareness, treatment, and control of hypertension, 1988\u20132008. _JAMA_. 2010;303:2043.\n\nEpstein SS. Losing the war against cancer: Who's to blame and what to do about it. _Int J Health Serv_. 1990;20:53.\n\nEpstein SS. Environmental and occupational pollutants are avoidable causes of breast cancer. _Int J Health Serv_. 1994;24:145.\n\nFee E, Krieger N. Thinking and rethinking AIDS: Implications for health policy. _Int J Health Serv_. 1993;23:323.\n\nFrieden TR et al. Reducing childhood obesity through policy change. _Health Aff (Millwood)_. 2010;29:357.\n\nGrundy SM et al. Implications of recent clinical trials for the National Cholesterol Education Program Adult Treatment Panel III guidelines. _Circulation_. 2004;110:227.\n\nHayward RA et al. Optimizing statin treatment for primary prevention of coronary artery disease. _Ann Intern Med_. 2010;152:69.\n\nHeron MP et al. Deaths: Final data for 2006. _Natl Vital Stat Rep_. 2009;57(14):1\u2013134.\n\nKeys A. Coronary heart disease in seven countries. _Circulation_. 1970;41(suppl 1):11.\n\nLandrigan PJ. Commentary: Environmental disease: A preventable epidemic. _Am J Public Health_. 1992;82:941.\n\nLing PM, Glantz SA. Using tobacco industry marketing research to design more effective tobacco control campaigns. _JAMA_. 2002;287:2983.\n\nLloyd-Jones DM et al. Applicability of cholesterol-lowering primary prevention trials to a general population. _Arch Intern Med_. 2001;161:949.\n\nLoucks EB et al. Life-course socioeconomic position and incidence of coronary heart disease. _Am J Epidemiol_ 2009;169:829.\n\nMannino DM. Why won't our patients stop smoking? _Diabetes Care_. 2009;32(suppl 2):S426.\n\nMcKeown T. Determinants of health. In: Lee PR, Estes CL, eds. _The Nation's Health_. Boston, MA: Jones & Bartlett; 1990.\n\nMcKinlay JB et al. A review of the evidence concerning the impact of medical measures on recent mortality and morbidity in the United States. _Int J Health Serv_. 1989;19:181.\n\nNutting PA, ed. _Community Oriented Primary Care: From Principle to Practice_. Albuquerque, NM: University New Mexico Press; 1990.\n\nOgden CL, Carroll MD. Prevalence of overweight, obesity, and extreme obesity among adults: United States, trends 1976\u20131980 through 2007\u20132008. June 2010. www.cdc.gov.\n\nPharoah PD, Hollingworth W. Cost effectiveness of lowering cholesterol concentration with statins in patients with and without pre-existing coronary heart disease. _Br Med J_. 1996;312:1443.\n\nPollan M. You are what you grow. _N Y Times Mag_. April 22, 2007.\n\nRavdin PM et al. The decrease in breast-cancer incidence in 2003 in the United States. _N Engl J Med_. 2007;356:1670.\n\nRose G. Sick individuals and sick populations. _Int J Epidemiol_. 1985;14:32.\n\nSchroeder SA, Warner KE. Don't forget tobacco. _N Engl J Med_. 2010;363;201.\n\nSigerist HE. _Medicine and Human Welfare._ New Haven, CT: Yale University Press; 1941.\n\nStamler J. Established major coronary risk factors. In: Marmot M, Elliott P, eds. _Coronary Heart Disease Epidemiology_. New York: Oxford University Press; 1992a.\n\nStamler R. The primary prevention of hypertension and the population blood pressure problem. In: Marmot M, Elliott P, eds. _Coronary Heart Disease Epidemiology._ New York: Oxford University Press; 1992b.\n\nSteinberg D, Gotto AM. Preventing coronary artery disease by lowering cholesterol levels. _JAMA_. 1999;282:2043.\n\nTerris M. The changing relationships of epidemiology and society: The Robert Cruikshank Lecture. _J Public Health Policy_. 1985;6:15.\n\nTerris M. What is health promotion? _J Public Health Policy_. 1986;7:147.\n\nTerris M. Concepts of health promotion: Dualities in public health theory. _J Public Health Policy_. 1992a;13:267.\n\nTerris M. Healthy lifestyles: The perspective of epidemiology. _J Public Health Policy_. 1992b;13:186.\n\nUS Department of Health and Human Services. Health United States. 2009. www.cdc.gov.\n\nUS Preventive Services Task Force. _Guide to Clinical Preventive Services_. 2010\u20132011. August 2010. www.ahrq.gov.\n\nWallinga D. Agricultural policy and childhood obesity. _Health Aff (Millwood)._ 2010;29:405.\n\nWarner KE. Smoking and health: A 25-year perspective. _Am J Public Health_. 1989;79:141.\n\nWinslow CEA. Who killed Cock Robin? _Am J Public Health_. 1944;34:658.\n\nXu J et al. Deaths: final data for 2007. _Natl Vital Stat Rep_. 2010;59(19):1\u2013135.\n\n## **12 Long-Term Care**\n\n_Eddie Taylor awoke one morning at his home in California unable to speak or to move the right side of his body, but able to understand other people around him. After 3 terrifying days in a hospital and 3 frustrating weeks in a stroke rehabilitation center, Mr. Taylor failed to improve. Because he no longer required hospital-level care, he became ineligible for Medicare hospital coverage. Since Mrs. Taylor was wheelchair-bound with crippling rheumatoid arthritis and unable to care for him, he was transferred to a nursing home. Medicare did not cover the $220 per day cost. After 2 years, Medicaid began to pick up the nursing home bills. Much of the family's life savings\u2014earned during the 50 years Mr. Taylor worked in a men's clothing store\u2014had been spent to allow Medicaid eligibility. Because Medicaid paid only $140 per day, few recreational activities were offered, and Mr. Taylor spent each day lying in bed next to a demented patient, who screamed for hours at a time. Unable to voice his complaints at the inhuman conditions of his life, he became severely depressed, stopped eating, and within 3 months was dead._\n\n_On high school graduation night, Lyle celebrated with a few drinks and drove to his girlfriend's house. He lost control of the car, hit a tree, and suffered a fractured cervical spine, unable to move his arms or legs. After 9 months in a rehabilitation unit, Lyle remained quadriplegic. He returned home, with a home care agency providing total 24-hour-a-day care at a cost of $300 per day, not covered by insurance. Lyle's father, a businessman, became increasingly angry at his wife, the principal flutist in the city's professional orchestra, because she refused to leave the orchestra to care for Lyle. After 1 year and $110,000 in long-term care expenses, Lyle's parents were close to divorce. One night Lyle's father awoke in a cold sweat; in his dream, he had placed a plastic bag over Lyle's head and suffocated him._\n\nTime and again physicians and other caregivers witness the tragedy of chronic illness compounded by the failure of the nation's health care system to meet the social needs created by the illness. The crisis of long-term care is twofold: Thousands of families each year lose their savings to pay for the chronic illness of a family member, and long-term care often takes place in dehumanizing institutions that rob their occupants of their last remaining vestiges of independence.\n\nLong-term care includes those health, social, housing, transportation, and other supportive services needed by persons with physical, mental, or cognitive limitations sufficient to compromise independent living. The need for long-term care services is usually determined by evaluating a person's impairment of activities of daily living (ADLs; eg, eating, dressing, bathing, toileting, and getting in or out of bed or a chair) and in instrumental activities of daily living (IADLs; eg, laundry, housework, meal preparation, grocery shopping, transportation, financial management, taking medications, and telephoning) (Table 12\u20131). Twelve million people in the United States require assistance with one or more ADLs or IADLs, and can therefore be considered as needing long-term care services (Kaye et al, 2010).\n\n**Table 12\u20131.** Activities requiring assistance in long-term care\n\nProjections of growth for the elderly population in the United States are startling. In 2000, the population 65 years of age and older numbered 35 million; this figure is expected to reach 72 million by the year 2030. The number of people 85 years and older will more than double from 4.2 million in 2000 to 8.7 million in 2030. Those 80 years and older are most likely to need long-term care because 56% have severe disability (Administration on Aging, 2010). As more and more people need long-term care, the answers to two questions become increasingly urgent: How shall the nation finance long-term care? Should most long-term care be delivered through institutions or in people's homes and communities?\n\n### **WHO PAYS FOR LONG-TERM CARE?**\n\n_Phoebe McKinnon was in good health until she fell, broke her hip, and suffered a postoperative joint infection. She was placed on complete bed rest with oral antibiotics for 3 months, after which time she would have another surgery. Widowed, Ms. McKinnon lived alone; her only daughter lived 1500 miles away. Because Ms. McKinnon required 24-hour-a-day help, the social worker, after carefully researching the financial options, reluctantly suggested that Ms. McKinnon spend the 3 months in a nursing home. Ms. McKinnon and her daughter agreed but were shocked when the social worker explained that the cost would be $220 a day, for a total bill of $19,800._\n\nThe United States spent $205 billion on long-term care in 2009, including $137 billion on nursing home care (Martin et al, 2011). In 2006, a 1-year nursing home stay cost an average of $76,000.\n\nIn 2009, direct out-of-pocket payments by patients and their families financed 22% of long-term care services in the United States. A common scenario is that of Eddie Taylor: After a portion of their life savings are spent for long-term care, families finally become eligible for Medicaid long-term care coverage. Medicaid pays for 34% of US long-term care expenditures (Table 12\u20132). Many people expect the Medicare program to pay for nursing home stays, and like Phoebe McKinnon and her daughter, are surprised and shocked when they find that Medicare will not assist them. Only 28% of long-term care costs are financed by Medicare (Martin et al, 2011).\n\n**Table 12\u20132.** Long-term care financing, 2009 _a_\n\nAverage out-of-pocket expenses for health care paid by the Medicare beneficiaries amounted to 20% of family income in 2005. One-fifth of these expenses went to nursing homes (Kaiser Family Foundation, 2009).\n\nWhat are the precise roles of Medicare, Medicaid, and private insurance in the financing of long-term care services?\n\n#### **Medicare Long-Term Coverage**\n\n_Glenn Whitehorse, who was a diabetic, developed gangrene of his right leg requiring above-the-knee amputation. He was transferred from the acute care hospital to the hospital's skilled nursing facility, where he received physical therapy services. Because he was generally frail, he was unable to move from bed to chair without assistance. Mr. Whitehorse's physical and occupational therapists felt he might do better at home, where he could receive home physical therapy and nursing care. All these services were covered by Medicare._\n\n_Mrs. Whitehorse had Parkinson's disease and was unable to assist her husband in bathing, getting out of bed, and going to the bathroom; she was forced to hire someone to assist with these custodial functions, which were not covered by Medicare. When Mr. Whitehorse no longer showed any potential for improvement, Medicare discontinued coverage of his home health services. The situation became too difficult, and he was placed in a nursing home for custodial care. Medicare did not cover the nursing home costs._\n\nWhich services provided in a nursing facility or at home are covered by Medicare? The key distinction is between \"skilled care,\" for which Medicare pays, and \"custodial care,\" which is usually not covered. A related issue is that of postacute versus chronic care. Medicare usually covers services needed for a few weeks or months after an acute hospitalization but often does not pay for care required by a stable chronic condition.\n\nWhat are some examples of skilled care versus custodial services? Registered nurses in a hospital nursing facility, nursing home, or home care agency provide a wide variety of services, such as changing the dressing on a wound, taking blood pressures, listening to the heart and lungs to detect heart failure or pneumonia, reviewing patient compliance with medications, and providing patient education about diabetes, hypertension, heart failure, and other illnesses. Physical and occupational therapists work with stroke, hip fracture, and other patients to help them reach their maximum potential level of functioning. Speech therapists perform the difficult task of teaching stroke patients with speech deficits how to communicate. These are all skilled services, usually covered by Medicare.\n\nCustodial services involve assistance with ADLs and IADLs rather than treatment or rehabilitative care related to a disease process; these are tasks such as cooking, cleaning house, shopping, or helping a patient to the toilet. These services, usually provided by nurses' aides, home health aides, homemakers, or family members, are considered unskilled and are often not covered by Medicare.\n\n#### **Medicaid Long-Term Coverage**\n\n_Willie Robinson, who lived alone, suffered from deforming degenerative arthritis and was unable to do anything more active than sitting in a chair. Because Mr. Robinson had no skilled care medical needs, Medicare would not provide any assistance. Medicaid and the county welfare agency paid for a homemaker to provide 20 hours of help per week, but that was not sufficient. Mr. Robinson had no choice but to enter a nursing home, because that was the only way he could obtain 24-hour-a-day help paid for by Medicaid._\n\nMedicaid differs from Medicare in paying the costs of nursing home care. For home health care, however, Medicaid generally does not cover 24-hour-a-day custodial services for people unable to care for themselves. The completeness of Medicaid's nursing home coverage, in contrast to the limited nature of Medicaid-financed home health care, forces many low-income disabled people like Willie Robinson to go into nursing homes unless they have families capable of providing 24-hour-a-day custodial care. In order to qualify for Medicaid nursing home coverage, families may be forced to spend their savings down to low levels, although in some states, Medicaid allows spouses of nursing home residents to keep some of their assets.\n\nMedicaid's coverage of home health services has increased as a result of home- and community-based care 1915(c) waivers, initially authorized in 1981 (Ng et al, 2010). This program, which attempts to prevent nursing home admissions, allows Medicaid recipients to receive more home care services than previously. Whereas Oregon allocated 71% of its long-term care Medicaid dollars to this program in 2005, nationally Medicaid spent 35% of long-term care dollars for home and community-based care (Kaiser Family Foundation, 2006a).\n\n#### **Private Long-Term Care Insurance**\n\n_Sue and Lew MacPherson, both age 72, were worried about their future. They remembered their cousin, who was turned down for private long-term care insurance because of his high blood pressure and later spent his entire savings on nursing home bills. Hoping to protect their $32,000 in savings, they decided to apply for long-term care insurance before an illness would make them uninsurable. Their insurance agent calculated the cost of two policies at $6000 per year, or 30% of their $20,000 per year income. At that price, Sue and Lew would spend most of their savings on insurance premiums within a few years. They declined the insurance._\n\nPrivate insurance plays a minor role in long-term care financing, with only 8% of long-term care costs covered by private policies (Table 12\u20132). Experience rating (see Chapter 2) has had a profound effect on the dynamics of private long-term care insurance. The largest market for this type of insurance is the elderly population. Under experience-rated insurance, the elderly are charged high premiums because they are at considerable risk of requiring long-term care services. The 2009 median income of people over 65 is $31,000 (US Census Bureau, 2010). Only 17% of households with the head of the household aged 70 to 74 years could afford the average long-term care insurance policy (Kaiser Family Foundation, 2006b). The major attractive market for long-term care insurers is the younger employed population, but only a tiny fraction of this group is interested in long-term care insurance because the prospects of needing such care are so remote.\n\nPeople purchasing long-term care insurance may find it to be a poor investment. Some private policies specify that a policyholder must be dependent in three or more ADLs before receiving benefits for home health services. Yet many people with fewer than three ADL impairments need long-term care services; for these people, their insurance may pay nothing.\n\nLong-term care policies usually have a large deductible (measured in nursing home days) for nursing home care, and most policies pay a fixed daily fee rather than reimbursing actual charges. A typical policy might provide $150 per day after a 90-day deductible. The 2006 average daily nursing home charge was $220, meaning that $70 would be the patient's responsibility. Thus a year's stay would require out-of-pocket expenditures totaling over and above payment of the insurance premium. Most policies limit their coverage to a few years, which places a cap on how much the insurance will pay.\n\n### **WHO PROVIDES LONG-TERM CARE?**\n\n#### **Informal Caregivers**\n\n_Since her husband died, Mrs. Dora Whitney has lived alone. At age 71, she became forgetful and one day left the gas stove on, causing a fire in the kitchen. Two months later, she was unable to find her way home after going to the store and was found by the police wandering in the streets. Her daughter, Kimberly, brought her to the university hospital, where she was diagnosed with Alzheimer's dementia. After a team conference with her mother's physicians, nurses, occupational therapist, and social worker, Kimberly admitted that her only option was to abandon her career as a teacher to care for her mother. Kimberly refused to place her mother in a nursing home, and funds were not available to hire the needed 24-hour-a-day help._\n\nMost people needing long-term care services receive them from their family and friends. In 2007, about 52 million people served as unpaid family caregivers, of whom the majority were women. For men, their wives often provide long-term care, and for women, their daughters are frequently caregivers. A growing number of the elderly do not have family living near enough to them to provide informal care; the absence of an informal caregiver is a common reason for nursing home placement. On average, informal caregivers provide 20 hours per week of care, and the estimated economic value of their unpaid contributions was approximately $375 billion in 2007. Thirty-seven percent of informal caregivers to persons age 50 and older reported quitting their job or reducing their work hours in order to assist their family members. Elderly people with care-givers have shorter hospital stays, fewer readmissions, and lower inpatient expenses, demonstrating that unpaid caregivers create a great deal of value for the health care system (Levine, 1999; AARP, 2008).\n\n#### **Community-Based and Home Health Services**\n\n_Ana Dominguez insisted that her daughter Juana accept the Yale scholarship. Though at age 49 Ms. Dominguez was bed- and wheelchair-bound with multiple sclerosis, she would feel too guilty if Juana remained in San Antonio, TX, just to care for her. Before Juana left, she arranged with the home care agency to have her mother transported to an adult day health center 3 days a week; for nursing, physical, and occupational therapy 3 times a week; and for meals-on-wheels. Medicare paid for these services. But Ms. Dominguez needed someone at home 24 hours a day, a service not covered by Medicare. For $15 a day, Juana was able to hire Vilma, an undocumented teenager from El Salvador, to live at home. Adding Vilma's pay and the cost of her food, Juana figured they would spend $35,000 of their $42,000 in savings by the time she graduated from Yale._\n\nCommunity-based long-term care is delivered through a variety of programs, such as home care, adult day care, assisted living settings, home-delivered meals, board and care homes, hospice care for the terminally ill, mental health programs, and others. During the 1970s, the independent living movement among disabled people created a strong push away from institutional (hospital and nursing home) care toward community-based and home care that fostered the greatest possible independence. During the 1980s, AIDS activists furthered the development of hospice programs that provide intensive home care services for people with terminal cancer and AIDS. The home is usually a more therapeutic, comforting environment than the hospital or nursing home.\n\nAs a product of the intersection of the popular movement toward home care and the DRG-created incentive to reduce Medicare hospital stays, home health services expanded rapidly from 1980 to 1997. This, in turn, prompted changes in Medicare payment policies to rein in home care expenditures. After concerns were voiced about excessive cuts in Medicare home care payments, in 2000 Medicare instituted a prospective payment system for home care based on the episode-of-illness model (see Chapter 4). Home care agencies are paid a lump sum (which, like DRG hospital payments varies with the severity of the illness) for 60 days of care.\n\nMany categories of health caregivers function in teams to perform home care, including nurses, physical, occupational, speech, and respiratory therapists, social workers, home health aides, case managers, and drivers delivering meals-on-wheels. Yet home care, designed to help fill the once low-tech niche in the health care system that assists the disabled with ADLs and IADLs, has become increasingly specialized. Home care agencies now offer intravenous antibiotic infusions, morphine pumps, indwelling central venous lines, and home renal dialysis, administered by highly skilled intravenous and wound care nurses, respiratory therapists, and other health care professionals. These developments are a major advance in shifting medical care from hospital to home, but they have not been matched by growth in the paid personal custodial care needed to allow disabled people to remain safely in their homes. Similarly, hospice care, while providing excellent nursing services for patients with terminal illnesses, is limited in the ADL support it provides. Hospice programs may not accept terminal patients without an informal caregiver at home; thus, the people who may need home hospice services the most cannot receive them.\n\nAssisted living, which provides housing with a graded intensity of services depending on the functional capabilities of its residents, has been growing rapidly. However, the average annual cost in 2009 was $34,000, most of which comes from out-of-pocket payments, thereby pricing assisting living out of the reach of low- and moderate-income families (Stevenson and Grabowski, 2010).\n\n#### **Nursing Homes**\n\n_Each morning, more than one and a quarter million Americans awaken in nursing homes. Most of them are very old and very feeble. Most will stay in the nursing home for a long time. For most, it will be the last place they ever live.... [Nursing home] residents live out the last of their days in an enclosedsociety without privacy, dignity, or pleasure, subsisting on minimally palatable diets, multiple sedatives, and large doses of television\u2014eventually dying, one suspects at least partially of boredom. (Vladeck, 1980)_\n\nOften, informal help and formal home health services are unable to provide the care required for severely disabled people. Such people may be placed in nursing homes with 24-hour-a-day care provided by health aides and orderlies under the supervision of nurses. In 2007, 1.8 million people resided in US nursing homes. Sixty-seven percent of nursing home residents are women, who more often outlive their spouses (Kaye et al, 2010). Frequently, after caring for a sick husband at home, women will themselves fall ill and be placed in a nursing home because no one is left to care for them at home. People who reach the age of 65 have a 40% chance of entering a nursing home at some time in their life (www.medicare.gov\/longtermcare\/static\/home.asp).\n\nSeventy-six percent of nursing home residents have cognitive impairment and 93% have restricted mobility (Kaye et al, 2010). There are two main differences between the chronically ill inside and outside nursing homes: Nursing home residents have no family able to care for them, and a far larger proportion of nursing home patients suffer from dementia, a condition whose care is extremely difficult to provide at home by family members.\n\nNursing homes vary widely in quality. The Omnibus Budget Reconciliation Act of 1987 set standards for nursing home quality and mandated surveys to enforce these standards. Serious quality problems persist; the average number of deficiencies per facility increased from 5.7 in 1999 to 7.1 in 2005. Only 9% of nursing homes had no deficiencies in 2005. The most frequently cited deficiencies in 2005 were inadequate food sanitation, quality of care, professional standards, accident prevention, housekeeping, comprehensive care plans, infection control, pressure sores, and dignity (Harrington et al, 2006; Werner and Konetzka, 2010). Compared with non-Hispanic whites, Hispanics requiring nursing home are more likely to be placed in low-quality facilities (Fennell et al, 2010).\n\nLower-income people are housed in close quarters with several other patients and become totally dependent on an underpaid, inadequately trained staff. Hour after hour may be spent lying in bed or sitting in a chair in front of a TV. While quality of life varies between one nursing home and another, placement in a nursing home almost always thwarts the human yearning for some degree of independence of action and for companionship. A sense of futility overwhelms many nursing home residents, and the desire to live often wanes (Vladeck, 1980).\n\nTo keep down costs, most care in nursing homes is provided by nurse's aides, who are paid very little, receive minimal training, are inadequately supervised, and are required to care for more residents than they can properly serve. The job of the nursing home aide is very difficult, involving bathing, feeding, walking residents, cleaning them when they are incontinent, lifting them, and hearing their complaints. In 2005, 66% of all nursing homes were under for-profit ownership, many operated by large corporate chains (Harrington et al, 2006). For-profit ownership has been associated with lower staffing levels and poorer quality of care compared with nonprofit ownership (Comondore, 2009).\n\nOffering a humane existence to severely disabled people housed together in close quarters is a nearly impossible task. One view of nursing home reform holds that only the abolition of most nursing homes and the development of adequately financed home and community-based care can solve the nursing home problem.\n\n### **IMPROVING LONG-TERM CARE**\n\n#### **Financing Long-Term Care**\n\n_Boomer was mad. As a self-employed person, his family's health insurance coverage was costing $600 each month, in addition to his out-of-pocket dental bills. To make matters worse, a big chunk of his social security payments went to Medicare each year, not to mention federal and state income taxes and sales taxes going to finance Medicare and Medicaid, so that other people could get health care. While spending all this money, Boomer was healthy and had not seen a physician for 6 years._\n\n_One day Boomer's father, Abraham, suffered a devastating stroke. After weeks in the hospital, largely paid for by Medicare, Abraham was transferred to a nursing home. Because Medicare does not cover most long-term care, Boomer's mother paid the bills out of her savings until most of the money ran out. Abraham then became eligible for Medicaid,which took care of the nursing home bills. After Abraham's illness, Boomer stopped complaining about his social security and tax payments going to medical care. Even though Boomer was paying more than he was receiving, Abraham was receiving far more than he was paying. Boomer was grateful for the care his father received and figured that he might be in Abraham's shoes some day._\n\nIn the early 1960s, it was recognized that private insurance was unable to solve the problem of health care financing for people older than 65. The costs of health care for the elderly were too great, making experience-rated health insurance premiums unaffordable for most elderly people. Accordingly, Medicare, a social insurance program, was passed (see Chapter 2). An identical problem confronts long-term care financing: As shown earlier in this chapter, most people who might wish to purchase long-term care insurance are unable to afford an adequate policy. Table 12\u20133 lists some proposals for improving long-term care.\n\n**Table 12\u20133.** Proposals for improving long-term care\n\nThe Pepper Commission (1990) recommended that the nation institute a social insurance program to finance long-term care. This program, like Medicare Part A, could be financed by an increase in the rate of social security contributions by employers and employees. It would pay for caregivers to provide those services not currently covered by Medicare, especially in-home help in feeding, dressing, bathing, toileting, housework, grocery shopping, transportation, and other assistance with ADLs and IADLs. A similar proposal was offered by Physicians for a National Health Program (Harrington et al, 1991).\n\n#### **Providing Long-Term Care**\n\n_Mei Soon Wang was desperate to go home. Since a brain tumor had paralyzed her left side and left her incontinent, she had been confined to a nursing home because she had no family in San Francisco to care for her. Her daughter, visiting from Portland, heard of On Lok Senior Health Services, which cared for the frail elderly in their homes. On Lok accepted Ms. Wang, placed her in adult day care, arranged for meals to be delivered to her home, and paid for part-time help on evenings and weekends._\n\nBecause a reasonable quality of life and personal independence, within the confines of a patient's illness, are so difficult to achieve in the nursing home environment, long-term care reformers often advocate that most long-term care be provided at home. The first step toward deinstitutionalizing long-term care is a financing mechanism that pays for more comprehensive community-based and home long-term care services.\n\nThe ideal long-term caregivers are the patient's family and friends; thus, it can be argued that long-term care reform should support, assist, and pay informal caregivers, not replace them. Teams of nurses, physical and occupational therapists, physicians (who often know the least about long-term care), social workers, and attendants can train and work with informal caregivers, and personnel can be available to provide respite care so informal caregivers can have some relief from the 24-hours-a-day, 7-days-a-week burden. If informal caregivers are not available, all possible efforts can still be made to deliver long-term care in people's homes rather than in nursing homes (Harrington et al, 1991).\n\nAn innovative long-term care program that has achieved great success is the On Lok program in San Francisco. Translated from Chinese, On Lok means peaceful, happy abode. Begun in 1971 in San Francis-co's Chinatown, On Lok merges adult day services, in-home care, home-delivered meals, housing assistance, comprehensive medical care, respite care for caregivers, hospital care, and skilled nursing care into one program. Persons eligible for On Lok have chronic illness sufficiently severe to qualify them for nursing home placement, but only 15% ever spend time in a nursing home. Services for each participant are organized by a multidisciplinary team, including physicians, nurses, social workers, rehabilitation and recreation therapists, and nutritionists.\n\nIn 1983, On Lok became the first organization in the United States to assume full financial risk for the care of a frail elderly population, receiving monthly capitation payments from Medicare and Medicaid to cover all services. Whereas 45% of US personal health care expenditures go to hospital and nursing home services, On Lok spent a mere 17% on these items, making 83% of the health care dollar available for ambulatory home- and community-based services. While its services are far more comprehensive, On Lok's costs are no higher than those for a similar frail elderly population under traditional Medicare and Medicaid (Eng et al, 1997; Bodenheimer, 1999). Seventy-five On Lok \"look alikes\" now exist in 29 states under the Program of All-Inclusive Care for the Elderly (PACE). However, PACE sites care for fewer than 25,000 of the 3 million frail elderly and disabled people in the United States.\n\nThe United States has not implemented a social insurance program for long-term care. However, other nations have been more proactive in addressing the needs of their aging populations. In 1995, Germany enacted a system of near-universal social insurance for long-term care\u2014a program that the public has accepted as both affordable and beneficial (Harrington et al, 2002). A major expansion of the PACE concept combined with comprehensive social insurance for long-term care could provide a badly needed solution to the problems of long-term care in the United States.\n\n### **REFERENCES**\n\nAARP. _Valuing the invaluable: the economic value of family caregiving, 2008 update_. AARP Public Policy Institute. November 2008. www.aarp.org\/ppi.\n\nAdministration on Aging. _A Profile of Older Americans, 2010_. US Department of Health and Human Services; 2010. www.aoa.gov\/aoaroot\/aging_statistics\/Profile\/2010\/docs\/2010profile.pdf. Accessed November 16, 2011.\n\nBodenheimer T. Long-term care for frail elderly people\u2014the On Lok model. _N Engl J Med_. 1999;341:1324.\n\nComondore VR. Quality of care in for-profit and notfor-profit nursing homes: Systematic review and meta-analysis. _BMJ._ 2009;339:b2732.\n\nEng C et al. Program of All-inclusive Care for the Elderly (PACE). _J Am Geriatr Soc_. 1997;45:223.\n\nFennell ML et al. Elderly Hispanics more likely to reside in poor quality nursing homes. _Health Aff (Millwood)._ 2010;29:65.\n\nHarrington C et al. A national long-term care program for the United States: A caring vision. _JAMA_. 1991;266:3023.\n\nHarrington C et al. Germany's long-term care insurance model: Lessons for the United States. _J Public Health Policy_. 2002;23:44.\n\nHarrington C et al. Nursing Facilities, Staffing, Residents, and Facility Deficiencies, 1999 through 2005. University of California at San Francisco, September 2006. www.nccnhr.org.\n\nKaiser Family Foundation. Medicaid and long-term care services. July 2006a. www.kff.org\/medicaid\/upload\/Medicaid-and-Long-Term-Care-Services-PDF.pdf. Accessed November 16, 2011.\n\nKaiser Family Foundation. Private long-term care insurance: A viable option for low and middle-income seniors? July 2006b. www.kff.org\/uninsured\/upload\/7459.pdf. Accessed November 16, 2011.\n\nKaiser Family Foundation. Revisiting \"skin in the game\" among Medicare beneficiaries. February 2009. www.kff.org\/medicare\/upload\/7860.pdf. Accessed November 16, 2011.\n\nKaye HS et al. Long-term care: who gets it, how provides it, who pays, and how much? _Health Aff (Millwood)._ 2010;29:11.\n\nLevine C. The loneliness of the long-term care giver. _N Engl J Med_. 1999;340:1587.\n\nMartin A et al. Recession contributes to slowest annual rate of increase in health spending in five decades. _Health Aff (Millwood)._ 2011;30:11.\n\nNg T et al. Medicare and Medicaid in long-term care. _Health Aff (Millwood)._ 2010;29:22.\n\nPepper Commission. _A Call for Action_. Washington, DC: US Government Printing Office; 1990.\n\nStevenson DG, Grabowski DC. Sizing up the market for assisted living. _Health Aff (Millwood)._ 2010;29:35.\n\nUS Census Bureau. _Income, Poverty, and Health Insurance Coverage in the United States, 2009_. P60\u2013238, September, 2010.\n\nVladeck BC. _Unloving Care: The Nursing Home Tragedy._ New York: Basic Books; 1980.\n\nWerner RM, Konetzka RT. Advancing nursing home quality through quality improvement itself. _Health Aff (Millwood)._ 2010;29:81.\n\n## **13 Medical Ethics and Rationing of Health Care**\n\nFor those who work in the healing professions, ethical values play a special role. The specific content of medical ethics was first formulated centuries ago, based on the sayings of Hippocrates and others. The refinement of medical ethics has continued up to the present by practicing health caregivers, health professional and religious organizations, and individual ethicists. As medical technology, health care financing, and the organization of health care transform themselves, so must the content of medical ethics change in order to acknowledge and guide new circumstances.\n\n### **FOUR PRINCIPLES OF MEDICAL ETHICS**\n\nOver the years, participants in and observers of medical care have distilled widely shared human beliefs about healing the sick into four major ethical principles: beneficence, nonmaleficence, autonomy, and justice (Beauchamp and Childress, 2008) (Table 13\u20131).\n\n**Table 13\u20131.** The four principles of medical ethics\n\n_Beneficence_ is the obligation of health care providers to help people in need.\n\n_Dr. Rolando Bueno is a hard-working family physician practicing in a low-income neighborhood of a large city. He shows concern for his patients, and his knowledge and judgment are respected by his medical and nursing colleagues. On one occasion, he was called before the hospital quality assurance committee when one of his patients unexpectedly died; he agreed that he had made mistakes in his care and incorporated the lessons of the case into his future practice._\n\nDr. Bueno tries to live up to the ideal of beneficence. He does not always succeed; like all physicians, he sometimes makes clinical errors. Overall, he treats his patients to the best of his ability. The principle of beneficence in the healing professions is the obligation to care for patients to the best of one's ability.\n\n_Nonmaleficence_ is the duty of health care providers to do no harm.\n\n_Mrs. Lucy Knight suffers from insomnia and Parkinson's disease. The insomnia does not bother her, because she likes to read at night, but it irritates her husband. Mr. Knight requests his wife's physician to order strong sleeping pills for her, but the physician declines, saying that the combination of sleeping pills and Parkinson's disease places Mrs. Knight at high risk for a serious fall._\n\nThe modern array of medical interventions has the capacity to do good or harm or both, thereby enmeshing the principle of nonmaleficence with the principle of beneficence. In the case of Mrs. Knight, the prescribing of sedatives has far more potential for harm than for good, particularly because Mrs. Knight does not see her insomnia as a problem.\n\n_Autonomy_ is the right of a person to choose and follow his or her own plan of life and action.\n\n_Mr. Winter is a frail 88-year-old found by Dr. James Washington, his family physician, to have colon cancer, which has spread to the liver. The cancer is causing no symptoms. An oncologist gives Mr. Winter the option of transfusions, parenteral nutrition, and surgery, followed by chemotherapy; or watchful waiting with palliative and hospicecare when symptoms appear. Mr. Winter is terrified of hospitals and prefers to remain at home. He feels that he might live a comfortable couple of years before the cancer claims his life. After talking it over with Dr. Washington, he chooses the second option._\n\nThe principle of autonomy adds another consideration to the interrelated principles of beneficence and nonmaleficence. Would Mr. Winter enjoy a longer life by submitting himself to aggressive cancer therapy that does harm in order to do good? Or, does he sense that the harm may exceed the good? The balance of risks and benefits confronts each physician on a daily basis (Eddy, 1990). But the decision cannot be made solely by a risk\u2013benefit analysis; the patient's preference is a critical addition to the equation.\n\nAutonomy is founded in the overall desire of most human beings to control their own destiny, to have choices in life, and to live in a society that places value on individual freedom. In medical ethics, autonomy refers to the right of competent adult patients to consent to or refuse treatment. While the physician has an obligation to respect the patient's wishes, he or she also has a duty to fully inform the patient of the probable consequences of those wishes. For children and for adults unable to make medical decisions, a parent, guardian, other family member, or surrogate decision maker named in a legal document becomes the autonomous agent on behalf of the patient.\n\n_Justice refers_ to the ethical concept of treating everyone in a fair manner.\n\n_Joe, a white businessman in the suburbs, suffers crushing chest pain and within 5 minutes is taken to a nearby private emergency department, where he receives immediate coronary angioplasty and state-of-the-art treatment for a heart attack. Five miles away, in a poor neighborhood, Josephine, an African American woman, experiences severe chest pain, calls 911, waits 25 minutes for help to arrive, and is brought to a public hospital whose emergency department staff is attending to five other acutely ill patients. Before receiving appropriate attention, she suffers an arrhythmia and dies._\n\nThe principle of justice as applied to medical ethics is newer, more controversial, and harder to define than the principles of beneficence, nonmaleficence, and autonomy. In a general sense, people are treated justly when they receive what they deserve. It is unjust not to grant a medical degree to someone who completes medical school and passes all the necessary examinations. It is unjust to punish a person who did not commit a crime. In another meaning, _justice_ refers to universal rights: to receive enough to eat, to be afforded shelter, to have access to basic medical care and education, and to be able to speak freely. If these rights are denied, justice has been violated. In yet another version, justice connotes equal opportunity: All people should have an equal chance to realize their human potential. Justice might be linked to the golden rule: Treat others as you would want others to treat you. While there is no clear agreement on the precise meaning of justice, most people would agree that the differential treatment of Joe and Josephine is unjust.\n\n#### **Distributive Justice**\n\nIn exploring the concept of justice, one area of concern is the allocation of benefits and burdens in society. This realm of ethical thinking is called _distributive justice_ , and it involves such questions as: Who receives what amount of wealth, of education, or of medical care? Who pays what amount of taxes?\n\nThe principle of justice is linked to the idea of fairness. In the arena of distributive justice, no agreement exists on what formula for allocating benefits and costs is fair. Should each person get an equal share? Should those who work harder receive more? Should the proper formula be \"to each according to ability to pay,\" as determined by a free market? Or \"to each according to need?\" In allocating costs, should each person pay an equal share or should those with greater wealth pay more? Most societies construct a mixture of these allocation formulas. Unemployment benefits consider effort (having had a job) and need (having lost the job). Welfare benefits are primarily based on need. Job promotions may be based on merit. Many goods are distributed according to ability to pay. Primary education in theory (but not always in practice) is founded on the belief that everyone should receive an equal share (Beauchamp and Childress, 2008; Jonsen et al, 2010).\n\nHow is the principle of distributive justice formulated for medical care? Throughout the history of the developed world, the concept that health care is a privilege that should be allocated according to ability to pay has competed with the idea that health care is a right and should be distributed according to need. In most developed nations, the allocation of health care according to need has become the dominant political belief, as demonstrated by the passage of universal or near-universal health insurance laws. In the United States, the failure of the 100-year battle to enact national health insurance, and the widely divergent opinions on the 2010 Affordable Care Act, attest to the ongoing debate between health care as a privilege and health care as a right (see Chapter 15).\n\nIf the overwhelming opinion in the developed world holds that health care should be allocated according to need, then all people should have equal access to a reasonable level of medical care without financial barriers (ie, people should have a right to health care). In this chapter, therefore, we consider that the principle of distributive justice requires all people to equally receive a reasonable level of medical services based on medical need without regard to ability to pay.\n\n### **ETHICAL DILEMMAS, OLD AND NEW**\n\nEthical dilemmas (Lo, 2009) are situations in which a provider of medical care is forced to make a decision that violates one of the four principles of medical ethics in order to adhere to another of the principles. Ethical dilemmas always involve disputes in which both sides have an ethical underpinning to their position. Financial conflicts of interest on the part of physicians (see Chapters 4 and ), in contrast, pit ethical behavior against individual gain and are not ethical dilemmas.\n\n_Anthony, a 22-year-old Jehovah's Witness, is admitted to the intensive care unit for gastrointestinal bleeding. His blood pressure is 80\/60 mm Hg, and in the past 4 hours, his hematocrit has fallen from 38% to 21%. The medical resident implores Anthony to accept life-saving transfusions, but he refuses, saying that his religion teaches him that death is preferable to receiving blood products. When the blood pressure reaches 60\/20 mm Hg, the desperate resident decides to give the blood while Anthony is unconscious. The attending physician vetoes the plan, saying that the patient has the right to refuse treatment, even if an avoidable death is the outcome._\n\nIn Anthony's case, the ethical dilemma is a conflict between beneficence and autonomy. Which principle has priority depends on the particular situation, and in this case, autonomy supersedes beneficence. If the patient were a child without sufficient knowledge or reasoning capability to make an informed choice, the physician would not be obligated to withhold transfusions, even if the family so demanded (Jonsen et al, 2010).\n\n_Pedro Navarro has lung cancer that has metastasized to his brain. No effective treatment is available, and Mr. Navarro is confused and unable to understand his medical condition. Mrs. Navarro demands that her husband undergo craniotomy to remove the tumor. The neurosurgeon refuses, arguing that the operation will do Mr. Navarro no good whatsoever and will cause him additional suffering._\n\nThe case of Mr. Navarro pits the principle of autonomy against the principle of nonmaleficence. Mr. Navarro's rightful surrogate decision maker, his wife, wants a particular course of treatment, but the neurosurgeon knows that this treatment will cause Mr. Navarro considerable harm and do him no good. In this case, nonmaleficence triumphs. Whereas patient autonomy allows the right to refuse treatment, it does not include the right to demand a harmful or ineffectual treatment.\n\nThe traditional dilemmas described in many articles and books on medical ethics feature beneficence or nonmaleficence in conflict with autonomy. In two famous ethical dilemmas, the families of Karen Ann Quinlan and Nancy Cruzan, young women with severe brain damage (persistent vegetative state) asked that physicians discontinue a respirator (in the Quinlan case) and a feeding tube (in the Cruzan case). Both cases were adjudicated in the courts. The Quinlan decision promoted the right of patients or their surrogate decision makers to withdraw treatment, even if the treatment is necessary to sustain life. The outcome of the Cruzan case placed limits on autonomy by requiring that life-supporting treatment can be withdrawn only when a patient has stated his or her wishes clearly in advance (Annas, 2005).\n\nIn 2005, the case of Terri Schiavo, for 15 years in a persistent vegetative state similar to the situations of Karen Ann Quinlan and Nancy Cruzan, made national headlines. In spite of multiple decisions of state and federal courts\u2014up to the Supreme Courts of Florida and the United States\u2014supporting the right of Terri Schiavo's husband to discontinue Ms. Schiavo's feeding tube, the US Congress, encouraged by President George Bush, passed legislation reopening the option of reinserting the feeding tube. Eventually, based on the precedents of the Quinlan and Cruzan cases, the courts prevailed and Ms. Schiavo died (Annas, 2005).\n\nOverall, medical ethics has moved in the direction of giving priority to the principle of autonomy over that of beneficence.\n\nIn the late twentieth century, a new generation of ethical dilemmas emerged, moving beyond the individual physician\u2013patient relationship to involve the broader society. These social\u2013ethical problems derive from the new reality that money may not be available to pay for a reasonable level of medical services for all people. When money and resources are bountiful, the issue of distributive justice refers to equality in medical care access and health outcomes (see Chapter 3). Is it fair that some people are unable to receive needed care because they lack money and insurance? When money and resources become scarce, the issue of justice takes on a new twist. Should limits be set on treatments given to people with high-cost medical needs, so that other people can receive basic services? If not, might health care consume so many resources that other social needs are sacrificed? If limits should be set, who decides these limits?\n\n_Angela and Amy Lakeberg [actual names] were Siamese twins sharing one heart. Without surgery, they would die shortly. With surgery, Amy would die and Angela's chance of survival would be less than 1%. On August 20, 1993, a team of 18 physicians and nurses at Children's Hospital of Philadelphia performed an all-day operation to separate the twins. Amy died. The cost of the treatment was_ $ _1 million. The Medicaid program covered_ $ _700 to_ $ _1000 per day, and the hospital underwrote the balance of the costs. On June 9, 1994, Angela died; she had spent her brief life on a respirator in the hospital._\n\nThe new fiscal reality has spawned two related dilemmas.\n\n1. The first involves a conflict between the duty of the physician to follow the principles of beneficence and nonmaleficence and the growing sentiment that physicians should pay attention to issues of distributive justice. In the case of the Lakeberg twins, the hospital and the surgeons adhered strictly to the principle of beneficence: Even a remote chance of aiding one twin was seen as worthwhile. The hospital could have balked, arguing that its funding of the surgery would be unfairly shifted to other payers. The surgeons could have declined to operate on the grounds that the money spent on the Lakebergs could have been better used by patients with a greater chance of survival. But, the surgeons could argue, who can guarantee that the money saved would have gone to better use?\n\n2. The second category of social\u2013ethical dilemma is the conflict between the individual patient's right to autonomy and society's claim to distributive justice. In the Lakeberg case, individual autonomy won out. The Lakeberg parents could have decided that spending $1 million of society's money on a less than 1% chance of saving one of two infants was excessive and could indirectly harm other patients. On the other hand, would not most parents have done what the Lakebergs did?\n\nPhysicians take up the practice of medicine with a recognition that they have a duty to help and not harm their patients. Individuals claim a right to health care and do not want others to restrict that care. Yet the principle of distributive justice (recognizing that resources for health care are limited and should be fairly allocated among the entire population) might lead to physicians denying legitimate services or patients setting aside rightful claims to treatment.\n\nThe basis for the principle of justice is the desire shared by many human beings to live in a civilized society. To live in a state of harmony, each person must balance the concerns of the individual with the needs of the larger community. There is no right or wrong answer to the question of whether the Lakeberg surgery should have been done, but the surgery must be seen as a choice. The $1 million spent on the twins might have been spent on immunizing 10,000 children, with greater overall benefit. When health resources are scarce, the principle of justice creates ethical dilemmas that touch many people beyond those involved in an individual physician\u2013patient relationship. The imperatives of cost control have thrust the principle of justice to the forefront of health policy in the debate over rationing.\n\n### **WHAT IS RATIONING?**\n\n_Dr. Everett Wall works in a health maintenance organization (HMO). Betty Ailes came to him with a headache and wanted a magnetic resonance imaging (MRI) scan. After a complete history and physical examination, Dr. Wall prescribed medication and denied the scan. Ms. Ailes wrote to the medical director, complaining that Dr. Wall was rationing services to her._\n\n_Perry Hiler arrives at Vacant Hospital with fever and severe cough. His chest x-ray shows an infiltrate near the hilum of the lung consistent with pneumonia or tumor. Since Mr. Hiler has no insurance, the emergency department physician sends him to the county hospital. At the time, Vacant Hospital has 35 empty beds and plenty of staff. When he recovers, Mr. Hiler calls the newspaper to complain. The next day, a headline appears: \"Vacant Hospital Rations Care.\"_\n\n_Jim Delacour is a 50-year-old man with terminal cardiomyopathy. His physician sends him to a transplant center, where an evaluation concludes that he is an ideal candidate for a heart transplant. Because the number of transplant candidates is larger than the supply of donor hearts available, Mr. Delacour is placed on the waiting list. After waiting 6 weeks, he dies._\n\n_When the emergency department called, Dr. Marco Intensivo's heart sank. The eight-bed intensive care unit is filled with extremely ill patients, all capable of full recovery if they survive their acute illnesses. He has worried all day about another patient needing intensive care: a 55-year-old with a heart attack complicated by unstable arrhythmias. Which one of the nine needy cases will not get intensive care? Dr. Intensivo needs to make a decision, and fast._\n\nThe general public and the media often view rationing as a limitation of medical care such that \"not all care expected to be beneficial is provided to all patients\" (Aaron and Schwartz, 1984). Such a view only partially explains the concept of rationing. More precisely, rationing means a conscious policy of equitably distributing needed resources that are in limited supply (Reagan, 1988) (Table 13\u20132). Under this definition, only the last two cases presented above can be considered rationing. In the first case, Dr. Wall did not feel that the MRI was a resource needed by Betty Ailes. In the second, Vacant Hospital's refusal to care for Perry Hiler was simply a decision on the part of a private institution to place its financial well-being above a patient's health; there was no scarcity of health care resources. In the heart transplant and intensive care unit cases, in contrast, donor hearts and intensive care unit beds were in fact scarce. For Mr. Delacour, the scarcity was nationwide and prolonged; for Dr. Intensivo, the scarcity was within a particular hospital at a particular time. In both cases, decisions had to be made regarding the allocation of those resources.\n\n**Table 13\u20132.** Two definitions of rationing\n\nDuring World War II, insufficient gasoline was available to both power the military machine and satisfy the demands of automobile owners in the United States. The government rationed gasoline, giving priority to the military, yet allowing each civilian to obtain a limited amount of fuel. In a rural area, there may be a shortage of health care providers; in an overcrowded urban public hospital, there may be an insufficient number of beds; in the transplant arena, donor organs are truly in short supply. These are cases of commodity scarcity, wherein specific items are in limited supply.\n\nThe United States is a nation with an adequate supply of hospital beds and physicians in most communities; commodity scarcity in health care is the exception (eg, scarcity of primary care resources is becoming a reality). But a different kind of health resource is becoming scarce, and that is money. Those who pay the bills are insistent that the flow of money into the health sector be restricted. Most discussions of health care rationing presume fiscal scarcity, not commodity scarcity.\n\nIn summary, rationing in medical care means the limitation of resources, including money, going to medical care such that not all care expected to be beneficial is provided to all patients, and the fair distribution of these limited resources.\n\n### **COMMODITY SCARCITY: THE CASE OF ORGAN TRANSPLANTS**\n\nWhile fiscal scarcity is the more common form of resource limitation, commodity scarcity provides an instructive example of the interaction of ethics and rationing.\n\n_Mr. George Olds is a 76-year-old nonsmoking retired business executive with end-stage heart failure. He has good pulmonary and renal function and is not diabetic; thus, he is medically a good candidate for a heart transplant. His life expectancy without a transplant is 1 month. He has a loving family, with the resources to pay the $300,000 cost of the procedure._\n\n_Mr. Matt Younger is a 46-year-old divorced man who is unemployed, having lost his job as an auto worker 3 years ago. He has a history of smoking and alcohol use. He suffers a heart attack, develops intractable heart failure, and will die within 1 month without a heart transplant. He has good pulmonary and renal function and is not diabetic, making him a good candidate for the procedure._\n\n_Mr. Olds and Mr. Younger are in the same hospital and cared for by the same cardiologist, who applied for donor hearts on behalf of both patients on the same day. The cardiologist receives a call that one donor heart\u2014histocompatible with both patients\u2014has become available. Who should receive it?_\n\nIn 1951, the first kidney transplant was performed in Massachusetts. But it was in 1967, when Dr. Christiaan Barnard sewed a living heart into the chest of a person suffering end-stage cardiac disease, that modern medicine fully entered the age of transplantation. Since that time, thousands of people have been kept alive for many years by transplantation of the kidneys, hearts, lungs, and livers of their fellow human beings. In 2010, 17,000 kidney, 2300 heart, 6300 liver, and 1800 lung transplants were performed in the United States. Transplants are truly life saving in most cases. Seventy to eighty percent of patients receiving heart, liver, or kidney transplants survive at least 5 years after transplantation ().\n\nTransplantation of organs is both a medical miracle and an ethical watershed. It has generated debate on such questions as these: When are people really dead (so that their organs can be harvested for use in transplantation)? What is the responsibility of the families of brain-dead people to allow their organs to be harvested? Who pays and who is paid for organ transplants? Who should receive organs that are in short supply? We will focus only on the last of these issues.\n\nThe number of persons on the national waiting list for organ transplants rose from 16,000 in 1988 to 111,000 in 2010, yet the pool of organ donors has been estimated at 14,500. Even if all potential donors became actual donors, the number of organs that could be harvested each year falls far short of the required number. Nineteen patients in the United States die each day awaiting organs (www.mayoclinic.org\/transplant\/organdonation.html).\n\nTransplantation presents a classic case of commodity scarcity: There is insufficient supply to meet demand. Explicit rationing, which is a system that determines who gets organs and who does not, is inevitable. For heart, lung, and liver transplants, rationing is all or nothing: Those who receive organs may live, while those who do not will die.\n\nGiven the supply and demand imbalance, which potential transplant patients actually receive new organs? In the early 1980s, the major heart transplant center at Stanford University excluded people with \"a history of alcoholism, job instability, antisocial behavior, or psychiatric illness,\" and required transplant recipients to enjoy \"a stable, rewarding family and\/or vocational environment.\" Stanford's recipients had a better than 50% chance of surviving 5 years, signifying that acceptance or rejection from the program was a matter of life and death. The US Department of Health and Human Services was concerned about Stanford's selection criteria, which favored those middle-class or wealthy people with satisfying jobs. Moreover, the $100,000 cost restricted heart transplants to those with insurance coverage or ability to pay out of pocket. Both the social and economic criteria for access to this life-saving surgery raised serious issues of distributive justice.\n\nFollowing the passage of the National Organ Transplantation Act of 1984, the federal government designated the United Network for Organ Sharing (UNOS) as a national system for matching donated organs and potential recipients (www.unos.org). According to the Task Force on Organ Transplantation (1986), organ allocation should be governed by medical criteria, with the major factors being urgency of need and probability of success. The Task Force recommended that if two or more patients are equally good candidates for an organ according to the medical criteria, length of time on the waiting list is the fairest way to make the final selection. In 2006, the US Department of Health and Human Services issued updated guidelines and in 2007 the Medicare program promulgated conditions that hospitals with transplant programs needed to follow.\n\nOverall, UNOS follows these recommendations, placing potential recipients of organ transplants on its computerized waiting list. Recipients are prioritized according to a point scale based on severity of illness, time on the waiting list, and probability of a successful outcome. A serious attempt has been made to allocate scarce organs on the basis of justice criteria. But haunting the ethics of the prioritization process is the issue of ability to pay. In 2008, the average kidney transplant cost $259,000 and liver transplant $534,400. Persons needing a transplant are often rejected if they lack health insurance coverage (Laurentine and Bramstedt, 2010).\n\n### **FISCAL SCARCITY AND RESOURCE ALLOCATION**\n\nDuring the 1980s, technologic advances in medicine combined with the rapid rise in health care costs led to the belief that medical care rationing was upon us. The ethical issues raised by organ transplantation have thereby become generalized to all medical care. However, great differences separate the case of organ transplants from that of medical care as a whole.\n\n1. Medical care in general is not a scarce resource; in many geographic areas, facilities and personnel are overabundant.\n\n2. Whereas a nationwide structure is in place to decide who will receive a transplant, no such structure exists for medical care as a whole.\n\n_Dr. Ernest, who works in a for-profit HMO, wants to do her part to keep medical costs down. She prescribes low-cost amoxicillin at 20 cents per capsule rather than ciprofloxacin, which is priced at_ $ _1.50 for each dose. She teaches back pain patients home exercises at no cost rather than sending them to physical therapy visits at_ $ _75 per session. At the end of each year, she enjoys calculating how many thousands of dollars she has saved compared with one of her colleagues, who ignores costs in making medical decisions. Because of her efforts and those of other cost-conscious physicians, the HMO's pharmacy bill goes down, and HMO management is able to lay off one physical therapist, thereby raising its profit margin._\n\nWhile Dr. Ernest can be praised for attempting to reduce costs without sacrificing quality, her cost savings had no impact on overall national health care expenditures. Nor were the savings used to provide more childhood immunizations or to hire a physician assistant for a nearby rural community without any health care provider. In the United States, there is no structure within which to effect a trade-off between savings in one area and benefits in another. According to analyst Joshua Wiener. (1992)\n\n_In countries that have a socially determined health budget, cuts in one area can be justified on the grounds that the money will be spent on other, higher-priority services. This closed system of funding provides a moral underpinning for resource allocation across a range of potentially unlimited demands. In the United States, it is difficult to refuse additional resources for patients, because there is no certainty that the funds will be put to better use elsewhere (Wiener, 1992)._\n\nIn the United States, persuading physicians to save money on one patient in order to improve services for someone else is as illogical as telling a child to eat all the food on the plate because children in Africa are starving (Cassel, 1985).\n\nIn order for health care providers like Dr. Ernest to make their cost savings socially useful, two things are needed: a closed system of health care funding, whether governmental through a global budget or private through a network of HMOs; and a decision-making structure controlling such funding that has the responsibility to allocate budgets to health care interventions in a fair manner.\n\nFor the purposes of the following discussion, let us assume that the United States is in a position of fiscal scarcity and that a mechanism exists to fairly allocate medical care resources from one individual or population group to another. Which ethical conflicts arise between beneficence, nonmaleficence, and autonomy on the one hand and justice (equitable distribution of resources) on the other?\n\n### **THE RELATIONSHIP OF RATIONING TO COST CONTROL**\n\n_Assume that Limittown, USA, has a fixed budget of_ $ _250 million for medical care in 2011. Limittown has three imaging centers, each with an MRI scanner that is used only 4 hours each weekday. None of the medical facilities perform bone marrow transplantation, a procedure that can markedly prolong the lives of some leukemia patients. In 2010, Limittown spent_ $ _5 million to pay for bone marrow transplants at a university hospital 50 miles away._\n\n_Limittown's health commissioner projects that 2011 medical care expenditures will be_ $ _5 million over budget; she must implement cost savings. She considers two choices: (1) Two of the three MRI scanners could be closed, allowing the remaining scanner's cost per procedure to be drastically reduced or (2) Limittown could stop paying for bone marrow transplantation for leukemia patients._\n\nIs rationing the same as cost containment? We have defined rationing in medical care as the limitation of resources, including money, going to medical care such that not all care expected to be beneficial is provided to all patients, and the fair distribution of these limited resources. While the limitation of money going to medical care is cost containment, not all cost containment reduces beneficial care to patients. In the case of Limittown, both options for saving $5 million can be considered cost containment, but only denial of coverage for bone marrow transplants requires rationing. Consolidating MRI scanning at a single facility would allow the same number of scans to be performed but at a substantially lower cost. Rationing is associated with painful cost control (reducing effective medical care), but cost containment (see Chapter 8) can be either painful or painless (not reducing effective medical care) (Table 13\u20133). The extent of unnecessary care and administrative waste has led many health experts over the past three decades to conclude that the United States may not need to ration effective medical services (Brook and Lohr, 1986; Relman, 1990). Other health policy experts feel that rationing is likely to take place, and the issue is whether rationing is rational, based on the most effective medical interventions, or irrational, based on income or health insurance (Dranove, 2003). Whether or not rationing is needed today, advances in medical technology guarantee that rationing of medically efficacious services will be necessary in the future. But to maximize beneficence and autonomy without violating distributive justice, no rationing of beneficial services should take place until all wasteful practices are curtailed; painless cost control should precede painful cost control.\n\n**Table 13\u20133.** Rationing and cost control\n\n#### **Care Provided to Profoundly Ill People**\n\n_Lula Rogers is an 84-year-old diabetic woman with amputations of both legs; multiple strokes have rendered her unable to move, swallow, understand, or speak. She has been in a nursing home for three years during which time her medical condition has slowly deteriorated. Ms. Rogers' son wishes to remove her feeding tube, but her physician and the nursing staff feel it is cruel to cause her death by malnutrition and dehydration. Ms. Rogers continues to live for 3 more years, costing_ $ _300,000._\n\nA hotly debated issue is the amount of health care that should be provided to the profoundly and incurably ill. Were Lula Rogers' caregivers right to prolong a life that had value to her? Or were they prolonging Ms. Rogers' suffering and denying her a peaceful death? Should cost be a factor in such decisions, or should such matters of life and death be governed by autonomy, beneficence, and nonmaleficence alone (Luce, 1990)?\n\nTwenty-seven percent of Medicare's budget is spent on people in their last year of life, with almost half of those funds ($68 billion in 2009) spent in the final 60 days (Lubitz and Riley, 1993; Hogan et al, 2001) (Figure 13\u20131). In 2000, an estimated 67% of people who died had their last place of care in the hospital or nursing home; 33% died at home, with half of patients dying at home cared for by hospice programs (Teno et al, 2004). Patients in hospice programs have lower end-of-life costs than those not in hospice programs (Emanuel et al, 2002), and family members of patients receiving hospice care at home are more satisfied with the care than families of patients dying in hospitals or nursing homes (Teno et al, 2004). Thus, a strong possibility exists that reduced expenditures can go hand in hand with better care. If these savings could be transferred to more efficacious therapies for other people, then improving the care of the incurably ill could promote all four of the ethical principles\u2014beneficence, nonmaleficence, autonomy, and distributive justice.\n\n**Figure 13\u20131.** Medicare funds spent at the end of life.\n\n### **RATIONING BY MEDICAL EFFECTIVENESS**\n\nWe have seen that cost containment does not necessarily equal rationing and that eliminating administrative waste, medical waste, and unwanted interventions for the profoundly and incurably ill before rationing needed services best realizes the principles of beneficence and justice. However, if rationing of truly beneficial services is needed, the issues become even more difficult. If a health care system or program must compromise beneficence because of true fiscal scarcity, how can this compromise be made in a manner that yields the least harm and allocates the harm in the fairest possible way? In 2009 and 2010, federal legislation created a new structure for performing research on the comparative effectiveness of medical interventions (Benner et al, 2010)\n\n_Joy Fortune develops Hodgkin's disease, or cancer of the lymphatic system; she receives radiation therapy and is cured. Jessica Turner is moribund from advanced metastatic cancer of the pancreas. She undergoes chemotherapy and dies within 3 days._\n\nIn the event of rationing, science is the best guide: The providing or withholding of care is ideally determined by the probability that the treatment will maximize benefits and minimize harm, that is by the criterion of medical effectiveness. Radiation therapy can often cure Hodgkin's disease, but chemotherapy is unlikely to provide much benefit to people with very advanced pancreatic cancer. If rationing is needed and only one of these therapies can be offered, a decision based on the criterion of medical effectiveness would allow for the treatment of Hodgkin's disease but not of terminal pancreatic cancer.\n\nIf intervention A increases person-years of reasonable-quality life more than intervention B, intervention A is more medically effective. The cost of the two interventions is not considered. Cost-effectiveness adds dollars to the equation: If intervention A increases person-years of reasonable-quality life per dollar spent more than intervention B, it is more cost-effective. Which is a better standard for rationing medical care: medical effectiveness or cost-effectiveness?\n\nIf money were not scarce, medical effectiveness (maximizing benefit and minimizing harm) would be the ideal standard upon which to ration care (ie, the less effective the therapy, the lower its priority on the list of treatments to be offered). But if money were not scarce, we would not need to ration. It is unrealistic to pretend that costs can be ignored (Garber and Sox, 2010). Suppose that bone marrow transplantation saves as many person-years of life by treating advanced cancers as does doxycycline by curing pneumonia. The former costs $150,000, while the latter can be obtained for $10. There is no reason to ration doxycycline, as its cost is negligible, whereas to make bone marrow transplantation similarly accessible is costly. Thus consideration of costs is essential as a means of deciding which services to ration.\n\n#### **Rationing for Society as a Whole**\n\n_Mrs. Smith's breast cancer has spread to her liver and bone. She has been told that her only slim hope lies in high-dose chemotherapy with autologous bone marrow transplantation (HDC-ABMT), costing $250,000. Even with the optimistic assumption that HDC-ABMT has a 5% cure rate, screening mammography is eight times as cost-effective as HDC-ABMT in person-years of life saved._\n\nIn 1991, Dr. David Eddy (1991a) published a compelling article entitled \"The individual vs society: Is there a conflict?\" Dr. Eddy poses the preceding case of Mrs. Smith. If medical care must be rationed, it seems logical to spend funds on mammography rather than HDC-ABMT because the former intervention is more cost-effective. Dr. Eddy does not confine his analysis to cost-effectiveness, however, but moves on to the ethical issues.\n\n_Each of us can be in two positions when we make judgments about the value of different health care activities. We are in one position when we are healthy, contemplating diseases we might get, and writing out checks for taxes and insurance premiums. Call this the \"first position.\" We are in a different position when we actually have a disease, are sitting in a physician's office, and have already paid our taxes and premiums (the \"second position\").... Imagine that you are a 50-year-old woman employed by Mrs. Smith's corporation.... [The company] is considering two options: (1) cover screening for breast cancer.... or (2) cover HDCABMT.... Now imagine you are in the firstposition.... as long as you do not yet have the disease (the first position), option 1 will always deliver greater benefit at lower cost than option 2.... Now, let us switch you to the second position. Imagine that you already have breast cancer and have just been told that it has metastasized and is terminal.... The value to you of the screening option has plummeted because you already have breast cancer and can no longer benefit from screening...._\n\n_Maximizing care for individual patients attempts to maximize care for individuals when they are in the second position. Maximizing care for society expands the scope of concern to include individuals when they are in the first position. As this example illustrates, the program that delivers the most benefit for the least cost for society (option 1) is not necessarily best for the individual patient (option 2), and vice versa. But as this example also illustrates, individual patients and society are not distinct entities. Rather, they represent the different positions that each of us will be in at various times in our lives. When we serve ourselves in the second position, we can harm ourselves in the first. (Eddy, 1991a)_\n\nPhysicians generally care for patients in Dr. Eddy's second position\u2014when they are sick. But if the cost of treating those in the second position reduces resources available to prevent illness for the far larger number of people in the first position (who may not be seeing physicians because they feel fine), the individual principles of beneficence and autonomy are superseding the societal principle of justice. One could even say that choosing for individuals in the second position violates beneficence for those in the first position. On the other hand, if all resources go to those in the first position (eg, to cost-effective screening rather than highly technical treatment for those with life-threatening disease), injustice is committed in the other direction by ignoring the costly needs of the very ill.\n\nClearly, no ideal method of rationing medical care exists. The use of cost-effectiveness as a measuring stick raises ethical problems, and because of the difficulty in determining the cost-effectiveness of different interventions, has scientific limitations. All efforts should be made to control costs painlessly before resorting to the painful limitation of effective medical care. But if rationing is inevitable, a balance must be struck among many legitimate needs: The concerns of healthy people for illness prevention, the imperative for acutely sick people to obtain diagnosis and treatment, and the obligation to provide care and comfort to those with untreatable chronic illness.\n\n#### **Rationing within One Health Program: The Oregon Health Plan**\n\nThe previous discussion of rationing medical care nationwide presumes a mechanism that redirects savings from interventions not performed toward more cost-effective services. In fact, such a mechanism does not exist nationwide. Only in specific medical care programs do we find a decision-making apparatus for allocating expenditures. One example is the Oregon Health Plan (Bodenheimer, 1997).\n\nIn 1994, Oregon added 100,000 poor uninsured Oregonians to its Medicaid program. To control costs, a prioritized list of services was developed, and the state legislature decided how many services would be covered. The prioritized list was based on how much improvement in quantity and quality of life the treatment was likely to produce. The final list contained 745 condition\u2013treatment pairs, and the State of Oregon paid for items above line 574 on the list; conditions below that line were not covered (Kilborn, 1999). What are some of the Oregon Health Plan's ethical implications?\n\n1. The plan was more than a rationing proposal; its chief feature was to extend health care coverage to 100,000 more people. That aspect of the Oregon plan promotes the principle of justice.\n\n2. Another positive feature of the plan was its attempt to prioritize medical care services on the basis of effectiveness, which, if rationing is needed, is a reasonable method for deciding which services to eliminate.\n\nOther features of the Oregon plan must be viewed as negatively impacting distributive justice, or equal access to care without regard for ability to pay.\n\n1. In 1996, 12% of beneficiaries reported being denied services because they were below the line on the priority list. Of those, 78% reported that the denial had worsened their health (Mitchell and Bentley, 2000). Medical services were rationed for Oregon's poor but not for anyone else.\n\n2. The plan targeted beneficial medical services in a state with considerable medical waste. In 1988, many areas of Oregon had average hospital occupancy rates below 50%. The closing of unneeded hospital beds could have saved $50 million per year, enough to pay for some of the treatments eliminated in the plan (Fisher et al, 1992). Oregon did not exhaust its options for painless cost control before proceeding to potentially painful rationing.\n\nBy 2004, the Oregon Health Plan had unraveled (Oberlander, 2006). The state entered a period of budgetary woes, new premiums and copays were instituted, and Oregon Health Plan enrollees responded by dropping out of the program. The rate of uninsurance climbed from 11% to 17%. But the bold experiment in rational rationing remains alive in the minds of health care policymakers.\n\n#### **Rationing within One Institution: Intensive Care**\n\n_Ms. Wilson is a 71-year-old woman with a recently diagnosed lung cancer. Obstructing a bronchus, the tumor causes pneumonia, and Ms. Wilson is admitted to the hospital in her rural town. She deteriorates and becomes comatose, requiring a respirator. By the eighth hospital day, she is no better. On that day, Louis Ford, a previously healthy 27-year-old, is brought to the hospital with a crushed chest and pneumothorax suffered in an automobile accident. Mr. Ford is in immediate need of a respirator. None of the six patients in the intensive care unit can be removed from respirators without dying; of the six, Ms. Wilson has the poorest prognosis. She has no family. No other respirators exist within a 50-mile radius (Jonsen et al, 2010). Should Ms. Wilson be removed from the respirator in favor of Mr. Ford?_\n\nResources may be scarce throughout an entire nation or within a small hospital. _Macroallocation_ refers to the amount and distribution of resources within a society, whereas _microallocation_ refers to resource constraints at the level of an individual physician or institution. Macroallocation decisions may be more significant, affecting thousands or millions of people. Microallocation choices can be more acute, bringing ethical dilemmas into stark, uncompromising focus and placing issues of resource allocation squarely in the lap of the practicing physician. The microallocation choice involving Ms. Wilson incorporates all four ethical principles, which must be weighed and acted on within minutes: (1) Beneficence: For whom? This ideal cannot be realized for both patients. (2) Nonmaleficence: If Ms. Wilson is removed from the respirator, harm is done to her, but the price of not harming her is great for Mr. Ford. (3) Autonomy: Withdrawal of therapy requires the consent of the patient or family, which is impossible in Ms. Wilson's case. (4) Justice: Should resources be distributed on a first-come first-served basis or according to need?\n\nThese are tragic decisions. Many physicians would remove Ms. Wilson from the respirator and make all efforts to save Mr. Ford. The main consideration would be medical effectiveness: Ms. Wilson's chance of living more than a few months is slim, while Mr. Ford could be cured and live for many decades.\n\nLess stark but similar decisions face physicians on a daily basis. On a busy day, which patients get more of the physician's time? In a public hospital with an MRI waiting list, when should a physician call the radiologist and argue for an urgent scan, thereby pushing other people down on the waiting list? Situations involving microallocation demonstrate why, in real life, health care professionals are forced to balance the interests of one patient against those of another and the interests of individuals against the imperatives of society.\n\n### **A BASIC LEVEL OF GUARANTEED MEDICAL BENEFITS**\n\n_Don Rich is a bank executive who receives his care through a New York City HMO. He develops angina pectoris, which remains stable for over a year. An exercise treadmill test suggests mild coronary artery disease. Although this evaluation indicates that Mr. Rich's condition can be safely managed with medications, he asks his cardiologist to arrange a coronary angiogram with an angioplasty or coronary bypass if indicated. He is told that the HMO has finite resources for such procedures and limits their use to patients with unstable angina or highly abnormal treadmill tests, for whom the procedures are more efficacious. Mr. Rich flies to Texas, consults with a private cardiac surgeon, and receives a coronary angiogram at his own expense._\n\nMost people in the United States believe that health care should be a right. But how much health care? If every person has a right to all beneficial health care, the nation may be unable to pay the bill or may be forced to limit other rights such as education or fire and police protection. One approach to this problem is to limit the health care right to a basic package of services. (In the case of Don Rich's HMO, angiography for stable angina pectoris is not within the basic package.) Any services beyond the basics can be purchased by individuals who choose to spend their own money. This solution creates an ethical problem. If a service that does produce medical benefit is not included in the basic package or is denied by an insurance company medical director, that service becomes available only to those who can afford it. Where should society draw the line between a basic level of care that should be equally available to all, and \"more than basic\" services that may be purchased according to individual ability and willingness to pay (Eddy, 1991b)? Unless the basic package covers all beneficial health services, the principle of distributive justice, that all people equally receive a reasonable level of medical services without regard to ability to pay, will be compromised.\n\n### **THE ETHICS OF HEALTH CARE FINANCING**\n\n_Yoshiko Takahashi's first heart attack came at age 59. It was minor, and she felt well the next day. Then came the real shock: because of her high blood pressure, her private insurance policy considers disease of the cardiovascular system a preexisting condition and will not cover costs for complications of hypertension. She demands to go home to limit her hospital bill. Twelve hours later comes the second heart attack, which is severe. She is readmitted to intensive care and remains in the hospital for 8 more days. Because of persistent pain, she is a candidate for coronary angiography, which she refuses on account of the cost. When she purchased the insurance, Ms. Takahashi had not understood its terms because her English skills were poor._\n\nDecisions by physicians encompass only one aspect of resource allocation; the payers of health care have great power in the distribution of medical care. The policies of the private insurance industry, which covers the largest number of people in the United States, raise important ethical issues. In the case of Yoshiko Takahashi, the insurance company, rather than her physicians, largely determined what kind of medical care she received.\n\nPrivate insurance may be experience rated (see Chapter 2), with premiums costing more for people or groups with a higher risk of illness. Under the practice of experience rating, people who need health care the most (because they have a chronic illness) are less likely to be able to purchase affordable health insurance. Many people feel that private insurers violate the justice principle because those most in need of services have the least chance of gaining coverage for those services.\n\nHealth insurance executives, however, have a different view, believing that private health insurance is fair. An advertisement sponsored by the insurance industry argued,\n\n_If insurance companies didn't put people into risk groups [experience rating], it would mean that low-risk people would be arbitrarily mixed in with high-risk people.... and [low-risk people] would have to pay higher rates. That would be unfair to everyone. (Light, 1992)_\n\nAccording to this notion, it is unfair to force one person or group to pay for the needs or burdens of another. An alternative view, citing the principle of distributive justice, holds that young and healthy people should pay more in health costs than they use in health services so that older and less healthy people can receive health services at a reasonable cost. Even from the perspective of one's own long-term self-interest, it makes sense to pay more for health care while young and healthy, and to pay less when advanced age creates a greater risk of becoming sick.\n\nA much-discussed issue involves individuals whose behavior, particularly smoking, eating unhealthy diets, and drinking alcohol in excess, is seen as contributing to their ill health.\n\n_Jim Butts, a heavy smoker, develops emphysema and has multiple hospitalizations for respiratory failure, including many days on the respirator. Randy Schipp, a former shipyard worker, develops work-related asbestosis and has multiple hospitalizations for respiratory failure, including many days on the respirator. Should Jim pay more for health insurance than Randy?_\n\n_Gene eats a low-fat diet, exercises regularly, but has a strong family history of heart disease; he suffers a heart attack at age 44. Mac eats fast food, does not exercise, and has a heart attack at age 44. Should Mac pay more for health care coverage than Gene?_\n\nOne view holds that individuals who fall sick as a result of high-risk behavior such as smoking, substance abuse including use of alcohol, and consumption of high-fat foods are entirely responsible for their behavior and should pay higher health insurance premiums. Opponents of this idea see it as \"blaming the victim\" and argue that high-risk behaviors have a complex causation that may involve genetic and environmental factors including uncontrollable addiction. They cite a number of facts to support their position. The food industry spends billions of dollars each year on television advertising; the average child sees thousands of food commercials each year, most of them for products with poor nutritional value. The tobacco industry heavily advertises to teenagers. Illegal drug use is associated with poverty, hopelessness, and easy availability of drugs. Some evidence finds a genetic predisposition to alcoholism. To the extent that individuals are not entirely at fault for their high-risk behavior, it would be unfair to charge them more for health insurance. On the other hand, it seems sensible that users of tobacco and alcohol pay through taxes on those products.\n\n### **WHO ALLOCATES HEALTH CARE RESOURCES?**\n\nThe predicament of limited resources has been likened to a herd of cattle grazing on a common pasture. The total grazing area may be regarded as the entirety of economic resources in the United States. A smaller pasture, the _medical commons_ , comprises that portion of the grazing area dedicated to health care. The herd represents the nation's physicians, using the resources of the commons in the process of providing care to patients. Physicians, guided by medicine's moral imperative to \"do everything possible for the patient,\" continually attempt to extend the borders of the medical commons. But communities outside the medical commons have legitimate claims to societal resources and view the herd as encroaching on resources needed for other pursuits (Grumbach and Bodenheimer, 1990).\n\nWho decides the magnitude of the medical commons, that is, the resources devoted to health care? Physicians and other health care providers, whose interventions on behalf of their patients add up to the totality of medical resources used? The sum of individual consumer choices operating through a free market? Health insurance plans, watching over their particular piece of the commons? Or government, using the political process to set budgetary limits on the entire health care system?\n\nTraditionally, physicians and patients have had a great deal to say about the size of the medical commons. In the United States, the medical commons traditionally has been an open range. The quantity and price of medical visits, hospital days, surgeries, diagnostic studies, pharmaceuticals, and other such interventions determine the total costs of medical care. This is not the case in other nations, where government health care budgets constitute a \"fence\" around the medical commons, setting a clear limit on the quantity of resources available. Some advocates of fence-building in the United States have considered parceling the medical commons into numerous sub-pastures, each representing an HMO or Accountable Care Organization (see Chapter 6) working within the constraints of fixed, prepaid budgets. Not all pastures would be equal in size, and the fences would have holes that allow patients to purchase additional services outside of the organized systems of care.\n\nEthical considerations play a role in both open and closed medical care systems. In the US open range, the principles of beneficence and autonomy have the upper hand, tending toward an expanding, though not equitable, system. Fenced-in systems, in contrast, balance the more expansive principles of beneficence and autonomy with the demands of distributive justice in order to allocate resources within the medical commons.\n\nIf the United States moves toward a more fenced-in medical commons, decisions will be needed about who gets what. Do all 90-year-old people with multiple organ failure receive kidney dialysis that may extend their lives only a few months? Are very low-birth-weight infants afforded neonatal intensive care even with a small chance of leading a normal life? Do individual physicians, interacting with their patients, have the final say in making these decisions? Should societal bodies such as government, commissions of interested parties, or professional associations set the rules?\n\nMicroallocation issues come down to daily clinical decisions about which individual patients will receive what types of care (Lo, 2009). Physicians and other caregivers may well recoil from the prospect of \"bedside rationing,\" believing that allocative decision making unduly compromises their commitment to the principles of beneficence and autonomy. Levinsky (1984) has argued that physicians must maintain their single-mindedness in maximizing care for each patient:\n\n_There is increasing pressure on doctors to serve two masters. Physicians in practice are being enjoined to consider society's needs as well as each patient's needs in deciding what type and amount of medical care to deliver.... When practicing medicine, doctors cannot serve two masters. It is to the advantage both of our society and of the individuals it comprises that physicians retain their historic single-mindedness. The doctor's master must be the patient. (Levinsky, 1984)_\n\nYet if physicians abstain from the arena of macroal-location decision making, who is to decide? Currently, these decisions are often made by medical directors of private insurance companies and the leaders of the Medicare and Medicaid programs. Studies have documented that such decisions vary from plan to plan, and even within a single insurance plan, a medical director may make different decisions on different days for similar patients (Light, 1994). If physicians refuse to accept two masters, then medicine will be granting allocation decisions to insurance company and governmental officials. The physician of the twenty-first century will continue to face individual patient responsibilities but will find it difficult to escape the obligation to balance the wishes of individual patients against the larger needs of society (Cassel, 1985; Morreim, 1989).\n\nIf physicians are to serve two masters (ie, to maintain their dedication to individual patients while at the same time responsibly managing resources), they need rules to assist them. These rules should operate at both a population and an individual level. At the population level, society should ideally decide which general treatments are to be collectively paid for through the process of universal health insurance. At the individual level, rules are needed to guide decisions about the prioritization of resources for specific patients. The workings of organ transplantation provide a model of how physicians can serve two masters: They do everything possible to procure organs for their transplant patients, but also accept the rules of the system that attempt to allocate organs in a fair manner (Benjamin et al, 1994). The modern health care professional is caught in a global ethical dilemma. On the one hand, patients and their families expect the best that modern technology can offer, paid for through private or public insurance. The imperatives of beneficence, nonmaleficence, and autonomy rule the bedside. On the other hand, grave injustices take place on a daily basis: An uninsured young person with a curable illness is unable to pay for care, while an insured, bedridden individual who had a stroke incurs vast medical bills during the last weeks of her ebbing life. Should not the physician at the stroke patient's bedside be concerned about both patients? However this dilemma is resolved, the principle of justice will relentlessly peek at the physician from under the bed.\n\n### **REFERENCES**\n\nAaron HJ, Schwartz WB. _The Painful Prescription_. Washington, DC: The Brookings Institution; 1984.\n\nAnnas GJ. \"Culture of life\" politics at the bedside\u2014the case of Terri Schiavo. _N Engl J Med_. 2005;352:1710.\n\nBeauchamp TL, Childress JF. _Principles of Biomedical Ethics_. 6th ed. New York: Oxford University Press; 2008.\n\nBenjamin M et al. What transplantation can teach us about health care reform. _N Engl J Med_. 1994;330:858.\n\nBenner JS et al. An evaluation of recent federal spending on comparative effectiveness research. _Health Affairs._ 2010;29:1768.\n\nBodenheimer T. The Oregon Health Plan: Lessons for the nation. _N Engl J Med_. 1997;337:651, 720.\n\nBrook RH, Lohr KN. Will we need to ration effective health care? _Issues Sci Technol_. 1986;3:68.\n\nCassel CK. Doctors and allocation decisions: A new role in the new Medicare. _J Health Polit Policy Law_. 1985;10:549.\n\nDranove D. _What's Your Life Worth? Health Care Rationing... Who Lives? Who Dies? And Who Decides?_ Upper Saddle River, NJ: Prentice Hall; 2003.\n\nEddy DM. Comparing benefits and harms: The balance sheet. _JAMA_. 1990;263:2493.\n\nEddy DM. The individual vs society: Is there a conflict? _JAMA_. 1991a;265:1446.\n\nEddy DM. What care is \"essential?\" What services are \"basic?\" _JAMA_. 1991b;265:782.\n\nEmanuel EJ et al. Managed care, hospice use, site of death, and medical expenditures in the last year of life. _Arch Intern Med_. 2002;162:1722.\n\nFisher ES, Welch HG, Wennberg JE. Prioritizing Oregon's hospital resources. _JAMA_. 1992;267:1925.\n\nGarber AM, Sox HC. The role of costs in comparative effectiveness research. _Health Aff_. 2010;29:1805.\n\nGrumbach K, Bodenheimer T. Reins or fences: A physician's view of cost containment. _Health Aff_. 1990;9:120.\n\nHogan C et al. Medicare beneficiaries' costs of care in the last year of life. _Health Aff_. 2001;20:188.\n\nJonsen AR et al. _Clinical Ethics: A Practical Guide to Ethical Decisions in Clinical Medicine_. 7th ed. New York, NY: McGraw-Hill; 2010.\n\nKilborn PT. Oregon falters on a new path to health care. _New York Times_. January 3, 1999.\n\nLaurentine KA, Bramstedt KA. Too poor for transplant: Finance and insurance issues in transplant ethics. _Prog Transplant_. 2010;20:178.\n\nLevinsky NG. The doctor's master. _N Engl J Med_. 1984;311:1573.\n\nLight DW. The practice and ethics of risk-rated health insurance. _JAMA_. 1992;267:2503.\n\nLight DW. Life, death, and the insurance companies. _N Engl J Med_. 1994;330:498.\n\nLo B. _Resolving Ethical Dilemmas. A Guide for Clinicians_. 4th ed. Baltimore, MD: Lippincott Williams & Wilkins; 2009.\n\nLubitz JD, Riley GF. Trends in Medicare payments in the last year of life. _N Engl J Med_. 1993;328:1092.\n\nLuce JM. Ethical principles in critical care. _JAMA_. 1990;263:696.\n\nMitchell JB, Bentley F. Impact of Oregon's priority list on Medicaid beneficiaries. _Med Care Res Rev_. 2000;57:216.\n\nMorreim EH. Fiscal scarcity and the inevitability of bedside budget balancing. _Arch Intern Med_. 1989;149:1012.\n\nOberlander J. Health reform interrupted: The unraveling of the Oregon Health Plan. _Health Affairs._ 2006:w96\u2013w105.\n\nReagan MD. Health care rationing: What does it mean? _N Engl J Med_. 1988;319:1149.\n\nRelman AS. Is rationing inevitable? _N Engl J Med_. 1990;322:1809.\n\nTask Force on Organ Transplantation. _Issues and Recommendations_. Washington, DC: US Department of Health and Human Services; 1986.\n\nTeno JM et al. Family perspectives on end-of-life care at the last place of care. _JAMA_. 2004;291:88.\n\nWiener JM. Rationing in America: overt and covert. In: Strosberg MA et al, eds. _Rationing America's Medical Care: The Oregon Plan and Beyond_. Washington, DC: The Brookings Institution; 1992.\n\n## **14 Health Care in Four Nations**\n\nThe financing and organization of medical care throughout the developed world spans a broad spectrum. In most countries, the preponderance of medical care is financed or delivered (or both) in the public sector; in others, like the United States, most people both pay for and receive their care through private institutions.\n\nIn this chapter, we describe the health care systems of four nations: Germany, Canada, the United Kingdom, and Japan. Each of these nations resides at a different point on the international health care continuum. Examining their diverse systems may aid us in our search for a suitable health care system for the United States.\n\nRecall from Chapter 2 the four varieties of health care financing: out-of-pocket payments, individual private insurance, employment-based private insurance, and government financing. Germany, Canada, the United Kingdom, and Japan emphasize the last two modes of payment. Germany finances medical care through government-mandated, employment-based private insurance, though German private insurance is a world apart from that found in the United States. Canada and the United Kingdom feature government-financed systems. Japan's financing falls between the German method of financing and the government model of Canada and the United Kingdom. Regarding the delivery of medical care, the German, Japanese, and Canadian systems are predominantly private, while the United Kingdom's is largely public.\n\nAlthough these four nations demonstrate great differences in their manner of financing and organizing medical care, in one respect they are identical: They all provide universal health care coverage, thereby guaranteeing to their populations financial access to medical services.\n\n### **GERMANY**\n\n#### **Health Insurance**\n\n_Hans Deutsch is a bank teller living in Germany. He and his family receive health insurance through a sickness fund that insures other employees and their families at his bank and at other workplaces in his city. When Hans went to work at the bank, he was required by law to join the sickness fund selected by his employer. The bank contributes 7.3% of Hans's salary to the sickness fund, and 8.2% is withheld from Hans's paycheck and sent to the fund. Hans's sickness fund collects the same 15.5% employer-employee contribution for all its members._\n\nGermany was the first nation to enact compulsory health insurance legislation. Its pioneering law of 1883 required certain employers and employees to make payments to existing voluntary sickness funds, which would pay for the covered employees' medical care. Initially, only industrial wage earners with incomes less than $500 per year were included; the eligible population was extended in later years.\n\nAlmost 90% of Germans now receive their health insurance through the mandatory sickness funds, with 10% covered by voluntary insurance plans (Figure 14\u20131). Several categories of sickness funds exist. Thirty-seven percent of people (mostly blue-collar workers and their families) belong to funds organized by geographic area; 33% (for the most part the families of white-collar workers) are in nationally based \"substitute\" funds; 21% are employees or dependents of employees who work in 700 companies that have their own sickness funds; and 6% are in funds covering all workers in a particular craft (Busse and Riesberg, 2004; Busse, 2008).\n\n**Figure 14\u20131.** The German national health insurance system.\n\nIn 2010, the proportion of earnings going to a sickness fund was set at 15.5%, with employers paying 47% and employees 53% of that amount. These contributions formerly went directly to the sickness funds, which are nonprofit, closely regulated entities that lie somewhere between the private and public sectors. Since 2009, employee and employer contributions are collected by a government-run health fund, which then distributes the money to health funds based on a risk-adjusted (more for older and sicker people) amount per insured person (Ornyanova and Busse, 2009). The number of sickness funds is shrinking, down from 1000 to less than 200 in 2011. The funds are not allowed to exclude people because of illness, or to raise contribution rates according to age or medical condition; that is, they may not use experience rating. The funds are required to cover a broad range of benefits, including hospital and physician services, prescription drugs, and dental, preventive, and maternity care. Because wages supporting health care financing are declining relative to health care costs, employers are proposing that their contribution be capped at 7% of earnings so that further increases are borne by employees (Zander et al, 2009).\n\n_Hans's father, Peter Deutsch, is retired from his job as a machinist in a steel plant. When he worked, his family received health insurance through a sickness fund set up for employees of the steel company. The fund was run by a board, half of whose members represented employees and the other half the employer. On retirement, Peter's family continuedits coverage through the same sickness fund with no change in benefits. The sickness fund continues to pay approximately 60% of his family's health care costs (subsidized by the contributions of active workers and the employer), with 40% paid from Peter's retirement pension fund._\n\n_Hans has a cousin, Georg, who formerly worked for a gas station in Hans's city, but is now unemployed. Georg remained in his sickness fund after losing his job. His contribution to the fund is paid by the government. Hans's best friend at the bank was diagnosed with lymphoma and became permanently disabled and unable to work. He remained in the sickness fund, with his contribution paid by the government._\n\nUpon retiring from or losing a job, people and their families retain membership in their sickness funds. Health insurance in Germany, as in the United States, is employment based, but German health insurance, unlike in the United States, must continue to cover its members whether or not they change jobs or stop working for any reason.\n\n_Hans's Uncle Karl is an assistant vice-president at the bank. Because he earns more than 49,500 Euros per year, he is not required to join a sickness fund, but can opt to purchase private health insurance. Many higher-paid employees choose a sickness fund; they are not required to join the fund selected by the employer for lower-paid workers but can join one of 15 national \"substitute\" funds._\n\nTen percent of Germans, with incomes more than 49,500 Euros per year (2011), choose voluntary private insurance. Private insurers pay higher fees to physicians than do sickness funds, often allowing their policyholders to receive preferential treatment. In summary, in Germany 88% of the populace belong to the mandatory sickness fund system, 10% opt for private insurance, 2% receive medical services as members of the armed forces or police, and less than 0.2% (all of whom are wealthy) have no coverage.\n\nGermany finances health care through a merged social insurance and public assistance structure (see Chapters 2, , and for discussion of these concepts), such that no distinctions are made between employed people who contribute to their health insurance, and unemployed people, whose contribution is made by the government.\n\n#### **Medical Care**\n\n_Hans Deutsch develops chest pain while walking, and it worries him. He does not have a physician, and a friend recommends a general practitioner (GP), Dr. Helmut Arzt. Because Hans is free to see any ambulatory care physician he chooses, he indeed visits Dr. Arzt, who diagnoses angina pectoris\u2014coronary artery disease. Dr. Arzt prescribes some medications and a low-fat diet, but the pain persists. One morning, Hans awakens with severe, suffocating chest pain. He calls Dr. Arzt, who orders an ambulance to take Hans to a nearby hospital. Hans is admitted for a heart attack and is cared for by Dr. Edgar Hertz, a cardiologist. Dr. Arzt does not visit Hans in the hospital. Upon discharge, Dr. Hertz sends a report to Dr. Arzt, who then resumes Hans's medical care. Hans never receives a bill._\n\nGerman medicine maintains a strict separation of ambulatory care physicians and hospital-based physicians. Most ambulatory care physicians are prohibited from treating patients in hospitals, and most hospital-based physicians do not have private offices for treating outpatients. People often have their own primary care physician (PCP) but are allowed to make appointments to see ambulatory care specialists without referral from the primary care physician. Fifty-one percent of Germany's physicians are generalists, compared with only 35% in the United States. The German system tends to use a dispersed model of medical care organization (see Chapter 5), with little coordination between ambulatory care physicians and hospitals (Busse and Riesberg, 2004).\n\n#### **Paying Physicians and Hospitals**\n\n_Dr. Arzt was used to billing his regional association of physicians and receiving a fee for each patient visit and for each procedure done during the visit. In 1986, he was shocked to find that spending caps had been placed on the total ambulatory physician budget. If in the first quarter of the year, the physicians in his regional association billed for more patient services than expected, each fee would be proportionately reduced during the next quarter. If the volume of services continued to increase, fees would drop again in the third and fourth quarters of the year. Dr. Arzt discussed the situationwith his friend Dr. Hertz, but Dr. Hertz, as a hospital physician, received a salary and was not affected by the spending cap._\n\nAmbulatory care physicians are required to join their regional physicians' association. Rather than paying physicians directly, sickness funds pay a global sum each year to the physicians' association in their region, which in turn pays physicians on the basis of a detailed fee schedule. These sums have been based on the number of patients cared for by the physicians in each regional association, but in 2007, a risk-adjustment factor is being introduced that increases payments for populations with greater health problems. Since 1986, physicians' associations, in an attempt to stay within their global budgets, have reduced fees on a quarterly basis if the volume of services delivered by their physicians was too high. Sickness funds pay hospitals on a basis similar to the diagnosis-related groups used in the US Medicare program. Included within this payment is the salary of hospital-based physicians (Busse and Riesberg, 2004).\n\n#### **Cost Control**\n\nThe 1977 German Cost Containment Act created a body called Concerted Action, made up of representatives of the nation's health providers, sickness funds, employers, unions, and different levels of government. Concerted Action is convened twice each year, and every spring, it sets guidelines for physician fees, hospital rates, and the prices of pharmaceuticals and other supplies. Based on these guidelines, negotiations are conducted at state, regional, and local levels between the sickness funds in a region, the regional physicians' association, and the hospitals to set physician fees and hospital rates that reflect Concerted Action guidelines. Since 1986, not only have physician fees been controlled, but as described in the above vignette about Dr. Arzt, the total amount of money flowing to physicians has been capped. As a result of these efforts, Germany's health expenditures as a percentage of the gross domestic product actually fell between 1985 and 1991 from 8.7% to 8.5%.\n\nIn 1991, however, German health care costs resumed an upward surge, paving the way for a 1993 cost control law restricting the growth of sickness fund budgets. In 2004, Germany raised copayments, ceased coverage of over-the-counter drugs, and enacted new controls on pharmaceutical prices (Stock et al, 2006). While Germany's 2008 health care expenditures as a percent of GDP was the fifth highest among developed nations, this figure has remained stable since 2000, indicating that cost control measures limiting the size of sickness fund budgets are having success.\n\n### **CANADA**\n\n#### **Health Insurance**\n\n_The Maple family owns a small grocery store in Outer Snowshoe, a tiny Canadian town. Grandfather Maple has a heart condition for which he sees Dr. Rebecca North, his family physician, regularly. The rest of the family is healthy and goes to Dr. North for minor problems and preventive care, including children's immunizations. Neither as employers nor as health consumers do the Maples worry about health insurance. They receive a plastic card from their provincial government and show the card when they visit Dr. North._\n\n_The Maples do worry about taxes. The federal personal income tax, the goods and services tax, and the various provincial taxes take almost 40% of the family's income. But the Maples would never let anyone take away their health insurance system._\n\nIn 1947, the province of Saskatchewan initiated the first publicly financed universal hospital insurance program in North America. Other provinces followed suit, and in 1957, the Canadian government passed the Hospital Insurance Act, which was fully implemented by 1961. Hospital, but not physician, services were covered. In 1963, Saskatchewan again took the lead and enacted a medical insurance plan for physician services. The Canadian federal government passed universal medical insurance in 1966; the program was fully operational by 1971 (Taylor, 1990).\n\nCanada has a tax-financed, public, single-payer health care system. In each Canadian province, the single payer is the provincial government (Figure 14\u20132). During the 1970s, federal taxes financed 50% of health services, but the federal share declined to 22% by 1996, generating acrimony between the federal and provincial governments. In response to this political debate, the federal contributions began to increase in 2001. Currently, the federal government funds approximately one-third of provincial health expenditures (Canadian Institute for Health Information, 2010). Provincial taxes vary in type from province to province and include income taxes, payroll taxes, and sales taxes. Some provinces, for example British Columbia and Alberta, charge a compulsory health care premium\u2014essentially an earmarked tax\u2014to finance a portion of their health budgets.\n\n**Figure 14\u20132.** The Canadian national health insurance system.\n\nUnlike Germany, Canada has severed the link between employment and health insurance. Wealthy or poor, employed or jobless, retired or younger than 18, every Canadian receives the same health insurance, financed in the same way. No Canadian would even imagine that leaving, changing, retiring from, or losing a job has anything to do with health insurance. In Canada, no distinction is made between the two public financing mechanisms of social insurance (in which only those who contribute receive benefits) and public assistance (in which people receive benefits based on need rather than on having contributed). Everyone contributes through the tax structure and everyone receives benefits.\n\nThe benefits provided by Canadian provinces are broad, including hospital, physician, and ancillary services. Provincial plans also pay for outpatient drugs, although the scope of drug coverage\u2014and also long-term care benefits\u2014varies across provinces.\n\nThe Canadian health care system is unique in its prohibition of private health insurance for coverage of services included in the provincial health plans. Hospitals and physicians that receive payments from the provincial health plans are not allowed to bill private insurers for such services, thereby avoiding the preferential treatment of privately insured patients that occurs in many health care systems. Canadians can purchase private health insurance policies for gaps in provincial health plan coverage or for such amenities as private hospital rooms.\n\n#### **Medical Care**\n\n_Grandfather Maple has had intermittent sensations of palpitations in his chest for a few weeks. He calls Dr. North, who tells him to come right over. An electrocardiogram reveals rapid atrial fibrillation, an abnormal heart rhythm. Because Mr. Maple is tolerating the rapid rhythm, Dr. North starts treatment with metoprolol in the office to gradually slow his heart rate, tells him to return the next day, and writes out a referral slip to see Dr. Jonathan Hartwell, a cardiologist in a nearby small city._\n\n_Dr. Hartwell arranges a stress echocardiogram at the local hospital to evaluate Mr. Maple's arrhythmia, finds severe coronary ischemia, and explains to Mr. Maple that his coronary arteries are narrowed. He recommends a coronary angiogram and possible coronary artery bypass surgery. Because Mr. Maple's condition is not urgent, Dr. Hartwell arranges for his patient to be placed on the waiting list at the University Hospital in the provincial capital 50 miles away. One month later, Mr. Maple awakens at 2 AM in a cold sweat, gasping for breath. His daughter calls Dr. North, who urgently sends for an ambulance to transport Mr. Maple to the University Hospital. There Mr. Maple is admitted to the coronary care unit, his condition is stabilized, and he undergoes emergency coronary artery bypass surgery the next day. Ten days later, Mr. Maple returns home, complaining of pain in his incision but otherwise feeling well._\n\nApproximately half of Canadian physicians are family physicians (contrasted with the United States, where only 35% of physicians are generalists). Canadians have free choice of physician. As a rule, Canadians see their family physician for routine medical problems and visit specialists only through referral by the family physician. Specialists are allowed to see patients without referrals, but only receive the higher specialist fee if they specify the referring primary care physician in their billing; for that reason, most specialists will not see patients without a referral. Unlike the European model of separation between ambulatory and hospital physicians, Canadian family physicians are allowed to care for their patients in hospitals. Because of the close scientific interchange between Canada and the United States, the practice of Canadian medicine is similar to that in the United States; the differences lie in the financing system and the far greater use of primary care physicians. The treatment of Mr. Maple's heart condition is not significantly different from what would occur in the United States, with the exception that high-tech procedures such as cardiac surgery and magnetic resonance imaging (MRI) scans are regionalized in a limited number of facilities and performed far less frequently than in the United States. In 2007, Canada had 6.7 MRI scanners per million inhabitants compared with 25.9 in the United States (OECD, 2010).\n\nCanadians on average wait longer for elective operations than do insured people in the United States and also have slightly more difficulty accessing primary care physicians (Schoen et al, 2010). Over the past ten years the federal and provincial governments have implemented successful policies to reduce elective surgery delays (Ross and Detsky, 2009). The median 2005 wait time for nonemergency surgery in Canada was 4 weeks (Willcox et al, 2007). Despite queues for elective procedures, only a tiny number of Canadians cross the border to seek care in the United States (Katz et al, 2002).\n\nCanada's universal insurance program has created a fairer system for distributing health services. Canadians are much less likely than their counterparts in the United States to report experiencing financial barriers to medical care (Schoen et al, 2010). Low-income Canadians receive almost the same amount of medical services as Canadians from higher-income groups, whereas in the United States higher-income groups receive more health services than lower-income groups (Sanmartin et al, 2006). Nonetheless, inequities in care according to socioeconomic status remain in Canada despite universal insurance coverage (Guilfoyle, 2008).\n\n#### **Paying Physicians and Hospitals**\n\n_For Dr. Rebecca North, collecting fees is a simple matter. Each week she electronically bills the provincial government, listing the patients she saw and the services she provided. Within a month, she is paid in full according to a fee schedule. Dr. North wishes the fees were higher, but loves the simplicity of the billing process. Her staff spends 2 hours per week on billing, compared with the 30 hours of staff time her friend Dr. South in Michigan needs for billing purposes._\n\n_Dr. North is less happy about the global budget approach used to pay hospitals. She often begs the hospital administrator to hire more physical therapists, to speed up the reporting of laboratory results, and to institute a program of diabetic teaching. The administrator responds that he receives a fixed payment from the provincial government each year, and there is no extra money._\n\nMost physicians in Canada\u2014primary care physicians and specialists\u2014are paid on a fee-for-service basis, with fee levels negotiated between provincial governments and provincial medical associations (Figure 14\u20132). Physicians participating in the provincial programs must accept the government rate as payment in full and cannot bill patients directly for additional payment. Because fee-for-service payment emphasizes volume over quality of care and makes cost control difficult (see Chapter 9), Canadian provinces are experimenting with alternative forms of payment such as salary or capitation for physicians in group practice and clinic settings. By 2009, many primary care physicians in the province of Ontario were being paid capitation with bonuses for high quality (Collier, 2009).\n\nCanadian hospitals, most of which are private nonprofit institutions, negotiate a global budget with the provincial government each year. Hospitals have no need to prepare the itemized patient bills that are so administratively costly in the United States. Hospitals must receive approval from their provincial health plan for new capital projects such as the purchase of expensive new technology or the construction of new facilities. Canada also regulates pharmaceutical prices and provincial plans maintain formularies of drugs approved for coverage.\n\n#### **Cost Control**\n\nThe Canadian system has attracted the interest of many people in the United States because in contrast to the United States, the Canadians have found a way to deliver comprehensive care to their entire population at far less cost. In 1970, the year before Canada's single-payer system was fully in place, Canada and the United States spent approximately the same proportion of their gross domestic products on health care\u20147.2% and 7.4%, respectively. By 1990, Canada's health expenditures had risen to 9% of the gross domestic product, compared with 12% for the United States. In 2008, Canada dedicated 10.4% of its gross domestic product to health care while the United States reached 16% (OECD, 2010). The differences in cost between the United States and Canada are primarily accounted for by four items: (1) administrative costs, which are more than 300% greater per capita in the United States; (2) more widespread use of expensive high-tech services in the United States; (3) cost per patient day in hospitals, which reflects a greater intensity of service in the United States; and (4) physician fees and pharmaceutical prices, which are much higher in the United States (Anderson et al, 2003; Woolhandler et al, 2003; Reinhardt, 2008).\n\nWhile 2008 Canadian per capita health care costs ($4079) were far lower than those in the United States ($7538), Canada was the fifth highest on that measure among developed nations (OECD, 2010). Canadian concern with cost increases began in the 1990s, when Canadian provinces put into effect caps on physician payments similar to those used in Germany (Barer et al, 1996).\n\nHowever, the Canadian federal government's fiscal austerity policies of the 1990s appear to have shaken the public's traditionally high level of confidence in the Canadian health care system. In 2010, about one-quarter of Canadians were not confident that they would receive the care they needed (Schoen et al, 2010). This unrest in public opinion has prompted vigorous debate in Canada about whether to allow greater private financing of health care, raise taxes to increase public financing, or restructure services to improve efficiency (Steinbrook, 2006). By 2010, Canada had opted for the latter two options: a commitment of substantial increases in federal funds for provincial health plans coupled with reform of the organization of primary care and other services (Hutchison et al, 2011).\n\n### **THE UNITED KINGDOM**\n\n#### **Health Insurance**\n\n_Roderick Pound owns a small bicycle repair shop in the north of England; he lives with his wife and two children. His sister Jennifer is a lawyer in Scotland. Roderick's younger brother is a student at Oxford, and their widowed mother, a retired sales-woman, lives in London. Their cousin Anne is totally and permanently disabled from a tragic automobile accident. A distant relative, who became a US citizen 15 years before, recently arrived to help care for Anne._\n\n_Simply by virtue of existing on the soil of the United Kingdom\u2014whether employed, retired, disabled, or a foreign visitor\u2014each of the Pound family members is entitled to receive tax-supported medical care through the National Health Service (NHS)._\n\nIn 1911, Great Britain established a system of health insurance similar to that of Germany. Approximately half the population was covered, and the insurance arrangements were highly complex, with contributions flowing to \"friendly societies,\" trade union and employer funds, commercial insurers, and county insurance committees. In 1942, the world's most renowned treatise on social insurance was published by Sir William Beveridge. The Beveridge Report proposed that Britain's diverse and complex social insurance and public assistance programs, including retirement, disability and unemployment benefits, welfare payments, and medical care, be financed and administered in a simple and uniform system. One part of Beveridge's vision was the creation of a national health service for the entire population. In 1948, the NHS began.\n\nThe great majority of NHS funding comes from taxes. As in Canada, the United Kingdom completely separates health insurance from employment, and no distinction exists between social insurance and public assistance financing. Unlike Canada, the United Kingdom allows private insurance companies to sell health insurance for services also covered by the NHS. A number of affluent people\u201412.5% of the population in 2007\u2014purchase private insurance in order to receive preferential treatment, \"hopping over\" the queues for services present in parts of the NHS. Some employers offer such supplemental insurance as a perk. People with private insurance are also paying taxes to support the NHS (Figure 14\u20133).\n\n**Figure 14\u20133.** The British National Health Service: traditional model.\n\n#### **Medical Care**\n\n_Dr. Timothy Broadman is an English GP, whose list of patients numbers 1750. Included on his list is Roderick Pound and his family. One day, Roderick's son broke his leg playing soccer. He was brought to the NHS district hospital by ambulance and treated by Dr. Pettibone, the hospital orthopedist, without ever seeing Dr. Broadman._\n\n_Roderick's mother has severe degenerative arthritis of the hip, which Dr. Broadman cares for. A year ago, Dr. Broadman sent her to Dr. Pettibone to be evaluated for a hip replacement. Because this was not an emergency, Mrs. Pound required a referral from Dr. Broadman to see Dr. Pettibone. The orthopedist examined and x-rayed her hip and agreed that she needed a hip replacement, but not on an urgent basis. Mrs. Pound has been on thewaiting list for her surgery for more than 6 months. Mrs. Pound has a wealthy friend with private health insurance who got her hip replacement within three weeks from Dr. Pettibone, who has a private practice in addition to his employment with the NHS._\n\nPrior to the NHS, most primary medical care was delivered through GPs. The NHS maintained this tradition and formalized a gatekeeper system by which specialty and hospital services (except in emergencies) are available only by referral from a GP. Every person in the United Kingdom who wants to use the NHS must be enrolled on the list of a GP. There is free choice of GP (unless the GP's list of patients is full), and people can switch from one GP's list to another.\n\nWhereas the creation of the NHS in 1948 left primary care essentially unchanged, it revolutionized Britain's hospital sector. As in the United States, hospitals had mainly been private nonprofit institutions or were run by local government; most of these hospitals were nationalized and arranged into administrative regions. Because the NHS unified the United Kingdom's hospitals under the national government, it was possible to institute a true regionalized plan (see Chapter 5).\n\nPatient flow in a regionalized system tends to go from GP (primary care for common illnesses) to local hospital (secondary care for more serious illnesses) to regional or national teaching hospital (tertiary care for complex illnesses). Traditionally, most specialists have had their offices in hospitals. As in Germany, GPs do not provide care in hospitals. GPs have a tradition of working closely with social service agencies in the community, and home care is highly developed in the United Kingdom.\n\n#### **Paying Physicians and Hospitals**\n\n_Dr. Timothy Broadman does not think much about money when he goes to his surgery (office) each morning. He receives a payment from the NHS to cover part of the cost of running his office, and every month he receives a capitation payment for each of the 1750 patients on his list. Ten percent of his income has been coming from extra fees he receives when he gives vaccinations to the kids; does Pap smears, family planning, and other preventive care; and makes home visits after hours. Recently, he also received a substantial bonus from the new pay-for-performance system._\n\nSince early in the twentieth century, the major method of payment for British GPs has been capitation (see Chapter 4). This mode of payment did not change when the NHS took over in 1948. The NHS did add some fee-for-service payments as an encouragement to provide certain preventive services and home visits during nights and weekends. Consultants (specialists) are salaried employees of the NHS, although some consultants are allowed to see privately insured patients on the side, whom they bill fee-for-service.\n\nIn 2004, a major new payment mode began for GPs: pay for performance (P4P) (see Chapter 10), known in the United Kingdom as the Quality and Outcomes Framework. NHS management negotiated the program with the British Medical Association (BMA), and the success of the negotiations was in large part because of the government's policy of increasing payment to GP, whose average income rose by 60% from 2002 to 2007, with GP incomes approaching those of hospital specialists (Doran and Roland, 2010). The NHS and BMA agreed on dozens of clinical indicators measuring quality for preventive services and common chronic illnesses such as coronary heart disease, hypertension, diabetes, and asthma. In addition, physician practices are measured on practice organization\u2014involving such measures as documentation in medical records, ability of patients to access the practice by phone, computerization, and safe management of medications\u2014and on the patient experience as measured by patient surveys. Physician practices were awarded a maximum of 1050 points for GPs who performed well on all these measures. In 2005, each point was worth approximately \u00a3120 annually (more than $200). GP practices achieving maximum quality could potentially increase earnings by approximately $77,000 per physician (Roland, 2004).\n\nIn preparation for P4P, UK GP practices employed more nurses, established chronic disease clinics, and increased use of electronic medical records. In the first year of the program, practices in England scored a median of 1003 points, suggesting that a high level of quality was achieved. Moreover, performance improved faster among lower-quality practices, which narrowed inequalities in care. As a result, GP income increased markedly and the cost to the NHS was far greater than expected.\n\nThe extent to which actual quality was improved is unclear; successes may have been related in part to improved documentation rather than improved quality. Practices were allowed to exclude certain patients in the performance calculations on the basis of repeated no-shows, serious comorbidities, and other factors, introducing the possibility of \"gaming\" the system. An analysis of performance improvement prior to and following the introduction of P4P suggests that performance had been increasing before P4P, but that quality increased slightly faster after P4P for some chronic conditions. Nurses in GP practices were responsible for much of the quality improvement, as GPs delegated many preventive and chronic care tasks to them.\n\nBy 2009, an evaluation of the Quality and Outcomes Framework revealed that the rate of improvement in the quality of care increased for asthma and diabetes from 2003 to 2005, but not for heart disease. By 2007 the rate of improvement had slowed for all three conditions. Many practices had reached the quality benchmarks, which meant that the financial incentive to continue improving was blunted. Moreover, performance for quality measures removed from the Framework fell in some cases, suggesting that practices might neglect quality of care unassociated with financial rewards. No significant changes were found in patient reports of access to care and interpersonal aspect of care, but continuity of care decreased after the introduction of the Framework (Campbell et al, 2009; Doran and Roland, 2010).\n\n#### **Cost Control**\n\nHealth expenditures in the United Kingdom accounted for 7.0% of the gross domestic product (GDP) in 2000, far below the US figure of 13.4%. Believing that the NHS needed more resources, the government of Prime Minister Tony Blair infused the NHS with a major increase in funds. Between 1999 and 2004, the number of NHS physicians increased by 25%. In addition, the pay-for-performance system channeled the equivalent of several billion new dollars into physician practices (Roland, 2004; Klein, 2006). By 2008, health expenditures as a proportion of the GDP had risen to 8.7% and per capita spending had increased from $1837 (2000) to $3129 (2008), a 37% increase (OECD, 2010). In 2005, as a result of this large growth in health expenditures, the NHS found itself in a serious deficit and scaled back some of the increase in NHS staffing (Klein, 2006).\n\nIn spite of these developments, the United Kingdom continues to have a relatively low level of per capita health expenditures. Two major factors allow the United Kingdom to keep its health care costs low: the power of the governmental single payer to limit budgets and the mode of reimbursement of physicians. While Canada also has a single payer of health services, it pays most physicians fee-for-service and had to create physician expenditure caps (like Germany) in an attempt to control the inflationary tendencies of fee-for-service reimbursement. In contrast, the United Kingdom relies chiefly on capitation and salary to pay physicians; payment can more easily be controlled by limiting increases in capitation payments and salaries. Moreover, because consultants (specialists) in the United Kingdom are NHS employees, the NHS can and does tightly restrict the number of consultant slots, including those for surgeons. As a result, queues have developed for nonemergency consultant visits and elective surgeries (Hurst and Siciliani, 2003). From 2005 to 2007, 30% of patients with cerebrovascular events and an indication for carotid artery surgery experienced a delay of over 12 weeks in spite of national guidelines recommending surgery within 2 weeks of the onset of symptoms (Halliday et al, 2009). In 2006, the United Kingdom had 5.6 MRI scanners per million population compared with the US rate of 25.9 (OECD, 2010). Overall, the United Kingdom controls costs by controlling the supply of personnel and facilities and the budget for medical resources, and by investing heavily in a primary care system that has achieved some of the best quality measures in the developed world (Doran and Roland, 2010).\n\nThe United Kingdom is often viewed as a nation that rations certain kinds of health care. In fact, primary and preventive care are not rationed, and average waiting times to see a GP in the United Kingdom are significantly shorter than those for people in the US seeking medical appointments (Schoen et al, 2010). Overall, a striking characteristic of British medicine is its economy. British physicians simply do less of nearly everything\u2014perform fewer surgeries, prescribe fewer medications, order fewer x-rays, and are more skeptical of new technologies than US physicians (Payer, 1988).\n\n#### **Reforms of the National Health Service**\n\nA series of dramatic structural changes have been introduced into the NHS over the past 2 decades. In 1991, the Conservative government of Margaret Thatcher implemented market-style reforms requiring hospitals to compete for business by reducing delays for specialty and surgical care, and introducing general practitioner fundholding, by which GP practices could choose to receive a global budget to purchase all care for their panel of patients. In 1997, Tony Blair's Labor government abolished GP fundholding and replaced it with primary care trusts\u2014a network of GPs working in the same district. All GP practices were required to join a primary care trust, which was given the responsibility for planning primary care and community health services in its area, contracting with hospitals and hospital consultants for specialty care, scrutinizing GP practice patterns, and implementing quality improvement activities. The average primary care trust had approximately 50 GP members, as well as additional primary care representatives from other professions, and covered a population of approximately 100,000 enrolled patients (Figure 14\u20134). Eighty-five percent of NHS funding flowed through the trusts, which were responsible for contracting for specialty and hospital services (Klein, 2004). As a result of the package of reforms (primary care trusts, the Quality and Outcomes Framework, and increased NHS funding), waiting times dropped, primary care access increased, chronic disease outcomes improved, and patient satisfaction grew.\n\n**Figure 14\u20134.** The British National Health Service: Recent reforms.\n\nIn 2010, the new coalition government proposed yet another major structural reform, abolishing the primary care trusts but strengthening the policy of giving groups of GPs large budgets from which they will fund primary care and buy specialty care for their patients. These GP commissioning groups will receive up to 70% of the NHS budget. GPs will either organize consortia to receive their budgets or be assigned to a consortium. This reform is touted as a shift in control from managers to physicians even though it is not clear that GPs want to manage budgets. As of early 2011, 170 consortia have been formed and 100 more are emerging. As the third major upheaval in 20 years, with each turnaround requiring several years to implement, it is unclear how health care providers and patients will fare in this constantly changing environment (Roland and Rosen, 2011), with critics complaining about a pattern of repeated \"redisorganization\" of the NHS from one governing party to the next.\n\n### **JAPAN**\n\n#### **Health Insurance**\n\n_Akiko Tanino works in the accounting department of the Mazda car company in Tokyo. Like all Mazda employees, she is enrolled in the health insurance plan directly operated by Mazda. Each month, 4% of Akiko's salary is deducted from her paycheck and paid to the Mazda health plan. Mazda makes an additional payment to its health plan equivalent to 4% of Akiko's salary._\n\n_Akiko's father Takeshi recently retired after working for many years as an engineer at Mazda. When he retired, his health insurance changed from the Mazda company plan to the community-based health insurance plan administered by the municipal government where he lives. Mazda makes payments to this health insurance plan to help pay for the health care costs of the company's retirees. In addition, the health insurance plan requires that Takeshi pay the plan a premium indexed to his income._\n\n_Akiko's brother Kazuo is a mechanic at a small auto repair shop in Tokyo. He is automatically enrolled in the government-managed health insurance plan operated by the Japanese national government. Kazuo and his employer each contribute payments equal to 4.1% of Kazuo's salary to the government plan._\n\nAlthough Japanese society has a cultural history distinct from the other nations discussed in this chapter, its health care system draws heavily from European and North American traditions. Similar to Germany, Japan's modern health insurance system is rooted in an employment-linked social insurance program. Japan first legislated mandatory employment-based social insurance for many workers in 1922, building on preexisting voluntary mutual aid societies. The system was gradually expanded until universal coverage was achieved in 1961 with passage of the National Health Insurance Act. The Japanese insurance system differs from the German model by having different categories of health plans with even more numerous individual plans and less flexibility in choice of plan (Figure 14\u20135).\n\n**Figure 14\u20135.** The Japanese health system.\n\nEmployers with 700 or more employees are required to operate self-insured plans for their employees and dependents, known as \"society-managed insurance\" plans. Although these plans resemble the German industry-specific sickness funds, each company must operate its own individual health plan. Approximately 1800 different employer-based plans exist. Eighty-five percent of these society plans are operated by individual companies, with the balance operated as joint plans between two or more employers, although none involve as many companies as the typical German sickness fund. The boards of directors of society plans comprise 50% employee and 50% employer representatives. Employees and their dependents are required to enroll in their company's society plan, and the employee and the employer must contribute a premium to fund the society. Because each plan is self-insured, the premium rate varies (from 3% to 9.5% in 2006) depending on the average income and health risk of the company's employees, creating considerable inequities (Imai, 2002; Kemporen, 2007). Society-managed insurance plans cover 24% of the Japanese population.\n\nEmployees and dependents in companies with fewer than 700 employees are compulsorily enrolled in a single national health insurance plan for small businesses that is operated by the national government. This government-managed insurance plan, primarily financed by a premium (8.2% in 2006) on employers and employees, covers 28% of the population. The federal government also uses general tax revenues to subsidize the government-managed insurance plan.\n\nYet a third type of health insurance, community-based health insurance (also called citizens' health insurance), covers self-employed workers and retirees (41% of the population). Each municipal government in Japan administers a local citizen's insurance plan and levies a compulsory premium on the self-employed workers and retirees in its jurisdiction. In addition, each employer-operated society-managed insurance plan and the single government-managed insurance plan must contribute payments to subsidize the costs for retirees. Approximately 40% of the financing for the citizens health insurance program comes from contributions from the society-managed and government-managed insurance plans, making employers liable for a large portion of the costs of their retirees' health care. Additional funds for the community-based health insurance plan come from general tax revenues.\n\nA smattering of smaller insurance programs exist for government employees and other special categories of workers, and resemble the society-managed insurance plans. Persons who become unemployed remain enrolled in their health plan with the payroll tax waived. All plans are required to provide standard comprehensive benefits, including payment for hospital and physician services, prescription drugs, maternity care, and dental care. In addition, in 2000 Japan implemented a new long-term insurance plan, financed by general tax revenues and a new earmarked income tax, which provides comprehensive benefits to disabled adults, including payment for home care, case management, and institutional services.\n\nBecause Japan's society is aging more rapidly than any other developed nation, inequities and imbalances have developed in the financing of care for the most expensive patients\u2014those at the highest age levels. In 2006, a new law was passed creating a more rational financing plan for retirees older than 75 (Kemporen, 2007).\n\nIn summary, Japan\u2014like Germany\u2014builds on an employment-based social insurance model, using additional general tax subsidies to create a universal insurance program. Compared with Germany, the national and local governments in Japan are more involved in directly administering health plans and a majority of Japanese are covered by government-run or government-managed plans rather than by employer-managed private plans (Kemporen, 2007).\n\n#### **Medical Care**\n\n_Takeshi Tanino's knee has been aching for several weeks. He makes an appointment at a clinic operated by an orthopedic surgeon. At the clinic Takeshi has a medical examination, an x-ray of the knee, and is scheduled for regular physical therapy. During the examination the orthopedist notes that Takeshi's blood pressure is high and recommends that Takeshi see an internist at a different clinic about this problem._\n\n_Six months later, Takeshi develops a cough and fever. He makes an appointment at the medical clinic of a nearby hospital run by Dr. Suzuki, is diagnosed with pneumonia, and is admitted to the medical ward. He is treated with intravenous antibiotics for 2 weeks and remains in the hospital for an additional 2 weeks after completing antibiotics for further intravenous hydration and nursing care._\n\nHealth plans place no restrictions on choice of hospital and physician and do not require preauthorization before using medical services. Most medical care is based on three types of settings: (1) independent clinics, each owned by a physician and staffed by the physician and other employees, with many clinics also having small inpatient wards; (2) small hospitals with inpatient and outpatient departments, owned by a physician with employed physician staff; and (3) larger public and private hospitals with outpatient and inpatient departments and salaried physician staff. Facilities are organized by specialty, with larger hospitals having a wide range of specialties and smaller hospitals and clinics offering a more limited selection of specialty departments. Care is delivered in a specialty-specific manner, with a few organizations using a primary care-oriented gatekeeper model (Reid, 2009).\n\nPhysician entrepreneurship is a strong element in the organization of health care in Japan. Most clinics and small hospitals are family-owned businesses founded and operated by independent physicians. Unlike clinics in the United States such as the Mayo Clinic and Palo Alto Medical Foundation that began as family-owned institutions but evolved into nonprofit organizations with ownership shared among a larger group of physician partners, most clinics in Japan have remained under the ownership of a single physician, often passed down within a family from one generation to another. Many physicians expanded their clinics to become small hospitals, but the government builds and operates the larger medical centers. The distinction between clinics and hospitals in Japan is not as great as in most nations. Clinics are permitted to operate inpatient beds and only become classified as hospitals when they have more than 20 beds. Approximately 30% of clinics in Japan have inpatient beds. Virtually all physicians either own clinics and hospitals or work as employees of a clinic or hospital, and practice only within their single institution. Although many physician-owned clinics and hospitals are modest facilities, others are larger institutions offering a wide array of outpatient and inpatient services featuring the latest biomedical technology, electronic medical records, and automated dispensing of medications.\n\nRates of hospital admission are relatively low in Japan and rates of surgery are approximately one-third the rate in the United States. A cultural norm that makes patients reluctant to undergo invasive procedures in part explains the low surgical rate in Japan. When hospitalized, patients remain unusually long compared with most developed nations; average lengths of stay vary by hospital from 16 to 29 days. Patients are allowed long periods to convalesce while still in the hospital (Ikegami and Campbell, 2004).\n\n#### **Paying Physicians and Hospitals**\n\n_One month after returning home from the hospital, Takeshi Tanino develops stomach pain that awakens him several nights. He makes an appointment at a general medical clinic run by Dr. Sansei. Dr. Sansei performs an endoscopy, which reveals gastritis. Dr. Sansei prescribes an H 2 blocker and arranges for Takeshi to return to the clinic every 4 weeks for the next 6 months. Takeshi's stomach ache improves after a few days of using the medication. At each follow-up visit, Dr. Sansei questions Takeshi about his symptoms and dispenses a new 4-week supply of medications._\n\nUntil recently, insurance plans paid both physicians and hospitals on a fee-for-service basis. In 2003, a per diem hospital payment based on diagnosis was introduced (Nawata et al, 2009) while physicians continue to be paid fee-for-service. The government strictly regulates physician fees, hospital payments, and medication prices, which are very low by US standards. The fee schedule is in many ways the opposite of US fees: In Japan, primary care services tend to command higher fees than do more specialized services such as surgical procedures and imaging studies. Services such as MRI scans that have shown large increases in volumes have had substantial cuts in fees (Ikegami and Campbell, 2004). Based on fee schedules in place in 2007, a family physician office visit might be reimbursed $5 or $10, one night's stay in a hospital $11, and a brain MRI $105 (Reid, 2009). Physicians make up for low fees with high volume, at times seeing 60 patients per day. In 2007 the number of physician visits per capita was 13.4, compared with 4.0 for the United States (OECD, 2010). Physicians are permitted to directly dispense medications, not just to prescribe them, and make a profit from the sale of pharmaceuticals. The government recently restricted how much physicians could charge patients for medications (Kemporen, 2007), but many physician visits are solely for the purpose of refilling medications. Quality of care in Japan is not systematically measured and is believed to vary greatly among physicians and hospitals (Henke et al, 2009).\n\n#### **Cost Control**\n\nHealth care costs in Japan were only 8.1% of GDP in 2007. However this is a considerable rise from 1990's 6.0%, and concerns are mounting due to Japan's demographics. The health care system relies heavily on payroll taxes and thus requires a large employed population. But with, a plummeting birth rate and the longest life expectancy in the world, Japan's population is aging faster than that of other developed nations. The proportion of Japanese older than 65 years is projected to increase from 12% in 1990 to 40% in 2050 (Kemporen, 2007). In comparison, the proportion of the US population older than 65 years will increase much more modestly, from 12% to 21%, during this same period.\n\nThrough its fee schedule, the government has kept medical prices low, which is the main cost containment strategy. But physicians are unhappy and see too many patients for short visits, while many hospitals are old and underfunded. The stresses resulting from Japan's demographic reality and its overstretched health care providers make for an uncertain future (Reid, 2009).\n\n### **CONCLUSION**\n\nKey issues in evaluating and comparing health care systems are access to care, level of health expenditures, public satisfaction with health care, and the overall quality of care as expressed by the health of the population. Germany, Canada, the United Kingdom, and Japan provide universal financial access to health care through government-run or government-mandated programs. These four nations have controlled health care costs more successfully than has the United States (Tables 14\u20131 and 14\u20132), though all four face challenges in containing their spending.\n\n**Table 14\u20131.** Total health expenditures as a percentage of gross domestic product (GDP), 1970\u20132008\n\n**Table 14\u20132.** Per capita health spending in US dollars, 2008\n\nSixteen percent of US adults surveyed in 2007 felt that the health system works well with only minor changes needed; 48% felt that fundamental change is needed, and 34% wanted the system rebuilt completely. Adults in Germany, Canada, and the United Kingdom had somewhat more favorable views of their health systems, though the majority in those countries also felt that major changes were needed (Schoen et al, 2007). Adults in the United States were much more likely than adults in Germany, the United Kingdom, and Canada to report problems with access to medical services due to costs (Figure 14\u20136).\n\n**Figure 14\u20136.** Problems accessing medical services due to costs.\n\nCrossnational comparisons of health care quality are treacherous since it is difficult to disentangle the impacts of socioeconomic factors and medical care on the health status of the population. But such comparisons can convey rough impressions of whether a health care system is functioning at a reasonable level of quality. From Table 14\u20133, it is clear that the United States has an infant mortality rate higher than Germany, Canada, the United Kingdom, and Japan, with the Japanese rate being the lowest. Japan also has the highest male and female life expectancy rates at birth. The life expectancy rate at age 65 is believed by some observers to measure the impact of medical care, especially its more high-tech component, more than it measures underlying socioeconomic influences. Even by this standard, the United States ranks below the other four nations (OECD, 2010). Researchers have developed another metric intended to assess the functioning of national health care systems, known as \"mortality amenable to health care\" (Nolte and McKee, 2008); the United States performs poorly on this metric as well relative to other nations (Table 14\u20133).\n\n**Table 14\u20133.** Health outcome measures\n\nJust as epidemiologic studies often derive their most profound insights from comparisons of different populations (see Chapter 11), research into health services can glean insights from the experience of other nations. As the United States confronts the challenge of achieving universal access to high-quality health care at an affordable cost, lessons may be learned from examining how other nations have addressed this challenge.\n\n### **REFERENCES**\n\nAnderson GF et al. It's the prices, stupid: Why the United States is so different from other countries. _Health Aff._ 2003;22(3):89.\n\nBarer ML, Lomas J, Sanmartin C. Re-minding our Ps and Qs: Medical cost controls in Canada. _Health Aff._ 1996;15(2):216.\n\nBusse R. The health system in Germany. _Eurohealth_. 2008;14(1):5.\n\nBusse R, Riesberg A. _Health Care Systems in Transition: Germany_. Copenhagen: WHO Regional Office for Europe; 2004.\n\nCampbell SM et al. Effects of pay for performance on the quality of primary care in England. _N Engl J Med_. 2009;261:368.\n\nCanadian Institute for Health Information. Health Care in Canada, 2010, December 2010. www.cihi.ca.\n\nCollier R. Shift toward capitation in Ontario. _Canadian Med Assoc J_. 2009;181:668.\n\nDoran T, Roland M. Lessons from major initiatives to improve primary care in the United Kingdom. _Health Aff._ 2010;29:1023. . Accessed November 17, 2011.\n\nGuilfoyle J. Prejudice in medicine. Our role in creating health care disparities. _Can Fam Physician._ 2008;54:1511.\n\nHalliday AW et al. Waiting times for carotid endarterectomy in UK. _BMJ._ 2009;338:b1847.\n\nHenke N et al. Improving Japan's health care system. _McKinsey Q._ 2009.\n\nHurst J, Siciliani L. _Tackling Excessive Waiting Times for Elective Surgery_ : _A Comparison of Policies in Twelve OECD Countries_. Paris: Organisation for Economic Co-operation and Development; 2003.\n\nHutchison B et al. Primary health care in Canada: Systems in motion. _Milbank_ Q. 2011:89(2):256.\n\nIkegami N, Campbell JC. Japan's health care system: Containing costs and attempting reform. _Health Aff._ 2004;23(3):26.\n\nImai Y. Health Care Reform in Japan. Organisation for Economic Co-operation and Development, February 2002. www.oecd.org.\n\nKatz SJ et al. Phantoms in the snow: Canadians' use of health care services in the United States. _Health Aff._ 2002;21(3):19.\n\nKemporen (National Federation of Health Insurance Societies). Health Insurance, Long-Term Care Insurance and Health Insurance Societies in Japan, 2007. Kemporen, 2007.\n\nKlein R. Britain's National Health Service revisited. _N Engl J Med._ 2004;350:937.\n\nKlein R. The troubled transformation of Britain's National Health Service. _N Engl J Med._ 2006;355:409.\n\nNawata K, et al. Analysis of the new medical payment system in Japan, July 2009. www.mssanz.org.au\/modsim09\/A2\/nawata.pdf.\n\nNolte E, McKee CM. Measuring the health of nations: Updating an earlier analysis. _Health Aff_. 2008;27:58. Erratum in: _Health Aff_. 2008;27:593.\n\nOrnyanova D, Busse R. Health Fund now operational Health Policy Monitor, May 2009. www.hpm.org.\n\nOrganisation for Economic Co-operation and Development. OECD Health Data, 2010. www.oecd.org.\n\nPayer L. _Medicine and Culture._ New York: Henry Holt; 1988. Reid TR. _The Healing of America_. New York: The Penguin Press; 2009.\n\nReinhardt U. Why does US health care cost so much? _Economix_. November 14, 2008. .\n\nRoland M. Linking physicians' pay to the quality of care\u2014a major experiment in the United Kingdom. _N Engl J Med._ 2004;351:1448.\n\nRoland M, Rosen R. British NHS embarks on controversial and risky market-style reforms in health care. _N Engl J Med_. 2011;364:1360.\n\nRoss JS, Detsky AS. Choice? Making health care decisions in the United States and Canada. _JAMA_. 2009;302:1803.\n\nSanmartin C et al. Comparing health and health care use in Canada and the United States. _Health Aff._ 2006;25:1133.\n\nSchoen C et al. Toward higher-performance health systems: Adults' health care experiences in seven countries, 2007. _Health Aff._ 2007;26:w717.\n\nSchoen C et al. How health insurance design affects access to care and costs: by income, in eleven countries. _Health Aff_. 2010;29:2323.\n\nSteinbrook R. Private health care in Canada. _N Engl J Med._ 2006;354:1661.\n\nStock S et al. The influence of the labor market on German health care reforms. _Health Aff._ 2006;25:1143.\n\nTaylor MG. _Insuring National Health Care. The Canadian Experience._ Chapel Hill, NC: University of North Carolina Press; 1990.\n\nWillcox S et al. Measuring and reducing waiting times: A cross-national comparison of strategies. _Health Aff._ 2007;26:1078.\n\nWoolhandler S et al. Costs of health care administration in the United States and Canada. _N Engl J Med._ 2003;349:768.\n\nZander B et al. Health policy in Germany after the election. _Health Policy Monitor_. November 2009. www.hpm.org.\n\n## **15 Health Care Reform and National Health Insurance**\n\nFor 100 years, reformers in the United States have argued for the passage of a national health insurance program, a government guarantee that every person is insured for basic health care. Finally in 2010, the United States took a major step forward toward universal health insurance.\n\nThe subject of national health insurance has seen six periods of intense legislative activity, alternating with times of political inattention. From 1912 to 1916, 1946 to 1949, 1963 to 1965, 1970 to 1974, 1991 to 1994, and 2009 to 2010, it was the topic of major national debate. In 1916, 1949, 1974, and 1994, national health insurance was defeated and temporarily consigned to the nation's back burner. Guaranteed health coverage for two groups\u2014the elderly and some of the poor\u2014was enacted in 1965 through Medicare and Medicaid. Expansion of coverage to over 30 million uninsured people was legislated with the Patient Protection and Affordable Care Act of 2010. National health insurance means the guarantee of health insurance for all the nation's residents\u2014what is commonly referred to as \"universal coverage.\" Most of the focus, as well as the political contentiousness, of national health insurance proposals tends to concern how to finance universal coverage. Because health care financing is so interwoven with provider reimbursement and cost containment, national health insurance proposals usually also address those topics.\n\nThe controversies that erupt over universal health care coverage become simpler to understand if one returns to the four basic modes of health care financing outlined in Chapter 2: out-of-pocket payment, individual private insurance, employment-based private insurance, and government financing. There is general agreement that out-of-pocket payment does not work as a sole financing method for costly contemporary health care. National health insurance involves the replacement of out-of-pocket payments by one, or a mixture, of the other three financing modes.\n\nUnder government-financed national health insurance plans, funds are collected by a government or quasigovernmental fund, which in turn pays hospitals, physicians, health maintenance organizations (HMOs), and other health care providers. Under private individual or employment-based national health insurance, funds are collected by private insurance companies, which then pay providers of care.\n\nHistorically, health care financing in the United States began with out-of-pocket payment and progressed through individual private insurance, then employment-based insurance, and finally government financing for Medicare and Medicaid (see Chapter 2). In the history of US national health insurance, the chronologic sequence is reversed. Early attempts at national health insurance legislation proposed government programs; private employment-based national health insurance was not seriously entertained until 1971, and individually purchased universal coverage was not suggested until the 1980s (Table 15\u20131). Following this historical progression, we shall first discuss government-financed national health insurance, followed by private employment-based and then individually purchased universal coverage. The most recent chapter of this history is the enactment under the administration of President Obama of the Patient Protection and Affordable Care Act of 2010, a pluralistic approach to national health insurance that draws on all three of these financing models: government financing, employment-based private insurance, and individually purchased private insurance.\n\n**Table 15\u20131.** Attempts to legislate national health insurance\n\n### **GOVERNMENT-FINANCED NATIONAL HEALTH INSURANCE**\n\n#### **The American Association for Labor Legislation Plan**\n\nIn the early 1900s, 25% to 40% of people who became sick did not receive any medical care. In 1915, the American Association for Labor Legislation (AALL) published a national health insurance proposal to provide medical care, sick pay, and funeral expenses to lower-paid workers\u2014those earning less than $1200 a year\u2014and to their dependents. The program would be run by states rather than the federal government and would be financed by a payroll tax\u2013like contribution from employers and employees, perhaps with an additional contribution from state governments. Payments would go to regional funds (not private insurance companies) under extensive government control. The funds would pay physicians and hospitals. Thus, the first national health insurance proposal in the United States\u2014because the money was collected by quasi-public funds through a mandatory tax\u2014can be considered a government-financed program (Starr, 1982).\n\n_In 1910, Edgar Peoples worked as a clerk for Standard Oil, earning $800 a year. He lived with his wife and three sons. Under the AALL proposal, Standard Oil and Mr. Peoples would each pay $13 per year into the regional health insurance fund, with the state government contributing $6. The total of $32 (4% of wages) would cover the Peoples family._\n\nThe AALL's road to national health insurance followed the example of European nations, which often began their programs with lower-paid workers and gradually extended coverage to other groups in the population. Key to the financing of national health insurance was its compulsory nature; mandatory payments were to be made on behalf of every eligible person, ensuring sufficient funds to pay for people who fell sick.\n\nThe AALL proposal initially had the support of the American Medical Association (AMA) leadership. However, the AMA reversed its position and the conservative branch of labor, the American Federation of Labor, along with business interests, opposed the plan (Starr, 1982). The first attempt at national health insurance failed.\n\n#### **The Wagner\u2013Murray\u2013Dingell Bill**\n\nIn 1943, Democratic Senators Robert Wagner of New York and James Murray of Montana, and Representative John Dingell of Michigan introduced a health insurance plan based on the social security system enacted in 1935. Employer and employee contributions to cover physician and hospital care would be paid to the federal social insurance trust fund, which would in turn pay health providers. The Wagner\u2013Murray\u2013Dingell bill had its lineage in the New Deal reforms enacted during the administration of President Franklin Delano Roosevelt. President Roosevelt had initially considered including a national health plan as part of the Social Security Act, but facing resistance from the AMA decided to omit health reform from the New Deal legislative package.\n\n_In the 1940s, Edgar Peoples' daughter Elena worked in a General Motors plant manufacturing trucks to be used in World War II. Elena earned $3500 per year. Under the 1943 Wagner\u2013Murray\u2013Dingell bill, General Motors would pay 6% of her wages up to $3000 into the social insurance trust fund for retirement, disability, unemployment, and health insurance. An identical 6% would be taken out of Elena's check for the same purpose. One-fourth of this total amount ($90) would be dedicated to the healthinsurance portion of social security. If Elena or her children became sick, the social insurance trust fund would reimburse their physician and hospital._\n\n_Edgar Peoples, in his seventies, would also receive health insurance under the Wagner\u2013Murray\u2013Dingell bill, because he was a social security beneficiary._\n\n_Elena's younger brother Marvin was permanently disabled and unable to work. Under the Wagner\u2013Murray\u2013Dingell bill he would not have received government health insurance unless his state added unemployed people to the program._\n\nAs discussed in Chapter 2, government-financed health insurance can be divided into two categories. Under the social insurance model, only those who pay into the program, usually through social security contributions, are eligible for the program's benefits. Under the public assistance (welfare) model, eligibility is based on a means test; those below a certain income may receive assistance. In the welfare model, those who benefit may not necessarily contribute, and those who contribute (usually through taxes) may not benefit (Bodenheimer and Grumbach, 1992). The Wagner\u2013Murray\u2013Dingell bill, like the AALL proposal, was a social insurance proposal. Working people and their dependents were eligible because they made social security contributions, and retired people receiving social security benefits were eligible because they paid into social security prior to their retirement. The permanently unemployed were not eligible.\n\nIn 1945, President Truman, embracing the general principles of the Wagner\u2013Murray\u2013Dingell legislation, became the first US president to strongly champion national health insurance. After Truman's surprise election in 1948, the AMA succeeded in a massive campaign to defeat the Wagner\u2013Murray\u2013Dingell bill. In 1950, national health insurance returned to obscurity (Starr, 1982).\n\n#### **Medicare and Medicaid**\n\nIn the late 1950s, less than 15% of the elderly had health insurance (see Chapter 2) and a strong social movement clamored for the federal government to come up with a solution. The Medicare law of 1965 took the Wagner\u2013Murray\u2013Dingell approach to national health insurance, narrowing it to people 65 years and older. Medicare was financed through social security contributions, federal income taxes, and individual premiums. Congress also enacted the Medicaid program in 1965, a public assistance or \"welfare\" model of government insurance that covered a portion of the low-income population. Medicaid was paid for by federal and state taxes.\n\n_In 1966, at age 66, Elena Peoples was automatically enrolled in the federal government's Medicare Part A hospital insurance plan, and she chose to sign up for the Medicare Part B physician insurance plan by paying a $3 monthly premium to the Social Security Administration. Elena's son, Tom, and Tom's employer helped to finance Medicare Part A; each paid 0.5% of wages (up to a wage level of $6600 per year) into a Medicare trust fund within the social security system. Elena's Part B coverage was financed in part by federal income taxes and in part by Elena's monthly premiums. In case of illness, Medicare would pay for most of Elena's hospital and physician bills._\n\n_Elena's disabled younger brother, Marvin, age 60, was too young to qualify for Medicare in 1966. Marvin instead became a recipient of Medicaid, the federal\u2013state program for certain groups of low-income people. When Marvin required medical care, the state Medicaid program paid the hospital, physician, and pharmacy, and a substantial portion of the state's costs were picked up by the federal government._\n\nMedicare is a social insurance program, requiring individuals or families to have made social security contributions to gain eligibility to the plan. Medicaid, in contrast, is a public assistance program that does not require recipients to make contributions but instead is financed from general tax revenues. Because of the rapid increase in Medicare costs, the social security contribution has risen substantially. In 1966, Medicare took 1% of wages, up to a $6600 wage level (0.5% each from employer and employee); in 2004, the payments had risen to 2.9% of all wages. The Part B premium has jumped from $3 per month in 1966 to $115.40 per month in 2011.\n\n#### **The 1970 Kennedy Bill and the Single-Payer Plan of the 1990s**\n\nMany people believed that Medicare and Medicaid were a first step toward universal health insurance. European nations started their national health insurance programs by covering a portion of the population and later extending coverage to more people. Medicare and Medicaid seemed to fit that tradition. Shortly after Medicare and Medicaid became law, the labor movement, Senator Edward Kennedy of Massachusetts, and Representative Martha Griffiths of Michigan drafted legislation to cover the entire population through a national health insurance program. The 1970 Kennedy\u2013Griffiths Health Security Act followed in the footsteps of the Wagner\u2013Murray\u2013Dingell bill, calling for a single federally operated health insurance system that would replace all public and private health insurance plans.\n\n_Under the Kennedy\u2013Griffiths 1970 Health Security Program, Tom Peoples, who worked for Great Books, a small book publisher, would continue to see his family physician as before. Rather than receiving payment from Tom's private insurance company, his physician would be paid by the federal government, perhaps through a regional intermediary. Tom's employer would no longer make a social security contribution to Medicare (which would be folded into the Health Security Program) and would instead make a larger contribution of 3% of wages up to a wage level of $15,000 for each employee. Tom's employee contribution was set at 1% up to a wage level of $15,000. These social insurance contributions would pay for approximately 60% of the program; federal income taxes would pay for the other 40%._\n\n_Tom's Uncle Marvin, on Medicaid since 1966, would be included in the Health Security Program, as would all residents of the United States. Medicaid would be phased out as a separate public assistance program._\n\nThe Health Security Act went one step further than the AALL and Wagner\u2013Murray\u2013Dingell proposals: It combined the social insurance and public assistance approaches into one unified program. In part because of the staunch opposition of the AMA and the private insurance industry, the legislation went the way of its predecessors: political defeat.\n\nIn 1989, Physicians for a National Health Program offered a new government-financed national health insurance proposal. The plan came to be known as the \"single-payer\" program, because it would establish a single government fund within each state to pay hospitals, physicians, and other health care providers, replacing the multipayer system of private insurance companies (Himmelstein and Woolhandler, 1989). Several versions of the single-payer plan were introduced into Congress in the 1990s, each bringing the entire population together into one health care financing system, merging the social insurance and public assistance approaches (Table 15\u20132). The California Legislature, with the backing of the California Nurses Association, passed a single-payer plan in 2006 and 2008, but the proposals were vetoed by the Governor.\n\n**Table 15\u20132.** Categories of national health insurance plans\n\n### **THE EMPLOYER-MANDATE MODEL OF NATIONAL HEALTH INSURANCE**\n\nIn response to Democratic Senator Kennedy's introduction of the 1970 Health Security Act, President Nixon, a Republican, countered with a plan of his own, the nation's first employment-based, privately administered national health insurance proposal. For 3 years, the Nixon and Kennedy approaches competed in the congressional battleground; however, because most of the population was covered under private insurance, Medicare, or Medicaid, there was relatively little public pressure on Congress. In 1974, the momentum for national health insurance collapsed, not to be seriously revived until the 1990s. The essence of the Nixon proposal was the employer mandate, under which the federal government requires (or mandates) employers to purchase private health insurance for their employees.\n\n_Tom Peoples' cousin Blanche was a receptionist in a physician's office in 1971. The physician did not provide health insurance to his employees. Under Nixon's 1971 plan, Blanche's employer would be required to pay 75% of the private health insurance premium for his employees; the employees would pay the other 25%._\n\n_Blanche's boyfriend, Al, had been laid off from his job in 1970 and was receiving unemployment benefits. He had no health insurance. Under Nixon's proposal, the federal government would pay a portion of Al's health insurance premium._\n\nNo longer was national health insurance equated with government financing. Employer mandate plans preserve and enlarge the role of the private health insurance industry rather than replacing it with tax-financed government-administered plans. While the Nixon plan preserved existing government programs such as Medicare and Medicaid, it proposed to expand coverage for the uninsured through a widening role for private, employment-based insurance. Government's new role under the Nixon plan would be to enforce private health insurance as a required benefit for employed people. The Nixon proposal changed the entire political landscape of national health insurance, moving it toward the private sector. In later years, Senator Kennedy embraced the employer mandate approach himself, fearing that the opposition of the insurance industry and organized medicine would kill any attempt to legislate government-financed national health insurance.\n\nDuring the 1980s and 1990s, the number of people in the United States without any health insurance rose from 25 million to more than 40 million (see Chapter 3). Approximately three-quarters of the uninsured were employed or were dependents of employed persons. The rapidly rising cost of health insurance premiums made insurance unaffordable for many businesses. In response to this crisis in health care access, President Clinton submitted legislation to Congress in 1993 calling for universal health insurance through an employer mandate, as well as broadened eligibility for Medicaid. Like the Nixon proposal, the essence of the Clinton plan was the requirement that employers pay for most of their employees' private insurance premiums.\n\nA variation on the employer mandate type of national health insurance is the voluntary approach. Rather than requiring employers to purchase health insurance for employees, employers are given incentives such as tax credits to cover employees voluntarily. The attempt of some states to implement this type of voluntary approach has failed to significantly reduce the numbers of uninsured workers.\n\n### **THE INDIVIDUAL-MANDATE MODEL OF NATIONAL HEALTH INSURANCE**\n\nIn 1989, a new species of national health insurance appeared, sponsored by the conservative Heritage Foundation: the individual mandate. Just as many states require motor vehicle drivers to purchase automobile insurance, the Heritage plan called for the federal government to require all US residents to purchase individual health insurance policies. Tax credits would be made available on a sliding scale to individuals and families too poor to afford health insurance premiums (Butler, 1991). Under the most ambitious versions of universal individual insurance proposals, neither employer-sponsored group insurance nor government-administered insurance would continue to play a role in financing health care. These existing financing models would be dismantled and replaced by a universal, individual mandate program. Ironically, the individual insurance mandate shares at least one feature with the single-payer, government-financed approach to universal coverage: Both would severe the connection between employment and health insurance, allowing portability and continuity of coverage as workers moved from one employer to another or became self-employed.\n\n_Tom Peoples received health insurance through his employer, Great Books. Under an individual mandate plan, Tom would be legally required to purchase health insurance for his family. Great Bookscould offer a health plan to Tom and his coworkers but would not be required to contribute anything to the premium. If Tom purchased private health insurance for his family at a cost of $8000 per year, he would receive a tax credit of $4000 (ie, he would pay $4000 less in income taxes). Tom's Uncle Marvin, formerly on Medicaid, would be given a voucher to purchase a private health insurance policy._\n\nWith individual mandate health insurance, the tax credits may vary widely in their amount depending on characteristics such as household income and how much of a subsidy the architects of individual mandate proposals build into the plan. In a generous case, a family might receive a $10,000 tax credit, subsidizing much of its health insurance premium. If the family's tax liability is less than the value of the tax credit, the government would pay the family the difference between the family's tax liability and $10,000.\n\nA related version of the individual mandate is a voucher system. Instead of issuing tax credits, the federal government would issue a voucher for a fixed dollar amount that could be used toward the purchase of health insurance, just as some local government jurisdictions issue vouchers that may be used to enroll children in private schools. In the most sweeping proposals, a tax-financed voucher system would completely replace existing insurance programs directly administered by government as well as employer-sponsorship of private insurance (Emanuel and Fuchs, 2005). Another version of individual health insurance expansion is the voluntary concept, which was proposed by President George W. Bush. Uninsured individuals would not be required to purchase individual insurance but would receive a tax credit if they chose to purchase insurance. The level of the tax credits in the Bush plan and similar proposals have been small compared to the cost of most health insurance policies, with the result that these voluntary approaches if enacted would have induced very few uninsured people to purchase coverage.\n\n#### **The Massachusetts Individual Mandate Plan of 2006**\n\nNearly 20 years after the Heritage Foundation drafted a proposal for a national individual mandate, Massachusetts enacted a state-level universal health coverage bill implementing the nation's first legislated individual mandate. The Massachusetts plan, enacted under the leadership of Republican Governor Mitt Romney, mandates that every state resident must have health insurance coverage meeting a minimum standard set by the state. Individuals are required to provide proof of coverage at the time of filing their annual tax return, and face a financial penalty for failing to provide evidence of coverage. The state provides subsidies for purchase of private health insurance coverage to individuals with incomes below 300% of the federal poverty level if they are not covered by the state's Medicaid program or through employment-based insurance.\n\n_Brian Mayflower earns $16,000 a year as a waiter to support himself as an aspiring actor in Boston. He has chronic asthma, with his inhaler medications alone costing more than $1,000 annually. He is not eligible for Medicaid and is required under the Massachusetts Plan to purchase a private health plan. As a low-income person, Brian receives a state subsidy for most of the premium cost of the plan. The plan has a $500 per year deductible but pays for most of Brian's medications once he meets the annual deductible._\n\n_Brian's sister Dorothy Mayflower is a self-employed accountant living in Springfield and earning $58,000 a year. At her income level, the Massachusetts state subsidy for insurance coverage would leave her having to pay $3000 per year toward the premium for a plan that has a $2000 per year deductible. Dorothy is in good health and is having trouble paying the mortgage on her house, which recently ballooned. She decides she will not enroll in a health insurance plan and instead pays the $900 fine to the state for not complying with the individual mandate._\n\nLike the Nixon employer mandate proposal, the Massachusetts individual mandate does not eliminate existing government insurance programs; it extends the reach of private insurance through a government mandate, in this case for individually purchased private insurance. State government provides an income-adjusted subsidy for individual coverage for people not eligible for employer-sponsored insurance and limits the degree to which private plans can experience-rate their premiums. The Massachusetts plan allows insurers to offer policies with large amounts of cost-sharing in the form of high deductibles and coinsurance. The plan also includes a weak employer mandate, requiring employers with more than 10 employees to either contribute toward insurance coverage for their employees or pay into the state fund that underwrites public subsidies for the individual mandate and related programs.\n\nThe Massachusetts Health Plan of 2006 is credited with reducing the uninsurance rate among nonelderly adults in Massachusetts from 13% in 2006 to 5% in 2009 (Long and Stockley, 2010). Some residents of the state, such as Dorothy Mayflower, continue to have trouble affording private insurance even with some degree of state subsidy, and the high levels of cost-sharing allowed under the minimum benefit standards leave many insured individuals with substantial out-of-pocket payments. In 2008, 18% of low-income people in Massachusetts reported unmet health care needs due to costs (copayments, deductibles, uncovered services) and about 20% of the entire population experienced difficulty accessing primary care due to the primary care shortage (Clark et al, 2011).\n\n### **THE PLURALISTIC REFORM MODEL: THE PATIENT PROTECTION AND AFFORDABLE CARE ACT OF 2010**\n\nFollowing a year-long bitter debate, the Democrat-controlled House of Representatives and Senate passed the Affordable Care Act (ACA) without a single Republican vote. President Obama, on March 23, 2010, signed the most significant health legislation since Medicare and Medicaid in 1965 (Morone, 2010). Although the ACA was attacked as \"socialized medicine\" and a \"government takeover of health care,\" its policy pedigree derives much more from the proposals of a Republican President (Nixon) and Republican Governor (Romney) than from the single-payer national health insurance tradition of Democratic Presidents Roosevelt and Truman. The pluralistic financing model of the ACA includes individual and employer mandates for private insurance and an expansion of the publicly financed Medicaid program. Ironically, despite the ACA's close resemblance to the Massachusetts Health Plan of 2006, Mitt Romney, the former Governor of Massachusetts who supported and signed that state's reform bill, upon turning his sights to his candidacy for the Republican nomination for the 2012 presidential election, called for repeal of the ACA.\n\n_In 2013, Mandy Must is uninsured and works for a small shipping company in Texas that does not offer health insurance benefits. In 2014, if the ACA survives legal and political challenges, she would be required to obtain private insurance coverage. Mandy earns about $35,000 per year, and in 2014 would receive a federal subsidy of about $2000 toward her purchase of an individual insurance policy with a premium cost of $5000._\n\n_In 2013, Walter Groop works full-time as a salesperson for a large department store in Miami which does not offer health insurance benefits to its workers. In 2014, he begins to apply for an individual policy to meet the requirements of the ACA, but his employer informs him that the department store would start contributing toward group health insurance coverage for its employees to avoid paying penalties under the ACA._\n\n_In 2013, Job Knaught has been an unemployed construction worker in St. Louis for over 18 months and, aside from an occasional odd job, has no regular source of income. Because he is not disabled, he does not qualify for Medicaid despite being poor. In 2014, Job becomes eligible for Missouri's Medicaid program._\n\nThe ACA has four main components to its reform of health care financing:\n\n1. _Individual mandate_ : Beginning in 2014, the ACA requires virtually all US citizens and legal residents to have insurance coverage meeting a federally determined \"essential benefits\" standard. This standard would allow high-deductible plans to qualify, with out-of-pocket cost-sharing capped at $5950 per individual and $11,900 per family, in 2010 dollars. Those who fail to purchase insurance and do not qualify for public programs such as Medicaid, Medicare, or veteran's health care benefits must pay a tax penalty which would be gradually phased in by 2016, when it would equal the greater of $695 per year for an individual (up to $2085 for a family) or 2.5% of household income. Individuals and families below 400% of the Federal Policy Level are eligible for income-based sliding-scale federal subsidies to help them purchase the required health insurance.\n\n2. _Employer mandate_ : Also beginning in 2014, employers with 50 or more full-time employees face a financial penalty if their employees are not enrolled in an employer-sponsored health plan meeting the essential benefit standard and any of their employees apply for federal subsidies for individually purchased insurance. While this measure does not technically mandate large employers to provide health benefits to their full-time workers, it functionally has this effect by penalizing employers who do not provide insurance benefits and leave their employees to fend for themselves to comply with the individual mandate.\n\n3. _Medicaid eligibility expansion_ : As discussed in Chapter 2, Medicaid eligibility has traditionally required both a low income and a \"categorical\" eligibility requirement, such as being a child or an adult with a permanent disability. Effective in 2014, the ACA eliminates the categorical eligibility requirement and requires that states make all US citizens and legal residents below 133% of the Federal Poverty Level eligible for their Medicaid programs. In 2011, 133% of the Federal Poverty Level was $14,484 for a single person and $29,726 for a family of 4. The federal government pays states 100% of the Medicaid costs for beneficiaries qualifying under the expanded eligibility criteria for 2014 through 2016, with states contributing 10% after 2016. The benefit package is similar to current Medicaid benefits.\n\n4. _Insurance market regulation_ : The ACA also imposes some new rules on private insurance. One of the first measures of the ACA to be implemented in 2010 was a requirement that private health plans allow young adults up to age 26 to remain covered as dependents under their parents' health insurance policies. The ACA also eliminates caps on total insurance benefits payouts, prohibits denial of coverage based on preexisting conditions, and limits the extent of experience rating to a maximum ratio of 3-to-1 between a plan's highest and lowest premium charge for the same benefit package. The ACA also establishes state-based insurance exchanges to function as a clearing house to assist people seeking coverage under the individual mandate to shop for insurance plans meeting the federal standards (Kingsdale and Bertko, 2010). The benefit packages offered by plans in the exchanges would vary depending on whether individuals purchase a low-premium bronze plan with high out-of-pocket costs, a high-premium platinum plan with low out-of-pocket costs, or the intermediate silver or gold plans. These regulatory measures were deemed by many to be essential to the feasibility and fairness of an individual mandate. For example, mandates cannot work if insurers may deny coverage to individuals with preexisting conditions or steeply experience rate premiums. The insurance industry, for its part, balks at these types of market reforms in the absence of a mandate, fearing adverse disproportionate enrollment of high-risk individuals when coverage is voluntary.\n\nThe major coverage provisions of the ACA and their timeline for implementation are summarized in Table 15\u20133.\n\n**Table 15\u20133.** Key coverage measures and implementation timeline for the Affordable Care Act of 2010\n\nIf the ACA is implemented in its entirety, 32 million of the 51 million uninsured Americans are expected to receive insurance coverage, an estimated 16 million through Medicaid expansion and 16 million through the individual mandate (Kaiser Family Foundation, 2010). None of the coverage expansion measures would benefit undocumented immigrants; they would not be eligible for federal premium subsidies under the individual mandate nor for Medicaid except for emergency care.\n\nThe ACA is expected to cost $938 billion over 10 years, with most of the costs associated with Medicaid expansion and individual mandate subsidies. The law is financed by a combination of new taxes and fees and by cost savings in the Medicare and Medicaid programs. Individuals with earnings over $200,000 and married couples with earnings over $250,000 would pay more for Medicare Part A. Health insurance companies, pharmaceutical firms, and medical device manufacturers would pay yearly fees. Medicare Advantage insurance plans and hospitals would receive less payment from the Medicare program. The Congressional Budget Office estimated that the new law would reduce the federal deficit by $124 billion over 10 years, though the CBO projection is not universally accepted.\n\nIn developing a proposal to expand coverage by building on the existing pluralistic funding model rather than turning to a single-payer model, President Obama and his congressional allies successfully calculated that they would be able to garner the political support of some powerful interest groups, such as the American Medical Association and pharmaceutical industry, that had been stalwart opponents of health reform proposals in prior eras (Morone, 2010). However, some conservative groups that opposed the ACA did not relent after the Act's passage, and the ACA has come under political and judicial threats since its enactment. One of the first acts passed by the House of Representatives in its 2011 session after Republicans regained a majority of seats in the House was repeal of the ACA. The ACA remained law because the Senate, with a Democratic majority, did not vote to repeal. Republican Governors and Attorney Generals in many states filed suits against the ACA, challenging the constitutionality of the federal government's mandating of individuals to purchase a private product. Federal judges in district courts have issued different rulings on the constitutionality of the ACA, and the case will ultimately be heard by the US Supreme Court.\n\n### **SECONDARY FEATURES OF NATIONAL HEALTH INSURANCE PLANS**\n\nThe primary distinction among national health insurance approaches is the mode of financing: government versus employment-based versus individual-based health insurance, or a mixture of all three. But while the overall financing approach may be considered the headline news of reform proposals, some of the details in the fine print are extremely important in determining whether a universal coverage plan will be able to deliver true health security to the public (Table 15\u20134). What are some of these secondary features?\n\n**Table 15\u20134.** Features of national health insurance plans\n\n#### **Benefit Package**\n\nAn important feature of any health plan is its benefit package. Most national health insurance proposals cover hospital care, physician visits, laboratory, x-rays, physical and occupational therapy, inpatient pharmacy, and other services usually emphasizing acute care. One important benefit not included in the original Medicare program was coverage of outpatient medications. This coverage was later added in 2003 under Medicare Part D. Mental health services have often not been fully integrated into the benefit package of universal coverage proposals, a situation that has in part been addressed by the Mental Health Parity Act of 1996 and Mental Health Parity and Addiction Equity Act of 2008 which apply to group private health insurance plans. Neither the ACA nor most of its reform proposal precursors have included comprehensive benefits for dental care, long-term care, or complementary medicine services such as acupuncture.\n\n#### **Patient Cost Sharing**\n\nPatient cost sharing involves payments made by patients at the time of receiving medical care services. It is sometimes broadened to include the amount of health insurance premium paid directly by an individual. Naturally, the breadth of the benefit package influences the amount of patient cost sharing: The more the services are not covered, the more the patients must pay out of pocket. Many plans impose patient cost sharing requirements on covered services, usually in the form of deductibles (a lump sum each year), coin-surance payments (a percentage of the cost of the service), or copayments (a fixed fee, eg, $10 per visit or per prescription). In general, single payer proposals restrict cost sharing to minimal levels, financing most benefits from taxes. In comparison, the individual mandate provisions of the Massachusetts Health Act and the ACA include considerable amounts of cost sharing. The ACA, for example, would require an individual such as Mandy Must with an income between 300% and 400% of the federal poverty level to pay up to 9.5% of her income toward a health insurance premium, in addition to having to potentially pay thousands of dollars per year in deductibles and copayments at the time of service. Critics have argued that this degree of out-of-pocket payment raises questions about whether the Affordable Care Act is a bit of a misnomer and that people of modest incomes will continue to be underinsured and subject to large amounts of out-of-pocket expenses. The arguments for and against cost sharing as a cost containment tool are discussed in Chapter 9.\n\n#### **Effects on Medicare, Medicaid, and Private Insurance**\n\nAny national health insurance program must interact with existing health care programs, whether Medicare, Medicaid, or private insurance plans. Single-payer proposals make among the most far-reaching changes: Medicaid and private insurance are eliminated in their current form and are melded into a single insurance program that resembles a Medicare-type program for all Americans. The most sweeping versions of individual mandate plans, such as that proposed by the Heritage Foundation, would dismantle both employment-based private insurance and government-administered insurance programs. Employer mandates, which extend rather than supplant employment-based coverage, tend to have the least effect on existing dollar flow in the health care system, as do pluralistic models such as the ACA that preserve and extend existing financing models through mandates for private insurance and broadened eligibility for Medicaid.\n\n#### **Cost Containment**\n\nBy increasing people's access to medical care, national health insurance has the capacity to cause a rapid increase in national health expenditures, as did Medicare and Medicaid (see Chapter 2). By the 1990s, policymakers recognized that an increase in access must be balanced with measures to control costs.\n\nDifferent national health insurance proposals have vastly disparate methods of containing costs. As noted above, individual- and employment-based proposals tend to use patient cost sharing as their chief cost control mechanism. In contrast, government-financed plans look more to global budgeting and regulation of fees to keep expenditures down. Single-payer plans, which concentrate health care funds in a single public insurer, can more easily establish a global budgeting approach than can plans with multiple private insurers.\n\nProposals that build on the existing pluralistic financing model of US health care, such as the Clinton health plan and the ACA, face challenges in taming the unrelenting increases in national health care expenditures that seem to be endemic to a fragmented financing system. One of the items that contributed to the demise of President Clinton's health reform proposal before it could even be formally introduced as a bill in Congress was the inclusion of a measure to allow the federal government to cap the annual rates of increases in private health insurance premiums. President Obama eschewed such a regulatory approach in developing the ACA, and the ACA includes much weaker language about private insurance plans needing to \"justify\" premium increases to be able to continue to participate in state health insurance exchanges. In an effort to control costs, the ACA limits the percentage of health insurance premiums that can be retained by an insurance company in the form of overhead and profits (a concept known as the \"medical loss ratio,\" whereby a greater loss ratio means more premium dollars being \"lost\" by the company in the form of payments for actual health care services). The ACA also caps the amount that an employer can contribute toward a health insurance premium as a nontaxable benefit to the employee ($10,200 for an individual policy and $27,500 for a family policy), in an attempt to discourage enrollment in the most expensive plans. Many of the savings in the ACA are expected to come from slowing the rate of growth in expenditures for Medicare through measures such as reducing payments to Medicare Advantage HMO plans and appointing an Independent Payment Advisory Board to recommend methods to contain Medicare costs. Yet another strategy of the ACA for addressing costs is to redesign health care delivery to achieve better value, discussed next.\n\n#### **Reform of Health Care Delivery**\n\nThroughout the history of national health insurance proposals in the United States, reformers viewed their primary goal as modifying the methods of financing health care to achieve universal coverage. Addressing how providers were paid often emerged as a closely related consideration because of its importance for making universal coverage affordable. However, intervening in the way in which health care was organized and delivered was typically not something that featured prominently in reform proposals. Reformers tended to have their work cut out to overcome the strong opposition of the AMA and hospital associations to health insurance reform without further antagonizing those interests by challenging professional sovereignty over health care organization and delivery. Even many advocates of single payer reform in the United States looked to the lessons of the introduction of government insurance programs in the Canadian provinces, where until recently government took great pains to largely focus on insurance financing and payment rate regulation and not on reforming models for care delivery.\n\nThe ACA went considerably farther than most previous major reform proposals in the United States in including measures to shape health care delivery. The ACA created an Innovation Center in the Centers for Medicare and Medicaid Services to spearhead efforts to redesign care models in the United States. One of the charges to the Innovation Center is to promote Accountable Care Organizations. As discussed in Chapter 6, Accountable Care Organizations are intended to be provider-organized systems for delivering care that can emphasize more integrated and coordinated models of care for defined populations of patients, with financial incentives to reward higher value care. The Innovation Center also has responsibility for encouraging development of primary care Patient-Centered Medical Homes, also discussed in Chapter 5. Other measures in the ACA call for pilot programs to expand the roles of nurses, pharmacists, and other health care professionals in redesigned care models.\n\n### **WHICH FINANCING MODEL FOR NATIONAL HEALTH INSURANCE PLAN IS BEST?**\n\nHistorically, in the United States the government-financed single payer road to national health insurance is the oldest and most traveled of the three approaches. Advocates of government financing cite its universality: Everyone is insured in the same plan simply by virtue of being a US resident. Its simplicity creates a potential cost saving: The 25% of health expenditures spent on administration could be reduced, thus making available funds to extend health insurance to the uninsured. Employers would be relieved of the burden of providing health insurance to their employees. Employees would regain free choice of physician, choice that is being lost as employers are choosing which health plans (and therefore which physicians) are available to their workforce. Health insurance would be delinked from jobs, so that people changing jobs or losing a job would not be forced to change or lose their health coverage. Single-payer advocates, citing the experience of other nations, argue that cost control works only when all health care moneys are channeled through a single mechanism with the capacity to set budgets (Himmelstein and Woolhandler, 1989). While opponents accuse the government-financed approach as an invitation to bureaucracy, single-payer advocates point out that private insurers have average administrative costs of 14%, far higher than government programs such as Medicare with its 2% administrative overhead. A cost-control advantage intrinsic to tax-financed systems in which a public agency serves as the single payer for health care is the administrative efficiency of collecting and dispensing revenues under this arrangement.\n\nSingle-payer detractors charge that one single government payer would have too much power over people's health choices, dictating to physicians and patients which treatments they can receive and which they cannot, resulting in waiting lines and the rationing of care. Opponents also state that the shift in health care financing from private payments (out of pocket, individual insurance, and employment-based insurance) to taxes would be unacceptable in an antitax society. Moreover, the United States has a long history of politicians and government agencies being overly influenced by wealthy private interests, and this has contributed to making the public mistrustful of the government.\n\nThe employer mandate approach\u2014requiring all employers to pay for the health insurance of their employees\u2014is seen by its supporters as the most logical way to raise enough funds to insure the uninsured without massive tax increases (though employer mandates have been called hidden taxes). Because most people younger than 65 years now receive their health insurance through the workplace, it may be less disruptive to extend this process rather than change it.\n\nThe conservative advocates of individual-based insurance and the liberal supporters of single-payer plans both criticize employer mandate plans, saying that forcing small businesses\u2014many of whom do not insure their employees\u2014to shoulder the fiscal burden of insuring the uninsured is inequitable and economically disastrous; rather than purchasing health insurance for their employees, many small businesses may simply lay off workers, thereby pitting health insurance against jobs. Moreover, because millions of people change their jobs in a given year, job-linked health insurance is administratively cumbersome and insecure for employees, whose health security is tied to their job. Finally, critics point out that under the employer mandate approach, \"Your boss, not your family, chooses your physician\"; changes in the health plans offered by employers often force employees and their families to change physicians, who may not belong to the health plans being offered.\n\nAdvocates of the individual mandate assert that their approach, if adopted as the primary means of financing coverage, would free employers of the obligation to provide health insurance, and would grant individuals a stable source of health insurance whether they are employed, change jobs, or become disabled. There would be no need either to burden small businesses with new expenses and thereby disrupt job growth or to raise taxes substantially. While opponents argue that low-income families would be unable to afford the mandatory purchase of health insurance, supporters claim that income-related tax credits are a fair and effective method to assist such families (Butler, 1991).\n\nThe individual mandate approach is criticized as inefficient, with each family having to purchase its own health insurance. To enforce a requirement that every person buy coverage could be even more difficult for health insurance than for automobile insurance. Moreover, to reduce the price of their premiums, many families would purchase \"bare-bones insurance\" plans with low-cost, high-deductible coverage and a scanty benefit package, thereby leaving lower- and middle-income families with potentially unaffordable out-of-pocket costs.\n\n### **CONCLUSION**\n\nThe concept of national health insurance rests on the belief that everyone should contribute to finance health care and everyone should benefit. People who pay more than they benefit are likely to benefit more than they pay years down the road when they face an expensive health problem. In 2009, national health insurance took center stage in the United States with the fierce debate over health reform legislation that resulted in the ACA. This debate revealed a wide gulf between those who believe that all people should have financial access to health care and those who do not. The fate of the ACA will determine which of those two beliefs holds sway in the United States, until now the only developed nation that does not insure virtually all its citizens for health care.\n\n### **REFERENCES**\n\nBodenheimer T, Grumbach K. Financing universal health insurance: Taxes, premiums, and the lessons of social insurance. _J Health Polit Policy Law_. 1992;17:439.\n\nButler SM. A tax reform strategy to deal with the uninsured. _JAMA_. 1991;265:2541.\n\nClark CR et al. Lack of access due to costs remains a problem for some in Massachusetts despite the state's health reforms. _Health Aff._ 2011;30:247.\n\nEmanuel EJ, Fuchs VR. Health care vouchers\u2014a proposal for universal coverage. _N Engl J Med_. 2005;352:1255.\n\nHimmelstein DU, Woolhandler S. Writing Committee of Physicians for a National Health Program: A national health program for the United States: A physicians' proposal. _N Engl J Med_. 1989;320:102.\n\nKaiser Family Foundation. Summary of New Health Reform Law, 2010. . Accessed August 22, 2011.\n\nKingsdale J, Bertko J. Insurance exchanges under health reform: Six design issues for the states. _Health Aff._ 2010;29:1158.\n\nLong SK, Stockley K. Sustaining health reform in a recession: An update on Massachusetts as of Fall 2009. _Health Aff_. 2010;29:1234.\n\nMorone J. Presidents and health reform: From Franklin D. Roosevelt to Barack Obama. _Health Aff._ 2010;29:1096.\n\nStarr P. _The Social Transformation of American Medicine._ New York: Basic Books; 1982.\n\n## **16 Conflict and Change in America's Health Care System**\n\nAs this book enters its closing chapters, it is worth stepping back from the detailed workings of the US health care system to view the system as a larger whole. Who are the major actors? How have they interacted over the past few decades? What might the future bring?\n\n### **THE FOUR MAJOR ACTORS**\n\nThe health care sector of the nation's economy is a 2.5 trillion dollar-plus system that finances, organizes, and provides health care services for the people of the United States. Four major actors can be found on this stage (Table 16\u20131).\n\n**Table 16\u20131.** The four major actors\n\n1. The _purchasers_ supply the funds. These include individual health care consumers, businesses that pay for the health insurance of their employees, and the government, which pays for care through public programs such as Medicare and Medicaid. All purchasers of health care are ultimately individuals, because individuals finance businesses by purchasing their products and fund the government by paying taxes. Nonetheless, businesses and the government assume special importance as the nation's _organized_ purchasers of health care.\n\n2. The _insurers_ receive money from the purchasers and reimburse the providers. Traditional insurers take money from purchasers (individuals or businesses), assume risk, and pay providers when policyholders require medical care. Yet some insurers are the same as purchasers; the government can be viewed as an insurer or purchaser in the Medicare and Medicaid programs, and businesses that self-insure their employees can similarly occupy both roles. (In previous chapters, we have used the term \"payer\" to refer to both purchasers and insurers.)\n\n3. The _providers_ , including hospitals, physicians, nurses, nurse practitioners, physician assistants, pharmacists, social workers, nursing homes, home care agencies, and pharmacies, actually provide the care. While health maintenance organizations (HMOs) are generally insurers, some are also providers, owning hospitals and employing physicians.\n\n4. The _suppliers_ are the pharmaceutical, medical supply, and computer industries, which manufacture equipment, supplies, and medications used by providers to treat patients.\n\nInsurers, providers, and suppliers make up the health care industry. Each dollar spent on health care represents an expense to the purchasers and a gain to the health care industry. In the past, purchasers viewed this expense as an investment, money spent to improve the health of the population and thereby the economic and social vitality of the nation. But over the past 35 years, a fundamental conflict has intensified between the purchasers and the health care industry: The purchasers wish to reduce, and the health care industry to increase, the number of dollars spent on health care. We will now explore the changing relationships among purchasers, insurers, providers, and suppliers.\n\n### **THE YEARS 1945 TO 1970: THE PROVIDER\u2013INSURER PACT**\n\nDuring this period, independent hospitals and small private physician offices populated the US health delivery system (see Chapter 6). Some large institutions existed that combined hospital and physician care (eg, the Kaiser\u2013Permanente system, the Mayo Clinic, and urban medical school complexes), but these were the exception (Starr, 1982). Competition among health care providers was minimal because most geographic areas did not have an excess of facilities and personnel. The health care financing system included hundreds of private insurance companies, joined by the governmental Medicare and Medicaid programs enacted in 1965. The United States had a relatively dispersed health care industry.\n\n_Bert Neighbor was a 63-year-old man who developed abdominal pain in 1962. Because he was well insured under Blue Cross, his physician placed him in Metropolitan Hospital for diagnostic studies. On the sixth hospital day, a colon cancer was surgically removed. On the fifteenth day, Mr. Neighbor went home. The hospital sent its $1200 bill to Blue Cross, which paid the hospital for its total costs in caring for Mr. Neighbor. In calculating Mr. Neighbor's bill, Metropolitan Hospital included a small part of the cost of the 80-bed new building under construction._\n\n_At a subsequent meeting of the Blue Cross board of directors, the hospital administrator (also a Blue Cross director) was asked whether it was reasonable to include the cost of capital improvements when preparing a bill. Other Blue Cross directors, also hospital administrators with construction plans, argued that it was proper, and the matter was dropped. In the same meeting, the directors voted a 34% increase in Blue Cross premiums. Sixteen years later, a study revealed that the metropolitan area had 300 excess hospital beds, with hospital occupancy down from 82% to 60% over the past decade._\n\nA defining characteristic of the health care industry was an alliance of insurers and providers. This provider\u2013insurer pact was cemented with the creation of Blue Cross and Blue Shield, the nation's largest health insurance system for half a century (see Chapter 2). Blue Cross was formed by the American Hospital Association, and Blue Shield was run by state medical societies affiliated with the American Medical Association. In the case of the Blues, the provider\u2013insurer relationship was more than a political alliance; it involved legal control of insurers by providers. As in the example of Metropolitan Hospital, the providers set generous rules of reimbursement, and the Blues made the payments without asking too many questions (Law, 1974). Commercial insurers usually played by the reimbursement rules already formulated by the physicians, hospitals, and Blues, paying for medical services without asking providers to justify their prices or the reasons for the services.\n\nBy the 1960s, the power of the provider\u2013insurer pact was so great that the hospitals and Blue Cross virtually wrote the reimbursement provisions of Medicare and Medicaid, guaranteeing that physicians and hospitals would be paid with the same bountiful formulas used for private patients (Law, 1974). With open-ended reimbursement policies, the costs of health care inflated at a rapid pace.\n\nThe disinterest of the chief organized purchaser (business) stemmed from two sources: the healthy economy and the tax subsidy for health insurance. From 1945 through 1970, US business controlled domestic and foreign markets with little foreign competition. Labor unions in certain industries had gained generous wages and fringe benefits, and business could afford these costs because profits were high and world economic growth was robust (Kuttner, 1980; Kennedy, 1987). The cost of health insurance for employees was a tiny fraction of total business expenses. Moreover, payments by business for employee health insurance were considered a tax-deductible business expense, thereby cushioning any economic drain on business (Reinhardt, 1993). For these reasons, increasing costs generated by providers and reimbursed by insurers were passed on to business, which with few complaints paid higher and higher premiums for employees' health insurance, and thereby underwrote the expanding health care system. No countervailing forces \"put the brakes\" on the enthusiasm that united providers and the public in support of a medical industry that strived to translate the proliferation of biomedical breakthroughs into an improvement in people's lives.\n\n### **THE 1970S: TENSIONS DEVELOP**\n\n_Jerry Neighbor, Bert Neighbor's son, developed abdominal pain in 1978. Because Blue Cross no longer paid for in-hospital diagnostic testing, his physician ordered outpatient x-ray studies. When colon cancer was discovered, Jerry Neighbor was admitted to Metropolitan Hospital on the morning of his surgery. His total hospital stay was 9 days, 6 days shorter than his father's stay in 1962. Since 1962, medical care costs had risen by approximately 10% per year. Blue Cross paid Metropolitan Hospital $460 for each of the 9 days Jerry Neighbor spent in the hospital, for a total cost of $4140. The Blue Cross board of directors, which in 1977 included for the first time more business than hospital representatives, submitted a formal proposal to the regional health planning agency to reduce the number of hospital beds in the region, in order to keep hospital costs down. The planning agency board had a majority of hospital and physician representatives, and they voted the proposal down._\n\nIn the early 1970s, the United States fell from its postwar position of economic dominance, as Western Europe and Japan gobbled up markets (not only abroad but in the United States itself) formerly controlled by US companies. The United States' share of world industrial production was dropping, from 60% in 1950 to 30% in 1980. Except for a few years during the mid-1980s, inflation or unemployment plagued the United States from 1970 until the early 1990s.\n\nThe new economic reality was a critical motor of change in the health care system. With less money in their respective pockets, individual health care consumers, business, and government became concerned with the accelerating flow of dollars into health care. Prominent business-oriented journals published major critiques of the health care industry and its rising costs (Bergthold, 1990). A new concern for primary care, which seemed underemphasized in relation to specialty and hospital care, spread within the health professions. These developments produced tensions within the health industry itself.\n\nFaced with Blue Cross premium increases of 25% to 50% in a single year, angry Blue Cross subscribers protested at state hearings in eastern and midwestern states and challenged hospital control over Blue Cross boards (Law, 1974). Some state governments began to regulate hospital construction, and a few states initiated hospital rate regulation. The federal government established a network of health planning agencies, in an attempt to slow hospital growth. Peer review was established to monitor the appropriateness of physician services under Medicare. Thus, the purchasers took on an additional role as health care regulators. But the health care industry resisted these attempts by purchasers to control health care costs. Medical inflation continued at a rate far above that of inflation in the general economy (Starr, 1982).\n\nNonetheless, these early initiatives from the purchasers made an impact on the provider\u2013insurer pact. As pressure mounted on insurers not to increase premiums, insurers demanded that services be provided at lower cost. Blue Cross, widely criticized as playing the role of an intermediary that passed increased hospital costs on to a helpless public, legally separated from the American Hospital Association in 1972 (Law, 1974). State medical societies were forced to relinquish some of their control over Blue Shield plans. Conflicts erupted between providers and insurers as the latter imposed utilization review procedures to reduce the length of hospital stays. Hospitals, which had hitherto purchased the newest diagnostic and surgical technology desired by physicians or their medical staff, began to deny such requests because insurers would no longer guarantee their reimbursement. Moreover, the glut of hospital beds and specialty physicians, which had been produced by the attractive reimbursements of the 1960s and the influence of the biomedical model on medical education (see Chapter 5), turned on itself as half-empty hospitals and half-busy surgeons began to compete with one another for patients. Strains were showing within the provider\u2013insurer pact.\n\nBy the late 1970s, the deepening of the economic crisis created a nationwide tax revolt. As a result, governments attempted to reduce spending on such programs as health care (Kuttner, 1980). But major change was still awaiting the arrival of the other powerful purchaser: business.\n\n### **THE 1980S: THE REVOLT OF THE PURCHASERS**\n\n_In 1989, Ryan Neighbor, Jerry Neighbor's brother, became concerned when he noticed blood in his stools; he decided to see a physician. Six months earlier, his company had increased the annual health insurance deductible to $1000, which could be avoided by joining one of the HMOs offered by the company. Ryan Neighbor opted for the Blue Cross HMO, but his family physician was not involved in that HMO, and Mr. Neighbor had to pick another physician from the HMO's list. The physician diagnosed colon cancer; Ryan Neighbor was not allowed to see the surgeon who had operated on his brother but was sent to a Blue Cross HMO surgeon. While Mr. Neighbor respected Metropolitan Hospital, his surgery was scheduled at Crosstown Hospital; Blue Cross had refused to sign a contract with Metropolitan when the hospital failed to negotiate down from its $1800 per diem rate. Ryan Neighbor's entire Crosstown Hospital stay was 5 days, and the HMO paid the hospital $7500, based on its $1500 per diem contract._\n\nThe late 1980s produced a severe shock: The cost of employer-sponsored health plans jumped 18.6% in 1988 and 20.4% in 1989 (Cantor et al, 1991). Between 1976 and 1988, the percentage of total payroll spent on health benefits almost doubled from 5% to 9.7% (Bergthold, 1991). In another development, many large corporations began to self-insure. Rather than paying money to insurance companies to cover their employees, employers increasingly took on the health insurance function themselves and used insurance companies only for claims processing and related administrative tasks. In 1991, 40% of employees receiving employer-sponsored health benefits were in self-insured plans. Self-insurance placed employers at risk for health care expenditures and forced them to pay more attention to the health care issue. These three developments (ie, a troubled economy, rising health care costs, and self-insurance) catapulted big business into the center of the health policy debate, with cost control as its rallying cry. Business, the major private purchaser of health care, became the motor driving unprecedented change in the health care landscape (Bergthold, 1990). Business threw its clout behind managed care, particularly HMOs, as a cost-control device. By shifting from fee-for-service to capitated reimbursement, managed care could transfer a portion of the health expenditure risk from purchasers and insurers to providers (see Chapter 4).\n\nIndividual health care consumers, in their role as purchasers, also showed some clout during the late 1980s. Because employers were shifting health care payments to employees, labor unions began to complain bitterly about health care costs, and major strikes took place over the issue of health care benefits. More than 70% of people polled in a 1992 Louis Harris survey favored serious health care cost controls (Smith et al, 1992). The growing tendency of private health insurers to reduce their risks by dramatic premium increases and policy cancellations for policyholders with chronic illnesses created a series of horror stories in the media that turned health insurance companies into highly unpopular institutions.\n\nDuring the 1980s, the government was facing the tax revolt and budget deficits, and it took measures designed to slow the rising costs of Medicare and Medicaid, with limited success. The 1983 Medicare Prospective Payment System (diagnosis-related groups DRGs]) reduced the rate of increase of Medicare hospital costs, but outpatient Medicare costs and costs borne by private purchasers escalated in response. In 1989, Medicare physician payments were brought under tighter control, resulting in Medicare physician expenditures growing at only 5.3% per year from 1991 to 1993, compared with 11.3% per year from 1984 to 1991 (Davis and Burner, 1995). Numerous states scaled back their Medicaid programs, but because of the economic recession and the growing crisis of uninsurance (see [Chapter 3), the federal government was forced to expand Medicaid eligibility, and Medicaid costs rose faster than ever before. Governments began to experiment with managed care for Medicare and Medicaid as a cost-control device.\n\nThe most significant development of the 1980s was the growth of selective contracting. Purchasers and insurers had usually reimbursed any and all physicians and hospitals. Under selective contracting, purchasers and insurers choose which providers they will pay and which they will not (Bergthold, 1990). In 1982, for example, California passed a law bringing selective contracting to the state's Medicaid program and to private health insurance plans. The law was passed because large California corporations formed a political coalition to challenge physician and hospital interests, and because insurers deserted their former provider allies and joined the purchasers (Bergthold, 1990). The message of selective contracting was clear: Purchasers and insurers will do business only with providers who keep costs down. This development, especially when linked with capitation payments that placed providers at risk, changed the entire dynamic within the health care industry. For patients, it meant that like Ryan Neighbor, they had lost free choice of physician because employers could require employees to change health plans and therefore physicians. For the health care industry, selective contracting meant fierce competition for contracts and the crumbling of the provider\u2013insurer pact.\n\nAs a result of the purchasers' revolt, managed care became a burgeoning movement in US health care. By 1990, 95% of insured employees were enrolled in some form of managed care plan, including fee-for-service plans with utilization management, preferred provider organizations (PPOs), and HMOs. The growth of managed care plans, especially HMOs, competing against one another for contracts with business and the government, changed the entire political topography of US health care (Table 16\u20132).\n\n**Table 16\u20132.** Historical overview of US health care\n\n### **THE 1990S: THE BREAKUP OF THE PROVIDER\u2013INSURER PACT**\n\n_In 1994, Pamela Neighbor, Ryan's cousin, developed constipation. Earlier that year, her law firm had switched from Blue Cross HMO to Apple a Day HMO because the premiums were lower; all employees of the firm were forced to change their physicians. Apple a Day contracted only with Crosstown Hospital, whose rates were lower than those of Metropolitan, resulting in Metropolitan losing patients and closing its doors. Ms. Neighbor's new physician diagnosed colon cancer and arranged for her admission to Crosstown Hospital for surgery. The physician's office was across the street from the now-closed Metropolitan Hospital. Four days before the procedure, a newspaper headline proclaimed that Apple a Day and Crosstown had failed to agree on a contract. The colonoscopy was canceled. Pamela Neighbor waited to see what would happen next._\n\nDuring the 1990s, many metropolitan areas in the United States, and some smaller cities and towns, experienced upheavals of their medical care landscape. Independent hospitals began to merge into hospital systems. In the most mature managed care markets, three or four health care networks were competing for those patients with private insurance, Medicare, or Medicaid. Selective contracting allowed purchasers and insurers to set reimbursement rates to health care providers. HMOs that demanded higher premiums from employers did not get contracts and lost their enrollees. Providers who demanded higher payment from HMOs were cut out of HMO contracts and lost many of their patients.\n\nSelective contracting tended to disorganize rather than organize medical care patterns. Physicians were forced to admit patients from one HMO to one hospital and those from another HMO to a different hospital. Laboratory, x-ray, and specialist services close to a primary care physician's office were sometimes not covered under contracts with that physician's patients' HMO, forcing referrals to be made across town. In one highly publicized case with a tragic outcome, the parents of a 6-month-old infant with bacterial meningitis were told by their HMO to drive the child almost 40 miles to a hospital that had a contract with that HMO, passing several high-quality hospitals along the way (Anders, 1996).\n\nThe 1990s was a period of purchaser dominance over health care. The federal government stopped Medicare inflation in its tracks through the tough provisions of the Balanced Budget Act of 1997. The average annual growth in Medicare expenditures declined from 12% in the early 1990s to zero in 1999 and 2000. On the private side, employers bargained hard with HMOs, causing insurance premium annual growth to drop from 13% in 1990 to 3% in 1995 and 1996. In California, employer purchasers consolidated into coalitions to negotiate with HMOs. The Pacific Business Group on Health, negotiating on behalf of large companies for 400,000 employees, and California Public Employee Retirement System (CalPERS), representing a million public employees, forced HMO premiums to go down during the 1990s. Enrollment in HMOs grew rapidly in the 1990s, expanding from 40 million enrollees in 1990 to 80 million in 1999.\n\n### **THE NEW MILLENNIUM: PROVIDER POWER RE-EMERGES**\n\n_In 2005, Pamela Neighbor, who was feeling well, made an appointment for her yearly colon cancer follow-up. The IPA in which her physician practiced had recently gone bankrupt and closed its doors. Ms. Neighbor's employer had switched its employees from Apple a Day Insurance Company's HMO product to Apple a Day PPO, allowing patients to access most of the physicians and all the hospitals in town. Ms. Neighbor had a difficult time finding a new primary care physician, and when she found one, it took several weeks to get an appointment. Eventually, a colonoscopy was scheduled at a diagnostic center owned by a group ofgastroenterologists. She was diagnosed with a second colon cancer and her primary care physician arranged for her admission to Crosstown Hospital. Ms. Neighbor never saw her primary care physician in the hospital; a surgeon plus a salaried inpatient physician called a hospitalist cared for Ms. Neighbor during her 4-day hospital stay. Apple a Day paid Crosstown Hospital $7200, $1800 per diem._\n\nSeveral trends characterize the first decade of the twenty-first century: the counter-revolution by providers, consolidation in the health care market, growing power of specialists and specialty services, increasing physician\u2013hospital tensions, an emerging crisis in primary care, growing criticism of pharmaceutical companies, and a steady increase in the uninsured and underinsured population.\n\n#### **The Provider Counter-Revolution**\n\nIn the mid-1990s, most health care analysts were certain that tightly managed care\u2014with purchasers and insurers dominating health care providers\u2014had become the new paradigm for health care in the United States. By 2001, this certainty had evaporated (Robinson, 2001). From 2000 to 2010, HMO enrollment dropped from 32% to 19% of insured employees, with only 19% of HMOs affiliated with an integrated delivery system. During those years, preferred provider organization (PPO) enrollment grew from about 30% to 60% of insured employees (Claxton et al, 2010). Tightly managed care was faltering.\n\nThe first decade of the twenty-first century could be called the era of the provider counter-revolution. Hospitals consolidated into hospital systems and demanded large price increases from insurers. Physicians balked at tight managed care contracts. Negotiations between health care providers and insurers became increasingly hostile, with one side or the other often refusing to sign contracts. As hospitals and providers gained an upper hand in negotiations with health plans, HMOs in turn demanded more money from employers. Insurance premiums for family coverage went from an average of $6,000 per year in 2000 to almost $14,000 in 2010, with virtually no difference between HMO and PPO premiums (Claxton et al, 2010). Purchasers lost faith that HMOs could control costs.\n\nAt the same time, individuals have been stuck with a greater proportion of health care costs. Twenty percent of insured employees, up from 10% in 2006, have a deductible of $1000 or more for individual coverage. The percent of insured employees with high-deductible plans has risen from 5% in 2006 to 13% in 2010; in a typical high-deductible plan, employees pay over $3000 for their portion of the premium plus a deductible of $4000 (Claxton et al, 2010). Employees' out-of-pocket health care costs increased 34% from 2004 to 2007 (Gabel et al, 2009).\n\n#### **Consolidation in the Health Care Market**\n\nThe intense competition of the 1990s stimulated consolidation among insurers and providers, as each vied to improve its bargaining power. Large HMOs bought up smaller ones and merged with one another. In most states, three large insurance companies control more than 60% of the market (Robinson, 2004). These companies generally offer a variety of products including HMO, PPO, high deductible, and Medicare Advantage plans. Three huge insurers, all for-profit, are Wellpoint with 34 million enrollees in 2010, United Healthcare with 32 million, and Aetna with 18 million.\n\nProviders also consolidated. By 2001, 65% of hospitals were members of multihospital systems or networks (Bazzoli, 2004), and consolidation continued, though at a slower pace, through 2008. In many cities, two or three competing hospital systems encompass all hospitals. Hospital prices often rise rapidly after consolidation has taken place since payers are forced to contract with dominant hospital systems (Vogt, 2009). Specialists increasingly joined single-specialty groups, with the majority of cardiologists or orthopedists in some cities belonging to a dominant group (Liebhaber and Grossman, 2007). Private primary care and specialty practices are being acquired by hospital systems hoping to increase their market clout (Iglehart, 2011).\n\nConsolidation went hand in hand with organizations converting from nonprofit to investor-owned \"for-profit\" status as they sought to raise capital for buy-outs, market expansion, and organizational infrastructure. For decades, for-profit companies have played a prominent role in health care, with the rise in the 1970s of the \"medical\u2013industrial complex\" (Relman, 2007). For-profits, which owned 35% to 40% of health care services and facilities in 1990, expanded their reach during the 1990s. Nine of the largest 10 HMOs were for-profit by 1994. HMO stocks soared in the early 1990s and executives were rewarded with enormous compensation packages (Anders, 1996). Already in 1990, 77% of nursing homes and 50% of home health agencies were for-profit. Between 1993 and 1996, more than 100 nonprofit hospitals were taken over by forprofit hospital chains, though several financial scandals slowed down this trend. For-profit hospitals provide less charity care, treat fewer Medicaid patients, have higher administrative costs, and lower quality than nonprofit hospitals (Relman, 2007). By 2009, most specialty hospitals, imaging centers, ambulatory surgery centers were investor-owned (Relman, 2009).\n\n#### **The Quest for Profitability and the Growing Power of Specialists and Specialty Services**\n\nThat hospitals, physicians, and other providers respond to financial incentives is hardly a new phenomenon. As discussed in Chapter 5, more lucrative third-party payment for procedurally oriented specialty care has been one of the key factors shaping a physician workforce weighted toward nonprimary care fields and a hospital sector filled with tertiary care facilities. However, twenty-first-century health care in the United States is becoming characterized by a single-minded quest for profitability that is threatening traditional notions of professionalism and community service. Emblematic of this trend is the emergence of a new type of for-profit hospital, the specialty hospital fully or partially owned by groups of specialist physicians. As of 2010, 265 of these hospitals existed in the United States, typically limiting their services to cardiac and orthopedic procedures\u2014service lines that are well reimbursed (Perry, 2010). Physician owners of these hospitals doubly benefit financially, receiving income from both the payment for the services they directly provide and their share of hospital profits. Moreover, physician owners often channel well-insured patients from nonprofit general hospitals to their own for-profit specialty hospitals. In one example, 16 cardiac surgeons and cardiologists shifted their patients with heart disease from a university medical center to a new hospital only caring for patients with heart disease; the number of cardiac surgeries performed at the university medical center dropped from over 600 to below 200 between 2002 and 2004, resulting in the loss of $12 million in revenues. Uninsured patients continued to have cardiac procedures at the university hospital (Iglehart, 2005). For the community as a whole, the opening of a cardiac hospital is associated with increased rates of coronary revascularization (coronary artery bypass surgery and angioplasty), raising questions about whether all the additional procedures are medically appropriate (Nallamothu et al, 2007). Because of these problems with specialty hospitals, the Affordable Care Act of 2010 barred new specialty hospitals from receiving Medicare payments (Perry, 2010).\n\nSimilar financial incentives have attracted specialist physicians to set up many thousands of ambulatory surgery, diagnostic, and imaging centers that they own. A growing proportion of profitable services\u2014cataract surgery and orthopedic procedures, diagnostic studies such as colonoscopies, and CT or MRI studies\u2014have been shifted from hospital facilities to these physician-owned ambulatory centers. As with specialty hospitals, physicians earn income from both the services they directly provide and the facility's profits. Because general hospitals formerly earned considerable income from these procedures, this phenomenon has created major tensions between hospitals and specialists (Berenson et al, 2006b). The common practice of physicians referring patients for imaging tests at a facility owned by the same physician is associated with higher volumes of imaging services, increasing costs, and exposing patients to unnecessary radiation (Sunshine and Bhargavan, 2010).\n\nSingle-specialty groups have grown markedly since the late 1990s. Two major drivers of this growth are (1) the ability of organized specialists with market power in a local area to negotiate for high reimbursement rates from insurers and (2) the bringing together of capital to invest in specialist-owned surgery, diagnostic, and imaging centers. As a result of these trends, the income of specialists who offer procedural or imaging services has far outpaced the growth in earnings for primary care physicians (Bodenheimer et al, 2007). Multispecialty groups, which include primary care physicians and tend to have the best scores on quality report cards, are not growing in part because specialist physicians in multispecialty groups are expected to share their high revenues with lower-reimbursed primary care physicians (Casalino et al, 2004; Mehrotra et al, 2006).\n\nNonprofit community hospitals are responding to competition from specialist physicians by creating \"specialty service lines\" to attract specialist physicians and well-insured patients to their institutions. To create capacity for these profitable service lines, hospitals are de-emphasizing traditional medical-surgical wards. Whether the hospital is a nonprofit community hospital, a for-profit hospital, or a physician-owned specialty hospital, filling a hospital bed with a patient receiving an organ transplant or spine surgery is much more financially rewarding than filling the same bed with an elderly patient with pneumonia and heart failure, even if the latter patient has insurance. Strategic planning by hospitals increasingly focuses on how to maximize the most profitable service lines, rather than on how to provide the services most needed in the community (Berenson et al, 2006a).\n\nCompounding this situation is the weakening claim hospitals can make on physicians for community service. Surgeons, diagnostic cardiologists, gastroenterologists, ophthalmologists, and radiologists can successfully run a medical practice without ever setting foot in a hospital by focusing their work on ambulatory centers of which these physicians are owners. Because these specialists no longer need the hospital, they feel little obligation to be on call for hospital emergency departments or for patients in intensive care. Hospitals are forced to pay specialists large sums to provide nighttime emergency department backup or are employing specialists to perform the duties formerly done for free by specialists on the hospital medical staff. The divorce of physicians from the community hospital is not limited to specialists. As a result of the hospitalist movement, many primary care physicians are never seen in a hospital. Hospitalists are physicians who specialize in the care of hospitalized patients. Most are employees of a hospital or hospital system; others are members of single-specialty hospitalist groups, which contract with hospitals to supply hospitalist physicians. Hospitalists are the fastest growing specialty in the history of medicine in the United States; the 500 hospitalists existing in 1997 have multiplied into about 30,000 hospitalists in 2010.\n\nThe quest for profitability is further aggravating the primary care-specialist imbalance in the physician workforce in the United States. In 2007, only 7% of US medical school graduates planned careers in adult primary care, with an adult primary care physician shortage projected at about 40,000 by 2020. Nurse practitioners and physician assistants help mitigate this shortage but their numbers are not sufficient to solve the problem. As a result, patients are having increasing difficulty gaining timely access to primary care or finding a new primary care physician. Although the causes of the declining career interest in primary care are multifactorial, the gap between primary care and specialty incomes is one reasons why growing numbers of US medical students and residents\u2014many of whom have more than $150,000 in personal debts from medical school expenses\u2014have turned away from careers in primary care (Bodenheimer and Pham, 2010).\n\nThese trends pull health care in the United States farther away from a primary care-based, community-responsive model. Evidence suggests that this trend will fuel continued inflation in health care costs without yielding commensurate benefits for the health of the public. A major reform of payment policies in the United States, along with a rethinking of the role of investor-owned enterprises in health care, will be required in order to realign financial incentives with the values that make for a well-functioning system.\n\n#### **The Pharmaceutical Industry Comes Under Criticism**\n\nThe rising tensions among purchasers, insurers, and providers spilled over to engulf health care's major supplier: the pharmaceutical industry. In 1988, prescription drugs accounted for 5.5% of national health expenditures. With 71% of drug costs borne out of pocket by individuals and only 18% paid by private insurance plans, these costs had little impact on insurers. In contrast, by 2009, prescription drug costs had risen to 10.1% of total health expenditures, with only 21% paid out of pocket, the rest covered by employers, insurers, and governmental purchasers. The growing cost of pharmaceuticals for the elderly became a major national issue. Because of its unaffordable prices and high profits, the pharmaceutical industry was becoming public enemy number one (Spatz, 2010).\n\nFor years, drug companies have been the most profitable industry in the United States, earning net profits after taxes close to 20% of revenues (19.3% in 2008), compared with 5% for all Fortune 500 firms. The pharmaceutical industry argues that high drug prices are justified by its expenditures on research and development of new drugs, yet the National Science Foundation estimates that true R&D spending is half of what the pharmaceutical industry claims (Congressional Budget Office, 2006). R&D for the largest drug companies consumed 14% of revenues in 2002, while marketing and administration accounted for 33% and after-tax net profits 21% (Reinhardt, 2004). Unlike many nations, the US government does not impose regulated prices on drugs; as a result of drug industry lobbying, the Medicare prescription drug coverage law passed in 2003 forbid the government to regulate drug prices (see Chapter 2). From 2006 to 2008, the health industry spent more money on lobbying than any other sector of the economy, and the drug industry was the largest contributor within the health industry (Steinbrook, 2008).\n\nCompanies developing a new brand-name drug enjoy a patent for 20 years from the date the patent application is filed, during which time no other company can produce the same drug. Once the patent expires, generic drug manufacturers can compete by selling the same product at lower prices. Some drug companies have waged legal battles to delay patent expirations on their brand name products or have paid generic drug manufacturers not to market generic alternatives (Stolberg and Gerth, 2000; Hall, 2001). In addition, the industry attempts to persuade physicians and patients to use brand-name products, spending $7 billion in 2009 on sales representatives' visits to physicians, journal advertising, and sponsorship of professional meetings, plus $4 billion on direct-to-consumer television ads (Kaiser Family Foundation, 2010). The federal Food and Drug Administration (FDA) has sent hundreds of letters to drug manufacturers, citing advertising violations including minimizing side effects and exaggerating benefits (Donohue et al, 2007). Four out of five physicians have some type of financial relationship with the pharmaceutical industry, ranging from accepting gifts to serving as a paid lecturer on behalf of a company. These physician\u2013industry relationships influence physicians to prescribe new drugs that are the most expensive and whose safety has not been adequately evaluated (Campbell, 2007). Drug firms may pay medical school faculty physicians tens of thousands of dollars to participate on their corporate boards (Lo, 2010).\n\nMost trials to determine the efficacy of prescription drugs are funded by that drug's manufacturer, and trials funded by industry are more likely than those with nonindustry funding to report results favorable to the funding company (Bero, 2007). Yet physicians base treatment decisions on these trials, which inform clinical practice guidelines. Authors of clinical practice guidelines often have ties to the pharmaceutical industry (Abramson and Starfield, 2005). From 2006 to 2010, at least 18 relatively new drugs were removed from the market because of serious side effects; in some cases, the manufacturer knew of the problems but hid them from the FDA and the public; in other cases, the FDA ignored the evidence. Some members of FDA committees recommending approval of a drug have ties to that drug's manufacturer, and these members are often not recused from the process (Angell, 2004).\n\nThese revelations have tainted the image of the pharmaceutical industry in the eyes of the medical profession and the public. Private health insurance companies have mounted the most effective response to the drug industry by creating tiered formularies in which generic drugs have lower copayments than brand-name drugs. As a result, 75% of all prescriptions filled in the United States in 2009 were for generic products (Spatz, 2010). This development has slowed the rate of growth of pharmaceutical costs. However, some brand-name drug companies are starting to produce generics, and the generic industry is starting to consolidate into fewer and larger companies; these trends could mean that generic prices may rise to levels not far below brand name prices.\n\n### **THE CHALLENGE**\n\nThe health care system has been dominated by a series of unstable power relationships among purchasers, insurers, providers, and suppliers. One of these actors may take center stage for a time, only to be pushed into the corner by another actor. Which entity has the leverage to get its way varies from city to city, depending on who has consolidated into larger institutions. Larger institutions can (in the case of providers and suppliers) demand to receive more money, or (in the case of purchasers and insurers) succeed in paying out less money. Patients continue to be at the mercy of these powerful institutions, as health care costs rise and as individuals bear a greater share of those costs.\n\nInequities in insurance coverage and in access to care continue, and cost control remains elusive. Whether the Affordable Care Act of 2010 will succeed in improving access and containing costs remains to be seen. The drive to make money\u2014whether for specialist physicians, for-profit and nonprofit hospitals, insurers, or pharmaceutical companies\u2014increasingly determines what happens in health care. For physicians, this economic motivation may clash with the professional commitment to patient welfare. The commitment of all health care professionals to the ethical principles of beneficence, nonmaleficence, patient autonomy, and distributive justice is tested on a daily basis in the profit-oriented environment of twenty-first century health care in America.\n\nChapter 1 introduced the paradox of excess and deprivation: Some people get too little care while others receive too much, which is costly and may be harmful. The first decade of the twenty-first century saw a sharpening of this paradox, with the number of uninsured climbing from 40 million to 50 million at the same time as the increasing number of specialist physicians owning their facilities was associated with growing volumes of expensive procedures, many of questionable appropriateness (Brownlee, 2007; Welch et al, 2011). Overcoming this paradox remains the fundamental challenge facing the health care system of the United States.\n\n### **REFERENCES**\n\nAbramson J, Starfield B. The effect of conflict of interest on biomedical research and clinical practice guidelines. _J Am Board Fam Pract_. 2005;18:414.\n\nAnders G. _Health Against Wealth: HMOs and the Breakdown of Medical Trust._ Boston, MA: Houghton Mifflin Company; 1996.\n\nAngell M. _The Truth About the Drug Companies_. New York: Random House; 2004.\n\nBazzoli G. The corporatization of American hospitals. _J Health Polit Policy Law_. 2004;29:885.\n\nBerenson RA et al. Specialty service lines: Salvos in the new medical arms race. _Health Affairs Web Exclusive_. 2006a; 25(5):w337.\n\nBerenson RA et al. Hospital\u2013physician relations: Cooperation, competition, or separation? _Health Aff Web Exclusive_. 2006b;26(1):w31.\n\nBergthold L. _Purchasing Power in Health._ New Brunswick, NJ: Rutgers University Press; 1990.\n\nBergthold L. The fat kid on the seesaw: American business and health care cost containment, 1970\u20131990. _Annu Rev Public Health_. 1991;12:157.\n\nBero L et al. Factors associated with findings of published trials of drug\u2013drug comparisons. _PLoS Med_. 2007;4:e184.\n\nBodenheimer T et al. The primary care-specialty income gap: Why it matters. _Ann Intern Med_. 2007;146:301.\n\nBodenheimer T, Pham HH. Primary care: Current problems and proposed solutions. _Health Aff_. 2010;29:799.\n\nBrownlee S. _Overtreated. Why Too Much Medicine Is Making Us Sicker and Poorer_. New York, NY: Bloomsbury; 2007.\n\nCampbell EG. Doctors and drug companies\u2014scrutinizing influential relationships. _N Engl J Med_. 2007;357:1796.\n\nCantor JC et al. Business leaders' views on American health care. _Health Aff_. 1991;10(1):98.\n\nCasalino L et al. Growth of single-specialty medical groups. _Health Aff_. 2004;23(2):82.\n\nClaxton G et al. Health benefits in 2010: Premiums rise modestly, workers pay more toward coverage. _Health Aff_. 2010;29:1942.\n\nCongressional Budget Office. _Research and Development in the Pharmaceutical Industry_. 2006. www.cbo.gov\/ftpdocs\/76xx\/doc7615\/10-02-DrugR-D.pdf. Accessed November 26, 2011.\n\nDavis MH, Burner ST. Three decades of Medicare: What the numbers tell us. _Health Aff_. 1995;14(4):231.\n\nDonohue JM et al. A decade of direct-to-consumer advertising of prescription drugs. _N Engl J Med_. 2007;357:673.\n\nGabel JR et al. Trends in underinsurance and the afford-ability of employer coverage, 2004\u20132007. _Health Aff_. 2009;28:w595.\n\nHall SS. Prescription for profit. _New York Times Magazine._ March 11, 2001.\n\nIglehart JK. The emergence of physician-owned specialty hospitals. _N Engl J Med_. 2005;352:78.\n\nIglehart JK. Doctor-workers of the world, unite! _Health Aff_. 2011;30:556.\n\nKaiser Family Foundation. Prescription drug trends, 2010. www.kff.org.\n\nKennedy P. _The Rise and Fall of the Great Powers._ New York: Random House; 1987.\n\nKuttner R. _Revolt of the Haves._ New York, NY: Simon & Schuster; 1980.\n\nLaw SA. _Blue Cross: What Went Wrong?_ New Haven, CT: Yale University Press; 1974.\n\nLiebhaber A, Grossman JM. Physicians moving to mid-sized, single-specialty practices. Tracking Report No. 18. Washington, DC: Center for Studying Health System Change; August 2007.\n\nLight DW, Warburton R. Demythologizing the high costs of pharmaceutical research. _Biosocieties_. 2011;6:34.\n\nLo B. Serving two masters\u2014conflicts of interest in academic medicine. _N Engl J Med._ 2010;362:669.\n\nMehrotra A et al. Do integrated medical groups provide higher-quality medical care than individual practice associations? _Ann Intern Med_. 2006;145:826.\n\nNallamothu BK et al. Opening of specialty cardiac hospitals and use of coronary revascularization in Medicare beneficiaries. _JAMA_. 2007;297:962.\n\nPerry JE. A mortal wound for physician-owned specialty hospitals? 2010. www.academia.edu.\n\nReinhardt UE. Reorganizing the financial flows in US health care. _Health Aff_. 1993;12(suppl):172.\n\nReinhardt UE. An information infrastructure for the pharmaceutical market. _Health Aff_. 2004;23(1):107.\n\nRelman AS. _A Second Opinion. Rescuing America's Health Care._ New York, NY: Public Affairs; 2007.\n\nRelman AS. The health reform we need & are not getting. _New York Rev Books_. 2009;56(11):38.\n\nRobinson JC. The end of managed care. _JAMA_. 2001;285:2622.\n\nRobinson JC. Consolidation and the transformation of competition in health insurance. _Health Aff_. 2004;23(6):11.\n\nSmith MD et al. Taking the public's pulse on health system reform. _Health Aff_. 1992;11(2):125.\n\nSpatz ID. Health reform accelerates changes in the pharmaceutical industry. _Health Aff_. 2010;29:1331.\n\nStarr P. _The Social Transformation of American Medicine._ New York: Basic Books; 1982.\n\nSteinbrook R. Campaign contributions, lobbying, and the US health sector. _N Engl J Med_. 2008;359:1313.\n\nStolberg SG, Gerth J. How companies stall generics and keep themselves healthy. _NY Times._ 2000.\n\nSunshine J, Bhargavan M. The practice of imaging self-referral doesn't produce much one-stop service. _Health Aff_. 2010;29:2237.\n\nVogt WB. _Hospital Market Consolidation: Trends and Consequences_. National Institute for Health Care Management. November 2009. . Accessed November 26, 2011.\n\nWelch HG et al. _Overdiagnosed. Making People Sick in the Pursuit of Health_. Boston, MA: Beacon Press; 2011.\n\n## **17 Conclusion: Tensions and Challenges**\n\nThe perfect health care system is like perfect health\u2014a noble aspiration but one that is impossible to attain. In the preceding chapters, we have discussed many fundamental issues and principles involved in formulating health care policy. A recurrent theme has been the notion that \"magic bullets\" are hard to come by. As stated in Chapter 2, policies tend to evolve in a cyclic process of finding solutions that create new problems that require new solutions. Policy changes may offer a degree of relief for a pressing problem, such as inadequate access to care, but frequently also give rise to various side effects, such as stimulating health care cost inflation.\n\nAll health care systems face the same challenges: improving health, controlling costs, prioritizing allocation of resources, enhancing the quality of care, and distributing services fairly. These challenges require the management of various tensions that pull at the health care system (O'Neil and Seifer, 1995). The goal of health policy is to find the points of equilibrium that produce the optimal system of health care (Table 17\u20131).\n\n**Table 17\u20131.** Major tensions in health care\n\n_Dr. Madeleine Longview is chief resident in critical care medicine and supervises the intensive care unit of a large municipal hospital. It's 5:30 AM, and the intensive care unit team has finally stabilized the condition of a 15-year-old admitted the previous evening with gunshot wounds to the abdomen and chest. Dr. Longview sits by the nursing desk and surveys the other patients in the unit: a 91-year-old woman admitted from a nursing home with sepsis from a urinary tract infection, a 50-year-old man with shock lung caused by drugs ingested in a suicide attempt, and a 32-year-old woman with lupus erythematosus who is rejecting her second kidney transplant. Dr. Longview feels personally responsible for the care of every one of these patients. She tells herself that she will do her best to help each of them survive._\n\n_As Dr. Longview gazes out of the windows of the intensive care unit, the apartment houses surrounding the hospital take shape in the breaking dawn. She wonders: Which block will be the scene of the next drive-by shooting or episode of spouse abuse? Which window shade hides a homebound elder lying on the floor dehydrated and unable to move, waiting for someone to find him and bring him to the emergency department? Which one of the unvaccinated kids in the neighborhood will one day be rushed into the unit limp with meningitis? In which room is someone lighting up the first cigarette of the day? Dr. Longview somehow feels responsible for all those patients-to-be, as well as for the patients lying in the hospital beds around her. After these sleepless nights on duty, the doubts about the value of all the work she does in the intensive care unit creep into her thoughts. She has visions of shutting down the unit and putting all the money to work hiring public health nurses in the community, or maybe just paying for a better grammar school in the neighborhood. But then what would happen to the patients needing her care right now?_\n\nOne of the most basic tensions affecting physicians and other caregivers is the tension between caring for the individual patient and caring for the larger community or population. Many of the most important decisions to be made in health policy\u2014decisions such as allocating health care resources, addressing the social context of health and illness, and augmenting activities in prevention and public health\u2014depend on broadening the practitioner's view to encompass the population health perspective. The challenge for physicians and other clinicians will be to make room for this broader perspective while preserving the ethical duty to care for the individual patients under their charge.\n\nLike Dr. Longview, the health care system as a whole will continue to struggle over finding the proper balance between the provision of acute care services and preventive and chronic care services, as well as striking the right balance between the levels of tertiary and primary care. Few observers would encourage Dr. Longview to succumb to her despair, close all the intensive care units, and expel all the critical care sub-specialists from the health care system. Yet most would agree that health care in the United States has drifted too far away from the primary care end of the tertiary care\u2013primary care axis.\n\n_Dr. Tom Ransom has performed what he believes to be a reasonably thorough workup for Zed's abdominal pain and decreased appetite, including a detailed history and physical examination, blood tests, and abdominal ultrasound\u2014all of which were normal. When Dr. Ransom tells Zed that they will have to work together to manage Zed's symptoms, Zed tells Dr. Ransom that he wants one more test, an abdominal CT scan. Zed says that he had a cousin with similar symptoms who was eventually diagnosed with advanced-stage lymphoma after complaining of pain for over a year._\n\n_Dr. Ransom is in a quandary. He believes it extremely unlikely that Zed has serious pathologic changes in his abdomen that will be detected on CT scan. He could order the scan, but then there's the issue of the cost. He can't recall whether Zed is covered by a fee-for-service plan or by one of the health maintenance organizations (HMOs) that pays on a capitated basis and puts Dr. Ransom at financial risk for all radiologic tests ordered. He starts to ask Zed about his coverage but feels a pang of guilt that he should allow these economic considerations to intrude into his clinical judgment._\n\nThe desire (and in many instances, expectation) of patients to receive all potentially beneficial care, and the unwillingness of these same individuals in their role as purchasers to spend unlimited amounts to finance health care, creates a strain for all caregivers and systems of care. Physicians increasingly are being called upon to incorporate considerations of costs when making clinical decisions. Debate will continue about the best ways to encourage physicians to be more accountable for the costs of care in a manner that is socially responsible and does not unduly intrude on the physician's ability to serve the individual patient. Is it necessary to use payment methods that place physicians at individual financial risk for their treatment decisions in order to control costs? Are more global methods available to induce physicians and other care-givers to practice in a more cost-conscious manner? If Zed does not get a CT scan, does that constitute painless or painful cost control?\n\n_On the eve of his retirement, Dr. Melvin Steadman reminisces with his son, Dr. Kevin Steadman. The elder Dr. Steadman has practiced as a solo pediatrician for more than 40 years in the same town. The only boss he has known in his professional life has been himself. He has served as president of the local medical society, helped spearhead efforts to build a special children's wing of the local hospital,and antagonized several of his colleagues when he pushed for a change in hospital policy that required physicians to attend extra continuing medical education courses in order to maintain their hospital privileges. Mel swore that he'd never retire; but he also swore that he'd never let the insurance companies \"tell me how to practice medicine.\" He has refused to sign any managed care contracts. Facing a dwindling supply of patients, Mel has decided to call it quits._\n\n_His son Kevin is also a pediatrician, working as a staff physician for a large for-profit multispecialty group that recently opened up an office in town. Kevin remembers the many nights when his father didn't get home from work until after he had gone to bed. Kevin's work hours are more regular at the group practice, and he is on call for only one weekend every 2 months. He considers his father's approach to medicine old-fashioned in many ways\u2014excessively paternalistic toward patients and irrationally scornful of the pediatric nurse practitioners who work with Kevin. He does, however, envy his father's professional independence. Just this week, the group practice notified Kevin that he would have to divide his time between his current office and a new site that would soon open in a suburban mall. His schedule will be limited to 10-minute drop-in appointments at the new site, rather than the style of practice that promotes a sense of continuity, one that allows him to get to know his patients over time._\n\nA system of health care formerly managed according to a professional model by independent practitioners is being pulled toward a corporate model of care featuring large organizations managed by administrators. As the role of corporate entities expands, traditional responsibilities toward patients and local communities are vying with new obligations to shareholders. Power relationships are changing, with insurance companies and organized purchasers challenging the dominance of the medical profession. A shift toward multidisciplinary group practice may provide more opportunity for health care professionals to work collegially and implement new approaches to quality improvement to elevate the competence of all health care providers. At the same time, a competitive, forprofit health care environment may induce physicians to compromise their humanity and turn toward the \"homo economicus\" model, basing clinical decisions in part on monetary considerations.\n\n_Aurora can't wait any longer in the crowded county hospital emergency department. She's already been there for 6 hours, and the physician hasn't seen her yet. Her lower abdomen still hurts, but she figures she'll just have to put up with it for a few more days. She really doesn't have much choice. Poor and uninsured, where else could she go? Aurora has two young children at home who need to be put to bed. In half an hour, their father has to get to his night job as a security officer. As she enters her apartment, she collapses, the pregnancy in her fallopian tube having ruptured, producing internal hemorrhage. Her husband frantically dials 911, praying that his wife won't die._\n\nPerhaps no tension within the US health care system is as far from reaching a point of satisfactory equilibrium as the achievement of a basic level of fairness in the distribution of health care services and the burden of paying for those services. Many more people in the country were uninsured in 2011 than in 1991. Because of persistent financial barriers, patients do not benefit from early detection of potentially curable cancers, patients with chronic diseases are hospitalized because of lack of timely primary care, hypertensive patients forego the medications that might avert the occurrence of strokes and kidney failure, and babies are born prematurely and spend their first weeks of life in a neonatal intensive care unit. The poor pay a greater proportion of their income for health care than do more affluent families. The Affordable Care Act of 2010 would greatly reduce the number of uninsured. However, implementation of the Act faces political, judicial, and financial challenges, and coverage would fall short of truly universal even if fully implemented.\n\nPeople providing and receiving care in the United States must work together to achieve a brighter future for the nation's health care system. Changing the future will require that people look beyond their immediate self-interest to view the common good of a health care system that is accessible, affordable, and of high quality for all. A heightened level of public discourse will be needed, with a populace that is better informed and more actively engaged in shaping the future of their health care system. Concepts in health policy based on established facts rather than ideologically driven myths will need to be discussed and debated in a manner that connects with the daily realities experienced by patients and caregivers. The attitudes and actions of physicians and other health care professionals will play a major role in determining the future of health care in the United States. With leadership and foresight among the community of health care professionals, our nation may yet achieve a system that allows the most honorable features of the healing professions to flourish.\n\n### **REFERENCE**\n\nO'Neil E, Seifer S. Health care reform and medical education: Forces towards generalism. _Acad Med_. 1995;70:S37.\n\n## **18 Questions and Discussion Topics**\n\n### **CHAPTER 2: PAYING FOR HEALTH CARE**\n\n1. What are the four modes of financing health care? Describe each.\n\n2. Describe regressive, proportional, and progressive financing. Explain how each of the following is regressive, proportional, or progressive: out-of-pocket payments, experience-rated individual private insurance, community-rated individual private insurance, health insurance purchased 100% by the employer (assuming that employees actually pay for health insurance as explained in the text), and the federal income tax.\n\n3. Harvey, who has worked all his life for General Electric, reaches 65 years of age. He does not retire. Is he eligible for Medicare Part A? Part B? Six months later, his wife, who has never worked, reaches 65 years of age. Is she eligible for Medicare Part A? Part B? How are Parts A and B paid for?\n\n4. Hubert has received social security disability for 24 months because he has AIDS. Is he eligible for Medicare?\n\n5. Rena developed chronic renal failure and started renal dialysis 2 weeks ago. She feels fine and is working. Is she eligible for Medicare?\n\n6. Heidi, aged 72 years, on Medicare Part A and B without Medicaid or a Medigap policy, is hospitalized for a stroke complicated by a deep vein thrombosis of the leg and a pulmonary embolus. She is in the acute hospital for 70 days and cared for by a family practitioner and a neurologist. She improves somewhat and is then transferred to the skilled nursing facility (SNF) for rehabilitation. She remains in the SNF for 30 days and is still severely disabled and unable to go home. She is sent to a nursing home for custodial care, where she stays for 3 months. Surprisingly, she improves and goes home, where she receives skilled physical therapy services from a home care agency and also has a homemaker come in for 4 hours a day to buy food, cook, and clean the house. She is on three prescription medications at home. What does Heidi pay and what does Medicare pay? Acute hospital? SNF? Nursing home? Home care? Physicians? Prescriptions while in hospital? Prescriptions while at home?\n\n#### **Discussion Topics**\n\n1. Discuss your experiences with health insurance that was provided through a job. How did you obtain the insurance? Did you pay part of the premium? Were there deductibles or copayments? How many choices of plans did you have? What happened if you left your job?\n\n2. Divide into two groups: one insurance company selling community-rated health insurance policies and the other selling experience-rated policies. Each side should try to convince the instructor to buy its policy, first with the instructor as a young, healthy person, and then with the instructor as an older person with diabetes. Which policy is the young person more likely to choose, and which the older person?\n\n### **CHAPTER 3: ACCESS TO HEALTH CARE**\n\n1. Describe the two main categories of people without health insurance.\n\n2. Why did uninsurance increase during the period 1980 to 2010?\n\n3. Compare access to health care for people with private insurance, for Medicaid recipients, and for people without insurance. Give examples.\n\n4. Compare health outcomes for people with private insurance, for Medicaid recipients, and for people without insurance. Give examples.\n\n#### **Discussion Topics**\n\n1. What are some explanations as to why Ace Banks was healthy at age 48 while Bill Downes died at that age?\n\n2. Women on average have more visits than men to physicians. Does that mean that women receive better health care than men?\n\n3. Discuss possible reasons why minority patients receive poorer quality of care than white patients for many diseases.\n\n4. What is the relationship between socioeconomic status (including factors such as income, education, and occupation) and health? Why does such a relationship exist?\n\n5. What would be the best strategies to improve the health status of African Americans in the United States?\n\n### **CHAPTER 4: REIMBURSING HEALTH CARE PROVIDERS**\n\n1. Explain each mode of physician reimbursement: fee-for-service, episode of illness, capitation, and salary. Explain each mode of hospital reimbursement: fee-for-service, per diem, episode of illness (diagnosis-related group [DRG]), and global budget.\n\n2. How does capitation payment free insurers of risk? How does capitation payment shift risk to providers of care?\n\n3. What are the arguments for risk-adjusting capitation payments?\n\n#### **Discussion Topics**\n\n1. You are a primary care physician (PCP) caring for a young woman with new onset of severe headaches and amenorrhea and a normal physical examination. What are the financial incentives and disincentives that would lead you to order or not to order a magnetic resonance imaging (MRI) scan in a case in which the need for the MRI was equivocal?\n\n(a) under traditional fee-for-service practice;\n\n(b) under fee-for-service practice with utilization review;\n\n(c) under an independent practice association (IPA)-model health maintenance organization (HMO) in which you receive a capitation payment that places you at risk for laboratory and x-ray studies and specialty referrals;\n\n(d) under a staff model HMO that has a two-month waiting list for elective MRI scans?\n\nIn the case of the staff model HMO, what would you do if you felt you needed to obtain the MRI within 48 hours?\n\n2. You are a hospital administrator and your hospital is in financial difficulty. You are about to address the medical staff, imploring them to help the hospital financially. In the old days, all you had to say was, in effect: \"Admit as many patients as possible and keep them in the hospital as long as you can,\" but times have changed. For some methods of reimbursement, you want physicians to admit more patients; for others, you don't. For some methods, you want patients to stay long, for others, you don't. What do you tell the medical staff regarding the following:\n\n(a) Medicare (DRG) patients\n\n(b) Medicaid (per diem) patients\n\n(c) HMO (per diem) patients\n\n(d) HMO (capitated) patients\n\nFor each of these categories of patients, does it help or hurt the hospital for physicians to\n\n(a) admit more patients;\n\n(b) keep them in the hospital more days;\n\n(c) order more diagnostic studies?\n\n### **CHAPTER 5: HOW HEALTH CARE IS ORGANIZED\u2014I: PRIMARY, SECONDARY, AND TERTIARY CARE**\n\n#### **Discussion Topics**\n\n1. You are 63 years old and you begin to experience chest pain when walking. You do not have a physician. A friend suggests that you need a coronary artery bypass and recommends a cardiac surgeon at the medical school. What do you do\n\n(a) under a dispersed model of health care delivery?\n\n(b) under a regionalized model?\n\n2. Give some examples of the statement, \"Common disorders commonly occur and rare ones rarely happen.\" What are the implications of this statement for the ratio of generalist to specialist physicians in the United States?\n\n3. In Great Britain, 65% of physicians are general practitioners. In Canada, 50% of physicians are generalists. In the United States, approximately one-third of physicians are generalists (general and family practitioners, general internists, and general pediatricians). Assume you are Chair of the Health Subcommittee of the US House of Representatives Ways and Means Committee. What legislation might you propose to increase the proportion of generalist physicians?\n\n4. Discuss the pros and cons of requiring everyone to enter the health care system through a \"gatekeeper\" health care provider (generalist physician, nurse practitioner, or physician assistant).\n\n5. What are some advantages of a primary-care-based health system?\n\n### **CHAPTER 6: HOW HEALTH CARE IS ORGANIZED\u2014II: HEALTH DELIVERY SYSTEMS**\n\n1. What are the two generations of HMOs? Give examples of each (if possible, in your community).\n\n2. What is vertical integration? What is virtual integration?\n\n3. What is an ACO? What is a medical home and a medical neighborhood? Is a medical neighborhood the same as an ACO?\n\n### **CHAPTER 7: THE HEALTH CARE WORKFORCE AND THE EDUCATION OF HEALTH PROFESSIONALS**\n\nDescribe past and future trends in the physician, \"mid-level,\" nursing, and pharmacist workforce.\n\n### **CHAPTER 8: PAINFUL VERSUS PAINLESS COST CONTROL**\n\n1. Give examples of medical interventions that lie on the steeper portions of the cost\u2013benefit curve, and p>interventions that lie on the flatter portions. Is the elimination of the latter painful or painless cost control?\n\n2. Give examples of painless cost control. Are these painless for everyone?\n\n#### **Discussion Topics**\n\n1. CABGville has four cardiac surgery units; one unit performs 300 coronary artery bypass graft (CABG) surgeries each year, and the other units perform an average of 40 per year. Cardiac surgeons can schedule a CABG anytime they wish. The small units have an operative mortality of 7% compared with 4% for the large unit. To control costs, the health planning council of CABGville closes the three less productive cardiac surgery units. Elective CABG surgeries now have a 1-month waiting list, and because of tight scheduling, surgeons are less likely to operate; the number of CABGs goes down from 420 to 340 per year; both the overall costs of CABG surgery and the unit cost per CABG operation drop, as does the mortality rate. Did CABGville achieve painful or painless cost control?\n\n2. Pretend that total US health care expenditures have been capped and are controlled by a health services commission. Because of tight budgetary constraints, the commission must decide whether to fund an all-out program of mammography or to limit mammography and finance in its place high-cost chemotherapy regimens for patients with metastatic breast cancer, treatments whose effectiveness has not been proven, but which might help certain subgroups of women. Under the first option, several thousand cases of early-stage breast cancer could be treated with curative surgery each year, but women currently suffering from advanced-stage breast cancer would receive no benefit. Which is the more painful cost control option from the point of view of women without breast cancer? From the perspective of women with metastatic breast cancer? From the perspective of society as a whole? Which of these two groups of women should have priority in this decision?\n\n### **CHAPTER 9: MECHANISMS FOR CONTROLLING COSTS**\n\n#### **Discussion Topics**\n\n1. You are chair of the health planning council of CABGville, a town that continues to have a health care cost crisis. The town has 30 physicians, each seeing 30 patients a day at a cost of $30 per visit. Total daily cost is 30 \u00d7 30 \u00d7 30 = $27,000. What methods are available to reduce the total cost of physician services? Would it work to reduce the fee per visit from $30 to $20? If an expenditure cap strategy (tying fees to volume) were used, how would it work?\n\n2. The CABGville health planning council changes the mode of physician reimbursement from fee-for-service to capitation: $20 per patient per month to PCPs, with 20 PCPs each having 2000 patients. (PCPs pay specialists from the $20 capitation.) Total cost per month = $800,000 (approximately $27,000 per day). How could the health planning council reduce the monthly cost? Could physician costs still increase despite this method of cost control? Why or why not?\n\n3. You have finished your residency in internal medicine and have the choice to work at Kaiser or at a private practice that is part of an IPA. You are particularly concerned about your ability to order laboratory tests and x-rays and to obtain specialty consultations. At Kaiser, you learn that you have freedom in ordering tests and obtaining consultations, but that patients may have to wait (except in urgent situations) because of the limited supply of such equipment as MRI scanners and of specialty appointments. At the IPA, you must request prior authorization for expensive diagnostic studies and for specialty consultations, but once prior authorization has been obtained, waiting periods are fairly short. Which work situation would you prefer, and which do you think has the better chance of controlling costs?\n\n4. What are the arguments pro and con patient cost sharing as a cost control strategy?\n\n5. You are the President of the United States, and your first term ends in a year. The cost-control mechanism you instituted 2 years ago, based on patient cost sharing and managed competition, has not worked, and the American people are upset about persistent health care inflation. You are preparing for a major television address on health care costs. What will you propose? Can you convince the public that yours is a painless cost-control strategy?\n\n### **CHAPTER 10: QUALITY OF HEALTH CARE**\n\n#### **Discussion Topics**\n\n1. Have you ever experienced or witnessed a medical care encounter of poor quality? What did you do about it? What should you have done?\n\n2. In the vignette about Shelley Rush, who do you think was responsible for the error in giving insulin to the wrong patient?\n\n3. In the vignette about Nina Brown, had the physician been working in a fee-for-service environment rather than a cost-conscious HMO, do you think he or she would have admitted Ms. Brown to the hospital?\n\n4. Reread the example of the 23-year-old graduate student whose x-ray report was lost. If you were the administrator of the hospital, what would you do to prevent such an error from taking place again? If you were the office manager of the internist's office that never received the x-ray report, what would you do to avoid a recurrence of this problem?\n\n5. What is wrong with the malpractice system? What would you do to fix it?\n\n### **CHAPTER 11: PREVENTION OF ILLNESS**\n\n1. Why did tuberculosis (TB) decline prior to the identification of the TB bacillus? Why did polio morbidity and mortality decline? Why did Hodgkin disease mortality fall in the late twentieth century?\n\n2. What are the first and the second epidemiologic revolutions?\n\n#### **Discussion Topics**\n\n1. Two people are campaigning for the consumer board of their group practice. The incumbent is running on a platform of charging tobacco users higher premiums than nonusers, because their use of tobacco costs the group practice more money. The opponent believes that society rather than the individual is responsible for tobacco addiction and that the group practice should become involved in social action against cigarette smoking. Conduct a debate between these two views.\n\n2. How do you explain the fact that a large number of heart attacks occur at early ages in people with cholesterol levels below the median level for the United States? That heart attacks seldom occur at these ages in Japan? What is the implication for primary prevention of coronary heart disease?\n\n3. You are named as head of the breast cancer prevention section of the US Centers for Disease Control and Prevention. What primary and secondary prevention programs would you favor to reduce the incidence of and mortality from breast cancer?\n\n### **CHAPTER 12: LONG-TERM CARE**\n\n1. What are activities of daily living and instrumental activities of daily living?\n\n2. What percentage of long-term care services are funded by which funding sources?\n\n3. Which long-term care services are covered by Medicare and which are not? Which are covered by Medicaid?\n\n#### **Discussion Topics**\n\n1. You are president of LTC Insurance Company and are testifying before a Senate committee on long-term care. You are asked two questions: Why do only a few million people carry private long-term care insurance? How do you answer the complaints that senior citizen advocacy groups make about the terms of private long-term care insurance policies? What do you say to the committee?\n\n2. Your mother's Alzheimer's disease is getting worse; she wanders around the neighborhood, sometimes unable to find her way home; she sleeps during the day and stays up most of the night; and she has become incontinent. Your father died 2 years ago. You and your spouse both work, you have three school-aged children, and you have an extra room in your home. The hospital social worker calls and says that your mother needs 24-hour-a-day help. Your choices are:\n\n(a) hiring a homemaker to live with your mother at $16,000 per year;\n\n(b) placing your mother in a nursing home whose bill will be paid by Medicaid;\n\n(c) taking your mother home with you. What do you decide?\n\nWhat reforms in the US long-term care system would have benefited you in this situation? How should such reforms be financed?\n\n### **CHAPTER 13: MEDICAL ETHICS AND RATIONING OF HEALTH CARE**\n\n#### **Discussion Topics**\n\n1. Pretend that the Lakeberg family discussed in this chapter belongs to an HMO, and that you are the HMO's medical director. The Lakeberg parents want surgery to separate the Siamese twins at the cost of $1 million. The list of benefits covered in the Lakebergs' HMO policy neither affirms nor denies their right to the surgery, so the responsibility to approve or deny the surgery falls on you. What do you decide? If you approve the surgery, who will end up paying for it? Is an ethical dilemma involved or not?\n\n2. You are Dr. Marco Intensivo, as described in the vignette in the section \"What is Rationing?\" What do you do?\n\n3. In the case of Mr. Olds and Mr. Younger described in the organ transplant section, which patient should receive the donor heart?\n\n4. You are the PCP for Rodolfo, a 58-year-old man who suffered a cerebral hemorrhage and has been in a persistent vegetative state for 18 months. He lives in a nursing home, requires tube feedings and round-the-clock nursing attention, and his care is paid for by Medicaid. Rodolfo's daughter is a nurse in the intensive care unit of your hospital. Rodolfo's wife is deeply religious and has faith that Rodolfo will get better.\n\nApproximately every 6 weeks, Rodolfo develops a urinary tract infection with septicemia and must be admitted to the hospital\u2014often to the ICU\u2014for treatment. Over the course of 2 years, Rodolfo's care has cost $260,000. The hospital ethics committee discussed the case and recommended that tube feedings be withdrawn, or that the next episode of septicemia not be treated, thereby allowing Rodolfo to die. When you discussed the ethics committee recommendations with the family, the daughter agreed but the wife demanded that everything possible be done to continue Rodolfo's life. As Rodolfo's physician, what do you do? Which ethical dilemmas are involved? Autonomy versus beneficence? Autonomy versus nonmaleficence? Autonomy versus distributive justice? Beneficence versus distributive justice? If Rodolfo's care were withdrawn, what would happen to the money saved?\n\n5. Evidence from public opinion polls suggests that people in the United States want the right to health care but don't want to pay for it.\n\nAt midnight, a new mother awakens to hear her 2-week-old infant scream. The mother and baby are Medicaid recipients. If she were experienced, the mother would know that the scream is normal, but she is frightened. She phones the emergency department and asks to bring the baby in to be seen. No amount of telephone advice seems to reassure her. Does the right to health care include society paying for her visit to the emergency department? Who is actually paying? Should the mother be advised to come into the emergency department if she is uninsured and wealthy? Uninsured and poor?\n\n6. In Oregon, the Medicaid program was extended to thousands of Oregonians who had previously been uninsured. To help pay for this extension, the breadth of services available to Medicaid recipients was reduced such that recipients lost access to some care that might have been beneficial. You are the Governor of Oregon and you have to testify in a lawsuit alleging that the program is unfair because it deprives Medicaid recipients of certain services enjoyed by privately insured people. What is your response?\n\n7. Should physicians be responsible to serve one master\u2014their patient\u2014or two masters\u2014their patient and the broader needs of society? In your discussion, draw from the examples of the Lake-bergs, Dr. Intensivo, and Rodolfo. How has the distribution system for organ transplantation tried to balance these two masters?\n\n### **CHAPTER 14: HEALTH CARE IN FOUR NATIONS**\n\n1. You are a secretary in a large company in Germany (Canada, United Kingdom, or Japan). How is your health care paid for? You become sick and are forced to retire from your job. How is your health care paid for in Germany (Canada, United Kingdom, or Japan)?\n\n2. If you developed a urinary tract infection, what would you do in Germany (Canada, United Kingdom, or Japan)? What if you needed cataract surgery? What if you had a sudden abdominal pain in the middle of the night? What if you developed leukemia and needed a bone marrow transplant? In each of these cases, which physician would care for you and where would you be cared for?\n\n3. You are a general practitioner in Germany (Canada, United Kingdom, or Japan). How are you paid? You are a specialist in Germany (Canada, United Kingdom, or Japan). How are you paid? You are a hospital administrator in Germany (Canada, UK, or Japan). How is your hospital paid?\n\n4. How are costs controlled in the four countries?\n\n### **CHAPTER 15: HEALTH CARE REFORM AND NATIONAL HEALTH INSURANCE**\n\n1. Describe how a government-financed national health insurance plan, an employer mandate plan, and an individual mandate plan would work.\n\n2. What is the difference between a social insurance and a public assistance approach to government-financed national health insurance? Use Medicare and Medicaid as examples.\n\n3. What are the main features of the 2010 Patient Protection and Affordable Care Act (ACA)?\n\n#### **Discussion Topics**\n\n1. You are the speech writer for two candidates for the Democratic presidential nomination. One candidate favors a mixed employer and individual mandate and the other a single-payer approach. What points would you have each candidate make about the strengths of his or her position and the weaknesses of the other candidate's position?\n\n2. Why do you think that there has been such a polarized debate over the ACA?\n\n### **CHAPTER 16: CONFLICT AND CHANGE IN AMERICA'S HEALTH CARE SYSTEM**\n\n1. Describe how the payers of health care services increased their power between 1945 and 1995.\n\n2. Describe changes in the relationships between physicians and insurance companies between 1945 and 1995.\n\n3. Describe the 1995\u20132000 backlash against managed care.\n\n4. Describe the recently growing power of specialty-oriented providers of care.\n\n#### **Discussion Topics**\n\n1. Discuss potential conflicts between the profit motive and the principles of beneficence and nonmaleficence in the following situations:\n\n(a) a private surgeon receiving fee-for-service reimbursement;\n\n(b) a primary physician in a small group practice that receives capitation payments covering primary care, laboratory, x-ray, and specialty referrals;\n\n(c) a physician who is the utilization manager of a large for-profit HMO receiving requests from her employed physicians to authorize expensive MRI scans for their patients;\n\n(d) the administrator of a nonprofit hospital who has calculated that a new cardiac surgery unit will be profitable even if only one surgery is performed each week;\n\n(e) the CEO of an HMO deciding whether to accept Medicaid patients, for whom the state government is paying premiums 30% lower than premiums paid for private patients.\n\nWhat changes in the organization of health care could be made that would minimize such conflicts?\n\n2. Discuss how health care is organized in your community\u2014who are the payers, insurers, and providers? To what degree has your local health care system moved from a dispersed set of institutions to a small number of vertically or virtually integrated health care conglomerates?\n\n3. Where in the health care system of the twenty-first century would you like to be\u2014as a provider and as a patient? What are yours fears and hopes for the future?\n\n## **Index**\n\nPage numbers followed by f refer to figures; page numbers followed by t refer to tables.\n\n**A**\n\nAccreditation Council for Graduate Medical Education (ACGME), ,\n\nActive practitioners in the U.S., , t\n\nActivities of daily living (ADL), , t\n\nAdequate access to care,\n\nAdequate scientific knowledge, t\n\nAgency for Health Care Policy and Research\/Agency for Health Care Research and Quality (AHRQ), ,\n\nAmbulatory care physicians, ,\n\nAmerican Association for Labor Legislation (AALL),\n\nAmerican Medical Association (AMA), ,\n\nAntismoking campaigns, \u2013140\n\nAutonomy, \u2013154\n\nnonmaleficence, comparison with,\n\n**B**\n\nBalanced Budget Act of 1997,\n\nBeneficence,\n\nBeveridge, William,\n\nBiomedical model, ,\n\nBlue Cross subscribers,\n\nBlue Cross, \u2013203\n\nBlue Shield,\n\nBreast cancer mortality rates, \u2013143\n\nBreast Cancer, \u2013143\n\nmultiple risk factors,\n\nBritish Medical Association (BMA),\n\nBritish National Health Service (NHS), \u201346, \u2013176\n\nreforms of, \u2013180\n\nBritish system\n\nof capitation payments,\n\nof health care, \u201346\n\n**C**\n\nCalifornia Medical Injury Compensation Reform Act,\n\nCanada's universal insurance program,\n\nCanadian family physicians,\n\nCanadian health care system,\n\nCanadian Hospital Insurance Act,\n\nCarve-outs,\n\nCHD. _See_ Coronary heart disease Chief organized purchaser, disinterest of,\n\nChronic disease prevention,\n\nmedical model,\n\nClinical practice guidelines, \u2013123\n\nClinton, Bill (U.S. President), health insurance plan of,\n\nCommercial insurers,\n\nCommodity scarcity through transplantation of organs, \u201359\n\nCommunity health centers, \u201362\n\nCommunity rating, . _See also_ Experience rating\n\nCommunity-based long-term care,\n\nCompetent health care providers, \u2013117\n\nCompetitive strategies\n\nfor cost control, \u2013106\n\nComputerized information systems,\n\nComputerized physician order entry (CPOE),\n\nConcerted action,\n\nContinuous quality improvement (CQI), ,\n\nCoronary artery bypass graft (CABG) surgery, \u2013120\n\nCoronary heart disease (CHD), , \u2013143\n\nage-adjusted mortality rates, \u2013138,\n\ndue to cigarette smoking, \u201340\n\ndue to hypertension, \u2013142\n\ndue to rich diet, \u2013141\n\nContinuous quality improvement (CQI) model,\n\nsystematically monitoring of health care providers, , t\n\nCQI. _See_ Continuous quality improvement\n\n_Curing Health Care_ ,\n\nCustodial services,\n\n**D**\n\nDepartment of Health and Human Services (U.S.), \u2013159\n\nDiagnosis-related group (DRG) method of payment, \u201340\n\nDispersed Model, \u201347, , , . _See also_\n\nRegionalized Model Distributive justice, \u2013155\n\nDRG. _See_ Diagnosis-related group\n\n**E**\n\nEarmarked health tax,\n\nEmployer mandate approach, ,\n\nEmployer mandate plans,\n\ncriticism of,\n\nEmployment-based private insurance, \u201310\n\nEmployment-based social insurance model,\n\nEmployment-based privately administered national health insurance proposal,\n\nEnd-of-life costs\n\nof patients in hospice programs,\n\nEnterprise liability,\n\nEthical dilemmas, \u2013157\n\nExperience rating, . _See also_ Community rating\n\nExperience-rated insurance, ,\n\nExplicit rationing,\n\n**F**\n\nFederal social insurance trust fund, \u201389\n\nFee-for-service payment,\n\nFee-for-service reimbursement, , , ,\n\nFinancially Neutral Clinical Decision Making,\n\nFinancing controls\n\nfor health care, \u2013106\n\nweaknesses,\n\nFinancing medical education,\n\nFirst epidemic revolution,\n\nFiscal reality,\n\nFiscal scarcity\n\nresource allocation, correlation with, \u2013159\n\nFrontier Nursing Service,\n\n**G**\n\nGatekeeping\n\nin primary care,\n\nGerman Cost Containment Act of 1977,\n\nGerman national health insurance system, f\n\nGood quality care,\n\nGovernment financed insurance, \u201316, f\n\nGovernment-financed national health insurance plans, \u2013190\n\npublic assistance (welfare) model,\n\nGPs and specialists\n\nBritish, pay for performance (P4P), \u2013178\n\nCanadian, fee-for-service basis,\n\nGerman, detailed fee schedule,\n\nJapanese, per diem hospital payment based on diagnosis,\n\nGriffiths, Martha, Michigan Representative,\n\nGuaranteed medical benefits,\n\n**H**\n\nHarvard Medical Practice study,\n\nHealth care \"report cards,\" 125\n\nHealth care financing ethics, \u2013166\n\nHealth care financing system, \u2013198\n\nHealth care financing,\n\nHealth care institutions,\n\nHealth care market, consolidation in,\n\nHealth care outcomes\n\norganization of health care systems and institutions,\n\nHealth care providers\n\nsocially useful cost savings, \u2013160\n\nHealth care quality, crossnational comparisons,\n\nHealth care resources, \u2013167\n\nHealth care sector\n\nof U.S. economy,\n\nHealth care system, \u20133,\n\nin the U.S.\n\nperformance analysis,\n\norganization of, \u201353\n\npublic's view of,\n\nquality of care, correlation with, \u2013120\n\ntraditional quality assurance, \u2013122\n\nweaknesses, understanding of, \u20133\n\nHealth care\n\nadministrative overheads of, \u201399\n\nbalance in levels of, \u201348, t\n\ncost control\n\nin Canada,\n\ncategories, t\n\nin Germany,\n\nin Japan,\n\npainless, \u2013102, t\n\nrequirement of,\n\nrequirements for,\n\nstrategies, \u2013101\n\nin United Kingdom,\n\ncost effectiveness\n\nprioritization and analysis of, \u2013101\n\ncost saving\n\nthrough disease prevention,\n\nthrough innovation,\n\ncost sharing, \u2013109\n\ncosts, description of,\n\ndeprived patients, ,\n\nexcess care,\n\npatients with, \u20132\n\nexpenditures, , t\n\nfinancial incentives, \u201355\n\nfinancing of\n\nin the U.S., , t\n\nmodes, \u201315\n\ngeneralism, role of, \u201354\n\nhistorical overview, t\n\nineffective, elimination of, \u201398\n\ninflation,\n\ncontrol of,\n\nneed and cost of, \u20136\n\npatient access to, ,\n\ngender and race factors, \u201326\n\npayment modes, \u201315\n\nphysician recommendations, implications of, \u20137\n\nrelation with health status, \u201328\n\nsocioeconomic status, influence of, \u201329\n\nresource input and outcomes, correlation in, \u201397\n\ntraditional structure of, \u201360\n\nHealth care, tensions, t\n\nHealth insurance premiums,\n\nHealth insurance\n\nin Canada, \u201373\n\ntax-financed, public, single-payer health care system,\n\nessence of,\n\nin Germany, \u2013171\n\nmerged social insurance and public assistance structure,\n\nhealth care services, implications of,\n\nin Japan, \u201382\n\nlack of, , \u201324\n\nfactors for, \u201319\n\nlimitations of,\n\nnonfinancial barriers, \u201326\n\nsource of, , t\n\nin United Kingdom, \u201376\n\nHealth Maintenance Organization Act,\n\nHealth maintenance organizations (HMO), ,\n\ncontracting of, \u2013205\n\ndecline of,\n\nIndependent practice associations (IPA) model, \u201367\n\nprepaid group practice, basis on, \u201363\n\nsecond generation, \u201367\n\nHealth maintenance,\n\nHealth plan employer data and information set (HEDIS),\n\nHealth policy\n\nabstract concepts in, \u2013216\n\ngoal of,\n\nconcerns of,\n\nHealth reform law, Massachusetts,\n\nHealth workers\n\nsupply of, \u201383\n\nunderrepresented minorities, representation of, \u201387, f\n\nwomen, \u201386, f\n\nHealth Commissioner projects, Limittown, U.S.A.,\n\nHeritage plan,\n\nHigh blood pressure, primary prevention,\n\nHigh-deductible health plan (HDHP),\n\nHigh-dose chemotherapy with autologous bone marrow transplantation (HDC-ABMT),\n\nHigh-quality health care, \u2013120\n\nHMO. _See_ Health maintenance organization Home care agencies,\n\nHome health services medicaid coverage of,\n\nHospital payment methods, \u201340\n\nHospital Quality Initiative,\n\nHSA. _See_ Health savings account Hybrid national health insurance proposals,\n\nin California,\n\n**I**\n\nIdeal long-term caregivers,\n\nIllegal drug use,\n\nIndependent practice associations (IPAs),\n\nIndiana Medical Malpractice Act,\n\nIndividual health care consumers,\n\nIndividual mandate health insurance, \u2013193\n\nIndividual mandates, \u2013194\n\nIndividual patients, maximizing care,\n\nIndividual physician report cards,\n\nIndividual private insurance, , f,\n\nInfant mortality\n\nrace, correlation with, \u201329, t\n\nInfectious disease mortality rates,\n\nInformal caregivers,\n\nIn-hospital medical errors,\n\nInstitute for Healthcare Improvement (IHI),\n\nInstrumental activities of daily living (IADLs),\n\nInsurance co-payments,\n\nInsurance deductibles,\n\nInsurers and providers of care, alliance of,\n\nInsurers,\n\nIntegrated organizations,\n\nIntegrated Healthcare Association (IHA) program,\n\nInvestor-owned \"for-profit\" status,\n\nIPA. _See_ Independent practice associations\n\n**J**\n\nJapan's community-based health insurance\/citizens' health insurance,\n\nJoint Commission on Accreditation of Hospitals\/Joint Commission,\n\n_Journal of the American Medical Association_ , \u201361\n\nJustice,\n\n**K**\n\nKaiser Health Plan, , \u201364\n\nKaiser\u2013Permanente medical care program, \u201365\n\nKennedy, Edward, Massachusetts Senator,\n\nKennedy-Griffiths Health Security Act of 1970,\n\nKnifeless' gamma ray surgery,\n\n**L**\n\nLCME. _See_ Licensing Council on Medical Education Licensing agencies,\n\nLicensing Council on Medical Education (LCME),\n\nLife-expectancy\n\nrace, correlation with, , t\n\nLong-term care policies,\n\nLong-term care,\n\nactivities requiring assistance, t\n\ndeinstitutionalizing,\n\ndirect out-of-pocket payments,\n\nfinancing of, \u2013151\n\nout-of-pocket expenses, ,\n\nLower-income people, nursing home care, ,\n\n**M**\n\nMacroallocation, ,\n\nMagic bullets, ,\n\nMalpractice liability system,\n\nnegative side effects on medical practice, \u2013129\n\nManaged care plans, , , ,\n\nMaximizing care for each patient, physicians' single-mindedness, \u2013167\n\nMayo clinic, , , ,\n\nMedicaid long-term coverage, \u2013148\n\nMedicaid public assistance model,\n\nMedicaid, , t\u201312t, , ,\n\nhealth care access through,\n\nMedicaid\/SCHIP (beneficiaries), 2007, , t\n\nMedical care\n\nin Canada, \u201374\n\ncosts of, influence of prevention,\n\nfinancial considerations and quality,\n\nin Germany,\n\nin Japan,\n\nin United Kingdom, \u201376\n\nMedical commons,\n\nMedical education, \u201375\n\nMedical ethics,\n\nprinciples of, \u2013154\n\nMedical injury, alternatives to jury trials,\n\nMedical insurance. _See_ Health insurance Medical negligence, , \u2013120\n\nMedical reimbursements,\n\nMedicare Advantage program, , ,\n\nMedicare coverage disparities, t,\n\nMedicare long-term coverage,\n\nMedicare Modernization Act of 2003,\n\nMedicare Part A, , t\n\nMedicare Part B, , t,\n\nMedicare Part D, \u201311, ,\n\nMedicare Prospective Payment System (diagnosis related groups [DRGs]) of 1983,\n\nMedicare, ,\n\nMedicare's budget,\n\nMedicine, commercialization of,\n\nMethods of payment, \u201332\n\nMicroallocation,\n\ndaily clinical decisions,\n\nMortality rates\n\nin the U.S.,\n\nMultidisciplinary group practice,\n\nMultispecialty group practice, \u201361\n\nMultispecialty groups,\n\n**N**\n\nNational Cholesterol Education Program,\n\nNational Committee for Quality Assurance (NCQA), ,\n\nNational health expenditures per capita, , f,\n\nNational Health Insurance (U.S.), \u2013199\n\nbenefit package,\n\ncost containment, \u2013197\n\npatient cost sharing, \u2013196\n\nNational Organ Transplantation Act of 1984,\n\nNHS. _See_ British National Health Service NIH cholesterol reduction strategy,\n\nNixon, U.S. President, \u2013191\n\nNo-fault malpractice reform,\n\nNonmaleficence, ,\n\nNurse practitioners, \u201380\n\nNurses\n\ndemand and need, \u201381\n\nsupply of, , f\n\nNursing homes, \u2013150\n\n**O**\n\nOmnibus Budget Reconciliation Act of 1987,\n\nOpen and closed medical care systems, role of ethical considerations,\n\nOregon Health Plan, \u201364\n\nOrgan allocation,\n\nOrganizing care, models of,\n\nOutcome measures,\n\nOutcomes,\n\nOut-of-pocket payments, , , , , f\n\nOveruse\u2013underuse spectrum,\n\n**P**\n\nPanel management\n\nin primary care, \u201352\n\nPatient-centered medical home, \u201352\n\nPatients' persistent financial barriers,\n\nPay for Performance (P4P), \u2013128\n\nPay for Reporting, \u2013127\n\nPaying physicians and hospitals\n\nin Canada, \u2013175\n\nin Germany,\n\nin Japan,\n\nin United Kingdom,\n\nPayments\n\nfor all services,\n\nper diem, ,\n\nper episode of illness, , \u201334\n\nper hospitalization episode, \u201340\n\nper institution,\n\nper patient\n\nto hospital,\n\nto physician, ,\n\nper procedure\n\nfor hospital,\n\nfor physician, \u201333\n\nper time,\n\nPCP. _See_ Primary care physician\n\nPeer review, \u2013122\n\ntheory of bad apples,\n\nPepper Commission of 1990,\n\nPharmaceutical industry (U.S.), \u2013210\n\nPharmacists,\n\nshortage of,\n\nsupply of, \u201381, f\n\nPharmacy technician,\n\nPhysician assistant education, \u201378\n\nPhysician assistants (PAs), \u201377\n\nPhysician entrepreneurship in Japan,\n\nPhysician organizations, funding of,\n\nPhysician payment methods, \u201333\n\nPhysician practices,\n\nPhysician Quality Reporting Initiative, ,\n\nPhysician\u2013industry relationships,\n\nPhysicians for National Health Program, ,\n\nPhysicians, \u201374\n\nfinancial incentives, \u201355\n\nin traditional fee-for-service, \u201360\n\nprofessionalism, \u201356\n\nsupply of\n\nrelation with mortality,\n\nPostdoctoral education, \u201376\n\nPower relationships, ,\n\nPPO. _See_ Preferred Provider Organization Practice organizations, patterns of care,\n\nPreferred Provider Organization (PPO), \u201333, \u201369\n\nPrepaid group practice, \u201363\n\nPrescription drug, efficacy and clinical trials of, \u2013211\n\nPrevention, strategies, \u2013136\n\nPrice controls, , , t, ,\n\nPrice, \u201397\n\nPrimary care physician (PCP), , , ,\n\nPrimary care physician,\n\nPrimary care trusts,\n\nPrimary prevention,\n\nthrough public health action,\n\nPrivate insurance,\n\nPrivate insurers,\n\nPrivate long-term care insurance,\n\nProcess measures,\n\nProcess of care,\n\nProfitability, \u2013210\n\nProfoundly ill people, medical care,\n\nPromotion of good health and the prevention of illness,\n\nProvider-insurer pact, \u2013202, \u2013206\n\nProviders,\n\nProviding or withholding of care,\n\nPurchaser dominance over health care,\n\nPurchasers,\n\nrevolt, \u2013205\n\n**Q**\n\nQuality improvement organizations (QIOs),\n\nQuality monitoring, contemporary approach,\n\nQuality of care,\n\nproposals for improvement, \u2013121\n\npublic reporting of, \u2013126\n\nQuantity,\n\n**R**\n\nRand Health Insurance Experiment, ,\n\nRationing\n\nin medical care, ,\n\nby medical effectiveness,\n\nfor society as a whole,\n\nrelationship to cost control,\n\nwithin one institution,\n\nRegionalized Model, , f. _See also_ Dispersed Model Registered nurse education,\n\nRegistered nurses, \u201378, ,\n\nRegulatory strategies for cost control,\n\nReimbursement controls for health care, , \u2013107\n\nRisk adjusted capitation, , ,\n\n**S**\n\nSalaried payment, , f\n\nSan Francisco, On Lok program,\n\nSan Joaquin Foundation for Medical Care, \u201366\n\nSCHIP. _See_ State Children's Health Insurance Program\n\nSecond epidemic revolution,\n\nSecondary care,\n\nSecondary prevention,\n\nSelective contracting,\n\nSelf-insurance placed employers,\n\nSickness funds, \u2013171\n\nSingle-payer detractors,\n\nSingle-payer proposals, \u2013197\n\nSingle-specialty groups,\n\nSkilled care versus custodial services,\n\nSmoking cessation, physician counseling,\n\nSocial insurance model, \u201314,\n\nSocial insurance,\n\nSocial\u2013ethical dilemma, \u2013156\n\nSociety-managed insurance plans,\n\nSpecialist physicians and specialty services, financial incentives,\n\nSpecialty service lines,\n\nSponsored collaboratives,\n\nStarfield, Barbara\n\nprimary care tasks, formulation of,\n\nState Children's Health Insurance Program (SCHIP), , , , , , ,\n\nStatin drugs, treatment for hyperlipidemia,\n\nSuppliers,\n\nSupply limits, \u2013202,\n\ncontrol of, \u2013112\n\n**T**\n\nTask Force on Organ Transplantation (1986),\n\nTax credits, \u2013192\n\nTertiary care, \u201344, , , \u2013219\n\nThree-tiered capitation, \u201336, f. _See also_ Two tiered capitation\n\nTobacco,\n\nTort reform,\n\nTraditional insurers,\n\nTruman, Harry S., U.S. President, ,\n\nTwo-tiered capitation, \u201336. _See also_ Three-tiered capitation\n\n**U**\n\nU.S. health care system\n\n1945\u20131970,\n\n1970s,\n\n1980s: purchasers' revolt, \u2013205\n\n1990s: provider\u2013insurer pact, breakup of, \u2013206\n\nmanaged care,\n\nnew millennium: re-emergence of provider power, \u2013207\n\nunstable power relationships,\n\nUnderinsurance, , t\n\neffects of,\n\nUnemployment benefits,\n\nUninsured people, , f, , f\n\nhealth outcomes, \u201323\n\nUnit of payment, , \u2013108,\n\nUM, relation with,\n\nUnited Network for Organ Sharing (UNOS),\n\nUnited States system\n\nof capitation payments, \u201336\n\nof health care,\n\nUnits of payment, \u201332\n\nUniversal health insurance through an employer mandate, , ,\n\nUniversal individual insurance proposals, ,\n\nUtilization (quantity) controls, \u2013112\n\nUtilization management (UM)\n\nin health care,\n\n**V**\n\nVertical integration, , f,\n\nVoluntary approach,\n\n**W**\n\nWagner\u2013Murray\u2013Dingell Bill, \u2013189\n\nWelfare benefits,\n\nWorld War II, rationing,\n\nWorthy, Joshua, \n","meta":{"redpajama_set_name":"RedPajamaBook"}} +{"text":"\n\n\u00c9pict\u00e8te\n\nManuel\n\nLibrio\n\nJ'ai Lu\n\nFlammarion\n\n\u00a9 \u00c9ditions J'ai lu, 2014\n\n\u00a9 E.J.L, 2014 pour la s\u00e9lection\n\nD\u00e9p\u00f4t l\u00e9gal : janvier 2014\n\nLe livre a \u00e9t\u00e9 imprim\u00e9 sous les r\u00e9f\u00e9rences : \nISBN : 9782290082751\n\nISBN num\u00e9rique : 9782290088074\n\nISBN PDF web : 9782290088166\n\nOuvrage compos\u00e9 et converti par PCA (44400 Rez\u00e9)\nPr\u00e9sentation de l'\u00e9diteur :\n\nPour atteindre le bonheur, tout est question d'attitude. Sur l'avis des autres, la richesse, la chance ou la mort, nous n'avons pas de prise. C'est donc aux op\u00e9rations de l'\u00e2me qu'il faut accorder tous nos soins, et apprendre \u00e0 jouir des choses mat\u00e9rielles sans nous y attacher. D\u00e9sirer que les choses arrivent comme elles arrivent, voil\u00e0 la clef pour \u00c9pict\u00e8te.\n\nEn illustrant cette voie de mani\u00e8re concr\u00e8te, il expose avec simplicit\u00e9 et sagesse une mani\u00e8re diff\u00e9rente de voir le monde, qui affranchit l'homme des angoisses et des pr\u00e9occupations de sa condition.\n\nCompil\u00e9 par Arrien, le Manuel est suivi de morceaux choisis parmi les Entretiens avec ce m\u00eame disciple : \u00ab \u00c0 ceux qui craignent la pauvret\u00e9 \u00bb, \u00ab Contre les gens querelleurs et m\u00e9chants \u00bb ou \u00ab Pour ceux qui parlent trop ais\u00e9ment d'eux-m\u00eames \u00bb...\n\n\u00c9PICT\u00c8TE (VERS 50 \u2013 120 AV. J.-C.)\n\nEsclave, puis affranchi dans des conditions mal connues, \u00c9pict\u00e8te, dont le nom signifie \u00ab homme achet\u00e9 \u00bb, v\u00e9cut \u00e0 Rome avant d'en \u00eatre chass\u00e9 sous Domitien, qui craignait l'influence des sto\u00efciens.\n\nIllustration de couverture : Portrait imaginaire d'\u00c9pict\u00e8te. Frontispice de l'Enchiridion d'\u00c9pict\u00e8te, traduit par Edward Ivie et imprim\u00e9 \u00e0 Oxford en 1751. Biblioth\u00e8que John Adams, Boston \nDANS LA M\u00caME S\u00c9RIE\n\nL'Art d'aimer, Librio no 11\n\nLe Banquet, Librio no 76\n\nLe Prince, Librio no 163\n\nDiscours de la m\u00e9thode, Librio no 299\n\nL'Utopie, Librio no 317\n\nDiscours sur l'origine et les fondements de l'in\u00e9galit\u00e9 parmi les hommes, Librio no 340\n\nLettres et maximes, Librio no 363\n\nSi la philosophie m'\u00e9tait cont\u00e9e, Librio no 403\n\nLe Bonheur, d\u00e9sesp\u00e9r\u00e9ment, Librio no 513\n\nFragments et aphorismes, Librio no 616\n\nApologie de Socrate, Librio no 635\n\nDe la vie heureuse et de la tranquillit\u00e9 de l'\u00e2me, Librio no 678\n\nNi Dieu, ni ma\u00eetre ! De Diderot \u00e0 Nietzsche, Librio no 812\n\nSur le mensonge, Librio no 1074\n\nGorgias, Librio no 1075\n\nL'Art d'avoir toujours raison, Librio no 1076\n\nPens\u00e9es, Librio no 1078\n\nDiscours de la servitude volontaire, Librio no 1084\n\nDu contrat social, Librio no 1085\n\nTrait\u00e9 sur la tol\u00e9rance, Librio no 1086\n\nEssai sur l'art de ramper, \u00e0 l'usage des courtisans, Librio no 1096\nManuel\n\nI. DISTINCTION ENTRE CE QUI D\u00c9PEND DE NOUS ET CE QUI NE D\u00c9PEND PAS DE NOUS\n\n1. Des choses les unes d\u00e9pendent de nous, les autres ne d\u00e9pendent pas de nous. Ce qui d\u00e9pend de nous, ce sont nos jugements, nos tendances, nos d\u00e9sirs, nos aversions, en un mot tout ce qui est op\u00e9ration de notre \u00e2me ; ce qui ne d\u00e9pend pas de nous, c'est le corps, la fortune, les t\u00e9moignages de consid\u00e9ration, les charges publiques, en un mot tout ce qui n'est pas op\u00e9ration de notre \u00e2me.\n\n2. Ce qui d\u00e9pend de nous est, de sa nature, libre, sans emp\u00eachement, sans contrari\u00e9t\u00e9 ; ce qui ne d\u00e9pend pas de nous est inconsistant, esclave, sujet \u00e0 emp\u00eachement, \u00e9tranger.\n\n3. Souviens-toi donc que si tu regardes comme libre ce qui de sa nature est esclave, et comme \u00e9tant \u00e0 toi ce qui est \u00e0 autrui, tu seras contrari\u00e9, tu seras dans le deuil, tu seras troubl\u00e9, tu t'en prendras et aux dieux et aux hommes ; mais si tu ne regardes comme \u00e9tant \u00e0 toi que ce qui est \u00e0 toi, et si tu regardes comme \u00e9tant \u00e0 autrui ce qui, en effet, est \u00e0 autrui, personne ne te contraindra jamais, personne ne t'emp\u00eachera, tu ne t'en prendras \u00e0 personne, tu n'accuseras personne, tu ne feras absolument rien contre ton gr\u00e9, personne ne te nuira ; tu n'auras pas d'ennemi, car tu ne souffriras rien de nuisible.\n\n4. Aspirant \u00e0 de si grands biens, songe qu'il ne faut pas te porter mollement \u00e0 les rechercher, qu'il faut renoncer enti\u00e8rement \u00e0 certaines choses et en ajourner d'autres quant \u00e0 pr\u00e9sent. Mais si outre ces biens tu veux encore le pouvoir et la richesse, peut-\u00eatre n'obtiendras-tu m\u00eame pas ces avantages parce que tu aspires en m\u00eame temps aux autres biens, et, en tout cas, ce qu'il y a de certain, c'est que tu manqueras les biens qui peuvent seuls nous procurer la libert\u00e9 et le bonheur.\n\n5. Ainsi, \u00e0 toute id\u00e9e rude, exerce-toi \u00e0 dire aussit\u00f4t : \u00ab Tu es une id\u00e9e, et tu n'es pas tout \u00e0 fait ce que tu repr\u00e9sentes. \u00bb Puis examine-la, applique les r\u00e8gles que tu sais, et d'abord et avant toutes les autres celle qui fait reconna\u00eetre si quelque chose d\u00e9pend ou ne d\u00e9pend pas de nous ; et si l'id\u00e9e est relative \u00e0 quelque chose qui ne d\u00e9pende pas de nous, sois pr\u00eat \u00e0 dire : \u00ab Cela ne me regarde pas. \u00bb\n\nII. APPLIQUER LA DISTINCTION AU D\u00c9SIR ET \u00c0 L'ACTION\n\n1. Souviens-toi que ce que le d\u00e9sir d\u00e9clare qu'il veut, c'est d'obtenir ce qu'il d\u00e9sire, que ce que l'aversion d\u00e9clare qu'elle ne veut pas, c'est de tomber dans ce qu'elle a en aversion ; et quand on n'obtient pas ce qu'on d\u00e9sire, on n'est pas heureux, quand on tombe dans ce qu'on a en aversion, on est malheureux. Si donc tu n'as d'aversion que pour ce qui est contraire \u00e0 la nature dans ce qui d\u00e9pend de toi, tu ne tomberas dans rien de ce que tu as en aversion ; mais si tu as de l'aversion pour la maladie, la mort ou la pauvret\u00e9, tu seras malheureux.\n\n2. Cesse donc de donner pour objet \u00e0 ton aversion rien de ce qui ne d\u00e9pend pas de nous, transporte-la sur ce qui est contraire \u00e0 la nature dans ce qui d\u00e9pend de nous. Quant au d\u00e9sir, supprime-le absolument pour le moment. En effet, si tu d\u00e9sires quelque chose qui ne d\u00e9pende pas de nous, infailliblement, tu ne seras pas heureux ; et quant aux choses qui d\u00e9pendent de nous, qu'il est beau de d\u00e9sirer, il n'en est aucune qui soit encore \u00e0 ta port\u00e9e. Borne-toi \u00e0 tendre vers les choses et \u00e0 t'en \u00e9loigner, mais l\u00e9g\u00e8rement, en faisant des r\u00e9serves, et sans ardeur.\n\nIII. CE QU'EST CE QUI EST D\u00c9SIR\u00c9\n\n\u00c0 propos de tout objet d'agr\u00e9ment, d'utilit\u00e9 ou d'affection, n'oublie pas de te dire en toi-m\u00eame ce qu'il est, \u00e0 commencer par les moins consid\u00e9rables. Si tu aimes une marmite, dis : \u00ab C'est une marmite que j'aime \u00bb ; alors, quand elle se cassera, tu n'en seras pas troubl\u00e9 : quand tu embrasses ton enfant ou ta femme, dis-toi que c'est un \u00eatre humain que tu embrasses ; et alors sa mort ne te troublera pas.\n\nIV. ENTREPRENDRE UNE ACTION EN PR\u00c9SERVANT SON CHOIX\n\nQuand tu entreprends quelque chose, rappelle-toi ce que c'est. Si tu t'en vas te baigner, repr\u00e9sente-toi ce qui arrive tous les jours au bain, les gens qui vous jettent de l'eau, qui vous poussent, qui vous injurient, qui vous volent ; tu seras plus s\u00fbr de toi en allant te baigner, si tu te dis aussit\u00f4t : \u00ab Je veux me baigner, mais je veux aussi conserver ma volont\u00e9 dans un \u00e9tat conforme \u00e0 la nature. \u00bb Et de m\u00eame en chaque occasion. Ainsi, s'il te survient au bain quelque contrari\u00e9t\u00e9, tu auras aussit\u00f4t pr\u00e9sent \u00e0 l'esprit : \u00ab Mais je ne voulais pas seulement me baigner, je voulais conserver aussi ma volont\u00e9 dans un \u00e9tat conforme \u00e0 la nature ; et je n'y r\u00e9ussirai pas, si je m'irrite de ce qui arrive tous les jours. \u00bb\n\nV. NOS JUGEMENTS NOUS TROUBLENT, NON LES CHOSES\n\nCe qui trouble les hommes, ce ne sont pas les choses, ce sont les jugements qu'ils portent sur les choses. Ainsi la mort n'a rien de redoutable, autrement elle aurait paru telle \u00e0 Socrate ; mais le jugement que la mort est redoutable, c'est l\u00e0 ce qui est redoutable. Ainsi donc quand nous sommes contrari\u00e9s, troubl\u00e9s ou pein\u00e9s, n'en accusons jamais d'autres que nous-m\u00eame, c'est-\u00e0-dire nos propres jugements. Il est d'un ignorant de s'en prendre \u00e0 d'autres de ses malheurs ; il est d'un homme qui commence \u00e0 s'instruire de s'en prendre \u00e0 lui-m\u00eame ; il est d'un homme compl\u00e8tement instruit de ne s'en prendre ni \u00e0 un autre ni \u00e0 lui-m\u00eame.\n\nVI. CE QUI EST \u00c0 TOI : L'USAGE DES REPR\u00c9SENTATIONS\n\nNe t'enorgueillis d'aucun avantage qui soit \u00e0 autrui. Si un cheval disait avec orgueil : \u00ab Je suis beau \u00bb, ce serait supportable ; mais toi, quand tu dis avec orgueil : \u00ab J'ai un beau cheval \u00bb, apprends que tu t'enorgueillis d'un avantage qui appartient au cheval. Qu'est-ce qui est donc \u00e0 toi ? L'usage de tes id\u00e9es. Quand tu en uses conform\u00e9ment \u00e0 la nature, alors enorgueillis-toi ; car tu t'enorgueilliras d'un avantage qui est \u00e0 toi.\n\nVII. LA VIE COMME NAVIGATION\n\nIl en est de la vie comme d'une navigation. Si l'on rel\u00e2che, et que l'on t'envoie faire de l'eau, accessoirement tu pourras sur ta route ramasser un coquillage ou un oignon, mais il faut toujours avoir l'esprit tendu vers le navire, te retourner sans cesse pour voir si le pilote ne t'appelle pas, et, s'il t'appelle, laisser tout cela pour ne pas te voir li\u00e9 et jet\u00e9 \u00e0 bord comme un mouton : de m\u00eame dans la vie, si au lieu d'un coquillage ou d'un oignon, tu as une femme et un enfant, rien n'emp\u00eache ; mais si le pilote t'appelle, cours au vaisseau, en laissant tout cela, sans te retourner. Si tu es vieux, ne t'\u00e9loigne pas trop du navire, pour ne pas risquer de manquer \u00e0 l'appel.\n\nVIII. D\u00c9SIRE QUE LES CHOSES ARRIVENT COMME ELLES ARRIVENT\n\nNe demande pas que ce qui arrive arrive comme tu d\u00e9sires ; mais d\u00e9sire que les choses arrivent comme elles arrivent, et tu seras heureux.\n\nIX. LE CORPS N'EST PAS UNE SOURCE DE CONTRARI\u00c9T\u00c9\n\nLa maladie est une contrari\u00e9t\u00e9 pour le corps, mais non pour la volont\u00e9, si elle ne veut pas. \u00catre boiteux est une contrari\u00e9t\u00e9 pour la jambe, mais non pour la volont\u00e9. Dis-toi la m\u00eame chose \u00e0 chaque incident ; tu trouveras que c'est une contrari\u00e9t\u00e9 pour autre chose, mais non pour toi.\n\nX. NOUS AVONS LA FORCE POUR R\u00c9SISTER AUX REPR\u00c9SENTATIONS TROUBLANTES\n\n\u00c0 chaque occasion qui se pr\u00e9sente, replie-toi sur toi-m\u00eame et cherche quelle facult\u00e9 tu as en toi-m\u00eame pour te conduire : si tu vois une belle femme, tu trouveras en toi la facult\u00e9 de la continence ; s'il se pr\u00e9sente une fatigue \u00e0 supporter, tu trouveras celle de l'endurance ; une injure, tu trouveras celle de la patience. Si tu prends cette habitude, tes id\u00e9es ne t'emporteront pas.\n\nXI\n\nNe dis jamais de quoi que ce soit : \u00ab Je l'ai perdu \u00bb, mais : \u00ab Je l'ai rendu. \u00bb Ton enfant est mort : il est rendu. Ta femme est morte : elle est rendue. \u00ab On m'a enlev\u00e9 mon bien. \u00bb \u2014 Eh bien ! il est rendu aussi. \u2014 \u00ab Mais c'est un sc\u00e9l\u00e9rat que celui qui me l'a enlev\u00e9. \u00bb \u2014 Eh ! que t'importe par qui celui qui te l'a donn\u00e9 l'a r\u00e9clam\u00e9 ? Tant qu'il te le laisse, occupe-t'en comme de quelque chose qui est \u00e0 autrui, ainsi que les passants usent d'une h\u00f4tellerie.\n\nXII\n\n1. Si tu veux faire des progr\u00e8s, laisse l\u00e0 toutes ces r\u00e9flexions, comme : \u00ab Si je n\u00e9glige ma fortune, je n'aurai pas de quoi manger \u00bb ; \u00ab Si je ne ch\u00e2tie pas mon esclave, il sera vicieux. \u00bb Il vaut mieux mourir de faim, exempt de peine et de crainte, que de vivre dans l'abondance et le trouble ; il vaut mieux que ton esclave soit vicieux, et que tu ne sois pas malheureux.\n\n2. Commence donc par les petites choses. On laisse couler ton huile ; on vole ton vin : dis-toi \u00ab C'est \u00e0 ce prix que se vend l'impassibilit\u00e9, c'est \u00e0 ce prix que se vend le calme \u00bb. On n'a rien pour rien. Quand tu appelles ton esclave, pense qu'il peut ne pas r\u00e9pondre \u00e0 ton appel, et, y r\u00e9pondant, ne rien faire de ce que tu veux, mais que sa situation n'est pas assez belle pour qu'il d\u00e9pende de lui que tu ne sois pas troubl\u00e9.\n\nXIII\n\nSi tu veux faire des progr\u00e8s, r\u00e9signe-toi \u00e0 passer pour un idiot et pour un imb\u00e9cile dans les choses du dehors, consens \u00e0 passer pour n'y rien entendre ; et si quelques-uns te croient quelque chose, d\u00e9fie-toi de toi-m\u00eame. Sache qu'il n'est pas facile de conserver sa volont\u00e9 dans un \u00e9tat conforme \u00e0 la nature, et en m\u00eame temps de veiller sur les choses du dehors ; mais n\u00e9cessairement, on ne peut s'occuper de l'un sans n\u00e9gliger l'autre.\n\nXIV\n\n1. Si tu veux que tes enfants, ta femme, tes amis vivent toujours, tu es un imb\u00e9cile ; tu veux que ce qui ne d\u00e9pend pas de toi, d\u00e9pende de toi ; tu veux que ce qui est \u00e0 autrui soit \u00e0 toi. Ainsi, si tu veux que ton esclave ne commette pas de fautes, tu es fou ; tu veux que le vice ne soit pas le vice, mais autre chose. Mais si tu veux ne pas manquer ce que tu d\u00e9sires, tu le peux ; applique-toi donc \u00e0 ce que tu peux.\n\n2. On est toujours le ma\u00eetre d'un homme, quand on a le pouvoir de lui donner ou de lui \u00f4ter ce qu'il veut ou ce qu'il ne veut pas. Si l'on veut \u00eatre libre, qu'on n'ait ni d\u00e9sir ni aversion pour rien de ce qui d\u00e9pend d'autrui ; sinon, il faut \u00eatre esclave.\n\nXV\n\nSouviens-toi que tu dois te comporter dans la vie comme dans un festin. Le plat qui circule arrive \u00e0 toi : \u00e9tends la main et prends avec discr\u00e9tion. Il passe plus loin : ne le retiens pas. Il n'est pas encore arriv\u00e9 : ne le devance pas de loin par tes d\u00e9sirs, attends qu'il arrive \u00e0 toi. Fais-en de m\u00eame pour des enfants, pour une femme, pour des charges publiques, pour de l'argent ; et tu seras digne de t'asseoir un jour \u00e0 la table des dieux. Mais si l'on te sert et que tu ne prennes rien, que tu d\u00e9daignes de prendre, alors tu ne seras pas seulement le convive des dieux, tu seras leur coll\u00e8gue. C'est en se conduisant ainsi que Diog\u00e8ne, qu'H\u00e9raclite et ceux qui leur ressemblent ont m\u00e9rit\u00e9 d'\u00eatre appel\u00e9s des hommes divins, comme ils l'\u00e9taient en effet.\n\nXVI\n\nQuand tu vois quelqu'un qui pleure, soit parce qu'il est en deuil, soit parce que son fils est au loin, soit parce qu'il a perdu ce qu'il poss\u00e9dait, prends garde de te laisser emporter par l'id\u00e9e que les accidents du dehors qui lui arrivent sont des maux. Rappelle-toi sur-le-champ que ce qui l'afflige ce n'est pas l'accident, qui n'en afflige pas d'autre que lui, mais le jugement qu'il porte sur cet accident. Cependant n'h\u00e9site pas \u00e0 lui t\u00e9moigner, au moins des l\u00e8vres, ta sympathie, et m\u00eame, s'il le faut, \u00e0 g\u00e9mir avec lui ; mais prends garde de g\u00e9mir du fond de l'\u00e2me.\n\nXVII\n\nSouviens-toi que tu es l'acteur d'un r\u00f4le, tel qu'il pla\u00eet \u00e0 l'auteur de te le donner : court, s'il l'a voulu court ; long, s'il l'a voulu long ; s'il veut que tu joues un r\u00f4le de mendiant, joue-le na\u00efvement ; ainsi d'un r\u00f4le de boiteux, de magistrat, de simple particulier. C'est ton fait de bien jouer le personnage qui t'est donn\u00e9 ; mais de le choisir, c'est le fait d'un autre.\n\nXVIII\n\nQuand un corbeau pousse un cri de mauvais augure, ne te laisse pas emporter par ton id\u00e9e ; distingue aussit\u00f4t en toi-m\u00eame, et dis : \u00ab Dans tout cela il n'y a point de pr\u00e9sage pour moi, il ne peut y en avoir que pour mon corps, ma fortune, ma r\u00e9putation, mes enfants, ma femme. Quant \u00e0 moi, tout est de bon augure, si je veux ; car quel que soit l'\u00e9v\u00e9nement, il d\u00e9pend de moi d'en tirer profit. \u00bb\n\nXIX\n\n1. Tu peux \u00eatre invincible, si tu ne t'engages dans aucune lutte, o\u00f9 il ne d\u00e9pend pas de toi d'\u00eatre vainqueur.\n\n2. Quand tu vois un homme rev\u00eatu d'honneurs extraordinaires ou d'un grand pouvoir ou de toute autre illustration, prends garde de le proclamer heureux et de te laisser emporter par ton id\u00e9e. Si la substance du bien est dans les choses qui d\u00e9pendent de nous, il n'y a pas de place pour l'envie ni pour la jalousie ; et toi-m\u00eame, tu ne voudras pas \u00eatre strat\u00e8ge, prytane ou consul, tu voudras \u00eatre libre. Or il n'y a qu'une route pour y arriver : m\u00e9priser ce qui ne d\u00e9pend pas de nous.\n\nXX\n\nSouviens-toi qu'on n'est pas outrag\u00e9 par celui qui injurie ou qui frappe, mais par le jugement qu'ils vous outragent. Quand quelqu'un te met en col\u00e8re, sache que c'est ton jugement qui te met en col\u00e8re. Efforce-toi donc avant tout de ne pas te laisser emporter par ton id\u00e9e ; si une fois tu gagnes du temps, quelque d\u00e9lai, tu seras plus facilement ma\u00eetre de toi.\n\nXXI\n\nAie tous les jours devant les yeux la mort, l'exil et tout ce qui para\u00eet effrayant, surtout la mort, et jamais tu ne penseras rien de bas, ni ne d\u00e9sireras rien avec exc\u00e8s.\n\nXXII\n\nSi tu d\u00e9sires \u00eatre philosophe, attends-toi d\u00e8s lors \u00e0 \u00eatre un objet de d\u00e9rision, \u00e0 \u00eatre en butte aux moqueries d'une foule de gens qui disent : \u00ab Il nous est revenu tout \u00e0 coup philosophe \u00bb et \u00ab D'o\u00f9 vient cet air renfrogn\u00e9 ? \u00bb Toi, n'aie pas l'air renfrogn\u00e9 ; mais attache-toi \u00e0 ce qui te para\u00eet le meilleur, avec la conviction que la divinit\u00e9 t'a assign\u00e9 ce poste : souviens-toi que si tu restes fid\u00e8le \u00e0 tes principes, ceux qui se moquaient d'abord de toi, t'admireront plus tard ; mais si tu es vaincu par leurs propos, tu te rendras doublement ridicule.\n\nXXIII\n\nS'il t'arrive de te tourner vers l'ext\u00e9rieur par complaisance pour quelqu'un, sois s\u00fbr que tu as perdu ton assiette. Contente-toi donc, partout, d'\u00eatre philosophe. Si de plus tu veux le para\u00eetre, parais-le \u00e0 toi-m\u00eame ; et c'est suffisant.\n\nXXIV\n\n1. Ne t'afflige pas par des raisonnements comme : \u00ab Je vivrai sans consid\u00e9ration et je ne serai rien nulle part. \u00bb Si le manque de consid\u00e9ration est un mal, tu ne peux souffrir de mal par le fait d'autrui, non plus que de honte. Est-ce que c'est quelque chose qui d\u00e9pend de toi, que d'obtenir une charge ou d'\u00eatre invit\u00e9 \u00e0 un grand repas ? nullement. Comment est-ce donc une humiliation ? Comment ne seras-tu rien nulle part, toi qui ne dois \u00eatre quelque chose que dans ce qui d\u00e9pend de toi, l\u00e0 o\u00f9 tu peux avoir le plus grand m\u00e9rite ?\n\n2. Mais tu ne viendras pas en aide \u00e0 tes amis. Qu'est-ce que tu dis l\u00e0, ne pas venir en aide ? Tu ne leur donneras pas de monnaie ? Tu ne les feras pas citoyens romains ? Et qui donc t'a dit que ce sont l\u00e0 des choses qui d\u00e9pendent de nous, et non d'autrui ? Qui est-ce qui peut donner \u00e0 un autre ce qu'il n'a pas lui-m\u00eame ? \u00ab Acquiers, dira l'un d'eux, pour que nous ayons. \u00bb\n\n3. Si je puis acqu\u00e9rir en restant discret, s\u00fbr, magnanime, montre-moi le moyen, et j'acquerrai. Si vous exigez que je perde les biens qui me sont propres pour vous acqu\u00e9rir ce qui n'est pas un bien, voyez vous-m\u00eames comme vous \u00eates injustes et d\u00e9raisonnables. Et que pr\u00e9f\u00e9rez-vous donc ? de l'argent qu'un ami loyal et r\u00e9serv\u00e9 ? Aidez-moi donc plut\u00f4t \u00e0 acqu\u00e9rir ce bien-l\u00e0, et n'exigez pas que je fasse ce qui me le fera perdre.\n\n4. \u00ab Mais, dira quelqu'un, ma patrie, je ne lui viendrai pas en aide, autant qu'il est en moi. \u00bb Encore une fois, quelle aide ? Elle ne te devra pas de portiques, de bains ? Et qu'est-ce que cela ? Tes concitoyens ne sont pas non plus chauss\u00e9s par l'armurier, ni arm\u00e9s par le cordonnier ; il suffit que chacun remplisse sa t\u00e2che. Si tu procurais \u00e0 la patrie quelque autre citoyen loyal et r\u00e9serv\u00e9, ne lui aurais-tu rendu aucun service ? \u2014 \u00ab C'est vrai. \u00bb \u2014 Eh bien ! alors, tu ne lui seras pas non plus inutile.\n\n5. \u00ab Quelle place aurai-je donc dans l'\u00c9tat ? \u2014 Celle que tu peux avoir en restant homme loyal et r\u00e9serv\u00e9. Mais si pour venir en aide \u00e0 ta patrie, tu perds ces biens, de quelle utilit\u00e9 peux-tu lui \u00eatre quand tu seras devenu impudent et d\u00e9loyal ? \u00bb\n\nXXV\n\n1. On t'a pr\u00e9f\u00e9r\u00e9 quelqu'un, soit pour l'inviter \u00e0 un repas, soit pour le saluer, soit pour l'appeler \u00e0 une d\u00e9lib\u00e9ration ? Si ce sont l\u00e0 des biens, tu dois te r\u00e9jouir de ce qu'il les a obtenus ; si ce sont des maux, ne t'afflige pas de n'en avoir pas ta part : souviens-toi que quand tu ne fais pas la m\u00eame chose que les autres pour avoir ce qui ne d\u00e9pend pas de nous, tu ne peux pas pr\u00e9tendre en avoir autant.\n\n2. Et comment celui qui ne fait pas une cour assidue \u00e0 un grand peut-il \u00eatre trait\u00e9 comme celui qui la fait ? celui qui ne lui fait pas cort\u00e8ge, comme celui qui le fait ? celui qui ne le loue pas, comme celui qui le loue ? Tu seras injuste et insatiable si, sans avoir pay\u00e9 le prix, tu veux recevoir pour rien ce qu'il vend.\n\n3. Voyons, combien se vend la laitue ? Supposons que ce soit une obole. Quand quelqu'un a de la laitue en donnant son obole et que toi tu n'en as pas en ne donnant pas la tienne, ne crois pas \u00eatre moins bien trait\u00e9 que celui qui en a. S'il a sa laitue, toi, tu as ton obole, que tu n'as pas donn\u00e9e.\n\n4. De m\u00eame ici. Quelqu'un ne t'a pas invit\u00e9 \u00e0 un repas ? C'est que tu n'as pas pay\u00e9 le prix auquel il vend son repas ; il le vend pour des compliments, il le vend pour des soins. Paye le prix auquel il vend, si tu y trouves un avantage ; mais si tu veux \u00e0 la fois ne pas payer et recevoir, tu es insatiable et imb\u00e9cile.\n\n5. N'as-tu donc rien \u00e0 la place du repas ? Oui, tu as quelque chose, tu as de ne pas louer qui tu ne veux pas, tu as de ne pas essuyer les insolences des esclaves qui gardent sa porte.\n\nXXVI\n\nOn peut reconna\u00eetre ce que veut la nature aux choses sur lesquelles nous ne diff\u00e9rons pas d'avis entre nous. Ainsi, quand l'esclave d'un autre casse sa coupe, nous avons aussit\u00f4t sur les l\u00e8vres : \u00ab Cela se voit tous les jours. \u00bb Sache donc que quand on cassera ta coupe, tu dois \u00eatre tel que tu es quand on casse celle d'un autre. Applique cette r\u00e9flexion \u00e0 des \u00e9v\u00e9nements plus importants. Quelqu'un perd son fils ou sa femme ? Il n'est personne qui ne dise : \u00ab C'est la condition de l'humanit\u00e9. \u00bb Mais quand on fait cette perte soi-m\u00eame, aussit\u00f4t de dire : \u00ab H\u00e9las ! que je suis malheureux ! \u00bb Il faudrait pourtant se rappeler ce qu'on \u00e9prouve en l'entendant dire d'un autre.\n\nXXVII\n\nComme on ne place pas de but pour qu'on le manque, de m\u00eame le mal de nature n'existe pas dans le monde.\n\nXXVIII\n\nSi on confiait ton corps au premier venu, tu serais indign\u00e9 ; et toi, quand tu confies ton \u00e2me au premier venu, pour qu'il la trouble et la bouleverse par ses injures, tu n'en as pas de honte ?\n\nXXIX\n\n1. Dans toute affaire, examine bien les ant\u00e9c\u00e9dents et les cons\u00e9quents, et alors entreprends. Sinon, tu seras d'abord plein de feu, parce que tu n'as pas r\u00e9fl\u00e9chi \u00e0 l'encha\u00eenement des choses ; et plus tard, quand quelques difficult\u00e9s se produiront, tu renonceras honteusement.\n\n2. Tu veux \u00eatre vainqueur aux jeux Olympiques ? Et moi aussi, de par les dieux ; car c'est une belle chose. Mais examine bien les ant\u00e9c\u00e9dents et les cons\u00e9quents, et alors entreprends. Il faut ob\u00e9ir \u00e0 une discipline, manger de force, t'abstenir de g\u00e2teau, faire des exercices forc\u00e9s, \u00e0 des heures r\u00e9gl\u00e9es, par le chaud, par le froid, ne boire ni eau fra\u00eeche ni vin indiff\u00e9remment, en un mot, te mettre entre les mains du dresseur comme entre celles d'un m\u00e9decin ; puis, dans l'ar\u00e8ne, il faut creuser des fosses, quelquefois se d\u00e9mettre un bras, se donner une entorse, avaler force poussi\u00e8re, quelquefois \u00eatre fouett\u00e9, et avec tout cela \u00eatre vaincu.\n\n3. Quand tu auras bien pes\u00e9 tout cela, si tu persistes, fais-toi athl\u00e8te. Sinon, tu seras comme les petits enfants qui jouent tant\u00f4t au lutteur, tant\u00f4t au gladiateur, qui tant\u00f4t sonnent de la trompette, tant\u00f4t d\u00e9clament ; de m\u00eame, tu seras tant\u00f4t athl\u00e8te, tant\u00f4t gladiateur, puis rh\u00e9teur, ensuite philosophe, et jamais rien du fond de l'\u00e2me ; tu imiteras comme un singe tout ce que tu verras faire, et chaque chose te plaira \u00e0 son tour. C'est qu'avant d'entreprendre tu n'as pas bien examin\u00e9, retourn\u00e9 la chose sous toutes ses faces ; tu vas au hasard et sans d\u00e9sirer vivement.\n\n4. C'est ainsi que certaines gens pour avoir vu un philosophe, pour avoir entendu parler comme parle Euphrate (et pourtant qui peut parler comme Euphrate ?), veulent aussi \u00eatre philosophes.\n\n5. Mais, pauvre homme, examine d'abord ce que c'est que d'\u00eatre philosophe ; ensuite \u00e9tudie ta propre nature, pour voir si tu es de force. Tu veux \u00eatre pentathle ou lutteur ? Consid\u00e8re tes bras, tes cuisses, examine tes reins. L'un est dou\u00e9 pour une chose, l'autre pour une autre.\n\n6. Crois-tu qu'en te faisant philosophe tu peux manger et boire de la m\u00eame mani\u00e8re, avoir les m\u00eames d\u00e9sirs, les m\u00eames aversions ? Il faut veiller, peiner, te s\u00e9parer des tiens, t'exposer au m\u00e9pris d'un petit esclave, aux ris\u00e9es des passants, avoir le dessous partout, en honneurs, en dignit\u00e9s, devant les juges, enfin en toute chose.\n\n7. P\u00e8se bien tout cela. Maintenant si tu tiens \u00e0 avoir en \u00e9change l'impassibilit\u00e9, la libert\u00e9, le calme, c'est bien ; sinon, retire-toi. Ne fais pas comme les enfants ; ne sois pas maintenant philosophe, ensuite percepteur, puis rh\u00e9teur, puis procurateur de C\u00e9sar. Tout cela ne saurait s'accorder. Il faut que tu sois un, ou vertueux ou vicieux ; il faut cultiver ou ton \u00e2me ou les choses du dehors, l'appliquer ou aux choses int\u00e9rieures ou aux choses ext\u00e9rieures, c'est-\u00e0-dire, rester ou philosophe ou non-philosophe.\n\nXXX\n\nPour faire son office, il faut se r\u00e9gler ordinairement sur les rapports de corr\u00e9lation. C'est ton p\u00e8re ; il t'est prescrit d'en prendre soin, de lui c\u00e9der en tout, de supporter ses injures, ses coups. \u2014 \u00ab Mais c'est un mauvais p\u00e8re. \u00bb \u2014 Est-ce avec un bon p\u00e8re que la nature t'a mis en rapport intime ? C'est avec un p\u00e8re. \u2014 \u00ab Mon fr\u00e8re me fait tort. \u00bb \u2014 Eh bien, alors observe les rapports qui sont \u00e9tablis entre toi et lui ; ne t'occupe pas de ce qu'il fait, mais de ce que tu dois faire pour que ta volont\u00e9 soit dans un \u00e9tat conforme \u00e0 la nature : un autre ne te nuira pas, si tu ne veux pas ; mais on t'aura nui, si tu juges qu'on te nuit. De m\u00eame avec les autres : si tu prends l'habitude de consid\u00e9rer les rapports de corr\u00e9lation qui sont entre toi et un autre en tant que voisin, concitoyen, pr\u00e9teur, tu trouveras quel est ton office.\n\nXXXI\n\n1. Sache que le fond de la pi\u00e9t\u00e9 envers les dieux, c'est d'en juger sainement, de penser qu'ils existent et qu'ils gouvernent l'univers avec sagesse et avec justice, et en cons\u00e9quence de te donner le r\u00f4le de leur ob\u00e9ir, de leur c\u00e9der et de les suivre en tout ce qui t'arrive, dans la pens\u00e9e que c'est arrang\u00e9 pour le mieux. Ainsi tu ne t'en prendras jamais aux dieux, et tu ne te plaindras pas d'en \u00eatre n\u00e9glig\u00e9.\n\n2. Or tu ne peux le faire qu'en \u00f4tant les biens et les maux de ce qui ne d\u00e9pend pas de nous pour les placer uniquement dans ce qui d\u00e9pend de nous. Si tu crois que quelque chose qui ne d\u00e9pend pas de nous est bon ou mauvais, infailliblement, toutes les fois que tu manqueras ce que tu veux et que tu tomberas dans ce que tu ne veux pas, tu t'en prendras aux auteurs responsables et tu les prendras en haine.\n\n3. En effet, tout \u00eatre anim\u00e9 est naturellement port\u00e9 \u00e0 fuir et \u00e0 \u00e9viter ce qui lui para\u00eet un mal et ce qui le cause, et d'autre part, \u00e0 rechercher et \u00e0 aimer ce qui lui para\u00eet un bien et ce qui le procure. Il est donc impossible \u00e0 celui qui croit qu'on lui nuit, d'aimer ce qui para\u00eet lui nuire, comme il est impossible d'aimer le dommage en lui-m\u00eame.\n\n4. De l\u00e0 les injures que le fils adresse au p\u00e8re, quand le p\u00e8re ne lui fait pas part de ce qui passe pour un bien. C'est ce qui fait que Polynice et \u00c9t\u00e9ocle sont devenus ennemis : ils croyaient que la royaut\u00e9 est un bien. C'est pourquoi le laboureur, le matelot, le marchand, ceux qui perdent leur femme ou leurs enfants, injurient les dieux. La pi\u00e9t\u00e9 est fond\u00e9e sur l'int\u00e9r\u00eat ; par cons\u00e9quent, quand on s'applique \u00e0 donner la direction qu'il faut \u00e0 ses d\u00e9sirs et \u00e0 ses aversions, on s'applique par l\u00e0 m\u00eame \u00e0 \u00eatre pieux.\n\n5. Quant aux libations, aux sacrifices, aux offrandes, il faut toujours suivre les lois de sa patrie, \u00eatre en \u00e9tat de puret\u00e9, n'avoir pas de nonchalance ni de n\u00e9gligence, ne pas rester trop en de\u00e7\u00e0 de ses moyens ni aller au-del\u00e0.\n\nXXXII\n\n1. Quand tu as recours \u00e0 la divination, souviens-toi que, si tu ne sais pas quel sera l'\u00e9v\u00e9nement, puisque tu viens aupr\u00e8s du devin pour l'apprendre, tu sais, avant de venir, de quelle nature sera cet \u00e9v\u00e9nement, si du moins tu es philosophe. Si c'est quelque chose qui ne d\u00e9pend pas de nous, il faut de toute n\u00e9cessit\u00e9 qu'il ne soit ni bon ni mauvais.\n\n2. N'aie donc, en te pr\u00e9sentant au devin, ni d\u00e9sir ni aversion ; ne tremble pas en approchant, sois convaincu que l'\u00e9v\u00e9nement quelconque qui sera annonc\u00e9 est chose neutre qui ne te regarde pas, que, quel qu'il puisse \u00eatre, il sera possible d'en tirer un bon parti, sans que personne au monde t'en emp\u00eache. Aie donc confiance en recourant aux conseils des dieux ; et quand tu auras re\u00e7u ces conseils, il ne te restera plus qu'\u00e0 ne pas oublier quels sont ceux qui te les ont donn\u00e9s et \u00e0 qui tu d\u00e9sob\u00e9irais, si tu ne les suivais pas.\n\n3. Maintenant ne consulte les devins, comme le voulait Socrate, que sur les choses o\u00f9 tout se rapporte \u00e0 l'issue, et pour lesquelles il n'y a ni raisonnement ni art quelconque qui donne le moyen de conna\u00eetre ce qu'on veut savoir ; ainsi, quand il faut se risquer pour un ami ou pour sa patrie, il ne faut pas demander au devin s'il faut se risquer. En effet, si le devin te d\u00e9clare que l'\u00e9tat des entrailles de la victime n'est pas favorable, il est \u00e9vident que cela pr\u00e9sage ou la mort ou une mutilation en quelque partie du corps ou l'exil, mais la raison prescrit, m\u00eame avec cette perspective, de venir au secours d'un ami et de se risquer pour sa patrie. Ob\u00e9is donc au plus grand devin, \u00e0 Apollon Pythien, qui chassa du temple celui qui n'\u00e9tait pas venu au secours de son ami, qu'on assassinait.\n\nXXXIII\n\n1. Retrace-toi d\u00e8s maintenant un genre de vie particulier, un plan de conduite, que tu suivras, et quand tu seras seul et quand tu te trouveras avec d'autres.\n\n2. Et d'abord garde ordinairement le silence, ou ne dis que ce qui est n\u00e9cessaire et en peu de mots. Il pourra arriver, mais rarement, que tu doives parler quand l'occasion l'exigera ; mais ne parle sur rien de frivole : ne parle pas de combats de gladiateurs, de courses du cirque, d'athl\u00e8tes, de boire et de manger, sujets ordinaires des conversations ; surtout ne parle pas des personnes, soit pour bl\u00e2mer, soit pour louer, soit pour faire des parall\u00e8les.\n\n3. Si tu le peux, ram\u00e8ne par tes discours les entretiens de ceux avec qui tu vis sur des sujets convenables. Si tu te trouves isol\u00e9 au milieu d'\u00e9trangers, garde le silence.\n\n4. Ne ris pas beaucoup, ni de beaucoup de choses, ni avec exc\u00e8s.\n\n5. Dispense-toi de faire des serments, en toute circonstance, si cela se peut, ou au moins dans la mesure du possible.\n\n6. Refuse de venir aux repas o\u00f9 tu te trouverais avec des \u00e9trangers qui ne sont pas philosophes ; et si l'occasion l'exige, fais bien attention \u00e0 ne pas tomber dans leurs mani\u00e8res. Souviens-toi que quand ton compagnon est sale, tu ne peux pas te frotter \u00e0 lui sans te salir, quelque propre que tu sois toi-m\u00eame.\n\n7. Ne prends pour les besoins du corps que ce qui est strictement n\u00e9cessaire, en fait de nourriture, de boisson, de v\u00eatement, de logement, de domestiques. Tout ce qui est d'ostentation et de luxe, supprime-le.\n\n8. Si l'on vient te dire qu'un tel dit du mal de toi, ne cherche point \u00e0 te justifier sur ce qu'on te rapporte ; r\u00e9ponds seulement : \u00ab Il faut qu'il ne soit pas au courant de ce qu'on peut encore dire sur mon compte ; autrement il ne se serait pas born\u00e9 l\u00e0. \u00bb\n\n9. Il n'est pas n\u00e9cessaire d'aller souvent au spectacle. S'il le faut, ne t'int\u00e9resse s\u00e9rieusement qu'\u00e0 toi-m\u00eame, c'est-\u00e0-dire, d\u00e9sire simplement que les choses arrivent comme elles arrivent et que celui-l\u00e0 soit vainqueur, qui est vainqueur ; ainsi tu ne seras pas contrari\u00e9. Abstiens-toi enti\u00e8rement de crier, de rire de tel acteur, de partager les passions des spectateurs. Quand le spectacle est termin\u00e9, ne parle pas beaucoup de ce qui s'est pass\u00e9, sauf en ce qui peut contribuer \u00e0 te rendre meilleur ; autrement il serait \u00e9vident que tu as \u00e9t\u00e9 frapp\u00e9 du spectacle.\n\n10. Ne te d\u00e9cide pas \u00e0 la l\u00e9g\u00e8re et facilement \u00e0 assister \u00e0 des lectures publiques. Quand tu y viens, garde une attitude grave et calme qui n'ait pourtant rien de d\u00e9sagr\u00e9able.\n\n11. Quand tu dois avoir affaire \u00e0 quelqu'un, particuli\u00e8rement \u00e0 quelqu'un de puissant, repr\u00e9sente-toi ce que Socrate ou Z\u00e9non aurait fait en pareil cas, et tu ne seras pas embarrass\u00e9 pour te comporter convenablement dans la circonstance.\n\n12. Quand tu fais des visites \u00e0 un homme puissant, repr\u00e9sente-toi d'avance que tu ne le trouveras pas chez lui, qu'on ne t'admettra pas, qu'on te fermera la porte sur le nez, qu'il ne se souciera pas de toi. Et si avec cela c'est ton office d'y aller, vas-y et supporte ce qui arrive, sans jamais te dire en toi-m\u00eame : \u00ab Ce n'\u00e9tait pas la peine ; \u00bb car cette r\u00e9flexion est d'un homme qui n'est pas philosophe et qui se met en col\u00e8re pour les choses du dehors.\n\n13. Dans la conversation, \u00e9vite de parler beaucoup et sans mesure de ce que tu fais ou des dangers que tu as courus. Si tu as du plaisir \u00e0 te souvenir des dangers auxquels tu as \u00e9t\u00e9 expos\u00e9, les autres n'ont pas autant de plaisir \u00e0 t'entendre raconter ce qui t'est arriv\u00e9.\n\n14. \u00c9vite aussi de chercher \u00e0 faire rire. On est induit par l\u00e0 \u00e0 glisser dans le genre de ceux qui ne sont pas philosophes, et en m\u00eame temps cela peut diminuer les \u00e9gards que les autres ont pour toi.\n\n15. Il est facile aussi de se laisser aller \u00e0 tenir des propos obsc\u00e8nes. Quand il arrive quelque chose de pareil, tu peux, si c'est \u00e0 propos, aller jusqu'\u00e0 faire des reproches \u00e0 celui qui se le permet ; sinon, t\u00e9moigne au moins par ton silence, ta rougeur, ton visage s\u00e9v\u00e8re, que cette conversation te d\u00e9pla\u00eet.\n\nXXXIV\n\nQuand une id\u00e9e de plaisir se pr\u00e9sente \u00e0 ton esprit, fais comme pour les autres, prends garde de te laisser emporter, diff\u00e8re d'agir, et obtiens de toi-m\u00eame quelque d\u00e9lai. Puis repr\u00e9sente-toi les deux moments, celui o\u00f9 tu jouiras du plaisir et celui o\u00f9, apr\u00e8s en avoir joui, tu te repentiras et t'accableras toi-m\u00eame de reproches ; mets en balance la joie que tu \u00e9prouveras \u00e0 t'abstenir et les f\u00e9licitations que tu t'adresseras. Si les circonstances exigent que tu agisses, fais attention \u00e0 ne pas te laisser vaincre par ce que la chose offre de doux, d'agr\u00e9able et d'attrayant : mets en balance l'avantage qu'il y a \u00e0 avoir conscience que tu as remport\u00e9 cette victoire.\n\nXXXV\n\nQuand tu fais quelque chose, apr\u00e8s avoir reconnu qu'il le faut faire, ne crains pas d'\u00eatre vu le faisant, quelque d\u00e9favorablement que le vulgaire en doive juger. Si tu as tort de le faire, \u00e9vite l'action elle-m\u00eame ; si tu as raison, pourquoi crains-tu ceux qui auront tort de te bl\u00e2mer ?\n\nXXXVI\n\nDe m\u00eame que les propositions \u00ab il fait jour \u00bb et \u00ab il fait nuit \u00bb ont une grande valeur pour une proposition disjonctive et n'ont pas de valeur pour une proposition copulative, ainsi dans un festin choisir la plus forte part peut avoir de la valeur pour le corps, mais n'a pas de valeur pour l'observation des pr\u00e9ceptes qui r\u00e8glent la mani\u00e8re dont on doit se conduire avec les autres dans un repas. Quand tu manges avec un autre, souviens-toi de ne pas consid\u00e9rer seulement la valeur de ce qu'on sert par rapport au corps, mais aussi de garder les \u00e9gards que l'on doit \u00e0 celui qui donne le festin.\n\nXXXVII\n\nQuand tu as pris un r\u00f4le au-dessus de tes forces, non seulement tu y as fait une pauvre figure, mais encore tu as laiss\u00e9 de c\u00f4t\u00e9 celui que tu aurais pu remplir.\n\nXXXVIII\n\nDe m\u00eame qu'en te promenant tu prends garde \u00e0 mettre le pied sur un clou ou \u00e0 te donner une entorse, de m\u00eame fais attention \u00e0 ne pas nuire \u00e0 la partie sup\u00e9rieure de ton \u00e2me. Si nous prenons cette pr\u00e9caution en chaque affaire, nous serons plus s\u00fbrs de nous en l'entreprenant.\n\nXXXIX\n\nLes exigences du corps sont la mesure de ce que chacun a besoin de poss\u00e9der, comme le pied est la mesure de la chaussure. Si tu t'en tiens l\u00e0, tu resteras dans la mesure ; si tu d\u00e9passes, infailliblement, tu ne feras plus que rouler dans le pr\u00e9cipice : de m\u00eame pour la chaussure ; si tu vas au-del\u00e0 de ce qu'il faut pour chausser ton pied, tu prends d'abord des chaussures dor\u00e9es, puis de pourpre, puis brod\u00e9es. Une fois qu'on a d\u00e9pass\u00e9 la mesure, il n'y a plus de limite.\n\nXL\n\nLes femmes aussit\u00f4t apr\u00e8s leur quatorzi\u00e8me ann\u00e9e, sont appel\u00e9es madame par les hommes ; alors elles commencent \u00e0 se parer et mettent l\u00e0 toutes leurs esp\u00e9rances. Il faut donc faire attention \u00e0 ce qu'elles sentent que rien ne peut leur attirer de la consid\u00e9ration que de para\u00eetre d\u00e9centes et r\u00e9serv\u00e9es.\n\nXLI\n\nC'est la marque d'un manque de disposition pour la vertu que de donner une grande place aux choses du corps, comme de donner beaucoup de temps \u00e0 faire de la gymnastique, \u00e0 manger, \u00e0 boire, \u00e0 excr\u00e9ter. Il ne faut faire tout cela qu'accessoirement, et appliquer toute son attention \u00e0 son esprit.\n\nXLII\n\nQuand on te maltraite ou qu'on t'injurie, souviens-toi que celui qui parle ou agit ainsi, croit que c'est son office. Il ne peut pas suivre ta mani\u00e8re de voir, il ne peut que suivre la sienne ; en sorte que s'il a tort, c'est pour lui qu'il y a dommage, puisque c'est lui qui est dans l'erreur. En effet, si l'on juge fausse une proposition copulative qui est vraie, il n'y a pas de dommage pour la proposition copulative, mais pour celui qui s'est tromp\u00e9. Si tu te fondes l\u00e0-dessus, tu seras indulgent pour celui qui te dit des injures. R\u00e9p\u00e8te chaque fois : \u00ab Il en a jug\u00e9 ainsi. \u00bb\n\nXLIII\n\nToute chose a deux anses, l'une, par o\u00f9 on peut la porter, l'autre, par o\u00f9 on ne le peut pas. Si ton fr\u00e8re a des torts, ne le prends pas par ce c\u00f4t\u00e9-l\u00e0, qu'il a des torts (c'est l'anse par o\u00f9 on ne peut porter) ; prends-le plut\u00f4t par cet autre c\u00f4t\u00e9, qu'il est ton fr\u00e8re, qu'il a \u00e9t\u00e9 nourri avec toi, et tu prendras la chose par o\u00f9 on peut la porter.\n\nXLIV\n\nCes raisonnements ne sont pas concluants : \u00ab Je suis plus riche que toi, donc, je te suis sup\u00e9rieur \u00bb ; \u00ab Je suis plus \u00e9loquent que toi, donc je te suis sup\u00e9rieur. \u00bb Mais ceux-ci sont plus concluants : \u00ab Je suis plus riche que toi, donc ma fortune est sup\u00e9rieure \u00e0 la tienne \u00bb ; \u00ab Je suis plus \u00e9loquent que toi, donc ma parole est sup\u00e9rieure \u00e0 la tienne. \u00bb Mais toi, tu n'es ni fortune ni parole.\n\nXLV\n\nQuelqu'un se baigne de bonne heure : ne dis pas que c'est mal ; dis que c'est de bonne heure. Quelqu'un boit beaucoup de vin : ne dis pas que c'est mal ; dis qu'il boit beaucoup de vin. Car avant d'avoir reconnu comment il en juge, d'o\u00f9 peux-tu savoir si c'est mal ? Ainsi il ne t'arrivera pas d'avoir des id\u00e9es \u00e9videntes de certaines choses et d'acquiescer \u00e0 d'autres.\n\nXLVI\n\n1. Ne te donne jamais pour philosophe et le plus souvent ne parle pas maximes devant ceux qui ne sont pas philosophes ; fais plut\u00f4t ce que les maximes prescrivent : ainsi, dans un repas, ne dis pas comment on doit manger, mais mange comme on le doit. Souviens-toi que Socrate s'\u00e9tait interdit toute ostentation, au point que des gens venaient le trouver pour se faire pr\u00e9senter par lui \u00e0 des philosophes ; et il les menait, tant il lui \u00e9tait \u00e9gal qu'on ne f\u00eet pas attention \u00e0 lui.2. Si, entre gens qui ne sont pas philosophes, la conversation tombe sur quelque maxime, garde le plus souvent le silence ; tu cours grand risque de rendre aussit\u00f4t ce que tu n'as pas encore dig\u00e9r\u00e9. Quand on te dit que tu ne sais rien, si tu n'en es pas piqu\u00e9, sache qu'alors tu commences \u00e0 \u00eatre philosophe. En effet, ce n'est pas en rendant leur herbe aux bergers, que les brebis leur montrent combien elles ont mang\u00e9 ; mais quand elles ont bien dig\u00e9r\u00e9 leur p\u00e2ture au dedans, elles produisent au dehors de la laine et du lait : de m\u00eame ne fais pas \u00e9talage des maximes devant ceux qui ne sont pas philosophes, mais commence par les dig\u00e9rer pour les produire en pratique.\n\nXLVII\n\nQuand tu es parvenu \u00e0 satisfaire \u00e0 peu de frais \u00e0 tous les besoins du corps, ne fais pas tes embarras, et si tu ne bois que de l'eau, ne dis pas \u00e0 tout propos que tu ne bois que de l'eau. Si tu veux t'endurcir \u00e0 la peine, fais-le pour toi et non pour les autres, ne tiens pas les statues embrass\u00e9es ; mais quand tu as soif, prends dans ta bouche un peu d'eau fra\u00eeche, rejette-la et n'en dis rien.\n\nXLVIII\n\n1. Conduite et caract\u00e8re de celui qui n'est pas philosophe : il n'attend pas de profit ni de dommage de lui-m\u00eame, mais de l'ext\u00e9rieur. Conduite et caract\u00e8re du philosophe : il n'attend de profit ni de dommage que de lui-m\u00eame.\n\n2. Signes de celui qui est en progr\u00e8s : il ne bl\u00e2me personne, il ne loue personne, il ne se plaint de personne, il n'accuse personne, il ne parle jamais de lui-m\u00eame comme de quelqu'un d'importance ou qui sait quelque chose. Quand il se sent contrari\u00e9 ou emp\u00each\u00e9, il ne s'en prend qu'\u00e0 lui-m\u00eame. Quand on le loue, il se moque \u00e0 part soi de celui qui le loue, et quand on le bl\u00e2me, il ne se justifie pas. Il fait comme les gens relevant de maladie qui se prom\u00e8nent avec pr\u00e9caution pour ne pas d\u00e9ranger ce qui se remet, avant que cela ait pris de la consistance.\n\n3. Il a supprim\u00e9 en lui tout d\u00e9sir, et il a transport\u00e9 toutes ses aversions sur ce qui est contraire \u00e0 la nature dans ce qui d\u00e9pend de nous. En toutes choses ses tendances sont mod\u00e9r\u00e9es. S'il para\u00eet b\u00eate ou ignorant, il ne s'en inqui\u00e8te pas. En un mot il se d\u00e9fie de lui-m\u00eame comme d'un ennemi dont on craint les pi\u00e8ges.\n\nXLIX\n\nQuand un homme est tout fier de pouvoir comprendre et expliquer les livres de Chrysippe, dis en toi-m\u00eame : \u00ab Si Chrysippe avait \u00e9crit clairement, cet homme n'aurait pas de quoi se vanter. Pour moi, qu'est-ce que je veux ? conna\u00eetre la nature et la suivre. Je cherche donc quel en est l'interpr\u00e8te ; j'apprends que c'est Chrysippe et je vais \u00e0 lui. Mais je ne comprends pas ce qu'il a \u00e9crit ; alors je cherche quelqu'un qui me l'explique. Jusque-l\u00e0 il n'y a rien de bien extraordinaire. Mais quand j'ai trouv\u00e9 l'interpr\u00e8te, reste \u00e0 mettre en pratique les pr\u00e9ceptes, et c'est cela seulement qui est beau. Mais si c'est pr\u00e9cis\u00e9ment l'explication des pr\u00e9ceptes que j'admire, n'est-il pas arriv\u00e9 que je suis devenu grammairien au lieu de philosophe ? Seulement : au lieu d'Hom\u00e8re j'explique Chrysippe. Aussi quand on me dit : \"Explique-moi Chrysippe\", si je rougis, c'est plut\u00f4t de ne pas pouvoir montrer une conduite qui soit semblable et conforme \u00e0 ses pr\u00e9ceptes. \u00bb\n\nL\n\nObserve tout ce qu'enseigne la philosophie comme des lois que tu ne peux violer sans impi\u00e9t\u00e9. Quoi qu'on dise de toi, ne t'en inqui\u00e8te pas ; cela ne d\u00e9pend plus de toi.\n\nLI\n\n1. Combien de temps encore diff\u00e8res-tu de te juger propre \u00e0 ce qu'il y a de meilleur et de ne d\u00e9sob\u00e9ir \u00e0 rien de ce que la raison prescrit ? Tu as re\u00e7u les maximes envers lesquelles il fallait s'engager, et tu t'es engag\u00e9. Quel ma\u00eetre attends-tu donc encore pour lui transf\u00e9rer le soin de t'amender ? Tu n'es plus un jeune homme, tu es un homme fait. Si tu t'abandonnes maintenant \u00e0 la n\u00e9gligence et \u00e0 la paresse, si tu introduis sans cesse d\u00e9lais sur d\u00e9lais, si tu remets d'un jour \u00e0 l'autre de faire attention \u00e0 toi-m\u00eame, tu ne t'apercevras pas que tu ne fais pas de progr\u00e8s, et tu ne seras jamais philosophe de ta vie, y compris le moment de ta mort.\n\n2. Prends donc d\u00e8s maintenant le parti de vivre en homme fait et qui est en progr\u00e8s, que tout ce qui t'est d\u00e9montr\u00e9 bon soit pour toi une loi inviolable. S'il se pr\u00e9sente quelque chose qui soit p\u00e9nible ou agr\u00e9able, avantageux ou nuisible \u00e0 ta consid\u00e9ration, souviens-toi que le jour de la lutte est venu, que tu es maintenant dans l'ar\u00e8ne d'Olympie, que tu ne peux plus diff\u00e9rer et qu'il ne tient qu'\u00e0 un seul jour, \u00e0 une seule action que tes progr\u00e8s soient assur\u00e9s ou compromis \u00e0 tout jamais.\n\n3. Si Socrate est devenu ce qu'il a \u00e9t\u00e9, c'est qu'en toute rencontre il ne faisait attention qu'\u00e0 la raison. Quant \u00e0 toi, si tu n'es pas encore Socrate, tu dois vivre comme si tu voulais \u00eatre Socrate.\n\nLII\n\n1. La premi\u00e8re partie de la philosophie et la plus essentielle, c'est de mettre en pratique les maximes, par exemple de ne pas mentir ; la seconde, ce sont les d\u00e9monstrations, par exemple, d'o\u00f9 vient qu'il ne faut pas mentir ; la troisi\u00e8me est celle qui confirme et \u00e9claircit les d\u00e9monstrations elles-m\u00eames ; par exemple d'o\u00f9 vient que c'est une d\u00e9monstration ? Qu'est-ce qu'une d\u00e9monstration ? Qu'est-ce que cons\u00e9quence, incompatibilit\u00e9, vrai, faux ?\n\n2. Ainsi donc, la troisi\u00e8me partie est n\u00e9cessaire \u00e0 cause de la seconde, et la seconde \u00e0 cause de la premi\u00e8re ; mais la plus n\u00e9cessaire, celle au-del\u00e0 de laquelle on ne peut plus remonter, c'est la premi\u00e8re. Nous, nous agissons au rebours. Nous nous arr\u00eatons \u00e0 la troisi\u00e8me partie ; toute notre \u00e9tude est pour elle, et nous n\u00e9gligeons compl\u00e8tement la premi\u00e8re. Aussi nous mentons, mais nous savons sur le bout du doigt comment on d\u00e9montre qu'il ne faut pas mentir.\n\nLIII\n\nIl faut \u00eatre pr\u00eat \u00e0 dire en toute rencontre\n\n1. Emm\u00e8ne-moi, Jupiter, et toi, Destin\u00e9e, l\u00e0 o\u00f9 vous avez arr\u00eat\u00e9 que je dois aller. Je vous suivrai sans h\u00e9siter et quand m\u00eame j'aurais la folie de ne pas le vouloir, je ne vous en suivrai pas moins.\n\n2. Quiconque se soumet de bonne gr\u00e2ce \u00e0 la n\u00e9cessit\u00e9 est sage \u00e0 notre avis et sait les choses divines.\n\n3. Mais, Criton, si telle est la volont\u00e9 des dieux, qu'elle s'accomplisse.\n\n4. Anytus et M\u00e9litus peuvent me tuer, ils ne peuvent pas me nuire.\nEntretiens \n(extraits)\n\nLe combat n'est pas \u00e9gal entre une jolie fille \net un jeune apprenti philosophe\n\nLivre premier\n\nII. COMMENT ON PEUT CONSERVER SA DIGNIT\u00c9 EN TOUTE CHOSE\n\nPour l'\u00eatre dou\u00e9 de la vie et de la raison, il n'y a d'impossible \u00e0 supporter que ce qui est contre la raison, mais tout ce qui est conforme \u00e0 la raison se peut supporter. Les coups par eux-m\u00eames ne sont point impossibles \u00e0 supporter. \u2014 Comment cela ? \u2014 Vois comme les Lac\u00e9d\u00e9moniens se laissent battre de verges, sachant que cela est conforme \u00e0 la raison. La pendaison elle-m\u00eame se peut supporter. Lorsque quelqu'un croit qu'elle est conforme \u00e0 la raison, il s'en va et se pend. En un mot, si nous y faisons attention, nous trouverons que l'\u00eatre dou\u00e9 de la vie ne souffre de rien tant que de ce qui n'est pas raisonnable ; et qu'en revanche il n'est attir\u00e9 par rien autant que par ce qui est raisonnable.\n\nMais ce qui para\u00eet raisonnable ou d\u00e9raisonnable \u00e0 l'un, ne le para\u00eet pas \u00e0 l'autre. Il en est de cela comme du bien et du mal, de l'utile et du nuisible. Et c'est pour ce motif surtout que nous avons besoin d'instruction pour apprendre \u00e0 mettre d'accord avec la nature, dans chaque cas particulier, notre notion a priori du raisonnable et du d\u00e9raisonnable.\n\nOr, pour juger de ce qui est conforme ou contraire \u00e0 la raison, nous ne nous bornons pas \u00e0 appr\u00e9cier les objets ext\u00e9rieurs, nous tenons compte encore de notre dignit\u00e9 personnelle. L'un, en effet, trouve conforme \u00e0 la raison de pr\u00e9senter le pot de chambre \u00e0 quelqu'un, parce qu'il ne voit qu'une chose : que, s'il ne le pr\u00e9sente point, il recevra des coups et ne recevra pas de nourriture ; tandis que s'il le pr\u00e9sente, il n'aura \u00e0 supporter rien de f\u00e2cheux ni de p\u00e9nible. L'autre, non seulement trouve intol\u00e9rable de le pr\u00e9senter lui-m\u00eame, mais encore ne saurait souffrir qu'un autre le lui pr\u00e9sente. Si tu me fais cette question : \u00ab Pr\u00e9senterai-je ou non le pot de chambre ? \u00bb Je te dirai que recevoir de la nourriture vaut mieux que n'en pas recevoir, et qu'il y a plus de d\u00e9sagr\u00e9ment \u00e0 \u00eatre frapp\u00e9 de verges qu'\u00e0 ne pas l'\u00eatre ; de sorte que, si tu calcules d'apr\u00e8s cela ce qui te convient, va pr\u00e9senter le pot de chambre. \u2014 Mais la chose est indigne de moi. \u2014 C'est \u00e0 toi de faire entrer cela en ligne de compte, et non pas \u00e0 moi, car tu es le seul qui sache combien tu t'estimes, et combien tu veux te vendre. Chacun se vend un prix diff\u00e9rent.\n\nAussi quand Florus demanda \u00e0 Agrippinus s'il devait descendre sur la sc\u00e8ne avec N\u00e9ron pour y jouer un r\u00f4le lui aussi, \u00ab Descends-y \u00bb fut la r\u00e9ponse. Et \u00e0 cette question : \u00ab Pourquoi, toi, n'y descends-tu pas ? \u00bb Il r\u00e9pondit : \u00ab Parce que, moi, dit-il, je ne me demande m\u00eame pas si je dois le faire. \u00bb C'est, qu'en effet, celui qui s'abaisse \u00e0 d\u00e9lib\u00e9rer sur de pareilles choses et qui p\u00e8se les objets ext\u00e9rieurs avant de se d\u00e9cider, touche de bien pr\u00e8s \u00e0 ceux qui oublient leur dignit\u00e9 personnelle.\n\nQue me demandes-tu en effet ? Qui vaut le mieux de la mort ou de la vie ? Je te r\u00e9ponds, la vie. De la souffrance ou du plaisir ? Je te r\u00e9ponds, le plaisir. \u2014 Mais si je ne joue pas la trag\u00e9die, dis-tu, j'aurai la t\u00eate coup\u00e9e. \u2014 Va donc, et joue la trag\u00e9die. Pour moi, je ne la jouerai pas. \u2014 Pourquoi ? \u2014 Parce que toi, tu ne te regardes que comme un des fils de la tunique. \u2014 Que veux-tu dire ? \u2014 Que d\u00e8s lors, il te faut chercher \u00e0 ressembler aux autres hommes, de m\u00eame qu'aucun fil ne demande \u00e0 \u00eatre sup\u00e9rieur aux autres fils. Mais moi, je veux \u00eatre le morceau de pourpre, cette petite partie brillante qui donne aux autres l'\u00e9clat et la beaut\u00e9. Que me dis-tu donc de ressembler aux autres ? Comment serais-je pourpre alors ?\n\nC'est ce qu'avait bien vu Priscus Helvidius ; et il agit comme il avait vu. \u2014 Vespasien lui avait envoy\u00e9 dire de ne pas aller au s\u00e9nat : \u2014 Il est en ton pouvoir, lui r\u00e9pondit-il, de ne pas me laisser \u00eatre du s\u00e9nat ; mais tant que j'en serai, il faut que j'y aille. \u2014 Eh bien ! Vas-y, lui dit l'empereur, mais tais-toi. \u2014 Ne m'interroge pas, et je me tairai. \u2014 Mais il faut que je t'interroge. \u2014 Et moi, il faut que je dise ce qui me semble juste. \u2014 Si tu le dis, je te ferai mourir. \u2014 Quand t'ai-je dit que j'\u00e9tais immortel ? Tu rempliras ton r\u00f4le, et je remplirai le mien. Ton r\u00f4le est de faire mourir ; le mien est de mourir sans trembler. Ton r\u00f4le est d'exiler, le mien est de partir sans chagrin. \u00c0 quoi servit cette conduite de Priscus, seul comme il \u00e9tait ? Mais en quoi la pourpre sert-elle au manteau ? Que fait-elle autre chose que de ressortir sur lui en sa qualit\u00e9 de pourpre, et d'y \u00eatre, pour le reste, un sp\u00e9cimen de beaut\u00e9 ? Un autre homme, si C\u00e9sar, dans de pareilles circonstances, lui avait dit de ne pas aller au s\u00e9nat, aurait r\u00e9pondu : \u00ab Je te remercie de m'\u00e9pargner. \u00bb Mais C\u00e9sar n'aurait pas emp\u00each\u00e9 un tel homme d'y aller, sachant bien qu'il y devait rester immobile comme une cruche, ou que, s'il y parlait, il dirait ce qu'il savait d\u00e9sir\u00e9 de l'empereur, et que m\u00eame il rench\u00e9rirait encore dessus.\n\nDe m\u00eame cet athl\u00e8te qui \u00e9tait en danger de mourir, si on ne lui coupait pas les parties sexuelles. Son fr\u00e8re vint le trouver (l'athl\u00e8te \u00e9tait philosophe) et lui dit : \u00ab Eh bien ! fr\u00e8re, que vas-tu faire ? coupons cette partie, et retournons encore au gymnase. \u00bb Mais celui-ci refusa, tint bon, et mourut. Quelqu'un demandait \u00e0 quel titre il avait agi ainsi : \u00e0 titre d'athl\u00e8te ou \u00e0 titre de philosophe ? \u2014 \u00c0 titre d'homme, r\u00e9pondit \u00c9pict\u00e8te ; au titre d'un homme qui avait \u00e9t\u00e9 proclam\u00e9 \u00e0 Olympie apr\u00e8s y avoir combattu, d'un homme qui avait pass\u00e9 sa vie sur ce terrain-l\u00e0, et non \u00e0 se faire parfumer d'odeurs chez Baton. Un autre se serait fait couper jusqu'\u00e0 la t\u00eate m\u00eame, s'il avait pu vivre sans t\u00eate. Voil\u00e0 ce que c'est que le sentiment de notre dignit\u00e9. Voil\u00e0 la force qu'il a chez ceux qui ont l'habitude de le faire entrer en ligne de compte dans leurs d\u00e9lib\u00e9rations. \u2014 Va donc, \u00c9pict\u00e8te : fais-toi raser. \u2014 Si je suis philosophe, je r\u00e9ponds : \u00ab Je ne me ferai pas raser. \u00bb \u2014 Mais je t'enl\u00e8verai la t\u00eate. \u2014 Enl\u00e8ve-la, si cela te semble bon.\n\nQuelqu'un lui demandait : Comment sentirons-nous ce qui est conforme \u00e0 notre dignit\u00e9 ? \u2014 Comment le taureau, dit-il, \u00e0 l'approche du lion, sent-il seul la force qui est en lui, et se jette-t-il en avant pour le troupeau tout entier ? Il est \u00e9vident que d\u00e8s le premier instant, avec la force dont il est dou\u00e9, se trouve en lui le sentiment de cette force. Eh bien ! de m\u00eame chez nous, nul de ceux qui seront ainsi dou\u00e9s ne restera sans le savoir. Mais ce n'est pas en un jour que se fait le taureau, non plus que l'homme d'\u00e9lite ; il faut s'exercer et se former \u00e0 grand-peine, et ne pas s'\u00e9lancer \u00e0 l'\u00e9tourdie vers ce qui n'est pas de notre comp\u00e9tence.\n\nVois seulement \u00e0 quel prix tu vends ton libre arbitre. Au moins, mon ami, vends-le cher. \u2014 Ce prix \u00e9lev\u00e9 et exceptionnel convient peut-\u00eatre \u00e0 d'autres (diras-tu), \u00e0 Socrate et \u00e0 ceux qui lui ressemblent ? \u2014 Pourquoi donc, puisque nous naissons tous semblables \u00e0 lui, un si petit nombre plus tard lui sont-ils semblables ? \u2014 Tous les chevaux deviennent-ils donc rapides, et tous les limiers bons chasseurs ? \u2014 Eh bien ! parce que je suis d'une nature ingrate, faut-il me refuser \u00e0 tout effort ? \u00e0 Dieu ne plaise ! \u00c9pict\u00e8te n'est pas sup\u00e9rieur \u00e0 Socrate, mais qu'il ne lui soit pas inf\u00e9rieur, et cela me suffit. Je ne deviendrai pas non plus un Milon, et cependant je ne n\u00e9glige pas mon corps ; un Cr\u00e9sus non plus, et cependant je ne n\u00e9glige pas ma fortune. Il n'y a aucune autre chose en un mot, dont nous nous refusions \u00e0 prendre soin, parce que nous y d\u00e9sesp\u00e9rons du premier rang.\n\nVIII. LES TALENTS DES IGNORANTS NE SONT PAS SANS P\u00c9RILS\n\nAutant il y a de mani\u00e8res de varier les propositions \u00e9quivalentes, autant il y en a de varier dans nos raisonnements la forme des \u00e9pich\u00e9r\u00e8mes et des enthym\u00e8mes ; comme dans celui-ci, par exemple : Si tu m'as emprunt\u00e9 et ne m'as pas rendu, tu me dois de l'argent ; or, tu ne m'as ni emprunt\u00e9 ni rendu, tu ne me dois donc pas d'argent. Et c'est ce qu'il n'appartient \u00e0 personne plus qu'au philosophe de faire habilement. Car si l'enthym\u00e8me est un syllogisme incomplet, il est \u00e9vident que celui qui est exerc\u00e9 au syllogisme complet ne sera pas moins habile \u00e0 l'incomplet. Pourquoi donc ne pas nous exercer en ce genre, seuls ou avec d'autres ? \u2014 Parce que aujourd'hui que nous ne nous y exer\u00e7ons pas, et que, autant que nous le pouvons, rien ne nous distrait de l'\u00e9tude de la morale, nous ne faisons cependant pas de progr\u00e8s dans la vertu. \u00c0 quoi ne devrions-nous pas nous attendre alors, si nous y ajoutions cette distraction ? d'autant plus que ce ne serait pas seulement une distraction des choses plus n\u00e9cessaires, mais encore une cause non commune de pr\u00e9somption et d'orgueil. C'est une grande puissance, en effet, que l'art d'argumenter et de persuader, surtout quand il se fortifie par la pratique et qu'il emprunte au style un certain prestige. De plus, toute puissance, en g\u00e9n\u00e9ral, est dangereuse aux mains des ignorants et des faibles, car elle les porte \u00e0 s'enorgueillir et \u00e0 faire les fiers. Comment, en effet, persuader au jeune homme qui se distingue par ces talents que ce n'est pas lui qui doit leur appartenir, mais eux qui doivent lui appartenir \u00e0 lui ? Ne foule-t-il pas aux pieds toutes ces observations ? Et ne s'en va-t-il pas tout fier et tout plein de lui-m\u00eame, repoussant quiconque s'attacherait \u00e0 lui, pour lui repr\u00e9senter ce qu'il quitte, et o\u00f9 il va \u00e0 la d\u00e9rive ?\n\n\u2014 Mais quoi ! Platon n'\u00e9tait-il pas philosophe ? \u2014 Eh bien ! Hippocrate n'\u00e9tait-il pas m\u00e9decin ? Et tu vois comment sait parler Hippocrate. Or, est-ce en tant que m\u00e9decin qu'Hippocrate parle ainsi ? Pourquoi donc confonds-tu des choses qui se trouvent dans le m\u00eame homme \u00e0 des titres diff\u00e9rents ? Si Platon avait \u00e9t\u00e9 beau ou fort, me faudrait-il rester l\u00e0 \u00e0 me fatiguer pour devenir beau ou fort moi aussi, comme si cela \u00e9tait n\u00e9cessaire pour \u00eatre philosophe, parce qu'un philosophe aurait \u00e9t\u00e9 \u00e0 la fois beau et philosophe ? Ne veux-tu donc pas voir et distinguer ce que les gens sont en tant que philosophes, et ce qui est chez eux \u00e0 d'autres titres ? Si, par exemple, moi j'\u00e9tais philosophe, faudrait-il donc que vous, vous devinssiez boiteux comme moi ? Mais quoi ! est-ce que je pr\u00e9tends supprimer ces talents ? \u00e0 Dieu ne plaise ! pas plus que la facult\u00e9 de voir. Mais cependant si tu me demandes quel est le bien de l'homme, je ne puis te r\u00e9pondre que ceci : une certaine fa\u00e7on d'user des id\u00e9es.\n\nXXI. CONTRE CEUX QUI VEULENT SE FAIRE ADMIRER\n\nLorsque quelqu'un dans cette vie est ce qu'il doit \u00eatre, il ne s'extasie pas devant les choses du dehors. Homme, que souhaites-tu qu'il t'arrive ? Pour moi il me suffit que mes d\u00e9sirs et mes aversions soient conformes \u00e0 la nature ; que, dans mes vouloirs ou dans mes refus, dans mes projets, dans mes efforts, dans mes jugements, je sais que je suis n\u00e9 pour \u00eatre. Pourquoi marches-tu aussi raide que si tu avais aval\u00e9 une broche ? \u00ab Je veux que tous ceux qui se trouvent sur mon chemin m'admirent et me suivent en criant : \"Quel grand philosophe !\" \u00bb \u2014 Eh ! qui sont ces gens dont tu veux te faire admirer ? Ne sont-ce pas ceux dont tu as l'habitude de dire qu'ils sont fous ? Et c'est par des fous que tu veux \u00eatre admir\u00e9 !\n\nLivre deuxi\u00e8me\n\nIV. SUR UN HOMME QUI AVAIT \u00c9T\u00c9 SURPRIS EN ADULT\u00c8RE\n\nUn jour qu'il soutenait que l'homme \u00e9tait n\u00e9 pour l'honn\u00eatet\u00e9, et que m\u00e9conna\u00eetre ce principe c'\u00e9tait m\u00e9conna\u00eetre le caract\u00e8re essentiel de l'humanit\u00e9, survint un de nos pr\u00e9tendus lettr\u00e9s, qui avait \u00e9t\u00e9 autrefois surpris \u00e0 Rome en adult\u00e8re. Que faisons-nous, dit alors \u00c9pict\u00e8te, lorsque, renon\u00e7ant \u00e0 cette honn\u00eatet\u00e9 pour laquelle nous sommes n\u00e9s, nous nous attaquons \u00e0 la femme de notre voisin ? Ce que nous faisons ? Nous perdons et d\u00e9truisons... Quoi donc ? Notre honn\u00eatet\u00e9, notre retenue, notre puret\u00e9. Est-ce l\u00e0 tout ? Ne d\u00e9truisons-nous pas encore les rapports de bon voisinage ? Et l'amiti\u00e9 ? Et la soci\u00e9t\u00e9 civile ? Quel r\u00f4le nous donnons-nous \u00e0 nous-m\u00eames ? \u00d4 homme, quelles relations entretiendrai-je avec toi ? des relations de voisin ? d'ami ? De quoi, enfin ? de citoyen ? Quelle confiance puis-je avoir en toi ? Si tu \u00e9tais un vase en si piteux \u00e9tat, que tu ne pusses servir \u00e0 rien, on te jetterait dehors, sur un tas de fumier, et personne ne t'y ramasserait. Si tu es un homme, et que tu ne puisses jouer aucun des r\u00f4les de l'homme, que ferons-nous de toi ? Car, si tu ne peux \u00eatre \u00e0 ta place comme ami, y pourras-tu \u00eatre comme esclave ? Mais l\u00e0 encore, qui se fiera \u00e0 toi ? Et tu ne veux pas qu'on te jette toi aussi sur un tas de fumier, comme un vase inutile, aussi sale que le fumier !\n\nPuis tu viendras dire : \u00ab Quoi ! personne ne fait cas de moi qui suis un lettr\u00e9 ! \u00bb C'est que tu es un m\u00e9chant, dont il n'y a rien \u00e0 faire. C'est comme si les gu\u00eapes s'indignaient de ce qu'on ne fait pas cas d'elles, de ce qu'on les fuit, et de ce qu'on les frappe et les abat, quand on le peut ! Tu as un dard qui porte le chagrin et la douleur partout o\u00f9 il frappe. Que veux-tu que nous fassions de toi ? Il n'y a pas de place o\u00f9 te mettre. \u00ab Comment ! dis-tu. Est-ce que la nature n'a pas fait les femmes communes \u00e0 tous ? \u00bb Et moi je te dis : Le cochon de lait lui aussi est commun \u00e0 tous les invit\u00e9s. Mais, quand il a \u00e9t\u00e9 partag\u00e9, avise-toi d'aller prendre de force la part de ton voisin, ou de la lui d\u00e9rober ; ou bien encore, mets la main dans son assiette pour go\u00fbter de ce qui est dedans, et, si tu ne peux lui enlever sa viande, tra\u00eene tes doigts dans sa graisse, et l\u00e8che-les. Quel honn\u00eate convive ! Quel disciple de Socrate \u00e0 table ! Le th\u00e9\u00e2tre lui aussi n'est-il pas commun \u00e0 tous les citoyens ! Eh bien ! lorsqu'ils sont assis, va t'aviser de chasser l'un d'eux de sa place. C'est de cette fa\u00e7on-l\u00e0 que les femmes sont communes. Lorsque le l\u00e9gislateur, comme un ma\u00eetre de maison, les a partag\u00e9es entre tous, toi, plut\u00f4t que de chercher \u00e0 en avoir ta part \u00e0 toi, aimeras-tu mieux voler la part de ton voisin et y porter la dent ? \u2014 Mais je suis un lettr\u00e9, dis-tu, et je comprends Arch\u00e9d\u00e9mus ! \u2014 Eh bien ! toi qui comprends Arch\u00e9d\u00e9mus, sois d\u00e9bauch\u00e9, sois sans honneur ; au lieu d'\u00eatre un homme, sois un loup ou un singe. Car en quoi diff\u00e8res-tu d'eux ?\n\nXV. SUR LES GENS QUI PERSISTENT OBSTIN\u00c9MENT DANS CE QU'ILS ONT D\u00c9CID\u00c9\n\nIl est des gens qui, pour avoir entendu dire qu'il faut \u00eatre ferme, que notre facult\u00e9 de juger et de vouloir est de sa nature ind\u00e9pendante et libre, que tout le reste, pouvant \u00eatre entrav\u00e9 ou contraint, est esclave et ne nous appartient pas, s'imaginent qu'ils doivent persister obstin\u00e9ment dans toutes les d\u00e9cisions qu'ils ont pu prendre. Mais, avant tout, il faut que ta d\u00e9cision soit saine. Je veux que ton corps ait de la force, mais une force due \u00e0 la sant\u00e9 et au travail. Si la force que tu m'\u00e9tales est celle de la fr\u00e9n\u00e9sie, et si tu t'en vantes, je te dirai : \u00ab Mon ami, cherche un m\u00e9decin ; ce n'est pas l\u00e0 de la force, mais un manque de force \u00e0 un autre point de vue. \u00bb Tel est au moral l'\u00e9tat de ceux qui comprennent mal les pr\u00e9ceptes dont nous parlions. C'est ainsi qu'un de mes amis r\u00e9solut, sans aucun motif, de se laisser mourir de faim. Je l'appris quand il y avait d\u00e9j\u00e0 trois jours qu'il s'abstenait de manger ; j'allai le trouver, et lui demandai ce qu'il y avait. \u2014 Je l'ai r\u00e9solu, me dit-il. \u2014 Mais quel est le motif qui t'y a pouss\u00e9 ? Car, si ta r\u00e9solution est raisonnable, nous allons nous asseoir pr\u00e8s de toi, et t'aider \u00e0 sortir de cette vie ; mais, si elle est d\u00e9raisonnable, changes-en. \u2014 Il faut \u00eatre ferme dans ses d\u00e9cisions. \u2014 Que dis-tu l\u00e0, mon ami ? Il faut \u00eatre ferme, non dans toutes ses d\u00e9cisions, mais dans celles qui sont raisonnables. Quoi ! si, par un caprice, tu avais d\u00e9cid\u00e9 qu'il faisait nuit, tu ne changerais pas, tu persisterais en disant : \u00ab Je persiste dans mes d\u00e9cisions ! \u00bb Que fais-tu, mon ami ? Il ne faut pas persister dans toutes. Ne consentiras-tu pas \u00e0 poser d'abord solidement ta base et tes fondements, \u00e0 examiner si ta d\u00e9cision est bonne ou mauvaise, avant de lui faire porter le poids de ta fermet\u00e9 et de ta constance ? Si les fondements que tu poses sont d\u00e9fectueux et sans solidit\u00e9, plus ce que tu y \u00e9tabliras sera fort et massif, plus ce sera prompt \u00e0 s'\u00e9crouler. Vas-tu, sans aucune raison, nous enlever un homme que la vie a fait notre ami et notre compagnon, notre concitoyen dans la grande comme dans la petite patrie ? Tu commets un meurtre, tu tues un homme qui n'a fait aucun mal, et tu dis : \u00ab Je suis ferme dans mes d\u00e9cisions ! \u00bb Mais, s'il te venait l\u00e0 volont\u00e9 de me tuer, serait-ce un devoir pour toi d'\u00eatre ferme dans ta d\u00e9cision ?\n\nNotre homme se laissa dissuader, mais non sans peine ; et, de nos jours, il en est plus d'un qu'on ne peut faire changer. Aussi, crois-je savoir aujourd'hui ce que j'ignorais auparavant, le sens de ce dicton : \u00ab On ne persuade pas plus un sot qu'on ne le brise. \u00bb Dieu me pr\u00e9serve d'avoir pour ami un philosophe qui ne soit qu'un sot ! Il n'y a rien de plus difficile \u00e0 manier. \u00ab J'ai d\u00e9cid\u00e9 \u00bb dit-il ! Mais les fous aussi d\u00e9cident ; et plus ils persistent dans leurs d\u00e9cisions erron\u00e9es, plus pr\u00e9cis\u00e9ment ils ont besoin d'ell\u00e9bore. Ne consentiras-tu pas \u00e0 te conduire comme un malade, \u00e0 appeler le m\u00e9decin ? \u00ab Je suis malade, ma\u00eetre (lui dit-on) : viens \u00e0 mon secours ; examine ce que je dois faire ; je n'ai, moi, qu'\u00e0 t'ob\u00e9ir. \u00bb De m\u00eame ici : \u00ab Je ne sais pas ce que je dois faire (devrait-on lui dire) ; je suis venu pour l'apprendre. \u00bb Au lieu de cela, on lui dit : \u00ab Parle-moi d'autre chose ; quant \u00e0 cette question-l\u00e0, je suis d\u00e9cid\u00e9. \u00bb \u2014 Et de quelle autre chose veux-tu qu'on te parle ? Car qu'y a-t-il de plus important et de plus utile que de te convaincre qu'il ne suffit pas d'avoir d\u00e9cid\u00e9 et de ne point varier dans sa d\u00e9cision ? C'est le d\u00e9ploiement de force d'un fou et non pas d'un homme de bon sens. \u2014 Je suis r\u00e9solu \u00e0 mourir, si tu veux me contraindre \u00e0 cela ! \u2014 Pourquoi, mon ami ? Qu'est-il arriv\u00e9 ? \u2014 Je l'ai d\u00e9cid\u00e9 ! \u2014 Je suis bien heureux que tu n'aies pas d\u00e9cid\u00e9 de me tuer ! \u2014 Je ne veux pas de ton argent ! \u2014 Pourquoi ? \u2014 Je l'ai d\u00e9cid\u00e9. \u2014 Sache donc que la force que tu d\u00e9ploies pour refuser, rien ne garantit que tu ne la d\u00e9ploieras pas un jour pour prendre avec aussi peu de raison, et que tu ne diras pas encore : \u00ab J'ai d\u00e9cid\u00e9. \u00bb Dans le corps d'un malade qu'assi\u00e8gent les rhumatismes, les humeurs se portent tant\u00f4t sur un point, tant\u00f4t sur un autre ; de m\u00eame une \u00e2me faible se porte d'un c\u00f4t\u00e9 sans savoir pourquoi ; puis, quand \u00e0 cette inclinaison et \u00e0 ce mouvement vient s'ajouter la force, il n'y a plus contre le mal qui en r\u00e9sulte ni secours ni rem\u00e8de.\n\nXVIII. COMMENT IL FAUT LUTTER CONTRE LES ID\u00c9ES DANGEREUSES\n\nToute habitude, tout talent, se forment et se fortifient par les actions qui leur sont analogues : marchez, pour \u00eatre marcheur ; courez, pour \u00eatre coureur. Voulez-vous savoir lire ? Lisez. Savoir \u00e9crire ? \u00c9crivez. Passez trente jours de suite sans lire, \u00e0 faire tout autre chose, et vous saurez ce qui en arrivera. Restez couch\u00e9 dix jours, puis levez-vous, et essayez de faire une longue route, et vous verrez comme vos jambes seront fortes. Une fois pour toutes, si vous voulez prendre l'habitude d'une chose, faites cette chose ; si vous n'en voulez pas prendre l'habitude, ne la faites pas, et habituez-vous \u00e0 faire quoi que ce soit plut\u00f4t qu'elle. Il en est de m\u00eame pour l'\u00e2me : lorsque vous vous emportez, sachez que ce n'est pas l\u00e0 le seul mal qui vous arrive, mais que vous augmentez en m\u00eame temps votre disposition \u00e0 la col\u00e8re : c'est du bois que vous mettez dans le feu. Lorsque vous avez succomb\u00e9 aux attraits de la chair avec quelqu'un, ne vous dites pas qu'il n'y a l\u00e0 qu'une d\u00e9faite, mais que vous avez du m\u00eame coup aliment\u00e9, fortifi\u00e9 votre penchant au plaisir. Il est impossible, en effet, que les actes en analogie avec quelque habitude et quelque disposition, ne les fassent point na\u00eetre, si elles n'existent pas avant, et ne les d\u00e9veloppent point, ne les fortifient point, dans l'autre cas.\n\nC'est certainement ainsi, au dire des philosophes, que se forment jour \u00e0 jour nos maladies morales. Convoitez une fois de l'argent, et qu'il vous arrive ensuite un raisonnement qui vous fasse sentir votre mal, votre convoitise cesse, et votre partie ma\u00eetresse est r\u00e9tablie dans son premier \u00e9tat ; mais que rien ne vienne la gu\u00e9rir, elle ne redeviendra pas ce qu'elle \u00e9tait ; bien loin de l\u00e0, qu'une apparition du m\u00eame genre l'excite une seconde fois, et la convoitise s'allumera en elle bien plus vite que la premi\u00e8re. Que ceci se reproduise d'une mani\u00e8re suivie, le calus se forme \u00e0 jamais, et la cupidit\u00e9 devient en nous une maladie durable. Celui qui a eu la fi\u00e8vre, et qui a cess\u00e9 de l'avoir, n'est pas dans le m\u00eame \u00e9tat qu'avant de l'avoir eue, \u00e0 moins qu'il n'ait \u00e9t\u00e9 gu\u00e9ri compl\u00e8tement. La m\u00eame chose arrive pour les maladies de l'\u00e2me. Elles y laissent des traces, des meurtrissures, qu'il faut faire dispara\u00eetre compl\u00e8tement ; sinon, pour peu qu'on re\u00e7oive encore quelque coup \u00e0 la m\u00eame place, ce ne sont plus des meurtrissures, ce sont des plaies qui se produisent. Si donc tu ne veux pas \u00eatre enclin \u00e0 la col\u00e8re, n'en entretiens pas en toi l'habitude ; ne lui donne rien pour l'alimenter. Calme ta premi\u00e8re fureur, puis compte les jours o\u00f9 tu ne te seras pas emport\u00e9. \u00ab J'avais l'habitude de m'emporter tous les jours, diras-tu ; maintenant c'est un jour sur deux, puis ce sera un sur trois, et apr\u00e8s cela un sur quatre. \u00bb Si tu passes ainsi trente jours, fais un sacrifice \u00e0 Dieu. L'habitude, en effet, commence par s'affaiblir, puis elle dispara\u00eet enti\u00e8rement. Si tu peux te dire : \u00ab Voici un jour que je ne me suis pas afflig\u00e9 ; en voici deux ; puis voici deux mois, voici trois mois ; j'ai veill\u00e9 sur moi, quand il s'est pr\u00e9sent\u00e9 des choses qui pouvaient me contrarier \u00bb, sache que tout va bien chez toi. Si je puis me dire : \u00ab Aujourd'hui, \u00e0 la vue d'un beau gar\u00e7on ou d'une belle femme, je ne me suis pas dit : \"Pl\u00fbt aux dieux qu'on couch\u00e2t avec elle !\" ni \"Bienheureux son mari !\" (car celui qui dit cela, dit aussi : \"Bienheureux son amant !\"). Je ne me suis pas non plus repr\u00e9sent\u00e9 tout ce qui s'ensuit, cette femme pr\u00e8s de moi, se mettant nue, se couchant \u00e0 mes c\u00f4t\u00e9s \u00bb, je me caresse la t\u00eate, et je me dis : \u00ab C'est bien, \u00c9pict\u00e8te ! Tu es venu \u00e0 bout d'un beau sophisme, d'un sophisme bien plus beau que celui qu'on nomme le Dominateur. \u00bb Et si cette femme ne demandait pas mieux, si elle me faisait des signes, si elle venait vers moi, si elle me touchait et se mettait tout pr\u00e8s de moi, et que je me dominasse encore et triomphasse d'elle, ce serait venir \u00e0 bout d'un sophisme bien au-dessus du Menteur et de l'Endormi. Voil\u00e0 ce dont on a le droit d'\u00eatre fier, et non pas d'avoir propos\u00e9 le Dominateur !\n\nMais comment en arriver l\u00e0 ? Veuille te plaire \u00e0 toi-m\u00eame ; veuille \u00eatre beau aux yeux de Dieu ; veuille vivre pur avec toi-m\u00eame qui resteras par, et avec Dieu. Puis, quand il se pr\u00e9sentera \u00e0 toi quelque apparition de ce genre, Platon te dit : \u00ab Recours aux sacrifices expiatoires ; recours, en suppliant, aux temples des dieux tut\u00e9laires ; mais il te suffira de te retirer dans la soci\u00e9t\u00e9 de quelqu'un des sages, et de rester avec lui en te comparant \u00e0 lui ; qu'il soit un de ceux qui vivent, ou un de ceux qui sont morts. \u00bb Va vers Socrate, vois-le, couch\u00e9 pr\u00e8s d'Alcibiade, triompher de sa beaut\u00e9 en se jouant ; songe quelle grande victoire, quelle victoire olympique, il e\u00fbt alors conscience d'avoir remport\u00e9e. Fut-il en ce moment beaucoup au-dessous d'Hercule ? De par tous les dieux ! on put, \u00e0 bon droit, le saluer de ces paroles : \u00ab Salut, \u00f4 l'homme incroyable ! \u00bb Ceux que tu as vaincus, ce ne sont pas ces mis\u00e9rables histrions ou h\u00e9ros du pancrace, ni ces gens bons \u00e0 une seule lutte qui sont de la m\u00eame famille que les autres ! Si tu te repr\u00e9sentes tout cela, tu triompheras de l'apparition, et tu ne seras pas entra\u00een\u00e9 par elle. Commence par r\u00e9sister \u00e0 son impression trop vive, et dis : \u00ab Attends-moi un peu, id\u00e9e ; laisse-moi voir qui tu es et sur quoi tu portes. Laisse-moi te juger. \u00bb Puis ne la laisse pas faire de progr\u00e8s, et retrace \u00e0 ton imagination tout ce qui la suit ; sinon, elle va t'entra\u00eener partout o\u00f9 elle voudra. Appelle bien plut\u00f4t \u00e0 sa place quelque autre id\u00e9e honn\u00eate et noble, et chasse ainsi l'image impure. Si tu t'habitues \u00e0 ce genre de lutte, tu verras ce que deviendront tes \u00e9paules, tes tendons et tes muscles ; mais pour aujourd'hui, ils n'existent qu'en parole, et rien de plus.\n\nVoil\u00e0 le v\u00e9ritable lutteur : c'est celui qui s'exerce \u00e0 combattre ces id\u00e9es. R\u00e9siste, \u00f4 malheureux ! ne te laisse pas entra\u00eener ! Importante est la lutte, et elle est le fait d'un Dieu : il s'agit de la royaut\u00e9, de la libert\u00e9, de la vie heureuse et calme. Souviens-toi de Dieu, appelle-le \u00e0 ton secours et \u00e0 ton aide, comme dans la temp\u00eate les navigateurs appellent les Dioscures. Est-il, en effet, temp\u00eate plus terrible que celle qui na\u00eet de ces id\u00e9es, dont la force nous jette hors de notre raison ? La temp\u00eate elle-m\u00eame, en effet, qu'est-elle autre chose qu'une id\u00e9e ? Enl\u00e8ve la crainte de la mort, et am\u00e8ne-nous tous les tonnerres et tous les \u00e9clairs que tu voudras, et tu verras quel calme et quelle tranquillit\u00e9 il y aura dans notre \u00e2me. Mais, si tu te laisses vaincre une fois, en te disant que tu vaincras demain, et que demain ce soit la m\u00eame chose, sache que tu en arriveras \u00e0 \u00eatre si malade et si faible qu'\u00e0 l'avenir tu ne t'apercevras m\u00eame plus de tes fautes, mais que tu seras toujours pr\u00eat \u00e0 trouver des excuses \u00e0 tes actes. Tu confirmeras ainsi la v\u00e9rit\u00e9 de ce vers d'H\u00e9siode :\n\n\u00ab L'homme irr\u00e9solu lutte toute sa vie contre le malheur. \u00bb\n\nXXIV. \u00c0 QUELQU'UN QU'IL N'ESTIMAIT PAS\n\nQuelqu'un lui dit un jour : \u2014 Je suis venu souvent vers toi dans le d\u00e9sir de t'entendre, et jamais tu ne m'as r\u00e9pondu. Aujourd'hui, au moins, si faire se peut, dis-moi quelque chose, je t'en conjure. \u2014 Ne crois-tu pas, lui dit \u00c9pict\u00e8te, qu'il y a l'art de parler, comme il y a l'art de telle autre chose ? Ceux qui poss\u00e9deront cet art parleront en gens qui s'y connaissent, et en ignorants, ceux qui ne le poss\u00e9deront pas. \u2014 Je le crois. \u2014 Eh bien ! ceux qui en parlant se font du bien \u00e0 eux-m\u00eames et peuvent en faire aux autres, ne sont-ils pas ceux qui parlent en s'y connaissant ? Et ceux, au contraire, qui se font du tort \u00e0 eux-m\u00eames et aux autres, ne sont-ils pas ceux qui ne se connaissent pas \u00e0 cet art de parler ? Or, il est facile de trouver des gens qui se font du bien en parlant, et d'autres qui se font du tort. \u00c0 leur tour, ceux qui \u00e9coutent tirent-ils tous quelque profit de ce qu'ils \u00e9coutent ?\n\nEt ne peut-on pas, parmi eux aussi, trouver des gens qui profitent, et des gens qui p\u00e2tissent ? \u2014 Oui, parmi eux aussi. \u2014 Eh bien ! l\u00e0 aussi tous ceux qui \u00e9coutent en s'y connaissant ne sont-ils pas ceux qui profitent, tandis que tous ceux qui \u00e9coutent en ne s'y connaissant point, p\u00e2tissent ? \u2014 D'accord. \u2014 N'y a-t-il pas alors un art d'\u00e9couter, comme il y a un art de parler ? \u2014 Il semble que oui. \u2014 Si tu le veux bien, pense encore \u00e0 ceci. \u00c0 qui appartient-il, suivant toi, de faire de la musique ? \u2014 Au musicien. \u2014 Faire une statue comme elle doit \u00eatre faite, \u00e0 qui crois-tu encore que cela appartienne ? \u2014 Au statuaire. \u2014 Et pour la regarder en connaisseur, crois-tu qu'il n'y ait besoin d'aucune science ? \u2014 Il y en a besoin. \u2014 Eh bien ! s'il faut un homme exerc\u00e9 pour parler comme on le doit, ne vois-tu pas qu'il faut aussi un homme exerc\u00e9 pour \u00e9couter avec profit ? Ne parlons pour le moment, si tu le veux, ni de perfection ni de profit complet, car tous les deux nous sommes \u00e0 grande distance de quoi que ce soit de ce genre. Mais voici, ce me semble, une chose que tout le monde m'accordera : c'est que, pour \u00e9couter un philosophe, on a besoin de quelque pr\u00e9paration. N'est-ce pas vrai ?\n\nDe quoi donc te parlerai-je ? Sur quel sujet peux-tu m'\u00e9couter ? Sur le bien et le mal ? Mais de qui ? Du cheval ? \u2014 Non. \u2014 Du b\u0153uf ? \u2014 Non. \u2014 De qui donc ? De l'homme ? \u2014 Oui. \u2014 Savons-nous donc ce que c'est que l'homme, quelle est sa nature, quelle est son intelligence ? Avons-nous les oreilles familiaris\u00e9es avec cette question, au moins dans une certaine mesure ? Comprends-tu ce que c'est que la nature ? Ou pourras-tu me suivre dans une certaine mesure, si je te le dis ? Puis me servirai-je avec toi de d\u00e9monstrations ? Et comment le ferai-je ? Car te rends-tu au moins compte de ce que c'est qu'une d\u00e9monstration, de la mani\u00e8re dont elle se fait, des moyens qu'elle emploie, des cas o\u00f9 il y a semblant de d\u00e9monstration, sans d\u00e9monstration r\u00e9elle ? Sais-tu, en effet, ce que c'est que la v\u00e9rit\u00e9, ce que c'est que l'erreur, ce que c'est que cons\u00e9quence, opposition, d\u00e9saccord, contradiction ? Puis te pousserai-je vers la philosophie ? Comment le ferai-je ? En te montrant les oppositions et les divergences de la plupart des hommes sur le bien et le mal, l'utile et le nuisible ? Mais tu ne sais m\u00eame pas ce que c'est qu'opposition ! Montre-moi donc ce \u00e0 quoi je puis arriver en causant avec toi. Donne-moi le d\u00e9sir de le faire. Que la brebis aper\u00e7oive une herbe qui lui convient, cela lui donne l'envie d'en manger ; mais place aupr\u00e8s d'elle une pierre ou du pain, elle y sera indiff\u00e9rente. Il y a de m\u00eame en nous une certaine propension naturelle \u00e0 parler, quand celui qui doit nous entendre nous fait l'effet de quelqu'un, quand il a en lui-m\u00eame quelque chose qui nous y invite ; mais, quand il n'est l\u00e0 pr\u00e8s de nous que comme une pierre ou comme une botte de foin, quelle envie peut-il donner \u00e0 un homme ? Est-ce que la vigne dit \u00e0 l'ouvrier des champs : \u00ab Cultive-moi ? \u00bb Non ; mais, rien qu'\u00e0 la voir, il est clair que celui qui la cultivera s'en trouvera bien ; et elle invite ainsi d'elle-m\u00eame \u00e0 la cultiver. Un petit enfant, charmant et vif, ne vous donne-t-il pas l'envie de jouer avec lui, de marcher avec lui sur les mains, de balbutier avec lui ? Mais qui a jamais l'id\u00e9e de jouer avec un \u00e2ne ou de braire avec lui ? C'est que, si petit qu'il soit, il n'est jamais qu'un \u00e2non.\n\n\u2014 Pourquoi donc ne me dis-tu rien ? \u2014 Je ne puis te dire qu'une chose, c'est que l'homme qui ignore ce qu'il est et pourquoi il est n\u00e9, qui ne sait ni ce qu'est ce monde o\u00f9 il est, ni ce que sont ses compagnons, ni ce qui est bon, ni ce qui est mauvais, ni ce qui est beau, ni ce qui est laid, qui ne comprend ni un raisonnement, ni une d\u00e9monstration, ni ce que c'est que la v\u00e9rit\u00e9, ni ce que c'est que l'erreur, et qui ne sait pas les distinguer, ne se conformera \u00e0 la nature ni dans ses d\u00e9sirs, ni dans ses craintes, ni dans ses vouloirs, ni dans ses entreprises, ni dans ses affirmations, ni dans ses n\u00e9gations, ni dans ses doutes. En somme, il s'en ira \u00e0 droite et \u00e0 gauche sourd et aveugle ; on le prendra pour quelqu'un, et il ne sera personne. Est-ce en effet la premi\u00e8re fois qu'il en est ainsi ? Est-ce que, depuis que la race humaine existe, toutes nos fautes et tous nos malheurs ne sont pas d\u00e8s ce moment venus de notre ignorance ? Pourquoi Agamemnon et Achille se sont-ils disput\u00e9s ? N'est-ce point faute de savoir ce qui est utile et ce qui est nuisible ? L'un ne dit-il pas qu'il est utile de rendre Chrys\u00e9is \u00e0 son p\u00e8re, et l'autre que cela est funeste ? L'un ne dit-il pas qu'il doit recevoir la r\u00e9compense qui a \u00e9t\u00e9 donn\u00e9e \u00e0 un autre ; et l'autre, qu'il ne le doit pas ? N'est-ce pas pour cela qu'ils ont oubli\u00e9 qui ils \u00e9taient, et pourquoi ils \u00e9taient venus l\u00e0 ? Homme, pourquoi es-tu venu ? pour gagner des ma\u00eetresses ? ou pour combattre ? \u2014 Pour combattre. \u2014 Qui ? les Troyens ou les Grecs ? \u2014 Les Troyens. \u2014 Eh bien ! laisseras-tu l\u00e0 Hector pour tirer l'\u00e9p\u00e9e contre le roi ? Et toi, mon cher, oublieras-tu ton r\u00f4le de roi, toi \u00e0 qui les peuples sont confi\u00e9s, \u00e0 qui tant d'int\u00e9r\u00eats sont remis ; et te disputeras-tu pour une femme avec le plus vaillant de tes alli\u00e9s, que tu devrais entourer de toute sorte d'attentions et d'\u00e9gards ? Seras-tu au-dessous de l'habile grand-pr\u00eatre, qui a toute esp\u00e8ce de soins pour les grands guerriers ? Vois-tu ce que peut produire l'ignorance de ce qui est vraiment utile ?\n\n\u2014 Mais, moi, dis-tu, je suis riche ! \u2014 Es-tu donc plus riche qu'Agamemnon ? \u2014 Mais, moi, je suis beau ! \u2014 Es-tu donc plus beau qu'Achille ? \u2014 J'ai de plus une chevelure magnifique ! \u2014 Est-ce qu'Achille n'en avait pas une plus belle encore, et une blonde ? Et il ne la peignait ni ne l'arrangeait avec \u00e9l\u00e9gance ! \u2014 Mais, de plus, je suis fort ! \u2014 Peux-tu donc soulever une pierre telle que celle que soulevait Hector ou Ajax ? \u2014 Mais, de plus, je suis de noble race ! \u2014 As-tu donc une d\u00e9esse pour m\u00e8re ? As-tu pour p\u00e8re un fils de Jupiter ? Et de quoi tout cela servait-il \u00e0 Achille, quand il \u00e9tait assis \u00e0 pleurer pour une femme ? \u2014 Mais je suis orateur ! \u2014 Est-ce qu'il ne l'\u00e9tait pas lui aussi ? Ne sais-tu pas comment il s'est tir\u00e9 d'affaire avec les plus habiles parleurs de la Gr\u00e8ce, Ulysse et Ph\u00e9nix ? Comment il les a r\u00e9duits au silence ?\n\nVoil\u00e0 tout ce que je puis te dire, et encore sans plaisir. Pourquoi ? parce que tu ne m'as pas inspir\u00e9. Que puisse en effet regarder en toi qui m'excite, comme la vue d'un cheval de bonne race excite un \u00e9cuyer ? Ton corps ? Mais tu en as soin d'une fa\u00e7on honteuse. Tes habits ? Mais eux aussi sont ceux d'un d\u00e9bauch\u00e9. Ta tournure ? Ton regard ? Rien. Quand tu voudras entendre parler un philosophe, ne lui dis pas : \u00ab Tu ne me dis rien \u00bb ; borne-toi \u00e0 lui montrer que tu es digne de l'entendre, que tu as ce qu'il faut pour cela ; et tu verras quelles paroles tu lui inspireras.\n\nLivre troisi\u00e8me\n\nIV. CONTRE CEUX QUI, AU TH\u00c9\u00c2TRE, DONNENT DES MARQUES INCONVENANTES DE FAVEUR\n\nUn procurateur de l'\u00c9pire avait favoris\u00e9 un histrion d'une mani\u00e8re inconvenante, et le public lui avait dit des injures ; il \u00e9tait venu alors raconter ces injures \u00e0 \u00c9pict\u00e8te, et il s'indignait contre ceux qui les lui avaient adress\u00e9es. \u2014 Qu'ont-ils fait de mal ? lui dit celui-ci. Ils ont donn\u00e9 des marques de leur faveur, tout comme toi. \u2014 Mais peut-on en donner de pareilles ? dit notre homme. \u2014 Quand ils te voyaient, r\u00e9pliqua \u00c9pict\u00e8te, toi leur magistrat, toi l'ami et le procurateur de C\u00e9sar, t\u00e9moigner ainsi ta faveur, ne pouvaient-ils pas de m\u00eame t\u00e9moigner la leur ? Car, si l'on ne doit pas t\u00e9moigner ainsi sa faveur, commence par ne pas t\u00e9moigner la tienne ; ou, si on le doit, pourquoi leur en veux-tu de t'avoir imit\u00e9 ? Qui la multitude peut-elle imiter, si ce n'est vous qui \u00eates au-dessus d'elle ? Et, quand elle va au th\u00e9\u00e2tre, sur qui a-t-elle les yeux, si ce n'est sur vous ? \u00ab Vois, dit-on, comme l'intendant de C\u00e9sar regarde le spectacle ! Il a cri\u00e9 ! Je crierai donc, moi aussi. Il tr\u00e9pigne d'enthousiasme ! Je tr\u00e9pignerai donc aussi. Ses esclaves, assis \u00e0 ses c\u00f4t\u00e9s, poussent des clameurs ! Moi, je n'ai pas d'esclaves ; je vais \u00e0 moi seul, si je le puis, en pousser autant que tous. \u00bb Il te fallait savoir, quand tu es entr\u00e9 au th\u00e9\u00e2tre, que tu y entrais pour servir de r\u00e8gle et d'exemple aux autres, sur la mani\u00e8re dont on doit regarder. Pourquoi donc t'ont-ils injuri\u00e9 ? parce que tout homme hait ce qui le contrarie. Ces gens voulaient qu'un tel f\u00fbt couronn\u00e9 ; toi tu voulais que ce f\u00fbt un autre : ils te contrariaient, tu les contrariais. Tu t'es trouv\u00e9 le plus fort ; ils ont fait ce qu'ils pouvaient faire : ils ont injuri\u00e9 qui les contrariait. Que voudrais-tu donc ? que tu fisses ce que tu veux, et que ces gens ne pussent m\u00eame pas dire ce qu'ils veulent ? Qu'y a-t-il d'\u00e9tonnant qu'ils aient agi ainsi ? Les laboureurs n'injurient-ils pas Jupiter, quand il les contrarie ? Les matelots ne l'injurient-ils pas aussi ? Cesse-t-on jamais d'injurier C\u00e9sar ? Eh bien ! est-ce que Jupiter ne le sait pas ? Est-ce que les paroles qu'on a dites ne sont pas rapport\u00e9es \u00e0 C\u00e9sar ? Que fait-il donc ? Il sait que, s'il punissait tous ceux qui l'injurient, il n'aurait plus sur qui r\u00e9gner.\n\nQue conclure de l\u00e0 ? Que tu devais te dire, en arrivant au th\u00e9\u00e2tre, non pas : \u00ab Il faut que Sophron soit couronn\u00e9 \u00bb ; mais : \u00ab J'aurai soin dans cette occasion que ma volont\u00e9 soit conforme \u00e0 la nature. Personne ne m'est plus cher que moi-m\u00eame. Il serait donc ridicule de me nuire \u00e0 moi-m\u00eame, pour faire triompher l'un des com\u00e9diens. Quel est donc celui que je veux voir vainqueur ? Celui qui le sera. De cette fa\u00e7on celui qui vaincra sera toujours celui que j'aurai voulu. \u00bb \u00ab Mais je veux, dis-tu, que la couronne soit \u00e0 Sophron ! \u00bb Fais c\u00e9l\u00e9brer alors dans ta maison tous les jeux que tu voudras, et proclame le vainqueur aux jeux N\u00e9m\u00e9ens, aux Pythiens, aux Isthmiques et aux Olympiques. Mais en public pas d'empi\u00e9tements : ne t'arroge pas ce qui appartient \u00e0 tous. Sinon, supporte les injures ; car, lorsque tu agis comme la multitude, tu te mets toi-m\u00eame \u00e0 son niveau.\n\nV. CONTRE CEUX QUI PARTENT PARCE QU'ILS SONT MALADES\n\n\u2014 Je suis malade ici, dit quelqu'un ; je veux m'en retourner chez moi. \u2014 Est-ce que chez toi tu ne seras plus malade ? Ne veux-tu pas te demander si tu ne fais pas ici quelque chose qui serve \u00e0 l'am\u00e9lioration de ta facult\u00e9 de juger et de vouloir ? Car, si tu ne fais pas de progr\u00e8s, c'est inutilement, en effet, que tu es venu. Va-t'en, et occupe-toi de ta maison. Car, si ta partie ma\u00eetresse ne peut \u00eatre conforme \u00e0 la nature, ton champ du moins le pourra ; tu augmenteras tes \u00e9cus ; tu soigneras ton vieux p\u00e8re ; tu vivras sur la place publique ; tu seras magistrat ; et, corrompu, tu feras en homme corrompu quelqu'une des choses qui sont la cons\u00e9quence de ce titre. Mais si tu avais la conscience de t'\u00eatre d\u00e9livr\u00e9 de quelques opinions mauvaises, et de les avoir remplac\u00e9es par d'autres ; si tu avais fait passer ton \u00e2me de l'amour des choses qui ne rel\u00e8vent pas de ton libre arbitre \u00e0 l'amour de celles qui en rel\u00e8vent ; si, quand tu dis : \u00ab H\u00e9las ! \u00bb tu ne le disais ni \u00e0 cause de ton p\u00e8re, ni \u00e0 cause de ton fr\u00e8re, mais \u00e0 cause de ton moi, est-ce que alors tu te pr\u00e9occuperais encore de la maladie ? Ne sais-tu pas, en effet, qu'il faut que la maladie et la mort viennent nous saisir au milieu de quelque occupation ? Elles saisissent le laboureur \u00e0 son labour et le marin sur son navire. Que veux-tu \u00eatre en train de faire quand elles te prendront ? Car il faut qu'elles te prennent en train de faire quelque chose. Si tu sais quelque chose de meilleur que ceci \u00e0 faire au moment o\u00f9 elles te prendront, fais-le.\n\nPour moi, puisse-t-il m'arriver d'\u00eatre pris par elles ne m'occupant d'autre chose que de ma facult\u00e9 de juger et de vouloir, pour que, soustraite aux troubles, aux entraves, \u00e0 la contrainte, elle soit pleinement libre ! Voil\u00e0 les occupations o\u00f9 je veux qu'elles me trouvent, afin de pouvoir dire \u00e0 Dieu : \u00ab Est-ce que j'ai transgress\u00e9 tes ordres ? Est-ce que j'ai mal us\u00e9 des facult\u00e9s que tu m'avais donn\u00e9es ? Mal us\u00e9 de mes sens ? De mes notions a priori ? T'ai-je jamais rien reproch\u00e9 ? Ai-je jamais bl\u00e2m\u00e9 ton gouvernement ? J'ai \u00e9t\u00e9 malade, parce que tu l'as voulu. Les autres aussi le sont, mais moi je l'ai \u00e9t\u00e9 sans m\u00e9contentement. J'ai \u00e9t\u00e9 pauvre, parce que tu l'as voulu, mais je l'ai \u00e9t\u00e9, content de l'\u00eatre. Je n'ai pas \u00e9t\u00e9 magistrat, parce que tu ne l'as pas voulu ; mais aussi je n'ai jamais d\u00e9sir\u00e9 de magistrature. M'en as-tu jamais vu plus triste ? Ne me suis-je pas toujours pr\u00e9sent\u00e9 \u00e0 toi le visage radieux, n'attendant qu'un ordre, qu'un signe de toi ? Tu veux que je parte aujourd'hui de ce grand spectacle du monde ; je vais en partir. Je te rends gr\u00e2ce, sans r\u00e9serve, de m'y avoir admis avec toi, de m'avoir donn\u00e9 d'y contempler tes \u0153uvres et d'y comprendre ton gouvernement. \u00bb Que ce soit l\u00e0 ce que je pense, \u00e9crive ou lise, au moment o\u00f9 me prendra la mort !\n\n\u2014 Mais, dans ma maladie, ma m\u00e8re ne me tiendra pas la t\u00eate ! \u2014 Va-t'en donc pr\u00e8s de ta m\u00e8re, car tu m\u00e9rites bien qu'on te tienne la t\u00eate, quand tu es malade. \u2014 Mais chez moi j'\u00e9tais couch\u00e9 dans un lit \u00e9l\u00e9gant ! \u2014 Va donc trouver ton lit ; tu m\u00e9rites de t'y coucher en bonne sant\u00e9. Ne te prive pas de ce que tu peux te procurer l\u00e0-bas.\n\nEt que dit Socrate ? \u00ab Comme un autre, dit-il, est heureux des progr\u00e8s qu'il fait faire \u00e0 son champ, et tel autre \u00e0 son cheval, ainsi moi je suis heureux chaque jour quand je sens les progr\u00e8s que je fais. \u00bb \u2014 En quoi donc \u00e9taient ces progr\u00e8s ? Dans l'art des jolies phrases ? \u2014 Tais-toi, mon cher ! \u2014 Dans l'\u00e9tude de la Logique ? \u2014 Que dis-tu l\u00e0 ? \u2014 Je ne vois pourtant pas autre chose dont s'occupent les philosophes. \u2014 N'est-ce donc rien \u00e0 tes yeux que de n'adresser jamais de reproches \u00e0 personne, ni \u00e0 la divinit\u00e9, ni \u00e0 l'homme ? Que de ne bl\u00e2mer personne ? Que d'avoir toujours le m\u00eame visage, en sortant comme en rentrant ? C'\u00e9tait l\u00e0 ce que savait faire Socrate ; et jamais cependant il ne se vanta de savoir ou d'enseigner quelque chose. Si quelqu'un lui demandait l'art des jolies phrases ou la science de la Logique, il le conduisait \u00e0 Protagoras ou \u00e0 Hippias, comme il aurait conduit \u00e0 un jardinier celui qui serait venu lui demander des l\u00e9gumes.\n\nOr, quel est celui de vous qui a de pareils principes ? Si vous les aviez, vous seriez heureux d'\u00eatre malades, d'\u00eatre pauvres, et m\u00eame de mourir. S'il est quelqu'un de vous qui soit amoureux d'une jolie fille, il sait que je dis vrai.\n\nX. COMMENT DOIT-ON SUPPORTER LES MALADIES ?\n\nQuand vient le moment d'appliquer quelques-uns de nos principes, il faut toujours les avoir l\u00e0 pr\u00e9sents : \u00e0 table, ceux qui sont pour la table ; aux bains, ceux qui sont pour le bain ; au lit, ceux qui sont pour le lit.\n\nQue tes yeux trop faibles ne donnent jamais entr\u00e9e au sommeil, avant que tu n'aies pass\u00e9 en revue toutes tes actions de la journ\u00e9e. Quelle loi ai-je viol\u00e9e ? Quel acte ai-je fait ? \u00c0 quel devoir ai-je failli ? Pars de l\u00e0 et continue. Puis, si tu as fait du mal, reproche-le-toi ; si tu as fait du bien, sois-en content.\n\nVoil\u00e0 des vers qu'il faut retenir pour les mettre en pratique, et non pas pour les d\u00e9biter \u00e0 haute voix, comme on d\u00e9bite le P\u00e9an \u00e0 Apollon !\n\nDans la fi\u00e8vre \u00e0 son tour, ayons pr\u00e9sents les principes qui sont faits pour elle, bien loin de les laisser de c\u00f4t\u00e9 tous en masse et de les oublier, parce que nous avons la fi\u00e8vre. \u00ab M'arrive ce qui voudra, t'\u00e9cries-tu, si je m'occupe encore de philosophie ! Je m'en irai quelque part, o\u00f9 je ne songerai qu'aux soins de mon corps, et o\u00f9 la fi\u00e8vre ne me viendra plus ! \u00bb Mais qu'est-ce que s'occuper de philosophie ? N'est-ce pas se pr\u00e9parer contre les \u00e9v\u00e9nements ? Ne comprends-tu pas alors que tes paroles reviennent \u00e0 dire : \u00ab M'arrive ce qui voudra, si je me pr\u00e9pare encore \u00e0 supporter avec calme les \u00e9v\u00e9nements ! \u00bb C'est comme si quelqu'un renon\u00e7ait au jeu du pancrace, parce qu'il y aurait re\u00e7u des coups. Encore est-il tout loisible dans ce cas de cesser la lutte et de ne plus \u00eatre battu ; tandis que nous, si nous cessons de nous occuper de philosophie, qu'est-ce que nous y gagnerons ? Que doit donc dire le philosophe, \u00e0 chaque chose p\u00e9nible qui lui arrive ? Voil\u00e0 ce \u00e0 quoi je me suis pr\u00e9par\u00e9, ce en vue de quoi je me suis exerc\u00e9. Dieu te dit : \u00ab Donne-moi une preuve que tu t'es pr\u00e9par\u00e9 \u00e0 la lutte suivant toutes les r\u00e8gles, que tu t'es nourri comme on doit le faire, que tu as fr\u00e9quent\u00e9 le gymnase, que tu as \u00e9cout\u00e9 les le\u00e7ons du ma\u00eetre. \u00bb Vas-tu maintenant mollir \u00e0 l'instant d\u00e9cisif ? Voici le moment d'avoir la fi\u00e8vre ; qu'elle vienne, et sois convenable. Voici le moment d'avoir soif ; aie soif, et sois convenable. Voici le moment d'avoir faim ; aie faim, et sois convenable. Cela ne d\u00e9pend-il pas de toi ? Quelqu'un peut-il t'en emp\u00eacher ? Le m\u00e9decin t'emp\u00eachera de boire ; mais il ne peut t'emp\u00eacher d'\u00eatre convenable en ayant soif. Il t'emp\u00eachera de manger, mais il ne peut t'emp\u00eacher d'\u00eatre convenable en ayant faim.\n\n\u2014 Mais, en cet \u00e9tat, je ne puis pas \u00e9tudier ! \u2014 \u00c0 quelle fin \u00e9tudies-tu donc, esclave ? N'est-ce pas pour arriver au calme ? \u00e0 la tranquillit\u00e9 ? N'est-ce pas pour te mettre et te maintenir en conformit\u00e9 avec la nature ? Or, quand tu as la fi\u00e8vre, qui t'emp\u00eache de mettre cet accord entre la nature et ta partie ma\u00eetresse ? C'est ici le moment de faire tes preuves ; c'est ici l'\u00e9preuve du philosophe ; car la fi\u00e8vre fait partie de la vie, comme la promenade, les travers\u00e9es, les voyages par terre. Est-ce que tu lis en te promenant ? \u2014 Non. \u2014 Eh bien, c'est la m\u00eame chose quand tu as la fi\u00e8vre. Si tu restes convenable, en te promenant, tu es ce que doit \u00eatre un promeneur ; si tu es convenable, en ayant la fi\u00e8vre, tu es ce que doit \u00eatre un fi\u00e9vreux. Qu'est-ce donc qu'\u00eatre convenable en ayant la fi\u00e8vre ? C'est de ne t'en prendre ni \u00e0 Dieu ni aux hommes ; c'est de ne pas te d\u00e9soler de ce qui arrive ; c'est d'attendre dignement et convenablement la mort ; c'est de faire tout ce que l'on t'ordonne ; c'est de ne pas t'effrayer de ce que va dire le m\u00e9decin, quand il arrive, et de ne pas te r\u00e9jouir outre mesure, quand il te dit : \u00ab Tu te portes bien. \u00bb Qu'est-ce l\u00e0, en effet, te dire de bon ? Car, lorsque tu te portais bien, qu'y avait-il l\u00e0 de bon pour toi ? C'est encore de ne pas te d\u00e9sesp\u00e9rer, quand il te dit : \u00ab Tu te portes mal. \u00bb Qu'est-ce, en effet, que se mal porter ? Approcher du moment o\u00f9 l'\u00e2me se s\u00e9pare du corps. Qu'y a-t-il donc l\u00e0 de terrible ? Est-ce que, si tu n'en approches pas maintenant, tu n'en approcheras pas plus tard ? Est-ce encore que le monde doit \u00eatre boulevers\u00e9 par ta mort ? Pourquoi donc flattes-tu le m\u00e9decin ? Pourquoi lui dis-tu : \u00ab Si tu le veux, ma\u00eetre, je serai en bonne sant\u00e9 \u00bb ? Pourquoi lui donner un motif de porter haut la t\u00eate ? Pourquoi ne pas l'estimer juste ce qu'il vaut ? Le cordonnier est pour mon pied, le charpentier pour ma maison, et le m\u00e9decin, \u00e0 son tour, pour mon mis\u00e9rable corps, c'est-\u00e0-dire pour quelque chose qui n'est pas \u00e0 moi, pour un \u00eatre mort-n\u00e9. Voil\u00e0 ce qu'a \u00e0 faire le fi\u00e9vreux ; et, s'il le fait il est ce qu'il doit \u00eatre. La t\u00e2che du philosophe, en effet, n'est pas de sauvegarder les choses du dehors, son vin, son huile, son corps, mais de sauvegarder sa partie ma\u00eetresse. Comment se conduira-t-il vis-\u00e0-vis les choses du dehors ? Il s'en occupera dans la mesure que la raison comporte. Et alors quand aura-t-il encore \u00e0 s'effrayer ? Quand aura-t-il encore \u00e0 se mettre en col\u00e8re ? Quand aura-t-il encore \u00e0 trembler pour des choses qui ne sont pas \u00e0 lui, et qui ne m\u00e9ritent pas qu'il en fasse cas ? Voici, en effet, les deux pens\u00e9es qu'il faut avoir toujours pr\u00e9sentes : c'est qu'en dehors de notre libre arbitre, il n'y a rien de bon ni de mauvais, et qu'il ne faut pas vouloir conduire les \u00e9v\u00e9nements, mais les suivre. Mon fr\u00e8re ne devait pas se conduire ainsi avec moi. Oui ; mais c'est \u00e0 lui d'y voir ; et quant \u00e0 moi, de quelque fa\u00e7on qu'il se soit conduit, j'agirai envers lui comme je le dois. Car voil\u00e0 ce qui me regarde, tandis que l'autre chose ne me regarde pas ; voil\u00e0 ce que nul ne peut emp\u00eacher, tandis qu'on peut emp\u00eacher l'autre chose.\n\nXII. DE L'EXERCICE\n\nIl ne faut nous exercer \u00e0 rien qui soit extraordinaire et contre nature ; autrement, nous qui nous disons philosophes, nous ne diff\u00e9rerons pas des faiseurs de tours. Il est difficile, en effet, de danser sur la corde ; et non seulement cela est difficile, mais cela est encore dangereux. Est-ce une raison cependant pour que nous aussi nous apprenions \u00e0 danser sur la corde, \u00e0 y \u00e9lever en l'air une branche de palmier, \u00e0 y tenir embrass\u00e9es des statues ? Pas le moins du monde. Tout ce qui est difficile et p\u00e9rilleux n'est pas un bon objet d'exercice ; il n'y a de tel que ce qui nous conduit au but qui est propos\u00e9 \u00e0 nos efforts. Quel est donc le but propos\u00e9 \u00e0 nos efforts ? De n'\u00eatre jamais entrav\u00e9 dans ce que l'on d\u00e9sire ou cherche \u00e0 \u00e9viter. Et qu'est-ce que n'y \u00eatre pas entrav\u00e9 ? C'est ne jamais manquer ce qu'on d\u00e9sire, ne jamais tomber dans ce qu'on veut \u00e9viter. C'est l\u00e0 le seul but en vue duquel nous devions nous exercer. Car, sache-le, comme ce n'est que par un exercice s\u00e9rieux et soutenu qu'on peut arriver \u00e0 ne jamais manquer ce qu'on d\u00e9sire, \u00e0 ne jamais tomber dans ce qu'on veut \u00e9viter, tu ne saurais, si tu te laisses aller \u00e0 t'exercer \u00e0 des choses ext\u00e9rieures qui ne rel\u00e8vent pas de ton libre arbitre, arriver \u00e0 ne jamais manquer ce que tu d\u00e9sires, \u00e0 ne jamais tomber dans ce que tu veux \u00e9viter. Et, comme la force de l'habitude est souveraine, et que ce n'est qu'aux choses du dehors que nous sommes habitu\u00e9s \u00e0 appliquer notre puissance de d\u00e9sirer ou de fuir, il nous faut donc opposer \u00e0 cette habitude une habitude contraire, opposer l'exercice le plus soutenu l\u00e0 o\u00f9 la s\u00e9duction des apparences sensibles est la plus grande.\n\nJe penche vers la volupt\u00e9 : je vais me jeter du c\u00f4t\u00e9 contraire, et cela avec exc\u00e8s, afin de m'exercer. J'ai le travail en aversion : je vais habituer et accoutumer ma pens\u00e9e \u00e0 n'avoir plus jamais d'aversion pour lui et ce qui lui ressemble. Qu'est-ce, en effet, que s'exercer ? C'est s'appliquer \u00e0 ne jamais rien d\u00e9sirer, et \u00e0 n'avoir d'aversion que pour des choses qui rel\u00e8vent de notre libre arbitre, et s'y appliquer de pr\u00e9f\u00e9rence l\u00e0 o\u00f9 il nous est le plus difficile de r\u00e9ussir. D'o\u00f9 il r\u00e9sulte que les choses contre lesquelles on doit s'exercer le plus varient avec chacun. Or, \u00e0 quoi bon pour cela \u00e9lever en l'air une branche de palmier, et promener partout une tente de cuir, un mortier et un pilon ? Homme, si tu es prompt \u00e0 la col\u00e8re, exerce-toi \u00e0 supporter les injures, et \u00e0 ne pas t'irriter des outrages. Et tes progr\u00e8s iront si loin ainsi, que tu te diras, si quelqu'un te frappe : \u00ab Suppose que tu as voulu embrasser une statue. \u00bb Puis exerce-toi \u00e0 bien te comporter en face du vin, ce qui n'est pas t'exercer \u00e0 en boire beaucoup (comme plus d'un le fait malheureusement), mais, avant tout, \u00e0 t'en abstenir ; exerce-toi apr\u00e8s cela \u00e0 te passer de femme et de friandises. Ensuite, pour t'\u00e9prouver, si une heureuse occasion se pr\u00e9sente, va de toi-m\u00eame au p\u00e9ril, afin de savoir si les sens triompheront de toi comme auparavant. Mais, au d\u00e9but, fuis loin des tentations trop fortes. Le combat n'est pas \u00e9gal entre une jolie fille et un jeune apprenti philosophe : \u00ab Cruche et pierre, dit-on, ne peuvent aller ensemble. \u00bb\n\nApr\u00e8s le d\u00e9sir et l'aversion, la seconde chose qu'il nous faut travailler c'est notre fa\u00e7on de vouloir les choses ou de les repousser. Il faut que ces volont\u00e9s soient conformes \u00e0 la raison, qu'elles ne soient \u00e0 contresens ni du moment ni du lieu, qu'elles ne violent enfin aucune convenance de ce genre.\n\nLa troisi\u00e8me chose \u00e0 travailler est l'assentiment que nous donnons \u00e0 ce qui persuade et entra\u00eene. Socrate disait que l'on ne pouvait vivre sans examiner ; de m\u00eame, on ne doit accepter aucune apparence sans l'examiner. On doit lui dire : \u00ab Attends ; laisse-moi voir qui tu es, d'o\u00f9 tu viens \u00bb ; comme les gardes de nuit disent : \u00ab Montre-moi le signe convenu. \u00bb As-tu re\u00e7u de la nature le signe que doit avoir toute id\u00e9e pour se faire accepter ?\n\nEn dernier lieu, il faut nous exercer aussi \u00e0 tout ce que les ma\u00eetres de gymnastique prescrivent au corps, pourvu que cela tende \u00e0 nous exercer au sujet du d\u00e9sir et de l'aversion. Mais, si ce qu'ils prescrivent ne tend qu'\u00e0 la montre, c'est l'affaire de l'homme qui se penche au-dehors pour chercher autre chose, et appeler des spectateurs auxquels il entendra dire : \u00ab Quel grand homme ! \u00bb Aussi Apollonius disait-il avec raison : \u00ab Veux-tu t'exercer ? Quand il fait chaud et que tu as soif, mets dans ta bouche une gorg\u00e9e d'eau fra\u00eeche, puis rejette-la, et ne le conte \u00e0 personne. \u00bb\n\nXIII. QU'EST-CE QUE C'EST QUE L'ABANDON ? ET QU'EST-CE QUI EST ABANDONN\u00c9 ?\n\n\u00catre abandonn\u00e9, c'est se trouver sans appui. Un homme qui est seul n'est pas dans l'abandon pour cela ; par contre, on peut \u00eatre au milieu de beaucoup d'autres, et n'en \u00eatre pas moins abandonn\u00e9. C'est pour cela que, quand nous perdons un fr\u00e8re, un fils, un ami qui \u00e9tait notre appui, nous disons que nous restons abandonn\u00e9s, bien que souvent nous soyons \u00e0 Rome, en face d'une si grande foule, au milieu de tant d'autres habitants, et parfois m\u00eame que nous ayons \u00e0 nous un si grand nombre d'esclaves. Car celui-l\u00e0 se dit abandonn\u00e9, qui, dans sa pens\u00e9e, se trouve priv\u00e9 d'appui, \u00e0 la merci de qui veut lui nuire. C'est pour cela qu'en voyage nous ne nous disons jamais plus abandonn\u00e9s qu'au moment o\u00f9 nous tombons dans une troupe de voleurs ; car ce n'est pas la pr\u00e9sence d'un homme qui nous sauve de l'abandon, mais la pr\u00e9sence d'un homme s\u00fbr, honn\u00eate, et pr\u00eat \u00e0 nous venir en aide. Si la solitude suffisait \u00e0 faire l'abandon, il faudrait dire que Jupiter est dans l'abandon lors de l'embrasement du monde, et qu'il y g\u00e9mit ainsi sur lui-m\u00eame : \u00ab Malheureux que je suis ! je n'ai plus avec moi Junon, ni Minerve, ni Apollon ; je n'ai plus, enfin, ni fr\u00e8res, ni fils, ni petit-fils, ni parent d'aucune sorte. \u00bb C'est pourtant l\u00e0 ce que quelques-uns disent qu'il fait, quand il est seul lors de cet embrasement. C'est qu'ils ne comprennent pas comment on peut vivre seul ; et il faut avouer qu'ils partent d'un principe naturel, car la nature nous a faits pour vivre en soci\u00e9t\u00e9, pour nous aimer les uns les autres, pour \u00eatre heureux de nous trouver avec des hommes. Mais cependant il faut que chacun ait en lui les moyens de pouvoir se suffire, et de pouvoir vivre seul ; de m\u00eame que Jupiter vit seul, jouissant tranquillement de lui-m\u00eame, songeant \u00e0 la fa\u00e7on dont il gouverne, et tout entier aux pens\u00e9es qui conviennent \u00e0 sa divinit\u00e9. Il faut que nous aussi, \u00e0 son exemple, nous puissions converser avec nous-m\u00eames ; nous passer des autres ; n'avoir besoin d'aucune distraction ; r\u00e9fl\u00e9chir au gouvernement divin et \u00e0 nos rapports avec le reste du monde ; songer \u00e0 la conduite que nous avons tenue en face des \u00e9v\u00e9nements, et \u00e0 celle que nous tenons aujourd'hui ; chercher quelles sont les choses qui nous g\u00eanent encore, comment on peut y porter rem\u00e8de, comment on peut les faire dispara\u00eetre ; et, si quelque c\u00f4t\u00e9 en nous a besoin d'un perfectionnement, le lui donner conform\u00e9ment \u00e0 la raison.\n\nVoyez quelle large paix C\u00e9sar semble nous avoir faite : plus de guerres, plus de combats, plus de grandes troupes de voleurs, plus de pirates. On peut se mettre en route \u00e0 toute heure ; on peut naviguer de l'orient \u00e0 l'occident. Mais C\u00e9sar a-t-il pu nous garantir \u00e9galement de la fi\u00e8vre ? des naufrages ? des incendies ? des tremblements de terre ? de la foudre ? Allons plus loin : de l'amour ? Il ne le peut. De la douleur ? Il ne le peut. De la jalousie ? Il ne le peut. Il ne peut rien contre aucune de ces choses. Or, la philosophie s'engage \u00e0 nous garantir de celles-l\u00e0 aussi. Et que nous dit-elle \u00e0 cet effet ? \u00ab \u00d4 hommes, si vous vous attachez \u00e0 moi, en quelque lieu que vous soyez, et quel que soit votre sort, il n'y aura pour vous ni douleur, ni col\u00e8re, ni contrainte, ni entraves ; vous serez affranchis de tout, vous serez libres partout. \u00bb Celui qui jouit de cette paix, que C\u00e9sar n'a pas promulgu\u00e9e, car comment le pourrait-il faire, mais qu'a promulgu\u00e9e Dieu lui-m\u00eame avec l'aide de la raison, a-t-il donc besoin d'autre chose, quand il est seul ? Il n'a qu'\u00e0 ouvrir les yeux et qu'\u00e0 se dire : \u2014 Maintenant rien de mauvais ne peut m'arriver ; il n'y a pour moi ni voleurs, ni tremblement de terre ; partout la paix et la tranquillit\u00e9. Il n'est pas une route, pas une ville, pas un compagnon de voyage, pas un voisin, pas un associ\u00e9 qui puisse m'\u00eatre fatal. Il est quelqu'un qui prend soin de me fournir ma nourriture et mes v\u00eatements ; il est quelqu'un qui m'a donn\u00e9 mes sens et mes notions a priori. Lorsqu'il ne me fournit pas ce qui m'est n\u00e9cessaire, c'est qu'il me sonne la retraite, qu'il ouvre la porte, et qu'il me dit : \u00ab Viens. \u00bb \u2014 O\u00f9 cela ? \u2014 Vers rien qui soit \u00e0 craindre ; vers ce dont tu es sorti ; vers des amis, vers des parents, vers les \u00e9l\u00e9ments. Tout ce qu'il y avait de feu en toi s'en ira vers le feu ; tout ce qu'il y avait de terre, vers la terre ; tout ce qu'il y avait d'air, vers l'air ; tout ce qu'il y avait d'eau, vers l'eau. Il n'y a pas de Pluton, pas d'Ach\u00e9ron, pas de Cocyte, pas de Phl\u00e9g\u00e9ton en feu ; non : tout est peupl\u00e9 de dieux et de G\u00e9nies. Quand on peut se dire tout cela, quand on a devant ses yeux le soleil, la lune et les astres, quand on a la jouissance de la terre et de la mer, on n'est pas plus abandonn\u00e9 que l'on n'est sans appui. \u2014 Mais quoi ! si quelqu'un me surprenait seul et me tuait ! \u2014 Imb\u00e9cile ! ce ne serait pas toi qu'il tuerait, ce serait ton corps !\n\nQu'est-ce donc que l'abandon ? Qu'est-ce donc que le d\u00e9nuement ? Pourquoi nous faire inf\u00e9rieurs aux enfants ? Quand on les laisse seuls, que font-ils ? Ils prennent des coquilles et de la terre, et font des maisons, qu'ils renversent ensuite pour en faire d'autres. De cette fa\u00e7on les moyens de passer le temps ne leur manquent jamais. Vais-je donc, moi, si vous faites voile au loin, m'asseoir en pleurant, parce que vous m'aurez laiss\u00e9 seul et dans l'abandon ? Est-ce que je n'ai pas comme eux des coquillages ? Est-ce que je n'ai pas de la terre ? Et, quand ils agissent ainsi faute d'avoir la raison, nous qui avons la raison serons-nous malheureux par elle ?\n\nToute grande puissance est un p\u00e9ril au d\u00e9but. Il faut en porter le poids suivant ses forces, mais d'une mani\u00e8re conforme \u00e0 la nature... mais non pas pour le poitrinaire. \u00c9tudie-toi parfois \u00e0 te conduire comme si tu \u00e9tais malade, pour savoir un jour te conduire comme un homme bien portant. Je\u00fbne, bois de l'eau, interdis-toi toute esp\u00e8ce de d\u00e9sir, pour savoir un jour d\u00e9sirer conform\u00e9ment \u00e0 la raison. Et, quand tu d\u00e9sireras conform\u00e9ment \u00e0 la raison, quand le bien sera ainsi en toi, tes d\u00e9sirs seront bons. Mais ce n'est pas l\u00e0 ce que nous faisons : d\u00e8s le premier jour nous pr\u00e9tendons vivre comme des sages et servir l'humanit\u00e9. Eh ! Comment la sers-tu ? Que fais-tu ? Quels services, en effet, as-tu commenc\u00e9 par te rendre \u00e0 toi-m\u00eame ? Tu veux les exhorter au bien ! Mais t'y es-tu exhort\u00e9 toi-m\u00eame ? Tu veux leur \u00eatre utile ! Montre-leur par ton propre exemple quels hommes la philosophie sait faire, et ne bavarde pas inutilement. Par ta fa\u00e7on de manger, sois utile \u00e0 ceux qui mangent avec toi ; par ta fa\u00e7on de boire, \u00e0 ceux qui y boivent : c\u00e8de-leur ; fais abn\u00e9gation de toi-m\u00eame ; supporte tout d'eux ; sois-leur utile ainsi, et ne crache pas sur eux.\n\nXVI. QU'IL FAUT Y REGARDER \u00c0 DEUX FOIS AVANT DE SE LAISSER ENTRA\u00ceNER \u00c0 UNE LIAISON\n\nDe deux choses l'une : ou celui qui se laisse entra\u00eener souvent \u00e0 causer, \u00e0 d\u00eener, et g\u00e9n\u00e9ralement \u00e0 vivre avec d'autres, leur deviendra semblable ; ou il les convertira \u00e0 ses m\u0153urs. Placez, en effet, un charbon \u00e9teint aupr\u00e8s d'un charbon allum\u00e9, le premier \u00e9teindra le second, ou le second allumera le premier. En face d'un semblable p\u00e9ril, il faut y regarder \u00e0 deux fois avant de se laisser entra\u00eener \u00e0 de pareilles liaisons avec les hommes ordinaires ; il faut se rappeler qu'on ne saurait se frotter \u00e0 un individu barbouill\u00e9 de suie, sans attraper soi-m\u00eame de la suie. Que feras-tu, en effet, s'il te parle de gladiateurs, de chevaux, d'athl\u00e8tes, ou, ce qui est encore pis, s'il te parle des hommes ; s'il te dit : \u00ab Un tel est un m\u00e9chant homme ; un tel est honn\u00eate ; ceci a \u00e9t\u00e9 bien fait ; cela l'a \u00e9t\u00e9 mal \u00bb ? Et si c'est un moqueur, un persifleur, une mauvaise langue ? Avez-vous donc les ressources du musicien, qui, d\u00e8s qu'il a pris sa lyre, et qu'il en a touch\u00e9 les cordes, reconna\u00eet celles qui ne sont pas justes, et accorde son instrument ? Avez-vous donc le talent de Socrate, qui, dans toute liaison, savait amener \u00e0 ses sentiments celui avec qui il vivait ? Et d'o\u00f9 vous viendrait ce talent ? Forc\u00e9ment, ce serait vous qui seriez entra\u00een\u00e9s par les hommes ordinaires.\n\nEt pourquoi sont-ils plus forts que vous ? Parce que toutes ces sottises, c'est avec conviction qu'ils les disent ; tandis que vous, toutes ces belles choses, c'est des l\u00e8vres seulement que vous les dites. Aussi sont-elles dans votre bouche sans force et sans vie ; aussi prend-on en d\u00e9go\u00fbt les exhortations qu'on vous entend faire, et la mis\u00e9rable vertu que vous vantez \u00e0 tort et \u00e0 travers. C'est l\u00e0 ce qui fait que les hommes ordinaires vous battent. Car partout la conviction est forte, partout la conviction est invincible. Jusqu'au moment donc o\u00f9 tous ces beaux principes seront profond\u00e9ment grav\u00e9s en vous, et o\u00f9 vous serez devenus assez forts pour n'avoir rien \u00e0 craindre, je vous conseille d'y regarder \u00e0 deux fois avant de descendre au milieu des hommes ordinaires ; autrement, tout ce que dans l'\u00e9cole vous aurez \u00e9crit en vous, s'y fondra jour \u00e0 jour comme la cire au soleil. Tenez-vous donc bien loin du soleil, tant que vos principes seront de cire. C'est pour cela encore que les philosophes nous conseillent de quitter notre patrie, parce que les vieilles habitudes nous entra\u00eenent, et ne nous permettent pas de prendre d'autres plis ; parce que aussi nous ne savons pas r\u00e9sister \u00e0 ceux qui disent, en nous rencontrant : \u00ab Regarde donc ! Un tel est philosophe, lui qui \u00e9tait ceci et cela. \u00bb C'est ainsi encore que les m\u00e9decins envoient dans un autre pays, et sous un autre ciel, ceux qui sont malades depuis longtemps ; et ils ont raison ! Vous aussi, inoculez-vous d'autres m\u0153urs, gravez profond\u00e9ment en vous les principes, exercez-vous \u00e0 les appliquer. Ce n'est pas l\u00e0 ce que vous faites : vous allez d'ici au spectacle, aux combats de gladiateurs, aux galeries des athl\u00e8tes, au cirque ; puis de l\u00e0 ici, et d'ici l\u00e0, toujours de m\u00eame. Point de noble habitude en vous, point d'application, point de s\u00e9v\u00e9rit\u00e9 pour vous-m\u00eames, point d'attention \u00e0 vous dire : \u00ab Quel usage fais-je des objets qui se pr\u00e9sentent \u00e0 mes sens ? Est-il conforme \u00e0 la nature ou lui est-il contraire ? Comment suis-je vis-\u00e0-vis d'eux ? Comme je dois \u00eatre ou comme je ne dois pas \u00eatre ? Dis-je bien aux choses qui ne rel\u00e8vent pas de mon libre arbitre, que je n'ai rien \u00e0 faire d'elles ? \u00bb \u2014 Tant que ce n'est pas encore l\u00e0 ce que vous \u00eates, fuyez vos anciennes habitudes, fuyez les hommes ordinaires, si vous voulez jamais commencer \u00e0 \u00eatre quelque chose.\n\nXXIII. CONTRE CEUX QUI LISENT OU DISCUTENT PAR D\u00c9SIR DE SE MONTRER\n\nCommence par te demander ce que tu veux \u00eatre ; puis, apr\u00e8s cela, fais ce que veut ton m\u00e9tier. Car, dans les autres parties, c'est presque toujours ainsi que nous voyons les choses se passer. Ceux qui se destinent \u00e0 l'ar\u00e8ne commencent par d\u00e9cider ce qu'ils veulent \u00eatre, puis, apr\u00e8s cela, ils agissent en cons\u00e9quence. Si tu veux fournir la grande course, voici ta nourriture, voici tes promenades, voici tes frictions, voici tes exercices ; si tu ne veux courir que le stade, tout cela changera ; si tu veux \u00eatre pentathle, tout changera encore. Tu trouveras la m\u00eame chose dans les arts. Si tu veux \u00eatre charpentier, voici ce que tu auras \u00e0 faire ; si tu veux \u00eatre fondeur, voil\u00e0. Car, si nous ne rapportons pas chacune de nos actions \u00e0 un but, nous agissons au hasard ; et, si nous la rapportons \u00e0 un autre but que celui qu'il faudrait, nous nous \u00e9garons.\n\nReste \u00e0 d\u00e9terminer le but g\u00e9n\u00e9ral et les buts particuliers. Le premier, c'est d'agir comme un homme. Qu'est-ce que cela implique ? De ne pas agir comme un mouton, tout en \u00e9tant bon, ni comme un m\u00e9chant, \u00e0 la fa\u00e7on des b\u00eates fauves. Quant aux buts particuliers, ils varient avec la profession de chacun, et avec la vie qu'il a choisie.\n\nQue le musicien agisse comme un musicien ; le charpentier, comme un charpentier ; le philosophe, comme un philosophe ; l'orateur, comme un orateur. Lors donc que tu nous dis : \u00ab Venez ici, et entendez-moi vous faire une lecture \u00bb, prends garde d'abord d'agir ainsi sans but ; puis, si tu trouves un but \u00e0 ton acte, prends garde qu'il ne soit pas celui qu'il faut. Cherches-tu \u00e0 \u00eatre utile ? Ou ne cherches-tu que des \u00e9loges ?\n\nD\u00e8s que l'on parle ainsi, on entend le personnage nous dire : \u00ab Que m'importent les \u00e9loges de la multitude ! \u00bb Et il a raison. Car ces \u00e9loges ne sont rien non plus pour le musicien, en tant que musicien ; pour le g\u00e9om\u00e8tre, en tant que g\u00e9om\u00e8tre. Tu veux donc \u00eatre utile ! Mais \u00e0 quoi ? Dis-le-nous, pour que nous aussi nous courions t'entendre. Et maintenant ; quelqu'un peut-il faire profiter les autres, s'il n'a pas commenc\u00e9 par profiter lui-m\u00eame ? Non. Celui qui n'est pas charpentier ne peut nous aider \u00e0 devenir charpentier, ni celui qui n'est pas cordonnier, \u00e0 devenir cordonnier.\n\nVeux-tu donc savoir si tu as profit\u00e9 ? Philosophe, apporte-nous ici tes principes. Que se propose-t-on, quand on d\u00e9sire une chose ? \u2014 De ne pas la manquer. \u2014 Et quand on cherche \u00e0 l'\u00e9viter ? \u2014 De ne pas y tomber. \u2014 Eh bien ! nous, r\u00e9alisons-nous ce que nous nous proposons dans ces deux cas ? Dis-moi la v\u00e9rit\u00e9. Si tu me trompes, je te dirai : \u00ab Tel jour, parce qu'on avait \u00e9t\u00e9 moins empress\u00e9 \u00e0 venir t'entendre, moins empress\u00e9 \u00e0 t'acclamer, tu t'es retir\u00e9 tout honteux. Tel autre, parce que tu avais \u00e9t\u00e9 applaudi, tu te promenais par l'assembl\u00e9e, en disant \u00e0 chacun : \"Comment m'as-tu trouv\u00e9 ? \u2014 Admirable, ma\u00eetre, par mon salut ! \u2014 Et comme j'ai dit ce passage ! \u2014 Lequel ? \u2014 Celui o\u00f9 j'ai fait le portrait de Pan et des Nymphes. \u2014 Merveilleusement.\" \u00bb Et tu viendras me dire que tu ne d\u00e9sires et ne redoutes rien que conform\u00e9ment \u00e0 la nature ! Va-t'en le faire accroire \u00e0 un autre. L'autre jour n'as-tu pas lou\u00e9 tel individu contrairement \u00e0 ce que tu en pensais ? N'as-tu pas adul\u00e9 le fils de tel s\u00e9nateur ? Voudrais-tu donc que tes enfants lui ressemblassent ? \u2014 \u00c0 Dieu ne plaise ! \u2014 Pourquoi donc tant de flatteries et tant d'attentions pour lui ? \u2014 C'est un jeune homme merveilleusement dou\u00e9, et un auditeur tr\u00e8s intelligent. \u2014 Comment le sais-tu ? \u2014 Il m'admire. \u2014 Tu nous as dit ta vraie raison. Mais que te figures-tu donc ? Crois-tu que ces gens-l\u00e0 ne te m\u00e9prisent pas en secret ? Quand un homme qui a la conscience de n'avoir jamais rien dit ni rien pens\u00e9 de bon, trouve un philosophe qui lui dit : \u00ab Quelle nature d'\u00e9lite ! Quelle honn\u00eatet\u00e9 ! Quelle puret\u00e9 ! \u00bb que crois-tu qu'il se dise, si ce n'est : \u00ab Voici un homme qui a besoin de moi ? \u00bb Si je me trompe, dis-moi ce qu'il a fait qui soit l'\u0153uvre d'une nature d'\u00e9lite. Voici ce qu'il a fait : il a \u00e9t\u00e9 assidu pr\u00e8s de toi pendant un certain temps. Il t'a \u00e9cout\u00e9 parler ; il t'a \u00e9cout\u00e9 lire. Mais en est-il plus modeste ? A-t-il fait un retour sur lui-m\u00eame ? A-t-il le sentiment de sa mis\u00e8re ? S'est-il d\u00e9pouill\u00e9 de sa pr\u00e9somption ? Cherche-t-il un ma\u00eetre ? \u2014 Il en cherche un, dis-tu. \u2014 Pour lui enseigner comment il faut vivre ? Non pas, sot que tu es ; mais pour lui enseigner comment il faut parler ; car c'est pour ta fa\u00e7on de parler qu'il t'admire. \u00c9coute ce qu'il dit : \u00ab Voici un homme qui \u00e9crit avec la derni\u00e8re habilet\u00e9, beaucoup mieux que Dion. \u00bb C'est l\u00e0 tout. Dit-il : \u00ab Voici un homme plein de retenue et de probit\u00e9, un homme que rien ne trouble \u00bb ? S'il parlait ainsi, je lui dirais : \u00ab Puisque cet homme est si probe, qu'est-ce donc en lui que la probit\u00e9 ? \u00bb Et, s'il ne pouvait me le dire, j'ajouterais : \u00ab Commence par apprendre ce que tu dis ; et ne parle qu'apr\u00e8s. \u00bb Et c'est dans cette triste situation d'esprit, c'est quand tu t'extasies devant ceux qui t'applaudissent, c'est quand tu comptes tes auditeurs, que tu pr\u00e9tends \u00eatre utile aux autres ! \u2014 Aujourd'hui, dis-tu, j'ai eu beaucoup plus d'auditeurs ! \u2014 Oui, beaucoup. Cinq cents, ce me semble. \u2014 Vous ne savez ce que vous dites ! Mettez-en mille. \u2014 Jamais Dion n'en a eu autant. \u2014 Et comment les aurait-il ? Puis, comme ils \u00e9coutent ma parole ! C'est que le beau, monsieur, agit jusque sur les pierres elles-m\u00eames ! \u2014 Et c'est l\u00e0 le langage d'un philosophe ! Ce sont l\u00e0 les sentiments du futur bienfaiteur de l'humanit\u00e9 ! C'est l\u00e0 l'homme qui a \u00e9cout\u00e9 la raison, qui a lu les livres socratiques comme on lit des livres socratiques, et non pas comme on lit des livres de Lysias ou d'Isocrate ! Au lieu de lire : \u00ab Je me suis souvent demand\u00e9 avec surprise par quels raisonnements... \u00bb C'est ceci qu'il faut lire : \u00ab Par quelle raison... ? \u00bb Car cet ouvrage-ci vaut mieux que l'autre. Et ces livres socratiques, les avez-vous lus d'une autre fa\u00e7on qu'on ne lit des chansonnettes ? Si vous les lisiez comme il faut, vous ne vous attacheriez pas \u00e0 toutes ces frivolit\u00e9s ; mais vous fixeriez plut\u00f4t votre attention sur ceci : \u00ab Anytus et Melytus peuvent me tuer ; ils ne peuvent me nuire \u00bb ; et sur ceci encore : \u00ab Je suis de nature \u00e0 ne m'attacher qu'\u00e0 une seule chose en moi, \u00e0 la raison qui, bien consid\u00e9r\u00e9e, me para\u00eet la meilleure. \u00bb Aussi, quelqu'un a-t-il jamais entendu dire \u00e0 Socrate : \u00ab Je sais et j'enseigne ? \u00bb Loin de l\u00e0 : il avait pour chacun un ma\u00eetre \u00e0 qui l'adresser. Les gens venaient donc le prier de les pr\u00e9senter \u00e0 des philosophes ; et il les y menait et les recommandait. Est-ce que cela n'est pas vrai ? Est-ce qu'il leur disait, en les reconduisant : \u00ab Viens m'entendre parler aujourd'hui dans la maison de Codratus \u00bb ? Eh ! pourquoi irais-je t'entendre ? Veux-tu me montrer que tu sais disposer les mots \u00e9l\u00e9gamment ? Tu sais les disposer, \u00f4 homme ! Mais quel bien cela te fait-il ? \u2014 Applaudis-moi. \u2014 Qu'entends-tu par t'applaudir ? \u2014 Dis-moi : \u00ab Ah ! \u00bb et \u00ab C'est merveilleux ! \u00bb \u2014 Eh bien ! je le dis. Mais, si les applaudissements doivent porter sur quelque chose que les philosophes placent dans la cat\u00e9gorie du bien, qu'est-ce que j'ai \u00e0 applaudir en toi ? Si c'est une bonne chose que de bien parler, prouve-le-moi, et je t'applaudirai. \u2014 Quoi donc ! serait-ce qu'il doit m'\u00eatre d\u00e9sagr\u00e9able d'entendre bien parler ? \u2014 \u00c0 Dieu ne plaise ! Il ne m'est pas d\u00e9sagr\u00e9able non plus d'entendre jouer de la lyre ; mais est-ce une raison pour que je doive me tenir l\u00e0 debout \u00e0 jouer de la lyre ? \u00c9coute ce que dit Socrate : \u00ab Hommes, il ne convient pas \u00e0 mon \u00e2ge de me pr\u00e9senter devant vous en arrangeant mes discours, comme le fait un jeune homme. \u00bb Il dit comme le fait un jeune homme. C'est qu'en r\u00e9alit\u00e9, c'est une jolie chose que de savoir choisir et disposer ses mots, que de savoir apr\u00e8s cela les lire ou les d\u00e9biter avec gr\u00e2ce, que de s'interrompre enfin au milieu de sa lecture pour s'\u00e9crier : \u00ab Par votre salut ! ce sont l\u00e0 des choses que peu de gens peuvent comprendre. \u00bb\n\nEst-ce que le philosophe prie les gens de venir l'entendre ? Est-ce que par le seul fait de son existence il n'attire pas \u00e0 lui, comme le soleil, comme la nourriture, ceux \u00e0 qui il doit \u00eatre utile ? Quel est le m\u00e9decin qui prie les gens de se faire soigner par lui ? J'entends dire, il est vrai, qu'aujourd'hui \u00e0 Rome les m\u00e9decins prient les malades de venir \u00e0 eux ; mais, de mon temps, c'\u00e9tait eux qu'on priait. Je t'en prie, viens apprendre que tu n'es pas en bon \u00e9tat, que tu t'occupes de tout autre chose que ce dont tu dois t'occuper, que tu te trompes sur les biens et sur les maux, que tu es malheureux, que tu es infortun\u00e9. La charmante pri\u00e8re ! Et cependant, si la parole du philosophe n'a pas r\u00e9ellement ces effets, elle n'est qu'une parole morte, et c'est un mort qui parle. Rufus avait l'habitude de dire : \u00ab S'il vous reste assez de libert\u00e9 d'esprit pour m'applaudir, c'est que je ne dis rien qui vaille. \u00bb Il parlait de telle fa\u00e7on que nous, qui \u00e9tions assis l\u00e0, nous croyions chacun lui avoir \u00e9t\u00e9 d\u00e9nonc\u00e9s ; tant il mettait le doigt sur ce qui \u00e9tait, tant il nous pla\u00e7ait \u00e0 chacun nos mis\u00e8res sous les yeux.\n\nHommes, c'est la maison d'un m\u00e9decin que l'\u00e9cole d'un philosophe. Avant d'en sortir, il vous faut, non pas jouir, mais souffrir ; car vous n'y entrez pas bien portants, mais l'un avec une \u00e9paule d\u00e9mise, l'autre avec un abc\u00e8s, celui-ci avec une fistule, celui-l\u00e0 avec des maux de t\u00eate. Et moi, vais-je m'asseoir l\u00e0 \u00e0 vous d\u00e9biter de belles sentences et de belles paroles, pour que vous partiez m'ayant applaudi, mais en remportant, l'un son \u00e9paule telle qu'il l'avait apport\u00e9e, l'autre sa t\u00eate dans le m\u00eame \u00e9tat, celui-ci sa fistule, celui-l\u00e0 son abc\u00e8s ? Et ce serait pour cela que les jeunes gens se d\u00e9rangeraient ! Ils quitteraient leurs parents, leurs amis, leur famille, leur h\u00e9ritage, pour venir te dire bravo ! pendant que tu leur d\u00e9bites de belles paroles ! Est-ce l\u00e0 ce que faisait Socrate, ce que faisait Z\u00e9non, ce que faisait Cl\u00e9anthe ?\n\n\u2014 Mais quoi ! l'exhortation n'est-elle pas un genre oratoire sp\u00e9cial ? \u2014 Qui dit le contraire ? C'est ainsi qu'il y a le genre de la r\u00e9futation, et celui de l'enseignement. Mais qui donc a jamais parl\u00e9 d'un quatri\u00e8me genre apr\u00e8s ceux-l\u00e0, le genre de l'ostentation ? En quoi consiste le genre de l'exhortation ? \u00c0 pouvoir montrer \u00e0 un individu ou \u00e0 plusieurs dans quelle m\u00eal\u00e9e ils se trouvent emport\u00e9s, et comment ils sont sans cesse en qu\u00eate de tout autre chose que ce qu'ils veulent. Car ce qu'ils veulent, c'est ce qui conduit au bonheur, et ils le cherchent o\u00f9 il n'est pas. Et pour faire cette d\u00e9monstration, il te faudrait commencer par disposer un millier de si\u00e8ges, et inviter les gens \u00e0 venir t'entendre, puis, \u00e9l\u00e9gamment drap\u00e9 dans ta robe ou dans ton manteau, te jucher sur des coussins, et raconter de l\u00e0 la mort d'Achille ! Cessez, par tous les dieux ! de d\u00e9shonorer, autant qu'il est en vous, de grands noms et de grandes choses. On dirait que les exhortations ne sont jamais plus efficaces que lorsque l'orateur laisse voir \u00e0 ses auditeurs qu'il a besoin d'eux !\n\nEt qui, dis-moi, en t'entendant lire ou parler, a con\u00e7u des inqui\u00e9tudes sur lui-m\u00eame ou est descendu au fond de son c\u0153ur ? Qui a dit, en sortant : \u00ab Le philosophe a bien mis le doigt sur mes plaies ; je ne dois plus me conduire ainsi \u00bb ? Personne ; mais, quand tu as eu du succ\u00e8s, l'un dit : \u00ab Il a bien parl\u00e9 de Xerx\u00e8s ! \u00bb l'autre : \u00ab Non, mais du combat des Thermopyles. \u00bb Est-ce donc l\u00e0 l'auditoire d'un philosophe ?\n\nXXVI. \u00c0 CEUX QUI CRAIGNENT LA PAUVRET\u00c9\n\nN'as-tu pas honte d'avoir moins de courage et moins de c\u0153ur que les esclaves fugitifs ? En quel \u00e9tat fuient-ils, quand ils abandonnent leurs ma\u00eetres ? Quels domaines, quels serviteurs ont-ils pour se rassurer ? Ils d\u00e9robent le peu qu'il leur faut pour les premiers jours, puis ils se lancent \u00e0 travers les terres, et m\u00eame \u00e0 travers les mers, se procurant habilement les moyens de subsister : aujourd'hui celui-ci, demain celui-l\u00e0. Et qui d'entre eux est jamais mort de faim ? Mais toi, tu trembles de manquer du n\u00e9cessaire ; et te voil\u00e0 passant tes nuits \u00e0 veiller ! Malheureux, es-tu donc si aveugle ? Ne vois-tu pas le chemin ? Et ne sais-tu pas o\u00f9 nous conduit le manque du n\u00e9cessaire ? O\u00f9 nous conduit-il donc ? O\u00f9 nous conduit la fi\u00e8vre, o\u00f9 nous conduit une pierre qui nous tombe sur la t\u00eate : \u00e0 la mort. N'est-ce pas ce que tu as dit cent fois toi-m\u00eame \u00e0 tes amis ? Ne l'as-tu pas lu cent fois ? Ne l'as-tu pas \u00e9crit cent fois aussi ? \u00c0 combien de reprises ne t'es-tu pas vant\u00e9 d'attendre la mort avec calme ?\n\n\u2014 Mais les miens mourront de faim ! \u2014 Eh bien ! Est-ce que la faim les conduit ailleurs que toi ? Est-ce que la descente n'est pas la m\u00eame pour eux ? Est-ce qu'en bas ils ne trouveront pas les m\u00eames choses ? Ne peux-tu donc pas, sans t'effrayer du d\u00e9nuement et de la disette, fixer un \u0153il calme sur le lieu o\u00f9 doivent descendre les plus riches, les magistrats les plus \u00e9lev\u00e9s, les rois et les tyrans eux-m\u00eames ? Peut-\u00eatre y descendras-tu d'inanition ; ils y descendront, eux, crevant d'indigestion et d'ivresse. Mais ne sera-ce pas toute la diff\u00e9rence ? Que de mendiants n'as-tu pas vus arriver \u00e0 la vieillesse ! Combien m\u00eame \u00e0 l'extr\u00eame vieillesse ! Ces gens transis de froid et le jour et la nuit, ces gens qui gisent sur le sol, et qui ne mangent que bien juste leur n\u00e9cessaire, ces gens-l\u00e0 arrivent presque \u00e0 ne pouvoir mourir. Ne peux-tu donc pas faire un m\u00e9tier ? Ne peux-tu pas \u00eatre copiste ? \u00catre pr\u00e9cepteur ? Garder la porte d'autrui ? \u2014 Mais c'est une honte, d'en venir \u00e0 cette extr\u00e9mit\u00e9 ! \u2014 Eh bien ! commence par apprendre o\u00f9 est la honte, et alors seulement dis-toi philosophe. Pour le moment, ne permets m\u00eame pas \u00e0 un autre de t'appeler de ce nom.\n\nEst-ce que c'est une honte pour toi que ce qui n'est pas ton \u0153uvre, que ce dont tu n'es pas l'auteur, que ce qui t'arrive par hasard, comme le mal de t\u00eate, comme la fi\u00e8vre ? Si tes parents \u00e9taient pauvres, ou si, riches, ils ont laiss\u00e9 leur h\u00e9ritage \u00e0 d'autres, ou si encore, de leur vivant, ils ne te donnent rien, est-ce une honte pour toi ? Est-ce l\u00e0 ce que tu as appris chez les philosophes ? Ne leur as-tu pas entendu dire que ce qui est bl\u00e2mable est seul honteux, et que ce qui est bl\u00e2mable c'est ce qui est digne de bl\u00e2me ? Et qui peux-tu bl\u00e2mer de ce qui n'est pas son \u0153uvre, de ce qu'il n'a pas fait lui-m\u00eame ? Est-ce donc toi qui as fait ton p\u00e8re tel qu'il est ? Ou bien t'est-il possible de le corriger ? Est-ce l\u00e0 une chose qui soit en ta puissance ? Eh bien ! dois-tu vouloir ce qui n'est pas en ta puissance ? ou rougir quand tu n'y r\u00e9ussis pas ? Est-ce la philosophie qui t'a fait prendre cette habitude d'avoir les yeux sur les autres, et de ne rien attendre de toi-m\u00eame ? G\u00e9mis donc, lamente-toi, et ne mange qu'en tremblant de n'avoir pas de quoi vivre demain. Tremble que tes esclaves ne te volent, ne s'enfuient ou ne meurent. Que ce soit l\u00e0 ta vie, et qu'elle ne cesse jamais, puisque c'est de nom seulement que tu t'es approch\u00e9 de la philosophie, puisque tu d\u00e9shonores son enseignement autant qu'il t'est possible de le faire, toi qui montres qu'il est sans utilit\u00e9 et sans profit pour ceux qui l'ont re\u00e7u. Jamais tu n'as souhait\u00e9 le calme, la tranquillit\u00e9, l'impassibilit\u00e9 ; jamais tu ne t'es attach\u00e9 \u00e0 personne pour y arriver ; mais que de gens auxquels tu t'es attach\u00e9 par amour pour les syllogismes ! Jamais pour aucune des choses qui apparaissaient \u00e0 tes sens tu ne t'es demand\u00e9 \u00e0 toi-m\u00eame : \u00ab Pourrai-je, ou ne pourrai-je pas supporter cela ? Que me reste-t-il \u00e0 faire ? \u00bb Mais, comme si tout ce qui est \u00e0 toi \u00e9tait en bon \u00e9tat et \u00e0 l'abri de tout p\u00e9ril, tu t'occupais de ce qui ne doit venir qu'apr\u00e8s tout le reste, de l'immutabilit\u00e9 ! Et qu'avais-tu donc \u00e0 rendre immuable ? Ta l\u00e2chet\u00e9, ta couardise, ton admiration pour les riches, tes d\u00e9sirs avort\u00e9s, tes efforts inutiles pour \u00e9viter les choses ? Voil\u00e0 ce que tu voulais mettre \u00e0 l'abri de tout p\u00e9ril !\n\nNe devais-tu pas commencer par acqu\u00e9rir ce que la raison te conseillait, puis songer alors seulement \u00e0 mettre tes acquisitions en s\u00fbret\u00e9 ? Qui as-tu vu construire un couronnement autour de sa maison, sans placer ce couronnement sur un mur ? Quel est le portier que l'on \u00e9tablit o\u00f9 il n'y a pas de porte ? Ta pr\u00e9occupation \u00e0 toi, c'est d'\u00eatre capable de d\u00e9montrer ; mais de d\u00e9montrer quoi ? Ta pr\u00e9occupation, c'est de ne pas te laisser entra\u00eener par les sophismes ; mais entra\u00eener loin de quoi ? Montre-moi d'abord ce qui est l'objet de tes soins, ce que tu mesures, ou ce que tu p\u00e8ses ; puis ensuite montre-moi ta balance ou ta mesure. Jusques \u00e0 quand ne mesureras-tu que de la cendre ? Ce que tu dois d\u00e9montrer, n'est-ce pas ce qui rend l'homme heureux, ce qui fait que les choses lui arrivent comme il les d\u00e9sire, ce qui est cause qu'il doit ne bl\u00e2mer personne, n'accuser personne, mais se conformer \u00e0 la mani\u00e8re dont le monde est gouvern\u00e9 ? Voil\u00e0 ce qu'il te faut me montrer. \u2014 Voici, dis-tu, ma d\u00e9monstration : je vais t'analyser des syllogismes. \u2014 Esclave, c'est l\u00e0 ta mesure ; mais ce n'est pas ce que tu mesures ! Voil\u00e0 comment tu es puni aujourd'hui d'avoir n\u00e9glig\u00e9 la philosophie : tu trembles, tu ne dors pas, tu consultes tout le monde ; et si les r\u00e9solutions que tu prends ne conviennent pas \u00e0 tout le monde, tu crois avoir eu tort de les prendre. Tu crois aujourd'hui redouter la faim ; mais non : ce n'est pas la faim que tu redoutes. Ce que tu crains, c'est de n'avoir plus de cuisinier, de n'avoir plus personne pour tes sauces, personne pour t'attacher tes chaussures, personne pour te passer tes habits, personne pour te frictionner, personne pour te faire cort\u00e8ge. Tu veux pouvoir aux bains te d\u00e9pouiller, t'\u00e9tendre \u00e0 la fa\u00e7on de ceux qu'on met en croix, puis te faire frotter et de ci et de l\u00e0 ; tu veux que le ma\u00eetre baigneur, qui pr\u00e9side \u00e0 l'op\u00e9ration, dise ensuite : \u00ab Passe ici ; montre-nous le flanc ; prends-lui la t\u00eate ; pr\u00e9sente ton \u00e9paule \u00bb ; puis, rentr\u00e9 chez toi apr\u00e8s le bain, tu veux crier : \u00ab Ne m'apporte-t-on pas \u00e0 manger ? \u00bb Et apr\u00e8s cela : \u00ab Enl\u00e8ve la table ; passe l'\u00e9ponge. \u00bb Ce que tu crains, c'est de ne pouvoir plus mener la vie d'un malade. Quant \u00e0 la vie de ceux qui se portent bien, apprends \u00e0 la conna\u00eetre : c'est celle que m\u00e8nent les esclaves, les ouvriers, les vrais philosophes ; c'est celle qu'a men\u00e9e Socrate, quoique avec femme et enfants ; c'est celle de Diog\u00e8ne, celle de Cl\u00e9anthe, qui tenait une \u00e9cole et \u00e9tait porteur d'eau. Si tu veux mener cette vie, tu la pourras mener partout, et tu vivras dans une pleine assurance. Fond\u00e9e sur quoi ? Sur la seule chose \u00e0 laquelle on puisse se fier, sur la seule qui soit s\u00fbre, qui soit sans entraves, que nul ne puisse t'enlever, sur ta propre volont\u00e9. Pourquoi par ta faute es-tu si inutile et si impropre \u00e0 tout, que personne ne veut te prendre chez lui, ne veut se charger de toi ? Un vase intact et propre au service aura beau \u00eatre jet\u00e9 dehors, quiconque le trouvera l'emportera, et croira que c'est tout profit ; toi, au contraire, chacun croira que c'est tout perte. Ainsi tu ne peux m\u00eame pas rendre les services d'un chien et d'un coq, et tu veux encore vivre, tel que tu es !\n\nLe Sage craindra-t-il que les aliments viennent \u00e0 lui manquer ? Ils ne manquent pas \u00e0 l'aveugle ; ils ne manquent pas au boiteux ; et ils manqueraient au Sage ! Un bon soldat trouve toujours qui le paye ; un bon ouvrier, un bon cordonnier aussi ; et celui qui est l'homme parfait ne le trouverait pas ! Dieu sera-t-il si insoucieux de ses propres affaires, de ses ministres, de ses t\u00e9moins, de ceux qui lui servent \u00e0 prouver par des faits aux hommes ordinaires, qu'il existe, qu'il gouverne sagement ce monde, qu'il ne n\u00e9glige pas l'humanit\u00e9, et qu'il n'y a jamais de mal pour le Sage, ni de son vivant, ni apr\u00e8s sa mort ? \u2014 Mais lorsqu'il ne me fournit pas de quoi manger ? \u2014 Que fait-il autre chose que de me donner le signal de la retraite, comme un bon g\u00e9n\u00e9ral ? Je lui ob\u00e9is alors ; je le suis, en chantant les louanges de mon g\u00e9n\u00e9ral, en approuvant bien haut tout ce qu'il fait. Je suis venu, en effet, quand il l'a voulu ; je m'en irai de m\u00eame, quand il le voudra ; et, de mon vivant, qu'avais-je pr\u00e9cis\u00e9ment \u00e0 faire, que de chanter les louanges de Dieu, seul avec moi-m\u00eame, en face d'un autre, ou de plusieurs ? Il me donne peu, il ne me donne pas en abondance, il ne veut pas que je vive dans la mollesse ; mais il n'a pas donn\u00e9 davantage \u00e0 Hercule, son propre fils. C'\u00e9tait un autre qui r\u00e9gnait sur Argos et sur Myc\u00e8nes ; la part d'Hercule \u00e9tait l'ob\u00e9issance, les travaux, les \u00e9preuves. Mais Eurysth\u00e9e \u00e9tait ce qu'il \u00e9tait, et ne r\u00e9gnait pas plus r\u00e9ellement sur Argos et sur Myc\u00e8nes qu'il ne r\u00e9gnait sur lui-m\u00eame ; tandis qu'Hercule, par toute la terre et par toute la mer, \u00e9tait v\u00e9ritablement roi, v\u00e9ritablement chef, r\u00e9parant les iniquit\u00e9s et les injustices, amenant avec lui la justice et la pi\u00e9t\u00e9 ; et tout cela il le faisait nu et seul. Quand Ulysse fut jet\u00e9 \u00e0 la c\u00f4te par un naufrage, se laissa-t-il abattre par son d\u00e9nuement ? Perdit-il courage ? Non : voyez comme il va demander \u00e0 des vierges ces v\u00eatements indispensables, que nous trouvons si honteux de demander \u00e0 un autre.\n\nIl allait comme un lion nourri dans les montagnes et qui se confie en sa force.\n\nQu'est-ce qui faisait donc sa confiance ? Ce n'\u00e9tait ni la r\u00e9putation, ni la richesse, ni le pouvoir ; c'\u00e9tait sa force int\u00e9rieure, c'est-\u00e0-dire, ses convictions sur ce qui d\u00e9pend de nous et sur ce qui n'en d\u00e9pend pas. Ce sont elles seules, en effet, qui nous font libres et ind\u00e9pendants, qui font relever la t\u00eate \u00e0 celui qu'on humilie, qui nous font regarder en face et d'un \u0153il fixe les riches et les puissants. Voil\u00e0 la part du philosophe. Mais toi, tu sortiras comme un l\u00e2che, tremblant de peur pour tes manteaux et pour ta vaisselle d'argent ! Malheureux, est-ce ainsi que tu as perdu ton temps jusqu'\u00e0 pr\u00e9sent ?\n\n\u2014 Mais si je suis malade ? \u2014 Tu seras ce que tu dois \u00eatre dans la maladie. \u2014 Mais qui me soignera ? \u2014 Dieu, et tes amis. \u2014 Je serai durement couch\u00e9. \u2014 Comme doit l'\u00eatre un homme. \u2014 Je n'aurai pas de maison commode. \u2014 Eh bien ! tu seras malade dans une maison incommode. \u2014 Qui me donnera les moyens de vivre ? \u2014 Ceux qui les donnent aux autres. Tu seras comme Man\u00e8s dans ta maladie. \u2014 Mais quelle sera la fin de cette maladie ? \u2014 La mort, et quoi de plus ? Ne sais-tu donc pas que la source de toutes les mis\u00e8res pour l'homme, la source de toutes ses faiblesses et de toutes ses l\u00e2chet\u00e9s, ce n'est pas la mort, mais bien plut\u00f4t la crainte de la mort ? Exerce-toi donc contre cette crainte ; crois-moi, que ce soit l\u00e0 que tendent tous tes raisonnements, tout ce que tu \u00e9coutes, tout ce que tu lis, et tu reconna\u00eetras que c'est par l\u00e0 seulement que les hommes s'affranchissent.\n\nLivre quatri\u00e8me\n\nII. SUR NOS LIAISONS\n\nVoici un point auquel il te faut faire attention avant tout : ne te lie avec aucun de tes habitu\u00e9s ou de tes amis d'autrefois, jusqu'\u00e0 descendre o\u00f9 il en est descendu ; sinon, tu te perdras. Si l'id\u00e9e te vient qu'il te trouvera d\u00e9plaisant, et qu'il ne sera plus pour toi ce qu'il \u00e9tait auparavant, rappelle-toi que l'on n'a rien pour rien, et qu'on ne peut pas, en n'agissant plus de m\u00eame, rester l'homme qu'on \u00e9tait jadis. D\u00e9cide donc lequel tu pr\u00e9f\u00e8res : ou de garder intacte l'affection de ceux qui t'aimaient auparavant, en demeurant ce qu'auparavant tu \u00e9tais ; ou de ne plus obtenir d'eux la m\u00eame affection, en devenant meilleur. Si c'est ce dernier parti qui vaut le mieux, il faut le prendre, et sur-le-champ, sans t'en laisser d\u00e9tourner par d'autres consid\u00e9rations. Il n'est pas possible d'avancer, quand on va tant\u00f4t d'un c\u00f4t\u00e9, tant\u00f4t d'un autre. Si tu as jug\u00e9 que ce parti valait mieux que tous les autres, si tu veux t'attacher \u00e0 lui seul, et ne travailler que pour lui, laisse-moi l\u00e0 tout le reste. Sinon, ces tergiversations auront pour toi ce double r\u00e9sultat, que tu ne feras pas les progr\u00e8s que tu devrais faire, et qu'on ne t'accordera plus ce qu'on t'accordait auparavant. Auparavant, quand tu d\u00e9sirais franchement des objets sans valeur r\u00e9elle, tu \u00e9tais agr\u00e9able \u00e0 tes amis ; mais tu ne peux pas r\u00e9ussir aux deux choses \u00e0 la fois : il faut n\u00e9cessairement que ce que tu gagneras d'un c\u00f4t\u00e9, tu le perdes de l'autre. Tu ne peux pas, si tu cesses de boire avec qui tu buvais, para\u00eetre \u00e0 ces gens aussi agr\u00e9able qu'alors. D\u00e9cide donc ce que tu pr\u00e9f\u00e8res : ou de t'enivrer et de leur \u00eatre agr\u00e9able, ou de leur d\u00e9plaire en \u00e9tant sobre. Tu ne peux pas, si tu cesses de chanter avec qui tu chantais, rester aussi cher \u00e0 ces gens. Choisis donc encore ici le lot que tu voudras. S'il vaut mieux \u00eatre temp\u00e9rant et r\u00e9gl\u00e9, que de faire dire de soi : \u00ab Quel homme agr\u00e9able ! \u00bb, laisse-moi l\u00e0 tout le reste ; renonces-y ; d\u00e9tourne-t'en ; n'y touche plus. Si ce parti-l\u00e0 ne te pla\u00eet pas, donne-toi tout entier au parti contraire : sois un de nos hommes-femmes ; sois un de nos coureurs d'aventures ; fais tout ce qui s'ensuit, et tu arriveras \u00e0 ce que tu veux. N'oublie pas aussi de tr\u00e9pigner des pieds en acclamant le baladin. Mais on ne peut pas r\u00e9unir en soi ces deux personnages si diff\u00e9rents : on ne peut pas jouer \u00e0 la fois le r\u00f4le de Thersite et celui d'Agamemnon. Si tu veux \u00eatre Thersite, il te faut \u00eatre bossu et chauve ; si tu veux \u00eatre Agamemnon, il te faut \u00eatre beau, et de haute taille, et aimer ceux qui te sont subordonn\u00e9s.\n\nV. CONTRE LES GENS QUERELLEURS ET M\u00c9CHANTS\n\nLe Sage ne se querelle jamais avec personne, et, autant qu'il le peut, emp\u00eache les autres de se quereller. Sur ce point, comme sur tous les autres, la vie de Socrate est l\u00e0 pour nous servir d'exemple. Non seulement il a partout \u00e9vit\u00e9 de se quereller, mais il a emp\u00each\u00e9 les autres de le faire. Vois chez X\u00e9nophon, dans le Banquet, que de querelles il a apais\u00e9es ; vois d'autre part sa patience avec Thrasymaque, avec Polus, avec Callicrate ; vois cette m\u00eame patience avec sa femme, avec son fils, quand celui-ci essayait de le r\u00e9futer par ses sophismes. C'est qu'il savait de science trop certaine que nul n'est le ma\u00eetre de l'\u00e2me d'autrui ; et qu'en cons\u00e9quence il n'avait de volont\u00e9 que pour lui-m\u00eame. Et qu'est-ce que cela ? C'est ne pas avoir la pr\u00e9tention de faire agir les autres conform\u00e9ment \u00e0 la nature, car cela ne d\u00e9pend pas de nous ; mais s'attacher, tandis que les autres agissent pour leur compte comme bon leur semble, \u00e0 vivre et \u00e0 agir soi-m\u00eame conform\u00e9ment \u00e0 la nature, seulement en faisant tout ce qui d\u00e9pend de soi pour qu'eux aussi vivent conform\u00e9ment \u00e0 la nature. Car tel est le but que se propose toujours le Sage. Veut-il \u00eatre g\u00e9n\u00e9ral ? Non, mais si son lot est de l'\u00eatre, il veut dans cette position maintenir pure en lui sa partie ma\u00eetresse. Veut-il se marier ? Non, mais si son lot est de le faire, il veut dans cette position se maintenir en conformit\u00e9 avec la nature. Quant \u00e0 vouloir que son fils ou sa femme ne fissent jamais rien de mal, ce serait vouloir que ce qui ne d\u00e9pend pas de lui en d\u00e9pend\u00eet. Or, s'instruire n'est autre chose qu'apprendre \u00e0 distinguer ce qui d\u00e9pend de vous et ce qui n'en d\u00e9pend pas.\n\nQuelle occasion de dispute y a-t-il donc encore pour celui qui est dans ces sentiments ? Rien de tout ce qui arrive l'\u00e9tonne-t-il ? Rien lui para\u00eet-il extraordinaire ? Est-ce qu'il ne s'attend pas toujours, de la part des m\u00e9chants, \u00e0 des choses plus f\u00e2cheuses et plus tristes que ce qui lui arrive ? Est-ce qu'il ne regarde pas comme autant de gagn\u00e9 tout ce qui manque au malheur complet ? \u2014 Un tel t'a injuri\u00e9, dit-il, sache-lui gr\u00e9 de ne pas t'avoir frapp\u00e9. \u2014 Mais il m'a frapp\u00e9 ! \u2014 Sache-lui gr\u00e9 de ne pas t'avoir bless\u00e9. \u2014 Mais il m'a bless\u00e9 ! \u2014 Sache-lui gr\u00e9 de ne pas t'avoir tu\u00e9. En effet, quand, ou de qui, a-t-il appris qu'il est un animal sociable, fait pour aimer les autres, et que l'injustice est un grand mal pour qui la commet ! Et, puisqu'il ne l'a pas appris, et qu'il ne le croit pas, comment ne suivrait-il pas ce qui lui semble son int\u00e9r\u00eat ? \u2014 Mon voisin m'a jet\u00e9 des pierres ! \u2014 Eh bien ! as-tu pour ta part commis quelque faute ? \u2014 Tout ce qui est dans ma maison a \u00e9t\u00e9 bris\u00e9 ! \u2014 Serais-tu donc toi-m\u00eame un meuble ? Non : tu es un jugement et une volont\u00e9. Qu'est-ce qui t'a donc \u00e9t\u00e9 donn\u00e9 contre ce dont tu te plains ? En tant que tu tiens du loup, il t'a \u00e9t\u00e9 donn\u00e9 de mordre \u00e0 ton tour, et de jeter un plus grand nombre de pierres. Si tu cherches ce qui t'a \u00e9t\u00e9 donn\u00e9 en tant que tu es homme, regarde dans ta bourse, et vois quelles ressources tu avais en venant ici. Serait-ce la f\u00e9rocit\u00e9 ? Serait-ce l'esprit de vengeance ? Quand un cheval est-il malheureux ? Quand il a perdu ses facult\u00e9s naturelles ; non quand il ne peut point chanter comme le coq, mais quand il ne peut plus courir. Et le chien ? Non quand il ne peut point voler, mais quand il ne peut plus suivre la piste. Eh bien ! n'est-il pas pareillement vrai que l'homme malheureux n'est pas celui qui ne peut \u00e9trangler des lions, ou embrasser des statues (nul n'est venu au monde en tenant de la nature des moyens pour cela), mais celui qui perd sa bienveillance et sa loyaut\u00e9 ? Voil\u00e0 celui sur qui devraient g\u00e9mir ceux qui le rencontrent, \u00e0 la vue des maux dans lesquels il est tomb\u00e9. Par Jupiter ! il faut le plaindre, non pas d'\u00eatre n\u00e9 ou d'\u00eatre mort, mais d'avoir perdu de son vivant ce qui lui appartenait en propre : non point son patrimoine, son champ, sa maison, son h\u00f4tellerie, ses esclaves (rien de tout cela n'appartient \u00e0 l'individu ; ce sont toutes choses en dehors de lui, au pouvoir et \u00e0 la merci d'autrui, que donnent tant\u00f4t \u00e0 l'un, tant\u00f4t \u00e0 l'autre, ceux qui en sont les ma\u00eetres), mais ce qui est vraiment de l'homme, la marque qu'il portait dans son \u00e2me, lorsqu'il est venu au monde, marque semblable \u00e0 celle que nous cherchons sur les monnaies, pour les juger bonnes quand nous l'y trouvons, pour les rejeter quand nous ne l'y trouvons pas. \u2014 Quelle marque (disons-nous) a cette pi\u00e8ce de quatre as ? \u2014 La marque de Trajan. \u2014 Apporte. \u2014 Elle a la marque de N\u00e9ron. \u2014 Jette-la ; elle est de mauvais aloi ; elle est alt\u00e9r\u00e9e. \u00bb Il en est de m\u00eame ici : \u2014 Quelle marque portent ses fa\u00e7ons de penser et de vouloir ? \u2014 Celle d'un \u00eatre doux, sociable, patient, affectueux. \u2014 Apporte. Je le re\u00e7ois ; j'en fais mon concitoyen ; je le re\u00e7ois pour voisin, et pour compagnon de travers\u00e9e. Prends garde seulement qu'il ne porte pas la marque de N\u00e9ron. Ne serait-il pas col\u00e8re, rancunier, m\u00e9content de tout ? Ne serait-il pas sujet, quand l'id\u00e9e lui en vient, \u00e0 casser la t\u00eate de ceux qu'il rencontre ? Si cela est, pourquoi l'appelais-tu un homme ? Ce n'est pas \u00e0 la forme seule qu'on distingue chaque esp\u00e8ce d'\u00eatres. \u00c0 ce compte, en effet, il faudrait dire qu'une pomme en cire est une vraie pomme, tandis qu'il y faut encore et l'odeur et le go\u00fbt, la configuration ext\u00e9rieure n'y suffisant pas. De m\u00eame, pour faire un homme il ne suffit pas des narines et des yeux ; il y faut encore des fa\u00e7ons de penser et de vouloir qui soient d'un homme. Un tel n'\u00e9coute pas la raison ; il ne se rend pas, quand on l'a convaincu d'erreur : ce n'est qu'un \u00e2ne. Toute retenue est morte chez cet autre : il n'est bon \u00e0 rien ; il n'y a rien qu'il ne soit plut\u00f4t qu'un homme. Celui-ci cherche \u00e0 rencontrer quelqu'un afin de ruer ou de mordre : ce n'est pas m\u00eame un mouton ou un \u00e2ne ; c'est une b\u00eate sauvage.\n\n\u2014 Quoi donc ! veux-tu que je me laisse m\u00e9priser ? \u2014 Par qui ? Par ceux qui s'y connaissent ? Eh ! comment ceux qui s'y connaissent m\u00e9priseraient-ils un homme pour sa douceur et sa retenue ? Par ceux qui ne s'y connaissent pas ? Que t'importe ! En dehors de toi, quel homme expert dans un art s'inqui\u00e8te des ignorants ? \u2014 Mais ils s'en acharneront davantage apr\u00e8s moi ! \u2014 Comment dis-tu apr\u00e8s moi ? Peut-on donc alt\u00e9rer ton jugement et ta volont\u00e9, ou t'emp\u00eacher de faire de toutes les id\u00e9es qui t'arrivent un emploi conforme \u00e0 la nature ? \u2014 Non. \u2014 De quoi donc te troubles-tu ? Et pourquoi tiens-tu \u00e0 te montrer redoutable ? Pourquoi plut\u00f4t ne pas t'avancer en public et proclamer que tu vis en paix avec tous les hommes, quoi qu'ils puissent faire ? Pourquoi ne pas rire surtout de ceux qui croient te nuire ? \u00ab Ces esclaves (dirais-tu) ne savent ni qui je suis, ni en quoi consistent pour moi les biens et les maux. Ils ignorent qu'ils ne sauraient atteindre ce qui m'appartient. \u00bb\n\nC'est ainsi que les habitants d'une ville bien fortifi\u00e9e se rient de ceux qui l'assi\u00e8gent. \u00ab Qu'est-ce qu'ont ces gens, disent-ils, \u00e0 se donner tant de peine pour rien ? Nos murailles sont solides ; nous avons des vivres pour longtemps ; nous sommes bien munis de tout. \u00bb Avec ces moyens, en effet, une ville est forte et imprenable ; mais l'\u00e2me humaine ne l'est que par ses principes. Car, pour la rendre telle, quel mur serait assez solide, quel corps assez de fer, quelle fortune assez s\u00fbre, quel rang assez au-dessus de toutes les attaques ? Toutes ces choses sont partout p\u00e9rissables et promptes \u00e0 succomber. Celui qui s'y attache doit n\u00e9cessairement se troubler, esp\u00e9rer \u00e0 tort, s'effrayer, g\u00e9mir, \u00e9chouer dans ses d\u00e9sirs, tomber dans ce qu'il veut \u00e9viter. Et nous ne prenons pas le parti de fortifier la seule chose solide qui nous ait \u00e9t\u00e9 donn\u00e9e ! Et nous ne nous arrachons pas aux choses p\u00e9rissables et d\u00e9pendantes, pour donner tous nos soins \u00e0 celles qui, de leur nature, sont imp\u00e9rissables et ind\u00e9pendantes ! Nous ne songeons point que personne ne peut faire du mal ou du bien \u00e0 un autre, et que les opinions de chacun \u00e0 l'\u00e9gard de tout cela sont la seule chose qui nuise et qui bouleverse ; la seule cause des querelles, des dissensions, des guerres ! Qu'est-ce qui a fait \u00c9t\u00e9ocle et Polynice ? Rien autre chose que leurs opinions sur la royaut\u00e9 et sur l'exil. Celui-ci leur paraissait le dernier des maux, et celle-l\u00e0 le plus grand des biens ; or, la nature de tous les \u00eatres est de chercher le bien et de fuir le mal, et de regarder comme un adversaire et comme un ennemi quiconque veut leur enlever l'un et les jeter dans l'autre, f\u00fbt-il leur fr\u00e8re, leur fils ou leur p\u00e8re. Rien, en effet, ne nous tient de plus pr\u00e8s que le bien ; et de l\u00e0 suit que, si les choses ext\u00e9rieures sont des biens ou des maux, le p\u00e8re n'est plus l'ami de ses enfants, le fr\u00e8re n'est plus l'ami de son fr\u00e8re ; partout il n'y a plus que des ennemis, des tra\u00eetres et des calomniateurs. Si, au contraire, le bon \u00e9tat de la facult\u00e9 de juger et de vouloir est le seul bien, son mauvais \u00e9tat le seul mal, que deviennent les querelles et les invectives ? \u00c0 propos de quoi existeraient-elles ? Pour des choses qui nous sont indiff\u00e9rentes ? Et contre qui ? Contre des ignorants et des malheureux qui se trompent sur les choses les plus importantes ?\n\nC'est parce que Socrate savait tout cela, qu'il demeurait dans sa maison, en supportant la plus m\u00e9chante des femmes et un fils ingrat. \u00c0 quoi aboutissait, en effet, la m\u00e9chancet\u00e9 de sa femme ? \u00e0 lui verser sur la t\u00eate toute l'eau qu'elle voulait, et \u00e0 tr\u00e9pigner sur son g\u00e2teau. \u00ab Qu'est-ce que cela me fait, disait Socrate, d\u00e8s que je le regarde comme indiff\u00e9rent ? Or, ceci d\u00e9pend de moi : il n'y a ni tyran ni ma\u00eetre qui puisse m'en emp\u00eacher, si je le veux ; la multitude ici est impuissante contre l'individu, le plus fort contre le plus faible. L'ind\u00e9pendance sur ce point est un don de Dieu \u00e0 chacun de nous. \u00bb\n\nVoil\u00e0 les principes qui mettent l'amiti\u00e9 dans une famille, la concorde dans une ville, la paix entre les nations. Par eux, on est reconnaissant pour Dieu, et toujours sans crainte, parce qu'il n'y a jamais en question que des choses qui ne nous appartiennent pas et qui sont sans valeur.\n\nQuant \u00e0 nous, nous sommes bons pour \u00e9crire ou lire tout cela, et pour l'approuver quand nous l'avons lu ; mais que nous sommes loin de nous en p\u00e9n\u00e9trer ! Aussi ce qu'on disait des Lac\u00e9d\u00e9moniens, qu'ils sont des lions chez eux, des renards \u00e0 \u00c9ph\u00e8se, peut s'appliquer \u00e0 nous aussi : \u00ab Lions dans l'\u00e9cole, renards dehors. \u00bb\n\nXI. DE LA PROPRET\u00c9\n\nIl est des gens qui doutent que la sociabilit\u00e9 soit dans la nature de l'homme ; mais je ne vois pas ces gens eux-m\u00eames douter que la propret\u00e9 soit r\u00e9ellement dans notre nature, et qu'\u00e0 d\u00e9faut d'autre trait, il y ait l\u00e0 du moins quelque chose qui nous distingue des animaux. Lorsque nous voyons un animal se nettoyer, nous avons l'habitude de dire avec surprise : \u00ab C'est comme un homme \u00bb et, par contre, si l'on reproche \u00e0 un animal sa malpropret\u00e9, nous avons l'habitude de dire aussit\u00f4t, comme pour le d\u00e9fendre : \u00ab Ce n'est pas un homme. \u00bb Nous croyons donc qu'il y a l\u00e0 quelque chose de sp\u00e9cial \u00e0 l'homme, et ce quelque chose c'est des dieux m\u00eames que nous le tirons tout d'abord. Les dieux, par leur nature, sont purs et sans taches ; autant donc l'homme se rapproche d'eux par la raison, autant il devra s'efforcer d'\u00eatre pur et sans souillure. Il est impossible \u00e0 son \u00eatre de se trouver jamais compl\u00e8tement pur, avec les mat\u00e9riaux dont il est compos\u00e9 ; mais la raison, qui lui a \u00e9t\u00e9 donn\u00e9e, essaye du moins de le rendre pur dans la mesure du possible. La premi\u00e8re puret\u00e9, la plus noble, est celle de l'\u00e2me ; et r\u00e9ciproquement pour l'impuret\u00e9. On ne d\u00e9couvre pas les impuret\u00e9s de l'\u00e2me aussi ais\u00e9ment que celles du corps ; mais que peuvent \u00eatre ces impuret\u00e9s de l'\u00e2me, si ce n'est ce qui l'encrasse et la g\u00eane dans ses fonctions ? Or, les fonctions de l'\u00e2me sont de vouloir, de repousser, de d\u00e9sirer, de fuir, de se pr\u00e9parer, d'entreprendre, de donner son adh\u00e9sion. Qu'est-ce donc qui nuit chez elle \u00e0 ces fonctions, en la salissant et la rendant impure ? Rien autre chose que ses m\u00e9chants jugements. L'impuret\u00e9 de l'\u00e2me, ce sont donc ses opinions d\u00e9fectueuses ; et le moyen de la purifier, c'est de lui faire des opinions telles qu'elle en doit avoir. L'\u00e2me pure est celle qui a les opinions qu'elle doit avoir ; car c'est la seule dont les fonctions ne soient troubl\u00e9es par aucune salet\u00e9.\n\nIl y a quelque chose de pareil \u00e0 faire pour le corps \u00e0 son tour, autant qu'il s'y pr\u00eate. Il \u00e9tait impossible que les narines ne coulassent pas, l'homme \u00e9tant compos\u00e9 comme il l'est. C'est pour cela que la nature lui a fait dos, mains et les narines elles-m\u00eames, esp\u00e8ces de canaux pour mettre dehors les humeurs. Si donc quelqu'un ravale ces humeurs, je dis qu'il n'agit pas comme doit le faire un homme. Il \u00e9tait impossible que les pieds ne fussent jamais boueux, jamais sales d'aucune fa\u00e7on, avec les choses sur lesquelles nous marchons. C'est pour cela que la nature nous a donn\u00e9 de l'eau ; c'est pour cela qu'elle nous a donn\u00e9 des mains. Il \u00e9tait impossible qu'apr\u00e8s que nous avons mang\u00e9, quelque salet\u00e9 ne nous rest\u00e2t pas aux dents. C'est pour cela qu'elle nous dit : \u00ab Lavez vos dents. \u00bb Et pourquoi ? Pour \u00eatre des hommes, et non des b\u00eates sauvages ou des cochons. Il \u00e9tait impossible avec la sueur et les habits que nous portons, qu'il ne rest\u00e2t pas sur le corps quelque salet\u00e9 qui e\u00fbt besoin d'\u00eatre nettoy\u00e9e. C'est pour cela que nous avons l'eau, l'huile, les mains, le linge, les brosses, la soude, avec tout le reste de l'attirail pour nettoyer le corps. \u00ab Non \u00bb, dis-tu. Mais quoi ! l'ouvrier qui travaille les m\u00e9taux nettoiera le fer et aura des instruments faits pour cela ; toi-m\u00eame, lorsque tu seras pour manger, tu laveras ton plat de bois, si tu n'es pas compl\u00e8tement sale et malpropre ; et tu ne laverais ni ne nettoierais ton corps ! \u00ab Pourquoi le ferais-je ? \u00bb dis-tu. Je te r\u00e9pondrai : \u00ab D'abord pour te conduire en homme ; puis pour ne pas incommoder ceux qui se trouvent avec toi. \u00bb Car c'est l\u00e0 ce que tu fais maintenant, sans t'en apercevoir. Tu trouves convenable de t'empester toi-m\u00eame ; soit ! Je veux bien que ce soit convenable. Mais l'est-il \u00e9galement d'empester ceux qui s'asseyent pr\u00e8s de toi, ceux qui couchent avec toi, ceux qui te baisent ? Ou va-t'en dans un d\u00e9sert, ce qui est ta place ; ou vis seul, \u00e0 n'empester que toi ! Il est bien juste que tu aies seul la jouissance de ta malpropret\u00e9. Mais, quand tu es dans une ville, vivre avec cette n\u00e9gligence et cette stupidit\u00e9, de qui crois-tu que ce soit le fait ? Si la nature t'avait confi\u00e9 un cheval, le laisserais-tu ainsi sans soins ? Regarde aujourd'hui ton corps comme un cheval qu'on a remis entre tes mains ; lave-le, essuie-le ; fais que personne ne s'en d\u00e9tourne, que personne ne s'en recule. Qu'est-ce qui ne se recule pas d'un homme sale, d'un homme qui sent, d'un homme qui pue, encore plus que d'un individu couvert d'ordures ? La puanteur dans ce dernier cas nous vient du dehors ; mais celle qui na\u00eet de notre incurie vient de nous : elle ressemble \u00e0 celle d'une charogne.\n\n\u2014 Mais Socrate se lavait rarement ! \u2014 Oui, mais son corps reluisait ; mais ce corps \u00e9tait si agr\u00e9able et si attrayant, que les plus jeunes et les plus nobles s'en \u00e9prenaient, et auraient mieux aim\u00e9 coucher avec lui qu'avec les plus beaux gar\u00e7ons. Il aurait eu le droit de ne pas se baigner, de ne pas se laver, s'il avait voulu ; et, si peu qu'il le fit, le r\u00e9sultat y \u00e9tait. Si tu ne veux pas qu'il se baign\u00e2t \u00e0 l'eau chaude, il se baignait du moins dans l'eau froide. \u2014 Mais, il y a contre lui le mot d'Aristophane : \u00ab Je parle de ces gens p\u00e2les et sans chaussures. \u00bb\n\n\u2014 Mais Aristophane a dit aussi que Socrate marchait dans l'air, et volait les habits dans les gymnases ! Et tous ceux qui ont \u00e9crit sur Socrate en rapportent tout le contraire, qu'il n'\u00e9tait pas seulement s\u00e9duisant \u00e0 entendre, mais encore \u00e0 voir. On a \u00e9crit la m\u00eame chose sur Diog\u00e8ne aussi. C'est qu'en effet il ne faut pas \u00e9loigner le vulgaire de la philosophie par l'aspect de notre corps, mais nous montrer \u00e0 ses yeux dispos et heureux dans notre corps comme dans le reste. \u00ab Voyez, \u00f4 mortels, que je n'ai rien et que je n'ai besoin de rien ! Voyez comment sans maison, sans patrie, exil\u00e9, s'il le faut, et sans feu ni lieu, je vis plus heureux et plus calme que tous vos Eupatrides et tous vos riches. Voyez aussi mon corps, qui ne souffre en rien de ma vie s\u00e9v\u00e8re. \u00bb Si quelqu'un me parlait ainsi avec l'air et la mine d'un condamn\u00e9, quel est le Dieu qui pourrait me persuader de m'attacher \u00e0 un philosophe qui rendrait les gens tels ? Que le ciel m'en pr\u00e9serve ! Je m'y refuserais, alors m\u00eame que je devrais y devenir un sage.\n\nPour moi, par tous les dieux ! j'aime mieux que le jeune homme qui vient \u00e0 moi pour la premi\u00e8re fois, s'y pr\u00e9sente bien fris\u00e9, que sale et les cheveux en d\u00e9sordre. On voit du moins en lui quelque id\u00e9e du Beau, quelque amour de ce qui sied. Il le cherche o\u00f9 il croit qu'il est. On n'a plus qu'\u00e0 lui montrer o\u00f9 il est, et \u00e0 lui dire : \u00ab Jeune homme, tu cherches le Beau, et tu fais bien. Sache donc qu'il est pour toi o\u00f9 est ta raison. Cherche-le o\u00f9 est ta facult\u00e9 de vouloir et de repousser, de d\u00e9sirer et de fuir. Car c'est l\u00e0 chez toi ce qui a de la valeur ; pour ton corps, il n'est que boue de sa nature. \u00c0 quoi bon te donner pour lui des peines inutiles ? Le temps, \u00e0 d\u00e9faut d'autre chose, t'apprendra qu'il n'est rien. Mais si celui qui vient \u00e0 moi est couvert d'ordures et de salet\u00e9s, avec une barbe qui lui descend jusqu'aux genoux, que puis-je lui dire ? Par quelles analogies l'amener o\u00f9 je veux ? Apr\u00e8s quoi a-t-il couru qui ressembl\u00e2t au Beau, pour que je n'aie qu'\u00e0 le changer de direction, et \u00e0 lui dire : \u00ab Le Beau n'est pas l\u00e0, mais ici ? \u00bb Veux-tu que je lui dise : \u00ab Le Beau n'est pas dans la salet\u00e9, mais dans la raison ? \u00bb Est-ce qu'il se soucie du Beau ? Est-ce qu'il en a en lui quelque id\u00e9e ? Va-t'en donc disputer avec un pourceau, pour qu'il ne se roule pas dans la fange ! C'est gr\u00e2ce \u00e0 cela que les discours de X\u00e9nocrate ont touch\u00e9 Pol\u00e9mon : le jeune homme aimait le Beau. Quand il entra dans l'\u00e9cole, il avait en lui le principe de l'amour du Beau ; seulement, il cherchait le Beau o\u00f9 il n'\u00e9tait pas.\n\nIl n'y a pas jusqu'aux animaux qui vivent avec l'homme, que la nature n'ait faits propres. Est-ce le cheval qui se roule dans la fange ? Est-ce un chien de noble race ? Non, mais le pourceau, mais les sales oies, mais les vers, mais les araign\u00e9es, tout ce qu'il y a de fait pour vivre le plus loin de l'homme. Et toi, qui es un homme, voudras-tu n'\u00eatre m\u00eame pas un des animaux qui vivent avec l'homme ? Aimeras-tu mieux \u00eatre un ver ou une araign\u00e9e ? Ne te laveras-tu donc jamais, quel que soit le mode que tu pr\u00e9f\u00e8res ? Ne te baigneras-tu jamais ? Ne voudras-tu pas nous arriver propre, pour que l'on soit heureux d'\u00eatre avec toi ? Entreras-tu avec nous en pareil \u00e9tat dans ces temples, o\u00f9 il n'est permis de cracher ni de se moucher, toi qui n'es que morve et que crachat ?\n\n\u2014 Quoi donc ! doit-on vouloir se faire beau ? \u2014 \u00c0 Dieu ne plaise ! si ce n'est dans ce qui est nous par nature, dans notre raison, dans nos jugements, dans nos actes ; quant au corps, il ne faut s'en occuper que pour qu'il soit propre et ne choque personne. Parce qu'on t'aura dit qu'il ne faut pas porter de v\u00eatements \u00e9carlates, vas-tu couvrir ton manteau d'ordures ou le mettre en loques ? \u2014 Et d'o\u00f9 pourrais-je avoir un beau manteau ? \u2014 Homme, tu as de l'eau ; laves-y le tien. \u00d4 l'aimable jeune homme ! \u00d4 le vieillard fait pour aimer et pour \u00eatre aim\u00e9, \u00e0 qui on am\u00e8nera son fils pour qu'il l'instruise, que les jeunes filles et les jeunes gar\u00e7ons viendront trouver au besoin, et qui leur fera la le\u00e7on sur un tas de fumier ! Toute aberration a sa source dans quelque c\u00f4t\u00e9 de la nature humaine ; mais celle-ci est bien pr\u00e8s de n'avoir rien d'humain.\n\nXII. DE L'ATTENTION\n\nSi tu te rel\u00e2ches un instant de ton attention sur toi-m\u00eame, ne t'imagine pas que tu la retrouveras, lorsque tu le voudras. Dis-toi, au contraire, que, par suite de ta faute d'aujourd'hui, tes affaires d\u00e9sormais seront forc\u00e9ment en plus mauvais \u00e9tat. Car d'abord, et c'est ce qu'il y a de plus triste, l'habitude nous vient de ne pas veiller sur nous-m\u00eames, puis l'habitude de diff\u00e9rer d'y veiller, en remettant et reportant sans cesse \u00e0 un autre jour d'\u00eatre heureux, d'\u00eatre vertueux, de vivre et de nous conduire conform\u00e9ment \u00e0 la nature. S'il est utile de le remettre, il sera bien plus utile encore d'y renoncer compl\u00e8tement ; et, s'il n'est pas utile d'y renoncer, pourquoi ne pas continuer \u00e0 veiller constamment sur soi ? \u2014 Aujourd'hui je veux jouer ! \u2014 Eh bien ! ne dois-tu pas le faire en veillant sur toi ? \u2014 Je veux chanter. \u2014 Qu'est-ce qui t'emp\u00eache de le faire en veillant sur toi ? Est-il dans notre vie une chose exceptionnelle, \u00e0 laquelle l'attention ne puisse s'\u00e9tendre ? En est-il une que nous g\u00e2tions par l'attention, que nous am\u00e9liorions en n'\u00e9tant pas attentif ? Est-il quoi que ce soit, dans la vie, qui gagne au d\u00e9faut d'attention ? Le charpentier construit-il plus parfaitement en ne faisant pas attention ? Le pilote, en ne faisant pas attention, conduit-il plus s\u00fbrement ? Est-il quelqu'un des travaux les moins importants qui s'ex\u00e9cute mieux sans l'attention ? Ne sens-tu pas qu'une fois que tu as l\u00e2ch\u00e9 la bride \u00e0 tes pens\u00e9es, il n'est pas en ton pouvoir de les reprendre en mains, pour \u00eatre honn\u00eate, d\u00e9cent et r\u00e9serv\u00e9 ? Loin de l\u00e0 : tu fais d\u00e8s lors tout ce qui se pr\u00e9sente \u00e0 ton esprit, tu c\u00e8des \u00e0 toutes tes tentations.\n\n\u00c0 quoi donc me faut-il faire attention ? D'abord \u00e0 ces principes g\u00e9n\u00e9raux, qu'il te faut avoir toujours pr\u00e9sents \u00e0 la pens\u00e9e, et sans lesquels tu ne dois ni dormir, ni te lever, ni boire, ni manger, ni te r\u00e9unir aux autres hommes : \u00ab Personne n'est le ma\u00eetre du jugement ni de la volont\u00e9 d'autrui ; et c'est dans eux seuls qu'est le bien et le mal. \u00bb Il n'y a donc pas de ma\u00eetre qui puisse me faire du bien ou me causer du mal ; sur ce point je ne d\u00e9pends que de moi seul. Puis donc qu'il y a s\u00e9curit\u00e9 pour moi sur ce point, qu'ai-je \u00e0 me tourmenter pour les choses du dehors ? Pourquoi craindre un tyran, la maladie, la pauvret\u00e9, un \u00e9cueil quelconque ? Je n'ai pas plu \u00e0 un tel ! Est-ce donc lui qui est ma fa\u00e7on d'agir ? Est-ce lui qui est ma fa\u00e7on de juger ? Non. Que m'importe d\u00e8s lors ! Mais il para\u00eet \u00eatre un personnage ! C'est son affaire, et celle des gens qui le prennent pour tel. Pour moi j'ai \u00e0 qui plaire, \u00e0 qui me soumettre, \u00e0 qui ob\u00e9ir : c'est Dieu, et ceux qui viennent apr\u00e8s lui. C'est moi-m\u00eame que Dieu a pr\u00e9pos\u00e9 \u00e0 ma garde ; c'est \u00e0 moi seul qu'il a soumis ma facult\u00e9 de juger et de vouloir ; et il m'a donn\u00e9 des r\u00e8gles pour en bien user. Lorsque je les applique aux syllogismes, je ne me pr\u00e9occupe pas de ceux qui parlent autrement ; lorsque je les applique aux raisonnements \u00e9quivoques, je ne m'inqui\u00e8te de personne ; pourquoi donc dans les choses plus importantes les critiques me font-elles de la peine ? Qu'est-ce qui fait que je me trouble ainsi ? Une seule chose : c'est que je ne me suis pas exerc\u00e9 sur ce point-l\u00e0. Quiconque sait, en effet, d\u00e9daigne l'ignorance et les ignorants ; et je ne parle pas seulement des savants, mais aussi des gens de m\u00e9tiers. Am\u00e8ne-moi le savetier que tu voudras, et dans ce qui est de son art il se moquera de tout le monde. Am\u00e8ne-moi de m\u00eame le charpentier que tu voudras.\n\nIl faut, avant tout, avoir ces id\u00e9es pr\u00e9sentes \u00e0 la pens\u00e9e, et ne rien faire qui soit en contradiction avec elles ; il faut bander son \u00e2me vers ce but, de ne poursuivre aucune des choses qui sont hors de nous, aucune de celles qui ne sont pas \u00e0 nous. Acceptons-les comme en dispose celui qui a pouvoir sur elles. Les choses qui rel\u00e8vent de notre libre arbitre, il faut les vouloir sans restriction, mais les autres, comme on nous les donne. Il faut de plus nous rappeler qui nous sommes, et quel est notre nom, et nous efforcer de faire ce qui convient dans chaque situation. Demandons-nous quand il est \u00e0 propos de chanter, \u00e0 propos de jouer, et devant quelles personnes ; qu'est-ce qui est hors de saison ; qu'est-ce qui nous ferait m\u00e9priser des assistants ou prouverait de notre part du m\u00e9pris pour eux ; quand faut-il plaisanter ; qui faut-il railler ; en quoi et pour qui faut-il avoir de la condescendance ; puis dans cette condescendance comment faut-il faire pour sauver notre dignit\u00e9 ? Quand tu te seras \u00e9cart\u00e9 des convenances sur un de ces points, le ch\u00e2timent te viendra tout de suite, non pas du dehors, mais de ton acte m\u00eame.\n\nQuoi donc ! peut-on \u00eatre infaillible ? Non pas ; mais il est une chose que l'on peut, c'est de s'efforcer constamment de ne pas faire de faute. Et il faut nous trouver heureux, si, en ne nous rel\u00e2chant jamais de cette attention sur nous-m\u00eames, nous \u00e9chappons \u00e0 un certain nombre de fautes. Mais dire maintenant : \u00ab Je ferai attention demain \u00bb, sache que c'est dire : \u00ab Aujourd'hui je serai sans retenue, sans convenance, sans dignit\u00e9 ; il sera au pouvoir des autres de me faire de la peine ; je vais \u00eatre aujourd'hui col\u00e8re et envieux. \u00bb Vois que de maux tu attires l\u00e0 sur toi ! Si l'attention doit t'\u00eatre bonne demain, combien plus le sera-t-elle aujourd'hui ! Si demain elle doit t'\u00eatre utile, elle le sera bien plus aujourd'hui. Veille sur toi aujourd'hui pour en \u00eatre capable demain, et ne pas le remettre encore au surlendemain.\n\nXIII. POUR CEUX QUI PARLENT TROP AIS\u00c9MENT D'EUX-M\u00caMES\n\nLorsque quelqu'un semble nous parler de ses affaires \u00e0 c\u0153ur ouvert, nous sommes entra\u00een\u00e9s, nous aussi, \u00e0 lui r\u00e9v\u00e9ler nos secrets ; et nous croyons que cela est tout simple : d'abord parce qu'il nous para\u00eet contraire \u00e0 l'\u00e9quit\u00e9 d'\u00e9couter les affaires de notre prochain, sans lui faire part \u00e0 son tour des n\u00f4tres ; puis, parce que nous croyons que nous ne ferions pas aux autres l'effet d'un homme franc, si nous nous taisions sur nous-m\u00eames. Que de fois certes on nous dit : \u00ab Moi, je t'ai dit toutes mes affaires ; et toi, tu ne veux me rien dire des tiennes ! D'o\u00f9 cela vient-il ? \u00bb Ajoutez-y qu'on croit pouvoir se confier en toute s\u00fbret\u00e9 \u00e0 qui vous a d\u00e9j\u00e0 confi\u00e9 ses affaires ? Car la pens\u00e9e nous vient que cet homme ne contera jamais les n\u00f4tres, de peur que nous aussi nous ne contions les siennes. C'est ainsi qu'\u00e0 Rome les gens trop prompts \u00e0 parler se font attraper par les soldats. Un soldat vient s'asseoir aupr\u00e8s de toi sous l'habit d'un bourgeois ; il se met \u00e0 parler mal de C\u00e9sar, et toi, comme s'il t'avait donn\u00e9 un gage de sa bonne foi, en \u00e9tant le premier au d\u00e9nigrement, tu dis \u00e0 ton tour tout ce que tu penses ; on te garrotte alors, et on t'emm\u00e8ne. C'est l\u00e0 l'image de ce qui nous arrive \u00e0 tous. Parce qu'un homme s'est confi\u00e9 \u00e0 moi en toute s\u00fbret\u00e9, puis-je de m\u00eame, moi, me confier au premier venu ? Si je suis ce que je suis, je me tais, moi, sur ce qu'il m'a dit. Mais lui, il va conter \u00e0 tout le monde ce que je lui ai dit. Cela fait, si je lui ressemble, je veux me venger, quand j'apprends la chose, et je conte ses affaires ; je l'ab\u00eeme, et il m'ab\u00eeme. Si je me dis, au contraire, que personne ne peut nuire \u00e0 un autre, et qu'il n'y a que nos actes propres qui nous nuisent ou qui nous soient utiles, je parviens bien \u00e0 ne pas faire comme lui, mais ce qui m'est arriv\u00e9 par suite de mon bavardage, ne m'en est pas moins arriv\u00e9.\n\n\u2014 Soit ! Mais il est contraire \u00e0 l'\u00e9quit\u00e9 d'\u00e9couter les secrets de son prochain, sans lui faire part \u00e0 son tour de quoi que ce soit ! \u2014 \u00d4 homme, est-ce que je t'ai provoqu\u00e9 \u00e0 parler ? Lorsque tu m'as livr\u00e9 tes secrets, y a-t-il eu convention que tu entendrais les miens \u00e0 ton tour ? Si tu es un bavard, et si tu prends pour des amis tous ceux que tu rencontres, veux-tu que je te ressemble ? Quoi donc ! si tu as pu sans danger te confier \u00e0 moi, mais si l'on ne peut sans danger se confier \u00e0 toi, veux-tu que je tombe dans le pi\u00e8ge ? C'est comme si j'avais un tonneau bien solide, toi un tonneau perc\u00e9, que tu vinsses m'apporter ton vin pour le mettre dans mon tonneau, et que tu t'indignasses ensuite de ce que je ne voudrais pas te confier mon vin. Ma raison serait que tu as un tonneau perc\u00e9. Comment y aurait-il \u00e9galit\u00e9 ? Tu te livres \u00e0 un homme s\u00fbr, \u00e0 un homme honn\u00eate, qui croit que ses actes seuls peuvent lui \u00eatre utiles ou nuisibles, et que toutes les choses du dehors ne sont rien ; et tu veux que je me livre \u00e0 toi, qui tiens pour rien ton libre arbitre, qui veux arriver \u00e0 la fortune ou \u00e0 une magistrature, ou bien faire ton chemin \u00e0 la cour, quand tu devrais pour cela \u00e9gorger tes enfants, \u00e0 la fa\u00e7on de M\u00e9d\u00e9e ? Quelle \u00e9galit\u00e9 y a-t-il l\u00e0 ? Montre-moi que tu es un homme s\u00fbr, honn\u00eate, in\u00e9branlable ; montre-moi que tes id\u00e9es sont bienveillantes ; montre-moi que ton vase n'est pas perc\u00e9 ; et tu verras que je n'attendrai pas que tu me confies tes secrets, mais que j'irai moi-m\u00eame vers toi pour te prier d'\u00e9couter les miens. Qui, en effet, ne voudrait pas se servir d'un vase en bon \u00e9tat ? Qu'est-ce qui fait fi d'un conseiller bienveillant et s\u00fbr ? Qu'est-ce qui n'accueillerait pas volontiers celui qui vient pour ainsi dire prendre sa part du fardeau de vos affaires, et vous le rendre plus l\u00e9ger par cela seul qu'il en prend sa part ?\n\n\u2014 Oui ; mais, quand j'ai confiance en toi, n'auras-tu pas confiance en moi ? \u2014 D'abord, tu n'es pas un homme qui ait confiance en moi ; mais un bavard, qui ne peut rien garder. Car, s'il en \u00e9tait ce que tu dis, tu ne confierais tes secrets qu'\u00e0 moi seul. Or, aujourd'hui, d\u00e8s que tu vois quelqu'un inoccup\u00e9, tu vas t'asseoir \u00e0 ses c\u00f4t\u00e9s et tu lui dis : \u00ab Fr\u00e8re, je n'ai personne qui m'aime plus que toi ni qui me soit plus chef ; je te prie donc d'\u00e9couter mes secrets. \u00bb Et cela, tu le fais \u00e0 des gens que tu ne connais pas le moins du monde.\n\nSi tu as cependant confiance en moi, il est \u00e9vident que c'est parce que je suis s\u00fbr et honn\u00eate, et non point parce que je t'ai cont\u00e9 mes affaires.\n\nLaisse-moi donc \u00eatre dans les m\u00eames id\u00e9es. Montre-moi que, par cela seul que l'on conte ses affaires, on est s\u00fbr et honn\u00eate. Car, en ce cas, je m'en irais partout dire \u00e0 tout le monde mes secrets, si je devais \u00e0 ce prix \u00eatre s\u00fbr et honn\u00eate. Mais les choses ne vont pas ainsi ; et ce qu'il faut pour \u00eatre tel, ce sont des principes qui ne sont pas les premiers venus. Si donc tu vois quelqu'un s'attacher aux choses qui ne d\u00e9pendent pas de son libre arbitre, et leur soumettre ce libre arbitre m\u00eame, sache que cet homme a des milliers d'individus qui peuvent le contraindre ou l'emp\u00eacher d'agir. Il n'y a pas besoin d'employer la poix ou la roue pour lui faire dire ce qu'il sait ; un signe d'une femme le fera parler au besoin, ou bien les caresses d'un ami de C\u00e9sar, le d\u00e9sir d'une charge, d'un h\u00e9ritage, et mille autres choses de cette esp\u00e8ce.\n\nIl faut donc se rappeler, comme r\u00e8gle g\u00e9n\u00e9rale, que les secrets demandent un homme s\u00fbr, avec des principes qui le soient aussi. Mais o\u00f9 trouver cela facilement aujourd'hui ? Que l'on me montre un homme capable de dire : \u00ab Je ne m'inqui\u00e8te que des choses qui sont \u00e0 moi, que nul ne peut emp\u00eacher, et qui sont libres de leur nature ; c'est l\u00e0 qu'est pour moi le bien r\u00e9el ! Que les autres arrivent comme elles se trouvent ; j'y suis indiff\u00e9rent. \u00bb\n\u00c9pict\u00e8te\n\n\u00c9pict\u00e8te na\u00eet \u00e0 Hi\u00e9rapolis, en Phrygie (actuellement en Turquie), en 50 ; d\u00e8s son enfance, il est emmen\u00e9 \u00e0 Rome comme esclave, d'o\u00f9 lui vient son nom qui en grec signifie \u00ab homme achet\u00e9 \u00bb, \u00ab serviteur \u00bb. Selon la l\u00e9gende, il se fait remarquer en se laissant arracher la jambe par son ma\u00eetre \u00c9paphrodite, premi\u00e8re preuve de fermet\u00e9 et de ma\u00eetrise de soi. Apr\u00e8s avoir \u00e9t\u00e9 affranchi \u00e0 la mort d'\u00c9paphrodite, il s'adonne \u00e0 l'\u00e9tude du sto\u00efcisme, mais doit quitter Rome \u00e0 la suite d'un d\u00e9cret de l'empereur Domitien contre les philosophes en 90.\n\nIl s'installe alors \u00e0 Nicopolis, en \u00c9pire (les Balkans), o\u00f9 il commence une vie plus calme, en compagnie de sa femme. Il se fait conna\u00eetre en enseignant la philosophie sto\u00efcienne, fond\u00e9e par les philosophes grecs Chrysippe et Z\u00e9non ; parmi ses \u00e9l\u00e8ves, on compte Julius Rusticus, qui deviendra le ma\u00eetre de l'empereur et philosophe sto\u00efcien Marc Aur\u00e8le. Il meurt \u00e0 Nicopolis en 125 ou 130.\n\nComme Socrate, \u00c9pict\u00e8te n'a laiss\u00e9 aucun \u00e9crit, mais un de ses \u00e9l\u00e8ves, Arrien (il sera plus tard consul et historien), se charge de transcrire sa doctrine, avec concision dans le Manuel (l'Enchiridion) qui se pr\u00e9sente comme un ensemble de r\u00e8gles pratiques, et d'une fa\u00e7on beaucoup plus d\u00e9taill\u00e9e dans les huit livres des Entretiens, dont quatre seulement nous sont parvenus. Ces deux ouvrages, \u00e9crits en grec, ne seront pas traduits en latin avant la fin du Moyen \u00c2ge.\n\nLa vie d'\u00c9pict\u00e8te reste tr\u00e8s mal connue et son \u0153uvre, qui op\u00e8re la synth\u00e8se de diff\u00e9rentes tendances, sera quasiment oubli\u00e9e au Moyen \u00c2ge ; cependant, il redevient un mod\u00e8le de sagesse avec Montaigne et Pascal. \u00c0 la diff\u00e9rence des autres grands repr\u00e9sentants du sto\u00efcisme \u00e0 Rome, l'orateur Cic\u00e9ron, le pr\u00e9cepteur de la famille imp\u00e9riale S\u00e9n\u00e8que ou l'empereur Marc Aur\u00e8le, \u00c9pict\u00e8te a pass\u00e9 sa vie loin des fastes et des honneurs, dans la pauvret\u00e9 et l'humilit\u00e9. De m\u00eame que ses deux mod\u00e8les, Socrate et le cynique Diog\u00e8ne, il repr\u00e9sente ainsi le type du philosophe dont l'existence fut en ad\u00e9quation avec les principes, qui ne se soucia pas de devenir c\u00e9l\u00e8bre et chercha surtout \u00e0 proposer une sagesse pratique et exemplaire.\n\nTable\n\nDans la m\u00eame s\u00e9rie\n\nManuel d'\u00c9pict\u00e8te\n\nEntretiens (extraits)\n\nLivre premier\n\nII. Comment on peut conserver sa dignit\u00e9 en toute chose\n\nVIII. Les talents des ignorants ne sont pas sans p\u00e9rils\n\nXXI. Contre ceux qui veulent se faire admirer\n\nLivre deuxi\u00e8me\n\nIV. Sur un homme qui avait \u00e9t\u00e9 surpris en adult\u00e8re\n\nXV. Sur les gens qui persistent obstin\u00e9ment dans ce qu'ils ont d\u00e9cid\u00e9\n\nXVIII. Comment il faut lutter contre les id\u00e9es dangereuses\n\nXXIV. \u00c0 quelqu'un qu'il n'estimait pas\n\nLivre troisi\u00e8me\n\nIV. Contre ceux qui, au th\u00e9\u00e2tre, donnent des marques inconvenantes de faveur\n\nV. Contre ceux qui partent parce qu'ils sont malades\n\nX. Comment doit-on supporter les maladies ?\n\nXII. De l'exercice\n\nXIII. Qu'est-ce que c'est que l'abandon ? Et qu'est-ce qui est abandonn\u00e9 ?\n\nXVI. Qu'il faut y regarder \u00e0 deux fois avant de se laisser entra\u00eener \u00e0 une liaison\n\nXXIII. Contre ceux qui lisent ou discutent par d\u00e9sir de se montrer\n\nXXVI. \u00c0 ceux qui craignent la pauvret\u00e9\n\nLivre quatri\u00e8me\n\nII. Sur nos liaisons\n\nV. Contre les gens querelleurs et m\u00e9chants\n\nXI. De la propret\u00e9\n\nXII. De l'attention\n\nXIII. Pour ceux qui parlent trop ais\u00e9ment d'eux-m\u00eames\n\nFiche biographique\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}} +{"text":" \nWHY?\n\nExplaining \nthe Holocaust\n\nPETER HAYES\n\nW. W. NORTON & COMPANY\n\n_Independent Publishers Since 1923_\n\n_New York_ | _London_\nIN GRATEFUL MEMORY OF INSPIRING TEACHERS:\n\n_Mary Faherty_\n\n_James McGillivray_\n\n_Athern Park Daggett_\n\n_John C. Rensenbrink_\n\n_Timothy W. Mason_\n\n_Henry Ashby Turner Jr_.\nCONTENTS\n\nINTRODUCTION: Why Another Book on the Holocaust?\n\n1. TARGETS: Why the Jews?\n\n_Antisemitism_\n\n_Emancipation and Backlash_\n\n2. ATTACKERS: Why the Germans?\n\n_Nation and_ Volk\n\n_Hitler's Opportunity_\n\n3. ESCALATION: Why Murder?\n\n_From Aryanization to Atrocity_\n\n_Gentile and Jewish Responses_\n\n4. ANNIHILATION: Why This Swift and Sweeping?\n\n_From Bullets to Gas_\n\n_Perpetrators: the \"generation without limits\"_\n\n_Enslavement_\n\n5. VICTIMS: Why Didn't More Jews Fight Back More Often?\n\n_Compliance and Resistance_\n\n_The World of the Camps_\n\n6. HOMELANDS: Why Did Survival Rates Diverge?\n\n_Varieties of Behavior_\n\n_The Case of Poland_\n\n7. ONLOOKERS: Why Such Limited Help from Outside?\n\n_Prewar Evasions_\n\n_Wartime Priorities_\n\n8. AFTERMATH: What Legacies, What Lessons?\n\n_Return, Resettlement, Retribution, and Restitution_\n\n_Memory, Myths, and Meanings_\n\nACKNOWLEDGMENTS\n\nNOTES\n\nSELECTED BIBLIOGRAPHY\n\nINDEX\nLIST OF FIGURES\n\n1. THE OVERLAPPING LAYERS OF ANTISEMITISM\n\n2. ANTISEMITIC VOTING IN IMPERIAL GERMANY\n\n3. ANTISEMITIC VOTING FOR THE REICHSTAG IN POST\u2013WORLD WAR I GERMANY\n\n4. THE GEOGRAPHICAL COMPRESSION OF THE HOLOCAUST\n\n5. THE FATE OF A GHETTO: LODZ, 1940\u201344\n\n6. TWO PHOTOS FROM _T HE STROOP REPORT_\n\n7. GOVERNANCE AND HOLOCAUST MORTALITY RATES\n\n8. ROMANIA, 1941\u201344\nINTRODUCTION\n\n[Why Another Book \non the Holocaust?](contents.xhtml#intro_1)\n\nSEVENTY YEARS AFTER the Holocaust ended, it continues to resist comprehension. Despite (or maybe because of) the outpouring of some sixteen thousand books cataloged at the Library of Congress under the heading, despite the proliferation of museums and memorials, despite the annual appearance of new cinematic treatments, and despite the spread of educational programs and courses devoted to the subject, a coherent explanation of why such ghastly carnage erupted from the heart of civilized Europe in the twentieth century seems still to elude people. Indeed, perhaps the adjectives most frequently invoked in connection with the Holocaust are \"unfathomable,\" \"incomprehensible,\" and \"inexplicable.\" These words attest to a distancing reflex, an almost instinctive recoiling in self-defense. To say that one can explain the occurrence of the Holocaust seems tantamount to normalizing it, but professing that one cannot grasp it is an assertion of the speaker's innocence\u2014of his or her incapacity not only to conceive of such horror but to enact anything like it. Small wonder that incomprehension is the default position in the face of the enormity of the Holocaust, even though that stance blocks the possibility of learning from the subject.\n\nSelf-protection is not the only reason for the enduring difficulty of coming to intellectual grips with the Holocaust, however. Another is the complexity of the task. To understand the Holocaust requires solving multiple puzzles that surround it. In the course of teaching the subject to American undergraduates for almost three decades and lecturing widely to both academic and general audiences, I have come to recognize eight central issues that people grappling with the subject find most perplexing. Some of these issues involve acts of commission, some concern acts of omission, and still others entail both. All require interlocking clarification before a mind can comprehend and account for the cataclysm. Each chapter of this book examines one of those eight central issues, raised in the form of a question, and the book as a whole reflects my conviction that the Holocaust is no less historically explicable than any other human experience, though the job is not easy.\n\nIn answering these questions, I bring to bear expertise that is unusual among students of the Holocaust. I am by training an economic historian. That does not mean that I foreground material motivations for murder (in fact, I contend that they were secondary to ideological motives). My background makes me, however, alert to numbers and their significance, and I deploy them frequently in order to demonstrate their powerful interpretive effects. A second distinguishing feature of my account is its dialectical origins. This is not a book driven by a thesis that the author wants to prove but rather a work that emerged out of the give-and-take of many years of teaching and public speaking, during which I learned which aspects of the subject people most want clarified and why, directed my reading and thinking toward identifying the most reliable responses that scholarship can offer, and then honed ways to make that knowledge as accessible and memorable as I could.\n\nAlongside its explanatory purpose, this book also has another goal: to set the record straight. As the late historian Tony Judt observed, \"Impossible to remember as it really was, [the Holocaust] is inherently vulnerable to being remembered as it wasn't.\" Numerous myths have grown up around the subject, many designed to console us that things could have gone much differently if only some person or entity had acted more bravely or wisely, others intended to cast new blame on favorite or surprising villains or even on historians of the subject. This book dispels many such legends\u2014from the notion that antisemitism brought Adolf Hitler to power in Germany to the belief that a large number of major perpetrators of the Holocaust escaped punishment afterward\u2014and the final chapter reviews and debunks the most prevalent ones, including the loud and recurrent claim that the Holocaust never happened.\n\nThe arc of the book's argument is as follows. The Holocaust was the product of a particular time and place: Europe in the aftermath of the Industrial Revolution and the upheavals of World War I and the Bolshevik Revolution. These were the contexts in which ancient hostilities toward Jews and Judaism, deeply rooted in religious rivalry but updated with the trappings of modern science, turned into a fixation on removing Jews from civil society as a magical solution to all social problems. Germany was where the fault lines of disruption lifted this belief to power during the 1930s, but the murder of the Jews of Europe was neither pre-programmed by German history nor an exclusively German project. The massacre took shape under specific political and military conditions and intensified in part because it suited the objectives of many other Europeans, at least during the short, ferocious period when most of the killing occurred. In the face of the slaughter, the victims were largely powerless and the onlookers preoccupied with their own, to them more pressing concerns. The parts of the trap that clicked into place around European Jews during the Nazi era fit so tightly that only a minority could escape, most only just barely and in the nick of time. Afterward, most countries of the old continent delayed acknowledging what they had participated in yet also constructed numerous barriers to its repetition, barriers that now, seventy years later, are under stress.\n\nThe pace of specialist research on the Holocaust has outrun most people's ability to keep up and to integrate new findings into a general interpretive picture. Even as misleading notions about the subject have gained ground, antiquated ones linger. Given this, people interested in the subject need a comprehensive stocktaking directed squarely at answering the most central and enduring questions about why and how the massacre of European Jewry unfolded. That is what this book offers.\nWHY?\nCHAPTER 1\n\n[TARGETS: \nWhy the Jews?](contents.xhtml#ch_1)\n\nOUTBREAKS OF HOSTILITY against minorities are almost always rooted both in ideas\u2014what the majority thinks about the minority\u2014and circumstances: the ways in which or the terms on which the two groups are interacting at a particular time. In order to explain why the Jews became objects of murderous intentions in the twentieth century, one has to look at both sorts of roots.\n\nANTISEMITISM\n\nNowadays the term usually used to describe hostility to Jews is antisemitism. A professor of mine in graduate school used to say that the problem with the word is that it is a single term for lots of different attitudes\u2014covering everything from telling crude jokes about Jews to desiring to kill them. He had a point, but a working definition is nonetheless possible. Mine goes like this: Antisemitism is a categorical impugning of Jews as collectively embodying distasteful and\/or destructive traits. In other words, antisemitism is the belief that Jews have common repellent and\/or ruinous qualities that set them apart from non-Jews. Descent is determinative; individuality is illusory.\n\nThat attitude has a long history; indeed, a famous book by Robert Wistrich and a widely seen documentary film on the subject are called _The Longest Hatred_. But that title is misleading for two reasons. First, if Jews have long been hated in Western culture, they have not been equally hated at all times and in all places; and second, the hatred has exhibited a good deal of shape-shifting. In fact, the name that we now use to describe prejudice against or hatred of Jews illustrates both points. The word \"antisemitism\" appeared only in 1879, and its popularization is usually ascribed to Wilhelm Marr, a German agitator who intended it to describe something new and different from previous forms of hostility toward Jews. Like other \"-ism\" words that appeared in abundance in the nineteenth century, the word was chosen to suggest that this new hostility was about politics and science. Note what the word claimed to be against: Not Jews but something called Semitism. What was that? Unlike the other targets of nineteenth century \"anti-\" movements (for example, antisocialism, anticommunism, anti-Catholicism, antivivisectionism, even antidisestablishmentarianism), this term did not combat a belief system that had named itself, but instead invented the phenomenon being opposed. Self-styled \"antisemites\" borrowed a category from linguistics, and they did so misleadingly. They claimed to combat Semites\u2014speakers of the Semitic family of languages, which differed in syntax and grammatical structure from the so-called Indo-European family of languages that predominated in Europe. But actually not all Semites were targeted, since Arabs generally were not included, though Arabic is a Semitic language. Neither were the modern speakers of Aramaic, the language Jesus spoke, though it is also a Semitic tongue. In the late 1930s and early 1940s, the Nazi regime implicitly conceded that the new term was a lie, as Germany took pains to reassure Arab governments that it regarded their inhabitants as neither threatening nor inferior.\n\nThe target of the new \"-ism\" was Jews, and by focusing on their ancestral language and using an abstract, pseudoscientific euphemism to group them, the antisemites purported to (a) differentiate Jews authoritatively from everyone else, (b) root their difference in their very nature and thought processes, and thus (c) assert that opposition to Jews was not a mere prejudice, but a response to a demonstrable reality that had to be dealt with politically.\n\nUntil recently, English spelling has unwittingly accepted the antisemites' case, since the customary insertion of a hyphen and a capital letter in \"anti-Semitism\" implies that there is something called \"Semitism\" somewhere. The original language of the term, German, doesn't make this mistake; the spelling is _Antisemitismus_ , all one word. Nowadays, people and institutions alive to this subtle fact, such as the United States Holocaust Memorial Museum, insist on the one-word spelling of antisemitism. Neither Microsoft Word's spellcheck function nor the _Oxford English Dictionary_ has caught on yet.\n\nThe principal way in which antisemitism has evolved and varied over time has to do with the relative strength of its xenophobic and chimerical forms, as identified by Gavin Langmuir, a distinguished medievalist, but slightly redefined here. The xenophobic form sees Jews as _different_ from others in some _observable_ respects, and its adherents exhibit varying degrees of _discomfort_ with this difference. The chimerical form sees Jews as _dangerous_ to others in some _imagined_ ways, and its exponents advocate _doing something_ in response. The origins of the words highlight the distinction: _Xenos_ is Greek for \"stranger, guest;\" _chimera_ is Greek for a mythical fire-breathing monster with a lion's head, a goat's body, and a serpent's tail.\n\nAncient Roman attitudes toward Jews best illustrate the consequences of the distinction. The Roman writer Tacitus criticized the Jews for a supposedly \"stubborn attachment to one another . . . which contrasts with their implacable hatred for the rest of mankind,\" and the Romans did not like or understand such Jewish customs as monotheism, which entailed refusing to worship the emperors as gods; the Sabbath, which amounted to taking only one and the same day off from work every week; endogamy, which mandated marrying only other Jews; and circumcision of male infants as a symbol and reminder of a special arrangement with God. But the Romans did not see the Jews as especially or intrinsically dangerous, except when they resisted the empire's authority. Even after the destruction of the temple in Jerusalem in 70 CE by the army of the future Emperor Titus and the suppression of the three successive revolts against Roman rule that led to the almost complete dispersal of the Jews from ancient Judaea after 136 CE, individual Jews could and did still become Roman citizens, and they populated many different walks of life.\n\nAlthough some ancient Egyptian and Greek texts express animosity toward Jews, the rise of intense hostility to and fear of them largely coincides with the rise of Christianity. The relationship between adherents of the two religions always has reflected a paradox: The two faiths were both very similar and very different, which created intense competition. Jews saw the new religion as essentially a heresy, an erroneous variation on their theology. Christians saw themselves as embracing a new, improved version of that theology, one that superseded the old, which should be cast off as a relic of an earlier era.\n\nChristians both took over and then deviated from the central tenets of Judaism. First, they proclaimed monotheism but declared Jesus the son of God and thus divine and advanced the doctrine of the Trinity, one God in three forms. Second, Christians accepted the Hebrew Bible as revealing the word of God and incorporated it into their Bible as the Old Testament, but then added the Gospels (the \"Good News\") and other books as new revelations of God's will. Third, Christianity adapted the Jewish ideas of election and the Covenant between God and His people to new purposes. Jews believed that they and God had concluded a series of special agreements or pacts, the most famous of which are the ones with Abraham and Moses in which God promised to make the Jews his Chosen People and \"a light unto the nations\" in return for obedience to his laws. These consisted initially of the Ten Commandments, but became elaborated into the 613 central laws or _mitzvot_ \u2014actually 248 commandments and 365 prohibitions\u2014set forth in the Torah, the first five books of the Bible, which Christians call the Pentateuch. These laws covered everything from what one may eat or wear to how one should wash and worship. Christianity declared that Jesus heralded a new Covenant that replaced Moses's, that the old laws were now obsolete, and that election to the status of Chosen People was now open to anyone who accepted Christ and the teachings of the Bible and the new scriptures.\n\nOne way of understanding what followed is to recall that Jews were the people who said no. Offered a new form of relationship with God, they said they preferred the one they had, and this rejection set off several hundred years of rivalry and mutual recrimination, as the two groups competed for followers until the fourth century of the common era, when Christianity became the official religion of the Roman Empire and thus seemed to win the battle.\n\nThat brings us to figure 1, which tries in schematic fashion to capture three interwoven matters: (1) the evolving and overlapping forms of animosity to Jews that developed in Europe during successive eras once Christianity had become dominant, (2) the changing definitions of the problem Jews supposedly represented, and (3) the shifting prescriptions for fixing the situation.\n\nThe time frames specified on the chart indicate that distinct frameworks for criticizing Jews developed in those periods, but that does not mean that the new ones erased the old ones entirely. Some people remained antisemitic in the 1940s for reasons devised during the first period; in fact, some people are still antisemites for these supposed reasons. One of the most interesting recent books on the Holocaust, Alon Confino's _A World Without Jews_ (2014), in fact argues that a secularized version of the Christian claim to historical supersession was at the heart of the Nazi drive to eradicate the Jews. Instead of advancing a new religion of supersession, the Nazis saw themselves as promoting an entirely new conception of morality. Confino's claim is not entirely original. Sigmund Freud and Maurice Samuel argued similarly about the roots of antisemitism on the eve of the Holocaust, L\u00e9on Poliakov and Norman Cohn about those of Nazi racism shortly afterward. But these thinkers understood that the Nazis were not so much trying to supersede Judeo-Christian morality as to nullify or repeal it. Nazi morality was of the \"back to the future\" sort; it demanded acknowledgment that the only governing principle of life is the primordial law of the jungle and that the only measure of goodness is physical survival.\n\nFIGURE 1: THE OVERLAPPING LAYERS OF ANTISEMITISM\n\nThat the justifications of antisemitism changed over time also does not mean, despite the prestige of science, that the prejudice became more intellectually sophisticated\u2014that the later phases were more informed and intelligent. They only posed as such.\n\nThe first horizontal block in figure 1 concerns the long era of European history in which the dominant framework of thought was religious and the central question that determined or legitimized ideas and policies was \"What does God want or demand?\" The governing dilemma of Christianity during this long period of discrimination against Jews is that the Church had to do a theological balancing act between two contradictory obligations toward them, as stipulated in the \"doctrine of Jewish witness\" devised by St. Augustine, the Bishop of Hippo in North Africa, in the early fifth century: persecution and preservation. On the one hand, Augustine taught that the Church had to demonstrate \"the negation of the Jews\" and \"the election of the Christians\" by, first, emphasizing the Jews' responsibility for Christ's death as alleged in the later Gospels and, second, making the Jews' existence on earth ever more isolated and miserable as a physical representation of the consequences of rejecting Christianity. Thus, according to this part of Christian theology, the Jews had to suffer because they were religiously benighted\u2014spiritually in darkness.\n\nOn the other hand, Jesus was a Jew, and the Jews once had been God's Chosen. Augustine taught that they could not be massacred, unlike every other religious group that denied or deviated from the claim to truth of Catholic or Orthodox Christianity. Indeed, they had to be allowed to live, albeit in misery, until the wondrous day when they saw the light and converted, for that development would herald the Last Judgment and the coming of the Kingdom of Heaven. This is the explanation for a remarkable irony of this story, the survival of the Jews. They were the only religious minority whose faith remained legal in Christian Europe, whose adherents were not automatically and always slaughtered, as the Cathars, Lollards, and other dissenters were, until the Reformation split Western Christendom, and the catastrophically bloody and ultimately stalemated Wars of Religion of the sixteenth and seventeenth centuries taught Catholics and Protestants the necessity of coexistence.\n\nThe results of Augustine's dicta included, for hundreds of years, Christian condemnation of the Jews, supplemented by constant establishment of barriers to contact with Christians so that their faith could not be subverted. The relegation of Jews to pariah status, the Church hoped, would induce them to convert. In the final centuries of the Roman Empire, Jews lost the rights, first, to acquire, and later even to hold, Christian slaves, which broke the back of Jews' wealth. One after another, new laws barred Jews from proselytizing, from reversing baptisms, from cohabiting with or marrying Christians, from holding public office, and from building synagogues. Enforcement of separation between Christians and Jews was uneven in Christian Europe, but it steadily increased, eventually creating a pattern of occupational and residential ghettoization that confined Jews to certain usually despised or dangerous activities, such as moneylending or leather tanning, and certain permissible locations, also usually less than desirable ones.\n\nAs all this unfolded, the Church found that it could not quite get away with simultaneously fostering hostility to Jews and forbidding violence toward them. Ordinary people periodically lost sight of the theological reasons why adherents of this religion that denied Christ's divinity should be treated differently from all other heretics and infidels and therefore periodically lashed out at Jews, especially in times of adversity. Forced conversions and expulsions flared up as early as the seventh century, then faded away until a rash of outbreaks following the millennial year 1000 and a widespread surge of attacks surrounding the First and Second Crusades (1095\u20131149). These were generally mob actions opposed by local priests and nobles, but they became common responses to crisis events, and in twelfth-century England they acquired a legitimizing legend, namely the blood libel or ritual murder accusation that ascribed disappearances or deaths of Christian children to an alleged Jewish need for their blood to make matzoh bread for Passover or other ritual purposes. Precisely because the charge was a blatant projection onto Jews of a corrupted form of the Catholic belief in transubstantiation\u2014the creed that the communion wafer and wine become the real flesh and blood of Christ during the Mass\u2014the charge stuck, and it became the pretext for numerous massacres, first in England and later in much of Europe.\n\nBy the late Middle Ages, a correlation between social crisis and the slaughter and expulsion of Jews had become firmly established. Whenever adverse developments occurred that people could not otherwise account for, they identified Jews as the agents of Satan who had caused the problems. Massacres of Jews followed the Italian famine of 1315\u201317 and the outbreak of the Black Death in the Rhineland in 1347, for example. Such popular panics, combined with monarchical desires to confiscate Jewish property, resulted in expulsions of Jews from England and southern Italy in 1290, from France in 1306 and again in 1394, from Spain and Portugal in 1492 and 1497, and from many German cities during the fifteenth century.\n\nAs segregation and degradation increased, so did the penetration of popular culture by denigrating images of Jews. The Passion Plays performed at Easter across Christian Europe highlighted the supposed role of Jews, rather than that of Pontius Pilate, in ordering Christ's crucifixion. Chaucer's _Canterbury Tales_ , written around 1386, include a story of ritual murder by Jews in \"The Prioress's Tale.\" The story of Shylock seeking his pound of flesh that is at the core of Shakespeare's _The Merchant of Venice_ , written in the final years of the sixteenth century, had appeared in Italy more than two hundred years earlier. After 1400, church decorations included an increasing number of depictions of Jews being nursed by sows, and the first printed version of what later became the stereotypical caricature of a Jew with a beaked nose and hunched back appeared in a book of 1493.\n\nBy the time of the Reformation in the sixteenth century, hatred of Jews was widespread, and it had crystallized around two central generalizations: (1) that Jews were parasitic profiteers, intent on extracting wealth from Christians, and (2) that Jews were incorrigible instruments of Satan, intent on serving his purposes and afflicting the pious. Martin Luther gave the most extreme voice to these prejudices when he discovered that Jews were no more willing to convert to his version of Christianity than the one he claimed to have reformed. He urged Christians to burn Jews' synagogues, schools, and homes and subject Jews who would not convert to forced labor. In fact, as David Nirenberg has pointed out, \"Like so many prophets before him,\" Luther literally \"died in combat with the Jews.\" In the winter of 1546, he traveled to Eisleben, the city of his birth, in order to dissuade the town from giving refuge to Jews who had fled other cities. He promptly came down with a chill, nonetheless delivered several angry sermons that turned out to be the last ones of his life, and passed away. Even Luther's contemporary, the learned sixteenth-century humanist Erasmus, a man generally regarded as one of the most open-minded thinkers of his day, wrote, \"If hatred of Jews makes the Christian, then we are all plenty Christian.\"\n\nAnd yet, we all were not, so to speak. In the mid-1500s, the Netherlands welcomed the Jews expelled from Spain, just as the kings of Poland earlier had encouraged the Jews driven from the Rhineland to migrate eastward. In the 1600s, England reversed its policy and opened its borders to Jews again. Despite the exiles from many of the territories that today compose Italy and Germany, Jews never entirely disappeared from all of them. And for all of Luther's fury, other Protestants, especially Calvinists, were respectful toward the people they saw as their religious forebears.\n\nIn other words, if hostility toward Jews was sometimes xenophobic and sometimes chimerical, it was also sometimes dormant. By the eighteenth century, hostility to Jews on various religiously rooted grounds, now fortified by centuries of segregation and condemnation, was widespread and habitual in Europe but not universal. And it was at least theoretically not murderous. The perceived problem was what the Jews believed or chose not to believe; the solution was for them to change their minds, to convert. The means to get them to do so was cruelty toward them, but generally not murder. They were to suffer, but they were also to survive.\n\nThat brings us to the second horizontal block in figure 1, the era in which the domination of thought by religion began to come to an end in the European world. The transition is beautifully and best expressed in a couplet from Alexander Pope's poem \"An Essay on Man\" (1734): \"Know then thyself, presume not God to scan; \/ The proper study of mankind is man.\" This is a fitting epigraph for the Age of Enlightenment, also known as the Age of Discovery, and the precursor to the Age of Revolution. In these new eras, the driving question was \"How can people improve the world?\" Of course, labels and generalizations of this sort should be treated warily, but, broadly speaking, the eighteenth century ushered in an age whose watchwords were freedom and liberation or, as the French Revolution's slogan went, \"liberty, equality, fraternity.\" Liberty not only from political tyranny, however, but also from intellectual restrictions, such as those set by religious authority and tradition. Alexander Pope's admonition amounted to a call to stop concentrating intellectual energy on matters like theology and to focus attention instead on the human and natural world. Of course, distinguished individuals had been doing that to some degree since the beginning of the Renaissance, but Pope's clarion call heralded a shift in emphasis, a change in the intellectual center of gravity in the Western world.\n\nThe emblematic figure in this sense was the French philosopher Fran\u00e7ois-Marie Arouet, better known by his pen name of Voltaire, who mocked those who believed they inhabited \"the best of all possible worlds\" and goaded his readers to use their heads to improve society. A fierce critic of all traditional religions, he attacked both the Catholic Church and traditional Judaism with equal vehemence for confining people's thinking and insisting on the continued practice of ancient rituals. In championing the human capacity for improvement, Voltaire embodied the optimism of the age. He also endorsed a new form of hostility to Jews, one that hoped in a figurative sense to \"kill them with kindness\"\u2014to end their difference from the rest of society by freeing them from inherited restrictions, such as ghettos and confinement to certain occupations.\n\nIn theory, this form of hostility to Judaism closely resembled its predecessor in everything but its religious foundation and method. The problem was no longer what Jews believed, though many eighteenth-century thinkers criticized Judaism as overly fixated on obeying old laws and rituals. The problem was the supposedly backward culture of intense Talmud study and rigid observance of traditional practices that prevented Jews from being free and full contributors to society. And the remedial method was no longer cruelty and suffering but kindness and opportunity\u2014the carrot, not the stick. Jews were to be enticed out of their distinctness and into a secular form of conversion: not necessarily a change of religion but a change of everything unique about the religion's adherents, until they became indistinguishable from everyone else. Making them useful was the initial goal of Emancipation\u2014the abolition of residential and occupational restrictions on Jews\u2014by the Austrian Emperor Joseph II in the 1780s, and by the French Revolution and Napoleon Bonaparte in subsequent decades, but making them similar was the ultimate purpose.\n\nThis strategy had notable successes, at least in Western Europe. But even there Judaism and differences between the customs and marriage patterns of Jews and non-Jews did not disappear. Many emancipators were disappointed with the results, even as many people who disliked Jews were, too, though for entirely different reasons.\n\nThat brings us to the third horizontal bar in figure 1, the bar concerning the period following the invention of the new word _antisemitism_. The invention marked an ominous qualitative change, in that the new form of hostility focused not on what Jews believed or how they behaved but on what they intrinsically and unchangeably supposedly are. Antisemites generally agreed, as the classification of Jews by their original language implied, that the nature of Jews, their inherited and common qualities, made them not only incapable of becoming like other people, but also fundamentally subversive of other peoples and their societies. Jews could not be changed, but only contained and then eliminated. We might call this the biologization of antisemitism, and it coincided with a shift in the central questions of intellectual and public life from \"What does God want?\" and its successor \"How can people improve the world?\" to \"What material or physical laws govern us?\"\n\nProponents of this depiction of Jews drew on both old and new forms of science. The old one was basically animal husbandry, the science of breeding. Such people argued that peoples or nationalities were essentially like breeds of horses or dogs, each with special qualities that were passed on from generation to generation and that could be enhanced by selective mating. Thus Germans, like their German shepherds, were good fighters; the French, like their poodles, were showy; and the British, like their bulldogs, were tenacious. The nineteenth century was the great age of generalization of this sort, as each competing European nationality strove to define what made it distinct and great and what made its rivals inferior. As the historian Albert Lindemann has pointed out, \"beliefs in racial or ethnic determinism were the norm in most countries\" in the nineteenth century, even among Jews.\n\nThis old science was reinforced by vulgarized understandings of a new one, Darwinism, the science that argued that animal and plant species survive by random, perhaps accidental, but definite adaptations to their environments. As populations of flora or fauna spread out, they become increasingly different from each other by virtue of this adaptation. Many nationalists argued that their fellow Frenchmen or Germans or Jews and so on were like species, specifically adapted to their historically different surrounding conditions and profoundly different from each other as a result. As Julius Langbehn, a widely read German antisemite, put the matter: \"A Jew can no more become a German than a plum can turn into an apple.\"\n\nThe other newer sciences were often belief systems that we no longer consider scientific at all, but until they were invalidated, they also sustained a line of thought that exaggerated the differences among groups of human beings. These belief systems played to a widespread desire in Europe during the heyday of colonialism to show that descriptive or horizontal differences among peoples in such things as skin color and eye shape in fact denoted qualitative or vertical differences in ability\u2014that is, superiority and inferiority. Arthur de Gobineau's three-volume _Essay on the Inequality of the Human Races_ , which appeared from 1853 to 1855, became the authoritative text of this sort. It divided humanity into three great racial blocs: the white peoples, who were supposedly spiritual and creative; the yellow peoples, who were allegedly materialist and imitative; and the black ones, who were reputedly sensual and primitive. Even worse than this sort of global categorization, if that is possible, were Gobineau's warnings against race mixing. He linked the existence and endurance of a civilization with the purity of its dominant race and thus, although he was not an antisemite, provided arguments that such people later could use.\n\nWhat Gobineau's and other nineteenth-century pseudosciences had in common was the claim that external qualities indicated internal ones. Among the other prototypical schools of thought were: physiognomy, the invention of Johann Lavater (1741\u20131801), who claimed that the shape of faces, notably the straightness of the line from brow to chin, denoted superior traits; and phrenology, the creation of Franz Joseph Gall (1758\u20131828), who claimed that the shape of heads did the same thing because that shape determined the configuration of the brain, and the size of its various parts determined humans' capacities. Gall's follower Anders Retzius (1796\u20131860) devised a system of measuring skulls and a formula for expressing the relationship between his findings that he called the cephalic index. Unsurprisingly, given his European origins, Retzius concluded that the longer and narrower the head, the more superior the person. Finally, a more legitimate field of study called philology focused not on people's appearance but on their speech. As practiced in the nineteenth century, philology traced the origins and historical relationship of languages. By the beginning of that era, scholars had established that most European languages\u2014the exceptions are Basque, Hungarian (Magyar), and Finnish\u2014descended from ancient Sanskrit, which had been carried from southern Asia to Europe by a people called Aryans. The point of origin and the destination are what account for titling this family of languages Indo-European.\n\nThe person who turned the descriptive classification of tongues into a hierarchical one was the German philologist Friedrich Schlegel (1772\u20131829), who became the godfather of the Aryan theory of transmission in a book published in 1808. He and his followers described the grammar of Sanskrit-based languages as more precise and subtle than that of other language families, especially the Semitic one that included Arabic and Hebrew, and thus as proof of higher imagination, reasoning, and intellectual growth potential on the part of those who spoke Indo-European tongues. The basis of modern antisemitism became the claim that Jews had been shaped over time\u2014not only by their language but also by their original desert environment\u2014into a species of people fundamentally and unchangeably different from all Europeans, who had been molded differently by the wooded and fertile setting that most of Europe provided. Moreover, because Jews were immutably alien, they had to be contained and expelled, not converted or absorbed, because\u2014this is where the animal husbandry and Darwinism got mixed into a witches' brew\u2014peoples could thrive, compete, and adapt only by preserving their purity, by inbreeding. Ethnic mixing inevitably corrupted the special qualities associated with each breed or nation and led to decline because the traits of the inferior partner always predominated in the offspring.\n\nOf course, this is nonsense as genetics; even as aesthetics, it is accurate only for the sort that prevail at the Westminster Kennel Club, where a winning dog must conform perfectly to an idealized image of its breed. Nowadays we know, partly as a result of such thinking, that inbreeding actually can be harmful. Pursued obsessively, it leads in humans, as in dogs, not to greater perfection but to a host of congenital ailments and to heightened vulnerability to illness.\n\nBut the appeal of breeding as a form of public policy increased in the final decades of the nineteenth century because of widespread anxiety over what industrialization and urbanization were doing to European populations. A buzzword of the time was \"degeneration,\" and its signs were supposedly everything from the mounting incidence of tuberculosis, alcoholism, and venereal diseases that went with crowded and poor urban conditions to the supposed brutishness and ineducability of the rapidly multiplying working classes. In this climate, support for ideas of improving the human stock by selective breeding increased rapidly; indeed, such ideas were considered the cutting edge of sophistication. Their chief proponent in the English-speaking world was Francis Galton (1822\u20131911), who coined the term \"eugenics\" for his program of human betterment. In Germany, the equivalent figure was Alfred Ploetz (1860\u20131940), who preferred the term \"racial hygiene\" to describe his system of defending the development of the \"West Aryan\" or \"Germanic race\" from the supposedly counterproductive consequences of what he termed the \"growing protection of the weak.\" Chief among the protective measures he advocated was the killing of deformed or handicapped children, so that they would not burden healthy people or reproduce physical or mental defects.\n\nAlthough these doctrines spoke a language of racial \"improvement,\" the measures they proposed were profoundly fatalistic and reactionary. The message of Galton and Ploetz was that throwing money at poor people's problems was pointless; they existed because poor people were less able or, in pseudo-Darwinian language, \"fit\" for survival in life's struggles. Thus, if one wanted to improve humanity, these eugenicists or racial hygienists contended, the way forward was not to help the downtrodden by building better housing, improving working conditions, and raising the level of public health, for example, but instead to reduce reproduction by the poor and diseased and to increase it by their betters. Galton's successors labeled these two processes \"negative\" and \"positive\" eugenics.\n\nNeither these doctrines nor their founders were necessarily or explicitly antisemitic, but their conceptions of what needed to be improved in various populations and what needed to be bred out of them swiftly spilled over into the arguments of racists and became adapted to their purposes. This reinforced the pseudoscientific pose that bigotry toward Jews assumed with the coinage of the word \"antisemitism.\" Once Jews were defined as distinct from others, then their presence could be depicted as an invitation to destructive cross-breeding; once they were declared the embodiments of unwanted characteristics, their removal from the national body could be justified as a form of racial hygiene.\n\nThus, by the late nineteenth century, European antisemitism had a long and varied history. The persecution of Jews had been recurrent but far from universal or continuous. Attacks on them had evolved over time from ones ostensibly inspired by religious differences to ones that expressed physical fear. Of course, the overlapping phases of Jewish stigmatization always had one constant element: a depiction of Jews as contaminating or corrupting. Their proximity was seen as potentially undermining: first to Christians' faith, then to liberals' belief in human improvement, and finally to the strength and health of other populations.\n\nYet persecution appeared to be on the wane at the end of the nineteenth century, even though new justifications for it had come into being. The expansion of Jews' rights contained the seeds of boisterous backlash that reinforced old prejudices but failed to erase Jews' gains. At the same time that antisemitism seemed to surge and swell, it remained politically largely impotent.\n\nEMANCIPATION AND BACKLASH\n\nIn order to explain why Jews encountered a resurgence of agitation against them in the late nineteenth and early twentieth centuries, we must shift our attention away from the ideas that supposedly legitimated hostility and toward the circumstances that made certain groups of people receptive to it. The result is an ironic and somewhat contradictory story of widening opportunities and rights for Jews accompanied by ever more fervent and frustrated attempts to reverse this process.\n\nUntil what historians call the \"long nineteenth century\"\u2014the 125-year period between the outbreak of the French Revolution in 1789 and the onset of World War I in 1914\u2014most Jews lived in very confined worlds. Jews could be moneylenders, tavern keepers, itinerant peddlers, or cattle buyers who came in contact with non-Jewish customers; in parts of Eastern Europe Jews often managed estates for noblemen and thus dealt with tenants; and observant Jews may have had a reliable non-Jewish employee who came in to light their fires and do any other work that was forbidden on the Sabbath by the 613 laws. Otherwise Jews had very limited interaction with and visibility to non-Jews.\n\nBoth of these circumstances began to change in the 1780s. The first crack in the wall of religiously inspired restrictions on Jews came via the succession of Patents of Toleration issued by Austrian Emperor Joseph II for disparate parts of his realm between 1781 and 1789. Of these, the most famous was the Edict of Tolerance of January 2, 1782 governing Vienna and its environs, which set forth the general purpose of \"making the Jews useful to the state.\" To that end, the edict opened Christian schools and universities to Jews, along with numerous trades and commercial occupations previously denied to them; permitted them to employ Christian servants; and relieved them of two conspicuous burdens: a special tax and the obligation of men to wear beards. But the edict also severely restricted Jews' abilities to settle and worship in and around the Austrian capital and to enforce documents written in Hebrew or Yiddish. The point of that last prohibition was to make Jews learn to read and write German, and it succeeded to a remarkable degree. In the German-speaking lands of the early nineteenth century, Jews enjoyed a higher literacy rate than even their gentile neighbors, who were relatively well educated by European standards.\n\nMuch more far-reaching than Joseph II's edict was the enactment of the Declaration of the Rights of Man on August 26, 1789, during the heady first days of the French Revolution. That document declared, \"Men are born and remain free and equal in rights. Social distinctions may be founded only upon the general good,\" and went on to proclaim that all citizens are equal in the eyes of the law and therefore equally entitled to hold office and to do \"everything that injures no one else.\" But it took another two years, until September 27, 1791, for the National Assembly to pass a law making Jews full citizens of France. Although Napoleon backtracked to some degree on Jewish equality in France over the next few decades, his armies spread French ideas and practice across much of Europe, tearing down ghetto walls and removing occupational and political restrictions. He thus set in motion both the modern process of Jewish emancipation and the backlash against it that produced the modern form of antisemitism. As noted earlier, the roots of modern antisemitism are in religious differences: Christianity caused Jews both to suffer and to survive for centuries in Europe. But the form of hostility toward Jews that arose in the late nineteenth century and that called itself antisemitism is fundamentally a political movement, an expression of resistance to the emancipation of Jews that began in the late eighteenth century, gathered strength throughout Western and to a lesser degree Central Europe in the nineteenth century, and then reached even into the eastern parts of the continent with the Russian revolution in 1917.\n\nFormally speaking, emancipation was the process by which Jews were freed of all occupational, residential, and political restrictions and placed on a legal status of equality with all other citizens of a state. But to put it that way is too abstract; that definition ignores what emancipation meant in human, day-to-day terms, including what it felt like to the non-Jews who experienced it. It meant the emergence of Jews from pariah status; it meant almost literally their \"entrance\" into society and into regular contact with non-Jews; and, above all, it meant two possibilities that aroused opposition: first, people who had previously been kept from competing with the practitioners of certain trades and professions now could do so; and second, people who had previously been derided as benighted and backward, as dirty and superstitious, could ascend to positions of authority over non-Jews, over people accustomed to seeing themselves as \"better\" than Jews. Fear of this second possibility was particularly pronounced in a by no means unusual Bavarian petition of January 10, 1850, opposing equality for Jews. In that document, eighty-three citizens of the town of Hilders in the province of Lower Franconia, which eighty years later became a Nazi stronghold, pleaded for the repeal of emancipation and, in particular, \"that . . . no Jew be admitted to a judicial or revenue office, lest we have to humble ourselves before the Jews.\"\n\nThese emotional and practical effects of emancipation go a long way toward explaining the intense resistance it encountered and the halting and erratic nature of its course. After the fall of Napoleon in 1815, the Austrian Empire retained the reforms that Joseph II had introduced, but France was the only other state in Europe that did not turn the clock back; the only legal difference that remained there between Christians and Jews was that the state paid priests and ministers, but not rabbis. In 1830, that distinction disappeared, too. But everywhere else where the French had brought emancipation the new or restored rulers rolled it back, even if sometimes only briefly. Then, between 1830, when Belgium established civil equality upon achieving its independence, and 1871, when newly unified Germany did so, every state that had once been under French domination, along with a few countries in Western and northern Europe that had not, such as Great Britain, Sweden, and Switzerland, reversed the rollback and completed the emancipation process.\n\nEmancipation did not extend, however, to the lands of the Russian Empire, including the largest population of Jews in Europe in the Pale of Settlement, the parts of today's Poland, Lithuania, Belarus, and Ukraine to which most Jews were confined until the revolution that overthrew the tsars in 1917. Neither were the Jews emancipated in Romania until the end of World War I, and then only at the insistence of the victorious Allies. The late onset of emancipation in these regions and the strong resistance to it there are significant, for these are the areas where the Nazis later found most of their victims and received the most widespread local assistance in their murder.\n\nEmancipation was the political project of people called liberals, and it rose or fell everywhere according to their strength. Who were they? The word \"liberal\" derives from the Latin word _liber_ , which means \"free.\" They were the advocates of political and economic freedom, of (a) the rule of law, as created through constitutions and popular elections, not by royal fiat; (b) open and competitive markets, as opposed to guilds that restricted access to an economic activity and tolls and tariffs that restricted the movement of goods; and (c) the importance of ability over birth, as opposed to the aristocratic principle. Uniting the liberals' political and economic tenets was a general openness to change expressed by the French phrase _laissez faire_ , \"allow to do\" or, more figuratively, \"let happen,\" the phrase connoting a willingness to permit economic events to take their course and to generate a continuous process of what Joseph Schumpeter later dubbed \"creative destruction.\"\n\nThe liberals' heyday in Europe was exactly the period when emancipation triumphed, the years between 1830 and 1870, but the strength of liberalism, like the pace of emancipation, declined from west to east, from Britain and France to Russia. The farther west, the quicker the liberals' ascent to power, and the quicker emancipation came; the farther east, the less influence they exerted and the less change occurred in the legal position of Jews and their interaction with gentiles. In England, a man of Jewish descent, Benjamin Disraeli, could become prime minister in the 1860s. But in the Russian Empire such a thing was unthinkable, the religiously rooted condemnation of Jews remained the official doctrine of the state, and violent attacks on Jews remained an ever-present possibility. As we shall see, Germany was \"the land in the middle,\" both geographically and with regard to the pace and extent of emancipation.\n\nThe liberals' triumph was gradual and incomplete because it encountered resistance almost everywhere, though to varying degrees. To understand why, we need to look at what else was happening while emancipation was spreading. In the nineteenth century, six sweeping trends transformed European society.\n\nFirst, Europe experienced a population explosion from about 190 million people in 1800 to about 420 million in 1900. In some places, the increases were even greater: The total inhabitants of England, Scotland, and Wales tripled from 1821 to 1911; the populations of the Netherlands, Denmark, Norway, and Germany almost did so from 1816 to 1909\/10; and those of Belgium and Sweden grew by 250 percent. Amid this massive upheaval, the European Jewish population multiplied even faster, from 1.5 million in 1800 to 8.7 million in 1900 (an almost sixfold increase). And it multiplied fastest where it was poorest and most persecuted, in the Russian Empire, which created enormous pressure on Jews to get out somehow to somewhere.\n\nSecond, Europe underwent widespread industrialization, which transformed landscapes, created massive factories, provided employment for those surging numbers of people, multiplied goods, and in the process extinguished entire lines of work. Factories, not cobblers, came to produce most shoes. Textile mills turned out cloth far more rapidly and cheaply than individual weavers at home. Whole trades disappeared\u2014how many people today know what a \"cooper\" is or a \"wainwright\"?\u2014and the skilled workers who populated them, known as artisans, lost their livelihoods and social standing. But mass production was sensitive to fluctuations of supply and demand, and mill owners tended to push the consequences of these fluctuations onto workforces, with the result that industrialization created cycles of boom and bust, widespread resentment, efforts to push back in the form of unions and organized socialist movements, and enormous social tensions.\n\nThird, with industrialization came urbanization: The population of London grew from 900,000 to 4.7 million between 1800 and 1900, that of Paris from 600,000 to 3.6 million, and that of Berlin from 170,000 to 2.7 million. In 1800, only two European cities had more than half a million inhabitants, London and Paris; in 1900, twenty-three cities did, including seven with more than one million people. Everywhere Jews were conspicuous participants in this migration from the countryside to the cities, and their share in the urban populations, along with their visibility, generally rose dramatically, especially in Vienna, Berlin, Warsaw, and Budapest.\n\nFourth, extensive improvements in transportation, notably the railroad and steam shipping, accelerated trade and opened Europe to increased competition, especially in agriculture, from newly developing regions, such as the Great Plains of the United States and the pampas of Argentina. This put substantial downward pressure on the prices European farmers could get for their harvests. It also meant that the handicrafts of some regions could be wiped out by the industrial production of others. Increasing exposure to market forces bred widespread insecurity and free-floating desire to blame someone for it.\n\nFifth, increasing democratization occurred in the forms of successive extensions of voting rights, though as yet only to men, and progressive though incomplete reductions in the privileges and political powers of aristocrats. The results included the rise of mass politics and political parties and of the popular press, much of it of the tabloid sort. Political agitation became a more regular feature of life, as newspapers sought to whip up circulation through sensational accounts, especially of mysterious, behind-the-scenes wire pulling. The term \"muckraking\" is a creation of the era, and there was plenty of it going on in the last thirty years of the nineteenth century, when one financial and\/or political scandal after another occurred.\n\nSixth, though religious observance remained important, the nineteenth century saw considerable secularization in thought and education, and the trends were resisted fiercely by the papacy, many Protestants, and the Orthodox Church in the East. In fields as disparate as theology, where David Friedrich Strauss launched the critical historical study of Jesus, or biology, where Darwin advanced his theory of the long-term evolution of all life through adaptation, the Christian worldview and traditional piety came under attack and became increasingly regarded in sophisticated quarters as pass\u00e9. Perhaps the most advanced state of secularization was reached in France, which passed the Ferry Laws between 1879 and 1886, removing elementary education from the purview of the Catholic Church and setting up an explicitly anticlerical school system.\n\nIn short, the nineteenth century was an era of rapid, constant, and often bewildering change, and change always unnerves and\/or harms some people. The \"losers\" were clear: clergy who experienced declining deference to their persons and views; nobles who no longer monopolized office or found their lands a guarantee of great relative wealth; conservatives who disliked change in principle and parliamentary government in practice; farmers who faced international competition and thus downward pressure on their incomes; artisans driven out of business by factory production; property owners who feared the growing strength, as the century progressed, of workers' unions and workers' political movements, notably socialism; and even university graduates, who faced steep competition for professional positions. Of course, not every member of these groups experienced a decline of wealth or status during the nineteenth century, but a good many of them did.\n\nMembers of all these groups sought explanations for what was happening, and more importantly for what was going wrong for them. In such a context, conspiracy theories found an audience. They were easy to understand, and, then as now, no matter how convoluted, such theories were precise about who to blame for events, namely whoever is apparently benefiting from them. The perpetual motto of conspiracy mongers is the Latin phrase _cui bono_. Who benefits? Or in modern parlance, \"Follow the money.\"\n\nMany Jews were among the conspicuous and principal beneficiaries of the open and competitive universe that liberalism fostered. Many Jews also remained grindingly poor, especially the farther east one looked in Europe. But the number who became prosperous during the nineteenth century, the number who seized on the opportunities that came with emancipation, was real and striking. This was especially true in the spheres of banking and commerce and the professions of law and medicine. In a sense, the Jews of nineteenth-century Europe engaged in what sociologists and historians think of as classic first-generation, upwardly mobile \"immigrant\" behavior in the United States. Newly emancipated Jews sought out and strove for places in lucrative and secure walks of life, activities that would make their and their children's existences reliably better than their parents'. And, indeed, most of these Jews were immigrants or at least internal ones. Massive numbers of Jews from the far eastern provinces of the Austro-Hungarian monarchy (Galicia, Ruthenia, and Bukovina) migrated to Vienna and its environs, where their traditional garb and their Yiddish speech, which sounded to German ears like a corrupted and grammatically simplified form of their language, later aroused the ire of Adolf Hitler. In Paris, much of the Jewish population arrived in the nineteenth century from Alsace, the border province that Germany took away from France in 1871. In Berlin, a similar inflow came from Posen, a largely rural eastern province that Prussia had stripped from Poland in the late eighteenth century.\n\nInvisible among college students, lawyers, and doctors and rare among business leaders in 1800, Jews seemed disproportionately present in all these prized roles by the 1880s in many places, and even more so by the early 1900s. Here are some illustrative figures from Central Europe:\n\nIn the 1880s, Jews accounted for only 3\u20134 percent of the Austrian population, but 17 percent of all university students and one-third of those at the University of Vienna; in Hungary, Jews constituted 5 percent of the population, but 25 percent of the university students and 43 percent of those at the leading technological university; in Prussia, the biggest state in the German Empire, Jews made up less than 1 percent of the population in 1910\u201311, but 5.4 percent of the university students, and 17 percent of those at the University of Berlin.\n\nAt the turn of the century in Vienna, 62 percent of the lawyers, half the doctors and dentists, 45 percent of the university medical faculty, and one-fourth of the total faculty were Jews; so were some 55 percent of the professional journalists, 40 percent of the directors of publicly traded banks, and 70 percent of the board members of the Vienna stock exchange. In Hungary at the same time, Jews accounted for 34 percent of the lawyers and 48 percent of the physicians.\n\nIn 1912, 20 percent of the millionaires in Prussia were Jews; in Germany as a whole, Jews came to 0.95 percent of the population but made up 31 percent of the wealthiest families.\n\nOf course, this surge of success was not simply explicable as standard, upwardly mobile immigrant behavior; it also had specific cultural origins. Much of Jews' initial success in commercial activities represented an extension of the few economic roles previously permitted to them. Moneylenders became bankers; peddlers became shopkeepers and later owned and ran department stores; and cattle traders became brokers of commodities and stocks. And the ascent of Jews in the professions certainly drew on the premium their families and faith placed on learning. The discipline in childhood of religious study with heavy doses of memorization and debate over the meaning of texts is not bad training for going into medicine and law. That may have been what Albert Einstein had in mind when he supposedly quipped that the extent of Jewish academic success in nineteenth-century Europe suggested that the Jews had spent the last two thousand years preparing for university entrance exams.\n\nIn nineteenth-century Europe, most Jews did not become successful and\/or rich in the ways just listed, but the number and percentage of Jews rose among the people who did achieve these forms of success. This pattern was noticed, envied, and resented by the social groups that felt and often were disadvantaged or threatened by the change and competition that liberalism favored. Unlike some disappointed emancipators who thought Jews had not taken enough advantage of liberalism by becoming just like everyone else, members of the declining groups argued that Jews had taken too much advantage of the opportunities liberalism opened up. The tendency within these groups often was to confuse correlation with cause, to conclude that the rise of some Jews resulted from a conspiracy by all Jews. A group that benefited from modernization became pilloried as its destructive driving force. Of course, one can hear in these charges echoes of the medieval tradition of blaming Jews for plagues or other catastrophes. But the linkage also echoed the modern socialist movement, which posited a conspiracy on the part of the capitalists to maximize their wealth at the expense of the proletariat. In fact, leftists derided antisemitism as \"the socialism of fools,\" the belief system of people who mistook the identity of their real exploiters by focusing on Jews instead of capitalists. Whatever its medieval or modern inspirations, the connection between the incidence of antisemitism and the extent of perceived economic crisis is close; a clich\u00e9 of the subject is that the appeal of antisemitism rises and falls in inverse relationship with the stock market.\n\nCountering the antisemites' association of Jews with commercial corruption was made more difficult by the fact that many of the late nineteenth century's worst economic and political scandals did involve noticeable numbers of Jews. The most notorious instance in France, the Panama Scandal of 1888\u201392, centered on widespread bribery of French officials and parliamentarians in order to obtain loans to finance a French company seeking to build a canal through Panama. In the end, more than one hundred deputies, senators, ministers, and ex-ministers were exposed as corrupt, and thousands of small investors lost their savings. The bagmen who bought and paid these politicians were almost exclusively Jews, and the case was grist for antisemitic propaganda that attacked their supposed greed and selfishness.\n\nIn sum, the more liberalism triumphed, the more visible and successful Jews became, and the more groups that felt endangered or harmed by economic and political trends lent an ear to a convenient explanation of their troubles. That explanation blamed the Jews and promised relief by repealing emancipation and relegating them to their former contained status. The prevalence of such views seemed to grow with the rise of mass politics and the popular press, both of which encouraged agitators and ideologues. Antisemitism became vocal and loud in many parts of Europe after 1879, and the number of its spokespersons multiplied. Wherever they appeared, such figures as \u00c9douard Drumont in France, Georg von Schoenerer in Austria, and Hermann Ahlwardt in Germany had one thing in common. They came from and spoke to the social groups described here as susceptible to discontent with the direction of the modern world. Wilhelm Marr, the man most responsible for popularizing the word \"antisemitism,\" almost prototypically embodied the frustration and downward mobility that characterized those who found solace in attacking Jews. By the late 1870s, he had failed in succession as a businessman, a journalist, a politician, and a husband, in the last case to a succession of Jewish and half-Jewish wives.\n\nAnd yet the story of emancipation during the long nineteenth century ends with a paradox. Despite their volubility, antisemitic political parties and movements had very little to show for their agitation prior to World War I. Yes, Karl Lueger campaigned on an antisemitic platform, got elected Lord Mayor of Vienna repeatedly, and served from 1897 till his death in 1910. But he also did the Jews of the city no practical harm\u2014in fact, they experienced a sort of golden age during his time as mayor\u2014and his popularity was atypical. At the same time, Budapest elected a Jewish mayor, Adam Vazsonyi, and in 1895, the Hungarian parliament enacted a law that placed the Jewish and Christian faiths on the same legal footing. Indeed, after 1870, emancipation was not rolled back in a single European state. And in some countries, such as France, Italy, and Austria, Jews gained access to the historic bastions of the aristocracy in the diplomatic and officer corps and the university professorships. The reason for this is that, despite all the disruptive effects of modernization and change, the trajectory for most people in Western and Central Europe during the decades preceding World War I was steadily upward as standards of living improved. Occasional recessions were sharp but usually brief or merely sectoral; they hit particular economic sectors, usually farmers, harder than others, but scarcely affected everyone else. In this context, the laments of the pessimists and their claims that the Jews were at the root of all evil never stopped, but these cries also never gained a wide enough following to change laws.\n\nWhatever the popular strength of antisemitism anywhere, it proved really dangerous to Jews only when powerful officials or elites set out to exploit it or harness it to their purposes. The most famous examples are the Dreyfus affair of 1894\u20131906, in which conservative and self-serving army officers tried to pin spy charges on a Jewish colleague, and the ritual murder trial of Mendel Beilis in Kiev in 1913. But Dreyfus ultimately was exonerated, though the effort took years, and a jury of non-Jews, half of whom belonged to an antisemitic organization called the Union of the Russian People, actually acquitted Beilis. Even when power holders sought to exploit antisemitism for their own purposes, an aroused or embarrassed public could and did fight back successfully.\n\nStill, the message conveyed by both the Dreyfus and the Beilis affairs regarding the strength of antisemitism was ambiguous. The evidence is strong, as Barbara Tuchman pointed out in the 1960s and several other scholars have since, that Captain Alfred Dreyfus of the French army General Staff did not come under suspicion of being a spy for the German embassy in Paris solely or even primarily because he was a Jew. Equally important in leading to his indictment were two other facts: His handwriting strongly resembled that of the most incriminating document in the matter, the famous _bordereau_ found in a wastebasket of that embassy by a cleaning woman; and he was a rather remote and condescending person, much given to bragging about his wealth. The French officer corps was monarchist, Catholic, and antiliberal, but an average of 3 percent of the officers were Jews at any given time during the half-century leading up to World War I, which was thirty to sixty times their share in the total French population in that era, so the institution was not overtly antisemitic. In other words, Dreyfus's military peers and superiors turned on him initially and impetuously in 1894 because they needed a culprit, the handwriting evidence seemed plausible, and they disliked him personally. They persisted in professing his guilt because they feared that backtracking would embarrass the army to whose prestige they were devoted. The antisemitic gutter press turned Dreyfus's heritage into the central issue in the case, not the army, and the prosecutors at his trial did not even mention the subject. At the moment of Dreyfus's conviction in shamelessly manipulated proceedings, even prominent Jewish leaders, as well as Jean Jaur\u00e8s, the leading French socialist who later was one of Dreyfus's most vigorous defenders, believed in his guilt.\n\nAnother disconcerting fact is that the man who first identified another, more plausible spy within the General Staff and whose efforts ultimately led to Dreyfus's vindication was exactly the sort of person usually depicted as having persecuted him. The hero's name was Colonel Georges Picquart, and he was a conservative Catholic with distinctly negative attitudes toward Jews. So were Captain Louis Cuignet and Minister of War Godefroy Cavaignac, the men who later exposed the perjurer who had deflected attention from Picquart's alternative suspect. Finally, \u00c9mile Zola, the famous writer who led the crusade to free Dreyfus, articulated crude forms of racial determinism of the sort discussed earlier in this chapter, and these sometimes bordered on antisemitism. He fought for Dreyfus not to defend a Jew from persecution but to combat the Catholics, reactionaries, and militarists he held responsible for Dreyfus's prosecution. In the words of one sharp observer, Zola and the leading Dreyfusards were \"enemies of the antisemites, not of antisemitism.\" The Dreyfus affair stirred up and bequeathed a great deal of antisemitism, but it did not play out along strict party lines, and its resolution was not an unqualified victory over prejudice.\n\nMendel Beilis appears to have been set up by a local group of prosecutors interested in placating public opinion in Kiev and by several ministers in Moscow who were playing to the deep-seated antisemitism of Tsar Nicholas II. These people connected Beilis to the murder of a thirteen-year-old boy named Andrei Yushchinsky, whose body was discovered in a cave just outside Kiev, for two purely circumstantial reasons: first, the body had been stabbed in ways that supposedly facilitated the draining of blood, as in the sort of ritual murder connected to the blood libel, and second, Beilis owned a brick factory located near the cave and was a Jew. But, unlike in the Dreyfus case, the frame-up took in almost no one. From the beginning, local newspapers questioned the allegations, and a municipal detective swiftly produced evidence that linked a local gang to the murder. Apparently, that gang had gathered up a great deal of loot during the pogroms in Kiev in 1905\u201306 and hoped to instigate a new round by butchering a body in a manner intended to suggest a ritual murder and cast suspicion on a Jew. Once again, as in the Dreyfus affair, many of Beilis's local defenders were antisemites who simply hated those attacking Beilis more than they hated Jews and thought that the integrity of their own kind was more at stake than the rights of Jews.\n\nIn the decades leading up to World War I, the prevailing combination of constant antisemitic agitation, on the one hand, and general growth of Jews' rights and opportunities, on the other, goes a long way toward explaining twin developments among Jews that were the mirror image of what was happening among other Europeans. I am referring to the launching by Theodor Herzl in 1897 of the movement called Zionism, the drive for a Jewish homeland that soon centered on Jerusalem, which occurred in reaction to enduring antisemitism, and to the very limited success of this movement in winning support from Jews in the early decades of this century. Although obsessive and noisy, antisemitism not only generally failed to bend governments to its will, but also generally failed to panic Jews into thinking that their only sustainable future lay in founding their own country. Persistent antisemitism drove millions of Jews to leave Eastern Europe between 1880 and 1910, but rarely for Palestine. Instead, they came overwhelmingly to the United States.\n\nTo return to the question with which this chapter began: Why the Jews? Because an ancient tradition of blaming them for disasters, both present and prospective, a tradition deeply rooted in religious rivalry and superstition, persisted into the modern world and even assumed new forms during the eighteenth and nineteenth centuries. That tradition and its adaptations remained available to wax and wane as the impulse to blame did. In the decade immediately preceding World War I, the blaming impulse seemed to course primarily through other channels, especially those of class warfare, and antisemitic outbursts generally were held in check. At the middle of the continent, the territories that became the German Empire in 1871 and the Republic of Austria in 1918 remained for antisemites epicenters of agitation but also of frustration. We will see next why that was so and why the situation changed for the worse during that war and in its aftermath.\nCHAPTER 2\n\n[ATTACKERS: \nWhy the Germans?](contents.xhtml#ch_2)\n\nANY EUROPEAN ASKED in the immediate aftermath of the Dreyfus and Beilis affairs to identify the country most likely to persecute Jews in the future surely would have named France or Russia. Yet Germans became the principal tormenters of Europe's Jews in the second quarter of the twentieth century. Explaining how this happened involves examining a highly contradictory history.\n\nNATION AND _VOLK_\n\nPerhaps one way of approaching the contradictions is to remember that Germany is the land in the middle of Europe. In the nineteenth century, this was true not only geographically, looking west to east, but also with regard to political structure and the relative strength of antisemitism. The states to Germany's west, notably Great Britain, France, Holland, and Belgium, were all more democratic countries than the German Empire that came together in January 1871. They were constitutional monarchies or republics in which parliaments elected by steadily expanding sectors of the population chose the cabinets and prime ministers that made the major decisions, not kings or queens. To the east, the Russian Empire, on the other hand, was the last great autocracy in Europe, a state in which the tsar claimed to rule alone by divine right, and where a parliament did not exist until 1906. Even thereafter, the tsar claimed the right to dismiss that body whenever he chose and to appoint his ministers without regard to its preferences.\n\nUnder the German constitution of 1871, that nation was a political and constitutional hybrid, a mix of these two systems. On the one hand, it had a parliament (the Reichstag) chosen by the broadest electorate then allowed in Europe, all male citizens over the age of twenty-five voting by nominally secret ballot. On the other hand, the parliament had very restricted powers: it could set the national budget annually, but the 75\u201380 percent of expenditures that went to the military could be debated and authorized only once every seven, later five, years, and the government could take out loans without parliament's permission. In other words, the power of the purse that is the foundation of legislative authority was severely circumscribed. Parliament did not select the prime minister, called the chancellor; the kaiser (emperor) did, and he had exclusive power to declare war in response to an attack and to command the army. In short, the German Empire that lasted from 1871 to 1918 was an authoritarian, militarized country with the trappings of democracy, one that blended elements of the form of governance that had prevailed in Europe before the French Revolution and still prevailed in Russia with the newer form of parliamentary rule that had developed in Great Britain during the eighteenth century and on the continent after 1789.\n\nSomething similar can be said about antisemitism in this newly unified state. If we describe the period of post-Napoleonic emancipation as extending from 1815 to 1918, then Germany's enactment of equality for Jews before the law, which occurred in 1869 for the northern two-thirds of the country and in 1871 for the entire realm, falls almost precisely at the midpoint. The breakthrough came after emancipation in virtually every country to Germany's west or north and before emancipation in most of the lands to its south and east, Austria-Hungary being the exception. Germany occupied a middle point not only temporally but also in the forms and extent of emancipation, which were more complete than to the country's east but less than to its west or south.\n\nAnother distinct feature of nineteenth-century Germany both determined the timing and influenced the extent of emancipation there. Germany was not only the land in the middle, it was also, in the eyes of its citizens, \" _die versp\u00e4tete Nation_ ,\" the delayed nation. Like Italy, which also completed national unification only in 1870, the word \"Germany\" was only a geographical term prior to that year. An entity called, in English, the \"Holy Roman Empire of the German Nation\" had existed until 1806, but a truer description of reality would have been \"of the German nations.\" It was a very loose association under a single monarch of many highly autonomous entities, 1,789 of them, in fact, in 1789. Most Germans thought of themselves as Bavarians, Prussians, Swabians, Hessians, Westphalians, and so on, and most of these names of duchies and kingdoms derived from the Latin names of the tribes that had inhabited each centuries earlier. Bavaria comes from _Bajuvarii_ , and Prussia from _Borusii_. Insofar as a sense of German nationalism developed during the nineteenth century, it did so in reaction to and rejection of the French conquest and occupation under Napoleon, and it crystallized around the only idea that could unite so much difference, the notion that all the tribes were related and parts of a common people, or _Volk_.\n\nThe founding father of this line of thought was Johann Gottfried Herder, who did his most significant work before the armies of the French Revolution got to Germany and died while they were there. He maintained that nationalities are \"wonderfully separated . . . by languages, inclinations, and characters,\" and that each has an essence, a special set of core characteristics possessed by nearly all people born into it. He was not hostile to Jews, and though he insisted on enduring national differences rooted in different languages, he refused to postulate hierarchies of languages and peoples. \"Every nation bears within itself the standard of its perfection,\" he said. But his sentimental glorification of the unchanging virtues of the German _Volk_ , along with his insistence that \"every human perfection is national,\" encouraged a self-exalting quality in German nationalism.\n\nEstablishing precisely what this _Volk_ had in common was the great task of German nationalist thinkers during the early nineteenth century. They labored to identify, some would say \"invent,\" a collective German nature, and they began by defining it around what Germans in the early 1800s were collectively against: the conquering French and the ideas they had brought with them and stood for. Since emancipation of the Jews was a French import, many German nationalists rejected it as the product of an alien spirit. One of the earliest exponents of this rejection was Johann Gottlieb Fichte, a philosopher who in 1808 delivered a series of lectures published under the title _Addresses to the German Nation_. Fichte's animosity toward Jews predated his nationalism; in the late 1790s, he had called Jews \"a state within a state\" and had spoken out against their emancipation. Now he flatly argued that \"making Jews free German citizens would hurt the German nation\" and identified antisemitism with German patriotism. As for the nature of Germanness, he located it in the heroic and martial virtues that Tacitus had ascribed to the German tribes seventeen centuries earlier.\n\nDuring the later years of the French occupation, the Brothers Grimm began collecting folktales as sources for the essence of Germanness. Though not explicitly anti-French or antisemitic, their enterprise was implicitly exclusionist. The goal was to establish the human qualities that were intrinsically, continuously, and definitively German, qualities that, drawing on Herder, could not be possessed or combined in the same way by any other nationality. Ironically, the most famous tales that the Grimms reproduced\u2014the ones known as Snow White, Red Riding Hood, and Sleeping Beauty\u2014were, in fact, French in origin. The brothers learned them from Hessians descended from Huguenots\u2014that is, from French Protestant immigrants. This telling fact highlights the artificiality of the Grimms' quest to maximize national and ethnic differences. Nonetheless, by the middle of the nineteenth century, this sort of thinking produced Richard Wagner's pamphlet \"Jewishness in Music.\" It asserted that genuine musical works of art were products of the profound German spirit, to which Jews had no access, which is why they supposedly could produce only shallow and artificial works. The very notion of \"German culture\" ( _Kultur_ ) had become a prized family birthright that no outsider could inherit or exercise.\n\nAll of this made the German sense of nationality somewhat different from that which developed in Britain and France. In Great Britain, the cohering principle was a Protestant monarchy, and it embraced and pulled together different ethnicities\u2014English, Scottish, Welsh, and Scotch Irish. In France, the glue after 1789 was allegiance to the nation\u2014whether it was republic, empire, or kingdom\u2014and citizenship was open to any free person, regardless of race, creed, or color. French reading primers may have begun with the words \"Our ancestors the Gauls,\" but loyalty, not lineage, determined citizenship, and anyone born on the soil of France was, in principle, equal in its eyes. In Germany, and in the multiple states that preceded its unification, citizenship was more exclusive; it derived from one's parents, not the accident of where one was born, and was generally difficult for immigrants or outsiders to acquire.\n\nThese conceptual developments help explain the contested status of emancipation in Germany after Napoleon's fall, and the halting and relatively slow pace of its progress between 1828 (when the Kingdom of W\u00fcrttemberg became the first German state to enact lasting emancipation) and 1864 (when the city of Frankfurt became the last to do so before the spread of civil equality to all of the north of Germany in 1869 and to the south two years later). The process required forty-three years from beginning to end because resistance was considerable. It sometimes took violent form, as in the Hep-Hep riots, which began in W\u00fcrzburg and Frankfurt in 1819, spread to thirty other cities, and lasted for two months. The instigators were small-scale craftsmen and merchants angry at the prospect of competition if Jews were made citizens. One of the rioters' spokesmen, the writer Hartwig von Hundt-Radowsky, declared that the Jews' \"freedom to choose their own trades . . . is also a license to plunge Christians into misery.\" Usually the resistance remained rhetorical; nonetheless, it was impassioned. Representative examples are some of the poems of Heinrich Hoffmann von Fallersleben, who also wrote the words to what has been Germany's national anthem ever since 1922, and the numerous petitions against Jewish equality that were submitted to the Frankfurt parliament when it met in 1848 to write an ultimately abortive constitution for a united Germany. Most of these pleas came from small towns and rural farming communities, and most emphasized traditional complaints about supposed Jewish profiteering. Wherever and whenever it occurred, however, resistance to emancipation had a unifying theme: They are fundamentally different from us\u2014less honest and less spiritual\u2014and can never become like us.\n\nBut emancipation came, and it came in tandem with national unification in 1867\u201371, because liberals were the chief parliamentary patrons of both causes. Achieving one meant, to liberals, insisting on the other. Otto von Bismarck, the conservative Prussian leader who masterminded the three wars against Denmark, Austria, and France that forged German unification, initially found working with the liberals convenient, so he accepted the establishment of full civil and political equality for Jews at the time. But Bismarck was no liberal himself, and he was no fan of political equality in general. He was a fierce defender of his aristocratic caste, the Prussian Junkers, and determined to protect its economic interests and to preserve its near monopoly on leading positions in the government and the military.\n\nIf emancipation rode to success on the back of national unification, the backlash against emancipation gained strength when the economic consequences of unification began to look adverse. In 1873, the German stock markets, which had been driven upwards by an inflow of investment capital in the form of enormous indemnity payments from the defeated French, abruptly plummeted. The event has gone down in German history as the founders' crash ( _Gr\u00fcnderkrach_ ), since it came so soon after the founding of the unified empire. The trigger was the collapse of some railroad shares promoted by a baptized entrepreneur of Jewish descent named Bethel Henry Strousberg. A year later, a journalist named Otto Glagau published a series of articles in the popular weekly magazine _Die Gartenlaube_ (The Garden Bower) alleging that the crisis had been brought on by stock manipulators, \"ninety percent\" of whom were Jews. The Catholic newspaper _Germania_ soon spread the charges, and in 1877 Glagau republished his articles as a book, adding an introduction that read, in vitriolic part:\n\nNo longer should we tolerate Jews pushing themselves everywhere to the foreground. . . . They push us Christians continuously aside, they press us to the wall, they take away the air we breathe. In fact, they exercise domination over us . . . and they exert an extremely unwholesome influence. . . . The whole history of the world knows no other example of a homeless, definitively physically and psychically degenerate people, simply through fraud and cunning . . . ruling over the orbit of the world.\n\nMeanwhile, Germany's leading conservative newspaper, the _Kreuz-_ _zeitung_ , had gotten into the act. In mid-1875, it published a series of five articles that purported to disclose how the policies of German government and business were conducted \"almost exclusively in favor of our co-citizens of the Mosaic faith and Jewish nationality,\" largely because these policies were secretly directed by a Jewish banker in Berlin, Gerson von Bleichr\u00f6der, who was Bismarck's personal advisor. And, finally, in 1876, the first general secretary of the German Conservative Party, the political vehicle of landowners and agricultural regions, a man named Carl Wilmanns, gave this school of antisemitism a popular catchphrase when he titled a book _The \"Golden\" International_. Accusing the Jews of constituting a rich, self-interested, unpatriotic, and transnational conspiracy to promote their own wealth, the work went through six editions within a few months.\n\nIn short, the 1870s illustrated the force of the remark that antisemitism rises and falls in inverse relationship to the stock market. In that decade, when the market crashed, bigotry rose. The economic fallout of the stock market crash, in the forms of increased unemployment and lost savings, was substantial. The downturn coincided with a crisis in German agriculture brought on by an influx of cheap wheat and corn from the United States that pushed prices down and made large landowners and marginal farmers clamor for tariff protection. All of this created an audience for simple explanations, and the antisemitic agitation of the 1870s provided them. Thus the emergence in February 1879 of Wilhelm Marr and the new word \"antisemitism\" came as the culmination of a decade of rising reaction against emancipation.\n\nTwo other significant events in the history of antisemitism also occurred in 1879. In September, Adolf Stoecker, the Protestant chaplain to the Emperor and his court, added an antisemitic plank to the platform of the Christian Social Workers' Party, which he had formed to strengthen religious feeling and combat socialism among the working classes of Berlin. His motive was more pragmatic than ideological. His party had failed to win a large following through religious appeals, so he now sought a more attractive vote-catching strategy\u2014namely, the claim that an alien minority of greedy and immoral materialists was threatening to take over and corrupt Germany. In December, Heinrich von Treitschke, a professor of history at the University of Berlin, published an essay that praised the antisemitic agitation of the 1870s as a \"natural reaction of the Germanic national consciousness against an alien element that has taken too much space in our life.\" Near the end of his text he lamented, \" _Die Juden sind unser Ungl\u00fcck!_ \" \"The Jews are our misfortune!\" Antisemites soon turned the phrase into an accusation. In their hands, the words came to convey something like \"The Jews are the cause of our misfortune,\" and that was the message heard when the Nazis turned the phrase into a slogan emblazoned on the mastheads of their newspapers and the banners at Party rallies during the 1920s and 30s.\n\nThe repeated invocation of Treitschke's words demonstrates the lasting legacy of the antisemitic wave of the 1870s, but its immediate impact was not so great. In 1880\u201381, 265,000 German men signed the Antisemites' Petition, the centerpiece of a campaign to repeal emancipation by prohibiting immigration by Jews, compiling a census of those in the country, and removing all Jews as teachers, judges, and civil servants. But the drive was a political failure. Chancellor Bismarck refused even to respond to the petition, and the number of signatures collected disappointed its initiators. That Treitschke declined to sign showed that the document went too far for even critics of Jewish influence. Stoecker's party was overwhelmed in Berlin in the election of 1881 by the pro-emancipation Progressive Party, whose popular vote nationally almost doubled that of the previous elections in 1878 and raised the party's delegation to the second largest in the Reichstag.\n\nThat was the story of antisemitism in Germany before World War I in microcosm: The movement was loud, quotable, recurrent, but it had little political traction or legislative success. From 1887, when Otto B\u00f6ckel won election to the Reichstag from the city of Marburg, to 1912, the last election prior to World War I, a bewildering series of leaders and political parties dedicated to reversing emancipation came and went without attracting very large followings or enacting a single restriction on Jews' civil rights. At the polls, these parties largely flopped, as shown in figure 2. In seven parliamentary elections from 1887 to 1912, antisemites won only 78 out of a total of 2,779 seats, or 2.8 percent of the whole. They never won more than 4 percent of the popular vote or more than 5.5 percent of the parliamentary seats in any single election. Not only was their electoral base small, it was remarkably narrow: 35 of those 78 seats, or 45 percent of them, were won in the same area that elected B\u00f6ckel: Electoral Hesse, a small province in the west\/center of the country, north of the city of Frankfurt, that Prussia had conquered and annexed in 1867. By the 1880s, the region was economically depressed, and B\u00f6ckel and his followers thought they knew who was responsible. His party ran on the slogan \" _Gegen Junker und Juden_ ,\" \"Against the Prussian nobles and the Jews.\" Notice the order. Moreover, of the 44 men who ever held those 78 seats, 1 was a peasant, 2 were aristocrats, and 41 belonged to what Germans call the _Mittelstand_ , which means they were mostly artisans and shop owners, people who worked for themselves and were struggling against competition from factories and department stores.\n\nFIGURE 2: ANTISEMITIC VOTING IN IMPERIAL GERMANY\n\nfor the Reichstag (the national Parliament)\n\nfor the Prussian Landtag (the largest state legislature)\n\nThese data suggest that as an electoral phenomenon antisemitism was largely a vehicle of economic protest and not sufficiently popular in its own right to sustain a political movement. So do two other interesting pieces of electoral sociology. First, the only other part of Germany where antisemites did unusually well was the Kingdom of Saxony, along the border with today's Czech Republic, which elected another quarter of those antisemitic Reichstag deputies. But of the six seats they had won in the election of 1893, they lost five even before 1903, when all of them went to the left-wing Social Democrats. Second, notice in figure 2 what happened to the Conservatives after they added an antisemitic plank to their platform at the Tivoli Convention of 1892. At the national level, their vote rose slightly in 1893, but it then declined steadily thereafter, falling by more than one-third by 1912. In voting for the Prussian parliament their support dropped by an even steeper 41 percent. In Imperial Germany, antisemitism was hardly a ticket to electoral success.\n\nWhy could German antisemites generate a series of bestselling books, such as Julius Langbehn's _Rembrandt als Erzieher_ (Rembrandt as Educator) in 1890 and Houston Stewart Chamberlain's _The Foundations of the Twentieth Century_ in 1899, but not a sustained national political movement or any legislative victories? One reason was that the leaders of the antisemitic parties were often incompetent and corrupt, which generated scandals that undercut their popularity. Another was that these leaders had trouble working together, so the history of antisemitism in Imperial Germany is a history of constant mergers and splits and little stability to even the names of the groupings. Wilhelm Marr, the so-called patriarch of German antisemitism, was so disputatious that he ended up quarreling with virtually every other leader of the movement in the 1880s and then repudiating antisemitism altogether. In parting, he mocked the ideology as \"a business\" that blamed Jews for social problems created by industrialization.\n\nBut the more fundamental problem for the antisemitic parties was that the discontent they mobilized was always sectoral; it was generally confined, in the period 1887\u20131912, to one or two particular parts of the country at a time or to one or two particular social groups. Broadly speaking, when Hesse was hurting, Bavaria or Brandenburg was not suffering as badly or in the same ways; when artisans and farmers were complaining, the fortunes of workers were improving. So long as discontent was not general or other groups offered responses to it that some people found more persuasive, as the Center Party did to devout Catholics, the socialists did to industrial workers, and the Conservative Party did to landowners and pious Lutherans, political antisemitism could not thrive. Intellectual antisemitism, however, was another matter; it had a broader, more constant audience and reflected a persistent unwillingness to see Jews in Germany as Germans.\n\nIn Imperial Germany, a peculiarity of the electoral process erected one additional barrier to political antisemitism. The German constitution that governed national elections mandated universal manhood suffrage, but the separate states of Germany had their own electoral systems that often privileged wealth. Two states, Prussia until 1918, which made up over 60 percent of the country, and Saxony from 1896 to 1909, weighted votes in parliamentary and local elections according to the taxes on property and income that men paid. Basically, those who paid the top third of taxes in each election district chose one-third of the electors for a seat, the men who paid the next third chose the second set of electors, and the remaining male taxpayers chose the third set. People too poor to pay direct taxes could not vote at all. This system awarded disproportionate influence to the prosperous. In Essen, Alfred Krupp, a vastly wealthy coal and steel magnate, cast the only vote for the first third of the electors from 1886 to 1894, so he in effect chose them. In Berlin 10 percent of the population chose the first third, and the usual breakdown in election districts was something like 3\u201310 percent\/10\u201315 percent\/75\u201387 percent. This meant that local and state elections in Prussia and Saxony were decided by the richest quarter or less of the electorate, which consisted of only 15\u201320 percent of the adult male population. Because Jews were disproportionately well represented in the top two tax groups in most cities, their votes carried extra weight in urban districts and municipal elections. For example, in Frankfurt in 1900, Jews made up 63 percent of the people who chose the first third of the electors. Distributions like this worked against antisemitic candidates and encouraged others to support Jews' rights or at least to pay lip service to them. After the German Empire fell in 1918, the new republican regime made German elections more uniformly democratic, and the electoral prospects of antisemitic candidates actually benefited.\n\nAs political antisemitism both ebbed and flowed during the lifespan of the German Empire, another contradictory set of trends developed\u2014namely, a transformation of the German Jewish population that made Jews both more like and more unlike the rest of the nation's citizens. On the one hand, Jewish distinctness seemed on the way to disappearing and Jews on the way to fitting into German society in three senses. First, the Jews of Germany constituted a steadily declining share of the population (from 1.25 percent of the national population in 1871 to 0.95 percent in 1910) and, after 1910, when 615,000 Jews lived in Germany, a steadily declining number of people, too. The cause was not conversions to Christianity, as only about 34,000 of these took place from 1800 to 1918. Rising rates of intermarriage also played but a small role, as they began to jump only at the end of the imperial period, when the ratio of mixed to all-Jewish marriages rose from 1:5 in 1901\u201305 to 2:5 in 1916\u201320. The main reason was the drop in the Jewish birthrate to just above replacement level. If almost 80,000 Jews had not immigrated to Germany under the empire, the Jewish population would have barely grown at all between 1871 and 1910. Despite an inflow of a roughly comparable number of Jewish immigrants in the years surrounding the end of World War I, the Jewish population in Germany continued to fall; in 1933, it was almost 20 percent smaller than in 1910.\n\nSecond, German Jews became increasingly acculturated, demonstrating great enthusiasm for German literature, art, and philosophy and eagerly participating in the German glorification of _Bildung_ , or cultivation. One consequence was the steep and rapid decline of Jewish schools and the use of Yiddish, and the nearly total integration of Jews into the German educational system. Third, Jewish religious practices also moved in a somewhat syncretic direction, as Germany became the homeland of Reform Judaism. That movement relaxed observance of many of the 613 laws, abandoned routine rituals and customs that seemed to smack of non-European origins, and introduced new forms of observance, including the seating of men and women together in the synagogue, the use of choirs and music during worship services in German, and sometimes the designation of Sunday rather than Saturday as the Sabbath. Though Jewish synagogue architecture remained quite distinct, favoring Moorish towers and domes, in other ways the observable differences between Christian and Jewish practice clearly diminished.\n\nOn the other hand, Jews continued to stand out from other Germans, sometimes increasingly, in four conspicuous ways. First, Jews left the eastern and rural parts of the country\u2014places like Posen, Prussia, Hesse, and southwestern Germany\u2014and migrated to cities even faster than did non-Jews. Between 1871 and 1910, the percentage of all German citizens living in cities with more than 100,000 residents rose from almost 5 percent to over 21 percent; for Jews in Germany, the corresponding figures were 20 percent to 58 percent. By 1910, almost 28 percent of Germany's Jews lived in Berlin, where they made up about 4 percent of the capital's population; in Frankfurt, their share exceeded 6 percent. Moreover, they tended to cluster in particular neighborhoods in each big city\u2014for example, Mitte, Charlottenburg, and Wilmersdorf in Berlin.\n\nSecond, the traditional concentration of Jews' occupations in trade and commerce grew steadily more pronounced, and within those spheres the patterns of Jewish employment were quite distinct. German Jews were three times more likely than all Germans to own their own businesses. Of the roughly one-quarter of Jews categorized as working in manufacturing, more than half were tailors. By the turn of the twentieth century, Jews owned 80 percent of the nation's department stores, 70 percent of its metal wholesalers, and 60\u201370 percent of the ready-to-wear clothing stores, and had preponderant positions in the advertising and printing industries. Regionally, Jews constituted 75 percent of the livestock dealers in Franconia, Westphalia, and Hesse, and half the grain dealers in Hesse and Baden. Finally, in 1910, when Jews made up less than 1 percent of the national population, they were 15 percent of the lawyers, 6 percent of the doctors and dentists, and 10 percent of the law school students and 14 percent of the medical students. As a result of the declining birthrate, all of these figures trended downward after World War I, as did the margin between the average income of Jews and non-Jews in Germany, but that did little to offset the general identification of Jews with non-manual labor and prosperity.\n\nThird, Jewish immigrants under the empire bulked larger because they concentrated in cities. Jewish immigrants from Poland, who were often far more traditional in dress and religious practice than German Jews, came to only 13 percent of the Jews in all of Germany in 1910 but to much larger shares of those in certain municipalities: 67 percent of the Jews in Leipzig, for example, 53 percent in Dresden, and 15 percent in greater Berlin. They stood out and created an illusion of a massive influx of alien people. After 1914, only 90,000\u2013100,000 more Jews gained entrance to Germany, but their visibility and even greater concentration in places like Berlin and Leipzig had the same effect and gave rise to a veritable psychosis of \"inundation\" by Jews that antisemites cultivated.\n\nFinally, German Jews stood somewhat politically apart from their fellow citizens, voting noticeably more frequently for the moderate left than most Germans. In the empire, this meant that they consolidated increasingly behind the Progressives; after World War I, it meant that they voted mostly for the Democratic Party; and as that party declined during the Depression, they gravitated toward the Social Democratic Party (SPD). Amos Elon writes of the Jewish bourgeoisie, \"They lived like bankers but voted like hard-pressed workers and leftist intellectuals.\"\n\nOnce again, Germany was the land in the middle, the country in which native-born Jews were less integrated in society than to the west but more so than to the east. Despite a great deal of acculturation, the separateness of Jews from other Germans remained apparent in certain respects. Similarly, despite the electoral and legislative failure of antisemitism, it enjoyed administrative and sociocultural successes. An example was the tight limits on immigration and naturalization that the Reich imposed. Most of the Jews who migrated to the United States from Eastern Europe in the 1890s and early 1900s embarked from the ports of Hamburg and Bremen. To get there, they traveled in trains that were sealed the moment they crossed from Russian Poland into Germany and that arrived at long piers, built out into the harbors alongside ocean liners. Steel doors were locked behind the last car before the passengers could descend and board the ships. The goal was to make sure that no one could alight in Germany along the way. Almost 80,000 Jews from Eastern Europe got into Germany between 1871 and the onset of World War I in 1914, but the Reich labored hard to limit the number and to restrict the immigrants' chances of becoming citizens.\n\nAnother manifestation of lingering antisemitism was the way Germany mixed formal legal equality with a great deal of social and professional discrimination. The Antisemites' Petition may have been a political bust, but 41 percent of the students at the University of Berlin signed it, and it led to the founding of the League of German Students (Verein deutscher Studenten), an increasingly popular organization that promoted the exclusion of Jews and the children of converts from Judaism from much of student life. By 1896, the national association of German university fraternities banned the initiation of Jews. In 1910, the Austro-Hungarian army had 2,000 Jewish officers; the French army, 720; and the Italian army, 500. The Prussian army, which made up the great majority of the Reich's forces, had none and refused to let Jews become officers in reserve units, as well. Jews were largely kept out of prestigious teaching positions: Prussia's secondary school faculties included only 12 Jews in 1910. In the same year, only 2 percent of the professors in all of Germany were Jews, almost all of them in medicine and the sciences. Antisemitism became institutionalized in elite and conservative society rather than in laws. As Shulamit Volkov has demonstrated, it became part of the \"cultural code\" of German conservatives and right-wingers, part and parcel of their self-described responsibility to uphold traditional values against the ideologies of liberalism, materialism, and internationalism.\n\nNonetheless, on the eve of World War I, the trend of events seemed to favor German Jews. The three-class voting system assured that attempts to exclude Jews from professions at the local level\u2014for example, as teachers in elementary schools\u2014were much less successful than snobbish barriers at the elite governmental level. The Prussian state had taken firm action against the last outbreaks of ritual murder accusations at Xanten in 1891 and Konitz in 1900, even dispatching troops to put down antisemitic riots in the latter case, and the accused Jews had been acquitted. Prominent Jewish industrialists such as Walther Rathenau, the head of the German General Electric corporation, and Alfred Ballin, the chief of the Hamburg-America Line, were becoming part of the kaiser's entourage (though Jewish wives still were not invited to court). The election of 1912 routed the antisemitic parties and sent more Jews to Parliament than in the preceding thirty years. Not only did the number of deputies of Jewish descent reach nineteen, but also some of them belonged to the National Liberal and Progressive parties, which had not even nominated Jews during the previous two decades. Many people, Jews and sympathetic non-Jews alike, confidently likened antisemitism to a _Kinderkrankheit_ , a childhood disease that German society was outgrowing.\n\nThe force that shattered these expectations was the cataclysm of World War I, and the turning point came in 1916, when the German High Command, desperate to divert blame for the murderous military stalemate, authorized the infamous \"Jew count,\" or _Judenz\u00e4hlung_. The generals hoped to prove the charge made by antisemites in Parliament that Jews were shirking their military duty and thus to provide an excuse for the army's failure to win the war. In fact, the census showed a slight overrepresentation of Jews in the military compared to their share of the national population: 100,000 served in the German army, 80,000 in combat; 35,000 were decorated and 12,000 killed. The disappointed military leaders thereupon concealed the results; declined to contradict partial, leaked figures that appeared in the antisemitic press; and allowed the army's political arm, the Fatherland Party, to revive accusations of Jewish draft dodging. In a sense, this episode was Germany's Dreyfus affair, another instance of an elite institution, again the army, trying to use antisemitism to conceal its own failures by spreading vitriol against Jews. However, unlike in the Dreyfus affair, a countermovement did not arise to expose and discredit the lie in public, so it had even more lasting effects. Not the least of them is a paragraph in _Mein Kampf_ in which Hitler claims that Germany would have won World War I with less loss of life if only 12,000 or 15,000 more Jews had faced and succumbed to poison gas at the front.\n\nThe toxic effects of this new libel against the Jews\u2014indeed, of the preceding forty years of ceaseless agitation and vilification\u2014became apparent even before World War I ended in Germany's defeat and humiliation. By February 1918, Kaiser Wilhelm II had convinced himself that an international Jewish conspiracy controlled all the forces arrayed against him. Meanwhile, Erich Ludendorff, one of Germany's two principal military commanders, had begun contemplating the expulsion of two million supposedly politically unreliable Jews from the part of Poland he planned to annex upon winning the war. After Germany's collapse in the fall of 1918, those desperate to blame the outcome on anything or anyone other than the nation's leaders or armed forces echoed the charges that provoked the Jew count and scapegoated the Jews, along with liberals and leftists, for undermining the war effort.\n\nThe audience for such claims grew wider than ever before because the sense of crisis was no longer sectoral but had become national. The combined effect of the Versailles Treaty terms and of the huge debt the country had run up to fight the war, the difficult process of demobilizing the army and converting to a peacetime economy, and the huge burden of supporting veterans and widows led swiftly to rising unemployment and runaway inflation. By 1923, the German currency was not worth the paper it was printed on, and the nation was in turmoil. The opportunity this presented for political antisemitism in Germany is reflected in figure 3, which shows the vote for openly antisemitic political parties rising from 10.3 percent in 1919 (a little less than where it stood in 1912) to 26 percent in early 1924. At the same time and continuing into the late 1920s, the incidence of violent acts against individual Jews increased, along with that of politically motivated violence in general. Yet figure 3 also records that the opportunity passed, and the vote for antisemites fell again, only to surge once more in 1930 after the onset of the Depression and then to spike in 1932, after the nation's largest banks had failed and unemployment peaked at 36 percent of the workforce.\n\nFIGURE 3: ANTISEMITIC VOTING FOR THE REICHSTAG IN POST\u2013WORLD WAR I GERMANY\n\nELECTION| NAZI % OF VOTES| NATIONALIST %| COMBINED % \n---|---|---|--- \n1919| \u2014| 10.3| 10.3 \n1920| \u2014| 15.1| 15.1 \n1924| 6.5| 19.5| 26.0 \n1924| 3.0| 20.5| 23.5 \n1928| 2.6| 14.2| 16.8 \n1930| 18.3| 7.0| 25.3 \n1932| 37.4| 6.2| 43.6 \n1932| 33.1| 8.9| 42.0\n\nHITLER'S OPPORTUNITY\n\nThe statistics in figure 3 suggest that antisemitism acquired a new lease on life in Germany during the 1920s, a development that raises two challenging questions: How could a political fixation on rolling back Jewish emancipation go from a prevalent but unsuccessful movement before 1918 to a victorious one in 1933? And how could such hatred succeed at a time when the relative position and even the raw number of Jews in Germany were in decline?\n\nThe answers to these questions lie primarily in the changed nature of Germany's problems after 1918: They were no longer episodic and sectoral, they became continuous and national, and they therefore generated a pervasive sense of crisis that fostered support for extremist positions and simplifying explanations. The remainder of this chapter concentrates on that crisis and on how the Nazi Party and Adolf Hitler became its ultimate beneficiaries.\n\nBut the answers do not lie only in the depth and breadth of Germany's crisis. One other, vital impetus to antisemitism's resurgence both there and in many European countries after World War I emerged from that conflict: the linking of Jews to the specter of communist revolution. In 1917, when the Bolsheviks came to power in Russia, a number of Jews were prominent among their leaders. Leon Trotsky is the most famous, but he was not alone, and supporters of the tsar, including thousands who fled the revolution into Central and Western Europe, played up this fact. Jews such as Rosa Luxemburg, Kurt Eisner, and Bela Kun assumed leading roles in the revolutions in Germany and Hungary in 1918\u201319, and opponents trumpeted this as proof that these regime changes were alien impositions that confirmed the menace Jews embodied. As a result, a new variation crystallized on the old practice of demonizing Jews as agents of destructive change, and a new kind of fear\u2014fear of communism\u2014became available for antisemites to exploit.\n\nOf course, some conservatives long had linked Jews to the political left, but symptomatic of the new virulence and its appeal was the sudden popularity of a failed prewar fabrication, the infamous _Protocols of the Elders of Zion_. Largely unknown outside Russia before World War I, the _Protocols_ purported to be the transcripts of meetings among nefarious Jewish leaders intent on fomenting discord within all nations in order to increase Jews' power and wealth. In the aftermath of the Russian Revolution, tsarist loyalists brought the _Protocols_ west, and translations into most European languages found a large and credulous audience. The first German edition in 1920 sold 120,000 copies, for instance. In 1921, the _Times_ of London conducted a thorough debunking that exposed the _Protocols_ as an invention\u2014in fact, a pastiche of plagiarism from two works of fiction of the 1860s, Hermann Goedsche's German novel _Biarritz_ and Maurice Joly's French political satire _Dialogue betweenMachiavelli and Montesquieu in Hell_. But these disclosures made no difference to the _Protocols_ ' fervent devotees. Adolf Hitler spoke for them in _Mein Kampf_ , insisting that \"the moans and screams\" about the falseness of the _Protocols_ actually constituted \"the best proof that they are authentic after all.\"\n\nThe pervasive sense of crisis that afflicted Germany after World War I was both emotional and material. The emotional part was a product of the way the war ended, which Germans found profoundly humiliating and unfair. When they asked for an armistice, overthrew the imperial regime, and drove the kaiser into exile in 1918, Germans thought they would get a negotiated peace from the victorious Allies, and that it would be based on Woodrow Wilson's Fourteen Points, which promised \"no annexations, no indemnities.\" Instead, the Germans got the Versailles Treaty, which the Allies worked out among themselves and presented in 1919 on a \"take it or leave it\" basis. Not only was this what the Germans called a _Diktat_ , a dictated peace, but it stripped the country of 10 percent of its territory and most of its armed forces, stigmatized Germany with a clause that assigned it sole responsibility for the outbreak of the war, and imposed an at first unspecified but ultimately staggering monetary penalty in the form of a bill of reparations for the damages inflicted on France and Belgium. Germans of all political stripes felt, as their expression goes, _belogen und betrogen_ (lied to and deceived). But they already had demobilized their army as required by the armistice, so the German government had no choice but to sign a document that its people never viewed as legitimate. This gave rise to a kind of siege mentality among the Germans after 1918, an attitude of \"it's us against the cruel and unjust world.\"\n\nThe material part of Germany's postwar crisis was the result of the combined challenges of paying reparations, trying to undermine them at the same time, servicing the huge debt that the nation had run up to fight the war, converting from a war economy to a peacetime one, and supporting large numbers of disabled veterans and widows. The reparations came to either $12.5 billion, which was what the Allies actually expected the Germans to pay, or $35 billion, which was the amount the Allies nominally imposed to please their own electorates, and the payback period was estimated to last from seventeen to thirty-six years. The sums due yearly came to about 5 percent of the average real annual German national income between 1918 and 1931 ($11 billion), which may not sound particularly onerous, but the Reich's debts, mostly to Germany's own citizens who had bought government bonds, came to another $41.5 billion at the end of World War I. Germany's debt burden, in other words, amounted to 38 percent of the country's total national income during these thirteen years. The repayment obligations, when added to recurrent government expenditures, swamped revenues: In 1922, the Reich collected less than one-fifth of its budgeted outlays. The government could not borrow (who would lend to such a debtor?); it feared raising taxes (which might generate a revolution or help one of the country's recurrent putsches to succeed); and it could not earn funds from exports because of foreign tariffs that priced German goods out of other markets.\n\nThe government's only recourse was simply to print more money, and the result was runaway inflation. By 1923, the exchange rate had reached 4.2 trillion reichsmark to the dollar, meaning the German currency was worthless, and the nation was in turmoil. During that year, leftists rebelled in Hamburg; Hitler's Nazis staged the unsuccessful Beer Hall Putsch in Munich; the Lithuanians marched into East Prussia and annexed the city of Memel; and the French occupied Germany's industrial heartland, the Ruhr region, in order to force Germany to keep up with reparations payments and meanwhile to collect their equivalent directly. And, in Berlin, one reflection of the tensions was the Scheunenviertel (Barn District) riot in early November, a small-scale pogrom directed at Jewish immigrants from Eastern Europe who had set up shops in the nation's capital.\n\nThe Weimar Republic, the democratic regime that replaced the monarchical German Empire, survived this crisis, thanks to a brief period of military dictatorship and an influx of billions of dollars in loans from the United States attracted by fatefully high interest rates. But both before and after 1923, the nation was deeply polarized over who was to blame for its miseries. On the one side were the political left and the supporters of the republic who said that everything was the consequence of the old regime that had plunged the nation into war in 1914 and then led it badly and to defeat. On the other side were the political right and supporters of the old monarchy, many of them still entrenched in the judiciary, civil service, and the military, who said that the root of all evil was a supposed sinister conspiracy of Marxists and Jews that had undermined the war effort from within, overthrown the kaiser in 1918, and introduced an incompetent parliamentary government. For most of the life of the Weimar Republic, the two groups were fairly evenly balanced, but also internally divided. On the left, the communists and socialists fought each other, and on the right the old-line nationalists competed with other groups, including the fledgling National Socialist German Workers' Party (NSDAP), the Nazis. This situation did not make for stable or effective leadership. Twenty-two governments came and went in the short fourteen years of the Republic, and the constant wrangling and instability undermined the popularity of democracy.\n\nAdolf Hitler was the ultimate beneficiary of this stalemated blame game, though he failed in his first bid for power in 1923. What did he offer Germans, and why and how did he succeed? The core of Hitler's message was a flattering explanation of what ailed Germans and of why they deserved so much better. Flattering because the central claim was that Germans had not brought their troubles on themselves by following a blundering imperial government or fighting a war they could not win. No, the disasters had been done _to_ the Germans, not _by_ them. Who were the culprits? Above all, the duplicitous Allies, the delusional Marxists, and the debilitating Jews. Central to Hitler's narrative were the claims that Germany was a victim and thus entitled to lash back by all means necessary. In other words, \"they did evil to us, so we get to pay them back.\" Hitler believed profoundly and unshakably in this narrative because it performed the same function for his wounded psyche after 1918 that the message did for the people attracted to it. It explained his nation's unjust fate, exonerated him and his compatriots for bringing it on, exposed the villains, and exhorted Germans to fight back. The psychiatrist James Gilligan argues that all violence results from the attempt to replace shame with self-esteem. Whatever the general validity of that remark, it brilliantly captures the motivation behind the violence of Hitler's ideology toward Jews, communists, and foreigners. Shame at the defeat in 1918 generated a furious determination to punish the alleged authors of that defeat so as to expunge it and restore national pride.\n\nThe rhetorical centerpieces of this story were the phrases \"stab in the back\" and the \"November criminals.\" The first asserted that the German army had not lost the war but had been undermined at home by an insidious coalition of Jews and leftists; the second labeled the people who had overthrown the monarchy in November 1918 as traitors. Both claims diverted attention from the German military's and the German people's roles in the defeat and the revolution. After all, the General Staff had begged for the armistice in the fall of 1918 as the only way of preventing the retreating German army from completely breaking up, and many Germans were war-weary and welcomed the kaiser's overthrow. But the Nazis explained these facts away by treating them as creations of the forces that had conspired to undermine the war effort and overthrow the old regime. Both phrases became key components of Hitler's and Nazism's insistence that antisemitism was a defensive, not an offensive, stance. This is a central theme in the history of the Holocaust. The argument that persecution was an act of self-defense was so essential as a justification for what the Nazis wanted to do that it repeatedly appears in ever new forms: They threaten us, so we must strike to protect ourselves.\n\nHitler tricked his message out with a synthesis of pseudoreligion and pseudoscience that may be aptly dubbed a \"theozoology\": On the one hand, he posed as an evangelist of the _Volk_ , the person who would lead a national revival by making the German people sense its own power and, as the Nazi slogan \" _Deutschland Erwache_ \" said, \"Awaken Germany.\" Hitler presented himself as the one person, singled out by providence and arising from modest origins, who could deliver Germans from their afflictions and, indeed, from division, controversy, and internal conflict altogether. \"His speeches,\" an early biographer observed, \"begin always with deep pessimism and end in overjoyed redemption, a triumphant happy ending.\" At the same time, he claimed to be the eugenicist of the race, the person tough enough to purge the German people of defective and degenerate elements and maximize its purity and strength through selective breeding. Together, evangelism and eugenics promised to create a rejuvenated, unified, and healthy people that would shape its own destiny, and the offer had widespread appeal in a defeated, economically troubled, and politically polarized country that felt battered by the demands of the nations that had triumphed over it.\n\nHitler based his claim to be able to accomplish all this on the assertion that he alone grasped the fundamental laws and processes that govern history. What were these? Basically they amounted to a kind of bastardized Marxism that substituted race for class. Whereas Karl Marx taught that all history is the struggle among classes to control the means of production and distribute wealth in the victorious class's favor, a process he called dialectical materialism, Hitler taught that all history is the struggle among races to control space or territory from which to generate food and wealth that will support further expansion. In 1949, the first postwar president of West Germany, Theodor Heuss, aptly described this doctrine as \"biological materialism\" because it so perfectly parallels Marxian notions of class struggle. In a nutshell, Nazism is an ideology of feed and breed or race and space that posits a permanent struggle to the death among ethnic groups. Hitler insisted that perpetual struggle is \"the law of nature,\" but a more fitting term would be \"the law of the jungle.\"\n\nBecause struggle is perpetual, Hitler insisted that Germans lived in a permanent state of emergency. Although they deserved to succeed by virtue of the cultural superiority he claimed they possessed, they were not necessarily destined to triumph, as Marx had claimed that the proletariat is or as Christ promised Christians in the Sermon on the Mount by predicting that the meek will inherit the earth. The only assurances of success were fertility, military strength, and racial purity. The state's job is to promote these and to destroy anything that works against them. Morality is defined not by principles or commandments but by service to these goals. What promotes them is good and praiseworthy, what impedes them is evil and traitorous. In other words, Nazism combined arrogance about Germany with anxiety about its future, and the combination translated into virtually unlimited aggressiveness.\n\nGiven these premises, Nazi ideology was thoroughly and unabashedly self-centered. Hitler openly and repeatedly proclaimed: \"We know only one people for whom we fight, and that is our own. Perhaps we are inhumane! But if we save Germany, we have accomplished the greatest deed in the world. Perhaps we perpetrate injustice! But if we save Germany, we have abolished the greatest injustice of the world. Perhaps we are immoral! But if our people is saved, we have paved the way again for morality.\" This absolute ethical solipsism is a\u2014perhaps _the_ \u2014central article of faith of Nazism. That the philosopher Hannah Arendt, herself a refugee from Nazi Germany, believed she had discovered the distinguishing attribute of Adolf Eichmann, the quintessential Nazi \"desk murderer,\" in his supposed \"thoughtlessness,\" which she defined as his inability to see the world through the eyes of others, always has struck me as puzzling. His supposed inability was, in fact, a refusal, and it was not a characteristic feature of the task-fulfilling automaton that she saw Eichmann as, but rather the cultivated trait of a fervent adherent to Nazism. To subscribe to Hitler's ideology was to affirm that only the views and only the fates of Germans mattered; swearing never to put oneself in the place of non-Germans was part and parcel of being a National Socialist.\n\nIn this thought system, the greatest enemy of the Germans is _der Jude_ (\"the Jew\") and _das Judentum_ (Jewry), generally referred to with these singular nouns in order to deny any variation among Jews and to assert their homogeneity. That people is supposedly like no other in that it has no country of its own, but instead lives as a parasite within other societies. And, like a parasite, \"the Jew\" allegedly drains the strength of the host. Jews, said Hitler, unalterably seek to undermine Germans' fertility, military strength, and purity in order to make them too weak to cast Jews off and out. Thus, Jews were behind prostitution and venereal disease, delusional notions like international law and human rights, and softhearted ideas about the equality and brotherhood of peoples. Like Nietzsche, Hitler thought Jews had introduced the debilitating language of morality, ethics, compassion, and empathy into the world. Their idea of conscience was, he insisted, choosing his simile deliberately, \"a blemish like circumcision\"\u2014a supposedly unnatural alteration of how human beings are created.\n\nLogically, then, \"the Jew\" must be contained and ultimately \"removed\" from the German sphere if Germany is to succeed in the struggle for \"living space\" ( _Lebensraum_ ) and survival. Hitler therefore promised, in _Mein Kampf_ , to roll back emancipation and drive Jews into their own world or abroad by expelling them first from German political life, then from the nation's cultural life, and finally from its economic life. Most of the Nazi Party's public statements and private planning documents prior to 1933 followed this three-stage format. The Party platform, the Twenty-five Point Program of 1920, for instance, contained Point 4, calling for the denial of citizenship to Jews and their descendants; Point 5, demanding their classification as \"resident aliens\"; Point 6, excluding them from public office; Point 8, blocking further immigration by Jews and expelling all who had entered the country since the beginning of World War I; and Point 23, barring them from owning newspapers. These were all political and cultural restrictions. The programs laid down by the Legal and Domestic Policy sections in the Party headquarters in Munich in 1931 included these intentions, as well as removing Jews from the civil service and banning intermarriage with non-Jews. In June 1932, Hermann G\u00f6ring of the NSDAP made a speech envisioning these actions plus an exclusion of Jews from all prominent positions in the press, theaters, film, universities, and schools, all of which are cultural institutions. But he also said that in a future Nazi state every Jewish business person who stayed as an alien \"will remain able to operate his business undisturbed and under the protection of the law.\" So far as the Party let on prior to 1933, the goal was separation of Jews from non-Jews, reduction of Jews' capacity to influence non-Jews, expulsion of immigrant and naturalized Jews, and making the life of the rest so difficult that they gradually would leave.\n\nAlthough Nazi leaders did not talk openly of murder, let alone en masse, they made plenty of threats of violence and organized occasional local assaults on Jews, such as the bloody riot of 1932 on Berlin's elegant boulevard, the Kurf\u00fcrstendamm. And the storm troopers (SA) sang a marching tune with the words \"when Jewish blood spurts from the knife.\" Moreover, Nazi antisemitism always was implicitly murderous because of the metaphors it used: Jews were likened to vermin, parasites, germs, and cancer and called carriers of \"racial tuberculosis.\" These are things to kill or cut away, and Hitler dubbed himself more than once the Robert Koch of politics, referring to the famous bacteriologist who discovered the bacilli that cause anthrax and tuberculosis and thus greatly reduced their incidence. Above all, Hitler was always more dedicated to the goal than to any particular means; it was fixed, they were changeable. At the core of the Nazi vision was an unwavering dream of a Jew-free environment, since that was a precondition of German strength and happiness. This is extremely important because, as will become apparent, the combination of the appeal of that dream and its frustration by events drove the Nazis to consider ever more radical means of pursuing Hitler's goal.\n\nIn sum: Nazi ideology was a witches' brew of self-pity, entitlement, and aggression. It was also a form of magical thinking that promised to end all of Germans' postwar sufferings, the products of defeat and deceit, by banishing their supposed ultimate cause, the Jews and their agents.\n\nYet the centrality of the so-called Jewish problem was much more important and obvious to Hitler than to the average German voter. We have no reason to think that the antisemitic nucleus of his ideology propelled Hitler's rise to power. It played an important role in attracting many of the core believers to the Nazi Party, but not the mass of the Nazi electorate. Hitler was a product of crisis and opportunity, and Germans seem to have been drawn to him out of desperation and a sense that only the Nazis were energetic and organized enough to deal with the nation's woes. In 1928, before the Great Depression struck, the NSDAP received only 2.6 percent of the votes in the national parliamentary election, which was less than half the Party's share in the first of two rounds of voting four years earlier. Clearly, antisemitism alone had, as before, little political traction. As always, it could gain mass support only in tandem with a crisis that antisemites could exploit.\n\nAfter 1930, by which time Germany's economic difficulties had intensified and the Nazis' share of the national vote had jumped to over 18 percent, Hitler and the Nazis actually steadily downplayed antisemitism as a campaign issue, knowing that it already had attracted as many followers as it could. Instead, the Nazis concentrated on attacking what they called \"the System,\" by which they meant parliamentary democracy and free market capitalism, both of which they wanted to replace with more authoritarian political and economic arrangements. The platform they ran on was summarized succinctly by Gregor Strasser, the day-to-day director of Party operations in the early 1930s, when he defined National Socialism as \"the opposite of what exists today.\" And their method in state and national parliaments, as well as in municipal councils, was to disrupt democratic government, make it dysfunctional, and thus \"prove\" its ineffectiveness in meeting Germans' needs. In a fundamental sense, this highly partisan political force ran against politics, with all its messy compromises, disagreements, and imperfections, and promised to replace it with order and strength. National Socialism promised Germans both radical change and reassuring return to old certainties, and the mix appealed to many people in the atmosphere of anxiety that the Depression spread. In short, dissatisfaction with the nation's political and economic condition, along with fear of communism, the votes for which also were rising, clearly had more to do with Hitler's ascent than hatred of Jews.\n\nYet, as before 1918, antisemitism continued to have social traction. Various forms of discrimination escalated in the 1920s, including physical attacks on German Jews. In part, these developments were continuations of the earlier backlash against emancipation, since the Weimar Republic had removed the last forms of professional discrimination against Jews. Many of them became professors, judges, and civil servants during the 1920s; a few even became military officers and diplomats. Ironically, even as German Jews' birthrates continued to fall and their rates of intermarriage rose, Jews seemed to loom ever larger in public perception. The 100,000 or so Jews from Eastern Europe who managed to get into Germany during the years of weak border enforcement between 1916 and 1920 became the subject of paranoid fears of \"overforeignization\" ( _\u00dcberfremdung_ ), particularly because of their concentration in Berlin, and the prominence of Jews in the arts became an excuse to blame them for the alleged \"corruption\" of German culture during the Roaring Twenties. The nation's moralists had a field day with the fact that the leading proponent of sex education and research and of gay rights, Magnus Hirschfeld, was a Jew, as was the owner of Germany's preeminent manufacturer of condoms, Julius Fromm, an immigrant from Poland who had changed his first name from Israel. At a time when Jews were actually a declining presence in German life, whether measured by their total number, their share of the population, their representation in the professions and among university students, and their incidence among the very wealthy and in corporate boardrooms, Jews remained the subject of a persistent fixation on the part of many other Germans who were dissatisfied with the nation's condition.\n\nThis fixation eased the Nazi Party's electoral ascent in 1930\u201332 but did not propel it. The real driving force of Hitler's rise was the widespread and increasingly desperate desire in Germany for deliverance from the Depression and its unsettling effects. The nation's catastrophic economic situation increased receptivity to the Nazi movement and reduced antisemitism as a disqualifier for office. The context of Hitler's rise between 1928 and 1932 was unemployment that exceeded one-third of the workforce, a drop in industrial production by 42 percent, the collapse of the value of stock market shares by 60 percent, a decline in farm prices by 38 percent, a fall in total national income by 41 percent, and a reduction in real wages for people who still had jobs by 15 percent. The parliamentary system seemed incapable of devising policies that would reverse the crisis, and in fact the national legislature was stalemated after 1930 and unable to form a coalition that commanded a majority of the votes in Parliament. The president therefore exercised his power under Article 48 of the Constitution to appoint the prime minister and cabinet and rule by decree. These governments chosen by President Paul von Hindenburg adopted first a policy of deflation, which is to say cutting government expenditures\u2014what we nowadays call austerity\u2014and that only worsened the crisis. The cabinets that followed Heinrich Br\u00fcning's then switched in 1932 to a version of supply-side economics, by reducing the tax burden on firms, and that had only slightly positive effects.\n\nBut why were the Nazis the principal beneficiaries of the crisis? Why did they alone seem to capitalize on it? Actually, they were not alone; the communists also gained greatly in strength during the death rattle of the Weimar Republic, though not nearly as much as the Nazis did. But this fact actually worked in the Nazis' favor; it seemed to confirm what they constantly reiterated: Germans' choice came down to us or them, brown or red, and no middle ground remained. When the nation's options were reduced to these, the Nazis were bound to benefit. This is why they actively sought to generate street fights with leftist groups; every such battle strengthened the Nazi claim that the nation was on the verge of civil war, and that citizens therefore had to choose up sides between Hitler and the Commies ( _die Kozis_ ).\n\nIn addition, the competing political parties all seemed tired and unwilling to reach beyond their natural bases; for the Social Democrats these were unionized workers; for the Center Party, Catholics; for the People's Party, business leaders; for the Democrats, educated professionals; for the Nationalists, primarily aristocrats and farmers; and for the communists, the unskilled and largely nonunionized parts of the labor force. None of these parties had any imaginative or creative response to the Depression beyond waiting it out or, in the case of the communists, nationalizing everything. Even the Social Democrats in the summer of 1932 voted down the so-called WTB-Plan, named for the first initials of the last names of its authors, a massive government spending program that represented the only potentially successful way to jump-start the economy. The Nazis were fortunate in their opponents.\n\nThe way the Nazis campaigned\u2014relentlessly and energetically\u2014exploited the contrast between those groups and Hitler's party. Nazism described itself as a movement, a _Bewegung_ , and it was indeed a kind of political perpetual motion machine that never shut down. Nazis did not campaign just during elections but constantly. In the little north-central German town of Northeim, whose 10,000 citizens did not have a lot of entertainment options, the Nazi Party held an average of three meetings per month between 1930 and 1933. And the gatherings had the format and impact of religious revival meetings, with plenty of military music, well-trained speakers brought in from outside the town, and pageantry. Often the speakers were former war heroes or, in Protestant areas, Lutheran pastors who railed against \"the Godless left.\" Always there was emphasis on the youthful nature of the Party, its disproportionate appeal to the young\u2014over 40 percent of Nazi Party members prior to 1933 were thirty years old or younger\u2014in order to demonstrate that Nazism represented Germany's future. Moreover, the Party did not just campaign via meetings, rallies, and street fights. Its brown-shirted members were constantly visible taking up collections for the destitute or opening soup kitchens for the unemployed, thus giving the impression that they had the _will_ , a very important word in the history of Nazism, to fix things. All of this made a huge political impact in places like Northeim, which became a bedrock of Nazi support. Long before Adolf Hitler visited the town for the first time in mid-1932, almost two-thirds of its citizens were voting for the Nazi Party.\n\nRelated to this was a special feature of Nazi campaigning, the Party's propaganda, which was carefully tailored to disparate audiences. In working-class districts, the Party played up its populist streak, attacked the selfishness of aristocrats and big business, and posed as the defender of the little guy. In traditional areas like Northeim, the Party berated the unions, spoke up for family values, and emphasized patriotism. Nazism was, in other words, not quite all things to all people but agile in adjusting to its immediate surroundings.\n\nThe contrast between the lethargy and stasis of the old parties and the dynamism and youthfulness of the Nazis opened the way for them to offer Germans something distinct and appealing: unity. Alone among the parties, the Nazis could claim to draw followers from every social class and every part of the nation (other than Jews). Even though German Protestants were more inclined to join than Catholics, rural and small-town residents more than city dwellers, middle-class people more than workers, and men more than women, significant numbers of people in all these groups were members of the Party. It alone could claim to be gathering Germans of all walks of life into a \"People's Community\" ( _Volksgemeinschaft_ ). Unity is a very seductive word when people are tired of or frustrated with politics, and the Nazis seemed convincing in their pledge to wipe out divisions, if necessary by force.\n\nThe promise of Nazism was to restore all that was best in Germany's traditions yet also to revolutionize the country at the same time. Perhaps the best way to grasp how this worked is to look at what the Party held out to women: On the one hand, it promised to \"emancipate women from woman's emancipation\"\u2014that is, to restore their primary field of activity as the home and childbearing; on the other hand, it enlisted women in all sorts of paramilitary, athletic, and productive activity from which they had been largely excluded previously and told them they could be just as important as men in building the People's Community, only in different roles.\n\nThese circumstances explain why the Nazis became the largest political party in Germany by 1932, but they were not enough to propel Hitler to a majority in Parliament. At the end of 1932, German politics was deadlocked. Between the parliamentary elections in July of that year and those in November, Hitler had lost 2 million votes, 4 percent of the total cast. His electoral march seemed to have crested, and the Party faced a severe financial crisis because the membership dues and fees on which it depended had fallen off sharply. On New Year's Day 1933, the humor magazine _Simplicissimus_ ran a poem with the final line, \"This 'F\u00fchrer's' time is up,\" and the more sober _Frankfurter Zeitung_ congratulated Germans on having survived the Nazi onslaught. Elections had carried Hitler to the threshold of power but not across it. For that, he needed the help of an elite conservative group around Franz von Papen, a former chancellor. The conspirators wanted to engineer and serve in a cabinet backed by Hitler's large block of votes in Parliament and expected to be able to control Hitler because the Nazi leader had little formal education and had never held major office. Article 48 left the choice of chancellor to President von Hindenburg, and a clique of aristocrats and landowners went to work on him in January 1933. Led by Papen, they persuaded the aged president to offer the prime minister's position to Hitler.\n\nThe Jews of Germany were almost powerless to affect the course of events. Far from being the fantastically controlling wire-pullers of Hitler's feverish imagination, they were too few and isolated and their resources too limited to make any difference. Their fate depended on a German population that contained a minority deeply hostile to them and a vast majority that was indifferent or unsympathetic to them.\n\nIndifference and lack of sympathy were the principal effects of the combination of many decades of vocal German antisemitism and a decade and a half of intense German crisis. Both processes reduced the number of vigorous antiantisemites, the people who were willing to defend Jews or who thought that Nazi threats to Jews made Nazis unacceptable as leaders. If the Nazis' antisemitism was not a major contributor to their victory, neither was it a significant barrier to it. Even sympathetic non-Jews were inclined to understate the menace the Nazis represented by quoting the old German maxim, \"Nothing is eaten as hot as it is cooked.\"\n\nStill, probably 55 percent of the Germans had never voted for Hitler or the Nazis by the time he came to power. A majority had remained loyal to their traditional political allegiances: the Center Party for the Catholics, the socialists and communists on the left for most workers. As a result, William Sheridan Allen has ventured the observation that more Germans \"were drawn to antisemitism because they were drawn to Nazism, not the other way around.\" He probably is correct, and the observation is a reminder that the key to understanding what happened in Germany after 1933 is not so much events and attitudes that predated that turning point but ones that developed after it. The short answer to the question \"why the Germans?\" is \"because Hitler came to power,\" but it is too short an answer.\nCHAPTER 3\n\n[ESCALATION: \nWhy Murder?](contents.xhtml#ch_3)\n\nHITLER AND THE NAZI PARTY came to power having declared their intention to strip Germany's Jews of citizenship and the right to hold office, to exclude them from the civil service, journalism, education, and the arts, and to ban intermarriage with non-Jews. The new masters of the nation expected these measures would reduce not only Jews' influence over the rest of the population but also their very numbers in Germany, and the regime planned to accelerate the latter process by barring Jewish immigration and expelling all Jews who had entered the country since the outbreak of World War I. In short, the Nazis set out to degrade, segregate, and diminish Germany's Jews but not yet to kill them, let alone all the Jews of Europe. Although Nazi rhetoric toward Jews regularly employed implicitly murderous metaphors, likening Jews to pests or diseases to eradicate, official policy initially concentrated on harassment, intimidation, isolation, and dispossession but generally stopped short of organized and widespread physical violence. The individual Jews subjected to brutality during the formative months of the Nazi dictatorship were usually officeholders or politicians, executives of firms that attracted the Nazi Party's attention for various reasons, or people who dared to object to Nazi actions, and those did not yet include large-scale destruction of property and roundups of Jews, let alone mass murder. Why not? And why did the situation escalate thereafter?\n\nIn answering these questions, much depends on whether the intervals January 1933\u2013November 1938 and January 1933\u2013June\/October 1941 seem like short or long times. Does the first interval, the time between Hitler's appointment as chancellor and the onset of systematic assaults upon and arrests of Jews during the Crystal Night pogrom, amount to \"merely\" or \"more than\" five and a half years? Does the second interval, the time between Hitler's appointment and the beginnings of mass murder of the Jews, come to \"merely\" or \"more than\" eight and a half years? If the answer is \"merely,\" the implication is that the Nazi leaders moved fast and probably knew what they were up to rather quickly. If the answer is \"more than,\" that suggests that the regime actually proceeded gradually and may have changed its objectives along the way. Either way, an important question arises about each of the turning points of 1938 and 1941, a question that a good historian has to ask about every significant event she or he studies: Why now?\n\nMy answer to that question is that the Nazi regime engaged in a three-stage discovery or learning process between 1933 and 1941. In the first phase, which lasted for just over five years from the time Hitler came to power until the annexation of Austria in March 1938, the so-called Third Reich learned what it could do, namely persecute the German Jews without encountering serious resistance from Germany's other inhabitants or from other countries. In the second phase, which lasted for a bit more than three years from the takeover in Austria until the invasion of the Soviet Union in June 1941, Nazi Germany learned what it nonetheless could not achieve\u2014namely, the complete \"removal\" or expulsion of Jews from its territory. In the third phase, which lasted only five months from the attack on the Soviets to the fall of 1941, Hitler and his most important advisor and executor in this matter, Heinrich Himmler, recognized that they possessed not only the motive but also the means and the opportunity to murder the Jews in not only the newly occupied territories of Serbia and the Soviet Union under cover of war, but also in all of Europe.\n\nFROM ARYANIZATION TO ATROCITY\n\nThe first phase of the Nazi assault on the German Jews proceeded under the euphemistic watchword of \"Aryanization\" ( _Arisierung_ ), which referred to the process of transferring Jews' jobs and property in Germany into the hands of non-Jews. Since Nazi ideology depicted Jews as parasites who had acquired what they had by draining it from the gentile majority of the population, the Party faithful regarded this process as simple payback for decades of deception and theft and thirsted to begin the repossession immediately after Hitler took office. But the Nazi leaders were more cautious.\n\nHitler and his principal advisors could not yet be sure in 1933 of how much persecution domestic or foreign opinion would accept, and they had other issues on their minds. After all, over half of the German population had never voted for Hitler, Nazis had only three of twelve seats in the cabinet President von Hindenburg appointed, and the authority to rule by decree was initially the president's, not Hitler's. Hitler knew that even many German antisemites were not as \"scientific,\" by which he meant categorical, in their hatred as he was. He frequently complained that the problem with most Germans is that each had a \"good Jew,\" a friend or acquaintance who did not conform to antisemitic stereotypes and who therefore should be treated as an exception to the general condemnation. The Nazi F\u00fchrer knew he would need time to win over most of the population to his conviction that persecuting all Jews was a necessary act of self-defense and indispensable to national survival. Moreover, the new regime needed stability and an appearance of moderation while generating a recovery from the Depression that would secure Hitler's hold on power. Finally, the Reich had to lull Britain and France into tolerating the military buildup that was a prerequisite for the conquest of \"living space\" for Germany in Eastern Europe. Hitler the fanatic was obsessed with Jews, but Hitler the politician and expansionist dared not let the obsession show too much or too early.\n\nCaught between the ideological fervor of its followers and the practical requirements of economic and foreign policy, the Nazi regime devised a two-tier approach to the so-called Jewish question. At the national, official, and public level, a stop-and-go policy allowed the regime to feel its way forward from 1933 to 1937 and to test the limits of public acceptance, both domestic and foreign. Overt displays of organized antisemitism were confined to the boycott of Jewish stores and businesses on April 1, 1933, which the regime depicted as merely a reprisal to a supposed wave of \"atrocity propaganda\" that Jews had instigated abroad. Otherwise, the Nazis generally contented themselves at the national level with a few well-spaced decrees that enacted the sorts of measures the Party long had advocated. Conspicuous outbreaks of overt violence against Jews, such as two more nasty rampages on Berlin's Kurf\u00fcrstendamm, in March 1933 and July 1935, were exceptional.\n\nIn 1933, the regime focused its efforts on driving Jews from political and cultural life. That goal led to four laws that: (1) purged Jews from the German civil service, including the law courts and hospitals, since these were state institutions in Germany, unless a Jew had held his position since before World War I, served in the army during that conflict, or had a father or son who died while doing so (these were the so-called Hindenburg exceptions, adopted in order to placate the field marshal-turned-president); (2) allowed the government to denaturalize people who had become citizens since that war began; (3) excluded Jews from cultural institutions, such as theaters, orchestras, and newspapers; and (4) imposed a _numerus clausus_ restricting Jews' share of students in German secondary schools and universities to 1.5 percent. In the economy, on the other hand, the Nazis treaded somewhat carefully, harassing individual Jewish executives and demanding their removal in many cases, but not actively trying to drive out Jewish owners except in sectors that were particularly important to the Party rank-and-file, notably department stores and breweries. The one major legislative initiative in this sphere in 1933 was a prohibition on Jews owning farmland, a particularly sensitive issue for a Nazi Party that proclaimed the twin pillars of Germandom to be _Blut und Boden_ , Blood and Soil.\n\nThe year 1934 brought a legislative lull, as the Third Reich decreed only one major antisemitic measure, a law that increased the government's ability to deport people it denaturalized. Then, in 1935, the regime closed the ranks of the German military to Jews, forbade them to display German flags on national holidays, and completed the agenda laid down before 1933 by stripping Jews of German citizenship and reducing them to resident \"subjects\" of Germany, banning new intermarriage and all extramarital sexual relations between Jews and non-Jews, terminating the Hindenburg exceptions, now that the aged president had died, and dismissing the last Jews from the civil service. Nineteen thirty-six saw another lull, as the regime downplayed its antisemitism in the run-up to the Summer Olympics in Berlin, lest other nations decline to participate, and little happened at the national level in 1937, either, aside from a prohibition on granting doctoral degrees to Jews.\n\nThis staccato pattern of increasing persecution at the national level from 1933 to 1937 belied, however, a continuous \"squeezing\" of Jews by Nazi activists at the local or street level, usually out of ear- or eyeshot of foreign reporters. Even in big cities, but especially in small towns, pressure was brought to bear on Jews' jobs and businesses in a host of ways. While the brown-shirted storm troopers of the SA threatened or inflicted harm on Jews' property or children, Nazi officeholders canceled contracts or refused to entertain bids from firms owned or led by Jews, welfare agencies prohibited the use of payments or vouchers at Jewish-owned shops, local leaders forbade their employees to buy at such establishments or to patronize Jewish professionals and publicly posted lists and\/or pictures of people who did so, Jews' businesses became identifiable by the absence of \"German firm\" ( _Deutsches Gesch\u00e4ft_ ) window signs that now proliferated among competitors, municipal councils banned Jews from having stands at public market halls or using public swimming pools, local and regional savings banks and credit unions stopped making loans to Jews or their businesses, branches of the Nazi labor union (the NSBO) insisted on dismissals of managers, tax authorities seized ledger books and charged Jews with tax evasion or illegal transfers of money abroad, and in many places, Nazi stalwarts intimidated non-Jewish shopkeepers, especially sellers of foodstuffs, into refusing to accept Jewish customers, thus forcing them either to move or to shop far from home where they would not be recognized. All the while, the Nazi-controlled press kept up a barrage of allegations regarding supposed Jewish criminality and deceitfulness.\n\nDiscriminatory actions of these sorts gave the Party's radical antisemites satisfaction and habituated Germans to Jews' suffering. As the forms of exclusion and persecution multiplied, the Nazi regime also learned that almost no non-Jew would stick up for Germany's Jews; instead, most people and institutions hastened to adapt to the way the wind was blowing. Across the country in 1933, clubs, singing groups, bowling leagues, and similar organizations began restricting membership to so-called Aryans and thus inflicting on Jews what Marion Kaplan has called \"social death.\" Jews found themselves increasingly abandoned and alone.\n\nThis popular acceptance, even adoption and internalization, of Nazi antisemitism was not the only success of the two-tier policy between 1933 and 1937. Another was the regime's dispelling or deflecting of international opposition. Despite some vocal criticism overseas, notably at large rallies in New York City, and attempts to boycott the sale of German goods abroad, the persecution of the Jews did little to dampen the urge in Britain and France to avoid war by appeasing Hitler or to undermine the resurgence of the German economy that increased his popularity at home.\n\nYet from the Nazis' point of view, their success was incomplete and the Jewish problem only half solved by late 1937. On the one hand, they had achieved the isolation of Jews from the rest of the population and gone a long way toward impoverishing them. Since 1933, up to 40 percent of the indeterminate number of businesses that Jews owned in Germany and between 40 percent and 50 percent of their wealth had become the property of someone else or the German state. In addition, most of the Jews still in Germany had been relegated to a destitute economic ghetto, as they scraped together meager livings working for themselves or each other.\n\nOn the other hand, the Jewish population in Germany had fallen by only about 35 percent, and the slow pace was an increasing irritant to Hitler. The F\u00fchrer held the Jews responsible for Germany's defeat in World War I; he did not intend to let them undermine the nation if World War II occurred, and as the 1930s passed, he saw that conflict coming ever closer. Already in mid-1936, he had written a memorandum that laid the basis for an economic Four Year Plan. The document called for an economy impervious to blockade and an army capable of war within four years and included a passage demanding \"A law making the whole of Jewry liable for all damage inflicted upon the German economy by individual specimens of this community of criminals.\" That Hitler inserted a provision about Jews in a long discussion of military and economic preparations shows how seriously he took the stab-in-the-back legend from World War I and how determined he was to hold the Jewish community collectively liable for perceived acts of sabotage.\n\nOn November 5, 1937, the F\u00fchrer presided over a meeting of his foreign and war ministers and his principal military commanders and gave a speech, summarized in a memorandum by his adjutant Colonel Friedrich Hossbach, that Hitler described to them as \"in the event of his death, his last will and testament.\" The gist was that Germany had to fight for living space by 1943\u201345 at the latest, when the Reich's window of opportunity would close. By then, the greater resources of the British and French empires would have enabled them to catch up with the momentary advantage in armaments that Germany had achieved by breakneck spending since 1933. Meanwhile, however, opportunities to annex Austria and wipe out Czechoslovakia might arise with surprising speed, and the Nazi regime would have to seize them, even at the risk of conflict coming earlier. Hitler said nothing about Jews at this meeting, but its aftermath had a great deal to do with Jews.\n\nClearly, Hitler's remarks alarmed the people assembled at the conference, as well as a few people outside it. Economics Minister Hjalmar Schacht, who was not there, already had cautioned Hitler that the pace of German rearmament had to slow down lest inflation get out of hand. Now, War Minister Werner von Blomberg, Commander in Chief of the Army Werner von Fritsch, and Foreign Minister Constantin von Neurath warned against precipitous action because they feared the army was unready and the British and French too strong. In the ensuing months, Hitler fired all of these men, Schacht promptly and premeditatedly in November, the others opportunistically the following February when a series of perceived sexual scandals created room for political intrigue.\n\nMoreover, the following months saw the enactment by Acting Economics Minister Hermann G\u00f6ring of a series of measures designed to speed up the process of driving Jews out of the German economy and then out of the country altogether. The fact that the regime went over to a program of forced, accelerated Aryanization at this juncture was not coincidental. The move reflected Hitler's conviction that the Jews were disloyal and sure to be saboteurs and fifth columnists when a conflict came. Nazi planners knew that at the current rate of attrition through emigration and death, the German Jewish population would disappear in fifteen to twenty years, but that was well beyond Hitler's time horizon for war. More pressure had to be put on Jews to leave, lest they once more have the supposed opportunity to stab the nation in the back.\n\nA crushing avalanche of new decrees designed to pauperize the Jews of Germany and convince them that they had no future in the country began with the definition of Jewish firms as ones with even a single senior Jewish executive or more than 25 percent of their stock in the hands of Jews. Such enterprises became immediately ineligible to receive government contracts, foreign currency to pay for necessary imports, and rationed raw materials, without which those enterprises could not operate. In March 1938, Jewish communities lost their legal status and the right to own property, which opened the way to the confiscation of their synagogue and school sites. In April, all Jewish-owned enterprises were required to register and all Jews were ordered to fill out an itemized census of their property and its value, down to the last teaspoon. To make Jews instantly recognizable, they had to add uniform middle names\u2014Israel for men, Sarah for women\u2014to all their identity papers in August, and in October to have a large red letter _J_ stamped onto the front page of their passports. In July, Jews were forbidden to work as traveling salespeople, which cost 30,000 their jobs; in September, Jewish doctors were forbidden to treat non-Jews; and in November, Jewish lawyers were demoted to legal counselors whose sole permitted task was to help other Jews wind up their businesses and dispose of their property. In December, a new law permitted local governments to ban Jews from the public streets on certain days of the week.\n\nAs if all this were not enough, now the Nazis decided also to terrorize and deplete the Jewish population. In annexed Austria, unfettered intimidation characterized the German occupation from its earliest days, as Nazis broke into Jews' homes, looting and smashing with impunity. The violence spilled over into Germany during the summer of 1938, producing the destruction of the main synagogue in Munich in June and its counterpart in Nuremberg in August. In both the old and the new parts of the Reich, about 5,000 Jews were carted off to concentration camps on various pretexts, sometimes on none at all. Meanwhile, forced expulsions of foreign Jews occurred, beginning with those who held Soviet citizenship in February and culminating with the deportation of some 18,000 Polish Jews in late October, mostly to the village of Zbaszyn on Germany's eastern border. The Poles had triggered this action by announcing their intention to bar any Polish citizen residing abroad from ever reentering the country unless he or she quickly applied for and obtained an authorization to return by November 1 for passport revalidation. Because the Polish government was trying to shed these citizens, its representatives in Germany dragged their feet in granting the necessary authorizations, obviously trying to run out the clock on the return visits. Because the Nazis wanted Polish Jews in Germany to have somewhere to go, either immediately or eventually, which cancelation of their passports would make more difficult, the Reich wanted to expedite the revalidations. Thus the Germans rounded up thousands of Polish Jews in Germany and delivered them to the border, the Poles simply refused to let the Jews in, and many of them languished miserably in no-man's-land well into 1939. Among the deportees were the parents of Herschel Grynszpan, a young Jew living illegally with relatives in Paris, and he took his revenge by walking into the German embassy in Paris on November 7, 1938 and shooting the third secretary, a young diplomat named Ernst vom Rath.\n\nHitler and the Nazi regime seized on vom Rath's death two days later as the pretext for a vicious collective assault on the German Jewish population, once more billed as an act of self-defense against Jewish hostility. This was the so-called _Kristallnacht_ (Crystal or Broken Glass Night) pogrom, a wave of destruction and plundering that swept over most remaining Jewish-owned homes and businesses and nearly all of the nation's synagogues. The storm troopers disguised as civilians who spearheaded this operation found a good many fellow citizens, especially teenagers, willing to join in the violence. By the time the mayhem stopped, the perpetrators had killed at least 91 Jews but perhaps many more, driven at least 300 people to commit suicide, and rounded up some 36,000 Jewish men across the country. About 26,000 of them were exposed the following day to public humiliation as they marched to trains and buses destined for the concentration camps at Buchenwald, Dachau, and Sachsenhausen. At least 600 and perhaps up to 1,000 of these men died from brutal treatment in subsequent months at these places, from which a person could obtain release only by signing over virtually everything he owned and promising to emigrate. In addition, in the aftermath the German government seized a fraction of the payments due to the Jews for insured damage and allowed the German insurance companies to renege on paying the rest. Then the Nazi state imposed a collective fine of one billion reichsmark on the Jews, payable in part by the confiscation of all their possessions containing precious metals, except for wedding rings and one table setting per person. And in April 1939, after having stripped nearly all Jews of gainful employment and all unemployed Jews from the welfare rolls, the Nazi state imposed compulsory labor on all Jewish males below the age of sixty-five, tens of thousands of whom now had to clean streets, shovel snow, and work in factories at discriminatory wages.\n\nEven before this cascade of cruelty, few Jewish Germans needed much convincing that they had to leave. By early 1938, more applications for visas to get into other nations were on file at their consulates and embassies in Germany than Jews were left in Germany. But getting out was difficult, especially because the German policy of stripping Jews of all they owned made them unattractive immigrants in the eyes of many foreign governments. Besides, the Depression was not over in most countries, and resistance to immigration fed in many places on fear of competition for jobs. Nonetheless, about 60 percent of the Jews of Germany and 67 percent of those in Austria managed to escape by the time World War II began.\n\nThe Nazi regime remained unsatisfied, largely because its foreign policy victories were negating its racial policy successes. As of September 1939, the annexations of Austria in March 1938, of the Sudetenland border region of Czechoslovakia in October after the Munich Conference, and of the remainder of the Czech provinces of Bohemia and Moravia in March 1939 had offset much of the reduction in the Jewish population of Germany proper by that date. On the eve of World War II, the Greater German Reich, including the annexed areas, contained around 350,000 Jews, far too many for the comfort of a Nazi regime that was about to invade Poland, the home of 3.3 million more Jews.\n\nAll of this was foreseeable in advance, but the fateful mathematics of German expansionism, the fact that the Reich could not drive out Jews faster than it planned to conquer them, seems to have dawned on German policymakers only during 1938 and contributed heavily to both the accelerated persecution of Jews at the time and to the regime's turn to overt violence. The math also accounts for the emergence in Nazi circles of a new vocabulary about the destiny of the Jews. Given the prospect of not being able to get rid of Jews faster than the Reich added them, officials began to give voice to the previously unthinkable.\n\nThe first documented emergence of a new word and a new prediction comes from a report of a Swiss diplomat in Paris of a conversation on November 14, 1938, less than a week after _Kristallnacht_ , in which the number two man in the German Foreign Ministry, Ernst von Weizs\u00e4cker, said, \"The remaining . . . Jews in Germany should immediately be deported somewhere. . . . If . . . no country will take them in, they surely are going sooner or later toward their complete annihilation.\" Ten days later, on November 24, _Das Schwarze Korps_ , the publication of the Nazi Party's elite SS formation, which now also controlled all German police, editorialized as follows: \"The German people are not in the least inclined to tolerate in their country hundreds of thousands of . . . impoverished Jews. . . . In such a situation we would be faced with the hard necessity of exterminating the Jewish underworld . . . by fire and sword. The result would be the actual and final end of Jewry in Germany, its complete annihilation.\" Finally, in January 1939, Hitler made the new vocabulary his. On the twenty-first, he told the Czech foreign minister, Frantisek Chvalkovsky, that the Jews of Germany would be \"annihilated\" unless other nations cooperated in deporting them. Nine days later, in a speech to the Reichstag on the sixth anniversary of his appointment as chancellor, he predicted \"the annihilation of the Jewish race in Europe\" in the event of a new world war. As yet, these remarks mentioned annihilation only as something that would happen under certain conditions, but for the first time the thought was out in the open. So much so, in fact, that the U.S. consul general in Germany, Raymond Geist, prematurely concluded and told the State Department on December 6\u2014weeks before Hitler uttered even the threat\u2014that \"[t]he Germans. . . . have embarked on a program of annihilation of the Jews.\"\n\nOn September 1, seven months after Hitler's address to the Reichstag, he launched the Second World War in Europe by invading Poland. Less than four weeks later, on the day that Warsaw finally surrendered, the SS created a new subdivision, the Reich Security Main Office (Reichssicherheitshauptamt, or RSHA), under the direction of Reinhard Heydrich and including a Jews Department ( _Judenabteilung_ ) headed by Adolf Eichmann. Initially, the German invaders shot fewer Jewish Poles than non-Jewish ones; as at home in 1933, the Nazi rulers were more concerned with punishing potential political opponents and resisters than attacking Jews per se. But sporadic attacks on Jews occurred and worse followed. The victorious Germans ordered ghettoization and the almost complete abandonment of Jews' possessions on September 21 as a means of concentrating Jews along railroad lines for eventual deportation and seizing most of their property in the meantime. Where were they to go? To what the Nazis explicitly called, in imitation of the history of the United States' policies toward Native Americans, a \"reservation\" ( _Reservat_ ), but where was it to be located? Initially, in an area called Nisko on the San River at the western edge of the Lublin district of the General Government (GG), the rump of Poland that Germany occupied but did not annex. Then, early in 1940, Hitler hoped to talk Stalin into accepting the more than two million Jews still in Greater Germany and occupied Poland, but the effort came to nothing. Finally, after France fell in June, the Germans focused on the French colony of Madagascar, a destination for Jews favored by European antisemites since the late nineteenth century. German planners actually began to work out how many ships would be needed over how long a time to deport the 3.25 million Jews now in Hitler's hands, and the Gestapo compelled several German Jewish leaders to find out whether American Jewish organizations would help to finance the exodus. But the German failure in the aerial Battle of Britain made transportation impossible, so a fourth destination gained prominence as 1940 turned into 1941: Siberia above the Arctic Circle, after victory in the impending invasion of the Soviet Union.\n\nThis transition in German policy from encouraging emigration, which remained possible, to planning deportations, marked a major turning point, for it amounted to a first step toward implementing annihilation. To be sure, the regime had not yet decided to kill every Jew, but it had chosen a course that entailed the death of a great many, either in the poorly provisioned ghettos or in the inhospitable designated destinations, for which the Jews would be physically unprepared and materially unequipped. In confirmation of this shift in policy, from September 1939 to June 1941, ghettos and forced labor camps took the lives of more than half a million Polish Jews. Still, as late as May 1940, Heinrich Himmler called \"the bolshevist method of the physical destruction of a people . . . un-German and impossible.\" He was referring in that remark to the treatment of Poles, but events soon made him reconsider whether such inhibitions applied also to Jews.\n\nSeveral developments in 1940\u201341 created incentives to find a \"total\" or \"overall solution\" ( _Gesamtl\u00f6sung_ ) to the Jewish question sooner rather than later. While the Nazi Gauleiters, the regional Party bosses, especially Joseph Goebbels in Berlin, clamored to begin deportations of the remaining German Jews to Poland, partly to free up housing for other Germans, partly as an end in itself, a logistical logjam developed in that conquered country. Himmler, freshly appointed as the head of the National Commission for the Strengthening of Germandom (the Reichskuratorium f\u00fcr die Festigung deutschen Volkstums, or RKFDV), had embarked on a massive program of demographic engineering there. It entailed the repatriation to the German-annexed parts of Poland of 500,000 _Volksdeutsche_ , people of German descent, from the Soviet Union and the Baltic states, pursuant to the agreement by which Hitler and Stalin had partitioned Eastern Europe in 1939. Simultaneously, at least twice as many Jews and Poles were to be expelled into the non-annexed but German-occupied General Government. But Hans Frank, the Nazi governor there, pushed back, claiming that his fiefdom should not and could not become a \"dumping ground\" for Jews, since conditions in the ghettos already were awful, and those conditions endangered the health of nearby populations. Himmler's views on the future of the General Government also appear to have evolved during the run-up to the invasion of the USSR. Since the occupied Polish region would no longer be at the periphery, but rather at the center of an expanded German empire, the GG now became in his eyes a potential area for Germanization, instead of the demographic \"trash heap\" that he had considered it earlier, and that meant the expansion of his aspirations for ethnic cleansing to the whole of Poland.\n\nMeanwhile, the victories in southeast Europe in early 1941, which brought German occupation of Serbia and parts of Greece and cemented the Reich's alliances with Hungary, Romania, and Bulgaria, multiplied the number of Jews within the Nazi ambit, thus increasing the pressure to do something about them on a continent-wide basis. The decision to attack the Soviet Union in 1941 threatened to do the same and to expose German troops to the imagined possibility of Jewish sabotage behind the advancing lines. Finally, that decision, plus Hitler's expectation that America would soon join the alliance against him\u2014the United States had broken off diplomatic relations with Germany in June 1941, and President Franklin Delano Roosevelt and Prime Minister Winston Churchill just had signed the Atlantic Charter, an implicit alliance, in mid-August\u2014meant that European Jews no longer had any value as hostages whose fate could be used to pressure the Allies or to intimidate other Jews abroad. In the euphoria of the initial German victories in Belarus and Ukraine, any reason for restraint toward the Jews fell away.\n\nIn other words, a combination of impatience, frustration, and hubris convinced the Nazi leaders that they had much to gain and nothing to lose by proceeding more radically now against the Jews rather than waiting until the victorious conclusion of the war. As a result, Hermann G\u00f6ring charged Reinhard Heydrich on July 31, 1941, to prepare \"an overall solution of the Jewish question in the German sphere of influence.\" But by the time he did so, the Germans already had taken the first step toward mass murder. In the course of July, their initial \"pacification\" efforts in conquered territory developed into a resolve to avoid a repetition of the Polish logjam and the administrative problems that the ghettos presented by bringing death to the Jews in the Soviet Union. The dispensers of this death began as four _Einsatzgruppen_ (Operations Groups), comprising fewer than 3,000 men, who were subdivided into eighteen mobile _Kommando_ units assigned to advance behind the German troops, foment pogroms, and shoot potential \"partisans\" and communists. Initially the victims were mostly Jewish men of military age, but the killing spread to women and children in late July 1941, only a month after the invasion began, and two brigades of 10,000 SS men plus 30,000 German Order Police were sent to the East to help do the job. Militias drawn from the local populations, the so-called _Schutzmannschaften_ , and security divisions of the regular German army stationed in areas behind the front supplemented these forces.\n\nThese disparate killing units slaughtered more than half a million Jews in the last six months of 1941 and perhaps one million by early 1942. Whereas in Poland some Jews had been killed and most ghettoized, in Ukraine, Belarus, and the Baltic states the pattern was reversed: Death became the norm, anything more than brief ghettoization the exception. The murders reached a crescendo between late August and late September, when 24,000 Jews were killed at Kamenets-Podolsk, 28,000 Jews at Vinnytsia, and nearly 34,000 at Babi Yar outside Kiev, in each case over two days. At these places and elsewhere in the occupied Soviet Union, most victims died, in a sense, one by one, by single shots to the back of the head or neck, not by machine-gun fire, because the killers wanted to be as sure as possible that they had not missed or wasted ammunition.\n\nEverywhere the cover for murder was _Partisanenbek\u00e4mpfung_ , combating partisans, even though few were at work in the early months of the war on the Eastern Front. And, everywhere from August on, the Germans claimed that women and children had to die, too, because they served as the eyes and ears of snipers and other guerrillas resisting the German advance. Alongside this military justification stood an ideological one, the Nazi conviction that Jews were the masterminds and wirepullers of Bolshevik rule. General Walter von Reichenau's order to his Sixth Army of October 10, 1941, rolled the legitimations together in the pronouncement that \"the soldier must have full understanding for the necessity of harsh but just punishment of the Jewish sub-humans. It has the broader objective of nipping in the bud any uprisings in the Wehrmacht's rear, which experience shows always to have been instigated by Jews.\" Hitler considered Reichenau's order outstanding and had it distributed to every German unit fighting on the Eastern Front.\n\nThe rapid growth in the number of victims is not surprising when one recalls that the Germans entered the USSR with a _Hungerplan_ that called for feeding their armies off the land and letting upwards of twenty million Soviet citizens starve to death. In keeping with the plan, the Nazi regime fed and provisioned the prisoners of war from the Red Army so poorly that 58 percent of those captured during the war, more than three million people, died in captivity, more than half of them in the first seven months of the German invasion. Reducing the number of mouths to feed in the conquered East was a consistently high priority for the Germans, and spreading the killing to ever more numerous groups of Jews aligned perfectly with military planning. But the food supply was a reinforcing, not a primary, motive for murder. Events proceeded along parallel lines in German-occupied Serbia during the summer and fall of 1941, with Jews being shot en masse in reprisal for partisan attacks even where provisions were not scarce. The Jews did not have to die because Nazi officials kept finding justifications for murder; the causal process ran the other way around.\n\nBy late summer 1941, Nazi policy had evolved from driving Jews from Germany's space to virtually forcing them to leave, to concentrating them for deportation and assuming many deaths along the way, to bringing death to the newly conquered. Only one step remained, bringing the already conquered Jews to death. The first tentative step in that direction occurred in mid-September, perhaps triggered by Stalin's decision to deport the remaining Germans in southern Russia to the country's interior. Hitler now finally agreed to require German Jews to wear a distinguishing Star of David on their clothing, a measure long since enacted in occupied countries but not yet in Germany itself. Marking the Jews was a preliminary to their deportation, as the Gauleiters had been demanding with increasing insistence, and all that stood in the way of making their departures end in immediate eradication was figuring out how to kill them en masse.\n\nGENTILE AND JEWISH RESPONSES\n\nHow was it possible for the Nazis to radicalize their assault on German and later European Jewry without appreciable interference? The question requires breaking down into three more precise subquestions: (1) Why did non-Jewish Germans, most of whom had not accepted antisemitism as a powerful political motive before 1933, act afterward as if it was exactly that? (2) Why did Jews in Germany not organize more effective countermeasures or at least all flee? (3) Why did foreign powers or entities not intervene on humanitarian grounds?\n\nThat more than half the Germans had not voted for Hitler by 1933 does not mean that these people rejected antisemitism. Some were faithful Catholics who had voted for the Center Party that was closely associated with the Church but also had absorbed its religiously based hostility toward Jews. Others did not believe that the Jews were the chief cause of the nation's troubles, as the Nazis insisted, but did not particularly like Jews, either. The main problem was that the number of antiantisemites was limited, and most non-Jewish Germans thought the fate of the Jews was secondary to their own concerns. Indifference and self-interest created opportunities for the Nazis to change people's behavior by a combination of carrot and stick, rewards for endorsing the new regime's ideology and punishment for not doing so. The punishment might be, but was not necessarily, violent. It might be only a slowing or blocking of a person's advancement in his or her career. The Nazi regime had numerous mechanisms that promoted conformity and corruption, and one of the most alarming features of the Holocaust is not only the rapidity with which these worked their effects on Germans but also the way in which these effects were replicated in virtually all the Nazi-occupied and -allied lands of Europe later, with disastrous consequences for Jews.\n\nEven among the segments of the German population that were best educated, most cosmopolitan, and most averse to violence, a process that could be called self-coordination ( _Selbstgleichschaltung_ ) set in remarkably rapidly in 1933 and led to a swift abandonment of organized efforts to protect Jews as a group. Several senior German diplomats, for example, considered resigning in early 1933 in protest against Nazi discrimination and brutality, but only one of them, Friedrich von Prittwitz und Gaffron, the German ambassador in Washington, actually did so. A group of leading business executives, including Carl Friedrich von Siemens, of the giant corporation that bore (and still bears) his family name, and Carl Bosch, of IG Farben, the huge chemicals conglomerate, met a few times during the year to draft a document intended to dissuade Hitler from antisemitic actions, but they never actually submitted it. Instead, the typical response of corporate executives was to knuckle under to Party attacks on Jews while seeking, at most, to shield a few valued individuals. Thus, Gustav Krupp von Bohlen und Halbach, the head of the National Association of German Industry, caved in to the demands of storm troopers who occupied his office in Berlin on April 1, 1933, and agreed to dismiss the Jews employed by his organization, along with anyone else the Nazi Party deemed politically unacceptable. Thus, too, Degussa, a firm that refined precious metals, responded to insinuations by Nazi newspapers that it was under Jewish influence by issuing notarized announcements that it had never employed Jews. In this fashion, the firm hoped to divert attention from the facts that several Jews had played a significant role in founding the enterprise seventy years earlier and six Jews still sat on the company's supervisory board in 1933.\n\nWhy did prominent, successful, established Germans fail to take a moral stand in 1933? There were many reasons. For one thing, the Nazi regime rapidly acquired a monopoly on political discourse and changed the moral valence of hatred from bad to good. Prior to 1933, antisemitism seemed crude and shameful in many quarters; now it was identified with patriotism everywhere. Conversely, expressing sympathy for Jews was now an unpatriotic act that could attract suspicion or condemnation. Attacking Jews was of far greater importance to the Nazis than defending them was to other Germans, so most such people decided that discretion was the better part of valor and said nothing. Besides, even Germans who found Nazi antisemitism distasteful approved of other aspects of the Party's program\u2014in other words, shared a partial identity of interests with Hitler's movement. Diplomats and military officers, for example, generally longed for the revision of the terms of the Versailles Treaty and the resurgence of the German army, and Hitler promised to deliver these things. Many corporate executives hoped for the suppression of the trade unions and recovery from the Depression, and Hitler embraced these goals, too.\n\nAbove all, in the rapidly changing and violent context of early 1933, most upper-class Germans took refuge in a delusional mix of fear for their livelihoods and a misplaced sense of responsibility. As Ernst von Weizs\u00e4cker of the Foreign Ministry wrote at the time, \"the specialist cannot simply quit the field.\" Instead, he had to give the regime \"all forms of support and experience . . . and help see to it that the . . . current revolution becomes genuinely constructive.\" Of course, Weizs\u00e4cker did not find this particularly difficult because he also deplored what he imagined as the nation's \"inundation with Jews\" since 1919. But even a somewhat more liberal figure, Fritz Roessler, the head of the supervisory board of Degussa, tried to put the best face on things in 1933 and concluded that people like him \"should recognize the good in the movement, ignore the human deficiencies associated with every revolution, and do one's bit so that this wild-grown juice becomes wine.\"\n\nAs in Weizs\u00e4cker's and Roessler's cases, a strong sense of duty in 1933 often blinded people in high places to the implications of their choices. In the long run, helping the Nazis \"in order to avoid worse,\" as the phrase of the day went, merely made them stronger and more dangerous. As the pastor of Kurt Gerstein, who later served in the SS and procured some of the Zyklon gas used at Auschwitz, told him in the 1930s, when he decided to join the Nazi Party and try to influence it from within, \"you reckon that you can still have a say in things. . . . [But] he who enters this tumbling avalanche only increases the plunging mass.\" Very few Germans were this farsighted.\n\nAmong younger people still working their way up professionally, the way the Nazis mixed intimidation and indoctrination in 1933 comes across very powerfully in Sebastian Haffner's memoir, called in English _Defying Hitler_. That title is melodramatic and sensational, but Haffner did flee his homeland in 1938 and make a new career in England as a journalist before returning to Germany after World War II. In his native language, he called his book simply _Geschichte eines Deutschen_ (Story of a German). Haffner, whose real name was Raimund Pretzel, was twenty-six years old when Hitler came to power and a law student preparing to take his bar exams. He paints vivid pictures of the marauding storm trooper units that beat up anyone in the streets who failed to raise a hand in the Nazi salute when a swastika flag passed by and of the day in the spring of 1933 when these thugs broke into the law library where he was studying, asked all the patrons, _\"Sind Sie arisch?\"_ (\"Are you an Aryan?\"), and assaulted those who said no or appeared to be lying. He walked away from that occasion deeply ashamed of himself for having answered the question truthfully with a yes. This was far from the only or even the first such assault on the legal system that spring.\n\nThat was the intimidation side of 1933; the indoctrination side came later, when the new regime ordered Haffner and all the other bar candidates to spend the summer at a kind of boot camp for future lawyers, where they were taught the Party's racist ideology and drilled endlessly. Searching for a word to describe what the experience had done to him and his peers, Haffner coined a neologism based on one of the words Party members used to address each other, _Kamerad_ , or comrade. He said the camp had _verkameradet_ these young men, which means \"comraded\" or \"comradified\" them. The militarization of German life by institutions and practices such as this discouraged critical thinking during the 1930s and encouraged group identification, solidarity, and obedience. So did the regime's relentless emphasis on the People's Community ( _Volksgemeinschaft_ ) and insistence that German citizens had moral obligations only to \"Us\" and no one else. A great, intoxicating glorification of \"belonging\" began to grip German life. Bernhard Rust, the new Nazi minister for education, explained the intellectual obligations this imposed in 1933, when he told a group of professors in Munich, \"From now on, it is not up to you to decide whether something is true, but whether it is in the interests of the National Socialist Revolution.\"\n\nThese stories illustrate how power magnifies the ideas of those who hold it because of the human tendency to seek safety in conformity. The only antidotes are conviction\u2014loyalty to a strong countervailing ideology\u2014and the freedom to express it. Where these are lacking, as was the case in Germany after 1933, ideologues quickly get the upper hand and call the tune for behavior. A minority of haters, backed by the authority of the state, thus becomes free to drive events forward, to make the lives of any targeted group ever more miserable.\n\nSelf-interest dictated to most other Germans that they should ignore what was happening to the Jews or treat it as merely the price of the apparently good things the Nazi regime was bringing. After all, by 1936, the Depression was over and unemployment a thing of the past; the Nazi regime had achieved the fastest economic revival in the world. By the same year, Germany had recovered the Saar region, which France had administered since 1919; had renounced the military limits imposed on the Reich by the Versailles Treaty; and had sent its soldiers back into the Rhineland, the formerly demilitarized western strip of territory that bordered the Netherlands, Belgium, Luxembourg, and France. Within the next two years, Hitler annexed Austria, brought the German speakers of the Sudetenland \"home\" to the nation of which they had never been a part, and occupied the rest of today's Czech Republic, all without firing a shot or losing a soldier in battle.\n\nSelf-interest also encouraged some people to seek to benefit from the persecution, which accounts for the many eager lawyers and brokers who acted as middlemen in the sale of Jews' assets and the numerous willing graspers for their medical and legal practices, their artwork, their houses and apartments, their furniture and carpets, and so on. Many non-Jews concluded that they could not stop the persecution, so they might as well get something out of it. Even Germans who did not exploit the situation in this fashion increasingly looked out for themselves by cutting off contact with Jewish friends and neighbors, thus both increasing their isolation and becoming deaf and blind to their suffering.\n\nBesides, the gradual nature of the Nazi escalation raised the general problem of seeing ahead that affected all parties, including non-Jewish Germans, Jewish Germans, and foreigners. Had people known that cruelty and discrimination would become starvation and slaughter, more might have balked. But even the Nazis did not know this in 1933, so why should anyone else have been sure? Instead of imagining where persecution might lead, Germans got caught up in the completely self-referential intellectual world that the Nazis created, where public information was tightly controlled, foreign publications were banned, the adjective \"cosmopolitan\" was a preferred term of abuse, and people were constantly reminded to \"work towards the F\u00fchrer,\" to imagine what Hitler would want them to do and then to do it. The public mind was, in other words, methodically poisoned, and the measuring stick of morality systematically shifted from general ethical principles like the Golden Rule to the specific matter of whether an action strengthened Germany or did not. This warping of people's thinking worked especially powerfully on young people coming of age during the 1930s, who seldom had an independent frame of reference. Teenagers were at the forefront of violent attacks on individual Jews and attempts to humiliate them and those who consorted with them during that decade.\n\nTwo caveats about this collective brainwashing need stressing. First, it did not necessarily change what older people thought, but it decisively changed what they would say or do. The Nazis defined public discourse and controlled the social reward system, and that was enough to limit open disagreement or dissent. Second, the corruption of people's sense of decency toward Jews did not happen overnight or without the occasional application of extra pressure. Another story about Degussa illustrates the latter point. The tale concerns the behavior of Ernst Busemann, the head of Degussa's managing board, toward two Jewish families, the Meyers and the Margulieses, who had sold majority interests in their firms to Degussa during the 1920s but retained 26 percent of the stock and management positions in the respective subsidiaries. In November and December of 1937, the NSBO chapter at the first of these subsidiaries wanted it to compete for the honor of being a National Socialist Model Factory. The union therefore petitioned the managing director to make this possible by buying out the Jewish members of the Meyer family and removing them from management. The director dutifully wrote to Busemann and asked what to do. Busemann's reply survives, and it is a remarkable document that begins with fulsome praise for the members of the Meyer family as old friends and upstanding businessmen, then expresses regret that he holds their fate in his hands, and finally delivers the crushing judgment that, nonetheless, \"it is pointless to swim against the stream\"\u2014the Meyers and their shareholding would have to go. To soften the blow, Busemann contrived a way to pay for their stock in the subsidiary with stock in IG Farben that had the same face value but a considerably higher market worth. In contrast, only a few months later, in April 1938, after the Nazis demanded the expulsion of the Jewish minority owners of the Degussa subsidiary in Austria, Busemann offered the members of the Margulies family only a fraction in cash of what the stock was worth and told them to take it or leave it. Why the abrupt and extreme change? The political danger of being discovered acting generously or even sympathetically toward Jews had become much greater in the aftermath of the annexation of Austria (the _Anschluss_ ) and G\u00f6ring's decrees accelerating Aryanization, and Busemann adjusted his behavior accordingly.\n\nThese stories from Haffner's memoir and the history of Degussa are indicative, but are they representative? What do we know about what most Germans thought about the persecution of the Jews, and how do we know it? We actually have quite a few sources, of which the following four are among the most important: the _Sopade-Berichte_ , periodic reports smuggled out to the Social Democratic Party in Exile in Prague by leftist opponents of the regime still in Germany; the _Stimmungsberichte_ , or \"mood reports,\" collected by Gestapo (secret state police) agents among the public and published decades later as _Meldungen aus dem Reich_ (an English translation of passages concerning Jews comes to 657 pages); numerous diaries kept by non-Jewish Germans and gleaned effectively by Peter Fritzsche in _Life and Death in the Third Reich_ ; and a brilliant, poignant diary kept by a baptized Jew with a non-Jewish wife, Victor Klemperer, who survived the Nazi regime, which has been published in English in two volumes under the title _I Will Bear Witness_.\n\nThese sources paint a complex, inconsistent picture, in which acts of kindness mix with extraordinary callousness, but the overall portrait is of a public split into three groups: people who endorsed the persecution of the Jews, people who merely accepted it, and people who disliked it but saw little point in protesting, even though they frequently expressed reservations or felt embarrassed about specific actions. The Gestapo reports vividly record both the bullying and harassment that Jews experienced on a daily basis and the distaste that such actions sometimes aroused. In September 1934, the office in Potsdam regretted to relate that \"the Jewish question is not the main problem of the German public. . . . Utterances on the Jewish peril are played down, and those engaged in enlightening the population are depicted to a certain extent as fools.\" The following July, the office in Kiel stated, \"It is noteworthy that, whenever there are actions against the Jews, these emanate chiefly from members of the Party and its affiliated organizations, whereas the majority of the population shows little participation.\" In October 1935, the office in Magdeburg had this to say about the public response to the Nuremberg Laws, enacted that year: \"all in all, it is accurate to note that the new laws have been received in part with indifference, and in part with very little appreciation and understanding outside the solid National Socialist-oriented population.\" Other offices reported precisely the opposite, however, contending that the public welcomed the Nuremberg Laws as finally creating clarity about the position of Jews in Germany.\n\nInsofar as one can generalize about the indications of adult German public opinion prior to _Kristallnacht_ , they record general acceptance of antisemitic policies except when they threatened the self-interest of non-Jews. Thus many Germans resented the Party's appeals to stay away from Jewish-owned shops, because they were perceived as often offering better goods at better prices, and many farmers had to be forced to break off their relationships with trusted Jewish livestock dealers. Similarly, but in a more abstract sense, many Germans feared damage to the nation's image abroad from antisemitic actions and from the prominence of virulent antisemitic publications, notably Julius Streicher's _Der St\u00fcrmer_. In the mid-1930s, it was posted conspicuously every day in special glass display cases in many towns and villages, but as time passed, public discomfort led to a decline in that practice. The reaction to _Kristallnacht_ itself reaffirmed this pattern, as numerous people in the street expressed shame and disgust on the morning after, though as much at the wasteful destruction of property and appearance of disorder as at the harm done to Jews. Whatever the mix of attitudes among the non-Jewish population, the decisive point is that violence and viciousness toward Jews increased steadily during the 1930s in Nazi Germany and in full public view, especially in small cities and the countryside, yet the pattern gave rise to too little rejection or revulsion to make the Nazi regime change course.\n\nOnce the war began, hostility toward Jews hardened. Rumors were rife about ghettoization and impending deportation from Germany to Poland, and little dissent emerged in the early years of the fighting. Many Gestapo reports in 1940 stressed the powerful impact on the populace of the propaganda film _Jud S\u00fcss_ , which clearly strengthened antisemitic feeling. In July 1941, the Gestapo office in Berlin commented on the public response to the first round of newsreels from the Eastern Front in these words, \"The images of the arrest of the Jews . . . have met with enthusiastic approval, and people say that the Jews here [that is, in Germany] are being treated with far too much leniency. The series of pictures on the forced deployment of the Jews in clearing operations were greeted everywhere with great delight.\" And, in September, local offices from all around the country chorused that the order for German Jews to start wearing identifying Stars of David on their clothing had been greeted with \"genuine satisfaction\" and \"gratification.\" Nonetheless, a month later, Goebbels lamented in his diary, \"our intellectuals and high society have once again suddenly discovered their humane feelings for the poor Jews.\" He therefore made sure that an announcement of new punishments for \"Jew-friendly behavior\" accompanied the next set of monthly ration books delivered to every German home.\n\nDespite such reminders, the Nazi regime felt the need to take additional precautions against sympathy arising toward German Jews after their deportation began. In Berlin during October 1941, the first contingents of people being sent \"to the east\" reported to the Levetzowstrasse synagogue in the Moabit neighborhood near the center of the city and then had to walk six kilometers in broad daylight to the loading point at the Grunewald freight railroad station on the far west side of town. When the shipments resumed in mid-1942, the authorities decided to expose fewer witnesses to the spectacle by conducting the marches in the middle of the night. Similarly, later that year, when Berlin Jews began being shipped from another, even more centrally located collection point in the Grosse Hamburgerstrasse to the ghetto camp at Theresienstadt in Bohemia, they made the first legs of the journey before dawn via streetcar to trains that left from the not yet bustling Anhalter railroad station.\n\nThe use of Theresienstadt (Terezin in Czech) as a destination for elderly and decorated German Jews also attested to the Nazi regime's residual desire to disguise what was happening. A formerly Austrian garrison town in Bohemia that had been turned into a holding pen for Czech Jews in November 1941, the site became after mid-1942 a supposed refuge for German Jews unable to perform the \"work in the east\" to which most deportees allegedly were being sent. In reality, this \"old people's ghetto\" proved to be a mere way station on the road to death for most of the 58,000 Jews from Greater Germany ever confined there. About 41 percent of them were sent on to death camps; a slightly larger share died on the site from cold, hunger, and disease; and only about 7,000 remained at liberation in 1945. Meanwhile, however, Nazi propaganda highlighted the mythical comforts of the installation in order, first, to mislead Germans about the regime's intentions, and then, in 1944, to delude the gullible representatives of the International Red Cross whom the Reich allowed to tour the temporarily prettified grounds.\n\nAt least with regard to two categories of German Jews, those descended from or in marriages to non-Jewish Germans, the regime was for a time also cautious. It introduced restrictions on them more slowly than on \"full Jews\" ( _Volljuden_ ) or people the Nazis counted as such because of their marital status or religious affiliation ( _Geltungsjuden_ ) and deferred expulsion, lest numerous non-Jewish German relatives protest. Still, essentially the same long discovery process that had occurred in making policy toward German Jews took place, only at a slower pace, toward those Germans who had one Jewish grandparent (second-degree _Mischlinge_ ; 40,000 people in 1939) or two but no other connection to Jews or Judaism (first-degree _Mischlinge_ ; 64,000 in 1939). Thus _Mischlinge_ , unlike Jews, had neither lost German citizenship nor been forbidden to have sexual relations with so-called Aryans under the Nuremberg Laws, though future marriages between Aryans and _Mischlinge_ were banned. _Mischlinge_ were barred from military service later than Jews and exempted from concentration in so-called Jew Houses and from deportation to ghettos or death camps until 1943. Thereafter the regime cracked down, exhibiting increased confidence or fanaticism in sweeping potential objections aside and encountering, in fact, very few. Roundups for incarceration in forced labor camps began in the spring of 1944, followed in early 1945 by the planned deportation to Theresienstadt of all remaining _Mischlinge_ and all of the approximately 21,500 German and Austrian Jews who still clung to precarious existence as part of mixed marriages. Victor Klemperer, the now famous diarist, escaped deportation in early 1945 only because the firebombing of Dresden, where he lived, occurred just before his scheduled departure, and the resulting chaos enabled him to conceal his identity as he and his wife fled the city. Many other prospective deportees were not so lucky. Although shipments from Berlin were impeded by the last Soviet offensive, trains from places like Frankfurt and Leipzig arrived at Theresienstadt, and some of those aboard did not survive the few remaining months of the war; many barely did so. Had the Third Reich endured or even lasted a little longer, most of the first-degree _Mischlinge_ and the Jewish spouses of non-Jews appear to have been destined for death and most of the second-degree _Mischlinge_ for sterilization.\n\nAs Germans grew steadily more hard-hearted toward Jews and steadily more receptive to Nazi propaganda about them after 1933, why did the Jewish community in Germany not defend itself better or at least get entirely out of harm's way? In a sense, the question, like the one often raised about Jewish behavior in the ghettos of Poland examined later in this book, is terribly na\u00efve and cruel. Jews were up against a Nazi movement that was both ruthless and shameless in what it would say about and do to them. They constituted a tiny share of the German population to begin with in 1933 and became ever fewer as time passed. They shared with everyone else an inability to see what was coming, all the more so as it involved behavior unprecedented on the part of a civilized country.\n\nAbove all, Germany's Jews, like those of occupied Europe later, were not monolithic and conspiratorially united, as the Nazis claimed, but divided among themselves about what the Nazi onslaught signified and therefore how to respond to it. About two-thirds of them were liberal, acculturated, often somewhat secular or entirely non-observant Jews, either members of or in sympathy with the Centralverein deutscher Staatsb\u00fcrger j\u00fcdischen Glaubens, the Central Association of German Citizens of the Jewish Faith, a name that signified their desire to be integrated into the German nation and to have the same rights as all other German citizens. For this group, the Nazi attack was difficult to comprehend and especially painful to experience because it seemed such an unwarranted rejection of their loyalty to Germany.\n\nThe other two principal groups, the Orthodox, who accounted for perhaps 20 percent of Jews in Germany in 1933, and the Zionists, who then constituted 5\u201310 percent, were not so hurt by the Nazis' hostility, because they expected it. To the Orthodox, it was the work of an inscrutable God but probably a punishment for the apostasy of so many German Jews. The answer was to pray harder. To the Zionists, advocates of settling and founding a Jewish state in Palestine, endemic hostility of gentiles toward Jews was the assumption on which their movement rested. The Zionist answer to German persecution was to work with the Nazis on the basis of a common conviction that Jews and Germans constituted separate nationalities in order to achieve one partially shared objective: emigration of Jews from Germany. Partially shared, not identical\u2014because the Nazis wanted to drive all Jews out of Germany; but the Zionists knew that the _Yishuv_ , the Jewish settler community in Palestine, could afford to take only some Jews in, preferably young and fit ones who could speak Hebrew and were willing to do hard physical work on collective farms, the _kibbutzim_. Moreover, even while the Nazis promoted the Zionist goal of Jewish immigration to Palestine, they opposed the Zionist objective of founding a Jewish state there.\n\nThis overlapping interest in emigration resulted in the controversial _Ha'avara_ , or Transfer Agreement, of August 1933. It created a modest escape route for German Jews during the 1930s, eventually financing the emigration to Palestine of about 20,000 of the 52,000 German Jews who got there by 1939, and contributed to the increasing popularity of Zionism among German Jews during the 1930s. But the Transfer Agreement was morally questionable, hotly debated at the time, and consequently not generally imitated by or for Jewish communities elsewhere. Basically, the agreement set up a system by which Jews seeking to leave Germany for Palestine had their possessions in Germany appraised and then handed them over to the German state. Germany thereupon paid some individual \u00e9migr\u00e9s whose wealth exceeded a certain minimum in reichsmark at least 1,000 Palestinian pounds sterling, the threshold level of cash assets for unrestricted entrance to Palestine. The Reich then was supposed to pay the remainder of all admitted German Jewish emigrants' wealth to the Jewish Agency in Palestine in the form of German goods that the agency could sell for the benefit of these or other new settlers. The _Yishuv_ got people out, and Germany got most of the Jews' property, along with increased production for export that buoyed employment at home and thus strengthened the Nazi regime. For a while, this system seemed to offer something to both Zionists and Nazis, but its economic value to Germany declined rapidly. Beginning in 1935, the Nazi regime steadily raised the reichsmark minimum for issuing Palestinian currency and reduced the range of goods available for resale. In the end, exiting German Jews got to retain less than 1.5 percent of their property via the agreement, and it remained operative until World War II began only because Hitler did not wish to abandon any device that might encourage Jews to leave Germany.\n\nAs to other forms of emigration, German Jewish leaders at first hesitated. The Central Association discouraged emigration in 1933\u201335 because leaving amounted to surrendering claims to Germanness and abandoning those Jews who could not get out. But after the enactment of the Nuremberg Laws, the Central Association changed its attitude and began to encourage emigration. The organization remained a proponent of the diaspora over a Jewish state but conceded that German Jewry had no long-term future by renaming itself in 1936. Dropping the reference to German citizenship, the group became simply the J\u00fcdischer Centralverein, the Jewish Central Association. By 1937, as noted above, most German Jews had begun seeking an exit, even though they had limited prospects. But some people had better chances of being accepted elsewhere than others. Broadly speaking, age worked against a person, youth worked for him or her: 84 percent of German Jews under the age of twenty-four in 1933 got out alive, compared to 60 percent of the total population; by 1939, one-third of the remaining Jews in Germany were sixty or more years old, and just more than half were over fifty. A person possessing the few skills needed elsewhere had a better chance than someone whose abilities threatened to compete with residents of other countries. This often meant that people with artisanal or agricultural training had better prospects than professionals. Wealth sometimes enabled people to leave early because other countries were more open to persons bringing money, and in the early years of Nazi rule, Jews could take a larger share of their assets with them than later was allowed. But wealth also tempted people to remain, since the Nazis generally targeted large firms and their owners last, and then such people generally lost almost everything. Men had better chances of going abroad than women, but the fact that women made up 60 percent of the remaining German Jewish population in 1939 almost certainly reflected something else\u2014namely, that they more often assumed caretaker responsibilities for aged parents or disabled or handicapped relatives than did men, given the prevailing gender roles of the day. A good many of the Jews still in Germany in 1939 simply could not leave someone behind who depended on them.\n\nThe legend that German Jews faced the persecution passively or incredulously is just that, a legend. They fought back the only way they collectively could: by equipping as many people as possible with skills that would help them get out and by mutually sustaining all those who remained. Already in 1933, they organized a Central Welfare Office of German Jews and a national organization, the Reichsvertretung der deutschen Juden, the National Representation of German Jews. These groups collected and disbursed funds for labor offices, cars for traveling salesmen, legal aid clinics, and the like. From 1933 to 1937, such welfare offices at the national and local levels spent 26.3 million reichsmark from their own resources, plus 7.5 million donated from abroad. Special groups for doctors, lawyers, and artists came into being and worked to find new positions for unemployed colleagues abroad or at Jewish institutions within the country. Some 140 retraining institutes were established, through which 30,000 people passed by 1938, two-thirds of them younger than twenty. As the government cut Jews off from the state welfare system, they depended increasingly for support on contributions from the shrinking population of Jews who still had work. Already in 1935, one-third of the Jews in Germany relied on such help, and Jewish soup kitchens across the country dispensed 2.5 million meals. But in subsequent years, both the number of people and the percentage of the population that the community could sustain dropped along with the size of the remaining population. Jewish self-help was fighting a losing battle, but it was an effort that did credit to the people who undertook it.\n\nIn 1939, the Nazi regime dissolved the Reichsvertretung and all other Jewish communal organizations and replaced them with a new entity that enrolled all remaining Jews in Germany. This was called the Reichsvereinigung der Juden in Deutschland, the National Union of Jews in Germany. Its leaders continued the heroic struggle to sustain Germany's remaining and increasingly aged and impoverished Jewish population, but the effort proved hopeless as the persecution escalated toward murder. By the time deportations began in October 1941, most of the remaining Jews in Germany were in a beleaguered and wretched state. Crammed at least two people to a room into Jew Houses with communal kitchens and baths and scattered around the worst neighborhoods of the big cities, deprived of their radios and even their pets, allowed fewer ration coupons for food and other goods than those allotted to non-Jewish Germans, and permitted to shop only in the final hours of specified days, by which time stores often had sold out, many Jews were on the edge of starvation and despair. Their leaders in the National Union were subordinated directly to the RSHA and sought to protect themselves, as did their counterparts in occupied Europe, by carrying out the SS's instructions. The Reichsvereinigung thus degenerated into an instrument by which the Nazi regime kept track of all Jews left in Germany, plundered what was left of their possessions, and then managed many aspects of the deportations, including identifying eligible Jews according to selection criteria set by the Reich Security Main Office. In 1942\u201343, the National Union even sent its own personnel, called _Ordner_ , or auxiliaries, to collect the people assigned to each transport if they did not comply with a summons to report on the preceding day. In Vienna the name given to these Jewish helpers of the SS was more descriptive; they were called \"lifters\" ( _Ausheber_ ).\n\nAs was the case in the ghettos further east, such submissiveness resulted from desires for both self-preservation and amelioration. Cooperation with the SS seemed the only available way for Jewish leaders to stay alive and to alleviate the plight of deportees by providing food and blankets to them at the collection and departure points. But behind the actions of the Reichsvereinigung was something else that also operated farther east: direct intimidation. The Nazis took fierce reprisals against recalcitrance or resistance. Emblematic of the viciousness were the actions that followed an attempt in May 1942 by a group around a Jew named Herbert Baum to burn down a propaganda exhibition against the Soviet Union in Berlin. The Gestapo caught thirty-three conspirators almost immediately and executed not only those people but another 250 Jewish men, who were rounded up and sent to Sachsenhausen, just outside the city. Another 250 Jewish males then also disappeared into that camp, the families of all 500 men immediately were deported \"to the east,\" and Goebbels stepped up the timetable for making Berlin \"Jew-free.\"\n\nHow did Hitler manage to ratchet up the persecution of Germany's Jews during the 1930s without provoking foreign interference or even intervention? He did this, in part, by phasing in restrictions and even occasionally holding out the prospect that some Jews could remain in Germany\u2014or at least in Theresienstadt\u2014in the long run. Nazi leaders kept people guessing about their intentions and said just enough contradictory things to make at least some outsiders believe that the worst would not happen. That some outsiders wanted to believe this is the second key piece of an explanation. In Britain and France, the two nations best positioned to alter Hitler's behavior before German rearmament had reached dangerous proportions, homegrown antisemitism combined with wishful thinking to argue for noninterference in Germany's internal affairs, however barbaric they might seem. That wishful thinking propelled the policy of appeasement, which amounted to the belief that protecting the rights of others, whether German Jews or the Czechs at the time of the Munich Conference, was not worth the risk of another world war and the terrible carnage that Britain and France had experienced. Until _Kristallnacht_ and sometimes beyond, many appeasers actually were inclined to blame Jews for poisoning relations with Germany rather than to blame Germany for persecuting Jews.\n\nHitler played brilliantly throughout the 1930s on fear of war in the Allied nations, and he invited them repeatedly to buy him off with concessions that he later announced were insufficient. This tactic worked so well for him until he occupied the Czech provinces in March 1939, only six months after he promised to leave them independent at the Munich Conference, that the Allies recurrently declined to let the fate of Germany's Jews upset the quest for peace. Hitler and his propaganda agencies also played shrewdly on the antisemitism present in Britain, France, and also the United States. He blackmailed these countries into reticence or silence by the simple trick of claiming that they were tools of the Jews and then citing any protest on their behalf by these countries as proof of his charge. Fearing to seem to confirm his propaganda and thus to arouse domestic antisemites, the Allied governments generally fell into a Nazi trap and pulled their punches, at least until _Kristallnacht_. Even after the pogrom\u2014in fact, less than a month later, on December 6, 1938\u2014France signed a new treaty with Germany reaffirming the integrity of the border between the two countries. Meanwhile, Joseph Lyons, the prime minister of Australia and a vigorous proponent of appeasement, resolutely refused to condemn the atrocities in Germany, lest doing so interfere with his efforts to head off war. Only one nation, the United States, exercised the usual diplomatic form of expressing revulsion at the Nazi rampage by calling America's ambassador home \"for consultations.\"\n\nThe Nazi regime was adept before the war at constructing choices that looked bleak either way for both Germany's Jews and the Allied states. German Jews rapidly recognized that they faced a constant choice between complying with Nazi actions and making them worse. They opted to do the best they could under barbaric circumstances and to play for time. The Allies constantly had to choose, at least after 1936, between accepting both Nazi territorial demands and Nazi mistreatment of German Jews or a bloody war that Britain and France expected would weaken their holds on their empires and, as Neville Chamberlain explicitly predicted, put them hopelessly in debt to the United States. Even if they won, the Allies stood to lose, and after 1945, they did, as Chamberlain's fears came true. Britain, after all, suffered under food rationing into the early 1950s because the war did such damage to its economy, and the British, French, Dutch, and Belgian empires melted away following the war.\n\nAs the persecution of the Jews escalated, the Nazi regime presented another group with unpalatable alternatives, namely the owners of foreign direct investment in Nazi Germany. Ford, General Motors, IBM, Standard Oil, and many other American corporations all possessed significant German subsidiaries during the 1930s, as did several large Dutch, Swedish, and Swiss enterprises. Recent books have criticized the American firms for not divesting their holdings in protest against mounting Nazi discrimination and brutality and, instead, letting their offshoots in Germany become complicit in German rearmament and in some cases in the persecution of the Jews. Some authors have spoken of a \"strategic alliance\" between American corporations and Hitler, of corporate \"collaboration\" and \"pacts\" with the Nazis.\n\nSuch overblown charges overlook a number of aspects of the situation the parent companies confronted. Divestment for political or moral reasons was a virtually unknown practice in the 1930s, which is the principal reason why almost no major corporation with holdings in Germany, regardless of the country in which its headquarters stood, suspended operations or sold out and withdrew. The exceptions were a few of the Hollywood film distribution companies, notably Warner Bros., which closed its German sales operations in 1933, and United Artists, Universal, RKO, and Columbia Pictures, which followed suit fairly quickly. But only Warner's gave up completely on the German market. The other four companies preserved special arrangements with German partners, and MGM, Paramount, and Twentieth Century-Fox kept trying to get their films introduced into German theaters and making the compromises that seemed necessary to that objective right up until World War II began. And these were not manufacturing firms with major fixed investments. Though businessmen like to say that \"all past costs are sunk,\" meaning that the chief criterion for continuing an enterprise is its future returns, not the capital previously committed to it, few firms in this or any other era have found acting on that maxim palatable, especially when the asset in question is even only slightly profitable. The prevailing tendencies among businessmen were to retain what they had in the hope that political conditions would improve in the future and meanwhile to try to extract the returns that had been the goal of the original investment.\n\nMoreover, financial controls established by the Nazi regime blocked the repatriation of income earned in Germany. Both while a company continued and in the event of its sale, all net proceeds had to be reinvested in the Reich or converted into government bonds. This reinforced the reluctance to divest, since the only comparably profitable investments were likely to be at least as implicated in German government policies as the ones the parent companies already possessed. In the event of divestment, then, the foreign investor faced not quite a total loss, but declining control over assets without appreciable moral gain. Finally, most foreign-owned companies in Germany spent the 1930s fighting and largely losing a rearguard action precisely against this declining control. In almost every case, managers from the owning country gave way to Germans, who took pains to position the subsidiaries as German firms in order to hold onto business and who acted increasingly independently of their home offices, not least because Nazi mandates regarding economic secrecy restricted what the local managers even could report about their activities. As a result, headquarters in Detroit in the case of the car companies and New York in the case of IBM had little influence over day-to-day operating decisions in their German affiliates after 1938 or 1939 at the latest; the same was true of Lever Brothers of the Netherlands, one of the largest foreign investors in Nazi Germany. For all of these reasons, as well as the general difficulty of seeing ahead, the Nazi persecution of the Jews did not encounter the sort of economic pressures successfully brought to bear some fifty years later on the apartheid regime in South Africa.\n\nIn any case, IBM's rebellious subsidiary, the _Deutsche Hollerith Maschinen Gesellschaft_ , managed by the spiteful German who formerly owned it, did not play the roles in identifying and later rounding up the German Jews or in managing slave labor that the parent firm's critics have maintained. From 1933 to 1943, the Gestapo used the card files efficiently compiled and regularly updated by the Reichsvertretung and its successor to keep tabs on the nation's Jews and their residences. The SS experimented briefly in 1944 with using Hollerith cards and tabulators to steer the deployment of camp inmates to work sites but soon gave up on the idea. GM's Opel division became complicit, in that it began building thousands of trucks for the German army and then aircraft engines for the Luftwaffe well before the United States and Germany went to war in 1941 and the company's plants were placed under a German trustee. But this acceptance of government contracts began only after the Nazi regime threatened to expropriate the firm. In 1939\u201341, Ford-Werke in Cologne produced fewer trucks for the Wehrmacht than Opel, but some were used in the invasions of Austria and Bohemia-Moravia, and the local management gave in, not to threats but to desperation to offset declining sales of civilian vehicles.\n\nAfter World War II ended, the economic appeasers who clung to their foreign investments in Germany could claim, like the political appeasers but with more positive results, that events had vindicated their course. The parent companies recovered their assets west of the iron curtain and even such profits as these had accumulated. Perhaps alone among all the groups confronted with poor options by the Nazi regime during the 1930s, the American owners ultimately found playing for time a successful strategy.\n\nIn sum, the years 1933\u201341 taught Hitler and his followers that neither Germans nor foreigners were inclined to interfere with Nazi actions toward Jews. In the context of the regime's inability to expel them faster than it conquered them, such passivity added impetus to ever more radical persecution.\nCHAPTER 4\n\n[ANNIHILATION: \nWhy This Swift \nand Sweeping?](contents.xhtml#ch_4)\n\nONE OFTEN AND surprisingly overlooked feature of the Holocaust is its combination of shocking temporal and spatial compression with sweeping extent. Although the Nazis kept killing Jews until the Third Reich crumbled, and although it rounded them up all over Europe to kill them, several striking fractions give a sense of how concentrated the time and place of the massacre was, as well as how encompassing. Three-quarters of the nearly six million victims were killed within only twenty months, from June 1941 to February 1943, and half of the total victims died within only the last eleven months of that time frame. Moreover, three-quarters of those killed lived before the war in only three countries: Poland, Lithuania, and the USSR (mostly in the northeast quadrant of the European continent, demarcated on figure 4 by dotted horizontal and vertical lines going east and north from Vienna), and probably nine-tenths of the victims died in those places, since that is where the _Einsatzgruppen_ , the Order Police, the Reserve Police Battalions, and the great bulk of the Wehrmacht operated and where Germans placed the death camps. Altogether, at least three-quarters of the Jews who ever came within reach of Nazi Germany and its allies were killed, constituting in the end two-thirds of the Jews of Europe (six million out of nine million when World War II began; the oft-quoted total of eleven million given in the minutes of the Wannsee Conference was an exaggeration or included converts as well as their children and grandchildren). For Jewish children sixteen or younger, the mortality rate was almost nine-tenths.\n\nWhy was the Holocaust so concentrated in these ways? How could the Nazis come so close to killing all the European Jews\u2014and do so at the average rate of 225,000 people per month, from mid-1941 to early 1943, and 325,000 per month (more than 10,000 _per day_ ), at the frenetic peak of the Holocaust in 1942\u201343?\n\nFIGURE 4: THE GEOGRAPHICAL COMPRESSION OF THE HOLOCAUST\n\nFROM BULLETS TO GAS\n\nIn a sense, the heart of the answer to these questions is technical. By 1941, the Nazis had a motive to kill the Jews of Europe\u2014namely, the deep-seated ideological conviction that they were implacable enemies\u2014and an opportunity to do so\u2014the chance to slaughter under the guise of military action. The expansion of the conflict added new rationales to remove Jews from German territory, such as the desire to resettle the _Volksdeutsche_ and the need to conserve scarce food supplies, and removed restraints on Nazi action, since a regime already or about to be at war with the whole world had little left to lose. Considerations like these led the Third Reich to decide on a policy of mass murder in the occupied Soviet Union, but there alone, at first.\n\nWhat seemed to be missing before Nazi Germany took the final, fateful step toward total annihilation of the Jewish population in Europe was the means to accomplish it. But during the late summer and early fall of 1941, in the months of September and October, Hitler and Himmler came to recognize that they already had these. The war had spawned possibilities for carrying out mass killing, and all that remained lacking were installations at which to apply these possibilities. The Nazi leaders knew that they could not employ the methods being used in the lands conquered from the USSR in Central or Western Europe. Simply shooting Jews and burying them in pits was likely to arouse revulsion and opposition there and thus to increase resistance to German rule, which would raise the military costs of maintaining it. Besides, Himmler quickly came to fear the effect on his men of having to shoot women and children hour after hour, day after day. Indeed, at least one of the _Einsatzgruppen_ commanders, Erich von dem Bach-Zelewski, later suffered a nervous breakdown, though only briefly. What the Nazi regime needed was a way of killing people that was inconspicuous or, as the SS planners put it, \"noiseless\" ( _ger\u00e4uschlos_ ), and that was more, again as they put it, \"humane\" . . . to the perpetrators.\n\nThis is the context in which to interpret the letter of July 31, 1941, that G\u00f6ring sent to Heydrich, authorizing him to find \"an overall solution to the Jewish question in the German sphere.\" Heydrich already had authority over \"emigration and evacuation,\" as the letter noted. He had no need for new authority unless he was being given a new assignment, and this document extended his competence to the entire \"German sphere\" and asked him to identify a \"total solution,\" implicitly in addition to the partial solution already being enacted in Russia. In other words, this letter is the surest sign that the Nazi state already was looking for a comprehensive method to apply continent-wide, and Heydrich's task was to find it.\n\nIn fact, the regime had possessed such a method since early in 1940. Beginning in 1938 with a single case, Hitler had authorized subordinates in his personal Chancellery to grant parental petitions to kill German children born mentally handicapped, and his regime followed this up in August 1939 with a decree requiring hospitals to report all births of deformed, paralyzed, or mentally deficient children to Berlin. During the same summer, he directed his staff to consult with medical doctors and professors from some of Germany's leading universities to devise a procedure for extending euthanasia to adults in the event of war, when Germany would need to free up hospital beds for military casualties. None of the experts balked at participating, but the matter was legally murky, since Hitler declined to issue a law justifying the killing, lest Germany's enemies use that as propaganda against the Reich. So the Chancellery officials felt the need for a way to assure the people involved of immunity from criminal liability and therefore asked Hitler for some form of written authorization. The result was a letter, signed by the Nazi F\u00fchrer on his personal, not his official, stationery and backdated to the opening day of World War II, directing the leader of his personal medical staff, Karl Brandt, and the head of the personal Chancellery, Philipp Bouhler, to expand the practice of granting a \"mercy death\" to irreversibly disabled people in state institutions. This written instruction, unlike any document ever discovered about the Holocaust, connects Hitler directly and in writing to a murder operation, the so-called Euthanasia Action. Known bureaucratically as T4, an abbreviation of the street address of its main office after April 1940, at Tiergartenstrasse Nr. 4, in the center of Berlin, this heavily camouflaged program proceeded under the day-to-day direction of Viktor Brack, one of Bouhler's aides. Though the Euthanasia Action continued for the duration of the Third Reich, the operation had two distinct phases, the first of which, from October 1939 to August 1941, was a direct forerunner of the Holocaust, the second, from 1942 to 1945, an extension of it.\n\nThe Nazi regime had prepared the German public for such an action by a propaganda campaign in the 1930s that stressed the drain that handicapped people, described as \"useless eaters\" and \"life unworthy of life,\" represented for the national economy and food supply. But Hitler favored the program, the evidence indicates, not so much for practical as for ideological reasons. He wanted to cancel out the \"negative selection\" that wartime casualties would mean for the Aryan race by accompanying the inevitable attrition of young, vital, and fit Germans with the compulsory reduction of the number of those who were genetically deficient. With the onset of war, he expected potential religious objections to decline or fall away.\n\nThe children who became the first victims of T4 perished from overdoses of medicines that induced illnesses or physical conditions on which deaths could be blamed. As the program expanded, starvation and direct injection into the heart of poisonous substances, usually phenol, became supplementary murder methods. Usually doctors did the killing in this initial phase. The general procedure was to move the institutional inhabitants whom physicians selected for death to one of six designated sanatoria spread around the country\u2014no more than four of which were operational at the same time\u2014and then to carry out the executions there. By January 1940, the MDs in charge of the program had decided that establishing small gas chambers at the institutions would be more efficient\u2014that is, would allow them to kill more people in less time and with fewer personnel than the injection method. They soon rigged up rooms that resembled shower facilities with piping that carried carbon monoxide (CO) instead of water into the chambers. An institute within the Reich Security Main Office bought the carbon monoxide in large metal flasks from the BASF division of the IG Farben conglomerate and supplied them to the sanatoria, where they merely had to be hooked up to the pipes. The gassing method was not Hitler's idea but adopted upon the recommendation of an advisory group of pharmacologists. In fact Brandt, the F\u00fchrer's physician, initially opposed using gas and argued for death by \"medical means.\" This information undermines the causal connection drawn, in the filmed German dramatization of the Wannsee Conference (1984) and the televised and star-studded Anglo-American version (2001), between the use of gas chambers and Hitler's remark, in _Mein Kampf_ , that more Jews should have been exposed as soldiers in World War I to poison gas.\n\nIn the annexed parts of Poland, an SS man named Herbert Lange soon modified the killing process in a consequential way. His assignment was to empty formerly Polish mental institutions and sanatoria, and he had no interest in going to the trouble of shipping the condemned residents to the six killing facilities in Germany. So he retrofitted large moving vans, disguised as Kaiser's Coffee delivery trucks, with space near the driver for the flasks of CO, which were connected by tubes to the rear compartments of the vehicles. He used these during 1940 to pick up the targeted people and kill them as the trucks drove away toward mass gravesites in concealed forest areas or toward local crematoria that burned the bodies.\n\nIn April 1941, a new operation with the code number 14f13 extended the T4 program to inmates of Germany's concentration camps judged unable to work. This extension caused the construction of relatively small gas chambers at Dachau, Sachsenhausen, Mauthausen, and several other sites. But most of these installations gassed people only infrequently until the frenetic final months of the war, when overcrowding, epidemics, and food shortages led to increased utilization. Until early 1945, for example, the Dachau camp appears to have used its gas chamber primarily to fumigate clothing. Most of the 20,000 camp inmates who perished in Action 14f13 were transported to die at Sonnenstein, Bernburg, and Hartheim, three of the sanatoria where T4 executions were carried out.\n\nThe continued operation of T4, combined with the new 14f13 program, created a supply and a secrecy problem for the Nazi state. BASF could barely keep up with the demand for bottled CO as of 1941, and applying it on a large scale in occupied Poland or further east raised transportation issues. Persuading BASF to increase production would require giving it some assurance that demand was likely to continue, and that might lead to awkward questions about what the product was being used for. These concerns prompted the SS and the T4 operation to explore jointly whether carbon monoxide produced from stationary internal combustion engines could kill patients as efficiently as, and perhaps more cheaply than, bottled carbon monoxide. Tests on mental patients in the conquered Belarusian cities of Minsk and Mogilev in September gave an affirmative answer.\n\nThe transfer of the euthanasia killing system to the murder of the Jews already was being prepared at about the same time as these tests. By early 1941, rumors about the euthanasia program had spread across Germany, and many of the relatives of the victims had grown suspicious of the standardized postcards that notified survivors of the deaths. The usual cause was given as pneumonia or appendicitis, even when the victims no longer had appendixes, and the notifications invariably included the statement that the body had been cremated to avoid the danger of an epidemic. A number of Protestant leaders began to speak up, and then so did Clemens Graf von Galen, the Catholic Bishop of M\u00fcnster. Because mercy killing violates Catholic teaching that only God may give and take life, Galen issued pastoral letters and delivered sermons denouncing the practice, something he never did with regard to the deportation of the Jews, whom he regarded in the typically Catholic fashion of the time as dangerous agents of modernity and Bolshevism. Fearful that Galen's protest would arouse public opinion and harm the war effort, Hitler formally put an end to the first phase of the T4 Action on August 24, 1941. By then it had taken the lives of between 71,000 and 80,000 people. Murders of disabled and handicapped people resumed a few months later but now in more widely dispersed locations and on a slower, better concealed basis until the end of the war. The killers usually reverted to their original methods of injection or overdosing, but gassings did not entirely cease; indeed, camp inmates, some Gypsies, some half-Jewish children, select groups of forced laborers, and even some Germans driven mad in bombing raids subsequently died that way in several sanatoria. In this second and longer phase of the T4 operation, nearly as many people perished as during the first.\n\nLess than three weeks after Hitler acted, Himmler approved the transfer to the command of SS Police Leader Odilo Globocnik, in Lublin, of many of the operational T4 personnel, eventually some 121 men who had been responsible for bringing handicapped victims to their places of execution and for disposing of the bodies. Most of these men did not begin applying their murderous technical expertise in the conquered areas of Eastern Europe until early 1942, following brief service in hospitals on the Eastern Front, but already by October 25, 1941, officials in the Ministry for the Occupied Eastern Territories were discussing setting up \"gassing devices\" in Riga and Minsk and murdering in them deported German Jews found incapable of work. Around the end of the same month, construction work began on the Belzec death camp, in a village southeast of Lublin that already was the center of a complex of forced labor camps populated mostly by Jews. The new camp's first commander was Christian Wirth, a veteran of not only the T4 Action but of its first gassing, at Brandenburg in 1940, who came under Globocnik's authority on October 14. Also in October 1941, Herbert Lange, the inventor of the gas vans, identified the derelict manor house that soon became the center of the Chelmno death camp, thirty miles northwest of Lodz, in the so-called Warthegau, part of the land annexed from Poland.\n\nAt just about the same time, the mechanics of the Security Police motor pool in Berlin, working under the direction of Walter Rauff, solved the carbon monoxide supply problem. They demonstrated the ease with which a T joint could connect the exhaust pipes under a van to its rear compartment and thus replace bottled CO with that produced by the vehicle's motor. In early November, the SS tested the process on forty Soviet prisoners of war at Sachsenhausen; all of them died within half an hour. The motor pool then ordered from a local supplier thirty converted trucks with rear compartments measuring about sixteen feet long by six and a half feet wide, plus a few smaller versions. Some of these were intended for and briefly used by the _Einsatzgruppen_ in Belarus and Ukraine, but that practice proved short-lived because the generally poor roads there led to frequent breakdowns and because the German personnel actually preferred shooting people to the gut-wrenching process of unloading the gassed bodies from the vans.\n\nWhile Himmler, Heydrich, and their henchmen were discovering that T4 had provided them with one means of disposing of Europe's Jews, a group of SS men recognized that Germany's chemical industry had supplied them with another. The setting was Auschwitz, where a concentration camp for Polish political prisoners had existed since May 1940 on the site of a former Polish military base. In August 1941, anticipating the arrival of increasing numbers of Soviet prisoners of war, Commandant Rudolf H\u00f6ss apparently told one of his subordinates, Karl Fritzsch, to explore means of killing sick or weak prisoners in bulk. Fritzsch hit on the idea of applying Zyklon, a powerful vaporizing pesticide that the camp normally used to fumigate barracks, as a potential poison. He knew that it was lethal to humans\u2014in fact, seventy milligrams, or 1\/3000th of an ounce, which is to say a whiff, will kill a 150-pound person within two minutes\u2014and he already had a substantial quantity of the stuff on hand. Although this product is generally referred to as Zyklon B, the label usually said only Zyklon. The \"B\" sometimes appeared on sales invoices, but the designation was mostly internal to the manufacturers, since it merely distinguished the product's chemical formula from an earlier, very short-lived one.\n\nIn the early days of September 1941, Fritzsch tested the efficacy of Zyklon on two groups of Soviet prisoners of war locked into the basement of one of the original stone barracks at Auschwitz. He learned that Zyklon killed reliably, but for maximum effectiveness, the product needed a more open space than the subdivided barracks basement. Quickly thereafter, he and H\u00f6ss discerned two more things: The substance was evidently plentiful and rather cheap in relation to the damage it could do. At 5 reichsmark per kilogram (that is, per 2.2 lbs.) and an overdosage of 5\u20137 kilograms for each group of 1,500 victims, the usual practice according to the postwar testimony of Commandant H\u00f6ss, the average cost of murder per head ultimately came out to about two German pfennig (pennies) a person, which is to say less than one U.S. cent in 1942.\n\nThus by late October, the SS leaders knew that they had not one but two effective ways of killing large numbers of people, and this knowledge set in motion a series of pivotal events that mark the onset of what the Nazi regime called the Final Solution of the Jewish Question ( _die Endl\u00f6sung der Judenfrage_ ). First, on October 23, 1941, Himmler issued an instruction to the Gestapo and the SS that forbade further emigration of Jews from the European continent. This document clearly signaled the end of the policy of driving Jews away, either now or later, and suggested that the Nazis had found a new approach to the Jewish problem.\n\nSecond, in November, the regime expanded upon the initial deportation of about 20,000 German Jews to Lodz in mid-October by beginning to ship even more German Jews to the Baltic states and Belarus, where some of them were shot upon arrival, and subjected virtually all the Jews already in concentration camps to the 14f13 murder program. Then, on the twenty-ninth, Reinhard Heydrich issued invitations for various ministerial representatives to gather at a villa along the Wannsee, a lake on the western edge of Berlin, to discuss the final solution. He enclosed with the invitations copies of G\u00f6ring's letter to him of July 31. Originally scheduled for December 9, the meeting did not take place then because of two surprising events that threw the German capital into confusion, the Soviet counterattack around Moscow on December 5 and the Japanese bombing of Pearl Harbor two days later. Beyond the addition of two officials of the General Government to the initial invitation list, no evidence suggests that the agenda for that meeting changed between its canceled and actual occurrence, on January 20, 1942, when Heydrich laid out a plan \"to comb Europe from west to east\" of Jews, deport them to Poland, put them to work at hard labor, and subject those who survived this ordeal to \"special handling\" ( _Sonderbehandlung_ ). The SS leadership probably already envisioned all or most of this when the first round of invitations went out. Eleven days earlier, on November 18, Alfred Rosenberg, the Minister for the Occupied Eastern Territories, had briefed trusted German reporters on deep background and spoken of the \"biological eradication of the entire Jewry of Europe.\"\n\nThird, on December 12, 1941, the day after Hitler declared war on the United States in solidarity with his Japanese ally, he met with the Nazi Party Gauleiters at his private apartment in Munich and informed them that the Jews would have to pay \"with their lives\" for the war they had inflicted on Germany\u2014indeed, that they already were doing so. As historian Peter Fritzsche has remarked, \"this is as close to a Hitler order as historians will get,\" meaning the closest counterpart to the euthanasia letter he signed that we are likely to find to connect Hitler personally with the command to kill the Jews.\n\nTo sum up what we know about the decision-making process that produced the Holocaust: By August 1941, the Germans were engaged in slaughtering the Jews of the Soviet Union, including women and children. By October, the Nazi leaders knew they had the means to kill people en masse in gas chambers, began constructing sites to do so, and tried to close the escape hatches from Europe. By November, the key figures were ready to bring the German press on board by leaking what was coming and to inform\u2014and implicate\u2014the bureaucracy while seeking its cooperation at the Wannsee Conference. And, in December, Hitler let the Gauleiters in on the change in policy. The Final Solution, the annihilation of the Jews of Europe, was in motion.\n\nImplementation became the next order of business. As it happened, Europe was not combed from west to east, as Heydrich predicted at Wannsee, but almost in the reverse direction. Nazi ideology designated Eastern Europe as the destined living space for an ever growing German _Volk_. Propelled by this expansionist vision, murder gathered mass and momentum faster there than elsewhere because a majority of the continent's Jews still lived in or around the old Pale of Settlement, and it was the conquered region where Germany had least reason to worry that killing Jews would arouse objections from other inhabitants. Accordingly, the six death camps set up in 1941\u201342 were all within prewar Poland, and each initially concentrated on killing the Jews who lived in its vicinity. Nor was this the only respect in which developments did not proceed according to plan. As the pioneering scholar of the Holocaust Raul Hilberg emphasized, the Holocaust bore the characteristic features of many Nazi initiatives: little foresight or preparation, rocky coordination of participating agencies, and even haphazard budgeting. Yet the killing of millions of people turned out to require no better.\n\nThe six death camps divided into two groups. Group 1, consisting of four camps that employed carbon monoxide gas, like the T4 Action in which all of their initial commanders had participated, predominantly killed Jews from designated portions of Poland's prewar territory and operated only as long as any of these people were left. The first such camp was Chelmno, and it was the only one to use gas vans exclusively, certainly because Lange, its first commander, was so experienced with them. Chelmno began operations with two converted Dodge trucks on December 8, 1941, and then expanded its fleet and its death toll. Each truck carried fifty to seventy people and made five to ten return trips per day from the camp to body disposal areas in a nearby forest. By December 31, 1942, the operation had killed 145,301 people, according to the SS's tabulation, nearly all of them drawn from the part of western Poland annexed to Germany. Closed in March 1943, Chelmno reopened in 1944 to liquidate about 7,200 inhabitants of the Lodz ghetto, and that brought total documented mortality on the site to 152,000 people. This is surely the minimum figure; itemizations of the transports that went to Chelmno suggest at least 20,000 more victims, and the most recent Polish research indicates that the number may have reached 225,000.\n\nThe other death camps that used carbon monoxide became operational in 1942 and used stationary gas chambers, initially rather jerry-rigged edifices made airtight by a layer of sand between the inner and outer wooden walls and an external coating of tar paper. Soon, brick or concrete buildings replaced these, but the new structures were still very simple, easy to put up and easy, later, to take down. Like Chelmno, the three sites were chosen for their remoteness, but unlike it, all were located on branch rail lines on the eastern edge of the General Government. In order of foundation and coincidentally from south to north, these were Belzec, which began gassings in March 1942; Sobibor, which came on line in May of that year; and Treblinka, which followed in July. All were conceived as the instruments of Operation Reinhard, the name the Nazis gave at mid-year, following the assassination of Reinhard Heydrich in Prague, to the annihilation of the Polish Jews. All three camps used captured Soviet tank engines to generate the carbon monoxide, and all usually operated with stunning efficiency under the initial direction of veterans of the T4 operation.\n\nBelzec killed at least 434,000 people, but perhaps as many as 600,000, in the only ten months that it was open, an average of up to 2,000 people per day, more than two-thirds of them from southern and southeastern Poland and the rest Jews from other parts of Europe who had been deposited in ghettos in and around Lublin. Sobibor consumed between 167,000 and 200,000 people during its seventeen-month life span, most of them from Poland, but some from prewar Czechoslovakia, many from France and the Netherlands, and a few from Greater Germany, Belarus, and Lithuania. Treblinka, the last to close, wiped out up to 925,000 people in the eighteen months before Operation Reinhard ended in November 1943, which made it almost as lethal as Belzec on a daily basis. However, at the time of its peak murderousness, from July 22 to August 27, 1942, Treblinka killed 280,000 people, an average of 56,000 per week, or 8,000 per day. During one of those five weeks, the daily average reached 10,000. Nearly all of Treblinka's victims came from central and northern Poland, but about 32,000 Czech, Greek, and Macedonian Jews also died there. All three Reinhard camps were in operation simultaneously for only four out of the six months from July to December 1942, yet in that half-year the three sites killed more than one million Jews, which is more Jews and nearly as many people all told as Auschwitz-Birkenau wiped out in four years. Altogether, these three places devoured between 1.5 and 1.8 million human beings. Including Chelmno, the four CO camps killed up to 2 million people. Fewer than 400 Jews ever emerged alive from all four sites, and of these only somewhere between 90 and 150 outlived World War II.\n\nThe second group of death camps consisted of only two installations: Auschwitz-Birkenau and Majdanek, which were different from Group 1 in three respects: (1) they primarily used Zyklon, not CO, to kill people (though Majdanek also sometimes employed bottled CO and had a gas van); (2) they were dual-purpose camps\u2014both death and slave labor camps\u2014and thus had larger ongoing inmate populations; and (3) they were not closed until the Soviet armies approached, and thus they were the principal destinations, Auschwitz especially, for Jews from outside of Poland, and the only death camps still open in 1944. Their importance as labor reservoirs began as part of the plans for an SS agricultural research center at Auschwitz and a complex of SS-owned factories at Majdanek, but then greatly expanded because they stood at or near the ends of the Polish part of Durchgangstrasse IV, the long highway that the Nazis were building from Silesia to the Caucasus as the lifeline of their power in conquered Ukraine. The need for labor for this project inspired the addition of Birkenau to Auschwitz in October 1941 and of Majdanek to a nearby preexisting labor camp in Lublin. Both were built for Soviet prisoners of war as the prospective workforce but later populated primarily by Jews\u2014in Auschwitz's case throughout its existence, in Majdanek's only until November 1943, when nearly all of the remaining Jews were shot to death, not gassed.\n\nOver time, other labor needs arose to sustain the importance of each camp. Auschwitz became the geographical center of frantic industrial development because it was near coal and water supplies and out of reach of Allied bombers based in Britain. The biggest plant located in the region was the IG Farben installation three miles east at Monowitz, where at least 27,000 prisoners died while constructing a factory for synthetic fuel and rubber and operating the firm's coal mines in the area. But tens of thousands of other inmates labored at more than forty branch camps in the region, including the giant synthetic fuel factories at Blechhammer and Heydebreck and numerous mines. Majdanek became the hub of the SS's plans to process and recycle the goods collected from the people killed at Reinhard camps, especially the hundreds of thousands of leather shoes that one can still see filling warehouse barracks on the site.\n\nBoth Auschwitz and Majdanek had another distinguishing feature: gas chambers built to last. The three at Majdanek were relatively small, solid stone structures, and they are still there. Auschwitz initially turned a small munitions magazine at the edge of the main, original camp into a crematorium that also could be used for gassings, then converted two peasant cottages on the plain near Birkenau into gas chambers, and finally constructed and opened in 1943 four large brick buildings in the Birkenau or Auschwitz II camp that contained both gas chambers and crematoria to dispose of the bodies at the rate of 4,000 to 8,000 per day. The first, small chamber at the main camp remains today; the cottages are gone, and only ruins of the brick buildings survive. Since the chambers could kill faster than the crematoria could incinerate, bodies sometimes also burned on open-air pyres. If the Nazis had been able to ship all the remaining Jews in Europe to Auschwitz in 1945\u201346\u2014that is, if the war had gone on and the impediments to rounding up the remaining Jews been pushed aside\u2014Auschwitz could have killed them all by early in the latter year. As it was, some 1.3 million people arrived at the camp between its opening in May 1940 and its evacuation in January 1945, of whom approximately 1.1 million died\u2014there or in one of the subcamps. Perhaps half of the survivors died at other installations before World War II ended, and only 100,000 emerged alive from the conflict. Majdanek was far less lethal, especially for Jews, and recent research findings have driven down the probable number of Jews killed there from some 145,000 to about 59,000, perhaps one-third of whom were gassed.\n\nThe death camps were distinct from three other principal kinds of Nazi camps:\n\n(1) The main SS concentration camps, such as Dachau and Buchenwald in Germany and places like Natzweiler in Alsace-Lorraine and Stutthof along the Baltic coast and their more than 1,100 satellite installations. These were murderous places, especially in the last year of the war, but they were not \"factories of death,\" and their populations did not consist primarily of Jews. A partial exception in the former regard was Mauthausen, founded in 1938 near the city of Linz, Austria, and the subcamps it spawned at nearby military production installations. Reserved for \"incorrigible . . . and barely educable\" political opponents and criminals, this harshest of the concentration camps killed 52 percent of its almost 16,000 inmates in 1941 and almost 10 percent of its constantly replenished prisoner population every month between mid-1941 and April 1943. By the time American troops liberated the camp, more than half of the almost 200,000 people ever held there had perished, some of them in a gas chamber on the premises that used Zyklon, some in gas vans, and others at the nearby Hartheim sanatorium. About 25 percent of the victims were Jews.\n\n(2) So-called transit camps, where particular groups of people were collected, generally in order to be exchanged later with the Allies. The best known of these was Bergen-Belsen, in northern Germany. These were relatively benign places until the German supply system collapsed toward the end of the war, at which point they became murderous centers of infection and starvation.\n\n(3) Labor camps, of which tens of thousands came into existence by the end of the war. Here, the inmates could be and were treated viciously, but the goal of production exerted some, though limited, protection against large-scale killing. But one must draw a distinction within that distinction. Labor camps in the East were far worse than those even in the General Government, let alone Germany. The ones established for Jews along Durchgangstrasse IV were little more than delayed murder stations, and the same was true of Janowska, a camp in Lviv that began as a labor site in late 1942 but turned into such a voracious venue for the mass shooting of Jews that it may have taken more of their lives, all told, than Majdanek, without possessing a gas chamber.\n\nOne of the most chilling aspects of the history of the Holocaust is that so much carnage could occur without any serious ill effect on the German war effort, in fact with little diversion of manpower, mat\u00e9riel, and money. Aside from the loss to Germany and the gain to Britain and the United States of the Jewish scientists and other loyal citizens driven to flee during the 1930s, the Reich hardly had to pay a price for all the pain and suffering it dispensed. Yes, it came to miss the labor of most of the last remaining Polish Jews, some 300,000 of whom the SS massacred in 1943, but otherwise the balance sheet of murder was strikingly favorable to Germany, at least in the short run, which was the only time horizon that mattered to the Nazi regime.\n\nIn the first place, the camps took in enormous plunder as well as payments for the laborers that they leased to industries and government offices. Although enslaved camp inmates were not paid, they were paid for. In part as a result, Auschwitz, the biggest labor vendor, made a profit of 100 percent in this fashion from 1941 to 1945. It took in sixty million reichsmark in fees for workers, whereas it spent only thirty million, all told, to feed and house them. Globocnik's office in Lublin calculated that its net loot from Operation Reinhard, after deducting all the personnel and other costs associated with the deportations and murder, came to almost 179 million reichsmark, conservatively estimated, including more than 80 million reichsmark in cash, 52 million reichsmark in jewelry and precious metals, and 46 million in recyclable clothing. Because the T4 people at those camps sent additional amounts directly to the F\u00fchrer's Chancellery and to the Reichsbank, the overall income clearly was much more. We do not know how much plunder Chelmno collected because that booty was shipped to the Lodz ghetto administration and mixed in with its other extortions from Jews. But one document from May 1942, a year before Chelmno suspended operation, speaks of needing 900 trucks to transport clothing the camp delivered for reuse.\n\nIn the second place, the Germans did not pay for transporting the Jews to the camps; they made Jewish community administrations do this, just as they usually made them, at least in Western Europe, Germany itself, and the Polish ghettos, do the dirty work of identifying potential deportees or even drawing up lists of them whenever the Nazis dictated a certain number or category to be shipped out. Even the offices from which the killing orders came had been Jewish-owned property, now repurposed for murder. SS-Obersturmbannf\u00fchrer Adolf Eichmann coordinated deportations from the former premises of the Jewish Brethren Society, a charitable organization, in Berlin's Kurf\u00fcrstenstrasse 116; the euthanasia program's headquarters at Tiergartenstrasse 4 had belonged to relatives of Max Liebermann, a famous Jewish German painter. Meanwhile, the Reich raked in enormous sums in confiscated bank accounts, jewelry, art works, and other fungible forms of property from Jews in all the occupied countries. As a result, the Holocaust as a whole was not only a self-financing but also, like Auschwitz standing alone, a profit-making enterprise. Consider the example of the Netherlands. Here, Nazi Germany reaped more than one billion guilders from the sale of property stolen from Jews, most of which went directly into the occupation administration's coffers or into the accounts of front organizations that bought German and Dutch government bonds and thus helped support the German war effort. A tiny share of the income\u201425 million guilders, or less than 2.5 percent of the total\u2014was expended to expand and maintain the two Dutch transit camps at Vught and Westerbork and to pay for the roundups and deportations. About 75 percent of the Dutch Jews were killed, some 105,000 people, at a cost that came to a small fraction of what the German state seized from them.\n\nIn the third place, the costs of murder that Jews did not pay were quite low. Aside from Auschwitz from 1943 on, the death camps were remarkably low-tech, non\u2013capital intensive entities. Chelmno consisted of a rundown manor house surrounded by a wooden fence. Jews came in the main gate on trucks, descended from these on one side of the house, lost what was left of their possessions as they passed through its basement, and walked into vans at the back entrance. After Jewish prisoners closed and barred the doors, the vehicles either idled or drove until the people inside died of the exhaust fumes, and then delivered the bodies to a forest clearing, where other Jewish prisoners burned the corpses in pits. Even at Auschwitz, the first gas chamber was a preexisting building, and much of the construction material for the barracks and fencing was obtained from IG Farben in a barter agreement of steel, bricks, and barbed wire for workers and gravel. So improvised were the first two gas chambers at Birkenau, the converted peasant cottages, that they lacked mechanical ventilation and thus could not function in rapid succession. Treblinka and the other Reinhard camps were Potemkin villages of building facades, plus a few workshops, around square reception areas where the arriving victims disembarked and undressed. The rear side of the square led to a \"tube\" ( _Schlauch_ ), a narrow passageway framed by wooden sawhorses, covered in barbed wire and pine boughs, and into the gas chambers. None of this cost much to erect.\n\nMoreover, the operating costs of these sites were low. Gasoline to generate carbon monoxide was inexpensive and not in short supply until after Operation Reinhard ended, and the motors used came from captured Soviet tanks. The nearly 32 metric tons of Zyklon sold to Auschwitz and Majdanek in 1942\u201344 cost just under 160,000 reichsmark, or about 64,000 U.S. dollars at the time. Only about one-fifth of this Zyklon actually was used for gassing\u2014the rest went to fumigating barracks and the like or simply spoiled on the shelves\u2014so the quantity needed for murder was even less expensive, probably costing only about 30,000 reichsmark, or U.S. $12,000 at the time.\n\nNor was it expensive to staff the places. Auschwitz was the only camp with a large, nearly all German guard force. Its average size during the life of the camp was about 2,500, and some 7,000 Germans served there from 1940 to 1945, which is about one-third as many men as the German armed forces shot for desertion during World War II. But Belzec and Sobibor needed only about twenty Germans at any one time, as did Treblinka initially, though its German staff rose in 1943 into the thirties. The rest of the guards, 90\u2013130 men at each site, were Eastern European _Hiwis_ (an abbreviation of _Hilfswillige_ , which means \"volunteer helpers\"), usually recruited from starved Soviet prisoner-of-war camps, and offered uniforms, room and board, low pay, and the chance to pillage the Jews in return for serving as support troops for the Germans. A special camp at Trawniki in Poland trained 4,750 of these people by the time it closed in September 1943. The Germans at the death camps earned substantially more, in fact something like ten times their nominal monthly pay of fifty-eight reichsmark, thanks to a special daily allowance of eighteen marks, a loyalty bonus, and a \"Jew murder supplement\" paid from the budget of the T4 program. Even so, that program also operated in the black thanks to the proceeds on the dental gold extracted from victims' mouths and the practice of routinely waiting for about ten days after an execution before entering the death notice in the records, meanwhile assessing per diem charges to the person, agency, or insurance company responsible for maintaining the now deceased disabled person.\n\nFinally, legend has it that the deportation trains to the camps must have impeded the German war effort. Nothing could be further from the truth. Very few deportation trains were in transit at any one time, and they had the lowest priority on German railroads, which means they were never allowed to obstruct or delay troop movements or supply trains. That is one reason why the trips from Western Europe to the death camps, and even the ones in the early stages of the deportations from Warsaw to Treblinka in 1942 that traveled only sixty miles, often took as long as three or four days and arrived carrying numerous suffocated, starved, parched, and in winter frozen corpses. Boxcars usually were used in the East, and either closed cargo wagons or third-class passenger cars from Western Europe, but in both cases, the transports nearly always consisted of dilapidated equipment. Even the locomotives were relics. Loading each transport of 1,000 people or more generally required only ninety Germans, and the guard personnel en route usually consisted of only fifteen, since sealed boxcars required little supervision. Indeed, the Germans preferred them in part for that reason.\n\nAll told, the Germans used about 2,000 trains to move three million people to camps over thirty-three months in 1942\u201344, which works out to sixty trains per month or two departing per day, on average. In contrast, the German Reichsbahn carried 6.6 _billion_ passengers in 1942\u201343 and ran 30,000 trains _per day_ in 1941 and 1942 and about 23,000 per day in 1944. In that final year, the Nazi regime needed only 147 trains over eight weeks, an average of fewer than three per day and never more than six, to deport almost 440,000 Hungarian Jews. Allocating even that many trains in a short time for a murder operation was unprecedented, and it happened only because the deportations had a subsidiary purpose directly tied to the war effort. Auschwitz was supposed to extract 100,000 able-bodied workers from the deportees, 10\u201315 percent of the initially anticipated total, and ship them immediately on to the Reich, where they were to labor on the massive effort to put Germany's war production plants underground. Even so, at the height of the deportations from Hungary, those trains constituted no more than 1\u20132 percent of the daily railroad traffic in that country. They employed an infinitesimal one-fifteenth of 1 percent of the functioning locomotives and one-tenth of 1 percent of the operating rolling stock under the jurisdiction of the German Armaments Ministry at the time. Clearly, the shares of German railroad equipment and activity devoted to the Holocaust were tiny, both in total and at any particular time.\n\nAs sometimes happens in historical writing, the most conclusive demonstration that the deportations had no significant impact on the German war effort is a book that purports to prove the opposite, Yaron Pasher's _Holocaust vs. Wehrmacht_ (2014). Pasher examines four German military defeats, each of which occurred at approximately the same time as a wave of deportations: the failure to take Moscow in 1941 as the first transports of Jews were leaving Germany; the failure to relieve Stalingrad in 1942\u201343 during Operation Reinhard; the debacle of the Battle of Kursk in the summer of 1943 shortly after the suppression of the Warsaw Ghetto Uprising; and the successful Allied invasion of and breakout from Normandy in June to August 1944, which partially overlapped with the massive deportations from Hungary. At each juncture, Pasher asserts that the primary reason more troops and supplies did not arrive at the German front lines was a shortage of trains to carry what was needed, that shortage having been caused by the use of rolling stock to deport Jews. In the process, he makes statistical calculations that count, for example, each journey by the same one or two slow and rickety trains with sixty boxcars going back and forth between Bialystok and Treblinka and between Theresienstadt and Auschwitz every two to three days in early 1943 as potentially the trip of a fully loaded and speedy supply train to the Eastern Front. His estimates, in each instance, of what the trains that carried Jews instead could have brought to the German armies still fall drastically short of his own understated tallies of what the Reich's soldiers lacked in men and mat\u00e9riel compared to their adversaries. In all four cases, Germany's defeat was massively overdetermined. For that reason, repeatedly asserting, as Pasher does, that \"every train counted\" is not nearly the same thing as showing that every train mattered.\n\nSo how could the Nazis achieve such an extensive massacre in so short a time? The first piece of the answer is: because they perfected a low-cost, low-overhead, low-tech, and self-financing process of killing with great speed. We turn now to a second component of the answer: because the Nazi movement and state generated and unleashed remarkably dedicated killers.\n\nPERPETRATORS: THE \"GENERATION WITHOUT LIMITS\"\n\nThe Holocaust involved tens of thousands of people who participated in it directly\u2014the SS guards, the _Einsatzgruppen_ , the Order Police, the regular military units that often helped with rounding up Jews and killing them, and the thousands of bureaucrats and officials who planned the murders and made the death-dispensing system function\u2014as well as hundreds of thousands of German civilians who facilitated the persecution at some remove from the process. How can we explain their behavior? How could they do these things?\n\nBroadly speaking, two schools of thought dominate attempts to answer these questions, the volitional and the situational schools, and each gets deployed to account for behavior at two different levels, subordinate and senior. The volitional school holds that people persecuted and killed because they chose to; the situational school argues that they acted in response to their immediate context, not their convictions. Recently, a number of German-educated authors have laid the basis for a powerful and persuasive synthesis of the two points of view.\n\nWith regard to the subordinates who actually did the killing in the shooting units and gassing installations, the classic formulations of the volitional and situational points of view are, respectively, Daniel Goldhagen's _Hitler's Willing Executioners_ (1996), a bestseller that the public loved and most historians panned, and Christopher Browning's _Ordinary Men_ (1992), which won academic acclaim as well as a popular audience. Goldhagen insists that Germans killed Jews because they wanted to; they wanted to because they universally hated Jews; and they hated Jews because Germans always had\u2014their nation's culture had been thoroughly and pervasively antisemitic for hundreds of years. On the basis of postwar trial testimony by the former shooters of Reserve Police Battalion 101, Browning maintains that antisemitic convictions had little to do with the readiness of Germans to commit murder; rather, they acted out of loyalty to one another. Their sense of group solidarity made them unwilling to let one another down by showing weakness. In making his case, Browning draws heavily on two sets of social psychology experiments. The first, conducted by Stanley Milgram in New Haven in 1961, led volunteers to believe that they were administering electric shocks upon orders from a supposed scientist. The second, Philip Zimbardo's at Stanford in 1971, simulated relations between prison guards and inmates. Each experiment highlighted human tendencies to defer to or to abuse authority.\n\nBoth Goldhagen's and Browning's analyses have drawbacks: Goldhagen's picture is static; it contains no change over time; the attitudes of Germans in 1642 are identical to the attitudes of Germans in 1942 and just as uniform, which is implausible. Moreover, Goldhagen makes no allowance for the stark fact that power magnifies the ideas of those who hold it. Thus he underplays the decisive importance, with regard to German behavior, of the period after Hitler came to power. Browning, on the other hand, relies a good deal on his protagonists' descriptions of their motives, which he acknowledges is a risky practice. Not for nothing do lawyers say that \"no one can be a witness in his or her own cause.\" In this instance, as is usually the case in court proceedings, some of the protagonists had a reason to lie. They gave their testimony in a potential West German murder investigation. Under German law, a murder conviction required proof that a person acted with a \"base motive,\" such as greed or hatred, or exhibited sadistic zeal. That made former shooters reluctant to admit to antisemitism or to ascribe it to a comrade, even though anyone indicted was likely, given German legal practice at the time, to be charged as an accessory to murder, which entailed less severe punishment.\n\nAlthough Browning has the better of the argument with regard to the men of Police Battalion 101, and his findings have chilling implications regarding the general susceptibility of men in certain situations to inflict hideous violence, two sorts of considerations, one theoretical and one empirical, suggest that more remains to be said about why most German killers acted as they did in 1941\u201345. In the first (theoretical) place, Goldhagen and Browning probably tried to be too precise in capturing motives that may have been diverse, mixed, and variable over time. Moreover, given the assignments doled out to the shooting units and the ideological environment in which they lived, many shooters may have embraced antisemitism at the time as a conveniently available form of legitimizing what they had been ordered to do. In other words, they did not kill because they hated their victims, but they decided to hate them because they thought they had to kill them. Psychologists call this sort of mental mechanism, in which beliefs conform to behavior rather than the other way around, a response to \"cognitive dissonance,\" and it may have been just as important as animosity or sadism in explaining why so many Germans showed or expressed pleasure in torturing and killing Jews. Hatred and even glee became ways to ease the task at hand, and we know that it was not easy, at least not initially. Himmler called what he thought the SS had to do \"a repulsive duty\" and \"a horrible task.\" The _Einsatzgruppen_ and the foreign auxiliary units often had to get drunk to carry out the slaughter. German women serving as nurses and soldiers' aides on the Eastern Front reported repeatedly that men who returned from massacres \"all had an intense need to talk\" about what they had done.\n\nIn the second (empirical) place, Edward Westermann has demonstrated conclusively that Police Battalion 101 was not typical of the police units sent east to kill Jews. Some 80 percent of these personnel were not reservists at all, as the men in Police Battalion 101 were, and most units did not consist, as it did, of middle-aged men who had matured before Hitler came to power. On the contrary, the battalions generally comprised young, heavily indoctrinated career policemen who saw themselves as \"political soldiers\" in service to Nazi racial ideology. They were not workaday civilians placed in unfamiliar and extreme circumstances or typical Germans of an earlier era but rather military creations of the Nazi regime, schooled in the need for racial purification. Although Browning's work shows that even \"ordinary men\" were prepared to kill in the German-occupied east, most of the killers there in the early 1940s were not ordinary men.\n\nLike Omer Bartov, who wrote _The Eastern Front, 1941\u201345_ , Westermann stresses the role of ideological indoctrination in shaping the behavior of the police shooters, but unlike Bartov, Westermann thinks these units did not need to experience the increasing barbarization of warfare over time in order to become hardened killers. They were ready to act as such from the first day of their arrival in the east. Waitman Beorn's _Marching into Darkness_ (2014) reaches a similar conclusion about the regular German army units that began massacring Jews in Belarus in the fall of 1941, well before encountering serious resistance or partisan activity. His close examination of several mass shootings reveals that relentless propaganda about the Jewish-Bolshevik menace disposed most of these men to slaughter virtually from the beginning of the invasion of the USSR.\n\nAmong the most insightful works on this topic are studies by the German scholars Harald Welzer ( _T\u00e4ter_ , 2005), Felix R\u00f6mer ( _Kameraden_ , 2012), and Thomas K\u00fchne ( _Belonging and Genocide_ , 2010). (Thomas K\u00fchne now teaches in the United States.) Using somewhat different sources and analytical approaches, these scholars agree in highlighting the Hitler regime's success in developing, among Germans, \"a Nazi self\" with an inverted value system that offered a host of justifications for cruelty. The Third Reich redefined morality and turned humiliation, persecution, and murder into virtues. Overcoming scruples against inflicting pain became a sign of moral progress, not of indecency. If a person had difficulty doing this, so much the better, since that engendered more self-pity than pity for the victims, and thus made Germans more willing to lash out at the people whose existence was causing such discomfort. K\u00fchne provides some memorable phrases to describe what happened: He says that the Nazis created a \"dichotomist ethics\" of Us vs. Them and a \"moral grammar of comradeship\" that glorified acts of solidarity and shamed ones born of individualism. He reminds us of Browning's observation that even the members of Police Battalion 101 who said that they could not shoot did not raise moral objections; instead, they just said that they were \"too weak\" to do what was asked. The same pattern held for the exceptional regular army personnel who asked to be excused from killings, according to Beorn. As he writes, \"By claiming weakness or sentimentality as their reason for non-participation, soldiers . . . avoided directly challenging the actions of their comrades. This allowed them to remain within the community of their peers.\" Even when saying no, individual Germans partially affirmed the collective purpose. And most Germans ordered to kill did not say no. R\u00f6mer's study, which is based on the bugged conversations of some 3,000 German prisoners of war who passed through Fort Hunt outside of Washington, DC, concludes that even when they professed to believe that \"extreme violence against defenseless civilians, women, and children . . . cross[ed] a line, they were always capable of such violence, the minute group pressure or the circumstances demanded it.\" That reflexive readiness owed a good deal to the pervasiveness among admired and veteran junior and noncommissioned officers of a \"particular military mentality\" that, in turn, reflected what Welzer calls a \"particular National Socialist morality.\"\n\nWelzer's, R\u00f6mer's, K\u00fchne's, and Beorn's findings demonstrate that the situational can become volitional; beliefs adjust to circumstances, and power magnifies the ideas of those who wield it. Ordinary Germans could and did _become_ willing executors of Nazi persecution and even in many cases willing executioners. In a book called _Experten der Vernichtung_ (Experts of Annihilation, 2013), Sara Berger has reached a similar conclusion regarding the T4 participants who went on to staff the Operation Reinhard camps. Having closely studied their records and postwar testimonies, she stresses that they did not become killers on their own initiative, but that they became willing and increasingly identified with the Nazi regime's justifications for murder. As a result, none of them exercised their option to transfer back to their previous postings.\n\nThe treatment of the two known railroad officials who declined to participate in transports confirms a point established in numerous postwar examinations of the military and SS records and reinforced by Browning's research: Opting out of the killing process went unpunished in Nazi Germany but was nonetheless rare. Richard Neuser, a conductor based in Bialystok, asked to be relieved of having to work on transports to camps, and he was reassigned without penalty. Alfons Glas worked in the main passenger train office at Gedob, the organization that ran the railways in the General Government, and he learned from subordinates in the field enough about what was happening to trains carrying Jews that he asked for a transfer, which he received without any disadvantage to his career. But these were highly exceptional cases. The behavior of German railroad personnel paralleled that of police and other uniformed organizations. The senses of group solidarity and\/or professional obligation and\/or ideological conviction outweighed any reservations or compunction people might have felt. Similarly, the German postwar trials of Operation Reinhard death camp personnel produced only two documented instances of SS guards who asked to transfer out of direct involvement with the killing process. Both succeeded without adverse consequences.\n\nOskar Gr\u00f6ning, an SS bookkeeper at Auschwitz who was interviewed by the BBC in 2005, explained his own behavior with reference to compartmentalization and indoctrination. He had volunteered for the SS in 1940 and then worked in a paymaster's office until transferred to Auschwitz in 1942, when he was twenty-one. There he tallied the money taken from the camp's victims. Although upset by instances of brutality that he saw, he generally endorsed the need to wipe out the Jews as Germany's mortal enemies who had defeated it in World War I and would try to do so again. He therefore regarded their murder as necessary. But he felt detached from the killing; his unit carried out the desk job side of life at Auschwitz, not the murders, and he considered the two activities more or less separate. So he stayed until September 1944, when the SS granted his request for transfer and sent him to a Waffen-SS unit that later fought in the Battle of the Bulge.\n\nAt the camps, key elements in explaining guard behavior are the small numbers of perpetrators involved, the sorts of people they were, and the ways they delegated the worst of the killing process and thus distanced themselves from it. The small number of perpetrators required made them easy to find. Remember, only twenty Germans and Austrians were at Belzec at any one time; fewer than five hundred ever were at all three camps of Operation Reinhard. Each camp crematorium required only five to twelve German supervisors. Even if we assume that they were all psychopaths, we have to concede that recruiting that few cannot have been difficult. The guard staffs were often very poorly educated; for example, at Auschwitz only 30 percent of the SS men who served in the garrison ever got beyond grade school. Except at Auschwitz, the guard personnel consisted largely of foreign auxiliaries\u2014the _Hiwis_ \u2014with a strong interest in satisfying their German masters. Each Operation Reinhard camp had 90 to 130 such men. At Auschwitz, _Volksdeutsche_ eager to prove that they were just as tough and German as their native-born comrades made up a large percentage of the garrison.\n\nMoreover, the Germans were adept at insulating themselves from the worst aspects of the killing processes. In the ghettos, they often made the Jewish police forces do the dirty work of rounding up people who did not appear for deportation when scheduled to do so. In the camps, they used other Jewish prisoners in _Sonderkommandos_ to empty the gas chambers, burn the bodies, and, in the case of Crematorium III at Birkenau, to hold open the heavy lid of the chute through which an SS man poured the Zyklon pellets. Finally, among the camp guards, as in the shooting squads, a fateful element was self-centeredness, a preoccupation with one's own challenges rather than the pain being inflicted. For the guards, the daily problem was to manage large numbers of prisoners, and brutality was always the easiest method available. The basest elements of people's temperaments were elicited by the nature of the camp system, where the rules encouraged such things as goading prisoners into trying to escape so a guard could shoot them and thus earn an extra day's leave.\n\nSo what do all these explanations add up to? Why was there no shortage of Germans ready to participate in the torture and killing of Jews? Above all, because the Nazi regime succeeded in creating a closed mental world, an ideological echo chamber in which leaders constantly harped on the threat the Jews supposedly constituted and the need for Germans to defend themselves against it. The war itself, the air raids on German cities, the snipers at sentries in the occupied east\u2014everything was the work of the Jews. At the same time, the regime degraded the Jews so thoroughly in ghettos, camps, and transports that they came to resemble the vile picture that the regime painted of them as dirty, disease-bearing, self-seeking, and uncivilized creatures, which fostered German contempt for them and readiness to inflict harm. Nazi propaganda and power combined to turn antisemitism into a relentlessly self-fulfilling feedback loop, and rank-and-file Germans behaved accordingly.\n\nWendy Lower's _Hitler's Furies_ (2013) reinforces this line of analysis with evidence concerning a previously largely unstudied group: the half-million German women sent into occupied Eastern Europe as wives, secretaries, nurses, teachers, settlers, Red Cross volunteers, and radio operators, and in many other capacities. Some 300,000 German women served as auxiliaries in Gestapo and police offices and in prisons in the occupied east, another 10,000 in the German civil administration, and 3,500 more as camp guards. Almost all of these women were between the ages of seventeen and thirty. They saw a very great deal of persecution and murder; most of them facilitated it in one respect or another, such as typing up liquidation orders, and some of them perpetrated it, entering ghettos and shooting inhabitants or helping men root Jews out of hiding. As Lower points out, \"In favoring perceived duty over morality, men and women were more alike than different.\" They were also alike in succumbing to the temptations of absolute power that Germans enjoyed in occupied Eastern Europe. One of the killers, Erna Petri, who presided with her husband over a confiscated estate in eastern Poland from 1942 to 1944, summed up many of their intertwined motives when she said after the war, \"I did not want to stand behind the SS men. I wanted to show them that I, as a woman, could conduct myself like a man. So I shot four Jews and six Jewish children. I wanted to prove myself to the men. Besides, in those days in this region, everywhere one heard that Jewish persons and children were being shot, which also caused me to kill them.\"\n\nWhen all these impulses to conformity failed, when expressions of human sympathy or solidarity somehow asserted themselves among Germans in uniform, the Nazi regime resorted to violent retribution. Opting out of murder qualified as understandable weakness, and a German officer could get away with arguing, as Major Karl Plagge did as head of a repair yard for army vehicles in Vilna, that military needs justified keeping Jewish workers and their families alive for the moment. But overt assistance to Jews constituted sabotage punishable by death. Thus, on April 9, 1942, Anton Schmidt, a forty-two-year-old member of a rear echelon ( _Landessch\u00fctz_ ) battalion, wrote a farewell letter to his wife shortly before his execution in Vilna. He told her that, shocked by the massacres there, including the killing of babies by slamming them against tree trunks, he had used his position as leader of a straggler collection point throughout the fall of 1941 to facilitate the escape of more than 100 Jews from the city's ghetto (postwar research established that the real number may have exceeded 300). Exposed in January 1942 and court-martialed, he explained to his wife, \"you know how it is with me and my soft heart,\" adding that \"in my room are six men aged 17 to 23 who have the same fate. Condemned for desertion and cowardice in the face of the enemy. Jews too are the enemy\u2014that's just the way it is.\" Though outcomes of this sort were rare, even their possibility dampened humane inclinations on the part of Germans in the field.\n\nIf the most convincing explanations of the readiness of subordinate Germans to behave viciously toward Jews blend situational and volitional elements, that is not the case with regard to the senior figures who designed and gave the orders for the Final Solution. Fifty years ago, Hannah Arendt tried to use the figure of Adolf Eichmann to argue that these people were faceless and, as she put it, \"thoughtless\" bureaucrats who acted out of personal ambition more than ideological conviction and thus represented what she called \"the banality of evil.\" Almost no historian believes this anymore. As Tom Segev wrote in his study of concentration camp commanders, _Soldiers of Evil_ (1987), what characterized them was not banality, \"but rather inner identification with evil.\"\n\nDetailed prosopographical studies (collective biographies) have shown that perpetrators at this level were almost all highly educated, enthusiastic, and conscious proponents of murder and true believers in Nazi ideology. The most powerful such study is a book by Michael Wildt available in English under the title _An Uncompromising Generation_ (2009). This is a rather wan translation of the German title, _Generation des Unbedingten_ , which means something like the \"generation without limits or restraints.\" Wildt examined the life histories of 221 people who occupied leading positions in the RSHA, the SS office most responsible for carrying out the Holocaust, either during 1939\u201341, when the organization took shape, or for at least eighteen months in a later period. He found that 60 percent of them were born between 1900 and 1910, and another 17 percent were even younger. That means that most of them were in their thirties or, at most, their early forties during the Holocaust. In this respect, they were like their most prominent leaders. Heinrich Himmler, dubbed the \"architect of genocide\" by one scholar, was born in 1900, as was Rudolf H\u00f6ss, the commandant at Auschwitz during most of the camp's existence. Ernst Kaltenbrunner, the head of the Reich Security Main Office from 1943 to 1945, came into the world in 1903; Reinhard Heydrich, Kaltenbrunner's predecessor and the man who launched the Final Solution, in 1904; Adolf Eichmann, who arranged many of the deportation trains, in 1906; and Joseph Mengele, the doctor who sorted arrivals at Auschwitz between life and death and who conducted vicious medical experiments on them, in 1911. Similarly, those 121 T4 personnel who helped staff the Reinhard camps were remarkably young: more than 83 percent of them were born between 1900 and 1914.\n\nThe RSHA leaders were usually youthful, upwardly mobile men on the make, eager to prove themselves, to make a mark and a difference. Most were well educated\u2014one-third of them had PhDs, as did all four of the first _Einsatzgruppen_ commanders, and many had studied at Germany's best universities, notably Heidelberg, Leipzig, and T\u00fcbingen. Most had long records, dating to their student years in the 1920s, of involvement in extreme nationalist, antisemitic, and violent politics, and most had dedicated themselves to remaking the world by avenging the wrongs supposedly done to Germany. Imbued with a romantic view of war and a hunger for action, they were men on a mission who scorned sentimentality. _Gef\u00fchlsduselei_ was the word they used for all forms of human empathy, and the literal translation into English, which is \"spraying of feelings,\" conveys a sense of the contempt they expressed. They knew exactly what they were doing, and they believed completely in their utterly Germanocentric vision of national redemption through revenge and racial cleansing. Although the T4 personnel generally came from lower down the social scale, they also constituted a highly indoctrinated group.\n\nTo this mix of idealism and careerism, the RSHA and T4 people added a hard-hearted form of professionalism, a cold-blooded determination to do their jobs well. The phrase in German for someone who will stop at nothing is \" _sie\/er geht \u00fcber Leichen_ ,\" \"s\/he walks over corpses,\" and it applies literally as well as figuratively to these men. They found the language of \"duty\" very convenient; in its name, almost anything was justifiable as long as it served the German _Volk_. Invoking duty not only relieved them of personal responsibility, it made murder into a higher calling. Higher because they also claimed that the Reich's expansion to the east was part of a civilizing process that expanded European culture at the expense of supposedly barbaric Asia. Hitler once called Eastern Europe \"our India,\" and on more than one occasion he likened Germany's eastward expansion to America's westward one. The men atop RSHA believed deeply in this missionary vision, and they expected to have to kill millions of people to realize it.\n\nIn short, most German perpetrators of the Holocaust fit a pattern of militarily inspired, nationalist young men who seized on the opportunities for advancement and fulfillment that were created by the enormous increase in the ranks of the SS in the late 1930s, especially as it absorbed the police, and by Germany's expansion. They also hailed disproportionately from areas the Reich had lost after World War I or from border regions, which is to say from environments that heightened senses of national consciousness and competition. Very few of them were new to political violence or mere draftees. As the sociologist Michael Mann has summarized the evidence, \"the majority of Nazi genocide . . . was accomplished by ideological, experienced Nazis. . . . The vast majority of those involved in actual killing knew what they were doing, [and] most thought there was a good reason for it.\"\n\nZealotry is especially characteristic of the chief perpetrators, namely Himmler, Heydrich, Eichmann, H\u00f6ss, Kaltenbrunner, and two men not previously mentioned, Oswald Pohl and Hans Kammler, the leaders of the SS Economics and Administration Main Office (WVHA), the organization that ran the Nazi slave labor system. Heinrich Himmler rose to power and exercised it as the embodiment of the SS's motto: \"My honor is called loyalty\" ( _Meine Ehre hei\u00dft Treue_ ). Even though he became a Party member before the Beer Hall Putsch of 1923, having already imbibed the mixture of romanticism about Germany and animosity toward foreigners, Jews, and leftists that characterized the movement, he was initially closer to other early Nazi leaders than to Hitler. But after becoming head of the F\u00fchrer's personal bodyguard in 1929, Himmler made himself into Hitler's reliably ruthless agent in dealing with people or groups that Nazism defined as enemies. As a result, the main paper trail that connects Hitler to the Holocaust runs through Himmler's appointment books for 1938\u201342. They reveal how closely radicalizations of Nazi policy toward Jews followed meetings between the two men.\n\nNearly everyone who ever met Himmler\u2014or who has written about him since\u2014has commented on his unprepossessing appearance and colorless personality. Short, unathletic, and nearsighted, he hardly embodied the Nazi ideal. In fact, a Gauleiter once remarked, \"If I looked like him, I would not speak of race at all.\" Yet beneath the exterior lurked two driving, apparently contradictory characteristics that also have struck most observers: his absorption in a fantasy world\u2014including faith in astrology and herbalism, pleasure in torchlight rituals, the conviction that he was the reincarnation of German Emperor Heinrich I (\"the Fowler,\" who died in 936), and the dream of populating the colonized German East with interlinked settlements of Teutonic warrior-soldiers ( _Wehrbauern_ )\u2014and his methodical attention to practical bureaucratic details. The combination underpinned his \"success\" as a mass murderer. He demanded organized, thorough, and \"merciless\" translation of his Aryan supremacist dream world into reality, and that demand animated the organizations responsible to him: the SS, the RSHA, the German police, the _Einsatzgruppen_ , and all their auxiliaries.\n\nReinhard Heydrich came to the Nazi Party relatively late for one of its main leaders. He initially pursued a naval career that was cut short by his court-martial for having an affair with a woman other than his fianc\u00e9e. The woman's family turned out to have powerful connections, but what really led to his dismissal from the navy was the arrogance he displayed before the court. Though he had been active in conservative nationalist groups in the 1920s, the fianc\u00e9e, whose portentous name was Lina von Osten (Lina from the East), pushed him, beginning in 1931, toward the Nazi Party and membership in the SS. Physically, he was the model SS man: tall, blond, blue-eyed, long and thin in the face, athletic, and graceful. As one of his German biographers noted, \"[I]f National Socialism had looked in the mirror, Reinhard Heydrich would have looked back.\" Emotionally, too, he fit the mold: tough, decisive, hard-driving, persistent, relentless, and risk-taking. His favorite adjective was \" _unerh\u00f6rt_ ,\" unheard-of or unprecedented, and he strove to make his actions earn that description. Carl Jacob Burckhardt, a Swiss diplomat and historian, said after their first meeting that Heydrich was \"a young, evil god of death.\" At his funeral after his assassination by Czech resistance fighters in 1942, Hitler called him \"the man with the iron heart.\"\n\nHeydrich's manifest conceit and cold-bloodedness increased as time passed, probably in compensation for his late entrance into the Party, his scandalous eviction from the navy, and the persistent rumors, which proved false but led to a humiliating internal Nazi investigation in 1932, that his mother's parentage was Jewish. These impetuses, along with his close personal friendship with Himmler and his considerable organizational talent, turned him into a murderous executor of Nazi ideology. He became the epitome of the Nazi belief that only Germans counted; everyone else was simply outside his moral universe and expendable. Some of his biographers contend that he adopted Nazi ideology merely as a vehicle for his urge to power, but this is too simple. His conviction was genuine, as was his emotional addiction to military life and violence. Also genuine was his belief, typical of Nazi perpetrators, that he was innocent, high-minded, and self-sacrificing in fulfilling the tasks assigned to him. As he reportedly told his wife, \"I feel free of all guilt. I make myself available; others pursue egotistical goals.\"\n\nAdolf Eichmann was in some respects a rather pathetic figure, which did not prevent him from becoming an extraordinarily destructive one. His family moved from an industrial city not far from D\u00fcsseldorf to Austria in 1913, when he was seven years old, and he never managed to graduate from either an academic or a vocational high school. Helped by his father's business contacts and by his stepmother's Jewish relatives, he got jobs as a salesman for first an electric company and then an oil firm, but lost the latter post in May 1933 during the Depression. By then, he already had joined the Nazi Party, impelled largely by his Protestant, pro-German family milieu, and this new political affiliation led him to return to Germany after the Austrian government cracked down on the Nazis in mid-1933. There he volunteered for the SD, the Security Service of the SS that Heydrich had started a few years earlier, and received an assignment to keep tabs on Freemasons in Germany. He moved on to the SD's Jewish Department in 1935, and by 1938 was in charge of the agencies in Vienna dedicated to driving the Jews out of the city and confiscating their wealth as they left. Responsibility to handle Jewish affairs within the RSHA followed in 1939, along with the task of arranging the deportations of the Poles and Jews that Himmler wished to push out of the parts of Poland annexed to Germany into the General Government. Later, his portfolio expanded to include transportation to the ghettos and\/or death camps of all the Jews of Europe except those already inside the General Government\u2014their removal was the job of another SS officer, Hermann (Hans) H\u00f6fle.\n\nWhen Hannah Arendt described Eichmann as the embodiment of the \"banality of evil,\" the bureaucrat without convictions who saw no difference between shipping cargo and shipping people, she fell for the cover story that he constructed for himself in preparation for, during, and after his trial in Jerusalem in 1961. He knew that his only conceivable defense was to portray himself as a mindless cog in the machine, someone who merely had obeyed irresistible orders. In truth, he had come, during the 1930s, to believe deeply in Germany's need to fight the Jews. That belief hardened, following the conquest of Poland, into a readiness to kill and then into a determination to do so that had become so intense by November 1944 that he circumvented a direct order from Himmler to stop renewed deportations from Hungary. Thanks to the assiduous research of Bettina Stangneth, who examined Eichmann's many recorded utterances to fellow Nazis and their sympathizers while he was hiding in Argentina between 1950 and 1960 and assembled them in _Eichmann Before Jerusalem_ (2014), we now know how proud he was of his SS service in retrospect and how thoroughly he rationalized it, not as conscientiously carrying out an allotted task as a dutiful civil servant but as creatively and energetically defending his nation against perfidious attacks by Jews. Antisemitism was a means to his advancement, but it was not only that.\n\nRudolf H\u00f6ss was a rather different sort of person and perhaps the one leading perpetrator who most closely resembled Arendt's depiction of a \"thoughtless\" Nazi desk killer. H\u00f6ss came from a deeply religious family and felt attracted to military life because it offered a comradely antidote to his lonely upbringing and temperament. After service in World War I as only a teenager, he joined a right-wing paramilitary unit, got caught up in the murder of a comrade, served five years in jail, and emerged in 1928 lost and adrift. Hoping to start a farm, he joined a rather mystical agricultural group called the Artamanen, where he met Heinrich Himmler. In 1934, his invitation drew H\u00f6ss into the SS and concentration camp work, which offered the attraction of a quasi-military life. Once in, H\u00f6ss sought to win commendation by accomplishing whatever he was asked to do, without regard to its content. He remained Himmler's man throughout his career, not least because others in the camp system hierarchy disliked H\u00f6ss intensely.\n\nOne of his biographers has called him \"a functionary in the true sense,\" a man so empty that he found meaning only in carrying out directions and serving values that were created for him. He was a monument to what Germans call \"the secondary virtues\": selflessness, loyalty, diligence, helpfulness, and order, all displayed without reflection on the purposes to which they were being put. In a succession of camp command posts, culminating with Auschwitz, H\u00f6ss demonstrated neither pleasure nor discomfort in inflicting suffering. Whether his many victims deserved their fate was a subject on which he, in his own devastatingly self-incriminating words, \"had never really wasted much thought.\" Duty was his only concern, and at the end he therefore depicted himself, not the people he killed, as a victim of the fate that had cast him in the role of Auschwitz camp commander. None of these comments means that H\u00f6ss was a robot. Within given policy parameters, he was inventive and energetic. But he appears to have thought entirely within the box of pleasing his superiors and performing his assigned tasks\n\nErnst Kaltenbrunner, who imbibed his politics, including intense and vocal antisemitism, from his extremely right-wing father, joined the Nazi Party in 1930, eight years before his native Austria became part of the Reich, and the SS only a year later. He promptly recruited Adolf Eichmann, and then spent the 1930s brawling with political opponents and agitating for the _Anschluss_. In its aftermath, he became the chief of the SS and the police in Vienna from 1938 to 1943, when he moved up to succeed Heydrich as head of the Reich Security Main Office. Kaltenbrunner referred to Himmler as his _\"\u00dcbervater\"_ \u2014that is, his ideal and role model. Even after the war, Kaltenbrunner gave vent to his fervent Nazism by attesting that the Party presented \"a world view encompassing life in its entirety,\" that the idea of race constituted \"the divinely inspired building block of mankind,\" and that the Jews, especially in Eastern Europe, were \"really the only stratum that possessed enough intellectuality to provide the enemy with the necessary actors to execute his plans.\"\n\nThe last two figures in this rogues' gallery, Oswald Pohl and Hans Kammler, directed the murderous slave labor system, the former with responsibility for administration and finance, the latter for engineering and construction. Both were veterans of right-wing paramilitary formations, and both had joined the Nazi Party before Hitler became chancellor in 1933. They harbored dreams of an industrial empire that would provide building materials for the massive architectural expressions of the new Germany, furniture and knickknacks for German settlers in the conquered East of the continent, and roads to connect them. All of this was to contribute to the demographic transformation of Europe and the creation of a new state-owned economic sector. Like the other killers described above, Pohl and Kammler were ideologically inspired creators of a Nazi New Order, imbued with a spirit of activism and with \"ideals\" of racial supremacy. Neither had any hesitation in carrying out Himmler's instruction to work the concentration camp inmates like the Pharaoh's slaves.\n\nPerhaps the most remarkable feature of the mentality of the Nazi perpetrators was their self-delusion, their capacity to distract themselves from what they were doing by calling it something else. Perpetrators never owned up to torturing and slaughtering; they always professed to be serving a sanctified purpose that immunized them from the charge of immorality. The epitome of this stance was Himmler's speech to the assembled SS commanders at Posen in October 1943. He summarized the philosophy of the SS bluntly: \"honest, decent, loyal, and comradely must we be to members of our own blood and to nobody else.\" He congratulated his men for having waded through gore but nonetheless \"remained decent.\" He referred to their deeds as a \"never to be written page of glory\" in Germany's history. Of course, he was not just saying that the end justified the means, though he was saying that. He was also congratulating his subordinates on being people who could, to use contemporary phraseology, bite the bullet and do what had to be done. He was praising them for understanding that \"winning is the only thing.\" When we put his language into ours, we are reminded of how common such self- and principle-abandoning thinking is in the world. Maybe Stanley Milgram and Philip Zimbardo were right after all in suggesting that \"How could people do such things?\" is a na\u00efve question.\n\nA nation is not only what it does, Kurt Tucholsky, the great German satirist, wrote in 1934, but also what it puts up with. What of the ordinary Germans who did not carry out the killings directly but witnessed the deportations, sometimes photographed them for local histories, frequently took over the possessions left behind, and heard the rumors that abounded about the fate of not only Germany's Jews but also those in the east? What did they know of the murders, and how did they respond? Awareness of the self-dug graves and shootings by the _Einsatzgruppen_ , the Order Police, the foreign auxiliaries, and the German army was widespread, thanks to letters home and troops on furlough. Indeed, such information was sufficiently plentiful as time passed that more and more Germans spoke with open dread of the reprisals or retribution that they expected to experience once the tide of the war turned. A representative expression of this view, as well as of slightly more complete knowledge, is this diary entry by Curt Pr\u00fcfer, a semiretired diplomat and an antisemite who had purchased property formerly owned by Jews. On November 22, 1942, he wrote the following\u2014mostly in French, as if to conceal what he was saying: \"Men, women, and children have been slaughtered in large numbers by poison gas or by machine guns. The hatred that inevitably must arise from that will never be appeased. Today every child knows this in the smallest detail.\" Already in March of that year, the isolated diarist Victor Klemperer recorded that he had heard of a place called Auschwitz where Jews were worked to death rapidly; by October, he could describe it as a \"swift-working slaughterhouse.\" Meanwhile, in April 1942, he had written that his wife heard an eyewitness report of mass murders of Jews in Kiev, a reference to the killings at Babi Yar seven months earlier. Several months before Germany surrendered, Klemperer, again quoting only what his \"Aryan\" neighbors had told him, knew even the approximate death toll of the Holocaust. He wrote in his diary for October 24, 1944, \"six to seven million Jews . . . have been slaughtered (more exactly: shot and gassed).\"\n\nKnowledge of the Holocaust in Germany was extensive because, as Peter Fritzsche shrewdly has noted, \"the Nazis wanted to manage, but not entirely conceal, the facts.\" After all, Goebbels announced in the journal _Das Reich_ on November 16, 1941, that \"world Jewry . . . is now gradually being engulfed by the same extermination process that it had intended for us.\" On April 30, 1942, the _V\u00f6lkischer Beobachter_ , the official mouthpiece of the Nazi Party, reported \"the rumor\" that \"it is the task of the Security Police to exterminate the Jews in the occupied territories. The Jews were assembled in the thousands and shot; beforehand they had to dig their own graves.\" Hitler reminded Germans of his prophecy that a world war would bring about the annihilation of the Jews in no fewer than seven major speeches: on January 30, 1941; on January 30, February 24, October 1, and November 8, 1942; on February 25, 1943; and on January 1, 1945. By one scholar's count, the F\u00fchrer referred to the wiping out of the Jews in at least a dozen wartime public speeches or pronouncements. If such partial revelations had a purpose, it was to secure loyalty by reminding people of their complicity. Having allowed such brutality, Germans could expect nothing but reprisal, so they had best fight tooth and nail to sustain the Third Reich. For the most part, this strategy succeeded.\n\nWhatever the state of their knowledge, the German public's willingness to help Jews was exceedingly limited. Jews who went underground, who refused to answer the order to appear for deportation and then hid their identities and tried to survive within the Reich, were called U-boats, after the German word for submarines. Perhaps 10,000 people tried this means of outliving and outwitting the Nazis, about half of them in the city of Berlin; both there and nationwide, somewhere between 30 percent and 50 percent of them made it to 1945. The mortality rate was high, and the numbers involved small. But for every person who did survive, the number of non-Jewish Germans who helped at one time or another had to be substantial. Sometimes that help was active, as in the creation of forged identity documents or the offer of a place to live; sometimes it was passive, as when an old acquaintance recognized a U-boat on the street but did not expose the person. Konrad Latte, whose father and mother had converted to Protestantism, spent the months from March 1943 to May 1945 as a U-boat. Before he died in 2005, he named fifty people who protected him in one way or another, only one of whom was ever caught and punished. Arthur Arndt, a Jewish doctor hidden with his wife and two children in Berlin, cited exactly the same number of non-Jews on whom the family's survival had depended. Max Krakauer, still a third successful U-boat, put the number in his case at sixty-six.\n\nDespite such numbers, heroic behavior of this sort was rare. That makes it both admirable and a standing reproach to the general attitude of the German population, which to the very end of the war viewed the persecution of the Jews only through the self-interested lens of the benefit or punishment Germans were likely to receive because of it. Until the fronts began to close in on the Reich in 1944, the benefits greatly outweighed the potential costs, as Germans profited from the Holocaust in ways that ranged from the government revenue obtained from stealing the Jews' precious metals to the individual allocation of the furniture from their apartments to Reich citizens who had been bombed out. In Hamburg alone from 1941 to 1943, the authorities auctioned off some 4,000 shipping containers holding the goods of Jews who had emigrated, and the Nazi state took in 7.2 million reichsmark in proceeds. In 1942\u201343, forty-five shiploads of goods taken from Dutch Jews went to the same German port city. A German scholar of the subject estimates that between 1941 and 1945 \"at least one hundred thousand\" inhabitants of the town and its environs bought household property confiscated from Jews. Similarly, the occupation of Europe generated enormous benefits to Germans in the form of food and goods shipped home by far-flung troops, most of it bought by soldiers flush with local currency but some of it stolen. Returns on conquest and murder such as these did much to preserve loyalty to the Nazi regime well into 1945.\n\nHow little effective help most Jews could expect from the German public is demonstrated by an event that some commentators cite as evidence for not only the opposite point, but also the potential of popular opposition to alter Nazi racial policy. To be sure, the Rosenstrasse protest of February 27 to March 6, 1943 in Berlin constituted the only outbreak of popular resistance to deportations of Jews in the history of Nazi Germany, but the events were far less consequential than legend has it. The trigger was a push finally to make Germany virtually _judenrein_ (cleansed of Jews) by rounding up all remaining Jewish forced laborers at their workplaces and then deporting all except those in mixed marriages to Auschwitz or Theresienstadt. Across Germany, Jewish spouses in those marriages who got caught up in the Gestapo raids were released immediately, and in Berlin more than three-quarters of them were. But the SS detained some 2,000 such men in a Jewish community building on the Rosenstrasse in the middle of the city in order to double-check their marital status against the records deposited there and to identify personnel suitable for future assignment to nearby Jewish institutions as replacements for _Volljuden_ , or \"full Jews,\" the regime intended to (and did) deport a few weeks later. As these processes dragged on, several hundred of the worried non-Jewish wives and female in-laws of the detained men gathered around the building seeking information about their relatives, occasionally crying out for their release but mostly, as one participant reported, standing in \"silent protest,\" and in defiance of repeated police efforts to disperse the crowd. Once the releases from the building gathered pace, the assembly dwindled, and the episode came to a close.\n\nThe Rosenstrasse protest required considerable courage of the women who carried it out, but two telling aspects of it deserve emphasis. First, it was limited to a few hundred relatives of the rather small number of men affected, not joined by other so-called Aryans, and not accompanied by any popular resistance to the deportation of thousands of other Jews from Berlin and the Reich at this time. Second, the protest accomplished little in protecting men whom the SS intended to use in the near term and dispense with later. With the partial exceptions of the spouses transferred to Jewish community institutions, the Jewish parties to mixed marriages became subject to increasingly severe measures in the ensuing months. Not allowed to return to their factory jobs, they were condemned \"to the hardest manual labor,\" increasingly driven with their non-Jewish spouses into vacated Jew Houses, shipped off to work camps, and finally included in the directive that consigned all part-Jews, or _Mischlinge_ , to Theresienstadt in early 1945. The scant remaining statistical evidence suggests that fewer than half the mixed marriages of 1943 still existed when the war ended.\n\nFar from demonstrating what greater popular resistance to Nazi persecution might have accomplished, the Rosenstrasse incident showed that overt protest had little impact on the direction or pace of the regime's relentless course. Ironically, in Nazi Germany the prospect of popular opposition sometimes stayed the regime's hand, as clearly happened in dissuading Hitler and his entourage from promulgating a law automatically dissolving all mixed marriages, but the reality of resistance generally goaded the Reich into more radical action, not only in Germany but also in occupied countries.\n\nAs World War II drew toward its close, and the returns on persecution turned adverse for Germans, few of them paused to reflect on the horror they had inflicted. Instead, they devoted most of their attention to the supposed injustice of their own suffering, either at the hands of Allied air raids or as the likely result of the regime's crimes when vengeful enemy troops rolled in. The self-pity and sense of victimization that gave rise to Nazi rule also outlasted it.\n\nENSLAVEMENT\n\nThe most drawn-out and agonizing, though numerically least lethal, form of murder, after gassing and shooting\u2014namely, the system of slave labor\u2014accounted for at least one-half million deaths in the Holocaust. Why and how did the Nazis develop this system? Why did they bother to keep some Jews alive for labor for at least some period of time? Why did they treat such laborers so apparently counterproductively? With regard to these questions, perhaps more misinformation has accumulated than concerning any other aspect of the Holocaust. This is so for an ironic reason: Many of the lawyers who in recent decades worked hard to obtain compensation for former slave laborers sullied a good cause by frequently misrepresenting how the system came into being and how profitable it proved.\n\nSlave and forced labor were two parts of a common system. Forced laborers were non-Jews recruited or rounded up in occupied countries during World War II and brought to work in Germany for nominal wages. They were often, though not always, badly fed, housed, and treated and kept segregated from the German population. Collected and supervised by a Nazi _Gauleiter_ named Fritz Sauckel, they made up 15 percent of industrial workers in Germany in 1942, a figure that rose to 30 percent in 1944, but 20\u201350 percent of the labor force in the largest and most militarily important German firms during that time frame, and more than half the people working in German agriculture. In August 1944, they included almost 1.3 million French people, over 580,000 Italians, almost 2.8 million Soviet citizens, and almost 1.7 million Poles. Their number peaked at 6.8 million at the end of 1944, but altogether 13 million people did forced labor in Germany from 1939 to 1945, 4.6 million of them prisoners of war and 8.4 million of them civilians. This was a staggeringly large system of exploitation.\n\nSlave laborers, who numbered some 1.1 million people during the whole of World War II, of whom about 714,000 were toiling at the beginning of 1945, were inmates of ghettos and concentration camps, mostly but not always Jews. Indeed, the number of non-Jews among them rose during the final year of the war, especially as more and more Eastern European women were put into camps like Ravensbr\u00fcck and Sachsenhausen and then parceled out to labor sites. The Economics and Administration Main Office of the SS supervised and controlled most slave laborers and leased them to government agencies or private industries for a set price per person per day. In other words, they were not paid, but they were paid for. And, contrary to legend, they were not necessarily cheap. The SS charges in many cases exceeded what a German civilian laborer, especially a construction worker, would have received; even when this was not the case, the productivity of slave laborers often was so limited that it offset their low wages. After all, few slave laborers had done manual labor before, especially on construction projects. All this gave the employers a perverse incentive to economize on food and housing for slave laborers, to drive them excessively hard, and to work them long hours, at least so long as ample numbers were available to replace people who died of maltreatment. In this respect, the term \"slave labor\" is actually misleading. Slaves are bought, and their owners thus acquire an economic interest in their survival. But camp and ghetto inmates were rented by the day. The renter had little interest in their long-term survival unless they were highly skilled. Meanwhile, the employer could send any flagging workers back to the camp they came from and trade them for fitter workers.\n\nWhat difference this made is illustrated by the relative fates of the men and women slave laborers used at a plant of a Degussa subsidiary at Gleiwitz in Upper Silesia between 1943 and 1945. Only two females out of 209 in the workforce died, and they committed suicide the day the SS took over supervision of the laborers' barracks in 1944. But a substantial share\u2014probably about one-third\u2014of the more than 1,000 men put to work on the site expired. Why? The men worked on construction and were of no interest to the firm once the plant was finished, but the women were indispensable to its manufacturing, at least as long as the war lasted. The surviving statistics indicate that the company was indifferent to the working conditions and fates of the men but very concerned to retain the women. Food and medical help must have been better for the women than the men, because relative working conditions alone\u2014the men did heavy labor outside, the women worked indoors mostly packaging the output\u2014are not enough to explain the discrepancy in survival rates. One scholar who has examined mortality rates of slave laborers meticulously has concluded that those put to work on construction projects were five to ten times more likely to die than those employed on assembly lines, but at Gleiwitz the men were 150 times more likely to die. This example illustrates a general pattern: Firms could make a difference to the survival chances of their slave laborers, but tried to do so only when self-interest commanded such action.\n\nHow and why did these systems come into being? The forced labor system was rooted in the mathematics of the German labor force during World War II: The Reich called up eleven million men for military service, and a larger percentage of German women were employed already in 1939 than was ever the case in Britain or the United States during the war, which meant that relatively few German women were available as replacements. Yet the war created demand for enormous increases in output. Germany therefore faced a choice between farming out production to the occupied countries or importing replacement workers. For the most part, the Reich opted for the latter to guard against sabotage and\/or the loss of industrial secrets. The forced labor program that developed drew on two precedents for compulsory work in Nazi Germany: first, the conscription of unemployed Germans and those with jobs in nearby factories into work columns to build new highways (the Autobahnen) and fortifications in the Rhineland (the Westwall) in the 1930s; and second, the use of primarily Polish and French POWs as supplementary workers beginning in 1940.\n\nThe slave labor system, however, had its roots in Nazi antisemitism, which contended that Jews avoided manual labor and thus should be forced to do it, and in the economic interests of the SS, which wanted to become financially self-sustaining. The system had two forerunners. First, the Reich inaugurated a compulsory labor program for German Jewish males on Hitler's birthday in April 1939. Cut off from other gainful employment and the German welfare system, Jews were supposed to \"earn their keep\" in road-building and street-cleaning projects and in private industrial assignments, notably at the large Siemens plant in Berlin. The Nazi regime extended this program to all Jews in Poland in October 1939, and many died when forced to do river dredging and straightening projects and airfield and road construction under brutal overseers. Two of the later death camps, Belzec and Treblinka, began as labor camps for Jews put to work digging tank traps and other fortifications along the nearby border with the Soviet Union following the partition of Poland. Second, in 1936\u201339, the SS set up a web of its own companies that used camp inmate labor to generate revenue. The holding company was called the German Economic Plants (DWB). One of its subsidiaries was the German Equipment Works (DAW) that made weaponry. Another, the German Earth and Stone company (DEST), made bricks at most camps in Germany and operated an infamous quarry at Mauthausen in Austria that supplied much of the building material for the Nazi Party Grounds in Nuremberg.\n\nAlthough these were the precedents, they were not the actual triggers for the vast expansion of the slave labor system during World War II. Three developments set that process in motion. First, ghettoization created both labor pools that attracted German firms and an incentive for the Nazi administrators to develop revenue-generating initiatives that would make the ghettos pay. Second, the Reich decided in the Fall of 1940 to build a road in southern Poland that would link the Autobahn from Berlin to Upper Silesia with the highway they envisioned (Durchgangstrasse IV) running across Ukraine all the way to the Black Sea after the invasion of the U.S.S.R. This was the impetus for the formation of the Organisation Schmelt, named after the SS officer who commanded it, which developed the system of wage rates, barracks, underfeeding, and severe treatment that later characterized the slave labor program everywhere. A simultaneous project connecting Berlin to Lodz led to the first use of slave labor by German private industry, in this case the construction firm Philipp Holzmann. The route through southern Poland stimulated the expansion of the Auschwitz and Majdanek camps, which were to be the sources of labor for the construction. When Heydrich referred, at the Wannsee Conference, to the use of able-bodied Jews on road building in the East, Durchgangstrasse IV was the undertaking he had in mind. It ultimately consumed the lives of at least 25,000 Jewish construction workers, who toiled without the assistance of machinery and were brutally mistreated. Third, in early 1941 two of the largest corporations in the Reich, Volkswagen and IG Farben, chose to deviate from industry's earlier refusal to hire camp inmates. Volkswagen agreed to set up a concentration camp on the factory grounds in Wolfsburg, in northwest Germany, ultimately to construct an aluminum foundry, and Farben agreed to build a huge synthetic rubber plant just east of the town of Auschwitz and to lease camp inmates as construction workers.\n\nFrom these small beginnings, the use of slave labor mushroomed, especially after September 1942, when the SS broadened its usual policy of hiring out inmates only for production in and around concentration camps by agreeing to expand upon the Volkswagen precedent with the establishment of satellite sites near important factories. The most voracious consumers of slave labor became the Eastern Front, for military installations and factory reconstruction; the French Atlantic coast, where inmates built most of the defenses; Upper Silesia, the preferred location of massive new factories for fuel and rubber because Allied bombers could not reach the area from bases in Great Britain; and the Project Giant ( _Projekt Riese_ ) site in the Owl Mountains of Lower Silesia, a warren of underground passages that began as a huge prospective bombproof headquarters for Hitler and then morphed into military production lines. All told, slave labor camps in Nazi-occupied Europe numbered in the tens of thousands, and they extended from the island of Alderney, in the English Channel, to the farthest reaches of German penetration into the Soviet Union.\n\nDespite (or because of ) the size of the system, it was never efficient or well managed. Half the inmates of Auschwitz never even got labor assignments. The SS companies were neither profitable nor usually successful in their joint ventures with private enterprises to manufacture military equipment inside camps, though one initiative, involving the construction of fighter planes at Flossenb\u00fcrg with Messerschmitt, made money and contributed to the German war effort. Until the turn of 1943\/44, Jewish slave laborers were generally kept out of Germany proper, and projects were brought to them in occupied or annexed territories\u2014the inmates supplied to Volkswagen in 1941 had included only a few Jewish political prisoners\u2014but the Nazi regime reversed this practice as labor shortages mounted and British and American air raids took a toll on the Reich. This change in German policy had major consequences for the survival chances of people arriving at Auschwitz. Up until late 1943, the camp primarily wanted men to work on construction, so fewer women were selected for admission to the camp than men. Combined with the fact that women generally outnumbered men on transports into the camp, this meant that the female mortality rate upon arrival was much higher than the male. Beginning in late 1943, however, demand rose for women to work on assembly lines, so the number of them selected caught up with and sometimes exceeded the number of men. This is why the young women from Hungary who arrived at Auschwitz in 1944 had better chances of surviving the camp than almost any other Jewish group.\n\nOne can get a sense of the variable horrors of the slave labor system by looking at two examples from occupied Poland of production sites that continued to operate long after most other installations using Jews had been liquidated: Starachowice and Skarzysko-Kamienna. Both lay south of Radom in the middle of the General Government, and both endured because they produced munitions. The two sites also had a common and peculiar feature that proved decisive in the survival of some of their slave laborers: Neither plant was incorporated into the SS camp system under the jurisdiction of the WVHA. Both remained largely governed by pragmatic factory managers who behaved erratically and unpredictably but, on the whole, less ruthlessly than the SS.\n\nStarachowice came into existence as a camp on October 27, 1942, following the liquidation of the surrounding ghettos, during which two-thirds of the inhabitants were sent to death at Treblinka and one-third brought into the new, hastily constructed camp. The installation lasted twenty-one months, until July 28, 1944, and the transport of the workers to Auschwitz. Shifts were either eight or twelve hours long, depending on the strenuousness of the work and not counting marching time to and from the camp to the factory. Output quotas were high, but working conditions depended heavily on the character of the German or Polish foremen, which varied immensely. The predominant recollection of survivors is of the extreme filth that prevailed. One former inmate said that when he got to the Monowitz camp alongside the IG Farben factory near Auschwitz, he found that wretched site much cleaner than what he was used to at Starachowice, where he had not showered for months on end.\n\nSkarzysko-Kamienna operated from April 1942 to August 1944 on the site of several former Polish state ammunition plants that a German munitions producer, the Hugo Schneider AG (HASAG), had taken over. Some 25,000 Jews passed through the camp during its life, and about four-fifths of them perished. The inmates worked in two shifts, day and night, without special work clothes or adequate sanitary conditions. In the shell department, women workers were expected to carry 180 nine-pound shells to each polishing machine during each hour in a ten-hour shift\u2014in other words, three per minute. In the antiaircraft department, supervisors whipped workers who produced defective pieces. In the mine department, packers had to cram explosives into the shells by hand with no gloves or aprons to protect them. Those who worked with picric acid found that their hands turned black and their hair turned green; those who worked with TNT saw their skin turn reddish-pink. As the TNT was being prepared, women had to stir it while it boiled at a rate of 1,800 stirs per hour and 21,600 stirs in a twelve-hour shift, all while standing. The food was the usual watery soup, ersatz coffee, and crusts of bread. The workers wore wooden clogs. The bunk beds had no mattresses or blankets and were lice-infested. Typhus was rampant, and weak prisoners were shot weekly until the spring of 1943, when the Germans began to worry about running out of them. Yet the camp and its factories continued to operate and some Jewish inmates continued to live until the Soviet army appeared on the horizon in late July 1944, when the sick were killed and everyone else shipped out to HASAG plants in Germany. Their survival was highly unusual, and it resulted from the fact that even Himmler could not bring himself to order the deaths of the people who in 1944 produced one-third of the German infantry's ammunition.\n\nBeginning in late 1943 and with gathering speed during 1944, the Germans reversed the policy of keeping Jews out of the Reich and began to bring more and more concentration camp inmates to work in Germany. The model for what could be done became the Dora-Mittelbau site in the Harz Mountains, where V-rockets were to be produced. Sixty thousand prisoners passed through this installation in 1943\u201345, and more than 40 percent of them had died by the time the war ended. The SS installed an assembly line in two large shafts that had been driven into Kohnstein Mountain since 1937 to create a huge storage installation for aircraft fuel. The shafts were not straight lines, but S-shaped, each about a mile long, about 30 feet wide and 23 feet high, just under 300 feet apart, and connected every 100 feet or so by somewhat smaller cross-shafts, creating a sort of curved ladder design. The original idea was to lay railroad lines down in each shaft, to put the storage tanks in the cross-shafts, then to bring trains into the mountain to fill their barrel cars with fuel, and finally to drive the trains out the other side. This conception lent itself readily to conversion to rocket assembly lines, with weapons moving down the tracks and the 20,000 different components of each, which were stored in the cross shafts, applied in succession. But only one of the shafts actually passed completely through the mountain and out the other side, so the assembly-line plan had to be altered somewhat. Even so, the scale of the underground production facility was vast, coming to more than one million square feet.\n\nThe project began in August 1943, with a non-Jewish labor force drawn from Buchenwald concentration camp, and the camp sent an additional 800 workers per week in the first months. Fed a meager and watery diet, they were quartered in the stinking, dusty, lice-ridden, and overcrowded cross-shafts, which were never quiet because the assembly work was continuous in two twelve-hour shifts. The prisoners had no safety gear and little protection against outbreaks of disease. Oxygen in the interior of the shafts was in short supply, cold water accumulated on the floors and chilled the bootless workers, and the interior temperature never rose above 59 degrees Fahrenheit. When production of rockets began in January 1944, the Dora camp adjacent to the site held more than 10,000 inmates, and about 4,000 of them were working in the shafts. Over the next three months, they turned out about 300 rockets, most of them defective because of design flaws, and the camp acquired a terrible reputation for mortality. By the beginning of April 1944, when most of the production problems were ironed out, 34 percent of the cumulative population to date of just over 17,000\u2014that is, almost 6,000 people\u2014had died, at a rate that had reached 20 to 25 inmates per day. That death rate was the highest of any concentration camp at the time. Another 20,000 prisoners died in the next twelve months, making total mortality among the people who assembled V-1 and V-2 rockets about two-thirds greater than the total number of English and Belgian citizens killed by them.\n\nDora-Mittelbau was the prototype of what became, in March 1944, the Fighter Staff Program ( _J\u00e4gerstab-Programm_ ), a massive effort to bury German arms-producing factories so that Allied bombing could not damage them. Junkers aircraft motor factories and assembly lines were installed in the northern end of the Mittelbau shafts and in other caves in nearby mountains. By September 1944, 12,000 Dora inmates, now including Hungarian Jews, were at work on these sites in central Germany, and tens of thousands of laborers from other camps were carving caves into the steep banks of the Rhine or pouring and camouflaging concrete hangers in open Bavarian fields. As the workforce at Dora grew, the SS created a new concentration camp consisting of some ninety buildings outside the south end of one of the tunnels, where the production of V-1 rockets was concentrated after September 1944.\n\nNo one has ever succeeded in precisely tabulating the total number of people killed across Germany in this crazed attempt to protect plants from bombing, or in the slave labor program altogether. One-half million deaths is the best rough estimate. We know that mortality rates fluctuated: They were high in 1942\u201343, when laborers seemed so numerous as to be expendable; they dropped in 1943\u201344, and then they surged again during the Fighter Staff Program and the collapse of Germany. The number of former Jewish slave laborers still alive in 1945 may have been as low as 150,000.\n\nThis was a state-driven system, not one propelled by private greed, as is often implied. The principal reason private enterprises asked for slave laborers was the absence of alternative ways to meet rising production targets or, late in the war, to salvage their machinery by getting it underground. Of course, if Germany won the war, the companies expected to gain in the form of having new plants that slave laborers had helped construct. But corporate executives generally fixed their eyes on more short-term objectives: doing their national duty, protecting their market or political positions, and continuing to produce. At Auschwitz, IG Farben remained wedded to using and paying for slave labor, even though camp inmates accomplished only about 15 percent of the construction work, primarily as a disguised form of bribing the SS for future favors. Very few companies, in the end, made much money off the system, not least because much of what they built was lost in the war or afterward. For example, IG Farben's plant near Auschwitz and Degussa's at Gleiwitz produced for only a few months before being overrun by the Red Army. The beneficiary of both turned out to be the Polish Communist state, which nationalized the factories after 1945 and operated them until the fall of communism in 1989.\n\nIn other words, the use of slave labor by German companies was criminal, but not because it was profitable, which it often was not. The principal profiteer from the slave labor program, as from Aryanization in general, was the German state, which collected fees on the labor, estimated at 600 to 700 million reichsmark in 1943\u201344 alone, commissioned most of the projects into which that labor went, and consumed most of the products that the labor ultimately generated.\n\nProbably the most murderous phase of the slave labor program was the final one, the interval between January and May of 1945, when the retreat of the German army on all fronts prompted the regime to try to salvage slave labor for the Reich by marching camp inmates back into Germany. This effort to save the laborers turned into a massive destruction process, to which about 35 percent of the people involved succumbed in the last five months of World War II. Often columns of prisoners set out from camps with no clear sense of how to get where they had been told to go, and confusion was compounded by the rapid movement of Soviet or other Allied forces that often blocked previously open escape routes. Most of these evacuation columns carried little food with them and consisted of ill-shod and ill-clad people marching in the dead of winter. Massive casualties resulted, as the guards, terrified of being captured if the columns slowed down, shot anyone who faltered or straggled. In the evacuation of Auschwitz in January 1945, prisoners marched through the snow on either of two routes, both over thirty miles long, until they reached a passable railroad line and were loaded onto open freight cars in subfreezing temperatures. Amazingly, the death toll in this first round of retreat was relatively modest, something like 7,000 out of 56,000. But 15,000 of those survivors went to the Gross-Rosen camp, which was abandoned, in turn, in February, and in that retreat the mortality rate was much higher, amounting to perhaps 50 percent of the 97,000 prisoners marched out.\n\nAmong the most horrible of the death marches were those launched from the Stutthof camp, on the Baltic coast, near Danzig, in January 1945. Almost 69,000 prisoners, most of them Jews and over half of them women, each carrying only eighteen ounces of bread and four ounces of margarine, and many of them barefoot, left the camp in six marching columns on the morning of January 25. The lucky columns were the ones the Soviets caught up with; the others marched westward for hundreds of miles, growing thinner by the day, or suffered a more swiftly murderous fate. In Palmnicken, a village thirty-one miles west of K\u00f6nigsberg where one of the prisoner columns with about 3,000 inmates bivouacked in a factory building for a few nights, the guard force and the local Nazi Party leader decided that they did not want the town to contain inmates when the approaching Russians arrived. So the Germans marched the prisoners, most of them women, some three miles to a row of high bluffs overlooking the Baltic seashore and machine-gunned them into the freezing water below.\n\nThese ghastly retreats also had calamitous ripple effects, as the camps that received retreating prisoner groups swiftly became unable to feed or otherwise maintain them. Throughout the concentration camp system, all semblance of sanitation and sustenance collapsed. As a result, the overwhelmed guard staff at Neuengamme, near Hamburg, began killing the sick with poison injections; at least 8,000 died this way from February to April 1945. At Dachau, conditions became so horrendous that 4,000 inmates died of typhus in February of that year. At Buchenwald, the prisoners arriving from elsewhere were stuffed into a sector called the Little Camp; its population rose from 6,000 in January to 17,000 in April, even as some 5,200 inmates of the site died during that time span. At Ravensbr\u00fcck, Sachsenhausen, and Mauthausen, and possibly at Dachau, the response was to set up gas chambers or to activate the small ones left from the 14f13 operation for sick, infirm, and troublesome prisoners. About 10,000 inmates were asphyxiated at these places from February to April 1945, mostly with Zyklon but some perhaps in gas vans. The last gassing of the Holocaust occurred at Mauthausen on April 29, the day before Hitler killed himself.\n\nBy far the worst conditions prevailed at Bergen-Belsen, in northwestern Germany, not far from Hanover. Once a small site that held Jews who were to be exchanged for Germans in Allied hands, the camp ballooned to 15,000 inmates by November 1944, most of them sick prisoners dumped there from other camps. By March 31, 1945, the population had reached 44,060, even though the mortality rate had averaged between 250 and 300 people per day in the preceding four weeks. And this was before six convoys containing 20,000 prisoners arrived from Dora-Mittelbau in early April. As far as historians can reconstruct from the records, approximately 35,000 people died from disease and starvation at Bergen-Belsen in the final months of the war. This was an instance of what one might call unplanned annihilation, though one should add that intention was clearly present, since the humane thing to do was to leave the prisoners in camps for the Allies to capture. The decisions to withdraw them under horrendous conditions and then to try to maintain control of them for as long as possible were, in effect, murderous, and the number of resulting deaths was about as large as in the massacre of Hungarian Jewry in the spring of 1944.\n\nThese decisions were Heinrich Himmler's. Hitler had favored slaughtering all the camp inmates and blowing up the sites, but the leaders of the WVHA, the SS economics division, who had been put in charge of jet aircraft as well as V-rocket production, wanted to hold on to the labor supply as long as possible, and Himmler hoped to keep some Jews alive as a bargaining chip with the Allies. Although he vacillated somewhat between preservation and massacre during the final months, evacuation was a way of accomplishing both goals simultaneously, and his instruction of mid-April that \"no inmate may fall into the enemy's hands alive\" left the final choice up to camp commanders. Not all of them opted to send their inmates meandering around the Reich's shrinking territory, but most did. Thus, Bergen-Belsen fell to the British in mid-April without prisoners having been pulled out. But Buchenwald's 48,000 inmates were sent off in trains and on foot to Dachau and Flossenb\u00fcrg only a few days after Belsen was captured; at least one-third perished in the following three weeks. At Neuengamme the SS also began dispersing the prisoners, sending 10,000 of them to the port city of Neustadt, where they were loaded onto three vessels anchored in the harbor. When the British bombed the city on May 3, the ships caught fire, and at least 7,000 of the prisoners burned, drowned, or were shot by German guards while trying to swim to shore. At Flossenb\u00fcrg, evacuations began even as prisoners from Buchenwald were coming in; of almost 46,000 inmates sent mostly southward toward Dachau, at least 7,000 died in the ensuing three weeks. Finally, in late April, the SS emptied Sachsenhausen and Ravensbr\u00fcck and forced the prisoners to march northwest toward the Baltic coast. American troops liberated approximately 20,000 of them in Schwerin. About 40,000 others endured the last days of the war in an open-air camp in a nearby forest, where thousands died of exposure and starvation.\n\nAt the end of April 1945, Bavaria and Austria remained under Nazi control, and they contained two large camp complexes, those of Dachau and Mauthausen, each with numerous subcamps. As the SS abandoned the most distant of Dachau's satellites, the guards simply burned down the barracks, with the sickest and weakest inmates inside. But once the dispersals from the main camp began in late April, the columns had no place to go, and those that set out were intercepted so quickly by American troops that only about 1,500 inmates died en route. Back in the Dachau camp, however, conditions were far worse for those inmates left behind or arriving on trains pouring in from elsewhere, and the last-minute ravages of hunger and disease were considerable. As for Mauthausen, it and its subcamps at Gusen, Ebensee, and Gunskirchen contained some 85,000 prisoners in early 1945, even though the mortality rates on the marches from other camps toward these sites had been devastating. Of 76,000 Jews handed over to Germany by Hungary at the former Austrian border at the turn of 1944\/45, for example, at least 45,000 died on the way to Mauthausen in early 1945, often at the hands of civilians along the route or the _Volkssturm_ militia units assigned to guard the prisoner columns. Those who reached the camp still faced long odds against survival. About 15,000 Jews were sent from Mauthausen to Gunskirchen in late April 1945; on May 4, the arriving American forces found 5,419 survivors, and of them, a soldier bleakly wrote, \"many of the living people look dead. Bones covered in skin with almost no sign of flesh, sunken cheeks and deeply sunken eyes and a glassy expression, the expression of the living dead.\" One of those people was Theodore Zev Weiss, the man whose name is on the professorship I held at Northwestern for sixteen years. As of this writing, he is eighty-five years old and living in Wilmette, Illinois. Neither he nor any other inmate was supposed to survive. If the GIs had arrived even a week or two later, none would have.\nCHAPTER 5\n\n[VICTIMS: \nWhy Didn't More Jews \nFight Back More Often?](contents.xhtml#ch_5)\n\nWE TURN NOW from the perpetrators of the Holocaust, and the questions of why and how they killed so many, to the victims of the Nazi murder campaign, the surrounding populations, and the international community, and the question of why these groups could or did not do more to stop the carnage. First, the sensitive and controversial matter of the response of the Jews themselves: \"Why didn't more Jews fight back more often?\" is a common question that succeeding generations have posed from the comfort of living in liberal and law-observing societies. Why did the Warsaw ghetto's inhabitants not rebel against the Germans until April 1943, the inmates of Treblinka and Sobibor until August and October of that year, and the _Sonderkommando_ of Jews detailed to operate the crematoria at Auschwitz until the fall of 1944\u2014in each case only after the Germans' intention to kill the last of them became unmistakable?\n\nThe question is not quite fair, since flare-ups of armed resistance did occur when the Nazis began deportations from particular places. For example, in Cracow in December 1942, a Jewish group blew up a caf\u00e9 favored by German officers in an effort to slow the transports. Two months earlier, an inmate killed a German staff member named Max Bialas at Treblinka. Jews were involved in an attempt to free the roughly 1,000 deportees on the twentieth transport from Belgium to Auschwitz on April 19, 1943, which actually enabled seventeen people to escape from one boxcar, ten of whom were not recaptured. But, as that small number suggests, these incidents had very limited consequences and were easily snuffed out. Perhaps, as various scholars contend, armed Jewish underground movements came into existence in five to seven of the large Polish ghettos, forty-five smaller ones, five death or concentration camps in Poland, and eighteen forced labor sites, but even so, their ultimate effectiveness was slight. On the whole, the Jewish response to the Nazi onslaught was to comply with German demands and orders in hopes of preventing them from getting worse.\n\nCOMPLIANCE AND RESISTANCE\n\nTowering figures in the study of the Holocaust, notably Raul Hilberg and Hannah Arendt, have addressed the matter of this Jewish response in highly provocative form. The very first page of Hilberg's monumental _The Destruction of the European Jews_ , both when first published in 1961 and again when an expanded third edition appeared forty-two years later, speaks of \"the Jewish collapse\" in the face of the Nazi assault and calls this \"a manifestation of failure.\" He goes on to argue that the efforts of Jewish communities to sustain themselves, to maintain order, and to placate the Nazis actually helped the Germans to achieve annihilation. Hannah Arendt's famous work of 1963, _Eichmann in Jerusalem_ , pushed this point further. She called \"the role of the Jewish leaders in the destruction of their own people . . . the darkest chapter of the whole dark story.\" In her opinion, \"without Jewish help in administration and police work . . . there would have been either complete chaos or an impossibly severe drain on German manpower.\"\n\nThe charge by both Hilberg and Arendt has two parts. The first is that Jews did not resist because the only meaningful resistance would have been armed action, in which relatively few Jews engaged. The second is that Jews actually made things worse by trying to survive in ways other than fighting. The Israeli historian Yehuda Bauer has rejected both arguments emphatically. He defines Jewish resistance as any undertaking designed to frustrate the Germans' purpose of harming or killing the Jewish people. He invokes the Hebrew word _amidah_ to describe the many forms this unarmed resistance could take, from smuggling food to organizing schools and cultural events, and he insists that all of these actions were the best Jews could do with a hopeless situation, a testament to their dignity and will to live against enormous odds. Bauer is determined to avoid blaming the victims for their fates, and if that means he sometimes stretches the definition of resistance to include ordinary acts of self-preservation, his position is nonetheless preferable to Hilberg's and Arendt's. Their harsh accusations have not stood up to historical analysis over the past forty years. Above all, they underestimate the forms of resistance that Jews participated in, and they overestimate the possibilities of armed resistance or even noncooperation that were available to Jews, either upon initial contact with the Nazis or later.\n\nWith regard to the underestimation, Hilberg is probably right that the Germans lost no more than a few hundred men, dead and wounded, in the course of the destruction process, and that one cannot identify many incidents in which Jewish resistance appreciably slowed or impeded the killing machinery. Still, up to 25,000 Jewish fighters operated in Lithuania, Belarus, and the occupied Soviet Union, and several thousand more in the mountains of Greece and Yugoslavia. Diverse estimates describe the share of Jews among French resistance fighters as declining from 40 percent early in the occupation to 15\u201320 percent later, partly as a result of attrition, partly because more non-Jews began to take up arms. At either figure, the overrepresentation of Jews in comparison to their proportion of the French population (less than 1 percent) is striking. Jews in Charles de Gaulle's Free French Forces were six times more numerous than predicted by that figure. These are not huge numbers, but they are not nothing, either. Even so, Benjamin Ginsberg, the author of _How the Jews Defeated Hitler: Exploding the Myth of Jewish Passivity in the Face of Nazism_ , apparently knows that such statistics are not enough to make his case. He includes Jews in the American, British, and Soviet armies and intelligence services among his Jewish resisters, which is rather like moving the goalposts, since these people were not subject to anything like the same constraints, were not generally acting as Jews but rather as parts of national war efforts, and were not necessarily volunteers.\n\nTo understand the degree of overestimation, one has to begin by looking at how the Germans proceeded against the Jews in occupied Eastern Europe, where the bulk of the killing occurred. In the first place, the Nazis applied in remarkably short order all the lessons they had learned from the persecution of the German and Austrian Jews in prior years. Even before the conquest of Poland was complete, on September 21, 1939, Chief of the German Security Police Reinhard Heydrich directed his subordinate offices in occupied Poland to enforce \"the concentration of the Jews from the countryside into the larger cities. . . . which either are railroad junctions or at least lie on railroad lines.\" So far as possible, responsibility for the implementation of not only this policy but also all subsequent German orders regarding the new areas of residence was to be imposed upon Jewish Councils of Elders ( _Judenr\u00e4te der \u00c4ltesten_ ), modeled on the body that the Nazis had created in Vienna in 1938 and composed of community leaders. Six days later, as we have seen, Heinrich Himmler, acting as head of the SS and of the German police, created the Reich Security Main Office under Heydrich and entrusted him with overall responsibility for the Jewish question. Within this organization, Adolf Eichmann, who recently had supervised the fleecing and expatriation of thousands of Austrian Jews, assumed control of the \"Jews Department\" that was to handle the logistics of the incipient ghettoization.\n\nIn short, less than one month after invading Poland and adding approximately two million Jews to its realm, the Nazi regime had devised a system of segregating these people from the surrounding population, positioning them for swift roundup later, stripping them of all their immovable and most of their movable property in the process, and turning the leaders of their communities into the executors of German policy.\n\nThis last feature of German policy, the assignment of responsibility for carrying out German instructions to Jewish Councils, was a diabolically effective means of minimizing the resources Germany would have to use to police the Jews and making them complicit in their own persecution. In effect, the Nazis applied the tried-and-true colonial practice of indirect rule through favored natives who got privileges or exemptions from punishments in exchange for helping to control everyone else. This tactic of dividing and conquering turned out to be almost impossible to resist, because it was coupled with force. When the first Jewish Councils were appointed in towns and villages, often before ghettoization had begun, those who declined to carry out distasteful or cruel German orders were simply shot on the spot; sometimes the first round of council members was shot for no reason other than to intimidate the members of the second round. In Lodz, for instance, twenty-two of the first thirty council members were killed to set an example. Serving on the councils and executing German orders were the first iterations of the \"choiceless choices\" (Lawrence Langer) with which the German occupation repeatedly confronted Jews. The appointees could refuse and die now or consent and perhaps die later or not at all. Almost everywhere, the designated members of the councils, like the inhabitants of the ghettos as a whole, chose the latter and played for time.\n\nAlthough the Germans conceived of the ghettoization program rapidly, they translated it into practice in Poland unevenly and haltingly. Events moved fastest in the regions annexed to the Reich, namely eastern Upper Silesia, West Prussia, and the Warthegau. The last-named included the city of Lodz, which became, on May 1, 1940, the site of the first large-scale ghetto to be sealed off from the outer world. Initially with 163,177 residents crammed into 2.4 square miles of a slum district that mostly lacked indoor plumbing and sewers, the Lodz ghetto became, under Hans Biebow, its German administrator, and Chaim Rumkowski, its Eldest of the Jews, the most self-sustaining and the longest-lasting of the Jewish population centers, even though it was now officially within Germany. In contrast, Warsaw's ghetto in the General Government was not closed off until November 1940, and its staggering congestion\u2014by March 1941, more than 460,000 Jews were confined to less than one square mile\u2014made it less manageable and more lethal. Further to the east, in Lublin, the gates did not shut on some 40,000 Jews until April 1941, and the permeability of the ghetto boundaries remained much greater. This was also the case at many of the smaller sites and even at Czestochowa, the second largest ghetto in the General Government. In fact, in the many villages of the largely rural Lublin district of the GG, a majority of the Jews were still in their own homes in 1942.\n\nThe variability of ghettoization had a number of causes. The chaotic and contradictory nature of German population policy in Poland slowed implementation, as did shortages of personnel and transportation. German administrators squabbled about whether and how ghetto inmates were to be kept alive and thus about how much trade they could do with their environs and with the occupying regime. Among the SS men put in charge of the ghettos, the group scholars call \"attritionists\" thought that the inhabitants should just be allowed to die off, whereas the so-called productionists encouraged economic activities by which the ghettos could earn their keep. Above all, no German planner seemed to know how long the ghettos were to last and where the denizens someday were to be sent. Nazi leaders spoke consistently of eventually consigning the Jews to a \"reservation\" but kept changing its location. With each change of venue came a deferral of deportation and thus, among the German occupiers, a declining sense of urgency about completing the ghetto system\u2014but also declining patience with its existence.\n\nAmong the Jewish inhabitants, the result was the opposite: The longer the ghettos lasted, the more the illusion of their permanence developed, and people settled into the hope that they could create sustaining institutions that could preserve at least some share of the population. By early January 1942, before the liquidations began, the populations of the Lodz and Warsaw ghettos were about the same size as when their gates closed in May and November of 1940, respectively. As figure 5 indicates regarding Lodz, new arrivals had offset thousands of deaths from starvation and cold, especially from the three predominant diseases in the ghettos: tuberculosis stoked by hunger and dank conditions; typhoid caused by contaminated food or water; and typhus or spotted fever, spread by the ubiquitous lice. Although conditions were wretched, the possibility of sustenance seemed real, at least for those who had money, jobs, or positions in the ghetto administration. Such prospects became weapons in the hands of the Germans, however. Subjected to ever-mounting scarcity, Jews were pitted against each other in the struggle for food, clothing, shelter, and sheer survival, and their ability to sustain one another both materially and morally eroded steadily. On May 30, 1942, Dawid Sierakowiak, an eighteen-year-old boy trapped in the Lodz ghetto, recorded one of the more extreme consequences in his diary, as he told of how his own father seized and ate both Dawid's and his mother's bread rations, and then devoured all of the family's small allotment of meat and whey. If even family ties snapped, imagine what happened to solidarity among unrelated people.\n\nFIGURE 5: THE FATE OF A GHETTO: LODZ, 1940\u201344\n\nDATE| POPULATION| DEVELOPMENTS \n---|---|--- \nMay 1, 1940| 163,177| Ghetto sealed; in an area of only 2.4 square miles, the Jewish population exceeded that of either Bohemia-Moravia or the Netherlands when World War II began. \nMarch 31, 1941| 150,436| \nMay 1, 1941| 148,547| On October 9, the daily mortality rate dropped to its lowest point to date: 11. From 16 October to 3 November, 21 transports arrived, bringing 19,883 Jews from Germany, Austria, and Bohemia. \nJanuary 1, 1942| 162,681| \"Resettlement\" began on 16 January; 55,000 people deported by the end of May. \nJune 1, 1942| 104,469| More than 15,000 people sent to Chelmno in the second large round up, September 1942. \nJanuary 19, 1943| 87,164| \nJuly 1, 1943| 84,495| \nFebruary 8, 1944| 79,777| \nJuly 1, 1944| 73,217| Liquidation of the ghetto began on 23 June. \nJanuary 19, 1945| 877|\n\nApproximately 45,000\u201350,000 of the people who entered the ghetto died of starvation, disease, brutality, or some combination of those causes in a period of forty months. At least 140,000 were massacred, mostly at Chelmno but in the final stage also at Auschwitz, during the nine months when deportations took place (January to May and September 1942, and June to August, 1944).\n\nInternal disunity among Jews aggravated the situation. They were even more divided in the ghettos than were the Jews in Germany during the 1930s, though along somewhat different lines. The major cleavages ran between (1) the secular, socialist, nonseparatist, primarily urban populations associated with the Bund (Alliance) Party; (2) the Zionist groups, which splintered among religious, secular, general, revisionist, and Marxist factions; (3) the traditionalist Orthodox groups; (4) the ecstatic Hasidim; and (5) a smattering of communists. All of these strains of opinion had their own institutions, networks, and longstanding difficulties in communicating with each other, and their differences did not disappear in the crucible of Nazi persecution. Indeed, as ghetto communities tried to decide on the proper response to the German onslaught, groups took divergent positions. Unlike Zionists, Bund members generally rejected service in the ghetto administrations, but also discouraged attempts at overt or armed resistance, unless these occurred in partnership with gentile groups outside the ghettos. Generational differences overlaid the political and religious ones, with the various forms of Zionism, especially the more militant ones, gathering a growing following among younger ghetto inhabitants.\n\nFinally, class and regional conflicts arose: Working-class Jews tended to resent the dominance of middle-class and elite professionals in certain councils; people who had been sent to ghettos from elsewhere sometimes felt disadvantaged by the original residents, especially in obtaining favorable work assignments. Professionals felt a loss of status unless they could obtain positions in the ghetto administrations, while a frequently ostentatious clique of nouveau riche smugglers and traders sprang up and aroused envy. Even before the deportations began, gradations in wealth and status frequently made the difference between survival and starvation. According to Mordechai Lensky, a physician who survived the Warsaw ghetto by escaping in the nick of time with his family to the \"Aryan side\" of the city, \"When deportations started in July 1942 . . . the community's social structure disintegrated [and] the upper economic and social classes sacrificed the lower classes to save themselves.\"\n\nInternal competition to survive was perhaps the strongest impediment to organized resistance within the ghettos, but it was not the only one. Just as the Germans staggered or spaced out the formation of ghettos and thus prevented Polish Jews from grasping exactly what was happening to them in 1940, so were the liquidations of the ghettos done in such a fashion during 1942\u201343 that word of them spread slowly, and Jews could not immediately recognize that wholesale massacre was unfolding. Deportations from Lodz and Lublin, at opposite corners of German-dominated Poland, began in early 1942, but Warsaw's turn did not come until the summer of that year, and Bialystok's not until February 1943. There followed the liquidation of Cracow's ghetto in March, Lviv's in June, Minsk's and Vilna's in September, Riga's in November 1943, and finally the ghettos of Kovno and Lodz in the summer of 1944. Even when inhabitants of one place got wind of murderous events in another, people could cling to the hope that their fate would be different and continue to play for time. The same delusion operated even within ghettos once the liquidations began, because they were carried out in phases, as figure 5 (above) shows in the case of Lodz.\n\nThe urge to grasp at straws of hope was powerful within the ghettos because mass murder seemed not just unimaginable but downright irrational. Why would the Germans kill people who could be useful, especially in those ghettos that set up productive workshops and factories? This conviction that the Germans would not act against their own interests has much to do with the remarkable refusal of ghetto residents in both Lodz and Bialystok to believe that deportees were being killed, even after the trains that took them away returned to the ghettos with the clothing and personal identification cards of inhabitants who had left only recently. Calel Perechodnik, who served for a time as a Jewish ghetto policeman and later briefly as a resistance fighter, left behind at his death in 1944 a remarkable testament that includes this vivid passage conveying the extent of Jews' denial in his little town in central Poland at the beginning of 1942:\n\nThey say . . . that in Slonim they gathered in the town square fourteen thousand people\u2014women, children, men\u2014and all were machine-gunned.\n\nI ask you . . . is it possible to believe such a thing? To shoot without reason women, innocent children just like that in full daylight? After all, even the worst female criminal cannot be sentenced to death if she is pregnant\u2014and here they apparently killed small children. Where can you find people, fathers of families, who would have the courage to aim their machine guns at helpless, small children? Where is the opinion of the cultured world? . . . How can the world remain silent? It is probably not true.\n\nFollowing this news comes another one, even more monstrous: In Wilno [Vilna], they killed sixty thousand people; in Baranowicze, twenty thousand. People stop understanding this; in truth, they believe it, but they can't visualize that one day someone can come to murder my two-year-old daughter, who scarcely can talk yet, only because she was born to a Jewish mother and Jewish father.\n\nFinally, we hit on an explanation. Those Jews were killed because they were Soviet citizens and probably because they fought against the Germans. But we are citizens of the _General Gouvernement_ ; such a thing cannot happen here. Moreover, there is martial law there, whereas here we have a civil administration.\n\nIn the space of these four paragraphs, disbelief turns into discounting, as people desperately sought and found ways to immunize themselves to the rumors that flew.\n\nWhat else stood in the way of organized resistance to the deportations, once they began? First, the Nazis went to great lengths to camouflage what they were doing. Sometimes they exempted the ill and hospitalized ghetto residents from deportations in order to imply that the people being shipped out really were going to work camps further east; sometimes they made a great show of exchanging local or ghetto currency for other forms of money before people got on the trains; and sometimes the Germans even sent postcards back to the points of departure, supposedly from recent deportees, to reassure those left behind about the destinations. This was a particularly common practice with regard to deportees from Western Europe; most of the cards were postmarked from Leipzig in Germany, but some even came from Auschwitz and Birkenau.\n\nSecond, the Nazis mixed the carrot and the stick, bait and threats, to assure compliance with deportation orders. Soup, bread, and jam were offered at assembly points and prospective deportees were told they could have certain privileges with regard to baggage and rations if they showed up when ordered to, but that these would be taken away if they failed to appear. Conversely, Adam Czerniakow, the Jewish Council head in Warsaw, was told that his wife would be shot if he impeded deportations in any way; Joseph Parnes, a Jewish leader in Lvov, was killed when he declined to designate people for deportation to a labor camp. In Amsterdam, the Nazis told the Jewish Council that failure to cooperate in the deportations \"for labor in the East\" would result in shipment to concentrations camps such as Mauthausen instead, which initially sounded much worse.\n\nThird, the delegation of the dirty work to the Jewish Councils gave them an illusion of some control over what was happening and saddled them with responsibility to minimize the damage. In most ghettos, just as was the practice in Amsterdam and the Westerbork transit camp, the Nazis simply told the Jewish Council how many people to assemble for deportation on a given day and left the choice of the people, until the very end of most liquidations, to the council. In Warsaw, the SS demanded delivery of the first 6,000 people on July 22, 1942, and the same number by 4:00 every subsequent afternoon until further notice. We know in what sort of position this put the Jewish authorities in the Lodz ghetto because of a remarkable record of a visit by a German named Friedrich Hielscher in the spring of 1942. He talked with the head of the Jewish police, Leon Rosenblatt, who admitted to knowing that the deportees were being sent to be gassed and then spoke as follows:\n\nI have to choose people for this. If not, I will be shot. That for me would be a simple solution. What will they do then? The SS already told me. Then _they_ will choose. That is, the strong ones, the pregnant women, the Rabbis, the learned ones, the professors, the poets\u2014they will be the first for the oven. But if I stay where I am, I can take the volunteers. Often they demand to be taken, and sometimes I have as many as I have to deliver, and sometimes they are few, and then I take the dying that Jewish doctors tell me about, and if these do not suffice, I take the seriously ill. If these, too, are not enough, what shall I do? I can take the criminals. . . . Who will be the judge? I asked the heads of the community, the Rabbis, the learned people; all of them said: You did the right thing by staying at your post. . . . Tell me\u2014should I remain at my post, or should I prefer to be killed?\n\nThe same logic of \"better us than them\" propelled the conduct of Abraham Asscher and David Cohen, the co-chairs of the Jewish Council in Amsterdam, when confronted in May 1943 with a German request for a list of 7,000 council employees (about 40 percent of the total) who were to be deported next. Ignoring the pleas of colleagues to destroy their central card registry of all remaining Jews rather than comply, the council leaders and some staff members worked frantically for two straight days and nights to designate and provide the names.\n\nRosenblatt, Asscher, and Cohen may have thought that they were adhering to Jewish religious law, but they were not. According to David Daube's careful examination of pertinent passages of the Talmud, handing over a person specifically demanded by name by an oppressive power is permissible as compliance with a threatening order, but handing over \"simply any odd person for execution\" is not because that involves choosing the victim and thus amounts to taking on personal guilt.\n\nThe most extreme example of the excruciating position in which German procedures put the councils occurred on September 4, 1942, when Chaim Rumkowski addressed the assembled ghetto population in Lodz:\n\nA severe blow has befallen the ghetto. They are asking from it the best it possesses\u2014the children and old people. . . . I never imagined that my own hands would have to deliver the sacrifice to the altar. In my old age, I must stretch out my hands and beg: Brothers and sisters, give them to me! Fathers and mothers, give me your children! . . . Yesterday during the day, I was given a command to send twenty-odd thousand Jews out from the ghetto; if not\u2014\"We will do it.\" And the question arose: \"Should we take it over and do it ourselves, or leave it for others to carry out?\" But being dominated not by the thought \"How many will be lost,\" but by the thought \"How many can be saved,\" we, i.e., I and my closest co-workers, came to the conclusion that as difficult as this will be for us, we must take into our own hands the carrying out of the decree. I have to carry out this difficult and bloody operation. I must cut off limbs in order to save the body! I must take the children because, if not, others could also, God forbid, be taken. . . . One needs the heart of a bandit in order to ask for what I am asking of you. But put yourself in my position and think logically, and you yourself will come to the conclusion that you cannot act differently because the number of the portion that can be saved is much larger than the part that must be surrendered.\n\nThis passage illustrates how perfectly and diabolically the system of divide and conquer worked, and so do the lists of people excepted from the initial deportation orders and from all others prior to the very end. Most of these were people performing valuable labor functions for the Germans, but two other groups were conspicuous among those exempted: people who worked for the Jewish Councils in the administration of the ghettos, of whom there were almost 13,000 in Lodz and 6,000 in Warsaw, and members of the Jewish Order Service, the police force in the ghettos, who totaled 800 men in Lodz and 2,000 in Warsaw. Some of the police were recruits from the prewar Jewish communities of these cities, but most had arrived from elsewhere and thus had few local ties to inhibit them. Joseph Szerynski, the head of the Jewish police in Warsaw, had converted to Catholicism, did not consider himself a Jew, and had no bonds with the local community. He and his men were so hated that the underground in the ghetto wounded him badly in an attempted assassination and succeeded in killing his successor. Paid little or nothing, ghetto policemen became increasingly corrupt and extortionist, demanding bribes to keep people off the lists for forced labor or deportation or routinely seizing attractive possessions. When the final stages of a liquidation arrived, and the Germans began instructing the councils not to select the deportees by name but rather to round them up from particular portions of a ghetto that were being cleared, these police did the footwork in most cases. They did so on the calculation that they and their families would remain alive as long as they were useful, and the Germans sometimes deceptively promised that they would survive even longer. The same motives account for the cooperation of the Jewish _Ordedienst_ from the Westerbork camp in rounding up Jews in Apeldoorn and Amsterdam in 1943 and in loading and sealing the deportation trains from Holland in 1942\u201344. And the _Ordedienst_ men shared the distancing social profile of their counterparts in the Polish ghettos: About half of them, including their commander, were not Dutch Jews but German or Austrian Jewish refugees in the Netherlands.\n\nA fourth impediment to resistance was the weakened condition of the ghetto inhabitants, which is something that cinematic depictions of the Holocaust generally cannot convey. Usually, as in Lodz and Kovno, the Germans sited the ghettos in the most miserable parts of a city, without sewers or much running water. Daily food intake for most ghetto inhabitants, aside from the privileged ones who worked in the administration or war production, hovered between 400 to 1,000 calories per day; in the largest ghettos, it usually averaged far less; in Warsaw in 1941, the daily allocation per person was between 180 and 220 calories. In Perechodnik's ghetto at Otwock, most of the 14,000 Jews barely subsisted in early 1941 on a weekly ration of 1.5 pounds of bread per person; there was no allotment of meat, eggs, or vegetables. Hunger, rampant disease, cold, filth, overcrowding, and enervation all took terrible tolls and undermined any desire to fight back. By July 1941, 5,550 people were dying in the Warsaw ghetto per month, almost 200 per day. One hundred thousand Jews died in the ghetto between its inception and the onset of the great deportations in mid-1942. The Germans designed the ghettos to confirm the picture of Jewish degradation, dirtiness, and disease that Nazi ideology posited; they were in this sense the fulfillment of an ideological prophecy, and cramming people together was part of the plan. In Warsaw, for instance, the ghetto inhabitants outnumbered the available rooms 7 to 1, and the population density worked out to 200,000 people per square kilometer in April 1941; in Kovno, 30,000 people lived where 7,000 had previously, in Vilna, 29,000 where 4,000 had. Under such conditions, people often lost the ability to think ahead, and those ghetto inhabitants who still could do so generally calculated that endurance was preferable to resistance or flight because of concern about family members and dependents who would be put at risk. Fleeing from the ghettos was also not an appealing prospect for most people because the topography of Poland was not conducive to hiding, and neither were the prevailing attitudes among the surrounding non-Jewish population.\n\nFifth and finally, the viciousness of the reprisals the Germans took was a powerful deterrent to resistance. These occurred not only in the form of individual beatings and shootings but also at the collective level. After all, putting aside the heroism of the people who launched the Warsaw Ghetto Uprising, the military balance sheet of trying to fight the Germans was catastrophic. In the course of suppressing the uprising, the Germans and their auxiliaries suffered somewhere between 110 casualties (seventeen dead and ninety-three wounded, according to the official figures) and three times that many (according to the resisters). Either way, the figure is tiny compared to the 56,065 people the Germans captured or killed. A few weeks later, the poorly prepared, more spontaneous resistance against the German drive to empty the Bialystok ghetto resulted in exactly nine German soldiers wounded, compared to the deaths of about 30,000 Jews either in the fighting or as a result of deportation to Auschwitz and Majdanek. Also in August 1943, the attempted breakout at Treblinka enabled only fifty to seventy inmates to survive the war. The camp uprising at Sobibor that October led to the deaths of only eleven or twelve SS men and two _Hiwis_ and the survival of only forty-seven inmates out of the 650 or so present when the fighting began. Moreover, the price of these events was Operation Harvest Festival in the fall of 1943, when Himmler ordered the liquidation of nearly all the remaining Polish work camps and ghettos in reprisal for these acts of Jewish resistance and as a sure method of preventing more of them. The shooting on November 3\u20134, 1943, of 42,000 Jews in the Lublin district, most of them at Majdanek and nearby Poniatowa, constitutes the largest single massacre of the Holocaust. At Poniatowa, one men's barracks tried to resist; the Germans locked the doors from outside and set it on fire, incinerating everyone inside. The only known survivors of those two days of killing were three women who had been left for dead in a mass grave. Only lightly wounded, they crawled away under cover of darkness, received succor from a Polish woman, and lived to see the end of the war.\n\nGiven all these circumstances, about the only effective form of resistance the Jews in ghettos could exercise in the short run was to defeat the Nazi effort to starve them to death. The principal way of doing this was through smuggling, and both the Warsaw ghetto administration and individual informal networks developed that into a fine art, which undoubtedly prolonged the lives of many people. Smuggling could not stop deportations once they began, though it sometimes became a means of helping individuals escape the trains. Otherwise, the only other form of resistance that offered a prospect of success was itself an expression of hopelessness. This was the attempt in almost all ghettos to leave a record of the Nazi crimes and a proof that the Jews had existed and struggled to survive. Both at Warsaw and Lodz, networks of people collected and hid extensive archival records and compiled a chronicle of the major events in the ghetto's history. Emanuel Ringelblum, a historian, and an organization called Oyneg Shabes, independent of the ghetto administration, did this work in Warsaw and buried the resulting documents under basements in the ghetto, where most of them were recovered after the war. At Lodz, the chronicle was the work of historians in an official ghetto archive that the Jewish Council established, and about two-thirds of these records were unearthed under various ghetto buildings after 1945. Along with a number of surviving individual diaries, these are our primary sources of information on the internal conditions of the ghettos during the Holocaust.\n\nPerhaps the most conclusive demonstration of Jews' limited ability to affect their own fates, whether they chose to resist or not, is that several Jewish ghetto administrations adopted different survival strategies, different mixes of compliance and resistance, but regardless of what they chose, they ultimately came to the same end. In Warsaw, Adam Czerniakow, the head of the Jewish Council, followed a strategy of placating the Germans until he realized that they intended to kill all the ghetto inhabitants, so he committed suicide in July 1942 rather than cooperate further. His act did nothing to head off the massacres of most of Warsaw's Jews that summer and of the last remnant in May and June of 1943. In Vilna, Jacob Gens, the head of the Jewish Council, tried to walk both sides of the street, providing the Germans labor and cooperation but also aiding the resistance in and around the city. The Nazis nevertheless liquidated the Vilna ghetto without interference from that resistance in September 1943. In Minsk, the two ghetto leaders Eliyahu Mushkin and Moshe Yaffe did not defy the Germans, but they were among the Jewish Council leaders who were most supportive of armed resistance, perhaps because some 10,000 Jews were actively fighting back in the nearby forests of Belarus. Still, that availed them little as the ghetto's population dropped from 100,000, in October 1941, to 12,000, in August 1942. By then both men were dead, too, and what was left of the ghetto was liquidated in October 1943. Finally, Chaim Rumkowski, the Jewish Council leader in Lodz, was the most persistent proponent of satisfying the Germans' every whim as the only way to survive. The strategy probably helped his ghetto endure longer than any other one, but it did not prevent the liquidation of its last 70,000 inhabitants in August 1944. In short, whatever the Jewish leaders did\u2014kill themselves, aid the resistance, appease the Nazis\u2014the outcome was the same. Historian David Silberklang's judgment, on the basis of studying hundreds of ghettos in the Lublin district of the General Government, holds throughout Eastern Europe and probably throughout the continent as a whole: \"No Jewish action caused any significant difference for large groups of people in terms of survival, though certain actions could make a difference for individuals.\"\n\nThe Jews had almost no control over their collective fate. Individuals could flee to the forests in some cases and try to survive, but whole communities could not. Neither could they devise a strategy that could accomplish anything more than to delay their deaths. It is unfair and inaccurate to hold the Jewish victims responsible for what happened to them. Whether they lived or died depended on two things alone: the actions of the Nazi regime and the progress of the Allied armies.\n\nTwo incidents\u2014one of resistance and one of compliance\u2014demonstrate this emphatically. The first, already discussed, is the Warsaw Ghetto Uprising of 1943, which failed utterly because it could not withstand the armed might the Nazis could bring to bear. The second illustration of the dependence of the Jews on the progress of the Allies is the fate of the Lodz ghetto. When its liquidation began, Soviet forces stood only seventy-five miles away to the east, where they had stopped their offensive at Warsaw in order to regroup and resupply their forces and to allow the Nazis to crush a rising that Polish nationalists launched in the city as the Red Army approached. Stalin planned to install a communist government in Poland after the war, and he thought that the suppression of this rising by the Germans would make that goal easier to achieve. Had he not made this cynical calculation and instead quickly resumed his military advance, the Germans probably would not have had time to liquidate the remaining Jewish population in Lodz, and Rumkowski's gamble of exchanging Jewish compliance with Nazi wishes for long-term survival might have paid off. Of course, the Nazis might have marched the survivors west, and many would have died in the process, which is what happened in July 1944 to the inhabitants of Kovno, the only other remaining ghetto of significant size. But the number of people to emerge from the Lodz ghetto alive at the end of the war probably would have been larger than turned out to be the case. Throughout the Holocaust, the Jews were at the mercy of the decisions of others.\n\nLeo Baeck, the rabbi and leader of the German Jews who ultimately survived the war in Theresienstadt, knew this, and his knowledge informed his most controversial decision. Even after he learned that most Jews were being gassed or shot, he refused to admit what he knew to the people around him and persisted in concealing their likely fates. Why? Because he believed that \"living in the expectation of death . . . would . . . be . . . harder\" than living with an illusion. His position may have been highly humane, but it was also debilitating. As the late dates of the ghetto and camp uprisings show, hope of surviving was the enemy of fighting back. So long as hope remained, people generally chose not to charge the German guns and urged their fellow Jews not to do so lest that provoke reprisals. Everywhere, Jews took arms only when they knew the alternative already was certain death.\n\nIn May and June of 1944, Hungary's Jewish leaders, principally Samu Stern, the head of the national Jewish Council that had come into existence when Germans occupied the country that March, similarly learned of what was happening at Auschwitz and chose to keep the information to themselves as the deportations from the Hungarian countryside began. Stern believed he was \"running a race against time,\" in which his job was to keep at least some Jews alive until the progress of Germany's enemies cut off the transports. For Jews to survive that long, he insisted they had to obey the authorities, and to make sure that Jews did, he withheld from them his knowledge of the fate that awaited those who boarded the trains. The result was the almost complete and almost completely unresisted annihilation of Hungarian Jewry, as the Nazis deported faster than the Russians advanced.\n\nIn assessing Jewish resistance to the Holocaust, some comparisons are instructive. Did any other persecuted group act more forcefully as the Holocaust proceeded? Consider the behavior of the 5.7 million captured Soviet prisoners of war, 3.3 million of whom died in German captivity, a mortality rate of 58 percent. Confined to prison camps or labor brigades, the survivors staged no important uprising until the very end of the war, even though they, unlike the Jews in the ghettos and occupied states, consisted almost entirely of young men with military training. Or consider the behavior of the occupied European peoples, among whom resistance movements generally became significant only after Stalingrad, by which time most of the European Jews already were dead. Even in 1943\u201344, according to the most authoritative estimate, only 2 percent of the French population consisted of active resisters. Finally, in assessing the conduct of the Jewish Councils, bear in mind that the Dutch civil servants who remained at the head of ministries in the Netherlands protested the deportations and threatened to resign if they continued but never could summon up the courage actually to do so. Instead, Dutch police, along with Dutch transport and railway workers, frequently helped the Germans carry out their plans.\n\nWhere resistance movements flourished, they generally had four advantages: favorable topography in mountainous and\/or heavily forested places like Yugoslavia and central France; sympathetic local populations; trained military veterans of the sort that the Germans usually killed upon arrival in the East; and equipment supplied by the Allies. The Jews of the ghettos, especially those in Poland and Ukraine, lacked all of these things.\n\nIn sum, why didn't more of them fight back more often? Because the odds were stacked against them, because they could not see or could not bear to see what was going to happen to them, because the slimmest chance that some might survive tempted them to avoid committing suicide by fighting back, and because they clung to life as best they could in ever more adverse circumstances. We have no right to expect or demand that they should have behaved more forcefully or heroically. They were, in the end, subjected to excruciating torture and confronted with \"choiceless choices\" in which all alternative courses of action seemed to present more danger than relief.\n\nLet me drive home this point about our obligation to withhold judgment with reference to a pair of iconic photographs that are reproduced in figure 6, the first of the little boy wearing a cloth cap and holding his hands up after being captured in the Warsaw ghetto, and the second of a group of Warsaw ghetto inhabitants being marched away by the Germans, with a young girl in the front row on the right. German photographers took both pictures, and they surfaced after the war in a few extant copies of an album made to commemorate the suppression of the uprising and later published under the title _The Stroop Report_ , using the last name of the German commander of that operation. Often overlooked because of the poignancy of these pictures is the most remarkable feature about them: that each shows the presence of a child under the age of ten as the ghetto was being liquidated. In fact, the first picture shows three or four more young children in the background. Yet the population of the ghetto had fallen from almost 460,000 at its peak in March 1941 to about 53,000 just before the Ghetto Uprising broke out in the spring of 1943; the corresponding figures for young children are from approximately 51,000 to fewer than 500, 255 boys and 243 girls. Moreover, the uprising occurred seven months after Rumkowski told the Jews of Lodz that they had to give up their children under the age of ten. The only children spared in Lodz were those of the Jewish policemen and firemen who helped round up all the others and those of the Jewish ghetto administrators. Almost certainly, the same was true in Warsaw, and the children you see in those two photographs are the offspring of people in or well connected to the Jewish Warsaw ghetto administration, people with enough clout there to shield their children from deportation, people who at the same time had benefited from the German exemptions and perhaps had helped organize the deportation of others, people who probably had argued against resistance so long as their positions gave them a chance of survival. We actually know who the girl was, unlike the boy in the first photo. Her last name was Neyer, and she is walking beside (from left to right) her mother, Yehudit, her paternal grandmother, and her father, Avraham, who was a member of the Bund Party and the only person in this family who survived the war. Now that you have the backstory of these children and what their parents may or may not have done, do you feel any less sympathy for any of them than you did when you first saw their images? I hope not.\n\nFIGURE 6: TWO PHOTOS FROM _T HE STROOP REPORT_\n\nCredit: United States Holocaust Memorial Museum\n\nImmediately after the Holocaust, its survivors had trouble understanding this point, not least because their pain was so fresh and their desire to imagine different outcomes so intense. Ad hoc Jewish Courts of Honor sprang up to unmask and punish alleged Jewish \"collaborators,\" especially former so-called _Kapos_ (who led work details) and members of Jewish councils or police forces. One such body in Italy convicted two former Jewish Council members in Lviv and Bedzin and banned them from \"any position in the public life of the Jews\"; a similar court in the American occupation zone of Germany handed down an identical judgment and penalty for a former Jewish Council member in Upper Silesia. In Amsterdam, another such court reached the same decision and imposed the same punishment in 1947 on the former co-chairs of the Jewish Council there, Abraham Asscher, who had survived Bergen-Belsen, and David Cohen, who outlived Theresienstadt. But a vote of the Permanent Committee of the Netherlands Israelite Church Organization vacated that decision three years later. Asscher died shortly thereafter, completely alienated from the Jewish community. Cohen continued defending his conduct until his demise in 1967, never abandoning the implausible claim that he first learned of the death camps after he got to Theresienstadt and thus did not know until then that the people on the deportation lists he compiled were almost certain to die.\n\nThe case of Rezso or Rudolf Kastner in postwar Israel was even more divisive, and its outcome more violent. A Zionist official in wartime Budapest who had helped Jews from elsewhere find refuge in Hungary, he knew by May 1944 that the Jews about to be deported from that country were likely to die, but he undertook no effort to warn them and instead began negotiating with Eichmann to let a limited number of them escape in return for cash payments. He thus managed to save 1,625 people, including several hundred from his hometown, his mother, wife, and siblings, and many dedicated Zionists. Because he was a parliamentary candidate of the governing Labor Party in Israel and the press spokesman for a government ministry, his conduct became an issue in Israeli partisan and identity politics. In 1952, an elderly journalist named Malchiel Gruenwald published a pamphlet attacking Kastner as a collaborator with the Nazis who had saved some of his family and friends in return for allowing the Hungarian Jews to cling to a false sense of security as they boarded the trains going northward. The Israeli attorney general insisted on suing Gruenwald for libel on Kastner's behalf, only to have the presiding judge conclude that Kastner had \"sold his soul to the devil . . . by deliberately avoiding his duty . . . to reveal to the Jews the fate awaiting them.\" The Israeli Supreme Court overruled that verdict by a vote of 4\u20131 in January 1958, on the grounds that Kastner's \"thoughts were directed to good and not to evil, to rescue and not to extermination.\" But the vindication was no help to Kastner, who had been assassinated ten months earlier.\n\nThe few surviving Jewish ghetto or council leaders who fell into the hands of the Soviet Union also faced summary justice. Moshe Kopelman, the Kovno ghetto Jewish police chief from 1941 to 1943, managed to escape as the ghetto was liquidated in July 1944. Captured two months later by the Red Army, he was tried for collaboration and sentenced to fifteen years of hard labor despite a plea for clemency from more than seventy other Kovno survivors. Almost exactly a year later, he died in a Siberian camp. The Soviets condemned Walter Lustig, the last leader of the remnant Reichsvereinigung in Berlin, as a collaborator and executed him in December 1945.\n\nAlthough such sentences came readily to Soviet judges, David Ben-Gurion, the first prime minister of Israel, struck a wiser note in two letters he wrote, one right after the final court verdict on Kastner, one almost five years later. He said, \"I would not take it upon myself to judge any Jew who was there. The Jews who lived in safety during the time of Hitler cannot judge their brothers who were burned and slaughtered or those who were saved. . . . The tragedy is deeper than the abyss, and the members of our generation who did not taste that hell would do best (in my modest opinion) to remain sorrowfully and humbly silent.\" As time passed, most Israelis came to accept Ben-Gurion's point. Although the Knesset, Israel's parliament, had enacted the Nazis and Nazi Collaborators (Punishment) Law in 1950, the last prosecution of a Jew for violating it occurred in 1964.\n\nTHE WORLD OF THE CAMPS\n\nOne cannot write about the Holocaust or about compliance and resistance without discussing the concentration camp system, but it is a confusing, dismal, and often inaccessible subject. It is probably also the aspect of the Holocaust about which people have the most misleading images in their heads, not least because most films dare not represent a reality that was so repellent, so the great majority are distortions. Probably the greatest offender of this sort is the Oscar winner _Life Is Beautiful_ , but even _Schindler's List_ misrepresented the Plaszow camp for artistically symbolic reasons (in reality, Commandant G\u00f6th's villa was somewhat below, not above, most of the camp site, and he aimed his rifle up, not down, at the prisoners).\n\nA great many types of camps existed, and a few, especially Auschwitz and Majdanek, combined all the different sorts. Indeed, the sheer number of camps is staggering. It used to be said that more than 1,000 of them dotted the German landscape by 1945. But if one includes all the sites identified by the United States Holocaust Memorial Museum, which is publishing a massive, multivolume encyclopedia of camps and ghettos, the number of camps established at one time or another in Germany and occupied Europe runs to about 40,000. They were, in short, neither rare nor invisible, but in fact constant, frequent presences across the continent and within the Reich. And they were not completely closed off from their surrounding areas but penetrable in many cases, and even sometimes visited and inspected by local dignitaries.\n\nAt the core of the system were camps established for political prisoners, people who were regarded as threatening or disloyal to the Nazi regime, at first in Germany (for example, Dachau, Buchenwald, Sachsenhausen, Gross-Rosen, Flossenb\u00fcrg, and Ravensbr\u00fcck) and then in the annexed and occupied regions (for example, Mauthausen in Austria, Westerbork in the Netherlands, Natzweiler in Alsace, Theresienstadt in Bohemia, Stutthof in northern Poland). Such installations and their satellites held fewer than 22,000 people when the war began in 1939 but then metastasized during the war until their population peaked at more than 714,000 in January 1945 (of this figure, 28 percent were women). Not including Jews, about 1.65 million people passed through the camp system between 1933 and 1945; about one million died. For Jews, the survival rate was far worse; at most, about 150,000 Jewish veterans of these core camps, probably fewer, emerged alive at the end of the war out of the nearly four million Jews sent there. From the camps dedicated solely to murder, the \"death factories,\" survival rates were infinitesimal: perhaps seven people sent to Chelmno and only two sent to Belzec\u2014all of them men\u2014were still alive when World War II ended.\n\nWithin the camps, a highly stratified system of indirect rule developed in which the Nazi officers were a feared but usually distant presence, and they delegated the management of the prisoners to privileged figures among them. These prisoner functionaries often occupy much more vivid\u2014and hated\u2014places in the memories of survivors than the SS personnel. _Kapos_ led work details, and a Block Senior or Elder ruled over each barracks. The SS usually selected these people from among the inmates incarcerated for political or criminal offenses. In fact, a hierarchy of prisoner categories developed in the camps, with each group clearly designated by the color of the triangles sewn onto their uniforms or clothing, next to each prisoner's identification number. Political prisoners wore a red triangle, criminals green, so-called asocials black, homosexuals pink, Jehovah's Witnesses purple, Sinti and Roma brown, and Jews yellow, sometimes by itself, sometimes in combination with a triangle of another color.\n\nIn the context of scarce food and grueling work, competition for favors was rife and corruption endemic. The reds, greens, and blacks fought a constant struggle to control the most important trustee assignments\u2014not only as _Kapos_ and block elders but also as clerical personnel in central offices, where a prisoner could gather important intelligence, and as workers in the kitchens, where an inmate could obtain extra food. Generally, when the reds were in charge, conditions improved, especially for fellow prisoners of the same political persuasion. Hermann Langbein, a political prisoner at Dachau, Auschwitz, and Neuengamme, has left a vivid account of the jockeying for position and its consequences in _People in Auschwitz_. But no matter who the trustees were, they were almost never Jews, who were at the bottom of the social pyramid, along with gays. Here, as in the ghettos, the lowest instincts of self-preservation were encouraged by the system of constant fear and deprivation, and survival seemed often to require sacrificing others.\n\nThis is probably where this book should address the matter of \"other victims\" of the Holocaust, even though doing so involves a partial digression from the central theme of this chapter. Up until now we have talked exclusively about Jews and the people with disabilities targeted by the T4 campaign as victims of the Holocaust, even though most museums and memorials in the United States also refer to the other groups that had their own camp triangle colors, especially the Jehovah's Witnesses (called by the Germans _Bibelforscher_ , Bible researchers), Sinti and Roma people (colloquially referred to as Gypsies), and gays. Although it is true that Nazism attacked these groups, it did not do so for the same reason it attacked Jews or with the same intensity or to the same extent. The Nazis did not consider any of these groups nearly as threatening to German power as Jews supposedly were and thus did not set out to kill every one of them. The Nazis also thought that the offenses of these groups were ones of behavior, not essence, so if they changed the offending behavior, they often were spared, whereas people of Jewish descent had no such option. Thus the Third Reich attacked Jehovah's Witnesses because they were pacifists; if they recanted and agreed to serve in the army, they were welcomed, though few, if any, took advantage of this opportunity.\n\nMost Sinti and Roma were racially impure in German eyes, but not all of them, so some were killed and some allowed to live. The Nazi regime even allowed some German Gypsies to remain in the German army into 1943; ultimately the Reich appears to have deported and killed about two-thirds of them and left one-third of them alone. In most of the occupied countries, the Gypsies rounded up were the itinerate ones; people with stable and continuous residences were not molested. Deportation rates of Sinti and Roma from Western Europe were thus not very high. Further east, inconsistency prevailed, but the proportion of the Gypsy population murdered was much lower than that of the Jewish population. Some 5,000\u20137,000 out of almost 12,000 Gypsies identified in the German-controlled Protectorate of Bohemia and Moravia by 1943 were put in camps and killed; in occupied and annexed Poland, the death rate appears to have been 8,000 out of 28,000; in occupied Serbia, 20,000 of 150,000; in Hungary, perhaps 30,000 out of 300,000. The most murderous place was the occupied Soviet Union, but even here policy varied by both time and place. Nearly all Gypsies died in the Crimea, but sedentary ones tended to survive further north, and Muslim Gypsies sometimes were treated differently than others, as was the case in Croatia. Two-thirds of Lithuania's Gypsies survived the German occupation, but virtually none of Latvia's or Estonia's did. As an indication of the arbitrary and capricious nature of Nazi policy toward Gypsies, consider the example of the six Roma deported from the border region between the Warthegau and the General Government in 1940 and sent to a labor camp at Belzec. Instead of being put to work and later gassed at the death camp that arose next door the following year, these Gypsies were released with a warning that they would be arrested again if ever found within Germany's borders without official permission. They lived not only to return to Germany after the war but also to petition a state government for restitution of lost property. Overall, then, the Nazis murdered Gypsies, including 20,000 of them sent to Auschwitz, but not systematically. Estimates of the death toll for all of Nazi Europe run from 200,000 to 500,000, but historians are not sure of the total population in 1939. The proportion that perished, however, was certainly well below the two-thirds figure for Europe's Jews, probably between one-fifth and one-quarter.\n\nThe treatment of gays was also far less harsh and sweeping as a rule than the treatment of Jews. In the first place, the Nazi regime cared almost exclusively about gay German men and their same-sex partners. The number of people prosecuted in occupied countries was tiny. In the Netherlands from 1940 to 1943, for example, only 138 court cases occurred, and those resulted in 90 convictions. The German legal prohibition on same-sex acts, Paragraph 175 of the Criminal Code, did not extend to women, so relatively few lesbians attracted the regime's notice and hostility. And, outside of Germany, the Nazis reacted positively when a client regime issued new regulations that criminalized gay male sex, as Vichy France did in 1942, but Germany did not pressure governments to do this; it did, of course, pressure them to deport Jews. Even inside Germany, persecution was uneven. The Nazi authorities estimated the number of gay males in Germany in 1933 at two million, or about 6.25 percent of the German male population of almost 32 million at the time. That population increased to almost 38 million by 1939, thanks largely to the annexations of Austria and what had been western Czechoslovakia, and the same percentage of that population works out to almost 2.4 million gay men. But the Third Reich arrested only 100,000 men under Paragraph 175 between 1933 and 1945, convicted only 50,000 of them, and sent only about 10,000 of them to camps, where 6,000 of them perished.\n\nThe gay men caught in this system suffered excruciating punishments, but they constituted a tiny portion of the target population. Why? Because the Nazis cared only about their behavior, not their nature. Himmler actually believed until at least 1943, when some documentary evidence suggests that he began to have his doubts, that most gay men were \"curable\" if given the right incentives. The goal was to eliminate their behavior among Germans through intimidation, punishment, and reeducation and, in so-called incorrigible cases of repeat offenders, castration or death. Put in contemporary phrasing, Himmler believed that most gay men could be \"scared straight.\" Killing them all was simply not necessary. It was even less necessary in the occupied countries, because the chief offense of a German gay man was non-procreation, but in the occupied countries, non-procreation of the native population was desirable. That is why the German authorities briefly toyed with the idea of decriminalizing sex among men in occupied Poland in 1939. A clear sense of what drove Nazi policy toward gay men comes across in the title of the Nazi organization founded in 1936 to conduct their persecution, the Reichszentrale zur Bek\u00e4mpfung der Homosexualit\u00e4t und Abtreibung, the Central Office for Combating Homosexuality and Abortion. Unlike the Jews, all gay men did not have to die because they were immutable enemies of the German _Volk_. Closeted gay German men could live; foreign gay men who stayed away from German civilians or military personnel were a matter of indifference to the Nazis.\n\nAll of this amounts to saying that Nazism targeted many groups, but it did not target them all in the same way. But if Jehovah's Witnesses, Gypsies, and gays did get caught up in the camp machinery, they had much in common with each other and with Jews. These groups were the most exploited, the ones treated consistently worst, and the ones with the fewest ways of bettering their situation.\n\nAnother group whose numbers in the camp system increased exponentially as time passed was Slavs, but they did not have a distinguishing color for their triangles, largely because they were usually considered \"political\" prisoners or \"asocial\" ones. Their presence has led some observers to lump them with other victims of the Holocaust, most famously in Simon Wiesenthal's formulation that the Holocaust had eleven million victims, six million Jews and five million others, mostly Slavs. But that number is fictional\u2014Soviet civilian casualties alone came to more than ten million people\u2014and not all Slavs were the same in German eyes. Nazi theory doomed some\u2014mostly Poles, Russians, and Serbs\u2014to ultimate extinction, but only over time, as German settlers multiplied and their need for native slaves in the conquered East declined to the vanishing point. Himmler's General Plan East foresaw the reduction of the Polish population by 85 percent, the Belarussian by 75 percent, the Ukrainian by 65 percent, and the Czech by 50 percent. But the Nazis considered other Slavs valuable \"racial\" allies, notably Bulgarians, Croats, Slovaks, and some Ukrainians. Hitler and Himmler even considered many Czechs and some Poles to be capable of \"Germanization\" ( _Germanisierung_ ), that is, of being turned into German speakers and racially assimilated.\n\nIn short, though the camps contained many different sorts of people, all of whom were subjected to terrible suffering, no other group was attacked as thoroughly and systematically as the Jews. And not even the population of Germany's mental institutions and sanatoria experienced a mortality rate comparable to that of Europe's Jews.\n\nIn discussing the camps, one must begin by remembering how people arrived at them: usually parched and starving, after train trips that lasted for days in stifling and overcrowded cars filled with often wailing, sometimes crazed people, many of them dying or already dead. The deportees from Polish ghettos endured all this after debilitating months of clinging to life under only slightly better conditions. Some of these people welcomed deportation as a relief, even when they feared the worst, and more or less embraced the German proverb \"better an end with horror than a horror without end.\" To put the matter bluntly, little fight was left in people who had been subjected to this sort of brutal treatment prior to and during deportation. To expect mass resistance on their part as they debarked at Auschwitz or Belzec or Treblinka is utterly to fail to imagine what they had been through.\n\nTo understand the behavior of people admitted to the camps, one needs to remember the fundamental observation by historian Michael Marrus that they were \"the most complete totalitarian structure to have been devised by man.\" Inmates were crushed beneath this structure, worn down by exhaustion, starvation, extreme heat and cold, and disease and wholly cut off from outside help. Any infraction of even the most trivial rules resulted in the application of the Nazi doctrine of collective responsibility\u2014vicious punishment of whole groups of inmates, not just those who had stepped out of line. The punishments included beatings and floggings, endless roll calls in all kinds of weather, group hangings, and two particularly gruesome acts: throwing people alive into the crematoria and in wintertime tying inmates to posts or to suspended ropes and spraying them with water, which turned to ice and froze the victims to death. Not for nothing did one survivor call Auschwitz \"a mixture of Hell and an insane asylum.\"\n\nThe inmates, like the inhabitants of the ghettos, were cowed and divided against themselves by the constant fear that any resistance to or even evasion of the Germans' mandates would provoke more suffering than was already prevalent. This has a lot to do with the hesitations that surrounded every underground organization in the camps and with their constant caution and changes of plan. For example, the inhabitants of the \"family camp\" for Czech Jews at Auschwitz, which lasted for several months, made elaborate preparations for launching an uprising at the moment when word came that gassing was imminent. But when it did come, the planners' will faltered because they feared harm to the children in a pitched battle. In the end, they went to the gas without incident.\n\nAll would-be resisters had to contend with the ubiquitous presence of spies, motivated by the prospect of extra bread or sleep or cigarettes or exemption from a selection in return for providing the German guards with information about plots. The prisoners also had to cope with their national and linguistic differences, which made coordination tense and communication difficult. And, as the Czech family camp example suggests, the Nazi guards were relatively few but vastly powerful, so highly intimidating. It is worth recalling that no camp rebellion ever really succeeded. Even at the very end of the war, on February 2, 1945, when 419 albeit much weakened Soviet POWs succeeded in breaking out of Mauthausen, the Nazi regime still hunted nearly all of them down. Only eleven were still alive when the war ended eight weeks later. We already have noted how few people survived the rebellions at Sobibor and Treblinka in 1943 and at Auschwitz in late 1944.\n\nThe only successful form of resistance in the camps was escape, although the odds were long. Only five people are known to have escaped Belzec, and the two of them who survived the war did not actually escape from the camp. Rudolf Reder got away when he was sent to a nearby town to collect building materials and left in the care of one sleepy guard while the rest went to dinner. Chaim Hirszman jumped from a train that was taking him from Belzec to Sobibor. Treblinka was more porous because its fencing was neither electric nor equipped with alarms, but only a handful of the dozens of people who got away survived for very long, either because they fled to ghettos that later were liquidated or because the Germans quickly recaptured them. The Germans surrounded Sobibor, on the other hand, with a minefield that made that camp particularly difficult to escape. At Auschwitz, the most extensively guarded of the death camps, prisoners made no fewer than 802 escape attempts, of which at least 144 succeeded. Jews, who made up half the camp's population in the second half of 1942 and a majority thereafter, accounted for only 115 of the tries (14 percent) and 4 of the known successes (3 percent). Those numbers tell a lot about the hierarchy, the gradations of treatment, and the limited role of Jews in resistance groups in the camps. At all of these sites, the reprisals for trying to escape were fierce, ranging from public beatings and hangings of recaptured prisoners or alleged planners of new attempts to the simple practice of killing ten inmates, or even every tenth remaining inmate, for each missing escapee. Such ratios might make the balance sheet of escape attempts seem problematic, except for one consideration: Escapees from the carbon monoxide death camps were virtually the only people remaining after World War II who could provide eyewitness testimony against the murderers and thus get some of them convicted and punished.\n\nWithin the camps, the Nazis could rely on three circumstances beyond firepower to retain complete control. The first was the way camp conditions were designed to strip people of their sense of dignity, indeed their sense of self, and to dehumanize them so that they became fatalistic and resigned. Everything from the insistence that inmates be addressed and identify themselves always by number, not by name, to the incessant verbal abuse by the _Kapos_ and guards, to the refusal to let people go to the latrines when in need, to the filthy and lice-ridden clothing and bedding\u2014all these things were intended to produce just such a degrading result. People so changed were called _Muselm\u00e4nner_ , which literally means Muslims, apparently because inmates who invented the term thought that Muslims were similarly accepting of all that happened to them. Once people lost the active will to live, they were useless to any potential resistance movement but also useless to the Nazis themselves, and thus destined for certain death. After the war, Hanna L\u00e9vy-Hass, who spent 1944\u201345 at Bergen-Belsen, recalled that camp life deadened people, even to their own memories. She wrote, \"We no longer even remember our own past. No matter how hard I strive to reconstruct the slightest element . . . not a single human memory comes back to me. . . . They've managed to kill in us not only our right to life in the present . . . but . . . all sense of a human life in our past. . . . I turn things over in my mind, I want to . . . and I remember absolutely nothing.\"\n\nThe second key weapon in the hands of the guards was their ability to drive inmates to exhaustion. That was the point of the long marches to and from work, the even longer hours at the labor sites, the assignments to ditch digging and industrial construction, the endless roll calls, the overcrowded bunks, and the compulsory calisthenics at the beginning or end of the workday, all coupled with malnutrition. Prisoners were made too exhausted to think, let alone to plan resistance efforts.\n\nA third instrument of the Nazi masters, some psychologists maintain, was inmates' awareness that they had been consigned to an arbitrary and negative universe through no fault of their own. This explains the powerful impact of the famous incident in which Primo Levi, a new arrival at Auschwitz, asks, \"Why?\" when faced with an act of unfathomable meanness and is told, \"Here there is no why.\" Though prisoners were right to think they did not deserve their fates, that knowledge often provoked self-pity and paralysis. Obsessing about the injustice of the situation and recognizing that the Nazis were impervious to persuasion led to despair and abandonment of the desire to survive.\n\nWho did survive, then? Broadly speaking, the late, the lucky, and the well connected. Those who entered the camps in, say, 1944 and were relatively (but not too) young had the best chance of emerging alive. So did those who drew fortunate labor assignments, such as the women sent to the Degussa subsidiary at Gleiwitz mentioned earlier in this book. And those who had allies in important trustee positions, which means that non-Jews survived more than Jews did. Non-Jews generally looked out for themselves and did little to aid Jews. Rudolf Vrba, who in April 1944 became one of the very few Jews who escaped from Auschwitz, recalled bluntly, \"the Resistance in the camp is not geared for an uprising but for the survival of the members of the Resistance.\" Even where a camp underground existed, it did little to impede the Holocaust. Auschwitz consumed 75,000 Poles, which is perhaps one-third of those sent there, but it took the lives of probably four-fifths of the Jews ever registered in the camp. If we include the unregistered Jews killed immediately upon arrival, then the mortality rate of Jews at Auschwitz was over 90 percent.\n\nOf course, we can never know the full reality of life in the camps or gauge accurately what it took to survive. In the first place, the evidence available is partial; it springs from the writings and testimony of people who did survive and thus may be skewed in some ways. What worked for them may not have worked for countless others, but we do not know how many tried the same methods and failed. In the second place, it is clear that survival was often arbitrary and purely fortuitous, a matter of having a skill the Germans desired at a certain moment, landing a particular work detail by some stroke of luck, or enjoying the favor of a key official in the camp or a pivotal _Kapo_ for some reason or as a result of a whim on that person's part. Zev Weiss survived Auschwitz, he says, because he sensed something fishy about a particular assembly call for his barracks, so he wriggled through a crack in the wall of that building, mingled with another barrack's population, and got himself registered there in place of a missing or dead prisoner, which actually was a not uncommon form of camp resistance activity. To this day, he cannot say what made him act as he did at that moment, but he is sure it saved his life, because the call to assemble led to the gassing of nearly all the population of the barracks he had fled. As G\u00f6ran Rosenberg, whose parents both survived Auschwitz, notes, \"There are no roads from Auschwitz but those of improbability.\"\n\nThe most persuasive insights we have on this subject are still those of Terrence Des Pres, in _The Survivor_ , published in 1976. Des Pres saw four key elements that determined who outlived the camps. The first was discovery of purpose\u2014bearing witness. Recording daily events helped inmates both to preserve senses of the future and of hope and to transcend the horror around them. Thinking ahead also was an act of resistance; the Nazis repeatedly mocked the inmates by saying that no one would ever know what had happened to them. Simply proceeding as if the Nazis could be proven wrong may have helped inmates maintain a will to live and a self-respect born of their defiance of anonymity.\n\nA second determinant of survival was the recognition that preserving appearances was essential. Central to this was appreciating that one purpose of the camps was to degrade people, to make them filthy and ashamed, and then to punish them for being those things. Facilities for washing were almost nonexistent, and latrines were both crude and withheld. Inmates were tortured by being denied the chance to relieve themselves except during two permitted times per day, but were fed and worked in a manner that made dysentery rampant. People either concealed their excrement in their own clothing or surreptitiously tried to deposit it in the only receptacles available, their eating dishes, and their thoughts became focused on controlling their bowels. Such an environment sapped self-respect and made daily urgencies loom so large that few people had the mental energy to contemplate overt resistance. Des Pres puts all of this under the heading of \"excremental assault,\" and he argues that those who saw through it had the best chance of living. They washed, even in filthy water. They kept their wretched wooden clogs tied securely and bound up their ragged clothes, not only to avoid selection by the SS but also to preserve a sense of self.\n\nA third key element in enduring the camp system was coping with the initial shock of arrival. Studies of prisoner mortality undertaken since Des Pres confirm his point: Those who made it through the first three months had an above-average chance of survival. If mourning or disgust did not produce a rejection of existence, a person might have time to pull together or, to put things another way, if fate saved a person long enough to recover from mourning and disgust, survival became possible. All of this was very difficult, since most people dumped into the unfamiliar environment of the camps were prone to deny its reality, to experience it as if it were a nightmare. This often proved fatal. Vigilance was the best protection; not giving in to shock became indispensable to existence. Those who managed this transition had a chance of developing the capacity to operate alertly and without illusion\u2014to take each day as it came. The difference between living and dying was sometimes between those who calculated the odds and despaired and those who thought that one chance of survival in one hundred or one thousand was good enough. Unsurprisingly, statistical and memoir-based evidence indicates that the chances of surviving the initial shock were better if one arrived in relatively favorable weather\u2014during the spring or the summer rather than in the depth of winter.\n\nDes Pres argues that a fourth determinant of surviving the camps was the discovery of what he calls a means of living simultaneously with and against the terms of existence: with them enough to avoid being snuffed out, against them enough to do the same. One or the other extreme\u2014complete abandonment to the rules or complete defiance\u2014meant death. One had to learn to operate at the margins, \"to organize,\" as camp jargon had it. One had to learn to use bribery, to smuggle, to carry out useful forms of barter, and all of these depended upon the ability to create or join little networks of prisoners who helped each other. In Auschwitz, survival networks generally involved some participant attached to the Canada detail. This was the group that sorted the goods of gassed deportees in giant warehouses along the edge of the Birkenau camp that the inmates called Canada because they imagined that land as overflowing with natural resources. Every day, these workers managed to pilfer food, clothing, and valuables despite at least three rounds of searches by the SS guards. Such thefts, along with corrupt deals with some of the guards, were the principal basis of the extensive black market within the camps, usually involving small possessions of value\u2014everything from needles and knives to tablets of sugar, saccharine, boullion, and the like.\n\nAnother key aspect of learning to live with and against the system was knowing when to lie in order to get the sort of administrative post or desirable job in a warehouse that could keep one alive. When the Germans asked arriving prisoners if they were chemists or tailors or carpenters or machinists, an inmate had to be ready to step forward whether she or he was or not\u2014virtually the only people to survive were those who eluded the hard physical work that either consumed prisoners, given the prevailing rations, or led to beatings or shootings at work sites. Still another form of with-and-against behavior was that of _Kapos_ who learned to appear vicious to prisoners in front of the SS personnel, thereby shielding them from the latter. Memoirs tell of a system of counteradministration in which prisoner trustees in offices and camp hospitals appeared to follow SS orders to the letter but found ways to conceal or change prisoners' identities or falsify diagnoses. Of course, not every prisoner in a key position helped his fellow inmates, but the system of solidarity was enforced by the knowledge that prisoners could exact revenge on toadies when the SS was not looking. A favorite tactic at Auschwitz was to push collaborators into the open latrines, where they would drown, apparently by accident. Within virtually all the camps, an intelligence system also sprang up, consisting of prisoners working as clerks in the SS offices and specialists among the prisoners who did repair work on the barracks. While seeming to serve the camp administration, they gathered information about the workings of the camp or events in the wider world and spread it around.\n\nFinally, Primo Levi and others stress that survival often depended on \"pairing\" with another inmate, looking out for each other, and simultaneously holding at bay the camp's multiple ways of crushing any sense of human solidarity.\n\nDes Pres and survivors who have given us memoirs to these effects may be right. But numerous accounts also come from survivors who say that they do not know how they got through the camps, cannot remember adopting a strategy for survival, and cannot say why they lived and others died. They recall instead a kind of endless numbness, a state of almost suspended mental animation that was broken only by liberation at the end of the war.\n\nIn short, the camp inmates developed a host of survival mechanisms, but the odds always were stacked against them, just as they were in the ghettos. In both settings, the likely outcome was death, sooner or later, unless the Allies arrived first. The most important explanation of why resistance never crystallized into a form that interrupted the killing machinery or threatened German control is that the system of divide and conquer functioned in the camps to the same diabolical effect that it operated in the ghettos. Inmates were not only outgunned but also atomized and generally resigned. The Germans exploited internal divisions and individuals' will to live right up until the dissolution of the camps. As Imre Kertesz, a Hungarian Jewish novelist and survivor of Auschwitz, writes, \"Provided that under the conditions of totalitarianism a person wants to remain alive, he will contribute with such an attitude to the preservation of totalitarianism: this is the simple trick of the organization.\"\nCHAPTER 6\n\n[HOMELANDS: \nWhy Did Survival \nRates Diverge?](contents.xhtml#ch_6)\n\nIF JEWS COULD do relatively little to deflect or break the force of the Holocaust, what of their non-Jewish fellow citizens in the countries affected? What did they try or fail to do, and why? Does the relative incidence of courage or lack thereof on the part of individuals explain why the survival rates of Jews diverged so widely by country?\n\nEveryone knows or should know that freedom is indivisible; when taken away from someone, it can be taken away from anyone. But few people dare act on that principle\u2014or think they need to do so\u2014even under the best of circumstances. The temptation in times of persecution is for those not immediately subject to it to try to ride it out until the horrors end, and in the meantime to look away or to take advantage. This was all the more true in German-occupied Europe, because the Nazi regime made sure people understood the risks of helping Jews. In Western Europe, these included being sent to a concentration camp. In Eastern Europe, concealing or hiding Jews could result in the execution of one's entire family. Such penalties lie behind one of the most uncomfortable truths of the Holocaust. For all our appropriate attention to the Righteous Among the Nations memorialized by Yad Vashem and the brave individuals who risked their lives to hide or otherwise save people, no more than 5\u201310 percent of the Jews who survived the Holocaust did so by virtue of someone's individual heroism.\n\nVARIETIES OF BEHAVIOR\n\nAlthough readiness to help Jews emerged within every country in Europe, the number of people willing to help, their proportion of the local population, and their attributes and motives varied greatly from place to place and over time. Germans may have hidden and saved 5,000\u201310,000 Jews during the war years, not counting those protected by marriages to non-Jews and other special provisions; Dutch people 7,000\u20138,000; and Poles anywhere from 20,000 to 65,000. But those totals represent much smaller shares of the native Jewish populations than those saved by Danes or Italians. Humanity was not the special property of any one or two nationalities nor altogether absent among any, but neither was it evenly distributed by place or over time. Especially in the first year and a half of the mass killing, when the carnage was at its peak and Jews most needed help, willingness to give aid was generally rare, including in areas where it later mounted.\n\nWhere such willingness did appear, it generally had its roots in one of three sorts of convictions: political, religious, and personal. Leftists tended to be more likely to help Jews than conservatives, in part because communist and socialist thought discouraged racist thinking, and in part because Communist Party discipline called for resisting all Nazi actions after the invasion of the Soviet Union. Minority religious status sometimes fostered identification with persecuted Jews. For example, Polish and Ukrainian Catholics living in western Ukraine were more likely to aid Jews there than were either Polish co-religionists in overwhelmingly Catholic Poland or the more numerous Orthodox Ukrainians who predominated further east. Similarly, Quakers and Baptists in Germany were much more active in smuggling Jews out of the country before 1939 and hiding them thereafter than their Catholic or Lutheran fellow Germans. And in Catholic France, the remote, predominantly Protestant village of Le Chambon-sur-Lignon and the surrounding region managed to save about 3,500 Jews, many of them children (along with 1,500 other people being pursued by the Gestapo), though one should note in this connection that some Catholics in the region also helped.\n\nMinority status was not always necessary to remind the pious to stand up for the persecuted. The Protestant bishops of Lutheran Norway protested collectively as deportations from that country were being prepared in November 1942. The Orthodox Primate of Bulgaria\u2014the head of the official church of that country\u2014played a central role in preventing deportations there. And although the Pope refused to speak out forcefully against the treatment of Jews and most Catholic cardinals and archbishops in majority Catholic countries remained silent, not all did. Cardinal Jozef-Ernest van Roey in Belgium and Cardinal Pierre Gerlier of Lyon, Archbishop Jules-G\u00e9rard Sali\u00e8ge of Toulouse, and Bishop Pierre Th\u00e9as of Montauban in France were among the Catholic prelates who openly denounced German racism.\n\nAs for the personal motives that led to attempts to protect Jews, certain character traits and behavioral records were better predictors of willingness to act than others. The sociologist Nechama Tec's _When Light Pierced the Darkness_ concluded, on the basis of a study of 754 Polish rescuers of Jews, that people with a strong sense of individualism and empathy and long records of helping the needy were more protective of Jews than people who took their behavioral standards from their environment and were more self-centered. That judgment is, perhaps unavoidably, rather circular. But another close examination of rescuers, Samuel and Pearl Oliner's _The Altruistic Personality_ , which rests on a similarly small sample but includes rescuers from across occupied Europe, reinforces Tec's conclusions in one important respect: People who rescued Jews tended to come from families that instilled strong moral and ethical values, empathy, and concern for the common good. Both these studies suggest that solidarity and courage were not spontaneous, as they often appeared to be, but rather the long-gestating products of a person's upbringing. Otto Jodmin, a German superintendent of an apartment building in Berlin who hid Jews in its cellars and vouched for others as bombing victims so they could obtain identity and ration cards, attributed his actions to the way he had been brought up, which made him think, \"I simply had to do it. . . . I just couldn't act in any other way.\" A Polish researcher named Teresa Prekerowa took issue with this line of analysis in the late 1990s on the basis of a much bigger sample of 3,300 people she considered \"typical\" of those who helped Jews. She concluded that helpers \"were ordinary people who differed greatly from each other, as ordinary people do, and I do not think it is possible to find any characteristics they shared in common.\" Perhaps, but a great deal of social psychology research reinforces the idea that altruism has to be ingrained and practiced or it atrophies. Think of it as a kind of muscle memory. When a person wonders, \"What would I have done?,\" the best clue to the answer may be his or her record of putting time and energy into helping people at risk.\n\nWe do not know enough about the youths of several diplomats who aided Jews in 1940\u201342 to say that they all fit this overall pattern, but some of them did. As the Nazis were sweeping over Europe, virtually the only effective rescue stemmed from the swift and unauthorized decisions of exceptional foreign diplomats to issue entry visas to their homelands to fleeing Jews. A notable example from mid-1940 was Aristides de Sousa Mendes, the Portuguese consul in Bordeaux, who, defying direct and repeated orders from his government, signed thousands of such documents as the German army bore down on that city. He was a _marrano_ , a Catholic whose ancestors had converted from Judaism under duress centuries earlier, and he was deeply devoted to the adopted faith. The combination of religious conviction and family sentiment may have accounted for his remarkable display of courage and empathy. A similar, almost simultaneous, and remarkable tag-team effort occurred just beyond the opposite end of the Nazi empire at the time. This was the joint action of the Dutch and Japanese consuls Jan Zwartendijk and Chiune Sugihara, in Kovno, Lithuania, which the Soviets recently had occupied, to provide partially specious documentation that enabled almost 2,000 Jews to escape across the USSR to Shanghai and other destinations. Another celebrated individual, quasi-official rescue effort was that of the American journalist Varian Fry, who went to France on behalf of the newly formed Emergency Rescue Committee and funded the escapes of some 2,000 people, most of them Jews and many of them famous artists and intellectuals, across the Pyrenees Mountains into Spain in 1940\u201341. A Swiss consular official in Austria named Ernest Prodolliet helped Jews gain admission to his homeland in 1938. Reprimanded and transferred to Amsterdam, he once more deftly evaded his orders by issuing a number of transit visas through Switzerland to Dutch Jews after Germany occupied the Netherlands. Just before his office was closed in 1942, he turned over the remaining available consular funds, worth about $180,000 in 2014 dollars, to Gertrude van Tijn, the head of the Dutch Jewish Council's still-functioning emigration department. In return, he sought only an unenforceable (but ultimately honored) promise of repayment by the representatives of the American Jewish Joint Distribution Committee in Switzerland.\n\nSad to say, not enough people in Europe possessed the same humanitarian reflexes as these individuals did, and there were not enough people like Oskar Schindler, either, the Sudeten German opportunist and would-be war profiteer who took over an enamelware factory in Cracow and gradually resolved to save the lives of some 1,300 Jews who worked for him. His heroism is inexplicable because it seems out of keeping with his self-indulgent and disorderly life both before and after. But he displayed great ingenuity and nerve in outwitting the SS. His story stands out not just because he tried to help Jews but also because he succeeded. He did so primarily because he owned his company and did not have to explain or justify his actions to any superiors who might have tipped the Nazis off to what he was up to. In contrast, Berthold Beitz, a German who managed a Karpathian Oil Corporation drilling site in Boryslaw in eastern Galicia, where he protected hundreds of Jews who worked for him for almost two years, did not work for himself. He could not arrange for the Jews' withdrawal to another factory when the Germans retreated in 1944, lest some superior denounce him to the Gestapo for that action. The most he could do was warn his employees just before the SS moved against them, enabling many of them to go into hiding. Alfred Rossner, a German who ran a number of uniform factories in Bedzin in eastern Upper Silesia, bribed and wheedled the local Nazi authorities successfully from May 1942 until August 1943 to keep his Jewish workers from the deportation trains, sometimes even hiding them in his shops. But in the end, not only were most rounded up and deported in the final clearing of the Upper Silesian ghettos, but also the Gestapo caught on to Rossner. Arrested in December 1943, he died in prison in 1944, hanged either by his warders or by his own hand.\n\nPerhaps the most remarkable story of an employer who tried and often succeeded in rescuing his Jewish workers comes from the very heart of the Third Reich, the capital city of Berlin, from a workshop on Rosenthaler Strasse in the middle of town. There Otto Weidt managed an operation that made brushes and brooms and employed at any one time about thirty deaf and mute Jews from a local home. Altogether during the war some fifty-six people from his workshop were slated for destruction by the Nazi state on grounds of both disability and heritage. He argued with the Gestapo each time one of his worker's names appeared on a deportation list, insisting that their work was vital to the war effort, and even bribed Nazi officials to get their names removed. In the end, half of his workers outlived the war, and so did he.\n\nIndividual heroism could achieve only so much in the face of the Nazi onslaught, yet about one-quarter of the European Jews in Nazi-occupied or -allied states and one-third of all the European Jews as of 1939 survived the Holocaust. How and why? We can begin to answer that question by looking at figure 7. It sorts Nazi-occupied or -allied countries in Europe according to two characteristics: whether more or less than the continent-wide average of two-thirds of the Jewish inhabitants were killed in each place, and whether each was ruled directly by the Germans or by a collaborating government.\n\nThe pattern that emerges is unmistakable, but not quite self-explanatory. That the most lethal parts of the continent were those directly occupied and administered by German officials does not mean that collaboration there was unimportant. In Serbia and Greece, veteran military leaders agreed to head puppet regimes that carried out German orders; similar arrangements arose in the Baltic states. In all of these areas, local police forces and\/or militias continued to function and often to participate in rounding up Jews, and residents eager to denounce Jews in hiding were numerous. A particularly notorious example was the Dutch staff of an organization called the Recherchegruppe (or Colonne) Henneicke, which tracked down and turned in 8,000\u20139,000 Jews who tried to hide in the Netherlands\u2014that is, more Jews than survived under cover in that country. Conversely, the generally lower death rates under indigenous collaborating governments do not imply that their personnel or citizens refrained from persecuting Jews. On the contrary, Vichy France under Philippe P\u00e9tain, Hungary under Regent Miklos Horthy, Romania under Marshal Ion Antonescu, and Bulgaria under Tsar Boris III independently enacted virulently antisemitic legislation, stripped many Jews of citizenship, and delivered certain groups of Jews to Germany and\/or engaged in killing them.\n\nFIGURE 7: GOVERNANCE AND HOLOCAUST MORTALITY RATES\n\n| DEATH RATE OVER 2\/3| DEATH RATE UNDER 2\/3 \n---|---|--- \nUnder \nGerman \nAdministration| Baltic states, Belarus, Holland, Germany, Greece, Belgium,Luxembourg, Czech Protectorate, Poland, Serbia, Ukraine| Belgium \nUnder \nCollaborating \nGovernments| Slovakia, Croatia, Hungary in 1944| Bulgaria, Romania, Denmark, Finland, Norway, France, Italy, Hungary until 1944\n\nThe decisive variable that determined the mortality rate in any given country was usually time\u2014more specifically, whether the Nazi state attacked the resident Jews in 1941\u201342. Where Germans ruled directly, they almost always mobilized in pursuit of Jews promptly and thoroughly, unencumbered by an interest in preserving smooth working relations with local governments and populations. An exception was Belgium, and it was not really much of one. Though under German administration, authority over the police and so-called racial policy was in the hands of the German army, not the SS or the Nazi Party, until May 1942. And though a collaborating government did not actually rule Belgium, since the country's cabinet (but not its monarch) had fled to Britain, the indigenous civil service continued to function, and the Germans found working with it convenient and worth preserving. Still, Belgium did prove a great anomaly in one sense: More than 90 percent of the Jews there were foreigners, the sort of people usually deported first from most places, yet half of them survived. Another exception of a quite different sort was Greece, where a puppet government nominally existed after the country capitulated to Germany and Italy but where German control of the Nazi-occupied regions was very tight. Nonetheless, the Germans did not begin to deport Jews until March 1943; the somewhat smaller number in the Italian-occupied zone were not deported until 1944. Despite the delays, the death toll ultimately came to between 80 and 90 percent.\n\nWhere local administrations remained in place and were more autonomous, however, the Germans at first preferred to let native antisemitism run its course while they concentrated on the larger populations of Jews in the Reich's grasp elsewhere. By late 1942, when most of those other Jews were dead and Germany became insistent, the tide of war was turning and affiliated governments were growing wary of further persecution, since they might have to answer to the Allies if Hitler went down to defeat. Emblematic of the changing climate are the deportation statistics from two countries where the final death toll proved relatively low: The majority of the Jews ever deported from both France and Belgium departed in 1942, and then the pace from both places slowed. Equally telling was the behavior of Hitler's Balkan allies. Bulgaria, Hungary, and Romania each handed over to the Nazis some or all of the Jewish populations of regions taken in 1938\u201341 from neighboring states under German auspices but declined to turn over the Jewish inhabitants of their core territories in 1942\u201343. Even Slovakia, which in early 1942 eagerly agreed to deport most Jews\u2014in fact, actually paid Nazi Germany for taking them\u2014also had second thoughts toward the end of that year and suspended the deliveries, most of which had gone directly to Auschwitz.\n\nIn other words, the four chief determinants of the differing rates of Jews' rescue and mortality in Nazi Europe were: (1) how swiftly the Germans moved\u2014if they began massively killing in 1941\u201342, they got almost all the Jews in any given area; (2) how long the Germans remained\u2014their presence enabled the events in Hungary in May\u2013July 1944 and would have enabled the slaughter of the French Jews had D-Day not interfered with their general deportation, which the Germans finally mandated less than two months earlier; (3) whether the Germans had to deal with an indigenous and at least quasi-autonomous government interested in surviving the war; and (4) whether most Jews were still alive in an area by the time of the battles of El Alamein and Stalingrad 1942\u201343\u2014the interval that Winston Churchill called \"the hinge of fate\"\u2014and the onset of Germany's forced labor drafts. These are the moments when the likelihood that the Third Reich would win the war dwindled, and the Jews' interests and national interests began to coincide around resistance to Germany. If enough Jews were still alive, these developments began working to their benefit.\n\nThe importance of national and Jewish interests seeming congruent emerges clearly from the fate of Jews in an area where and at a time when the opposite was the case\u2014where national interests seemed to favor cooperation with the Nazis and the sacrifice of the Jews, notably the Baltic states and Ukraine in 1941\u201343. Ukrainian nationalists had seen their aspirations for independence crushed by the Bolshevik regime in 1919\u201321 and had endured a series of famines and purges in the 1930s that had deepened alienation from the Soviet state. Lithuanian, Latvian, and Estonian nationalists had lost their independence to occupation by the USSR in 1940. To them, the Germans came as potential liberators from Soviet enslavement, all the more so as Germany had allowed various national liberation groups to set up offices in Berlin and thus implied support for their goals. But because nearly all of these nationalist movements were historically antisemitic, Soviet rule in 1940\u201341 had an upside to the Jewish citizens of these states, some of whom found opportunities for advancement when the less discriminatory Soviet regime arrived, even though many other Jews suffered from the nationalization of their property by communism and from the deportations to Siberia that the Soviets conducted. As a result, Jews were overrepresented compared to their share of the Lithuanian population not only in the Communist Party and secret police there in 1940\u201341 but also among the people the communists sent to the Russian interior.\n\nThe consequence in most of the former Pale of Settlement was a situation in which Jewish and local nationalist interests seemed to conflict. To the Jews, the Soviet occupation of 1940 seemed the lesser of two possible evils; as one Jew from the region said at the time, the Soviet Union brought life in prison, but Nazi Germany brought the death sentence. But to Ukrainian and Baltic nationalists, even initially to the Catholic Metropolitan Archbishop Andrei Sheptytsky, who later tried to protect Jews, occupation by Germany appeared the lesser evil. These seekers of independence were only too ready to cooperate in removing a population that they disliked anyway in order to curry favor with the Germans. Well before the German invasion of Ukraine, both factions of the Organization of Ukrainian Nationalists (OUN), the Banderites and the Melnykites, had branded Jews as the allies of Bolshevism and endorsed killing Jewish males. On July 1, 1941, the day after the Germans occupied Lviv, the OUN issued a leaflet calling on Ukrainians to \"destroy\" Jewry, and a pogrom took place. Similarly, in March 1941, the Lithuanian Activist Front declared that Jews had \"betrayed\" the country and thus had no future there.\n\nAlthough the SS had some initial difficulty carrying out its orders to stimulate pogroms in Baltic cities captured by German troops, local militias soon recognized what the Germans wanted them to do to Jews and began bludgeoning and hacking them to death in Vilna, Kovno, and Riga, the largest cities of Lithuania and Latvia. Lithuanian militiamen probably killed more of the 180,000 Jews who died in that country by the end of 1941 than Germans did. Ukrainian police and militias played an active part in the massacres in their homeland during 1941, including at Babi Yar in September, even though by that time the Germans already had made their opposition to Ukrainian independence clear and even arrested Stepan Bandera, the leader of one wing of the OUN. So long as Germany remained on the offensive on the Eastern Front, it had no shortage of willing local volunteers for militias and security forces that hunted and killed Jews. In 1943, in fact, Himmler had about 300,000 mostly cooperative local policemen under his command in the occupied East, and Russian scholars have put the number of citizens of the occupied USSR who served in Wehrmacht and SS units during the war at 1.2 million. By the time the tide of the war turned, and the Baltic and Ukrainian nationalists finally recognized that a Nazi New Order in Europe was not going to restore their independence, nearly all of the Jews in Ukraine and the Baltic states were dead. Even then, many of the collaborators continued to fight for Hitler because they had become so complicit in his crimes that they had no alternative and, in the case of the Ukrainians, because they still aspired to eliminate Poles and Jews from their lands. These men and their families retreated with the German armies in 1944\u201345 and made up a significant proportion of the people in displaced persons camps there after the war ended.\n\nThis is not to say that national interests were the only driver of widespread popular participation in killing Jews in the occupied East. The availability of plunder also served as a strong inducement. A Pole who lived on the outskirts of Vilna and witnessed the massacres there observed, \"For the Germans 300 Jews are 300 enemies of humanity; for the Lithuanians they are 300 pairs of shoes, trousers, and the like.\" Nonetheless, the force that unleashed and claimed to legitimize covetous motives was perceived national interest.\n\nAnother sign of the decisive importance of national political considerations to the fate of Jews was the way that states officially or tacitly allied with Nazi Germany drew policy distinctions, albeit to varying degrees, between native-born Jewish citizens of their countries, especially veterans, and immigrant Jews of different nationalities. Vichy France, for example, was willing, even eager, to treat alien Jews as fit subjects for deportation but more resistant to giving up French citizens, though it did so on some occasions. Of the roughly 350,000 Jews in France in 1940, more than half had immigrated or fled illegally to the country since the beginning of the twentieth century. Drawing on mounting xenophobia during the 1930s and the convenient scapegoating of Jews for France's defeat in 1940, the collaborationist government headquartered in Vichy enacted antisemitic legislation voluntarily and in a form that was in some respects even more restrictive than Germany's. Vichy also accepted the arrests of foreign Jews in the occupied northern part of France beginning in October 1940; in fact, French police often carried out the roundups. French police also collected and handed over to the Germans 10,000 foreign Jews from unoccupied France during the summer of 1942. Deportations had begun in March of that year and resulted eventually in the transportation of approximately 76,000 Jews in France to concentration camps; only about 2,500 survived, and more than two-thirds of the deportees were foreigners. In the end, more Polish Jews who had sought refuge in France died at Nazi hands than did French Jews. The survival rate among foreign Jews in France ultimately came to about 50 percent, whereas for Jews who had French citizenship it came to a little less than 90 percent. But once more than half of the unfortunates left in 1942, the French government dragged its feet, partly as a matter of asserting its status as an independent, sovereign entity and partly as a matter of hedging its bets about the outcome of the war.\n\nSimilarly, the three German allies in southeastern Europe\u2014Romania, Bulgaria, and Hungary prior to 1944\u2014drew distinctions between Jews Germany could have and ones it could not and tailored national policies toward Jews to each country's political interests. The three states had leaders who held antisemitic views of varying intensity, and all three agreed to hand over Jews in lands acquired from neighboring countries in 1938\u201341 under German auspices. Thus in August 1941, Hungary pushed 17,000 Jews from the parts of Slovakia it had annexed in 1938\u201339 across its borders into German-occupied Poland and Ukraine, where the SS massacred 11,000 of them. Early in 1942, Hungary killed another 1,000 or so Jews in territory acquired during the dismemberment of Yugoslavia in April 1941. After Bulgaria received Thrace from Greece, Macedonia from Yugoslavia, and Dobrudja from Romania in 1940\u201341, it delivered 11,384 Jews from these regions to the Germans in early 1943. In all these instances, the principal and cynically self-interested motivation was demographic. Where killing Jews would reduce the size of the non-Hungarian and non-Bulgarian populations and speed the absorption of territory into their states, the acquiring countries were ready to cooperate.\n\nBoth Bulgaria and Hungary also enacted antisemitic laws designed to strip Jews of their property and exclude them from the civil services; Hungary's legislation went so far as to restrict the share of Jews in any profession to 6 percent and forbade sexual relations and new marriages between Jews and Magyars. But the countries' policies diverged even in 1941\u201342 regarding further deportations, with the Bulgarians promising to begin them in 1943 and the Hungarians steadfastly refusing, although they drafted adult Hungarian Jews for labor service on the Russian front, where about 42,000 of them died or were murdered. The Bulgarians bought the Germans off for a time by conscripting Bulgaria's Jews for work in the countryside but then reversed their position on deportations in March and April of 1943, partly because of intense and widespread domestic opposition and partly because of mounting concern that Germany might not win the war. Nearly all of the Bulgarian Jews ultimately survived because the Germans did not force the issue by occupying the country.\n\nIn Hungary, however, opposition to deportations crumbled after March 1944, when German troops poured into the nation, ostensibly to defend it from an impending Soviet invasion but really because Hungary was considering following Italy's example and finding a way out of the Axis alliance. Following an initial period of isolating and pillaging the Hungarian Jews, almost 60 percent of them, approximately 437,000 people, were deported in the space of only fifty-five days, between May 15 and July 9. At Auschwitz-Birkenau, about 25 percent of these people were selected for work and generally shipped onward to labor in Germany; perhaps half of them, about 55,000 people, survived the war. The rest of the deportees, more than 325,000 people, perished in the gas chambers upon arrival, making the total death toll from this round of Hungarian deportations roughly 380,000 and the total number of Hungarian Jews killed in the Holocaust, following another round in late 1944 and then a series of horrific death marches, between 500,000 and 565,000.\n\nEven though most of the victims died at the hands of Germans, not Hungarians, the thoroughness of this operation, which netted 97 percent of the Jews in the Hungarian countryside and annexed areas and left virtually the only survivors in the Hungarian capital of Budapest, was largely homegrown. Only 150\u2013200 German SS personnel were involved in the Hungarian roundups that the nation's own national and local police forces, supplemented by civil servants and volunteers, carried out under the direction of a Ministry of the Interior led and staffed by extreme right-wing Magyar antisemites. The concentration of Jews across the country into fifty-five short-lived ghettos proceeded according to a plan for the successive clearing of six different sections of the country. Finalized by Adolf Eichmann and several Hungarian policemen, that plan merely refined and made more specific a program that two nationalist Hungarian generals had devised in 1942, almost two years before the German occupation. In the apt summation of Peter Kenez, a historian of modern Hungary and himself a refugee from that country, \"The German role in the destruction of Hungarian Jewry is best understood as giving an opportunity to some determined [Hungarian] antisemites to carry out a policy that they had long desired and planned.\"\n\nHow can one explain the scale and speed of the carnage in Hungary, comparable only to the liquidation of most of the Warsaw ghetto in fifty-three days during the summer of 1942, and so fast that Commandant H\u00f6ss at Auschwitz repeatedly sought to slow the overwhelming pace? One part of the answer is that the RSHA could focus its efforts\u2014after all, most of the other Jews of Europe were dead or out of reach by May\u2013July 1944\u2014and the killing center at Birkenau was very close by and more swiftly murderous than ever, thanks to the recently completed rail spur that ran into the camp almost to the doors of two of the gas chambers. A second, somewhat technical, explanation is that the deportation had a war-related purpose, which helps to account for so many trains\u2014147 over the course of the operation, 3\u20136 per day\u2014being made available. Auschwitz was supposed to cull 100,000 able-bodied workers, 10\u201315 percent of the anticipated deportees, and send them on immediately to Germany to labor on the massive effort to put war-producing factories underground. But still a third important component of a response lies in the history of antisemitism in Hungary. As in Germany, prior to World War I Jews in Hungary enjoyed apparently ever-expanding acceptance, opportunities, and prosperity, only to experience rising hostility following defeat and territorial losses in 1918-19 and the bloody suppression of a communist revolution in which Jews played conspicuous parts. In interwar Hungary, as in Germany, nationalist forces harped continuously on a supposed link between Jews, disloyalty, and unrest and fanned resentment at the prominence of Jews in commerce, industry, law, and medicine. As a result of such agitation, the installation of an authoritarian government, the inauguration of antisemitic restrictions, and the rise of a mass antisemitic political movement, the Association of Awakening Hungarians (EME), all occurred more than a decade before Hitler came to power in Germany.\n\nWhen German diplomatic successes in 1938\u201341 led to the dismemberment of first Czechoslovakia and later Yugoslavia and meanwhile to the arbitration of Hungary's and Romania's competing claims to Transylvania, the Hungarian rulers were happy to express their gratitude for the pieces of territory that Hitler threw their way with the enactment of further restrictions on Jews' civil rights and economic activities. But the new territories nearly doubled the Hungarian Jewish population, from 401,000 to 725,000 (or from 491,000 to 825,000, counting converts of Jewish descent), making it larger than the entire Jewish population of Western Europe. That the additional Jews were much less likely to speak Magyar, and to resemble in dress and religious practices their co-religionists inside the previous borders, stoked the antisemitism that already prevailed in military and some government circles, including around the head of state, Admiral Horthy. When he finally capitulated in March 1944 to Nazi demands that he furnish 100,000\u2013300,000 \"Jewish workers for German war production purposes,\" the first two parts of the nation combed were the annexed regions to the northeast that happened to be closest to the advancing Russian troops; the last region scheduled for purging was the capital city, where the most assimilated and economically valuable Jews lived. In short, Hungarian desires, as well as Hungarian personnel, not only accelerated the deportations but also determined their course. Conversely, when Hungarian officials withdrew their cooperation with the Germans between July and October 1944, Eichmann and his aides could achieve almost nothing further except the deportation of 2,700 prisoners already interned in camps on Hungarian soil.\n\nBy far, the most contradictory and confusing policies toward Jews were those carried out in and by Romania in 1940\u201345, and to understand them, one needs to pay close attention to figure 8. Marshal Antonescu, the country's dictator, was an inveterate antisemite who blamed Jews for all of his country's weaknesses. In particular, he claimed that they had welcomed Romania's losses, in 1940, of the northern province of Bukovina, the northeastern province of Bessarabia (also then called eastern Moldova), and of northern Transylvania. The first two losses came as a result of a Soviet ultimatum and the third because of a German-Italian arbitration of border claims that gave this region to Hungary. Determined to regain all three territories, Antonescu joined in the invasion of the USSR in June 1941 and set out to murder the Jews of Bukovina and Bessarabia, both in punishment for their supposed pro-Soviet stance and as a means of speeding the Romanianization of the regions. He also hoped to use willingness to deport and kill Jews as leverage in persuading Hitler to give him back northern Transylvania. As a result, Romania acquired the dubious honor of becoming the German ally that killed the largest number of Jews, about 400,000 in Bukovina and Bessarabia and a territory the Romanians called Transnistria, the portion of Ukraine that Hitler awarded Antonescu in compensation for his loss of part of Transylvania.\n\nFIGURE 8: ROMANIA, 1941\u201344\n\nBut in 1942, when the Germans began requesting a schedule for the deportation of the Jews from the core provinces of Romania, an area called the Regat, Antonescu demurred. His generals on the Eastern Front already had begun warning him of impending disaster there, and he wanted to extort every last penny from the local Jews before he sent them off. So he played for time, deferring the first deportations scheduled for October 1942 until spring 1943, at which point he joined the Bulgarians in reneging on his earlier promises. He never gave up his crazed dream of ultimately sending the Romanian Jews to Transnistria and thus creating an ethnically pure Romanian heartland, but he did not hand over the Jews in the Regat to the Nazis. This engendered one of the greatest ironies of the Holocaust: The nation that next to Germany killed the largest number of Jews also was the nation that had the largest surviving Jewish population in Europe in 1945. Whereas 80 percent of the Jews in Bukovina, Bessarabia, and Transnistria died at Romanian hands, 80 percent of the Jews in the Regat remained alive at the end of the war. Their situation was wretched, as the Romanian regime had terrified and impoverished them, but they still lived.\n\nIn all these instances, cynical and practical politics played a greater role in deciding the fate of Jews than moral considerations. In fact, taking a strong moral stand in solidarity with Jews proved counterproductive when the timing was not right. Early mass resistance to Nazi discrimination could backfire, as it did in Holland. A general strike there in February 1941 in protest against the persecution process led to its acceleration, and the objection of Catholic bishops to the deportation of Jewish converts to Catholicism triggered expedited arrests of such people. Provoked by this opposition, the German administrators in the Netherlands, many of whom had acquired experience with persecuting Jews in Austria in 1938\u201339, now moved with a fury unmatched in any other occupied Western European country. Between July 1942 and September 1943, they rounded up and deported 110,000 Dutch Jews out of a total population of 140,000.\n\nFinally, as a demonstration of both the importance of politics to the fate of Jews and the difficulties of individual rescue efforts, consider what happened in the two countries where Jews enjoyed unusually high survival rates thanks largely to popular solidarity, namely Denmark and Italy. Denmark is, of course, famous for concealing almost all the nearly 8,000 resident Jews and ferrying them across the narrow strait between that country and Sweden, and Italy for resolutely refusing, even while Mussolini was enacting antisemitic laws, to let anyone be deported, not only out of Italy but also out of the parts of France, Yugoslavia, Albania, and Greece that Italy occupied and administered. In both countries, roundups of the Jews were delayed by special political circumstances. Until 1943, a Danish government continued to function and to cooperate with the German occupation, Mussolini's regime was an Axis ally, and the Germans thought preserving these arrangements more important than forcing these countries even to compile lists of Jews or make them wear identifying badges, let alone to begin deportations. Only when the political conditions changed in late 1943\u2014in Denmark when the cabinet resigned because the Germans imposed martial law in response to mounting popular resistance, in Italy when the Germans rushed to occupy the country because the king had dismissed the Duce and his successor had concluded an armistice with the Allies\u2014only then could and did the Germans strike against Jews. All this meant that the assaults occurred precisely at the moment when helping Jews became an act of national resistance against an oppressive foreigner.\n\nMany other pieces of good fortune were involved: the Danes had the virtual collusion of the leading German administrator in the country, Werner Best, and the opposition to the operation of both the military commander in Denmark, General Hermann von Hanneken, and the Gestapo chief there, Rudolf Mildner. Caught between Himmler's impatience to begin deportations and his own belief that they would complicate the job of managing the occupation, Best played a double game. He advocated action to the SS in Berlin in order to please Himmler, but he leaked news of the impending roundups to the Danish Jews via a German named Georg Duckwitz four days in advance in order to preserve a good working relationship with the Danish bureaucracy and police. This gave the Jews time to get away from their homes before the German police came to apprehend them on the evening of October 1, 1943. During the ensuing weeks, the Jewish Danes had two more priceless advantages in making their escape: an offer of asylum from the Swedish government for anyone who got to its shores, and only a narrow body of water to cross. Even so, most of them got away only because German navy patrol boats off the Danish coast made no attempt to interfere with the roughly 700 vessels\u2014mainly fishing boats\u2014that carried the exodus. Only 284 Danish Jews fell into Nazi hands in the roundup of October 1, and 22 more drowned while trying to escape. Ultimately, 7,742 reached Sweden, including 1,376 German Jewish refugees, along with 686 non-Jewish spouses. Perhaps even more amazing than these numbers is the fact that after the war, when these people returned to Denmark, they found their homes and property not only unmolested but also often carefully tended in their absence.\n\nIn Italy, Mussolini had just announced the imposition of forced labor on Italian Jews and was on the verge of commanding deportations when he was overthrown in July 1943, and the Jews in Italian-occupied Croatia already had been interned in a potential transit camp. But in the chaos that surrounded the influx of the German army, the remaining 2,600 Croatian Jews could escape, and most of the 32,000 Italian ones remaining in the northern two-thirds of the country had time to go underground. They then had the geographical advantages of rugged mountains\u2014 namely, the Apennines in the center of the country and the Alps in the north that provided ideal hiding places\u2014and proximity to the Allied front lines advancing up the Italian boot.\n\nNevertheless, the latest studies suggest that about one-quarter of those still vulnerable Jews were killed during the Holocaust, that many of those deported after the Germans occupied the country were apprehended by Italian collaborators, and that in regions where the German presence was greatest, for example around Trieste in the northeast, Italian attitudes hardly mattered and 90 percent of the Jewish community perished. One reason for the German thoroughness in that region is that the SS units from Operation Reinhard were transferred there when those camps in Poland closed. Local heroism was no match for German ruthlessness, as indicated by the fate of Giovanni Palatucci, a commissioner in the police headquarters in nearby Fiume. He used his office to impede roundups and help Jews escape by boat to southern Italy until the Gestapo arrested him in September 1944 and sent him to Dachau, where he died shortly before the liberation of that camp.\n\nTiming, geography, and small numbers were factors that favored relatively high rates of survival among Jews in Denmark and Italy. So did the fact that by the fall of 1943, the likelihood that Germany would lose the war became strong. But other circumstances played a role, and they were important in Bulgaria, too. In none of these countries were Jews notably prominent in commercial or cultural life or in communist politics and thus representable as profiteers or threats. Moreover, all three Jewish communities were highly acculturated. The members of all three routinely spoke the national language and dressed like the majority population. Unlike a majority of the Jews of Ukraine, Poland, Lithuania, Romania, and rural Hungary, Yiddish was not their lingua franca, and traditional Orthodox garb was rare. In Italy in 1938, more than one-third of all married Jews had non-Jewish spouses. Sometimes, the fate of Germany's and Hungary's urbanized Jews is invoked as a warning against acculturation, a proof that it does not protect Jews from the hostility of others. Certainly in those instances, it did not. But acculturation did not always fail as a safeguard. The different experiences of the highly integrated Jewish communities of Bulgaria, Denmark, and Italy, where much of the gentile population mobilized to shelter Jews, bear paying attention to as well.\n\nTHE CASE OF POLAND\n\nProbably no subset of issues related to the behavior of non-Jewish populations during the Holocaust is touchier than those surrounding what happened in Poland. The country was, after all, the epicenter of the Holocaust: the location of the death camps, the homeland before World War II of half the victims, and the graveyard of fully 90 percent of the Jews who lived there in 1939. Among the Jews who survived, the sense that Christian Poles had done little to help, indeed in many cases had favored and even encouraged the outcome, has been strong.\n\nThat perception drew strength from Claude Lanzmann's monumental, nine-hour-long documentary film of the 1980s, _Shoah_ , which showed some Polish peasants near Chelmno explaining that the Jews died because they had killed Christ centuries before and others near Auschwitz grinning as they recalled the arrival of deportation trains there. In 2001, the publication of Jan Gross's book _Neighbors_ brought renewed attention to the fraught nature of communal relations in Poland under the Nazis. Gross recounted a savage massacre of Jews by Christian Polish peasants in the village of Jedwabne, in the Soviet-annexed part of Poland, just after the Germans arrived in 1941. He overstated the numbers of both victims and perpetrators and minimized the instigating role of the Germans but established that local residents did the killing, often in bestial fashion. Works such as Lanzmann's and Gross's may have something to do with a curious feature of survivors' testimonies that Christopher Browning came across when he studied the Starachowice labor camp: Accounts of Poles' behavior given by Jews immediately after the war were generally far milder and less angry than those collected later. The sense of bitterness and betrayal grew more intense with the passage of time.\n\nAt the same time, aside from Belarus, Poland was the place where the German occupation was worst for everyone subjected to it. During the invasion, German troops repeatedly machine-gunned civilians and prisoners of war. In the parts of Poland seized by Germany, the death toll came to at least 50,000 people and perhaps more than 60,000 by December 1939 alone. Before the massacre of the Jews began, the Germans intentionally liquidated much of the Polish intelligentsia, killing one-third to two-thirds of the professors, journalists, lawyers, priests, leading politicians, and so on and thus greatly weakening the country that later emerged from World War II. The concentration camp at Auschwitz actually was established initially for these people, not for Jews. A few statistics give a sense of how complete the purge at some local levels was: In the Catholic diocese of Poznan (Posen in German), in the Warthegau, 77 percent of the priests were put in concentration camps, deported, or killed outright between 1939 and 1945. In the six dioceses that Nazi Germany annexed from Poland, the death rate for priests during the war varied from a low of 30 percent to a high of just over 50 percent. Moreover, Himmler established a program that essentially stole thousands of blond-haired, blue-eyed, racially German-looking children from their Polish parents and placed them in adoptive Nazi families within the Reich. His demographic engineering efforts in the Warthegau involved the displacement and impoverishment of 300,000 Poles, and the arbitrary nature of Nazi rule in the General Government assured that thousands more died at German hands. About two million more Poles were brought to the Reich for forced labor and exploited and mistreated there to varying degrees.\n\nThe lot of those left at home was not much better; hundreds at a time were shot and their farms or villages burned in reprisal actions against any resistance to German measures or harm to German troops. Secondary education for Poles was almost entirely banned, and all universities closed and plundered. Hans Frank, the Nazi master of the General Government, openly declared that he did not care whether Poles \"had anything to eat or not.\" As a result, in 1941 official rations provided Poles with only 29 percent of the daily calorie norms set by the League of Nations; in 1943, the figure was 17 percent. Most Poles survived by buying on the black market, but the search for food was time-consuming and exhausting, and the prices exorbitant. Martin Winstone, who has written the most thorough recent study of the GG, reports that \"bread prices\u2014the supreme barometer of the black market\u2014hovered at around 4,000 per cent of prewar levels\" from 1941 until the end of the German occupation. In just the first few months of German rule, the resources and machinery of the GG were stripped so clean that Frank described the region in March 1940 as \"economically speaking, an empty body. What there was. . . has, as far as possible, been taken out by the [German] Four Year Plan.\" Though the Nazis came to recognize a contradiction between keeping order in the country and impoverishing it, they never resolved the conflict, and living standards continued to plummet during the occupation.\n\nPolicy was one reason the annexed and occupied parts of Poland suffered grievously, but personnel was another. Germany generally staffed the GG with district leaders who were long-time, deeply committed National Socialists, usually of the most racist sort, and often the most incompetent, greedy, or scandal-ridden. Frank, the governor general, was prototypical in all these respects. Although this pattern made Nazi administrators occasionally susceptible to bribery that might ameliorate conditions, it also made the new overlords even more determined to extract everything they could, both for the Reich and for themselves, from the occupied region. Fritz Cuhorst, the first Nazi-appointed head of the city of Lublin, spoke for many of them in December 1939 when he remarked, \"[W]e have decided to behave as officials exactly the opposite of at home, that is, like bastards.\" As a result, according to the contemporary account of an anonymous Polish doctor, \"[I]t was like living in a country where all the thieves and gangsters had been let loose and the operation of the law entirely suspended.\"\n\nPerhaps one can get the best sense of the extent of the damage to Poland in World War II by comparing events there to two of the war's greatest conflagrations: the air wars against Germany and Japan. More Poles died in the bombing of Warsaw in 1939 than Germans in the firebombing of Dresden in 1945; in fact, more inhabitants of Warsaw alone, about 720,000 people, perished in the Second World War than did Germans in all Allied air raids. Even more shockingly, more Poles may have been killed in the suppression of the Warsaw Rising of 1944 than Japanese people were in the atomic bombing of Hiroshima and Nagasaki a year later. In the end, approximately two million non-Jewish Polish citizens perished during the Nazi occupation between 1939 and 1945, a staggering total, but not as many as the number of Polish Jews who were killed, and hardly comparable as a percentage of the prewar population. The oft-quoted statement that as many Christian as Jewish Polish citizens died under Nazi rule is false. Jakub Berman, a Polish communist leader who was Jewish, simply cooked up the number in December 1946 for political reasons, and it has been disproven by close statistical analysis. Even so, a great many non-Jews died in annexed and occupied Poland. Moreover, while this was happening, Polish gentiles also hid and saved tens of thousands of Jews. To put the matter in a superficially surprising form: The Jewish survival rate in Warsaw was equal to that in Amsterdam.\n\nSince 1945, competing claims to greater suffering and mutual indifference under the Nazis have perpetuated the sense of distance between the Jewish and non-Jewish Polish communities. To a regrettable degree, many Jews have talked as if the Poles were worse than the Germans during the Holocaust, and many non-Jewish Poles have treated every criticism of their behavior as a treasonous and ungrateful insult to a beleaguered nation. In April 2015, the Polish foreign minister went so far as to summon the American ambassador to complain that a reference to \"the murderers and accomplices of . . . Poland\" in a speech and newspaper column by FBI Director James Comey\u2014a remark that this chapter shows is entirely justified\u2014constituted an insult to the many heroic Poles who had resisted Nazi Germany. Of course, Poles exhibited both complicity and heroism during World War II; to cite one is not to deny the other. Moreover, as we will see, resistance to Nazism in Poland and implication in the Holocaust sometimes went hand in hand. But Comey decided to do the politic thing and apologize. Even in contemporary academic circles outside of Poland, these sensitivities have surfaced in the critical reaction to Timothy Snyder's important book _Bloodlands_ , which appeared in 2010. Snyder juxtaposed the suffering of Jews at Nazi hands with the suffering of Poles, Ukrainians, and other Eastern Europeans primarily at Soviet hands in the former Pale of Settlement between 1933 and 1945. In response, a number of prominent Jewish scholars, both here and in Israel, charged him with downplaying Polish antisemitism and generally presenting an excessively pro-Polish account of the carnage.\n\nHow can we sort through the mutual recrimination fairly and come to a measured assessment of what happened? I think such an effort has to keep seven essential and somewhat contradictory sets of facts in mind.\n\nFirst, antisemitism in Poland was considerable before 1939 and on the rise. To be sure, it was not universal. The popular Peasant Party and the elitist Democratic Party that emerged just before World War II advocated toleration and discouraged persecution. But the chief proponent of discrimination against Jews was Roman Dmowski. His National Party (known until 1928 as the National Democrats, or Endecja) gained influence from 1935 on, as the government that succeeded that of the deceased Marshal Jozef Pilsudski adopted a series of measures aimed at driving Jews out of the Polish economy and, indeed, the country. Recurrent small-scale pogroms flared up in the late 1930s, most of them in small towns in the center of Poland, resulting in the deaths of fourteen Jews and the wounding of 2,000 more. A government decree requiring businesses to post the full names of their owners facilitated boycotting Jewish enterprises, as did the division of municipal market halls into Jewish and non-Jewish sections. Discriminatory admissions policies drove down the share of Jews among university students in Poland by almost two-thirds (from 20.4 percent to 7.5 percent) between 1928 and 1938. Those who enrolled had to sit on specially designated benches in the lecture halls after 1937 and sometimes were subjected to violent attack. Between 1936 and 1939, the Polish parliament first limited, and then banned, kosher slaughtering. Meanwhile, virtually no Jews held positions in Polish governmental or municipal offices, the railway and postal systems, and the government monopoly industries, such as tobacco, alcohol, and lumber. Aside from two baptized generals of Jewish descent, the few Jews in the Polish army were almost all medical doctors. National legislation restricted Jewish actors to Yiddish theaters and Jewish journalists to Jewish-owned newspapers, and various professional associations, including those for electrical engineers and physicians, voted to exclude Jews henceforth. The political organization founded in 1936 to support the post-Pilsudski regime, the Camp of National Unity (OZON), also barred Jewish members.\n\nIn 1937, the Conservative Party leader, Prince Janusz Radziwill, endorsed the \"forcible emigration of the Jews,\" and the Polish government actually sent a delegation to Madagascar to explore the possibility of sending Jews there. The Polish foreign minister even discussed the idea with his French counterpart the following year and tried to lease roughly a million acres of land on the island to support the emigration of 30,000 Jewish families per year during the next five or six years, some 500,000 to 600,000 people all told. Shortly thereafter, the Polish ambassador in the United States opened talks with a group of wealthy and influential American Jews about their purchasing the Portuguese colony of Angola as a \"supplemental Jewish homeland.\" So eager was the Polish government to drive out Jews that it actually trained right-wing Zionist fighters in Poland in 1938\u201339 and then sent them off to Palestine. The hope was that they would perpetrate enough violence to persuade the British either to leave that territory or relax restrictions on immigration to it.\n\nThe depth and breadth of Polish antisemitism reflected the close link between Polish nationalism and Polish Catholicism. In the eyes of many Poles, one simply could not be Polish without being Catholic, and the nation's priests heartily concurred. As the most outspokenly antisemitic Catholic clergy in Europe, they usually threw their political weight behind Dmowski and his Nationalists. Church leaders and publications tied the Jews repeatedly to every alien and supposedly corrupting or polluting force in modern life and thus to every current of opinion or behavior that threatened the authority, power, and income of the Church. Poland's cardinals were particularly unabashed about blaming Jews for the nation's problems. The Church's official stance toward Jews remained unchanged since the Middle Ages: They were evil and seditious people who should be shunned, but not harmed physically.\n\nTypical of the entrenched animosity toward Jews among Catholic leaders in Poland was the pastoral letter entitled \"On Catholic Moral Principles\" that Cardinal August Hlond, the Primate of Poland, issued in February 1936. It read, in part, as follows:\n\nIt is a fact that the Jews . . . constitute the avant-garde of godlessness, the Bolshevik movement, and revolutionary activities. It is a fact that Jewish influence on morality is pernicious, and that their publishing houses spread pornography. It is true that the Jews permit fraud and usury. . . . But let us be fair. Not all Jews are like that. . . . One may love one's own nation more, but one may not hate anyone. In commercial matters it is good to prefer your own ahead of others, avoiding Jewish stores and Jewish booths at the market, but one may not plunder Jewish shops. . . . One must close oneself off to the harmful moral influences of Jewry . . . in particular boycott the Jewish press and corrupting Jewish publishing houses, but it is not permitted to attack the Jews, beat them, wound them, injure them, defame them.\n\nTwo years later, Father Jozef Kruszynski, the former rector of the Catholic University of Lublin and the chief intellectual propagator of the _Protocols of the Elders of Zion_ in interwar Poland, summarized the ambivalent teachings of the Church toward Jews. He described their persecution in Germany as barbaric but added, \"Hitler called the Jews the microbe of the world. The accusation is unusually harsh but we must admit that it is correct.\" In short, traditional religious antisemitism remained vivid and strong in Poland, and very few Catholic priests, especially at the parish level, spoke up in defense of the Jews or urged their parishioners to help them. On the contrary, the Church hierarchy repeatedly excused what it called the \"regrettable excesses\" of Polish antisemites by depicting them as understandable reactions to the Jews' disrespect \"for the faith and traditions of Christians.\"\n\nSecond, Jews and Poles tended to live as separate ethnic communities in much of prewar Poland, and there was little sense of solidarity between them. Only 12 percent of Polish Jews described Polish as their native language in a survey conducted before the war. The ambiguous phrasing of the questions asked suggests that this figure may be an understatement, and records of library borrowing indicate that Jews were reading more in Polish than in Yiddish. Nonetheless, most Jews spoke Yiddish primarily, and most who spoke Polish did so with an identifiable accent. Intermarriage and conversion were rare. Jews operated their own choirs, cooperatives, credit unions, cultural societies, hospitals, orchestras, orphanages, newspapers, publishing houses, sports clubs, and theater companies.\n\nIn much of the country, Jews and Poles were divided by residence and occupations. Although Jews came to 10 percent of the prewar national population, they made up 33 percent of the urban dwellers in western and central Poland and between 40 percent and 60 percent in different parts of the eastern half of the country, the area that fell to the Soviet Union in 1939. Although only 1 percent of the Jews were professionals, these Jews accounted for 63 percent of the people employed in commerce as of 1921, and 56 percent of the MDs ten years later, along with 43 percent of the teachers, 33.5 percent of the lawyers, and 22 percent of the journalists and publishers. On the eve of World War II, firms owned by Jews employed more than 40 percent of the Polish workforce, and Jews paid 35\u201340 percent of Poland's taxes. Class resentment and envy, in other words, along with ethnic distinctness and religious differences, created distance between the two communities. In Poland, the belief flourished that Jews had grown disproportionately wealthy by unfair collusion and thus that Poles were justified in repossessing what remained rightfully theirs. Yet much of the Jewish population remained poor, sometimes grindingly so, in part as a result of discriminatory employment and taxation policies enforced by the government. When World War II began, perhaps one-third of the Jews in Poland were dependent on relief aid, most of it coming from Jewish organizations in the United States.\n\nAll of this was less true in Warsaw than elsewhere: Jews in the nation's relatively cosmopolitan capital were more acculturated, more likely to speak Polish and interact with non-Jews, and less uniformly envied or resented. In fact, the relative frequency of contacts across communal lines largely accounts for the number of Jews concealed and saved in the city, which came to around 11,500, perhaps more. Gunnar S. Paulsson, the closest student of the rescue of Jews in Warsaw, has calculated that 70,000\u201390,000 non-Jews must have been involved in the effort. There were so many Jews successfully hiding in Warsaw in June 1943\u2014probably more than 25,000\u2014that the Nazis resorted to a trick. They claimed to have entry documents to various Latin American countries and to be willing to sell them to Jews who would then be exchanged with the Allies for German nationals abroad. The Germans even installed a number of Jews to live comfortably in the Hotel Polski, supposedly the collection point for the exchange. About 3,500 Jews emerged from hiding to fall into this trap and then die at Auschwitz.\n\nThird, politics divided Poles and Jews, too. As early as the Polish-Soviet War of 1919\u201320, the Polish army interned its Jewish troops in a detention camp as security risks. During the interwar period, a greater percentage of Jews than non-Jews stood on the political left in Poland, and in the 1930s, Jews composed more than half of the Polish Communist Party's local leaders and most of the members of its Central Committee, though most Jews did not belong to the Party. Most Poles believed that Jews were pro-communist in 1939\u201341, and, given the Germans' intentions toward Jews, the belief was not unfounded. Yehuda Bauer's book on the Soviet-annexed portion of Poland shows that Jews there did recognize that for them the Russians presented the lesser of two evils in Eastern Europe and generally behaved with corresponding cooperativeness when the Soviets arrived. Writing of 1939 in 1943, Calel Perechodnik described the \"immense happiness\" with which Jews greeted the Soviet occupation of eastern Poland and added, \"This is nothing to be surprised at. From one direction a German invaded, proclaiming slogans of merciless destruction and murder of all Jews. From the other direction, a Bolshevik invaded, proclaiming slogans that for him all people were equal under the law. There was nothing to compare here.\"\n\nGiven the depth of Polish hatred of Russia, born of both the long tsarist occupation in the nineteenth century and the Soviet deportation of some one-half million Poles from the annexed regions to Siberia in 1940\u201341, the general Jewish stance was bound to split the two communities even wider apart. In the case of Jedwabne, allegedly pro-Soviet behavior by local Jews in 1939\u201341 provided the pretext for the murders. And the massacre there was hardly an isolated occurrence: It was one of sixty-six nearly simultaneous such attacks in the province of Suwalki alone and some two hundred similar incidents in the Soviet-annexed eastern provinces. As Stefan Rowecki, a general in the Polish resistance, reported to the Polish government-in-exile in London on July 4, 1941, while the German armies were sweeping across the formerly Polish territory taken by the Soviets, many Poles were ready to offer \"administrative and economic cooperation with the Germans in these areas . . . [as] a knee-jerk reaction of gratitude to their liberators from the Bolshevist oppression in which the Jews had played a big part.\" Ironically, as Yehuda Bauer has argued, two forces\u2014the appeal of Soviet society to Jews in eastern Poland, especially to younger elements of the population, and the economic and antireligious measures that the communists adopted\u2014undermined the cohesion of the Jewish community there between 1939 and 1941. This sapped its capacity to resist the Germans when the invasion came, just as divisions over the Soviet occupation eroded solidarity between Jews and Poles in the region.\n\nJan Kozielewski, a valiant non-Jewish resister during the war who operated under the code name Jan Karski, wrote a report to the government-in-exile in London in February 1940 that made clear how deep this sort of division between Jewish and non-Jewish Poles already had become. He concluded that \" '[t]he solution of the Jewish question' by the Germans. . . . is creating something of a narrow bridge upon which the Germans and a large portion of Polish society are finding agreement.\" In consequence, almost everywhere outside of Warsaw, the principal resistance organization, the Armia Krajowa (AK), or Home Army, excluded Jews from its ranks on the assumption that they were security risks and potentially pro-Soviet. As the Russian armies pushed the Germans back toward Poland's borders, this attitude made some Home Army commanders, who were now fighting a two-front war against the Germans and the pro-Soviet Polish People's Army, even more hostile to Jews. As a result, no fewer than twenty-two of the Jewish inmates who escaped Sobibor during the uprising there in October 1943 died at Polish hands in subsequent days, at least eight of them killed by a unit of the Armia Krajowa. In August 1944, the Barwy Biale detachment of the Home Army, now part of the 2nd Legions Infantry Regiment, discovered three to four dozen Jewish escapees from the Skarzysko-Kamienna munitions factory hiding in a forest and slaughtered every one of them in cold blood.\n\nWhether for religious, social, personal, or political reasons, many peasants and even resistance units in rural areas also routinely killed or turned in hundreds of Jews who tried to hide from the Nazis. In fact, these Poles did this increasingly as time passed and right up until the end of the German occupation. Zygmunt Klukowski, a physician in a small town near Lublin, wrote in his diary on November 26, 1942:\n\nThe farmers are seizing the Jews hiding in the villages, out of fear of possible reprisals, and are taking them to the town, or sometimes simply killing them on the spot. In general, there has been a strange brutalization in relation to the Jews. A psychosis has seized hold of people, and, following the German example, they do not consider the Jews to be human, regarding them rather as an injurious pest that must be exterminated using all available means, like a dog sick with rabies or a rat.\n\nOn many occasions, the so-called Blue Police, the remaining Polish cops on the beat, and local units of voluntary firefighters were also involved in flushing out hidden Jews or picking them up after locals reported their locations. Whenever a successful hunt for Jews in hiding occurred, the local Polish leaders who led it got the right to distribute any property obtained, including the clothing of the victims. Meanwhile, the Germans offered rewards for each Jew delivered up, sometimes kilograms of sugar, sometimes money, and sometimes vodka, and threatened communities in which Germans found concealed Jews with collective punishment. Fear of such punishment had a lot to do with the collective psychosis that Klukowski noticed.\n\nThe evidence that is accumulating suggests that, at a conservative estimate, at least as many non-Jewish Poles turned Jews in as hid Jews from the Nazis. The vast majority of Polish Christians did neither of these things, but the minorities that helped or harmed Jews appear to have been unevenly balanced. Particularly in rural areas, the chances of being protected long enough to survive were slight. One study of what happened in Dabrowa Tarnowska County, about fifty miles east of Cracow, traced in the Polish and German archives and the records of postwar trials the destinies of some 337 Jews who tried to hide there after the liquidation of the ghettos. Fifty-one succeeded and emerged alive after the Soviet armies arrived, but 286 perished between 1942 and 1945. Among those killed, people who died at the hands of Polish civilians and police outnumbered those murdered by Germans 122 to 105. Tellingly, the underground press divided sharply in commenting on this sort of collaboration, with some resistance papers condemning it as shameful while others proclaimed, \"[W]e have to punish those who want to hide Jews and declare them [that is, the protectors] traitors.\" That divergence may account for something else apparent from those figures regarding Dabrowa Tarnowska: Most people who hid Jews there did so in return for money or other payments, yet very few of the Jews hidden on that basis\u2014only 9 percent\u2014actually managed to survive the war. This suggests that they were turned in when they ran out of valuables to exchange for protection.\n\nFourth, during the German occupation, the Polish resistance did rather little to help Poland's Jews, even though it was fully and quickly informed, first, about the conditions in the ghettos and, later, about the deportations and the death camps. The AK did pass its knowledge, including specific references to gassings, to the Polish government-in-exile in London, which publicized it, and did make sure that the underground press within the country disseminated the information. Official proclamations warned Poles against collaborating with the persecution of Jews or blackmailing those in hiding, and late in the war AK units carried out executions for such offenses. Moreover, that government sent Jan Karski to Britain in November 1942 and on to the United States in July 1943 to brief leaders on what was happening in Poland. But in keeping with the Home Army's strategy of hoarding strength until Nazi rule in Poland was on the verge of collapse, the AK made no effort to impede the transports from Warsaw or to blow up the rail lines to Belzec, Sobibor, and Treblinka. For the same reason, the AK provided only modest support for the Warsaw Ghetto Uprising in the spring of 1943: a total of fifty pistols, fifty hand grenades, about ten pounds of explosives, two unsuccessful attempts during the fighting to blow holes in the ghetto walls, and several sniper attacks on German guards. This level of assistance actually marked the high point of AK help to Poland's Jews. After Tadeusz Komorowski succeeded the captured General Rowecki as commander of the Home Army in July 1943, its willingness to aid Poland's few surviving Jews declined. Instead, the AK displayed greater eagerness to combat the so-called banditry of Jewish fugitives and partisan units that survived by requisitioning from peasants.\n\nPerhaps the most powerful demonstration of the halfhearted nature of the Polish resistance's support for Jews is the story of the organization that the government-in-exile formed for that very purpose. Zegota, the Committee to Aid the Jews, was supposed to do so by funding forged papers and hiding places. Not only did it come into existence rather late, in the autumn of 1942, after most Polish Jews had been killed, but its effectiveness was limited. Estimates of how many people the group actually saved vary widely but top out at several thousand, most of them children. Even the leaflet that called for public protests against the deportations and led to the formation of Zegota betrayed the ambivalence toward rescue that undermined such efforts. Zofia Kossak, the author, could not resist noting: \"Our feelings toward the Jews have not changed. We continue to deem them political, economic, and ideological enemies of Poland.\" The Catholic Church hierarchy provided no support and took little notice of the organization, and most of its funds\u2014certainly most of those that actually got to Poland\u2014came from Jewish sources abroad, not the London-based Polish government. One reason for this was that the faction of the National Party in the Polish government-in-exile refused to join or support Zegota and kept up a constant drumbeat of underground antisemitic propaganda.\n\nThe National Party's presence and actions undermined the government-in-exile's declarations that postwar Poland would be a state in which all citizens had equal rights. In fact, one of the most remarkable aspects of the Holocaust in Poland is how little impact the carnage had on the attitudes that had prevailed toward the nation's Jews before World War II. A survey of the leaders of thirteen political groups in the Polish resistance at the end of 1943 established that they favored liquidation or emigration of the Jews over integration and equality in a postwar state by a ratio of nine to four. Even among political prisoners in the concentration camps, according to Hermann Langbein, the persistent antisemitism of the Poles stood out.\n\nFifth, precisely because there were more Jews in Poland than anywhere else, more people stood to gain by their disappearance than anywhere else, and this, too, undercut any sense of solidarity. In fact, the Nazis consciously set out to buy the loyalty of non-Jews, especially in the regions conquered from the USSR in 1941, by swiftly giving away Jews' household possessions to the local populations and by turning schools, community houses, synagogues, and hospitals into service facilities for the local non-Jewish population. And even the death camps became sources of local enrichment, since the surrounding villages profited from the spending of guards and sometimes from the black market in goods taken from the people killed. Thanks to rumors that spread through the occupied country, currency and jewelry dealers set up shop around Treblinka in 1942\u201343, and prostitutes were drawn to the area. The extent to which many ordinary Polish citizens benefited from the murders is apparent even today in the Polish restitution laws, which stipulate that no one can reclaim property stolen from Jews unless the applicant is a resident of Poland. Given that most surviving Jews left after 1945 or were driven out by the late 1960s, this law effectively protects a massive degree of theft, and it was designed to do so precisely because the theft was so massive.\n\nSixth, antisemitism in Poland outlasted the Holocaust and continued afterward. Nechama Tec, who studied rescuers and who survived wartime Poland thanks to several of them, recalls in her memoir, _Dry Tears_ , that the first thing her protectors asked of her after the Russians liberated their town was not to tell anyone who had hidden her. Many other hidden children in Poland have reported the same thing, and when the Jewish Historical Commission in Cracow began to publish the names of rescuers in 1947, many of them requested that the organization omit them in the future. Polish rescuers often expected disdain from their neighbors for their kindness, not praise or respect, and often got what they expected. The only family to hide Jews from the massacre in Jedwabne in 1941 experienced so much disapproval after the war that most of the members emigrated to Chicago.\n\nMultiple pogroms broke out in Poland after the war; one of them took the life of Chaim Hirszman, one of only two survivors of Belzec. The Kielce pogrom of 1946 erupted when someone claimed that the Jews who had returned to live in the city's Jewish community center had kidnapped and killed a gentile child, just as the ancient blood libel alleged. Jan Gross, who has closely studied what happened in a book called _Fear_ , argues that the pogrom was not just an outbreak of endemic Jew hatred but also an attempt to remove witnesses to the extent of previous Polish complicity in the Holocaust. Indeed, a good deal of prosopographical evidence suggests that some of the most enthusiastic Polish antisemites in 1942 and 1946 tried to cover their tracks after the war by becoming some of the most enthusiastic Polish collaborators with communism. Meanwhile, other antisemites simply carried on the old tradition of depicting Jews as people ready to betray Poland to the Reds and spread stories\u2014once more exaggerated, but not completely unfounded\u2014of Jewish \"overrepresentation\" in the communist security services.\n\nAs a result of the pervasively antisemitic atmosphere, some 250,000 Jews, including many who had just returned to the country from their earlier refuge in the Soviet Union, fled westward from Poland in the first years after the war ended. In the late 1960s, the communist government of postwar Poland organized a so-called anti-Zionist campaign, largely in order to divert mounting popular unrest but also to disprove the regime's lingering reputation with the Polish public as a tool of Jews. After 1989 and the fall of communism in Poland, Lech Walesa, the hero of the Solidarity movement, showed how persistent antisemitism remained in parts of Polish society. He tried to discredit a competing candidate for the presidency of Poland by asserting that he was of Jewish descent and later dismissed Jan Gross's findings about Jedwabne as the work of \"a Jew who tries to make money.\"\n\nSeventh, more Poles are commemorated at Yad Vashem for saving Jews than any other nationality. In part, this simply follows from the fact that Jews were more numerous in Poland than anywhere else, so even a lower than average number of would-be rescuers would have saved more people or died trying than elsewhere. Still, Tec's studies and those of other scholars contain thousands of stories of Polish courage in defense of Jews in a place where this was especially dangerous. Emanuel Ringelblum, the creator of the Oyneg Shabes archive in the Warsaw ghetto, was hidden, along with his wife and son and thirty-four other Jews, in a hideout prepared and owned by a non-Jew named Mieczyslaw Wolski. He and his nephew were killed with those Jews when the Germans discovered the bunker in March 1944. Polish and German researchers have identified reliably almost 1,000 cases of Poles executed for helping Jews in hiding, nearly all of them not included in the Yad Vashem total.\n\nSo, to what conclusion do these seven points lead? Above all, to a call for understanding and for suspending the mutual blaming and competing claims to having suffered worst. The key point to remember is that the Nazis created a Hobbesian world in annexed and occupied Poland, where no indigenous government existed to exert restraint and different parts of the population were constantly pitted against each other in a desperate struggle to survive. This was not fertile ground for the growth of a sense of common interest. The environment was far more conducive to preoccupation with one's own interests and taking advantage of opportunities. That is why the fact that the Nazis victimized Poles in many ways did not immunize some Poles from several sorts of complicity in the Germans' crimes.\n\nAchieving this kind of balanced perspective has been and will continue to be much harder for people in Poland than it should be for those living elsewhere. This is because interpreting a nation's history is generally a high-stakes political game to its citizens; as William Faulkner famously observed, \"The past is never dead. It's not even past.\" Ever since 1945, descriptions of non-Jewish Poles' behavior toward Jewish ones during the Holocaust have provoked considerable controversy in Poland. People on the left, considering themselves increasingly secular and progressive and perhaps less embittered by the experience and memory of communism, have dwelled upon the hostility or indifference of the Catholic Church and the antipathy of the Home Army toward Jews. People on the right, closely identified with Catholicism and traditional values and still mistrusting the supposed Jewish-Bolshevik connection, have focused on every remotely demonstrable image of Christian charity and national generosity toward persecuted people in wartime Poland. Because these views of history function as sources of both identity and legitimacy in the present, Poles will continue to argue intensely over who did what to whom on their soil in the years 1939\u201345.\n\nBut the rest of us should have less difficulty acknowledging and empathizing with the suffering of nearly all parties, without asserting a false equivalence. The fate of the non-Jews who largely survived the war was not the same as that of the Jews who were, at least in Poland, virtually wiped out. Jews in occupied Poland were fifteen times more likely to be killed than non-Jews. And an ideological reason accounts for the difference: In the Nazi New Order, the Jews were destined for swift death, the Poles for enslavement and exploitation, but for extinction only when the time came that Germany no longer needed their labor.\nCHAPTER 7\n\n[ONLOOKERS: \nWhy Such Limited \nHelp from Outside?](contents.xhtml#ch_7)\n\nIF THE NAZIS meant what they said during the 1930s, namely that their goal was to \"remove\" Jews from German territory, then the best chance to save large numbers of Jews from what later became the Holocaust lay in their escape to other countries. As it happened, 60 percent of the Jews of Germany did get away in this fashion, along with 67 percent of the Jews of Austria, and about 25 percent of the Jews in Bohemia and Moravia. But very few Jews in Hitler's path toward living space could emigrate in the 1930s\u2014even fewer once the murders began. Instead, Jews discovered that no outside power would or later could offer them much beyond rhetorical support and promises of reprisal, and even this sort of backing was quite measured. Why couldn't more people get out of harm's way? Why did the Jews receive such limited help?\n\nThe short answer is that a combination of antisemitism and economic and political interests worked to restrict the admission of Jews to other countries throughout the Holocaust and to inhibit other action on their behalf. Sooner or later, every nation that might have helped decided that it had higher priorities than aiding or defending Jews. So did the League of Nations, headquartered in Geneva; most non-governmental organizations, such as the International Olympic Committee and the International Committee of the Red Cross; and almost every transnational religious institution, including the Catholic Church. The result was an erratic line of possibility for those persecuted within Germany's borders. Opportunities to get out of Nazi hands were widest in 1933\u201334 and again in 1938\u201339, but very narrow in the years in between or afterward. For the Jews of Eastern and southeastern Europe in the 1930s, the prospects were even worse.\n\nPREWAR EVASIONS\n\nThe initial opportunity for Germany's Jews stemmed from the hospitality of four democracies on the Reich's periphery: France, the Netherlands, Belgium, and Czechoslovakia. Revulsion at Nazi brutality arose in all four places, reinforced, in the case of the French, who took in 55,000 Jews between 1933 and 1939, by the same receptiveness to other immigrants that France had shown since the Great War depleted the nation's population. But in all four instances, sympathy declined over time, especially in France, where the late onset of the Depression compared to other countries meant that hostility to economic competition from refugees peaked just as the need for asylum did. By the mid-1930s, France enacted various rules that made immigration less appealing, restricting the practice of medicine to citizens and placing quotas on foreign artisans who could enter the country, for example.\n\nAfter 1936, four other arguments arose and gradually narrowed the passage through France's gates. First, the popularity of the policy of appeasing Nazi Germany\u2014that is, allowing changes to some of the terms of the Versailles Treaty in order to avoid war\u2014made the presence of Jewish refugees politically inconvenient. Second, the election in 1936 of L\u00e9on Blum, a Jew, as prime minister of a leftist government called the Popular Front mobilized antisemitic feeling among French conservatives. Third, opponents of immigration stoked suspicion that escapees from Nazi Germany would include spies who would undermine French security. And fourth, critics pointed out that letting in German refugees would lead to admitting far more numerous Jews seeking to escape from Poland. The influential journalist Emmanuel Berl, himself a Jew, argued in November 1938 that such people were \"taken as a whole not very desirable.\" Opening the borders to them, he said, would be an act of \"crazy generosity.\" Even the French Jewish Committee split over the advisability of increasing Jewish immigration and failed to make a strong case for it. After _Kristallnacht_ , France actually made getting into and staying in that country more, not less, difficult for refugees, imposed prison sentences on illegal residents, and even sentenced the aunt and uncle of Herschel Grynszpan to six months in prison for having let him live in their home as an unauthorized immigrant.\n\nThe trend of events in the Netherlands, Belgium, and Czechoslovakia was similar, though not for exactly the same reasons. In Holland, the general policy was to take in any refugees who crossed the border but to make sure that the local Jewish community, through its Jewish Refugee Committee, formed in 1933, paid for their upkeep, and then to hasten them on their way. Thus the Dutch government progressively narrowed work opportunities for Jews in Holland during the 1930s. The 22,000 German Jews who were in or had passed through the Netherlands by the end of 1937 depended almost completely on charity, much of it obtained from the American Jewish Joint Distribution Committee, usually referred to as \"the Joint.\" The name reflected the organization's origins as an alliance of philanthropic groups associated with differing strains of political and religious opinion within the American Jewish community. Following the German pogrom of November 1938, the Dutch government decided to intern all new immigrants in camps for which the Refugee Committee would have to provide a million guilders ($550,000 at the time) in construction and maintenance funds. Though Holland generally did not enforce its threat to turn away all refugees at its borders beginning in December 1938, construction of a central internment camp at Westerbork began for the estimated 23,000 to 30,000 German Jewish refugees in the country as of early 1939. Of that number, 7,000\u20138,000 got away before the Germans invaded in May 1940. Belgium also took in about 30,000 German Jews between 1933 and 1939, about half before and half after the pogrom, but also made entering the country and staying progressively more difficult. Statistics from Czechoslovakia highlight its hardening policies: 60,000 Austrian Jews applied for residence following the pogrom of November 1938, but only about 6,000 got in, most of them illegally. In short, the chances of finding refuge in Western Europe declined as desperation to escape Nazi Germany rose.\n\nSwitzerland provided the most glaring illustration of this tendency to close the escape hatches precisely when they were most needed. Never very receptive to Jewish refugees during the 1930s, thanks largely to the efforts of an antisemite named Heinrich Rothmund, who headed the Swiss Federal Police for Foreigners, the country actually sealed its borders on August 19, 1938, and deployed troops to catch and repatriate anyone trying to enter from Nazi-occupied Austria without the appropriate visa. Paul Gr\u00fcninger, a courageous police captain in the frontier canton of St. Gallen, was one of the rare officials who refused to comply with these instructions. As a result, about 1,000 Jewish refugees slipped into Switzerland via his jurisdiction before his conduct cost him his job early in 1939. Meanwhile, in order to preserve normal tourist traffic with Nazi Germany while keeping Jews from there out, the Swiss persuaded the Germans to stamp the passports of German Jews with a large _J_ and began categorically refusing them admission. One month later, in October 1938, the Swedes adopted the same practice.\n\nThroughout the 1930s, the USSR declined to offer a haven to all but a handful of ranking Jewish communists. The Soviet Union declared that Jewish refugees were unsuited to life in an unfamiliar socialist society and, in any case, not the USSR's responsibility, since their persecution was a product of capitalist quarreling. From September 1935 on, Jews entering the socialist motherland had to satisfy several discouraging and, at least in part, mutually contradictory preconditions: proletarian ancestry, possession of substantial amounts of money, and willingness to become Soviet citizens and perform manual labor on construction sites in northern or eastern parts of the USSR.\n\nBritain saw its role in the crisis of German Jewry as that of a \"transit nation,\" one that might allow refugees to land on the \"tight little island\" but not to stay very long. Given the reluctance of the Dominions, such as Canada, Australia, New Zealand, and South Africa, to provide destinations\u2014South Africa, for example, accepted only 6,000\u20137,000 Jewish refugees during the 1930s and Canada fewer than 5,000 in the entire period of 1933\u201345, including a grand total of only 23 from Germany and Austria during 1938, the year of the _Kristallnacht_ pogrom\u2014this self-image meant that Britain was not a frequent refuge for Jews seeking to get out of Germany. The country took in only about 70,000 European Jews during the 1930s, only 10,000 of them up until the end of 1937, and fully 50,000 of them just in the short time from January to September 1939, including some 10,000 young people on the famous _Kindertransport_ , the Children's Transport.\n\nMoreover, Britain pursued similarly restrictive policies in Palestine, the territory in the Middle East from which the Romans had driven most Jews in ancient times and to which Zionists wanted to return, but which for the present was subject to British rule under a mandate from the League of Nations. To be sure, in the Balfour Declaration of 1917, the British government had declared its rhetorical support for a future \"national home for the Jewish people\" in the region, largely and cynically for two reasons. First, Britain wanted to bolster its claims to the territory after the war. Second, the British cabinet hoped the declaration would prompt allegedly influential Jews in the United States and elsewhere to support the countries then at war with Germany and Austria. Ironically, Edwin Montagu, the only Jewish member of the cabinet at the time, voted against the declaration, in part because he was appalled by the antisemitic overtones of this fantasy of Jewish power, in part because he rejected Zionism as likely to bring strife and misery to Palestine. Partly in consequence of the declaration, the Jewish population in the region rose to around 400,000 by the mid-1930s. This growth set off the violent backlash that Montagu had predicted in the form of the Arab Revolt (1936\u201339). Henceforth, the British thought that their control, not only of Palestine but also of the Suez Canal, the jugular vein of the British Empire, depended on placating Arab opinion, so they reduced the already low permissible annual quota of Jewish immigrants. Whereas 149,076 Jews from all countries got into Palestine from 1933 to 1935, only 54,899 did in 1936\u201338. The annual intake rose again in 1939, but only to 31,195, and a government white paper of May 1939, issued just after the revolt finally was suppressed, set the quota for the ensuing five years at 15,000 per annum for a total of only 75,000 more places, after which Jewish immigration to the region would cease altogether. The British explicitly stated that their goal was to confine the Jewish proportion of the population in Palestine to one-third.\n\nIn Britain, as virtually everywhere else, the people who made these restrictive policies claimed to be haunted by the specter of what might happen if they were more generous. They feared that more open borders would prompt Eastern European governments, especially those in Poland, Hungary, Romania, and Lithuania, to enact even more antisemitic measures than they already had and thus to set off an exodus of almost five million Jews, which is to say 5.5 times the combined Jewish population of Germany, Austria, and western Czechoslovakia in 1933. Such concerns were not imaginary. At the meeting of the Council of the League of Nations in May 1938, Poland and Romania explicitly expressed the desire to reduce the size of their Jewish populations and requested aid in doing so. The following October, Poland's ambassador in London tried to blackmail Britain into allowing 100,000 Polish Jews into its colonies per year by stating that otherwise his government would be \"inevitably forced to adopt the same kind of policy as the German government.\"\n\nThe fate of these Eastern European Jews also haunted the Jewish Agency in Palestine, to which the British occupation authority delegated the distribution of the annual allotment of legal entry permits, and that fact further narrowed the access of German Jews to the region. Because the Jews of Poland and Romania seemed equally in danger but were historically more pro-Zionist and currently had fewer avenues of escape than the Jews of Germany, the agency restricted the latter population's share of the permits awarded annually from 1933 to 1938 to an average of only 22 percent. The proportion exceeded one-third in only the last of those years, when it topped 40 percent. In short, Nazi racism set off a vicious circle in which partial success in driving Jews out of Germany encouraged imitation by bigots in countries to the east that, in turn, pressured potentially hospitable places into scaling back possible exit opportunities for German Jews.\n\nFear of a refugee flood also prevailed in the United States, along with other obstacles to generosity toward refugees from Nazi persecution. The problem was not at the top of the American government; President Franklin D. Roosevelt was no antisemite\u2014indeed, he appointed more Jews to senior positions in government than any president before him. He was anti-German, an attitude born of both bad experiences with a German governess in his youth and his service as secretary of the navy while the United States fought Germany in 1917\u201318. But existing American law and public opinion hamstrung him from doing much to help Jewish refugees during the 1930s, and he avoided taking political risks for the sake of Jews abroad.\n\nThe legal obstacles stemmed from the quota system of immigration that the United States introduced during the 1920s. It set a maximum total of 150,000 legal entrants to the country annually and apportioned that figure almost entirely among European nations according to the share of the American population that traced its descent to each of them in the census of 1890. The year was not an accidental or arbitrary choice but an intentional and eugenicist one. Congress selected 1890 because it antedated a great influx of immigration from Italy, the Balkans, and Russia around the turn of the century. America's legislators wanted to give preference to predominantly White Anglo-Saxon Protestants over all others. Ironically, this resulted in a relatively large permissible number of arrivals from Germany per year, 25,957 people\u2014relatively large as a share of the number of entrants allowed annually (more than one-sixth) but, of course, tiny in relation to the need in the 1930s, since Germany had 560,000 inhabitants whom the Nazis considered Jews when Hitler came to power and later added some 300,000 Jews in Austria, the Sudetenland, and Bohemia-Moravia. At almost 26,000 German Jewish immigrants a year, admitting everyone in this population would have required thirty-three years; admitting even the almost 310,000 German, Austrian, and Czech Jews who actually applied for entrance by 1939 would have required almost twelve years.\n\nBut the United States had no intention of admitting the full quota of German immigrants annually, let alone of allowing Jews to fill the full allotment. In fact, from 1933 to 1939, when the quota permitted the admission of up to 156,000 people, the United States let in only 77,000, including about 65,000 Jews. From all of Europe in this interval, the United States took in only 92,000 Jews. If one extends the time frame to 1933\u201344, the immigration total for America is probably about 225,000 Jews from all of Europe, including 120,000 Jews from Germany and Austria, and the number of unused German quota slots rises to 190,000.\n\nThese admission figures are paltry compared to the European Jewish population of nine million in 1939 or the six million Jews killed in the Holocaust. But 225,000 is three times the number of people Great Britain took in and almost fifty times the number that Canada accepted; 120,000 Jews is more than any other country admitted from the German Reich; the number the United States took rose annually from 1933 to 1940; and the total admitted from 1937 to 1941 was more than four times the total of the preceding four years. In 1938, President Franklin Roosevelt combined the immigration quotas for Germany and Austria to raise the permissible number of entrants to 27,370 and thus improve the odds of Jews getting out, and in 1939, he issued an executive order indefinitely extending the visitors' visas of all Jews then in the United States, thereby saving another 15,000 Jews from repatriation and death. The most authoritative study of FDR and the Jews concludes that between 1937 and 1941, \"FDR's second-term policies likely helped save the lives of well over 100,000 Jews.\" In other words, America performed terribly in the face of the crisis of European Jewry, except in comparison to every other country. Moreover, at least until World War II broke out, American receptivity gradually increased, precisely as refuge in Europe was growing generally harder to find.\n\nWhy didn't the United States do better? The short answer is that both powerful individuals and public opinion opposed doing better, with the results that no serious move to change the immigration quotas arose, and strict enforcement of immigration rules held down the numbers admitted until very late during the 1930s. American policy was not as harsh as British, which assured that that country admitted more than five-sevenths of its total refugees at the last minute before World War II began. Still, in the United States, the comparable figure was one-half in the short interval between _Kristallnacht_ and mid-1939.\n\nOpposition to immigration fed on three primary causes: fear of economic competition, popular nativism and isolationism, and elite antisemitism. The fear of economic competition was expressed in many quarters. For example, the dentists of Westchester County lobbied FDR's political advisor Samuel Rosenman to prevent the admission of any more refugee dentists into the country, and the national conventions of the Veterans of Foreign Wars and the American Legion passed resolutions against further immigration so long as unemployment persisted in the United States. Such resistance resulted in strict enforcement of the Likely to Become a Public Charge, or LPC, rule, which denied immigration to people considered so lacking in funds that they would become dependent on welfare. U.S. consular officials abroad, who received applications to immigrate, required extensive data on each person's likely financial resources after arrival and generally set, as their superiors in Washington required, a high standard for economic security before granting a visa. Frances Perkins, the secretary of labor, argued vehemently for a relaxation of these standards, but the State Department argued equally forcefully in their favor. FDR sided with State except in two brief intervals, one during late 1936 and the second in 1938\u201339 after the _Anschluss_ and _Kristallnacht_.\n\nThe rationale behind FDR's stance was straightforward: The American public opposed letting more people in. Throughout the 1930s, every national public opinion poll on the question showed that two-thirds to three-quarters or more of Americans rejected the relaxation of the quotas and the admission of more refugees. As a result, early in 1939, Congress defeated the Wagner-Rogers Bill, which would have admitted 20,000 Jewish children under the age of fourteen to the United States. Not only was the general public hard-hearted, but student opinion followed: the _Daily Northwestern_ of December 13, 1938, reported that 68 percent of American students were against more admissions for fear of \"imperiling U.S. living conditions.\" The sad truth is that virtually no politician outside of a few urban centers on the East Coast could get elected in the United States during the 1930s on a platform of offering asylum to the Jews of Europe. FDR was a politician who had to win across the nation, not just in these pockets of empathy.\n\nNativism and isolationism were revved up by the radio broadcasts of a Detroit-area priest named Father Charles Coughlin, who peddled the same charge against Jews that others had leveled at his Irish forebears decades earlier, namely that they could not fit into American life. He had an audience of three million fans listening weekly and cited (and republished) the infamous _Protocols of the Elders of Zion_ to make his point. The Catholic hierarchy did not muzzle him until 1942. Meanwhile, far more mainstream figures played on American antisemitism, including those leaders of the Republican Party who declared that, thanks to FDR's appointment of many Jews to office, he was offering the public, not the New Deal but the Jew Deal. Given such attitudes, perhaps one should not be surprised that a poll in 1938 found that 58 percent of Americans considered Jews in Europe at least \"partly\" at fault for their own persecution. Another survey in July 1939 showed that 32 percent of Americans believed Jews had too much influence in business, while another 10 percent favored deporting Jews. The prevalence of these views caused even the proponents of helping Jewish refugees to sanitize their vocabulary and to speak of \"persecutees\" who needed help, not of Jews.\n\nA telling illustration of the forms antisemitism could take comes from the history of the university where I taught for thirty-six years, which had restrictions on the number of Jews admitted annually until the 1960s. In January 1939, the _Daily Northwestern_ ran an article about a class in the Medill School of Journalism that had compiled a list of the ten greatest news stories of 1938. The group considered worthy of inclusion such by now long forgotten events as the wrong-way flight of Douglas Corrigan\u2014he ostensibly set out to fly to California from New York but went to Ireland instead\u2014and the Lima Conference of Pan-American states, but not the _Kristallnacht_ pogrom, even though the burning of the synagogues in Germany in November 1938 had made the front page of the _Chicago Tribune_. On the almost universally white and predominantly Protestant students of Medill, the attack on Jews in Germany seems hardly to have registered.\n\nAttitudes at Northwestern testified to the power of antisemitism among American elites in the 1930s. Especially important as a representative of this current was an assistant secretary of state with the sonorously white Anglo-Saxon Protestant name of Breckenridge Long. A former ambassador to Italy and a fervent admirer of Mussolini, he fought hard to keep down the number of Jews admitted to the United States annually. His most convenient and effective argument was the possibility that relatives remaining in Germany could be used to blackmail refugee Jews into becoming spies for the Reich. Every train or boat carrying Jews out of Nazi Europe, he said, \"is a perfect opening to Germany to load the United States with agents.\" As a result, early in June 1941, the U.S. State Department instructed its consuls worldwide to deny visas to foreigners who had close relatives in Germany or the countries it controlled.\n\nAgainst these impediments to accepting more refugees, no political force arose that was strong enough to prevail. FDR chose at key moments to offer modest help, but he did not want to incur political costs. Immediately after the _Anschluss_ , he asked his cabinet whether Congress would support an expansion of the German immigration quota but backed off when the members answered in the negative. He became even more cautious after the Republicans, including many isolationists, gained eighty-one seats in the House and eight in the Senate the following November, a few days before _Kristallnacht_. In 1939, he encouraged several Latin American states, notably Bolivia, Brazil, the Dominican Republic, and Paraguay, to admit more Jewish refugees. But he also refused to issue a special order to let the refugee ship _St. Louis_ land in the United States or even in the U.S. Virgin Islands, as offered by that territory's governor and legislative assembly and advocated by two members of FDR's cabinet. He also declined to endorse the Wagner-Rogers Bill. Taking the long view, he decided that such actions would undermine his effort to get Congress to repeal America's Neutrality Acts and thus would prevent him from helping countries to resist Hitler's aggression later. Shortly thereafter, he backed away from the idea of using Alaska as a refuge for Jews once he learned that he would then have to set up new restrictions on travel between there and the continental United States. At a press conference in June 1940, he even repeated Long's claim that the German government was threatening to shoot the relatives still in the Reich of refugees who declined to work as spies for Germany. Finally, in March 1941, FDR did nothing to reverse the U.S. Maritime Commission's denial of permission for an ocean liner, the _S.S. Washington_ , with a capacity of 1,700 passengers, to add a direct route to New York from Lisbon, almost the last escape hatch from Europe.\n\nInterestingly enough, German intelligence offices in occupied Holland did concoct a scheme for smuggling agents into the Americas under the cover of releasing Jews, and some 486 of them got permission to leave the Netherlands for Spain, the Caribbean, and South America between May 1941 and January 1942. But in view of the time frame, this project clearly was an outgrowth of Long's often publicly expressed fears as much as a vindication of them.\n\nThe American Jewish community proved on its own incapable of rallying popular solidarity with Europe's Jews. Though Jews then constituted a slightly larger percentage of the U.S. population than they do now, they were divided by heritage between the American Jewish Committee, whose members traced their lineage back to Germany for the most part, and the American Jewish Congress, headed by Stephen Wise, whose members hailed primarily from Eastern Europe. The former group feared that too much agitation for Jewish immigration would stoke antisemitism and preferred behind the scenes, high-level efforts to exert influence. The latter group favored public rallies and boycotts of German goods to put pressure on the Nazi regime. Neither organization had much effect on U.S. policy toward refugees. The two groups also differed in their attitudes toward the creation of a Jewish state: the committee's leaders were non-Zionist or sometimes anti-Zionist, whereas the congress favored settlement and eventually a Jewish state in Palestine. That stance on the part of the congress meant that it had conflicting priorities: Escape to the United States was not escape to the prospective Jewish homeland, so it was both desirable and not. Even David Ben-Gurion, the principal Jewish leader in Palestine, feared that too much receptivity to Jewish refugees elsewhere would endanger the Zionist project. Conversely, the American Jewish Joint Distribution Committee, dominated by the American Jewish Committee, preferred to concentrate its resources on supporting the increasingly impoverished German and Eastern European Jewish communities and on sustaining refugees in Europe and the Western Hemisphere rather than on aiding immigration to Palestine.\n\nThe difficulties of getting out of Germany gave rise to an improbable escape route in late 1938, one of the few that remained available in the early years of World War II. Perhaps 17,000\u201320,000 European Jews found refuge in Shanghai, on China's east coast, more specifically in the part of the city called the International Settlement, which a consortium of eleven countries ruled until December 1941, when Japanese troops marched in. People who reached the International Settlement needed no visa to enter, just transit visas through any countries en route, which were usually the USSR plus the Japanese puppet state of Manchukuo, in northern China, and both were happy to collect the fees. The Japanese occupiers confined the Jews to a slum neighborhood called Hongkew, where most of them survived the war through complicated transfers of aid from the Joint Distribution Committee.\n\nThe futile Evian Conference of 1938 reflected a widespread tendency of nations to pass the buck when it came to offering a haven to Jews. The outcome confirmed the observation two years earlier by Chaim Weizmann, the president of the World Zionist Organization, that Jews confronted a globe \"divided into places where they cannot live and places they cannot enter.\" During the late 1930s, as historian Bernard Wasserstein writes, \"Dutch Guiana, Angola, Cyprus, the Philippines, the Belgian Congo, the Dominican Republic, Mexico, Haiti, Ethiopia\u2014each was broached, researched, and hailed as a potential haven. In each case obstacles were discovered and globes twirled again, until eyes fixed anew on the latest, ever more improbable land of redemption.\" In view of all this, the remarkable thing is how many people got out, not how few. The fate of the passengers on the refugee ship _St. Louis_ was prototypical, in this sense. Of the 937 Jews on board, 28 got off in Cuba, and one committed suicide. Of the 908 remaining, 620 were admitted to France, Belgium, and Holland, where 365 survived the war. Britain admitted 288, all of whom were similarly fortunate. In short, about three-quarters of the once apparently doomed passengers in fact found secure refuge from the Germans, and about half eventually found their way to the United States. But these were German Jews, and the large share of them who got out mirrored the relatively high overall escape rate of their fellows. Farther east, the chances were far worse. In the year 1937, for example, only 9,000 Jews managed to emigrate legally from Poland to any new homeland, and the annual U.S. immigration quota for that country came to only about 6,000 people.\n\nA similar sense of ambiguity surrounds the actions of another international player that might have done more to help the Jews in the 1930s, namely the Roman Catholic Church, especially its spiritual head in the Vatican. The Church's record has both up and down sides, but in the end, like most of the countries discussed thus far, it looked out for itself first and did not do nearly as much not only as it could have done but also as its leaders at one time or another actually thought of doing.\n\nFrom the start, the Church's leaders in Rome recognized that Nazism represented a barbaric force. They had to be talked into signing the Concordat with Hitler in July 1933 by Germany's Catholic cardinals, a majority of whom favored the agreement as the only way to limit Nazi incursions on Church activities. Almost immediately after Hitler came to power, a group of Jesuits began drafting a condemnation of Nazism for the pope to issue. Four years later, this document became the encyclical entitled _Mit brennender Sorge_ ( _With Burning Sorrow_ is the official translation, but _Sorge_ actually means \"concern\" or \"anxiety\"). Despite that dramatic label, the wording was considerably watered-down from what the Jesuits had prepared. Though it denounced the glorification of race and nation as \"idolatrous,\" the text did not mention Nazism by name and was far less critical of Germany's ruling regime than another encyclical condemning communism issued only a few days earlier. The juxtaposition was telling: However much the Church's leaders in Rome despised Nazism, they always hated and feared communism more, and this fact consistently caused the Vatican to pull its punches in dealing with Hitler's regime. When the Nuremberg Laws appeared in 1935, the Church said nothing. It was likewise silent in 1938, when Nazi mobs burned synagogues, smashed homes and shops, and arrested and beat Jews.\n\nChurch authorities hardly could mount a full-throated defense of Jewry from persecution because they had so long advocated and indeed enforced forms of it. As Mussolini pointed out when he inaugurated Italy's first antisemitic laws in 1938\u2014which excluded Jews from the Fascist Party, the military, and public education, ejected them from honorary societies, revoked grants of citizenship to them since 1919, and limited the size of businesses or estates they could own\u2014these restrictions were not as severe as the ones the popes had imposed in the lands they ruled until 1870, including the city of Rome. Moreover, the limits on Jewish activity were quite similar to those that major Catholic publications, including _La Civilt\u00e0 Cattolica_ , the biweekly Jesuit publication whose contents had to be cleared by the Vatican before printing, had been advocating for more than fifty years. Behind that advocacy stood the Church's traditional conviction that contact with Jews could corrupt the faith of believers, now reinforced by the view that Jews, Freemasons, and Bolsheviks constituted an unholy modern and conspiratorial alliance against all that the Holy Mother Church stood for. Shortly before the publication of _Mit brennender Sorge_ , _La Civilt\u00e0 Cattolica_ published \"The Jewish Question,\" an article that denounced the Jews' supposed \"domination over money and their preponderance in socialism and communism\" and concluded by suggesting that the only way to contain their influence was to strip them of citizenship in Christian nations. In May 1937, another article began with this sentence, \"It is an evident fact that the Jews are a disruptive element due to their spirit of domination and their preponderance in revolutionary movements,\" and went on to advocate segregating Jews from Christians. And, in mid-July 1938, the journal wrote of the Jews' \"messianic craving for world domination\" before endorsing the recent enactment of laws in Hungary designed to restrict Jews' professional opportunities.\n\nNonetheless, Pope Pius XI, who reigned until February 1939 and grew increasingly appalled at Nazi violations of the terms of the Concordat and glorification of the Aryan race, contemplated speaking out. In September 1938, he told a group of Belgians visiting the Vatican, \"antisemitism is inadmissible. Spiritually, we are all Semites,\" and just before Christmas that year, he referred to the Nazi swastika, for which the German word is _Hakenkreuz_ (hooked cross), as \"a cross that is the enemy of the Cross of Christ.\" But these remarks remained known only to the few who heard them and passed them on. Temporizers and outright antisemites in the Vatican not only made sure of that but also undercut an initiative the pope had taken the previous June, when he asked an American Jesuit named John LaFarge to draft an encyclical tentatively titled \"The Unity of the Human Race.\" Wlodzimierz Ledochowski, the vehemently antisemitic head of the Jesuit order, first assigned two more traditionalist clergymen to assist LaFarge and then held up the completed text for months before reluctantly passing it on in January 1939, less than a month before the pope died. The document went no further than his bedside table. The text has survived, however, and it suggests that neither the Church nor the reigning pope could break free of the policy of caution toward Nazism or the doctrinal inheritance of certain forms of antisemitism. The text contained an ambivalent argument that on the one hand condemned racism as heretical and called for an end to the persecution of the Jews, but on the other, repeatedly referred to the supposed moral failings of Jews and the danger to the faithful of paying attention to or coming into close contact with them.\n\nEven that was too much criticism of Nazi racism for Pius XI's successor, the former Vatican secretary of state Cardinal Eugenio Pacelli, who took the papal name of Pius XII. In 1936, Pacelli had opposed issuing _Mit brennender Sorge_ as an encyclical to be read in German churches and suggested a mere pastoral letter to just the German bishops. Not without reason was he the candidate the Nazi envoys in Rome hoped would prevail in the conclave that chose the new pope. After his election on the third ballot, in March 1939, he destroyed not only all the copies of his predecessor's draft encyclical that his aides could find, but also the very plates on which a speech Pius XI had been planning on the subject had been printed. A Roman aristocrat by descent and a diplomat by training, Pius XII was a deeply cautious and conservative man with regard to both politics and theology. As the papal nuncio in Germany prior to Hitler's accession, he had grown fond of the country and critical of the Catholic Center Party for its participation in democratic politics, which he disliked, and for its longtime opposition to becoming part of a coalition government with the Nazis, which he advocated. Though not happy with the form the Concordat took and disappointed that it contained no protection for Jewish converts to Catholicism, he negotiated the agreement and then clung to it in subsequent years as the best hope for the Church's survival in the Third Reich. Above all, he abhorred \"godless communism\" and sought to coexist with any political regime that combated it.\n\nPius XII considered his chief duty to be to the Church and to Catholics, not to suffering people in general, and some critics have reproached him therefore for practicing parochial rather than pastoral politics. Not all of these critics have spoken in hindsight or from outside the Church. Some of the leading German clerics, notably Bishop Konrad von Preysing in Berlin, advised him to display greater firmness in dealing with the Nazi regime and to express greater solidarity with German Jews, but others overrode such arguments. The dominant voice was that of Cardinal Adolf Bertram in Breslau, who feared provoking a renewed persecution of the Catholic Church in Germany along the lines of Chancellor Otto von Bismarck's _Kulturkampf_ during the 1880s. Thus, although Preysing, for example, organized various efforts to help Jews in Berlin, especially ones who had converted to Christianity, the Vatican encouraged neither these actions nor their imitation by other dioceses.\n\nTo appreciate the constraints that Church leaders felt in the 1930s and 1940s, one has to grasp the importance of the sacraments in Catholic religious teaching. Broadly speaking, the Church's doctrine was \"no salvation without the sacraments\"; that is, one cannot go to heaven without having had access to baptism, communion, confirmation, confession, marriage, holy orders, and extreme unction or the last rites (now called the anointing of the sick). Moreover, one cannot have the sacraments without the clergy who administer them: no salvation without priests. Graham Greene's novel _The Power and the Glory_ (1940) provides an excellent testimony to the importance of the sacraments to Catholic doctrine. The flawed, alcoholic, and unchaste \"whisky priest\" at the heart of the plot goes underground in a corner of Mexico during the 1930s, when that country's government was trying to suppress the Church. In the story, his determination to risk his life to keep the means of salvation available to the faithful overrides his personal flaws and transforms a sinner into a holy man. The deliverer is essential; the power of the sacraments outweighs and cancels both his sins and those of his flock.\n\nThis indispensability of the sacraments is a doctrine that can disarm the Church in the face of ruthless and violent political movements. The threat to remove priests and suppress the Church is, given Catholic doctrine, a threat to deprive all Catholics in the areas affected of any hope of salvation. Fear of this threat propelled both the Church's anticommunism, since communists appeared intent on suppressing the Church, and its caution in challenging Nazism, lest it resort to suppression. And the Nazi regime proved adroit at exploiting these fears. It displayed animosity toward the Church on occasion, especially when it tried to remove crucifixes from school classrooms and engaged in show trials of supposed immorality among monks in the 1930s. But it always stopped just short in Germany of wholesale persecution. Like many other Germans and German institutions, as we have seen, Catholic leaders therefore tried to steer as clear as possible of challenging the regime \"in order to avoid worse.\" But worse for whom or what? Generally for them and their Church.\n\nWARTIME PRIORITIES\n\nAs German policymakers inched toward the decision to murder the European Jews, the worldwide failure to rally to their defense largely continued. To be sure, the Nazis permitted emigration from the continent until October 1941, and approximately 72,000 Jews got out of Greater Germany between the outbreak of the war and that date. But that was a relatively small number compared to the need. And after the slaughter began, the countries allied against Nazi Germany, the papacy, neutral countries, and Jews abroad did little to contain the carnage. Why?\n\nAllied passivity was not the result of ignorance of what was happening in Nazi-occupied Europe. Diplomats and journalists from neutral countries, a category that included the United States until December 1941, could read about ghettos in the German press, and businessmen from these nations, such as Swiss factory owners in Poland, witnessed the condition of Jews there and reported on the matter to their head offices. Once the killing began, the Allies learned of it almost immediately. British intelligence intercepted the reports of mass shootings in Russia by the Order Police as they began and passed summaries on to Winston Churchill every morning during the summer of 1941. Both he and the code breakers concluded on September 12 that continuing to do so was superfluous because he could do nothing in response. He could not even reveal the gruesome information for fear of tipping off the Germans to the deciphering of their messages. In October of that year, apostolic delegate Giuseppe Burzio, the Vatican's ambassador in Slovakia, reported the killings to the pope. By March of 1942, a representative of the Joint Distribution Committee in Budapest described the mass murders in Russia at a press conference in New York City, and many American newspapers picked up and publicized the information. That same month marked the beginnings of large-scale deportations to the first death camps, and the papal nuncio in Bern, Switzerland, gathered enough information about what was happening to report to the Vatican that deportation was tantamount to execution. The first published report of murders by gas appeared in the _Times_ of London on March 10, 1942, very shortly after they began on a large scale.\n\nBy May 1942, the flow of information was thickening. Father Pirro Scavizzi, a military chaplain with the Italian troops that participated in the invasion of the Soviet Union, came home on leave, secured an audience with the pope, and told him of the mass shootings. Meanwhile, the Jewish Bund Party in Poland smuggled out to the Polish government-in-exile in London an account of the gassings at Chelmno in particular and the massacre of the Polish Jews in general, which was then publicized by the British government and picked up by the American press. By June and July of 1942, the BBC and American newspapers were carrying fairly frequent reports of mass murder, though knowledge of gassing was not yet widespread, and Auschwitz had not been identified publicly by name.\n\nBut the information ran into considerable emotional and psychological resistance, and even among Jewish leaders disbelief prevailed for a long time. Mass annihilation seemed inconceivable, and those who did not want to accept its occurrence recalled the exaggerated stories that had been told in 1914\u201318, about the brutality of the German occupation of Belgium, as an example of how inflated tales circulate in wartime. As late as December 1944, a majority of the British public did not believe in the so-called atrocity reports coming out of occupied Europe. Incredulity persisted, even though all grounds for it fell away in the period between August and November 1942. In August, Gerhart Riegner, the representative of the World Jewish Congress in Switzerland, told the American vice consul there and Jewish leaders in London that he had received a reliable report that the Nazis planned to \"exterminate\" all the Jews in the East in one swift operation using prussic acid gas, which is the generic name for Zyklon. His source for this broadly but not completely accurate information, not revealed until long after the war, was Eduard Schulte, the chief executive of a German steel and mining firm headquartered in Upper Silesia, not far from Auschwitz, whose talkative second-in-command was a close friend of the region's Gauleiter. Schulte was an unlikely person to pass information to foreigners, an old-fashioned German nationalist whose sons were fighting on the Eastern Front. But he had grown so disgusted by the conduct of the Nazi regime that he decided to use his periodic business trips to Switzerland to begin transmitting intelligence to his country's enemies, not only about mass murder but later also about troop movements and the like.\n\nIn November 1942, confirmation of the revelations about gassing in the East came from three unimpeachable sources: The first was Carl Burckhardt, then the vice chairman and later the chairman of the International Committee of the Red Cross in Geneva, who relayed his knowledge to the U.S. State Department, and the second was a series of leaks from the officially secretive Vatican. The third source swept away the last doubts among Jewish leaders; it consisted of a group of Palestinian Jews whom the Germans had interned in Europe and then exchanged for Germans captured by the Allies. These repatriated Jews recounted what they had seen with their own eyes in ghettos and camps. The result was the United Nations declaration of December 17, 1942, in which the Allies acknowledged and denounced the massacre, thus informing every neutral or Nazi-allied government in Europe of what was happening and of the Allies' intention to exact postwar punishment. This was followed by a radio broadcast in German on the BBC by Thomas Mann, the exiled Nobel Prize\u2013winning novelist, that notified anyone listening of what the Nazi regime was doing to Jews.\n\nEven so, Allied operatives had trouble grasping the reality. This is most strikingly apparent with regard to a message intercepted by British intelligence on January 11, 1943. It contained a tally by H\u00f6fle, the Eichmann of the General Government, of the total number of Jews killed at the Operation Reinhard camps to date: a figure of 1,274,166. That's a stunning number, considering that it refers to the deaths at only Belzec, Sobibor, and Treblinka, none of which had been in operation for even ten months. But the code breakers did not know the names of the camps or recognize the initials that identified them in the document, so the information remained unpublicized and classified for the duration of the war.\n\nGrasping was not the only problem; deciding what to do with the information was also difficult. The principal Allies worried that making too much of Jewish suffering would play into the claims of Nazi propaganda that Churchill, Roosevelt, and Stalin were fighting for the Jews, indeed were their agents. Thus the Allies were reluctant to see, let alone to stress, the special nature of the Nazi attack on Jews, preferring to talk always\u2014and this continued even after the war\u2014about the suffering of \"citizens\" in the occupied states. Given the speed of the German onslaught and its geographical concentration out of the reach of Allied aircraft, there is little the Allies could have done to impede the process except to spread the word, mobilize neutral states to help, and encourage resistance to what the Germans were doing, but worries about seeming too pro-Jewish inhibited such actions.\n\nNor was that the only constraint on Allied rhetoric. The USSR continued to refuse to recognize ethnic distinctions among the victims of Nazism. Stalin spoke only once during the entire Second World War about the fate of the Jews\u2014in a speech on November 7, 1941, accusing the Germans of having carried out pogroms. Neither he nor any other Soviet military leader encouraged partisan units to aid Jews or to interfere with attacks on them. The USSR never considered the idea of bombing railroads to the camps or launching offensives aimed in their direction. The Russians first learned of Auschwitz by name in November 1943, about eight months before the Western Allies acknowledged its existence. By August 1944, when Soviet troops were only 160 kilometers (about 100 miles) away, the leaders of the NKVD, the Soviet secret police, were well versed in the camp's operations and tasks. Yet that information did not flow down the chain of command, and capturing the camp did not become a military objective. In 1944, various Jewish representatives lobbied Soviet diplomats to encourage a raid on the site. At the time on the Eastern Front, the USSR's planes outnumbered the Germans' by seven to one, and Auschwitz was within range of all Soviet light bombers, including the Pe-2 dive-bombers that were ideally suited to hitting narrow targets like the camp's crematoria. But nothing happened, and the principal reason appears to be that Stalin did not want to highlight the situation of the Jews. The Soviets' attitude was summed up by their behavior after they liberated Auschwitz in January 1945\u2014they maintained complete silence about the camp until May, when they issued a report and broadcast a description that did not even mention the word \"Jew.\"\n\nMeanwhile, the top ranks of Great Britain's government exhibited sympathy for the plight of the Jews but remained constrained by worries about holding the loyalty of Arabs. Churchill, who had opposed the white paper of May 1939 that limited Jewish immigration to Palestine, was the most vocal in urging British action to impede the Holocaust. The United Nations' declaration of December 1942 that condemned the murders occurred largely at Britain's instigation, and Churchill insisted that the British seriously consider how to attack Auschwitz in 1944. But he got equivocal support from his own foreign secretary Anthony Eden, who was personally antisemitic and very pro-Arab, and almost no backing from lower down in the British bureaucracy, where Air Secretary Sir Archibald Sinclair blocked the bombing plans and other officials stymied assorted relief efforts. In 1943, one British official called the possible \"release\" of 70,000 Jews from Romania a \"frightful prospect\" because they might go to Palestine and upset the delicate political balance there. The same worry made the British strong opponents of various plans toward the end of the war to ransom Jews still in the Nazi grip. Only 37,451 Jews succeeded in legally immigrating to Palestine between the outbreak of World War II and the end of 1944, and more than a third of them got in only in that last year. Most remarkably, the British reluctance to seem to be defending Jews accounts for the fact that both the United States and the UK purposefully ignored for more than a year after April 1943 consistent reports from Polish intelligence identifying Auschwitz as a site of mass murder.\n\nOne of the reasons the British and other Allies hesitated to emphasize Jewish suffering and the need for Jews to escape Europe was a man named Hajj Muhammad Amin al-Husseini, the Grand Mufti or highest authority on Islamic law in Jerusalem since 1921. One of the leaders of the Arab revolt in Palestine from 1936 to 1939, he had been driven by the British first out of Palestine, then Iraq, and finally Iran. In November 1941 he found asylum, along with several other Arab nationalists, in Berlin. His propaganda writings and broadcasts depicting America, Britain, communism, and the Jews as the common enemies of Arabs and the Axis powers had little practical effect, except possibly in provoking a spike in desertions by Palestinians from British army units in the run-up to the Battle of El Alamein in the fall of 1942. Whether exiled or not, Arab leaders were so internally divided and contentious during World War II that none of them, including the Mufti, could speak for very large segments of the public in the Middle East. But the British especially feared that demonstrative support for Jews and their interests might change the situation, provoking anything from increased sabotage of Allied military units and operations in the region to an uprising that would divert precious troops and resources from the war effort.\n\nLike the Eastern European nationalists who imagined that allying with the Nazis would increase their chances for future independence, and the several hundred thousand Muslims of the southern Soviet Union who joined the Wehrmacht and the SS in order to shake off Stalin's yoke, the Mufti had his hopes for affiliation with the Axis disappointed. Hitler put off giving him anything more than oral support for an end to colonial rule in the Middle East, not least because the Reich's Italian ally intended not only to retain Libya and Ethiopia after the war but also to expand its holdings in the region. The Germans also steadily refused to sanction an Arab-led army to fight with them. Instead, they confined their support to the raising of a few Muslim Bosnian and Albanian SS units (with German officers) and a tiny German-Arab Battalion that by August 1942 had attracted a mere 243 volunteers. These forces proved ineffective. Transferred in 1943 to North Africa and swollen, through local recruitment efforts, to about 2,000 men, the German-Arab Battalion fought so badly that the officers broke it up into labor units. By the fall of 1944, rampant desertion led the Germans to disband the Bosnian and Albanian forces, which the Mufti had encouraged and helped to organize. The desperate Third Reich finally announced its \"recognition of the independence of the Arab countries\" in November 1944, but the moment for rallying Arab or Muslim support to the Nazi cause long since had passed. All told, far more Arabs fought for the Allies in World War II than for the Axis, and probably more Muslims did, too.\n\nIn the meantime, the Mufti scored a few victories, notably in late 1942, when he used his influence with Himmler to block an exchange of Jewish children from Slovakia, Poland, and Hungary for German civilians in Palestine under the auspices of the Red Cross. Husseini also in the following months successfully discouraged the German allies Romania and Bulgaria from accepting monetary payments in return for permitting thousands of Jews to emigrate to Palestine. He even suggested to the Bulgarian foreign minister that the children should go to Poland instead, even though the Mufti had learned directly from Himmler what happened to Jews transported there. But these were limited, short-term triumphs, and the Mufti's association with the Axis ultimately had larger, long-term, and disastrous consequences for his goal of a Palestinian state, let alone one over which he would rule, as he hoped to do after the war. Although he managed to escape to Cairo after Germany's defeat and thus live on, his wartime linkage of Arab and Nazi interests helped move the United Nations toward the partition of Palestine in 1947 and the United States and UK toward siding with the Jews in the civil war there in 1948. Husseini's active political career came to an end due to his adamant refusal to accept partition, his disastrous leadership during the fighting that year, and his unacceptability to the former allies against Nazi Germany. King Abdullah of Jordan replaced him as Grand Mufti in December 1948.\n\nIf the British were more worried about their position in Palestine and the Middle East than the fate of the Jews, Pope Pius XII was more worried about protecting the city of Rome, finding a way to mediate an end to the war before the atheistic Soviets penetrated to the heart of Europe, and preserving his standing as the \"Common Father\" of Catholics everywhere, even those perpetrating atrocities. He therefore carefully maintained public silence about the killing of the Jews, on which he was thoroughly informed. He also did not speak publicly about the arrests of Catholic priests in various parts of occupied Europe or the murders of the Sinti and Roma and of the Soviet prisoners of war. He was only slightly more overt in opposing the German euthanasia campaign, even though some of the victims were Catholics and Bishop von Galen condemned it from within the Reich. His only public utterance on the subject of the Jews came in his Christmas message of 1942, which made oblique and brief reference (twenty-seven words in a document of twenty-six pages) to the tragedy of hundreds of thousands of innocent people dying on account of their race. Quietly and behind the scenes, however, he tried to exert some influence against German persecution. He had his ambassador to Vichy France, for instance, tell Marshal Philippe P\u00e9tain in July 1942 that the pope did not approve of deportations, and a few months later, P\u00e9tain agreed to limit them to foreign Jews in the German-occupied parts of the country. But the achievement evaporated when the Germans extended their occupation to all of France in November 1942. In August 1943, following an audience with a remarkable French monk and rescuer named Father Marie-Ben\u00f4it, Pius XII used Vatican diplomatic channels to persuade Spain to grant entry visas to and repatriate all Spanish Jews in occupied France, even those who had fought against the ruling regime in the Spanish Civil War a few years earlier.\n\nBut for the most part, the pope left decisions about whether and how to aid Jews to individual bishops, abbots, prioresses, and nuncios throughout the Catholic world, while he at the same time withheld from them the information he was accumulating about the murders. Even more strikingly, he declined to intervene with the Germans when they began deporting Jews from Rome in 1943, although he did get the Nazi representatives to promise to respect sanctuary in many Roman churches. Pius also put off pressing the Catholic ruler of Hungary, Admiral Horthy, to stop the deportations from that country in 1944 until after the Allies liberated Rome. By the time the pope finally appealed to Horthy on June 25\u2014three full weeks after U.S. troops entered Vatican City\u2014115 trains carrying more than 340,000 Hungarian Jews already had arrived at the selection ramp of Birkenau. When the deportations resumed after Horthy's overthrow that fall, Pius refused to send another protest.\n\nTimid, prudent, and hoping to play the role of intermediary who could broker a peace that would end the fighting, Pius XII behaved more like a politician\u2014and a rather petulant one at that\u2014than a prelate, more like the keeper to the keys to Rome's sumptuous churches than the keeper of the keys to the Kingdom of Heaven, and thus more like a provincial Roman than a prince of the universal church. Whether one endorses or condemns his conduct depends to a large degree on what one considers his primary responsibilities or obligations. He thought they were to Catholics and the accumulated patrimony of St. Peter. How one judges him also depends on whether one agrees with the Polish president-in-exile, Wladislaw Raczkiewicz, who contradicted Pius XII's refusal to denounce Nazi atrocities by asserting that \"divine law knows no compromise.\" Does it? Perhaps it does not to a religious figure who sees himself as the Vicar of Christ on earth, but surely it does to someone who sees himself as the CEO of a morally and materially valuable institution. Pius XII's actions suggest that he saw himself more as the latter than the former. The historian Michael Bess has summed up the pope's conduct: \"Push never came to shove because the Nazis pushed, but the Vatican did not shove back.\"\n\nThe upside of Pius's generally hands-off policy regarding the persecution of the Jews is that his conduct opened the way for some of the clerics to whom he left the matter to behave better than he did. As we have seen, several French and Dutch bishops spoke out against the mistreatment and murder of Jews. These brave souls came to nowhere near a majority of the bishops in either country, most of whom remained silent, but they demonstrated that at least some prelates' priorities differed from the pope's. We also already have encountered Metropolitan Sheptytsky in Lviv, who tried to impede Ukrainian collaboration in the murders in two extraordinary ways. First, he wrote to Himmler in February 1942 to request that Ukrainian Catholic police not be used in actions against Jews. Second, he issued a pastoral letter the following month that deprived parish priests of the power to absolve parishioners of murder after confession. Reserving that power to himself alone, Sheptytsky directed the faithful to treat Catholics involved in killing with \"the disgust and disgrace they deserve.\"\n\nArchbishop Aloysius Stepinac of Zagreb in Croatia also spoke out, complaining to that country's leader in November 1941 about the \"inhuman and cruel treatment of non-Aryans\" and delivering sermons in subsequent years that forbade participation in killing Roma and Jews and condemned racism. Going beyond words, Stepinac also provided baptismal certificates and work permits to Jews and hid many of them in Catholic buildings. In France, the first roundups and deportations in mid-1942 triggered a protest on the part of the Assembly of French Catholic bishops, in the form of a letter from Cardinal Emmanuel C\u00e9lestin Suhard, the head of the assembly, to Marshal P\u00e9tain. It read, in part, \"Deeply moved by the information reaching us about the massive arrests of Israelites that took place last week and by the harsh treatment inflicted upon them . . . our voice is raised to protest in favor of the inalienable rights of human beings. It is also an anguished call for pity for . . . mothers and children.\" But the letter asked for nothing more specific, and the papal nuncio in Vichy, Monsignor Valerio Valeri, dismissed the document as \"platonic.\" The church hierarchy in Slovakia was even slower to perceive its Christian duty. A pastoral letter distributed by the bishops of that country in 1942 defended only Jews who had converted to Catholicism and otherwise invoked antisemitic arguments to justify deportations. But the authors reversed themselves less than a year later. In March 1943, they issued a new letter that denounced the shipments as an unwarranted application of collective guilt that violated the Golden Rule. The deportations from Slovakia already had stopped, but the clerics' intervention tied the hands of the Slovak president, Jozef Tiso, an antisemitic Catholic priest who periodically toyed with resuming the deliveries, and no more occurred until the death rattle of Tiso's regime in the latter part of 1944.\n\nThe further down the Catholic Church hierarchy one looks, the more brave and principled behavior one finds. In Poland, for example, about two-thirds of the convents offered refuge to Jewish children and adults. The nuns saved, according to the best estimate, no fewer than 1,500 people in this fashion. In Lithuania during the carnage of 1941, several parish priests berated their flocks for having brutalized and stolen from Jews, even though their bishops took at first an equivocal stance and did not begin to condemn the persecution and organize rescue efforts until 1943. In Belgium, a network led by two Catholic laymen, Albert van den Berg and Georges Fonsny, and supported by Capuchin and Franciscan friars and the Sisters of St. Vincent de Paul, placed about 400 children in various Catholic institutions and thus rescued them. It was one of at least six such primarily Catholic groups that worked to conceal Jews in Belgium during the war. And in Rome, perhaps 4,000 Jews found refuge in monasteries, convents, and churches in 1943\u201344. But much more could have been achieved had the pope made rescue the Church's official policy, enjoined the faithful to carry it out, and used some of the Holy See's convertible currency to aid operations aimed at saving Jews, all of which the Vatican steadfastly declined to do.\n\nWhen pressed by foreign diplomats as to why he did not do more, Pius XII always emphasized the dangers that might flow from speaking out. Perhaps it would only enrage the Germans and provoke even more violence, like\u2014although the pope never explicitly drew this comparison\u2014the Dutch bishops' protest against the deportation of Jewish converts to Catholicism that resulted in an acceleration of the practice. Yet neither the Dutch prime minister nor the queen of the Netherlands was cowed into silence. Only days after the deportation trains began to roll east from Holland in July 1942, the prime minister condemned the practice, and a follow-up broadcast on Radio Orange, the voice of the Dutch government in exile, spoke of gas chambers; in October, Queen Wilhelmina told her people over the same sender that she felt \"personally affected [by] the systematic extermination\" of Dutch Jewry. Still, these leaders spoke from the safety of exile, and events in Germany highlighted the risks of reprisal. Although the Gestapo left Bishop von Galen alone after he spoke out against the euthanasia campaign, police arrested thirty-seven clerics in his diocese and sent them to camps, where six of them died.\n\nAlternatively, the pope sometimes claimed that an open protest might cause patriotic Germans to desert their faith, either during the war or in anger after a German defeat, for which they might blame him. That is what some of the German bishops and cardinals feared, and, like them, Pius XII opted to avoid confrontation \"in order to avoid worse.\" Once again, the sacraments functioned as an inhibition: The pope felt responsible not only to keep them available to the faithful but also to keep the faithful open to them and thus to salvation. But such fears did not stop Bernhard Lichtenberg, the provost of the Catholic cathedral in Berlin, St. Hedwig's, from praying publicly for the Jews being deported and from drafting a denunciation of a pamphlet written by Goebbels that condemned any expression of sympathy for Jews. For these offenses, the Gestapo arrested Lichtenberg in October 1941, jailed him for two years, then put him in a work camp, and finally shipped him to Dachau. Incarceration and mistreatment had shattered his health, and he died in transit on November 5, 1943. To Lichtenberg, unlike the pope, the prospect of mass apostasy could not outweigh the commandment to love one's neighbor here and now. Far better than Pius XII himself, Lichtenberg lived up to the ringing words of the new pope's first encyclical, \" _Summi Pontificatus_ : On the Unity of Human Society\" (October 1939): \"In the fulfillment of this, Our duty, we shall not let Ourselves be influenced by earthly considerations nor be held back by mistrust or opposition . . . nor yet by fear of misconceptions or misinterpretations. . . . The Ecclesiastical Hierarchy . . . in union with the Successor of Peter . . . [is] firm when, even at the cost of torments or martyrdom, it has to say: _Non licet_ ; it is not allowed!\"\n\nWhen all other arguments for inaction failed, the pope played his final card in fending off Allied calls for greater forthrightness: the assertion that if he criticized Nazi crimes, he would have to criticize Soviet ones, and the Allies could not want that while the fighting still raged. But of course he never relented in his vocal opposition to communism, so Allied representatives found this a particularly infuriating pretext.\n\nThe best hope for outside help for Jews caught up in the Holocaust was probably the United States, but it responded to the crisis hesitantly, partly for reasons already apparent before the war, partly for new ones. First, the incidence of antisemitism in the American population actually increased during the conflict. A Gallup poll in July 1942 found that 44 percent of the respondents thought that Jews had too much power and influence; two years later, another such survey reported that the same proportion\u201444 percent\u2014considered Jews \"a threat\" to the United States. Second, Breckenridge Long remained in office until 1944 and in opposition to anything that would increase the flow of Jews into the United States, and FDR, who had known Long since their common service in the Department of the Navy in World War I, remained deferential toward him, especially toward his argument that refugees presented security risks. As a result, in 1940\u201341, the two years before America entered World War II, only about 30,000 German Jews got into the country, along with perhaps an equal number from elsewhere in Europe. Third, military planners, in particular, took the line that the only way to help Jews was to win the war as quickly as possible, and then used this argument to rule out small actions that might have helped at the margins. For example, suggestions to transport more immigrants to the United States were rejected repeatedly from 1941 to 1944 because of an alleged shortage of shipping. But 400,000 German POWs were brought across the Atlantic to the United States, and many munitions ships returned empty from Europe.\n\nWashington's attitude softened in late 1943 as a result of several developments. Jan Karski, the Polish resister who earlier had reported to his government-in-exile about rising antisemitism in occupied Poland, also briefly had managed to smuggle himself into the Izbica ghetto, near Lublin, to see what was happening. When he met with FDR personally and told him of what he had witnessed in 1943, the president did not display particular interest in the subject, but the meeting lasted twice as long as planned, and after the conversation FDR seemed aroused as never before by the killings. At the same time, the turn of the tide of war in the Allies' favor indicated that action was becoming possible: Pressure from Congress began to mount, prompted partly by the activism of the outspoken representative of right-wing Zionism in the United States, Peter Bergson; and Treasury Secretary Henry Morgenthau and others persuaded FDR that Long had to go after he gave clearly misleading testimony to a congressional committee. In January 1944, a War Refugee Board came into existence, equipped with large amounts of money, almost all of it provided by the Joint Distribution Committee and other American Jewish organizations. The funds aided Jews through means that ran from bribing Nazi officials to financing the creation of protective identities for individuals, lobbying foreign and neutral governments to help Jews, and supporting escape efforts. Although the board could not stop the Hungarian deportations in the first half of 1944, it did underwrite the efforts of Raoul Wallenberg, the Swedish businessman-turned-diplomat who went to Budapest later that year and organized efforts to issue thousands of Swedish and Swiss protective documents to Jews in that city. In early 1945, the food that the board paid for and stockpiled in Swedish ports saved thousands of lives when Himmler briefly tried to open a negotiating channel to the Western Allies by allowing the International Red Cross to provision prisoners at Ravensbr\u00fcck and other camps in northern Germany. In retrospect, it seems clear that the board was created relatively late, but perhaps as early as it could make any difference to people's fates.\n\nEven the War Refugee Board's growing influence in Washington could not suffice, however, to persuade the War Department to order the bombing of Auschwitz and the train lines to the camp. The United States first recognized its function and location only in March 1944, despite earlier Polish underground attempts to alert Americans to the camp's operations. This was at about the same time that the refitting of captured air bases in Italy first permitted aircraft to get from American or British lines to the camp and back without running out of fuel. The first bombing proposals by Jewish groups in the United States reached the State Department and the Department of the Army between May 16 and June 2, 1944, and John Pehle, the director of the War Refugee Board, forwarded other such requests to U.S. Assistant Secretary of War John McCloy in late June. A few weeks later, on July 7, British Foreign Secretary Anthony Eden asked British Air Secretary Sinclair whether bombing runs could stop the murder of Hungary's Jews. It was, in fact, already nearly over, since the last deportation trains from Hungary departed on July 9. In early August, the chief of the U.S. Air Staff requested reconnaissance photos of the Auschwitz-Birkenau area, which had been taken on April 4, May 31, June 26, and July 8 but not yet developed. A formal request to bomb Auschwitz, again to McCloy, came from the World Jewish Congress in New York on August 9, but John Pehle undermined the request on August 11 for fear that large numbers of inmates would die in any air raid. Three days later, McCloy turned down the proposal, saying that it was \"impracticable,\" even though low-flying bombers that could have hit the crematoria were at Italian bases at the time. McCloy also maintained that the idea represented an unwarranted diversion of military assets to a nonmilitary target.\n\nThe military argument actually had some force, especially as things looked at the time. One needs to remember that Allied airpower had three principal preoccupations in the months just prior to September 1944: smashing development and launching sites for the German V-1 and V-2 rockets that were terrorizing England; aiding the Allied advance up the Italian boot, which had been very slow; and breaking German resistance in Normandy, which finally occurred on August 12, more than two months after D-Day and just as McCloy was considering the WJC's proposal. Thereafter, while the Allies were hurtling toward the German borders on the west and east, and when Pehle first on October 1 transmitted a Polish request that Auschwitz be targeted and then on November 8 added his own such plea, most American bombers were fixed on smashing Germany's fuel production in order to bring the Reich's armies to a halt. That was, in fact, the mission of American bombers that flew directly over and took pictures of the Auschwitz camp on August 20 and September 13 on their way to hitting the nearby IG Farben factory at Monowitz, only three to four miles east of the gas chambers.\n\nIn the end, bombing the camp might not have saved many lives. By the time those planes appeared, about 90 percent of Auschwitz's victims already were dead. The SS transferred more than half of the population of the camp complex\u2014and of the core sites at Auschwitz-Birkenau in particular\u2014to camps further inside Germany between July 1944 and the end of the year. Though transports of Jews continued to arrive and to provide, along with inmates, victims for the gas chambers, the Germans could have murdered the numbers involved (30,000 in October, for example) by other means without difficulty. And the gassings were almost over, in any case: Himmler terminated them at Auschwitz on November 2. Had Allied warplanes attacked the camp, collateral damage would have occurred, as it did when U.S. aircraft bombed a V-2 guidance factory adjacent to Buchenwald on August 24, 1944, and 315 prisoners died, and in early 1945, when planes hit suspected atomic energy facilities near Sachsenhausen and killed some 250 prisoners. The fortified perimeter of Auschwitz was so wide that few people could have broken out while the crematoria were being hit.\n\nBut planners did not know all this at the time, so the question remains, why did they not try? The answer with regard to the military authorities, as with regard to the governments herein described and the papacy, is simple: Trying just was not important enough to them; other needs or goals always took precedence. Even with regard to the one thing that the Allies and the pope might have done in 1942 that would have worked\u2014publicizing Nazi crimes against Jews more\u2014political and theological inhibitions prevailed.\n\nShould we include the behavior of American or Palestinian Jewry among the reasons for the insufficiency of the world's response? It is true that the American Jewish community was divided and did not concentrate its lobbying effort. Stephen Wise of the American Jewish Congress held FDR in awe and refused to pressure him. Similarly, Yehuda Bauer, the author of a multivolume history of the Joint Distribution Committee, describes the people who worked at its headquarters in New York as \"constitutionally incapable of serious questioning, let alone serious criticism, of an administration that stood between the Jewish community and antisemitism or worse.\" But Jews came to only 3.6 percent of an increasingly antisemitic American population during the war. They plainly lacked the power antisemites constantly ascribed to them. Something similar must be said about the Jews of the _Yishuv_ , who also largely failed to rise to the challenge. They, too, were a relatively small population, around 400,000 people in 1940, more than 85 percent of them living in only three urban areas: Haifa, Tel Aviv, and Jerusalem. Also in an exposed and vulnerable position, Jewish leaders in Palestine recognized that they had few material resources to bring to bear in rescue attempts and achieved little success with these until 1944. All told, their clandestine program of _Aliyah Bet_ , or illegal immigration to Palestine, managed to smuggle no more than 19,000 Jews into the territory between 1939 and 1945. One problem was fragmentation: The _Yishuv_ was as deeply politically divided as the Jews in the ghettos, in part along the same lines, so coordination around a coherent strategy was lacking. A second problem was mounting fatalism. The advocates of a future Jewish state understood by 1943 the depressing implications for them of what the Nazis already had accomplished, namely that the population the Zionists had counted on to provide the overwhelming majority of future settlers had been largely eradicated. Henceforth, the demographic future of a Jewish homeland seemed to them to lie with the then 800,000 Jews of North African and Arab lands and to depend, even more than before, on not alienating the Allies by protesting against their perceived inaction in bringing the Nazi killings to a stop.\n\nThe tragedy of 1939\u201345 is that the fate of the Jews of Europe was always a matter of secondary importance to everyone but themselves and the regime that wished to kill them. This was especially true of Switzerland, almost the last remaining potential refuge for Jews left in a Nazi-dominated Europe. The Alpine confederation's official policy during World War II read, \"Refugees who have fled purely on racial grounds, e.g., Jews, cannot be considered political refugees.\" But enforcement was inconsistent. About 2,000 Jews got into the country legally between 1939 and 1945. Almost 20,000 more were admitted and held in internment camps, while approximately 24,500 were turned back at the borders, even though the Swiss government possessed ample information about their likely fate thereafter.\n\nA similarly uncaring attitude initially prevailed in the one country that ultimately rose to the challenge: Sweden. Until late 1942 and the Nazi roundup of Norway's Jews, Sweden and its diplomats were distinctly indifferent to the Jewish catastrophe and intent on not jeopardizing their country's neutrality during the war and its lucrative sales of such items as iron ore and ball bearings to Nazi Germany. But thereafter the Swedish government began successively extending its protection to ever-widening groups of Jews. Its first move in this direction came in December 1942, when the Swedish cabinet informed the German government that Sweden would open its borders to all remaining Jews in Norway, regardless of their country of citizenship, and offer asylum. Almost exactly eleven months later, in early October 1943, Sweden announced the same policy regarding all Jews in Denmark, which opened the way for their mass flight across the Baltic Sea. The Swedes even tried unsuccessfully to persuade the German government to divert its ship carrying some of the few arrested Danish Jews to a Swedish port.\n\nAs the exodus from Denmark began, the Swedish embassy in Copenhagen commenced issuing provisional passports to Jews who could establish some connection with Sweden, and these documents sometimes sufficed to prevent Germans from detaining the bearers. That precedent proved highly important in March 1944, when Germany occupied Hungary and fearful Jews began besieging the Swedish embassy. But provisional passports required some commercial or residential basis and approval in Stockholm; they could not be issued to just anyone, and thus were inadequate to the need. Ambassador Carl Ivar Danielsson and his chief aide, Per Anger, now improvised a hierarchy of protective documents modeled on ones called _Schutzp\u00e4sse_ , which the Swiss vice consul in Budapest, Carl Lutz, had been issuing since 1942. These were simply official-looking pieces of paper with the Swedish coat of arms in color and assorted stamps, all designed to impress upon Hungarian police that the bearer was exempt from deportation by virtue of not being a Hungarian citizen. But the Hungarian deportations started in the outer provinces, far away from the Swedish embassy, so this sort of protection initially was not of much help; neither was the Swedish decision, in mid-June, to allow the embassy to issue provisional passports and entry visas without prior approval from Stockholm.\n\nMore efficacious was a letter from King Gustav V of Sweden to Miklos Horthy, delivered on July 3, urging him to put a stop to the deportations. Along with messages from the pope and the U.S. government, this document helped to push Horthy, on July 7, into ordering the suspension of the deportations, thus providing breathing room for the not yet arrested Jews of Budapest. Two days later Raoul Wallenberg arrived in the Hungarian capital, and his continuation of the practices that Swedish diplomats had developed proved instrumental in saving thousands of Jews after Horthy's overthrow on October 15, 1944. These heroic efforts were the culmination of a two-year-long process of recognizing Sweden's responsibility to act that was virtually unparalleled by any other nation.\n\nThat is not to say that the Swedes were the only diplomats who tried to impede the Holocaust in Hungary. From Switzerland, an equally extraordinary effort was led by George Mantello, a Romanian Jewish refugee serving as the first secretary in the Salvadoran consulate in Geneva. In 1943, backed by his superiors, he began issuing at no cost Salvadoran citizenship papers to between 20,000 and 30,000 mostly Jewish applicants in Hungary and Romania, thus obstructing their deportation. Then, in the late spring of 1944, he sent an emissary to Budapest who obtained copies of two eyewitness reports on the functioning of Auschwitz and the extent of the Hungarian deportations. He immediately released these to Swiss newspapers and with the help of four prominent Swiss theologians whipped up a press campaign exposing and denouncing the murders. To claim that he thus became \"the man who stopped the trains to Auschwitz,\" as the title of a recent study of his actions asserts, goes too far, but the outcry surely played a part in persuading Admiral Horthy to suspend the deportations in July. After they resumed the following fall, the papers Mantello had issued became significant again, not because of further action on his part but because Carl Lutz of the Swiss consulate took over representing Salvadoran interests in Hungary and persuaded the new Arrow Cross regime to honor the documents. They were then supplemented by thousands of comparable papers being issued by the Portuguese and Vatican representatives in Budapest and by a remarkable Italian, Giorgio Perlasca, who had obtained political asylum in the Spanish embassy in Hungary's capital. From November 1944 to January 1945, he posed as a Spanish diplomat, and in that capacity he issued thousands of safe conduct documents, supposedly on the basis of a Spanish law that extended citizenship to Jews descended from those expelled from Spain in the fifteenth century. Of the 140,000\u2013150,000 Jews who survived the Holocaust in Budapest, roughly 120,000 owed their lives, at least in large part, to the protective papers that Salvadoran, Spanish, Swedish, and Swiss diplomats provided.\nCHAPTER 8\n\n[AFTERMATH: \nWhat Legacies, \nWhat Lessons?](contents.xhtml#ch_8)\n\nTHE TRAGEDY OF THE Holocaust did not end with Germany's surrender in May 1945. Conditions on the death marches from abandoned camps and in the camps that continued to operate until the Allies arrived were so atrocious that tens of thousands of Jews died even afterward. The toll at Belsen was prototypical: 35,000 inmates expired in the final few weeks of the war, including Anne Frank, and more than 14,000 died after liberation, sometimes from disease but mostly because their bodies could no longer absorb the food that now was available to them. As a result, by mid-1945, only about 200,000 Jews had survived the camps, roughly 100,000 of whom had been at Auschwitz at one time or another.\n\nRETURN, RESETTLEMENT, RETRIBUTION, AND RESTITUTION\n\nEven this number was too much for the unprepared Allies to cope with. They had set up the United Nations Relief and Rehabilitation Administration (UNRRA) in 1943, but neither this organization nor the military units that liberated the camps in 1945 were ready for what they found. Indeed, the shock to the U.S. troops that liberated Dachau at the end of April 1945 was so profound that some of them went on a rampage, murdering between 40 and 122 of the guards discovered on the site. Two weeks earlier, when British units had reached Bergen-Belsen, they used bayonets and rifle butts to make the remaining German guards collect and bury the bodies strewn around the camp. Of the treatment meted out, an accompanying journalist wrote, \"The punishment they got was in the best Nazi tradition, and few of them survived it; but it made one pensive to see British soldiers beating and kicking men and women, even under such provocation.\"\n\nBut the object of Allied revulsion soon changed from the perpetrators to the victims, since many of the camp inmates had been demoralized, in both senses of the word, by what they had experienced and now behaved in ways that aroused more antipathy than sympathy on the part of those who had freed them. Such behavior prompted General George Patton, the commander of U.S. troops in southern Germany, to call \"the Jewish type of Displaced Person . . . a sub-human species without any of the cultural or social refinements of our time.\" But Patton was a notorious loudmouth and bigot who was looking for such opportunities. He later called the Nuremberg trials of Nazi war criminals a \"semitic\" event, accused the American press of being under \"semitic influence,\" and stated that the purpose of that influence was to \"implement Communism.\"\n\nPatton's antisemitism was more overt than most, but prejudice played a part in shaping the initial American and British incomprehension of the differences among the two million displaced persons (DPs) in Germany at the end of 1945. Many of them were survivors of Nazi labor camps, but many others were refugees who had fled Eastern Europe with the German armies, including 600,000 people from the Baltic states alone, among whom were numerous former collaborators with the Germans. Once more, Jews were treated as just one persecuted group among many. At some 2,500 UNRRA installations, a lot of them identical to places where the Nazis had caged people, all DPs were initially thrown in together, without regard to whether they had been victims or servants of the Nazis. The tensions that arose were particularly high between Jews and the far more numerous Christian refugees from Eastern Europe who feared to return to their now Soviet-occupied homelands.\n\nSo bad were the conditions in the refugee camps at the middle of 1945 that Earl Harrison, President Harry S. Truman's inspector, issued a scathing report that included a passage that, in retrospect, is astounding: \"As matters now stand, we appear to be treating the Jews as the Nazis treated them except that we do not exterminate them. They are in concentration camps in large numbers under our military guard instead of SS troops. One is led to wonder whether the German people, seeing this, are not supposing that we are following or at least condoning Nazi policy.\" Truman instructed General Eisenhower, the overall commander of American forces in Europe, to improve the situation of Jewish DPs. Some progress was made, including the establishment of thirteen separate camps for Jews, no fewer than twelve of them in the American occupation zone, but U.S. policy remained constrained by its chief goals: (1) to keep the residents confined so that the German populace, who feared them, would not be alienated; and (2) to keep conditions uncomfortable enough to encourage DPs to return to their lands of origin.\n\nReturn proved short-lived for many Jews because of events like the pogrom at Kielce in Poland in July 1946 and because of the hardships that accompanied the tightening Soviet grip on Eastern Europe. As a result, the problem of Jewish refugees actually increased after 1945. Whereas only 18,000 Jewish survivors were in UNRRA camps in Germany and Austria in December 1945, that number swelled to over 97,000 a year later and to more than 167,000 at the end of 1947. Many of these people were Polish Jews who had escaped the Nazis by retreating with Soviet forces in 1941, returned to Poland in 1945\u201346, and then decided to flee westward. Others were Hungarian and Romanian Jews who had survived homegrown persecution and now took the first chance to get away. Still others were Jews from Ukraine who had fled with the Red Army in 1941, returned with it in 1944, and encountered the same sort of reluctance to vacate their former residences that the new occupants showed virtually everywhere else. The large numbers were increasingly expensive to support\u2014UNRRA spent almost $4 billion in the late 1940s, a staggering sum at the time, most of it from the United States. Yet the refugees seemed to have nowhere else to go. Because the United States continued to enforce its quota system of immigration, only 15,000 Jewish DPs were admitted between May 1945 and June 1947. Meanwhile, Britain and the Dominions proved more hospitable to non-Jewish refugees from Eastern Europe than to Jewish ones, though Australia was a partial exception, and the British continued to enforce the white paper limit of 15,000 Jewish immigrants to Palestine per year.\n\nThe United States, to which Britain was heavily indebted at the end of World War II, tried to persuade Prime Minister Clement Attlee to allow 100,000 Jewish refugees into Palestine, but he set two preconditions: that the United States pay to transport and support them, and that the Jewish fighters then seeking to drive Britain out of Palestine lay down their arms. Neither party cooperated, so no deal occurred. But the point became moot when the British decided in early 1947 to turn the question of Palestine over to the fledgling United Nations. It voted nine months later to partition the region between Jews and Arabs. Between 1945 and 1951, by a combination of legal and illegal means, somewhere between 133,000 and 200,000 European Jewish refugees got to the territory that became the State of Israel in 1948. They were often received somewhat insensitively\u2014either suspected of having collaborated with the Nazis in order to survive, or pitied as the remnant of a weak and failed diaspora. Either way, they were regarded as living proof of the need for Zionism and an independent and self-reliant Jewish state.\n\nMeanwhile, resistance to the immigration of DPs to the United States softened somewhat as a result of growing public sympathy with their plight and an increased perception among American officials that their continued presence on German soil was politically inconvenient and embarrassing. Even so, the price of greater openness was readiness to turn a blind eye to earlier collaboration with Nazism on the part of many of the tens of thousands of non-Jewish Latvian, Lithuanian, Ukrainian, and _Volksdeutsche_ residents of German DP camps, who were also allowed into the United States under the terms of the Displaced Persons Act of 1948 and the amendments passed in 1950. Now admired as refugees from communism, these people actually outnumbered the Jews who gained entrance to America under the new legislation. Altogether, the United States admitted between 80,000 and 137,000 Jewish Displaced Persons by 1953, a not inconsiderable total but about the same share of the total number of Jews who found refuge worldwide as in the 1930s. Think about that for a minute: The United States actually was, in relative terms, no more open to Jewish immigrants from 1945 to 1953 than from 1933 to 1939. Of course, most of them were no longer in mortal danger after World War II, so one might argue that their need had become less acute, but they were often homeless and destitute.\n\nThose who got here faced incomprehension of their experience akin to the attitudes exhibited earlier by the U.S. military commanders in Germany. Most Americans simply could not imagine what survivors had been through. The sense of isolation that many of them felt was compounded by two policies on the part of well-meaning Jewish social service agencies. The first was a conscious decision to disperse survivors among disparate Jewish communities that volunteered as sponsors. Many Jewish survivors thus found themselves in places, such as Columbia, South Carolina, and Denver, Colorado, that were worlds apart from their places of origin. Once there, they ran up against another conscious policy that the agencies encouraged sponsors to adopt: an emphasis on persuading survivors to \"move on\" and not dwell on or talk about the past and its losses. A great deal of emotional and psychological pain remained unresolved in the process of rebuilding lives.\n\nIt is, of course, not true that the Holocaust was forgotten in the 1950s and 1960s. By the time I graduated from high school in 1964, the subject was already a conspicuous topic in American popular culture. I first learned about it in junior high in the late 1950s from reading Leon Uris's novel _Exodus_ , the biggest bestseller in the United States since _Gone with the Wind_. This highly fictionalized account of the voyage of a real illegal refugee ship to Palestine became a film starring Paul Newman. I saw it, as I did _The Diary of Anne Frank_ (1959). When I was in high school, _Judgment at Nuremberg_ with Spencer Tracy was a box-office success, and so was _The Pawnbroker_ , with Rod Steiger, during my first year in college. But it is true that the Holocaust did not yet stand out sharply from the enormous cataclysm of World War II. This is an example of what I call the optic of history. Most Americans thought, after 1945, that the real story of World War II was the story they had been part of\u2014namely, the war in the Pacific and the invasions of Europe, not what had occurred in Poland and Ukraine. According to family lore, my father ended World War II on the island of Tinian, having been trained in New Mexico in 1944\u201345 to drop an atomic bomb on Japan. Fortunately, he did not in the end do so because there were only two bombs for five trained crews, but I grew up hearing a lot more about defeating Japan than fighting Nazis, even though almost twice as many Americans died in the war against Germany than in the Pacific theater. Besides, the cold war put a damper on paying attention to the Holocaust, since most Germans were now America's allies, and political convenience argued for not raking up the past. Decades had to pass before survivors felt that they had an audience for their recollections.\n\nPostwar politics also worked against extensive retribution to perpetrators and demands for restitution to the victims, but in both instances, as with the myth of silence about the Holocaust following the war, more happened than people tend now to remember. It is simply untrue that many major perpetrators of the Holocaust escaped punishment afterwards, just as it is untrue that Germans, especially those in the eastern half of their country, paid little or no price for what their nation had done. Both legends are the opposite of reality. Germany was a badly damaged country in 1945, and it remained that way for many years after, despite the economic revival of the 1950s. When I first visited, in 1968\u201369, I saw numerous empty, bombed-out lots in D\u00fcsseldorf, as well as trees growing from the roofs of the Frankfurt Opera House and the twin churches on the Gendarmenmarkt in Berlin. Those edifices were ruins twenty-four years after World War II ended.\n\nSo tenacious is the legend of perpetrators escaping punishment that it seems to blind the people who retell it to even their own evidence. A case in point is Donald McKale's _Nazis after Hitler: How Perpetrators of the Holocaust Cheated Justice and Truth_ , an extended philippic about Nazis who supposedly avoided punishment after World War II and advanced self-justifications that aided the cause of Holocaust denial. Yet of the thirty-one individuals on whom the author focuses to make his case, his own text shows that twelve were executed for their deeds, two committed suicide, four died in captivity, two died as they were about to be arrested and prosecuted, one died on the run, and four went to jail. Only six went unpunished. The mortality rate came to two-thirds by the time Adolf Eichmann was executed in Jerusalem in 1962.\n\nIn fact, in the early postwar years, the reckoning was pretty intense. Altogether, European courts condemned and sentenced approximately 100,000 Germans and Austrians for wartime criminality of one sort of another. The four victorious Allies convicted another 8,812 Germans and Austrians in proceedings held in occupied Germany. American prosecutions of 1,030 camp officials and guards on atrocity charges in 1945\u201347 produced 885 convictions; 261 of the 432 defendants condemned to death for these offenses or for harming American military personnel ultimately died on the gallows. The hanging at Dachau on May 27\u201328, 1947, of forty-eight German personnel from Mauthausen constitutes the largest mass execution in American history. Among those also executed by the United States were Paul Blobel, who had commanded _Sonderkommando_ 1005, the unit charged with exhuming the bodies of camp victims, burning them, and destroying all traces of Belzec, Sobibor, and Treblinka; Otto Moll, among other things the commandant of several gassing installations and of the slave labor camps at IG Farben's murderous mines near Auschwitz; and Oswald Pohl, the head of the SS Economics and Administration Main Office and, as such, the architect of the SS camp labor system. The British tried 989 people on war crimes charges and hanged eleven members of the camp administration at Belsen. The executees included Franz H\u00f6ssler, who had commanded the first gas chambers at Auschwitz and later supervised the exhumation and burning of 100,000 bodies there, as well as two businessmen who sold Zyklon to the SS. The Soviets hanged Friedrich Jeckeln, the SS man who presided over the murders at Babi Yar, on the site of the Riga ghetto in 1946. They also executed six of the former _Hiwis_ at Sobibor even before the war ended and ten more of them after a trial in 1962. Altogether, Soviet courts convicted almost 26,000 Germans and Austrians and about 11,000 local collaborators.\n\nThe Poles tried 5,358 German nationals between 1945 and 1957. Among the people executed were Rudolf H\u00f6ss, the longest-serving commandant of Auschwitz; J\u00fcrgen Stroop, the SS commander who put down the Warsaw Ghetto Uprising; Hans Biebow, the German administrator of the Lodz ghetto; Amon G\u00f6th, the sadistic commandant of the Plaszow concentration camp made famous in _Schindler's List_ ; Arthur Greiser, the Nazi governor of the Warthegau; the two top-ranking officials of the General Government; the four senior German figures in occupied Warsaw; Heinrich Josten, the commander of the SS guard force at Auschwitz; Erwin von Helmersen, an SS doctor at Birkenau; Werner H\u00e4ndler, the man in charge of food for the inmates of both those camps; and Maximilian Grabner, the head of the Political Section of the Auschwitz-Birkenau camp administration from 1940 to 1943, which was the subunit responsible for torture and executions. Poland also sentenced two of the men who poured Zyklon into the gas chambers to long prison terms; one of them died in his cell in 1955, and the other was released in 1958.\n\nArthur Seyss-Inquart, the German administrator of the occupied Netherlands, was condemned to death at the Nuremberg trials and executed. The Dutch followed up by putting to death forty Nazi officials and collaborators, including Hanns Rauter, the SS chief in Amsterdam. The death sentences of Ferdinand aus der F\u00fcnten, who directed the deportations from Holland, and of Willy Lages, the chief of the SS Security Service there, were commuted in 1951 to life in prison. Lages served fifteen years, then died five years later. Aus der F\u00fcnten served thirty-nine years until the Dutch released him on grounds of ill health two months before he died. Albert Gemmeker, the commandant of the camp at Westerbork from which most Dutch Jews were sent to their deaths, got off more lightly with a ten-year prison term, of which he spent six behind bars before his release in 1955.\n\nIn general, the chances of high-ranking perpetrators being punished were quite high. Consider the fates of the sixteen people who at one time or another had independent command of a death camp: thirteen were killed one way or another during the 1940s, one received a death sentence in 1954 and promptly died of a heart attack, and two escaped justice for a time, only to be caught ultimately and given life sentences. The figures for the fourteen people who commanded an _Einsatzgruppe_ are similar: Seven perished during the war, two committed suicide in custody, and three were executed, for a total of twelve fatalities. The remaining two were sentenced to prison, albeit for what turned out to be only brief terms. In both categories, no one got off scot-free. Of the forty-two individuals who ever commanded one of the thirteen most notorious concentration camps\u2014Bergen-Belsen, Buchenwald, Dachau, Dora-Mittelbau, Flossenb\u00fcrg, Gross-Rosen, Mauthausen, Natzweiler, Neuengamme, Ravensbr\u00fcck, Sachsenhausen, Stutthof, and Theresienstadt\u2014fourteen died before 1945, eighteen were executed or committed suicide, four served prison terms, and only six (14 percent) went unpunished or unaccounted for after the war. Nikolaus Wachsmann, the author of a definitive history of the Nazi concentration camps published in 2015, says that only seven of the former wartime commandants of the twenty-seven main SS concentration camps were still alive in 1950. Though not a perfect record, this is hardly a terrible one.\n\nWith regard to euthanasia killings and slave labor, the attrition among order-givers was also extensive. The three chief figures in T4, Philipp Bouhler, Viktor Brack, and Karl Brandt, died shortly after the war, Bouhler by suicide after his capture by the United States in 1945 and Brack and Brandt by hanging in 1948 after their condemnation by an American court. Both Albrecht Schmelt, who devised the SS's sliding charges for different categories of Jews leased out as slave laborers, and Hans Kammler, the SS man in charge of the enslaved workforce at Dora-Mittelbau and the other Fighter Staff Program factories, perished as the war was ending.\n\nIn addition, courts in their homelands condemned to death the Vichy prime minister Pierre Laval and the Romanian dictator Ion Antonescu, both of whom had delivered Jews to the Nazis and then thought better of it. Vichy's chief of state Philippe P\u00e9tain escaped the same fate only because Charles de Gaulle commuted his sentence to life in prison, where he died. Norway's collaborationist leader Vidkun Quisling was shot by a firing squad; Slovakia's Jozef Tiso was hanged; and so was Hungary's Ferenc Szalasi, who took over its government and resumed the deportation of Jews in late 1944, along with the three chief figures of the Hungarian Interior Ministry who organized the mass deportations earlier that year. In the meantime, every one of the German envoys to Croatia, Slovakia, Hungary, Bulgaria, and Romania who lobbied those governments to kill or deport Jews during the war had been killed either upon capture or following a trial in 1945\u201347.\n\nFinally, the Germans themselves accounted for a considerable number of prosecutions. West Germany sent 6,479 people to prison between 1945 and 1986, and the East Germans convicted 12,861 individuals between 1945 and 1976. Still, there were notable omissions and lapses, especially during the 1950s. Only about 10 percent of the Germans who ever worked at Auschwitz went on trial anywhere after the war, and the mid- to low-ranking personnel at most concentration camps were largely ignored later or given light sentences\u2014at least by American standards\u2014when tried. Of the 50,000 members of the Police Battalions that killed about half a million people in occupied Eastern Europe, only 64 men were ever charged and 41 ever sentenced. And most of the SS officers who were imprisoned shortly after the war were out by 1958. But exceptions occurred: Hermann Krumey, a key figure in the deportations from Hungary, got a life sentence in the late 1960s and served it. Hans H\u00f6fle, Globocnik's chief of staff during Operation Reinhard and the man who drew up the infamous statistical tally of the killings during 1942, evaded justice until his arrest in 1961 but killed himself the following year. In 1969, East Germany executed Josef Bl\u00f6sche, the SS man pointing a machine gun toward the boy with a cloth cap (see chapter 5, figure 6) during the suppression of the Warsaw Ghetto Uprising. Though most of the district-level Nazi administrators who presided over deportations in the General Government went unpunished, two who were investigated in the 1960s also committed suicide. Only a few of the high-ranking figures at Auschwitz got away, notably Josef Mengele, the doctor who conducted selections at the arrival ramp and vicious experiments on inmates and who hid out in South America until he drowned in 1979. Wilhelm Boger, the infamous interrogation officer in the Political Section of the Auschwitz camp, was not quite as lucky. After he escaped from American custody in 1947, he managed to remain at large until 1965, when the German authorities caught up with him. Sentenced to life in prison, he died there twelve years later.\n\nThe record also is mixed, but not negligible, regarding the 121 men from T4 who staffed the Operation Reinhard camps. Forty-two (that is, more than a third) of them died during the war, in Soviet captivity, or immediately after 1945, mostly by their own hands. Twenty-two were sentenced after the war, nine to life in prison, twelve to terms of three to twelve years, and one, who committed suicide, to death. One other former T4 man killed himself during the preliminaries to his trial in 1965. Among those caught and punished was Hermann Bauer, who called himself the \"gas master\" of Sobibor. Condemned to death in 1950 but saved by the abolition of capital punishment in West Germany, he served out a life sentence, dying in West Berlin's Tegel prison in 1980. Still, about fifty-seven (47 percent) of these participants in murder escaped punishment. According to Michael Bryant, the most recent and careful student of their prosecutions, German courts would have convicted as many as twenty-one more of them as accomplices to murder in the three successive trials of Reinhard camp guards in 1963\u201366 and another of Majdanek personnel in 1966\u201371 if more eyewitness testimony, to which those courts generally were deferential, had been available. Where relevant survivors who could implicate individual guards in cruelty or killing were in short supply, however, the courts had no alternative under West German law than to give defendants the benefit of the doubt regarding their claims to have \"inwardly opposed\" Nazi actions and been uninvolved in the gassings.\n\nOf course, a number of infamous figures did escape punishment, many of them through the efforts of an entity that also had been less than consistently helpful to Jews during the Holocaust: the Roman Catholic Church. Driven by the view that any ally against communism was worth assisting, Catholics developed several escape routes, known colloquially as ratlines, for Nazis and their European allies. All of these itineraries ran out of Germany through South Tyrol, the German-speaking area in northeastern Italy, and then either directly to the port city of Genoa or first to Rome and then to that port. From there, the escape routes went either to Franco's Spain via Barcelona and sometimes then on to Juan Peron's Argentina or directly to Buenos Aires. Along the way, the International Red Cross and the Vatican Relief Commission, the latter run by Monsignor Giovanni Montini, the future Pope Paul VI, provided new identities and travel documents, and Giuseppi Siri, the Archbishop of Genoa, furnished food and shelter there. Most of the money for the operation, about $5 million at the time, came from unwitting donations on the part of the National Catholic Welfare Committee in the United States, spurred on by Cardinal Francis Spellman of New York City.\n\nSome of that American funding also went to a Nazi sympathizer in Rome, an Austrian bishop named Alois Hudal, the rector of a seminary for German-speaking priests. Among the notorious criminals that Hudal helped get away were Josef Mengele and Adolf Eichmann; Gerhard Bohne, one of the principal organizers of the T4 program, who ultimately returned to Germany and escaped punishment; and Eduard Roschmann, the vicious commandant of the Riga ghetto, who lived in Argentina from 1948 to July 1977. The prospect of extradition drove Roschmann to Paraguay, where he died within about a month of his arrival. Another beneficiary of Hudal's aid was Erich Priebke, who led a massacre of 335 Italians during the German retreat from Rome and then spent fifty years in Argentina before finally being extradited to Italy in 1995, tried, and sentenced to life imprisonment under house arrest, which is where he died, in 2013, at the age of 100. Hudal also hid Franz Stangl, the former commander of the Sobibor and Treblinka death camps, at the seminary until he could make his escape to Syria and later to Brazil. He finally was arrested there in 1967, extradited to West Germany, and sentenced to life imprisonment in 1970. Stangl's escape, as well as that of one of his deputies at Sobibor, Gustav Wagner, was the work of another old Nazi who worked out of Hudal's seminary, Walter Rauff, the inventor of the motor-fed gas van. They, too, initially went to Syria, but, unlike Stangl, both Wagner and Rauff lived out their lives in safety, the former in Brazil, where he died in 1980, the latter in Chile until he succumbed to lung cancer in May 1984.\n\nA similar ratline emanated from a pontifical college for Croatian priests in Rome, where Father Krunoslav Draganovic funneled both the remaining funds of the brutal Ustasha state to the Vatican Bank and thousands of Ustasha veterans and a few Nazis to safety abroad. Two of his more infamous successes were Klaus Barbie, known as \"the butcher of Lyon\" for his role as a wartime torturer for the SS in that city, and Ante Pavelic, the former ruler of Croatia, who had presided over the slaughter of thousands of Jews, Sinti and Roma, and Serbs. Justice did not catch up to Barbie until 1983, partly because he was protected by both American and West German intelligence agencies and a succession of military rulers in Bolivia. When that country returned to democracy that year, the new government arrested and extradited Barbie to France, where he was sentenced to life in prison four years later. He died there of multiple cancers in 1991. Meanwhile, Pavelic had escaped retribution far more briefly, despite the fact that he, too, had additional help from a Western intelligence agency, in his case Britain's. After an assassin sent by the Yugoslavian secret police nearly killed Pavelic in Argentina in 1957, he fled to Chile and then Spain and died of his wounds two years later.\n\nEven more Nazis and their allies would have escaped if Catholic leaders had gotten their way. Pius XII pleaded repeatedly for clemency for condemned war criminals, both in general and in specific instances. He thus held to the theory of pastoral responsibility that he had followed throughout World War II, a theory that historian Jacques Kornberg has shown assigned less importance to condemning sin that to keeping open possibilities of forgiveness and redemption. Most German bishops, including Clemens von Galen, the man who had criticized the euthanasia program, went even further by denouncing the war crimes trials themselves as unjust. So did Bishop (later Cardinal) Aloisius Muench, the antisemitic, German-speaking son of Bavarian immigrants to the United States who served both as liaison between the American occupation administration and the German Catholic Church and as the papal representative in occupied Germany. He wrote a pastoral letter that contrasted \"Christ's law of love\" with the \"Mosaic idea of an eye for an eye.\" But when the pope and German Catholic leaders pressured John McCloy, the U.S. High Commissioner for Germany from 1949 to 1952, to commute the sentence of Otto Ohlendorf, who had commanded an _Einsatzgruppe_ and a section of the RSHA, their advocacy was too much for even Muench. Quietly but firmly, he advised the German prelates and the Vatican to back off, lest their stance become public and embarrass the Church.\n\nOne other famous group long supposed to have enabled escapes by war criminals, the _Organisation der ehemaligen SS-Angeh\u00f6rigen_ (Organization of Former Members of the SS), known by the acronym ODESSA, appears to have been largely mythical. That did not prevent the famous \"Nazi hunter\" Simon Wiesenthal from believing it was real and from encouraging the novelist Frederick Forsyth to place it and the aforementioned Eduard Roschmann at the center of a gripping bestseller called _The Odessa File_. In 1974, it became a hit movie of the same name starring Jon Voight. Although grist for a vivid story, ODESSA was the sort of fantasy that fevered postwar imaginations conjured up, and its nonexistence helps explain why, in the end, few major Nazi war criminals got away. An independent historians' commission entrusted in the late 1990s with an exhaustive examination of Nazi activities in Argentina combed the archival record there and in Europe and reached the conclusion that only 180 likely war criminals or collaborators had gained entry to that South American country, of whom about 100 were French and Belgian, about 50 Croat, and only 23 German or Austrian. That assessment gains credibility from a recent detailed study of the case of Aribert Heim, an SS doctor who killed prisoners at Mauthausen and got away to live out his years in Cairo until 1992. The authors, two investigative journalists, argued that the success of people like Heim in eluding capture owed much more to the efforts of their friends and families than to any organization's support. A peculiarity that helped make this so was the provision in postwar German law that barred charging close relatives of suspects with aiding and abetting their escape. In consequence, immediate families could refuse to cooperate with investigators without fear of punishment.\n\nThat the Western powers did not hold more people responsible was partly a matter of cold war politics. To combat the Soviet Union, the United States wanted to exploit the expertise of some compromised individuals, not only people like Barbie and Pavelic but also scientists such as Wernher von Braun of Germany's V-2 rocket program. Braun's connection to the use of slave labor at Dora-Mittelbau seemed less important to America after 1945 than his ability to design ballistic missiles and, ultimately, the spacecraft that took John Glenn into orbit around the earth. More generally, the United States sought to embed West Germany in the West and the NATO alliance and considered continuing prosecutions counterproductive to that purpose. But leniency also reflected a domestic German democracy-building strategy. Konrad Adenauer, West Germany's first postwar leader and a man with impeccable anti-Nazi credentials, believed that integrating former Nazis into the new political order was the best way of reconciling them to a democratic system and alliance with the West. He wished to prevent the rise of a sense of victimization comparable to the one Germans had nursed after World War I. Thus he accepted prosecution of only the most obviously criminal actors, but argued for forbearance otherwise and for a kind of collective amnesia about the degree to which Germans had supported the Nazi F\u00fchrer and shared his hatreds. Adenauer saw to it that former perpetrators and their widows received state pensions and that some even returned to positions in the West German government. His own right-hand man in the 1950s was Hans Globke, who had written the manual for implementing the Nuremberg Laws.\n\nAdenauer's strategy largely succeeded in political terms but only temporarily in historical ones. Beginning in the late 1950s and accelerating thereafter, pressure to confront the details of the past and in some cases the perpetrators themselves rose dramatically in West Germany, as East German propagandists exposed the questionable pasts of many officials and people born after the war reached maturity and began posing painful questions. By the time they did this, the country's democratic institutions were strong enough to withstand the call for honesty about the past. Since the 1970s, openness about what Germans did and about the reality of the Holocaust has been part of what Germans call their \"constitutional patriotism,\" and memorials reminding Germans of the worst of which they were capable now dot the nation's capital, as well as most of its large cities.\n\nGerman restitution policy toward victims of the Holocaust followed a similarly halting course toward a similarly accepting endpoint. From 1945 on, the Germans conceded that they had to pay something in the form of restitution or compensation for all the misery that they had caused, but they sought consistently to keep the bill as low as possible. As a result, every concession came in response to outside pressure and was confined to relieving it, but the pressure never really stopped, and the ultimate bill came to a staggering sum. Since 1945, total payments to survivors, their heirs, and the state of Israel have come to more than $100 billion, not counting the value of returned objects, such as art works. Yet certain categories of victims benefited disproportionately, some received nothing at all because they died before compensation was extended to them, and even $100 billion falls well short of the worth of the damages the Germans inflicted.\n\nThe survivors who came off best were Jewish Germans who managed to flee the country before the Holocaust or who survived it somehow on German soil. Under Allied occupation rules as well as laws passed by the fledgling German state in the early 1950s, such people were entitled to the return of their old property, such as homes, businesses, furniture, jewelry, and other assets, or a cash payment equal to its worth. The total payout for identifiable and lost property came to 7.5 billion deutsche mark by the mid-1960s, which was just shy of two billion U.S. dollars at the time. German Jews whose careers had suffered by virtue of being driven out of the country were entitled to a lump-sum payment of 10,000 deutsche mark for \"damage to education,\" and those expelled from the practice of law or university faculties were granted the lifelong pension of someone who had reached the senior ranks of the judiciary or the professoriate. Hannah Arendt lived comfortably in New York City in part off of such income. In the first years of this century, 100,000 people worldwide were still receiving such payments.\n\nOther categories of victims came away with much less, if anything. Jewish refugee organizations got 120 million marks worth of German foreign assets to use in the immediate postwar years to aid resettlement of Jewish survivors, along with the proceeds of refining and selling the gold that had been shipped to Berlin from the death camps in Poland but not smelted by the end of the war. Israel received three billion marks to support survivors as a result of the Luxembourg Agreement of 1952, and the Jewish Claims Conference another 450 million marks for the same purpose. Between 1958 and 1961, in accordance with the usual rule in international law that only countries, not individuals, can get compensation from other countries, West Germany signed treaties with sixteen non-communist European states that provided them with 2.5 billion marks to distribute to Holocaust survivors within their borders.\n\nSubstantial as these amounts were, they excluded large groups of survivors, primarily those in Eastern Europe. The West Germans insisted until unification in 1990 that they were responsible only for survivors who met two criteria: (1) they had lived within Germany, defined by its borders in 1937, at some time between 1933 and 1945 or moved to the Western occupation zones or West Berlin between 1945 and 1952; and (2) they currently resided in West Germany or in a country that had diplomatic relations with West Germany. The second rule disqualified most Eastern European survivors until the 1970s, since most of their countries had diplomatic ties only with communist East Germany until then. The first rule excluded many survivors in Eastern Europe altogether.\n\nA second major category of survivors who were left out of the German compensation schemes comprised people who had been slave laborers for German private industry. For many years, German courts refused to hold German firms liable to pay such people, contending that the companies had been acting on government orders and that the state, not the firms, was the proper address for claims. Largely in order to limit damage to their reputations in foreign markets, a few companies made token postwar payments to former slave laborers as a gesture, not an admission of obligation or guilt. The legal remnant of IG Farben, along with the Krupp, Siemens, AEG, and Rheinstahl companies gave the Jewish Claims Conference 51.5 million marks between 1957 and 1962 to aid approximately 15,000 Jews who had toiled for these enterprises during World War II, but the average payout converted to a rather paltry $850 at the time. Two decades later, Daimler-Benz and Volkswagen did something similar.\n\nBoth gaps in the German compensation system were filled in the 1990s. The German government first extended payments to survivors who had fled Eastern Europe after 1965 and to severely injured survivors still in the region. Then, in 1999, Germany worked out a deal with the United States that traded the suspension of class-action suits in American courts\u2014legal efforts to seize German assets in the United States to pay survivors\u2014in return for the establishment of a fund to pay their claims. The fund would contain ten billion marks, half from the German government and half from private German corporations. Some of that money collected for this German Foundation Initiative went to non-Jewish Eastern European forced laborers, but three billion marks, or about 1.5 billion U.S. dollars at the time, went to Jewish survivors in compensation for both former slave labor and confiscated monetary assets, notably insurance policies.\n\nIn short, the history of recompense by Germany for the crimes of the Holocaust is an ambiguous one. On the one hand, buoyed by the extraordinary postwar revival of its economy and motivated by an initially self-interested desire for integration into NATO and the new Europe, Germany consented to pay an overall indemnity for the Holocaust that no one would have thought possible in 1945. On the other hand, the German record is clouded by the highly variable support provided to individuals, along with the halting and grudging way in which compensation expanded, which meant that hundreds of thousands of victims died before they became eligible.\n\nSomething similar happened in other European countries. Throughout Eastern Europe, of course, the record was far worse, as communist governments nationalized property rather than returned it, and most of them quickly drove out their surviving Jewish populations. Ninety percent of Bulgaria's remaining Jews had emigrated by 1949, nearly all of Romania's and Poland's by the 1960s. After the fall of communism, most of these countries then established residency and citizenship requirements for restitution of confiscated possessions, mostly real estate, which conveniently meant that they would not have to give anything back, since few Jews wished to return and many had lost their citizenship automatically upon emigrating.\n\nIn Western Europe, an initial flurry of attention to restoring homes and physical assets soon gave way to insensitivity and indifference that lasted into the 1990s. Backed by the Vatican, Catholic religious institutions and orphanages in Holland and France often declined to relinquish the Jewish children consigned to them by parents who had perished to other relatives or Jewish community institutions. So-called heirless assets\u2014ones whose owners never came back\u2014notably thousands of art works, remained in the hands of whatever person or institution held them when the war ended. Only in the 1990s were the postwar deficiencies made up. For example, the Dutch state provided compensation for the stocks and bonds that had been seized from Jews in Holland in the 1940s and sold to Dutch citizens, and the French government endowed a new Foundation for the Memory of the Shoah with 2.5 billion francs, a sum thought to be the value of property formerly owned by Jews in France that had remained unclaimed after the war.\n\nSwitzerland presented a particularly awkward case of restitution because it had been formally neutral during World War II but had purchased considerable quantities of plundered gold from the Third Reich and had served as the salesroom for much of the art, furs, jewelry, and commercial paper that the Nazis stole from Jews. Moreover, Switzerland's banks were suspected of having pocketed the contents of numerous \"dormant\" accounts opened by Jews who later were killed in the Holocaust. These issues were largely swept under the rug in the immediate postwar years. The United States acceded to the Washington Agreement of 1946, by which the Swiss promised to liquidate frozen German assets in their country, transfer half their value to a fund for stateless Nazi victims, and hand over one-sixth of the gold acquired from Nazi Germany in return for rehabilitation as an acceptable trading partner. Although the Swiss government passed a law in 1946 that ordered restitution of stolen art even if the purchase had been made in good faith, the legislation allowed only a short interval for making claims and applied only to works bought after 1939 in occupied areas, not in Germany proper.\n\nDuring the 1990s, the World Jewish Congress succeeded in turning a spotlight on Switzerland's involvement with the Nazi regime, especially the issues of stolen gold, dormant bank accounts, production of war materials, and hostility to refugees. A series of commissions of inquiry were named, notably one under Paul Volcker on the conduct of Swiss banks and another led by Jean-Fran\u00e7ois Bergier on the broad subject of Swiss policy and actions during World War II. The associated research teams demonstrated that the number of bank accounts opened by Jews during the Nazi era, unclaimed after the war, and then drained by the banks through fees probably was lower than Switzerland's critics had claimed, but that Swiss banks had conspired to frustrate postwar inquiries about them. The Bergier Commission also found that the Swiss National Bank knowingly had accepted plundered gold from the Nazi regime and afterward repeatedly mischaracterized\u2014that is, lied about\u2014its policies and conduct.\n\nAs these findings emerged, they played a significant role in setting the terms of the settlement of a U.S. court case against the Union Bank of Switzerland in 1999 by which the bank agreed to pay $1.25 billion into a fund administered by the Jewish Claims Conference: $800 million for restitution of dormant bank accounts; $100 million for compensation for looted assets; and $325 million for payments to former slave laborers at Swiss-owned companies in occupied Europe or at German firms that had put their revenue in Swiss banks and for refugees mistreated by the Swiss. By January 2005, the Claims Conference had disbursed $690 million, mostly in the latter two categories.\n\nIn retrospect, the recurrent pursuit of recompense for the victims of the Holocaust has proven both impossible and necessary\u2014impossible because so much of what was lost was intangible and irremediable, necessary because so little of what could be given back or paid for was treated as such in the early postwar years, when every European nation was preoccupied with reconstruction. And because thousands of victims died before being able to benefit, the justice achieved was incomplete. Moreover, the monetization of loss is always approximate and grows more so as the interval between offense and redress increases, and many of the countries where the thefts were most extensive, notably Poland and Romania, have yet to grapple seriously with their obligations.\n\nSo despite enormous expenditures, gaps still yawn between what people suffered and what they got back and between what a perpetrating entity did or gained and what it ultimately paid. Every major restitution or compensation settlement since 1950 has been an instance of \"negotiated justice,\" in which the amounts made available have had less to do with what real compensation required or real criminality deserved than with the momentary bargaining strength of the parties. This was as true of the sums distributed pursuant to the Luxembourg Agreement of 1952 as of those raised by the German Foundation Initiative of 2000. Political realities also explain why Switzerland never has been forced to indemnify any person or agency for the agreements that the Alpine republic signed with the governments of Poland and Hungary shortly after World War II. These deals allowed Switzerland to seize the heirless Swiss assets of dead Polish and Hungarian citizens, most of whom were Jews, as compensation for the nationalization of Swiss property in these newly communist states.\n\nMoreover, the settlements involving corporations have been instances of rough justice: The enterprises bought valuable advantages by paying arbitrarily determined sums that bore no relation to the firms' earlier conduct, while sometimes guiltier parties walked away untouched. The Union Bank of Switzerland, in effect, purchased the right to complete a merger and do further business in the United States in return for a payment that vastly exceeded the value of all Holocaust-related dormant bank accounts and gold deposits in the country's commercial banks. Yet the National Bank of Switzerland, the recipient of 92 percent of the gold in Switzerland that came from Nazi Germany, escaped with its underpayment under the Washington Agreement because it had no business interests in the United States that later could be threatened. German companies are not obligated to contribute to the Foundation Initiative, regardless of their involvement in slave labor or other dimensions of the Holocaust, and the extent of each voluntary contribution is pegged to a company's recent annual sales, not its degree of culpability, and is tax-deductible.\n\nThese are not the only blemishes on the quest for recompense. Although lawyers for restitution plaintiffs provoked numerous German firms into opening their archives and thus precipitated many significant historical studies, these advocates also spread a lot of misconceptions about the origin and worth of several forms of spoliation. Historians will be busy correcting the record for a long time. The admiring accounts of class-action suits that have been published also warrant rebuttal, not least because several lawyers in those and other restitution proceedings of the 1990s turned out to be awful role models for their profession. A number of them were censured, disbarred, forced to resign their positions, or sentenced to jail in subsequent years for legal and financial misconduct. Finally, recent settlements have opened old wounds within the Jewish community worldwide regarding the propriety of accepting money as indemnification for death and whether funds received should go exclusively to survivors or, at least in part, to Jewish cultural undertakings.\n\nAll that said, hundreds of thousands of survivors and heirs have benefited from the persistence of people who refused to settle for the first round of restitution and compensation in the immediate postwar years, and, as with regard to the relentless pursuit of the last Nazi war criminals at large, an important point has been made. That point is that statutes of limitations do not apply to the crimes of mass murder and mammoth larceny. Sooner or later, the repressed returns and, contrary to the legal axiom, justice delayed is not necessarily justice denied.\n\nMEMORY, MYTHS, AND MEANINGS\n\n_Why?_ has examined a subject full of pain: pain of separation and exile, of persecution and torture, of degradation and murder, and of harrowing and haunted survival. To enter into the Holocaust is to risk enormous disillusionment with human beings and to awaken deep anxiety about how badly things can go wrong in this world. How can we sum up what we can and should learn from putting ourselves through this experience? What are the lessons and legacies of the topic?\n\nIn seeking an answer to that large question, perhaps we should begin by asking why anyone should study the Holocaust. The answer is not self-evident, and many people criticize our culture's fascination with the topic. In fact, Peter Novick's bestselling _The Holocaust in American Life_ insisted that we can learn almost nothing useful from human conduct in so extreme a historical situation. Elsewhere, specialists in the field have been charged with engaging in \"shoah business,\" and courses like the one on the Holocaust that I taught at Northwestern for many years have been derided as a form of special pleading that puts the miseries of Jews above those of many other populations that have suffered grievous onslaughts. Essentially, most responses to these criticisms stress that what makes the Holocaust stand out from other mass murders of the twentieth century is the sort of place that perpetrated it (an advanced and ostensibly civilized country) and the cause that propelled it (race, the most pressing issue of our time, not just in a polyglot country like the United States but also in a globalizing world). One should study the Holocaust, in other words, because its setting and impetus are highly relevant to the modern world.\n\nThe implicit corollary to that argument is that the Holocaust is a deadly precedent (after all, anything that has happened once can happen again), so we must learn about it in order to act effectively to prevent a recurrence. This practical argument can come in both universal and parochial variations. Some evidence vindicates the universal one, which emphasizes how learning can impede genocide, since the memory of the Holocaust helped impel Americans and Europeans to intervene, however belatedly, to stop the killing in Bosnia and Kosovo during the 1990s. But examples from outside Europe suggest that learning goes, literally, only so far. It clearly made no difference to the course of events in Rwanda in the 1990s, only slightly more to that in Darfur in the 2000s, and, thus far, very little to what has happened in Syria in the 2010s. The parochial version of the practical justification for studying the Holocaust\u2014because the Holocaust is a warning against Jews depending on others\u2014has been far more consistently consequential, but for both good and ill. It has stiffened the resolve of the citizens of a Jewish state in a hostile region, but it also has reinforced condescension toward Jews in the diaspora and an \"us alone against the world\" attitude that threatens to become self-fulfilling.\n\nA related and important preliminary question is: How should we study the Holocaust? I have tried to indicate that I think the answer is \"carefully and soberly,\" with a mix of precision and feeling, and without engaging in sentimentality or sanctification. Unfortunately, a certain amount of sanctification is built into the word \"Holocaust,\" which derives from the ancient Greek term for \"an offering totally consumed by fire\"\u2014in other words, a religious sacrifice. But many of those who were killed would have rejected an attribution of religious meaning to their deaths. To avoid this sort of ascription of meaning, even holiness, to mass murder, the Hebrew word \" _Shoah_ ,\" which means \"destruction,\" probably would be preferable. But the biblical uses of that word also are religiously inflected. In any case, the terminology has become firmly entrenched. Despite that, I hope readers of _Why?_ come to see what happened as a set of historical events, to be recovered, studied, and comprehended by the usual historical means. We have to approach the record neither in awe nor in anger if we hope to learn anything valuable, rather than merely to have our preconceptions confirmed and our righteousness aroused.\n\nYes, the subject challenges our sense of the comprehensible, but that is because of our revulsion. We reflexively call the Holocaust unfathomable or unbelievable as a way of distancing ourselves from it and expressing our disgust. Nonetheless, the _Shoah_ is comprehensible in the same way that any other catastrophic human or life experience is: with difficulty, patience, and application to the task. To say that the subject is incomprehensible is to despair, to give up, to admit to being too lazy to make the long effort, and, worst of all, to duck the challenge to our most cherished illusions about ourselves and each other that looking into the abyss of this subject entails. And the alternative to trying to understand how and why the Holocaust happened is to capitulate to a belief in fate, divine purpose, or sheer randomness in human events.\n\n_Why?_ has approached the problem of comprehensibility by breaking the topic down into four primary questions:\n\n1. _Why the Jews?_ Because their emancipation in the nineteenth century from centuries of residential and occupational confinement aroused a backlash that gave new impetus and new form to a chimerical hatred\u2014that is, to a belief that they constituted the single cause for everything that others opposed and feared.\n\n2. _Why the Germans?_ Because a massive and multidimensional national crisis, a perfect storm of economic, political, cultural, and social upheavals, opened the way for believers in this hatred to acquire power in Germany and to reinforce or indoctrinate others in their views.\n\n3. _Why murder and with these means?_ Because of a process of problem-solving mission creep, a cumulative radicalization of policy, as increasingly harsh efforts to \"remove\" Jews from German territory proved insufficient or unworkable and gave way to ever more extreme methods of \"elimination.\"\n\n4. _Why was the eradication of the Jews so nearly successful, resulting in the deaths of two-thirds of those in Europe and at least three-quarters of those within reach of the Nazis?_ Because indifference and self-interest in Germany and then the occupied or satellite states during World War II cleared the way for the haters; because the logistics of murder proved uncomplicated and self-financing; because the Nazis' ferocious onslaught peaked during the period of their greatest military success; and because most of the killing was done when the Allies against Germany could neither observe nor interdict it.\n\nAlong the way, we have debunked or at least complicated a number of myths. A few years back, I developed a lecture about how wide the gap has grown between what specialists know and what much of the public believes about the Holocaust. I was not alone in sensing this problem. Paul Levine also has perceived a growing \"gap between scholarship and public memory\" and called it a veritable \"clash between 'town' and 'gown.' \" My talk listed nine prevailing myths and misconceptions about the Holocaust and tried to explain why they are not so. As I list the first eight erroneous propositions, you will notice that much of this book has sought to undermine them:\n\nFirst, that antisemitism played a primary or decisive role in bringing Hitler to power; it did not. Its persistence undermined and corroded a sense of solidarity between Jewish and non-Jewish Germans, but belief in an international Jewish conspiracy or a need to \"remove\" Jews from the German body politic was never strong or widespread enough in Germany to propel Hitler to high office. Without the Depression and the collusion of conservative leaders who expected to use Hitler for their purposes, he would not have come to power.\n\nSecond, that Hitler planned to murder the Jews from the day he took office, if not before; as far as historians can tell, he did not. Massacre was always a possibility implicit in Nazi ideology, but only gradually became a semi-explicit policy of the German state\u2014as a result of the clash between the ethnic mathematics of Hitler's drive for living space and his conviction that military victory depended on the disappearance of Jews from his realm.\n\nThird, that the Allies could have done much to impede the killing once it began; given where and when most of the slaughter took place\u2014in the northeast quadrant of the European continent and in the eighteen months following Germany's invasion of the Soviet Union, when the Reich was continuously on the offensive and winning\u2014they could not. As David Cesarani has shown, the only ways Germany's opponents could have reduced the carnage significantly were for the British and\/or the Soviets to lose the war in 1941, thus salvaging the prospect of deporting Jews elsewhere, or for the Allies to win the conflict in 1942\u201343, which was clearly beyond their power.\n\nFourth, that greater passive or active resistance by Jews could have reduced the death toll considerably; not realistically speaking. Such behavior would have required an almost unimaginable degree of clairvoyance on the part of Jews, an equally unimaginable degree of solidarity among them, and a far different balance of forces between Jews and their Nazi captors.\n\nFifth, that popular attitudes toward Jews, rather than political structures and interests, were the principal determinants of survival; not in the aggregate. More courage to help on the part of non-Jews would have produced more survivors, but nowhere near as many as remained alive because of the cynical political and personal calculations of collaborationist regimes in Europe.\n\nSixth, that the Holocaust diverted resources from the German war effort and weakened it in significant ways; not really. Germany sent more trains to the staging areas of Operation Barbarossa, the invasion of the Soviet Union, every day in mid-1941 (2,500) than the SS deployed to transport Jews to camps during the entire Holocaust (2,000), so clearly the deportations did not stress the capacity of the Reich's railroads. The nation's reliance on forced and slave labor would have been just as chaotic, inefficient, and insufficient with the retention of the murdered Jews as it proved to be without them.\n\nSeventh, that the slave labor system was driven principally by greed; it was not. It was the creation of a regime that lacked the population to sustain the massive war on which Germany had embarked and the imagination and generosity to enlist enough other Europeans in the cause.\n\nEighth, that most of the leading perpetrators of the Holocaust escaped punishment after World War II; in fact, the great majority of the vilest ones were already dead by 1945 or caught and penalized fairly shortly thereafter. To be sure, the Germans and the victors of World War II could have tried harder to find the killers who got away and could have rested less content with punishment that concentrated more on the order givers than the order executers in the camps and the shooting units. But the reckoning after 1945 for the Holocaust was more comprehensive than for any other modern instance of genocide.\n\nAt one point or another, the chapters in _Why?_ have presented detailed evidence that refutes each of those assertions. Nonetheless, I do not expect them to disappear. Sometimes historical work is an extended game of whack-a-mole.\n\nBut _Why?_ has said little thus far about the ninth common misconception. This is the idea, associated with the widely read books of Zygmunt Bauman and Detlev Peukert, that the Holocaust was a product of modernity and a demonstration of its dangers. The prevailing image is of mechanized murder, epitomized by ubiquitous references to \"factories of death.\" But, although Auschwitz was a human disassembly line, it resembled a nineteenth-century slaughterhouse more than a modern manufacturing plant, and the other killing centers, with the partial exception of Majdanek, were ramshackle affairs. Most of the camps killed with a rather simple and at most early industrial device: a gasoline engine. Even the designation of the intended victims was done the old-fashioned way: by drawing up deportation lists with ink on paper, a task that was usually delegated to Jewish organizations in Western Europe and Germany and to Jewish Councils in the ghettos, if it was done at all. In occupied Russia and Ukraine, the non-Jewish locals just pointed out the Jewish ones. Finally, almost half of the killing occurred by starvation and exposure or by one-on-one bludgeoning or shooting\u2014in short, by rather primitive means.\n\nNeither is the broader form of this identification of the Holocaust with modernity accurate\u2014namely, that the Holocaust represented the modern world's aspiration and achievement of the means to carry out vast forms of social engineering. The ambition to wipe out a whole group is not specifically modern: The goal is as old as the Israelites' extirpation of the Amalekites and the Romans' erasure of the Carthaginians, both of which were more complete than the Nazi murder of the Jews, despite being accomplished with mere fire and sword. Moreover, the pseudoscience that gave a supposedly modern gloss to the attempt at racial purification\u2014eugenics\u2014was, in fact, the very opposite of modern. It was the application of animal husbandry to human society, an argument that people can and should be bred like racehorses, and nationalities can and should be considered as breeds. Nazi racism was fundamentally rooted in an agricultural, not an industrial, world, and in an understanding of genetics that approximated that of the medieval or pre-modern eras. Furthermore, in scientific terms, eugenics was a fraud. Far from being modern in either conception or means, the Holocaust was an outbreak of extraordinary primitivism, a fitting product of an ideology that believed that all life is governed by the law of the jungle. In the astute words of Dan Stone, \"Modernity was less the driving force of the Holocaust than the setting for it.\"\n\nFinally, of course, the biggest myth about the Holocaust is another one not yet discussed, the claim that it never happened, and this book cannot close without discussing how ridiculous this claim is and why it continues to get made. On the ridiculousness, the first point to make emphatically is that the Holocaust is, quite simply, one of the most amply documented events in world history. To be sure, historians had to dig for several decades to arrive at as complete a picture of what happened as we now have, and, along the way, interpretations evolved as the state of our knowledge did. After all, the perpetrators went to considerable trouble to destroy the evidence of what they had done, though it was fortunately too voluminous to eradicate. Thus, to cite the most significant examples, we still have many of the passenger lists of the deportation trains, the prisoner death registries at Auschwitz and Mauthausen, some of the receipts for orders of Zyklon, most of the _Einsatzgruppen_ reports that itemized and categorized the dead, photos of the victims' belongings piled up at Babi Yar, Lodz, and Birkenau, the minutes of the Wannsee Conference, H\u00f6fle's tally of the death toll at the Reinhard camps, SS statistician Richard Korherr's report from the spring of 1943 on the extent of the final solution to date, a vinyl recording of Himmler delivering his speech at Posen in 1943, Joseph Goebbels' extensive diaries, Alfred Rosenberg's somewhat more episodic ones, the postwar confessions of Rudolf H\u00f6ss and numerous other killers, and so on.\n\nYet a vocal group of deniers persists in asserting that gas chambers did not exist and genocide did not occur during the Third Reich, that the number of Jews who died in World War II was small and an incidental outcome of the fighting, that the evidence mentioned above consists of forgeries or coerced testimonies, and that Jews and communists contrived the \"hoax\" of the Holocaust after World War II in order to discredit Germany, extract money from it, and gain support for a Jewish state in Palestine. Calling themselves \"revisionists,\" these deniers drape themselves in the trappings of scholarship, but their strained arguments so clearly resemble the conspiracy theories that animated nineteenth-century antisemitism that their role as the real driving force behind denial shows through.\n\nA British judge examined the claims of David Irving, perhaps the leading Holocaust denier of recent decades, during a libel trial in 2000 and pronounced them a deliberate falsification of the historical record. More recently, Bettina Stangneth's _Eichmann Before Jerusalem_ has shown how a group of Nazi exiles and sympathizers in Argentina assembled most of the core arguments of denial and published them under a pseudonym in an \u00e9migr\u00e9 German journal called _Der Weg_ (The Way) in 1954. Their article, \"On the Streets of Truth,\" bearing the byline of a fictitious American journalist named Warwick Hester, who revisionists later claimed was the equally imaginary \"American jurist Stephen F. Pinter,\" still circulates on the Internet. Stangneth considers it \"the principal source text\" for Holocaust deniers. Arguing with people who believe this nonsense is pointless, because the real source of their belief is not evidence or reasoning but incorrigible and circular fantasies about Jewish power and malevolence.\n\nThe title of a fine book by Eva Hoffman, the daughter of Holocaust survivors, is _After Such Knowledge_ , and a way to bring this book to an end is to put that title in the form of a question. What should we do \"after such knowledge\"? What are the implications of all that we have learned about the Holocaust? Few subjects seem to cry out more for an attempt to establish their \"meaning\" or \"message,\" and few subjects can make the person trying to formulate such conclusions feel so inadequate. Raul Hilberg often said that he was afraid to address the big questions raised by the _Shoah_ for fear of giving small answers. Can we nonetheless draw any larger conclusions from our examination of these terrible events, despite all appropriate cautiousness about the attempt?\n\nI think we can, but before I try, let me underline three features of our world that have profoundly changed since the end of the Second World War and that have affected the potential for renewed outbreaks of antisemitism.\n\nFirst, the European world of the first half of the twentieth century was caught up in a kind of civil war between ideologies that prized individualism, discussion, and fulfillment\u2014such as liberalism, representative government, and free enterprise\u2014and ones that prioritized collectivism, obedience to group goals, and submission to authority: for example, fascism, Nazism, communism, and in those days most forms of Christianity. Many of the dangers to the Jews arose out of the way they were used in this conflict, as symbols of individualism or wealth or communism or freethinking and unbelief. These days are largely gone. Western nations are nearly all individualist, secular, and capitalist now. It is difficult to depict Jews as threats because the ideologies once tied to them have either triumphed and become generally shared or, in the case of communism, collapsed. That does not mean that antisemitism has disappeared, only that it has become, for the moment, largely powerless in the Western world.\n\nOther parts of the globe are another matter, however. Wherever individualism, religious or ethnic pluralism, and enterprise remain ideologically suspect or are perceived as alien, Jews and all other minorities remain endangered, and the wretched lies of the _Protocols of the Elders of Zion_ continue to be circulated and believed, as they are currently in Russia and many predominantly Muslim lands and among non-Jewish immigrants from these lands to other places. Just as in nineteenth- and twentieth-century Central and Eastern Europe, in the rest of the world today, the security of Jews, like that of most minorities, is least wherever the liberal values of toleration, coexistence, and openness to change are weak. To prevent other Holocausts, it is not enough to combat antisemitism; one has also to fight for these broader values, and not only at home. This is one of the central insights of the much-maligned European Union, which has insisted on increased protections of minority rights, especially for Roma and gay people, as preconditions for admitting countries to its ranks. In an increasingly globalized world, the obligation to combat and reduce parochialism and intolerance is an increasingly global matter.\n\nSecond, even in Western Europe and North America, the lessons drawn after 1945 about the world that spawned the Holocaust and the countermeasures put in place then are now under attack. In economically difficult times, Europe is experiencing a widespread resurgence of nationalism in the forms of hostility to foreigners, especially immigrants and the bureaucrats of the European Union, and a retreat from the welfare state under the cover of reducing debts. The current condition of Greek and Hungarian politics gives us a sense of what we have to look forward to if these trends continue and strengthen: the rise of neo-fascist parties and the enlistment of the energies of young and often unemployed men in brutality. Antisemitism, remember, rises and falls in inverse relationship to the stock market. Moreover, in this country and in Europe, economic inequality is growing, as the proponents of a certain version of free market capitalism increasingly lose sight of the implicit contract that most Western nations made with their populations in response to World War II. That contract traded a promise that governments would provide basic services and security in return for citizens abandoning political extremism. Communism and fascism were the outgrowths of societies in which the distribution of wealth and opportunity were massively unequal, and the postwar architects of European unification and social safety nets knew that reducing inequality was the essential prerequisite for social peace. As governments cease to keep their part of this bargain, they invite citizens to cease to keep theirs, and in such a context, no minority (and perhaps no democracy) will be safe. This is why the sort of political rhetoric that categorizes people as \"makers vs. takers,\" often implying that the latter group consists heavily of immigrants, is a profoundly dangerous and ignorant throwback to a vastly destructive era.\n\nThird, something else has changed in the past sixty years that may provide cause for worry. Ironically, that something else is the existence of a Jewish state. Nowadays, hostility to the existence and policies of the state of Israel tends in some quarters to slide into hostility to Jews in general and to the revival of vicious stereotypes about them. A potential for antisemitism to grow, a temptation to depict Jews once more as aliens with different purposes and priorities from those of their fellow citizens, exists in Europe (and to a lesser extent in the United States) because of a gap between European and Israeli interests and sensitivities.\n\nFor non-Jews in Europe, the top priority in the Middle East is not the survival of a Jewish state; the top priorities are political calm, access to oil, and sufficient economic development of the region so that its burgeoning and overwhelmingly young population does not swamp Europe's declining and aging one. Non-Jewish Europeans would prefer the problem of who gets what parts of what used to be Palestine simply to go away because the problem not only poses a threat of war on their doorstep, but also engenders militancy and unrest among the millions of Muslim immigrants already in Europe. At the same time, many American and European Jews, for emotional and practical reasons, including the memory of the Holocaust and the view that Israel is a potential home of last resort in the event of new eruptions of antisemitism, do not think they can afford such indifference.\n\nIn this division of interest lies a danger that demagogues may arise to accuse Jews of divided loyalties and of dragging the entire nations of which they are part into conflicts in which more is at stake for them than for those nations. Should that happen, a situation not unlike that of the 1930s will arise, antisemites will have an opening, and the strength of inclusive and liberal values will face powerful challenges. In other words, the existence of a Jewish state, especially one in which the most insular segments of the population play an increasingly decisive role, presents dangers to Jews elsewhere as well as benefits. In 2003, Tony Judt, a distinguished historian of Jewish descent, aroused an explosive debate by implicitly asking, in the _New York Review of Books_ , \"Is Israel good for the Jews?\" and more or less answering no. Peter Beinart, the author of _The Crisis of Zionism_ and himself an Orthodox Jew, has now taken Judt's place in highlighting the divergence between many Israeli practices and the liberal values that protect Jews elsewhere and in urging Jews to face up to the implications of this divergence as a matter of both principle and prudence.\n\nIn short, current conditions differ in hopeful and worrisome ways from those that produced the Holocaust. What, then, are the implications today of what we have learned for those of us fortunate enough to be living in relatively free societies? I think the Holocaust has two important lessons for minorities in the United States in general and for Jews in particular.\n\nLesson One is: Be alert but not afraid. Some degree of antisemitism is ineradicable for the foreseeable future; it has too long a pedigree and is too much the dark side of apartness and normal social frictions to disappear. But antisemitism is not necessarily always dangerous; it made Hitler possible, but it did not make him succeed. An irony of the history of antisemitism is that this ideology that called Jews parasites always has been a parasitic issue. To succeed, it has needed a host that it can exploit\u2014a pervasive sense of crisis and victimization that allegedly justifies lashing out in reprisal. That is the essential prerequisite for widespread demonizing of Jews as the root of all evil, and the presence of this sort of sweeping crisis is what brought Hitler to power.\n\nOne can argue that the Great Recession sorely tested America, yet demonstrated that the will to demonize cannot get the upper hand in this country for several reasons. First, we have the example of the Holocaust to serve as a warning of what happens when such demonization triumphs. If Holocaust education has any prophylactic value, it probably lies in dampening impulses to attack Jews and in multiplying the number of antiantisemites. Second, the spread of education and of more complicated notions of causation may have made more people resistant to simplified blame games. I hope this is so. Third, we benefit from the freedom of the media to expose stereotyping, but with the fragmentation of news outlets and market segments into increasingly walled-off camps, this protection may be waning. Fourth, and above all, the internal diversity of American culture is a form of protection against demonization. We have no dominant faith or ethnic group anymore\u2014in a sense, we are all members of some minority or another. As a result, many and perhaps most of us should and often do behave like those groups during the Holocaust whose own minority status led them to sympathize with Jews.\n\nConsider what has NOT happened regarding Jews in the United States in recent years. At the heart of three of the nation's most sensational recent corruption scandals were three Jews, Andrew Fastow at Enron, Jack Abramoff of the congressional lobbying payoffs, and Bernard Madoff, who perpetrated the largest Ponzi scheme of all time. Moreover, a majority of the heads of the big banks and brokerage houses that recklessly sold derivative contracts and mortgage-backed securities and thus brought on the recent recession also were Jews. Yet outside of several neo-Nazi websites, no one has been idiotic enough to advance the proposition that these people are typical of American Jewry, and no political movement has arisen around a program of reforming Wall Street by \"cleansing\" it and the nation of Jews. Look how far we have come from the Strousberg and Panama affairs of the late 1800s.\n\nSome commentators have noted the prominence of Jews among the proponents of an unnecessary and enormously costly war in Iraq, figures such as Paul Wolfowitz, Scooter Libby, and former senator Joseph Lieberman. Their role has given rise to muted suggestions that such people have been motivated by a desire to protect Israel and to muffled debates about the supposed power of pro-Israeli lobbying groups, such as AIPAC (American Israel Public Affairs Committee), to suppress discussion in the United States over Israeli policy in the occupied territories. But none of this conversation has turned virulent or violent. Most importantly, no political movement has emerged that presents any person's behavior as the expression of categorical beliefs and flaws rather than individual ones.\n\nAnd consider the situation of the American minority that is in some ways most comparable to the Jews in Germany prior to 1933: gay people. They, too, say no to fundamental beliefs\u2014in this case about gender roles; they, too, are present in small enough numbers to be easily attackable; they, too, have been depicted as degenerate and corrupting\u2014the Bible contains scriptural passages frequently invoked to stigmatize gay people just as the Gospel of John and the Easter service used to be quoted to stigmatize Jews; they, too, are simultaneously derided for supposedly hanging together but also trying to blend in; and above all, gays, too, are often depicted as threatening\u2014associated with child molestation and AIDS, as Jews once were with the blood libel and plague, and decried as people so unclean that they subvert and sully marriage by seeking to engage in it. Yet a \"chimerical\" image of gay people has failed to take hold in the United States, and resistance to a supposed \"homosexual agenda\" appears to be losing even its primary function of rallying the religious right now that \"godless communism\" is no longer available. The so-called defense of marriage and the hypocritical and cruel policy of \"don't ask, don't tell\" are gone.\n\nNonetheless, to expect bigotry to evaporate as if it had never existed would be foolhardy. Look at the way anti-immigrant feeling, much of it echoing the rhetoric of the nineteenth-century Know-Nothing Party, has surged in America in recent years, now turned not against Irish, Italians, and Jews but against Latinos, Muslims, and people of color. Overcoming these ugly repetitions of American exclusionism will take time, but it will happen, and in the same way it always has in the past, by sheer force of numbers. So, the first lesson of the study of the Holocaust for all minority groups in American society is \"be alert, but not afraid.\" The general trend in America remains toward pluralism, freedom, and Jefferson's right to \"the pursuit of happiness\" for each person in his or her own way. We all have a responsibility to see to it that the trend continues; its opposite is the oppression, the stasis, and the homogeneity that Nazism prized.\n\nLesson Two of the Holocaust for minority groups in America and Jews in particular is: Be self-reliant but not isolationist. That means taking care with two very dangerous and common words nowadays: memory and identity. We tend to glorify both with cries such as \"never again\" or \"never forget\" and assertions of our heritage or loyalties before every utterance. But both practices have downsides.\n\nThis may sound like an odd, even heretical thing for a historian to write, but there is such a thing, in every culture as in every life, as too much memory. It can block learning, change, and trust. Looking perpetually back can seem to justify endless bitterness and to authorize fatalism. But, as Susan Sontag once wrote, \"To make peace is to forget.\" The assumption that the future cannot help but resemble the past, that people who once hated me always will hate me, is often self-confirming. George Mitchell, the architect of the Good Friday agreement that brought peace to Northern Ireland, notes in his memoirs that on the eve of the deal, 83 percent of the people there thought a resolution of their civil war impossible. Sometimes one of the most valuable skills in life is the ability to think outside the box of the past, as Mitchell did. Not always, of course, but sometimes, and one of the purposes of studying history is to acquire a feel for the difference.\n\nThere is also such a thing, in every culture as in every life, as too much pride in what one's kind has been and currently is, rather than in what it can achieve in concert with others. The history of the Holocaust suggests that minorities run risks when they depend too much on others, since the others generally will be guided by self-interest, but also that cutting oneself off from others poses its own, perhaps equal, dangers. Groups, like individuals, cannot make their ways alone; they need friends.\n\nIn addition to those two lessons with special relevance to members of minority groups, I believe that the study of the Holocaust has three broad implications for all citizens, whether members of minorities or not. First, the Holocaust highlights the primacy of avoiding situational causes. The veneer of civilization is thin, the rule of law is fragile, and the precondition of both is economic and political calm. This means that politics matters, and none of us can ever afford to fail to participate in making responsible public policy. Nazism stemmed from German racism, but that ideology would never have become national policy without the presence of an economic, national, and ideological crisis that fostered demagoguery and irresponsibility. Everyone's first goal in a decent society must be to avoid contributing to such a crisis or to those responses. Remember, more Germans became antisemites because they became Nazis than vice versa. Some of them became Nazis before 1933 because of the mess and the impasse in which their country found itself, and even more of them did so later because the Nazis became the apparently rather successful holders of power.\n\nI have often thought that one of the great injustices of the Nuremberg trials was that Franz von Papen, the man who did more to bring about Hitler's appointment as chancellor of Germany than anyone else, was acquitted because he had not committed an actual war crime. True, but he had made all the war crimes possible. The court held that political misjudgment is not a criminal offense, and I concede the point. But historians rightly vilify Papen, and his name will be forever odious. Something similar can be said of the financiers and bankers whose recklessness brought on the stock market crash of 1929 and the collapse of Germany's banks in 1931. Like Papen, they are permanent reminders that our first responsibility as citizens, regardless of our walk of life, is to do no harm. That is not a doctrine of passivity. _Why?_ has shown how much harm doing nothing can do. It is a doctrine of activity informed by seriousness, prudence, restraint, and unselfishness. These were not Franz von Papen's chief attributes, any more than they were the chief attributes of the German financial wizards of the 1920s or, for that matter, of the American ones during the first decade of this century.\n\nSecond, the Holocaust illustrates the fundamental importance and difficulty of individual courage and imagination. This dreadful history shows the necessity of standing up to categorization and conspiracy peddling, of refusing to turn a blind eye or a deaf ear to defamation. There can be no drawing of distinctions between citizens when it comes to fundamental human rights, no hair-splitting about who gets to have them and who does not. In fact, such rights are for the people whom we fear or dislike because they are the people who need them. But this dreadful history also shows that doing the right thing can have costs that are multiplied by the unwillingness of most people to pay them, so bravery is not enough\u2014wit, wiliness, shrewd judgment, persistence, and creativity in challenging evil are also indispensable. Resistance is never easy and seldom comfortable, and compassion has to be practiced in order to hold up when challenged. Rising to that challenge begins with a refusal to be cowed, followed by alertness to opportunity. According to the philosopher Philip Hallie, who several decades ago wrote a powerful account of the villagers in Le Chambon-sur-Lignon, their principal leader, Pastor Andr\u00e9 Trocm\u00e9, \"believed that if you choose to resist evil, and you choose this firmly, then ways of carrying out that resistance will open up around you. His kind of originality generated originality in others.\"\n\nThird, the Holocaust testifies to the need to preserve the essential distinction between means and ends. Antinomianism\u2014the idea that moral restrictions do not apply to us because of some special nobility or necessity of our purposes\u2014is the fatal temptation that the Nazis proffered and the fateful rationalization they used. Still endemic, it always feeds on fear. In times of extreme crisis, the history of the Holocaust demonstrates, a person's most profound moral commitments\u2014to family, faith, community, country, organization, party, and principle, for example\u2014can be made to seem like reasons to choose to do great harm, can be deeply corrupted. Franz Neumann, one of the pioneer analysts of National Socialism, highlighted its adeptness at \"surrounding every perfidy with the halo of idealism.\" The dreadful events of the Holocaust should be a reminder that calls to self-defense and for retribution are among the most corruptible of ideals. As William Pitt, a British prime minister in the mid-eighteenth century, once warned: \"Necessity is the plea for every infringement of human freedom. It is the argument of tyrants; it is the creed of slaves.\" The politics of division and emergency, of bullying and rage\u2014the politics that says desperate times require the political equivalent of \"stand-your-ground\" laws\u2014that sort of politics always deserves opposition and scorn because it is the politics that is just itching to get out of hand.\n\nThe Holocaust was not mysterious and inscrutable; it was the work of humans acting on familiar human weaknesses and motives: wounded pride, fear, self-righteousness, prejudice, and personal ambition being among the most obvious. Once persecution gathered momentum, however, it was unstoppable without the death of millions of people, the expenditure of vast sums of money, and the near destruction of the European continent. Perhaps no event in history, therefore, better confirms that very difficult warning embedded in a German proverb that captures the meaning I hope readers will take away from this book: _Wehret den Anf\u00e4ngen_ , \"Beware the beginnings.\"\n\nThat proverb comes to mind whenever I am asked at public forums when and how I think the Holocaust could have been prevented or stopped. My response is to name a time and place exactly: April 1\u20135, 1933, in Berlin. April 1 is well known as the date of the Nazi boycott of Jewish-owned shops across Germany. But something else occurred that day, the occupation by a company of Nazi storm troopers of the offices of the National Association of German Industry, headed by Gustav Krupp von Bohlen und Halbach, who also was the leader of the Krupp armaments and steel firm. The thugs made clear their intention to stay and disrupt the association's work until it dismissed all its employees who were Jews or affiliated with other political parties. When Krupp, who was a very powerful and prominent man, tried to persuade Hitler to call off his dogs, the Nazi F\u00fchrer simply declined, explaining that he could not restrain the enthusiasm of people who had been through thick and thin with him as he rose to power. Krupp then gave in, firing everyone of whom the Nazis disapproved on April 5 and thus breaking his contracts with each of those people.\n\nOne of the members of the National Association's governing board, a man named Georg von M\u00fcller-Oerlinghausen, wrote a prophetic protest to Krupp eight days later, saying that his actions amounted to capitulation to bullying and that they deprived the organization of all basis for future noncompliance with Nazi demands. If the German industrialists would not stand up for the contractual legal rights of their own personnel, M\u00fcller-Oerlinghausen asked, for whom would they stand up and on what grounds? He was right, and the more powerful the Nazis became, the more irreversibly right he was.\n\nBeware the beginnings.\nACKNOWLEDGMENTS\n\nNaturally, a book that took shape over almost thirty years owes a lot to a great many people, too many to list most of them by name. I have the privilege of working in a vibrant field, full of indefatigable and intelligent researchers from whom I have learned something new every day, so I want to express my appreciation to the scholars on whose publications I have drawn heavily in this book, both explicitly and implicitly. I also thank the thousands of Northwestern University students who completed History 349 between 1987 and 2015 and whose curiosity and eagerness to learn motivated me to keep trying, year after year, to make my presentation of the evidence and my reasoning from it ever clearer and tighter. Many colleagues and graduate students contributed valuable input at a History Department workshop on chapter 3 in Evanston in May 2015. I am especially indebted to Professor Amy Stanley for prompting me to rethink one minor and one major issue and to Professor Robert Lerner for alerting me to a significant omission. Thanks, too, to the participants in the Silberman Seminar at the United States Holocaust Memorial Museum in June 2015, who heard these chapters as lectures and provided useful feedback. Several generous colleagues and friends read the manuscript and offered numerous suggestions that enhanced it: Christopher Browning, Benjamin Frommer, Richard Levy, Wendy Lower, Thomas Lys, and Michael Marrus. The Jewish Foundation for the Righteous and Oxford University Press kindly granted permission to reprint several passages that I wrote for the chapter introductions in _How Was It Possible? A Holocaust Reader_ (2015) and Chapter 35 of _The Oxford Handbook of Holocaust Studies_ (2010). My agent, Peter Bernstein, and my editor, John Glusman, saw exactly the virtues in the manuscript that I tried to put there, and that carried me over the finish line. Everyone mentioned here earned and has my gratitude, but I alone am responsible for any errors or defects that remain.\n\nSpecial thanks to Volt and the dogs: Offsetting the bleakness of what I was working on, they made each day a pleasure.\nNOTES\n\nPage numbers listed correspond to the print edition of this book. You can use your device's search function to locate particular terms in the text.\n\nINTRODUCTION\n\n(xiv) **\"Impossible to remember,\"** Judt, _Postwar_ , 830.\n\nCHAPTER 1: TARGETS\n\n() **A professor of mine** ; he put the matter more elegantly in writing: \"antisemitism is a cluster of behaviors with a single name;\" Gay, _Freud, Jews and Other Germans_ , 13.\n\n() **Germany took pains** , Motadel, _Islam_ , 56\u201360.\n\n() **xenophobic and chimerical forms** , Langmuir, _Toward_ , 306, 328\u201352.\n\n() **Ancient Roman attitudes** , Lindemann and Levy, _Antisemitism_ , 38\u201340.\n\n() **Freud, Samuel, Poliakov, and Cohn** , Hand and Katz, _Post-Holocaust France_ , 177\u201384.\n\n() **\"doctrine of Jewish witness,\"** Lindemann and Levy, _Antisemitism_ , 64\u201365; Bauer, _History_ , 9.\n\n() **surge of attacks . . . blood libel** , Lindemann and Levy, _Antisemitism_ , 68\u201370, 74.\n\n() **Luther and Erasmus** , Nirenberg, _Anti-Judaism_ , 254, 262, 266.\n\n() **\"beliefs in racial or ethnic determinism,\"** Lindemann, _Esau's Tears_ , xiv.\n\n() **\"A Jew can no more** , **\"** Stern, _Politics_ , 141.\n\n() **On Gobineau and Schlegel** , see Weitz, _Century_ , 33\u201335; Arvidsson, _Aryan_ , 26\u201330.\n\n() **On Galton and Ploetz** , see Burleigh and Wippermann, _Racial State_ , 29, 32.\n\n() **Patents of Toleration** , Beller, _Antisemitism_ , 33\u201334; Meyer, _German-Jewish History_ , v. 2, 16\u201317.\n\n() **Bavarian petition . . . opposing equality for Jews** , Hochstadt, _Sources_ , 24.\n\n() **Country and city population figures** , Mitchell, _Statistics_ , 3\u201315.\n\n() **Jews seemed disproportionately present . . . illustrative figures** , Slezkine, _Jewish Century_ , 47\u201350; Pulzer, _Rise_ , 12; Hamann, _Hitler's Vienna_ , 327\u201328; Elon, _Pity_ , 259.\n\n() **That may have been what Albert Einstein had in mind** , Elon, _Pity_ , 274.\n\n() **Dreyfus and Beilis** , Lindemann, _Jew Accused_ , passim.\n\n() **3 percent** , ibid., 60.\n\n() **\"enemies of the antisemites, not of antisemitism,\"** ibid., 126.\n\nCHAPTER 2: ATTACKERS\n\n() **On Herder** , see Burleigh and Wippermann, _Racial State_ , 25; Smith, _Handbook_ , 242\u201343; Arvidsson, _Aryan_ , 26, 29, 74\u201375.\n\n() **On Fichte** , see Katz, _Prejudice_ , 57\u201359; Smith, _Handbook_ , 245\u201346.\n\n() **the most famous tales that the Grimms reproduced** , Smith, _Handbook_ , 263.\n\n() **\"Jewishness in Music,\"** Katz, _Darker Side_ , 33\u201346.\n\n() **On Hep-Hep** , see Hoffmann et al., _Exclusionary Violence_ , 23\u201342\n\n() **\"freedom to choose their own trades,\"** Aly, _Why_ , 34.\n\n() **\"No longer should we tolerate,\"** Katz, _Prejudice_ , 252.\n\n() \" **almost exclusively in favor of our co-citizens,\"** ibid., 253.\n\n() **six editions** , ibid., 256.\n\n() **\"natural reaction. . . . misfortune,\"** Hochstadt, _Sources_ , 27.\n\n() **265,000 German men signed** , Levy, _Antisemitism_ , v. 1, 21.\n\n() **Stoecker's party was overwhelmed** , Pulzer, _Rise_ , 87.\n\n() **Progressive Party gains** , Ritter, _Wahlgeschichtliches Arbeitsbuch_ , 39.\n\n() **figure 2** , ibid., 40\u201341, 146.\n\n() **electoral base small . . . narrow** , Pulzer, _Rise_ , 189\u201390; Lindemann and Levy, _Antisemitism_ , 130.\n\n() **\" _Gegen Junker und Juden_ ,\"** Levy, _Downfall_ , 58.\n\n() **Wilhelm Marr . . . \"a business,\"** Zimmermann, _Marr_ , 103.\n\n() **Krupp and the usual breakdown in election districts** , Ritter, _Wahlgeschichtliches Arbeitsbuch_ , 133\u201335 (Prussia), 164\u201366 (Saxony).\n\n() **63 percent** , Lindemann, _Esau's Tears_ , 149.\n\n() **Jewish population, birthrate, intermarriage, immigration, urbanization** , Richarz, _Leben_ , v. 2, 12\u201323.\n\n() **concentration of Jews' occupations** , ibid., 23\u201334.\n\n() **Jewish immigrants from Poland** , ibid., 18\u201319.\n\n() **\"They lived like bankers,\"** Elon, _Pity_ , 255.\n\n() **Jewish officers** , Vital, _People Apart_ , 135.\n\n() **2 percent of the professors** , Richarz, _Leben_ , v. 2, 32.\n\n() **\"cultural code,\"** Volkov, _Germans_ , 115.\n\n() **The Prussian state had taken firm action** , see Smith, _Butcher's Tale_.\n\n() **Prominent Jewish industrialists** , Elon, _Pity_ , 265\u201367.\n\n() **election of 1912** , ibid., 293; Levy, _Downfall_ , 250.\n\n() **\" _Kinderkrankheit_ ,\"** Gay, _Freud, Jews, and Other Germans_ , 15.\n\n() **\"Jew count,\"** Levy, _Antisemitism_ , v. 1, 371\u201372; Pulzer, _German State_ , 205\u20136; Elon, _Pity_ , 338\u201339; Rosenthal, _Ehre_ , passim.\n\n() **if only 12,000 or 15,000 more Jews** , Hitler, _Mein Kampf_ , 679.\n\n() **Wilhelm II and Ludendorff** , Tooze, _Deluge_ , 135; Liulevicius, _War Land_ , 198.\n\n() **the incidence of violent acts** , see Walter, _Kriminalit\u00e4t_ , and Hecht, _Juden_.\n\n() **figure 3** , J. Falter et al., _Wahlen_ , 44.\n\n() **On the origins of the _Protocols_** , Segel, _A Lie_ , 65\u201369; Bronner, _Rumor_ , 80\u201388.\n\n() **\"the best proof,\"** Hitler, _Mein Kampf_ , 307.\n\n() **Reparations and war debts** , Tooze, _Deluge_ , 369, 444; Balderston, _Economics_ , 20\u201321.\n\n() **Scheunenviertel riot** , Hoffmann et al., _Exclusionary_ , 123\u201340.\n\n() **\"theozoology,\"** one of Hitler's ideological forerunners in Vienna, Jorg Lanz von Liebenfels, coined the term as the title of a book he published in 1904.\n\n() **\"His speeches,\"** Konrad Heiden, quoted in Rees, _Charisma_ , 28.\n\n() **\"biological materialism,\"** Aly, _Why_ , 11.\n\n() **\"We know only one people,\"** Hamann, _Hitler's Vienna_ , 212.\n\n() **\"thoughtlessness,\"** Arendt, _Eichmann_ , 49, 287\u201388; compare Stangneth, _Eichmann_ , 201\u20132, 217\u201319.\n\n() **\"a blemish,\"** Rauschning, _Voice_ , 220.\n\n() **Twenty-five Point Program** , Noakes and Pridham, _Nazism_ , v. 1, 14\u201316.\n\n() **programs laid down in 1931 and G\u00f6ring's speech** , Adam, _Judenpolitik_ , 26\u201331.\n\n() **\"racial tuberculosis\" . . . Robert Koch** , Fest, _Hitler_ , 212; Kershaw, _Hitler_ , v. 2, 470.\n\n() **downplayed antisemitism** , Allen, _Seizure_ , 142; Wistrich, _Hitler_ , 45.\n\n() **\"the opposite of what exists today,\"** Fest, _Face_ , 296.\n\n() **On Hirschfeld** , see Beachy, _Gay Berlin_ , 160\u201386; on Fromm, Aly and Sontheimer, _Fromms_.\n\n() **unemployment . . . 15 percent** , Balderston, _Economics_ , 79; Overy, _Recovery_ , 20; and James, _Slump_ , 357.\n\n() **The way the Nazis campaigned** , Bracher, _Dictatorship_ , 179, 182.\n\n() **Northeim** , Allen, _Seizure_ , 88\u201389, 126, 142.\n\n() **40 percent** , Noakes and Pridham, _Nazism_ , v. 1, 84.\n\n() **Alone among the parties** , M\u00fchlberger, _Social Bases_ , 71\u201380.\n\n() **\"emancipate women,\"** Stibbe, _Women_ , 17.\n\n() **New Year's Day 1933** , Turner, _Thirty Days_ , 1.\n\n() **\"were drawn to antisemitism,\"** Allen, _Seizure_ , 84.\n\nCHAPTER 3: ESCALATION\n\n() **two-tier approach** , Barkai, _Boycott_ , Chapter 2.\n\n() **\"social death,\"** Kaplan, _Between_ , 5.\n\n() **Since 1933 . . . working for themselves or each other** , Barkai, _Boycott_ , 106\u20138, and Bajohr, _Aryanisation_ , 108.\n\n() **\"A law making the whole of Jewry liable,\"** Tooze, _Wages_ , 221.\n\n() **\"in the event of his death,\"** Noakes and Pridham, _Nazism_ , v. 3, 73.\n\n() **The gist was** , ibid., 72\u201379.\n\n() **carted off to concentration camps** , Cesarani, _Final_ , 164\u201365.\n\n() **willing to join in the violence** , see Steinweis, _Kristallnacht_ , and as background, Wildt, _Volksgemeinschaft_ , chapters 5\u20137.\n\n() **36,000, 26,000, and 600 men** , W\u00fcnschmann, _Before Auschwitz_ , 197, 204.\n\n() **allowed the German insurance companies to renege** , Feldman, _Allianz_ , 221, 227.\n\n() **emergence of a new word** , Hayes, _How_ , 172\u201373.\n\n() **he told the Czech foreign minister** , Adam, _Judenpolitik_ , 235.\n\n() **\"[t]he Germans. . . . have embarked,\"** Breitman and Lichtman, _FDR_ , 120.\n\n() **German planners actually** , Browning, _Origins_ , 86; Gerwarth, _Hangman_ , 179\u201381; Cesarani, _Final_ , 300\u20131.\n\n() **\"the bolshevist method,\"** Breitman, _Architect_ , 119.\n\n() **Hermann G\u00f6ring charged Reinhard Heydrich** , Roseman, _Wannsee_ , 53.\n\n() **murders reached a crescendo** , Burds, _Rovno_ , 20\u201321.\n\n() **\"the soldier must have full understanding,\"** Megargee, _Annihilation_ , 125.\n\n() **Events proceeded along parallel lines in German-occupied Serbia** , Browning, _Origins_ , 334\u201346.\n\n() **_Selbstgleichschaltung_ and the diplomats**, Hayes, _How_ , 111\u201317.\n\n() **group of leading business executives and Krupp** , Berenbaum and Peck, _Holocaust and History_ , 198\u201399.\n\n() **Degussa** , Hayes, _From Cooperation_ , 38.\n\n() **Weizs\u00e4cker's remarks** , Hayes, _How_ , 113.\n\n() **Roessler's remarks** , Hayes, _From Cooperation_ , 26.\n\n() **\"you reckon that you,\"** ibid., viii.\n\n() **\" _Sind Sie arisch?_ ,\"** Haffner, _Defying_ , 150\u201351.\n\n() **comrade** , ibid., 290\u201391.\n\n() **\"From now on,\"** Noakes and Pridham, _Nazism_ , v. 2, 252.\n\n() **cutting off contact with Jewish friends and neighbors** , Fulbrook, _Dissonant_ , 103\u201313.\n\n() **\"work towards the F\u00fchrer,\"** Kershaw, _Hitler_ , v. 1, 529.\n\n() **warping . . . worked especially powerfully on young people** , Fulbrook, _Dissonant_ , 136\u201339.\n\n() **behavior of Ernst Busemann** , Hayes, _From Cooperation_ , 88\u201390, 93\u201398 (\"it is pointless,\" 90).\n\n() **Potsdam and Kiel** , Bankier, _Germans_ , 70\u201371.\n\n() **Magdeburg** , Kulka and J\u00e4ckel, _Jews_ , 155.\n\n() **many farmers had to be forced** , Stephenson, _Home Front_ , 139\u201340.\n\n() **shame and disgust on the morning after** , Schrafstetter and Steinweis, _Germans_ , 9, 60, 67\u201368.\n\n() **\"The images of the arrest of the Jews,\"** Kulka and J\u00e4ckel, _Jews_ , 529.\n\n() **local offices from all around the country** , ibid., 537\u201342.\n\n() **\"our intellectuals,\"** Browning, _Origins_ , 390.\n\n() **\"Jew-friendly behavior,\"** ibid; Stargardt, _German War_ , 242; Bajohr and Pohl, _Holocaust_ , 56.\n\n() **the first contingents. . . . When the shipments resumed** , Morehouse, _Berlin_ , 168, 171.\n\n() **those descended from or in marriages to non-Jewish Germans** , B\u00fcttner, _Not_ , 11\u201371; Tent, _Shadow_ , 1\u201319; Meyer, _Balancing_ Act, 346; and Gruner, _Widerstand_ , 178\u201389.\n\n() **Transfer Agreement, 20,000, and 1.5 percent** , Bauer, _Brother's_ , 128\u201329; Barkai, _Boycott_ , 51\u201353, 100\u20134; Barkai, \"German Interests,\" 245, 251\u201352, 261\u201366; Yisraeli, \"Third Reich,\" 139, 141\u201342, 147.\n\n() **some people had better chances of being accepted elsewhere than others** , Barkai, _Boycott_ , 55, 153\u201354; Kaplan, _Between_ , 138\u201344; Richarz, _Leben_ , v. 3, 49, 51\u201352; Wasserstein, _Eve_ , 417.\n\n() **They fought back the only way they collectively could** , Bauer, _Brother's_ , 105\u201337, 257\u201358; Barkai, _Boycott_ , 85\u201399; Barkai, _Centralverein_ , 307\u201317; Richarz, _Leben_ , v. 3, 42\u201347; Benz, _Juden_ , Chapter 4.\n\n() t **he effort proved hopeless. . . . The Reichsvereinigung thus degenerated** , Meyer, _Balancing_ , chapters 1\u20132; Richarz, _Leben_ , v. 3, 58\u201364; Benz, _Juden_ , 71\u201374.\n\n() **In Vienna** , Rabinovici, _Eichmann's_ , 2\u20133, 119, 129\u201331.\n\n() **Emblematic of the viciousness** , Meyer, _Balancing_ , 158\u201361; Moorhouse, _Berlin_ , 268\u201371.\n\n() **appeasers actually were inclined to blame Jews** , Cesarani, _Final_ , 216.\n\n() **France signed a new treaty** , Caron, _Uneasy_ , 196\u2013200; McCullough and Wilson, _Violence_ , 54\u201369.\n\n() **Joseph Lyons . . . resolutely refused** , ibid., 144.\n\n() **the Hollywood film distribution companies** , Doherty, _Hollywood_ , 38.\n\n() **the Gestapo used the card files** , Meyer, _Balancing_ , 127.\n\n() **The SS experimented briefly in 1944** , Wachsmann, _KL_ , 453.\n\n() **GM's Opel division** , Turner, _General Motors_ , 42\u201344, 86\u2013103.\n\n() **Ford-Werke in Cologne** , Ford Motor Co., _Findings_ , 35\u201340.\n\nCHAPTER 4: ANNIHILATION\n\n() **how concentrated the time and place** , Browning, _Ordinary Men_ , xv; Hilberg, _Destruction_ , 1321; Stargardt, _Witnesses_ , 9; Dwork, _Children_ , xi.\n\n() **the effect on his men** , Roseman, _Wannsee_ , 63\u201364; Rhodes, _Masters_ , 150\u201354, 167\u201368, 223\u201328.\n\n() **None of the experts balked. . . . to assure the people involved of immunity** , Bryant, _Confronting_ , 37\u201338.\n\n() **propaganda campaign in the 1930s** , Proctor, _Racial Hygiene_ , 181\u201385.\n\n() **he expected potential religious objections to decline** , Bryant, _Confronting_ , 27.\n\n() **the MDs in charge of the program had decided** , ibid __., 43\u201344.\n\n() **Lange soon modified the killing process** , Browning, _Fateful_ , 59.\n\n() **14f13** , Wachsmann, _KL_ , 250\u201358.\n\n() **Dachau. . . . Most . . . were transported** , Morsch and Perz, _Studien_ , 241, 338\u201340.\n\n() **Tests on mental patients** , Browning, _Origins_ , 283, 304.\n\n() **whom he regarded in the typically Catholic fashion** , Griech-Polelle, _Bishop_ , 107\u20138, 113\u201314, 118, 150\u201351.\n\n() **Less than three weeks. . . . did not begin applying** , Berger, _Experten_ , 30, 34\u201336; Bryant, _Eyewitness_ , 3, 54, 78, 151, 159, 161; Arad, _Belzec_ , 17\u201319.\n\n() **were discussing setting up \"gassing devices,\"** Hochstadt, _Sources_ , 116\u201317.\n\n() **work began on the Belzec death camp** , Browning, _Origins_ , 360\u201365; see also Witte et al., _Dienstkalendar_ , 233\u201334.\n\n() **identified the derelict manor house** , Montague, _Chelmno_ , 49\u201353.\n\n() **solved the carbon monoxide supply problem** , Browning, _Fateful Months_ , 57\u201362; C\u00fcppers, _Rauff_ , 109\u201318.\n\n() **1\/3000th of an ounce** , Hayes, _From Cooperation_ , 273.\n\n() **the average cost of murder per head** , ibid., 293, 296\u201397.\n\n() **an instruction . . . that forbade further emigration** , Arad et al., _Documents_ , 153\u201354.\n\n() **Beyond the addition of two officials** , Cesarani, _Final_ , 454\u201355.\n\n() **Heydrich laid out a plan** , Hochstadt, _Sources_ , 132\u201336.\n\n() **Rosenberg . . . had briefed trusted German reporters** , Browning, _Origins_ , 403\u20134; Matth\u00e4us and Bajohr, _Political Diary_ , 385\u201389, quotation at 388.\n\n() **\"this is as close . . . as historians will get,\"** Fritzsche, \"The Holocaust,\" 604.\n\n() **As . . . Raul Hilberg emphasized** , Hilberg, _Destruction_ , v. 1, 49\u201359.\n\n() **Mortality at Chelmno** , Montague, _Chelmno_ , 185\u201388.\n\n() **jerry-rigged edifices** , Berger, _Experten_ , 49, 96; Arad, _Belzec_ , 25; Kuwalek, _Belzec_ , 61\u201362, 66\u201367.\n\n() **Death tolls and survivors of the Operation Reinhard camps** , Berger, _Experten_ , 9, 52, 64, 116, 140\u201341, 177, 252\u201355, 272, 276, 388; Bryant, _Eyewitness_ , 5\u20137, 99, 110, 113, 125; Arad, _Belzec_ , 84, 87, 99, 127\u201330, 258\u201369, 341\u201348; Schelvis, _Sobibor_ , 197\u201398; Kuwalek, _Belzec_ , 14, 170, 225\u201327, 244\u201346.\n\n() **second group of death camps** , Gruner, _Jewish_ , 217\u201329, 255\u201356; Gutman and Berenbaum, _Anatomy_ , 114; Hayes, _Industry_ , 347\u201360; Dlugoborski and Piper, _Auschwitz_ , v. 2, 100\u2013136; Megargee, _Encyclopedia_ , v. IB, 875\u201388; Morsch and Perz, _Studien_ , 219\u201327; Mail\u00e4nder, _Female_ , 172\u201373.\n\n() **deaths at and survivors of Auschwitz** , Hayes, \"Capital,\" 330.\n\n() **Majdanek was far less lethal** , Mail\u00e4nder, _Female_ , 44.\n\n() **Mauthausen** , Wachsmann, _KL_ , 163\u201366, 214; Morsch and Perz, _Studien_ , 126\u201328, 244\u201359; Caplan and Wachsmann, _Camps_ , 131; Jardim, _Mauthausen_ , 54\u201356; Megargee, _Encyclopedia_ , v. IB, 900\u2013907; Horwitz, _Shadow_ , 17\u201318.\n\n() **Durchgangstrasse IV camps and Janowska** , Brandon and Lower, _Shoah_ , 190\u2013223, 324.\n\n() **the camps took in enormous plunder** , Hayes, \"Capital,\" 337; Hochstadt, _Sources_ , 170\u201378; Arad, _Belzec_ , 154\u201364; Berger, _Experten_ , 180; Montague, _Chelmno_ , 88.\n\n() **the example of the Netherlands** , Dean, _Robbing_ , 285.\n\n() **Chelmno consisted** , Montague, _Chelmno_ , 76\u201384.\n\n() **obtained from IG Farben** , Dwork and van Pelt, _Auschwitz_ , 207\u20138.\n\n() **Potemkin villages** , see the keyed maps in Arad, _Belzec_ , 34\u201335, 38\u201339.\n\n() **Quantity and cost of the Zyklon used at Auschwitz** , Hayes, _From Cooperation_ , 295.\n\n() **some 7,000 Germans** , Dlugoborski and Piper, _Auschwitz_ , v. 5, 102.\n\n() **about one-third as many** , Hagen, _German History_ , 343.\n\n() **Germans and _Hiwis_ at Belzec, Sobibor, and Treblinka**, Berger, _Experten_ , 138, 218; Arad, _Belzec_ , 19, 22; Kuwalek, _Belzec_ , 79\u201380, 111.\n\n() **4,750 of these people** , Black, \"Foot Soldiers,\" 7.\n\n() **earned substantially more** , Berger, _Experten_ , 329\u201330.\n\n() **operated in the black** , Bryant, _Confronting_ , 39.\n\n() **Priority, pace, and equipment of deportation trains** , Mierzejewski, _Asset_ , v. 2, 117\u201319; Hilberg, _Sonderz\u00fcge_ , 59, 81\u201382, 86; Gerlach and Aly, _Kapitel_ , 273; Lichtenstein, _Tod_ , 22, 34, 51\u201353, 96, 105, 135.\n\n() **German and Hungarian train statistics** , Mierzejewski, _Asset_ , v. 2, 127, 166; Gall and Pohl, _Eisenbahn_ , 228, 239; Lichtenstein, _Tod_ , 14; P\u00e4tzold and Schwarz, _Bahnhof_ , 104\u20136.\n\n() **the same one or two slow and rickety trains** , Hilberg, _Sonderz\u00fcge_ , 208\u201312; Arad, _Belzec_ , 52, 65\u201366; Mierzejewski, _Asset_ , v. 2, 117; Lichtenstein, _Tod_ , 67.\n\n() **On the doctrine of \"base motive,\"** see Bryant, _Eyewitness_ , 92\u201394.\n\n() **\"cognitive dissonance,\"** see Newman and Erber, _Understanding_ , 52\u201354.\n\n() **\"repulsive duty\" . . . \"horrible task,\"** Breitman, _Architect_ , 196.\n\n() **\"all had an intense need to talk,\"** Lower, _Furies_ , 93.\n\n() **\"political soldiers,\"** Westermann, _Hitler's_ , 15.\n\n() **\"dichotomist ethics,\" \"moral grammar,\" \"too weak,\"** K\u00fchne, _Belonging_ , 59, 87, 167.\n\n() **\"By claiming weakness,\"** Beorn, _Marching_ , 241.\n\n() **\"they were always capable of such violence,\"** R\u00f6mer, _Kameraden_ , 465.\n\n() **\"particular National Socialist morality,\"** Welzer, _T\u00e4ter_ , 31.\n\n() **they became willing . . . identified** , Berger, _Experten_ , 312.\n\n() **Neuser and Glas** , Mierzejewski, _Asset_ , v. 2, 125\u201326.\n\n() **Oskar Gr\u00f6ning** , Rees, _Auschwitz_ , 155\u201358.\n\n() **only 30 percent of the SS men . . . _Volksdeutsche_ made up a large percentage**, Hayes, \"Capital,\" 336. On the predominance of _Volksdeutsche_ and _Hiwis_ in the guard force at Majdanek, see Mail\u00e4nder, _Female_ , 67, 146\u201347.\n\n() **Numbers of German women in the occupied East** , Lower, _Furies_ , 6\u20137, 21.\n\n() **\"In favoring perceived duty over morality,\"** Lower, ibid., 111. On the propensity for violence among the women guards at Majdanek, see Mail\u00e4nder, _Female_ , 71\u201372, 274\u201379.\n\n() **\"I did not want to stand behind the SS men,\"** Lower, _Furies_ , 155.\n\n() **Major Karl Plagge** , Hayes, _How_ , 658\u201374.\n\n() **Anton Schmidt** , Wette, _Feldwebel_ , 234\u201335\n\n() **outcomes of this sort were rare** , ibid., 139\u201342.\n\n() **\"inner identification with evil,\"** Segev, _Soldiers_ , 214.\n\n() **60 percent . . . and another 17 percent** , Wildt, _Generation_ , 23, 458.\n\n() **more than 83 percent of them** , Berger, _Experten_ , 292\u201393.\n\n() **upwardly mobile, well educated, and with long records of involvement** , Wildt, _Generation_ , 38\u201347, 429\u201332. See also Ingrao, _Believe_ , 17\u201331, and Perz, \"Austrian Connection,\" 418\u201319.\n\n() **the T4 personnel . . . constituted a highly indoctrinated group** , Berger, _Experten_ , 302\u20134, 316\u201318.\n\n() **\"our India,\"** Kershaw, _Hitler_ , v. 2, 401\u20132.\n\n() **likened Germany's eastward expansion to America's westward one** , Tooze, _Wages_ , 8\u201311, and Fritzsche \"The Holocaust,\" 601.\n\n() **\"the majority of Nazi genocide,\"** Mann, _Dark Side_ , 276, 278.\n\n() **\"If I looked like him,\"** Breitman, _Architect_ , 4.\n\n() **\"[I]f National Socialism had looked in the mirror,\"** Smelser and Zitelmann, _Braune Elite_ , 100.\n\n() **His favorite adjective was \" _unerh\u00f6rt_ ,\"** ibid., 105.\n\n() **Burckhardt's and Hitler's descriptions of Heydrich** , Fest, _Face_ , 100, 110.\n\n() **\"I feel free of all guilt,\"** Smelser and Zitelmann, _Braune Elite_ , 111.\n\n() t **heir removal was the job of another SS officer** , Berger, _Experten_ , 79\u201381; Arad, _Belzec_ , 44\u201345.\n\n() **he had come, during the 1930s, to believe deeply** , Cesarani, _Becoming_ , Chapter 2; Rabinovici, _Eichmann's_ , 35\u201336.\n\n() **how proud he was of his SS service in retrospect and how thoroughly he rationalized it** , Stangneth, _Before_ , 221\u201330, 242\u201381, 302\u20137.\n\n() **\"a functionary in the true sense. . . . had never really wasted much thought,\"** Fest, _Face_ , 277, 284.\n\n() **\" _\u00dcbervater_ ,\"** Smelser and Zitelmann, _Braune Elite_ , 167.\n\n() **Kaltenbrunner gave vent to his fervent Nazism by attesting** , Mann, _Dark Side_ , 244\u201345.\n\n() **The last two figures** , the best treatments of these men are in Allen, _Business_ , passim; the most thorough studies of the SS economic empire are Kaienburg, _Wirtschaft_ , and Naasner, _SS-Wirtschaft_.\n\n() **Himmler's speech to the assembled SS commanders** , Hochstadt, _Sources_ , 163\u201365.\n\n() **A nation is not only what it does** , Craig, _Germany_ , 638.\n\n() **Awareness . . . was widespread. . . . Germans spoke with open dread** , Bajohr and Pohl, _Holocaust_ , 59\u201372; Longerich, _Davon_ , 223\u201340.\n\n() **diary entry by Curt Pr\u00fcfer** , McKale, _Rewriting_ , 11.\n\n() **Klemperer diary entries** , Klemperer, _I Will_ , v. II, 28, 41, 155, 371.\n\n() **\"the Nazis wanted to manage,\"** Fritzsche, _Life_ , 286.\n\n() **Goebbels announced in the journal _Das Reich_. . . . the _V\u00f6lkischer Beobachter_ . . . reported,** Bajohr and Pohl, _Holocaust_ , 57; Friedl\u00e4nder, _Nazi Germany_ , v. 2, 276, 337\u201338.\n\n() **Hitler reminded Germans** , Longerich, _Davon_ , 201.\n\n() **If such partial revelations had a purpose** , ibid., 325\u201326.\n\n() **Perhaps 10,000. . . . Konrad Latte** , Schneider, \"Saving,\" 52\u201357.\n\n() **Arndt and Krakauer** , Moorhouse, _Berlin_ , 180, 295.\n\n() **plundered Jewish property sent to Hamburg** , Bajohr, _Aryanization_ , 279; Aly, _Beneficiaries_ , 127\u201329. See also Mierzejewski, _Asset_ , v. 2, 127; Fritzsche, _Life_ , 258\u201359.\n\n() **food and goods shipped home by far-flung troops** , Aly, _Beneficiaries_ , 94\u2013152.\n\n() **\"silent protest\" and \"to the hardest manual labor,\"** Gruner, _Rosenstra\u00dfe_ , 139, 200. See also Friedl\u00e4nder, _Nazi Germany_ , v. 2, 425.\n\n() **the reality of resistance generally goaded the Reich** : the classic example is the crackdown precipitated by the Dutch general strike of 1941; see Moore, _Victims_ , 72\u201373; Presser, _Ashes_ , 56\u201357.\n\n() **Forced laborer statistics** , Hayes, _How_ , 315\u201330.\n\n() **they were not necessarily cheap** , Tooze, _Wages_ , 534\u201337; Spoerer, _Zwangsarbeit_ , 183\u201390; Hayes, _From Cooperation_ , 262\u201364, 268\u201371; Wachsmann, _KL_ , 452.\n\n() **Relative fates of male and female slave laborers at Gleiwitz** , Hayes, _From Cooperation_ , 267\u201368.\n\n() **five to ten times more likely to die** , Neander, _Beispiel_ , 59.\n\n() **the mathematics of the German labor force during World War II** , Overy, _War_ , 291\u2013311.\n\n() **compulsory labor program for German Jewish males** , Barkai, _Boycott_ , 159\u201362.\n\n() **extended this program to . . . Poland** , Browning, _Nazi Policy_ , 61\u201365.\n\n() **Autobahn and Organisation Schmelt** , Gruner, _Jewish_ , 214\u201329; Gutterman, _Narrow_ , 43\u201355.\n\n() **first use of slave labor by German private industry** , Pohl, _Holzmann_ , 264\u201365.\n\n() **Volkswagen and IG Farben** , Mommsen, _Volkswagenwerk_ , 433\u201341, 496\u2013515, and Hayes, _Industry and Ideology_ , 347\u201353.\n\n() **Project Giant** , Pohl, _Holzmann_ , 266\u201367; Gutterman, _Narrow_ , ch. 8.\n\n() **Alderney** , Deak, _Europe_ , 59.\n\n() **Half the inmates of Auschwitz never even got labor assignments** , Hayes, \"Capital,\" 337.\n\n() **SS companies were neither profitable, nor usually successful in their joint ventures . . . though one initiative . . . made money** , Wachsmann, _KL_ , 405\u20136; Tooze, _Wages_ , 630.\n\n() **Changing female selection and mortality rates at Auschwitz** , Dlugoborski and Piper, _Auschwitz_ , v. II, 180\u201382; Gutman and Berenbaum, _Anatomy_ , 466; Wachsmann, _KL_ , 353, 455, 477\u201378.\n\n() **On Starachowice** , see Browning, _Remembering_ , passim.\n\n() **On Skarzysko-Kamienna** , see Ofer and Weitzman, _Women_ , 285\u2013309.\n\n() **one-third of the German infantry's ammunition** , Karay, _Death_ , 70.\n\n() **On the design, conception, dimensions, and working conditions at Dora** , Sellier, _History_ , 31\u201332, 511\u201315; Neander, _Mittelbau_ , 179\u201384, 189\u201395; Allen, _Business_ , 222\u201332; Neufeld, _Rocket_ , 208\u201313, 224\u201328; Wachsmann, _KL_ , 444\u201347; and Neander, _Beispiel_ , passim.\n\n() **Rocket production and deaths at Dora (26,500) and deaths from rockets (15,386)** , Wachsmann, _KL_ , 453\u201354, and Seiler, _History_ , 398, 403\u20134. On the additional murderous effects of V-2 production at Mauthausen, see Horwitz, _Shadow_ , 20-21.\n\n() **Fighter Staff Program** , Allen, _Business_ , 232\u201339; Tooze, _Wages_ , 627\u201334; Wachsmann, _KL_ , 448\u201351.\n\n() **mortality rates fluctuated** , Buggeln, _Slave_ , 27\u201332.\n\n() **to salvage their machinery** , Gregor, _Daimler-Benz_ , 194\u201396, 221\u201352.\n\n() **about 15 percent of the construction work** , Hayes, \"Capital,\" 347.\n\n() **principal profiteer from the slave labor program** , Neander, _Mittelbau_ , 55;\n\n() **estimated at 600 to 700 million reichsmark** , Wachsmann, _KL_ , 410.\n\n() **Marches from Auschwitz and Gross-Rosen** , Blatman, _Marches_ , 81\u2013105; Rees, _Auschwitz_ , 264; Wachsmann, _KL_ , 554\u201357.\n\n() **Marches from Stutthof and killing at Palmnicken** , Blatman, _Marches_ , 111\u201325.\n\n() **the camps that received retreating prisoner groups** , Morsch and Perz, _Neue Studien_ , 25; Stangneth, _Eichmann_ , 53; Bessel, _1945_ , 50; Blatman, _Marches_ , 127\u201332; Buggeln, _Slave_ , 60\u201361.\n\n() **Bergen-Belsen** , Blatman, _Marches_ , 132\u201336; Rees, _Auschwitz_ , 265\u201367; Stone, _Liberation_ , 83.\n\n() **These decisions were Heinrich Himmler's** , Blatman, _Marches_ , 53\u201354, 137; Wachsmann, _KL_ , 572\u201376.\n\n() **\"no inmate may fall into the enemy's hands alive,\"** Blatman, _Marches_ , 154, 181.\n\n() **Not all of them . . . but most did** , Blatman, _Marches_ , 155\u201379; Wachsmann, _KL_ , 580.\n\n() **Deaths at Buchenwald** , Blatman, _Marches_ , 152.\n\n() **When the British bombed the city** , Bessel, _1945_ , 52.\n\n() **Dachau, Mauthausen, and their subcamps** , Blatman, _Marches_ , 197\u2013217; Jardim, _Mauthausen_ , 59\u201360.\n\n() **\"many of the living people look dead,\"** Blatman, _Marches_ , 242.\n\nCHAPTER 5: VICTIMS\n\n() **in Cracow** , Henry, _Resistance_ , 51.\n\n() **inmate killed a German** , Arad, _Belzec_ , 98\u201399.\n\n() **twentieth transport** , Henry, _Resistance_ , 129\u201330, and Gilbert, _Holocaust_ , 574\u201375.\n\n() **Number of underground movements in Polish ghettos and camps** , Gutman and Krakowski, _Unequal_ , 106.\n\n() **Arendt quotations** , _Eichmann_ , 117.\n\n() **_amidah_** , Bauer, _Rethinking_ , 120.\n\n() **Hilberg is probably right** , Hilberg, _Destruction_ , 1106.\n\n() **Numbers of Jews in resistance units** , Henry, _Resistance_ , xix, xxv, xxvii, xxxiii, 142\u201357, 168\u201375, 201\u201319, 432\u201337; Bauer, _Rethinking_ , 137\u201339.\n\n() **\"the concentration of the Jews,\"** Hochstadt, _Sources_ , 87\u201389.\n\n() **modeled on the body . . . in Vienna** , Rabinovici, _Eichmann's_ , 40.\n\n() **In Lodz, for instance, twenty-two of the first thirty council members were killed** , Dobroszycki, _Chronicle_ , xlvi; Trunk, _Judenrat_ , 23; Trunk, _Lodz_ , xxxiii, 34.\n\n() **Lodz ghetto area** , ibid., 16\n\n() **Lodz ghetto population** , Dobroszycki, _Chronicle_ , xxxix, but Horwitz, _Ghettostadt_ , 335, gives 163,777, and Trunk, _Lodz_ , xxx, says \"about 164,000.\"\n\n() **Warsaw ghetto population and area** , Engelking and Leociak, _Warsaw_ , 49; Gutman, _Jews_ , 63\n\n() **permeability . . . remained much greater. . . . at many of the smaller sites** , Perechodnik, _Am I?_ , 68.\n\n() **villages of the largely rural Lublin district** , Silberklang, _Gates_ , 29, 212\u201314.\n\n() **attritionists and productionists** , Browning, _Path_ , 28\u201356.\n\n() **figure 5** , Dobroszycki, _Chronicle_ , xxxix, lxvi, 50, 52, 107, 193, 314, 352, 444, 519; Trunk, _Lodz_ , xlvi\u2013xlvii.\n\n() **pitted against each other** , Redner, _Policeman_ , 86, 106.\n\n() **his own father seized and ate** , Adelson, _Diary_ , 176\u201377.\n\n() **Internal disunity among Jews** , Trunk, _Judenrat_ , 29\u201335, 368\u201387; Corni, _Ghettos_ , 172\u201389; Wasserstein, _Ambiguity_ , 154.\n\n() **\"the community's social structure disintegrated,\"** Lensky, _Physician_ , 163.\n\n() **Why would the Germans kill people who could be useful** , Redner, _Policeman_ , 127\u201328.\n\n() **refusal of ghetto residents in both Lodz and Bialystok** , Hayes, _Lessons I_ , 11.\n\n() **the extent of Jews' denial** , Perechodnik, _Am I?_ , 12, with slight corrections to the translation from Polish by my colleague Jacek Nowakowski. See also Redner, _Policeman_ , 175.\n\n() **Nazi camouflage measures** , for example, Horwitz, _Ghettostadt_ , 283\u201384; Wasserstein, _Ambiguity_ , 141\u201342; Redner, _Policeman_ , 166.\n\n() **mixture of bait and threats** , Dawidowicz, _War_ , 301; Dobroszycki, _Chronicle_ , 125, 164\u201365; Horwitz, _Ghettostadt_ , 277\u201379; Corni, _Ghettos_ , 69; Wasserstein, _Ambiguity_ , 141.\n\n() **procedure re deportations from the Netherlands** , Moore, _Victims_ , 91\u201397, 109; Wasserstein, _Ambiguity_ , 138\u201339, 193, 195.\n\n() **Rosenblatt quotation** , Bauer, _Rethinking_ , 80\u201381.\n\n() **the conduct of Asscher and Cohen** , Wasserstein, _Ambiguity_ , 174\u201376.\n\n() **According to David Daube's** , ibid., 251.\n\n() **Rumkowski's \"give me your children!\" speech** , Trunk, _Lodz_ , 272\u201375.\n\n() **Administration and police numbers in Lodz** , ibid., 38, 40, 44; **in Warsaw** , Gutman, _Encyclopedia_ , 1609; Engelking and Leociak, _Ghetto_ , 409. See also Corni, _Ghettos_ , 74.\n\n() **On Szerynski and the Warsaw ghetto police** , Trunk, _Judenrat_ , 475\u201394, 498\u2013501, 552\u201353; Gutman, _Jews_ , 88\u201390, 237\u201340; Corni, _Ghettos_ , 107\u201311; Perechodnik, _Am I?_ , 104.\n\n() **increasingly corrupt and extortionist . . . these police did the footwork** , Redner, _Policeman_ , 130\u201335, 155\u201360.\n\n() **Composition of the Dutch Jewish police** , Moore, _Victims_ , 220\u201321; Wasserstein, _Ambiguity_ , 190\u201391; Cesarani, _Final_ , 679, 681\u201382.\n\n() **Daily food intake . . . hovered** , ibid., 274\u201375.\n\n() **in Warsaw in 1941** , the lower estimate is in Gutman, _Encyclopedia_ , 1609, the higher in Bauer, _History_ , 170.\n\n() **at Otwock** , Perechodnik, _Am I?_ , 232\u201333.\n\n() **5,550 people were dying in the Warsaw ghetto per month** , Gutman, _Jews_ , 64.\n\n() **German and Jewish casualties in the Warsaw Ghetto Uprising** , Engelking and Leociak, _Ghetto_ , 51; Henry, _Resistance_ , 31 (Hilberg, _Destruction_ , 1105, gives slightly lower German figures, so does Friedl\u00e4nder, _Nazi Germany_ , v. 2, 526).\n\n() **resistance against the German drive to empty the Bialystok ghetto resulted in** , Bender, _Bialystok_ , 258\u201365.\n\n() **Outcomes of Treblinka and Sobibor uprisings** , Arad, _Belzec_ , 363\u201364; Schelvis, _Sobibor_ , 168, 175, 231\u201342.\n\n() **Shootings at Majdanek and Poniatowa** , Silberklang, _Gates_ , 402\u20137.\n\n() **On the extent of smuggling in and out of the Warsaw ghetto** , see Hilberg et al., _Czerniakow_ , 306; Cesarani, _Final_ , 435\u201336.\n\n() **On Ringelblum and Oyneg Shabes** , see Kassow, _Who?_\n\n() **On Lodz** , Dobroszycki, _Chronicle_.\n\n() **several Jewish ghetto administrations adopted different survival strategies . . . but . . . they ultimately came to the same end** , Bauer, _History_ , 157\u201367; Polonsky, _Jews_ , v. III, 479\u2013500.\n\n() **\"No Jewish action caused any significant difference,\"** Silberklang, _Gates_ , 440.\n\n() **\"living in the expectation of death,\"** Arendt, _Eichmann_ , 119.\n\n() **\"running a race against time,\"** Vagi et al., _Hungary_ , 256.\n\n() **2 percent of the French population** , Paxton, _Vichy_ , 294\u201395.\n\n() **Dutch civil servants and police** , Wasserstein, _Ambiguity_ , 143; Romijn et al., _Persecution_ , 13\u201326.\n\n() **Population figures for the Warsaw ghetto** , Engelking and Leociak, _Warsaw Ghetto_ , 50\u201351; Gutman, _Jews of Warsaw_ , 270\u201371.\n\n() **Her last name was Neyer, and she is walking beside** , according to Yisrael Gutman in Laqueur, _Encyclopedia_ , 693.\n\n() **Courts of Honor in Italy, Germany, and the Netherlands** , Trunk, _Judenrat_ , 553\u201355; Jockusch and Finder, _Honor_ , 107\u201336; Wasserstein, _Ambiguity_ , 253\u201354.\n\n() **Kastner case** , Segev, _Seventh_ , Part V; the quotations appear on pp. 283 and 318, respectively.\n\n() **who fell into the hands of the Soviet Union** , Anonymous, _Clandestine_ , xv; Meyer, _Balancing_ , 359.\n\n() **Ben-Gurion's letters** , Segev, _Seventh_ , 294.\n\n() **the last prosecution** , Jockusch and Finder, _Honor_ , 320\u201321.\n\n() **the number of camps** , van Pelt, \"Nazi,\" 150.\n\n() **Such installations and their satellites . . . about one million died** , Wachsmann, _KL_ , 627.\n\n() **at most, about 150,000 Jewish veterans** , estimated from Wachsmann, _KL_ , 771.\n\n() **survival rates were infinitesimal** , Montagu, _Chelmno_ , 126\u201341, 195; Arad, _Belzec_ , 258\u201369; Bryant, _Eyewitness_ , 35, 42\u201343; Kuwalek, _Belzec_ , 14, 170, 225\u201327.\n\n() **a hierarchy of prisoner categories developed . . . constant struggle to control the most important trustee assignments** , Orth, _System_ , 57\u201361, Wachsmann, _KL_ , 122\u201335.\n\n() **Hermann Langbein . . . has left a vivid account** , Langbein, _People_ , 12\u201314.\n\n() **Sinti and Roma** , Hayes and Roth, _Handbook_ , 275\u201381; Lewy, _Persecution_ , 221\u201326; Hayes, _How_ , 495\u2013505; Weiss-Wendt, _Genocide_ , 2, 16\u201317; Weiss-Wendt, _Murder_ , 144\u201348; Bryant, _Eyewitness_ , 41; Deletant, _Forgotten_ , 187\u201396.\n\n() **The treatment of gays** , Hayes and Roth, _Handbook_ , 281\u201383; Jellonek, _Homosexuelle_ , 19\u201336, 327\u201332; Longerich, _Himmler_ , 231\u201340; Gellately and Stoltzfus, _Outsiders_ , 233\u201355; Berenbaum and Peck, _Holocaust_ , 338\u201357; Wachsmann, _KL_ , 127\u201328, 665.\n\n() **not all Slavs were the same in German eyes** , Hayes and Roth, _Handbook_ , 283\u201387.\n\n() **how people arrived at them** , Gigliotti, _Train_ , especially chapters 4\u20135.\n\n() **\"the most complete totalitarian structure to have been devised by man,\"** Marrus, _History_ , 147.\n\n() **\"a mixture of Hell and an insane asylum,\"** Langbein, _People_ , 477.\n\n() **the inhabitants of the \"family camp\" for Czech Jews** , Henry, _Resistance_ , 584.\n\n() **when 419 . . . Soviet POWs succeeded in breaking out of Mauthausen** , Blatman, _Marches_ , 400\u2013401; Horwitz, _Shadow_ , 124\u201343.\n\n() **The only successful form of resistance in the camps was escape, although the odds were long** , Bryant, _Eyewitness_ , 42\u201343; Arad, _Belzec_ , 258\u201369; Schelvis, _Sobibor_ , 135\u201342; Hayes, \"Capital,\" 340; Wachsmann, _KL_ , 534\u201336.\n\n() **Hanna L\u00e9vy-Hass . . . wrote** , Confino, _World_ , 203.\n\n() **\"Here there is no why,\"** Levi, _Survival_ , 25.\n\n() **\"the Resistance in the camp is not geared for an uprising,\"** Henry, _Resistance_ , 587.\n\n() **Auschwitz consumed 75,000 Poles . . . but it took the lives of probably four-fifths of the Jews ever registered** , Hayes, \"Capital,\" 330, 332.\n\n() **\"There are no roads from Auschwitz but those of improbability,\"** Rosenberg, _Brief_ , 106.\n\n() **\"excremental assault,\"** Des Pres, _Survivor_ , 51\u201371.\n\n() **\"pairing,\"** Henry, _Resistance_ , 566\u201367.\n\n() **As Imre Kertesz . . . writes** , ibid., 580.\n\nCHAPTER 6: HOMELANDS\n\n() **Minority religious status** , Bauer, _History_ , 286; Bauer, _Death_ , 93\u201395, 106\u20137, 111; Engelmann, _Hitler's_ , 71\u201378; Henry, _We Only_ , 9\u201340.\n\n() **Minority status was not always necessary** , Petrow, _Bitter_ , 116; Todorov, _Fragility_ , 9, 25, 97\u2013101; Bar-Zohar, _Beyond_ , 167\u201377; Rhodes, _Vatican_ , 319; Marrus and Paxton, _Vichy_ , 271\u201373; Dwork and van Pelt, _Holocaust_ , 332\u201333.\n\n() **certain character traits . . . were better predictors** , Tec, _Light_ , 152\u201354, 188\u201391; Oliners, _Altruistic_ , Chapter 6.\n\n() **Otto Jodmin . . . \"I simply had to do it,\"** Moorhouse, _Berlin_ , 297.\n\n() **Teresa Prekerowa . . . \"ordinary people who differed greatly,\"** Libionka, \"Polish Literature,\" 61\u201362.\n\n() **Aristides de Sousa Mendes** , Gutman, _Encyclopedia_ , 1381\u201382.\n\n() **the Dutch and Japanese consuls . . . in Kovno** , Hayes, _How_ , 648\u201357.\n\n() **Varian Fry** , Wyman, _Paper_ , 142.\n\n() **Ernest Prodolliet** , Independent Commission, _Switzerland_ , 109; Bauer, _Jewry_ , 276; Wasserstein, _Ambiguity_ , 165.\n\n() **Berthold Beitz** , K\u00e4ppner, _Beitz_ , 47\u2013113.\n\n() **Alfred Rossner** , Fulbrook, _Small Town_ , 156\u201358.\n\n() **Otto Weidt** , Moorhouse, _Berlin_ , 296.\n\n() **puppet regimes that carried out German orders** , Pavlowitch, _Disorder_ , 58\u201359; Mazower, _Inside_ , 18\u201322; M\u00fcller, _Seite_ , 159, 168\u201369, 174.\n\n() **Recherchegruppe (or Colonne) Henneicke** , Dean, _Robbing_ , 283; Moore, _Victims_ , 207\u201310; Presser, _Ashes_ , 354, 366, 392\u201393.\n\n() **The majority of the Jews ever deported from both France and Belgium** , Benz, _Dimension_ , 124, 127\u201328, 132\u201333, 135.\n\n() **Even Slovakia . . . had second thoughts** , Ward, _Priest_ , 224\u201335.\n\n() **Germany had allowed various national liberation groups to set up offices in Berlin** , Friedl\u00e4nder, _Nazi Germany_ , v. 2, 220; Polonsky, _Jews_ , v. III, 409\u201311.\n\n() **Jews were overrepresented compared to their share of the Lithuanian population** , Kosmala and Verbeeck, _Facing_ , 79; Barkan et al., _Shared_ , 380\u201381.\n\n() **the Soviet Union brought life in prison, but Nazi Germany brought the death sentence** , Bauer, _Death_ , 37\u201338.\n\n() **to Ukrainian and Baltic nationalists . . . Germany appeared the lesser evil** , Snyder, _Bloodlands_ , 190\u201394, 397; Gitelman, _Bitter_ , 67.\n\n() **calling on Ukrainians to \"destroy\" Jewry, and a pogrom took place** , Petrovsky-Shtern and Polonsky, _Polin_ 26, 339; Lower, _Empire-Building_ , 94\u201395; Redner, _Policeman_ , 34\u201337; Bartov and Weitz, _Shatterzone_ , 371\u201373.\n\n() **Lithuanian Activist Front declared that Jews had \"betrayed,\"** Dieckmann, _Litauen_ , 252\u201353; see also K\u00fchne, _Belonging_ , 81; Polonsky, _Jews_ , v. III, 406.\n\n() **Ukrainian police and militias played an active part** , Struve, _Herrschaft_ , passim.\n\n() **Himmler had about 300,000** , Cesarani, _Final_ , 382.\n\n() **\"For the Germans 300 Jews are . . .,\"** ibid., 394.\n\n() **Of the roughly 350,000 Jews in France in 1940, more than half** , Marrus and Paxton, _Vichy_ , 364.\n\n() **Statistics on deportations from France** , Benz, _Dimension_ , 127, 133\u201334; Paxton, \"Jews: Vichy,\" 40\u201343.\n\n() **the French government dragged its feet** , Marrus and Paxton, _Vichy_ , 372.\n\n() **Hungarian deportations and death toll** , Braham, _Politics_ , 153, 251; Wachsmann, _KL_ , 460.\n\n() **the thoroughness of this operation . . . was largely homegrown** , Braham, _Studies_ , 71\u201378, 86.\n\n() **the apt summation of Peter Kenez** , Kenez, _Coming_ , 250.\n\n() **H\u00f6ss . . . repeatedly sought to slow the overwhelming pace** , Wachsmann, _KL_ , 459.\n\n() **the deportation had a war-related purpose** , Buggeln, _Slave_ , 46\u201349.\n\n() **history of antisemitism in Hungary** , Braham, _Politics_ , 20\u201325; Vagi et al., _Holocaust_ , xxxviii\u2013xliv.\n\n() **the new territories nearly doubled the Hungarian Jewish population** , Vagi et al., _Holocaust_ , 368\u201369.\n\n() **\"Jewish workers for German war production purposes,\"** Braham, _Politics_ , 59.\n\n() **Antonescu's policies and motives** , Hayes, _How_ , 445\u201365, excerpting the fundamental study by Ancel; Ioanid, _Romania_ , especially 271\u201381; Deletant, _Forgotten_ , 209\u201314.\n\n() **a strong moral stand . . . proved counterproductive when the timing was not right** , Moore, _Victims_ , 73, 79\u201390; Presser, _Ashes_ , 56\u201357; Friedl\u00e4nder, _Nazi Germany_ , v. 2, 410\u201311.\n\n() **Many other pieces of good fortune were involved** , Lidegaard, _Countrymen_ , 31\u201335, 44\u201351, 65\u201373, 96\u201397, 154, 289, 329\u2013332, 339\u201340; Friedl\u00e4nder, _Nazi Germany_ , v. 2, 545\u201347.\n\n() **In Italy, Mussolini had just announced** , Knox, \"faschistische Italien,\" 56, 61, 65, 79; Schlemmer and Woller, \"italienische Faschismus,\" 182\u201387.\n\n() **around Trieste . . . 90 percent of the Jewish community perished** , Zimmerman, _Italy_ , 247\u201351.\n\n() **Giovanni Palatucci** , Bess, _Choices_ , 81.\n\n() **In Italy . . . more than one-third** , Sarfatti, _Jews_ , 27\u201328; Zuccotti, _Italians_ , 20.\n\n() **Gross . . . overstated . . . but established** , Gross, _Neighbors_ , 73\u201389; Bikont, _Crime_ , 521\u201324; David-Fox, _Holocaust_ , 19\u201320; **a curious feature of survivors' testimonies** , Browning, _Remembering_ , 50.\n\n() **50,000\u201360,000 by December 1939 . . . liquidated much of the Polish intelligentsia** , Matth\u00e4us et al., _War_ , 3; Rossino, _Hitler_ , 234; Snyder, _Bloodlands_ , 126\u201327, 153\u201354; Gross, _Neighbors_ , 7.\n\n() **how complete the purge at some local levels was** , Libionka, \"Church Hierarchy,\" 86; see also Phayer, _Pius XII_ , 23\u201324; and Huener, \"Kirchenpolitik,\" 113\u201316, 128\u201329.\n\n() **official rations provided Poles** , Winstone, _Dark_ , 115.\n\n() **\"bread prices . . . hovered,\"** ibid., 118\u201319.\n\n() **\"economically speaking, an empty body,\"** ibid., 73.\n\n() **\"[W]e have decided. . . . [I]t was like living in a country,\"** ibid., 50, 53.\n\n() **More Poles died in the bombing of Warsaw . . . more Poles may have been killed in the suppression** , Snyder, _Bloodlands_ , 405\u20136.\n\n() **about 720,000 people** , Paulsson, _Secret_ , 1.\n\n() **Jakub Berman . . . simply cooked up the number** , Snyder, _Bloodlands_ , 356\u201357, 407; Gross, _Fear_ , 4.\n\n() **The Jewish survival rate in Warsaw was equal to that in Amsterdam** , Paulsson, _Secret_ , 2, 5, 229\u201331.\n\n() **newspaper column by FBI Director** , James Comey, \"Why I Require FBI Agents to Visit the Holocaust Museum,\" _Washington Post_ , April 16, 2015.\n\n() **a number of prominent Jewish scholars** , e.g., the reviews by Dan Diner in _Contemporary European History_ 21 (2012), 125\u201331, and Omer Bartov in _Slavic Review_ 71 (2012), 424\u201328.\n\n() **antisemitism in Poland was considerable before 1939 and on the rise** , Bauer, _Brother's_ , 194; Polonsky, _Jews_ , v. III, 80\u201381, 85\u201388; Blobaum, _Antisemitism_ , 158\u201370; Mendelsohn, _Jews_ , 71\u201376; Zimmerman, _Underground_ , 16\u201320; Watt, _Bitter_ , 361-66.\n\n() **\"forcible emigration of the Jews\" . . . sent a delegation to Madagascar** , Wasserstein, _Eve_ , 40, 359; Hamerow, _Why_ , 62.\n\n() **The Polish foreign minister even discussed . . . and tried to lease . . . \"supplemental Jewish homeland,\"** Bauer, _Brother's_ , 193; Zimmerman, _Contested_ , 22\u201323; Hamerow, _Why_ , 62, 87.\n\n() **trained right-wing Zionist fighters in Poland** , Snyder, _Black_ , 64\u201366, 281.\n\n() **Church leaders and publications. . . . \"It is a fact that,\"** Polonsky, _Jews_ , v. III, 81\u201384; Libionka, \"Church Hierarchy,\" 77\u201386.\n\n() **\"Hitler called the Jews the microbe of the world,\"** Blobaum, _Antisemitism_ , 261.\n\n() **\"regrettable excesses\" . . . disrespect \"for the faith and traditions of Christians,\"** Libionka, \"Church Hierarchy,\" 81.\n\n() **separate ethnic communities . . . survey conducted before the war** , Wasserstein, _Eve_ , 224, 330.\n\n() **Jews and Poles were divided by residence and occupations** , Mendelsohn, _Jews_ , 23\u201332, 42\u201343; Bauer, _Brother's_ , 180\u201389; Watt, _Bitter_ , 365.\n\n() **11,500, 70,000\u201390,000, 25,000, 3,500** , Paulsson, _Secret_ , 229\u201331, 236.\n\n() **the Polish army interned** , Hamerow, _Why_ , 44.\n\n() **Jews composed more than half** , Gross, _Fear_ , 195\u201397.\n\n() **Jews there did recognize** , Bauer, _Death_ , 35\u201341; Zimmerman, _Contested_ , 61\u201368.\n\n() **\"This is nothing to be surprised at,\"** Perechodnik, _Am I?_ , 2, with slight corrections to the translation from the Polish by my colleague Jacek Nowakowski.\n\n() **the massacre there was hardly an isolated occurrence** , Polonsky, _Jews_ , v. III, 421, 425; Bauer, _Death_ , 92\u2013120; Zimmerman, _Underground_ , 95\u201398; Barkan et al., _Shared_ , 306, 316.\n\n() **Stefan Rowecki . . . reported** , Kosmala and Verbeeck, _Facing_ , 66.\n\n() **\"is creating something of a narrow bridge,\"** Polonsky, _Jews_ , v. III, 408; Zimmerman, _Underground_ , 74\u201375.\n\n() **Jewish inmates who escaped Sobibor** , Schelvis, _Sobibor_ , 181\u201382.\n\n() **Barwy Biale detachment . . . slaughtered** , Mazurek and Skibinska, \"Barwy Biale,\" 433\u201380; Zimmerman, _Underground_ , 290.\n\n() **\"The farmers are seizing the Jews,\"** Polonsky, _Jews_ , v. III, 450.\n\n() **the so-called Blue Police. . . . Germans offered rewards . . . and threatened** , Grabowski, _Hunt_ , 101\u201320.\n\n() **what happened in Dabrowa Tarnowska County** , ibid., 61.\n\n() **\"[W]e have to punish those who want to hide Jews,\"** ibid., 58.\n\n() **Most people who hid Jews there did so in return for money** , ibid., 135\u201348.\n\n() **The AK did pass its knowledge . . . warned Poles against . . . blackmailing . . . and . . . carried out executions** , Fleming, _Auschwitz_ , 27; Zimmerman, _Underground_ , 84, 129\u201331, 134\u201339, 141\u201350, 154\u201360, 162, 224, 227, 264, 300\u2013302; Polonsky, _Jews_ , v. III, 461.\n\n() **made no effort to impede the transports . . . provided only modest support for the Warsaw Ghetto Uprising** , Zimmerman, _Underground_ , 54, 161, 167\u201368, 179, 197\u2013209, 214\u201317, 241; Fleming, _Auschwitz_ , 254\u201355; Polonsky, _Jews_ , v. III, 463, 511\u201312; Friedl\u00e4nder, _Nazi Germany_ , v. 2, 523.\n\n() **Komorowski . . . banditry** , Zimmerman, _Underground_ , 251\u201356, 262, 267\u201386, 297\u201398, 417\u201318.\n\n() **Zegota . . . \"Our feelings toward the Jews have not changed.\" . . . most of its funds** , Zimmerman, _Underground_ , 175\u201378, 184, 303\u201312; Bauer, _American_ , 332\u201333.\n\n() **favored liquidation or emigration . . . by a ratio of nine to four** , Polonsky, _Jews_ , v. III, 445.\n\n() **Even among political prisoners in the concentration camps** , Langbein, _Against_ , 146, and _People_ , 75\u201376.\n\n() **currency and jewelry dealers set up shop around Treblinka** , Gross and Gross, _Golden_ , 28\u201338.\n\n() **the first thing her protectors asked** , Tec, _Tears_ , 214.\n\n() **omit them in the future** , David-Fox, _Holocaust_ , 13.\n\n() **emigrated to Chicago** , Gross, _Neighbors_ , 131; Polonsky, _Jews_ , v. III, 424.\n\n() **Hirszman. . . . Kielce pogrom. . . . tried to cover their tracks** , Bryant, _Eyewitness_ , 35; Gross, _Neighbors_ , 152\u201367; Grabowski, _Hunt_ , 86. Cf. Kuwalek, _Belzec_ , 315\u201317.\n\n() **Jewish \"overrepresentation,\"** Gross, _Fear_ , 220\u201322, 226\u201331\n\n() **to discredit a competing candidate . . . \"a Jew who tries to make money,\"** Gross, _Fear_ , 30; Judt, _Postwar_ , 827.\n\n() **Ringelblum . . . was hidden . . . by a non-Jew** , Gutman and Krakowki, _Unequal_ , iii; Kassow, _Who?_ , 362\u201365, 383\u201385.\n\n() **almost 1,000 cases of Poles executed for helping** , Wette, _Feldwebel_ , 154; Grabowski, _Hunt_ , 56, gives \"slightly more than seven hundred\" as the figure arrived at by Polish researchers.\n\n() **fifteen times more likely** , Snyder, _Bloodlands_ , 406.\n\nCHAPTER 7: ONLOOKERS\n\n() **France enacted various rules that made immigration less appealing** , Caron, _Uneasy_ , 28\u201333.\n\n() **After 1936, four other arguments . . . narrowed** , Hamerow, _Why_ , 72\u201389; Weber, _Hollow_ , 87\u2013110; Caron, _Uneasy_ , 187\u2013205.\n\n() **\"taken as a whole not very desirable,\"** Wasserstein, _Eve_ , 218.\n\n() **even sentenced the aunt and uncle** , McCullough and Wilson, _Violence_ , 59.\n\n() **The trend . . . in the Netherlands, Belgium, and Czechoslovakia** , Bauer, _Brother's_ , 170\u201372, 177, 243, 267; Hamerow, _Why_ , 61.\n\n() **Switzerland provided the most glaring illustration** , Independent Commission, _Switzerland_ , 105\u20139, 128\u201330; Caestecker and Moore, _Refugees_ , 82\u2013102; Bauer, _Brother's_ , 172\u201376, 239\u201340, 267\u201368.\n\n() **several discouraging and . . . mutually contradictory preconditions** , David-Fox, _Holocaust_ , 37.\n\n() **Britain saw its role . . . as that of a \"transit nation,\"** London, _Whitehall_ , chapters 3\u20135; Hamerow, _Why_ , 90\u2013119, 156\u201361; McCullough and Wilson, _Violence_ , 108\u201350; Abella and Troper, _None_ , xx, 6\u20139, 48\u201349.\n\n() **Britain pursued similarly restrictive policies in Palestine** , Dwork and van Pelt, _Flight_ , 28\u201351; Bauer, _History_ , 127\u201328; Wasserstein, _Eve_ , 339, 363, 413.\n\n() **haunted by the specter of what might happen** , Hamerow, _Why_ , 104, 112, 114\u201316; Caestecker and Moore, _Refugees_ , 64; London, _Whitehall_ , 95.\n\n() **Poland's ambassador in London tried to blackmail Britain** , Wistrich, _Hitler_ , 21; Hamerow, _Why_ , 63; London, _Whitehall_ , 91.\n\n() **an average of only 22 percent** , Bauer, _Brother's_ , 163.\n\n() **the almost 310,000 . . . Jews who actually applied for entrance by 1939** , Breitman and Kraut, _American_ , 74.\n\n() **probably about 225,000** , Friedl\u00e4nder, _Nazi Germany_ , v. 2, 783, says 211,000 as of the end of 1943; London, _Whitehall_ , 12, says \"no more than 250,000 . . . in the years 1933\u201345.\" Wyman, _Paper_ , 218\u201319, calculates that just over 250,000 \"refugees from Nazism\" got into the U.S. by the autumn of 1944, but not all of these people were Jews.\n\n() **\"FDR's second-term policies likely helped save the lives of well over 100,000 Jews,\"** Breitman, _FDR_ , 317.\n\n() **more than five-sevenths of its total refugees at the last minute** , London, _Whitehall_ , 11\u201312, 103, 115\u201318, 131\u201334, 141; Bauer, _Brother's_ , 270\u201371.\n\n() **fear of economic competition** , Wyman, _Paper_ , 3\u20139; Breitman and Kraut, _American_ , 11\u201317, 21\u201322, 33\u201337, 49\u201350; see Hamerow, _Why_ , 252\u201353.\n\n() **The American public opposed letting more people in** , Breitman and Kraut, _American_ , 58; Breitman and Lichtman, _FDR_ , 116.\n\n() **Father Charles Coughlin. . . . Jew Deal . . . polls of 1938 and 1939** , Wyman, _Paper_ , 17\u201319, 22; Breitman and Lichtman, _FDR_ , 75\u201377; Hamerow, _Why_ , 251 (on the Jew Deal).\n\n() **\"is a perfect opening to Germany to load the United States with agents,\"** Hamerow, _Why_ , 281; Breitman and Kraut, _American_ , 112\u201345.\n\n() **State Department instructed its consuls worldwide** , Wasserstein, _Ambiguity_ , 110.\n\n() **FDR's caution** , Breitman and Kraut, _American_ , 222\u201335.\n\n() **a scheme for smuggling agents into the Americas under the cover of releasing Jews** , Wasserstein, _Ambiguity_ , 118.\n\n() **The two groups also differed in their attitudes toward the creation of a Jewish state. . . . Joint Distribution Committee . . . preferred** , Bauer, _Brother's_ , 157\u201366.\n\n() **Shanghai** , Hochstadt, _Exodus_ , especially chapters 3\u20134; Caestecker and Moore, _Refugees_ , 109\u201321.\n\n() **\"divided into places where they cannot live and places they cannot enter\" . . . \"Dutch Guiana, Angola, Cyprus,\"** Wasserstein, _Eve_ , 360, 403.\n\n() **Statistics on the _St. Louis_** , Vincent, \"Voyage,\" 255, 270\u201371, 274, 288; Breitman and Lichtman, _FDR_ , 138.\n\n() **Emigrants from Poland in 1937 and U.S. quota** , Bauer, _Brother's_ , 194, 249.\n\n() **the Church's leaders in Rome recognized . . . glorification of race and nation as \"idolatrous,\"** Wolf, _Pope_ , 230, 268; Kornberg, _Dilemma_ , 228\u201329; Godman, _Vatican_ , 102\u20136, 129, 141\u201353.\n\n() **As Mussolini pointed out** , Kertzer, _Mussolini_ , 307\u201315.\n\n() **\"The Jewish Question,\" . . . \"It is an evident fact.\" . . . \"messianic craving for world domination,\"** Kertzer, _Mussolini_ , 211, 289\u201391.\n\n() **\"Spiritually, we are all Semites\" . . . \"the enemy of the Cross of Christ,\"** Kertzer, _Popes_ , 280.\n\n() **Content and fate of \"The Unity of the Human Race,\"** Kertzer, _Popes_ , 280\u201382; Wolf, _Pope_ , 206\u201312; Passelecq and Suchecky, _Hidden_ , passim.\n\n() **Pacelli had opposed issuing _Mit brennender Sorge_ . . . suggested a mere pastoral letter**, Wolf, _Pope_ , 265\u201368.\n\n() **was he the candidate the Nazi envoys in Rome hoped would prevail . . . destroyed the copies and plates** , Kertzer, _Mussolini_ , 370\u201381; Ventresca, _Soldier_ , 130\u201332, 134\u201335.\n\n() **A Roman aristocrat by descent . . . and critical of the Catholic Center Party** , Ventresca, _Soldier_ , 7\u201318, 38\u201365, 72\u201384.\n\n() **Though not happy with the form the Concordat took** , Wolf, _Pope_ , 170\u201378.\n\n() **Pius XII considered his chief duty to be to the Church and to Catholics** , Kornberg, _Dilemma_ , 4\u20136, 255\u201367; Godman, _Vatican_ , 82\u201383.\n\n() **Preysing vs. Bertram** , Hayes and Roth, _Handbook_ , 238\u201341; Phayer, _Catholic_ , 67\u201381.\n\n() **On the importance of the sacraments to the Church's political conduct** , Kornberg, _Dilemma_ , 3\u20134, 272\u201373; Spicer, _Resisting_ , 6\u20139.\n\n() **approximately 72,000 Jews** , Bauer, _Jewry_ , 66.\n\n() **Swiss factory owners in Poland . . . reported** , Straumann and Wildmann, _Schweizer_ , 116\u201320.\n\n() **British intelligence intercepted** , Breitman, _Official_ , 89\u201398.\n\n() **the Vatican's ambassador in Slovakia** , Kornberg, _Dilemma_ , 81.\n\n() **Father Pirro Scavizzi . . . the Jewish Bund Party in Poland** , Phayer, _Catholic_ , 47\u201348.\n\n() **As late as December 1944, a majority of the British public did not believe** , see Hamerow, _Why_ , 410; Stone, _Liberation_ , 68.\n\n() **Gerhart Riegner and Eduard Schulte** , Breitman and Laqueur, _Breaking_ , passim; Riegner, _Never_ , 35\u201343, 50.\n\n() **three unimpeachable sources** , Friedl\u00e4nder, _Nazi Germany_ , v. 2, 458\u201361; Riegner _Never_ , 48\u201350.\n\n() **BBC broadcast by Thomas Mann** , Longerich, _Davon_ , 240\u201345.\n\n() **a tally by H\u00f6fle** , Friedl\u00e4nder, _Nazi Germany_ , v. 2, 479\u201380.\n\n() **making too much of Jewish suffering would play into the claims of Nazi propaganda** , Aronson, _Hitler_ , passim; Hamerow, _Why_ , 398, 400\u2013403, 409, 411\u201312, 414.\n\n() **the USSR never considered the idea** , David-Fox, _Holocaust_ , 31\u201336.\n\n() **The Russians first learned. . . . But nothing happened** , Orbach and Solonin, \"Calculated,\" 90\u2013113.\n\n() **Churchill, Eden, and Sinclair** , Wasserstein, _Britain_ , 307\u201320; Neufeld and Berenbaum, _Bombing_ , 261\u201371.\n\n() **\"frightful prospect,\"** see Wasserstein, _Britain_ , 340\u201341.\n\n() **Only 37,451** , Ofer, _Escaping_ , 319.\n\n() **both the United States and the UK purposefully ignored** , Fleming, _Auschwitz_ , 167\u2013218.\n\n() **The Mufti and his effects on the British** , Wasserstein, _Britain_ , 28\u201329, 71, 79\u201380.\n\n() **The Mufti's hopes and disappointments** , Motadel, _Islam_ , 41\u201343, 87\u201392, 96\u201397, 107\u20138, 113\u201314, 188\u201394, 226\u201335, 250, 274\u201382; Nicosia, _Nazi_ , 71, 267, 276\u201379.\n\n() **the Mufti scored a few victories . . . [his] association with the Axis ultimately had . . . disastrous consequences** , Motadel, _Islam_ , 43\u201344; Nicosia, _Nazi_ , 242\u201357; Achcar, _Arabs_ , 150\u201373.\n\n() **Pope Pius XII was more worried about** , Phayer, _Catholic_ , 57\u201366; Kornberg, _Dilemma_ , 253.\n\n() **He had his ambassador to Vichy . . . tell its leader** , Marrus and Paxton, _Vichy_ , 262.\n\n() **used Vatican diplomatic channels to persuade Spain** , Zuccotti, _P\u00e8re_ , 127\u201328; Ventresca, _Soldier_ , 199\u2013200.\n\n() **left decisions . . . withheld . . . information** , Phayer, _Catholic_ , 43, 46, 49.\n\n() **declined to intervene . . . put off pressing . . . Horthy . . . refused to send another protest** , ibid., 104\u20139; Phayer, _Pius_ , 91\u201393.\n\n() **\"divine law knows no compromise,\"** \"Pius XII,\" 16; see also Ventresca, _Soldier_ , 174.\n\n() **\"Push never came to shove,\"** Bess, _Choices_ , 86.\n\n() **Sheptytsky . . . tried to impede Ukrainian collaboration . . . in two extraordinary ways** , Petrovsky-Shtern and Polonsky, _Polin_ 26, 347\u201349.\n\n() **\"Deeply moved. . . . platonic,\"** Friedl\u00e4nder, _Nazi Germany_ , v. 2, 420.\n\n() **Catholic hierarchy in Slovakia** , Ward, _Tiso_ , 225\u201328, 232\u201339; Kornberg, _Dilemma_ , 78\u201386.\n\n() **In Belgium, a network** , Moore, _Survivors_ , 276\u201395.\n\n() **in Rome** , Schlemmer and Woller, \"italienische Faschismus,\" 195.\n\n() **emphasized the dangers that might flow** , Ventresca, _Soldier_ , 162\u201370, 174\u201376.\n\n() **although the pope never explicitly drew this comparison** , Kornberg, _Dilemma_ , 253.\n\n() **neither the Dutch prime minister nor the queen of the Netherlands was cowed** , Wasserstein, _Ambiguity_ , 244.\n\n() **arrested thirty-seven clerics . . . six of them died** , Griech-Polelle, _Galen_ , 217; Spicer, _Resisting_ , 137.\n\n() **such fears did not stop Bernhard Lichtenberg** , Hayes and Roth, _Handbook_ , 239; Spicer, _Resisting_ , 171\u201382.\n\n() **\"In the fulfillment of . . . Our duty . . . it is not allowed,\"** Kornberg, _Dilemma_ , 266.\n\n() **the incidence of antisemitism in the American population actually increased during the conflict** , Dinnerstein, _Antisemitism_ , 128\u201349; Wyman, _Abandonment_ , 14\u201315, Dinnerstein, _Survivors_ , 6; Hamerow, _Why_ , 311.\n\n() **in 1940\u201341 . . . only about 30,000 German Jews got into the country** , Bauer, _Jewry_ , 66.\n\n() **many munitions ships returned empty from Europe** , Wyman, _Abandonment_ , 335.\n\n() **Karski in Izbica and with FDR** , Karski, _Story_ , 368\u201384, 419, 446\u201347.\n\n() **War Refugee Board funding** , Bauer, _American_ , 407\n\n() **food that the board paid for and stockpiled in Swedish ports saved thousands of lives** , Rosenberg, _Brief_ , 139\u201340, 149\u201350.\n\n() **The non-bombing of Auschwitz** , Hamerow, _Why_ , 402\u201318; Neufeld and Berenbaum, _Bombing_ , passim, but especially 249\u201360, 271\u201380.\n\n() **bombing the camp might not have saved many lives** , Rees, _Auschwitz_ , 246\u201347.\n\n() **The SS transferred** , Czech, _Kalendarium_ , 701, 821.\n\n() **30,000 in October, for example** ; Steinbacher, _Auschwitz_ , 124.\n\n() **collateral damage would have occurred** , Wachsmann, _KL_ , 586; Hayes, _From Cooperation_ , 256.\n\n() **Stephen Wise . . . refused to pressure him** , Hamerow, _Why_ , 269, 345; Riegner, _Never_ , 71.\n\n() **\"constitutionally incapable of serious questioning,\"** Bauer, _American Jewry_ , 52.\n\n() **the _Yishuv_** , Porat, _Blue_ , 251, 256\u201358, 261\u201362; Ofer, _Escaping_ , 23\u201331, 318\u201319.\n\n() **\"Refugees who have fled purely on racial grounds,\"** Independent Commission, _Switzerland_ , 114.\n\n() **enforcement was inconsistent** , ibid., 110, 117.\n\n() **Evolution of Swedish policy** , Hayes, _How_ , 735\u201352, excerpting the fundamental work of Paul Levine.\n\n() **George Mantello** , Kranzler, _Man_ , chapters 7\u201311.\n\n() **Giorgio Perlasca** , Levine, _Wallenberg_ , 310\u201311, 324\u201348.\n\nCHAPTER 8: AFTERMATH\n\n() **The toll at Belsen** , Stone, _Liberation_ , 83\u201385, 107\u20138, 111\u201312.\n\n() **some of them went on a rampage . . . used bayonets and rifle butts** , Abzug, _Inside_ , 93; Bessel, _1945_ , 162\u201364.\n\n() **\"The punishment they got,\"** Stone, _Liberation_ , 100.\n\n() **the object of Allied revulsion soon changed** , Fritz, _Endkampf_ , 53\u201356, 227\u201338; Abzug, _Inside_ , 154\u201355.\n\n() **Patton's remarks** , ibid., 157.\n\n() **\"As matters now stand,\"** Brenner, _After_ , 11; Fritz, _Endkampf_ , 236\u201337.\n\n() **18,000\/97,000\/167,000** , Wyman, _DPs_ , 149.\n\n() **encountered the same sort of reluctance** , Petrovsky-Shtern and Polonsky, _Polin_ 26, 368\u201379.\n\n() **UNRRA spent almost $4 billion** , Gutman, _Encyclopedia_ , 1540.\n\n() **They were often received somewhat insensitively** , Hayes, _How?_ , 775\u201387.\n\n() **readiness to turn a blind eye** , Douglas, _Right_ , 28; Dinnerstein, _America_ , 251\u201371.\n\n() **faced incomprehension of their experience** , Cohen, _Case_ , passim.\n\n() **the reckoning was pretty intense** , Frei, _Transnationale_ , 31\u201332; Heberer and Matth\u00e4us, _Atrocities_ , 49\u201371; Jardim, _Mauthausen_ , 1, 197.\n\n() **The British tried** , Wachsmann, _KL_ , 608.\n\n() **Soviet courts convicted** , Frei, _Transnationale_ , 193; Bazyler and Tuerkheimer, _Forgotten_ , 40\u201341.\n\n() **The Poles tried** , Wachsmann, _KL_ , 608\u20139.\n\n() **The Dutch followed up** , Wasserstein, _Ambiguity_ , 223\u201324; Deak, _On Trial_ , 204.\n\n() **only seven . . . were still alive in 1950** , Wachsmann, _KL_ , 612.\n\n() **Only about 10 percent of the Germans who ever worked at Auschwitz** , Dlugoborski and Piper, _Auschwitz_ , v. 5, 102\u20133, 108, 116 (789 out of perhaps 7,200 Germans ever stationed there).\n\n() **In 1969, East Germany executed** , Raskin, _Child_ , 94\u201398.\n\n() **The record . . . regarding the 121 men from T4** , Berger, _Experten_ , 363\u201371; Bryant, _Eyewitness_ , 13\u201319.\n\n() **Catholics developed several escape routes** , Phayer, _Pius XII_ , 173\u201394;\n\nPhayer, _Catholic_ , 165\u201375; Stangneth, _Eichmann_ , 79, 90\u201392, 292\u201393.\n\n() **Alois Hudal** , Phayer, _Pius XII_ , 195\u2013207.\n\n() **Draganovic, Barbie, Pavelic** , ibid., 208-51; Deak, _On Trial_ , 217\u201318.\n\n() **Pius XII . . . held to the theory of pastoral responsibility that he had followed** , Kornberg, _Dilemma_ , 235, 255\u201374\n\n() **Galen . . . went even further** , Phayer, _Catholic_ , 139, 162\u201363.\n\n() **Muench . . . wrote a pastoral letter that contrasted** , Brown-Fleming, _Conscience_ , 5\u20136.\n\n() **their advocacy was too much for even Muench** , Phayer, _Pius XII_ , 165.\n\n() **ODESSA, appears to have been largely mythical** , Stangneth, _Eichmann_ , 89\u201390.\n\n() **An independent historians' commission . . . reached the conclusion** , Schneppen, _Odessa_ , 208\u20139. For evidence of a higher number of Croatian criminal immigrants, see Phayer, _Pius XII_ , 246.\n\n() **owed much more to the efforts of their friends and families** , Kulish and Mekhennet, _Eternal_ , 83\u201386, 91\u201394, 210\u201314.\n\n() **total payments . . . have come** , Marrus, _Measure_ , 68\u201376; Dean et al., _Robbery_ , 99\u2013133; Goschler, _Schuld_ , 474\u201375, 539.\n\n() **The survivors who came off best** , Hayes and Roth, _Handbook_ , 548\u201350.\n\n() **Other categories of victims came away with much less** , ibid., 551\u201354.\n\n() **a few companies made token postwar payments** , Ferenc, _Less_ , 188.\n\n() **German Foundation Initiative** , see Spiliotis, _Verantwortung_ , passim; Eizenstat, _Imperfect_ , 243\u201378; Dean et al., _Robbery_ , 128.\n\n() **Catholic religious institutions and orphanages . . . often declined** , Marrus, \"Custody,\" 378\u2013403; Ventresca, _Soldier_ , 222\u201327.\n\n() **the French government endowed a new Foundation** , Dean et al., _Robbery_ , 139.\n\n() **Switzerland and its payments** , Eizenstat, _Imperfect_ , 90\u2013186; Independent Commission, _Switzerland_ , 274\u201379, 442\u201349.\n\n() **\"negotiated justice,\"** Barkan, _Guilt_ , 309.\n\n() **The Union Bank of Switzerland, in effect . . . the National Bank of Switzerland, the recipient of 92 percent of the gold** , Spiliotis, _Verantwortung_ , 54; Independent Commission, _Switzerland_ , 238, 252\u201353.\n\n() **these advocates also spread a lot of misconceptions . . . awful role models** , Marrus, _Measure_ , 124\u201326; Petropoulos and Roth, _Gray_ , 7\u20139; Bazyler and Alford, _Restitution_ , 197\u2013204; Eizenstat, _Imperfect_ , 182.\n\n() **\"gap\" . . . \"clash,\"** Levine, _Wallenberg_ , 12\u201313.\n\n() **As David Cesarani has shown** , Cesarani, _Final_ , passim.\n\n() **more trains to the staging areas of Operation Barbarossa . . . every day** , Gall and Pohl, _Eisenbahn_ , 227.\n\n() **\"less the driving force . . . than the setting,\"** Stone, _Histories_ , 126.\n\n() **A British judge** , Lipstadt, _History_ , passim; Evans, _Lying_ , 104\u201348; van Pelt, _Case_ , 488\u2013506.\n\n() **\"On the Streets of Truth\" . . . \"principal source text,\"** Stangneth, _Before_ , 152\u201353, see also 142\u201344.\n\n() **Tony Judt . . . aroused an explosive debate** , reprinted in Judt, _Change_ , 115\u201323; see also Judt, _Reappraisals_ , 286\u201395.\n\n() **\"To make peace is to forget,\"** Sontag, _Pain_ , 103.\n\n() **83 percent of the people there thought a resolution . . . impossible** , Mitchell, _Negotiator_ , 315.\n\n() **\"believed that if you choose to resist evil . . . ways . . . will open up around you,\"** Bess, _Choices_ , 129.\n\n() **\"surrounding every perfidy,\"** Neumann, _Behemoth_ , 379.\n\n() **wrote a prophetic protest** , Hayes \"Industry under the Swastika,\" 28.\nSELECTED BIBLIOGRAPHY\n\nABELLA, IRVING, AND HAROLD TROPER. _None Is Too Many: Canada and the Jews of Europe, 1933\u20131948_. 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New Haven, CT: Yale University Press, 2005.\n\n________. _German Big Business and the Rise of Hitler_. New York: Oxford University Press, 1985.\n\n________. _Hitler's Thirty Days to Power_. Reading, MA: Addison-Wesley, 1996.\n\nUNITED STATES HOLOCAUST MEMORIAL MUSEUM. _Historical Atlas of the Holocaust_. New York: Macmillan, 1996.\n\nVAGI, ZOLTAN, ET AL. _The Holocaust in Hungary_. Lanham, MD: AltaMira Press, 2013.\n\nVAN PELT, ROBERT JAN. _The Case for Auschwitz: Evidence from the Irving Trial_. Bloomington: Indiana University Press, 2002.\n\n________. \"Nazi Ghettos and Concentration Camps: The Benefits and Pitfalls of an Encyclopedic Approach.\" _German Studies Review_ 37 (2014): 149\u201359.\n\nVAN RAHDEN, TILL. _Jews and Other Germans: Civil Society, Religious Diversity, and Urban Politics in Breslau, 1860\u20131925_. Madison: University of Wisconsin Press, 2000.\n\nVENTRESCA, ROBERT. _Soldier of Christ: The Life of Pope Pius XII_. Cambridge, MA: Harvard University Press, 2012.\n\nVINCENT, C. Paul. \"The Voyage of the St. Louis Revisited.\" _Holocaust and Genocide Studies_ 25 (2011): 252\u201389.\n\nVITAL, DAVID. _A People Apart: The Jews in Europe 1789\u20131939_. Oxford: Oxford University Press, 1999.\n\nVOLKOV, SHULAMIT. _Germans, Jews, and Antisemites_. New York: Cambridge University Press, 2006.\n\nWACHSMANN, NIKOLAUS. _KL: A History of the Nazi Concentration Camps_. New York: Farrar, Straus and Giroux, 2015.\n\nWALLER, JAMES. _Becoming Evil: How Ordinary People Commit Genocide and Mass Killing_. New York: Oxford University Press, 2002.\n\nWALTER, DIRK. _Antisemitische Kriminalit\u00e4t und Gewalt: Judenfeindschaft in der Weimarer Republik_. Bonn: J. H. W. Dietz Nachfolger, 1999.\n\nWARD, JAMES MACE. _Priest, Politician, Collaborator: Josef Tiso and the Making of Fascist Slovakia_. 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Ithaca: Cornell University Press, 1998.\n\nYISRAELI, DAVID. \"The Third Reich and the Transfer Agreement.\" _Journal of Contemporary History_ 6 (1971): 129\u201348.\n\nZIMMERMAN, JOSHUA D. _The Polish Underground and the Jews, 1939\u20131945_. New York: Cambridge University Press, 2015.\n\nZIMMERMAN, JOSHUA D., ED. _Contested Memories: Poles and Jews during the Holocaust and Its Aftermath_. New Brunswick: Rutgers University Press, 2003.\n\n________. _The Jews in Italy under Fascist and Nazi Rule, 1922\u20131945_. New York: Cambridge University Press, 2005.\n\nZIMMERMANN, MOSHE. _Wilhelm Marr: The Patriarch of Antisemitism_. New York: Oxford University Press, 1986.\n\nZUCCOTTI, SUSAN. _The Italians and the Holocaust_. New York: Basic Books, 1987.\n\n________. _P\u00e8re Marie-Benoit and Jewish Rescue_. Bloomington: Indiana University Press, 2013.\n\n________. _Under His Very Windows: The Vatican and the Holocaust in Italy_. New Haven, CT: Yale University Press, 2000.\nINDEX\n\nPage numbers listed correspond to the print edition of this book. You can use your device's search function to locate particular terms in the text.\n\nacculturation\n\nof Jews in Imperial Germany, 48\u201349\n\nas protective factor for Jews, 239\n\nand resistance to immigration, 269\n\nAction 14f13, 119\u201320, 124, 172\n\nadaptation, to concentration camps, 215\u201316\n\nAdenauer, Konrad, 315\u201316\n\nage, and Jewish emigration, 105\n\nAge of Enlightenment, 13\u201315\n\naid to Jews from general public; _See also_ prewar aid to Jews from general public; wartime aid to Jews from general public\n\nby businessmen, 222\u201324\n\nin Denmark, 237\n\nin Europe, 219\u201320\n\nin Italy, 237\n\nin Poland, 256\n\nand survival rates, 218\u201324\n\nAK (Armia Krajowa), 250, 252\u201353\n\nal-Husseini, Muhammad Amin, 283\u201385\n\nAllen, William Sheridan, 71\n\nAllied nations\n\nability of, to stop Holocaust, 328\n\nappeasement of Hitler by, 109\u201310\n\npassivity of, 278\u201382\n\nAmerican businesses, in Germany, 110\u201311\n\nAmerican Jewish Committee, 271\u201372\n\nAmerican Jewish Congress, 271\u201372\n\nAmerican Jewish Joint Distribution Committee, 272\n\nAmsterdam, 187\u201389, 191, 222, 243, 308\n\nAnger, Per, 297\n\nannihilation, 114\u201360\n\nconsequences of German resistance to, 142, 145\u201346\n\nemergence of concept, 84\u201388\n\nemergence of death camps, 125\u201331\n\nlack of effect of, on war effort, 131\u201336, 328\u201329\n\nmeans for, 116\u201317\n\npolitical evolution of, 121\u201325\n\npsychological explanations for German compliance with, 137\u201342\n\nself-delusion of public about, 154\u201360\n\nsenior officers' roles in, 146\u201354\n\nand T4 program, 117\u201321\n\nantinomianism, 341\u201343\n\nAntisemites' Petition, 44, 52\n\nantisemitism, 3\u201335\n\nand appeasement, 108\u201310\n\nas backlash against Jewish success, 27\u201331\n\nand Christianity, 6\u20137\n\ncultural basis for, 13\u201315\n\ndefining, 3\u20135\n\nand European societal changes, 24\u201327\n\nhistorical evolution of, 7\u20139\n\nin Imperial Germany, 40\u201348\n\nin interwar Germany, 66\u201367\n\nin interwar Hungary, 233\n\nand Jewish emancipation, 20\u201324\n\nin politics, 30\u201335\n\nin postwar Poland, 255\u201356\n\nin prewar Italy, 274\u201375\n\nin prewar Poland, 244\u201347\n\nin prewar United States, 109, 269\u201370\n\npotential for resurgence of, 332\u201336\n\nreligious basis for, 9\u201313\n\nrole of, in Hitler's rise, 54\u201355, 65\u201367, 327\n\nsupposed genetic basis for, 15\u201320\n\nin wartime United States, 291\u201392\n\nxenophobic and chimerical forms of, 5\u20136\n\nAntonescu, Ion, 225, 234\u201336, 309\n\nArendt, Hannah\n\non Adolf Eichmann, 62, 146, 152\n\n_Eichmann in Jerusalem_ , 177\n\non lack of Jewish resistance, 177\u201378\n\narmed Jewish resistance, 178\u201379\n\nArmia Krajowa (AK), 250, 252\u201353\n\nArndt, Arthur, 157\n\nAryanization, 75\u201379\n\nAsscher, Abraham, 188\u201389, 200\n\nAttlee, Clement, 303\n\nAugustine, Saint, 9\u201310\n\nAuschwitz-Birkenau concentration camp, 127\u201329, 133, 231, 233, 308\n\nAuschwitz concentration camps\n\nescapes from, 211\n\nevacuation of, 171\n\ngarrison of, 134, 310\n\nlack of resistance at, 210\n\nmortality rates at, 213\n\noriginal inhabitants of, 241\n\nplunder from, 131\n\nrejection of plans to bomb, 293\u201395\n\nas slave labor camp, 165\u201366\n\nsmuggling at, 215\u201316\n\nSoviet information about, 282\u201383\n\nuse of Zyklon at, 122\u201323\n\nAustria, 81\u201382\n\nBabi Yar, 89, 156, 228, 307, 331\n\nBach-Zalewski, Erich von dem, 116\n\nBaeck, Leo, 196\n\nBalfour Declaration, 263\n\nBalkan states\n\ndeportation of Jews from, 226\n\nforeign vs. native-born Jews in, 230\u201331\n\nsurvival rates in, 114\n\nBaltic states, 87, 89, 124, 224, 227\u201329\n\nBandera, Stepan, 228\n\nBarbie, Klaus, 313\n\nBartov, Omer, _The Eastern Front, 1941\u201345_ , 140\n\nBauer, Hermann, 311\n\nBauer, Yehuda, 178, 249\u201350, 295\n\nBaum, Herbert, 108\n\nBauman, Zygmunt, 329\n\nbehavior\n\nof perpetrators, 137\u201339\n\nof German public, 92\u2013101, 156\u201360\n\nwith-and-against in camps, 215\u201317\n\nBeilis, Mendel, 32\u201334\n\nBeinart, Peter, 335\u201336\n\nBeitz, Berthold, 223\n\nBelarus, 23, 88\u201389, 122, 124, 127, 140, 178, 194, 208, 224, 240\n\nBelgium, 23, 177, 224\u201326, 260\u201362, 273, 289\n\nBelzec death camp\n\nconstruction of, 121\n\ndeath toll of, 126\u201327\n\nescapes from, 210\n\nas labor camp, 163\n\nBen-Gurion, David, 202\n\nBeorn, Waitman, 140\u201341\n\nBergen-Belsen camp, 130\n\ndeath toll at, 300\n\nand evacuation of labor camps, 172\u201373\n\ninmate conditions at, 212\n\nBerger, Sara, _Experten der Vernichtung_ , 142\n\nBergson, Peter, 292\n\nBerl, Emmanuel, 261\n\nBerlin, 87, 108, 306\n\ndeportations from, 101\u20132\n\nJews in, 25, 28, 50\u201352, 58, 66, 76, 157\u201359, 163, 201, 221, 223\n\nBerman, Jakub, 243\n\nBertram, Adolf, 277\n\nBess, Michael, 287\n\nBessarabia, 234\u201336\n\nBest, Werner, 237\n\nBialas, Max, 177\n\nBialystok ghetto, 185, 192\n\nBiebow, Hans, 181, 307\n\nbigotry, 336\u201339\n\n_Bildung_ , 49\n\n\"biological materialism,\" 61\u201362\n\nBismark, Otto von, 41, 44\n\nBlobel, Paul, 307\n\nblock elders, 203\u20134\n\nBl\u00f6sche, Josef, 310\n\nB\u00f6ckel, Otto, 44, 46\n\nBoger, Wilhelm, 311\n\nBosch, Carl, 92\n\nBouhler, Philipp, 117\u201318, 309\n\nboycotts, of Jewish businesses, 76\n\nBrack, Viktor, 118, 309\n\nBrandt, Karl, 117\u201318, 309\n\nBraun, Wernher von, 315\n\nbreeding, as public policy, 18\u201319\n\nBritain, 263\u201365, 283, 307\n\nBrothers Grimm, 39\u201340\n\nBrowning, Christopher, 137\u201339, 240\n\n_Ordinary Men_ , 137\n\nBryant, Michael, 311\n\nBuchenwald concentration camp, 83, 129, 168, 172, 173, 295, 309\n\nBukovina, 28, 234\u201336\n\nBulgaria, 87, 208, 220, 224, 226, 230\u201331, 231, 239, 285, 310, 319\n\nBund members, 184, 200, 279\n\nBurzio, Giuseppe, 279\n\nBusemann, Ernst, 97\u201398\n\nbusinesses\n\nAmerican, in Germany, 110\u201311\n\nboycotts of Jewish, 76\n\nrestitution by, 321\u201323\n\nbusinessmen, aid from, 222\u201324\n\nCanada (Birkenau), 216\n\nCanada and refugees, 263, 267\n\ncarbon monoxide (CO)\n\ndeath camps, 125\u201327\n\ndemand for, 120\n\nfrom exhaust pipes, 122\n\nin T4 program, 119\n\nCentral Association of German Citizens of Jewish Faith, 103, 105\n\nCentral Office for Combatting Homosexuality and Abortion, 207\n\nCentral Welfare Office of German Jews, 106\n\nChamberlain, Houston Stewart, 46\n\nChamberlain, Neville, 110\n\nChelmno death camp, 122, 133, 183, 279\n\ndeath toll of, 127\n\ngas vans used in, 126\n\nplunder from, 131\u201332\n\nsurvivors of, 203\n\nchildren, in ghettos, 198\u2013200\n\nchimerical antisemitism, 5\u20136\n\nChristianity, antisemitism and, 6\u20137\n\nChristian Social Workers' Party, 43\n\nChurchill, Winston, 227, 279, 283\n\nChvalkovsky, Frantisek, 85\n\n_Civilta Cattolica, La_ , 274, 275\n\nCO, _See_ carbon monoxide\n\ncognitive dissonance, 139\n\nCohen, David, 188\u201389, 200\n\ncollaborators, 198\u2013202\n\ncollective punishments, 209\n\nComey, James, 243\u201344\n\ncommunism\n\nand aid to Jews, 219\n\nand German invasion of Baltic states, 227\u201328\n\nin interwar Germany, 68\n\nJewish involvement in, 56\n\nRoman Catholic Church concern over, 274\u201375\n\ncompulsory labor, 83\n\nconcentration camps, 202\u201317; _See also specific camps_\n\nbombing of, 295\n\ncosts of building and operating, 133\u201334\n\ndeath camps, 125\u201331\n\nfates of commanders of, 308, 309\n\nhierarchy within, 203\u20134\n\ninmate conditions in, 209\u201310\n\nNazi control in, 211\u201312\n\nnon-Jewish victims of, 204\u20138\n\nnumber of, 202\u20133, 309\n\noriginal use of gas chambers at, 119\u201320\n\nprimitivism of , 329\u201330\n\nresistance in, 210\u201311\n\nsatellite, 165\n\nSS firms at, 163, 165\n\nsurvival in, 212\u201317\n\ntransit camps, 130\n\ntypes of, 129\u201331\n\nConcordat with Third Reich, 276\u201377\n\nConfino, Alon, _A World Without Jews_ , 7, 9\n\nconspiracy theories, about Jews, 27\u201330, 331\n\nCoughlin, Charles, 269\n\ncourage, individual, 341\n\nCracow, Poland, 177\n\nCroatia, 206, 224, 238, 288, 310, 313\n\nCrystal Night, 82\u201383\n\nCuhorst, Fritz, 242\n\ncultural basis, for antisemitism, 13\u201315\n\ncultural roles, Jews expelled from, 76\u201377\n\nCzech family camp, 210\n\nCzechoslovakia, 80, 84, 127, 260, 262\n\nCzerniakow, Adam, 187, 194\n\nDabrowa Tarnowska County, Poland, 252\n\nDachau concentration camp, 83, 119\u201320, 172\u201374, 203, 239, 290, 301, 307, 309\n\n_Daily Northwestern_ , 268\u201369\n\nDanielsson, Carl Ivar, 297\n\nDaube, David, 189\n\ndeath camps, 125\u201331\n\ndebt burden, of Germany after World War I, 57\u201358\n\nDeclaration of the Rights of Man, 21\n\n_Defying Hitler_ (Sebastian Haffner), 94\n\nDe Gaulle, Charles, 309\n\nDegussa, 97\u201398, 162, 170\n\ndehumanization, of concentration camp inmates, 211\u201312\n\ndelegation, of unpleasant duties, 143\u201344\n\ndemocratization, in nineteenth-century Europe, 26\n\ndenaturalization, of Jewish citizens, 76, 77\n\nDenmark, 236\u201338\n\ndeportation of Jews\n\nafter gas chamber development, 123\u201324\n\nafter Polish invasion, 86\n\nfrom Balkan states, 226\n\nfrom Germany, 82, 90\u201391, 101\u201302\n\nfrom ghettos, 185\n\nfrom Hungary, 195, 231\u201334\n\nJewish attempts to disrupt, 177\n\nfor _Mischlinge_ , 102\n\nNazi attempts to disguise, 100\u2013101\n\npayment for, 132\n\nDes Pres, Terrence, 214\u201317\n\n_The Survivor_ , 214\n\n_Destruction of the European Jews, The_ (Raul Hilberg), 177\n\n_Deutsche Hollerith Maschinen Gesellschaft_ , 112\n\ndiplomats, 221\u201322, 297\u201399\n\ndiscrimination, _See_ Jewish discrimination\n\ndisease, in Polish ghettos, 182\n\ndisplaced persons (DPs), 301\u20135\n\nDisraeli, Benjamin, 24\n\nDmowski, Roman, 244, 246\n\n\"doctrine of Jewish witness,\" 9\u201310\n\ndocumentation, of Holocaust, 331\n\nDora-Mittelbau concentration camp, 168\u201369, 309, 315\n\nDPs (displaced persons), 301\u20132\n\nDraganovic, Krunoslav, 313\n\nDreyfus, Alfred, 32\u201333\n\nDreyfus affair, 32\u201333\n\n_Dry Tears_ (Nechama Tec), 255\n\n_Duckwitz, Georg, 237_\n\nDurchgangstrasse IV, 128, 130, 164\n\nduty, sense of\n\nNazi manipulation of, 93\u201395\n\nof senior officers, 148\n\nand willingness to kill, 140\u201341\n\nEastern Europe\n\nescalation of violence in, 88\u201390\n\nJewish refugees from, 302\u20133\n\nlocal Nazi collaboration in, 228\u201329\n\nrestitution to Jews in, 318, 319\n\neconomic competition, 268\n\neconomic crises\n\nand antisemitism, 30\n\ncurrent, 334\n\nand German unification, 42\n\nGreat Recession, 336\n\nin interwar Germany, 54, 55, 57\u201359\n\nand rise of Nazi Party, 67\u201368\n\nEden, Anthony, 283\n\nEdict of Tolerance, 21\n\neducation\n\nfor Jews, 28\u201329, 49, 76\u201377\n\nof senior officers, 146, 147\n\nEichmann, Adolf\n\nage of, 147\n\ndeportations coordinated by, 132\n\nas head of Jews Department, 85, 180\n\nin Hungary, 201, 232, 234\n\nself-centeredness of, 62\u201363\n\nzealotry of, 151\u201352\n\n_Eichmann Before Jerusalem_ (Bettina Stangneth), 152, 332\n\n_Eichmann in Jerusalem_ (Hannah Arendt), 177\n\n_Einsatzgruppen_ , 88\u201389\n\nfates of commanders of, 308\u20139\n\ngas trucks used by, 122\n\nmotivations for behavior of, 139\n\nsurvival rates in areas with, 114\u201315\n\nEinstein, Albert, 29\n\nElectoral Hesse, 46\n\nelectoral process, in Imperial Germany, 47\u201348\n\nemancipation, Jewish, 14\u201315, 20\u201324, 37\u201338\n\nemigration, _See_ Jewish emigration\/immigration\n\nErasmus, 12\n\nescalation of violence, 73\u2013113\n\nand Aryanization, 75\u201379\n\nin Eastern Europe, 88\u201390\n\nemergence of annihilation concept, 84\u201388\n\nand expansion of Germany, 79\u201380\n\nexpulsion of Jews, 80\u201384\n\nand general public's attitude toward Jews, 97\u2013102\n\nindoctrination of non-Jews, 94\u201396\n\ninternational response to, 108\u201313\n\nintimidation of non-Jews, 90\u201394\n\nJewish response to, 102\u20138\n\nescapes\n\nfrom concentration camps, 210\u201311\n\nof war criminals, 311\u201315, 329\n\n\"Essay on Man, An\" (Alexander Pope), 13\n\n_Essay on the Inequality of the Human Races_ (Arthur de Gobineau), 16\n\nEstonia, 206, 227\n\neugenics\n\nemergence of, 19\n\nin Hitler's political platform, 61\n\nprimitivism of, 330\n\nEurope; _See also_ Eastern Europe\n\ndemocratization of, 26\n\nJewish emancipation in, 23\n\npolitical changes in, 332\u201333\n\nprewar immigration to, 260\u201365\n\nsocietal changes in, 24\u201327\n\nWestern, 319\u201322\n\nEuthanasia Action, _See_ T4 program\n\nevacuations, of labor camps, 170\u201375\n\nEvian Conference of 1938, 272\u201373\n\nexhaustion, of concentration camp inmates, 212\n\n_Exodus_ (Leon Uris), 305\n\n_Experten der Vernichtung_ (Sara Berger), 142\n\nfarmland, prohibition on Jews owning, 77\n\n_Fear_ (Jan Gross), 255\n\nFerry Laws, 26\u201327\n\nFichte, Johann Gottlieb, 39\n\nFighter Staff Program, 169\u201370, 309\n\nFinal Solution of the Jewish Question, 123\u201331\n\nFlossenb\u00fcrg concentration camp, 173\u201374\n\nfood\n\nfor German army, 90\n\nin ghettos, 191, 193\n\nsmuggling, 193\n\nfor Soviet prisoners of war, 90\n\nin wartime Poland, 241\u201342\n\nforced labor\n\ndevelopment of, 162\u201363\n\nby Poles, 241\n\nslave vs., 160\u201361\n\nforeign-owned companies, in Germany, 110\u201313\n\nForsyth, Frederick, _The Odessa File_ , 314\n\nFrance; _See also_ Vichy France\n\nimmigration to, 260\u201361\n\nJewish armed resistance in, 178\u201379\n\nrestitution by, 320\n\nsurvival rates in, 229\u201330\n\nFrank, Hans, 87, 241\u201342\n\nFrench Revolution, 21\u201322\n\nFritzsch, Karl, 122\u201323\n\nFritzsche, Peter, 98, 124, 156\n\nFromm, Julius, 67\n\nFry, Varian, 222\n\nF\u00fcnten, Ferdinand aus der, 308\n\nGalen, Clemens Graf von, 120\u201321, 286, 290, 314\n\nGall, Franz Joseph, 17\n\nGalton, Francis, 19\n\ngas chambers, 119\u201320, 126\u201329, 133, 172\n\ngas vans , 119, 122, 126, 130, 133, 172\n\nGeist, Raymond, 85\n\nGemmeker, Albert, 308\n\ngender\n\nat Auschwitz, 165\u201366\n\nand propensity for violence, 144\u201345\n\nand slave labor system, 162\n\nand Jewish emigration, 105\u20136\n\nGeneral Government (occupied Poland), 86\u201387, 124, 126, 130, 142, 151, 156, 181, 241\u201342, 281, 307, 310\n\nGeneral Plan East, 208\n\ngeneral public\n\naid from, _See_ aid from general public\n\nattitude toward Jews of, 91, 97\u2013102, 328\n\nindoctrination of, 94\u201396\n\nintimidation of, 90\u201394\n\nknowledge about Holocaust of, 155\u201356\n\nprofit made from Holocaust by, 157\u201358\n\nself-delusion about annihilation of, 154\u201360\n\ngenetic basis, for antisemitism, 15\u201320\n\ngenocide, since Holocaust, 325\n\nGens, Jacob, 194\n\nGerman compliance, explanations of, 137\u201342\n\nGerman military, Jews banned from, 77\n\nGerman restitution, 316\u201319\n\nGerman women, in war effort, 144\u201345\n\nGermany, 36\u201372\n\nexpansion of, and escalation of violence, 79\u201380\n\nHitler's political platform, 59\u201362\n\nImperial, _See_ Imperial Germany\n\nin interwar period, 53\u201355, 57\u201359\n\nNazi ideology, 62\u201365\n\npopular support of Nazi party in, 65\u201371\n\npostwar condition of, 306\n\npostwar rebuilding of, 315\u201316\n\nprofitability of Holocaust for, 132\n\nrise of Hitler in, 55\u201359\n\nunification of, 36\u201338\n\nwar criminals tried by, 310\n\nGerstein, Kurt, 93\u201394\n\nGestapo, 98\u2013100\n\nghettoization; _See also specific ghettos_\n\ninternal disunity in, 184\u201385\n\nand Jewish resistance, 179\u201383\n\nof Jews in Poland, 85\n\nghetto police forces, 190\u201391\n\nGilligan, James, 60\n\nGinsberg, Benjamin, _How the Jews Defeated Hitler_ , 179\n\nGlagau, Otto, 42\n\nGlas, Alfons, 142\n\nGleiwitz concentration camp, 162\n\nGlobke, Hans, 316\n\nGlobocnik, Odilo, 121, 310\n\nGobineau, Arthur de, 16\u201317\n\n_Essay on the Inequality of the Human Races_ , 16\n\nGoebbels, Joseph, 100, 156\n\nGoldhagen, Daniel, 137\u201339\n\n_Hitler's Willing Executioners_ , 137\n\nG\u00f6ring, Hermann\n\nas Acting Economics Minister, 80\n\non Jewish segregation, 64\n\non overall solution to Jewish question, 88, 117\n\nG\u00f6th, Amon, 202, 307\n\nGreat Depression, 65\u201366\n\nGreat Recession, 336\n\nGreece, 87, 178, 224\u201326, 231, 237\n\nGreene, Graham, _The Power and the Glory_ , 277\u201378\n\nGreiser, Arthur, 307\n\nGrimm Brothers, 39\u201340\n\nGr\u00f6ning, Oskar, 142\u201343\n\nGross, Jan\n\n_Fear_ , 255\n\n_Neighbors_ , 240\n\nGross-Rosen concentration camp, 171, 203, 309\n\nGruenwald, Malchiel, 201\n\nGr\u00fcninger, Paul, 262\n\nGrynszpan, Herschel, 82\n\nGustav V, king of Sweden, 298\n\nGypsies, death toll, 205\u20136\n\nHaffner, Sebastian, _Defying Hitler_ , 94\n\nHallie, Philip, 341\n\nhandicapped people, 117, 118\n\nHanneken, Hermann von, 237\n\nHarrison, Earl, 302\n\nHasidic Jews, 184\n\nHeim, Aribert, 315\n\nHep-Hep riots, 41\n\nHerder, Johann Gottfried, 38\u201339\n\nHeuss, Theodor, 61\n\nHeydrich, Reinhard\n\nage of, 147\n\nghettoization of Polish Jews by, 179\n\nas head of Reich Security Main Office, 179\u201380\n\nrole in annihilation of, 117\n\nat Wannsee conference, 124\n\nzealotry of, 150\u201351\n\nHielscher, Friedrich, 188\n\nhierarchy, in concentration camps, 203\u20134\n\nHilberg, Raul\n\n_The Destruction of the European Jews_ , 177\n\non resistance, 178\n\non scholarship about Holocaust, 332\n\nHimmler, Heinrich\n\nage of, 147\n\non annihilation of Jews, 86, 139\n\non \"Germanization\" of Slavs, 208\n\non philosophy of SS, 154\u201355\n\nin Poland, 87, 241\n\nslave labor evacuations ordered by, 173\n\nT4 program personnel transferred by, 121\n\nzealotry of, 149\u201350\n\nHindenbrug, Paul von, 67, 71, 75\u201377\n\nHirschfeld, Magnus, 67\n\nHirszman, Chaim, 210, 255\n\nhistorical narratives, compiled by Jews, 193, 214\n\nHitler, Adolf\n\naccusations of Jewish draft dodging by, 54\n\nacknowledgment of annihilation by, 156\n\non annihilation of Jews, 85, 124\n\nantisemitism of, 59\u201361, 63\u201365, 327\u201328\n\non German expansion, 79\u201380\n\non \"Germanization\" of Slavs, 208\n\n_Mein Kampf_ , 54\n\non Nazi morality, 62\n\npolitical platform of, 59\u201362\n\npolitical rise of, 55\u201359\n\non _Protocols of the Elders of Zion_ , 57\n\nT4 program initiated by, 117\u201318\n\nand violence toward Jews, 64\u201365\n\n_Hitler's Furies_ (Wendy Lower), 144\u201345\n\n_Hitler's Willing Executioners_ (Daniel Goldhagen), 137\n\n_Hiwis_ , 134, 143, 192, 307\n\nHlond, August, 246\n\nHoffman von Fallersleben, Heinrich, 41\n\nH\u00f6fle, Hans, 151, 281, 310\n\nHolland, 236\n\nHolocaust, public awareness of, 278\u201382\n\nHolocaust deniers, 330\u201332\n\nHolocaust scholarship, 324\u201343\n\nmyths debunked by, 327\u201332\n\npurpose of, 324\u201327\n\n_Holocaust vs. Wehrmacht_ (Yaron Pasher), 136\n\nhomosexuals, 206\u20137, 338\n\nhopelessness, of Jewish deportees, 209\n\nHorthy, Miklos, 225, 234, 298\n\nH\u00f6ss, Rudolf, 122, 123, 147, 152\u201353, 307\n\nHossbach, Friedrich, 79\n\nH\u00f6ssler, Franz, 307\n\nHotel Polski, 248\n\n_How the Jews Defeated Hitler_ (Benjamin Ginsberg), 179\n\nHudal, Alois, 312\u201313\n\nHundt-Radowsky, Hartwig von, 41\n\nHungary, 28\u201329, 56, 205, 230, 264, 275, 309\u201310, 322\n\ndeportations from, 135, 166, 174, 196, 224\u201326, 231\u201334, 294\n\nrescue in, 297\u201399\n\nIG Farben, 128, 164, 170, 294\n\nimmigration, _See_ Jewish emigration\/immigration\n\nImperial Germany, 38\u201353\n\nantisemitism in, 40\u201348\n\nJews in, 48\u201353\n\nnationalism in, 38\u201340\n\nindifference, of non-Jews, 71\u201372, 91\n\nindividual courage, 341\n\nindoctrination\n\nof German youth, 140\n\nof non-Jews, 94\u201396\n\nof senior officers, 147\u201348\n\nindustrialization, of nineteenth century Europe, 25\n\ninflation, in interwar Germany, 58\n\ninmate conditions, in concentration camps, 209\u201310\n\nintelligentsia, massacre of, 241\n\nInternational Jewish organizations, 295\u201396\n\ninternational response, to escalation of violence, 108\u201313\n\nInternational Settlement (Shanghai), 272\n\nintimidation, of non-Jews, 90\u201394\n\nIrving, David, 332\n\nisolationism, 339\n\nIsrael\n\ncreation of, 303\u20134\n\nand Middle Eastern politics, 334\u201336\n\nrestitution to, 317\n\nItaly, 31, 224,236\u201339, 274\u201375\n\nJanowska labor camp, 130\u201331\n\nJeckeln, Friedrich, 307\n\nJedwabne, 240, 249, 255\u201356\n\nJehovah's Witnesses, 205\n\n\"Jew Count,\" 53\u201354\n\nJewish Central Association, 105\n\nJewish Claims Conference, 317, 321\n\nJewish communities\n\ndeportation of Jews paid for by, 132\n\ndisunity of, 103, 271\u201372, 295\u201396\n\nin ghettos, 184\u201385\n\nJewish cooperation, 107\u20138\n\nJewish Councils of Elders, 179, 180, 187\u201391, 200\n\nJewish Courts of Honor, 200\n\nJewish discrimination\n\nat local level, 77\u201379\n\nat national level, 80\u201381\n\nas patriotic, 92\u201393\n\nJewish emancipation, 14\u201315, 20\u201324, 37\u201338\n\nJewish emigration\/immigration\n\nadvantages in, 105\u20136\n\nbanning of, 123\n\nand Central Association of German Citizens of Jewish Faith, 105\n\ndifficulties with, 272\u201373\n\nas goal of Zionist Jews, 103\u20134\n\nin Imperial Germany, 50\u201352\n\nto Palestine, 34\u201335, 264\u201365, 303\u20134\n\nin prewar Europe, 260\u201365\n\nas result of treatment in Nazi Germany, 83\n\nto United States, 265\u201372, 304\n\n\"Jewishness in Music\" (Richard Wagner), 40\n\nJewish refugees, 300\u2013305, 317\n\nJewish resistance, 176\u2013202\n\narmed, 178\u201379\n\nand German ghettoization of Jews, 179\u201383\n\nGerman punishment for, 108, 192\u201393\n\nand internal disunity in ghettos, 184\u201385\n\nand Jewish Councils of Elders, 187\u201391\n\nand Jewish postwar response to perceived collaborators, 200\u2013202\n\nlack of organized, 177\u201379\n\nin Lodz ghetto, 195\u201396\n\nmyths about, 328\n\nand physical condition of Jews, 191\u201392\n\nand resistance movements by other groups, 196\u201397\n\nand self-delusion, 185\u201387\n\nsurvival strategies practiced by, 193\u201395\n\nJewish segregation\n\nhistorical, 10\u201311\n\nby Nazi Party, 73\n\nin Nazi Party platform, 63\u201364\n\nin prewar Poland, 245, 247\u201348\n\nJewish success\n\nbacklash against, 27\u201331\n\nin Imperial Germany, 50\n\nJews\n\ncommunist, 184, 249\n\nexpulsion from Germany of, 80\u201384\n\nGerman public's attitude toward, 97\u2013102\n\nHasidic, 184\n\nhistorical narratives compiled by, 193, 214\n\nin Imperial Germany, 48\u201353\n\nin Nazi ideology, 63\u201364\n\nOrthodox, 103, 184\n\nphysical condition of, 191\u201392\n\npolitical views of Poles and, 249\u201350\n\npostwar response of, to perceived collaborators, 198\u2013202\n\nrecent scandals and political controversies involving, 337\u201338\n\nrescuers of, 219\u201324\n\nresponse of, to escalation of violence, 102\u20138\n\nrestitution to, 316\u201317\n\ntrying to escape \"underground,\" 156\u201357\n\nvilification of, in Middle Ages, 11\u201312\n\nviolence in Poland toward, 250\u201355\n\nZionist, 103\u20134, 184\n\nJews Department, 85, 180\n\nJodmin, Otto, 221\n\nJoint Distribution Committee, 261, 272, 279, 292, 295\n\nJoseph II, Holy Roman Emperor, 21\n\n_Judenz\u00e4hlung_ , 53\u201354\n\n_Jud S\u00fcss_ , 100\n\nJudt, Tony, xiv\u2013xv, 335\n\nKaltenbrunner, Ernst, 147, 153\u201354\n\nKamenets-Podolsk, 89\n\nKammler, Hans, 149, 154, 309\n\n_Kapos_ , 200, 203\u20134, 211, 213, 216\n\nKarski, Jan, _See_ Kozielewski, Jan\n\nKastner, Rudolf \"Rezso,\" 200\u2013201\n\nKenez, Peter, 232\n\nKertesz, Imre, 217\n\nKielce pogrom, 255, 302\n\n_Kindertransport_ , 263\n\nKlemperer, Victor\n\n_I Will Bear Witness_ , 91\n\nknowledge about Holocaust of, 155\u201356\n\nas _Mischling_ , 102\n\nKlukowski, Zygmunt, 251\n\n_Kommando_ units, 88\u201389\n\nKopelman, Moshe, 201\n\nKorherr, Richard, 331\n\nKossak, Zofia, 253\n\nKovno, 185, 191, 195, 201, 222, 228\n\nKozielewski, Jan, 250, 292\n\nKrakauer, Max, 157\n\n_Kristallnacht_ , 82\u201383, 99, 109, 261, 267\u201370\n\nKrumey, Hermann, 310\n\nKrupp, Gustav, 48, 92, 342\u201343\n\nKruszynski, Jozef, 247\n\nK\u00fchne, Thomas, 140\u201341\n\nlabor\n\ncompulsory, 83\n\nforced, _See_ forced labor\n\nslave, _See_ slave labor system\n\nlabor camps\n\nAuschwitz-Birkenau and Majdanek as, 128\n\ndefined, 130\u201331\n\nevacuation of, 170\u201375\n\nfor slave labor system, 166\u201370\n\nLaFarge, John, 275\u201376\n\nLages, Willy, 308\n\nLangbehn, Julius, 16\n\nLangbein, Hermann, _People of Auschwitz_ , 204, 254\n\nLange, Herbert, 119, 122, 126\n\nLanger, Lawrence, 180\n\nLangmuir, Gavin, 5\n\nlanguages\n\nhierarchical classification of, 17\u201318\n\nand nationalism, 38\u201339\n\nof Polish Jews, 247\n\nLanzmann, Claude, 240\n\nLatte, Konrad, 157\n\nLatvia, 206, 227\u201328, 304\n\nLaval, Pierre, 309\n\nLavater, Johann, 17\n\nLeague of German Students, 42\n\nLe Chambon-sur-Lignon, 220, 341\n\nLedochowski, Wlodzimierz, 275\u201376\n\nLensky, Mordechai, 184\u201385\n\nLevi, Primo, 212, 217\n\nLevine, Paul, 327\n\nL\u00e9vy-Hass, Hanna, 212\n\nliberal politics, 23\u201324, 333\n\nLichtenberg, Bernard, 290\u201391\n\n_Life Is Beautiful_ , 202\n\nLikely to Become a Public Charge (LPC) rule, 268\n\nLindemann, Albert, 16\n\nLithuania, 23, 114, 127, 178, 206, 227\u201329, 239, 264, 289\n\nLodz ghetto\n\ncreation of, 181\n\nfate of, 194\n\nhistorical narratives compiled at, 193\n\nJewish Council of Elders members of, 180\n\nJewish resistance in, 195\u201396\n\npopulation in, 182\u201383\n\nLong, Breckenridge, 270\u201371, 291\u201392\n\n_Longest Hatred, The_ (Robert Wistrich), 4\n\n\"long nineteenth century,\" 20\u201324\n\nLower, Wendy, _Hitler's Furies_ , 144\u201345\n\nLPC (Likely to Become a Public Charge) rule, 268\n\nLublin ghetto, 127, 181, 185, 193\n\nLudendorff, Erich, 54\n\nLueger, Karl, 31\n\nLustig, Walter, 201\u20132\n\nLuther, Martin, 12\n\nLutz, Carl, 298\u201399\n\nLviv, 131, 185, 200, 228, 288\n\nLyons, Joseph, 109\n\nMadagascar as destination for Jews, 86, 245,\n\nMajdanek concentration camp, 127\u201329, 193, 311, 329\n\nMann, Michael, 148\n\nMantello, George, 298\u201399\n\nMarie-Ben\u00f4it, Father, 286\n\nMarr, Wilhelm, 4, 31, 43, 47\n\nMarrus, Michael, 209\n\nMarx, Karl, 61\n\nMauthausen concentration camp, 120, 130, 163, 172, 174, 187, 210, 307, 309, 315, 331\n\nMcCloy, John, 293\u201394\n\nMcKale, Donald, _Nazis after Hitler_ , 306\n\n_Mein Kampf_ (Adolf Hitler), 54\n\nMengele, Joseph, 147, 310\u201311\n\nMiddle Ages, 11\u201312\n\nMiddle Eastern politics\n\nand Israel, 334\u201336\n\nin prewar period, 264\n\nand wartime aid from general public, 283\u201385\n\nMildner, Rudolf, 237\n\nmilitarization, of German life, 94\u201395\n\nMilgram, Stanley, 138\n\nminority groups\n\nand bigotry, 336\u201339\n\nand liberal politics, 333\n\nself-reliance vs. isolationism for, 339\n\nMinsk, 120\u201321, 185, 194\n\n_Mischlinge_ (partial Jewish ancestry), 101\u20132, 159\n\n_Mit brennender Sorge_ (With Burning Sorrow), 274\u201376\n\nMitchell, George, 339\n\nmodernization, 329\u201330\n\nMoll, Otto, 307\n\nMontagu, Edwin, 264\n\nMontini, Giovanni (Pope Paul VI), 312\n\nmorality, 341\u201343; _See also_ Nazi morality\n\nMorgenthau, Henry, 292\n\nmortality rates\n\nin concentration camps, 203\n\nin Polish ghettos, 182\u201383\n\nfor slave laborers, 169\u201370\n\nMuench, Aloisius, 314\n\nM\u00fcller-Oerlinghausen, Georg von, 343\n\nmunitions, 167\u201369\n\nMushkin, Eliyahu, 194\n\n_Musselm\u00e4nner_ , 211\u201312\n\nMussolini, 238\n\nNapoleon, 22\n\nnationalism\n\nin Hitler's political platform, 61\n\nin Imperial Germany, 38\u201340\n\nNational Representation of German Jews, 106, 112\n\nNational Socialist German Workers' Party (NSDAP), _See_ Nazi Party\n\nNational Union of Jews in Germany, 107\u20138\n\nNazi morality\n\nand antinomianism, 341\u201343\n\ndefining, 62\n\npromotion of, 7, 9\n\nand willingness to kill, 140\u201341\n\nNazi Party\n\nbehavior of public and policies of, 97\u201398\n\ncampaigning of, 68\u201370\n\nideology of, 62\u201365\n\nand Pope Pius XII, 285\u201387\n\npopular support for, 65\u201371\n\nNazis\n\ncollaboration in Eastern Europe with, 228\u201329\n\ncontrol in concentration camps by, 211\u201312\n\ndeportation of Jews by, 100\u2013101\n\nmanipulation of sense of duty by, 93\u201395\n\ntreatment of Jews in Poland by, 85\u201386\n\n_Nazis after Hitler_ (Donald McKale), 306\n\nNazis and Nazi Collaborators Law (Israel), 202\n\n_Neighbors_ (Jan Gross), 240\n\nNetherlands\n\nimmigration to, 261\u201362\n\nlack of resistance to deportations from, 196\n\nrestitution by, 320\n\nsurvival rates in, 236\n\nwar criminals tried by, 308\n\nNeuengamme concentration camp, 172, 173\n\nNeumann, Franz, 341\u201342\n\nNeuser, Richard, 142\n\nNirenberg, David, 12\n\nNisko, 86\n\nnon-Jewish victims\n\nof concentration camps, 204\u20138\n\nin Poland, 242\u201344\n\nNortheim, 69\n\nNorthwestern University, 175, 268\u201370, 324\n\nNorway, 220, 224, 297, 309\n\nNSDAP, _See_ Nazi Party\n\n_ODESSA (Organisation der ehemaligen SS-Angh\u00f6rigen)_ , 314\u201315\n\n_Odessa File, The_ (Frederick Forsyth), 314\n\nOliner, Samuel and Pearl, _The Altruistic Personality_ , 221\n\n\"On the Streets of Truth,\" 332\n\nOperation Harvest Festival, 192\n\nOperation Reinhard death camps\n\ndeath toll of, 126\u201327, 281\n\nplunder from, 131\n\nstaff of, 239, 311\n\n_Ordinary Men_ (Christopher Browning), 137\n\n_Organisation der ehemaligen SS-Angh\u00f6rigen_ (ODESSA), 314\u201315\n\nOrganization of Ukrainian Nationalists (OUN), 228\n\nOrthodox Jews, 103, 184\n\nOyneg Shabes, 193, 256\n\nPacelli, Eugenio, _See_ Pius XII, Pope\n\nPalatucci, Giovanni, 239\n\nPale of Settlement, 23, 125, 228\n\nPalestine\n\nunder British rule, 263\u201364\n\nimmigration to, 104\u20135, 265, 303\u20134\n\nPalmnicken, 171\u201372\n\nPanama Scandal, 30, 337\n\nPapen, Franz von, 71, 340\n\nParnes, Joseph, 187\n\n_Partisanenbek\u00e4mpfung_ , 89\n\nPasher, Yaron, _Holocaust vs. Wehrmacht_ , 136\n\npassivity\n\nAllied, 278\u201382\n\ndangers of, 340\n\nof non-Jews, 71\u201372, 91\n\nby Pope Pius XII, 289\u201390\n\nPatents of Toleration, 21\n\nPatton, George, 301\n\nPavelic, Ante, 313\n\nPehle, John, 293\u201394\n\n_People of Auschwitz_ (Hermann Langbein), 204\n\nPerechodnik, Calel, 185\u201386, 191, 249\n\nPerlasca, Giorgio, 299\n\nP\u00e9tain, Philippe, 225, 286, 288, 309\n\nPetri, Erna, 145\n\nPeukert, Detlef, 329\n\nphilology, 17\n\nphrenology, 17\n\nphysiognomy, 17\n\nPitt, William, 342\n\nPius XI, Pope, 275\u201376\n\nPius XII, Pope, 276\u201377\n\non clemency for war criminals, 313\u201314\n\nlack of resistance to Nazi regime by, 285\u201387\n\nreasons for passivity of, 289\u201390\n\n\" _Summi Pontificatus:_ On the Unity of Human Society,\" 291\n\nPlaszow concentraton camp, 202, 307\n\nPloetz, Alfred, 19\n\nplunder, from concentration camps, 131\u201332\n\npogroms, 255\n\nPohl, Oswald, 149, 154, 307\n\nPoland, 240\u201358\n\naid to Jews from general public in, 256\n\nantisemitism in, 244\u201347, 255\u201356\n\ndeath camps in, 125\n\nghettoization in, 181\u201382\n\nJewish citizenship in, 82\n\nJewish segregation in, 245, 247\u201348\n\nNazi treatment of Jews in, 85\u201386\n\nand political views of Jews and Poles, 249\u201350\n\nviolence toward Jews in, 250\u201355\n\nwar criminals tried by, 307\u20138\n\nwartime conditions in, 240\u201344\n\nPolice Battalion 101, 138\u201340\n\nPolish People's Army, 250\n\npolitical evolution, of annihilation, 121\u201325\n\npolitics\n\nantisemitism in Imperial Germany, 31\u201335\n\nexpulsion of Jews from, 76\u201377\n\nof Jews in Imperial Germany, 51\n\nliberal, 23\u201324, 333\n\nMiddle Eastern, _See_ Middle Eastern politics\n\nin nineteenth-century Europe, 26\n\nviews of Jews vs. Poles about, 249\u201350\n\nPoniatowa concentration camp, 193\n\nPope, Alexander, \"An Essay on Man,\" 13\n\npopulation density, of ghettos, 191\n\npopulation growth\n\nand German expansion, 84\n\nin Imperial Germany, 48\u201349\n\nin nineteenth-century Europe, 24\u201325\n\npostwar period, 300\u2013323\n\nescapes by war criminals in, 311\u201315\n\ngeneral restitution in, 319\u201323\n\nGerman restitution in, 316\u201319\n\nJewish refugees in, 300\u2013305\n\nrebuilding of Germany in, 315\u201316\n\nRoman Catholic Church in, 311\u201314\n\nwar crime punishments in, 305\u201311\n\n_Power and the Glory, The_ (Graham Greene), 277\u201378\n\nPrekerowa, Teresa, 221\n\nPretzel, Raimund, 94\n\nprewar aid to Jews from general public, 259\u201378\n\nAmerican immigration, 265\u201372\n\nEuropean immigration, 260\u201365\n\nand immigration difficulties of Jews, 272\u201373\n\nPalestinian immigration, 265\n\nand Roman Catholic Church, 273\u201378\n\nPreysing, Konrad von, 277\n\nPriebke, Erich, 312\n\nPrittwitz und Gaffron, Friedrich von, 92\n\nProdolliet, Ernest, 222\n\n_Protocols of the Elders of Zion_ , 56\u201357, 247, 269\n\nPr\u00fcfer, Curt, 155\n\npsychological explanations\n\nfor aid from general public, 220\u201322\n\nof German compliance, 137\u201342\n\npunishment(s)\n\nfor aid to Jews, 218\u201319\n\ncollective, 209\n\nfor Jewish resistance, 108, 192\u201393\n\nfor war crimes, 305\u201311\n\nQuisling, Vidkun, 309\n\nracial hygiene, 19\n\nRaczkiewicz, Wladislaw, 287\n\nRathenau, Walther, 53\n\nratlines, 312\n\nRauff, Walter, 122, 313\n\nRavensbr\u00fcck concentration camp, 161, 174, 203, 293, 309\n\nRecherchegruppe Henneicke, 225\n\nRed Cross, International Committee of, 101, 260, 281, 293, 312\n\nReder, Rudolf, 210\n\nReformation, 12\u201313\n\nReform Judaism, 49\n\nrefugees, 300\u2013305, 317\n\n_Reich, Das_ , 156\n\nReichenau, Walter von, 89\n\nReich Security Main Office (RSHA), 85, 146\u201348, 179\u201380\n\nReichstag, 37\n\nantisemites and elections to, 44\u201348, 54\u201355\n\nReichsvereinigung der Juden in Deutschland, _See_ National Union of Jews in Germany\n\nReichsvertretung der deutschen Juden, _See_ National Representation of German Jews\n\nrelationships, as survival tactic, 217\n\nreligion\n\nand aid from general public, 219\u201320\n\nantisemitism based on, 9\u201313\n\nresistance; _See also_ Jewish resistance\n\nto annihilation, 142\u201346\n\nin concentration camps, 210\u201311\n\nindividual, 341\n\nby public, 196\u201398\n\nrestitution, in postwar period, 319\u201323\n\nRetzius, Anders, 17\n\nRiegner, Gerhart, 280\n\nRiga, 121, 185, 228, 307, 312\n\nRingelblum, Emanuel, 256\n\nRoessler, Fritz, 93\n\nRoma and Holocaust, 205\u201306\n\nRoman Catholic Church, 10\u201311, 14, 27, 42, 260, 319\u201320\n\nand escape of war criminals, 311\u201314\n\nand euthanasia, 120\u201321, 286\n\nGerman massacre of Polish priests, 241\n\nin Poland, 241, 246\u201347, 253, 257\n\nand prewar aid from general public, 269, 273\u201378\n\nprotests by bishops, 220\n\nand wartime aid from general public, 219\u201320, 236, 285\u201391\n\nRomania, 239, 264\u201365, 283, 285, 303, 309\u201310, 319, 322\n\nlate emancipation in, 23\n\nmurders in or by, 224\u201326, 230, 234\u201336\n\nterritorial gains and losses, 231, 233\u201334\n\nRome, ancient, 5\u20136\n\nR\u00f6mer, Felix, 140\u201341\n\nRoosevelt, Franklin Delano\n\nand American immigration, 265\u201368, 270\u201371\n\nand Breckenridge Long, 291, 292\n\nRoschmann, Eduard, 312, 314\n\nRosenberg, Alfred, 124\n\nRosenberg, G\u00f6ran, 214\n\nRosenblatt, Leon, 188\n\nRosenman, Samuel, 268\n\nRosenstrasse protest, 158\u201359\n\nRossner, Alfred, 223\n\nRothmund, Heinrich, 262\n\nRowecki, Stefan, 249\u201350, 253\n\nRSHA (Reich Security Main Office), 85, 146\u201348, 179\u201380\n\nRumkowski, Chaim, 181, 189\u201390, 194, 198\n\nRussian empire, 23\n\nRust, Bernhard, 95\n\nSachsenhausen concentration camp, 83, 108, 119, 122, 161, 172, 174, 203, 295\n\nsacraments, of Catholic Church, 277\u201378\n\nsatellite concentration camps, 165\n\nSauckel, Fritz, 160\n\nSaxony, 46\u201348\n\nScavizzi, Pirro, 279\n\nSchacht, Hjalmar, 80\n\nSchindler, Oskar, 222\u201323\n\n_Schindler's List_ , 202, 307\n\nSchlegel, Friedrich, 17\n\nSchmelt, Albrecht, 164, 309\n\nSchmidt, Anton, 145\u201346\n\nSchulte, Eduard, 280\n\nSchumpeter, Joseph, 24\n\n_Schutzmannschaften_ , 89\n\n_Schwarze Korps, Das_ , 84\u201385\n\nsecularization, 26\u201327\n\nSegev, Tom ( _Soldiers of Evil_ ), 146\n\nsegregation, _See_ Jewish segregation\n\nself-delusion\n\nabout Holocaust, 280\n\nand lack of Jewish resistance, 185\u201387\n\nself-dignity, in concentration camps, 214\u201315\n\nself-interest\n\nof German public, 95\u201396\n\nof Jews in ghettos, 184\u201385\n\nself-reliance, 339\n\nsenior officers, roles of, 146\u201354\n\nSerbia, 75, 87, 90, 205, 224\n\nSeyss-Inquart, Arthur, 308\n\nShanghai, China, 272\n\nSheptytsky, Andrey, 228, 288\n\n_Shoah_ (documentary), 240\n\nshock of arrival, at concentration camps, 215\n\nSiemens, Carl Friedrich von, 92\n\nSierakowiak, Dawid, 182\n\nSilberklang, David, 194\u201395\n\nSinclair, Sir Archibald, 283\n\nSinti and Holocaust 205\u20136\n\nSiri, Giuseppi, 312\n\nsituational explanations\n\nof the Holocaust, 339\u201340\n\nof perpetrator behavior, 137\u201338\n\nSkarzysko-Kamienna labor camp, 166, 167, 250\n\nskin color, 16\u201317\n\nslave labor system, 160\u201375\n\ndevelopment of, 161\u201363\n\nand evacuation of labor camps, 170\u201375\n\nexpansion of, 164\u201366\n\nand forced vs. slave labor, 160\u201361\n\nmyths about, 329\n\nPolish labor camps in, 166\u201370\n\nrestitution for, 318\n\nSlavs, 208\n\nSlovakia, 224, 226, 231, 279, 285, 288\u201389, 309\n\nsmuggling, in concentration camps, 215\u201316\n\nSobibor concentration camp, 134, 253, 307, 311\u201313\n\ndeath toll of, 126\u201327, 281\n\nescapes from, 211\n\nuprising at, 192, 250\n\nsocial class, in ghettos, 184\u201385\n\nsocial groups, Jews banned from, 78\n\n_Sonderkommando_ 1005, 307\n\nSontag, Susan, 339\n\nSousa Mendes, Aristides de, 221\u201322\n\nSouth America, 312\u201315\n\nSoviet prisoners of war\n\nescaped from Mauthausen concentration camp, 210\n\nfood for, 90\n\nlack of resistance among, 196\n\nas volunteer guards at concentration camps, 134\n\nSoviet Union, _See_ USSR\n\nSpellman, Francis, 312\n\nspies\n\nin concentration camps, 210\n\nfear of, 270, 271\n\n_St. Louis_ (ship), 273\n\nStab in the back legend, 60, 79\n\nStalin, Joseph, 282\n\nStangl, Franz, 312\u201313\n\nStangneth, Bettina, _Eichmann Before Jerusalem_ , 152, 332\n\nStarachowice labor camp, 166\u201367\n\nStepinac, Aloysius, 288\n\nStern, Samu, 195\n\nStoecker, Adolf, 43\n\nStone, Dan, 330\n\nStrasser, Gregor, 66\n\nStreicher, Julius, 99\n\n_Stroop Report, The_ , 198\u2013200\n\nStroop, J\u00fcrgen, 307\n\nStrousberg, Bethel Henry, 42, 337\n\n_St\u00fcrmer, Der_ , 99\n\nSugihara, Chiune, 222\n\nSuhard, Emmanuel Celestin, 288\n\n\" _Summi Pontificatus:_ On the Unity of Human Society\" (Pope Pius XII), 291\n\nsurvival\n\nin concentration camps, 212\u201317\n\nas Jewish resistance, 193\u201395\n\nsurvival rates, 218\u201339\n\nand aid from general public, 218\u201324\n\nin Baltic states and Ukraine, 227\u201329\n\nin Denmark, 236\u201338\n\nin Hungary, 231\u201334\n\nin Italy, 236\u201339\n\nfor native vs. foreign-born Jews, 229\u201331\n\nby region, 224\u201327\n\nin Romania, 234\u201336\n\n_Survivor, The_ (Terrence Des Pres), 214\n\nSweden, 23, 237\u201338, 296\u201399\n\nSwitzerland\n\nemancipation in, 23\n\nimmigration to, 262, 296\n\nrestitution by, 320\u201322\n\nSzalasi, Ferenc, 309\u201310\n\nSzerynski, Joseph, 190\n\nTacitus, 5\n\nTec, Nechama\n\n_Dry Tears_ , 255\n\n_When Light Pierced the Darkness_ , 220\n\nTheresienstadt concentration camp, 101\u20132, 108, 136, 158\u201359, 196, 200, 203, 309\n\nTiso, Jozef, 289, 309\n\nT4 program, 117\u201321, 134\n\npersonnel in, 142, 147\u201348, 309, 311\u201312\n\ntrain traffic, 134\u201336\n\nTransfer Agreement, 104\u20135\n\ntransit camps, 130\n\nTransnistria, 235\u201336\n\ntransportation, 26\n\nTrawniki concentration camp, 134\n\nTreblinka concentration camp\n\ndeath toll of, 126\u201327\n\nescapes from, 210\u201311\n\nJewish resistance in, 177, 192\n\nas labor camp, 163\n\nPolish communities profiting from, 254\n\nTreitschke, Heinrich von, 43\u201344\n\nTrieste, 239\n\nTrocm\u00e9, Andr\u00e9, 341\n\nTruman, Harry S., 302\n\nTucholsky, Kurt, 155\n\nTwenty-five Point Program, 63\u201364\n\nU-boats (underground Jews), 156\u201357\n\nUkraine, 128, 219\u201320\n\ncollaboration in, 227\u201329, 303, 330\n\nJews in, 89, 197, 239\n\nkilling in, 122, 224, 230\n\n_Uncompromising Generation, An_ (Michael Wildt), 146\n\nunification, of Germany, 36\u201338\n\nUnited Nations Relief and Rehabilitation Administration (UNRRA), 300\u2013303\n\nUnited States\n\nantisemitism in wartime, 291\u201392\n\nimmigration policy of, 266\n\npostwar immigration to, 304\n\nprewar immigration to, 265\u201372\n\nwar criminals tried by, 307\n\nwartime aid to Jews , 291\u201395\n\nunity, 70\n\nurbanization, 25\n\nurban migration, of German Jews, 49\u201350\n\nUris, Leon, _Exodus_ , 305\n\nUSSR\n\nimmigration to, 263\n\nwar criminals tried in, 307\n\nwartime aid to Jews , 282\u201383\n\nValeri, Valerio, 288\n\nvan Tijn, Gertrude, 222\n\nVazsonyi, Adam, 31\n\nVersailles Treaty, 57\n\nVichy France, 206, 225, 229\u201330, 286, 288, 309\n\nVilna, 145, 186, 191\u201392, 194, 228\u201329\n\nVinnytsia, 89\n\nviolence\n\nescalation of, _See_ escalation of violence\n\nin Nazi platform, 64\u201365\n\nvolitional explanations, of perpetrator behavior, 137\u201338\n\n_Volk_ , 38\u201340\n\n_V\u00f6lkischer Beobachter_ , 156\n\n_Volksdeutsche_ , 87, 116, 143, 304\n\n_Volksgemeinschaft_ , 70, 95\n\nVolkswagen, 164\u201365, 318\n\nVoltaire, 14\n\nvolunteer guards, at concentration camps, 134\n\nvom Rath, Ernst, 82\n\nVrba, Rudolf, 213\n\nWachsmann, Nikolaus, 309\n\nWagner, Gustav, 313\n\nWagner, Richard, \"Jewishness in Music,\" 40\n\nWagner-Rogers Bill, 268, 271\n\nWalesa, Lech, 256\n\nWallenberg, Raoul, 292\u201393, 298\n\nWannsee Conference, 115, 119, 124-25, 164, 331\n\nwar crimes, punishments for, 305\u201311\n\nwar criminals, escapes by, 311\u201315, 329\n\nwar effort\n\neffect of annihilation, on, 131\u201336, 328\u201329\n\nGerman women in, 144\u201345\n\nWar Refugee Board, 292\u201393\n\nWarsaw ghetto, 182, 188, 193, 307\n\nchildren in, 198\u2013200\n\nconsequences of uprising in, 192, 195, 310\n\neventual fate of, 194\n\nghetto police in, 190\n\nhistorical narratives compiled at, 193\n\nmortality rates in, 191\n\nPolish aid to, 253\n\nsealing of, 181\n\nsurvival in, 243, 248\n\nWarsaw Rising, 243, 253\n\nWarthegau (annexed Poland), 122, 181, 241, 307\n\nwartime aid to Jews from general public, 278\u201399\n\nand Allied passivity, 278\u201382\n\nInternational Jewish organizations, 295\u201396\n\nand Middle Eastern politics, 283\u201385\n\nand Roman Catholic Church, 285\u201391\n\nin Sweden, 296\u201399\n\nin Switzerland, 296\n\nin United States, 291\u201395\n\nin USSR, 282\u201383\n\nWasserstein, Bernard, 273\n\nwealth, Jewish immigration and, 105\n\nWeidt, Otto, 223\u201324\n\nWeimar Republic, 59\n\nWeiss, Theodore Zev, 175, 213\u201314\n\nWeizmann, Chaim, 273\n\nWeizs\u00e4cker, Ernst von, 84, 93\n\nwelfare states, 334\n\nWelzer, Harald, 140\u201341\n\nWestermann, Edward, 139\u201340\n\nWestern Europe, 319\u201322\n\nWesterbork transit camp, 132, 187, 203, 262, 308\n\n_When Light Pierced the Darkness_ (Nechama Tec), 220\n\nWildt, Michael, _An Uncompromising Generation_ , 146\u201347\n\nWilhelm II, Kaiser, 54\n\nWilhelmina, queen of Netherlands, 290\n\nWilmanns, Carl, 43\n\nWinstone, Martin, 242\n\nWirth, Christian, 121\u201322\n\nWise, Stephen, 271, 295, 363\n\nWistrich, Robert, _The Longest Hatred_ , 4\n\nwith-and-against behavior, 215\u201317\n\nWolski, Mieczyslaw, 256\n\nWorld War I, 53\u201354\n\n_World Without Jews, A_ (Alon Confino), 7, 9\n\nxenophobic antisemitism, 5\u20136\n\nYad Vashem, 219, 256\n\nYaffe, Moshe, 194\n\n_Yishuv_ , 104, 295\u201396\n\nZegota, 253\u201354\n\nZiimbardo, Philip, 138\n\nZionism, 34\u201335, 264, 292, 304\n\nZionist Jews, 103\u20134, 184\n\nZola, \u00c9mile, 33\n\nZwartendijk, Jan, 222\n\nZyklon, 93, 122\u201323, 127, 130, 133, 172, 280, 307, 308, 331\nALSO BY PETER HAYES\n\n_How Was It Possible? A Holocaust Reader_ (editor)\n\n_Frankreichforum XI: Universit\u00e4tskulturen\/L'Universit\u00e9 \nen perspective\/The Future of the University_ (coeditor)\n\n_The Oxford Handbook of Holocaust Studies_ (coeditor)\n\n_Das Amt und die Vergangenheit: Deutsche Diplomaten im \nDritten Reich und in der Bundesrepublik_ (coauthor)\n\n_From Cooperation to Complicity: Degussa in the Third Reich_\n\n_The Last Expression: Art and Auschwitz_ (coeditor)\n\n_\"Arisierung\" im Nationalsozialismus: Volksgemeinschaft, \nRaub und Ged\u00e4chtnis_ (coeditor)\n\n_Lessons and Legacies III: Memory, Memorialization, \nand Denial_ (editor)\n\n_Lessons and Legacies I: The Meaning of the Holocaust \nin a Changing World_ (editor)\n\n_Industry and Ideology: IG Farben in the Nazi Era_\n\n_Imperial Germany_ (coeditor)\nCopyright \u00a9 2017 by Peter Hayes\n\nAll rights reserved\n\nFirst Edition\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 Marysarah Quinn\n\nJacket design by Evan Gaffney\n\nJacket photograph by Paula Salischiker \/ Millenium Images, UK\n\nDescription: Young girl reading list of names of Holocaust victims inscribed on wall. Shoah Memorial, Paris, France\n\nProduction manager: Anna Oler\n\nThe Library of Congress has cataloged the printed edition as follows:\n\nNames: Hayes, Peter, 1946 September 7\u2013 author \nTitle: Why? : explaining the Holocaust \/ Peter Hayes. \nDescription: First edition. | New York ; London : W. W. Norton & Company, \nindependent publishers since 1923, [2017] | Includes bibliographical \nreferences and index. \nIdentifiers: LCCN 2016031588 | ISBN 9780393254365 (hardcover) \nSubjects: LCSH: Holocaust, Jewish (1939\u20131945) | Holocaust, Jewish \n(1939-1945)\u2014Causes. | Antisemitism\u2014Germany\u2014History\u201420th century. | \nJews\u2014Germany\u2014History\u201420th century. | Jews\u2014Persecutions\u2014 \nEurope\u2014History\u201420th century. | Germany\u2014History\u20141933\u20131945. | \nGermany\u2014Ethnic relations. \nClassification: LCC D804.3 .H387 2017 | DDC 940.53\/1811\u2014dc23 LC \nrecord available at https:\/\/lccn.loc.gov\/2016031588\n\nISBN 978-0-393-25437-2 (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":" \nTHE NEW QUISLINGS\n\nHow the International Left Used the Oslo Massacre to Silence Debate About Islam\n\nBruce Bawer\n\nContents\n\nThe New Quislings\n\nAbout the Author\n\nCopyright\n\nAbout the Publisher\nThe New Quislings\n\nI\n\nOn the morning of Friday, July 22, 2011, I was in a friend's house in the United States chatting on Skype with my partner in Norway. \"Oh my God,\" he suddenly said. \"There's been an explosion in Oslo.\"\n\nHe had a newspaper website open. I went at once to the same site and saw a giant headline and a horrific picture. I immediately opened the website of Norwegian state television, NRK, and began to watch its live coverage online.\n\nThe images of devastation were staggering. A government building, one of the tallest structures in Oslo, had sustained major damage, and the streets around it were filled with debris. There were reports of casualties, though the numbers were, as yet, unknown. The explosion had been so powerful that windows had been blown out of stores and offices blocks away.\n\nI was stunned. The government building that had been damaged was right down the street from where I had, until recently, lived. I had passed it almost every day for years, either on foot or on a bus. It was numbing to see Oslo, my longtime home, suffering a fate so similar to that which my native city, New York, had suffered in 2001.\n\nThe first thing I did was to contact my friends in Oslo, to ensure they were all okay. They were, although a couple of them had been very close to the explosion when it took place, and several of them had felt the power of the blast, even from some distance away.\n\nThen I began to look at every Norwegian news website I could think of, and watched NRK, Norwegian TV2, CNN, and Al Jazeera online. (I didn't have a TV at hand.) Within an hour, it was confirmed that the explosion had been the result of a bombing. Initial reports said that it bore all the earmarks of a jihadist attack. Nobody publicly disputed this conclusion.\n\nThen, suddenly, came reports of another event. Shots had been fired a half hour or so west of Oslo, on an island called Ut\u00f8ya, where the annual summer camp for Workers Youth League\u2014the Labor Party youth organization\u2014was under way. The first details were sketchy. Young campers on the island, apparently, were telephoning their parents and begging them frantically to call the police. A gunman, they said, was shooting their friends down in cold blood. When he had first come ashore, wielding a huge gun, he had pretended to be a cop, come to safeguard them in the wake of the explosion in Oslo. Then he had started firing at unarmed teenagers. According to an article by \u00c5sne Seierstad published weeks later in Newsweek, he \"shout[ed] 'Hurray!' 'Bull's-eye!' or 'Got you!' as he slew his victims.\" Kids were running, hiding in the woods, hysterical, in shock. There was nobody there with a gun to protect them, and no easy means of getting off the island. One minute they had been living in a pastoral idyll: the next minute they had been plunged into a nightmare. And they had no idea why it was happening.\n\nSince I had lived in Norway for many years and had written a great deal about Islam, my inbox soon began to fill with emails from editors asking me to write about this atrocity. I agreed to submit pieces both to the Pajamas Media (now PJMedia) website and to the Wall Street Journal opinion page. I began working on the Pajamas piece while listening to the Norwegian news and crying incessantly.\n\nI had already finished a draft when the news came that the attacks were not, after all, the work of jihadists. Instead, the perpetrator was an ethnic Norwegian named Anders Behring Breivik, who claimed that his actions were motivated by anti-jihadist sentiment.\n\nIn the piece I ended up sending to Pajamas Media I noted that one Norwegian newspaper had observed that the July 22 death toll was higher than at Columbine and Virginia Tech combined. \"The Norwegian media,\" I pointed out,\n\nhave always reported on mass murders by lone gunmen in the U.S. as if they were things that could never happen in Norway: rather, they were symptoms of a sick society that Norwegians could never possibly understand. In Norway, they use the term \"amerikanske tilstander\"\u2014American conditions. It never means anything good. Yesterday's nightmare, from a Norwegian perspective, was the most American of American conditions.\n\nI also wrote that while virtually everybody had assumed at first that the attacks in Oslo were the work of jihadists, \"it would've been just plain dumb for Islamists to make an enemy of Norway,\" given that the Norwegian government and cultural elite had been making friendly gestures for years to even the most extreme elements of Islam: they'd treated their Jews shabbily, coddled resident terrorist Mullah Krekar, squelched domestic criticism of Islam, dropped Muslim riots down the memory hole, and openly supported terrorist groups. I concluded the piece as follows:\n\n. . . it is deeply depressing to see this evil, twisted creature become the face of Islam criticism in Norway. Norwegian television journalists who in the first hours of the crisis were palpably uncomfortable about the prospect of having to talk about Islamic terrorism are now eagerly discussing the dangers of \"Islamophobia\" and \"conservative ideology\" and are drawing connections between the madness and fanaticism of Breivik and the platform of the Progress Party. Yesterday's events, then, represent a double tragedy for Norway. Not only has it lost almost one hundred people, including dozens of young people, in a senseless rampage of violence. But I fear that legitimate criticism of Islam, which remains a very real threat to freedom in Norway and the West, has been profoundly discredited, in the eyes of many Norwegians, by association with this murderous lunatic.\n\nAs the day wore on, it quickly emerged that Breivik had been an avid reader of a website called document.no, where he had posted a number of comments.\n\nThere are a couple of major websites that regularly address Norway's immigration and integration policy and its attendant problems. One of them is rights.no, the site of Human Rights Service, a small Oslo-based think tank for which I have worked on and off for several years as a writer, editor, translator, and consultant. Their focus is on the rights of women and girls in Norway and Europe, especially in Muslim communities, and their mission is to develop proposals for new laws and government programs. HRS's information director, Hege Storhaug, has appeared countless times on Norwegian TV debate programs, and has become a very familiar\u2014and polarizing\u2014figure; while many ordinary citizens have relished her bluntness about the failures of Norwegian immigration and integration policies, members of the cultural elite have tended to balk at her blithe violation of long-standing boundaries as to what can and cannot be said. For years, multiculturalists who frown on any mention of Muslim community problems have savaged HRS as racist and \"Islamophobic\" and have battled to remove its government funding. HRS's website features regular news commentaries in Norwegian and English by Hege and managing director Rita Karlsen; it has also published original articles about Islam, immigration, and integration by contributors from around the world, such as Robert Redeker in France and Henryk Broder in Germany.\n\nAnother website that addresses immigration and related issues is document.no, edited by my friend Hans Rustad. Unlike HRS's website, document.no is not connected to any larger organization, and its focus is not on the rights of Muslim women and girls (although this is certainly among its concerns) but on the threat that unreconstructed Islam and failed immigration and integration policies pose to the West. Like rights.no, it is a serious, intelligent, and respectable site that respects the facts and has never dealt in vulgar Muslim-bashing. Unlike HRS's website, it allows readers to post comments on its articles.\n\nWhen I looked at document.no, I found that Hans, in reaction to the atrocities, had already compiled all of the comments Breivik had ever posted on the site, forming a useful package for the edification of journalists and anyone else who was interested. The first thing I did was to search Breivik's comments, which had been posted between 2007 and 2009, for my name. It came up three times.\n\nOn September 14, 2009, apropos of the need to form an alliance between anti-jihadists and cultural conservatives, Breivik had written: \"Bawer is probably not the right person to work as a bridge-builder. He is a liberal anti-jihadist and not a cultural conservative in many areas. I have my suspicions that he is TOO paranoid (I am thinking of his homosexual orientation). It can seem that he fears that 'cultural conservatives' will become a threat to homosexuals in the future. He refuses therefore to take the opportunity to influence this in a positive direction. This seems entirely irrational.\"\n\nOn October 31, 2009, Breivik wrote that several things needed to be done in the next twenty years to prevent the Islamization of Norway, among them: \"Initiate a collaboration with the conservative forces in the Norwegian church. I know that the libertarian forces in the European anti-jihad movement (Bruce Bawer among others, and some other libertarians) will have a problem with this, but conservative forces in the church are in fact one of our best allies. Our main opponents must not be jihadists but the jihadists' facilitators\u2014namely the multiculturalists.\" And on November 6, 2009, he wrote: \"It is tragicomic that an important NGO like Human-Etisk Forbund [the Norwegian Humanist Association] has been taken over by a cultural Marxist when it should be run by a liberal anti-jihadist like Bruce Bawer.\"\n\nTo discover that this murderer knew who I was and had read my work filled me with a feeling that is hard to describe. As a professional writer for almost three decades, I have met or received communications from hundreds if not thousands of my readers, and while most of them have been very nice, there has always been a sprinkling of nuts. When you're a writer you never know who may be reading you. You get used to the idea. But this was new territory for me. I was chilled\u2014sickened.\n\nIt was interesting to note that, while Breivik preferred me to a \"cultural Marxist,\" he still found me too liberal for his tastes.\n\nAs many people on the left don't realize, there is a very broad range of views among the critics of Islam.\n\nStill, until Breivik came along, it had all been about debate. The violence had all been on the side of the jihadists\u2014the major right-wing extremists of our time\u2014and guilt had stained their apologists on the left. Now the tables were turned. Someone who claimed to be, broadly speaking, on my side\u2014the anti-jihadist side\u2014had committed a massive atrocity.\n\nThen it emerged that Breivik had written a 1,500-page manifesto which he had e-mailed to hundreds of recipients only moments before setting out on his murder spree. It was online. I found it easily. At the outset, Breivik summed up his argument:\n\n. . . the root of Europe's problems is the lack of cultural self-confidence (nationalism). Most people are still terrified of nationalistic political doctrines thinking that if we ever embrace these principles again, new \"Hitler's\" will suddenly pop up and initiate global Armageddon. . . . Needless to say; the growing numbers of nationalists in W. Europe are systematically being ridiculed, silenced and persecuted by the current cultural Marxist\/multiculturalist political establishments. This has been a continuous ongoing process which started in 1945. This irrational fear of nationalistic doctrines is preventing us from stopping our own national\/cultural suicide as the Islamic colonization is increasing annually. This book presents the only solutions to our current problems.\n\nThe book made for exceedingly creepy reading. Well, not the first half\u2014the first half was, in large part, a surprisingly sane-sounding take on modern society and politics. Had the killer actually written this? If so, I thought after skimming through it, he was a very well-read and thoughtful mass murderer indeed. Later, a closer perusal revealed that much of the book consisted of texts that Breivik had borrowed from various writers. Indeed, in a passage I had missed on my first read-through, Breivik acknowledged that he had \"written approximately half of the compendium myself\" and that the rest was \"a compilation of works from several courageous individuals throughout the world.\" Many of these borrowed works were credited to their authors; others were not.\n\nThe book's first few pages, for example, which linked political correctness to \"cultural Marxism,\" turned out to be the text of a 2004 Free Congress Foundation pamphlet titled Political Correctness: A Short History of an Ideology, by William Lind. This led off an introductory section in which Breivik went on to discuss the domination of Western Europe by political correctness; trace the development of PC to the rise of critical theory and the Frankfurt School of philosophers (each of whom he profiled at some length); and describe the assault on Western values by Jacques Derrida and radical feminism.\n\nThen came Book One, about \"our falsified history.\" He covered the current whitewashing of the history of Islam, served up some of the basics of Islam (sharia law, jihad, al-taqiyya, Koranic abrogation, dhimmitude), and quoted passages about Islamic history and theology from Robert Spencer, Walid Shoebat, Serge Trifkovic, and Bat Ye'or. He recounted the history of the Hindu Kush; the Crusades; the Ottoman Empire and the Armenian genocide (here he included an article by Andrew G. Bostom); the fall of Christian Lebanon; the defeat of the \"first Islamic wave\" by Charles Martel at the Battle of Tours in 732 (here he quoted dozens of historians); and the defeat of the \"second Islamic wave\" at the Battle of Vienna in 1683. He cut-and-pasted an article by Baron Bodissey about Crusader heroes and one by the pseudonymous Norwegian essayist \"Fjordman\" on Western versus Islamic science; wrote about Bosnian history (quoting at length from the Encyclopaedia Britannica, a Time-Life book about the Balkans, and other sources); and reprinted a speech by Bat Ye'or about Yugoslavia and an article by Daniel Pipes titled \"Palestine for the Syrians?\"\n\nIn Book Two, about \"Europe's current problems,\" Breivik moved from history to the present day. He wrote about jihad in today's Muslim and Western world. The section included many complete essays by \"Fjordman\" about such topics as media bias, the relative birthrates of native Europeans and Muslims, the role of the European Union, the United Nations, and feminism in aiding the Islamization of the West, the unreliability of moderate Muslims, Europe's demographic crisis, the capitulation of Europe to sharia law, and Norway's anti-discrimination act.\n\nWhich brought us to the halfway point of the manifesto\u2014and to Book Three, \"A Declaration of Pre-Emptive War.\" It was thoroughly, and stunningly, different from everything that had preceded it. It was, quite explicitly, the work of a madman. There was no smooth transition, either. One minute Breivik was writing seriously\u2014or cobbling together the work of other people who had written seriously\u2014about various strains of modern Western thought and their consequences for liberal democracy and individual freedom.\n\nThe next minute he was talking about killing people.\n\nYes, that was what the second half was about. Killing people. Where to get weapons and ammo. Where to acquire body armor. How to commit acts of terrorism. How many people to kill.\n\nBreivik claimed to be part of a movement\u2014a revival of the medieval Knights Templar\u2014that had re-formed that band of brothers with the goal of saving Europe from its fate. And how would they save it? By killing\u2014killing on a massive scale. And not by killing Muslims, but by killing the members of Norway's left-wing political class and, even more monstrously, their children. For his chief enemy was not Muslims themselves\u2014it was the socialist politicians whose multiculturalism had shaped the immigration and integration policies of modern Norway (and of modern Western Europe as a whole) and their comrades-in-arms, the left-wing academics, journalists, authors, and others who had abetted what he saw as their treason.\n\nThis second half of his manifesto, then, was addressed in particular to his fellow Knights, with an eye to preparing them to commit violence against the multiculturalist enemy.\n\nTo this end, he listed the preferred \"equipment for urban operations,\" including \"HK416 assault rifle with 'redpoint' optics (4 extra long clips),\" a \"Glock handgun with silencer and laser (2\u20134 extra long clips),\" \"3 splint grenades,\" \"1\u20132 shock grenade,\" \"2 x arm defensive devices (knives),\" a \"Gas mask,\" and \"Ammo (clip administration).\" He went into great detail about the different kinds of body armor, which provide various levels of protection against different types of ammunition. He recommended \"a small police version shield. . . . An optimal size for our purpose (and provided you have leg armour) would be any size from 50 x 50 to 70 x 70 cm, although most shields on the market are larger. You may have to re-equip with handles, a carrying strap (so you may carry it on your back), police insignia, one or two 10 cm spikes and\/or razorblades on the front of the shield (primarily as a deterrent to prevent people from jumping you from behind).\"\n\nIn later passages, he explained in great detail how to make bombs, how to obtain and deploy weapons of mass destruction, and how to acquire anthrax. He listed the nuclear reactors in Europe, all of them potential targets for the Knights' operations. And he offered strategic advice, which he himself would follow on July 22: \"Make a sound in the east, then strike in the west.\" (Ut\u00f8ya is west of Oslo.) He elaborated: \"In any battle the element of surprise can provide an overwhelming advantage. Even when face to face with an enemy, surprise can still be employed by attacking where he least expects it. To do this you must create an expectation in the enemy's mind through the use of a feint.\" In short: \"get the enemy to focus his forces in a location, and then attack elsewhere which would be weakly defended.\"\n\nHe also offered advice on how to choose targets. He divided the multiculturalist \"traitors\" into three categories, A, B, and C, noting that such traitors could be found in heavy concentrations at \"[a]nnual gatherings for journalists\" and at \"[l]iterature conferences and festivals.\" He did not mention the Labor Party summer camp at Ut\u00f8ya.\n\nIn the first part of his book, then, Breivik presented himself as a man concerned about the survival of Western civilization. It was, essentially, a thumbs-up to much of the best that has been thought and said, from the Enlightenment on, about the nature of freedom and of tyranny. The Breivik of these pages would have the reader believe that he was a true disciple of those great men and women who stood for liberal values against the encroachments of totalitarianism, religious or otherwise.\n\nThe second half of his book was an utter betrayal of the first\u2014a leap from civilized reason into the depths of barbarism, from logical sanity into pure madness.\n\nMuch has since been made of Breivik's references to this or that writer. In fact, his manifesto is a veritable phone book of names. Among the hundreds of people whom he mentions or quotes are Mark Twain, George Orwell, John Stuart Mill, John Locke, Henry David Thoreau, Bernard de Clairvaux, Bernard Lewis, Ayaan Hirsi Ali, Christopher Caldwell, Samuel P. Huntington, and Mohandas Gandhi. He quotes Andrew Jackson: \"One man with courage makes a majority.\" Winston Churchill: \"A man does what he must\u2014in spite of personal consequences, in spite of obstacles and dangers and pressures\u2014and that is the basis of all human morality.\" Thomas Jefferson: \"The tree of liberty must be refreshed from time to time with the blood of patriots and Tyrants.\" Robert Frost: \"Don't ever take a fence down until you know why it was put up.\" Winston Churchill again: \"An appeaser is one who feeds a crocodile, hoping it will eat him last.\" Hillel the Elder: \"If not us, who? If not now, when?\" He quotes the opening sentences of the Declaration of Independence and a couple of dozen militaristic verses from the Old Testament. Apropos of his need for prayer on the fast-approaching day of his \"mission,\" he cites the Twenty-Third Psalm (\"Though I walk through the valley of the shadow of death, thou art with me\") and the gospels (\"I am the resurrection and the life. He that believeth in me though he were dead, yet shall he live\"). And he lists his favorite books: Nineteen Eighty-Four, Leviathan, On Liberty, Essay Concerning Human Understanding, The Wealth of Nations, Reflections on the Revolution in France, Atlas Shrugged, The Fountainhead, Pragmatism, On War, and Fjordman's own immense, self-published manifesto Defeating Eurabia.\n\nTo this list one might add the manifesto of the Unabomber, Ted Kaczynski\u2014for, as it turned out, much of Breivik's book was in fact lifted from the Unabomber's. (Indeed, given that Kaczynski alone, of all the people whose writings are quoted by Breivik, actually came anywhere near to proposing or executing the kind of \"solution\" Breivik carried out on July 22, it would seem reasonable to suggest that Kaczynski was the only one of the hundreds of people mentioned in Breivik's text who can truly be considered an influence on his actions.)\n\nAnd what about me? The first thing I did when I found Breivik's manifesto online was to search for my name. I got twenty-two hits, and though I saw that some of these hits were accounted for by appearances of my name in essays lifted from \"Fjordman,\" I spent the next several weeks under the impression that Breivik himself had also quoted me a number of times. As we shall see, many members of the Norwegian cultural elite quickly embraced the idea that Breivik had quoted me frequently in his book, and that I therefore loomed large among his supposed influences\u2014for which, they made clear, I deserved contempt and condemnation.\n\nWhen I went carefully through Breivik's manifesto at the end of August, however, I discovered that every time my name appears in his book, it is in one or another of the essays by \"Fjordman.\" \"Fjordman\" quoted me on multiculturalism and its consequences, on Pim Fortuyn and Theo van Gogh, on jizya (the tax levied on infidels living in Muslim countries), the fact that some immigration-related issues are not open to debate in Sweden, on editor Vebj\u00f8rn Selbekk's apology for reprinting the Muhammad cartoons in his tiny Christian newspaper Magazinet, on a bombing outside the Danish embassy in Islamabad in retaliation for the Muhammad cartoons, on Norwegian novelist Dag Solstad's distaste for free speech, on the targeting by left-wing thugs of members of the Sweden Democrats Party, on Danish author Lars Hedegaard's grim prediction of Europe's future, and on the censorship of European media. But Breivik himself, it turned out, never once mentioned me in his manifesto.\n\nWho is Anders Behring Breivik? What makes him tick? His manifesto includes a Q&A in which he discusses his childhood and family. His father, a diplomat in London (and, later, Paris) \"had three children from a former marriage\"; his mother had \"a daughter from a past relationship.\" They divorced when he was a year old, and he moved back with his mother and stepsister to Oslo, where she married an army captain. \"My parents,\" he writes, \"were not politically active but supported the policies of the Norwegian Labour Party which was common for most individuals working in the public sector.\" His father broke off contact with him when Anders was fifteen because the boy had become involved with kids who were into hip-hop music and who committed graffiti vandalism. Anders tried to contact him five years ago but \"he said he was not mentally prepared for a reunion due to various factors, his poor health being one.\" Despite this nonrelationship with his father, Breivik writes,\n\nI feel I have had a privileged upbringing with responsible and intelligent people around me. I do not approve of the super-liberal, matriarchal upbringing though as it completely lacked discipline and has contributed to feminise me to a certain degree.\n\nBreivik had many Muslim friends growing up. In his Q&A, he asks himself: \"Why did you have so many non-ethnic Norwegian friends?\" The answer: \"pride and certain moral codexes\/principles have always been very important to me\"; in times of trouble, \"I expected my friends to back me up 100% without submitting or running away. . . . Very few ethnic Norwegians shared these principles. They would either 'sissy out,' allow themselves to be subdued or run away when facing a threat.\" So he hung around with Muslim kids who \"shared these principles of pride.\"\n\nNot that having Muslim friends saved Breivik from run-ins with Muslims. He recalls a dozen or so occasions on which he was the target of Muslim aggressiveness during his youth and young manhood in Oslo. And he writes about forming tactical alliances with Muslim gangs in secondary school:\n\nIn Oslo, as an ethnic Norwegian youth aged 14\u201318 you were restricted if you didn't have affiliations to the Muslim gangs. Your travel was restricted to your own neighbourhoods in Oslo West and certain central points in the city. Unless you had Muslim contacts you could easily be subject to harassment, beatings and robbery. Our alliances with the Muslim gangs were strictly seen as a necessity for us, at least for me . . . As a result of our alliances we were allowed to have a relaxing and secure position on the West side of Oslo among our age group.\n\nAfter graduating from school, Breivik became an entrepreneur. But then, in 2006, having decided to become a Knight Templar and to dedicate his time to preparing for armed action and to writing his manifesto, he moved back home with his mother:\n\nThis wouldn't have worked in my old life, when I was an egotistical career cynic as it would devastate my social image. However, individuals who choose to become a Justiciar Knight cares [sic] little about image. . . . Sure, some people will think you are a freak for living with your parents at the age of 31 but this is irrelevant for a Justiciar Knight.\n\nBreivik writes affectionately and at length about his four best friends, \"all of whom are now in the process of settling down\" with wives or girlfriends, but never mentions having any special someone in his own life. (He does record spending much of a year playing the computer game World of Warcraft.) And he mentions plans \"to meet my stepmom, Tove \u00d8vermo,\" a former director at UDI (Utlendingsdirektoratet, the Norwegian Directorate of Immigration), noting the irony that \"UDI is a highly valued target for Knights Templar in Norway as it is an essential tool and facilitator for the Norwegian multiculturalist regime. . . . Although I care for her a great deal, I wouldn't hold it against the KT if she was executed during an attack against UDI, as she used to be a primary tool and category B traitor for the multiculturalist regime of Norway.\"\n\nOther parts of the manifesto also shed light on Breivik's personality. In one passage, for example, he recalls \"mandatory knitting and sewing courses\" in primary school, the goal of which, in his view, was \"to feminise European boys\" and thus serve the cause of \"Marxist utopia\" and \"true equality between the sexes.\" Now, however, \"I am grateful for having received this insight into sewing and stitching as this knowledge is an essential skill when constructing and assembling modern ballistic armour. . . . It is quite ironic and even hilarious when reflecting on the fact that a skill which was intended to feminise European boys can and will in fact be used to re-implement the patriarchy by overthrowing the Western European cultural Marxist\/multiculturalist regimes.\"\n\nAt one point, apropos of the dilution of the Nordic gene pool, Breivik maintains that \"Nordic entertainment super-stars like Scarlett Johansson (60\u201370% Nordic purity), Gwyneth Paltrow (70\u201380%), Pamela Anderson (90\u201395%), Paris Hilton (70\u201380%), Taylor Swift (80\u201390%) would have never been where they are today hadn't it been for their distinct Nordic physical characteristics.\" And then there is a curious passage in which he advises potential Knights Templar on how to prepare for a photo shoot. \"As a Justiciar Knight you will go into history as one of the most influential individuals of your time. So you need to look your absolute best and ensure that you produce quality marketing material prior to operation.\" Therefore:\n\n\u2022 Take a few hours in a solarium to look fresher.\n\n\u2022 Train hard (work out) at least 7 days prior to photo session\n\n\u2022 Cut your hair [and] shave\u2014Visit a male salon if possible and apply light makeup. Yes, I know\u2014this might sound repulsive to big badass warriors like us, but we must look our best for the shoot.\n\n\u2022 Use your best clothing\u2014you can f[or] example bring 3 different sets of clothing to the sho[o]t location\u20141. Dress, tie etc. 2. Casual wear 3. Sporty wear 4. Militaristic wear (obviously, you can't bring your guns or anything indicating that you are a resistance fighter).\n\nHe even recommends musical artists whose work he considers suitable listening for Knights Templar.\n\nAnd he advises his fellow Knights on how to prevent friends and family \"from digging too much or ask too many questions\" while you are planning a secret military operation. His suggestions:\n\n\u2022 Say you play WoW (World of Warcraft) or another MMO and have developed an addiction for it. Say that are going to play hardcore for the rest of the year and it is no point trying to convince you otherwise. . . .\n\n\u2022 Say you think you are gay and are in the process of discovering your new self and that you don't want to talk any more about this issue. . . . Make them swear to not tell anyone!\n\nBreivik notes that he bought \"three bottles of Ch\u00e2teau Kirwan 1979 . . . at an auction 10 years ago\" and is saving the last remaining bottle \"for my last martyrdom celebration,\" during which he plans to \"enjoy it with the two high class model whores I intend to rent prior to the mission.\" He explains that in his view, the concept of the \"Perfect Knight\" \"should not include celibacy, although some of my KT peers might disagree with me on this point. . . . A pragmatic approach, which involves acknowledging the primal aspects of man for the purpose of preparing him for a martyrdom operation, should always take precedence over misguided piety, which only increases the chance of jeopardizing the execution of the operation. And I believe the majority of war strategy analysts will agree with me on this.\"\n\nBreivik also wonders what will happen if \"I survive a successful mission and live to stand a multiculturalist trial.\" He takes stock of himself: \"I have an extremely strong psyche (stronger than anyone I have ever known) but I am seriously contemplating that it is perhaps biologically impossible to survive the mental, perhaps coupled with physical torture, I will be facing without completely breaking down on a psychological level. I guess I will have to wait and find out.\" Yet whatever happens,\n\nI will always know that I am perhaps the biggest champion of cultural conservatism, Europe has ever witnessed since 1950. I am one of many destroyers of cultural Marxism and as such; a hero of Europe, a savior of our people and of European Christendom\u2014by default. A perfect example which should be copied, applauded and celebrated. The Perfect Knight I have always strived to be. A Justiciar Knight is a destroyer of multiculturalism, and as such; a destroyer of evil and a bringer of light. I will know that I did everything I could to stop and reverse the European cultural and demographical genocide and end and reverse the Islamisation of Europe.\n\nIt is perhaps appropriate here to say a few words about my own introduction to European Islam. In 1998 I moved to Amsterdam\u2014a beautiful, civilized city with which I had fallen in love. But after living there for a while I discovered a side of it I'd never heard about. The city center\u2014the part the tourists see\u2014was liberalism, or libertarianism, itself: a place of real diversity, tolerance, and freedom. But just outside it was another Amsterdam\u2014a parallel society, insular and Islamic.\n\nThis, I soon learned, was not unique to the Netherlands. In cities across Western Europe, Muslims inhabited isolated, intolerant enclaves, living in extended patriarchal families governed by sharia law. In countries whose residents were supposedly free, children were growing up in communities where the very idea of an individual's right to shape his or her destiny was alien and anathema. All too many of these kids were being brought up to disdain the countries they lived in, the values on which those countries were founded, and those countries' non-Muslim natives. And to disdain, especially, people whose identity or behavior was seen as specially offensive to Islam\u2014among them gays, Jews, and women who dressed \"immodestly.\" In many cities, as a result, rapes, gay-bashings, and attacks on Jews were on the rise.\n\nSome observers took it for granted that the children of Muslim immigrants would grow up to be more integrated into European society and values than their parents, and that the grandchildren would be even more integrated. But over the years evidence mounted that things were going the other way\u2014that the children of immigrants were more hostile to their surroundings than their parents had been. And the larger their enclaves grew, the less they felt any need to relate to European culture and values.\n\nWorst of all, the governments of these countries either didn't see what was going on or didn't want to. They'd based their immigration and integration policies on the idea of multiculturalism\u2014which is another way of saying that they didn't really have immigration and integration policies. Not only didn't they encourage newcomers to adopt liberal democratic values and become full, free members of their societies\u2014they actively discouraged it. Why? Part of the reason was a contempt for their own societies, founded in a multicultural guilt over the Western heritage of colonialism and imperialism. Part was a misguided, romantic notion that everything about the cultures that Muslim immigrants brought with them was colorful and enriching. And part was a deep-seated, unacknowledged, and largely unconscious unwillingness to see these newcomers become full members of their societies.\n\nIt was one thing to have, say, Pakistanis in one's capital city, sporting exotic costumes and serving exotic dishes at restaurants. It was another thing to think of these aliens as, say, Dutchmen or Norwegians. Better to let them marry one another and stay put in their own neighborhoods\u2014veritable outposts of their home countries\u2014than to let them blend in; better to encourage them to live indefinitely on welfare, supported by the state, than to welcome them into the workforce, where they might take jobs from natives.\n\nThe multicultural mentality of many elite Europeans was epitomized by a 2001 remark by Norwegian cultural anthropologist Unni Wikan. When informed that a colossal percentage of rapes in Norway were committed by \"non-Western\" men (Norwegian code for Muslims), and that almost all the victims were ethnic Norwegians, Wikan placed blame largely on the victims. These women, she insisted, must realize that theirs was now a multicultural society. Their new Muslim countrymen considered their attire provocative; if only they'd dress more conservatively, they'd be less likely to be raped. (In the United States, such comments would be roundly denounced as blaming the victim; among the Norwegian cultural elite, they were taken as sage guidance on how to achieve intercultural harmony.)\n\nIt is important to underscore here that while multiculturalism may seem to be rooted in respect for other cultures, it in fact often masks a distaste for the idea of sharing one's culture and national identity with foreigners. After I moved to Norway in 1999, I soon lost count of the number of times I heard ethnic Norwegians, in barroom conversations with the Norwegian-born children of immigrants (who spoke Norwegian just as well as they did), ask: \"No, but where are you really from?\"\n\nNorway has, after all, historically been an ethnically homogeneous society. Cultural elite types are very aware of that fact\u2014and are, consequently, desperately eager to show they are not racists and to advertise their love for people of different races. They often do this in artificial ways that can seem positively antiquated. After the atrocities of July 22, for example, several newspapers in Norway reported on an ethnic Norwegian woman who had been wounded in the explosion in Oslo and had received help from one Christian Armando Clementsen. Though born in Colombia, Clementsen had been raised in Norway from infancy as the adopted son of ethnic Norwegians. The media instantly declared a picture of him helping the woman \"iconic,\" and hailed his act as a triumph of common humanity over xenophobia. \"People were surprised that a dark-skinned person managed to show compassion,\" said Clementsen.\n\nConsider, too, the following passage from a Dagbladet piece published on July 30 by columnist Marte Michelet:\n\nHere in Gr\u00f8nland [a largely Muslim neighborhood in Oslo], where we live, it is as if a heavy and long-lasting air pressure has disappeared, dissolved by the 200,000 people who filled Oslo with love and roses held high. The Somalis walk a little more lightly. The hijab girls no longer look nervously up at passersby. The Pakistani shopkeepers smile a bit more broadly. A feeling of being encamped and of constant criticism has been replaced with a new sense of security. They don't hate us after all, these Norwegians. The fantastic celebration of community, solidarity, and tolerance that has manifested itself from one end of the country to another will have a long-lasting effect on the life of people with minority backgrounds in this country.\n\nReading this girlish drivel, all one could think was how remote Michelet was from the reality surrounding her. Did she really believe for one instant that her Muslim neighbors had been sitting around worrying all these years about what ethnic Norwegians think of them? (And notice how it didn't even occur to her to think of them as Norwegians; no, to her, she's the Norwegian\u2014they're still Somalis and Pakistanis.) She seemed barely aware that Norway, compared to the places they came from, is a playground. They may have feared Saddam Hussein, or Pervez Musharraf, or some local chieftain or warlord back in the old country. They most assuredly do not fear Norway's prime minister, Jens Stoltenberg, and they certainly don't tremble with worry about the opinion of some bubbleheaded columnist who is impressed with herself because she lives in a largely Muslim part of town.\n\nHow far has multiculturalism gone in Europe? So far that Muslim leaders who call for the execution of homosexuals are not considered to have overstepped the bounds of decency. At a 2007 debate, the deputy chairman of Norway's Islamic Council, who was also a high-ranking official in both the Labor Party in Oslo and the nation's leading trade union as well as an advisor to the government's Equality and Anti-Discrimination Ombud, refused to reject the death penalty for gays. Mohammed Usman Rana, then head of the Muslim Student Association, now a physician and a prize-winning columnist for Norway's leading newspaper, would not say whether he supported or opposed executing homosexuals.\n\nThe hostility toward gays in the Muslim community is so intense that until recently there was not a single publicly gay Muslim in all of Norway. When I did finally meet the country's first openly gay male and lesbian Muslims, they proved to be two of the most emotionally battered people I've ever encountered. Not only were they despised by their communities, but the institutions of mainstream Norwegian society that should have been giving them support had turned their backs on them, afraid to offend Muslim leaders. This, too, is the result of multicultural thinking on the part of the European establishment: all too often, gay Muslims, apostate Muslims, and Muslim women who've divorced their husbands are left to fend for themselves.\n\nIn the last few years, I've written regularly about such issues for the website of HRS, whose feminist founders, Hege Storhaug and Rita Karlsen, established it because they were outraged by the deprivation of women's and girls' rights in Norwegian Muslim communities. When they started HRS in 2001, it was still highly taboo in the Norwegian media to criticize Islam, multiculturalism, or Norway's immigration and integration policies. Criticizing Islam, it should be emphasized, is not about putting down Muslims as people\u2014it is about recognizing Islam as a severe, totalizing ideology of which Muslims are the first and greatest victims. It is about acknowledging that imams lauded as \"moderate\" by multiculturalists are in fact supporters of forced marriage, female genital mutilation, the execution of homosexuals, and the systematic subordination of women. Yes, there are liberal and kind-hearted Muslims\u2014but they are liberal and kind-hearted precisely to the degree that they reject the counsel of their religion's leading theologians, such as Yusuf al-Qaradawi, and choose instead to embrace the image of peaceful, loving deity.\n\nWhile Norway's multicultural media, politicians, and academics were systematically ignoring or explaining away or quite simply lying about various uncomfortable developments related to Islam, immigration, and multiculturalism, Hege and Rita, along with a number of others, were doing their best to speak the truth about such issues and to put them on the Norwegian national agenda. In doing so, they endured a great deal of abuse by the political and media establishment. I played my own small part in their effort, and received my share of unfair abuse.\n\nBut it paid off. Over the years, Norwegians became less uneasy about airing their concerns. Public support for the upstart Progress Party, with its vocal criticism of multiculturalism, climbed steadily. Islam and immigration became acceptable topics of discussion in newspaper opinion pages and on prime-time debate programs. In 2006, most Norwegians told pollsters that they opposed Selbekk's republication of the Danish cartoons; in early 2011, a majority took the opposite view.\n\nNot all critics of Islam in Europe, alas, oppose it in the name of tolerant, pluralist democracy. Soon after I became aware of Islam in Europe, I became aware too that there were far-right parties, groups, and individuals across the continent that opposed Islam in the name of narrow religious, ethnic, and\/or nativist values. From the time I first began writing about these issues, I warned that if Europe's cultural elites did not address them responsibly, they'd face an increasing danger of right-wing extremists taking matters into their own hands. The terrible events of July 22 only confirmed the Norwegian establishment's failure in this regard.\n\nThose events shocked Norwegians to the depths of their being.\n\nThe national display of grief was unlike anything I had ever seen. On 9\/11 the United States had been attacked by foreigners; on June 22, Norwegians had been victimized by one of their own. This fact shattered many Norwegians' sense of themselves as a homogeneous, peaceful, virtuous people who had nothing to fear from one another.\n\nA larger country\u2014a different country\u2014would not have responded to the massacres by Breivik in the way Norway did. The Oklahoma City bombing in 1995, which killed 168 people, including nineteen children under six, and injured 680, caused shock and horror around the world, but it did not engender anything remotely like the national existential crisis that the July 22 atrocities led to in Norway.\n\nAt the bottom of this existential crisis were two questions: How can anybody do this to us? and even more, How can one of us do this to us?\n\nAll their lives, Norwegians had been taught to think of their country as a virtuous actor on the international stage. A terrorist attack against Norway was unthinkable\u2014though such attacks against other countries, above all the United States, were thoroughly understandable. After 9\/11, Norwegian newspapers, like those in other Western countries, ran articles gleefully explaining why the United States had had it coming; after atrocities like Columbine, the reflexive response in the Norwegian media has been to attribute them to something unhealthy in the American soul. Yet many of the same people whose instinct, in the face of American national tragedies, has been to sneer at the Great Satan, haughtily condemned as tasteless any expressions of concern, after July 22, about the potential impact of Breivik's actions on Norwegian multiculturalism, immigration policy, and criticism of Islam. Few seemed to notice the inconsistency: it's perfectly fine to blame Americans and their culture for atrocious acts that have caused them to suffer; but when Norwegians are suffering, even to express legitimate concerns while one is patently hurting for the victims is considered an act of sacrilege. Underlying this moral contradiction is a belief, encouraged by the left-wing cultural elite, in Norway's unique virtue and innocence\u2014in the idea that it exists on a somewhat higher moral plane than the rest of the world, certainly the United States.\n\nThe reality of Norway, to be sure, differs considerably from its official self-image. Norway belongs to NATO; it has troops in Afghanistan and Libya. There's something else, too. On weekdays Norwegians present an exceedingly peaceable and sober image, but on weekends (as Norwegians themselves acknowledge) an extraordinary number of them drink to excess and\u2014well\u2014misbehave. It is as if they are letting out all the emotion they have repressed all week; it is as if, possessed all week by their overly self-disciplined inner Lutherans, they have been repressing their inner Vikings. Since July 22 I have wondered whether some Norwegians have seen in their homegrown terrorist a nightmare image of that inner Viking writ large\u2014and I have wondered whether the country's reaction to Breivik's acts\u2014a reaction that, viewed from abroad, may seem, in many ways, bizarre\u2014can be explained by many Norwegian's deep sense of guilt about their own inner Vikings.\n\nThe thrust of all this is that many Norwegians, in response to Breivik's atrocities, plainly felt a desperate need to restore their sense of themselves as a peaceable people. In the days after July 22, they held one memorial tribute after another. The streets of Oslo were filled with flowers and candles. Norwegian TV endlessly repeated something a girl belonging to Workers Youth League had said in a CNN interview: \"If one man can show so much hate, think of how much love we can share together.\" Somebody else said: \"This will make us be nicer to each other.\" The message sent out by official Norway was that \"all Norway is one\" and that the country was being \"united in love.\" A picture on one of the major newspaper's websites showed a string of Norwegians of various colors holding hands.\n\nYou have to be familiar with Norwegians to know how odd all this was. Norwegians never talk about love. On English-language TV shows, the subtitles typically water down expressions of emotion to make them sound more naturally Norwegian, so that \"I love . . .\" almost invariably becomes \"I like . . .\"\n\nYet the massive display of love clearly didn't fill the void inside. Norwegians wanted desperately to find some meaning in what had happened. That was, in large part, why they amassed so many flowers and lit so many candles\u2014as if they hoped that the scale of the tribute alone could somehow overcome the senselessness and evil of the whole terrible thing. But of course it couldn't. For however peaceable they liked to think of themselves as being, they were human. They wanted a sense of meaning, wanted something to do. And, last but not least, they wanted revenge.\n\nAfter 9\/11, Americans could at least look forward to sending soldiers to Afghanistan to take out the terrorists who had ordered the hit on their country. But the arrest of Breivik\u2014one solitary lunatic\u2014provided precious little revenge. Norwegian justice being what it is, the man had been put in a very comfortable cell. The law, needless to say, didn't permit the possibility of execution\u2014indeed, given current sentencing guidelines, it was entirely possible he would one day be released. It is only normal for the human mind to rebel at such facts, to long for proportionate justice.\n\nBreivik's atrocities, then, left Norwegians in a complicated and intensely emotional state of mind. And the country's left-wing cultural elite\u2014that grab bag of professors, authors, journalists, bureaucrats, and politicians, all too many of them pompous, self-regarding mediocrities who share essentially the same lockstep politics\u2014exploited it with breathtaking cynicism.\n\nFirst, they seized on a single detail in Breivik's Facebook profile\u2014he had put himself down as a Christian\u2014and suggested that his rampage had been carried out in the name of religion. On July 23, document.no noted that the assistant police chief of Oslo, Roger Andresen, had called Breivik a \"Christian fundamentalist,\" and that journalists had eagerly embraced this description. (This in a country where the media are at pains to avoid identifying evildoers as Muslims.) On August 2, NRK's website ran a long article coauthored by an American \"terrorist expert,\" Mark Juergensmeyer, affirming that Breivik was a \"Christian terrorist.\" The purpose here was obvious: to play down the threat of Islamic jihad by pretending that Breivik's actions had amounted to a kind of Christian jihad, and had proven that terrorism in the name of Christianity is at least as serious a menace as terrorism in the name of Allah. (Another approach was taken by Thorbj\u00f8rn Jagland, head of the Norwegian Nobel Committee, who wrote in a July 29 Aftenposten op-ed: \"After this . . . there should be a complete end to calling terrorism committed by Muslims 'Islamic terrorism.' No one would think of calling Anders Behring Breivik a Christian terrorist just because he has described himself as a Christian.\")\n\nTo anyone who had actually read Breivik's manifesto, this thesis was transparent hogwash: his book makes it clear that he is not a religious man in the slightest and that he views Christianity in an entirely abstract way, as a foundation of the Western civilization that he purported to be trying to rescue. \"I'm not going to pretend I'm a very religious person,\" Breivik wrote,\n\nas that would be a lie. I've always been very pragmatic and influenced by my secular surroundings and environment. . . . Religion is a crutch for many weak people and many embrace religion for self serving reasons as a source for drawing mental strength. . . . Since I am not a hypocrite, I'll say directly that this is my agenda as well. . . . I'm pretty sure I will pray to God as I'm rushing through my city, guns blazing, with 100 armed system protectors pursuing me with the intention to stop and\/or kill.\n\nElsewhere in the manifesto he declared: \"European Christendom isn't just about having a personal relationship with Jesus or God. It is so much more. Christendom is identity, moral, laws and codexes which has produced the greatest civilisation the world has ever witnessed.\" And, apropos of the rules for being one of his fellow Knights:\n\nIt is not required that you have a personal relationship with God or Jesus in order to fight for our Christian cultural heritage and the European way. In many ways, our modern societies and European secularism is a result of European Christendom and the enlightenment. It is therefore essential to understand the difference between a \"Christian fundamentalist theocracy\" (everything we do not want) and a secular European society based on our Christian cultural heritage (what we do want).\n\nFinally, Breivik declared emphatically that \"it is essential that science takes an undisputed precedence over biblical teachings. Europe has always been the cradle of science and it must always continue to be that way.\"\n\nBreivik is a Christian, in short, in much the same way that most other Norwegians are Christians\u2014purely nominally. It was easy for non-Norwegians, especially Americans, to fail to understand this, especially when members of the Norwegian cultural elite were telling them (with an air of absolute authority) that the murderer was, indeed, a Christian jihadist\u2014a theme that was quickly picked up by American leftists.\n\nNorway's cultural elites didn't stop at identifying Breivik with Christianity. They also strove to suggest that his atrocities had been motivated in large part by a fanatical devotion to Israel and identification with Jews. Among Breivik's \"core values,\" reported a Dagbladet article on July 23, was his \"Pro-Israel\" stance. On the same day, the Swedish newspaper Nationell referred darkly to his \"extreme Zionist sympathies.\" In a July 26 Dagbladet article about Breivik's manifesto, Andreas Malm (a supporter of Hamas and Hezbollah) zeroed in on the murderer's complaint \"that our leaders, instead of supporting 'our cultural cousin' Israel, which is only defending itself against the jihadists, side with the Palestinians\"\u2014and identified this view of Israel as one of \"the standard elements of Islamophobic ideology.\"\n\nThis effort to link Breivik with Israel and the Jews was a natural outgrowth of many Scandinavian leftists' pathological anti-Semitism and loathing of Israel, which they consider the modern-day equivalent of Nazi Germany. Norway is, alas, a country that has never been kind to Jews. Its 1814 constitution explicitly denied Jews entry into the realm. During the Nazi occupation, Norway (unlike all other Nordic countries) readily handed its Jews over to the Nazis\u2014and when some of those Jews returned after the war, Norway refused to restore to some of them their homes and property, while charging others an exorbitant \"administration fee\" to recover what was rightly theirs. In recent years there has been no Western nation in which editorial cartoons in the mainstream media have consistently gone as far in depicting Jews and Israelis in ways that bring to mind the virulent propaganda of Der St\u00fcrmer. The most familiar premise in these cartoons is that Israelis are today's Nazis, and Muslims are today's Jews.\n\nOnly days before the atrocities of July 22, it was reported that Norway was planning to support the Palestinian bid to be recognized as a separate state by the UN; and on the very day before the attacks, the young people at Ut\u00f8ya, as Alex Weisler of the Jewish Telegraph Agency pointed out on July 26, had discussed \"the organization of a boycott against Israel\" and pressed Norway's foreign minister \"to recognize a Palestinian state.\" A picture was widely circulated on the Internet (but not in the mainstream Norwegian media) of Ut\u00f8ya campgoers hoisting a huge \"Boycott Israel\" banner.\n\nAs it turns out, Israel-bashing at Ut\u00f8ya has a long history. In 2007, Stoltenberg gave a speech on the island praising Workers Youth League for their campaign to win the release of Palestinian prisoners in Israel; among the prisoners of special concern to the youth at Ut\u00f8ya, he noted, was Hussam Shaheen of Fatah Youth, \"who has been here at Ut\u00f8ya several times and who has many friends in Workers Youth League.\"\n\nA year earlier, on July 15, 2006, a local Norwegian newspaper, Adresseavisen, reported from Ut\u00f8ya that Workers Youth League had demanded Stoltenberg take \"a tougher line on Israel\" following that country's actions in Gaza and Lebanon. One campgoer, who accused the party of cowardice for failing to come down harder on Israel, \"gave Stoltenberg a Palestine scarf and a T-shirt reading 'free Palestine,' and encouraged Jens to become 'Palestine Jens.' \" Stoltenberg defended his government by pointing out that Norway had \"gone further than other countries in letting Hamas have contact\" with its political officials. That year, according to Adresseavisen, Palestinian scarves and flags were \"a conspicuous element of camp life,\" and guests on the island included Hassan Faraj of Fatah Youth, who handed Stoltenberg a T-shirt reading \"Tear down the wall.\"\n\nApparently the tone at the 2011 camp did not differ greatly from that in earlier years. Indeed the tragic irony of Ut\u00f8ya may be that before the slaughter started, the island was the setting for speeches in which politicians, and aspiring politicians among the campers, encouraged participants to give their heart and soul to groups that engage in exactly the kind of monstrous terrorism that would mow dozens of them down before the week was out. They were, in short, urged to support groups that are internationally recognized as terrorists\u2014and, therefore, to embrace the idea that the cold-blooded murder of innocent men, women, and children is a legitimate means of achieving an ideological end. As one Facebook user put it: \"Before the terrorist murdered the kids at the camp, the kids at the camp were lending their support to terrorists who murder kids.\"\n\nThe attempt to associate Breivik with Israel and the Jews, then, was just the latest chapter in a long history of Norwegian establishment anti-Semitism and hostility to Israel. In fact, Breivik had no special devotion to Jews (he described the United States as having a \"Jewish problem\") and supported Israel principally because it is a victim of and ally against jihad; in his manifesto he expresses admiration for Jews who shared his anti-jihadist sentiments, but denounces Jewish multiculturalists, imploring his readers to \"learn the difference between a nation-wrecking multiculturalist Jew and a conservative Jew.\"\n\nSo intense was the effort by the Norwegian cultural elite to link Breivik with Jews and with Israel, however, that Norwegian Jews were seriously concerned about blowback. \"If the Norwegian public is looking for a larger villain than Breivik,\" reported Weisler, \"Jews here [in Norway] are worried that Zionism and pro-Israel organizations may be singled out.\" Weisler quoted a chemistry professor who noted that in Norway, supporters of Israel are demonized: \"The Jews . . . have a lot of friends in Norway, but the Norwegian politicians are not our friends.\"\n\nAs if to underscore this point, the news came on July 26 that Norway's ambassador to Israel, Svein Sevje, had chosen to draw a distinction between the massacres in Norway and terrorist attacks by Palestinians in Israel. In an interview with the Israeli newspaper Maariv, Sevje said that while Breivik \"had an ideology that says that Norway, particularly the Labor Party, is forgoing Norwegian culture\"\u2014an ideology that Sevje plainly considered absurd\u2014terrorist attacks by Palestinians in Israel were caused by Israel's occupation of Palestinian land. In other words, Israelis were responsible for the terrorism inflicted upon them; Norwegians weren't.\n\nThree days later, Michal Rachel Suissa of Norway's Center against Anti-Semitism sent out an alarming bulletin that began: \"We are experiencing a macabre attempt to gag the opposition and a uniformity of debate by means of a tendentious representation of a person whose mental illness, directly or implicitly, is claimed to be the responsibility of the 'right.' \" After the atrocities of July 22, she noted, NRK and other media had lost no time in \"identify[ing] the political opposition in this country as both direct and indirect accomplices in terror and mass murder\"\u2014and in the days that followed had intensified this effort.\n\nThe message we others have received is clear: in the name of the new free speech, no one in Norway from now on will be able to dare to remind anyone that since September 11, 2001, the world has been subjected to more than 17,000 terrorist attacks, almost all carried out in Allah's name. This fact is no longer a part of the politically correct \"truth.\"\n\nSuissa expressed concern that if the media succeeded in their effort to impose this new \"politically correct 'truth' \" and to make the murderer the face of the Norwegian political opposition, \"Norway will emerge as the one-party state that some of the participants in the debate have shown that they admire.\" This state of affairs, Suissa enjoined her readers, obliged Norwegian Jews \"to be on guard against all tendencies to engage in the 'blame game' and in the apportioning of guilt. We know all too well that when the game is over, we Jews can risk being blamed for this, too.\" (Indeed, by the time Suissa voiced her concern, Ellie Merton, the chair of the Waltham Forest Palestinian Solidarity Campaign in Britain, had already publicly called Breivik's massacre \"an Israeli government-sponsored operation.\")\n\nEven as the Norwegian cultural elite strove to link Breivik to Christianity and Israel, it sought as well to use his atrocities to reimpose the ideological control that had lately been slipping away from them. Politicians, authors, academics, and journalists alike put forth the idea\u2014unargued, as if it were self-evident\u2014that the proper way to respond to the atrocities of July 22 was to drop all of the problems with current Norwegian immigration and integration policies down the memory hole and embrace Islam without qualification.\n\nDuring the days after the atrocities, the Norwegian TV news showed leading politicians and other prominent figures making pilgrimages to mosques and hugging imams. On July 26, for example, Crown Prince Haakon, the heir to the throne, participated in a memorial service for Breivik's victims at the World Islamic Mission, along with Oslo bishop Ole Christian Kvarme, Oslo mayor Fabian Stang, Minister of Children, Equality and Social Inclusion Audun Lysbakken, U.S. ambassador Barry White, and Foreign Minister Jonas Gahr St\u00f8re, who recited from the Koran. The message clearly being conveyed by this display was that any concerns about the loyalty of some Muslim leaders to Norway\u2014however much evidence there might be to back up those concerns\u2014were now inoperative, and to fail to abide by this new order of things would be received as a betrayal of the memory of all those young people who had died on July 22.\n\nWhile they were being cozy with imams, moreover, Norway's cultural-elite types were simultaneously encouraging their countrymen to turn on the critics of Islam. In other words, even as a sermon of universal love, harmony, and solidarity was being preached in the Norwegian media, the message was being sent out that certain persons\u2014namely, all those who had ever dared to criticize Islam\u2014were henceforth to be excluded from this glorious new circle of love. Imams, whatever their positions on forced marriage, female genital mutilation, the execution of homosexuals, and the systematic subordination of women, were now officially considered to have been washed clean in the blood of the victims of July 22.\n\nMeanwhile those who had criticized Islam, however legitimately concerned they might be with the denial of basic human rights and individual liberties within Muslim communities, had been deemed officially anathema. They were Islamophobes\u2014racists, bigots, extremists. They were the danger. They were the threat. They had fertilized the soil in which the mass murderer had grown.\n\nIn short, because one maniac had gone on a murderous rampage, Norwegians were expected to forget everything they knew about their country's immigration and integration fiascoes, and to drown all the facts in a sea of love\u2014and retribution. The premise was entirely illogical\u2014it was, in fact, sheer madness. But such was the atmosphere of intimidation that hardly anybody in a position to be heard dared challenge it openly.\n\nMost foreign observers, not being fluent in Norwegian, missed this story entirely. \"Even in their deepest sorrow,\" wrote Anna Reimann in Der Spiegel, \"the Norwegians don't get hysterical. They resist hate. It is amazing to see how politicians and the whole country react. They are sad to the deepest thread of their souls. They cry in dignity. But nobody swears to take revenge. Instead they want even more humanity and democracy. That is one of the most remarkable strengths of that little country.\"\n\nI have no trouble believing this to be essentially true of the Norwegian people as a whole, for whom I have a great deal of affection and respect. But the political leaders? The national media? The professoriat? In the wake of the murders of July 22, as we shall see, all too many members of the Norwegian cultural elite made use of this atrocity as an opportunity to launch personal attacks against their longtime ideological adversaries\u2014whom they unhesitatingly linked to the perpetrator of these unspeakable crimes.\n\nOf course it is a common practice on the far left, not only in Norway but elsewhere, to use guilt by association to smear one's opponents and delegitimize their views, even as one hypocritically refuses to \"jump to conclusions\" in obvious cases of Muslim fanaticism. For years one element of politically correct leftist orthodoxy has been that it is unfair to suggest that the \"root cause\" of acts of jihadist terrorism has anything to do with Islam or the contents of the Koran. Such argumentation is simply not permissible in politically correct circles\u2014not even if a terrorist shouts \"Allahu akbar\" as he shoves a knife into someone's throat, sets off explosives, or drives a plane into a building.\n\nNo, one must never connect such acts to Islam; if one dares to mention Islam at all, one must be sure to echo the mantra that the terrorist has \"misinterpreted\" his religion\u2014even if thousands upon thousands of other terrorists have \"misinterpreted\" it in precisely the same way, and even if untold numbers of their co-believers around the world have taken to the streets to cheer their terrorist acts because they, too, are guilty of theological \"misinterpretation.\" No matter that the Koran, a book presented to the world by Muhammad\u2014a warrior who created the Muslim world through brutal armed conquest\u2014actually does call on believers to use violence to expand the rule of Islam.\n\nFor an example of this stubborn refusal to connect the dots, consider the aftermath of the 2009 killing spree at Fort Hood, Texas. A Muslim U.S. Army major and psychiatrist, Nidal Malik Hasan, shouted \"Allahu akbar\" as he gunned down people at the military base, killing thirteen and wounding thirty-two. Evidence abounded that Hasan was a devout Muslim and that this had been an act of jihad: he had written about martyrdom and jihad on an Islamist website; he had posted a Muslim prayer on his apartment door; in conversations, he had expressed sympathy for suicide bombers and complained that the United States was making war on his faith; hours before the massacre, he had given his neighbors copies of the Koran. Yet the media refused to look these facts in the eye. Instead they attributed Hasan's actions to his colleagues' Islamophobia or to fear of deployment in a war zone, or acted as if his motives were utterly inscrutable. Months later an army major suggested in all seriousness that \"we may never know\" why Hasan did what he did.\n\nThen there was Faisal Shahzad, who attempted a car bombing in Times Square in 2010. Though he admitted he had been inspired by al-Qaeda operative Anwar al-Awlaki, New York mayor Michael Bloomberg suggested he may have been motivated by anxiety about the economy.\n\nBut let an Anders Behring Breivik cobble together a \"manifesto\" in which he approvingly quotes innumerable writers, thinkers, and politicians from across the generations who are advocates of peace and freedom and enemies of violence\u2014and then proceed to commit unspeakable atrocities in the name of opposition to jihad\u2014and suddenly a small army of left-wing multiculturalists start arguing that everyone the murderer has ever read shares responsibility for his actions.\n\nSuch irresponsible \"analysis\" is entirely of a piece with the left-wing tendency to jump to conclusions about the supposed propensity for violence on the right and to assume that any domestic American terrorist must by definition be motivated by racism, bigotry, or Christian fanaticism. The shooting in Arizona in which six people were killed and thirteen wounded, including Representative Gabrielle Giffords, was immediately ascribed\u2014by former MSNBC host Keith Olbermann, among others\u2014to Tea Party rhetoric. Sarah Palin was also apportioned a degree of the blame for having published an electoral map with targets placed on certain districts, including Giffords's own in Arizona. It quickly turned out that the perpetrator, Jared Loughner, was a crazed loner who had no politics to speak of and no affiliation with the Tea Party; nor was he a reader of Palin's website. (Needless to say, no apology was forthcoming from Olbermann or his allies on the left.)\n\nTo understand the vehemence with which Norway's left-wing cultural elite, headquartered in the Labor Party, sought to crush its enemies after July 22, it will help to go over a bit of history.\n\nThe Labor Party, it should be mentioned at once, is not just Norway's largest party. Ever since World War II, it has been first among equals\u2014primus inter pares. During its golden age in the immediate postwar years, it stood alone as the single dominant force in Norwegian politics. It is only a slight exaggeration to say that the Labor Party created postwar Norway\u2014the welfare state par excellence, with extraordinarily high taxes and an extraordinarily large government bureaucracy. (In few other countries does the public sector account for as large a percentage of the national economy as in Norway.) Labor instituted a political system in which the government has, by American standards, a mind-blowing degree of involvement and intrusiveness in virtually every aspect of every individual's life.\n\nToday, though it is no longer the unconquerable colossus it was during the decades after World War II, when Labor prime minister Einar Gerhardsen became known as \"the father of his country,\" and though actual workers (as opposed to public-sector paper-pushers) now tend to prefer the Progress Party, Labor remains unique. It's more than just a party. Government bureaucrats are overwhelmingly Laborites, as are the country's most prominent academics. Under the guise of supporting \"media diversity,\" the government, following a policy set down long ago when Labor reigned supreme, subsidizes a number of left-wing newspapers, including the Communist daily Klassekampen. There is no major conservative daily in the country, and the evangelical Christian paper that comes closest to being one lost its subsidy in 2008. The state television network, NRK, to which every owner of a TV set in the country must pay a license fee (now $460 a year), is widely recognized as a vehicle for Labor Party propaganda.\n\nThen there's LO (Landsorganisasjon i Norge, or Confederation of Trade Unions), Norway's answer to the AFL-CIO, with just under nine hundred thousand members in a country of fewer than five million. Though LO is officially nonpartisan, and though Norway's working class has increasingly recognized over the years that Labor's policies are not necessarily in its best interests, LO is intimately tied to the Labor Party\u2014an alliance that affords the party an immense advantage, pecuniary and otherwise, over its competitors. (The island of Ut\u00f8ya was, in fact, a gift from LO to the Workers Youth League. I do not know of any other islands given to political parties or their youth groups by LO.)\n\nThe attitude of many Labor loyalists toward their party is demonstrated by an exchange I had with a friend of mine one day when I'd been living in Norway for about a year. I ventured a mild criticism of the Labor Party; he was outraged. \"The Labor Party built this country after the war!\" he exclaimed angrily. \"We owe everything to the Labor Party!\" I was stunned: I'd known a lot of fervent political types back in the United States but had never heard any American talk this way about any political party. This wasn't the way people in a democracy talked about political parties; this was Soviet-style thinking.\n\nMany people outside of Norway have been puzzled, even shocked, by the very idea of a Labor Party youth camp. It should perhaps be explained that all of the political parties in Norway have youth divisions. They're not like the Young Republicans or Young Democrats. No, this is more like being a Cub Scout before you're old enough to be a Boy Scout. Or perhaps it makes more sense to compare the youth divisions to farm teams. In any case, these groups are very closely tied to their respective parties, and their members come into frequent and close contact with party leaders. Indeed, the leaders of these youth divisions are already active in real party politics: the head of a party's youth division at the county level, for example, automatically receives a seat on the party's county board. Thus a kid who joins the Workers Youth League, say, at age fourteen can make a smooth transition into a political career without any experience whatsoever of any kind of adult life other than politics. One of the things that make American politics look so alien to Norwegians is that people who have spent most of their adult lives in business or banking (or the movies) can become governors and presidents. Instead of finding this an admirable example, they mock it as typical of American unseriousness and fatuity.\n\nThere are other radical differences between American and Norwegian politics. In the United States, the two major political parties have their platforms and daily \"talking points,\" but at bottom (as the presidential primary campaigns demonstrate every four years) they're grab bags of individuals with a range of opinions and ideas; in Norway, with the conspicuous exception of the Progress Party, each of the seven or eight major parties is a group of people who are expected to be exceedingly loyal to the party line. In the United States, voters can choose among their parties' candidates in primary elections; in Norway, each party draws up its list of candidates, and you can either vote for the names listed on your party's slate or go elsewhere. No wonder, then, that meetings of Norwegian party youth groups are essentially indoctrination sessions: here's the party line, kid; if you want to become a successful politician in our ranks, learn to argue for it effectively. The aspiring politico who has fresh ideas and thinks for herself is not especially welcome.\n\nBreivik's targeting of Ut\u00f8ya was no accident, of course. He sees the Labor Party as the national headquarters for \"cultural Marxism,\" multiculturalist thinking, and the appeasement of Islam. In his manifesto he complained particularly about Labor's protective stance toward \"extreme Marxist movements\" such as Blitz, which has committed acts of vandalism at the Norwegian Parliament and at the government office building that Breivik bombed, but which still receives government support. Breivik recalls \"crackdowns on right-wing youth movements\" when he was young: \"The police raided them several times, called their parents and invested a lot of resources on [sic] squashing the right-wing movement all over Norway.\" Yet a far-left group like Blitz, which is \"often referred to as the 'storm troops' of the Norwegian Labour Party,\" receives public funds: the government subsidy for the building in which Blitz has been squatting for years amounts to \"more than 3 million USD per year alone. The violent Marxist group 'SOS Rasisme' receives 2\u20133 million NOK annually.\" That Breivik is a mass murderer does not make these facts any less true, or any less deplorable.\n\nLabor, to be sure, is not Norway's only major party of the left. Many Laborites, who were fonder of the Soviet Union than of the United States, opposed Norwegian NATO membership so strongly that, in 1961, they formed the Socialist People's Party, the forerunner of today's Socialist Left Party (which is part of Norway's current ruling coalition, and which still despises NATO and the United States). Others who also considered Labor insufficiently radical established the Arbeidernes Kommunistparti (marxist-leninistene), a Maoist party that promoted communist dictatorship, in 1972. This party, known familiarly as AKP (m-l), fielded candidates in elections via the R\u00f8d Valgallianse (Communist Party), which was formed in 1973 and was the forerunner of today's R\u00f8dt (Red) party.\n\nThough AKP (m-l) and the R\u00f8d Valgallianse were small parties, they included among their members many men and women who were, or who would become, influential writers, academics, journalists, and politicians. One former head of AKP (m-l) is Aslak Sira Myhre, who now runs Litteraturhuset, a major institution in Oslo that hosts lectures and other cultural and political events. Another former party head is Sigurd Allern, who went on to become Norway's first professor of journalism. A previous leader of the R\u00f8d Valgallianse, Hilde Haugsgjerd, is Allern's ex-wife and is now editor in chief of Aftenposten, which is generally regarded as Norway's newspaper of record and has traditionally been considered its most conservative major national daily. Kjersti Ericsson, head of the AKP (m-l) from 1984 to 1988, is now a professor of criminology at the University of Oslo and the author of dozens of books. Celebrated novelist Jon Michelet is a card-carrying communist, as is his daughter, Dagbladet columnist Marte Michelet, who once ran the youth division of the R\u00f8dt (Communist) party and whose Iranian-born fianc\u00e9, Ali Esbati, works for Manifest, a far-left think tank.\n\nAnd let's not forget revered \"peace researcher\" and founder of the discipline of peace studies Johan Galtung, a passionate admirer of Mao and booster of the Cultural Revolution. (When Galtung was presented with a prestigious award in early September, he delivered an acceptance speech in which he compared Norwegian soldiers in Afghanistan to Anders Behring Breivik.)\n\nThe list goes on. In Norway, in short, being an ardent supporter of left-wing totalitarianism is no impediment to becoming a respected and powerful member of the cultural elite.\n\nWhat of Norway's purportedly non-socialist mainstream parties? They differ with the socialists on some issues\u2014the Conservatives represent big business, the Christian People's Party advocates for churchgoers, the Liberals are green, the Center Party speaks for farmers and the provinces. But none of them opposes the welfare state and big, intrusive government in any serious way, and none of them makes any serious effort to exclude out-and-out totalitarians from public debate. And for a very long time, none of them challenged the government's disastrous multicultural approach to immigration from the Muslim world.\n\nFor years, then, multiculturalism reigned unquestioned in Norway. The situation was essentially the same, of course, throughout most of Western Europe. Immigration from the Muslim world proceeded apace, and though it became increasingly clear that integration was not taking place as expected, the pressure not to say anything was overwhelming. In 1968, British Conservative MP Enoch Powell became anathema overnight after giving his \"Rivers of Blood\" speech. He was a distinguished classical scholar who had once been the youngest professor in the Commonwealth, and an honored military man who during the war rose from private to become the youngest brigadier in the British Army; yet after his speech, in which he cited the concerns of a great many ordinary British citizens about the influx into their country of immigrants who did not share and did not want to share their values, his name became synonymous with racist hatred, and he was ejected from the Shadow Cabinet. (It was, admittedly, misguided of him to focus in his rhetoric on race rather than on the differences in culture and values that, as the substance of his speech makes clear, were plainly his real concern.)\n\nFor decades, hardly anyone in Europe dared follow in Powell's footsteps. Immigration was debated, if at all, in the most careful and euphemistic ways; it was de rigueur to refer to Muslim immigrants in an enthusiastic tone as \"our new countrymen\" (even if they despised the values of their new country) and to congratulate them for having enriched Europe by their presence (even if they had, in fact, been bleeding it dry by collecting massive welfare payments ever since their arrival).\n\nThe advent of the Progress Party changed all that. Founded in the 1970s by people fed up with sky-high taxation and devoted to individual rights, it is a classical liberal party that questions some of the fundamental assumptions of the social-democratic welfare state. It calls for lower taxes, a smaller government bureaucracy, stronger U.S. ties, support for NATO and Israel, reduced payments to the UN, and a more responsible approach to foreign aid (much of which currently goes into the pockets of African autocrats).\n\nIt also supports major immigration and integration policy reform. Though it believes in easing restraints on labor immigration, it does not support immigration by persons who are obviously going to go immediately onto Norway's welfare rolls. It also calls for a halt to the notorious \"fetching marriages\" whereby Norwegian Muslims wed their cousins abroad so as to provide the latter with Norwegian visas\u2014a practice that has crushed hopes of integration and also robbed so many individuals in the Muslim community of their freedom.\n\nIt is, in short, the only party that considers individual liberty more important than welfare state solidarity, and its positions have struck a chord with many ordinary Norwegians, who turned this once-small and isolated group into a major party. (For some years it was the second largest in parliament; in the days before July 22, it was third.) Many of its rank-and-file supporters are former members of Labor\u2014once the party of industrial workers, now largely a party of government bureaucrats whose main interest is in preserving the sprawling bureaucracy on which they thrive.\n\nHow did Norway's cultural elite respond to Progress Party's rise to power? Two words: sheer panic. Though perfectly willing to welcome into its ranks supporters of Mao and Stalin, the cultural elite consistently attempted to paint the Progress Party as a gang of dangerous extremists. All the major mainstream parties went along with this effort and joined in vilifying the Progress Party as unserious, irresponsible, \"populist,\" far right, and racist. This rhetoric, moreover, was picked up around the world by journalists who, utterly ignorant of Norwegian language and politics, simply echoed their sources in the Labor, Conservative, and other parties. (Not long ago, for instance, the Daily Telegraph labeled the Progress Party a \"fringe group\" and suggested its platforms were \"neo-Nazi.\") The fact is that on the American political spectrum the Progress Party would fall somewhere to the left of the Republican center. In any event, for all the vilification and demonization over the years, the party's enemies failed to vanquish it.\n\nNor could they silence a new willingness by many Norwegians, both in and outside the Progress Party, to speak home truths about Islam and multiculturalism. Over the last decade or so, as 9\/11 and later acts of jihad served as reminders that Islam was not all sweetness and light, and as the baleful consequences of multicultural policy in Europe reached such critical proportions that they could no longer be denied, the critics of Islam and of multiculturalism found their voices\u2014and found an audience. Parties that questioned the multicultural experiment won voters. In the months before July 22, the leaders of Western Europe's three largest countries, Angela Merkel, Nicolas Sarkozy, and David Cameron, pronounced the failure of multiculturalism. The multiculturalists, who a decade ago had held total sway over public opinion, were now on the defensive, the tide turning against them.\n\nAnd they were livid. In recent years, as they saw it, too many flowers had bloomed. Voices that dissented from the official consensus\u2014a consensus long set by them\u2014had multiplied. And this, as they saw it, was dangerous. It was sheer \"populism\"\u2014a phenomenon that in the view of elite Norwegian circles is a very, very bad thing. Part of the job of a left-wing cultural elite, they believed, was to protect a country from the boneheaded beliefs, attitudes, and prejudices of many members of the general public. In their view, America's whole problem was that the opinions of ordinary people were given too much weight.\n\nThat's not all. In disdaining \"populism,\" Norway's left-wing elite exhibits not only its class prejudice but its fear that the danger of a fascist resurgence is always lying just below the surface of the civilized, orderly society it has created. What today's left, not just in Norway but all over Western Europe, actually wants is to separate the socialist and nationalist aspects of fascist ideology, salvaging the one and jettisoning or suppressing the other.\n\nNorway's cultural elite, in short, didn't see the rise of the Progress Party as a positive, democratic development, a case of a party gaining power because it honestly and effectively represented the views of its supporters. No, they saw the Progress Party as a group of protofascists who were cynically exploiting fear and prejudice to gain power. Implicit in the elite view of these matters, obviously, is that ordinary people are too stupid to be trusted with democracy. Of course, in its contempt for the idea of government of the people\u2014as well as its distaste for American-style liberty\u2014the European elite is very much in sync with the Islamists, just as it was, before the fall of the Berlin Wall, very much in sync with Soviet communism.\n\nSo things stood as of July 21, 2011. And then along came Anders Behring Breivik. After countless acts of terror in the name of Allah, one Norwegian man had taken violent action against multiculturalism\u2014and thereby handed the multiculturalists a remarkable opportunity to regain the power that had slipped away from them in recent years and to vanquish those who had eroded their authority. With a single day's evil work, Breivik provided the multicultural left with the means to wrest back their power\u2014by linking their political opponents to the mass murderer.\n\nThe world has rarely seen such callousness so cynically disguised as a sensitive concern for social harmony.\n\nThe multiculturalists acted\u2014and with lightning speed. They lost no time in turning Breivik's actions to their advantage, brazenly maintaining that because he had opposed multiculturalism and the Islamization of Europe, everyone else who held such views also bore a share of the responsibility for his monstrous actions\u2014unless they repented now. The only proper response to Breivik, the Norwegian people were told in TV interviews and newspaper op-eds, was to reject criticism of Islam and accept multiculturalism wholeheartedly. Only in that way could they truly defeat Breivik\u2014and only in that way could they remove any suspicion that they, too, harbored the ugly prejudices that infected the murderer's soul.\n\nIt was an absurd line of argument, but\u2014to an amazing extent\u2014it worked. To understand why, we need to look at the kinds of ideas on which Norwegians have been nurtured throughout the modern era.\n\nThree weeks after Breivik's atrocities, on August 12, an article in Dagsavisen paid tribute to the Norwegian philosopher Arne N\u00e6ss, who died in 2009 at age ninety-six. N\u00e6ss was a rarity among philosophers, in that his was, during his lifetime, a household name in his own country. More than that, he was widely beloved (or at least was represented in the media as being widely beloved), for he was seen as having articulated the authentically Norwegian view of life\u2014good-hearted, great-spirited, nature-loving, and supremely unpretentious (and proud of it). It would not be unfair to call him \"the philosopher of niceness.\" In the Dagsavisen article in question, Truls Gjefsen recalled with admiration that during the Nazi occupation, N\u00e6ss had publicly criticized the Norwegian Resistance for stirring up hatred against Germans. Instead of treating the enemy as, um, the enemy, N\u00e6ss insisted, resistance fighters should be \"polite,\" \"friendly,\" and \"respectful\" to the occupiers, and should try to put themselves in the Nazis' shoes. After all, Nazis were people, too.\n\nAs Gjefsen summed it up, N\u00e6ss believed in \"the good in people.\" Gjefsen (who didn't explain how to reconcile N\u00e6ss's wisdom with the deportation of Norwegian Jews to death camps) proposed that Norwegians today, in the wake of July 22, should learn from N\u00e6ss's attitude toward the Nazis. Gjefsen didn't mean\u2014heavens, no\u2014that the left should stop demonizing the Progress Party and critics of Islam. What he meant was that everyone should shut up about Islam, link arms with people who, if given the power, would oppress or even kill them, and join in choruses of \"Kumbaya.\" (Gjefsen didn't seem to realize that he was implicitly accepting the idea, ordinarily anathema in the pages of Dagsavisen, that Islam can indeed legitimately be compared to Nazism, and that some Muslim leaders in Norway share certain traits with German officers who were stationed in Norway during the occupation.)\n\nGjefsen's article underscored a salient fact: that all too many members of the Norwegian cultural elite are inclined, like Arne N\u00e6ss, to be passive in the face of totalitarian tactics, and to attack those who dare to stand up to them. These multicultural elitists, who have elevated being \"nice\" to the status of an absolute moral principle, feel perversely compelled to befriend terrorist groups that target civilians and children and to give Nobel Peace Prizes to murderers. Also like N\u00e6ss, they tend to confuse this cowardice with a virtuous love for their fellowman.\n\nOnly to the untutored eye do these seem like contradictions. As we shall see, these and certain other ideas form the core of an ideology of politically correct leftism that appears to have first taken root in Norway in the years after the war, and that has now, mainly through the medium of universities and elite communications media, spread throughout the educated classes of Western Europe and North America.\n\nOne of this movement's core ideals is the exaltation of \"peace\" over all other values\u2014including freedom, which Americans consider the core ideal of their own political philosophy.\n\nTo live in Norway is to constantly hear Norway referred to as the \"peace nation.\" The country's cultural elite encourages Norwegians to be proud of the Nobel Peace Prize and of their country's role in international peace negotiations. And it has sought to foster a national self-image founded in the idea that, as former prime minister Gro Harlem Brundtland once famously put it, \"It is typically Norwegian to be good.\" Though most Norwegians have long since given up religious belief, deep down inside many of them, especially in the cultural elite, is, as I have suggested, an inner Lutheran\u2014a stiff-necked, holier-than-thou moralist missionary imbued with an overweening, unconscious condescension toward members of the lesser races, whom they are eager to help \"save\" in order to revel in their own virtue. (A leading embodiment of this principle is the insufferably pietistic and self-righteous Kjell Magne Bondevik of the Christian People's Party, an ordained minister who served two terms as prime minister between 1997 and 2005.)\n\nI have never seen a country's elite make more of a spectacle out of its foreign aid and charity efforts. Polls have shown that no people in the world have more faith in the United Nations as a promoter of international peace and goodwill than Norwegians do. (I blame this less on the Norwegian public, however, than on their media, which present them with an absurdly glowing picture of the UN.) Nowhere in the world, moreover, is peace studies taken more seriously than in Norway, where Johan Galtung, the aforementioned Maoist \"father of peace studies,\" is a veritable secular saint. This is, note well, a man who has said that the destruction of Washington, D.C., would be fair payback for American arrogance, who has praised Castro's Cuba for \"break[ing] free of imperialism's iron grip,\" who has mocked the West's preoccupation with \"persecuted elite personages\" such as Solzhenitsyn and Sakharov, and who has argued that while Mao's China was \"repressive in a certain liberal sense,\" it was \"endlessly liberating when seen from many other perspectives.\" In a commentary posted on August 29, French blogger Diane Berbain described an initiative that Norway's Anti-Racist Center had begun some months earlier and that (as I can attest) had been promoted ever since in splashy ads all over the public transit system in Oslo. The idea was simple: in order to overcome intercultural friction, non-Muslims should meet their Muslim neighbors over a cup of tea. \"After the murders of July,\" wrote Berbain, \"the 'Tea Time' crystallized all of the Norwegians' desires for harmony and for redemption.\" In her view, this harebrained scheme was a typical example of \"the cheerful nihilism of Norwegians\"\u2014a \"nihilism lite\" whose adherents, uneasy with social friction or debates about values, prefer to go with the politically correct flow, seeking only comfort, convenience, and unity (or a pretty illusion thereof).\n\nThis doesn't describe all Norwegians\u2014far from it\u2014but Berbain was on to something. America is a country of people from wildly different backgrounds\u2014E pluribus unum; Norway, until recently, was extraordinarily homogeneous ethnically, culturally, and religiously. Norwegians were like each other\u2014and they liked being alike. (One of the many strange things that I noticed during my first months in Norway was that when Norwegian audiences applaud, they applaud in unison: the sound of a Norwegian audience clapping is utterly unlike the sound of an American audience clapping.) Being different was not encouraged\u2014and being better at anything other than sports not only went unrewarded; it was often actively punished. (In 1933, this national pressure for conformity was given a name\u2014the \"Jante Law\"\u2014by novelist Aksel Sandemose, who codified it, in part, as follows: \"Don't think you're anything special. . . . Don't think you're smarter than us. . . . Don't think you are more important than us.\")\n\nHow to respond, then, to an influx into Norway of people so different from Norwegians in so many ways? On the one hand, Norwegians are not, by nature, comfortable with difference; on the other hand, they're also deeply uneasy about serious conflict. For years, while immigration proceeded apace, they chose conflict avoidance, keeping their heads down while the multiculturalists called the shots. Gradually, though, as I've noted, they began to stand up against looming differences they perceived as dangerous. The events of July 22 unsettled them profoundly, however, and when the cultural elite decreed that the appropriate Norwegian response\u2014the \"nice\" response\u2014to these atrocities would be a return to good old Norwegian unity and a papering over of differences, many were quick to acquiesce.\n\nTo a great extent, this response seems to have reflected a deep-seated need to believe, with Gro Harlem Brundtland, that \"it is typically Norwegian to be good.\" Earlier incidents had shown that when something happens to shake Norwegians' sense of their own virtue, the result is an earthshaking display of \"love\" and togetherness\u2014and uniformity\u2014by a constitutionally aloof people ordinarily not inclined even to smile at one another on the street or to say \"please\" when ordering a drink.\n\nCase in point: in 2001, not long after I moved to Norway, a neo-Nazi killed a Muslim teenager named Benjamin Hermansen in the Oslo neighborhood of Holmlia. Over the years a number of Norwegian teenagers have been killed by Muslims, but none of these murders has occasioned the kind of seismic reaction that Hermansen's death did\u2014for none of them made Norwegians feel as if their own goodness as a people had been called into question. After Hermansen's murder, massive candlelight processions were held across the country; the one in Oslo drew forty thousand people. The motive of these demonstrations was not only to express sympathy with the victim and hostility to the murderer, but to reassert Norwegian virtue, to make the statement: \"See? The murderer was Norwegian, but he didn't act in our names! We're good!\"\n\nNorway wasn't the only Western European country to lurch to the left after World War II. But in Norway the shaping of the \"peace country\" image, the notion of intrinsic Norwegian goodness promoted by the likes of Arne N\u00e6ss and Gro Harlem Brundtland, can be understood as a reaction not just against the wartime collaboration with Nazi brutality but against the inner Viking. What was so threatening to many left-wing Norwegians about the killing of Benjamin Hermansen, and what was many times more threatening about the atrocities of Breivik, is that these murderers seemed like the very personification of the inner Viking, that creature who would seem, in the view of many of them, to have always been threatening to arise out of the evil depths of \"nice\" Norway's national soul. To the collective-minded Norwegian, Breivik was a disconcerting phenomenon not only because he managed to do so much harm, but because his very existence seemed to give the lie to the myth of distinctive Norwegian goodness. And indeed that was how he presented himself in his manifesto\u2014as a rebel against the Labor Party image of what it means to be a Norwegian. He was a knight, resurrected out of the mists of the past, to lead a new generation of Viking warriors in a crusade against everything Gro and Arne stood for.\n\nRather than facing this challenge squarely and honestly, however, the multicultural left chose to account for Breivik in other terms: as the carrier\u2014and victim\u2014of a foreign infection, an ideological bacillus from without, imported by reactionaries, populists, conspiracy theorists\u2014and Jews.\n\nII\n\nThus far I have spoken a good deal about the cultural elite\u2014the left-wingers, the multiculturalists. But a better name for them\u2014especially in Norway\u2014would be Quislings. I am aware that this term is likely to strike the members of the Norwegian governing and media elite as highly insulting. That is precisely how I mean them to take it. And I will prove my case in the pages that follow.\n\nThe original Quisling was a Norwegian official who welcomed the Nazi occupiers during World War II and whose name, after the war, became synonymous with treason, collaboration, and appeasement. The New Quislings similarly welcome a new breed of totalitarian occupiers\u2014and seek to impose their own regime of ideological orthodoxy, ruthlessly using every tool at their disposal to silence their critics. What's more, they are in many cases the direct descendants of the ruling elite that followed Quisling during the war.\n\nWho was Vidkun Quisling?\n\nBorn on July 18, 1887, into an old and distinguished family in the county of Telemark, the son of a Church of Norway pastor, Quisling spent his teens in Skien (famous as the hometown of Henrik Ibsen). An exceptional student, he earned the top score on the entrance examination for the Norwegian Military Academy and graduated with unprecedentedly high scores from the Norwegian Military College. Joining the General Staff of the Norwegian Army at twenty, he was dispatched after the Bolshevik Revolution to Russia, where he admired the new rulers' control over their subjects, comparing Lenin's regime favorably with the more democratic government of Alexander Kerensky, which the Bolsheviks had so handily dispatched.\n\nLike many members of today's Norwegian elite, Quisling was, in a seeming paradox, a sincere humanitarian who at the same time admired autocracy and disdained individual liberty. In the 1920s he worked closely with Fridtjof Nansen, one of the true Norwegian heroes of the twentieth century, to try to ease the impact of the famine in the Ukraine. (Nansen considered Quisling's contribution vital to his efforts.) After living for a while in Paris\u2014where he developed a political philosophy, drawing on both Christianity and modern science, that he called Universism\u2014Quisling, who by now had left the army as a major, returned to Norway and joined the communist movement. Largely owing to his intense nationalism, his sympathies for communism soon soured, and in 1929 he founded a quasi-militaristic political organization, Norsk Aktion (Norwegian Action), modeled on the French fascist group Action Fran\u00e7aise. When Nansen died in 1930, Quisling suggested that Norway would best honor Nansen's legacy by focusing more on racial identity and instituting a strong government; not long after, serving as defense minister in an Agrarian Party cabinet (1931\u201333), the now firmly anti-Soviet Quisling called for war against the communists and for a ban on left-wing parties. And on Constitution Day 1933 (May 17), Quisling and a colleague formed a national-socialist party, Nasjonal Samling (National Unity), or NS, which was notable not only for its Nazi-like focus on the importance of a strong leader but also for its heavy emphasis on propaganda and for the strong support given to it by many members of the Oslo upper classes\u2014the denizens of the capital's posh west side.\n\nThis is a detail worth pausing over, because if one were to accept the received wisdom in Norway\u2014namely that the country's current left-wing elite is ideologically the polar opposite of Quisling and his NS\u2014it would seem more than a little puzzling that the leading lights of the present elite hail disproportionately from that same neighborhood. How, one might ask, could it be possible that the children, grandchildren, and great-grandchildren of people who may have supported Quisling during the war went on to fill the ranks of their country's postwar\u2014and very left-wing\u2014cultural establishment, the power base for the Labor Party and its multiculturalist agenda? Remember that Norway is a traditionally agricultural country whose politics, major media, and intellectual and cultural circles alike have been heavily dominated by a relatively small group of people who grew up on the same streets and went to the same handful of schools. To an American, any party called the Labor Party naturally sounds as if its power base lies in the working class; in today's Norway, however, Labor is the party of the chattering classes and power brokers. In the same way, while one would think that something calling itself the Progress Party would be a vehicle of the left-wing elite, it's in fact a party of the kind of people a certain U.S. president was referring to when he coined the term \"silent majority.\"\n\nAll of which is by way of pointing out that at least one curious aspect of the history of postwar Norway is the fact that many members of the country's contemporary cultural elite were raised in the same neighborhoods that once cheered Quisling. And this, in turn, is perhaps not unconnected to the fact\u2014also often obscured\u2014that Quisling was as much a socialist as he was a nationalist (Nazi, after all, is short for National Socialist), with a political philosophy that drew on both Hitler and Mussolini. Like many members of today's Norwegian elite, moreover, he held Jews in contempt and saw a baleful Jewish influence everywhere. While his effort to compel children to join the party's youth organization intensified public hostility toward him, he met with little resistance when he restored the constitutional ban on Jews in Norway that had been lifted in 1851, and, not long after, in quick succession, ordered the registration, rounding up, confinement in concentration camps, and, finally, deportation of Norwegian Jews.\n\nThe story of Quisling's path to power can be briefly told. When war broke out in September 1939, he and the Nasjonal Samling had already established very firm ties with the German regime. While the Nazis provided the NS with generous subsidies, Quisling provided the Nazis with information on Norwegian defenses. On April 9, 1940, the day the Germans invaded, Quisling became the first person in history to announce a coup on the radio. Hitler waffled, first recognizing his government, then (days later) withdrawing support and appointing a German Reichskommissar, then (two months on) naming Quisling head of government. (In 1942 he was granted even more power and the title of minister-president.) While the monarchy was officially abolished and all parties other than the Nasjonal Samling banned, party membership increased only minimally (indeed, it never exceeded forty thousand), suggesting that Quisling's ruling elite was, like today's, not necessarily representative of the Norwegian people at large.\n\n(It must be added, however, that in Norway, as in other European countries, the extent of wartime collaboration has long been downplayed; a rare honest look at this topic is Jo Nesb\u00f8's recent thriller, Redbreast, a story of elderly Norwegians who fought alongside the Germans and who resent the postwar cultural elite's denigration of what they still regard as having been a patriotic effort.)\n\nQuisling's regime was brutal. Striking workers and communist leaders were arrested, a Gestapo-like national police established, radios confiscated, and opponents of the regime executed. Though he pressed for greater Norwegian independence from Germany, the Germans exercised increasing authority as the war wound down, practicing greater brutality and causing resistance ranks to swell. At his war crimes trial in August 1945, Quisling argued that he had acted in his country's best interests and had fought to keep occupied Norway as independent as possible; but in the end he was sentenced to death (the government-in-exile having instituted the death penalty precisely for this purpose). He was executed by firing squad that October.\n\nQuisling was, naturally, often compared to Hitler, but personally and professionally he seems to have resembled, in many ways, some of the more typical members of today's political establishment in Norway\u2014which is to say that he came off (to many if not all observers) as civilized, dignified, sanguine, moderate, personable, hardworking, and modest and undemanding in his lifestyle. Though the New Quislings do not share his nationalism and his love of Norwegian mythology (while Hitler thought Germans were the master race, Quisling had similar ideas about Norwegians), they do share with him a belief in a strong state and a respect for the power of propaganda\u2014as well as a readiness to demonize Jews. And their apparently sincere tenderness for those they regard as helpless victims (in Quisling's case, the starving Ukrainians; in today's world, the starving Africans and put-upon Muslims) coexists with a pitiless readiness to crush all those whom they perceive as standing in the way of their efforts to translate ideology into reality.\n\nIt is important to note that while most Norwegians out in the hinterlands had little affection for Quisling, many members of the country's wartime establishment were ready enough to fall into line when it suited them. More than a few exulted, in particular, in the Nasjonal Samling's anti-American, anti-British, pro-Nazi posture. Among them was Nobel Prize\u2013winning novelist Knut Hamsun, who shared with a considerable number of his fellow Norwegian writers and intellectuals a contempt for the United States based entirely on reactionary grounds: in his eyes, America was a mongrel country and the wellspring of modern technology, and thus a threat to Norway's ethnic purity and traditional values. During the occupation, the octogenarian author publicly praised the Nazis, sent Goebbels his Nobel Prize medal as a gift, and had an audience with Hitler. (After the F\u00fchrer's death, Hamsun eulogized him as \"a warrior for mankind, and a prophet of the gospel of justice for all nations.\") Hamsun was far from alone. To read wartime issues of Aftenposten and other newspapers is to see politicians, professors, and journalists eager to praise their Nazi overlords, to echo the Germans' poisonous anti-Allied and anti-Semitic propaganda, and to convey the idea that Norwegian culture not only could continue to exist but could thrive under the benign oversight of the occupiers.\n\nYes, there were heroic Norwegian writers and intellectuals such as Sigrid Undset, who had always been a gutsy contrarian and who during the occupation, true to form, traveled around the United States as an eloquent spokeswoman for all of her fellow Norwegians who yearned to breathe free again. And there was a real Norwegian Resistance, whose valiant members deserve to be remembered forever. Indeed, the most successful Norwegian film ever made, Max Manus (2008), celebrates Norway's most famous resistance hero, and its popularity would appear to reflect Norwegians' pride in, and eagerness to identify with, those who stood up to Hitler. But to reside in Oslo during the war was to live in the midst of a cultural and political elite that showed few outward signs of resistance to Nazi tyranny.\n\nThe war's end marked an end to the Nasjonal Samling. Quisling's admirers disappeared quickly into the woodwork; today no name brings more shame to Norwegians than his. He is viewed as the country's single great historical embarrassment. But not everything associated with him and his party disappeared from the scene. Among the nation's cultural elite, the fondness for ideological conformity, the weakness for totalitarianism, and the belief that it is not only necessary but virtuous, in the name of national identity and social harmony, to destroy those who dissent from the party line\u2014which is to say, those who threaten peace and harmony\u2014all this endured, though in a somewhat muted form. Though ordinary Norwegians were, overwhelmingly, genuine lovers of freedom who would always remain grateful to the Allies for having made possible their liberation from Nazi tyranny, the people who came to form the country's postwar cultural elite were disproportionately enemies of America and romantic admirers of tyranny\u2014fans of Mao and Stalin, of Castro and Che, cheerleaders for communism, devotees of a lockstep, severely enforced \"solidarity\" that had little tolerance for independent thought or serious dissent. And those who did not sincerely admire tyranny were feckless appeasers of it.\n\nYet because they were of the left and not the right (though it has always been highly arguable whether a philosophy called National Socialism can be considered strictly a phenomenon of the right), the members of this new elite were viewed\u2014especially once the leftist propaganda had begun to be spread in the schools and official media\u2014as a triumphant rejection of everything Quisling had stood for. Similar political shifts took place across postwar Europe. The right had been discredited; the left was now in power\u2014except that many of the new leaders who presented themselves as heroes of anti-fascism had rather shaky credentials. (Kurt Waldheim, anyone?) In Norway many politically ambitious individuals who had, during the war, been cheerleaders for Quisling found a new and rewarding home on the left. Some may have been driven by shame over their parents' wartime histories and by a sincere need to redeem themselves, their families, and their class\u2014even if not all of them, perhaps, had shaken off all the beliefs and values that had driven their parents to support Quisling in the first place.\n\nThe result of all this is that the leading lights of the postwar Norwegian left, the stars of its cultural elite, proved in their own way to be Quislings, too\u2014only Quislings of the left, not the right. They were the bearers of a new ideology of politically correct leftism that bore significant earmarks of the fascist outlook and its propaganda methods. And the schools and universities, the rhetoric of politicians and the news media (hardly any less slanted, it could sometimes seem, than in Nazi times) passed on this Quisling-like mentality to the children and grandchildren of the nation's elite. The result: a governing class capable of giving a peace prize to Yasir Arafat and of embracing Hamas as a friend and partner.\n\nYes, there were always Norwegians\u2014many of them far from the corridors of power in Oslo, living in remote valleys and on rugged mountain farms where their ancestors had tilled the land from time immemorial\u2014who looked upon such activities with deep suspicion. But they had a long cultural history of shyness and aloofness, of provincial awkwardness, of deference to the educated and supposedly intelligent urban elites who shaped their country's fortunes. Not until the Progress Party came along and spoke for many of them did they find their voices, embrace their power as citizens of a free country, and pose a real challenge to their left-wing leaders in Oslo.\n\nBut then came July 22, 2011. And in the days and weeks that followed, those leaders\u2014the New Quislings\u2014saw their chance to retake the reins of power from people they saw as rubes and yokels, as provincial bigots and the ignorant pawns of \"populists.\" And they acted with a vengeance. In the electronic media and the pages of the newspapers, this was anything but a beautiful new era of love and harmony; it was an inquisition\u2014a time of brutal repudiations and denunciations, of ritual confession and capitulation. People who had been treated as respectable citizens a few days earlier were now being savaged as mentors to a murderer.\n\nIt was a classic example of what might be called the \"germ theory\" of ideological transference, straight out of the old fascist playbook. Unable to rebut opposing arguments using facts and logic, the multiculturalist left stigmatizes its ideological opponents by warning darkly that those opponents' views are \"extreme\" and \"dangerous\" and lead to violence. Such rhetoric, in the days after July 22, seemed to be applied disproportionately to foreigners, to people like me who lived in Norway but had come from somewhere else, and most especially, to Jews and their ideological sympathizers. We \"foreigners\" had introduced a deadly bacillus into pure, good Norway. It was all unsettlingly reminiscent of the anti-Semitic propaganda of the Nazis. The multiculturalists' highly disciplined prosecution of this theory\u2014as well as their effortless mastery of Goebbels-like propaganda techniques\u2014were on full display in the wake of the massacre. Watching Norwegian TV and reading Norwegian newspapers in the first days and weeks after the events of July 22 was a chilling experience for anyone who cherished freedom of speech. Opposition writers, politicians, and editors were now being portrayed by the media essentially as accomplices in the murders in Oslo and at Ut\u00f8ya, and \"interviews\" with them were not really interviews at all. These people were being called on the carpet to account for themselves\u2014to show contrition and to promise to reform. It was like something out of Stalinist Russia\u2014which was no surprise, given that many of the people who were running this inquisition were indeed communists.\n\nSo it was that in the days after July 22, Norway's newspapers and TV and radio news programs rang with mea culpas. Men and women who had criticized Islam or expressed doubts about Norwegian immigration policy or questioned multiculturalism now \"crept to the cross,\" to use a Norwegian expression\u2014meaning that they fell into line, writing articles to indicate that they were on the \"right side,\" and insisting that if they had ever written or said anything to indicate otherwise, they now repudiated it. It was clear that many people in the public eye felt pressured to make a great show, as it was repeatedly said, of \"looking into their hearts\" and \"examining their souls,\" and of admitting to what George Orwell would have called Thoughtcrimes.\n\nIn a piece that was typical of the kind of journalism that filled the Norwegian newspapers in the days after July 22, Roar Helgheim, a journalist for Dag og Tid, recalled on August 12 that when the attacks first took place, he had written that if the perpetrator turned out to be an immigrant or a Muslim, it would be devastating to those in Norway who had been too na\u00efve about the dangers of Islamic terrorism. But the fact that the perpetrator turned out to be \"one of 'us,' \" he wrote, made him think again. Now his view was that the warmth and love that had blossomed in Norway in the days after July 22 had dealt \"a crushing blow to the perpetrator who wanted to destroy diversity and the multicultural Norway.\"\n\nThis was the new meme. But there was no logic in it. To suggest that Breivik could have destroyed \"diversity and the multicultural Norway\" is to make him into something larger than he was. People wrote about Norway's response to Breivik's actions\u2014the flowers and candles, the crowds in the streets\u2014as if it were a calculated political statement. No, it was anything but political: it was a very natural, human, and (as I have suggested) psychologically complex and culturally rooted outpouring of sympathy and solidarity\u2014which certain people then began to exploit politically by carrying out a witch hunt.\n\nLike everyone else, needless to say, I was crushed at the news of the attacks of July 22. But when I learned that they were the work of a native Norwegian who claimed to have acted in opposition to his country's multicultural policies, I was even more devastated because I saw at once what this would mean. Norway, I knew, was now in an even greater danger than before\u2014in danger of losing its chance at a free, secure, and prosperous future. As I wrote in my Wall Street Journal piece, \"to speak negatively about any aspect of the Muslim faith has always been a touchy matter [in Norway], inviting charges of 'Islamophobia' and racism.\" But it would, I feared, \"be a great deal more difficult . . . now that this murderous madman has become the poster boy for the criticism of Islam.\"\n\nThis statement would later come in for harsh criticism from the multicultural left in Norway and elsewhere: how dare anyone speak of such issues at a time like this! But this was not an abstract consideration. This, like the atrocities in Oslo and at Ut\u00f8ya, was a matter of real human lives\u2014perhaps millions of them. For Breivik had committed acts that, thanks to the cultural elite's cynical efforts, would indeed come to be widely seen as discrediting honest and open discussion of Islam, immigration, integration, and multiculturalism\u2014a discussion whose whole point was to ensure that young Norwegians like those kids at Ut\u00f8ya, and even younger Norwegians like my two-year-old nephew in a small town in Telemark, would live their lives in freedom and prosperity.\n\nAnd if those critics protested against the witch hunt, then they were further accused of dishonestly representing themselves as victims of a witch hunt. And, moreover, of rank insensitivity: how dare they represent themselves as being victims, when the real victims were those poor kids at Ut\u00f8ya\u2014and, of course, the Muslims? Instead of representing themselves as victims, the critics of Islam and of multiculturalism were told, they should be looking into their hearts, doing some serious self-searching. (This Stalinist formulation appeared again and again.) How dare they take up such divisive issues as multiculturalism and Islam, thundered the scolds of the left, when the country was so beautifully united in grief? Thus did the New Quislings cynically use the image of a country \"united in grief\" as a pretext to attack and destroy their ideological opponents.\n\nWho, exactly, are the New Quislings?\n\nWell, one of them is a young Dagbladet opinion editor, Simen Ekern, who during the week after the murders wrote a piece headlined \"We can't ignore the ideological mudbath from which this murderer emerged.\" In his piece, Ekern dragged readers into his own slimy mudbath of scurrilous accusation and innuendo. The murderer, he wrote, \"has several role models.\" My name came first; it was followed by several others. Ekern summed us up as \"the new extreme European right.\" He cited, as if in approval, the official government directive that nobody should \"make political hay\" out of the events of July 22\u2014and then proceeded to do precisely that. He claimed that he intended \"to avoid demonizing all criticism of immigration and multicultural challenges\"\u2014then demonized away.\n\nIt was necessary, Ekern argued, \"to take the clear ideological elements of the killer's disgusting project seriously\" and \"to warn against the tendencies to turn the mass murderer into a lone wolf.\" Well, of course\u2014if he's a lone wolf, he's no use as a club with which to batter one's ideological opponents. Ekern proceeded to question freedom of speech\u2014or, as he put it, the \"instinct, in a free society, to air all views.\" Such an \"instinct,\" he declared, \"is a nice thought,\" but a dangerous one, for \"our society is not improved by cultivating ever more 'honest' and 'brave' warlike Crusader rhetoric directed against Islam. . . . Much 'Islam criticism' is . . . screaming dressed up as debate. . . . It is dangerous to spread fear.\"\n\nI would say in response to this slick formulation that it is not dangerous, but salutary, to spread correct information about a genuinely fearful enemy\u2014and that it is evil to demonize people who have done good and important work in the cause of freedom, and to try to exploit the murders of young people to silence debate about an issue of national importance. As a rule, moreover, it is not the Norwegian critics of Islam and of multicultural immigration policies who are engaged in \"screaming dressed up as debate.\" It is Ekern and his ilk who routinely reply to the cogent arguments of their opponents not with serious arguments of their own but with personal attacks (\"warlike Crusader rhetoric\").\n\nBy accusing his ideological opponents of wallowing in an \"ideological mudbath,\" even as he himself proffered nothing but character assassination, Ekern established himself as a worthy member of the New Quisling crowd.\n\nNot content to restrict their efforts to Norway itself, some of the New Quislings exported their scapegoating. Petter Nome, a longtime journalist and host on Norwegian national television who is now (oddly enough) head of the Norwegian Brewers' Association, chose to focus on the fact that Breivik, from 1997 to 2007, had been a member of the youth movement of the Progress Party. In an article charmingly titled \"To You Who Nourished the Killer,\" published on July 25 in the Spanish daily El Mundo, of all places, Nome, like Ekern, refused to dismiss the murderer as a lunatic, purportedly because \"making this a mental issue is a dangerous dead end road\" (yes, a dead end\u2014once again\u2014for those out to make political hay of Breivik's actions).\n\nThough the murderer's views were \"obviously extreme and pervaded with hate,\" Nome maintained, they were \"not obscure nonsense in the mind of a freak. They are all too well present in everyday conversations in streets and pubs\u2014and mainstream politics.\" He then listed several of the murderer's political views, none of which I have ever heard anyone express in a Norwegian street or pub: a desire to \"[r]eplace western democracies with administrative monarchies or republics,\" to \"increase the birth rate in western countries by banning abortion,\" to give \"[m]ore cultural power to the church,\" to put criminals to death \"after three criminal convictions,\" to put drug addicts in concentration camps, and to carry out \"[f]orced reeducation of Marxists.\"\n\nNome noted that Breivik had belonged to the Progress Party and had said that he \"identifies himself with Christian fundamentalism and strongly supports the state of Israel\"\u2014all of which, in official Norway, are of course highly suspicious associations. Nome further observed that Progress Party leader Siv Jensen \"claims she was shocked\" that Breivik had been a party member. \"But,\" asked Nome, \"did she ever carry one single brick to the bridge most of us are trying to build between people and cultures?\" To which I would say: the Progress Party has encouraged immigration by individuals who want to come to Norway to enjoy its freedoms and contribute to its economy; it opposes immigration by people who are hostile to Western values and seek to exploit the welfare state. The Progress Party recognizes that there are certain aspects of certain cultures to which one should not want to build bridges.\n\n(In a magnificent piece of reportage for the Weekly Standard that went online on July 30, James Kirchick exposed the outright lies told by Nome and others about the Progress Party. Kirchick talked with several Muslim members of the party, including Farida Amin, a Norwegian-Pakistani who was drawn to the party by, in her words, \"[i]ts emphasis on assimilation, and its concern for the harsh treatment that many Muslim women in Norway receive at the hands of their male relatives.\" This is the Progress Party I know, and that Nome and other Norwegian socialists have labored so hard to misrepresent and demonize.)\n\n\"Did she ever try,\" Nome thundered, referring to Siv Jensen, \"to make electoral catches [sic; I am quoting Nome's official English version of his text] with her talk about 'islamisation' and 'national' and 'Christian' values?\" Jensen has indeed talked unapologetically about Norwegian values\u2014by which she means things like individual liberty and sexual equality\u2014and about the danger posed to them by cultures in which women are systematically treated as inferiors and threatened with abuse or worse if they get out of line.\n\nThough Nome acknowledged that Jensen and \"most of her colleagues in populist and right wing parties in Europe\" are not supporters of violence, they \"carry profound responsibility for actively creating a climate where hate and violence appear as options for their most impatient followers.\" The Progress Party has not \"creat[ed] a climate,\" however much that may seem to be the case in the eyes of cultural elite types like Nome, who still cannot figure out how the Progress Party has managed to gain so much support and mount a challenge to their power. Rather than \"creat[e] a climate,\" the party has listened to, and articulated, the concerns of very many Norwegian people about a range of issues that the cultural elite long refused to address.\n\nAfter taking a break from his nasty screed to paint a pretty picture of how Norway was, at the moment, \"united in grief and sorrow,\" the streets filled with flowers and candles, Nome added darkly that \"a tomorrow must also come, for the vital questions of responsibility and lessons learned.\" This rhetoric was not only ugly but more than vaguely threatening: today we mourn together\u2014tomorrow comes the reckoning. Wait for the knock at the door\u2014we'll be there soon.\n\nNome's readiness to accuse his ideological opponents of creating an atmosphere of hatred and thereby \"nourishing\" a killer\u2014when in fact they had done no such thing\u2014and to hint at future retribution certainly recalls the kind of rhetoric once served up by the head of the Nasjonal Samling.\n\nOn July 28, two men who are very famous in Norway, novelist Jostein Gaarder and professor Thomas Hylland Eriksen, contributed an op-ed to the New York Times titled \"A Blogosphere of Bigots.\" Rejecting the idea that Breivik was \"an isolated case of pure evil,\" they wrote that \"the hatred and contempt from which he drew his deranged determination were shared with many others throughout the international right-wing blogosphere,\" which they described as consisting of \"loosely connected networks of people\u2014including students, civil servants, capitalists, and neo-Nazis,\" many of whom \"do not even see themselves as 'right-wing,' but as defenders of enlightened values, including feminism.\"\n\nBut those who see themselves as defending enlightened values are wrong, according to Gaarder and Hylland Eriksen. The authors did not take the trouble to explain how this curious business works psychologically\u2014how does it happen that we benighted souls think we are actually liberals concerned about things like human rights, sexual equality, and individual liberty, when in fact we are loathsome far-right bigots? Oh well. The first writer they named as an example of this execrable \"right-wing blogosphere\" phenomenon was me. Saying that the murderer \"has praised writers like Bruce Bawer,\" they characterized me and others as consisting of a \"new right\" in Europe that \"has swapped anti-Semitism for Islamophobia. . . . Traditional racism may actually be waning in several European countries, but hostility toward Islam and animosity toward Muslim immigrants and their children is on the rise.\"\n\nIn a masterpiece of understatement, Gaarder and Hylland Eriksen wrote that \"Norwegian society is changing, and rapid immigration has no doubt led to tensions.\" And in a statement that in any other context would be laughable, and that provides a useful example of the Norwegian habit of using \"immigrant\" as a synonym for \"Muslim,\" they referred to \"[t]he perception that immigrants are patriarchal and insular\"\u2014a \"perception,\" they went on to say, that \"has sparked controversies over everything from school excursions to swimming lessons to disrespect for female teachers.\"\n\nAn interesting sentence, in which the authors, in deft multicultural fashion, neatly avoided explaining what they were actually referring to. For example, \"disrespect for female teachers\" refers to the fact that all too many Muslim boys are taught at home to show no respect for non-Muslim adults, especially women\u2014a state of affairs that results in a great deal of unpleasantness for their female teachers. These \"controversies\" have nothing to do with \"perception\"\u2014they're about hard realities that many Norwegian schoolteachers (unlike, say, celebrity professors) have to deal with every day.\n\n\"Conceding that a culturally diverse society raises knotty and complex social and political questions is one thing,\" wrote Gaarder and Hylland Eriksen in a sentence rife with euphemistic abstraction. \"It is quite another to state that a multicultural society is impossible, or that Islam is incompatible with democracy.\" Tell that to Angela Merkel, David Cameron, and Nicolas Sarkozy, all of whom have declared the failure of multiculturalism. Are they far-right bigots, too?\n\n\"We hope,\" Gaarder and Hylland Eriksen went on to say, \"that Norway's longstanding consensus about immigration and integration policies will not be eroded.\" Consensus? There has been no such consensus for many years, except among elites like Gaarder and Hylland Eriksen. This is precisely why these men were exploiting July 22 to the hilt: it provided a wonderful opportunity to bring the country back into line and take back their power.\n\n\"Until last week,\" Gaarder and Hylland Eriksen concluded, \"Norwegian authorities did not see the far right as a security threat. Mr. Breivik has now shown that those who claim to protect the next generation of Norwegians against Islamist extremism are, in fact, the greater menace.\" This breathtaking last line pointed directly at these men's dark agenda: to convince as many people as possible, not only in Norway but internationally, that the real danger to Norway and other Western countries comes not from jihadists of the sort that carried out 9\/11, London, Madrid, Bali, Mumbai, and innumerable other terrorist attacks in the name of their faith\u2014or from the expansion in Western cities of communities of people whose imams preach the oppression of women, equate Jews with pigs and dogs, and call for the execution of gays\u2014but from writers who dare to warn of these terrible things.\n\nWho exactly, you might ask, are Jostein Gaarder and Thomas Hylland Eriksen? The contributor's note to the Times op-ed identifies the former as \"the author of 'Sophie's World' and many other books\" and the latter as \"a professor of social anthropology at the University of Oslo.\" In fact, these men who are so eager to push the meme that Muslims are Europe's new Jews happen to be two of the vilest anti-Semites in Norway.\n\nIn a now-famous 2006 op-ed for Aftenposten, sarcastically titled \"God's Chosen People,\" Gaarder, writing in the royal \"we,\" consigned Israel to the dustbin of history. Through its actions, he insisted, it had lost its right to exist: \"We no longer recognize the state of Israel. . . . The state of Israel in its current form is history. We don't believe in the idea of God's chosen people. . . . To present oneself as God's chosen people is not just stupid and arrogant, but a crime against humanity. We call it racism.\"\n\nGaarder went on: \"There are limits to our patience and there are limits to our tolerance. We do not believe in divine promises as a justification for occupation and apartheid. We have placed the Middle Ages behind us. We laugh uneasily at those who still believe that the God of the flora, the fauna, and the galaxies has chosen a certain people as his favorites and given them funny stone tablets, burning bushes, and a license to kill.\" (In the Norwegian original, those last three words are in English.) \"We laugh at this people's caprices and weep over their misdeeds,\" he wrote, and suggested that many Israelis celebrate the deaths of Lebanese children just \"as they once cheered the plagues of the Lord as 'fitting punishment' for the Egyptian people.\" He further envisioned \"little Israeli girls writing hateful greetings on the bombs to be dropped on civilian populations in Lebanon and Palestine\" and \"strut[ting] with glee over the death and torment.\" As if it were Israelis who were putting guns into the hands of children!\n\nThere was much more in this vein. To see such a screed in the pages of Norway's newspaper of record\u2014even after decades of Nazi-style Jew-baiting cartoons\u2014was jaw-dropping. Author Mona Levin, perhaps the best-known member of Norway's small Jewish community, wrote that Gaarder's piece was \"the ugliest thing I have read since Mein Kampf.\" But Hylland Eriksen, along with several other high-profile Norwegian writers and intellectuals, rushed to second Gaarder's twisted remarks. Gaarder suffered no negative repercussions for his op-ed; on the contrary, a year or so later I saw his bearded face, two or three feet high, beaming out at me from the side of a bus stop shelter. It was an ad for a children's book club.\n\nGaarder and Hylland Eriksen, in short, are Quislings of the first order, soft-pedaling the menace of Islamic totalitarianism while casting its opponents, liberal and otherwise, as the real threat.\n\nThere was more. On August 1, the Norwegian newspaper VG published an interview with Lars Gule, the former head of the Norwegian Humanist Association and a very high-profile figure in Norway. The headline: \"We need an oppgj\u00f8r with Breivik's 'heroes.' \" I have not translated oppgj\u00f8r because in this context the word can mean anything from \"We need to have a sharp talk with Breivik's 'heroes' \" to \"We need to do something about Breivik's 'heroes.' \" In other words, it can have a very dark meaning\u2014it can conjure images of the Gestapo knocking on one's door in the middle of the night. \"It is obvious,\" said Gule, \"that certain groups, persons, and communities have contributed to Breivik's warped view of reality, and these people need to take a good look at themselves. If not, others must help them.\"\n\nOthers must help them. This sentence immediately summoned visions of Maoist reeducation camps, and its sinister invocation is an authentic mark of the New Quisling. \"For they have contributed,\" Gule went on, \"to an extreme world view that has led this particular person to an extreme act. No one other than Breivik has any criminal responsibility for his actions. But if you run, in a cowardly and pathetic way, from the moral responsibility of having contributed to Breivik's world view, you're out of touch with reality.\"\n\nVG identified Gule as a \"professor and expert on multiculturalism.\" In fact, Gule (born in 1955) is a considerably more colorful figure than that description would suggest, and a man whose career is one that could hardly have happened anywhere but in Norway. In 1977, in his early twenties, he joined the Democratic Front for the Liberation of Palestine, a terrorist group responsible for the 1974 Maalot massacre in which more than one hundred Israeli children were taken hostage and twenty-two of them killed. Gule was delegated to set off a bomb in Israel on the tenth anniversary of the Six-Day War. At the Beirut airport, however, he was caught with 750 grams of explosives hidden in books in his backpack.\n\nHe was imprisoned for six months in Lebanon, then returned to Norway\u2014where he proceeded to enjoy an illustrious career, holding prominent positions at the Centre for the Study of the Sciences and Humanities, in the human rights program of the Christian Michelsen Institute, and at the Centre for Development Studies at the University of Bergen. In 2000, he was appointed general secretary of the Norwegian Humanist Society. During his years in that post he was, not surprisingly, a harsh critic of Israel (which he called a racist state) and a staunch defender of Muslims' right to discriminate against women and gays\u2014this at a time when he was his nation's leading official spokesman for secular values.\n\nOn August 16, another high-profile Oslo figure had his say. Anders Heger is the longtime head of Cappelen, one of Norway's three major publishing houses. He is also a columnist for Dagsavisen, where he described Breivik's rampage as an act of \"violence from the extreme right\" and expressed astonishment that while this violence \"has killed 77 people . . . what people are upset about is that they can't criticize Islam the way they did before.\" He assured readers that \"the political debate will continue\" but added that it would not be the same debate as it had been: \"Nothing will be entirely as before. The debate will be more open and inclusive, and less judgmental.\"\n\nWhy? Heger didn't say. He acted as if this new style of debate were a foregone conclusion, a natural consequence of what had happened on July 22. In fact, this new style of debate was plainly a new Norwegian order that the political and media establishment, led by the Labor Party, was trying to ram down the country's throat in the wake of the atrocities. It was also clear that while the new debate would indeed be \"more open and inclusive, and less judgmental\" of Islamists, it would be decidedly less open and inclusive, and more judgmental, when it came to critics of Islam. Indeed, it was now open season on Islam critics.\n\nHeger noted the calls by Stoltenberg and other political leaders in the wake of July 22 for civility\u2014no one, they decreed, should try to exploit Breivik's actions to score political points or carry out witch hunts. Heger said that this sounded good yet he complained that this enforced civility would make it harder to address xenophobia and intolerance. In other words, should we be civil to bigots, too? His unspoken assumption here was that criticism of Islamic theology amounts to bigotry, to \"hate rhetoric,\" to \"xenophobia,\" to \"contempt for human beings\"\u2014all of which, he said, must not be treated with civility but must rather \"be confronted and fought against.\"\n\nHe made no mention of the fact that it is the theology being criticized by these so-called bigots that is, in fact, rife with bigotry\u2014bigotry toward women, bigotry toward Jews, bigotry toward gays, and others. It has always been a cornerstone of cultural-elite Norwegian thinking to turn a blind eye to the monstrous bigotry that is part and parcel of orthodox Islamic theology; now the cultural elite was seeking to impose this willful blindness upon the entire country\u2014using the events of July 22 as a stick with which to pummel Progress Party supporters, among others, into an ideological retreat.\n\nHeger said that those on the \"extreme right\"\u2014meaning critics of Islam and of current Norwegian immigration policies\u2014viewed themselves as being \"forced into silence\" by the standard-bearers of \"political correctness\" who label all \"legitimate skepticism about immigration\" as racist. Heger dismissed this view even as he confirmed it by his own action of calling these critics \"extreme right.\" As for being \"forced into silence\"\u2014yes, a few years ago it was almost impossible for dissenters from official policy on Islam and immigration to be heard in Norway's mainstream media. More recently\u2014at least until July 22\u2014it was more possible, though in order to be heard, those critics had to deal with abuse by people like Heger, who instead of answering their reasonable arguments with reasonable arguments of their own, chose to demonize them and to depict their concern about real threats to liberal values (such as the equality of women) as far-right racism.\n\nHeger then brought me into the picture\u2014\"[t]he chronically Islam-hating American Bruce Bawer, who for unclear reasons is viewed as an important social critic in large areas of the Norwegian commentariat\"\u2014and quoted from my Wall Street Journal piece: \"It was immediately clear to me that [Breivik's] violence will deal a heavy blow to an urgent cause.\" Heger agreed that Breivik's actions had threatened \"an urgent cause,\" but he claimed that the \"urgent cause\" being threatened was, in fact, \"the open society, a democratic sense of community, social democracy, or diversity.\"\n\nNote how Heger praised \"the open society\" and \"diversity\" even as he was busy dismissing and demonizing an ideological opponent. This is Quisling thinking in a nutshell: they claim to believe, and many of them actually appear to think they believe, in an \"open society\" and \"diversity\"\u2014but it is clear that by these things they mean, in a profoundly Orwellian sense, a society of lockstep support for Norwegian-style social democracy (which they regard as goodness set in system) and a readiness to crush any opposition thereto. Heger sneered when he quoted my prediction that it would be harder to discuss Islam and immigration after July 22, but his entire article only proved me right; his whole point was that if people like me could not be ridiculed and intimidated into staying silent in the wake of July 22, then Norwegians as a whole needed to take action to silence us, in the name of Norwegian virtue and social harmony.\n\nOne of the New Quislings' chief targets was my friend Hege Storhaug of Human Rights Service, who has devoted most of her adult life to the struggle to secure greater individual liberty and a higher quality of life for Muslim women and girls in Europe. On July 27 she was invited to call in to the NRK news interview program Dagsnytt, which is broadcast simultaneously on television and radio. Hege's fellow participants, she was told, would be Per Fugelli, who had already blamed the Progress Party in part for the murders, and Magnus Marsdal, a young communist who had written a witheringly snide book about the Progress Party. (Marsdal exemplifies a true only-in-Norway situation: the classical liberal Progress Party receives about twenty to forty times as many votes as his totalitarian party does, but he's treated by the media and the rest of the cultural elite as a respectable mainstream voice while members of the Progress Party are reviled as extremists.)\n\nThe host promised Hege that it would be a decent discussion with no personal attacks. Yet, as she wrote on August 1, it turned out to be \"the worst debate I have ever experienced.\" Marsdal was permitted to \"Nazify HRS and me.\" He had brought along quotations from work by Hege and her HRS colleague Rita Karlsen, which he distorted in good Stalinist fashion: for example, an article in which Rita expressed concern about young girls being forced to wear hijab was twisted, in effect, into a suggestion that the girls were wearing hijab voluntarily in order to serve as \"warriors for Islam.\" We are talking here about two women who have striven most of their adult lives to help ensure that Muslim girls and women can live in the West as freely as their non-Muslim counterparts.\n\nOn August 1, Dagsnytt again welcomed Marsdal, who this time went on the attack against Aftenposten opinion page editor Knut Olav \u00c5m\u00e5s for having published too many articles critical of Islam and immigration\u2014and, in particular, for having praised my book While Europe Slept. \u00c5m\u00e5s, who participated in the discussion on Dagsnytt by telephone (and who has published op-eds by me), defended me, saying that I'm not an \"extreme writer\" but a writer who is concerned about the dangers posed to liberal society by fundamentalist religion. He insisted that all the writers that the Norwegian terrorist happened to read cannot be held responsible for his actions. But \u00c5m\u00e5s was clearly on the defensive\u2014for Norway had entered a new era. The program host had introduced the segment by saying, chillingly: \"Many people have felt that Aftenposten has let a hundred flowers bloom. But some of those flowers will be clipped now.\" Mao, too, one recalled, had let flowers bloom, and then clipped them ruthlessly.\n\nThere are, insisted Marsdal, connections between opinions and actions. So opinions must be confronted. \"Xenophobia and fascism\" must be confronted, he said\u2014and it was clear that by \"xenophobia and fascism\" he meant dissent from the far-left line, and that by \"confronted\" he meant crushed. He savaged me, Hege, and others for creating a picture of Muslims as the enemy; now he was doing his best to try to persuade all of Norway to see critics of Islam as the enemy.\n\nHege came under attack again on August 5, when Jens Brun-Pedersen, press representative for the Humanist Society, wrote a piece in Dagbladet in which he listed quotations from her about Islam\u2014and, instead of disputing what she had written, simply acted as if it were obvious that such outrageous statements could not be true. For example: \"Child marriages happen in Norway, too, in imitation of Muhammed's example.\" And: there are aspects of Muslim congregations that represent \"a threat to human rights and democracy.\"\n\nThese are statements of fact. Yet after July 22, such truths were not to be spoken. Instead, people like Hege were supposed to begin examining their \"use of words and tendency to generalize that may have helped fertilize the soil in which the terrorist was nourished.\" Hege, suggested Brun-Pedersen, should be \"looking into her heart.\" As if she had not been listening to her heart all along!\n\nIn an article published in Dagsavisen on August 8, a young man named Aksel Kj\u00e6r Vidnes, who (as a quick Google search revealed) received a master's degree in sociology in 2008, called Hege \"totalitarian and fanatical,\" attributed to her \"the same attitudes that characterize parts of Breivik's mental universe,\" and accused her of \"spread[ing] one-sided representations of the danger of Islam by presenting the victims of assault as anecdotal evidence.\"\n\nHege, Vidnes wrote, \"criticizes religion and the religious, often justly, but without taking into consideration that the groups she contributes to prejudice against consist of people. Individuals with feelings.\" As if there were any ethnic Norwegian more familiar with those individuals' feelings than Hege! Vidnes thundered that \"Human Rights Service is not about human rights. . . . the organization's primary work is to be critical of immigration and critical of the multicultural society. . . . Hege Storhaug . . . apparently forgets that human rights are not just for her. They're for everybody.\" Vidnes closed by suggesting that Hege \"engage in some self-reflection.\"\n\nThis was elite Norway's new Maoist mantra: \"reflect\"\u2014and reform. Or else.\n\nAlso on August 8, Marte Michelet, appearing on the radio program Radioselskapet on NRK P2, linked Hege to Oriana Fallaci and Ayaan Hirsi Ali, describing all three women as having inspired Breivik. Michelet echoed Marsdal's calumny that Hege \"speaks of all children in hijab as warriors for Islam,\" and the program host, Nina S. Martin, referring to \"Hege Storhaug and that gang,\" described her as part of a faction of people who hate immigrants. (It is perhaps worth noting that many Norwegians refer to NRK as ARK\u2014\"Arbeiderpartiets Rikskringkasting,\" or the Labor Party Broadcasting System.)\n\nA personal note: In the weeks before Breivik's atrocities, I had traveled to Ottawa, Montreal, and Philadelphia to introduce Hege at gatherings where she gave talks about her work in Norway. At these events she had shown pictures of Muslim girls with Norwegian citizenship who, as a consequence of the hands-off, see-no-evil multiculturalism of Norwegian authorities, were being \"educated\" in horrible, warehouse-like Koran schools in Pakistan and living in equally execrable situations in Gambia. The point was that no Norwegian government would ever have allowed ethnic Norwegian girls to suffer such fates: for these girls, multiculturalism was lethal, for it meant that in the eyes of many of the authorities who should be protecting them, their \"cultures\" deserved more respect than they did.\n\nHege's feeling for those girls was palpable. The high and noble purpose of her work, and her life, came through powerfully.\n\nI have known Hege for a decade now, and I know her through and through. Her organization, Human Rights Service, has formulated legislation that has palpably improved the lives of Muslim women and girls in both Norway and Denmark. I know as well as I know anything that Hege's heart bleeds more every day for individual Muslim women and girls than the typical member of Norway's preening, self-righteous cultural elite\u2014so ready to spew out vapid rhetoric about diversity and \"the colorful society\"\u2014could ever imagine.\n\nIn short, I have never known anyone as genuinely and selflessly dedicated as Hege is to securing freedom and human rights for Muslim women and girls. She has put herself in harm's way in some of the diciest places on earth in order to help people whom she did not know personally and who would probably never know what she had done on their behalf. She is extraordinarily strong and courageous, but I have seen her break into tears over the fate of Muslim women and girls who have crossed her path. I know she is haunted by the memory of Anooshe, an extraordinarily young Muslim woman who came to Norway and divorced her husband only to be gunned down by him outside a courthouse, where they had an appointment to discuss child custody.\n\nYet now, in the wake of July 22, the word was being spread that a woman like Hege\u2014a woman of whom Norwegians should be as proud as they are of their resistance heroes\u2014was to be viewed as a heartless monster.\n\nPerhaps it is no coincidence that Hege\u2014who knows what it means to stand up for freedom when so many of those around her seem to have forgotten the meaning of the word\u2014is, in fact, the daughter of a hero of the Norwegian Resistance.\n\nThe New Quislings are not confined to Norway. They are also to be found on the other side of the Atlantic, among some of America's leading journalists and intellectuals.\n\n\"Bawer vs. Bawer\" ran the headline on my old friend Andrew Sullivan's blog on July 25. It was followed by quotations from my Pajamas Media piece, which had appeared on July 23, and my Wall Street Journal piece, which had run two days later. Here are the quotations:\n\nWhen I first heard the news of the explosions at those buildings, my first thought, of course, was that it was a jihadist attack. But it wasn't: it was a right-wing lunatic. It wasn't jihad. It was a meaningless killing spree by a madman, like the ones at Columbine and Virginia Tech.\n\nIn bombing those government buildings and hunting down those campers, Breivik was not taking out people randomly. He considered the Labor Party, Norway's dominant party since World War II, responsible for policies that are leading to the Islamization of Europe\u2014and thus guilty of treason. The Oslo bombing was intended to be an execution of the party's current leaders. The massacre at the camp\u2014where young would-be politicians gathered to hear speeches by Prime Minister Jens Stoltenberg and former Prime Minister Gro Harlem Brundtland\u2014was meant to destroy its next generation of leaders.\n\nAndrew asserted that these quotations contradicted each other. \"How can a mass murder be both right-wing and meaningless?\" he asked, and articulated his main point in his next sentence: \"There you have the cognitive dissonance of someone devoted to stopping terrorism only to find his own rhetoric may have played a part in motivating a terrorist.\"\n\nAndrew quoted from my article, pointed out what he considered a contradiction, then\u2014instead of making an argument for this assertion\u2014proceeded to make a psychological diagnosis to account for my self-contradiction: cognitive dissonance arising from guilt. In his view, the murders by Breivik could not be meaningless because Breivik had a clear rationale: the victims weren't randomly chosen, they were selected because he held them responsible for treason. As I made clear in my articles, I agree that that was his intention. But his acts were nonetheless insane\u2014meaningless, if you will\u2014because it was obvious, as I also made clear, that the result of his atrocities would be the exact opposite of what he wanted.\n\nLike others, Andrew was eager to identify Breivik's atrocities as the work of what he called a \"Christianist terrorist\"\u2014though it seemed that Andrew had, at best, taken only the briefest of looks at Breivik's manifesto (in which Breivik called himself Berwick, a name by which Andrew chose to refer to him). Insisting, for some reason, on calling Breivik by the British-sounding pseudonym under which he wrote his manifesto, Andrew characterized me as acknowledging, in my Journal article, \"that the Christianist terrorist had been deeply influenced by the anti-Jihadist blogosphere and his own work, quoted 22 times by Berwick,\" but condemned me for \"fail[ing] to assess for a second whether the rhetoric used by him and so many others, was an inspiration for this political mass assassination.\" Talk about \"cognitive dissonance\": Andrew said flat-out in one sentence that I acknowledged having inspired this murderer, and a couple of sentences later accused me of not having considered for an instant whether I had done so.\n\nA more careful reading would have revealed (as I mentioned above) that the twenty-two supposed quotes from me were in fact indirect quotes appearing in essays by Fjordman that Breivik reproduced wholesale in his manifesto. I was never quoted directly by Breivik, who, as I also mentioned earlier, considered me too liberal for his tastes. One might think that in a sensitive case such as this, Andrew would have taken the time to ascertain exactly which statements of mine had been quoted approvingly by Breivik before rushing to collapse the ground between us. But this was apparently too much to expect of someone who declared himself my friend and ally.\n\nIt also needs to be pointed out that \"Christianism\" has become one of Andrew's major hobbyhorses. He appears to believe that there is a thing out there called Christianism that is every bit as dangerous as Islamism. Having staked his reputation on this notion, he was therefore quick to seize on the idea that Breivik's actions had proven his case. For Andrew, the need for a moral equivalency between \"Christianism\" and Islamism is obviously a matter of conscience, and for Andrew (a devout Catholic) his own conscience is privileged over everything else. He loves to put his conscience on display, to wrestle with it in a brightly lit ring in view of all his fans. To Andrew, this kind of casuistical wrestling is the epitome of intellectual virtue, embodying the best of the Western tradition. Yet lost in all of this is any notion of a patient and (yes) conscientious study of the evidence. Instead he prefers to play the Grand Inquisitor, appearing before us every day to serve as the Chief Justice of the Blogosphere, rendering moral judgment on every matter big and small. In his mind he is a champion of rectitude and virtue; in reality he is a reckless purveyor of conspiracy theories, drawing moral lines in the sand as fast as Picasso could scribble a profile on a napkin, hysterically jumping to conclusions, compulsively exaggerating arguments for effect, and habitually pretending that the extremists on the other side speak for all his opponents. He is a master of the straw man and slippery-slope fallacies.\n\nSo it was that Andrew, having invented the \"Christianist\" category, leapt at the opportunity when Breivik said he was a defender of Christian values. Unfortunately Andrew didn't bother to read the manifesto with any care before presenting himself as an expert on its author's motivations. Nor did he have the slightest clue what the average Norwegian (as opposed to, say, Jerry Falwell or Pat Robertson) might mean when he speaks of Christianity.\n\nAndrew claimed that I contradicted myself when I called the murderer a \"madman\" yet described him as \"both highly intelligent and very well read in European history and the history of modern ideas.\" Andrew's comment: \"It is precisely this blind spot by the anti-Islamist right that made me and others get off the train.\" In Andrew's view, apparently, if you accept the argument that Europe is endangered by a rapid growth in the Muslim population that is being \"aided and abetted\" by multicultural European leaders, there is no option other than \"the fascist solutions he [the murderer] recommends and the neo-fascist violence he unleashed.\"\n\nWhen an entire population in your midst is the enemy within and your government is acquiescing to it and your entire civilization is thereby doomed, what does Bruce think a blue-eyed patriot like Berwick should do? Is the leap to violence so obviously insane? Or is it actually the only logical conclusion to the tyranny Berwick believed he faced?\n\nYes, the leap to violence is obviously insane\u2014for no sane person would ever have thought such actions would have accomplished what he wanted. Note Andrew's argument here: that Breivik, in terms of his own premises, was behaving rationally. But we don't judge the sanity of an action by its premises but by the actor's expectation of its results. Since Breivik's means would not lead to his desired ends, he was not only a bloodthirsty monster but obviously insane. (The Holocaust is perhaps the best example of a murderous program, based on irrational premises, that was carried out by thoroughly rational people. Creating a world without Jews is an insane goal. The fact that it was rationally pursued\u2014and justified with rational-sounding arguments\u2014doesn't make it any less so.)\n\nMore important, perhaps, Andrew's argument undermines any possibility of a nuanced middle ground in the debate about radical Islam. Since Andrew, by his own account, is a defender of nuance and a partisan of the liberal tradition of free and open debate, one might think that he would see the value in separating Breivik's social diagnosis from his murderous actions. Surely it is possible for people to agree that a problem exists, and yet disagree in the most fundamental way about how to address it. Writers like myself eschew violent solutions and declare our hope for a reasonable approach to the problem within the confines of a healthy liberal-democratic politics. To claim that Breivik's actions were somehow inevitable given his views is to wipe out this hope and to declare, in effect, that there is no reasonable solution to the problem, and therefore no grounds for a rational distinction between those who wish to discuss and debate it politically and those who wish to answer the problem with violence.\n\nThis is what makes Andrew Sullivan a New Quisling of the first order. Today's Quislings insist that \"fighting words\" are dangerous, that criticism of Islam is \"hate speech\" that necessarily incites violence, and that a liberal society therefore has a vital and legitimate interest in stigmatizing, marginalizing, and suppressing it. This is now Andrew's position, and it flies in the face of his supposed commitment to nuance and open debate.\n\nShortly after 9\/11, Andrew had written a stirring essay titled \"This Is a Religious War,\" acknowledging that Islam was indeed at the root of the jihad that had destroyed the Twin Towers. In 2006 I wrote a book called While Europe Slept: How Radical Islam Is Destroying the West from Within, and Andrew wrote about it on his blog:\n\nThere is no more important issue than that of religious fundamentalism's current battle with liberal democracy. And no one has confronted this issue as forthrightly as Bruce Bawer. I re-read his last book, \"Stealing Jesus,\" as essential background for my next book, \"The Conservative Soul,\" and was struck again by its rigor and passion. Now, Bruce, who's an old friend and ally, has written a clarion call for the West to understand the radical threat to our freedoms from politicized fundamentalist Islam. He writes from the belly of the beast, Norway, where he has lived for several years with his husband. I wish he'd toured the U.S., but he's so enmeshed in the fight in Europe that he has stayed put. I know of very few as close up to what we face as Bruce is; and very few as brave and as eloquent on confronting it.\n\nAt that time, Andrew and I seemed to be on the same page when it came to the danger of radical Islam. But over the years, his views underwent a major shift. At first a staunch defender of what was called the \"war on terror,\" he became one of its fiercest critics. In \"This Is a Religious War\" he had pointed out the menacing nature of Islamic ideology; now he soon began to savage many of those who did exactly the same thing. He and I faced the same challenge: how far to go in making alliances with people who share our view that a certain problem exists but whose method of argument or ideology or practical vision of how to deal with it is not entirely congenial. I have to admit that I often had a hard time making such choices myself when it came to the anti-jihadist cause. For me the question was always whether I felt their rhetoric and actions helped or harmed the cause. Only in a handful of cases did I publicly dissociate myself from anyone by name. I might add that some people from whom I distanced myself before 9\/11 over other matters I have since tacitly reconciled with, because we agree about things that now seem more urgent. Such is always the case with profoundly divisive political and moral questions. We cannot always choose our allies\u2014and we cannot afford to make an absolute morality out of our own motives and preferences.\n\nAndrew has taken a different path. He has turned every break with an old friend into an opportunity to showcase his own righteousness. And he has used the issue of torture at Guant\u00e1namo and Abu Ghraib to ply a moral-equivalency line that is frankly obscene. In what twisted moral universe does America's selective and highly restricted use of enhanced interrogation techniques make us the same as people who saw the heads off innocent civilians? Were the values of Jefferson and Madison really threatened more by George W. Bush than by Osama bin Laden? To make such an argument is (as one might put it) to torture logic and morality beyond all recognition.\n\nStill, Andrew continued to link approvingly to my own writings on radical Islam, usually suggesting, when he did so, that I differed from other writers on the subject in a positive way. In 2006, when a Seattle weekly, the Stranger, dared to reprint the Danish Muhammad cartoons, which appeared alongside an article by me, Andrew praised the Stranger for running the cartoons and agreed with me that doing so was a matter of standing up to Muslim bullies: \"Do we need now to be 'sensitive' toward Wahhabist Islam's treatment of women?\" he asked sardonically.\n\nIn 2009, he praised both me and Christopher Caldwell, saying that we were \"not Steynian hysterics [the reference is to Mark Steyn, author of America Alone and far less of a hysteric than Andrew]; and not authoritarian conservatives\" but rather \"liberal-minded conservatives who are deeply alarmed at the enabling of Islamist illiberalism in Europe.\" And in the same year he praised a piece I had written on the Fort Hood shootings, quoting a passage in which I noted that the killer had, among other things, \"repeatedly expressed sympathy for suicide bombers\" and \"handed out copies of the Koran to neighbors.\" Andrew called the piece \"powerful,\" yet he expressed the following concern:\n\nBut what does Bruce want the US to do in response to an incident like this?\n\nScreen all potential Muslim soldiers in future? Have special surveillance of such soldiers? It's easy to see how this might make matters worse just as it might make them better. Michelle Malkin, remember, favored interning Japanese-Americans during the Second World War. Is that what the anti-Jihadists now want for American Muslims? Or what, exactly?\n\nThis was classic straw-man argumentation: Andrew was using the tired old debater's trick of pushing the argument to an extreme, mentioning far-out \"solutions\" and suggesting that his opponent would support them, instead of acknowledging that the Fort Hood massacre could have been avoided if the army simply took action against soldiers who repeatedly express sympathy for suicide bombers. The army's failure to do so reflects the extent to which politically correct sensitivities have permeated the entire military governing structure.\n\nBut it is Andrew's method to hammer a theme into the ground, and so the day after his \"Bawer vs. Bawer\" posting, he was back with more.\n\nBy far the most destructive terrorist attack in Norway was carried out not by Islamists but by a Christianist fanatic. Per capita, it was more destructive of human life in Norway than 9\/11 or 7\/7 in the US and UK. My problem with Bruce Bawer's WSJ piece is that it didn't seem even to reflect on that astonishing fact. When you have been pointing out the danger of terror from Islamism for years, and it turns out that the terror comes from someone who is on the fringe of your side of the debate, I think it's worth taking some stock.\n\nAndrew went on to contrast my Journal piece, with its lack of the kind of self-flagellation he apparently expected, with an essay by Nils August Andresen, the editor of a Norwegian website, Minerva, who, as Andresen himself put it, had \"been forced to confront the fact that . . . the mass murderer of my countrymen, has visited our website and posted comments in our forum. Though it was impossible to detect this extremism in his comments at the time, I have often worried about the increasingly aggressive tone that characterizes too many not only in our forum, but everywhere that the multicultural society is debated.\"\n\nIt is no wonder that Andrew loved Andresen's piece. He loved the self-criticism; he loved the fact that Andresen made himself the bad guy. Andrew is very experienced in this line. Indeed, his ultimate purpose as a writer is apparently to show that he is too pure for any party. Thus he commits himself heart and soul to a position, attacking everyone on the other side with everything he has, then switches sides and sets to work attacking his former allies with equal gusto. Just beneath the surface it is clear that the whole business is really not so much about examining the issues themselves but about showcasing that exquisite thing, Andrew's moral conscience. The more he berates his former self, the more he expects to be admired for the moral honesty and courage that has led him, through immense and extremely dramatic inner struggle, to assume his present position. He may oppose torturing terrorists, but he loves to make a spectacle of torturing himself ethically. Thus it is that the workings of his conscience become elevated to a moral principle in and of themselves. Andrew praised Andresen's piece for its \"nuances\" and for the suggestion that \"extremist rhetoric\" had some responsibility for the murderer's actions. He plainly agreed with Andresen's assertion that many in Norway \"have Islam as their only concern, their only evil\" and that they are therefore \"willing to accept ever more illiberal measures against Muslims.\"\n\n\"This is the Christianist temptation,\" Andrew went on to say: \"to be so convinced of your own good intentions and culture that you become blind to the fact that you too can spawn and enact evil.\"\n\nThat's how the US came to adopt a torture program based on those once used by Nazis and Communists. That's how Israel can look at the dead bodies of children buried in Gaza rubble and accept no blame or responsibility at all. That's how Bill O'Reilly can simply assert that a confessed Christian simply cannot be one because he is a mass-murderer. And that's how some neocons can regard an Iraq invasion based on false premises that resulted in the deaths of tens of thousands of innocents as a success worth repeating.\n\nThis is, of course, an offensive riot of false equivalency. The Israel line is an obscene misrepresentation of an embattled society that nonetheless has a lively capacity for self-criticism\u2014unlike its jihadist enemies. The United States did not base its interrogation practices on those of the Nazis\u2014any more than the internment camps for Japanese Americans resembled those at Dachau and Auschwitz. Apropos of O'Reilly, jihad is enshrined in the Koran; Christians have committed heinous crimes in the name of faith but mass murder violates the essence of the gospel. Moreover there are innumerable Islamist terror organizations around the world; where, exactly, are their Christianist counterparts? Where are the Christians threatening to decapitate artists who create works like Piss Christ or directors who make movies like Life of Brian? This is sheer hyperbole.\n\nThe task for us is to fight extremist terror\u2014Islamist and Christianist\u2014while retaining common decency, the Geneva Conventions, respect for moderate Islam and apolitical Christianity, and bedrock commitments to free speech, however inflammatory.\n\nThis is a set of sentiments with which I agree entirely. But it is clear that the key words here, for Andrew, were \"Islamist and Christianist.\" These three words summed up Andrew's objective in these postings: to make the argument that there really is something that can be called \"Christianist\" terror out there in the world, and that it represents as much of a threat to the free world and democratic values as does the Islamist variety.\n\nAndrew quoted one last line from Andresen: \"Read American counter-jihadi blogger Pamela Geller's comments on the attacks. That attitude must be confronted.\" \"Indeed it must,\" wrote Andrew. \"And I have little doubt that my friend, Bruce Bawer, will do so once the dust settles.\"\n\nWell, as it happens, I have been confronting the rhetoric of Geller and others ever since I started writing about this topic. If you look at the archives of the website Little Green Footballs, for example, you'll discover the chronicles of a major dustup in the blogosphere that took place a few years ago after I publicly made clear my distaste for the far-right, nationalist, and anti-Semitic Vlaams Belang party in Belgium. It began with a blog posting by me about Charles Johnson, a celebrated anti-jihadist blogger who had publicly distanced himself from friends and allies whose embrace of Vlaams Belang and other illiberal groups and individuals troubled him. In my posting I said that I shared Johnson's dismay, noting that people I had thought of as liberals were now opposing jihadism on what sounded to me like not very liberal grounds\u2014or were rallying around others who did. After my comments were reposted at Little Green Footballs, there ensued a ferocious online conversation. Many well-known Islam critics who had forged close ties with Vlaams Belang were incensed by my position.\n\nIt's all there on the Internet, for anyone willing to do the least bit of Google research. To this I will only add that (a) for all my discomfort with certain elements of the anti-jihad movement, I have subordinated my criticisms of them to what I consider to be the far more urgent problem, namely jihadism itself, and (b) I find it offensive to suggest that any honest writer on this subject, however objectionable in his ideas or language, has the slightest responsibility for Breivik's actions.\n\nAs it turned out, Andresen felt the same way. In a passage that Andrew chose not to quote, Andresen wrote:\n\n. . . from what we know today, it appears that he was not a product of the increasingly hostile Internet debate over the last few years. He was a part of the discourse, but it was not what radicalized him. . . . [T]hese atrocities . . . were not caused by the websites or the rhetoric that we have seen in recent years.\n\nIn the end, Andrew's postings on me were all about justifying his change of position in the war on Islamic terror. He came to feel he had defended a bad cause, that the greater threat was from his former allies, and that moral decency required him to attack them as viciously as he had previously attacked the other side. In this regard he expressed not so much his moral rectitude as an extreme and somewhat narcissistic sensitivity about his political and intellectual bedfellows. I have put some distance between myself and some former anti-jihadist allies out of concern that they were fighting for the wrong reasons and alienating people who might be persuaded to fight for the right ones, but I have not made a habit of attacking them. Andrew, on the other hand, has made a morality\u2014a veritable religion\u2014out of his political alignments and realignments, so that his own moral posturing and preening on life-or-death issues, and not the issues themselves, take center stage. In doing this he habitually erases the middle ground, presenting a false moral choice between extremes, in much the way that Quisling picked Hitler as if he were the only alternative to Stalin.\n\nAndrew wasn't the only New Quisling in the American media. Roger Cohen, too, had a hand in. Cohen lives in London and writes columns for the New York Times in which he reliably bashes Israel (he is a British-born Jew of South African extraction). As someone who has spent much of his career in the capitals of Europe, Cohen thoroughly subscribes to the European liberal line and typically exalts the multicultural elite.\n\nOn July 25, Cohen wrote in the Times that Breivik was, on one level, \"just a particularly murderous psychotic loner\"\u2014or, at least, \"that is how Islamophobic right-wingers in Europe and the United States who share his views but not his methods will seek to portray Breivik.\" But Cohen, like so many others, refused to buy the lone-wolf interpretation. Recalling the shooting of Representative Gabrielle Giffords by Jared Loughner, Cohen insisted that Breivik's atrocities were \"brewed in a specific European environment that shares characteristics with the specific American environment of Loughner: relative economic decline, a jobless recovery, middle-class anxiety and high levels of immigration serving as the backdrop for racist Islamophobia and use of the spurious specter of a 'Muslim takeover' as a wedge political issue to channel frustrations rightward.\"\n\nCohen's sentence reflected the orthodox left-wing notion\u2014which began, perhaps, with the cockamamie idea that the JFK assassination was caused by Dallas's \"climate of hate\"\u2014that the environment, particularly the economic environment, is always the \"root cause\" of any malefaction. Not so: people are responsible for what they do. Individuals who share the same environment and economic background take wildly different paths. As we all know, the atrocities of al-Qaeda have been blamed on poverty in the Arab world, despite the wealth of the bin Ladens and the high educational level of many 9\/11 terrorists. And of course if the root cause is economic, then it is always the Western world and predatory, exploitative Western capitalism that are at fault.\n\nThere were, in any case, a couple of slight flaws in Cohen's argument. To begin with, Norway, more than probably any other country in the Western world, had been left unscathed by the economic downturn that had scarred Europe and North America; its people are not afflicted by \"middle-class anxiety\"; unemployment levels there remained extremely low. As for \"racist Islamophobia,\" hostility to Islam is not racism because Islam is not a race\u2014it is a religion, one whose ideology, in a free country, should be subject to legitimate examination and criticism, which is what had finally begun to happen in Norway in recent years. To accuse Islam's critics of racism is the way of the Quisling: it seeks to stigmatize an argument by discrediting its motives. Furthermore, the rise of Islam in Europe, and the creation of increasing numbers of no-go zones, is not a \"spurious specter\" but an objective fact. The amount of sheer disinformation in Cohen's piece was overwhelming.\n\nIt was reprehensible to see an insulated elitist member of the fourth estate depicting millions of people in Europe as mindless pawns who had been fooled into buying \"the spurious specter of a 'Muslim takeover' \" by politicians seeking to \"channel frustrations rightward.\" To the contrary, people who are concerned about Islam are preoccupied with these matters for one reason only: because they have experienced things that Roger Cohen has not. They've witnessed the duplicity of political leaders who have imposed upon them, without ever asking their permission, a mass immigration into their countries that has radically altered their lives and seriously imperiled their futures, and they've experienced the condescension of those politicians (not unlike Cohen's condescension) in the face of their own thoroughly legitimate concerns.\n\n\"What has become clear in Oslo and on Ut\u00f8ya Island,\" Cohen wrote, \"is that delusional anti-Muslim rightist hatred aimed at 'multiculturalist' liberals can be just as dangerous as Al Qaeda's anti-infidel poison.\" Note Cohen's use of language here: in using the word delusional and hatred, he was clearly referring to the delusions and hatreds of the insane Breivik\u2014but he also seemed to be implying that any criticism of Islam is equally delusional and amounts to hatred. As if to confirm this interpretation, Cohen, in his very next sentence, expressly linked Breivik to others:\n\nBreivik has many ideological fellow travelers on both sides of the Atlantic. Theirs is the poison in which he refined his murderous resentment. The enablers include . . .\n\nThere followed a list of people and parties with a range of political stances\u2014Geert Wilders, Marine Le Pen, \"far-rightist parties in Sweden and Denmark and Britain,\" Newt Gingrich and Representative Peter King, \"who have found it politically opportune to target 'creeping Shariah in the United States' at a time when the middle name of the president is Hussein.\" Note the no-holds-barred terminology: \"Fellow travelers.\" \"Poison.\" \"Enablers.\" These are terms that deeply alarmed and offended liberals like Cohen when they were used by the likes of Senator Joseph McCarthy. Yet now they were at the tip of Cohen's tongue.\n\nA sensible response to Cohen's catalogue of suspect political figures might be to wonder: do all these names actually appear in Breivik's manifesto, or was Cohen simply listing people whom he didn't like? Cohen's article was a fine example of how PC works: not through fair argument with respect for the truth, but through moral intimidation, the demonizing of one's opponents, and the characterization of their opinions as dangerous.\n\nNot that I would defend Le Pen or some of the far-right parties of Europe (though I would not tar them with responsibility for Breivik's crimes, either). But it is repulsive to link Breivik with, for example, the courageous congressman Peter King, who, far from doing something \"politically opportune,\" has stepped in where many of his House colleagues have feared to tread. How? By holding hearings to discuss the influence of organizations like the Council on American-Islamic Relations, organizations that, despite their proven terrorist connections, have continued to be treated with respect\u2014by, not least, incidentally, craven newspapers like Cohen's.\n\n(By the way, King's name does not appear anywhere in the manifesto.)\n\n\"Muslims over the past decade,\" continued Cohen, \"have not done enough to denounce those who deformed their religion in the name of jihadist murder. Will the European and U.S. anti-immigrant Islamophobic crowd now denounce what Breivik has done under their ideological banner? I doubt it.\" In fact, everyone whom Cohen smeared in this piece did condemn Breivik's actions.\n\nCohen proceeded to condemn \"the widespread condoning of an anti-Muslim racism once reserved for the Jews of Europe.\" Never mind that in Europe today, the incidence of anti-Jewish aggression (generally by Muslims) far exceeds anti-Muslim aggression, and that, especially in Norway, the kind of \"racism\" most widely condoned by mainstream journalists, politicians, and academics is not targeted at Muslims but at Jews.\n\nIn a single short piece, Cohen managed to throw around a great many ugly labels, apparently hoping that some of them would stick and mark the individuals in question as being beyond the pale. His repeated use of the words Islamophobe and Islamophobia, and of related terms like right-wing and anti-immigrant, was pure name-calling, a matter of hurling invective at everyone he disagreed with about these subjects. Cohen has for some time, it should be recalled, been quick to slap the label Islamophobe on anyone and everyone who dared to criticize Islam. (In April 2011 he described Oklahoma, ridiculously, as \"a state where Islamophobia is rampant.\") He is every bit as quick to smear as bigots people with legitimate concerns about Islam as he is to soft-pedal the open and brutal hostility of many Muslims toward groups of people whom they deem worthy of contempt and even of summary execution.\n\nThe word Islamophobia is, of course, a cant term, invented by the Muslim Brotherhood and now used by Muslims and multiculturalists alike to tar any critic of Islam as a bigot. They analogize Islamophobia and homophobia, but this is a false equivalence. Homosexuality is an orientation; Islam is an ideology. There is no reason to fear homosexuality, but there are very good reasons to fear Islamic ideology. Yet the word carries a powerful heft. Journalist Juan Williams was punished for daring to admit that he was scared of Muslims. The term also came in handy during the Ground Zero mosque controversy in New York. New Quislings, with Mayor Bloomberg in the lead, rushed in to silence legitimate criticism of the mosque plans not through sensible argumentation but simply by charging mosque opponents with Islamophobia.\n\nThis is a favorite term of the New Quislings because it serves to shut down debate on issues where facts and evidence do not support their views.\n\nAnother New Quisling of note is the American writer Chris Hedges, who has been published for years by the Nation and other left-wing organs. He has also been a New York Times reporter, although he left the paper after being censured for antiwar remarks at a college commencement in 2003.\n\nHedges agreed with Sullivan and Cohen about the supposed danger of Christian jihad\u2014only he went even further, indicting \"secular fundamentalists\" as well: \"The gravest threat we face from terrorism . . . comes not from the Islamic world but the radical Christian right and the secular fundamentalists who propagate the bigoted, hateful caricatures of observant Muslims and those defined as our internal enemies.\" Writing at the left-wing Truthdig website on July 26, Hedges went on:\n\nThe caricature and fear [of Islam] are spread as diligently by the Christian right as they are by atheists such as Sam Harris and Christopher Hitchens. Our religious and secular fundamentalists all peddle the same racist filth and intolerance that infected Breivik. This filth has poisoned and degraded our civil discourse. The looming economic and environmental collapse will provide sparks and tinder to transform this coarse language of fundamentalist hatred into, I fear, the murderous rampages experienced by Norway. I worry more about the Anders Breiviks than the Mohammed Attas.\n\nThis was beyond absurd: in the post-9\/11 era, in a time when Islamic terrorists have murdered untold numbers of people around the world\u2014their coreligionists included\u2014in cold blood, Hedges was making the argument that the more sinister threat originates largely in religious groups that have overwhelmingly been victims of terrorism and in the anti-fanatical language of secular critics of religion like Harris and Hitchens. This wasn't Christianist\/Islamist moral equivalence; Hedges was actually saying that we have more to fear from fans of Christopher Hitchens than from the kind of people who brought down the Twin Towers. But one could hardly expect more from the execrable Hedges, an apologist for Hamas whose systematic anti-Israeli distortions are the stuff of legend and who, in 2001, accused the IDF of \"murder[ing]\" children \"for sport.\"\n\nIt did not take long for Norway's New Quislings to rally around a new label for their enemies: \"Eurabia writers,\" \"Eurabia conspiracists,\" \"Eurabia propagandists,\" or some variation thereupon. One of the first to employ this label was Sindre Bangstad, a social anthropologist at the University of Oslo, who in the Danish newspaper Politiken used Breivik's atrocities as a club with which to beat not only a wide range of adversaries. He went, for example, after Walid al-Kubaisi, an Iraqi Norwegian writer whose website is called Opplyste Muslimer (Enlightened Muslims). Walid is one of the bravest people in Norway and one of its most fervent defenders of individual liberty\u2014yet Bangstad smeared him as a \"Eurabia literature propagandist\" and mocked him for having assumed, in the first moments after the explosions in Oslo, that they were the work of Islamic terrorists.\n\nWhat is \"Eurabia\"? The word refers to the book of that title by the scholar Bat Ye'or, who describes how various obscure European commissions, committees, and such have smoothed the way for the Islamization of Europe. Since July 22, the book Eurabia has repeatedly been characterized in the Norwegian media as pure fantasy; on the contrary, it is a sober work of solid documentation, and anyone who wishes to try to refute it should do so by resorting to facts, not by smearing it as baseless propaganda. Ye'or has studied a small library of obscure agreements produced by diplomat meetings, conferences, conventions, and the like over recent decades, and has found what she considers an unsettling pattern of \"informal alliances\" between European officials and their Mediterranean Arab counterparts that take place under the umbrella of something called the Euro-Arab Dialogue, which dates back to 1974. Bat Ye'or considers these alliances to be characterized by a European deference toward Muslim values, sensibilities, and sensitivities, a pattern she likens to the historical subordination of non-Muslims in Islamic countries. These agreements, in her view, have been instrumental in producing an increasingly Islamized Europe in which government leaders are quick to give way to Muslim wishes and demands and loath to defend Western values and principles\u2014thus, Eurabia. Ye'or is no shrill self-promoter, and her books are hardly the punchy screeds they have been made out to be; on the contrary, they are dry, sober, and packed with long, thoroughly footnoted quotations. The serious and responsible way for an opponent to respond to such work is by challenging the facts or the interpretations thereof; it is not to name-call, to describe her as a street-corner hatemonger or a reckless peddler of baseless conspiracy theories.\n\nBangstad also went after the Progress Party and its former head, Carl I. Hagen: \"We don't know when Carl I. Hagen began to read the type of Eurabia literature that Anders Behring Breivik has also read, but it is well documented that he and several of his fellow party members have read precisely that type of literature.\" As in any totalitarian society, it was now apparently an offense in Norway simply to have read certain books that the country's new Public Enemy Number One had also read. (By the same logic\u2014in fact, by far better logic\u2014one would expect that after 9\/11, Madrid, London, Beslan, Bali, Mumbai, and so forth, it would be forbidden to read the Koran.)\n\nBangstad had more to say:\n\nAnders Behring Breivik . . . has read widely in the racist and Islamophobic literature, from the Israeli right-wing extremist Bat Ye'or by way of the American-Norwegian neoconservative Bruce Bawer to the Norwegian professor emeritus in sociology Sigurd Skirbekk. . . . Hereafter it will be difficult for editors and intellectuals to minimize the existence of Islamophobia in Norway, and it will, if possible, be even more difficult to claim that racists' and Islamophobes' words are just words. . . . Anders Behring Breivik has, by his actions, set himself up against history. Multicultural Norway has come to stay. No pasaran\u2014the line is drawn here.\n\nThose last words, of course, were a quote from Che Guevara\u2014which gave a helpful hint as to exactly where, ideologically, these nasty lucubrations had their origin. (Though it should have been No pasar\u00e1n, with an accent over the third a.)\n\nLate August saw the publication of more thorough, sustained attacks on the critics of Islam and of multiculturalism\u2014attacks that had obviously been in the works for some time. They reached new level of propagandistic poison. The malice and mendacity were palpable; it was now no longer disputable that the New Quislings were out to destroy\u2014nothing less. By now it was also clear that many of them considered the \"Eurabia\" line of attack a winner.\n\nOn August 19, the Norwegian weekly Morgenbladet ran a long, mischievous article by Maren N\u00e6ss Olsen and Anders B. Bisgaard titled \"The Eurabian Verses.\" It was yet another attempt to link critics of Islam with Breivik\u2014and yet another in a years-long list of attempts by left-wing journalists across the Western world to dismiss concerns about the Islamization of Europe as the product of misinformation by a few nutty extremists. Indeed, even given the ostrich-like attitude of the Western mass media generally toward the darker facts about Islam, Olsen and Bisgaard's article was well-nigh breathtaking in its utter refusal to acknowledge basic realities about the world we live in.\n\nAccording to the tale spun by Olsen and Bisgaard, pretty much all of Islam criticism, it seems, can be traced to the work of one whacked-out lady, and then, in turn, to the work of another whacked-out lady. \"The mass murderer Anders Behring Breivik, the blogger Fjordman, the Dutch politician Geert Wilders, the Oslo-based author Bruce Bawer, the Libyan dictator Muammar al-Gaddafi, and [Norwegian] Conservative Party veteran Hallgrim Berg,\" the Morgenbladet authors wrote, \"are among those who have embraced all or parts of Bat Ye'or's imaginative Eurabia universe, which in turn is built on the thoughts of the Italian journalist and author Oriana Fallaci. Ye'or's book, in time, has in time spawned an entire genre, with titles like Londonistan, The Last Days of Europe, Defeating Eurabia, and While Europe Slept.\"\n\nOlsen and Bisgaard called me, Wilders, Ye'or, Berg, Walter Laqueur, Melanie Phillips, and other critics of Islam \"conspiracy theorists.\" But it was Olsen and Bisgaard's article that was framing a conspiracy theory. For their whole agenda was to dismiss all concerns about the Islamization of Europe as the product of a cultish conspiracy by us, a group of loony right-wing bigots who are the disciples of a lone crackpot and whose views have gained no traction whatsoever among academic \"experts\" in Islam and immigration\u2014but who, for some mysterious reason, have managed to convince millions of readers that there is something to what we say.\n\nOlsen and Bisgaard spoke of the \"parallel universe\" of Islam critics and interviewed a professor at Uppsala University who dismissed our work as \"Islamophobia\" and \"pure conspiracy theories\" and compared Ye'or's views to \"the theory that flourished before the Second World War of a Jewish world conspiracy.\" (In a thoroughly redundant sidebar, Lena Lindgren helpfully described Eurabia as \"a fantasyland\" and reassured Morgenbladet readers that all is well.) Olsen and Bisgaard refused to acknowledge that the books of Islam criticism that they mentioned, far from being the work of a pack of anyone's disciples, are in fact the work of a very non-uniform group of writers who have a wide range of backgrounds and political convictions; they refused to recognize that these books, far from being faint echoes of Ye'or's works, approach the subject of Islam in Europe from diverse perspectives and with their own special emphases, all of them written not out of discipleship of anybody (or, for that matter, out of loyalty to some academic orthodoxy, such as the see-no-evil orthodoxy that has hobbled Middle Eastern studies) but, in very large part, out of the authors' own observations and experience. And they refused to acknowledge that if many readers are receptive to what these authors have to say, it is largely because those readers have had observations and experiences similar to those that the authors relate, and have come to similar conclusions.\n\nThis latter point is one that Olsen and Bisgaard do not wish to grant, for the entire premise of their article is that those of us who have written books critical of Islam have somehow convinced millions of people in the West to feel trepidation about something that they would otherwise not be worried about. On the contrary, our books have struck a chord precisely because they have addressed concerns that are already out there that are founded in people's very real worries about social changes that they have witnessed firsthand and that the political establishment and mass media have largely refused to address. (How far do you think you'd get if you tried to whip up anxiety in the West about, say, Hinduism or Buddhism?)\n\nOlsen and Bisgaard devoted a good deal of space to me. They described me as a \"self-appointed victim of the Islamic threat\" who \"never gives interviews in the Norwegian media\"\u2014implying, apparently, that I've been hiding from the press. This is a hilarious charge, given that, as Hans Rustad noted at document.no on August 24, I've been pretty much \"boycotted by the Norwegian media\" throughout my years in Norway. By contrast, during those same years I've done innumerable interviews with non-Norwegian journalists\u2014a long sit-down with Bill Moyers on his TV series, a couple of radio chats with Dennis Miller, several radio interviews in Canada, podcasts with Shire Network News in Australia and with John J. Miller of the National Review, as well as interviews with the Danish papers Berlingske Tidende and Jyllands-Posten (twice), the Dutch newsweekly Elsevier, L'Occidentale and Il Foglio in Italy, and the Jerusalem Post, plus many, many others. The fact that I've never been interviewed in the Norwegian media, in short, says nothing about me but a great deal about the Norwegian media, which can sometimes seem to think that the only people in Norway who know anything about anything are either members of Parliament or professors at the University of Oslo. (Or, of course, journalists.)\n\nDespite my putative refusal to be interviewed by the Norwegian media, Olsen and Bisgaard went on, \"Bawer does not hesitate . . . to use his experiences in Oslo as the basis for the Eurabia arguments he sets forth in right-wing American media . . . and in his book While Europe Slept.\" Huh? Was I now being attacked for basing my arguments about Europe on my experiences in Europe? Isn't this how these things are supposed to work? Clearly not in the New Quislings' world, where ideology trumps reality, and where the first rule of journalism is not to believe your lying eyes. This is how they think. Their need to deny reality in order to preserve the edifice of multiculturalism is that powerful.\n\nJust for the record, I didn't read a word by Bat Ye'or, and knew nothing of her work, until I'd already finished writing a near-final draft of While Europe Slept; after I read Eurabia I just barely had time to insert a short passage about it\u2014less than a page\u2014into my book. A search of the dozens of articles I've written about these issues since reveals exactly one fleeting mention of Ye'or. You'd never know this from reading Olsen and Bisgaard.\n\nUnsurprisingly, Olsen and Bisgaard turned to academic \"experts\" on immigration and Islam (the same kind of people who assured us before 9\/11 that there was no \"Islamic threat\") for reassurance that there is nothing of value in the work of any of us critics of Islam. The Dutch academic Cas Mudde, who teaches at DePauw University in Indiana and is identified as an expert on the European far right, told the Morgenbladet reporters that when he moved to the United States \"he was surprised by how deeply the Eurabia conspiracy and people like Bruce Bawer had penetrated.\"\n\n\"I met conservatives who obviously were not crazy who asked about Eurabia,\" said Mudde. \"What I hadn't grasped when I lived in Europe was how mainstream people like Bruce Bawer are in the U.S. Those who make the Eurabia arguments are influential people in the conservative movement. Bawer is considered an expert on European conditions because he lives there.\"\n\nIndeed, one of Olsen and Bisgaard's main contentions was that these books about the Islamization of Europe have only been popular in America and haven't gotten (as they put it) \"the same foothold in Europe,\" apparently because Europeans, being closer to the real situation, know better. (There was also an insinuation that we Islam critics are peddling nonsense that only stupid, bigoted Americans would buy, and that Europeans are too sophisticated and intelligent to fall for.) In fact, a host of European writers have also written important and widely read books in this genre\u2014among them Henryk Broder and Thilo Sarrazin in Germany (Sarrazin's book Deutschland schafft sich ab, or Germany Does Away with Itself, was the biggest-selling political book in Germany in a decade), Lars Hedegaard and Helle Merete Brix in Denmark, and Guy Milli\u00e8re, Robert Redeker, and Chahdortt Djavann in France. Olsen and Bisgaard did interview Hallgrim Berg, author of a Norwegian bestseller about the Islamization of Europe, but, curiously, when it came around to talking about the European versus American markets for these books, dropped him down the memory hole. They also mentioned Fallaci, of course, but neglected to mention that her first book on Islam, The Rage and the Pride, was the largest bestseller ever in her native Italy, and a big hit across Europe as well.\n\nThe title of Olsen and Bisgaard's piece was disconcerting. Plainly, it was meant to pun on Salman Rushdie's The Satanic Verses\u2014which, as the world knows, resulted in a fatwa that forced Rushdie to go into hiding for years. What were Olsen and Bisgaard trying to say by linking the \"Eurabia writers,\" as they called us, to Rushdie? That the fatwa against him was, like our work, nothing more than the product of a fevered imagination?\n\nOn August 22, Sindre Bangstad, who, as we have seen, had already savaged the critics of Islam and of multiculturalism in the Danish newspaper Politiken, published an even longer and even more vicious piece on the Open Democracy website. He began by referencing a \"peaceful Muslim\" demonstration that had taken place some months earlier at University Square in Oslo, then criticized Hege for writing a negative piece about that demonstration in which \"incitement to violence\u2014or 'fighting words'\u2014directed against Norwegian Muslims hover below the surface.\" What Bangstad neglected to say about these demonstrators is that they had cheered a call for a new 9\/11 in Norway, and that the whole point of Hege's piece had been to highlight, and to condemn, the explicit incitement to violence that had taken place at that event.\n\nAfter smearing Hege, Bangstad proceeded to call me a \"Eurabia author\" and to attack Aftenposted opinion editor Knut Olav \u00c5m\u00e5s, whom he described as an \"ardent fan of Ayaan Hirsi Ali and Alan Dershowitz, and a personal friend of the Danish editor Flemming Rose (who commissioned the cartoons which provoked the global cartoon crisis in 2005\u201306)\"\u2014all of which facts Bangstad presumably considers suspicious. This was classic McCarthyism\u2014attacking \u00c5m\u00e5s for his friendships and other associations. Also to his apparent discredit, \"\u00c5m\u00e5s has on no less than two occasions recommended Bruce Bawer's Eurabia-books to his readers in editorial columns.\" No less than two occasions? Does this mean \"two\"? What kind of a McCarthyite locution is this? Liberals are supposed to recoil at such rhetoric\u2014but Bangstad was no liberal, rejecting \u00c5m\u00e5s's enthusiasm for freedom of expression and his belief, paraphrased by Bangstad, \"that most\u2014if not all\u2014opinions should be aired, so that they may be 'debated.' \" Bangstad's response:\n\nThis is a mistaken view, inasmuch as it is based on the contention that most, if not all, opinions will be challenged and contested in the public square. Norwegians who have become used to more and more vile public expressions of Islamophobia and racism in recent years would recognise the futility of debating publicly with Ms Storhaug and others who have pushed the limits of acceptable speech to extremes.\n\nBangstad cast his net wide. He went after Walid al-Kubaisi, for his 2010 documentary about the Muslim Brotherhood. Bangstad claimed that \"echoes of Eurabia-literature were more than evident in the 'documentary' \" and found it deplorable that the film \"was co-financed by the prestigious Fritt Ord Foundation and TV2, and earned public plaudits and recommendations from several Norwegian professors, among them Prof. Terje Tvedt at the University of Bergen and Prof Unni Wikan at the University of Oslo.\" Bangstad was also appalled that major Oslo publishers issued Norwegian editions of books by Oriana Fallaci and Ayaan Hirsi Ali, both of whom, after all, were on Breivik's reading list. Breivik, it seemed\u2014this solitary madman\u2014was henceforth to be the touchstone of what could and couldn't be said and written in the Kingdom of Norway.\n\nOne sentence of Bangstad's stood out in particular:\n\nIn recent years, I have often been approached by young, well-educated and upwardly mobile Norwegian Muslims after public lectures, with the question as to why a society which from government level to that of the media and civil society opposes anti-Semitism in all its forms with all its might, can still deem Islamophobic speech and utterances quite acceptable. I have not been able to provide any plausible answer\u2014let alone comfort for them in their despair.\n\n\"Opposes anti-Semitism in all its forms with all its might\"? \"Deem[s] Islamophobic speech and utterances quite acceptable\"? This is Orwellian, period.\n\nHege Storhaug, Ayaan Hirsi Ali, Oriana Fallaci, Walid al-Kubaisi, me: Bangstad accused us and many others of having \"provided the echo chambers for Anders Behring Breivik's thoughts and ideas.\" And after serving up the rote acknowledgment that we don't \"share any direct responsibility for his deeds,\" he made it clear that he considers us indirectly responsible\u2014for \"mass murder . . . requires ideological preparation. And that ideological preparation involves de-humanizing the 'other'\u2014whether she be a social democrat, a Muslim or both.\" Of course the whole point of my writing about Islam and related issues, and of everything I have read by people like Hege Storhaug and Ayaan Hirsi Ali, is to attend to the dignity and rights of the human individual\u2014as opposed to the cultural or religious group. The fundamental problem with multiculturalism is its exaltation of the group over the individual; our purpose is to reverse that emphasis. Dehumanization is the very opposite of what we are about.\n\nFor all his concerns, Bangstad was positive about Norway's future. Henceforth, he trusted, the gatekeepers of the Norwegian media would be less inclined to give a platform to \"Eurabia writers\" and other \"Islamophobes.\" Bangstad set himself against what he calls \"the ultra-liberalist champions of free speech\": in his view, censoring certain opinions is vital to the creation of the kind of society he wants.\n\nThe \"Eurabia\" pieces were now coming fast and furious. On August 29, not one but at least two opinion pieces pushed the meme. In Aftenposten, Mohammed Usman Rana (who, as head of the Muslim Student Association at the University of Oslo, had refused in 2007 to say whether he supported the death penalty for homosexuality) argued that some of those who had \"helped shape Behring Breivik's thoughts\" had failed to publicly distance themselves from him. His prime example was me, whom he described as having lamented in the Wall Street Journal that Breivik's atrocities \"would be a shot across the bow for [my] Eurabia theories.\" I had not even mentioned Eurabia in my Journal piece, but this hardly mattered: Eurabia was now the weapon of choice for those out to silence critics of Islam.\n\nEurabia\u2014this was the new \"big lie.\" People like me and Hege and Ayaan Hirsi Ali were \"Eurabia writers\"\u2014disciples of Bat Ye'or, members of an intellectual conspiracy. On the contrary, what was striking about While Europe Slept and so many of the other books of Islam criticism that were published around the same time was that writers with such a range of backgrounds had independently and (roughly) simultaneously decided to write about the rise of Islam in the West, unaware that other writers were doing the same thing, and that all had come to pretty much the same conclusions. This was not a conspiracy\u2014it was a case of several people responding sincerely to things that had deeply troubled them.\n\nThese New Quislings did not even try to engage with our ideas, to defeat us by quoting our arguments fairly and challenging them honestly. No, they were simply smearing us as bigots\u2014a nefarious cabal of bigots who had inspired a murderer, and who were that murderer's moral and spiritual kin. We had voiced our views, and now these tools of the Islamists were out to punish us for it\u2014by seeking to bring about our utter personal and professional destruction.\n\nIt did not seem a coincidence that the woman who was deemed the leader of this conspiracy was Jewish. Do we need to point out the parallel between the New Quislings' \"germ theory\" of ideological transmission and the Nazi war against \"Jewish thinking\"?\n\nOn August 5, Peder Jensen, a thirty-six-year-old man from \u00c5lesund, living in Oslo, told VG that he was the anonymous blogger known as \"Fjordman,\" whom Breivik had identified as his favorite writer and thirty-nine of whose essays Breivik had reprinted in his manifesto.\n\nI knew Jensen. A few years earlier he had contacted me and we had met for drinks a couple of times. We also e-mailed frequently, and when I was attending the Pim Fortuyn Memorial Conference in The Hague in 2006, I phoned him and suggested he come down, which he did. (It was at that conference that he first met Islam critics like Robert Spencer and Bat Ye'or, with whom he would become friendly.)\n\nJensen had studied Arabic, worked in the Norwegian foreign service, and been stationed in Hebron. I was impressed. He was polite, friendly, smart, serious, obviously very widely read, and knew what he was talking about when he talked about Arabic culture and about Islam. I never knew him to say anything racist or otherwise offensive, except perhaps for a few comments about women: he thought feminism had been the ruin of Norway, destroying the once proud Viking nation's willingness to stand up for itself. As for Islam, he recognized it as a threat to Europe, but I don't remember him ever talking about Arabs or Muslims personally in an offensive way; on the contrary, he was steeped in Arabic culture and in many ways admired it.\n\nI liked him, in short, and was happy to have made a new acquaintance with whom I could talk intelligently about matters of common concern. I did think the nickname Fjordman was terribly silly and told him so, and encouraged him to write under his own name. But he was determined to stay anonymous, certain that he would otherwise be in terrible peril.\n\nIndeed he was certain, I soon learned, about almost everything. His self-assurance proved daunting. He was convinced, for example, that his magnum opus about Islam in Europe, when it came out, would save Europe.\n\nI tried to give him advice about publishing, but he seemed uninterested. Indeed he had a lack of social antennae that gave me pause: virtually every time we spoke, he disparaged my work, comparing it unfavorably with his own, and doing so in a matter-of-fact tone that made it clear he had no idea he was giving offense.\n\nIn time it became clear to me that Jensen was rather too comfortable with nationalist groups of the kind I didn't want to have anything to do with. In the end, he was one of those with whom I broke off contact in connection with the Vlaams Belang matter. Like many of the others with whom I broke off contact, he still had my respect; I just didn't respect some of the people he was willing to associate with, and I didn't think that those associations could lead to anything good\u2014though, I might add, I wasn't about to trumpet that concern to the world at every opportunity or make it the centerpiece of my worldview.\n\nHe seemed truly hurt when I cut him off, and while others with whom I broke off contact responded with hostility and vindictiveness, Jensen remained friendly and solicitous. He repeatedly tried to reestablish contact, and if I remember correctly I backslid once or twice, replying to messages from him and thanking him for forwarding items of interest to me, because I still felt kindly toward him: to judge by what I knew of him and what I had read by him, he did not seem to me in any way to be a bad person, just someone who was inclined to fall too precipitously into the ideological orbit of bad people to achieve what he saw as a good end. (Making an alliance with Stalin may be necessary, but don't do it until you're sure you have to.)\n\nIn any event, at the time of the atrocities I had not been in contact with him for a couple of years, had seen virtually none of his recent work, and had actually forgotten his real name. As soon as it emerged that he was Breivik's \"hero,\" I was overcome with sympathy for him. As far as I knew, he had never written anything that any sane person would read as encouraging violence. In my view, he was an earnest, intelligent young man for whom there had been no place in a foreign service that treats Hamas with more respect than it does Israel, and who therefore sought to serve his country's interests in the only way he could come up with\u2014by writing about the truth as he saw it.\n\nNow he was being tarred mercilessly by the Norwegian media because his writing had been admired by a mass murderer. And not just tarred. After he came forward voluntarily, the police treated him, he felt, like an accomplice to murder. A dozen cops spent hours ransacking his apartment, going through everything\u2014including photographs and kitchen supplies\u2014and confiscated books, clothing, and electronic equipment, including a laptop. According to Jensen, his lawyer said their conduct was, at best, just barely on the borderline of legality. Jensen concluded that the Norwegian authorities wanted to scour his laptop for information on critics of Islam.\n\nIt was not only Peder Jensen who was demonized. After Hege publicly acknowledged that she had met him once, at his request, several years earlier, the New Quislings began attacking her for that. In short, it was okay that leading Norwegian politicians had friendly ties with Hamas and Hezbollah terrorists\u2014but not okay for someone like Hege to have agreed, years ago, to meet a writer whose work would later turn out to have been read with enthusiasm by a terrorist.\n\nThere were, to be sure, sane voices amid the madness.\n\nA noted Norwegian critic of Islam wrote to me a few days after the murders that \"NRK and the Labor Party and the Socialist Left Party intend to use this event politically. . . . they are not in grief, they are going for our throats. . . . It started just hours after the carnage. Their dishonesty never stops amazing me. Civil debate is what they do not want. Silencing debate has been their objective since I was [a] student in the Sixties.\"\n\nOn August 13, in a long and searching article in Klassekampen, Walid al-Kubaisi, too, expressed alarm that leading figures in Norway were now discussing \"ways to limit free speech,\" with a special focus on speech about immigration and Islam. But he pointed out that Breivik had not attacked a mosque or other Islamic target\u2014he had gone after the Labor Party. Should we therefore, Kubaisi asked, \"forbid criticism of Labor Party policy?\"\n\nIn any event, he added, concern about the airing of right-wing extremist ideas was irrelevant to the case of Breivik, for the views expressed in his writings were not extremist but \"moderate.\" Kubaisi further noted that, despite all the post\u2013July 22 rhetoric to the contrary, nobody had ever bashed Muslims in Norway's mainstream media, for the media had never allowed it. The people now being accused of bashing Muslims had, in fact, presented facts and arguments\u2014and their opponents, instead of characterizing them as mudslinging bigots and trying to silence them, should answer their facts with facts and their arguments with arguments.\n\nWere the critics of Islam veritable accomplices of Breivik, as was now being argued? On the contrary, said Kubaisi: it was the opponents of open debate who were \"Breivik's indirect accomplices.\" For Breivik's goal, after all, had been to shut down democracy. And that was exactly what they were now doing. And the more successful they were at stifling free speech, the more right-wing extremism would grow.\n\nInterestingly, it was a week to the day after the murders that several of my Norwegian Facebook friends and friends-of-friends broke their silence and began to rebel against the exploitation of July 22 by the New Quislings. One of those Facebook friends posted a grim comment about \"NRK's Gestapo hunt for all 'right-wing extremists.' \" Here are some comments by others:\n\n[P]eople on the left are now doing exactly what they have accused others of, namely stigmatizing. Now everyone who is against immigration is more or less an accomplice. They are of course using this unique chance to gag immigration opponents as much as possible, especially those who point to the danger of a large-scale influx of Muslims. Now all opposition will be crushed, once and for all. May it not happen!\n\nIndoctrination will continue as before, if not even worse.\n\nThe press's aggression in this case is, in my view, already a warning. . . .\n\nThe forces that are using this tragedy to further prevent freedom of speech must be totally ignored.\n\nNow we can see what a mourning period means for the blood red [i.e., the socialist, or \"red,\" parties, as opposed to the nonsocialist, or \"blue,\" parties].\n\nThe Labor Party's new motto must be \"Strike while the corpses are hot!\"\n\nAt the end of August, one of my Norwegian Facebook friends wrote that he had spent almost the entire month in Italy, where as a Norwegian had been asked many questions about July 22. The Italians he met viewed Breivik not as a right-wing extremist but as a madman. They were astonished, moreover, by the idea of a political summer camp for teenagers, which reminded them of Nazi and Stalinist brainwashing. And they all had great sympathy for my Facebook friend for having to live in a country where the debate was now being controlled by what they saw as a \"red-brown Communist regime\"\u2014for the kind of harassment of the opposition that was under way in Norway, the Italians pointed out, only takes place in dictatorships. Were they wrong?\n\nIt was hard not to conclude that the full-court press by the New Quislings was working like a charm. Normally tough opposition figures were plainly shaken. Formerly unblinking critics of Islam and multiculturalism were now mewling out meek mea culpas. It was dispiriting to read, on August 2, that Progress Party leader Siv Jensen, a strong, articulate woman whom I had compared in the British magazine Standpoint to Margaret Thatcher, was now confessing to the media she regretted things she had said in the past about Islam and immigration. Although she did, to her great credit, acknowledge the \"witch hunt\" to which she and her party were being subjected, she promised to alter her tone in debates.\n\n\"I think we have all already changed our behavior, and we will not be the same again,\" she said. Asked whether there were specific words she regretted using, she said: \"Yes there definitely are. But I do not think I should subject myself to this witch hunt. I think I will focus on contributing to the dignity that the nation needs to get through this grief.\"\n\nThe same article reported that many local Progress Party leaders had now fallen into line, distancing themselves from their party's rhetoric on immigration and integration, such as the word snikislamisering, meaning stealth Islamization. The party leader for Oslo said that policy positions must be expressed in a way that does not \"offend anyone\"; the Progress Party mayor of Mandal called for a change in the party's rhetoric about immigration; the party leader for Ask\u00f8y looked forward to \"another kind of discussion\" about immigration and condemned an article written in 2010 by a Progress Party MP who had criticized multiculturalism as \"idiocy.\" Such rhetoric, he said, should now be abandoned.\n\nBut none of this kept the media from demonizing the Progress Party in a way that went far beyond the levels of pre\u2013July 22 stigmatizing. At document.no, Hans Rustad reported that Siv Jensen, asked by an NRK correspondent during a hospital visit about an op-ed she and her predecessor, Carl I. Hagen, had written in 2005 about \"the overrepresentation of Muslims among terrorists,\" said that serious discussion of such matters could not be terminated because of July 22. When NRK aired this exchange on August 15, it was followed by an interview with Hajo Funke, a German political scientist who writes about right-wing extremism in Europe. \"Politicians who bash Islam are taking on a great responsibility,\" pronounced Funke, who went on to compare Jensen's party to neo-Nazis; by way of illustration, NRK created a montage in which images of marching neo-Nazis were shown alongside images of Carl I. Hagen.\n\nWith such taxpayer-funded propaganda supporting the New Quisling agenda, who would dare to dissent? On August 15, an article by Eirik Bergesen was posted on the website of the intellectual journal Minerva. Bergesen, who at the time was on leave from the Norwegian foreign service, noted that since July 22, \"Some individuals on the left want to spread responsibility for the tragedy to those whose words helped shape the context within which the perpetrator acted.\" He reported that on August 1, during an appearance on Dagsnytt, Magnus Marsdal had said that while he was a \"free-speech fundamentalist,\" he wondered, apropos of Knut Olav \u00c5m\u00e5s's words of praise in Aftenposten for my book While Europe Slept, whether it should any longer be permitted to say that one feared or opposed the influence of Islam. Bergesen's point was not to defend my work (he appeared to accept the notion that I am a far-right extremist) but to defend free speech: \"Right-wing and Islamist extremists are evil stepsisters,\" he wrote, but the way to fight them both is with more speech, not less.\n\nYet few dared agree out loud. And that was the whole idea. The New Quislings wanted their opponents to feel imperiled. They wanted to tame us, and if possible silence us entirely. Siv Jensen was right: a full-scale witch hunt was indeed under way in Norway. I had read about such witch hunts in history books, but had never experienced one in person before. It was profoundly unsettling. Conservatives and liberals alike were being put on the defensive by radical leftists. And all too many of the targets of this witch hunt were indeed running scared\u2014confessing their past sins, regretting their use of language, changing their opinions. It was tragic to see this happening in a supposedly free country. And it all just proved that when I worried, in my pieces for Pajamas Media and the Wall Street Journal, that the atrocities of July 22 would make it impossible to have an honest debate about Islam in Norway, I was even more correct than I thought.\n\n(Perhaps a gentle reminder is in order that, when it comes to Islam and all of Norway's failed multicultural policies, everything is, in fact, just as it was before. The imams who were embraced by Norwegian politicians and by members of the Norwegian royal family after July 22 still have the same views they had before July 22.)\n\nOn August 21 a national memorial ceremony for the victims of July 22 was held at Oslo Spektrum, Norway's answer to Madison Square Garden. The king and prime minister spoke, the names of the dead were read aloud, the NRK orchestra played Mozart and Beethoven, the 1980s Norwegian boy band a-ha performed, and someone sang \"Bridge over Troubled Water.\" And in a taped segment, the current head of the Norwegian Humanist Society, \u00c5se Kleveland, joined a group of religious leaders in speaking words of healing and unity into the camera. The leaders included a Church of Norway pastor, a Hindu priest, a rabbi, a Buddhist monk\u2014and the head of the Islamic Council of Norway, an imam named Senaid Kobilica. The segment plainly sought to suggest that all of these religious and secular leaders stood for the same loving, humane values. The organizers had apparently chosen to overlook the fact that Kobilica, not very long ago, made headlines by refusing to condemn the death penalty for homosexuality. What was presented as a display of diversity and tolerance, then, was in fact a confirmation of the new official determination to sweep the disturbing reality of Islamic theology under the rug.\n\nWhat I viewed at the time as the New Quislings' ultimate Statement of Principles appeared on August 22. It came in the form of an Aftenposten op-ed coauthored by Bangstad, Hylland Eriksen, University of Oslo philosophy professor Arne Johan Vetlesen, and a young woman named Bushra Ishaq, who is a medical student, a board member of the Anti-Racist Center, a member of a Conservative Party committee on religious policy, a \"peace worker\" for something called the Youth Global Harmony Association, a member of the women's panel of the Ministry of Children, Equality, and Inclusion, a participant in ecumenical dialogue between the Islamic Council and the Norwegian Church Council, an official Norwegian participant (nominated by Crown Prince Haakon) in an international forum called One Young World, and a former head of the Muslim Student Association.\n\nThe op-ed, titled \"Hateful Utterances,\" was a strident call for tighter limits on free speech in the wake of July 22.\n\n\"Certain hateful utterances,\" the four coauthors insisted, \"are legally and morally unacceptable. . . . Neither freedom of speech or the right to express oneself are absolute in any existing human society. Nor does freedom of speech stand above other rights and compliance with key human rights declarations.\" Although \"freedom of speech is a key value in a free and democratic society,\" they wrote, it is not \"unlimited.\"\n\nDecrying \"free speech absolutism,\" and explicitly rejecting the United States (\"the country in the world that goes the furthest in protecting the right to expression\u2014including hateful expression\") as a \"role model,\" the authors noted that the European Human Rights Convention \"is clear that the exercise of freedom of speech imposes a responsibility upon the person expressing himself.\" Yet in the last decade, they complained, \"the limits to hateful speech . . . have been stretched very far\" in Norway. This, they insisted, needed to be reversed in the aftermath of July 22\u2014the implication being that Breivik's atrocities were the consequence of the exercise of free speech by certain individuals. In the coming years, the op-ed argued, \"Norwegian editors as well as politicians\" needed to make it clear that \"it is not a human right to express oneself in public; and that certain hateful utterances . . . are not acceptable.\"\n\nThe New Quislings' determination to use the events of July 22 to silence their political opponents had never been more blatant. Hateful utterances, indeed!\n\nAnthropologist Runar D\u00f8ving was quick to agree, declaring flatly, in a September 2 interview with Morgenbladet, that the \"conspiracy theories\" of people like Hege Storhaug should be censored. As an example of these \"conspiracy theories\" D\u00f8ving cited Hege's January 6 Aftenposten op-ed, \"A Growing Unease,\" in which she reflected soberly and sensibly on the call for a Norwegian 9\/11 at that Muslim demonstration in University Square, the planning of a new mosque in Troms\u00f8 led by a woman whose husband has trained as a terrorist, and Iran's dispatching of imams to Norway to carry out such training. In D\u00f8ving's twisted view, Hege was not reporting in this op-ed on hatred; no, she was expressing a hatred\u2014and paranoia\u2014that, D\u00f8ving argued, is now ubiquitous in Norway and, in the wake of July 22, needed to be quashed. D\u00f8ving admitted readily that his view of the public square was, in a sense, \"authoritarian\"\u2014it should simply not be allowed, he said, to express certain ideas\u2014and that he was \"entirely in favor of what many people are now describing as a witch hunt,\" for \"there needs to be an investigation of what was written before July 22 so that we can see the connection between words and actions. Those who have demanded all along that Islam should be held responsible for Islamism must take responsibility for Islamophobic Christian hate violence.\"\n\nOn September 4, the Norwegian journalist \u00c5sne Seierstad\u2014whom I had admired for her readiness, in her international bestseller The Bookseller of Kabul, to take on political correctness by exposing the harsh patriarchal mentality of even a highly cultured and supposedly Westernized Afghani bookstore owner\u2014published in Newsweek a piece of outright character assassination. After dismissing the Norwegian public's concern about immigration and Islam as \"xenophobia\" and saying that the cause of this concern \"is somewhat of a puzzle\" given that \"[t]here has been no Islamic terror in Norway\u2014a picture that also fits the rest of increasingly immigrant-skeptic Europe\" (here she seemed to drop the London and Madrid bombings, among other atrocities, down the memory hole), Seierstad portrayed Hans Rustad and Siv Jensen as hard-hearted right-wing extremists and Marte Michelet as a gentle, diversity-loving soul.\n\nAs one Norwegian wrote in an online reader comment on Seierstad's article: \"Dear Americans. Please be aware that there is an information war going on in Norway.\" The reader noted that while document.no publishes \"articles that would be considered moderate in the USA, the left[-wing] Norwegian press is constantly trying to convince [its] audience that the topic[s], facts and opinions expressed on sites like document.no are right[-]wing extremism.\" By attempting \"to link Hans Rustad with the hideous terrorist attack,\" the mainstream Norwegian media were out \"to demonize all politics that are not left[-]wing enough for [them].\" The reader called the situation \"shameful\": in place of journalism, there was now \"pure propaganda.\"\n\nIn Aftenposten on September 6, Per Fugelli, a professor of social medicine at the University of Oslo, served up more of what was now becoming a tiresomely familiar dish: \"If we want to answer Behring Breivik with the opposite of his goal, answer number one must be to make life for our fellow citizens of the Muslim faith safer, more dignified, freer. . . .\" We must ask ourselves \"how they are doing\u2014how this evil deed has affected their security, sense of belonging, and sense of peace about the future.\" And we must do more to make them feel welcome: \"Have we met Fatima and Ali with the openness and trust they deserve?\"\n\nFugelli made the now-standard claim that he did not mean to blame any political party or person for Breivik's actions\u2014then, in now-standard fashion, started blaming away. \"The need for self-criticism,\" he thundered, \"is especially great in the Progress Party,\" which he accused of having spread the raw materials of \"fear\" and \"hate.\" But he lamented that \"there's little hope\" that the party would change its spots.\n\n\"For me,\" wrote Fugelli, \"this is a danger warning. We must . . . go into ourselves and ask: Have we stood up strongly enough against the forces that create enemy images of 'the others'?\" After all, \"Behring Breivik killed 68 people at Ut\u00f8ya because Workers Youth League is multicultural Norway in a nutshell. Workers Youth League is anti-racism in its heart and in its actions. Workers Youth League believes, with Nordahl Grieg: If we create equality, we create peace.\" In sum: \"Now we must emerge from the sweet narcotic of the roses and fight for what the young people at Ut\u00f8ya lived and died for. . . .\"\n\nNowhere in Fugelli's piece was there any recognition that trying to create \"equality\" for women and girls in Muslim communities is one of the goals of the Progress Party, and that you can't have both equality and multiculturalism\u2014for the latter obliges you to put the group's welfare and the preservation of its culture and values above the rights of individuals within that group. Nor, of course, was Fugelli prepared to acknowledge that more than a few Norwegian Muslims have opinions and beliefs that they might do well to reexamine. No, for Fugelli, it seemed, the Muslims around him were one-dimensional figures\u2014pure, unsullied, harmless victims\u2014who might almost have been dropped down in Norway to serve as reflections of non-Muslim Norwegians' goodness or evil, love or hatred.\n\nLife went on. On August 26 it was reported that several residents of an asylum center in western Norway had beaten, tortured, and poured boiling water over a fellow resident because he had converted from Islam to Christianity (which is, of course, a capital offense under sharia) and was therefore not fasting during Ramadan. The story served as a timely reminder that there are, indeed, aspects of Islam that merit criticism\u2014and that those who would silence such criticism are not heroes of diversity but enemies of freedom and enablers of Islamist tyranny.\n\nOn August 27, blogger Brage Baklien reported an incident he had witnessed earlier that day in downtown Oslo. It was now campaign season, during which each of the political parties sets up a little covered stand along Oslo's main street, Karl Johans Gate, not far from the Parliament building, where they hand out literature and discuss issues with passersby. While Baklien was minding the Progress Party stand, a group of \"20 enraged demonstrators from SOS Racism\" approached the person who was monitoring the adjacent stand for Kristent Samlingsparti, a tiny party of \"Bible-believing Christians,\" and kicked and tried to punch him. (This report, according to Dagbladet, was confirmed by several witnesses.) Baklien phoned the police, but by the time they arrived most of the thugs had disappeared. SOS Racism later claimed that they had simply held a peaceful, nonviolent demonstration. \"The event,\" observed Baklien on his blog, \"is an important reminder that powerful forces in Norwegian society want to curtail freedom of speech and democracy.\"\n\nThis was hardly the first time that members of SOS Racism had behaved like fascist gangsters. Still, it seemed more than likely that these \"demonstrators,\" in carrying out this particularly audacious assault (in the shadow of the Norwegian Parliament, no less) on freedom of expression and of assembly, had been inspired by the post\u2013July 22 atmosphere\u2014by, specifically, the New Quislings' message that it was high time to still Norway's non-PC voices.\n\nWhile the New Quislings, then, were claiming, against all reason, that the violent actions of a lone, insane terrorist had been influenced by scores of passionate, freedom-loving opponents of violence and terrorism, the New Quislings themselves, with their overheated rhetoric, were now successfully instigating precisely the kind of actions they wanted. Multicultural hooligans had begun to take action to cow, and to silence, the critics of multiculturalism\u2014just as Hylland Eriksen, Bangstad, and others so devoutly desired.\n\nAnd speaking of inspiration: in September 3, after weeks and weeks of shrill assertions throughout the Norwegian media that Breivik's atrocities had been inspired by a range of anti-jihadist writers, it was announced that the murderer, under interrogation by police, had said that his actions had in fact been inspired by the Red Army Faction, the far-left German terrorist group of the 1970s and '80s that was also known as the Baader-Meinhof Group. As Hans Rustad noted at document.no, Breivik shared the RAF's \"total ruthlessness in making war against society\"\u2014a modus operandi also evinced, he noted, by al-Qaeda.\n\nThis news might have silenced warriors with less determination\u2014or a greater capacity for shame\u2014than the New Quislings. But Bangstad, Hylland Eriksen, and company were unwavering. They knew what they wanted\u2014and, to a remarkable extent, they were getting it. Many Norwegians did indeed seem to be intimidated. They were doing penance for having thought for a couple of hours on July 22 that the bombings in Oslo were the work of Islamists. They were doing penance for having shared with this insane murderer\u2014and with Angela Merkel, David Cameron, and Nicolas Sarkozy\u2014the awareness that multiculturalism in Europe has been a failure. They were doing penance for having recognized that their country has critical problems with Islam, immigration, and integration that need to be addressed before it is too late.\n\nThey could not, one hoped, remain cowed forever. But it was unclear how long this state of affairs would continue. It was frightening to think that as a result of these atrocities, real discussion of multiculturalism, immigration, and Islam might remain in limbo for so long that Norway would miss its chance to save itself.\n\nI had hoped that in the September 12 local elections, Norwegian voters would prove their mettle and react strongly against the left's poisonous post\u2013July 22 rhetoric by voting in large numbers for the unfairly smeared Progress Party. But no. The election results showed that the poisonous rhetoric had, in fact, done its work. The Progress Party suffered its greatest losses ever, dropping from 17.5 percent of the vote in 2007 to 11.4 percent. Siv Jensen acknowledged what everyone knew\u2014that the fanatical effort by the party's opponents to link it to Breivik had driven supporters away in droves. It was depressing to discover that so many Norwegian voters could, indeed, be so easily swayed by what they well knew was a mendacious campaign of personal destruction.\n\nThe months went by. And the Norwegian media kept up their assault on the critics of Islam and multiculturalism. Nearly every day, one or another of the major newspapers ran an opinion piece making exactly the same charges that had already been made by Ekern, Nome, Bangstad, and company. My name came up constantly, but nobody, as far as I noticed, ever fairly represented my views or made a serious attempt to challenge any argument I had ever made; instead they tirelessly demonized me. It was pure name-calling: I grew used to seeing myself labeled a \"right-wing extremist,\" \"conspiracy theorist,\" and \"Eurabia writer.\" Nor was I alone in being targeted in this way.\n\nEvery now and then there was a brief flare-up of resistance. In November, for instance, in an intense exchange with Prime Minister Stoltenberg in front of the Norwegian Parliament, Per Sandberg, deputy chairman of the Progress Party, complained that the Labor Party had been playing the victim card ever since July 22. Stoltenberg replied heatedly that the Labor Party had, in fact, been the victim on July 22. Sandberg might have responded: \"I am not talking about those young people who were senselessly killed at Ut\u00f8ya. I am talking about the way in which the Labor Party and its supporters have exploited their deaths in the month since.\" Instead, Sandberg hastily withdrew his remarks and issued an abject apology.\n\nNovember also saw the publication of the first book about July 22.\n\nBefore Anders Behring Breivik came along, \u00d8yvind Str\u00f8mmen, born in 1980, had been a left-wing Norwegian blogger known mainly to a small coterie of like-minded readers. In the weeks after Breivik's atrocities, however, he won a degree of national attention as one of the leaders of the charge in the mainstream press against the critics of Islam. In the autumn of 2011, he was named \"freelancer of the year\" for \"his work monitoring Islam-hating environments on the net.\" When his book, The Dark Net: On Right-Wing Extremism, Contra-Jihadism and Terror in Europe, came out, it proved to be exactly what one expected.\n\nPublished by Cappelen, Str\u00f8mmen's book did not deviate in the slightest from the new party line. Str\u00f8mmen equated \"Fjordman\" with al-Qaeda's intellectual hero Sayyid Qutb. He described Serge Trifkovic, a serious critic of Islam, as \"one of the ideologues behind the modern Islam-hate.\" He lumped me and several other authors together as disciples of Bat Ye'or, writers of \"Eurabia literature\" and purveyors of conspiracy theories. He called Bat Ye'or \"a central source\" of my book While Europe Slept, and claimed that I treated her in that book \"as a guru.\" While admitting that \"there are immigrant-heavy areas in [Sweden] that have obvious social problems,\" Str\u00f8mmen insisted that my accounts of such problems \"are blown out of all proportion.\"\n\nBut what did he himself have to say about Islam in Europe? Virtually nothing. It was a subject on which he plainly did not wish to dwell. He had written an entire book about what he manifestly considered a dangerous epidemic of Islam-hatred in Europe\u2014but he was not interested in exploring the question of why, exactly, so many writers from so many different backgrounds had chosen to become, in his eyes, haters of Islam. To ask such a question would be to look into a reality Str\u00f8mmen was plainly out to avoid.\n\nHence he repeatedly quoted factual statements about Islam in a dismissive manner, as if they were so obviously untrue and rooted in bigotry as not to require refutation. For example, he scorned the notion of \"stealth Islamization\" without deigning to examine the evidence. He mockingly rejected the idea that Osama bin Laden, far from perverting his religion, had in fact been acting according to its dictates\u2014but did not dare to look at the relevant Koranic texts. And he described taqiyya as \"the Muslim-haters' answer to The Protocols of the Elders of Zion\"\u2014as if this Muslim doctrine, which permits believers to lie to infidels, were an invention of Islam's critics rather than an integral part of Islam itself. I may have missed it, but I don't remember seeing the word sharia once in St\u00f8mmen's entire book.\n\nThe first sentences of The Dark Net prefigured the way in which Str\u00f8mmen treated Islam throughout. He used to live, he told us, in a Muslim neighborhood in a Belgian town, where he had a perfectly nice relationship with the local merchants, who sold him \"Moroccan cakes . . . dripping with honey.\" This opening flourish was right out of the European cultural elite's ragged old playbook: Muslims are an exotic, enriching addition to European culture and cuisine\u2014period.\n\nNaturally, Str\u00f8mmen called for censorship\u2014while insisting that he was doing nothing of the kind. \"Freedom of expression and of the press,\" he wrote in his concluding pages, \"should be key elements of every democratic society. But they place us under an obligation, too. Good democratic debates required good soil and fresh water. If they lack this, seething pools of hate will develop. As citizens, we have a responsibility.\" It wasn't exactly clear what he was calling for here, but it certainly didn't sound good. In fact, it sounded downright totalitarian.\n\nHow did the Norwegian media receive The Dark Net? Guess. \"A strong contribution to the soul-searching the Norwegian community must go through after July 22,\" wrote NRK's reviewer. \"The year's most important book,\" pronounced Dagsavisen.\n\nBack in 1998, when I moved from America to Europe, I was on a personal, moral, and spiritual journey. I was in search of a home that was both liberal and willing to defend its liberal values. In Norway, alas, the defenders of liberal values now seemed to be in full retreat. In the weeks after July 22, for the first time in my life, I found myself breathing a sigh of relief that I was not in Norway\u2014and feeling a deep concern for all of the people I love and respect who were there, many of whom had now been placed in the position of having to defend liberalism from savage attacks by the New Quislings of the Norwegian left.\n\nThose New Quislings had been waiting for years for an opportunity to destroy their enemies utterly\u2014and they had now found it. They have been brutal in their attempt to stamp out the truth and to silence the truth-tellers, and so I will be brutal in my bluntness: speaking practically, Breivik is the best thing that has ever happened to them. As Walid al-Kubaisi has so cogently argued, if Breivik had \"indirect accomplices,\" they were not the critics of Islam and defenders of freedom, but the multiculturalist opponents of open democratic debate about Islam. Or as another wise observer of the situation put it to me recently: \"If you want to know who is responsible for Breivik, it's not the people whose books he read. It's the people who refused to debate and discuss the contents of those books and instead chose to stigmatize their authors\u2014and who in the aftermath of the Oslo massacre decided that this was an opportunity to win the argument without having to address the evidence. They're exploiting this episode as viciously as they can to try to restore their control over the parameters of public debate\u2014not understanding that that is precisely what caused the problem in the first place. And not understanding, either, that their 'solution' will only make things worse.\"\nAbout the Author\n\nA native New Yorker who has lived in Norway since 1999, Bruce Bawer has written several influential books on a range of issues. A Place at the Table: The Gay Individual in American Society (1993) was named by columnist Dale Carpenter as the most important nonfiction book about homosexuality published in the 1990s; Publishers Weekly called Stealing Jesus: How Fundamentalism Betrays Christianity (1997) \"a must-read book for anyone concerned with the relationship of Christianity to contemporary American culture\"; While Europe Slept: How Radical Islam Is Destroying the West from Within (2006) was a New York Times bestseller and a National Book Critics Circle Award finalist; and Surrender: Appeasing Islam, Sacrificing Freedom (2009) was hailed by Booklist as \"immensely important and urgent.\" He has also published several collections of literary and film criticism, including Diminishing Fictions and The Aspect of Eternity; and a collection of poetry, Coast to Coast, which was selected by the Dictionary of Literary Biography Yearbook as the best first book of poems published in 1993. He is a frequent contributor to such publications as The Hudson Review, City Journal, The American Scholar, Wilson Quarterly, and The Chronicle of Higher Education, and has reviewed books regularly for the New York Times Book Review, the Washington Post Book World, and the Wall Street Journal. His website is www.brucebawer.com.\n\nVisit www.AuthorTracker.com for exclusive information on your favorite HarperCollins authors.\nCopyright\n\nBroadside Books\u2122 and the Broadside logo are trademarks of HarperCollins Publishers.\n\nTHE NEW QUISLINGS. Copyright \u00a9 2012 by Bruce Bawer. All rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the non-exclusive, non-transferable right to access and read the text of this e-book on-screen. No part of this text may be reproduced, transmitted, down-loaded, decompiled, reverse engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of HarperCollins e-books.\n\nEPUB PUBLISHED FEBRUARY 2012\n\nISBN: 9780062188694\n\n12 13 14 15 16 EPUB 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\n\nCanada\n\nHarperCollins Canada\n\n2 Bloor Street East - 20th Floor\n\nToronto, ON, M4W, 1A8, Canada\n\n\n\nNew Zealand\n\nHarperCollins Publishers (New Zealand) Limited\n\nP.O. Box 1\n\nAuckland, New Zealand\n\n\n\nUnited Kingdom\n\nHarperCollins Publishers Ltd.\n\n77-85 Fulham Palace Road\n\nLondon, W6 8JB, UK\n\n\n\nUnited States\n\nHarperCollins Publishers Inc.\n\n10 East 53rd Street\n\nNew York, NY 10022\n\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}} +{"text":"\n\nThe \nMEDICINAL \nCHEF\n\n# Heart disease\n\nEat your way to better health\n\nDALE PINNOCK\n\n# CONTENTS\n\n 1. Title Page\n 2. 3. An introduction to cardiovascular disease \n 4. Stress and diet \n 5. The cardiovascular system: what it is and how it works \n 6. Key physiology \n 7. The cholesterol conundrum \n 8. Cardiovascular disease processes \n 9. The role of nutrition in heart health \n 10. Key heart-healthy ingredients \n 11. References, contacts and resources \n 12. 13. RECIPES \n 14. Breakfast \n 15. Weekday lunches \n 16. Weekend lunches \n 17. Quick dinners \n 18. Fancy dinners \n 19. Drinks, desserts and snacks \n 20. 21. Index \n 22. Copyright\n\n# An introduction to cardiovascular disease\n\n## CARDIOVASCULAR DISEASE IS NOW THE BIGGEST KILLER IN THE DEVELOPED WORLD. FACT!\n\nIn this day and age you would think that it would be something apocalyptic, such as famine or war, that would be humanity's downfall. But, alas, it seems that in terms of our health, we are the victims of our own 'progress'.\n\nWhen you look at the numbers in the UK alone, you start to see a very scary picture forming. According to The British Heart Foundation's latest set of released figures, this is the scale of the problem: coronary artery disease will kill one in six men and one in 10 women... and there are more than 2.3 million people living with the condition in the UK. There are 103,000 heart attacks and 152,000 strokes in the UK each year, while 750,000 people are living with heart failure.\n\nAs we look across to the United States, an even more grim tale unfolds. In the US someone dies from a heart attack every 33 seconds! More than 920,000 heart attacks are recorded annually and more than 80 million people have cardiovascular disease. This is a staggering picture. This is epidemic proportions.\n\nThe sad thing is, in these modern times, we are not seeing a decline. In fact, quite the opposite. Heart disease, its deaths and complications are on a rapid rise and are set to become the leading cause of death on the planet.\n\nPerhaps what is most alarming of all, though, is the fact that the highest proportion of these numbers come from avoidable circumstances. Granted, there are hereditary factors that can increase our risk of cardiovascular disease but, in the main, we are looking at a lifestyle condition. This means that we really can be in the driving seat here. Of course there are no guarantees in life, but if you don't want to get run over, then jogging blindfolded round the M25 is possibly not the best of ideas. Right? Making a few small changes to your lifestyle will be like whipping that blindfold off and getting on the footpath. Something may swerve off the road, but it's fair to say you are doing all you can to stay in the clear.\n\nOur lives today, with their stresses and strains and weird and wonderful habits, are driving the rapid movement towards a cardiovascular epidemic. There are some factors that are really fanning the flames, so let's examine them and see what we can do to help ourselves.\n\n# Stress and diet\n\n## STRESS\n\nModern life is pretty insane. I think that's a reasonably fair assessment of things. The pressures imposed upon us by this life we have created for ourselves here on 21st century planet Earth are overwhelming. The worries that accompany our financial ebbs and flows are ever growing and, let's face it, listening to the news doesn't help. Juggling home and career is like an insane science. Raising a family. Moving home. Modern life is filled with things that can take their toll on us and cause us to become so far detached from how we are supposed to live.\n\nNow, I'm a realist and I happen to enjoy modern life, so I'm not suggesting you should sell your house, buy a yurt and set up a commune in Glastonbury. But learning to manage stress can have a huge impact on many aspects of our health, especially the health of the circulatory system.\n\nStress can seriously send up our blood pressure, increase inflammation and play havoc with blood sugar balance. As you will see later in the book, these are important factors in heart health and disease.\n\n## DIET\n\nThis is where things have taken a massive nose dive. The modern diet in the West doesn't remotely resemble what we are meant to eat. The utter rubbish that has somehow become staple food makes the mind boggle. The consumption of processed food, refined food, fast food and \u2013 frankly \u2013 non-food is off the scale. There are people out there \u2013 I speak to a lot of them \u2013 who don't eat _any_ fruit and veg, unless you count cider and chips. This is a serious issue and it affects a big proportion of the population.\n\nThere are a lot of others who are trying to be health-conscious and make changes based on outdated and falsified information and guidelines (see page ). Their good intentions are actually putting them at greater risk of disease.\n\nThese factors can be addressed. With a little clarity, focus and effort, you don't have to become another statistic. We can all move away from this epidemic... it is not an inevitability.\n\nI will keep the information you need clear and to the point, but I won't skimp on detail. I want you to read this book without getting bored senseless, but also to learn enough from it to understand what is happening in your body, and how the food you eat can directly impact that for good and for bad. Then, best of all, I'll give you inspiration and ideas about how to put this picture all together rather deliciously.\n\n\"If you eat the standard Western diet that most people eat in the modern world, it is likely you will develop heart disease.\"\n\nDR JOEL FUHRMAN\n\n# THE CARDIOVASCULAR SYSTEM: WHAT IT IS AND HOW IT WORKS\n\nHaving a basic understanding of the cardiovascular system will enable you to start to build a clear picture of what is going on in your body, how small changes in your diet and lifestyle will have a great impact upon it, and especially how your current diet and changes you make to it may affect your specific issues. 'The cardiovascular system' refers to the heart, the blood vessels and their contents.\n\n### THE BLOOD\n\nThe most obvious place to start. This tissue is the whole reason the circulatory system exists and finding out what it is, the components in there and which plays what roles will be useful later.\n\nOne of the primary functions of the blood is as a transport system. It brings oxygen and nutrients to the cells and tissues of the body. The nutrients we take in \u2013 vitamins, minerals, amino acids, fats, glucose, or their by-products \u2013 play vital roles in the daily operations of every cell in every tissue in every system. These nutrients and their by-products get where they need to go via the blood. The blood also carries away waste. Our cells are very good at housekeeping; they process waste and throw it out as rubbish to be carried away in the circulatory system.\n\nThe blood is made up of several components:\n\n### PLASMA\n\nThis is the liquid portion of the blood, and makes up around 55 per cent of blood volume. It has very little colour \u2013 just a subtle pale yellow tinge \u2013 and is mostly water with a bundle of proteins, clotting factors and nutrients suspended in it. It also carries antibodies and other important elements for our immune function.\n\n### ERYTHROCYTES\n\nOtherwise known as red blood cells. These are the familiar disc-like cells that we often see in images and animations of the blood. Their main job is to transport oxygen to our tissues. Red blood cells contain a protein-based structure called haemoglobin. This is known as a metalloprotein (a protein that binds to metal), as iron makes up an important part of its structure. The iron in haemoglobin actually binds to oxygen to carry it around the bloodstream, where it can be deposited to cells and tissues. This is why people who have serious anaemia or iron deficiency become very tired and fatigued, as their capacity to deliver vital life-giving oxygen to cells is diminished. If cells don't get enough oxygen, their ability to create energy and perform many important functions is greatly impaired and severe fatigue and malaise soon set in.\n\n### LEUKOCYTES\n\nOtherwise known as white blood cells, these are the second most prominent type of cell in our blood. They are essentially the army of our immune systems, patrolling the body on the look out for anything that is upsetting the peace.\n\nThey can rapidly identify invaders that are trying to cause infection or damage. They can also identify our own cells that are suffering for whatever reason. They can tell if one of our cells has become infected and is in trouble. Or they can identify cells that are going through pathological changes, such as the changes that occur during the initiation of cancer. When they make this identification, they can set about a series of events that can deal with it. Some incidents can be dealt with by leukocytes there and then; others may require the leukocytes to recruit help and back-up.\n\nThere are several different types of white blood cells that do slightly different jobs. I won't go into all the details now but, as we go on, I will touch on the subtle differences as they become relevant to the whole picture of cardiovascular health.\n\n### THROMBOCYTES\n\nAlso called platelets, these are the third cell type that make up the non-liquid portion of our blood. Their role is to carry out what is called haemostasis. This is basically stopping bleeding at sites of injury. When you cut yourself, the blood doesn't keep oozing out of your body without stopping; we'd soon be in trouble at a very young age if that were the case. This is all thanks to our thrombocytes.\n\nThey stop the bleeding by rushing to the area of damage and forming a platelet plug. This is as it sounds, a clumping together of these cells to plug the wound. When this occurs, platelets send out a series of chemical messengers. Clotting factors (substances that assist with the clotting process) that are circulating in the plasma are sensitive to these signals and, when they get to the area of the platelet plug, they begin to lay down a fibrous structure called fibrin, which forms a mesh around the plug and strengthens it.\n\nThis series of events is an important thing to remember, as it is a vital part of understanding some of the things that take place in the body in cardiovascular disease.\n\n### THE HEART\n\nThis astounding pump system is so complex that it is beyond even the best human engineers. There have been numerous attempts to replicate it, all of which have failed miserably. There are artificial systems that can do its job during surgical procedures, but nothing that comes close to mirroring its functionality. About the size of a closed fist, the heart takes the deoxygenated blood (blood that has delivered all of its vital oxygen to the tissues) that is in your veins to the lungs to become oxygenated, before it is taken back off to the tissues of our body once again. It is divided into four chambers: two atria and two ventricles. Between each atrium and ventricle there is a one-way valve that prevents the backward flow of blood, ensuring the pump works as an effective continuous one-way system, with blood flowing in, then out in a perfectly orchestrated fashion.\n\nThe heart is divided in half, with two chambers per half. The right and left sides of the heart have two distinct jobs to do. The right side brings in blood that has low levels of oxygen and sends it to the lungs to get its oxygen levels topped up and also to remove its carbon dioxide. The left side of the heart takes the blood that has been freshly oxygenated and pumps it back out to the rest of the body, sending vital oxygen to our cells.\n\n### THE BLOOD VESSELS\n\nOur blood vessels (arteries and so on) resemble a network of incredibly complex plumbing. Thousands of vessels run through our body, some as thick as a hose pipe, others thinner than a single hair, delivering blood, oxygen and nutrients to our tissues. The thicker ones are called arteries, the next size down are arterioles, with the smallest and finest being capillaries.\n\nUnderstanding the structure of the blood vessels and how they work is a vital part of understanding the events that take place in cardiovascular disease, and to start seeing how diet and lifestyle may offer both prevention and intervention.\n\nBlood vessels are made up of several layers that all have different functions to carry out. Of these layers, the two I want you to become most familiar with \u2013 and those that I am going to discuss most frequently \u2013 are the endothelium and the smooth muscle layer.\n\n### SMOOTH MUSCLE\n\nThe bulk of our blood vessel walls is formed from smooth muscle. Smooth muscle is a type of involuntarily muscle (that means it reacts to environmental and chemical changes, rather than our conscious choice to move it, as we would a muscle in our legs).\n\nBlood vessels need to be incredibly responsive to the constantly changing environment of our bodies and the continual fluctuation in our tissues' needs for oxygen and nutrients. To be this responsive they must change size and shape very quickly.\n\nThe smooth muscle in the blood vessel walls can rapidly contract and relax to allow this change to occur. This has great relevance to heart disease as will be described in the next section (see page ).\n\n### ENDOTHELIUM\n\nThe endothelium is an incredibly thin yet unbelievably supple and complex inner skin that lines our blood vessels.\n\nAt face value level, the endothelium acts as a physical barrier between the blood vessel's contents and the rest of the vessel structure. This in itself is vitally important, as there are many potentially damaging components that can be in our circulation that could affect the health of the vessel.\n\nThe endothelium also regulates many aspects of blood vessel function, anything from responding to hormonal signalling to even controlling the activity of the smooth muscle described above. The health of the endothelium is of vast importance to cardiovascular health in general, and will be a recurrent theme in this book.\n\n# KEY PHYSIOLOGY\n\nOK, so I'm not going to expect you to suddenly become a heart surgeon or anything, but I do firmly believe that you should be as informed as possible about the inner workings of your body. There are a few aspects of how the cardiovascular system works that will really help you both to start to put the information in this book into some kind of context and to bring the bigger picture together. The more you can grasp what is going on, the better you can understand how nutrition is a key part of the solution. There are a few main areas, things that you would have heard about over and over \u2013 either from your doctor or in the news \u2013 that are vital to understand. The first of these is:\n\n### BLOOD PRESSURE\n\nWe all know by now that, if blood pressure is too high, we have a problem. The British Heart Foundation estimates that there could be in excess of five million people in the UK with undiagnosed high blood pressure. That is a serious number. But what is blood pressure? What does the term mean and why is it such a big deal?\n\nBlood pressure is basically the pressure that circulating blood places on the blood vessel walls. There has to be a certain amount of pressure in our vessels so that each contraction of the heart can push the blood to where it needs to go. As the blood is moved along, it exerts pressure against the vessel wall. That's it. So, why does it matter how high it is?\n\nThe higher your blood pressure is, the more force is placed upon the blood vessel walls, which figures. The vessel walls are designed to withstand a vast amount of force, but not a limitless amount. We get to a point where too much pressure is a problem. If your blood pressure is high enough for long enough, then the vessel becomes more susceptible to damage and any areas that have already been damaged from previous events run the risk of getting worse.\n\n### WHAT THE NUMBERS MEAN\n\nWhen we have our blood pressure checked, we are told (sometimes) a couple of numbers, then there is usually a suggestion as to whether they are good or bad. But what do the numbers mean? Following a blood pressure measurement, you will be told that your blood pressure is 'something over something'; 120 over 80 for example. The first number (systolic), represents the pressure exerted on the vessel wall when the heart squeezes and a large volume of blood is forced through the blood vessel. The second (diastolic) number points to the pressure put upon the vessel wall when the heart is at rest, or between beats. So which number is the most important? Well, in most cases it is the first one, the number that shows how much of a beating the vessel wall is taking under the most amount of force. The higher this is, the more risk there is of vascular injury, heart attack or stroke. More pressure = more chance of damage. (However, according to Blood Pressure UK, recently it has been thought that in those less than 40 years old, diastolic pressure is a greater predictor of risk. This may be because it can show that there is less flexibility in the vessel wall than expected, or that a kind of hardening has started to take place.)\n\n### WHAT IS NORMAL AND WHAT ISN'T?\n\nThere isn't really a gold standard perfect reading as such, and the old saying that your blood pressure should be 100 plus your age doesn't actually stand up to a lot, but these ranges (see right) should give you a rough idea.\n\n### WHAT CONTROLS BLOOD PRESSURE\n\nThe main driver of blood pressure is a pair of responses called vasoconstriction and vasodilation. Vasoconstriction makes the vessel get smaller and narrower. Vasodilation makes it get larger and wider. As a vessel constricts and gets narrower, the pressure within it gets much higher, as a specific volume of blood is having to be forced through a smaller space. When a vessel dilates and gets wider on the other hand, the pressure within it drops as there is more space for the blood to fill, hence less force exerted on the blood vessel wall. A healthy vessel is moving between these two states constantly to keep our blood flowing along nicely.\n\n\"Think about it. Heart disease and diabetes, which account for more deaths globally than anything else combined, are completely preventable by making comprehensive lifestyle changes. Without drugs or surgery.\"\n\nDR DEAN ORNISH\n\nEverything from physical activity to the health of our blood vessels will determine the rate and extent to which these changes occur. In a healthy individual, the move between vasodilation and vasoconstriction should be smooth and highly responsive. If you recall from the previous section (see page ), the bulk of our blood vessel walls are made up of layers of smooth muscle. This smooth muscle is the key component of the blood vessels that allows them to widen and constrict so readily. The muscle contracts; the vessel gets narrower. The muscle relaxes; the vessel gets wider.\n\nThink back to the previous section and you will recall that I placed great importance on the endothelium, that thin but vitally important inner skin that lines our blood vessels. Well, this seemingly simple structure is one of the major controllers of vasodilation. This is due to a highly active chemical that the endothelial cells produce, called nitric oxide. When the endothelial cells produce nitric oxide, it leaves the endothelial cells and then migrates out deep into the vessel walls, where it encourages the smooth muscle of the blood vessel to relax, which then allows the vessel to widen. Vasoconstriction is caused by a number of chemical factors, from calcium flowing into the muscle, through to neurotransmitters. Calcium, for example, causes muscle fibres to move together and muscles to contract, which in turn will narrow the blood vessel wall.\n\nIn terms of how nutrition can influence this whole picture, nitric oxide production and vasodilation is the most relevant part. The section of this book that describes cardiovascular disease processes will give you a good idea of how things can start to go wrong with the endothelium and the knock-on effects of that through the system as a whole, then when we get to explore the role of nutrition in cardiovascular health, we will begin to see how certain nutrients and dietary patterns can influence this. Hopefully, by building the picture piece by piece, you will finish reading the book with a better understanding of what's going on and what you can do about it.\n\n### BLOOD CLOTTING\n\nThis is a normal and absolutely vital response to injury and, without it, we would be in big trouble. It basically describes a series of events that stem bleeding when a vessel is injured. This can be the obvious type of injury, such as when you cut yourself, or silent internal injuries, such as damage to a blood vessel wall or a ruptured plaque (more on that later).\n\nSticking with the cut finger, you will realise that, when you cut yourself, you don't carry on bleeding to the point that you turn white and keel over. After a minute or so, the bleeding stops; give it a good few hours and a scab will start to form. This is the whole process of coagulation in action.\n\nWhen an area of a blood vessel becomes damaged, whether a kitchen knife ploughs through it or an atherosclerotic plaque ruptures (again, more on that later), a response is set in motion. Thrombocytes (platelets) begin to clump together around the site of injury to form a platelet plug. When this occurs, platelets send out a series of chemical messengers. Clotting factors that are circulating in the plasma are sensitive to these signals and, when they get to the area of the plug, they begin to lay down a fibrous structure called fibrin, which forms a mesh around the plug and strengthens it. This is essentially like a scab; the masking tape over the leak in the hose pipe. A makeshift repair job that stems the bleeding, while your body begins to repair damaged tissues. While this sequence of events is designed to save your life, as we'll see, when the process happens within a blood vessel, there is the potential for it to be life threatening.\n\n# THE CHOLESTEROL CONUNDRUM\n\nI don't think there is any greater area of misunderstanding, confusion, contradiction and outright panic than the area of cholesterol and heart health. So many clients that I have worked with over the years have been almost fixated on their cholesterol levels... and terrified by the numbers.\n\nMassive health campaigns that cross continents and span generations have been among the public health front runners. There have been TV campaigns, funny little drinks, you name it. And there have been many efforts to get us all to reduce our cholesterol and, of course, many a commercial opportunity, too. But how many of us actually understand our cholesterol, know what it is, or even know what half the terminology means?\n\nLet's get one thing perfectly clear from the off: cholesterol is a vital substance. The fact that our body produces up to one gram of it per day, regardless of what our diet looks like, is a pretty good indicator that it may actually need to be there and may not be the murderous villain that we are led to believe it is. Mechanisms that lead to that scale of production inside all our bodies cannot be some unfortunate physical flaw; they exist because the substance is vital to our health. Cholesterol is needed for the manufacture and maintenance of cell membranes.\n\nEach cell in our body consists of hundreds of different pieces of machinery and a host of biochemical signalling and relaying systems. These substances are held in place by a double-layered fatty bubble called the membrane. The membrane also helps our cells to function, as it actively gets involved in carrying messages from outside the cell to the inside, and moving things in and out. So, all in all, pretty important! Cholesterol helps strengthen the bonds within this structure, so it is more resilient. It also helps to secure specific proteins found within cell membrane walls that are involved in relaying signals between the inside and outside of cells.\n\nCholesterol plays an incredibly important role in digestion, too, as it is also used to create the bile acids that are released from the liver during digestion. Specifically they are involved in the breakdown of fats into smaller, more manageable particles, ready for absorption. Cholesterol has another vital role to play. It is the metabolic precursor (the chemical building block) for some vital substances in the body. One of the most important is vitamin D.\n\nAs you may be aware, vitamin D is the latest nutritional darling and the centre of a huge amount of research. What we are discovering about it is truly remarkable. We all know that it is important for maintaining healthy bones, because it helps the body to use calcium properly. But its benefits don't stop there. It has been shown to affect both mind and mood and it also regulates immunological responses.\n\nWhere does it come from? Well, a certain amount can come from our diet, from foods such as oily fish and offal. But the primary source of vitamin D for humans is the conversion of cholesterol into vitamin D when our skin is exposed to ultraviolet radiation (the sun)! Here in good old England, the sun is little more than a rumour for most of the year, so if the benefits of what little we do get are stifled by having very low levels of vitamin D precursors, we are in a bit of trouble.\n\nCholesterol is also the precursor for our main sex hormones: oestrogen and testosterone. The body needs cholesterol to make these. I don't know about you, but I for one don't want to see my levels of testosterone plummet any time soon! So, with all this in mind, I think it is clear that cholesterol isn't the demonic destroyer of health that we automatically suppose.\n\n### LDL AND HDL\n\nI guess you might have heard the terms LDL cholesterol and HDL cholesterol. These are sometimes also called 'bad' and 'good' cholesterol. What do they mean? Well, to start with, there is only one type of cholesterol. Cholesterol = cholesterol. It is a thick waxy substance. As such, it doesn't mix well with our blood (oil and water just won't mix) so, left to its own devices, it wouldn't get very far or fare very well just bobbing around fattily in our bloodstreams.\n\nTo this end, the body has its own transport system to shuttle cholesterol around the body. These are like two different bus routes and the bus is called a lipoprotein, a protein that can give fatty substances a piggy back. LDL = Low Density Lipoprotein, HDL = High Density Lipoprotein. LDL carries cholesterol out into our bodies' tissues via the bloodstream. HDL returns cholesterol from the blood to the liver for recycling and breakdown.\n\nSo, the theory goes that if your LDL (bad) cholesterol is high, then you are at greater risk of heart disease, but if your HDL is high, then it's good news. The basic proposition was \u2013 for a very long time \u2013 that an excess of cholesterol in the blood would begin to deposit itself in the walls of the blood vessels and cause a plaque. It was as simple as that. The more there was, it was thought, the greater your risk of developing heart disease, as more cholesterol would be getting deposited out into your tissues. The resulting medical treatment protocol for the prevention of cardiovascular disease focused upon getting levels of cholesterol down.\n\nFor a couple of decades, the medical profession were happily able to target a specific biochemical marker (cholesterol) in an attempt to decrease cardiovascular disease. Very soon there was a multi-billion dollar market in pharmaceuticals such as statins, while the functional foods market (those funky little cholesterol-lowering drinks) was also beginning to boom.\n\nBut, as the years went by and research moved forward, this clear picture and perfect theory began to fall apart. Did you know that almost 75 per cent of people who have a heart attack have clinically 'normal' cholesterol levels? Normal, as in at healthy levels... now there's a conundrum! Recent studies have even shown that a large proportion of patients with heart disease have a lower-than-average level of cholesterol. If it were purely a numbers game, then this simply would not be the case. Something else must be going on.\n\nPerhaps we have missed the trick! There are populations on the Earth, such as the Inuit people, that have been shown to have staggeringly high cholesterol levels, yet heart disease in their communities is as good as non-existent. Is there something else in their environment, diet or even their genes that offers them protection? If cholesterol were the single pathogenic factor that we have been looking for, then this lack of clarity as well as the seemingly outright contradiction would simply not exist.\n\n# CARDIOVASCULAR DISEASE PROCESSES\n\nHeart disease isn't something you just get struck down with. It is the result of a lot of small changes coming together over time. Many of the things that you have heard of as being risk factors for heart disease \u2013 such as high blood pressure, or smoking \u2013 essentially set the stage for a series of events to occur which can lead to the condition ultimately known as heart disease.\n\nThere are several pathological (disease causing) events that take place. There isn't always a specific order for these occurring and, very often, one gives rise to another in a vicious cycle. Here are the main events, the key pathological processes and certainly the ones that have been at the centre of most studies. Understanding them properly will help you get a better grasp on how nutrition and lifestyle can be one of the most powerful parts of your armoury against cardiovascular disease.\n\n### ENDOTHELIAL DYSFUNCTION\n\nThe endothelium, as we have seen earlier in this book, is the skin that lines the inside of our blood vessels. This thin skin is absolutely vital in maintaining the function of the rest of the blood vessel and, when it goes wrong in some way, the consequences can be devastating to cardiovascular health.\n\nAnd one of the fundamental areas where the endothelium can start to malfunction is when it has a reduced production or utilisation or release of nitric oxide. This is a chemical that is naturally made by \u2013 and released by \u2013 the endothelial cells, and which controls several aspects of vascular biology. It reduces blood clotting, reduces the movement of white blood cells into the vessel walls (see plaque formation, page 30), and also reduces the oxidation of LDL cholesterol. The major and most widely understood role for nitric oxide, however, is vasodilation, which means the widening of blood vessels (remember from when we explored blood pressure, see page ). This is the relaxation of the muscular walls of the blood vessels. It is normally stimulated mostly by sheer stress placed on the vessel walls by blood flowing through. Nitric oxide is produced and released by endothelial cells in response to immediate localised changes that signal a need for a change in blood pressure or vessel function.\n\nProblems arise when nitric oxide release or utilisation is impaired. The first and most obvious consequence is the reduced capacity for the vessels to widen under stress. Lets use the analogy of a hose pipe to give some clarity to this. Imagine you have two hose pipes. One is made of flexible, responsive rubber, the other of stiff, inflexible plastic. When water runs through them at a normal and steady pace, they both perform perfectly well. But, what happens if you should turn on the taps at full pelt? The rubber hose, when faced with the sudden rise in pressure, will simply stretch and expand and 'go with the flow'. The plastic one has no give, so begins to crack and split under the pressure.\n\nWell, this gives you an idea of how things go awry when our blood vessels are less responsive to changes in pressure. Suddenly we are at more risk of damage to the endothelium from the increased pressure against the vessel walls, and any areas of repaired damage and plaques (see page 30) are more susceptible to further damage and rupture. The initial stages of endothelial damage, whether induced by physical stress or other events, involve and lead to...\n\n### INFLAMMATION\n\nWe are now at the point, with research going in the direction that it is, where we can say with certainty that, despite whatever else is going on in the body, cardiovascular disease is essentially an inflammatory condition. Inflammation that causes damage to the vital endothelium and then fuels further pathophysiological changes in the body.\n\nInflammation is a normal, natural and vital thing. It aids our immune system in dealing with pathogens, infection or damaged tissue. In cardiovascular disease, it seems there is a two-way street, or a vicious cycle. Inflammation can cause endothelial dysfunction; endothelial dysfunction can cause inflammation.\n\nOne of the main and most widely established causes of inflammation within the endothelium is the oxidation of LDL cholesterol. LDL cholesterol can become damaged by circulating free radicals (reactive oxygen molecules that cause damage) and be readily oxidised (damaged and chemically altered). When this happens, the oxidised cholesterol can cause damage to the endothelium. Oxidation makes LDL far more able to penetrate the endothelium and cause some of the damage outlined over the page (plaque formation). Other factors that can trigger inflammation are smoking, high insulin levels \u2013 caused by eating too many fast-release carbohydrates for too long \u2013 and stress.\n\nBut, probably, the biggest factor of all for most of us in the Western world is the wrong types of fats in our diet. I am going to go into much more detail in the next chapter on nutrition and heart disease and heart health, but in the Western world we are eating too much of a type of fat that may be killing us slowly. Now, before you think this may be that old-school message about saturated fat that you have heard a million times for decades and that has now been proven to be wrong, think again. Listen up: saturated fat is not the villain that you might have thought.\n\nIn fact, the wrongdoer was the thing we moved over to when we all began abandoning butter for 'heart-healthy' margarines! We are consuming too much of something called omega 6 (see page for more on this). Omega 6 is a polyunsaturated fatty acid that, when metabolised by the body in more than minuscule amounts, actually exacerbates inflammation.\n\nWhen inflammation arises within the endothelium, the next series of events that can occur are...\n\n### PLAQUE FORMATION\n\nPlaques are the things that form in the blood vessels walls during what is called atherosclerosis. This is what people are referring to when they use the rather crude terminology of 'furring up', or 'clogging', of the arteries. They are the result of a series of events and knowing some of the stages will allow you to understand those elements of diet and lifestyle that may offer help in the prevention and management of the condition.\n\nThe first stage of this process stems from damage to the endothelium, that thin inner skin that lines the blood vessel. This can be susceptible to damage, given the right circumstances (such as those that stem from endothelial dysfunction described on page 27). When this damage occurs, circulating materials in the bloodstream \u2013 such as cholesterol and fats \u2013 can get trapped in the area of damage. Cholesterol that has become trapped in this area suddenly becomes more susceptible to oxidation and damage, due to the array of chemical responses taking place as all this circulatory junk accumulates. When the cholesterol oxidises, it triggers an inflammatory response. This then alerts circulating white blood cells, which move to the site of vascular injury. White blood cells, being what they are, wade in and try to help clean up the mess because, after all, this consistent oxidation of cholesterol can cause untold damage if not managed. So, in order to contain this, the white blood cells begin to swallow up oxidised cholesterol.\n\nWhen white blood cells do this, they begin to change and transform and become what is known as foam cells. When they have changed in this way, their normal ability to move and circulate disappears and they stay put at the site of injury. This is the first stage of what is termed a fatty streak, or fatty build-up within the blood vessel wall.\n\nAs this matures, smooth muscle cells from the muscular walls of the vessel also begin to move into the mass of foam cells and aid in supplying a matrix of fibres that can make this fatty streak more stable. It becomes a collection of fatty material, topped with a fibrous cap that is essentially holding everything in place.\n\nThis plaque can sometimes be very stable and lay unaffected in the blood vessel for a whole lifetime. At other times, plaques can be very unstable, meaning that the slightest increase in blood pressure, or increased force on the vessel wall from blood flow, can cause the plaque to rupture, which gives rise to a thrombus (see below). Also, chronic inflammation that has built up over a long time can give rise to the release of enzymes that can break down the fibrous cap, again leading to rupture. When this happens, the next stage is...\n\n### THROMBUS FORMATION\n\nWhen atherosclerotic plaques rupture, a blood clot quickly forms around the site of rupture. This can be likened to the formation of a scab when you cut your finger. When damage occurs, the affected area sends out chemicals that activate platelets in the blood. Platelets are known as cell fragments. They are cells without a nucleus and contain much less complicated machinery than most cells in our bodies.\n\nWhen the platelets are activated, they become sensitive to the effects of different clotting factors. These varying clotting factors come into play, binding platelets together using a fibrous mesh called fibrin. This ends up almost like a layer of netting that holds everything in place.\n\nThis clot can grow quite large, sometimes large enough to completely block the blood vessel it is inside. When this happens, the tissue it supplies becomes oxygen starved. Depending on how long this state lasts, the tissue may lose some of its function, or die completely. This is what is known as infarction.\n\nIn a heart attack, this occurs in a vessel that supplies blood to the heart muscle. In a stroke, this happens in a vessel that supplies blood to the brain.\n\nSometimes, the clot forms in a relatively wide blood vessel and is in no way big enough to cause occlusion (blockage) of the vessel. But, with changes in blood pressure and the force exerted on to the blood vessel wall from blood flow, clots (thrombus) can be dislodged. They can then move through the circulatory system.\n\nAs the blood vessels get closer to key tissues, they get smaller and smaller and their networks more intricate. As a thrombus moves through this seemingly endless network, sooner or later it will end up reaching a vessel that is just too small to accommodate it, where it will then cause a blockage.\n\n# The role of nutrition in heart health\n\n## THE ROLE OF NUTRITION IN HEART HEALTH, DISEASE PREVENTION AND DISEASE MANAGEMENT\n\nIt is a certainty that 90 per cent of cases of cardiovascular disease are ultimately preventable. That sounds like a bold statement, I know, but one I stand by. They are a result of our environment. This is of course partly the external environment we live in, such as stress, pollution and so on. But, when we talk of environment, we are referring to the internal biochemical terrain of the body. There is nothing that can influence this biochemical terrain more than our diet. With a few simple changes, we can guide our diet towards being cardioprotective. This means it can support cardiovascular health, potentially prevent the damaging issues and play a role in the management of any existing cardiovascular issues.\n\n### OMEGA 3, OMEGA 6 AND A QUESTION OF BALANCE\n\nMany of you that are familiar with my work will have probably twigged by now that I am a little obsessed with dietary fats. It is my belief that the fat composition of our diet is one of the key factors in cardiovascular health and disease. The fixation with dietary fats and cardiovascular health is, however, the reason we have got into such a mess in the first place, with the huge prevalence of this disease globally. In the last four or five decades, the patterns of fat intake in our diet has changed drastically. This is mostly thanks to the work of a man by the name of Ancel Keys.\n\nKeys was an American physiologist who came up with a hypothesis that the cause of cardiovascular disease was saturated fat intake. He was a very ambitious chap and set out to prove this hypothesis with vigour. He designed a 22-country study. It literally was as the name suggests, a study of 22 countries, searching for a correlation between saturated fat intake and cardiovascular disease. Now, the odd thing was, when this study was published, it was as 'The Seven Countries Study'; only seven of the 22 countries' results were used and the results looked very impressive indeed. The data produced a beautiful positive curve and essentially proved Keys's hypothesis that saturated fat intake was indeed associated with cardiovascular disease. But hang on a minute. What about the other 15 countries? What's going on here?\n\nAs it turns out, the seven countries selected were those that actually supported his theory. Had Keys used all 22 countries, the data would have shown absolutely no correlation whatsoever between saturated fat intake and cardiovascular disease. What was published was basically a fraudulent and engineered piece of reporting. Selective inclusion and exclusion of data that 'proved' something that didn't exist. But, alas, this study was taken on board around the world and Keys became a hero.\n\nBefore long he appeared on the cover of _Time_ magazine and his misleading study became the inspiration for the biggest public health cock-up known to man. In no time at all, the American government were developing a public health campaign that encouraged the population to ditch saturated fat and move towards a diet that was high in starchy foods and the supposed 'heart-healthy' oils such as sunflower oil and margarine. The same public health message made it to the UK soon after and then began to dominate the Western world. We took it on board. Didn't we just! This is where the problem began.\n\nYou can actually see, by looking at data from institutions such as the World Health Organisation, that as these changes in our diets occurred and we moved towards more starchy foods and more polyunsaturated oils, the incidence of cardiovascular disease, type-2 diabetes and cancer began to soar and, all of a sudden, we saw an obesity epidemic.\n\nSo, why are these dietary changes an issue? Well, I will talk about the starchy foods in greater detail when I discuss the glycaemic effects of foods and their relevance to heart health. But, for now, let's look at the oils that we started to consume in place of saturated fats. The message was that we needed to move towards a higher intake of vegetable oils, so sunflower, corn and soy oils and spreads became super-popular. Sickeningly they began (and still do) adding the 'heart-healthy' label to their products and advertising.\n\n### OMEGA 6\n\nThe problem that was completely overlooked in those early days was that most vegetable oils are incredibly high in things called omega 6 fatty acids. These are essential fatty acids that are vital to the body and must come from the diet as our body can't make them itself. All good so far. The snag is, however, that we only need a very small and finite amount of omega 6. Once we go over this level, the body metabolises it in a slightly different way than it would when we are at safe levels and changes it for the worse. Fatty acids in the diet are the metabolic building blocks for several important structures and compounds in the body. One of the big and vital groups that they give rise to are a group of communication compounds called prostaglandins. One of the main roles that prostaglandins carry out in the body \u2013 and this is important \u2013 is in the management of inflammation.\n\nThere are three different types of prostaglandins: Series 1, Series 2 and Series 3. Series 1 are mildly anti-inflammatory. Series 3 are strongly anti-inflammatory, switching off or down and regulating inflammation, and regulating pain signalling. Series 2 prostaglandins, on the other hand, actually switch on and exacerbate inflammation. This isn't a bad thing per se, providing that the body can be in a state of flux and produce sufficient amounts of these compounds to manage inflammation adequately.\n\nBut the balance of dietary fats in our bodies can disrupt this process. Different dietary fats are metabolised to form different series of prostaglandins. Omega 6 fatty acids are the metabolic precursors to \u2013 you guessed it \u2013 the Series 2 prostaglandins that switch inflammation on. The drastic shift in dietary fat intake in the last decades has meant that in the UK we take in almost 23 times more omega 6 fatty acids than we need _per day_!\n\nWe are essentially force-feeding metabolic pathways that manufacture prostaglandins, and our body's expression of the pro-inflammatory Series 2 goes into overdrive. The end result is a state of subclinical (your foot doesn't suddenly swell up, this is happening on a microscopic level within tissues), chronic (consistent and long-term) inflammation in the body. These compounds travel around the body in our circulation, so one of the first tissues to take a battering is, of course, the endothelium, as it is the tissue that is immediately exposed to their changing levels.\n\nIf you recall from the previous chapter, inflammatory damage within the endothelium sets the stage for plaque formation and, in essence, cardiovascular disease. The dietary change that was supposed to bring down cardiovascular disease ended up killing us faster. It was akin to trying to put out a bonfire with petrol.\n\n### OMEGA 3\n\nThis is the perfect time to bring in the other big dietary fatty acid, one you have probably heard a great deal about: omega 3 fatty acids. The benefits of omega 3 on heart health are well documented and have been studied widely for at least 20 years. These amazing fatty acids are the antidote to what we have just learned. There are three main types of omega 3: ALA, EPA and DHA. EPA and DHA are metabolised to form Series 3 prostaglandins (EPA more so). These are the most potently anti-inflammatory. So, eating good amounts of omega 3 fatty acids encourages our body to produce more anti-inflammatory prostaglandins.\n\nA growing body of evidence is showing that fish and fish oil consumption appears to offer significant protective benefit against heart disease; indeed, several studies have shown that fish consumption is directly related to a reduced risk of heart disease. A review of three large-scale epidemiological studies found that men who ate at least some fish per week had lower incidence of heart disease than those who ate none . Similar patterns were also found in women. A 2002 report from The Nurses' Health Study showed that, compared to women who ate no fish, risk of cardiovascular disease deaths were 21 per cent, 29 per cent, 31 per cent and 34 per cent lower for a fish consumption of respectively one to three times per month, once per week, two to four times per week, and more than five times per week .\n\n\"I'm not comfortable recommending people eat saturated fat with abandon, but it is clear to me that sugar, flour and oxidised seed oils create inflammatory effects in the body that almost certainly bear most of the responsibility for elevating heart disease risk.\"\n\nDR ANDREW WEIL\n\nOmega 3 fatty acids have also been shown to reduce levels of triglycerides. These are fats in the blood that can arise from dietary fat intake and from eating very high-GI foods (see page 41). These fats are believed to be very susceptible to oxidative damage, which could cause or aggravate endothelial inflammation and oxidise LDL cholesterol. A 1997 review of human studies found that around 4g per day of marine-derived omega 3 fatty acids reduced triglycerides in the blood by 25\u201330 per cent .\n\nPost-prandial triglyceridemia is the elevation of fats in the blood following a meal. This elevation in triglycerides appears to be very sensitive to omega 3 fatty acids, with a dose of around 2g per day reducing it significantly . These kinds of doses would come from supplementation. (See page for recommendations.) My approach \u2013 and my advice to you \u2013 is to eat fish and plenty of it and take supplements, too. Omega 3 fatty acids have also been shown both to deliver a dose dependent (that is, greater intake = greater result) reduction in blood pressure , and to reduce clotting factors that may offer some protection against thrombus formation .\n\n### THE BALANCING ACT\n\nSo, as you can see, omega 3 fatty acids are a pretty important part of the picture, while too much omega 6 can cause a problem. So it is therefore vital that we get the balance right. With the current trends arising from research, the recommendation now is to aim for a 2:1 ratio in favour of omega 3. That basically means that you need to be eating twice as much omega 3 as omega 6 in order to maximise the potential benefits, and counteract any negative effects of omega 6 in the body. Thankfully, this is pretty easy to manage in practice.\n\nThe first step is to avoid most vegetable oils like the plague. These are the so-called 'heart-healthy' oils such as sunflower oil, corn oil or the generic vegetable oil. These are basically pure omega 6 and will send your levels rocketing up very fast.\n\nIn place of these oils there are two cooking oils to choose from. In most of my cooking I use olive oil. The dominant fatty acid in olive oil is something called oleic acid which comes into a third category: omega 9. Omega 9 fatty acids have zero influence on omega balance, so don't particularly present a problem at all.\n\nThe other oil I use is coconut oil. This is best for high temperature cooking as it is completely heat stable. Also the fatty acids found in there, medium chain triglycerides, are rapidly broken down and used as an energy source, so their impact on postprandial lipaemia (elevation of blood fats after a meal) is minimal.\n\nThe next step in aiming for omega balance is to drastically cut back on processed foods. This is good advice for a million and one reasons but, in terms of omega balance, many processed foods use untold amounts of vegetable oils. They are cheap as chips and, for decades, food manufacturers have been under pressure to reduce saturated fat in foods, so have moved over to cheap vegetable oils as an alternative. Most ready meals, pre-made sauces, biscuits, cakes and so on will have a lot of omega 6 in them. Get back to basics, as we do in the recipes in this book, and get cooking from scratch as much as you can.\n\nThe second part of the solution is to up the levels of omega 3 in your diet. The first and most obvious place to start is by making sure you eat oily fish around three times every week. Then you could also consider taking supplements. I personally take an omega 3 supplement that contains 750mg of EPA and 250mg of DHA twice daily. (But if you are taking medication such as warfarin, or if you have recently had a heparin injection, please check with your doctor before using high-dose fish oil supplements as there is potential for interaction here.)\n\n### KNOW YOUR NUMBERS\n\nFor those of you that really want to be serious about getting your omega balance in check, there is now a home test available online that you can carry out which essentially tells you the ratio between omega 3 and omega 6 in your tissues.\n\n### THE GLYCAEMIC RESPONSE OF FOODS\n\nOne area that is very often overlooked in cardiovascular health is the glycaemic response of foods. This basically describes the rate and extent to which a food raises our blood sugar. Different foods, because of their composition, will release their energy at different rates. Pure glucose, for example, will send blood sugar up very rapidly and vigorously. Glucose is actually the benchmark against which all other foods are measured. It is the simplest form of sugar, so requires no digestive effort. It is consumed, then goes straight into circulation.\n\nFoods vary in their make up and complexity and certain factors will influence how rapidly foods release their energy. Fibre is one of the biggest factors. Let's compare white and brown bread, for example. Brown bread has all the fibre from the wheat husk and many brown breads have additional seeds and fibres added to them. White bread, on the other hand, has had all of the wheat husks removed and so the fibre content is drastically lower. The fibre in the brown bread will simply make the sugars in the bread harder to get to and will require more digestive effort to release. With the refined white bread, on the other hand, the lack of fibre makes the sugar much easier to get at. In the higher fibre food, the sugar is released at a more slow and steady pace, whereas with refined foods (anything white is usually a culprit) it is released at a very rapid pace as it takes far less digestive effort in the gut to liberate the easy-to-get-at glucose.\n\nAnother influence on glycaemic response is the combinations in which you eat certain foods. Adding protein to your carbohydrates, for example, will require a great deal more digestive effort to liberate the glucose. This is because proteins are digested more slowly, so there is a lot more work for the digestive system to do when you eat a combination of protein and carbohydrate. The end result is that you will get a slow, even drip-feeding of glucose into the bloodstream, rather than the giant surge you get when eating refined carbohydrates.\n\nBut why does any of this matter? Well, an obvious reason is that it will greatly influence your energy levels and mood stability, but that is by the by for your heart, which is what we are concerned with in this book. The glycaemic response of your diet over the long term is of great importance to cardiovascular health.\n\nWhen our blood sugar rises, our bodies secrete a hormone called insulin. This hormone basically tells our cells to take in glucose for converting into a substance called ATP, the energy unit that cells run off. So, the first reason insulin is secreted is so that the cells know there is glucose available for use. But the other factor to consider is that our blood sugar must stay at a very precise level. If it gets too high or too low, both states are potentially life threatening. In light of this, there are very precise balance homeostatic (homeostasis = the physiological control of balance in the body) mechanisms in place that control blood sugar. If it drops too low, the secretion of hormones that stimulate appetite is upregulated. Another hormone called glucagon is secreted from the pancreas which encourages the body to release glycogen, the storage form of glucose, for immediate use. If blood sugar gets too high, insulin production goes up, so at the same time our cells' uptake of sugar increases.\n\nHowever, this is where things can begin to go awry. Our cells only have a certain capacity for how much glucose they can take up at any given time, because if they take in more than they can readily metabolise and change into ATP, what is left over can oxidise and cause damage inside the cell. They can get full. If our cells are full to capacity and our blood sugar remains high, the excess sugar must be dealt with somehow and got out of the system as painlessly and effectively as possible before it does damage.\n\nAfter filling cells up to their maximum with glucose, the next most satisfactory way of dealing with it is via a reaction called de novo lipogenesis. This is where the glucose gets converted into a fatty substance called triacylglycerol, a fat that can be taken to the adipose tissue (our bodies' fat cells) for storage and taken away as rapidly as possible.\n\nAnother word for triacylglycerol is triglycerides... Sound familiar? They are often measured during routine blood tests that monitor cholesterol and other cardiovascular disease markers. These are the fats that, when in circulation, are susceptible to oxidative damage which can then cause damage to the endothelium. They also encourage oxidation of LDL, which can further damage the endothelium. Further, they make the LDL particles more susceptible to penetrating the endothelium as in the description of plaque formation (see page ). The clincher is that insulin also increases the likelihood of LDL oxidation, so you get a double whammy blow here. Higher blood sugar on a consistent basis means more triglycerides plus higher levels of insulin. None of this is good news!\n\nNow if you recall from my discussion of fatty acids above, following the 'healthy heart' public health campaigns that arose from Ancel Keys's work, we were all encouraged to fill up on more fruit and veg (that's a good thing) and more starchy foods (that's not a good thing). We started eating more and more bread, potatoes, pasta, grains and so on, every day and at every meal.\n\nNow, before anyone thinks I'm trying to get everyone on the Atkins diet, there is nothing wrong with these foods, but in general in the Western world we are eating way more than we should and, in essence, the balance on our plates is all wrong.\n\nOur preoccupation with fat and the advice to veer away from it and eat more starchy foods mean that we are eating a level that is harmful. These foods can raise our blood sugar notably. Now, once in a while that is no big deal. You will simply send out a bit more insulin, your cells will take in more glucose, problem solved. But, our intake isn't just every now and again.\n\nLet's see if this sounds like an inaccurate or extreme picture: how many people would have cereal and a slice of toast for breakfast? A sandwich for lunch? Then perhaps meat, vegetables and potatoes \u2013 or maybe pasta \u2013 for dinner? That sounds pretty common, right? Well, do that every day for a week, a month, a year and you will soon find your body's blood sugar staying consistently high and more insulin being produced, meaning more lipogenesis, more LDL oxidation, more endothelial damage. Nasty!\n\nThis situation is such an easy thing to remedy though, using a few simple steps:\n\n### REDUCE YOUR INTAKE OF STARCHY FOODS\n\nOK, so this may sound a bit obvious, but this is the place to start. For breakfast, go for a good source of protein such as eggs, smoked salmon or kippers. Ditch the cereal most days and, when you are craving cereal, opt for porridge, as oats have a low glycaemic response (see page ).\n\nLunches should be built around a good protein source, vegetables and salads. One of my lunch staples is salmon salad with a bit of feta and an olive oil-based dressing.\n\nThe evening meal is one where you can afford to have a bit of carbohydrate, as the carbs help the brain to take up the amino acid tryptophan, which helps us sleep. But this doesn't mean scoffing a bowl of pasta or a big jacket spud. Instead, go for choices such as roasted squash or sweet potato. Maybe add some high-protein quinoa or fibre-filled bulgar wheat, or brown rice. These are all very low-GI options and will fill you up and satisfy your appetite. Still, I would advise you only have a very small portion.\n\n### WHEN YOU HAVE CARBS, ALSO HAVE PROTEIN AND FAT\n\nThis is one of the real keys to buffering the effects of the carbohydrates on blood sugar as much as possible. Both protein and fat really slow down the digestion of a meal, meaning that available sugar will be freed slowly and blood sugar will be drip-fed. This is really easy in practice. You could have poached egg and avocado on toast (delicious, trust me). Maybe a piece of grilled fish with roasted sweet potato and some buttered greens. It truly is pretty straightforward.\n\nBy making these simple changes, you prevent the blood sugar roller coaster that, aside from making you feel rubbish, can completely destroy your long-term health. From damaging your cardiovascular system, to causing long-term insulin resistance. See my book _Diabetes: Eat Your Way to Better Health_ to see how this starch-laden diet that dominates the West is causing an epidemic of type-2 diabetes.\n\n### DIETARY FIBRE\n\nWhile we are on the subject of such foods, I wanted to add a little note on dietary fibre. We have all heard of the importance of dietary fibre. It is obviously beneficial for digestive health, but we won't go into that now, because there are also many benefits for what we are concerned with here: the cardiovascular system.\n\nNow, due to the conundrums surrounding cholesterol, I am sitting on the fence and watching what happens with the evidence as it unfolds. But, for many, lowering cholesterol is an important goal and until I can be more certain about what the evidence is really telling us, I won't argue against that, despite what my own personal convictions may be. Well, dietary fibre is a useful tool here. As we have already discussed, cholesterol is made in the body naturally. A small amount of this cholesterol leaves the liver and goes straight into circulation. Most of it, however, takes a bit of a scenic route. It is released from the liver with bile, where it enters the digestive tract. Once it gets in there, it is then re-absorbed back into the circulation.\n\nCertain types of fibre, known as soluble fibre, actually form a gel-like substance in the digestive tract which binds to this cholesterol and carries it away via the bowel before it gets the chance to be absorbed. As there is less cholesterol being absorbed, the liver takes more from the blood to make bile acids and for metabolic usage. This takes blood cholesterol levels down. This has been clinically proven with the fibre from oats, a particularly effective soluble fibre called beta glucan. The recipes in this book have a good fibre content and ingredients such as oats are well represented.\n\n### THE MAGIC OF MINERALS\n\nWe always hear so much about the array of vitamins in our foods. Weird and wonderful fats and fatty acids (guilty) and more antioxidant compounds than you can shake a stick at. However, a group of nutrients that are often overlooked are minerals, some of the substances so vital for human health that even the most tiny microgram difference in intake can be detrimental to our health. In terms of heart health, there are four minerals that are relevant and three of which, if you increase your intake, can have a positive impact on the health of your heart and blood vessels.\n\n### SODIUM\/POTASSIUM\n\nSodium and potassium are two of the most important minerals to be aware of in your daily diet, especially when it comes to managing blood pressure. Sodium has been at the forefront of heart health campaigns for many years and rightfully so. We have for a long time been encouraged to reduce our intake of salt. Why? Well, cheap table salts and most refined sea salts are predominantly composed of sodium chloride. Sodium is an important mineral in the body and we cannot be without it. An excess, however, can create real problems in the body.\n\nSodium has an important role to play in kidney function. Different minerals in different concentrations affect the rate at which fluids move through our kidneys. Sodium basically slows the movement down. When there are high concentrations of sodium in the body, the movement of fluid through the kidneys slows down sufficiently to cause the body to start retaining water. When this happens, the watery portion of our blood, the plasma, begins to increase in volume. This of course increases the volume of blood within the vessel. Which then increases the pressure against the vessel walls, simply because there is more blood in that tight space pushing against it.\n\nAdd to this picture the fact that the substance sodium can in itself be vasoconstrictive (cause contraction of the blood vessels, that makes them narrower) and it soon becomes a serious situation, where the risk of endothelial damage or the rupture of a plaque becomes very real.\n\nPotassium, on the other hand, is like the mirror to this. It is a mineral that we certainly don't get enough of, because its common sources are dark green leafy vegetables, and \u2013 admit it \u2013 these are definitely not top of the list of British favourites. Potassium can speed up the movement of fluid through the filtration mechanism of the kidneys (called the nephron). This can give a diuretic effect and soon begin to reduce plasma volume. In turn, this can take some burden off the vessel walls. The less volume within the vessels, the lower the pressure will be in them, as there is physically less pressing against the vessel wall. Potassium can also help to relax the blood vessel walls, giving a vasodilatory effect.\n\nTo reduce sodium intake, do not use table salt and also avoid refined sea salts. A natural sea salt should have a dull, dirty grey colour. Refined white sea salts have all the other vital minerals removed and are no better than table salt. One option is to go for a low-sodium\/high-potassium salt. There is a subtle taste difference, but when you are using so many fantastic flavoursome ingredients \u2013 such as those in this book \u2013 you will never know about it and you will benefit your health in a big way. Also, again, avoid processed foods such as ready meals and shop-bought sauces. These contain untold amounts of hidden sodium. Get back to cooking from scratch as much as possible.\n\nAt the same time, increase your intake of potassium-rich foods. The best sources are bananas, sweet potatoes, greens, mushrooms, dairy produce, tomatoes and some fish, such as tuna and halibut.\n\n### MAGNESIUM\/CALCIUM\n\nMagnesium and calcium are joined at the hip! Well, not quite, but they do work in tandem with each other all day and all night. This partnership is particularly important in muscle, where they are potent partners in crime. Calcium stimulates muscle fibres to contract, whereas magnesium causes muscle fibres to relax. The two move back and forth, allowing muscles to contract and relax all day. So notable is the effect of these two minerals that they are often used therapeutically. For example, a popular drug for stubborn hypertension is a class of drug called a calcium channel blocker. This reduces the amount of calcium that can get into muscle cells and reduces contraction, in short encouraging muscles to relax.\n\nMagnesium has also been studied as a potential hypotensive agent (something that lowers blood pressure). A 2012 meta-analysis (study of many other studies to determine the significance of results) of 1,173 people found that magnesium supplementation gave a reduction of both diastolic and systolic blood pressure, with the greatest reduction in intakes of over 370mg\/day .\n\nI feel that supplements are worth considering here. As always, it's all about the food and your emphasis should be on that, but a little extra magnesium in supplement form really wouldn't hurt. (See page , for more information.)\n\nIn terms of foods, greens are definitely at the top of the list. Green vegetables are rich in something called chlorophyll. This is what makes them green. Chlorophyll has a significant amount of magnesium bound to it by its very nature. So, if it's green, it has got decent levels of magnesium in it. Nuts and seeds, oily fish and pulses are other rich sources, but greens definitely rule the roost!\n\n### FLAVONOIDS\n\nThere are a very exciting group of compounds that are rapidly becoming the new superstars of nutritional research in the field of heart health. These are the flavonoids. They are phytochemicals, biologically active, non-nutrient compounds derived from plants. (When I say non-nutrients, I mean that there is no recognised deficiency disease attached to them. Things such as vitamin C, for example, you can be deficient in. Its intake is absolutely essential for our body to function.)\n\nPhytochemicals, on the other hand, are non-essential. You won't die if you don't consume them. But please don't for a moment think that they are not useful! In fact, when it comes to heart health, I would go as far to say that they are essential. I think they have a vital role to play in a healthy diet.\n\nThe richness of phytochemicals in fruit and veg and the power of their activity is part of the motivation behind the five-a-day campaign and a big contributor to the disease protection that is observed in high fruit and veg consumers. Phytochemicals are biologically active, which means that they can directly affect cells, tissues, genes, hormones, enzymes, reactions... you name it! These compounds literally are like nature's medicine cabinet. There are thousands of phytochemicals in plant foods and many that are being researched for every imaginable aspect of health. But, in terms of cardiovascular health, it seems to be the flavonoids that have come up trumps!\n\nFlavonoids are very broadly distributed phytochemicals, found in almost all plants. In short, they are colour pigments and are responsible for colours in plants from yellow and orange, through to deep red and purple. For the most part, flavonoids are known to be powerful antioxidants, helping to protect cells and tissues from free radical damage. However, in recent years, research has uncovered that they may prove to be superheroes in the fight against heart disease. The initial observations here come from meta-analyses of epidemiological data that has, for example, found a correlation between tea drinking and reduced incidence of cardiovascular disease . It has been found that there is an average 11 per cent reduction in risk for every three cup increase of tea each day .\n\nProbably the most well known of epidemiological observations is the curious relationship between wine consumption and heart disease. Many studies have shown that there is an observable dose-related (more intake = greater response, although with wine there is a fine line between benefit and risk) benefit to heart health from the regular consumption of wine . This is where the model of the French Paradox came from, the observation that the French, despite a diet high in dairy, meat and foods high in the saturated fats that were dietary heresy in terms of heart health, had a notably lower risk of cardiovascular disease than did the English, for example .\n\nWhile these observations of association were being made decades ago, it is only in recent years that we have started to figure out how flavonoid-rich foods may actually be delivering their benefits to our hearts and protecting them from disease.\n\nThink back to earlier in this book, when I described the structure of our blood vessels and the role that these structures played. We now understand that flavonoids interact with the endothelium and that is how the above results are most likely achieved.\n\nWe know that flavonoids actually get taken up by the endothelial cells. Once inside, they cause a little bit of chaos and act almost like an irritant. When this happens, the endothelial cells begin to secrete higher levels of nitric oxide .\n\nIf you recall (see page ), nitric oxide is a powerful vasodilator. The nitric oxide moves from the endothelial cells into the muscular walls of the blood vessel and causes the smooth muscle to relax.\n\nAs the muscle relaxes, the vessel dilates and gets bigger. As it gets bigger, the pressure within it drops. Evidence now tells us that consistent, regular consumption of flavonoids can have a notable lowering effect upon blood pressure.\n\n# KEY HEART-HEALTHY INGREDIENTS\n\nThis is by no means an exhaustive list, but below are some of the everyday ingredients that I think are the real heroes when it comes to keeping our hearts healthy. The good news, too, is that there's nothing obscure here: they are all regular and familiar ingredients that you can find at your local grocery shop.\n\n##### APPLES\n\nThis fruit is a very simple, easily accessible and versatile heart-healthy food. Why are they so good? Apples are very rich in a soluble fibre called pectin. Any of you that make jam will know that pectin is an effective gelling agent. This gel-like soluble fibre will bind to cholesterol in the digestive tract and carry it off before it gets the chance to be absorbed.\n\n##### AVOCADOS\n\nFor years, people thought of avocado as a fattening food. This was back in the days when we were completely obsessed with fat and the merest mention of it would strike fear into the hearts of many. This is, of course, ridiculous. The fats in avocado are unique, amazing for our health and they should be seen as nothing other than a health food. The fruit is very high in a group of compounds called phytosterols. These are the same compounds that you find in those little cholesterol-lowering drinks. They have been shown clinically to reduce cholesterol, by blocking the absorption of cholesterol through the gut wall (similar to soluble fibre). Avocados are also very rich in vitamin E, which is a powerful antioxidant nutrient. Vitamin E can actually protect LDL cholesterol from oxidation. As we have explored earlier in the book, this can be one of the early factors that triggers endothelial damage, so any protection against this is a vital part of looking after your heart.\n\n##### BEETROOT\n\nOK, so I admit that beetroot is definitely one of those love\/hate foods. Personally I am a huge fan and can't get enough of the stuff. Luckily, in recent years it has been found to have many health benefits. One of the areas that has attracted a lot of attention is the effect beetroot has on blood pressure. It is very high in natural nitrates, a type of mineral salt. This is then converted by the body into nitric oxide, which is naturally produced to regulate blood pressure. Nitric oxide causes the smooth muscles in the blood vessel walls to relax, which widens the vessels and in turn reduces blood pressure within them. Some small-scale studies have confirmed this effect. This doesn't mean you can throw your medicine in the bin and eat beetroot all day, though, it just highlights a powerful ingredient that we can consume more of to benefit our heart health.\n\n##### BLACKBERRIES\n\nBlackberries are incredibly rich in the flavonoid compounds called anthocyanins. These potent compounds are responsible for their deep dark purple colour, and are one of the most bioactive flavonoids in terms of stimulating endothelial function. They are known to be taken up into the endothelium, where they can stimulate nitric oxide release.\n\n##### BLUEBERRIES\n\nBlueberries, like blackberries, are high in the antioxidant compounds anthocyanins. These are the compounds that give them their deep purple colour, and have been shown to cause relaxation of blood vessels, protect vessel walls against damage, even reduce cholesterol slightly. Many studies have shown significant benefits to patients with cardiovascular disease, even vascular dementia.\n\n##### BROWN RICE\n\nOK, it's a health food staple and a lot of people still see it as a bit hippyish, but brown rice has benefits for heart health. It is mostly the high fibre content that makes brown rice useful. It helps move cholesterol out of the digestive tract, reducing the amount absorbed into the bloodstream. It also contains a compound known as gamma-oryzanol, that is linked with reducing levels of bad (LDL) cholesterol.\n\n##### BULGAR WHEAT\n\nThe fibre content of bulgar wheat makes it an ideal ingredient for digestive and heart health, as high-fibre foods will help remove cholesterol from the digestive tract before it can be absorbed. There are also a lot of B vitamins and magnesium in bulgar wheat, which have a soothing and relaxing effect. This may have knock-on effects for stress-induced high blood pressure, for example. Magnesium also works against calcium in smooth muscle, aiding relaxation and therefore vasodilation.\n\n##### CACAO\/COCOA\n\nCacao is packed to the hilt with flavonoids. As we have seen, these compounds have been very widely researched and are known to cause the cells that line our blood vessels to release high levels of a compound called nitric oxide, which in turn causes the muscles in the blood vessel walls to relax. When they relax, the blood vessel widens, which reduces the pressure within it. Cacao has been the focus of a great deal of research here in the UK, with many studies confirming its benefits \u2013 albeit transient \u2013 for elevated blood pressure and enhanced peripheral circulation (a marker of increased vasodilation). Cacao is also very high in magnesium, which also encourages relaxation of the smooth muscle in vessel walls.\n\n##### CHILLIES\n\nChillies contain a phytochemical called capsaicin, which gives them their intense heat. Capsaicin causes the cells that line the inside of our blood vessels to secrete a chemical called nitric oxide, which as we have seen is naturally produced by these cells (chilli just gives them a kick in the right direction). Nitric oxide tells the muscles in the blood vessel walls to relax, so the vessel widens. This has two benefits: firstly, the wider the blood vessel, the lower the pressure within it; secondly, circulation to the extremities is improved. Have you (or anyone you know) turned red-faced after eating something particularly hot and spicy? Well, this is that very vasodilatory response in action!\n\n##### GARLIC\n\nGarlic has long been championed for keeping the heart healthy. It contains a potent compound \u2013 ajoene \u2013 which interacts with a compound in the body that regulates the rate and extent to which blood clots. As we have seen, excessive clotting can be a very high risk for cardiovascular incidents, while keeping clotting at a reasonable level may deliver several benefits. Some surgeons even advise their patients against eating garlic before surgery, just in case it increases their bleeding. On a day-to-day basis, it can protect from clotting, so is a weapon against strokes and heart attacks.\n\n##### GREEN TEA\n\nAnother of those healthy staples. I remember 10\u201315 years ago, when I would drink green tea, that friends, family, all and sundry looked at me like I'd just stepped out of a spaceship. Well, how times have changed. Green tea has quite the reputation these days as a healthy ingredient, and justifiably so in my view. Green tea has some potentially powerful benefits for the heart. This is thanks to the presence of a group of compounds called catechins. These have been shown to reduce platelet (thrombocyte) adhesion, so may offer protection against clots. There are also other flavonoids present in green tea that can stimulate nitric oxide release and therefore increase vasodilation.\n\n##### MACKEREL\n\nThe omega 3 fatty acids in mackerel have a very favourable effect upon cholesterol levels, and can also protect blood vessel walls from inflammatory damage. Omega 3 also delivers antithrombotic activity and can help to reduce blood pressure. What's not to like? Prolonged regular intake of oily fish, as well as fish oil supplements, has been shown in numerous studies to be associated with a decreased incidence of heart disease and has also been shown clinically to improve several of the clinical markers for cardiovascular disease.\n\n##### OATS\n\nOats have become one of the most famous of all 'heart healthy' foods today. They contain a soluble fibre called beta glucan. This has been clinically proven to reduce cholesterol in the digestive tract. It does this by forming a gel-like substance, which then binds to cholesterol that has been released from the liver. Once bound to it, it carries the cholesterol out the body through the bowel before it has had a chance to be absorbed back into the bloodstream.\n\n##### OLIVE OIL\n\nOlive oil has long been known to be beneficial to heart health. The Mediterranean diet is believed to be one of the healthiest ways to eat in the world, and has an exceptional track record for protecting the heart and circulatory system. One of the main protective elements within that diet is, of course, olive oil. The fatty acids in olive oil have been shown on many occasions to increase the levels of HDL cholesterol, and decrease LDL. Oleic acid, the most abundant fatty acid in olive oil, seems to have a beneficial effect upon blood pressure, with some subtle vasodilatory function.\n\n##### QUINOA\n\nAs I have explained earlier in the book, a high-GI diet is a fast track towards cardiovascular problems. Unlike many grains (which tend to be total starch bombs) quinoa is very, very low in carbohydrates and is very low GI. This means it will release its energy slowly and won't cause blood sugar spikes, making it a perfect alternative to rice for anyone wishing to stabilise their blood sugar more effectively. It also naturally has a high protein content which will aid satiety and slow down digestion of a meal, giving that all-important drip-feeding of blood sugar.\n\n##### RED LENTILS\n\nRed lentils are another ingredient with a high percentage of soluble fibre. I know I may sound like I am repeating myself a little bit, but I really want to drive the point home. This soluble fibre helps remove cholesterol from the gut, reducing the amount that gets absorbed into the bloodstream through the digestive tract.\n\n##### RED ONIONS\n\nAll onions are amazing for you, but red onions in particular are extra-special for the health of the heart. This is because they are particularly high in flavonoids, part of the cocktail of chemistry that gives them their deep purple colour. So, again, these will enter the endothelial cells in our vessels and increase their expression of nitric oxide, aiding vasodilation (widening of the blood vessels) and helping to protect the endothelium from damage.\n\n##### RED PEPPERS\n\nRed peppers are definitely up there with my favourite heart-healthy ingredients. Their deep red colour is given by a reasonably high concentration of flavonoids, offering protection to the endothelium and enhancing vasodilation again by \u2013 you guessed it \u2013 increasing nitric oxide expression by the endothelium.\n\n##### RED WINE\n\nAnd you thought it was all bad news, didn't you! Red wine consumption has been shown, in dozens of population studies, to be associated with reduced incidence of cardiovascular disease and many of the clinical markers associated with cardiovascular disease risk. It is believed that this is again due to the flavonoid content and also a compound called resveratrol. Both of these compounds are known to induce vasodilation, have anticoagulant properties, reduce inflammation and have positive effects upon cholesterol levels. The bad news is... only two glasses a day.\n\n##### SALMON\n\nOily fish are definitely at the top of the healthy heart food chain and are big players in my recipes, as you will see in the next section. Salmon is packed with omega 3 fatty acids, all-important good fats. These help maintain healthy cholesterol levels and protect blood vessels from long-term, persistent inflammatory damage, which can be the first step in the process that leads to heart attacks. Omega 3 is also beneficial for the rate and extent to which blood clots, offering a reduction in clotting.\n\n##### SWEET POTATOES\n\nAnother of my staple ingredients and rightly so, I think. These are packed with anti-inflammatory beta carotene. This is the thing that makes the flesh orange and which may offer some anti-inflammatory protection when consumed regularly. Sweet potatoes also give a much lower glycaemic response than the regular spud, so they are a perfect alternative to chips, mash, shepherd's pie, the lot!\n\n##### TROUT\n\nTrout is a fish that has very good levels of the anti-inflammatory omega 3 fatty acids.\n\n##### TUNA STEAK\n\nSeveral studies have found that tuna positively affects cholesterol levels. This is most likely due to the high omega 3 levels in fresh tuna. Canned tuna, although it's a great lean protein, is not a good source of omega 3, as all of the oils have already been pressed out and sold (ironically) to the nutritional supplements industry.\n\n# References, contacts and resources\n\n### REFERENCES\n\n### Omega 3:\n\n Stone NJ. Fish consumption, fish oil, lipids, and coronary heart disease. _Circulation_. 1996; 94: 2337\u20132340\n\n Hu FB, Bronner L, Willet WC. Fish and omega 3 fatty acid intake and risk of coronary heart disease in women. JAMA. 2002; 287: 1815\u20131821\n\n Harris WS. N-3 fatty acids and serum lipoproteins: Human studies. _Am J Clin Nutr_. 1997; 65 (5 Suppl): 1645S\u20131654S\n\n Roche HM, Gibney MJ. Postprandial triacylglycerolaemia: the effect of low-fat dietary treatment with and without fish oil supplementation. _Eur J Clin Nutr_. 1996; 50: 617\u2013624\n\n Howe PR. Dietary fats and hypertension: focus on fish oil. _Ann N Y Acad Sci._ 1997; 827: 339\u2013352\n\n Agren JJ, Vaisanen S, Hanninen O, et al. Hemostatic factors and platelet aggregation after a fish-enriched diet or fish oil or docosahexaenoic acid supplementation. _Prostaglandins Leukot Essent Fatty Acids._ 1997; 57: 419\u2013421\n\n### Magnesium:\n\n Kass L, Weekes J, Carpenter L. Effect of magnesium supplementation on blood pressure: a meta-analysis. _Eur J Clin Nutr_ 2012; 66: 411\u20138\n\n### Flavonoids:\n\n Peters U, Poole C, Arab L. Does tea affect cardiovascular disease? A meta-analysis. _Am J Epidemiol._ 2001; **154** : 495\u2013503.\n\n Di Castelnuovo A, Rotondo S, Iacoviello L, Donati MB, DeGaetano G. Meta-analysis of wine and beer consumption in relation to vascular risk. _Circulation._ 2002; **105** : 2836\u201344\n\n Renaud S, de Lorgeril M. Wine, alcohol, platelets, and the French paradox for coronary heart disease. _Lancet_. 1992; **339** : 1523\u20136\n\n Fisher, Naomi DL; Hughes, Meghan; Gerhard-Herman, Marie; Hollenberg, Norman K. Flavanol-rich cocoa induces nitric-oxide-dependent vasodilation in healthy humans. _Journal of Hypertension._ 2003; 21 (12): 2281-2286\n\n### USEFUL CONTACTS AND RESOURCES\n\n### Organisations\n\n### British Heart Foundation\n\nProbably the best known organisation championing heart health. The British Heart Foundation offers a vast amount of information about heart health, keeping your heart healthy, heart disease, statistics, and everything in between. bhf.org.uk\n\n### Heart UK\n\nHeart UK have a big focus upon cholesterol. There is lots of interesting information on their website regarding cholesterol, medications, and diet and cholesterol. heartuk.org.uk\n\n### Blood Pressure UK\n\nAs the name suggests, Blood Pressure UK are a great resource for all things high blood pressure. Whether it is advice about knowing your numbers, or how your lifestyle may be affecting your blood pressure, their website has plenty on offer. bloodpressureuk.org\n\n### Nutritional resources\n\n### British Dietetic Association\n\nThe BDA have a series of fact sheets available on their website. The factsheet regarding blood pressure has some useful guidelines in it. bda.uk.com\/foodfacts\/hypertension\n\n### Nutritional supplements\n\n### Viridian Nutrition\n\nThis company make an extensive range of the cleanest supplements around. Over 180 products including vitamins, minerals, herbs, oils and specific formulae made from the purest ingredients, with no additives, nasty fillers or junk. viridian-nutrition.com\n\n# RECIPES\n\n# Breakfast\n\n###### VITAMIN E\n\nGARLIC (ANTICOAGULENT) \nSOLUBLE FIBRE \nLOW GI\n\n**Avocado and poached egg rye toast topper** I concocted this addictive dish one morning when faced with nothing but half an avocado and some eggs in the fridge. Oh, what a happy discovery it was!\n\n#### SERVES 1\n\n\u00bd ripe avocado\n\n1 garlic clove, finely chopped\n\njuice of \u00bd lemon\n\nlow-sodium salt and freshly ground black pepper\n\n1 slice of pumpernickel bread\n\n2 eggs\n\nScoop the avocado flesh into a bowl and add the garlic, lemon juice and low-sodium salt and pepper to taste. Mash the avocado mixture and mix well.\n\nToast the bread while you poach the eggs; the whites should be set but the yolks still runny.\n\nSpread the avocado mixture over the bread, top with the eggs and sprinkle on a little pepper.\n\n###### OMEGA 3 FATTY ACIDS\n\nMAGNESIUM \nLOW GI\n\n**Salmon, pea and asparagus frittata** I am a real frittata freak. They are so satisfying when you are really hungry and are a great way to throw all manner of flavour combinations together. Use pre-cooked salmon (but not canned) from the supermarket, to save time. It's easy to scale up the recipe, as we have done in the photo, to serve more people.\n\n#### SERVES 1\n\n\u00bd tbsp coconut oil\n\n2 tbsp peas\n\n3\u20134 large asparagus stalks, each cut into 3 or halved lengthways\n\n2 eggs, lightly beaten\n\n1 small cooked salmon fillet, flaked\n\nlow-sodium salt and freshly ground black pepper\n\nPreheat the grill.\n\nHeat the coconut oil in a small ovenproof frying pan over a medium heat. Add the peas and asparagus and saut\u00e9 for four to five minutes, until the vegetables have turned a brighter green and are beginning to soften.\n\nAdd the eggs to the frying pan and cook for a couple of minutes, until the edges of the egg have started to cook well, but the middle is still raw. Add the salmon, low-sodium salt and pepper and cook for another minute.\n\nPlace under the grill until all the egg is cooked; this should take three or four minutes max. Ready to serve.\n\n###### FLAVONOIDS\n\nOMEGA 3 FATTY ACIDS \nSOLUBLE FIBRE\n\n**Oat and berry layer** This is a gorgeous, speedy breakfast. I find it especially refreshing in the summer months.\n\n#### SERVES 1\n\n2 tbsp blueberries\n\n3 tbsp porridge oats\n\n2 tbsp blackberries\n\n1 tbsp natural live probiotic yogurt\n\n1 tsp flax seeds\n\nIn a tall glass tumbler, layer up the dish: begin with a layer of blueberries, then oats, then blackberries, then oats and so on. You should finish with a layer of oats.\n\nSpoon the yogurt on top, then finish with a sprinkling of flax seeds.\n\n###### FLAVONOIDS\n\nMAGNESIUM \nOMEGA 3 FATTY ACIDS \nSOLUBLE FIBRE\n\n**Mixed seed and blackberry bowl** This is a lovely refreshing breakfast. It's a really thick smoothie\/pudding\/parfait-type vibe.\n\n#### SERVES 1\n\n200g natural live probiotic yogurt\n\n2 tbsp blackberries, plus more for the top (optional)\n\n1 tbsp vanilla protein powder (optional)\n\n1 tsp ground flax seeds\n\n1 tsp sunflower seeds\n\n1 tsp pumpkin seeds\n\nPlace the yogurt, berries and protein powder into a food processor and blend into a thick, creamy, smoothie-type texture.\n\nTransfer to a serving bowl and sprinkle with the flax, sunflower and pumpkin seeds. You could also place a few whole berries on top for added colour, if you like.\n\n###### OMEGA 3 FATTY ACIDS\n\nFLAVONOIDS \nMAGNESIUM\n\n**Kippers, boiled egg and watercress salad** OK, so I know having salad at breakfast may seem a bit alien. But in many parts of the world it is the norm and, on my travels, I have become very fond of the idea. Give it a try. Break the mould. You will soon see how refreshing it is, not to mention a great opportunity to get more of the good stuff into your body.\n\n#### SERVES 1\n\n1 kipper fillet\n\n2 large eggs\n\nsmall bunch of watercress\n\n1 tbsp olive oil\n\nFor packaged kippers, cook (usually boil) according to the manufacturer's instructions. If the kipper fillet is unpackaged from a fishmonger, grill it for eight to 10 minutes.\n\nHard-boil the eggs according to how you like them. I prefer an eight-minute egg that is still moist in the centre, but whatever floats your boat... Peel them, then slice.\n\nArrange the kipper and eggs on a plate, add the watercress and sprinkle over the olive oil.\n\n###### BETA GLUCAN\n\nLOW GI \nMEDIUM CHAIN TRIGLYCERIDES\n\n**Creamy coconut porridge** Oats are a great ingredient for heart health, thanks to the presence of the soluble fibre beta glucan (see page ). Oats and coconut are a marriage made in heaven. Give this one a bash!\n\n#### SERVES 1\n\n50g porridge oats\n\n200ml coconut milk\n\n\u00bc tsp stevia\n\n3\u20134 drops vanilla extract\n\n1 tsp desiccated coconut\n\nPlace the oats, coconut milk and stevia into a saucepan, pour in 100ml of water and simmer for five to six minutes, until the oats are soft and a creamy texture has formed.\n\nAdd the vanilla extract and desiccated coconut and stir well, before serving.\n\n# Weekday lunches\n\n###### NITRATES\n\nFLAVONOIDS \nSOLUBLE FIBRE \nMAGNESIUM\n\n**Beetroot, bean and rocket salad with orange dressing** This may sound like a peculiar mish-mash of flavours... until you taste it. The orange and beetroot work beautifully together and the pepperiness of the rocket cuts straight through the middle. All this, plus it is a heart-healthy dynamo to boot. Magic!\n\n#### SERVES 1\n\n_For the salad_\n\n2 large or 3 medium cooked beetroot (not in vinegar)\n\n400g can of mixed beans, drained and rinsed\n\nlarge handful of rocket\n\n_For the dressing_\n\n1 tbsp orange juice\n\n1 tbsp olive oil\n\n1 tsp white wine vinegar\n\npinch of low-sodium salt\n\nAssemble all the salad ingredients in a bowl and mix well.\n\nWhisk the dressing ingredients thoroughly until emulsified.\n\nDress the salad and serve.\n\n###### FLAVONOIDS\n\nESSENTIAL FATTY ACIDS\n\n**Goat's cheese, pomegranate and olive salad** This just oozes Mediterranean delight, with a fresh but indulgent flavour. Nutrient-dense, flavour-packed and easy to make. Does it get much better? Pomegranate is now available pre-prepared, so is also hassle free.\n\n#### SERVES 1\n\n_For the salad_\n\n2 handfuls of mixed salad leaves\n\n2 tbsp kalamata olives\n\n\u00bd red pepper, finely chopped\n\n2 tbsp pomegranate seeds\n\n75\u201380g goat's cheese, crumbled\n\n_For the dressing_\n\n1 tbsp olive oil\n\n1 tsp balsamic vinegar\n\nlow-sodium salt and freshly ground black pepper\n\nCombine the leaves, olives, pepper and pomegranate in a salad bowl.\n\nWhisk the dressing ingredients thoroughly until emulsified, then pour the dressing over the salad and toss well.\n\nCrumble the cheese over the top.\n\n###### SOLUBLE FIBRE\n\nMAGNESIUM \nFLAVONOIDS \nLOW GI\n\n**Herbed chickpea salad with sun-dried tomatoes and spinach** This is such a flavourful little treat. Easy to prepare, filling and packed with the good stuff!\n\n#### SERVES 1\n\n\u00bd tbsp olive oil, plus more to dress\n\n3 handfuls of baby spinach\n\nleaves from a few sprigs of parsley\n\nleaves from a few sprigs of thyme\n\n400g can of chickpeas, drained and rinsed\n\n1 spring onion, finely chopped\n\n8 sun-dried tomatoes, chopped\n\njuice of \u00bd lemon\n\nfreshly ground black pepper\n\nPour the oil into a saucepan placed over a medium heat, then saut\u00e9 the spinach for one or two minutes, just until it wilts.\n\nMix the cooked spinach with the herbs, chickpeas, spring onion and sun-dried tomatoes. Add a little olive oil, the lemon juice and pepper and mix well.\n\n###### LYCOPENE\n\nFLAVONOIDS\n\n**Speedy tomato and paprika soup** This is a seriously speedy soup. Canned tomatoes really aren't that bad as long as they are pure and don't have added sugar (just read the label). And oddly enough, when tomatoes are cooked and processed, though the vitamin C may be destroyed the heart-healthy carotenoid lycopene actually becomes more bio-available to the body! This is a doddle to make and is a speedy lunchtime fix.\n\n#### SERVES 1\n\n1 tbsp olive oil\n\n1 red onion, finely chopped\n\n2 garlic cloves, finely chopped\n\nlow-sodium salt and freshly ground black pepper\n\n400g can of chopped tomatoes\n\n1 tsp smoked paprika\n\nPour the olive oil into a saucepan placed over a medium heat. Saut\u00e9 the onion and garlic, with a good pinch of low-sodium salt, until the onion has softened and is turning translucent.\n\nAdd the tomatoes and paprika, bring to the boil, then reduce the heat and simmer for 10 minutes.\n\nPour into a blender (or use a hand-held blender) and blitz into a rich, smooth soup. Season to taste and serve.\n\n###### OMEGA 3 FATTY ACIDS\n\nCAROTENOIDS \nSOLUBLE FIBRE\n\n**Spinach and anchovy pitta pizzas** Pitta pizzas are quick-fix gold. Out the pack, whack some bits on, under the grill, bang... Lunch. That's what you need!\n\n#### SERVES 1\n\n1 tsp tomato pur\u00e9e\n\n1 wholemeal pitta bread\n\n8 baby spinach leaves, torn\n\n4\u20135 anchovy fillets\n\n50g feta cheese\n\nPreheat the grill to its highest setting.\n\nSpread the tomato pur\u00e9e over the pitta. Add the baby spinach, scattering it over. Lay the anchovy fillets haphazardly on top, then crumble over the feta cheese.\n\nPlace under the grill for a few minutes, until the feta begins to brown at the edges, then serve.\n\n###### NITRATES\n\nOMEGA 3 FATTY ACIDS \nSOLUBLE FIBRE\n\n**Smoked salmon, beetroot and minted yogurt wrap** This is a wonderful portable lunch and much lighter \u2013 with a lower GI \u2013 than your average sandwich.\n\n#### SERVES 1\n\n2 tbsp natural live probiotic yogurt\n\n6\u20137 mint leaves, shredded\n\n1 small cooked beetroot (not in vinegar), chopped\n\nlow-sodium salt and freshly ground black pepper\n\n1 wholemeal tortilla wrap\n\n3 slices of smoked salmon\n\na few rocket leaves (optional)\n\nYou choose how to assemble this; it's your lunch, after all. You can mix the yogurt, mint and beetroot together in a bowl and season to taste, or you can keep all the elements separate.\n\nPlace the wrap on a work top and add the salmon, yogurt, mint and beetroot in the centre, with the rocket leaves (if using), then roll it up.\n\n###### FLAVONOIDS\n\nSOLUBLE FIBRE \nCAROTENOIDS \nAJOENE\n\n**Roasted onion and cannellini bean houmous with vegetable crudit\u00e9s** This can be a super-quick option. The onions can be roasted the day before so it is quick to throw together. Or, if you have more time on your hands, they can be done there and then, giving a nice warmer houmous which is an interesting variation.\n\n#### SERVES 1\n\n_For the houmous_\n\n1 small red onion, thickly sliced\n\n1\u00bd tbsp olive oil\n\nlow-sodium salt\n\n1 garlic clove, finely chopped\n\n400g can of cannellini beans, drained\n\n_For the crudit\u00e9s_\n\n2 carrots, cut into batons\n\n1 celery stick, cut into batons\n\n4\u20135 whole radishes... or any combination of vegetable crudit\u00e9s you would like!\n\nPreheat the oven to 200\u00b0C\/400\u00b0F\/gas mark 6.\n\nPlace the sliced onion in a small roasting tin and drizzle with about 2 tsp of the olive oil, with a pinch of low-sodium salt. Roast in the oven for 20\u201325 minutes, until the onions are soft and beginning to caramelise.\n\nPlace the roasted onion, garlic, cannellini beans, remaining oil and a pinch of low-sodium salt into a food processor and process into a thick houmous. Transfer to a serving bowl and set in the centre of a plate.\n\nSurround with the vegetable crudit\u00e9s and serve.\n\n###### FLAVONOIDS\n\nMAGNESIUM \nCAROTENOIDS\n\n**Red cabbage and carrot salad with creamy orange dressing** This may sound really bizarre at first but believe me, when you taste it, all will make perfect sense. This is great just as a main course salad, as it's very dense, or is a wonderful side salad for things such as chicken or other white meats.\n\n#### SERVES 1\n\n_For the salad_\n\n\u00bc red cabbage, finely grated\n\n1 large carrot, finely grated\n\nleaves from a small bunch of flat-leaf parsley, torn\n\nhandful of baby spinach, torn\n\n_For the dressing_\n\n1 tbsp tahini\n\n2 tbsp fresh orange juice\n\n1 tsp cider vinegar\n\nlow-sodium salt\n\nCombine the grated vegetables, parsley and spinach and mix thoroughly.\n\nCombine all the dressing ingredients and mix well, before using to dress the salad.\n\n# Weekend lunches\n\n###### BETA CAROTENE\n\nOMEGA 3 FATTY ACIDS \nMAGNESIUM\n\n**Roasted squash, rocket and sun-dried tomato salad** This is a gorgeous, colourful and nutrient-packed salad. It is a perfect lunch when you fancy lots of contrasts in flavour, yet still want something light.\n\n#### SERVES 1\n\n_For the salad_\n\n\u00bd small or \u00bc large butternut squash, chopped, skin left on\n\n\u00bd tbsp olive oil\n\n8 sun-dried tomatoes\n\nlarge handful of rocket\n\n1 tbsp walnuts\n\n_For the dressing_\n\n1 tbsp olive oil\n\n1 tsp balsamic vinegar\n\n\u00bc tsp ground cumin\n\nlow-sodium salt and freshly ground black pepper\n\nPreheat the oven to 200\u00b0C\/400\u00b0F\/gas mark 6. Place the squash on a baking tray, drizzle with olive oil and toss with your hands to coat. Bake at the top of the hot oven for about 30 minutes, stirring occasionally, until soft and roasted and the skin has turned crispy.\n\nCombine the squash with all the other salad ingredients.\n\nWhisk the dressing ingredients thoroughly until emulsified.\n\nDress the salad and toss well.\n\n###### NITRATES\n\nFLAVONOIDS \nAJOENE\n\n**Bold beetroot and horseradish soup** This is an awesome soup with a real bolshy flavour.\n\n#### SERVES 1\u20132\n\n1 tbsp olive oil\n\n1 large red onion, finely chopped\n\n2 garlic cloves, finely chopped\n\ngood pinch of low-sodium salt\n\n3 large raw beetroot, skins on, chopped\n\nup to 1 litre vegetable stock\n\n2 tbsp horseradish sauce\n\nPour the olive oil into a large saucepan placed over a medium heat. Saut\u00e9 the onion and garlic with the low-sodium salt, until the onion is nicely softened.\n\nAdd the beetroot and pour in just enough vegetable stock to cover. Allow to simmer for about 30 minutes, until the beetroot is tender to the point of a knife.\n\nTransfer to a blender (or use a hand-held blender). Add the horseradish sauce, then blend into a smooth soup.\n\n###### RUTIN\n\nERITADENINE \nFLAVONOIDS \nAJOENE\n\n**Soba noodle vegetable stir-fry** Soba noodles are an amazing source of the flavonoid rutin, which has been shown to be especially good for the health of blood vessels, protecting the walls from inflammatory damage.\n\n#### SERVES 1\n\n1 bundle of soba noodles (they come ready-portioned)\n\n1 tbsp olive oil\n\n1 large red onion, finely chopped\n\n2 garlic cloves, finely chopped\n\n1 small chilli, finely chopped\n\n1 carrot, cut into thin julienne\n\npinch of low-sodium salt\n\n2 spring onions, chopped lengthways\n\n5 shiitake mushrooms, sliced\n\n2 handfuls of baby spinach\n\n3 tsp low-sodium soy sauce\n\n2 tsp sesame oil\n\nCook the noodles according to the packet instructions, then drain and set aside. Pour the olive oil into a small wok or saut\u00e9 pan set over a medium heat. Saut\u00e9 the onion, garlic, chilli and carrot, with the low-sodium salt, until the onion is soft and the carrot is beginning to soften.\n\nAdd the spring onions and shiitake mushrooms and saut\u00e9 for five to eight minutes, until the mushrooms are cooked. Throw in the baby spinach and saut\u00e9 just until it wilts.\n\nFinally, tip in the drained noodles and mix everything together well with the low-sodium soy sauce and sesame oil.\n\n###### BETA CAR OTENE\n\nFLAVONOIDS \nAJOENE\n\n**Roasted sweet potato and coconut soup** This recipe is off the charts in the tasty spectrum. Deep, rich, creamy and decadent.\n\n#### SERVES 1\u20132\n\n1\u00bd large sweet potatoes, skin-on, chopped\n\n1 tbsp olive oil, plus more to serve (optional)\n\n1 large red onion, finely chopped\n\n2 garlic cloves, finely chopped\n\nlow-sodium salt and freshly ground black pepper\n\n400g can of coconut milk\n\n500ml vegetable stock\n\ncoriander leaves, to serve (optional)\n\nslivers of red chilli, to serve (optional)\n\nPreheat the oven to 200\u00b0C\/400\u00b0F\/gas mark 6. Place the sweet potatoes on a baking tray and roast at the top of the oven for about 30 minutes, until they have started to soften and the skins are beginning to caramelise.\n\nPour the olive oil into a saucepan set over a medium heat. Saut\u00e9 the onion and garlic, with a good pinch of low-sodium salt, until the onion just softens.\n\nAdd the roasted sweet potato and pour in the coconut milk with enough of the vegetable stock to completely cover the sweet potatoes.\n\nSimmer for about 10 minutes, then blend into a luxurious soup. Serve scattered with coriander and chilli and\/or drizzled with a little more oil (if using).\n\n###### BETA CAROTENE\n\nLOW GI \nOMEGA 3 FATTY ACIDS \nMAGNESIUM\n\n**Black olive and anchovy-stuffed chicken breast, sweet potato mash and wilted greens** This is a lovely, strongly flavoured dish that is a dinner party favourite.\n\n#### SERVES 1\n\n1 large skinless chicken breast\n\n3 anchovy fillets\n\n4 kalamata olives, sliced\n\n\u00bd sweet potato, peeled and chopped\n\n\u00bd tbsp olive oil\n\nlow-sodium salt and freshly ground black pepper\n\nhandful of curly kale\n\nhandful of purple basil leaves (optional)\n\nPreheat the oven to 200\u00b0C\/400\u00b0F\/gas mark 6.\n\nSlice a pocket into the chicken breast, cutting into the thickest part. Open up this pocket and add the anchovies and olives, then close and seal with cocktail sticks.\n\nPlace on a baking sheet and bake at the top of the oven for 25 minutes.\n\nMeanwhile, place the sweet potato in a pan and cover with just-boiled water. Simmer for 15 minutes, until tender. Drain and mash with the oil, low-sodium salt and pepper. Keep warm until the chicken is ready.\n\nFive minutes before you're ready to serve, cook the greens by lightly steaming them until they soften slightly and turn a brighter green. Serve with the chicken and sweet potato mash, sprinkled with purple basil (if using).\n\n###### LOW GI\n\nBETA CAROTENE \nFLAVONOIDS \nAJOENE\n\n**Roasted vegetables with quinoa salad** A gorgeous dish that is both filling and light, as well as nutrient packed.\n\n#### SERVES 1\u20132\n\n1 large courgette, sliced\n\n1 large red pepper, sliced\n\n1 large red onion, halved, then sliced\n\ndrizzle of olive oil\n\n1 tsp garlic powder\n\n1 tsp smoked paprika\n\nlow-sodium salt and freshly ground black pepper\n\n150g quinoa\n\nleaves from a few sprigs of parsley\n\n1 tsp capers, drained and rinsed\n\nPreheat the oven to 200\u00b0C\/400\u00b0F\/gas mark 6.\n\nPlace the sliced vegetables in a roasting tin with the oil and mix well. Add the garlic powder, smoked paprika, a little low-sodium salt and pepper and mix again. Roast at the top of the hot oven for about 35 minutes, stirring occasionally so the edges don't catch.\n\nTip the quinoa into a saucepan and cover with boiling water. Simmer for about 20 minutes, until the grains have softened and what looks like a small 'tail' has appeared on the side of each. Drain.\n\nFinely chop the parsley and capers together and mix with the cooked quinoa.\n\nAdd the roasted vegetables. Ready to serve.\n\n###### B VITAMINS\n\nBETA CAROTENE \nLYCOPENE \nFLAVONOIDS \nAJOENE\n\n**Mixed bean chilli with baked sweet potato** The classic(ish) chilli! This gorgeous dish is often served with rice, but following a low-GI diet means that you need to give white rice a really wide berth. (The odd bit of brown rice is fine.) This is a bit of a twist on the classic baked potato with chilli con carne. Sweet potatoes have a much lower glycaemic impact than regular potatoes, plus are packed with beta carotene, so are a great option.\n\n#### SERVES 1\n\n1 sweet potato\n\n\u00bd tbsp olive oil\n\n1 red onion, finely chopped\n\n1 garlic clove, finely chopped\n\n1 red chilli, finely chopped\n\n\u00bd red pepper, finely chopped\n\nlow-sodium salt and freshly ground black pepper\n\n400g can of mixed beans, drained and rinsed\n\n400g can of chopped tomatoes\n\n1 tsp ground cumin\n\n1 heaped tsp smoked paprika\n\nPreheat the oven to 200\u00b0C\/400\u00b0F\/gas mark 6. Make a few holes in the sweet potato with the tines of a fork and place in the top of the hot oven for about one hour. Keep checking on it, waiting until it has fully softened.\n\nPour the olive oil into a large saucepan set over a medium heat. Saut\u00e9 the onion, garlic, chilli and red pepper, with a little pinch of low-sodium salt, until the onion softens.\n\nTip in the beans and tomatoes and bring to the boil, then reduce the heat and simmer for about 10 minutes. Add the spices and simmer for a further 15 minutes, until the sauce has reduced and thickened. Season to taste.\n\nOpen up the baked sweet potato and spoon a generous amount of chilli over it.\n\n###### NITRATES\n\nAJOENE \nFLAVONOIDS \nSOLUBLE FIBRE\n\n**Baked beetroot wedges with white bean houmous** Baked beetroot has become like a slightly odd alternative to jacket potato or chips here at Pinnock HQ. It all stemmed from having a large amount of unused beetroot in the fridge that needed eating and a moment of creativity\/boredom. The result was very pleasing indeed. It is too firm to serve whole, but in big baked wedges it is pretty special.\n\n#### SERVES 1\n\n1 large beetroot, skin-on, cut into wedges\n\n2 tbsp olive oil, plus a tiny amount more for the beetroot\n\n400g can of cannellini beans, drained\n\njuice of \u00bd lemon\n\n1 garlic clove, finely chopped\n\nlow-sodium salt\n\nhandful of parsley leaves, chopped, to serve (optional)\n\nPreheat the oven to 200\u00b0C\/400\u00b0F\/gas mark 6.\n\nPlace the beetroot on a baking sheet and drizzle with a tiny amount of olive oil. Toss well to coat the wedges. Bake at the top of the hot oven for about 40 minutes, until the wedges are soft, turning occasionally.\n\nPut the beans in a blender with the 2 tbsp of olive oil, the lemon juice, garlic and low-sodium salt. Blend into a thick houmous.\n\nPlate up the beetroot wedges, add a generous helping of the houmous and sprinkle with parsley (if using). Serve with a green salad.\n\n###### CAROTENOIDS\n\nAJOENE \nFLAVONOIDS\n\n**Squash, goji berry and red onion soup** OK, fruit in a savoury soup. That sounds like I have finally tipped over the edge. But trust me, there is something so special about squash and goji berries. They intensify each other's flavour beautifully. Give it a go. You'll be glad you did!\n\n#### SERVES 2\n\n1 large red onion, finely chopped\n\n2 garlic cloves, finely chopped\n\n\u00bd tbsp olive oil\n\nlow-sodium salt\n\n1 small butternut squash, chopped, skin-on\n\n2 handfuls of goji berries\n\nup to 500ml vegetable stock\n\nSaut\u00e9 the onion and garlic in the olive oil, with a pinch of low-sodium salt, until the onion is nice and soft. Add the squash and goji berries.\n\nAdd enough vegetable stock to just cover all of the ingredients, then simmer until the squash is soft and falls apart when prodded.\n\nBlend into a thick, bright orange soup.\n\n###### FLAVONOIDS\n\nAJOENE \nCAROTENOIDS\n\n**Balsamic caramelised pepper soup** This one is just a bit special. It takes a little time to make but is seriously worth it for the deep, lingering flavour you get in return.\n\n#### SERVES 1\u20132\n\n2 red peppers, deseeded and sliced\n\n2 yellow peppers, deseeded and sliced\n\n1\u00bd tbsp olive oil\n\n2 tbsp balsamic vinegar\n\n1 large onion, finely chopped\n\n1 garlic clove, finely chopped\n\n\u00bd small sweet potato, peeled and chopped\n\n200\u2013300ml vegetable stock, plus more if needed\n\nPreheat the oven to 200\u00b0C\/400\u00b0F\/gas mark 6.\n\nPlace the peppers in a roasting tin and drizzle \u00bd tbsp each of the olive oil and balsamic vinegar over them. Roast in the oven for 30\u201340 minutes. Every 10 minutes, take them out, add another \u00bd tbsp of the balsamic vinegar, stir, then return to the oven. By 40 minutes, the balsamic vinegar should have caramelised around the peppers and the smell will be divine.\n\nMeanwhile, saut\u00e9 the onion and garlic in the remaining 1 tbsp of olive oil, just until the onion has softened.\n\nTransfer the caramelised peppers to the cooked onion, add the sweet potato, then enough vegetable stock to half-cover all the ingredients. Simmer until the sweet potato has softened.\n\nBlend into a smooth soup. If you find it needs thinning out slightly, add a little more stock.\n\n# Quick dinners\n\n###### BETA CAROTENE\n\nFLAVONOIDS\n\n**Sweet potato wedges with red pepper-walnut dip** An absolute flavour bomb and, though it sounds simple, even light, its nutritional density means it will seriously fill you up. We're talking for hours.\n\n#### SERVES 1\n\n1 large sweet potato, skin-on, cut into wedges\n\n2 tbsp olive oil, plus a drizzle for the sweet potatoes\n\n1\u00bd large red peppers, roughly chopped\n\n80g walnuts\n\n1 garlic clove\n\nlow-sodium salt\n\nPreheat the oven to 200\u00b0C\/400\u00b0F\/gas mark 6.\n\nPlace the sweet potato wedges on a baking sheet and drizzle with a little olive oil. Stir so all of the wedges are coated with oil. Bake at the top of the hot oven for about 20 minutes, until the wedges are soft but with crispy skins, turning them over occasionally.\n\nAt the same time, roast the peppers in the hot oven for about 12 minutes. I like to roast them without oil so the skins get a bit charred on the edges and give a beautiful char-grilled flavour. Once they are turning and beginning to soften, remove them from the oven.\n\nPlace the roast peppers, walnuts, garlic, the 2 tbsp of olive oil and a good pinch of low-sodium salt in a blender or food processor and blend at full power to make a houmous-like dip.\n\nDip the wedges into the walnut mixture, it is heaven! Serve with a good side salad.\n\n###### FLAVONOIDS\n\nAJOENE \nBETA CAROTENE \nLYCOPENE \nSOLUBLE FIBRE\n\n**Stuffed aubergine** This is such a treat, I love all the flavours. Filling, sumptuous and easy to make. Doesn't get much better if you ask me!\n\n#### SERVES 1\u20132\n\n1 tbsp olive oil\n\n1 large red onion, halved, then sliced\n\n2 garlic cloves, finely chopped\n\n1 large red pepper, deseeded and chopped\n\n1 large courgette, sliced\n\nlow-sodium salt and freshly ground black pepper\n\n400g can of chopped tomatoes\n\n1 large aubergine\n\n2 tbsp porridge oats\n\n3 tsp grated Parmesan\n\nPreheat the oven to 200\u00b0C\/400\u00b0F\/gas mark 6.\n\nPour the olive oil into a saucepan set over a medium heat. Saut\u00e9 the onion, garlic, red pepper and courgette, with a good pinch of low-sodium salt, for about eight minutes, until they all begin to soften.\n\nAdd the tomatoes and simmer for 15\u201320 minutes, until the tomatoes have reduced right down and you have a thick ratatouille. Season further, if desired.\n\nCut the aubergine in half. Scoop out the flesh from each half, leaving a rim of about 0.5cm of flesh. Lay the aubergine halves face down on a baking tray and pour in a little water. Bake for about 12 minutes, until they start to soften. Turn over and bake for another five minutes.\n\nMix the oats and Parmesan together and season to taste. Spoon the ratatouille mixture into the aubergine halves, pressing it down firmly. Divide the Parmesan topping between them.\n\nReturn to the oven for another 12 minutes. Serve with a side salad.\n\n###### BETA CAROTENE\n\nFLAVONOIDS \nAJOENE \nLOW GI\n\n**Chicken and tarragon-stuffed peppers with greens** This super-tasty and unusual dish is set to become a favourite. It can work as a lighter dish with a side salad, or as a heartier dinner with sweet potato mash and greens.\n\n#### SERVES 1\n\n1 large skinless chicken breast\n\n\u00bd tbsp olive oil\n\n1 garlic clove, finely chopped\n\npinch of low-sodium salt\n\n1 heaped tsp soft cheese\n\nleaves from 2\u20133 sprigs of tarragon, roughly chopped\n\n1 large red pepper, halved and deseeded\n\nlarge handful of curly kale\n\nPreheat the oven to 200\u00b0C\/400\u00b0F\/gas mark 6.\n\nPlace the chicken in a food processor and process on a low speed to create coarsely minced meat.\n\nPour the olive oil into a saucepan set over a medium heat. Saut\u00e9 the garlic, with a pinch of low-sodium salt, for two or three minutes. Add the chicken and continue to cook, stirring and turning the meat, for about 12 minutes, until thoroughly cooked.\n\nMix in the soft cheese and tarragon and use the mixture to stuff the pepper halves. Place on a baking sheet. Add a small amount of water around the pepper, then bake at the top of the hot oven for 15\u201320 minutes, until the pepper has softened and there is a light golden crust on top of the stuffing.\n\nFive minutes before you're ready to serve, cook the greens by lightly steaming them until they soften slightly and turn a brighter green. Serve with the stuffed peppers.\n\n###### MAGNESIUM\n\nFLAVONOIDS \nAJOENE \nCAPSAICIN\n\n**Chicken and green vegetable nutty stir-fry** This is a great super-quick fix after a long day. A quick-fire, nutrient-dense, one-pot wonder. Almond butter is available from most health food stores and some larger supermarkets. If you can't find it, peanut butter will do fine.\n\n#### SERVES 1\n\n1 tbsp olive oil\n\n2 garlic cloves, finely chopped\n\n1 large leek, sliced\n\n1 red chilli, finely chopped\n\n1 large skinless chicken breast, chopped\n\n1 small courgette, sliced handful of curly kale\n\n2 handfuls of baby spinach\n\n1 heaped tbsp almond butter\n\n2 tsp soy sauce\n\n1 tsp honey\n\n1 tbsp flaked almonds\n\nPour the oil into a saucepan set over a medium heat. Saut\u00e9 the garlic, leek and chilli for about five minutes. Add the chicken and stir-fry for eight to 10 minutes, until it is cooked. (You can cut one of the large pieces in half to check, if you want to be sure; you should see no trace of pink.)\n\nAdd the courgette, kale and spinach and stir-fry for a further five minutes, then measure in the almond butter, soy sauce and honey. Mix well. Serve sprinkled with the flaked almonds.\n\n###### B VITAMINS\n\nFLAVONOIDS \nAJOENE \nSOLUBLE FIBRE\n\n**Peppered king prawn skewers with tarka dal** I'm a complete freak for Indian flavours. I find it some of the most divinely flavoured food on the planet and, when you push aside those weird takeaway staples that have been invented for the British palate (such as chicken tikka masala), you'll find it some of the healthiest in the world, too. The combination of vegetables, pulses and antioxidant-dense spices create dishes that are an edible medicine chest.\n\n#### SERVES 1\n\n1 tbsp olive oil\n\n\u00bd red onion, finely chopped\n\n1 large garlic clove, finely chopped\n\nlow-sodium salt and freshly cracked black pepper\n\n75g red lentils\n\n500ml vegetable stock (you may not need it all)\n\n\u00bd tsp ground cumin\n\n\u00bd tsp turmeric\n\n12 king prawns, shelled and deveined\n\n3 wooden skewers, soaked for 30 minutes\n\nPour the oil into a saucepan over a medium heat. Saut\u00e9 the onion and garlic, with a good pinch of low-sodium salt, until the onion softens.\n\nAdd the lentils and a small amount of vegetable stock and simmer. As if you were making a risotto, keep adding stock as the liquid reduces, until the lentils are cooked. The finished texture should be like a thin porridge. Stir in the cumin and turmeric, mixing well.\n\nPlace a griddle pan over a medium-high heat. Thread four prawns on to each skewer, sprinkle with cracked black pepper and place in the griddle pan for three minutes each side.\n\nServe the dal in a bowl with the skewers.\n\n###### NITRATES\n\nOMEGA 3 FATTY ACIDS \nFLAVONOIDS\n\n**Salmon and beetroot wasabi stacks** This is a rather odd but incredible (and stunning looking) combination that is fantastic as a summer evening dish, because it is served cold. You could also have it as a starter.\n\n#### SERVES 1\n\n2 small cooked beetroot (not in vinegar), finely chopped\n\n1 tbsp mayonnaise\n\n2 tsp wasabi\n\n4 slices of smoked salmon, cut into small pieces\n\njuice of \u00bd lemon\n\nfreshly ground black pepper\n\nhandful of rocket leaves\n\nMix the beetroot, mayonnaise and wasabi in a small bowl.\n\nIn a separate bowl, mix the salmon, lemon juice and black pepper.\n\nTo assemble, place a ring mould in the centre of a plate. Put the beetroot mix in first and push it down well so it is pressed into the shape of the mould. Top with a layer of the salmon, again pushing down well so the salmon takes the shape of the mould. Or you could make more, thinner layers, if you want.\n\nCarefully lift off the ring mould and top the stack with a few rocket leaves.\n\n###### OMEGA 3 FATTY ACIDS\n\nSOLUBLE FIBRE \nFLAVONOIDS \nAJOENE\n\n**Tapenade salmon with borlotti bean crush** This is a seriously filling dinner in a hurry, if you use canned beans. It is perfect after a long day at work, when you could eat anything that stays still for long enough. Find black olive tapenade in any supermarket.\n\n#### SERVES 1\n\n1 salmon fillet\n\n1 garlic clove, finely chopped\n\n1\u00bd red onions, finely chopped\n\n\u00bd tbsp olive oil\n\nlow-sodium salt and freshly ground black pepper\n\n400g can of borlotti beans, drained\n\n1 tsp capers\n\n\u00bd tbsp black olive tapenade\n\nPreheat the oven to 200\u00b0C\/400\u00b0F\/gas mark 6.\n\nPut the salmon fillet on a baking tray and place in the oven for around 10 minutes.\n\nMeanwhile, saut\u00e9 the garlic and onions in the olive oil, with a pinch of low-sodium salt, until the onion has softened. Add the beans to the onion and garlic and saut\u00e9 for another minute or two. Using a potato masher, roughly crush the beans; they should be semi-mashed. Add the capers and mix well.\n\nRemove the salmon from the oven, top with the tapenade, then return to the oven for a final 10 minutes, until the edges of the tapenade get firmer and almost crisp up.\n\nPlace the bean crush in the centre of the serving plate, then top with the salmon.\n\n# Fancy dinners\n\n###### OMEGA 3 FATTY ACIDS\n\nMAGNESIUM \nAJOENE \nOLEIC ACID \nNITRATES \nSOLUBLE FIBRE\n\n**Grilled trout with root vegetables and salsa verde** This is such a vibrant dish and is awash with fresh flavours and beautiful colours.\n\n#### SERVES 1\n\n\u00bd raw beetroot, cut into wedges\n\n1 large carrot, cut into wedges, or a handful of baby carrots\n\n1 small parsnip, cut into wedges\n\n2 tbsp olive oil, or more to taste, plus more for the root vegetables\n\nlow-sodium salt and freshly ground black pepper\n\nsmall bunch of parsley\n\nsmall bunch of mint\n\nsmall bunch of basil\n\n1 garlic clove, finely chopped\n\n2 tsp capers, drained and rinsed\n\n1 tsp white wine vinegar\n\n1 large trout fillet\n\nPreheat the oven to 200\u00b0C\/400\u00b0F\/gas mark 6.\n\nPlace the chopped root vegetables into a roasting tin. Drizzle with a little olive oil, add a generous pinch of low-sodium salt and pepper and mix well. Roast in the top of the hot oven for about 30 minutes, until they are all soft and beginning to turn golden.\n\nMeanwhile, tip the parsley, mint, basil, garlic, capers, vinegar and the 2 tbsp of olive oil into a blender and blend at a slow speed to maintain a coarse texture. Add more oil, if you prefer. Preheat the grill. Place the trout under the hot grill for 15 minutes, turning occasionally, until a golden crispiness begins to form on the fillet.\n\nStack the root vegetables in the centre of a plate. Top with the trout fillet (or just serve the trout and vegetables alongside), then drizzle a generous amount of salsa verde over the top.\n\n###### LOW GI\n\nB VITAMINS \nBETA CAROTENE \nFLAVONOIDS \nSOLUBLE FIBRE\n\n**Mediterranean brown rice risotto** This is real comfort food and a great way to get the heft and substance you need on a cold winter's evening, without it sticking to your waistline!\n\n#### SERVES 2\u20133\n\n1 tbsp olive oil\n\n1 large red onion, finely chopped\n\n2 garlic cloves, finely chopped\n\nlow-sodium salt\n\n60g sun-dried tomatoes\n\n250g short-grain brown rice\n\n400g can of chopped tomatoes\n\n1 litre vegetable stock (you may not need it all, but it's always best to have plenty)\n\n1 courgette, sliced\n\n1 red pepper, sliced\n\nPour the olive oil into a large saucepan set over a medium heat. Saut\u00e9 the onion and garlic with a pinch of low-sodium salt, until the onion starts to soften.\n\nAdd the sun-dried tomatoes, rice and canned tomatoes. Simmer until the liquid is notably reducing, stirring very frequently.\n\nAt this stage, begin adding stock little and often, topping it up when you notice the liquid beginning to reduce. Keep this up until the rice is virtually cooked.\n\nNow add the courgette and red pepper and continue pouring in the stock until the rice is cooked and the vegetables have softened.\n\n###### LOW GI\n\nFLAVONOIDS \nSOLUBLE FIBRE\n\n**Wholewheat pasta with roasted pepper sauce** OK, so as you have probably gathered by now, I'm not a massive fan of heavy amounts of carbs. But we all crave these foods from time to time. Rather than depriving ourselves, we may as well make the best version of these treats that we can. This is a prime example and it just so happens that this sauce tastes awesome! Just saying...\n\n#### SERVES 1\n\n1 red pepper, deseeded and sliced\n\n1 yellow pepper, deseeded and sliced\n\n1\u00bd tbsp olive oil\n\nlow-sodium salt\n\n1 large red onion, finely chopped\n\n2 garlic cloves, finely chopped\n\n65g (dry weight) wholewheat fusili pasta\n\n50g feta cheese\n\nPreheat the oven to 200\u00b0C\/400\u00b0F\/gas mark 6.\n\nPlace the peppers in a roasting tin, drizzle with \u00bd tbsp of the oil and a pinch of low-sodium salt. Roast in the oven for about 30 minutes, turning occasionally. This may seem a long time but you want some of the edges to char slightly, as this will give amazing flavour later on. Meanwhile, saut\u00e9 the onion and garlic in the remaining 1 tbsp of olive oil, with a pinch of low-sodium salt, until the onion has softened.\n\nTip the pasta into a pan, cover with boiling water and boil for 10\u201312 minutes, or according to the packet instructions. Meanwhile, place the peppers and the onion mixture into a food processor and process into a smooth sauce.\n\nDrain the pasta and stir the sauce through it. Finally, top with the crumbled feta.\n\n###### OMEGA 3 FATTY ACIDS\n\nBETA CAROTENE \nLOW GI \nMAGNESIUM\n\n**Salmon with pea pur\u00e9e and roasted butternut squash** When I first discovered a simply seasoned pea pur\u00e9e it blew my mind. So simple but a real treat. This combination is a regular feature at Pinnock HQ.\n\n#### SERVES 1\n\n\u00bc large butternut squash, skin-on, chopped\n\n\u00bd tbsp olive oil\n\nlow-sodium salt and freshly ground black pepper\n\n160g frozen peas\n\n1 salmon fillet\n\nmixed salad leaves, to serve\n\nPreheat the oven to 200\u00b0C\/400\u00b0F\/gas mark 6.\n\nPlace the squash in a roasting tin and drizzle with a little olive oil, a pinch of low-sodium salt and pepper and mix well. Roast at the top of the hot oven for 20\u201325 minutes, until it has softened and the skins are turning golden and crispy.\n\nPlace the peas in boiling water and simmer until soft, but still bright green. If they are dull they are dead! You need them soft enough to semipur\u00e9e. Drain and mash with a potato masher, or put them into a food processor and process on a low setting to get a rough, chunky pur\u00e9e.\n\nSeason the salmon with a little low-sodium salt and pepper, place on a baking sheet and bake at the top of the oven for about 20 minutes, until well-cooked with a crispy skin and marginally crisped edges.\n\nPut a dollop of the pea pur\u00e9e, off centre, on the plate, then arrange a stack of roasted squash next to it. Place the salmon on the pea pur\u00e9e and top with salad leaves.\n\n###### B VITAMINS\n\nLYCOPENE \nCURCUMINOIDS \nAJOENE \nLOW GI\n\n**King prawn and spinach curry with herby brown rice** A quick, simple curry. It isn't particularly fiery and is very straightforward.\n\n#### SERVES 2\n\n150g brown rice\n\n1 tbsp coconut oil\n\n1 red onion, finely chopped\n\n2 garlic cloves, finely chopped\n\n2cm piece of root ginger, peeled and very finely chopped\n\n1 cinnamon stick, broken\n\nlow-sodium salt\n\n\u00bd tsp turmeric\n\n\u00bd tsp ground coriander\n\n200g (\u00bd can) of chopped tomatoes\n\n\u00bd tsp garam masala\n\n\u00bd tsp chilli flakes\n\n150g raw king prawns, shelled and deveined\n\n3 handfuls of baby spinach\n\nsmall bunch of chopped coriander leaves\n\nsmall bunch of chopped parsley leaves, plus more to serve (optional)\n\njuice of \u00bd lime, plus lime wedges to serve\n\nIn a saucepan, cover the rice with just-boiled water and simmer for 25\u201330 minutes.\n\nHeat the coconut oil in a large saucepan set over a medium heat. Cook the onion, garlic, ginger and cinnamon with a good pinch of low-sodium salt, until the onion has softened and the flavour of the ginger has died down a little.\n\nTip in the turmeric and ground coriander and cook for two minutes, stirring continuously. Add the tomatoes and simmer for 10\u201315 minutes, until the sauce thickens considerably.\n\nAdd the garam masala, chilli flakes and prawns and cook for about five minutes, until the prawns are cooked. Throw in the spinach and cook just until it wilts.\n\nDrain the rice, add the herbs and lime juice and mix. Serve with the curry, scattered with more herbs and lime wedges if you like.\n\nWe've shown it here served with a king prawn in its shell, which notches the presentation up a gear for a really fancy affair!\n\n###### OMEGA 3 FATTY ACIDS\n\nB VITAMINS \nMAGNESIUM \nLOW GI\n\n**Tuna steak with mango salsa, wilted greens and quinoa verde** A seriously nutrient-packed, fresh, vibrant and very satisfying dinner. For some reason it reminds me of holidays in far-off places. Maybe that's just me!\n\n#### SERVES 1\n\n\u00bc mango, finely chopped\n\n\u00bc small red onion, very finely chopped\n\n\u00bc small red chilli (deseeded if you want it less hot), very finely chopped\n\n1 tsp white wine vinegar\n\n70g quinoa\n\nleaves from a few sprigs of flat-leaf parsley, chopped\n\n1 tsp chopped capers\n\n\u00bd tbsp olive oil\n\n1 tuna steak\n\nlarge handful of spring greens, or similar\n\nCombine the mango, onion, chilli and vinegar, mix well and set aside.\n\nPlace the quinoa in a saucepan and cover with just-boiled water. Simmer for about 20 minutes, until the grains have softened and what looks like a little 'tail' has formed on the side of each. Drain and stir in the parsley and capers.\n\nSet a griddle pan over a high heat and add the oil. Place the tuna steak on the hot, oiled griddle pan and griddle for about three minutes on either side to get a pink middle. If you prefer it more well done, cook for a little longer.\n\nMeanwhile, cook the greens by lightly steaming them until they soften slightly and turn a brighter green.\n\nPlate up the quinoa first, top with the steamed greens, then finish with the tuna and salsa.\n\n###### OMEGA 3 FATTY ACIDS\n\nB VITAMINS \nSOLUBLE FIBRE \nFLAVONOIDS\n\n**Grilled salmon with red barlotto** Barlotto is basically a risotto made from pearl barley. Barley is a very nutrient-rich grain that is incredibly low GI and full of B vitamins.\n\n#### SERVES 1\n\n1 tbsp olive oil\n\n\u00bd red onion, finely chopped\n\n1 garlic clove, finely chopped\n\n\u00bd red pepper, finely chopped\n\nlow-sodium salt and freshly ground black pepper\n\n75g pearl barley\n\n500ml vegetable stock (you may not need all of this)\n\n1 salmon fillet\n\njuice of \u00bd lemon\n\nPour the olive oil into a saucepan set over a medium heat. Saut\u00e9 the onion, garlic and red pepper with a good pinch of low-sodium salt until the onion and pepper have softened.\n\nAdd the barley and a little vegetable stock. Simmer until the stock begins to reduce, then stir in a little more. Repeat this over and over until the barley is cooked and a creamy risotto-like texture has been reached.\n\nMeanwhile, preheat the grill. Season the salmon with low-sodium salt, black pepper and a squeeze of lemon juice and place under the hot grill for 10\u201315 minutes, turning halfway through. This should give you salmon that is still a little soft in the middle. If you prefer it more well done, simply cook it for a little longer.\n\n###### MAGNESIUM\n\nSOLUBLE FIBRE \nLOW GI\n\n**Sea bass with salsa verde and tabbouleh** Fresh, herby and wholesome. This dish is equally at home on a summer afternoon or a winter day.\n\n#### SERVES 1\n\n40g bulgar wheat\n\n15g flat-leaf parsley leaves\n\n3\u20134 mint leaves\n\n5\u20136 basil leaves\n\n1 tbsp capers, drained and rinsed\n\n1\u00bd tbsp olive oil\n\n1 sea bass fillet\n\nlow-sodium salt and freshly ground black pepper\n\nPlace the bulgar wheat in a pan and cover with boiling water. Simmer for around 20 minutes, until it swells and softens.\n\nPlace one-third of the parsley, the mint, basil and capers into a food processor, along with 1 tbsp of the olive oil, and process in bursts to create a coarse salsa.\n\nGently fry the sea bass in the remaining \u00bd tbsp of olive oil for five to seven minutes, turning occasionally.\n\nDrain the bulgar wheat, add the remaining parsley, roughly chopped, a pinch of low-sodium salt, and some black pepper and mix well.\n\nPlace the bulgar in the centre of the plate, lay the fish on top, then drizzle over the salsa verde.\n\n###### SOLUBLE FIBRE\n\nBETA CAROTENE \nFLAVONOIDS \nAJOENE\n\n**Chickpea and red pepper stew with sweet potato mash** This flavoursome, filling and wholly satisfying dish is super-convenient and nutrient-dense, with a great depth of flavour.\n\n#### SERVES 2\n\n1 large sweet potato, skin-on, chopped\n\nlow-sodium salt and freshly ground black pepper\n\n1 tbsp olive oil\n\n1 large red onion, finely chopped\n\n2 garlic cloves, finely chopped\n\n1 red pepper, finely chopped\n\n400g can of chickpeas, drained\n\n400g can of chopped tomatoes\n\n1 tsp ground cinnamon\n\n1 tsp smoked paprika\n\nhandful of coriander leaves, to serve (optional)\n\nPlace the sweet potato in a pan and cover with boiling water. Simmer for 15\u201320 minutes, until the potatoes are soft and almost falling apart. Perfect for mashing! Drain, mash and season.\n\nMeanwhile, pour the olive oil into a saucepan set over a medium heat. Saut\u00e9 the onion, garlic and red pepper with a pinch of low-sodium salt, until the onion and pepper start to soften.\n\nTip in the chickpeas and tomatoes and simmer for about 15 minutes, until the sauce reduces. Add the cinnamon and paprika and season further if required. Simmer for another five to eight minutes.\n\nServe a dollop of the mash with a generous helping of the stew poured over it. Sprinkle with coriander leaves (if using).\n\n###### FLAVONOIDS\n\nOMEGA 3 FATTY ACIDS \nAJOENE\n\n**Tuna steak with chilli-blueberry compote and roasted celeriac** I love this. Tuna steak and fruity sauces are a match made in heaven. Mango is a traditional pairing, but I have made blueberries the star of the show here because of their high concentration of flavonoids.\n\n#### SERVES 1\n\n\u00bc small celeriac, peeled and chopped\n\n1 tbsp olive oil\n\nlow-sodium salt\n\n150g blueberries\n\n\u00bd garlic clove, finely chopped\n\n\u00bd red chilli (deseeded if you want it less hot) finely chopped\n\n1 tuna steak\n\nPreheat the oven to 200\u00b0C\/400\u00b0F\/gas mark 6.\n\nPlace the celeriac in a roasting tin, drizzle over \u00bd tbsp of the oil and season with a little lowsodium salt. Roast for 20\u201325 minutes, until soft and golden.\n\nMeanwhile, put the blueberries, garlic and chilli in a saucepan with 1 tbsp of water, add a good pinch of low-sodium salt and simmer for about 12 minutes, until the blueberries burst and the sauce starts to resemble a thin jam.\n\nPan-fry the tuna steak in the remaining oil for one or two minutes max on each side, or more if you don't want it too pink.\n\nPlace the celeriac in the centre of the serving plate, place the tuna on top, then drizzle the spicy blueberry compote over the fish.\n\n# Drinks, desserts and snacks\n\n###### FLAVONOIDS\n\n**Berry protein smoothie** I have mixed feelings about fruit smoothies. Most of those you can buy are essentially just sugar bombs and can cause many of the issues we covered earlier, when blood sugar levels get too high. But there is a way around that: add protein to the equation. The added protein will slow down the release of the sugars, drip-feeding your blood sugar rather than carpet-bombing it.\n\n#### SERVES 1\n\n\u00bd punnet mixed summer berries (such as blackberries, raspberries or blueberries)\n\nlarge scoop of vanilla protein powder\n\nPlace the berries and protein powder into a blender and pour in 150ml of cold water.\n\nBlend on full speed into a thick smoothie.\n\n###### NITRATES\n\nFLAVONOIDS \nPOTASSIUM\n\n**Beetroot, blackberry, celery and ginger juice** This slightly weird-sounding combination works a treat from both a flavour perspective and also from a nutritional one.\n\n#### MAKES 1\n\n1 large raw beetroot, washed, skin left on\n\n2 celery stalks\n\n3 tbsp blackberries\n\n3cm piece of root ginger\n\nRun all the ingredients through a juicer.\n\n###### MAGNESIUM\n\nFLAVONOIDS \nLOW GI\n\n**Nutty chocolate smoothie** This simple smoothie has a lovely luxurious flavour, so much so that you may take some convincing that it is actually really rather good for you!\n\n#### SERVES 1\n\n150ml coconut water\n\n1 heaped tbsp cocoa powder\n\n1 scoop of low-carb chocolate whey protein powder\n\n1 heaped tsp peanut butter\n\nPlace all the ingredients into a blender, and blitz for about one minute.\n\nI suggest this much time just make sure all the peanut butter is fully broken down. Blenders vary in power, so this amount of time should cover everyone.\n\n###### OMEGA 3 FATTY ACIDS\n\nFLAVONOIDS \nCAROTENOIDS \nSOLUBLE FIBRE\n\n**Pomegranate goji omega smoothie** This is a great smoothie for days when you want to go a little lighter but don't want to skimp on nutrition. This one is nutrient-dense. I know that pomegranate juice can be a bit pricey in some places, but market demand is pushing the cost down. Shop around and you will get it at a reasonable price.\n\n#### SERVES 1\n\n150ml pomegranate juice (not anything labelled 'juice drink')\n\n1 tbsp frozen blueberries\n\n2 tbsp goji berries, soaked in water for 30 minutes to soften, water reserved\n\n1 tbsp ground flax seeds\n\nPlace all the ingredients into a food processor \u2013 including the goji berry soaking water \u2013 and blitz on full power until all the ingredients have blended well.\n\n###### VITAMIN E\n\nMAGNESIUM\n\n**Nutty chocolate avocado pots** OK, I know avocado and dessert don't seem as though they belong in the same sentence, but trust me. When making healthy desserts, avocados can be your best friend. They provide a creamy texture without the need to add any nasties... and happen to be packed with heart-healthy nutrients to boot!\n\n#### SERVES 2\n\n1 very ripe avocado\n\n1 tbsp almond butter\n\n1 tbsp maple syrup, or \u00bd tsp stevia if you want to keep the sugar down\n\n1\u20132 tbsp cocoa powder, to taste, plus more to serve (optional)\n\nScoop the avocado flesh into a blender or food processor. Add the remaining ingredients with 1\u20132 tbsp of cold water.\n\nProcess on full speed until all the ingredients have mixed into a smooth chocolatey dessert.\n\nSpoon the mixture into ramekins and chill in the fridge for two to three hours before serving, sprinkled with cocoa powder, if you like.\n\n###### FLAVONOIDS\n\nCAROTENOIDS\n\n**Tonic tipple** Surprise... it's not all about staying on your best behaviour. Sometimes we need a little treat. When it comes to heart health, a bit of red wine here and there can be your friend. This summery drink is very refreshing and full of important compounds for heart health, too.\n\n#### SERVES 1\n\npomegranate juice (not anything labelled 'juice drink')\n\nblood orange juice\n\nred wine\n\nTake a red wine glass, fill one-quarter with pomegranate juice, one-quarter with blood orange juice, then top up with a red wine of your choice.\n\nYou can add a little ice, too, if you like.\n\n###### BETA GLUCAN\n\nFLAVONOIDS \nOMEGA 3 FATTY ACIDS\n\n**Oaty flax berry crumble** This is a tasty and simple dessert that takes very little time and is a perfect piece of guilt-free indulgence.\n\n#### SERVES 1\n\n200g mixed berries\n\n3 tbsp porridge oats\n\n1 tbsp ground flax seeds\n\n\u00bd tsp ground cinnamon\n\nPreheat the grill on its highest setting.\n\nPlace the berries and 1 tbsp of water in a saucepan and set over a high heat; the maximum the hottest ring will go on. Stew the berries until they start to burst and, before long, a thicker jam-like texture will form. Place in an oven\u2013 and flameproof serving bowl.\n\nSprinkle the oats, ground flax and cinnamon evenly over the top, then place the bowl under the grill for a few minutes until the oaty topping begins to turn golden.\n\n###### FLAVONOIDS\n\nSOLUBLE FIBRE\n\n**Pears poached in spiced red wine** This is a lovely recipe that has a great seasonal festive vibe to it, but is just as at home served cold in the summer.\n\n#### SERVES 3\n\n250ml red wine\n\n3 ripe pears, peeled\n\n1 large cinnamon stick\n\n4\u20135 cloves\n\n2 slices of root ginger\n\n1 tsp vanilla extract\n\n\u00bd tsp stevia, or honey if you prefer, to taste\n\nPlace all the ingredients in a saucepan and bring to a gentle simmer (not a boil). Cook for 25\u201330 minutes. The pears should be tender to the point of a knife.\n\nFish out the pears and place in a serving bowl, one per person. Strain the wine through a sieve over the fruits and serve.\n\n###### OMEGA 3 FATTY ACIDS\n\nMAGNESIUM\n\n**Mackerel and caper p\u00e2t\u00e9** This is a gorgeous snack. Spread on an oatcake or use it as a dip for raw veggies such as carrots and celery.\n\n#### SERVES 1\n\n2 smoked mackerel fillets\n\n4 tbsp natural live probiotic yogurt\n\n1 tbsp extra virgin olive oil\n\njuice of \u00bd lemon\n\n2 tsp capers, drained and rinsed\n\nlow-sodium salt and freshly ground black pepper\n\nPlace all the ingredients into a food processor and process at full power until a smooth p\u00e2t\u00e9 has been formed.\n\nPlace in a bowl and snack at will.\n\n###### FLAVONOIDS \nMAGNESIUM \nVITAMIN E \nSTEROLS \nOMEGA 3 FATTY ACIDS\n\n**Heart-healthy trail mix** One thing that I hear very often from friends or clients is that they wish they had healthier snacks to hand when they are sitting at their desk. Workplace vending machines are kryptonite to many people, dangling temptation before us, so making your own snacks to take with you is an obvious solution. This little trail mix is tasty, portable and \u2013 most importantly \u2013 contains a broad array of heart-healthy nutrients. This makes enough for at least three or four days. Remember, this is for between-meal nibbling!\n\n#### MAKES 3\u20134 SNACKS\n\n1 tbsp pumpkin seeds\n\n1 tbsp sunflower seeds\n\n1 tbsp flax seeds\n\n1 tbsp goji berries\n\n1 tbsp dried blueberries\n\n1 tbsp dark chocolate chips\n\nCombine all the ingredients together and store in a sealable plastic container.\n\n# INDEX\n\n 1. ajoene 1\n 2. anchovies: black olive and anchovy-stuffed chicken breast 1\n 1. spinach and anchovy pitta pizzas 1\n 3. anthocyanins 1\n 4. apples 1\n 5. arteries 1, 2\n 6. arterioles 1\n 7. asparagus: salmon, pea and asparagus frittata 1\n 8. atherosclerosis 1, 2\n 9. ATP 1\n 10. aubergine, stuffed 1\n 11. avocados 1\n 1. avocado and poached egg rye toast topper 1\n 12. nutty chocolate avocado pots 1\n\n 1. barlotto, grilled salmon with red 1\n 2. beans: beetroot, bean and rocket salad 1\n 1. mixed bean chilli 1\n 3. beetroot 1\n 1. baked beetroot wedges with white bean houmous 1\n 2. beetroot, bean and rocket salad 1\n 3. beetroot, blackberry, celery and ginger juice 1\n 4. bold beetroot and horseradish soup 1\n 5. salmon and beetroot wasabi stacks 1\n 6. smoked salmon, beetroot and minted yogurt wrap 1\n 4. berries: berry protein smoothie 1\n 1. oat and berry layer 1\n 2. oaty flax berry crumble 1\n 5. beta carotene 1\n 6. beta glucan 1\n 7. blackberries 53 \n 1. beetroot, blackberry, celery and ginger juice 1\n 2. mixed seed and blackberry bowl 1\n 8. blood 1, 2, 3, 4, 5\n 9. blood clotting 1, 2, 3, 4\n 10. blood pressure 1, 2, 3, 4, 5\n 11. blood vessels 1, 2, 3, 4, 5\n 12. blueberries 1\n 1. chilli-blueberry compote 1\n 13. borlotti bean crush, tapenade salmon with 1\n 14. bread: avocado and poached egg rye toast topper 1\n 15. bulgar wheat 1\n 1. sea bass with salsa verde and tabbouleh 1\n\n 1. cabbage: red cabbage and carrot salad 1\n 2. cacao 1\n 3. calcium 1, 2, 3\n 4. cannellini beans: roasted onion and cannellini bean houmous 1\n 1. white bean houmous 1\n 5. capers: mackerel and caper p\u00e2t\u00e9 1\n 6. capillaries 1\n 7. capsaicin 1\n 8. carbohydrates 1, 2\n 9. cardiovascular system 1\n 10. carrot salad, red cabbage and 1\n 11. catechins 1\n 12. celery: beetroot, blackberry, celery and ginger juice 1\n 13. cheese: goat's cheese, pomegranate and olive salad 1\n 14. chicken: black olive and anchovy-stuffed chicken breast 1\n 1. chicken and green vegetable nutty stir-fry 1\n 2. chicken and tarragon-stuffed peppers with greens 1\n 15. chickpeas: chickpea and red pepper stew 1\n 1. herbed chickpea salad 1\n 16. chillies 1\n 1. chilli-blueberry compote 1\n 2. mixed bean chilli 1\n 17. chlorophyll 1\n 18. chocolate: heart-healthy trail mix 1\n 1. nutty chocolate avocado pots 1\n 2. nutty chocolate smoothie 1\n 19. cholesterol 1, 2, 3, 4\n 1. food that reduces 1, 2, 3, 4\n 20. clotting factors 1, 2, 3, 4\n 21. cocoa 1\n 22. coconut milk: creamy coconut porridge 1\n 1. roasted sweet potato and coconut soup 1\n 23. coconut oil 1\n 24. compote, chilli-blueberry 1\n 25. curry, king prawn and spinach 1\n\n 1. de novo lipogenesis 1\n 2. diet 1\n 3. dietary fibre 1, 2\n 4. digestion 1\n 5. dip, red pepper-walnut 1\n 6. dressings: creamy orange 1\n 1. orange 1\n\n 1. eggs: avocado and poached egg rye toast topper 1\n 1. kippers, boiled egg and watercress salad 1\n 2. endothelium 1, 2, 3, 4, 5, 6\n 1. endothelial damage 1\n 2. endothelial dysfunction 1, 2\n 3. and flavonoids 1, 2, 3\n 3. erythrocytes 1\n\n 1. fats 1, 2, 3, 4, 5\n 2. fatty acids 1, 2, 3, 4\n 3. fibre 1, 2\n 4. fibrin 1, 2, 3\n 5. fish 1, 2, 3, 4, 5\n 1. see also mackerel; salmon, etc\n 6. flavonoids 1, 2\n 7. flax seeds: oaty flax berry crumble 1\n 8. frittata, salmon, pea and asparagus 1\n\n 1. gamma-oryzanol 1\n 2. garlic 1\n 3. ginger: beetroot, blackberry, celery and ginger juice 1\n 4. glucagon 1\n 5. glucose 1\n 6. glycaemic response of foods 1\n 7. goji berries: pomegranate goji omega smoothie 1\n 1. squash, goji berry and red onion soup 1\n 8. green tea 1\n\n 1. haemoglobin 1\n 2. HDL cholesterol 1, 2\n 3. heart attacks 1\n 4. horseradish: bold beetroot and horseradish soup 1\n 5. houmous: roasted onion and cannellini 1\n 1. white bean 1\n\n 1. infarction 1\n 2. inflammation 1, 2, 3\n 3. ingredients 1\n 4. insulin 1, 2, 3\n 5. iron 1\n\n 1. juice, beetroot, blackberry, celery and ginger 1\n\n 1. Keys, Ancel 1, 2\n 2. kippers, boiled egg and watercress 1\n\n 1. LDL cholesterol 1, 2, 3, 4, 5, 6\n 2. lentils 1\n 1. tarka dal 1\n 3. leukocytes 1, 2\n 4. lipogenesis 1\n 5. lipoprotein 1\n\n 1. mackerel 1\n 1. mackerel and caper p\u00e2t\u00e9 1\n 2. magnesium 1, 2\n 3. mango salsa, tuna steak with 1\n 4. Mediterranean brown rice risotto 1\n 5. minerals 1\n 6. muscles 1\n\n 1. nitric acid 1\n 2. nitric oxide 1, 2, 3, 4, 5\n 3. nuts: chicken and green vegetable nutty stir-fry 1\n 1. nutty chocolate avocado pots 1\n 2. nutty chocolate smoothie 1\n\n 1. oats 1, 2\n 1. creamy coconut porridge 1\n 2. oat and berry layer 1\n 3. oaty flax berry crumble 1\n 2. occlusions 1\n 3. olive oil 1, 2\n 4. olives: black olive and anchovy-stuffed chicken breast 1\n 1. goat's cheese, pomegranate and olive salad 1\n 5. omega 3 1, 2, 3\n 6. omega 6 1, 2, 3\n 7. onions 1\n 1. roasted onion and cannellini bean houmous 1\n 2. squash, goji berry and red onion soup 1\n 8. oranges: creamy orange dressing 1\n 1. orange dressing 1\n\n 1. pasta: wholewheat pasta with roasted pepper sauce 1\n 2. p\u00e2t\u00e9, mackerel and caper 1\n 3. pearl barley: grilled salmon with red barlotto 1\n 4. pears poached in spiced red wine 1\n 5. peas: salmon, pea and asparagus frittata 1\n 1. salmon with pea pur\u00e9e and roasted butternut squash 1\n 6. pectin 1\n 7. peppers 1\n 1. balsamic caramelised pepper soup 1\n 2. chicken and tarragon-stuffed peppers 1\n 3. chickpea and red pepper stew 1\n 4. red pepper-walnut dip 1\n 5. roasted pepper sauce 1\n 6. stuffed aubergine 1\n 8. phytochemicals 1, 2\n 9. phytosterols 1\n 10. pitta pizzas, spinach and anchovy 1\n 11. plaque 1, 2, 3\n 12. plasma 1, 2, 3\n 13. platelets 1, 2, 3\n 14. pomegranate: goat's cheese, pomegranate and olive salad 1\n 1. pomegranate goji omega smoothie 1\n 2. tonic tipple 1\n 15. porridge, creamy coconut 1\n 16. post-prandial lipaemia 1\n 17. post-prandial triglyceridemia 1\n 18. potassium 1\n 19. prawns: king prawn and spinach curry 1\n 1. peppered king prawn skewers 1\n 20. processed foods 1\n 21. prostaglandins 1\n 22. protein 1, 2\n\n 1. quinoa 1\n 1. roasted vegetables with quinoa salad 1\n 2. tuna steak with quinoa verde 1\n\n 1. red blood cells 1\n 2. resveratrol 1\n 3. rice 1\n 1. king prawn and spinach curry with herby brown rice 1\n 2. Mediterranean brown rice risotto 1\n 4. rocket: beetroot, bean and rocket salad 1\n 1. roasted squash, rocket and sun-dried tomato salad 1\n 5. rye bread: avocado and poached egg rye toast topper 1\n\n 1. salads: beetroot, bean and rocket 1\n 1. goat's cheese, pomegranate and olive 1\n 2. herbed chickpea 1\n 3. kippers, boiled egg and watercress 1\n 4. red cabbage and carrot 1\n 5. roasted squash, rocket and sun-dried tomato 1\n 6. roasted vegetables with quinoa 1\n 2. salmon 1\n 1. grilled salmon with red barlotto 1\n 2. salmon and beetroot wasabi stacks 1\n 3. salmon, pea and asparagus frittata 1\n 4. salmon with pea pur\u00e9e and roasted butternut squash 1\n 5. smoked salmon, beetroot and minted yogurt wrap 1\n 6. tapenade salmon with borlotti crush 1\n 3. salsa, mango 1\n 4. salsa verde: grilled trout with root vegetables and salsa verde 1\n 1. sea bass with salsa verde and tabbouleh 1\n 5. salt 1\n 6. sea bass with salsa verde and tabbouleh 1\n 7. seeds: heart-healthy trail mix 1\n 1. mixed seed and blackberry bowl 1\n 8. skewers: peppered king prawn skewers with tarka dal 1\n 9. smooth muscle 1, 2, 3, 4, 5\n 10. smoothies: berry protein 1\n 1. nutty chocolate 1\n 2. pomegranate goji omega 1\n 11. soba noodle vegetable stir-fry 1\n 12. sodium 1\n 13. soups: balsamic caramelised pepper 1\n 1. bold beetroot and horseradish 1\n 2. roasted sweet potato and coconut 1\n 3. speedy tomato and paprika 1\n 4. squash, goji berry and red onion 1\n 14. spinach: herbed chickpea salad with sun-dried tomatoes and spinach 1\n 1. king prawn and spinach curry 1\n 2. spinach and anchovy pitta pizzas 1\n 15. squash: roasted squash, rocket and sun-dried tomato salad 1\n 1. salmon with pea pur\u00e9e and roasted butternut squash 1\n 2. squash, goji berry and red onion soup 1\n 16. starchy foods 1\n 17. stew, chickpea and red pepper 1\n 18. stir-fries: chicken and green vegetable nutty stir-fry 1\n 1. soba noodle vegetable stir-fry 1\n 19. stress 1, 2\n 20. strokes 1\n 21. sweet potatoes 1\n 1. roasted sweet potato and coconut soup 1\n 2. sweet potato wedges with red pepper-walnut dip 1\n\n 1. tabbouleh, sea bass with salsa verde and 1\n 2. tarka dal 1\n 3. tea 1\n 4. thrombocytes 1, 2\n 5. thrombus formation 1, 2\n 6. tomatoes: herbed chickpea salad with sun-dried tomatoes and spinach 1\n 1. roasted squash, rocket and sun-dried tomato salad 1\n 2. speedy tomato and paprika soup 1\n 7. tonic tipple 1\n 8. trail mix, heart-healthy 1\n 9. triacylglycerol 1\n 10. triglycerides 1, 2, 3\n 11. trout 1\n 1. grilled trout with root vegetables and salsa verde 1\n 12. tryptophan 1\n 13. tuna 1\n 1. tuna steak with chilli-blueberry compote and roasted celeriac 1\n 2. tuna steak with mango salsa, wilted greens and quinoa verde 1\n\n 1. vasoconstriction 1, 2, 3\n 2. vasodilation 1, 2, 3, 4, 5, 6\n 3. vegetable oils 1\n 4. vegetables: chicken and green vegetable nutty stir-fry 1\n 1. grilled trout with root vegetables and salsa verde 1\n 2. roasted vegetables with quinoa salad 1\n 3. soba noodle vegetable stir-fry 1\n 4. vegetable crudit\u00e9s 1\n 5. vitamin C 1\n 6. vitamin D 1\n 7. vitamin E 1\n\n 1. walnuts: red pepper-walnut dip 1\n 2. watercress: kippers, boiled egg and watercress salad 1\n 3. white blood cells 1, 2\n 4. wine 1, 2\n 1. pears poached in spiced red wine 1\n 2. tonic tipple 1\n\n 1. yogurt: mixed seed and blackberry bowl 1\n 1. oat and berry layer 1\n 2. smoked salmon, beetroot and minted yogurt wrap 1\n\n# Copyright\n\nClare Hulton \u2013 we are really cooking on gas now! Amazing work. Thank you! Jenny Liddle \u2013 you are tireless at what you do! Tanya Murkett \u2013 as always, supporting me and putting up with me no matter what! A big thank you to all the team at Quadrille, Smith & Gilmour, Martin Poole, and Aya Nishimura. Catherine Tyldesley, Gaby Roslin, and all of the wonderful people that have supported my work and career. Ramsay and Candy. Mum and Dad.\n\nEditorial director: Anne Furniss \nCreative director: Helen Lewis \nProject editor: Lucy Bannell \nArt direction and design: Smith & Gilmour \nPhotography: Martin Poole \nIllustration: Blindsalida \nFood stylist: Aya Nishimura \nProps stylists: Polly Webb-Wilson & Wei Tang \nProduction: Tom Moore\n\nFirst published in 2015 by Quadrille Publishing Limited \nwww.quadrille.co.uk\n\nQuadrille is an imprint of Hardie Grant. \nwww.hardiegrant.com.au\n\nText \u00a9 2015 Dale Pinnock \nPhotography \u00a9 2015 Martin Poole \nDesign and layout \u00a9 2015 Quadrille Publishing Limited\n\nThe rights of the author have been asserted. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, electrostatic, magnetic tape, mechanical, photocopying, recording or otherwise, without the prior permission in writing of the publisher.\n\nCataloguing in Publication Data: a catalogue record for this book is available from the British Library.\n\n978 184949 658 2\n","meta":{"redpajama_set_name":"RedPajamaBook"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzryis b/data_all_eng_slimpj/shuffled/split2/finalzzryis new file mode 100644 index 0000000000000000000000000000000000000000..4b0ad539c7b0d5ccf15df094a1e46432a9e21c60 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzryis @@ -0,0 +1,5 @@ +{"text":" \n**Alan Hunter** was born in Hoveton, Norfolk in 1922. He left school at the age of fourteen to work on his father's farm, spending his spare time sailing on the Norfolk Broads and writing nature notes for the _Eastern Evening News_. He also wrote poetry, some of which was published while he was in the RAF during the Second World War. By 1950, he was running his own bookshop in Norwich. In 1955, the first of what would become a series of forty-six George Gently novels was published. He died in 2005, aged eighty-two.\nThe Inspector George Gently series\n\n_Gently Does It_\n\n_Gently by the Shore_\n\n_Gently Down the Stream_\n\n_Landed Gently_\n\n_Gently Through the Mill_\n\n_Gently in the Sun_\n\n_Gently with the Painters_\n\n_Gently to the Summit_\n\n_Gently Go Man_\n\n_Gently Where the Roads Go_\n\n_Gently Floating_\n\n_Gently Sahib_\n\n_Gently with the Ladies_\n\n_Gently North-West_\n\n_Gently Continental_\n\n_Gently at a Gallop_\n\n_Gently in the Trees_\n\n_Gently French_\n\n_Gently Where She Lay_\n\n# Gently French\n\nAlan Hunter\n\nConstable & Robinson Ltd\n\n55\u201356 Russell Square\n\nLondon WC1B 4HP\n\nwww.constablerobinson.com\n\nFirst published in the UK by Cassell & Company Ltd., 1973\n\nThis paperback edition published by C&R Crime,\n\nan imprint of Constable & Robinson Ltd., 2013\n\nCopyright \u00a9 Alan Hunter 1973\n\nThe right of Alan Hunter to be identified as the author of this work has been asserted by him in accordance with the Copyright, Designs & Patents Act 1988.\n\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, or to actual events or locales is entirely coincidental.\n\nAll rights reserved. This book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, re-sold, hired out or otherwise circulated in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser.\n\nA copy of the British Library Cataloguing in Publication\n\nData is available from the British Library.\n\nISBN 978-1-47210-870-8 (paperback)\n\nISBN 978-1-47210-878-4 (ebook)\n\nTypeset by TW Typesetting, Plymouth, Devon\n\nPrinted and bound in the UK\n\n1 3 5 7 9 10 8 6 4 2\n\nCover image by David Woodroffe; Cover by JoeRoberts.co.uk\n\n## CHAPTER ONE\n\nCRIME SOMETIMES PAYS: but it has its casualties, too.\n\nI was sitting behind a clear desk and smoking the last pipe of the day. Everything was tidy; my reports were in, and I was waiting only to give Dutt a lift to Tottenham. Since this was the new-New Scotland Yard I couldn't see the Thames from my window any more; but I could see, far away over the roofs, a row of tall, graceful steel storks, dipping and raising their intelligent beaks and performing slow ballet-movements among themselves: dock cranes. They assist my thinking when I'm engaged in a two-pipe problem. Because the street, on the other hand, fails to do this, I have had the bottom of the window masked with hard-board; and, still in pursuit of a climate for thought, have smuggled all my old furniture into this otherwise soulless cubicle. Fertile irregularity. Like the unofficial patina my pipe has begun to lay on the upper paintwork.\n\nA moment of peace, with the blower resolutely silent: just the click of Dutt's typewriter from next door. But then a step in the passage and the door opening without a knock. Only one man does that: the Assistant Commissioner.\n\n'Ah \u2013 Gently. Don't get up.'\n\nI hadn't been going to. He whisked in.\n\n'I've just come back from that Angry Brigade conference. Thought I had better look in if I was to catch you.'\n\nHe sat; he bowled me over with a smile.\n\n'Are you feeling like a trip into the country?'\n\n'It depends on the weather.'\n\n'Capital. I have something here that will just suit you.'\n\nWhen he beams like that, watch out. I moved my feet under the desk. Dutt's final barrage sounded off-stage, followed by the squeal of a sheet being whipped from a typewriter.\n\n'Would that be Len?'\n\n'I am waiting for him.'\n\n'Call him in. He'll be going with you.'\n\nI rose and obeyed. The A.C. exploited the interval by humming a snatch from _Pinafore_.\n\n'Now look, you two. Frederick Albert Quarles. What do you know about that gentleman?'\n\nDutt looked at me for a lead.\n\n'Isn't he the villain they call Flash Freddy?'\n\n'The same.' The A.C. beamed at us. 'He's the boss of a snatch gang located in Hammersmith. The Met have been after him for four or five years. He has been one of their real headaches. Well, no longer. Freddy is dead. Apparently a confederate slipped a knife in him.'\n\n'Have they got the man?'\n\n'Yes. He is cooling his heels in a cell in Norchester.'\n\n'Then where do we come in?'\n\n'A simple double check. You are just to go over the locals' lines with them.'\n\nPause for gentle laughter.\n\nI scratched a match for my pipe, which hardened the gleam in the A.C.'s eye. It is two years now since he gave up smoking, but the old Adam still twitches.\n\n'Don't they have a case, then?'\n\n'Of course they have a case. The fellow's name is Stanley Rampant. A local, he acted as a nose for Quarles. Freddy's gang had just done a job in Norchester. Rampant gave them the tip and Freddy set it up, but somebody put in a squeak to Met. The Met boys stopped the gang at a roadblock. They nicked four out of five of them, but missed the money.'\n\n'How much?'\n\n'Thirty-five thousand.'\n\nI considered. 'Isn't Freddy a big operator?'\n\n'One of the biggest. He may have slipped up this time. Perhaps Rampant's information wasn't reliable.'\n\n'Then what happened?'\n\nA comic shrug from the A.C. 'My guess is that Freddy wouldn't pay up. He gave Rampant a pourboire for expenses and told him he would have to try again. So buying it. They got on to Rampant through the car he had supplied for the getaway. It was clean, you see, it had to be legitimate. He gave a false name but the dealer knew him. When they nicked the mug he was still wearing the suit he had been wearing when he killed Freddy. Blood-spots on the sleeve. He's a petty villain with minor form.'\n\n'But he will have a story.'\n\n'He says he got there second.'\n\n'I can't see that inhibiting a jury.'\n\nThe A.C. made staccato popping sounds. 'Very well, then! Perhaps the case does have a few wrinkles. For one thing, somebody shopped the gang, and that somebody would scarcely have been Rampant. I.e. there was another villain around who wanted to put a spoke in Freddy's wheel. Then there are the injuries. He was badly cut up. There were thirteen stab-wounds in the back and neck. Any one of seven of them could have been fatal, and the rest weren't exactly acupunctures. How does that strike you?'\n\n'Unprofessional.'\n\n'A panicky amateur. Or else?'\n\n'A hate killing.'\n\n'Or else?'\n\n'Cherchez la femme.'\n\n'Aha.'\n\nThe A.C. had been selling it. And I'd bought it.\n\nHe pulled out some paper.\n\n'Our dossier on Quarles. Freddy wasn't a common villain. Father a senior civil servant, deceased; a cousin in the Foreign Office, attached Washington. Prep school, Merchant Taylors', Magdalen, called to Bar '61, disbarred '64: interfered with witness in murder trial. Not convicted. No form. Associate of villains listed hereunder. Suspected complicity in fifty-six snatch jobs, proceeds totalling \u00a32,357,025, in part recovered. Alibi specialist. No person participating in robberies.'\n\nI delivered a smoke-ring. 'A steady performer.'\n\n'The Met boys won't shed any tears, sir,' Dutt said.\n\n'Never mind that.' The A.C. waved at my smoke. 'Listen to what comes now. August '69 Quarles went to Paris. There was a snatch job done at the Renault works. No known complicity. What Quarles came back with was a Frenchwoman, Mimi Deslauriers. She has been living with him since then and she was staying with him in Norchester. Mimi Deslauriers, who was tried in Paris for the stabbing-to-death of her husband, Charles.' He rustled the paper. 'A nice coincidence?'\n\n'I like the sound of Rampant better.'\n\n'Neater, of course. It will please everyone. But meanwhile, Mimi has a lousy alibi.'\n\n'Where were they staying?'\n\n'At a place called the Barge-House. A riverside hotel outside Norchester.'\n\n'Where was he killed?'\n\n'He was killed in his car. It was parked on heathland near the city.'\n\n'What type of car?'\n\nThe A.C. gleamed again. 'Not your or my sort of car, Gently. He didn't get his sobriquet for nothing. The devil owned a Bugatti racer.'\n\n'A which?'\n\n'A Bugatti racer. One of those cars they sold to Maharajahs. A hundred and twenty in the shade. They were seeing off Bentleys when you were still at school.'\n\n'It's an open two-seater?'\n\n'Right. You must allow that Freddy had flair.'\n\n'He would be wearing his shoulders handy for a knife.'\n\n'Well, that sort of thing didn't happen to Louis Chiron.'\n\n'Huh.' I stirred my feet. 'So Mimi is what's bothering them up there?'\n\n'Principally Mimi. I hear she's flamboyant, is sort of giving the picture some colour. But don't overlook the other angle. Freddy must have made a lot of enemies. His just sitting back and using catspaws couldn't have made him terribly popular.'\n\n'Who is handling the case?'\n\n'Norchester and Mid-Northshire. But don't bother to phone them, I already have.' He dropped the paper on the desk. 'It's quite a simple case, really.'\n\n'Oh quite,' I said. To the cranes.\n\nHe headed for his Bentley.\n\nDutt came round the desk and we skimmed through the bumf together.\n\n'Len,' I said, leering at him. 'Len. Since when were you on first-name terms with His Nibs?'\n\nDutt coloured. 'He must have had my docs, sir.'\n\n'And that means one of two things.'\n\n'Well, I hope it's the right one, sir. With Terry going to Cambridge I could use the lolly.'\n\nSix foot of cockney, that's Dutt. Born in Seven Sisters Road. Lifelong supporter of the Spurs, brown ale and small Fords. Not so much thick as slow: he's got a brain that won't be hurried. Hence missing preferment's eye. Preferment being the loser.\n\n'Then we had better make a good impression with this one. I can get you a mug-shot in the local press.'\n\n'Don't suppose His Nibs will see it.'\n\n'You are underestimating His Nibs.'\n\nAs I chanced to know, one of His Nibs' disbursements went to a cuttings agency in Chancery Lane. No press acquired by a Central Office lackey escaped the eye of Big Brother.\n\nAlong with the CR mish were copies of photographs of Quarles (all highly confidential, of course, since Quarles had never been convicted). A handsome, long-featured man with a romantic black mane, smiling dark eyes, set close, and thin lips parted over ferret's teeth. Forty-five. Slim, tallish. Spoke with a public school accent. Charm that triggered-off women. A numbered bank-account; a Bugatti.\n\n'Would you buy a used car from him?'\n\nDutt sniggered. 'He wouldn't be selling my kind of car.'\n\n'I'm keen to see his choice in women.'\n\n'Bet you she looks like Ursula Andress.'\n\nThe address given was a flat in Upper Cheyne Row ('Where they half-inched the posh paving-stones': Dutt), and alongside the Bugatti he had run a Citro\u00ebn Pallas: for when it rained, no doubt.\n\nAttached to the rest, a r\u00e9sum\u00e9 of the snatch job and the Met C.I.D.'s commendable action. The villains involved were named Norton, Elsing, Wicken, Lound and Fring. A specialist mob. They all had form; three had done time for GBH. Fring was the one who had got away, taking with him the loot in a black suitcase. Named i\/c case, Chief Inspector Dainty. The switchboard got him at the third attempt.\n\nBurning question: 'Who gave you the tip-off?'\n\nDainty's answers were evasive.\n\n'A regular?'\n\n'Not as far as we know.'\n\n'Man or woman?'\n\n'We think it was a man.'\n\n'You are not sure?'\n\n'Pretty certain. But he was talking through his scarf.'\n\n'When did it come through?'\n\n'At fifteen-five. We only just had time to set up the block.'\n\n'How did you come to lose Fring?'\n\n'They pulled up short of us. Fring was out of the car like a rabbit. He hooked on to a bus turning out of a junction. By the time we stopped it he had vanished.'\n\nThey had had a dust-up with the other four, no doubt laid on to give Fring his start. Fring, of course, was removing the evidence. It was tucked away in the black suitcase.\n\n'What are you holding them on?'\n\n'An offensive weapon charge. But that will change when we catch Fring. We have a stake-out at his house in Battersea and a watch on all his known haunts.'\n\n'Well, he won't be strapped for a night's lodging.'\n\nDainty's laugh sounded sour. 'We have had information coming in. I don't think you need worry about Fring.'\n\nI nagged him again about the tip-off, which had come from a call-box. The informant had named two of the men, Lound and Fring, and had referred to the gang as 'Flash Freddy's mob'. He had also described the car accurately, except for transposing numerals in the registration.\n\n'A local call?'\n\n'No way of telling.'\n\n'Who would have it in for Flash Freddy?'\n\n'That's what the snouts aren't telling us. When they do, you will be informed.'\n\nI hung up and exchanged looks with Dutt, who had been listening on the extension.\n\n'So. What do we make of that?'\n\nHe rumpled his face. 'It beats me, sir. It can't have been Rampant who put the squeak in. He'd be cutting his own throat.'\n\n'Suppose he had reasons.'\n\n'Like what, sir?'\n\n'Like trying to put the squeeze on Freddy.'\n\nDutt shook his homely bonce. 'Wouldn't be a sensible thing to do, sir.'\n\nNo, it wouldn't. But villains are stupid, especially little-leaguers like Rampant. And if it wasn't Rampant who put in the squeak, then we were groping around already.\n\nAh, well. Blessings on snouts.\n\n'First thing in the morning then, Dutt.'\n\n'Perhaps we'll have had a tinkle by then, sir.'\n\nI'm not an optimist, but I like them round me.\n\nLiving my life, and not theirs, I spent the evening with Brenda Merryn. Why aren't we married? We prefer it that way, and Brenda would make a wretched housewife. It was May and sweet weather so we took a stroll along the Embankment, had a couple of drinks at her favourite pub, then returned to her flat to grill two steaks.\n\nWith Brenda, I am indiscreet (she first came my way as a murder suspect). I mentioned Flash Freddy's sad end, introducing the Bugatti and Mimi Deslauriers.\n\n'She's a raving blonde,' Brenda said promptly.\n\n'Is this psychic vision or have you seen her?'\n\n'Seen her, met her, watched her operate. I've always moved in exalted circles.'\n\nWhich didn't altogether surprise me. Brenda works in Chelsea and has friends and a relative there.\n\n'Where did you meet her?'\n\n'At one of Siggy's parties. He never did sail round the world, you know. She's a busty bitch with a snub nose and dimples. If you disappear I shall know what has happened.'\n\n'She was accused of stabbing her first husband.'\n\n'Ha,' Brenda said. 'Then watch your back. I was going round telling myself all evening that Mimi Deslauriers had probably stabbed her first husband.'\n\n'Was Quarles with her?'\n\n'Tall, dark and sneaky?'\n\n'That's the man.'\n\n'He was there. He made a teeny-weeny little pass at me, and then keeled over when she looked at him.'\n\n'She was jealous.'\n\n'Possessive.'\n\n'What about him?'\n\n'I don't think he had much say in the matter. Mimi was lining them up in a queue, but that was her pre-rogative. Not Sneaky's.'\n\n'Interesting.'\n\nBrenda went out and returned wearing something more comfortable. I got back to Elphinstone Road at about ohone-hundred hours: not the best of preparations for a trip to the country.\n\n## CHAPTER TWO\n\nNO TINKLE IN the morning. Just an electricity bill and a letter in pencil, signed Justice: a threat to bomb the Bank of England unless we released a felon called Dakin. Worth a try, I suppose. I made arrangements to have it collected. Outside, a brilliant day, with a scent of lime-flowers coming from the Gardens.\n\nDutt arrived in time to drink coffee. We fetched my Lotus from the garage and locked up his Escort. Dutt had been brooding over the tip-off mystery and had reached the conclusion that, after all, the squeaker must have been Rampant.\n\n'It's the timing, sir. It had to come from someone who knew the job had been pulled. Then there's the car, he knew all about that. Even got the number nearly right.'\n\n'Rampant bought the car. Wouldn't he have got it quite right?'\n\n'It couldn't have been long in his possession, sir. And me, I always have to think twice when I'm asked for my number.'\n\nWell. But if Rampant had planned a tip-off, he would surely have made a note of the number. Also, he would probably have named all the gang. The message to Met had been less than explicit.\n\n'It might have been a snap decision by Rampant, but more likely it was a grass from some other ill-wisher.'\n\n'But how did they know about the car, sir?'\n\n'Simple. They saw it. Crooks can put on a tail as well as we can.'\n\n'You mean someone was out there keeping tabs on Freddy?'\n\n'Right. And with luck he'll have left a trail.'\n\n'So like that it could have a connection with the killing?'\n\nI grinned. 'Get in the car. We'll go and find out.'\n\nDriving fast.\n\nWe picked up the A1 and switched to the A505 at Baldock. The Lotus's virtues are wasted on dual carriageways and their semi-legal eighty. Jigging by puffing transporters, swooping round coveys of hard-driving reps. Slinking through bends with a steady clock. Here and there brushing the ton. The Lotus is a naughty car which has always a train to catch somewhere. Dutt, the perfect passenger, loves it: sits loose and dreamy, watching the road perpetually opening for us.\n\n'Wonder what that Bugatti's like to drive, sir.'\n\nI nod. 'It's been crossing my mind, too.'\n\nDutt gives me a glance. 'Perhaps we'll get a whirl in it.'\n\n'Perhaps,' I say, savouring my hypocrisy.\n\nWe slotted in at Norchester police H.Q., which is a wing of the big, pinkish City Hall. The press were waiting outside and I introduced Dutt to them as our leading expert on knife-killings. They took appropriate photographs. Then we were ushered in to the office of C.I.D. Chief Inspector Hanson. This was my fourth time with Hanson, who is not an unmixed admirer of mine. But today he was affable enough; I believe he thought he had the case licked.\n\n'Rampant's going to crack.'\n\n'That will be nice.'\n\nHanson flicked his grey eyes at me. Hack-faced Hanson. He's not so tough really; there's a soft centre under the chromium plate.\n\n'He's admitted he was after his cut. Quarles brushed him off with two hundred nicker. Quarles would never have seen him again unless Rampant was threatening him. Chummie's got no answer to that one.'\n\n'Where is Rampant now?'\n\n'I've got him downstairs.' Hanson hesitated. 'Do you want to have a go at him?'\n\n'First, you'd better put me in the picture.'\n\n'Yeah, well. It makes quite a story.'\n\nWe seated ourselves. Also in the office was Hanson's lieutenant, Sergeant Opie, a short, solid, dark-haired man with an empty face but alert eyes.\n\n'Let's start at the beginning. Whose money was it?'\n\nHanson gave a little snatch with his head. 'Bryanston Shoes. Big footwear people. They have a factory on the outskirts.'\n\n'Wages?'\n\n'Yep. They draw them on Thursdays to give the clerks time to make them up. Collect them at Lloyd's branch on The Walk. The car, driver, and one guard.'\n\n'Just one guard?'\n\n'One guard. And don't think we haven't talked to them about security. But this is Norchester, not London. Here they don't believe it till it happens.'\n\n'What about their route?'\n\n'They use two, through the centre and by Unwin Road. The trouble is they just alternate them, one this week, the other the next. So they were sitters for a villain like Quarles. He set it up at the quiet end of Unwin Road.'\n\n'How long had Quarles been in the district?'\n\n'He was out at the Barge-House all week. It's on the river, you know, a holiday spot. Quarles just acted as though he were on holiday.'\n\n'Where was he when the job was pulled?'\n\n'In a launch on the river, along with his woman and two others they'd invited. When we heard from Met we went out and questioned him, but he just laughed in our bloody faces. Then the next evening, he was dead.'\n\n'Tell me about that.'\n\nHanson heaved rough breath. He pulled open a drawer, took out a folder and slid it to me across the desk. The photographs. Not very pretty, but I've spent much of my lifetime studying such things. They showed Flash Freddy with a faceful of steering-wheel and a ventilated back and a bloody neck. Also the car, the lovely car. It was standing on a rough track, surrounded by trees. Just where it was parked was a large, jungly hawthorn with bracken growing round its skirts.\n\n'Where was this?'\n\n'Part of Mussel Heath. It flanks the city to the north.'\n\n'It looks more like a wood.'\n\n'There's plenty of cover there. No doubt why chummie picked it for a meeting.'\n\n'Give me the timetable.'\n\n'Quarles left the hotel around twenty hundred hours Friday evening. It's a seven-mile drive. E.T.D. between twenty hundred hours and midnight. Reported oh-eight-twenty-five Saturday by Samuel Trivett, labourer. Trivett lives in a road adjacent to the heath, was taking his dog for a stroll.'\n\n'Wasn't Quarles reported missing by Madame Deslauriers?'\n\n'Nope.'\n\n'Did you ask her why?'\n\n'You bet I asked her. She said that Quarles had gone out on business, and when that happened she expected him when she saw him.'\n\nI hesitated. 'Did she know what business?'\n\n'If she did, she's not admitting it.'\n\n'Where was she during the rest of that evening?'\n\n'In her room is what she says.'\n\n'But no proof?'\n\nHanson swept his bony hand. 'All right, I thought about that! But I couldn't believe it. Not with the lab report coming in about Rampant's jacket, and him with a motive as big as a house. Believe me, I know that bastard \u2013 he could do it and not lose any sleep.'\n\nPerhaps, perhaps. I pointed to the photographs. 'Nobody's mentioned the weapon yet.'\n\nHanson got red. 'Because we haven't found it. I'd say chummie took it with him and threw it in the river.'\n\n'Do we know what it was like?'\n\n'Yes, a short-bladed knife, blade not longer than four inches. A straight back with a curved edge. Could be a small kitchen knife.'\n\n'Commonly of French manufacture.'\n\n'Yeah, well! That's a point. But you can buy them here in town, so I don't see where that gets us.'\n\nI hunched. 'Had the body been frisked?'\n\n'If it had, the chummie missed five hundred nicker.'\n\nHanson lifted a plastic bag from his drawer and decanted its contents on the desk. Out came a fat wallet, keys, change, pens, a platinum cigarette-case, matching lighter, pen-knife, nail-file, comb and a rabbit's foot. I chivvied them around. The cigarette-case and lighter may have been worth another five hundred. In the wallet, mostly twenty-pound notes, bank-fresh, very handsome. Driving-licence, virgin. Insurance and M.O.T. certificates for 3.3 litre Bugatti (1932). Membership card the Dolly Club, Chelsea, receipt for jacket (\u00a3132.13), stamps, three credit cards, two theatre-ticket stubs, two gilt-edged visiting cards.\n\n'Personal jewellery?'\n\nHanson opened an initialled envelope. 'These came back from the mortuary.'\n\nHe shook out a Longines watch with a platinum case and expanding band and a solitaire diamond ring in the same metal.\n\n'Clothes?'\n\nHanson signalled to Opie, who fetched another bag, from a metal cabinet. He spread out a grey light-weight suit, silk shirt, socks and underwear and a pair of handmade shoes in natural camel-skin. The shirt and jacket were ripped and stained: very butcher-like exhibits.\n\n'A well-turned-out corpse.'\n\n'Yeah,' Hanson said. 'You'd have thought he was worth taking away.'\n\n'Somebody just wanted him dead.'\n\n'Somebody like Rampant.'\n\n'But wasn't Rampant's quarrel with him about cash?'\n\nHanson made noises. 'So the chummie panicked. Hell, he wasn't so relaxed when he spoiled that shirt. Perhaps something disturbed him, like a car passing close. There's a lot of necking goes on out there.'\n\nI grunted. 'It'll keep till I've viewed the scene. Now tell me something about Mimi Deslauriers.'\n\nHanson rolled his eyes. 'It'll be a pleasure. But if you think she's chummie, you must be slipping.'\n\nHe lit one of his cheroots, long, black, and puffed coarse smoke over my head.\n\n'Around thirty,' he said. 'Blonde. Green eyes. Say thirty-six, twenty-four, thirty-six. Tallish. Moves like a cat. Husky voice with an accent. Nearly knocks you down when she looks at you. Fancy dresser. Smells like honey.'\n\n'I'll recognize her,' I said. 'Where was she on Friday evening?'\n\nHe leached smoke. 'In her room. She went up after dinner, after Quarles left.'\n\n'And stayed there?'\n\n'That's what she says. Had a bath and went to bed.' Hanson's eyes were dreamy. 'It's a hell of a world,' he said.\n\n'Any corroboration?'\n\n'None.'\n\n'Does the room have a phone?'\n\n'Not an outside line.'\n\n'Could she have left unnoticed?'\n\n'With a lot of luck. You can get down backstairs to the kitchen end.'\n\n'Does she have alternative transport?'\n\n'Nothing we know about. But there'd be a rush if she wanted to borrow some. Only it's crazy, quite crazy. That doll wouldn't have to murder anyone.'\n\nI threw him an empty look. 'They did tell you her form?'\n\nHe pulled in a contemptuous lungful. 'Sure, sure. But she got off, didn't she? That was just her bit of hard luck.'\n\n'And this would be another bit?'\n\n'Why not? She's the sort of dame things happen around. But if she sneaked out and filled-in Quarles, I'll eat a year's supply of these things.'\n\nWell . . . he'd met her, and I hadn't. 'What was her reaction to the killing?'\n\n'She was concerned, you could say that. But she wasn't washing out her hankie.'\n\n'Scared?'\n\n'Could have been scared. She was all round me with a lot of questions.'\n\n'About what you were thinking?'\n\n'That sort of thing. I didn't get the impression it was the end of her world.'\n\n'So Quarles was just another mug.'\n\nHanson scowled through his smoke. 'Why should she break her heart over a jerk like that?'\n\nWe drank coffee, the canteen kind, then drove out to Mussel Heath. I had seen it before, on a previous case, but just to admire it in passing. The city fingers its suburbs into the edge of the heath, which rises above it in broken lines; from up there the city is mapped below you with its landmarks of churches, tower-blocks, the castle and the cathedral. The heath is a hilly and holey place where you could lose an armoured division. Parts are open, parts scrubby and wooded, with precipitous dells and overgrown hollows. It is criss-crossed by stony tracks, going nowhere in particular, and divided by a road that snakes up to join a ring-road.\n\nWe arrived by the dividing road. It rose beside a bare hill-slope, topped by an empty Victorian barracks; passed some pre-war council estates, then wound its way fenceless into high, woody heath. Hanson pulled into an official parking-place; from the far side a track dipped sharply; we bumbled down it, brushing bracken and birch-twigs, and levelled off in one of the dells. Another hundred yards brought us to the hawthorn which I had noticed in the photographs. We climbed out. Hanson pointed to four pegs hammered firmly into the ground.\n\n'That's where Quarles parked his heap. A nice, quiet spot for a villains' conference.'\n\n'But how would Quarles know about it?'\n\n'Rampant showed it to him. This is where he gave Freddy the dope. Freddy wasn't keen on being seen with Rampant, so he picked him up at the parking-place and drove down here.'\n\n'Does Rampant admit that?'\n\n'Sure. He's given us the whole tale as far as the hold-up. Then we know that Rampant called Freddy here for a second meeting. It's after that when the edges get blurred.'\n\nI poked around. The sides of the dell were fledged with tall-growing birches and sycamores. Where we'd come down was hidden by a turn and the dell came to an end just past the hawthorn. The ground was stony and didn't take tracks. There were a couple of footpaths leading off. One climbed out at the end of the dell, the other at the side, starting by the hawthorn. I scrambled up the latter. It took me nowhere, just through the trees into open heath. I came down again; and paused for some moments beside the hawthorn, which was in flower.\n\n'Give me your version of what happened.'\n\n'Huh?' Hanson stared scowlingly. 'There's only one version I know about. Rampant got Quarles to meet him here, didn't he?'\n\n'But then?'\n\n'Then he'd get in the jalopy beside him, start trying to pressure him to cough up.'\n\n'Go on.'\n\n'So Quarles wouldn't play, so there was a barney and Rampant pulled a knife. If he was planning to put the black on Quarles, he'd have been a mug to come bare-handed.'\n\n'And all this took place in the cockpit of a Bugatti?'\n\nHanson held back, glowering. 'You reckon it couldn't have?'\n\n'I reckon there wasn't much room to draw a knife, and none to use it in the way it was used. Is Rampant some sort of Samson?'\n\n'Not so as you'd notice.'\n\n'Then he wasn't in the car when he stabbed Quarles. He couldn't have put the knife in once, not reaching round Quarles while sitting beside him.'\n\nHanson looked savage. 'So how did he do it?'\n\n'I think Quarles' attacker came round this bush. He jammed Quarles' face into the wheel with one hand while he stabbed his back with the other.'\n\n'Oh fine,' Hanson said. 'Very clever.'\n\n'Which suggests something vital about the attacker. He is probably left-handed. Is Rampant left-handed?'\n\nHanson glared awhile. Then he got in the car.\n\n## CHAPTER THREE\n\nSO FAR SO good: I felt now I had earned a look at the Bugatti. We drove back to H.Q. in silence, then I put my request to Hanson.\n\nWe found the Bugatti sitting in a corner of the H.Q. garage, reverentially draped in a plastic dust-sheet. Lab had finished with it. Most of Quarles' bleeding had been soaked up by his clothing. A few smears on the wheel and the white leather bucket-seat had been photographed then cleaned off, while the recognizable latents were either Quarles' or off-record, probably innocent.\n\nHanson called over a mech. The mech started it for us and drove it out into the yard. It stood there growling in a chesty way, like a leopard meditating its spring. A marvellous blue shape. Beginning at the rad, an ellipse perhaps borrowed from Leonardo da Vinci; carried on through the delicate humping of the louvred bonnet, completed in the powerful signature of the fish-tail. Ettore had reached for one of Plato's patterns, and it had come to his hand like a pint pot.\n\n'What's the price of a heap like this?' Hanson asked.\n\nI didn't hear him; I was walking round it. Whatever Flash Freddy's sins had been, I felt I owed him gratitude for the Bugatti. The French racing-blue enamel was stove-hard everywhere, no hint of rust or tarnish. The cockpit appointments were immaculately original, so too were the strap-spoked aluminium wheels. The seats were new, but gave immediate conviction that they had been scrupulously copied from the originals. And the note of the engine, a precise, clear grumble, needed no connoisseur to confirm its tune.\n\nThe mech gently revved it, bringing in the supercharger. Faces appeared at a few nearby windows. Other mechs, who had been working in the garage, came out to stare at Ettore's car.\n\nRampant wasn't left-handed.\n\nI had ordered coffee before they fetched him to the office. When I offered him a cup, he shifted it to his left hand and then stirred it with his right.\n\nA frightened little villain. Aged about thirty, five-foot-seven, slim build; a blotchy ferret-face, long, lank fair hair and a soupy, unwashed appearance. Dress, a scruffy sweater, poncy jacket, dirty jeans and cheap suede sneakers.\n\nA petty villain; mostly a nuisance; sometimes useful to the villainocracy.\n\n'You knew Frederick Quarles, Rampant?'\n\n'Well yes, I had to, didn't I?'\n\n'Where did you meet him?'\n\n'Well, I didn't sort of know him, just met a bloke who was working for him.'\n\n'Where?'\n\n'Well, in the nick, wasn't it?'\n\n'I'm asking you.'\n\n'Yeah, in the nick.'\n\n'What was his name?'\n\n'It was Wickey, wasn't it? Him what was in there for knocking-off cars.'\n\n'Are you referring to Alfred Wicken?'\n\n'Bleeding Wickey is all I know. Wish I'd never listened to the frigger. Wouldn't've been here now, would I?'\n\nHe lapped up coffee, hands a-tremble. Very scared was Stanley Rampant.\n\n'What did he want?'\n\nRampant clattered the cup and saucer. 'Said I could be a nose for a big boy, didn't he? Wasn't no risk, it was money for jam. Just give him a tinkle when I was on to something.'\n\n'And you tipped him off about the Bryanston wages collection.'\n\n'Well yes, I did, didn't I?'\n\n'Then you actually met Quarles.'\n\n'Yeah, all right, I met him. I ain't trying to hide nothing, I'm being straight.'\n\nStraight as a meat-hook.\n\n'What happened at that meeting?'\n\nRampant clutched cup and saucer together. 'Bleeding business, that's what it was. You can't make nothing else of it.'\n\n'Quarles gave you money?'\n\n'Yeah, for the car\u2014'\n\n'Just for the car?'\n\n'Ain't I bleeding telling you! He gave me the price of a '65 Jag and a bundle for me, isn't that right?'\n\n'How much for you?'\n\n'He give me a bundle.'\n\n'How much?'\n\n'Two hundred nicker!'\n\nI clicked my tongue. 'That wasn't much, Stanley. I would have reckoned your cut at about fifteen hundred.'\n\n'Yeah, but that was on account\u2014!'\n\n'On account of what?'\n\nRampant juggled with the cup, got coffee on his jeans.\n\n'It was on account of what you'd get from the share-out,' I said. 'Only there wasn't any share-out. The two hundred was all.'\n\nDutt divested him of the cup and saucer. Rampant's blotches were mottling unhealthily. Too many bellyfuls of crisps and beer and morning lie-ins with the tarts. I let him sweat while I skimmed through his statement, a tiresome document in policese. Then I lit my pipe. He was watching me hungrily; there were yellow stains on his shaking fingers.\n\n'Want a fag, Rampant?'\n\n'Yeah. Yeah, I want one.'\n\nI nodded. 'Tell me what happened Friday evening.'\n\nHis eyes glazed. 'You got it down there.'\n\n'Not what I want to hear from you, Rampant.'\n\n'It's the truth, isn't it?'\n\n'Not all the truth. Quarles didn't just meet you for a social chat. I want to know what it was that fetched him there. What you had to say to each other.'\n\n'But we didn't say nothing\u2014!'\n\n'What fetched him there?'\n\n'I don't bleeding know! He was dead, wasn't he?'\n\n'Then why did you want to meet him?'\n\n'I frigging didn't. It was his idea.'\n\n'He rang you to arrange it?'\n\n'That's right!'\n\nHanson said: 'You're not on the phone, you stupid bastard.'\n\n'So like it was a message, then \u2013 yeah, a note\u2014'\n\n'And you're a pig's arse,' Hanson said.\n\nRampant breathed fast. I mouthed a smoke-ring.\n\n'Listen,' I said. 'You're in a hole, Rampant. Your part in the wage-snatch was worth a twicer. But unless you help us to come up with a better answer, that blood on your sleeve could make it life. Is that what you want?'\n\n'I never killed Freddy.'\n\n'There's nobody else under suspicion.'\n\n'But I frigging didn't!'\n\n'Who's going to believe you?'\n\n'They got to believe me!'\n\n'Shit,' Hanson said.\n\nRampant whimpered, screwing himself away from us. Suddenly he looked about fifteen. He'd got thick, ugly hands and bony wrists. Just a nasty little kid with a too-old face.\n\nHe moaned. 'I wanted my pay-off.'\n\n'Now he tells us,' Hanson sneered.\n\n'He was dead. He was frigging dead. It's true. He was all over blood when I got there.'\n\n'Just you and him.'\n\n'I wanted my cut. It wasn't my fault the job went wrong.'\n\n'So you put the black on him.'\n\n'It was what he owed me.'\n\n'Oh Christ,' Hanson said. 'They come thicker and thicker.'\n\nI signalled to Dutt; he gave Rampant a fag. Rampant's mouth quivered as he dragged on it. Hanson stuffed a cheroot into his own mouth and sat gazing hood-eyed at Rampant.\n\nSo now Rampant was giving.\n\nI took him through his statement detail by detail, trying to squeeze out some small fact that would give us a hint, a direction. But Rampant didn't know much. His contact with the gang had stopped short at Wicken and Quarles; the Bryanston job was his first tip-off: he had stepped up a league to come an immediate cropper. Still, I kept leaning on him. We worked through the wage-snatch and came to his phone-call to Quarles on the Friday. Rampant was sweating. This was the part he'd got to get us to believe somehow.\n\n'What happened on Friday?'\n\n'Well, on Friday I saw the papers, didn't I?'\n\n'You saw the papers and phoned Quarles.'\n\n'Yeah, I wanted to know what was going on.'\n\n'And Quarles was happy about you phoning him?'\n\n'No, he bleeding wasn't! He chewed me up, told me I'd got to forget I'd ever heard of him.'\n\n'Then you threatened to grass.'\n\n'Well, he asked for it, didn't he? It wasn't like he hadn't got the loot. One of his blokes dodged off with it. I'd read all that in the papers.'\n\n'Who suggested the meeting?'\n\n'Like I did.'\n\n'And Quarles agreed?'\n\n'Bleeding had to, hadn't he? I'd got my monkey up. All he wanted was to get me off the frigging phone.'\n\n'He didn't shout back at you?'\n\n'No, he never. He was talking like he'd got a cop at his elbow.'\n\n'There was someone with him?'\n\n'Well, talking like that. But bleeding nasty, all the same.'\n\nPoint to check at the Barge-House.\n\n'What time was this?'\n\n'Oh Christ knows. Maybe half-past ten, eleven.'\n\n'Where were you phoning from?'\n\n'The box near my flat.'\n\n'Just you on your tod?'\n\n'Yeah, I'm telling you.'\n\nThe meeting had been timed for eight-thirty p.m. He had spent the rest of the day in agitation. Successive editions of the evening papers made no mention of Fring's having been arrested. At opening-time he'd gulped down two pints at his local, but swore he hadn't mentioned the meeting to anyone. From seven-thirty to eight-fifteen he'd been alone in his flat. Then he'd got in his car and driven to the heath.\n\n'Nobody tailed you?'\n\n'I wouldn't know, would I? I had to go all through the city.'\n\n'What about where you go up by the barracks?'\n\n'Wasn't nobody behind me there.'\n\nSo he'd arrived at the parking-place and parked there. Several other cars were present, but not the Bugatti. Rampant had waited a few minutes in case Quarles was late, then he'd set off to the rendezvous on foot.\n\n'Can you remember those other cars?'\n\nRampant dusted sweat. 'I see a couple smooching in a red A40. Then there was a Cort parked next to me. And a clapped-out Consul, all over stickers.'\n\n'Did you notice anyone sitting in them, apart from the couple?'\n\nRampant shook his head reluctantly.\n\n'What colour was the Consul?'\n\n'Black, I reckon.'\n\n'And the Cortina?'\n\n'Green. A Mark 2.'\n\nAnd there had been six or seven cars more, of which he remembered not a single feature.\n\n'Let's take it slowly, now. You went down that track, from which there's a good view round about. What did you see?'\n\n'Well nothing, did I? It's just a lot of trees and sort of ferns.'\n\n'Think carefully. Did you hear anything?'\n\n'There were cars going along the road.'\n\n'No sounds from below?'\n\nHis ferret eyes were helpless. 'Honest. I never heard a thing.'\n\n'All right. Keep going.'\n\nHis mouth quavered. 'So I come round the corner and see the car. At first I think the bleeder's dozed off, that's how it looks, him hung over the wheel. Then I come up. His frigging arm is dangling and there's blood all over his back. So I gets hold of the bastard and tries to pull him up. Then I can see he's bloody deado.'\n\n'Did you see a knife?'\n\n'There wasn't no knife.'\n\n'Sure?'\n\n'Of course I bleeding am! If there was a knife it wasn't stuck in him. I ain't saying it wasn't around.' 'Did you touch or move anything?'\n\n'No I never. I just got out of there sharp.'\n\n'You heard, saw, nobody?'\n\n'I keep telling you, don't I? I never see nobody at all.'\n\nI fed him another fag, then we went through it once again. Nothing fresh, except now he remembered two more of the cars, a Minx and a Herald. By this time Rampant was drying-up, so I let them march him back to the cells. His information had been mostly negative: but that can be important, too.\n\nFive minutes' thought, while Hanson fumigated his office with a fresh cheroot. He had been quiet during the later stages of my examination of Rampant. Now he hosed smoke at me.\n\n'So what's the verdict from the big man?'\n\nI slid him a grin. 'Better tell me yours. You've been longer on the case.'\n\n'I've seen more of Rampant.'\n\n'Granted.'\n\n'That bastard's been in my hair a long time. He's owing for six or seven jobs that I know about but can't pin on him.'\n\n'One of the facts of life.'\n\n'Yeah. And now I've got the sod dangling. You could do me an Old Pals' Act and he would be out of my hair for good.'\n\n'Is that what you're suggesting?'\n\n'I'm bloody tempted. Except I know you won't play. And except I've been listening to you turning him over, and I'm not so sure about him any more.'\n\n'He's still the lad with the blood on his sleeve.'\n\n'Yeah, but he was telling his story better. At times I was almost just going to believe him, had to keep reminding myself he was Rampant.'\n\nI got up. 'Too soon,' I said. 'We may circle round and come back to Rampant. It could be that he saw that job done, even though he didn't use the knife himself.'\n\nHanson huffed smoke. 'Is that your verdict?'\n\nI hunched a shoulder. 'Rampant will keep. Meanwhile I have another appointment.'\n\nHanson nodded slowly. 'And she smells sweeter.'\n\n## CHAPTER FOUR\n\nRAMPANT WASN'T LEFT-HANDED: which helped his case, though it didn't exonerate him. On the other hand, I knew he wasn't Dainty's squeaker the first time he opened his mouth. He spoke with a Norchester street-accent, a debased form of the more vigorous Northshire; no Met officer would have missed it, though Rampant had talked through a dozen scarves.\n\nIn fact, Dainty had made no mention of accent: a negative point that was slightly suggestive. What accent, or nuance, wouldn't register with a Met man? Quick answer: his own. Thus the squeaker most likely was a Londoner, though not one with a coarsely cockney accent. A man indifferently educated, perhaps a rival gang-leader \u2013 in which case the snouts should be able to finger him.\n\nThough they hadn't, yet. My next move was to ring Dainty, who had no news: he sounded uffish.\n\nLunch. I invited Hanson, but he had got hung up with some petty villainery \u2013 two chummies who were impersonating council rent-collectors, and making a good thing of it. I took Dutt to The Princess, a cellar-like establishment in the neighbourhood of the provision market, known to me from that early case involving a Dutch timber-importer and his ingenious manager. The Princess had changed little that I could see. The same dimly-lit cosiness and competent waitresses. We had their mixed grill followed by gateau, and the years had altered the quality of neither.\n\n'Done any thinking?' I asked Dutt.\n\n'No sir. Excepting I don't like Rampant.'\n\n'You don't think he came clean?'\n\n'That's the trouble, sir. I've a nasty feeling that he did.'\n\n'So leaving the field open.'\n\nDutt chewed and nodded. 'I reckon it's a queer old job altogether. I can see another villain putting a squeak in, but knocking off Freddy would be just stupid.'\n\n'Especially the way it was done.'\n\n'Exactly, sir. It isn't the style of our villains. They'd have picked up Freddy coming out of a pub, not tailed him out to no heath.'\n\n'Somebody with more than a professional motive.'\n\n'Yes, but the snouts would know about that, sir. And we don't hear nothing. I'm getting the notion the bloke we're looking for is a strict amateur.'\n\nI gnawed some liver. 'Yet there was a squeak. The professionals come into the act somewhere.'\n\nWe ate up and paid. Our waitress was elderly, and my vanity hoped that she might remember me. She didn't, of course. I was just a stranger, betrayed by the unwonted size of my tip.\n\nI collected the Lotus and we drove out of Norchester by a route made unfamiliar by one-way systems. The Barge-House was at Haughton St John (pronounced Hoffen-John by Hanson), a riverside village eight miles distant. This too I had known in past times; it is a sort of hire-boat metropolis. But in autumn, when the motor-cruisers have stopped knocking value off each other, it is also a station for pike and bream.\n\nThe road led through a gentle, paintable country of suave undulations and psychic trees, with once a glimpse of a square flint tower to give a fix in centuries. Followed the ribbon-development of half-comely Wrackstead, which lives across the river from Haughton; and finally the vandalized humpback bridge, a victim of traffic and official callousness.\n\nA few years ago you could have parked by the bridge while you strolled back to admire the boating scene. Now you crawled over a bumping Bailey structure to be marshalled through yellow lines to a suicidal crossways. One of the roads I remembered had vanished, its place taken by a sprawling Superstore. Another had been widened in an unlovely way, hastening more traffic to the inevitable jam. No logic, no way out, except possibly a surgical use of the bomb. Many years behind need the local authorities would doubtless concede a new river-crossing.\n\nWe negotiated the jam, then turned left into a road that paralleled the river. The Barge-House, an Edwardian pub enlarged into a hotel, occupied a site opposite to a bank. It was a heavy, redbrick building, set flush to the pavement, with a small forecourt intended for horse-gigs; also a yard to one side, which was presently resembling a car-dealer's lot. Not a love-some place; but what you didn't see was that it had lawns running through to the river. A sign-board, not yet modernized, offered launches, skiffs and row-boats for hire.\n\nI parked and sat appraising the scene. The road was called Bylore Road. Traffic was queued all along it, waiting to break into the chaos at the crossroads. Adjacent to the Barge-House were three sad terrace cottages, apparently built with bricks left over; then a slightly more engaging, white-plastered building, exhibiting the sign of Three Tuns. I nudged Dutt.\n\n'Think like a villain who wants to keep an eye on Flash Freddy.'\n\nDutt grinned. 'He couldn't have hired the bank, sir.'\n\n'So go buy yourself a pint before the bar closes.'\n\nDutt went. I parked the Lotus in the forecourt, where there was space beside a badge-heavy Alfa. Two good-looking cars: though one had stood there lately which \u2013 for looks \u2013 would have smeared either into the woodwork.\n\nThe exterior of the Barge-House did it an injustice; once through the swing-doors things became plusher. You stepped into a long hall with concealed wall-lighting, a spongy carpet and a smell of old brandy. Left was the reception office: empty. I pressed the bell-push and waited. Across the end of the hall passed deft young waiters, presumably en route between kitchen and dining-room. A door to the bar was opposite the office and from that way came conversation, clinking and laughter. Finally, from kitchenwards, hurried a man in a lounge-suit. He glided up to me, smiling apologetically.\n\n'So sorry to keep you. We've been busy.'\n\n'Mr Frayling?'\n\n'What can I do for you?'\n\n'Shall we go in the office?'\n\nHis smile slipped a notch; but only for a moment. We went into the office.\n\nFrayling was the manager. He had guessed who I was, but I told him all the same. He was a slim, willowy type, mid-forties, a lined face and conciliatory eyes. Somehow he suggested to me a busted school-master, but I didn't go into his record. He flicked the glass shutter across the reception window and we sat down on two well-padded chairs.\n\n'I'm here about the killing, not the robbery, though there's a probable connection. I intend to make the hotel my base. I'll need rooms for myself and my Inspector.'\n\n'Of course, I'd like to help you, but\u2014'\n\n'Good. You can begin by booking us in.'\n\nHe shrugged but pulled over the register. A couple of the existing guests got shunted.\n\n'Now tell me about Quarles.'\n\nFrayling's eyes jumped. 'I \u2013 I made a full statement to Inspector Hanson.'\n\n'I know.' I patted the brief-case I had with me. 'And now I want to hear it at first-hand. Have you any objection?'\n\n'No, of course not!'\n\n'You had no connection with Quarles, for instance?'\n\n'Good lord, no!'\n\n'Then give me a quick run through. I'd like to know why he chose the Barge-House.'\n\nFrayling couldn't tell me that, and the point was probably of no importance. But it got Frayling going on the wrong foot, which is the first move in interrogation. Quarles had booked by phone a fortnight ahead, no doubt when he'd received the tip from Rampant. He had booked for a week, from Saturday to Saturday, just like any other vacationist. He had arrived at mid-day, making some stir with his car and his companion. Frayling had given them his best room, overlooking the river. Quarles had registered Mimi as his wife.\n\n'Were you happy about that?'\n\nFrayling's smirk was edgy. 'I can't ask too many questions, can I? Most of our guests are married couples. You take them on trust if it isn't too obvious.'\n\n'Wasn't this obvious?'\n\n'No, I wouldn't have said so. Even ladies like this one can get married. And they were matter-of-fact enough with each other, they'd obviously been together for a while.'\n\n'Matter-of-fact?'\n\n'I think that describes it. They weren't rushing to jump into bed. Actually, I was wondering if they'd had a tiff. But they were just the same later on.'\n\nInteresting.\n\n'Did they have any quarrels?'\n\n'No. Not to my knowledge.'\n\n'Displays of jealousy?'\n\n'She could be catty, but there wasn't any real venom behind it.'\n\n'What about Quarles?'\n\nFrayling fidgeted. 'I have to admit that I rather liked him. He was a well-educated man, you know, very polite, a lot of charm. But subdued, that's how he struck me. As though he might have some sadness in his background. I was never more taken aback in my life than when I learned he was a crook.'\n\n'Did _he_ show any jealousy?'\n\n'He wasn't easy to read. He had that sort of public-school veneer. But it didn't seem to bother him what the lady got up to. I suppose he must have been used to her winsome ways.'\n\nI stared. 'Tell me about them.'\n\nFrayling hefted his neat shoulders. 'Don't forget she is still a guest here.'\n\n'And don't you forget we're talking about murder.'\n\nHe sulked a bit; flipped the leaves of the register. 'All right. But don't let on that I told you. Anything in trousers was getting an eye from her. Some of the wives here weren't so tranquil.'\n\n'How far did it go?'\n\n'I don't think it went anywhere. The lady is just fond of exciting attention.'\n\n'Was she needling Quarles?'\n\nFrayling hesitated. 'I don't think I'm qualified to pronounce on that.'\n\n'So he just took it in good part.'\n\n'Yes, as far as I could see. But like I said, he wasn't easy to read. A quiet sort of character. He would sit chatting racing while the lady was preening herself with the men.'\n\n'Quiet.'\n\n'Yes.'\n\nWell, he'd have things on his mind; which weren't necessarily connected with Mimi Deslauriers.\n\nI fetched up my brief-case.\n\n'Then from what you're saying, neither Quarles nor Deslauriers were particularly stand-offish.'\n\nFrayling nodded. 'They were both good mixers. Quarles arranged several picnics to go on the river.'\n\n'Any special friends?'\n\n'None I noticed. I can give you some names of the people they invited.'\n\n'Did you notice if there was anyone they specially avoided?'\n\nHe shook his head. 'I don't think so.'\n\n'Did they have any visitors?'\n\n'Not to my knowledge. But of course, strangers use the public rooms. The saloon and bar are packed every night. It goes on like that till late September.'\n\n'People off the boats?'\n\n'Mostly from the boats. Some of them drive out here from town. Then you get day-trippers in from the coast. We have a big turnover of casuals.'\n\n'Anyone like this?'\n\nI took out a mug-shot of Rampant, part of the bumf I'd collected from Hanson. Frayling stared at it curiously, but apparently it didn't ring a bell.\n\n'You'd better show that to the bar-tenders. I'm not often in the bar. Who is he?'\n\n'It doesn't matter.' I returned the photograph to the brief-case. 'Now: about Quarles' movements. What can you tell me?'\n\nFrayling made an act of weaving his head. 'You can't expect me to keep tabs on the customers. For one thing I'm too busy running the place.'\n\n'You'd know about meals: who was in, who was out.'\n\n'Not necessarily with guests who are on full board. But I can tell you that Quarles lunched out most days. We packed him baskets to take on the launch.'\n\n'How many is most days?'\n\n'All the week except Friday. Going on the river is why people stay here.'\n\n'Including, say, the Sunday?'\n\n'Not including the Sunday. That day he must have gone out in his car.'\n\nHis contact with Rampant, checking of the route.\n\n'Where do the guests make their phone-calls?'\n\n'There's a pay-box down the hall. We switch incoming calls through to it.'\n\n'Did Quarles use it often?'\n\nFrayling shrugged. 'That's something I wouldn't know, isn't it? But he had an incoming call on Friday. I called him down from his room to take it.'\n\n'At what time?'\n\n'Half-past elevenish.'\n\n'Were you around when he took the call?'\n\n'I didn't listen-in, if that's the idea, but I was in the hall talking to the housekeeper. It was a short call, two or three minutes, and the caller was a man with a local accent. Quarles came out of the box looking vexed. He went straight past us up to his room again.'\n\nRampant, of course.\n\n'You don't remember any calls made?'\n\n'No. But he could have made a dozen.'\n\n'Madame Deslauriers?'\n\nFrayling checked. 'Yes, I saw her in the box once.'\n\n'Which day?'\n\n'The same day, Friday.'\n\n'Friday! Before or after Quarles?'\n\nHe puckered his eyes. 'Must have been after. I'd've been going to the dining-room for my pre-lunch inspection.'\n\n'That you can swear to?'\n\n'Yes. Is it important?'\n\nI let my eyes empty. I didn't know. Just that the information touched a nerve which good detectives keep near the surface.\n\n'Where is Madame Deslauriers now?'\n\n'I rather think she's taken a launch out. Incidentally, she's miffed at being made to stay on here. Apparently she has business to see to in London.'\n\n'I'll take that into consideration. Let me see the register.'\n\nFrayling obligingly turned it towards me. During the previous week there had been eighteen other guests and five transients staying at the Barge-House. None of the names meant anything to me. The addresses ranged from Kent to Glasgow. Nearest to Chelsea were a couple called Stanwick who lived in Garden Lane, Chiswick. I shoved the book back.\n\n'Any more to tell me?'\n\nFrayling's eyes jerked, steadied.\n\n'If I remember any more, I'll tell you.'\n\n'Thank you,' I said. 'Please do.'\n\n## CHAPTER FIVE\n\nFRAYLING DEPARTED TO make apologies to two guests and to prepare their rooms for the gendarmerie. I sought the lounge, a long, sunny room with a view down the lawn to the river. An entertaining place. The two walls facing south and east had been glassed, so that besides the river you could see the bridge, which on this side had its parapet intact. The parapet was of local brick, stone-capped and very elbowable \u2013 or would have been, except for the traffic which crashed over the bridge in fractious queues. Rubber-tyred bulldozers, impatient to level it. No more elbows on the comfortable stones. While below blundered other queues, motor-cruisers, battering and scarring brick and stone. Give it two years? Five? They don't really care in these parts. The National Parks Commission turned its back and the joyful exploiters were soon in business.\n\nSeveral of the late-lunching guests were taking their coffee in the lounge: middle-aged respectables, business-workers, bank-employees, small traders. An atmosphere of defensive politeness, into which Quarles would probably have merged neatly. With Deslauriers in tow, they would perhaps have put him down as a TV wheel or something in films. Speak nicely, dress well and only a cop can smell your b.o. I took a seat at the back of the house and helped the atmosphere with my pipe.\n\n'Coffee, sir?'\n\nThe waiter was intriguing: a long-haired youth with a bush of beard. And a blush.\n\n'Do I know you?'\n\n'N-no, sir. I don't think so.'\n\nBut he knew me. Which suggested that Frayling had already briefed his staff.\n\n'What's your name?'\n\n'Bavents, sir.'\n\nIt was not a name I would be likely to forget.\n\n'Bring me coffee for two.'\n\n'Yes, sir.'\n\nDutt arrived as Bavents left.\n\nDutt was looking pleased. He glanced round the company, then took a seat in my lee.\n\n'I've been having a chat with the landlord, sir. Very interesting it was.'\n\n'Did he have a guest last week?'\n\n'As a matter of fact he had five. But four we don't have to bother with. They were youngsters down here for a bit of sailing.'\n\n'So get to the other one.'\n\n'Yes, sir. He's a lad who registered as Peter Robinson. And he gave an address in Finsbury Park, which I just happen to know doesn't exist.'\n\n'You're sure of that?'\n\n'Dead sure. I go past the place when I drive to work. And this is what makes it really interesting: chummie stayed here for only the one night.'\n\n'Which night?'\n\n'Thursday.'\n\nI clicked my tongue. 'After the snatch, before the kill.'\n\n'Yes, sir. Arrived here during the evening, driving a pale blue Viva. Called himself a speciality salesman and said he might be staying a couple of days. But something must have changed his mind, because he checked out again on Friday morning.'\n\n'What time did he leave?'\n\n'After breakfast.'\n\n'Was he away from the pub at any time during his stay?'\n\n'That evening, sir. He took his bag to his room, went out and didn't return till closing-time.'\n\n'You're right, it is interesting.'\n\nDutt nodded happily. 'I reckon he could be our killer, sir. A pro they sent up here for the job, some lot who wanted to fix Freddy for good. Just shopping his mob wasn't enough, because Freddy could soon have whipped together a fresh bunch. So they hired this pro. A quiet, out-of-town killing, no strings, no come-back.'\n\n'That's how it could have been.'\n\n'It fits the facts, sir.'\n\nI rasped the bristle on my chin. If that's how it was, then we might as well pack up. The loose ends, if any, were all back in town.\n\n'Did you get a description?'\n\n'A pretty fair one. Aged between thirty and thirty-five. About five-foot-ten, strong build, fair hair with short side-boards, pale eyes. London accent. Wearing a fawn sports jacket with an open-necked shirt and grey slacks.'\n\n'Does it bring anyone to mind?'\n\n'It'd fit Jack Straker.'\n\n'Straker's doing a niner on the Moor.'\n\n'Well, perhaps Fring will talk when they catch him, sir.'\n\n'It's a big perhaps. And they haven't caught him.'\n\nDutt hunched a little. 'Still, Met may know him. It may give them a lead to the gang who worked it.'\n\n'And that's probably where it will end. In Met's files.' Dutt put on a glum look and stayed silent.\n\nI worked my chin again. 'Just answer me one thing. Is this a job that carries a Straker-type signature?'\n\n'Perhaps not, sir.'\n\n'You know it isn't. There was nothing professional about that knife-attack.'\n\n'But then who is this Peter Robinson joker, sir? Him being here could hardly be a coincidence.'\n\n'The odds are against it. But they're even longer against him being a professional killer. So we'll just hand him on to Hanson and Dainty \u2013 and bear him in mind when we're asking questions.'\n\nDutt looked unconvinced; but Bavents chose that moment to return with the coffee. He poured it with a nervous sort of obsequiousness, his hair weeping round his eyes. Dutt took his cup. I considered Bavents. I was still finding him an intriguing subject. He draped a napkin round the plated coffee-pot, then referred to his little pad.\n\n'Will that be all, sir?'\n\n'No. Tell me where I should look for Madame Deslauriers.'\n\n'M-Madame Deslauriers?'\n\n'You know her, don't you?'\n\nHe was going hot amongst his hair.\n\n'She went out in the l-launch, sir.'\n\n'So I was told. But where do you think she was heading?'\n\n'W-well, you could try the Broad, sir. I heard her asking where she could pick some w-water-lilies.'\n\nI tipped him ten pence. He departed pinkly, doubtless to report to Mr Fr-Frayling. I sipped coffee, then went to the french windows, from which I could see the hotel's boat shed and mooring cut. I saw a runabout moored with three skiffs. I drank up my coffee and returned to Dutt.\n\n'Get off Peter Robinson's description. Then you can make a start with checking statements.'\n\n'Yes, sir,' Dutt said. 'Will you be around?'\n\nI shook my head. 'I'm going on the river.'\n\nThe runabout was called _Frolic III_. It was a fibreglass twelve-footer powered by a 20 hp Mercury outboard; it displayed a certain degree of zip as I dodged it downstream among the cruisers. At first, a suicidal-seeming project. The river here was narrow and weltering with craft: clumsy, steep-sided motorised barges that rolled and moaned and crunched their topsides. No effective right and wrong side: just a weak side and a strong side. _Frolic III_ was hip, however, and we skated through to less-saturated reaches.\n\nThere followed a mile of plugging. I hadn't lost the traffic, but below the village it fell into queues. We grumbled along resignedly between bungalow lawns and saluted moored craft with grinding wash. 200-seater trip-boats thundered by with their 200-sitters waving and jeering. We met one piece of flotsam, a misguided yacht, and left it with its gear bucking and slatting. Then we reached the Broad and the battle of the gateway: when finally the mess could spread out.\n\nI zimmed _Frolic III_ across to the reed-beds. I was beginning to wonder if I'd been a fool to come chasing Mimi. The Broad was larger than I had remembered, full of ranging bays and hidden corners. Yachts, allowed a little breeze, loitered across it. I could see numerous launches, moored and manoeuvring. On the opposite bank lay the yacht club, bristling with moorings, where a dozen Mimis might be lurking. I tried to formulate a plan of action that would maximize my chances of meeting her: none occurred to me. I set out to plod conscientiously round the perimeter.\n\nI knew which launch I was looking for because I had asked the boatman at the Barge-House. Mimi had taken the hotel's best boat, _Osprey_ , a mahogany, slipper-sterned, twenty-footer. She had left a short while before noon, but had taken no lunch with her; which suggested that either she had stuck among the water-lilies, or had steered for one of the down-river hostelries. Of the latter the nearest was at Harning, about four miles distant. But if she had gone there, then it was likely she would have returned before now. So look out for water-lilies. I pedalled circumspectly, scanning each inlet for white blossoms.\n\nTwenty minutes later I had reached the south end, where the reeds thinned out into beach-fringed meadows. Beyond these, to the west, was a small concealed bay. And in there were the lilies and Mimi's launch.\n\nMimi wasn't in the launch.\n\nIt was moored to a landing-stage that had been built out on piles from the bank. Open water led up to the stage through acres of lilies and their brawling roots. I tinkered _Frolic III_ up to the stage; the scent of the lilies was stupefying; alder carrs were hedging them from the modest breeze and they simply lay sunning and disgorging their odours. There was nothing in _Osprey_ except her cushions, no sounds but the murmur of distant motors. A still-life of fire-white, saffron-hearted stars, scenting yellow cups, an empty launch and flashing water.\n\nI looped _Frolic III's_ painter over a post and climbed out on the stage. A footpath led from it across the meadow and disappeared between high, flowering hawthorn hedges. I followed it. It led to a stile beyond which was a narrow lane. On each side of the lane were fields of green crops, bounded by hedgerows and mature woodlands. Here and there were glimpses of pantile; a long way off stood a sad church-tower. It all basked sleepily in afternoon sunlight and whiffs of fragrance from the hawthorns.\n\nBut where was Mimi?\n\nI mounted the stile, intending to continue down the lane. Not necessary. As I straddled the stile, a figure came into view ahead. It had to be Mimi. At a hundred yards range you could feel the vibrations beginning. Seeing me, she hesitated, but then came on. I settled on the stile and waited.\n\n'Madame Deslauriers?'\n\n'Yes, okay?'\n\nThe sun wasn't sunnier than the smile she gave me. She had come swinging up to the stile, her bag dangling, and had halted in a playful, swayed stance. Her hair was a warm straw blonde and her eyes a startling emerald green; her features had that majestic, sleeping, symmetry one sometimes finds in Greek marbles.\n\n'Who are you \u2013 shall I guess?'\n\n'I'm a policeman.'\n\n'Oh yes, but there are policemen and policemen, huh?'\n\n'My name is Gently. Chief Superintendent.'\n\n'A famous one, yes. That is what I was guessing.'\n\nShe dredged up a throaty little laugh. To describe her voice as husky was a simplification. It had an ecstatic, caressing quality that seemed to go straight to the base of your spine.\n\n'You are famous, aren't you?'\n\n'You seem to have heard of me.'\n\n'But no, it is just guessing. Poor Freddy now, he would have heard of you. But the policemen were in his line of business.'\n\n'Is business how you thought of it?'\n\nShe made a mouth, and dimpled. 'That's the way it goes, my friend. Cops and robbers. Freddy was a robber. At least you must agree he had talent for it.'\n\n'That didn't help him in the end.'\n\n'No. But I think poor Freddy had got careless. Going out to meet a little rat like Rampant, and at such a place. It wasn't wise.'\n\n'Didn't you try to dissuade him?'\n\n'How could I? Freddy didn't tell me his affairs.'\n\n'Not that Rampant was making trouble?'\n\nHer eyes widened. 'But no! He told me nothing at all. Freddy was a lawyer, you understand? Perhaps that was why he was such a good crook. He told people only what they needed to know, then the policemen can get nothing out of them. And I, what did I need to know? I am just his woman, that's all. He would say, It has been a good week, we have cleared so-and-so, you can read about it in this paper. Okay?'\n\n'And you didn't want to know more?'\n\n'Ha, ha. Why should I be interested in crookery? Do you think I am a crook?'\n\n'You have been associating with one.'\n\n'No, Monsieur. Just with a man.'\n\nWell, it was believable. I eyed her clinically. Not only the face was borrowed from Praxiteles. With a body as regal as that one might not feel the need for additional excitements.\n\n'Were you fond of Quarles?'\n\nShe swirled her hair. 'He was a man of great _savoir faire_ , Freddy. He could talk about this, about that. It did not matter what company he was in.'\n\n'But you were fond of him.'\n\n'He was fun to be with.'\n\n'That isn't really answering the question.'\n\nShe warmed her smile for me. 'Perhaps I don't like the question. So perhaps I'm not going to give you an answer.'\n\n'Then I'll draw my own conclusion. You weren't fond of him.'\n\nShe pouted prettily. 'Perhaps, I said. Maybe I don't know myself, exactly. That sometimes happens to one, huh?'\n\n'At least, you're not grief-stricken.'\n\n'I am sad, oh yes. After all, we had been together three years. But grief-stricken, no. I have had one big grief, and after that\u2014' She gestured. 'So say I am sad.'\n\n'Or even less than that?'\n\nHer eyes narrowed slightly. Then she thrust her bag at me.\n\n'Here \u2013 hold this! It is time I permitted myself a cigarette.'\n\nWhich brought us closer: I sitting on the stile, Mimi extracting a cigarette from the bag I was holding. Then, strangely, she couldn't find her matches, had to beg a light from me: and steady my hand. Fingers of character. Not anonymously feminine, but made to do something more than caress. And I caught that scent which Hanson had likened to honey, and which I immediately qualified: heather honey.\n\n'I am told you wish to go back to London.'\n\n'Oh, perhaps. It is not important.'\n\n'Some business was mentioned.'\n\n'Not true. Just some parties, a first night.'\n\n'Then you won't mind staying here a little longer.'\n\nShe blew me a tender stream of smoke. 'No. It will not be so boring. It is the small men I find tiresome.'\n\n'After all, you seem able to amuse yourself.'\n\n'Aha. So you have noticed.'\n\n'Picking water-lilies. Where did you put them?'\n\n'They have horrible stems, my friend. No good.'\n\n'Yet you found them so intriguing that they made you miss lunch.'\n\n'Is that why my tummy feels empty?'\n\n'Did you miss lunch?'\n\nShe blew smoke pettishly. 'This is a foolish conversation.'\n\n'But did you miss it?'\n\nShe took the bag. 'I must have done. I am suddenly bored. So now I go straight back to have a meal. Au r'voir, my friend. Enjoy the daisies.'\n\n## CHAPTER SIX\n\nSHE MOUNTED THE stile with quick grace and jumped down lightly into the meadow.\n\n'Wait,' I said.\n\n'Why should I wait?'\n\n'I have some more questions to ask you.'\n\nShe slitted her eyes. 'And if I am not in the mood? If I do not choose to be pestered?'\n\nI shook my head. 'You are too intelligent. You would never take up a foolish attitude.'\n\n'Foof!' But her mouth twitched. 'You know that you have no right to detain me. And I am not very pleased to be harassed like this, to be chased by a policeman when I stretch my legs.'\n\n'Is that what you were doing?'\n\n'Of course. Do you doubt it?'\n\n'I don't doubt you could find a way to be more helpful.'\n\n'Huh-huh. And why should I?'\n\n'Because it would amuse you. And I make a change from the clientele at the Barge-House.'\n\nShe drew herself up. 'Monsieur, what vanity!'\n\n'Also, you're not yet sure if I admire you.'\n\nShe gave a throbbing chuckle. 'I think you are a devil. What a good thing I find you unattractive.'\n\nWe stared at each other. She was smiling now.\n\n'Okay, okay, we will play the game. I find I am not hungry after all. It must be the scent of so many flowers.'\n\n'Shall we go back to the launch?'\n\n'I prefer not. It will be more comfortable out of the sun.' She glanced around casually. 'Perhaps beneath that hawthorn. It seems unlikely that we shall be disturbed.'\n\nShe stubbed her cigarette and made for the hawthorn. It was the most spreading of several that fringed the meadow; a handsome pyramid of milky blossom, throwing broken shade on the grass beneath. Mimi selected her spot and sat down; I selected mine, leaving turf between us. From there you saw a steely slice of the Broad with distant sails moving slowly upon it. Mimi plucked a stalk of plantain and chewed it appreciatively. She had turned towards me and was leaning on her elbow. Two harnessless breasts were moulding themselves sweetly, one drooping, one pouted by its neighbour.\n\n'Are you married, my friend?'\n\n'Not entirely.'\n\n'Hah. Such wisdom in two words. Is she so beautiful?'\n\n'I don't carry her photograph.'\n\n'Then she is either very beautiful or very plain. I wonder which.'\n\nI let her wonder. 'How did you come to take up with Quarles?'\n\n'Oh, I was unhappy. It was after my trouble. Two million Frenchmen wanted to marry me.'\n\n'And you didn't want to marry?'\n\nShe bit off some stalk. 'I am rich too, that is the trouble. My late husband was an industrialist. If he had been poor there would have been no trial. La Famille. His poisonous mother. No doubt you are provided with the details. Afterwards, who cares about marrying Mimi? The bride is so many million francs.'\n\n'And Quarles was so different?'\n\n'Oh, but yes. I think you do not understand. Freddy also was rich, very rich. It is all tucked away in a little Swiss bank. So what did Freddy care about marrying rich girls? No, my friend, this was love. He picked me up one night in Montparnasse. He was a thief. I let him steal me.'\n\n'Did you know he was a thief?'\n\n'Not that first night. But he didn't know I was a rich girl, either. Then it was too late. We make these grand discoveries when it would be disagreeable to let them interfere. So, we ignore them. He doesn't want my money, nor do I wish to reform Freddy. After all, he is brilliant in his line. What chance has he ever given you to catch him?'\n\n'Somebody did catch him.'\n\n'Aha. But that somebody was not a policeman. In the end perhaps he catches himself. Or it is just that the little crooks grow envious.'\n\n'Which little crooks?'\n\n'Why not Rampant?'\n\n'It wasn't Rampant who tipped the police.'\n\n'No, you are sure?'\n\nI popped the head of a daisy. 'So if not Rampant, who would it be?'\n\nShe drew her stalk through her teeth. 'Well, it wasn't me. I had no reason to shop Freddy. Was it a woman's voice?'\n\n'That's not important. It wouldn't exclude a woman's having been behind it.'\n\n'Ah, ah, it was a man, then. The rest is guessing. You are just trying it on, my friend. I think you had better stick to this little pig, Rampant. Because, after all, who is going to believe him?'\n\nShe elevated a knee, and admired it. The action slid her hem down her thigh. She had a strong, distinctive leg that flowered from an athletic foot and ankle. She smiled and let the knee slowly unflex: leaving the hem where it was.\n\n'You think I was tired of Freddy, huh?'\n\n'Were you ever really in love with him?'\n\nShe made a small mouth. 'I think so, at first. Those first few weeks were formidable. It was like bubbles up my nose, I could scarcely get my breath. Better than my husband, oh yes. It is a shame to kill a man like Freddy.'\n\n'But you were through with him by Friday.'\n\n'You are right.' She sighed. 'So then it may not really have been love. I am swept off my feet, as you say. Freddy took me on the bounce. But still I am fond of him, huh? He was such an interesting man to live with. Such a wide acquaintance. They knew he was a crook, but it didn't matter. He was always welcome.'\n\n'He was jealous of you.'\n\nShe gave her gurgling chuckle. 'All men are jealous, more or less.'\n\n'Perhaps you'd given him cause.'\n\n'But why not? We are only young for you once, my friend.'\n\n'Then he resented it.'\n\n'And I killed him?'\n\n'Well?'\n\nShe rolled on her stomach and squirmed closer to me. 'No.' It was spoken as though to a child. 'You are trying so hard, petit. So hard.'\n\nI thrummed another daisy-head at the meadow. Mimi picked daisies and thrummed one, too. Hers landed squarely on my chin. She giggled and lined up another. Two bull's-eyes. I shifted further off; Mimi squirmed after me like a seal. She rested her chin in her hands and stared up at me, her breasts pendant among the daisies.\n\n'Forget it,' she said. 'I didn't kill him. Even though he was so stupidly jealous. Even though he threatened me with violence. There he was weak. And he knew I knew it.'\n\n'And you, of course, weren't jealous of him.'\n\n'Shall I tell you the truth?'\n\n'If it isn't being old-fashioned.'\n\n'Yes, I was jealous. Isn't that strange? I couldn't bear him eyeing another woman.'\n\n'Which sometimes he did?'\n\nShe nodded. 'Sometimes. And that made me so angry. Perhaps I am thinking I am much the most beautiful, so why does he insult me like that, huh?'\n\n'Was there any particular woman?'\n\n'Oh no. I would have left him on the spot.'\n\n'Please think carefully. It could be important.'\n\n'I tell you for certain. No particular woman.'\n\n'Just you.'\n\n'Wouldn't you say I was enough? I never grew stale with poor Freddy. And I didn't need to kill him, that's certain too: if I had grown tired of him, I could have left.' She let fly with a daisy. 'So you had better believe me, instead of thinking up such useless questions.'\n\n'I believe anything I can prove.'\n\n'Oh, foof.' She plucked and loaded a fresh daisy.\n\nI grabbed her firing-hand. She liked that, and let the daisy fall to the grass. The hand had a cool, consenting feel; it moved lazily under mine. But I dropped it. She lay still, leaving the hand where it fell.\n\n'Tell me about your stay here.'\n\n'Must you waste our time, my friend?'\n\n'Did you know that Freddy had come on a job?'\n\nShe sighed expressively. 'He didn't tell me.'\n\n'But you knew?'\n\n'Okay, I knew. Freddy would not have come here just for pleasure. A bourgeois inn wasn't his style. It isn't my style, either.'\n\n'How did he propose it?'\n\n'Oh, very politely. He is thinking we would like a week out of town.'\n\n'It didn't arise from some . . . earlier circumstance?'\n\nShe stared. 'Of course, he had the tip from Rampant.'\n\n'But nothing else?'\n\n'What should there be?'\n\nI shrugged. 'The Bryanston job wasn't a grand one. I would like to know why Freddy bothered with it. Whether there was something else in the wind.'\n\nShe gazed for a while. 'You are subtle,' she said. 'This is why they make you top man.'\n\n'Have you any suggestions?'\n\n'None, my friend. Unless it is that this Rampant misleads Freddy.'\n\nI shook my head. 'Freddy was a specialist. He could cost a job like an accountant. He would have checked the size of the Bryanston labour-force and multiplied it by the average wage-rate. Add a percentage for over-time and N.H.I., deduct a percentage for the sick and absent. The result would give him a minimum figure, probably accurate within a few thousands.'\n\n'Freddy did all that?'\n\n'On his cuff. He'd know exactly what he was going for.'\n\nShe giggled. 'I think I'm proud of Freddy. I think he really was a clever man.'\n\n'Not so clever with this job, though.'\n\n'Perhaps he does it just to show his skill.'\n\n'You can't help me.'\n\n'I am so sorry.'\n\n'Right. Now let's talk about Peter Robinson.'\n\nHer eyes widened; were suddenly empty.\n\n'Why should we talk about a shop?'\n\n'Not the shop. A man. A man who was at Haughton Thursday evening.'\n\n'But I do not know any Peter Robinson.'\n\n'A man of about your own age. Five-foot-ten, fair hair with sideboards, comes from town, drives a blue Viva.'\n\n'But no, I don't know him.'\n\n'He spent the night at the Three Tuns.'\n\n'I have never visited that place.'\n\n'He was out during the evening. Perhaps paying a call.'\n\n'I cannot help it \u2013 I didn't see him!'\n\nI paused, holding her eyes. 'Where were you Thursday evening?'\n\nNow she was sitting up straight in the grass. 'In the hotel, of course \u2013 at first on the lawn\u2014'\n\n'With Quarles?'\n\n'Yes! Why should I not say true?'\n\n'And after that?'\n\n'Then we go into dinner \u2013 and in the bar \u2013 and watch TV\u2014'\n\n'Still with Quarles?'\n\n'Of course! With Freddy.'\n\n'Until you went to bed, never alone?'\n\nShe drew quick breaths, her eyes glinting. Her hands were clasping her flexed knees. I had her going; but suddenly she realized it: suddenly let the tension go. She gave a breathless chuckle.\n\n'Ha-ha! You are trying to bulldoze me, huh?'\n\n'Were you alone?'\n\n'You are fierce, my friend. I adore a man with a touch of steel.'\n\n'Please answer the question.'\n\n'I grow so weak. A man like that can do what he wants with me. I melt for him, huh? A couple of times I go to the loo.'\n\n'Twice?'\n\n'It may be three times. Why do you bore me with such nonsense?'\n\n'Then you could have been available for a brief interview.'\n\n'I prefer the longer ones. All night.'\n\nI gave it up. She'd turned on her back, with a knee crooked and waving. Her arms were folded behind her head, her eyes thinned, lips parted. Venus inviting. And I couldn't be certain if she was covering-up or not.\n\n'Have you been to this part of England before?'\n\n'I am a Parisienne, Monsieur.'\n\n'Meaning yes, or no?'\n\n'Would that be likely? I have not even heard of it before this time.'\n\n'Then you have no friends here?'\n\n'None.'\n\n'Nobody to speak to on the phone.'\n\nShe hesitated. 'Now you ask something different. It is not only to friends that one speaks on the phone.'\n\n'Then who was it on Friday?'\n\nShe re-composed her legs; crooking both knees, letting them spread.\n\n'Don't you want to answer?'\n\n'Just thinking, Monsieur. Let us say it was Friday when I phoned the theatre.'\n\n'The theatre!'\n\n'But yes. They have a theatre in the town. One day I feel desolate, think it will amuse me. Perhaps Friday, I do not know.'\n\n'Only, of course, there were no suitable seats.'\n\nHer lips twitched. 'Monsieur knows.'\n\n'And you gave no name, so they wouldn't remember you.'\n\nShe released a hand to make a gesture.\n\n'And I am supposed to believe this.'\n\nShe came coiling across to me. 'Monsieur will believe what he likes, won't he?' She hung on my shoulder. 'But it doesn't matter. Because perhaps it was another day, after all.'\n\n'Though having no connection with Peter Robinson.'\n\n'Aha! I think that man makes you jealous. But there is no need, my fierce friend. I can truthfully say I have not met him.'\n\n'Not then or later.'\n\n'Not at all.'\n\n'Not, for example, today at lunch.'\n\nI felt her tense: the weight of flesh grew a little less on my arm.\n\n'Now I think you are teasing me.'\n\n'Really? How long has your launch been moored over there?'\n\n'One hour, two. How would I know? I am beginning to feel it is too long.'\n\n'Where does the lane lead?'\n\n'You must ask a map.' She broke from me quickly and got to her feet. 'This talk of lunch makes me hungry again, Monsieur. It is sad, but I fear our game is over.'\n\nI didn't budge. 'Au r'voir, Madame.'\n\nShe paused to give me a sharp stare. Then she tossed her hair with superb disdain and set off for the staithe. She didn't look back.\n\n## CHAPTER SEVEN\n\nTHE LAUNCH LEFT; I watched it make its turn and go creaming away up the Broad; then I sat beneath the hawthorn a few minutes longer, moodily sorting out my results.\n\nThey were not encouraging.\n\nIn the first place, I couldn't link Peter Robinson to the crime. He had turned up _a propos_ , giving a false name and address, but otherwise he wasn't implicated. True, I had made a pass with him at Madame Deslauriers and seemed to have got a small bite; but it was a very small one, and the reaction may not have been due to Peter Robinson.\n\nBringing me to the second place. If Madame Deslauriers had a secret, it didn't necessarily link with the crime either. In fact it probably didn't, because she had no motive: Quarles had been no obstruction to her. Her secret, if she had one, was probably a lover whom she felt it injudicious to produce at this moment: whether Peter Robinson or another villain who might come gratefully to our hand.\n\nAll very semi-innocent. And yet . . .\n\nI rose and went back to stare at the lane.\n\nIt was such an _excessively_ discreet place for a rendezvous. You would almost say it would be wasted on a pair of lovers.\n\nI got over the stile and continued to the bend. Beyond it the lane entered a plantation; then it stretched away between ranks of wild parsley to meet a minor road a quarter of a mile distant. The surface was dried, rutted mud, and the straggling parsley suggested little use. But here and there a frond was broken, and the damaged leafage had not yet shrivelled. A car? A car must have turned. The only place for that would be the plantation. I checked back till I found a gap between trees, then the plain marks of wheels in grassed leaf-mould. I followed them. They entered the plantation; stopped and criss-crossed in a little clearing. Here the car had parked, out of sight from the lane, the precise spot shown by the deeper indentations. I prowled around. Cores, apple-peelings; screwed-up wrappings from chocolate biscuits. Fresh: the peel hadn't browned, the wrappings had taken no damp from the ground. The car-tracks were unidentifiable, but the car had not been a large one, credibly a Viva. And along with the tracks were a number of footprints: these similarly unidentifiable.\n\nSo what more had I now?\n\nA small matter of confirmation: that Mimi was in contact with a person unknown; and whom she wanted to keep unknown.\n\nAnd whom she was probably dashing back to warn by phone.\n\nI handed in the runabout, collected the Lotus and drove through rush-hour traffic to Norchester. I found Hanson in his office; he was drinking beer and eating fish-and-chips from a newspaper package. I went over my facts. Hanson listened, scowling.\n\n'That lane would be Sallowes way,' he said. 'Are you saying there's a chummie hiding out there?'\n\n'It's a possibility. And he could be the man who stayed at the Three Tuns on Thursday.'\n\n'You think he's the killer?'\n\n'We don't know that. We do know he's in contact with Deslauriers.'\n\nHanson worried a chip. 'I still fancy Rampant. I wish I could believe he's a brilliant liar.'\n\nHe fetched a map and we found the lane. It connected with a back road between Sallowes and Wrackstead. By water about two miles from Haughton, by road nearer seven, when you knew the way. In the vicinity were two farms and a scattered handful of farm cottages; Sallowes village was two miles one way, Wrackstead village four miles in the other.\n\n'Is there a pub at Sallowes?'\n\n'Yeah, The Peal of Bells.'\n\nHanson reached for the phone and talked to the switchboard. Two minutes later he was connected; they had had no guests at The Peal of Bells.\n\n'Any guest-houses? Private lodgings?'\n\n'There's nothing of that sort at Sallowes. A bit of housing development, mostly commuters. Perhaps chummie is camping in a field.'\n\n'He'll be close to a telephone.'\n\n'Well, that should help. I'll ask the County to do some checking. Only if he isn't the chummie with the blue Viva, how are we going to know him when we find him?'\n\nA good question.\n\n'He'll have been around since Friday, possibly all the preceding week. A man on his own, no apparent business. Most likely from London or that direction.'\n\nHanson hefted a shoulder. 'So we'll look. But it could be Timmy from Timbuctoo.' He ate a few chips. 'Meanwhile there's Rampant. You haven't got closer than him yet.'\n\nI used Hanson's phone to ring Dainty. Dainty had a tale of woe to tell. He had just missed laying hands on Fring at the staked-out house in Battersea. At about 2 p.m. a Ford Zephyr came by with a driver resembling Fring. It had slowed, pulled in, then departed in haste, the driver obviously having smelled a rat. Alarms and excursions. They had found the Zephyr (it was stolen) across the river in Chelsea, but no Fring, no money; and now the stake-out had been blown.\n\nI made sympathetic noises. 'What about our Peter Robinson?'\n\nDainty sounded less than interested. 'You have to admit your description is vague.' I was getting that reaction from everywhere.\n\n'This chummie has been missing from his usual haunts.'\n\n'So have half the chummies we know.'\n\n'The description would fit someone like Jack Straker.'\n\n'Straker's away. Hadn't you heard?'\n\nI passed on my little bit of information about Quarles' deposits in a Swiss bank. That didn't cheer Dainty either: but I hadn't supposed it would. He came back with something else.\n\n'We found Quarles' will in his safe deposit.'\n\n'He left a will?'\n\n'It's dated last August. It leaves his whole estate to Mimi Deslauriers.'\n\nI chewed that over as I drove back to Haughton. It had a chilling sort of ring to it. By her own account, Mimi was a rich woman, but her account was all I had. And even if it were true, this was motive. The rich are not averse to becoming richer. Nor must I forget that previous occasion when a man had died to Mimi's profit.\n\nA second shake with the same dice?\n\nBut that would mean she had known about the will. Quarles, ex-lawyer, master-criminal, would surely have kept his counsel in a matter so sensitive. And supposing she had known: then I had still to construe the crime as a plot devised by her, whereas the principal circumstances were arranged by Quarles, and the murder apparently a piece of opportunism. To make it credible, one would have to assume communication and conspiracy between her and Rampant: not to mention the shadowy Peter Robinson, necessary if Rampant jibbed at the killing. Possible, but highly unlikely: it would have left her at the mercy of two con-federates. Mimi was much too _au fait_ for that. A simple jostle at a tube station would have served her better.\n\nAnd yet . . . Quarles must have been worth a great deal of money.\n\nIf it hadn't been Mimi, then perhaps a secondary operator?\n\nFor example, Peter Robinson, with a hold on Mimi, working through her to net Freddy's jackpot . . . ?\n\nI shook my head: this was thinking like Hanson \u2013 trying to angle it away from Mimi! Mimi, who had no need to murder anyone, who could do it all with the drop of a bra. Not practical thinking. Mimi could kill, perhaps had blood on her hands already. The field was open.\n\nAnd now I knew of one lode-stone that could have applied a fatal deflection.\n\nI parked in the yard at the Barge-House and went in to confer with Dutt. Dutt was refreshing himself in the lounge, where Mimi, with a group of admirers, was also installed. She favoured me with a vivacious wave and a cooing 'Hallo!' \u2013 which I acknowledged with a dead-bat nod; her appetite, officially unlunched, appeared to have been satisfied with toast and jam.\n\nI joined Dutt, who was sitting alone and looking every inch a copper. A waiter, not Bavents, came up and took my order for tea and toast.\n\nDutt nodded towards Mimi. 'I see you clicked, sir.'\n\nI grunted. 'And what have you been up to?'\n\nDutt looked sly. 'There's a little maid called Nancy. We spent quite a time going over her statement.'\n\n'And what did you get \u2013 in the way of business?'\n\n'In the way of business, not very much, sir. But the head-waiter, Colby, remembered something.'\n\n'Save him till after I've had my cuppa.'\n\nI drank and ate, while up the lounge Mimi continued to glamorize the peasants. She was clever with it: she talked to the wives and left the husbands to drink her in. She had changed out of the shortie dress she'd been wearing and put on a clinging gown with a split skirt. Most of one clamorous leg was on view, and though the bust was now harnessed, it was cleft to infinity.\n\n'She does fetch them in, sir,' Dutt murmured wistfully.\n\nI crunched some toast. 'You keep your heart for Nancy.'\n\n'Yes, sir. But you can't help admiring it. I reckon you admire the mostest in anything.'\n\nI finished, and lit my unromantic pipe. 'Now, if we can, let's get back to Colby.'\n\nDutt sighed and dragged his eyes away. He cleared his throat, trying to sound like business.\n\n'Colby is the big, bald-headed man, sir. I got him remembering about last Thursday. How the deceased and the lady went out in a launch with a couple called Silverman, man and wife. They came back again about four-ish and sat in the lounge, like now. Then, after dinner, Colby went for a drink and remained in the bar for half-an-hour. He says Quarles was in there along with the Silvermans, but he doesn't recall seeing the lady.'\n\n'He could scarcely have overlooked her.'\n\n'It seems the bar was pretty full, sir. Colby was sitting with a mate in a corner.'\n\n'And everyone else would have been sitting round Mimi.'\n\n'So there it is, sir. She was missing.'\n\nBut missing where?\n\n'What time was this?'\n\n'Colby says from nine till half-past.'\n\n'Did anyone else see her during that time?'\n\n'Nobody I've had a talk with yet, sir.'\n\nI puffed expansively. It was fitting all right. At eight Peter Robinson had arrived at the Three Tuns. Had booked in and gone out, say at eight-thirty. Half-an-hour to contact Mimi. How had he done it?\n\n'Are any of the staff very friendly with the lady?'\n\n'Reckon all of them are, sir. The men especially.'\n\n'These young waiters. Is there one with a crush?'\n\nDutt looked blank. 'As much one as another, sir. Though I did hear of one she sort of makes use of, gets to run errands, fetch things to her room.'\n\n'Who?'\n\n'The Bavents kid, sir. But he was off this afternoon. I haven't talked to him yet.'\n\nDutt was using the reception office for interrogation, and there I had Bavents brought when he returned. He was looking even furrier in a T-shirt and jeans: like a narrow-faced Jesus fresh back from the wilderness. I pointed to a chair and he sat nervously. I had his statement to Hanson on the desk before me.\n\n'You are Adam Bavents?'\n\n'That's me.'\n\n'I see it says here that you are a student.'\n\nBavents flicked back a lock from his nose. 'I told the other man all about that.'\n\n'Now tell me.'\n\n'So that's w-what I am, then. A third-year student at Norchester U.'\n\n'If you are a student, what are you doing here?'\n\n'I'm just filling in time till next term.'\n\nOh yes. 'And why is that?'\n\nHe jerked a look over his shoulder. 'I got sent down. It isn't a secret. They said I was ring-leader of a demo.'\n\n'And were you?'\n\n'I might have been. I'm not ashamed of it. They were trying to sack a tutor for speaking out against racialism.'\n\n'Wasn't that when the students smashed up a lecture-hall?'\n\nHe stared through his hair. 'We had to make our point.'\n\n'But by violence.'\n\n'If you call that violence.'\n\nI nodded. 'Yes, I call that violence. And violence is what I have come here about. So I seem to have reached the right quarter.'\n\nHis tresses rustled. 'But that's just talk! I don't know anything about the other.'\n\n'But about anti-racialism you know something. Tell me, what are your feelings towards the French?'\n\nA sweaty silence. His hair fondled the T-shirt, showed his nose through a Gothic window. A pink nose: and pink hands rucking the fade-spots in his jeans. Then his mouth loosened.\n\n'I didn't kill him!'\n\n'Fine. What happened on Thursday evening?'\n\n'Th-Thursday?'\n\n'In the evening. When the men wanted a word with Madame Deslauriers.'\n\n'I \u2013 I\u2014'\n\n'Where were you that evening?'\n\n'I \u2013 I was w-working on my car!'\n\n'You have a car?'\n\n'Yes! A Mini\u2014'\n\n'And you were working on it \u2013 in the yard?'\n\n'Yes, but\u2014'\n\n'You were handy then. Handy for this man coming into the yard. Who wanted a message slipped to Madame Deslauriers. Fair hair, sideboards. What name did he give?'\n\n'He d-didn't \u2013 I wasn't\u2014'\n\n'Oh come on, now. He was staying at the Tuns. Did you know that?'\n\n'I tell you\u2014'\n\n'Drives a blue Viva. Come on, the name's on the tip of your tongue.'\n\n'But I s-swear\u2014'\n\n'You say you didn't kill Quarles?'\n\n'No! I don't know anything about it!'\n\n'So then let's have the name of this man.'\n\nHe went into a huddle with his hair.\n\n'Listen,' I said. 'You've let something slip. Now I know you ran the message for that man. And if you get nicked on a conspiracy charge you'll be filling in time for longer than a term. So you'd better talk while you still have the chance.'\n\n'I d-don't have anything to tell you.'\n\n'Because you love Madame so much?'\n\n'That isn't t-true!'\n\n'I'll remember to ask her.'\n\nHe jumped up from his chair.\n\n'Hold it,' I said. 'Is this your signature on the statement?'\n\n'Of course it's m-my signature!'\n\n'That surprises me. Just do a specimen underneath.'\n\nHis eyes sparkled through his mane, baffled. Then he grabbed a pen from the desk and jerked off a signature. The same, of course, less a margin for nerves. He slammed down the pen in feeble triumph.\n\n'Now may I go?'\n\n'For the present.'\n\nHe towed his hair out of the office. Dutt, a silent spectator, gave me a wink. I fanned myself with the twice-signed statement.\n\n'An interesting customer.'\n\n'Yes, sir. I'd say that ties in our Peter Robinson.'\n\n'There's something else.'\n\n'The signature, sir?'\n\nI shook my head. 'He's left-handed.'\n\n## CHAPTER EIGHT\n\nLEFT-HANDED; BUT SO is every tenth person, according to a reliable set of statistics; and adding it together, there didn't seem much ground for placing Bavents on the list of suspects. He might have loved Mimi and loathed Quarles, but that scarcely qualified as a live motive. He had no prospect of stepping into Quarles' shoes, and without such bait his interest was marginal.\n\nOr did he have a prospect. . .?\n\nI played with the thought, giving it a chance to attract credibility; trying to visualize his hairy highness as a demon lover for whom Mimi would be content to risk her all. But it wouldn't focus. Mimi was too sophisticated. She had too much emotional poise. She might give him a tumble for the novelty of it, but that would be the summit for Master Bavents. The Quarleses were her taste, suave and tough: men who didn't know how to stutter. The rest were to run and serve: lackeys and go-betweens: Baventses.\n\nWhich didn't mean I had lost interest in Bavents, who certainly hadn't told us all he knew; or that it would be unprofitable to probe there a little, seeking out a perhaps-unsuspected conjunction.\n\nI met Frayling in the hall and invited him into the office.\n\n'How did you come to employ Adam Bavents?'\n\nFrayling flickered me his harassed, ingratiating smile: a promise of satisfaction in exchange for modest patience.\n\n'He applied for the job. I'm always short of waiters.'\n\n'How did he know the job was vacant?'\n\n'Oh, they run an employment section in the students' magazine. It lists details of jobs going in the vacations.'\n\n'You knew why he was sent down?'\n\n'Of course. I asked him. But things like that don't count much these days. He seemed a decent sort of youngster, and I haven't had any complaints.'\n\n'What are his hours?'\n\n'Seven to eight-thirty. Two afternoons and one evening off.\n\n'Which evening?'\n\n'The evening varies.'\n\n'Which was it last week?'\n\nFrayling wriggled. 'Friday.'\n\nOne conjunction.\n\n'Would you know if he went out?'\n\nFrayling's smile became more harassed. 'I imagine he did, that's what one would expect. But he might well have been studying in his room.'\n\n'Where's his room?'\n\n'It's off the back landing. A room we keep free for temporary staff.'\n\n'How close to Madame Deslauriers' room?'\n\nWell . . . next-door, I suppose! But a door shuts-off the landing.'\n\nTwo conjunctions?\n\n'Isn't Bavents Madame's favourite?'\n\n'No, really! That's putting it too strong. He serves at her table, that's all. Guests tend to adopt their regular waiter.'\n\n'But something of that sort?'\n\n'No, I protest. You must have been listening to staff gossip.'\n\n'Wouldn't the staff know?'\n\nBut Frayling still protested, so I let him go to get on with ushering dinner.\n\nAt dinner I had an opportunity of studying Madame and her waiter together. Mimi had one of the best tables, with a view of the river: rather remote from our late-comer's corner. She sat alone, but this didn't prevent her from conversing merrily with her nearest neighbours. Bavents came, went, and did his duty: if anything special passed, I failed to notice it. Had Frayling cautioned him? More than likely. But Frayling could scarcely have cautioned Mimi. Mimi must have taken her own counsel to preserve distance, a circumstance not without significance.\n\nOnce or twice she glanced at our table, but each time managed to avoid my eye. Then she would eat silently for a few moments before engaging in some fresh sally with the neighbours. She was as conscious of me as I of her; her rattle of small talk was a screen; through the subdued busyness of the peopled room a strand of tension stretched between us. Excellent: it gave me appetite. Madame was not so confident after all. I had begun to tread a little on her skirt, might have set it fraying at the edges.\n\nBeside me Dutt ate ploddingly and well, though not without his own eye for the lady. He nudged me once:\n\n'Do you reckon it's all natural?'\n\n'Get on with your dinner.'\n\nHe chortled into his trifle.\n\nWe had coffee on the lawn, from where we could watch late motor-cruisers raising wash over the quay-headings. While we drank I pondered the utility of tackling Mimi then or of letting her sleep on it. On the whole I favoured the latter (it had been a long day); so I went in to ring Brenda: who for the second time surprised me with quite unpredictable information.\n\n'George. I've been talking to Siggy about your corpse.'\n\n'Thank you. But it's still eating hot dinners.'\n\n'Not that one, idiot! Flash Freddy. Did you know he was going to retire?'\n\n'Retire?'\n\n'That's what I said. He'd been talking of giving up business. He'd bought a villa in the South of France, Cap Ferrat way. Hadn't you heard?'\n\n'No, I hadn't heard.'\n\n'Well, it's true, because Siggy borrowed it for a week last summer. He says it's a super place, perched on a cliff, with a private beach and all the etceteras.'\n\n'How nice for Siggy. He knows nice people.'\n\n'George, I think you ought to be grateful. If one of your relatives is chummy with crooks, the least you can do is to profit by it.'\n\nI grunted. John Sigismund Fazakerly is a relative only by marriage. My first act on meeting him was to arrest him, which doesn't make him my favourite in-law.\n\n'What was the retiring bit?'\n\n'Just what I said. Siggy and he were chatting about Riviera properties. About Somerset Maugham, the English set. Freddy said soon, he was going to retire there.'\n\n'Did he mention a date?'\n\n'Stupid. He just said he was getting bored with business. I suppose it can happen to crooks like everyone else. The day comes when it doesn't switch them on. He was rich enough, wasn't he?'\n\n'Oh quite.'\n\n'There you are, then. He wanted to relax. If some imbecile hadn't gone sticking a knife in him, Freddy would soon have been out of your hair.'\n\nA comforting thought.\n\n'Only it isn't quite like that. Crooks don't find it so easy to retire.'\n\n'Why not?'\n\n'They tend to have disapproving associates \u2013 men who can make their point with a knife.'\n\n'Hah.' She was silent for a moment. 'Are you saying that's what happened to Freddy?'\n\n'I wish I knew. But what you've told me does suggest the possibility.'\n\nA further silence. 'That's disappointing.'\n\n'Why?'\n\n'Because I've got a bet on with Siggy. A fiver on Mimi Deslauriers' nose. Siggy's fiver is on Rampant.'\n\n'You could both be wrong.'\n\n'Not me. Never. Don't you remember my intuition?'\n\n'Not as a viable force.'\n\n'Nuts to you. Just remember to keep your eye on Mimi.'\n\nFollowed a slightly more pregnant silence.\n\n'How are you doing with her?'\n\n'You could say we understand each other.'\n\n'Pig! Is she making a play for you?'\n\n'That wouldn't single me out in a crowd.'\n\nBrenda made ferocious noises. 'You listen, George Gently! That woman'll be poison if she gets you to bed with her. She'll have you doing somersaults to keep her out of it. And then bang will go my fiver.'\n\n'Why make these rash bets?'\n\n'Do you hear me talking to you? Just take a tip from someone who knows.'\n\n'I'll keep it in mind when the lights go out. What else did Siggy have to tell you?'\n\nBrenda fumed again. 'I'm not sure I'll tell you. I wouldn't if I thought it was at all important. It's just that Freddy mentioned some other properties he owned, two in Scotland and one on the Broads.'\n\n'What!'\n\n'You needn't get excited. He didn't tell Siggy where they were. I expect they're little bolt-holes where he could lie low when people like you were being unfriendly.'\n\n'Was there any description?'\n\n'No, there wasn't. Only exactly what I've told you.'\n\n'A cottage \u2013 bungalow?'\n\n'Three properties. Think about that when the lights go out.'\n\nI got from her the address of the Cap Ferrat villa, then sat for a while, my hand on the phone. But first things first. I rang H.Q., who gave me Hanson's private number.\n\n'Any progress re Peter Robinson?'\n\n'Hell, I'm watching a programme,' Hanson said indignantly. 'There's nothing in.'\n\n'Then listen to this. I've had a tip that Quarles owned property on the Broads.'\n\n'What sort of property?'\n\n'That's the bonus question. But it will be one only occasionally inhabited. Owned by a Londoner, sure to have a phone, probably in a remote situation.'\n\n'But the Broads are lousy with properties like that. Three parts of the bungalows are owned by foreigners.'\n\n'This one will have Peter Robinson living in it. And a blue Viva in the garage.'\n\n'You reckon so?'\n\n'Look at the facts. He booked in at the Three Tuns for an indefinite stay. Then he made contact with Deslauriers, that's been established, and checked out of the Three Tuns, but remained in the district. So where else?'\n\n'You think she hid him there?'\n\n'Can you suggest a better prospect? If the hideaway was good enough for Quarles, then it would surely be good enough for Peter Robinson.'\n\n'So then maybe it'll take a bit of finding.'\n\n'We have one fix. It's handy for Sallowes. That suggests it's on the south side of the river, because Haughton's the only crossing for miles.'\n\n'That still leaves it open. It could be on the South River.'\n\n'Either way, we have to find it. I want County to treat this as a matter of urgency. Picking up Robinson is top priority.'\n\n'You want immediate action.'\n\n'I want that.'\n\nI heard Hanson sigh. 'Will do.'\n\nThen I rang London, where after a ten-minute delay they connected me with Dainty. Similarities in background noise suggested he had been occupied in the same way as Hanson. I told him about Quarles' villa, about Quarles' talk of retiring.\n\n'Who do we know who would take that to heart?'\n\nDainty hedged. 'I could name you a dozen. Quarles has master-minded for several of the gangs as well as running his own mob. He could put away some top villains and none of them would stop at having him done.'\n\n'But you can narrow that down.'\n\n'Do you think I haven't been trying? I've had whispers about O'Leary and Whitey Ferrier. I have men out now trying to get the dope on them. We haven't been sitting on our hands all day.'\n\n'Have you had another session with Quarles' boys?'\n\n'Don't make me laugh. They've clammed solid.'\n\n'You could let it out that Fring was playing them double, was hooked in with other interests.'\n\n'It's been tried. They were there. They're not going to believe that Fring planned his skarper.'\n\n'A few days in the cells would give them time to wonder.'\n\n'Have it your way. But they haven't talked yet.'\n\nI shrugged to myself. 'Let's get to Fring, then. How sure was the identification today?'\n\nDainty hesitated. 'Are you questioning that?'\n\n'I'm thinking you won't be the only ones looking for him. The story has been blown for two days. A man on the loose with thirty-five grand. Fring might have come cruising by his house today, but so might some other interested parties.'\n\n'Sergeant Dymock thought it was Fring.'\n\n'Has Dyrnock met him?'\n\n'He's studied photographs.'\n\n'Then there's room for error. Fring may be still with us, or he may be catching a tan on a private beach.'\n\nI heard Dainty suck breath. 'Quarles' villa?'\n\n'It's only a short flight from Heathrow. And today's was your only sighting of Fring. Unless you're positive, I'd say he's skipped.'\n\n'You could be right. We've had no whispers.'\n\n'Which could also mean he's floating down the Thames.'\n\n'I like the first idea best. You'd better give me particulars of the villa.'\n\nI dictated them, against background music and a voice that sounded like Harry Corbett's.\n\n'Fine,' Dainty said. 'I'll ring Interpol. How are things shaping at your end?'\n\n'Packed with psychological interest.'\n\nHe gave a dirty laugh. 'I've heard about her. When do you get to the ooh la la?'\n\nI hung up: but sat a little longer, very silent, on the padded desk-chair. From across the hall was coming the murmur of the bar, but that wasn't the only sound I had been hearing. Now I rose quietly and moved to the door. Nothing to see through the glass panels. I whipped open the door and stepped out. Bavents was pressed against the wall alongside.\n\n'Were you able to hear, then?'\n\nHe started away from me, but I was blocking his retreat down the hall. He stood frightened-eyed, breathing quickly, looking oddly Victorian in his waiter's tails.\n\n'I \u2013 I wasn't listening!'\n\n'Pull the other one.'\n\n'I was w-waiting to use the phone.'\n\n'What's wrong with the call-box?'\n\n'I don't have any change!'\n\n'But you do get free use of the office phone?'\n\nBavents was trembling. Just then, however, some people came out of the bar behind him. I had to make way for them, way for Bavents. He grabbed the opportunity and went.\n\nNot that it mattered. As Joe Louis once said, they can run but they can't hide.\n\n## CHAPTER NINE\n\nBUT THIS INCIDENT settled one thing: my session with Mimi could not now wait till morning. I didn't know whether or not she had set Bavents to spy on me, but there was a strong possibility that she would get his report. I went into the bar. Drinkers were grouped round the piano, at which someone sat playing _La Mer_. I pushed up till I could see the pianist: it was Mimi, of course: her Parisian thing. She was playing _La Mer_ with a languorous unction and no mean skill in the fingering, evoking, with poised, resonant octaves, an American's idea of the mood of the boulevards. It was musical ham, but being played with a conviction that stripped the tinsel from the clich\u00e9. And the group of drinkers hummed it with her, swaying slowly to the wave-like rhythm.\n\nI eased my way to the piano and found a corner on which to lean my elbow. Mimi's eyes connected with mine: she hooded them provocatively and leaned towards me. She began singing in French. Her hoarse contralto was thrillingly suited to the nostalgic melody. It came over powerfully, and was sensuously supported by the humming of the group. The rest of the bar was still and listening. Even the bar-tender had interrupted his serving. He stood with a glass in his hand, the other hand on a pump-handle, and his eyes intent upon the singer. Mimi had created something. With a pub-piano, she had built an experience with an echo.\n\nThe song ended; but ignoring applause, she blended the signature notes into _Clopin Clopant_. The transition was faultless, and instead of singing this one she hummed it with a sort of affectionate abandon. Everyone joined her except the bar-tender and me. Nobody now was sitting down the room. They were clustering together with linked arms, swaying and crooning around Mimi. Occasionally she would throw in a few vibrant words, as though the music were recalling to her some blissful memory; and all the time her eyes remained linked to mine, making me an audience of one.\n\n_Clopin Clopant_ finished in a volley of clapping and raucous pleas for more, but this time Mimi rose from the piano, laughing, and picked up a glass that was standing on it.\n\n'That is all, my friends. I have business.'\n\nShe pointedly raised the glass to me. I said nothing; she tossed off the drink; it was she who led me through the disappointed customers.\n\nWe went out into the hall and she laid her hand on my arm.\n\n'So. Shall we go to your room, or mine?'\n\nI shook my head. 'This isn't social.'\n\n'Not social?'\n\n'No. I have some questions to ask you.'\n\nShe gestured carelessly. 'What else? But mustn't we be comfortable while you're asking them?'\n\n'Not so comfortable as you're suggesting.'\n\n'But, my friend, that's a matter of taste. If you don't want to make love to me, foof, foof. But at least, let us discuss it sitting on a bed.'\n\n'We'll discuss it in the office.'\n\n'I don't like the office. To begin with, there is not even a couch. Then there are windows in the door and at the counter. We may as well go back into the bar.'\n\n'At the moment the office happens to be my office.'\n\n'Then I do not admire your taste.'\n\n'I am filled with regret.'\n\n'Huh-huh. How does one fill you with something else?'\n\nI opened the office door and she went in distastefully. But then she noticed the curtains for the glass panels.\n\n'Aha. This is not so bad. All it needs now is a soft mattress.'\n\n'Kindly wait here while I fetch my Inspector.'\n\n'But, my friend, for what do we need him?'\n\n'Let us just say to preserve the proprieties.'\n\nShe put out her tongue and gave me a V-sign.\n\nI collected Dutt. When we returned, Mimi had drawn the office curtains. She was sitting on a chair turned back-to-front so that her skirt was pushed up to her crotch. She had her arms folded on the back of the chair and was staring maenad-eyed at space. She remained so while we took our seats and while I was leafing through her statement.\n\n'Madame Deslauriers?'\n\n'Uhuh?'\n\n'It would be nice to have your attention.'\n\nShe hoisted her shoulders. 'I didn't get yours. Why do you expect me to give you mine?'\n\n'Doesn't the death of your friend matter to you?'\n\n'Can you bring him back to life?'\n\n'I can perhaps discover who killed him. Or don't you really want me to do that?'\n\nShe swivelled the chair heavily. 'Monsieur, what good will it do? If this little Rampant killed Freddy, no doubt he is sorry enough now.'\n\n'You don't think he should be punished?'\n\n'He is punished already. He was thinking that Freddy would make him rich. But \u2013 pfft! \u2013 the job went sour. There was no money for little Rampant.'\n\n'So he was justified in killing Freddy?'\n\n'I do not know about justifications. But he is punished twice, because if there is no Freddy there are no more jobs to do for Freddy. Also, it is in part Freddy's fault. He should not have taken such risks in dealing with Rampant. He should not have made him blow his top, huh? I think that Freddy was a lot to blame.'\n\n'Did Freddy have a temper?'\n\n'Who has not? And it was all the worse because he controlled it. A lawyer, you see. He did it with words. Ah, I can well understand what happened.'\n\n'You are saying he provoked Rampant.'\n\n'But yes. He was very angry after Rampant rang him. He lay on his bed up there, brooding, planning all he is going to say to him. At the time I am thinking this is perhaps not wise, better let a hard boy handle it for him. But I could not advise Freddy. My wisdom was not to interfere.'\n\n'And this was his mood when he set out.'\n\nShe nodded. 'And the rest I understand so well. He skinned this little man, this Rampant; his words were like claws into his brain. He was going to crush him, ha, ha. This Rampant will never cross him again. But they are all alone there. Nobody about. And little Rampant has a knife.'\n\n'You could foresee that.'\n\n'Trouble I saw. I couldn't know it would be so bad.'\n\n'You feared violence.'\n\n'Yes, violence. Freddy's mood is very black.'\n\n'And so he went out on this dangerous mission. He went out, and he didn't come back.'\n\n'Exactly so.'\n\n'Yet you raised no alarm. Why was that, Madame Deslauriers?'\n\nShe was still for perhaps two seconds, gazing emptily into nothing. Then she pulled back on the chair in cowboy style and smiled up into my face.\n\n'A trap, huh, pardner?'\n\n'I would like an answer to my question.'\n\n'I think it is a pity you couldn't have questioned Freddy. My God! That would have been a treat. Do you know, in a way you are reminding me of him?'\n\n'You may have time to think, if you wish.'\n\n'He would have said that. He would have been thinking of ways to put me down, make me say what he wanted. Isn't that bizarre?'\n\nI shook my head.\n\n'And you a policeman, he a thief. Uhuh, what is the difference? It is just two teams who play one game.'\n\n'No, Madame Deslauriers.'\n\n'You do not like my paradox?'\n\n'I am afraid this won't do. Either you have your cake or eat it. If you don't decide, I shall.'\n\n'Cake? What cake is that?'\n\n'The cake is your ignorance of the risk that Freddy was taking. That could explain why you raised no alarm. But it excludes you from persuading me that Rampant was the killer.'\n\nShe gurgled throatily. 'You think I try to do that?'\n\n'I think you have been trying for the past ten minutes.'\n\n'But it is logical, my friend. And you must admit, convenient. It is not insulting the credibilities.'\n\n'Then why no alarm?'\n\n'Oh foof. Perhaps I am giving it too much drama. I was uneasy, yes, but not too worried. This sort of thing has happened before. Freddy goes away for one, two days. If anyone asks me, I have an excuse. He has gone to view property in Wales, in Cornwall. Freddy was fond of little deals in property.'\n\n'He owned properties?'\n\n'Oh yes, one or two.'\n\n'In Wales, in Cornwall?'\n\nShe pulled a glum face. 'Haven't I been telling you that Freddy was secretive? I do not know where his little pieces are.'\n\n'Buying property is legal. Wouldn't he have taken you to visit them?'\n\n'No. His business was not my affair.'\n\n'He just bought them and forgot them?'\n\n'Oh, Freddy forgot nothing. Especially when not to open his mouth.'\n\n'You begin to make me think it's catching,' I said. 'Freddy had property in Scotland. Also a nice villa at Cap Ferrat.'\n\n'The villa, oh yes, the villa. I was thinking you meant in this country.'\n\n'Also in this country. We know of another property.' She drooped her mouth and humped her shoulders.\n\n'Would you like to know where?'\n\n'Should it matter to me?'\n\n'At present it has an occupant.'\n\n'Ah. Freddy let it.'\n\n'I didn't say that. I doubt if this occupant pays any rent.'\n\nShe looked askew for a moment. Then she sighed sadly. 'My friend, I can guess what you are trying to tell me.'\n\n'I was sure you could.'\n\n'But I am quite resigned. I have long felt there was another woman.'\n\n'Another woman!'\n\n'But yes. It is not a thing a man can hide. Not even Freddy. I had intimations, you know? The way he was to me in bed.'\n\n'I am not referring to another woman!'\n\n'You try to spare me. You are so kind. But, my friend, it doesn't matter now. All jealousy was over when Freddy died.'\n\nFor a space I was silent. I couldn't help it; I had to admire that splendid foil. With style of such an order it was no wonder that Mimi had triumphed over the machinations of her mother-in-law. Nor did she rub it in; she sat mournfully glum, as though bravely accepting her sad thoughts. Not a flicker or gleam in her downcast eye. Nothing to give scepticism a chance.\n\nI threw a look at Dutt: he was studying his notebook.\n\n'Very well,' I said. 'We'll come back to that later.'\n\n'But Monsieur, her name?'\n\n'Never mind her name! What I'd like to talk about now is Bavents.'\n\nHer surprise was perfect. 'The waiter?'\n\n'Indeed yes. The waiter. Who sleeps in a room next to yours. Who blushes when your name is mentioned.'\n\nShe gave a little chuckle. 'His name is Adam.'\n\n'I am well aware his name is Adam.'\n\n'He is a dear. A furry animal. I am quite fond of my Adam.'\n\n'So he is your Adam?'\n\n'Am I not saying so? He is feeding out of this hand. It is not a bad idea to enchant the waiter. It ensures the service is dependable.'\n\n'And precisely how far does the enchantment go?'\n\nShe did her smiling pull-back on the chair. 'Do not bother to be jealous, my friend. I do not admit Chi-Chi between the sheets. One would need to be butch, huh? All that hair falling over one's face. Oh no. He is for those little thin girls with their drain-pipe bodies and sparrow's legs.'\n\n'But still, the enchantment is pretty strong.'\n\nShe gestured. 'Each one has his talent.'\n\n'He would run little errands and keep his mouth shut.'\n\n'How else could he be of service to a lady?'\n\nI drew closer to her. 'And on Thursday evening. Didn't he run a little errand then?'\n\n'An errand for me?'\n\n'An errand to you. From someone requesting a private interview.'\n\nShe pretended to think. 'It is not easy to remember. All sorts of funny things go on. One half of the gentlemen staying here keep hinting that they would like a private interview. What would I be doing on Thursday evening?'\n\n'At first you were in the bar with Freddy and the Silvermans.'\n\n'Ah yes, the Silvermans. Tiresome people. Freddy liked them because they were in racing.'\n\n'But then Bavents entered and attracted your attention.'\n\nHer eyebrows lifted. 'You are telling me this?'\n\n'Yes, I'm telling you. And we have a witness. Bavents brought you a message. You went out.'\n\nShe swished her hair. 'That is very likely. Yes, I think it may have happened.' She hesitated. 'On Thursday, was it? I wonder what it could have been about?'\n\n'I am sure you know very well what it was about.'\n\n'But no. You had better jog my memory.'\n\n'When you were gone from the bar for over half-an-hour?'\n\n'As long as that?' She made a mouth. 'Where was I, then?'\n\n'You went out in the yard. The man who sent the message was waiting there. You took him aside, perhaps behind those garages, so that you could talk without being seen.'\n\n'Does Adam say this?'\n\n'You had got rid of Adam. Nobody was to hear that conversation. The man had just arrived, he had booked at the Three Tuns. You had to fix him up with a less obvious address.'\n\n'Oh yes, that's certain! I put him under my bed.'\n\n'In point of fact you probably gave him a key.'\n\n'A bedroom key?'\n\n'The key of a front door. With instructions where the door was to be found.'\n\nShe shook her head. 'A strange story, my friend. It would surely need proof to make it stand up?'\n\n'Proof \u2013 like a statement from the man involved?'\n\nHer eyes flashed quickly, were calm again. 'That would be his word against mine, huh? And how could the word of this man be trusted?'\n\n'You think it couldn't be?'\n\n'Me, I know nothing. These are all your ideas, Monsieur. And I think I hear Freddy say to me, Ignore him, my dear, this is just a try-on.'\n\n'Do you wish to gamble on that?'\n\nHer eyes were hard: then they smiled. 'Yes. I call you.'\n\n'Freddy owned some property just down the river. That was where the door-key fitted.'\n\nOur eyes locked. For a short second I was gazing into the eyes of a furious animal. It passed; she gave her breathless chuckle, dredging it up from deep down.\n\n'I still call you, huh? What do I know of Freddy's properties? And now, if our interesting conversation is to continue, you must be a gentleman and order drinks.'\n\n## CHAPTER TEN\n\nI ORDERED DRINKS. Frayling brought them himself; he was doubtless abreast of affairs in the office. He received his reward, if that is what it was, in a melting smile from Madame Deslauriers. He seemed embarrassed. He slopped Dutt's brown ale and got himself tangled with a chair; then he backed out, ducking and grinning. You would almost have taken him for a hot suspect.\n\nMimi was smiling creamily to herself.\n\nI nodded after Frayling. 'Do you like him better than Bavents?'\n\n'Would it show poor judgement if I did?'\n\n'He has a wife. It could be stupid.'\n\n'Ha, ha, a wife.' She sipped her Martini.\n\n'Perhaps wives don't count in your book.'\n\n'But yes, my friend. I find wives interesting. They are often more bored than the husbands. Sometimes I can put a man in their path.'\n\n'I am sure you favour advanced methods.'\n\n'I prefer not to create a vacuum, huh? After all, I am a woman too. I have every wife's interest at heart.'\n\nI gulped lager. 'And the system works?'\n\nShe skewed her mouth. 'Of course, there are failures. Wives who have been paralysed by their one success. These you will find in the divorce courts. But mostly, no. They are simply bored. The big adventure has become routine. And when there are also so many bored husbands, it would be a shame not to spread some happiness. That is logical?'\n\n'It overlooks a few factors.'\n\n'I talk of adventuring, my friend. There is also the grand passion, huh? The love that is proof against all adventures? But that is enfranchising, it does not imprison. The love that imprisons is mere possession. The grand passion has open doors by which we go out and may return.'\n\n'And that was the state of affairs with Freddy?'\n\nShe swung her shoulders. 'You know it was not. He was a fascinating man, a tender lover, a good friend. Not more.'\n\n'With your husband, then?'\n\n'Alas no. My marriage to Charles was arranged. He was very charming, but a man of business. I am just his pretty wife waiting on the doorstep.'\n\n'So with whom?'\n\n'Perhaps I am waiting.'\n\n'Or perhaps it is just something you've read in books.'\n\nShe drank. 'You are unkind, Monsieur. Why should I tell you all my secrets?'\n\n'Then there is such a man?'\n\n'That is always possible.'\n\n'A man who would undertake grave risks.'\n\n'Maybe.'\n\n'And who is not far away.'\n\nShe gazed at me fixedly. 'He may be as close as that chair you are sitting on.'\n\nI grunted and drank more lager. She lifted her glass to me and sipped.\n\nIt was growing dark now. I signalled to Dutt, who rose and switched on the light.\n\n'Let us get to something more interesting,' I said. 'When was Freddy proposing to retire?'\n\nMimi, Madame Deslauriers, had changed her perch on the chair. It could have been that she had judged I had seen enough of her legs. She looked beautifully blank.\n\n'Who says he was retiring?'\n\n'One John Sigismund Fazakerly says it.'\n\n'Oh . . . Siggy.'\n\n'Who you obviously know.'\n\n'But yes.' She faded in a dreamy smile. 'He is the yachtsman who lives in Chelsea. Freddy and he were good friends. Very good-looking, quite rich, but with strange ideas about women.'\n\n'And he says that Freddy was talking about retiring.'\n\n'Aha. But why is that so interesting?'\n\n'Because crooks who retire are a problem to other crooks. Who sometimes make fundamental adjustments.'\n\nShe looked studiedly dumb. 'You think that may have happened?'\n\n'First, I would like to hear of his plans for retirement. When; who knew about it; what he had arranged about his mob.'\n\n'But, my friend, you ask the wrong person.'\n\nI shook my head. 'I think you would know.'\n\n'It is a matter of business . . .'\n\n'I am not accepting that. My guess is that you were Freddy's reason.'\n\nShe smirked faintly. 'You are obstinate, Monsieur. But it comes to the same thing in the end. I do not know what Freddy has planned, or all these other things you are asking.'\n\n'But you knew he was retiring?'\n\n'Very well. It is something we had discussed. After all, Freddy had made his pile. With him, it was getting to the point of being pure art.'\n\n'When did you discuss it?'\n\n'Oh, many times. I had always made my position clear. I did not like Freddy for ever taking such risks. Also, I wished to return to France.'\n\n'So it was largely on your account he bought the villa?'\n\n'But yes. And that was two years ago.'\n\n'He bought it to retire to?'\n\n'That was certainly in his mind. We have never talked of living anywhere else.'\n\n'And you, of course, never urged him to fix a date.'\n\nShe flicked her hand. 'Just when he was ready. A little influence, perhaps: I too am a woman. But I am not so na\u00efve as to nag.'\n\n'This summer? The autumn?'\n\n'I think this year. Earlier or later I cannot say.'\n\n'Cannot?'\n\nHer shoulders moved; she tinked the rim of her glass with her nail.\n\n'All right,' I said. 'But this you can tell me. Who did Freddy use to invite to his villa?'\n\nShe shifted pose slightly. 'All sorts of people! Once, I believe, he lent it to Siggy.'\n\n'People from London?'\n\n'Not so often. He preferred to mix with the English who live down there. People with money, cars, yachts. Freddy had the style for it. He was liked.'\n\n'Respectable people.'\n\n'Oh yes. Down there he is just another rich man. He is creating an image, you understand. For when all the business is behind him.'\n\n'Yet sometimes he would take acquaintances from London.'\n\n'Well, he knew honest people there, too.'\n\n'Not honest people.'\n\n'He would not take crooks.'\n\n'For example, associates. Like Wicken. Or Fring.'\n\n'Fring?'\n\n'Jimmy Fring.'\n\nShe hung on momentarily, her eyes searching. Then she chuckled. 'Ha! Now I understand. It is Jimmy who has the money, huh?'\n\n'You know Fring?'\n\nHer eyes sparkled. 'Of course. I have met them all at some time. Jimmy is a funny one. He is well-educated. He wrote me a poem on a bank-note.'\n\n'And he has been to the villa?'\n\n'I do not know that. No, I should think it is unlikely.'\n\n'Why wouldn't you know?'\n\n'Why should I? I am only at the villa now and then.'\n\nI stared hard at her. 'You'd know, Madame Deslauriers, because the villa was bought on your account. You were the hostess there. It is very improbable that people would visit it without your knowledge.'\n\nShe pouted. 'Very well, then. I say no. None of the boys have been to the villa.'\n\n'Not Fring.'\n\n'Not any of them. They perhaps do not know that the villa exists.'\n\nAnd she looked me straight in the face with frank eyes. The liar.\n\nI took her through her statement. Either it was true or she had the memory of the devil. Nothing was added to that beguiling sketch of innocent crookery gone astray. Quarles had received the message from Rampant; he had majestically programmed and mounted the crime; then disaster had struck. The gang had been shopped, and Quarles slain in a quarrel with Rampant. Simple and logical. What more did I want? Rampant had been taken with blood on his sleeve. Fring and the loot were adrift, certainly, but that was a loose end to tie up at leisure; while sooner or later, suppose it mattered, I might learn the identity of the squeaker.\n\nSimple and logical! Then why confuse it with Peter Robinsons and that malarkey? Which would only turn out, if I chose to pursue it, as an amorous intrigue of poor Mimi's? Mimi had secrets, very well, but it was only the turn of events that had made them seem sinister.\n\nAnd the more we went over it, the better it sounded. You could feel her poise, her confidence, hardening. This was the picture: but if it wasn't, who was ever going to prove different?\n\nNobody, of course.\n\nSo we talked it over, her cigarette to my pipe.\n\n'It is truly an irony, my friend. Poor Freddy deserved his success.'\n\n'He was a thief. He left behind him a trail of injured guards and bank officers.'\n\nShe issued smoke. 'And that makes him different? Worse than other rich men? The capitalist who steals from the workers and injures them with industrial diseases? No, my friend. Freddy is in step with the moral climate of his culture. His exactions are perhaps less harmful, his initiative more to be admired.'\n\n'An honest thief in a theft-society.'\n\n'That is very nicely put. I am sure you would have liked Freddy. He was a subtle man too.'\n\nI took some puffs. 'I lack his initiative. I am not in the same financial league.'\n\nShe sighed. 'No. He was very successful. I think he was perhaps a millionaire.'\n\n'That means there is a lot of money going loose.'\n\n'Oh, I expect he made arrangements. A lawyer, huh? It is not very often that you catch them with their trousers down.'\n\n'A will?'\n\n'That is almost certain.'\n\n'Did Freddy have any close relatives?'\n\n'Uhuh. He had a married sister.' She gave a throbbing laugh. 'I foresee a moral problem there, my friend.'\n\n'How do you mean?'\n\n'She is the wife of a clergyman, I think you call him a rural dean. I have met her, a great hypocrite. She regarded Freddy as dirt.'\n\n'Perhaps she won't inherit.'\n\n'Oh yes, I think so. It would appeal to Freddy's sense of humour. So what is she going to do, poor lady, with all that dirty money in her lily-white hands?'\n\n'Give it away.'\n\n'Aha. Not so easy! She is one with an eye to the main chance. No, I think she will find a way to double-cross her conscience. After all, she is Freddy's sister.'\n\nI floated a smoke-ring. 'Would the money hurt your conscience?'\n\nShe chuckled. 'No. But I am a sinner.'\n\n'He may not have made his will for the laughs.'\n\nShe hesitated, quizzing me; then shook her head. 'No. Not possible.'\n\n'Why not possible?'\n\n'You do not understand. Money was never a thing between us. We had a relationship where money was nothing.'\n\n'Yet Freddy loved you.'\n\n'So I expect something, a memento of my poor friend. Shall I tell you what? It will be the Bugatti. And I shall keep it the same, just like Freddy.'\n\nI sped another ring. 'You could never drive it.'\n\n'My friend, I have an eye that creates drivers. The Bugatti will be better than money to me. Perhaps some day soon you are going to return it?'\n\nI shrugged. 'Perhaps. But it isn't yours yet.'\n\n'Oh, I am certain. I shall get the Bugatti.'\n\n'And certain there is a will?'\n\nShe leaned her head to one side. 'I do not know that. But wouldn't you?'\n\nEnd of session.\n\nI sat for a time sucking comfort from a dead pipe, while Dutt lit one of his rare cigarettes (five a day: Iron Len). It was nearly midnight. Three hours of Mimi, and she had left as sprightly as she had come. And why not? On the judges' cards she had probably earned a majority verdict. I tried a fresh match, got a raw, wet taste, and certified the pipe as a goner.\n\n'Did you notice anything useful?'\n\nDutt carburetted smoke. 'She struck me as being a cool one, sir.'\n\n'I don't need Einstein to tell me that. What did you spot that I might have missed?'\n\n'Well, sir.' Dutt eased his seat. 'I thought you had her going a couple of times. Once about her failing to raise the alarm, and once when you were sprucing her about a statement from chummie.'\n\n'But especially the latter.'\n\n'As you say, sir. And the two do go together. Though I reckon we shan't be getting much ahead until we can lay our hands on him.'\n\nI nodded. 'That's the break we need.'\n\n'It will clear it up, one way or the other.'\n\nI stared. 'Are you going along with the lady?'\n\nDutt humped his shoulders and looked stupid.\n\nThen the phone went. I hooked it up; Dainty was at the other end.\n\n'Hallo? Haven't you managed to get to bed yet?'\n\n'Cut the comedy,' I said. 'It's too late.'\n\nI could hear his mates cackling in the background.\n\n'No, listen,' he said. 'We've something for you. This Peter Robinson. There's a chummie called Bilney. He's been adrift a few days. He could fit.'\n\n'Does he match the description?'\n\n'Who doesn't? But I'd say he matches it as well as most. Thirty, fair hair, sideboards. I'm sending you the bumf on teletype.'\n\n'What makes him a candidate?'\n\n'He's missing, for one. For two, he's an associate of Wicken's. May have done a job or two with Quarles. He hangs around the fringe of the gangs.'\n\nIt sounded promising. 'What's his form?'\n\n'He's done some porridge for G.B.H. I'm told he has a nasty temper, has been known to use a knife.'\n\n'Is he left-handed?'\n\n'Get knotted,' Dainty said. 'The fine print is coming over the wire. I just thought I might catch you between rounds.'\n\n'Ha, ha,' I said. 'Go to hell.'\n\n## CHAPTER ELEVEN\n\nBUT GIVE DAINTY a mark for prescience: there was indeed a late-night comedy interlude.\n\nWhen we broke up, Dutt went for a bath; I decided mine could wait till morning.\n\nI got into bed and lay brooding a while over the twists and nuances of the case. I find that when I'm relaxed, horizontal, and about to drop off, the facts will sometimes sort themselves without help from me. It is as though, at that point, they take on a life of their own, and begin to exhibit aspects that till then I've been blind to; but it may be simply that I am resting the intellect and permitting the intuition to have free play. Moments of sartori, involuntary Zen: a genius is a man who has learned to switch off.\n\nSo I was lying like that, trying to be a genius, when I heard Dutt's slippers slopping back along the corridor; then the sound of him opening his door, which apparently he had left unlocked (a policeman's mind is never still). Followed confused sounds, and a tap at my door. I switched on the bed-light. Dutt entered. His face was pink, and he was grappling his dressing-gown round him with a curiously intent modesty.\n\n'Sir . . . could you spare a moment?'\n\nI climbed into a dressing-gown and we went to his room. Sitting-up in the bed, and sensationally naked, was Mimi, Madame Deslauriers. She regarded us with mild surprise.\n\n'This is flattering. But shouldn't one of you gentlemen retire?'\n\n'Get out of it,' I said. 'You've picked the wrong room. Mine is across the corridor.'\n\n'This is not your room?'\n\n'It's the Inspector's.'\n\nShe said something rapidly, in French. 'It is that stupid yak, he tells me wrong. I will give him a haircut with a blow-torch.'\n\n'So kindly hook it.'\n\nShe got out of the bed and stood for a moment, nudely glaring. But then she burst into gurgling laughter.\n\n'This is formidable, don't you agree?'\n\n'I don't agree.'\n\n'The poor Dutt. And he is a family man, huh?'\n\n'Never mind about Dutt. Just slip into this.' I picked a frothy black night-dress from a chair by the bed.\n\nShe took the night-dress but didn't slip into it.\n\n'Monsieur, it is the mistake which has made you angry. But that is easily put right. I will cross the corridor. Let us leave the poor Dutt to his honest slumbers.'\n\n'I just want you to scram.'\n\n'Oh, but no. When I am so convenient and agreeable. All day you are making the impression, huh? You must not be impolite now.'\n\n'But I haven't been making the impression!'\n\n'Oh yes, yes. In your policeman's way.'\n\n'Look,' I said. 'Put it on or leave it off, but get back to your room \u2013 or I'll call the manager.'\n\nShe looked at me sadly. 'That is not being serious.'\n\n'Yes it is. I want you gone.'\n\n'No. The mistake, that is the trouble. You have wished my visit to be more discreet. But now I tell you. I will go away. I will do exactly as the man says.' She gave me a lightning wink. 'Poor Mimi. This has not been her lucky day.'\n\nShe slid the night-dress over her head \u2013 which was an erotic act on its own \u2013 smiled apologetically at Dutt, and cruised regally out of the room.\n\nDutt goggled after her.\n\n'Do you think she'll be back, sir?'\n\n'That appeared to be the message. We had better bolt our doors.'\n\n'You bet, sir!'\n\nBut Mimi didn't come back.\n\nHanson's messenger delivered the Bilney dossier at breakfast the next morning. It made no mention of Bilney's being left-handed, but the other details fitted rather well.\n\nBilney, Thomas Henry. Age 30. 5\u00b4 10\u00bd\u00b4\u00b4, strong build. Fair hair, grey eyes, narrow features, small ears. 2\u00b4\u00b4 scar left cheek. Missing top joint of little finger, left. London accent. Born, Lambeth. P.O.A., Shepherd's Bush. Last seen, Thursday. Total of six years for G.B.H. and robbery with violence.\n\nThe photographs showed a good-looking villain, one who might well appeal to the ladies; but there was violence in the mouth, which was small, and in the prominence of the blunt chin (check fifty or so photographs of convicted murderers and you will find that Lombroso wasn't far out). The eyes were glazed-looking, avoiding the camera. He had thick eyebrows but scanty lashes. The scar, nearly vertical, was certainly a knife-slash, and he may have lost the finger-joint in parrying the attack.\n\nI showed the photographs to Dutt.\n\n'Would you let him buy you a drink?'\n\nDutt grinned. 'Only for a cover while I was getting out the cuffs, sir.'\n\n'Do you know him?'\n\n'No sir. But I know a lot like him. And when his type are around I'm careful not to turn my back.'\n\nMimi was seated at her table, looking gorgeous in white leather hot pants. I took the photographs along to her and sat myself in the chair opposite. She was eating grapefruit. She gave the grapefruit a dig, sending a spurt in my direction. I gravely blotted the juice with a napkin before exhibiting the photographs.\n\n'A friend sent me these.'\n\nShe gave them a glance. 'Monsieur enjoys a distinguished acquaintance.'\n\n'His name is Tom,' I said. 'I am wondering if you can guess his age.'\n\nShe took a longer look; but if there was a tremor of recognition I failed to detect it. Or anything else. She was keeping her face completely vacant, an unregistering mask.\n\n'I would guess he was seven.'\n\n'That's his mental age.'\n\n'So then. You will have much in common.'\n\n'Have you any message for him?'\n\n'Please go away,' she said. 'I wish to continue with my breakfast.'\n\nSo I switched to Bavents, who I waylaid as he came through the swing doors from the kitchen. He was juggling with a tray and a covered dish: I shepherded him into the chef's corner.\n\n'Take a look at these.'\n\nI made a fan of the photographs and held them close to his pink nose. The tray and the dish chattered.\n\n'I \u2013 I don't know anyone like that!'\n\nI clicked my tongue. 'You were talking to him on Thursday.'\n\n'No! I've n-never seen him before.'\n\n'Not Tom Bilney? Who slipped you the quid?'\n\n'No, it's the truth! I've never m-met him.'\n\n'But he did slip you a quid?'\n\n'He d-didn't, I tell you!'\n\n'So how much was it? Fifty pence?'\n\n'I \u2013 no, n-nothing! I d-didn't see anyone!'\n\nI left off before he dropped the tray.\n\nBut I was luckier after breakfast, when I paid a visit to the Three Tuns. Both Eddie Jimpson, the licensee, and his wife Doris had had avowed contact with 'Peter Robinson'. On Thursday Eddie had been serving in the bar, and he had passed on the man to Doris. Doris had booked him in and taken him up to show him his room.\n\n'Could this have been the man?'\n\nThey went into a huddle over the photographs.\n\n'It's like him,' Eddie said. 'He's fair, isn't he?'\n\n'Fair. Grey eyes. About five feet ten.'\n\n'This one was big with it,' Eddie said. 'Looked as though he could be a rum customer.'\n\n'This one is big with it. He can be rum.'\n\n'Then I reckon it's the same man.'\n\nI looked at Doris. 'What do you say?'\n\nDoris, plump and curly, was frowning.\n\n'I don't know what to say. It could be him, but it isn't easy to tell from a photograph.'\n\nI whipped the photographs away. 'Describe your man.' Doris leaned her haunch against the bar. 'Well, he was fair all right, and I didn't much like him. He'd got dead sort of eyes. You were just muck to him.'\n\n'Any special features?'\n\n'Not that I remember. Though of course you could tell he was a cockney.'\n\n'Eddie?'\n\nEddie shook his head. 'That's what I was going to say,' he said.\n\n'Try thinking about his face. Just let it come to mind, don't force yourself into seeing it.'\n\n'He was looking a bit scruffy,' Doris said, after a moment. 'Sweaty. Like he might have been driving all day.'\n\n'Sweaty and grimy?'\n\n'A bit of that too. You'd have thought he would have washed before he went out.'\n\n'But in the morning, at breakfast, he would be tidied up?'\n\n'Well yes, he was smart enough then.'\n\nCould they have missed the scar? It wasn't very prominent, except perhaps to an eye conditioned like mine: it followed the natural lines of the face, it might register without at first being recognized. As for the missing joint, he would keep that inconspicuous.\n\n'Did you watch him sign the book?'\n\n'Of course.'\n\n'Did he use his right hand or his left?'\n\nDoris gestured helplessly. 'If he had used his left hand, I should think I would've noticed that.'\n\n'Anything else about his hands?'\n\n'They weren't very clean.'\n\n'Do you mind if I see the book?'\n\nDoris fetched it. The 'Peter Robinson' entry had been made in bold but back-slanted writing. No visible dabs, and a poor paper for latents: not much to hope for from that.\n\n'What I would like to see now is the room where he slept.'\n\n'The room has been let again, you know.'\n\nI sighed to myself. 'Never mind. Just ask the occupant if I may step in.'\n\nIn fact, the occupant was out. Doris used her pass-key to admit me to a small, pleasant room, the single window of which was framing a view of a giant chestnut in lavish bloom. It was fitted with a wash-basin, mirror, a glass shelf and a tooth-glass located in a chromium-plated holder. The paint was clean and shiny on the frame of the sash-window, a white-enamelled dressing-table, and the door.\n\n'Who serviced the room after he left?'\n\n'I did,' Doris said.\n\n'Tell me what you did.'\n\n'I changed the bed-linen, hoovered, dusted and gave it all a wipe over.'\n\n'How much is all?'\n\n'Well, the wash-basin mostly; the shelf, the mirror. And I changed the glass.'\n\n'Did you touch the paintwork?'\n\n'Only with a duster. The paint was washed a fortnight ago.'\n\nWhich sounded like a frost; but to turn every stone, I rang Hanson to send out a dabs team. They arrived within half-an-hour. I gave them the register and turned them loose in the little bedroom. A lot of insufflating and snazzy camera-work and paint left looking as though the devil had stroked it; then Eddie, Doris and the apprehensive room-occupant were check-printed for comparisons. Results: nil. Bilney wasn't yet a certainty, just a hot front-runner. One witness liked him, one was cautious. But I felt the wind was blowing his way.\n\nAnd the more so when I returned to the Barge-House, where Dutt was just putting down the phone.\n\n'That was Dainty, sir.'\n\n'Has he collared Fring yet?'\n\n'No sir. But he's been chatting-up Bilney's girl-friend.'\n\nI shrugged and sat. The girl-friends of villains are a highly variable quantity. Even when they are jealous their information is suspect, and in the normal way they simply go dumb.\n\n'Why has this one suddenly turned chatty?'\n\n'Dainty says it's because she's scared.'\n\n'Scared of what?'\n\n'Of Bilney's being missing, sir. She reckons he ought to be back by now.'\n\nI grunted. But somebody might love Bilney.\n\n'What's this girl-friend's name and trade?'\n\n'Name is Mavis Treadwell, sir, and she claims to be a photographer's model. It seems she had a date with Bilney for Friday. She has a key to his flat in the Bush. When she arrived there she found he'd left a note for her saying he'd been called away on a job.'\n\n'On a job?'\n\n'Well, that's what she infers, sir. And she's the one who should know. But the point is he's been gone for three or four days now, and she reckons that something must have happened to him.'\n\n'Then she knows more than she's saying.'\n\n'Dainty thinks not, sir. She's sure Bilney would have rung her before now. All the other jobs he's done have been in the locality. He's never been away so long before.'\n\n'Any inkling of what job?'\n\n'Afraid not, sir.'\n\n'Had she any idea of where he was heading?'\n\nDutt shook his head. 'I did put some questions, sir. But what I told you was all Dainty had got.'\n\nI drew invisible lines on Frayling's desk. The picture was growing.\n\n'It would be on Thursday that Bilney left the note.'\n\n'Yes, sir. Dainty did go into that. Treadwell says the note wasn't dated. She'd last seen Bilney on Wednesday evening.'\n\n'So on Thursday someone called him up here, and we assume he was the man who booked in at the Three Tuns. His first move then was to contact Deslauriers. There can't be much doubt about what the job was.'\n\n'Not very much, sir.' Dutt looked glum.\n\n'Then Deslauriers installed him in Freddy's hideaway. On Friday the trouble with Rampant provided an opportunity, and Deslauriers phoned Bilney with instructions. Straightforward so far?'\n\nDutt nodded.\n\n'Bilney did the job and returned to the hideaway. But now we have a problem. Bilney stayed put. He didn't hurry back to home and Mavis. Why would that be?'\n\nDutt puckered his eyes. 'Could he have been on the same lark as Rampant?'\n\n'You mean blackmail?'\n\n'He might have had a bash at it. His sort don't go in much for brains.'\n\nI considered it. 'It's the simple answer, and it fits with Deslauriers secretly meeting him. But she must have warned him yesterday that I nearly caught her with him. You would think he would be back home by now.'\n\nDutt hunched a shoulder. 'Some of them are thick. Perhaps he thinks the pressure will make her cough up.'\n\n'The alternative is that Bilney is her boy-friend. Which raises another problem. I can't believe it.'\n\nA tap at the door interrupted the conference; semi-handsome Hanson stalked in. He was looking happy. He leered at each of us before sprawling himself on the third chair.\n\n'Are you still wanting Bilney?'\n\nI stared. 'Have you got him?'\n\n'Well, maybe not yet under lock and key. But we've found his little paw marks at Raynham. I thought you might like to come along.'\n\n## CHAPTER TWELVE\n\n'PAW MARKS' WAS a metaphor: at that precise juncture all Hanson had was a missing person. But the person was missing from a riverside inn, and the description mentioned a scar and an amputated finger-joint. Furthermore, Raynham was the next village downstream from Sallowes; the inn, the Reed-Cutters, stood opposite the staithe. Bilney, now using the pseudonym of H. Wilson, had booked in there on the Friday evening.\n\n'One of the County men called there yesterday,' Hanson explained. 'We thought the description sounded interesting. But chummie was out. When we called this morning the publican told us he hadn't come back.'\n\n'Did he leave his gear?'\n\n'So I'm told. He must have got word from the lady and skarpered. But this description of Bilney is a snap fit. There can't be two like him swanning around. Is Friday right?'\n\n'Friday is right.'\n\n'Then it looks like this case is falling together.' He plucked his lip. 'A pity, really. I was tipping you to get round to Rampant in the end.' He got to his feet. 'Shall we go?'\n\n'First, I want a man and a car.'\n\n'Huh?' Hanson looked aggrieved.\n\n'We'll need to leave this place covered. If Bilney's on the loose, this is where he may head for.'\n\nIn the end I got a van and two men, with a third man to cover the river approach. Dutt I left with the special mission of keeping his willing eye on Mimi.\n\nRaynham was nine country miles from Haughton. It was a small village on a bluff by a broad; its handsome church tower stood high among trees, with below it pantiled cottages and infillings of bungalows. The broad was small and fly-blown with hire craft, a handful of yachts in a slum of motor-cruisers. An ugly beard of battered craft fringed the tiny staithe, which was itself parked solid with the cars of day-visitors. Facing the stage was a cramped junction and in one of its angles stood the Reed-Cutters. Two police cars were parked on the handkerchief of frontage, leaving bare room for us to slot in too. Hanson rammed us home. We got out.\n\n'Welcome to Mug's Corner,' Hanson said sourly. 'Once I kept a half-decker in the dyke here. The second time they sank it I gave up.'\n\n'But it looks a good spot for a chummie to hole-up in.'\n\n'Oh sure.' He stared about savagely. 'These are the conditions of crime, sonny. Greed-pollution. Maybe it's time we had another war.'\n\nWe pushed into the bar, where there was standing-room only, and through a door to the back premises. Hanson introduced me to the licensee, Silkin, and the County C.I.D. man, Inspector Breckles. Silkin was a heavy, fresh-faced countryman, Breckles a cherub with watchful eyes. We pulled up seats round a massive table; I spread out the Bilney photographs in front of Silkin.\n\n'Is this your customer, H. Wilson?'\n\nSilkin looked them over. 'Yes, that's him, sir.'\n\n'What made you notice his little finger?'\n\nSilkin blew out his cheeks. 'Don't rightly know, sir.'\n\n'Did you see it when he filled in the register?'\n\n'No. Because I filled it in myself.'\n\n'Did you see him write anything?'\n\nSilkin looked puzzled. 'Now you mention it, I don't think I did. I reckon I noticed that finger when he was sinking a pint, that's the time when I saw most of him.'\n\nA left-handed drinker.\n\n'Was he in the bar a lot?'\n\n'I'm telling a lie, sir. He wasn't in often. Just last thing he'd come in for a couple, and then go straight up to bed.'\n\n'Did he talk to anyone?'\n\n'That's difficult to say, with all the crowd we get in here. But I can't say I noticed he was very sociable. He never said much to me or the missus.'\n\n'What about phone calls?'\n\n'He didn't make any here.'\n\n'Did he receive any?'\n\nSilkin shook his head. 'But there's a phone-box a few yards up the road. He could have used that if he wanted to be private.'\n\nFor out-going calls . . . but the others?\n\n'What did he do with himself all day?'\n\nSilkin puffed his cheeks. 'I reckon you'll have to tell me, sir. He was out of here each day after he'd had his breakfast.'\n\n'He went out in his car?'\n\n'He did if he had one.'\n\n'But surely you know if he had a car?'\n\n'The guests park over the way, sir,' Inspector Breckles put in. 'There's no room this side. I've sent a man to make enquiries.'\n\n'They leave their cars on the staithe?'\n\n'That's right,' Silkin said. 'And that's where this man would've left his. But there's always a dozen or more left across there, so whether he had one I couldn't say.'\n\n'A blue Viva,' Hanson said. 'Of course the bloody car was across there.'\n\nSilkin stuck out his chin mulishly. Hanson never had charm to spare for the natives.\n\n'Let's go back to Friday,' I said hastily. 'When did this fellow arrive here?'\n\n'It was three to three-thirty,' Silkin said grumpily. 'We were having a bite after the bar closed. He came through the yard and knocked on the door. Asked if we could put him up for a few days.'\n\n'A few days?'\n\n'Those were his words. Told me he was here on a bit of business. He offered to pay me in advance, but like a fool I didn't accept it.'\n\n'Did he have his case with him?' Hanson snapped.\n\nSilkin sniffed. 'He fetched it afterwards.'\n\n'So like that wouldn't he have had a car outside?'\n\nSilkin humped his shoulders. 'It must have been my dull day.'\n\nHanson snorted: I shot him a quick look.\n\n'What happened after you had booked him in?'\n\nSilkin sniffed again. 'He went out, didn't he? Said he'd see us later, then he went out.'\n\n'When did he come back?'\n\n'Well, it was latish. He came into the bar near closing-time. He had his couple, the way I told you, and went up while I was still doing the till.'\n\nI paused. 'You are sure you didn't see him earlier?'\n\nHis eye met mine. 'Quite sure of it, sir.'\n\n'Your wife?'\n\n'She wouldn't have seen him till closing. She was in the back when he came in.'\n\n'You understand what these questions are about. That you may have to repeat what you're telling me on oath?'\n\n'Yes, sir. Inspector Breckles informed me. But that fellow wasn't back here till turned ten.'\n\nSo there it was: barring an alibi, which Bilney wouldn't find easy in a strange manor.\n\n'Did you notice anything special about him that night?'\n\nSilkin hesitated. 'He may have looked a bit untidy.'\n\n'How, untidy?'\n\n'Well, his hair was ruffled, and maybe his clothes a bit creased.'\n\n'No blood on his sleeve?' Hanson cut in.\n\nSilkin looked shocked. 'I didn't see blood. I'm trying to tell you what I can remember. I can't do better for you than that.'\n\n'What about his manner?' I asked.\n\nSilkin blew into his cheeks. 'I just served him. I didn't notice.'\n\n'Was his hand trembling?'\n\n'I didn't notice. When I think of any more, I'll tell you.' I silently cursed Hanson.\n\n'Now I'd like you to tell me about yesterday. What time did Bilney go out?'\n\nSilkin's eyes were sullen. 'His usual time. After breakfast.'\n\n'Wasn't it a little later yesterday?'\n\n'No, it wasn't. It was how I said.'\n\n'Didn't he receive a phone call?'\n\n'I told you he didn't. Now I want to go through there and help the missus.'\n\n'You'd better go then.'\n\nSilkin hesitated briefly before hauling himself up and clumping out. Hanson stared after him evilly. Breckles gave me a quizzical glance.\n\nWe went up to the bedroom by a crooked stair that had a rope for a hand-rail. A different dabs-team was at work there, and this time no check-printing was necessary. On the door, the tooth-glass and Bilney's Remington razor were prints matching those that had come over the wire. Bilney was in. Hanson had already sent out a general W.F.Q. alert.\n\nWe turned over Bilney's gear, which suggested that he hadn't anticipated a lengthy stay out of town. In a squash-top suitcase were a soiled change of underwear, a screw of bennies and two pornographic paperbacks. No spare shoes, ties or socks. A raincoat he might have kept in the car. His toilet stuff was the bare minimum and didn't run to talcum or after-shave.\n\nHanson sprawled gracelessly on the bed. 'What does Scotland Yard make of it?'\n\nI shrugged and fed Erinmore into my pipe. I wasn't quite sure what I was making of it: my intuition was failing to click. But I had a feeling of sadness about that little room, about the paltry possessions Bilney had abandoned there. Almost feeling sorry for the stupid jerk: an emotion he wouldn't have wasted on me.\n\nHanson lit a cheroot. 'Do you want my opinion?'\n\nI borrowed his matches. 'Why not?'\n\n'I'd say chummie came out here to do a quick job, but then he got hooked with a different angle.'\n\nHanson and Dutt, both.\n\n'Of course, you mean blackmail.'\n\nHanson horsed smoke. 'What's wrong with that? Deslauriers has money. She ordered a killing. Which left her wide open for a big touch.'\n\nI tossed back the matches. 'No touch is worth a lifer. Bilney could only shop her by shopping himself.'\n\n'Yeah, that's how you think, that's how I think. But we're talking of a buster with his brains in his knuckles.' He spat some cheroot. 'Look, chummie does his job, but he doesn't go home the way he planned to. Then what's holding him here? What's the attraction? With the police busting a gut all around?'\n\nI puffed twice. 'Say he's sweet on the lady.'\n\n'Ha bloody ha,' Hanson jeered. 'You know there's only one attraction for a louse like Bilney, and that's the stuff that comes out of banks. He's putting the black on, and this is the place for it, where the lady has to act all sweet and innocent. She's wanting him gone and long gone. Every day he stops here is a boost to the pressure.'\n\n'It fits some of the facts.'\n\n'You bet it fits them. Bilney wasn't risking any lifer. She had to pay him to go away, to stop giving us notions. That was the deal he was sitting in with.'\n\n'And now he's gone.'\n\n'Yeah, now he's gone.' Hanson sucked and spat out more leaf. 'So either the lady paid him off, or more likely your coming on the scene scared him.'\n\n'You're saying she warned him.'\n\n'Yeah.'\n\n'How?'\n\nHanson stared. 'How should I know how? You let her go back to the hotel before you. She would have had time to put in a call.'\n\nI shook my head decidedly. 'It can't be that simple. Bilney didn't return to base after meeting her. And he wouldn't spend his days camping-out in a call-box, waiting for the lady to ring him warnings.'\n\n'Then she used a messenger. Maybe that long-hair.'\n\n'Bavents?'\n\n'Yeah. Didn't I hear he was wet on her?'\n\nI sieved a puff; Bavents was a possible. I wasn't at all sure of my ranking of Bavents.\n\n'It would mean letting someone else into the know, and the lady is too intelligent to want that to happen. In fact, the lady is too intelligent, period. She would never let Bilney get away with blacking her.'\n\n'Balls. She would be in a cleft stick.'\n\n'So she would get on the phone. But not to Bilney.'\n\nHanson smoked ferociously, but it was a point. Cheaper to buy muscle than pay black.\n\n'All right, then. Suppose I'm wrong. You tell me why chummie hangs on here.'\n\nI launched smoke at the small lattice window that overlooked the road, the jammed staithe, the jammed broad.\n\n'I don't know. I've been wrong too. Deslauriers didn't send Bilney to Freddy's hideaway. And according to Silkin Bilney received no phone calls, yet Deslauriers must have phoned him at least once. That could have been a call by appointment, but if so the timing was strangely felicitous. And if it happened again yesterday, after she met me, then felicitous stops describing it.'\n\nHanson wriggled. 'So what's the next move?'\n\n'Bilney hadn't returned to town this morning.'\n\n'Meaning he's still here?'\n\n'We had better assume that. And assume also that he's still in touch with the lady.' I puffed. 'What would you do in her place?'\n\n'Me?' Hanson champed on the cheroot. 'I'd get him out of circulation fast. It's too late in the game to leave him around loose.'\n\n'And where would you put him?'\n\nHe sighed smoke. 'This time it has to be Hernando's Hideaway. But for crying out loud, we've been doing our nuts over it. Maybe you'd better call in the Army.'\n\nWe went back down into the parlour. Hanson had a map fetched from the car. We spread it out on Silkin's great mahogany dining-table and clustered round it in a hopeful seance. Breckles, the local man, pointed out the venues of holiday-bungalow development. They peppered the river-banks for miles and choked minor backwaters and tributaries. Then there were boatyards and mooring dykes where house-boats were lodged in their dozens. The Army wasn't such a bad idea; a thorough check of the riverside might take weeks.\n\n'Have you been in touch with the rating department?'\n\n'Yes, sir,' Breckles said. 'They are getting out lists for us, all the properties with registered owners in the London district.'\n\n'Roughly how many?'\n\n'Over two thousand, sir. And we're getting lists of house-boat owners from the River Commissioners. But it's all taking a bit of time. I reckon the men on the spot have got the best chance.'\n\nPerhaps.\n\n'You're a native here, aren't you?'\n\n'That's right, sir. Born in Haughton.'\n\n'Right. Now forget the map. Just close your eyes and think of the river. The quiet, hidden places. Places that for some reason missed being devel- oped. Maybe ruinous, ramshackle places. Silted-up dykes, too shallow to navigate. Lonely; poor access; barely good enough to get a car down. Are you doing that?'\n\n'I'm trying, sir.'\n\n'Then make me a list of all those places.'\n\n'Yes, sir. I'll certainly try.'\n\nHanson's expression said I'd never rated lower.\n\n## CHAPTER THIRTEEN\n\nSILKIN'S WIFE SUPPLIED us with sandwiches and we took them, with bottles, to a bench across the road; not on the staithe, but beside a mooring cut used by the trip-boats to decant their pay-load. A path led to the cut through a grove of alders, and the bench stood in the shade of the grove. Except for a mound of dredged mud that lay steaming on the bank the spot was pleasant, being screened from the broad.\n\nWe ate and drank silently. Hanson had a dreamy expression. He was beginning to see the end of this case. Our discoveries at Raynham had reduced it to a routine-matter \u2013 time-wasting, of course, but no longer speculative. Sooner or later, most probably sooner, we would have Bilney tucked away in the cooler; and with any sort of policeman's luck, enough hard evidence for a copper-bottomed case. Like the knife, like blood on sleeves. Bilney would be fortunate if we didn't find something. And with Bilney in the cooler we could play him against Deslauriers, and Deslauriers against him \u2013 routine, routine!\n\nThen why wasn't I feeling happy too, who didn't have to bother even with the routine? A few loose ends? But every case has them. Otherwise defence counsels would go out of business. So I didn't know how Deslauriers communicated with Bilney \u2013 well, no doubt I would know, later. And I didn't know why Bilney stayed around after the job \u2013 well, there were a couple of theories covering that. Then there was the right-hand, left-hand business: wasn't I attaching too much importance to it? If it wouldn't throw a jury (and it wouldn't), had I any right to let it throw me? No: when you added it all together, I had no grounds for feeling so pensive over my sandwiches. From the moment we had tied Bilney into the case its main outlines were cut and dried.\n\nI finished my bottle, and Hanson offered me one of his sin-black Burmese cheroots. While I was lighting it we were joined by the man who had been sent to make enquiries at the staithe. He had had no luck. About twenty cars were left parked on the staithe every night, some from the guest-house up the road, some belonging to vacationists who rented the cottages. Nobody specifically remembered a blue Viva, though some remembered cars that were blue. Among them Bilney's, without doubt. Only a Viva-driver notices another Viva.\n\n'Did anyone remember seeing Bilney himself?'\n\n'Yes, sir. He used the shop on the staithe a few times. He bought his fags and newspapers there. The lady who runs it gave a good description.'\n\n'At what times was he in there?'\n\n'In the morning, sir. And it was the local paper he bought.'\n\nNaturally. 'Did she notice his hand?'\n\n'Yes, sir. Also his scar.'\n\nI ran it through my mind: Bilney buying a paper, feeling in his pocket for a coin. If he had felt with his right hand, holding the paper in his left, would the lady have been able to see that finger? But he had bought cigarettes there, too, putting out his left hand as he tended with his right \u2013 or vice versa: and either way, giving her a sight of the finger.\n\n'Is there a garage in Raynham?'\n\n'No, sir. The nearest is in Sallowes.'\n\n'Call in and enquire if he bought petrol there. With special reference to Friday evening.'\n\nThe man ducked and went; Hanson dragged smoke; a trip-boat came nosing up the cut. We watched her naval-suited crew moor her to two posts, then got out ahead of the crowd.\n\nIn Silkin's parlour Breckles was still sitting with the map spread out before him, but now a number of red ball-pen carrots had been neatly marked upon it. Breckles rose as we entered.\n\n'I've had a shot at your idea, sir.'\n\n'Are these your probables?'\n\n'I wouldn't like to say that, sir. But there's a couple we could take a look at. I've just been checking with the Rates Department and two of these places have London-registered owners. One is a private person with a Kensington address, the other is a holding company in Balham.'\n\n'And where are those places?'\n\n'Both near here, sir. This one is Blackdyke Fen, at Beastwick. Then there's Turnpudden Hole, between Sallowes and Wrackstead. Both of them are pretty well off the map.'\n\n'Which is your choice?'\n\nBreckles shrugged embarrassedly. 'I'd say it was all a bit of guess-work, sir. Turnpudden Hole is nearest to Wrackstead, but I can't think how a Londoner would get to know about it.'\n\n'It's part of the old Gifford estate,' Hanson said. 'The estate was sold up after the war. A development company bought a lot of it, all the fens down that side.'\n\n'What about the other place?'\n\n'Perhaps more likely,' Breckles said. 'But I wouldn't care to bet on that, either. It's a converted mill right out on the marshes. As far as I know it isn't being lived in.'\n\n'But that one is privately owned?'\n\n'Yes, sir.' Breckles took out a notebook and flipped the pages. 'E. V. Selkirk, 73 Glebe Road, Kensington. He's been the owner since '68.'\n\nI looked at Hanson. 'Any preference?'\n\nHanson chewed his cheroot unhelpfully.\n\n'Right then,' I said. 'We'll try the mill.'\n\nHanson opened and closed his bony hand.\n\nWe collected a fourth man and drove to Beastwick, a pretty village with its back to the river. The cottages were styled in Art Nouveau rustic, but grouped with a keen eye for effect. We entered a skein of jumbled lanes, with the marsh and carr hazy below us, and came at last to a humpy marsh track, where marl combined with flints and brickbats.\n\n'How much further?' I asked Breckles.\n\n'It'll be about another quarter of a mile, sir.'\n\n'Any cover?'\n\n'There's alder carrs, sir. But you'll maybe go in up to your backside.'\n\n'Is there any other way out besides this?'\n\n'No sir, unless chummie has got a boat. But I passed by on a River Patrol launch last week, and the cut was empty then.'\n\n'Suppose he is a swimmer?'\n\nBreckles shook his head. 'It's all carrs and marshes, both sides. He might lose himself in there for a couple of days, but he would be damned glad to come out after that. If he takes to the marshes, we'll have him.'\n\n'Unless he steps into a mud-hole,' Hanson said.\n\nWe bumbled on a short way further, then I halted and parked to block the track. Just there it was running through thick groves of alder in-filled with willow brush and sedge. Off the track it was sloughy black peat-mud; the air was sweatily humid and smelling of mint. The four of us alighting disturbed a jay, which blundered off through the twigs with klaxon-like cries.\n\n'Christ,' Hanson muttered. 'That should tell him!'\n\nWe waited by the car for a couple of minutes. Once the jay had settled the carrs fell silent: just the murmur of mosquitoes that had come to inspect us.\n\n'You lead,' I said to Breckles.\n\nWe followed him down the track at twenty yards distance. The track made a slow turn through the alders and brought into view the tops of giant willows. Breckles signalled us to wait. He edged cautiously forward, was lost to sight behind a screen of scrub willow. We stood moistly flipping at the mosquitoes for what seemed an unnecessary interval. Then Breckles reappeared, waving to us. We joined him beside the scrub willow. Peering round it, we could see the brick mill-tower standing among the tall willows, with the river beyond.\n\n'I don't think he's at home,' Breckles whispered. 'I've been up to have a squint in the garage.'\n\nHe indicated a sagging out-building with a roof of reed thatch, which was beginning to shed.\n\n'Any signs of use?'\n\n'None I've seen. But you would expect chummie to play it clever. He may have parked his car on the hard-standing. You would never spot it from the river.'\n\nI grunted and took in the scene. Once, someone had spent a lot of money on the mill. Fresh windows had been pierced at each of its four stories and a circular, white-painted verandah constructed around the cap. Once, too, there had been a lawn under the willows, trellised roses, a quay-heading. The mill-dyke had been enlarged and piled and had doubtless housed a launch or a motor-cruiser. Once. But not now. Now, the jungle was taking it back. The white paint was flaking, the quay-headings ruinous, and persicaria blooming in the silted-up dyke. And it gave an impression of intense loneliness, of a far-off outpost that had died. If it wasn't haunted, it ought to be. A place fit only for ghosts.\n\n'Where is the door?'\n\n'It faces the river.'\n\n'Let's spread out and take a look.'\n\nDyke, marsh and undergrowth prevented us from surrounding the mill, but we did our best with what was left. I crossed a shaky bridge and followed a tiled path, of which the pemmons were sinking and choked with grass. It brought me to a shabby door. The door was secured with a massive rusty chain and a rusty padlock. Breckles joined me.\n\n'Is this the only entry?'\n\n'There's a ground-floor window, sir. But it looks intact.'\n\n'Would you say this door had been unlocked since Christmas?'\n\nBreckles poked the padlock, and swallowed. 'No, sir.'\n\nBut we were there, so we went through the motions. Hanson thumped the door and called on Bilney to come out. He disturbed the jay again. It went clamouring through the carrs like a panicky blackbird with roup. Then silence.\n\n'We have tools, sir,' Breckles ventured. 'I could get that lock off in two minutes.'\n\nI looked at Breckles, Breckles looked at his feet.\n\nWe went back to the car.\n\nBefore we set out on our second goose-hunt I rang Dutt from a box in the village. Dutt had seen no more of Bilney than we had and could offer only minor and marginal information. Dainty had rung. The French police at Cap Ferrat had paid a call at Freddy's villa. It was empty, but they had found signs of a very recent occupation. The caretaker, a retired procuress from Marseilles, had attempted to explain this by admitting to the illicit entertainment of friends there; the French police had pretended to accept the explanation. They were now keeping a close watch on the villa.\n\n'Any word of Bilney from Shepherd's Bush?'\n\n'No, sir. But they've posted a man at his flat.'\n\n'What has the lady been doing?'\n\n'She's been shopping, sir. She bought two blouses and a George Formby record. Then she went up to her room and played the record, and about half-past twelve she must have rung for a drink. Bavents fetched it, a Dubonnet and lemon, and he was in her room about twenty minutes.'\n\n'Was your ear to the key-hole?'\n\n'Well, actually, yes, sir. But all I could hear was that blooming record. First it was _If Women Like Them_ and then _Swimmin With The Women_.'\n\nI clicked my tongue. 'She's adding to her repertoire. Has Bavents gone out or made any phone calls?'\n\n'No, sir. He was serving at lunch, and now he's in the kitchen manicuring vegetables.'\n\nWhich sounded innocent enough, unless one remembered that he would be fixing the veg with his left hand.\n\nI rejoined the others in the car and we went on our way to Turnpudden Hole. Nobody was saying much. Breckles in particular had a droopy expression on his round-cheeked face. Hanson was silently savaging a cheroot. The D.C., who was driving, stared over his bonnet. I chewed my pipe-stem. We passed through Sallowes and turned once more into the lanes.\n\n'What sort of place is Turnpudden Hole?'\n\nBreckles made a little gesture with one shoulder. 'It's just a small broad, sir. Mostly grown over. The old blokes used to say it was bottomless.'\n\n'But is there a house or something?'\n\nHanson gnashed smoke. 'There's a shack they put up for Clytie Gifford. She was a weirdo, an eccentric. Liked to live alone with the birds.'\n\n'And it's been empty for some time?'\n\n'Yeah. Clytie pegged out soon after the war. Then the estate was sold up. Only a nut could live out there.'\n\nI left it at that. But what I had been noticing were telephone posts marching beside us. They stayed with us through a plantation of larches before turning right, across a field. In a little distance we also turned right, to find our way barred by a ramshackle field-gate; the D.C. made to get out to open it: I stopped him and climbed out myself. I prowled round the gate. At first, I saw nothing. The gate was secured by a chain and staple; a shag of ivy had grown up the hinge-post and was sending a tendril along the top bar. Then I spotted a scrape, slight but definite, where the gate had brushed the crown of the track; and when I came to unhook the chain there was a glint of silver where it left the staple.\n\nI got back in the car.\n\n'The gate has been used lately.'\n\nHanson sniffed. 'So what does that tell us?'\n\n'It tells us the gate has been used lately.'\n\n'Oh great. Maybe we'll pinch ourselves a poacher.'\n\nWe drove through and refastened the gate. Here the track was descending through a belt of elms. At the foot of the descent lay a wash of mud, and in the mud was a clear imprint of tyres.\n\n'Is your poacher mechanized?'\n\n'Yeah, well,' Hanson said. 'It could be the farmer has business down here.'\n\n'Park the car,' I told the D.C. 'Perhaps we need a walk to clear our brains.'\n\nWe left the car. Now the track climbed again, with the elms still tall on either hand; but then it levelled suddenly and made a shallow turn; and there, at the turn, stood a blue Viva.\n\n'Hell!' Hanson breathed.\n\nWe hastened up to it. It was unlocked, and the keys were missing. It carried a West Essex registration, and the licence had been issued in Harlow. In the glove compartment were maps, pressure-gauge, duster and service records from a Harlow garage. They were made out to K. Stillwell, Orchard Croft, Harlow. The driving seat was pushed back. The boot was locked.\n\n'Pinched,' Hanson said.\n\n'If it's Bilney's.'\n\n'Yeah, but it has to be,' Hanson snapped. He hoisted the bonnet, popped open the distributor and dropped the rotor arm in his pocket. 'So now he won't be travelling far, and all we have to do is grab him.'\n\n'Unless he has pinched another car.'\n\n'Oh sure, they grow on trees out this way.'\n\nHe bustled away up the track: I signalled the others to follow. A hundred yards further on the belt of elms ceased abruptly. Beyond lay a slope of shaggy pasture, running down to reed-beds and a weed-choked pool; but to the right, nestling under the trees, stood a timber chalet and a cluster of sheds. Seeing it, Hanson broke into a run, and there seemed little point in bawling to him to wait. We raced after him. Breckles had the good sense to shepherd the D.C. to the rear of the chalet. Hanson vaulted the rail of a verandah which enclosed the front of the building and launched his shoulder at the door. But the door didn't give.\n\nHanson thumped it. 'Come out, Bilney. We've got your hidey-hole surrounded!'\n\nNo response. Hanson thumped again; then ran to a window and peered in through his hands.\n\n'Jesus Christ!'\n\nHe backed away from the window, his lantern-jaw sagging. I jumped up beside him.\n\n'What's the matter?'\n\n'There's a bleeding body in there!'\n\n## CHAPTER FOURTEEN\n\nTHE D.C. FETCHED a tool-roll from the car and Breckles expertly jemmied the door. The body was in a room to the right of a hall that extended from the front of the chalet to the kitchen. The room was a bedroom; it was sparsely furnished with a Safari camp bed and a folding chair; the body was lying beside the bed on the side that was furthest from the door. It lay on its back. The arms were bent, the fingers hooked, the legs folded sideways. It was terrifyingly dead: wide-eyed and snarling. There was a lot of blood on the board floor.\n\nHanson sent air hissing through his teeth. 'Hell oh hell. The bloody bastard.'\n\nBreckles and the D.C. were staring pop-eyed; most likely they hadn't seen a killing before.\n\n'Is this \u2013 is he the chummie?' Breckles ventured.\n\nSomehow that horror defied identification. The face was now the simple face of humanity, shocked and outraged by a hideous dying. But there was the blunt finger, clawing at air.\n\nI nodded. 'He's Bilney.'\n\n'But who . . . what happened?'\n\nI hunched a shoulder. 'Someone stabbed him. Can't you see?'\n\nPerhaps Breckles couldn't see. Bilney's shirt-front looked just a chewed-up, bloody mess. I kicked the bed aside and approached the body, taking care to keep my shoes from the blood. There were multiple stab-wounds in thorax and abdomen, defensive cuts on each hand. Bilney had fought, but it hadn't helped him. The attack had been too strong, too fierce. It had ended in a frenzy of superfluous stabbing as Bilney lay dying beside the bed. When? I stooped to manipulate a leg; rigor mortis was complete. The blood-puddle was largely congealed, though still liquid towards the centre. Sometime yesterday: perhaps mid-afternoon, when Bilney had returned from his rendezvous with Deslauriers. The killer had been waiting for him: he may have suspected it, have left his car down the track while he reconnoitred.\n\nBut now . . . what?\n\nOne killer, or two?\n\nThis time there was no clue of sinistrality. Just a knife going in with mortal hatred: someone who couldn't kill Bilney enough.\n\nI fished for his wallet, a smart lizard-skin number. Like Freddy's wallet, it hadn't been robbed. Forty-seven pounds in fives and ones, a twice-endorsed driving licence, stamps, receipts. A revenge killing? One for one? But somebody had known where to look for Bilney. Had tracked him to this obscure place: maybe with a little help from his friends.\n\nI stood back from the body and looked round the room, but the room was emptier than a punishment cell. The chair, the camp-bed, the naked floor: boards of which creaked under my foot. I stooped to finger one. It pulled up easily, the securing nails rusted through. Underneath, an empty air-space and the puggy smell of dry-rot. A futile gesture. I dropped the board back. All that room really contained was the body.\n\n'Come on. There has to be a phone here somewhere.'\n\n'Yeah, but it doesn't make bloody sense!' Hanson yapped.\n\n'It made sense to someone,' I said. 'And he was no playboy. This chummie we need behind bars.'\n\n'But who'd want to do it?'\n\nI stamped down the board. 'Perhaps who wanted it done is a better question. Only standing here won't get any answers, so let's call in the people who may have some.'\n\nNobody was sorry to get out of the bedroom. We found the phone, as I knew we would. This was clearly where Bilney had spent his time during his absences from the Reed-Cutters. No doubt he had jibbed at staying at a place that was so remote and uncomfortably furnished; but here was his point of contact, and here each day he would have to come. In fact, we found the evidence by the phone: a chair and a tin-lid of cigarette-ends. Sitting there, he would have been told of Freddy's appointment with Rampant, would have received his summons to meet Deslauriers .. .\n\nI left Hanson to do the phoning and joined the others in a search of the premises. What the theory of a killer waiting in ambush needed was evidence of a break-in, and that we found in a forced window. But not very much else. In the bleak little kitchen was a cache of empty cans and soiled picnic plates; half a sliced loaf, a lump of cheese, tea, sugar and tinned milk. In the Elsan closet we found a local paper with Freddy's demise in the stop-press, and outside a few spots of oil showing where Bilney had parked the Viva.\n\nWhen we returned to the parlour we found Hanson seated by the phone with an inspired light in his grey eye. He had a cheroot going; he waved it at us, adding ash of his to ash of Bilney's.\n\n'Sit down. I've got the whole picture.'\n\nThere was only one other chair: I took it.\n\n'Look, I've been thinking about what you said. It's not who did it, but who wanted it done.'\n\n'I said that might be the angle.'\n\n'But yeah. It fits all down the line. From somebody shopping Freddy's mob to us finding chummie behind the what-not.'\n\nI shrugged, guessing pretty well what was coming. That notion had jumped into my mind, too. But there were snags. It wasn't nearly as simple as Hanson was now proposing to make it seem.\n\n'Let's have it, then.'\n\n'It's like this. We've got it all happening round Deslauriers. Nothing goes on but she has a link with it. Unless she's in the middle, it doesn't work.'\n\nI nodded. 'Go on.'\n\n'Take it back to the beginning. Freddy coming up here to do a job. It goes like clockwork, but surprise, surprise \u2013 someone blows the gaff to Met.'\n\n'You can take it back further,' I said.\n\n'How's that?'\n\n'Someone influenced Freddy to do that job. It wasn't in his class. Below a hundred grand, Freddy wouldn't have wasted a week in the country.'\n\nHanson paused. 'You're saying she persuaded him?'\n\nI trailed my hand. 'You're telling me.'\n\n'Yeah, well, why not? He was loopy over her. If she said jump, he'd fall off a cliff.' He fizzed smoke. 'So take it all the way: she had this caper planned from the start. A job out of London with a hideaway handy, where the killer can wait till she turns him on. And that's how it worked out. The mob pulled the robbery, Deslauriers arranged for Met to be waiting. Because why? Because the mob might make her trouble if they got the idea she'd bumped-off Freddy. Are you happy with that?'\n\n'Who phoned in the tip-off?'\n\n'What's wrong with chummie next door?'\n\n'He'd be in London.'\n\n'Who says she didn't phone him?'\n\nI nodded reluctantly: it was possible.\n\n'Fine,' Hanson said. 'That was the mob fixed. Now she could whistle up her killer. He picked himself a nice car and arrived in Haughton Thursday evening. But the lady didn't like him staying in the open so she sent him on here. Only this place is damp or it has draughts, so chummie took a room at a cosy pub.'\n\n'Hold it,' I said. 'I don't like that part.'\n\n'Huh?' Hanson's thick brows registered surprise.\n\n'If Deslauriers had planned this, Bilney wouldn't have gone to Haughton and taken the risk of meeting her at the Barge-House.'\n\nHanson dragged on the cheroot. 'But that's what happened.'\n\n'That's what happened, and it doesn't fit. Bilney should have been briefed to come straight out here, and not to have shown in Haughton at all.'\n\n'Perhaps he lost his way, sir,' Breckles suggested. 'A stranger wouldn't find Turnpudden Hole in a hurry.'\n\n'Yeah,' Hanson said. 'Yeah, that has to be it. He lost his way and had to get instructions.' He gave me a leer. 'Okay?'\n\n'He didn't need to book a room to get instructions.'\n\n'So he was fed up with horsing around. And that makes sense to me, anyway.'\n\nHe stared around, daring comment. I let it go. Hanson dragged smoke.\n\n'Now we have him sitting by this phone. Rampant made his play for Freddy. Deslauriers saw it as an opportunity, got on the blower and alerted Bilney. Deslauriers knew where the meeting would take place because Freddy described it to her from last time. Bilney was waiting when Freddy turned up and he came in behind and let him have it. So that was part two over. The mob was inside, Freddy was cooling on the heath. What should have happened next was Bilney going home and leaving the coppers to chase their arses. Only Bilney doesn't do that. And why not? Because he's caught the sweet smell of money. If Deslauriers wanted Freddy dead the odds are she stands to collect his dough. So Bilney stays and acts tough, and Deslauriers has to get him off her back.' He drew breath. 'What's your guess at the E.T.D.?'\n\n'Near enough to the time you're after.'\n\n'Three or four p.m.?'\n\n'About that. I'm not a professional, but I've watched them at work.'\n\n'And that would be the time you met Deslauriers yesterday? When she was moored at this end of the Broad?'\n\nI nodded. 'All that part fits. Deslauriers could have been setting him up.'\n\n'So that's it,' Hanson said. 'What we're looking at in there is part three. Bilney tried too hard, and the lady countered with another rough boy from the Smoke. And this time I'll bet it was a quick, smart job, with chummie taking off for home straight afterwards. Which is where we'll find him, if we find him. This end the case has gone cold.'\n\nHe shot me a fierce look, backed with smoke.\n\n'It does seem to make sense, sir,' Breckles ventured.\n\n'Of course it does,' Hanson snorted. 'Deslauriers has to be the one behind it.'\n\nI wriggled a shoulder. 'I give you that. Deslauriers is in it up to her neck. But there is one thing that goes on bothering me about this phenomenal eruption of pro killers.'\n\n'And what's that?'\n\n'The level of ferocity.'\n\n'Huh?'\n\n'We've just been looking at another example. And it's like the first: too many blows struck. In fact, too like the first for comfort.'\n\nHanson, about to jump in, checked himself. 'Are you trying to tell me it's the one man?'\n\nI nodded. 'I think it has to be. And that one man is no pro. He's an amateur, probably a psychopath, paranoid, his temper on a hair-trigger. A hate-killer. A man with grudges against both Bilney and Freddy.'\n\nHanson stared, his eyes rimmed. 'Oh, come on now! What's his motive?'\n\n'At a guess, I'd say Madame Deslauriers. As you were saying, she must be behind it.'\n\nHanson did his famous impression of a man watching a giraffe turn into an elephant: then he came down on the tin-lid with his cheroot stub and mashed the stub into shreds.\n\n'It won't work!'\n\nI shrugged. 'It will. It fits the evidence better than the other way.'\n\n'Like hell it does. Just ask yourself this \u2013 why was Bilney here, if not to do Freddy?'\n\n'He would be here because he was sweet on Deslauriers.'\n\nHanson hooted. 'That's so likely! And him leaving a note in his pad to tell his girl-friend he was on a job.'\n\n'It's the sort of note he might have left a girl-friend. No doubt he would still have uses for Mavis. And the note troubles me a good deal less than Bilney's turning up at Haughton unannounced.'\n\n'Yeah, yeah,' Hanson said. 'Big point taken. And the mob getting shopped \u2013 how about that?'\n\n'Call it part of the same deal. Malice towards Freddy. Followed by a knife when opportunity offered.'\n\nHanson threw up his hands. 'It stinks. And all this time the lady sits by smiling.'\n\n'She may not know, or not know all of it. Then she would behave in just the way she's behaving.'\n\n'It still stinks.'\n\n'Listen,' I said. 'Let's give it a run-through my way. The evidence points to the sort of killer I've described. At least we have to try it from that view-point.'\n\nHanson clawed his hand across his face. 'Okay, you try it.'\n\n'First we know that Bilney was acquainted with Wicken. Deslauriers admits that she has met all the gang, and so through Wicken she could have met Bilney. It doesn't have to follow that Bilney was her boy-friend, but it could very well follow that he was attracted by her. If he learned from Wicken that she was staying at Haughton he might have been foolish enough to come out after her.'\n\n'Him having such a sentimental record,' Hanson said sourly.\n\n'Perhaps. But the lady has a lot of horse-power. And it squares with Bilney taking a room at the Three Tuns and bribing a waiter to smuggle her a message. A paid killer wouldn't have run those risks, but they would be part of the fun for a roughneck Romeo. And Deslauriers was game. She didn't want him next door, but he was welcome to stay around to brighten up the scenery.'\n\nHanson sniffed. 'Yeah, that sounds like her. A lover boy in every bush.'\n\n'So she sent him here, where he could nurse the phone and be available for romps. Bilney didn't fancy sleeping here, but he wasn't hiding, so there was no reason why he should. Even Freddy's murder probably didn't worry him, and may have given him a motive for staying on. Now he was free from competition and could expect a readier response from Deslauriers.'\n\n'It's lovely,' Hanson said. 'I'm almost sold. But then why does Bilney finish up so dead?'\n\n'For the same reason that Freddy did,' I said. 'They were both Deslauriers' lovers.'\n\n'You mean there's a mad lover in the scene somewhere?'\n\n'For the moment I'm merely reading the evidence. It suggests that Bilney didn't come here as a paid killer, but that it was his connection with Deslauriers that led to his death.'\n\nHanson glared at Breckles. 'Which side are you on?'\n\nBreckles looked flustered and rolled his shoulders.\n\n'Me too,' Hanson said. 'It's a way of looking at it, but I always gag at demon lovers.'\n\n'And the mode of the killings?'\n\n'I'll swallow them,' Hanson said. 'What's wrong with some pro killings being messy? Bilney's job may have been his first, and what happened here a tit for tat.'\n\n'That was never a pro job.'\n\n'You're forgetting who paid for it. The lady could have ordered fancy trimmings. And I would sooner go along with that than with a leching Bilney and a demon lover.' He hesitated, eyes suddenly small. 'Or did you have a candidate in mind?'\n\nI let my face go blank. 'Just someone in the know. Who is left-handed.'\n\n## CHAPTER FIFTEEN\n\nAN AMBULANCE AND two Wolseleys came up the track and halted smartly where Bilney had been parking the Viva. Hanson strutted out to take command, I strolled away down the rough meadow.\n\nDown there by the pool was a different world from the shabby chalet with its busy policemen. All fresh growth, thrusting new reeds, tender-leaved alders and polleny willows. Water-hens foraged among the rushes; a handsome male grebe fished the open water; reed-warblers flirted among last-year's reeds and made creaky, dripping comments. A world that couldn't care less about the mind-ridden animals up the slope: the perverted, self-doomed animals who killed each other when they weren't hungry.\n\nI lit my pipe and continued to stroll, switching my mind off for later. I came to an old, ruinous hide, sited to overlook the pool and the reed-beds. A trace of Clytie Gifford, no doubt. I squatted for some minutes on its crude bench. The grebe came close, and I spotted its mate seated haughtily on their sloppy nest. Then a true breath-catcher: reed-pheasants: two pairs, swinging musically through the fawn, dead stems. I sat as still as the rotting bench and let my pipe go cold in my mouth.\n\nWhat a hell of a place for Bilney to have come to: a hell of a place for him to die in!\n\nThe pool, the lodging, were for gentle people, gentle as the birds and the tranquil trees.\n\nYet here he had come, loutish, insensible, locked in some grubby prison of intent: basely living and basely dying, with his sneering eyes staring blind.\n\nWhat did he want? What did any of them want?\n\nThere was nothing they were capable of receiving.\n\nIn or out of the Scrubs, the Moor, they were prisoners, the key turned by their own hand.\n\nI got up angrily, disturbing the reed-pheasants and sending the grebe into an instant dive. I stamped back to the chalet, and stood watching policemen puffing their powder and flashing cameras.\n\nTwo hours later they were through, and the corpse had already departed for town. The results were debatable, due very largely to the unhelpful character of the chalet. It was a bad place for dabs. The paintwork was rough and the metalwork corroded. Whether chummie had been careful with his break-in or not, there were no traces of latents round the forced window. Even Bilney's dabs were found only once, some faint impressions on the bedroom door. And in the kitchen, and again in the Viva, were slight indications that wiping had taken place.\n\nHanson received this intelligence with gloomy satisfaction.\n\n'At least it proves one thing \u2013 chummie was a pro.'\n\nI couldn't contradict that. My enthusiastic amateur would scarcely have bothered with such refinements.\n\n'But what was he doing in Bilney's car?'\n\nHanson flipped his hand. 'Seeing what he could nick. There was a car with no owner. Chummie just couldn't help giving it a frisk.'\n\n'He didn't rob the body.'\n\n'Maybe he was squeamish.'\n\n'I'm told the knife went in eleven times.'\n\n'Yeah, well.' Hanson squirmed. 'Then there'd be blood about, wouldn't there? He wouldn't be keen to get mixed up with that.'\n\nI shrugged and didn't push the point; yet still it seemed a little odd. Chummie had finished his job and was on his way out, then he stopped to enter the blue Viva. Something obvious that took his eye? A clue that might have led us to him? Ah well: we would know one day, or know we didn't need to know.\n\nI drove back to the Norchester H.Q. with Hanson and put through a call to Dainty. I got his assistant, Inspector Jason, a dapper young man with a cooing accent. Jason had no news for me. I told him my news; he listened with little dove-like murmurs. He liked the bit about the multiple stab-wounds and the blood pooled round the body.\n\n'Do you have a sus, sir?'\n\n'Not exactly a sus. There are two schools of thought going here. One says the killer is a pro from town. That's the school you had better follow.'\n\n'The Super is betting on Whitey Ferrier, sir. We've had whispers from the snouts. Quarles set up a snatch for Ferrier last month and the job went sour. We made three arrests.'\n\n'Great,' I said. 'Great. But now we want whispers about a hard boy. He would certainly have been missing from the scene all yesterday. He uses a knife, and he's far from stupid. Are you taking notes?'\n\n'Wait \u2013 yes, sir!'\n\n'Here is the thing that may catch him. He has probably had connections with the Quarles gang: enough so that Deslauriers would know how to contact him.'\n\n'Is she your sus for setting it up, sir?'\n\n'She's my nothing. Just get me some action.'\n\nJason cooed assurances, and I hung up. Hanson, who'd been listening, fingered his chin.\n\n'I'd say the lady was our next move,' he said. 'Maybe we could spare her some photographs of Bilney.'\n\nI rocked my chair and gazed over his head. 'Would you have finished with Freddy's car?'\n\n'What's that got to do with it?'\n\n'She wants it back. And letting her have it would be a nice gesture.'\n\nHe was gazing hard at me. 'You want us to give it to her?'\n\nI twitched a shoulder. 'It's been in my mind. Wondering what she intends to do with it. Who'll be sitting behind the wheel.'\n\n'You mean she's planning a skip?'\n\n'I don't know that. But just now she lacks the number one requisite.'\n\nHanson rasped his chin again.\n\nFive minutes later, I was sitting in the Bugatti.\n\nOr more or less in it.\n\nOne's first impression was of putting on a one-man roller-coaster: of sitting far too high and naked, and much too close to the shoulder-width wheel. The pedals were quaint but fairly available, the instrumentation on the light side; the gear-shift came to hand, though it felt grotchy, and one realized in a flash why the handbrake was outboard. That's where the space was, not in the cockpit. The cockpit was strictly something to wear. At a guess, Louis Chiron was a short, wiry man, and his mechanic an under-nourished midget.\n\nI started the engine and let it roll while I got used to my surroundings. Straight ahead was the curved scuttle with a half-moon windscreen mounted above it. Then a mile of curved, tapering bonnet, secured by two leather straps, and finally the geometry of the out-rigged, narrow-tyred wheels with bicycle-type mudguards curling round them. Well, it was built to drive. I clonked off the handbrake and fed in some power. Ettore's car took a bound across the M\/T yard, screamed disapproval, then settled for a jerky, restive walking-pace.\n\nWe made the street. By good fortune the teatime rush-hour was nearly over. I slanted the wheels towards the Ring Road and double-declutched cautiously through the gears. Soon we were doing a dyspeptic thirty and I was getting a bit of feel. Wind was fingering my hair and cooling the elbow I was trailing nonchalantly over the side. The wheel bothered me. It was giving me the impression that I had hold of the helm of the _Queen Mary_. Also the steering was surprisingly heavy, and less connected with direction than one might have supposed. I checked the brakes. In good time, they worked, though not as though they intended to waste any rubber. On the other hand the accelerator seemed set on a watch-spring, while the clutch came in like a Mills bomb.\n\nI reached the Ring Road, where the roundabout junction taught me some more about vintage steering. It taught me also that I was driving a car that would get more consideration than any ambulance. The last hell-bent commuters stood on their noses to let Louis Chiron go through, then happily fell into procession behind him as he cantered along at the legal forty. A classless society? Not among car-owners. The same thing happened at successive roundabouts. When I came to my turn-off the queue was solid behind me, while ahead stretched a relatively empty road.\n\nBut now my moment had come. We cruised past the delimit signs. The road ahead was reasonably straight. I squeezed a little; Ettore's car pricked up its ears and began to surge. There was an Escort behind me, and for a languid moment it seemed likely to cling to my tail; then it walked backwards in the rear-view mirror and disappeared behind some trees. Small fry. Up front, an S-bend. I jigged the reluctant brakes in good time. No other traffic. I went straight through, hugging the wheel in Segrave-style. A clear road. I let her scream. The needle drifted over the ton. My hair was getting torn out by the roots, my trailed elbow was trying to flap. Bryan de Grineau should have seen this. I could feel visible slipstream shaling off me. Le Mans, Brooklands, the great days: watch me drift the next bend.\n\nIt was nearly my last. I hit it at a speed that the Lotus would scarcely have noticed. The next moment I was doing one of those celebrated fighting-the-wheel acts straight out of a pre-war _Modern Boy_. Now I knew why they had those wheels. I was losing the rear-end in gigantic hops. At each hop the tyres gave a baleful screech and spirted distress-signals of black smoke. The real McCoy. I steered-in, steered-out for it might have been the next hundred yards, and just when it seemed it was going on for ever, the car gave a brutal lurch and came back on course. A man's car, keep it in. The tape on the wheel was sodden with sweat. I crept along at a painful sixty, waiting for the jelly to drain out of my arms.\n\nBut strangely, I felt happier after that, as though now the car and I had got to know each other. So it didn't handle like a Lotus \u2013 never mind: it still had the heart and sinew of a lion. The next firm bend I got right, putting her into it slower, coarser: a problem solved. I was learning. No doubt Chiron had had his troubles, too. We came through Wrackstead in a growling amble, nosing aside the peasant traffic. When we crossed the bridge I touched my horn: shrill trumpets blasted 'Colonel Bogey'.\n\nI parked the Bugatti in the hotel forecourt, where it sat simmering like a contented cat. The time was between tea and dinner and the Barge-House had an unpeopled air. Dutt was in the lounge, reading an evening paper. Through the open french windows he had a view of the lawn. At the bottom of the lawn, decorating a sun-bed, lounged Mimi, Madame Deslauriers.\n\nI nodded to Dutt and took a seat by him.\n\n'The lady looks lonely down there.'\n\nDutt gave me a slow grin. 'Not so lonely, sir. She's been having a teatime chat with Bavents.'\n\n'How many does that make?'\n\n'It makes three. She had him up in her room again this afternoon. He was in there for nearly an hour, but he didn't come out looking very chuffed.'\n\n'Was he happier at teatime?'\n\n'Not so as you'd notice, sir. But with all that hair it's hard to tell. My hunch is the lady wants him to do something which he isn't very keen on.'\n\n'I'd have thought she could talk him into anything.'\n\n'Yes, sir, you get that impression. And as a matter of fact I got round to wondering just how well they do know each other.'\n\nI leered. 'You did, did you?'\n\nDutt had the grace to turn pink. 'Not like that, sir! What I mean is whether they'd met before she came here.'\n\nA shrewd point. 'What was your conclusion?'\n\n'Well, sir, I don't know about a conclusion. But I did give the University a tinkle to find out Bavents' home address.'\n\n'And that was?'\n\n'Chelsea, sir. Vought Street, Chelsea. His father keeps The Peacock pub. That's a few blocks away from Upper Cheyne Row, but not outside the lady's district.'\n\n'Well, well,' I said. 'So they could have met.'\n\nDutt looked pleased. 'It's on the cards, sir. If you ask me the lady has a taste for pubs, so she could have spent an evening in The Peacock.'\n\n'And now she wants him to do something he doesn't fancy.' I gazed down the lawn at the recumbent Mimi. She was draped elegantly in the evening sunlight, gazing at nothing through tinselly-framed sun-glasses. 'Has Bavents used the phone?'\n\n'Not to my knowledge, sir. I've been keeping an eye on the phone.'\n\n'Has he been out?'\n\n'Don't think so, sir. Not since yesterday afternoon.'\n\nI hesitated. 'He was out then?'\n\n'Yes, sir. If you remember, I couldn't check his statement. I'd say he went out after they finished serving coffee, and it was after you came back when he returned.'\n\nIt was indeed. I did quick arithmetic. It must have taken me an hour to find Mimi's launch. In half that time Bavents could have driven to the chalet, hidden his Mini and forced the window. And Friday evening too he had been at liberty . . . while there were plenty of chef's knives in the kitchen! Dutt was eyeing me thoughtfully.\n\n'Does that fit in, sir?'\n\nI nodded and briefly brought him up to date. Dutt listened stolidly. I sketched Hanson's theory without bothering to throw in comment. Dutt wrinkled his nose.\n\n'I like Bavents better, sir. Those rough boys are all for keeping it simple.'\n\n'Where is Bavents now?'\n\n'He's in the kitchen. Do you want me to fetch him out for you?'\n\nI shook my head; first things first. I rose and went down the lawn. Mimi received me with a melting smile and was gracious enough to remove her sun-glasses.\n\n'You are back, my friend! Was the day tiresome?'\n\nI took the Bugatti's key from my pocket. It was a custom-made, gold-plated key, attached to an enamel badge by a snake-chain. Not to be mistaken. I let it dangle. Mimi's green eyes fixed on it covetously.\n\n'Ha! You have brought me Freddy's car.'\n\nI gave the badge a flip. 'I've brought it from town.'\n\n'But for me, huh?'\n\n'Perhaps. At the moment it is just part of Freddy's estate.'\n\nShe sat up on the bed. 'But he has left it to me! You will find it is so in his will.'\n\n'I haven't seen his will.'\n\n'But yes, it is true! You have only to ring up Freddy's solicitors.'\n\nI flipped the badge again. 'Of course, if you have seen it . . .'\n\n'It is the same. I shall have the car.'\n\n'But if I could have your word for it?'\n\nHer eyes narrowed; she stared at the key, then back at me.\n\n'You are mean, Monsieur. You know quite well that the car will be mine. But it doesn't matter, I will wait. Only I shall not think you are very generous.'\n\nI moved my shoulders. 'That's too bad. Especially since I bring unpleasant news.'\n\n'Unpleasant news? How?'\n\nI dropped the key in my pocket. 'I think we had better talk about that in private.'\n\n## CHAPTER SIXTEEN\n\nI GAVE DUTT the pleasure of escorting her to the office and myself went in search of Frayling. I found him in the dining-room, decanting spirit into the chef's stove from a Winchester bottle. It was near the door of the kitchen; I drew him further off.\n\n'Did Bavents have permission to go out yesterday?'\n\nFrayling's tentative smile became anxious. 'It was all right, wasn't it? He didn't get into any trouble?'\n\n'Please answer the question.'\n\n'Well \u2013 yes, I suppose so. He told me he had to attend a Student Union meeting. Apparently the meeting had been called unexpectedly and was to do with his being sent down.'\n\n'Have you a private phone, other than in the office?'\n\n'There's one in my flat upstairs.'\n\nHe took me there. I rang the University; I was handed around between secretaries. Eventually I contacted the Student Union liaison officer, who informed me there had been no meeting for three weeks. I asked him if Bavents' rustication was an item on their agenda, and he informed me that that was unlikely. The matter had been discussed at a previous meeting, when no motion had been put before the committee.\n\nFrayling was concerned. 'I just don't understand it. He must have some personal problem he wants to keep quiet.'\n\nI threw him a look. 'Don't mention this conversation to him. And don't let him out again without telling me.'\n\n'But look \u2013 what has he done?'\n\n'Just do as I say.'\n\nI left him gazing after me with wretched eyes. Outside, across the road, was parked an electrician's van with two of Hanson's men yawning inside it. I went to it and slid back the door.\n\n'Do either of you two know the waiter, Bavents?'\n\nThey didn't, so I gave them a _portrait parl\u00e9_ : which in Bavents' case wasn't difficult.\n\n'If he leaves, detain him. I want him for questioning.'\n\nThey looked uncertain. 'Will there be a charge, sir . . . ?'\n\n'I want him stopped. Understood?'\n\nApparently it was. I slammed the door.\n\nI had brought back copies of the Bilney photographs and I took them with me into the office. Mimi had appropriated the swivel chair and sat smoking, her sandalled feet on the desk. She was wearing her hot pants, along with a sleeveless top in a black, clinging jersey material; she looked politely bored, and didn't bother to glance up as I entered the room. I adjusted the curtains; Dutt, in his corner, sat quietly sharpening a pencil. I perched on a corner of the desk, remote from the sandals, and laid down the photographs with their backs uppermost.\n\n'Have you been in a blue Viva car lately, Madame?'\n\nMimi considered it. She flicked ash. 'Do you think I have, Monsieur?'\n\n'I don't know for certain. But later I shall need your fingerprint sample.'\n\n'Aha. Then you have such a car.'\n\n'We have both the car and the driver.'\n\nMimi sat still, appraising her toes. Not a twitch of emotion on her magnificent face.\n\n'Perhaps you should ask him if I have been in his car. After all, I know nothing of the makes and colours. The cars today are so much the same. It is only the men that one remembers.'\n\n'Unfortunately, this driver is answering no questions.'\n\n'No?' The corners of her mouth dimpled. 'Possibly he is nervous, being questioned by policemen. I believe it happens with many people.'\n\n'I am sure he isn't nervous.'\n\n'Then it may be stubborn? In some subtle way you have hurt his feelings?'\n\n'This man doesn't have feelings, Madame Deslauriers. I have pictures of him here. Would you care to see them?'\n\nHer eyes flickered to mine. She could sense a trap, but there was now no way for her to avoid it; and doubtless she felt confident she could control herself and subdue any sign of recognition. She reached for the top photograph and turned it over.\n\nHer cry had the thrill of mortal agony.\n\nHer legs jerked from the desk: she sprang up, gasping, and stood with her face turned to the wall. She hugged herself, her breasts, her stomach, fighting to hold her hysteria in check. The sense of violence was awesome. She should have fainted; instead, there was this brief, epileptic-like struggle; then the frantic breathing began to subside, and her arms sank shakily to her sides. She pulled round to face me again, her cheeks flushed, her eyes smouldering.\n\n'You . . . _pig_!'\n\nI reached for the photograph, but she lunged forward and seized it with eager greed. She gazed at it angrily, triumphantly. Then she threw it in my face.\n\n'That was cheap, you pig \u2013 cheap! How dare you show me such a picture?'\n\nI returned the photograph to the pile. 'I certainly agree it should not have been necessary.'\n\n'You take advantage. That is how my husband died. It is what they did to poor Freddy. You say \"Aha, aha, this will break her down. This is just the thing for little Mimi\".'\n\n'You are over-reacting, Madame.'\n\n'Pig!'\n\n'You were less disturbed when the victim was Freddy.'\n\n'Did I see such photographs of what happened there? Were they pushed under my nose in this fashion?'\n\n'Freddy was your lover.'\n\n'Ha, ha, lover! In the end we were just friends.'\n\n'You had no grief for him. Surely this emotion is excessive for a mere stranger?'\n\n'It is not for a stranger. It is for shock. It is for that horror you make me see. It is unfair, a low trick. I am angry: I despise you.'\n\nI shook my head. 'Too uncharacteristic.'\n\n'Ha?'\n\n'Madame Deslauriers has more poise.'\n\n'Beast and pig!'\n\n'I think Madame Deslauriers could have seen that picture without turning a hair.'\n\n'Am I a butcher then? An executioner?'\n\n'You are a person of coolness and resource.'\n\n'Ha-ha, flattery will not do either.'\n\n'Nor will any further denial that you know that man.'\n\nShe snatched her head and glared at the pile of photographs. Her colour had slowly been returning to normal. Her breathing was well in check again and her hands trailing loose. Now she let her eyes die, too, the lids relaxing and hooding. The passionate set of her lips began to soften, to curve.\n\n'You are a bastard, Monsieur. Which you know very well.'\n\n'Perhaps you will take your seat again.'\n\n'But understand that I hate you.'\n\nShe sat however, and replaced her immaculate feet on the desk. Then she took cigarettes from her pants-pocket, lit one and blew caressing smoke.\n\n'Of course, I do know that man. I knew him when you showed me the picture this morning. But I did not choose to acknowledge that. Which you will admit is my privilege.'\n\n'Why didn't you choose to acknowledge it?'\n\nShe cocked a shoulder. 'Let us say you were interrupting my breakfast. Also I did not know him well. I didn't wish to answer questions.'\n\n'You know his name?'\n\n'Why not? Bilney.'\n\n'How did you come to make his acquaintance?'\n\n'One evening I was in the Hammersmith Feathers with Freddy and Wicken and we were joined by this man. We went on to other pubs. He came with us. I think he was hoping for some business with Freddy. There were plenty of hangers-on like that. Doing a job for Freddy was a safe thing.'\n\n'Did Freddy employ him?'\n\n'You must ask Wicken. It was he who introduced him to Freddy.'\n\n'But after that you would see him around again?'\n\n'Oh no. I saw him just that once. He was not my style, you understand. He was a dull-witted, uncouth boy. If you ask me I do not think Freddy would have used him. Freddy was a man who required intelligence.'\n\n'But he was a friend of Wicken's. And Wicken you did see.'\n\n'Wickey, yes. Not Wickey's friends.'\n\n'So you could have got in touch with Bilney through Wicken.'\n\n'I suppose I could. But why should I?'\n\n'Say a personal interest.'\n\nShe laughed derisively. 'I tell you already he is not my style. I like some subtlety with my beef. Even among policemen one can find it.'\n\n'Then what about Bilney?'\n\n'What about him?'\n\n'Didn't Bilney take a personal interest in you?'\n\nShe gestured with the cigarette. 'He was lecherous, of course. But it goes no further with that sort of animal.'\n\n'He didn't try to see you?'\n\n'I think you are joking.'\n\n'Didn't follow you around? Angle for attention?'\n\n'Not that one. Never.'\n\n'Yet we find him in Haughton. Staying next door. Sending you a message.'\n\nShe breathed out a long miasma of smoke. 'Does that make sense to you, my friend? That I would play footsy with such a dumb ox, when there were as good or better right on my doorstep?'\n\n'Bilney was there. He was no mirage.'\n\n'So. I am not responsible for that.'\n\n'You spoke to him. Directed him to Freddy's hideaway. He could have learned of that only from you.'\n\n'You are forgetting Wickey. Wickey would know of it.'\n\n'Perhaps. But Bilney learned of it from you. His first move was to book at the Three Tuns. He didn't change quarters till he had talked to you.'\n\n'I do not admit that.'\n\n'Then tell me something else. What was Bilney's purpose in hanging around here? Sitting every day in Freddy's chalet, smoking, waiting for the phone to ring?'\n\nShe gestured peevishly. 'Why should I know this?'\n\n'Because yesterday the phone rang, and the caller was you. You told Bilney to drive to the lane leading to the Broad, and you met him there and spent two or three hours with him.'\n\n'I did all that?'\n\n'You sat in the car with him. The car was pulled off into the trees. You had a picnic of sorts, fruit, chocolate \u2013 I can be more precise after the post- mortem. Because then Bilney died: straight after that. He went back to the chalet and met his killer. As though the real purpose of getting him away from the chalet had been to give the killer an opportunity for ambush.'\n\n'And you are accusing me?'\n\n'The killer knew where to go. He knew that Bilney would be absent. Two things that you knew and nobody else knew \u2013 just as there were two things that you knew about Freddy.'\n\nShe flicked ash on the floor. 'Poor Mimi. This is quite a formidable indictment.'\n\n'I think you had better help us.'\n\n'That was always inevitable. You are a man of such persuasion, my friend.'\n\nShe wanted a drink, but I wouldn't permit it; she sat awhile with a sulky expression. Though I had drawn the curtains of the door and counter windows, we could hear chattering people passing along to the dining-room. The sound accentuated the arrest of time which is the peculiar quality of interrogation. My refusing the drink had been a symbol. Now we were remote from the world of innocence.\n\nAt last she folded her legs with a sigh.\n\n'Monsieur's psychology is impeccable. I am a creature exposed to perpetual temptation. How sad if I spent my time rejecting it.'\n\n'Bilney was your lover?'\n\nShe made a faint mouth. 'I would rather not award him that title. He was \u2013 what shall we say? \u2013 a taste for garlic. He served as a purge for the coarser emotions.'\n\n'How long was he put to these medicinal uses?'\n\n'Oh, he has been around since Easter. I met him as I told you, in the Hammersmith Feathers. Next day he rang me. It developed from there.'\n\n'Did you go to his flat?'\n\nShe hesitated. 'No. There is a discreet hotel in Kensington. Not that we used it very often. One takes purges only occasionally.'\n\n'He was keen?'\n\n'Naturally.'\n\n'Wanted more than you would give him?'\n\n'Yes. It is the character of the type. Because of that we had a disagreement, which is why he followed me up here.'\n\n'What was he trying for?'\n\nHer hand lifted. 'Some more artichokes on the same basis He was stupid, but not so stupid as to suppose I would leave Freddy. Of course, I wouldn't let him stay in Haughton, but I was tickled to think he had come after me. So I told him he would have to lie low in the chalet, and perhaps I would ring him, perhaps I wouldn't.'\n\n'And he settled for that?'\n\n'He was sure I would ring him. And he may not have been entirely wrong. It was dull here; Freddy was boring. I think Bilney may have gone to bat.'\n\nI nodded. It was fitting pretty well; I could believe in Bilney playing along. He had tasted the honey and it was some honey: worth a little patience for another dip.\n\nBut that had been Thursday.\n\n'How often did you ring him?'\n\n'I gave him a call every day. It was amusing, like teasing a pet. He tried lots of tricks to make me say yes.'\n\n'When on Friday?'\n\n'You know when. My famous call to the theatre.'\n\n'After Rampant's call.'\n\n'I do not deny it. But I said nothing of that matter to Bilney.'\n\nShe faced me with frank eyes; it was either true or cleverly untrue. By freely conceding a critical point she was leaving me no room for manoeuvre. And alas, she was skilful enough to have done that. I could read nothing from her eyes. A frank look is a frank look, besides being the hallmark of accomplished liars.\n\n'Wouldn't Freddy's absence have given you a chance to meet Bilney?'\n\n'Ha-ha, do you think I lacked chances? I was not married to Freddy, you know. I do not recognize a monopoly.'\n\n'Still, you would have looked for a discreet occasion?'\n\n'Any day I could take a launch down the river. No, no, I wasn't giving it to Bilney so easily. A little waiting would improve his manners, ha?'\n\n'But you did know Freddy would be out when you rang him.'\n\n'I have told you, yes. I knew when and where. But I did not tell Bilney. It was not his business. It would have encouraged him to come here, and I didn't want that.'\n\nAnother frank stare, with a flash of indignation.\n\n'Very well then. You rang, but you didn't tell him. The next day you learned what had happened to Freddy. Wasn't it risky to let Bilney hang around after that?'\n\nShe threw up her hands. 'Are you telling me! It was more than risky, it was suicidal. With the police running about spending the tax-payers' money and looking for just such a boy as little Bilney. But he wouldn't go. He thought now was his chance. No longer did Mimi have to hoodwink her Freddy. He was more than stupid, he was mad. I do not wonder he finished up like this.'\n\n'Did you see him?'\n\n'No! Do you think I am mad too? I could scarcely push past the police to the telephone. It was not till yesterday that they went away, that I could arrange to give Bilney a lecture.'\n\n'That was your object yesterday?'\n\n'What else?'\n\n'You seem to have spent several hours with Bilney.'\n\n'Because he is an imbecile! It was like drilling concrete. Surely you have met these cretins before?'\n\n'So the picnic was fortuitous.'\n\n'It was food that he brought. He has not been living on fresh air. And I got hungry talking to the ape. I didn't expect it would take me so long.'\n\n'And what was the result?'\n\n'Not any result. He would not promise to go away.'\n\n'But after so much oratory? Three hours?'\n\n'Pyuh!' It was a noise like a cat's.\n\nI let my eyes drift, then snapped them back suddenly.\n\n'Tell me, Mimi. Who killed Bilney?'\n\nHer eyes were steady. 'Someone with a knife.'\n\n'His name.'\n\nHer eyes mocked me. 'Why should I tell you?'\n\nI got up and walked over to the window. The Bugatti was sitting proudly where I had parked it. Across the way lurked the tradesman's van and the shadowy faces of the two D.C.'s. Some traffic was crawling across the junction, but the road in front of me was empty; sunlight was slanting on the bank opposite and lighting the windows of a flat built above it. I watched it as I talked.\n\n'Listen carefully. You've told me too much and too little. You have admitted sending Bilney to the chalet and calling him out to a meeting yesterday. We are back where we began. You knew he was there. You knew he would be absent for several hours. They are two things which only you knew. You must also know who killed Bilney.'\n\nShe gave a little low chuckle. 'A logical Englishman. And I thought you trusted only the intuition.'\n\nI turned from the window. 'I need a logical answer. Or you may be spending tonight in a cell.'\n\n'Aha, a threat.' She leaned back in her chair and hooked her thumbs in the sleeveless top. 'Yet the cells are no strange thing to me, my friend. And I am told they are better furnished over here.' She lowered her lids with their perfect lashes. 'So then. Let us ventilate your logic. Would it be surprising if the man who killed Freddy was also the man who killed Bilney?'\n\nI said nothing. She nodded emphatically.\n\n'Oh yes. Oh yes. The same man. You show me the photograph. It is done with a knife. Unhappily, I know about these things. So, one man. He has killed Freddy. He is perhaps not a stranger in this district. He doesn't go away. He is here, watching. He has seen Bilney. He has tracked him home. Then, where is the difficulty? He wishes to kill Bilney, decides he will lay for him in the chalet. Now he watches till Bilney goes away, which by chance is to meet me.' She held out her hand. 'Is this impossible? Does it not fit the facts as well? Would a jury prefer your version to mine? And so bang goes your logic.'\n\n'Not quite,' I said. 'There is a matter of motive.'\n\n'Oh, motive! That is for counsels.'\n\n'In this case a motive of massive gain. Freddy had more to leave than the Bugatti.'\n\nHer eyes widened. 'You have seen the will?'\n\nI nodded. 'And so, I suspect, have you.'\n\n'You are wrong, my friend.'\n\n'It amounts to the same thing. Freddy's whole estate is willed to you.'\n\n'This is true?'\n\n'Yes.'\n\nShe looked away. There was a sparkle between her lashes. 'But how sad. I didn't need it, and he was just going to enjoy his life.'\n\nI watched her hungrily. They were real tears; if it was an act it had the stamp of sincerity. I didn't think it was an act. She wasn't giving it enough emphasis. From where he was sitting, Dutt probably couldn't see it.\n\n'All the same, it leaves you the gainer.'\n\nShe twisted her mouth. 'Does that make me guilty?'\n\n'It will weigh with juries.'\n\n'We have not come to juries. We have not come to anything but so-called logic.'\n\n'Then we will follow that. How long have you known Bavents?'\n\nShe was silent for a moment, still looking away from me.\n\n'What makes you ask that?'\n\n'A piece of information. Bavents' father keeps a pub in Vought Street, Chelsea.'\n\n'Aha. The Peacock.' She gestured wearily. 'All right. I admit it. I know the yak. But he is not my lover, has not been my lover. Even with me such things are possible.'\n\n'He was in your room twice today.'\n\n'So then. I have to talk to someone.'\n\n'Once for an hour.'\n\n'It was a longer talk.'\n\n'And again at tea, on the lawn.'\n\nShe tossed her hair irritably. 'What is all this about? I find the yak an interesting subject. He is full of fire, full of passion. Simply he dare not say boo.'\n\n'You were merely teasing him?'\n\n'Not merely, my friend. Teasing a man is a great art. He must always be having a little hope, a belief that his reward will come in the end.'\n\n'And that was Bavents' situation?'\n\n'Every man's. What do you think a woman is made of?'\n\n'But Bavents is infatuated.'\n\nShe kissed a finger. 'At his age, surely it does no harm.'\n\nI stared into her eyes. 'Only that is what I'm wondering. Whether it couldn't do him a great deal of harm. Whether it couldn't lead him into some fantasy world where right and wrong are not clearly defined.'\n\nShe pouted. 'You are not serious?'\n\n'Very serious. His record doesn't suggest a stable character.'\n\n'But Monsieur, I have just amused myself.'\n\n'No more than that?'\n\nShe wriggled and tossed her hair again.\n\n'What were you wanting him to do today?'\n\nShe checked fractionally. 'Who says I did?'\n\n'You had long talks with him. He was reluctant. You were perhaps pushing him too far.'\n\n'Huh.' She made a sweep with her hand. 'Now it is you who are fantastical, Monsieur. He is a moody yak, that is all. I think you had better stick to logic.'\n\n'Is that all you are telling me?'\n\n'Isn't it enough?'\n\nI came off the desk suddenly and stepped to the door. I threw it open and stood beside it. She watched me with an expression of mocking surprise.\n\n'It is time to go?'\n\nI held the door and said nothing. She rose disdainfully and marched from the office. I slammed down in the swivel-chair and lit my pipe. I jerked a hand to Dutt.\n\n'Fetch Bavents.'\n\n## CHAPTER SEVENTEEN\n\nTHERE WAS A delay, probably occasioned by Frayling's reluctance to lose Bavents during dinner; then he arrived, hot-faced and nervous, with Dutt nudging him from behind. I pointed to a chair. Bavents sat or sprawled; his blushing hands clutched his knees. He was either sweatily, meltingly innocent, or aware that a point of no return was fast being reached. I tried to fix his staring eye.\n\n'Madame Deslauriers has just been helping us. She says she was acquainted with you in Chelsea and she admits to several recent conversations. Now I am going to ask you a frank question, and I would like you carefully to consider your reply.\n\nHis eyes rolled; his body was trembling.\n\n'Did you kill Frederick Quarles and Thomas Bilney?'\n\nHis mouth worked and he made a gulping sound; the flush ebbed suddenly from his cheeks.\n\n'Did you?'\n\n'N-n-no! I didn't!'\n\n'It would be best to get it over now.'\n\n'P-please, no!'\n\n'It will save you some distress.'\n\n'B-but I didn't. I didn't!'\n\n'Think carefully.'\n\n'No!'\n\nI checked. He was swaying dangerously, teetering on the rim of a faint. Push him some more, and he would go over; he wasn't ready to confess yet. I swung in the swivel-chair.\n\n'Very well then. Perhaps you can help us in other ways. We know now you took a message from Bilney to Madame Deslauriers. Were you in the yard when she came out?'\n\n'I was w-working on my car\u2014'\n\n'And you returned to it after you had taken in the message?'\n\n'Well, y-yes. But I didn't hear anything. They went over behind the garages.'\n\n'You saw them meet?'\n\n'Yes.'\n\n'Would you say they seemed fond of each other?'\n\nThe flush began again. 'He kissed her. She was worrying about him being seen.'\n\n'But she was fond of him too?'\n\n'Well, I'd s-say so.'\n\n'In fact, they met like two lovers?'\n\nBavents chewed his lip. 'Yes, I suppose so. But she w-wasn't so keen on him as he was on her.'\n\n'How long were they talking?'\n\nHe dragged at his knees. 'About quarter of an hour, twenty minutes.'\n\n'But you heard nothing of it.'\n\n'No! Why should I l-listen to what they were saying?'\n\n'That's fairly obvious. You would be jealous.'\n\n'I wasn't jealous and I wasn't listening!'\n\n'You just went back to tinkering with your engine. While she was in his arms behind the garages.'\n\nHe punished the knees. 'I d-didn't want to listen! I made a lot of noise revving the engine. When they came out she said something about ringing him, that's all I heard. Then she went in.'\n\n'What was wrong with your engine?'\n\n'The s-slow-running is dicey\u2014'\n\n'Skip it,' I said. 'Now tell me about Friday.'\n\nHe licked his lips with small conviction. 'I d-don't know anything about Friday. I didn't see Mr Quarles except at meal-times. I think he spent a lot of time in his room.'\n\n'Which I am told is next door to yours.'\n\n'I can't help that! I didn't see him.'\n\n'But you could have heard him. Heard him discussing a certain matter with Madame Deslauriers.'\n\n'No! It isn't true.'\n\n'It could easily be true. You are not particular about loafing near doors. And if Quarles was excited and raising his voice you could very likely have heard him through the wall. Wasn't that what happened?'\n\n'No, it wasn't!'\n\n'Where were you in the evening?'\n\n'I was here \u2013 in the hotel\u2014'\n\n'On your evening off?'\n\nHis eyes popped at me.\n\n'You had time and opportunity,' I said. 'You could very well have known where Quarles was going. And I daresay what you heard going on in the next room was motive enough for wanting him away. At least you could stop that. There were knives in the kitchen. All you had to do was follow in your Mini. If you want me to believe different you'll need to come up with something pretty convincing.'\n\n'But I was here \u2013 in my room!'\n\n'Your room is no alibi.'\n\n'Yes, but I w-was. Ask Mimi!'\n\n'Madame Deslauriers was with you?'\n\n'Yes \u2013 no!' He clawed his hands together desperately. 'She could have heard me m-moving about. She was in her room too, she might remember.'\n\n'I'll certainly ask her,' I said.\n\nBavents groaned and wrestled his hands.\n\nI gave my chair another swing. 'Next, you would want to settle Bilney.'\n\nBavents shuddered.\n\n'To make time for that you had to fake an excuse to Mr Frayling.'\n\n'But I didn't go there!'\n\nI hesitated. 'Where?'\n\nHe gasped as though I'd punched him in the wind.\n\n'Look, I was in Norchester! It's true, I was! I just had to go into town yesterday.'\n\n'So you weren't where?'\n\n'Not anywhere! I had to meet a m-man in Norchester.'\n\n'What man?'\n\n'This man\u2014'\n\n' _What man?_ '\n\n'He \u2013 I d-don't know what his name is!'\n\n'Just a man with no name.'\n\n'Yes! No name! You meet him in the shelter in Chapel Field Gardens.' He was breathing jerkily, his colour draining. 'Ask any student, they'll t-tell you!'\n\n'Do you mean he's a pusher?'\n\nBavents nodded.\n\nI rose. 'Come on. We'll take a look in your room.'\n\nThe pot was there: about half an ounce of it, packed in an OHMS envelope. Also five crudely-rolled joints and the butt-ends of two more. We searched the room. It was a tiny place with a slanted roof and a dormer window; bare space for a bed, a chair, a chest-of-drawers and a hanging wardrobe behind the door. Books were piled on the chest-of-drawers, mostly works on politics and economics, and on the chair were four or five notebooks, filled with neat, small writing. Bavents didn't interfere. He stood out in the passage, watching us through the open door. Apart from the pot we found nothing. I tapped the dividing-wall: it was lath-and-plaster.\n\n'You say you bought this stuff yesterday?'\n\nBavents shrugged his narrow shoulders.\n\n'Can you prove that?'\n\n'There's the m-man\u2014'\n\n'Forget it. The pusher won't be giving evidence.' I took the pot and squeezed out of the room. 'Now. I'm going to charge you with possession. You had better pack a few things in a case, because you won't be coming back here tonight.'\n\n'You're going to charge me with p-possession?'\n\n'Does this look like a plant?'\n\nHe stared for some moments through his mane. Then he turned into the room, in a beaten sort of way, and began stuffing toilet gear in a zip-bag.\n\nI left him in Dutt's charge while I went to ring Hanson and to order a car. In the hall I intercepted Madame Deslauriers on her way to dinner, dressed now in her slinky, slit-skirt gown.\n\n'Just a moment.'\n\n'Please, Monsieur. You have made me late already.'\n\n'This won't take long. It's about Friday evening. You told us you retired to your room after dinner.'\n\n'So?'\n\n'Did you order up any drinks?'\n\n'Yes. You may confirm that with the waiter.'\n\n'Bavents?'\n\nShe smiled insolently. 'No. It was the German boy, Fritz.'\n\n'Did you see or hear anything of Bavents?'\n\n'I was watching television in my room.'\n\n'His room is next door. The wall is thin.'\n\n'Nevertheless, Monsieur, I heard nothing.'\n\nShe waltzed away, to be met at the dining-room door by the smiling, ducking head-waiter: Mimi, Madame Deslauriers, with a lot of leg showing through her slit-skirt.\n\nWe had Bavents brought into Hanson's office after I had briefed Hanson on the development. Hanson had tried to preserve his incredulity, but it was wilting under the impact of vulgar fact. He gazed indignantly as Bavents entered, still wearing his neat waiter's garb; he didn't want Bavents, and if Bavents was chummie, Hanson was going to take it as a personal affront. Bavents barely glanced at him. He shuffled in forlornly and dropped on the chair Dutt had placed for him.\n\n'Adam Bavents.'\n\nHe stared at me wildly.\n\n'Please listen to me carefully. You have been charged with possession of cannabis resin, but now I am going to ask you some further questions. You don't have to answer them, but if you do your answers will be taken down in writing and may be used in evidence. Is that clear?'\n\nHis eyes rolled horribly, his hands moved in fluttery gestures.\n\n'Please! I just w-want to get it over. I'll tell you anything you want to know.'\n\n'But you understand the warning?'\n\n'Oh God . . . ask me!'\n\n'I wish to have your answer on record.'\n\nHe made a wretched gasping sound. 'Yes \u2013 I understand! And it's true \u2013 I killed those two men!'\n\n'You're lying!' Hanson snarled, jumping up.\n\nBavents sobbed and clutched his hair-cocooned head.\n\nI ordered coffee to give Bavents time to digest his moment of hysteria; also Hanson, who was raving quietly and spitting cheroot on the office linoleum. Then I took Bavents through it, beginning with straight questions about his acquaintance with Deslauriers in Chelsea, but nudging him into taking the initiative when we came to the killings. Bavents answered well. Now he had let go the pressure, some of his hang-dog attitude had left him.\n\n'Where did the knife come from?'\n\n'I got it from the cook's box. One of my jobs is sharpening his knives.'\n\n'How big?'\n\n'Well, not very big. I wanted a knife that would go in my pocket.'\n\n'Where is it now?'\n\n'I put it back. Otherwise the chef would have missed it. I washed it, naturally.'\n\n'Can you show me the knife?'\n\n'Of course. It's the only one of that size.'\n\nAll straight-forward. But sometimes questions can be too easy to answer.\n\n'Tell me what happened on Friday evening, beginning when you left the hotel.'\n\nBavents hesitated for several seconds, his eyes fixed on his knees.\n\n'W-well, I was waiting in the yard. I knew what time he had to get there. When I heard the Bugatti's engine start I started mine and went after him. I left the Mini on the car park. Freddy was down there, sitting in his car. I c-came up behind him and let him have it. He just fell forward over the wheel.'\n\n'What clothes were you wearing?'\n\n'Clothes\u2014?'\n\n'Stabbing gives rise to spurts of blood.'\n\nBavents paled. 'I-I was wearing an old windcheater, one I keep for working on the car.'\n\n'Where is it now?'\n\n'I th-threw it in the river. I thought I'd b-better, with all that blood on it. And I washed the knife with that special cleaner which is supposed to take out blood-stains.'\n\nI nodded. 'Go on.'\n\n'Then I just left him. I went back to the Mini and drove back here.'\n\n'Seeing nobody.'\n\n'N-no, nobody. There wasn't a soul about anywhere.'\n\n'Nobody coming down the track?'\n\nHe shook his head.\n\n'No other cars on the park?'\n\nHe hesitated. 'Y-yes, I think there were some. But I was too worked up really to notice.'\n\nI asked him some more, noticing specially the questions that brought about the stammer; then switched him quickly to the Bilney killing and his knowledge of the location of the chalet.\n\n'Did Madame Deslauriers tell you about it?'\n\n'No! But I heard her giving directions to Bilney.'\n\n'How do you get there?'\n\n'It's the b-back road to Sallowes. There's a turning off th-through a gate.'\n\n'Could you drive me there?'\n\nHe swallowed. 'Yes. But I've only b-been there once.'\n\nI was tempted to try him with a map, but had my reasons for not pressing him.\n\n'Right. Describe what happened yesterday.'\n\n'M-Mimi wanted me to drive her out there.'\n\n'She did?'\n\nHe nodded in his hair. 'Only I couldn't get the time off. But I knew where she'd gone when she took the launch, and I thought it would give me a chance at Bilney. So I asked Mr Frayling for the afternoon and d-drove round to the chalet.'\n\n'What time would that be?'\n\n'I'm not s-sure. I'd say I got there about three-thirty. Anyway, he wasn't there, and the place was locked up. I had to break in through a b-back window.'\n\n'Using what?'\n\nHe made a reaching gesture. 'I'd b-brought a tyre-lever with me.'\n\n'Not, for example, a large screw-driver?'\n\n'I \u2013 n-no! A tyre-lever.'\n\n'A tyre-lever,' I said.\n\nBavents' eyes swivelled and he breathed a little faster. Then he started off again, filling in the words in jerks.\n\n'I waited for him behind the door. I thought I'd hear his car pull up. Only he must have guessed I was there b-because he left his car down the track. Then he came in right fast, slamming the door back, so I couldn't get him as he came in. I had to ch-chase him into the bedroom. I g-got him down behind the bed.'\n\n'Was he scared?'\n\nBavents' throat worked. 'I g-guess so.'\n\n'Bilney was a criminal who knew about knives.'\n\n'He tried to keep me off, but I g-got him all right. He went down. I f-finished him off.'\n\n'Which made a lot of blood.'\n\nBavents nodded stupidly.\n\n'What did you do with your clothes this time?'\n\n'I-I'd remembered about Freddy, I was wearing a boiler suit. Then I threw that in the river too.'\n\nI didn't ask him if he had washed the knife again. I felt I knew the answer to that one. I signalled to Dutt. He escorted Bavents out. There was a silence broken only by Hanson's gnashing of a cheroot.\n\nI lit my pipe too and contributed my quota to the office atmosphere. Hanson, who had been hovering tigerishly in the background, now advanced and dumped himself on the desk.\n\n'Look, it sounds as phoney as hell. But that dozey bastard just _must_ have done it!'\n\n'Why do you say that?'\n\n'Because, Christ alive, he knows what only the chummie could know. He knows where the car was left, how the window was forced, that Bilney was knifed behind the bed. And he would probably have told us a whole lot more if you hadn't played soft and laid off him.'\n\n'And knowing these things makes him chummie?'\n\n'Yeah,' Hanson said. 'It flaming has to. Because we haven't leaked information at this end, and you're the tightest-mouthed sod I ever met.'\n\nI grinned but shook my head. 'You were right the first time. He's phoney as hell.'\n\n'But how in blazes can he be, when he sits there coughing up chapter and verse?'\n\n'Simple. He's been briefed.'\n\n'Briefed\u2014?' Hanson's horse-teeth showed in a gape.\n\n'By our ingenious friend, Mimi Deslauriers. He was in conference with her today.'\n\nHanson gurgled. 'But for crying aloud. You're not saying he would turn himself in for Mimi?'\n\n'Why not?'\n\n'Why not? What good will it do him if he ends up sitting out a lifer?'\n\n'He won't.' I feathered smoke. 'I'm sure that's not the plot at all. Once Bavents has served his turn we shall find he has alibis a yard long. Mimi can clear him of the Quarles killing whenever her memory starts to improve, and there will be fireproof witnesses around somewhere to place him in Nor-chester yesterday afternoon. All Bavents is risking is a charge of obstruction, for which he'll probably get off with a wigging. And the possession charge is a first offence. You can guess who'll be picking up the fine.'\n\n'But for Christ-sake, why?'\n\n'Mimi wants the heat off. Our camping on her doorstep is cramping her style.'\n\n'Like a bear's backside!' Hanson snarled. 'From now on she'll only draw breath when we do.'\n\nI gentled smoke at him. 'No.'\n\n'You aim to let her get away with it?'\n\n'Mimi has made her first mistake. Bavents wasn't up to the job of conning us.'\n\n'So we give her three cheers?'\n\n'What she has let out is that she's in touch with the real killer. Before, we suspected it. Now we know it. We are going to play the game from there.'\n\n'Like pulling her in!'\n\n'Not like that. She doesn't know we've seen through Bavents.'\n\nHanson spat cheroot. 'Fine. But I can't see a sweat-session doing any harm.'\n\nI aimed more smoke at him. 'What we're going to do is withdraw the police presence from Haughton. Bavents can go up on the possession charge, and you'll see the beak and get a remand. There mustn't be as much as a traffic cop at Haughton. Mimi will be free to go or stay. Free to meet or contact whom she pleases. Nobody will bother her at all.'\n\nHanson traded smoke for smoke. 'And meanwhile you'll go chase your tail?'\n\nI shook my head. 'I'll be sitting on it. But on the outside, looking in.'\n\n## CHAPTER EIGHTEEN\n\nI GAVE HER the key of the Bugatti the next morning, in what turned out to be a touching ceremony. It took place after breakfast, in the hall, where Dutt's bag and mine were already waiting. Mimi dropped a tear and pressed the key to her bosom. The gesture was so natural that one could almost believe it. Then she stood by sadly, the key in her hand, while I wrote out a chit for Frayling.\n\n'Of course, my friend, I understand what this means. The poor yak didn't serve at table this morning. And now you are leaving. It is a melancholy moment. I wish I could think that you were wrong.'\n\nI bowed my regrets.\n\n'Are you so certain?'\n\n'I'm afraid I mustn't discuss the matter.'\n\nShe nodded. 'But you have given me the key. You would not do that if there were still doubts.' She eyed me earnestly. 'Is that not so?'\n\n'You must draw your own conclusions, Madame. We shall probably need your testimony later. But just now we have no further business with you.'\n\nShe sighed. 'Yet I wish I could help him.'\n\n'I don't think you should waste your sympathy.'\n\nShe sighed at this, too. 'Yet it is so sad. My only consolation is to have met you, Monsieur.'\n\nI should have kissed her hand, but anyway she waved it as Dutt and I drove off in the Lotus. When I saw in the rear-view mirror that she had gone back into the hotel, I made a left turn and drove round the block. Hanson was waiting at the rear of the bank. He had brought an unmarked Capri, which he had borrowed from Traffic. I parked the Lotus, leaving the engine running, and Hanson slipped me the key of the Capri. 'Everything fixed?'\n\n'Roger. Now I'm off back to process Bavents.'\n\n'Watch the Lotus. It can go to your head.'\n\nHanson grunted, got in and gunned away.\n\nWe entered the bank by a service door and were met inside by the manager. He introduced us to his head cashier, who was the tenant of the flat above. Hanson had done all the explaining. The cashier led us up outside steps. We were admitted to the flat by his wife, a snub-nosed woman with ginger hair.\n\n'Come this way, sir.'\n\nShe showed us into a bedroom furnished with a pleasant-looking teak suite. Venetian blinds were dropped over the two windows, before each of which had been placed a chair. I went to the nearest one. The blind-slats were slanted to give a view across and below; I found myself staring at the blue Bugatti and, behind it, Frayling, sitting in his office.\n\n'Is this how you want it, sir?'\n\n'Exactly right. Have you any objection to our smoking?\n\nShe smiled. 'No, sir. We're both of us smokers.'\n\nShe brought us ash-trays. And we sat.\n\nIt was a number of years since I had been on a stake-out, and my ennui-index had risen in the interval. To a certain extent I had lost the faculty of watching while allowing my mind to pursue its courses. Not much was happening down there below. I soon tired of trying to memorize the traffic. The shadow of the bank, tucked in beneath us, seemed never to shorten as the minutes limped by. Three guests fetched their cars, four set out on foot; a butcher, a baker, a green-grocer delivered. Two guests returned, one carrying a parcel. Frayling appeared and disappeared in his office. Out of sight in the kitchen, minus Bavents, they would be busy preparing food for lunch, while maids were hoovering in the bedrooms and barmen washing and stacking glasses. But Mimi? The only token of Mimi was the Bugatti basking in the sun. Was it possible she had left it there for me to gaze at, while she slipped away in a launch?\n\n'Sir!'\n\nDutt was poking towards his slats; Mimi had appeared, and I had nearly missed her. She was seated in the office in her usual style, her feet up, the phone in her hand. She was laughing, trilling strings of words; in her other hand a cigarette. Relaxed, unhurried; filling in the small-talk; somebody she liked: somebody she loved.\n\n'What do you make of that, sir?'\n\nMimi had hung up and taken some dancing steps from the office.\n\n'It could have been lover boy.'\n\n'It looked like the green light, sir.'\n\n'So let us hope there is nothing holding him back.'\n\nTime, eleven-twenty-one. I tried to think of some circumstance suggested by the time for lover boy's availability. I failed, but it didn't bother me: didn't spoil the picture of Mimi's sweet confidence.\n\n'I'd say it was a local call, sir. I was watching her dial. Don't think it was more than six figures.'\n\n'She might have been gassing to a girl-friend.'\n\n'Not that one, sir. She doesn't go in for them.'\n\nWe settled again, with renewed alertness. But that was the high spot of the morning. The next excitement was opening time with its quick build-up of male custom. Some of the patrons were familiar to us, regulars from nearby shops and offices; but there was a healthy residuum of casuals, holiday-makers and visiting businessmen. We tried to filter them. Reps you could identify fairly easily by their cars. Older men were probably out, also men of poor physique and provincial style. What emerged were three possibles, young men with looks, bodies and arguable panache. One arrived on foot, one was driving a Fiat, and one a much-accessoried M.G. Each of them stopped to admire the Bugatti but they were not alone in this; as a custom-maker a Bugatti would be worth its weight to any brewery.\n\n'Any preferences?' I asked Dutt.\n\n'I like the bloke who walked in, sir. I don't think chummie would just drive up here. He would want to prowl around first.'\n\n'Did you think the Fiat-driver looked familiar?'\n\n'Bit of the pop-singer about him, sir. Maybe we've seen him on the telly.'\n\n'Maybe,' I said. 'Speak for yourself.'\n\nIn the event the Fiat-driver had stopped only for a quick one; while the pedestrian, who soon followed him out, was claimed by a girl driving a Volks. The M.G.-decorator was the stayer; he had probably decided to wait for lunch. He was lowest on the list: no vulnerable chummie would have given much time to a comic M.G.\n\n'Excuse me, sir.'\n\nIt was the cashier's wife, who had come in with a tray of beer and sandwiches. She was smiling embarrassedly: because, after all, she was entertaining two strange men in her bedroom.\n\n'Nobody said anything about your meals . . .'\n\nThey were hefty sandwiches of ham and tomato. I made her an offer of a subsistence payment, which only embarrassed her the more.\n\n'No, we are happy to help the police \u2013 my husband being what he is.'\n\n'But we pay for our rations.'\n\n'No \u2013 please! We would like to do our bit to help.'\n\nStrange attitude from a member of the public. We ate her sandwiches and drank her beer. At two-forty the M.G.-driver claimed his car and drove off alone.\n\nWe waited some more.\n\nNow the sun had worked round to our side of the flat and the Bugatti was inching its way into shade. The bedroom was muggy. We couldn't open the windows without raising the blinds, which we daren't risk. The scene below had fallen asleep; no traffic passed for minutes at a time. Frayling visited his office at ten minutes past three, but then was absent for two long hours. Entertainment unlimited; my pipe tasted vile and the beer and close atmosphere made me feel torpid. I kept awake mainly by debating with myself whether it was I or Mimi who was being the smart one.\n\nDutt, too, was doing his policeman's fret.\n\n'Do you think we should have started at the other end, sir?'\n\n'Dainty is on the job. I rang him last night. Any information will come straight here.'\n\n'Has he any ideas?'\n\n'Just Whitey Ferrier.'\n\nDutt sniffed. 'Whitey wouldn't do the job himself.'\n\n'So we're watching for who did.'\n\nDutt pulled out a sigh. 'It would still be nice to know who we're expecting.'\n\nHe didn't add, or if he is coming: which was what the conversation was really about.\n\nAt four-thirty two guests returned, breaking a dead-spell of fifty minutes. The shadow on the Bugatti had reached the cockpit-cover, which I had buttoned-on last evening. A newsboy pushed his bike along the pavement and stopped to make a delivery at the Barge-House; Fritz, the German waiter, ran after him and returned with a copy of a sporting paper. Life beginning again. By five, an intermittent stream of traffic had developed. Five-ten, Frayling reappeared to talk patiently and lengthily into the telephone. Five-twenty, two more returning guests. Five-thirty, a sudden explosion of giggling shop-girls. It had been a rough afternoon, but the swinging evening was at hand.\n\n'Excuse me, sir.'\n\nOur embarrassed lady was back again with tea and cake. We munched and drank, keeping only half an eye on the glaring slats and the view between them. Were we making fools of ourselves? It had begun to seem like it, after a solid eight hours of sitting and watching. Trying to feel like cops on a job while the working world went on around us. Mimi hadn't been bluffed. I had taken the bait too easily, hadn't probed and questioned in the way she had expected. She had borrowed the office, rung her man, but hadn't signalled him to come running. Non, Monsieur, non. Why had I thought it would be so simple?\n\n'Car pulling on to the apron, sir.'\n\nI swallowed cake and took a look. The car was a commonplace green 1100, rather dusty, with a J registration. A man got out. He was something of an eyeful, dressed in the full boutique gear: a draping floral shirt-jacket over matching bell-bottom trousers, with suede boots. He had shoulder-length black hair and wore baroque sun-glasses.\n\n'Chelsea, here I come,' Dutt murmured.\n\n'Know him?'\n\n'No, sir. But he looks a good spec. I could see him stepping out with the lady.'\n\nWe watched. He paused beside the Bugatti. I judged his height at five-eleven; strongly-built, with good shoulders; handsome features, lightly tanned. He glanced around him with quick alertness, then un-buttoned one corner of the Bugatti's cover. He stared inside, at the dials, the controls; then re-buttoned the cover and went into the Barge-House.\n\n'A bit of a cheeky chummie, sir.'\n\nI shrugged. 'I think I would have done the same myself. But the 1100 does have a London area registration. There'll be no harm in giving it a check.'\n\nDutt slipped out to use the cashier's phone. I drank tea with my eye on the Barge-House. Lucky after all? Or was this one more holiday-maker, ordering dinner, waiting for opening time? The clothes didn't match what I was expecting, but the clothes could be a clever disguise; nobody would expect a fugitive killer to dress like a Carnaby Street peacock. And there would be Mimi's brain behind that one: Mimi, who didn't miss a trick. So why not? What would look more natural than Chelsea Joe turning up at the Barge-House?\n\nDutt came back. 'It's pinched, sir. On the Met list for last Friday.'\n\n'Friday?'\n\n'Yes, sir. Some time in the P.M. From Norland Road, off Holland Park.'\n\nThe district meant nothing. Dutt's eye had a gleam in it.\n\n'We could go down there and nab him, sir.'\n\n'For what?' I said. 'Stealing a car? That's all we have on him at the moment.'\n\n'It would make a start, sir.'\n\nI shook my head. 'I think we'll wait a little longer. Just to see if Mimi will tip us her hand. When she does will be the time to move.'\n\nWe stood now at the one window, waiting, willing events to happen. But once more time began to build up, minute laying itself to minute. The bar had opened, a little cluster of custom had formed from leaving-off shop and office staff. The commuter traffic had passed its peak and pedestrians stood chatting on the pavement. Behind us we could hear a faint clinking of dishes as the cashier and his wife ate their evening meal, along with the low monotone of the television rounding up the regional news. The day relaxing. And now the Bugatti was sitting full in the advancing shade.\n\nThen Frayling backed into view in the office, a slip of paper in his hand. A cheque: and he was chatting animatedly to someone as he unlocked a drawer of the desk and took out a cash-box. He nodded and smiled and nodded again before turning his attention to the cash-box. A few moments later the swing-door opened: and out stepped Mimi and Chelsea Joe.\n\nHe was carrying her luggage: two neat, jazzy suitcases and a coat of featherweight blue mink. Mimi was laughing. She was wearing a dusty pink two-piece, the least casual garb I had seen her in yet. They went to the Bugatti. He stripped off the cover, tilted forward the seats and loaded the suitcases. Then he held the coat for Mimi to slip into, a proceeding which she delayed by giving him a kiss.\n\n'Let's go!' Dutt muttered.\n\n'Wait.'\n\nNow Chelsea Joe had gone to the 1100. He unlocked the boot and took from it a canvas holdall and a black suitcase. The suitcase fitted in with Mimi's cases, but the holdall was too bulky. He took a spider from the Bugatti's luggage-hold and secured the holdall to the car's small grid.\n\n'Right.'\n\nWe hustled out of the bedroom and down the outside stairs to the Capri. I started the engine, drifted the Capri forward, and halted it, still in the cover of the bank.\n\n'Report in.'\n\n'Shouldn't we grab him now, sir?'\n\n'I want to see which way they are heading.'\n\nReluctantly Dutt picked up the transceiver and made contact with HQ. I waited, listening. The roll of the Bugatti's engine broke in suddenly on the Capri's murmur. Chelsea Joe gunned it three or four times, then I could hear it shifting in the direction of the junction. I sneaked the Capri out. The Bugatti was turning right towards the bridge and Norchester. I let a couple of cars go by, eased the Capri on to the Bugatti's tail.\n\n'They're off back to town, sir.'\n\n'We shall see.'\n\nWe trailed the Bugatti through Wrackstead. But beyond the village his winker went and he turned left into the road to Sallowes. This lost me my cover. I lingered on the turn, letting him go another hundred yards up. He wasn't racing. I had the impression that the Bugatti was as novel to him as it had been to me.\n\n'They wouldn't be heading for the chalet, sir?'\n\n'Report his direction.'\n\n'Yes, sir. The patrols are converging.'\n\n'No contact yet. Tell them to stay clear.'\n\nDutt spoke his piece and was grittily answered.\n\nNot the chalet. The Bugatti passed that turning and rumbled on into Sallowes village. There it hesitated at a cross-roads, and eventually turned left. The signpost said: Ockley.\n\n'Check with the map.'\n\nDutt took a map from the glove-locker. The road we had joined was some sort of B road, but it was tracking purposefully across the open country. There were fast stretches, tempting Chelsea Joe to get the feel of a vintage sixty. My nose said we were pointing eastwards, and a church across the fields offered confirmation.\n\n'About six to Ockley, sir.'\n\n'Then?'\n\n'Ockley is on the main Norchester-Starmouth road. So he could go either way there, except it would be a roundabout way to Norchester.'\n\n'But it would be the direct way to Starmouth.'\n\n'Yes, sir. Couldn't be more direct.'\n\n'What's marked at Starmouth?'\n\nDutt peered at the map. 'There's an airfield and a roll-on, roll-off ferry to Rotterdam.'\n\n'A ferry!'\n\n'Yes, sir.'\n\n'Buzz control. We'll have the patrols pick him up now.'\n\nI closed on the Bugatti, which was whipping along at plus-sixty. I could perhaps have taken him, perhaps not: anyway, I decided not to try. Heroic measures make good film but tend to prejudice public safety. I needed only to keep my eye on him. The patrols would do the rest.\n\nWe reached Ockley, where Chelsea Joe had to halt at the main road junction. I looked round for the patrol cars, but apparently we had got there first. Joe surged off again; then, as I followed him, caught sight of the Capri in his mirror. Mimi's scarfed head jerked round: her big eyes stared at me. Directly, the Bugatti began drawing away.\n\nI cursed to myself and squeezed the Capri. I knew the stretch from Ockley to Starmouth. Dead level, it stretched across the marshes with only one bend in ten miles. The Bugatti could lose me by sheer horse-power if Chelsea Joe kept it booming. And there was a side-road, just one, where the main road made its bend.\n\n'Report in.'\n\nWe came out of the village trailing the Bugatti by a hundred yards. It doubled that distance in the next half-mile down the flat, pollard-willow-lined straight.\n\n'Two patrols heading this way from Starmouth, sir.'\n\n'Tell them to switch on lights and sirens.'\n\nThe Capri was revving its sophisticated heart out and still the Bugatti was growing smaller. It reached an almost-level bridge, where a dyke passed under: I saw daylight briefly beneath its four wheels. We hit the bridge and skipped too, and probably landed a lot lighter. But it made no odds: we were being distanced; Ettore was having the last laugh. Quarter of a mile had stretched into a half, and soon the half would be three-quarters.\n\n'Lights over there, sir.'\n\nAway across the marsh were a pair of faintly sparking roof-lights. They were hurrying along on a diagonal towards the dog-leg bend, about a mile ahead.\n\n'What do you think they will do, sir?'\n\n'Get them a message. If chummie sees them he'll take the side-road.'\n\n'He's probably spotted them already, sir.'\n\n'Tell them just to haul up and make a block.'\n\nI lost sight of them. Perhaps they had switched off their lights. But half a minute later it was academic. I saw Chelsea Joe's brake-lights glow at the bend. Then he crossed the verge and hit a tree.\n\nWe screamed in from one direction and the patrol cars from the other. The Bugatti had bounced clear of the tree and slipped nose-down into a dyke. Chelsea Joe was still with it; he was flaked out over the big wheel. Mimi had landed in a thicket of bush-willow, from which, amazingly, she was beginning to crawl. She was nearest to the patrols, so we left her to them and went down to rescue Chelsea Joe.\n\nHe had got a cut forehead, which was bleeding prettily, but a quick check revealed no broken bones. He was out cold. We lifted him up the dyke-bank and stretched him gingerly on the verge. I felt a twinge of recognition as I stared down at the blood-smeared features: the good-looking lines of the nose and cheekbones, the primitive chin and the loose-lipped mouth. Dutt knelt to dab the gashed forehead, then gave a startled exclamation.\n\n'This is a wig, sir!'\n\nHe grabbed the black locks: they came away in his hand. Underneath was pale, golden-brown hair, cut medium-length, and short side-boards. Now I knew who he was.\n\n'Would you call that hair fair?'\n\nDutt gazed incredulously. 'Holy Jesus! Could this be the original Peter Robinson?'\n\n'I think it could. And the driver of the Viva. Which is why we found it wiped clean.'\n\n'So Bilney was a con!'\n\n'Bilney was a mug. There's not much doubt what brought him up here.'\n\nI scrambled back down the bank to the tilted Bugatti and salvaged the black suitcase from the luggage-hold. It was locked, but there were tools handy and I burst the catches with a screwdriver. The suitcase was stuffed with bank-notes. Most of them were still in wrappers. Some were spotted with dark stains and stains had been scrubbed from the lining of the lid. I slammed the lid shut and returned to Dutt.\n\n'It's all there in one parcel. And chummie's tan is a home-grown product.'\n\nDutt nodded dully.\n\n'It's Fring.'\n\nA sharp cry behind us made us turn: Mimi was standing there between two patrolmen. Her eyes were fixed fascinatedly on Fring, who still lay senseless and lazily bleeding. She made a sudden move forward, but I got in front of her.\n\n'No. It is best that you don't see him.'\n\n'But I must go to him!'\n\n'No. There is nothing you can do for him now.'\n\nI drew her away to one of the patrol-cars. She wailed touchingly, but didn't resist. Her Balmain suit was rent and muddy, but otherwise she appeared in fair fettle. I sat her in the rear of the car, which faced away from the crash, and went round to take a seat beside her.\n\n'Now, Madame Deslauriers, I need some answers. Fring's troubles are settled, yours are just beginning.'\n\n'Oh Monsieur, you are heartless!'\n\n'The first question is this. Do I charge you as accessory to one or both killings?'\n\n'I, Monsieur!' She jerked indignantly upright. 'But I had nothing to do with either of them.'\n\n'It's your choice,' I shrugged. 'But you had better explain. Because only you can help yourself now.'\n\nMimi, Madame Deslauriers, glared at me. 'This is altogether too much! I am the person to whom all this is happening, and now you tell me that I am to blame?'\n\n'You sent Fring after Freddy.'\n\n'Oh, it isn't true! Did I know that Jimmy was going to kill him?'\n\n'Didn't you?'\n\n'No! I thought he would beat him up. Are you telling me now that Freddy didn't deserve it?'\n\n'Was it Freddy who shopped them?'\n\n'Who else? It was how he planned to get rid of Jimmy. He couldn't bear me loving Jimmy better than him. He wished to get rid of him, to close up business. And all that he boasted to me after the hold-up. Is it any wonder I helped Jimmy? But not if I knew he was going to kill him, oh no! You cannot blame me for that.'\n\n'Yet it didn't seem to trouble you.'\n\n'What could I do? You would not expect me to shop Jimmy.'\n\n'And the second killing?'\n\n'Oh, Monsieur! Who asked the stupid Bilney to steal money? He was a burglar, a common thief. It was no matter what happened to him.'\n\n'So Fring caught him at it.'\n\n'Just so. He had found the money under the floorboard. He had his dirty fingers in it when Jimmy went in there. That is all there is about that.'\n\n'Which Fring explained when you phoned him.'\n\n'Yes, I rang to warn him about you.'\n\n'And you told him to get out, and to take Bilney's car.'\n\n'Was it not right I should help to think for him?' She sniffed feelingly. 'And it was all going well. He had even succeeded in buying a passport. Oh, Monsieur, it is desolating. This is not how such a boy should have died.'\n\n'Look in the mirror,' I said.\n\nShe stared at me suspiciously before craning her head to look. What she saw was James Fring being helped into an ambulance by SJAB men. Mimi exploded. I took a fist in the mouth, then she kicked me and went for the door. No use: I'd bolted the child-lock. Mimi roared at the top of her lungs.\n\n'Pig! Pig! Pig!'\n\n'Cool down,' I said. 'We'll convict him anyway.'\n\n'You have tricked me. Oh, I hate you! I wish that you were dead too.'\n\n'But you are going to give me a statement.'\n\n'No \u2013 never!'\n\nI nodded. 'Yes, I think you will. You being such a sensible, logical Frenchwoman. And after all, you're not very much in love with Fring.'\n\nShe spoke rapid, idiomatic French.\n\n'Listen,' I said. 'This is why you will. I am going to charge you in any case, but if you give me the statement the charge will be accessory after the fact. That is not so serious, and with a clever counsel you will probably get off with a suspended sentence. But if I have to charge you as an accessory before and after the fact, then you will be in the same boat as Fring. That will mean a life-sentence. You will be over forty when they let you out of Holloway.'\n\n'I tell you, never!'\n\n'Think about it, Mimi. Make your gesture but pull the chain.'\n\nShe spat at me, but missed. We sat gazing into each other's eyes.\n\n_Rigby House, Norwich,_\n\n_November 1971\u2013March 1972_\n","meta":{"redpajama_set_name":"RedPajamaBook"}} +{"text":" \n# ALSO BY \nANDREA CAMILLERI\n\n_The Revolution of the Moon_\n\n_The Sacco Gang_\nEuropa Editions \n214 West 29th St. \nNew York NY 10001 \ninfo@europaeditions.com \nwww.europaeditions.com\n\nThis book is a work of fiction. Any references to historical events, real people, or real locales are used fictitiously.\n\nCopyright \u00a9 2011 by Sellerio Editore, Palermo \nFirst publication 2019 by Europa Editions\n\nTranslation by Stephen Sartarelli \nOriginal Title: _La setta degli angeli \n_ Translation copyright \u00a9 2019 by Europa Editions\n\nAll rights reserved, including the right of reproduction in whole or in part in any form.\n\nCover Art by Emanuele Ragnisco \nwww.mekkanografici.com \nCover image: Fauk74 \/ Alamy Stock Photo\n\nISBN 9781609455149\n\nAndrea Camilleri\n\nTHE SECT OF ANGELS\n\n_Translated by Stephen Sartarelli_\n\n# **THE SECT OF ANGELS**\n\n## CHAPTER I \nTHE MATTER OF THE MARBLES\n\nGentlemen! Members of the club! A moment of your attention, please!\" said don Liborio Spart\u00e0, president of the Honor and Family Social Club. \"I will now open the urn and begin counting the marbles.\"\n\nThe buzz of voices in the drawing room gradually hushed to a relative silence. Relative, that is, because don Anselmo Buttafava had, as usual, fallen asleep in the damask armchair he'd sat in for the past thirty years and more, and was snoring so loud that the windowpanes giving onto the balcony in front of him were rattling lightly. Although the club had changed all the furniture some ten years earlier, they could do nothing about that armchair: they'd had to leave it in its place, for the exclusive use and enjoyment of don Anselmo.\n\n\"What's that burning smell?\" Commendatore Paladino asked aloud just as the president opened the urn.\n\n\"So you smell it, too?\" retired Colonel Petrosillo asked the commendatore in turn.\n\n\"I do too!\" said Professor Malatesta.\n\n\"It's true!\" said many of those present. \"There's something burning!\"\n\nAs everyone was wrinkling his nose and looking around left and right, trying to determine where the burning smell was coming from, don Serafino Labianca cried out:\n\n\"There's smoke coming from don Anselmo!\"\n\nThey all turned to look at don Anselmo, who kept on snoring, head hanging down to his chest. And indeed they saw an ever so fine column of smoke rising up from the armchair towards the ceiling, which had been frescoed by Angelino Vasalic\u00f2, a carriage painter and local celebrity, and dubbed \"better than the Sistine Chapel!\" by the mayor, Nicol\u00f2 Calandro.\n\nThe first to pinpoint the source of the smoke was don Stapino Vassallo, perhaps because he was the youngest member present and still had good eyesight, being only forty-two years of age, whereas the average age of the others was around sixty.\n\n\"The cigar!\"\n\nHe ran up to the damask armchair.\n\nDon Anselmo's cigar had in fact slipped from his sleeping hand and fallen onto his trousers, in the very spot where the male pudenda are usually tucked away. The ember had already burnt through the dense English fabric of his trousers and was now attacking the thick wool of his underpants.\n\nAs don Stapino was dashing over to the president's table to grab the pitcher of water on it, Colonel Petrosillo, a man of action, quickly crouched down between don Anselmo's legs and with his left hand snatched up the cigar, throwing it to the floor, while with his right hand he began vigorously patting the areas in danger of catching fire.\n\nAwakened by a sudden blow to his cojones, don Anselmo Buttafava, seeing the colonel between his legs, got the wrong idea. For some time now, nasty rumors had been circulating about town concerning the excessive fondness that Amasio Petrosillo, who had never married, seemed to have for a certain Ciccino, the twenty-year-old son of the colonel's farm overseer. Instinctively, therefore, don Anselmo pushed the colonel's face away brusquely, causing him to fall backwards, then got up and ran towards the president's table yelling like a madman.\n\n\"I've always known that Petrosillo is a big pervert! Out of this club, now!\"\n\nPresident Spart\u00e0 tried to clear things up.\n\n\"Don Anselmo, there's been a mistake! The colonel, you see . . . \"\n\nBut don Anselmo, who by habit lit up like a match at the slightest provocation, was by this point extremely worked up and wouldn't listen to reason.\n\n\"Either he goes or I go!\"\n\n\"But, don Anselmo, if you would just listen to me for a second . . . \"\n\n\"Then I'll go myself!\"\n\nAnd with a violent sweep of the arm he swatted the urn, which fell to the floor and, having already been opened, sent all the marbles rolling across the room, as don Anselmo ran into the privy, cursing like a Turk, and locked the door.\n\nBetween one thing and another\u2014with the colonel shouting and bleeding from the nose after don Anselmo's shove, the president wanting to resign immediately, and the secretary scrambling about trying to pick the marbles up off the floor\u2014a squabble arose between those who thought don Anselmo was in the right and those who thought he was in the wrong. It took a good half an hour to settle things down again.\n\n\"We must all recast our votes. The gentlemen members must vote whether or not to admit Attorney Matteo Teresi to the club. A black marble means no, a white marble means yes. There are twenty-nine members present, since Baron Lo Mascolo sent word that he couldn't take part, and Doctor Bellanca did the same, and don Anselmo Buttafava is now\u2014\"\n\n\"\u2014is now present. And so there are thirty members voting,\" said don Anselmo, appearing in a secondary doorway in the salon.\n\nColonel Petrosillo, still holding a wet handkerchief over his nose, stood up and said:\n\n\"I man the peak.\"\n\nEveryone fell silent in bewilderment, wondering what bizarre military fantasy the colonel might have of sending a garrison to some unnamed mountaintop. The only person to grasp the situation was, as usual, don Stapino Vassallo.\n\n\"Colonel, please be so kind as to lower your handkerchief and repeat your statement.\"\n\nThe colonel complied.\n\n\"I demand to speak.\"\n\n\"Please go ahead,\" said the president.\n\n\"I hereby publicly declare that don Anselmo should consider himself slapped by me, and therefore challenged to a duel. So, for my seconds I should like to name\u2014\"\n\n\"Can't we talk about this later?\" asked the president.\n\n\"All right,\" said the colonel.\n\nThey cast their votes.\n\nWhen the urn was reopened, out came twenty-nine black marbles, signifying twenty-nine \"no\" votes, and one white marble, signifying one \"yes.\" Since the vote had not been unanimous, the matter had to be raised again for discussion and then voted on a second time, as every decision concerning a potential new member had to be unanimous.\n\nDon Liborio Spart\u00e0 decided to intervene.\n\n\"Gentlemen. Since today is Sunday, the midday Mass will be starting in half an hour. And we must all go. I therefore propose a waiver of the rule concerning abbreviations of procedure. Are you all in agreement?\"\n\n\"Yes, yes,\" said many voices.\n\n\"As we know, gentlemen, every new candidate for membership must, according to the rules, be presented by two associates of the club with more than five years' membership. In the present case, those sponsoring the candidacy of Attorney Matteo Teresi were Baron Lo Mascolo, absent, and Marquis don Filadelfo Cammarata, here present. Clearly, the white marble could only have been put in the urn by my lord Marquis Cammarata, to whom I politely request\u2014\"\n\n\"Clearly, my arse!\" said the angry marquis.\n\nDon Filadelfo Cammarata was about fifty, skinny as a rail, married and the father of eight daughters, all fine churchgoing young ladies, and he was always upset about something, always arguing with someone and quick to resort to vulgar language. Even when alone, he could often be seen gesticulating animatedly\u2014arguing with himself.\n\n\"My good marquis, simple logic leads me to\u2014\"\n\n\"Simple logic leads you up your own anus,\" the marquis retorted, standing up. \"And I'm saying that, both the first and the second time, the vote I cast was a black marble!\"\n\nEveryone looked bewildered.\n\n\"What?\" they said. \"But it was you who presented him for membership!\"\n\n\"And then I changed my mind, all right? Is a man not free to change his mind?\"\n\n\" _I_ can tell you why you changed your mind!\" don Serafino Labianca said with an insinuating smile from the far end of the room.\n\nIt was well known that the two men were not fond of each other. Don Serafino was a liberal and a Freemason, the marquis was a Papist and man of the Church, and they were also at odds over a lawsuit more than twenty years old concerning the disputed ownership of a cherry tree.\n\nAll at once the marquis's face, already red, turned green. Traffic lights did not exist at the time, otherwise the similarity would have been striking.\n\n\"And just what are you, Serafino by name but a horned devil in fact, trying to insinuate?\"\n\n\"Please, gentlemen, for pity's sake!\" the president beseeched them.\n\nDon Serafino took no offense.\n\n\"I'm not insinuating anything. You sued Father Raccuglia, claiming he'd taken possession of a piece of your land exactly the same way you've taken people's cherry trees away, and so you turned to the lawyer Teresi, who eats priests for dinner, roasted, fried, or topped with tomato sauce . . . Is that true or not?\"\n\n\"Yes, it's true! So what? What the hell is your point? It doesn't mean that when somebody turns to a lawyer he has to embrace his political ideas as well!\"\n\n\"Let me finish. The lawyer agreed to sue on your behalf, but he also asked you to support his candidacy for membership in the club. Which you did.\"\n\n\"I certainly couldn't refuse a common courtesy such\u2014\"\n\n\"Courtesy, my eye! The lawyer told you that if you supported his candidacy he would handle your case free of charge. And you, despite your wealth, are as stingy as a dried-up riverbed, you couldn't believe your ears!\"\n\n\"So then why did I vote against him, can you tell me that?\"\n\n\"Of course I can. The lawsuit had barely begun when Father Raccuglia let himself be persuaded, by someone acting on your behalf, that he should admit he was in the wrong. And so, just like that, no more lawsuit. So you, who had turned to Attorney Teresi\u2014the only man in town with the cheek to sue a priest\u2014immediately turned your back on him. Therefore, as you can see, I didn't insinuate anything.\"\n\n\"No, you are insinuating that I had someone act on my behalf! So, why don't we start with you naming his name?\"\n\n\"Oh, no you don't! No names! That's enough of that! Let's end this! It's getting late!\" shouted a number of voices.\n\nIt was of utmost importance that the person's name not be revealed. The arguments were starting to take a dangerous turn. And the name that mustn't be mentioned was that of _'u z\u00f9_ Carmineddru, the town's Mafia chieftain, a man of honor and consequence.\n\n\"In that case, gentlemen, after the marquis's declaration, I have no choice but to address my words to the unknown member who . . . \"\n\n\"So how do you explain that two noblemen, Baron Lo Mascolo and Marquis Cammarata, turned to a lawyer like Teresi, who's a known instigator?\" asked don Serafino with his usual smile, taking advantage of the momentary silence to slip in a question that everyone, truth be told, had been asking themselves.\n\n\"I will break your bones, God help me!\" exclaimed the marquis, jumping up from his chair and hurling himself at his adversary.\n\nHe never reached him, however, as three men managed to restrain him. Frothing at the mouth like an enraged bull, the marquis stormed out of the meeting.\n\n\"Gentlemen, gentlemen, please! Let's get this over with quick. The bell has already rung for the Mass. I therefore address myself to the unknown\u2014\"\n\n\"And when are we going to talk about the duel?\" asked Colonel Petrosillo, whose nosebleed wouldn't stop, enraging him further with every passing minute.\n\n\"Later, later,\" they all said in a sort of chorus.\n\n\"Then I beg the unknown member who voted for admission please to explain to the rest of us\u2014\" the president began.\n\n\"There's no damn need to beg,\" said don Anselmo Buttafava. \"I was the one who voted yes.\"\n\n\"Why?\" asked the president. \"I believe you said several times in the past you never wanted to see Teresi in here, not even dead.\"\n\n\"And in fact in the first round of voting I voted no.\"\n\n\"So why did you change your mind?\"\n\n\"Because if there's a pervert like Colonel Petrosillo in this club I don't see why we can't admit a Bakuninist like the lawyer Teresi.\"\n\n\"Good point,\" commented don Serafino, who that Sunday morning seemed determined to get on the nerves of all of creation.\n\nColonel Petrosillo shot to his feet, as pale as a corpse.\n\n\"Consider yourself slapped likewise!\" he said to don Serafino.\n\n\"I don't consider myself anything at all. Come over here and slap me in person, if you're man enough. And since your bum has already been thrashed, I'll get to work on your face, just to finish what don Anselmo started.\"\n\nThe colonel opened his mouth to reply, but at that moment a nervous twitch came over his face. He immediately stiffened, eyes rolling back into his head, and fell backwards. He suffered from occasional epileptic fits. A good fifteen minutes were lost in the efforts to revive him and accompany him to his carriage.\n\n\"May I have permission to speak, Mr. President, sir?\" asked Giallonardo the notary.\n\n\"You may.\"\n\n\"Just now you said that the sponsors of Attorney Teresi's candidacy were Marquis don Filadelfo Cammarata and Baron Lo Mascolo, correct?\"\n\n\"Correct.\"\n\n\"Then, since don Filadelfo admitted to having twice cast a black marble vote, the fact of having repeated the same gesture would serve to invalidate, essentially, his prior sponsorship\u2014indeed annul it entirely. Therefore, things being so, Attorney Teresi's candidacy can be considered to have been advanced by only one signature, that of Baron Lo Mascolo. Now, according to the rules, only one sponsor is not enough. Ergo, it is as if Attorney Teresi never made any request for admission.\"\n\n\"Well, I'll be damned! Brilliant!\" don Stapino Vassallo said in admiration.\n\n\"Seems to make perfect sense to me,\" said the president. \"Are the gentlemen members in agreement that . . . ?\"\n\n\"Yes! Yes!\"\n\nThe chorus was unanimous.\n\n\"Then the session is adjourned,\" said the president.\n\nThere was a mad dash for the door, legs flying, arms pushing, as the members of the club ran to catch the last Mass in their respective churches.\n\n*\n\nA town of seven thousand inhabitants located right in the middle of the great _latifondi_ estates, Palizzolo, in the year 1901, could boast of having two marquis, four barons, a one-hundred-and-two-year-old duke who no longer set foot outside his castle, and an anti-Bourbon martyr, attorney Ruggero Colapane, hanged in the public square for having supported the Parthenopean Republic.\n\nBut its greatest source of pride was its eight churches, each endowed with a bell tower and bells so powerful that when they rang all together in unison, it felt just like an earthquake inside people's houses.\n\nSeven of these eight churches had been divvied up between the nobility and landowners on the basis of mutual antipathies and sympathies, familial relations acknowledged or denied, longstanding resentments and quarrels dating back to the times of Carlos Quinto, and civil suits hailing from the age of Frederick II of Swabia and carried on even after the Unification of Italy, as well as undying hatreds and changing attachments.\n\nAnd so, for example, you would never have seen, say, someone like don Stapino Vassallo and someone like don Filadelfo Cammarata attending Mass together at the church of Our Lady of Sorrows, whose parish priest was Patre Don Angelo Marraf\u00e0.\n\nIn 1514, an ancestor of don Stapino\u2014more specifically the beautiful young Attanasia\u2014had been married at age sixteen to an ancestor of the Marquis Cammarata, a forty-year-old by the name of Adalgiso. After two years of a marriage ratified but never consummated, owing to a case of _impotentia coeundi_ on the husband's part, Attanasia could no longer stand living like a cloistered nun despite being married, so she started looking around. And soon, by dint of looking, she found herself impregnated, apparently by a stable hand. Adalgiso sent his wife back to her parents, calling her a trollop. Attanasia riposted by saying that her husband couldn't perform his husbandly duties, because his thingy was as soft as ricotta cheese. This gave rise to lawsuits, trials, and litigations, the result of which was that the two families not only ceased greeting each other in public but indeed never missed an opportunity to do one another a bad turn.\n\nThe eighth church, the Church of the Most Holy Crucifix, with the seventy-year-old Don Mariano Dalli Cardillo as its priest, was attended neither by nobles nor by landowners, and not even by the bourgeoisie. It was the church of the peasants and the poor, of those who lived on bread and air.\n\n*\n\n\"Beloved children,\" said Don Alessio Terranova, priest of the parish of San Giovanni, opening his Gospel sermon. \"I find myself obliged today to talk to you about a serious matter. A petty newssheet edited and paid for by a local lawyer whose name I won't mention, since it would sully my tongue, and distributed here and in nearby towns, featured an article in this morning's edition in which, on top of the customary insults hurled at the Holy Mother Church and at those of us who represent her in all our unworthiness, the writer mocks the sacrament of holy matrimony and the virginity of maidens and ridicules modesty, chastity, and feminine virtue . . . And so I exhort you, my beloved children\u2014especially my beloved daughters\u2014not to listen to such iniquities, which are clearly inspired by the devil. Virginity is the noblest gift a young bride can give to her legitimate spouse; it is in every way comparable to a flower, which . . . \"\n\n*\n\nPatre Raccuglia, head of the town's Mother Church, its most ancient, also said during his sermon that Palizzolo was facing a grave danger, that of ending up just like Sodom and Gomorrah, if the sacrilegious ideas of a little lawyer who loved to pretend he was the people's advocate, when he was in fact the devil's advocate, continued to spread. This man\u2014if a Godless person who disdained family, religion, and country, and every divinely blessed thing on earth, could really be called a man\u2014this man had written in his newssheet that virginity, that supreme gift of maidens, was merely a commodity for purchase! Something that a man could simply buy for cash with a wedding ring! Blasphemy! Whereas virginity was, in fact . . .\n\n*\n\nThat Sunday, at the end of the Mass, Giallonardo the notary stopped to talk with don Liborio Spart\u00e0 outside the church of San Cono, the patron saint of Palizzolo, whose parish priest was Don Filiberto Cusa.\n\n\"There's something I don't understand,\" said the notary. \"Why did Teresi request admission to the club when he must have known he would never be accepted?\"\n\n\"In my opinion,\" said don Liborio, \"he wants to brag about it.\"\n\n\"With whom?\"\n\n\"With the clients he defends. The ragamuffins, shirkers, and subversives who don't have an honorable bone in their bodies . . . He'll probably tell them: 'See? The nobles, the bourgeois, the landowners don't want me in their club. And that proves that I'm one of you!'\"\n\n\"I just can't imagine what that man's got in his head,\" the notary said pensively. \"He let his father, don Masino, who was a very fine person, die of a broken heart. What? You studied to become a pharmacist and that's not enough for you? Nosirree. He goes and gets that law degree, disowns his family and the class he belongs to, and starts doing what he's doing now. The guy's stirring up the riffraff to the point that one of these days, a revolution's going to break out in Palizzolo!\"\n\n\"Well, the man is certainly dangerous, as far as that goes,\" said don Liborio.\n\n\"Maybe we should give the matter some thought,\" said the notary, seeing Don Filiberto, the parish priest, come out of the church and begin to approach them, waving his hands in the air by way of greeting.\n\n\"I saw you, you know!\" said Don Filiberto. \"You came late to Mass! Why's that?\"\n\n\"We had a trying morning at the club,\" replied don Liborio.\n\n\"Why, what happened?\"\n\n\"We voted on whether to accept lawyer Teresi's request for admission,\" said the notary.\n\n\"And how did it turn out?\" asked the priest, his jocund face turning suddenly serious.\n\n\"It was considered invalid.\"\n\n\"And a good thing too! If you'd accepted it I would have denied you the sacraments! And you know what else? This Teresi, when he dies, will even have trouble getting into Hell! The devil won't want him!\"\n\nThey all laughed.\n\n*\n\nHaving just come out of the church of the Heart of Jesus, whose priest was Don Alighiero Scurria, Commendatore Paladino and don Serafino Labianca set out, as they did every Sunday morning, on their way to the Gran Caff\u00e8 Garibaldi to drink their customary glass of malmsey before going to lunch. While don Serafino was certainly a liberal and a Freemason, deep down he feared that God might exist after all, and so, taking the good with the bad, he made a point never to miss a Sunday Mass.\n\nThey sat down at a table and started talking, inevitably, about Matteo Teresi.\n\n\"His request for membership was just a ruse to provoke us,\" said the commendatore.\n\n\"That's clear,\" don Serafino agreed.\n\n\"But it would be a mistake to react to his provocations, don't you think?\"\n\n\"Yes, I completely agree.\"\n\n\"On the other hand, we can hardly put up with this forever.\"\n\n\"Patience does have its limits.\"\n\n\"And I'm afraid that sooner or later this man will do some damage, some very grave damage. Don't you think?\"\n\n\"Absolutely!\"\n\n\"Don Serafino, at the club you asked an intelligent question but you never gave us the answer.\"\n\n\"I don't remember. What was it?\"\n\n\"How is it that two noblemen sponsored Teresi's candidacy?\"\n\nDon Serafino smiled.\n\n\"But it's precisely because of what you've just finished saying! They're afraid that lawyer will stir up the riffraff to the point where all hell breaks loose. So, just to be safe, they want to keep him close.\"\n\nThe waiter brought the two glasses of malmsey, and the men drank in silence.\n\n\"Maybe,\" don Serafino resumed, \"we need to discuss this with some other friends of ours. I'd say it's rather urgent. And then we'll meet back up at my house.\"\n\n\"Sounds to me like a good idea,\" said the commendatore.\n\nProfessor Ubaldo Malatesta, superintendent of the local elementary schools\u2014the only schools in Palizzolo\u2014walked into the sacristy of the church of the Most Holy Virgin as the parish priest, Don Libertino Samon\u00e0, was removing his vestments with the help of a little boy.\n\n\"Why didn't you come to serve Mass today?\" asked Don Libertino.\n\nThe professor, a shy man, blushed in shame.\n\n\"I'm here to apologize. I was detained at the club, and\u2014\"\n\n\"What?! So you've come to tell me that your gambling vice kept you away\u2014\"\n\n\"No, Father, there were no games this morning. We were voting on whether to admit Teresi the lawyer to the club.\"\n\nPatre Samon\u00e0 was a good six feet tall and six feet wide. Pointing a finger that looked like a cudgel at Professor Malatesta, he asked in a Last-Judgment tone of voice:\n\n\"And how did _you_ vote?\"\n\n\"I . . . I voted 'no.'\"\n\n\"Well, you should know that had you voted 'yes,' not only would I never again have let you serve Holy Mass, I would have chased you right out of the church with so many kicks in the pants you wouldn't be able to sit down for a week!\"\n\n A short-lived, French-inspired and French-supported republican revolution that seized the city of Naples, under Bourbon rule, in 1799, only to be put down later that year.\n\n## CHAPTER II \nTHE PASSION AND FLIGHT OF DON ANSELMO\n\nDon Anselmo, for his part, was unable to attend Mass at the church of Saints Cosma and Damiano, whose parish priest was Don Ernesto Pintacuda, because he'd had to go home to change his singed trousers.\n\nAnd so, since there was still a while to go before lunchtime, he decided to pay a call on Baron Lo Mascolo, first of all to find out whether he'd recovered from the touch of flu that he said he'd come down with two days before, and secondly to ask him to explain why he'd supported Teresi's request.\n\nDon Anselmo, being a good friend of the baron's even though don Fof\u00f2 was a good twenty years his junior, knew him inside and out, and, truth be told, didn't believe for a minute this business about him having the flu. It was well known that the baron was the picture of good health and had never spent a day in his life in bed, had never had a toothache, never a bellyache, even though he was capable of eating two roast suckling goats with a few kilos of potatoes all by himself.\n\nAnd so? What's two plus two? Clearly don Fof\u00f2, after first supporting lawyer Teresi's request for membership, came to regret it and changed his mind, just as his friend, Marquis Cammarata, had done, and instead of going to the club and voting with a black marble, he decided to pretend he was sick.\n\nDon Anselmo had just raised his hand to lift the heavy door knocker when a small door cut out of one wing of the great door of Palazzo Lo Mascolo opened and out came Doctor Bellanca, medical bag in hand.\n\n\"I've been here all morning, which is why I couldn't come to the club,\" he said, shaking don Anselmo's hand. \"How'd it turn out?\"\n\n\"The request was deemed invalid.\"\n\n\"So much the better,\" said the doctor, and he started to close the small door behind him.\n\n\"Please leave it open,\" said don Anselmo.\n\n\"Do you want to go in?\"\n\nHe'd asked the question without budging a single millimeter from the doorway, so that don Anselmo couldn't get by him.\n\n\"Yes.\"\n\n\"Did you want to see the baron?\"\n\nWhat was with all the questions?\n\n\"Yes.\"\n\nThe doctor closed the small door decisively.\n\n\"He's in no condition to receive you, believe me.\"\n\nDon Anselmo balked. So the baron really was sick after all!\n\n\"Is it anything serious?\"\n\n\"Well, yes and no.\"\n\n\"Does he have the flu? . . . \"\n\n\"No, it's not the flu.\"\n\n\"Then what's wrong with him?\"\n\nBellanca seemed a little uneasy.\n\n\"It's . . . how shall I say? An unusual case.\"\n\n\"Oh, well. I'll just go and say hello to the baroness and\u2014\"\n\n\"She can't receive you, either.\"\n\n\"She's caught it too?\"\n\n\"Well . . . yes, in a manner of speaking.\"\n\n\"What about _Baronessina_ Antonietta?\"\n\nDr. Bellanca made a strange face.\n\n\"Well, let's just say she . . . is the origin of the illness.\"\n\nHow could that be? The baronessina was an eighteen-year-old girl as beautiful as the sun and more bursting with health than even her father!\n\n\"Listen, doctor. If this illness spreads so easily . . . \"\n\n\"Please don't be alarmed, and, most importantly, don't go spreading the word. You mustn't needlessly stoke people's fears. The baron and his family are in a kind of quarantine in there. They need only avoid direct contact with others. In a few days it'll all be over.\"\n\nDon Anselmo remembered he'd shaken Bellanca's hand. He shuddered. He was deathly afraid of illness.\n\n\"Er, doctor, did you by any chance wash your hands?\"\n\nWithout answering, Bellanca walked away cursing. As he began to head home, don Anselmo turned around to look at Palazzo Lo Mascolo. All the shutters over the windows and balconies were closed. As if the family were in mourning. There was no visible sign of life inside. At one o'clock on a Sunday afternoon? With a sun hot enough to split rocks? What, were they all dead or something?\n\nTo go home, don Anselmo had no choice but to walk past Palazzo Cammarata, which stood all by itself on a street that was also called Cammarata. Nobility dictated that the palazzo mustn't have any other constructions around it. The great house took up the whole street and had only its private garden, the _firriato_ , in front.\n\nReaching the end of the street, which led into Piazza Unit\u00e0 d'Italia, don Anselmo stopped and looked around in bewilderment. Something had unnerved him at Palazzo Cammarata, but he couldn't figure out what it was.\n\nThe silence! That's what it was!\n\nMarquis Filadelfo head eight daughters, the youngest being five years old and the oldest eighteen, a wife\u2014Marquise Ernestina\u2014who was vociferous by nature, and two maids. The only man amidst eleven women who were always quarreling one minute, laughing the next, crying one minute, chattering the next, cursing each other one moment, raising hell the next, the marquis would sometimes lose his head, and the nervous agitation from which he suffered even while asleep would become so great that he would go outside dressed just as he was and, to let off steam, pick a fight with the first person he passed on the street. Everything that went on inside the palazzo always became instantly known to whoever happened to be walking along Via Cammarata. The chatter of the eleven women, who habitually spoke loudly, would waft out of the windows, which were perpetually open, rain or shine, bounce off the stones outside and back into the windows through which it had just exited.\n\nSo, why was the palazzo now as silent as the grave? Looking up, don Anselmo noticed that all the shutters over the windows were closed, something he'd never seen before. What could have happened?\n\n\"No, no, no . . . \" he said to himself. \"They're not playing straight with me here, neither the baron nor the marquis!\"\n\nAnd he turned and went back, determined to knock on the door and demand an explanation. But after he'd taken barely three steps, he froze.\n\nBreathlessly approaching from the other end of the street was Dr. Bellanca, with his medical bag in hand.\n\n\"Were you going to the marquis's?\"\n\n\"Yes.\"\n\n\"Did he send for you?\"\n\n\"No, but as I was passing by I noticed\u2014\"\n\n\"Please, don Anselmo, just go home.\"\n\n\"But why?\"\n\n\"Because I don't think the marquis is in any condition to see you,\" said the doctor, knocking at the door.\n\n\"Is he sick?\"\n\n\"Yes.\"\n\n\"But I saw him at the club this morning!\"\n\n\"That doesn't mean anything. The . . . er . . . In short, it came on all of a sudden.\"\n\nThe explanation dawned on don Anselmo, as sudden and swift as a punch in the stomach.\n\n\"Accompanied by diarrhea?\" he asked in terror.\n\n\"Among other things.\"\n\n\" _O matre santissima!_ So it's an epidemic!\"\n\nThe great door came open and the doctor went inside. The door then closed again.\n\nFor the second time, don Anselmo asked himself what was two plus two.\n\nAnd he answered his own question. And \"four,\" this time, could equal only one terrible thing: cholera. There'd been a wave of cholera a few years back that had ushered half of the town into the cemetery. He stood there staring at the closed windows for a moment; then, leaning even more on his cane, since his legs were trembling more than usual, he walked quickly home, opened the door, went in, sat down in a chair in the vestibule, and was unable to go any further.\n\nHis wife, Signura Agata, who'd heard the door open, came out into the vestibule and saw her husband there, looking as pale as a corpse and fanning his face with his hat. She got worried.\n\n\"'Nzelm\u00f9, what happened? Are you unwell? Why do you look like that, eh?\"\n\n\"Just be quiet for a minute and let me catch my breath, dammit!\"\n\nBut the signura couldn't restrain herself.\n\n\"Talk to me, 'Nzelm\u00f9, you're frightening me! _Matre santa_ , what's wrong?\"\n\n\"Nothing's wrong! And stop buzzing around me like some kind of mosquito! Where's Girolamu?\"\n\n\"The coachman? I don't know.\"\n\n\"Tell the maid to go and find him. I want him to hitch up the big carriage.\"\n\n\"Why? Are you going somewhere? May I ask where?\"\n\n\"Agat\u00ec, you're coming with me, and we're leaving straight away.\"'\n\n\"What! And where are we going?\"\n\n\"To the country!\"\n\n\"Back to San Giusippuzzo? We were there less than a week ago!\"\n\n\"And now I feel like going back, devil take it all!\"\n\n\"All right, all right, there's no need to curse. But how long will we stay?\"\n\n\"I figure about a month.\"\n\n\"What! So long? Why?\"\n\n\"Agat\u00ec, there's something going on in town I don't like the look of. Baron Lo Mascolo's whole family is sick, all of 'em, and everyone in Palazzo Cammarata's sick too.\"\n\n\"The flu's been going around.\"\n\n\"What flu? The main thing that's going around is Doctor Bellanca! And the bastard won't tell me anything! But I figured it out all by myself. There's an epidemic, Agat\u00ec! Maybe cholera!\"\n\n\" _Matre tutta santa e biniditta!_ I'll go and start packing the trunks!\"\n\nThey left two hours later, and it took the usual hour to reach San Giusippuzzo. The road was in such bad shape that several times the carriage very nearly fell into a ditch. Finally, by the grace of God, Girolamu halted the two horses inside the enclosure that contained the Buttafavas' villa, the wine vat, the stables, the carriage house, and the cottage of 'Ngilino the overseer, who lived there with his wife Catarina and daughter of seventeen, Totina. From the window of his coach, don Anselmo noticed that there didn't seem to be anyone in the house, since the door as well as all the shutters were closed.\n\nSince the overseer hadn't been expecting his masters' arrival, he must surely be out in the fields somewhere. And Catarina and Totina must have gone into town, since it was Sunday.\n\nDon Anselmo climbed down from the carriage and went and opened the front door of the villa. As his wife was going inside, he said to his coachman:\n\n\"Try calling 'Ngilino. If he's somewhere nearby, he can help you bring the trunks inside.\"\n\nThe master bedroom was upstairs. Don Anselmo lay down with all his clothes on; the journey had worn him out and, on top of that, he hadn't been able to have his usual afternoon nap.\n\n\"I'm going to have a little rest,\" he said to his wife, who was bustling from room to room.\n\nHe fell asleep at once and slept for two straight hours.\n\n*\n\nHe was awakened by Signura Agata, his wife.\n\n\"Time to wake up. Girolamu and 'Ngilino are bringing up the trunk with the clothes.\"\n\nHe went into the privy.\n\nWhen he reemerged, his wife was taking the clothes out of the trunk, and he could tell she was angry because she was whining with her mouth closed. Agata was a good-hearted woman, but she liked to be served. She normally wouldn't even bend down to pick up a pin that had fallen to the floor.\n\n\"What need is there for you to do that work yourself? You could have asked Catarina and Totina to do it when they got back from town.\"\n\n\"'Ngilino told me they didn't go into town.\"\n\n\"Then where did they go?\"\n\n\"They didn't go anywhere. They're right here, at home.\"\n\n\"At home? Then why didn't they come out when we arrived?\"\n\n\"Because they're sick.\"\n\n\"Both of 'em?\"\n\n\"Both of 'em.\"\n\n\"But were they in church this morning?\"\n\nAt don Anselmo's personal request, Don Ernesto Pin\u00adtacuda, priest of the church of Saints Cosma and Damiano, had accepted Catarina and Totina as members of his parish, when they should by rights have gone to the church of the Most Holy Crucifix, the peasants' church. The fact of the matter was that don Ansemlo was terribly fond of Totina. The girl was a sight to behold, and her cheerful disposition was contagious. Don Anselmo would sometimes spend hours out on the balcony, watching the girl performing her chores in the farmyard. And, unbeknownst to Signura Agata, he'd even given her money to buy herself some nice clothes so she would look good at Sunday Mass.\n\n\"No, they weren't.\"\n\nA thought flashed into don Anselmo's mind.\n\n\"Shit!\"\n\n\"What's with you?\"\n\n\"Shit shit shit!\"\n\n\"Don't use obscenities! What's wrong?\"\n\n\"The clo . . . the locked doors! The closed windows! Just like Palazzo Lo Mascolo! And . . . and Palazzo Cammarata! Put all the clothes back in the trunk!\"\n\n\"Have you lost your mind?\"\n\n\"Agat\u00ec! The epi . . . the epidemic has spread here too!\"\n\nHe went out of the room, raced down the stairs and into the courtyard, headed for the stables, went upstairs, and promptly kicked open the door to the room where the coachman slept.\n\nGirolamu, who was in his underpants, very nearly had a heart attack.\n\n\"Wha\u2014what is it, sir?\"\n\n\"Hitch the carriage back up! We're leaving!\"\n\n\"Where to, sir?\"\n\n\"To La Forcaiola!\"\n\nGirolamu looked bewildered.\n\n\"But, sir, that'll take a two and a half hours at the very least! And it'll be dark soon.\"\n\n\"I don't give a damn. Hitch 'em up! Then come and get the trunks!\"\n\n\"Could I ask 'Ngilino to give me a hand, sir?\"\n\n\"No! You mustn't so much as look at his shadow!\"\n\n\"Sir, can I tell you something, since your missus isn't present?\"\n\n\"Tell me.\"\n\n\"You should know that people are saying Salamone the brigand's been hanging around Forcaiola way.\"\n\nThat was all they needed! Salamone the brigand not only stripped any nobleman or bourgeois he encountered of everything he had, leaving him as naked as Adam, but he never passed on any woman he crossed paths with either. He did them all, from age fifteen to fifty, right before the eyes of their husbands, fathers, and brothers, whom his henchmen would restrain. Anselmo's wife was already past sixty, and so wasn't in any danger.\n\nThe problem was that Salamone was liable to make off with the carriage itself, leaving both of them\u2014actually, all three of them, since the brigand certainly wouldn't spare even Girolamu\u2014naked on a godforsaken country road in the middle of the night.\n\nBut, between cholera and the brigand, the choice was clear.\n\n\"Hitch 'em up! Hitch 'em up!\"\n\n*\n\nLa Forcaiola was an estate belonging to a first cousin of don Anselmo, don Lovicino Scattola, who at present was in prison in Palermo, serving a seven-year term for having killed don Michelangelo Fichera during a hunting party, after the latter had claimed, just minutes before, that don Lovicino had never in all his life managed to shoot a rabbit or hare because he was incapable of hitting even an elephant from two feet away. And so don Lovicino shot him from thirty feet away, just to show, in the presence of witnesses, that the man was wrong.\n\nUpon hearing news of his boss's sentence, Benuzzo Cogliastro, don Lovicino's farm overseer, had felt his heart fill with joy. For seven years he would be the real owner of the estate. But then one day don Anselmo had shown up with full power of attorney granted by his cousin, and Benuzzo had sworn to get even with him. For this reason don Anselmo didn't show his face much around there, if at all, and only went when he absolutely had to. As in the present instance.\n\n*\n\nThey arrived late at night, luckily without having crossed paths with Salamone the brigand. The situation at the estate was almost exactly the same as at San Giusippuzzo. The shutters of Benuzzo's house seemed to be open, but there wasn't a hint of any light inside. Surely the whole family was asleep. Girolamu took the cart lamp out out from under the carriage and lit the way for don Anselmo, who was holding the keys, to unlock the main door of the villa. Signura Agata hadn't wanted to get out of the carriage before the all the oil lamps in the entrance hall were lit.\n\nHands trembling from fatigue, don Anselmo had to try three times before successfully putting the key into the door. And at that exact moment, a rifle shot rang out, splitting his eardrums. The large boarshot of the _lupara_ blasted several holes in the great door, just a few inches from his head. And the two horses, frightened by the blast, started running towards the farmyard exit, with Signura Agata screaming wildly. But when the animals took a turn a little too sharply, the left wheel crashed against the wall and the carriage flipped.\n\n\"I'm dying!\" cried Signura Agata, before fainting.\n\n\"Get out of here or I'll kill you all!\" a man shouted angrily.\n\nDon Anselmo, dropping to the ground and shaking in terror, recognized the voice of Benuzzo, the farm manager.\n\n\"Benuzzo! It's me, don Anselmo! Don't shoot!\"\n\nBy way of reply a flash went off in one of the windows of Benuzzo's house, and don Anselmo closed his eyes.\n\n\"I'm a dead man!\" he thought.\n\nThe shot hit the great door again.\n\n\"You're not don Anselmo, you're Salamone the brigand and you take me for a fool!\" said Benuzzo.\n\n\"Get me out of here! Help! Somebody get me out of here!\" Signura Agata shouted in the meantime, having regained consciousness.\n\nSince Girolamu had dropped the cart lamp in terror and was now spread out belly-down on the ground, praying to the Madonna aloud, don Anselmo got a crazy idea.\n\nHe reached out, grabbed the lamp, and held it next to his face.\n\n\"Take a good look at me, you stupid shit! I'm don Anselmo!\"\n\n\"Oh! So it's you? I din't rec'nize you, sorry. You coulda told me you was comin'! I'll be right down.\"\n\nAt that moment don Anselmo realized, from Benuzzo's tone of voice, that the overseer had known perfectly well from the moment the carriage had entered the courtyard that it was him and not Salamone the brigand.\n\nAnd he'd shot at him on purpose, the bastard!\n\nBut when he tried to stand up, don Anselmo was unable. His body ached all over.\n\n\"Go and help the signora!\" he yelled at Girolamu.\n\nBy this point the voices of the farmer's wife Ciccina, son Paolino, and daughter Michilina could be heard inside the house, as they hurriedly got dressed to go and help the masters who had just arrived.\n\nBenuzzo came down out of breath and, with a lamp in his hand, bent down to look at don Anselmo. He still had his rifle in his other hand.\n\n\"Wha'd I do, hit you?\"\n\n\"No.\"\n\n\"Well, I'm glad for that! Here, lemme help you up.\"\n\nAnd he held out his hand. Don Anselmo didn't take it at once, but instead asked Benuzzo a strange question:\n\n\"Everyone in the family all right?\"\n\n\"Everyone's fine, thanks be to God.\"\n\nOnly then did don Anselmo grasp the farm manager's hand. If they were all fine, it meant that the cholera, luckily, hadn't spread to that area yet.\n\n## CHAPTER III \nDON ANSELMO'S CHOLERA AND OTHER COMPLICATIONS\n\nAt around the same time that don Anselmo finally managed to fall asleep at La Forcaiola\u2014that is, around four in the morning\u2014the great door of Palazzo Lo Mascolo was carefully opened, and a man's head poked out and looked both ways to make sure there wasn't anyone around.\n\nReassured, the man came out, closing the door behind him.\n\nHe was completely masked, wearing a cloak tossed over his left shoulder in such a way that it covered his face, leaving only the eyes visible, since the beret on his head was pulled down over his brow.\n\nOn his feet the man had an old pair of hobnailed boots like the kind the peasants wore. In his right hand he was holding a shepherd's staff.\n\nWalking away, he didn't encounter another living soul. But even if he had seen someone, the person was unlikely to recognize that bundled-up peasant as Baron don Fof\u00f2 Lo Mascolo.\n\nArriving at the home of Teresi the lawyer, a lone, freestanding house near the top of the hill whose slopes the town was built on\u2014indeed behind the lawyer's house was a drop of some two hundred feet\u2014the baron stopped, raised his walking stick, and knocked hard on the wooden door. Nobody came to open it.\n\nTeresi was not married, and lived with a lad of twenty, Stefano Pillitteri, son of his sister, who had married a ne'er-do-well and died young. The lawyer was very fond of his intelligent nephew and kept him around as an apprentice, paying for him to study law at the University of Palermo.\n\nThe baron resumed his assault. As he was slamming the knocker with all his might with his right hand, he rapped his cane against the door with his left, all the while kicking the door with his hobnailed boots. You could have opened a tomb with all the racket he was making. And, indeed, through the shutter slats over one of the upstairs windows a dim light came on, the window opened, and Teresi the lawyer appeared, reciting his customary formula:\n\n\"My door is open to everyone. Therefore, whoever you are, you are welcome in this house. I'll be right down to open the door.\"\n\nFive minutes later, the man entered the house. At first Teresi didn't recognize him. But as soon as the man removed his barracan and cap, the lawyer balked in surprise.\n\n\"Baron! So it's you? Why are you dressed like that?\"\n\n\"I didn't want anyone to recognize me.\"\n\n\"Why not? You've certainly never put on a disguise to come to my house before.\"\n\n\"Well, this time I did.\"\n\n\"Let's go into my study.\"\n\nTeresi sat down behind the desk, while the baron settled into the armchair opposite it.\n\n\"Shall I make some coffee?\"\n\n\"No.\"\n\nThere was a silent pause. The lawyer knew from experience that it was always best to let the person in front of you make the first move.\n\n\"Is your nephew here?\" the baron asked after a spell.\n\n\"Stefano? Yes, he's in his room, sleeping.\"\n\n\"And why didn't he wake up?\"\n\n\"No idea. Maybe because kids sleep deeply. May I ask why you came here at this hour of the night?\"\n\n\"To kill your nephew Stefano,\" said Baron Lo Mascolo, taking a revolver out of his pocket and setting it down on the desk. \"Shall we wake him up?\"\n\n*\n\nSignura Agata, meanwhile, had confided to her chambermaid Suntina the reason why they were in such a hurry to get away from Palizzolo.\n\n\"My husband don Anselmo says it looks like there's cholera going around. But he doesn't want anyone to know.\"\n\nThe last time cholera had passed through town, Suntina had lost her father, mother, all four grandparents and her only brother. Afterwards, she was taken in by her father's brother, Tamazio, a peasant who ended up treating her like a servant (which was normal), deflowered her at age thirteen (which was also normal), but also demanded that the girl wash his feet every Sunday. This was not normal, and Suntina would not stand for it. So she ran away and knocked at the first door she saw. Which was the front door of Palazzo Lobue, where lived Galatina and Natale Lobue, Agata's young parents. Suntina helped raise the little girl, and when Agata got married to don Anselmo, she brought Suntina with her.\n\n\"Do you want to come with us, Sunt\u00ec?\"\n\n\"No, ma'am. I'd rather stay here and wait for you to come back.\"\n\n\"But that may be dangerous, you know.\"\n\n\"I know, but if I stay, I can watch the house for you.\"\n\nAnd this was a good idea, since during the last wave of cholera many houses had been robbed and ransacked.\n\n\"As you wish.\"\n\nAs soon as the masters had left, Giseffa, the other housekeeper, who wasn't yet twenty years old, went into such a song and dance that Suntina was forced to tell her the reason for their departure.\n\n\" _Matre santa!_ Cholera! I'm leaving too! Right now!\" said Giseffa, scared out of her wits.\n\n\"And where you gonna go?\"\n\n\"To my father's house.\"\n\n\"But your father's house is also here in town! Listen to me: just stay here, that would be best.\"\n\n\"Why would that be best?\"\n\n\"First, because cholera never attacks the rich, only the poor. If we stay here, in the house of rich people, it's possible that when the cholera passes through it'll be in a hurry and mistake us for rich people too. Secondly, because here there's flour, cheese, salted sardines, tomatoes, and all the water we need. We could hole up here for at least three months without ever having to go out. We'll lock the great door and not open for anyone.\"\n\n\"No. I want to go to my father's house.\"\n\n\"Listen, tell you what. Since don Anselmo doesn't want the news of the cholera to get out right away, you'll sleep here tonight, and tomorrow morning, you can get up at the crack of dawn and go to your father.\"\n\n\"Is this some kind of joke, Baron?\"\n\n\"I'm warning you, Teresi, if you piss me off, I'll shoot you too.\"\n\n\"All right, all right. But can you at least tell me the reason?\"\n\n\"Shall we set something straight, first?\"\n\n\"If you think we need to, then yes, of course.\"\n\n\"How would you characterize the relations you and I have always had?\"\n\n\"I would say they've always been good.\"\n\n\"And I would say excellent. I'll cite just one example. Didn't I entrust you with my lawsuit against Baron Mostocotto instead of giving it to the lawyer Moschino, who was very keen on having it?\"\n\nThe reason for the dispute between the two barons was that one day, Baron Mostocotto, who had a weak bladder and therefore was always having to pee, was caught by don Fof\u00f2 urinating against a corner of Palazzo Lo Mascolo. The baron took umbrage.\n\n\"Look,\" Mostocotto had said to him, to defuse the situation, \"if you want some kind of compensation, you can come and pee on my palazzo whenever you like.\"\n\nBut there was no settling the dispute, not even with the authoritative intervention of Giallonardo the notary. Baron Lo Mascolo finally sued his fellow baron for damages to his building.\n\n\"Yes, that's true,\" Teresi admitted.\n\n\"And didn't I pay you, with no questions asked, the rather considerable advance you asked of me?\"\n\n\"Yes, sir, you did.\"\n\n\"And when you asked me to support your request for membership to the club, did I support you or not?\"\n\n\"Of course you did.\"\n\n\"And did I not allow your nephew, Stefano, to come and call at our house whenever he liked?\"\n\n\"Yes. And I am very grateful to you for your generosity.\"\n\n\"But he's not.\"\n\n\"I'm sorry, he who?\"\n\n\"Your nephew.\"\n\n\"He hasn't been grateful to you?\"\n\n\"No.\"\n\n\"And that's why you want to shoot him?\"\n\n\"Stop speaking twaddle, Teresi.\"\n\n\"Then why?\"\n\n\"Three days ago, my daughter Antonietta felt unwell. For the first time in eighteen years. And so my wife sent for Dr. Bellanca. Ever since, my house has been in deep mourning.\"\n\n\"Oh my God, is her illness really so serious?\"\n\n\"Serious? My daughter is dead!\"\n\nThe lawyer stood up.\n\n\"Please allow me to embrace you, Baron,\" he said sincerely. \"So terrible a misfortune warrants\u2014\"\n\n\"Just remain seated or a terrible misfortune will befall you instead. Until tonight, despite my wife's prayers, my daughter hadn't wanted to talk about it.\"\n\nTeresi broke into a cold sweat. Baron Lo Mascolo had surely lost his mind; there could be no other explanation. One branch of that family did have a history of madness. Hadn't the baron's sister, donna Romilda, become a nun? And hadn't she one fine day, after twenty years of cloistered life, come out of the convent and start dancing naked?\n\n\"Well, the problem, my dear baron, is that, normally, we can pray all we want, but the dead don't\u2014\"\n\n\"What dead?\"\n\nTeresi wiped the sweat from his brow with one hand.\n\n\"Baron, unless I'm mistaken, just a moment ago you told me your daughter was dead, and so\u2014\"\n\n\"For me, she's dead. For her mother, she's not.\"\n\nSo had the baroness lost her mind too? Both husband and wife, stark raving mad? That sort of thing does happen sometimes, in families. Didn't Signura Rossitano think she was a wasp and her husband a hornet, and they communicated by buzzing?\n\n\"Listen, Baron, maybe you'd better go home and\u2014\"\n\n\"I'll go home after I've shot your nephew.\"\n\nThis time Teresi snapped. He was fed up.\n\n\"Would you please be so kind as to tell me why the hell you want to kill Stefano? What's he got to do with this whole business?\"\n\n\"He's got everything to do with it. He's the only person who could have made my daughter Antonietta pregnant.\"\n\n*\n\nGiseffa the maid arrived at her father's house in Vicolo Raspa as the town-hall belfry was ringing four o'clock in the morning. At 4:05 A.M., Giseffa's mother, Nunziata, opened the window and started shouting:\n\n\"Cholera! Cholera!\"\n\nSince the street was narrow, her shouts were heard in all twenty-five of the residences situated on it. The first family to head out to the country were the Cumellas, then the Licatas, the Bonacci\u00f2s, the Gaglios, the Bonadonnas, the Restivos . . . In short, by five o'clock the only ones left in Vicolo Raspa were seven cats, two dogs, and Tano Pullara, who, being ninety years old, didn't feel like going anywhere with the others and said that he welcomed the cholera because he was tired of living.\n\nAt ten past five, Gesummino Torregrossa, who every morning on his way to work came by at five to pick up his friend, Girlanno Tumminia, found nobody at home and saw only Tano Pullara, sitting outside his hovel. He asked him what had happened.\n\n\"There's cholera about,\" said the old man.\n\n\"Says who?\"\n\n\"Don Anselmo Buttafava.\"\n\nGesummino turned around and raced back to Vicolo Centostelle where he lived. By five-thirty, half of that street, too, was empty.\n\nAt the day's first Mass, at six A.M., the priests all seemed to have spread the word among themselves.\n\nFaces that had never been seen in the churches before suddenly appeared: servants, coachmen, stable boys, hard laborers, wet nurses, housemaids, cooks from the noble houses, had all sat themselves down beside their masters, and all were praying to the little lord Jesus to save them from cholera.\n\nThen there were the people just passing through who were ready to run away to the countryside but first wanted the Lord's blessing. But in the various different churches, three families were noticeably absent: those of don Anselmo Buttafava, Marquis Cammarata, and Baron Lo Mascolo.\n\nIt's a well-known fact that there's no sermon at the day's first Mass.\n\nAnd yet on this occasion the priests all stepped up to their pulpits, but, instead of preaching the sermon, they hurled insults and curses.\n\nPatre Eriberto Raccuglia warned:\n\n\"Didn't I tell you that this town would end up like Sodom and Gomorrah? You must drive out the devil, who has taken the form of Teresi the lawyer . . . \"\n\nPatre Alessio Terranova said:\n\n\"It's no use crying and beseeching God to save you from cholera! First you must free the town!\"\n\nPatre Filiberto Cusa exclaimed:\n\n\"The poison plant must be uprooted!\"\n\nPatre Alighiero Scurria scoffed:\n\n\"So now you're crying, eh? So now you're praying, eh? You're all a bunch of sheep crawling on all fours! And what did you do when I told you Teresi was the devil incarnate? Nothing! But maybe there's still time . . . \"\n\nPatre Libertino Samon\u00e0 proclaimed:\n\n\"It's time to embark on a holy crusade!\"\n\nPatre Angelo Marraf\u00e0 threatened:\n\n\"I swear that no survivors of the cholera shall ever set foot again in this church if they haven't first got rid of Matteo Teresi!\"\n\nPatre Ernesto Pintacuda heroically offered his services:\n\n\"I'll lead the charge and hold the Cross high!\"\n\nOnly Patre Mariano Dalli Cardillo didn't preach that day. He limited himself to praying, along with his flock, for the Lord to save them all from the cholera looming at the city gates like a terrible scourge.\n\n*\n\nThe mayor, Nicol\u00f2 Calandro, was woken up by a great deal of shouting under his windows. His wife, Filippa, who was as deaf as a doorpost, kept right on sleeping. He immediately thought it was something he'd been fearing would happen sooner or later: a popular uprising unleashed by that incorrigible sonofabitch, Matteo Teresi.\n\nAnd he imagined himself strung up, head down, from the tree in the middle of the public garden, as had happened thirty years earlier to his predecessor, Mayor Bonifazi.\n\n\"They'll never take me alive!\" he shouted, getting out of bed and grabbing the revolver he kept in the drawer of his nightstand.\n\nBarefoot as he was, and still in his nightshirt, he went up to the window and looked through the shutters, which luckily were not fully closed.\n\nHe was flabbergasted to see what he saw.\n\nAn endless stream of men, women, old folks, youngsters, and children leading a procession of goats, sheep, chickens, and rabbits, running along as they pulled small handcarts or an occasional donkey with household objects piled on top, mattresses, cooking pots, water jugs, chests filled with clothing . . .\n\nBut it wasn't a revolution. They were not angry at him. The people were fleeing. But why? What was happening in town? He opened the shutter, stuck his head out, and asked:\n\n\"What on earth is going on?\"\n\n\"Cholera! Cholera!\" said many voices as one.\n\nWhat the hell were they saying? Cholera?\n\n\"Who told you there was cholera?\"\n\n\"Don Anselmo Buttafava,\" said a woman's voice.\n\nDon Anselmo was generally considered a sensible person, and should therefore be taken at his word. But then why hadn't Dr. Bellanca said anything about it to him, the mayor?\n\nMayor Calandro got dressed in a flash and went out of the house without bothering to wake his wife. Five minutes later he was knocking on the doctor's door.\n\nA window opened.\n\n\"My husband's gone out looking for you, Mr. Mayor,\" Signura Bellanca said from the window.\n\nCity Hall was still closed at that hour, which meant that the doctor must be headed for the mayor's house. And in fact he found him there at the door, knocking pointlessly, since Signura Filippa's deafness was so great she would even miss an earthquake.\n\n\"Why didn't you tell me there was cholera about?\" the mayor asked angrily.\n\n\"Calm down! And don't speak to me in that tone of voice!\"\n\n\"But aren't you aware of my responsibilities as mayor of this town?\"\n\n\"Of course!\"\n\n\"So why didn't you tell me anything about the cholera? It's obviously been festering for days, and you\u2014\"\n\n\"Oh, enough of this cholera nonsense!\" the doctor interrupted him.\n\nThe mayor thought he'd heard wrong.\n\n\"What did you say? You mean there's no cholera?\"\n\n\"Precisely! Just to be safe, before coming here, I went and woke up my colleague, Dr. Palumbo, and he too was taken completely by surprise.\"\n\n\"So then how do you explain that don Anselmo Buttafava . . . \"\n\nPeople kept streaming past them at a run. One of them, holding a sickle in his hand, stopped.\n\n\"So you rich folk aren't running away, eh? Cholera never attacks you bastards!\"\n\n\"Get out of here or I'll kill you!\" shouted the mayor, pulling out the revolver he'd put in his pocket.\n\nA woman grabbed the man by the arm and pulled him away with her.\n\n\"Don't go getting into trouble, Nin\u00f9,\" she said.\n\n\"Bastards!\"\n\n\"I can explain what happened,\" the doctor said as soon as the man was gone, \"but not in the middle of the street like this, with all these people around. It's a very confidential thing.\"\n\n\"Let's go to City Hall.\"\n\nBut after they'd taken just a few steps they were hailed by Tot\u00f2 Carrubba, who had a little food shop. The man was pulling his hair out in despair.\n\n\"They're cleaning me out! I'm ruined!\"\n\n\"What happened, Tot\u00f2?\" asked the mayor.\n\n\"They broke down the door of my shop! They're stealing everything.\"\n\nSo now there was looting? Calandro made a snap decision.\n\n\"Doctor, you can tell me about don Anselmo later. We need some law and order here! I have to go and talk to the carabinieri.\"\n\n*\n\n\"My good baron, let me remind you, before you enter my nephew's bedroom, that you gave me your word of honor that you would not shoot him before I've had a chance to talk to him.\"\n\n\"And I will keep my word.\"\n\nThey went in. The lad was sleeping like a baby. Teresi was holding an oil lamp, the baron his revolver. Firmly convinced of his nephew's innocence, the lawyer was extremely tense and ready to throw the oil lamp in the baron's face the moment the latter made any move to start shooting.\n\nTeresi approached the bed, while don Fof\u00f2, in keeping with the agreement they'd come to in the lawyer's study after two hours of negotiation, stood fast in the doorway.\n\n\"Stefan\u00f9, wake up,\" said Teresi, shaking the young man's shoulder.\n\nThe nephew opened his eyes and immediately shielded them with his arm, as the lamp was right in front of his face\n\n\"What time is it anyway?\" he asked in a hoarse voice.\n\n\"I don't know. Six-thirty, seven o'clock . . . \"\n\n\"Has something happened?\"\n\nAnd he made as if to get up. But if he got out of bed, he would surely notice the baron.\n\n\"Stay in bed. I just need to ask you something.\"\n\n\"So ask.\"\n\n\"Do you swear you'll tell me the truth?\"\n\n\"Of course!\"\n\n\"Swear on your mother's soul?\"\n\n\"I swear on my mother's soul. What do you want to know?\"\n\nTeresi swallowed and then spat out the question in a loud voice, so the baron could hear him clearly.\n\n\"Did you do anything with the daughter of Baron Lo Mascolo?\"\n\n\"With Antonietta? Do what?\"\n\nTeresi was worried that if he said even one wrong word, the baron might feel offended and start firing wildly. And he ended up doing even worse.\n\n\"Do 'it.'\"\n\nAnd just so as not to be mistaken, he made a fist and pumped it back and forth.\n\n\"Know what I mean?\"\n\nBut then, realizing the gesture he'd just made without thinking, he closed his eyes and waited for a bullet to shatter his skull.\n\n## CHAPTER IV \nWHAT DR. BELLANCA TOLD MAYOR CALANDRO\n\nThe lad reacted with unexpected violence, his right hand shooting straight out of from under the cover and striking his uncle's left cheek hard.\n\n\"Don't you ever dare say anything like that about Antonietta.\"\n\nHe was now sitting up in bed, trembling with indignation and white as a sheet.\n\n\"You must tell me who it was who said those vile things about her! I'll kill him with my bare hands!\"\n\nTeresi, who as a lawyer had a way with words, was suddenly speechless. To his immense satisfaction, and despite his burning cheek, he was becoming acquainted with his nephew's true nature, which until that moment had remained hidden.\n\n\"Calm down, Stefan\u00f9!\" he managed to say.\n\n\"No, I won't calm down! You must tell me who told you that calumny!\"\n\nThere was no longer any reason for the baron to remain in the shadows outside the door.\n\n\" _Signor barone_ , please, if you will . . . \"\n\nThere was no reply.\n\n\"The baron is here?\" Stefano asked in shock.\n\nWithout answering, Teresi stood up, went out of the room, and managed just in time to see don Fof\u00f2 opening the front door to go outside.\n\n\" _Signor barone!_ \"\n\nDon Fof\u00f2 turned around, looked at him, stood there in silence for a moment, then said:\n\n'\"Your nephew convinced me.\"\n\nAnd he left, closing the door behind him.\n\nTeresi had just turned round to go back to his nephew's bedroom when he again heard knocking at the door, accompanied by kicks and cane-blows, just like a few hours earlier.\n\nIt could only be the baron again, newly prey to the whims of his folly.\n\n\"Somebody's knocking,\" Stefano said from his room.\n\nTeresi didn't move. He didn't know what to do. Wasn't it too dangerous to let that raging madman back into the house?\n\n\"Please open the door, for Christ's sake, they're coming!\" said the baron from outside the door.\n\nAnd who could it be that was coming? Surely these were fantasies that existed only in the baron's fevered brain. At any rate, weighing his options, Teresi decided it was best to find out what was going on.\n\n\"Stefan\u00f9, look out the window and tell me if you see any people coming.\"\n\nHe heard the window opening, followed by the frightened voice of his nephew.\n\n\"There's hundreds of them!\"\n\nBut who the hell were all these people coming? Since the baron clearly was not seeing ghosts but real people in the flesh, Teresi raced downstairs and opened the door for him. Don Fof\u00f2 rushed in, out of breath and panting.\n\n\"They're on their way here!\"\n\n\"But _who_ is on their way here?\"\n\n\"How the hell should I know? Men and women armed with clubs, pitchforks, and hoes and being led by a priest carrying a cross! I don't want them to recognize me!\"\n\n\"But what do they want?\"\n\n\"How the hell should I know?\" the baron repeated, more alarmed than ever.\n\nAt that moment they heard the first shouts.\n\n\"Death to Teresi! Death to the devil incarnate!\"\n\n\"Give me your revolver and come with me,\" said the lawyer, who had turned white as a sheet.\n\nAt the back of the entrance hall was a window that Teresi opened. The morning light poured in.\n\n\"You can go out this way.\"\n\n\"But there's an overhang!\"\n\n\"No, that's what it looks like, but there's a very narrow pathway that leads down below. You just need to be a little careful.\"\n\nClosing the window, Teresi went into his study, took his revolver, and when upstairs. Stefano looked as if he was in a daze and no longer understood what was going on. His uncle handed him the baron's gun.\n\n\"If they try to break down the door, fire a shot from your window. But in the air, first time around. I mean it. If they don't run away, get back inside and shut yourself up. I'm going into my room.\"\n\nBut there weren't hundreds of them. There were about sixty, which was quite enough to do damage. At that moment they were all kneeling, and the priest with the cross was giving them benediction.\n\n\"O my holy crusaders,\" he said. \"You, my beloved children, who revere the sanctity of the family and keep watch over the virtues of the home . . . \"\n\nTaking advantage of the fact that they were all looking at the priest, Teresi opened the shutters slowly, just enough to slip his hand with the revolver out the window.\n\nThen, all at once, the priest, whom Teresi recognized as Patre Raccuglia, turned around, raised the cross in the air, and said:\n\n\"Go! Let's rid ourselves of the demon!\"\n\nIn a flash Teresi realized these people would break down the door on their first try.\n\n\"Shoot!\" he shouted to Stefano, as he did the same.\n\nThe echoes of the two shots hadn't yet faded before there was nobody left in front of the house. Or, at least, there was nobody still standing. Because there were in fact a man and a woman lying on the ground.\n\nTeresi felt his blood run cold. But he'd shot up into the air! He was positive! He ran into his nephew's bedroom.\n\n\"I told you to shoot in the air!\"\n\n\"I did!\"\n\nThey looked back outside. The man, by now, had stood up, and the woman was getting to her feet. They'd fainted in fear and were now running away.\n\nAt around eight o'clock that morning, the marshal of the Palizzolo Carabinieri station, Vitangelo Sciabbarr\u00e0, in direct consultation with the mayor, declared that the situation was worsening.\n\nIndeed three shops had already been looted; some burglars had tried to enter Palazzo Spart\u00e0 before they were driven away by a pair of rifle blasts fired by don Liborio and his wife Vetusta, who was the best shot in town; and there had been an attempted assault on the Veronica brothers' mill.\n\nThe brothers had also defended themselves with rifles, with one fatality. True, the victim was a delinquent with five or six burglary convictions already; but he was still dead.\n\nWhat was happening was that many men, having accompanied their families out to the country, had come back into town to take advantage of the situation and steal what there was to be stolen.\n\nThe mayor was normally supposed to have six municipal policemen at his disposal, but hadn't seen a single one of them that morning. No doubt they had all run away. And what could the ten carabinieri at the station do by themselves? At half past eight, Marshal Sciabbarr\u00e0 phoned the main headquarters at Camporeale, which was some twenty kilometers away, to ask for reinforcements.\n\nAnd by twelve-thirty the reinforcements arrived, in the form of a squadron of mounted carabinieri under the command of Captain Eugenio Montagnet, who declared martial law and took over all local operations.\n\nAt two o'clock that afternoon, a half-witted wretch who got by on begging for alms and whose name nobody could really remember, since everyone knew him only as _'u cani_ (that is, \"the dog\"), was \"made to suffer the consequences\" after being caught with a kilo of potatoes without being able to explain where he'd got them. Nobody witnessed the execution. Twelve carabinieri shot him against the wall of the ancient convent. _'U cani_ died laughing, convinced up to the very end that it was all some kind of joke, one of the many that the townfolk were always playing on him for their amusement.\n\nBy four o'clock that afternoon, absolute calm reigned in Palizzolo.\n\nAt half past four, Captain Montagnet had an idea: to send his men into the countryside to inform those who had run away that there was no danger of cholera and they could therefore safely return to town.\n\n\"I don't think your soldiers will have any success convincing them,\" said the mayor.\n\n\"Why not?\" asked Montagnet.\n\n\"Because they're carabinieri,\" replied the mayor.\n\n\"Shall we make a bet?\" said the captain.\n\nThen, turning to a reed-thin lieutenant by the name of Villasevaglios who was always at his side, he said:\n\n\"They'll be under your command. And don't make me lose that bet.\"\n\n\"Yessir!\" said the lieutenant, snapping to attention.\n\nAnd he left the room. The captain turned towards the mayor, lighting a cigar.\n\n\"I've heard a report that this morning there was an attempted attack on the home of a lawyer whose name I can't remember . . . \"\n\n\"Teresi.\"\n\n\"Yes, right. Apparently this lawyer, along with a relative of his, fired on the attackers. Is that true?\"\n\n\"Well, in a sense . . . \"\n\n\"Mr. Mayor, is it true or not?\"\n\n\"It's true. But, you see, this lawyer\u2014\"\n\n\"Is it also true that the attackers were led by a priest brandishing a large cross?\"'\n\n\"So I've been told. But, you see, for quite some time, this lawyer\u2014\"\n\n\"Would you please be so kind as to tell me this priest's name?\"\n\nHow was he going to lie to him? By telling him he didn't know his name? This captain was jovial and polite, but he seemed to have a chip on his shoulder the size of Mount Etna. If he didn't answer the man's question, he was liable to be made to \"suffer the consequences\" like that poor wretch _'u cani_. The mayor took a deep breath.\n\n\"Don Eriberto Raccuglia, priest of the parish of the Mother Church.\"\n\n\"Listen, I would like to avoid any malicious gossip or speculation . . . Could you summon this priest here to City Hall at nine o'clock tomorrow morning?\"\n\n\"Could _I_ summon him? Why me?\"\n\n\"Don't you see? If I have him taken to the carabinieri station, who knows what kind of reaction that would unleash.\"\n\nThe captain was right.\n\n\"All right.\"\n\n\"Thank you. And now would you please tell me who spread the rumor that there was cholera about, and why?\"\n\nMayor Calandro broke into a cold sweat. If they started dragging the town's big cheeses into this mess, there could be serious complications.\n\n\"Apparently . . . it was all a very big misunderstanding.\"\n\n\"You think? So the public unrest was not, in your opinion, intentional?\"\n\n\"I don't think so.\"\n\n\"That would be the second part.\"\n\nThe mayor felt flummoxed.\n\n\"I'm sorry, Captain, but the second part of what?\"\n\n\"Of my question. You didn't answer the first part.\"\n\n\"And what was that?\"\n\n\"The name of the person who started the rumor.\"\n\n\"To be honest . . . a few names were mentioned and until I am absolutely certain, you must understand that I\u2014\"\n\n\"We'll discuss this again in the morning,\" said Montagnet, rising to his feet. \"Would you please tell me where this lawyer Teresi lives?\"\n\nThis captain, the mayor thought bitterly, will end up doing more damage than the nonexistent cholera.\n\nBut the captain found no one at home at Teresi's.\n\nAfter three full hours of discussion with his nephew, the lawyer had decided to go to Palazzo Lo Mascolo, to talk to the baron. He wanted an answer to a specific question: Was it Antonietta who had named Stefano as her lover, or was this just an idea don Fof\u00f2 had got into his head? Stefano, who'd wanted to see the girl again, took him there in the tilbury.\n\nBut they never reached their destination.\n\nTo save time on their way into the center of town, they avoided the main road and took a sort of country trail that shortened the distance. There were no buildings or farms along this track, and hardly anyone ever took it. At the exact point where the trail crossed a road of beaten earth leading into the main square of the town, Stefano saw a large, fat sack in the middle of the path. He pulled on the horse's reins to avoid running over it.\n\n\"The sack is moving!\" shouted Teresi.\n\nIndeed, something inside the sack seemed to have shifted. They climbed down to have a look. The sack was closed with string wrapped several times around. Then they both heard, very faintly, a catlike sort of wail.\n\n\"There must be a cat in there,\" said Stefano.\n\n\"A cat the size of a tiger?\" Teresi asked doubtfully.\n\nThen he made up his mind. Pulling a hunting knife out from a pocket of his coat, he cut the string and opened the sack to empty it. Out came the blond head of a young man of about twenty, with his face so battered from punches and whacks that his eyes were little more than two fissures. Blood was running from his nose and ears. His lips were so swollen that his mouth looked like an open pomegranate. And past them they could see that his blood-filled mouth was missing three teeth. Clearly the young man had also been kicked in the face.\n\nTeresi lowered the sack a bit more. The shoulders appeared. The young man was not a peasant. The clothes he was wearing, though tattered, were made of fine fabric.\n\n\"Do you know him?\" Teresi asked his nephew.\n\n\"I think so.\"\n\n\"And who is he?\"\n\n\"I think he's the son of a cousin of the Marquise Cammarata. He's not from around here, though he's often at the marquis's house. I met him at the ball on\u2014\"\n\n\"Fine, fine,\" said the lawyer, cutting him off. \"Help me put him back in the sack.\"\n\n\"What?! He's a relative of the Cammaratas! We should take him to their palazzo and tell them how we found him and\u2014\"\n\n\"\u2014And they'll just thank us and finish him off.\"\n\nStefano was dumbstruck.\n\n\"C'mon, give me a hand,\" said his uncle. \"We'll take him to our house and then decide what to do.\"\n\n\"But let's at least take him out of the sack!\"\n\n\"Not on your life. If we run across the carabinieri, they'll just think it's a sack of potatoes. Actually, let's do this. While I'm taking him home, you go and get Dr. Palumbo.\"\n\nTeresi did not want to let this opportunity slip away. This kid was a relative of the Cammaratas. His lordship the marquis must surely know the reason for this brutal beating. His lordship! The great big son of a bitch who'd first supported his request for membership and then just dropped him. No, this was an opportunity not to be missed.\n\nThe very same people who were first to run away\u2014the inhabitants of Vicolo Raspa, Giseffa included\u2014got back into town around seven that evening, and Giseffa returned to the Buttafava home.\n\n\"You see? I won the bet! We managed to persuade them!\" Captain Montagnet said in triumph to the mayor.\n\nThe carabinieri had indeed used the most persuasive of arguments: blows with the flat of their swords, lashes of the riding whip, and threats of arrest, all tactics in the subtle dialectics of the forces of order.\n\n\"May I come in? Am I bothering you?\" asked Dr. Bellanca, poking his head into the mayor's office.\n\n\"No, quite the contrary, please sit down!\" said Calandro.\n\n\"I've come to tell you how, in my opinion, don Anselmo was misled.\"\n\n\"You've come just in time.\"\n\n\"Why do you say that?\"\n\n\"Because I believe that Captain Montagnet wants to charge him with disturbing the peace.\"\n\nBellanca couldn't hold back a curse.\n\n\"We really don't need this ball-busting captain at the moment!\" he said.\n\n\"I agree,\" said the mayor. \"And so, you wanted to tell me . . . \"\n\n\"Could we close the door?\"\n\n\"Is it a delicate matter?\"\n\n\"Rather.\"\n\n\"I'll get it myself.\"\n\nBefore closing and locking the door, Calandro told the usher, who'd returned about an hour earlier after running away with two municipal policemen:\n\n\"Pippin\u00e8, I'm not here for anyone.\"\n\n\"Not even for the captain?\"\n\n\"Especially not for the captain.\"\n\nHe went and sat back down in the armchair behind the desk and waited for his interlocutor to begin talking.\n\n\"Let me preface by saying that I am speaking to the mayor in my capacity as the municipal doctor.\"\n\n\"And so? What is that supposed to mean?\"\n\n\"It means that I, as an ordinary doctor, would never tell you what I am about to tell you. But in my capacity as municipal doctor invested with an official responsibility, and faced with what happened in town today, I am forced to speak to you.\"\n\n\"So speak, for Christ's sake!\"\n\n\"Three days ago,\" the doctor began, \"I was summoned to an emergency at the home of Baron Lo Mascolo, when his daughter fell ill.\"\n\n\"Antonietta? She's the very picture of health! So what was she suffering from?\"\n\n\"Maybe from too much health.\"\n\n\"I don't understand.\"\n\n\"She's pregnant.\"\n\n\"Pregnant?!\"\n\n\"Two months pregnant.\"\n\n\"Two months pregnant?!\"\n\n\"Antonietta.\"\n\n\"Antonietta?!\"\n\n\"Would you please stop repeating everything I say? You're starting to get on my nerves.\"\n\n\"And do we know who the father is?\"\n\n\"Antonietta didn't want to say. And I won't even go into what happened when I was forced to tell the baron and his wife that their daughter . . . Swoons, fainting fits, the baron going out of his mind and breaking chairs, vases, and anything else he could get his hands on . . . The following day, Sunday, I had to go back to give the baron some tranquilizers and the baroness some stimulants. But on my way out, I ran into don Anselmo. And that's when the trouble began.\"\n\n\"Why?\"\n\n\"Because I couldn't very well tell don Anselmo the real reason I'd gone there. And so I answered his questions rather vaguely, and he ended up thinking I was treating something contagious.\"\n\n\"But that's a huge leap, from 'something contagious' to cholera!\"\n\n\"Wait, I haven't finished. Right after leaving the baron's, I had to go back to Palazzo Cammarata.\"\n\n\"Go back to? You mean you'd been there earlier?\"\n\n\"Yes indeed.\"\n\n\"When?\"\n\n\"The day before.\"\n\n\"For what reason?\"\n\n\"Their eldest daughter, Paolina, had taken ill.\"\n\nJust from the doctor's expression, the mayor understood everything. He was in shock. He sat there for a moment with his mouth agape, then asked:\n\n\"Pregnant?\"\n\n\"Pregnant.\"\n\n\"She too?\"\n\n\"She too.\"\n\n\"Jesus bloody goddamn Christ!\"\n\n\"I concur,\" said the doctor.\n\n\"How long?\"\n\n\"Two months. Just like Antonietta.\"\n\n\"And Paolina doesn't want to say who the father is either?\"\n\n\"She won't say a word.\"\n\n\"But how about these girls, anyway! All church and home and hearth, and suddenly they're all getting pregnant like nobody's business!\"\n\n\"And, unfortunately,\" the doctor resumed, \"as I was about to knock on the front door of Palazzo Cammarata, don Anselmo saw me and asked me whether somebody was sick in there. I said yes, and that was when he must have concluded that it was cholera.\"\n\n\"And now what are we going to tell our ball-busting captain?\" asked the mayor. \"If he discovers the truth, it's going to make the Sicilian Vespers look like a picnic!\"\n\n\"I have an idea. For the good of the town\u2014and also because I consider myself partly responsible for the misunderstanding\u2014we'll have to lie.\"\n\n\"Meaning?\"\n\n\"I could say that both the Lo Mascolo and Cammarata families came down with a particularly virulent form of influenza. And that was how don Anselmo got the wrong idea.\"\n\n\"Wait a second. But wasn't Marquis Cammarata at the club on Sunday morning?\"\n\n\"We'll say that the poor man went just to be a good sport. That he was already sick on Sunday morning, but stubbornly refused to stay home and then got much sicker in the afternoon. As you can well imagine, neither the Lo Mascolos nor the Cammaratas would have any reason to want to contradict this story.\"\n\n\"Very well, then,\" said Calandro, expecting Bellanca to stand up.\n\nBut the doctor remained seated.\n\n\"Is there something else?\" the mayor asked, starting to feel weary from a dreadful day.\n\n\"Yes, but I'm not sure it's relevant.\"\n\n\"To what?\"\n\n\"To my public function.\"\n\n\"If you think it's something you should talk to me about, then talk, otherwise . . . \"\n\n\"In short, we could consider even this a kind of epidemic,\" the doctor said, as though to himself.\n\n\"No, no,\" Calandro interjected, \"if you think there really is an epidemic, then it is your duty to fill me in on it!\"\n\n\"Do you know who the widow Cannata is?\"\n\n\"How could I not? A fine-looking dame like that? And a serious, devout woman as well. Lost her husband three years ago, poor thing.\"\n\n\"Her too.\"\n\n\"I'm sorry, her too what?\"\n\nWith his right hand, the doctor mimed a gesture indicating a large belly.\n\n\"Pregnant?!\" the mayor asked in astonishment. \"I don't believe it.\"\n\n\"You'd better believe it.\"\n\n\"Don't tell me she's two months pregnant!\"\n\n\"I'm afraid I have to. And that's not all.\"\n\n\"It's not?\"\n\n\"No. Do you know Totina, the daughter of 'Ngilino, don Anselmo's farm overseer?\"\n\n\"Her too?\"\n\n\"Yessirree, two months pregnant. And, like the widow Cannata, she doesn't want to reveal the father's name either.\"\n\n\"So, in all, there are four women who are presumably pregnant?\"\n\n\"As far as I know. But it's possible my colleague Palumbo knows of a few more. Though if there are any, he's not required, like me, to reveal their names.\"\n\nThe mayor sat there in thoughtful silence for a moment. Then he spoke.\n\n\"I don't think you can call it an epidemic. And, even it was one, how would we avoid its spread? Would we send the town crier out to warn all the woman to steer clear of all penises? Separate all the men from all the women? No, I really don't think so.\"\n\n\"I don't either. Also because nobody's ever heard of an epidemic of pregnancies,\" said the doctor.\n\n*\n\nAt eleven o'clock that evening, a few of Captain Montagnet's carabinieri started making the rounds of the town's streets, announcing that the curfew would begin at twelve midnight sharp, that anyone who was found wandering outside after midnight would \"suffer the consequences . . . \"\u2014 _That captain is obsessed_ , thought the mayor, who heard the announcement as he was on his way home\u2014and that, in addition, all public assembly was prohibited until further notice.\n\nWhich meant, in plain language, that there would be no Masses, no classes at school, no markets in the square, and that even the Honor and Family Social Club would have to remain closed.\n\nFinally, with God's help, at midnight what came to be known in Palizzolo as the day of don Anselmo's cholera came to an end.\n\nThe mayor got into bed beside his wife, Filippa, who was indeed deaf, but also young and pretty. But he didn't lie down. He remained sitting up, back propped against the pillows, and started counting on his fingers how many town hall employees were still absent.\n\n\"What are you counting?\" asked Filppa. \"When the last time we made love was? Well, you can stop counting, 'cause I can tell you myself. The last time was exactly two months ago.\"\n\nAnd she sighed. The mayor had a troubling thought.\n\n\"Don't tell me you're pregnant too!\"\n\n\"What do you mean, me too? What does that mean?\"\n\n\"Nothing. Are or aren't you?\"\n\n\"No, I'm not! What's got into you?\"\n\nThe mayor didn't answer. He just lay down, and five minutes later was already snoring. Signura Filippa stayed up a long time, sighing in frustration. \n\n## CHAPTER V \nTHE CONSEQUENCES OF THE CHOLERA \nAND OTHER MATTERS\n\nDr. Enrico Palumbo was, in a sense, the poor people's doctor, just as the lawyer was their defender. The difference between the two was that the doctor didn't do what he did for any political ideal, but just because he wanted to.\n\nOftentimes, after examining some sick child from a penniless family, he would leave the mother as much money as she needed to buy medicine or cook up a dish of pasta. He hated priests perhaps even more than Teresi did. The only man of the cloth in town he respected was Patre Dalli Cardillo.\n\nAnd the doctor was a man with guts. He never said a word to anyone about any of the things he came to know.\n\nAt dawn on the day after the cholera brouhaha he went knocking on Teresi's door. The lawyer answered the door himself.\n\n\"Did he make it through the night all right?\" he immediately asked.\n\n\"He was delirious.\"\n\n\"Fever?\"\n\n\"A hundred and four.\"\n\n\"Let's go and see him.\"\n\nAs he was climbing the stairs, the doctor asked:\n\n\"How about your nephew?\"\n\n\"He's gone to bed. We took turns.\"\n\nThey'd put the injured young man in a small room barely big enough to fit a single bed, a chair, and a bedside table.\n\n\"Would you like some coffee?\"\n\n\"Yes, thanks.\"\n\nTeresi went into the kitchen, made the coffee, and then brought it to him. The doctor took a sip.\n\n\"What do you think?\"\n\n\"About the coffee?\"\n\n\"No, about the kid.\"\n\n\"I haven't had a good look at him yet. But the main thing is that he made it through the night.\"\n\nThe doctor handed the little cup back to Teresi.\n\n\"I'll be waiting in my study,\" said Teresi. \"Call me if you need anything.\"\n\nHe sat down and started thinking about the things he'd heard coming from the delirious young man's mouth during the night. Not that he really understood any of it; all that came out were truncated words mumbled by a mouth missing three teeth, through swollen lips as big as cucumbers.\n\n_\" . . . no . . . no . . . pleeathe . . . no more . . . no . . . no . . . \"_\n\nThis part, unfortunately, was all too comprehensible.\n\n_\" . . . on flefo . . . pleeathe . . . on flefo . . . \"_\n\nHe'd also said other things Teresi couldn't understand. _Flefo. Flefo._\n\nMaybe he was saying _no flefo_. What did it mean? Nothing. How about _Don't flay me_? No, that didn't really fit. Perhaps _Don thay tho_ \u2014that is, _Don't say so_? But maybe the _. . . on_ didn't mean \"don't\" at all. Wait a second. Maybe it was a mangled way of saying _I'm not afraid of you_ : _. . . no' fray 'f'ou_. No, that didn't work either. If the lad wasn't afraid, then why would he plead for the person to stop?\n\nWait. What if it was _Don Flefo_?\n\nThe doctor came in.\n\n\"Do you know anyone by the name of Don Flefo?\"\n\n\"No. What an odd name. Listen, do you know how to give someone a shot?\"\n\n\"Yes.\"\n\n\"Then give him this in four hours. I'll leave you the syringe and the medicine. I don't want to come here too often, otherwise people might notice.\"\n\n\"How did you find him?\"\n\n\"Better. He's a strong, healthy young man. He'll definitely recover. I'll be back to give him a third shot this evening, around nine.\"\n\nAfter the doctor left, Teresi started thinking again about what the lad had said during the night.\n\nLet's put aside this _flefo_ for a minute. What did he say after that?\n\n_. . . no . . . no . . . tu ariddru . . . ttu ariddru. . . no . . . cent . . . gno . . . no . . . cent . . . gno . . ._\n\nThe lawyer stopped at this point, leaving aside the other things he'd heard the boy say. He grabbed a sheet of paper and a pencil. Best go about this in orderly fashion. He was going to work something out of this muddle, even if it took him all day. And he was convinced that whatever it was, it would allow him to screw don Filadelfo Cammarata, the marquis.\n\nWait a second, Matt\u00e9, he said to himself.\n\nDon Filadelfo. _Don Flefo_. That could be it. _Don Flefo_ might very well be don Filadelfo.\n\n*\n\nAt eight o'clock the following morning, as the mayor was going up to his office in City Hall, the usher warned him:\n\n\"The captain's here, you know.\"\n\nWhat a constant pain in the arse that man was! With some effort, the mayor managed to smile as he entered his office.\n\n\"My good captain!\"\n\n\"Good morning,\" the other said drily.\n\nHe was sitting in the armchair in front of the desk. The mayor took possession of his own chair.\n\n\"A quiet night?\" he asked.\n\n\"Quiet as can be,\" Montagnet replied. Then he said: \"Did you summon Don Raccuglia?\"\n\n\"I'll do so immediately.\"\n\n\"There's no longer any need.\"\n\nMayor Calandro felt a little relieved. The fewer priests around, the better for all involved.\n\n\"Well, I'm gla\u2014\"\n\n\"I changed my mind,\" the captain continued. \"I went and paid him a call at church.\"\n\n\"When?\"\n\n\"At six o'clock this morning. I waited for him to finish saying the first Mass, which was a mass of thanksgiving for the passing of the cholera menace. The church was packed.\"\n\nWell, thanks a lot, captain dickhead! Might as well have gone there with a brass band and woken up the whole town! First he says he wants no speculation or malicious gossip and then he goes and shows up in uniform in front of all of Don Raccuglia's parishoners!\n\n\"Did you speak to him?\"\n\n\"Of course. In the sacristy.\"\n\n\"And what did you say to him?\"\n\n\"That I'd denounced him to the court of Camporeale for sedition and inciting public disorder.\"\n\nWell at least he didn't put him up in front of a firing squad! But did this captain see things backwards or something?\n\n\"But, Captain, that's not exactly the way things went.\"\n\n\"Oh, no? Then how, exactly, did things go?\"\n\nSo now the man was going to play the wise guy?\n\n\"If there's anyone in town inciting people to rebel, it's the lawyer Teresi! You have no idea what the man says and writes!\"\n\n\"You're wrong, I do have an idea. His weekly comes to Camporeale, and I read it. As part of my duty, of course.\"\n\n\"So you know perfectly well which of the two men is the real subversive. It's Teresi!\"\n\n\"Allow me please to firmly disagree. In this specific case, it wasn't the lawyer leading the unrest, but the priest.\"\n\n\"But you must try to understand Don Raccuglia! Teresi advances ideas which\u2014\"\n\n\"\u2014Which in no way authorize Don Raccuglia to stir up the population against him.\"\n\n\"But he dared fire a gun at the crucifix!\"\n\n\"He fired in self-defense. If I take any action against him, I must also take action against don Liborio Spart\u00e0 and his wife, and against the Veronica brothers. What do you think about that?\"\n\nClever man, this captain. The mayor weighed his options.\n\n\"I think you're right.\"\n\n\"Thank you. And as for the fact that he shot at the crucifix, I would advise you not to insist too much on that version of events. First, because the lawyer shot in the air. And second, because, even if he had fired at the crowd, he would have been firing at a priest using the crucifix to break down his door.\"\n\nThe mayor said nothing. But the captain didn't let up.\n\n\"And now, would you please give me the name of the person who sowed the rumor about the cholera?\"\n\nMayor Calandro had given this a lot of thought after his long discussion with Dr. Bellanca.\n\n\"All right,\" he said. \"I checked into the matter and came up with the name of a very stable person considered by all to be a serious man of great integrity. And not only, but\u2014\"\n\n\"Are you talking about don Anselmo Buttafava?\" asked the captain, interrupting him.\n\nCalandro's eyes opened wide.\n\n\"But how . . . how did you know that?\"\n\n\"We're carabinieri, my good man. At five o'clock this morning I sent five of my men out to arrest him.\"\n\n\"But do you know where he is?\"\n\n\"Of course. I've learned that Signor Buttafava owns an estate called San Giusippuzzo. And he also manages another, La Forcaiola, for a cousin of his who's presently in jail. He's either at San Giusippuzzo or La Forcaiola, there's no two ways about it. I think he'll be back here in town by eleven at the latest.\"\n\nDon Anselmo, arrested by the carabinieri? Was this asshole trying to trigger a revolution or something?\n\n\"In . . . handcuffs?\"\n\n\"No. There's no reason for that. Don't worry, he'll be treated with the utmost consideration. Along with the two carabinieri I also sent Lieutenant Villasevaglios.\"\n\n\"Speaking of consideration . . . \"\n\n\"Tell me.\"\n\n\"The municipal physician, Dr. Bellanca, told me last night that he can explain how don Anselmo came to misconstrue the situation. May I have him come?\"\n\n\"Of course,\" said the captain, rising to his feet. \"And now, if you'll allow me, I have to leave you. I have a few little things to attend to. I'm going to ring His Excellency the Prefect by telephone to set his mind at rest. And while I'm at it, I'm going to ring His Excellency the Bishop as well.\"\n\nWas this man insane? The bishop? So now he was going to go and bust the bishop's chops as well? Why? A bout of genuine cholera might really have been less of a plague than this goddamn captain!\n\n\"I'm sorry, but why the bishop?\"\n\n\"Well, you know, it may not be in my line of duty, but I think it's only common courtesy to inform him that I have reported one of his priests. I'll see you back here at eleven.\"\n\n_I hope you break your neck_ , the mayor said to him in his mind.\n\n*\n\nHis Excellency the Most Reverend Monsignor Egilberto Martire put down the telephone receiver and grabbed a notepad he had on the table. He opened it, thumbed through it, tore out a sheet, and then rang the little bell he kept within reach.\n\n\"You rang, Your Excellency?\" said his secretary, Don Marcantonio Panza.\n\n\"Don Marcantonio, on this sheet of paper are the names of the priests of the eight parishes in Palizzolo. I haven't yet made my episcopal visits, so I don't know them yet. I want them all here, at the bishopric, at four o'clock this afternoon. I will not tolerate any absences.\"\n\n\"Yes, sir.\"\n\nWorking with His Excellency Egilberto Martire was worse than living in a barracks. Physically, he looked more like a sergeant major than a bishop. Fat, short, and with a red face that, when he got upset, started to turn purple. In the six months since he arrived at Camporeale, he'd instituted a military-style discipline. The best of it was that, being Roman, every so often he would speak in obscenities.\n\n\"The stupid shits!\" Don Marcantonio heard him in fact say as he was leaving the room.\n\n*\n\nDon Anselmo Buttafava did not come in at eleven o'clock, for the simple reason that before reaching La Forcaiola, Lieutenant Villasevaglios and the two carabinieri ran into some trouble. And therefore don Anselmo didn't find out that day that he was supposed to report to Captain Montagnet to defend himself against the accusation of disturbing the peace. Right where the trail to La Forcaiola branched off into a smaller trail leading into the Galluzzo district, there stood a giant olive tree. Up in the boughs of this tree a brigand from the Salamone gang was hiding, a man known as _Savaturi 'u pecuru_ \u2014\"Salvatore the Sheep\"\u2014because the great quantity of hair he had all over his body. He was standing guard. Seeing the three soldiers approach, he gave a shepherd's whistle to sound the alarm.\n\nSalamone the brigand, who the night before had seized three women fleeing from Palizzolo, was having his way with them inside a nearby cave. Outside the cave, keeping watch, were two trusted brigands, Arelio the Hare and Pancrazio the Snake.\n\nThe rest of the gang had headed on towards the Arbanazzo woods a few miles away.\n\nHearing the whistle, Arelio and Pancrazio ran over to the olive tree and took up position. As soon as the carabinieri were within range, they started shooting. The three soldiers dismounted, took cover behind some trees, and returned fire. After five minutes of firing back and forth, the lieutenant signaled to the two carabinieri and, slithering along the ground through the grass, he came close to the tree from which Savaturi _'u pecuru_ was firing, took careful aim, and cut him down with one shot.\n\nSavaturi's dead body fell right onto Pancrazio's head, whereupon the bandit shot to his feet only to fall back down again a second later, struck by another bullet from the lieutenant, who was a peerless marksman. Scared to death, Arelio the Hare started running towards the cave, yelling. The carabinieri gave chase, and two things happened at once. As Arelio fell down, dead, at the mouth of the grotto, out of the same grotto came Salamone the brigand, naked and a little woozy from having repeatedly ravished the three women. As soon as he saw the three carabinieri, he realized he was done for and put his hands up. Behind him, in tears, appeared the three women, also naked, who ran into the carabinieri's arms saying they'd been sent by God to deliver them from a certain death at the hands of the bandits. After giving the women a chance to put their clothes back on, the lieutenant let them go as they wished. That way they could tell everyone that Salamone had treated them like queens and respected them like Madonnas.\n\nThe only person Lieutenant Villasevaglios had to escort back to Palizzolo was therefore Salamone, and he forced the brigand to trudge into town in his underpants, which he'd been allowed to put back on so as not to offend common decency and bound in ropes so tight he looked like a walking salami being pulled along by a horse.\n\nLess than an hour after reaching Palizzolo, Salamone was \"made to pay the consequences\" by order of Captain Montagnet. The corpses of the other three brigands were hung from nearby trees.\n\nThis time there were a lot of people attending the execution in front of the wall of the ancient convent.\n\n*\n\nThe clock was striking the twelve chimes of midday when Stefano woke up. _Matre santa!_ He'd fallen asleep instead of taking over for his uncle!\n\nHe got dressed in a hurry and went into the young man's room. His uncle wasn't there, but the injured youth was sleeping and didn't seem agitated. Stefano placed his palm on the sleeper's forehead: the fever seemed to have dropped a little.\n\nHe went downstairs and into his uncle's study. Teresi was sleeping, mouth open, head leaning against the back of the armchair. On the table he noticed some sheets of paper with writing on them. As he turned to leave, to let his uncle sleep, Teresi opened his eyes and called to him.\n\n\"Stefano, come here!\"\n\nThe tone of his voice sounded pleased.\n\n\"Sit down.\"\n\nAnd he handed him a sheet of paper.\n\n_. . . no . . . no . . . pleeathe . . . no more . . . no . . . no . . . on flefo . . . pleeathe . . . on flefo . . . no . . . no . . . tu ariddru . . ._ _no_ _. . ._ _cent_ _. . ._ _I_ _. . ._ _no_ _. . ._ _cent_ _. . ._ _no_ _. . ._ _cent_ _. . . . . ._ _ahhh_ _. . ._ _ahhh_ _. . ._ _ndu_ _. . ._ _nuthin_ _. . ._ _olina_ _. . ._ _oww_ _. . ._ _no_ _. . ._ _no_ _. . ._ _nuthin_ _. . ._ _o_ _. . ._ _olina_ _. . ._\n\n\"So? That's the stuff the guy was saying last night, all strung together. But it doesn't make any sense.\"\n\nTeresi chuckled.\n\n\"Did you manage to make some sense of it?\" Stefano asked.\n\n\"In fact I think I did,\" said his uncle.\n\nHe handed him another sheet of paper.\n\n_. . . no . . . no . . . please . . . no more . . . no . . . no . . . don Filadelfo . . . please . . . don Filadelfo . . . no . . . no . . . 'z\u00f9 Carmineddru . . . I'm innocent . . . I'm innocent . . . innocent . . . ahhh . . . ahhh . . . I didn't do nothing to Paolina . . . ahhh . . . no . . . no . . . nothing to Paolina . . ._\n\nStefano turned pale. Then Teresi slapped himself hard on the forehead.\n\n\"The shot!\"\n\nHe grabbed the medicine and syringe and raced down the stairs.\n\n\"Except for that one . . . what's your name?\" Bishop Egilberto Martire asked the oldest of the eight priests lined up in front of him.\n\n\"Mariano Dalli Cardillo, Your Excellency.\"\n\n\"How old are you?\"\n\n\"Seventy.\"\n\n\"Now, going from right to left, the names and ages of everyone else.\"\n\n\"Alessio Terranova, forty-three years old.\"\n\n\"Eriberto Raccuglia, forty years old.\"\n\n\"Filiberto Cusa, forty-one years old.\"\n\n\"Libertino Samon\u00e0, forty-five years old\"\n\n\"Angelo Marraf\u00e0, forty years old.\"\n\n\"Ernesto Pintacuda, thirty-nine years old.\"\n\n\"So, with the exception of Don Dalli Cardillo, the rest of you are young. Too young to understand in full the responsibilities that rest on the shoulders of a parish priest\u2014and not just religious responsibilities. This was a grave mistake my predecessor made. He should have taken care to ensure that it didn't happen. I will try to set things right as quickly as possible. But let's get to the point. This morning I was told by Carabinieri Captain Montagnet, who was sent to Palizzolo over that cholera idiocy, that you, Don Raccuglia, were so goddamn stupid as to get yourself reported for sedition. Nice going! A seditious priest!\"\n\n\"I'd like to explain, Your Excellency . . . \" Don Raccuglia began.\n\n\"You're not going to explain anything to me, got that?\"\n\n\"But . . . \"\n\n\"Shut up! Here you talk only when I say so! And that's not all! The good captain also informed me that in all the churches, except for that of Don Dalli Cardillo, you incited the faithful, from your pulpits, against Teresi the lawyer! You incited the faithful to hatred! Come on! Where are you, anyway! You're in Church, for Christ's sake! In church you're supposed to preach love, and only love should spread outward from the church! Have you forgotten what Jesus said? Love thy neighbor as thyself! That's what Jesus said! You say: But Teresi the lawyer wants to destroy the family! So what! And you, instead of preaching love for the family and the sanctity of matrimony, you start preaching against some lawyer? Are you shitting me? Fight that lawyer with your own weapons, not with his! Take one Sunday, any Sunday, when the sun is shining bright, and declare it Family Day! Have all the married couples with all their children come to the church. Then organize a big feast in the churchyard! With singing and dancing! And everyone will smile and laugh and say: What a great thing the family is! What a beautiful thing is the sacrament of holy matrimony! And then you all eat and drink your fill to spite Teresi the lawyer! Do we understand one another now?\"\n\n\"There is no other explanation, Stefan\u00f9, believe me.\"\n\n\"But that's not possible, Zio! So, in your opinion, Paolina, the Marquis Cammarata's eldest daughter, is also pregnant, like Antonietta?\"\n\n\"That's exactly right.\"\n\n\"And don Filadelfo, thinking that his daughter's lover was that relative of his wife, and knowing that the kid was coming to see them, sent for _'u z\u00f9_ Carmineddru and had him beat him within an inch of his life?\"\n\n\"That's exactly right.\"\n\n\"I don't believe it.\"\n\n\"Do you have any other way to explain what happened?\"\n\n\"Well, it's exactly like what happened to me, except that the baron wanted to shoot me.\"\n\n\"I agree with you there.\"\n\n\"So what do we do now?\"\n\n\"For the moment, nothing. If the kid confirms what I've been thinking\u2014and let's hope he recovers soon\u2014then I'll see to finishing off the marquis myself!\"\n\nOne of the three women that Salamone the brigand had enjoyed in the grotto was a pretty, firm-fleshed girl by the name of Rosalia Pampina. The orphaned daughter of a peasant couple, she worked as a maid at the home of Giallonardo the notary. Since the notary was a parishioner of Don Filiberto Cusa's church, the girl had obtained permission to attend her employer's church, the church of San Cono.\n\nOn the afternoon of the same day she was freed by the carabinieri, she went back to work at the notary's house. But she said nothing to him about what the brigand had done to her. She merely told him she'd run off to the country to flee from the threat of cholera. At vespers, however, she asked Signura Romilda Giallonardo for permission to go to church to thank the Lord for delivering her from danger. Rosalia was a very devout girl and an avid churchgoer, so much so that Signura Romilda often told her she should become a nun.\n\nShe would recite all the decades of the rosary upon awaking each morning and in the evening when she went to bed, and she never missed the day's first Mass. Oftentimes Patre Cusa, who had noticed her great religious devotion, would call her into the sacristy to explain the catechism to her. For this reason Signura Romilda saw no reason not to allow her to go out now.\n\n\"I would like to confess,\" Rosalia said to Don Filiberto, calling him aside as he was closing the great door of the church.\n\n\"It's late now. Come back tomorrow morning.\"\n\n\"No, sir. You need to hear my confession right away.\"\n\nThe priest looked at her and saw that she was crying.\n\n\"All right,\" he said, leading her to the confessional.\n\nHe sat down inside, made the sign of the cross, put on his stole, said a prayer, and opened the little door over the grate.\n\n\"What happened to you, my child?\"\n\n\"I lost my virginity.\"\n\nAnd she started weeping audibly. Luckily there was nobody else in the church.\n\n\"How did that happen? So first you go out and have a good time, then you come in hear and start crying, you wretch?\" the priest said in indignation.\n\n\"No, no! I didn't have a good time at all! Salamone the brigand did it to me!\"\n\nAnd she started telling him what had happened to her. The priest, as was his duty, interrupted her every so often, asking her for more details.\n\n\"Twice from the front and twice from behind?! The horror! The horror!\"\n\n\"So he really hurt you?\"\n\n\"And you, did you enjoy it when he was . . . \"\n\nIn the end, Rosalia started shouting.\n\n\"My soul is damned! Damned for eternity! Even though you had me drink holy water to protect me, I've damned myself just the same!\"\n\n\"No, Rosalia, don't say that. The holy water springing from my body was to protect you from yourself, from the temptations you might have. But this is another matter! You were forced. You did not do it of your own free will! You are not to blame!\"\n\n\"Is that really true?\"\n\n\"Yes, it's true. Your soul is unblemished, but your body has been gravely contaminated. Sullied. We must make it pure and clean again.\"\n\n\"But how, Father?\"\n\n\"Through the penance I will have you do.\"\n\n## CHAPTER VI \nTHINGS GET COMPLICATED\n\nFour days after the famous day of cholera, at eight o'clock in the morning, don Anselmo Buttafava and his wife got in their carriage to head back to Palizzolo from La Forcaiola. Lieutenant Villasevaglios, who would escort them back to town along with the two mounted carabinieri, had managed to convince him that there had never been any cholera. He also told him that Captain Montagnet wanted urgently to talk to him, but despite all of don Anselmo's questions, he never explained why.\n\nSince to return to Palizzolo they had no choice but to pass by San Giusippuzzo, don Anselmo was granted permission to stop at his villa for a moment to fetch a pair of eyeglasses he'd left there the night they'd fled to La Forcaiola. When he entered the compound, he noticed that the front door of farm overseer 'Ngilino's house was closed, as were all the shutters on the windows. 'Ngilino must surely have been making the rounds of the estate, but how could Catarina and Totina still be sick? Don Anselmo went into the villa, retrieved his glasses, and just as he was climbing back into the carriage, saw a wisp of smoke rising from the overseer's house. So there was indeed someone inside!\n\n\"Just one minute,\" he said to the lieutenant.\n\nAnd he went and knocked on the overseer's door. Catarina, who'd been watching him from behind the shutters, remained absolutely silent and didn't move.\n\n\"Ma, you can't not open the door for him,\" said Totina, who was standing beside her.\n\n\"Why?\"\n\n\"Because he's with the carabinieri.\"\n\nCatarina came downstairs and opened the door.\n\n\"Good morning, sir.\"\n\n\"Good morning, Catar\u00ec. Why wouldn't you open the door?\"\n\n\"I have the flu, sir, and had to get out of bed.\"\n\n\"What about Totina?\"\n\n\"She's also in bed.\"\n\n\"I'll go and comfort her,\" said don Anselmo, putting his handkerchief over his face to avoid catching germs and advancing towards the entrance.\n\nCatarina blocked his path.\n\n\"No, sir, you can't come in!\"\n\nDon Anselmo got incensed. How dare this peasant woman talk to him like that? He shoved her aside and went in. Totina was standing next to the window in her bedroom.\n\nThere are women who can be pregnant even up to eight months without anyone noticing, and there are others who already at two months have a belly so big that they look like they might give birth at any moment. Totina belonged to the latter category.\n\nDon Anselmo, having entered on the run, froze in his tracks. Behind him he heard Catarina weeping. Then he took two steps forward and dealt the girl a hard slap in the face. But she didn't budge, didn't raise her arm to protect herself, and only stood there, immobile, staring at him.\n\n\"Who was it?\"\n\nShe didn't answer.\n\n\"Who was it?\" he repeated, raising his hand again.\n\n\"It was the Holy Spirit.\"\n\nSo the slut wanted to joke around, did she? Don Anselmo could barely restrain the urge to start kicking her in the belly.\n\n\"Strumpet!\"\n\nThen, to Catarina:\n\n\"Tomorrow morning I want to see 'Ngilino!\"\n\nHe turned his back, went down the stairs, and got back in the carriage.\n\n*\n\n\"Where am I?\n\nThese were the first words that came out of the young man's mouth as he opened his eyes and saw someone he thought he recognized sitting in the chair beside the bed.\n\n\"At the home of friends.\"\n\n\"How long have I been sick?\"\n\nSick? Didn't he remember any of what had happened to him?\n\n\"Just a minute,\" said Stefano. Then he called loudly: \"Zio, come upstairs! The kid is awake!\"\n\nTeresi climbed the stairs two at a time.\n\n\"How long have I been sick?\" the lad asked the new arrival.\n\n\"A few days,\" Teresi answered vaguely.\n\n\"Has my mother been informed?\"\n\n\"Young man,\" said the lawyer, \"we found you in the street.\"\n\nHe decided it was best to spare the lad the detail about the sack. And he continued:\n\n\"And we didn't find anything in your pockets: no papers, no money. So how could we have informed your mother?\"\n\n\"My name is Luigino Chiarapane, and I live in Salsetto, in the palazzo next to town hall.\"\n\n\"I'll get on it at once,\" said Teresi.\n\n\"But what happened to me?\"\n\n\"I really don't know,\" said Teresi. \"But don't strain yourself trying to remember. You've talked enough. For now you should just go back to sleep.\"\n\nThey shut his window and went down to the study.\n\n\"I'll go and get the horse. I'll be back in an hour,\" said Stefano.\n\n\"Where are you going?\"\n\n\"To Salsetto, what do you think?\"\n\n\"To do what?\"\n\n\"What do you mean, 'to do what'? To go and tell his mother . . . \"\n\n\"You're not going to tell anyone.\"\n\n\"But, Zio, the poor woman must be worried to death!\"\n\n\"Stefan\u00f9, until the kid begins to remember everything, nobody must know he's here! I smell something fishy in this whole affair. Something very fishy. This is a card to play against the marquis, and I don't want to waste it!\"\n\n\"Could someone please tell me why I've been summoned here?\" don Anselmo asked at once, still in a rage from the sight of the pregnant Totina.\n\nThey were in Mayor Calandro's office in City Hall. The captain had taken an armchair for himself and placed it next to the mayor's. Don Anselmo was sitting opposite them, across the desk.\n\n\"I can tell you straightaway,\" said Montagnet. \"Everyone I questioned named you as the person who first spread the false information that there was a cholera outbreak in town. It is my duty to ascertain whether you did so intentionally or by mistake. That's all.\"\n\n\"The only person I told was my wife. And in private. In our home. As you see, I didn't spread a goddamn thing!\"\n\nThe mayor squirmed in his chair. From the morning sky you can tell what kind of day it's going to be, and in this case the sky was already full of dark storm clouds.\n\n\"I beg you please to temper your language,\" the captain said frostily.\n\n\"May I explain what actually happened, so we can clear things up and stop wasting my time?\" said don Anselmo.\n\n\"Do you think this is a waste of your time?\" asked Montagnet.\n\n\"I don't _think_ it's a waste of my time, it _is_ a waste of my time.\"\n\nWithout saying a word, the captain got up and headed for the door.\n\n\"Where are you going?\" asked the mayor in alarm. This guy was liable to have even don Anselmo put up against the wall to \"pay the consequences.\"\n\n\"I'm going to call the lieutenant and turn him over.\"\n\n\"Turn me over?\" said don Anselmo, springing to his feet. \"And how would like me cooked? Rare? Well done?\"\n\nThe mayor ran over to the captain and practically knelt down in front of him.\n\n\"Oh, please, for God's sake, don't do that! I will take it upon myself personally, as mayor, to vouch for don Anselmo Buttafava's good behavior! And you, don Anselmo, what are you doing? Trying to ruin us all?\"\n\n\"I apologize,\" said don Anselmo.\n\nThey all sat back down.\n\n\"Please tell me your version of events,\" Montagnet said to don Anselmo. \"But I'm warning you: if I'm not convinced by what you say, I will arrest you for willful disturbance of the public order!\"\n\nDon Anselmo turned red in the face and opened his mouth to answer in kind, but a powerful kick from the mayor under the table persuaded him to sit tight.\n\n\"Go on, speak,\" the captain urged him.\n\nDon Anselmo, who on his way to City Hall had stopped at home to change his clothes and learned the maid Giseffa's side of the story, told the captain how he'd become convinced that there was cholera about and had said so to his wife, who then repeated it to their aged housekeeper Suntina, who then confided this news to the young maid, Giseffa. The girl had then run home to her father's house and spilled the beans, the rumor began to spread, and, in the end, all hell broke loose.\n\n\"There's still one point that needs to be cleared up,\" said the captain. \"Which is: What made you think there was an outbreak of cholera?\"\n\nWith saintly patience, don Anselmo explained how it had been Dr. Bellanca who had first raised his suspicions. Seeing the doctor running between Palazzo Lo Mascolo and Palazzo Cammarata, he'd asked him what was going on, and Bellanca had replied in such a way that he could only conclude . . .\n\n\"All right,\" the captain said when don Anselmo had finished, \"you're dismissed.\"\n\nDon Anselmo stood up, held out his hand to the mayor, and Montagnet suddenly tensed up.\n\n\"You're not free to go yet, you know,\" he said icily. \"I merely meant you're dismissed to go and wait in the room next door.\"\n\n\"Of all the goddamn . . . \" don Ansemo began.\n\nBut the mayor put a hand over his mouth and pushed him into the next room. The captain hadn't noticed anything, because he'd stood up and gone out of the office. He returned a moment later with Dr. Bellanca, sat him down in don Anselmo's place, and said:\n\n\"Doctor, Signor Buttafava just now told us that you were the cause of his misunderstanding, because when he saw you making house calls at both Palazzo Lo Mascolo and Palazzo Cammarata, you told him that the baron and his entire family, as well as the marquis and his entire family, were sick, but you didn't specify what the illness was. Is this correct?\"\n\n\"Yes, I confirm in full.\"\n\n\"It seems to me that at this point there's no further need . . . \" said the mayor.\n\nBut the captain seemed not to have heard him.\n\n\"Why didn't you explicitly tell Signor Buttafava that it was a simple case of influenza, however serious? If you had, Signor Buttafava would not have misconstrued the situation.\"\n\n\"Well, he was irritating me with his insistent curiosity. And anyway, professional ethics don't\u2014\"\n\n\"I see. So was it really a grave form of flu?\"\n\n\"Of course!\"\n\n\"Do you remember what Marquis Cammarata's temperature was on Sunday morning?\"\n\n\"Not really . . . but it was at least 101 or 102 . . . \"\n\n\"So then why did the marquis go to the club that morning?\"\n\n\"He's a stubborn man, you know. They were voting on whether to admit the lawyer Teresi for membership . . . I'd told him not to get out of bed, but . . . In fact, when he got back home, his condition worsened.\"\n\n\"Whereas Baron Lo Mascolo took your advice and stayed in bed.\"\n\nWhat the hell was the guy getting at? the mayor wondered. Luckily, however, Dr. Bellanca didn't crack.\n\n\"He couldn't have even if he'd wanted to. He was the sickest of them all!\"\n\n\"He had a high fever?\"\n\n\"Very high. A hundred and four.\"\n\n\"On Monday morning too?\"\n\nThe question caught the doctor by surprise.\n\n\"I . . . really don't know . . . I don't remember . . . \"\n\n\"Please try to remember.\"\n\n\"Let me think . . . Monday morning, you say? Well, if it wasn't a hundred and four, it was around a hundred and three, I'd say.\"\n\n\"And how do you explain that around seven o'clock on Monday morning, or shortly thereafter, he was stopped by one of Marshal Sciabarr\u00e0's carabinieri on the walkway behind Teresi's residence?\"\n\nBellanca looked at him with his mouth agape. The mayor turned pale.\n\n\"They . . . arrested him?\"\n\n\"No. When the marshal learned that a number of ruffians being led by a priest were laying siege to the lawyer's house, he sent two of his men. One of them saw a peasant having some trouble descending a narrow alley, and so he followed him and then stopped him. But then he recognized the baron and let him go.\"\n\n\"I'm sorry, but why did you say the carabiniere had seen a peasant?\"\n\n\"Because the baron was disguised as a peasant.\"\n\nNow the mayor's jaw dropped in surprise. He no longer understood anything. What on earth was happening in that accurs\u00e8d town?\n\n\"And what did the baron say to the carabiniere?\" asked the doctor.\n\n\"He said he'd gone out for a breath of air.\"\n\n\"And why was he disguised?\"\n\n\"He didn't want to be bothered by any acquaintances during his walk.\"\n\n\"Maybe he'd gone to see Teresi.\"\n\n\"That's his business,\" the captain said by way of conclusion. \"But the upshot of all this, doctor, is that you're clearly lying. I'll give you five minutes' time to make up your mind; if, by then, you haven't told me the truth, I will have you arrested.\"\n\nAnd so it was that, as don Anselmo was on his way home, cursing out loud like a madman, Dr. Bellanca emerged from City Hall in handcuffs, flanked by two carabinieri.\n\nThe captain had charged and released on bail don Anselmo, and imprisoned the doctor, for \"working together on a criminal project the purpose of which is not yet clear.\" They had \"created a grave public disturbance, artfully spreading alarming rumors designed to sow panic among the local population.\"\n\nAs news of Dr. Bellanca's arrest and the charges against don Anselmo began to spread, a number of things happened.\n\nDon Liborio Spart\u00e0 sent the club's manservant Casimiro from house to house to summon the executive committee for a meeting to be held at five o'clock that afternoon. But the committee members had to come one at a time, without attracting any attention, since martial law was in effect.\n\nMayor Calandro, for his part, got on the phone to talk to the prefect. He told His Excellency, Commendator Eustachio Benincasa, that the town had been left without a municipal doctor after the\u2014in his opinion\u2014arbitrary arrest of the holder of that title, and it therefore behooved His Excellency to appoint a physician from the provincial capital and send him to Palizzolo. He also told him that seeing Dr. Bellanca, a man loved and esteemed by all, walk down the street in handcuffs had been a terrible blow for the whole town. In short, His Excellency should be aware that there was general resentment over Captain Montagnet's way of going about things.\n\nHe'd just set the phone down when, without bothering to knock, don Serafino Labianca, Commendator Agusto Paladino, and Patre Alighiero Scurria, the parish priest of the Heart of Jesus church, walked in.\n\n\"Is it true . . . \"\n\n\" . . . that the captain . . . \"\n\n\" . . . arrested the doctor?\"\n\nThis was the first tripartite question asked.\n\n\"Yes, it's true.\"\n\n\"Is it true . . .\n\n\" . . . that don Anselmo . . . \"\n\n\" . . . has been charged and released on his own recognizance?\"\n\nThis was the second tripartite question.\n\n\"Yes, it's true.\"\n\n\"That captain is a maniac!\" said Patre Scurria.\n\n\"And why did he arrest Bellanca?\" asked don Serafino.\n\n\"Because the doctor didn't want to tell him what the marquis and the baron were sick with. Or, more precisely, he told them they had the flu, but Montagnet didn't believe him.\"\n\n\"Sheer lunacy!\" said Patre Scurria. \"But doesn't the good captain know that a doctor is like a priest? We can't just tell others what we're told in the confessional, and doctors are not free to tell other what their patients are sick with.\"\n\n\"This is a clear abuse of power!\" don Serafino exclaimed.\n\n\"It's a classic example of the Piedmontese disdain for us Sicilians,\" the commendatore proclaimed. \"But I'm not going to let this asshole off easy. I'm going to my office now to call Ciccino Barrafranca in Rome and tell him the whole story, and ask him to take immediate action.\"\n\nThe honorable parliamentary deputy Francesco Barrafranca, a first cousin and very good friend of the commendatore's, had owed his resounding electoral victory in Palizzolo to none other than Commendatore Agusto Paladino.\n\n\"And I,\" said the mayor, \"have just finished speaking with the prefect. I told him there was a lot of resentment building up around town.\"\n\n\"And this state of martial law certainly can't last until the next cholera outbreak,\" said don Serafino. \"It must be lifted at once, no later than tomorrow.\"\n\nAt seven o'clock that evening, the executive committee of the Honor and Family Social Club, consisting of don Liborio Spart\u00e0, don Stapino Vassallo, Colonel Amasio Petrosillo, and Professor Ubaldo Malatesta, finished drafting a petition to the prefect of the provincial capital, Camporeale, in which they claimed that the entire population of Palizzolo was indignant over the charges brought by Captain Montagnet against three individuals so beloved and venerated as don Anselmo Buttafava, Don Raccuglia, and Dr. Bellanca, the latter having even been unjustly incarcerated. The town's citizens therefore demanded:\n\n\u2014that all charges against said persons be dropped;\n\n\u2014that Dr. Bellanca be freed at once;\n\n\u2014that the declaration of martial law be revoked, as there was no longer any reason for it.\n\nThe club's manservant, Casimiro, was assigned the task of collecting the signatures of not only the club's members, but also anyone else who wished to add his or her name.\n\nThat evening, Dr. Girlanno Presti arrived from Camporeale to fill in for the municipal doctor. The first thing Presti did was to introduce himself to Captain Montagnet, telling him he needed to confer with Dr. Bellanca in order to have a better sense of the health conditions of the townfolk. Montagnet granted him permission to talk to the doctor at eight o'clock the following morning.\n\n*\n\n\"So, now that I've told where and how we found you, and I've explained to you why I thought it would be better if I brought you back to my house instead of to Palazzo Cammarata, are you ready to tell us everything?\"\n\n\"Yes,\" said Luigi Chiarapane.\n\nThe lad had recovered fairly well, the compresses had reduced the swelling in his lips, his fever was now under 100, but his three broken ribs still hurt whenever he made the slightest movement.\n\n\"I want to see them both in jail: the marquis and _'u z\u00f9_ Carmineddru,\" said Luigino, almost as if to himself.\n\n\"I do too,\" Teresi smiled. \"So please tell me everything from the very beginning.\"\n\n\"My mother is a cousin of Filadelfo Cammarata's wife, and while we were still living in Palizzolo\u2014up until I turned fifteen\u2014our families spent a lot of time together. I grew up with Paolina, the marquis's eldest daughter, though I'm three years older than her. Even after we moved to Salsetto, I've always kept coming back here, at least twice a week, to see her. I'm an only child, and Paolina for me was the sister I never had. She's a jewel of a girl\u2014religious, good-hearted and unselfish. I have no idea how she got into this situation!\"\n\n\"We'll talk about that later,\" said Teresi.\n\n\"The other day when there was all the cholera confusion, my mother herself told me to come to Palizzolo to see how the Cammaratas were doing. First she'd sent a servant to ask the marquis if I could drop by his house in the afternoon, and he'd answered yes. I started to get worried as soon as I saw the front door closed and the windows shuttered. It looked like they were in mourning, so I was afraid somebody had died. I knocked on the door and Gnazina, their eleven-year-old daughter, came and told me that all the servants had run away and Paolina and everybody else in the house were sick with the flu. Then she took me into the marquis's study and told me to wait there. The house was as quiet as a graveyard, when normally it's a pretty noisy place. About a half an hour later, the marquis came in. He was angry and more nervous than usual. He told me to come with him down to the cellar for a minute to fetch a bottle of wine, and so I followed behind him.\n\n\"'How is Paolina?' I asked.\n\n\"He didn't answer, but just opened the cellar door. It was already lit up in there, with oil lamps.\n\n\"'I have to do something first,' he said. 'You go on downstairs, I'll be along in a minute.'\n\n\"As soon as I got to the bottom of the stone staircase, which is quite long, I heard the door close. I thought it was a gust of wind. Then suddenly, standing before me was _'u z\u00f9_ Carmineddru.\"\n\n\"Did you already know him?\" Teresi asked.\n\n\"Yes. He would come and call on the marquis, and they would shut themselves up in his study.\"\n\n\"Did you know who he was?\"\n\n\"How could I not know? Everybody in town knows he's a man of influence.\"\n\n\"And what did he say to you?\"\n\n\"Say to me? He didn't say a word.\"\n\n\"What did he do?\"\n\n\"He laid me out on the floor with his first punch in the face. Then it was all kicks and blows with a kind of club . . . I was yelling and screaming, but who's gonna hear me there in the cellar? After he'd been beating me for about ten minutes, don Filadelfo came in. 'So you had a good time with my daughter Paolina, eh, you pig? Got her pregnant, eh, you dog? Well, you're a dead man now.' I swear before God that this news hurt me more than any of the things _z\u00f9_ Carmineddru was doing to me. At that point they both started pummeling me. And then I fainted and don't remember anything else.\"\n\n\"They thought you were dead,\" said Stefano.\n\n\"And we're going to let them keep thinking you're dead,\" said Teresi.\n\n## CHAPTER VII \nTHE DAY OF DENUNCIATION\n\nGirlanno Presti was a good doctor, but in Camporeale he was known as someone who was afraid of his own shadow.\n\nHe lived with a man who was actually more of a walking tree trunk than a man. His name was Costantino, and he was so big and so broad that he was frightening just to look at. He was always at the doctor's disposal, ready to accompany him out on nighttime house calls, since never in a million years would the doctor have gone out into the darkness alone.\n\nHe would get scared out of his wits at the slightest thing, and already the mere fact of having to go to the carabinieri station at eight o'clock in the morning had made him break out in a cold sweat. But what surprised him most was that he found his colleague, Dr. Bellanca, as fresh and calm as if he'd just spent the night at the Grand Hotel. Bellanca knew Presti and was relieved they'd chosen him to replace him. Montagnet had set no time limit on the meeting between the two doctors and had made a room with a table and two chairs available to them. The first thing Bellanca said was:\n\n\"Did you bring a lot of changes of clothes?\"\n\n\"No. Why do you ask?\"\n\n\"Because I don't think they'll free me any time soon. Yesterday evening the captain came and told me that I could be in here till the end of my days if I didn't start talking. And I won't talk. So . . . \"\n\n\"But what does he want to know?\"\n\n\"He wants to know what Baron Lo Mascolo and Marquis Camarata are sick with. I told him the flu, but he won't believe it.\"\n\n\"So what are they sick with?\"\n\n\"They're not sick at all. But I can hardly throw mud on the honorable names of two families!\"\n\nGirlanno Presti turned pale. What was this new complication? He knew his colleague was charged with disturbing the peace, and now it's a question of the honor of two families? The word _honor_ , in Sicily, was a dangerous matter, one that almost always led to bloodshed.\n\n\"Is this something I should know?\" he asked, secretly hoping Bellanca would say no.\n\n\"Of course! Are you not fulfilling the function of town doctor now?\"\n\nBellanca told him everything.\n\n\"Four pregnant women, none of whom is married? And all of them two months pregnant? How do you explain that?\" Presti asked in bewilderment.\n\n\"There is no explanation. That's the problem. And I'll have you know that these four pregnancies already raise the annual average in Palizzolo. They're a surplus, a bonus. A flowering out of season. Know what I mean? But not a word about this to anyone, do you hear?\"\n\nPresti looked offended.\n\n\"No need to remind me of that.\"\n\n\"And now let's discuss the town's health situation. Around here, tracoma and malaria . . . \"\n\nThey talked for about an hour, after which they shook hands and a carabiniere came to take Dr. Bellanca away. As Presti was gathering up the papers on which he'd been taking notes, the door opened again.\n\nHe looked up to see Montagnet staring at him like a cat eyeing a mouse.\n\n*\n\nDr. Palumbo, for his part, arrived late for his morning visit to Teresi's house. He found the young man quite improved, and told him that in three days or so he would be able to get out of bed and walk around the house a little.\n\n\"But Teresi's nephew told my mother I was here, and she'll be coming to see me this afternoon. She wants to take me back to Salsetto.\"\n\n\"For the moment that's out of the question. You're not strong enough to handle a journey in a carriage.\"\n\nAfter the examination, Teresi offered the doctor a cup of coffee.\n\n\"Sorry I got here late, but I was called to Giallonardo's house.\"\n\n\"Is the notary sick?\"\n\n\"No, he's fine, as is his wife.\"\n\n\"So what was the problem? They haven't got any children!\"\n\n\"It's the maid. A pretty twenty-five-year-old by the name of Rosalia Pampina\u2014or, so Signura Giallonardo told me, since the girl has stopped talking.\"\n\n\"What do you mean, she's stopped talking?\"\n\n\"The girl ran away when she heard about the cholera. She spent one day and one night away, then came back the following afternoon. Ever since, she's stopped talking, eating, and drinking. Or, rather, when she got back she asked her mistress if she could go to church, and when she returned a few hours later she'd stopped talking.\"\n\n\"And how do you explain that?\"\n\n\"Well, I examined her, unfortunately. She'd been torn to shreds.\"\n\n\"Raped?\"\n\n\"In every way possible and imaginable. In my opinion, she'd run into some bad people during her night away. If she's not any better by tonight, I'll have her taken to Camporeale hospital.\"\n\n\"Did the signura tell you which church the girl had gone to?\"\n\n\"The same one where they themselves go: San Cono.\"\n\nDr. Presti's stay in Palizzolo as substitute town doctor turned out not to be as long as Bellanca had expected.\n\nIn fact, it lasted only until eleven o'clock that morning, because at ten-thirty, the door of the holding cell opened and the captain said to the doctor:\n\n\"You're free to go. I've exonerated you of the charges. Have a good day.\"\n\nHe turned his back and went out. Bellanca was so surprised he didn't even say goodbye.\n\nAt more or less the same time of day, Lieutenant Villasevaglios went to don Anselmo Buttafava's house.\n\n\"It is my pleasure to inform you that Captain Montagnet has withdrawn the charges against you.\"\n\nHaving thought, upon seeing the lieutenant, that Villase\u00advaglios had come to arrest him, don Anselmo very nearly fainted in relief.\n\nAt half past eleven, His Excellency Eustachio Benincasa, prefect of Camporeale, rang Mayor Calandro on the telephone. The mayor was so excited he immediately gushed with gratitude.\n\n\"Thank you so much, Your Excellency, for having so quickly intervened to rel\u2014\"\n\n\"Would you let me speak first, for the love of God?\"\n\n\"My apologies, Your Excellency.\"\n\n\"I wanted to tell you that I just now received a petition signed by a hundred or so citizens of Palizzolo, demanding the release of Dr. Buzzanca . . . \"\n\n\"Bellanca, Your Excellency.\"\n\n\"Yes, right, Bellanca. I'm telling you so that you can communicate to the signatories that my reply is to be patient for a few more days. Captain Montagnet is acting in a perfectly lawful manner to restore order in Palizzolo. And you, as mayor, must cooperate with him unconditionally. Have I made myself clear?\"\n\n\"Perfectly clear, Your Excellency.\"\n\n\"What did you want to tell me?\"\n\n\"Nothing, Your Excellency.\"\n\nSo it wasn't the prefect who'd ordered Montagnet to step back. And the captain was not the type of man to change his mind about any action of his own. Then the mayor remembered that Commendatore Padalino had said he would talk with the Honorable Barrafranca. It was possible Barrafranca had intervened immediately. The captain's surprise move made the mayor feel uneasy. He went out of his office, telling the usher he'd be right back. On mornings when the weather was good, the commendatore liked to sit out on the balcony of his home and watch the people passing by.\n\nAnd there he was, in fact. Calandro called up to him from the street.\n\n\"Commendatore, have you heard the news?\" he asked.\n\n\"That they let Bellanca out of jail? Yes.\"\n\n\"I want to thank the Honorable Barrafranca for\u2014\"\n\n\"But, Mr. Mayor, I wasn't able to talk to Ciccino.\"\n\nThen why had Montagnet decided to release Dr. Bellanca?\n\n'Ngilino the overseer pulled up at the Buttafava house just before midday. From the mule he unloaded rounds of _tuma_ , _primosale_ , and ricotta cheese, fruit, vegetables, pork sausages, a just-slaughtered suckling lamb, and four rabbits, and brought them into the larder. Then he went upstairs to don Anselmo's office.\n\n\"I beg your pardon for the other day, sir. But I'd just found out about Totina and felt like I was going crazy . . . \"\n\n\"Why didn't you say anything to me at the time?\"\n\n\"I was too ashamed. My wife told me you slapped Totina around a little. You were right.\"\n\n\"She's like a daughter to me.\"\n\n\"I know, don Anselmo.\"\n\n\"And you know what made me lose my head, 'Ngil\u00ec? She wouldn't give me a straight answer! She just pulled my leg and said it was the Holy Spirit that got her pregnant!\"\n\n\"But, sir, with all due respect, you're wrong. She wasn't trying to pull your leg. She really believes it.\"\n\n\"Believes what?\"\n\n\"That it was the Holy Spirit. She really means it.\"\n\n\"Has lost her mind?\"\n\n\"No, sir, she's a normal girl. She just says it was the Holy Spirit.\"\n\n\"But what do you think? Have you any idea who it could have been?\"\n\n\"None whatsoever. My wife's got no idea either. You see, Catarina never lets Totina out of her sight. We're scared, with all these criminals roaming round the countryside . . . . Totina's a beautiful girl and somebody might try and take advantage of her.\"\n\n\"So we're supposed to believe it was the Holy Spirit?\"\n\n'Ngilino shrugged.\n\n\"And what about when Catarina and Totina come into town for Mass on Sundays?\" don Anselmo continued.\n\n\"Nah, she keeps 'er eye on 'er even worse than at San Giusippuzzo. Totina and Catarina get there early in the morning, say their confessions, and then take communion. Round four in the afternoon, they head back home to San Giusippuzzo.\"\n\n\"Wait a second,\" said don Anselmo. \"And what do they do between the end of the Mass and four o'clock?\"\n\n\"They go an' eat at my sister-in-law Clarizza's house. She's Catarina's big sister.\"\n\n\"And does this sister-in-law have any sons?\"\n\n\"Yessir, she's got two. But they're in America.\"\n\n\"And how old is her husband?\"\n\n\"'E's eighty. Turiddru was twenty years older than Clarizza when they got married.\"\n\nThere seemed to be no answer. Could it really have been the Holy Spirit?\n\nAt that moment the voice of the town crier rose up from the street.\n\n\"Citizens of Palizzolo!\" he shouted in Italian, for anyone who might understand. \"The state of martial law has ended! The curfew and prohibition of public assembly have also been lifted!\"\n\nThis was immediately followed by the Sicilian translation:\n\n\" _Cumpaisani palizzoloti!_ No more martial law! You can stay out all night if you like and get together with as many people as you want!\"\n\n*\n\nAt four o'clock that afternoon Casimiro the manservant intercepted the mayor as he was coming out of his home on his way to City Hall.\n\n\"Don Liborio asked if you could drop in at the club for a minute.\"\n\nAs soon as Mayor Calandro entered the salon, everyone present started clapping.\n\n\"Long live our mayor!\" cried don Stapino Vassallo.\n\nAlmost all the members were there, even Dr. Bellanca, who hardly ever went to the club. Only Baron Lo Mascolo and Marquis Cammarata were missing.\n\n\"Are we all present?\" President Spart\u00e0 asked the secretary.\n\n\"All present except for the sick.\"\n\n\"Casimiro!\" don Liborio called.\n\nThe waiter came in with four bottles of champagne fresh out of the icehouse. A small table had already been set in one corner of the great room and was covered with glasses. The bottles were uncorked and the glasses filled.\n\n\"Gentlemen, please serve yourselves,\" said don Liborio Spart\u00e0. \"But first I should like to propose a toast of thanks to Mayor Calandro and all those who supported our initiative to free Dr. Bellanca. The pressure the mayor and all of us put on the prefect has achieved the desired result. To your health, Dr. Bellanca!\"\n\nThey all drank. The mayor didn't feel like telling the truth, which was that the prefect had nothing whatsoever to do with the matter.\n\n\"Another round?\" don Serafino Labianca asked.\n\n\"Of course!\" said don Liborio. \"Whom would you like to toast?\"\n\n\"I propose a toast to the unfaithful wife of Captain Mon\u00adtagnet!\"\n\nEveryone laughed. At a certain point don Anselmo approached Dr. Bellanca, put an arm around his shoulders, and pulled him a short distance away from the others.\n\n\"There's something I wanted to ask you.\"\n\n\"Go right ahead.\"\n\nBefore speaking, don Anselmo pulled him even farther aside. He didn't want anyone to hear him.\n\n\"Can an eighty-year-old man get a girl pregnant?\"\n\n\"Apparently there have been some such cases. But it's extremely rare. Why do you ask?\"\n\n\"Because my overseer's daughter, Totina\u2014\"\n\n\"I know the whole story, don Anselmo. Her mother brought her to me for an examination.\"\n\n\"The only male that Totina could have had any contact with was Zia Clarizza's husband, but he's eighty years old.\"\n\n\"You're referring to Turiddru Cannizzaro?\"\n\n\"Precisely.\"\n\n\"But do you know Cannizzaro?\"\n\n\"No, sir.\"\n\n\"Cannizzaro is a patient of mine. He suffers from catarrh, but is otherwise a strong, healthy man.\"\n\n\"So in fact you're saying it _could_ have been him!\"\n\n\"Not on your life, don Anselmo! I didn't say that!\"\n\n\"You just don't want to compromise yourself,\" don Anselmo said in disappointment.\n\nAnd, sidling up to the lawyer Sciortino, he took him by the arm and pulled him aside.\n\n\"I would like you to draft a statement of denunciation for me, which I'll drop by later to sign.\"\n\n\"I'm at your service, don Anselmo. Whom do you want to denounce?\"\n\n\"Captain Montagnet, for abuse of power.\"\n\n\"You can't do that, don Anselmo. It's true that the prefect has proved us right, but that would be pissing outside the pot!\"\n\n\"Well, maybe you happen to piss outside the pot, seeing that your hands shake! And if your hands shake that bad, we can only imagine your cock!\"\n\nSciortino decided it was best to turn his back and walk away. It was not a day for squabbling.\n\nSignura Albasia Chiarapane arrived in Palizzolo from Salsetto. A woman of fifty, five-foot-eleven, and blonde, she had a baritone voice, was authoritarian and brusque in manner, and looked a little like an ostrich. Even Teresi felt a tad intimidated by her. She didn't even embrace her son and didn't bother to ask what had happened to him. Instead, she went immediately on the attack:\n\n\"What is this? All these days without any news from you? Is that any way to treat your mother?\"\n\n\"Mam\u00e0 . . . \"\n\n\"You're just like your father! Both of with your heads in the clouds, and I always have to take care of everything!\"\n\n\"Mam\u00e0 . . . \"\n\n\"What did you do to your lips?\"\n\n\"Mam\u00e0 . . . \"\n\n\"I bet you didn't even go to the Cammaratas'!\"\n\n\"Mam\u00e0 . . . \"\n\n\"The man you sent to Salsetto told me you had a bout of the flu. Well, it looks to me like you're all better. Now get dressed and let's go!\"\n\n\"Mam\u00e0 . . . \"\n\n\"Signora, your son has suffered a concussion, lost three teeth, broken three ribs, and I don't know how many\u2014\"\n\n\"Didn't I just say he always has his head in the clouds? You went and got run over by a carriage!\"\n\nLuigino, her son, became disheartened, closing his eyes and lying down in bed. Teresi grabbed the enraged ostrich by the arm and dragged her into his study.\n\n\"And who are you, may I ask?\" asked Signura Albasia.\n\n\"I'm Matteo Teresi, I'm a lawyer, and it was I who, together with my nephew Stefano, found your son on the street. He'd been savagely beaten, left for dead, put in a large sack, and dropped by the side of the road like an animal.\"\n\nThe lawyer made a point of telling her exactly what had happened, without diluting a thing. He wanted to make her as incensed as possible.\n\n\"And did Luigino recognize the assailant?\"\n\n\"The assailants, Signora. There were two of them: a mafioso, and the Marquis Cammarata.\"\n\n\"I don't think this is any time for jokes! Shame on you! Marquis Cammarata is a gentleman who would never hurt a fly!\"\n\n\"Just ask your son, Signora.\"\n\n\"But why would he do anything like that?\"\n\n\"Because he's convinced Luigino got his eldest daughter, Paolina, pregnant, and that\u2014\"\n\nThe signora jumped out of her chair, dashed straight for the staircase, went upstairs, entered her son's room, and dealt him a hard slap in the mouth.\n\nBlood immediately started flowing from the reopened wounds.\n\nBut the lad's mother didn't even notice.\n\n\"You disgusting cad! Taking advantage of my cousin's innocent daughter!\"\n\n\"Grab her,\" Teresi said to his nephew.\n\nAnd together they seized her and dragged her downstairs again to the study.\n\nTeresi locked the door and put the key in his pocket.\n\n\"Now try to calm down, Signora. Your son spent the whole night in a state of delirium. I wrote down what he was saying. And you must know that people speak the truth when they're delirious.\"\n\nHe handed her a sheet of paper.\n\n\"Would you please read this?\" he said.\n\n*\n\nAlso at four o'clock that afternoon, Dr. Palumbo, seeing that Rosalia Pampina still wouldn't make up her mind to drink so much as a drop of water and insisted on remaining silent with her eyes popping out of her head, loaded her into the carriage with the help of the notary's wife and a maid, and took her to the hospital in Camporeale. There a doctor examined her and then said to his colleague from Palizzolo:\n\n\"I'm going to have to report this.\"\n\n\"So go ahead and report it,\" said Palumbo.\n\nHalf an hour later, the report, which described severe, repeated acts of sexual violence and sodomy inflicted by unknown parties upon the person of young Rosalia Pampina, residing in Palizzolo at the home of Domenico Giallonardo, notary, was brought to the attention of one Lieutenant Di Lullo, commanding officer of the Carabinieri station in Camporeale. The lieutenant then duly passed it on, according to protocol, to Captain Montagnet.\n\nWhen the report arrived on the captain's desk, he was not surprised. He already knew Rosalia Pampina's story, having been told it by Lieutenant Villasevaglios, who had obtained the names of the three women raped by Salamone the brigand before letting them go free.\n\nFor the sake of investigative thoroughness, the captain dropped in at the Giallonardo home. The notary was out, but his wife, Signura Romilda, told him everything he wanted to know.\n\n\"But who could it have been?\" Signura Romilda asked when she had finished. \"She didn't tell us anything when she got back here. And it could only have happened during the night she spent out.\"\n\n\"Right,\" said the captain, refraining from telling her about the brigand.\n\nHe thanked her and left. But if Rosalia was speaking before she went to church, why did she stop when she got back from church? Of course it was possible that the trauma from the violence she'd suffered had a delayed onset. Or else, since she was a churchgoing girl, maybe when she went to confess, the priest hadn't wanted to grant her absolution? And if so, why deny her that?\n\nThe girl had not consented to the act\u2014on the contrary, according to Villasevaglios, she had taken the whole thing much harder than the other two girls, and it had required a lot of effort to reassure her.\n\nCaptain Montagnet decided to go and speak directly with the priest of San Cono.\n\nThe first thing Patre Filiberto Cusa said was that all he could tell him about Rosalia Pampina was that she was a serious girl with a healthy fear of God who went to confession and took Communion every week.\n\n\"And did she confess the evening she came here?\"\n\n\"That was her reason for coming.\"\n\n\"And did she tell you about the rape?\"\n\n\"I can't answer you, as I'm sure you're well aware.\"\n\n\"One last question, Reverend. Did you grant her absolution?\"\n\n\"You're very clever, Captain. If I answered your question, I would be implicitly admitting that Rosalia had confessed to something so dire as to jeopardize her chance of absolution. But I want to tell you something that might be of help to you. For us, no sin is committed if a person is forced to sin through violent coercion. I hope I'm being clear.\"\n\n\"Perfectly clear.\"\n\nTherefore, Rosalia had been granted absolution. So then why had she fallen into despair? He had an idea.\n\n\"One more question, Father. How long was she here in church, do you remember?\"\n\n\"I would say not more than twenty minutes.\"\n\nThere was something that didn't add up. Signura Giallonardo had told him that Rosalia returned home two hours after she'd gone out. Granting that she'd spent half an hour in church, where had she spent the other hour and a half? And, more importantly, with whom?\n\n*\n\nWhen the captain returned to the station, the carabiniere on duty told him there was a couple, a man and a woman, waiting to see him. He'd shown them into the captain's office. As Montagnet entered, the two stood up.\n\n\"Good afternoon. My name is Matteo Teresi, and I'm a lawyer here in town,\" said the man.\n\n\"And I am Signora Albasia Chiarapane.\"\n\n\"Pleased to meet you.\"\n\nSitting down he realized that he'd left the hospital doctor's report on his desk. He grabbed it and put it in a drawer, not knowing, however, that Teresi had had all the time in the world to commit it to memory.\n\n\"What can I do for you?\"\n\nThe lawyer and the lady exchanged a glance of consultation. She went first.\n\n\"The lawyer and I are here to report the attempted murder of my son, Luigi.\"\n\n\"Did this happen here?\"\n\n\"Yes.\"\n\n\"Is it an accusation against an unknown party?\"\n\nNow it was Teresi's turn to speak.\n\n\"No. We are accusing Marquis Filadelfo Cammarata and a noted local mafia chief known in town as _'u z\u00f9_ Carmineddru, but whose surname is unknown to us.\"\n\n\"I know his surname,\" said the captain. \"His name is Carmine Pregadio. And where is her son at present?\"\n\n\"At my house,\" replied Teresi. \"We picked him up off the street. They'd put him inside a sack, apparently thinking he was dead.\"\n\n\"I would like to hear his testimony first. Can he come here?\"\n\n\"Doctor Palumbo, who has been attending to him, has forbidden him to get out of bed. But if you'd like to come to my house\u2014\"\n\n\"Let's go,\" said the captain, getting up.\n\n*\n\nAfter more than an hour, Teresi came out with the captain and accompanied him to Dr. Palumbo's office, as Montagnet wanted to hear his testimony. As they were walking in silence, Teresi came out with an expression intended to prick the captain's curiosity.\n\n\"How odd, though!\"\n\n\"What's odd?\" asked Montagnet.\n\n\"That Baron Lo Mascolo's daughter is also two months pregnant!\"\n\nBut he didn't seem so interested.\n\n\"He came charging over to my house,\" Teresi continued, \"accusing my nephew of having seduced his daughter.\"\n\n\"Did he threaten him?\"\n\n\"He wanted to shoot him dead!\" the lawyer said, laughing.\n\n\"Threatening with a firearm. Would you like to press charges?\"\n\n\"No. He was finally convinced it wasn't my nephew. But don't you find it strange that two unmarried girls are both exactly two months pregnant?\"\n\n\"As far as that goes, the number of pregnant women who don't want to reveal the name of the culprit\u2014for lack of a better term\u2014is four.\"\n\nTeresi was stunned, stopping dead in his tracks in the middle of the square. He didn't know that Dr. Presti, after half an hour of interrogation by the captain, spiced with the threat of a ten-year prison sentence, had cracked and told him everything he'd been told by Dr. Bellanca.\n\n\"But . . . how did you find out . . . ?\"\n\n\"We're carabinieri, aren't we?\"\n\nHe'd just dropped the captain off at Dr. Palumbo's and was on his way home, where Luigino's mother was waiting for him, when he was stopped by don Anselmo Buttafava.\n\n\"I have a request, my good man.\"\n\n\"I'm at your service, don Anselmo.\"\n\n\"I need you to draft an accusation for me . . . \"\n\n\"I'm sorry, don Anselmo, but isn't Sciortino your regular lawyer?\"\n\n\"Yes, and in fact I approached him first, but he wouldn't hear of it.\"\n\n\"Whom are you accusing?\"\n\n\"Captain Montagnet, for abuse of power.\"\n\n\"That's a rather groundless accusation.\"\n\n\"You think? So his charge against me had deep foundations?\"\n\n\"Listen, could we talk about this tomorrow morning at nine? I'll come to you, if you prefer.\"\n\n\"All right, I'll be waiting. Can I ask you one more thing?\"\n\n\"I'm in a bit of a rush, don Anselmo. Go ahead.\"\n\n\"In your opinion, can an eighty-year-old man get a young girl pregnant?\"\n\nThe lawyer froze in his tracks for the second time in ten minutes.\n\n\"Why are you asking me that?\"\n\n\"Because Totina, the daughter of my farm overseer, 'Ngilino, is two months pregnant and saying it was the Holy Spirit that did it. But I think it was an uncle of hers, except for the fact that he's eighty years old. He must have taken advantage of her when she came into town for Holy Mass.\"\n\nThe lawyer was barely even listening to him anymore. He was wondering whether Totina was already one of the four pregnant girls, or the fifth. \n\n## CHAPTER VIII \nATTORNEY TERESI STARTS THINKING ABOUT THINGS\n\nSignura Albasia Chiarapane headed back to Salsetto as the sun was setting, but before leaving announced that she would be back the following afternoon. After eating what his daytime housekeeper had made, and serving some as well to Luigino, Teresi told his nephew not to go upstairs to chat with the lad, but to come into his study.\n\n\"I want to ask you something.\"\n\n\"Go ahead.\"\n\n\"It's about Antonietta Lo Mascolo.\"\n\n\"I've already told you what I know about her. But if you want to keep talking about her, I can do that too.\"\n\n\"Stefan\u00f9, I've been giving this whole affair a lot of thought. You have maintained, and continue to maintain, that Antonietta was in no way the kind of girl who would drop her knickers for somebody she'd just met the day before, correct?\"\n\n\"Correct. But not even for somebody she met three years before, either.\"\n\n\"I would like to know what it is that makes you so sure.\"\n\n\"It's the way she acts, Zio. The way she talks. And she wasn't just making small talk. She was convinced in her heart of everything she did and said. One time we even talked about when she would get married. She had a very clear idea of the kind of man she would choose: he had to be serious, and honest. Just like her. She didn't care whether he was rich or not. Last year the baron told her that Baron Piscopo's son, Arrigo, had expressed interest. She replied that it was completely out of the question. She'd seen this Arrigo once, and that had been enough.\"\n\n\"So you would rule out some secret boyfriend?\"\n\n\"Absolutely. And I can also tell you that even if she had one, she would never make love to him until after they got married. I would bet my life on it.\"\n\n\"So the baron's reasoning concerning you wasn't incorrect.\"\n\n\"What do you mean?\"\n\n\"That is, being the only man his daughter frequented and in whom she confided . . . \"\n\n\"Yes, in that sense, he was right. But I've never touched Antonietta.\"\n\n\"Well, somebody did. And how!\"\n\n\"But, Zio, where did it happen? How did he manage to persuade her? How did he find the time to be with her?'\"\n\n\"You told me Antionietta was all church and hearth and home, right?\"\n\n\"Right.\"\n\n\"And what did Luigino say to us about Paolina? Didn't he describe her with the same words as you just now did with Antonietta? Don't the two girls seem like carbon copies?\"\n\n\"They are like carbon copies.\"\n\n\"And now let me tell you the story of a third girl I learned about on my way home. Her name is Totina, and she's the daughter of don Anselmo Buttafava's overseer. Like the others, she's also two months pregnant, and says the father is the Holy Spirit.\"\n\n\"What are you saying?\" Stefano said in dismay.\n\n\"What I'm saying is true.\"\n\n\"An overseer's daughter!\"\n\n\"As you can see, cocks don't care about class. Now, back to Totina. When she and her mother come to town for Sunday mass, the mother never lets her out of her sight. The only time the girl is alone is when she goes to confession. So, another case of a girl who's all church and hearth and home.\"\n\n\"But that doesn't seem to be of any help to us.\"\n\n\"Oh, it's a help, Stefan\u00f9, it's a big help!\"\n\n\"How?\"\n\n\"Let's think it through. If we rule out the possibility that these girls screwed up at home, what's left?\"\n\n\"Church.\"\n\n\"There you go!\"\n\n\"But, Zio, what's going through your head? How would it be possible to do anything like that in church?\"\n\n\"Well, technically, it _is_ possible, as far as that goes. Have you ever been, for example, at midnight mass on the last day of the year? It's so crowded you couldn't fit a pin in there! And that was how my friend Geg\u00e8 Pirrotta fucked his girlfriend the first time! Standing up, right there in church!\"\n\n\"Zio, I think you're letting your imagination run away with you. Here we're talking about three women who\u2014\"\n\n\"Four.\"\n\n\"What do you mean, four?\" Stefano asked in shock.\n\n\"There are four of them, all two months pregnant. Montagnet told me.\"\n\n\"And who's the fourth?\"\n\n\"He didn't tell me her name.\"\n\n\"All right, but can you imagine four women, in four different churches but on the same occasion, letting someone lift their skirts without making a peep? Your friend's case was a little different, he and the girl were a couple. But I can assure you that Antonietta did not have a secret boyfriend. Nor did Paolina, I don't think.\"\n\n\"As far as you know.\"\n\n\"Of course, as far as I know!\"\n\n\"What if it was just one man?\"\n\n\"But, Zio! Are you trying to tell me there's some magic dick wandering from church to church?! Whether it was one man or four, the girls would have rebelled just the same!\"\n\n\"Maybe they didn't open their mouths because they were ashamed of what was happening to them.\"\n\n\"Knowing Antonietta as well as I do, I think she would have started screaming so loud they would've heard her all the way to Palermo!\"\n\n\"All right, let's make another hypothesis. I never go to church, and neither do you. However, the last time a procession went by, I saw that it wasn't just old and young women and old men behind San Cono, there were also middle-aged and young men. Some of them were wearing rosettes in their lapels.\"\n\n\"That's the symbol of the congregation of San Cono.\"\n\n\"And who are the ones with the cowls on Good Friday?\"\n\n\"They're the congregation of the Passion.\"\n\n\"Do you see what I'm getting at? There isn't a single church that doesn't have its congregation. And they are composed of more or less young men who go to those churches. Who share the same devout sentiments as Antonietta, Paolina and Totina. Who share spiritual interests. Isn't it possible that one of the girls could have a secret lover among these men?\"\n\n\"But I've already told you, Zio, that before she's married, Antonietta will never\u2014\"\n\n\"But how do you know they're not already married?\" Teresi asked, wild-eyed.\n\n\"Married? Without anyone knowing?\"\n\n\"What need is there to tell anyone? They could have got married in secret before God! In their minds and consciences they're married! And in that case Antonietta could very well make love with the man she considered her husband!\"\n\n\"And where would they have consummated the marriage, in your opinion? On the main altar?\"\n\nTeresi didn't answer.\n\n\"I'm a little tired,\" said Stefano, getting up. \"I'm gonna go talk to Luigino a little and then go to bed. Good night.\"\n\nBut Teresi did not have a good night. He spent three quarters of it in his study, racking his brains and taking notes.\n\n*\n\nThe chief clerk at the registry office of Palizzolo, Cosimo Spartipane, opened the office at eight o'clock sharp, as he did every morning except weekends and holidays. He went in, doffed his hat, bent down to open the bottom drawer of his desk, where he kept his pen and ink pot, and when he stood back up, he found Captain Montagnet before him. He nearly had a heart attack. First, because, though he was a perfectly honest man, the mere sight of a carabiniere always scared him to death. And, second, because he hadn't heard him come in.\n\n\"Good morning,\" said the captain.\n\n\"Good morning. Do you need something?\"\n\n\"Yes. Two family status certificates.\"\n\n\"For whom?\"\n\n\"For Baron Alfonso Lo Mascolo and Marquis Filadelfo Cammarata.\"\n\n\"I'm not sure whether regulations allow\u2014\"\n\n\"As you may have gathered, I am not requesting them for my own personal pleasure. I need them for an investigation. And I don't think family status certificates are confidential. Therefore . . . Unless you intend to put up an obstruction, in which case\u2014\"\n\n\"When do you need them by?\"\n\n\"Within the hour.\"\n\nAs soon as Montagnet left, Spartipane dashed into the office of the mayor, who had just arrived.\n\n\"The captain wants the family status certificates for Cammarata and Lo Mascolo!\"\n\n\"Why?\"\n\n\"How should I know?\"\n\nWant to bet that Montagnet was itching again to throw someone in jail? Perhaps it was best to alert both men, the marquis and the baron, of this new development. The mayor wrote two identical notes in which he changed only the name of the addressee, and sent them off with two municipal policemen in two open envelopes, so that the policemen could read them and tell everyone in town about the captain's latest move.\n\n*\n\nAt nine o'clock that morning Teresi came knocking at don Anselmo's door and was immediately shown into his office.\n\n\"So, as I was saying yesterday evening, I would like to denounce Captain Montagnet for . . . \" don Anselmo began, trailing off when he saw the lawyer raise his hand.\n\n\"I spent all light researching whether there were any legal precedents,\" said the lawyer.\n\nThis was a big fat lie, but Teresi had no reason for wanting to turn Montagnet against him. But neither did he want to displease don Anselmo.\n\n\"Why the hell should I care about precedents?\"\n\n\"You may not care, but we lawyers do! And I have to tell you that I found no precedents for this kind of case.\"\n\n\"Oh, yeah? So if someone steals my shit, I can't do anything about it because nobody has stolen anyone's shit before?\"\n\n\"Your comparison is not exactly apropos, don Anselmo. The fact is that the captain has been acting in accordance with the powers granted him by the state of emergency.\"\n\n\"And has the state of emergency been lifted?\"\n\n\"It has.\"\n\n\"Then why doesn't this goddamn captain get the hell out of our hair instead of going around and asking left and right for everybody's family status certificates?\"\n\n\"Everybody who?\"\n\n\"Everybody like Baron Lo Mascolo and Marquis Cammarata, for example.\"\n\n\"And when did he request them?\"\n\n\"Somebody came and told me just a minute before you got here.\"\n\nWhat could this mean? Teresi would give this some thought later.\n\n\"That's why, don Anselmo, I'm sorry I can't help you. But I do think I could be some use to you in Totina's case.\"\n\n\"Really?\"\n\n\"Yes. I would need to talk to\u2014\"\n\n\"The girl isn't talking; all she says is that it was the Holy Spirit that did it.\"\n\n\"All I would need is five minutes with her mother.\"\n\nDon Anselmo took his watch out of his pocket.\n\n\"Are you free at five o'clock this afternoon?\"\n\n\"Yes.\"\n\n\"Then come back at five, and Catarina will be here.\"\n\nThe explanation for Montagnet's requesting the family status certificates came to Teresi as he was eating a morning cannolo at the Caf\u00e8 Esperia. Dropping the cannolo halfway through, he dashed back home.\n\nThere he found Stefano and Luigino chatting and laughing.\n\n\"Stefan\u00f9, do you know exactly how old Antonietta is?\"\n\n\"Seventeen years and seven months old.\"\n\n\"And how old is Paolina?\" Teresi asked Luigino.\n\n\"Sixteen and a half. Why do you ask?\"\n\n\"Because they're both minors, that's why!\"\n\nThen he got into his carriage and galloped off to Camporeale, where he had to argue a case in court.\n\n*\n\nAnd so he missed the greatest spectacle ever seen in Palizzolo.\n\nAt ten-thirty that morning, two carabinieri corporals and one marshal, led by Lieutenant Villasevaglios, who was looking even taller and thinner than usual\u2014looking indeed like death itself on the march\u2014left their compound and headed towards the center of town. Feeling curious, a few layabouts started following them. When the carabinieri reached the main square, the number of rubberneckers doubled. In short, by the time the carabinieri turned onto Via Cammarata, there were some fifty people behind them. Lieutenant Villasevaglios knocked on the great door. They went inside. The door closed. Taking advantage of the pause, a few people ran off in search of other townfolk.\n\nThen, all at once, despite the fact that all the windows were closed, a terrible buzz of voices burst forth inside the palazzo, shouts, wails, cries that could be heard all the way down in the street.\n\nThen the door opened, and out came, in the following order: Lieutenant Villasevaglios, the carabinieri marshal, the Marquis Filadelfo Cammarata in handcuffs, and, bringing up the rear, the two carabinieri corporals. The marquis's face was green and he was trembling like a leaf. But clearly it was not from fear or shame, but from rage. The instant he came out the door, all the windows on the front of the palazzo opened suddenly, and seven of the marquis's eight daughters appeared, shouting and cursing at the carabinieri.\n\nAt that exact moment the marquis leapt forward and in a flash sunk his teeth into the ear of the marshal in front of him, not letting go until the lieutenant, having unsheathed his sword, whacked him in the head with the side of it.\n\nIt was estimated that some two hundred people accompanied the carabinieri to the station.\n\nLess than fifteen minutes later, the lawyer Sciortino arrived. He was greeted by Lieutenant Villasevaglios.\n\n\"Mind telling me what my client is charged with?\"\n\n\"Attempted murder, in complicity with Signor Carmine Pregadio.\"\n\n\"Have you also arrested Pregadio?\"\n\n\"He has fled and is at large.\"\n\n\"And whom did they try to kill?\"\n\n\"A young man by the name of Luigi Chiarapane.\"\n\n\"And why?\"\n\n\"I don't know the answer to that question.\"\n\n*\n\nAt noon there was a special meeting at the castle of the Duke Ruggero d'Altomonte, attended by all the nobility of Palizzolo, namely: Marquis Spinotta, Baron Piscopo, Baron Roccamena, and Baron Lo Mascolo (who had gone out of his house for the first time since the start of the confusion, owing to the gravity of the situation). Unavoidably absent was the Marquis Cammarata.\n\nThe meeting was held in the duke's bedroom. The duke sat in an armchair with a heavy woolen blanket over his legs. At a hundred and two years of age, he was always cold. And continuously talking to himself.\n\n\"There is no more religion!\"\n\n\"There is no more respect!\"\n\n\"There is no more order!\"\n\n\"There are no more manners!\"\n\n\"What have we come to?\"\n\n\"A marquis handcuffed like a common delinquent!\"\n\n\"Held up to public ridicule!\"\n\n\"To the mockery of the low-born rabble!\"\n\nWhen the duke had finished his rant, Marquis Spinotta said that they had to prevent the captain from doing any more harm.\n\n\"Can he do any more than he's done?\" asked Baron Lo Mascolo.\n\n\"Absolutely!\" replied Baron Piscopo. \"And his next victim will be you!\"\n\n\"Me?!\"\n\n\"Yes, sir, you! Did you know that Montagnet has asked to see your family status certificate?\"\n\n\"Yes, I know. But why did he do that?\"\n\n\"How should I know? The fact remains that he requested the Cammarata certificate and then immediately arrested the marquis. Therefore . . . \"\n\nBaron Lo Mascolo turned pale.\n\n\"What sort of rapport do you have with your cousin, the duke of San Loreto?\"\n\nDuke Simone Loreto di San Loreto was the highest functionary of the court of His Majesty the King.\n\n\"I have an excellent rapport with him. Why do you ask?\"\n\n\"Could you ring him in Rome and describe to him the situation that has developed here? If the duke could just say a few words to the General Commander of the Carabinieri . . . \"\n\n\"I can try,\" said Marquis Spinotta.\n\nAt that moment Duke Ruggero d'Altomonte opened his mouth to speak.\n\n\"Friends . . . \"\n\nSince he spoke in only the faintest of voices, everyone drew near to listen.\n\n\"Would you like to know who's to blame for all this?\"\n\nAmidst the respectful silence of everyone present, the duke, after taking a short breath, pronounced his verdict:\n\n\"It's all the fault of the French Revolution!\"\n\n*\n\nTeresi wasn't at the Camporeale courthouse more than half an hour, because the hearing was postponed. The hospital was right next door to the courthouse.\n\nWithout thinking twice, he decided to go and inquire about the condition of the girl who had been raped, the one mentioned in the doctor's deposition he had read while waiting for Captain Montagnet. Her story was of interest to him as a journalist. He wanted to write an article about it. Luckily he remembered her name.\n\n\"My name is Stefano Torrisi,\" he said to the nun at the entrance desk. \"I would like some news about a relative of mine.\"\n\n\"What is her name?\"\n\n\"Rosalia Pampina.\"\n\nWhy did the nun seem a little awkward?\n\n\"I don't know whether . . . Well, please take a seat in the waiting room.\"\n\nAfter a short wait, a doctor in a white coat came in.\n\n\"Signor Torrisi?\"\n\nThere were three men in the room, and none of them moved.\n\n\"Is there a Signor Torrisi here?\" the doctor asked again.\n\nAll at once Teresi remembered he'd given the receptionist that name.\n\n\"I'm sorry,\" he said, standing up. \"My mind was elsewhere.\"\n\n\"Please follow me,\" said the doctor.\n\nHe led him into an office, sat him down, and closed the door.\n\n\"In what way were you related to Rosalia Pampina?\"\n\n_Were_ related? Why didn't he say \" _are_ related\"?\n\n\"I'm a second cousin. Why do you ask?\"\n\n\"Because Rosalia Pampina committed suicide at dawn this morning, throwing herself out a fourth-floor window. My deepest condolences.\"\n\n\"But . . . wasn't anyone keeping an eye on her?\"\n\n\"Why should we have been keeping an eye on her? Yesterday evening we had the impression her condition was starting to improve.\"\n\n\"In what sense?\"\n\n\"She spoke. She made a perfectly comprehensible statement, even though the meaning was a little obscure.\"\n\n\"What did she say?\" the lawyer asked, feeling, for reasons unknown, a lump in his throat. As if Rosalia really were a relation of his.\n\n\"She said: 'The penance is like the sin.' And she repeated it twice. Then fell back into her catatonic state. But there's a problem you could perhaps help us with.\"\n\n\"What is it?\"\n\n\"We've been unable to inform her family because we have no address for them. Are you from Palizzolo?\"\n\n\"Yes.\"\n\n\"All right, then, if you could inform Dr. Palumbo, I'm sure he could\u2014\"\n\n\"I'll do so at once.\"\n\n*\n\nThe penance is like the sin. What could it mean? Why not discuss it with Montagnet? Two heads might stand a better chance of grasping the meaning.\n\nHe dropped in at the printworks where they published the newssheet he edited and wrote, which came out once a week.\n\n\"We're still missing the lead article,\" said the printer. \"And if you don't send me something by tomorrow, or day after tomorrow at the latest, we'll have to come out late.\"\n\nTeresi headed back to Palizzolo, but when he got there he found some fifty people still gathered outside the carabinieri station.\n\n\"What's going on?\"\n\n\"The Marquis Cammarata's been arrested.\"\n\nHe turned around and went to inform Dr. Palumbo of the death of Rosalia Pampina.\n\n## CHAPTER IX \nWHAT'S TWO PLUS TWO?\n\nThe investigating magistrate, Artidoro Tommasino, arrived in Palizzolo at the break of dawn and set up shop in a room at the carabinieri compound, along with the court clerk he'd brought with him.\n\nFirst he sat and spoke face to face with the captain, and then he sent a carriage to fetch Luigino Chiarapane. He listened to the young man for an hour, then sent him home again.\n\nAfter this, he sent for Dr. Palumbo and drew up a report of all the wounds the doctor had found on the lad's body.\n\nAnd after this he had Stefano, Teresi's nephew, brought in and asked him to describe how they had happened to find Luigino inside a sack, and what they had done as soon as they realized the youth was still alive.\n\nThen, to everyone's surprise, he sent for Teresi.\n\nThe moment he saw him enter the building, the lawyer Sciortino, who was standing outside the judge's room, went in together with Teresi.\n\n\"Which of you is Matteo Teresi?\" asked the judge.\n\n\"I am,\" said Teresi.\n\n\"And who are you?\"\n\n\"My name is Sciortino, and I'm also a lawyer and represent the Marquis Cammarata. I am here to file a formal protest.\"\n\n\"Why?\"\n\n\"Because you are seeking the help of the accuser's lawyer, while the case is still in the investigative stages!\"\n\n\"For your information, I am not seeking the help of anyone! And I refuse to consider your statement any further! I have not summoned Matteo Teresi here in his capacity as the accuser's lawyer, but as a witness. You can verify this later, when you read the minutes of our meeting. Now leave this room at once!\"\n\nUpon hearing the news of the marquis's arrest, Teresi had congratulated the captain in his mind for having had the courage to do such a thing; he was thus now further delighted to learn that Judge Tommasino wasn't afraid of anyone.\n\nAs soon as Sciortino left, the judge began speaking.\n\n\"Let me start by saying, as I have just said, that you are here solely as a witness. Your nephew, Stefano, told me how you managed to find young Chiarapane on the street. He said he'd wanted to take him to the nearest house, which was Palazzo Cammarata, and that you were against this and brought him instead to your house. Is this correct?\"\n\n\"Yes, that's correct.\"\n\n\"Then my question is: Why?\"\n\nTeresi hadn't been expecting this dangerous question, and had a moment of hesitation.\n\n\"I don't quite understand,\" he said to buy time.\n\n\"Your nephew was very clear on the matter. He told us that when he'd suggested taking the injured young man to Palazzo Cammarata, you replied, saying, in so many words, that you didn't want to give the marquis the chance to finish the job. So my question is quite simple: Since you had never seen that young man before and therefore knew nothing about him, what made you think at once that he would be risking his life by returning to the Cammarata home?\"\n\nThe question was truly dangerous. If Teresi said he knew that the marquis was not to be trusted and had voted against his entry into the club, the judge might get the wrong idea. In the meantime, however, Teresi remembered what he'd been thinking when he found the young man in a sack on the street.\n\n\"Your honor, it was intuition, pure and simple, but still based on certain known patterns of behavior. You see, Stefano told me he thought the lad was a nephew of the marquis and often went to Palazzo Cammarata. Bear in mind that whoever beat the young man up believed him already dead. They then wrapped the body up in the sack and tossed it out just a few hundred yards away from Palazzo Cammarata. It was a precise message. A Mafia message. A corpse dumped far enough away from Palazzo Cammarata to rule out any suspicion that the Cammaratas were in any way involved. And I also remembered that the marquis had often been quick to solicit the services of the local Mafia chieftain, Carmine Pregadio, known as _'u z\u00f9_ Carmineddru. That's all.\"\n\nAfter dismissing Teresi, the judge summoned Marquis Cammarata. Since the marquis was still under arrest, he was flanked by a marshal and an unranked carabiniere, who stood on either side of his chair.\n\nThe marshal's ear was wrapped in a bandage.\n\n\"Did you injure yourself?\" the judge asked.\n\n\"No, this man here bit me.\"\n\n\"So we must add the charge of resisting arrest and assaulting an officer of the law.\"\n\n\"I don't give a flying fuck,\" said the marquis, whose face was now sea-green.\n\n\"Listen, my lord marquis, you've been accused of attempted murder. What have you got to say for yourself?\"\n\n\"That I did it, and you should stop busting my balls.\"\n\n\"Please have the accused's attorney come in,\" the judge said to the two guards.\n\nThe lawyer entered.\n\n\"Marquis, would you kindly repeat what you just said to me?\"\n\n\"I said it was I who tried to kill that little son of a bitch.\"\n\nAttorney Sciortino's heart sank into his shoes.\n\nAn hour later, Marquis Filadelfo Cammarata, now in chains \"due to the dangerousness of the detainee,\" was taken to the prison of Camporeale.\n\n*\n\nAt three o'clock that afternoon, with the authorization granted him by Judge Tommasino in hand, Captain Montagnet knocked on the great door of Palazzo Lo Mascolo.\n\n\"There's a carabinieri captain here wants to talk to you,\" Filippa the housekeeper said to the baron.\n\nDon Fof\u00f2 got worried. Not only did he not want to see so much as the shadow of Captain Montagnet, he also hadn't forgotten the words of Marquis Spinotta: \"His next victim will be you!\" And this time the could not escape out the back window, as he'd done at Teresi's house. There was nothing to be done.\n\n_Calati junco ca passa la china_ , he reminded himself. Then he said to Filippa:\n\n\"Show him into the office\u2014actually, no. Show him into the salon.\"\n\nIn the salon hung the painted portraits of some twenty ancestors; they would let the captain know who he was dealing with. Don Fof\u00f2 took off his dressing gown, and as he was getting dressed his wife, the Baroness Marianna, arrived in a flurry.\n\nAlready when they got married the baroness wasn't exactly a raving beauty. But now, between the aging process and her anguish over their daughter Antonietta, she'd become frighteningly ugly. She took one look at Fof\u00f2 and started crying.\n\n\"They're going to take you to jail! I just know it! This time you're going to end up in jail!\"\n\nThe baron pushed her aside with one hand, and with the other he grabbed his cojones firmly, to ward off back luck, then left the room, descended the stairs, and went into the salon.\n\nThe captain, who was standing and looking at the family portraits, gave the military salute and handed him a document. Don Fof\u00f2 felt his heart sink. It was surely an arrest warrant. He started sweating as the room began to spin around him.\n\n\"I haven't got my glasses,\" he said, worried that his voice was trembling.\n\n\"Shall I read it for you?\"\n\n\"Yes.\"\n\nThe captain read it.\n\nWhen he'd finished, the baron very nearly embraced him. He was not going to jail after all, at least not this time. And he was especially glad not to be going to jail because of his slut of a daughter, whom he alternately considered dead and then alive but a slut. He decided to put up a little resistance, for form's sake, only to give in at once.\n\n\"If I've understood correctly, you are authorized to question my daughter, Antonietta, in the presence of her mother.\"\n\n\"That's correct.\"\n\n\"May I ask why you wish to speak with my daughter? And why in the presence of her mother? Because, on top of everything else, my daughter is sick in bed.\"\n\n\"My good baron, I could have sent two carabinieri here to bring your daughter in for questioning. Out of respect for your feelings as her father, I didn't do that.\"\n\nSo the captain knew that Antonietta was pregnant! His \"feelings as her father\" couldn't mean anything else. Therefore there was no need to playact any longer with him.\n\n\"I appreciate your courtesy, but you haven't answered my question.\"\n\n\"I shall do so at once. Your daughter is a minor, Baron. And I need to know whether she was the victim of a rape or a consenting partner. And I must proceed in the presence of her mother precisely because she is a minor.\"\n\nCourteous but firm. On top of everything else, if the captain succeeded in making Antonietta talk, the whole thing could prove useful. The baron would find out the lover's name.\n\n\"All right,\" he said, leaving the room.\n\n*\n\nOne hour later, the captain was knocking on the great door of Palazzo Cammarata.\n\n\"Bastard! Son of a bitch! Bugger!\" the marquise started shouting as soon as she saw him.\n\nAnd the seven daughters, including a tiny little girl who to the naked eye looked to be barely five years old, repeated in chorus:\n\n\"Bastard! Son of a bitch! Bugger!\"\n\nWhile the maids, from the kitchen, echoed:\n\n\" . . . ugger!\"\n\nLuckily Sciortino the lawyer was there in the house and was able to bring the situation under control and calm the marquise down. And so the captain was able to speak with the underage pregnant girl.\n\n*\n\nAt five o'clock sharp, Teresi showed up at don Anselmo's house.\n\n\"Catarina is here.\"\n\n\"Don Anselmo, I'd like to ask a favor of you.\"\n\n\"Go ahead.\"\n\n\"I would like to speak with Catarina alone.\"\n\n\"Why can't I be present too?\" asked don Anselmo, feeling offended.\n\n\"Because there may be some things she won't want to say to me with her boss present. Please do me this favor. It's in your own interest.\"\n\n\"All right, as you wish. Please go into my office, and I'll send her straightaway.\"\n\n\"I din't do nothin',\" said Catarina as soon as she came in.\n\nShe was scared to death. Her hands were trembling.\n\n\"Nobody's saying you did anything.\"\n\n\"Are you a lawyer?\"\n\n\"Yes.\"\n\nCatarina started crying and shouting:\n\n\"Oh my God, my God, I'm done for! Oh the pain and trouble! Jesus, Mary and Joseph, take pity on my unhappy soul!\"\n\n\"Why are you acting this way?\"\n\n\"'Cause don Anselmo wants to take me to court!\"\n\n\"What on earth is going through your head? Why would he want to take you to court!\"\n\n\"'Cause I din't watch over my daughter and now she went and got pregnant!\"\n\n\"Listen, Catarina. Don Anselmo can't do anything to you, believe me. Anyway, I'm not here as a lawyer, but as don Anselmo's friend.\"\n\n\"Really?\"\n\n\"Really.\"\n\n\"No papers wit' writing on them?\"\n\n\"No papers.\"\n\nThe woman calmed down a little.\n\n\"What do you want to know?\"\n\n\"I'd like to try to find out who got your daughter pregnant.\"\n\n\"She says it was the Holy Spirit.\"\n\n\"Listen, the Holy Spirit is a spirit. Which means he doesn't have a body, do you understand? I want you to tell me what the two of you do on Sundays, after you go to Mass.\"\n\n\"We go an' eat at my sister's place.\"\n\n\"And do you stay at your sister's until four o'clock?\"\n\n\"Sometimes . . . sometimes . . . \"\n\nTeresi felt a kind of tingling along his spine. He pretended not to be too interested, lighting a cigar and taking a few puffs.\n\n\"So, you were saying that sometimes . . . \"\n\n\"Sometimes, right after we eat, she goes back to the church. But I take her there.\"\n\n\"And do you stay and wait for her?\"\n\n\"No sir. She says: 'Mam\u00e0, come back an' get me in about a hour or hour an' a half.' An' so that's what I do.\"\n\nTeresi, choking on his cigar smoke, started coughing.\n\n\"Now listen. When you go back to get her, are there sometimes other people with her?\"\n\n\"Bah . . . sometimes there's another girl.\"\n\n\"How about boys?\"\n\n\"Boys? Never!\"\n\n\"And has Totina sometimes stayed at the church for more than an hour or hour and a half?\"\n\n\"No, not at the church. But there was that time when they went on a retreat for a half a day.\"\n\n*\n\nOn his way to the carabinieri station to talk to the captain, Teresi couldn't get the words Rosalia had said to the hospital doctor out of his head. _The penance is like the sin_.\n\nAnd all at once the meaning of those words flashed before him, blinding and paralyzing him to the point that he almost got run over by a carriage. He recovered only because he'd managed for a moment to obliterate that meaning from his mind. He didn't want to believe it.\n\n*\n\n\"If you've come here to talk to me about Marquis Cammarata, you know as well as I, Signor Teresi, that the matter is no longer in my hands,\" the captain said, to stay on the safe side.\n\n\"That's not why I came. And I thank you for agreeing to see me.\"\n\n\"What can I do for you?\"\n\n\"Have you been informed that Rosalia Pampina committed suicide this morning at dawn?\"\n\n\"Oh my God, no,\" said Montagnet. \"The poor girl!\"\n\nHe looked intently at Teresi.\n\n\"But how did you find out about this unlucky young woman?\"\n\n\"I committed an indiscretion.\"\n\n\"Did you read the report I'd left on my desk?\"\n\n\"Yes. I'm also a journalist, you know.\"\n\n\"I know. I read your articles, to keep myself informed. It's part of my job.\"\n\n\"Just to keep yourself informed?\"\n\nThe captain pretended not to have heard, and kept on talking.\n\n\"That girl . . . Rosalia . . . was kidnapped and repeatedly raped by the brigand Salamone. She was freed by Lieutenant Villasevaglios, but apparently she never\u2014\"\n\n\"First,\" said Teresi.\n\n\"I'm sorry, I don't understand.\"\n\n\"First she was raped by the brigand Salamone, and then, when she felt guilty about it, she requested permission to confess her sins to a priest.\"\n\n\"I know. I spoke to her parish priest.\"\n\n\"And what did he tell you?\"\n\n\"He said she arrived just as he was closing the church, insisted that she make a confession, and then left after she'd done this. But according to my research, she did not go straight back home, but returned about an hour and a half later. Since the woman she works for told me that as of that moment she no longer said a word and didn't want to eat or drink anything, there are two possibilities: either she didn't really start feeling the effects of the outrage done to her until after the confession, or, on her way home, had another unpleasant encounter that proved fatal to her.\"\n\n\" _Tertium non datur_?\" asked the lawyer.\n\n\"I don't see how . . . \"\n\n\"Captain, the hospital doctor told me that just the night before, Rosalia had seemed to be improving, to the point that she began to speak again. She repeated twice, in dialect, a statement that I'll translate for you: 'the penance is like the sin.'\"\n\nMontagnet looked at him, not knowing what to say.\n\n\"The penance is like the sin,\" he repeated softly.\n\nThen he understood.\n\nHe bolted upright and suddenly lost all his Piedmontese military aplomb.\n\n\"Oh, shit!\" he said.\n\nThen he sat back down and ran his hand over his brow.\n\n\"You must excuse me,\" he said, slightly embarrassed at having uttered an obscenity. \"If you don't mind . . . \" he said, undoing his tie and unfastening the top button of his shirt.\n\n\"Actually, that's not all I have to tell you,\" Teresi resumed. \"A short while ago I spoke to the mother of one of the mysteriously pregnant girls.\"\n\n\"Who?\"\n\n\"Her name is Totina; she's the daughter of don Anselmo Buttafava's overseer.\"\n\n\"Ah, yes, I know.\"\n\n\"Every Sunday Totina comes into town to attend Mass and then, afterwards, she sometimes goes back to the church and spends some time alone with the priest. She says her baby was conceived by the intercession of the Holy Spirit.\"\n\n\"More or less the same thing the other two girls I questioned today said,\" the captain observed. \"One of them told me it was the will of God, and the other said that the fruit of her womb was willed by the Lord.\"\n\n\"So, then, Captain, shall we add things up? All of these pregnant girls assiduously frequented their respective churches. The only men they ever met with face to face were their priests. So, what's in the _cavagna_?\"\n\n\"What is a _cavagna_?\"\n\n\"It's a small wicker basket shaped like a cannolo but closed at one end, whose only purpose is to contain a small amout of ricotta cheese. So let's try again, Captain. What's in the _cavagna_?\"\n\n\"Ricotta,\" Montagnet replied through clenched teeth.\n\n\"So we agree,\" said Teresi. \"But we have no proof.\"\n\n\"But there _is_ something we could do, just to get started,\" said the captain. \"The overseer's daughter is also two months pregnant, isn't she?\"\n\n\"Yes.\"\n\n\"Like the other three.\"\n\n\"Have you questioned the fourth girl as well?\"\n\n\"No. She's a legal adult. The matter is out of my hands. And I can't tell you her name. But there is a very specific question we must ask: What happened in the churches of Palizzolo two months ago?\"\n\n\"If we could only find out . . . \"\n\nMontagnet had an idea. He got up, opened the door, and went out. He returned five minutes later.\n\n\"I've just spoken with Marshal Sciabbarr\u00e0.\"\n\n\"Is he going to conduct an investigation?\"\n\n\"Unofficially. A sort of home-cooked investigation. His wife is very devout and goes to church every day.\"\n\n\"I'm sorry, Captain, but if his wife is so devout, maybe she's not the right person for the task.\"\n\n\"No, I think she's fine. She's fifty years old and she's . . . not very attractive.\"\n\n\"Let's assume the woman tells us what happened. Surely she will not have been present when it happened. So we will have an additional element, but still no concrete proof.\"\n\n\"That's true. But I don't think it's going to be so easy to find any concrete proof.\"\n\n\"So we'll have nothing to show for our efforts.\"\n\n\"Perhaps we should do something to alarm the culprits and then await their next move. Know what I mean?\"\n\n\"Perfectly. But what can we do to alarm them?\"\n\nMontagnet gave a sly smile.\n\n\"You're a journalist, aren't you?\"\n\n\"Wait a second. If I write what's really been happening I'll be hit with at least eight or nine lawsuits for libel.\"\n\n\"But who ever said you should write what's really been happening? It's up to you and your skill to hint, make allusions, let it be surmised that . . . You journalists are masters at that sort of thing. The point is to set the alarm bells ringing, that's all.\"\n\nThis was also true. There was a knock at the door, and Marshal Sciabbarr\u00e0 came in and gave a military salute. Only the captain asked questions.\n\n\"Did you speak with your wife?\"\n\n\"Yes, sir.\"\n\n\"What did she tell you?\"\n\n\"She said she'd heard mention, but she couldn't remember by whom, that there'd been a select gathering at the Benedictine convent, which has been empty for the past year.\"\n\n\"Did this event take place two months ago?\"\n\n\"More or less.\"\n\n\"What was its purpose?\"\n\n\"It was a kind of award given to the most devout women of the parish.\"\n\n\"What did it involve?\"\n\n\"A half day of spiritual exercises conducted by the priests from the parishes in town.\"\n\n\"And did the priests have the convent reopened for the specific purpose of this gathering?\"\n\n\"Yes, sir.\"\n\n\"Can you tell me anything else?\"\n\n\"No, sir.\"\n\n\"Forgive me, but I must ask you a personal question, Marshal.\"\n\n\"Yes, sir.\"\n\n\"How is it that your wife, who seems to me an extremely devout woman, wasn't invited to this gathering?\"\n\n\"The gathering was limited to young women only, married and unmarried, between the ages of sixteen and twenty-five.\"\n\n A Sicilian saying, the literal translation of which is: \"Bend, reed, until the spate passes.\" It means, basically, \"give in to adverse cirumstances while awaiting better times.\"\n\n Is there no third possibility? (Lat. loc.)\n\n## CHAPTER X \nTHE LAWYER LAYS A TRAP\n\nTeresi left the station well after Vespers had rung, and headed for Piazza Garibaldi, where the church of San Cono stood. When he got there the church door was already locked. He looked at his watch: almost seven o'clock. It was just then starting to get dark. The parish priest, Don Filiberto Cusa, had told the captain that Rosalia arrived just as he was closing up, and that she'd left just after confessing herself. Calculating that she'd taken half an hour to tell the priest what Salamone the brigand had done to her, it must still have been light outside when she came out. This made it unlikely she'd run into any troublemakers, as the captain had conjectured. That wasn't yet the hour for troublemakers to come out on the street. There were still too many people about at that time of day; all those who had fled because of the cholera scare were returning. The house of Giallonardo the notary was barely fifty yards away.\n\nThe grocery shop directly opposite the church was still open, and a man, perhaps the owner, was sitting on a wicker chair right beside the entrance . . .\n\nPerhaps he was also there the same accursed evening Rosalia went into the church? There was no harm in asking. He had nothing to lose. The sign over the shop entrance said: GERARDO PACE GROCER.\n\n\"Good evening, Signor Pace.\"\n\n\"Good evening,\" the man replied, looking confused.\n\nThere was nobody inside the store. On the counter Teresi saw three or four rounds of tumazzo and other cheeses, including a caciocavallo. It must have been a house specialty.\n\n\"I'm looking for some caciocavallo di Ragusa. My good friend Giallonardo, the notary, just told me you might have some.\"\n\nThe man stood up. He was fat and sweaty.\n\n\"Of course I've got some. I'm the only person in town who's got it.\"\n\nHe went into the store, with Teresi following behind.\n\n\"How much would you like?\"\n\nIt was best to get on his good side.\n\n\"A whole round, if that's all right.\"\n\nGerardo Pace's eyes glistened. He probably didn't do a great deal of business. Clearly this lone sale would make up for the whole day.\n\nAs the grocer was weighing the cheese, Teresi was racking his brains trying to figure out a way to broach the subject. But then Gerardo Pace asked him a question that took him by surprise.\n\n\"Do you know if there's any news about Rosalia?\"\n\nSince Teresi had said he was a close friend of the notary, it only stood to reason that . . .\n\n\"I'm very fond of that girl,\" the shopowner continued. \"She does all her food shopping here. What do they say at the hospital?\"\n\n\"They're not saying anything yet.\"\n\n\"I knew it was probably something serious! I walked her back to the notary's house after I saw her come out of the church.\"\n\n\"You saw her come out of the church in front?\"\n\n\"She didn't really come out of the front door, but out the little door to the side, the one that leads to the sacristy. And, believe me, the girl couldn't stand up! And she wouldn't talk. I asked her over and over: 'What's wrong, Rosal\u00ec?' And she'd say nothing! Poor kid, I felt so bad for her!\"\n\n\"Do you remember what time it was?\"\n\n\"It was probably round eight-twenty, something like that, 'cause I always close at eight-thirty, and I remember that after I walked her home, I came back here and closed up. Will you be needing anything else?\"\n\n\"Yes,\" Teresi said on impulse. \"Another whole round, this time of sweet provolone. And give me that leg of prosciutto as well.\"\n\n\"But how are you going to carry all this stuff? Want me to help you carry it home?\"\n\nSignor Pace would have attracted more attention than a brass band, walking him home with so much food.\n\n\"Tell you what. Please wrap it all up for me, I'll pay for it, and tomorrow morning my nephew will drop by and pick it all up. But tell me something. Where does Don Filiberto Cusa live?\"\n\n\"He's got three rooms above the sacristy. There's a wooden staircase leading straight up there from the sacristy.\"\n\n*\n\n\"Do you know Don Filiberto Cusa?\" Teresi asked Stefano as they were eating with Luigino, who by now was getting up out of bed whenever he felt like it. Dr. Palumbo had said that he could go home to Salsetto in two days' time.\n\n\"No. He doesn't know me and I don't know him. Who is he?\"\n\n\"The priest of San Cono parish. Do you know at least where the church is?\"\n\n\"Yes, that I know.\"\n\n\"Good. Do you have a piece of black cloth?\"\n\n\"I think so.\"\n\n\"Good. Cut a strip of it and sew it onto the left sleeve of your jacket.\"\n\n\"Mourning?\"\n\n\"Yes indeed. And if you have a black tie, put that on, too.\"\n\n\"So I'm in mourning.\"\n\n\"Yes, you are.\"\n\n\"So who died?\"\n\n\"Your cousin, Rosalia Pampina, the daughter of your mother's sister. She killed herself while she was staying at the hospital.\"\n\n\"Why'd the poor thing kill herself?\"\n\nTeresi told him everything about the girl, and even told him about the talk he'd had with the grocer.\n\n\"What Pace told me confirms the captain's and my suspicions. Rosalia suffered two sexual aggressions: first at the hands of the brigand Salamone, and second, at the hands of Patre Filiberto Cusa.\"\n\n\"Inside the church?\" asked Stefano, who couldn't bring himself to believe it.\n\n\"I found out that you can go upstairs to the priest's apartment directly from the sacristy. He must have taken her home.\"\n\n\"And what do you want me to say to the priest, all dressed up in mourning?\"\n\n\"I want you to wait for him to finish saying the Mass, then go into the sacristy and, observing him very carefully, tell him that Rosalia killed herself. Once he's swallowed this news, you must tell him you want to talk to him in private, in a safe place, because you have something important to tell him. Try to get him to take you upstairs to his apartment. Then, when you're alone, you'll reveal to him that the night before she threw herself out the window, Rosalia talked to you and told you everything. And say there was also a nurse present.\"\n\n\"Then what?\"\n\n\"Then you'll blackmail him. You'll say that, for starters, he must give you two thousand lire.\"\n\n\"And what if the guy's innocent and calls the carabinieri?\"\n\n\"He won't, rest assured. If, at any rate, that were to happen, I'll explain the whole thing to Captain Montagnet.\"\n\nAt this point Luigino, who hadn't uttered a word throughout, said:\n\n\"I want to go with Stefano.\"\n\n\"And who will you be?\"\n\n\"I'll be the nurse who heard what Rosalia said to her cousin. I'll be Stefano's accomplice. That way it'll all be more believable, I'm sure of it.\"\n\n\"All right,\" Teresi consented.\n\n\"And at what time should we go to the church?\"\n\n\"At six in the morning, for the first Mass.\"\n\n\"Shit, why so early?\"\n\n\"Because it's dangerous, Stefano. If, for example, Signora Giallonardo in the meantime goes and tells the priest that Rosalia is dead, we're screwed. Ah, and since there'll be two of you, there's a grocery shop right in front of the church, and I want you go there to pick up a round of caciocavallo and another of provolone, and a leg of prosciutto.\"\n\n*\n\nAt a quarter to six the following morning, the two young men left the house to go to church. Teresi accompanied them to an appointed intersection. He was too nervous to sit tight at home waiting for them; it would have driven him crazy.\n\nHe went to the Burruano pastry shop and scarfed down three ricotta cannoli fresh out of the oven. In fact he'd wanted to eat only one, but the aroma was so heavenly he couldn't resist. When he came back out he had the feeling that if someone stuck a finger down his throat, they would touch the creamy ricotta with which he'd filled his stomach.\n\n_If I don't drink some coffee right away, I'm going to get heartburn so bad it'll kill me_ , he thought.\n\nBut all the caf\u00e9s in town were still closed at that hour. He had no choice but to go home and make his own coffee. When he was done, he fired up a cigar and started wondering whether or not he should inform Montagnet of the trap he'd set for Don Filiberto. But he came to the conclusion that it would be best to talk to him afterwards and present him with a fait accompli. It was ninety-nine percent certain he wouldn't agree with the idea; he would say it was illegal.\n\nBut Teresi couldn't stay at home. He felt like he was suffocating. He glanced at his watch. An hour had gone by without him even noticing. He decided to leave, and as soon as he was out the door, he saw Stefano and Luigino at the far end of the street, returning home. He went back inside and, feeling his throat parched, drank a glass of water.\n\n\"It's done!\" Stefano cried out loudly.\n\nIt was all Teresi could do not to start dancing.\n\n\"Did he give you the money?\"\n\n\"No, Zio. He didn't have that much, which makes perfect sense. He said to come back later, around one, and he would have it for us.\"\n\n\"Tell me everything.\"\n\nStefano did the talking.\n\n\"When the priest went into the sacristy, we followed him and approached him just as he was taking off his vestments. As soon as he saw us he said: 'If this is going to take a while, please come back in an hour. I have to give last rites to a dying man.' I replied that it wouldn't take but a minute. 'Then go ahead and speak,' he said. But with a glance I let him understand that I didn't want to talk in front of the sacristan. He immediately got my message and ordered him to leave. As soon as it was just the three of us, I simply said: 'Rosalia killed herself.' He didn't say anything. Didn't ask when or where. Nothing. I got the impression he already knew. He leaned against the back of a chair with both hands, hung his head, and stayed that way for a minute, still without saying anything. I said I wanted to talk to him, but not in the sacristy, because other people might come in.\"\n\n\"And how did he react?\"\n\n\"Want to know something strange, Zio? He didn't even ask what I wanted to talk to him about. He just nodded 'yes,' and walked towards the staircase, still keeping his head down.\"\n\n\"So he already knew! I'd bet the family jewels he already knew!\" said Teresi.\n\n\"I was thinking the same thing,\" said Luigino.\n\n\"When we went upstairs, I told him that Rosalia had said what had happened first with Salamone and then with him. And when I finished, before I could even ask for the money, and without raising his head, he asked: 'How much?' I was so shocked I couldn't answer.\"\n\n\"So I answered for him,\" said Luigino. \"'Two thousand,' I said.\"\n\n\"And what did he say?\"\n\n\"He said simply: 'Come by later at one o'clock. I'll have the money for you. Now please leave by way of the sacristy door, and when you return, come back the same way.' And that was all. We went back down the stairs, but the priest stayed where he was.\"\n\nTeresi sat there, looking pensive.\n\n\"What is it, Zio?\"\n\n\"There's a problem that just occurred to me. From what you've just told me, it's clear the priest feels responsible for the girl's death. Caught by surprise, he agreed to give you the blackmail money. But can we trust him? If he talks about it with any other priests, they're sure to make him change his mind. And that's what could ruin us all. Or else he could change his mind on his own.\"\n\n\"And not give us the money?\"\n\n\"He might even give it to you. But when I weigh in by writing an article about the whole thing in my weekly, he can still claim you guys made the whole thing up, that you tried to blackmail him, but he didn't give you one lira because he had nothing whatsoever to do with Rosalia's death. And if he finds out that you, Stefano, are not Rosalia's cousin but my nephew, and on top of that, that Luigino has never worked as a nurse at Camporeale hospital, then all three of us will end up in jail.\"\n\n\"So, what should we do?\" asked Stefano.\n\n\"I'm going to tell the whole story to Montagnet. That should give us cover. Did you bring back the cheese and other stuff?\"\n\nStefano slapped himself in the forehead.\n\n\"Damn! We completely forgot!\"\n\n*\n\nTeresi dashed off to the carabinieri station, but Marshal Sciabbarr\u00e0 told him the captain had just left for Camporeale, where he'd been summoned to report to the provincial commander, Colonel Chiaramonte.\n\n\"Do you know when he'll be back?\"\n\n\"I can't really say.\"\n\n\"I'm sorry, but is it some sort of state secret?\"\n\n\"No, sir, but the fact is that the colonel summoned him to a meeting in the early afternoon, and the captain decided to take advantage of the situation to drop in and see his family.\"\n\nTeresi balked. Montagnet had a family? Seeing him always in uniform, with never a button out of place, elegant, impeccably groomed, inflexible, polite but aloof, Teresi had come to think of him as a kind of machine, not a man capable of the same feelings as other men.\n\n\"Is he married?\"\n\n\"Yes, and he has two children. The boy is seven, the girl five. Is there anything I should tell him when he returns?\"\n\n\"No, thank you, Marshal. I'll drop in again later.\"\n\n*\n\nSo, what could he dream up to help the time pass? He went and paid a call on Giallonardo the notary. He wanted to know what he and his wife had decided about Rosalia. And if Giallonardo asked him why he was so interested, he would reply that he wanted to write an article about it. But there was no need to ask anything.\n\n\"My husband's not in,\" said Signura Romilda.\n\nHer eyes were red. It was clear she'd been crying.\n\n\"When will he be back?\"\n\n\"He's gone to Camporeale to bring Rosalia back here. Did you know she killed herself?\"\n\nBig tears began to roll down her cheeks.\n\n\"Yes, I was told.\"\n\n\"I'm sorry, but we were very fond of her, my husband and I. She was a poor orphan girl. We took her in when she wasn't even ten years old, poor little thing. Tomorrow, since the funeral can't be held in a church, I'm going to have Don Filiberto give his benediction outside the church of San Cono. He was so fond of Rosalia himself! He was always saying how devout she was! How powerful her faith!\"\n\n\"And at what time will he give his blessing?\"\n\n\"Tomorrow morning at nine.\"\n\n\"I'll be there.\"\n\nHe wouldn't have missed Don Filiberto Cusa's blessing for Rosalia for all the gold in the world.\n\n*\n\nWhile stepping out of the notary's house, he heard someone calling him. It was don Anselmo.\n\n\"How are we coming along?\"\n\n\"On Totina's case?\"\n\n\"Of course!\"\n\nTeresi decided to tell him a lie to keep him in check.\n\n\"It couldn't have been the husband of the sister of the wife of your overseer.\"\n\nA complicated sentence, but he'd forgotten all those people's names, except for Totina's.\n\n\"Why not?\"\n\n\"It's true he's eighty years old, as you say, but to look at him you'd think he's at least ninety. The guy can barely even breathe.\"\n\n\"But have you seen him in person?\"\n\n\"Of course. With these two eyes. I always serve my clients honestly.\"\n\n\"But are you starting to get any ideas as to who it might have been?\"\n\n\"I'm gathering information, don Anselmo.\"\n\n\"Well, I'm telling you: if and when you find out who did it, I want to be the first to know.\"\n\n\"But why are you so keen to know?\"\n\n\"So I can shoot him.\"\n\n\"I'm sorry, but what has this got to do with you? You're not her father, husband or brother . . . \"\n\n\"You're right! But I'll shoot him just the same! Come on! I've been raising the kid for twenty years, buying her things, giving her money without telling my wife, and the girl could never spare me even a caress or a little peck on the cheek . . . And now the first son of a bitch to come along suddenly gets her pregnant?\"\n\n*\n\nTeresi made a plan. Go home, prepare a liter of chamomile tea, drink the whole thing, take a bath, change all his clothes because he was all sweaty, then go to the station at twelve-thirty and ask after Montagnet. If he happened to be in\u2014which was impossible because the colonel had summoned him for an early afternoon meeting\u2014he would tell him everything. If he wasn't, the only thing to do was to wait in front of the church, stop Stefano and Luigino when they arrived, and wait for Montagnet to return.\n\nThe lads weren't at home. Stefano's jacket with the mourning band on the sleeve was hanging from the coatrack. He would have to drop by to put it on. Beside it was the black tie. All at once, Teresi felt a chill run down his spine. _Matre santa_ , what a terrible mistake they'd made that morning! Good thing it was still early and there were no people out on the street. Because anyone who knew Stefano, seeing the youth on the street, dressed in mourning, would surely have asked him who in his family had died! Teresi went into the lad's bedroom, took an overcoat from the armoire, and brought it into the entrance hall. Then he did what he'd decided to do, and as he was coming out of the bathroom, he heard Stefano return. He got dressed in a hurry. It was half past twelve.\n\n\"Where's Luigino?\"\n\n\"He's waiting for me near the church.\"\n\n\"I'm going to the carabinieri station to see if Montagnet's there. And, listen: I want you to wear an overcoat. I put yours in the vestibule.\"\n\n\"Why?\"\n\nTeresi explained why.\n\n\"And if anyone asks why I'm wearing an overcoat?\"\n\n\"Tell them you have the flu. Everyone's been getting the flu in this town, so why can't you?\"\n\n*\n\n\"No, there's been no news from the captain.\"\n\nTeresi became discouraged. He was sure that Don Filiberto would give the money to Stefano, but also that as soon as he broke the story in his newssheet, the priest would claim that none of it was true, and that it was a scheme hatched up by the notorious anticlerical lawyer Matteo Teresi in cahoots with his nephew Stefano to drag the church's good name through the mud. His brain was telling him to dash over to San Cono and stop the two lads. But his instinct told him to let things take their course. His instinct won out.\n\nHe raced back home, got undressed down to his underpants, and lay down in bed with his head under the pillow.\n\nThen, a little while later, he heard the front door of the house open and close. He pulled his head out from under the pillow. He could hear the two youths in the kitchen, but they weren't talking or laughing. What had happened?\n\nHe went downstairs dressed just as he was. Stefano hadn't even taken his coat off and was sitting in a chair, drinking a glass of water. He looked pale. Luigino was also sitting down, his head in his hands.\n\nNeither of the two seemed to have noticed Teresi.\n\n\"So what happened?\"\n\nThey said nothing.\n\n\"Jesus Christ, would you tell me what happened?\" said Teresi, raising his voice.\n\n\"The priest hanged himself,\" said Luigino.\n\nTeresi felt the ground give out from under his feet. The trap he'd laid for the priest had worked all too well. Damn the moment he got that brilliant idea!\n\n\"Did anyone see you go in or come out?\"\n\n\"No.\"\n\n\"Tell me about it.\"\n\n\"We entered by way of the sacristy door, which was open,\" said Luigino. \"We went upstairs, and there he was, in the first room. Hanging from the ceiling. It was . . . ghastly. There was an envelope on the table.\"\n\n\"Did you take it?\"\n\n\"Yes. And I put it in Stefano's pocket. I had to literally drag him out of there. He was in shock and couldn't move.\"\n\nTeresi looked over at his nephew. The lad's eyes were open wide and staring into space. He went up to him, stuck his hand in the youth's pocket, took the envelope out, and opened it.\n\n_You won't get the money you wanted, because I was unable to find anyone to lend me such a sum. In exchange, I give you my confession. I abused Rosalia Pampina, my parishioner, for a long time, and in unnatural ways, making her believe that what we were doing were secret practices to ward off temptation and to allow her to remain pure until marriage. But the evening she came to confess about having been raped by Salamone the brigand, I don't know what got into me. What Rosalia said isn't exactly correct\u2014that is, that the penance is like the sin. In fact, the penance was worse than the sin. You can sell this letter of mine to a newspaper, if you like. They will surely pay you more than what you asked of me._\n\nThis was followed only by the man's signature.\n\n\"Do me a favor,\" Teresi said to Luigino. \"Go and look for Dr. Palumbo and bring him back here. I'm getting worried about Stefano.\"\n\n## CHAPTER XI \nAN INCONVENIENT DEATH\n\nIt was the sacristan, Virgilio Bellofiore, who discovered the body of Patre Filiberto, whereupon the whole town did a repeat\u2014that is, descended into a pandemonium almost exactly like the one unleashed on the day of don Anselmo's cholera.\n\nSpooked as he was, the sacristan, dashing out of the house, missed a step and rolled all the way down the staircase, smashing his nose. Then, picking himself up, he went out into the street with his face covered in blood and shouting desperately:\n\n\"Don Filiberto killed himself!\"\n\nThese words were quickly passed from mouth to mouth by hundreds of people. Those in the street repeated them to those at their windows, while those at their windows shouted them at those on their balconies, and those on their balconies yelled them at those on their terraces, while those on their terraces shouted them in turn at the wind, and the wind soon carried the news out into the countryside around Palizzolo.\n\nWhat happened next was that whoever was eating stopped eating; whoever was sleeping woke up; whoever was breastfeeding laid the crying baby down; whoever was working in the vegetable garden set the hoe aside; whoever was dying managed to stave off death; and whoever was making love stopped in the very midst.\n\nAnd all those who could do so ran towards the church of San Cono, filled up Piazza Garibaldi, and clogged the nearby streets.\n\n\"Is it really true he killed himself?\"\n\n\"Apparently.\"\n\n\"But is it true or not true?\"\n\n\"It's true.\"\n\n\"And how did he kill himself?\"\n\n\"With rat poison.\"\n\n\"He shot himself.\"\n\n\"He threw himself off the balcony.\"\n\n\"He inhaled the smoke from the bedwarmer.\"\n\n\"He hung himself from a ceiling rafter.\"\n\n\"He stabbed himself in the heart.\"\n\n\"But why?\"\n\n\"He'd gone crazy.\"\n\n\"He was a gambler. He'd lost a lot of money playing _zicchinetta_.\"\n\n\"Come on! The man had never seen a playing card in his life!\"\n\n\"He was sick.\"\n\n\"He had debts.\"\n\n\"He'd quarreled with the bishop.\"\n\n\"He didn't believe in God anymore.\"\n\n\"Did he leave any note?\"\n\n\"Nothing.\"\n\n\"What do you mean, nothing? When someone kills himself, he always leaves a note saying why!\"\n\n\"This whole thing is very strange!\"\n\n\"Extremely strange!\"\n\n\"Maybe he wrote to the bishop.\"\n\n\"Maybe he did write a letter, which they then destroyed.\"\n\n\"Who?\"\n\n\"I dunno! The sacristan, for one.\"\n\n\"And why would he destroy it?\"\n\n\"Maybe it said some compromising things.\"\n\n\"Bah!\"\n\n\"Now I'm having my doubts.\"\n\n\"And what if he didn't kill himself?\"\n\n\"What do you mean, didn't kill himself?\"\n\n\"I mean, what if he was killed and then they made it look like he killed himself?\"\n\n\"And why was the sacristan's face all bloody?\"\n\n\"Maybe he caught the killers in the act.\"\n\n\"Then why was he yelling that the priest had killed himself?\"\n\n\"'Cause they threatened him. They would've killed him too, if he didn't say what he did.\"\n\n\"That's bullshit!\"\n\n\"Who would have wanted to kill Don Filiberto?\"\n\n\"He didn't have any enemies.\"\n\n\"All he ever did was good.\"\n\n\"He helped everybody.\"\n\n\"He always had a good thing to say about everyone.\"\n\n\"He would take from himself to give to others!\"\n\n\"He was an honest man! A great man!\"\n\n\"A great man? He was a saint!\"\n\n\"A saint! A saint! A saint!\"\n\nGrowing more and more excited, the throng began to move forward, perhaps to go into the church to get a glimpse of the saint's mortal remains, or else to give vent to all the agitation they'd been subjected to of late, from the cholera scare to the arrest of Marquis Cammarata.\n\n\"Saint! Saint! Saint!\"\n\n\"Let's break down the door of the church!\"\n\n\"Let's grab the saint for ourselves!\"\n\n\"We'll have a procession and march him through town!\"\n\nThe six carabinieri who'd formed a cordon in front of the church started backpedaling.\n\nMarshal Sciabbarr\u00e0 felt lost. If the crazed mob actually did manage to get their hands on the corpse, they would surely tear it immediately to pieces, each trying to get his own personal relic.\n\nWithout thinking twice, he cocked his revolver and fired twice in the air. Everyone fled. Everyone, that is, except eighty-year-old _ragioniere_ Michele Orlando, who lay on the ground in the middle of the piazza, cut down by a heart attack.\n\n*\n\nThe sacristan, meanwhile, had raced over to the nearest church, which was San Giovanni. The main door was half closed. Dashing in, he nearly crashed into Don Alessio Terranova, the parish priest, who was just coming out to close up.\n\n\"Don Filiberto killed himself!\"\n\nDon Alessio froze with his left foot in midair, unable to complete his step.\n\n\"What the hell are you saying?\"\n\n\"He killed himself! Hung himself from a rafter! I saw him with my own eyes!\"\n\nDon Alessio set his left foot down.\n\n\"Did he leave any kind of written message?\"\n\n\"I didn't see anything! But it really spooked me!\"\n\n\"Go and wash your face!\"\n\nThese words took the sacristan by surprise. He didn't understand.\n\n\"What did you say?\"\n\n\"Wash your face. It's all covered with blood.\"\n\n\"I'll go into the sacristy.\"\n\n\"No, don't waste any time. Wash it right here, with the holy water in the baptismal font. Then go and tell Patre Raccuglia, Patre Scurria, Patre Samon\u00e0, Patre Marraf\u00e0, and Patre Pintacuda.\"\n\n\"You forgot Patre Dalli Cardillo.\"\n\n\"No, I didn't forget him. There's no need to go and talk to Patre Dalli Cardillo. But you must tell all the others to meet here, in no more than fifteen minutes.\"\n\n*\n\n\"Listen, Marshal, they told us at the courthouse that no judges are available at the moment.\"\n\n\"What do they mean 'at the moment'?\" Marshal Sciabbarr\u00e0 asked his colleague at the other end of the telephone line.\n\n\"They mean that before tomorrow no magistrate from Camporeale can come to Palizzolo.\"\n\n\"So I'm supposed to leave the priest dangling from the rafter until tomorrow?\"\n\n\"I have an idea. Cut the rope he's hanging from, and later, when they ask you about it, tell them you did it because you thought the priest was still alive.\"\n\n\"All right, but then what am I going to do with the corpse?\"\n\n\"Have somebody fashion a catafalque from the bedclothes and posts, then put it out on display in the church.\"\n\n\"What the hell are you saying?\"\n\n\"Why, what's wrong with that?\"\n\n\"If the bishop comes and sees it, he'll break my neck! Don Filiberto is excommunicated, since he killed himself!\"\n\n\"You're right. Wait, let me ask the captain.\"\n\nThree minutes went by during which Marshal Scibbarr\u00e0 damned his soul by dint of curses.\n\n\"Sciabbarr\u00e0? The captain wants to know if there are any chests in the sacristy.\"\n\n\"Yes, there are two or three.\"\n\n\"Wait just a second.\"\n\nThe marshal had all time he needed to utter every curse he knew.\n\n\"Sciabbarr\u00e0? The captain says you should bring the body down into the sacristy and put it temporarily in one of the chests.\"\n\n\"What about afterwards?\"\n\n\"We'll see about what to do afterwards. And don't let anyone into the sacristy.\"\n\n*\n\nHis Most Reverend Excellency Egilberto Martire, bishop of Camporeale, normally took a half-hour nap after eating. That day, before dozing off, he'd given an order to his staff.\n\n\"You mustn't wake me for any reason! I don't want to be bothered for anything, even if you start hearing the goddamn trumpets of the Apocalypse!\"\n\nTherefore his secretary, Don Marcantonio Panza, solved the problem by calling his second secretary, Don Costantino Perna.\n\n\"Listen, Don Costantino, I just now got a phone call from the mayor of Palizzolo. Apparently the priest of San Cono parish, Don Filiberto Cusa, has killed himself.\"\n\n\"Killed himself?! _O Madonna benedetta!_ How very strange! Are they sure?\"\n\n\"That's exactly why I've decided to go straight to Palizzolo myself. I'd like to confirm things in person. I'll inform His Excellency by telephone. And when he wakes up, you must tell him everything, but with the utmost caution.\"\n\n*\n\nWith the help of two carabinieri and Lance Corporal Magnacavallo, Marshal Scibbarr\u00e0 did what the captain had said and, just to be sure, not only covered the chest with the priestly vestments he'd found inside, but placed four heavy bronze candelabra on top as well. Then he left the lance corporal and carabinieri on guard outside the sacristy door, to prevent anyone from going in, and headed back to the station.\n\nHalf an hour later, the lance corporal found six priests he already knew standing before him.\n\n\"We've come to bless the mortal remains of our unfortunate brother,\" said a sorrowful Don Alessio Terranova, opening his cloak so the guard could see the aspergillum and basin he'd brought.\n\nLance Corporal Magnacavallo broke out in a cold sweat. Now what was he going to tell these priests? Could he possibly tell them they'd put the body inside a chest? Then he had an idea.\n\n\"He's no longer here.\"\n\n\"Then were is he?\"\n\n\"He was taken . . . to the station.\"\n\n\"And where is the train headed?\"\n\n\"No, sir, I meant to our station, the carabinieri compound. But you can't see him.\"\n\n\"And why not?\"\n\n\"I have no idea. By order of the judge in Camporeale.\"\n\nThe six priests stepped back and started conferring amongst themselves.\n\nThen Patre Pinta went back on the attack.\n\n\"We need to go into the home of our poor late brother.\"\n\n\"It's not possible. I have orders to\u2014\"\n\n\"You can't treat us this way!\" screeched Patre Marraf\u00e0.\n\n\"We're not common thieves! We're priests!\" shouted Patre Scurria.\n\n\"And you, corporal, you know us perfectly well! You know who we are!\" yelled Patre Raccuglia.\n\nThe windows of the house opposite opened, and some faces appeared.\n\nAll they needed was more chaos.\n\n\"All right, go on in,\" said the lance corporal.\n\n*\n\nFive minutes after the social club opened for the afternoon, as scheduled, at three o'clock, the salon was already mobbed. Giallonardo the notary was receiving the members' condolences as if he had been a relative of Don Filiberto.\n\n\"But, the last time you spoke to him, Signor Giallonardo, how did he seem?\" asked don Liborio Spart\u00e0, the president.\n\n\"Well, the last time . . . he started crying.\"\n\n\"Crying? Don Filiberto seemed like such a strong man . . . \"\n\n\"Thirty-nine years old, poor man!\" said Colonel Petrosillo.\n\n\"What's that got to do with anything? Thirty-nine years old or forty, the fact is the man was crying!\" don Anselmo Buttafava retorted.\n\n\"Gentlemen, I would like to clarify that it was a rather unusual occasion,\" Giallonardo resumed speaking.\n\n\"And what was that? Can you tell us?\" asked Professor Malatesta.\n\n\"It's no secret. When my housekeeper Rosalia killed herself the day before yesterday by throwing herself out a fourth-floor window at Camporeale hospital . . . \"\n\n\"Your housekeeper killed herself?\" asked don Stapino Vassallo.\n\n\"That's what I just said, isn't it?\"\n\n\"Yes, but why did she do it?\"\n\n\"Nobody knows.\"\n\n\"But would you just let him finish speaking without interruption?\" said don Serafino Labianca.\n\n\" . . . I went to see Don Filiberto,\" the notary resumed, \"and I asked him if he would be so kind as to give the dead girl benediction. He said yes, and then started crying.\"\n\n\"But the question remains: Why did he start crying?\" asked don Serafino.\n\n\"Rosalia was a parishioner of his.\"\n\n\"But, my good notary, if a priest cried over every one of his parishioners who died, he'd go blind in a month, believe me.\"\n\n\"But he was particularly fond of Rosalia!\"\n\n\"Oh, was he?\"\n\n\"Yes, he was! He cared a great deal for her and admired her. He always used to talk about what a good girl she was, so respectful and devout . . . He would often keep her a long time in the sacristy . . . \"\n\n\"In the sacristy?\" President Spart\u00e0 repeated.\n\n\"Yes, what's so strange about that? Isn't catechism taught in the sacristy?\"\n\n\"Bah!\" said don Serafino.\n\n\"And what is 'bah' supposed to mean?\"\n\n\"It means, Mr. Notary, that two plus two makes four!\"\n\n\"I agree!\" Colonel Petrosillo chimed in.\n\n\"But you agree with what, exactly?\"\n\n\"Mr. Notary, it's quite simple: Don Filiberto killed himself because he was in love with Rosalia,\" don Serafino said bluntly.\n\n\"And Rosalia killed herself because she herself was in love with Don Filiberto!\" the colonel said, smiling. \"An impossible love!\"\n\n\"Colonel, you know as well as anybody that there's no such thing as an impossible love,\" said don Anselmo.\n\nThe colonel took umbrage.\n\n\"And just what are you insinuating?\"\n\n\"I'm merely saying that if they loved each other so much, the priest could easily have taken his frock off and hooked up with the girl. It certainly wouldn't have been the first time, nor the last!\"\n\n\"Ah, the flesh is weak!\" the colonel sighed.\n\n\"And yet,\" said President Spart\u00e0, \"we mustn't necessarily dismiss the possibility that they were in love. Was Rosalia by any chance pregnant?\"\n\n\"Oh, stop speaking twaddle!\" the notary snapped. \"Patre Filibeto was a saint, just as people say.\"\n\n\"Sainthood and earthly love can easily coexist,\" the colonel proclaimed.\n\n*\n\nAn hour later another meeting was held at the castle of Duke Ruggero d'Altomonte. Except for Marquis Cammarata, all the local nobles were there.\n\n\"What's this about the priest of San Cono parish?\" asked Baron Roccamena.\n\n\"Just now at the club, people were saying that he killed himself because he was in love with a girl and got her pregnant,\" Baron Piscopo replied.\n\nHearing the word _pregnant_ , Baron Lo Mascolo turned pale.\n\n\"Why did you summon us here?\" the Baron Roccamena asked Marquis Spinotta.\n\n\"Because the other day you asked me to telephone my cousin, Duke Simone Loreto di San Loreto.\"\n\n\"And did you?\"\n\n\"Of course I did.\"\n\n\"And what did the duke say?\"\n\n\"He said he would look into the matter immediately. And indeed he called me back just two hours ago.\"\n\nHe paused for effect. And amidst the silence one could hear the hoarse voice of Duke Ruggero in the background saying:\n\n\"It's all the fault of the French Revolution!\"\n\n\"And so?\" Baron Roccamena pressed the marquis.\n\n\"He told me the provincial commander of the carabinieri, Colonel Chiaramonte, has summoned Captain Montagnet to tell him he must return immediately to Camporeale. So now we've finally got him out of our hair, once and for all,\" the marquis concluded, to the general exultation of all present.\n\n*\n\nThey didn't know, however, that Captain Montagnet was in fact on his way back to Palizzolo.\n\nWhat had happened was that around three o'clock that afternoon, as the captain was waiting for the colonel's call, another phone call came in, this one from Marshal Sciabbarr\u00e0.\n\n\"Ciaramiddaro, I urgently need to speak with Captain Montagnet.\"\n\n\"It's not possible. He's in the colonel's antechamber.\"\n\n\"Is the adjutant Sinibaldi there?\"\n\n\"Yes, I'll put him on.\"\n\n\"Hello, Sciabbarr\u00e0, how are you?\"\n\n\"Major, sir, Captain Montagnet at the moment is in the colonel's anteroom. I need to inform him that the situation here in Palizzolo is becoming difficult again.\"\n\n\"How?\"\n\n\"Don Filiberto Cusa, a parish priest, has killed himself.\"\n\n\"So what?\"\n\n\"There've been clashes between some of Don Filiberto's parishioners and people from other churches in town. The latter group claims that Don Filiberto seduced a young female parishioner of his, and Don Filiberto's faithful are up in arms. So far we've had two stabbings. So far.\"\n\n\"Do you fear further complications?\"\n\n\"As surely as death.\"\n\n\"All right, thanks.\"\n\nThe adjutant knew already what the colonel was going to say to Montagnet, and so, instead of speaking with the captain, he thought it best to mention the phone call directly to Commander Chiaramonte.\n\nAs a result, when he was finally received, the captain was told by the colonel that, although the order had come from \"higher up\" for him to return at once to Camporeale, the situation had at Palizzolo had changed again, due to the priest's suicide, and so he was granted a week's extension.\n\n*\n\nBefore leaving for Palizzolo, Don Marcantonio Panza had obtained from the courts of Camporeale a document written and signed by President Onorio Labarbera, which went as follows: \"Don Marcantonio Panza, secretary to His Excellency Egilberto Martire, bishop of Camporeale, is hereby granted full access to the Church of San Cono and adjacent rooms (sacristy, priory, etc.), to allow him to catalogue all objects belonging to the late Don Filiberto Cusa and arrange for the shipment of said objects to the priest's family.\"\n\nUpon arrival, Don Marcantonio presented the document to Lance Corporal Magnacavallo, who let him in. But less than five minutes later, the guard heard the envoy call him from inside the sacristy.\n\n\"Corporal, please come here for a moment.\"\n\nThe young man shuddered. The envoy had probably discovered the corpse inside the chest! But that wasn't the case. The chest was just as they'd left it.\n\n\"Would you please follow me?\" said the priest.\n\nFollowing behind him, he climbed the wooden staircase and entered the room where Don Filiberto had hanged himself.\n\nThe lance corporal froze in the doorway. It looked as if a cyclone had torn through the room. Drawers, cabinets, glass cupboards, and everything else had been opened, and all their contents thrown onto the floor.\n\n\"Go and have a look in the other room and the bedroom.\"\n\nIn the second room, a desk had been overturned, its feet now in the air, drawers open and totally empty. The parish's registers, papers, and documents were all gone. There wasn't a single sheet of paper to be seen anywhere.\n\nIn the bedroom, even the mattresses had been torn open and gutted.\n\n\"What reason could your people have possibly had to do all this?\" asked Don Marcantonio.\n\n\"My people?!\" the corporal began, trembling with rage.\n\n\"Who, then?\"\n\n\"It was those other priests, whom I, like an idiot, was stupid enough to let in here!\"\n\nThey went back into the first room.\n\n\"Are you sure about what you just said?\"\n\n\"And what was that?\"\n\n\"That it was the other parish priests who took all the papers away.\"\n\n\"Yes, sir, I'm absolutely certain. And I'm going to report it immediately to the marshal.\"\n\nDon Marcantonio looked up and cut the remaining rope still dangling from the rafter.\n\n\"Where have you taken him?\"\n\n\"To the carabinieri station.\"\n\n\"You were right to do so. The body won't be allowed into a church, and therefore cannot have a funeral mass said for it, and cannot be buried in consecrated land.\"\n\nThe lance corporal made such a bewildered face that Don Marcantonio couldn't help but notice. He threw up his hands.\n\n\"Are you sorry? There are rules, however, and they must be respected. Suicide is an act against God.\"\n\n\"And what about Count Mortillaro?\"\n\nThe lance corporal bit his lip. The question had just slipped out. Two years earlier, Count Mortillaro had shot himself in the head. He'd been given a solemn funeral and buried in the family vault.\n\n\"That was a very different case,\" Don Marcantonio said brusquely.\n\nWant to bet, thought Lance Corporal Magnacavallo, that they would end up having to take the dead priest to the station after all, and hide him in the closet?\n\n*\n\nMatteo Teresi was at home, thinking about the article he had to write that night, when a carabiniere came to tell him that the captain wanted urgently to see him.\n\n\"What's your part in all this confusion?\" was Montagnet's first question.\n\n\"Well, Captain, you'd suggested that I write an insinuating article, but in the meantime I was lucky enough to run into a witness, someone who'd seen Rosalia come out of the church after eight o'clock, and so . . . \"\n\nHe told him everything, even about the blackmailing charade. As he was talking, the captain's face turned darker and darker.\n\n\"I ought to arrest you for disturbing the peace. And this time I wouldn't be wrong, as I was with Dr. Bellanca. However, I believe you're right.\"\n\n\"About what?\"\n\n\"Just after the news of the suicide began to spread, six parish priests dashed over to Don Filiberto's residence, turned the place upside down, and took away all the papers they found there. Who knows what they were looking for.\"\n\n\"They were looking for this,\" said Teresi, taking out the letter written by Don Cusa and setting it down on the desk. \n\n## CHAPTER XII \nFOUR ARTICLES, TWO MONOLOGUES, AND ONE DIALOGUE\n\nTwo days after the the death of the parish priest of San Cono, Matteo Teresi published in his newssheet an article he'd written after coming to an agreement with Captain Montagnet, the title of which was \"The Penance Is Like the Sin,\" with, as subhead, \"The Truth on the Suicide of Don Cusa.\"\n\nIt went as follows:\n\n_There have been many diverse and conflicting rumors circulating among the population of Palizzolo (and among those of the nearby towns, even in Camporeale, the provincial capital) concerning the reasons that may have driven thirty-nine-year-old Don Filiberto Cusa, priest of the local parish of San Cono, to commit the tragic act that has created such a stir._\n\n_We are now able to reveal to our readers the truth of the matter, thanks to a handwritten letter from Don Cusa himself, drafted just minutes before he took his life. We were able to read this letter before turning it dutifully over, as we have done, to the proper authorities at the Court of Camporeale._\n\n_In just a few brief lines, Don Cusa confesses to having deceived, over a certain period of time, a na\u00efve young member of his parish, Rosalia P., subjecting her to such unnatural practices as masturbation and fellatio, which he presented as magical religious rites designed to protect the young woman from the temptations of the flesh. We will not dwell here on the tawdry details._\n\n_On the day when a rumor spread throughout Palizzolo that cholera had descended upon the town, the young woman fled to the countryside with two female friends. But during the night the three women had the misfortune of crossing paths with the noted brigand Salamone, who set upon Rosalia with particular ferocity and at great length, keeping her prisoner for an entire night and the following morning, until he was captured by the valorous Lieutenant of the Royal Carabinieri Rodolfo Villasevaglios._\n\n_Returning that same day to her place of residence, where she worked as a housemaid (but was treated like a daughter), the young woman asked that evening for permission to go to the church of San Cono, where Vespers had just rung, so she could meet with Don Filiberto. After hearing the girl's confession, and her description of what the brigand had put her through, the priest, blinded by his passions, convinced her to come with him into the sacristy and then to his apartment upstairs, where he subjected her to a series of \"penances\" that were in no way any less cruel and ferocious than the turpitudes of the brigand Salamone. When she came back out through the sacristy door an hour and a half later, shaken and upset, Rosalia was aided and escorted back to her residence by an acquaintance. As of that moment, she refused to speak, eat, or drink._\n\n_Dr. Palumbo of Palizzolo was promptly summoned to examine the young woman, and after administering first aid decided it was best for her to be admitted to Camporeale hospital._\n\n_After verifying the terrible abuse the girl had undergone and assessing her mental state, the hospital's chief physician dutifully reported the matter to the Royal Carabinieri. The investigation was assigned to Captain Eugenio Montagnet, who noticed, after questioning Don Filiberto, that the priest's claim to have seen Rosalia leave the church right after her confession was inconsistent with the time of the girl's return home, at 8:30_ P.M. _, as reported by her patroness. The testimony of the acquaintance who had come to her aid instead indicated that Rosalia had remained inside the church until that time\u2014a period lasting about an hour and a half. This was as far as the investigation had got at the moment when the unhappy Rosalia unexpectedly took her own life, throwing herself out of a fourth-floor window of the hospital where she was staying. The previous evening, however, she had in fact resumed talking, only to utter, in the presence of the doctor and the nurse, this terrible statement: \"The penance is like the sin.\"_\n\n_The horrific meaning of these words will surely not escape our readers' comprehension._\n\n_At this point Captain Montagnet resorted to a strategy to corner the priest. Finding himself with no way out, and gripped by remorse, the priest decided to take his own life._\n\n_Don Filiberto's mortal remains have been reclaimed by his brother, Orazio, who lives in Quattrocastagni._\n\n_Such, then, are the facts concerning this suicide._\n\n_There is, however, another episode, in itself quite alarming, which has come to our attention. Shortly after the news of Don Filiberto's tragic death began to spread, the priests of the other parishes of Palizzolo (namely, Don Alessio Terranova, Don Eriberto Raccuglia, Don Alighiero Scurria, Don Libertino Samon\u00e0, Don Angelo Marraf\u00e0, and Don Ernesto Pintacuda), with the sole exception of Don Mariano Dalli Cardillo, priest of the parish of the SS. Crocefisso, presented themselves to the lance corporal of the Royal Carabinieri assigned to guard the door to the sacristy, and asked to be granted entry to Don Filiberto's apartment in order to bless his mortal remains. Upon being categorically refused, the priests began to raise such a row that the carabiniere, to avoid further tension, let them in. The six priests spent a good deal of time, unsupervised, in the apartment, and then left. Shortly thereafter, Don Marcantonio Panza, secretary of His Most Reverend Excellency, Bishop Egilberto Martire, equipped with a lawful authorization from the Court of Camporeale, appeared before the same lance corporal. Upon going upstairs into the late Don Filiberto Cusa's apartment, however, he immediately summoned the corporal into the residence and showed him that the apartment had been turned upside down, apparently as the result of a frantic search, and that every document, including private letters, receipts and the like, had been removed, apparently by the other priests._\n\n_Upon learning that the secretary of His Excellency the Bishop was in Palizzolo, Captain Montagnet held a meeting with the emissary in the course of which he informed him that he intended to request the authorization of the Court of Camporeale to institute legal action against the six priests for theft of materials placed under restriction by the authority of the judiciary._\n\n_We will keep our readers informed as this case develops._\n\n_A few questions remain, however. What were the reverend fathers looking for in their confr\u00e8re's home? Were they perhaps afraid that Don Filiberto had left behind some compromising materials? And, if so, compromising for whom?_\n\n*\n\n\"What a fine-looking bunch you are, all six of you! Young, healthy, strong, vivacious, full of fire, initiative, and will to live . . . Real soldiers of Christ! Good for you! The problem is you haven't got a goddamn thing in your heads! Now, I've summoned you here to tell you something really quick. But let me preface by saying that, though my family name may be Martire, I have no desire whatsoever to become a martyr for your sakes. Got that? I have to tell you that this morning I got a phone call from the Presiding Judge of the Court, the eminent Commendatore Onorio Laberbera, who is such a pants-shitting coward . . . Anyway, he says to me: You must understand my position, your excellency, I cannot ignore Captain Montagnet's request . . . I have no choice but to grant him authorization and so on and so forth . . . And so, in the end I said to him: Who's asking you for anything? The law must take its course, says he. So I says: Then let it take this famous course! Do you understand what that means? If they have to arrest you, they're gonna arrest you. All of you. An' I won't lift a finger. I don't want any trouble. You break something, you fix it yourselves. What need was there for all six of you to trudge over there to that wretch's home? One of you woulda been enough. You have a look at whatever there is to look at, you take whatever there is to take, and you leave everything just the way it was. Nice and neat. But since you're all stupid young shits, you broke all the eggs. You even took the parish registers! What the hell were you looking for? Wait! Don't tell me! Don't tell me! I don't want to know! That's your goddamn business, not mine. You've all got pumpkins for heads! For now, all I'm gonna say is that, starting tomorrow, you'll all be replaced by other priests from the diocese, at least until this whole story blows over. Only Don Dalli Cardillo will keep his position. No! Not a word! Cuz if you start talking I'll kick your asses from here to kingdom come! Now, get the hell outta here, all of you! Hop to it!\"\n\n*\n\nTwo days after the first article, Teresi wrote another, which he published in a special one-page edition of his newsheet.\n\nThe article was entitled _Let's Venture an Hypothesis_.\n\nIt went as follows:\n\n_We know from an unimpeachable source that His Excellency, the Most Reverend Egilberto Martire, bishop of Camporeale, when faced with the request to institute legal proceedings against six Palizzolo parish priests for removing and absconding with documents under legal sequester (a crime calling for the arrest of the culprit(s)), declared his willingness not to obstruct the pursuit of Justice, adding that he has relieved the six priests temporarily of their duties, replacing them with six other priests from the diocese who will lead their respective parishes. Such a gesture once again underscores the great wisdom of His Excellency the Bishop, which we have seen at work on other occasions of considerably less gravity._\n\n_What the bishop refrained from doing was instead done by the Camporeale correspondent for the main newspaper of Sicily, who fiercely defended the actions of the six priests, asserting that it was fully within their rights to remove the parish documents so that the activities of the parish might not suffer any interruptions or slowdowns due to the tragic death of their spiritual leader._\n\n_But, if this were really the case, what need was there to deceive the carabiniere on duty? Perhaps all the six priests had to do was to say this, in order to persuade the officer of the law, who then would no doubt have gladly accompanied them into the late Don Filiberto's apartment and requested and obtained a proper receipt for any registers taken away._\n\n_Or else they could have addressed the Court (as indeed the bishop's secretary, Don Marcantonio Panza, had done) to request and obtain the necessary authorization._\n\n_But the six priests did not proceed in this fashion, my distinguished colleague in journalism. They didn't want any prying eyes to see them as they ransacked the apartment._\n\n_And even if they now are anxiously declaring that they have turned over to Captain Montagnet everything they spirited away, hidden under their robes, what guarantee do we have that this is true? And if it happens that they have not returned absolutely everything, what might these priests have wanted to keep for themselves?_\n\n_In my capacity as a journalist, I have conducted a modest investigation that has yielded an interesting result. I requested from the Court, and obtained, permission to enter the late priest's domicile. Nothing had been touched since the ransacking. The apartment was still in a state of indescribable disarray, such as the the six priests left it after their visit. In one corner of the dining room stood a painter's easel, beside a small overturned table that had once held the paints and brushes now scattered across the floor. Don Filiberto was indeed known as an amateur painter, and a number of paintings of religious subjects were hanging on the walls of his apartment._\n\n_This triggered a suspicion in me. During a conversation with the sacristan, Virgilio Bellofiore, I found my suspicion confirmed. Signor Bellofiore told me that Don Filiberto used to carry around some notebook pages on which he would sketch in pencil the things that struck him most over the course of the day, and he usually would store these drawings in the drawers of his desk. The priest's daytime housekeeper, Signora Amelia Putifarro, also confirmed this fact._\n\n_Let's venture a hypothesis: Is it possible that the six priests were not looking for documents at all, but for compromising drawings? Perhaps the drawings he kept in his drawers were not compromising, but Don Filiberto may have hidden other more salacious ones in more secret places in his apartment. This would explain the need to search everywhere in his home._\n\n_I have dutifully brought this hypothesis to the attention of Captain Montagnet. So far, however, the captain has not deemed it necessary to arrest the six priests._\n\n_It is therefore with some regret that I must conclude that our hypothesis is henceforth almost certainly destined to remain nothing more than that, since by now it is unlikely any trace remains of such drawings._\n\n__\n\n_*_\n\n\"This is the first time I've come to confess to you, Patre Dalli Cardillo. I used to always go to the Church of the Heart of Jesus and confess to Patre Alighiero Scurria, but I don't want to do that anymore. I need more than just an absolution, Don Mariano. I need some advice. I haven't been able to sleep for the past few nights, ever since I read in the paper that Rosalia Pampina killed herself over what Don Filiberto did to her. I met Rosalia a little more than a couple of months ago, when the priests took us out to the Benedictine convent, which was empty, for a day of retreat and spiritual exercises. There was me, Rosalia, Baron Lo Mascolo's daughter Antonietta, the daughter of don Anselmo Buttafava's overseer, Totina, Marquis Cammarata's daughter Paolina, Lorenza Spagna, who was the youngest of all of us, since she's only fifteen and a half, and Filippa Lanza, who's the daughter of the bank president. There was one of us from each church, chosen by her own parish priest. I'm a widow, and I'm twenty-four years old and have no children. Ever since my husband died I've really been missing him and suffering a lot, and I confessed to Patre Scurria that I often have naughty dreams, and sometimes I touch myself . . . And he told me he would do an exorcism on me, which we would have to renew once a week, and that would keep me pure.\n\n\"He had me look at an ancient book, all written in Latin, with pictures. One showing a devil doing it with a naked lady . . . An' he explained that when I touch myself, though I think I'm alone, the devil is always there, taking me like he's taking the woman in the picture. An' he also said that it wasn't the thing itself that was a sin, but the intention you do it with. An' if the intention is right, it can change the sin into a purification. Anyway, he convinced me. An' then there was the day of spiritual exercises. With all the consecrated wine we were drinking, we all got drunk. An' two hours later we were all naked, men and women alike . . . An' as soon as one priest was finished with one of us, another would pick her up . . . As for Rosalia, Patre Filiberto ordered everyone to preserve her virginity, but for all the rest of us . . . Anyway, so they got me pregnant, and a few other girls, too, I'm sure. An' it's true what the newspaper says: Don Filiberto was fucking and drawing pictures. I'm so mad, Don Mariano, so mad and desperate. They took advantage of me and my trust, my honesty, and my faith most of all. Now I've got a baby in my belly an' I don't even know who the father is, 'cause they all had a turn with me. I read in the paper today about the drawings . . . An' I had an idea: I'm gonna go to the Carabinieri an' tell 'em everything, I thought. An' if they don't believe me, I'll tell 'em Patre Scurria's got a red spot on his bum, Patre Raccuglia's got a great big wart under his belly button, and Don Libertino\u2014 Wha', Father? That's enough, you say? Okay, I'll stop. What are you doing, Don Mariano, are you crying? I know how you feel! You're the only real priest in this town! But what do you think I should do? Should I go to the Carabinieri?\"\n\n*\n\n_Just two days after our special edition, we find ourselves again faced with the need to publish another, to inform our readers of the incredible developments emerging in Carabinieri Captain Montagnet's investigation into the actions of the six Palizzolo priests, who are, by name: Don Alessio Terranova, Don Eriberto Raccuglia, Don Alighiero Scurria, Don Libertino Samon\u00e0, Don Angelo Marraf\u00e0, and Don Ernesto Pintacuda._\n\n_They were all arrested yesterday evening, not only for removing documents under sequester and absconding with them, but for far more serious charges, including the sexual assault and rape of seven women belonging to their parishes (including no fewer than three minors!), whom they morally subjugated, through dubious \"purification rites,\" into consenting to their lusts. Indeed, so subjugated have these women been, that those who are now pregnant continue to claim that the infants in their wombs are the work of the Holy Spirit or the will of God. In short, our fine parish priests had created a veritable sect\u2014which we'll here call, ironically, \"The Sect of Angels\"\u2014in which they passed off patently obscene acts as mystical religious rites._\n\n_This all reached an apex of depravity a little more than two months ago in a group orgy (which the priests called a \"spiritual retreat\"), lasting an entire day, at the Benedictine Convent, which was reopened for the occasion. The profusion of spirituality yielded concrete results: four of the seven women participating in the retreat came away pregnant. And thus these Fathers became fathers! Except that none of the four women (including two of the three minors) will ever know which priest sired her child, since they were all subjected to abuse by more than one priest that day._\n\n_Upon hearing the news that the six priests had admitted guilt to all of the crimes with which they were charged, and further explained that they had ransacked Don Cusa's apartment to remove some sketches bearing witness to their orgy\u2014as I had earlier conjectured\u2014His Excellency the Bishop of Camporeale suspended them of their religious duties \"a divinis.\"_\n\n_What does my eminent journalistic colleague, who defended them with sword drawn in Sicily's most important newspaper, think of all this?_\n\n_On the matter of these drawings, a clarification is in order._\n\n_The priests did not find them in their search, despite turning the apartment upside down looking for them._\n\n_They were located by Captain Montagnet in a hollow carved into the hearth of Don Filiberto's kitchen and then covered up with an earthenware tile._\n\n_And in the detailed, meticulous rendering of the faces, they drawings serve as incontrovertible evidence of the priests' guilt._\n\nMatteo Teresi then published a fourth article on the matter in the regular edition, rather than a special one, of his newssheet, which went by the name of _The Battle_.\n\nBy this point there was no longer anything so special about any of it. Or so, at least, he believed.\n\n_On more than one occasion, I, and those associated with this journal, have been accused of being instigators, subversives, and prejudiced anticlerics._\n\n_I would like to point out to our readers that on the occasion of the ridiculous cholera outbreak, which proved nonexistent, I was fingered by seven of the eight pulpits of Palizzolo as the sole person responsible for the supposed epidemic._\n\n_I was guilty, said the priests, of having brought the wrath of God upon our town._\n\n_There was even a priest who organized and personally led an attack upon my home, which was fortunately aborted. Yet their declared purpose was to kill the devil, whom I had supposedly come to incarnate._\n\n_Even now, after the priests have fully admitted to committing their odious acts, malicious rumors about me persist, insinuating that the whole affair was an underhanded maneuver motivated by my insatiable hatred of the Church!_\n\n_And that is not all._\n\n_There is even one person who has dared write that an \"accurs\u00e8d alliance\" [sic!] has been created in Palizzolo between a freebooting lawyer who disparages all that is sacred in his quest for notoriety and an officer of the Carabinieri who has been granted too much freedom of action and has exploited this fact to take measures well beyond the limits of his legitimate duties._\n\n_In other words, Captain Montagnet and the present writer are engaged in an iniquitous conspiracy._\n\n_Another has claimed that the captain's manner of proceeding has actually been dictated by the disdain the Piedmontese feel towards Sicilians._\n\n_Pure humbug._\n\n_What is, in essence, being absurdly and blindly maintained in such accusations is the premise that the lawyer and the officer of the law are in cahoots to deliver a mortal blow to the summit of our social system, as represented by the Aristocracy and the Church._\n\n_The supposed attack upon the Aristocracy\u2014let us not forget\u2014is represented by the \"unjust\" (!) arrest of the Marquis Cammarata and the public outcry said to have been purposely created by the very manner in which the marquis was arrested._\n\n_What the rumormongers omit from their story, however, is that the clamor arose directly from the actions of the marquis's own family, who began shouting obscenities at the carabinieri carrying out the arrest, and from the behavior of Marquis Cammarata himself, who, though handcuffed, managed, in a surge of bestial rage, to bite the ear of the carabiniere marshal and draw blood._\n\n_And they omitted one scarcely negligible detail above all, which is that the marquis himself has confessed to the crime of attempted murder with the complicity of a noted local Mafia chieftain still at large._\n\n_Captain Montagnet has therefore done nothing more than fulfill his duty. Conscientiously, I would add. With little fear of anyone. As is the custom for those who have the honor of belonging to the Royal Order of Carabinieri._\n\n_As for the accusation of an attack on the Church, let us say, once and for all: Enough!_\n\n_To this end, I present below, verbatim, the indignant words an illustrious priest, Don Luigi Sturzo, has written for_ Il Sole del Mezzogiorno _, the newspaper published in Palermo, in its July 15\u201316, 1901, edition:_\n\n__\n\nReaders may not know that in the town of Palizzolo, there exists a sect, derisively said to be \"of angels,\" composed of a number of degenerate priests unworthy of their holy ministry, unworthy even to be called men. These sectarians, resorting to mystical Gnostic precepts and abusing the Holy Sacrament of Confession, have taken to misleading a number of their female penitents into believing that the ignominious acts into which they've been initiated are conduits of divine grace and paths to the highest degrees of perfection. This sect is enveloped in a veil of extreme mystery; the sectarian priests pretend to be men of prayer, while the most sanctimonious of their female parishioners are the most assiduous participants in the long, all-too-long, rituals of piety practiced right there in church.\n\nThe delivery of these priests into the hands of justice for corruption of minors has shed light on the shameful sect of Palizzolo and revealed to all its secret purpose.\n\n__\n\n_I have only one thing to add to Don Luigi Sturzo's statement. At a certain point he writes that the priests were acting in accordance with \"mystical Gnostic precepts.\" In so doing, Don Luigi to some degree ennobles them. Whereas they haven't acted in accordance with any precepts, or even basic human decency!_\n\n_The Palizzolo scandal is beginning to resonate across the entire country. A number of high-ranking politicians, such as Turani, Tasca, and others, have even weighed in on the question. We, however, prefer to bring to your attention only the words of a priest of Don Luigi Sturzo's stature, as we believe that such words, coming from such a source, vindicate us against all malicious gossip and base insinuations._\n\n_There is, therefore, no conspiracy. Only a love of Truth and Justice._\n\n*\n\n\"I was told you're going back to Camporeale this evening, so I've come to say goodbye.\"\n\n\"Thank you, Signor Teresi.\"\n\n\"Captain, if I may: Since I go rather often to Camporeale, I would like to call on you every now and then. Why are you laughing?\"\n\n\"I just now received a telephone call from my commander. He told me there's a big surprise in store for me, which he'll reveal to me tomorrow, when I return to headquarters. Except that it won't be a surprise to me, since I already know what it is. I'm going to be promoted and transferred.\"\n\n\" _Promoveatur ut amoveatur_.\"\n\n\"Exactly.\"\n\n\"I'm not sure whether I should congratulate you or be disappointed.\"\n\n\"You can do both. Oh, and listen: I also read that article attacking us and insinuating that we were plotting together . . . \"\n\n\"Ignoble.\"\n\n\"Exactly. I just hope my uncle doesn't read it, since it was written in a national daily. He's getting on in years, and it would upset him greatly.\"\n\n\"Excuse me for asking, Captain, but who is your\u2014\"\n\n\"Oh, just a country priest. I lost my father before I was ten. We were poor, and it was this uncle who brought me up and saw me through my studies . . . Everything I have I owe to him, even my character. Well, I'd better be going now, counsel. And be careful . . . I mean it.\"\n\n\"Careful about what?\"\n\n\"You're Sicilian and you have to ask me, who is from the Piedmont? For now you've won, and they may even hoist you up onto their shoulders . . . \"\n\n\"You're right, you know. The president of the social club has asked me to resubmit my request for admission. He assured me that the club will be honored to have me as a member. And the mayor has even recommended to the prefect that I be given a knight's cross.\"\n\n\"You see? And yet I'm convinced that, starting tomorrow, you'll be entering the most difficult period of your life. A backlash will come. It's inevitable. I wish you the best of luck.\"\n\n In the original text, the Bishop, who speaks in Roman dialect, says that Labarbera \" _\u00e8 tanto, ma tanto scacarcione,_ \" a phrase clearly echoing a line by 19th-century _romanesco_ (Roman dialect) poet Giuseppe Gioacchino Belli, a man of severely anticlerical sentiment. Writing of Pope Gregory XVI, Belli writes: \" _Povero frate! \u00e8 ttanto scacarcione \/ Che ssi una rondinella passa e fischia \/ La pijja pe 'na palle de cannone._ \" (\"Poor brother! He's such a pants-shitting coward \/ That if a swallow passes and tweets \/ He takes it for a cannonball.\")\n\n Latin expression meaning: \"Have him promoted, to get him out of the way.\" Nowadays we sometimes say that such a person is \"kicked upstairs.\"\n\n## CHAPTER XIII \nTHE WHEEL CHANGES DIRECTION\n\nOne week later, on the quiet, the bishop of Camporeale sent for Don Mariano Dalli Cardillo. When the aging priest was shown in by Don Marcantonio, His Most Reverend Excellency Egilberto Martire got up from his armchair and greeted him with arms raised to the heavens, as if they were old friends from the seminary.\n\n\"Our dear Don Mariano!\"\n\nHe rested his hands on the other's shoulders, looked him in the eye with one half of his mouth smiling, the other not, then sat him down on the sofa and sat himself down beside him.\n\n\"How are things, my dear friend, how are things with you? Not too good, I take it? My own wounds have not yet healed, and I imagine it's the same for you! At any rate, with God's help, we can say we overcame this terribly difficult ordeal the Lord has put before us!\"\n\nDon Mariano thought that since His Excellency was speaking to him in Italian, and not Roman dialect, it must mean he was not angry at him.\n\n\"And now, to us. I wanted to see you in person, you know, so I could thank you!\"\n\n\"For what, Your Excellency?\"\n\n\"For what?! What do you mean, 'for what?' For having demonstrated\u2014by your presence, by your daily practice\u2014that not all the priests of Palizzolo were made of the same matter as those seven base individuals unworthy of their office as shepherds of souls!\"\n\n\"But, Your Excellency, I\u2014\"\n\n\"No, no\u2014no modesty, let me tell you outright! You were like the luminous beam of a lighthouse as all the world around you fell into darkness!\"\n\n\"But, Your Excellency, I did nothing special! I merely kept on doing what I've always done, hearing confessions, comforting the faithful\u2014\"\n\n\"Giving counsel . . . \"\n\n\"Also, yes, as needed.\"\n\n\"Well, come to think of it, on the subject of counsel, do you remember the words of our Lord Jesus, when he said: \"Give unto Caesar what is Caesar's, and to God what is God's?\"\n\n\"Of course I remember those words!\"\n\n\"Have you always kept them foremost in your mind?\"\n\n\"Yes, sir, I have!\"\n\n\"Then why is it that when that widow came to you to confess, and asked for your fatherly advice, you consented that she should give to Caesar what should in fact have been given to God?\"\n\nPatre Mariano was totally flummoxed.\n\n\"But, Your Excellency, I don't know what you're\u2014\"\n\n\"Let me explain. Unless I am mistaken, when that unfortunate woman, the widow, revealed to you, during confession\u2014during con-fes-sion, mind you\u2014the turpitudes of your confr\u00e8res, you allowed her, consented, permitted, paved the way for her to go straight to the Royal Carabinieri to report their actions, causing what happened to happen.\"\n\n\"And what should I have done?\"\n\n\"But, my blessed son, it's priests we're talking about! Ministers of the faith! Anointed by the Lord! Men of God! Priests who had, yes, erred from the straight and narrow path\u2014I'm the first to admit it\u2014but still priests nonetheless! _In aeternum!_ You should have given to God what was God's; you should have told that woman to come to me and tell me that a few soldiers of Christ were sullying their cassocks! You forgot, Don Mariano, that they were wearing frocks, not the uniforms of\u2014I dunno, the royal army or carabinieri! I myself would have taken care of banishing those scoundrels, but with the proper care, and the necessary caution, over time, without creating a scandal . . . Because, let's be frank, the scandal you so carelessly triggered risked shaking the very foundations of the Church!\"\n\n\"Please forgive me, Your Excellency, I beg you, I implore you to forgive me! But I was so upset by that woman's revelation that I didn't think for a moment that\u2014\"\n\n\"But I'm not reproaching you in any way! I understand you! I understand you perfectly!\"\n\n\"To this day, I swear, I still cannot fall asleep. Ever since that woman told me everything, I spend my nights awake, in prayer!\"\n\n\"Indeed when I saw you come in today, I got scared. I thought you were seriously ill.\"\n\n\"No, Your Excellency, I'm not ill, it's just that this whole business\u2014\"\n\n\"But you can't carry on like this! With no sleep for a whole week! You're at the end of your rope, my dear friend! You're urgently in need of help! Listen, Don Mariano, shall we do what's best?\"\n\n\"And what's that?\"\n\n\"Shall we have you take a nice, long period of rest? Don't say no; you really do need one. Tell you what: in the next two or three days I'll send another priest to relieve you. What do you say?\"\n\n\"God's will be done.\"\n\n\"Good for you, Don Mariano! Come, let's have a big hug!\"\n\n*\n\n\"Gentlemen, fellow members, your attention, please. In two days\u2014that is, this next Sunday\u2014at ten o'clock in the morning, all members, as is written on the flyer posted on the showcase window, are invited to vote on the admission to this club of the attorney, Matteo Teresi, who has resubmitted his request,\" said don Liborio Spart\u00e0.\n\n\"So we're starting all over with that same bloody headache?\" asked Commendatore Paladino.\n\n\"But do the rules allow that?\" asked Giallonardo in turn.\n\n\"The rules allow three admission requests,\" President Spart\u00e0 clarified. \"And this is Teresi's second request.\"\n\n\"Well, while we're talking about rules,\" don Anselmo intervened from his damask chair, \"I'd like to know whether abstention is allowed, or we must vote only yes or no.\"\n\n\"One who abstains is someone who hasn't the courage of his convictions,\" declared Colonel Petrosillo.\n\n\"And, since you have no convictions whatsoever, you have no need for courage, either,\" retorted don Anselmo.\n\n\"Well, dear sir, for your information, I have been awarded the bronze medal!\"\n\n\"What was that? I didn't quite hear. What kind of medal?\"\n\n\"The bronze!\"\n\n\"Ah, I'm sorry. I thought you'd said the 'pawn's medal.'\"\n\nIn an effort to wash this terrible slight away with blood, the colonel took off through the air, flying across the salon towards don Anselmo, but was intercepted in midflight by don Stapino Vassallo.\n\n\"Consider yourself challenged!\" shouted the colonel, foaming at the mouth as he struggled to free himself of don Stapino's embrace.\n\n\"Like the last time? When first you challenged me, then you disappeared from circulation?\"\n\n\"Gentlemen, gentlemen! For goodness' sake!\" said the president. \"Please calm down. And allow me to clarify something. It was I myself who personally solicited Attorney Teresi's new request.\"\n\n\"Why not just let sleeping dogs lie?\" queried don Anselmo.\n\n\"Because I consider it the highest of honors for this club to have, as a member, a person who did not hesitate to risk a great deal, to expose himself to personal danger, to\u2014\"\n\n\"Who's the other sponsor?\" Giallonardo interrupted him.\n\n\"Our dear mayor.\"\n\n\"I call to your attention that my question has not yet been answered.\"\n\n\"Yes, abstention is allowed.\"\n\n\"Well,\" said don Anselmo, \"I hereby declare that I will abstain.\"\n\n\"Whereas I, this time, will vote 'yes,'\" said don Serafino Labianca.\n\n\"Did the Grand Lodge order you to do that?\" asked Professor Malatesta.\n\n\"The Grand Lodge hasn't a bloody thing to do with it! And enough of your priestlike insinuations, you who used to serve Mass with Patre Samon\u00e0! And kneel before him to kiss his hand! I'm voting yes because Teresi helped send that renegade Marquis Cammarata to prison!\"\n\n\"And I'm going to vote 'no,' precisely _because_ I used to serve the Mass with Patre Samon\u00e0! But don't you realize that this is a plot against the Church?\" asked Professor Malatesta.\n\n\"Oh, come now! A plot?\"\n\n\"Gentlemen, gentlemen, this is no time to argue. The voting will take place Sunday morning. We have two more days to think it over. You should each take the time to reflect calmly, and\u2014\"\n\n\"Mr. President, if I may. Sunday morning is no good,\" Commendatore Paladino interjected.\n\n\"Why not?\"\n\n\"On my way here I saw some people posting announcements. On Sunday morning there's going to be a great procession of reconciliation, on the orders of the bishop of Camporeale.\"\n\n\"All right, then, we will postpone the meeting until five P.M. that evening. Is everyone in agreement?\"\n\n*\n\n\"Thank you for inviting me to lunch,\" said Luigino Chiarapane, whom Stefano had run into by chance that morning in Palizzolo.\n\n\"What did you come into town for?\" Teresi asked him.\n\n\"Well, there's something I didn't really understand, to be honest.\"\n\n\"What do you mean?\"\n\n\"Three days ago, Z\u00e0 Ernestina suddenly arrived at our house in Salsetto.\"\n\n\"The marquise?!\" Teresi and his nephew said in chorus.\n\n\"Yes.\"\n\n\"And what did she want?\"\n\n\"No idea,\" said the young man. \"At first my mother didn't even want to see her, but Z\u00e0 Ernestina insisted, and she was crying. So in the end they shut themselves up in Mamma's bedroom and were in there talking for two hours.\"\n\n\"And didn't your mother tell you anything afterwards?\" asked Stefano.\n\n\"No, nothing. Then, the day before yesterday she came here to Palizzolo.\"\n\n\"To talk to her cousin?\"\n\n\"Of course. Why else would she come here?\"\n\n\"Maybe her cousin wants your mother to withdraw her denunciation,\" said Stefano.\n\nTeresi started laughing.\n\n\"Stefan\u00f9, I get the feeling that your law school studies . . . Don't you know that at this point nobody can do anything anymore? At most, the marquise could ask the Chiarapane family not to press charges. Which means I would lose a job, since I'm her lawyer. Oh, well . . . \"\n\n\"But you still haven't told us why you came into town,\" said Stefano.\n\n\"My mother said I had to come to see Z\u00e0 Ernestina'cause she wants to talk to me. She's expecting me this afternoon at three.\"\n\n\"Just be sure that you don't run into _'u z\u00f9_ Carmineddru again!\" said Stefano.\n\nThey all laughed.\n\n\"Still, I'm dying of curiosity to know what she wants from you,\" Stefano added.\n\n\"Let's do this. After I go to see her, I'll come back here around five and tell you everything.\"\n\nBut Luigino never returned.\n\nAs soon as the procession emerged from the Mother Church, it was clear it was going to be a grand affair.\n\nPreceded by all the municipal police officers in full dress uniform, four priests came out hoisting up a large, gold-embroidered baldachin with His Excellency the bishop of Camporeale sitting inside, holding a monstrance, also gold, in his hand.\n\nBehind him came the four remaining priests of Palizzolo.\n\nAnd right behind them were Baron Lo Mascolo, Baron Roccamena, Baron Piscopo, and Marquis Spinotta.\n\nThen there was a short space between the nobles and the town council, and in this space was a lone man, all dressed in fustian, shod in boots, and carrying his _coppola_ beret in his hand.\n\nAfter him came Mayor Calandro with the town council and staff, followed by the town businessmen and bourgeois\u2014all of them, from don Liborio and don Anselmo to don Serafino, Giallonardo the notary, Professor Malatesta, and Colonel Petrosillo . . .\n\nAnd each\u2014whether noble or bourgeois, businessman or bureaucrat\u2014with his respective wife.\n\nThe municipal band separated this group at the head of the procession from the rest of the common folk. Almost three thousand in all, a first.\n\nAll the other people who had come out on their balconies and terraces, wearing their Sunday best, knelt down as the procession passed, showering the bishop's baldachin with roses and other flowers.\n\nThe procession then headed down the street on which Teresi's house stood. Everyone looked up.\n\nAnd they saw the lawyer on his balcony with his hat on. Was he trying to taunt them all by keeping his head covered in front of the Most Holy Sacrament? There wasn't a single person in the passing procession who wasn't staring at him. But then, the moment the baldachin was directly under his balcony, Matteo Teresi doffed his hat and made a deep bow.\n\nNot to the Most Holy Sacrament, however, but to the man in fustian walking alone between the nobles and the town council.\n\nAnd he called to him loudly, shouting above the blare of the band:\n\n\"When you see _'u z\u00f9_ Carmineddru, give him my fondest regards!\"\n\nThen he went inside, shutting the doors to the balcony.\n\n\"Gentlemen members, I hereby open the voting for the admission of lawyer Matteo Teresi into our club. I remind you that a black marble means 'no,' and a white marble means 'yes.'\"\n\n\"Please, if I may,\" said Giallonardo.\n\n\"Yes, go ahead.\"\n\n\"Mr. President, when you announced to us the other day that we would be holding this meeting, something unusual happened. According to the rules, the voting must be secret. Whereas two days ago, two members openly declared what their vote would be. You should have immediately disqualified them. But you didn't. So my question is: are their publicly admitted votes still valid?\"\n\n\"Please explain what you mean,\" the president said with some pique.\n\n\"I'll cite an example. The last time we met, Professor Malatesta, here present, declared that he would vote against admission. So I now ask the professor, is he still of the same opinion?\"\n\n\"Of course I'm still of the same opinion! All the more so after what the lawyer did when the procession passed by his house!\"\n\n\"Speaking of which, who was that man?\" asked don Liborio.\n\n\"Don't you know?\" asked don Serafino. \"You're probably the only person here who doesn't. That man is _'u z\u00f9_ Peppi Timpa, whom we could call _'u z\u00f9_ Carmineddru's temporary replacement.\"\n\n\"Well, to continue,\" Giallonardo the notary resumed, \"if that's the way it is, then it's clear that the voting will be invalid, since Professor Malatesta's pre-announced black marble will be counted and admission to the club must be based on unanimity. Therefore voting will only be a waste of time.\"\n\n\"So how do we get out of this predicament?\"\n\n\"I have a suggestion, if I may . . . \"\n\n\"Please go ahead, sir.\"\n\n\"The novelty of the other day\u2014meaning, the open declaration of a member's vote, which is not explicitly prohibited by the rules and therefore could be admissible\u2014could be of help to us here. You could ask the members how many of them intend to vote no, without them needing to say why.\"\n\n\"Would the gentlemen members who intend to vote no please raise their hands?\" asked the president.\n\nSome twenty hands went up. The president turned pale and said not a word. Aside from five or six strict Catholics, all the others must have been people who couldn't bring themselves to accept the public insult made to _'u z\u00f9_ Peppi Tinca, or whatever the hell his name was.\n\nGiallonardo the notary spoke for the president.\n\n\"As you can see, Mr. President, there is no point in voting. My advice is that Signor Teresi, if he's really so keen on it, should submit a third and final request.\"\n\nThe silence that descended upon the salon was broken by don Stapino's cheerful voice.\n\n\"Casimiro, bring out the playing cards!\"\n\n*\n\nAt seven o'clock Monday evening, the town council met to discuss the mayor's proposal to write to the prefect to have Matteo Teresi awarded the title of _cavaliere_ and given the knight's cross.\n\n\"I would like to speak in a personal capacity,\" said Mangiameli, a lawyer.\n\n\"Please go ahead,\" said President Burrano.\n\n\"I speak as a practicing, observant Catholic. I had been entirely in favor of underwriting the mayor's proposal because I was convinced that my legal colleague Teresi's action against the parish priests who had revoltingly betrayed their divine mission was dictated by a sincere desire for justice. But after what happened yesterday morning during the procession I had to revise my position. He offended the holy solemnity of the occasion! He started shouting in the presence of the Most Holy Sacrament! This I have taken as a clear sign that he hasn't the least bit of respect for our sacred religion!\"\n\n\"And neither for our sacred Mafia,\" someone said under his breath, though it was unclear who.\n\n\"And therefore,\" Mangiameli concluded, \"I will vote against rewarding Teresi, and nothing can make me change my mind!\"\n\n\"Permission to speak!\" said Pasqualino Marchica, a grain and fava bean merchant.\n\n\"Permission granted.\"\n\n\"With all due respect to our mayor, I wouldn't feel right voting yes, either. Matteo Teresi is a man whose opinions I respect, but he's also someone who always comes out guns blazing without thinking twice. He seeks to do the right thing, but without taking into account the harm it might bring to others.\"\n\n\"That's the absolute truth!\"\n\n\"I'll cite just one example. When he found out what those swinish priests were doing, he took that bucket of shit, and instead of dumping it into the pit, he threw it over the whole town! He covered us all in shit! The priests surely deserved it, but not everyone else. He ruined the lives of four girls who\u2014\"\n\n\"Five,\" said another voice.\n\n\" . . . five girls who\u2014\"\n\n\"There are seven of them,\" suggested another voice.\n\n\"Would somebody then please tell me how many goddamn girls there are?\" asked Pasqualino Marchica.\n\n\"Just one minute,\" said President Burruano, counting on his fingers. \"Paolina Cammarata, Antonietta Lo Mascolo, Totina Perricone, the widow Cannata, Lorenza Spagna, and Filippa Lanza. That makes six.\"\n\nPasqualino Marchica resumed speaking.\n\n\" . . . ruined the lives of six girls who\u2014\"\n\n\"Hey, Pasqual\u00ec, it doesn't add up!\"\n\n\"Why not?\"\n\n\"We're forgetting the dead girl, Rosalia Pampina.\"\n\n\"But she's already dead! Just let me finish! He's ruined the lives of six girls whose only fault was to have believed what their priests told them! These poor young women, whether noble or of humble station, can only become nuns now. They'll never find a husband anymore! Thanks to our fine lawyer friend, all over Italy everyone's talking about Palizzolo as if it was some kind of whorehouse! He's not the kind of man to do the right thing. And so I say no!\"\n\nAfter three hours of discussion, the town council decided to reject the mayor's proposal.\n\n\"Montagnet was right,\" Teresi said to Stefano at the dinner table. \"The wheel of fortune is already changing direction. The backlash has begun.\"\n\n\"But you didn't really believe him, since you requested admission to the club a second time. If you had, you wouldn't have made the request, because you would have known that in one way or another they would say no.\"\n\n\"You're right. I didn't believe Montagnet. I thought my fellow townsmen would be a little more grateful. When in fact they're not. No club membership, no knight's cross.\"\n\n\"But did you really care so much?\"\n\n\"Well, yes and no.\"\n\n\"Zio, you know what your worst fault is? Being an idealist.\"\n\n\"Is that a fault?\"\n\n\"Well if you don't like the word 'fault,' we can call it a 'shortcoming.'\"\n\n\"Oh, there's something else I wanted to tell you. I went to the bank today and they told me the manager wanted to talk to me. He never once looked me in the eye; he only said, 'Thank you.' And I said: 'For what?' And he said: 'For having ruined my life, and my family's life. I'm hoping to be transferred out of here as soon as possible.' The poor guy! I really felt sorry for him. But what do I have to do with any of it? I didn't even know that his daughter Filippa was one of the girls involved! It was the widow Cannata who revealed her name, but the fault is always mine!\"\n\nHe threw his napkin onto the table and went out onto the balcony.\n\nIt was a hot evening. Dark but starry. He pulled a cigar out of his waistcoat pocket and lit a match.\n\nThe bullet passed so close to him that it blew out the match.\n\n## CHAPTER XIV \nHOW IT ALL ENDED\n\nThe following morning was market day. \nAs he had always done every week, Teresi didn't miss it, despite the fact that the gunshot of the previous evening had cost him a few hours of sleep. You can be as brave as you want, but a bullet whizzing right past your head will never fail to rattle your nerves at least a little. But he did not feel afraid. It was something he'd been expecting, in a sense. _One of these days they're going to shoot me_ , he often used to think when some of the more fiery polemics he wrote in his newssheet touched upon untouchable local interests or roiled the already foul waters.\n\nAt the market he loved to browse from stall to stall, and especially to chat with the merchants and hawkers, who, in covering the entire province over the course of the week, knew more things than the prefect himself. And since they all knew Teresi well, they would tell him everything: all the stories of infidelity, theft, and fraud, as well as the marriages, births, and deaths that had occurred in the towns they'd just been through. They were better than any local news correspondents could be, of which he had none for his newssheet anyway. Some of these stories actually came in installments, and each week he would get the most recent updates.\n\nThat morning, however, as he walked among the people, stopping at each stall, he felt that something around him had changed. Something barely perceptible, but real. A darting glance perhaps, a half smile, a word left hanging . . .\n\nHe also noticed another difference. Whereas in the past he'd always had to shoulder his way through the throng, this time, as soon as people saw him, they stepped aside, almost as if to avoid coming into physical contact with him.\n\n_They know!_ he thought.\n\nThe night before, when he went out onto the balcony, he was absolutely certain there wasn't anybody in the street below. And right after the shot, he hadn't heard so much as a window open or close. So how was it that the news of the gunshot had reached the ears of everyone?\n\n\"Attorney Teresi!\"\n\nHe turned around. It was a carabiniere.\n\n\"I went to your house looking for you, and your nephew kindly told me I would find you here.\"\n\n\"What is it?\"\n\n\"Marshal Sciabbarr\u00e0 wants to see you.\"\n\n*\n\n\"Could you please explain to me why you went directly to the market this morning instead of coming to this station?\"\n\n\"Why should I have done that?\"\n\n\"To report what happened last night.\"\n\n\"And what happened?\"\n\n\"So nothing happened to you last night?\"\n\n\"Absolutely not,\" said Teresi with a questioning expression on his face.\n\n\"I get it,\" said the marshal. \"So I guess I'm just talking to hear the sound of my own voice.\"\n\n\"If you feel like talking and enjoy doing so, go right ahead.\"\n\n\"No, I don't feel like it, and I don't enjoy doing so. There's nothing enjoyable about it whatsoever. If you find a gunshot that actually blows out the match you just struck to light your cigar enjoyable, that's your business. To each his own. I'm just trying to do my job.\"\n\nTeresi was flabbergasted. How the hell did the marshal know even the detail about the match? There was no need to ask.\n\n\"This town, my good lawyer, is like a sleeping cat. Its eyes are shut, it doesn't move, and we think it's asleep. But in fact the cat is counting the stars in the sky. In this town, nothing ever remains secret. Everyone comes to know everything about everyone. For that reason, I understand perfectly well why you don't feel like reporting the incident. Shall I tell you why?\"\n\n\"Please.\"\n\n\"First of all, you are correctly convinced that if you did report it, and I began an investigation, it would be like trying to catch the wind. Secondly, any report on your part would, in one way or another, only increase the gossip about you, whereas you, at the present moment, need a little peace and quiet around you.\"\n\n\"You're a very intelligent man, Marshal.\"\n\n\"Thank you. But, to continue, and to keep hearing the sound of my own voice, even if you need a little peace and quiet around you, there's no saying that other people want to grant you this peace and quiet. Eternal peace, perhaps, yes, but a few months of quiet, probably not. Know what I mean?\"\n\n\"No, I don't really follow.\"\n\n\"My good man, whoever shot at you last night . . . Actually, no, let me rephrase that: Whoever shot at a man lighting his cigar on the balcony last night\u2014\"\n\n\"\u2014and missed . . . \"\n\n\"You really think so? Come on, Mr. Teresi, whoever shot at you missed on purpose! He needed only to fire a second time to kill the man on the balcony. But he didn't. He didn't, because his intention was to send you a message. That bullet whizzing past your head was talking to you. And it could only have been saying two things: 'Whatever you've done is done. But from this moment on, be careful what you do.' Is that a little clearer now?\"\n\n\"Perfectly clear. And what was the second thing?\"\n\n\"The second thing may have been the following: 'Pack your bags and get out of this town while there's still time.' Clear?\"\n\n\"Perfectly. And I thank you for your courtesy. Oh, and, listen: Have you got any news of Captain Montagnet?\"\n\n\"I'm told the good captain left yesterday for his new destination. He's been promoted to major. And he's moving to Alessandria, in the Piedmont. He'll be very happy, I'm sure.\"\n\nAnd thus they'd taken away the one friend he could always count on.\n\n*\n\n\"Have you heard the news?\" asked don Anselmo, entering the club in a rush.\n\n\"Yes, we have,\" don Serafino Labianca, don Stapino Vassallo, and Commendatore Paladino, the only people present, replied almost in unison.\n\n\"Attorney Teresi sent his regards to _'u z\u00f9_ Carmineddru through _'u z\u00f9_ Peppi Timpa, and _'u z\u00f9_ Carmineddru didn't waste any time replying,\" said don Stapino, laughing.\n\n\"You think so?\" asked don Serafino.\n\n\"Why, do you have a better explanation for what happened?\"\n\n\"I've got more than just one, my friend, as far as that goes. Let's begin by saying it could have been a shrewd move by a father whose daughter has been ruined by the scandal Teresi stirred up. So he shoots at him, unfortunately missing his target, but it's no sweat off his back, because afterwards we're all ready to say that it could only have been _'u z\u00f9_ Carmineddru or _'u z\u00f9_ Peppi Timpa that did it.\"\n\n\"It seems to me you're thinking of one person in particular.\"\n\n\"The Marquis Cammarata? No, I would rule him out. But it could have been\u2014just to name one possibility, for the sake of argument\u2014 _ragioniere_ Toto Lanza.\"\n\n\"The bank manager? No, come on!\"\n\n\"I'm sorry, but why does that surprise you? Didn't his daughter Filippa end up being the talk of the town because of Teresi? And then Toto Lanza's own history . . . ah, never mind!\"\n\n\"Oh, no you don't! Now you must tell us everything you know!\"\n\n\"Just the other day I spoke with the lawyer defending Patre Samon\u00e0, who took advantage of Filippa, Lanza's daughter. He said Don Samon\u00e0 had told him that once, right after he'd finished doing it with the girl, Toto Lanza himself had come into the sacristy. The priest had forgotten to lock the door! Luckily they'd both just put their clothes back on, but Filippa's face was as red as a tomato, one of her tits was hanging halfway out, and it was clear that something had happened between the two. But Toto Lanza said nothing. He just whispered, 'Excuse me,' and went back out.\"\n\n\"But that means he hadn't understood a thing!\"\n\n\"He'd understood perfectly well, my friend! In fact, one month later he went to Patre Samon\u00e0 and told him he'd discovered that the priest was a cousin of the bank's president. In short, he asked him to ask his cousin to promote him from teller to manager. And Don Samon\u00e0, who realized the man was trying to make a deal with him, did what he needed to do and, one month later, Toto Lanza became the branch manager.\"\n\n\"It's hardly surprising,\" said don Anselmo. \"Toto Lanza actually looks like a cuckold. Cuckolded by his own daughter, and probably also by his wife!\"\n\n\"That's something you'd have to ask don Cec\u00e8 Greco, who unfortunately is not present!\" said Commendatore Paladino.\n\nIt was well-known that Michela Lanza and Cec\u00e8 Greco had had a thing going for years.\n\n\"But there's another possibility,\" don Serafino resumed. \"Which is that nobody at all shot at Teresi.\"\n\n\"What are you saying?! All the neighbors heard the shot!\"\n\n\"Calm down. I'm saying that Teresi may have told his nephew, Stefano, to go down into the street and fire a shot at him.\"\n\n\"But why would he do that?\"\n\n\"Because, since nobody saw who fired the shot, Teresi can give the gunman whatever name he wants when he writes about it in his newspaper. And thereby ruin the life of whoever he's got it in for most at that moment. Someone like you, don Stapino, for example. If the man writes that it was you who fired the shot, how will you defend yourself? Sue him for libel? If you sue, then people will start to think it really was you. Take my word for it: there's only one thing we should be hoping for: that next time the gunman doesn't miss.\"\n\n\"But didn't you say you would have voted in favor of admitting him to the club? Have you changed your mind or something?\" asked don Anselmo.\n\n\"And would you yourself vote to admit a momentarily walking corpse?\"\n\n*\n\n\" _For pressing family reasons I find myself forced to terminate our association, as I am no longer in need of your services. Regards, Giovanni Galletto_.\"\n\nWith the above telegram, the fifth of its kind, Matteo Teresi lost the fifth of the six most important cases he was working on. Five telegrams in a single week, almost one a day, all the same, all using the same formula: \" _For pressing family reasons_ . . . \" This was to make him understand, if he hadn't already, that this was all by design, and he would never again be given the kind of case that allowed him to eat, to live his life, to publish his newssheet.\n\nThe Mafia, the priests, and the nobles, with these telegrams from his former clients\u2014whom they'd threatened and forced to withdraw their support\u2014were telling him that they intended to reduce him to abject poverty. Because all his other cases\u2014the ones involving the poor, the wretched, the peasants abused by their masters\u2014not only did they not bring him a single lira in revenue, but very often he had to pay for the officially stamped documents and the court fees out of his own pocket.\n\nHe had enough money in the bank to live for about two or three months. What would he do afterwards?\n\n\"Zio,\" said Stefano. \"I have something to tell you that will upset you, but I have to tell you just the same. You'd even predicted it, actually.\"\n\n\"Go ahead.\"\n\n\"I ran into Luigino.\"\n\n\"So why haven't I heard from him?\"\n\n\"You'll know in a minute. He told me the Chiarapana family has decided not to press charges.\"\n\nAnd there went his sixth and last important case!\n\n\"Was that why the marquise went to talk to Luigino's mother?\"\n\n\"It's not the only reason.\"\n\n\"There's more?\"\n\n\"Yes. The scheme's a little more complicated . . . \"\n\n\"Go on, don't be shy.\"\n\n\"Luigino's gonna go to the carabinieri and tell them it was him who got Paolina pregnant, letting Patre Terranova off the hook. Terranova will swear he never laid a hand on Paolina, and only abused the widow and Totina\u2014both legal adults\u2014on the day at the convent. That way he gets out of the more serious charge of corrupting a minor, and the marquis gets to claim the usual extenuating circumstance of defending family honor for the crime of attempted murder of Luigino, which on top of everything else he failed to complete.\"\n\nAttorney Teresi had become so pale that Stefano got scared he might be having a heart attack.\n\n\"Drink a little water, Zio.\"\n\n\"So what does Luigino get out of this?\"\n\n\"He gets married to Paolina and they make him a rich man. They pay off all his father's debts, which are considerable, he gets the Cammaratas' palazzo in Salsetto, as is, and they're going to give him their Zumm\u00eca estate in the dowry.\"\n\n\"So, I guess it all works out.\"\n\n\"And you want to know something else?\"\n\n\"Tell me.\"\n\n\"I ran into Baron Lo Mascolo by chance. He said he wants to talk to me.\"\n\n*\n\nThree mornings later, the postman delivered a stamped envelope from the Criminal Court of Camporeale. Teresi opened it. There was a letter dated September 10 and signed by the presiding judge, Gianfilippo Smecca, a man always ready to adjust his sails to catch the prevailing winds.\n\n_Illustrious sir,_\n\n__\n\n_This communication is to inform you that your presence is required on the 15th day of the current month, at 17:00 hours at the Court of Camporeale, via Regina Margherita 10, for a hearing with the Disciplinary Committee pursuant to the charge of professionally inappropriate behavior, as raised against your person by all of the criminal lawyers practicing in the City of Palizzolo._\n\n_The charge stems from the fact that, on the occasion of the well-known case that led to the arrest of Marquis Filadelfo Cammarata, you accumulated unto yourself a series of positions that lead one to conclude the existence of a pre-existing, personal hostility, on your part, towards the abovementioned Marquis._\n\n_They are, in order:_\n\n_Plaintiff (presenting yourself as such to the Royal Carabinieri);_\n\n_Witness for the prosecution (presenting yourself as such to the Investigating Magistrate);_\n\n_Attorney for Plaintiff (as designated by the Chiarapane family, a role accepted by you though later revoked by the same family)._\n\n_It behooves us to inform you, moreover, that Signora Albasia Chiarapane sent us, of her own accord a declaration in which she asserts that, to the end of filing together, as one, a denunciation with the Royal Carabinieri, you demanded of her a sum of ten thousand lire in cash, claiming that you would otherwise \"wash your hands of the whole affair.\"_\n\n_We must also remind you that the Disciplinary Committee has the right to proceed even in the absence of the person under disciplinary investigation._\n\n_Regards,_\n\nNaturally, Matteo Teresi had no desire to appear at the hearing. And even if he did appear, what could he possibly say in his defense? The most serious charge was not that of having played three roles in a farce\u2014which was actually true\u2014but of having taken ten thousand lire to go and report the attempted murder of Luigino. The accusation was patently false, but how would he ever prove it? By this point it was clear that they were all in agreement to get him out of their hair, in one way or another. But he still had his newssheet, and as long as he still had the money to publish it, no one would ever succeed in silencing him.\n\n*\n\nIn the evening after receiving the letter from the Criminal Court, he didn't feel like eating.\n\nClearly, it would turn out one of two ways: either they would impose a long period of suspension on him, or they would strike him from the lists. The second was the more likely.\n\nHe would have to abandon the paupers who turned to him for help, leave them to their fate as the wretched of the earth.\n\nNot that he'd won so many of his cases on behalf of the poor; the law always came down on the side of the rich. But at least they'd served to give a little hope to those who'd never known any hope at all.\n\nHe felt empty inside, however, and a little confused. The fact was that he was used to fighting out in the open, to going hand-to-hand, even to being insulted, but not to treacherous surprise attacks, stabs in the back in the dark and in secret. They were scorching the earth all around him, and to set it on fire they were using the hands of those who had nothing in particular against him but couldn't or didn't know how to say no to those who asked them to light the match.\n\nHe went to bed early, read a bit of _Don Quixote_ , which he always kept on his bedside table, then slowly, little by little, fell asleep with the light on.\n\nThe sound of the front door opening and closing woke him up. He looked at his watch: it was past midnight.\n\nWhere had Stefano been all this time?\n\n\"Stefano.\"\n\n\"I'll be right there, Zio.\"\n\nSeeing the young man come in, he knew at once from the look on his face that he had news.\n\n\"I've been with Baron Lo Mascolo. He invited me to dinner.\"\n\n\"With the whole family?\"\n\n\"No, just me and him.\"\n\n\"What did he want?\"\n\nStefano sat down on the edge of the bed.\n\n\"That baron's got a real poker face, but in the end he tells you exactly what's on his mind.\"\n\n\"And what's on his mind?\"\n\n\"Zio, it took him at least three hours to explain the whole business to me. He made the most convoluted arguments, going round and round in concentric circles, all the while zeroing in on the point he wanted to make.\"\n\n\"And what was that?\"\n\n\"The point was sort of a copy of what the Marquis Cammarata did.\"\n\n\"Explain.\"\n\n\"What's to explain, Zio? Can't you figure it out for yourself?\" said Stefano, a little irritated.\n\n\"I get it, Stefan\u00f9. Antonietta will testify that Patre Raccuglia wasn't the first man in her life. It was you who got her pregnant. Right?\"\n\n\"Right.\"\n\n\"So Patre Raccuglia wriggles out of the charge of corrupting a minor, just like Patre Terranova. And you take Antonietta, an only child, for your wife, and you get rich. Right?\"\n\n\"Right.\"\n\n\"And did you start slapping him?\"\n\n\"No.\"\n\n\"Did you laugh in his face?\"\n\n\"Not even.\"\n\n\"But, Stefan\u00f9, this stuff is straight out of the puppet theatre! Do you realize that?\"\n\nStefano stood up.\n\n\"Yes, I do, but there's something you don't realize.\"\n\n\"And what's that?\"\n\n\"That I actually love Antonietta. But I told the baron I couldn't accept. Out of respect for you, Zio.\"\n\n*\n\nThe following morning the postman handed him a letter from America.\n\nTeresi recognized the handwriting. It was his brother Agostino writing to him.\n\nTwo years older than Matteo, Agostino had married an American cousin of his, moved to New York, and made a fortune buying and selling homes. He had three children, all girls. The oldest, Carmela, had married an engineer who worked for her father, and had two children. Agostino and Matteo customarily wrote to each other once a month.\n\nIn the present letter, after giving the usual news about his wife, daughters and grandchildren, Agostino wrote:\n\n_And so, dear brother, chatting the other day with my wife, she asked me a question I had no answer for. She said: \"But what's your brother Matteo doing still hanging around Palizzolo? Since the death of your parents, he's more alone than a dog. If he came here to live, he would be among family again.\" I didn't know what to say. But I did think that you might have answered the question with a question: \"And what on earth would I do in New York?\" Dear Matteo, there's a great deal of things that someone like you could do here. There are so many wretchedly poor immigrants who are treated worse than our peasants in Palizzolo! You have no idea the kinds of conditions they're forced to live in! And there's something else, too. I have a great business opportunity within my reach. There's this big pharmacy, and . . ._\n\nRight, because Matteo had first taken a pharmaceutical degree before studying law. He'd almost forgotten.\n\n*\n\nThe real blow, however, the kind that lays you flat out on the ground to the point where you can't get back up, came in the form of seven lines signed by His Excellency the Prefect of Camporeale.\n\n_We inform you that we have ratified the request of the Commissioner of Police to revoke your authorization to publish the weekly newssheet entitled_ The Battle _, printed by Mazzullo & Sons Printworks, granted by the Court of Camporeale on February 12, 1897, and addressed to you, Matteo Teresi, as editor in chief and publisher. This revocation, effective immediately and for an indeterminate amount of time, is dictated by the fact that you have been distributing seditious tracts and passing them off as special editions of your weekly, without, moreover, the proper authorization for such editions_.\n\nAnd so, for the first time since the wheel of fortune had changed direction, Matteo Teresi felt his face wet with tears.\n\n*\n\nHe spent the whole day dawdling about the house. In shirtsleeves, hair disheveled, slippers on his feet, he wandered from room to room, adjusting a book on a shelf or a lamp on a table, straightening a picture on the wall, dusting off some old photos on the living-room bookcase. At half past noon he mechanically set the table for Stefano and himself\u2014even though he knew that nothing had been cooked because it was the housekeeper's day off, and he hadn't even lit the wood in the stove. So he just sat there staring at the empty plates.\n\nBut where was Stefano? Why hadn't he come home? Then Teresi remembered that his nephew had told him the night before that he was leaving for Palermo early that morning to take an exam and would be away for three days. The fact had entirely slipped from his mind. He went upstairs and into the young man's bedroom. The bed was unmade, the wardrobe was missing a suit, and the suitcase was gone. Yes, he'd gone away to take his exam.\n\nThen Teresi went into his own bedroom. Feeling a little feverish, he took the thermometer from the nightstand drawer, lay down, and took his temperature. 99.9. He didn't feel sick, however. It was only the effect of the terrible blow.\n\nHe felt a great weight on his eyes, and closed them.\n\nWhen he awoke, the sun was setting. So he got up and went out on the balcony. He needed some fresh air.\n\nJust some thirty yards past his house, the street he lived on began to descend towards the countryside, and so, at that hour, it was always busy with peasants who had come into town to sell fruits and vegetables and were now on their way home.\n\nHe knew them all, every single one of them, and every evening was a rich exchange of greetings. That evening, however, nobody looked up towards his balcony; it was as if he hadn't come out.\n\n\"Gnaziu!\" he called.\n\nGnaziu Pirrera was one of the poor devils he had helped out. A father of five, he managed to eat about every other day, and every so often Teresi would give him a little money to feed his children.\n\nGnaziu Pirrera seemed not to have heard, and kept on walking, eyes on the ground.\n\n*\n\nNight fell ever so slowly.\n\nAnd when it was completely dark he went back into his bedroom, grabbed the cigar box and a small box of matches, went back out on the balcony, and lit the first cigar, keeping the match burning as long as possible in front of his face.\n\nIf he was waiting, hoping, for the gunshot that this time would blow out not the match but his own life, he was disappointed. Nothing happened.\n\nThe night was still. It breathed slowly and restfully, as a scent of straw scorched all day by the sun rose up from the countryside.\n\n*\n\nBy one o'clock in the morning, he got tired of staying up. How long had it been since he'd eaten, anyway? He went into the bedroom, grabbed a chair, took it outside, and sat down. He wasn't thinking about the letter from the Prefect, nor about the one from the Criminal Court.\n\nOnly Stefano's words were pounding in his head.\n\n\"You don't realize that I actually love Antonietta.\"\n\nAnd also:\n\n\"I told the baron I couldn't accept. Out of respect for you, Zio.\"\n\nSo, Stefano needed to lose the respect he felt for him. If he disappeared from Stefano's life without a word of explanation, maybe the lad would feel betrayed. And that would free him to choose his own destiny. Yes. That was the only solution.\n\nLittle by little, the idea gained strength. When the sky began to lighten in the east, the idea became a firm decision.\n\nHe looked at his watch. Five o'clock in the morning.\n\nSo, if he immediately started packing his bags, then bathed and shaved and put on a nice suit, he could easily catch the coach to Palermo after dropping in at the bank and withdrawing all his money. It would be more than enough to buy a ticket to America on the first ship out.\n\n# AUTHOR'S NOTE\n\nThis novel purposely distorts real events that took place in a Sicilian town, Alia, at the start of the twentieth century, to the point of rendering them so unrecognizable as to border on sheer fantasy. A priest by the name of Rosolino Martino was arrested for corruption of underage girls. A former local pharmacist turned lawyer, Matteo Teresi\u2014who in the pages of his weekly newspaper, _La Battaglia_ , fought the abuses of mafiosi, landowners, and the clergy\u2014began an investigation of the case and came to the shocking discovery that the priests of Alia had founded a secret sect that \"brought together inexperienced, virgin young girls and young brides, deceiving them into believing that sexual relations, and the sexual practices preparatory to the sex act, were a means for acquiring divine indulgences and opening the gates of Heaven,\" as explained by Gaetano D'Andrea, ex-mayor of Alia.\n\nThe discovery of the sect and its practices, when publicized by Teresi, sent shock waves beyond the island of Sicily and elicited the indignation of many political and religious officials, including Filippo Turati and Don Luigi Sturzo. The priest Rosolino Martino confirmed what Teresi had written in his newsweekly.\n\nThe clergy, the landowners, and the Mafia, however, closed ranks. On the one hand they attacked Teresi, on the other they forced the population\u2014even the immediate families of the victims of the abuse\u2014to remain absolutely silent on the matter.\n\nDismayed by the lack of reaction on the part of his fellow townspeople, Teresi goaded them harshly:\n\n\"The men are now resigned to the religious prostitution of their women, since it is no longer admissible [ . . . ], after what has happened, to plead ignorance. Let us avoid this danger, let us open the eyes of husbands and fathers, and after we speak to them frankly and boldly, things will reacquire their proper names. _Divine grace_ will no longer be a cover for sexual relations; the mystical bride will seem a common prostitute; the husband, amidst the saints with their haloes, will see his horns sprout majestically twisted; the young woman already on the path to perfection, setting aside her saintly mask, if not yet the mother of some scion of angels will be like the half-virgin of the French, who has lost everything and given everything except her supposed honor, which resides in the simple physiological sign of her virginity.\"\n\nThe article, which appeared in the August 11, 1901, edition of _La Battaglia_ , achieved the opposite result, eliciting a wave of genuine hatred towards its author. The bishop of Cefal\u00f9 accused him of blasphemy and held a reparative procession onto which Teresi, from his balcony, dropped flyers further denouncing the misdeeds of the clergy.\n\nIt amounted to underwriting a death sentence for himself. After a warning, Teresi wrote a last article of goodbye in his newsweekly, and took ship for the United States.\n\nIn the town of Rochester, New York, he continued practicing his profession as a lawyer and wrote a tremendous number of articles in support of the Italian immigrant community and on such broad questions as divorce and abortion.\n\nHis \"American\" writings were published in 1925, by D'Antoni Editori, a Palermo publishing house, under the title _Con la patria nel cuore_ (\"With the Homeland in My Heart\"). They were republished in an anastatic printing, edited by the Comune di Alia, in 2001, with a preface by Gaetano D'Andrea.\n\nTo repeat: the reader should consider this novel a product of my own imagination. Only two things were drawn from reality: the names of the protagonist and his newssheet (I did this in homage), and the passage from the article by Don Luigi Sturzo.\n\nShould any reader encounter names or situations reminiscent of real-life names or situations, they should be attributed to chance.\n\nI dedicate this book to Rosetta, for the more than fifty years of life we have spent together.\n\nA.C.\n\n# ABOUT THE AUTHOR\n\nAndrea Camilleri is widely considered to be one of Italy's greatest living writers. His Montalbano crime series, each installment of which is a bestseller in Italy, is published in America by Penguin Random House, and several books in the series have been _New York Times_ bestsellers.\n","meta":{"redpajama_set_name":"RedPajamaBook"}} +{"text":"\n\n_Champagne Baby_ is a work of nonfiction. Certain names and identifying details have been changed throughout. If as depicted there is any resulting resemblance to other real people, it is coincidental and unintentional.\n\nCopyright \u00a9 2016 by Laure Dugas\n\nAll rights reserved.\n\nPublished in the United States by Ballantine Books, an imprint of Random House, a division of Penguin Random House LLC, New York.\n\nBALLANTINE and the HOUSE colophon are registered trademarks of Penguin Random House LLC.\n\nISBN 9781101884638\n\nebook ISBN 9781101884645\n\nIllustrations by Dowotaki\n\nrandomhousebooks.com\n\n_Book design by Elizabeth A. D. Eno, adapted for eBook_\n\nCover design: Victoria Allen\n\nCover illustration: Vikki Chu\n\nv4.1_r1\n\nep\n\n# Contents\n\nCover\n\nTitle Page\n\nCopyright\n\nIntroduction\n\nMap\n\nPart I: Planting \/ Racine\n\nChapter 1: Nobody Knows Everything\n\nChapter 2: Youth Has Its Virtues\n\nChapter 3: There's More to Wine Than the Grape\n\nChapter 4: Taste the Present\n\nChapter 5: What's Your Vintage?\n\nPart II: Cultivation \/ Culture\n\nChapter 6: There's Always Occasion for Champagne\n\nChapter 7: Even the Best Wine Can Go Bad\n\nChapter 8: You're Better Than the Cheapest Bottle\n\nChapter 9: Trust Your Palate\n\nPart III: Harvest \/ R\u00e9colte\n\nChapter 10: The Older the Vines, the More Complexity\n\nChapter 11: Wine Is Community\n\nChapter 12: Wine Is Travel\n\nChapter 13: It Always Comes Back to Terroir\n\nEpilogue\n\nDedication\n\nAcknowledgments\n\nAbout the Author\n\nWhen I was a baby in Paris, my mother would dip her finger in champagne and rub it on my lips, to acclimate me to the taste and smell, she told people, because our family was from Champagne. I have no memory of this. The modern parent would be horrified. _You are getting your baby drunk! You are giving her brain damage!_ But I'm sure she did it in good fun, and I don't feel too much the worse for it.\n\nMy mother's grandparents cultivated our family's first vines in Champagne in the 1930s before the war, and good wine still comes from our vineyard (a small one not found in many guides). My paternal grandfather managed all the wine distribution that came through Bercy during the war. My father went in a slightly different direction, selling fine spirits throughout France. His brother, my Uncle Alain, is a respected vintner in the esteemed Ch\u00e2teauneuf-du-Pape geographic appellation of the Rh\u00f4ne Valley. Wine is most definitely in my blood.\n\nSo you might think I am some kind of wine princess, that I was schooled in everything there was to know about wine from the moment I could speak, that in my youth I could have recited the great vintages of France the way the child of the Church knows her catechism. _Oh pfft,_ you say, _no wonder she is writing about wine: She's French! Her family is in the business!_ And you would be right about my family, but not about me. I don't deny that I grew up surrounded by wine. As a girl I watched my mother open a bottle of her family's champagne at nearly any excuse. A friend stopping by the house? _Champagne!_ The sun coming out after a little rain? _Alors! Champagne!_ But the truth is that I knew almost nothing about it, except that there was plenty in the pantry. My pedigree, as it were, came with no \"special\" understanding at all, simply proximity.\n\nAnd so what it comes to is this, and I have no compunction about sharing it with you right away: entitlement is the very opposite of what makes for the most rewarding, most open experience of wine. Growing up, I drank only one kind of champagne\u2014my family's\u2014and could not have described any other, or even what made ours any good (though it benefits from some of the best _terroir_ of Champagne, in Ambonnay).\n\nBeing French is not the answer, either. While France, in my opinion, possesses the tradition and know-how to make the best wines in the world, that doesn't mean every French citizen can tell the difference between a Chablis and a Pouilly-Fum\u00e9 (just as, despite what I learned as a child, from the TV show _Dallas_ , not every American knows how to herd cattle). Being French can, in fact, actually breed complacency. I've seen plenty of my fellow countrymen and -women act a certain way to suggest they're naturally gifted when it comes to wine, when that couldn't be further from the truth. Nationality alone is not enough.\n\nI'm a prime example. If you'd told me as a little girl that I would one day operate my own wine bar and store, premised on the pleasures of French wine, I'd have laughed out loud. It was the last thing I wanted to do, having grown up in a family like mine. And anyway, credentials guarantee nothing\u2014in life as in wine. By the time I was nearly finished with university, I was still na\u00efve about both. And I did not learn, really learn, what it means to _love_ both until I left my family and my country at twenty-three, and moved to New York City.\n\nThis is the story I want to tell you.\n\nI've always had the unfortunate tendency to leap before I look. It's not very French of me, but I can't help it. So when my uncle pulled me aside after a family dinner in the summer of 2006 to offer me the chance to work in the U.S., I didn't say I would think about it or consult with others. I said yes. He'd heard I wanted to learn English, and he needed someone to work with his new American wine importer, which was based in New York. Never mind that to do the job well I would need a minimum of two things, neither of which I had: a minimal competency in the English language, and a much deeper understanding of French wine. Never mind that I had a boyfriend in Paris, Jules, who had already patiently waited for me as I studied in Spain for a year. Never mind also that my uncle needed someone for only the first six months of 2007, after which I would be on my own.\n\nI said yes immediately, before I could think twice. Even my uncle was surprised; after a moment he asked if I'd heard correctly that this would mean traveling around _les \u00c9tats-Unis_ \u2014the whole country. I assured him I had, and he laughed and said that in that case we'd have to make our preparations.\n\nMy answer might have been different if he had told me that in January I'd find myself in front of a hundred and fifty sales representatives in Seattle, conducting a tasting of his wine with a shaky hand. I'd never spoken in front of even fifteen people, but there I was, not only expected to speak but to speak intelligently, while secretly knowing that I had only a fraction of the experience of every other person there. When it came to wine, all my uncle and I had in common was a last name.\n\nI stood outside the tasting room unable to feel my toes out of nervousness, mouthing a script I'd gone over a hundred times. The opening monologue describing Ch\u00e2teau la Nerthe and what made it special was easy enough. It was the prospect of then having to walk the sales reps through the wine itself that worried me. If you'd handed me a glass of wine that morning, I could have instinctively told you whether it was good or bad\u2014but not why. If you'd asked me to describe what I was tasting, I wouldn't have known where to start.\n\n> _Now we will taste the Ch\u00e2teauneuf-du-Pape red of Ch\u00e2teau la Nerthe. [Wait for others to pour. Hold up glass.] The color you see is a bright ruby with a raspberry tint. The nose is fresh with red fruit, a touch of black pepper. [Wait for others to smell.] The palate is frank. [Wait for others to sip.] What you are holding is a complex wine, with hints of cherry, blackberry, black currant, and spice. The tannins, as you can tell, are elegant. [Wait for others to spit.]_\n\nThere were a hundred ways for me to screw it up. The scripted pauses, which I needed to remember _not_ to say out loud. I didn't even know what fruit the English words \"black currant\" referred to. And what _was_ a ruby color, exactly? I'd never seen a ruby in real life.\n\nI'd have turned around and _walked_ back to France if my audience hadn't been expecting me. I opened the door, with only one thought echoing in my head: _What have you gotten yourself into?_\n\nAs I've said, it's how I've always been, pressing on first and asking questions later. I said yes to my uncle, and yes to myself, that summer, and decided I'd move to America early, in November. That way, I thought, I'd be able to find a place to live, learn English, study wine, and conquer New York City, all in the two months before my job was to start. I was certain it would be plenty of time.\n\nOver the years I've noticed that people commonly think about wine in one of two ways: Some believe that its secrets are possessed by a select few. For the French, who have been making wine since before the Roman Empire, this often takes the form of snobbery: _Only we (and maybe some Italians) know anything about wine_. All experts are tempted to believe that their knowledge is special and not easy to attain. Let's call this the \"a few people know everything\" camp.\n\nThe other extreme is a response to this elitism: a conviction that experts are a fraud, that the industry is a charade. You may have heard about taste tests that show no real difference between a one-hundred-dollar bottle of wine and a ten-dollar bottle of wine. Reports like this are meant to assure people that they shouldn't feel ashamed if they don't know much about wine\u2014after all, a ten-dollar bottle is all you ever need! We can call this the \"nobody really knows anything\" school.\n\nThis is what I want to say before I get to what happened in New York: Neither is right. The truth is in the middle. There is indeed a lot to learn, but anyone can do it. I once watched as a celebrated winemaker failed to detect his own wines in a blind taste test. His own wines! But did that mean he was a fraud? Hardly. He had been making quality wines for two decades. All it meant was that _tasting wine is complicated, even for experts_. It didn't mean that there is no such thing as an expert.\n\nThere are no flash cards here. There are no bite-sized facts to memorize and recite at dinner parties. If you are looking for either a complete encyclopedia or an easy cheat sheet, you can stop reading now. This is about my experience\u2014what I learned about wine and about myself when I moved to America. So a better approach might be to say that \"nobody knows everything.\" Wine is too mysterious, too influenced by nature and by the smallest decisions of the people who produce it, to be easily pinned down. There is always more to learn. But with a few specific principles and a commitment to take it seriously, anyone\u2014whether you have no experience at all, or are an elite pedigree but with no formal understanding\u2014can start a lifelong journey in the art of enjoying wine.\n\nI'm living proof.\n\nWhat would have happened if I'd turned my uncle down and stayed in France? Who knows? I certainly could never have predicted that I'd one day have the U.S. to thank for much of my success and happiness. Americans seeking the French secret to the good life have launched a thousand bestsellers\u2014but a Frenchwoman improving herself by way of America? Blasphemous! I might still have ended up working with wine, but without the influence of a culture that prizes ingenuity and experimentation, I'd be a very different person.\n\nI'd originally imagined living in England, not the U.S. London was both closer to home and culturally more familiar. I had yet to complete my undergraduate degree, but my time in Spain had only increased my wanderlust. New York was so far, and in the days after telling my uncle yes, I had second thoughts. Jules\u2014patient, encouraging Jules, my boyfriend of four years, who had already waited for me over one of them without complaint\u2014wouldn't hear of it. \"You have to go,\" he said. \"It's the chance of a lifetime. You'll be back soon.\" He would turn out to be right about one of those things.\n\nI'd never been so excited and scared at the same time\u2014it was confusing. I'd only just returned from Spain, and there was so much to do before I left again. I went to the south of France to visit my uncle, who walked me around the vineyards pointing out where the grenache was planted and where the syrah was, and in the cellar he let me taste some wine directly from the barrels that wasn't ready for bottling\u2014the kind of in-depth tour I'd never cared to do in all my childhood visits to my mother's family in Champagne. He even signed me up for a six-week wine course to at least reduce the likelihood that I would make a total fool of myself. Though you cannot become an expert in anything in six weeks, learning the basics gave me the slightest boost in confidence.\n\nBy the time I returned to Paris there was only a month left to spend with Jules. My closest childhood friend, Vera (who had been the one to introduce me to Jules), knew someone I could stay with in New York. I could not imagine being so far from both of them, but when it was time to leave I was too anxious to allow either to accompany me to the airport\u2014only my mother, whose relative silence I knew was for my sake.\n\nAs for the flight itself, my memory is a blank. I do remember my first sight of Manhattan through the clouded Plexiglas of the taxicab divider, the gray outline of buildings that rose up closer to the sky the closer we got. And then we were in the city. I'd always pictured Manhattan as an array of immaculate, shining skyscrapers from end to end but had never really imagined the streets themselves; it was like picturing a person only from the neck up and completely ignoring everything below. So I was surprised to be dropped off on a desolate, narrow stretch of East Eleventh Street, at the far reaches of the East Village. The sidewalk could hardly fit my two giant suitcases, myself, the tree planter dotted with dog excrement, and the trash cans so full the lids would not close. The plain brick and stone row houses lining the street were no taller than six stories. For a second, I could have believed I'd stepped into an alternate, dirtier version of Paris, not the glass and metal towers I thought would be awaiting me.\n\nThis was the neighborhood that introduced me to New York\u2014and, despite my first impressions, where I would eventually make my home, the place I chose over every other neighborhood in the city. I waited, biting my nails, for a few minutes before I spotted Vera's friend approach from down the street. Because she was the friend of a friend we kissed cheeks, but the greeting felt mechanical. Rose was French as well, and officially the first person I met in the U.S. She was also, it quickly became clear, my polar opposite, starting with appearance: she was tiny, blond, and sharply featured where I am tall and brown-haired and round-nosed. She was taking a break from work to let me in, and I could see the tension in her shoulders and lips, even though her eyes were cool and unreadable\u2014unnervingly so for an open book like me. (Pinocchio had his nose; I have my cheeks, which turn helplessly red at the smallest provocation.) \"Come, come,\" she said in her clipped Parisian French, beckoning me to follow her into the building. She was doing Vera a favor, it was obvious, and little more.\n\nI let out an involuntary moan of relief when I saw the building had an elevator, and Rose glanced back at me questioningly. \"My suitcases,\" I said. \"I brought too much!\" and laughed with embarrassment. There was barely room for the four of us; I had to stand very close to Rose and her stern face as we ascended the six flights.\n\nThe apartment was a two-bedroom she shared with her boyfriend. The d\u00e9cor was minimalist and modern\u2014black couch, black coffee table, uninterrupted white wall except for a few framed black-and-white photos. Everything was tidy; everything matched. \"This is very classy,\" I said. Her lip twitched in satisfaction. She led me to the spare bedroom, which looked into the concrete courtyard at the rear of the building, weeds sprouting from the cracks. This, at least for a short time, would be my view of the city.\n\n\"Here is your key,\" she said, waiting for me to turn from the window, \"and sometimes, just so you know to be careful, you have to flush the toilet twice. I must return to work. I will see you tonight.\"\n\nAnd with that, she was gone.\n\nThough a part of me wanted to draw all the shades and hide, I drummed up enough courage to return to the street to find something to eat. I had no way to orient myself, however\u2014I had been lulled into overconfidence by Manhattan's grid. But \"Eleventh Street\" meant nothing to me. Neither did \"Avenue C,\" where I next found myself. Had I known better I would have turned west toward the rest of the city, but instead I wandered eastward until I was suddenly facing a highway. Had I already reached the end of Manhattan? I backtracked and turned to walk along Avenue D, where a power station loomed to the north, and to the east housing projects blocked my view of the water. Figures in the twilight burst out of a corner store, startling me with their laughter. I walked a few more blocks, reminding myself not to be frightened by every noise. Everything was strange and growing stranger in the dark. I hurried back to Rose's apartment, in shock, and still hungry.\n\nThis was the first moment I realized how unprepared I was. I'd believed it would be better to come early, with many ideas but few concrete plans. It hadn't occurred to me that this was like bringing a bottle of wine home with no way to store it until you are ready to drink it. I spent three nights with Rose and her boyfriend, Nicolas, who was as warm and friendly as she was cool, and he helped me go through Craigslist looking for places to live. His English was very good, and he would launch into spontaneous and unprompted vocabulary lessons\u2014\"this is a _sofa_ \"; \"this is a _pen_ \"; _\"today, tonight, tomorrow\"_ \u2014most of which I immediately forgot, having no way to practice. He did teach me some basic phrases beyond the few that I knew, and told me if I ever wanted more time to think about what to say in English, I shouldn't make the usual raspberry sound, or shrug, like we do in France. His preferred method instead was to start sentences with, _\"The thing is...\"_\n\n\"This phrase tells your listener that what you are about to say is important, and they'll wait for you to complete the thought while you're coming up with what to say,\" he said to me in French. (From then on, whenever I heard him speak in linguistically mixed company I noticed every time he used his favorite trick. _The thing is...the chicken looks delicious,_ he'd say. Or _the thing is...the situation in Iraq is no good._ )\n\nRose tolerated our lessons, and sometimes even contributed a word or two at Nicolas's urging, but she seemed preoccupied. On the third day she told me, almost casually, that her parents were coming to visit the following week.\n\n\"How nice!\" I said, not sure what else to say.\n\nAfter a pause, she continued, \"I'd like for them to stay with me.\"\n\nI nodded. \"I understand.\"\n\n\"The only thing is we want to paint the room first.\"\n\nI kept nodding, even as my heart sank into my stomach. She didn't say when exactly she wanted me to leave\u2014and in fact she never actually said that I had to go at all\u2014but I packed up that morning regardless. (It was my first lesson in Rose's uncanny ability to make everything clear without having to spell it out.) I had nowhere to go. Nicolas told me I didn't have to leave right at that moment, but I was too proud not to\u2014a running theme in my story, I should let you know now.\n\n\"Call us if you need anything,\" he said. \"Where are you staying?\"\n\n\"Oh, I'll just go to a hotel,\" I lied. I had a very limited budget and didn't want to ask my parents for help. Over those three days, no permanent lodgings had materialized\u2014the few people I'd met through Craigslist wanted a lot of money up front, or proof of current employment, or both. I had neither. And I'm sure I made a poor impression, stuttering my way through the pocketful of English I knew, unable even to understand what \"security deposit\" meant.\n\nThe realization I'd had about my poor preparation was even truer than I'd imagined. I don't know how I got through the next two weeks; a mix of determination and luck, I suppose. From Rose and Nicolas's I moved to a youth hostel on 14th Street. I signed up for an English class near Madison Square Garden, an area that was more like the New York of my imagination\u2014bustling sidewalks, enormous buildings\u2014but the class itself was terrible. Most of the students were from China, South Korea, or India and had even less of a head start than I did. Yet the teacher would pair us off for hours at a time to practice our conversation, as if two bad English speakers would add up to one good one. Instead, as you might expect, one partner would thoroughly mangle the language, and the other, not knowing any better, would just cheer them on. The room was full of people sitting across from each other, one side spouting nonsense and the other saying, \"Good! Very good!\"\n\nAfter each day's class I met potential landlords and roommates, but the answer was always the same. Then every evening the hostel clerk would ask, \"Are you sure you don't want to reserve a few days in advance?\" and, ignoring his concern, I would always reply, \"No, this might be my last day here.\" I was stubborn. I talked to Jules on Skype using the common area Wi-Fi and admitted that things were a bit slow, while trying not to alarm him. I worried that he'd say to forget the whole thing and come home, and I'd be tempted to listen. It was not the advice I wanted to hear. At night I climbed into my bunk bed, envious of the excited whispers of the tourist girls around me, and cried quietly.\n\nOf course, the inevitable happened: one evening I returned from class to reserve my bed for the night and the clerk said, in a tone that was both sympathetic and I-told-you-so, \"We're booked up.\"\n\n\"Oh,\" I said, considering what he would do if I sat right there on the floor and refused to move.\n\n\"Let me call the one on Bowery and see if they have space.\"\n\nI pushed my two baby-elephant-sized suitcases down the sidewalk in the chilly November dusk. The beds at the new place were in individual cubicles with walls tall enough to block the view but not the sound. I lay down and listened to the ambient noise of the guests I couldn't see, the throat-clearing, the occasional laughter. The bathroom wasn't far from my cubicle, and I heard every flush of the toilet. I'm sure there were people staying there that night in far more dire straits than me, but I was too busy feeling sorry for myself to wonder about them.\n\nI dreamed I was back in Spain, in the small coastal town not far from Valencia where I had studied. I was on a hill overlooking the sea, with a picnic of bread, ham, and a floral Spanish white wine that replenished itself with every glass I drank. A cat came up to my blanket, but instead of mewing, it let out a blaring noise. The cat wouldn't stop making that noise. I'm not sure how much time passed before I woke up to the fire alarm screaming on the wall above my head. I hurried outside with the other guests, a coat hastily thrown on over my pajamas, as we waited for the fire department. My feet were freezing. Then it started to rain.\n\nAs I stood in the downpour, miserably waiting to return to a bed that was only mine for the night, my mind went, as it does in hard times, to the most popular comic strip in France. _Asterix and Obelix_ is to France what _Peanuts_ is to the U.S. But instead of a lonely boy and his dog, the heroes are mismatched friends in ancient Gaul who spend their days rebelling against the Roman Empire. All they have on their side is pluck and a magic potion. It's a very French comic, but it would take me too long to tell you exactly why.\n\nThe important thing to know is that Asterix is small and clever, while Obelix is big and strong, if a bit na\u00efve and impulsive. As a baby, clumsy from the start, Obelix fell into a vat of the village's magic potion, rendering him nearly indestructible. Obelix had his potion, and I had my champagne\u2014the point is that I am much more like Obelix than like Asterix. And when I find myself in difficult situations, I usually end up blaming our shared traits for my troubles. I, too, have always stumbled along belly first without thinking and can have a bit of a temper. Look where it had gotten me. The next time my family compared me to Obelix, how would I ever refute them?\n\nBut if Obelix benefits from the clever presence of his friend, I had no one to help me but myself. Something had to change if I was going to make it to January. I quit the English class. I had to forfeit the rest of the week's tuition, but I'd rarely been so sure about a decision. Sitting at a caf\u00e9, surrounded by New Yorkers productively typing away on their laptops\u2014how I wanted to be one of them!\u2014I Googled \"French restaurant East Village\" and wrote down every address. (I would keep that pen for years, as a good luck charm.) My plan: walk into every single one until I got a job. There's a reason restaurant work is the obvious path for new arrivals with poor English skills, and I wished I had thought of it earlier.\n\nIt was also my last, best hope, and I didn't dare imagine what I'd do if it didn't work out.\n\nAt the first restaurant on my list, a bistro and jazz bar on Second Avenue, the manager didn't even look up as he turned me down, just waved me away and kept going over his receipts. The whole thing took less than a minute, and then I was on the sidewalk again, trying to suppress my sense of panic, to keep from hailing a cab to JFK right there. It was probably half an hour before I started walking in the direction of the second restaurant.\n\nL'\u00c9l\u00e9phant, on First Street, was a Thai place with a charming blue-and-yellow awning, and I soon found out why it had come up in my search\u2014it was French-owned, and half the menu was French-influenced. I took a breath and approached the man standing in the back, and, taking a gamble, spoke to him in French. I told him I had a passion for Thai food (not true) and a strong work ethic (true).\n\nWhen I finished, he thought for a moment, and said only: \" _Mais, avez-vous de l'exp\u00e9rience?_ \"\n\nYes, I lied, I had experience.\n\nHe nodded.\n\n\"You're in luck,\" he said. \"Someone just quit and I need help with lunch. You can start now? Here's an apron.\"\n\nLuck had found me! And I'd done my best to give it a chance to. L'\u00c9l\u00e9phant was my first exposure to working in hospitality, and though I stayed there only until the end of December, the experience has paid off to this day. I loved the restaurant. It was my introduction to the multiplicity of New York: a French manager who oversaw a staff of French and American servers and a head chef who had emigrated from Mexico as a teenager and now cooked Thai food. I spoke French to the manager, teased the chef in Spanish, and repeated to customers the same three English phrases:\n\n\"How are you today?\"\n\n\"Do you know what you'd like to order?\"\n\n\"How is everything?\"\n\nEven now these are the three sentences I can say with the least trace of an accent. My English rapidly improved as I interacted with diners\u2014faster than I could have ever hoped had I remained in that overheated classroom by Madison Square Garden. I worked nearly every day, mostly the lunch shift and sometimes dinner. When it was quiet in the afternoon, I sat at a back table and studied the Ch\u00e2teau la Nerthe materials.\n\nMy biggest disappointment with L'\u00c9l\u00e9phant was that whenever I went down to the basement I saw the restaurant's wine stacked with the other dry goods in boxes next to the boiler. It was stifling hot and damp down there\u2014a terrible environment for wine, which is best kept sideways in a cool, dark spot. One afternoon I poured myself a glass, and it was awful, warm and vinegary. It had overoxidized, and every other bottle that was sitting by the boiler had likely suffered the same fate. But in all the time I was there, I never heard anyone complain.\n\nI said nothing to Jean, the manager, not wanting to risk my position. It wasn't worth it. The job had allowed me, finally, to find a room in a three-bedroom apartment off Second Avenue. The leaseholder, Carl, was about my age or a little older, from California and living in New York to finish a novel he was working on. He was the first Californian I'd ever met, and he looked the part: blond, tan, and always in knee-length shorts, even as winter approached. He interviewed me in his living room, arms thrown over the back of the couch and one leg propped up on the coffee table while I sat, hands in my lap, in a chair across from him.\n\nAfter we'd spoken for a while I asked, \"Your novel, what is it about?\" He yanked his leg off the table and sat up straight, and I realized that no other candidate had asked him this.\n\n\"It starts out a roman \u00e0 clef,\" he said, gesticulating urgently. \"You know what that means, of course, you're French. It's the life of a normal guy, normal childhood in the San Fernando Valley. Then\"\u2014he came even more alive, his hands moving wildly\u2014\"it turns into, like, a thriller. He discovers a secret organization that runs all of Hollywood, Silicon Valley, _and_ the agriculture industry, and a conspiracy involving movie revenues and the water utility. But the thing is, no one will believe him,\" he said, his voice dropped to a hush. He sat back again in satisfaction. \"It's a satire,\" he finished.\n\n\"Of what?\" I said. I recognized the word \"satire,\" which though pronounced differently, is the same in French.\n\n\"The _system,_ of course,\" he said, as if there were no other answer. He had only one more question for me. \"So what do you do?\"\n\nI could feel myself beaming uncontrollably. \"I'm a waitress.\"\n\nIn the larger scheme, getting an apartment was the most minor of accomplishments, something thousands of people do every day in New York. But I was ecstatic to finally have something good to tell Rose, who had texted me a few times to see how I was doing. I'd replied only _\"Bien!,\"_ not wishing to say any more, since nothing really was any good. The last thing I wanted was to confront her guilty and pitying gaze. But once I signed on the apartment, I agreed to meet for coffee and ended up telling her everything, not only about my improved situation, but the days I'd been stuck in that dead-end English class, sleeping at the hostel. She put her small hands over her mouth in horror. I told her things I hadn't even told Jules, and it all poured out in one breath. When I was done, she told me about her parents' visit and complained briefly about her job and how she hadn't been able to take any time off, then sighed as if she had just exerted herself. I didn't know if she'd really wanted to share, or just reciprocate a bit, but I appreciated it all the same.\n\nShe hesitated for a moment. \"This Friday my friend Maya is having a party,\" she said. \"You should come. She is French, too. There will be a lot of French people. And also some people who are not French.\"\n\nI tried not to show my excitement. \"That sounds very nice,\" I said.\n\nIn my new room in Carl's apartment, I made Jules, on Skype, help me choose a dress to wear. He didn't understand why I was trying so hard for a house party with people I didn't know. I told him that was exactly why I had to try so hard.\n\nI left my new apartment far too early to be fashionable and killed time by walking around the East Village. It was a mild evening and it felt like everyone was out, enjoying the last of autumn. On one street, I passed groups of young people in leather jackets and belt chains and piercings. Then I turned a corner and everyone was in skinny jeans and flannel shirts, buttoned all the way up. I reached the building, walked around the block twice more, and rang the bell. No one answered. I texted Rose, then called her. I checked and rechecked the little piece of paper on which I had written the address until the creases started to tear. Ten minutes passed, then twenty. I sat on the steps in a panic.\n\nHow quickly everything can turn! I'd been so hopeful, and now was overwhelmed by dark thoughts\u2014that the party had been canceled and she forgot to tell me or, in a nightmare right out of high school, that I'd been invited as a joke. Or did I have the wrong night?\n\nA couple approached the building and I frightened them by asking in garbled English, tears in my eyes, about the party whose hosts I did not know the names of. They had no idea what I was talking about, and went up without letting me in. I sat back down on the stoop and cried. After the job, the apartment, I'd been sure that a social life would be next, and now I was paying for that optimism.\n\nForty minutes after I first rang the bell, forty despairing minutes during which for some reason I didn't go home, Rose called me back. The background was loud; there was music. \"People have been coming up, I don't understand how we missed you,\" she said. Seconds later, she appeared\u2014across the street. I'd had the address wrong. She waved and smiled and laughed.\n\nWe often place too much emphasis on first impressions, when the second is truly more important. I met other people at that party who would become a part of my life, but I remember it mostly as the night that set Rose and me on the path to great friendship. For all I've lamented my poor planning, it's the surprises that have stayed in my mind. I'd come seeking a metropolis and, although I didn't know it yet, ended up in a village.\n\nMy roommate Carl, to whom I wrote the rent checks, was odd, but easygoing and clean. Our third roommate, Shaina, was a woman of Indian descent who worked in finance and had lived there only a month longer than me. One weekend early on, we all went out to bond as roommates, but the bar was a little too loud and crowded for me, and I left early. When Carl and Shaina returned late that night, they were drunk and laughing, and in the morning did a poor job of pretending they hadn't slept together. Then they stopped speaking completely. I had no way of telling whether this was just a normal part of living in New York or not.\n\nAt the end of December, I quit the restaurant. It had been a nearly perfect experience\u2014busy at nights and easy at lunch. I told Jean I'd be leaving by Christmas, and he was nonplussed. \"Sure, sure,\" he said. \"Thanks for letting me know.\" One thing he didn't know, however, was that in my last couple of weeks I started trying to steer customers away from the overheated wine. If a diner ordered a glass, I'd say very casually, \"We have nice beer, too; it will go with the food even better.\" Sometimes it worked, sometimes it didn't. If they still wanted wine I watched from a distance as they took a first sip, always hoping they would say, \"This is terrible! Let me talk to the manager!\" But no one did.\n\n\"You suddenly care so much about the wine,\" my mother said. We were talking regularly again, after a stretch of brief and inconsistent calls when I was having a hard time and didn't want to admit it.\n\n\"Well, I don't want anyone to have a bad experience.\"\n\n\"Is that all?\" she said cryptically.\n\n\"Plus, if I'm going to be helping Uncle Alain, I should at least play the part.\"\n\n\"If you say so.\"\n\nI knew what she was getting at, although we never said it out loud. Neither she nor my father (they divorced when I was a little girl but remain close) ever pressured me into seeking a career in wine. This was the first time she hinted that she may have hoped I would.\n\n# Thinking About Wine\n\nA bottle of wine is always a surprise. This is one of the reasons why nobody can know everything. Not even the winemaker knows what she'll find inside. You can make an educated guess; you can have expectations, based on the producer, the vintage, the grapes, the region of origin, and, most important, your past experiences. But you can't know for sure.\n\nSo when you greet a newly opened bottle of wine, do so with care and attention. Evaluate it the way you would anyone you've just been introduced to: with generosity, but a critical wariness. After all, any recent acquaintance can end up being a dud. Do the wine the honor of sizing it up. Don't just drink it blindly\u2014a lot has gone into giving it a specific personality, and who doesn't want to be recognized for who they are?\n\nPouring. Before you pour, make sure you have the right setup. Are you using wine glasses? Good. Have they been stored for a long time in a closet? Bad. Rinse them out. Take a look at the cork, if there is one. It won't tell you everything, but if it smells moldy or vinegary, that's not a good sign. Wine is not inert\u2014it is alive, always changing. These details are important.\n\nSmelling. This is an underrated step. Don't just sniff for the sake of sniffing; put your nose into the mouth of the glass and inhale deeply. You can tell so much about a wine by its aroma, before it ever touches your tongue. You often see people swirl their wine before smelling it. Next time, try taking a whiff _before_ swirling\u2014this is your first impression. See if the wine reveals anything to you so soon after pouring, if you can detect the aromas typical of the appellation, or of the grapes. (The \"appellation\" is a government-controlled designation based on geography and character\u2014there are appellations for cheese, as well, though in France wine is the largest and most famous system.) At this point, the wine might not have much to say, or it might have a lot. Now swirl it around the glass, helping it get a little air, so it can really begin to speak for itself. Bring it to your nose again\u2014if you still don't smell much, the wine probably isn't _open_ yet, and is a little shy! Give it time to reveal its character. You can either sit and wait patiently, or decant it in a carafe. A carafe can be a wine's best friend, whether the wine is red or white, especially if it is good quality and\/or a recent vintage\u2014if it's young, no wonder it's a little hesitant. If you don't have a carafe, pour some into your guests' glasses so there is more air in the bottle, letting more wine open up at once. (I hope you didn't open the bottle just five minutes before dinner.)\n\nLooking. All this time, you've also been taking in the wine with your eyes. You'll have noticed that a young red wine will be redder, brighter, more purply, and that older reds will have a slightly orange hue. Young whites will have a yellow-green glow, older wines softening into a brownish tint. Anything very dull and faded-looking may be past its prime\u2014or gone bad. But you won't know for sure until the next step.\n\nTasting. Now you will really see what a wine is made of. As you take your first sip, let it coat your entire tongue. You don't shake someone's hand with one finger, but with your entire palm. Let every one of your thousands of taste buds encounter the saltiness, the sweetness, the bitterness, the acidity. Let the wine move all around your mouth and notice how the flavor progresses with every passing second. Then swallow. Congratulations! Take another sip. Have another glass. The last glass in a bottle will taste different than the first\u2014with wine, time affects everything. Pay attention.\n\nRemembering. This is the step the experts don't talk about enough. _Remember_ what it is you tasted, and how you responded. What sort of fruit pleases you the most, and how much of it? Maybe you like earth tones, licorice, and spice. If you liked this wine, where is it from, who makes it, and how old is it? The most important thing is identifying what you like, so that you can purchase the right wine and drink it on the right occasions. This doesn't happen at once. Like all wisdom and self-knowledge, it takes time, and concentration, and care. Keep a wine journal. Take pictures of the labels with your phone. Anything. Just don't forget. The wonder of wine, like everything else in life, is in the details.\n\nThe French see their wine, much as they see themselves, as defined by elegance, refinement, tradition. That we have a long and proud tradition is indisputable, but the truth is that not all French wine is elegant, and not all French people are refined. I recently tried to help a Frenchman who drank only Burgundies look for something new. I recommended a juicy Sancerre made from pinot noir grapes, and he shook his head to cut me off. \"No, no,\" he said, \"I don't like pinot noir\"\u2014and I was obliged to inform him, as politely as I could, that all of Burgundy uses pinot noir.\n\nIt's easy to criticize youth for being brash, opinionated, and immature. But it's just as easy to find fault in the old for being condescending, unadventurous know-it-alls. I wanted to live in England more than the U.S. because it was more familiar, more refined. More _Old World_. America\u2014well! You know the stereotypes as well as I do. I'm not proud to admit it, but even though I was excited about what awaited me across the Atlantic, my expectations of a worthy cultural experience were not very high. If you are thinking, with a bit of irony, that this plays right into the stereotype of the snobby French girl, you're right. I wouldn't have understood this then, as I had very little conception of what it was _to be French,_ or how we looked to the outside world. After all, I'd been French all my life and had no reason to wonder what it meant.\n\nI would soon learn that there were many ways I wasn't so traditionally French after all, and had I known this earlier I may have been more open-minded. My family had always teased me about my outbursts, my impulsiveness, the way I sought out new things\u2014for being youthful, for being very \"Obelix,\" who is hardly _une personne typique fran\u00e7aise_. (My \u00e9migr\u00e9 life soon showed me there were others out there who also broke the French mold: a few months after I arrived in New York, Nicolas confessed his deepest shame one night over dinner. A perfect Parisian in every other way, the man hated cheese. Have you ever met a Frenchman who hates cheese? No, because they are either hidden in caves by the government or they run away to America.)\n\nWine also inspires stereotypes. One of the simplest is a preference for older vintages\u2014the idea that if you have to choose between a 2009 and a 2010 you should always go for the 2009. But this isn't true! There are many factors you must take into account, including the quality of the vintage, which we'll get into later, and, most important, what kind of wine it is and what kind of wine you _want_. All wines change as they age, but some change for the better over a long period, and some\u2014you may be surprised to discover\u2014hit their peak rather quickly.\n\nThere's more to youth than just being pre-old. Well-aged wines, like older people, are complex and often marvelous. But wines meant to be enjoyed young, like many of the varietals from Beaujolais, for example, are perfect with the right food and on the right occasion: fresh, fruity, and fun. And, like Beaujolais, youth can easily be misunderstood, dismissed offhandedly for the very qualities that make it great.\n\nNot so unlike the U.S.\n\nL'\u00c9l\u00e9phant was my first substantial glimpse into American culture. I saw all of New York pass through the doors in those weeks: people of every age, of every color and creed, people who I imagined lived rich American lives as bankers and lawyers and students and artists. In the interests of educating myself, I tried to eavesdrop as much as I could, but could understand only snippets that made no sense in isolation. It was overwhelmingly new, and so much to process that I would walk home at night in a daze, going over everything I had seen and heard.\n\nI was amazed at how fast everything moved. At lunch, people ate and left in a rush, and lingered only a little longer at dinner. It was nothing like France or Spain, where meals are long and leisurely. In Paris, if you sit down for dinner at eight you're free to stay until midnight. (The first time I was asked to leave a restaurant because the next party had reserved the table, I was shocked.) Here, we saw dozens of diners every night, and we turned over tables as quickly as possible.\n\nI was also confused by all the rituals of politeness. I had to ask, \"How are you today?\" and \"How is everything going?\" Every stage of the interaction between waiter and customer was acknowledged with a smile. The service in French restaurants is nothing like this, though I wouldn't go so far as to call it rude. They just have different priorities\u2014and most French diners, in fact, would see this level of attention as bordering on harassment.\n\nAfter I quit, I went home for several days for Christmas. It was a short break before the new year and my start with my uncle's new importer, the importer and champagne house Mo\u00ebt Hennessy. I almost didn't return to Paris at all, not wanting to spend the money or disrupt my acclimation in the States, and I felt sheepish flying back having accomplished nothing more than finding a place to live and waitressing for just over a month. But my father insisted on paying for one leg of the flight, so I gave in, admitting to myself how much I missed my family after so short a time.\n\nWe spent our usual Christmas Eve and morning together\u2014with great food, yes, but even better drink. Imagine, my mother bringing the champagne, my father the cognac, my uncle the Ch\u00e2teauneuf-du-Pape\u2014some of the best libations France has to offer on one dining room table. I answered their questions without lingering over my first miserable weeks. I didn't want them to know how close I'd been to giving up before the real job had even begun.\n\nJules came to pick me up to spend Christmas afternoon with his family. I can't talk about my homeland without talking about Jules, who was the most quintessentially _French_ person I had ever met: sardonic, intelligent, intense, proud. If I was a young spirit, he was an old one, cool, with strongly held beliefs that had grown and taken hold like tree roots over many years. And there he was waiting for me, standing beside his run-down Mobylette motorbike with his hands in his pockets and his brown leather jacket on, looking much like the first time I'd ever seen him: James Dean with a strong Gallic nose and deep-set eyes, his hair cut close to his scalp.\n\n\" _Salut, mon coeur_ ,\" he said, playing it as cool as if he'd just seen me yesterday, until I got close enough for him to throw his arms around me and lift me up.\n\nHe had just started design school when we'd met four years earlier. My friend Vera had invited him along to see a movie, and we watched him come chugging up on his motorbike in that same leather jacket. All the other students had fancy scooters they parked outside their pricey rented flats, but Jules had his old Mobylette that was a pain to start, and he lived with his parents. I was especially charmed by his slightly bashful demeanor, which would flash at moments when he was unsure of himself, his wide mouth breaking into a shy smile and his long lashes blinking rapidly.\n\nA few days after the movie we met again at a party, and a few days after that we had dinner together, just the two of us. We still didn't know much about each other, but I wondered what Vera had told him about my background because he made sure to hold the wine list himself, as if to impress me. He didn't know how far from a connoisseur I was and that he had no reason to try to prove himself. But it seemed presumptuous to say something, so I stayed silent as he scanned the list with his finger, making thoughtful sounds. He glanced up once to gauge my reaction, and I tried to convey with a smile that I knew no more than he did.\n\nThen his index finger stopped, and he said to the waiter, \"This, this one.\"\n\nThe waiter paused, and said, \"This one?\"\n\nJules looked briefly annoyed. \"Yes, yes,\" he said, and the waiter gave a tiny shrug and left. \"You'll like it,\" Jules continued. \"It's from Dordogne, where _my_ family is from,\" he said. The way he placed the emphasis on _my_ all but confirmed that he'd heard about my family's business.\n\nThe waiter returned and poured a taste of the white wine. I will never forget the look on Jules's face as he sipped it.\n\n\"No, no,\" he said to the waiter. \"This is very sweet.\"\n\nThe waiter presented the label to Jules again. \"It's a Monbazillac. You have chosen a dessert wine,\" he added, with a hint of glee.\n\n\"Well, we can't drink this. It's too sweet for our dinner. We must switch it for something else.\"\n\n\"Is the wine spoiled, _monsieur_?\"\n\n\"No, it's just too sweet.\"\n\n\"I'm sorry, _monsieur,_ if there is nothing wrong with the wine we cannot take it back.\"\n\nWe were nineteen-year-old students with no money\u2014ordering another bottle was out of the question. \"It's fine,\" I said in a hurry. \"We'll drink it.\" Jules's face steadily turned red as the waiter filled my glass and then his with the Monbazillac. I sipped and, wanting to reassure him, said, \"It's good.\" It was good\u2014sweet, of course, reminiscent of peaches and apricots with a hint of spice\u2014though I wouldn't have known at the time how to articulate all that. I hadn't even heard of Monbazillac, though I now frequently recommend it as an affordable alternative to France's most famous sweet wines: the Sauternes of Bordeaux, which are rich, floral, luscious.\n\n\"It doesn't go with the duck.\" Jules frowned.\n\nThis was an important moment in those early days of our romance. I was young, but there are some things a woman learns to spot very quickly, and one is the particular disappointment of men. Especially if paired with embarrassment, it can soon turn pouty, or angry, and then you know you are in the presence of a man who doesn't handle obstacles very well. I sipped my wine nervously as he took a slow draught and let it sit in his closed mouth for a few seconds before swallowing.\n\nThen his eyes relaxed and brightened, his forehead smoothed out, and his brow lifted in amusement. \"Oh well, _tant pis!_ \" he said. _Too bad!_ And we both laughed for a long time.\n\nIt was at that moment I felt he was someone I could love.\n\nAfter an early Christmas dinner in his family's apartment above the suburban school where his parents taught, we rode the Mobylette back into Paris. It was cold; Paris winters are not as cold as New York's, but the air is moist, which I've always found unpleasant. We parked near the H\u00f4tel de Ville so we could spend some more time together before I had to go home. Walking around the city had always been the easiest way for us to be alone when we were both living with our parents. He knew the Paris streets better than anyone else, and liked to point out to me the places where bits and pieces of history had happened.\n\nAs we began to walk, I tried to explain Manhattan to him, the way nearly all of its streets were arranged in a grid, making it impossible to get lost (for the most part).\n\n\"Sounds boring,\" he said.\n\nI found myself feeling a little defensive.\n\n\"It's elegant, in its way,\" I said.\n\nWe were holding hands, and he turned a little to look at me with disbelief. For most French people, complaining about everything, especially their own country, is a part of the national character. But Jules was a true patriot and believer in the French state as the most advanced society in the world, and he never failed to compare it favorably to the U.S. (which he had never visited). And whatever skepticism the French may have about the American way of life, they generally admire New York as a symbol of human potential, and as a place where people arrive from all over the world to make new lives. Jules was exempt. To him, New York was just a denser concentration of all the things that already made him suspicious about America; in particular, he railed against Wall Street as a glorified money-launderer for the military-industrial complex. Our generation had been sensitive teenagers on 9\/11\u2014Jules and I met just months afterward\u2014and we'd been in college during the disastrous American war in Iraq. It had left my peers trying to understand the world through a prism of disillusionment.\n\nThough we had discussed planning a visit during the six months I would be working for my uncle, we had yet to decide when. I needed to know my job's travel itinerary first. I also worried that he would back out, since he had already vowed (years before my uncle's offer ever came up) never to step foot or spend money in a country where George W. Bush was president.\n\nWe walked from Notre Dame to the Bastille. We saw each other again the next day, and the next\u2014the day on which it started to feel as though I maybe had never really left, like Alice waking up on her side of the looking glass. But the end of the week approached and I realized that I would soon have to step back through the mirror again, back to the view of bustling Second Avenue just outside my East Village window, to Rose and Nicolas, to the smell of laundry on my way to the subway, to the caf\u00e9 where I liked to get a chai latte on cool mornings.\n\nOnce again I didn't let Jules take me to the airport. \"But you are coming to visit, right?\" I asked pointedly the last night when he dropped me off at home.\n\n\"Six months isn't really so long, is it?\"\n\nHe was joking, but I wasn't, and I made him soothe me for a minute in his arms before letting him off the hook. I watched him round the corner on his Mobylette, feeling my chest tighten when I could no longer see him.\n\nThe first time I'd landed in New York, just two months before, I had been perfectly helpless, but now I knew where to go, and\u2014a fact that thrilled me when I remembered it\u2014in my purse was a key to the door of the New York City apartment where I lived.\n\nOne thing I needed to do before the new year was buy some professional clothes. My waitress uniform of black shirt and jeans wasn't going to cut it; neither were the shorter skirts I'd brought with me in anticipation of spring. I took the train uptown to the Fifth Avenue department stores but ended up just looking at the opulent Christmas window displays alongside the tourists, feeling happy not to be counted among their ranks. I'd never been on such crowded sidewalks, even on the Champs-\u00c9lys\u00e9es. It was all very lively, but I knew without even going inside that I wouldn't be able to afford anything at Bergdorf's or Saks. I walked all the way down Fifth to the Flatiron District. There, at a more modest store, I bought three simple outfits to rotate through until my first paycheck.\n\nBack at the apartment, I put one outfit on and walked from one side of the room to the other, sat on a corner of the bed, crossed my legs. Carl came in without knocking, as he often did, and sat down on the other corner eating an apple and complaining about the holidays. Not the upcoming holidays\u2014all holidays. He'd gone home to California for a short time; I didn't know what his relationship with his family was like, but it must have been pretty good if they were sending him enough money to write his novel and pay rent without starving. Shaina was in Virginia for another week; the two of them had yet to start speaking again. I was probably the only person he talked to regularly except for the coffee shop baristas.\n\nI ignored him for a few minutes and then, while examining my new professional look in the mirror, said, \"Carl?,\" interrupting his stream of thought.\n\n\"Yeah?\"\n\n\"You have to knock before you come in.\"\n\nHe looked around as if just realizing where he was. I'd never asked anything of him before, because he liked to remind me just how many people had applied for my room and that I'd been lucky to get it.\n\n\"Oh, sorry,\" he said. He was holding the apple core in his hand and moved to throw it away in my trash, but thought better of it and ambled out.\n\nFifth Avenue aside, nothing could have prepared me for the glamour and luxury I would encounter in my first two weeks of preparation for my role as Ch\u00e2teau la Nerthe's brand ambassador. The Tuesday after New Year's I went to the Mo\u00ebt offices in the West Village to meet my liaison and travel manager, Luc, the son of French immigrants whose nearly perfect French and totally perfect American English was a combination I'd never encountered. He would be my main contact and help me arrange everything with the regional wholesaler offices I'd be visiting\u2014the ones who'd be doing the actual on-the-ground selling of Ch\u00e2teau la Nerthe around the country.\n\nBut before I began my six-month tour, I had to do two weeks of training with Mo\u00ebt's top Manhattan salesperson, a woman named Elise who, like me, was born in Paris but had lived in the U.S. for five years. \"Have fun out there. You have a lot in common,\" Luc said as he introduced us. I couldn't tell if he was joking. Elise was blond and statuesque and intimidatingly beautiful. I thought at first glance she was about thirty, but after our first day together I realized she must have been only a year or two older than me\u2014perhaps twenty-four or twenty-five. Her assurance and polish had deceived me. She was one of those women who could get away with wearing black and white every day and look different every time. She favored sleeves with wide cuffs that floated around her wrists as she moved. I might as well have shown up to work in sweatpants and braces.\n\nElise's task was to introduce me to clients and show me how her job worked. How her job worked, I learned, mostly involved eating at the nicest restaurants in New York and chatting up the chefs and sommeliers so that they would buy more wine. And she was very good at it, seeming to be friends with everyone we met, never a word or strand of hair out of place. I didn't see how I could ever do what she did.\n\nThat first day, as we lunched at a three-star restaurant, she leaned over and whispered in our mother tongue, \"Rule number one, don't forget.\" I sat at attention, ready to absorb her hard-earned wisdom. \"Always. Be. French.\" I looked around to see if any of the waitstaff were in earshot, but they'd positioned themselves at a respectable distance.\n\n\"Pardon?\" I said, not understanding. I'd spent the meal being careful not to be poked by one of her high heels whenever she crossed her legs, and now I had to refocus in order to process this advice.\n\n\"You must use your Frenchness!\" she said emphatically.\n\n\"Use it?\"\n\nShe leaned over even closer.\n\n\"It's your secret power,\" she continued, raising her eyebrows.\n\n\"My power?\"\n\nShe sat back and looked at me with impatience. \"Do you know,\" she said, sipping her Chablis, \"my American accent has actually become quite good. But I would never use it at work. I have more _authenticity_ speaking with a heavy French accent. Your uncle's wine is expensive. Restaurants and stores want to trust who they buy it from. They will trust you. Why? Because you are French. And being French gives you authority.\"\n\nI was so surprised by what she'd said about her accent that I didn't speak for a few seconds. \"But I don't know much about wine,\" I blurted out, and immediately regretted it.\n\nShe waved off my concern as if it were a fly. \"That doesn't matter. The clients can taste the wine for themselves. There is plenty of good wine out there. What they want is to be reassured. And what you have to do is make sure they remember you and the wine you are representing. What are they going to remember, some wine with a pretty label and yet another high _Wine Spectator_ rating, or the producer who sent an actual French brand ambassador to meet with them? It won't be just what you say, but how you say it. Your selling point is authenticity. You're a beautiful French girl, your name is Dugas, and you're selling good French wine. What American wouldn't believe in you?\"\n\n\"Oh,\" I said, too unsure of myself to point out that this was hardly the definition of authentic.\n\nI didn't know how to feel. It was a relief to know that I had a head start and a cushion in case I floundered, which I was still sure I would. But it was not how I wanted to prove myself, and I worried I would always doubt my own skills if everyone I spoke to saw me first as a Frenchwoman and only second as someone with real knowledge to offer. But then\u2014I _didn't_ have any real knowledge to offer, beyond the talking points given to me by my uncle and the practiced tidbits of wine-speak I was picking up from Elise. Wouldn't it be foolish not to accept any advantage I had?\n\nAnd it wasn't hard to enjoy being Elise's shadow. Just two weeks earlier I'd been having pad Thai at staff meals at L'\u00c9l\u00e9phant and avoiding the rancid wine. Now, I was eating some of the best food of my life, day after day after day. You might think that in Paris it's fine dining all the time, but that's hardly the case. Like any other kid I mostly ate at home with my family. Now I was having three-course lunches paired with wine I couldn't afford to buy for myself. Warm winter squash salad and a creamy fish stew with a rich Chablis. Grilled quail with a Moulin-\u00e0-Vent\u2014one of the ten special \"crus\" of the Beaujolais region\u2014that was plummy and earthy and so much better than I would have expected based on the reputation of Beaujolais (shows how much I knew). Elise introduced me to a rarified realm of New York that I knew I would only experience because of my job. But that didn't mean I couldn't be seduced by it.\n\nWhen I was not tailing Elise my own foreignness was largely invisible to me. As long as I could order food and drink, make change at the Laundromat, keep track of my dollar denominations and the different coins, I felt blended in with the melting-pot fabric of New York\u2014especially since my friends were also French. I often ate long meals with Rose, Nico (we were close enough now that I called him by his nickname), Maya, whose front stoop I'd cried on all those weeks ago, and her boyfriend Alex\u2014the French way, until late, after which we'd stand on the sidewalk chatting before finally parting in a flurry of embraces. Once, Nico invited an American friend to dinner, forcing me to do my best to keep up in English. When we were all outside saying goodbye in our usual _d\u00e9nouement_ , the American friend left us to run an errand before a store closed. Half an hour later we saw him approaching from the other direction on his way home; his mouth dropped open and he said, \"What? You're still here?\"\n\nSo there were some moments in which I saw myself through American eyes. But there were also days I spoke basically no English\u2014working with a French rep to sell French wine through a French conglomerate to mostly French restaurants, eating dinner after work with my expatriate friends, coming home to Carl and Shaina, who weren't speaking English or anything else to each other. Then I would Skype with Jules or my family in French and go to sleep, and dream in French. It was enough to make me worry that my English wouldn't be good enough to get by with on the road.\n\nEven with the American-born Luc at Mo\u00ebt I spoke French. I found it strange and funny how he appeared so conspicuously American in the way he stood, the hand gestures he used\u2014but with perfect French coming out of his mouth. And then he would change again, walking the floor of his office as an all-American guy, with a big laugh and grin, patting the staff on the back, teasing everyone. It was uncanny, and I wondered if I could ever approach that sort of duality.\n\nWhen I was not with Elise in the field, I was with Luc in the office, planning my itinerary, making sure that I would hit both the big markets and the ones with growth potential for my uncle's wines, trying to learn a little about the wholesalers I'd meet. After that, it was up to me to make final arrangements, with the costs split between Mo\u00ebt and Ch\u00e2teau la Nerthe. When we finally had a calendar set, we went out to lunch at a Japanese restaurant by the office to celebrate. I'd hardly ever eaten Japanese food, except for one mediocre experience with sushi back in Paris. I was nervous about the food, but mostly about the fact that our preparations were complete and my job was about to really begin.\n\n\"The owner is a big name,\" Luc told me in French as we approached the restaurant. \"Great food, and he serves your uncle's wine! French wine and Japanese food. Always have a good relationship with your neighbors.\"\n\nThere's nothing like your first taste of excellent sushi, and it put at least some of my nerves at ease. (I'm only sad on behalf of my country that the Japanese discovered and perfected it first.) The tuna seemed to melt with the slightest pressure from my tongue, the rice was perfectly warm and tangy. We drank Ch\u00e2teau la Nerthe's standard white, called the Ch\u00e2teauneuf-du-Pape white (named after the appellation), which was perhaps a little full-bodied and rich for sushi, but I didn't mind. I would have been a fool to.\n\nLuc was clearly enjoying my reaction, and ordered fish after fish for me to try. Outside of the office he seemed more relaxed. He told me a little about what it had been like growing up speaking French at home and English everywhere else until he started working for Mo\u00ebt, and asked me what I thought of his French. I said it was very good.\n\n\"Your English is getting there,\" he offered. He meant it as a compliment, but my face must have seized up in anxiety, because he quickly apologized.\n\n\"I have an idea,\" he said. \"Why don't we switch to English, you and I? From now on, until the six months are over.\"\n\n\"I'd like that,\" I said.\n\nHe nodded. The next words he spoke were in flawless American English, and although there are many moments I could say really launched my wine journey, this is the one I return to most.\n\n\"We'll start right now,\" he said.\n\n# Thinking About Wine\n\nIf young wine is often misunderstood, then Beaujolais may be the most misunderstood of all. Some of it is due to marketing; some of it has to do with the lighter gamay grape and a widespread tendency to equate bold wines with better wines. I say it's a small national misfortune.\n\nWhen you're considering a wine, the grape variety is your first clue about what to expect: cabernet sauvignon, cabernet franc, and merlot in Bordeaux; malbec in Cahors; mourv\u00e8dre in the south\u2014these, for example, are grapes made for aging. The tannins\u2014the chemical compounds that come from the skin and stems and that give a wine fullness and that dry sensation on your tongue and gums\u2014are more concentrated. With enough time in the bottle the tannins evolve, growing subtler and bringing out unexpected flavors. The dominant grape in Beaujolais, however, is gamay, which is naturally light on tannins, and so it lends itself to fruity, refreshing wines best drunk young and slightly chilled. Though to many ears this might sound like \"unserious\" wine, that reputation is wholly undeserved.\n\nThe grape alone does not decide flavor. C\u00f4tes du Rh\u00f4ne red blends rely on grenache, syrah, mourv\u00e8dre, and cinsault, as does the nearby Ch\u00e2teauneuf-du-Pape, but while the former appellation is known for young, inexpensive wines, good wines from the latter can be aged for decades. So what you'll taste also depends on the terroir, the land and climate, which varies dramatically. And much also relies on the skills and strategies of the winemaker. Two producers in the same appellation, working with the same varietals on similar land, can make very different wines.\n\nBut back to Beaujolais. If you are even a casual wine drinker, you've likely heard of or drunk Beaujolais nouveau\u2014it's often the only wine that people know from the region. Beaujolais nouveau is actually a particular kind of wine called _vin de primeur_ , which is meant to be consumed the same year the grapes are harvested. Plenty of French people dismiss nouveau as undrinkable, saying it tastes and smells like bananas and is a vulgar excuse to party. But just as many French people participate in \"Beaujolais Nouveau Day,\" the third Thursday in November when that year's wine is released at midnight; for many bars and restaurants, their entire stock of nouveau is consumed that night. The truth is a little of both: there is plenty of bad nouveau, but it can also be quite respectable in the hands of the right winemaker.\n\nThe real pleasure of Beaujolais, however, is found in its other wines. Officially, there are three \"tiers.\" Nouveau generally comes from the first two: basic Beaujolais, and Beaujolais-Villages, a designation given to villages whose terroir has the potential to yield higher quality wines. Then there is Cru Beaujolais. In France a \"cru\" can refer to a village or even one specific vineyard\u2014the important thing is that a cru has been certified as producing wines distinct enough to receive special classification. A cru doesn't guarantee quality\u2014nothing can do that\u2014but it's a good sign that the history and geography of a particular place is extraordinary. Beaujolais has ten villages that have earned the cru label (the unshaded sections grouped below):\n\nThere's no magic pattern to the cru geography: Morgon produces fuller-bodied wines with good minerality thanks to its soil composition. Saint-Amour wines, just to the south of Burgundy, are among the lightest of the crus, but intensely fruity. Brouilly, the southernmost of the crus, is also light, yielding juicy wines, while those from Fleurie, true to their name, have floral notes.\n\nAnother interesting note about Beaujolais is that its producers are free to make different kinds of wines, which is not always the case. One fine winemaker is Jean-Marc Burgaud, who makes a remarkable nouveau while also producing great wine within the crus R\u00e9gni\u00e9 and Morgon. Another producer to look out for is Olivier Merlin, who makes Burgundies as well\u2014his Moulin-\u00e0-Vent wines are gloriously perfumed and complex. I also admire Marcel Lapierre, Jean Foillard, Jean-Claude Lapalu, Christophe Pacalet, Jean-Paul Brun\u2014I won't go on endlessly, but you get the point. The Beaujolais crus are real wines.\n\nThe only other quirk to note before you start shopping for Cru Beaujolais: the label will list the village name, such as Saint-Amour or Ch\u00e9nas or Brouilly\u2014rather than Beaujolais\u2014meaning there's a good chance the very people who turn up their noses at Beaujolais have actually been drinking it happily all along.\n\nThere are no shortcuts in wine. That is, there is no substitute for simply drinking and paying attention. Ratings and scores can't do it for you\u2014though these days it can feel like a risky proposition to purchase a bottle that doesn't have a big number written on its tag. Well, take that risk!\n\nThis is not to say that reviews can't play a useful role. The danger is when they start to influence\u2014or worse, override\u2014your own critical faculties. If you buy a 92-point bottle, how could you not have that number in your head the first time you take a sip? How confident would you feel disagreeing if that particular wine doesn't suit you? What if, horror of horrors, you prefer a wine that received an _80_? Taken alone, scores don't tell you anything about the type of wine, your personal taste, or who you're drinking with.\n\nScores are convenient shorthand that often say more about the person or institution doing the grading than the wine itself. The accompanying tasting notes are to my mind more valuable. If they say this bottle has some spice to it, think: Do you like a peppery kick? Do you prefer red fruit or dark fruit? Used wisely, the notes help remind you about your own preferences.\n\nAnother shortcut is the grape variety. I was surprised to discover that most New World wines are known by a single variety, and are even produced with the goal of achieving some \"true form\" of that grape. French wines are referred to by region and appellation, _not_ grape, since the French place greater value on terroir; some of their most prestigious wines, like Bordeaux, are blends of more than one grape variety. Wines made in the U.S., however, are generally what are called varietal wines, and are almost always purchased that way: _Oh, I'd like a glass of chardonnay\/cabernet sauvignon\/merlot._ You would never hear someone in France order wine that way. For one, it suggests that all, say, pinot noirs taste roughly the same, when that's not true. Though the grape certainly has specific qualities, a pinot noir from Oregon is different than one from California (or Burgundy!). You might fall in love with a delicate and refreshing Oregon pinot noir and believe you've found your favorite varietal wine. Imagine your surprise then when you drink one from California that is too bold for your tastes.\n\nFor a time, chardonnay, the most planted white grape in the world, developed a bad reputation for being overly oaked and bulky. So now you hear people in restaurants and bars murmur, \"I don't want chardonnay; it's too oaky.\" But many chardonnays are steel-aged, or aged only slightly in oak, resulting in a totally different wine. Don't let stereotypes about a certain grape variety make you prejudge wines that include it.\n\nBut have no fear. The good news is that there is more excellent wine in the world now than ever before. The bad news is that there is more excellent wine in the world than ever before\u2014more than you can drink in a lifetime. You may think this is reason enough to cheat a little, to outsource your own care and judgment. But in fact the opposite is true. Let the sheer volume of wine set you free; since there's no way you will try it all, take your time with it, and look beyond the label.\n\nMy first trip as brand ambassador was by far the longest I'd take during my six months at Ch\u00e2teau la Nerthe and in many ways the biggest test of the job\u2014three weeks on the West Coast. On my flight to Seattle, all I could think about was just how much I didn't know: not just the new cities and unfamiliar customs, but the very business I was supposed to be an expert in. The distance between what was asked of me and what I knew how to do had never been so great. Would I be able to perform the bare minimum, to come off as anything but an idiot? Would I somehow, in these three weeks, achieve an understanding of the wine I could still only imagine?\n\nElise's directive to _be French_ had stuck with me, but I could now see that her success was due to more than that, that for whatever reason she'd played down her professionalism, savvy, and work ethic. While in her company I'd been able to follow her cues and copy her winning smile as best as I could, but now I was on my own and, with nothing but my Frenchness to save me, would quickly be exposed as a fraud and sent home. There was no way that a foreign accent and a few talking points could cover up for a fundamental lack of experience.\n\nI would turn out to be both right and wrong. Wrong because for the most part I did fine on that trip, and when I faltered, people were\u2014in their American way\u2014forgiving and supportive. I also was able to fool a good number of people. But I was also right, too, because the entire time _I_ saw the gap between who I was pretending to be and who I really was. And it bothered me like little else ever has.\n\nI landed in Seattle and took a car to the hotel, and had only an hour to gather myself before a sales rep arrived to pick me up. This would be my routine on the road for the six months ahead: a full day of appointments with restaurants and stores, chaperoned by the rep, capped off by a dinner with our most prized clients in the area. There would be no sitting quietly and listening to others. I was going to have to charm, charm, charm. It was as Elise had said\u2014a personal touch could help a business owner cut through the noise and decide to carry our wines. I'd also overheard Luc on the phone saying, \"Ms. Laure Dugas is coming from France to visit only our best sellers and most valuable clients. You'll have her for three days. Select your top reps.\"\n\nIt was a lot to live up to.\n\nMy first two days I had two different sales reps take me around, both men. The first, Robert, was very much an old-school salesman, with a big smile, booming voice, and salt-and-pepper hair that must have been premature because his face was still unlined. I don't know if Luc had clued in any of the distributors about my inexperience, but the meetings were so painless that by the end of the day I began to feel a flush of confidence.\n\nThe first store was on a tree-lined street just outside of downtown. The owners were a husband and wife who both greeted me with a relaxed warmth. I'd stepped through the jingling door feeling on edge, unsure how I'd ever prove myself to them, but it was immediately clear that _they_ felt honored to meet _me_. And as I tried to form an adequate greeting in English, their faces lit up, as if my fumbling was actually charming.\n\nElise and Luc had been right! It was, as they'd both known, all about the power of expectations. This nice couple had been told they'd be getting a visit from a 92-point expert\u2014why would they ever have suspected that the young Frenchwoman walking through their door had no rating at all? But I also must have somehow managed to keep my cheeks from turning their usual beet red.\n\nAnd although Robert wasn't as charismatic as Elise, he was a pro; he kept up the small talk and helped move the tastings along so smoothly I wanted to ask if he could accompany me for the next three weeks. During that first sales call we developed a routine we then had no trouble repeating the rest of that day.\n\n\"And here I'm very happy to show to you our prize red, the Cuv\u00e9e des Cadettes,\" I'd say.\n\nAs the client's eyes widened in anticipation, Robert would say, as if for the first time, \"Ah, this is from the _vieilles vignes,_ is that right, Laure?\"\n\n\"Yes, that's quite right, Robert. Very old vines. A century, in fact, it's true. The grenache vines my uncle cultivates in this plot are more than a hundred years old. The syrah and mourv\u00e8dre vines are a bit younger, but still among the oldest of the estate, and it's from these three that the Cuv\u00e9e des Cadettes is made. The vines are the lifeblood of the wine. The nineteenth-century owner Joseph Ducos had to replenish his vines from American rootstock after the great phylloxera crisis of 1881.\"\n\nIt sounds good, no? But I was embarrassed by the fact that my job involved repeating these mini-speeches over and over again, and especially that the rep would be there listening each time while I tried to make it sound fresh and new. How do salespeople do it?\n\nBut there's nothing like a little success to take the edge off. The first two stores placed their orders right there in front of me, and as we left Robert winked at me. I'd be lying if I said the feeling wasn't a rush.\n\nRobert and I had our first client dinner with six men of around my father's age, one the owner of the restaurant. They all knew one another from the Seattle wine scene, so not only was I the odd person out, I was also the center of attention. I was very conscious of being the only woman\u2014and of being a _young_ woman\u2014but there was also a strange comfort in being so obviously an outsider. In fact, they seemed to love the ways in which I was different, not least was the fact that I was the only one to order Pacific salmon amid a roster of steaks. I've noticed that the volume of a man's laugh increases in proportion to the number of other men in the room, and these guys emitted belly laughs while I sat at the head of the table and smiled, toggling between honest curiosity at their industry gossip and an air of forbearance every woman is expected to put on when surrounded by men.\n\nI knew to pay the most attention to the restaurateur who was our host. I asked him about each dish, and received some teasing chuckles when I added, \"Egg and plant, what is that?\"\n\nThe lone French speaker among them explained, \"Eggplant. _Aubergine_.\"\n\nI felt myself turning red, until I remembered to be as Elise-like as I could. \"Ah,\" I said, raising myself up in my seat a little. \"Well, eggplant is a stupid name. You should change it.\"\n\nThis got bigger laughs, and my confidence grew. I pointed to the remains of a T-bone steak and said to its owner, \"In my country it is customary at the end of a meal to dip the bone in the wine and chew on it.\" He hesitated, appearing to consider doing it, and his peers around the table started to roar\u2014they wouldn't soon let him forget it.\n\nIn many ways the dinners were easier than the sales calls: for one, I could actually _drink_ the wine instead of spit it out, and the setting was more convivial and less overtly businesslike. Even a novice like me, though, could sense the transactional undercurrent at every meal. Ch\u00e2teau la Nerthe and Mo\u00ebt Hennessey were paying for it, and despite the jocularity, the men's faces would turn momentarily serious whenever they sipped my uncle's wine. When we were down to the last of it, the host raised his glass\u2014toward me. \"We thank you for coming all this way,\" he said. I realized he meant from France, not New York.\n\n_\"Sant\u00e9!\"_ I called out, thinking of Elise's advice. \"To your health.\" And everyone echoed, _\"Sant\u00e9!\"_\n\nThe waitress returned with the check and scanned the faces of all these accomplished older men. I can still picture the barely restrained look of surprise on her face when it was my hand that rose in the air to receive it.\n\nI would be lying if I didn't admit that as surreal and nerve-racking as the experience was, it was also heady and glamorous. Who wouldn't want to feel like a foreign dignitary there to close a high-profile negotiation by signing a restaurant bill for a couple thousand dollars? I suddenly understood how appropriate the title \"ambassador\" was for the job. It only further hit home when Robert drove me back to my five-star hotel, the fanciest place I'd ever slept. It boggled my mind that my employers would put me up here, but Luc had explained it this way: my uncle's wine was not cheap, and Mo\u00ebt was a luxury house. It wouldn't have befitted their joint reputations to have me slumming around in roadside motels\u2014which I would have been perfectly happy to do, though I wasn't about to complain. The hotel was unlike anything I had ever seen. I strode past the front gardens, beneath the glittering awning, through the glass doors held open for me by white-gloved men. My room had a silent pristine majesty about it, with floor-to-ceiling draperies and a marshmallow-soft bed wider than it was long. If I had really been a wine princess, maybe I would have felt at home in such extravagance. The day had been very convincing.\n\nWith an extra three hours between us, there was no way to get in touch with Jules at this hour, to reorient myself around the compass point of his personality. I fell asleep on that bed without changing my clothes or brushing my teeth, the cool touch of the white bedspread as powerful a sedative as the prick of a spindle.\n\nI woke just before dawn, confused by the darkness. I walked down to the water, only a few blocks away, as the sun was coming up. The air was cool and, though I had been warned about Seattle's weather, dry. Some boats were already on the water, or had been there overnight, their lights still on. Even in the dim sunrise the ocean was an immense and humbling presence. It was the first time I had ever seen the Pacific, and it made me realize just how big the country behind me truly was.\n\nI have a theory about cities with a coastline, as opposed to river cities like Paris and New York. People of the river are traders at heart and are more likely to abuse it the way you end up abusing any tool. People of the sea, meanwhile, have a broader perspective. You can never trick yourself into believing you own an ocean. Though I'd been in Seattle for only a day, I felt something in the locals that reminded me of my time in Spain. I looked at the ocean and felt suddenly breathless over how many thousands of miles\u2014with all of America's river people, ocean people, mountain people, desert people, everyone I hadn't met yet or never would\u2014I'd crossed to be standing there.\n\nThen it was Friday, and my magic carpet ride of pleasant sales calls and sociable lunches and dinners was over. It was time for my presentation to the local sales force. This was the real test\u2014no small talk, no helpful chaperone at my elbow. Just me and a hundred and fifty reps in person and on the other side of a video camera.\n\nI woke up that morning laughing. The thought of giving a presentation to a room full of professionals was so unreal, so hilarious, that I kept laughing in the shower, and while salting my eggs at the hotel breakfast. Then the laughter took a sharp left turn into nausea. One or two store owners was one thing; a half-dozen clients at a dinner was another; but a hundred and fifty? The most people I'd ever spoken in front of was maybe my classmates at school, but that could hardly count. I called Jules in a half-hiccupping, half-giggling fit, hoping to calm my nerves.\n\n\"Do you know what you're going to say?\" he asked me.\n\n\"Yes, I've practiced it many times, but\"\u2014hic\u2014\"I'm too anxious.\"\n\n\"Say it to me.\"\n\nI began to recite my opening, voice shaking.\n\n\"No, no, no,\" he said. \"Think about what you are doing. Think about the wine. Pay attention to what you're saying.\"\n\nThe French, for all their romantic reputation, are a relatively unsentimental and blunt people. But I couldn't tell right away if his advice was helping or hurting. I hadn't been worried about the speech itself, only my nerves, but now thanks to Jules I was worried about both.\n\n\"I just need a little reassurance,\" I bristled.\n\n\"I _am_ reassuring you. I know you can do this.\"\n\nI felt a burst of both gratitude and exasperation toward my boyfriend. Angry, frustrated, determined, I spat out the opening of my presentation so quickly\u2014and flawlessly\u2014that Jules couldn't keep up, even though his English was better than mine.\n\n\"That was good, you see?\" he said, satisfied.\n\nWe said our usual affectionate goodbyes, and I hung up with a sigh. I knew, despite my agitation, that he'd helped, and that I wouldn't have gotten so annoyed if we'd been able to stay in as close touch as we normally did when I was in New York. What did this mean for us over the next few months of heavy travel? I didn't know. And I had no time to think about it. At the table where I'd eaten I took a photo of a heart hastily constructed out of sugar packets and texted it to Jules. For the moment, it would have to do.\n\nSo there I was, standing on the other side of the door to the tasting room. The two office managers who'd greeted me and helped me get set up didn't need special powers to see how nervous I was. But they were kind enough not to say anything. My lips felt dry, even with gloss on, but I couldn't possibly drink any more water without risking an incident. I paced in a small circle until it was time.\n\nThen I was through the door and in front of a dozen rows of well-dressed men and women. I spotted the camera to one side, which stared at me unblinkingly. I tried to clear my throat but out came a little cough instead, which seemed to echo around the room. Everyone shifted in their seats. I realized that there was no cue\u2014no prompt for me to know when to begin. Walking in was my own cue. I was supposed to start\u2014now. I searched for the words I'd spent so long memorizing.\n\n> Hello, everyone! My name is Laure Dugas, and I'm so happy to be here.\n\nI took a deep breath as unobtrusively as I could and plowed ahead before my nerves had a chance to catch up.\n\n> I am the niece of Alain Dugas, the winemaker of Ch\u00e2teau la Nerthe. I'm traveling the U.S. for six months to tell our most important clients about the wine. First, I would like to describe for you Ch\u00e2teau la Nerthe and what makes it and the Ch\u00e2teauneuf-du-Pape village so special. Ch\u00e2teauneuf-du-Pape is the most prestigious cru from the southern Rh\u00f4ne. I probably don't have to tell you that. Gigondas, Vacqueyras, and Beaumes de Venise all make wonderful wine, but, let's be honest, they cannot compare with Ch\u00e2teauneuf-du-Pape, which is blessed with the superior terroir. And Ch\u00e2teau la Nerthe is one of the most prestigious ch\u00e2teaux from Ch\u00e2teauneuf-du-Pape. So you might say we have the best of the best.\n> \n> Now, I did not have anything to do with it\u2014I'm just a lucky girl to be representing such quality. My uncle is the one with the flair for winemaking. He was the one who convinced the current owners to acquire the ch\u00e2teau in 1985 when it was producing good but unexceptional wine, and worked hard to improve the vineyard, expand the cellars, and renovate the ch\u00e2teau. I was just a baby. It was he who on faith introduced a greater percentage of the mourv\u00e8dre grape than usual, to complement the all-important grenache base and give the wine great aging potential. He also had high enthusiasm for counoise, which provides structure and peppery, spicy aromas characteristic of a good Ch\u00e2teauneuf.\n\nI garbled a few words here and there, and was nearly derailed by a couple of faces in the crowd whose frowns set off flares of panic in my mind: did no one understand a word I was saying? I could see some pens moving on notepads, but others were still. I quickly scanned for friendlier faces, and located a woman who was smiling enormously at me\u2014but this was distracting, as well, so I ended up fixing my eyes at a point just above their heads.\n\nI finished the introduction and steeled myself for the difficult part, walking them through the tasting. Luc had told me over and over that these reps had at least four hundred wines, or \"references,\" to memorize\u2014it was therefore my job to make sure Ch\u00e2teau la Nerthe was not a name they'd forget. If I was already having trouble communicating, the tasting would truly expose me. Thankfully, no one asked difficult questions, only the occasional request for me to repeat a few facts, which I did happily. They were busy sipping the four Ch\u00e2teau la Nerthe wines (two white, two red), and I stood there feeling an inkling of relief, thinking it was a bit surreal to be surrounded by so many people sniffing, swirling, sipping\u2014and spitting\u2014all at once. They wrote things down and nodded as I recited the notes I'd memorized but still didn't understand.\n\n> Here is our flagship Cuv\u00e9e des Cadettes. My uncle says this release, 2005, is one of the best ever. It's characterized by a dark red color with an intense aromatic expression. It is rich, concentrated, and dense. The tannins are suave, and the wine has a long finish of fruits and spices. It is a wine you can easily age for ten or fifteen years before drinking. But as with all great wines, it can also be appreciated while young if you really can't wait.\n\nOnly once did all the reps pay perfect attention and begin scribbling simultaneously: when I shared the high Robert Parker and _Wine Spectator_ scores. So you see, scores are important to everyone, professionals and amateurs alike, even if there's so much the numbers can't capture.\n\nThe Cuv\u00e9e des Cadettes tasting marked the end of the presentation. I wanted to sink into a plush chair and take a nap, but I still had some socializing to do. The reps shook my hand and said brief thank-yous. Their meeting was to continue without me, so I stepped back out to the office area, where the managers patted my arm and said, \"Great job!\" in a tone that suggested that I hadn't really done a great job at all, or that they hadn't expected me to.\n\nAmerican optimism and cheerfulness are wonderful qualities that aren't celebrated enough. Americans are always lying through their teeth to tell you how nice you look or how wonderful their day is. How could this not make the world a better place, even when reality is in fact a little disappointing?\n\nBut sometimes a kind lie is the last thing you want to hear.\n\nI took a car back to the hotel to get my things before my flight to San Francisco, and could barely keep from bursting into tears when Jules called me. The texts I'd been sending him all morning had been alarming enough that he'd stepped away from dinner to use the phone. But as I explained what had happened, down to the office managers' responses, his soothing tone hardened a little, and he said, \"You are reading too much into things. It sounds like you did fine. Everyone tasted the right wines, yes? You did not mix anything up, any descriptions?\"\n\n\"Yes, no. I mean, yes, I got the order right.\"\n\n\"Then that is the most important thing. They will remember you. Who couldn't? You are very charming when you get nervous.\"\n\n\"I hate you.\"\n\nI wiped my nose with a cocktail napkin I'd taken from the distributor's office, but Jules had managed to thwart any more tears. By the time I landed in San Francisco, I felt better, my mood overwhelmed by the view of the Pacific coastline as we followed it to the nub of land that was our destination. I hadn't known San Francisco was a peninsula, and the sight of the water gave me a little lift. After checking into another luxurious hotel as the sun was setting, I walked up the nearest hill, thinking that would make the return route easier, but the land played a trick on me by falling again, then rising once more, until it was fully dark; and, tired and lost, I took a taxi back and ordered room service.\n\nI ordered room service the entire weekend, and when I wasn't outside I was in the hotel bathrobe. I indulged because I was alone, and because Seattle had worn me out, as new and challenging experiences do. I did touristy things like ride a cable car and take pictures of the sea lions at Fisherman's Wharf for Jules, but otherwise had little opportunity for conversation, which, for the first time since I'd landed in November, I was thankful for. San Francisco felt like a more weathered Seattle, dirtier but also livelier, _almost_ Old World (and at the risk of alienating one great city for another, possibly more beautiful). I'd been told it would be cold and windy, but just as I'd lucked out with no rain in Seattle, it was actually warmer in San Francisco, the air crisp and still, with blue skies.\n\nCome Monday I was ready to go again. As I waited in the hotel lobby for my chaperone, the embarrassment I'd felt on Friday was long behind me. The week went by quickly. My visits to the stores and restaurants went well\u2014I was in my element speaking to just one or two people at a time. There was one sales rep who was aloof and uninterested, despite the fact that I was there to help _his_ client relationships, but I managed to hold my own, even with the small talk and transitions that I'd been relying on my escorts for. Each night I returned to the hotel and sat for a minute trying to think about how I felt: Was I enjoying myself? Was I improving? Was I comfortable talking about wine? And each night, to my surprise, the answers were yes, yes, yes.\n\nMy Friday presentation to the entire sales staff was still a little herky-jerky, but better. I slowed down. I thought about what I was saying, instead of just discharging a rapid-fire barrage of words. I paused and smiled. Once, Luc had given me the strangest advice\u2014somewhat hesitantly, as if he didn't believe it himself\u2014that it could help to picture the audience naked. I didn't even try to follow it. Instead, I pretended the audience was a casual acquaintance I wanted to convert into a good friend\u2014someone I had to win over and flirt with, someone who would bring out, albeit in a temporary, superficial way, the best in me. _Remember me!_ was the command I tried to channel. _Remember me!_\n\nThere were no pitying pats on the arm from the office managers this time.\n\nThen I was on the plane again, to Los Angeles. I wasn't going to train reps this time, but to represent Ch\u00e2teau la Nerthe at Mo\u00ebt Hennessey's annual West Coast wine show. (The Midwest and East Coast shows were to follow in Chicago and New York.) My uncle had made it clear how important this was for publicity. As an event it was less about selling wine than about getting the ch\u00e2teau's name in front of new people in the industry\u2014a rare opportunity to represent ourselves directly to them, instead of relying on wholesalers. It was important enough that my uncle\u2014employer and kin\u2014was flying out himself.\n\nLet me tell you a little about my Uncle Alain. Though he's my father's brother and my grandfather's son, unlike both of them he ended up an expressively happy and positive person. It's possible this is what drew him to the south of France, where there are more people like him. The family members in Paris have more dour demeanors and higher blood pressures\u2014we love one another with little bites and snaps, a tradition Uncle Alain doesn't participate in, instead sitting slightly apart with a big smile on his face. He loves the good things in life, and it's no surprise that his calling was not only to sell wine, but to make it, and make it very well.\n\nJust standing outside the Roosevelt Hotel, right on the Walk of Fame, I felt a sense of d\u00e9j\u00e0 vu even though I'd never stepped foot there: this was Hollywood, which to anyone not raised underground needed no introduction. The giant movie theaters along the street, the palm trees, the sunshine, the warmth\u2014like a summer day in Paris but in late January. Los Angeles, more than any other city in the U.S., looked and felt exactly as I'd expected (even if I hadn't yet seen any movie stars in the hour since arriving). After eleven days on my own, I was hungry for a familiar face, and a little nervous: other than the short time before and after my six-week crash course in wine, I'd never spent much time alone with my uncle. As my father's elder brother, he had been a sage if playful presence in my life, someone I'd always felt like a little girl around (after all, I'd _been_ a little girl for most of our relationship), but now I had to pretend, every second we were together, to be a competent employee\u2014and an adult.\n\nI dropped my bags off and knocked on the door of his room just down the hall. After a minute he opened the door, a little bleary-eyed\u2014I'd woken him from a nap. I started to apologize but he embraced me. \"Here we are! California!\" he said, immediately launching into a string of stories\u2014for my uncle, life is a long comedy that it is his job to narrate\u2014about the guy at the airport who almost accidentally walked away with my uncle's luggage; how he was certain he'd seen Julia Roberts on the street but she wouldn't turn around when he called out; the very hip sunglasses he'd bought in the gift shop because he'd forgotten his, and had I noticed how _bright_ everything was?\u2014all while changing his shirt, pulling on his shoes, splashing water on his face and bald head. \"All right, let's go!\" he said, leading the charge.\n\nWhile the valet went for his rental car, he told me three times to \"just wait\"\u2014until a silver Mustang convertible appeared. \"California!\" he said, as if it were the explanation for everything. Laughing, we took a picture in front of the car and sent it to the rest of the family.\n\n\"Where are we going?\" I asked as we pulled out.\n\n\"To the beach, where else?\" he said. I protested that I wasn't prepared for the beach, but he waved his hand and explained that we weren't going _on_ the beach, just _near_ the beach.\n\nI have since confirmed that both Beverly Hills and Venice Beach are in Los Angeles, but the drive seemed to take forever. The city never ended. We stayed on one road for as far as it took us toward the ocean, and whenever we hit a red light I pushed off my seat a little to look around. My uncle loved driving with the top down, chuckling every so often at nothing I could see, happy behind his new sunglasses. After a while I couldn't hold it any longer and began to ask questions about the business\u2014the new world\u2014I'd entered just a matter of weeks ago.\n\n\"Hush, Laure, let's not talk about work today.\"\n\nBut I would not leave him alone. \"Sulfites,\" I said. \"Someone asked me how much sulfites you put in the wine and I didn't know the answer!\"\n\n\"Just a few milligrams a liter,\" he said with a sigh. \"We are trying to reduce the amount. You can say that. But a little bit is important to halt fermentation.\"\n\n\"Do we have to decant the 2005?\"\n\n\"Oh yes, the 2005 is still so young you must decant it!\" He looked over at me with alarm in his face for the first time, as if wondering if he'd made a mistake with me. \"Tell me you decanted it.\"\n\n\"We did,\" I said, and patted his hand.\n\nHis brow relaxed, but I would remember that moment of tension\u2014a reminder that this was serious stuff.\n\nWe had to speak loudly to be heard over the engine as I asked him more about his wine. He was openly amused, as these were questions I'd never asked before, though I'd had plenty of opportunity to. The top button of his shirt was undone and his collar flapped in the breeze. I'd never seen him quite like this. When we arrived in Venice we walked along the boardwalk and joked about my father, who had never been to California, wondering\u2014as we passed a unicyclist in cutoffs being pulled by a large dog shaved to look like a lion\u2014what he'd think of the place. How remarkable it was, he said, that the two of us were here together, five thousand miles from home. Though I'd always liked my uncle, we now had something else connecting us besides blood.\n\nThe sun was setting directly over the water, something I couldn't remember ever seeing before this trip, and I wondered if the daily sight of the sun being extinguished by the ocean had an effect on all the people who live on a western shore. It certainly made an impression on me. We turned around and drove back to the main commercial strip for dinner, where my uncle knew the owner of a popular restaurant called Primitivo, named after an Italian red wine grape. We sat on the rear patio in the warm night and ordered too much: steak, which my aunt doesn't normally let him eat, a goat cheese salad, Spanish-style potatoes. Alain acted as if he didn't want to think about work, but he handed out cards to everyone his friend introduced him to\u2014regulars at the bar, local businesspeople, a family of four.\n\nThe next day Alain made even less effort to suppress his networking instinct, stopping at restaurants he thought looked interesting as we drove semi-aimlessly around Beverly Hills and Hollywood and West Hollywood and Santa Monica. We ate tacos for lunch because he had a craving, then stopped at three French restaurants so he could meet the management and tell them about the upcoming trade fair. He stuck with ginger ale while I got a little drunk on cocktails.\n\n\"I'll tell you a secret,\" he said at one point. \"This is when I am at my best as a salesman. When I am relaxed and happy. It is infectious.\"\n\n\"It won't work with me,\" I said, feeling a little sorry for myself. \"My English is still too poor. Everyone speaks too quickly and they understand me even less.\"\n\nHe shook his head slowly and dramatically, and I was afraid for a second I was about to be reprimanded by my uncle and boss.\n\n\"That's not it at all. Laure, my dear,\" he said with a hand on my arm, \"everyone understands a smile.\"\n\nThe trade show was not quite what I expected. Mo\u00ebt had rented out a large restaurant and nightclub in which long rows of cloth-covered tables were set up for each of the fifty to sixty producers to be represented. At one end of the room was a temporary stage with a tall metal frame supporting what looked like theater lights. I didn't know what it could possibly be for. We set up early, at eight-thirty in the morning, unpacking the wines, opening them ahead of time to let them breathe and to make sure they weren't corked. We checked before cooling the whites, because, my uncle told me, bad wine is easier to detect at room temperature (this is why cheap wine is often served very cold). Other winemakers were setting up their booths, too, and occasionally my uncle would wander over to someone he knew to say hello.\n\nAfter an hour or two, people started to file in. They were ostensibly all folks in the industry who had signed up, with their identification on lanyards as proof, but I suspected that some were friends or dates tagging along for the free wine. A seller or sommelier there on business might taste dozens of wines, hence the spittoons at every booth. There was no way to stay clear-headed if you actually drank every pour.\n\nLet's just say that our spittoon did not get very much use.\n\nAround midday the purpose of the stage became ostentatiously clear: a DJ had set up while I wasn't paying attention and was now playing loud dance music that echoed through the hall. The lights began to shine bright primary colors over the booths, zipping around and spinning, as if it was Friday night on a dance floor, not Monday afternoon at a corporate, industry event. My uncle was in the middle of describing the color of his Clos de Beauvenir to a potential buyer when a green spotlight landed on the glass, turning the golden liquid an unappetizing neon. I watched as two beautiful young women I'd seen drinking at nearly every booth suddenly turned toward each other and started kissing.\n\nI looked at my uncle helplessly, and he seemed to shrug to me with his eyes: _What do you expect? It's Los Angeles!_\n\nIf he wasn't annoyed, then why was I? Mo\u00ebt had made the L.A. show fun and sexy, and I took it personally. I'd wanted it to be more serious, because _I_ wanted to be more serious. I'd said yes to my uncle's job offer on a lark\u2014free travel! free English!\u2014and had underestimated how much work and care would go into the actual wine. I could see\u2014and my nerves had made crystal clear\u2014that I wasn't going to be able to skate by on my Frenchness. Even if I could fool some people, I'd never fool myself. The bits of knowledge I'd accumulated thus far only made what I still had to learn that much more apparent.\n\nI was also starting to realize how much I missed New York.\n\n# Thinking About Wine\n\nThough a wine label cannot tell you everything, it does convey vital information: where it was made, when it was made, and who made it. French wine labels are among the most complicated, because they must follow strict rules\u2014far more stringent than in the U.S. They're not always artistically designed, the way many American labels are, and are often just-the-facts. What information is allowed or required on the label also varies from region to region, and while we don't have room here to list every combination, the major components of a French wine label are as follows: 1. Region, 2. Sub-region and\/or appellation, 3. Producer, 4. Vintage, and, often of interest, some 5. Miscellany.\n\nNotice: no grape varieties! With few exceptions, the type of grape is not listed on a French label (sometimes you'll see it included in an attempt to market a wine internationally). There are several reasons for this. For one, as I've said, wine is more than the grape. Wine can also be more than _one_ grape. Bordeaux is the most famous example, as the three grapes found in almost every Bordeaux red are cabernet sauvignon, merlot, and to a lesser extent cabernet franc. That's why it's impossible to be in Bordeaux and order a local \"merlot.\"\n\nThe Ch\u00e2teauneuf-du-Pape appellation, meanwhile, allows the blending of up to _thirteen_ specified grapes for red wine, the main ones being grenache, syrah, and mourv\u00e8dre. This astonishing number comes from the fact that the region has wonderful terroir but harsh (very hot!) conditions. Blending allows the producers flexibility to fine-tune their formulas, so to speak, based on what the land gives them. And the Ch\u00e2teauneuf-du-Pape land gives producers a wide mix of soil types to deal with, including the _galets,_ which translates roughly to \"pebbles.\" These stones, which form a natural layer over the sloping mounds of clay and sand rising above the Rh\u00f4ne River, help ripen the grapes by storing heat during the day and warming the vines at night. They also allow water to penetrate the soil, reducing stagnation and disease. The effects of the features unique to this region are really remarkable. And they've allowed great wine to flourish there since the fourteenth century, when the pope lived in Avignon instead of Rome (hence Ch\u00e2teauneuf-du-Pape, which means \"the pope's new castle\"), and even long before then.\n\nHere is a reproduction of Ch\u00e2teau la Nerthe's label for its flagship red, the Cuv\u00e9e des Cadettes:\n\nFrom top to bottom, you can see the information clearly conveyed: the producer (Ch\u00e2teau la Nerthe); the appellation, or geographical region (Ch\u00e2teauneuf-du-Pape); the tier designation of this particular French agricultural product (the highest being _appellation d'origine contr\u00f4l\u00e9e_ , AOC, as the apellation name indicates); the house name for this wine (Cuv\u00e9e des Cadettes); an assurance that the wine was bottled at the estate ( _Mis en bouteille au Ch\u00e2teau_ ); and the year of harvest, or _mill\u00e9sime._\n\nIt's a lot. And every detail, and the order they're in, changes from wine to wine, region to region. (Right now there are sixteen official wine regions in France, and, gulp, 383 appellations. On the next page is a map of the region most important to this chapter: the Rh\u00f4ne Valley, which you'll see is more complex than Beaujolais.) Not all houses have a special \"nickname\" for the wine. The estate may call itself a _ch\u00e2teau,_ or a domaine depending on the region\u2014either word, though, is a clue that this is the name of the producer. Sometimes, as with Cru Beaujolais, the cru is named but not the greater region. Sometimes, as with Bordeaux, the subregion is spelled out in big letters. In some places, like Burgundy, the finest crus are their own appellation, and their own vineyard, too. It's complicated. You'll also see official signifiers like _1er_ _cru_ ( _1er_ means \"premier\"\u2014very good) or grand cru (even better). There are also unofficial, unregulated signifiers that sound fancy but don't mean much, like _r\u00e9serve_ (special collection), _vieilles vignes_ (old vines), or _vieillit en f\u00fbt de ch\u00eane_ (aged in oak\u2014but what kind of oak?).\n\nWhew. Ready to just go back to ordering single varietal wines? But don't give up quite yet: take heart in that you don't need to know French to read a wine label. In many ways, a label is its own language. And to speak it, you need to know not its grammar, but its context: a little about the different regions, and the character of each and the viticulture there. How much of the story you want to hear is up to you, but once you have a grasp of the bigger picture, your trips to the wine store will become much more interesting.\n\nImagine a beautiful village of several thousand people, each interesting in his or her own way. You want to meet all of them, and not just _meet_ them, but get to know them a little, spend time with each and every one. Even if you shared every meal of every day with a new friend, though, it would take years. And how would you remember Julie as you sat down with Bob? What if you really liked Lucy and wanted to eat a second meal with her, even if that meant that everyone else had to wait? Consider also that people are always changing, so if you miss someone one year, well, they'll be slightly different by the time you do get around to them. This is the conundrum when it comes to tasting wines\u2014even just from Bordeaux, say, a huge region with thousands of producers. Imagine the task at hand if you wanted to meet all the wines of France, or all the wines of the world. It's impossible.\n\nYet it was a task I felt I had to at least attempt in good faith, even while acknowledging its futility. I mentioned this to my uncle on our flight to Chicago. We'd spent one more day in the silver Mustang meeting with clients in San Diego, a beautiful, relaxed city that felt like Los Angeles without Hollywood, and were on our way to the second leg of Mo\u00ebt's wine shows.\n\n\"No, no,\" he said. He leaned back against the headrest and closed his eyes, pursing his lips as he settled in to nap. \"You can't think of all the wines out there, like all the books you never got to read. You'll go mad. Think only about the wine in front of you.\" He smiled to himself as he fell asleep, as if he had a particular wine in mind that very moment\u2014possibly one of his own.\n\nIt was good advice ten years ago, and even better today. We are living in a golden age of wine in terms of more information available to the consumer than ever before, and the advancement of viticultural techniques that allow excellent wine to be produced in more places. Don't listen to the voices that say things were better _back then_. It's simply not true. I know an older man who is always nostalgic for the days when French wine mainly meant just Bordeaux and Burgundy. \"Laure,\" he has said with an astonished look on his face. \"I just had a wine from the Languedoc that was...superb.\" And he would shake his head, as if disappointed.\n\nYou may already know about the American wine boom that began with Napa Valley in the 1980s. But you might be surprised to find out that something similar was happening in Europe\u2014yes, even in France. In the seventies and eighties, Languedoc was little more than a giant wine factory, mass-producing cheap table wine. Even regions as historic as Ch\u00e2teauneuf-du-Pape and the Rh\u00f4ne Valley were turning out inconsistent products. For a long time, there was even _too much_ French wine on the market, which may not have been good for sellers but brought prices down.\n\nIn 1985, my uncle saw an opportunity. He loved the terroir of Ch\u00e2teauneuf-du-Pape, and at the time most of the grapes were being sold into trade, not made into wine at the estates. He saw great potential in Ch\u00e2teau la Nerthe in particular, which was a good value for its terroir, but still too expensive for him to buy. So he convinced the Richard family to buy it and hire him as winemaker. It took a decade for my uncle to prove himself to the wary community. Other producers started to take advantage of opportunities in \"up-and-coming\" regions like the Rh\u00f4ne, the Languedoc, and up north in the Loire. And now with great wine coming from all corners, the mark of a true enthusiast is no longer determined by who has a bottle of 1947 Ch\u00e2teau Margaux gathering dust in a dark cellar. That rarified world still thrives, but the truth is that there has never been a better time to fall in love with wine than today.\n\nThe West Coast was still shimmering in my mind, golden and bright like an overexposed photograph, when we landed in Chicago, and it remained the most vivid of all the places I would see in my remaining months as brand ambassador. We spent two days in Chicago before returning to New York. Our only full day we spent at the trade show, held in a hotel ballroom and a far more sedate affair than the Los Angeles disco ball. Nearly all the attendants were sober, literally and figuratively.\n\nChicago itself was beautiful, with a skyline that reminded me of Manhattan, but more manageable. I was charmed by the elevated trains, which made the city seem antiquated to me, but in an admirable way, like an old clock that has been lovingly preserved. The city was also cold. Bone-chillingly, mind-numbingly cold. I'd never been so cold in my entire life, especially having stepped directly out from under the California sun.\n\nThen, nearly three weeks after I'd left, I was back in New York. For someone who had lived in the city for only a couple of months, it was a strange experience, like coming back to someone you'd dated only a short while and weren't sure if you would still be attracted to. It was my third arrival, and my first at night, and my heart lifted when I saw the lights outlining Manhattan in the distance. When I reached the East Village I stood in the hall outside the apartment for a few nervous seconds. I pushed open the door and rolled my suitcase toward my bedroom. Shaina was in the living room watching a sitcom without laughing, and I said hello. She grunted. Carl's door was open, and he was at his desk in nothing but boxers, eating cereal. They weren't acknowledging each other's existence. I smiled to myself.\n\nI was home.\n\nMy uncle stayed another few days in New York. He attended the final show of the Mo\u00ebt tour, and a day later he had a black-tie charity event at the gorgeous main branch of the New York Public Library, also sponsored by Mo\u00ebt. He told me to come with him. It felt like a natural extension of the high-altitude schmoozing I'd done with Elise and on the West Coast (would I ever come down?). But I had nothing to wear. Rose and Maya ransacked their closets for me, but everything was too small\u2014in the case of Rose's clothes, comically so. Maya held up a finger and went into her roommate's closet and brought back a beautiful, simple black dress with a dark, shimmering blue lace overlay. I wore my favorite earrings of Rose's (who couldn't be seen without a pair on), shaped like diamonds surrounded by tiny black gems, and a simple necklace of Maya's.\n\nI met my uncle at the foot of the stairs to the library, the stone lions proudly guarding the entrance. Inside, as I stood awkwardly by the banquet tables in the main hall, there was no doubt that I was among the city's wealthy and elite. I couldn't spot anyone else even close to my age. My uncle seemed unfazed, even if it was not his natural crowd.\n\n\"There are a lot of people here like us,\" he whispered during dinner, as though reading my mind, \"invited to make the night more interesting. If you can't tell by the clothes, look here and here,\" he said, smiling, touching his neck, his ears, and his wrist. My skin suddenly felt bare and underadorned.\n\nThe night before Alain left he took me to dinner with a small, respected importer from upstate, a Qu\u00e9b\u00e9cois man named Mark Brodeur. It was possibly the most decadent French meal I've ever had, eaten the most French way\u2014five courses over four hours in a three-star restaurant overlooking Central Park. It was nearly too long for me, but the men, intent on catching up with each other, didn't notice. I enjoyed watching my uncle interact in French with an old acquaintance, especially someone who had as joyful an air as he did. They swooned collectively over a white Burgundy, praising its toastiness and firm acidity, while I remained silent, too intimidated by the depth of their knowledge to add anything. When we finally finished, it was late, but they insisted on spending a little more time walking up Broadway smoking cigars and interrupting each other. I stayed a few steps behind. When he had to leave to make his train, Mark gave me his card and told me to call if I needed anything.\n\n\"You see,\" my uncle said as we watched him dash off, \"I am making sure everyone is watching over you.\" He took me by my shoulders with a firm grip I had not felt from him before and kissed me loudly on each cheek, then looked at me for several seconds with a big smile, as if to imprint it on me. \"You're going to have a great time,\" he said. \"I can guarantee it.\"\n\nBut once he was gone, like a floating feather I slowly descended into a sort of lull. I spent a week in Houston and Dallas, which I hoped would add new exciting dimensions to the map of the U.S. that I was drawing in my head, the way my previous trip had, but I was disappointed. It was nice to have a warm reprieve from the New York winter, my sales calls and client meals all went smoothly, and both cities were impressively large, but I couldn't quite get a grasp on either. I knew Houston was near the gulf, but didn't get to see the water, and the parts of town where I spent my time remained anonymous to me. I was also likely affected by constant texts from Jules, warning me that I was in George W. Bush's home state.\n\nDallas was a letdown for one other reason. My late grandmother had watched _Dallas_ the TV show obsessively, and as a child I'd often sit beside her in the afternoons and watch along. So I wanted above all else to visit a ranch, to see horses and cowboys with big hats, and I thought for some reason it would be a natural part of my business there, though I was too embarrassed to ask Luc to confirm. Instead, I spent my whole time within sight of the downtown skyscrapers and felt cheated, as if I'd gone to San Francisco and been told the Golden Gate Bridge was unavailable for viewing. (Back in New York, Luc asked me if I'd had any trouble with the Texas accent, and, confused, I said, \"What accent?\")\n\nMy next trip was to Boston, which I loved. It felt almost European: I was at home among its narrow, crooked streets, its brick buildings and unmistakable sense of history. One client dinner was at a restaurant in a town house on a tree-lined street that could have been in a quiet London neighborhood, or even a Parisian one. It also made me homesick. The initial thrill of the job was over, it seemed\u2014and so soon. Boston was my third trip, and I was impatient that I hadn't made more progress with my wine education. I was a bit more polished, but I was building expertise solely in Ch\u00e2teau la Nerthe wine, nothing else.\n\nI was also lonely. Though I'd already spent a school year apart from Jules while in Spain, it turned out that being in the same time zone had kept us feeling much closer (and we were, of course, physically closer). I couldn't get used to the dramatic way our days were now misaligned. By the time I woke up, he'd finished lunch, and our goodnight calls had to happen before I went to dinner while he got ready for bed alone. I'd leave a restaurant with Rose and Nico, and Alex and Maya, the couples wrapped around each other, and grip my phone, knowing it was too late to talk to Jules.\n\nBut I loved spending time with the foursome. Even though I had no real alternative, socially, and was lucky to have friends given how much traveling I'd been doing, I genuinely liked them. I liked Rose's compact fierceness, Nico's frank gregariousness, Maya's devil-may-care attitude, and Alex's sense of play. Rose and Nico were Parisians, like me, but Alex and Maya were southerners, she from the coastal town of Marseilles and he from Aix-en-Provence. You could spot the differences immediately when we were together. Rose and Nico\u2014in fashion photography and advertising, respectively\u2014dressed sharply, Rose in sleek shoes and dark colors, Nico in skinny ties and trendy hats. Maya, meanwhile, was a publicist who wore dresses and leggings, and Alex was a Himalayan salt salesman and favored casual shirts with the top buttons unbuttoned. We grew close in the way expatriates are forced to be, but each couple had met and started dating before I'd known them\u2014the southerners in a bar in New York, and the Parisians going back to their freshman year of high school in France\u2014so there was always a secret history and a depth of intimacy among the couples that I couldn't come close to, or experience myself.\n\nThe next city on my itinerary was Las Vegas, and even that was hard to get excited about (my friends were excited enough on my behalf). But if many of the cities I saw in the U.S. have since blurred together in my mind, Las Vegas isn't one of them. How could it? From the window of my descending plane I saw the Great Pyramid of Egypt, the Space Needle (hadn't I just been there?), and, of course, the Eiffel Tower, all reconstructed at smaller scale on a single street. I checked into my hotel, but not before watching in amazement as a singing gondolier steered a middle-aged couple down the indoor, too-blue-to-be-real Grand Canal.\n\n\"It is like the gift-store version of everything that is wrong with America,\" Jules said when I called him. It was long after midnight his time. He was a night owl, but he yawned, bored perhaps by the pinnacle of American extravagance.\n\n\"I kind of like it,\" I said. This woke him up.\n\n\"How so?\"\n\n\"Because it's all out there. At least it's not pretending to be anything but entertainment.\"\n\nHe made a scoffing, dismissive noise. \" _Je t'aime, mon coeur,_ \" he said. \"I'm going to sleep.\"\n\n\" _Je t'aime, mon loup_ ,\" I said, and hung up. Outside my hotel window, it was as bright as a desert day can be, amplified by all the reflective surfaces up and down the Strip.\n\nThe sales rep, one of the few women I'd work with, picked me up in a car, though nearly everywhere we went was within walking distance. The restaurants for our client meetings were inside the hotels, the diners sitting among fine art or looking out on swimming pools made to look like a tropical resort's. Each time we walked across a casino floor and I heard the ringing of slot machines and the cheers of groups huddled around the craps tables, I felt a strumming of excitement\u2014not because of the games themselves, but because I wanted to believe, as everyone else did, that anything could happen.\n\nDinner that night was on the larger side, with ten clients, so there was less pressure for me to hold court. I spoke a little at the beginning about my uncle's wine and how happy we were for it to be served in such fine establishments, including the one we were sitting in. We toasted to Ch\u00e2teau la Nerthe, and to partnerships, and of course to lady luck. The sommelier to my right was a handsome brown-haired, blue-eyed man who couldn't have been older than thirty. While he was deep in conversation with the older man on his other side, I tried to remember if the rep had guided people to their seats, or if he'd sat there on purpose.\n\nWhen he paused and turned to look my way, as if to speak, I said, \"I'm sorry, there were so many introductions. Which restaurant do you work at again?\"\n\n\"This one.\" He smiled.\n\nI felt the red shoot straight into my cheeks. \"Forgive me. You are so young.\"\n\n\"I could say the same about you. So you've come from New York?\" he said\u2014not _France_. He'd been paying attention.\n\n\"Yes, I live there right now. But I was born in Paris.\"\n\n\"I never would have guessed. Your accent is undetectable.\"\n\nI snorted, and hated myself for laughing at such a cheap joke. He told me he'd gone to the French Culinary Institute in New York and had wanted to be a chef before switching to wine. I said I lived close by, in the East Village. He asked about a _pommes frites_ shop he loved and was relieved and pleased when I told him it was still there.\n\nI don't remember the rest of dinner. At the end I said another thank-you to all the guests, looking each client in the eye during the final toast (another tip from my uncle), and I stood as everyone began to leave, shaking hands all around, exchanging cards, joking that the 2005 vintage was going to be a tough year to beat but we'd ask the weather gods to do their best. When I shook hands with the young sommelier, he said, \"Would you like to try some of our other wines? They're not as good as your uncle's, but they're not bad.\"\n\nThe hardest times to say no are when you have more than one reason to say yes. It was too easy to convince myself that the opportunity to taste great wine was the real justification for accompanying him alone to the restaurant's temperature-controlled vestibule, next to the kitchen but tucked away, out of sight. He put two glasses on a narrow shelf and ran his fingers along a row of bottles whose labels I tried to read, then said, \"Hang on, there's an open bottle at the station from earlier today. It should have breathed enough by now.\"\n\nHe left me alone for a minute, during which I could feel all my muscles clench and unclench in a slow and steady rhythm. When he returned, I said quickly, \"I didn't realize how tired I was. I should go.\"\n\nHe studied my face for a second, and said, \"You have to try this, unless you know it already?\"\n\nI didn't\u2014it was Californian. \"Like me,\" he said, a little raffishly. \"It's a Bordeaux-style blend\u2014cabernet-merlot. I want you to try it and see what you think.\"\n\nI took in the aroma: fragrant, fruity, with a woodsy edge. I sipped. It was powerful, with a sweet spice that made me think of Moroccan food. I said some of this out loud, and he gave a gentle, appreciative smile and nodded. \"I get the nutmeg,\" he said, and I felt another flush spreading over my body, a more pleasant one this time. \"Sandalwood,\" he added. The wine made him extremely happy\u2014it was all over his face. And I was happy, too, to share it with someone who was affected so openly.\n\nHe held eye contact over our glasses, and I could see him hesitate. \"I have something else you have to try,\" he said, and turned back toward the bottles, breaking the spell.\n\n\"Thank you,\" I said, silently instructing my feet to move this time. \"This has been really nice, but I have to get back to Venice.\"\n\nHe laughed. We shook hands again, and I left the restaurant, and the bells of the casino floor were like Cinderella's chimes in the seconds before everything turns back the way it was before, her carriage into a pumpkin.\n\nI woke up the next morning in my hotel room feeling nothing but relief. I couldn't stop my mind from flashing to what might have happened, but I knew I hadn't really wanted it to. I sleepwalked through the rest of my Vegas obligations, with only one overriding conscious thought: the hope that I wouldn't see the sommelier again, even though I thought I might around every corner.\n\n_Good morning?_ My phone buzzed when I was already with the rep heading to the nicest wine store in town.\n\n_Sorry! Good morning. Bisous,_ I wrote back. I always texted right when I woke up, and if he had been the jealous type, some questions might have crept into his head. (I was the jealous one. If he'd ever gone a day without texting me, I probably would have called our friend Vera and demanded that she track him down by any means necessary.)\n\nBut my little t\u00eate-\u00e0-t\u00eate with the sommelier had given me an idea. I'd been thrilled by his attention, yes, but also by the wine. I'd felt a real tingle while holding a glass of something I'd never tasted before. The deep red swirling in the bowl and the aromas wafting up at me were a kind of flirtation even more alluring than the interpersonal one happening at the same time. I'd fallen for the wine, and for the experience of encountering it in a moment where all my senses were in tune. It had been nothing like ordering wine in a restaurant, or opening a bottle at home before putting a movie on. At those times my attention had been divided, the wine secondary.\n\nI realized how timid I'd been these last weeks, how little I'd really worked at either my wine appreciation or my English. In between appointments, I texted Rose and Maya: _Let's do a wine tasting. What do you think? I'm back Friday._\n\n_Yes!_ they both wrote back. Maya added: _We can do it at my place. Alex says we should make it a competition between French and American wines._\n\nMen!\n\nI invited Carl and Shaina, though I wasn't sure they'd come. As soon as I landed at LaGuardia I asked a cab driver to take me straight to a wine shop that Elise had once recommended, not too far from my apartment. With my travel bag in tow, I asked for Jacob, the name she had given me, and a friendly-faced, round little man stepped out from behind the counter. He had extremely hairy arms and a thick beard, and wore a T-shirt that read MEGADETH, a name that meant nothing to me. Was this really the person whose taste Elise trusted above all others?\n\nI told him I was looking for two good but affordable Burgundies, each around twenty dollars, one white and one red. Picking Burgundy was strategic. I knew Alex would choose bold California wines heavy on the fruit\u2014hot-headed, like himself\u2014and I was sure I could beat him with a subtler approach. As soon as I started speaking, Jacob's face lit up. \"Ah, _bienvenue! Je parle Francais. Vous \u00eates au bon endroit_.\" I was surprised by the quality of French coming out of his mouth! He told me that most stores didn't carry a lot of Burgundy, but that he personally loved it and bought as much as he could. He helped me pick two bottles. I made a point of writing the names down so I'd remember them, and these were the first entries in my American wine journal: a white M\u00e2con-Village from the Auvigue brothers, and a red C\u00f4te Ch\u00e2lonnaise from Fran\u00e7ois Raquillet.\n\nI must have been tired from my flight, and from my Las Vegas exertions, but I could hardly feel it. I went home and showered and then walked over to Maya's giant three-bedroom apartment, which she shared with two roommates (one of whom had lent me the dress I wore to the library gala). She was the only one of us who had a real kitchen, with counter space and room for a butcher-block island in the middle. I found her in there humming and unwrapping canap\u00e9s, bread, and cheese from Dean & DeLuca, taking samples for herself as she laid them out.\n\n\"You caught me!\" she said. She held her food-coated fingers away from her body as we kissed cheeks.\n\nAlex arrived straight from the wine store, too, and began cutting paper bags to tape over the labels. \"Don't look!\" he said to anyone who approached. I didn't have the heart to tell him you could tell the Burgundy and California wines apart by bottle shape (mine were low-shouldered and wider).\n\nTwo of Nico's friends were the next to arrive, Min-Ji and Peter, both of whom were Asian. \"Where do you guys live?\" I asked, and was about to add, \"And how long have you been together?\" before Min-Ji firmly but politely mentioned that her boyfriend was coming soon. (Have pity on this na\u00efve Frenchwoman!) It turned out she and Peter were co-workers at a film production company next to Nico's office. To hide my embarrassment and bright red cheeks I went to help Maya in the kitchen just as Rose and Nico arrived with Min-Ji's boyfriend, Derek (who for the record was very tall and very white). Carl showed up, too, but not Shaina, which was just as well, because he immediately and irrevocably fell in love with Rose and could hardly keep his eyes off her.\n\nAll together there were ten of us, including one of Maya's roommates. Alex, never forgetting the competitive aspect, tore slips of paper for everyone to use as voting ballots. We gathered around the island and realized we were split exactly: five French expats and five Americans. The four brown-papered bottles stood in the center. Alex raised his empty glass.\n\n\"Laure, why don't you get us started?\"\n\nAll eyes were on me. It was as though I was back in a distributor's tasting room in front of a dozen reps. I picked up one of the bottles of white, and poured a small taste into the ten glasses. I lifted mine. \"Color first,\" I instructed, and they followed my lead. \"Then look at the legs; they can give you some indication of the alcohol content\u2014the more alcohol, the more droplets. Now smell it\u2014before swirling. Smell it first, breathe it in deeply.\"\n\nEveryone breathed in deeply. \"Think about what you are smelling. Now swirl just a few times and smell again, and see if it's different this time.\n\n\"Remember, you are courting the wine,\" I said, and everyone laughed. It was not something I had ever said in a presentation to professionals. Something new was expanding inside me. I was already being more adventurous with my English, and with wine I'd never tasted before and did not already have the answers to. \"Clear your head. Don't think about where it might be from, what the label might look like, how much it costs. When you're about to kiss someone, you don't think about anything else but the feeling of wanting to kiss them.\n\n\"Clear everything away except the color you saw and the aroma you smelled, and then sip. When you sip, don't think to yourself, _Should I like this?_ Think, _Do I like this?_ Let the wine touch every inch of your mouth.\"\n\n\"Just like a kiss,\" Maya said, almost with a sigh, and there was more laughter around the table as everyone lifted their glasses.\n\nI took a sip, and it was like a liquid bite of fruit and flower, with balanced richness and minerality. I knew it was mine because of the shape of the bottle, but I was surprised and taken with it no less.\n\nWhen we had tasted all four wines, Alex tallied up the scores and looked up with a triumphant expression: the French white had beaten the American white, but the French red had lost to the American red, a victory he found more significant. I conceded\u2014but only as far as this test went. I'd quickly realized my mistake: given such small pours, of such young wine, the bold California red with strong undertones of oak and vanilla had overpowered the vibrant but velvety and earthy Burgundy. People are always seduced by big punches in little packages.\n\nWe began to have ap\u00e9ritifs every Friday. The regulars, aside from the French couples, were Nico's friends from the first tasting, who were great additions to the group: Peter a somewhat raucous free spirit, Min-Ji sweet and steely with a surprisingly filthy mouth, and Derek, for whom nothing was unworthy of irony. We always tried a few new bottles, from all over, plus at least one French wine\u2014I had decided that I couldn't dive into other countries' wines before learning more about my own. Wherever I was during the week\u2014Charleston or Atlanta or Denver or Portland\u2014by Thursday my mind was on the ap\u00e9ritif and what kind of wine I'd bring.\n\nAt the end of March, I went to Miami. My first destination was a private community outside the city whose purpose I couldn't understand when Luc tried to describe it to me at lunch before my flight. We'd nearly resorted to speaking in French before I finally got what he was saying. It was a place established by millionaires who wanted to spend their retirement with other millionaires, living in a private club with its own grocery store, high-end restaurant, and, naturally, security force. They wanted to serve Ch\u00e2teau la Nerthe at a special dinner, and when they'd heard the brand ambassador was in the country visiting, how could Luc say no?\n\nI gave a simplified version of my usual presentation to a room full of very rich old people and spent the rest of dinner pretending to recognize the names of all the companies and funds they used to run as I met the residents. For all my brushes with extreme glamour and wealth over the previous few months, this was the most surreal. I looked at everyone's wrists, and necks, and ears, and everything glittered.\n\nThey put me up in a guest bungalow for the night\u2014two thousand square feet just for me. I slept restlessly, eager to leave but worried there was a chance no one ever did. I sent a message to Luc blaming him for anything bad that had happened to me. (Now when I tell this story to people, they shake their heads and say only, \"Florida,\" as if it is explanation enough.)\n\nIn the morning a driver picked me up and I was finally free from the eerie enclave. Never before had I been so excited for my next destination: South Beach. After being dropped off, the first thing I noticed was that I was severely overdressed in my beige and gray silks. The buildings I could see were beautiful, and blindingly white. I worried I stood out for being just as pale. In the sun-drenched lobby a receptionist in a white uniform gave me a bright pink drink and told me to enjoy my stay. I had high hopes I would.\n\nI took the elevator to my room, where Jules was waiting for me.\n\nThree months was the longest we'd ever gone without seeing each other. Jules had wanted to go somewhere warm and was curious to see if he'd like Miami. (Spoiler: he didn't.) It had been all I could do to keep from thinking about it every day, every hour\u2014I kept telling myself not to live in the future, when I had so much going on. Each time we texted each other _Can't wait to see you in Miami,_ I immediately blocked it from my mind. It was the only way I could get on with my day.\n\nSo even as I put the key card into the door slot, fingers trembling, half of me believed he wouldn't actually be inside. He was. Suddenly he was enveloping me. And I couldn't think of anything but his smell\u2014a smell that was unlike all the thousands of strange and novel smells I'd sought out the last few months, in my unrelenting pursuit of the new, new, new. His smell surrounded me like everything comfortable and familiar at once. I wanted to drown in it.\n\nWhen we woke, it was nearly evening. He brought me out to the balcony where he'd had a bottle of champagne waiting on ice. The ice had long since melted. We waited for the wine to cool in fresh ice, and watched the endless ocean crash gently upon the shore, where colorful little dots of people were sunbathing and splashing in the last light of the day.\n\n\"So this is it,\" he said. He'd arrived just an hour before me, bought the champagne at the airport, and had seen nothing beyond the hotel lobby and this view.\n\nI brushed the hair out of my eyes to look at him more clearly. His hair was even and very short\u2014he must have gotten it cut just before leaving.\n\n\"What is it?\"\n\n\"America!\" he proclaimed in English, throwing his arms out wide. Some people are just good at languages. I'm not one of them. Even though I'd lived here for months now, his accent was better than mine.\n\n\"Well, we are facing the wrong way to see the rest of it, but yes, this is some,\" I teased.\n\nWe ordered room service and drank the champagne and didn't leave the room that night. In the morning we went for an early walk around the neighborhood for tea and croquettes. I kept holding his arm, or touching his shoulder, not only to remind myself of his physical presence, but because I also felt protective. The country, I knew, was something he was on guard against, and I wanted to protect him from it\u2014or it from him.\n\nIt was an unconventional slice of America to introduce him to\u2014the sandy streets, Spanish floating through the air, the ocean-lovers in sandals and cutoffs and neon bra tops, their bodies bulked up or otherwise augmented. There was more bare skin in three blocks than you'd find in the entire city of Paris.\n\nSoon it was time to get ready for work\u2014the rep was picking me up in thirty minutes. I hated to leave him so soon, but he said, \"I'm fine, I'll be fine.\"\n\n\"What will you do?\"\n\n\"Oh.\" He shrugged, and a glint came into his eye as he gave me that tentative smile I'd found so attractive the first time we'd met. \"Rob a bank, buy a lot of cocaine.\"\n\nI spoke a little faster to the customers that day, and the rep must have wondered at my nervous energy. Who comes to Miami and is unable to relax? Between my last sales appointment and the client dinner I stopped by the hotel, but Jules wasn't in our room. I texted him and got no response. I went out onto the balcony and looked up and down the beach, then wandered around the lobby, and was thinking about retracing our steps from that morning when the receptionist told me I might find what I was looking for out back.\n\nThere he was, lying on a floating mattress paddling slowly up and down the length of the infinity pool, sunglasses on, with a daiquiri in his hand.\n\n\" _Bonsoir, mademoiselle_ ,\" he intoned from his lazy perch. He lifted his drink in salute. \"This was twenty dollars.\"\n\nWe had the weekend all to ourselves, during which we did very little other than go to the beach and pool, walk a short distance for meals, and spend a lot of time in bed. Saturday night we took a very expensive taxi to the mainland for Cuban food, and on Sunday we went dancing at a club not far from the hotel. I had to twist his arm to get him to go, and had to physically push him toward the door of the club while he kept looking at me accusingly. Once we were inside, though, he suddenly became very passionate, holding me close, spinning me a few times almost roughly\u2014never in sync with the music, mind you\u2014before dipping me and giving me a long kiss that made me forget which side of the Atlantic we were on.\n\nThat night in bed, I fought sleep as I lay against him. Even though he was coming back with me to New York for a week, it was so rare for us to have space all to ourselves, even when we were in France, with nobody knocking or calling or needing anything. I was sure Miami would forever be a special place for me, because for a few days it had been our own strange and dazzling planet. Then I made the mistake of asking him if he liked it.\n\n\"I think it's a silly place.\"\n\n\"It's beautiful, isn't it?\"\n\nI felt him shrug with my head on his chest. That my disappointment also had a hint of fear meant only one thing: I was not only asking him about Miami, but about New York, too, before he'd ever seen it.\n\nHis reaction to the city that was currently my home could best be described as _scientific_. It could have been worse. He'd hated Alicante, where I'd studied in Spain\u2014the buildings were old and ugly, the marina an eyesore, the beaches no good. We once drove to the resort community of Benidorm for a weekend, and he hated it there even more. You must understand that Jules was a man for whom his dislike of some things was matched by his extreme passion for others. Viewed as a whole, you could even say he was quite balanced. And I have always been drawn to people with such concrete and articulated judgments\u2014they're appealing to someone like me who operates mainly on instinct and only later can explain my reasoning.\n\nSo _scientific_ was not so bad. In the cab ride it involved a lot of staring out the window and throaty noises that sounded like curious grunts\u2014 _\"Huggh\"_ \u2014and asking forward questions of our Middle Eastern driver about how long he'd lived in New York and what his family did. In Paris, I didn't care whether Jules liked what I ordered, or wanted to walk the direction I did, or felt the same way about a film. But in New York I was suddenly hypersensitive, wanting to defend both him and my temporary home. I was worried that they might be incompatible.\n\nI hid him from the others the first night we were back, but my friends made it clear that I wouldn't be able to do it again. I had to make some sales calls in nearby Westchester, a welcome break from flying, but no client dinner, so Rose and Nico, Maya and Alex, and Jules and I went out. Once we were all seated, for once equally paired, my anxieties melted away. It went even better than I'd hoped: Jules bonded with Nico over their hatred of George W. Bush, Rose over design, Maya over Disney cartoons (one of the things he loved with a true and abiding ardor), and Alex with jokes about Americans.\n\n\"So what did you do today while I was gone?\" I asked him as we walked home.\n\n\"I played basketball with your roommate.\"\n\nIt was the most Jules answer he could have given. I'd left him a guidebook but knew he would ignore it. Playing basketball with Carl was his form of tourism. He ended up doing it every day I had to work.\n\nFriday, I brought him to our ap\u00e9ritif, where he stood a little to the side and smiled warmly at everyone and shook hands with his long, loose arms. Peter had brought a girl he was dating, and Jules spent the most time of everyone interviewing her intently\u2014where she was from, what she did, how she met Peter (whom Jules had just met that same night), and what she thought of Maya's apartment.\n\nI liked her immediately; she admitted how little she knew about wine, but she was fearless. We were trying a Loire Valley red from the Chinon village and appellation, a wine that was fresh and round, perfect with the cured meats we were eating, and she was the first to speak up:\n\n\"I taste...blackberries,\" she said, a dawning awareness in her eyes.\n\n\"That's great,\" I encouraged her. \"I taste it, too.\" She beamed at me. Peter looked as if he couldn't decide whether to be proud or embarrassed. (This is the sad mentality of wine drinking today!) His expression became a little clearer at her next question.\n\n\"Does that mean there are blackberries in here?\"\n\nJules later repeated this question that night in bed, where we whispered to each other so Carl wouldn't be able hear even if he was hanging around outside the door. \"Who asks that?\" Jules said, as if she'd had one chance to do right by her entire country, and had failed.\n\n\"I think it was a good question,\" I said. \"You don't hear it much.\"\n\n\"Why does that make it good?\"\n\nI didn't have an answer right away. I was distracted, distressed: his Sunday flight back to France was approaching. \"Tomorrow,\" I said as we fell asleep. \"Tomorrow you will do everything and go everywhere I say.\"\n\n\" _Oui, madame_.\"\n\nI knew better than to try to take him to anything touristy. So instead we went to the places I liked best. A morning chai latte at the coffeehouse across the street. Russ & Daughters for bagels and lox. Tompkins Square Park to watch the morning dog run. Union Square Greenmarket for apples and cookies. The Whole Foods caf\u00e9 for lentil salad. The early spring day was wonderful, sunny and cool. Jacob's wine store. It was nearly three\u2014time was running out. Bubble tea near NYU. A single skewer of chicken skin on St. Mark's. The footbridge at Tenth Street that went over the FDR to the edge of the East River. A window seat at the Astor Place Starbucks where we could watch people come out of the subway.\n\nIn one long tour, I was trying to convince him of how great the city could be. We sat at the counter without speaking for a few minutes, looking out the window and then at each other, smiling. I imagined someone seeing us from the sidewalk as they walked by: me, freckled, red-faced, as readable as an epigraph etched in stone, while Jules was lithe, angular, fine-boned, long-lashed\u2014beautiful.\n\n\"I like that you have made a nice nest for yourself here. I like your friends. I like how exciting your job is,\" he said without being prompted. \"I like the passion you are developing for wine. I think that's a really good thing.\" He paused. \"And I feel very reassured knowing you'll be happy here until your job is finished and you come home.\"\n\nI started to cry, having fully processed now that he was leaving in a matter of hours. And I cried, too, knowing that I'd failed in one week to make him love New York as much as I did, and that come June I was not sure I'd be ready to leave.\n\n# Thinking About Wine\n\n\"Why is wine made from grapes?\"\n\nThis is the other way to phrase the question that Peter's date asked during that ap\u00e9ritif. I'm not qualified to answer it well, although it has to do with the particular combination of tannins, sugars, and acidity of grapes that allow both fermentation and aging. But when she asked, I told her that the first thing to know about French wine is that it's _only_ made from grapes. If it tastes like blackberries, it's a chemical effect of the fermentation alone, and has nothing to do with actual blackberries. It's all chemistry. Wine can taste like other fruit, or mushrooms, rocks (minerality), or smoke. It can taste like vanilla or butter, which can come from the oak barrel it's aged in.\n\nThe interesting thing to me, though, is that this question is rarely thought of, and never asked. The French either take it for granted or find the whole topic gauche. I'd gone twenty-four years without it popping into my head once. But only five months in the U.S., and there it was! It made me surprisingly happy. We're so lucky to have a strain of grapes that produces the wine we cherish. Here is a fruit that sucks up all the essence of the land and conveys those qualities to you in liquid form. This is terroir. It's important to remind oneself of this miracle once in a while.\n\nOf all the wine regions of France, Burgundy is the most obsessed with terroir, and is most defined by the way even the smallest differences in the terrain\u2014a different side of a hill, receiving different sunlight; being a few miles closer to a river or to a vein of limestone\u2014affects the wine. This, along with its historical ties to the French clergy, is why Burgundy is still one of the two most prestigious regions in France.\n\nIt's not a big region, but every patch of it is so meticulously identified and marked that in an area much smaller than Bordeaux, Burgundy has nearly twice the number of appellations (with many more sub-appellations, to boot). You can see this prioritization on its labels, where the appellation figures more prominently, and the producer is smaller and near the bottom. The message: terroir is key.\n\nWithin each of those five growing regions are appellations you might recognize, like Nuits-Saint-Georges or Gevrey-Chambertin in the C\u00f4te de Nuits region, or Meursault or Pommard in C\u00f4tes de Beaune. The premier and grand crus\u2014the best of the best\u2014have their own appellations. Hence a Nuits-Saint-Georges is different (and less expensive to buy) than a Nuits-Saint-Georges 1er cru.\n\nAll this famously documented variation in the terroir also helps to explain why Burgundy pretty much runs on two grapes: pinot noir for red and chardonnay for white. Within this narrow strip of land, those two grapes have created a lasting and rich variety. Burgundy wine is generally characterized by a complex minerality and richness, but within that is an entire world. It is not a wine that flexes its muscles, that shouts, that grabs you and says, _Love me!_ It seduces and enchants, demands your attention in order to appreciate its delicacy and power. Some would say that because of its subtlety, Burgundy is not a beginner's wine, but why not? For beginners who want to understand the relationship between wine and the land, there is no better place to start.\n\nEvery bottle of wine has a story, and some stories, if we're to be honest, are better than others. It would be nice to be able to love all wine equally, but this would be like loving all people equally. It's too much to ask. Some will have started out closer to your heart. Some will agree perfectly with your palate. Some will taste like vinegar. It can't be helped.\n\nBut there are a few ways to get an idea of what a wine will be like before opening it, and one of the most important is the vintage. Without it, you may know _who_ produced the wine and _where_ , but not _when_. A bottle's vintage is like its calling card\u2014it can tell you whether the wine was born in a good year and how long it has aged before you meet it. If you become an experienced and observant enough drinker, certain years from a certain region will evoke for you the place and time, the start of a season, the strength of the sun and rain, even if you weren't there to see it yourself.\n\nBut vintage cannot tell you everything. Just as family and schooling cannot guarantee a person's character, a well-known vintage is not enough to assure a bottle's integrity. You can always find bad wine in good years, and good wine in bad years. You might think this kind of inconsistency would be undesirable, that a wine should taste the same no matter when it was produced.\n\nBut that is not what terroir yields. And it is not what nature yields. Real-life stories never follow a straight line and rarely turn out as predicted. Where a wine starts at that moment of capture, the bottle sealed, is not where it ends up at the moment of encounter. Wine changes as it matures, the tannins mellow, flavors deepen and mutate. The same is true for me or you. The me who had just gotten back from Spain with no intention of moving to America\u2014let alone staying there longer than the allotted time\u2014was not the me lying in my Second Avenue bedroom listening to the street traffic. Your personality\u2014your desires and tendencies and traits\u2014is a message in a bottle. To find out what it says, you have to open it.\n\nMy time working for my uncle was coming to an end. It was May. I had just a few trips left, including one to Nashville, Tennessee, a city that I knew nothing about. Truth be told, I would have rather skipped it altogether; I would have rather returned to California, as I didn't know when I'd get a chance to see it again.\n\nMy job was ending in weeks\u2014and then what? All I had was a vague sense that my time in the U.S. wasn't finished, a buzzing that grew louder and louder in my inner ear with each passing day. That I had yet to talk to Jules about it was starting to eat away at me. I am not someone who holds on to secrets, but this felt less like a secret I was guarding and more like something I was scared to face. Perhaps I did not know myself as well as I thought I did.\n\n\"Do you know anything about Nashville?\" Jules asked the night before my flight. We were Skype messaging instead of talking, so that I could step away and keep stuffing things into my luggage.\n\n\"Not much,\" I said, though I knew less than that.\n\n\"There is nothing there,\" he said. \"It is surrounded by desert.\"\n\n\"Is that so?\"\n\n\"Totally barren,\" he said. \"Except for a great fort where America keeps all its gold.\"\n\nHe was teasing me, I sensed, but I was too distracted to play along. Telling him that I wanted to stay here, in this vast and strange country, was on the tips of my fingers that night, on everything I wrote. But I still couldn't say\u2014or type\u2014it out loud. I was afraid that my honeymoon with America might be over, and that we were about to see who each of us really was. I was afraid we would both end up disappointed.\n\n\"And not only is it barren, but it is full of _ploucs_.\" He said. Hicks.\n\n\"Stop,\" I said, not in the mood to be needled.\n\n\"It's only one week,\" he said cheerfully. \"Then soon you will be back here and everything will be fine.\"\n\nI suddenly didn't feel well.\n\nJules started complaining about one of his classes, and I snapped at him. I regretted it immediately but it was too late. The conversation cooled.\n\nIf you have ever been in a long-distance relationship, you recognize this moment, when it suddenly becomes so easy to turn against the person closest to you because of something you feel guilty for.\n\nWe made plans to talk again in the morning and logged off. But then I opened my computer again and began to type, \"You've been so patient with me. You will not like what I want to say...\"\n\nBut I deleted it without sending.\n\nOn the flight the next day, I took a short, fitful nap and woke feeling groggy and airsick and a little put out. I was still in a bad mood, and the only thing that had really started to grate on me about this job was the flying\u2014and the fact that I'd failed to get a good sense of some of the cities I'd visited. My hotel was an elegant, old-fashioned place downtown. The taxi took me right by what must have been an active district at night; looking out the car window at the unlit marquees, I felt a twinge of curiosity, but then went back to being irritable.\n\nI would be accompanying the same sales rep for both days. When I came down from my room at the appointed time I saw a man in a dark blue suit and tie who made no move toward me, though he looked like he was waiting for someone. There was no one else around.\n\n\"Jim?\" I called.\n\nHe looked up, startled.\n\n\"Oh,\" he said, shuffling over. \"I'd thought\u2014I'd\u2014the name I had was Lawrence Dugas.\"\n\n\"It's Laure,\" I said, perhaps a little short.\n\nIt didn't stop him from laughing. \"Laure! Yes, of course, that makes so much sense. I can see how that might happen.\" He chuckled to himself. His suit was three sizes too big, and he had a spot of dried mustard in one corner of his mouth. This was one of Nashville's top wine salesmen? \"Oh, boy,\" he said, shaking his head and letting out one last hoot. \"Well, you're here now, you made it. We found each other,\" he said, extending his hand.\n\n\"I'm sorry, Jim,\" I said. \"You have a little something,\" and gestured at my own mouth.\n\n\"My,\" he said, and retrieved a folded napkin from one of his cavernous pockets. He wiped at his face. \"Eat and run, you know,\" and gave another friendly chuckle.\n\nEating and running is something the French certainly never do. Looking at my disheveled companion through my emotional hangover, I felt a flash of both regret and relief\u2014maybe Jules was right, that I would be home soon.\n\nI gave Jim a tight smile and followed him out to his car.\n\nWe headed to our first appointment, a wine store in the West End, a neighborhood Jim made sure to tell me was a nice area. He'd seen the look on my face as I'd lowered myself into his beat-up Nissan\u2014at my feet were a few empty snack bags and candy wrappers that he must have forgotten to throw away before picking me up. At some point, I was fairly sure, this car had seen some very good wine paired with Funyuns and a Baby Ruth.\n\nThe sales call was pleasant enough, as was lunch with three other local business owners. Jim was friendly with everyone. He was a likable guy, eager to please, but different than the other chummy, backslapping salesmen I'd worked with\u2014Jim seemed to make his sales more through puppylike endearment than bulldog persistence. Maybe in a place like Nashville it was enough. At one point, as we drove along a particularly green stretch of road, I asked, \"How far out of the city is the desert? I haven't seen it yet.\" He paused only briefly before responding, but it was just long enough to cause a rush of red to flood my cheeks. I remembered now that Jules had only been teasing me about the desert.\n\n\"I don't think there are any too close,\" was all Jim said. I thought about explaining, but then realized how silly it would sound, and kept my embarrassment to myself.\n\nWe pulled up to our last call of the afternoon. In retrospect, there were a number of red flags from the start: my own poor mood, the store's dim interior, the dark gray walls and black metal racks making the labels hard to read unless you were standing directly beneath one of the lights hanging from the ceiling. A tall man in a suit that matched the walls stepped out from the back. \"That you in here, Jim?\" He reached over to pump Jim's hand without acknowledging me. \"What have you brought for me today?\"\n\n\"Humphrey, we've got a treat for you today. This is Laure Dugas, from Ch\u00e2teau la Nerthe. She has some very nice\u2014\"\n\n\"Don't know it.\"\n\n\"It is an estate in Ch\u00e2teauneuf-du-Pape,\" I began.\n\n\"I love a bold Burgundy,\" he said.\n\n\"Rh\u00f4ne Valley,\" I corrected, bristling somewhat.\n\nHe blinked at me. It was already obvious that this man knew nothing about wine. I'd encountered one or two others like him in my travels, people who owned a store as a business but had no love for their product. Most of the time, the distributors did a thorough job culling them from the list of people we were to meet, even if they were good clients.\n\n\"Do you have a price list?\" he asked before I'd even started my presentation. It was his right to ask, of course, but the clients generally knew this was an informational visit first and a sales visit second. I never turned down any sales, naturally, but my priority was educating and building a relationship. Selling a couple of cases was just a sign that I'd succeeded at my main job.\n\n\"Maybe we should try some first,\" I deflected. Humphrey turned and repeated the question to Jim, who said, \"Let's talk about that in a bit, Humphrey. We've got some great wine here to introduce you to. You're going to love it.\"\n\n\"I have four stores from here to Knoxville, and they'd close in an instant\"\u2014Humphrey snapped his fingers\u2014\"if I didn't know every single number associated with my business.\" But he added, \"All right then. Let me get some cups.\"\n\nI could feel the heat radiating from my chest and only hoped it wouldn't reach my face. The store owner returned with a stack of red Solo cups, peeled off three and set them up on the counter with the edges touching, as if he expected me to pour into all three without lifting the bottle.\n\n\"Do you have any glassware?\" I said through tightened lips. \"I believe glass would allow the wine to express itself better.\"\n\n\"Tennessee China,\" he said, waving a hand over the cups and giving a dry laugh that made my spine tingle.\n\nJim\u2014clearly sensing that things were about to go very wrong\u2014tried to call a truce by jumping in to say he thought he had a couple of glasses in the car. While he was gone, I didn't move a muscle, standing with both hands on the countertop, trying to look as placidly as possible to one side while Humphrey studied me with overt disdain.\n\nJim returned, huffing a little as though he had sprinted the fifty yards to the car and back, and looked relieved to have found things no worse than he'd left them. \"Got 'em!\" he said. \"Just got to give them a little rinse first.\"\n\nAs Jim prepared the glasses, I opened with a shortened version of my normal presentation. It calmed me down to settle into the well-worn words about Ch\u00e2teauneuf-du-Pape, the galets, the ch\u00e2teau, the vineyards. Humphrey was a little perfunctory as he tasted the first white, but seemed satisfied. \"My customers will like this?\" he asked Jim\u2014not me. I pretended not to notice the slight.\n\n\"They'll love it, Humph. It's a real winner.\"\n\nI cleared my throat. \"But most important, what do _you_ think?\"\n\nHe replied too quickly for me to understand, and I asked him to repeat himself. He cleared his throat and said, \"I'm just trying to do business, sweetheart.\"\n\nSomething in me snapped. \"Men have been saying that for a long time,\" I said. \"It's grown quite stale by now.\"\n\nAs I turned to leave, my elbow tipped one of Jim's glasses off the bar, but by the time I heard it shatter I was halfway to the door. I didn't look back. I'd had it with Humphrey, four stores or no, but also with myself, for being so combustible, hasty, and unsure. For being too Obelix. (Though I was admittedly a little proud I'd been able to express my anger so precisely in English.) I thought about my uncle's advice in Los Angeles, about how the best salesman is a happy salesman, and felt a pang of despair at having failed him.\n\nWhen Jim finally returned to the car, I'd managed to stop crying. With the windows up and the late spring sun pounding down, I had a sense of what it must have been like to be one of his candy bars melting on the seat.\n\n\"Oh, Jim,\" I said, hot with shame, when he opened the door. We may not have been friends, but at least by the nature of the partnership, we were on the same side, and I'd let him down in addition to myself. \"I'm so sorry. I'm sorry about the glass. I'm sorry about my behavior.\"\n\nLooking a bit weary, he put the tasting bag in the backseat and told me he had to make a call. Through the hazy window, I watched him pace the perimeter of the parking lot but couldn't hear what he was saying. Then he got in and started the car. My mind was racing. _Had he just gotten me blacklisted?_ _Would Luc cancel the rest of my trip?_ In a perverse way, that would have made the decision I was facing much easier.\n\nBut Jim said only, \"I've worked with Humphrey for years now but didn't know he could be such an idiot. I don't know how that happened back there. I'm sorry.\" We pulled out onto the street. \"You know what's funny?\" he continued. \"I think he's going to buy the wine. He doesn't know it yet, but I do.\"\n\n\"I don't want my uncle's wine in that store,\" I said.\n\nJim made a sympathetic but noncommittal sound\u2014which was, all things considered, generous of him.\n\nThere was only one thing to do when I reached my hotel room, and that was fall facedown on the bed in self-pity. Eventually the self-pity gave way to self-recrimination as it began to dawn on me how intolerable I'd been all day. Even worse\u2014as I'd been so busy reducing everything in Nashville to a stereotype ( _ploucs!_ ), I'd become a stereotype myself: the snotty French girl.\n\nI passed out where I lay on the still-covered bed, and slipped into a dream in which Jules and I were living in Nashville, in a house attached to the restaurant where I'd eaten a mediocre salad ni\u00e7oise at lunch. There was no separation between the home and the restaurant and only a single shared entrance, and every time we wanted to sleep I would see and hear the diners just a few feet away. But they wouldn't acknowledge me even as we were lying there, completely open to them.\n\nI woke to a knock on my door. In the way that dreams can run straight into reality, I was certain it was Jules. I first thought he'd just come home to our restaurant\/house. I then realized it had been only a dream, but still believed he'd flown in to surprise me. I lunged for the door and threw it wide open, nearly scaring the life out of the woman who had come to turn down my bed.\n\nJim was early picking me up for the client dinner, and I was late coming down. He was still his polite, smiling self\u2014making me feel relieved and guilty at the same time. \"You look lovely,\" he said when I got in the car, yet another kindness. It was still light out, but the sun was dipping in and out of the tree line. Soon the stores fell away, and the houses started to grow farther and farther apart, separated by wide swaths of land. One side of the road was walled off by trees. For the second time that day, faint alarms began to go off in my head. \"Are we not going to a restaurant?\" I finally asked.\n\nHe looked over with a wide smile and it was suddenly clear that he was holding back a surprise. \"Nope!\" he said. My heart sped up. When you start to feel a little frightened, smiles can actually make things worse. The French fairy tales I grew up with are similar to the ones American children hear, but with bloodier consequences (we are Catholic, after all). If Jim turned out to be a wine salesman and serial killer, I knew there'd be no woodsman to save me, and that if these really were my final moments, I'd probably done something to deserve it (again, Catholic).\n\n\"We're almost there, don't worry,\" Jim added, but somehow I wasn't reassured. \"And no, it's not a restaurant. It's the home of a collector. And a very good client. I wanted to surprise you.\"\n\n_A collector!_ I thought with relief. And then, because I could not help myself, said out loud, \"Of wine?\"\n\nJim laughed.\n\nThe driveway we pulled into took us past the longest, largest lawn I'd ever seen, short of Versailles. The house was nearly as big. Before we reached the end, I saw the front door open between enormous white columns, and a tall, trim man appeared, wearing a cardigan and no tie or jacket. He came down to the car and offered to help with the tasting case, and I wondered if he was a servant of some kind.\n\n\"Paul, this is Laure Dugas,\" Jim said, \"the grande ambassadrice from Ch\u00e2teau la Nerthe.\"\n\nThe tall man reached out his hand. \"Just last week I had the pleasure of opening a bottle of your uncle's 1995 Clos de Beauvenir and it was tremendous.\"\n\nI didn't know what impressed me more, Jim's French\u2014not bad at all!\u2014or Paul's familiarity with Ch\u00e2teau la Nerthe. Clos de Beauvenir is the white grand cuv\u00e9e of Ch\u00e2teau la Nerthe, a blend of the best rousette and clairette grapes of the estate. And 1995 was one of the finest years in recent memory, with flavors of marmalade and honeysuckle, spice and nougat. I felt Paul's words starting to charm me, and I had to work to maintain my wariness.\n\nI moved to follow the men toward the house but felt something catch and pull me down to the asphalt. I'd caught my skirt in the car door. The men rushed back to me and looked even more concerned when I only laughed a little. _What else could possibly happen?_ This trip had quite literally brought me to my knees. There was a long tear in my skirt, and I spent more time trying to keep myself covered than paying attention to my skinned knee. I told Jim I thought I should go back to the hotel, but Paul insisted that I stay, taking off his cardigan for me to wrap around my waist.\n\n\"Stay,\" he said again, his gray eyes leveling on me in a polite but firm way that made me think he was someone not used to hearing no. \"My friends have been waiting for you, and I understand this is your only night in Nashville. Please. We shouldn't waste any time.\"\n\nAgainst my better judgment, I followed him, still dazed by the fall. I expected to end up in some reception room with a couple dozen of his closest associates, like a smaller version of the millionaire commune outside Miami. But instead he led us to a kitchen the size of my entire apartment, where to my surprise there were only three people: an older man with a white beard and a cane hooked on one forearm, and two women standing close together, one about Paul's age and one much older. Everyone was hovering around the kitchen island.\n\nThe man, I learned, was Vernon, an old friend of Paul's, and the younger woman was Paul's sister, Jessica. Everyone laughed when he introduced the older woman, Judy, as Jessica's twin. Paul asked the women to help me find the bathroom to address my skinned knee. To my surprise they entered the room with me, where they fought over the peroxide as Julia, the older woman, repeated, \"This house, it's just too big!\" as if it were to blame for my fall.\n\nAfter I was patched up, we wasted no time in going through a tasting of all four of my uncle's wines. I was very informal with the presentation\u2014I had to be, with a sweater tied around my waist\u2014and it felt easy and comfortable, like one of my ap\u00e9ritifs at home (but better, with more expensive cheese). The older man, Vernon, was chatty throughout, the first to react to a wine and the first to say something. The women sipped, murmuring about the aromas and flavors and nodding to each other. Only Paul stayed silent, with a look of concentration as he pinched the stem of his glass, swirled the wine with three quick motions of his wrist, and immediately brought the rim up to his nose. When he sipped he closed his eyes, and sometimes kept them closed when the others spoke. After the Cuv\u00e9e des Cadettes, he paused for a little longer, and a small, private smile crept across his face.\n\nWhen he opened his eyes again, it was as if from a deep, refreshing sleep. \"That was marvelous,\" he said. \"Simply marvelous. But you know I'm already a fan. Now that you have so kindly shared some of your wine with us, we should try some of mine.\" The smiles on the twins' faces could not have been bigger. I looked over at Jim with confusion and he only gave me a reassuring lift of his eyebrows. Paul reached down to open a cooler built into the kitchen island and pulled out a bottle of champagne.\n\n\"Is that...?\"\n\nHe pretended to squint at the label.\n\n\"I believe it's a 1985 Krug.\"\n\nI've told you that even though I was raised on champagne, we drank only my family's. We were makers, not collectors\u2014it was like drinking our local water. But Champagne is not a very large place, and the famous houses are well known by the French people, who are as proud of good champagne as of anything else in our history. And Krug is one of the most illustrious of them all. I knew that 1985 had been a very good year for champagne. The bottle had been aging for twenty-two years, which sounded obscenely long to me but could have been just right for a barrel-aged wine of Krug's stature.\n\nI watched Paul carefully divide it among the glasses. The liquid was a deep straw gold, not the bright transparent yellow of the young champagne I was used to. We toasted, and, without further ado, sipped.\n\nHow do you choose the right moment to taste a wine that has been waiting for twenty-two years? In this case, the moment had chosen me, and I wasn't about to question it. I let the wine sparkle on my tongue, taking in its depths. It was rich and elegant\u2014aristocratic\u2014powerful but still fresh, deeper and more expansive than a young champagne, tasting of apples and toast and light citrus. It was a wonderful encounter, and one that would have been different a year before, or a year after that night, but always distinguished by the complexity that Krug is known for.\n\nAfter that first sip, any remaining jitters were gone. My embarrassment, my fears\u2014all of it disappeared. Not because of the alcohol. It was the pure pleasure flowing through my body. I glanced around the island and everyone had the same beatific smile, like in a Renaissance painting. Even Vernon was quiet.\n\n\"Can I finish this whole glass?\" I asked no one in particular.\n\nPaul laughed. \"I'm not going to finish it for you.\"\n\nI don't need to say again how shallow the depth of my knowledge was, how everything I knew about the great producers and vintages was simply from family conversations I'd overheard. But I didn't need any special knowledge tonight.\n\nIt was getting late, and I was ready to say thank-you and goodbye when Paul reached once again into his cooler\u2014the way a magician reaches into his hat\u2014and pulled out a red wine, turning the label to face me. I leaned in.\n\nIt was a 1961 Cheval Blanc.\n\nLet me explain briefly how wild this was. Cheval Blanc is one of only four premier grand cru class\u00e9 A producers in Bordeaux's Saint-\u00c9milion region. And 1961 is considered perhaps the finest vintage of the twentieth century, a year in which even the spring frost and the summer rainstorms were perfectly timed to produce a smaller, more concentrated yield.\n\nFirst a 1985 Krug. Now a 1961 Cheval Blanc.\n\nI can hardly describe it. Drinking the Cheval Blanc was one of the most overwhelming sensory experiences in my life. I tasted cherry, plum, tobacco, chocolate, damp earth, and more I couldn't even name, rising and falling and merging and separating all within seconds, drawing out into a velvety sweetness like a long violin note.\n\nI lost track of time. It felt like hours went by between sips, each one spreading through every nerve ending of my body. Paul brought out steak tartare and salad, which I nibbled on before I turned back to my glass, taking only tiny sips. Then, to my surprise, he tied an apron around his waist and in ten minutes produced a simple pasta with tomatoes and basil\u2014a perfectly light dish to complement the spectacular wine without trying to dominate it. Judy had finished her glass of Cheval Blanc and was reclining in a reading chair at one end of the kitchen, singing to herself.\n\n\"I think she's ready,\" Jessica said, nodding in my direction.\n\n\"Ready for what?\" I said. At that point I would have been game for almost anything, as long as I didn't have to stop drinking the Cheval Blanc.\n\n\"Paul has a tradition with new friends.\"\n\n\"I do,\" Paul agreed. \"Come with me, Laure.\"\n\nI followed him down a hallway I hadn't noticed, sweater still tied around my waist and apron still tied around his, and glanced nervously back at Jim, who waved me on ahead. My host entered a code into a keypad that opened a door, and we went down a set of stairs that glowed from within. At the bottom, there was another door with thick glass panels through which I could faintly see the outline of wine racks. He unlocked this door as well and we stepped inside to the climate-controlled room, also as big as my apartment, filled with wine racks containing hundreds of bottles. I turned toward him, wonderment on my face.\n\n\"It's custom-made,\" he said. \"It didn't have to be underground but I liked the tradition.\"\n\n\"How do you decide what to put in here?\" I said, a little breathlessly.\n\n\"What to collect? A lot of it is by reputation,\" he said, almost sadly. \"I wish every decision were personal. I have a penchant for the Rh\u00f4ne Valley. But I have learned that once you know your taste, you are made even happier by what you like. And it also makes you more forgiving of what you don't like, strange as it sounds.\" It felt true to me, if a little cryptic. \"I'm lucky to like a lot of things,\" he continued.\n\n\"You can afford to,\" I blurted out before I could stop myself.\n\nHe laughed. \"That was a lot of luck, too,\" he said. He'd been scanning his collection proudly, but now turned to face me. \"Tell me what year you were born.\"\n\n\" _Non_ ,\" I said, mostly out of surprise.\n\n\"Tell me.\"\n\n\"1983.\"\n\n\"1983,\" he repeated, and his eyes widened as an idea came to him. \"A fine year, though I'll bet you wish you were a year older,\" he teased. Some believe 1982 was an even better year than 1961 for French wine, so he was not the first to make that joke\u2014my family had always needled me about it.\n\n\"It's not something I had a lot of say in.\"\n\n\"This one,\" he said, and gently pulled a bottle from its resting place.\n\nIt was a Ch\u00e2teau de Beaucastel, one of the largest and most storied estates of Ch\u00e2teauneuf-du-Pape, just a few miles from Ch\u00e2teau la Nerthe. He handed the bottle to me, and I cradled it with two hands, afraid I would drop it. No, the vintage wasn't as renowned, but at that moment I wouldn't have traded the 1983 for a 1982 or anything else. This was my year. I was holding a piece of my homeland as it had been at the time of my birth.\n\nWhen the others saw the bottle Paul had picked out, the twins exclaimed cheerfully, Vernon mumbled approvingly, and Jim just smiled. Paul opened it carefully and decanted the wine before pouring. The '83 Rh\u00f4nes are known to be muscular and rustic, and this one was, despite its slightly faded color. It had a palate of dark fruit and pepper, as well as a meatiness, and the harsher edges it might have had earlier in its life had rounded out. I could see that it didn't compare to the Cheval Blanc in overall complexity and refinement, but I loved it all the more.\n\nEven after all this, there was still an encore. And what else but a Sauternes, France's most celebrated sweet wine? The night had been such a string of unbelievable wines I was almost\u2014almost\u2014unsurprised when Paul held up a bottle of Ch\u00e2teau d'Yquem. My first sip of this storied wine conveyed tropical fruit, light oak, a marmalade finish. It cast a buttery glow on the evening, which now felt both endlessly long and far too short.\n\nI could not give a warm enough farewell to the ebullient twins, to the dignified gentleman, and to Paul, whom I embraced at the bottom of his front steps under the warm Nashville night. All the pretense of my ambassadorship was long gone, evaporated with that first sip of the Krug champagne. He hadn't even needed to taste la Nerthe; I saw plenty of my uncle's wine in his cellar. He'd poured thousands of dollars' worth of wine tonight, much of it going to someone he'd just met.\n\n\"Why me?\" I asked him.\n\n\"Your uncle has given me so much joy,\" he said. \"Please tell him.\"\n\n\"I will,\" I promised.\n\nBack in the car, Jim said, \"I used to pick on that boy in high school. Can you believe it?\"\n\n\"Where did he make his money?\"\n\n\"Computers,\" he said, and shook his head in disbelief.\n\nWe were a few miles away when I cried out, \"The sweater!\" which was still around my waist. Jim told me not to worry; I could give it to him tomorrow. He'd mostly stopped drinking after the Cheval Blanc to be sure he could drive, but had tasted everything. We talked about the Beaucastel, and I found myself stumbling over my inadequate English trying to articulate its gamey, leathery flavor. But I kept going. Something had changed\u2014the cautiousness with which I'd always talked about wine, the measured, practiced, almost guarded way I'd approached it, was gone. I was\u2014there was no other way to describe it\u2014giddy.\n\n\"Do you have a Southern accent?\" I said, the thought just occurring to me.\n\n\"I guess you could say I do,\" Jim said.\n\nI started to giggle softly, and then laugh outright. \"I can't hear it,\" I said. \"I can't tell. Americans all sound the same to me. Isn't that awful?\"\n\nJim chuckled, too, and leaned down to retrieve something from the floor. \"Listen to this,\" he said. He inserted a cassette, of all things, into the stereo, and out came the warbling high voice of a man accompanied only by a guitar.\n\n\"I don't know what he's saying!\" I started to giggle again, but then, as I began to catch words here and there, I heard a sadness that conflicted with his jaunty strumming.\n\n\"This is strange music,\" I said.\n\n\"The greatest of all time, Laure,\" said Jim. The headlights ran over the trees along the side of the road.\n\n\"Elvis?\"\n\nHe laughed. \"I'll tell you this: that voice helped make this town, though he didn't live to see it. His name was Hank Williams and he was a better artist than a salesman. And I'm a salesman, not an artist. He had a bad back and a drinking problem and died before he was thirty. But you can hear what he knew. And this town sells what it learned from him. Nashville is full of salesmen, Laure, not just the wine sort. Everybody's selling.\"\n\n\"That's terrible.\"\n\n\"Naw, it's not bad. It's only bad if you're selling bad things. This is a good thing, music. Country music. Real country music. Wine is a good thing. You can't just give things away in this world. People have got to live.\"\n\nIt was the most philosophical I'd heard him get. It was the most philosophical I'd heard any sales rep get. I still felt ashamed for behaving so poorly that morning, but his kindness\u2014and the great wine\u2014had started to ease my guilt.\n\nThen we were downtown again, passing the neon signs. \"These places, they play music?\"\n\n\"Sure do.\"\n\n\"Let me out, Jim. I can walk from here.\"\n\nHe studied me for a few seconds as if waiting for me to change my mind. \"I'd go with you, but my family's going to be up early. If you need anything, go here and ask for Lois.\" He scribbled something on a scrap of paper and handed it to me.\n\n\"You have children?\" I asked him.\n\n\"Two girls.\"\n\n\"I could have guessed,\" I said, and he smiled.\n\n\"If the boys don't leave you alone,\" he said, \"pretend you don't speak English.\" He thought about it for a second. \"Or give 'em a good kick. That works, too.\"\n\nI was feeling brave, brave enough, at least, to go into two honky-tonks, sweater-skirt and all, order a root beer, sit at the end of the bar, and listen to the bands play. I didn't learn to love the music that night, I'm sorry to say. But I was happy to hear it.\n\nI walked down Broadway beneath the bright marquees, music drifting out every door. When I reached the hotel it was past midnight. The doorman seemed surprised to see me walk up alone, but my face must have reassured him that everything was all right. It was more than all right.\n\nMy room was eerily quiet. I thought about turning the television on for company, but then realized I didn't really want any. I sat for a minute on the edge of the bed, then made a call to France, where it was now after seven in the morning.\n\n\" _Tonton Alain_ ,\" I said.\n\n\"Yes, Laure, is everything all right?\" my uncle said, a rare strain of concern in his voice.\n\nI told him about the night, stumbling over my words, this time from excitement. I told him about Paul and the wine we'd had. He was impressed with the Krug and started to say, \"That's very good,\" but I cut him off to explain about the Cheval Blanc. He let out a low whistle. Then I told him about the Beaucastel, and he murmured an appreciative, \"Ah!\" When I got to the Yquem, he emphatically demanded, \"What year?\"\n\n\"1947,\" I said. I only later learned that 1947 is the most famous year for Sauternes.\n\nThere was a long pause, as if he was trying to imagine the experience for himself. Finally he said, \"Laure, even after you've worked in this business a long time, as long as me, you will have only a few nights like that. It is not just the wine\u2014though that was very good wine. It's the generosity.\"\n\n\"I understand.\"\n\n\"Remember this night. This quality, the passion and enjoyment you're feeling, it will have a positive effect on the rest of your career.\"\n\n_My career._ I didn't even think to contradict him. There was nothing to contradict.\n\nThen we hung up, and I counted to ten before calling Jules.\n\n# Thinking About Wine\n\nThe wine I drank that night in Nashville (not including the Ch\u00e2teau la Nerthe) had aged more than a hundred and fifty years combined. It's an impressive number, but on its own means nothing. I've said before that older wine is not always better wine. Wine can become _too_ old and start to fall apart\u2014its fruit fades and its complexity thins out into flatness. Some experts even say we're reaching the end of the era of aging wines for decades. In the old days, young wine was often extremely tannic and acidic, and needed many years to soften and round out. But viticulture has changed, and now wines aren't as harsh in their youth. In fact, the best window for opening most bottles is somewhere between now and ten to fifteen years from now\u2014that's it.\n\nKnowing this, you may start thinking about creating a little collection. It's not so hard! We can't all be like Paul. Keeping even half a dozen nice wines, each bought for what you'd spend on an inexpensive bottle at a restaurant, is a completely reasonable goal. If you drink one, that's simply a great opportunity to buy another.\n\nTo make a simple wine collection, there are only two things you really need to pay attention to: how you'll maintain the climate in your home, and the reputations of recent vintages for each region you're interested in. The easiest solution for the former is to buy a small wine cooler set to fifty-five degrees (your refrigerator is too cold\u2014it's not just heat that can damage wine). For the latter, you first need to know what kind of wine you enjoy, and, assuming you don't have tons of spare cash lying around, the best value for that type of wine. Don't think you have to start your collection with a Cheval Blanc that you'll hang on to for decades, taking it out only once every five years to stare at it longingly. For you and me, there's very little point. Of the places I've already discussed, Cru Beaujolais wines are often more than worth their cost, as are some of the southern subregions of Burgundy, like M\u00e2con. If you can wait even a year before opening that 2013 Burgundy, you may get a lot more out of it\u2014it's as simple as that. Which is why being able to store wines in your home for more than a short time may be valuable to you. And while you let your wines age longer, you can continue to enjoy the numerous young, drinkable wines that are out there, and open one of your older, more special bottles when the right meal calls for it, or the right occasion, or just on a whim. Overall it won't cost much more than just sticking with whatever you're used to drinking. It's more a matter of timing.\n\nAnother reason to have a few wines aging at the house, even just for one or two years, is that while it's true young wines are more balanced and more elegant now than in the past, they are still frequently sold too young. If you stop by the store for a dinner party you're hosting tonight, most of what's being sold will be one, two, three years old. That's because the pressures of the market have decreased the time between bottling at the estate to arriving at your home. Whereas in the past winemakers may have aged the bottles longer before releasing them to the distributors, they now get them to the stores as quickly as possible, in order to keep the cash flowing. And stores aren't going to age their inventory\u2014that would be a waste of retail space\u2014so they sell wine quickly, too. That last bit of helpful aging, then, falls on you. Short of finding that unicorn of a store that sells wine from 2011 and before (they are out there!), you will benefit from keeping your own wine around a little longer.\n\nThere is one alternative to putting the time and effort into storing wine for over a year: have a decanter at home, and use it liberally. If you open a bottle and pour a small bit to smell and taste, and find that the wine is tight, not very expressive, or gives off a limited bouquet, decant it! Just pouring a wine into a decanter will open it up, but an hour there will often do wonders.\n\nAs with all the other aspects of wine, mill\u00e9sime (vintage) is not everything. A great vintage from one place is not always great from another. Weather can cause similar patterns all over France, or influence one region very differently (or even an area within a region). And while a lot depends on the weather and terroir, yes, you also need a dedicated and expert winemaker.\n\nWith those caveats, here are some well-reputed French vintages. You can see that while there is some overlap from region to region, there are many dissimilarities. And as always, memorizing these years won't be enough to find the perfect glass; you must drink them!\n\nGOOD YEARS IN BURGUNDY:\n\n2010, 2009, 2005, 1999, 1996, 1995\n\nGOOD YEARS IN THE RH\u00d4NE VALLEY:\n\n2010, 2009, 2007, 2005, 2004, 1998, 1995\n\nGOOD YEARS IN CHAMPAGNE:\n\n2012, 2004, 2002, 1996, 1995\n\nGOOD YEARS IN BORDEAUX:\n\n2010, 2009, 2005, 2001, 2000, 1996, 1995\n\nGOOD YEARS IN ALSACE:\n\n2007, 2005, 1996, 1995\n\nGOOD YEARS IN LANGUEDOC:\n\n2011, 2007, 2000, 1998\n\nGOOD YEARS IN BEAUJOLAIS:\n\n2011, 2010, 2009, 2005, 2000, 1999, 1995\n\nMy mother has never been one to wait for the perfect moment. After all, how often does such a moment come around? There are no prizes in this life for denying yourself pleasure for an ideal that may never materialize, and this is why the worst bottle of wine is one that is never opened. My mother does not have this problem. The wine that most people save for special occasions is exactly the wine she had too much of around the house: champagne. To come home from school on a Tuesday afternoon and see her cooking, a glass of bubbly in one hand, was an education in how to enjoy life.\n\n\"Why are you drinking champagne, Maman?\"\n\n\"Why on earth not?\"\n\nNow, we don't _need_ champagne to appreciate the great luck of our very lives, but it certainly doesn't hurt. Like everything else, champagne as we know it is a product of chance and environment. The Champagne region has produced wine for hundreds of years (and developed its celebratory reputation because it was used to consecrate the Franc kings in the Middle Ages), but the wine was still, not sparkling. In the 17th century, when the extra fermentation that produces carbon dioxide became well known, it was considered an imperfection! Dom P\u00e9rignon himself had tried first to eliminate the bubbles, not refine them. Sparkling wine only grew popular in the 1800s, and that's when Champagne became synonymous with effervescent wine.\n\nSo you see, what is taken as fact now was not so obvious at one time, and was the product of a happy accident or two. I moved to the U.S. because my uncle overheard me say I wanted to go to England. I wanted to go to England because I'd loved my time in Spain. I went to Spain because while at university in Paris someone came to tell us about the EU's student exchange program, and that they were accepting only two spots from our school. I knew that all my classmates would consider it carefully and take their time in their French way; if I wanted to go, my advantage would be in filling out the forms immediately, without even telling my mother. I made a friend do it with me. We were the first to turn in our applications, and we were the ones selected. I didn't even look up what Alicante was like, or I might not have done it at all. I might have waited until I could go somewhere prettier, more famous.\n\nAnd so on. Each decision leading to another. If I had stayed in France, I would have gotten my economics degree and gone into NGO or social work. Would I still have fallen in love with wine? Or would I have felt oppressed by it, surrounded by the family business? Who knows! All I know is what happened. I went to the U.S., to my own surprise. And there, the love snuck up on me.\n\nNow the job was over. Like a whirlwind romance, it had left me wanting more\u2014of wine and of New York. I'd found something that suited me in this city, its energy and its feeling both old and constantly new. And it was full of people, brave and adventurous, who were at least a little like Obelix, fellow members of a tribe I hadn't known existed.\n\nBut just because I wanted to stay didn't mean I could. The expatriate life is full of goodbye parties. Getting a work visa is no easy task\u2014my first had been as a French employee working temporarily in the U.S.\u2014and it's even harder for people whose fields fall under the broad category of \"business,\" as sales, advertising, and marketing jobs do. Maya and Alex, whose positions were ending a month after mine, were having trouble finding work, and Nico's contract was almost up, too.\n\nI was the last to arrive, and it was increasingly looking like I would be the first to go. I had some interest from a mid-sized firm called Pringent, which like Mo\u00ebt owned a champagne house and also imported wine to the States. I was a nervous wreck waiting for them to call. I took a second sales trip to Boston, a city I'd really liked the first time, but wasn't able to enjoy it. I got back to New York just in time for Pringent to tell me they didn't have the budget to create the position they'd wanted to, and just like that, my last chance to stay was gone.\n\nIt seemed everyone I had met in New York came to the party Rose and Nico hosted for me. Even Carl and Shaina arrived together and politely stood side by side with their drinks. I felt fondness for them both. We had never been friends, but roommates do the important work of staving off the nightly loneliness that can haunt new arrivals in New York. A few glasses in, I threw my arms around them and demanded that they make up\u2014they only had each other now, I said, and shouldn't take each other for granted. I don't think my speech worked\u2014I heard Shaina moved out of the apartment not long after\u2014but my intentions were good.\n\nThere was beer and liquor for the Americans, but I insisted on sparkling wine. We couldn't afford champagne for everyone, so we bought prosecco instead, which is far more affordable, though less complex and generally sweeter, its bubbles closer to fizzy soda bubbles than the fine, long-lasting bubbles of champagne. But prosecco is perfectly pleasant and celebratory, which is what I wanted. I refused to even call it a goodbye party. But I could only hold the tears back so long. By the end of the night Rose and I were hugging and crying and promising to visit, wherever we ended up. I would never have expected the reserved, birdlike woman I'd found so cool and intimidating eight months ago to have become one of my closest friends.\n\nMaya and Alex were among the last to leave. They embraced me for a long time. \"Goodbye,\" they said.\n\n\"Don't say goodbye!\" Rose commanded.\n\n\"No, no, it's okay,\" I said. \"There's no point denying it now.\"\n\nI was embarrassed. To have spent all that time agonizing over whether to stay in New York, and now not to even have the option! When I'd called Jules after Nashville just a few weeks earlier he'd been supportive, but I knew he hadn't been happy. Now he was.\n\nEverything I owned besides the clothes I could fit into my two suitcases was gone: my bed and dresser, the little typewriter stand I'd found with Rose at a flea market, a stone vase where I'd kept pussy willow branches I wouldn't have to water while traveling. I had one bottle of my mother's champagne that my Uncle Alain had brought me as a token from home. I'd saved it since February, waiting for the right moment, hoping that moment would be when I found a permanent job in the U.S. Now I gave it to Rose. \"Drink it the next time you're happy,\" I said as we embraced.\n\nAnd only hours later, I was on board a flight back to France.\n\nJules met me at Charles de Gaulle, his face, when I spotted it, an endearing mix of joy, relief, and sympathy. I was lucky not to be stopped by security as I ran like a crazed woman into his embrace.\n\n\"If you're not happy to be home, don't tell me, I won't be able to take it,\" he said.\n\n\"No, of course I am,\" I said into his neck. And it was true. Rose had once told me that she always felt like she no longer belonged when she returned to Paris, but though I'd felt a little strange on my last visit, everything had still been comfortable and easy. What I felt now was disappointment; once I'd acknowledged my desire to stay in New York, I wanted it more than anything I could remember\u2014and I had failed.\n\nI planned to reregister for university in September and finish my economics degree. After that, Jules and I would see who found a job first; no matter where it was, the other would follow. We'd spent so much time apart in the past year and a half, making a promise seemed like the best way to create a life together. I wanted to go back to New York; he knew and accepted that. He surprised me by saying he'd been entertaining thoughts of finding graphic design work in China once he finished his degree that summer.\n\n\"You, China?\" I said. I had trouble picturing him living anywhere besides France.\n\n\"It's the future,\" he said solemnly. \"Why not see the future?\"\n\nBy Julesian logic, it was a very practical answer. I laughed, and for a few nights lay awake trying to picture myself in Shanghai.\n\nI mostly spent time with my mother and riding on the back of Jules's Mobylette to dinner or a movie or a party. When I'd been back for Christmas I'd felt d\u00e9j\u00e0 vu everywhere I went, as if my first two months in America had been a dream. But now there was no ignoring the impact the experience had on me. My friends all said I was the same, but I knew it wasn't quite true. They were most curious about what Americans were really like, how the cities I'd visited differed from Paris, and if the food was bad. No one asked how I actually felt about my time there, perhaps assuming they knew the answer or not wanting to hear anything that would contradict their preexisting notions. The few times I said that I would love to go back, I was met with surprise and exclamations. But the biggest change was simply having Jules always near\u2014I could reach for his face and touch warm skin, not a laptop screen. It was not so bad to be back, no. In nearly every way it was wonderful.\n\nMy mother knew better, though. \"I see you here but your brain is somewhere else,\" she said over breakfast one morning.\n\n\"I'm just getting used to things again.\" I smiled at her. But she didn't smile back, instead examining me skeptically over her glasses.\n\nThen late one night Jules and I were at a house party when I heard my phone ring. It's still a mystery how, because I never hear my phone ring, even when it's in my pocket\u2014this drove Jules crazy\u2014and that night my purse was buried in a pile of coats on a chair ten feet away. I hurried to retrieve it, my stomach in my shoes. The number was American. I ducked into the bathroom, the nearest place I might find some privacy, pressing \"answer\" just as I swung the door shut. \"Hello?\" I said as I crouched low in the bathtub to block as much noise as possible. It was in this dignified position that I learned Pringent had changed its mind and wanted me to start as soon as possible.\n\n\"How long will it take for you to come back?\" asked Rachel, the head of marketing and my new boss.\n\n\"I don't know, with the visa paperwork...maybe a few weeks?\" I said, hardly able to believe what I was hearing.\n\n\"Do it as quickly as you can,\" she said. \"We're excited to have you.\"\n\n\"Yes, yes, of course,\" I said. At that moment, all I could think to say was _yes_. It was not unlike the conversation I'd had with my uncle nearly a year before.\n\nSomeone knocked on the door. I threw my hand over the phone and shouted, \"One minute, I'm sorry!\" If my future employer heard anything untoward, she said nothing.\n\nWhen I left the bathroom the woman outside said, \" _Enfin!_ \" and slammed the door behind her. I wandered the party, stunned. I must have been pale, because my friends stopped what they were doing and called my name. \"Laure? Are you all right?\" I spotted Jules in the back of the room just as he saw me, and from the look on his face, the small but worried smile on his lips, I could tell he knew exactly what had just happened.\n\nJules skipped his classes the next day so we could meet at his favorite bar, in M\u00e9nilmontant. When I got there he was already chatting with the owner, who smiled at me and moved away.\n\n\"This is great news,\" he opened.\n\n\"For me, or for us?\" I asked, taking one of his hands in both of mine. So far every major choice we'd made as a couple had been to accommodate my ambitions. It had seemed, just two weeks ago, that we finally had a fifty-fifty chance at focusing on his instead, and I wouldn't have blamed him if he was now having second thoughts about our pact. Once again, the coin had landed in my favor.\n\n\"What's good for you is good for us,\" he said. I didn't think he was lying, but I knew there was more.\n\n\"You must be angry.\"\n\n\"Why would I be angry? I want to be where you are. That's all that matters.\"\n\n\"That's not _all_ that matters,\" I said to prod him. \"You don't like New York.\"\n\nFor the first time, he didn't have an immediate response. My heart began to sink.\n\n\"I have to finish school, then I have that project in the south\"\u2014a one-off design job he'd taken on\u2014\"so it's a difficult time...\" he trailed off. I braced myself for something along the lines of _So,_ _good luck without me_. _It's been a nice ride_. \"I won't be able to come until September,\" he finished.\n\nI nearly tipped the table over throwing my arms around him.\n\nMy mother had said very little that morning when I'd told her of my change in fortune. She didn't say anything when I got home that evening, either. She'd always been supportive, always believed my blundering ahead was ultimately for the better. But I knew it couldn't be easy for her.\n\n\"I will still be here for a few weeks,\" I said, smiling weakly. She turned back to her book. I told her Jules was going to move in a couple of months, too. Her frown grew slightly deeper. Then she walked over to the cooler for a bottle of our champagne.\n\n\"We should toast, at least for you,\" she said, popping the cork with a wry smile. \"I personally have nothing to celebrate.\"\n\nLater that week I stopped by Pringent headquarters to meet my new counterparts, even though I didn't expect to work directly with the Paris office. Everyone was welcoming and kind. It was still a family business\u2014three generations of Pringents\u2014and tiny compared to the giants of champagne like Mo\u00ebt. This seemed comforting.\n\nThen my mother and I drove to Champagne to see her family. When I was growing up we made regular visits to Ambonnay, within the subregion of Montagne de Reims\u2014the city of Reims being the heart of Champagne.\n\nHere's the thing about Champagne: it makes great wine, but it is a terrible place to live. Of the great winemaking regions of France it's the closest to Paris, only a hundred miles to the east, but you can leave a sunny day in Paris and arrive to clouds and chills in Champagne. The region is a network of small villages each about the size of a Parisian city block, in which you have one florist, one baker, one school, et cetera. There is something admirable about old towns that owe their existence to a single craft, where the traditions have changed only slightly over several generations, but they can also be stultifying.\n\nWe ate a long lunch in the garden of my uncle Charles, the oldest of the brothers and just a little younger than my mother. My other uncles came by to say hello, embracing me in the cool manner of my mother's side of the family. My uncles have some conflicts with one another, mainly about the land, but everyone likes my mother. The three men sat across from me, mostly silent, like the land itself\u2014a bit rough in manner, but proud and candid. Whenever they reached for a glass or a piece of cheese I was reminded of how large and callused their hands were. They rarely spent more than a few days a year outside Champagne, even for the holidays. The one time we convinced them to vacation in Paris for a week, Charles grew so antsy we had to let him leave early and return home. Now we ate bread and cheese and meat and drank\u2014what else\u2014champagne, while my mother and her middle brother jokingly traded barbs. \"You're looking old, Vincent.\" \"Is that a slight limp you have, Marie-Brigitte?\"\n\nWe said little about my past year and future plans\u2014if Paris was a foreign land to them, you can imagine how New York seemed. If I talked about how quickly things changed in New York, how fast everyone walked, how many restaurants were on every block, I might have given them a headache. I did tell them I'd be working for Pringent, thinking they might be intrigued or have some local gossip. Uncle Charles shrugged. \"I have never worked with them.\" Only my cousin Charles-Henri, the son and heir, seemed interested, telling me he once knew someone who worked there and would send me her name.\n\nThe vineyards of Montagne de Reims are mostly pinot noir. My family's wine, especially the _brut,_ meaning dry, is nearly all pinot noir, which lends a hint of red fruit and a fuller body to the wine while still being as refreshing as you would expect from champagne. That afternoon I walked out among the vines that rose up with the slope of the land behind the house. At the beginning of June is the blossoming, when flowers appear on the vines to indicate that the fruit is soon to follow. Thus the timing of the bloom gives the winemaker an important hint as to when the harvest will be. The flower clusters were out now, but I didn't have the skill to decipher what they meant.\n\nThis was Charles's plot. My mother's was near the village entrance, tended by her brothers, with most of the grapes sold to other producers for blending. My mother and her sister had both left Champagne as soon as they could, deciding that winemaking was not for them, and my uncles' daughters had done the same. So only the men were left, with their wives, caring for the land. All of them sons of sons.\n\nMy uncle Charles did little work in the field anymore\u2014most of that was left to Charles-Henri. Every September the family would hire a dozen people to help with the harvest and my aunt and my cousin's wife would make everyone lunch and dinner while the men worked. It was the same at the vineyards cared for by my mother's youngest brother, Fran\u00e7ois, and his son Pierre-Emmanuel. (Vincent, the middle brother, had no sons and had retired from wine production, selling his grapes instead.) I'd come here nearly every year around the harvest growing up, but it had always felt like just another holiday. I'd never really understood how important those days were to my family's business\u2014and as a girl I wasn't expected to have much curiosity about what went on with the land.\n\nI did now.\n\nI returned to New York in July, dropped off by taxi once again in front of the apartment on Eleventh Street. More than eight months ago I'd stood on this very spot, with these very suitcases\u2014but so much had changed since then! For one, the woman now walking quickly toward me was no stranger, but my friend and new roommate. Rose and I embraced warmly, and she helped wrestle my luggage into the apartment she had once politely kicked me out of. This time, the second bedroom would be mine for real.\n\nEverything looked the same, but I immediately felt an absence\u2014Nico was gone. Generous, gregarious Nicolas had been unable to find a new position in New York before his visa expired and had flown back to Paris just days earlier to continue the search from there. Rose was unhappy but not distraught\u2014after all, she kept saying like a mantra, it had worked out for me. Until he came back, though, it would be just us girls.\n\nWhen I finished showering and rejoined Rose in the living room, she brought out the bottle of my family's champagne I'd left with her.\n\n\"You still have it! You were supposed to drink it with the first good news.\"\n\n\"You're looking at it,\" she said. I knew what she meant: Maya and Alex were gone, too. Unable to find new jobs, they'd flown back to France just over a week before, unlikely to return. In the span of one month, our little social circle had scattered. There didn't seem to be a reason for a celebration. But it's nearly impossible to drink champagne and stay sad; it always does its best to lift you up. So we drank it with ambivalence, happy to be reunited in New York, and in honor of the friends we were missing.\n\nI slept restlessly that first night. In the morning I called Jules, and we sighed together\u2014here we were again, an ocean apart, but at least this time we knew we would be together in two months, this time for good.\n\nMy first day at Pringent, I decided on a conservative gray skirt and white shirt that I hoped projected seriousness and efficiency\u2014and adulthood. This was my first real job that I'd applied and interviewed for. An office! A commute! I took the elevator up to the top and was surprised when it opened not onto a hallway but directly onto the floor. I felt exposed\u2014I'd planned to at least smooth my hair and take a deep breath before opening the door, and yet here I was. The executives' doors were clustered by the front, and beyond them the space opened up to the cubicles. The tasting room lay in the center of the office, enclosed by glass so that wherever you were you could see all the bottles and glasses lined up, waiting to be opened and used. Thankfully, everyone was on the phone or typing away and paid me no attention.\n\nI might have stood there all morning if the office manager, Allison, who was also the CEO's assistant, hadn't come and found me, greeted me warmly, and showed me to my desk.\n\nI sat there\u2014at my first desk\u2014and fiddled with my hands until my boss, Rachel, the head of marketing, called me into her office. I noticed she was wearing a similar outfit to mine; from the way her eyes widened a little when we shook hands, I could tell she'd noticed, too, and wasn't pleased about it. Though I'd met her once before, at my interview, I'd forgotten how beautiful she was. She was, you could say, All-American\u2014blond, thin but athletic, in her early thirties. On her desk, angled so I could see it, was a photo of her with a man nearly as good-looking as she. No children.\n\n\"Is that your husband?\" I asked.\n\nShe held up a finger while she finished an email. Finally she said, \"And... _send,_ \" and turned to me with a wide smile as if I'd just walked in. \"It's so good you're finally here!\" she said. \"We've been waiting in excitement. Yes, that's my husband. Isn't he a cutie?\"\n\nShe paused, as if I should jump in. So I said, \"I'm very happy to be here. Pringent is a very respected champagne. But there's always room to grow,\" aiming for something that was least likely to be wrong.\n\n\"I like that attitude!\"\n\nMy title was marketing assistant. I'd be running campaigns to expand brand awareness of our champagne, but the truth was, I knew nothing about marketing. Pringent had more or less hired me because of the contacts I'd made during my quick tour of the U.S., and because I was French. I had no illusions about that. But I also wasn't sure how exactly they planned to make use of those things\u2014or, really, what I was supposed to do at all. But I couldn't bring myself to ask this rather basic question.\n\n\"I...can't wait to get going,\" I said.\n\n\"Neither can I!\" she said. \"I'll let you get geared up then. If you have any questions, don't hesitate to ask Marianne. Talk again soon!\"\n\nI left her office with no more direction than I'd come in with and went back to my new desk. The girl sitting at the desk next to mine was Marianne, the other marketing assistant. I introduced myself and was delighted to find out she was French, but as soon as I began to speak to her in our shared tongue she made a sharp cutting motion at her neck and hissed, \"Speak English.\" Before I could respond she handed me a giant binder titled \"Champagne\"\u2014an endless, haphazardly organized scrapbook of old campaigns and design standards, like the size of a banner, the logo, a list of everything needed to man a table at an event, how many bottles were needed for _x_ number of people, and dress codes for conducting the event. I started to understand her cool attitude\u2014the way she talked about the portfolio made it clear she'd previously worked on both the still wine and sparkling wine accounts, but now they'd split up the business and given the more \"glamorous\" half to me. \"The next big one is Fashion Week,\" she was saying. \"Since I've already put a lot of work into it, you have a good head start.\"\n\nI sat down with the binder until it was nearly one. \"Have you eaten?\" I asked Marianne. \"I was thinking of going out for food.\"\n\n\"Not yet,\" she said without taking her eyes from the computer screen.\n\nI lingered for another minute, uncertain how to interpret what she said, then left the office alone. Our building was in the Flatiron District, a neighborhood that I hadn't spent much time in. I wandered a few blocks until I happened on a bakery that made sandwiches, and I also bought myself a cookie for good luck.\n\nWhen I returned, Marianne was gone. I asked the man sitting two desks away if he knew where she was. \"She and some others went out for lunch just after you left,\" he said, and went back to whatever it was he was doing.\n\nI spent the next few days trying to pinpoint the exact number of the questions I could ask Marianne or Rachel before their answers became clipped and annoyed. I wasn't doing much more than making sure the right number of cases of champagne arrived from Paris and then went to the right places. I had to follow up with the printers, the vendors. It wasn't very interesting work, but I couldn't help but feel a smile creep across my face every morning. It was all still new to me.\n\nThe office itself was beautiful. There were only about fifteen of us, so the space felt open and uncrowded. Arched Beaux-Arts windows lined the walls, and from a couple of them you could see the Empire State Building ten blocks to the north. The real draw was the tasting room, a see-through bank vault that featured every bottle in Pringent's current portfolio, tucked into floor-to-ceiling shelves. In the center was a huge, austere tasting bench, loaded up with wine glasses hanging upside down.\n\nBut smiles seemed hard to come by, so I kept mine to myself as much as possible.\n\nWithout my noticing, New York had taken on a different character. It had grown smaller in some ways, because I was there every day instead of traveling during the week, and bigger in others, because without my friends the neighborhoods felt less accessible. Rose and I were an odd but inseparable pair. One short, one tall; one blond, one brunette; one quiet and one loud. We occasionally saw our American friends from the ap\u00e9ritifs, but those weekly nights had ended with my departure and we hadn't started them back up again.\n\nBecause Rose couldn't talk to Nico at night, when it was late in France, I tried to get her out of the house. She wasn't used to long-distance relationships, and I could tell she was having trouble handling it. Sometimes we would just walk around, ending up at a Japanese place on St. Mark's, or try to meet our few acquaintances at bars or parties. When she wasn't in a social mood I would pick up a bottle of wine from Jacob, who was always clad in a different T-shirt and always greeted me happily, and bring it home for us to share.\n\nWhat Rose really wanted to do, however, was go dancing. I know I've described her as guarded, even a little uptight\u2014in every way my opposite. But on the dance floor she was totally free and natural. I may be bold and outgoing most of the time, but I am a stilted, spasmodic dancer, like a marionette whose strings have come loose. Rose, however, danced like smoke in glass. She closed her eyes but never made an awkward move or bumped into anyone, even though everyone nearby was inexorably drawn toward her. I always wanted to stop and watch. How could a woman who cut her food into a hundred little pieces dance like that?\n\nThose two months, when all we had was each other, we went out at least twice a week. Rose could get us in anywhere. Not only was she stunning, wearing tops that threatened to slip off her shoulders but never did, she could also charm any bouncer in perfectly clear English laced with just enough sultry French. Inside, if we were approached by men (women, too, sometimes), we'd say that we had boyfriends\u2014Rose with a touch of melancholy that only I noticed\u2014and most would move on, but a few stayed and talked to us anyway. These were the people I loved meeting, the ones who genuinely seemed to want to get to know us. Every time, it felt like we were on the cusp of making friends. We'd exchange email addresses or phone numbers, dance in a group, and talk until it was time to go home.\n\n\"You are so nice!\" we'd shout.\n\n\"We should get brunch!\" they'd say. \"Have you been to such-and-such?\"\n\n\"No! We would love to go!\"\n\n\"It's always crowded. You have to get there by eleven. But we never get up before eleven.\"\n\n\"We would do it!\"\n\n\"We know this guy who would love you. Don't worry, he's gay. We'll bring him!\"\n\n\"Bring him! Bring anyone! We love to meet people!\"\n\n\"You guys are so funny!\"\n\nBut they never emailed or texted. This was perhaps the thing that surprised me most about Americans (New Yorkers, at least): how earnestly they seemed to want to befriend me, and how rarely they followed through. For all of the failings of the French, the one thing we would never do is not call you for brunch if we promised to. Is it the price of American outgoingness? Too many friendly people and not enough free time?\n\nBut it is to their credit\u2014and purity of intentions, I hope\u2014that even after months of these broken dates, Rose and I still believed them, every time.\n\nI had only slightly better luck with my officemates, most of whom I suspected believed I'd come to take Marianne's job. It was weeks before she eventually caved and had lunch with me but she spent the entire time chewing. (This is the French way of not being friends.) I thought once we were out of the building she would deign to speak French with me, which she did, but only in brief mutters\u2014and when I switched back to English, to see if I got a better response, she looked at me as if I were insane.\n\nThere were still plenty of small pleasures. Whenever a sales rep came to the office with samples, I would find a time near the end of the day when Rachel was gone to slip into the tasting room and pour myself a tiny glass. Never before had I such easy access to such a wide array of wines, and been able to compare them in such close proximity: the colors, ranging from light red to deep purple, pale yellow to dark gold. Red fruits, black fruits, stone fruits, citrus, spice, white and black pepper, smoke, tobacco, licorice, other herbs and plants, and, most intriguingly, the earth\u2014clay, chalk, dirt, hay, mud. Once or twice, when I was the last to leave the office (not often, because Marianne gave me dirty looks if it seemed like I was going to stay much later than her), I just stood in the middle of the room for a while, surrounded.\n\nI wanted to take advantage of my first summer in New York. It seemed a shame to go underground for the subway, where it was hot, sticky, and full of grumpy people, so I bought a blue bicycle for twenty bucks at one of the flea markets Rose was always dragging me around to. It was a rusty, squeaky cruiser with a basket, too heavy to carry up and down stairs. I rode it to work every day, locked it up on the sidewalk, and stored it by the recycling on the first floor of our apartment building (the super loved Rose). It was a tank, but I loved its wide handlebars, its fat pedals, its wheel guards. I took photos to send to Jules as if it were a new pet.\n\n_Here is the front!_ (Click click.)\n\n_Very nice._\n\n_Here is the side!_ (Click click.)\n\n_Also nice._\n\nOne evening a week I went up to Morningside Heights for English lessons with a retired Columbia music professor. They were ten times more useful than that first horrible class I'd taken by Madison Square Garden. My English was far better now, but since I was no longer touring on behalf of a single estate the amount of memorized and repeatable phrases in my daily conversation had shrunk greatly. I felt that there was no excuse for me not to be able to hold a long, confident, improvised conversation with anyone, so once a week I would leave work, ride the subway as far uptown as I'd ever been, and take a tiny elevator to the fifth floor of a cavernous old apartment building close to campus, where the professor would be waiting for me at the entrance to a four-bedroom apartment he had likely paid a paltry sum for decades before. His wife still taught music, and sometimes while he and I chatted in one room, I heard the muffled sounds of a student plucking out chords on a violin.\n\nThere was no lesson plan. We just talked. Each lesson would begin once he lowered himself into his upholstered reading chair, pushed his bifocals to the crown of his white-haired head, rubbed his eyes for a moment, and spoke. He asked me questions about my day, and then broader questions about my life, and as I answered he gently corrected my word choice, tense, and conjugation, without disrupting my train of thought. It was not only instructional, but therapeutic. After a few weeks I felt he knew more about me than all my friends, in Paris or here. He knew about my family, my relationship, my job, my childhood. He didn't know much about wine but saw how excited it made me to talk about it.\n\nI learned only a little about him, although I tried to pry once or twice. There were books everywhere in the apartment, shelves overflowing with them. His children were grown. One day in August he let slip that it was his forty-fifth wedding anniversary, so at our next appointment I brought a bottle of Drappier champagne. It was a little extravagant\u2014it had cost me about as much as the lesson itself\u2014but he and his wife were a lovely couple, a couple I aspired to be like, though I would never have said that out loud. I had no idea if they were champagne drinkers and chose a dry one that would be aromatic and rich and easy to like. They thanked me warmly, but I worried they wouldn't enjoy it.\n\nBut at our next lesson my teacher told me they had managed to get through most of the bottle in one sitting, and it had been quite a romantic evening. Forty-five years\u2014I felt myself tearing up.\n\nHe cleared his throat. \"And how is Jules?\" he said softly. \"He's supposed to arrive in two weeks, right?\"\n\n\"Two weeks, yes,\" I said, realizing I'd been trying to block the exact day out of my mind, so I wouldn't obsess over it. It couldn't come soon enough.\n\nThe next day I biked to work as usual, made my calls, suffered Marianne's silence, and responded to Rachel, who exclusively communicated over email even though she sat twenty feet away. As soon as the clock hit five, I packed up and walked out as if a bell had rung. It was a gorgeous day. The sun was high, but there had been a break in the humidity, and the sky was a watery blue and cloudless. I thought about going to the park and staying there until sunset, perhaps even leaving my bike for the night and walking all the way home. At the last second I decided to bring it with me, just in case someone chose this night to steal a rickety old cruiser. As I unchained it, something in the basket caught the sunlight. I leaned closer. It was a plain white card that said \"Je t'aime\" in block letters.\n\nMy heart stopped. My first thought was that I had a stalker, and that I was possibly in danger at that very moment. I didn't know whether to stand still or start to slowly walk away. I had no idea who it could be\u2014maybe one of those American \"friends\" I'd foolishly given my email address to! _What if they were watching me now? What if they were\u2014_\n\nThere was writing on the back of the paper. I flipped it over: \"Turn around.\" And then from across the avenue Jules was running to me, picking me up before I could even scream.\n\n\"Surprise!\" he said, being a man unable to speak anything but the truth.\n\n# Thinking About Wine\n\nMy great-grandmother, L\u00e9ona, was a family legend who lived up to her name. All she ever thought about was the family and the land. She and her husband Charles came to Ambonnay at the start of the twentieth century and expected their descendants to be farmers, not winemakers. The Montagne de Reims was always more valuable as farmland, even when my mother was a girl. It wasn't until the 1960s and 1970s, when the demand for wine grapes from the region\u2014which has forested, chalky slopes that are perfect for growing pinot noir\u2014skyrocketed, that the farms all turned into vineyards. Of the nearly four hundred villages in Champagne, seventeen are designated grand cru, and eleven of those are in Montagne de Reims. Ambonnay is one.\n\nBesides pinot noir, there are two other grapes that are used in Champagne: chardonnay and pinot meunier. Each gives a specific character to the wine. Pinot noir lends body and complexity, chardonnay elegance and freshness, pinot meunier\u2014the reliable little brother of pinot noir\u2014fruitiness. The average champagne will use more pinot noir than the other two, but you can find combinations using two of the three, or chardonnay alone (which you'll recognize on the label as blanc de blancs), or pinot noir alone (blanc de noirs), or, in a real feat for the underdog, the rare solo by pinot meunier. Blanc de blancs have a lighter golden color and a bouquet not unlike chardonnay\u2014rich and delicate, fruitful, finely structured. Blanc de noirs are a more yellow gold, and reflect the black grape characteristics you'd expect: red fruit, flowers, toast, and a creamy texture.\n\nBut wait! There's another variable to champagne, and that's the level of sweetness, or the _dosage;_ that is how much sugar is added to the wine. The most common level is _brut,_ which is your standard very dry champagne, to which only a little sugar is added to smooth out the acidity and any harshness. In my grandparents' day most champagne was on the sweeter side\u2014 _sec_ or _demi-sec_ \u2014but tastes have changed, and sweeter champagnes are harder to find now. In fact you're more likely to see wine that's even drier than dry: _extra brut,_ or the suddenly fashionable _brut nature_ , which has zero added sweetness and can be bracing to drink. Why champagnes used to be so much sweeter I can't say, although one reason is surely that poor quality is more easily masked with sugar. Standards are higher now\u2014as they are for all wine\u2014and quality has risen correspondingly, so it's possible to make a well-balanced, very dry champagne with the complexity of still wine that doesn't make you pucker up.\n\nAnd what about vintage? In fact, 90 percent of champagnes do not have a mill\u00e9sime and are generally marked NV, for non-vintage. This doesn't mean you can't age NV champagne. But it does mean the non-vintage wines won't be as \"representative\" of the year you purchase it. Because the northern weather can be less than cooperative from year to year, most producers blend grapes from different harvests in order to maintain a consistent house style. This is an exception to the standards for still wine, made out of necessity. If a house feels confident in that year's harvest, it may produce vintage champagne, meaning it all comes from that year's harvest\u2014a risky proposition. It takes more effort and time to produce, and is no guarantee of quality. But when it pays off, you're left with a richer, deeper wine. It also costs a lot more.\n\nBut you don't have to go broke buying champagne. The average person can't afford to drink it every day, but that doesn't mean you should write it off completely in favor of less expensive sparkling wines. (As a Frenchwoman it is my duty to mention that yes, only wine from Champagne can correctly be called champagne.) You've likely heard of the biggest names in the game, so here are a few smaller houses that are worth your time to seek out. None are cheap, but at $40 to $50 they all strike a good balance between price and quality.\n\n_Champagne Jacquesson_ \u2014From the Vall\u00e9e de la Marne, the largest of the major Champagne regions. Many champagne houses buy a lot of their grapes from other vineyards, but Jacquesson grows most of its own. These are elegant champagnes.\n\n_Champagne Gonet-M\u00e9deville_ \u2014From Montagne de Reims and C\u00f4te des Blancs, they use grapes from both premier cru and grand cru vineyards. I especially like their blanc de noirs for their vinosity\u2014the way the wines' complexity is reminiscent of regular wine. They also have a charming light pink color.\n\n_Champagne Drappier_ \u2014From L'Aube, a smaller, southern region that has only recently started to come into its own. The house is well known for its brut nature, or zero-sugar wine. They also produce a cuv\u00e9e without added sulfites, a difficult feat since sulfites help keep a wine from spoiling.\n\n_Champagne Legras_ \u2014From Chouilly in the C\u00f4te des Blancs region of Champagne. This grand cru vineyard specializes in blanc de blancs, which I adore for their bright apple taste and their minerality.\n\n_Champagne Charles Hubert_ \u2014From Ambonnay, in the Montagne de Reims. I would be remiss if I didn't at least mention the wine dearest to my heart: my family's. A fruity and rich grand cru champagne that is still fresh and elegant. You sadly may not be able to purchase it in the U.S., but that will make it even more rewarding to find. It is, without a doubt, my favorite wine in the world.\n\nSo don't shortchange champagne\u2014or yourself. Few things in this world can do so much with so little. Just one glass is enough to turn any night into a celebration. Champagne appeals to all the senses: the sight of bubbles suspended in golden transparency, the scent of freshness, the taste of fruit, the feel of sparkling foam on your tongue, and not least, the sound of the cork being removed. What other beverage has a built-in starter pistol for festivity? You just need to let the occasion choose you.\n\n_Sant\u00e9!_\n\nWine is an unsentimental teacher for those who, like me, would prefer to be in control all the time. Trying to impose your will on everything doesn't work with people, and it certainly doesn't work with wine. Someone can be described to you a hundred different ways and still surprise you when you finally meet face-to-face\u2014as can wine. A winemaker's power is limited by the land and the weather that comes. For a wine drinker, the notes, the scores, the recommendations, all of it means nothing as soon as you open that bottle. The wine might not only be different than you expect\u2014it might have gone bad.\n\nNew York is not a shabby teacher either. Just when I'd settled into the prewar tenement in the East Village I was sharing with Rose, the landlady told us she was selling the apartment. We would have to move by month's end.\n\nIt was nearly too much for Rose, who was unsettled enough as it was without Nico. Now we had to find a new home. We had no luck until with one week left we found a top-floor two-bedroom in a town house just a few blocks away on the same street. The new landlord was sympathetic to our situation, but firm about needing a guarantor since neither of us were permanent residents. I called our only American friends\u2014Peter, Min-Ji, Derek\u2014and they generously offered to act as a combined guarantor, but he still declined. At the last minute, through a network of phone calls, an old friend of my uncle's agreed to help. We were saved. (I later found out that these kinds of high-wire acts involving a deadline, confusion over bank hours, and a lot of awkward running down the sidewalk are extremely common for New Yorkers.)\n\nThose same friends helped us shuttle our furniture in the smallest U-Haul available. We didn't want to pay movers for a three-block move, but we quickly learned our lesson. It took several trips, three pizzas, and a few tears before I finally flopped down on our couch in the new space, among the boxes holding all of our possessions (mostly hers), with lamps and frames and shirt sleeves hanging over the sides.\n\nOnly then did Rose go from box to box and say, \"I don't need this. I don't need this, either.\" I wanted to kill her, until I realized what she was doing.\n\n\"Nico chose this,\" she said, taking out a creamer shaped like a cow. \"It's useless. I don't need it.\" She stood up and put her hands on her hips, gazing over the piles. \"Let's get rid of it all. I want us to start new.\"\n\nThis new apartment was the one I brought Jules to after he arrived in the U.S. early to surprise me at the end of August. Rose was in France to see her family and Nico (a trip she'd agonized over, worrying that it might make her feel worse in the end), so for a few days, the apartment was ours alone.\n\nJules and I had never lived together before. In the mornings we moved from room to room separately, one person making breakfast and the other taking a shower as if we were playing house, and when we found each other again we smiled stunned little smiles. It was so sweet and so simple to wake up in the same place every day. We were both acutely aware of having received this gift, and we didn't want to take it for granted.\n\nI don't know what I had expected. For two years I hadn't thought much about the details\u2014I'd just tried to imagine being in the same _country_ again, to start. I was blissfully unprepared, and surprised by how wonderful the actuality of his presence in my New York life was. It was as though he'd been my secret diary, to which I confided everything I thought about and cared about but kept in a private drawer, never seen by the people around me. Suddenly, instead of walking down Broadway and texting him about a water pitcher I really liked in a store window, I could just take him by the hand and pull him into the store with me.\n\nI wanted to show him everything. The coffee shop servers (who made my chai lattes) and the lady at the Laundromat all greeted him with some version of \"So, you _are_ real!\" while I blushed. I even made a point of introducing him to Jacob, the wine seller, who jumped at the chance to show off his French.\n\n\" _Et qui-est-ce? Est-ce ton petit-copain?_ \"\n\n\" _Oui, Jacob,_ \" I said, humoring him with a faire-la-bise, a cheek kiss. \" _Il s'appelle Jules._ \"\n\nAnd Jacob shook Jules's hand vigorously while Jules looked over at me with a bemused expression.\n\n\"Is this what you have been doing?\" he asked as we walked out with a bottle. I'd been making a point of buying wine regularly, so I could continue to teach myself about it, and because our new apartment was too hot to store bottles for very long\u2014too hot in the summer because we didn't have air-conditioning, and too hot in the winter when the heat was on full blast (another condition of living in New York). I'd been drinking my way through the Loire Valley but was having a hard time getting to know it well, as its wines are in general lighter and grassier than the wines I grew up with. Like Burgundies, they can confound the senses more used to full-bodied and richer wines. Jacob had turned me on to one of the more affordable Pouilly-Fum\u00e9 wines from the Loire called Petit Fum\u00e9, made by Michel Redde\u2014a fresh and fruity little white cuv\u00e9e for which I'd broken my rule of not buying the same wine twice in a month. I wanted to share all my good finds with Jules.\n\n\"Is this what you've been doing? Making poor American men fall in love with you?\"\n\n\"They always promise to call,\" I teased, \"but they never do.\"\n\nThe idea that we were always within walking distance of each other, even on workdays, was so novel that we met for lunch every afternoon at Whole Foods. I was happy to have a regular dining partner, since my attempts to ingratiate myself with my colleagues had stalled. I loved spotting him on the sidewalk, tall and long-limbed, with a neutral, slightly glazed look on his face. When he saw me his eyes would come alive, as though he'd just been switched on. In the store he was curious and tentative, stunned by the crowds even after we'd been there several days in a row\u2014almost paralyzed, actually, if I didn't urge him to keep moving. He poked at the mountains of quinoa and chard at the salad bar, looked critically at the cheese selection, and expressed wonderment at the color-coded, talking checkout lines.\n\n\"This place,\" he murmured. \"Is it the only grocery store in the city? I don't understand how it can be so crowded.\"\n\nIf we had been in France, we could have carried a bottle of wine to Union Square and taken time with our lunch. Instead we ate quickly in the seating area upstairs\u2014I spent half the time watching him chew the American food carefully, his eyes lowered in concentration\u2014and then I had to get back before Rachel's emails piled up. But Jules's presence meant I was no longer lingering in the Pringent tasting room at the end of the day; I had better places to be.\n\nThough it took some persuading to get Jules to come along, I finally got to act like a tourist in some ways. At the Statue of Liberty, his hair glistening from the salt water from the ferry ride, he looked up and joked, \"Much bigger than ours\"\u2014on a little island called \u00cele aux Cygnes in Paris, there's a replica only sixty feet tall. At the top of the Empire State Building he zipped his windbreaker up to the chin and made an obligatory loop of the observation deck, impervious to the sort of vertigo I was susceptible to. I could tell he thought it was a nice view, but my mind raced to think of something that would really wow him.\n\nI stamped an Empire State Building penny on one of the machines for him, and we took the long elevator ride back down to street level, where we spent a few minutes watching a man play a drum between his legs for the passing tourists. I thought Jules just wanted a moment to recover from the observation deck, but it was the most excited I'd seen him.\n\n\"You,\" he asked pointedly. \"What are you playing?\"\n\n\"This is a drum, my friend.\"\n\n\"What is it called?\"\n\n\"It is called a _djembe_ drum.\"\n\n\"It's very pretty. Where did you learn this?\"\n\n\"From my country.\"\n\n\"Where is your country?\"\n\n\"Trinidad.\"\n\n\"Ah, Trinidad.\"\n\nJules listened for another moment. Then the musician took a break and considered this Frenchman watching him so intently. \"If you like this music, my friends play in Brooklyn on Sundays after church. Just for fun. All different drummers, my country, Jamaica, Barbados, Saint Lucia. In Prospect Park. You know where this is?\"\n\n\"No, tell me, I will come,\" Jules said.\n\nThat was Jules: a man who barely lifted an eyebrow at two iconic landmarks, making fast friends with a street musician. That Sunday we were on the subway to Brooklyn. I'd been to Williamsburg, but this would be a new part of the borough for me. The train rose out of the tunnel and we were crossing a bridge, the river below us, and outside the window, downtown Manhattan on one side, and on the other, the Brooklyn piers. Jules and I pressed up against the glass until we dipped back below the surface. When we reached our station I held Jules's hand tightly and followed his lead across the street into the park and up the path. I didn't think he knew where he was going, but then we heard the sounds of the drums, a rhythmic clattering and booming. At a clearing we saw perhaps fifty people standing in a semicircle playing drums. That was it\u2014no other instruments. Just drums, most of them like the one Jules's new friend had played\u2014wood, with leather tops, but of all different sizes and tones, from short and high-pitched to low and resonant. Two men in the back were playing steel drums, and in the front, a small group of children were banging away at instruments their own size.\n\nWe spotted our friend somewhere in the middle, and eventually he noticed us and smiled and waved but didn't stop drumming. Most parkgoers stopped for a minute and then walked on, but a small audience stood just outside the circle. It wasn't long before I realized the music had hypnotized me. We watched without speaking. Jules's face was rapt. There didn't seem to be anyone directing the music, but somehow everyone followed the same basic beat. After several minutes the tempo would slow, and some of the drummers would drop off and rest, eventually leaving just two or three still going steady. Gradually they'd all join in again, this time creating a different song altogether.\n\nWe stayed for an hour, then walked through the park to a different subway station. When the train rose up again on the bridge over the river, Jules said quietly, \"This is actually something good about New York.\"\n\n\"What's good?\" I said. His words sparked a hope deep inside my stomach.\n\n\"The different people,\" he said. \"It's a good place to learn about other people.\"\n\n\"I couldn't agree more.\"\n\nAnd then we taught the Americans in our subway car a thing or two about French public displays of affection.\n\nThe more you want things to be a certain way, the more likely you are to believe them to be true. Your mind is always happy to play a trick or two on you. I'd desperately wanted Jules to settle in without a hitch. And the first few weeks were enough to convince me that he had, despite everything I'd learned about his personality in our years together.\n\nAnd I'd wanted the same for myself and Pringent. When I'd applied for the job, it had seemed on paper exactly what I wanted. I didn't have much of a choice, it's true, but I believed that I would love any position that involved wine and kept me in New York. It even involved making the wine more attractive to others\u2014perfect! Had I been more self-aware I might also have noticed the pleasant tingling I felt at the glamourous prospect of working for the New York arm of a champagne house. That sort of thing can get under your skin quicker than you think. Going all the way back to my first lunch with Elise, I'd become a little spoiled by the five-star hotels, the nonstop schmoozing, the pricey meals.\n\nThen there was Rachel, one of the prettiest and most stylish women I'd ever met, even more than Elise. She emitted a kind of charismatic sparkle that affected everyone within a certain radius. Walking down the sidewalk with her it was impossible not to notice the lingering glances, from both men and women. In short: she was born to convince people to buy things, and thus it was no surprise that she had been able to reach such a high position so young. I liked her for her style and her accomplishments, even if so much of what she did remained a mystery to me. I knew she spent a lot of time outside the office at meetings and lunches. She was admittedly not a warm person, and though she emailed a lot of piecemeal instructions, I'd yet to get a real sense of direction from her. But she was genteel and refined, and had a quick-draw smile, in that particularly American way, that accentuated her high cheekbones. After I'd been there a few weeks, I'd catch myself imitating the way she spoke, her calm, delicate enunciation, or trying to stifle my laugh so that it would come out as a quiet puff of air, the way hers did. We hadn't yet developed a close working relationship or started to brainstorm campaigns together, but it seemed only a matter of time.\n\nIt was this hope that made me continue to say hello to Marianne each morning, even though she appeared to seethe more and more as the weeks went by. And whenever I'd start to feel a prick of doubt, there would be an event at the Soho House or on the roof of a Meatpacking District hotel where I'd pour champagne and chat with A-list guests, feeling fortunate just to be there. There's nothing like handing a glass of wine to Christy Turlington and being rewarded with her thousand-watt smile to make you feel decent about life. And just in case I wasn't feeling lucky enough, there would be Marianne's bitterly jealous look to greet me the next morning.\n\nBut everything always looks better in the summer, and people hold on to it for as long as possible, until the day they can no longer deny that the sun is setting earlier and it's starting to get cooler at night. This was my first September in New York, and it brought a clearer, sobering view into focus, along with the crisp air. The month was also marked by my first corked bottle in New York.\n\nThe wine was from Jacob's store, though it wasn't Jacob's fault. Corked wine can't really be blamed on anyone other than the winemaker, and even she can only take some measures to reduce the chance of it happening. I'd started to shift with the season from the Loire's fresher wines to Burgundies, and it was one of these bottles that had spoiled.\n\nI could tell something was off as soon as I opened it. The cork had a musty, woodsy odor. I poured a small amount into a glass, and when I brought it up to my nose for a whiff, any fruit had been replaced by a dank odor. Just to be sure, I took a tiny sip, and it was like I'd put a sweaty sock in my mouth.\n\n\" _Tant pis,_ \" Jules said, just as he had with the mistaken Monbazillac all those years ago. Only now it was less charming, because I had no other wine in the house and Jacob had closed his store for the night. Jules shrugged.\n\nI won't say it was an omen, but it did come at the same time that there were other signs our New York honeymoon had come to a close. Rose returned from Paris after Labor Day, refreshed but heartbroken again to be separated from Nico. Now there were three of us in the apartment, and my roommates, though both near and dear to me, were little more than strangers to each other. Rose had left behind her boyfriend, with whom she'd once shared a home filled with her carefully selected furniture (and his impulsively selected knickknacks), and come back to New York to find another man in that space. It must have been odd for her, however much she tried to hide it.\n\nIt was less odd for Jules, but still odd. \"Everything has flipped,\" he observed one night when we were talking quietly in bed\u2014with Rose back, everything had to be quieter in bed now.\n\n\"What do you mean?\"\n\n\"I mean you were once the third wheel to Rose and Nicolas, and now she's the third wheel to us. Nicolas had to move back to France, and I had to move here.\" Perhaps it was unintentional, but I noticed that he described the move as an obligation.\n\nWe became a common sight as a trio. Whenever we ran into Peter, he'd say, \"Look, it's _Three's Company_!\" which made no sense to us until late one night Jules pulled up a few old episodes of the show online and we were entranced by the number of misunderstandings three people were capable of having. (But we were nothing like them!) On the nights I brought home a new bottle of wine, we'd pour three glasses, sit at the kitchen counter, and patter on in French about our jobs, and Jules's job search, while trying to find the words to discuss the surprising depth of a Corbi\u00e8res from Languedoc, or the underwhelming fruit of a Bordeaux. Rose was a delicate and patient sipper. Jules was an avid smeller, regularly lifting the glass to his long nose and inhaling for the pleasure of it.\n\nRose still liked to go out to distract herself, and Jules and I often accompanied her, but I preferred the nights we all stayed in and I could enjoy both the company and the wine. You usually think of a bottle of wine as perfect for two to share, and I would tend to agree (when I'm not hogging it all for myself). But I grew to love sharing a bottle among three. True, we didn't have much of a choice\u2014wine in the U.S. is much more expensive than it is in Europe, and we were young, and one of us didn't even have a job. But I also noticed that two people tend to agree on their response to a wine, maybe just out of habit, whereas a third has the potential to make things interesting.\n\nAnd even though Jules was a good sport for Rose's sake, the nights we did go out he usually lasted less than an hour before saying, \"It's too loud, I have to go home.\" Once or twice he said nothing at all and just left, and I later found him snoring in bed. It drove me crazy, but I liked his advance warnings only a little more. It was the noise he couldn't stand, and not just in bars and clubs. His most frequent observation was just how loud New York was, everywhere\u2014its stores, its sidewalks, its restaurants. You couldn't get away from it, he said, whereas in Paris, even the bars are quiet. The French talk so much, we consider it a crime against our nature to play loud music (unless it's in a dance club), or to raise our voices in public, forcing everyone else to raise their voices, too. To have to shout over your dinner is about as un-French as you can imagine but a normal New York experience. Jules couldn't get over it. \"How can it count as a conversation when you are just asking each other 'what?' all the time?\" he said incredulously. \"This is why Americans don't listen to each other. They are too busy yelling!\"\n\nOnce, he even left the theater in the middle of a movie. It was an action film, so not exactly something you needed perfect silence to appreciate, but less than halfway through he leaned over, whispered, \"I have to leave,\" and was gone. I fumed for a few minutes, trying to keep watching, before following him out to the sidewalk, where he was standing with his hands in his pockets and staring at nothing in particular. He gave me a sheepish smile.\n\n\"Why did you leave?\"\n\n\"The people are so loud in there! How can anyone concentrate on the movie?\"\n\nI tried to remember who was sitting around us. One woman eating popcorn (the French eat popcorn, too, but not as much), and some audible gasps with every explosion and fight sequence. How could the same man who followed a Trinidadian drummer all the way to Brooklyn be so skittish?\n\n\"You didn't have to leave,\" he said apologetically. \"I was just going to wait for you.\"\n\n\"How am I supposed to know,\" I said through clenched teeth, \"if you don't tell me?\"\n\nPart of it was just Jules being Jules. It had taken me time to adjust to the city's noise, but I'd never been as allergic to it as he was. But he wasn't stubborn about just his dislikes: once he did actually like something, there was no going back. In Paris he had one favorite bar and could go there four or five times a week, even by himself. If I had no idea where he was and couldn't reach him (he had an infuriating tendency to turn off his phone), I could find him there. He was not just obsessive, he was repetitively obsessive. He had watched every Disney animated film a dozen times with a designer's eye and memorized nearly every one, and he could listen to the same song thirty times in a row (especially Manu Chao), which I still found charming on paper, if less so in practice. And, in what I found a truly ironic twist, his biggest sports obsession since childhood was American in origin.\n\nBasketball.\n\nJules had always liked basketball, but now that he lived in the country of its birth, his fandom reached new heights. Once the season began in the fall, his life changed; for the first time he was able to go into any bar with a television and watch his favorite sport being played at its highest level. I like sports just fine\u2014I love tennis and appreciate soccer\u2014but his fixation became too much for me. One day I returned home during lunch to pick up something I'd forgotten, and found him in the bedroom with a conspicuously sweaty T-shirt. He had his hands behind his back.\n\n\"What were you just doing?\" I said, suddenly suspicious.\n\n\" _Rien,\"_ he said sheepishly. _Nothing_.\n\n\"Tell me,\" I said.\n\n\"Really, it's nothing.\"\n\n\"Show me your hands!\" I cried.\n\nReluctantly, he showed them to me. He was holding photographs of professional basketball players.\n\n\"They are cards,\" he explained. \"Collector's cards. There is a photograph on the front and statistics on the back, you see, look. They'll be worth a lot of money one day.\"\n\n\"You are buying these?\"\n\n\"Not so many,\" he said, avoiding my eyes.\n\n\"And why are you so sweaty? Tell me?\"\n\nHis long face grew even longer. \"I was...playing basketball.\"\n\n\"You were supposed to be working on your r\u00e9sum\u00e9 today! You were playing with who?\"\n\n\"With...Carl.\"\n\n\"Carl? My old roommate?\"\n\nJules shrugged in a quintessentially French way. \"He has nothing to do all day.\"\n\n\"But you do!\" I said, hitting him with a pillow. \"You have two months left to find a job before you are kicked out! And then what happens to us!\"\n\nHe ran away from me around the apartment.\n\n\"Admit it,\" I said, \"you hate it here!\"\n\n\"No, that's not it,\" he said, but now that I'd said it out loud I knew it was obviously true. The realization made me even more incensed because it meant he was lying, to me or to himself.\n\n\"You have no right to make me feel guilty,\" I said. \"We made a pact. You were supportive.\"\n\n\"I'm sorry,\" he said, dodging my blows. \"I'm sorry.\"\n\nIf you've been in love long-distance, you know the mantra that keeps you from abandoning all hope: _Everything will be better once we're in the same place._ You say it together out loud and in silence every single day until hopefully it comes true. Even though I knew that New York wasn't Jules's ideal city, I'd told myself that our being together would make up for it.\n\nDid I have any inkling that things would get tougher instead of easier? Is that why I had been nervous to tell him how much I'd fallen for New York in the first place? Either way, what made it worse was the fact that I was the one who'd brought him here, and now I both felt bad and resented that he hadn't been fully honest, either. The truth was that neither of us had known what would happen once we were here. Once we were together.\n\nIf Jules were a wine, he'd be a red with a lot of character and depth, intimidating on the outside but soft on the tongue, smooth and supple and a little sweet. The furthest thing from an easy-to-drink, flexible Beaujolais. He'd be a wine that takes time to open up, that isn't easy to understand when it's young. I can think of the perfect example: a Bandol from Provence, on the southern coast of France, made from 100 percent mourv\u00e8dre\u2014the same moody grape used in Ch\u00e2teauneuf that my uncle likes to highlight (Jules, all-or-nothing to the end, would never be a blend). Mourv\u00e8dre can be a hard grape to love, but it blooms with dark fruit, earth, game, and leather. You have to come to a Bandol on its terms, give it space to express itself, but when you do, the rewards are immense.\n\nI'm not very good at giving space, though. Now that Jules was here I wanted and expected to stay in easy reach, like any other couple. For the first couple of weeks at the end of August, when he had no cell phone, it was easy enough to just make a date and show up, but I assumed he would eventually get an American phone and was surprised when he refused.\n\n\"What do I need it for? Nobody needs to contact me but you, and you know where I am.\"\n\n\"I know where you are in general, but not where you are at any given minute.\"\n\n\"Don't worry about it. Everything's fine.\"\n\n\"What about job interviews?\"\n\nI learned the answer to that question the first time a strange number showed up on my phone\u2014he'd given it out as his contact, even though I was at work all day. Sometimes he'd be up early in the morning and on my phone in the bathroom, talking to family or potential employers (who never panned out). Then I began to lose track of him in the evenings. He was spending a lot of his free time with Peter, partially to get away from me and my various frustrations, but also because they had a lot in common I hadn't recognized until now: fiercely independent streaks above all. When Rose and I went dancing, instead of putting in his usual hour accompanying us he started to go directly to Peter's place uptown or to meet him for a late-night hamburger. (Yes, hamburgers! This Frenchest of Frenchmen, who could barely walk down the streets of New York without holding his hands over his ears\u2014not only was this man collecting basketball cards, he ate every hamburger he laid eyes on.) If when I got home there was no sign of Jules, I learned to text Peter first: _Do_ _you have him?_ and nine times out of ten he replied, _Right here._ A couple of times he wrote back, _Left a while ago!_ which meant Jules was now completely off the grid, like an untagged animal in the wild, and the best I could do was stick my head out the window and look up and down the street. Once, he walked home from the Upper East Side, and expected me to be happy that he'd enjoyed himself on the quiet late-night sidewalks, rather than angry I'd lost track of him hours before.\n\nBut I didn't like being angry with him. The very things that frustrated me were some of his most endearing traits. Once, about a month after he arrived, I woke to strange whispers in the living room. I bolted up right away, but all I could think to do was pull my sheets up to my neck. I was preparing to scream at the top of my lungs when a man's head appeared in the doorway. \"Hi,\" Jules said. \"Don't worry, but there are people here.\"\n\n\"Who is here?\"\n\n\"Two guys,\" he said, smiling, as if that would reassure me. \"Two guys from North Carolina. They hitchhiked here to go to Ground Zero to protest George Bush. They have nowhere to sleep tonight.\"\n\n\"You just met them?\"\n\n\"Yeah, I just met them on the street. They are really nice guys.\"\n\nHow could you not love someone like that, as long as he didn't get you murdered in your sleep? (And they _did_ turn out to be nice guys, who ate breakfast with us, exchanged numbers\u2014mine, of course\u2014and left to continue their hitchhiking tour of protest.) No, I wasn't in any danger of falling out of love with him. Rather, I worried constantly that the love wasn't enough to keep him with me. By October he had no good leads and only six weeks left on his tourist visa.\n\n\"What are you going to do if you don't find a job?\" I asked him.\n\n\"I'll find a job,\" he said. \"Don't worry.\"\n\n\"Jules,\" I said. \"But what if you _don't_?\"\n\nHe was sitting on the bed and I was standing in the doorway, upset enough that I had to maintain some distance in order not to break down.\n\n\"I'll find one.\"\n\n\"You can't talk as if it is a sure thing. It's not a sure thing. You have to be honest with me. What will we do if you don't find one?\"\n\nHe spread his hands out on his knees, a sign he didn't want to have this conversation.\n\n\"If I really don't?\" he said, his voice softer. \"What choice do I have then? I'd have to leave. But I'd come back and try again.\"\n\nI began to cry, but screwed up my face to try to keep the tears in. Something important had just been said and I had to make sure I had understood. \"So you wouldn't rather just go back and stay there? That wouldn't make you happier?\"\n\nHe shook his head vigorously, as if surprised I would even ask. \"No, I wouldn't be happier at all.\"\n\nI was still crying, but now in relief. He stood, and he came to me or I went to him, I can't remember which. More than likely, we met in the middle.\n\n# Thinking About Wine\n\nThere is always an element of risk when it comes to opening a bottle of wine. It may not taste the way you expect; that's a given. But it may also just stink. Ninety-nine percent of the time, your sense of smell is all you need to tell if a bottle is corked. A damp cork is not actually a sign of corked wine\u2014it just means the bottle was stored too hot, forcing the wine to expand and push against the stopper and become oxidized. These wines are often flawed, but not always corked.\n\nNo, to be corked means a specific chemical called trichloroanisole (TCA) has made it to the wine, most often from the cork, corrupting its flavor. TCA isn't toxic, but it is unpleasant, and humans are particularly sensitive to it. It's what gives that musty, damp-sock odor; the more TCA, the mustier. TCA comes from natural cork only\u2014if there were traces of a particular fungus in the cork when it was harvested, the cork sterilization actually triggers the creation of TCA. This happens in 1 to 5 percent of bottles stopped with natural cork. It's a cruel irony that the most common method we use to seal the bottle, allowing the wine to age and become sublime, can also be the source of its undoing. (It's a common misconception that at restaurants the standard practice is to pour you a small taste to see if you like the wine. They're actually giving you the chance to see if the wine has gone bad.)\n\nBut don't let this talk of bad wine dampen the mood. Corking happens infrequently enough that you shouldn't be too concerned. One piece of advice I like to give on this matter: if you like a certain wine that will benefit from aging, and you can afford it, consider buying three bottles. Then you can open one bottle at some point down the road and see how it tastes. If it's still too young, you have two bottles left for another few years down the road, almost ensuring that even if one is corked, you'll be able to enjoy the other. You can never plan for a corked bottle, but you can do your best to be prepared.\n\nIf you're just buying a bottle for tonight and discover it's corked, however, all you can do is laugh it off\u2014 _tant pis!_ \u2014and return it. A store should always refund you for a corked wine (as long as you bring it back).\n\nIt's also possible you are reading this twenty years from now and corked wine has become a thing of the past, in which case I'm sorry for wasting your time! As I mentioned, corked wine comes solely from natural cork: plastic corks and screw caps don't produce TCA. Plastic corks are often frowned upon because they don't allow any exchange of air (it's believed that wine ages well because natural corks let in minuscule amounts of oxygen), and are favored by mass producers churning out huge amounts of uninteresting wine. They're also just a total pain to remove.\n\nThe screw cap is a more interesting case. It has none of the inconvenience of a plastic cork and is just as resistant to spoilage. And most important, depending on the type and quality of the screw cap, there can be a tiny bit of air exchange, meaning that it might be the best option to replace natural cork. Every year that goes by, more and more winemakers are using them.\n\nThe place that seems the most reluctant to give the screw cap a decent shot, though, is (surprise!) France. You do see them, but rarely. Patrice Rion, who makes marvelous Burgundy wines in the prestigious Nuits-Saint-Georges, is a vocal advocate of the screw cap. He's so confident about their performance that he once showed me how he sealed some of his 2005 white Burgundy, the Nuits-Saint-Georges premier cru he calls Les Terres Blanches, using both methods\u2014some bottles with natural cork, and some with screw caps, and stored them side by side. He had us conduct a blind taste test right there.\n\nOne was clearly fresher, more pure tasting. You've already heard my thoughts on taste tests, but the results were indisputable: everyone preferred it. And, as you may have already guessed, it came from the screw-capped bottle. The natural-cork bottle was also good, but just the slightest bit duller in comparison. Everyone was shocked. Rion explained that he'd done many tests to find the best cap available, as not just any will do, just as not every natural cork is of the same quality.\n\nI'd perceived the difference with my own nose and tongue, but when I told my father, he refused to believe me. I explained the blind test and that there had been others present. I repeated Rion's explanation. _Non, non, non_ , my father said, shaking his head. He wouldn't hear of it. \"Why do all the grand crus of Bordeaux use a natural cork then?\" he asked.\n\nI had one word for him: \"Tradition!\"\n\nWhat I was too angry to tell him then was that the prospect of forgoing natural corks altogether saddens me\u2014not for any practical reason, if the success of screw caps bears out over further years of testing. And not for tradition's sake, either. I just find corks beautiful. Their texture is pleasing. They require a tool to remove, giving the act of opening a bottle a bit of pomp and ceremony. Twisting off a screw cap makes an unattractive cracking sound, a poor replacement for the gentle satisfying pop of a cork being pulled from the mouth of the bottle. It's like opening a gift\u2014a gift the cork can outlast, a piece of memory remaining long after the wine is gone.\n\nFew things sadden me more than when someone comes into my wine store and doesn't want to talk about wine. \"Can I help you?\" I ask, and with a shake of their head, or a polite refusal, they go back to scanning the labels and squinting at the prices, and then make a purchase, all without exchanging a single word. If that customer comes back, it's less likely I'll remember them than if we'd had even a brief conversation, and I'm always left wondering, Why _that_ wine? What's the occasion? Will they like it, based on their preferences? Was it a random choice? Even in the U.S., where people are generally unafraid to say they don't know much about wine, if you offer advice they'll still shrug and demur, \"Oh, I can't tell the difference anyway.\" Isn't that more of a reason to ask a question, not less? If it turns out that you don't like the bottle a store recommends, it's always possible the place doesn't know its wine, and you don't have to go back. But, even better, give them another chance\u2014tell them why you didn't like the wine, and they may find something that works for you.\n\nIt's no different with restaurants. Instead of scanning the wine list and settling on one randomly\u2014or worse, picking simply based on price\u2014ask for help. (I know that some people always choose the second-least-expensive bottle on the list, so that they don't _appear_ to be deciding based on cost alone. Don't do it!) You don't need to have a t\u00eate-a-t\u00eate with a master sommelier to find a match\u2014all waitstaff should at least know the general flavor profiles and tasting notes of the bottles on the list. But more often than not, you'll find that a good answer to your questions is another question: \"What do you like?\" Answer truthfully, and from your own experience. \"I like anything\" is perfectly valid, if not super helpful. \"I like reds\" at least cuts down the possibilities, and \"I like full-bodied reds\" is enough to receive a recommendation. \"I don't know\" is an honest, admirable answer. And whatever wine arrives, approach it with openness and curiosity. It may not be the perfect match. But it may be an unexpected one. Just remember: drink the wine that you have, not the wine you imagined. Most important, whether you are satisfied or not, preserve your reaction by writing it down, or taking a picture, or just concentrating, so that the next time someone asks you what you like your answer will be a little better informed.\n\nOne last thing: don't be afraid to spend a few extra dollars. I don't mean your life savings; I really mean a few extra dollars, a difference in price that we generally tolerate in a food entr\u00e9e but which makes us hesitate when it comes to ordering a drink. This is even truer in stores than in restaurants\u2014the difference in quality between a fifteen-dollar bottle and a ten-dollar bottle can be significant. Consider skipping your morning latte and get a nicer wine for dinner instead! I'm not saying that more expensive wine is always better\u2014that's definitely not the case. Depending on the region, vintage, and state of the market, the relationship between price and quality varies widely, and there are wonderful wines to be had at a good value. But _value_ is different than _cheap_. Small, conscientious growers (as opposed to the mass producers) often produce excellent wine at a relatively low cost, and you'll find that little step up is worth it. It's time to expect more from the establishments that sell you wine, and from yourself.\n\nUntil the end of summer I held out hope that Rachel would warm to me, and that even if she didn't become the mentor I wanted, she would at least start giving me more direction, enabling me to grow as a businesswoman and wine enthusiast. In person, she was always nice and never fumbled for her words, but the emails she sent me were often incomplete thoughts, or terse demands with no deadline attached, or sometimes, most unnervingly, just \"Can you come here?\" I always knew when Marianne got those, because in the middle of typing she would pause, puff a sigh from her nose, like steam from an iron, and walk over to Rachel's office.\n\nBut once I accepted that Rachel was a bad manager, I started to view her various charms as a smokescreen. Everyone besides me and Marianne was still fooled. The clients were always a little sad to find themselves talking to me instead of her. \"She's such an amazing woman. It must be so great to work for her,\" they said, as if reminiscing about a lover who had broken their hearts years ago. \"You have no idea,\" I'd reply. Even Paris headquarters would ask after her in an awestruck manner. _Rachel est trop charmante!_ She was beautiful and flatteringly dressed, always sporting an impeccable ivory manicure. She was also articulate, even when I didn't understand exactly what she was saying. It still _sounded good_.\n\nWhen she asked me to go over a marketing plan, I learned to expect one of two reactions. Sometimes, she said little as I went down my checklist, except to repeat certain key phrases. I would tell her the signage for a certain event had to be made from nylon, not paper, since it would be outdoors. And she would tell me to make sure the sign was made of nylon, not paper. She punctuated her comments with nods and ambiguous grunts, and when I left her office she looked lost in thought, dreaming up some strategy, it seemed, that I could hardly imagine.\n\nOnce, she asked me to order pens inscribed with \"Pringent USA\" for the goodie bags. But when I brought her the sample, she said, \"Who wants a pen? Nobody uses pens anymore. Whose idea was it to use pens?\" I was too stunned to say it had been her idea. Only by chance had I waited to place the final order, assuming that getting Rachel's approval would be straightforward. Now I realized how close I'd been to screwing up by doing exactly what she'd asked.\n\n\"We should do something cooler. What's cooler than pens? More _current_? Can we do bracelets? You know, the rubber bracelets for charitable causes.\"\n\n\"But what would our cause be?\"\n\nShe looked at me as if I was stupid, but her voice was as pleasant and even as always. \"It's not a real cause, Laure. Why would _we_ need a cause? We just need to pick a color.\"\n\nI took a deep mental breath.\n\n\"What color then?\"\n\nShe looked down at her own thin wrists. \"I don't know. What color hasn't been taken? White? Let's do white,\" she said.\n\nBut the next day she stopped by my desk and said, \"Laure, forget about the bracelets. Let's get those pens. The pens are fine.\" And she turned and walked back to her office without another word.\n\nThere's an English expression that captures perfectly how I felt at Pringent after my too-brief grace period: \"Rearranging the deck chairs on the _Titanic_.\" I didn't know if or when the iceberg would show up to tear the whole thing apart, so in the meantime it felt like our ship's fate was to sail onward, disaster on the horizon. Every so often the crew would say to one another, \"Are we going in circles?\" but no one could tell. We were all busy ordering pens.\n\nI knew I deserved better than what had turned out to be the bare minimum of a real job, but was trapped because of my visa and the knowledge that I was lucky to have it.\n\nAlso, despite the glamour of champagne, I began to miss working with still wine, with luscious reds and creamy whites, and knew I'd have to make up for it on my own, after work. The apartment I shared with Rose was too small for hosting ap\u00e9ritifs (we'd been spoiled by Maya's huge kitchen). But one night near the end of September when the weather was still pleasant, we saw the super coming down the emergency stairs connecting our top-floor hall to the roof. The door had a big red push-bar to open it and a sign that read, ALARM WILL SOUND. We'd been too scared to go near it.\n\n\"Izzy,\" we said. \"You were on the roof? The alarm didn't go off?\"\n\nHe glanced back at where he'd come from and said, \"Nah. There hasn't been an alarm on the door for years.\"\n\nAfter he went back downstairs, Rose and I immediately went over to the door and gave it a shove. It opened without complaint, and then we were outside on top of the East Village, on a clear night. The entire surface of the roof was coated in a silver, reflective material. We approached the edge, where the wall was just thigh high and easy to trip over and plunge six stories. Down below was Eleventh Street, the tops of people's heads bobbing by, and on the other side, the unruly garden space squeezed between the back of our building and one on Tenth. We could see the taller buildings on Third Avenue, and farther up, the colored tip of the Empire State Building. The ambient light of the city made a glowing horizon, even late at night. It was the perfect perspective on New York\u2014not so high that it becomes abstract, but high enough that you feel you've escaped the bustle below.\n\nIzzy told us we couldn't have parties there, but that we could bring a couple of people up once in a while. He might have wished he could take it back after we brought up beach chairs and ran extension cords from a utility outlet to put up a string of lights. We had no table, so we laid out a towel between the chairs, like a picnic blanket.\n\nOur first guest was Peter. It was a pleasantly warm evening, but he seemed uncharacteristically sullen. I'd gotten two bottles of a festive Beaujolais-Villages to inaugurate the rooftop, a deep, fruity natural wine from Marcel Lapierre that I'd chilled slightly. Halfway through the night, though, I realized that Peter wasn't sulking\u2014he was pensive, mesmerized by the sights and sounds of the city, like someone standing before the grandeur of nature, only instead of the Grand Canyon it was the East Village.\n\nThe next night he was himself again, joking, laughing twice as loud as anyone else, standing up to do small hip-shaking dances to prove a point. He chose the music on his iPod, plugged into portable speakers, and every twenty minutes took out a pack of cigarettes and considered it, usually but not always lighting one. Our other friends from the ap\u00e9ritifs, Min-Ji and Derek\u2014Peter's colleagues from the film production company\u2014had joined, too, so there were six of us, which is about as perfect a number as there is when it comes to company. When we toasted I made everyone make eye contact with everyone else.\n\n\"If you don't meet eyes it's bad luck,\" I said. \"And not just any bad luck. Seven years of bad sex.\" You've never seen six people look so intently at one another while clinking glasses. We were all low and close to the ground in our beach chairs. Only Jules perched in a taller lawn chair with cup-holders in the arms, and we teasingly toasted him as our king. As it grew later Min moved to Derek's lap to stay warm.\n\nMin and Derek had been dating since they'd first started working together a couple of years earlier\u2014he was a lanky soft-spoken computer animator, and she was a lighting specialist who'd grown up in Korea and moved here for college. English was her second language but she had no problem finding him funny. She would smile with her entire face. Some humor translates easily, for the right people. He was funny to us, too\u2014there was something a little French about his dry, gently sarcastic delivery. He especially liked to shut down any of our complaints by saying, \"Baguette about it,\" which would send Jules into fits of giggling.\n\n\"I'm serious,\" Derek would press on sternly. \"Let it go. Baguette about it.\"\n\nI was happy to be among friends in our own space, and Americans, too, which changed the tenor from when it was just three French people sitting around talking about home and all the things we couldn't understand about the U.S. Even Rose, who loved New York intensely, couldn't always avoid comparing it with Paris. When you live in a country that's not your own, every other thought, especially at first, has to do with that strangeness. Being around the Americans relaxed my brain. We talked about movies, or restaurants, or our childhoods in a way that felt less like comparing notes and more like an introduction, an explanation. _This is who I am._\n\nIt was a conversation, and a game\u2014trying to make everyone laugh, although half of us were more comfortable in a different language\u2014that we played over a nearly consecutive string of nights after we found the roof and before it got cold. Deep into October, three, four, sometimes five nights a week some combination of our group was up there. Peter moved to Fourteenth Street, close to Rose's old apartment, and with Min and Derek in NoHo, everyone was within walking distance. If anyone could help Jules feel more relaxed in New York, it was this group.\n\nI was so afraid of disturbing the fun that I didn't complain about work for the first week or two of roof nights. When I finally did, no one really had any advice, because they all wanted to leave their jobs, too. Rose was tired of doing all the busywork for her photographer boss and wanted more control of her time. Peter didn't want to be a middle manager; he wanted to lead his own projects. It was the same with Derek. Min liked her job but felt overworked. It was the blind leading the blind, all of us sure we wanted more but less sure of how to get it.\n\nThen Benji arrived. Benji was a freelance photo retoucher who'd been doing work for Rose's boss, and he and Rose had become friends. He was a one-man shop\u2014the only one among us to work for himself\u2014so his words had a little extra weight when, the second or third time he stopped by the roof, he finally spoke up and said something that might as well have been directed at me alone: \"Here's how I see it: nobody can really be happy working for someone else. No one likes to be told what to do. Some people just put up with it better than others. Some bosses are better than others. You just have to know your tolerance level. Otherwise you have to work for yourself.\"\n\nBenji was generally quiet, but he sometimes gave long, surprising monologues that made everyone pay attention. He was striking to look at, too, with a full beard and ponytail, his broad shoulders filling out his leather jacket. We eventually learned to recognize the growl of his Ducati motorcycle pulling up on the street below. In comparison to him, we must have seemed to be a ragtag group of misfits and outsiders. But he seemed to genuinely enjoy our company and antics, even though he was really Rose's guest, and (it quickly became clear) was there only for her.\n\nAfter a few nights during which he'd taken her out to fancy bars and restaurants we'd normally never go to, I asked her about it. \"You know he has a crush on you, right?\"\n\n_\"Non!\"_ She was adamant. \"He knows I have had the same boyfriend since I was fifteen years old.\"\n\n\"Just because he is respectful about it doesn't mean he doesn't have feelings for you.\"\n\n\"No,\" she said. \"It doesn't work like that. He knows nothing can happen. He wouldn't waste his time.\"\n\nI knew better than to argue. I could tell she missed Nico terribly, and none of us brought him up too often for fear of ruining the mood. Whatever Benji's intentions, I was glad she had someone else to spend time with. One night he called while we were on the roof and I heard her say, \"No, I haven't. That sounds fun.\"\n\n\"Where are you going?\" I asked her when she hung up.\n\n\"Benji's going to take me to Coney Island.\"\n\n\"Is it still open?\"\n\n\"He says there's stuff to do. I've never been!\"\n\nI heard his motorcycle pull up below, but this time he waited for her instead of coming upstairs. I went to the edge and watched her climb on the back of his motorcycle and put her arms around his waist as they rumbled away, like a movie\u2014or like me and Jules when we'd first met, if you replaced a delicately boned Frenchman on a Mobylette in Paris with a big American guy on a powerful Ducati in New York. They were spending so much time together that Jules and I found ourselves alone more often than we had since he'd first arrived. It was good for us, but it also meant I was more constantly reminded of what was at stake: he had just one month left to find a job. The chances were getting slimmer by the day.\n\nAt work I'd thought that once Marianne saw that I shared her view of Rachel, she would start to treat me like an ally. It didn't happen. I asked her about the trinkets on her desk and she only grunted in response. I still hoped we would at least bond over our shared nationality, the way we were the only ones in the office who didn't keep snacks at our desks. _Why do Americans do this, rather than just wait for mealtime?_ We would laugh together. Or how Rachel forbade us from speaking French. As Rachel put it, \"I just want to let you know this is an English-only office, Laure. We have to maintain our identity as the American division, servicing American clients.\"\n\n\"Sure, this will help me improve my English!\" I'd said cheerily at the time.\n\nI slipped up here and there, but couldn't understand why it was such a serious offense. If I was on the phone with Paris, checking on a shipment, or asking their sales department for tips about a common client, it made little sense to spend twenty minutes in English what would take five in French. If my counterpart slipped back into French (there was a wide range of English competence over there), it seemed only polite for me to speak in kind. But suddenly there'd be Rachel, standing in the doorway of her office, smiling at me with raised eyebrows and a very slight shake to her head.\n\n\"Oh, _d\u00e9sol\u00e9e_ , sorry,\" I'd stammer. She'd disappear back into her office while Marianne glowered at me. Marianne obviously thought I was making things worse for both of us.\n\nIt wasn't until weeks later that I wondered why Rachel was so adamant about English. Wouldn't it have been better for us to be as efficient, correct, and natural as possible? There was only one possible explanation. She didn't like me to speak French because _she couldn't understand what I was saying_. She worried I was talking about her, or making deals she didn't know about, or doing something, anything, however improbable, that might threaten her position. Me, a twenty-four-year-old she'd saved from deportation. It sounded like the silliest thing in the world, and yet as soon as I thought of it, I knew it was true, even though I had no way to confirm it.\n\nUntil, that is, I made friends with Allison, the office manager and executive assistant, a spectacularly well-organized girl slightly older than me with horn-rimmed glasses and an accent she said was unmistakably from New Jersey, although I couldn't hear it. I hadn't had much to say to her, since she worked on the other side of the tasting room and mainly with the CEO. She'd been very patient helping me figure out my insurance. But one day out of the blue she asked me to lunch. Why she chose that day, three months into my job, I still don't know. Either she'd taken pity on me as the odd woman out or, perhaps, she had something she wanted to get off her chest.\n\nShe started with harmless banter. \"I would love to go to Paris,\" she told me over soup and salad. \"I went once, for a junior-high band trip, but that doesn't count. You don't know what you're seeing when you're in junior high. You probably wouldn't even be able to pick yourself out of a lineup in junior high!\"\n\nShe drank Diet Coke like it was water. I ordered a glass of house white.\n\n\"Do you have wine with every meal?\"\n\n\"Never for breakfast,\" I lightly joked. \"But when I was growing up I had wine with most lunches and dinners. You can't do that here. It's too expensive.\"\n\n\"But weren't you just drunk all the time?\"\n\n\"No,\" I said. \"One glass at lunch is fine for me. It's for the pleasure.\"\n\nAs she contemplated her Diet Coke, I mentioned how much I liked her skirt.\n\n\"I get everything from Forever 21,\" she said, \"but only the good stuff.\" She laughed to herself. \"I'm Italian-Jewish, which means I'm cheap and have to celebrate a lot of holidays. That's all you need to know.\"\n\nI realized I liked her immensely.\n\n\"So how has it been? You have a fun job, doing champagne. Fancy.\"\n\n\"It's not bad. I like being in the business.\"\n\n\"You like working with Rachel?\"\n\nI was surprised by her bluntness. For a second I wondered if Rachel had sent her to spy on me. But she was waiting for my answer with such an open face I decided to take a chance.\n\n\"No. She doesn't give me much responsibility. I don't agree with her all the time, and yet she doesn't spend much time in the office, and when she's here she...how do you say this, she watches everything I do.\"\n\n\"Micromanages.\"\n\n\"Yes, that's it. It makes things harder. But you know, she's always at meetings, all the time.\"\n\nAllison took a sip from her Diet Coke without responding and watched the bubbles for a second before looking up with a slightly forced smile. I could tell she was holding something back, something she wished she didn't know.\n\n\"What about Eric?\" I said, referring to the CEO and her boss, to change the subject. She flinched, just briefly enough that I thought I might have imagined it.\n\n\"Eric's great,\" she said. \"Great boss. I really admire him. I feel like he's one of those guys who can be taken advantage of, though. Sometimes I worry about him.\" It sounded as if she was trying to tell me something, but before I could ask, she cleared her throat and asked for the check.\n\nThe new girl is always the last to find out. The clues were there in plain sight, and it took just the gentlest push from Allison to start seeing them everywhere. The meetings, the long lunches. The times Rachel hurriedly but in a strange hush left the office early. I noticed that Eric often would leave the office just a minute before her\u2014sometimes just seconds.\n\nWhen it hit me, I nearly dropped to the floor. Rachel had called her husband a \"cutie\" on my first day, and Eric had two children whose photographs smiled out from his desk. I didn't know how long it had been going on or if they'd been more careful about it in the beginning\u2014they must have. But at some point they'd started to get reckless, and by the time I picked up on everything, they were hardly bothering to get in the elevator at different times.\n\nIt went a long way toward explaining the tension around the office. It sounds entertainingly scandalous, but the entire balance of the company was thrown off by an affair between two of its top three executives (if the CFO was also involved somehow, I didn't want to know). Business was slow, sales were low, marketing deals were thin and spread out. Whenever the two of them came in the door, red-cheeked and with collars just recently adjusted, we would all glance at one another with withering looks in our eyes before turning back to our computer screens.\n\nAllison and I had lunch a few more times before I dared to bring it up. I knew I had to do it as soon as we sat down or I would put it off again.\n\n\"Allison. I know.\"\n\nShe smiled, sipped her Diet Coke, and innocently asked, \"What do you know?\" When I didn't respond, she said, \"Thank God. I couldn't stand you not knowing. But I couldn't just say it, you know?\"\n\n\"I know,\" I said.\n\nThe next time the waiter came around, she switched from soda to wine.\n\n\"What can we do?\" I asked.\n\n\"What can _we_ do? Nothing that I know of. Way over my pay grade. She's bleeding him dry,\" she said, looking a little angry, and I wondered if she wasn't a little in love with Eric herself.\n\nI lowered my voice. \"How long has it been?\"\n\nShe didn't hesitate. \"Six months? Could be longer.\"\n\n\"Well, I wish something could be done. It's poisoning the office. You said so yourself.\"\n\n\"Don't do anything rash, Laure.\" She gave me a look, like someone who had known me much longer than she had. Was I that transparent?\n\n\"I won't,\" I promised. \"Something will happen. It can't go on like this forever.\"\n\nBut I wasn't as sure of my words as I'd sounded.\n\n\"Isn't it none of your business?\" Derek said on the roof that night, in his contrarian way, after I told them that a co-worker had confirmed my suspicions. \"It's at work, sure, but it's still their personal lives.\"\n\n\"But it is making my job worse!\"\n\nOnly Rose was on my side, but for the wrong reasons. She didn't seem to care whether it was making my job harder (however ill-defined it may have been). Suddenly more Catholic than ever, she said what they were doing was immoral, even worse because they were our leaders. She thought I should take a public stand.\n\n\"What, you mean confront them?\"\n\n\"If you must. Otherwise you could send a letter to Paris.\"\n\nEveryone looked a little surprised by her vehemence. Back in the apartment, I told her, \"I don't know who would believe me. And if I have to leave the job, what then?\"\n\n\"We'll find you another job. Benji knows a lot of people.\"\n\nJules waited until we were in bed to say, \"This is not something you can fix, _mon coeur_. It is, what do you call it, _un mauvais bouchon_ \"\u2014a bad cork.\n\n\"But nobody forces you to drink bad wine. I'm forced to go to work. It's not like I have an alternative,\" I said.\n\nFor a second I thought he might say the alternative would be for us to return to France together, especially now that he was down to his last weeks, but to his credit he didn't.\n\nThat weekend, Jules and I walked to Chinatown for breakfast and talked more about his own looming job crisis, without reaching a solution. When we returned, Rose greeted us excitedly at the door and said, \"What are you doing for dinner? Benji is asking.\"\n\n\"We're having dinner with Benji?\" I was confused. \"What will you be doing?\"\n\n\"I'll be there, too,\" she said.\n\n\"Oh!\" I said, relieved\u2014and then alarmed. I looked over at Jules, standing uselessly with his hands in his pockets. What could we say but yes, even though it made me uncomfortable? Jules was not as close to Nico as I'd been, and he didn't know Rose as well as I did. She wouldn't accept it if I tried again to tell her that Benji was playing a game of chess against an opponent who was an ocean away.\n\nThat night, my stomach was in knots as we walked over to the restaurant, and Rose commented on my stiffness. \"Oh, just work stress,\" I said. In fact, I was dead set against being pleasant. When we sat down I gave Benji a long stare so he would know I was on to him. He didn't seem to notice, though, and was as affable as ever. He even ordered the wine (something that very few people did in my presence these days), explaining it was a bottle he'd had there before and liked\u2014even though, he was sure to make clear, he was no expert. It was such a charmingly American thing to say that I almost forgot to be angry at him.\n\nAfter the appetizers he cleared his throat and ran his fingers through his beard, and I knew it was coming: he was going to ask for our blessing to date Rose. And she had no idea what was about to hit her! I steeled myself to cut him off, and was so focused on Rose and Nico's relationship that I almost didn't grasp what he was saying.\n\n\"So there are these guys I've worked with, great guys.\" He turned toward Jules. \"They started a graphic design firm. They're really small still, but have a good client list.\"\n\nI suddenly understood. My grip on Jules's thigh tightened so dramatically he gasped and pried my hand off. The food came right then, but everyone waited for Benji to complete his thought.\n\n\"So if it's okay with you,\" he said, looking at Jules, \"I'd like to put in a good word. I know how talented you are. I think it'd be a great fit.\"\n\nAnd then, having said his bit, he dug into his coq au vin, leaving Rose beaming beside him, and Jules and me still speechless.\n\nThe design firm wanted Jules to do a trial run before they'd agree to hire him, so nothing was settled. But even the chance it presented completely changed the energy in the apartment. Jules put his basketball cards away and began to plan for the projects he'd be working on for the next three weeks. It was all he thought about.\n\nI, on the other hand, felt myself start to relax. If Jules didn't get the job, it would be crushing, but I was both confident in him and happy he had any chance at all. The next morning I sat down next to Marianne and told her about a man I'd seen on Broadway carrying a huge python around his neck\u2014and even when it was clear she wished I would stop talking, I kept going until I reached the end (the man got in a cab). If I was trapped at Pringent, I was at least going to make the best of it.\n\nRight around this time, Rachel began to ask me to attend more parties, launches, and shows, and even to present directly to stores and restaurants in the city. The duties were almost like the old days working for my uncle and Mo\u00ebt. Soon I was traveling with a tasting bag all around the city three or four days of the week, sometimes without even stepping foot in the office. Had I been savvier, I might have taken the change as a warning instead of a blessing\u2014that Rachel was trying to keep me away. I'd never actually argued with her, but it's possible she saw something new on my face, perhaps even some incriminating knowledge. I've always been a terrible poker player.\n\nBut ultimately, I didn't care what she thought. I felt strangely free, thanks to a combination of Benji's advice, Jules's opportunity, and the knowledge that Rachel and the CEO were sleeping together. I wasn't going to be hasty, as my visa was at stake. But I decided to look for a new job in the spring. I wanted more than this. I deserved more. And to be honest, there's only so much champagne you can drink.\n\nForgive me, Mother!\n\n# Thinking About Wine\n\nOne thing I would never expect the casual hobbyist to fully understand is the French system of wine classification. The term AOC, for _appellation d'origine contr\u00f4l\u00e9e,_ is used not only for wine, but for cheese, butter, honey, and other agricultural products. Getting an AOC stamp is a bureaucratic maze, but it does confer a distinct advantage to producers: the ability to charge more. This is both good and bad.\n\nFirst, appellations are extremely useful and aren't going away anytime soon. They contain an incredible amount of information: the land where the wine is from, the grapes that can be used, even (if you care about this sort of thing) the style of pruning allowed. The appellation can tell you if the grapes are blended from within the region, _villages_ style\u2014as in Beaujolais-Villages\u2014or if they come from a single vineyard. There are around 350 appellations in France, each one with its own set of rules and regulations. Generally speaking, when you pay for an AOC wine, you are also paying for a seal of approval\u2014to be able to trust the quality. (Whether the quality-to-price ratio is always worth it is another matter.)\n\nThere are also lower tiers of classification with less demanding restrictions. They are generally less expensive, but it's not a given they are lower in quality. Immediately below AOC is Vin de Pays, or \"country wine\" (you may also see the equivalent term _indication g\u00e9ographique prot\u00e9g\u00e9e_ ) _._ These regional categories are much broader: the largest is the Vin de Pays d'Oc, which covers the entire Languedoc region along the southern coast. Vin de Pays de l'Atlantique covers all of Bordeaux, along the southwest. This classification will still give you a general sense of geography and terroir, as well as the mill\u00e9sime\u2014Vin de Pays is still vintage wine. But while Vin de Pays still has a say over what grapes can be used in the region (no one would try to make a pinot noir wine in the brutally hot south, for example), its regulations are far less strict. You'll see varietal Vin de Pays\u2014single-grape wines\u2014from regions otherwise known for their patented blends; they're easier to market internationally, especially in the New World, where as I've said consumers are much more used to thinking of wines as \"chardonnay\" or \"merlot.\" The varietal wine Fat Bastard, for example, is one of the biggest success stories out of Vin de Pays d'Oc.\n\nBelow Vin de Pays is Vin de France, a category formed in 2009 for wines that were once just called table wines and were nearly unmarketable. Within Vin de France, you can use whatever grapes you want, from whatever regions, mixed from whatever vintages. You won't see a year on these labels, and the wine is usually pretty bad, although there are exceptions.\n\nThat's the important thing to remember: that there are exceptions to everything. AOC wine is, on paper, the best you can buy, because the classification is difficult to obtain if you aren't producing from a legacy vineyard or ch\u00e2teau, and because the rules are so comprehensive. But there's a lot of crappy AOC wine out there. It makes sense: if the AOC stamp alone is enough to keep up sales, some producers will do the bare minimum to keep that wine drinkable.\n\nMeanwhile, a small producer who wants to make a good Vin de Pays or Vin de France, who really cares about putting fantastic wine out there, not only has to ensure that the wine is of high quality, but also to hope that people will hear about it. Big companies use their marketing muscle to push well-known brands, but finding the real prizes within these less regulated classifications takes a little work. You might look for a recommendation on an oenophile's blog or a wine store's website or in a magazine. Price, too, can be an indication\u2014table wine in the traditional sense will be just a few bucks, but quality Vin de Pays or Vin de France will be priced similarly to AOC wine. If you're in a store, you have to ask.\n\nAnd you may have to pay a little more for a classification that otherwise wouldn't grab your attention. The lesson here is not that you should _always_ pay more; when I say to avoid \"cheap wine,\" I don't just mean the price. I mean a more insidious kind of cheapness, that is sloppy or indifferent or, at worst, trades on a name or label to appear more expensive than it's really worth.\n\nDiscovering the gems that don't lean on AOC and are good for the sake of being good is incredibly rewarding because no one is resting on their laurels. I'm lucky to be able to sell a few fantastic cuv\u00e9es in my store, but because there's almost no information allowed on the label, customers usually only find them by talking to me. Here are a few I would recommend:\n\n_Domaine Henri Milan, in Saint-R\u00e9my-de-Provence._ Milan makes one red, Le Vallon, and one white, Le Grand Blanc. The white is a standout, floral and creamy with a hint of the famous Proven\u00e7al lavender.\n\n_Domaine Claire Naudin-Ferrand, in Burgundy._ An aromatic and complex white she calls Le Clou 34, made in Bourgogne Aligot\u00e9.\n\n_Ch\u00e2teau Revelette, in Jouques, near Aix-en-Provence._ Le Grand Blanc is a Vin de Pays des Bouches du Rh\u00f4ne made from 100 percent chardonnay. It's firm and ripe with good acidity.\n\nI also carry wine made by my cousin Guillaume, the son of Alain. He manages a vineyard in Tavel called Prieur\u00e9 de Mont\u00e9zargues, known for its ros\u00e9. In fact, under AOC rules all Tavel wines have to be ros\u00e9, made from a minimum of 50 percent grenache. But my cousin also has a wonderful little red cuv\u00e9e of pure syrah, deep and spicy, that he has to bottle as Vin de France. He makes only eight hundred bottles each year. The wines are vintage, but he's not allowed to put the year on the bottle, so he stamps it on the cork instead. It's his little way of getting around the regulations, and a secret code passed from him to his customers. The good winemakers find a way.\n\nYou just have to find _them_.\n\nUltimately, in wine as in life, only you can know what you like. You may find the perfect moment, ask advice from others, drink with an open mind\u2014but none of it will matter if you don't know your own taste. When you meet someone new, even if you only speak for a minute, a voice in your head murmurs, \"I like her!\" or \"He gives me the creeps.\" This voice, if asked, would probably even be able to explain why. Only rarely does it have no opinion at all.\n\nOne advantage our instincts about people have over our instincts about wine is exposure. You meet far more people in your life than you do wines (one hopes!). But that doesn't mean it's too late to become a connoisseur, starting right now. With some practice and concentration, you'll be able to tell\u2014from a look, a sniff, a taste\u2014whether a wine is the right one for you. Just as a musician strengthens her ear like a muscle, learning about wine is a mix of experience, memorization, and attention. You must try a lot of wine. You must remember those you try and what you thought of them. You must focus as you drink. _It doesn't happen by accident_. Very little that is good does.\n\nIt takes work. It takes desire. It takes commitment. Any world that you are determined to explore seems intimidatingly wide before you start to map it the hard way, on foot.\n\nIn December, Marianne was fired. Though by then I was out of the office most days, I happened to be there to see it\u2014or at least the end of it. It had started harmlessly enough, with a regular meeting in Rachel's office. But after several minutes, the volume behind the closed door rose enough that the rest of us glanced at one another with question marks on our faces. The door opened, and Marianne walked out. Whatever the conflict was, it appeared to be over\u2014until, halfway between the office and our desks, she stopped, turned back, and said, loud enough for everyone to hear, \"You want to know your real problem? You can't make a good decision to save your life.\"\n\nWe held our collective breath. If Marianne had just sat down, whatever this was would probably have blown over and everything would have returned to its normal level of tension and discomfort. Instead, Rachel appeared in her doorway just as Marianne reached her desk. After a moment, she walked right up to Marianne\u2014that is, right next to me\u2014and said, \"Here's one good decision: you're fired. Pack your stuff.\"\n\nI'd never seen someone fired before, let alone so dramatically. Rachel remained there, with her rear end against my desk, arms crossed, gaze eerily placid, as it took Marianne a few seconds to process what had just happened. Marianne then began to pack up. I tried to look away but couldn't. Her stunned expression slowly tightened up into one of anger. Her clenched jaw quivered as she threw items into her purse, which was in no way big enough to fit all of her stuff\u2014the little toys, her collection of Japanese pens, all the things we scatter around to make ourselves feel at home.\n\n\"Are you sure that's yours?\" Rachel said at one point, making everyone go so quiet I swore I could hear several hearts beating. Instead of combusting once and for all, as I might have expected, Marianne just replied, firmly but at a normal volume, \"I'm sure.\" This triggered the final stage of her reaction: her face became dominated by a crazy, crooked grin that only grew wider when Ben, the chatty doorman from downstairs who always had a high-five ready, suddenly appeared from the elevator to escort Marianne out. I still have no idea who called him, and he seemed uncomfortable at having to play the heavy with one of our own.\n\nI didn't know what Marianne's expression meant, though I knew I would remember it forever. It might have been a nervous reaction, or it might have been something else: relief. The kind of instant relief that comes when you achieve the escape you always wanted but were too scared to go after. Clarity and horror at the same time. Marianne's eyes, just before the elevator doors closed, were bright and unblinking, and the grin had faded into a wan smile. Then she was gone, and I never spoke to her again.\n\nI also don't know to this day what became of Marianne's duties. It's not as though Pringent USA closed its normal wine business, but Rachel didn't give it to me either. In the days that followed, I kept waiting for her to call me into her office to discuss new arrangements, but the next time we were alone, she said only, \"Well, the holidays are coming, let's try to make it through the rest of the year without any surprises.\"\n\n\"Let's hope!\" I said with as much cheer as I could without veering into obvious sarcasm.\n\nThe truth was that even though my job was fine\u2014especially since I'd started to spend most of my days out of the office\u2014whatever tiny shred of respect I may have still had for Rachel had utterly disintegrated. Not that it was unreasonable for her to fire Marianne after such open defiance, but no one in the office had been on her side to begin with, and the public execution turned everyone even more against her while breeding an atmosphere of fear. The result was that everyone was outwardly more polite and accommodating to Rachel and the other executives, but inwardly mutinous. If nothing else, it taught me that most corporations survive by the grace of a peaceful society.\n\nBut if there was one thing Rachel was good at, it was having a bead on how people saw her. For a few weeks, she and Eric were more discreet, and every few days when I was in the office I'd watch her make a point of touring the floor desks, smiling her brilliant smile and pretending to care about the lives of everyone she said hello to. The bloodletting had also inspired her to new heights of efficiency. Her emails, sent from twenty feet away, were no longer signed with \"Thanks, dear,\" or \"Merci!\" There were no niceties at all anymore, just her initials: \"RM.\" Without compromising her genteel manners, every efficient motion and gesture said one thing: _I've done it before, and I'll do it again_. I requested a week of vacation between Christmas and New Year's to return to Paris, and she pretended to think about it before telling me, \"I'm sorry, but we've got to keep the engine running. You can have two days.\" Two extra days for an international trip\u2014for Christmas! But just when I'd gathered my senses and opened my mouth to protest, she said, \"Please close the door behind you when you leave.\"\n\nJules was in France to get his American work visa\u2014he'd gotten the job, to no one's greater relief than mine. It was the happiest possible reason for us to be apart, but it would take a month to process, and he wouldn't be back until after New Year's. Rose and I spent nearly every evening drinking a bottle between us and laughing, cathartically, deep into the night. It was like old times. She was angry at Nico for being unable to find a new position in New York, especially now that Jules had done it. She was bored and frustrated by her current job. Once, in the interest of trying something different, we went to a sake bar in the East Village and ordered a tasting of six sakes of different flavor profiles and opacities. (Japanese cuisine, including sushi, was only just starting to become popular in Paris at the time. Parisians don't like admitting this, but the city is always behind New York when it comes to trends.) The sake was delicious\u2014fragrant and complex\u2014though at home later that night we each had a glass of a Richaud C\u00f4tes du Rh\u00f4ne before bed. What can I say? I'm a loyalist.\n\nEach night I'd text Jules something for him to see when he woke up, as I'd done for months and months, and I'd wake in the morning to a text from him. The distance, ironically, returned a simple sweetness to our relationship.\n\nBut back in November, during his trial with Benji's friends' design firm, tensions had been high. He worked late into the night, and every morning I would wake instinctively ten minutes before the alarm and stare at his wide, fluttering nostrils. I've never known a deeper sleeper, and I constantly worried about his ability to get up. With one minute left I would shake his shoulder, gently at first, and then rougher. \"Jules, Jules, Jules, _attention_.\"\n\n\" _Quoi?_ \"\n\n\"Jules, don't be late.\"\n\n\"I won't be late.\"\n\n\"Get up, right now!\"\n\nWhat did he need an alarm for? He had me, throwing socks at the sleepy, silly half-smile on his face and pulling the shirt down over his head like a boy.\n\n\"You have to get this job!\"\n\n\"I'll get it, don't worry,\" he said. \"I'm the best designer there.\"\n\nHe only made me more anxious. \"Don't get cocky,\" I said, \"it'll make you screw up!\" and I would push him out the door with his shoes in his hand and his belt unbuckled.\n\nBut he was right. He was the best designer there, and just before Thanksgiving, with days to spare before his visa expired, the owners told him he had the job. It was exactly the kind of good news he would normally play a prank on me with, but he knew how lucky he was to get a full sponsored work visa in New York City, given how many friends and acquaintances had been unable to do it. He called me right away. When she heard me scream in joy, Rose came running into the room, and even she, whose boyfriend had spent months in employment limbo, teared up with happiness.\n\nBefore he left, we hosted Thanksgiving. Benji had gone home to Maine, Derek to Georgia, Peter to California, so it was Jules, Rose, Min, and me. Of the four of us, only Min had experience roasting a turkey. We purchased one with a built-in pop-up thermometer and spent the afternoon being scolded by Min for opening the oven every twenty minutes to look; when dinner came around, only the deepest of the thigh meat had any moisture left at all.\n\nI love many, many things about the U.S., but turkey is not one of them. I spent that Thanksgiving feeling sorry for us all. Mashed potatoes, baked sweet potatoes, string beans, roasted root vegetables\u2014all of these I like. But that turkey, the very centerpiece of the entire meal: do Americans eat it as a form of punishment? For a dish so proudly void of fat and flavor, there can be no other explanation\u2014an _homage_ to the early American struggle, perhaps, or the country's Puritan roots.\n\nBut I wouldn't dare suggest a major change to the holiday tradition. The only modification I'd propose\u2014just a small one, I promise\u2014is one we adopted that night. Your next Thanksgiving, don't think red or white: think champagne or ros\u00e9. As bland as turkey is, you don't have to worry too much about matching a wine. For once the meat will accompany your wine, rather than the other way around. (I've even heard of champagne-basted turkey, but that's a waste. You're better off drinking it.) We shared one good bottle of champagne, a Drappier brut nature, dry as can be, citrusy and floral, to toast Jules and our New York lives. And then we opened three bottles of Tavel ros\u00e9, deep-colored and peachy. Jules sat sheepishly at the head of the table, unused to the attention but red-faced and happy. I've heard my American friends describe Christmas as the holiday you spend with the family you're given, and Thanksgiving as the holiday you spend with the family you choose. This was the family I'd chosen.\n\nAs I looked around the table, feeling light-headed and appreciative, I thought that everyone reminded me of some wine or other that I had recently gotten to know. This game is more flattering to the wine\u2014people contain far more contradictions and complexities, and bring more pleasure, than any bottle of wine. But the comparisons still worked.\n\nRose loved champagne above all, but when I think of her I think of a red Burgundy in the C\u00f4te de Nuits, say from the Gevrey-Chambertin appellation: elegant, deep, with a personality that needs breathing room to be properly appreciated. But if you take your time with it, you're in for an unforgettable glass.\n\nMin gave us a hard time for ruining her turkey but never lost her good cheer. I knew she preferred full-bodied reds like Bordeaux, but I thought she was more like a Juran\u00e7on from the Sud-Ouest, an appellation where you can find both a wonderfully sweet golden wine, tropical-fruit-flavored and easy to drink, and a drier version that is subtle and complex, both honeyed and nutty. The main grapes, not as well known, are gros manseng and petit manseng.\n\nAnd while Jules might have reminded me of a tempestuous Bandol, like Min he preferred fuller reds (there's a lot to love!). And not just any red; he asked for Ch\u00e2teau la Nerthe every time we were in a restaurant. He didn't drink much wine before meeting me, but once he found out that Ch\u00e2teau la Nerthe was a part of my family, he never asked for anything else. If the waitstaff said they didn't serve it (which was usually the answer, and even if they did we generally couldn't afford it anyway), Jules would say, \"You should carry it. It's a very good wine. Her family makes it,\" and point at me. The server would feign interest, while I tried not to dive under the table. Then Jules would ask, \"Well, do you have anything at all from Ch\u00e2teauneuf-du-Pape?\"\n\nAs for me, my friends might say I am a northern Rh\u00f4ne, a syrah, with its spice and bold nature. The truth is that I like all wine\u2014a bit of a cheat, I know\u2014but the northern Rh\u00f4ne does have a special appeal. Given a choice I would be a C\u00f4te-R\u00f4tie, my favorite red wine from the region. Elegant, but not as subtle as Burgundy, it's full-bodied and well-structured, and forces you to wait before drinking it, giving the pepper time to soften. It's a wine that rewards patience, without hiding anything.\n\nBut enough about me.\n\nThe last half of December, Nico flew in to see Rose, so they could have a head start together before they both returned to France for the holidays. With Jules gone, our new apartment fell into a configuration that was familiar\u2014but different in important ways. While I was once again the tentative third wheel, not wanting to intrude, they were anything but the stable, easygoing couple I'd first met. There was an intensity to their interactions\u2014the same way Jules and I must have appeared to others after months of not seeing each other. Hungry, almost malnourished, as if the other person were the only source of food. This was how they looked at each other, how they locked fingers even sitting on the couch doing nothing. And in the fights I heard through two closed doors late at night.\n\nIf they weren't arguing in the apartment, they were arguing outside on the sidewalk and then filing into the apartment under a glaring, unmistakable silence. For the first week Rose and I never found ourselves alone so I couldn't ask her about it. Nico wasn't going to ask Rose to come back to France\u2014I didn't think he would, knowing how much she loved it in New York. But he also had to ask her for patience and assure her he was doing his best to return. It was the holidays\u2014no one was hiring now. If I'd had a moment with Rose I would have advised her to relax and just enjoy their time together until the new year. In the days before they flew back to Paris she seemed calmer, as if she had reached the same conclusion herself\u2014or had somehow discharged the bulk of her pent-up frustrations.\n\nTo give them space I went for long walks through the Village, something I hadn't done since Jules first arrived. It was dark before five o'clock, leaving long nights to fill. It had been a wet December, raining on and off for the whole month, and the sidewalks glistened from the constant drizzle. New York achieves another kind of beauty when it rains. The lights of the buildings appear in puddles, and while darkness usually makes the city contract, the water makes it expand, the way mirrors make a room look bigger.\n\nFor the first time, things were slow enough for me to look back on the year I'd had. What was I doing this very moment one year ago? Walking to my dinner shift at L'\u00c9l\u00e9phant, probably, dressed all in black beneath my coat buttoned up to the neck, having left the apartment I was sharing with Carl and Shaina (where were they now?). I still looked up into its windows whenever I passed on Second Avenue, but never saw anyone in them.\n\nThe decision I'd previously arrived at, to persevere at Pringent until the spring, had loosened me up in other ways. For one, though my budding friendship with Allison at work could have fallen by the wayside after the big reveal about Rachel and the CEO, I made an effort not to let that happen. And when Marianne was fired, we had more than enough material to talk about.\n\nI said I wouldn't have put up with Rachel's treatment if I had been Marianne. \"She'd been brainwashed,\" I said, \"working with Rachel too long. I would have made an even bigger scene, trust me!\" We were two cocktails in and I may have felt a little belligerent.\n\nAllison squinted at me affectionately, and said, \"You're really French.\"\n\n\"What do you mean? Are you insulting me?\"\n\n\"No! I don't know what it is, it's a lot of little things.\"\n\n\"My mother says I'm so American, because I'm always rude at home!\" I said, laughing. \"But I want to be even more American. Teach me to be more American.\"\n\n\"I promise,\" she nodded seriously. \"Just do as I do.\"\n\nEventually she even invited me to her place in Hoboken. \"You'll never be a true American unless you step foot in Jersey at least once,\" she said, teasing. I didn't have the standing to argue. We walked after work to a train station that I'd never noticed before, with a sign that read PATH, and I looked around in awe as though only a select few of us could even see it. \"Like _Harry Potter_!\" I whispered. Minutes later, Allison informed me that we were in Jersey. We'd gone under the river and I hadn't even noticed.\n\nAllison's street at first didn't look so different to me from some parts of Manhattan: row houses with apartments stacked over stores and restaurants. But then I started to notice the clean sidewalks, the concentration of chain restaurants, the seeming scarcity of non-white people. She lived above a tanning salon on a quiet side street. Her living space was on one floor and her tiny bedroom half a flight up, where the roof sloped. The bed was neatly made, a threadbare stuffed donkey propped on one of the two pillows. For some reason I'd expected girlish d\u00e9cor; instead, her living room was furnished with a brown leather couch and beaded throw blanket, a worn reading chair, and, most surprisingly, a small animal skull on the end table\u2014like that of a squirrel or weasel.\n\nAllison saw where I was looking. \"Oh that thing. It's my test. If a guy sits next to that thing for an hour and still wants to make out...\"\n\nShe didn't have any wine in the house. \"My secret,\" Allison said. \"Working for a wine company and knowing nothing about wine.\"\n\n\"I'm sure you know more than you think you do,\" I said, but she shook her head emphatically. \"Well, let's go shopping.\"\n\nThe wine store was a block away and we were the only customers dressed to stay in for the night. \"What do you feel like?\" I asked.\n\n\"I don't know.\"\n\n\"White? Red?\"\n\n\"Red, I guess.\"\n\n\"I hope you aren't this indecisive when it comes to men,\" I said as a joke, but her eyes widened, as if I'd stumbled upon the truth.\n\nWe settled on a Bourgueil from the Loire Valley. I usually associate the Loire with white wine, like Sancerre or Pouilly-Fum\u00e9, but I knew the region also had well-balanced reds that could be enjoyed without food or with a small snack. The wine needed to have enough body to stand on its own but be supple enough that we wouldn't have to fight through the tannins. You can get this balance from cabernet franc grapes.\n\n\"Let's try it,\" I said, \"and you tell me if you like it. If you don't, we'll pick something else next time.\"\n\n\"Deal,\" she said. She was normally so confident, but the talk of wine and men had clearly hit a soft spot. She told me she hadn't had a boyfriend in over a year, but I didn't know whether that meant she wasn't dating at all\u2014the American approach to dating has always been a subject of confusion for me. When I asked, she said in a way that was no help, \"I'm meeting people. I'm having a good time. But I love being single right now. Not having to answer any phone call or text message I don't want to.\"\n\n\"You are speaking about the opposite of my life,\" I said.\n\n\"I thought Jules is back in Paris right now.\"\n\n\"He is,\" I said. \"That makes things even worse for our texting.\" I scrolled through my messages with Jules. She was amazed at the quantity.\n\n\"It's like you guys would die without texting every few minutes.\"\n\n\"Yes.\" I nodded sadly. \"That's exactly what it's like.\"\n\nThe conversation, inevitably, returned to Rachel. Maybe it was the wine, maybe it was being at home, but Allison was more forthcoming about what she knew about \"the affair.\" She felt conflicted, having helped Eric, her boss, make and rearrange appointments to manage his time with Rachel, even though he'd never explicitly told her what was going on. She said he was really good to her and although she was angry, like everyone else, she felt bad that the situation he'd gotten himself into was eating away at the company. Rachel, however, she had less sympathy for.\n\n\"She has that mortician's smile.\"\n\n\"The what?\" I asked, and Allison pulled her cheeks into a frozen grimace with her fingers, her eyes flat and emotionless. I laughed until I lost my breath.\n\nMy flight back to Paris was the Friday evening before Christmas Eve. Since Rachel had allowed me only two days off, my return flight was less than a week later, on the 27th. The champagne business is dependent on the holidays, and Rachel said she wanted me to set up tables at the best Manhattan stores we'd sold to, leading up to New Year's, to help push stock before the big night. Even the morning of my flight, my schedule was full up until three or four in the afternoon\u2014four stores and two restaurant visits\u2014after which I was going to have to hurry to the airport. But Rachel called me just before lunch and asked me to come to the office between appointments.\n\nI texted Rose and told her that I had a feeling something was wrong.\n\n_You are overreacting,_ she wrote back. _They are probably finally assigning you Marianne's work._\n\nIt was the most likely explanation, so I tried to push the fears out of my head as I walked up to the building. Rachel was waiting for me in her office. Only once I sat down did I notice the CEO, Eric, standing by the far wall. Rachel wasted no time.\n\n\"Laure, your performance has been a disappointment. We're letting you go.\"\n\nI looked into her eyes for a good ten seconds before I could speak. She didn't lower her gaze once.\n\n\"This...this is a total shock,\" I said.\n\n\"I don't see how that can be. We've been unhappy with you for some time.\"\n\n\"You never said a word to me,\" I said. \"You never asked me to do anything different. I always did everything you asked. I completed every marketing plan. I've been meeting nonstop with clients. I've built relationships. I work well with Paris.\" (Only later would I suspect that some of those accomplishments were the very reasons she wanted me gone.)\n\n\"I didn't like that email you wrote to Maynesfeld House in September, as I told you on the record.\"\n\n_\"September,\"_ I couldn't stop myself from repeating incredulously. \"When did you know you wanted to fire me?\"\n\nI looked over at Eric, who was standing with his arms crossed, sporting his usual, flattering three-day beard and a grave, inscrutable expression. Unlike Rachel, he could hold my eyes for only a second before glancing away.\n\nA few strands of blond hair had fallen in front of Rachel's face and she swiped them away with an efficient flick of her hand.\n\n\"This isn't a negotiation,\" she said calmly. \"Today is your last day. Please gather your things. Don't make this a scene.\"\n\n\"But you turned down my request to take Christmas Eve off so I could take an earlier flight to be with my family. Why would you do that if you were going to fire me?\" I asked just to say it out loud, though there was only one possible answer. She'd done it out of spite.\n\nI've often thought about this moment, because it was the same chance not to go quietly that I envisioned for Marianne. In my mind, instead of walking out in stunned silence, I stand and tell Rachel that I know she is firing me not because I am bad at my job but because I am _good_ at it. Sometimes I bring up the affair that she may think she's doing such a good job hiding but which everyone on the floor snickers and groans about every single day. And I bring it up with Eric standing right behind her.\n\nBut I did neither of these things. In fact, I understood exactly what Marianne had felt in that moment. It was over, and Rachel\u2014all of this\u2014wasn't worth the calories it would have required to throw a fit. With every second that passed, Pringent and everything about it was shrinking in my mind, shrinking and shrinking until it was worth no more anguish than a pebble in my shoe. I had little to say to most of my colleagues, but from the disbelieving looks they gave me, I could see that whatever Rachel had hoped to gain by getting rid of me was unlikely to work. When I passed Allison's desk, she looked stricken. I mouthed, \"I'll call you.\"\n\nThe one and only thing to Rachel's credit is that she didn't have Ben the doorman escort me out.\n\nMy short and inglorious career at Pringent USA had lasted nearly six months, or about the same amount of time I'd spent touring the U.S. on behalf of my uncle. I spent much of the flight back to Paris thinking as little as possible, alternating between sleeping and watching movies. I knew that given the time pressure of my visa expiration I should have started planning what my next step would be, but I was too spent. At the airport, my mother seemed happier than usual to see me, putting her hands on my shoulders and smiling while looking me over as if to verify there was no appendage missing. \"There's my girl,\" she said, hugging and kissing me. We barely made it to the car before I started to cry and told her what had happened. Immediately, her goodwill disappeared. Instead of sympathizing, she began to berate me with peak exasperation.\n\n\"This is because you are too difficult!\" she said. \"You are too spirited, too loud, you never listen to what anyone says. You always think you know better than anyone else. I'm not at all surprised you were fired!\"\n\nNaturally, I cried even harder. When we reached the house, she apologized, but still made it clear that she thought I had somehow brought it on myself.\n\nWhen I saw my father, he told me that while he was certain I hadn't done anything out of line, he wouldn't have been surprised if I had somehow made the situation worse. (My parents are either very insightful, very patronizing, or both.)\n\n\"She fired the other French girl, too, you know,\" I protested. \"She was out to get the both of us. I felt it from the start; I just didn't listen to it.\"\n\n\"Why did she hire you then?\"\n\n\"She thought I was stupider than I turned out to be. I don't know.\"\n\n\"I don't know either. I do know this. You were not happy. You did not respect your boss. In those situations, it's only a matter of time, one way or another.\"\n\nAfter lunch, Jules came to pick me up. After months of listening to the roar of Benji's Ducati, seeing the chugging little Mobylette made me laugh. I'd missed it. He gave me a long kiss and held my head in his hands, and said, \"I'm so sorry, _mon coeur_ , but everything will be all right.\" There we were again, puttering along the streets of Paris as we did when we were both still in school. I thought if I wrapped my arms around his waist hard enough we might be transported back to that time. But instead of two Parisian students, we were now two New York residents just back for the holidays. We went to his favorite bar, but he seemed a little restless. When I asked him what was wrong, I almost expected him to suggest that he drop his job and for us to move back. I waited for it so intently that when he actually did speak I was so unprepared for what he actually said that it cheered me up for the rest of the day.\n\n\"The only thing this place is missing is a good hamburger.\"\n\nThirty people came to my father's house for Christmas dinner: his two sisters and Uncle Alain and all their children. I was nervous about seeing my uncle\u2014I didn't know what he'd say about my having been fired from the job I took after his. Would he think I'd impugned his good name? Question my general competency? But he was as accepting and encouraging as always, and I should have never doubted him.\n\nWe had a wondrous, enormous meal. The French are famous for moderation, it's true, but even moderation needs moderating\u2014you can't do it every time. Every member of my family is a big eater and, even more, a big drinker. Leave it to the Americans to cherish a nearly fatless bird on a national holiday; to the French, the real king of fowl is the duck, rich and fatty. The wine had been hand-delivered from Champagne and Ch\u00e2teauneuf-du-Pape, and my father donated the spirits. It was about as good a menu as you can have.\n\nAfter dinner, Uncle Alain finally pulled me aside, and changed my life for the second time.\n\n\"This just came to me, Laure. Do you remember the man we had dinner with in February, Mark Brodeur? He's an importer. He does everything himself and might need help. But he runs his business out of Westchester, so it would be far for you.\"\n\nI did remember him, a tall, white-haired man who seemed to have a hundred stories and a hundred laughs at the ready. I remembered his cigars. I remembered the dinner probably the best of all. At the end of the night Mark had given me his card but I'd never used it.\n\nMy uncle had his number, and I called the next afternoon. I didn't expect Mark to pick up the day after Christmas, but he did.\n\n\"I remember you,\" he said. I thought I detected a small note of amusement in his voice.\n\nThere was, of course, no way not to tell him that I was out of work, so I got right to the point, saying only that I hadn't been happy where I was and had recently parted ways with Pringent. I bit my thumb nearly to the bone as I waited for a response. After a beat, he said, \"I see. Well, I'm in Montreal until the new year. Would you like to come up on the third and talk? I'm always looking for help\u2014but the _right_ kind.\" I could tell he had a strong idea of exactly what the right kind of help was.\n\n\"Yes, of course, I can do that,\" I said.\n\nJules and I met Rose a couple of days later at the M\u00e9nilmontant bar, which was thankfully open. There was something strange but poignant about the three of us meeting in a Paris bar, when we would all see one another again in a week in our shared New York apartment. But there we were, sipping tea and coffee and eating _\u00e9clairs au caf\u00e9_ beside frosted glass as though we had never set foot in the New World. We were all aware of this oddness, which felt even more like a time warp than my first Christmas home, because both cities, Paris and New York, were equally corporeal to me now, with New York the more immediate and Paris the more natural. It felt a little like being in two places at once.\n\n\"I don't know what to do,\" Rose said, echoing my own dilemmas from another time. \"I want to live in New York, but Nico won't come back with me. He says it's too hard to find a job and he has so many more good opportunities here that he'll start to sabotage his career if he holds off any longer.\"\n\n\"It is hard, Rose.\"\n\nShe sniffed. \"I don't know if he is trying to call my bluff about my wanting to stay in New York. He says he is trying to come, he's still sending out many feelers, but that there is no demand. He says the economy is no good. I suppose I believe him, but I'm so angry anyway. Maybe I am trying to call my own bluff.\"\n\nJules and I held hands under the table and murmured encouraging things. The most we could do was offer support for whatever decision she made, as much as we wanted her and Nico to make it work together.\n\nThe rest of the week went by quickly. I was at ease, at my mother's apartment, sleeping in my old bed, while knowing that a good opportunity awaited me in New York. But I was also supremely nervous just below my relaxed exterior, _because_ I knew that a good opportunity, maybe the best I'd ever get, awaited me in New York.\n\nJules and I spent a subdued New Year's at a friend's house party. In our last minutes together in Paris the next morning, standing on my mother's stoop, Jules kissed me sweetly. I was definitely not going to let him come to the airport. He would be back in New York with his brand-new visa in just a few days, while now I was the one under threat of losing my ability to stay in the States. We'd switched places. We couldn't catch a break! But then again, we'd caught so many to get to this point. I needed just one more.\n\nJules was right, of course, though he couldn't have been as sure as he sounded. I got the job. But not before some trepidation and soul-searching. I'd thought hard about the hypothetical situation Jules had never actually raised\u2014should we just call America quits and come back? It seemed more appealing once I looked up the commute I was going to have to put up with each day. But in the end the decision wasn't really that hard. I wasn't ready to leave New York.\n\nI landed in the morning and only had time to stop by the apartment to change and shower before heading to Grand Central to catch the train to Westchester for lunch with Mark Brodeur. But just before I purchased tickets I received a text message: _My flight delayed. Let's do tomorrow if you are able._\n\nI was able, although it just meant another twenty-four hours of nerves eating a hole in my stomach. Instead of returning home straightaway I bought a sandwich and stood in the station's great hall, watching the business travelers back from the holiday crisscross in front of me. I'd only been there once or twice, and nearly as impressive as the hall itself was the number of people who were no longer impressed by it. I made a vow to always be impressed.\n\nI went back to my empty apartment, fell asleep on the couch, and woke six hours later not remembering where I was. I put on my coat and went outside to get something to eat. The ground was slick from light rain. It was the first Thursday of the new year, but the streets were oddly quiet. I couldn't remember ever feeling so alone in New York, not since my first few days in the city. Back then the solitude I'd felt was that of an alien. This was different. It felt like a city on hold\u2014almost as if I'd paused it while sorting out my life. What I wanted more than anything was the power to press PLAY again, or, if it was about to start up again, to follow along and not get left behind. What I had been doing was, I felt, essentially American\u2014rushing toward new experiences and challenges, as if life were a great experiment. I'd been following its loose, improvisational script.\n\nAs I turned the corner onto Fourteenth Street, backtracking the route I'd once pushed my giant suitcases down from one hostel to another, I started to cry. Not out of sadness, but from an intense feeling of belonging.\n\nI will tell you one last story about Pringent, with some ambivalence, because if I truly harbored no bad feelings and even felt some gratitude toward the company for forcing me to move on, then this story shouldn't make me happier to know it. But it does.\n\nIn late January, after I'd been working with Mark for a few weeks, my uncle called. I thought he just wanted to chat, but he was very excited. I started listening with polite curiosity, but it quickly turned into rapt attention.\n\n\"Laure! Listen,\" he said, sounding breathless. \"I just spoke to a friend who knows many people at Pringent. Your old boss Rachel and the CEO, Eric, they separated from their spouses and took a long vacation together to Thailand. But hold on, that is not the story. It just so happened that at this time Jean Pringent was in New York and stopped by the office. He was there less than an hour before everyone started telling him how unhappy they were, how the business was going down, and of course how the two _rongeurs_ had let their _feu_ consume them and fired two good workers because of pettiness and jealousy. And Pringent believed them! But the affair was not his real concern. He called in an auditor and sat with them day and night as they went over the records, the ledgers, until they had proof that it was not only the culture of the office that had been poisoned, but the business itself. Satisfied, he found the resort in Ko Samui where Rachel and Eric were staying\u2014with the help of the CEO's assistant, which should tell you how bad things had gotten\u2014and instructed them to fly directly to Paris. He dismissed both of them within five minutes of their arrival at headquarters. They were told they would have no further access to the New York office. That was two days ago, and I have no idea where they are now. Perhaps they saw this as a sign and now live together in Paris, yes?\"\n\nIt was a surprisingly romantic note for my uncle to finish on, but he can be forgiven. Our family tends toward optimism.\n\n# Thinking About Wine\n\nAs a celebration of family\u2014real or constructed\u2014Thanksgiving is a wonderful holiday. But I still don't understand why Americans insist, with 100 percent seriousness, on cooking the most boring bird on the planet.\n\nI'll at least repeat my small plea to kick up the occasion by avoiding the usual still wines we invariably serve at dinner, and try what the four of us did on the first Thanksgiving I ever celebrated: champagne and ros\u00e9, or just champagne, or just ros\u00e9. You can't go wrong.\n\nChampagne hardly needs further introduction or praise. Ros\u00e9, however, is perhaps the most misunderstood of the French wines, and for that reason alone deserves special attention. I've had people ask me if ros\u00e9 is made by mixing white and red wines. It's not (except for some ros\u00e9 champagnes). Ros\u00e9 is made from red wine grapes, but some producers add white grapes for a floral touch, as my cousin Guillaume does with the clairette grape. An extra step draws out the juice while limiting how long the juice stays in contact with the skins, depending on the producer's preference and the kind of wine desired. The two methods are _ros\u00e9 de presse_ (pressing the grapes to produce their tinted juice) and _ros\u00e9 de saign\u00e9e_ (bleeding off the juice after macerating the grapes with the skins). The balancing act is in achieving the right color and flavor without letting the tannins overtake the wine's delicacy and elegance. Given how specialized this process is, you shouldn't think of ros\u00e9 as something \"between\" red and white. It is its own affair altogether.\n\nRos\u00e9 also has a reputation as a sweet, girly summertime wine, which leads people to clearer, lighter varieties. Provence produces the prototypical ros\u00e9s\u2014lightly colored, fruity, and dry. And indeed in Paris when the weather turns warm, the caf\u00e9s all set up their outdoor tables and chairs and everyone sits in the sun and orders ros\u00e9 from Provence. It's a beautifully tinted three-month movie, and then it is over. Look: if you just want a refreshing drink, stick with lemonade. Ros\u00e9 isn't just there to cool you down. While it can't be aged long-term, it is more than capable of subtlety and flavor\u2014and ros\u00e9s can be, and frequently are, quite dry, not sweet. It allows you to taste the best parts of red wine grapes without the tannins. It is friendly and pleasant and more flexible than it is given credit for.\n\nAnd this brings us back to Thanksgiving. As far as poultry goes, people often have strong opinions on whether you should drink it with red or white. _Pinot noir! Sauvignon blanc!_ But I recommend a third way: try a darker ros\u00e9 like the Tavel I ended up choosing. Tavel is an appellation in the southern Rh\u00f4ne Valley, and some of its ros\u00e9s might surprise you\u2014rather than the light pink wine of Provence, Tavel makes a deep red one, fruity and spicy with peach flavors. Though most ros\u00e9 is best enjoyed within a year of harvest, Tavel ros\u00e9s, along with some from Bordeaux, are an exception, and actually evolve well with another two or three years of aging. Ros\u00e9 in fall and winter? Believe it. And try for yourself.\n\nWhen you meet a bottle of wine, everything that has made it what it is has already happened. The aging, the fermenting, the harvest, the growth, the soaking up of the land through its roots\u2014by the time it is opened by you, the wine's entire life cycle, leading up to tonight, is complete. It's the same when you meet a person\u2014the lines on their face say something about who they are, but not how they got there. That story takes much more time to tell.\n\nMy mother always told me that something marvelous happens to a person when they reach a certain age: they're more resilient, more _themselves_. The same can be said for grapevines. A root system that has developed over decades is hardier and less susceptible to swings in the weather, able to handle too much rain or too little. Older vines may produce less, having lost the boundless energy and freshness of youth. But the wine they are capable of producing is less reliant on sheer fruit, and can be more balanced, more layered, more complex.\n\nWhen does a vine qualify as an old vine? Nobody knows for sure. Or rather, there are plenty of opinions without consensus. Some say fifteen years and some say fifty. Even when you see _vieilles vignes_ , \"old vines,\" on a label, it doesn't tell you much. The grenache vines my uncle uses for his Cuv\u00e9e des Cadettes are more than a hundred years old, and I can tell you that the product is quite good, but it's impossible to know how much is due to the age of the vines, or to cultivation techniques, or to terroir. When it comes to the art of viticulture\u2014and life\u2014the value of age is real, but difficult to quantify.\n\nTime itself is no guarantee of quality. You can't simply tuck some vines away on deprived land, not prune them for half a century, and come back to make great wine, just as you can't drink wine your whole life without paying attention and expect to have magically gained wisdom. Though I'd grown up with wine, I'd only been paying attention for a year. Now it was the first week of 2008, and I was twenty-four. Who is wise at twenty-four? I knew what I _wanted_ to know, but it was like something on a shelf just out of reach. No matter how I stretched and grazed it with my fingertips, I could not will myself to be taller, just as I could not will myself to reap another year's experience in a day.\n\nI worried I would need it, though. Mark Brodeur had been importing French wines for longer than I'd been alive. At our dinner with my uncle, he'd been in a chatty, boisterous mood, but I'd also detected an intensity that my uncle didn't quite match, and I didn't know what to make of it. I didn't know what he would make of me.\n\nMy knees were weak as I got off Metro North in the small town an hour out of the city that was home to Mark Brodeur Imports. I'm not generally nervous in interviews, but things are always different when transatlantic stakes are involved. The fact that he was stubborn enough to run his business here, instead of in Manhattan, seemed an important clue to his character. My uncle told me Mark had long resisted expanding his business, but that he was one of the most passionate wine people he knew. I wondered if those two facts were related.\n\nThe village was so small that I'd barely stepped off the train when I found myself on the one-street downtown, a short strip of charming one- and two-story redbrick buildings trimmed in white. American flags were pinned to the telephone poles, and the shops had Christmas lights strung on their roofs. The only traffic signal I could see was a stop sign at the far end of the street where a few cars were making turns toward what I imagined were large beautiful houses with expansive lawns for their children to run around on.\n\nBrodeur Imports was on the first floor of a small gabled building. It had been nearly a year since I had met Mark, and all I could remember was that he was tall and white-haired, with a voice that carried, and that I hadn't been able to keep up with the conversation at the time, the names of wines and wine producers that floated in and out like old mutual friends.\n\nThe door, stenciled with his name, was as elegant and old-fashioned as you'd imagine: a wide wood frame with a single large glass panel in the middle, horizontal shades askew behind the glass, and a round brass doorknob. It was a door I would later go in and out of hundreds of times and, aside from my desk, the thing about the office I remember best. Now I opened it for the first time, and there was Mark, standing from his desk to take my hand. He was just as I remembered\u2014the white hair and glasses, the slight downturn of his eyes matched by a slight upturn to his mouth that gave him a look of permanent bemusement. But everything that came out of his mouth in Qu\u00e9b\u00e9cois French was immediately serious and to the point.\n\n\"When we met you were helping Alain. You were traveling the country,\" he began, as if to remind himself.\n\n\"Yes, I was touring Mo\u00ebt's distributors.\"\n\n\"Mo\u00ebt, ah right. Tell me about what it was like and everything you learned.\"\n\nI stammered a bit as I tried to describe the importance of making a personal connection with the sellers and restaurateurs. I said that I'd learned that every wine has a story, and that telling the story is invaluable to understanding the wine. I also said that the areas of the U.S. I'd seen were quite different from one another, but to my surprise nearly everyone I'd met was very enthusiastic and passionate about wine.\n\n\"Nearly everyone?\" he interrupted.\n\nI thought about the bad encounter I'd had in Nashville, but said only, \"It's a business. I understand it's possible to be in wine just as a business, for the money, but I also think you won't be able to run the best business that way.\"\n\nHe nodded thoughtfully. \"And Pringent. Tell me about Pringent.\"\n\nI laughed nervously. \"At Pringent I learned the details. I learned the difference between a medium-sized business and a giant business like Mo\u00ebt. It was not the best place for me, but it did help me realize I like smaller businesses, where you aren't just doing the same task over and over. You have wider responsibilities. And I learned what I want to do and what I want to be. It's not enough for me to be in the industry. I want to work with wine every single day.\"\n\nHe leaned forward, and his brow furrowed. \"So you weren't happy with Pringent.\"\n\nI shook my head. \"No, I wasn't very happy. My boss was young and glamorous\"\u2014as I said it I realized that what I'd originally admired about Rachel was now what I counted against her\u2014\"but she was more interested in how things looked than how they worked.\"\n\nHe grunted. \"You won't have to worry about youth and glamour with me,\" he said, and I laughed. He smiled in return, the corners of his eyes and his mouth nearly meeting.\n\nMark told me about his business. He _was_ his business. He traveled to France to tour the producers. He made the deals to import the wine. He built relationships with the wholesalers and the restaurants. When they called, he was the one to pick up the phone. When there was a problem, he was the one to solve it. He wrote the checks, cashed the checks, balanced the ledger.\n\n\"I do everything,\" he concluded. \"And I could use help,\" he said, opening up his palms. \"But I want the person I hire to be able to do everything, too. I don't want someone just to do the books, or just to do the calls, or just marketing, or just dealing with clients. I want someone who can also talk to winemakers, go to tastings, pour the wine\u2014and also clean the desk.\"\n\nHe gestured toward his expansive desk, which had neat stacks of paper and folders lined up next to his computer screen. \"I clean mine every night. You would also be expected to clean yours every night,\" he said, pointing across the room at an unoccupied, smaller desk. He leaned back in his chair, interlaced his fingers across his stomach and paused, as if to let it all sink in.\n\n\"Most important\"\u2014he raised a finger\u2014\"I don't want somebody who is just here for a job. I am not just here for a job. I want somebody who loves wine as much as I do, if not more.\"\n\nIf this was a negotiation, he'd certainly laid out demanding and detailed terms. My knees were still quaking slightly, but no longer from fear. I couldn't remember the last time I'd been so excited.\n\nFour hours after I'd arrived, I returned to the station and boarded a train heading home. I couldn't believe how long the interview had gone. I'd done my best to assure him that the person he was describing was me\u2014that I wanted to do it all, and out of love. My biggest concern was that I was too inexperienced for what he was looking for.\n\nMy head buzzed as the train slipped by the beautiful-sounding Westchester towns, crossed over the bridge into Manhattan, and went underground. If I'd convinced him, this would be my commute\u2014three hours a day in total\u2014but worrying about it now would have been like complaining about a gift I had yet to receive.\n\nI was still alone in the apartment, days I spent in nervous anticipation. I bought some colored lights half off at a hardware store and strung them around the living room for a touch of festivity. On Monday, Mark called to offer me the job. I was practically shaking with glee\u2014until he mentioned the salary, which was a little more than half of what I'd made at Pringent. He seemed to hear the surprise in my silence and added, with the firmness of a true dealmaker, \"It's all I can do.\" He didn't condescend to say he was doing me a favor, but he also knew that without the job, I would be on a plane back to Paris within weeks, this time for good.\n\n\"What about the train?\" I drew up the courage to ask. \"Could the company pay for my train tickets?\"\n\n\"I'm sorry, no,\" he said. \"But the company will pay for lunch. I take lunch very seriously. That should be familiar to you, as a Parisian, no?\"\n\nI accepted.\n\nMin and Derek invited me that night to a party at the Tribeca loft of someone they knew. I happily tagged along. At some point during the night, Min and Derek became uncharacteristically ardent, kissing each other as if I wasn't standing right there, so I walked alone around the duplex, dodging strangers who wanted to talk, looking at the giant canvases of abstract art, and then made my way back to the drinks table, where I found an uncommon C\u00f4tes du Vivarais red from the Rh\u00f4ne Valley\u2014more rustic in its hints of blackberry and spice than typical Ch\u00e2teauneuf wines.\n\nMin found me and asked how I was doing. \"Aren't you happy?\" she said. \"I'm happy you're still here. I'm happy you don't have to go back to that nasty place.\"\n\n\"I am,\" I assured her, though I was close to tears. Complicated tears, of relief, frustration, anxiousness, joy. I've always thought that Min was a mind reader and a fortune-teller, so I listened closely as she squeezed my arm and leaned in to say something. \"This is just the beginning,\" she said.\n\nThe first month with Mark was an adjustment, to say the least. On sunny mornings, I'd biked or even walked to Pringent. Now I woke up before sunrise, in full dread of the January air, and trudged through the slushy streets to Grand Central, where I rushed to buy a pain au chocolat and, if I didn't have a cheap French novel with me, the _New York Times,_ and fight against the wave of commuters heading into the city. I felt like I was swimming upstream.\n\nMy first day, I got up early enough to shower, dry my hair, and apply a full face of makeup. After a couple of weeks, I started to put on my makeup on the train, and then I simply began wearing less and less. I wish I could say I gave it up out of feminist ideals, but the truth is I was just exhausted. Eventually I was regularly walking through the jingling front door of Mark's office with no makeup on at all. If he noticed, he never said anything. (My mother would have been horrified. She never goes outside without lipstick and since my teens has been pressuring me to at least put a little on to \"emphasize the right things.\")\n\nThe return commute was even worse, especially those first days when it was new to me. I left the office between seven and eight and still had an hour and a half of travel ahead. By the time I stepped into the apartment I was too tired to eat, and skipped dinner or ate cereal or toast and fell asleep in my clothes. When Rose came back in the middle of January her desire to go out was undiminished, but she was disappointed to have lost her dance partner. The nights she decided to stay in, she sat on the couch with me and read until I fell asleep with my head on her shoulder.\n\nIt was grueling. But I never questioned it.\n\nJules, also back, was now a working New Yorker, a fact that made me smile the dozen times I thought of it each day. The only problem was that his schedule had shifted nearly as much as mine, but in the opposite direction. I woke up while it was still dark, whereas his days started at ten in the morning and often went to eleven or twelve at night\u2014the hours of a young start-up design firm scrambling to establish a client base. He was determined to prove himself, but was also working hard out of loyalty\u2014something we both felt toward our employers. As he saw it, the design firm had taken a double chance on him, a French national with whom they'd had no previous connection. He wanted them to know they'd made the right decision. So I left while he was still deep asleep. He woke up at nine and walked to Fifth Avenue. I returned late at night, barely able to keep my eyes open, and would frequently be asleep by the time he got home. Some days we didn't see each other awake at all.\n\nIt was almost as if we were long distance! I whispered my goodbyes in the morning to his unconscious face, not wanting to risk waking him with a kiss. And he came home wired, wrists sore from a long day of drawing, with a hamburger in a bag, awake for another two or three hours watching basketball highlights on his computer. Long gone were our leisurely people-watching Whole Foods lunches and walks around the city. In a matter of months our lives had shifted into a more serious, more demanding gear.\n\nAlso, poor Jules had never experienced the punishing winters of the American Northeast, and it made him return to his old favorite game: things about America to complain about. \"You didn't tell me it was colder here than Paris,\" he groaned. \"My coat is no good now,\" he said, referring to his prized all-weather leather jacket, which was warm enough in France even when it snowed. Even after we bought him a new coat he'd loved in the store, he still grumbled. \"There is something different about the air in New York,\" he said, sounding extremely certain for someone without any science to back him up. \"I can tell, it goes right through the cloth much easier than Parisian wind.\"\n\nBut secretly I was glad to hear his complaints about winter. It meant he was no longer complaining about the noise, or the food, or the people who made up the living, breathing city. He had started, against every fiber of his being, to get used to it.\n\nThe commute and loss of time with Jules was worthwhile for one reason: Mark Brodeur. I've never quite understood the phrase \"larger than life\"\u2014it seemed to be enough that he was more alive than almost anyone I've met. He had a deep-rooted awareness of life's gifts and grievances, and drew everyone's attention to them constantly. \"Isn't life grand?\" he liked to say. Or, in moments of frustration, \"I'm not going to waste my life doing this.\" He was in love with it; he flattered and scolded it. Aside from his Qu\u00e9b\u00e9cois accent, he could have been French, with the sort of attitude I was accustomed to from my friends and family back home: stubborn, opinionated, always relishing the moment.\n\nHe also would have preferred that the earth split in two beneath his feet rather than miss lunch. Each day at one o'clock, no matter what we were doing, he heaved a sigh, pressed his hands against his desk, and said with the utmost gravity, \"Time for lunch.\" My first day, I didn't know what to expect\u2014he'd mentioned lunch when he offered me the job, but I figured it would be delivery or a sandwich.\n\nHe stood and gestured for me to come with him, and we walked out into the Westchester winter, the well-heeled local residents blowing puffs into the air and dodging the light tufts of snow that had gathered by the curb. At the end of the next block was one of the few buildings in town with a dark wood frame. The name above the door read LA MAISON DU ROI. It was French owned and operated, Mark told me proudly. Inside, we stamped the dusting of snow from our shoes, and suddenly there were hearty greetings being thrown our way.\n\n\" _Monsieur Brodeur! Bienvenue. Pour deux?_ \"\n\nAs we sat down, Mark introduced me, and everyone from the owner to the servers welcomed me with enormous, knowing smiles\u2014it was clear he'd already clued them in to my arrival. As he unfolded his napkin he let out another, far more pleased sigh, and without cracking his menu ordered a French onion soup and a duck confit. He leaned forward to explain why he had ordered such a hearty meal, even though I hadn't asked.\n\n\"I go home now, there's no dinner. There's screaming, but no dinner. So this is all I have,\" he said, and spread his hands out over the place setting in front of him. He then told me that he had become a father for the first time in his fifties with his second wife, an American woman, and now had two small children. In fact, that's why he needed help. His wife had managed the office for years, but she had decided to stay home when the first child was born\u2014five years ago. It had taken him this long to admit that he would eventually have to trust someone else around the office.\n\nThe French onion soup, which he insisted I also order, was too cheesy and oily and salty to be authentic. But Mark dug in with relish. As someone who went to France twice a year on business, he must have known what good onion soup tasted like\u2014and yet here he was, smacking his lips over every spoonful. It had to be partially a form of pragmatism\u2014it wasn't as if there were a dozen bistros to choose from\u2014but his excitement was infectious, and so the soup wasn't so bad, really, I thought, as I took another sip.\n\n\"This life is what I want my daughters to have,\" he went on. \"The good food, the French way of thinking. I want them to go to Paris and learn French\u2014we don't speak Qu\u00e9b\u00e9cois at home. They should study abroad when they're in college. I lived there for a few years in my twenties, you know. I loved it. I worry about the American schools\u2014the children are too prized, they are like little kings.\" _Petits rois_ , he repeated in French, like an insult.\n\nThough I'd been working for Mark for less than a day, he had no trouble unloading long tirades of personal details and opinions. I'm straightforward, but I'm not a big sharer, so I mostly nodded and clucked sympathetically. I was also biting my tongue to avoid arguing with him so soon in our working relationship. He had something of the old-fashioned belief in the supremacy of the Old World and the shallowness of the New, and I wanted to tell him that as someone who'd been born in France, it wasn't so simple. Everything that was great about France, I would have explained, was a double-edged sword. Our pride, for example, makes us susceptible to becoming defensive and small-minded, whereas the American way is to welcome everything, devour everything. It could be confusing, at least to an expatriate like me, but also exciting.\n\nAt the same time, not a day went by that I didn't wish New York could be just a little bit more like Paris\u2014that it was easier to find a good croissant, that more things were built to be pleasing and to aid pleasure. The old ways, built on centuries of experience and experiment, _are_ often more rewarding. In short, my feelings were complicated, which I was only starting to be able to articulate, as I began my second year in America. But I wasn't going to say all this to Mark on my first day of work.\n\nLater, perhaps.\n\nThe next day at one o'clock on the dot, Mark sighed that same sigh and again we dropped what we were doing, retrieved our coats, and stepped out onto the chilly sidewalk. Only this time instead of La Maison du Roi, he took me to a nearby Japanese restaurant. Here, too, he was greeted by name, settled into a favorite table, and ordered a selection of sushi from memory. This time he talked less about himself and more about the business. He had a curious way of talking about the business\u2014not with numbers, but stories. He told me about a winemaking family he had great relations with in Saint-\u00c9milion, one of the more famous regions of Bordeaux\u2014a father and three daughters, one of whom, he said, looked a little like me: tall, brunette, freckled, and spirited. For as long as he'd been in business he had imported their wine: two wonderful Bordeaux sup\u00e9rieur, Ch\u00e2teau de Macard and Ch\u00e2teau de Ribebon. The father, he told me\u2014dropping his chopsticks to demonstrate\u2014had very big hands, weathered from working so hard on the vines. He smiled and picked up his chopsticks again, while I silently interpreted the lesson he was trying to get across. I'd been so used to having to read the subtext at work, but ultimately, I decided that Mark was just telling me this man and his family existed, made good wine, and were an important part of his long professional life.\n\nOn Friday, we returned to the French restaurant, and as we hurried down the main street, jackets pulled tight against the cold, I realized that aside from a pizza parlor, these were the only two restaurants downtown.\n\nAs for actual work, the first thing Mark asked me to do was to update the fact sheets for all sixty producers in his portfolio.\n\n\"Where are your previous ones, so I can see how they were done?\" I asked.\n\nHe hesitated, the first time I'd seen even a flicker of doubt on his face.\n\n\"Well, you see,\" he said, shuffling around a filing cabinet. \"My wife did them before she stopped working.\"\n\n\"Five years?\" I asked incredulously. \"You've been going without fact sheets for five years? There's no way they're still good.\"\n\n\"No need to get excited,\" he said. \"Some of the information I carry in my head, and I tell the sales reps everything I can. But you know how it is.\" He shrugged. I did know. Many sales reps had to be familiar with three to four hundred wines. If you didn't make things easier for them, your wines didn't sell.\n\nMaking the updates was not quite as straightforward as I hoped. The producers I needed to reach out to for information were in France, and busy. The winter is an important season\u2014the vines are dormant and need pruning, a skill that takes a great deal of experience to master. You have to trim aggressively to stimulate the coming summer's growth. So the winemakers were often in the fields, or it was dinnertime and they didn't want to be disturbed. Many weren't diligent about using computers and printers and email, if they even had them in 2008. (Whenever the subject came up, they'd ask me, in typical fashion, how could computers possibly help them make better wine?) I sometimes faxed them, if they had a fax. And I needed to make sure they sent me everything, from a basic description of the vineyards to the proportion of the current vintage's blend\u2014say, for example, a 2005 Bordeaux was 10 percent cabernet franc, 60 percent merlot, and 30 percent cabernet sauvignon. That gave a customer a clue about the balance of the wine. The more cabernet sauvignon, the richer, more robust and serious, and less exuberant the wine will be. When it is young, cabernet sauvignon can be severe, which is why it is often blended with merlot to make it fruitier and more drinkable.\n\nWe also included anything about the story of the vineyard to make it stand out. Family-run producers are often very good at making wine but don't always prioritize marketing. Their attitude is, \"I make the best wine there is; why would you need help to sell it? It should sell itself!\" They were proud of the histories of their ch\u00e2teaux, but didn't see what that had to do with sales.\n\nThey also hated sponsoring promotions and deals. At Ch\u00e2teau la Nerthe, I'd seen how effective incentives could be for our reps. Tell one that selling fifty cases will get them a free trip to the estate in France, and I guarantee you those fifty cases will get sold. But the wineries were reluctant to pay for the prizes.\n\nBy the end of January, I finally had the updates to show Mark, who nodded his approval and said to print them up.\n\n\"But this is just the information, it doesn't even look good,\" I said. I'd just typed up the facts in a long bulleted list. \"We need a photo of each winery. We need them to be designed.\"\n\n\"I don't see why,\" he said. \"The details are the details.\"\n\n\"No, no, no,\" I argued, used to explaining the ways of the contemporary marketplace to crotchety producers. \"If something looks good, it gets more attention. That is always the way it is.\"\n\n\"You can do what you like,\" he said, raising his voice a little and waving at the paper. \"I'm not paying for a graphic designer.\"\n\nI retreated to my desk, surprised at how little he cared about the ugly fact sheets. He was exacting about most things but strangely indifferent about others, and this was clearly a case of the latter. But I was determined to get it right. And I had a designer in mind.\n\n\"How much will you pay me?\" Jules said.\n\n\"I will pay you in smiles.\"\n\nHe considered. \"I can't feed myself with your smiles.\"\n\n\"You're going to have to learn how.\"\n\nHe was already coming home at eleven. For the next few weeks I stayed up waiting up for him to help me design the new sheets, at a pace of two or three a night. It was child's play from a design point of view; he could do one in twenty minutes. But the cumulative effect was not only mind-numbing for him, it took away from his sleep and basketball-watching hours, no matter how much I tried to make it seem like a fun way for us to spend some time together.\n\n\"I should break up with you over this,\" he said drily. It was true. He was doing more than a thousand dollars' worth of work, gratis, just so I could prove a point.\n\n\"That's fine,\" I said, putting my arms around his neck as he clicked away. \"But please wait until you finish the last ones.\"\n\nEventually I turned them in by dropping the stack triumphantly on Mark's desk and announcing that they hadn't cost him a dime. He flipped through them with his lips pursed, and my confidence began to droop. Finally he said, \"You were right. Very pretty. I think it will make a difference. I don't know if it was worth your time and thus my time, but it's very nice. We'll have them printed for the wholesalers.\"\n\nI told him my boyfriend had been helping me late at night, so I hadn't wasted any of his time at work.\n\nHe raised his eyebrows. \"Sounds like a nice man,\" he said. \"You also seem to be someone who it's hard to say no to.\"\n\nMark was careful with his money, to say the least, and his most common complaint was that everything was too expensive (everything except his wine, of course). I never went to his house but assumed it was large, and I knew both of his daughters were in private school and daycare. But he watched every penny, debating with me down to the thickness of the paper we used.\n\nHe was the opposite, however, when it came to knowledge, which he could never give away fast enough. Every day he was in the office, we went to lunch, where he would tell me about his wines and his trips to France and his crazier stories about disreputable wholesalers who liked to bounce checks. It was as though he was impatient because I did not know everything he did. Sometimes he'd say something like, \"Remember when it appeared we lost that whole shipment?\" referring to a time before I'd started. But when it came to the sixty producers he currently worked with, he never repeated himself. \"You need to know my wines in order to do your job,\" he reminded me sternly. But what began as instructional lectures would always end up as discursions on the beauty of a vineyard, of a certain vintage, of a part of France, which occasionally turned into gentle ribbing.\n\n\"You are the niece of a respected southern Rh\u00f4ne winemaker and you haven't seen the Dentelles de Montmirail?\" I shook my head apologetically, just the excuse he needed to keep talking. \"Limestone ridges, not so tall, but quite striking. And all around the base, the vineyards of the Gigondas. So beautiful. You know the Gigondas.\"\n\nI nodded, my other standard response when he was in the middle of a lecture. It was a Friday afternoon, and we had gotten back from lunch a few minutes before.\n\n\"Mmm,\" he said, a small smile on his face. As if the idea had just come to him, he reached into the cooler where he kept samples and retrieved a bottle, he told me, from a winemaker he had known for thirty years. He stared at it as if he could see the man's face in the glass.\n\n\"I don't want to tell you what to expect,\" he said. \"Let's just taste.\" I'd passed the test of my diligence with the fact sheets, but this was a test I was not only unprepared for, but certain I'd fail. It was like the nightmares students have about quizzes they haven't studied for, where the questions are written in a completely alien language.\n\nHe took out a butler's wine opener, the kind that is not a screw-pull but has two prongs of unequal length to both grip the cork and let a little air into the bottle so it can be removed. I've never been able to use one well, but they are effective for old corks that can fall apart easily. In just a few seconds, he had the bottle open and was pouring the wine into two glasses.\n\n\"Tell me what you detect,\" he said, and waited.\n\nMy throat was closed up. Among my friends I had no problem explaining every note; they didn't know any better than me. But now I was with an experienced professional, my boss, who had just presented me with a wine I had never tasted before and, it was clear, was not about to give me any hints. I worried that if I floundered, he would send me packing.\n\nI forced myself to lift the glass to my nose and sniff, registering a vaguely familiar profile; it was like meeting a cousin of my uncle's wine. It was elegant and supple, the fruit and the alcohol wafting gently into my nostrils. I swirled and smelled again, the red berries becoming more apparent. Then I took a sip and let the wine run all over my tongue, let it interact with the air in my mouth before swallowing.\n\nI nearly coughed out of nervousness and looked over at Mark, uncertain if he'd expected me to drink or spit. But he was happily drinking, too.\n\n\"Well?\" he asked.\n\n\"Cherry,\" I said tentatively, the one thing I knew for sure\u2014it's among the easiest and most common notes to detect. \"Plus darker fruits. Mineral. A little toast or smoke in the finish?\"\n\nMy hand was shaking. I put my glass down on a table.\n\n\"Mm,\" he muttered, the worst, most ambiguous thing he could have said. Then he nodded. \"There's more\u2014a little white pepper, a little leather and gravel. But I'll tell you what I really taste,\" and he took another sip and smiled. \"I taste the south of France. I taste the feeling of sitting outside in the dry weather, on a rock at the chalky feet of the Dentelles de Montmiral.\" He sipped again. \"Oomph,\" he said looking at the label. \"It needed more aging. Another two years at least. _C'est la vie_.\"\n\nStill staring down at the label, he grinned and told me about the first time he'd met this winemaker, back when nobody would give Gigondas the credit it has today. It was the eighties. The family had been selling its grapes to other producers, until the son\u2014a serious fellow, Mark explained\u2014decided they should try once more to make wine themselves. Their Grande R\u00e9serve, Mark said, comes from grenache vines that go back to 1930, all together on a small plot\u2014just a few acres!\u2014in soil marked with white limestone and clay. The mountains had crumbled over millions of years to make this terroir. Millions of years, and the vines had been in the ground just a tiny fraction of that time, though long enough to extend over several generations.\n\n\"We humans are nothing, eh?\" Mark said, laughing a little.\n\nIt had started to snow outside, dark flakes against a blue sky that was about to turn black. We both took another sip and looked out the window.\n\n\"That was a good beginning,\" he said, clearing his throat. \"But you have to learn more.\" He wiggled the cork partway back into the mouth of the bottle. \"This one I'll take home.\"\n\nI consider this the first real tasting of my life, because of the care I'd put into it and because of the company, and because of what it awoke within me. Mark later explained that he wanted me to taste all his wines, a prospect that excited me more than I could describe. How often do you get to try dozens of excellent wines in the company of the person who has hand-selected each one? I could sometimes recall our fact-sheet description of a wine as we opened it, but the percentages and _Wine Spectator_ ratings fell away as soon as the aroma hit my nose, and it would just be me, and the wine, and Mark intoning another one of his stories, in that little office in Westchester.\n\nMark was a firm teacher. Once I said I tasted oak, and he wasted no time telling me the wine was not barrel-aged at all. My face turned its usual Beaujolais red. To some extent a wine's notes can be subjective, but its prominent flavors are more objective (whether you like them is another matter). It's a matter of chemistry\u2014a wine that tastes of cherry actually has compounds like those found in cherries. When I say nobody can tell you what you taste, that is true; but it doesn't mean you can taste just anything in a glass of wine.\n\nGood wine will develop for a day or two after being opened, so sometimes we opened a bottle on a Thursday so we could try it again on Friday. I lived for those days. Every single one of my Mondays was oriented toward the end of the week when we would taste wine together. On Friday evenings I walked back to the train station drunk, not on the alcohol but on the wine itself, on the world of flavors and stories it contained, as if everything you could find on this earth could be captured in a bottle.\n\nIf I were putting together a satellite to send into space for the aliens to know who we are, I wouldn't send music and math and pictures. I would send wine. Those afternoons were more important to me than anything else I learned\u2014anything. I have often wanted to tell Mark this. Now I have.\n\n# Thinking About Wine\n\nWhat makes a wine complex? It's not just the number of flavors it contains, but how they are expressed. In the nose, or on the tongue? At the start, or at the finish? What kinds of fruit do you recognize? What other earthy or mineral aromas and flavors? It's like tasting food at a restaurant and trying to figure out everything that went into the recipe.\n\nTo develop a language for talking about wine, you must first build up your knowledge of scents. The more familiar you are with the ways wine can smell, the easier it will be for you to know what you detect. And for that, there's just one thing to do:\n\n_Smell everything._\n\nOut of all our senses, smell is the most underappreciated. I've read that most people say they would give it up before any of the other senses, and that is truly a shame. Smells are instinctive, primitive, they surprise you\u2014and we aren't fully aware of their power. Most of the time we think of smelling as a passive act, that odors just come to us. But without _actively_ smelling, you are cutting off at the start your ability to appreciate wine. It is the first important way you greet a wine and is inextricably linked to how it tastes. Cultivating your sense of smell, training it to be more active and precise than before, is the first step toward a greater acquaintance with wine.\n\nSo smell everything, and start to remember what you smell. Cut an orange in half and sniff it\u2014this is easy. Do it with cherries. Blackberries. Try to tell them apart by the smell alone. Sit in your car and breathe it in\u2014what do you smell? Leather, plastic, stale French fries? Every street you're on, every room you enter, take a big whiff. Go for a walk in the woods, or on the beach, and register every new odor. This practice has become second nature to me: when I'm doing laundry, I scoop up an armful of clean clothes and press it to my face. Every time I pick up my little dog to give her a kiss, I smell her at the same time.\n\nYour relationship to wine will change completely. I cannot overstate this. Think about how limited your appreciation for colors would be if you didn't know what to call them. That is why developing a vocabulary\u2014even a simple one\u2014is important. No one can taste exactly what you taste, but learning the language of wine allows you to share the experience as closely as possible. When someone else tells you what they detect in the _nose,_ you'll know they're talking about the aroma of the wine in the glass. When they go on to describe the _mouth,_ you'll know they are referring to the taste and effect of sipping the wine. The initial sensation on your tongue is called the _attack_ \u2014but it doesn't end there. The flavors continue to evolve _mid-palate_ as you hold the wine in your mouth. And even after you swallow, the taste and vapors will linger and develop for a few seconds, or even a minute\u2014this is called the _finish,_ and a long finish is for many people a sign of great quality and craft. Throughout all these stages the flavors, and the structure, develop and change.\n\nFirst we'll go over the flavors, which I've divided into twelve basic groups, starting with the most important:\n\n1. Fruit:\n\na. Red: cherry, strawberry, raspberry, cranberry\n\nb. Black: black currant, black cherry, plum, blackberry, blueberry\n\nc. Citrus: lemon, lime, orange, grapefruit\n\nd. Tropical: pineapple, banana, lychee, mango\n\ne. White: melon, pear, apple\n\nf. Yellow (or stone): apricot, nectarine, peach\n\ng. Nuts and dried fruits: almond, hazelnut, fig, prune, raisin\n\nh. Candied: compote, jam, cooked fruits\n\nA full-bodied red will tend toward black fruit flavors, while a lighter red will tend toward red fruit. White wine is most commonly characterized by citrus, white, or stone fruit, whereas nutty and candied flavors can be found in both red and white. And here are the rest of the flavor groups:\n\n2. Floral: orange blossom, iris, jasmine, rose, violet\n\n3. Fermented: butter, cream, cheese\n\n4. Wood: cedar, oak, pine, resin\n\n5. Green: lemongrass, fennel, hay, fern, herbs, lavender, mint, tobacco, tea\n\n6. Spice: anise, cinnamon, paprika, clove, coriander, ginger, nutmeg, white pepper, black pepper, licorice, rosemary, thyme, vanilla\n\n7. Toast: coffee, toffee, chocolate, smoke, mocha\n\n8. Candy: marshmallow, bubble gum\n\n9. Animal: beeswax, leather, game, musk\n\n10. Mineral: clay, graphite, chalk, gravel, flint\n\n11. Earth: mushroom, dead leaves, truffle\n\n12. Faults: cardboard, cork, mold, onion, rotten apple, wet mop, sweat, vinegar (may you encounter these the least!)\n\nOnce you've identified the flavors, you can seek to describe the structural identity of the wine: is it sweet or dry, fruity or earthy, light-bodied or full, crisp or creamy, expressive or closed? These impressions come from the tannins, the acid, and the alcohol, components the winemaker tries to keep in harmony, so that none overpowers the others. It's a delicate balancing act.\n\nTannins come from the skin and stalks of the grape, and you taste them as a dry, astringent sensation on your gums\u2014try chewing on grape seeds for the same feel. They give a wine body and structure (and aging potential, since they mellow over time), but too much can be harsh and obscure the wine's flavors.\n\nAcidity is expressed as the tartness on the sides of your tongue, and provides freshness to a wine, making the flavors bright and present. Acidity also allows a wine to age, preventing it from growing dull too quickly as it softens. Too little, and a wine will taste flat and \"flabby,\" the flavors muddled and out of focus. But too much, and a wine will taste sharp or even sour.\n\nAlcohol gives off heat on your tongue (think of how warm whiskey or other strong spirits taste). It helps give a wine body and weight\u2014a southern Rh\u00f4ne with 15 percent alcohol sits in your mouth very differently than a 12.5 percent Burgundy. But again, too much or too little in relation to the other components can make a wine either oppressive or thin.\n\nWhen all three elements are in balance, however, a wine is at its most pleasurable. Balanced wines can be velvety and supple, or rich and lush in fruit, or intense and opulent. The softer tannins and diminished acidity found in mature wines make them taste _round_. These are well-produced wines that have reached the right stage\u2014a wine doesn't need to be aged a long time to taste round, remember. Wines that lack roundness can taste sharp or hard, on the other hand, and you'll find these described as angular, or even austere.\n\nBut remember, as with all language, the best any of these words can do is approximate what you taste in a wine, not capture it exactly. You'll never stop discovering odd and obscure flavors found in small and remote terroirs for which you may not have the right words. But even a basic vocabulary, and even the _attempt_ to articulate the elusive pleasures of wine, will empower you to appreciate\u2014and share\u2014your experience, however imperfect the translation.\n\nThere are few objects that blend form and function as perfectly as the wine bottle. And when I say wine bottle I think first, as you likely do, of the standard \"Bordeaux\" bottle: a smidge under twelve inches high, two and three-quarters inches wide; straight sides with high rounded shoulders; tall enough to be elegant but not so tall that pouring is awkward; narrow enough to be easily gripped but wide enough to feel substantial; its neck a practical length without calling attention to itself. No one knows exactly how the shape was arrived at, but it can hardly be improved upon.\n\nIt's also a shape that invites sharing. On the table it cuts a dignified, unshowy profile\u2014within reach when you need it, but never in the way of a sight line or conversation. Beer bottles are portioned for an individual, and it's uncouth to have an entire bottle of spirits on the table. Only the wine bottle is welcoming and communal, its colors attractive and soothing. I would never say you can't enjoy a glass of wine alone (I've certainly been known to do so), but it tastes better in company. Beyond the science of the chemistry in flavor compounds, a chemical reaction occurs in you, too, when you drink. If you love a certain wine because the first time you drank it you were with a close friend in Provence, then don't let anyone take that away from you, even if _Wine Spectator_ pooh-poohs it. You can't remove memory from taste and sensation, and that sort of objectivity should never be your goal.\n\nWine is messy and imperfect and organic. Not organic as in the classification\u2014organic as in _alive_. Of the earth. There is no lab for making wine, and there's no lab for drinking it. That's one reason I've never paid much attention to \"taste tests\" (except in good fun, with friends). Wine differs from person to person, from bottle to bottle, from day to day. When we use the more technical language of wine to try to capture it, it's only so we can share the sensation of drinking it. The greater experience, however, involves the language of emotion, and of memory.\n\nAfter I first moved to the U.S., I started to make an effort to remember the wines I tasted. I wrote them down when I could, when a glass was notable, in a wine diary that did double duty as a journal of my life in New York. We drank that Marcel Lapierre Morgon, slightly chilled, when we first discovered the rooftop. I remembered the cool raspberry palate under the lights, with the city all around us, and Peter touchingly moved by it all. Or the minerality of the white Auvigue M\u00e2con-Villages I'd brought to face off with Alex's California wines, a flavor profile I remembered because I was worried whether everyone would appreciate it. In my store, I carry wines made by the Auvigue brothers, and whenever I drink one, I think back to Maya's kitchen, Alex taping over the labels, my mistaking Min and Peter for a couple. All of them stars\u2014the people and the wines\u2014in the constellation of my mind.\n\nThe last snow had just melted when Rose and Nicolas broke up. The distance was too much. Nico had been looking for work for more than half a year, and he finally took a job in Paris because he needed the money and a jump start to his career. She couldn't forgive him. He'd tried; she knew he'd tried. But to her heart it felt like a betrayal. They had been together for ten years.\n\nThe final decision came on a Saturday morning. There wasn't any shouting, but we heard Rose crying quietly. Jules and I stayed in bed. We had been nervously anticipating this for weeks, and he had a better idea of the other side of things than I did, having become friends with Nico when they'd overlapped in Paris. Rose eventually wandered in without knocking, looking as if she'd seen a ghost, and Jules made a hasty retreat. She climbed into bed with me and began to sob.\n\n\"Did I ever tell you?\" she said as I tried to reach the tissues without leaving the bed, because she had a firm hold of my leg. \"Nico and I made a pact when we were fifteen years old. We said we were going to move to New York together one day.\"\n\n\"You guys made it,\" I said, stroking her back. \"You made it happen.\"\n\n\"Yes but\"\u2014she honked into a tissue\u2014\"we were supposed to stay.\"\n\nI hadn't known until then that petite, inscrutable Rose was the most stubborn of all of us. But now that I saw it, it was obvious. Like so many other kids around the world, she'd had a dream of New York\u2014its excitement and glamour, its energy and drama\u2014and stuck to it, and nothing was going to pull her away, not even love. She hadn't just gotten off a bus from Indiana\u2014she'd left her country and flown across an ocean to make a new life. It was the first time I truly perceived this difference between us. I'd come to New York for an opportunity, not a dream\u2014and even though I loved the city, there was a limit to how much I would have been willing to sacrifice for it. As good friends as we were, she'd always been a bit closed off to me. Until now. I finally felt like I understood.\n\nWhat's more, she'd left her job with the fashion photographer back in January and had been without full-time work for two months.\n\n\"I can't have a boss,\" she had explained. \"Neither can you, you know.\"\n\nI'd suspected she was right but I didn't want her to be\u2014I loved working with Mark. \"What are you going to do?\" I asked. \"How are you going to stay in the U.S.?\"\n\nShe shrugged. \"I'll figure it out.\"\n\n_I'll figure it out._ It was this kind of tenacity that had taken me so long to fully recognize. Now it was two months later, and she was still here. I certainly wouldn't have pulled it off the previous summer if Pringent hadn't come through at the last minute. And I wouldn't have pulled it off this January if Mark hadn't taken me on. What did it say about how badly I really wanted to be in New York? I'd once believed I wanted nothing more, but if push had come to shove I wouldn't have given Jules up for it.\n\nFor Rose, there really was nothing she wanted more. Just when it seemed that she was going to have to move back to France, a move that would have reunited her with Nico, she wrote a short essay about her life in America for a magazine editor friend in Paris and then demanded, begged, and bribed her for help getting a journalist visa. Rose promised to handle the paperwork and the expense; she just needed the magazine to sign off on her as a correspondent.\n\n\"You wrote one article!\" I said to her.\n\n\"So? It counts. Maybe I will write more.\"\n\nAfter a relentless campaign, the friend came through, and Rose was safely a New Yorker for at least a little longer. She couldn't understand why I was so amazed. Her credo was that if you truly want something, you can always find a way to get it. She had succeeded; Nico had failed. Getting that journalist visa was probably the nail in the coffin for their relationship. It was all the proof she needed that he hadn't wanted it enough\u2014hadn't wanted her enough. And it broke her heart.\n\nEventually Rose fell asleep. I was immensely sad for her, and also a little guilty, as if my sympathy for Nico\u2014the realization that personality-wise I was more like him than like her\u2014was a form of betrayal.\n\nI quietly left her in my bed and went to the living room, where Jules was also asleep, his legs dangling over the arm of the couch. I got dressed and started my usual Saturday routine of walking to the Greenmarket in Union Square. Many mornings went this way, with the two of them unconscious, and me alone on the street, although normally I was rushing to the train station for work. Saturday mornings were among the most leisurely hours I had in the city, and the most anonymous, walking among the farm stands without having to say a word.\n\nI always felt the most connected to strangers when I was alone on the streets of New York. When you are new to any city, your anxiety and uncertainty can be as distracting as another person. _Which way is the traffic going? Why are some streets numbered and some not? Why is everyone walking so fast?_ When these questions quiet down, months or years later, you know you've become a part of the city.\n\nIt was early, but the usual mix of shoppers, people-watchers, students, and skateboarders were assembled on the southern steps of the square. I crossed over to the market. Though the air still had the snap of winter, hundreds of people were out among the booths and tables set up against the open backs of vans and trucks. The sun was out, and a big blue sky sat over the square. This, I think, is a big reason New Yorkers gravitate to public squares\u2014it's not just the ground that opens up, but the air above, too, uncrowded by the tall buildings that make you feel like you're in a canyon.\n\nThese days, I only had time to cook on the weekends. Weeknights were a free-for-all. If Jules didn't pick up a burger, he would just cook a fried egg on white rice, the two mild but distinctive smells usually wafting through the apartment around midnight. At the Greenmarket I went from stall to stall picking up an item from each: a small head of cabbage, a bunch of carrots, shell peas, parsnips, potatoes.\n\nI was relieved about Rose and Nico, because they hadn't been happy, but I was also scared. They were the first people I had met in New York, and the couple I'd hoped Jules and I could be. Their relationship was central to our circle of friends. It was Nico who'd introduced us to Peter and Min and Derek, whom he'd originally met on the soccer field of their corporate complex. I wouldn't have been surprised if they were still closer to him than to Rose. They may have been more sympathetic to him even, though they'd never tell her that. I didn't know what the breakup would mean, and it was strange to realize I'd now spent more time in the U.S. without Nico than with him.\n\nAs I circled back through the market one last time, I decided to invite everyone over for dinner that night to cheer Rose up, as if to declare that whatever happened, we would all still be friends. You may not be surprised to hear that I have a tendency to overcompensate. I walked to the wine store, deep in the kind of hyperpreparation I sometimes find myself in, and though it was still early, Jacob was in, reassuringly unshaven and in another heavy metal T-shirt.\n\n\"Where have I not tried yet, Jacob?\" I said, pronouncing his name the French way, which always made him blush.\n\nI asked _where_ instead of _what_ because I was more focused on the regions than on any particular producer. He scratched his chin, then pointed me to some Sud-Ouest wines. \"These you don't see too often in the U.S.,\" he said. They were from Bergerac, in the Dordogne region\u2014like Jules's family. Jules had taken me there several times as students to see the medieval castles overlooking the Dordogne River, and the famous Lascaux cave with its treasure of prehistoric human art. His grandfather still lived in the region, and Jules loved it. The Dutch tourists drove him crazy, but I didn't mind\u2014everything was so beautiful. It was also the home of Monbazillac, the sweet wine we'd shared (accidentally) at our first dinner out together. Jacob told me the wines were a good value, given the region's low profile. I bought one white and two reds.\n\nAt home, Jules had his headphones on, watching clips from the basketball playoff games he'd missed that week. I tapped him on the shoulder to show him the wine. He looked at me, confused why I'd returned with so many bags. \"From your ancestral home, remember?\" I said, placing the bottles on the counter. \"Jacob says this one is dry, with soft acidity,\" I said, pointing to the white. \"This red is medium-bodied, supple, half cabernet sauvignon, half merlot. And this one is fuller, richer, more earthy. We shall see.\"\n\n\"You're thinking of wine, now?\"\n\n\"Wine with my friends,\" I corrected him. \"I don't see why not. Rose needs support. We should all be together.\"\n\nRose was in her room, lying on her bed, but her eyes were open. I asked if she'd eaten breakfast and she said no. I told her I was asking the others to come over for dinner, that I would cook, and it would be fun. She nodded mutely.\n\nRose was still in her room when Min and Derek arrived that night, Derek with a kind of frozen grimace on his face, and Min's expression a little crumpled, as if she'd been crying on Rose and Nico's behalf. I went to Rose's door and knocked lightly. No answer. Min shook her head at me. \"It's okay,\" she whispered. Peter was late\u2014he was always either ten minutes early or an hour late, nothing between\u2014and seemed surprised to find us sitting quietly in the living room, which smelled of roasted chicken, not eating.\n\nI handed him a glass of the white and told him Rose was probably not coming out. He lowered his voice and asked, \"Have you talked to Nico?\"\n\nI shook my head, wondering if that was the wrong answer. \"No, have you?\"\n\nHe nodded, and I felt immediately embarrassed. It meant Min and Derek had talked to Nico, too. Maybe even Jules had. While I'd spent the day worrying too much about what was going to change, and trying to pretend it wouldn't, the others had done the appropriate thing and talked to their friend. Now, sitting quietly together around the coffee table, the evening felt like a wake.\n\nOver the next couple of months, things did change. We didn't get together very often, all of us, to my regret. I didn't know if it was because of the shift in the group dynamic, or if it was because we were all working more than ever. Mark steadily began to give me larger slices of the business. While he was reluctant to let me speak to the producers when deals were in mid-negotiation (if I picked up the phone first and one was on the line, he would make a rapid _\"fe-fe-fe-fe-fe\"_ noise from across the room while flapping his hand to signal that I should hand it over), he told a couple of wholesalers that I would be their main contact from then on.\n\n\"Gregory, I've hired a new associate. She'll be working with you. She's brilliant, and it'll be better for our relationship because I'm getting old and absentminded. You too? How would you know? Ah, ha, ha, ha.\" I would shuffle papers at my desk, pretending not to hang on to every word, trying not to blush with pride and nervousness. His _brilliant associate_. Mark had already begun taking me on sales calls, to dinners with wholesaler reps and restaurateurs. Following his lead, I didn't say too much. \"Listen first,\" he said, and then proceeded to work the table with good humor and deep knowledge, until by the time the coffees came there was no doubt in the room that the best wines\u2014at the best values\u2014could only be found through Brodeur Imports. It was as though a spell had been cast, at least long enough to net Mark some deals and, even better, relationships.\n\nTwice a month I went with him to the regional sales meetings, as I had with Mo\u00ebt. He spoke to the reps with no notes or preparation, conversationally rattling off the important facts as if they'd just occurred to him. Again, I stayed quiet, presenting the bottles to the audience, pouring tastes as I walked among the seats. I felt a distant kinship with the women on game shows who present the prizes, smiling not speaking. But I was watching everything he did, absorbing every word and gesture like a sponge. What I felt must be a common mix of emotions for an apprentice: envy, admiration, resentment, ambition. Looking back at my tour with Mo\u00ebt, I realized I must have come off to my audiences like a French wine robot, hardly comprehensible, repeating my presentation in an artificial singsong voice. Mark, however, spoke to a room as though he was having a conversation with each individual rep.\n\nI asked him one day how he did it. He said, \"No secret. Just remember it is always, always easier to do what you believe in. It takes energy to lie or to fake something. A tremendous amount of energy. Everything I say and do is true. I love everyone who loves wine. I love everyone who helps me get my wine out to as many people as possible. This is my joy. You understand.\"\n\nI said I thought I did.\n\n\"No,\" he said. \"That wasn't a question. I mean I know you already understand. This is why I hired you.\"\n\nAfter the sales meetings, we would drive back to the sleepy little village in Westchester, and the modest storefront of his office, and continue working until seven or so, when he would rise from his chair, hitch up his belt, and say it was time for dinner\u2014time, that is, for him to go home to his wife and daughters. He had a ten-minute drive. I had an hour-and-a-half commute. Although I could have tried to leave earlier, I didn't. I didn't want to miss anything. Sometimes he would open a bottle so that we could taste it that evening, and again the next day to see how it had changed. Wine doesn't have to be full-bodied to tolerate the exposure. It just has to be of high quality. Many perfectly enjoyable wines are meant to be consumed quickly, but to watch a good wine evolve over hours, over a day, is a marvelous thing. And so I stayed to the end every day of the week, on the chance it would turn into the master class I was always waiting for.\n\nIf it was early in the week there would likely be no tasting, and as we closed up I'd sigh to myself and board the train in the darkness and read my cheap French novel, or mentally go over everything that had happened that day. I never slept\u2014my mind was too active. But by the time I reached home the exhaustion would hit. I'd never concentrated so hard, so unceasingly, on anything.\n\nMost nights I was ready for a glass of wine, a small meal, and bed. Because of Mark's portfolio, I had been making my way through a swath of more affordable Bordeaux wines. Bordeaux is a huge, complicated region, with dozens of subregions and appellations. I'd tried several from the left bank of the Gironde River, in Margaux, which has bolder cabernet-based reds, and some from the right bank in Saint-\u00c9milion and its neighbors, which use fruitier merlot.\n\nBut that spring I would frequently return not to a bath and bed, but a roommate standing in front of the mirror, wearing a sparkly top, slim pants that tapered to her ankles, and eyeliner that gave her face a sallow, sharpened look. Since the breakup, Rose had decided she no longer wanted to do, well, most things. She was not interested in going out to eat or sitting at home with ice cream and Netflix. She still wanted to dance, though, four or five times a week. She hadn't been single since she was fifteen, and without a boyfriend, without a job, she had an abundance of energy. She danced as though she had poison to shake out of her system.\n\n\"Come, Laure, let's go,\" she pleaded with a manic intensity, the opposite of her usual _froideur._ \"Will you come with me tonight?\" More often than not I wanted to say no. I was tired; I _had_ a boyfriend who would soon come home. But I'd sit on the bed and think about how the night would be easier for Rose if I went, and begin to gauge just how little I would have to do to my appearance to avoid total embarrassment.\n\nMost of the time, I didn't have to stay out too long. If the night was slow, we danced for an hour or two and came home, laughing, expended. But the chances were decent that she would catch a boy's eye, and, after several minutes of careful vetting on my part, I would leave on my own. Sometimes she slipped back into the apartment a couple of hours later; sometimes not at all.\n\n\"Hey, I'm a free girl,\" she said, a point of pride and lamentation. She missed Nico; she was also determined to live her life in New York, which she'd sacrificed so much for. \"I can do what I like.\" She was right, of course.\n\nIt was, on the whole, a tumultuous spring. I don't just mean Rose's love life and its fallout. I was taking on more at work\u2014which I loved\u2014without yet feeling real empowerment. Jules and I were on our misaligned schedules, sleeping together every night but awake and in the same room a vanishingly small amount of time. And it had been more than a month since our friends had all gotten together in the same room.\n\nOne morning, in my usual rush to work, I ran into Peter in Grand Central. I was surprised to see him and wanted to talk to him enough that I missed my usual train. He told me he'd quit his job. \"Couldn't be a cog anymore,\" he told me. He'd joined a much smaller company, which he said was just a stepping-stone on his way to doing his own thing.\n\nHe'd also recently gone through a string of bad luck, he told me\u2014flights missed because of freak accidents, a mysterious stomach bug that felled him intermittently for several weeks, and an uncharacteristic dating dry spell. His long run of demi-girlfriends, as Rose and I had liked to call them\u2014women he was with long enough for us to learn their names and meet a few times before they disappeared\u2014had ended in March, and he hadn't seen anyone since then.\n\nI realized that quitting meant he no longer worked with Min and Derek. I asked if he still got to see them.\n\n\"Not a lot lately. We've all been busy! But it's not like quitting means we're not friends anymore. That's why I don't need a girlfriend,\" he added, just before we separated to get to our respective trains. \"I'm never lonely. I have you guys.\"\n\nMy only real window into mainstream American dating was through Allison, my old friend from Pringent. After I was fired, I didn't see her again until several months into the new year, when she called to say she'd gotten a new job, too, at New York University. When we finally met up at a West Village tapas restaurant, she launched immediately into everything I'd missed. It turned out my uncle had heard a mostly accurate version: Jean Pringent arrived on a surprise visit to discover a leaderless office. As people started telling him what was going on, his narrow mustache, Allison said, quivered in rage. In his dignified and calm way, he set up in the CEO's office and asked people to come and see him one by one. Each reemerged with a tentative smile; Pringent hadn't said what he planned to do, but he was clearly distressed. After he brought in the auditors, he sat Allison down and said he understood the difficult position she was in but had heard that Rachel and Eric the CEO were on a beach in Thailand. Which beach? he wanted to know. She had the resort written down on her calendar but felt conflicted. The best she'd been able to do, she told me, was open the calendar to the right page and leave it on her desk when she went to the bathroom.\n\nShe didn't know what had become of Rachel and Eric, just that they were gone.\n\n\"And what about you?\" I said.\n\n\"What about me?\" she said, feigning ignorance. \"Am I sleeping with a married CEO?\"\n\n\" _Are_ you?\"\n\n\"If I was, the difference is that nobody would know.\" This she said in a hush, and then broke into wild laughter. Though she had no problems expressing herself, she was, I realized, a very private person. But she told me she was seeing two men she'd met online and had just had drinks with a third but couldn't tell if it was really a date or not. Now she was at a decision point. She was trying to break down where she was with each of them based on a complicated set of codes and cues, like who asked whom more questions. But she hadn't slept with any of them yet\u2014she had a five-date rule\u2014and that made it harder to pick.\n\n\"Rule? Why a rule?\" I asked. I was mesmerized. Maybe this reveals my age\u2014I'm sure all the young Parisians are online-dating these days\u2014but my experience has been that there are no \"rules\" for dating in France. There isn't even a French word for \"dating.\" It doesn't mean you can't have a good time, the way Rose was as a single woman. But if you like someone, often someone you were introduced to through a mutual friend (like how Jules and I met through my friend Vera), you can take the time to explore those feelings without having to worry about the signals you are giving off, or whether the other person is seeing multiple people. Nobody _dates around_. You hang out, you hang out again, and if at some point you kiss, you are in a relationship. And if you don't, you decide to keep going, or to end it. That's it. The method has its downfalls, but I said a silent blessing for its simplicity.\n\nAllison buried her face in her hands. \"I need more time, but I'm running out. You get a certain number of punches in your card before it starts to become awkward, you know?\"\n\n\"Awkward how? Why do you need more time?\" I was getting more and more confused.\n\n\"I'm also Catholic!\"\n\n\"So am I!\"\n\n\"You're no help!\"\n\n\"I still don't understand what you mean about the punch card,\" I said.\n\nShe tried to explain it to me, but it came out a jumble of statements about shame, expectations, religiosity, and confidence. She may have believed them all at once.\n\n\"Allison, I love you, but that is the stupidest thing I have ever heard.\"\n\n\"You're right, but I can't help thinking that way.\" She asked me to come with her to a wine store before she headed back to Hoboken. It had recently rained, and we linked arms for stability.\n\n\"What are you drinking these days?\" she said. She knew I'd been making my way through the regions, but it had been months since I'd given her an update.\n\n\"Bordeaux,\" I said.\n\n\"Fancy!\"\n\n\"Not always,\" I said. \"Doesn't have to be. But it is probably the most famous, yes. My boss carries a lot of Bordeaux.\"\n\n\"Do you ever have a hard time deciding what bottle to get?\"\n\n\"If you're going to make a joke comparing my taste in wine to your taste in men, I'll save you the trouble,\" I said, and her laughter echoed off the buildings.\n\nWe walked slowly around the store waiting for her glasses to defog. It was an elegant shop, sparely stocked, with shelves that highlighted the bottles nicely. Each had a careful description attached, with far more detail than the usual recommendations. I was impressed by the amount of information, and how unguarded the shopkeepers were with it. You would not have found a store like this in Paris that I knew of; in my mind they were still dense warrens, with dim lighting and bottles everywhere. Because she'd perked up when I mentioned Bordeaux, I helped Allison choose a rich, merlot-based wine from the less well-known appellation C\u00f4tes de Castillon to take back with her to Hoboken. I'd never tasted it myself but had tried similar ones that were drinkable young, soft and round, even chocolatey.\n\nShe kissed my cheeks\u2014more firmly than a normal _faire la bise._ \"Laure,\" she called after she'd walked down the street a few paces. \"Now that we don't work together, who is going to pick my wine for me?\"\n\n\"You're going to have to figure it out yourself!\" I shouted back.\n\nThe first week in April the temperature topped seventy for a couple of days in a row and we knew it was time to open up the rooftop again. If I was a little trepidatious about everyone coming over, I shouldn't have been. We picked up where we'd left off, and no one seemed to blame Rose as I'd been afraid they would\u2014just as I'd be afraid of pointed fingers if anything happened to me and Jules. Most of us were still holding out hope that Nico would surprise us by landing at JFK with a freshly stamped visa in hand, and everything would go back to the way it had been those first six months when we were just beginning to rely on one another.\n\nAfter the long winter cooped up inside, I felt giddy to be again on the rooftop, looking around at these people who had become dear to me in the last year. We were all foreigners except for Peter and Derek\u2014Peter was the son of Korean immigrants, and Derek was a gangly nerd who you could tell had lived through a prolonged awkward phase, which gave him a special outsider pass.\n\nBenji's motorcycle may have disqualified him for outsider status, but he still pined in silence for Rose, especially now that she was single. He called almost every night to invite Rose to something or other\u2014an art show, the opera. It sometimes sounded like he was just crossing items off a long list of every possible thing you could invite a woman to. Anything she needed, he was there for, quietly and patiently. One day when I was at work she texted saying the kitchen sink was broken. I called the super, but he was nowhere to be found. When I returned home, Benji's legs were sticking out of the cabinet, a toolbox open beside him.\n\nRose just shrugged, smiling, as if to say, _Hey, he wanted to!_\n\nI woke up late that night and could hear Benji and Jules on the couch having a long, intense conversation I caught only snatches of\u2014female pronouns, long pauses from Benji, hushed but animated exposition from Jules. In the morning I asked Jules what they'd talked about.\n\n\"Oh, guy things,\" he said, just to annoy me.\n\nThose days, I was always trying to read Jules. With the arrival of the warm weather, he was complaining less than usual. I knew it didn't mean he was finally in love with New York, but he finally seemed satisfied, with friends he trusted, a job he was good at, and me. We had always talked about our lives as if they were inextricably connected, and made plans that way, but like Allison I was looking for signs and signals that we were on the right path. It was the planner in me. I've always felt that if I could guarantee my future, I'd be able to enjoy it more. Jules knew this about me, and liked to use it for his own amusement.\n\n\"Laure.\"\n\nThis was his favorite practical joke, which he played on me as often as once a week, and I fell for it every time. \"Laure,\" he said from behind me, while I was standing in the kitchen, or just out of the shower in a towel. \"Laure, I have something to ask you,\" he said in a serious voice that triggered in me a sudden desire to sit down. Every time, I thought he was going to ask me to marry him. And every time, his eyes watery and wide, his knee flexed as if he were about to lower himself onto it, he would pause for a heavy moment, and then ask something like, \"Do you know where the mustard is?\" And he'd laugh and laugh, uncontrollably, while I would pick up a pot holder, or anything, and hit him with it, chase him around the apartment, shouting, until we were on the bed and he had no choice but to wrestle the weapon away from me.\n\n# Thinking About Wine\n\nBordeaux is the most famous wine region in France, and has one of the longest, richest histories of wine production, distinctive because of its busy port and huge international appeal. It is simply massive, with sixty appellations and a complex classification system I won't try to detail here, under which more than seven thousand wineries, or ch\u00e2teaux, operate. That scope is why I've saved it until now.\n\nBut you shouldn't be intimidated. Yes, Bordeaux can be among the most expensive wine in the world (although you will find the rarest in Burgundy). Because Bordeaux is so old and so established, a lot of its pricing is based not on the wine itself, but on reputation. This is to be expected. The market is heavily influenced by the standing of both the ch\u00e2teau and the vintage. In Bordeaux, more than with most regions, mill\u00e9sime can have a big effect on price. With Mark I was selling 2005\u2014one of the best in recent history. Today, purchasing 2005 Bordeaux will cost you a pretty penny, far more than 2007 Bordeaux, for example, because of the former's reputation. But the benefit of having so many producers to choose from is that there are fabulous, small vineyards with affordably priced, good wine in any year.\n\nHere are a few other things to know about Bordeaux. It has a \"left bank\" (southern) and a \"right bank\" (northern) in relation to the Gironde River. The left bank uses more cabernet sauvignon in its trademark Bordeaux red wine blend\u2014as much as three-fourths\u2014while the right bank uses more merlot. This has to do with the nature of the terroir. Never doubt how different the land can be on either side of a river. (The third grape, always in the minority, is cabernet franc.) The left bank wines are drier, fuller-bodied, and dark, with black currant (at its most intense, a cassis flavor), pepper, and woodsy notes. Barrel-aged reds taste of vanilla and licorice. The older wines will have hints of tobacco, game, truffle, even pencil lead _._ Yes, you read that right\u2014this particular flavor compound is reminiscent of graphite. The right bank wines are fruitier: you'll taste prune, blackberry, and black cherry, as well as mint and violet flowers. In general\u2014though not as a hard rule\u2014these can be drunk younger.\n\nThe best ch\u00e2teaux of the left bank fall under a classification system that was organized in 1855 at the request of Napoleon III and is revered to this day. It's remarkable how long and how consistently the categories have lasted. And it's not just a rubber stamp\u2014the wines made by these houses have been tremendous, the few times I've had a chance to try them. You may have come across these names before.\n\n# PREMIER GRANDS CRUS CLASS\u00c9S FROM M\u00c9DOC, ON THE LEFT BANK\n\n\u2022 Ch\u00e2teau Latour\n\n\u2022 Ch\u00e2teau Lafite-Rothschild\n\n\u2022 Ch\u00e2teau Mouton Rothschild\n\n\u2022 Ch\u00e2teau Haut Brion\n\n\u2022 Ch\u00e2teau Margaux\n\nFive ch\u00e2teaux! That's it. They have formally been recognized as the best.\n\nThe right bank's classification is more recent, from 1955, with a more modern intention of updating it regularly.\n\n# PREMIER GRANDS CRUS CLASS\u00c9S FROM SAINT-\u00c9MILION, ON THE RIGHT BANK\n\n\u2022 Ch\u00e2teau Ausone\n\n\u2022 Ch\u00e2teau Cheval Blanc\n\n\u2022 Ch\u00e2teau Ang\u00e9lus\n\n\u2022 Ch\u00e2teau Pavie\n\nIf Cheval Blanc sounds familiar, it was one of the wines I had the fortune of drinking the night I was in Nashville. We'd finished that same meal with a sweet Ch\u00e2teau d'Yquem, which is classified as the _only_ premier grand cru sup\u00e9rieur in Sauternes. As I said, it's complicated!\n\nBut now that you have committed that to memory, forget it all. If you choose to spend your days chasing premier grands crus, you are welcome to. But I have also found something alluring about simply stumbling on them from time to time in particularly lucky moments.\n\nLook: if you really want the best deal, don't buy Bordeaux. But it's still possible to build a rewarding relationship with Bordeaux without breaking the bank. Don't feel pressured to spend money on the prestigious names. Seek out something from Lussac-Saint-\u00c9milion rather than Saint-\u00c9milion proper. Look for one of the \"lesser\" vintage years, such as 2007 or 2008, which still produced good wine you can drink relatively young. Most of us don't store wine for a decade or more anyway.\n\nIf you want to be adventurous (and impress those people always searching for developing regions), look to the Sud-Ouest, the southwest, a huge area that stretches from Bordeaux to the Pays Basque. This is where you'll find Bergerac\u2014the terroir is less highly heralded than Bordeaux's, but the same grapes thrive there: cabernet sauvignon and merlot. As you go deeper inland, such as Cahors, you find wines made 100 percent from malbec, a strong red grape that requires a lot of sun and yields robust wines. In Fronton, the grape is the n\u00e9grette, which has a similar personality.\n\nThe Sud-Ouest is more a collection of small regions than a strictly unified one, but the wines have a richness and a rawness in common. Because the terroir varies so much from one appellation to the next, it's impossible to take it all in at once. But don't be daunted. As always, be patient and open\u2014these wines are ready to make your acquaintance, new friendships that will only grow with time.\n\nCan people change? I'm not sure, but I do know that as a general rule your natural characteristics round out and soften with age, even as they increase in complexity. And you can learn to see things a new way.\n\nEach time I went back to France, my old friends would say the same thing: \"Laure, you haven't changed a bit!\" They meant it, I think, as a compliment\u2014that I still had my youthful looks and Parisian sophistication, that I hadn't yet turned into one of _them_. (It would have taken a long time to explain that _they_ weren't all that bad.) But I didn't want to be the same, and I felt my cheeks grow hot whenever anyone said it. What I really wanted was to be someone who had gained another sense, or power. I was definitely still French, but I was something else, too. I'd had an American infusion of the spirit and it had given me a newfound tolerance for risk, for uncertainty, for contrasting views. I'd left one job (if you count my brief tenure at L'\u00c9l\u00e9phant), had another expire, and been fired from the third. My closest friends were an assortment of people I would never have predicted. And I was the better for all of it.\n\nThis is the gift of travel. New places change your idea of what life can be. Our personalities have inertia\u2014they prefer to stay still and sideways, like a bottle of wine. But unlike wine we benefit from being shaken up once in a while. Travel makes you see things differently from the moment you step out of the car or off the plane. You take in new words, new manners, and reconcile them with what you thought you knew.\n\nIf you're lucky and the place you're going is wine country\u2014and more of the world than ever before counts as wine country\u2014then you'll always remember a special connection between the wine and the land. That ros\u00e9 you drank on vacation in Provence, for example; that wine will forever remind you of the place it's from, like a souvenir.\n\nBut the true power of wine is that it lets you travel without leaving home. I would never say you can only appreciate a wine by visiting its land\u2014not everyone has that opportunity. A well-made wine brings its terroir to you if you pay attention to where it's from, what geographical features set it apart, who makes it, and what grapes are used in order to get the most out of the land. And through its flavor, too: by having a mineral taste because the vineyard is beside the famous limestone ridge of Burgundy, or by tasting ripe because the grapes were under the burning sun of the south. Together, these factors give you a sense of place, and a rich and rewarding way to experience the land.\n\nMark, with all his stories about the winemakers he'd \"known forever,\" taught me the importance of intimacy. There's no other way to put it. The entire work of getting to know wine is removing the anonymity from it. You can easily drink a different bottle each night and forget the name of the producer by morning. But the rewards that come from closer consideration grow over time; drinking wine becomes more and more enjoyable. As long as you buy from small vineyards, you can rest assured that there is a person on the other side of the bottle, a family, a history tied to this wine.\n\nDiscovery takes work. And the more you do it, the more it teaches you about yourself.\n\nThe rest of 2008 went quickly\u2014a sign my life in New York had shifted into a steady gear. We went strawberry picking in the early summer and apple picking in the fall. On the Fourth of July we sailed around Long Island Sound with friends of Peter's. When my two-year anniversary with America arrived in November, I kept it to myself, thinking it was a silly thing to be proud of, but when I got home from work that day, Jules had left me a cupcake featuring an enormous candle shaped like the Empire State Building. The next week Min and Derek held an election-night viewing party, and after the results were in and the country had a new president we all ran outside to join the rest of the city in celebration.\n\nWinter was slow to come and mild when it arrived. Wanting to make the most of our weekends, Jules and I visited the overcrowded museums we hadn't been to before: the Frick, the Whitney, MoMA. Somehow, I'd gone two years without visiting the Met, and we spent an entire day there. I would make the rounds of an entire exhibition and return to find Jules standing in front of the same painting where I'd left him. It was as though he was a planet and I was the moon trying to circle him, or I was an indifferent comet and he was a star. It's not that we were unhappy. We were both stable, in our jobs and our rhythms, but there was something missing\u2014time, for one, since we didn't get much of it with each other during the week. Even though we finally shared a bed, it was still a little like being long distance. I always felt this the most strongly on Sundays, after we'd crammed the weekend full, and I'd start to miss him before it was even Monday. But then the week would start and we'd be back in our busy routines.\n\nWhatever I was feeling was subtle\u2014too subtle for me to put my finger on precisely\u2014but compared to everyone else, Jules and I were suddenly the most boring members of our friend group. A new pairing was taking up everyone's attention: at some point in the fall, Rose told me she and Benji had kissed, although she wouldn't say exactly when. (She didn't want to give me too much of a reason to say \"I told you so.\") She had cooled it off right away because she didn't feel ready, after a decade with Nico and a spring of brief encounters. That lasted maybe a week or two before they kissed again, giving her yet another reason to back off.\n\n\"What do I do?\" she asked me.\n\n\"I wouldn't think about it,\" I said, because it was the advice I had been giving myself.\n\nEven though it was too late to see any leaves, Rose and Benji took a weekend motorcycle trip to Vermont\u2014\"Just as friends,\" she told me\u2014and they came back a couple. I didn't talk to Nico directly, but the others had, and reported back that he was distraught and relieved, at the same time, and wished the best for her. It took me some getting used to, seeing them together, even though I'd been predicting it for a long time. They spent more time at his place than ours, but sometimes I'd come home to find them in the kitchen, saturated in that buzzy happy glow of a new couple, and feel a little jealous.\n\nOn New Year's Eve, Min was drunk, her face a bright strawberry pink, when she told me she was thinking about breaking up with Derek. I had just returned from Christmas with my family and asked if something had happened in the week I'd been gone.\n\n\"No, nothing happened,\" she said. \"But that's kind of the point. Have you noticed Derek and I have never gone on vacation?\" She continued before I could answer. \"Six years together, and zero vacations. Visiting family and going to weddings doesn't count.\" I tried to think back over the past two years I'd known them.\n\n\"That can't be true,\" I said.\n\n\"When then?\"\n\n\"There was the time...\" I started and then drifted off.\n\n\"I don't know what his priority is,\" she said. \"But it's not me.\"\n\n\"He loves you,\" I reassured her.\n\n\"Love's not enough sometimes. I feel like we are in a rut I don't know how to get out of.\"\n\nIt was an unexpected announcement coming from Min, who was always the most positive among us. But she was also a ruthlessly practical and efficient person. Once, we all showed up at a restaurant for Rose's birthday and realized we'd made a mistake with the reservation\u2014there was no table for us. While the rest of us agonized and passed the blame around, Min quietly disappeared and reappeared saying she'd talked us into a walk-in at Nobu, even though we were a party of seven on a Friday night.\n\nIf Min had known that I would immediately begin a personal crusade to force Derek into a vacation, roping everyone else into the effort, she probably wouldn't have told me anything. But as it was, the campaign worked. We nagged and shamed him until they made plans to travel in March. And where else? To France!\n\nThey told us over dinner one night, and I could see how excited they both were, how something quiet but charged had come back into their relationship. Derek was asking questions like, \"Is it mer-CI or MER-ci?\" Then I started to say that it was still cool weather in Provence in March and maybe they should consider June instead, until Min shot me a look that shut me up immediately. She was not about to risk losing this opportunity by messing around with the dates. They had to go on vacation, and go as soon as possible.\n\nThey were gone for two weeks, and we heard nothing the whole time\u2014radio silence. We knew they were spending a few days in Paris, going to see Maya and Alex in Aix, and then staying on the outskirts of a smaller Proven\u00e7al village. We waited. Even on the day they flew back\u2014no news. I waited a few days and then invited them over.\n\nThe people who came over looked a lot like Min and Derek but didn't sound exactly like them. Min was more or less herself, but at a slightly slower, less demonstrative pace. Derek, however, was more clearly different. He had never been abroad before, let alone on an extended vacation. He'd had a glimpse not only of a different life, but a version of himself who was happy and relaxed, who had the time to put his relationship first. It changed everything.\n\n\"Oh, man,\" he said three or four times in a row before responding to a question as simple as what they ate. \"Oh, man,\" and he ran his hands through his hair as if communing directly with his memories. He was still Derek, with the sardonic delivery and the low rolling laugh. But he looked about thirty pounds lighter\u2014and most of that was from stress. His eyes, usually pulled tightly inward beneath a furrowed brow, were wide open beneath a smooth, broad forehead.\n\n\"No phone, no email,\" he said in disbelief.\n\n\"I didn't think he could do it,\" Min said, smiling. But he had.\n\nThe first two days in Provence, she said, he was still himself. \"Where are we going now? What are we doing next?\" he kept saying. She refused to answer him, until he finally stopped asking. They did no sightseeing. They stayed in the village the whole time, walking to the center square for the farmer's market in the mornings, having a regular afternoon coffee, switching off between cooking and going out. One night, they ate only bread and cheese and drank wine\u2014the local ros\u00e9.\n\n\"Good food, good wine,\" Derek said in reverie.\n\n\"Good company,\" Min added, squeezing his hand. She was glowing.\n\n\"The best company,\" he said, looking at her. Suddenly, the rest of us were no longer in the room.\n\nThey were engaged a week later.\n\nIn the end, I was the one who went to France that June when the weather was warmer. Everything leading up to it felt of a piece. Like a pianist who has been practicing for years, I had some sense of my skill now. I knew my regions, I knew the producers I liked from my own explorations, plus the sixty in Mark's portfolio that I now knew very well. When I visited Jacob's wine store these days, it was like entering a room with hundreds of people, some of whom I'd gotten to know fairly well, some whom I could recognize but not quite place, and some whose names and faces were new to me. Some names, the big companies, rang a bell even if some of the individual labels were less familiar: Jadot (Burgundy), Bouchard (also Burgundy), Chapoutier (Rh\u00f4ne), Jaboulet (also Rh\u00f4ne).\n\nI'd grown in confidence. My tasting sessions with Mark were no longer tentative. I wasn't afraid of being combative, worrying that he'd think me amateurish and young. I was comfortable saying that a certain wine just needed to open; he would say nonsense, it needed more aging. I would say there was a touch of roasted coffee in the finish; he would pause, then shake his head and say it was cocoa.\n\nOur working relationship changed in other ways, too. I was spending more time out of the office\u2014either conducting Friday sales presentations by myself or going on wholesale tours at stores and restaurants in Westchester, western Connecticut, New Jersey, New York City, and Long Island. I had, in many ways, come full circle. But instead of making sales calls on behalf of one producer\u2014my uncle, or Pringent\u2014I was now representing several dozen, of which I only had time to talk up a few to each customer. They all bounced around my mind like the names of friends, with qualities and personalities and stories I knew by heart because they'd been told to me so often. I'd tasted nearly all of them by now. Passing them on was like recommending someone I knew. It was not just a job; I was proselytizing.\n\nHow far I'd come! Just two years earlier I'd been a bumbling brand ambassador with a few dozen English words and memorized talking points, relying on charm and a French accent to make my way. I was no longer pretending. I knew and meant every word I was saying. For one, now that I was in my second year with Mark, I could easily and directly compare the previous year's wines to this one\u2014and what a difference it was. The year before I'd been tasting and selling 2005 and 2006, both fantastic years in Bordeaux. Now we were working with 2007, too\u2014not a bad vintage, but not as good as 2006 and especially not 2005. Those two kept flying off the shelves by the power of their reputation (and largely living up to it). I tried to avoid making direct comparisons, but there's only so much you can do: the wine magazines and the ratings websites do it for you, and the customers pay attention. Once a year like 2005 is anointed as special, the price goes up and the wines sell fast, especially when it comes to Bordeaux. (The irony was that even for a year as special as 2005, the wines would not reach their full potential for a long time, whereas a vintage yielding more straightforward wines, like 2007, can mature sooner. Depending on the wine, it might have made sense to drink a 2007 _before_ a 2005.)\n\nAfter sales calls, I would return to the office exhausted, and Mark would remove his glasses to look at me and say, \"How did it go?\" I briefed him the way you respond to your father when he asks you about your day at school\u2014this many cases, that interesting new store, this enthusiastic rep, that unimpressed restaurant, all relayed as impartially as possible.\n\n\"Very good,\" he said, replacing his glasses. \"Write up a report for me by the end of tomorrow.\"\n\nI have many good qualities, I'd like to think, but completing tasks I don't see the point of has never been one of them. \"But I just told you what I did,\" I said.\n\n\"Yes, thank you,\" he said, his tone shifting from paternal to professorial. \"But I need the paper for my records. I can't remember everything you say.\"\n\n\"Pah.\" I slouched down in my chair. He went back to what he was working on, and it might have ended there, if it wasn't for my mouth: \"I don't know why it's so important.\"\n\nHe let out an impatient grunt.\n\n\"You don't have to know exactly why it's important. I do\u2014and it is my company, I hope you remember.\" He said that whenever we disagreed, and I hated it. \"And,\" he added, \"I don't know why you are wearing jeans here now. Don't you have nice slacks? You didn't used to always wear jeans to the office and I don't like it. We try to be proper here.\"\n\nClearly, cracks were starting to show in our working relationship. Part of it was that I was overly sensitive to any perceived slights or lack of trust. And Mark, as I've said, was a frank and emotive man who loved to talk and argue. It seemed like a recipe for disaster, leading to a Marianne-level blow-up that could easily have cost me my job. I didn't know why it had been so easy for me to talk back to Mark, when I'd been so acquiescent to Rachel\u2014possibly because I felt more comfortable around him (in no small part because of our shared language), even as I resisted his authority. But whenever it seemed like we were on the verge of a bigger argument, he would immediately clam up, study me intently with those brown, slightly downturned eyes, shrug, and turn his attention elsewhere. Only now do I see that this was an act of generosity, saving us from a bigger fight. By any measure, I was getting a little ahead of myself. A year of intense apprenticeship did not equal three decades in the business, no matter how much he wanted me to know what he knew.\n\nWhen we visited distributors, Mark often let me present to sales teams on my own, but on the days he did it himself, it was clear I was to play the assistant, pouring the wine, smiling at the reps, and saying nothing. It was frustrating to switch back and forth. Once, after a presentation in New Jersey in which I'd hardly said a word, I talked too much on the car ride back. It was uncharacteristic bluster about how much I'd learned, how good I was at making deals and communicating the wines' stories, how I always got great responses from customers but he didn't see it because it was only when he wasn't around. Then for some reason I started to talk about how much I missed my family. I told him about my older brother and his children, who lived in Luxembourg and with whom I Skyped whenever I could. I described my parents' supportiveness\u2014I knew they'd be happy no matter what I did but I thought they were secretly pleased I'd ended up in the family business.\n\nIt had never been like this, with me talking and Mark listening. And when I noticed how long the car had been filled with my voice, I went quiet. Mark said nothing for a minute and I watched the trees pass outside the passenger window, wondering if I'd said too much. Then he said, \"My next trip to France is at the start of June. Why don't you come with me? The company will pay. It will be good for you, and you can see your family as well.\"\n\nIt was the last thing I expected him to say. The last two times he'd gone to France I'd stayed behind, managing the office. I wasn't sure what had changed, but considering the added cost, the fact no one would be there to answer phones, it had to be a vote of confidence of some kind. As it always seems to, it took me only a millisecond to say yes.\n\nI flew to Paris a weekend early. It was like a warm bath of good feelings. Even though my mother was annoyed at me for waking up late the two days I was with her, I couldn't help but smile at her and tell her how happy I was to see her. (It only made her suspicious of me.) My cousin\u2014one of the daughters of my mother's clan\u2014came over for lunch, and filled us in on the latest news out of Champagne: Uncle Charles had all but completely handed the reins over to my cousin Charles-Henri.\n\n\"Well, it's your turn now,\" my mother said to me, and to my cousin, and from the tone in her voice I could tell she meant more than just the business.\n\nMark arrived on Monday. I picked him up at the airport in a rental car and we drove southeast, toward Aube, Champagne. Aube is on the southern end of the province, known for its C\u00f4te des Bar region\u2014too far from Montagne de Reims for me to see my ever-crankier uncles. I was nervous enough about traveling with my boss. We'd spent plenty of time together in the car around the tri-state area, but this was another level. It's hard enough to be with someone you're close to and trust when all your awkward and embarrassing habits and tendencies are inevitably exposed while on the road.\n\nBrodeur Imports did not have a big roster of sparkling wines, but you always want a few; otherwise you're practically begging clients to work with other importers. Our first visit was with a winemaker, Fran\u00e7oise, who had taken over the vineyards when her husband died, renamed the label after herself, and now ran the business with her children. Her signature blend was three-quarters pinot noir and one-quarter chardonnay\u2014the pinot gave it a nice structure, and it tasted of dried fruit, toast, and hazelnut, with a lovely yellow color. When we pulled up to the _maison_ , a modest stone building at the base of the hill, Mark and Fran\u00e7oise embraced like old friends. I was surprised. She was middle-aged, gray hair hanging in wiry trails down both sides of her face. Her hands were rough to the touch. I tried to remember if she had been one of the easier or harder people to work with on the fact sheets.\n\n\"My assistant,\" Mark said, proudly, I thought, and Fran\u00e7oise turned her appreciative eyes on me.\n\n\"Shall we walk out to the vineyard?\" she suggested. We climbed a gentle slope into the bright green rows. There is something beautiful about the neatly ordered vines on rolling land\u2014man's (or in this case, woman's) influence in an open, natural setting. It was remarkably cold for June, even by Champagne's standards. I hugged my cardigan closer to myself and had a hard time keeping up on the uneven dirt path. I stumbled and nearly dropped to a knee, gasping. Mark turned around in concern, took one look at my feet, and roared with laughter.\n\n\"You're definitely a beginner!\" he said. My face grew red\u2014like a fool I'd worn open-toed shoes, while Mark and our host were both in sturdy boots. They stopped to closely examine a chardonnay vine; satisfied, Mark stood up, murmuring something I couldn't hear. A small breeze caught his attention and he lifted his eyes over the vines and inhaled deeply, and smiled to himself. He was thoroughly in his element.\n\nMark always told me he had no dreams of making wine\u2014if wine was his religion, he was more of a missionary than a monk\u2014but visiting the producers he represented was as important a part of the process as drinking the wine itself. We were similar in that way\u2014my childhood visits to see my family in Ambonnay were all the evidence I needed that I was not made to be a winemaker\u2014but I was more drawn to city life than he was. For Mark, Westchester was close enough to Manhattan for him to conduct business without having to live there. And though he was a great salesman, a part of him liked the steadier, more routine pace of his life as it was, suburban, under his own control. It was in Aube that I realized I couldn't do exactly what he did. I didn't quite want to be the middleman, connecting winemakers and wholesalers, even as I found the most pleasure in discovering and maintaining relationships based around wine. I couldn't put it into words, but I knew I wanted to work directly with the average wine drinker.\n\nWe were many miles from my family's land, and the hills here were gentler than the Montagne de Reims but it felt familiar anyway to be among the vines. We spent another full day in Aube, during which we had lengthy meals with Fran\u00e7oise\u2014going over the business a bit, but mostly talking about the wine, how the winter had been and how the spring and summer were shaping up. We walked into the vineyard again, and this time I borrowed a pair of boots. Mark wanted to show me the differences, aside from color, between the smaller, compactly bunched pinot noir and pinot meunier grapes, and the larger chardonnay grapes, something I'd never paid much attention to even on my family's land.\n\nThen we drove on, further south into Burgundy to see a husband and wife team in the Nuits-Saint-Georges appellation. I'd never before stepped foot among Burgundy vines\u2014perhaps the best example of the heights a partnership between man and the land can reach. Nuits-Saint-Georges does not have any grand cru vineyards\u2014we'd passed the best of those just to the north, in the C\u00f4te de Nuits\u2014but the wine is still highly respected, and good bottles can be had at a value, at least in terms of Burgundy. We were far from Champagne now: the sky a deeper blue, the building roofs a glazed red.\n\nThe only night we spent in anything resembling a town was in Beaune, the wine capital of Burgundy. We rested for a bit and then met for dinner at the hotel restaurant, where Mark told me he dined every time he passed through. It was our first meal alone in France. We could have been back at Maison du Roi, by the office, except for the sheer quantity (and quality) of local delicacies piled on the table: snails with butter sauce, unpasteurized cheese you can't get in the States, a light and delicate mille-feuille (Napoleon) for dessert. It was the kind of meal you can't even find in Paris. Everything we consumed was sourced within miles of where we were sitting. Everything\u2014food, drink, environment\u2014was in perfect harmony. We ate four courses each and went through two bottles of local C\u00f4te de Beaune wine: a Pernand-Vergelesses white, lively and elegant with honey notes, from Domaine Vincent Rapet, and a Pommard red from Domaine Chantal Lescure that was deep, earthy, elegant, with a long finish.\n\nWe were, by the end, with no business to attend to in the morning, quite drunk.\n\n\"I come here every year, but always alone,\" he said, eyes moist with happiness. \"Even when my wife was helping with the company, she wouldn't come on these trips. This is my first time eating here with company. I'm glad to have it,\" he said, and touched his glass with mine.\n\n\"I've been worth every precious penny, no?\" I said to needle him.\n\n\"Every penny,\" he conceded.\n\nWe had never drunk this much together\u2014not simply tasting, but going through entire bottles. It had all the unexpected and liberating silliness, mixed with a slight fear of doing something wrong, of the first time I'd gotten inebriated in front of my parents.\n\n\"And you, Laure? Working with me has been worth your precious time?\"\n\n\"Every minute,\" I said right away. He raised his eyebrows as if he'd been expecting a sarcastic response and was surprised to receive an earnest one. I knew that I'd started to become impatient, had lost perspective on how much further I had to go, and how lucky I was to have a boss and mentor who wanted to see me get there. Just being in France with Mark had reminded me of this fact, and had revived more than a little gratitude in me. There was no real doubt in my mind that the job had been one of the greatest experiences of my life.\n\nWe had long finished everything on the table, but we lingered. There was something Mark wanted to say, I could tell, that he'd been working his way up to.\n\nHe cleared his throat. \"I've never liked having assistants,\" he said. \"I only have one when I desperately need it. I don't have a big business, but it's big enough for me. I like to work alone. And I may not be the easiest person to work for. I liked working with my wife\u2014I can trust her. It's hard to find someone you can trust. This, in the end, is why I don't hire many assistants. I'm too trusting. My company is like one of my children, and my assistant is like one of my children, too. Maybe a niece,\" he corrected himself. \"So that's why I don't always hire for the position. It's a big deal to add someone to your family and you can't do it lightly,\" he said with an air of finality. I had to blink to hold back the emotion that threatened to burst, especially after all the wine. I probably would have failed, if he did not then let out an immense belch, and sit back in his chair, looking as satisfied as I'd ever seen him.\n\nThe next day, after we recovered, we continued south along what the French call the Road of Great Wines, the Routes des Grands Crus, leaving behind the limestone ridge that is the backbone of Burgundy and the reason for its great terroir. We drove on through Beaujolais, past the steep slopes of the northern Rh\u00f4ne. We didn't have time to stop, but Mark took a route that added nearly an hour to our drive so that we passed more vineyards. We drove straight through the heart of the C\u00f4tes du Rh\u00f4ne Villages, along barely paved roads. The land was flatter here, and as far as I could see were rows of syrah vines. Vines upon vines.\n\nWe arrived, finally, at the feet of the Dentelles de Montmirail\u2014the Lace of the Mountains: lace because of the white limestone, the jagged, exposed range that catches the sun like a row of stone candles. It was the place Mark had described to me on our first lunch together, a year and a half before. Its Gigondas appellation is the chalkier, more rustic cousin of Ch\u00e2teauneuf-du-Pape, and the grenache, syrah, and mourv\u00e8dre vineyards were cut into the mountainside on small plots. The road narrowed into a groove in the rock. We bumped along in silence\u2014Mark's least natural state of being. Either he was still impressed by the sight he had seen many times, or was letting me take in the view unalloyed.\n\nI wondered if I would have had the same emotional reaction if I had never moved away from Paris and one day found myself here, perhaps having taken a drive with my uncle. I didn't think so. We would have traveled in a straight shot from north to south, and everything in between would have meant little to me, even as a child of wine country. I would have dozed off, daydreaming about whatever my job was\u2014writing briefs for a nonprofit, crunching numbers for a think tank. The vines, a constant fixture of my childhood, would have been nothing more than a fuzzy backdrop, like a pattern on wallpaper. \"Look, look,\" my uncle would have said, and I would have opened my eyes, maybe let out an appreciative murmur, and gone back to sleep.\n\nBut now the entire map of France had been remade in my mind; every square foot had taken on new meaning. We drank Gigondas and walked in the glimmering reflected light of the mountains. At this higher elevation, it was even cooler than Burgundy. The wine was plummy, earthy, spicy, powerful\u2014invigorating, like the land.\n\nAfter two days, we drove back to Paris, and after another weekend with my parents, I returned to New York\u2014not a changed woman, per se, but a humbled one. Even before we landed I knew where I would have to go next, that summer, even if I didn't know yet exactly why.\n\n# Thinking About Wine\n\nThe next time you visit a winery, don't just drink the wine. Walk around the vineyard, if you can. (Wear closed-toe shoes!) The appearance of the vines can tell you a lot. If they are pristine, with shiny leaves and clear earth around their base, that is a sign the winery is availing itself of modern fertilizer and pesticides to help maintain a consistent crop that can be harvested en masse. I've seen wineries like this in France, and these are likely the vineyards you imagine or have seen in pictures. They look sterile. They are the result of mankind imposing technology on the land, which has been the standard way of making wine for a very long time.\n\nBut I've also seen winemakers who let wild vegetation grow between the vines, who don't use synthetic sprays to eliminate insects, weeds, or microbes. The vines (and whatever other plants want to tag along) are fed with compost instead of industrial fertilizer. Compared with standard vineyards, these vineyards are unkempt, even _dirty_. But dirty in the best way, as this is a sign of organic grape cultivation. Yes! Everything else these days can be organic: why not wine? Organic wine is a complicated concept. As with all \"organic\" labels, qualifying for it requires following a set of protocols that differ from country to country. In fact, a lot of winemakers may use organic methods to grow their grapes without bothering to get the label. This is yet another reason to learn something about the producer\u2014the label can't tell you everything.\n\nThere are good reasons organic wines haven't yet taken the world by storm. It's difficult and expensive to care for vines without the help of synthetic fertilizer and pesticides. You have to watch them carefully, day and night, for signs of any of the big three diseases (downy mildew, Oidium, and Botrytis) instead of assuming the chemicals will take care of everything. In short: it's a pain. Is it worth it? It depends\u2014if you believe it's better for wine to speak to the character of the land, and that a land's character is more naturally expressed without presticides, then yes. In fact, 85 percent of the wine I carry in my store is organic.\n\nCultivation, though, is only part of what makes a wine. There's the actual production\u2014sorting the grapes, crushing them, fermenting the juice, adding sulfites to preserve and prevent microbial growth. Yes, sulfites; you might not have realized that almost every bottle of wine you drink has a tiny bit of sulfur in it. Even if no sulfites are added, a small amount will still occur naturally.\n\nThat leads us to another category: natural wine. You may have noticed this label showing up more frequently on wine bottles. If organic wine covers cultivation, natural wine includes the actual production, too. It means the grape juice ferments under its own natural yeast, with nothing added, and the wine is stored without added sulfites. Without added sulfites, wine is more fragile, vulnerable to spoilage and oxidation. Red wines can get away with it a little more because the tannins in grape skins are natural antioxidants. Some winemakers who avoid sulfites will add an inert gas like nitrogen or carbon dioxide to keep the wine away from oxygen; if you buy a bottle with added gas, it may appear to fizz for a few minutes. Don't worry! You can decant it if you wish\u2014once oxygen is introduced, it will calm down. Because you have to be so careful with natural wines, I currently only carry ten, out of three hundred total.\n\nNatural wine doesn't necessarily taste better, I should say. Some people choose to buy organic and natural wines because of health concerns; they think the sulfites are dangerous\u2014although in low amounts they're not\u2014or prefer to avoid herbicides and pesticides. I'll leave it to science and history to decide whether there really is anything to be worried about on these fronts. But I don't think it's a fad, either. It's part of a long-term trend, and a mark of progress. Organic and natural wine production is kinder to the land. Why not take care of the land that takes care of you? That more and more winemakers (and winesellers\u2014like me) are approaching their business this way is part of a smarter, more conscientious relationship with the earth. It allows the wine to be as direct and respectful an expression of the terroir as possible.\n\nEverything comes from the land and is of the land, whether it is a grape or you or me. Just as your environment and culture have helped make you who you are, so terroir does for wine. It can be easy to forget this in the U.S., where so many cultures and regions coexist within a hard-to-define sense of what America _is_. The wonderful way it is always changing, always shifting\u2014the very thing that drew me to it\u2014can also be frustrating, whereas a single strong tradition, like we have in France, provides guidance but also breeds conformity.\n\nSo what do I believe now, with the benefit of hindsight? Is one way better? The truth is, they're not as different as they seemed to me when I first arrived in New York. I could name a few stubborn and small-minded corners of America, and am often inspired by the innovations of French culture. At best, you can learn from both. And no matter where you are, you need a supportive community to flourish, and good wine is no different: it needs fair weather, a well-timed harvest, a careful attendant. There will be things you can control, like your methods, and things you can't, like the sun and the rain. The best winemaker in the world cannot make great wine from bad land, though a bad winemaker can squander even the most fertile soil. And the best wine is wine that has a sense of place. You can take the trimmings from the oldest, most heralded rootstock and transplant it to another country, and the wine won't taste the same\u2014it may be better, it may be worse, but it will most certainly be different.\n\nJust recently, I was working with the winemaker Sylvain Pataille, from Marsannay in Burgundy. He has two cuv\u00e9es whose plots are right next to each other: Clos du Roy and Longeroies. One is fresh, smooth, easy to drink, and the latter is deeper, full-bodied, stronger. I asked him what the difference was in his production techniques, to make such different wine.\n\n\"Just a road,\" he said.\n\n\"What do you mean?\" I asked, confused.\n\nWhat he meant was that his methodology for both cuv\u00e9es was exactly the same. All that separated the two plots of land was a narrow path. Same technique, different wines. That is the power of terroir.\n\nLike it or not, we are irrevocably a product of our time and place. True satisfaction comes from embracing it. It took me almost three years to learn this. I'm still learning it.\n\nMark Brodeur loved to work, and wanted the same from me. He'd made it clear from the start how he saw me as an extension of him. In the twenty months we had worked together, he'd allowed me one week off for Christmas and a few days here and there. I had no benefits (although he was reluctantly paying half my health insurance) and there was no official vacation or sick day policy. If I came in to work sick I could tell he was proud of me for doing it. So I wasn't sure what he'd say when I asked for a week and a half in August to tour California with Jules.\n\n\"What will you do?\" he asked, his eyes nearly closed in deliberation.\n\nI smiled. He was too easy to tease. \"I don't know. Just Disneyland.\"\n\n\"Humph,\" he said, shaking his head. \"If you are not going to drink the wine, then I don't know why you are going, and I can't let you go. I'm disappointed in you, Laure.\" He really did look like I had let him down, until I admitted to the prank and he gave me the time off.\n\nI hadn't flown within the U.S. since touring on behalf of Ch\u00e2teau la Nerthe, and I hadn't been to California since that first February, two and a half years before. I was thrilled about the prospect of visiting it again, after I had seen and learned so much. There was a natural symmetry to it.\n\nLately I'd been spending two or three days a week away from the office. I would go up to Westchester on Monday to collect several bottles, which I'd bring back on the train like a bag of rocks over my shoulder. At home, Rose or Jules would meet me on the landing, take hold of a strap, and help carry the bag up five floors to our apartment, where I'd carefully place the bottles below the air conditioner in the bedroom (I couldn't afford a wine cooler). On the days no one was at home, I had to lug the bag up the stairs myself.\n\nI went on sales calls, taking a few bottles with me\u2014in the summer no one wants full-bodied reds, so I carried plenty of white and ros\u00e9. To Queens, or Port Washington, or the Upper East Side, or the Hamptons, on a morning train. Once I went all the way to Montauk at the very tip of Long Island. I dressed in white linen. A sales rep picked me up at the station and we went around to the stores and restaurants. These were beach towns, with elegant main streets where the wealthy strolled, restaurants spilling over onto the sidewalk, and the smell of the sea. I thought back to when I'd first started these kinds of sales calls as a young woman who could barely put five English words together, with a raw palate and some memorized lines. Now I was confident, knowledgeable, opinionated, and able to make small talk\u2014even make others laugh (the test of true fluency I had set as a goalpost).\n\nSometimes, I just picked a sales destination myself and went on my own. The routine was, by now, second nature. Drop me off in front of any store with a bottle in my hand, and I'll walk in and sell a case. It's easier to pitch something when the product is good quality, and Mark's wines were. I never had to lie\u2014a luxury in any business\u2014and I never stopped being grateful to work for someone who cared more about spreading the gospel of great wine than making an easy buck.\n\nWherever I went, on Fridays I returned to Westchester, and Mark and I would put our feet up on our desks and I'd tell him about the week while he clasped his hands behind his head and nodded, his lips pursed: such and such cases of _x_. Such and such cases of _y_. Sold under this rep's name. Or with that rep at my side\u2014Who? You know, the guy who...Oh yes, he's good. Then, debrief complete, Mark would open a bottle for us to taste.\n\nWhen we tasted, there was nothing else in our minds but the wine at hand, and for a few moments it felt like we were in France again. Mark's face would change, soften, and the memories would flow about the winemaker and her family, about the first time he drank this wine, the weather that day, the shoes he was wearing, the look on his wife's face. Although I didn't quite have the memories he did\u2014and definitely not as far back in time\u2014I knew even then that these Friday afternoons would later be some of my best.\n\nAt the beginning of August, Jules and I left for San Francisco. Peter, who'd grown up in the Bay Area, was there for a long visit divided between work and family. He picked us up at the airport.\n\n\"Did you guys just come from the beach?\" he asked us, eyes wide.\n\n\"No,\" we said, smiling, arms around each other's waists. We thought it was just Peter being Peter. \"Why would you say that?\"\n\n\"You look like you're already on vacation!\"\n\n\"We are!\"\n\n\"No, I mean you _look_ like you're already on vacation.\" He meant we were relaxed; we had sunglasses on; our shirts were haphazardly buttoned; we were in sandals; we were smiling enormous smiles. Apparently it takes most New Yorkers\u2014and perhaps most overworked Americans\u2014a full day after landing anywhere to get into \"vacation mode.\" We had made the transformation before the plane had even backed away from the gate.\n\nPeter drove us to the South Bay. Jules and I both sat in the back, as a joke, but also to be close to each other. The last few months had been rough on us. We had been arguing, not unlike our first couple of months in New York, but for different reasons. In the early days of our life together in the city he'd been unmoored, anxious, resistant. A lot of tension came from his disdain toward the whole experience, how much he hated everything I loved\u2014the people, the bustle, the unpredictability. But as he approached his two-year anniversary as a New York resident, the things that had bothered him most he had grown, if not affectionate toward, at least accustomed to. He still disliked the noise of bars but no longer complained as much about it and could even tolerate it long enough to meet friends for a good portion of the night. He didn't whine about the cuisine, and his work schedule had eased enough that he was no longer subsisting on egg-on-rice. He frequently got home at a normal time, and because I was spending most of my days in or around the city, I, too, returned to the apartment in the early evening.\n\nBut our renewed closeness raised its own set of issues. Previously, the only quality time we spent together during the week was while we were asleep. It's very easy to be kind to each other in your sleep. Now, expectations were raised. We were suddenly an active couple again, and our preferences and rhythms often didn't line up. I had energy in the evenings. I wanted to try all the new restaurants popping up on our stretch of Eleventh Street, go to every rooftop bar, spend every night for a week in Williamsburg to see what the fuss was about, meet every friend of every friend. Jules wanted to watch basketball, develop a taste for Scotch with Peter, and go to the Bronx Zoo and the Coney Island Aquarium (he enjoyed animals so much I often thought he preferred their company). I didn't understand it, but we tried to compromise\u2014and I did truly love seeing his face light up at the sight of elephants. But more than once he didn't show up for dinner, and I texted, called, and finally apologized to our friends and, frantic with irrational fear, ran home to see if he was all right. He was invariably asleep, oblivious to all alarms and ring tones, and I shouted and shouted until he rolled off the bed onto the floor.\n\nI wasn't proud of us at this time. It seemed every few months there'd been a new twist, another challenge. _Would it ever be simple and easy?_ But our story was the story of every long relationship, I told myself, and we were still tender, and reassuring, and devoted. I was still his _coeur_ and he was still my _loup_. The same traits in him that drove me crazy\u2014stubbornness, a deep idiosyncrasy, single-minded pursuit of his passions\u2014were the same ones that made me laugh, that charmed me, that led me to desire him like no one I'd ever known. He still tricked me with false proposals, and on Saturday mornings, before I went to the Union Square Greenmarket and he fell back asleep, we told each other half-remembered, half-imagined dreams of ourselves in five years: children, bicycles, picnics (although we never mentioned in what city).\n\nI couldn't imagine a life without Jules.\n\nUnlike our friends back in Europe, we'd done little vacationing, having put everything into proving ourselves at work. This trip to California was important. I kept thinking about how well a vacation had worked for Min and Derek. So here we were, even _looking_ like we were on vacation. In the backseat we smiled at each other, unable to see each other's eyes behind the dark sunglasses. Once again, the bright California sky didn't fail to amaze me. In the distance were pale yellow mountains.\n\nPeter took us to his hometown. We ate at a sidewalk caf\u00e9 and marveled at our surroundings. It seemed idyllic\u2014the pedestrians moving at a leisurely pace, everyone in shorts and sandals, with bright smiles, casually exuding money. For each person who passed, Jules and I played a game: did they work for Google or Facebook? Peter was intensely embarrassed by us. We spent the night at his parents' house. They were gentle people who treated Peter with great love but a little gingerly, as if they hadn't raised him themselves but had found him in the woods.\n\nThen he drove us up to San Francisco, where we spent the night in an affordable bed and breakfast established in a traditional old Victorian house. It was adorable, and we slept well in a bright white double bed\u2014the only downside was a shared bathroom. On the whole, Jules seemed to find the city puzzling. He hadn't expected the hills, or the beauty of the bay, or its compactness. It was cool and windy. Overconfident in my memory and navigational instincts, I got us lost on a walk I swore I had taken the one other time I'd been there and we ended up at Fisherman's Wharf. Our feet hurt, but as soon as Jules heard the sea lions he went running to see them.\n\nThe real excitement for me began the next day, when we picked up our rental car and drove over the Bay Bridge and north, into Napa Valley, into the heart and soul of the American wine industry. It got much hotter as soon as we left San Francisco. It was a hundred degrees. The hills, yellow with green patches, were still all around us in the distance, but in all the spaces between, for miles in every direction as far as the eye could see, emerged the vines, the crooked trunks, the canopies well-kempt and bright with leaves.\n\nI was driving. Jules had never gotten a driver's license, even in France\u2014yet another source of tension. I looked away from the road in quick glances to see if I could spot the camouflaged grape bunches, the heavy green of the sparkling and white grapes that in this climate would be ready for harvest now, in early August. My heart drummed in tune with the thump of the tires on the road.\n\n\"The next winery, I'm pulling over,\" I said. I couldn't stand to be in the car any longer, with the grapevines\u2014and their end product\u2014so close. As soon as we stepped out of the car, the heat hit us. Even closing the car door nearly burned my hand. The tasting room was twenty feet away across the gravelly lot and it seemed like too far to cross under the desert sun.\n\n\"I can't drink wine in this weather, you are insane,\" Jules replied, but I wouldn't be dissuaded.\n\nThe tasting room wasn't like anything I had ever seen. In France, you often don't even find tasting rooms; at the older wineries in particular, you'll sometimes just get the winemaker in his or her office, paperwork everywhere and an antique computer in the corner, standing by a table that has been hastily cleared so a bottle can be retrieved and opened. This Napa room, however, was high-tech: enormous, with a large L-shaped marble-top bar and several employees behind it, all spread out, wearing matching shirts and smiles. And in the other corner a gift shop, where you could buy prepackaged cardboard boxes of wine, inscribed wine glasses and bottle openers, even shirts like the ones worn by the staff. I turned around slowly, openmouthed, taking it all in.\n\nThen we navigated the tasting fees\u2014tiered pricing for three, five, or seven wines. In France, no ch\u00e2teau would charge you to sample their wine (though this is starting to happen at a few famous estates); it's simply not the same touristic activity as it is in the States. Inside it was well air-conditioned and full of people\u2014boisterous, even, like the first minutes of happy hour, but at eleven in the morning. One couple seemed to be paying careful attention to the contents of their glasses, but everyone else drank quickly, nodding to each other, progressing from whites to reds without pause. Would they remember what they tasted tomorrow, or a year from now? I couldn't say. Everyone, at least, was enjoying themselves.\n\nBut I also hope they were able to find a way to give the wine their full attention. In all your days, so few of them will be spent drinking wine at its source. Those days are special.\n\nJules still refused to taste the wine. Even though most of the wineries were a pleasant sixty-eight degrees inside, he stood just beyond the line of customers at the bar, fanning himself with our road map. I tried to convince him that spitting the wine out would keep him from overheating, but he complained that even the tannins on his tongue and gums made him hotter. So I tasted and spat, and described my impressions to him, and he listened with the attention and amusement of a child being read a story.\n\n\"This is good, minerally, very dry,\" I said about a sauvignon blanc, and he smiled at me. I was the only customer using the spittoon.\n\n\"This is bold, plummy, so much alcohol, I think a little bit of chocolate,\" I said of a cabernet sauvignon.\n\n\"A lot of personality!\" I said of a Bordeaux-style blend. (And when the attendant turned away for a moment, I added in whispered French, \"But too young. Definitely needs to age.\")\n\nThe story I told was spinning out from my mind and senses with each new wine. We\u2014I say _we_ because it had started to feel like a joint endeavor, even though Jules wasn't putting anything in his mouth\u2014finished with a classic California zinfandel, which once dominated the California wine landscape but has for a while been pushed out in favor of cabernet sauvignon. \"Strong aroma,\" I said as I lifted the glass to my nose. \"High alcohol content. This will be interesting.\" I sipped. \"Dark berries, very peppery. So much berry. Jules, you must smell it.\" Like a mime, he sniffed and nodded.\n\nWe slept in a motel that night, the A\/C window unit on high. In the middle of the night I felt restless and slipped outside without waking Jules\u2014easy enough. It had cooled down after the sun set, and there was almost no moisture in the air. The sky was not totally black but a dark gray-blue. In the darkness I felt the presence of the vines and their precious cargo. I smelled the earth, the vegetation, carried lightly in the night breeze. And I felt a great richness in the land, but one that I didn't belong to. Does that make any sense? Even as I stood in the middle of it, admiring all of its grandeur and vitality, I was at a distance. It felt so big, all of it: California, the U.S., the land between here and the home I had made for myself. The cellars were big, the vats, the destemming machines I saw on a winery tour. Everything was new and clean, expansive and bold and young. There was so much happening, and so much being invented. I felt it deeply, but also knew that it was going to roll on with or without me.\n\nIt was only my first night in California wine country, but all the pieces were coming together\u2014why I wanted to come, why it meant so much to me, and why it felt so symmetrical with my first trip so early in my American journey. It was like a bookend. I'd wanted to touch this side of the country one more time before scooping up everything the U.S. had given me, and taking it back to France.\n\nI'd been considering it for months, I realized, but only half-consciously. The business trip with Mark had felt destined, putting me in closer touch with my terroir. For nearly three years, I'd been drinking French wine almost exclusively; I wanted to know it above all others. It was the wine that was mine, that was me. I hadn't anticipated this awareness dawning here, standing on a concrete deck along a roadside motel between two towns, surrounded by fifty square miles of grapes. But there it was.\n\nIt was time to go home.\n\nI didn't say any of this to Jules until our second night, which we spent in Sonoma. It was just one valley over from Napa, but it was a different world. The wineries were more relaxed, the environment more bucolic, slightly cooler. The zinfandel was still high in alcohol, but softer, more velvety. The unoaked chardonnay was crisp, with a greater portion of tree fruit like apple and pear. It's easy for oak to overpower a chardonnay, especially if the barrel is new, but I tried a couple that were beautifully balanced, buttery and rich without losing their delicacy and freshness.\n\nI'm sorry to say I didn't write any details down. I was enjoying the wine, but my mind was elsewhere. We ate dinner at a casual American bistro with garden seating\u2014desert shrubs of a green-gray color in pots between the tables, and lights strung overhead. I cut into my squab, feeling truly like an explorer in a strange land, while Jules poked at his seafood pasta and sipped his wine, which only now in the cool evening was he willing to drink, and waited for me to notice him smiling.\n\n\"What is it?\" I said, somewhat disingenuously.\n\n\"You tell me!\" he said.\n\nThere was no delaying any longer, though I knew it would mean something different as soon as I said it out loud.\n\n\"I'm ready to go back to France.\"\n\n\"You don't want to grow old here? Raise little American children? We can move to California.\"\n\nI looked around the garden, at the alien plants. \"I miss my family. I miss France.\" I shrugged a fundamentally French shrug that I had mostly dropped in the last couple of years.\n\n\"What do you want to do?\"\n\n\"I don't know. Something in wine, still.\"\n\n\"You came to America for wine,\" he said. He'd yet to say anything about his own feelings.\n\n\"Yes and no. I came for the experience and for the language. I stayed for the wine.\"\n\n\"And I came for you.\"\n\nThe way he put it in such stark terms broke my heart a little. It was true that between the two of us I had done more to shape our life together. If Pringent had never called me at that party in Paris, things might have turned out differently. Jules might have found a position he loved somewhere, and we could now be in England, as I'd originally planned, or even China. I might still have worked in wine, but for an old business, learning the old ways. Or I might have left the industry altogether.\n\nWe found another motel, and in bed before we fell asleep I asked him to assure me again that he was willing to leave New York, too, even though he'd spent two years assuring me that he was happy there.\n\n\"Yes,\" he said. \"I may be used to it, but it's still not for me.\"\n\n\"You'll have to watch your NBA at odd hours.\"\n\n\"I know.\"\n\nI was entirely wrapped up in his limbs, in the center of a sagging mattress. It was not a position I usually enjoyed for more than a few minutes, but tonight it felt like the most comfortable place in the world. I started to drift off, and Jules, his voice still clear and awake, said, \"How long are you thinking? There are a couple of projects at work I'm attached to the next few months and I would feel guilty leaving them in a bad position.\"\n\n\"I understand,\" I mumbled, and then drifted off. Unlike the previous night, I slept soundly the whole way through.\n\nI woke up as refreshed as if I'd slept for three days. All of my muscles felt loose, as if I'd had an intense massage.\n\nI wanted to spend one more day in Sonoma, which was bigger than I'd expected. We drove toward the coast where the cool-weather pinot noir was more likely to thrive. In the morning the remnants of fog retreated back over the hills separating us from the sea. The vineyards here were like patches of carpet on the steeper slopes. In France, California has a reputation (to its detriment) for brash reds, but these weren't it. The pinot noir was lush, dark, slightly vegetal.\n\nThe rest of the trip carried us along like a receding fog. We drove south not only because I thought Jules should see Los Angeles, but because I'd heard that some Central Coast wines were starting to make a name for themselves. On the famous Highway 1 somewhere between Monterey and Big Sur we pulled into a state preserve overlooking the ocean. It was dark, but the moon was bright.\n\n\"Do you want to sleep here?\" I said. It would save money, but mostly it just felt exciting.\n\n\"Right here in the car?\" Jules said. He shrugged in the glow of the dashboard. \"Why not?\"\n\nWe felt like children. A great weight had lifted, knowing now as we did that our time here was coming to a close, and we couldn't stop giggling, in relief and also a bit of hysteria. As strange as it had been to try to make a life in the U.S., it was also going to be strange to leave.\n\nWe tilted our seats all the way back and rolled toward each other so we could hold hands over the cup holders. Through the windows we'd left open a crack I could hear the ocean crashing against the cliffs. I'd just drifted off when I was woken by a bright light through my eyelids, and the tapping of metal on glass. A man in uniform was speaking to us. It took me a few seconds to reach full awareness, and I had to shake Jules violently until he woke up, too.\n\n\"You folks can't sleep here; are you crazy?\"\n\n\"It's just so beautiful here, officer. We are tired, we thought we would just sleep for the night and continue driving in the morning.\"\n\n\"I'm a ranger, not a cop.\"\n\n\"Ah, pardon us,\" I said. \"We were just so tired.\"\n\n\"Well, you can't sleep here. It's illegal,\" the ranger said, which was news to us. (We'd done it often in France and Spain.)\n\nSo we drove on in the dark, disappointed to be missing the view\u2014it was too late to find a motel, and we couldn't justify spending money on the few hours of night remaining. We exited at a small town to the south, and we slept until dawn in a parking lot, squeezed between two cars to hide from view. (This would be the last time I spent the night in a car. To this day I have fond associations with American motels: their spare, basic arrangements, the fundamental need for a warm bed they fulfill, the freedom they afford travelers who don't wish to plot out every last hour of a trip.) We spent the next day stopping at as many wineries as we could, first making our way through a cooler hilly area and then continuing eastward where the land was broader, drier, hotter. We sampled the California staples, bold cabernet sauvignons and chardonnays that were a little subtler than the Napa varieties. I tried a couple of Rh\u00f4ne-style wines, too. I was impressed by a few that I tried, which managed to balance the usual dark fruit with softer tannins and the spice and leather notes I always enjoy.\n\nIn the late afternoon we had a picnic at one of the wineries, under a trellis with flowering vines providing shade, in view of the acres and acres of pre-harvest, heavy grapes. The atmosphere was relaxed and uncrowded. We sat on opposite benches and the California light was as special as the clich\u00e9s indicate, and even more so here, softer, shining on every one of Jules's eyelashes.\n\n\"You really wouldn't want to raise kids here?\" he teased again.\n\n\"Here, maybe,\" I said. We had a bottle of one of the Rh\u00f4ne-style wines we'd bought in the tasting room. The wine was a dark violet with a radiant center where the sun pierced it. We sat across from each other, ankles intertwined. It seemed like we had stepped outside of time, entered a place where we could talk about anything without consequence. None of it, even the decision to leave, would be real until we returned to New York.\n\nWe spent another night in the area, and then drove south and met up again with the coastline before Santa Barbara. I thought Jules might hate Los Angeles, but he ended up loving it, the way a child loves the circus: the spectacle, the impossible standards of beauty, the never-ending Sunset Boulevard, and the Santa Monica pier. We drove up into the hills to meet friends of Jules's co-worker, a man and two women. First all three kissed Jules on the cheek one by one, and then each turned to me and kissed me on the lips. I looked at Jules with wide eyes\u2014 _What California custom is this?_ But he was too busy laughing to notice. They offered us pot, which we declined, and then invited us into their hot tub, where they kept suggesting we take off our suits and relax. Jules couldn't stop giggling and wouldn't listen to my increasingly alarmed suggestions that we head back. When we finally left, Jules wouldn't stop joking about it. \"What's wrong?\" he said. \"Two women together is a beautiful thing. You don't think it's beautiful?\"\n\nWe flew back to New York midweek. We landed in Newark and took the bus into Manhattan, running alongside the marshlands until the Empire State Building appeared in the distance. Jules had a catlike half smile on his face, as he slept in the seat beside me, and I wondered if he was dreaming about California\u2014the Los Angeles hills, perhaps. I, however, was wide awake.\n\nI didn't want to leave the U.S. right away. But I'm not a patient person, and once an idea gets into my head it invariably begins to grow in size like a snowball rolling down the mountain, triggering an avalanche of thoughts and actions. Back at work the next day, Mark greeted me in a restrained manner, as if he'd hardly noticed I'd been gone, but soon he was waving his arms, telling me that everything had fallen apart while I was gone, which I could see for myself: there were stacks of paper everywhere and two empty bottles on the floor by his desk.\n\n\"Give me a report,\" he said, with every ounce of seriousness, and put his hands behind his head and leaned back in the way he did when I told him about my sales calls. He wanted to hear about what I thought of California and its wines.\n\n\"If I'd known this was going to be for business research I would have made you pay for it,\" I said. He scoffed and smiled.\n\nThis is how fate works: chance conspires with opportunity. The next week I began a round of sales calls in Manhattan. I'd long planned to visit the stores near my apartment in the East Village, but had been saving it for after California. I avoided Jacob's, as I was too regular a customer to avoid a conflict of interest. But even so, canvassing my own neighborhood was both heartwarming and surreal. It had been a long time since I'd walked so many of these streets on a weekday morning. I ended up on Avenue C, just a couple of blocks from Rose and Nico's old apartment\u2014the location of my first, lonely night in New York.\n\nI knew nothing about the wine store I was standing in front of, which had opened after we'd moved to the new apartment, as I stepped inside with my heavy bag to a jingle of bells on the door. But it is a moment I will never forget. The space was tiny, just a few hundred square feet, but every inch was thoughtfully used. There were tall shelves against the walls and a row of shorter racks in the middle. I scanned the rows nearby and recognized only a couple of the labels; the owner was sourcing small producers. The signs for the regions were nicely designed, and every bottle had a handwritten tag around its neck. A solid wood table stood to one side for tastings. The owner stepped out from the office to greet me. I was surprised to see a young woman.\n\n\"Are you Laure?\" she said.\n\nShe couldn't have been too much older than me, meaning she was the youngest store owner I'd met\u2014and one of the few women. She was small, with red hair and large expressive eyes, and a big voice, a big laugh. She was pushy, I could tell, and frank. Just my kind of person. She bought three cases that day, one of ros\u00e9 and two of a Crozes-Hermitage Rh\u00f4ne (a medium-bodied syrah of soft tannins but good spice, fitting for summer barbecues), and asked if I would come back Friday to conduct a tasting for her customers. I agreed and gave her my direct line, marked her in my book, and left, trembling.\n\nI returned Friday evening, walking down the familiar streets with a small knot of anticipation in my stomach. Meeting the shopkeeper had excited me. She was my age, with a business she ran herself. It wasn't a possibility I'd ever seriously considered. It turned out she also owned a cheese store next door, and had brought over bread and cheese for the tasting. There was a small crowd, mostly younger, with one middle-aged couple. All the patrons seemed to live within a few blocks and know the owner. And everyone was excited, if tentative, smiling big smiles the way Americans do and nodding at one another. The owner seemed the happiest out of them all\u2014genuinely pleased to see people in her store having a good time. After the tasting, I asked her how long she'd been in business.\n\n\"Little over a year,\" she said. \"But I've aged ten.\"\n\nI laughed then, though I filed that unintended advice away.\n\nI couldn't sleep that night, or that weekend. Jules and I had planned to meet Min and Derek in Central Park, because he'd somehow gotten them to agree to go to the zoo with us, but I couldn't focus on what was happening around me. At home, Jules took one of my tasting bottles, a Bordeaux that would be too old to use as a sample by Monday, poured two glasses, sat me down, and said, \"What is it?\"\n\nHe'd reminded me at the start of the week that he had several months of projects lined up at work, and I'd reassured him that I understood. But I hadn't promised I'd stay in New York until he finished. And now I knew I couldn't.\n\n\"I am going to open a wine bar and store in Paris,\" I said.\n\n\"That's perfect!\"\n\n\"I think I have to go first, to prepare.\"\n\n\"Soon?\"\n\n\"Soon.\"\n\nHe sipped his wine, his lips moving as he let it swirl around in his mouth. \"You know I can't go yet,\" he said.\n\nI put my hands on his knees and kissed his cheek. \"I will get everything ready for us.\"\n\n\"Why now? You can't open something that quickly. Do you even know where? What it will look like?\"\n\nI shook my head. \"No,\" I said. \"That's exactly why I have to go now. So I can research, so I can be prepared. I don't even know how French business works, really. I only know American business. It will take time.\"\n\nHe exhaled. \"I need four, five months,\" he said. \"At least.\"\n\n\"Do you have to stay?\" Now it was my turn. \"They can find someone else. Your colleagues can pick up your work.\"\n\nHe shook his head more vigorously than I had. \"No, they took a chance on me two years ago. I can't leave them suddenly like this. I'll tell them my plans, but I have to help them finish. The work already has my personality in it.\"\n\nWe were quiet for a minute, then. Opposites attract because if one person happens to be headstrong and stubborn, the other often needs to be able to softly receive the blow, like a hammer hitting a pillow. When we first started dating, I'd thought it was obvious that I was the hammer and Jules the pillow, but at some point it had shifted, and for a while I thought it was the opposite. By now it was clear we were both hammers. That it had worked out this far was nothing short of a miracle.\n\nAnd aren't miracles often the most likely explanation?\n\nI told our friends on the roof one hot August night. I couldn't sit still, so this time I was the one pacing while Peter sat in my usual chair, selecting songs from his iPod. I kept waiting for the right moment, and the right moment kept passing. Just when there was a pause in the conversation, Derek stood up for a refill of wine, and the talk turned to weddings. I smiled and stayed silent, a boulder in my throat. Min and Derek began to make overlapping comments about their band versus DJ preferences. I'd told Rose the day before, and we cried and she admitted she wasn't surprised, that I was meant to go home and she was meant to stay, that was the difference between us.\n\nFinally, I just blurted it out, in the middle of an entirely different conversation. There was quiet, and then a hundred questions at once: _Did something happen with the job? What was Jules doing? What was Rose doing? Why?_ I answered as best as I could, even though I still didn't know many of the answers. I knew I wanted to leave in a month or less, sometime in September.\n\nIn the distance, just above the water towers, fireworks started going off, far enough that we couldn't hear them, small colorful starbursts on the horizon. They were clearly professional, though not as impressive as the ones on the Fourth of July. We lined up by the edge of the roof and watched without speaking until the last one faded from the sky. Even then, we lingered for a minute before going back to our seats and our wine. I felt embarrassed to have ruined the atmosphere. Everyone looked at me, and I felt tears gather in my eyes.\n\nPeter started laughing first. Soon we all were. The silence had become unbearable.\n\nBut I still had one person to tell.\n\nIt's never easy to quit. For every person who gleefully shouts, \"You can take this job and shove it!\" there are a hundred more nervously biting their tongues, breaking out into a sweat at the idea of letting down the company, their colleagues. It speaks to our better natures, how hard it is to leave even a place we complain about constantly.\n\nDays passed, and I didn't say anything to Mark. I was in the office three days in a row, which was unusual now, so that we could go over some business, and each day I would sit at my desk going over the words in my head while he hummed and flipped through papers across the office. \"Ah, _d\u00e9jeuner_ ,\" he said, like always, and we strolled together down the sleepy sidewalk to La Maison du Roi, for French onion soup for him and a salad ni\u00e7oise for me.\n\nThen it was Friday, and I knew it wouldn't be fair to let it go past the weekend. I was planning on leaving within a month, around the end of August, since nothing really happens in August, when the whole of France shuts down. The whole day I tried not to nibble my lower lip clean off. He went to fetch a bottle of wine, and before we even took the first sip, I told him quickly, like ripping off a bandage.\n\nHe put his glass down and let his head drop with a suddenness that made his glasses slide forward on his nose, and nearly fall off entirely. When he lifted his face again, it was red\u2014with anger.\n\n\"This is not cool. This is not a cool decision. I taught you so many things. I taught you everything you know!\" he said. \"I spent a lot of money on you. Training you, feeding you. I trusted you. Now you're leaving me like this. It's going to be very hard on me, very difficult to be by myself again...\" He drifted off. Now his glasses were in one hand while he massaged his brow with the other.\n\nI picked up my bag and left. I managed to hold back my tears until the train began to move out of the station and back toward New York.\n\nOn Monday I went to my appointments in Brooklyn as planned, over the bridge Jules and I had once taken to see drummers in Prospect Park. I didn't call to check in with Westchester. I imagined Mark would need another day to recover, but around lunchtime my cell phone rang.\n\n\"I'm sorry for my anger,\" he said. \"I did not think about why you wanted to leave. I was thinking about myself. Now I understand. I support you. Go, go be with your family. Go start your business. You learned everything so fast, it helps to be a natural. I knew in my heart I couldn't keep you. I will be there for anything you need.\"\n\nIt was better than I could have hoped for. I wasn't prepared for how touched I felt. I canceled my appointments for the following day and went up to Westchester instead. When I entered the office Mark lifted his head a little to acknowledge my presence, grunted, and turned back to his reading. I sat at my desk and conducted business as usual. I made a few calls to a wholesaler, tallying up the sales I'd made on their behalf (a mutually beneficial arrangement if there ever was one). As soon as I hung up, he turned to me. \"Let's eat early today,\" he said. \"I feel like sushi.\" We stood and met by the door, but before opening it, and without thinking, I gave him a hug. He patted my back paternally and said, \"Yes, yes. Yes, yes.\"\n\nThree weeks later, I was on a plane back to France.\n\nDrinking wine is a string of moments. Each sip is discrete\u2014you lift the glass, allow the wine to infiltrate your nose and run over your tongue, and put the glass down again\u2014but most people will remember it as one continuous experience, unless they're really paying attention. If you don't notice each moment, they blur, rather than connect, and instead of a braid of pleasures and observations, you are left with only a fuzzy memory. You remember you did _something_ \u2014but not what it felt like to do it.\n\nIt's the same with life. The years go by like _that_. If you don't pay attention to everything you see and do, every nerve tingling, every firing synapse, then the experience cannot touch you. You are cutting off your memories before they even have a chance to form.\n\nI wanted to fill my final month in New York. I spent many nights with my friends, laughing, dancing, and drinking wine. I didn't want a goodbye party. I told everyone just to remember the one I'd had before I got the Pringent job. And I bought champagne for them all, spending a whole paycheck on a different cuv\u00e9e for everyone\u2014I knew they would drink together and wanted them to be able to taste a few different ones: Drappier for Rose and Benji, Roederer for Min and Derek, Jacques Lassaigne for Peter, even a bottle for my old colleague Allison, who always complained that sparkling wine gave her a headache. I gave her a nice Larmandier-Bernier Cuv\u00e9e Longitude, and later she told me she felt fine after drinking it\u2014better than fine. There were thankfully few tears. I said I was coming back to visit soon, which was true. Jules would still be here, and Min and Derek's wedding was in the spring.\n\nRose moved in with Benji just days before I left, and maybe that was why there were more smiles than frowns this time. It felt as though I were giving her to him for safekeeping. I helped her carry some stuff to his apartment, which was tidy and spare, but not impersonal\u2014his own photography was on the wall, enough to get a brief glimpse of what it was like to see through his eyes, an intense, slightly off-center gaze. We cried a little, but the sadness paled in comparison to the wonder we felt when we thought back over our friendship, how unlikely it had been, something both of us could not have predicted and now couldn't live without. We'd been born only miles from each other, and there was a chance we'd never live in the same country again\u2014but we promised that would never get in the way.\n\nJules was moving into an affordable place on Fifth Street, rooming with a woman he'd found on Craigslist. We were giving up Eleventh Street completely, and the loss of the roof felt like an especially tough blow. For the first time, I let Jules come with me to the airport to help with my giant suitcases, help I hadn't had when I first arrived in New York.\n\n\"So?\" he said.\n\n\"Isn't this funny?\" I said. \"We've switched places. I would have never thought this would happen.\"\n\n\"I'll be back in Paris soon.\"\n\n\"Take care of the city for me.\"\n\n\"It's going to have to take care of me.\"\n\nWe kissed very hard at first, but ended with a light peck\u2014like a kiss good morning, the kiss you give when you know you're going to see that person soon. Then I was on the plane, holding my breath as it lifted off and up over the water. Only when we turned to point northeast and I could no longer see the skyline of Manhattan, did I really start to cry.\n\nThe journey is different for everyone. It had always felt as if I'd been going in one direction: up, up, and away. I'd gone to America and watched as France grew further and further behind, until suddenly it was right in front of me.\n\nThat's all I've really wanted to tell you. It's one story, looked at a dozen ways. I went across an ocean to find myself. Then I came home.\n\nI cannot tell you in this little space everything that has happened since I returned to Paris. Maybe I will another time. I have my own store and wine bar in the 17th Arrondissement, carved out of a converted carpenter's studio. I named it L'\u00c9b\u00e9niste du Vin\u2014\"the cabinetmaker of wine.\" Having grown up in the 12th Arrondissement, in Bercy, which was the center of Parisian wine trade for hundreds of years (and where, if you remember, my grandfather managed the wine trains coming in and going out of the city), I knew very little about the 17th and that side of Paris.\n\nIt was a few months after I returned to town. Maya was visiting from Aix and staying in the 17th with her cousin. \"Come have lunch with me here, it's a cute area!\" she said\u2014and who can refuse Maya? It was terrific to see her. She was as effervescent as always. We ate salad in a brasserie near the Batignolles market. Across the street, outside the window where I sat, I saw a FOR SALE sign on the industrial-looking fa\u00e7ade across the street. For a moment I could have been in New York.\n\n\"I knew you'd come back, but I didn't know when,\" she said. \"I thought it could have been another five years.\"\n\n\"How did you know I'd be back? We never really talked about it.\"\n\nShe looked at me as if I'd asked her how she'd added two and two. \"I knew your heart was here.\"\n\nAfter lunch we walked around arm in arm. We saw young professionals, families with strollers, a growing neighborhood. But there were still the signs, and sometimes the shops, of its past, too\u2014woodworking, leather making, cobbling, haberdashery. It was changing fast. I hoped there would still be room for me when I was ready.\n\nMonths passed. It felt as if I was building a nest, piece of straw by piece of straw. I worked as a rep selling Brazilian liqueur, saving what money I could. I looked for an apartment for me and Jules. It would be our first home together, just ours. My mother helped, tsk-tsking at every place we looked at. I thought she just didn't want me to move out, until we saw one place together, in the fashionable Marais, that was beautiful. The building reminded me of the West Village, with its big striking blue door and wide wood stairs going up to the apartment from the courtyard. It had an \"American kitchen,\" which in French means an open kitchen. It was January, and I felt the promise of the new year ahead of me.\n\nI kept finding straw. I furnished the apartment as affordably as I could and had time to be exacting\u2014Jules was taking longer than he thought and wouldn't be back until April. I sent him pictures. We talked every day about the time we would be together. It was what we held on to. Long distance again, there was no fighting. We texted good morning and good night, as always, only now I was the one several hours ahead. Our home was all I thought about, and how he was going to come soon and make it complete.\n\nWhen I went to pick him up at the airport, we held on to each other without letting go for twenty minutes. I took him back to the apartment he'd only seen in photos. We made love. Then, the sun bright through the window, we sat on the edge of the bed and, over the course of the afternoon and evening, until it was dark in the room and I had to stand up to turn on the light, we broke up.\n\nAt the same time that I'd been single-mindedly preparing for Jules's arrival, another feeling had been turning itself over deep in my breast, waiting to see how it might make itself heard: that we were not right for each other. I didn't know it, but a similar one had been speaking up in Jules's heart. And only when we saw each other again in the place where we were to truly start our lives together did those feelings recognize each other, and begin to talk, sadly but truthfully. That's the best explanation I can give. It will have to do.\n\nThere's no resealing a bottle of wine once it's been opened\u2014the wine is already changed, and changing. You can plug it, keep it for another day or two, but you cannot make the bottle whole again, the way it was before being opened. And that's okay! Most of the time, if you are lucky, the wine was enjoyable. Sometimes it was spectacular. But when it's gone, it's gone, and all you can do is remember it.\n\nI still see Jules once in a while around Paris, and we laugh about his eggs on rice, how stubborn he was, and how bossy I can be. The only one left in New York is Rose. We talk when we can. I picture her sometimes, chin up, walking briskly on our East Village streets, the way she did that November day we met. It makes me feel like a piece of me still has a claim on the city.\n\nIn April 2011, I walked into a real estate agency in the 17th Arrondissement, accompanied by my father, and asked them to show me some commercial spaces that afternoon. Nothing I saw felt quite right, and near the end of the day, the agent showed me yet one more old decrepit restaurant space. As I left, disappointed, I noticed an unusually wide storefront with tall windows across the street. It was a carpenter's studio. I made the real estate agent follow me inside. My father only smiled; he knew me. The space was beautiful, like a cave, the walls made of mortared stone. I knew instantly that it was my store.\n\n\"He's been here for thirty years,\" the agent said. \"It's not for sale.\"\n\n\"Why don't we ask?\" I said. The carpenter emerged from the rear, a tall, dignified looking man of about fifty, smelling of sawdust and sweat. He greeted us curiously; we didn't look like his normal customers. I asked him a little about his store and the neighborhood. He was friendly in his answers. Then I asked him point-blank if he was looking to sell his studio. His look of mild surprise deepened. \"How could you have known?\" he said. \"I've just started looking for a bigger space closer to the center of the city.\"\n\nHe wanted time, and I had plenty of that. But I had to fight resistance from everyone I knew. _Are you sure you want to do this? It sounds very hard. You have to close late. Do you really want to work until 2_ A.M. _if one day you have a family?_ Even some of my closest friends and family said to my face, \"You can't open a bar\u2014you're a woman.\"\n\nWe closed the sale on September 30, 2011. It was one of the best days of my life. I was determined to open before Christmas, which was going to require a lot of construction. Everything had to be perfect. I wanted to reuse all the old oak in the space, and pair it with clean steel. I wanted floor-to-ceiling shelves. I knew the types of tables and chairs that would be intimate and comfortable, solid feeling yet light, inviting people to linger. I wanted fresh flowers on the bar. I knew which wine glasses would be perfect\u2014and I wanted more than just your standard white and red glass shapes, but several kinds to suit each inimitable wine.\n\nAmericans love a happy ending, and so do I, so I'll say that just about everything went as I'd hoped. I opened l'\u00c9b\u00e9niste that December. The bar was all I thought about; the last thing I expected was any romance. But I met my husband, Max (this is another story), and in March 2013, we opened a seafood restaurant and oyster bar. The wine bar logo has my profile stamped on it; the restaurant, L'\u00c9cailler de l'\u00c9b\u00e9niste (the oyster seller of the cabinetmaker), has his. By the time you read this, our next restaurant, focusing on grilled meat and traditional French meals like _coq au vin, blanquette de veau, pot au feu, and cassoulet,_ will be open as well.\n\nI want to help my countrymen and -women learn about the wine that comes from their land, to understand and embrace the many changes happening around us every day without fearing the death of tradition, to not be afraid or too proud to appreciate creativity and trying new things. Chic wine bars are a dime a dozen in New York, but still a relatively new concept in Paris. And there are more places like mine now opening every day. I love it.\n\nIn French _tu_ is a casual way to address someone you know, whereas _vous_ is the more formal mode. At all of my establishments we use _tu_ with our customers; mostly they are charmed by the gesture of warmth and intimacy. (Sometimes they are put off. Not infrequently, these are the same customers who demand to talk to my general manager, a man who works for me, and ask if I am his wife.)\n\nHardly a day goes by that I don't think about my time in New York. The old ways are beautiful, time-tested; they are what make France great. The new ways will make us even better. My mother was right: I am a champagne baby, and that means that wine is my heritage, my passion. But she didn't know that one day it would be my mission to make champagne babies of us all. Without forgetting our rich history, we should always approach wine as we should approach life: with an open, willing heart, our five senses, our thirsty minds, ready for anything.\nFor my parents and my Uncle Alain\n\nI've changed names and identifying details of many people and some places in this book. It is a memoir, after all, which means it's about me. I don't need to drag others into my story more than they already have been. But all those who helped make this book possible know who they are, and I cannot thank them enough for being in my life.\n\nLAURE DUGAS was born in Paris to a family of winemakers and has worked for some of the biggest names in champagne. After three years in New York City, she moved back across the Atlantic to open a wine bar and store in the Batignolles neighborhood of Paris. L'\u00c9b\u00e9niste du Vin was such a success that a year later Laure and her husband opened an oyster bar next door to it, then opened another restaurant in early 2016. Laure Dugas lives with her husband and daughter in Paris.\n\n# _What's next on \nyour reading list?_\n\n[Discover your next \ngreat read!](http:\/\/links.penguinrandomhouse.com\/type\/prhebooklanding\/isbn\/9781101884645\/display\/1)\n\n* * *\n\nGet personalized book picks and up-to-date news about this author.\n\nSign up now.\n 1. Cover\n 2. Title Page\n 3. Copyright\n 4. Contents\n 5. Introduction\n 6. Map\n 7. Part I: Planting \/ Racine\n 1. Chapter 1: Nobody Knows Everything\n 2. Chapter 2: Youth Has Its Virtues\n 3. Chapter 3: There's More to Wine Than the Grape\n 4. Chapter 4: Taste the Present\n 5. Chapter 5: What's Your Vintage?\n 8. Part II: Cultivation \/ Culture\n 1. Chapter 6: There's Always Occasion for Champagne\n 2. Chapter 7: Even the Best Wine Can Go Bad\n 3. Chapter 8: You're Better Than the Cheapest Bottle\n 4. Chapter 9: Trust Your Palate\n 9. Part III: Harvest \/ Recolte\n 1. Chapter 10: The Older the Vines, the More Complexity\n 2. Chapter 11: Wine Is Community\n 3. Chapter 12: Wine Is Travel\n 4. Chapter 13: It Always Comes Back to Terroir\n 10. Epilogue\n 11. Dedication\n 12. Acknowledgments\n 13. About the Author\n\n 1. Cover\n 2. Cover\n 3. Title Page\n 4. Contents\n 5. Start\n\n 1. iii\n 2. iv\n 3. ix\n 4. x\n 5. xi\n 6. xiii\n 7. \n 8. \n 9. \n 10. \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 175. \n 176. \n 177. \n 178. \n 179. \n 180. \n 181. \n 182. \n 183. \n 184. \n 185. \n 186. \n 187. \n 188. \n 189. \n 190. \n 191. \n 192. \n 193. \n 194. \n 195. \n 196. \n 197. \n 198. \n 199. \n 200. \n 201. \n 202. \n 203. \n 204. \n 205. \n 206. \n 207. \n 208. \n 209. \n 210. \n 211. \n 212. \n 213. \n 214. \n 215. \n 216. \n 217. \n 218. \n 219. \n 220. \n 221. \n 222. \n 223. \n 224. \n 225. \n 226. \n 227. \n 228. \n 229. \n 230. \n 231. \n 232. \n 233. \n 234. \n 235. \n 236. \n 237. \n 238. \n 239. \n 240. \n 241. \n 242. \n 243. \n 244. \n 245. \n 246. \n 247. \n 248. \n 249. \n 250. \n 251. \n 252. \n 253. \n 254. \n 255. \n 256. \n 257. \n 258. \n 259. \n 260. \n 261. \n 262. \n 263. \n 264. \n 265. \n 266. \n 267. \n 268. \n 269. \n 270. \n 271. \n 272. \n 273. \n 274. \n 275. \n 276. \n 277. \n 278. \n 279. \n 280. \n 281. \n 282. \n 283. \n 284. \n 285. \n 286. \n 287. \n 288. \n 289. \n 290. \n 291. \n 292. \n 293. \n 294. \n 295. \n 296. \n 297. \n 298. \n 299. \n 300. \n 301. \n 302. \n 303. \n 304. \n 305. \n 306. \n 307. \n 308. \n 309. \n 310. \n 311. \n 312. \n 313. \n 314. \n 315. \n 316. \n 317. v\n 318. \n 319.\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}} +{"text":"\n\nAuthor's Note: This book is intended to provide helpful guidance and information on the subject of diet and exercise; it is not meant to be taken as medical advice or to replace the diagnostic expertise of a physician. You should aways refer any questions or concerns about your health to a trusted medical professional, particularly if you are pregnant, nursing an infant, or suffering from any medical condition or symptom. As with any diet or exercise program, stop immediately and consult a doctor if you experience pain or discomfort at any time.\n\n2009 Ballantine Books Trade Paperback Edition\n\nCopyright \u00a9 2006 by Harley Pasternak\n\nAll rights reserved.\n\nPublished in the United States by Ballantine Books, an imprint of The Random House Publishing Group, a division of Random House, Inc., New York.\n\nBALLANTINE and colophon are registered trademarks of Random House, Inc.\n\nOriginally published in hardcover in the United States by Meredith Books in 2006.\n\neISBN: 978-0-345-53506-1\n\nLibrary of Congress Control Number: 2006930029\n\nwww.ballantinebooks.com\n\nv3.1\nThis book is dedicated to all my clients. \nYou have been, and continue to be, my \nguinea pigs, my inspiration, and my friends.\n\n# ACKNOWLEDGMENTS\n\nMy parents. To whom I owe all my success.\n\nMy brothers, Jesse and Bobby. My best friends and my fountain of youth.\n\nMy manager, Kristin Giese. I treasure your leadership and strength.\n\nMy literary agent, Andrea Barzvi. The only agent I will ever have.\n\nMy cowriter, Myatt Murphy. For sharing my vision and being so professional.\n\nMy editor, Stephanie Karpinske. For polishing the words.\n\nPaola Patrella. For your delicious recipes.\n\nLogan Alexander. For your photography and humor.\n\nCarmen Bonicci. My Canadian strategist.\n\nMy commercial agent, Brittany Balbo, for your support.\n\nMy closest friends, Dave, Anne, Behzad, Sam, Rachel, Wendy, Michael, David, Jamie, Ricky, Jeff, Josh, David, Rick, John, Brian, Jen, Jodi, Vera, and Wil. For reminding me where I'm from and who I am.\n\nLucy and Viv. You are always in my heart.\n\n# Contents\n\n_Cover_\n\n_Title Page_\n\n_Copyright_\n\n_Dedication_\n\n_Acknowledgments_\n\nIntroduction\n\n1. A Fresh Start\n\n2. Fad Diets Don't Work\n\n3. Low-Carb Diets Don't Work\n\n4. The 5-Factor Diet Does Work\n\n5. 5 Meals a Day Are Key\n\n6. 5-Factor Golden Rules\n\n7. 5-Factor Must-Have Foods\n\n8. Shopping for 5-Factor Foods\n\n9. New 5-Factor Hollywood Workout\n\n10. 5-Factor Recipes\n\n11. 5-Factor Success Log\n\nRecipe Index\n\n#\n\n**T his book took me 15 years to write.**\n\nNot really 15 years of writing... more like a decade and a half of evolution.\n\nThe 5-Factor evolution started when I was a \"husky\" teenager (\"husky\" is simply a kinder way of saying overweight). I bought every diet book, fitness magazine, and exotic weight loss pill, powder, and bar! I tried weight training, step aerobics classes, Pilates, and yoga. I experimented with diets such as Pritikin, Body For Life, and the Zone. I became interested in exercise and nutrition as a way to make me look and feel better. However, it wasn't until both of my younger brothers were diagnosed with type 1 diabetes that I became interested in the science of food and how it affects our bodies. I spent eight years in university studying metabolism, biochemistry, nutrition, and physiology. I call this period my \"nerd years.\"\n\nWhile I was in graduate school, the 5-Factor evolution continued as I began to work as a nutrition scientist for the Canadian Department of National Defense. I learned a great deal about research. Not only did I perform my own nutrition studies (breaking many test tubes in the process), but I also learned how to assess existing nutrition research. Equipped with studies that could definitively support or refute popular diet info, I began to question many dietary practices. I thought more critically about claims I heard about popular diet programs and weight loss supplements.\n\nI reread all the diet books I had previously treated as gospel and underlined all of the \"facts\" the books used to support their claims. I then set out to find the research these claims were based on. To my chagrin, I realized most diets and weight loss strategies are quick-fix programs based on half-truths and flat-out fabrications.\n\nI knew there had to be a way of eating healthy that would also be realistic. I wanted a program based on real truth, real science, and real people's lifestyles, a sensible system that would promote fat loss _and_ enjoyment.\n\nMy 5-Factor evolution continued as my nutrition and fitness practice grew. I developed and refined my nutrition plan and applied it to clients with amazing results. My program garnered the attention of actors who wanted to tone up for upcoming roles. Nineteen films, nine television shows, and more than 50 actor and musician clients later, my 5-Factor Diet has helped the likes of Halle Berry, Alicia Keys, Kanye West, Mandy Moore, Eva Mendes, Rachel Weisz, Rick Fox, John Mayer, Brendan Fraser, Stephen Dorff, Robert Downey Jr., and Benjamin Bratt.\n\nIn 2005 my first book, _5-Factor Fitness_ , hit the top of two bestseller lists. Though _5-Factor Fitness_ was primarily an exercise plan, it contained a brief introduction to the 5-Factor Diet and offered a number of quick 5-Factor recipes. I received more than 5,000 emails from people who had purchased _5-Factor Fitness_ and had shed anywhere from 5 to 87 pounds! Nearly all of them requested more 5-Factor recipes and wanted to know more about the 5-Factor Diet.\n\nSo, with great pride, I present you with the last diet book you will ever purchase. Welcome to _The 5-Factor Diet_.\n\nHarley Pasternak, M. Sc.\n\n# CHAPTER 1\n\n# A Fresh \nStart\n\n**W e've met before.**\n\nI don't know your name or where you live, but it's safe to say that I do know you\u2014and I know why you're reading this book.\n\nYou're not happy with your body.\n\nIt doesn't matter whether you're looking to drop a few pounds, firm up, or improve your health so you can live a longer life; all of these goals start with eating right.\n\nMost likely you've already tried some\u2014it probably seems like _all_ \u2014of the different diets that are popular today. It's likely that you even lost some weight. But if you're like most people, you gained some or all of it back. Or worse, you regained it all plus a few extra pounds. You're sick and tired of fighting the \"fat war,\" running caloric calculations in your head and denying yourself entire food groups. You're fed up with weighing your food at every meal, scrimping on portions, and eating tasteless diet foods while salivating over the cooking shows on TV\u2014all in an effort to look as lean, fit, and glamorous as the movie stars you admire on the big screen.\n\nWell, I have some news for you. Television actors and movie stars don't make the same mistakes that you have. And with the 5-Factor Diet, you won't make those mistakes ever again. Absolutely anyone can get into better shape quickly\u2014the harder part, of course, is staying that way. 5-Factor can help.\n\n## MY HOLLYWOOD SECRETS ARE NOW YOURS\n\nWhen my first book, _5-Factor Fitness_ , became an overnight success in 2005, I was proud that the system I've been using for years with my Hollywood clients was finally available to everyone. In the book I outlined a 5-week program to jump-start readers to a better body and healthier lifestyle, focusing primarily on fitness and exercise.\n\nSince the book was published, I've been overwhelmed with requests from readers like you asking for more information on diet and nutrition. The sheer volume of letters I've received has made it clear to me that people want to know more about the nutritional plan I use with my clients\u2014particularly the 5-minute meals that are part of my program\u2014and how to incorporate it into their day-to-day lives.\n\nThe _5-Factor Diet_ book you're holding is my new nutritional bible, a complete guide to the simple yet extraordinarily effective diet program I've used for years with my celebrity clients to get them\u2014and keep them\u2014in great shape.\n\nMy 5-Factor Diet stands alone as a nutritional program that is simple yet comprehensive. More important, it actually works. Having this book on your shelf will be like having your very own round-the-clock personal nutritionist and chef on hand. It's the result of my many years of education and experience in the weight loss industry, with proven results you can see anytime one of my celebrity clients makes a movie, strolls down the red carpet, or poses for a magazine. The _5-Factor Diet_ is not just another diet book. I promise, this will be the last diet book you'll ever need.\n\n## THE 5-FACTOR DIET\n\nSo why do I call my program the 5-Factor Diet? That's easy! Or, I should say, it's to make things easier for you. Every nutritional and exercise principle I'm about to teach you breaks down into five easy-to-remember points:\n\n\u2022 The 5-Factor Diet is a 5-week diet plan.\n\n\u2022 There are 5 types of food you should eat in every meal.\n\n\u2022 The Diet incorporates a 5-phase 5-Factor Hollywood Workout.\n\n\u2022 My exercise routine, like my diet, is simple: 5 workouts a week, each consisting of five 5-minute phases.\n\n\u2022 To follow my diet, you'll want to try some of my 100-plus 5-Factor recipes, which require only 5 (or fewer) main ingredients. Each delicious recipe can be prepared in 5 minutes or less. (That doesn't count cooking time, of course; I said delicious, not miraculous!)\n\nI promise: If you can count to 5, the 5-Factor Diet will be the easiest diet you'll ever use to lose.\n\n* * *\n\n**\"I suffered from extremely poor health and almost died twice within the same year, a situation that left me too sick and unable to exercise for over three years. I had no energy and I gave up exercise altogether until I tried 5-Factor. The simplicity of the plan and, more important, seeing amazingly quick results made it easy for me to stay dedicated to the program. It's magic!\"**\n\n**Louise Meinardus AGE: 40s WEIGHT LOST SO FAR: 5 lbs.**\n\n* * *\n\n## YOU'RE READY TO START!\n\nBefore I get into detail about the simplicity and effectiveness of the 5-Factor Diet\u2014and teach you the scientific basis for it\u2014I want to talk to you about all the fad diets you may have tried in the past. Because you can't move forward without \u00adfiguring out why you've been falling backward, it's important to understand why every other diet has ultimately failed you. Turn the page to finally break yourself of the yo-yo diet cycle forever.\n\n* * *\n\nSophia Bush ** ACTRESS, STAR OF THE TV HIT _ONE TREE HILL_ **\n\n* * *\n\n**_\"To me, Harley's plan is an eating and fitness plan that makes sense, allows me to eat real food, and gives me compact, effective workouts. I actually crave a healthy lifestyle. It feels incredible!\"_ **\n\n# CHAPTER 2\n\n# Fad Diets \nDon't Work\n\n**A client came to me about 40 pounds overweight** and frustrated. Over the three preceding years, he had lost 200 pounds, desperately trying a variety of fad diets. Sounds impressive, right?\n\nIt wasn't.\n\nWhen I say he lost 200 pounds, that's counting the weight he had lost and gained back. He would drop 40 pounds, then gain 50 back. He would lose 50 pounds, then gain 60 back! It's called yo-yo dieting or weight cycling. It's not just a waste of your time; it could do your body harm.\n\nToday's fad diets are merely the latest in a long line of ineffective and often dangerous diet crazes. Over the past 20 years, Americans have been bombarded with one diet after another. Though the diets appear to be as different from each other as night and day, they all have something in common: They only work _up to a certain point_ , if they work at all.\n\nI believe that once you understand why these diets fall short, you can choose a more sensible plan and achieve the body and healthy lifestyle you've always hoped for. I don't want you to waste any more time or effort on diets that won't work. Failure is something your body simply can't afford.\n\nIn doing research for the 5-Factor Diet, I read dozens of diet books. As someone who makes a living educating people about health and fitness, I was shocked at how ridiculous many of these diets were.\n\nYou see, whenever you begin a low-calorie diet, your body notices that it's being fed fewer calories and immediately lowers your basal metabolic rate (BMR)\u2014the rate at which your body burns calories. That means the result of eating fewer calories is burning fewer calories all day long. Once you quit the diet\u2014and you will\u2014it takes a while for your body to bring your BMR back to normal.\n\nThat's why yo-yo dieters end up gaining more and more weight with each diet failure. If you go back to your old eating habits while your BMR is still low, you won't just regain the pounds you lost, you'll pack a few extra on top. Repeat this cycle a few times and you end up gaining more weight the more often you try to lose it. It's physically exhausting and emotionally frustrating.\n\nI want you to ask yourself what I consider the single most important health and fitness question: Do you want to look good tomorrow or do you want to look good for the rest of your life?\n\nYou need to think of your health and fitness goals as a marathon instead of a sprint. Most fad diets claim they can help you drop pounds fast, and I know that promise can be incredibly alluring. But that weight loss is usually the result of nutritional tactics that are not only unhealthy but also impossible to maintain.\n\nTo take pounds off for good, it's all about finishing the race. You need an effective and efficient plan of attack that lets you lose weight consistently, not just immediately. What differentiates the 5-Factor Diet from all the fad diets that I'm about to discuss is this: While all of these fad diets work for a short time, only the 5-Factor Diet will keep you lean and healthy for a lifetime.\n\n## BLOOD TYPE DIET\n\nThis diet claims it's your blood type, not just the calories you consume, that causes weight gain. According to the program, the secret to losing fat is to eat only specific foods that are compatible with your blood type. Eating the wrong foods is supposedly like receiving a transfusion of the wrong type of blood, causing substances from your food, called lectins, to enter your bloodstream. It's this flow of lectins that supposedly causes blood cells to clot, leading to a variety of health issues.\n\nThe diet's claims about blood type and weight loss are not backed up by relevant scientific research. Your blood type has nothing to do with your body's ability to burn excess fat. The plan restricts not only calories but food types as well. You're told not to eat certain healthy foods that are rich in antioxidants, vitamins, and minerals. And the diet recommends some unusual foods and supplements that are only available online.\n\n## CABBAGE SOUP DIET\n\nIt's easy to see why so many people have tried this strict, low-calorie program that has been around for decades. Its proponents claim you can drop up to 20 pounds in seven days by eating little more than cabbage-based soup several times a day. Cycling on and off the diet (7 days on, 14 days off) is said to promote rapid weight loss.\n\nThis diet can be harmful to your body because it restricts your caloric intake to less than 1,000 calories a day. It leaves you feeling perpetually hungry because you're basically forcing your body to live off nothing but fiber and water. There's no protein or fats and few vitamins or minerals. The weight loss most people see is almost always water and lean muscle mass because the lack of protein causes your body to cannibalize its own muscle tissue. You will also likely have uncomfortable side effects such as diarrhea, abdominal pain, lightheadedness, and flatulence.\n\n## GRAPEFRUIT DIET\n\nIn this popular diet, you're required to eat a whole grapefruit with every meal. Why a grapefruit? According to the diet, grapefruits contain a special fat-burning enzyme.\n\nThe negatives of this diet are identical to those of the Cabbage Soup Diet, even though the plan does allow small amounts of protein. It's not the grapefruit that deserves the credit for whatever weight is lost; this restrictive, 800-calorie diet basically starves you. No matter how much you may wish otherwise, there is simply no such thing as a superfood with magical abilities to make you lose weight.\n\n* * *\n\n**\"I tried practically every fad diet invented, but they never worked long term. The 5-Factor Diet opened my eyes to correct, healthy eating. By following Harley's plan, I was finally able to learn to change my eating habits. Diets come and go, but Harley's plan is for the rest of your life!\"**\n\n**Danielle Martin AGE: 37 WEIGHT LOST SO FAR: 77 lbs.**\n\n* * *\n\n## CAVEMAN DIET\n\nThe creators of this nutrition plan believe that cavemen and cavewomen were lean and healthy because of the all-natural foods they ate. According to this diet, processed and cultivated foods, including wheat and grains, are the true cause of all major disease and obesity. The diet requires you to return to your Neanderthal roots by eating only what your ancestors did. That means eliminating all processed foods in favor of natural foods such as fish, lean meats, berries, vegetables, fruits, nuts, and seeds.\n\nI can't argue with the premise that the less processed a food is, the healthier it is for your body. However, it is a stretch to claim that the lack of processed foods was the main reason our ancestors were leaner than we are. But they also had to spend many physically demanding hours hunting down or picking their own food.\n\nCavemen were so lean in part because they were much more physically active than we are today. Yet that factor is never considered in the caveman equation. Nor does the diet discuss the fact that our ancestors lacked convenient access to food and thus ate significantly less than we do. And there is no scientific research to date that links wheat or grains to obesity and resulting diseases\u2014yet this diet claims that corn is responsible for more cancer deaths than cigarettes. There is a more logical reason why our ancestors didn't suffer from cancer, heart disease, and other modern-day ailments: They never grew old because the average life span of a Neanderthal man was 20 years!\n\n* * *\n\nSanaa Lathan ** ACTRESS AND STAR OF THE MOVIE _LOVE AND BASKETBALL_ **\n\n* * *\n\n**_\"I was asked to lose some weight for my last film. Harley had me do his 5-Factor Diet and exercise program. Within weeks my body was transformed. Getting in shape was never this easy. And I just saw my movie. And if I may say so myself, my body looks better than it ever has on film. I'm a fan for life.\"_ **\n\n## NO SUGAR DIET\n\nThis diet eliminates foods that are high in refined sugar and carbohydrates that rank high on the glycemic index.\n\nBreak out your calculator because you'll have to make sure every meal is divided into 30 percent carbohydrates, 30 percent protein, and 40 percent fat. Not only are these calculations time-consuming, but your daily caloric consumption is limited to an unhealthy 1,200 calories. You don't lose weight because you're eating less sugar; you lose it because you're eating too few calories. The diet restricts many healthy-for-you foods, such as carrots, that contain ample amounts of essential vitamins and minerals. Instead, it claims you can lose weight while eating high-fat, low-sugar foods such as hamburger, steak, and cheese.\n\n## LIQUID DIETS\n\nThese diets make you forgo food in favor of a liquid meal replacement drink typically made from sugar, fat-free milk powder, fiber, vitamins, and minerals. On some versions of the plan you eat only shakes; on others you also have small, low-calorie meals.\n\nOne of my clients was on a shake diet for a while. Every day she drank five shakes instead of eating real food. She was miserable and depressed, especially when she went out to dinner at a five-star restaurant and had to bring a shake with her instead of eating the delicious meals.\n\nLiquid diets are antisocial, and they're not sustainable because they're not satisfying. Research has shown that liquids don't fill your stomach as effectively as solid foods. And most of these shakes are deficient in dietary fiber, so you never feel quite as full. These low-calorie diets\u2014as low as 700 calories\u2014can stress your kidneys because many liquid dieters end up dehydrated.\n\n## NEW BEVERLY HILLS DIET\n\nThis diet has you combine foods in particular ways in order to create a certain mix of enzymes that supposedly helps your body properly digest your food.\n\nAlthough combining certain types of foods can be beneficial for losing weight\u2014something I'll explain later as part of the 5-Factor Diet\u2014but it's not because of the enyme mix in food, as this diet claims. The truth is, the enzymes used to digest food are created by your body.\n\nThe theory that any food that can't be digested properly \"adds\" weight doesn't make sense either. If your body can't break down food, that means it has less chance to grab the calories and store them as body fat. Regardless of theories, this diet is also too low in protein, vitamins, and minerals to be considered healthy.\n\n## BODY FOR LIFE\n\nThis is a six-day-a-week diet and exercise plan whose creator promises that you'll be in the best shape of your life after 12 weeks.\n\nYou'll notice that Body For Life encourages you to use a lot of nutritional supplements. In fact, the program seems to be designed mainly to sell these supplements. One thing I do approve of about the Body For Life plan is that it encourages regular exercise.\n\n## HIGH-FIBER DIETS\n\nThe theory behind super-high-fiber diets is that if you overeat fibrous foods, your meals travel through your digestive system at an accelerated pace, preventing your body from absorbing all the calories.\n\nEating fiber daily offers many health benefits. But eating excessive amounts of fiber doesn't guarantee weight loss. Fiber has no absorbable calories, which simply means that high-fiber diets are lower in calories. That's the real reason you lose weight initially on these diets.\n\nHowever, eating excessive amounts of fiber can be rough on the digestive system. And it may push healthy, nutrient-rich foods out of your system with the fiber, preventing nutrients from being absorbed.\n\n* * *\n\n**\"Your book turned my life around after 15 years away from the gym had taken its toll. It was exactly what I needed to get back on track! I started using 5-Factor when I weighed 250 pounds and had a 42-inch waist. I am currently 185 pounds with a 32-inch waist. I feel amazing! Your book has given me the desire and discipline to attain a physical goal I thought was part of my past and never to be seen again!\"**\n\n**Andrew White AGE: 39 WEIGHT LOST SO FAR: 65 lbs.**\n\n* * *\n\n## ORNISH PLAN AND PRITIKIN DIET\n\nThe Ornish plan limits your protein intake to a mere 15 percent of your total daily calories. It also claims that any calories from fat cause you to get fat. The Pritikin diet forces you to limit fat consumption to less than 10 percent of your total daily calories.\n\nWith such low amounts of protein (Ornish plan) or fat (Pritikin diet), it's not likely that you'll feel full on either diet. That's why some people overeat on these plans or can't stick to the program for any great length of time.\n\n## POINT PLANS AND PREPARED MEALS\n\nSome weight loss plans limit the amount you eat by assigning you points based on your body weight and weight loss goals. The challenge with that is if you're not careful, you can gobble up all your points in one or two meals. That may leave you starving later in the day.\n\nThere are also diet plans that require you to buy packaged meals. On these plan, dieters often find themselves at a loss for what to eat when they aren't at home because they're not taught how to create their own healthy meals. Plus these plans can be pricey. Chances are, it's your bank account that will decide when you quit.\n\n# CHAPTER 3\n\n# Low-Carb \nDiets Don't \nWork\n\n**I believe that low-carbohydrate, high-protein diets**, such as the Atkins diet, the South Beach Diet, and the Zone Diet, are as unhealthy and dangerous as any fad diet. But because they have been immensely popular for the past decade, I feel they deserve a chapter of their own.\n\nSo what is a high-protein, low-carb diet? It's any diet that stresses eating lots of protein (such as meat and eggs) while severely limiting carbohydrates (such as bread, potatoes, pasta, and rice). Most low-carb diets also make you avoid fruits, vegetables, and other good-for-you foods.\n\n## WHAT YOU LOSE\u2014BESIDES WEIGHT\u2014ON THESE DIETS\n\nWhat's made low-carb diets so popular is that you do drop off pounds\u2014at least in the short term. But instead of losing fat, these are the five things you're losing on a low-carb diet.\n\n### 1. WATER\n\nAlthough low-carb, high-protein diets cause a sudden weight loss initially, much of what you're losing is water. When you starve yourself of carbs, your body is left with no choice but to use up its glycogen, which is the stored carbohydrates it keeps on reserve to fuel activity. Each gram of glycogen has 3\u00ad to 4 grams of water attached to it, so as your body uses it up, excess water is shed, and the needle on the scale starts to move downward. The problem is, as soon as you go back to eating normally, your body restocks glycogen\u2014and the excess water\u2014so the weight comes right back.\n\n### 2. MUSCLE\n\nAfter its initial water-weight loss, your body has to turn elsewhere to find calories to fuel activity. That's when it starts gobbling up any lean muscle and organ tissue it can find as a source of energy.\n\n### 3. NUTRIENTS AND FIBER\n\nMost low-carb diets limit the amount of fresh fruits and vegetables you can eat. This leaves your body severely deficient in vitamins and minerals, not to mention dietary fiber.\n\n### 4. INTEREST\n\nBecause so many foods (fruits, cereals, breads, grains, starches, baked goods, dairy products, starchy vegetables, and sweets) are eliminated or severely limited, this kind of diet is very hard to incorporate into life on a long-term basis. After a few weeks of following any low-carb regime, you'll lose interest in the diet because you're constantly feeling hungry and unsatisfied with the food you're allowed to eat.\n\n### 5. YOUR HEALTH\n\nSome low-carb diets let you eat large amounts of foods that are extremely high in saturated fats. That's why the American Heart Association warns that low-carb diets can raise your cholesterol levels and increase your risk of heart disease, stroke, and diabetes. Recent research suggests that low-carb diets may contribute to certain kinds of cancer.\n\nA low-carb diet can also put an enormous strain on your kidneys. Without carbohydrates to use for fuel, your body switches into a metabolic state called ketosis. When you're in ketosis, you get your energy from ketones\u2014a form of carbon that's created from the breakdown of fat. That sounds like exactly what you're looking for, right? Wrong! It's dangerous to your health. The more ketones you have in your system, the harder your kidneys have to work to filter them, and that can lead to kidney failure. If you already have kidney problems, the situation can be dire: A Harvard study published in the _Annals of Internal Medicine_ found that low-carb diets can cause a permanent loss of kidney function in people with reduced kidney function.\n\n## WHY LOW-CARB DIETS WON'T WORK LONG TERM\n\nMost people don't experience the negative long-term consequences of low-carb diets only because they quit the diets after just a few weeks. I'd hate to see you waste your time, so take a look at the top five reasons people stop following low-carb diets.\n\n### 1. THEY'RE FAR TOO COMPLEX\n\nHow did you fare in algebra back in high school? Adhering to a low-carb diet requires understanding your body's metabolism and calorie breakdowns, choosing the right portion sizes, and dividing up how many grams of protein, carbs, and fats are in every single food you eat. Some of the recipes in these diet books are so complex, they require more cooking skills\u2014and time\u2014than the average person has.\n\n**The 5-Factor Diet Difference:** With the 5-Factor Diet, the recipes are as tasty as those you'd find in the trendiest Hollywood restaurants, but they're still designed so that anyone\u2014no matter how limited his or her culinary skills\u2014can whip up nutritious meals and snacks with little effort. And as for math, you can count to 5, can't you? Because that's all I'll ever ask you to do.\n\n### 2. THEY TAKE UP TOO MUCH TIME\n\nBecause of their complexity, low-carb diets simply require too much time to think, organize, and implement. That's why many people give up on low-carb diets early on.\n\n**The 5-Factor Diet Difference:** My clients' time is extremely limited and incredibly valuable. So is yours. I realize that the only way to keep you eating right is to make it easier to eat right. That's why all of the recipes in the 5-Factor Diet take no more than 5 minutes of prep time before cooking.\n\n### 3. THEY AREN'T VERY SOCIAL\n\nEating a meal should be a social experience, yet low-carb diets leave many people feeling like the odd man out. You can't enjoy a meal at a restaurant with your friends when you're too busy trying to find a low-carb option on the menu and calculating protein grams. That's why most people end up cheating on these diets when they go out to eat.\n\n**The 5-Factor Diet Difference:** Most of my celebrity clients work in an industry that practically forces them to be sociable. At the same time, however, privacy is very important to them. Like you, they need a diet program they can use anywhere _without_ advertising the fact that they're dieting. The 5-Factor Diet works at home, on the road, or in any restaurant, so you'll never again have to choose between food and friends.\n\n### 4. THEY DON'T SHOW YOU HOW TO EXERCISE\n\nLow-carb diets may mention how essential exercise is for losing weight, but none of them go into detail about how you should exercise. That's like telling people a road trip will be much faster and smoother if they buy a faster engine, and then never saying where to find one!\n\n**The 5-Factor Diet Difference:** The 5-Factor Diet is one of the few nutritional programs out there that show you the right way to eat _and_ the right way to exercise.\n\n### 5. THEY SLOW YOU DOWN ON EVERY LEVEL\n\nYour brain relies on carbohydrates to help it function. Yet these low-carb diets reduce the amount of carbs you eat to a trickle of what your brain desperately needs. No wonder low-carb, high-protein dieters tend to have a tough time concentrating. They also end up suffering from fatigue, which leaves them with less energy for exercise.\n\n**The 5-Factor Diet Difference:** With the 5-Factor Diet, not only will you be free to eat carbohydrates, but you'll also learn which ones are best for your body. And while other diets may shock you when you see what you _can't_ eat, with my 5-Factor Diet, you'll be amazed at what you _can_ eat.\n\n## BEWARE OF THESE POPULAR LOW-CARB DIETS\n\nYou've just heard all the negatives that low-carb diets have in common, but each version is also controversial for a variety of its own reasons. Here are the facts you deserve to know.\n\n### ATKINS DIET\n\nThis popular low-carb diet contends that overconsumption of carbohydrates is the main reason for obesity. Bread, pastas, and potatoes are to be avoided on this plan. Therefore, the Atkins diet severely restricts how many carbohydrates you eat each day and limits your daily calories to between 1,200 and 1,800. The reason this diet is appealing to many people is that you do lose a certain amount of weight\u2014plus you can eat fatty meats, certain fried foods, high-fat dairy products, cheese, eggs, and even butter.\n\nAll the freedom the Atkins diet offers comes at a price. Because it's so anti-carb, the diet is lacking in fruits, whole grains, and fiber. Your body misses out on many important nutrients, including vitamin B, vitamin C, and other phytonutrients that boost your immune system. You may drop a few quick pounds by stripping most carbohydrates from your diet, but it's mostly water weight and muscle tissue\u2014and it may place you at risk for a series of short- and long-term health problems.\n\nYour risk of developing osteoporosis may increase because the diet lowers your calcium intake. Research published in the _American Journal of Kidney Disease_ found that healthy subjects who tried the Atkins diet experienced calcium losses that were 65 percent greater than normal.\n\nYour risk of heart disease may increase because the diet encourages people to eat fatty meats and certain cheeses, which are high in artery-blocking saturated fats.\n\nFurther, according to the National Weight Control Registry, which monitors the diets of more than 2,500 people who have maintained a 30-pound weight loss for at least a year, fewer than 1 percent of these successful dieters use a low-carb, high-protein plan that resembles the Atkins diet.\n\n### ZONE DIET\n\nThe Zone Diet is a rigid high-protein, low-carb diet. It requires you to divide every single meal you eat using a 40\/30\/30 ratio: 40 percent carbohydrate, 30 percent protein, and 30 percent fat.\n\nAccording to the Zone's creator, most people suffer from insulin imbalances that cause them to put on weight. By eating protein, carbs, and fats in the right proportions, you can correct this imbalance and drop the pounds, along with your risk of developing cardiac diseases, diabetes, depression, cancer, and even PMS.\n\nIn order to benefit from the diet, you must follow the calculations to the absolute letter. Dividing the components of every meal you eat can be incredibly complicated and takes the enjoyment out of eating, unless you find pleasure in having to pass a math exam each time you want a meal. This plan is too difficult for anyone with a life to manage, or just to put up with for very long.\n\nI'm sure you have friends who swear by the Zone. I will admit that I've seen people get excited when they lose some weight at the start of the plan. But it's the reduction of total daily calories\u20141,000 to 1,700 calories a day\u2014that's responsible for the weight loss, not the whole 40\/30\/30 breakdown. A good portion of the weight loss is water and muscle tissue\u2014two things your body can't afford to lose.\n\nThe portion sizes of the carbohydrates you're allowed to eat are so small, you'll forget you even ate them in the first place. Your body won't remember the carbs either, and as a result, you'll never achieve satiety. When you quit this diet you'll have better division skills, but don't count on having a leaner, healthier body.\n\n### SOUTH BEACH DIET\n\nThe South Beach Diet claims to be different from the Atkins diet because it's not completely anti-carbs; instead it encourages you to eat the \"right\" kinds of carbs.\n\nThe first stage of this three-stage diet requires you to stop eating potatoes, pasta, bread, candy, cookies, alcohol, ice cream, baked goods, and sugar. But giving up all of these vices at once is nearly impossible. It's ironic that the South Beach Diet starts off by saying how the Zone Diet is not the answer and the Atkins diet is too severe. Yet once you look at it carefully, you'll realize South Beach is a mix of Atkins and Zone. And the third and final stage is just the daily allowances recommended by the American Dietetic Association.\n\nSouth Beach also claims that you'll drop 8\u201313 pounds in its first two-week phase. But just like on the Atkins diet, you're losing water and not fat.\n\n# CHAPTER 4\n\n# The 5-Factor \nDiet Does \nWork\n\n**T ake a deep breath and relax.** I want you to know that you're finally about to embark on a diet plan that\u2014unlike the fad diets I've shown you don't work in the long term\u2014you can use for the rest of your life. The only side effect of the 5-Factor Diet is a healthier, fitter body. The only danger is that you may stop traffic when you walk downtown. If you're ready for those kinds of results, let's get started.\n\nMany celebrity trainers handle only the exercise portion of their clients' fitness programs. I incorporate both exercise and nutritional information. In fact, that's the core of my business as I work with thousands of people face-to-face, over the phone, and online. I work with people who live all over the world, from South America and Europe to North America, Asia, and Australia. That global perspective has shown me that people eat badly all over the world. I had to create a diet plan that would address the fitness and nutritional needs of anyone, no matter where he or she is from or what his or her nutrition habits and culture.\n\n## THE 5-FACTOR DIET IS BASED ON WELL-ESTABLISHED SCIENCE\n\nMost fad diets will base their entire eating plan on one study; other diets take one specific scientific finding and spin an entire eating plan around it. That's not very balanced, is it? Those kinds of programs can never deliver the realistic, healthy diet you need.\n\nThe 5-Factor Diet is different. Its components are based on well-established science and on many sound, time-honored studies. These aren't studies that will be refuted six months after this book is published. The 5-Factor Diet and 5-Factor Recipes are supported by unchallenged and rock-solid research that has existed for years\u2014in some cases, the studies were done long before I worked with my first client. (See \"5-Factor Golden Rules,\" for more on the science behind this diet.) And the 5-Factor Diet is easy to follow: Anyone can use it and see results\u2014no matter who you are or where you're from.\n\n## YOU'LL LEARN TO LOVE THE NUMBER 5\n\nThe 5-Factor Diet starts and ends with the number 5. In fact, everything in the 5-Factor Diet revolves around the number 5. Here's how my menu plan breaks down\u2014in 5s, of course\u2014to help you stay on track.\n\n### 5 IS THE NUMBER OF MEALS YOU'LL EAT EVERY DAY\n\nThere is no skipping meals on the 5-Factor Diet\u2014and with recipes as tasty as the ones I have developed for you, you won't want to pass up a single bite. Each day, you'll eat your typical three meals (breakfast, lunch, and dinner) plus two healthy snacks, one in midmorning and one in midafternoon.\n\nTo figure out when you should eat, start by adding up how many hours you're awake during the day\u2014from the time you get out of bed until you turn out the light at night\u2014and divide that number by 5. The resulting number is roughly the number of hours I want you to wait between meals. For example, if you get up at 7 a.m. and go to bed at 11 p.m., you're awake for 16 hours. Divide 16 by 5 and you get a little over three hours, so you should eat at 8 a.m., 11 a.m., 2 p.m., 5 p.m., and 8 p.m.\n\nAs you can already imagine, eating 5 meals a day ensures that you'll never feel hungry or deprived. (In \"5 Meals a Day Are Key,\" I'll get into more detail about the science behind why eating 5 meals is vital to seeing the best results.)\n\n### 5 IS HOW MANY ELEMENTS EACH MEAL SHOULD INCLUDE\n\nThat may sound daunting, but it's easier than it sounds. There are no particular foods I want you to eat. Rather, I want you to make sure every meal or snack you eat is a mix of five elements: protein, low- to moderate-glycemic carbohydrates, healthy fats, and fiber, along with a sugar-free beverage to wash it down. This is the heart of the 5-Factor Diet.\n\nI'll go into more detail soon, but for now just know that _you_ get to decide what foods to eat from these five categories. All I ask is that you incorporate all five on your plate at every meal. To make it effortless, the recipes in this book (see \"5-Factor Recipes,\") all meet the 5-Factor criteria. If you follow my menu, it'll be easy to stick to the 5-Factor Diet.\n\n### 5 IS THE MAXIMUM NUMBER OF STEPS, MINUTES, AND MAIN INGREDIENTS EACH RECIPE REQUIRES\n\nI know one of the hardest things about dieting is learning how to prepare what's healthy. With my 5-Factor Diet, I've removed that problem from the equation. Every one of my recipes uses 5 or fewer core ingredients, requires 5 steps or fewer to prepare, and takes only 5 minutes to prepare (not counting cook time). Now you'll always have time to watch what you eat.\n\n* * *\n\n** Ben Foster ACTOR, STAR IN THE MOVIE _X-MEN: THE LAST STAND_ **\n\n* * *\n\n**_\"Harley Pasternak has developed an extraordinary program for health and fitness. No trends, no gimmicks. Only serious results. It's user friendly, and it is the most effective way to ensure long-term health.\"_ **\n\n### 5 IS HOW MANY DAYS A WEEK YOU SHOULD EXERCISE\n\nExercise is the important factor that most diets gloss over, yet it's critical in the battle to lose weight and make your body healthy, strong, and more injury resistant. I said this in my first book, _5-Factor Fitness_ : Eating is 50 percent of the getting-fit equation, and exercise is the other 50 percent.\n\nTo get the full benefit of the 5-Factor Diet, you must exercise. That's why I've included a \"New 5-Factor Hollywood Workout\" with the 5-Factor Diet. It makes it easy to exercise five days a week, so you will always feel great while maximizing your progress.\n\n### 5 IS THE NUMBER OF FOOD TYPES YOU'LL STOCK IN YOUR KITCHEN\n\nAs part of the 5-Factor Diet, I have determined the 5 types of foods\u2014proteins, carbohydrates, condiments, snacks, and beverages\u2014that you should always have on hand. I've further broken this down into the 5-Factor Must-Have Foods, comprising are five of the best nutritional picks from each category, for 25 essential foods you'll always want to have on hand. (For details, see \"5-Factor Must-Have Foods,\".) There's no guesswork with my eating plan.\n\n## WHY THE 5-FACTOR DIET \nWILL WORK FOR YOU\n\nI call my plan the 5-Factor Diet, but truth be told, it's not as much a diet as a lifestyle. It works for a number of reasons, all of which should resonate with you, especially if you've ever failed at dieting in the past. It works because I can make the following five important promises to you.\n\n### 1. YOU'LL NEVER FEEL HUNGRY OR DEPRIVED\n\nHave you ever been on a diet that gives you a feeling of emptiness in your stomach, as if you haven't eaten for weeks? That sensation simply doesn't exist on my 5-Factor Diet for several reasons. One of the main principles behind the 5-Factor Diet is eating 5 times a day. It's hard to be hungry with 5 meals a day!\n\nEating 5 meals spaced evenly throughout the day also keeps your blood sugar (also called blood glucose) stable, which naturally keeps your appetite in check. I've found that eating frequent meals has an important effect on eating habits. My clients tell me that they always feel as if they're either just about to eat, eating, or just finishing a meal. That's good because it means they never feel desperate for food.\n\nAnother reason you won't feel hungry is that you'll combine five very important things\u2014protein, low- or moderate-glycemic carbs, fiber, healthy fats, and a sugar-free beverage\u2014in every meal. (See \"The Ideal 5-Factor Meal,\", for more details.) These five foods work with one another not just to improve your health and help you lose weight but also to leave you feeling fuller.\n\n* * *\n\nMY 5-FACTOR DIET PROMISES\n\n 1. You'll never feel hungry or deprived.\n 2. You'll enjoy a \"cheat\" day every week.\n 3. You don't have to buy supplements.\n 4. You won't spend hours in the kitchen.\n 5. You can use the 5-Factor Diet everywhere you go.\n\n* * *\n\nNever feeling hungry or deprived is exactly the condition I want for you, because that will keep you from eating more than you should. In studies, critical hospital patients who were given the option to administer their own morphine for pain actually chose to use less than what their doctors would have given them. Similarly, you may find you eat less on the 5-Factor Diet simply because it puts _you_ in control\u2014instead of dictating exactly what you can and can't have.\n\nThe 5-Factor Diet isn't based on restrictions. You won't be cutting carbs or eliminating sugar or any entire food group. In fact, half of this book is filled with recipes for delicious pizza, spaghetti, pancakes, burritos, and many other dishes that most people crave. And it's easy to fit the foods you like into the 5-Factor Diet even if you don't follow my recipes all the time.\n\n### 2. YOU'LL ENJOY A \"CHEAT\" DAY EVERY SINGLE WEEK\n\nOn the 5-Factor Diet, you're rewarded each week with the chance to eat whatever you want\u2014guilt free. Why would any diet allow you to do such a thing?\n\nYou must remember this: Living healthfully should not come at the expense of living well. I think it's awfully sad when somebody refuses to eat her own birthday cake. Or when someone visits the most delicious French restaurant in New York City but orders only a green salad because he's dieting. I would ask those people, \"Why are you alive? What good is your health if you aren't enjoying life?\" The 5-Factor program never forgets that.\n\n* * *\n\n** Common GRAMMY NOMINATED MUSICIAN, ACTOR **\n\n* * *\n\n**_\"Being trained by Harley has made me realize that physical fitness is part of the spirit, body, and mind connection. His insight and energy toward training and health have been a blessing toward improving my life.\"_ **\n\nEveryone needs a mini meal vacation, if you will. Taking off one day a week is a catharsis. It's a mental relief. It re-empowers you, so you never feel like you're in a diet prison. In fact, an occasional high-calorie day may be just what your body needs to lose weight. Researchers at the National Institutes of Health discovered that subjects who for one day ate twice as many calories as they do normally increased their metabolism by 9 percent in the 24 hours that followed. So cheating one day can help you burn calories\u2014as long as you return to the plan the next day.\n\nIf you're worried that cheating may make you want to stray the next day as well, don't fret. Most likely, the cheat day is exactly what you need to prove how well the 5-Factor Diet is working.\n\nIt's a lot like using premium fuel in your car. You don't realize how well your car runs on it until the day you decide to save a dime and pour in a few gallons of cheap fuel instead. Suddenly your car sputters and the motor doesn't seem to respond as well. It'll be the same for you when you put away the premium 5-Factor Recipes and fill up on junk.\n\nThat's one of the advantages to a cheat day. I want you to cheat so you can see how sluggish you feel when you slip back into old habits. I guarantee that after a few cheats, you'll begin to ask yourself afterward, \"Was that really worth it?\" You may find yourself craving pizza all week and then, after you have it, realize the main reason you craved it was because you weren't allowed to have it. I can tell you that after my cheat day, I feel mentally and physically different in a way that reminds me how much healthier my body feels when I follow the 5-Factor Diet.\n\nYou can pick any one day of the week to splurge, but I suggest you try to make it the same day each week so you always have something to look forward to. Of course, if you know there's a day in the week that's going to be more of a challenge foodwise\u2014perhaps you've been invited to a party or you have a work dinner\u2014then feel free to switch days.\n\nPersonally, I prefer Sunday for my cheat day because it's usually a day that's more social, and a day when I'm more likely to be around the sinful foods I've been dodging diligently all week. For me, Sunday also feels like the official end of the week. By making their cheat day Sunday, many of my clients feel as if they are building up to something that they earn through sticking to the plan.\n\n### 3. YOU DON'T NEED SUPPLEMENTS\n\nIt strikes me as odd that so many people are willing to shell out large amounts of money for flavorless pill and powder supplements to add nutrients to their diets\u2014nutrients they could be getting simply by eating the right combination of foods. Those foods would also fill them up so they wouldn't binge on the nutritionless fare that was causing them to be nutrient deprived in the first place.\n\nWhile writing this book, I looked up the word _supplement_ in the dictionary. By definition, it's \"something added to make up for a deficiency.\" If you're eating the right foods according to the 5-Factor Diet, you never have to worry about being deficient in any nutritional area, and you never have to spend a dime on supplements. The 5-Factor Diet is based on the inherent nutritional value of real foods. It is designed to give you the right amount of macro- and micronutrients that your body needs\u2014nutrients you may not be getting from your diet right now. (If you have an allergy or a religious obligation that prevents you from eating a particular food\u2014such as eggs, milk, or shellfish\u2014the 5-Factor Diet offers plenty of other options that are equally abundant in nutrients. That way, you can tailor the diet to accommodate your needs without missing out on nutrition.)\n\nIs it OK to take a multivitamin while on the 5-Factor Diet? Of course it is. In fact, I encourage you to take multivitamins, but not because the 5-Factor Diet is deficient. Certain foods\u2014especially fruits, vegetables, and meats\u2014can lose a percentage of their vitamins and minerals, depending on how they're prepared or when they're picked. A multivitamin serves as backup insurance for your body, just in case some of your foods have reduced levels of micro- and macronutrients.\n\nMost people can choose a regular over-the-counter multivitamin from the drugstore. However, women who are no longer menstruating, as well as men, should choose types that don't contain iron.\n\nThere are a few supplements that can make life easier because of their convenience. For example, if you simply can't get your hands on another protein source to include in a meal, I encourage you to use protein powders and high-protein RTD (Ready to Drink) meal replacement drinks. In fact, I put RTDs on my list of \"5-Factor Must-Have Foods\".\n\n* * *\n\n**\"I'm a baseball player with the Los Angeles Dodgers, and the 5-Factor program has changed my life. I ended up losing 40 pounds before spring training, and the level of fitness that I have achieved has helped me have the best season of my career. 5-Factor has become my off-season and in-season training program. Thanks, Harley!\"**\n\n**Casey Hoorelbeke WEIGHT LOST SO FAR: 40 lbs.**\n\n* * *\n\n### 4. YOU WON'T SPEND HOURS IN THE KITCHEN\n\nWho has time to bake a casserole? I don't, my clients don't, and I'm sure you don't either. Whether you're the head of a corporation or the head of your household, time is tight for everyone. Lack of time is a big reason why a lot of us eat badly to begin with. It often seems far easier and faster to swing by a drive-through or grab a packaged snack than it is to prepare a healthy dish from scratch.\n\nIt's a myth that cooking healthy has to take lots of time. To prove that fact, I designed all the 5-Factor Recipes in this book to be prepared in five minutes or less. How is that possible? Every recipe has a maximum of 5 key ingredients. Each recipe has 5 or fewer steps to follow. With the 5-Factor Diet, you can stop watching the clock and start watching your shrinking waistline.\n\nEven if you decide to create your own tasty dishes in addition to mine, you'll see how every food I've recommended for you to eat is easy to prepare. And because you don't count calories or portion sizes with the 5-Factor Diet, you don't waste time worrying about anything but enjoying your food.\n\n### 5. YOU CAN USE THE 5-FACTOR DIET EVERYWHERE\n\nSome of my clients are musicians, and when they're on tour they can be in a different city every night. So I needed to create a nutritional plan that they could stick to no matter where they landed the next day. That's perhaps the greatest gift the 5-Factor Diet offers anyone who wants to lose weight: It makes it easy to take your healthy habits with you outside your home and wherever you go.\n\nThe 5-Factor Diet doesn't require that you eat certain foods or certain portions. There are no strange supplements to order, no meetings to attend, no refrigerators filled with packaged meals. All you have to do in order to stick to this plan is eat five meals a day, with each meal containing a protein, a low- or moderate-glycemic carbohydrate, fiber, a healthy fat, and a sugar-free beverage.\n\nWith so few guidelines, you can see that it's easy to do the 5-Factor Diet anywhere. You can go to Jamaica and have jerk chicken with rice and peas and be eating the 5-Factor Diet. You can head to Spain and have brown rice with seafood. You can take the 5-Factor Diet with you as your own personal travel partner\u2014anywhere in the world\u2014and get results!\n\n# CHAPTER 5\n\n# 5 Meals a \nDay Are Key\n\n**R esearch has shown that eating five meals** a day rather than the traditional three (or two, for those who unwisely skip breakfast) is optimal for maintaining healthy and stable insulin levels.\n\nWhen I attended the University of Toronto years ago, my professors included Dr. David Jenkins and Dr. Thomas M. S. Wolever. Those names may not mean anything to you\u2014and if they do, then I'm proud of you\u2014but they did to me. You see, Jenkins and Wolever were two of the world's leading active glycemic index (GI) researchers. In fact, it was they who created the glycemic index\u2014the system that measures, on a scale of 0 to 100, the body's blood sugar response to carbohydrates. They also suggested eating smaller meals throughout the day\u2014\"grazing\" instead of gorging. A few years after that research was published, I was lucky, and honored, to study under both of them. They have had a profound \u00adinfluence on me, both personally and professionally. They're the reason you're holding this book.\n\n* * *\n\n**\"I wanted to lose my belly, but I was completely unmotivated to work out because most programs looked too long and complicated to start. I found 5-Factor to be well structured and extremely easy to maintain. In just eight weeks, I lost approximately 16 pounds and have managed to keep it off!\"**\n\n**Yvon Brunet AGE: 48 WEIGHT LOST SO FAR: 16 lbs.**\n\n* * *\n\n## WHY 5 MEALS A DAY WORKS\n\nIt started the day I decided to put my professors' theories to the test. At first I tried eating six meals a day, but I found it didn't feel very natural because I was used to eating breakfast, lunch, and dinner. Trying to squeeze in six meals felt like I was stuffing myself. Five meals is a lot easier and more sensible to maintain. I had my regular breakfast, lunch, and dinner and added snacks in between. As soon as I stuck with 5 meals, I felt\u2014and saw\u2014the results. From there, I researched ways of making my 5 meals even more effective at fighting fat, building muscle, and improving my overall health. The result is the 5-Factor Diet.\n\nBy following this 5-meal-a-day plan, you are changing the way you eat and the reasons why you are eating. If you eat on a schedule, rather than waiting until you're hungry and _must_ eat, you become proactive with your diet instead of reactive. _You_ are in control of what\u2014and how much\u2014you consume. Once you're in that driver's seat, you get to control how your body looks, feels, and performs.\n\n## 5-FACTOR DIET BENEFITS\n\nWhen you stick to my 5-meal-a-day 5-Factor Diet, you'll benefit from five important changes to your body that most diets simply can't offer.\n\n### 1. IT LOWERS YOUR INSULIN LEVELS\n\nEating 5 meals a day\u2014and eating the right combinations of foods\u2014can prevent your body from releasing excess insulin into its system. By eating 5 normal-size meals instead of the usual two or three big meals, you tend to eat less food at each meal. Eating less food at each meal means you naturally end up eating less sugar. As a result, less insulin is released and you store less fat.\n\nKeeping your insulin levels low all day long isn't important just for losing fat. It's also necessary if you want to avoid the dangerous medical condition hyperinsulinemia, which occurs when you have too much insulin in your blood too often. This condition is harmful to your long-term health. It also affects you daily by lowering your concentration, diminishing your memory, and causing headaches and dizziness. Sticking with the 5-meals-a-day rule of the 5-Factor Diet prevents all of the above problems, so all you have to do is eat instead of worry.\n\n### 2. IT GIVES YOU MORE ENERGY ALL DAY\n\nDespite all the other health benefits of the 5-Factor Diet, my guess is that your main goal is to lose weight. Following my system certainly will make that happen. But what really gives you an edge over other dieters is the enormous amount of energy you'll have on this plan. Eating 5 smaller meals a day keeps a nice, steady stream of calories flowing, so you feel more energized and less sluggish. Eating larger meals less often has the exact opposite effect\u2014Thanksgiving, anyone?\n\nYou also get an energy boost on the 5-Factor Diet because you're eating protein at all 5 meals. Here's why: One of protein's most important amino acids is tyrosine, which can increase your mental alertness and energy by elevating the brain chemicals dopamine and norepinephrine. By eating protein 5 times a day as opposed to two or three times a day, you release these chemicals twice as often for extra energy all day. That's a nutritional secret many fad dieters\u2014and even the general population, for the most part\u2014never take advantage of. How many times have you seen someone snack on pretzels, fruits, or other carbohydrates without eating anything else with it? When it comes to staying energized, that's a big no-no.\n\nWhat you decide to do with your newfound energy is up to you. Maybe you'll use it to exercise more effectively. Or maybe you'll use that extra burst to get more done at work or focus on a relationship. Whatever you do, I promise you'll have energy when you need it\u2014always.\n\n### 3. IT IGNITES YOUR METABOLISM\n\nDid you know that you burn more calories eating than when you're at rest? It's ironic but true. Every time you eat, your body uses up a certain amount of energy\u2014and calories\u2014digesting, absorbing, metabolizing, and storing your meal. In fact, about 5 to 15 percent of your total calories is spent on digestion alone. It's called the \"thermic effect of food\" (TEF): The more often you eat, the more often your metabolism revs up as your body processes the food. That's yet another scientific reason why there are five meals spaced throughout the day in the 5-Factor Diet.\n\nI like to think of the metabolism as a pinwheel\u2014you know, the toy that looks like a mini fan on a stick that spins when you blow air through it. Your metabolism is like a pinwheel, and you want to keep it spinning. The faster and longer you can make it spin, the more calories you burn.\n\nEach time you eat a meal, it's like blowing air on a pinwheel. If you wait too long before blowing again, the pinwheel starts to slow down. To keep your metabolism constantly spinning, you must time your meals so that just as your body begins to slow down, more food arrives to revive it. Eating 5 meals a day keeps these breezes flowing and your metabolism spinning.\n\n* * *\n\n5-FACTOR DIET BENEFITS\n\n 1. It lowers your insulin levels.\n 2. It gives you more energy all day.\n 3. It ignites your metabolism.\n 4. It improves your mood.\n 5. It reduces stress.\n\n* * *\n\nMy diet gives you even more of a TEF advantage because of the foods you eat at each meal. Protein has a TEF roughly twice as high as that of carbohydrates and fat. That's why simply raising the amount of protein you eat daily from 15 percent of your total calories (the amount most people eat) to 35 percent (the amount I want you to eat) will increase your TEF by 21 calories daily. That number may seem tiny, but remember, the effect is cumulative.\n\n### 4. IT IMPROVES YOUR MOOD\n\nHave you ever felt agitated, depressed, or irritable during the day, but you couldn't pinpoint what was causing it? It might be from eating less often than you should\u2014something eating 5 meals a day can fix.\n\nI've explained that eating less often and having bigger meals raises your insulin levels so you end up storing excess calories as fat. There's also an emotional downside to this situation. When you eat less often and have larger meals, your body not only releases insulin but also overcompensates by releasing too much insulin, just to be sure it's doing its job. The result is that your body removes more blood sugar than necessary, causing a net deficit in your body's supply of glucose. Having less energy leaves you feeling less happy and more miserable, no matter how happy you normally are. Eating 5 smaller meals a day can prevent this and improve your mood\u2014unless you have a good reason to be angry or upset!\n\nOn the 5-Factor Diet, you're also protecting yourself from mood swings by eating a low- to moderate-glycemic carbohydrate at every meal. Research performed at the Massachusetts Institute of Technology found that eating less than 50 grams of carbohydrates daily can cause a significant drop in the chemical serotonin, which your brain releases to help regulate mood and appetite.\n\nWhen your serotonin level dips, you're more susceptible to feeling depressed and anxious. Getting enough serotonin on a regular basis raises how much of the chemical your body produces. Eating 5 meals a day\u2014with carbs at each meal\u2014keeps your levels steady so you never encounter the kind of emotional highs and lows you may have felt on other diets.\n\nThe 5-Factor Diet and 5-Factor Recipes also incorporate plenty of foods that are rich in folate\u2014a mineral that helps lower homocysteine, an amino acid that's been shown to cause depression at high levels\u2014as well as healthy fats and essential fatty acids, which have been shown to help naturally treat depression.\n\n* * *\n\n** John Mayer GRAMMY-WINNING SINGER AND SONGWRITER **\n\n* * *\n\n**_\"5-Factor is not a diet, in the sense that there's nothing to fall off of. There's nothing to say good-bye to, and nothing to long for. It is almost too good to be true!\"_ **\n\n### 5. IT REDUCES STRESS\n\nEating is important for a reason most people don't quite understand: It's a time to relax and put your life on pause for a moment. A meal is a time for reflection for many people\u2014or at least it should be. It's a time to rest and think.\n\nI want you to look at each meal as \"your\" time. No matter how stressful your day is, or how angry your boss is making you, I want you to use your five mealtimes to simply pause and ponder, even if it's only for a few minutes. Taking a break isn't just healthy for your mind; it's also beneficial to your body. Taking the time to make a meal, then sit down and eat it, forces you to do something that you might not do otherwise during the day.\n\nAs a workaholic, I made a New Year's resolution a few years ago to find a balance in life. I wanted to speak to my parents more often. I wanted to read more. I wanted to spend more time focusing on me instead of all the work that was always piling up around me.\n\nI used my 5 meals a day as a no-excuse way to live up to that promise. They gave me five opportunities in the day to reach out and say hello to my mother or read a chapter in a book. They helped bring balance to my day. I felt calmer and less stressed by the time each meal was over.\n\nStudies have shown that the more stressful your life is, the higher your odds of being overweight. A study performed at the New York Academy of Sciences found that most women who face chronic stress suffer from a condition called stress overeating, caused by the hormone cortisol, which your body releases when under stress. Not only is cortisol toxic to your immune system, but it stimulates appetite, which may be why the study's subjects overate during stressful times.\n\nA study from Yale University found that women dealing with stress typically may develop excess fat around their waistlines and surrounding their organs. The study theorized that there are more cortisol-sensitive receptors within fat cells in your belly than any other areas of your body. That means stressing out about your belly could keep you from losing it\u2014if you don't find the time to unwind, that is.\n\nExercising regularly and adhering to a healthy diet can lower your stress and help keep your cortisol levels low. Of course, those are both things you'll be doing naturally when you follow my 5-Factor Diet. Taking the time to reflect with each meal can help curb your daily stress even further, helping you keep off the fat.\n\n# CHAPTER 6\n\n# 5-Factor \nGolden \nRules\n\n**W hen I was a graduate student doing nutrition research** for Canada's Department of National Defense, I learned all about the biochemistry of food and its effects on the body. What always amazed me was that amid all the science, there were 5 clear-as-day factors that were universally ideal for losing weight, maintaining lean muscle, and improving overall health. It is these 5 rules that became the scientific basis for my 5-Factor Diet.\n\n## THE SCIENCE BEHIND THE 5-FACTOR DIET\n\n**Scientific factor #1.** Protein is the building block of the most important parts of our bodies, from muscles, hormones, and enzymes to skin, organs, and blood.\n\n**Scientific factor #2.** All carbohydrates are not created equal, as proven by the glycemic index, which is the system that measures on a scale of 0 to 100 the body's blood sugar response to carbohydrates. It's healthier to avoid foods ranked high on the glycemic index and eat low-glycemic carbohydrates instead.\n\n**Scientific factor #3.** Fiber is vital to the body, as proven by overwhelming research: It has the power to lower everything from bad cholesterol and blood pressure to the risk of certain types of cancers. It also helps keep your digestive system regular.\n\n**Scientific factor #4.** Not all fats are evil. In fact, healthy fats are an important part of a good diet. Studies show that our hormones, nerves, reproductive system, skin, and hundreds of other parts of the body rely on fat to function properly\u2014yet our society desperately tries to remove every last bit of fat from our foods.\n\n**Scientific fact #5.** Water is essential to life. Unfortunately, many people use thirst as an excuse to consume sugar and excessive calories.\n\nUsing these scientific and nutritional facts, I created the 5-Factor Diet, which\u2014unlike fad diets\u2014is guaranteed to stand the test of time.\n\n## THE IDEAL 5-FACTOR MEAL\n\nMy diet works because it combines the right 5 types of foods\u2014protein, carbohydrates, fiber, healthy fats, and beverages\u2014in each meal. Each of these 5 Factors is critical to your nutritional success. It's simple: At every meal eat one food from each of the five categories. It's a program that's nutritionally sound and easy to use for the rest of your life.\n\n* * *\n\n5-FACTOR FOODS FOR EVERY MEAL\n\n 1. Protein\n 2. Low- to moderate-GI carbohydrates\n 3. Fiber\n 4. Healthy fats\n 5. Sugar-free beverages\n\n* * *\n\nThink of every 5-Factor Diet meal as a shopping trip to a five-story department store, which you can't leave until you've purchased something from all five floors. That's the simplicity of the 5-Factor Diet, only instead of a store, it's your plate. Instead of having to buy something from every floor, you must eat something from the five 5-Factor food categories. It's the easiest and simplest way to move toward a leaner, healthier body. Here is a closer look at each food category, with all the details on why it's important and information on proper portion size.\n\n### 1. PROTEIN\n\nEvery meal or snack should contain a low-fat protein such as chicken breast, fish, seafood, egg whites, or cottage cheese. Aim for one-third of your total calories to come from protein, which is vital for maintaining muscle tissue and regulating metabolism.\n\nProtein is No. 1 on this list for several good reasons. First, it helps you feel fuller longer. In recent studies, subjects who ate high-protein, moderate-carbohydrate meals (which is exactly what's recommended in the 5-Factor Diet and the breakdown of every 5-Factor Recipe you'll find in this book) had a greater feeling of fullness after meals that lasted longer during the day than did subjects who ate high-fat meals. That's because there's a certain amount of fat found in animal-based protein like chicken or fish. That may sound like a step in the wrong direction if you want to lose fat, but staying fuller for a longer period of time can curb your hunger in between meals.\n\n* * *\n\nEAT THE 5-FACTOR DIET ANYWHERE!\n\n**You don't have to stray from the 5-Factor Diet when dining out. Here are a few of my favorite combinations to order at different-style restaurants.**\n\n**IF YOU'RE DINING...** | **ORDER...** \n---|--- \n**American** | **Turkey burger** \n**Canadian** | **Grilled ostrich and a bowl of lentil soup** \n**Chinese** | **Black bean and shrimp stir-fry** \n**Cuban** | **Fish soup or grilled chicken with black beans** \n**Greek** | **Chicken shish kabob or Greek salad** \n**Indian** | **Tandoori chicken with basmati rice** \n**Italian** | **Branzino, minestrone soup, or tomato-basil salad** \n**Jamaican** | **Jerk chicken breasts or rice and beans** \n**Japanese** | **Seaweed salad, miso soup, sashimi, or teriyaki chicken** \n**Mexican** | **Chicken fajitas**\n\n* * *\n\nBecause protein helps you maintain muscle, it also helps raise your resting metabolism. It takes more calories to maintain muscle than fat, so the more muscle you have, the more calories you burn throughout the day. Eating plenty of protein and following the 5-Factor Hollywood Workout will help you build more muscle, thus revving up your metabolism.\n\nThe best perk about protein is that out of the three macronutrients\u2014protein, carbohydrates, and fat\u2014it's the most difficult to store as body fat. When you eat more fat than your body needs, your body stores it as fat. When you eat more simple carbohydrates than your body needs, blood sugar levels spike. This causes your body to release excess insulin, which helps speed up the conversions of carb calories into fat.\n\nHowever, when you eat more protein, your body doesn't require as much insulin to metabolize it. Having less insulin in your system lowers your odds of having any excess calories converted to fat. Plus your body has to convert the protein into carbohydrates before it can be converted into fat. All that takes a lot of work, which is why most excess protein leaves the body before it has a chance to become an extra pant size.\n\nThe one type of protein I don't recommend is nuts. Some nutritionists sing their praises because they are low in carbohydrates, but most nuts, in general, receive greater than three-fourths of their calories from fat. Although nuts are often considered a major protein source, in truth many of them contain only small amounts of low-quality protein that is incomplete (lacking one or more essential amino acids) or is not bio-available (that is, the body can't use it).\n\nInstead of nuts, pick lower-fat, more-complete protein sources such as egg whites, fish, lean beef, chicken breast, turkey breast, and fat-free milk. You'll maximize your intake of quality protein while minimizing your intake of bad fats.\n\n### 2. CARBOHYDRATES\n\nEvery meal should contain a carbohydrate that ranks low or moderate on the glycemic index. Good choices include vegetables, wild rice, beans, lentils, oatmeal, sweet potato, and quinoa.\n\nCarbohydrates have taken a lot of flak lately, thanks to poorly conceived fad diets. The truth is that carbs are responsible for fueling your body and providing most of the energy you need to live. That's why every meal you eat should have at least two portions (that's 50 percent of your total calories) of some type of low- to moderate-glycemic carbohydrate.\n\n* * *\n\n**\"I had always been interested in fitness, but it wasn't until reading your book that I finally corrected all my mistakes. I was overloading on carbohydrates, eating three meals a day, and didn't know the right balance of protein and carbs. Thanks to 5-Factor, I learned about eating the right types of foods and the benefits of having 5 small meals a day.\"**\n\n**Michael Bigman WEIGHT LOST SO FAR: 8 lbs.**\n\n* * *\n\nWhy am I not anti-carb like other nutrition experts? Because eating a mixture of fibrous carbohydrates and protein keeps you sharp. You see, carbohydrates are absorbed into the system much faster than protein is, so eating a mixture of protein and the right carbs increases your alertness by burning calories at staggered times. That gives you a feeling of satiety and an even release of energy throughout the day. That's energy your body can use to exercise later on. Carbs also help the fat in your diet be more efficiently metabolized. Basically, fat burns in a carbohydrate flame. Most low-GI carbohydrates also contain some soluble fiber (see \"Fiber,\"), which is also important.\n\nI've mentioned the glycemic index (GI), which is a system that rates carbohydrate foods based on how quickly your body converts them into glucose. Foods that break down rapidly\u2014such as starchy foods\u2014release glucose quickly into your blood and rank higher on the index. Foods that break down slowly\u2014such as spinach and cabbage\u2014slowly release glucose into your blood, so they rank lower on the index.\n\nThe problem with high-glycemic food is that when its sugar enters your blood, your pancreas immediately has to produce insulin to help regulate it. Your body's natural response to extra insulin in your system is to store whatever calories it can find\u2014whether from carbs, protein, or fat\u2014as unwanted body fat.\n\nLow- to moderate-glycemic carbs release glucose at a much slower pace, so your pancreas produces less insulin. Less insulin means less body fat\u2014need I say more? That's what makes low- to moderate-glycemic foods such a critical part of the 5-Factor Diet.\n\nTry to choose carbs with a glycemic level under 80. These foods can give your body enough all-day energy without causing an insulin surge that may store excess body fat. I prefer fruits and vegetables because they're nutrient rich, low in calories, and water based, which means they're packed with water that fills your stomach. Good picks that are low to moderate on the GI scale include apples, black beans, broccoli, cabbage, carrots, celery, cherries, chickpeas, cucumbers, grapefruits, green peas, lentils, lettuce, lima beans, mushrooms, onions, pears, peaches, peppers, plums, oatmeal, oranges, snow peas, spinach, strawberries, sweet potatoes, and wild rice.\n\n### 3. FIBER\n\nEvery meal should contain 5 to 10 grams of fiber. The health benefits of fiber are numerous: It reduces your risk of developing diabetes and some cancers and lowers your overall blood cholesterol. Fiber slows down the release of glucose (again, the substance your body uses for energy) into the bloodstream, preventing your body from burning through its energy stores too quickly. Fiber even increases how quickly your meals pass into your stomach. The faster you can move food through your digestive system, the less fat and calories you'll absorb. But most important, fiber leaves you feeling full, so you end up eating less at every meal.\n\nFiber comes in two forms: soluble and insoluble. Both are valuable assets, though, for entirely different reasons. Soluble fiber\u2014found in foods such as peas, oat bran, seeds, beans, barley, lentils, and apples\u2014is digestible and helps lower your risk of developing heart disease and high cholesterol. Insoluble fiber\u2014found in wheat bran, whole grains, vegetables, and beans\u2014is not digested or absorbed by your body but passes through instead, which helps improve the health of your digestive system and colon. Insoluble fiber can also help you drop a few extra pounds. A USDA study found that eating 36 grams of fiber each day can prevent your body from absorbing 130 calories a day.\n\n* * *\n\n** Kanye West SINGER\/SONGWRITER **\n\n* * *\n\n**_\"The 5-Factor Diet saved me on tour. I can't believe there is healthy food that tastes this good. I've never been in better shape!\"_ **\n\nYou should eat at least 20 to 30 grams of fiber each day. You can have even more than that\u2014if you can handle it\u2014but do make sure you're getting at least the bare minimum by eating 5 grams at each meal. Over your five meals, you'll ensure you're getting at least 25 grams daily. Ideally, I'd like you to eat 10 grams of fiber at breakfast, lunch, and dinner and 5 grams per snack, which would place you right around 40 grams of fiber a day.\n\nThat might sound like a lot, but simply throwing a few handfuls of fiber-rich beans (about 1\u20442 cup) into a meal adds around 8 grams of fiber. Some of my favorite fiber-rich foods include whole-grain cereal, brown or wild rice, beans and lentils, no-flour wheat breads, and whole veggies and fruits that have edible skins or seeds.\n\n### 4. HEALTHY FATS\n\nIf your meal contains any fat, it should always be a healthy one\u2014either monounsaturated or polyunsaturated. If you believe it's better to avoid eating fats altogether, think again. Your body needs it\u2014even if your No. 1 mission is to lose body fat. Fat is a major source of energy and helps the body absorb vitamins A, D, E, and K. It also provides taste and consistency, and it helps you feel full so you eat less. Research has even shown that having too little fat in the diet can cause clinical depression. That's because to function properly, your brain needs a certain amount of fat, especially the kind containing omega-3 and omega-6 fatty acids.\n\nBesides, I really don't need to remind you to eat fat because it's almost impossible to avoid. But when you're going to eat a food that contains fat or is cooked in fat, you should stick to the healthy kind\u2014or \"good fats,\" as nutritionists like to call them.\n\n* * *\n\nGOOD FATS\/BAD FATS: SIMPLE SUBSTITUTIONS\n\n**Replacing bad fats with good fats doesn't have to be difficult. Here are five ways to do it that your taste buds won't notice but your body will appreciate.**\n\n 1. Switch your cooking oil to grapeseed, canola, or extra-virgin olive oil. All three work well under extreme heat.\n 2. Toss a tiny amount of flaxseed meal on your veggies instead of butter or margarine.\n 3. If a recipe calls for vegetable shortening, substitute half as much virgin olive oil and a dash of salt.\n 4. Skip packaged snacks like potato chips and eat seeds instead.\n 5. Instead of using butter or margarine on your food, try extra-virgin olive oil or flaxseed oil mixed with a dash of salt.\n\n* * *\n\n**Good fats.** Monounsaturated fats are good fats because they don't increase your total cholesterol. In fact, they lower your LDL (bad cholesterol) while simultaneously increasing your HDL (good cholesterol). Monounsaturated fats are found in foods such as fish oil, peanut oil, olive oil, and canola oil.\n\nPolyunsaturated fats have the same positive effect, and they're found in a variety of foods, such as fattier fish like mackerel, albacore tuna, rainbow trout, herring, salmon, and sardines, as well as sunflower oil, canola oil, and flaxseed.\n\nBoth monounsaturated and polyunsaturated fats may be \"healthy,\" but they are still fat, and eaten in abundance, they will make you fat. To avoid that, limit your fats to 65 grams a day (or 100 grams maximum).\n\n**Bad fats.** Saturated fats raise your total blood cholesterol and LDL (bad cholesterol). These fats are hard to avoid; if you can't avoid them, eat them sparingly. Saturated fats are found mostly in animal products such as meat, poultry skin, whole milk, butter, milk chocolate, and egg yolks, as well as in coconut oil, palm oil, and palm kernel oil.\n\nTrans fats, or hydrogenated fats, have the same bad effect on your cholesterol. These are synthetic fats created to give a long shelf life to certain foods. You'll find them in processed foods, commercially prepared baked goods, stick butter, margarine, vegetable shortening, and every bad food you've ever seen made with the last two\u2014including french fries and microwave popcorn. I want you to eliminate trans fats from your diet.\n\n### 5. SUGAR-FREE BEVERAGES\n\nEvery meal should be accompanied by a sugar-free beverage such as water, sugar-free soda, tea, coffee, or an unsweetened energy drink. Your goal is to drink 8 to 12 ounces of a healthy beverage with every meal and snack.\n\nHydration is important for several reasons. First, for every ounce of excess liquid you drink with your meal, that's one ounce of real estate you steal away from food. More liquid in your belly leaves you feeling fuller and lessens your appetite for your next meal\u2014and throughout the day.\n\nSecond, you'll burn more calories all day long. Most people are dehydrated and don't even know it. That's because by the time your thirst mechanism kicks in, your body has already lost about 4 to 5 percent of its water. This condition\u2014called chronic mild dehydration\u2014can affect every biochemical function in your body, including digestion. When your body is well hydrated, it can digest your food with less effort, so even less of it gets stored as body fat. Keeping your digestive system running well also helps it absorb more nutrients as it processes your food.\n\nThird, being properly hydrated may prevent you from eating as much during your next meal or snack. Often people eat because they think they're hungry when they are actually thirsty. That's because thirst triggers the same physical responses as hunger. The next time you feel the urge to eat, try satisfying that urge with a sugar-free beverage instead.\n\nDrinking 8 to 12 ounces at each of your 5 meals guarantees that you'll drink between 40 and 60 ounces a day. But it's not enough to only drink at meals. I recommend that you drink a total of 10 to 12 glasses (roughly 96 ounces), spread throughout the day. If drinking straight water doesn't sound enticing, mix in a very small amount of fruit juice for flavor. Also opt for ice-cold water when possible. Ice-cold water forces your body to burn calories to heat the water up to your body temperature. The effect may be slight, but every little bit helps!\n\n# CHAPTER 7\n\n# 5-Factor \nMust-Have \nFoods\n\n**A lthough my 5-Factor Diet** neither prohibits nor advocates any one food or food group, I have scouted out foods that are ideally suited to the 5-Factor Diet. These foods will keep your diet varied, wholesome, and delicious. I call them the 5-Factor Must-Have Foods. If you keep your fridge and pantry stocked with at least a week's worth of these foods, you'll find that following the 5-Factor Diet is easy and convenient.\n\nThere are 5 categories of foods you should always have at the ready: proteins, carbohydrates, sugar-free beverages, snacks, and condiments. I've also selected the 5 best choices for each category. These 25 foods are the building blocks for many of the recipes in this book. (See \"5-Factor Recipes,\".) The beauty of the 5-Factor Must-Have Foods is that you can use your imagination and \u00adcreativity, combining them to make your own quick and healthy meals that match the 5-Factor Diet formula.\n\n## THE 25 ESSENTIAL 5-FACTOR FOODS\n\nOver the years I've figured out what works and what doesn't when it comes to diet. While there are no shortcuts in the pursuit of better nutrition and health, it is possible to keep your palate satisfied and your body in shape\u2014and these 25 foods will help you do just that.\n\n### PROTEINS\n\n**1. Egg whites.** Egg whites are often called the perfect source of protein because your body uses 100 percent of the nutrients they contain. They're also free of saturated fats, excess carbohydrates, and cholesterol, which many high-protein foods are laden with. But the main reason they rank high among my 5-Factor Must-Have Foods is the countless ways you can cook them.\n\nEgg whites are easier than ever to use when cooking, especially since you no longer have to separate them from the yolk yourself. Many grocery stores sell cartons of already-separated liquid egg whites. They're not only convenient but also pasteurized, so they last longer in the refrigerator and carry less risk of food poisoning.\n\n* * *\n\n5-FACTOR MUST-HAVE FOOD CATEGORIES\n\n 1. Proteins\n 2. Carbohydrates\n 3. Beverages\n 4. Snacks\n 5. Condiments\n\n* * *\n\n**My top picks:** Eggology 100% Egg Whites and Egg Beaters Egg Whites\n\n**2. Poultry.** I have only two rules when it comes to eating poultry: Choose white meat (which is leaner than dark meat) and remove the skin. Poultry is one of the few foods that you can easily strip fat from, so take full advantage of that perk. And remember that chicken isn't your only choice of poultry. Many people don't think of turkey except around the holidays, but it too is low in fat, high in protein, and loaded with ample amounts of zinc, iron, potassium phosphorus, and B vitamins.\n\nTo keep your taste buds interested, buy poultry in a variety of forms: Whole breasts are terrific, but so is ground poultry breast or sliced deli-style poultry (as long as it's all-natural and not heavily processed). Keep in mind that the packaged ground chicken and turkey you see in the grocery store often includes the skin so it can be as high in fat as ground beef. For that reason, it's best to have someone at the meat counter grind a skinless chicken or turkey breast for you.\n\n**My top picks:** I don't have a favorite brand, but I recommend that you become friendly with your local butcher. That way, you can specifically ask the butcher for the freshest and best cuts of poultry.\n\n* * *\n\n** Eva Mendes ACTRESS AND STAR OF THE MOVIE _HITCH_ **\n\n* * *\n\n**_\"Harley has changed my life. Not only do I feel better than ever, but I now can have guilt-free pizza anytime, and that has made me a happy girl.\"_ **\n\n**3. Seafood.** Seafood should be a staple in every kitchen. Why? It's very low in fat and is packed with protein. Fish also contains healthy omega-3 fatty acids, which research has shown can improve the overall health of your heart, joints, and immune system. Better yet, seafood can have a mood-elevating effect on the brain by boosting levels of dopamine and serotonin, two neurotransmitters that naturally help prevent depression. The only downside of seafood is that some fish, specifically tuna, swordfish, and mackeral, may contain high levels of mercury. It's best to limit your intake of these to twice a week, except for light tuna, which you can have up to three times a week (see below).\n\n**My top picks:** Salmon, cod, tuna, scallops, shrimp, lobster, squid, and crab. When buying canned tuna, pick the less-expensive versions (chunk light or flaked), which have less than half of the mercury of the more expensive white albacore variety. I love the convenience of StarKist Tuna Creations, which comes in a pouch instead of a can; it's easy to transport and easy to open, there's no water to drain, and it comes preflavored.\n\n**4. Dairy.** Dairy has unfairly gotten a bad reputation because of its fat content, but it's an excellent source of protein and bone-strengthening calcium. Dairy also helps quell your appetite, according to researchers from the University of California at Davis. They found that study participants who ate meals containing dairy products had a 20 percent increase in an appetite-suppressing hormone called cholecystokinin.\n\nRemember that dairy refers to more than just milk. It also comprises hard and soft cheeses (including cottage cheese and cream cheese), yogurt (plain with no sugar added), and sour cream. Always choose fat-free versions of these foods.\n\n**My top picks:** Fat-free cheese slices, cottage cheese, and yogurt. When it comes to yogurt, stick to plain, non-fat, Greek-style yogurt from companies like Fage, Oikos, and Siggi's.\n\n**5. Game meats.** While I was on a movie set a few years ago, I made a chili dinner for a few of the actors I was training. At the end of the meal, they all raved that it was the most delicious chili they'd ever eaten. It was only then that I revealed that the main ingredient was ground bison. They were stunned that its taste and texture were the same as those of regular beef\u2014and thrilled that the fat content was about half that of beef.\n\nMeats like ostrich, bison, elk, caribou, and venison may sound too exotic to eat, but _game_ is actually a relative term. Go to the Far East and some of the more common meats are frog and turtle. Go to Eastern Europe and the Caribbean and it's not unusual to eat ox. So be daring and try game meats, which are often leaner than red meat, very high in protein, and high in iron. Adding them to your diet will make a big difference in your life because they let you enjoy the same tastes and textures as those of traditional beef and fat-laden steaks\u2014without all the nutritional negatives.\n\nMost health food and grocery stores stock game meats in their frozen section. Or look online for dealers that specialize in game meats.\n\nThere are two important things to remember when choosing game. One, make sure to read the nutritional label because certain cuts are leaner than others. Two, because game has less fat than regular beef, it's easy to overcook. Shave a few minutes off your usual cook time, or you could turn that bison steak into shoe leather.\n\n**My top pick:** Intermountain Ostrich Cooperative ostrich burgers, Blackwing bison, Blackwing alligator, and Exotic Meats kangaroo loin fillet.\n\n### CARBS\n\n**1. Beans.** Beans are mathematically a perfect food. Not only are they a low-glycemic carbohydrate with a small amount of healthy fats, but they're also high in protein and belly-filling fiber. In fact, one serving of beans (about \u00bd cup) provides close to 8 grams of fiber, which will leave you feeling more satiated\u2014and less likely to overeat.\n\nWith so many varieties to choose from\u2014black, red, kidney, pink, garbanzo, and many more\u2014anyone can find a bean he or she likes. Beans are also perfect as a topping; sprinkle a handful on salads, chili, and soups to add extra fiber and protein to any meal.\n\n**My top picks:** Most of the brands on the market are good, so pick whatever suits your taste buds. Studies have shown that canned beans have the same nutrient profile as fresh beans, so feel free to choose fresh, dried, frozen, or canned depending on what best fits your lifestyle.\n\n**2. Grains.** Packed with fiber, grains are terrific because they fill you up and make a great companion to any protein. All forms of grain\u2014including oatmeal, oats, lentils, barley, and brown rice\u2014are good choices. One of my all-time favorites, however, is quinoa (pronounced \"keen-wa\"). This supergrain isn't a common staple\u2014in fact, it can be difficult to find if your local grocery store doesn't have a large health food section\u2014but it's loaded with about 50 percent more protein than most grains, and it's rich in calcium, iron, and the essential B vitamins.\n\n**My top picks:** Kashi Golean cereal, Kashi 7 Whole Grain Pilaf, Quaker Weight Control instant oatmeal, Amy's organic lentil soup, and Eden Organic quinoa.\n\n* * *\n\nCARBS\n\n 1. Beans\n 2. Grains\n 3. Breads\n 4. Vegetables\n 5. Fruit\n\n* * *\n\n**3. Breads.** I know what you're thinking: Bread is heavily processed, low in nutrients, and loaded with bad carbohydrates, so why is this a 5-Factor Must-Have Food? The problem with bread isn't bread itself but the ingredients that it's made from. I recommend that you avoid flour if possible. Luckily, there are several bread products such as tortillas, crackers, and flat bread that are made without flour. These products are made from sprouted grains that are not refined as much as flour. They're easy to spot because most brands will have the term _no-flour_ or _flourless_ right in the name of the product. These items may be located in the health food section of your supermarket. If your local stores don't carry no-flour breads, a second-best option is to choose breads made with whole grains.\n\n**My top picks:** Fitness Bread by Mestemacher, Food for Life Ezekiel 4:19 Organic Sprouted Flourless Whole Grain Tortillas, Food for Life Ezekiel 4:19 Sprouted Whole Grain Flourless Cinnamon Raisin Bread\n\n**4. Vegetables.** Low-calorie, low-glycemic, high in nutrients, and often packed with fiber, vegetables contain disease-fighting antioxidants (vitamins A and C) and potassium, which helps keep your muscles healthy. In short, you simply can't lead a healthy lifestyle if vegetables aren't a regular part of it. You can buy them fresh or frozen and eat them steamed, stir-fried, pureed, or grilled. Just remember that a healthy veggie quickly becomes unhealthy if it's batter-dipped or slathered with high-fat cheese sauce.\n\nWhich ones should you eat? Steer clear of avocados, olives, potatoes, and beets, which contain too much fat, carbohydrates, or sugars. All other vegetables are fair game. I prefer to buy mine frozen so I can stock up on all my favorites\u2014and they will keep for months, unlike fresh veggies.\n\n**My top picks:** Frozen mixed vegetables from Cascadian Farm or Westpac\n\n* * *\n\n**5-FACTOR DIET'S 5 FAVORITE VEGGIES**\n\n 1. Broccoli: Just \u00bd cup of this superfood provides 66 percent of the Recommended Daily Allowance (RDA) of vitamin C and 10 percent of the RDA for vitamin A. It's also rich in potassium and fiber, which helps you feel fuller longer.\n 2. Butternut squash: This tasty veggie is more healthy than most people realize, with more than 80 percent of your RDA for vitamin A, 20 percent of the RDA for vitamin C, and almost 3 grams of fiber per \u00bd-cup serving.\n 3. Cauliflower: Filled with vitamin A and belly-filling fiber, just \u00bd cup of cauliflower provides more than 33 percent of the RDA for vitamin C.\n 4. Spinach: It has everything: fiber, vitamins C and E, calcium, and folic acid, which is a vitamin that helps your body create healthy new cells.\n 5. Sweet potato: Loaded with vitamin C, \u00bd of a sweet potato yields close to 85 percent of the RDA for vitamin A.\n\n* * *\n\n**5. Fruit.** USDA research suggests that people who eat more fruit tend to have a lower body mass index (BMI) and lower total body weight than those who eat less fruit. Fruits are fat free and packed with fiber, vitamins, minerals, and antioxidants. Plus they let you enjoy sweet flavors without the empty calories of most sugary foods.\n\nNot all fruits are equal. Some types\u2014such as bananas\u2014are higher on the glycemic index, which means they cause blood sugar levels to surge, thus triggering your body to store body fat. And you don't want that! Don't worry. Here's an easy way to remember which fruits rank low on the glycemic index and are therefore your best choices: The next time you pick up a piece of fruit, ask yourself these three questions. If you answer yes to at least one of them, it's a smart fruit choice:\n\n**Does it have edible skin?** \n(Think of apples, pears, plums, and peaches.)\n\n**Does it have edible seeds?** \n(Think of pomegranates, blackberries, strawberries, and raspberries.)\n\n**Is it a citrus fruit?** \n(Citrus fruits include grapefruit, oranges, and tangerines.)\n\nThe only exception to this rule is grapes, which do have an edible skin but are not a good fruit choice due to their high dextrose levels.\n\n**My top picks:** Fresh fruit is best, but it's smart to keep a backup fruit handy in your freezer. Two of my favorites are Wyman's Quick-Frozen Mixed Fruit and Dole Mixed Berries.\n\n### BEVERAGES\n\n**1. Water.** There's no better beverage than plain, unsweetened water. However, plain, flat water can get boring. To keep things interesting, I tell my clients to buy water in as many different forms as possible. Try sodium-free seltzer and, if you want to spice it up a bit and you're out for a night at a restaurant, sparkling water. Between all the bubbles and the fizz, sparkling water really helps cleanse the palate and adds a different texture.\n\n**My top picks:** Bottled water (e.g. Dasani, Evian) and sparkling water (Pellegrino, Perrier)\n\n**2. Coffee.** Coffee may sound like an odd choice for my 5-Factor Must-Have Foods, but there's a very important reason I include it. When I was a scientist for the Defense and Civil Institute of Environmental Medicine in Canada, I ran and published scientific studies on the effects of caffeine on exercise. Research has shown that drinking a caffeinated beverage 30 to 90 minutes before exercise can boost your endurance and increase the rate at which your body burns fat. Just keep an eye on how much coffee you drink\u2014I would limit it to no more than three cups a day.\n\n**My top pick:** Although I don't have a favorite brand, I prefer espresso beverages, such as cappucino and macchiatto. They typically have less than half the caffeine content of regular drip coffee and significantly more taste. The addition of nonfat milk to these beverages adds protein and calcium to your diet. My daily wake-up is usually nonfat espresso macchiatto.\n\nIf you do choose to drink regular coffee, keep a close watch on what you put in your cup. Ordinary plain coffee has no calories or sugar, but if you want to sweeten it up, I suggest using fat-free dairy products and Splenda.\n\n* * *\n\n**\"I had a life-threatening illness and realized that to get through it, I needed to live a much healthier life. Because of its simplicity, 5-Factor allowed me to start while I was in treatment. The 5-Factor plan taught me how doing too much cardio and not eating enough of the right kinds of food combinations make your body hold on to the weight. All of the 5-Factor healthy eating habits helped me beat the illness, and I'm much healthier now.\"**\n\n**Trina Jones AGE: 25 WEIGHT LOST SO FAR: 21 lbs.**\n\n* * *\n\n**3. Tea.** Caffeinated tea is another must-have beverage because it offers the same endurance and metabolic benefits as coffee. But tea also comes with its own unique set of health advantages, so stock it in your kitchen, your desk drawer, and your purse or pocket so you always have a cup when you need it.\n\nCertain teas\u2014especially those that are rich in antioxidants such as polyphenols\u2014have been shown to boost the immune system, ward off colds, soothe aches and pains, and even reduce the risk of developing cancer. One Rutgers University study found that TF-2, a component of black tea, kills colorectal cancer cells without affecting normal healthy cells in the body. The antioxidant polyphenols in some teas can even prevent heart disease. In 2003, USDA researchers found that subjects who drank five cups (there's that magic number again!) of black tea a day for three weeks lowered their LDL (bad) cholesterol by 11 percent.\n\nAs with coffee, drink no more than three cups of caffeinated tea daily. If you drink both coffee and tea, limit your daily consumption of both beverages to three cups total. Once you reach that limit, switch from caffeinated to decaffeinated teas and coffees.\n\n* * *\n\nBEVERAGES\n\n 1. Water\n 2. Coffee\n 3. Tea\n 4. Sugar-free soda\n 5. Sugar-free juices\n\n* * *\n\n**My top picks:** Black tea is terrific, but green tea also gets a lot of praise, for good reason. The polyphenols in this centuries-old beverage have been shown to fight certain cancers, ease pain, and burn calories. That's not bad for a few leaves and some water!\n\nAny herbal tea will work fine too. Herbal teas are generally a combination of different herbs\u2014not tea leaves\u2014so they may not offer the same exact health benefits of tea. However, most are still calorie-free, contain different ratios of antioxidants, and offer health benefits that include everything from easing your stomach to relieving depression. Ice tea is a great option on a hot day. Try unsweetened Nestea, as it comes in many different flavors.\n\n**4. Sugar-free sodas.** Most people enjoy an ice-cold soda, and that's entirely fine. Not all soda is bad for you. The problem with most sodas is that they are loaded with sugar\u2014some have as much as 42 grams per serving\u2014which can add 100 to 200 unwanted calories to your diet with every can or bottle. Instead, I recommend a no-calorie, Splenda-sweetened soda. That way, you'll stay hydrated and enjoy some flavor with your meal\u2014without throwing on any extra calories. As part of the 5-Factor Diet, though, I would prefer that you limit yourself to one soda a day.\n\n**My top picks:** Diet 7-Up, Diet Rite, and diet Hansen's Soda, which contains zero caffeine, no sugar, no preservatives, and no artificial flavors or coloring\n\n**5. Flavored waters.** Like most sodas, many juice drinks contain excess sugar despite the fact that their product names sound healthy. That's why I recommend steering clear of any juices that have added sugar. All that sugar means excess calories that your body doesn't need.\n\n**My top picks:** Fuze Slenderize tops my list, as it is sweetened with Splenda, is vitamin enhanced, and comes in a bunch of great flavors.\n\n### CONDIMENTS\n\n**1. Fat-free mayonnaise.** Many healthy foods\u2014such as tuna and certain vegetables\u2014can be difficult to swallow because of their blandness. That's why fat-free mayo ranks high on the 5-Factor Must-Have Foods list. It's a \"consistency\" condiment, adding texture and taste to tuna dishes, chicken salad, salmon salad, and countless other meals.\n\nIf you stay away from fat-free mayonnaise because you don't like the taste, then you obviously haven't tried it in a while. Most of the fat-free brands available today actually taste good but without all the cholesterol and high amounts of saturated fat contained in regular mayo.\n\n**My top picks:** Hellmann's Reduced Fat Mayonnaise and Kraft Fat-Free Mayo\n\n**2. Salsa.** Just because you're used to eating salsa with bad-for-you foods like nachos doesn't mean this condiment should be banned from your eating routine. A healthy mix of tomatoes, onions, and other vegetables, salsa is all-natural, incredibly low in calories (as low as 4 calories per tablespoon), and a hands-down perfect substitute for high-fat dips and spreads. Salsa also contains lycopene, an antioxidant that may help prevent cancer, and it has absolutely no fat and only trace amounts of sodium.\n\nWhat I like most about salsa is that it has a zing that gives a kick to chili, soups, salads, or any other meal. Most salsas on the grocery store shelves are low calorie and low-fat. But shop wisely. You should still read the labels because a few of them have added sugar and higher calorie counts than you would expect. Avoid these at all costs!\n\n* * *\n\nCONDIMENTS\n\n 1. Fat-free mayo\n 2. Salsa\n 3. Mustard\n 4. Fat-free sauces\n 5. Mrs. Dash seasoning\n\n* * *\n\n**My top picks:** Pace salsas and Newman's Own salsas\n\n**3. Mustard.** Mustard has three qualities that make it an ideal food: It adds consistency when mixed with other foods, it has a definite taste, and it's fat-free. (Stay away from mustards like honey mustard and Dijonnaise, which have more sugar and excess fat.) Whether you like it hot, spicy, regular, or yellow, mustard adds a sour or sweet spike, giving blander foods a bit of a kick.\n\n**My top pick:** Gulden's Spicy Brown Mustard\n\n**4. Fat-free sauces.** There are plenty of tasty sauces to choose from, but here are three that I highly recommend: soy, Worcestershire, and Tabasco. These are head and shoulders above the rest because they are practically calorie free\u2014with no fat and no sugar\u2014yet each packs a huge punch when it comes to adding tang, color, and flavor to foods. I find Worcestershire is an amazing sauce to perk up the flavor of soup as well as animal protein such as chicken or fish.\n\nI'm not too concerned about whether you use a regular or low-sodium soy or Worcestershire sauce because most of the 5-Factor Must-Have Foods are low in sodium. Choose whichever one you think flavors your foods better.\n\nIf Tabasco is too intense for you, consider this: Research has shown that hot foods can mildly increase your metabolism. At the very least, a splash of Tabasco will encourage you to drink more water and fill up your belly even faster.\n\n**My top picks:** 365 Organic Everyday Value Soy Ginger Sauce from Whole Foods Market and Lea & Perrins Worcestershire Sauce\n\n**5. Mrs. Dash.** Why do I prefer this tried-and-true, sodium-free, sugar-free spice over all the other seasonings on the market? I'm not opposed to other brands, but I love that Mrs. Dash Seasoning Blend works with almost any food, making it the most versatile product I have in my kitchen. From fish and chicken to vegetables and soups, it's a great combination of herbs that turns even an amateur cook into a great chef. If you're not sure how to season something, just throw some Mrs. Dash on it and I guarantee it will taste terrific.\n\n**My top picks:** The Mrs. Dash Original Blend is tasty, and it also comes in delicious flavors like Tomato Basil Garlic, Onion & Herb, Southwest Chipotle, and Extra Spicy. Include a few in your spice rack and you'll be ready to flavor up any meal.\n\n### SNACKS\n\n**1. Jerky.** This superlean snack never needs to be refrigerated, so you can eat and transport it anywhere. Jerky is usually made with high-quality animal protein and very little fat or carbohydrates. With all the fat stripped out, you're left with pure, muscle-building protein\u2014and no worries about the calories.\n\nBecause jerky is a cured meat, it can be high in sodium. I'm not too concerned about the sodium levels, given that most of the foods in the 5-Factor Diet are naturally low in sodium. (If you're concerned about your sodium intake, drink plenty of water each time you eat jerky.) However, you should watch out for sugar. Some brands use it to make their jerky sweet; you'll find it in barbecue-flavored jerky, for example. Stick with the regular-flavor varieties or others without excess sugar.\n\n**My top picks:** Ostrim Ostrich Meat Sticks, Pemmican Turkey Jerky, and Pioneer Turkey Jerky\n\n**2. Oatmeal.** A bowl of oats can help you maintain an even level of energy throughout the day, according to research from Penn State University. That's because oats are loaded with extra soluble fiber, which slows down the release of sugar into the bloodstream.\n\nI prefer to buy boxes of individual-serving oatmeal packets so I can take and make it anywhere with just a little hot water. Read the labels if you're shopping for flavored oatmeal; it often has added sugar, which defeats the purpose of eating it. Look for flavored versions that are either sugar free or low in sugar.\n\n**My top picks:** Quick Quaker Oats and Quaker Weight Control oatmeal, which has fiber and protein added, comes in flavors like apple-cinnamon and banana and is very low in sugar.\n\n* * *\n\nSNACKS\n\n 1. Jerky\n 2. Oatmeal\n 3. Ready to Drinks (RTD)\n 4. Veggie meats\n 5. Non-flour crackers\n\n* * *\n\n**3. RTDs (Ready to Drinks).** RTDs are meal replacement drinks that come in convenient cans and drink boxes or in a mixable powder form. Essentially RTDs are complete meals in liquid form, fortified with vitamins, minerals, and enough calories to help sustain you. They can be as filling as an average meal, thanks to a great mix of protein, carbohydrates, and fat. The only thing missing typically is fiber, which is why I tell people to consume a fruit or a fiber cracker when having an RTD.\n\nA very important point to remember is that RTDs are not a \"liquid diet.\" In \"Fad Diets Don't Work,\", I told you that liquid diets\u2014in which you typically substitute shakes for meals\u2014fail mostly because the drinks are basically water with just enough sugar to keep you barely functioning. RTDs are completely different. I think they're perfect, especially as an on-the-go snack, but certainly not as a replacement for several meals in a row.\n\n**My top picks:** Lean Body Ready-to-Drink Shakes or RTDs from Met-Rx and Myoplex\n\n**4. Veggie Meats.** Veggie dogs, veggie burgers, veggie bologna\u2014today's grocery stores sell a wide assortment of faux meat products. They're smart substitutes for the real things because they contain very little fat, are typically as high in protein as real meat, and are extremely low in carbohydrates. They are also ideal to keep on hand in your fridge because they stay fresh as long as a month, which is a lot longer than fresh chicken, beef, or fish. If you doubt that they'll satisfy your taste buds, trust me when I say that food manufacturers have finally perfected the art of turning vegetables into a food that has all the flavor of real meat.\n\nIt's worth remembering that vegetarian meat isn't always healthier than regular meat. Certain brands of veggie burgers and veggie dogs are much higher in fat than I prefer. I suggest that you stick to products that are high in protein and get less than 20 percent of their calories from fat.\n\n**My top pick:** Yves Veggie Cuisine products\n\n**5. No-flour Crackers and Brown Rice Cakes.** No-flour crackers, made with whole grains instead of flour have roughly 5 grams of healthy fiber and less than 2 grams of fat per serving. Brown rice cakes are fat-free. These low-fat snacks are what I call the perfect \"transport mechanism\" for protein. Stack some turkey or smoked salmon on top and you'll get a high-protein, low-glycemic snack with a great crunch.\n\nYou'll find no-flour crackers and brown rice cakes in the health food section of your supermarket or right next to the less-healthy, flour-packed crackers and regular rice cakes.\n\nWhen buying non-flour crackers, always check the list of ingredients on the package. You shouldn't find the word _flour_ in any form\u2014no flour, rice flour, wheat flour, rye flour, etc. Also look for the word _oil_. If you can find a brand with neither flour nor oil, you have a winner.\n\n**My top picks:** Bran-a-crisp crackers, Quaker Rice Cakes (regular size)\n\n## 5-FACTOR MUST-HAVE FOODS SHOPPING LIST\n\n**These are my celebrity clients' go-to foods. Now they're all yours. To be sure that you always have these 25 essential foods in your kitchen, copy this page and post it on your fridge. Shop so that you have at least one week's worth of each item in your home at all times.**\n\n# CHAPTER 8\n\n# Shopping \nfor 5-Factor \nFoods\n\n**O ne of the greatest challenges** of following any diet program is figuring out exactly how to incorporate it into your personal day-to-day routine. In this chapter, I'm going to show you how seamless the transition to a healthy lifestyle can be. An important part of this is rethinking your relationship with your grocery store. Being a smarter, savvier, healthier shopper just takes a little understanding about all the foods vying for your attention.\n\n## SMART WAYS TO NAVIGATE A GROCERY STORE\n\nSticking to the 5-Factor Diet requires you to make healthy choices, but that's not always easy to do when visiting the supermarket. How you shop and where you shop play a big role in whether you'll cheat on your diet. Use my 5-Factor shopping rules to ensure that every trip to the market is a healthy one.\n\n### 1. SHOP EARLY\n\nEven if you're not a morning person, make an effort to do your grocery shopping early in the day. Your body will thank you for it. Getting to the market early offers more than just avoiding the late-afternoon crowd\u2014you'll also have your pick of the freshest foods available because most markets set out their produce and fresh meats in the morning. You can pick the best cuts of meat and choicest fruits and vegetables, thereby increasing your odds of choosing foods that are still packed with nutrients. Shop later in the day and you'll be stuck with older foods that have been picked through and, most likely, have lost nutrients.\n\n### 2. GO WITH A FULL STOMACH\n\nShopping for food when you're hungry is a recipe for disaster because your body desperately craves anything to fill its void, preferably something high in sugar and fat. That's why you should shop right after your first meal (breakfast) or your second meal (midmorning snack). That way, you'll feel satiated and be less tempted to pick up foods that aren't good for you.\n\n* * *\n\n5-FACTOR GROCERY STORE RULES\n\n 1. Shop early.\n 2. Go with a full stomach.\n 3. Stick to the outside aisles.\n 4. Always have a plan.\n 5. Shop at the same store.\n\n* * *\n\n### 3. STICK TO THE OUTSIDE AISLES\n\nMost grocery stores share similar layouts, keeping their most healthy and nutritious foods around the perimeter. Do a lap around the store\u2014going in one big rectangle\u2014and you'll likely find almost all the foods on the 5-Factor Diet, including your produce, dairy, and meats. Avoid the inside aisles as much as possible; this is where you'll find most of the foods that are higher in fat and lower in nutrients. The only exception to this rule is the frozen food aisle, which is usually not on the perimeter. This is one of my favorite aisles when shopping for the 5-Factor Diet. (See \"The Frozen Food Aisle is A Dieter's Best Friend,\".)\n\n### 4. ALWAYS HAVE A PLAN\n\nNever go to the supermarket without a well-thought-out list. Without a clear plan, you're more likely to buy bad foods on impulse. You may also forget to buy enough 5-Factor Foods. Remember, you need to eat foods from all five 5-Factor categories at each meal in order to achieve the best results, so missing even one of the five will hold back your progress.\n\nI know you're busy, so simply copy the \"5-Factor Must-Have Foods Shopping Checklist\" and you'll guarantee that you always have the right 5-Factor Diet foods in your kitchen.\n\n### 5. SHOP AT THE SAME STORE\n\nOnce you find a store that carries all of the 5-Factor Diet foods, avoid frustration and shop only at that one store whenever possible. Being familiar with the layout of a store makes it much easier to get what you need quickly and avoid the aisles with bad foods. If you're regularly popping into unfamiliar supermarkets, you'll increase your risk of getting lost and walking past unhealthy goodies that may tempt you.\n\n## THE FROZEN FOOD AISLE IS A DIETER'S BEST FRIEND\n\nMost dieters shy away from the frozen food aisle\u2014and why wouldn't they? It's home to some of the most tempting, fattening foods around, from ice cream and frozen pizzas to those man-size TV dinners whose packaging has more nutrients than the actual food inside. If dieters do venture into this aisle, they probably zip past all the frozen desserts en route to the packaged\u2014and pricey\u2014diet meals put out by Healthy Choice and Weight Watchers. If this sounds like you, then you're missing out on all the great things the frozen food aisle has to offer\u2014especially when it comes to implementing the 5-Factor Diet.\n\nEvery food has some sort of nutritional value, but its protein, vitamins, and minerals have a shelf life. From the moment a food is picked, caught, or killed, a nutritional clock starts ticking. Everything that happens to the food from that point forward ages it, affecting how much nutritional value you'll get from it when it finally finds its way to your mouth. As more people handle the food, the risk of it being tainted by things like bacteria or viruses increases. Every person that touches the food causes more bruising, which can make it spoil faster. The longer a food sits in a box or on a truck, the more it deteriorates. Even exposure to sunlight can degrade some of the important nutrients.\n\nThat's why the frozen food aisle is the 5-Factor dieter's greatest ally. Here are my top five reasons why you should no longer fear the frozen food aisle.\n\n### 1. YOU GET MORE NUTRIENTS\n\nWhen you buy frozen foods such as fruits and vegetables, they are flash-frozen almost immediately after they're harvested, so fewer people handle them. They're usually sealed in packages that are impervious to light. Time essentially stops, leaving all of the foods' nutrients sealed inside.\n\nThis means that when you buy a frozen strawberry at the grocery store, it's as nutritionally fresh as the day it was picked. After a good thaw, it is young and delicious again. On the other hand, a fresh strawberry in the produce aisle has an unknown history. It may have been picked out of state, then sifted, sorted, crated, warehoused, packaged, and trucked to a distribution center before being delivered to your local grocer and put out for display. It may have sat for as long as 10 days being touched by strangers and exposed to light. By the time you eat it, that strawberry is old. In fact, the FDA and the USDA have compared many fresh fruits and vegetables against frozen versions and found the two have relatively equivalent nutrient profiles. In fact, in some cases, certain nutrient levels are higher in the frozen foods.\n\n* * *\n\nBENEFITS OF FROZEN FOOD\n\n 1. You get more nutrients.\n 2. It's convenient.\n 3. You'll save money.\n 4. It offers great variety.\n 5. You'll always have more healthy food around.\n\n* * *\n\n### 2. IT'S CONVENIENT\n\nI love raw vegetables, but sometimes a bag of frozen mixed veggies or stir-fry veggies is more convenient. Fresh vegetables require a lot of cleaning and chopping. With frozen veggies, all that hard work has been done for you. Just grab a bag of mixed vegetables and they're ready to be added to a stir-fry or made into a side dish. Frozen meats and fish are even more of a time-saver. With no cleaning involved, just thaw and you're ready to cook\u2014mess free!\n\n### 3. YOU'LL SAVE MONEY\n\nA lot of my clients think frozen foods are more expensive, but ask yourself this: How many times have you thrown away fresh chicken, fish, fruits, or vegetables because they sat around in your fridge too long? We've all done it. With frozen foods, you rarely have to throw anything away because it has spoiled; most foods stay fresh for months in the freezer.\n\n* * *\n\n**TIPS FOR BUYING FROZEN FOODS**\n\n 1. Buy vegetables raw\u2014and never with cheese sauce or butter packets!\n 2. Before buying fruit, read the label. The only ingredient should be the fruit itself, not syrup.\n 3. When you pick up meat, gently squeeze the package. If you hear or feel a crunch, it's probably freezer-burned.\n 4. Always reach into the back of the freezer, where food is kept colder.\n 5. Read the cooking instructions. Some frozen foods are precooked, which means you could be getting more sugars and bad fats than if they were raw.\n\n* * *\n\n### 4. IT OFFERS GREAT VARIETY\n\nI'm big on berries, but they aren't in season as often as I'd like. Luckily, I can always find them in the frozen food aisle, along with any other out-of-season fruits I may be craving.\n\nI recommend you keep a big bag of stir-fry vegetables in the freezer. Why? Because a lot of dieters stick with eating the same veggies over and over again. Don't get me wrong; eating vegetables is good, but different vegetables often have different amounts of vitamins and minerals. If you're eating the same one or two vegetables, you may be getting a lot of, say, vitamin A while missing out on iron or vitamin C. Eating a handful of stir-fry veggies (I like to steam them) gives you an assortment every time, so you're guaranteed a balanced mix of nutrients.\n\n### 5. YOU'LL ALWAYS HAVE HEALTHY FOOD AROUND\n\nHaving lots of healthy food around is terrific, but it's not always practical to buy a 10-pound tray of fresh chicken breasts from the meat department! But you can find these foods in the frozen aisle, along with fruits and vegetables that come in 5- to 10-pound bags. Buying in bulk may seem like a space eater in your freezer, but it works to your advantage. It guarantees that you'll always have healthy foods to eat (and less space to stock unhealthy options). By keeping the freezer well stocked, you'll never run out and be left reaching for something off the 5-Factor Diet when you're hungry.\n\n## PICKING THE BEST SWEETENERS\n\nWhen you shop for healthy foods, you'll come across many low- or reduced-calorie foods. They may sound good, but it's important to know exactly what natural or artificial substitute you're eating in place of sugar. Some of these sweeteners may be great for adding calorie-free flavor to your food, but many come with health issues of which you should be aware.\n\n### THE SEVEN MOST COMMON SUGAR SUBSTITUTES\n\nI have to break from my use of the number 5 for a moment because we need to examine seven common sweeteners used today: sucrose, turbinado, honey, aspartame, saccharin, sucralose (Splenda), and stevia. You should make your own personal decision about which sweeteners to use, based on the facts about these products, including what the manufacturers won't tell you. Here's the truth\u2014and my recommendations\u2014about sweeteners.\n\n* * *\n\n**\"My biggest problem was finding a diet that accommodated my busy schedule. Between work, travel, and my family, I had no time to get in shape. Your diet was so simple to follow, with suitable food choices and simple substitutions. The rapid changes I experienced only increased my motivation to succeed. Thank you!\"**\n\n**David Widman, M.D. AGE: 41 WEIGHT LOST: 15 lbs. in 4 weeks**\n\n* * *\n\n**Sucrose (or sugar).** Sucrose is the most common food sweetener in the world. Extracted from sugar cane or sugar beets, it's purified and crystallized, then stripped of all vitamins, minerals, fiber, amino acids, and trace elements. It may be nutritionally worthless, but because of its taste and the quick hit of energy it provides, sugar is a hard habit to kick.\n\nUnfortunately, the fleeting burst of energy usually disappears as quickly as it came, leaving you feeling more sluggish than you did before eating. This effect is called \"reactive hypoglycemia.\" Think of it this way: Eating sugar is like accelerating your car by flooring the gas, then taking your foot off the pedal and letting the car go back to its normal speed. At a certain point, your car will slow down to the speed at which it was running before you hit the gas, then slow down even further (and eventually stop altogether). When you eat sugar, your energy levels ultimately dip below your baseline, which is why you end up craving even more sugar later.\n\nIt's a vicious cycle, and it's the reason why we obsess about sweet foods. But our consumption comes at a price. Just like highly glycemic carbohydrates, sugar causes an insulin surge that makes your body store calories as fat\u2014even if you're eating fat-free sweets. Research has shown that sugar has the same effect as other carbohydrates on blood sugar levels. Calorie for calorie, sugar raises blood glucose about the same amount as starches such as white bread and white potatoes.\n\n**Turbinado.** You may not recognize its real name, but you've seen it in the light brown packets. It's that dark brown, coarse, \"raw\" sugar that's supposed to be better for you because it's all-natural and chemical free. (White table sugar, by comparison, is processed with things like phosphoric acid, sulphur dioxide, and bleaching agents, to name just a few!) Turbinado is usually made by squeezing the juice out of crushed sugar cane, then spinning what's left after evaporation through a huge centrifuge. Because it's not chemically treated, it's supposed to be richer in vitamins and minerals.\n\nWrong! The big mistake most people make is assuming that turbinado is healthier than table sugar because it's unbleached. This assumption causes some people to use even more of it than they would regular sugar. People make a similar mistake when comparing white bread with breads that are dyed brown to seem more natural. But just because something is darker doesn't always mean it's better for you! White sugar and dark sugar may have different characteristics and tastes, but your body reacts to both in the same way, with a fat-storing insulin spike.\n\n**Honey.** I don't recommend honey. Honey producers make a lot of healthy promises, claiming honey can protect you against cancer and heart disease because it contains antioxidants and certain enzymes. The problem is, honey\u2014no matter how unrefined and all-natural its producers may say it is\u2014is still nothing more than pure sugar. To be exact, it's an invert sugar\u2014created by an enzyme in bee nectar\u2014that's an extremely dense, gelatinous form of easily absorbed sugar. At best, honey has only trace amounts of antioxidants, vitamins, and minerals. That's true whether you're talking about buckwheat honey, sunflower honey, or regular clover honey.\n\nHoney should never take the place of fruits and veggies, which are far richer in antioxidants and have much less sugar. You get far more antioxidants and nutrients from a single piece of fruit than you ever could from pouring on honey and adding excess sugar to your diet. It's the wrong approach.\n\n**Aspartame.** You may know it by other names (such as NutraSweet or Equal), but aspartame is a low-calorie sweetener that is about 200 times sweeter than sugar. However, it's not the best sweetener for use in hot drinks or cooking because it tends to lose its sweetness in high temperatures. That's why coffee or tea drinkers who use aspartame may find themselves pouring in extra packets when reaching for their morning pick-me-up.\n\nThere is a great deal of controversy over this sweetener, especially because it's made from methyl alcohol, which on its own is potentially toxic. Despite that, aspartame has been proven safe for human consumption. However, people with a rare hereditary metabolic condition called phenylketonuria (PKU) need to watch their intake of aspartame because it contains the enzyme phenylalanine, which they must avoid. That's why some product labels print \"This product contains phenylalanine\" on them. If you're pregnant, avoid aspartame because it's impossible to know if your baby has PKU.\n\nPersonally, I don't like the taste of aspartame but if you want to use it, I would prefer that you have it in small amounts only.\n\n* * *\n\n**WHAT IS \"PERCENT DAILY VALUE\"?**\n\n**Understanding Percent Daily Values\u2014which is often shortened to \"% Daily Value\" or \"%DV\"\u2014is key to deciphering any food label. These percentages tell you how a single serving of the food fits into a typical 2,000-calorie-a-day meal plan. (If you consume more or less than 2,000 calories, you need to adjust the Percent Daily Value accordingly.) At a glance, you can tell whether a food is high or low in a specific nutrient. For example, the label might tell you that a food provides 13% of your recommended daily value of carbohydrates or 35% of your fats. It's also a snap to compare the nutrients in products against each other\u2014just be sure the serving sizes match first. Sugars, protein, and trans fats don't have a Percent Daily Value, so you won't see a percentage listed for them. The FDA hasn't determined yet how much protein the average person should consume daily. As for sugar and trans fats, the FDA doesn't want you eating either, so it naturally doesn't recommend any set amount.**\n\n* * *\n\n**Saccharin.** Sweet'N Low and Sugar Twin are two brands of saccharin you're most likely familiar with. Saccharin is so popular because it can sweeten both hot and cold foods and is low calorie. Some people have had concerns about the sweetener because older studies found that rats who ingested large amounts of the sweetener were at risk for cancer. New research has found saccharin is safe in the small amounts most people use. But I'll be honest with you\u2014I'm not a big saccharin fan. In fact, following a 1977 study in which rats got bladder cancer after being fed saccharin, Canada, where I am originally from, banned it from being sold. It still is not sold there, which should say something in itself. If you choose to use saccharin in lieu of sugar, I recommend the smallest amounts possible, just to be safe.\n\n**Stevia.** Stevia is a natural dietary supplement extracted from the _Stevia rebaudiana_ plant, and it has been used for decades around the world, especially in Japan. It's about 300 times sweeter than sugar and is calorie-free. Go to any health food store and you'll see it touted as the most popular natural alternative to sugar. However, the FDA hasn't approved it for use as a sweetener. Why not? A few studies have shown that stevia may cause cancer and reproductive health problems, which is why Canada and some other countries won't allow it to be used as a sweetener. The FDA does state that when used sparingly, stevia is perfectly safe\u2014although the agency believes it could create health issues if approved as an artificial sweetener. Stevia is definitely an acquired taste; it can change the flavor of foods and beverages.\n\n**Sucralose (Splenda).** Sucralose is 600 times sweeter than sugar and the newest low-calorie sweetener on the market\u2014and is my sweetener of choice. It's basically regular table sugar that's been chlorinated, a process that tweaks it just enough so that it doesn't make your blood sugar rise. It also retains its sweetness in hot and cold foods.\n\nTo this point, there haven't been any negative findings in research on sucralose usage. In Canada, we've been using it for about 15 years.\n\n## DECIPHERING FOOD LABELS\n\nBefore I start working with clients, I give them reading material about nutrition. We talk everything through and I teach them how to cook, whether they want to or not. If they want to taste my food, they have to watch me cook it. Why? Because I want to empower them. Once they understand how their bodies work with the foods they eat, following the 5-Factor Diet is even easier. They gain a sense of confidence in the program, even when I'm not there. They can follow my advice without having to question why it works.\n\nA lot of diet books tell you what to eat and maybe a little bit about why you should eat that way. But they don't empower you to make good choices for yourself. Being able to decipher what's in every single food gives you power. You can finally look through your fridge and cupboards and understand\u2014maybe for the first time in your life\u2014what will work and what won't work for your diet.\n\n* * *\n\n**WHAT TO EXPECT ON A FOOD LABEL**\n\n**SERVING SIZE**\n\n**SERVINGS PER CONTAINER**\n\n**CALORIES**\n\n**Calories from fat**\n\n**TOTAL FAT**\n\n**Saturated fat**\n\n**Trans fat**\n\n**Polyunsaturated fat**\n\n**Monounsaturated fat**\n\n**CHOLESTEROL**\n\n**SODIUM**\n\n**TOTAL CARBOHYDRATES**\n\n**Dietary fiber**\n\n**Sugars**\n\n**Other carbohydrates**\n\n**PROTEIN**\n\n**VITAMINS AND MINERALS**\n\n**Vitamin A**\n\n**Vitamin C**\n\n**Calcium**\n\n**Iron**\n\n**Vitamin D**\n\n**Thiamin**\n\n**Riboflavin**\n\n**Niacin**\n\n**Vitamin B 6**\n\n**Phosphorus**\n\n**Magnesium**\n\n**Zinc**\n\n**LESS THAN SERIES**\n\n**LIST OF INGREDIENTS**\n\n* * *\n\n### WHAT YOU NEED TO KNOW\n\nIn this book you're already learning the science of nutrition. How you apply that knowledge starts with understanding the foods you eat. That's why knowing how to read a nutritional label is one of the most important lessons I can teach you. Here's the information you'll find on the label\u2014and what those numbers mean to you.\n\n**Serving size.** This number tells you what quantity of the food was used to determine the nutrition facts. To make it easy for you to compare it to other, similar foods, the measurements are usually standard: The label will first list the serving size in lay terms (such as 1\u20442 cup or 6 pieces), then give you the metric amount (122 grams, for example).\n\n**5-Factor Fact:** Most people eat a lot more than the recommended serving size. Try portioning out one serving size to get a better sense of how many servings you're really eating when you have that food.\n\n**Servings per container.** This number tells you the approximate number of servings the package contains.\n\n**5-Factor Fact:** This information is amazingly helpful. Some foods have very small serving sizes so that the amount of calories per serving seems low. That is, until you do the math. Multiplying the \"servings per container\" by \"calories per serving\" will give you the caloric content of the entire package.\n\n**Calories and calories from fat.** In addition to telling you how many calories you're getting per serving, the label also shows exactly how many of those calories are from fat.\n\n**5-Factor Fact:** Seeing big numbers in the \"calories from fat\" section shouldn't always scare you. Good-for-you fats such as olive oil get _all_ of their calories from fat.\n\n* * *\n\n** Alicia Keys GRAMMY-WINNING SINGER\/SONGWRITER **\n\n* * *\n\n**_\"Harley's style of working out is 100 percent my style. It doesn't take a lot of time out of your day, it's motivating, and you feel good (especially when people take notice!). The focus is not on starving yourself but on healthful living, so you don't feel like you're missing out on the foods you love. Once you get started, you get addicted to looking, feeling, and living your best.\"_ **\n\n**Total fat.** This number combines the fat grams from all four types of fats: saturated, trans, polyunsaturated, and monounsaturated.\n\n**5-Factor Fact:** The FDA suggests that you eat no more than 65 grams of fat per day.\n\n**Saturated fat.** This is the number of saturated fat grams contained in each serving.\n\n**5-Factor Fact:** Your daily maximum of this unhealthy fat is 20 grams, according to the FDA. I suggest keeping your intake of this dangerous fat even lower.\n\n**Trans fat.** This shows how many grams of trans fat are in each serving.\n\n**5-Factor Fact:** In January 2006, the FDA began requiring all food manufacturers to list trans fat on their labels. That's good news because before this it was tough to determine what foods contained this dangerous form of fat. Don't expect to find a Percent Daily Value listed for this bad-for-you fat, because your body doesn't need it. Keep your consumption of trans fat as close to zero as possible.\n\n* * *\n\n**GOOD NEWS FOR ALLERGY SUFFERERS**\n\n**Each year approximately 30,000 people in the United States require emergency room treatment and 150 die because of allergic reactions to food. Now, new food label laws may help prevent some of these problems. In January 1, 2006, the FDA began requiring that food labels clearly identify when ingredients contain protein derived from the eight major allergenic foods: milk, eggs, fish, crustacean shellfish, tree nuts, peanuts, wheat, and soybeans. If you're allergic to these foods, read the list of ingredients; you should find any troublesome ingredient listed along with the source of the food allergen.**\n\n* * *\n\n**Polyunsaturated fat.** This is the total number of polyunsaturated fat grams per serving.\n\n**5-Factor Fact:** Unlike saturated fat, polyunsaturated fat doesn't raise cholesterol levels. Rather, it actually lowers the amount of bad cholesterol lipids, called low-density lipoproteins (LDLs).\n\n**Monounsaturated fat.** This is the total number of monounsaturated fat grams in each serving.\n\n**5-Factor Fact:** Olive oil is an excellent source of monounsaturated fat, plus it makes a great base for salad dressing.\n\n**Cholesterol.** This is the total milligrams of cholesterol per serving.\n\n**5-Factor Fact:** Although your body needs cholesterol to assist with hormone production and other bodily functions, your liver manufactures cholesterol on its own. That's why you should limit your daily intake to 300 milligrams.\n\n**Sodium.** This is the total milligrams of sodium per serving.\n\n**5-Factor Fact:** I'm not overly concerned about excess sodium in the diet because only a very small percentage of the population is sodium sensitive. Sodium is relatively benign and passes out of the body fairly quickly. The FDA recommends keeping your daily intake below 2,400 milligrams. Of course, if you have high blood pressure or kidney issues, then you should monitor your sodium intake more closely.\n\n* * *\n\n**\"For me, starting a new program was less about losing a bunch of weight and more about wanting to finally tone and shape my middle-age body. My butt and thighs were beginning to make a world of their own! I didn't understand which foods were beneficial and which ones should simply be avoided, but the nutrition explanations in 5-Factor taught me what I should aim for with each and every meal.\"**\n\n**Andrea Cochran AGE: 44 WEIGHT LOST SO FAR: 7 lbs.**\n\n* * *\n\n**Total carbohydrates.** This number is the total grams of every type of carbohydrate\u2014dietary fiber, sugars, and other sources\u2014per serving.\n\n**5-Factor Fact:** The FDA suggests a daily total carb consumption of 300 grams or less. As you already know, I want you to eat only carbs with low to moderate glycemic levels. But food labels don't tell you what the carbohydrates' glycemic level is. To find out, go to the glycemic index database at www.glycemicindex.com.\n\n**Dietary fiber.** This is how many grams of fiber\u2014both soluble and insoluble\u2014are in each serving. This amount is included in the total carbohydrates measurement, but dietary fiber affects blood sugar less than other types of carbs do. That's why the American Diabetes Association suggests that if a food has 5 grams or more of fiber per serving, you can subtract this number from the carbohydrate total.\n\n**5-Factor Fact:** Fiber comes in two types\u2014soluble and insoluble\u2014but nutrition labels aren't required to list them separately. However, most manufacturers will tell you somewhere on the package how many grams of insoluble fiber their product contains.\n\n**Sugars.** This is where you'll see how many grams of sugar are in each serving. You may also see \"sugar alcohols\" or \"sugar replacers\" listed. Sugar alcohols don't affect your blood sugar levels as much as sugar does, but they have a caloric value 10 percent greater than other carbs.\n\n**5-Factor Fact:** If you see grams of sugar on the nutrition label but can't find the word _sugar_ on the list of ingredients, that's because sugar sometimes goes by different names. Check the list for names like fructose (fruit sugar), glucose (dextrose), galactose (milk sugar), lactose (a combination of glucose and galactose), and maltose (malt sugar).\n\n**Other carbohydrates.** This number\u2014which isn't on all labels\u2014is a catch-all category for any other types of carbohydrates that may be in each serving.\n\n**5-Factor Fact:** These trace carbs\u2014typically various organic acids and flavenoids\u2014don't raise your blood sugar level very much, so don't be concerned about them. Sometimes sugar alcohols are thrown into this category as well; sugar alcohols may include malitol, sorbitol, xylitol, and glycerine.\n\n**Protein.** This is how many grams of protein are in each serving.\n\n**5-Factor Fact:** Sometimes dairy protein appears on an ingredient list in the form of albumen, whey, or casein.\n\n**Vitamin and mineral percentages.** All food labels are required to list vitamin A, vitamin C, calcium, and iron content. Other nutrients\u2014such as vitamin D, thiamin, riboflavin, niacin, vitamin B6, phosphorus, magnesium, and zinc\u2014are shown only if they're added as a supplement.\n\nYou won't see how many grams or milligrams of each nutrient are in a serving. Instead you'll see what percentage of the recommended daily amount of that nutrient is contained in each serving.\n\n**Ingredients.** Finally, a label lists all the food's ingredients, arranged in descending order based on the weight of each ingredient.\n\n**5-Factor Fact:** Because of the ranking of ingredients by weight, the first few ingredients listed are typically the bulk of what's in the food. For a real eye-opener, compare the ingredients list of a processed food with its natural equivalent\u2014for instance, processed, sugary fruit drink versus 100 percent fresh-squeezed juice. The differences will startle you.\n\n* * *\n\n** Tracee Ross ACTRESS AND STAR ON THE TV SHOW _GIRLFRIENDS_ **\n\n* * *\n\n_\"Harley has taught me to love my body in a way I haven't since I was 18. I just want to run around naked with a tattoo on my ass that says, \"Body by Harley.\" Harley has taught me how to keep myself toned, lean, and strong and still have that perfect amount of womanly jiggle so I look and feel good on and off screen. Since I met Harley, I am never more than two weeks from my ideal.\"_\n\n## LEARN THE LINGO\n\n**If you see this word...** | **Then the food contains...** \n---|--- \n**LEAN** | **Less than 10 grams of fat, 4 grams of saturated fat, and 95 milligrams of cholesterol.** \n**EXTRA LEAN** | **Less than 5 grams of fat, 2 grams of saturated fat, and 95 milligrams of cholesterol.** \n**REDUCED FAT** | **25% less fat than the regular version.** \n**MORE** | **At least 10% more of a specific nutrient, compared to the regular version.** \n**GOOD SOURCE OF** | **10\u201319% of the Daily Value of a particular nutrient.** \n**HIGH IN** | **20% or more of the Daily Value recommended for that particular nutrient.** \n**LIGHT OR LITE** | **At least one-third fewer calories than the regular version of that food, or no more than half of the fat. If you see the word in reference to sodium, it means the food has at least 50% less sodium than the regular version.** \n**LOW** | **Less of a particular nutrient per serving than the regular version of that food. How much less depends on the nutrient.** \n|\n\n**If a food is \"low calorie,\" it has less than 40 calories per serving.**\n\n|\n\n**If a food is \"low fat,\" it has less than 3 grams of total fat per serving.**\n\n|\n\n**If a food is \"low in saturated fat,\" it has less than 1 gram of saturated fat per serving.**\n\n|\n\n**If a food is \"low cholesterol,\" it has less than 20 milligrams of cholesterol per serving.**\n\n|\n\n**If a food is \"low sodium,\" it has less than 140 milligrams of sodium per serving. \nIf a food is \"very low sodium,\" it has less than 35 milligrams per serving.**\n\n**FREE** | **Little or no trace of a particular nutrient per serving.** \n|\n\n**If a food is \"calorie-free,\" it has less than 5 calories per serving.**\n\n|\n\n**If a food is \"fat-free\" it has less than 0.5 gram of total fat per serving.**\n\n|\n\n**If a food is \"free of saturated fat,\" it has less than 0.5 gram of saturated fat per serving.**\n\n|\n\n**If a food is \"cholesterol-free,\" it has less than 2 milligrams of cholesterol per serving.**\n\n|\n\n**If a food is \"sodium-free,\" it has less than 5 milligrams of sodium per serving.**\n\n|\n\n**If a food is \"sugar-free,\" it has less than 0.5 gram of sugars per serving.**\n\n**% FAT-FREE** | **This designates the actual amount of a food that is not made up of fat. But don't be fooled. A product may be \"90% fat-free,\" but the other 10% might be loaded in calories.** \n**REDUCED** | **At least 25% less of a nutrient, compared to the regular product.**\n\n# CHAPTER 9\n\n# New 5-Factor \nHollywood \nWorkout\n\n**Y ou can't transform** your body through diet alone.\n\nBurning fat, shaping your muscles, feeling better, and being healthier\u2014it all starts with a smart eating plan and an equally smart exercise program. Other diets \"suggest\" exercise without giving specifics, or they prescribe a regimen that's too complex or too time-consuming for anyone with a life. The 5-Factor program is not like any other diet you've ever tried.\n\nMy first book, _5-Factor Fitness_ , focused more on exercise, while in this book I've been able to give you more nutritional information and exciting, delicious recipes to try. Still, exercise remains a major component of my program if you want the best results possible.\n\nIf you have my first book, you're in for a treat. The exercises and routines in this chapter are all new, yet equally effective, so you'll build even more lean muscle tissue and burn off even more body fat. I'll also show you how to extend the original five-week workout plan into a five-month fitness regime that will truly take your body to the next level. If you're brand-new to exercise and 5-Factor fitness, don't worry. My plan is the easiest, most effective exercise program you'll ever use.\n\n## 5-FACTOR HOLLYWOOD WORKOUT SECRETS\n\nThe 5-Factor Hollywood Workout routine, just like my 5-Factor Diet, is simple: You'll do 5 workouts a week, each 25 minutes long and broken into the following five 5-minute phases:\n\nPhase 1: 5 minutes of cardio warm-up\n\nPhase 2: 5 minutes of upper-body strength training\n\nPhase 3: 5 minutes of lower-body strength training\n\nPhase 4: 5 minutes of core training\n\nPhase 5: 5 minutes of fat-burning cardio work\n\nThat's it. If you can give me\u2014or should I say, your body\u2014125 minutes total of attention each week for a recommended five-week cycle, your results will amaze you.\n\nWith the 5-Factor Diet's 5-phase workout, I've had clients lose 5 or more pounds a month, without ever feeling like they're spending all their time working out. In fact, by tweaking the final cardio portion of the workout, you can burn off even more body fat, as I'll explain when I describe Phase 5 in detail.\n\nI'm sure you're wondering how a workout that takes so little time can be so effective. You shouldn't be surprised that I have five very good reasons!\n\n* * *\n\n5-FACTOR WORKOUT SECRETS\n\n 1. It never lets your muscles rest.\n 2. It's more intense.\n 3. It targets more muscle fibers.\n 4. It's perfectly balanced.\n 5. It makes you do more reps.\n\n* * *\n\n### 1. IT NEVER LETS YOUR MUSCLES REST\n\nThe 5-Factor Hollywood Workout uses an advanced technique called \"supersetting,\" in which you do two exercises back-to-back without resting in between. This makes the workout shorter but keeps your heart rate elevated longer, so you burn more calories.\n\n### 2. IT'S MORE INTENSE\n\nMost workout routines have you perform exercises exactly the same way every time. For instance, you may be asked to do three sets of 12 repetitions per exercise, with 60 seconds of rest between sets. The 5-Factor Workout, on the other hand, constantly changes the type of exercise, the number of repetitions, the rest time between supersets, and the resistance level of your workout. Because the workout is constantly changing, your body never gets bored so it keeps evolving, keeps burning fat, and never stops progressing.\n\n### 3. IT TARGETS MORE MUSCLE FIBERS\n\nMany workouts you see in magazines string together exotic exercises that isolate only specific, small muscle groups. The problem with that approach? To burn the most calories, you have to involve as many muscles as possible.\n\nThat's why the 5-Factor Workout targets large muscle groups, such as your chest, back, quadriceps, and hamstrings, twice a week. Smaller muscle groups, such as your biceps, triceps, and shoulders, get a workout once a week.\n\n### 4. IT'S PERFECTLY BALANCED\n\nThe muscles on the front of your body (chest, biceps, quadriceps) work in tandem with the muscles on the back of your body (back, triceps, and hamstrings). Most routines don't account for that fact, and they end up working one side of the body more than the other. With the 5-Factor Hollywood Workout, you work opposing muscle groups equally, so your body gets a balanced workout.\n\n### 5. IT MAKES YOU DO MORE REPS\n\nOther routines call for 8 to 12 repetitions of each exercise\u2014or maybe go as high as 15. The 5-Factor Hollywood Workout pushes your muscles beyond average levels of fatigue by sometimes requiring 15 to 25 reps. This technique uses more calories, so you end up burning off even more body fat.\n\n## THE 5 PHASES OF THE 5-FACTOR HOLLYWOOD WORKOUT\n\nMy workout breaks down into 5 phases, each of which lasts for 5 minutes. You'll always start with Phase 1, then move to Phase 2, then Phase 3, then Phase 4, and finish with Phase 5. From start to finish, the routine takes only 25 minutes. To give your body a chance to recover, you'll exercise five days a week and incorporate a rest day twice a week. (In this book, I've made Wednesday and Sunday rest days; feel free to choose whichever two days are best for your schedule.) To get the best results, follow my 5-week program, which builds up intensity gradually so that by week 5, your body is burning calories at its highest possible pace.\n\n### PHASE 1: 5-MINUTE CARDIO WARM-UP\n\nWarm up with 5 minutes of light cardio exercise. You can walk, cycle, stair climb, or use a cardio machine set on a low level. It doesn't matter what you do because the goal is just to get your blood flowing to warm up muscles, tendons, and joints.\n\nBegin at a low intensity. Gradually increase the intensity by speeding up the activity you're doing. By the end of the 5 minutes, I want your heart rate elevated so you're in a fat-burning zone when you start Phases 2 and 3.\n\n* * *\n\n5-FACTOR HOLLYWOOD WORKOUT\n\n**Phase 1: Cardio Warm-up**\n\n**Phase 2: Upper-Body Strength Training**\n\n**Phase 3: Lower-Body Strength Training**\n\n**Phase 4: Core Training**\n\n**Phase 5: Cardio Work**\n\n* * *\n\nAs you warm up, check your pulse by placing two fingers either on the side of your neck or on the front of your wrist just below your palm. Count the heartbeats for 10 seconds, then multiply that number by 6 to determine your pulse rate in beats per minute (BPM). By the end of your warm-up, your BPM should fall within the appropriate range below to burn fat efficiently. (If your BPM is less than suggested, up your intensity in the next workout; if it's higher than suggested, lower your intensity in the next workout.)\n\n**Age** | **Pulse** \n---|--- \n20\u201324 | 130\u2013170 \n25\u201329 | 127\u2013166 \n30\u201334 | 124\u2013162 \n35\u201339 | 120\u2013157 \n40\u201344 | 117\u2013153 \n45\u201349 | 114\u2013149 \n50\u201354 | 111\u2013145 \n55\u201359 | 107\u2013140 \n60\u201364 | 104\u2013136 \n65\u201369 | 101\u2013132 \n70\u201374 | 98\u2013128 \n75\u201379 | 94\u2013123 \n80+ | 91\u2013119\n\n### PHASES 2 AND 3: 10 MINUTES OF STRENGTH TRAINING (UPPER AND LOWER BODY)\n\nPhase 2 and Phase 3, which together work all of the upper- and lower-body muscles, are combined for a good reason\u2014to keep your heart rate elevated so you burn fat as you build muscle.\n\nFor 10 minutes, you'll do two different exercises back-to-back, resting only after completing a superset made up of both exercises. Refer to the charts below to see how many repetitions to perform (it varies by week) and how many seconds to rest between supersets. Repeat this cycle for the prescribed number of supersets.\n\nHere are the 10 exercises you'll use over the course of the week.\n\nAs you see, you'll be varying the repetitions and adding more supersets as the weeks progress. The rest time between supersets also decreases each week. The exercises themselves stay the same throughout this 5-week plan.\n\n* * *\n\n**THE ONLY EQUIPMENT YOU NEED**\n\n**To do the 5-Factor Workout, all you need is a set of dumbbells, a bench with an incline feature (if you don't have one, modify the exercises as described), and a stability ball.**\n\n**When using dumbbells, pick a weight that's heavy enough so you can just barely complete the prescribed repetitions with perfect form. For example, if an exercise calls for you to do 16 repetitions and you could have done 18, your dumbbell isn't heavy enough to work your muscles, and you're cheating yourself of results.**\n\n* * *\n\n## STRENGTH TRAINING EXERCISES \nDAY 1: MONDAY\n\n**INCLINE DUMBBELL FLYS**\n\nLie flat on your back on an incline bench with a dumbbell in each hand. Raise your arms above you so the weights come together directly above your chest, palms facing each other. Bend your elbows slightly and slowly lower your arms out to the sides until the weights are in line with your chest. Slowly sweep your arms back up until they are over your chest\u2014imagine you're hugging a wide barrel\u2014and repeat.\n\n_If you don't have an incline bench: Do this exercise while lying on a flat bench instead._\n\n**BALL WALL SQUATS**\n\nStand a few feet away from a wall with your back toward the wall. Tuck a stability ball between your back and the wall, then lean back against the ball until your entire upper body is supported by the ball and the wall. Maintaining your balance, cross your arms in front of your chest, then slowly squat down until your thighs are parallel to the floor. The ball should roll down the wall as you go. Slowly stand back up and repeat.\n\n## DAY 2: TUESDAY\n\n**REVERSE INCLINE DUMBBELL ROWS**\n\nLie facedown on an incline bench, with your chest flat against the elevated pad. Hold a dumbbell in each hand, letting your arms hang down to the floor, palms facing each other. Keeping your chest on the bench and your arms close to your torso, pull both dumbbells up to the sides of your chest. Slowly lower your arms back down and repeat.\n\n_If you don't have an incline bench: You can do the exercise one arm at a time. Stand with your right side toward a bench\u2014or bed\u2014and a dumbbell in your left hand. Rest your right hand and knee on the bench, bend forward at the waist, and let your left arm hang down toward the floor. Slowly pull the weight up to the side of your chest, then lower it. Repeat with other arm._\n\n**DUMBBELL DEADLIFTS**\n\nPosition a dumbbell on the floor along the outside of each foot, then stand tall. Bend your knees and grasp the dumbbells with your palms facing in. Keeping your head up and your back straight, slowly stand up until your legs are straight, knees unlocked. Make sure the weights stay close to your body as you stand. Slowly reverse the motion and place the dumbbells back down on the floor. Repeat.\n\n## DAY 4: THURSDAY\n\n**INCLINE DUMBBELL BICEP CURLS**\n\nLie faceup on an incline bench with a dumbbell in each hand, letting your arms hang straight down toward the floor. Your palms should face up toward the front. Keeping your upper arms stationary, slowly curl both weights up until they are in front of your chest\u2014remember to curl both weights up at the same time. Slowly lower the weights back down and repeat.\n\n_If you don't have an incline bench: Do this exercise standing up instead._\n\n**OVERHEAD DUMBBELL TRICEP EXTENSIONS**\n\nSit on a chair or an exercise bench with your back straight. Place your feet firmly on the floor and grasp a single dumbbell with both hands. Raise the weight above your head, rotating it so the top plate rests comfortably in the palms of your hands, with your thumbs around the handle. Slowly lower the weight behind your head until your forearms touch your biceps. Straighten your arms to raise the weight back over your head. Repeat.\n\n## DAY 5: FRIDAY\n\n**DUMBBELL LATERAL RAISES**\n\nStand with your arms in front of you with a dumbbell in each hand, palms facing each other. Keeping your arms straight and your wrists slightly bent, slowly raise the weights out to the sides until your arms are parallel to the floor (you'll look like the letter T). Pause for a second, then slowly lower your arms back down in front of you so the dumbbells touch each other right below your waistline. Repeat.\n\n**STEP-UPS**\n\nStand in front of an exercise bench (or a sturdy box or staircase). Let your arms hang at your sides. With your back straight, place your left foot on the bench and push yourself up onto the bench until your left leg is straight. You don't have to bring your right foot onto the bench unless you need to balance yourself. Reverse the exercise by stepping back down and placing both feet back on the floor. Repeat the exercise, using the same leg, for the number of repetitions prescribed. Then change positions to work the opposite leg, this time placing your right foot on the bench.\n\n_For added intensity, you may do this exercise while holding a dumbbell in each hand._\n\n## DAY 6: SATURDAY\n\n**BENT-OVER DUMBBELL ROWS**\n\nSit on the edge of a bench holding a dumbbell in each hand. Bend forward at the waist\u2014keeping your back flat\u2014until your back is almost parallel to the floor (your chest should come down as close to your thighs as possible). Let your arms hang straight down, with palms facing each other. Slowly draw your elbows up as high as you can, keeping your arms close to your sides. Pause, then slowly lower them back down until your arms are straight once again. Repeat.\n\n**LYING BALL HAMSTRING CURLS**\n\nLie flat on your back with your arms flat on the floor and your heels on top of a stability ball. Press your heels down onto the ball, then tighten your core muscles. Slowly raise your hips up and draw your heels\u2014and the ball\u2014toward your butt as far as you can. Pause, then roll the ball back by straightening your legs; your hips will naturally lower back to the floor as you reverse the motion. Repeat.\n\n### PHASE 4: 5 MINUTES OF CORE TRAINING\n\nPhase 4 targets all four muscle groups that make up your core. You'll do one abdominal exercise each day, but five different ones over the course of the week. Days 1\u20134 each focus on one individual muscle group plus the specific ab-toning move, and Day 5 works as many muscle groups as possible in one single exercise.\n\nHere's the plan:\n\nYou'll be doing more repetitions as the weeks go on. The rest time between sets also increases each week. The 5 core exercises\u2014just like the exercises in Phases 2 and 3\u2014will stay the same throughout the entire 5-week plan.\n\n* * *\n\n**I wanted to lose pregnancy weight, plus get in shape for health reasons. But I was getting tired of working out and never seeing results. I was never a couch potato, so I became frustrated when I wasn't seeing any change in my body. 5-Factor changed that. I started to finally see muscle definition, and I love how little time it takes to complete a workout.**\n\n**Holly Flom AGE: 37 WEIGHT LOST SO FAR: 29 lbs.**\n\n* * *\n\n## CORE EXERCISES \nDAY 1: MONDAY\n\n**BALL CRUNCHES**\n\nSit on a stability ball with your feet flat on the floor. Place your hands along the sides of your head. Keeping your feet flat on the floor, slowly lean back until your head, shoulders, and back are all touching the ball. This is the starting position. Slowly curl your shoulders and upper back up off the ball. Lower yourself back down on the ball and repeat.\n\n## DAY 2: TUESDAY\n\n**SEATED DUMBBELL SIDE BENDS**\n\nSit on a chair or bench, holding a dumbbell in your left hand, palm facing in. Rest your right hand on the top of your head and let your left arm hang straight down along your side. Keeping your left arm straight, take a breath and bend at the waist to the right as far as you comfortably can. Return to the starting position, then bend at the waist to the left. Return to the starting position and repeat the exercise for the prescribed number of repetitions. Then switch positions, placing the weight in your right hand and your left hand on top of your head, and repeat the exercise.\n\n## DAY 4: THURSDAY\n\n**REVERSE BALL CRUNCH**\n\nLie flat on the floor faceup and with your knees bent. Place a stability ball behind your knees and draw your feet toward your butt to tuck the ball in place. Extend your arms straight down at your sides, with your palms pressed flat on the floor. This is the start position. Keeping the ball tucked underneath your legs, slowly curl your knees toward your chest. Pause, lower your legs back down until the ball touches the floor, and repeat.\n\n## DAY 5: FRIDAY\n\n**BALL TWISTS**\n\nSit on a bench with your knees bent and your feet flat on the floor. Hold a stability ball with both hands and extend your arms above your chest. Keeping your arms straight, twist to the right. Bring the ball back to the front so it's directly in front of you. Then repeat the move, this time twisting to the left. Alternate right and left throughout the set.\n\n## DAY 6: SATURDAY\n\n**BALL TUCK CRUNCH**\n\nPosition yourself as if you were going to do a sit-up, but instead of keeping your feet on the floor, place them up on a stability ball. Your heels should press against the top of the ball. Keeping your arms bent behind your head, lift your hips and draw your knees toward your midsection\u2014the ball should naturally roll toward your head. Hold, then extend your legs back until they're back in the starting position. Repeat.\n\n### PHASE 5: 5 MINUTES (OR LONGER) OF CARDIO WORK\n\nFor the last phase, go back to whatever activity you were doing in Phase 1. This time, it should feel easy to work at the same high intensity you achieved at the end of Phase 1. Start exercising, bring your pulse rate back up to your target BPM, and maintain that pace for 5 minutes. If you can go longer and have the time, go for it. The longer you can exercise, the more calories you'll burn overall. Personally, I would go for no more than 10 minutes total so I'd have enough energy for the next day's workout.\n\n## SUMMARY OF THE 5-FACTOR HOLLYWOOD WORKOUT\n\n### PHASE 1\n\nFive minutes of cardio warm-up\n\n### PHASES 2 AND 3\n\nTen minutes of strength training\n\n### PHASE 4\n\nFive minutes of core training\n\n### PHASE 5\n\nFive minutes of cardio work\n\n## THE 5-MONTH 5-FACTOR CHALLENGE\n\nAfter you complete the 5-week 5-Factor program, you can repeat it for as long as you like. Its built-in variety makes it a constant challenge for your muscles, so they continue to reap the benefits with each and every cycle. If you're up for a new challenge, I've designed a 5-month plan that really keeps your body guessing\u2014and the results coming!\n\nFollow the same exercises in this chapter and do the required reps, sets, and rest intervals I've indicated in the chart. In the middle of the plan, you'll take a break from the strength training and core exercises by doing cardio for 25 minutes for all 5 workouts for a week. Are you up for my 5-Factor fitness challenge? Ready, set, go!\n\n# CHAPTER 10\n\n# 5-Factor \nRecipes\n\n**M ost diet books tell you exactly what** meals to eat and in what order. Who eats like that? I don't, and I know you don't either.\n\nAt the end of this book, I'll make suggestions for arranging my 120 5-Factor recipes into a weeklong plan. But whether you eat them in the recommended order is entirely up to you.\n\nI want you to be creative. That's what makes the 5-Factor Diet so effective\u2014and easy to stick to. Use these menus in whichever order keeps you coming back for more. When I work with Kanye West, for instance, he likes to eat the same thing for breakfast on most days: an egg white omelet with a little lean shredded beef and a bowl of berries. His favorite breakfast helps him stick to eating healthy.\n\nAlicia Keys, on the other hand, loves to cook, and that gives her the ability to be more experimental. She enjoys variety in her diet and loves to try different 5-Factor menus.\n\nIf you want to eat the same breakfast every single morning, I have no problem with that. And if you prefer to mix things up every day, that's fine too. Each of the 120 recipes in this book is balanced nutritionally and specifically to meet the 5-Factor criteria. That way, even if you chose to eat the same five meals every day\u2014or even the same meal five times a day!\u2014you'd never be left nutrient-deficient.\n\nI'm presenting you with 120 recipes\u2014nearly a month's worth of meals\u2014because variety is important to many dieters. These 120 fantastic recipes\u2014just like the 100 recipes in _5-Factor Fitness_ \u2014not only are delicious but also are hands-down some of the easiest and most convenient recipes you'll ever try. I should know because I've had to make them on the fly for my clients at the strangest places and times.\n\n## FAST, FUN, AND DELICIOUS\n\nWhen I'm on film sets with clients, I prepare their food quite often. I literally have minutes to make a meal when they unexpectedly take a break from filming. It was that kind of pressure that inspired my recipes.\n\nEach had to be very simple to make.\n\nEach had to use very few ingredients.\n\nEach had to be delicious\u2014I'm competing against catered food on sets, you know! (Halle Berry loves my 5-Factor fajitas, while Eva Mendes flips over my 5-Factor pizzas.)\n\nAnd finally, each had to meet my 5-Factor criteria.\n\nThat's exactly what these recipes deliver. Not only do they fulfill the 5-Factor criteria, but they can be prepared\u2014minus cook time\u2014in just five minutes. You need only five\u2014or fewer\u2014core ingredients (plus seasonings and oils). I kept the number of steps in each recipe to five as well. If you want even more variety, I encourage you to check out the recipes in _5-Factor Fitness_.\n\nIt's easy to make 5-Factor meals and enjoy the benefits of the 5-Factor Diet. So let's get cooking!\n\nNo more excuses.\n\n## Meal 1. Breakfast\n\n### Asparagus Crepes with Toast\n\n**1 bunch asparagus spears**\n\n**1 1\u20442 cups egg whites**\n\n**2 \u20443 cup nonfat milk**\n\n**4 slices whole grain bread, toasted**\n\n**_Salt and cracked black pepper to taste_ **\n\n**_Cooking oil spray_ **\n\n* * *\n\n**1.** Place the asparagus spears in a container with a little water. Microwave for 1 1\u20442 minutes, then drain and set aside.\n\n**2.** Whisk together the egg whites, milk, salt, and pepper.\n\n**3.** Coat a nonstick skillet with cooking spray and heat the skillet. Pour half of the egg whites into the skillet. When the egg whites begin to set, turn them over. Cook for 30 seconds and then slide the crepe onto a cutting board. Place half of the asparagus spears in the center of the crepe and roll tightly. Repeat but reserve a couple of asparagus spears for garnish.\n\n**To Serve:** Place the asparagus crepes on plates and serve with toast. Garnish with the reserved asparagus spears.\n\n**Servings:** 2\n\n### Frittata Italiana\n\n**1 1\u20442 cups egg whites**\n\n**1 \u20444 cup nonfat cream cheese, softened**\n\n**1 cup finely chopped sun-dried tomatoes**\n\n**4 leaves fresh basil, finely chopped**\n\n**4 slices whole grain bread, toasted**\n\n**_Salt and cracked black pepper to taste_ **\n\n**_Cooking oil spray_ **\n\n* * *\n\n**1.** Whisk together the egg whites, cream cheese, salt, and pepper.\n\n**2.** Spray a nonstick skillet with cooking spray and heat the skillet. Add the egg white mixture and cook until it begins to set. Immediately add the sun-dried tomatoes and basil leaves. Cover and cook about 2 minutes or until the eggs are completely set.\n\n**To Serve:** Slide the frittata onto a cutting board and cut into four wedges. Serve two wedges and two slices of toast on each plate. Garnish with pepper and additional fresh basil.\n\n**Servings:** 2\n\n### Breakfast Burritos I\n\n**3 \u20444 cup egg whites**\n\n**2 whole grain or whole wheat tortillas**\n\n**2 1\u20444 cups canned black beans, drained**\n\n**1 \u20442 cup shredded nonfat mozzarella cheese**\n\n**1 cup salsa**\n\n**_1 teaspoon ground cumin_ **\n\n**_1 teaspoon garlic salt_ **\n\n**_Cracked black pepper to taste_ **\n\n**_Cooking oil spray_ **\n\n* * *\n\n**1.** Preheat the oven or toaster oven to 350\u00b0F.\n\n**2.** Whisk together the egg whites, cumin, garlic salt, and pepper. Coat a nonstick skillet with cooking spray and heat the skillet. Add the egg white mixture. Cook and stir over low heat until egg whites are cooked. Set aside.\n\n**3.** Lay tortillas on a cutting board and sprinkle with black beans. Top with the egg whites and shredded cheese and roll tightly. Wrap the burritos in foil and bake for 2 minutes.\n\n**To Serve:** Unwrap the burritos, cut in half, and serve with salsa.\n\n**Servings:** 2\n\n### Breakfast Burritos II\n\n**1 cup nonfat ricotta cheese**\n\n**1 \u20444 cup egg whites**\n\n**4 cups diced tomatoes**\n\n**4 whole grain or whole wheat tortillas**\n\n**8 cups spinach leaves**\n\n**_2 teaspoons taco seasoning mix_ **\n\n**_1 teaspoon onion powder_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n**_Cooking oil spray_ **\n\n* * *\n\n**1.** Whisk together the ricotta cheese, egg whites, taco seasoning mix, onion powder, salt, and pepper. Stir in the tomatoes. Coat a nonstick skillet with cooking spray and heat the skillet. Add the egg mixture. Cook and stir until the egg whites are cooked. Set aside.\n\n**2.** Heat the tortillas in the microwave for 20 seconds and place on a cutting board. Place the scrambled eggs and spinach in the center of each tortilla. Roll tightly.\n\n**To Serve:** Cut the burritos in half and serve hot.\n\n**Servings:** 2\n\n### Breakfast Burritos III\n\n**2 cups egg whites**\n\n**2 large whole grain or whole wheat tortillas**\n\n**1 1\u20444 cups refried beans**\n\n**1 \u20444 cup shredded nonfat cheddar cheese**\n\n**1 cup salsa**\n\n**_Cooking oil spray_ **\n\n* * *\n\n**1.** Coat a nonstick skillet with cooking spray and heat the skillet. Add the egg whites. Cook and stir for 1 1\u20442 minutes. Set aside.\n\n**2.** Heat the tortillas in the microwave for 15 seconds. Spread the tortillas with refried beans and spoon the egg whites over the beans. Sprinkle with cheese and roll the tortillas tightly.\n\n**To Serve:** Cut the burritos in half and serve with salsa.\n\n**Servings:** 2\n\n### Broccoli-Cheddar Omelet\n\n**1 1\u20444 cups egg whites**\n\n**3 cups broccoli florets, coarsely chopped**\n\n**1 \u20444 cup shredded nonfat cheddar cheese**\n\n**4 slices whole grain bread, toasted**\n\n**_1 teaspoon Mrs. Dash seasoning mix_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n**_Cooking oil spray_ **\n\n* * *\n\n**1.** Whisk together the egg whites, Mrs. Dash, salt, and pepper. Coat a nonstick skillet with cooking spray and heat the skillet. Add the broccoli florets and cook and stir until they are bright green. Add the egg whites and cook while gently pushing them to the center with a rubber spatula. When the egg mixture begins to set on the bottom, turn it over. Sprinkle with cheese and cover the pan. Cook for 30 seconds or until the cheese begins to melt.\n\n**To Serve:** Slide the omelet onto a plate and fold in half. Cut in half and serve with toast.\n\n**Servings:** 2\n\n**NOTE:** You can use any green vegetable in your refrigerator in place of the broccoli.\n\n### Bell Pepper Pancakes with Mozzarella and Crisp Bacon\n\n**1 1\u20442 cups egg whites**\n\n**2 3\u20444 cups diced bell peppers**\n\n**1 tablespoon nonfat sour cream**\n\n**1 \u20444 cup shredded nonfat mozzarella cheese**\n\n**2 strips turkey bacon**\n\n**_Salt and cracked black pepper to taste_ **\n\n**_Cooking oil spray_ **\n\n* * *\n\n**1.** Preheat broiler. Whisk together the egg whites, bell peppers, sour cream, salt, and pepper.\n\n**2.** Coat a nonstick crepe pan with cooking spray and heat the pan. Ladle 1\u20444 cup of the egg white mixture into the pan and cook until it is partially set. Turn it over and cook until almost set. Repeat with the remaining egg white mixture. Place the pancakes on a nonstick baking sheet and sprinkle with the mozzarella cheese. Broil until the cheese is melted and golden brown.\n\n**3.** Microwave the turkey bacon for 3 minutes.\n\n**To Serve:** Transfer the pancakes and turkey bacon to serving plates.\n\n**Servings:** 2\n\n### The Cowboy Omelet\n\n**2 medium sweet potatoes**\n\n**1 cup egg whites**\n\n**5 cups sliced button mushrooms**\n\n**1 ounce Canadian bacon, cut into thin strips**\n\n**1 cup shredded nonfat cheddar cheese**\n\n**_1 teaspoon chili powder_ **\n\n**1 \u20442 _teaspoon garlic powder_**\n\n**_Salt and cracked black pepper to taste_ **\n\n**_Cooking oil spray_ **\n\n**_1 pinch ground cinnamon_ **\n\n* * *\n\n**1.** Microwave the sweet potatoes for 3 minutes each. Peel the potatoes and set aside.\n\n**2.** Whisk together the egg whites, chili powder, garlic powder, salt, and pepper. Coat a nonstick skillet with cooking spray and heat the skillet. Add the mushrooms and cook until most of the liquid has evaporated. Add the egg white mixture and cook until it begins to set. Add the Canadian bacon and cheddar cheese. Cover and cook until the cheese is melted.\n\n**To Serve:** Cut the sweet potatoes into cubes and gently toss with the cinnamon, salt, and pepper. Cut the omelet in half and serve with the sweet potatoes.\n\n**Servings:** 2\n\n### Egg and Veggie Muffins\n\n**1 1\u20448 cups egg whites**\n\n**1 3\u20444 cups broccoli florets, coarsely chopped**\n\n**3 \u20444 cup diced red and green bell pepper**\n\n**1 \u20442 cup shredded nonfat mozzarella cheese**\n\n**4 slices whole grain bread, toasted**\n\n**_Salt and cracked black pepper to taste_ **\n\n**_Cooking oil spray_ **\n\n* * *\n\n**1.** Preheat the oven to 350\u00b0F.\n\n**2.** Whisk together the egg whites, salt, and pepper. Coat 12 muffin cups with cooking spray. Pour the eggs into the muffin cups, filling each cup halfway. Drop the broccoli and bell pepper into the egg whites, dividing them evenly. Bake for 10 minutes or until the egg begins to set. Remove from the oven and sprinkle cheese over the top of each muffin. Return to the oven and bake until the egg has set completely and the cheese is melted and golden.\n\n**To Serve:** Slide a knife around the edge of each muffin and unmold onto a cutting board. Cut in half or leave whole. Place on plates and serve with toast.\n\n**Servings:** 2\n\n### Open-Face Egg and Bacon Sandwiches\n\n**2 strips turkey bacon**\n\n**1 1\u20444 cups egg whites**\n\n**4 slices whole grain bread, toasted**\n\n**1 \u20442 cup shredded nonfat cheddar cheese**\n\n**1 1\u20444 cups diced, seeded plum tomatoes**\n\n**_Salt and cracked black pepper to taste_ **\n\n**_Cooking oil spray_ **\n\n* * *\n\n**1.** Microwave the turkey bacon strips for 3 minutes or until crisp. Set aside.\n\n**2.** Whisk together the egg whites, salt, and pepper. Coat a nonstick skillet with cooking spray and heat the skillet. Add the egg white mixture. Cook and stir about 1 1\u20442 minutes or until the egg whites are set.\n\n**To Serve:** Spoon the egg whites on the toast. Top with cheese, turkey bacon, and diced tomatoes.\n\n**Servings:** 2\n\n**NOTE:** If you can't find nonfat cheddar cheese, you can substitute shredded part-skim mozzarella cheese.\n\n### Red Bell Pepper Frittata with Baked Yams\n\n**2 cups egg whites**\n\n**1 1\u20442 cups coarsely chopped roasted red peppers**\n\n**1 cup shredded nonfat mozzarella cheese**\n\n**2 large yams**\n\n**_2 teaspoons onion powder_ **\n\n**_1 teaspoon ground cumin_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n**_Cooking oil spray_ **\n\n* * *\n\n**1.** Whisk together the egg whites, onion powder, cumin, salt, and black pepper. Stir in the roasted peppers and shredded cheese. Coat a small glass baking dish with cooking spray. Pour the egg white mixture into the baking dish. Microwave for 4 minutes. Set aside. Microwave the yams for 31\u20442 minutes each. Cut the yams in half and season with salt and pepper.\n\n**To Serve:** Cut the frittata and serve with yams. Garnish with cracked black pepper.\n\n**Servings:** 2\n\n### Salmon-Leek Frittata with Whole Grain Toast\n\n**1 1\u20442 cups egg whites**\n\n**1 cup sliced leeks (white part only)**\n\n**2 tablespoons nonfat cream cheese**\n\n**2 ounces smoked salmon, chopped**\n\n**4 slices whole grain bread, toasted**\n\n**_Salt and cracked black pepper_ **\n\n**_Cooking oil spray_ **\n\n**_2 teaspoons dried parsley_ **\n\n* * *\n\n**1.** Whisk together the egg whites, leeks, salt, and pepper. Coat a small glass baking dish with cooking spray. Pour the egg white mixture into the baking dish. Cover with plastic wrap three-fourths of the way. Microwave for 4 minutes. Cool for 3 minutes.\n\n**To Serve:** Run a knife around the edge of the frittata and turn it over onto a cutting board. Spread the frittata with cream cheese and top with chopped salmon. Cut frittata in half and serve with toast. Garnish with parsley and additional pepper.\n\n**Servings:** 2\n\n### Scrambled Egg Casserole\n\n**1 plum tomato, seeded and diced**\n\n**1 tablespoon thinly sliced scallion, white part only**\n\n**3 \u20444 cup egg whites**\n\n**1 \u20442 cup shredded nonfat mozzarella cheese**\n\n**4 slices whole grain bread, toasted**\n\n**_Cooking oil spray_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n* * *\n\n**1.** Coat a nonstick skillet with cooking spray and heat the skillet. Add the tomato and scallion and cook until the scallion is light golden. Whisk in the egg whites and half of the shredded cheese. Cook and stir until the egg white mixture is almost set. Season with salt and pepper.\n\n**To Serve:** Spoon the scrambled eggs into a small casserole and sprinkle with the remaining cheese. Microwave until the cheese is melted. Serve with toast.\n\n**Servings:** 2\n\n### Scrambled Eggs with Toast and Grapefruit\n\n**1 \u20444 pound smoked chicken breast, cut into cubes**\n\n**1 \u20444 cup egg whites**\n\n**1 \u20442 cup shredded nonfat cheddar cheese**\n\n**4 slices whole grain bread, toasted**\n\n**2 grapefruit, cut in half and seeded**\n\n**_Cooking oil spray_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n* * *\n\n**1.** Coat a nonstick skillet with cooking spray and heat the skillet. Add the cubed chicken and egg whites. Season with salt and pepper and cook for 2 minutes. Add the cheese and cook until the cheese is melted.\n\n**To Serve:** Spoon the scrambled eggs onto plates and serve with toast and grapefruit.\n\n**Servings:** 2\n\n### Smoked Salmon Omelet with Cream Cheese and Whole Grain Toast\n\n**1 cup egg whites**\n\n**1 \u20444 cup nonfat cream cheese, softened**\n\n**2 ounces smoked salmon**\n\n**4 slices whole grain bread, toasted**\n\n**1 3\u20444 cups orange sections**\n\n**_Salt and cracked black pepper to taste_ **\n\n**_Cooking oil spray_ **\n\n* * *\n\n**1.** Whisk together the egg whites, cream cheese, salt, and pepper. Coat a nonstick skillet with cooking spray and heat the skillet. Pour the egg white mixture into the skillet. Gently push the egg whites toward the center as they cook. When they are almost set, place the smoked salmon on top. Cover the pan and cook for 30 seconds. Remove the lid and season with additional pepper.\n\n**To Serve:** Slide the omelet onto a cutting board and fold in half. Cut the omelet in half and serve with toast and orange segments.\n\n**Servings:** 2\n\n### Smoked Turkey and Tomato Scrambled Eggs with Toast\n\n**1 cup egg whites**\n\n**3 ounces deli-style fat-free smoked turkey, chopped**\n\n**1 1\u20442 cups chopped plum tomatoes**\n\n**1 \u20442 cup shredded nonfat mozzarella cheese**\n\n**4 slices whole grain bread, toasted**\n\n**_Cooking oil spray_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n* * *\n\n**1.** Coat a nonstick skillet with cooking spray and heat the skillet. Add the egg whites and cook for 30 seconds. Sprinkle with chopped turkey, tomatoes, and shredded cheese. Cook and stir about 2 minutes or until the egg whites are completely set. Season with salt and pepper.\n\n**To Serve:** Serve the scrambled eggs with toast.\n\n**Servings:** 2\n\n**NOTE:** For lunch, spoon the scrambled eggs on toast and eat it as an open-face sandwich.\n\n### Sweet Potato Home Fries and Scrambled Eggs\n\n**2 large sweet potatoes**\n\n**1 \u20442 cup diced Spanish onion**\n\n**1 bell pepper, seeded and diced**\n\n**1 cup egg whites**\n\n**1 cup shredded nonfat cheddar cheese**\n\n**_Cooking oil spray_ **\n\n**_1_ 1\u20442 _teaspoons garlic powder_**\n\n**_1 teaspoon paprika_ **\n\n**_1 teaspoon red pepper flakes_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n* * *\n\n**1.** Microwave the sweet potatoes for 31\u20442 minutes each or until tender. Peel off the skins and dice the potatoes. Coat a nonstick skillet with cooking spray and heat the skillet. Add the onion and cook for 1 minute. Add the sweet potatoes, bell pepper, garlic powder, paprika, and red pepper flakes. Toss gently and set aside.\n\n**2.** Coat a nonstick skillet with cooking spray and heat the skillet. Add the egg whites, cheese, salt, and cracked black pepper. Cook and stir until the egg whites are set.\n\n**To Serve:** Spoon the scrambled eggs and home fries onto plates. Garnish with cracked black pepper.\n\n**Servings:** 2\n\n**NOTE:** If you can't find nonfat cheddar cheese, use shredded part-skim mozzarella.\n\n### Ham Steaks with Applesauce and Toast\n\n**1 \u20443 pound extra lean ham, cut into two serving-size pieces**\n\n**2 3\u20444 cups peeled, cored, and cut-up Fuji apples**\n\n**1 \u20442 cup nonfat cottage cheese**\n\n**2 slices whole grain bread, toasted**\n\n**_1 teaspoon ground cinnamon_ **\n\n**1 \u20444 _teaspoon sugar substitute_**\n\n* * *\n\n**1.** Place the ham steaks in a hot nonstick skillet and cook for 1 minute on each side. In a bowl, toss together the apple pieces, cinnamon, and sugar substitute. Microwave for 2 minutes. Mash the cooked apples and the cottage cheese with a fork.\n\n**To Serve:** Serve ham steaks with applesauce and toast.\n\n**Servings:** 2\n\n### Bran Pancakes with Ricotta\n\n**1 \u20442 cup bran flakes**\n\n**1 \u20442 cup egg whites**\n\n**1 \u20442 cup nonfat sour cream**\n\n**1 3\u20444 cups nonfat ricotta cheese**\n\n**_1 tablespoon sugar substitute_ **\n\n**_1 pinch salt_ **\n\n**_Butter-flavor cooking oil spray_ **\n\n**_2 teaspoons ground cinnamon_ **\n\n* * *\n\n**1.** Whisk together the bran flakes, egg whites, sour cream, sugar substitute, and salt. Coat a nonstick skillet with cooking spray and heat the skillet. Ladle a thin layer of batter into the skillet. Cook until the batter begins to set. Carefully turn the pancake and cook on the other side until the pancake is completely set and is a light golden color. Repeat with the remaining batter.\n\n**To Serve:** Dust the pancakes with cinnamon and serve with ricotta.\n\n**Servings:** 2\n\n### Oatmeal-Berry Pancakes\n\n**1 1\u20442 cups egg whites**\n\n**1 1\u20443 cups chopped strawberries**\n\n**1 cup rolled oats**\n\n**1 cup nonfat sour cream**\n\n**1 cup blueberries**\n\n**_1_ 1\u20444 _teaspoons sugar substitute_**\n\n**_Butter-flavor cooking oil spray_ **\n\n* * *\n\n**1.** Beat together the egg whites, strawberries, rolled oats, and sugar substitute until smooth.\n\n**2.** Coat a nonstick skillet with cooking spray and heat the skillet. Ladle 1\u20444 cup of the batter into the skillet. Cook until the batter is set around the edges of the pan, then push it toward the center with a spatula. Cook until the batter begins to set in the center. Turn the pancake over or cover the pan. Cook for 1 minute. Repeat with remaining batter.\n\n**To Serve:** Slide the pancakes onto plates and top with sour cream. Garnish with blueberries.\n\n**Servings:** 2\n\n### French Toast with Ricotta\n\n**2 \u20443 cup egg whites**\n\n**2 \u20443 cup nonfat milk**\n\n**2 slices whole grain bread**\n\n**1 \u20448 cup nonfat ricotta cheese**\n\n**_1 teaspoon sugar substitute_ **\n\n**_1 pinch salt_ **\n\n**_Cooking oil spray_ **\n\n**_1 teaspoon ground cinnamon_ **\n\n* * *\n\n**1.** Whisk together the egg whites, milk, sugar substitute, and salt. Soak the bread in the egg white mixture. Drain the excess liquid.\n\n**2.** Coat a nonstick skillet with cooking spay and heat the skillet. Cook the bread, one slice at a time, until each side is set and bread is light brown.\n\n**To Serve:** Place the French toast on a plate and top with ricotta. Garnish with cinnamon.\n\n**Servings:** 1\n\n### Fully Charged Fruit Salad\n\n**4 oranges**\n\n**1 scoop protein powder (100% whey)**\n\n**1 cup nonfat cottage cheese**\n\n**2 Granny Smith apples, cored and cut into wedges**\n\n**2 cups quartered strawberries**\n\n**_1 teaspoon ground ginger_ **\n\n* * *\n\n**1.** Peel and section three of the oranges and squeeze the juice from the fourth orange. Whisk the orange juice, protein powder, and ginger into the cottage cheese.\n\n**To Serve:** Spoon the cottage cheese into bowls and top with orange sections, apple wedges, and strawberries. Serve chilled.\n\n**Servings:** 2\n\n### Cream of Wheat and Protein\n\n**2 1\u20444 cups nonfat milk**\n\n**3 \u20444 cup Cream of Wheat**\n\n**1 scoop protein powder (100% whey)**\n\n**_1 teaspoon ground cinnamon_ **\n\n* * *\n\n**1.** In a saucepan, combine the milk, Cream of Wheat, and protein powder and bring to a boil. Whisk until smooth and creamy.\n\n**To Serve:** Ladle into bowls and garnish with cinnamon.\n\n**Servings:** 2\n\n**NOTE:** If you can't find 100% whey protein, use soy protein.\n\n### Kashi GoLean with Nonfat Milk\n\n**2 cups Kashi GoLean cereal or other high-fiber whole grain cereal**\n\n**2 cups nonfat milk**\n\n* * *\n\n**To Serve:** Place 1 cup of cereal in each bowl and add the milk.\n\n**Servings:** 2\n\n**NOTE:** You can find Kashi GoLean cereal in the organic section of your local market.\n\n## Meals 2 and 4. Snacks\n\n### Apple-Turkey Roll-Ups with Relish and Mustard\n\n**5 ounces deli-style turkey breast, sliced**\n\n**3 Granny Smith apples, cored and thinly sliced**\n\n**2 tablespoons pickle relish**\n\n**1 tablespoon whole grain mustard**\n\n* * *\n\n**1.** Place the turkey slices on a cutting board. Lay the apple slices on the turkey and spread with relish and mustard. Roll tightly and secure each with a toothpick.\n\n**To Serve:** Make the roll-ups ahead, wrap in plastic wrap, and refrigerate.\n\n**Servings:** 2\n\n### Belgian Endive Stuffed with Cheesy Artichoke Spread\n\n**2 cups canned artichoke hearts, drained**\n\n**1 \u20442 cup nonfat cream cheese, softened**\n\n**2 tablespoons shredded nonfat mozzarella cheese**\n\n**1 whole Belgian endive**\n\n**_1 teaspoon dried parsley_ **\n\n**_1 teaspoon onion powder_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n* * *\n\n**1.** In a food processor, pulse the artichoke hearts, cream cheese, mozzarella cheese, parsley, onion powder, salt, and pepper.\n\n**To Serve:** Pull the leaves from the endive and arrange them on a serving platter. Spoon the artichoke spread into the leaves.\n\n**Servings:** 2\n\n### Bruschetta\n\n**3 \u20444 cup finely chopped sun-dried tomatoes**\n\n**4 whole grain crackers**\n\n**2 \u20443 cup shredded nonfat mozzarella cheese**\n\n**_1 teaspoon garlic powder_ **\n\n**1 \u20442 _teaspoon onion powder_**\n\n**_1 teaspoon Italian seasoning_ **\n\n**_Cracked black pepper to taste_ **\n\n* * *\n\n**1.** Preheat broiler to medium. Spoon the sun-dried tomatoes onto the crackers and top with the mozzarella cheese. Season with garlic powder and onion powder. Broil until the cheese is melted.\n\n**To Serve:** Garnish with Italian seasoning and pepper.\n\n**Servings:** 2\n\n### Cheese Course\n\n**2 pears, cored and cut into wedges**\n\n**1 cup nonfat ricotta cheese**\n\n**_Cracked black pepper_ **\n\n* * *\n\n**To Serve:** Arrange the pear wedges on a plate and spoon the ricotta over the pears. Garnish with cracked black pepper.\n\n**Servings:** 2\n\n### Chicken and Swiss Bites\n\n**1 ounce deli-style fat-free chicken breast, thinly sliced**\n\n**4 ounces nonfat Swiss cheese, cut into strips**\n\n**4 multigrain crackers**\n\n**1 cup salsa**\n\n* * *\n\n**To Serve:** Roll the chicken slices around the Swiss cheese and arrange on top of the crackers. Garnish with salsa.\n\n**Servings:** 2\n\n### Chicken Salad with Apples\n\n**2 3\u20444 ounces skinless, boneless chicken breast**\n\n**2 cups peeled, cored, and diced Granny Smith apple**\n\n**1 3\u20444 cups finely diced celery**\n\n**1 cup nonfat sour cream**\n\n**_1 \u20442 teaspoon celery seeds_ **\n\n**_1 tablespoon onion salt_ **\n\n* * *\n\n**1.** In a small saucepan, cook the chicken in water until the chicken is fully cooked. Drain, cool, and dice the chicken.\n\n**2.** In a mixing bowl, combine the chicken, apple, celery, sour cream, onion salt, and celery seeds. Cover and chill.\n\n**To Serve:** Spoon the chicken salad into small bowls.\n\n**Servings:** 2\n\n### Crunchy Celery Sticks with Roasted-Garlic Hummus and Smoked Turkey\n\n**1 garlic clove, peeled**\n\n**1 cup cooked garbanzo beans, drained and rinsed**\n\n**3 tablespoons freshly squeezed lemon juice**\n\n**1 stalk celery, cut into thick sticks**\n\n**3 ounces deli-style sliced fat-free turkey**\n\n**_1 \u20442 teaspoon olive oil_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n**_1 teaspoon dried parsley_ **\n\n* * *\n\n**1.** Preheat the oven or toaster oven to 350\u00b0F. Wrap the garlic clove in foil and roast for 10 minutes. For the hummus, in a food processor, combine the roasted garlic, garbanzo beans, lemon juice, and olive oil. Pulse until a smooth paste forms. Season with salt and pepper. (If the hummus is too thick, add a little water until it reaches the desired consistency.)\n\n**To Serve:** Arrange the celery sticks and turkey slices on plates. Serve the hummus in a small bowl and garnish with dried parsley.\n\n**Servings:** 2\n\n### Edamame and Tuna Sashimi with Ginger-Scallion Vinaigrette\n\n**1 \u20443 cup edamame beans, removed from pods**\n\n**3 teaspoons grated ginger**\n\n**3 teaspoons slivered scallion**\n\n**3 cups shredded carrots**\n\n**2 1\u20444 ounces sushi-grade yellowfin tuna, thinly sliced**\n\n**1 \u20442 _cup water_**\n\n**_1 tablespoon soy sauce_ **\n\n**_Salt to taste_ **\n\n* * *\n\n**1.** Cook edamame in boiling water for 2 minutes. Drain edamame and set aside. For vinaigrette, whisk together the water, soy sauce, ginger, and scallion. Toss the shredded carrots and tuna slices with the vinaigrette.\n\n**To Serve:** Place the warm edamame in the center of a plate and season with a little salt. Arrange the carrots and tuna slices around the edamame.\n\n**Servings:** 2\n\n**NOTE:** Combining warm edamame with chilled tuna makes a refreshing hot and cold snack.\n\n### Grilled Chicken Kabobs with Carrot-Ginger Vinaigrette\n\n**1 3\u20444 cups shredded carrots**\n\n**1 Granny Smith apple, shredded**\n\n**1 \u20442 cup rice wine vinegar**\n\n**5 ounces skinless, boneless chicken breast, cut into bite-size pieces**\n\n**1 bell pepper, seeded and cut into squares**\n\n**_1 teaspoon ground ginger_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n**_Cooking oil spray_ **\n\n* * *\n\n**1.** In a blender or food processor, combine the carrots, apple, rice wine vinegar, ginger, salt, and cracked black pepper. Pulse until smooth. Pour into a container and refrigerate. If the vinaigrette is too thick, add a little water.\n\n**2.** Alternately thread chicken and bell pepper pieces onto skewers. Coat lightly with cooking spray and season with salt and cracked black pepper. On a hot nonstick grill pan, cook the chicken skewers for 10 minutes or until the chicken is fully cooked.\n\n**To Serve:** Place the chicken kabobs on plates and drizzle with the vinaigrette. Serve warm.\n\n**Servings:** 2\n\n**NOTE:** If you double this snack, you'll have a great lunch.\n\n### Chicken Slices with Cheese and Crackers\n\n**6 whole grain crackers or any high-fiber, low-sugar crackers**\n\n**2 ounces deli-style smoked chicken, thinly sliced**\n\n**2 ounces nonfat cheddar cheese, thinly sliced**\n\n**1 \u20442 peach, cored and thinly sliced**\n\n**1 \u20442 pear, pitted and thinly sliced**\n\n* * *\n\n**To Serve:** Arrange the crackers on a plate and serve with chicken, cheese, and fruit slices.\n\n**Servings:** 2\n\n### Pear and Arugula Salad with Ricotta\n\n**1 strip turkey bacon**\n\n**1 1\u20442 cups arugula leaves**\n\n**2 pears, cored and thinly sliced**\n\n**3 \u20444 cup nonfat ricotta cheese, softened**\n\n**1 lemon, cut in half**\n\n**_Salt and cracked black pepper to taste_ **\n\n* * *\n\n**1.** Microwave the turkey bacon for 2 minutes or until crisp. Crumble the bacon.\n\n**To Serve:** Arrange the arugula on plates and top with the pear slices and crumbled turkey bacon. Spoon the ricotta around the plates. Squeeze the juice from the lemon over the salad. Season with salt and pepper.\n\n**Servings:** 2\n\n### Roasted Asparagus Spears with Turkey Slices\n\n**20 asparagus spears**\n\n**6 ounces deli-style fat-free turkey, thinly sliced**\n\n**2 cups shredded carrots**\n\n**1 \u20442 cup very thinly sliced red onion**\n\n**_Cooking oil spray_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n* * *\n\n**1.** Preheat the toaster oven to 350\u00b0F.\n\n**2.** Toss the asparagus spears with cooking spray, salt, and pepper. Roast in the toaster oven for 4 minutes.\n\n**To Serve:** Lay the turkey slices on a cutting board. Sprinkle with the shredded carrots and place roasted asparagus spears in the middle. Roll the turkey tightly around the carrots and asparagus. Garnish with sliced onion.\n\n**Servings:** 2\n\n### Smoked Turkey and Fruit Salad\n\n**1 1\u20442 cups quartered strawberries**\n\n**1 cup orange sections**\n\n**1 Granny Smith apple, cut into wedges**\n\n**1 ounce smoked turkey, cut into cubes**\n\n**1 \u20442 cup nonfat cottage cheese**\n\n* * *\n\n**1.** Gently toss together the strawberries, orange sections, apple wedges, and turkey cubes. Refrigerate until ready to serve.\n\n**To Serve:** Spoon the cottage cheese into bowls and top with the fruit mixture.\n\n**Servings:** 2\n\n### Salmon Sashimi with Plums\n\n**1 clove garlic, mashed**\n\n**12 ounces plums, pitted and thinly sliced**\n\n**3 ounces fresh salmon, sliced paper-thin**\n\n**1 scallion, thinly sliced**\n\n**1 \u20444 _cup low-sodium soy sauce_**\n\n**_1 teaspoon ground ginger_ **\n\n**1 \u20442 _teaspoon wasabi powder_**\n\n**1 \u20442 _teaspoon sugar substitute_**\n\n**_2 teaspoons sesame seeds_ **\n\n* * *\n\n**1.** Whisk together the soy sauce, garlic, ginger, wasabi powder, and sugar substitute.\n\n**2.** Arrange half of the plums on a plate. Arrange all of the salmon over the plums. Arrange the remaining plums over the salmon. Pour the soy mixture over the plums. Refrigerate.\n\n**To Serve:** Garnish with scallion and sesame seeds.\n\n**Servings:** 2\n\n### Egg Salad with Toast Points\n\n**4 hard-boiled eggs, yolks removed**\n\n**1 hard-boiled egg**\n\n**2 tablespoons nonfat mayonnaise**\n\n**2 stalks celery, finely diced**\n\n**2 slices whole grain bread, toasted**\n\n**_1 teaspoon onion powder_ **\n\n**_1 pinch celery seeds_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n* * *\n\n**1.** Chop the egg whites and whole egg and place in a mixing bowl. Stir in the mayonnaise, celery, onion powder, celery seeds, salt, and pepper. Mix well.\n\n**To Serve:** Cut the toast into quarters and serve with egg salad.\n\n**Servings:** 2\n\n### Egg and Celery Platter with Mustard-Balsamic Sauce\n\n**3 hard-boiled eggs, yolks removed**\n\n**1 3\u20444 cups nonfat sour cream**\n\n**1 \u20442 cup balsamic vinegar**\n\n**3 teaspoons Dijon mustard**\n\n**4 stalks celery, cut into small sticks**\n\n**_Salt and cracked black pepper to taste_ **\n\n* * *\n\n**1.** Quarter the egg whites and set aside. Whisk together the sour cream, balsamic vinegar, mustard, salt, and pepper.\n\n**To Serve:** Arrange the egg whites and celery sticks on a plate. Drizzle with the sauce and garnish with pepper.\n\n**Servings:** 2\n\n### Hard-Boiled Eggs Stuffed with Tuna Salad\n\n**1 \u20442 cup canned water-pack tuna, drained**\n\n**1 \u20442 cup nonfat sour cream**\n\n**1 \u20444 cup thinly sliced scallions**\n\n**2 hard-boiled eggs, halved and yolks removed**\n\n**3 cups shredded carrots**\n\n**_1 teaspoon onion powder_ **\n\n**_1 teaspoon garlic powder_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n* * *\n\n**1.** In a bowl, combine the tuna, sour cream, scallions, onion powder, garlic powder, salt, and pepper. Stir until the ingredients are combined. Stuff the egg whites with the tuna mixture.\n\n**To Serve:** Make a shredded carrot nest on a plate and top with the stuffed eggs. Garnish with pepper.\n\n**Servings:** 2\n\n### Spinach Frittata and Toast\n\n**1 \u20442 cup egg whites**\n\n**4 cups spinach**\n\n**2 tablespoons shredded part-skim-milk mozzarella cheese**\n\n**4 slices whole wheat bread, toasted**\n\n**_1 pinch onion powder_ **\n\n**_1 pinch garlic salt_ **\n\n**_Cracked black pepper to taste_ **\n\n**_Olive oil cooking spray_ **\n\n* * *\n\n**1.** Whisk together the egg whites, onion powder, garlic salt, and pepper. Coat a nonstick skillet with cooking spray and heat the skillet. Add the egg white mixture and cook until it begins to set. Add the spinach and cover the pan. Cook until the eggs begin to set on top. Sprinkle the cheese over the frittata and cook until cheese is melted.\n\n**To Serve:** Cut frittata into wedges and serve hot or cold with toast.\n\n**Servings:** 2\n\n### Hot Dog Skewers with Cherry Tomatoes and Pickles\n\n**4 veggie hot dogs**\n\n**3 cups halved button mushrooms**\n\n**2 cups cherry tomatoes**\n\n**1 cup pickles cut into chunks**\n\n**3 tablespoons Dijon mustard**\n\n* * *\n\n**1.** Microwave the veggie hot dogs for 1 1\u20442 minutes. Cut each hot dog into four pieces. Alternately thread hot dog pieces, mushrooms, tomatoes, and pickles on skewers.\n\n**To Serve:** Serve warm or cold with Dijon mustard.\n\n**Servings:** 2\n\n### Roast Beef with Carrot-Pear Slaw\n\n**1 1\u20442 cups shredded carrots**\n\n**1 pear, cored and chopped**\n\n**1 \u20444 cup nonfat sour cream**\n\n**1 tablespoon horseradish**\n\n**4 ounces deli-style roast beef, thinly sliced**\n\n**_1 teaspoon lime juice_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n* * *\n\n**1.** Lightly toss together the shredded carrots, pear, and lime juice. Combine the sour cream and horseradish and gently stir into the carrot mixture.\n\n**To Serve:** Season the roast beef with salt and pepper and serve with the coleslaw.\n\n**Servings:** 2\n\n### Spicy Jumbo Shrimp with Black Bean Dip\n\n**6 jumbo shrimp, peeled and deveined**\n\n**1 1\u20443 cups canned black beans, drained**\n\n**1 \u20444 cup finely diced red onion**\n\n**1 lime**\n\n**4 tablespoons whole cilantro leaves**\n\n**_1 teaspoon red pepper flakes_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n**_Cooking oil spray_ **\n\n* * *\n\n**1.** Season the shrimp with half of the red pepper flakes, salt, and cracked black pepper. Coat a nonstick skillet with cooking spray and heat skillet. Add the shrimp and cook for 2 minutes or until the shrimp turn opaque.\n\n**2.** Combine the black beans and red onion with the juice from one half of the lime, cilantro, remaining red pepper flakes, salt, and cracked black pepper.\n\n**To Serve:** Place the black bean mixture in a bowl and serve with the shrimp. Slice the remaining lime half and garnish with the lime slices.\n\n**Servings:** 2\n\n### Carrot Sticks with Onion Dip\n\n**3 \u20444 cup nonfat sour cream**\n\n**1 \u20442 cup nonfat cream cheese**\n\n**1 tablespoon dry onion soup mix**\n\n**12 ounces carrots, cut into sticks**\n\n* * *\n\n**1.** Beat together the sour cream, cream cheese, and onion soup mix until very smooth.\n\n**To Serve:** Arrange the carrot sticks on a plate and serve with the onion dip.\n\n**Servings:** 2\n\n### Spinach Dip with Carrot Sticks\n\n**1 1\u20444 pounds spinach leaves**\n\n**1 cup nonfat sour cream**\n\n**1 \u20444 cup shredded nonfat mozzarella cheese**\n\n**3 carrots, cut into 2-inch sticks**\n\n**_1 teaspoon onion powder_ **\n\n**_1 teaspoon garlic powder_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n* * *\n\n**1.** Combine the spinach, sour cream, mozzarella, onion powder, garlic powder, salt, and pepper in a plastic container. Microwave for 1 1\u20442 minutes and stir.\n\n**To Serve:** Serve the spinach dip with the carrot sticks.\n\n**Servings:** 2\n\n### Smoked Salmon Mousse with Crackers\n\n**2 ounces smoked salmon**\n\n**1 \u20444 cup nonfat cream cheese**\n\n**6 tablespoons freshly squeezed lemon juice**\n\n**6 multigrain crackers or any high-fiber, low-sugar crackers**\n\n**_Salt and cracked black pepper to taste_ **\n\n**_1 teaspoon dried dill_ **\n\n* * *\n\n**1.** In a food processor, combine the smoked salmon, cream cheese, lemon juice, salt, and pepper. Pulse until smooth.\n\n**To Serve:** Spoon the salmon mixture onto a plate and arrange the crackers around the plate. Garnish with dried dill and additional pepper.\n\n**Servings:** 2\n\n### White Bean Dip\n\n**1 \u20442 cup nonfat cream cheese**\n\n**1 \u20443 cup drained white navy beans**\n\n**1 tablespoon freshly squeezed lemon juice**\n\n**1 stalk celery, cut into sticks**\n\n**_Salt and cracked black pepper to taste_ **\n\n* * *\n\n**1.** In a food processor, combine cream cheese, navy beans, lemon juice, salt, and pepper. Pulse until a smooth paste forms.\n\n**To Serve:** Arrange the celery sticks on a plate and serve with bean dip.\n\n**Servings:** 2\n\n### Chips and Salsa\n\n**3 large whole grain or whole wheat tortillas, cut into triangles**\n\n**1 \u20442 tablespoon nonfat sour cream**\n\n**1 cup salsa**\n\n**1 \u20442 cup shredded nonfat cheddar cheese**\n\n**1 scallion, thinly sliced**\n\n* * *\n\n**1.** Preheat oven to 375\u00b0F. Put the tortilla triangles on a baking pan and bake until they are crisp. Set aside and cool.\n\n**To Serve:** Place the tortillas in two shallow bowls. Dollop sour cream on the chips. Spoon the salsa over the sour cream. Sprinkle the cheddar cheese on top and garnish with scallion.\n\n**Servings:** 2\n\n### Pesto Crisps with Tomatoes and Cheese\n\n**3 \u20444 cup nonfat ricotta cheese**\n\n**6 multigrain crackers (pesto or any Italian flavor)**\n\n**8 small tomatoes, sliced**\n\n**2 tablespoons basil leaves**\n\n**_Salt and cracked black pepper to taste_ **\n\n* * *\n\n**1.** Spread a generous amount of ricotta on each cracker. Season the tomatoes with salt and arrange them on top of the crackers.\n\n**To Serve:** Sprinkle the tomatoes with pepper and garnish with basil leaves.\n\n**Servings:** 2\n\n### Sauteed Apples over Rice Cakes\n\n**2 Granny Smith apples, peeled, cored, and cut into wedges**\n\n**4 rice cakes**\n\n**1 tablespoon low-fat cottage cheese**\n\n**1 1\u20442 ounces smoked turkey jerky, finely chopped**\n\n**_Cooking oil spray_ **\n\n**_1 pinch ground cinnamon_ **\n\n* * *\n\n**1.** Spray a nonstick skillet with cooking spray and heat the skillet. Add the apple wedges and cook until they begin to soften. Season apples with cinnamon.\n\n**To Serve:** Place the rice cakes on plates and top with the cottage cheese. Arrange the apples over the cheese. Garnish with chopped turkey jerky.\n\n**Servings:** 2\n\n**NOTE:** The saltiness of the turkey jerky balances with the sweetness of the apples.\n\n### Pears with Peanut Butter Dip\n\n**1 \u20443 cup nonfat cream cheese**\n\n**2 teaspoons peanut butter**\n\n**2 pears, cored and cut into wedges**\n\n* * *\n\n**1.** Combine the cream cheese and peanut butter.\n\n**To Serve:** Spoon the cream cheese mixture over the pear wedges.\n\n**Servings:** 2\n\n### Cottage Cheese and Pears\n\n**2 pears, cored and cut into wedges**\n\n**1 teaspoon freshly squeezed lemon juice**\n\n**1 1\u20444 cups nonfat cottage cheese**\n\n**_1_ 1\u20442 _teaspoons sugar substitute_**\n\n* * *\n\n**1.** Toss the pear wedges with the lemon juice. Combine the cottage cheese and the sugar substitute.\n\n**To Serve:** Serve the pear wedges with the cottage cheese as a dip.\n\n**Servings:** 2\n\n### Fruit Skewers with Cottage Cheese\n\n**1 pear, cored and cut into cubes**\n\n**1 teaspoon freshly squeezed lemon juice**\n\n**20 strawberries, stems removed**\n\n**1 peach, pitted and cut into cubes**\n\n**1 1\u20448 cups nonfat cottage cheese**\n\n* * *\n\n**1.** Lightly toss the pear cubes with the lemon juice.\n\n**2.** Thread the pear cubes, strawberries, and peach cubes onto skewers. Chill until serving.\n\n**To Serve:** Serve the fruit skewers with the cottage cheese.\n\n**Servings:** 2\n\n**NOTE:** You may also choose other 5-Factor-friendly fruits for this recipe.\n\n### Strawberry-Oatmeal Bars with Yogurt\n\n**1 cup sliced strawberries**\n\n**1 \u20442 cup rolled oats**\n\n**1 \u20442 cup egg whites**\n\n**1 \u20442 cup nonfat plain yogurt**\n\n**_2 teaspoons sugar substitute_ **\n\n**_Cooking oil spray_ **\n\n* * *\n\n**1.** Preheat the oven to 300\u00b0F. Combine the strawberries, rolled oats, egg whites, and 1 teaspoon of sugar substitute.\n\n**2.** Coat a shallow baking dish with the cooking spray. Pour the strawberry mixture into the baking dish. Bake for 15 to 20 minutes. Set aside and cool.\n\n**3.** Increase oven temperature to 425\u00b0F. Slice the cake into bars. Recoat the baking dish with cooking spray. Place the bars back in the baking dish. Bake for 5 minutes more or until crisp and golden. Cool.\n\n**To Serve:** Stir the remaining sugar substitute into the yogurt. Serve the bars with yogurt for dipping.\n\n**Servings:** 2\n\n### Toast with Berries and Cocoa Cottage Cheese\n\n**1 cup fresh berries**\n\n**2 slices whole grain bread, toasted**\n\n**1 \u20442 cup low-fat cottage cheese**\n\n**_1 tablespoon unsweetened cocoa powder_ **\n\n**_2 teaspoons sugar substitute_ **\n\n* * *\n\n**1.** Crush the berries with a fork and spoon them over the toast. Combine the cottage cheese, cocoa powder, and sugar substitute.\n\n**To Serve:** Serve the toast with berries with the cottage cheese.\n\n**Servings:** 2\n\n### Butterscotch and Apple Pudding\n\n**1 Fuji apple, peeled, cored, and diced**\n\n**1 packet sugar-free, fat-free butterscotch instant pudding mix**\n\n**1 1\u20442 cups cold nonfat milk**\n\n**1 cup nonfat ricotta cheese, softened**\n\n* * *\n\n**1.** Place the apple in a container with 2 tablespoons water. Microwave for 2 minutes and set aside to cool. Whisk the pudding mix and the milk until smooth. Fold in the apple.\n\n**To Serve:** Spoon 1\u20442 cup of ricotta cheese into each of two small glass bowls. Spoon the instant pudding on top.\n\n**Servings:** 2\n\n### Cheesecake\n\n**1 1\u20442 tablespoons unflavored gelatin powder**\n\n**1 \u20442 cup nonfat cream cheese**\n\n**1 \u20442 cup nonfat sour cream**\n\n**3 cups strawberries**\n\n**_1 \u20444 cup water_ **\n\n**_3 teaspoons sugar substitute_ **\n\n**_3 teaspoons vanilla extract_ **\n\n* * *\n\n**1.** In a bowl, dissolve the gelatin powder in the water. Stir in the cream cheese, sour cream, sugar substitute, and vanilla extract. Pour the gelatin mixture into a glass baking dish and refrigerate until set.\n\n**To Serve:** Spoon cheesecake onto plates and top with strawberries.\n\n**Servings:** 2\n\n### Chocolate-Berry Parfaits\n\n**1 package fat-free, sugar-free chocolate-flavor pudding mix**\n\n**1 cup nonfat milk**\n\n**3 \u20444 cup nonfat cottage cheese**\n\n**1 \u20442 cup nonfat plain yogurt**\n\n**1 1\u20444 cups fresh raspberries**\n\n**_1 teaspoon sugar substitute_ **\n\n* * *\n\n**1.** Whisk together the chocolate pudding mix and nonfat milk until smooth. In another bowl, whisk together the cottage cheese, yogurt, and sugar substitute.\n\n**To Serve:** Spoon 1 tablespoon of pudding into each serving glass. Top with 1 tablespoon of the cottage cheese mixture and several raspberries. Repeat until all ingredients are used. Garnish with additional raspberries.\n\n**Servings:** 2\n\n### Espresso Panna Cotta\n\n**1 cup nonfat plain yogurt**\n\n**1 cup nonfat sour cream or quark**\n\n**1 shot espresso coffee, chilled**\n\n**1 tablespoon finely chopped bittersweet chocolate**\n\n**_3 teaspoons vanilla extract_ **\n\n**_2 teaspoons sugar substitute_ **\n\n* * *\n\n**1.** Whisk together the yogurt, sour cream, coffee, chopped chocolate, vanilla extract, and sugar substitute until smooth. Pour into two small containers and chill until ready to serve.\n\n**To Serve:** Serve chilled.\n\n**Servings:** 2\n\n**NOTE:** If you want to avoid caffeine, use decaf espresso. You can also try adding a fat-free, sugar-free coffee flavoring such as hazelnut or Irish cream.\n\n### Fresh Figs with Balsamic Cream Sauce\n\n**1 cup nonfat cottage cheese**\n\n**2 tablespoons nonfat sour cream**\n\n**1 \u20442 tablespoon balsamic vinegar**\n\n**6 figs, quartered**\n\n**_1 teaspoon sugar substitute_ **\n\n**_Cracked black pepper to taste_ **\n\n* * *\n\n**1.** In a food processor, combine the cottage cheese, sour cream, balsamic vinegar, and sugar substitute. Process until smooth.\n\n**To Serve:** Arrange the fig quarters on plates and pour the sauce over the figs. Garnish with pepper.\n\n**Servings:** 2\n\n### Apple Wedges with Cinnamon Cream\n\n**1 Granny Smith apple, cored and cut into wedges**\n\n**1 teaspoon freshly squeezed lemon juice**\n\n**1 cup nonfat sour cream**\n\n**3 \u20444 cup nonfat cream cheese, softened**\n\n**_2 teaspoons ground cinnamon_ **\n\n**_2 teaspoons sugar substitute_ **\n\n* * *\n\n**1.** Toss the apple wedges with the lemon juice. Whisk together the sour cream, cream cheese, cinnamon, and sugar substitute. Microwave for 30 seconds.\n\n**To Serve:** Place the apple wedges on plates and top with the cinnamon cream.\n\n**Servings:** 2\n\n### Sauteed Peaches with Cheese\n\n**1 3\u20448 pounds peaches, pitted and cut into wedges**\n\n**1 cup nonfat ricotta cheese**\n\n**_Cooking oil spray_ **\n\n**_2 teaspoons sugar substitute_ **\n\n* * *\n\n**1.** Coat a nonstick skillet with cooking spray and heat the skillet. Place the peach wedges flat on the skillet and cook until they begin to soften. Turn each wedge over and cook other side.\n\n**To Serve:** Spoon nonfat ricotta into small bowls and top with the peach wedges. Sprinkle with the sugar substitute and serve immediately.\n\n**Servings:** 2\n\n**Note:** If you do not have peaches, substitute apples or pears.\n\n### Gelatin with Berries and Yogurt\n\n**1 package sugar-free raspberry-flavor gelatin**\n\n**2 \u20443 cup nonfat plain yogurt**\n\n**3 cups quartered strawberries**\n\n**2 cups raspberries**\n\n**1 \u20443 cup blueberries**\n\n* * *\n\n**1.** Prepare the gelatin according to the package instructions. Pour the gelatin mixture into ice cream bowls and refrigerate until set.\n\n**To Serve:** Spoon yogurt on top of the gelatin and serve with strawberries, raspberries, and blueberries.\n\n**Servings:** 2\n\n### Raspberry Gelatin with Cottage Cheese\n\n**2 packages sugar-free raspberry-flavor gelatin**\n\n**3 1\u20442 cups fresh raspberries**\n\n**1 cup nonfat cottage cheese**\n\n* * *\n\n**1.** Prepare the raspberry gelatin according to the package instructions. Pour the gelatin into pliable ice cube molds and place one raspberry in each cavity. Refrigerate until set.\n\n**To Serve:** Unmold the gelatin cubes into a serving bowl. Gently toss the cottage cheese and remaining raspberries with the gelatin. Serve immediately.\n\n**Servings:** 2\n\n### Lemon Yogurt with Kiwi\n\n**1 cup nonfat plain yogurt**\n\n**1 cup nonfat sour cream**\n\n**1 tablespoon freshly squeezed lemon juice**\n\n**2 sprigs fresh mint**\n\n**2 kiwifruits, peeled and diced**\n\n**_1 teaspoon sugar substitute_ **\n\n* * *\n\n**1.** Whisk together the yogurt, sour cream, lemon juice, and sugar substitute. Chop one of the mint sprigs and combine with the kiwi.\n\n**To Serve:** Place the yogurt mixture in small bowls and top with the kiwi. Garnish with the remaining mint leaves.\n\n**Servings:** 2\n\n### Lemon Pie\n\n**1 tablespoon unflavored gelatin powder**\n\n**1 \u20442 cup nonfat cream cheese, softened**\n\n**1 \u20442 cup nonfat sour cream**\n\n**_1 \u20444 cup warm water_ **\n\n**_2 teaspoons sugar substitute_ **\n\n**_2 teaspoons lemon extract_ **\n\n**_2 teaspoons lemon zest_ **\n\n* * *\n\n**1.** In a bowl, dissolve the gelatin powder in the warm water. Stir in the cream cheese, sour cream, sugar substitute, and lemon extract until the mixture is smooth. Spoon into a pie plate and refrigerate until set.\n\n**To Serve:** Spoon the lemon mixture into dishes and garnish with lemon zest.\n\n**Servings:** 2\n\n**NOTE:** Unflavored gelatin is sold in most markets in the pudding and gelatin section.\n\n### Chocolate-Mint Shakes\n\n**3 1\u20442 cups quartered strawberries**\n\n**2 1\u20444 cups nonfat milk**\n\n**3 \u20444 scoop protein powder**\n\n**1 tablespoon unsweetened cocoa powder**\n\n**1 sprig fresh mint**\n\n**_1 1\u20442 teaspoons sugar substitute_ **\n\n**_Crushed ice_ **\n\n* * *\n\n**1.** Place the strawberries, milk, protein powder, cocoa powder, sugar substitute, and mint in a blender and pulse until smooth.\n\n**To Serve:** Pour over crushed ice in tall glasses and serve immediately.\n\n**Servings:** 2\n\n### Passion Fruit and Tangerine Shakes\n\n**8 passion fruits**\n\n**1 cup tangerine sections**\n\n**2 scoops protein powder (100% whey)**\n\n**2 3\u20444 cups nonfat milk**\n\n**_2 teaspoons sugar substitute_ **\n\n**_Crushed ice_ **\n\n* * *\n\n**1.** Cut the passion fruits in half and scoop out the pulp and seeds. Place the passion fruit, tangerine sections, protein powder, and sugar substitute in a blender. Add the milk and pulse until smooth.\n\n**To Serve:** Pour over crushed ice in tall glasses.\n\n**Servings:** 2\n\n**NOTE:** If you are unable to find fresh passion fruits, buy passion fruit pulp from the frozen section of your local market.\n\n### Tropical Berry Protein Shakes\n\n**2 passion fruits**\n\n**1 cup raspberries**\n\n**3 \u20444 cup nonfat milk**\n\n**1 scoop protein powder (100% whey)**\n\n**_3 teaspoons sugar substitute_ **\n\n**_1 \u20442 cup water_ **\n\n**_Crushed ice_ **\n\n* * *\n\n**1.** Cut the passion fruits in half and spoon out the pulp. In a blender, combine the passion fruit pulp, raspberries, milk, protein powder, and sugar substitute. Add the water and pulse until smooth.\n\n**To Serve:** Pour over crushed ice in tall glasses.\n\n**Servings:** 2\n\n**NOTE:** If you are not able to find passion fruits in your local market, use 3 tablespoons freshly squeezed orange juice.\n\n### Berry Protein Shakes\n\n**2 cups nonfat milk**\n\n**1 1\u20442 cups raspberries**\n\n**1 cup strawberries**\n\n**1 \u20442 scoop protein powder (100% whey)**\n\n**_1 teaspoon vanilla extract_ **\n\n**_Crushed ice_ **\n\n* * *\n\n**1.** Combine the milk, raspberries, strawberries, protein powder, and vanilla extract in a blender. Blend until smooth.\n\n**To Serve:** Pour over crushed ice in tall glasses and serve immediately.\n\n**Servings:** 2\n\n**NOTE:** Look for 100% whey protein powder in health food stores or in the fitness section of your local market. Or use soy protein powder.\n\n## Meal 3. Lunch\n\n### Antipasto\n\n**4 cups canned artichoke hearts, drained**\n\n**2 cups diced tomatoes**\n\n**3 \u20444 cup cubed nonfat mozzarella cheese**\n\n**2 ounces deli-style fat-free turkey, cubed**\n\n**1 cup fat-free balsamic vinaigrette**\n\n**_Cracked black pepper to taste_ **\n\n**_1 teaspoon dried basil_ **\n\n* * *\n\n**1.** In a bowl, toss together artichoke hearts, tomatoes, cheese, turkey cubes, balsamic vinaigrette, and pepper. Garnish with dried basil.\n\n**Servings:** 2\n\n### Baked Chicken and Black Bean Quesadillas with Salsa\n\n**2 ounces skinless, boneless chicken breast**\n\n**2 whole grain or whole wheat tortillas**\n\n**1 cup canned black beans, rinsed and drained**\n\n**1 \u20442 cup shredded nonfat mozzarella cheese**\n\n**2 cups salsa**\n\n**_1 \u20442 tablespoon ground cumin_ **\n\n**_1 \u20442 tablespoon paprika_ **\n\n**_1 teaspoon garlic powder_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n**_Cooking oil spray_ **\n\n* * *\n\n**1.** Combine the cumin, paprika, garlic powder, salt, and pepper. Season the chicken breast with the spice mixture. Coat a nonstick skillet with cooking spray and heat the skillet. Add the chicken and cook on medium heat until brown, turning once. Cover the pan and cook for 3 minutes more or until the chicken is fully cooked. Cool the chicken and slice into strips.\n\n**2.** Preheat the toaster oven to 350\u00b0F. Place one tortilla on a cutting board. Arrange the sliced chicken on the tortilla and top with the black beans and mozzarella. Cover with the other tortilla and press down. Bake until the cheese is melted.\n\n**To Serve:** Cut the quesadilla into quarters and serve with salsa.\n\n**Servings:** 2\n\n### Baked Potato Skins with Sloppy Joe\n\n**2 large sweet potatoes**\n\n**6 ounces ground chicken breast**\n\n**1 \u20442 cup tomato sauce**\n\n**1 \u20442 cup ketchup**\n\n**2 cups diced tomatoes**\n\n**_2 tablespoons sloppy joe seasoning_ **\n\n**_1 tablespoon garlic powder_ **\n\n**_1 teaspoon onion powder_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n**_2 teaspoons dried chives_ **\n\n* * *\n\n**1.** Preheat the oven to 375\u00b0F. Wrap the sweet potatoes in foil and bake until tender. Remove potatoes from the oven and let cool for 10 minutes. Slice the sweet potatoes in half lengthwise and scoop out three-fourths of the pulp. The skins must remain intact. Place the potato skins in the oven and bake for another 8 minutes. Remove and set aside.\n\n**2.** In a saucepan, combine the ground chicken, tomato sauce, ketchup, sloppy joe seasoning, garlic powder, onion powder, salt, and pepper. Cook over medium heat for 15 minutes. Add the tomatoes; cook 5 minutes more.\n\n**To Serve:** Place the potato skins on plates and ladle the sloppy joe mixture over the potato skins. Garnish with dried chives.\n\n**Servings:** 2\n\n### Black Bean Gumbo\n\n**1 cup fat-free chicken broth**\n\n**2 ounces skinless, boneless chicken breast, cut into cubes**\n\n**3 cups canned black beans, drained**\n\n**2 cups diced tomatoes**\n\n**_3 cups water_ **\n\n**_2 tablespoons Cajun seasoning_ **\n\n* * *\n\n**1.** In a medium saucepan, combine the water, chicken broth, and cubed chicken breast. Simmer for 15 minutes. Add the black beans, tomatoes, and Cajun seasoning and cook for 3 minutes.\n\n**To Serve:** Ladle the soup into bowls.\n\n**Servings:** 2\n\n**NOTE:** If the soup seems too thick, add a little more chicken broth or water.\n\n### Chicken and Rice Miso Soup\n\n**4 cups fat-free chicken broth**\n\n**2 ounces skinless, boneless chicken breast**\n\n**2 tablespoons miso paste or instant miso soup**\n\n**1 3\u20444 cups cooked brown rice**\n\n**1 cup thinly sliced scallions**\n\n* * *\n\n**1.** Combine the chicken broth, chicken breast, and miso paste or miso soup packet. Simmer about 20 minutes or until the chicken breast is no longer pink. Remove the chicken breast from the broth and dice it into small pieces.\n\n**2.** Add the brown rice and the diced chicken to the soup and cook for 2 minutes.\n\n**To Serve:** Ladle the soup into bowls and garnish with scallions.\n\n**Servings:** 2\n\n**NOTE:** Brown rice can be purchased precooked and heated 1 minute in the microwave. This alternative will save you time.\n\n### Chicken Fingers and French Fries\n\n**1 large sweet potato, peeled and cut into sticks**\n\n**5 1\u20442 ounces skinless, boneless chicken breast, cut into strips**\n\n**3 egg whites**\n\n**4 slices stale whole grain bread, ground**\n\n**4 cups broccoli florets**\n\n**_Cooking oil spray_ **\n\n**_1 teaspoon ground cinnamon_ **\n\n**_Salt and cracked black pepper_ _to taste_ **\n\n**_2 tablespoons Mrs. Dash Original Blend seasoning_ **\n\n* * *\n\n**1.** Preheat the oven or toaster oven to 375\u00b0F. Spread the sweet potato sticks on a sheet pan and lightly coat with cooking spray. Season with cinnamon, salt, and pepper. Bake for 25 minutes.\n\n**2.** Dip the chicken strips into the egg whites, then drain off excess egg and coat with the ground bread. Coat a nonstick skillet with cooking spray and heat the skillet. Add the breaded chicken and cook until brown, turning once. Cook over medium-low heat for 5 minutes more.\n\n**3.** Place the broccoli in a bowl with a little water and salt. Microwave for 2 minutes. Remove from the microwave and season with Mrs. Dash.\n\n**To Serve:** Place the chicken fingers, sweet potato fries, and broccoli on plates.\n\n**Servings:** 2\n\n### Chinese Chicken Wraps with Peanut-Soy Sauce\n\n**5 ounces skinless, boneless chicken breast**\n\n**1 teaspoon unsalted peanut butter**\n\n** 3\u20444 cup shredded carrots**\n\n**4 large whole grain or whole wheat tortillas**\n\n**_3 \u20444 cup low-sodium soy sauce_ **\n\n**_1 teaspoon ground ginger_ **\n\n**_1 teaspoon ground coriander_ **\n\n**_1 teaspoon dried chives_ **\n\n**_1 teaspoon sugar substitute_ **\n\n* * *\n\n**1.** Place the chicken breast in a saucepan, cover with water, and simmer until fully cooked. Remove from the heat and cut the chicken into small cubes.\n\n**2.** Whisk together the soy sauce, ginger, coriander, chives, sugar substitute, and peanut butter. Place the chicken and shredded carrots in a Ziploc bag and add the soy sauce mixture. Seal and refrigerate for 15 minutes. Drain the chicken mixture.\n\n**To Serve:** Place some of the chicken mixture on each tortilla. Roll tightly and cut into pieces. Serve warm or cold.\n\n**Servings:** 2\n\n### Harley's Sweet Potato Melt\n\n**2 large sweet potatoes**\n\n**3 \u20444 cup water-pack canned tuna, drained**\n\n**1 \u20442 cup nonfat mayonnaise**\n\n**1 \u20442 cup shredded part-skim-milk mozzarella cheese**\n\n**_1 teaspoon Mrs. Dash roasted garlic and onion seasoning_ **\n\n**_Lemon pepper to taste_ **\n\n* * *\n\n**1.** Microwave the sweet potatoes for 31\u20442 minutes each or until tender. Cut in half and set aside.\n\n**2.** Preheat the broiler or toaster oven broiler to medium. In a mixing bowl, combine the tuna, mayonnaise, Mrs. Dash, and lemon pepper.\n\n**To Serve:** Place the tuna mixture on top of the sweet potato halves. Top with cheese and broil until the cheese has melted.\n\n**Servings:** 2\n\n**Note:** Tuna also comes with different flavorings. Try smoked tuna or tuna teriyaki by Starkist Creations.\n\n### Mediterranean-Style Chicken and Quinoa Salad\n\n**6 ounces skinless, boneless chicken breast**\n\n**3 \u20444 cup quinoa**\n\n**1 1\u20443 cups diced, seeded plum tomatoes**\n\n**1 cup chopped fresh parsley**\n\n**3 tablespoons freshly squeezed lemon juice**\n\n**_Salt and cracked black pepper to taste_ **\n\n* * *\n\n**1.** Place the chicken breast and 2 cups of water in a small saucepan and cook for 8 minutes. Let the chicken cool and dice it. Set aside.\n\n**2.** Put the quinoa and 11\u20442 cups of water into a saucepan and simmer about 15 minutes or until the liquid is absorbed. Stir occasionally. Combine the chicken, quinoa, tomatoes, parsley, lemon juice, salt, and pepper; toss gently.\n\n**To Serve:** Spoon the salad into shallow bowls.\n\n**Servings:** 2\n\n### Mexican Chicken Salad with Spicy Salsa Dressing\n\n**6 ounces skinless, boneless chicken breast**\n\n**1 cup nonfat sour cream**\n\n**1 cup salsa**\n\n**1 small head iceberg lettuce, coarsely chopped**\n\n**1 1\u20442 cups canned corn, drained**\n\n**_1 teaspoon fajita seasoning mix_ **\n\n**_1 pinch cumin_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n* * *\n\n**1.** Combine the fajita seasoning mix, cumin, salt, and pepper. Coat the chicken breast with the seasoning mixture. Microwave the chicken for 6 minutes. Remove from the microwave and set aside to cool slightly.\n\n**2.** In a blender, combine the sour cream and salsa. Pulse until smooth. If the dressing is too thick, add a little water.\n\n**To Serve:** Cut the chicken breast into 1\u20442-inch pieces and toss it with the lettuce, corn, and salsa dressing. Serve immediately.\n\n**Servings:** 2\n\n### Minestrone\n\n**4 cups chicken broth**\n\n**2 cups canned stewed tomatoes**\n\n**1 1\u20443 cups thinly sliced button mushrooms**\n\n**1 1\u20443 cups cooked cannellini beans**\n\n**1 cup diced smoked turkey breast**\n\n**_2 tablespoons dried basil_ **\n\n**_1 teaspoon sugar substitute_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n* * *\n\n**1.** In a saucepan combine the chicken broth, stewed tomatoes, sliced mushrooms, cannellini beans, turkey breast, basil, sugar substitute, salt, and pepper. Bring to a boil. Lower the temperature and simmer about 15 minutes or until the soup is reduced to half its volume.\n\n**To Serve:** Ladle into soup bowls and serve immediately.\n\n**Servings:** 2\n\n### Mixed Greens with Turkey and Cheese Quesadillas\n\n**1 \u20444 pound deli-style sliced fat-free turkey**\n\n**2 whole grain or whole wheat tortillas**\n\n**1 \u20442 cup shredded nonfat mozzarella cheese**\n\n**3 cups mixed greens**\n\n**1 cup fat-free blue cheese salad dressing or other fat-free dressing**\n\n**_Cooking oil spray_ **\n\n* * *\n\n**1.** Place the turkey slices on one side of each tortilla. Sprinkle with cheese and fold tortilla in half. Press tightly to secure the filling.\n\n**2.** Coat a nonstick skillet with cooking spray and heat the skillet. Cook the tortillas for 1 minute on each side or until the cheese is melted. Slide the quesadillas onto a cutting board. Slice each into three or four triangles. Set aside.\n\n**To Serve:** Toss the mixed greens with the salad dressing. Place the greens in the center of the plates. Arrange the quesadilla triangles around the salads.\n\n**Servings:** 2\n\n### Mushroom-Barley Risotto\n\n**3 cups sliced button mushrooms**\n\n**1 cup nonfat beef broth**\n\n**1 \u20442 cup pearl barley**\n\n**3 ounces shrimp, peeled, deveined, and cut in half**\n\n**1 cup nonfat sour cream**\n\n**_4 cups water_ **\n\n**_1 tablespoon dried sage_ **\n\n**_1 tablespoon garlic powder_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n* * *\n\n**1.** In a large saucepan, combine the water, mushrooms, beef broth, and barley. Simmer for 15 minutes or until most of the liquid has been absorbed. Stir in the shrimp, sour cream, sage, garlic powder, salt, and pepper. Simmer for 2 minutes.\n\n**To Serve:** Ladle the risotto into bowls and serve hot.\n\n**Servings:** 2\n\n### Open-Face Turkey BLT\n\n**2 strips turkey bacon**\n\n**1 head romaine lettuce, leaves washed and patted dry**\n\n**6 ounces deli-style fat-free turkey, thinly sliced**\n\n**2 tomatoes, thinly sliced**\n\n**1 tablespoon red wine vinegar**\n\n**_Salt and cracked black pepper to taste_ **\n\n* * *\n\n**1.** Microwave the turkey bacon for 3 minutes or until crisp. Crumble the bacon and set aside. Lay the romaine leaves flat on a plate. Layer with the sliced turkey, sliced tomatoes, and the turkey bacon. Season with salt and pepper and drizzle with red wine vinegar.\n\n**To Serve:** Place the bun-less BLTs on plates and serve immediately.\n\n**Servings:** 2\n\n### Pink Pizza\n\n**4 large whole grain or whole wheat tortillas**\n\n**1 cup tomato sauce**\n\n**3 \u20444 cup nonfat ricotta cheese**\n\n**1 cup chopped sun-dried tomatoes**\n\n**3 \u20444 cup shredded nonfat mozzarella cheese**\n\n* * *\n\n**1.** Preheat the oven to 375\u00b0F. Place the tortillas on a baking sheet and bake for 2 minutes. Remove from the oven. Ladle half of the tomato sauce over the tortillas and spread with the ricotta cheese. Ladle on the remaining tomato sauce and sprinkle with sun-dried tomatoes and shredded mozzarella. Bake until the cheese is melted.\n\n**To Serve:** Cut the pizzas into slices and serve immediately.\n\n**Servings:** 2\n\n### Portobello and Turkey Stacks\n\n**4 ounces skinless, boneless turkey breast**\n\n**4 portobello mushrooms, stems removed**\n\n**1 tomato, thinly sliced**\n\n**1 ounce fat-free mozzarella cheese, thinly sliced**\n\n**10 whole grain or multigrain crackers**\n\n**_Salt and cracked black pepper to taste_ **\n\n**_Olive oil cooking spray_ **\n\n**_1 teaspoon dried basil_ **\n\n* * *\n\n**1.** Preheat the broiler to medium. Season the turkey breast with salt and pepper. Coat a nonstick skillet with cooking spray and heat the skillet. Add the turkey breast and cook until fully cooked, turning once. Slice thinly and set aside.\n\n**2.** Lightly coat the mushroom caps with cooking spray and sprinkle with salt and pepper. Coat a nonstick skillet with cooking spray and heat the skillet. Add the mushrooms and cook until tender, turning once. Set aside.\n\n**To Assemble:** Place the mushrooms on a sheet pan and top with the turkey. Place the tomato on top of the turkey and season with salt and pepper. Top with mozzarella. Broil until the cheese is melted.\n\n**To Serve:** With a spatula, carefully slide the turkey stacks onto serving plates. Sprinkle with dried basil. Serve with crackers.\n\n**Servings:** 2\n\n### Salad Ni\u00e7oise\n\n**12 ounces sweet potato**\n\n**6 cups green beans, cooked**\n\n**1 1\u20442 cups water-pack tuna, drained**\n\n**2 hard-boiled egg whites, chopped**\n\n**1 \u20442 cup fat-free Italian salad dressing**\n\n**_Salt and cracked black pepper to taste_ **\n\n* * *\n\n**1.** Microwave the sweet potato for 3 minutes. Peel the potato and slice into 1\u20442-inch rounds. Set aside.\n\n**To Serve:** Place the sweet potato rounds on serving plates. Arrange the green beans beside the sweet potato. Sprinkle the tuna and chopped egg whites around the green beans and sweet potatoes. Season with salt and pepper and drizzle with the salad dressing. Served chilled.\n\n**Servings:** 2\n\n### Salmon Tartare with Arugula\n\n**6 ounces salmon fillets, finely diced**\n\n**1 \u20444 cup capers, rinsed and chopped**\n\n**2 lemons**\n\n**1 pound arugula**\n\n**12 multigrain crackers**\n\n**_1 tablespoon onion and garlic salt_ **\n\n**_1 tablespoon garlic powder_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n* * *\n\n**1.** In a bowl, combine the diced salmon, capers, juice from 1 of the lemons, onion and garlic salt, garlic powder, salt, and pepper.\n\n**To Serve:** Arrange the arugula on a plate and season with salt and pepper. Sprinkle the salmon mixture over the arugula. Cut remaining lemon into wedges and garnish with wedges. Serve with crackers.\n\n**Servings:** 2\n\n### Greek-Style Shrimp and Spinach Salad\n\n**1 \u20442 cup freshly squeezed lemon juice**\n\n**1 1\u20442 ounces feta cheese, crumbled**\n\n**5 ounces shrimp, peeled and deveined**\n\n**1 pound spinach leaves**\n\n**4 cups orange sections**\n\n**_2 teaspoons ground oregano_ **\n\n**_1 teaspoon ground coriander_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n**_1 1\u20442 tablespoons Mrs. Dash seasoning_ **\n\n**_Cooking oil spray_ **\n\n* * *\n\n**1.** Whisk together the lemon juice, feta cheese, oregano, coriander, salt, and pepper. Set aside. Season the shrimp with Mrs. Dash, salt, and pepper. Coat a nonstick skillet with cooking spray and heat the skillet until very hot. Add the shrimp and cook about 2 minutes or until shrimp are opaque.\n\n**To Serve:** Toss the feta mixture with the spinach and arrange on plates. Top with the shrimp and garnish with the orange sections.\n\n**Servings:** 2\n\n### Smoked Salmon Pizza\n\n**4 whole grain or whole wheat tortillas**\n\n**1 cup nonfat cream cheese, softened**\n\n**2 tomatoes, thinly sliced**\n\n**4 ounces thinly sliced smoked salmon**\n\n**2 \u20443 cup thinly sliced red onion**\n\n**_Salt and cracked black pepper to taste_ **\n\n* * *\n\n**1.** Preheat the oven to 375\u00b0F. Place the tortillas on a baking sheet and bake for 4 minutes or until crisp.\n\n**To Serve:** Spread the cream cheese on the tortillas and top with the tomato slices. Arrange the smoked salmon over the tomato and sprinkle with the red onion. Season with salt and pepper. Cut into wedges.\n\n**Servings:** 2\n\n### Snapper Ceviche with Sweet Potato Rounds\n\n**2 medium sweet potatoes**\n\n**10 ounces thinly sliced snapper fillet**\n\n**1 1\u20444 cups freshly squeezed lemon juice**\n\n**1 \u20442 cup thinly sliced red onion**\n\n**3 tablespoons chopped fresh cilantro**\n\n**_1 teaspoon ground cumin_ **\n\n**_1 pinch sugar substitute_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n* * *\n\n**1.** Microwave the sweet potatoes for 3 minutes each. Let the sweet potatoes cool, then peel. Slice into 1\u20442-inch rounds and set aside.\n\n**2.** In a Ziploc bag, combine the snapper, lemon juice, red onion, cilantro, cumin, sugar substitute, salt, and pepper. Marinate the fish in the refrigerator for 15 to 20 minutes or until it is completely pickled. (The fish will be white and firm to the bite.)\n\n**To Serve:** Arrange the sweet potato rounds on plates and spoon the snapper mixture on top.\n\n**Servings:** 2\n\n### Green Bean Salad with Tuna and Grapefruit-Scallion Vinaigrette\n\n**2 pounds green beans, stems removed**\n\n**1 cup rice wine vinegar**\n\n**2 small grapefruit, sectioned**\n\n**8 ounces canned tuna, drained and flaked**\n\n**1 \u20442 bunch scallions, bias sliced**\n\n**_1 teaspoon ground ginger_ **\n\n**_1 teaspoon garlic powder_ **\n\n**_1 teaspoon sesame seeds_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n* * *\n\n**1.** In a saucepan, cook the green beans with a pinch of salt in boiling water for 2 minutes. Drain the beans and place them in an ice bath until cool. Drain again.\n\n**2.** In a bowl, whisk together the rice vinegar, ginger, garlic powder, sesame seeds, salt, and pepper. Add two grapefruit sections to the vinaigrette and whisk until the segments fall apart. Place the green beans in a large bowl and toss with the tuna, scallions, and vinaigrette.\n\n**To Serve:** Place salad on plates and garnish with the remaining grapefruit segments.\n\n**Servings:** 2\n\n### Stuffed Mushrooms and Greens\n\n**2 cups lump crabmeat**\n\n**1 3\u20444 cups prepared bulgur**\n\n**6 large button mushroom caps**\n\n**5 cups mixed greens**\n\n**1 \u20442 cup fat-free red wine vinaigrette salad dressing**\n\n**_2 teaspoons paprika_ **\n\n**_1 teaspoon dried mint_ **\n\n**_1 teaspoon garlic powder_ **\n\n**_1 teaspoon onion salt_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n**_Cooking oil spray_ **\n\n* * *\n\n**1.** Preheat the oven to 375\u00b0F. Combine crabmeat, bulgur, paprika, mint, garlic powder, onion salt, salt, and pepper. It should be just moist enough to hold together. Pack the crab and bulgur mixture into each mushroom cap. Lightly coat with cooking spray. Place in a baking pan and bake for 10 minutes.\n\n**To Serve:** Season the greens with salt and pepper and toss with the red wine vinaigrette. Arrange the greens on plates and put the warm mushrooms on top. Serve immediately.\n\n**Servings:** 2\n\n**NOTE:** To prepare the bulgur, soak it in 3 cups water for 20 minutes, then drain.\n\n### Tuscan Tomato Soup\n\n**1 cup canned stewed tomatoes**\n\n**4 cups chicken broth**\n\n**3 cups nonfat sour cream**\n\n**_Cooking oil spray_ **\n\n**_1 tablespoon garlic powder_ **\n\n**_1 tablespoon onion powder_ **\n\n**_1 teaspoon sugar substitute_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n**_1 tablespoon dried basil_ **\n\n**1.** Preheat the oven to 400\u00b0F. Place tomatoes in a baking dish and lightly coat with cooking spray. Roast for 15 minutes. Remove from the oven and set aside.\n\n**2.** In a saucepan, combine the chicken broth, garlic powder, onion powder, sugar substitute, salt, and pepper. Simmer the mixture until it is reduced to half its volume. In a blender, pulse the tomatoes until chunky. Stir the tomatoes into the chicken broth mixture. Whisk in the sour cream.\n\n**To Serve:** Ladle the soup into bowls and garnish with dried basil.\n\n**Servings:** 2\n\n## Meal 5. Dinner\n\n### 5-Factor Lasagna\n\n**2 small eggplants, thinly sliced lengthwise**\n\n**1 cup tomato sauce**\n\n**1 pound tomatoes, thinly sliced**\n\n**1 cup nonfat ricotta cheese, softened**\n\n**3 \u20444 cup shredded nonfat mozzarella cheese**\n\n**_Salt and cracked black pepper to taste_ **\n\n**_Cooking oil spray_ **\n\n**_2 tablespoons dried basil_ **\n\n**_2 tablespoons Italian seasoning_ **\n\n* * *\n\n**1.** Preheat the oven to 400\u00b0F. Season the eggplant slices with salt and pepper and roast for 15 minutes.\n\n**2.** Coat a glass baking dish with cooking spray and cover the bottom with eggplant slices. Ladle some of the tomato sauce over the eggplant and top with some of the tomato slices. Sprinkle the tomato slices with basil, Italian seasoning, salt, and pepper. Spread ricotta over the tomatoes. Sprinkle with shredded mozzarella. Repeat the layers, ending with mozzarella.\n\n**3.** Bake for 20 minutes. Increase the heat to broil until the cheese turns golden.\n\n**To Serve:** Cut the lasagna into slices and garnish with additional Italian seasoning.\n\n**Servings:** 2\n\n### Argentine-Style Steak Salad with Watercress and Mustard-Cilantro Vinaigrette\n\n**1 \u20444 cup white wine vinegar**\n\n**1 1\u20442 teaspoons Dijon mustard**\n\n**6 ounces bison steak**\n\n**4 bunches watercress, washed and patted dry**\n\n**5 radishes, thinly sliced**\n\n**_2 tablespoons dried cilantro_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n**_1 teaspoon ground cumin_ **\n\n**_1 teaspoon ground coriander_ **\n\n**_Cooking oil spray_ **\n\n* * *\n\n**1.** In a mixing bowl, whisk together the vinegar, Dijon mustard, cilantro, salt, and pepper. Set aside.\n\n**2.** Season the bison steak with cumin, coriander, salt, and pepper. Coat a nonstick skillet with cooking spray and heat the skillet. Add the steak and sear on each side to desired doneness, turning once. Remove the steak from the skillet and let stand for 1 minute. Slice the steak.\n\n**To Serve:** Toss the watercress with the vinaigrette and place it on plates. Top with steak slices and garnish with radish slices.\n\n**Servings:** 2\n\n**NOTE:** Bison is a very lean meat, and has the best flavor and texture when cooked to medium-rare.\n\n### Chicken Chow Mein\n\n**6 ounces skinless, boneless chicken breast, cut into strips**\n\n**5 cups thinly sliced carrots**\n\n**3 cups snow peas, stems removed**\n\n**2 cups bean sprouts**\n\n**_Cooking oil spray_ **\n\n**_1 tablespoon sesame seeds_ **\n\n**_1 tablespoon garlic powder_ **\n\n**_1 \u20442 cup low-sodium soy sauce_ **\n\n* * *\n\n**1.** Coat a wok with cooking spray and heat the wok. Add the chicken strips and stir-fry for 2 minutes. Add the carrots, snow peas, bean sprouts, sesame seeds, and garlic powder. Stir-fry for 1 minute. Add the soy sauce and cook for 1 minute.\n\n**To Serve:** Ladle into shallow bowls and garnish with a few additional sesame seeds.\n\n**Servings:** 2\n\n### Chicken Ropa Vieja\n\n**7 1\u20442 ounces skinless, boneless chicken breast**\n\n**12 ounces tomato sauce**\n\n**1 red bell pepper, seeded and thinly sliced**\n\n**2 1\u20442 cups canned corn, drained**\n\n**2 tablespoons cilantro leaves**\n\n**_1 teaspoon ground cumin_ **\n\n**_1 teaspoon sugar substitute_ **\n\n**_1 bay leaf_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n* * *\n\n**1.** In a saucepan, bring salted water to a boil. Add the chicken and cook over medium heat for 10 to 15 minutes. Drain the cooked chicken and cool. Shred the chicken.\n\n**2.** In a saucepan, combine the tomato sauce, bell pepper slices, cumin, sugar substitute, bay leaf, salt, and cracked black pepper. Add the shredded chicken and cook for 5 minutes. Microwave the corn for 2 minutes. Season with salt and cracked black pepper. Discard bay leaf.\n\n**To Serve:** Spoon the chicken mixture into bowls and top with corn. Garnish with fresh cilantro.\n\n**Servings:** 2\n\n### Chiles Rellenos with Brown Rice\n\n**2 large poblano peppers**\n\n**6 ounces ground turkey breast**\n\n**1 \u20442 cup canned black beans, rinsed and drained**\n\n**3 tablespoons tomato paste**\n\n**1 1\u20442 cups cooked brown rice**\n\n**_Cooking oil spray_ **\n\n**_1 tablespoon ground cumin_ **\n\n**_1 pinch sugar substitute_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n* * *\n\n**1.** Preheat the oven to 400\u00b0F. Lightly coat the poblanos with cooking spray. Place them on a baking sheet and roast for 20 minutes or until the skins begin to char. Remove the peppers from the heat. Immediately place them in a bowl and cover with plastic wrap to cool.\n\n**2.** Carefully peel the poblanos and slit each one through one side. Remove the seeds with a paring knife and rinse the peppers under cold water to wash out any remaining seeds. Leave the peppers as intact as possible. Set the peppers aside.\n\n**3.** Coat a nonstick skillet with cooking spray and heat skillet. Add the ground turkey and cook until no longer pink. Add the black beans, tomato paste, cumin, sugar substitute, salt, and cracked black pepper. Stir until well mixed. Spoon the turkey filling into the peppers.\n\n**To Serve:** Spoon the cooked rice onto plates and top with the chiles. Garnish with fresh cilantro leaves, if desired.\n\n**Servings:** 2\n\n### Country-Style Ham Steaks with Yams and Corn on the Cob\n\n**9 ounces ham steaks**\n\n**2 large yams**\n\n**3 ears corn on the cob, husked and cut in half**\n\n**_Cooking oil spray_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n* * *\n\n**1.** Coat a nonstick skillet with cooking spray and heat the skillet. Add the ham steaks and sear on each side until golden brown. Microwave the yams for about 31\u20442 minutes each. Peel and slice them into rounds and season with salt and pepper. Cook the corn in boiling water for 3 minutes.\n\n**To Serve:** Place the ham steaks on plates and serve with sliced yams and corn on the cob.\n\n**Servings:** 2\n\n### Cream of Broccoli Soup with Sauteed Shrimp\n\n**2 1\u20448 cups chicken broth**\n\n**8 ounces broccoli florets**\n\n**2 1\u20444 cups chopped carrots**\n\n**3 \u20444 cup leeks, white part only, coarsely chopped**\n\n**8 ounces shrimp, peeled, deveined, and cut into pieces**\n\n**_1 tablespoon garlic powder_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n**_Cooking oil spray_ **\n\n* * *\n\n**1.** In a large saucepan, bring the chicken broth to a boil and add the broccoli, carrots, leeks, garlic powder, salt, and pepper. Cook for 3 minutes or until the broccoli is bright green and tender to the fork. Remove from the heat and let cool slightly.\n\n**2.** Ladle a portion of the broccoli mixture into a blender and pulse until it reaches a creamy consistency. Pour the blended soup into a large saucepan. Repeat until all the broccoli mixture is blended. Reheat the soup.\n\n**3.** Coat a nonstick skillet with cooking spray and heat the skillet. Add the shrimp and season with salt and pepper. Cook about 2 minutes.\n\n**To Serve:** Ladle the soup into bowls and garnish with shrimp.\n\n**Servings:** 2\n\n### Creamy Lemon-Ginger Halibut with Corn on the Cob\n\n**2 lemons**\n\n**3 \u20444 cup nonfat plain yogurt**\n\n**8 ounces boneless halibut fillet, cut into two portions**\n\n**2 ears corn on the cob, husks removed**\n\n**_1 teaspoon ground coriander_ **\n\n**_1 1\u20442 teaspoons ground ginger_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n* * *\n\n**1.** Zest one lemon and squeeze out the juice. Whisk together the lemon zest, lemon juice, yogurt, coriander, ginger, salt, and pepper. Place the halibut fillets in a Ziploc bag and pour three-fourths of the yogurt mixture over the fish. Marinate for 5 minutes.\n\n**2.** Meanwhile, place the corn in a plastic container with water. Microwave for 5 minutes. Season with salt and pepper and set aside.\n\n**3.** Remove the fish from the marinade and place in a plastic container. Cover and microwave for 6 minutes or until fish flakes when tested with a fork.\n\n**To Serve:** Place the fish and corn on plates. Spoon a little of the remaining yogurt mixture over the fish. Cut the remaining lemon into wedges and serve with fish.\n\n**Servings:** 2\n\n### Crispy Chicken Tostadas\n\n**6 ounces skinless, boneless chicken breast**\n\n**1 1\u20442 cups thinly sliced Spanish onion**\n\n**2 tablespoons freshly squeezed lime juice**\n\n**4 medium whole grain or whole wheat tortillas**\n\n**2 tablespoons nonfat sour cream**\n\n**_Salt and cracked black pepper to taste_ **\n\n**_1 teaspoon olive oil_ **\n\n**_4 teaspoons dried cilantro_ **\n\n**_1 teaspoon ground cumin_ **\n\n* * *\n\n**1.** Place chicken in large saucepan and add water to cover, salt, and pepper. Cook over medium heat for 25 minutes or until the chicken is fully cooked. Remove the chicken, cool, and shred.\n\n**2.** Heat the olive oil in a nonstick skillet. Add the onion and cook for 1 minute. Add the shredded chicken and stir constantly until it crisps. When most of the liquid has evaporated, drizzle the lime juice over the chicken and season with cilantro, cumin, salt, and pepper. Set aside. Bake the tortillas in a 350\u00b0 oven until they are crisp and light golden.\n\n**To Serve:** Place the tortillas on plates. Top with the chicken and sour cream.\n\n**Servings:** 2\n\n### Bison Steak with Cauliflower-Carrot Mash and Brown Rice\n\n**2 1\u20444 cups chopped carrots**\n\n**3 1\u20442 cups cauliflower florets**\n\n**2 tablespoons nonfat sour cream**\n\n**6 1\u20442 ounces bison steak**\n\n**1 1\u20444 cups cooked brown rice**\n\n**_1 tablespoon onion powder_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n**_2 tablespoons Montreal steak seasoning_ **\n\n**_Cooking oil spray_ **\n\n* * *\n\n**1.** In a medium saucepan, cook the carrots in lightly salted boiling water for 2 minutes. Add the cauliflower and cook for 3 minutes or until the cauliflower is tender. Drain the vegetables and place in a food processor. Pulse the vegetables with the sour cream, onion powder, salt, and pepper. Transfer to a bowl with lid and set aside.\n\n**2.** Divide the steak into two portions. Season the steaks with steak seasoning, salt, and pepper. Coat a nonstick skillet with cooking spray and heat the skillet until very hot. Add the steaks and sear on both sides. Then reduce heat to medium-high and cook until they reach the desired doneness (bison is best served medium-rare). Remove from the heat and let the steaks stand for 1 minute.\n\n**To Serve:** Place the cauliflower-carrot mash in the center of the plates. Slice the bison steaks and arrange the slices over the mash. Serve with brown rice.\n\n**Servings:** 2\n\n### Indian-Style Chicken with Curried Yogurt Sauce and Brown Rice\n\n**1 \u20442 cup nonfat plain yogurt**\n\n**1 teaspoon curry powder**\n\n**8 ounces skinless, boneless chicken breast, butterflied and thinly pounded**\n\n**2 cups cooked brown rice**\n\n**2 1\u20442 cups thinly sliced, peeled cucumber**\n\n**_1 \u20442 teaspoon ground coriander_ **\n\n**_1 \u20448 teaspoon ground paprika_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n**_Cooking oil spray_ **\n\n* * *\n\n**1.** Combine the yogurt, curry powder, coriander, paprika, salt, and pepper. Pour three-fourths of the mixture into a Ziploc bag. Add the chicken, seal, and refrigerate for 20 minutes. Drain the chicken and discard the marinade. Coat a nonstick skillet with cooking spray and heat the skillet. Add the chicken and sear on each side until golden brown. Cover the pan and reduce the heat to medium. Cook for 1 minute more and remove from the heat.\n\n**To Serve:** Place the chicken and brown rice on plates. Top the chicken with the remaining yogurt sauce and the sliced cucumbers.\n\n**Servings:** 2\n\n### Lobster and Peas with Tomato-Basil Sauce and Barley\n\n**1 1\u20442 cups tomato sauce**\n\n**1 1\u20444 cups cooked barley**\n\n**1 1\u20444 cups uncooked lobster, coarsely chopped**\n\n**1 \u20442 bunch fresh basil, chopped**\n\n**1 1\u20442 cups frozen green peas**\n\n**_1 teaspoon seafood seasoning mix_ **\n\n**_1 teaspoon sugar substitute_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n* * *\n\n**1.** In a soup pot, combine the tomato sauce, barley, lobster, basil, seafood seasoning mix, sugar substitute, salt, and pepper. Cook and stir for 4 minutes. Taste and adjust the seasonings. Add the peas and cook for 1 to 2 minutes.\n\n**To Serve:** Ladle into shallow bowls and garnish with additional fresh basil leaves.\n\n**Servings:** 2\n\n**NOTE:** For a less expensive dish, you can prepare this recipe with shrimp or chicken breast.\n\n### Salmon with Cucumber-Dill Salad\n\n**9 ounces salmon steak**\n\n**2 large sweet potatoes**\n\n**1 \u20442 cup nonfat sour cream**\n\n**2 tablespoons chopped fresh dill**\n\n**1 cup thinly sliced, peeled cucumber**\n\n**_1 \u20442 tablespoon lemon pepper_ **\n\n**_1 teaspoon paprika_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n**_Cooking oil spray_ **\n\n* * *\n\n**1.** Cut the salmon steak into two portions and season with lemon pepper, paprika, and salt. Coat a nonstick skillet with cooking spray and heat the skillet. Place the salmon steaks in the skillet, skin sides up. Sear for 1 minute, then turn and cook until the salmon flakes when tested with a fork.\n\n**2.** Microwave the sweet potatoes for 31\u20442 minutes each or until tender. Peel and slice into thick rounds. Season with salt and pepper and set aside.\n\n**3.** Meanwhile, whisk together the sour cream, dill, salt, and pepper. Stir in the cucumber slices.\n\n**To Serve:** Spoon the cucumber mixture over the salmon. Serve with sweet potato rounds.\n\n**Servings:** 2\n\n### Scallop Ratatouille\n\n**3 cups canned crushed tomatoes**\n\n**3 cups cubed zucchini**\n\n**3 cups cubed eggplant**\n\n**3 cups quartered button mushrooms**\n\n**1 \u20442 pound small scallops**\n\n**_1 1\u20442 cups water_ **\n\n**_3 tablespoons dried basil_ **\n\n**_2 tablespoons dried oregano_ **\n\n**_1 pinch sugar substitute_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n* * *\n\n**1.** In a soup pot, combine the tomatoes, zucchini, eggplant, mushrooms, water, basil, oregano, sugar substitute, salt, and pepper. Cover and cook over medium heat for 5 minutes. Add the scallops and cook for 21\u20442 minutes more.\n\n**To Serve:** Ladle the soup into bowls and garnish with additional dried basil.\n\n**Servings:** 2\n\n### Seared Halibut with Creamed Spinach and Brown Rice\n\n**5 ounces halibut fillets**\n\n**1 pound spinach leaves**\n\n**1 \u20442 cup nonfat cream cheese**\n\n**1 \u20444 cup nonfat sour cream**\n\n**1 2\u20443 cups cooked brown rice**\n\n**_1 teaspoon lemon pepper_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n**_Cooking oil spray_ **\n\n**_1 tablespoon onion powder_ **\n\n**_2 teaspoons garlic powder_ **\n\n* * *\n\n**1.** Season the halibut fillets with lemon pepper and salt. Coat a nonstick skillet with cooking spray and heat the skillet. Add the halibut and sear on each side. Then cover the pan and cook until the fish flakes when tested with a fork. Set aside.\n\n**2.** Cook the spinach in a nonstick pan over medium heat until wilted. Transfer the spinach to a strainer and press out as much liquid as possible.\n\n**3.** Return the spinach to the pan. Add the cream cheese, sour cream, onion powder, garlic powder, salt, and cracked black pepper. Cook and stir over medium heat until hot.\n\n**4.** Microwave the brown rice for 1 minute.\n\n**To Serve:** Place the brown rice on the center of each plate. Top with halibut and spoon the creamed spinach over the halibut.\n\n**Servings:** 2\n\n### Seared Scallops with Orange Sauce and Broccoli-Cauliflower Saute\n\n**10 ounces large scallops**\n\n**1 pound broccoli florets**\n\n**1 pound cauliflower florets**\n\n**1 cup freshly squeezed orange juice**\n\n**_Cooking oil spray_ **\n\n**_1 teaspoon curry powder_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n* * *\n\n**1.** Coat a nonstick skillet with cooking spray and heat the skillet. Add the scallops and season with the curry powder, salt, and pepper. Cook until scallops are golden brown. Add the broccoli, cauliflower, and orange juice. Cook until the broccoli is bright green and tender to the fork.\n\n**To Serve:** Spoon the scallops and vegetables into shallow bowls. Drizzle with the orange sauce.\n\n**Servings:** 2\n\n### Shrimp and Tofu Soup\n\n**1 1\u20442 cups cooked brown rice**\n\n**4 ounces shrimp, peeled, deveined, and cut in half**\n\n**4 ounces firm tofu cut into 1-inch cubes**\n\n**3 tablespoons miso paste or instant miso soup**\n\n**_4 cups water_ **\n\n**_1 cup low-sodium soy sauce_ **\n\n* * *\n\n**1.** In a large saucepan, combine the water, rice, soy sauce, shrimp, tofu, and miso. Simmer for 2 minutes or until the shrimp are opaque.\n\n**To Serve:** Ladle into soup bowls.\n\n**Servings:** 2\n\n### Shrimp and Rice Stir-Fry\n\n**1 pound shrimp, peeled and deveined**\n\n**1 1\u20442 cups cooked brown rice**\n\n**2 cups broccoli florets**\n\n**1 \u20444 cup slivered scallions**\n\n**_Cooking oil spray_ **\n\n**_1 \u20442 tablespoon garlic powder_ **\n\n**_1 \u20444 cup low-sodium soy sauce_ **\n\n**_2 teaspoons sesame seeds_ **\n\n* * *\n\n**1.** Remove the tails from the shrimp and cut the shrimp into bite-size pieces.\n\n**2.** Coat a nonstick skillet with cooking spray and heat the skillet. Add the shrimp and cook for 2 minutes. Remove from the heat and set the shrimp aside.\n\n**3.** Coat the skillet with cooking spray and heat the skillet. Add the rice and garlic powder and cook for 1 minute, stirring constantly. Add the broccoli and cook until it is bright green. Add the shrimp, scallions, soy sauce, and sesame seeds. Cook for 1 minute longer.\n\n**To Serve:** Spoon the stir-fried mixture onto plates.\n\n**Servings:** 2\n\n### Southern-Style Baked Chicken with Black-Eyed Peas and Collard Greens\n\n**6 ounces skinless, boneless chicken breast**\n\n**5 bunches collard greens, cut into strips**\n\n**2 cloves garlic, minced**\n\n**1 \u20444 cup balsamic vinegar**\n\n**2 3\u20444 cups canned black-eyed peas, drained and rinsed**\n\n**_1 tablespoon Lawry's Seasoned Salt_ **\n\n**_Cooking oil spray_ **\n\n**_1 \u20444 cup water_ **\n\n**_1 teaspoon sugar substitute_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n* * *\n\n**1.** Preheat the oven to 375\u00b0F.\n\n**2.** Sprinkle the chicken with seasoned salt. Coat a nonstick skillet with cooking spray and heat the skillet. Sear the chicken on both sides until golden and crisp around the edges. Put the chicken in a baking dish and bake for 10 minutes.\n\n**3.** Coat a large nonstick skillet with cooking spray and heat the skillet. Add the collard greens and garlic and cook for 4 minutes or until the greens are bright green. Add the vinegar and water. Cook until most of the liquid has evaporated. Add the sugar substitute, salt, and pepper and set the greens aside.\n\n**4.** Place the black-eyed peas in a bowl and season with salt and pepper. Microwave for 1 minute.\n\n**To Serve:** Transfer the hot chicken breasts to a cutting board and cut into 1\u20442-inch slices. Place the collard greens on the plates and arrange the chicken slices on top. Ladle the black-eyed peas over the chicken.\n\n**Servings:** 2\n\n### Soy-Poached Chicken with Vegetables and Brown Rice\n\n**8 ounces skinless, boneless chicken breast, pounded thin**\n\n**1 cup edamame beans**\n\n**2 cups thinly sliced carrots**\n\n**1 1\u20444 cups cooked brown rice**\n\n**_2 cups water_ **\n\n**_1 cup low-sodium soy sauce_ **\n\n**_1 teaspoon ground ginger_ **\n\n**_1 teaspoon garlic powder_ **\n\n**_Salt to taste_ **\n\n* * *\n\n**1.** In a saucepan, combine the chicken, water, and soy sauce. Simmer for 8 minutes or until chicken is no longer pink. Drain the chicken and set aside.\n\n**2.** Meanwhile, remove edamame beans from the pods and cook in boiling water for 2 minutes. Place the carrots, ginger, and garlic powder in a container and microwave for 2 minutes. Drain the edamame beans and stir into the carrot mixture. Season with a little salt.\n\n**To Serve:** Place the chicken on plates and spoon the cooked brown rice on top. Garnish with the vegetable mixture.\n\n**Servings:** 2\n\n### Spaghetti and Meatballs\n\n**2 small spaghetti squash, cut in half and seeded**\n\n**1 cup canned crushed tomatoes**\n\n**6 ounces ground turkey breast**\n\n**1 egg white**\n\n**4 slices whole grain bread, toasted and ground into crumbs**\n\n**_Cooking oil spray_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n**_1 bay leaf_ **\n\n**_1 tablespoon sugar substitute_ **\n\n**_2 teaspoons onion powder_ **\n\n**_2 teaspoons garlic powder_ **\n\n**_1 teaspoon tomato paste_ **\n\n**_Chopped fresh parsley (optional)_ **\n\n* * *\n\n**1.** Preheat the oven to 400\u00b0F. Lightly coat the squash with cooking spray and season with salt and pepper. Bake for 30 minutes or until tender. (Or microwave each squash half for 6 minutes).\n\n**3.** In a saucepan, combine the tomatoes, bay leaf, sugar substitute, 1 teaspoon of the onion powder, 1 teaspoon of the garlic powder, salt, and pepper. Bring to a simmer. In a mixing bowl, combine the ground turkey, egg white, bread crumbs, tomato paste, remaining onion powder, remaining garlic powder, salt, and pepper. Mix well. Roll 11\u20442-inch meatballs between the palms of your hands. Drop the meatballs into the tomato sauce and cook for 15 minutes.\n\n**To Serve:** Shred the spaghetti squash with forks and place on plates. Ladle meatballs and tomato sauce over the squash. Sprinkle with chopped parsley, if desired.\n\n**Servings:** 2\n\n### Turkey Fajitas\n\n**6 ounces skinless, boneless turkey breast, cut into strips**\n\n**1 cup sliced Spanish onion**\n\n**1 bell pepper, seeded and cut into strips**\n\n**2 large whole grain or whole wheat tortillas**\n\n**1 \u20442 cup nonfat sour cream**\n\n**_Cooking oil spray_ **\n\n**_2 tablespoons fajita seasoning mix_ **\n\n**_1 tablespoon garlic powder_ **\n\n**_2 teaspoons chili powder_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n* * *\n\n**1.** Coat a nonstick skillet with cooking spray and heat the skillet. Add the turkey strips and cook for 2 minutes. Add the onion and cook 1 minute longer. Add the bell pepper, fajita seasoning mix, garlic powder, chili powder, salt, and cracked black pepper. Stir well to mix and cook for 1 minute.\n\n**To Serve:** Heat the tortillas in the microwave for 15 seconds. Spoon the turkey mixture onto the tortillas and garnish with sour cream.\n\n**Servings:** 2\n\n### Warm White Bean, Beet, and Turkey Salad\n\n**8 ounces turkey breast cutlet, cut into 1-inch pieces**\n\n**1 2\u20443 cups canned white beans, rinsed and drained**\n\n**4 ounces veggie pepperoni, chopped**\n\n**1 \u20444 cup white wine vinegar**\n\n**6 ounces canned beets, drained and sliced**\n\n**_Cooking oil spray_ **\n\n**_1 teaspoon dried oregano_ **\n\n**_1 teaspoon dried basil_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n* * *\n\n**1.** Spray a nonstick skillet with cooking spray and heat the skillet. Add the turkey pieces and sear on both sides. Cook about 2 minutes or until they are golden brown.\n\n**2.** In a mixing bowl, combine the turkey, white beans, veggie pepperoni, vinegar, oregano, basil, salt, and pepper. Toss gently.\n\n**To Serve:** Arrange the beet slices on plates and spoon the turkey mixture over the beets.\n\n**Servings:** 2\n\n**NOTE:** Serve this salad hot or cold.\n\n### White Fish and Vegetables en Papillote with Brown Rice\n\n**8 ounces boneless snapper fillets**\n\n**2 cups carrots cut into strips**\n\n**1 \u20442 cup red bell pepper cut into thin strips**\n\n**2 tablespoons freshly squeezed lemon juice**\n\n**1 2\u20443 cups cooked brown rice**\n\n**_1 tablespoon onion powder_ **\n\n**_1 teaspoon paprika_ **\n\n**_1 teaspoon lemon pepper_ **\n\n**_1 teaspoon red pepper flakes_ **\n\n**_1 \u20442 tablespoon garlic powder_ **\n\n**_Salt and cracked black pepper to taste_ **\n\n**_Cooking oil spray_ **\n\n**_Lemon wedges (optional)_ **\n\n* * *\n\n**1.** Preheat the oven to 325\u00b0F. Combine the onion powder, paprika, lemon pepper, red pepper flakes, garlic powder, salt, and cracked black pepper. Sprinkle over the snapper. Cut two sheets of foil larger than the fish fillets in the shape of a heart. Lightly coat the foil with cooking spray. Place a piece of fish on each piece of foil and arrange the carrots and bell pepper around the fish. Drizzle the lemon juice over the fish and vegetables and carefully seal the edges of the foil.\n\n**2.** Place the packets in a shallow pan with 1\u20444 cup water. Cover the pan and bake for 10 minutes.\n\n**To Serve:** Place the packets on plates. Snip the top of the foil with scissors and let the steam escape. Serve the fish with the rice. Garnish with lemon wedges, if desired.\n\n**Servings:** 2\n\n## Your 5-Week 5-Factor Menu Plan\n\nThese menu plans are a great way to start the 5-Factor Diet. All of the foods listed are recipes from this book. Of course, you can always make up your own menus based on the 5-Factor foods you love.\n\n## WEEK ONE\n\n### DAY 1\n\n**Breakfast:** Open-Face Egg and Bacon Sandwiches\n\n**Snack 1:** Chocolate-Mint Shakes\n\n**Lunch:** Antipasto\n\n**Snack 2:** Spicy Jumbo Shrimp with Black Bean Dip\n\n**Dinner:** Southern-Style Baked Chicken with Black-Eyed Peas and Collard Greens\n\n* * *\n\n### DAY 2\n\n**Breakfast:** Oatmeal-Berry Pancakes\n\n**Snack 1:** Hard-Boiled Eggs Stuffed with Tuna Salad\n\n**Lunch:** Green Bean Salad with Tuna and Grapefruit-Scallion Vinaigrette\n\n**Snack 2:** Chicken and Swiss Bites\n\n**Dinner:** Scallop Ratatouille\n\n* * *\n\n### DAY 3\n\n**Breakfast:** Asparagus Crepes with Toast\n\n**Snack 1:** Apple Wedges with Cinnamon Cream\n\n**Lunch:** Snapper Ceviche with Sweet Potato Rounds\n\n**Snack 2:** Chicken Slices with Cheese and Crackers\n\n**Dinner:** Spaghetti and Meatballs\n\n* * *\n\n### DAY 4\n\n**Breakfast:** Ham Steaks with Applesauce and Toast\n\n**Snack 1:** Saut\u00e9ed Peaches with Cheese\n\n**Lunch:** Mushroom-Barley Risotto\n\n**Snack 2:** Hot Dog Skewers with Cherry Tomatoes and Pickles\n\n**Dinner:** Seared Scallops with Orange Sauce and Broccoli-Cauliflower Saut\u00e9\n\n* * *\n\n### DAY 5\n\n**Breakfast:** Smoked Salmon Omelet with Cream Cheese and Whole Grain Toast\n\n**Snack 1:** Roasted Asparagus Spears with Turkey Slices\n\n**Lunch:** Black Bean Gumbo\n\n**Snack 2:** Berry Protein Shakes\n\n**Dinner:** Chicken Ropa Vieja\n\n* * *\n\n### DAY 6\n\n**Breakfast:** Kashi GoLean with Nonfat Milk\n\n**Snack 1:** Roast Beef with Carrot-Pear Slaw\n\n**Lunch:** Minestrone\n\n**Snack 2:** Apple-Turkey Roll-Ups with Relish and Mustard\n\n**Dinner:** Chiles Rellenos with Brown Rice\n\n* * *\n\n### DAY 7\n\n**Cheat Day**\n\n## WEEK TWO\n\n### DAY 1\n\n**Breakfast:** Bran Pancakes with Ricotta\n\n**Snack 1:** Saut\u00e9ed Apples over Rice Cakes\n\n**Lunch:** Mixed Greens with Turkey and Cheese Quesadillas\n\n**Snack 2:** Bruschetta\n\n**Dinner:** Warm White Bean, Beet, and Turkey Salad\n\n* * *\n\n### DAY 2\n\n**Breakfast:** Fully Charged Fruit Salad\n\n**Snack 1:** Belgian Endive Stuffed with Cheesy Artichoke Spread\n\n**Lunch:** Smoked Salmon Pizza\n\n**Snack 2:** Grilled Chicken Kabobs with Carrot-Ginger Vinaigrette\n\n**Dinner:** Salmon with Cucumber-Dill Salad\n\n* * *\n\n### DAY 3\n\n**Breakfast:** Broccoli-Cheddar Omelet\n\n**Snack 1:** Apple Wedges with Cinnamon Cream\n\n**Lunch:** Chicken and Rice Miso Soup\n\n**Snack 2:** Chips and Salsa\n\n**Dinner:** Creamy Lemon-Ginger Halibut with Corn on the Cob\n\n* * *\n\n### DAY 4\n\n**Breakfast:** Red Bell Pepper Frittata with Baked Yams\n\n**Snack 1:** Cottage Cheese and Pears\n\n**Lunch:** Greek-Style Shrimp and Spinach Salad\n\n**Snack 2:** Crunchy Celery Sticks with Roasted-Garlic Hummus and Smoked Turkey\n\n**Dinner:** 5-Factor Lasagna\n\n* * *\n\n### DAY 5\n\n**Breakfast:** Breakfast Burritos I\n\n**Snack 1:** Tropical Berry Protein Shakes\n\n**Lunch:** Baked Potato Skins with Sloppy Joe\n\n**Snack 2:** Gelatin with Berries and Yogurt\n\n**Dinner:** Turkey Fajitas\n\n* * *\n\n### DAY 6\n\n**Breakfast:** Salmon-Leek Frittata with Whole Grain Toast\n\n**Snack 1:** Toast with Berries and Cocoa Cottage Cheese\n\n**Lunch:** Chicken Fingers and French Fries\n\n**Snack 2:** White Bean Dip\n\n**Dinner:** White Fish and Vegetables en Papillote with Brown Rice\n\n* * *\n\n### DAY 7\n\n**Cheat Day**\n\n## WEEK THREE\n\n### DAY 1\n\n**Breakfast:** French Toast with Ricotta\n\n**Snack 1:** Strawberry-Oatmeal Bars with Yogurt\n\n**Lunch:** Chinese Chicken Wraps with Peanut-Soy Sauce\n\n**Snack 2:** Chocolate-Berry Parfaits\n\n**Dinner:** Seared Halibut with Creamed Spinach and Brown Rice\n\n* * *\n\n### DAY 2\n\n**Breakfast:** Sweet Potato Home Fries and Scrambled Eggs\n\n**Snack 1:** Fruit Skewers with Cottage Cheese\n\n**Lunch:** Open-Face Turkey BLT\n\n**Snack 2:** Lemon Pie\n\n**Dinner:** Country-Style Ham Steaks with Yams and Corn on the Cob\n\n* * *\n\n### DAY 3\n\n**Breakfast:** Frittata Italiana\n\n**Snack 1:** Espresso Panna Cotta\n\n**Lunch:** Harley's Sweet Potato Melt\n\n**Snack 2:** Salmon Sashimi with Plums\n\n**Dinner:** Spaghetti and Meatballs\n\n* * *\n\n### DAY 4\n\n**Breakfast:** Cream of Wheat and Protein\n\n**Snack 1:** Egg Salad with Toast Points\n\n**Lunch:** Baked Chicken and Black Bean Quesadillas with Salsa\n\n**Snack 2:** Fresh Figs with Balsamic Cream Sauce\n\n**Dinner:** Cream of Broccoli Soup with Sauteed Shrimp\n\n* * *\n\n### DAY 5\n\n**Breakfast:** Scrambled Egg Casserole\n\n**Snack 1:** Chocolate-Mint Shakes\n\n**Lunch:** Pink Pizza\n\n**Snack 2:** Smoked Turkey and Fruit Salad\n\n**Dinner:** Crispy Chicken Tostada\n\n* * *\n\n### DAY 6\n\n**Breakfast:** Egg and Veggie Muffins\n\n**Snack 1:** Spinach Frittata and Toast\n\n**Lunch:** Salmon Tartare with Arugula\n\n**Snack 2:** Chicken and Swiss Bites\n\n**Dinner:** Bison Steak with Cauliflower-Carrot Mash and Brown Rice\n\n* * *\n\n### DAY 7\n\n**Cheat Day**\n\n## WEEK FOUR\n\n### DAY 1\n\n**Breakfast:** Ham Steaks with Applesauce and Toast.\n\n**Snack 1:** Berry Protein Shakes\n\n**Lunch:** Salad Ni\u00e7oise\n\n**Snack 2:** Egg and Celery Platter with Mustard-Balsamic Sauce\n\n**Dinner:** Indian-Style Chicken with Curried Yogurt Sauce and Brown Rice\n\n* * *\n\n### DAY 2\n\n**Breakfast:** Oatmeal-Berry Pancakes\n\n**Snack 1:** Apple-Turkey Roll-Ups with Relish and Mustard\n\n**Lunch:** Stuffed Mushrooms and Greens\n\n**Snack 2:** Chicken Salad with Apples\n\n**Dinner:** Lobster and Peas with Tomato-Basil Sauce and Barley\n\n* * *\n\n### DAY 3\n\n**Breakfast:** Scrambled Eggs with Toast and Grapefruit\n\n**Snack 1:** Raspberry Gelatin with Cottage Cheese\n\n**Lunch:** Mixed Greens with Turkey and Cheese Quesadillas\n\n**Snack 2:** Edamame and Tuna Sashimi with Ginger-Scallion Vinaigrette\n\n**Dinner:** Shrimp and Rice Stir-Fry\n\n* * *\n\n### DAY 4\n\n**Breakfast:** Breakfast Burritos II\n\n**Snack 1:** Carrot Sticks with Onion Dip\n\n**Lunch:** Antipasto\n\n**Snack 2:** Cheese Course\n\n**Dinner:** Southern-Style Baked Chicken with Black-Eyed Peas and Collard Greens\n\n* * *\n\n### DAY 5\n\n**Breakfast:** Smoked Turkey and Tomato Scrambled Eggs with Toast\n\n**Snack 1:** Pesto Crisps with Tomatoes and Cheese\n\n**Lunch:** Minestrone\n\n**Snack 2:** Cheesecake\n\n**Dinner:** Salmon with Cucumber-Dill Salad\n\n* * *\n\n### DAY 6\n\n**Breakfast:** Bell Pepper Pancakes with Mozzarella and Crisp Bacon\n\n**Snack 1:** Pears with Peanut Butter Dip\n\n**Lunch:** Mediterranean-Style Chicken and Quinoa Salad\n\n**Snack 2:** Chips and Salsa\n\n**Dinner:** Chicken Chow Mein\n\n* * *\n\n### DAY 7\n\n**Cheat Day**\n\n## WEEK FIVE\n\n### DAY 1\n\n**Breakfast:** Smoked Salmon Omelet with Cream Cheese and Whole Grain Toast\n\n**Snack 1:** Sauteed Apples over Rice Cakes\n\n**Lunch:** Black Bean Gumbo\n\n**Snack 2:** Crunchy Celery Sticks with Roasted-Garlic Hummus and Smoked Turkey\n\n**Dinner:** 5-Factor Lasagna\n\n* * *\n\n### DAY 2\n\n**Breakfast:** Open-Face Egg and Bacon Sandwiches\n\n**Snack 1:** Pear and Arugula Salad with Ricotta\n\n**Lunch:** Tuscan Tomato Soup\n\n**Snack 2:** Chicken Slices with Cheese and Crackers\n\n**Dinner:** Argentine-Style Steak Salad with Watercress and Mustard-Cilantro Vinaigrette\n\n* * *\n\n### DAY 3\n\n**Breakfast:** The Cowboy Omelet\n\n**Snack 1:** Passion Fruit and Tangerine Shakes\n\n**Lunch:** Mexican Chicken Salad with Spicy Salsa Dressing\n\n**Snack 2:** Bruschetta\n\n**Dinner:** Shrimp and Tofu Soup\n\n* * *\n\n### DAY 4\n\n**Breakfast:** Breakfast Burritos III\n\n**Snack 1:** Toast with Berries and Cocoa Cottage Cheese\n\n**Lunch:** Mixed Greens with Turkey and Cheese Quesadillas\n\n**Snack 2:** Butterscotch and Apple Pudding\n\n**Dinner:** Chicken Ropa Vieja\n\n* * *\n\n### DAY 5\n\n**Breakfast:** Asparagus Crepes with Toast\n\n**Snack 1:** Grilled Chicken Kabobs with Carrot-Ginger Vinaigrette\n\n**Lunch:** Mushroom Barley Risotto\n\n**Snack 2:** Berry Protein Shakes\n\n**Dinner:** Scallop Ratatouille\n\n* * *\n\n### DAY 6\n\n**Breakfast:** Broccoli-Cheddar Omelet\n\n**Snack 1:** Fruit Skewers with Cottage Cheese\n\n**Lunch:** Portobello and Turkey Stacks\n\n**Snack 2:** Lemon Yogurt with Kiwi\n\n**Dinner:** Soy-Poached Chicken with Vegetables and Brown Rice\n\n* * *\n\n### DAY 7\n\n**Cheat Day**\n\n# CHAPTER 11\n\n# 5-Factor \nSuccess \nLog\n\n**W riting down what you eat makes you think** about your food three times.\n\nFirst, you think about it as you eat it.\n\nSecond, you think about it when you write it down.\n\nThird, you think about it when you read later on.\n\nI've found that thinking three times about everything you eat gives you a sense of ownership of your actions. It also gives you a mini-assessment every day on how well you're doing with your diet. Rome wasn't built in one night. Seeing a few weeks' worth of food logs that show how much better you're eating can be the inspiration you need to stay the course. In fact it's been shown that people who keep track of what they eat are more successful with their nutritional goals.\n\nStill I know you don't have time to write down every calorie and every fat gram from every bite you eat. I wouldn't expect that from my clients, and I don't expect it from you. So here's the good news: You don't have to.\n\nKeeping track of how well you're doing on the 5-Factor Diet isn't painful. It doesn't require a calculator or more than a few seconds of your time.\n\nAll of the recipes in this book incorporate 5-Factor Diet requirements\u2014you don't even have to think about them.\n\nWhen you're ready to create your own meals and daily menus based on the 5-Factor principles\u2014and using the 5-Factor Must-Have Foods\u2014this easy-to-use chart will help you track your progress. (Make copies of it to use every day.)\n\nRecord the three foods\u2014a low-fat protein, a low- to moderate-GI carb, and a no- to low-sugar beverage\u2014you plan to eat at each of the five meals of the day. As for the fiber column, if you're eating a low-GI carb, I guarantee it contains the 5 to 10 grams of fiber required at each meal. If necessary, add another fibrous carbohydrate\u2014beans or spinach, for example\u2014to your meal so you can write \"yes\" in the fiber column. You must have 5 to 10 grams of fiber at every meal to follow the 5-Factor Diet.\n\nThe last column, healthy fats, needn't read \"yes\" at every meal. Just be sure you aren't eating any unhealthy saturated or trans fats.\n\nReady to start your 5-Factor day? Here you go!\n\n## YOUR 5-FACTOR WEEKLY PLAN\n\nOnce you get used to designing your 5-Factor meals, it's easy to design your own weekly meal plan, using many of the foods I suggested earlier\u2014especially the 5-Factor Must-Have Foods.\n\n_Clickhere to download a PDF of the charts in this chapter:_ 5-Factor Diet Day\u2014Sample, Your 5-Factor Diet Day, _and_ Your 5-Factor Diet Week.\n\n# Recipe Index\n\n## A\n\nApples\n\nApple-Turkey Roll-Ups with Relish and Mustard\n\nApple Wedges with Cinnamon Cream\n\nButterscotch and Apple Pudding\n\nChicken Salad with Apples\n\nFully Charged Fruit Salad\n\nHam Steaks with Applesauce and Toast\n\nSauteed Apples over Rice Cakes\n\nSmoked Turkey and Fruit Salad\n\nArtichokes\n\nAntipasto\n\nBelgian Endive Stuffed with Cheesy Artichoke Spread\n\nArugula\n\nPear and Arugula Salad with Ricotta\n\nSalmon Tartare with Arugula\n\nAsparagus\n\nAsparagus Crepes with Toast\n\nRoasted Asparagus Spears with Turkey Slices\n\n## B\n\nBacon. See Turkey bacon\n\nBarley\n\nLobster and Peas with Tomato-Basil Sauce and Barley\n\nMushroom-Barley Risotto\n\nBeans\n\nBaked Chicken and Black Bean Quesadillas with Salsa\n\nBlack Bean Gumbo\n\nBreakfast Burritos I\n\nBreakfast Burritos III\n\nChiles Rellenos with Brown Rice\n\nCrunchy Celery Sticks with Roasted-Garlic Hummus and Smoked Turkey\n\nEdamame and Tuna Sashimi with Ginger- Scallion Vinaigrette\n\nGreen Bean Salad with Tuna and Grapefruit-Scallion Vinaigrette\n\nMinestrone\n\nSalad Ni\u00e7oise\n\nSoy-Poached Chicken with Vegetables and Brown Rice\n\nSpicy Jumbo Shrimp with Black Bean Dip\n\nWarm White Bean, Beet, and Turkey Salad\n\nWhite Bean Dip\n\nBeef\n\nArgentine-Style Steak Salad with Watercress and Mustard-Cilantro Vinaigrette\n\nBison Steak with Cauliflower-Carrot Mash and Brown Rice\n\nRoast Beef with Carrot- Pear Slaw\n\nBeet, and Turkey Salad, Warm White Bean\n\nBelgian Endive Stuffed with Cheesy Artichoke Spread\n\nBerries. See also Raspberries; Strawberries Gelatin with Berries and Yogurt\n\nOatmeal-Berry Pancakes\n\nToast with Berries and Cocoa Cottage Cheese\n\nBeverages. See Shakes\n\nBison Steak with Cauliflower-Carrot Mash and Brown Rice, 10.1\n\nBlueberries\n\nGelatin with Berries and Yogurt\n\nOatmeal-Berry Pancakes\n\nBran Pancakes with Ricotta\n\nBreakfast recipes\n\nBroccoli\n\nBroccoli-Cheddar Omelet\n\nChicken Fingers and French Fries\n\nCream of Broccoli Soup with Sauteed Shrimp\n\nEgg and Veggie Muffins\n\nSeared Scallops with Orange Sauce and Broccoli-Cauliflower Saute\n\nShrimp and Rice Stir-Fry\n\nBruschetta\n\nBulgur\n\nStuffed Mushrooms and Greens\n\nButterscotch and Apple Pudding\n\n## C\n\nCarrots\n\nBison Steak with Cauliflower-Carrot Mash and Brown Rice\n\nCarrot Sticks with Onion Dip\n\nChicken Chow Mein\n\nCream of Broccoli Soup with Sauteed Shrimp\n\nEdamame and Tuna Sashimi with Ginger-Scallion Vinaigrette\n\nGrilled Chicken Kabobs with Carrot-Ginger Vinaigrette\n\nHard-Boiled Eggs Stuffed with Tuna Salad\n\nRoast Beef with Carrot-Pear Slaw\n\nRoasted Asparagus Spears with Turkey Slices\n\nSoy-Poached Chicken with Vegetables and Brown Rice\n\nSpinach Dip with Carrot Sticks\n\nWhite Fish and Vegetables en Papillote with Brown Rice\n\nCauliflower\n\nBison Steak with Cauliflower-Carrot Mash and Brown Rice\n\nSeared Scallops with Orange Sauce and Broccoli-Cauliflower Saute\n\nCelery\n\nCrunchy Celery Sticks with Roasted-Garlic Hummus and Smoked Turkey\n\nEgg and Celery Platter with Mustard-Balsamic Sauce\n\nCheese. See also Cottage cheese; Ricotta cheese\n\nAntipasto\n\nBelgian Endive Stuffed with Cheesy Artichoke Spread\n\nBell Pepper Pancakes with Mozzarella and Crisp Bacon\n\nBroccoli-Cheddar Omelet\n\nBruschetta\n\nCheesecake\n\nChicken and Swiss Bites\n\nChicken Slices with Cheese and Crackers\n\nThe Cowboy Omelet\n\n5-Factor Lasagna\n\nPink Pizza\n\nRed Bell Pepper Frittata with Baked Yams\n\nSweet Potato Home Fries and Scrambled Eggs\n\nCheesecake\n\nChicken\n\nBaked Chicken and Black Bean Quesadillas with Salsa\n\nBaked Potato Skins with Sloppy Joe\n\nBlack Bean Gumbo\n\nChicken and Rice Miso Soup\n\nChicken and Swiss Bites\n\nChicken Chow Mein\n\nChicken Fingers and French Fries\n\nChicken Ropa Vieja\n\nChicken Salad with Apples\n\nChicken Slices with Cheese and Crackers\n\nChinese Chicken Wraps with Peanut-Soy Sauce\n\nCrispy Chicken Tostadas\n\nGrilled Chicken Kabobs with Carrot-Ginger Vinaigrette\n\nIndian-Style Chicken with Curried Yogurt Sauce and Brown Rice\n\nMediterranean-Style Chicken and Quinoa Salad\n\nMexican Chicken Salad with Spicy Salsa Dressing\n\nScrambled Eggs with Toast and Grapefruit\n\nSouthern-Style Baked Chicken with Black-Eyed Peas and Collard Greens\n\nSoy-Poached Chicken with Vegetables and Brown Rice\n\nChocolate\n\nChocolate-Berry Parfaits\n\nChocolate-Mint Shakes\n\nEspresso Panna Cotta\n\nToast with Berries and Cocoa Cottage Cheese\n\nCorn\n\nChicken Ropa Vieja\n\nCountry-Style Ham Steaks with Yams and Corn on the Cob\n\nCreamy Lemon-Ginger Halibut with Corn on the Cob\n\nMexican Chicken Salad with Spicy Salsa Dressing\n\nCottage cheese\n\nChocolate-Berry Parfaits\n\nCottage Cheese and Pears\n\nFresh Figs with Balsamic Cream Sauce, 10.1\n\nFruit Skewers with Cottage Cheese\n\nFully Charged Fruit Salad\n\nRaspberry Gelatin with Cottage Cheese\n\nToast with Berries and Cocoa Cottage Cheese\n\nCrabmeat\n\nStuffed Mushrooms and Greens\n\nCream of Wheat and Protein\n\nCucumbers\n\nIndian-Style Chicken with Curried Yogurt Sauce and Brown Rice\n\nSalmon with Cucumber-Dill Salad\n\n## D\n\nDinner recipes\n\nDrinks. See Shakes\n\n## E\n\nEdamame\n\nEdamame and Tuna Sashimi with Ginger-Scallion Vinaigrette\n\nSoy-Poached Chicken with Vegetables and Brown Rice\n\nEggplant\n\n5-Factor Lasagna\n\nScallop Ratatouille\n\nEggs\n\nAsparagus Crepes with Toast\n\nBreakfast Burritos I\n\nBreakfast Burritos III\n\nBroccoli-Cheddar Omelet\n\nThe Cowboy Omelet\n\nEgg and Celery Platter with Mustard-Balsamic Sauce\n\nEgg and Veggie Muffins\n\nEgg Salad with Toast Points\n\nFrench Toast with Ricotta\n\nFrittata Italiana\n\nHard-Boiled Eggs Stuffed with Tuna Salad\n\nOpen-Face Egg and Bacon Sandwiches\n\nRed Bell Pepper Frittata with Baked Yams\n\nSalmon-Leek Frittata with Whole Grain Toast\n\nScrambled Egg Casserole\n\nScrambled Eggs with Toast and Grapefruit\n\nSmoked Salmon Omelet with Cream Cheese and Whole Grain Toast\n\nSmoked Turkey and Tomato Scrambled Eggs with Toast\n\nSpinach Frittata and Toast\n\nSweet Potato Home Fries and Scrambled Eggs\n\nEspresso Panna Cotta\n\n## F\n\nFigs with Balsamic Cream Sauce, Fresh\n\nFish. See also Salmon; Tuna\n\nCreamy Lemon-Ginger Halibut with Corn on the Cob\n\nSeared Halibut with Creamed Spinach and Brown Rice\n\nSnapper Ceviche with Sweet Potato Rounds\n\nWhite Fish and Vegetables en Papillote with Brown Rice\n\nFrench Toast with Ricotta\n\nFrittatas\n\nFrittata Italiana\n\nRed Bell Pepper Frittata with Baked Yams\n\nSalmon-Leek Frittata with Whole Grain Toast\n\nSpinach Frittata and Toast\n\nFruit. See also specific fruits\n\nFruit Skewers with Cottage Cheese\n\nFully Charged Fruit Salad\n\nSmoked Turkey and Fruit Salad\n\n## G\n\nGrapefruit\n\nGreen Bean Salad with Tuna and Grapefruit- Scallion Vinaigrette\n\nScrambled Eggs with Toast and Grapefruit\n\nGreen beans\n\nGreen Bean Salad with Tuna and Grapefruit- Scallion Vinaigrette\n\nSalad Ni\u00e7oise\n\nGreens. See also Spinach\n\nMixed Greens with Turkey and Cheese Quesadillas\n\nPear and Arugula Salad with Ricotta\n\nSalmon Tartare with Arugula\n\nSouthern-Style Baked Chicken with Black-Eyed Peas and Collard Greens\n\nStuffed Mushrooms and Greens\n\n## H\n\nHalibut\n\nCreamy Lemon-Ginger Halibut with Corn on the Cob\n\nSeared Halibut with Creamed Spinach and Brown Rice\n\nHam\n\nCountry-Style Ham Steaks with Yams and Corn on the Cob\n\nHam Steaks with Applesauce and Toast\n\nHot Dog Skewers with Cherry Tomatoes and Pickles\n\n## K\n\nKashi GoLean with Nonfat Milk\n\nKiwi, Lemon Yogurt with\n\n## L\n\nLasagna, 5-Factor, 10.1\n\nLemon Pie\n\nLemon Yogurt with Kiwi\n\nLobster and Peas with Tomato-Basil Sauce and Barley\n\nLunch recipes\n\n## M\n\nMeatballs, Spaghetti and\n\nMushrooms\n\nThe Cowboy Omelet\n\nHot Dog Skewers with Cherry Tomatoes and Pickles\n\nMushroom-Barley Risotto\n\nPortobello and Turkey Stacks\n\nScallop Ratatouille\n\nStuffed Mushrooms and Greens\n\n## O\n\nOats\n\nOatmeal-Berry Pancakes\n\nStrawberry-Oatmeal Bars with Yogurt\n\nOmelets\n\nBroccoli-Cheddar Omelet\n\nThe Cowboy Omelet\n\nSmoked Salmon Omelet with Cream Cheese and Whole Grain Toast\n\nOranges\n\nFully Charged Fruit Salad\n\nGreek-Style Shrimp and Spinach Salad\n\nSmoked Turkey and Fruit Salad\n\n## P\n\nPancakes\n\nBell Pepper Pancakes with Mozzarella and Crisp Bacon\n\nBran Pancakes with Ricotta\n\nOatmeal-Berry Pancakes\n\nPassion fruit\n\nPassion Fruit and Tangerine Shakes\n\nTropical Berry Protein Shakes\n\nPeaches\n\nFruit Skewers with Cottage Cheese\n\nSauteed Peaches with Cheese\n\nPeanut butter\n\nChinese Chicken Wraps with Peanut-Soy Sauce\n\nPears with Peanut Butter Dip\n\nPears\n\nCheese Course\n\nCottage Cheese and Pears\n\nFruit Skewers with Cottage Cheese\n\nPear and Arugula Salad with Ricotta\n\nPears with Peanut Butter Dip\n\nRoast Beef with Carrot- Pear Slaw\n\nPeas\n\nChicken Chow Mein\n\nLobster and Peas with Tomato-Basil Sauce and Barley\n\nPeppers\n\nBell Pepper Pancakes with Mozzarella and Crisp Bacon\n\nChiles Rellenos with Brown Rice\n\nEgg and Veggie Muffins\n\nRed Bell Pepper Frittata with Baked Yams\n\nTurkey Fajitas\n\nPizza\n\nPink Pizza\n\nSmoked Salmon Pizza\n\nPlums, Salmon Sashimi with\n\nPotatoes. See Sweet potatoes\n\n## Q\n\nQuinoa Salad, Mediterranean-Style Chicken and\n\n## R\n\nRaspberries\n\nBerry Protein Shakes\n\nChocolate-Berry Parfaits\n\nGelatin with Berries and Yogurt\n\nRaspberry Gelatin with Cottage Cheese\n\nTropical Berry Protein Shakes\n\nRice\n\nBison Steak with Cauliflower-Carrot Mash and Brown Rice\n\nChicken and Rice Miso Soup\n\nChiles Rellenos with Brown Rice\n\nIndian-Style Chicken with Curried Yogurt Sauce and Brown Rice\n\nSeared Halibut with Creamed Spinach and Brown Rice\n\nShrimp and Rice Stir-Fry\n\nShrimp and Tofu Soup\n\nSoy-Poached Chicken with Vegetables and Brown Rice\n\nWhite Fish and Vegetables en Papillote with Brown Rice\n\nRicotta cheese\n\nBran Pancakes with Ricotta\n\nBreakfast Burritos II\n\nButterscotch and Apple Pudding\n\nCheese Course\n\n5-Factor Lasagna\n\nFrench Toast with Ricotta\n\nPear and Arugula Salad with Ricotta\n\nPesto Crisps with Tomatoes and Cheese\n\nPink Pizza\n\nSauteed Peaches with Cheese\n\nRisotto, Mushroom-Barley\n\n## S\n\nSalads\n\nArgentine-Style Steak Salad with Watercress and Mustard-Cilantro Vinaigrette\n\nChicken Salad with Apples\n\nEgg Salad with Toast Points\n\nFully Charged Fruit Salad\n\nGreek-Style Shrimp and Spinach Salad\n\nGreen Bean Salad with Tuna and Grapefruit- Scallion Vinaigrette\n\nMediterranean-Style Chicken and Quinoa Salad\n\nMexican Chicken Salad with Spicy Salsa Dressing, 10.1\n\nMixed Greens with Turkey and Cheese Quesadillas\n\nPear and Arugula Salad with Ricotta\n\nSalad Ni\u00e7oise\n\nSmoked Turkey and Fruit Salad\n\nWarm White Bean, Beet, and Turkey Salad\n\nSalmon\n\nSalmon-Leek Frittata with Whole Grain Toast\n\nSalmon Sashimi with Plums\n\nSalmon Tartare with Arugula\n\nSalmon with Cucumber-Dill Salad\n\nSmoked Salmon Mousse with Crackers\n\nSmoked Salmon Omelet with Cream Cheese and Whole Grain Toast\n\nSmoked Salmon Pizza\n\nSandwiches\n\nOpen-Face Egg and Bacon Sandwiches\n\nScallops\n\nScallop Ratatouille\n\nSeared Scallops with Orange Sauce and Broccoli-Cauliflower Saute\n\nShakes\n\nBerry Protein Shakes\n\nChocolate-Mint Shakes\n\nPassion Fruit and Tangerine Shakes\n\nTropical Berry Protein Shakes\n\nShellfish. See also Shrimp Lobster and Peas with Tomato-Basil Sauce and Barley\n\nScallop Ratatouille\n\nSeared Scallops with Orange Sauce and Broccoli-Cauliflower Saute\n\nStuffed Mushrooms and Greens\n\nShrimp\n\nCream of Broccoli Soup with Sauteed Shrimp\n\nGreek-Style Shrimp and Spinach Salad\n\nMushroom-Barley Risotto\n\nShrimp and Rice Stir-Fry\n\nShrimp and Tofu Soup\n\nSpicy Jumbo Shrimp with Black Bean Dip\n\nSnack recipes\n\nSnapper\n\nSnapper Ceviche with Sweet Potato Rounds\n\nWhite Fish and Vegetables en Papillote with Brown Rice\n\nSoups\n\nBlack Bean Gumbo\n\nChicken and Rice Miso Soup\n\nCream of Broccoli Soup with Sauteed Shrimp\n\nMinestrone\n\nShrimp and Tofu Soup\n\nTuscan Tomato Soup\n\nSpaghetti and Meatballs\n\nSpinach\n\nBreakfast Burritos II\n\nGreek-Style Shrimp and Spinach Salad\n\nSeared Halibut with Creamed Spinach and Brown Rice\n\nSpinach Dip with Carrot Sticks\n\nSpinach Frittata and Toast\n\nSquash\n\nScallop Ratatouille\n\nSpaghetti and Meatballs\n\nStrawberries\n\nBerry Protein Shakes\n\nCheesecake\n\nChocolate-Mint Shakes\n\nFruit Skewers with Cottage Cheese\n\nFully Charged Fruit Salad\n\nGelatin with Berries and Yogurt\n\nOatmeal-Berry Pancakes\n\nSmoked Turkey and Fruit Salad\n\nStrawberry-Oatmeal Bars with Yogurt\n\nSweet potatoes\n\nBaked Potato Skins with Sloppy Joe\n\nChicken Fingers and French Fries\n\nThe Cowboy Omelet\n\nHarley's Sweet Potato Melt\n\nSalad Ni\u00e7oise\n\nSalmon with Cucumber-Dill Salad\n\nSnapper Ceviche with Sweet Potato Rounds\n\nSweet Potato Home Fries and Scrambled Eggs\n\n## T\n\nTangerine Shakes, Passion Fruit and\n\nTofu and Shrimp Soup\n\nTomatoes\n\nAntipasto\n\nBaked Potato Skins with Sloppy Joe\n\nBlack Bean Gumbo\n\nBreakfast Burritos II\n\nBruschetta\n\n5-Factor Lasagna\n\nFrittata Italiana\n\nHot Dog Skewers with Cherry Tomatoes and Pickles\n\nMediterranean-Style Chicken and Quinoa Salad\n\nMinestrone\n\nOpen-Face Egg and Bacon Sandwiches\n\nOpen-Face Turkey BLT\n\nPesto Crisps with Tomatoes and Cheese\n\nPink Pizza\n\nScallop Ratatouille\n\nSmoked Turkey and Tomato Scrambled Eggs with Toast\n\nTuscan Tomato Soup\n\nTortillas\n\nBaked Chicken and Black Bean Quesadillas with Salsa\n\nBreakfast Burritos I\n\nBreakfast Burritos II\n\nBreakfast Burritos III\n\nChinese Chicken Wraps with Peanut-Soy Sauce\n\nChips and Salsa\n\nCrispy Chicken Tostadas\n\nMixed Greens with Turkey and Cheese Quesadillas, 10.1\n\nPink Pizza\n\nSmoked Salmon Pizza\n\nTurkey Fajitas\n\nTuna\n\nEdamame and Tuna Sashimi with Ginger-Scallion Vinaigrette\n\nGreen Bean Salad with Tuna and Grapefruit- Scallion Vinaigrette\n\nHard-Boiled Eggs Stuffed with Tuna Salad\n\nHarley's Sweet Potato Melt\n\nSalad Ni\u00e7oise\n\nTurkey\n\nAntipasto\n\nApple-Turkey Roll-Ups with Relish and Mustard\n\nChiles Rellenos with Brown Rice\n\nCrunchy Celery Sticks with Roasted-Garlic Hummus and Smoked Turkey\n\nMinestrone\n\nMixed Greens with Turkey and Cheese Quesadillas\n\nOpen-Face Turkey BLT\n\nPortobello and Turkey Stacks\n\nRoasted Asparagus Spears with Turkey Slices\n\nSmoked Turkey and Fruit Salad\n\nSmoked Turkey and Tomato Scrambled Eggs with Toast\n\nSpaghetti and Meatballs\n\nTurkey Fajitas\n\nWarm White Bean, Beet, and Turkey Salad\n\nTurkey bacon\n\nBell Pepper Pancakes with Mozzarella and Crisp Bacon\n\nOpen-Face Egg and Bacon Sandwiches\n\nOpen-Face Turkey BLT\n\n## V\n\nVegetables. See specific vegetables\n\n## Y\n\nYams\n\nCountry-Style Ham Steaks with Yams and Corn on the Cob\n\nRed Bell Pepper Frittata with Baked Yams\n\nYogurt\n\nEspresso Panna Cotta\n\nGelatin with Berries and Yogurt\n\nLemon Yogurt with Kiwi\n\nStrawberry-Oatmeal Bars with Yogurt\n","meta":{"redpajama_set_name":"RedPajamaBook"}} +{"text":" \nLIGHT WHILE \nTHERE IS LIGHT\n\n_An American History_\n\nKeith Waldrop\n\n_Introduction by Jaimy Gordon_\n\nDedication\n\n_For Elaine_\nEpigraph\n\nThe overtakelessness of those \nWho have accomplished death...\n\n\u2014EMILY DICKINSON\n\nWork, for the night is coming....\n\n\u2014OLD HYMN\nContents\n\nCover\n\nTitle Page\n\nDedication\n\nEpigraph\n\nIntroduction\n\nA Pilgrimage\n\nTibet\n\nDiscerning of Spirits\n\nThe Call Asserts Nothing\n\nAbout the Author\n\nCopyright\n\nSelected Other Works by Keith Waldrop\nIntroduction\n\nI always knew I was not going to measure up as a literary giant, so from the start I put my hopes on making myself a name as a literary pygmy, that is, on writing one great but undoubtedly odd, sui generis, irreplaceable, one of a kind, modestly immortal book. The book I had in mind would be a novel along the lines of Richard Hughes's A High Wind in Jamaica, or Sybille Bedford's A Legacy, Elizabeth Bowen's The Death of the Heart, Walter Abish's How German Is It, Witold Gombrowicz's Ferdydurke, J. R. Ackerley's We Think the World of You, or Harry Mathews's Tlooth. True, each of these novels being by its nature unlike any other, they reveal nothing in common to the aspiring imitator, besides being the one book by their various authors (some prolific, most not) that one really couldn't live without. Oh yes, and I first heard of all of them from Keith Waldrop, the best person I know to talk about books with\u2014whatever it is, he's read it, has an opinion on it, and has it in his library on Elmgrove Avenue in Providence, in the next room, or just upstairs.\n\nTherefore it's both fitting and something of an impertinence that Keith Waldrop would end up writing such a novel himself. Beautiful, funny, wise, sad, endearingly economical in size, guaranteed one of a kind in voice and subject matter, inimitable, universally admired by that small percentage of the human race that has read it, Light While There is Light, which first appeared from Sun & Moon in 1993, was an instant eccentric classic. In protest I step out of the line where the would-be writers of one great but peculiar novel are waiting, to write this introduction. I will be brief.\n\nWhere was the need? Keith Waldrop is a poet with some fifteen volumes in that other genre to his credit. Both his first, A Windmill Near Calvary (1968), and his latest, Transcendental Studies: A Trilogy, were nominated for the National Book Award, which he won in 2009. He is the translator of another ten or twelve books, including an important, illuminating prose version of Les Fleurs du mal. As co-editor and publisher, along with his wife Rosmarie Waldrop, of Burning Deck Press since 1961, he has been a major influence on two or three generations of innovative writing in America, and was even knighted by the Republic of France in the year 2000\u2014named Chevalier des arts et des lettres\u2014for his contributions. In short, Keith Waldrop is a fully arrived international person of letters, a literary giant, there you have it, and a brilliant visual artist besides. So where was the need to toss off a great small novel like no other?\n\nLight While There is Light, though surely a work of imaginative fiction in design, is a bit shifty about its genre. It calls itself, not a novel, but An American History. A bookish, patient, witty, gently melancholic, ruminative narrator with the same name as the author and a rare gift for anecdote recalls his bizarre but profoundly American and Midwestern family, not from without\u2014and this is the special art of the book\u2014but from within, as a member, lifelong, like it or not, willy-nilly.\n\n\"All my family, and Julian is our type in this, have a streak of the unworldly,\" the narrator tells us. Long before Keith is born, just after the birth of the oldest of his three half siblings, Mother\u2014possibly in reaction to an offhand remark from that handsome wastrel, her first husband\u2014gets religion and passes it on as a lifelong quest to her young family, which, despite many obstacles, hard travels, and theological and financial gyrations, never entirely loses it again. The book's long opening chapter, which takes up almost half the novel, is entitled \"A Pilgrimage\"; in due course we realize that Mother's long search, at first with her one marriageable daughter in tow, for a church sufficiently homelike, unworldly, and doctrinally pure, will never end, not even when she comes to rest, in her last days, in a rented garage in Champaign, Illinois.\n\nThe terrible truth that haunts the family and the novel is that the world might simply be dull and meaningless:\n\nThe history of my mother's religious opinions should be told as the record of a pilgrimage. As I imagine most pilgrimages, it was less the struggle toward a given end than a continual flight from disappointment and unhappiness. Neither the joys of heaven nor hell's worst prospects provide as forceful a motive as the mere emptiness of the world.\n\nIt takes an eccentric, Waldropian sort of genius to see a weak attachment to the world-as-it-is as the common thread between Mother's wounded and wary fundamentalism, sister Elaine's cheerful obedience, brother Charles and brother Julian's talent (talent is not quite the word) for flimflam and otherwise illegal solutions to all of life's problems, and the narrator's troubled spirit: \"I was often afraid in those days, more than a little sometimes: afraid that there was no truth, or that there was one truth, only one, and that I had it.\" His faith wavers and slowly blinks out, becomes one of the book's many shadows of an absence.\n\nLight While There is Light takes its title from the Gospel of John, 12:35-36: \"Walk while ye have the light, lest darkness come upon you: for he that walketh in darkness knoweth not whither he goeth. While ye have light, believe in the light, that ye may be the children of light.\"\n\nAt rhythmic intervals\u2014in this respect as much like music as collage\u2014the novel revisits the theme of the narrator's own relations with light, in brief, image-rich variations throughout the text, each floating in its own shining white space. Is this the light of the title? The light of God? Of revealed truth about a God we once thought to grasp with our senses? Maybe it is just\u2014light. The physiological phenomenon of light, its perception and attendant sensations, is a subject of deep interest to Keith Waldrop the poet. Passages about it, images of it, are scattered all over his work. But in Light While There is Light, it is both itself\u2014light\u2014and the leitmotif for the origin and end of faith: what precedes it, what survives it.\n\nI remember, for some reason, a film I once saw, in which sequences resembling old, contrasty photographs faded, not into darkness like the usual fade, but into a bright white empty screen, so that the story seemed sketched in elaborate shadows against a field of perpetual light\u2014shining now through the pictures, illuminating them, and now supplanting them, shining on its own.\n\nThe odd balance of pleasures in a small, perfect, one-of-a-kind novel can be inventoried better than it can be explained. Some of its delights may sound like suffering when lifted out of the book for closer examination: for example, Light While There is Light is an exemplary Midwestern novel (though a substantial chunk of it unfolds at an unaccredited Bible College in Sharon, South Carolina) in its understanding of space as a baffling, featureless surplus\u2014what is a pair of towns like Champaign-Urbana, Illinois, doing in the so-called real world at all? And yet, only because Keith once tried, briefly, to go to university there, a good many family members get stuck in the homely sister cities seemingly for life\u2014even after the family used car lot is no more (see chapter three, \"Discerning of Spirits\"). I hate to accuse the narrator Keith Waldrop of anything as banal as a Midwestern trait, but what else, pray (I still live in the Midwest) is that polite hopelessness in the face of an ever receding normal that one eventually concludes was never there in the first place?\n\nThe narrator's voice in Light While There is Light, though it sometimes recalls that of Keith Waldrop the poet, is unlike any other in fiction, and likewise the character for whom it speaks. Always preserving its uniquely Waldropian distance, a sort of Olympian bemusement, quietly affectionate and without the slightest temptation to judge, what it tells, first of all, is the story of a family as it travels through place and time. In a tragicomic way, they are rather a close family\u2014they keep scattering and coming together again, like globules of stale milk in a cup of hot coffee. \"Pilgrimage\" yields to \"Tibet\"\u2014though no one in the family goes anywhere near Tibet\u2014as those travels grow stranger and stranger. In a compressed, musically segmented, anecdotal style of his own invention, but with all the building power of narrative, the author recounts each family member's religious experiences, flights from the law, nuptial pairings, and economic woes. His own troubles fade into the background, once he is an academic with a stable job and a place to live. But we can make out all the same what their youngest member begins to look like to the rest of the family: at once their salvation and their deserter, the one who flew the coop.\n\nIs it wrong for me to love madly and even wish to elope with his one slender novel, more than the entire sagging shelf of his poetical works? Remarkably, Keith Waldrop himself told an interviewer1 in 2008: \"I never wanted to be known as a poet. I'm in some ways more interested in writing prose than verse although verse is much easier. . . . Prose rhythm to me is very difficult and it's something I love. I like Henry James better than any poetry I can think of.\"\n\nHe went on to air another startling idea in that three-way conversation (Keith Waldrop, Rosmarie Waldrop, Jared Demick), as one sometimes hears oneself do, in a live interview, to one's own consternation. Even Rosmarie was taken aback:\n\nKeith: I think of my one novel as my major work. Out of all my work, that's the one I . . .\n\nRosmarie: Really?\n\nKeith: . . . put at the top.\n\nRosmarie: I'm surprised.\n\nPerhaps it was what he had been reading, or eating, that week, but\u2014having talked about books with Keith Waldrop for forty years, I'm inclined to believe he meant what he said. (He may have changed his mind later: There is something about winning the National Book Award, as he did, in poetry, the following year, that warms a writer to the work in question, whatever the genre.)\n\nI, however, was not completely astonished to hear Keith Waldrop speak of Light While There is Light as the best, or most lasting, thing he's done, for another reason: I have always thought that the soul of Keith Waldrop's verse, too, is, oddly enough, prose.2 Much as I admire them, I read his books of poetry as something like the collected marginalia of an ageless scribe, somewhat loopy on the fumes of decomposing paper and fermented printer's ink, who lives in the bottomless vault of a library. Keith Waldrop is, even in so-called real life, a poet who likes to say that he would rather read than write, and most of the books in the Waldrops' actual library\u2014of which belles lettres constitute only a part\u2014are works of prose. History, archaeology, architecture; epistemology and metaphysics, theology and religion; music; geology, botany, vulcanology; psychology, neurophysiology, the study of the brain and the senses\u2014these are some of the subjects that are well represented on those crammed shelves. They are also a fair sampling of the kinds of books\u2014whether antiquarian, crank or scholarly\u2014whose ghost prose echoes in the poetry.\n\nFrom his earliest book, A Windmill Near Calvary, which could almost pass for a conventional collection, to his latest, Transcendental Studies, whose avowed modus operandi was collage, Keith Waldrop's poems often seem to be built of sentences quarried out of faraway prose matrices\u2014used sentences from lost, antique, dog-eared, half-remembered tomes; sentences missing a piece here and there, logical edges not quite flush, patched and interpolated, but never quite losing the character of distant prose\u2014ghost prose, so to speak. Their prose origins linger in part because the poet makes us heavily aware of their periods\u2014not periods in the Ciceronian sense, but in the straightforward, punctuational sense of a full stop. In a Keith Waldrop poem, we hear the sentences as sentences (even when they're incomplete) because they so definitely end. One ends, and without transition another, probably from some other area of humane inquiry, begins. Medical advice, curious superstitions, Biblical exegesis, aphorism, architectural description, the crusty residue of an anecdote\u2014any of these might get its ghostly sentence in the poem. They lie like cantilevered architraves on top of one another, because that's how poems are shaped, but still they feel like prose.\n\nThis is from \"Beauty,\" a poem in A Windmill Near Calvary:\n\nAccording to a newspaper account, maybe distorted\n\nin my remembering (told me, come to think of it, by my\n\nbrother, who sometimes lies), a man carrying a shotgun\n\ndown Main Street in some town or other\n\nexplained to a policeman that an hour before, on the\n\nsame block, another man, a total stranger,\n\nspit in his eye and told him it was raining. It is\n\npossible to look, neither at surfaces nor beneath them,\n\nbut geometrically, squinting slightly to accommodate\n\nthings to our net of vision, robbing raw objects\n\nof their atrocity.\n\nOr, here is \"Real Motion,\" from Falling in Love Through a Description, the second book of Transcendental Studies:\n\nKeep well in mind that it is strangely possible\n\nfor us to oppose ourselves. An illustration: competing\n\nvisual fields. The projection room dark. The blue of the\n\nsky would not move us, were it a foot or so above\n\nour heads. Fear drives the body, looking for itself.\n\nSomeone lying in the roadway. About pain, we are\n\nall more or less agreed, but reflection is\n\nnecessary for such functions as urination, walking,\n\nwriting, sexual intercourse. A single, unified\n\njudgment establishes the matter as undecided.\n\nSweeps of the eye traverse and surmount\n\nsomething, the traversing and surmounting of which\n\nmight, in another way, be a matter of time, toil,\n\ndanger\u2014its very height suggesting the\n\nviolence of a fall. I am myself, but I develop.\n\nEven before (I think) Keith Waldrop began to identify himself as an artist in collage, his standard operating procedure was juxtaposition, the plane of the page holding, in experimental relation to one another, images or words from provenances many and mysterious. All of his work feels haunted\u2014haunted by shadows, silhouettes, moving veils; by the outline of absence as much as by actual specters, as he says again and again; haunted, you might say, by a literally sensational uncertainty, by a shaky sense of what's real that often finds its embodiment in tropes of shadow and light. A lot, perhaps most, of his sentences suggest unease (e.g., from \"Night Soil,\" in Falling in Love Through a Description: \"I'm in a bad mood, forever.\"3 ) And yet the prevailing temperament of his work\u2014all of his work, including Light While There is Light\u2014is pensive and calm, melancholic yet peculiarly tuneful\u2014I even want to say harmonious, in a haunted sort of way. It's easy for collage to be about grotesque juxtaposition itself\u2014to be no more than clowning in an attic full of junk. But Keith Waldrop's verbal and visual collages are, above all, eerily beautiful.\n\nKeith Waldrop has said in a number of places, among them this book, that he has little imagination. He must mean that his organ of fantasy, i.e., of invention of something out of nothing, out of the void, is undeveloped, for what could be more creative than his power of uncovering, by juxtaposition and combination, the secret properties of things? He also says that he throws nothing away, that, having no imagination, he is stuck with memory\u2014he has only memory, or maybe memory has him. Thus one of the novel's finest passages:\n\nMy imagination is poor. In my dreams, for instance\u2014where one would suppose wishes can be fulfilled without hindrance\u2014if I dream the events this account describes, they are not usually changed, but in what should be a world nearer to the heart's desire, they play again, just as I tell them here, exactly as already experienced. It is as if despairing, even of imaginary improvement, I contrive instead to set my affection on the damned world, this very world, as it was and as it is.\n\nI too put Light While There is Light at \"the top\" of Keith Waldrop's work\u2014but then, I'm a sucker for narrative, its willing slave. On the particulate level the novel has the richness of the poetry\u2014the ghost-laden sentences; the juxtapositional ironies, moody beauty and sly jokes, of beautifully made collage; and also, since Keith Waldrop made it, there is everywhere that sense that a vast library is dissolved in the ballast water and is somehow stabilizing (or unstabilizing) the vessel. But what Light While There is Light has that the rest of Keith Waldrop's work does not attempt is, simply, extended narrative. Poetry (this is my defect, I admit) I can pick up and put down. What I can't escape, once I am pulled into its clutches by sufficiently interesting prose, is narrative. Light While There is Light is a vehicle carrying human cargo that moves achingly through time, taking me with it\u2014and I want to be taken, I want to be entertained, in the raw etymological sense of that word, lifted out of my time into the novel's time, until it kicks me out in the end, because I need to understand what happens to these characters. Chief among them even as he hides among them, unique, irreplaceable, and one of a kind, is the narrator Keith Waldrop himself.\n\n**Jaimy Gordon, 2012**\n\n* * *\n\n1 \"Keith and Rosmarie Waldrop Interview,\" Jivin Ladybug, online at mysite. verizon.net\/vze8911e\/jivinladybug\/id53.html\n\n2 Of course, some books listed under poetry in Keith Waldrop's bibliography feel like prose because they are prose, discontinuous, floating in pieces, framed in white space, but formatted as prose nevertheless\u2014like about four-fifths of the charming Silhouette of the Bridge (Memory Stand-Ins), 1997.\n\n3 From the second book of Transcendental Studies.\n\n_Opal Mohler, my mother, with her parents_\n\n_(Leeton, Missouri)._\nA Pilgrimage\n\nI\n\nI've read many stories of revenants and apparitions, but my ghosts merely disappear. I never see them. They haunt me by not being there, by the table where no one eats, the empty window that lets the sun in without a shadow.\n\nFew memories give me a sense of my childhood\u2014perhaps, later, more will surface. Among those few is the darkened room from which proceed my mother's moans. This is not a particular moment that I remember; it is the background of many years, nearly all my early life. She moans for so many reasons that it will be difficult more than to suggest their range. Probably I am ignorant of her most exquisite pains. I know enough not to make light of lamentations.\n\nSometimes I could get her to play the piano. She sat at the battered old upright, her eyes shut, picking out what she could remember of a Chopin polonaise or some cheap waltz from 1920. And then\u2014what really moved her\u2014\"Brilliant Variations,\" by someone named Butler, on \"Pass Me Not\" or other hymn. I was fascinated by the way she kept her eyes closed. To glance at the music, just as to read a paragraph of print, gave her migraines.\n\nI knew, of course, the words to the hymns she played, and, whether or not I sang them, they sounded in my inner ear, even through Butler's brilliance.\n\nSome day the silver cord will break\n\nAnd I no more as now shall sing\n\nGhosts gather in such lines.\n\nBut O the joy when I shall wake\n\nWithin the palace of the King.\n\nIt is not for her that I write this. She is dead, safe at last, out of all relation. I can recall, still, what she looked like at particular times, how she moved in certain spaces. But little by little she fades, replaced by an unsubstantial description somewhere in the memory. Best to make it as definite as possible. All we remember, finally, is words.\n\n\"I was always so weak,\" she said. \"My heart.\" She held her throat between thumb and index finger, which is how she took her pulse. \"When I was sixteen, the doctor said\"\u2014an unaccustomed pleasure in her voice now\u2014\"I shouldn't ever have to work, I was made to sit on a velvet cushion.\"\n\nShe taught piano while still in high school. (How little I actually knew her, her life extending back into the blank before my time\u2014when I was asked for details, after she died, I put down a wrong place of birth.) As my father studied law and then went to work on the railroad, so she went to the conservatory, graduated, but then, fleeing her parents, married and, as they say, had children.\n\nIt was not my father that she married\u2014he came later. I have two pictures of her first husband. In both of them, the left arm is in a Napoleonic position, as if he were holding a glass in front of him, but the hand is empty. \"He posed that way,\" she told me. \"He was proud of his wrist watch.\" He showed up, years later, with a second wife named Bessie, a sad-faced, decent-seeming creature who apparently kept him with her, and under control, by means of small but frequent doles. He was then (I mean, at his re-entry) a barber in Hot Springs, Arkansas.\n\nMy mother's favorite image was that of the church considered as a great speckled bird, which she took as a simple parable. Alien down here, humiliated and despised, the saints would eventually, at the Rapture, soar. Her favorite color was green, signifying restfulness. She maintained that a room with red wallpaper would drive one crazy.\n\nShe grew up in Missouri, an only child, but moved with her parents to Redmond, Oregon, where she went through high school. Whether she was in fact content there, I have no way of knowing, but certainly ever after she looked back to those days as a lost happiness and Oregon as paradise. Just a few years before her death, when she realized, not only that nothing had turned out right, but that there was no longer time for any good to come\u2014no horizon left for any miraculous rescue\u2014she began to retrieve what memories she could of Oregon. There were many eligible young men in Redmond, though her parents were watchful. If she stayed out too late, her mother in a fret sent after her. Her father, mild but dutiful, would seek her out, take her home. When her affections settled too firmly on a certain Lindsay, they took panic, packed up their things, and fled with her back to Missouri.\n\nMy mother's high school graduation picture\n\n(Redmond, Oregon)\n\nAt Nebraska Wesleyan Conservatory of Music\n\nBut Lindsay came again to mind, and one day she wrote him at his old address in Oregon\u2014this must have been in 1972. He not only got the letter, but wrote back. And what he wrote was that he had never married but had waited for her. I was stunned when she talked, not altogether coherently, about going back to Oregon, to marry Lindsay.\n\nMy mother's first husband, Charles\n\n\"When did you actually last see him?\" I asked her. She had to think, to count it up.\n\n\"Nineteen seventeen.\" She was too ill by now to go anywhere.\n\nThe history of my mother's religious opinions should be told as the record of a pilgrimage. As I imagine most pilgrimages, it was less the struggle toward a given end than a continual flight from disappointment and unhappiness. Neither the joys of heaven nor hell's worst prospects provide as forceful a motive as the mere emptiness of the world.\n\nBefore her first marriage, she played the piano for Methodist services. Probably at that time she thought little about religious doctrine or religious experience. It was, she said later, \"an old formal M.E. church.\" But once she married handsome Charles\u2014under what circumstances I never heard\u2014and was delivered of their first child, also a Charles, her relation to those early services must have changed. Ill soon after giving birth, she was kept awake one night by sounds of a party in the next apartment. Charles senior, his hair slicked like Valentino's and his wrist watch gleaming, went to quiet them down, and joined the party.\n\nShe described the scene. It was one of those that stuck with her, humiliating still after thirty, after forty years. She got up, the noise having increased after his leaving. In her housecoat she went to the next apartment and knocked and asked for her husband. Charles, embarrassed in his turn by the appearance of obligation in the shape of this frail form at the door, went with her, but explained the exit to his friends of an hour with a wink and a formula: \"She's a Sunday School girl.\"\n\nWith their first child Charles, Jr.\n\nI am convinced that, at that moment, the formula was wide of the mark. Probably poor Charles never in his life figured anything quite correctly. But this must have been one of the incidents pushing her toward the church as a refuge from the world as represented by \"old painted women\" (her colloquial old not referring to age) and by the routines of a loveless marriage. By the time I could remember anything, she was taking me to the Free Methodist church at the corner of South Avenue and Commercial Street in Emporia, Kansas.\n\nThe Free Methodists split off from their parent church (the old formal Methodists) about the time of the Civil War. It was one of the many groups preaching a return to Wesley's doctrine of \"Christian Perfection.\" Sanctification, they teach, is a distinct act, subsequent to justification. To be justified, or \"saved,\" is to have one's sins forgiven, but to be sanctified is to have the carnal nature, the taint of original sin, removed. They also call this state \"holiness\" and they are aware that the world dismisses them as \"Holy Rollers.\"\n\nFor they have also kept the ecstatic side of Wesleyanism. What I retain most vividly of the church in Emporia (which I attended until I was fourteen) is the way services were always rescued from dullness by what I learned to call the demonstration of the Holy Ghost. What in fact happened, Sunday after Sunday (and at Wednesday night prayer meeting), was that two women\u2014I remember their names as Sister Eliot and Sister Faulkner, though it now sounds unlikely to me\u2014fell under the influence of the spirit and began to behave in exactly opposite ways. They were opposites already: Sister Eliot was strawberry-blond, open-faced, outgoing, and when the spirit hit her she ran down the aisle, shouting. Sister Faulkner shrank back, twisted, moaned, and often sank to the floor, a small, swarthy woman, weeping bitterly.\n\nMy father, Arthur Waldrop, at the Santa Fe yard office\n\n(Emporia, Kansas)\n\nTheir performance was joined in by the congregation in general, most of whom confined themselves to Amen's, shouted or murmured, but they were the natural leaders. What, I wonder, would they have done, have become, if the church had not been there? Perhaps it is well to add that these services had nothing Erskine Caldwell about them. Powered by sexual energy perhaps (what other source is there?) they were chaste and even, I would say, dignified. And they gave some meaning to lives otherwise lost in weekday blankness.\n\nMy father thought all females in terrible league against all males, but the center of the plot was among the Free Methodist women, whom he pictured as the hags from Macbeth sitting in unholy assembly to pass judgment on him. He felt them sitting; their weight bent his shoulders.\n\n\"Your mother,\" he would tell me, \"didn't have a dime when I married her.\" He always started that way. \"She had one damn cardboard suitcase.\" If he were drunk enough, he would go on, getting louder. \"Not a pot to piss in. And those three brats.\" Charles, Elaine, Julian: before Julian was born, the elder Charles had taken off. Julian was born in Leeton, Missouri, at his grandparents' house. My father had already two daughters and was close to twenty years older than his second wife. I have no idea how they met, let alone what drew them together. \"Now she runs off down to that damn church. They turn her against me.\"\n\nThe spookiest story I ever heard was told me by a friend, who may have written it down somewhere, but I know it from her directly.\n\nShe was in England, traveling with a boyfriend. At some point they found accommodations in one of those country houses where the family lets rooms to pay the monstrous upkeep on anachronistic grandeur. She and her friend were shown into the largest room they had ever seen, with high windows, oak paneling, huge four-poster, a room from what was to them a storybook era. A grandiose fireplace dominated the room, but there were none of the usual paraphernalia\u2014screen, fire-dogs, bellows. Instead there was only, half in the great fireplace and half out, a cradle. They wondered at the cradle\u2014of the old-fashioned kind, like the one Lillian Gish rocks in Intolerance\u2014and went to dinner.\n\nBut later, when ready for bed, they could not quite manage to disregard it. They tipped it, finding that it rocked with a sort of bulky motion, soon coming to rest again. It had somehow a great weightiness to it, a dense heaviness that struck them both as incongruous in a baby-bed.\n\nPerhaps what happened next was their effort to escape the fascination of the cradle there on the hearth (she never said so, made no attempt at explaining anything). In any case, they began horsing around and her friend, before she realized what was happening, picked her up and put her in the cradle. And then he ran across the room and turned the lights off.\n\nAnd it was dark then, of course, but it was not a darkness that she recognized. It was as though there lacked not light, but the flow of time. It was not, across the black room, a distance in steps, that even the blind might feel their way, but a space of centuries, a loss total and immeasurable. And she could not get out of the cradle, which she felt rocking. She could not even struggle. With the utmost effort, she managed to form her friend's name, but cried it so feebly that she knew it would never carry across the emptiness.\n\nHe meanwhile, as it turned out, was feeling much the same thing as she and was searching, terrified, for the light switch, which he could not find again. Finally his hand, groping blindly, hit the right spot and the room burst into light\u2014the same room, with its paneling, its four-poster, its cradle in the fireplace, and her, clambering out of the cradle. They were both terror-stricken and refused to stay the night in that room.\n\nYou should not suppose that I am writing this to judge between my father and my mother. It would hardly be reasonable, now that they are both gone, to decide their quarrel. In my mind it remains a given, and goes on, an eternal argument.\n\nMy father named me Bernard, after Shaw, and Keith, for Sir Arthur Keith (my father's name was Arthur). \"Two old atheists,\" my mother always said, certain he had picked the names to irritate her. (Her notion of atheist was a bit vague.) My father generally professed agnosticism, but in his last years\u2014especially while drinking\u2014insisted that he believed in God. \"Otherwise,\" he said, \"how could I be a Mason?\"\n\nOne of his favorite recollections (it must be remembered that he was half a century older than I) was the attempt of William Allen White, editor of the Emporia Gazette, to get into the local lodge. \"They came to the question,\" my father related, tapping my knee for emphasis, \"which every applicant has to answer. They asked him, Do you believe in God? He said, I believe in William Allen White \" A dramatic pause, whether of outrage or admiration I could never decide. It was certainly portentous, our most celebrated citizen hanging in the balance. \"And they turned him down!\" My father had gone on up, into the Scottish Rite and the Shrine.\n\nIn the era of the Civil War, the Methodists, already old and formal, charged rental on their pews. The seceding branch, preaching largely to outcasts and the needy, decided to abolish this charge and so denominated themselves Free. It was a common practice among congregations of that time, especially near the frontier, to hire musicians for Sunday services\u2014which meant, more often than not, bringing players from the local theater into the house of God. The Free Methodists, with all-or-nothing zeal, abolished instruments from their church, and in the early nineteen forties, when I attended and after much hesitation joined them, the singing was entirely congregational\u2014no choir\u2014and regulated by, at most, a pitch pipe. Sister Eliot or Sister Faulkner would lead (the one somewhat faster, the other somewhat slower, than expected tempo), not moving their hands, but simply by facing the congregation and singing out.\n\nSo when Sunday morning came around, or Sunday night, or Wednesday night, my mother would cease from her moanings, comb her hair, put on her best, and she and Elaine and I would hasten to South and Commercial for some a cappella praise, some middling preaching and, with luck, a breath of ecstasy. As long as she was there, among the saints, her life seemed clear and meaningful. Outside, in the world, she was a complex of miseries that I am still not sure I can quite sort out. \"My heart,\" she would say, fingers to her throat, but it stood for a whole existence.\n\nHer greatest pride was the smallness of her hands and feet. If one of us bought her bedroom slippers for her birthday\u2014demanding at the store their very most smallest size\u2014they were sure to be too large and she was sure to be enraged that we should think her feet so gross. On less emotional occasions, \"I have Cinderella feet,\" she would say, and it was terribly plain that, not only the prince, but the entire fairy-tale realm, had passed her by, leaving the most workaday ashes.\n\nHer first husband had been a deception. Her second she treated frankly as an enemy. It only gradually dawned on me, between battles, that I was disputed territory. Every time I went to church, it was a victory for her, and I came to regard my father as an alien power, sinister in behavior, but possessed of strange forces. His occupation itself was mysterious: as a railroader, he was fanatically punctual, continually checking his watch and angry if it got more than a few seconds from the official Santa Fe clock. But since he worked on freight trains, he was liable to appear at any hour of the night or day and just as arbitrarily to be called away. The unknown figure of the caller\u2014just a voice on the telephone\u2014made its way into my private mythology. Simply to answer the anonymous ring put one within the possibility of hearing, instead of an Hello that would connect with a remembered face, the disorienting but imperative \"This is the caller.\"\n\nBetween the living room, which for some reason or other was my bedroom for a time, and the room where my parents slept, there were huge sliding doors. (This was a house on Neosho Street, the most nearly permanent of our homes\u2014but there was inevitably, wherever we were, a sense of provisional arrangements, of waiting for better weather, a new government.) One night when my father was in, not likely to be called, I settled down on my convertible but could not enjoy my insomnia because of the argument from the other side of the door. An argument meant that my father's voice continued on and on into the night, occasionally raised to a shout or broken by a murmur from my mother, who for the most part maintained a dead silence. All their nights together spread out like this into an agonizing deadlock. I don't know what they argued about, or if indeed there was a subject. To escape from the oppressive sound, I set myself to wait for the unattainable moment of entering sleep. To be conscious for once at that magical transition seemed to me\u2014I don't know why\u2014a knowledge I would need, that I could not well do without.\n\nBut just when I had dozed off, slipping past the threshold I wanted so much to examine, I was jolted awake by doors slid open, then slid shut again. And then, in the dark, my father lay down beside me, breathing heavily. I made no sign of life and gradually he subsided into, I thought, a sleep of his own. But apparently he was listening, there in the dark beside me, for before I quite had a chance to miss again my moment of going to sleep, he had thrown off the cover and the great doors rolled back with a crash and he was swearing loudly. After he left his bed, Elaine had slipped in beside my mother. Now she raced back to her own room as he switched the lights on. \"This is what happens,\" he was yelling, \"as soon as I turn my back.\" The giant doors cracked shut again, leaving me dazzled with the light that was now shut out. His shouts continued on the other side, Elaine's voice sometimes chiming in from a distance. (Neither Charles nor Julian were there\u2014Charles was in the war in the South Pacific.) I lay tense while the shouts got louder. I heard Sister Eliot's name. Finally there were other sounds: movements, doors. Then a blow and my mother's scream and Elaine howling.\n\nI began to pray. I began, with an earnestness I have rarely recaptured in any action since then, to pray to God that he would strike my father dead. My prayer was answered, some dozen years later, after both my hatred and my faith had died long lingering deaths. The only immediate aftermath of that night was a peace-offering from my father, a new secondhand piano, at which my mother sat, eyes closed, playing what she could remember of something by Chopin, a syncopated waltz from 1920, brilliant variations on \"Pass Me Not.\"\n\nII\n\nOnce or twice I rode with my father on his waycar at the end of a mile-long freight. From the cupola one could see along the tops of a hundred cars to the stack of a 2900. It seemed to me a tedious trip, much of the time spent on sidings, waiting for the dinkiest passenger trains to zip by. Night came and my father lit the kerosene lamp over his jolting desk where he went through the paperwork for the hundred carloads. The lamp gave a brilliant white light from its ash mantle, but I dropped off before we made it into Newton and was only half awake to clamber across the immense freight yard where, strangely, there were blinding lights all through the air, and yet the crisscrossed rails seemed endless and unlit.\n\nOn one such trip\u2014when I was not along\u2014the waycar, caught suddenly in the collected slack of a hundred couplings, cracked whip-like and threw my father on the floor, where he found he could not rise again. His back was broken.\n\n\"It was those months in the hospital,\" he always said later, even much later, \"while I was helpless in a cast, while I was out of the way. That's when they really got control of your mother, those holy bitches.\" He was in the Santa Fe's hospital in Topeka, where I visited once a week. I am not sure, but possibly it was then I joined the church, a ceremony that my mother took as a triumph in the war against Satan and Arthur. At any rate the house was quieter. Then, finally, he came back, and gradually everything was as before, but a little worse, as things usually get worse.\n\nMy mother's health was a constant problem. She had had some of the best doctors: she went to Dr. Curran in Kansas City for her tinted glasses. A goiter that had developed about the time I was born was cut out by no less than Dr. Hertzler, the famous \"Horse-and-Buggy-Doctor\" of Halstead. He became, understandably, one of her heroes. In his hands, she felt safe. \"He's so good,\" she said, not alone I gather from her own experience but from a composite legend, \"he could just cut that old goiter out and then\" (her tiny hand in gesture of magic) \"just tie it all up with one hand.\"\n\nI might not have remembered that, except that some years later, when she went to Halstead for a checkup of some kind, I went with her and sat in the waiting room of the clinic and stared at the Mennonites in their bonnets and beards. And old Dr. Hertzler himself, who saw no patients anymore but wandered vaguely around the maze he had built, appeared with a fixed and benign smile. He had forgotten to put a shirt on, so his flannel underwear showed above baggy trousers. Stopping in front of me, he looked down from a great height and out of some pocket came the most enormous hand I had ever seen. He was a big man, but his hand was the hand of a giant, and in it was a yellow jawbreaker that he placed precisely in my palm. Then he wandered on to the next child. The nurses tolerated him.\n\nMy mother's favorite hymn, which she looked at long enough to memorize, at the price of a terrible migraine, was \"Where the Healing Waters Flow.\" She played it over and over. More and more her notion of deity became that of a healing god, the waters of salvation cleansing, not only moral stain, but physical sickness and deformity. But healing was always still to come\u2014meanwhile, she sat on the right side of the church, because her left eye was the stronger.\n\nIn one quarrel, when I was twelve, my father broke her glasses and she sued for divorce, getting an injunction that forced him to leave the premises. The final settlement by the court gave her the house and me, and he was to pay a small monthly allowance for my support.\n\nAbout the same time, Elaine was graduating from high school and was getting rather serious about a ministerial student at the College of Emporia. He wanted to marry her, but she decided that, in spite of her feelings for him, she could not marry a Presbyterian.\n\nThis was, I am convinced, her own seriously considered decision, whatever counsels Mother may have offered. But it pointed to a great problem that had obviously to be faced. The local church was small. There were many marriageable females, but no suitable, unattached man. The College of Emporia, to the extent that it was not simply secular, was the institution of a cold, formal, worldly church, worse than the Methodist if not quite Episcopalian. Kansas State Teachers College, on the other side of town, was of no religion at all\u2014they could well have atheists teaching there. There were two attempts to save the situation. The first was to explore other churches. (I should point out here that, though my father attributed everything to the influence of Free Methodist furies, the Free Methodists did not altogether approve of divorce and my mother was conscious of having fallen somewhat in their esteem.)\n\nWe visited the local Friends congregation, which I now think must have been rather degenerate\u2014with a hired preacher\u2014but my mother found it cold. From there we went straight to the Salvation Army, more to her taste, and once or twice I even played my violin in their street services. (I hope my father never found out.) But in spite of evangelical fervor, somewhat wasted on Emporia where there was only one bum and nothing stronger than 3.2 beer, except for those who could afford bootleg rates, their sense of doctrine was undeveloped, a bit Buddhist almost in its determination to rescue the perishing before specifying the works of grace. This irked my mother and, besides, though haphazard orchestration was at first a relief from the purely vocal strains we were used to, it hardly took conservatory training to be offended by the sound. Besides, there was hardly a male, over fourteen and under sixty-five, to be seen. We moved on to the Hardshell Baptists.\n\nAs I think back over all this it makes, alternately, too little and too much sense.\n\nThe Hardshell Baptists had a tiny church that had been a neighborhood grocery. The sign over the door, the only trace of recent paint, said \"Fundamental Baptist Church\" but my mother always referred to them as Hardshells. The preacher, who somehow made me think of a butcher, was a true ecstatic, carried off in turn by waves of joy and a pity for lost souls. While the latter was on him, during the altar call, he would sob uncontrollably as if his heart were broken. He radiated a sense of poverty. His church, a missionary effort, took all his time and energy and gave him only the most spiritual returns. I never saw more than six or eight attend any service and that was counting his wife and daughter. He wanted desperately for us to join his congregation and\u2014even though there was no young man for Elaine\u2014we might have done so, had he not insisted, with more sincerity than tact, on doctrines anathema to my mother. What annoyed her most was the claim that once saved, a person could never lose salvation, the doctrine of Eternal Security.\n\n\"Do you think,\" he asked Mother, \"that a sheep can become a goat?\"\n\n\"Once saved, always saved?\" she asked in return, daring him to affirm an outrage.\n\n\"When we are saved, we become God's lambs,\" he said, warming to his argument. \"And His grace is sufficient to keep us from falling.\"\n\n\"You mean you can go out and get drunk and be worldly and still be a Christian?\" Her voice was getting higher as his gestures took on more and more pulpit manner. While his left hand moved horizontally, as if smoothing something, the right made a sort of chopping motion.\n\n\"If you're really saved,\" he said, \"you won't do that. You'll live as a dove because the Lamb of God is in you. We're born again, we are his children. Do you think His children can become the children of the Devil?\"\n\n\"Well,\" she said, shouting by now, \"you can't tell me just because a man has been saved once, he can go out and drink and swear and murder someone and smoke old cigarettes and still be a born-again child of God.\" And as he raised his arms, doubtless to bring down his final and most crushing point\u2014he was now red in the face and his forehead was beading with sweat\u2014she bellowed, just before sweeping out the door, which she slammed behind her, \"And what about the backsliding heifer? \"\n\nGradually, after the Four-square Gospel and a few more, we found ourselves back in the old church at South and Commercial, and with a different solution: Elaine would go away to a holiness college. The Free Methodists have several schools, but the nearest was in Illinois, whereas a related sect, the Wesleyans, though unrepresented in Emporia, maintained a junior college in Miltonvale, Kansas, and in the fall Elaine entered Miltonvale as a freshman. My mother's inquiries into the faith of the Wesleyans brought forth not a single deviation from Free Methodist doctrine. Their services were identical. The only differences were, firstly, that the Wesleyans had church councils, but no bishops, and secondly, they had no prejudice against the use of musical instruments in their worship.\n\nMy father's mother at her house on Cottonwood Street, Emporia.\n\nMeanwhile my father, whom I saw more or less regularly\u2014given his completely random schedule\u2014had moved in with his mother and his brother Roscoe, who lived on Cottonwood Street. My memories of Emporia, not all pleasant, are lined with lovely elms arching and interlocking over streets of asphalt or brick. And when I try to think of the same town since the Dutch disease passed through (it has been over thirty years since I was last there, for my father's funeral) it comes before me as a wasteland of stumps and rotten trunks. Already fallen or still standing, they are dead, and in the middle of summer the sun shines through them. The cicadas (which we called locusts) must now have difficulty finding a patch of bark to discard their shells on, the great trees themselves skeletons.\n\nMy grandmother was in her nineties, almost totally deaf. She sat through most of each day rocking, wrapped in smelly woolens, nodding peacefully. My father hated her. He swore she could hear perfectly, but simply would not listen. His greatest fear was that she would outlive us all, me included. Occasionally she would struggle up out of her chair, make her way to the kitchen and do some mischief. Deciding to make coffee, for instance, she once brewed a good quarter pound of my father's Bond Street tobacco and if he had not come in while the reeking mixture was still percolating, she might have drunk it. Roscoe was her youngest child and had lived with her for sixty years.\n\nOne of my father's continual humiliations was Roscoe's presence at the depot. Freight crews assembled at the yard office, a mile out of town, but from time to time he would have some business or other at the passenger depot (which he always referred to as \"the levee,\" as though the tracks were a kind of Mississippi, floating streamliners between Chicago and California) and then he would invariably catch sight of Roscoe's grizzly form. Roscoe, with his antique Stetson and weary suspenders, made the most drab surroundings look opulent by simple contrast. He was a soup line figure, stiff, unshaven, with sunken cheeks and deep-sunken eyes. He rolled his own and had always a mutilated cigarette between two fingers,that had turned brown in consequence, or between his teeth, which were also heavily stained. He had a job at the Depot Hotel, where his official title was, unless someone was putting me on, \"dishwasher's assistant\"\u2014his main task, making soap for the kitchen\u2014and most of his working hours seemed to be spent across the street on the station platform. The baggageman and a couple of retired railroaders, now become whittlers, were his cronies, and they whiled away the afternoons telling adventures and arguing politics. It was not Roscoe's appearance that embarrassed my father\u2014though women tried to keep their children from staring at the little assembly\u2014but his strongly expressed Republican opinions.\n\nMore and more, as he got older, my father's defenses against the world were anger and eating, and often he indulged them concurrently. Eventually I hesitated, especially at meals, to introduce any subject whatever, the range of his irritation having increased along with the force of resentment. He listened to all the radio newscasts, and commentators, and read the Gazette from beginning to end. It all made him furious. \"You son-of-a-bitch!\" he would shout at Fulton Lewis, Jr., or just some neutral voice from the local station. Or reaching the editorial page of the paper, and seeing the signature \"William L. White,\" \"Young Bill isn't half the man his dad was. Runs off to New York and lets the paper go to hell.\" And then, putting down the paper a moment to refill his pipe, \"Of course, old White tried to get us all to vote for Cal Coolidge. The son-of-a-bitch.\"\n\nMy mother's favorite story was a simple anecdote. A woman\u2014\"and she wasn't an old woman either\"\u2014took down a dipper from its hook. And I was transported from the world I knew, a world of hot and cold faucets, from Emporia to Leeton, where the well water tasted awful to me\u2014or to Samaria. She took this dipper from the hook, without lighting the lamp. And in the shadows she scooped a drink from the water bucket and swallowed, along with the water, a black widow spider.\n\n\"No one knew this was going to happen,\" my mother always said at the end of this tale. \"Not even the angels knew. But the Holy Ghost knew.\"\n\n\"Didn't Jesus know?\" I asked.\n\n\"She thought she had a long life yet to live. And she went to meet her Judge, prepared or not. You never know, from one minute to the next.\"\n\nElaine, at Miltonvale, promptly attracted a young man, a prospective minister of the right faith, and at Christmas vacation brought him home. This was the first Christmas after the war. Charles was just out of the navy, Julian had just gone into the navy, but as it worked out, they were both home.\n\nMy mother's heart was set on having a preacher in the family. She had given up on Charles. He was not only headstrong, but a constant reminder of the Charles who had deserted her\u2014she always claimed the younger to be a spitting image of the elder. She decorated the walls with photographs of her son in uniform (and there was for some years a silver star in the front window, denoting a member of the family on overseas duty) but his presence was always a trial, from preschool age even, when from behind the post of a porch he hurled whatever swear words he knew down a steep terrace to startled passing strangers.\n\nShe had, by this time, just about given up on Julian, mainly because of his long-standing feud with the Emporia police, which had ended in the compromise of his recent enlistment in the navy. It was touch and go for a while, the police swearing they would have him in reform school. The balance shifted, however, with Julian's theft of their arsenal, at which point they showed a willingness to bargain, and, instead of threatening, offered him a position on the force. And they were again angered by the language of his refusal.\n\nMy case was not yet decided, which is to say that my mother had not given up, but as yet I had received no call to preach. All that could safely be said so far was that it was still not too late. The call was, of course, absolutely essential since, besides standing to reason, it has an unequivocal text (Romans 10:15). Elaine envisioned for herself the sort of double ministry in which both the preacher (male) and his helpmeet qualify as able workers\u2014in short, she felt called to be a preacher's wife.\n\nShe and mother consulted together and prayed long hours to know whether this particular conjunction was the will of God. Her young man, she insisted\u2014still insists, if his name comes up\u2014was a fervent Christian and a genius. The latter quality immensely pleased her, though I think in mother's view it suggested vain science and man's philosophy. In this case, at least, intelligence created a snare. While Elaine sought the counsel of God and parent, Charles and Julian were checking out her guest. Their conclusion, which clicked almost audibly as their eyes met after some reply, did not concern itself with matters of faith or morals, but with problems of coherence and comprehension. They had decided, simultaneously and irrevocably, that Elaine's suitor was mad.\n\nSo they began to treat him accordingly, taking his most commonplace remarks as full of strange meaning, echoing his words in slightly distorted senses. Having been a trifle nervous at meeting the family, he soon developed a case of jitters. But there was worse to come.\n\nCharles had brought home guns, several Japanese rifles and a German pistol. He had set up a target in the basement and every once in a while went down and blasted away with the pistol\u2014the only weapon he could find ammunition for. He and Julian now took the guest to the basement to show him around. \"They're shooting that old gun off,\" my mother said, red-eyed from prayer. The shots in fact were ringing out, a nerve-racking sound. Then they stopped. And soon, with laborious shuffling steps, Julian and Charles ascended, the third man more or less hanging between them. He was stunned, but soon recovered, without visible wound. Julian claimed he had\n\nsimply suddenly collapsed. He maintained, however, that Julian had hit him over the head. In any case, a few hours after what seemed full recovery, he became feverish and, by degrees, delirious. Dr. Hovorka examined him and pronounced with what seemed utter irrelevance\u2014we would not have been surprised at hearing a diagnosis of epilepsy, plague, Huntington's\u2014that the patient was suffering from a strep throat. He should be kept in bed.\n\nWe kept him in bed. In a few days he had recovered and gone home. And just before Elaine's vacation was up, she got a letter from him. Charles and Julian had both done their best to convince my mother that he was insane and Elaine's protests that he was an intellectual served merely to reinforce this idea. But even Elaine's sentiments were confused by the arrival of the letter. It started with compliments and went on to anecdotes and small talk, but one passage stood out and she went back to it again and again.\n\n. . . I felt you had somewhat cooled towards me. I don't mean at your home, where I was very sick, as you know, sicker perhaps than you know, but before, at Miltonvale where I will soon, I trust, see you soon again. I am afraid that they\n\n(Here he had apparently started to write some name, possibly Elaine's college roommate, but crossed it out and put \"they.\")\n\nhave told you evil stories about me.\n\nUnderstand, I don't mean to accuse anyone of willful lying. But they may be mistaken. It's hard to say what I mean, but they may have reported with the best of intentions a falsehood. Oh if you knew what it costs me to write this. Because, you see, I don't know if it is false or not. I have asked my Redeemer for forgiveness, even if I did not actually sin.\n\nWhat I am trying to say is that if it is a story about a woman they may have seen me with very late at night, long after hours, a few months ago, I would like to tell you the whole story. I did not know what kind of woman she was. Oh it is too painful, I cannot tell you how I came to be in her room that night. What I want you to know is that when I realized the sort she was, what she wanted of me, I started to leave.\n\nThen everything blacked out. Later I had no memory of what came after that moment. The next thing I knew was that we were walking along the street and she had hold of my arm. I pulled my arm loose and ran. It may be that they saw me walking with her arm. But what would they have been doing out at that hour.\n\nThat makes no difference. I only wish I knew what happened while I was out. In any case, it is under the blood. Believe me. . . .\n\nAnd on. And then to other things.\n\nElaine went back to Miltonvale, and might well have married him. But Mother had now decided he was possessed. The letter (she always liked things to be documented\u2014the letter went into her trunk) was a perfect admission of guilt. The demon had possessed his body while he was with some sinful creature. That was why he had no memory of the events. And if the devil had him once, what was to prevent its happening again? She decided that Elaine should be far away from him, at some other college. This was the sign that pointed us to Sharon.\n\nIII\n\nOne whole dinner party, once, I had to be polite and listen to a philosopher, whose name I can't recall, expand on the notion of divine omnipotence. No doubt Rosmarie was suffering more than I, since her Catholic background remains with her in the form of a distaste for religious conversation. (My mother, almost at the end of her life, was heard lamenting by telephone to her last pastor, \"I had three sons\u2014two of them married witches, and one married a Catholic.\" Her firmest conviction was that once a Catholic, always a Catholic.)\n\nThe miracles of the Bible are violations of the so-called laws of nature, but these laws are based on observation, are merely empirical, and deity is above them. When Jesus, after the resurrection, comes into the room without opening the door, it is a miracle\u2014in the sense that one cannot ordinarily do it. To raise the dead is miraculous and, likewise, to make the sun stand still. We cannot do these things, but God can, just as he performed the greatest miracle of all: the creation. He is omnipotent, which is to say, he can do anything. No\u2014he can do anything that can be done.\n\nMe, Charles (holding me), Julian, Elaine\n\n(Emporia)\n\nFor beyond these mere physical impossibilities\u2014saying to this mountain, Be removed\u2014there are the true impossibilities of logic. Even God cannot annul the law of identity, which says that whatever is, is. He can, of course, destroy what is, but cannot make it at the same time be and not be. He can create seven times seventy worlds, but cannot keep seven from being a prime number. Two plus two equals four in all possible worlds, created or uncreated.\n\nWell, I had heard all this before, and while he was talking, and on the way home, I was faintly amused by anyone taking such things so seriously. But later that night I was unreasonably angry, to think that people professing belief in God should turn and subject him to Copi or Quine. If I were to invent a god, I would make sure he didn't get stuck with the primeness of seven or have reason to feel threatened by Godel's proof. He might produce a square circle, if he felt so inclined. And, yes, he could, at one and the same moment, exist and not exist.\n\nElaine, me, Charles, and Julian on a visit to Leeton.\n\nElaine, Charles, me, and Julian (Emporia)\n\nSharon College, sitting on a hill in South Carolina, at a point where the Blue Ridge has petered out, was founded by Wesleyans shortly before the First World War. It is recorded (by the Reverend Eber Teter, a founding father whose health failed before he could accept the office of treasurer) that when the first spadeful of clay was turned, those present fell on their knees with one accord, sang repeatedly \"Praise God From Whom All Blessings Flow\" and prayed many a fervent prayer. \"How we felt our hearts,\" his account says, \"burn within us.\" When my mother and Elaine and I got to Sharon, walking the last half-mile through the dusk, the fragrance of apples gradually replacing the soot in our lungs, Wednesday night prayer meeting was on in the chapel and they were singing \"Jesus Paid It All.\"\n\nSharon was a junior college, but boasted a four-year program in theology\u2014leading to the degree Th.B.\u2014and it had a high school as well. We moved into three rooms of a building called Teter Hall, which had been the original men's dormitory but, converted now, housed married students and us. And the English teacher Miss Yodle who, to mutual regret, lived just below us. On each floor, at the end of the hallway, a john had been installed, though to shower one had to trudge to the basement. Miss Yodle would not use these inside facilities, I suppose because the rest of us did, and so she had a private outhouse\u2014not the only one around but the only one still in use, which every Halloween of course got tipped over, leaving a two-hole stool on exhibit.\n\nJulian was then stationed on the West Coast, having joined the navy on V-J Day. When his discharge was due, as I found out later, Charles had driven (from where, I have no idea) to California to get him (to take him where, Charles probably had no clear idea either). Julian, it turned out, was in the brig. He had bashed the mess cook with a serving spoon after the cook refused him seconds. I remember how once, when Julian was growing up, he ran to Mother crying and tried to get her to hide him.\n\n\"They're going to put me in jail,\" he kept screaming.\n\n\"What have you done?\"\n\n\"I pulled Swint's nose,\" he said and Mother, relieved, tried to console him.\n\n\"You shouldn't have done it, but they aren't going to put you in jail for pulling someone's nose.\" He looked doubtful, and it came out that he had pulled his friend, by the nose, from Fifth and Neosho to Twelfth and Main, close to a mile.\n\nThe town Sharon is three miles from the college. Its Main (and only paved) Street is the highway; on one side of it the railroad runs parallel and on the other there is a rampart of buildings about the length of a city block\u2014Sharon's business district. The buildings towards Greenville are old, the end one a large frame structure housing a general store; those the other direction are newer\u2014there is a movie house and, where the highway begins to curve, the quicker to reach Atlanta, a pink service station. (I say \"is\" because, although everything has doubtless changed since I left there, Sharon seemed to me at the time to present an aspect of eternity: it was so ugly.)\n\n_Charles, in front of the house at 614 Neosho Street._\n\nThere is one break in the rampart, one vacant lot between two edifices, where we held street services. J.W. would back his car over the sidewalk onto the packed clay. He had a loudspeaker on top of his car and after we had sung a few hymns he would preach into an elegant little microphone perched on a pole of aluminum. People just walking along Main Street would suddenly, when they were past Harvey's package store, find themselves confronted by an invitation of enormous volume. And coated with static, since J.W. yelled directly into the little cup and sometimes shook the rod in his enthusiasm.\n\nJ.W. was himself enormous, balloon-like, and when he was in the spirit it often seemed to me that he might bounce too high and be carried away. He was from somewhere in the hills\u2014the lower Blue Ridge\u2014and he had ministered to hill people for fifteen years before coming to Sharon to study homiletics. His reputation had, in fact, ranged from village to village over a surprising territory, mainly because of an exorcism he had performed in the late thirties. An old farmer, who produced principally moonshine, was dying in his shack and J.W., after climbing a mountain to see him, was greeted by a crone carrying a shotgun and told that the case was absolutely hopeless. The man, eighty-odd years old, had been hexed. A lady on a neighboring slope\u2014the shotgun-carrier suggested a broken engagement, but it could hardly have been recent\u2014was by some means or other causing his chest to constrict, a little more each day, and the end was near. J.W. went in, despite the smell, and found the man lying on a pallet, eyes bulging out of a dry skull, his arms locked around his chest as if he were holding himself to the bed. J.W. prayed, read the Twenty-third Psalm, prayed some more, put his hand on the old head, and prayed some more. The man seemed already in another world. J.W. leapt to his feet, took a charred stick from the stove and on the rough wall of the shack scratched a stick figure of more than human size.\n\n\"Where does it hurt?\" he shouted to the sick man, who in spite of himself had moved until he could see the drawing. \"Where does it hurt?\" J.W. shouted again and getting no reply shouted on, \"Here! Here's where the devil is!\" and scratched an X in the middle of his stick figure's chest. \"The devil is there! He's there!\" he shouted, pointing at the X. Then he grabbed the shotgun from the crone, and shouting \"In the blessed name of Jesus\" pulled the trigger and blasted the X, the devil, and a good portion of the wall. In an instant the man was on his feet, hopping mad and cursing, chasing J.W. halfway across the mountain while the crone, on her knees, screamed hallelujahs.\n\nI played the violin. I started in junior high and as in many of my more complicated projects, began with promise, a promise never fulfilled. But musicians were scarce at Sharon, so for street services a fragile girl named Stella played the accordion, and I fiddled away\u2014practically inaudible under Stella's vast and unarticulated sound. Often Evangeline stood next to me and held the hymnbook so that the pages would not flip in the wind. I tried hard to figure out Evangeline. Glancing at her (sideways, missing a few notes of \"Work For the Night is Coming\") I could sometimes suppose that she was different merely because she believed more fervently or lived more righteously or prayed more than all the others\u2014which she may have done.\n\n\"My text,\" J.W. roared on one such occasion, \"is from Revelations.\" It usually was. And he proceeded to read, in its entirety, what the Spirit said to the church of the Laodiceans: \"These things saith the Amen. . . .\" A train was pulling in across the street, stealing J.W.'s thunder and sending a shower of soot down on all of us. There were few listeners anyway, but some left amid the roar. A teenage girl began to laugh and shout into her boyfriend's ear\u2014he tried then to pull her away from the meeting, but she stayed and so did he. An old woman sitting on a package wrapped in brown paper slipped a little more snuff into her already bulging lip. \"I know thy works, that thou art neither cold nor hot.\" The train had stopped and was snorting. \"I would thou wert cold or hot.\" The black cloud that belched out now blotted the daylight and we could hear nothing but the engine puffing and grinding. Engineers do not like to stop at Sharon, going east; the rails are upgrade then all the way to Easley. The electric voice came through again as the last car rumbled past and the sky slowly reappeared. \"Behold I stand at the door and knock. . . .\" I could look at the scripture as it was read, because Evangeline had her blue leather Bible open, but I was gazing at my brothers, whom I had not seen for two years but who were now, suitcases and all, crossing the highway from the depot, followed by two of the toughest-looking characters I had ever seen.\n\n\"And I tell you,\" J.W. was roaring, \"when Jesus comes back to this earth with power and glory and sits down to judge the quick and the dead, is he going to find you with your lamps trimmed and waiting? Is he going to find you ready at the Rapture or will he say depart from me ye wicked into everlasting damnation I never knew ye?\" Stella started playing for the invitation. Evangeline opened the book to \"Just As I Am\" and I scratched out the melody underneath the heavy chords. Most of the sidewalk congregation moved away quickly, but those who did not were soon cornered and exhorted to give their hearts to Jesus. A tall black man with white hair and expensive clothes nodded yes he was already the Lord's and touched the hem of his jacket to make sure it still covered his hip pocket. The woman on the package was weeping and with a crooked index finger scooped a mass of snuff out of her jaw and slapped it on the sidewalk. When it was all over and the microphone and the accordion and my violin were packed away in J.W.'s car, neither Charles nor Julian was to be seen, so I rode back to the campus with the others.\n\nMy mother's first experience of divine healing was directly after separation from my father. It was rather an informal experience, without a service and without the shock of sudden recovery that is commonly reported. \"I told the Lord,\" she said, \"that I was going to do his will\u2014and he would just have to give me the strength for it.\"\n\nThe most obvious improvement was in her eyes. She now found it possible to read the Bible, in moderation, and to look at a certain amount of music. So she started again, after a lapse of twenty-odd years, teaching piano. At first she took on neighbor children, a few from the church, some friends of mine, and little by little built up a large class. The house was always ringing with some botched melody, which she would correct mercilessly, though often still she listened with her eyes closed. When we moved to South Carolina, what she regretted most leaving behind was her horde of keyboard thumpers who, twice a year, slicked up enough to be presented in public recital. Her students loved her, and some of them even learned to play.\n\nLiving on the campus of Sharon College was a problem for her teaching. The nearest town was three miles away; no one would likely come that far for lessons. So she became an itinerant. Swinging a pouch of music from her shoulder, she trudged past the orchard to the highway, hailed a bus, and twice a week made a circuit of Sharon, teaching mostly beginners, on their own pianos or, in some cases, a neighbor's instrument. And three times a week she would hail a bus going the opposite direction\u2014toward Greenville\u2014and get off at Liberty, where she did the same. These were long days; at one time she had more than fifty students, some in remote corners of one town or the other.\n\nMy father was furious at our moving so far away, but was unable to do anything about it. I went back to Kansas in the summers. He was at retirement age about the time we left, but he refused to retire, swearing at anyone who suggested it. In order, however, to make things a little lighter for himself, he left his waycar to a younger man and became (by a process trainmen call \"bumping\") flagman on the passenger streamliner El Capitan.\n\nSharon College was having all sorts of difficulties, but what bothered President Hatchel most was its lack of accreditation. The high school, where I was enrolled, was accepted by the state board of education and the Th.B. was respected throughout the church and even the holiness churches in general, but the liberal arts junior college did not impress the Southern Accrediting Association, not at all. After the war, enrollment had doubled, partly because of an influx of veterans. The number of veterans made Sharon eligible for government help, mainly in the form of housing. It was also rumored that Hatchel was buying up supplies of war surplus goods, though no one seemed to know exactly what or what for. There was also a new barracks-like dining hall, mainly from federal funds, allowing the kitchen to be removed from the basement of the administration building. But no accreditation.\n\nJulian and Charles arrived, as privately as strangers with flashy clothes and loud voices could\u2014Miss Yodle noticed them of course, and Netty Neal, the president's noseless secretary, walked up and down our sidewalk for a while, waiting for someone to come out so she could work around to the subject. But no one went so far as to ask anything. My curiosity was aroused too, I must admit, when I looked out the window and saw the two toughs who had arrived on the same train walking toward the administration building\u2014Netty forsaking her post to shadow them.\n\nCharles's car had collapsed in Arkansas and, where it stopped, they left it. It was probably just as well, because Julian had begun to see flying saucers. In Arizona, one had swept across the road in front of them, almost causing a wreck (Julian was driving) and another had chased them half the night across Texas, a quick trip that may have been what did the car in. Charles had seen nothing, but was equally shaken.\n\nWe rearranged our three rooms so that everybody had a place to sleep, and after talking late into the night (Miss Yodle tapped on her ceiling) we all went to bed, my bed being in the kitchen. I suppose I had been asleep two or three hours when there was a terrific crash and a woman's scream. It was Elaine screaming, because J.W. had just come barging through the back door of the apartment into her and mother's bedroom, sending the flimsy lock sailing through the air. By the time I was up and into the living room, where Charles and Julian were sitting up, startled into consciousness, J.W. was coming in the other door, his eyes closed and his arms raised. \"Julian!\" he said, so loudly it did not occur to me that he was without his speaker, \"Charles!\" (I never found out how he knew their names.) In our little apartment he looked bigger than usual. Miss Yodle pounded. \"I see your souls,\" J.W. shouted, \"descending into hell!\" They were on their feet\u2014my mother and Elaine too, by then, coming out of the room J.W. had barged through\u2014and before I knew it they were on their knees, all of them, all praying at the top of their voices. I crept back to bed. Miss Yodle, recognizing no doubt the familiar accents of prayer, stopped fussing. I found it easy, in that atmosphere, to drop back into sleep, the kitchen floating away with me.\n\nThere was no great change in my brothers, though I think they were genuinely converted. Preachers always exaggerate the effects of conversion\u2014nothing is changed but the direction. (Although, occasionally as with St. Paul, it releases a man from his inhibition, allowing a display of virtues that were hidden before.) Julian, for instance, saw the world, I am sure, the same way after as before that night. And he continued to pick up stray cats and feed them well\u2014steaks and kidneys\u2014and they shared his bed and sometimes peed in it.\n\nAll my family, and Julian is our type in this, have a streak of the unworldly. For us, God's sphere of action and the local scene rarely make contact, separated by the patrol of cyclonic storms marshaled overhead by the great westerlies, or directed\u2014we disagree on terminology\u2014by the Prince of those powers. Once in a while, however, a spark manages to pass from one sphere to the other. Adam, I think, must already have noticed that things were dull, treacherously dull, except where he could see, in flashes, cracks in the natural order. Adam soon realized (it was his fall) that everything was good, even very good, but only from God's point of view. Julian's favorite book used to be Ripley's Believe It Or Not; later he went through a time reading nothing but Confidential, and always looked for stray items on precognition and ESP. And he believed it all; it interested him. Or rather he believed it only if it interested him\u2014if it was enough out of the ordinary.\n\nIt was President Hatchel's little boy, Chigger, who first told me why the two strange toughs were prowling the campus. At that time, however, it was hard to talk to Chigger, because he refused to make any statement without adding \"I think.\" The other children teased him unmercifully and his parents beat him, but he still managed not to support any proposition unconditionally.\n\n\"Is this your marble?\" a girl was asking him as I came up.\n\n\"I think so,\" he said, reaching for it.\n\n\"If you just think so,\" she said, \"I'm going to keep it,\" and put it in her pocket. \"You can get anything out of him,\" she said to me, \"he's so afraid of lying.\" And she went away with her prize.\n\n\"I think those two men,\" Chigger volunteered (they were going into the library, a new building with few books and a heavy mortgage), \"are helping Daddy.\"\n\n\"What do they do for him?\" I decided Chigger talked to me simply because I didn't torment him.\n\n\"I don't think I know,\" he said, but added, \"They're getting credits, I think.\"\n\n\"Credits?\"\n\n\"I think so.\" One of them had now gone into the library; the other stood outside the entrance. \"They're from Chicago,\" Chigger said, and after a short but perceptible pause, \"I think.\"\n\nCharles and Julian enrolled, somewhat late, in the theology program. Possibly one or both of them meant to preach, though neither of them ever said so. Mother was ecstatic at the possibility. Charles was elected president of the Veterans Club. Julian began reading the prophetic passages of the Bible and listening to J.W.'s expositions.\n\nOne night during the fall revival, while an altar service was going on, sounding loud and mournful from where I was sitting\u2014toward the rear of the chapel\u2014seekers and altar workers were huddled indistinguishably around the altar rail and the first pews, on their faces, crying or praying. Julian suddenly raised his head, looking bewildered. Few of the others could have seen him and he made no sound, but with what seemed a single movement most of the huddled forms rose on their knees and looked at him. There was a hush. He was looking at the ceiling, which was plain, neither arched nor raftered, and he began to describe\u2014in words that I can't recall, words like those of his ordinary talk and unlike the religious utterances I was used to hearing\u2014what he was seeing.\n\nThere was a faint mist, that cleared. There was a sphere, like one of those glass paperweights that snow when you shake them, but this one was half full of water, some mountains rising behind the waters, an island in the middle from which a tall thin tower\u2014on a clearer look, the tower was a ladder\u2014ascended and somehow, without breaking the crystal sphere (I wish I could remember his words, but by a trick of memory it remains with me only visually, as though I were the one who saw it all), went through it and up to where it touched a face far above, which was round and yellow and wore a gold crown and became gradually so intensely bright that he had to look down again to where the sphere was changing. The surface was no longer transparent, but clouded, then blistered. For a moment he thought it was covered by fast-growing grass, that was, however, already brown and drying as it grew, but then he saw that what he had taken for blades of grass were actually tongues of flame licking upward. The ladder was gone and the bright face seemed to have merged with the now flaming ball. Then with a roar the ball of fire increased its size a hundred times, throwing off tremendous heat, and then slowly shrank until it was small and bluish. And then it was nothing but a smooth ball of copper, but he knew that it was now cold and perfect. The words The New Jerusalem were, later, all that remained of his vision. In fact, after the mourners resumed, I never heard anyone\u2014ever\u2014make the slightest allusion to the interruption. Maybe it was too unlike the expected message, too little edifying. For all I can tell, except for my remembering it from that back pew, it disappeared from time and might just as well never have taken place.\n\nIV\n\nI was often afraid in those days, more than a little sometimes: afraid that there was no truth, or that there was one truth, only one, and that I had it. It was not the same terror I had felt at the age of ten, trembling in the night, too scared to sleep\u2014afraid that the Lord would come before morning to tear the dark apart, cutting off suddenly the mercy of time. Older now, my fears were more indistinct and more meaningful, coming as vague unease, but also in roundabout yet concrete forms. There were several voices in me, not all with my accent, and the most distant of them, which hardly used words at all, told me that a few more years and I would not be the same person, would not have the same hopes, fears, attachments. Reduced to statement, this is the most banal of insights, but it was inexpressibly troubling.\n\nAnd there was Evangeline, in whom I seemed to see a resolution of my fears\u2014to which was then added the fear, amounting sometimes to panic, that I would never reach this resolution. All my fears, I now admit, were justified.\n\nEvangeline's brother Matthew was four years younger then Evangeline, which is to say, my age. I knew him first as a rather neutral fellow high school student. We were both given to night rambles. Not together. I, in fact, on occasion stalked him\u2014it was a kind of game I played.\n\nI used to think it would be nice to be invisible, and in the woods at night I was. Matthew, the only one who encroached on my nocturnal territory, was easy to follow. He was bulky. Besides, it would never have occurred to him that anyone else was out at those hours, let alone anyone interested in tracking him. He had special permission to leave his dormitory at any hour, for health reasons, these rambles being therapy for some nervous problem, probably simple insomnia.\n\nI'm not sure, to this day, whether my aim was to find out something I didn't know\u2014some fact unnoticeable in the light of day\u2014or rather to celebrate that no one knew I was there. Whether, that is, I was after a secret, or was simply trying to hide. At any rate, when I first heard those alien footsteps, I hid, recognized Matthew as one of the new students, and watched to see if his bland face would show some character when he thought himself unobserved. It was as bland as ever, so bland that people rarely noticed he was paying no attention to them. But he was obviously going somewhere, slowly, perhaps not quite sure of the way.\n\nI followed him. The way was across the campus, in front of the administration building (brightly lit\u2014I circled around) and past the dining hall into the woods. Then, in a large arc, twigs snapping under his tread, back to the edge of the campus where he stopped, still in the dark of the trees, from which point there was a moonlit view, full-face, of the back of the girls' dormitory.\n\nWhat puzzled me, as I watched him gaze at the moonlit bulk, was that there was really nothing to look at. I don't mean that none of the girls was worth a glance\u2014I mean there was, across the whole expanse, not one light burning. All the windows were dark and identical. And there was Matthew, just out of the moon's rays. And deeper in the obscurity, I watched him with, I must admit, no more profit. Perhaps both of us, adolescent, still supposed that the senses are made for taking things in, had not yet learned that more often they're for defense, for keeping things off. I tired first, and slipped off home and lay awake in bed, imagining.\n\nI still helped out in street meetings and sometimes a Sunday School for black children in Sharon and once or twice in a service for a nearby old folks home, my attendance at these things dwindling less rapidly than my faith. There were also services each first Sunday of the month at the county stockade, which I had never attended. One cold Sunday, however, I climbed, abstractedly, into the back of a car, alongside Evangeline. We were all bundled up and looking past her at the snowy campus I decided\u2014in words; it was a voice like my own\u2014I will know her, I will know what she is. And there was time to think how difficult the task would be, for she had a kind of seriousness about her that, without removing her, made her separate. And then old LeFebvre had started his car and she opened her eyes and must have seen me looking past her into the snow.\n\n\"You like all this white?\" she asked.\n\n\"No.\" After a moment.\n\n\"I like blue,\" she said.\n\n\"You should wear LeFebvre's glasses.\" LeFebvre was ancient and wore spectacles like thick blue magnifying glasses. He lived just off the campus. How he supported himself I have no idea, though he did plant yearly a minute patch of cotton. There was no snow on the highway and the snow had stopped falling. J.W., in front with LeFebvre, was reading aloud or quoting: \". . . the Lord himself shall descend from heaven with a shout, with the voice of the archangel, and with the trump of God: and the dead in Christ shall rise first. . . .\" Old LeFebvre said, \"Even so, come Lord Jesus.\"\n\n\". . . then we which are alive and remain. . . .\"\n\nLeFebvre turned his car, which seemed old as he, onto the highway.\n\n\"Why did you come today?\" she asked me and I heard my voice again, She doesn't know me, realizing suddenly that its inflections were slightly different from mine. \"You never came before.\" I did not answer, did not think of anything for a while, but gradually LeFebvre's words became words to me and suddenly even above the raucous motor they startled me:\n\n\"He says the evil spirit wanders about, seeking rest and finding none, then he comes back and finding his place swept and clean enters in again with seven others more terrible than himself. And the last state of that man is worse than the first. Unless you're on your guard every moment of the day the devil will come back. There's no Eternal Security.\"\n\n\"That's what I tried to tell him,\" J.W. said.\n\nI saw the old man's white hair shaking with belief and for an instant felt myself in the presence of unbelievable things\u2014spirits and spheres and personal fiery powers. Even Evangeline's face, beyond which white scenery was sliding past, was full and intent on fantastic existences. And then, while the car rattled through the harsh daylight and unusual snow (with an old man at the wheel who looked at things through thick blue lenses), while I tried to see Evangeline in some other light that would show me who she was and why\u2014Why? my voice asked in amazement, How possibly?\u2014LeFebvre told, above the roar, of Zella.\n\nI had heard of Zella, Zella and her demon. She was a legend on campus, a story I had thought from another time, or outside of time.\n\n\"The devil has great power,\" J.W. put in.\n\n\"He twisted her mind and tormented her little body,\" LeFebvre said. \"I seen her run screaming across campus like she was followed by hell itself. No rest night or day. Even when she was asleep she had to get hold of something. Mrs. Pildon found her once out in the orchard with her arms round a tree-stump and dead asleep. . . .\"\n\n\"How . . .\" I started to say, but only Evangeline heard, and turned.\n\n\"We prayed,\" LeFebvre said, \"for seven nights and seven days. We prayed till we shook the powers of hell and death and the Lord set her free from the curse of the demons.\"\n\nEvangeline was still looking at me and I thought of death and hell\u2014death! how could anyone want to die!\u2014and wondered what exorcism could drive out that desire. I looked into her eyes to see the awful power, but they were simply fixed on me. And I looked away, rather than commit myself to their view.\n\nThe other car was already there when we drove up. Stella stood shivering beside the huge accordion in its box. A guard, smiling deferentially, swung back an iron gate.\n\n\"You folks can go right on in,\" the guard said.\n\nSomeone behind me whispered. \"Don't ever do anything wrong in Pickens County.\" But there was only one stone wall\u2014I had imagined a labyrinth\u2014and then the little stone building. Faces were at all the windows, showing between the bars, looking curiously inert at the other side of a wall of stone three feet thick. Stella whispered a prayer and J.W. carried her accordion. I wanted to say something to Evangeline but could not, because of her look: that seriousness, not like Stella's prayer but like prayer all the same, some kind of prayer, a more serious kind.\n\nInside was a long tomb-like room\u2014bare stone\u2014with army bunks along the side with windows and in the middle of the concrete floor a coal stove, black, with a stovepipe going not straight up but in several angles toward the dingy ceiling, the stovepipe black too, but a burning red where it connected to the stove. \"Lord,\" Stella whispered, but the rest was unintelligible. On each bunk sat a prisoner and on some bunks several. Most were young, about two-thirds were black.\n\n\"Okay,\" said another guard, \"these people come from Sharon College for this service here and I don't want no talking or smoking or irreverence. I want all to pay close attention. Okay folks, y'all can take right over.\"\n\n\"Brethren,\" said J.W. while someone passed out hymn-books to the prisoners and Stella lifted the accordion and strapped it to her narrow chest, \"we're here bringing the word of God to whomever will listen. He says in his holy word he that cometh to me I will in no wise cast out.\" Stella's eyes were closed; she said, \"Lord,\" and the accordion made a wheeze. Several of the prisoners wore a chain from one leg to the other. \"Now turn with us if you will to page one hundred and fourteen,\" J.W. said, \"one one four.\" Stella began to play. Evangeline was holding a book in front of me. We sang (\"Won\u2014derful grace of our lov\u2014ing Lord. . . .\") the song J.W. picked to sing in the prison. The chorus (\"Grace\u2014grace\u2014God's grace\") the prisoners sang too, or at least their lips moved. The accordion dominated. Why did you come? my voice echoed her voice. Why did she come, I replied; but I knew why, remembering her face as she knelt by the altar, not screaming or even crying over the lost but utterly exposed with them. But I did not know, all the terms of my knowing in process of becoming meaningless. There was nothing. And yet there was Evangeline. Her voice was closer to me than the other voices but only the sound of the accordion was definite. (\"Grace that is great\u2014er than all our sins.\") Peter slept in prison. Stella and Betty Jo sang a special, \"I've anchored my soul.\" They huddled together so closely that both seemed to be seeking protection behind the instrument.\n\nThe guard, invisible, snapped his fingers. One of the blacks who had just lain back quietly on his bunk raised himself slowly to sitting position and rested his chin on his fist. He had leg-irons, the chain from ankle to ankle.\n\n\"They shouldn't keep him from lying down,\" Evangeline whispered behind the whine of the accordion. She still had her coat on as the coal stove, now red in a circle near the top, gave none of its warmth to our legs. J.W. got up to preach.\n\n\"Turn with us,\" J.W. said, \"to the Second Epistle of Peter, chapter three, verses three through thirteen.\" Stella was passing out black paperbound Testaments. \"Second Peter three three.\" Evangeline opened her Bible from the back and flipped a page too far. \"That's the wrong Peter,\" I whispered. She pointed to a verse and I read it, then looked up at her to see if she was laughing. Her face was serious so I read it again: \"By which also he went and preached unto the spirits in prison.\"\n\n\"Knowing this first,\" J.W. read, \"that there shall come in the last days scoffers, walking after their own lusts and saying, where is the promise of his coming? for since the fathers fell asleep, all things continue as they were from the beginning of the creation. . . .\" Her Bible was thin, bound in blue leather. My eye ran down the column, reading ahead of the heavy voice: \"Seeing then that all these things shall be dissolved. . . .\" We were seated behind the speaker, across from the row of windows behind the silent stone-like listeners. One bulb directly above J.W. gave a harsh downward light, leaving the ends of the vault in shadow. The windows were less bright than they had been and I noticed something like snow falling far on the other side of the bars. \"Seeing then,\" read J.W., \"that all these things shall be dissolved. . . .\"\n\n\"What have they done?\" I whispered to Evangeline. Stella turned and glared at us.\n\n\". . . wherein the heavens being on fire shall be dissolved, and the elements shall melt with fervent heat. Nevertheless, we, according to his promise, look for new heavens and a new earth, wherein dwelleth righteousness.\"\n\nShe closed the book. \"I don't know,\" she whispered.\n\n\"I wonder,\" said J.W. in the tone of a statement, \"if any of our little group would like to testify for the Lord's saving and sanctifying grace.\" Some of the prisoners rolled their eyes but not one seemed to move a muscle otherwise. Stella rose and stood under the discolored light, unsubstantial without her instrument, both her hands clutching a handkerchief.\n\n\"I'm s'glad,\" she said tearfully, \"that the Lord has washed away my sins and I'm a new creature of the Lord.\" Her legs trembled a little, though her voice grew more sure as she spoke. \"Oh I'm so glad I've been born again! born of the spirit, the holy spirit of God!\" Suddenly I noticed that one of the prisoners, a fat fellow with no hair, was smiling at me, grinning rather. And I realized that I had been watching Stella's thin legs. \". . . and I intend to make it through to heaven,\" Stella said and returned to her seat. It was a grin of complicity and I felt a little flush of shame on my face though my feet were still like ice against the concrete floor.\n\n\"Amen!\" shouted old LeFebvre and his ancient white head moved shining into the light. \"I tell you boys,\" raising his arm towards those in prison, \"Jesus is the best friend you'll ever have.\" LeFebvre would have slept in jail and waited for the angel of the Lord to deliver him\u2014and perhaps, I tried to think, the angel would come. \"For years and years I walked the downward path of sin to perdition and I gambled for the devil's pleasures and one day the Lord came to me and said I'll set you free LeFebvre and bless his name I been living for him ever since in glory hallelujah freedom.\" Some of the prisoners were smiling, either at LeFebvre's words or at his thick blue glasses. The luminous red had moved down on the stove until half the stove was the color of fire. Apparently it had just been lit for this service. \"Oh glory!\" LeFebvre shouted. \"Someday he'll come back to this old world and sinners will call for the rocks and the mountains to fall on them and hide them from the wrath of God. They'll weep and wail and gnash their teeth and wish they'd never never tempted God, it's a fearful fearful thing to fall into the hands of an angry God. This old world is going to rock and reel. . . .\"\n\n\"Amen!\" J.W. said.\n\n\". . . and the elements melt with fervent heat!\" Evangeline's face I saw was glowing though she made no sound. I wanted to touch the hand quietly holding the Bible.\n\n\"Amen,\" said Stella rather timidly. I looked at Stella and then saw that same prisoner smirking at me again, a smirk that said Why did you come?\n\n\"And listen to me you young men you may think because you once knew the grace of God that you'll be able to stand in the last day on your past righteousness. But listen to me the Lord says in his holy book if you become lukewarm I'll spew you out of my mouth.\"\n\n\"No Eternal Security!\" J.W. said.\n\n\"And oh, young men, think of the glorious reward in the last day.\" The old voice began to crack and became quieter. \"The righteous will ascend to the glories of heaven, the dead in Christ shall rise first and then we'll all go up and be forever with the Lord.\" Stella was weeping, her hand raised. LeFebvre sat down and J.W. came forward again.\n\n\"Now in the scripture that I've read to you. . . .\"\n\nAnd Evangeline was condemned. How could I know her in so short a time. I will, my voice said desperately, in unbelief. Some of the prisoners grew restless under the rhetorical tone of the preacher and changed position but none moved very far. Her eyes had rested in perfect peace and I could not believe what she told me. How could she sit there, knowing of her condemnation, knowing she was already cut off, enclosed in eternity. Why, she would lie down and turn to dust with the same serious smile as when around the altar rail the last seeker\u2014I had watched with them until the chilly morning\u2014the very last had raised an awkward hand in a triumph I could no longer understand. There was nothing there, nothing, not even a brass heaven to beat against. But she would die\u2014die well\u2014die and disappear altogether, still at the last moment conscious of . . . what? There was nothing. \"Listen,\" I had said to her, not knowing whether I said it in spite or in hope, \"The fear of the Lord prolongeth days: but the wicked shall be shortened. The wicked. It says so right here, read it, read it.\" She took the Bible\u2014it was her Bible\u2014and turned to another passage and handed it back to me already circled in blue: \"Delight thyself also in the Lord; and he shall give thee the desires of thine heart.\"\n\n'it's good to know,\" J.W. was saying, \"that the Lord's on your side. In the battle of life. . . .\"\n\nI wanted out. Suddenly I wanted only to be out of that prison. I raised my eyes and all I saw was dirty and dark. How could she\u2014how could anyone. . . . My voice was unable even to ask. I saw J.W.'s face running with sweat and suppressed an impulse to take Evangeline's arm and force her out the door.\n\n\"Amen!\" old LeFebvre shouted behind me.\n\n\"Amen,\" Stella said almost without tone. The sermon was over, but even then there was more. I stood with the rest. Several of the group, including Evangeline, crossed the chamber to talk with reluctant listeners. Stella sat down beside a pudgy prisoner, not seeming to sink the mattress where she sat, and over her shoulder the prisoner grinned at me. J.W. started a song without book or accordion:\n\nO come to the Savior, thou poor weary soul;\n\n'Tis Jesus invites thee to come. . . .\n\nEvangeline sat with the chained black, who smiled in embarrassment or delight and nodded to all her questions. Only J.W. and LeFebvre were singing. I stood foolishly alone until the group finally regathered. Someone was collecting the song books and Testaments. \"Lord,\" said J.W., \"we thank thee for this opportunity to bring thy message to these in need of thy great power and light. Thou hast said, I am the way the truth and the life. Go with us now and grant that we might all make it through to thy kingdom and be ready when the angel stands with one foot on the land and one on the sea and says, Time is no more. In thy most holy name we ask it. . . .\" I saw that the stove was almost burnt out and no one had made a move to replenish it. I picked up Stella's accordion and carried it out for her. The snow was falling in irregular hunks.\n\n\"Boy, I wouldn't want to do anything wrong in this county,\" someone murmured as we went out the gate.\n\n\"What have they done?\" I asked.\n\n\"Oh, stolen something, or got drunk, or drove too fast.\" I put the accordion in the back of the other car and went over to LeFebvre's. Betty Jo had climbed in back with Evangeline.\n\n\"Hey, come on and ride with us,\" the same voice shouted to me. The other car belonged to a student. I got in and as we pulled out I waved back to Evangeline, who smiled to me through the glass.\n\n\"Poor Evangeline,\" Stella said, wiping her eyes. This was a newer car and moved along the highway without seeming in motion.\n\n\"You know something,\" a girl said, \"this is the first time it's snowed here in three years.\"\n\n\"Rheumatic fever, isn't it?\" the driver said.\n\n\"No,\" Stella said, \"she had rheumatic fever and got over it, but it enlarged her heart\u2014I think that's what the doctor told her. Anyway it's her heart.\"\n\n\"Well, I'll tell you one thing,\" the driver said, \"she'll go straight to heaven.\"\n\n\"That's right,\" said Stella, and \"She sure will,\" the other girl added.\n\n\"That's the great thing about our religion,\" the driver said. \"It's something to hold on to in time of trouble. It's a mighty rock in a weary land.\"\n\n\"Yes,\" Stella said, then as an afterthought, \"Old Riley LeFebvre is going to pass on to his reward one of these days.\"\n\n\"He should have a lot of reward laid up by now,\" the other girl said and began to giggle.\n\n\"Listen,\" the driver said, amused also, \"did you ever know the story about old LeFebvre?\"\n\n\"Which one?\"\n\n\"About the fire.\"\n\n\"I have,\" said Stella. I tried to collect myself by gazing out across the bare white of the hills.\n\n\"Don't ruin it for her, she hasn't heard it,\" the driver said. \"One day old LeFebvre lit out across the campus and was shouting at the top of his lungs, Fire, Fire!\" Stella laughed. I thought, That couldn't be her real desire. \"And everybody run out of all the buildings to see where the fire was. Hatchel was shaving and came out all lathered.\" Stella laughed and the driver laughed too. \"And everybody run after him, and then when everybody had caught up with him and asked him where the fire was, he turns around and shouts, It's in my soul!\"\n\n\"Oh, I've heard that before about somebody else,\" the other girl said, but laughed with them anyway. I tried and finally decided (in words) Nobody, nobody could want to die. But the voice was artificial and almost extinguished, a voice rapidly dissolving away, a voice of my na\u00efvet\u00e9.\n\nV\n\nMy father's one great love was for Shakespeare. He had read all the plays several times and the more famous of them he had almost by heart. After my parents' separation, but while we still lived in Emporia, he took me to a production of Hamlet, the so-called \"G.I. Hamlet\" that Maurice Evans had worked up to give on far-flung army bases and was now, at the end of the war, offering to civilian audiences as remote as Topeka. My mother was sure that this was a trick to carry me off, and I am not absolutely certain there was no such thought behind it, but at the time all I was aware of was an excused absence from Lowther Junior High (away from the auditorium where the proscenium proclaimed in raised plaster letters, \"The days that make us happy make us wise\"), a trip to Topeka with my father, and then the revelation of that performance. People who should know (older people) have since told me that it was nothing exceptional, mediocre acting of a badly cut text\u2014and I remember the Edwardian costumes\u2014but for me it was a view into another realm, a realm infinitely appealing and, most surprisingly, available to me. I was, I think, different from that day on. I noted the way, common enough I now know, in which each scene, instead of being marked off by raising and lowering a curtain, was brought up out of the dark and at the end returned to dark, so that the entire play became a series of moments articulated by light on a background of darkness.\n\nAt Sharon, the category of \"worldly amusements\" included all films (I had seen a few while in the fourth and fifth grades, but saw no more until college) as it included dancing, card playing, and billiards. Plays somehow escaped this condemnation, perhaps not considered amusing, but of course there were none to go to. That is, so it seemed at first.\n\nAt just about this time, Bob Jones, founder of Bob Jones College in Cleveland, Tennessee, had the vision of a greater work, a gospel school on a larger scale, not bound to any particular denomination but solidly fundamentalist. Bob Jones College would become Bob Jones University. To emphasize, perhaps, the significance of this change, to point up that not mere growth but a new creation was taking place, he determined to leave the old campus behind and start afresh in some other territory. What directed the choice\u2014a prophecy? a gift of land?\u2014I never learned, but the new campus began to be built, from scratch, in Greenville, South Carolina. Acres of red clay were leveled and monumental buildings went up several at a time, and a body of students was there before grass had time to cover the torn soil.\n\nBob Jones was already of a certain age and, though perhaps still making the decisions, he invested the title of president on his son, Bob Jones, Jr. They were both, so my mother claimed, good enough men; both connected, she added\u2014her source of information as obscure as usual\u2014with the Christian and Missionary Alliance, a sect I never otherwise heard of. Actually, it passed her requirements only barely, and it must be said that at Sharon we considered Bob Jones a rather worldly place, though the world around us failed to make the distinction. Bob Jones, Jr., was a tall man, with great presence and a marvelous voice, of whose features I have no exact image, since I never saw him except in makeup. He was a Shakespearean actor, a good one\u2014to my unformed taste, anyway\u2014and this must have been why among the first buildings to go up was Rodeheaver Auditorium, with its equipped stage. At least once a semester the university would mount a Shakespeare play and almost always he was the lead. Richard III I remember particularly, with Bob Jones, Jr., limping in to declaim, \"Now is the winter of our discontent. . . .\"\n\nI never missed a production, though there were those at Sharon who thought I walked the broad way to perdition. I usually hitchhiked to get there. They presented also, at one point, The Barber of Seville (Bob Jones, Jr., was not in it) and so I came first to see an opera. One day, when I had gone over early for something or other, I noticed that on the second floor of the library (one of the first buildings, though not one of the largest) there was some kind of museum, so I wandered in. It was a museum of the Bible, artifacts from the Holy Land assembled by an archeologist named Barbara Bowen. No one was there\u2014I went back several times and no one was ever there\u2014except Barbara Bowen, a white-haired woman delighted to have a visitor. She gave me a book she had just published, Through Bowen Museum With Bible In Hand, and took me around her trove. I remember few items now, but still retain the image of some crudely blown glass flasks a few inches high. These, she explained, were tear bottles\u2014to be worn, tightly stoppered, around the neck\u2014used to collect tears, not only one's own, but those of friends and loved ones, so that after a time there would be mixed in one vial the aspects and occasions of a life's tragedies. Mary Magdalene did not, as I had supposed, weep, but washed the feet of Jesus by pouring over them the precious accumulated sorrows of her tear bottle, before drying them with her hair. This was pointed out in another of Barbara Bowen's books, Strange Scriptures That Perplex the Western Mind.\n\nThere were more books in the Sharon College library the spring after my brothers arrived, a lot more. Most of the bindings were so dirty and worn that they looked as if they had been there for years, and they were an odd assortment. There was a Life of Col. Jesse Harper. Next to this (most of the spines were numbered but the numbers seemed random) was The Great Pyramid Jeezah by Louis P. McCarty, Festus by Philip James Bailey, and a little book by a Dr. Frank B. Robinson called Blood on the Tail of a Pig, which I read with great interest. It was even odder, however, that sprinkled among the collection now were works by Darwin and even by Ingersoll. (Also a long eulogistic poem on Ingersoll's life called The Light-Bearer of Liberty.) One morning as a semi was being backed up to the library door, Netty Neal came in with Preacher Bamm, who lived with wife and two daughters on the other side of the apple orchard. His daughters went to high school at Sharon\u2014one them in my class\u2014and they figured largely in the best of my erotic dreams. Otherwise beautiful, they had spindly legs and enormous feet and it has always puzzled me how these characteristics, far from being corrected by my dream-work, were accentuated. Bamm was a traveling evangelist, coming home sometimes only three or four times a year. Whenever he arrived, he went through his daughters' closets and threw out any garment that might cover them too scantily, and also any that looked stylish to him. The current style happened to be longer skirts, so the girls hadn't a chance that year: their dresses were either stylish or immodest.\n\nHe was back now, and had apparently heard from Netty that the library kept foul books. He fumed at the librarian, a meek woman with her hair in a bun, who obviously hadn't the vaguest what was going on\u2014the new books were not catalogued. She was almost in tears as Bamm raged and two truckers brought in more books on a dolly, one of the now familiar toughs looking on. Netty tried to calm the librarian, with gossip, while Bamm searched the stacks and got in the way of the truckers who were filling more shelves.\n\nPresident Hatchel was looking more and more haggard those days. The spring revival had come round, with great hopes, but was not going well. The nightly services seemed to tire rather than revive. The evangelist who was holding the services (a preacher named Gant, from Ohio) was also disappointed in the results so far. The three weeks were to end on Sunday and Hatchel called off Friday classes. The day would be spent, he decreed wearily, in prayer and fasting. He asked Charles, Gant, Bamm, J.W., and a few others to meet for prayer in his office the first thing Friday morning. They met, all but Bamm, at the appointed hour and prayed, kneeling on the uncarpeted floor, for quite some time. Then J.W. got up and opened his Bible. The others rose, rubbed their knees, and sat, while J.W. read, probably where the book fell open, pacing the floor.\n\n\"Because thou sayest, I am rich, and increased with goods, and have need of nothing; and knowest not that thou art wretched, and miserable, and poor, and blind, and naked. . . .\" Gant said, \"Amen,\" but Hatchel was groaning. \"I counsel thee to buy of me gold tried in the fire, that thou mayest be rich. . . .\"\n\n\"The Lord is coming soon,\" Gant cried.\n\n\"I'm not ready,\" Hatchel said, tears streaming down his face. Everyone turned to him and the story began to roll out addressed more to God than to anyone in the room.\n\nHe had meant well, Hatchel had. He had wanted to raise Sharon out of the dust and make it worthy of the King whose children it served. He himself was from Oconee County. A short business venture in Cincinnati left him with a lasting and unreasoning awe of the northern states\u2014he fled back to South Carolina and started preaching, referring ever after to his time in Ohio in terms of Jonah's flight towards Tarshish. Reading the scriptures to a congregation, he exaggerated final r's; only when the spirit came on him did he revert to the accent of his native hills. He had risen through the ranks to be pastor of an urban church, the one in Anderson, and during the war he was named president of Sharon. His determination grew that by the time he retired Sharon would be accredited.\n\nBut he was too eager, and too na\u00efve for those postwar years. A letter from a Chicago firm (the embossed letterhead read \"Paideia, Inc.\") offered him \"academic efficiency experts\" and he had ended up with two thugs who were filling the library (some number of volumes per student), who were making deals that he was completely in the dark about, who had even\u2014so at least he was told\u2014made out and out threats against a certain Ph.D. teaching at Orangeberg because he would not come to Sharon. Hatchel, sounding as though he had just descended by mule-cart from the mountains, was lamenting all this in a prayer of misery and outrage when Bamm appeared at the door, his clothes filthy and his face covered with blood. He had tried to throw the Mistakes of Moses and some twenty other volumes out of the library. One of the academic efficiency experts slugged him.\n\nWhenever a revival was in progress, Matthew seemed on edge. Nerves began to show through his lack of expression; still inexpressive, he was tense. I tried to talk to him, once I knew he was Evangeline's brother. It was doubly difficult in that neither of us was interested. I can't say we ever developed, either of us, any great affection for the other, but eventually he talked to me, for lack of other friends. And what, after some clumsy approaches, he first confided, was a load of sin.\n\nIt was a confidence I was unprepared for, but I listened as he produced for my inspection one abstract fault after another. I began to feel priestly as he revealed his most persistent peccadillo, a lust for movies\u2014except that I had nothing to offer in the way of absolution. What could I say? I listened. Sometimes he retreated into a stubborn silence, which I no longer tried to break\u2014I had heard enough\u2014then without ceremony, without ornament, he would confess to me that he had slipped downtown and into a mystery. No details, no plot summary, just so.\n\nHe was not, I finally realized, after forgiveness, and realized by that how little I knew him. Him or, for that matter, his sister. For that matter, myself. There were moments when the very notion of sin meant nothing to me, my life a clear space, open, empty of any obstacle, boundless, centerless. Then I drew up lists and catalogues in a dream of order, but the only order was in the dream.\n\n\"Vange is four years older than me,\" Matthew would say, which I had already figured out. \"She was born before Mom was converted\"\u2014that kind of detail disoriented me, though I understood its importance later. \"Vange used to take baths in a big pan in the kitchen.\"\n\n\"How was the movie?\" I would ask, but could never get an answer.\n\nSunday morning service was not exceptional, and though it was the last day of the revival, the evangelist gave no altar call. That evening, the very last service, I sat across the aisle from Evangeline, a couple rows behind the Bamm sisters. I had not eaten all day\u2014it was not a fast, I was just given to extremes then\u2014and felt a bit dizzy. The face of Brother Gant kept reversing its color, a purple after image replacing the hue of flesh. I knew Evangeline was near, but couldn't have looked at her without an obvious turn, so I gazed at the auburn hair of the younger Bamm girl (her name was written Shelby Jean and pronounced Shebbajean)\u2014it was braided and twisted into a knot, as it always was while her father was home. Then the whole scene disappeared, sight and sound, and while I waited for it to come back (I didn't panic, knowing from experience that it would come back) I saw in front of me a simple wall, no color, quite flat, maybe forty feet away, like the wall of the church behind the evangelist if that wall were extended out of sight in all directions. Then it began to curve, as if I were facing a great bandshell, only still a little flattened directly in front of me. Then the wall disappeared altogether, but the neutral light remained, as if I were in a dense fog; there was sound, too, but indistinct. When the congregation stood up to sing I stood up too, gazing at Shebbajean's knot, and can recall no transition.\n\nIt was the most pitiful altar call I ever heard, \"Just As I Am\" from us, while Brother Gant, shouting above the song, described scene after scene of hardened hearts, sudden death, the terrors of judgment, the lake of fire, but hardly anyone went forward. Then the pianist switched tunes and we were singing J.W.'s favorite:\n\nTo be lost in the night, in Eternity's night,\n\nTo sink in despair and in woe\u2014\n\nBut such is thy doom if thou turn from the light,\n\nRefusing his Mercies to know.\n\nAmid shrieks and howls, a good quarter of the congregation surged to the mourner's bench. The very atmosphere seemed to open. I glanced across the aisle and saw Evangeline looking upward, but with closed eyes. Her face was glowing and her lips moved ever so slightly, murmuring, I think, \"Thank you.\" \"Prayer really does change things,\" she said to me once; \"you know, He just couldn't refuse us anything.\" Some years later I felt a chill around my heart when I heard she was dead, but it was not pity.\n\nMiss Yodle was at the altar; Hatchel was there too. Every once in a while a seeker would leap to his feet, shouting for joy or singing. I sat for a while in a deserted pew, then crept out into the dark, strolling out by the apple orchard, until the volume of prayer following me was muted and then lost. Past midnight I was wandering home. Chigger was standing alone in the dark.\n\n\"Your brother's going to kill someone,\" he said, \"I think.\"\n\n\"Who?\" He pointed across the campus toward the boys' dormitory and I saw a light go on in a second story window. It went out and in a moment the window next to it was lit. Then the next, and so on, with quick regularity, until there was a longer pause of darkness and then a light on the third floor commenced its round. I started up the steps to our apartment, but on the porch there was a dead cat. Even in the moonlight I could see that it was Julian's favorite brindle.\n\n\"Where's Charles?\" I asked Chigger.\n\n\"I think he went after him,\" he said.\n\n\"You mean after Julian?\"\n\n\"I think so.\" And after a moment, \"I think he went after him because he took his gun.\"\n\n\"Who killed the cat?\" The dormitory was all dark now.\n\n\"I think he kicked her, one of those men I think works for Daddy. I don't like them.\" I found my mother worried sick and told her nothing would happen. Julian, I said, would chase after the men a while, then get tired and come home.\n\nAs it happened, the two hoods (who had a basement room in the boys' dormitory) apparently heard that someone was after them and were already out the back door as Julian was clopping down the stairs. They lit out, down the road, and into the apple orchard. I know where they came out of the orchard, because they cut across Bamm's place at about one o'clock\u2014the preacher was celebrating the great concluding revival service by a bonfire of his daughters' vanities, burning with particular relish a pair of toeless shoes (huge things, they must have looked like landing craft launching into the lake of fire). Bamm at first thought the fiends had come to black his other eye, but they went puffing on and were followed in a while by Julian waving a Luger, Charles almost on his heels.\n\nThey can't have run all the way. There must have been periods of rest, of hiding out. But cross-country it is a good twenty miles to the North Carolina border and they passed sometime that morning into Transylvania County, where Julian gave up and, Charles having reclaimed his pistol, they hitchhiked back to Sharon. They were both expelled over the affair, not for chasing off the toughs, but because the dean of men claimed that his apartment had been broken into and he himself faced with the bad end of a gun. He still had a worried look weeks later when Charles and Julian had transferred their G.I. Bill and were commuting to a school of cosmetology in Greenville. Chigger buried the brindle, with a long but noncommittal sermon.\n\nHatchel was later fired or, rather, given early retirement. His health failed as the college debt grew, and the faculty and trustees were upset when they found the basement of the administration building (Netty Neal got hold of the key and couldn't help remarking what she saw when she looked in) stacked from floor to ceiling with rubber life rafts, pup tents, gas masks, Arctic underwear, and ike jackets, all of it suffering from mildew, mice, and rust.\n\n\"I can tell,\" I said, \"in his letters, the point at which he begins to get angry. Usually toward the end of the letter, sometimes about the middle. There'll be a change in the handwriting. Then right at the very end there's ordinarily something added after he's calmed down again.\"\n\n\"I'll never write you a letter,\" Evangeline said. \"You'd read it too well.\" I knew she'd never write, though that wasn't why.\n\nThe rule at Sharon was one date a week, on Friday evening, from supper until ten o'clock (though couples could sit together at Sunday evening service), but of course I was not dating Evangeline. We simply talked when we met, in moments that seemed to me broken off from routine.\n\n\"But in this week's letter, there isn't any anger at all. You could think he'd lived through it all, into some sort of serenity. It won't last, though, in the next letter he'll be angry at something else.\"\n\n\"I wish Matthew would get right,\" Evangeline said. She was not changing the subject\u2014we always talked of several things at once. \"What does he say?\"\u2014back to my father's letter.\n\n\"My grandmother just died. She was almost a hundred.\" She was the last of my grandparents, by a good decade.\n\n\"You know what it is. He can't settle down\"\u2014she was talking about Matthew\u2014\"because he knows he was consecrated to some great purpose.\"\n\n\"What purpose?\"\n\n\"It was before he was born. Mom\" (something in her intonation suddenly reminded me of Matthew, maybe because I had never heard her talk about her mother) \"didn't think she would have any more children, after me\u2014she was very sickly\u2014but she dreamt that she would have a son.\" We were standing under a tremendous maple, just to one side of the administration building, and Miss Yodle, passing on the sidewalk, gave us a sour look. She disapproved of Evangeline's hair, not its length, but that it fell in waves down her back. Miss Yodle's own was in tight little braids twisted around her skull, giving her head the air of a victim.\n\n\"Yesterday,\" I said, \"while I was sitting doing nothing\u2014about eight\u2014I thought I heard you say something. It was so real, I looked around to make sure you weren't there.\"\n\n\"About eight? I was in my room.\"\n\n\"I listened, and you seemed to be talking to Betty Jo.\" \"She stopped in. She often does.\"\n\n\"It was absolutely your voice, your way of talking, all perfectly clear and distinct. But it was like a dream\u2014even a moment afterward, with the sound still in my ears, I couldn't recall what you said.\"\n\n\"Probably wasn't important.\"\n\n\"But so real. I kept glancing around. I couldn't believe I was alone.\"\n\n\"How's your mother?\"\n\n\"Better.\" My mother's shoulder had developed a terrible pain, apparently from the satchel of music that she always lugged. An X-ray showed a bone bent from the continual strain.\n\n\"I've heard of cases,\" Evangeline said, \"where a person was in danger or needed help, and the Lord brought someone from an impossible distance. Maybe, you know, if you really needed to get in touch with me, it\"\u2014for the moment of her pause, my breath caught\u2014\"it might be permitted.\" And the hairs at the back of my neck stood up as I realized, even while refusing to think of it, where I would have to go to contact her. But she went on, as if she had been talking about Matthew all the time. \"It's because he knows Mom gave him to the Lord before he was born that he finds it difficult. On the one hand he rebels against the whole idea and tries to assert his own will.\" Try as I would, I could not see Matthew as fallen angel. \"On the other, he can't apply himself, because\u2014well, he simply waits for the revelation, some sign that will show him what he's to do.\"\n\n\"So in the meantime. . . .\"\n\n\"He's just waiting. But that means he makes no preparations, not knowing what to prepare for.\" So he was a latent messiah, wondering which dove was the one meant to descend on him, holding out for a scripture where he would read his mission already written, already well-writ-ten, from miraculous birth to culminating sorrow and fulfillment. Into thy hands\u2014the only voices I ever found in me now were parodies, and I pushed them all aside to try once more to fathom Evangeline, who saw all this clearly\u2014as I thought I was beginning to see it\u2014but saw it seriously, and with some kind of love. I decided, while feeling disgust for thinking it, that to witness her last moment\u2014to see her die\u2014would tell me finally . . . whatever it was I wanted to know. I no longer knew how to put the question.\n\nAs it happened, she left Sharon a year before I did, and died in the fall of 1954. Some months later, it occurred to Matthew to send me a picture postcard of the Merchandise Mart with, as might have been expected, only the essential information: she was dead. No details, not even the exact date. But most likely it was while I was in Venice, on leave from the army. My mother used to quote me the dying words of atheists: \"Draw the curtain, the farce is played\" or \"What a fool I have been,\" and then the less interesting agonies of Christians. She liked Wesley's defense of his church: \"Our people die well.\" I don't know how Evangeline died or even, those last years, how she lived. In my own mind, I can never make a proper connection between her life and her death. To do so would be to tell a story like that of Abraham and the sacrifice of Isaac, but leaving out the angel. It doesn't make sense as a story\u2014the story requires an angel, and a beast as substitute.\n\nWhen I think of her, it brings back Sharon, the whole complex of Sharon, but particularly the shared belief that set it apart from the world, above or below the world according to one's point of view. When I try to think of her dying, as I used to try for the moment of falling asleep, I find myself somehow with an image of the Piazza San Marco. Saint Mark's lion flaps in the wind. Pigeons resting on the frieze of Napoleon's Wing appear as irregular dots on bright filigree. God in shards, shining with gold leaf, receives with absolutely no expression the news of Abel's murder. Sun changes, unceasingly, the orange and white diamonds of the Doge's Palace. Considering the number of pigeons, I wonder why there is not more shit around. The square seems to absorb it without effort, along with the pigeons, the tables and chairs, the tourists, the band. A cat curls asleep at the base of the bell tower. The piazza is still crowded, even in my memory, men and women standing and moving, their faces blank, their hips at all possible angles to the edge of the lagoon.\n\nI have not heard from Matthew for many years now, but I suspect he is doing well. He was drafted at the very outbreak of the Korean conflict, became a hero and an officer all at once, and a year or so before his sister's death, returned to the states to be decorated.\n\nAt Sharon College, in front of the administration building (South Carolina)\nTibet\n\nVI\n\nAt Sharon, my junior year, I began writing. The first product that I remember was a pair of narrative poems in heroic couplets, on the antediluvian world and the deluge. Not a line remains, or remains with me, all those rhymes washed away without, I think, another soul ever seeing them. It is curious that, while I always intended to write, from before the time when I could reproduce the characters of the alphabet, I never thought of myself as a writer. In grade school I declared that I would be a chemist, and by the time I left Sharon had decided on psychiatry. But underneath these choices (which, along with others, came to nothing) the idea of writing remained, unchosen, unsupplanted.\n\nThe apartments of Teter Barn, as it was popularly known, were entirely unplastered, the walls and ceilings of board and the floors of wide softwood planks, irregular and half rotten. Miss Yodle's nerves, adjusted to quietness, were put on edge by squeaks from our walking over her head, let alone from my mother's \"Pass Me Not.\" Miss Yodle would stand on her porch, grimly, as Charles and Julian came clomping down to go to classes, each with an enormous Bible (a kind they had sold at one point) in the crotch of the arm. Charles would bow to her, with his professional smile.\n\nJulian was in her freshman English course. She graded simply and mechanically: certain errors, like a comma-fault or an incomplete sentence, meant an automatic fail, lesser punishments being reserved for lesser sins. He wrote weekly themes on her prescribed topics (\"My Summer Vacation,\" \"The Most Interesting Character I've Met\") and was rewarded by a series of D's and E's. He tried the exotic approach: when assigned an autobiographical sketch, he turned in \"My Life as a Bloomer Girl.\" She refused to grade it. I had heard her master's thesis was on Byron and the Bible and wonder just how she read Don Juan. And if she dreamt at night\u2014one of those nights when I was restless and, without thinking, paced the kitchen floor\u2014of a dazzling young man, hero or poet, approaching her bed as an angel of light with cloven hoof. She often thumped her ceiling with, I suppose, her broomstick. The dust on our floor jumped.\n\nJulian, in front of Teter Hall on the campus of Sharon College\u2014the porch on the left we shared with Miss Yodle.\n\nAssigned to write three hundred words on \"My Hobby,\" Julian wrote as follows: \"My hobby is stamp collecting. I have stamps from Aden, Afghanistan, Albania, Algeria, Andorra, Angola, Argentina . . .\" and so on for three hundred countries, all in alphabetical order. It was his only C that year; there was not a comma-fault or a misspelled word in it. Her only criticism was \"Lacks interest.\"\n\nI wasted then what seemed like ages\u2014and now I think perhaps it was\u2014looking out windows across the shallow valleys. The foothills visible to the north suggested an invitation, though I knew they were, at best, another Carolina. Nights I went for long strolls, from the ill-lit campus down the road into dark countryside till I could smell, in season, apples from the pitch black orchard\u2014returning, frightened by imaginary wild animals or highwaymen, by a shorter way and creeping as quietly as possible up the stairs, which even so creaked enough to rouse Miss Yodle. I think she slept just at the threshold of sleep, sensitive like the wise virgins to the slightest summons.\n\nShe went somewhere for Christmas vacation. A relief at first, though it made the cracks in our floor sinister. I remember a particularly large crack right in front of our refrigerator, because during that vacation a bottle of syrup fell out of the refrigerator and broke and Julian swept the smaller fragments of glass along with most of the syrup into the crack. I have no idea how much seeped through or where. On Christmas Eve it turned wickedly cold and our little kerosene stove refused to draw without the draft from hers. We almost wished her back. Charles wrapped up, went out behind Teter Barn, and shot her outhouse full of holes for target practice. President Hatchel's wife, next door, gaped over the hedge with a wide grin: \"Is she in there?\" Chigger, who had been allowed one shot, said, \"I don't think so.\"\n\nMany names that might have found a place in this account have dropped somehow out of mind. For example, there was a rather small, tow-headed, farm-bred ministerial student at Sharon. By one of memory's common tricks I remember him as witty but cannot produce an example of his wit\u2014which has, all the same, a definite quality for me, a light understatement that was effective partly because set in a matter-of-fact, open personality.\n\nGod, in response to long-continued, if not violent, prayers, told Elaine (so clearly that there was no doubt at all left in her mind) that what's-his-name was her destiny. They were, together, to preach the gospel. She had dated several students, including him\u2014indeed it was a rare week that someone or other did not try for her company on the coming Friday evening. True to her earlier call, she had no qualms on going out with anybody she thought presentable, but never went a second time with any but solid preacher-material. She knew how tricky the affections can be and had no desire to be caught in a conventional but unsanctioned union. Now everything changed. She determined to accept no date except with her ordained mate, and was pleased, but not really surprised, when the first to ask was in fact him. For twelve Fridays they met just after supper (since Elaine did not take meals in the dining hall), went to a program, and then chatted until ten. And every Sunday night they sat side by side at church.\n\nFriday evening entertainments were provided by the two campus literary societies, Moodys one week and Sankeys the next. They ranged from lectures to talent shows, with fierce competition between the two societies as to which could produce the most interesting or edifying program. Everyone came, all being members of one or the other and obliged to support their own and to discourage the opposition. Besides, there was no place else to go.\n\nIt was a long-established custom once a year to give a combined Sankey-Moody program in which the faculty was travestied. Every time this came around there was talk of abandoning the practice\u2014it was, so J.W. argued, both irreverent and unkind. But still it always took place. The high school principal was represented in ratty clothes, spoke delicately, and kept his head tilted to one side. (He was also the school's chief carpenter and it was noticed that in the newer buildings all the light switches were on a bias.) Professor Hocks, our only Ph.D., was made to give an incomprehensible announcement at which he himself then giggled hysterically. A marvelous replica of Hatchel stood like a toad ready to leap over the lectern and told a joke with the punch line, \"I got zay-roe, but zay-roe's better than nothing.\" And poor Miss Yodle, played by a male in a straight brown dress, sniffed sideways while giving out elaborate prohibitions.\n\nAll this, and church too, Elaine and what's-his-name took in together. They were steadies, known to be going together in a place that took such things seriously, and as such were watched more carefully than others on the few special occasions when they might meet\u2014at street services, the library, wherever. I wonder, now, if Elaine remembers his name.\n\nHe had begun to cool, probably precisely because they were taken seriously, and because he feared Elaine was becoming serious. The week after the faculty travesty, he neglected to make arrangements for the following Friday and when the subject came up, he was going home for the weekend. When he got back, he made a date with someone else, the someone else's surprise being communicated immediately the length of the campus. Elaine's first word of it was, unpleasantly enough, Netty Neal's sidelong questions about what was wrong. Elaine was stunned, but faithful; she waited for explanations, without a murmur.\n\nNo explanation came, for a time. Eventually, meeting by accident under the campus trees, they talked, and the gist was that he liked her and liked going with her, but felt that it needed to be clear that there should be no \"attachment.\" She must have smiled inwardly, knowing what she knew. They dated again. But again he grew anxious.\n\nThen Elaine made a mistake, which a child of this world, wiser in her generation, would not have made. She admitted that giving him perfect freedom bothered her not in the least. She was already sure of him, strain as he might, because\u2014she let it all out\u2014he was promised to her by a promise more unbreakable than a wedding vow. One who cannot lie had presented him to her, a marriage arranged before the foundation of the world. I do not know how long it took him to recover from this revelation.\n\n\"He . . . hasn't said anything to me about it,\" is what he finally replied\u2014with a whimper, because I think he really liked her.\n\nNot long before my brothers were expelled, Julian took up chicken farming. He bought, by mail, a hundred chicks and put them in a little shed, formerly a latrine, near Miss Yodle's privy. (Like Miss Yodle's, this backed onto a slight decline and no doubt the college farmer used to come and shovel out the accumulation, as he still did occasionally for Miss Yodle's.) They existed on whatever he fed them, for a few days, until it rained and water came through the roof of the shed and put an end to about a quarter of them. I have no statistics on the ordinary mortality rate for baby chicks, but after another week and another rain he had less than thirty left and a freak frost (this was early spring) completed the annihilation. The structure of the shed provided simple and appropriate means of disposal.\n\nJulian ordered another hundred chicks and, to avoid a repeat of his fiasco, brought them indoors, upstairs, to a corner of the living room, where he penned them carefully, food sprinkled on the floor and a supply of water in an ill-balanced receptacle. All day those yellow balls of fluff ate themselves silly, pecking away,, devouring the grain, devouring what little paint was on the floor, some of the wood, too, snuffing up their own white droppings with the rest. They must have sounded weird from below. Some died, but not many. Charles, from his wide reading, revealed that chickens can be made to grow faster by leaving a light on at night, roosting time indefinitely postponed by a crude and stationary imitation of the sun. From then on they pecked twenty-four hours a day, rain or shine, whether we were home or in classes or at church or wandering out by the apple orchard. The powdery grain covered, with a thin yellow film, everything in the living room, including the couches my brothers slept on. Probably more than a hint of it sifted through the flooring. Every once in a while the water vessel toppled.\n\nJulian got tired of it. I complained. Charles had asthma-like attacks at night. The eternal light bothered everyone. It irritated Julian, especially, that his cats were necessarily exiled, and finally he sold his still immature brood to Chigger, who got a whipping for buying them and, in revenge, murdered the chicks one by one on the front porch of the president's house, smearing their blood along the banisters, across the steps, up and down the doorposts and onto the threshold, where suddenly stricken with guilt he threw himself down, howling for punishment. Elaine mopped up the remains of uneaten grain and chicken turds and there was nothing to indicate we had ever been involved with fowl.\n\nVII\n\nThe bus for Liberty had to be hailed, but if hailed it would stop on the orchard side of the highway, just across from the dilapidated sign that said \"SHARON COLLEGE\u2014A GOOD CHRISTIAN EDUCATION.\" There was, I suppose, at one time an arrow on the sign, in the faded area between \"College\" and \"A Good,\" pointing across the highway, away from the railroad tracks, into the orchard. The road leading to the college was inconspicuous and the college itself, from this vantage, invisible. Three times a week, my mother, music hanging from her shoulder, hailed this bus and in ten minutes disembarked in Liberty.\n\nAfter my brothers were expelled, they continued to live with us, on campus, but scouted around for employment. For a time they went to school in Greenville. For a time they sold encyclopedias. For a time\u2014a few months\u2014they ran the bus station in Liberty. It was not profitable, partly because the owners charged too much rent and partly because Julian, who handled the bus tickets, leaving Charles the snack bar, never managed to keep the money straight. He was aware, on a superficial level, that the blank tickets were numbered consecutively and that some account had to be given for each one. But in practice, he was profoundly uninterested in this serial arrangement and if he made an error (as he was particularly prone, for some reason, to write \"Greensburg\" for Greenville) he wadded the ruined ticket and threw it behind the desk. I sometimes saw him use the pad of tickets for scratch paper.\n\nThey feared it would take a bit of salesmanship to unload the bus station. It was also, by this time, quite dirty. But for once they were lucky. The owners, who had operated the station for many years, found retirement unpalatable and begged my brothers to let them break the lease and come back. Mrs. Himble was so happy to be eased into her old chair\u2014she was a cripple\u2014that she made no fuss at all about the wastepaper behind her desk, the mess at the snack bar, the window that Julian had smashed out one morning when he couldn't find his key. She simply ordered Mr. Himble to get everything in shape and he set to it, in spite of arthritis that bent him forward at the hips and made it painful for him to use a broom.\n\nMrs. Himble wanted to return to her job selling tickets because at home she had no one to talk to. Here behind her desk, with her rack of rubber stamps, her time tables, she gossiped with the seriousness and enthusiasm of those with only one, but one unquestionable, talent. All the drivers stopped, as long as they dared, to banter with her. The woman who ran a beauty shop next door spent most of her time in the bus station, an eye toward the window for any of her own infrequent customers. My mother came in regularly, on arrival or waiting for her return bus\u2014enough to keep up on the scandals of all Liberty and environs.\n\n\"She's a wicked woman,\" my mother said, because\u2014I first supposed\u2014of the fact that Mrs. Himble was a chain-smoker. But since my mother continued, under her breath, \"A world of iniquity, a world of iniquity,\" it became obvious that she was referring to Mrs. Himble's tongue.\n\nAnd while that tongue was tearing up local reputations, Mr. Himble puttered around behind the counter of his snack bar. He had found that this was as far away as he was likely to get from his wife, a rack of Ritz crackers providing the greatest shelter he would ever know. At the bus station, unlike home, her attention was continually drawn away from him to her visitors, her drivers, her customers. He polished a particular part of the counter, the part most distant from her, with a rag always at hand. If someone wanted waiting on, for more than cookies or crackers, he looked surprised, stuffed an end of the rag into his hip pocket. But generally it was in use, shining again the one shiny spot on the formica. At closing time (Liberty is not big enough for the bus station to stay open past business hours\u2014my brothers, indeed, had often closed early) Mr. Himble would hoist his wife, help her into the car, turn the lights off, and lock up. Then they roared off down the street with, strangely enough, Mrs. Himble at the wheel.\n\nMy mother's greatest pleasure those days, when not too tired getting home from lessons, was to sit at the piano to work at her own variations for \"Where the Healing Waters Flow.\" They got, by and by, so brilliant that the melody was, to my ears at least, quite dissipated. I never said that to her, but she took it ill that I failed to follow the tune when, at her insistence, I tried to sing the text. She had formed the hope, without any encouragement from outside her imagination, that I was called to be a song evangelist. No doubt in her prayers she had offered me up.\n\nHer notions of salvation must have been changing, even then, though it became clear to me only later. She grew up in an ethical religion where righteous meant, more or less simply, moral. Gradually a pre-millenarian apocalypse won her over, and with it the vision of a world full, not simply of evil, but of false doctrines, evil in the guise of holiness. What she had considered the basest of mistaken dogmas\u2014predestination, purgatory, even Eternal Security\u2014faded into mere symptoms, as time itself went from bad to worse. She began to collect indications of approaching calamity, evidence that the second coming could not long be delayed.\n\n\"Some people claim,\" she said bitterly, \"the world is getting better and better.\" Her tone made the point perfectly clear, but she went on, perhaps for my sake, but perhaps only to bring her thought to clarity. \"They say all these highways and modern conveniences\"\u2014she put a ghastly tone to modern\u2014\"are what we'll use when the whole world is at peace. And that will bring in the millennium. And then after a thousand years of peace, Jesus will come back.\" I waited for an explosion, but she went on quietly, as if worn out, disgusted with childish error. \"That's wrong. He says in the last days perilous times. Evil is increasing. Don't we see sin all around us? Finally, when it's too bad, He'll just have to come.\" She paused so long I thought she was through, but she murmured, \"And the worst. . . .\"\n\n\"The worst?\" I said. It seemed to bring her back.\n\n\"Old Antichrist,\" she said. \"He'll make everybody believe a lie. And anybody who preaches the truth will have their heads cut off. The tribulation comes first, then Jesus comes back, and then brings the millennium. But until that happens\"\u2014her voice gaining back its strength\u2014\"the devil is getting stronger. He's deceiving more and more people. He deceives, if possible, the very elect. Don't you know\"\u2014only when she opened her eyes I realized she had all this time had them shut\u2014\"that witchcraft is gaining new converts? The devil is possessing more souls than ever. And he's doing it in the name of religion. There are devil cults in California. And even right around here. . . .\"\n\n\"Around here?\" I said.\n\n\"He's subtle,\" she went on. \"He's in cahoots with churches that call themselves Christian. He confuses them. He takes over and makes them babble. They gargle\u2014and they call it speaking in tongues. And they're deceiving millions.\" I conceived then and there a desire to hear the tongues she attributed to Satan, but in the meantime\u2014it was several years before I got to a Pentecostal service, and then under unexpected auspices\u2014I went through the New Testament looking for light on this strange phenomenon. My mother was delighted that I took such an interest. She wanted to compile\u2014the two of us together, that is\u2014a scriptural refutation of all false doctrines. She had already sketched out an argument to be used against Calvinists, but tongues now took precedence. While she stood behind an infant who was ruining some simple harmony, crying out from time to time, \"B flat, not B, B flat\" or \"tum-tum-ta-tum,\" her eyes were closed on more than the printed staves. She was searching for what she might say in reply, in case one of the demon-possessed should quote St. Paul, saying, \"I thank my God, I speak with tongues more than ye all.\"\n\nAnd she meditated along these lines while sitting in the bus station waiting to come home\u2014having accustomed herself to listen only minimally to the garrulous abuse that filled the smoky air, meanwhile winning intense inner debates. But there was something decidedly different this time, that she couldn't quite put her finger on\u2014until at the moment she realized that today there was, in fact, no smoke, she also became aware with a jolt that Mrs. Himble had said, with tears in her eyes and in her voice, \"I'd like to ask you, Sister, if you'd please pray for me.\" My mother was almost on her knees before Mrs. Himble could stop her, making it clear that she meant the prayer to be elsewhere and that, besides, the bus was just arriving. The secret, we learned a few days later, was that a famous healer was coming soon to pitch his tent in Liberty and Mrs. Himble was preparing herself. My mother certainly prayed for her, before and after finding this out, but refused to go with her to the tent when it came round, the preacher being a Pentecostal.\n\n\"If she's not real sincere, she'd better not go near that place,\" my mother told me, having told Mrs. Himble more or less the same. \"If they're speaking in tongues, there'll be a host of demons around and she'd better be right with the Lord before she sets foot in that tent.\"\n\nMy mother's view of evil was coherent and well-defined. In the world before this one\u2014the world between Genesis 1:1 and Genesis 1:2\u2014there was a species of angels who in their extreme wickedness dreamt of changing the creation into their own image. Their world was therefore, and justly, destroyed.\n\nPhysically destroyed, that is; for their souls, like all souls, are immortal. Now they scour our earth, disembodied imaginations, desperately wicked, searching for bodies to enter and pollute. It takes a battery of good angels to withstand their assault, and a heart filled with the Holy Ghost. I have a weakness for this kind of world view, these constructions of a passionate will that refuses to compromise with actuality and lifts no finger to save appearances. I'm fond of them, though I've lost the talent to believe.\n\nShortly before leaving Sharon, I happened to pass through Liberty and, without getting off the bus that was taking me to Greenville, caught a glimpse of Mr. Himble polishing his lunch counter and Mrs. Himble, in the midst of her tickets and timetables, surrounded by her usual audience. Her crutches hung from a nail behind her, like an ex-voto. The driver was held for some time by whatever she was telling and the cloud of smoke from which she spoke gave her the aspect of an oracle.\n\nMy mother avoided her more than ever, generally bought tickets at Sharon\u2014which was less convenient\u2014got off and on the bus without going inside the depot at Liberty.\n\n\"She just went for what she could get. She thought she could be healed by magic, without changing her heart. That's tempting the Lord and He may have given her over to the devil.\" It took me a long time to piece together the events behind this explanation.\n\nWhatever her motive, Mrs. Himble had certainly tried to put herself in the proper state for healing. For two weeks, she gave up smoking and swearing, read the Bible, almost stopped gossiping. (It might have been possible for her to quit completely if Miss Rigsbee, Dr. Karlson's receptionist, had not been suspected to be pregnant. \"Poor girl,\" Mrs. Himble said, recalling the conversation to a religious tone, \"she'll have to pay for this in hell.\") The last three days before the great tent meeting, Mrs. Himble fasted, praying a great deal\u2014even at the bus station, which began to lose its appeal for most of her regulars, especially those she insisted on praying for. The day of the meeting, she did not open up at all and the drivers fussed about having to write out tickets.\n\nThe tent, pitched on the outskirts of Liberty, would have held a small circus. To approach it meant parking at some distance and threading one's way through an acre of cars, suggesting immediately the impossibility of leaving early. When the Himbles arrived, two hours in advance, it was already crowded. One had to be there early to be sure to get into the healing line. Mrs. Himble, on her crutches, and Mr. Himble, bent and following, were received by a young man in a double-breasted suit, who asked them brief questions about their religious experience (when Mrs. Himble seemed ready to testify at length, he cut her off) and gave them cards to fill out\u2014name, address, church, salary, and complaint. In spite of daggers from his wife's eyes, Mr. Himble neither filled out his card nor replied to the questions. He was, he said, merely in attendance on her. I think nothing in his life had given him the notion that anything could be changed. They sat in a section of the tent reserved for those seeking healing, a large group of chairs to the left of the mounted podium, and there they waited for her name to be called, Mr. Himble touching the back of his chair only with the lower quarter of his spine. Mrs. Himble was praying, sometimes silently, sometimes not, on edge in spite of her weight, as if alert for the call of a trumpet.\n\nWithin the next two hours, the tent had almost filled. A choir from one of the local Pentecostal churches began singing while a few still straggled in. Bare bulbs were strung between tent poles and the light was harsh. A wiry song evangelist started things off then by getting the whole crowd to sing \"My God is Real.\" His own voice came out of speakers suspended high on the poles, came out over any other sound, but the space was so immense and the crowd so various (in voice and in motivating force) that there was always a drag somewhere, a slack not quite taken up, however the song leader might charge ahead a tempo.\n\nMy God is real \nFor I can feel \nHim in my soul.\n\nHis solo special was more successful\u2014especially the chorus to the last stanza, which he repeated again and again, pouncing on the da capo until the rhythm found its response in a muscular movement of the crowd as a whole, communicating itself as a throbbing sound, audible even through the combined music of voice, piano, and electronic organ, all amplified. The moment the singer stopped, amid a crescendo of shouts of praise, the evangelist raced to the microphone\u2014no one had heard his voice until then\u2014and told with tears of joy running down his face how the Lord had told him, just during the last chorus, that fifty persons among those now under the sound of his voice were going to pledge one hundred dollars to this ministry.\n\n\"Where are you?\" he shouted, and his ushers were already in the aisles with pledge cards and pens. He had exaggerated, but ten or fifteen people did write something on the cards and this was followed by a general offering while the choir sang. From the reports I gathered (this is all second hand, I was not there; perhaps I have embroidered this account from later, similar services) this evangelist did not compare in sheer take with Brother Allen\u2014whom I did witness, later, in Detroit,.raking in fantastic amounts.\n\nOf the sermon, reports were scanty and confused, though it was what my mother most enquired about. It seems to have been based on a series of texts in Exodus starting with this from the third chapter: \"And now let us go, we beseech thee, three days' journey into the wilderness, that we may sacrifice to the Lord our God.\" The first day's journey, through the wilderness, which is this present wicked world, is\u2014said the preacher\u2014being saved from our sins. The journey of the second day (except for false doctrines, nothing angered my mother more than allegory) represents the experience of sanctification. (\"The Pentecostals are right,\" she always said, \"about getting rid of the old sin-root. They just go too far.\") The third.day's journey is to be filled with the Holy Ghost, a third work of grace, attended in these last times by a rebirth of the gift of tongues. Doubtless at that moment in the service, the tongues descended.\n\nTo some, the rapid spread of speaking in tongues in America\u2014from a few students at Bethal Bible College in Topeka, 1900, to the millions of Pentecostals in this last gasp of the century\u2014is a symptom of religion in a decadent phase. I take it rather as evidence of religion springing up among the ruins of a decaying ethic. Two things in it particularly intrigue me, both noted by Saint Paul in a letter to the church at Corinth, where already this phenomenon was raising questions. It is an unknown tongue (not, therefore, to be confused\u2014in spite of the term pen-tecostal\u2014with the tongues of fire on the day of Pentecost, when the disciples spoke languages they had not learned) and whoever speaks it \"speaketh not unto men, but unto God: for no man understandeth him.\"\n\nThen, also, it is a gift, a kind of ornament in the economics of the spirit, something presented from outside, unaccountable.\n\nAt the peak of excitement, a big-boned woman rose to her feet, arched backward until she seemed about to fall on her head into the row behind, and spoke in a voice that without benefit of microphone carried to the ends of the tent, with syllables that could have belonged to some dialect (it was not the throaty, inarticulate sound that sometimes comes out on such occasions) but here kept their mystery\u2014until interpreted by the preacher to mean \"Trust in me, saith the Lord of Hosts, yea mightily, and I will pour out upon thee waters of healing, yea even rivers of mercy.\" Then the healing line formed.\n\nTwo assistants brought the seekers, sometimes helping them up and supporting them, to the steps of the platform. At that point, another took their cards (which they were supposed still to be holding) and read out loudly\u2014there was unimaginable din\u2014their names and complaints, the latter in brief. From the moment they were on their feet, someone's hands were on them: another assistant received them as they made it up the three steps to the platform, or in some cases pulled them up, and after embracing them with a shout, sent them flying across to the healer, who, grasping the sufferer firmly in his right arm (his arms seemed suddenly mightier) and crying \"Do you believe Jesus heals you?\" (one could see but not hear the affirmative answer), stretched his left arm out and upward and then, with a majestic motion, brought the hand down and across to fasten with the authority of a trap on the streaming face, a visible tremor marking the moment of contact. \"Jesus HEALS you!\" And the healed was propelled forward from his embrace to a different set of steps, these leading down to the center aisle. Two more assistants shouted blessings there and a steady flow of happy screamers, waving their arms and praising the Lord, walked or danced from the platform to the egress. Some spoke in tongues, as did many in the audience, on their feet, laughing, clapping, weeping.\n\nIt was a long line, but an efficient system. Mrs. Himble, breathing fragments of remembered prayers, was lifted bodily from her chair, projected along the stations I have described until the hand of the healer came down on her face with such spiritual force that she shrieked for joy, shaking her arms in the air. The next thing she realized was a shout, as from many loudspeakers, \"There goes a woman who hasn't walked in thirty-seven years!\" and she was moving swiftly down the aisle, sustained by two assistants holding her firmly between them. Another shout announced, \"There goes a man carrying his crutches!\" and she was not aware until later, in her own car where she had been gently but firmly placed, that her husband had slipped along the edge of the platform and was following her, bent forward and bearing her crutches.\n\nBright moonlight picked out the curves of the tent from among the dark fields. Waves of shouting still reached the couple in the car, but the distance took the volume down, made it remote enough that they could also hear from another direction (though in fact it made no impression on them) J.W., who had driven his car up just outside the parking area and, with his P.A. system at the full, was hurling prophecies against the beast and its deceptions.\n\nAs for the Himbles, neither said a word on the way home\u2014she drove\u2014and all the next day he creaked about the house as if nothing had happened, while she swore in a voice that carried the length of the block and, hobbling across the kitchen, smashed a broken set of dishes inherited from her mother.\n\nI left Sharon the day after my high school graduation, 1950. My father had driven from Kansas to be there for the ceremony, and I went back with him to go to college in Emporia. This was a triumph for my father, a great blow to my mother, for me a disappointment. I had taken a fancy\u2014from what sources of information I can't now say\u2014to go to the University of Chicago, but though I had applied there, and was accepted, it was not easy to see where the money would come from. Neither at the time, nor later, was I quite happy with myself for accepting the easy solution of Kansas State Teachers College and living with my father in a tiny two-room apartment on Commercial Street\u2014across from a turn-of-the-century Moorish edifice that had been a vaudeville house but was now a movie palace, the Granada. At the south end of Commercial, the Free Methodists still worshipped; to the north, closer, was the campus. My business was all in that direction\u2014I never again set foot in my sometime church\u2014but my theater of discovery was just across the street. In films, my principle of selection was simple: I went to everything.\n\nI doubt that Sharon ever got its accreditation. (Old Bob Jones, by the way, is reported to have scorned even applying, saying \"Bob Jones University is already accredited\u2014by a Higher Authority.\") Meeting a man who had worked with the Southern Accrediting Association for years, I brought up the subject, in a roundabout way, fully supposing he would echo the procurator of Judea when questioned about Jesus of Nazareth, \"No, I cannot bring him to mind.\" But, on the contrary, he struck his forehead and said (he was quite sober), \"Sharon! Good God, yes, I know it well. Second worst college in the South.\" I was prepared to digest that, but he went on: \"We managed to close the worst.\" This man had no drawl, though he was sipping bourbon without ice. \"There was a high school at Sharon College,\" he said, wrinkling his forehead to remember, \"yes\u2014and we closed that.\" And he told me how the girls' dormitory had burned down, killing half a dozen occupants.\n\nBut as for my own stay there, and any regrets\u2014well, I suppose Julian should not have thrown the firecracker under Miss Yodle's privy, not at least while she was in it. I think on the whole we treated Miss Yodle badly, and episodes like that of the firecracker she was too modest to report. During one revival, the last of my time there, when no one in Teter Barn could sleep because J.W. prayed nightlong, and loudly (while the most unlikely people saw visions and chronic seekers, who could never in the world forgive themselves, beat against a closed heaven) Miss Yodle came\u2014ashen faced\u2014to each of us (me, Julian, Charles, Mother, Elaine) asking our forgiveness. She had, she said, had evil thoughts about us. She even went to Clyde, to whom Elaine had just gotten engaged, explaining that she had nothing against him personally, only he was connected with (\u2014a slight sniff\u2014) \"that family.\" The day before my father arrived for my graduation, Julian, offended by something she had done, as if he were making a huge omelet, sat on the floor and broke a dozen eggs into the crack in front of our refrigerator.\n\nVIII\n\nAll his years in the navy, Charles bought books\u2014and read them. He began making outlines and charts for each book, then for aggregates of books. Finally\u2014I have a photo of him sitting outside a thatched hut in the Admiralty Islands, in the webbed helmet of the Amphibious Corps, no shirt; a nurse is visible on the other side of the hut, wearing what looks like her officer's cap and otherwise nothing but a man's T-shirt, but he is gazing straight into the camera with the sort of total smile professional men usually have to work for\u2014finally deciding to graph the entire range of human knowledge. Most of his schemes resembled the traditional family tree of genealogists. There was one for the pure, another for the applied sciences, one for history, and a large one for psychoanalysis, with a Freudian trunk and limbs of Adler, Jung, Rank, and Sullivan, lots of smaller twigs, named and unnamed. In 1945 he was Chief Petty Officer and then, quite suddenly, a civilian. Fie had picked up a degree in meatcutting, the first of his many diplomas, from a navy school, but that counted for little. So he went to Boston and took a course at Fanny Farmer's cooking school\u2014on the G.I. Bill\u2014then went in search of a job. He worked a single season as chef at a resort hotel on Cape Cod, then came to Kansas just as my mother, Elaine, and I were leaving for South Carolina. Mother's parting words to him were, \"Search the scriptures, son, and let the Lord have his way.\"\n\nElaine and Clyde at the time of their engagement (Teter Hall, Sharon College)\n\nNow Kansas was dry, legally, while in the surrounding states booze flowed like milk and honey. Charles followed a northerly route driving twice a week from St. Joseph, Missouri, to Emporia, where he unloaded at the alley entrance to a Greek restaurant heavy cardboard boxes marked \"Kosher Dills.\" Steady as this was, it was tiring work and somewhat dangerous. Charles always had the dream of a business whose profits would engender profits, so that the longer it went on the more profitable it would be, spreading geometrically. Nothing like that was happening\u2014obviously, large trucks would not do. Since he still had thirty four months of G.I. Bill, he enrolled in flying school, a three month course. The idea of bringing liquor in by airplane excited his imagination till at night he would study the most enormous maps he could find, plotting paths for his new air service, drawing intricate patterns of lines that would guide his fleet of flying boxcars from St. Louis, from Kentucky, from Canada, into Wichita, Dodge City, Emporia. In the middle of one lesson his mind wandered off into an elaborate scheme of distribution and finance, a whole complex of technicalities, till his instructor (he had not yet soloed) reminded him he should get set to land, and having landed he discovered that the state legislature had voted in local option. He did not bother to finish the course.\n\nIn 1950, when I was leaving Sharon to go back to Kansas with my father, Elaine and Clyde were in the process of moving to Atlanta, where Clyde had gotten a church. I thought that good for a first assignment, but it turned out to be a congregation so small and so poor that the church conference listed it as missionary work and the salary was intermittent. My mother went to live with them, but soon moved to manage a boarding house. Charles and Julian had taken off toward New Orleans, but Charles showed up soon after in Atlanta, alone and penniless.\n\nHis desperation showed in his willingness to return to a chef's job, which he has always regarded as the lowest sort of drudgery. The truth is, I think, that besides being a great deal of work, it is a job with little possibility of large, dishonest gain. Honest gain does not interest Charles\u2014he would rather earn less, so it be ill-gotten. And for a miserable return, with the right smack of illegality about it, he will work himself to the bone. At present, however, he simply needed cash.\n\nAs he walked down Peachtree Road, he noted the Hotel Doric, a recent building in plantation style, with six columns, somewhat Corinthian, and an acre of funereal lawn. He hesitated, as there was no walk, only a semicircular drive and the clipped grass, then set out in a grim line to the door, changing his path only to avoid a sprinkler and an old lady reclining in a low-cut dress who stared at him over her Bhagavad Gita. The girl at the desk, dismayed at his clothes but taken with his smile, directed him to a Mrs. Skye, who without looking up told him there were no jobs.\n\nAtlanta is the quack capital of the South. St. Petersburg is more famous as a town for the old, but, just for this reason, has the aura of an immense boneyard, and the better hypochondriacs prefer Atlanta. Once there, they go from doctor to doctor in search of an ailment worth its antidote. In many cases the only symptom of life is an oppressive force on their livers or their kidneys or closing around the diseased heart, and if this consolation is denied them they are lost. Physician after physician tells them there is nothing wrong or offers the ghastly comfort that they are simply old, when they know that inside them something is growing, branching into their veins, and collecting its own strength to topple them.\n\nBut there are other doctors, less hateful to them, whose science has differently developed. Theo Sansom had his degree (D.C.) from an institute in Nebraska. He was a brilliant student, especially for one who never finished high school, absorbing in record time, not only the therapeutic theory and practices of chiropractic, but its advanced sales approaches also. In fact these latter he radically improved and had lectured about his improvements back at the alma mater. He had the first essential for a healer: charisma. He had come to Atlanta to found a chiropractic hospital and found that the state of Georgia would not license it, so he bought (on credit, which is easy with charisma) a slightly run-down hotel, planting his clinic with its X-rays, neurocalometers, graphs, and gadgets in the basement. The ground and second floors accommodated whatever tourists or businessmen happened in and on the third and fourth Dr. Sansom's patients flourished. It was rather an expensive place, but they could afford it. And the clinic was free; that is to say, there was no charge, either for diagnosis or treatment. Gifts were possible, of course, and gratefully received, and all the old women who inhabited air-conditioned rooms on the top two floors were regular donors. In point of fact, the doctor's hold on them was complete, since if they discontinued their charitable contributions (untaxed for him, deductible for them) he cheerfully proclaimed their recovery, the news of which was usually received with less joy than had he predicted their demise. The important, the necessary thing was that their bones, their nerves, their blood, above all their digestive tract, be in the hands of the wise man (for his radio lectures he was announced as the \"Miracle Man of Peachtree Road\"). As Charles turned to leave Mrs. Skye's office, Dr. Sansom stood in the doorway, raising his arm to point at Charles. He said,\n\nCharles, Mother, and Julian (Atlanta)\n\n\"You've been sent.\"\n\nSome months later, when I came on a visit\u2014the summer between my freshman and sophomore years at college\u2014Charles still had no very exact idea how or for what he had been sent (he was chef of the hotel dining room) but he had learned something of Dr. Sansom.\n\n\"But I don't understand him,\" he said; \"he's got some kind of power that I don't understand.\" We went down into the clinic while the doctor was away. It was labyrinthine. In one room there was an over-lifesize replica of the human backbone. From each set of foramina, plastic cords extended, representing the spinal nerves. Punching any of the vertebrae caused a cord to be pinched and it lit up with a red glow. There was an enormous room with a podium at one end, but no chairs. This, said Charles, was to be a lecture hall. There were rooms for taking and for developing X-ray pictures. In the doctor's inner office Charles made me recline on a chaise longue and showed me how my heels did not come together; obviously my legs were of different length. Moving an unseen lever with his foot, something in the couch moved and my legs matched perfectly. \"Before and after,\" Charles said. On the desk was a book called The Bigness of the Fellow Within.\n\nMrs. Skye took an immediate liking to Charles. She was fortyish, had at various times been a nurse and an accountant, so she was well equipped to keep track of patients and taxes. Actually she owned a half-interest in the hotel and gained from it a livelihood and easy access to the drugs she required (the patients were not, of course, allowed any medicines).\n\n\"Do you believe in all this?\" Charles asked her.\n\n\"In what?\"\n\n\"All this. The patients. The adjustments.\"\n\n\"He gives them what they want, doesn't he?\"\n\n\"But he's a fake.\"\n\n\"He's a great man.\"\n\n\"He's a fake.\"\n\n\"You don't understand.\"\n\nAll the heat of that hot summer seemed packed into my mother's boarding house, which was not hers, of course, except in the sense that she ran it. There was also a cook, who always seemed hurried, and in fact was, because the owner had hired the one woman for this and another house, across the street, which she herself ran. On account of this arrangement the other house ate early and at what I call my mother's house the guests had to wait until the cook, bustling in and usually in a bad temper, got some mess onto the table. Otherwise, it was an ordinary boarding house, with the ordinary assortment of boarders. My mother, who lived in a tiny room at the back of the ground floor, knew all their histories, the knowable parts, that is\u2014that Marlene, for instance, was twenty-seven and was staying only until her fianc\u00e9 got enough money to send for her (he was in Washington, her family in Birmingham, which gave a hopeful direction to her itinerary). Most of the occupants were old, and hopeless.\n\nThe owner of the two houses was a woman with, as far as I could see, no thought on earth beyond taking in the rent (she made the weekly collections herself) and spending as little as possible. The only time I remember her to have expressed a concern unconnected with the operation of her business was once when she drew me aside\u2014she had come over, rather steamily, because one of the boarders was complaining about the lack of a window blind; the afternoon sun, so he maintained, made his room \"a furnace, a real fiery furnace.\"\n\n\"Your mother,\" she said to me in a tone of puzzlement, \"she's I think maybe not quite right.\" A trickle of sweat down my spine made it hard for me to react. \"I mean,\" she went on, \"right in the head.\" At that, I focused enough to reply.\n\n\"How do you mean?\"\n\n\"Well,\" she said, \"she's always back in that there room\u2014oh, she does her work alright.\" I knew how little my mother was paid. \"But then she goes back into her room, and\u2014well, she's always, like, playing the piano.\"\n\nI was a little slow again, ready enough to say how natural it was, that being a pianist\u2014but I remembered, as the owner, still looking perplexed but aware of things to do in her other source of income, hurried off, that in the whole house here, there was no piano. The bare table, however, which in the little back room served all the purposes for .which a dresser and nightstand might have been expected, was my mother's ghostly keyboard, where she practiced (really practiced, starting always with scales and arpeggios\u2014I watched her struggle over certain difficult passages in a polonaise), running her variations along the grain of a peeling veneer. She asked me to sing along, and it was not so difficult.\n\n\"Some day the silver cord will break. . . .\"\n\nJulian arrived. He had been tending a firehouse in some small Colorado town and was stricken by one of those fits of homesickness that are so difficult and so mysterious for people who have never lived any place long enough to call it home. Charles gave him a job as fry cook and he stayed with me on the second floor of the Hotel Doric\u2014there was no vacant room at the boarding house\u2014bringing stray cats up to our room to feed with scraps from the kitchen. Charles went to live for a while with Mrs. Skye and her husband. I remember her husband only vaguely\u2014I do remember that the Christmas before, she had given him a Nash, and coming out of a New Year's party he reclined the driver's seat, lay down, and kicked all the upholstery from the roof.\n\nIn Atlanta\n\n\"Read this,\" she told Charles. \"You think the doctor is a fake because he takes in the old biddies. That's for their own good, but this is what he really believes in.\" It was Borderlands of Eternity by Edwin J. Dingle, who, leaving his body in the throes of malaria, had traveled incorporeally to the Orient and was told many secrets, among them that he was himself a great guru, no more to be called Dingle, but Ding Le Mei, and that he should go forth and teach \"the strange power that knowledge gives.\" Laid in was an advertisement for Ding Le Mei's Institute of Mentalphysics, clipped from some newspaper:\n\nForty-three years ago, he was as sick as a man could be and live. Once his coffin was bought. Years of almost continuous tropical fevers, broken bones, near blindness, privation and danger had made a human wreck of him, physically and mentally. He was about to be sent home to die, when a strange message came\u2014\"They are waiting for you in Tibet.\"\n\nThis whole area Charles had missed in drawing up his charts (which had got no farther since his discharge from the navy). \"What you really want,\" Mrs. Skye said, \"you can have, that's the gist of it.\"\n\n\"Anything?\" he said, because he really did not know what to say, and it made no difference, since Mrs. Skye's eyes were changing, becoming directionless, remote.\n\nDr. Sansom filled his lecture hall with very expensive theater seats, the kind that get up when you do, so that he could lecture on yoga and related topics. The salesman he got the seats from was a young man, his first day with the company, who had come to the clinic on the merest hunch and was so excited over the purchase that he left his sample. Charles put it in the trunk of my car (sitting out in back, an obvious flaw in the line of Cadillacs and Lincolns). The lectures were much the same as his radio talks, except for occasional references to Ramakrishna. It was perfectly apparent, said the doctor (who, in this his final incarnation, remembered sitting by the River Jordan, feeding on locusts and wild honey, now satisfied at last, perhaps, that he was nobody's forerunner, but the real thing), that we are meant to enjoy the good things of the earth, foremost of which are vigorous health and a healthy fortune. We can have all these things if we think right, and we can grow into right thinking. From rather simple beginnings, one thought can lead to another until we are, so to speak, one of the world's power centers, a sort of focal point, a switchboard for the vast network of forces which is the universe. The old women were rapt. Charles was taking notes.\n\nJulian was cooking. He was marvelous with leftovers, which he mixed together and gave fancy names like \"Persian Delight.\" He was also a bit careless and the kitchen got dirtier and dirtier. The day the health inspectors came there was a burnt steak behind the grill and they changed the restaurant rating from A to C. Not that it mattered much; the ladies from the upper stories still ate there regularly, one sipping bouillon and telling how much better she was since the doctor had taken her colon in hand. Another, her jaw wet with Persian Delight, described in detail the aberrations of her respiratory tract. No matter what the ailment, it was always in terms of cancer, though the word itself was taboo. For some the evil lodged inside, a ball. For most it had tentacles or extended, fibrous, through the system, living its own life and gorging itself strong from their weaknesses. It absorbed whatever was sweet and good into its own system and left on the surface dry brown husks.\n\nI still have the theater seat, almost my only memento from that time. It looks a little odd in our living room, but not very. (Jokers ask what theater I stole it from or sit in it and call for the movie to begin.) It was Mrs. Skye who told Charles things were ending. The doctor had invested, under various names, in several well-rewarding ventures, including a pecan farm in southern Georgia. As for Mrs. Skye's interest in the hotel, it was somehow gone, and Dr. Sansom was declaring bankruptcy. All his investments were sure to prosper, and he himself was going to the West Coast (somewhere, I understood, in a desert outside Los Angeles) to sit in a Perceptatorium, and perceive. (He had always claimed to be sixty, an impressive example of his own efficient science, as he looked scarcely fifty, his actual age.) Mrs. Skye was not destitute; she owned her house and could easily get a job. Her husband disappeared about this time, but she seemed not to notice. Deciding Charles was not serious, she no longer showed him her occult books, but kept them in some secret closet.\n\n\"She thinks,\" Charles told me, \"she can have anything she wants.\"\n\n\"What does she want?\" I asked.\n\nThe day we left Atlanta we went to say goodbye\u2014just after seeing the last bunch of trembling patients disperse after surer medicine\u2014but no one came to the door. I went back to Kansas, back to college, Charles and Julian to Miami, where Charles set up office as a \"psychological physician.\"\n\nIX\n\nSomehow my mother got a job in Roanoke, Virginia. Later, she thought of Roanoke as her last not-entirely-unhappy time. Her job consisted partly in being companion to a retired spinster school teacher, but more in taking care of the woman's mother, an ancient invalid with no world outside her helpless body. My mother got along with her\u2014nobody else could, including her daughter\u2014by humoring her with small amounts of the candy she was not supposed to have, and by playing the piano to her much of the day. Except that she was paid rather little, it was a job for which she was well suited (though, to be honest, I must add that while she had it, she was miserable).\n\nI, meanwhile, had been drafted, only a few semester hours from my bachelor's degree. The truce in Korea was not yet signed, though the shooting was over, so technically it was wartime. I was sent to a camp in Missouri for eight weeks of infantry training, then shifted a few blocks for another eight weeks of basic, this time to be made into an engineer. It was fearfully hot when I started training, or seemed so to me, even after living in the South, but by the end it was ferociously cold. A friend, drafted at the same time, was put into personnel after his first eight weeks and I sometimes ran into him at the PX or the service club. One evening, after a particularly grueling exercise in laying mines (we studied nothing long enough to learn it properly) I saw him at a film and he told me that in three days my company would be given a battery of tests. A relief, since it meant we would be indoors, warm, not huddling in a latrine or running to press against each other\u2014the horde of us in dirty fatigues\u2014into a disappearing spot of sunshine.\n\n\"It's to decide who should go on to the Engineer School,\" he said, and explained that only two courses were actually open: refrigeration and water supply. I thought nothing of it at the moment, but when, three days later, we were marched into a hall with desks and I was handed not only a test (one of the standard IQ type, nothing specific) but also an educational questionnaire, with no hesitation I recorded a fantastic number of semesters of chemistry and physics. Elementary physics, in fact, was\u2014with tennis\u2014the only course I ever flunked. Nevertheless, when, in December, I left Fort Leonard Wood, it was for the Engineer School in Fort Belvoir, Virginia, to learn to purify water, and from there I managed a trip Shenandoahwards to see my mother.\n\nCharles was there, with a new mustache and several new doctor's degrees, of which only the M.D. struck me as potentially dangerous. His office in Miami had failed.\n\n\"It's not illegal,\" he said, \"as long as you call yourself psychological physician, it doesn't come under the medical board\u2014perfectly legal.\" My mother was lying on a couch, the blinds being drawn against the sun far in the south and brilliant. I was reminded of old scenes. Her fingers were on her throat.\n\nCharles as psychological physician\n\n\"Why does my heart beat so hard?\" she said.\n\n\"You can call yourself psychoanalyst Charles was saying, \"as long as you don't say psychiatrist\u2014that comes under the board.\"\n\nI tried to convince her her heart was beating as it always did, but that she simply, being nervous, felt it more.\n\n\"Psychological physician is good. It doesn't come under the state medical board, and it's broad-sounding. I treated anything.\"\n\n\"You mean it's my nerves?\"\n\n\"Vocational guidance, for instance, or marriage problems.\"\n\n\"Why did you close?\" I asked.\n\n\"Money.\"\n\n\"I think it's my heart.\"\n\n\"What I needed was an office\u2014I mean you need an office that inspires trust. A confidence-inspiring office. You sit there with no nice furniture, no nice waiting room, doesn't matter what you say then, they've got no confidence in you.\"\n\n\"My eyes have been bad lately.\" She moaned a little.\n\n\"An office like Sansom's,\" he said dreamily, but at that moment, the old woman, in a hospital bed on the other side of the room, woke up and called out.\n\n\"Opal. Opal.\" My mother was up in an instant, beside her, adjusting the bedclothes. \"Opal, there's something under me.\"\n\n\"No, there's nothing under you\"\u2014fidgeting some more with the covers.\n\n\"Yes there is. Yes there is. There's something terrible under me!\" So she had to be lifted slightly, the terrible thing being a crease in the sheet, after which she was placid. \"Doctor,\" she called then, meaning Charles. \"Doctor, could you come over here?\" He went to her, putting on his reassuring smile as he crossed the room. \"I have a problem, doctor,\" she said. \"Can you look at it?\"\n\n\"If it's anything in my specialty,\" he said with the same smile. It struck me as a good answer.\n\n\"Come closer,\" she said. Mother had sat down at the piano. \"Doctor, I have the most terrible hemorrhoids. On both sides.\"\n\nMy mother had already started some waltz tune, and I noticed her eyes were closed, and that she was trying desperately to keep from laughing.\n\nThe year I got out of the army\u20141955\u2014on Thanksgiving Day, my father died. I had gotten my discharge in May, finished over the summer what I lacked to graduate from college, and gone to Illinois\u2014where my brothers had then followed me. In three months we had lived in four different dwellings, each more squalid than the last, and only did not know yet (we had our suspicions) that the series had not ended. We scraped up between us (I think I have nowhere noted, but may do so appropriately at this point, my brothers' generosity) my bus fare to Emporia. Uncle Roscoe, who had sent me the telegram announcing his brother's death, looked exactly as he had from the first time, as a child, I had seen him and been frightened. He was lugubrious, but dry-eyed, and kept telling stories, the import of which, either because of the telling or because I wasn't listening well, was never clear to me. He handed over to me the old Hamilton, which had always, at night, whether in a hotel or at home, gone under my father's pillow along with his wallet. I was allowed, when I was old enough, to wind it. When I told my father that my draft notice had come, I remember that he took out his watch, without looking at it, turning it in his hand like an amulet, saying nothing for a while. As much as he hated everyone in authority, from the president on down, he was very much for law and order and, in a manner already a bit old fashioned, patriotic. So it astonished me when, nudging the gold timepiece face up, face down, he said quietly\u2014I think he was embarrassed\u2014that if I didn't want to go into the army (and there was something funny, he had decided, about this war in Korea) we could then and there, the two of us, sell all that we had and move to Canada. I turned down his offer, but I was touched. He must have been relieved. And selling everything at that point\u2014except, possibly, his railroad watch\u2014would hardly have gotten us farther than Omaha.\n\nMy army I.D. photo\n\nThe two daughters of my father's first marriage were both present, half-sisters of mine whom I hardly knew. The three of us and Roscoe were all the family there, but the pallbearers were Masons and there was a Masonic service at the graveside performed by old men in ceremonial aprons. Probably none of them knew him well\u2014he had seen almost no one for years, living with me and his anger, or alone with his anger, slowly consuming enormous platefuls of beans while the news on the radio filled him with rage. A short man, fine-boned, he had gone over two hundred pounds, from those endless limas, and from shorter sessions of peppermint schnapps.\n\n\"Keith,\" Roscoe said tearfully, but without tears, \"what will happen to you now?\" And I was wondering what would come of him. After his mother's death, he had determined to fix up the house on Cottonwood Street, perhaps with the idea of selling it. He pulled up linoleum and started steaming off wallpaper, took down various doors to be sanded or replaced, and somehow got only so far. Nothing was ever replaced or refinished. Softwood, laid half a century before, displayed its rot, catching the sift of disturbed plaster from the ceiling. Stripped walls gave an impression of moisture and some were not fully stripped. The circulation in Roscoe's limbs, never good, had worsened, and his legs, a dull gray, were becoming useless. He offered me two fifty-dollar bills (it never occurred to me that I might refuse them), which reminded me how my father had always believed that Roscoe was salting away a fortune, presumably from my grandmother's Social Security checks.\n\nUncle Roscoe on the day of my father's funeral\n\nI never saw any of them again. And after one letter from each sister, neither of which I answered, never heard anything. Roscoe, according to one of those letters, was going to the West Coast to live with my aunt, his sister, Lena. She was the youngest of the three, as my father had been the oldest, and had been sufficiently unfortunate in her marriage to change her name from Waldrop to Stallcup. \"Old Bill\" Stallcup, as my father always referred to him\u2014invariably adding, \"the son of a bitch\"\u2014was in show business; that is to say, for years he and Lena toured with a medicine show, living off the miracle of Indian River Medicine. I knew little about them, since my father would not allow them to visit us, but word sifted down of two sons, child prodigies, who played a dozen instruments each, and years later I ran across, second hand, a record\u201445 rpm, extended play\u2014and recognized the stage name of one of the younger Stallcups. He sings four parts and plays all the instruments of the orchestra simultaneously. The music is bad. I recently saw his name again, in an old Columbia catalog, listed under \"SPECIALTY,\" between Danny Kaye and Liberace.\n\nJust before I left Emporia, Roscoe delivered into my hands the Waldrop family Bible, one of the heavy kind, with na\u00efve illustrations stamped in gold on raised panels at the edges of the leather cover. The Complete Domestic Bible, it says inside, on bad paper, \"Beautifully Illustrated\"\u2014it has some unevenly printed Dor\u00e9 and Holman Hunt. It contains the Apocrypha, gives at the top of every column the date of the action for that particular narrative (starting things off in 4004 B.C.) and between Second Maccabees and Matthew has a place for the \"Family Record.\" The earlier entries there are in a green ink, faded enough to inspire confidence, if only the volume had not been printed two years after my father's birth. There is only one death recorded, that of my grandfather (less than a year after I was born). The last birth given is Lena's; her marriage and my father's (his first, that is) are dated, but no issue indicated. Roscoe had not entered his mother's death, nor have I added anything at all, but he thought the book should be in my possession, since I am the last of the family to have the name. Since that visit, more than thirty years ago, I have not set foot in Kansas.\nDiscerning of Spirits\n\nX\n\nI came to Urbana\u2014though I admitted it only later, and then only to myself\u2014to get away from my family. My mother was in Virginia, nursing her moribund, immortal patient. Elaine and Clyde had moved to Indianapolis, to take over a church that was bankrupt. Julian was in the air force, in Europe. Charles I had lost track of, but had the idea he was shipping out with the merchant marine. It may not be obvious that getting out of a net so loosely drawn would present the slightest problem, but this account should make it clear how well-founded my fears were.\n\nI wanted to get away from my father, too\u2014he was in Kansas\u2014though I never really considered him part of the family.\n\nMy first semester as a graduate student is marked in my memory by two events\u2014in addition, that is, to my father's death. One of them happened (as usual, I am not telling them in order) at the last meeting of a seminar on the novels of Henry James. It was taught by Cornelia Cramer, a woman of great age and strength of mind, whose method of teaching was to abstract, precisely and sarcastically, the more important criticism of the novel we were reading, consigning it all, mercilessly, to the realm of error. She aimed, I think, at disposing of any theory whatever, thinking James's fictions fine enough that no idea should be allowed to penetrate them. I respected her, less for that than for her immense and detailed knowledge of the work. Her own book, a standard study of the early tales, she had kept out of print for decades.\n\nAs the end of the term approached, she seemed content, satisfied with her demolition. But fifteen minutes before the last bell of the last class, a student with the Jamesian name of Moreen, sitting in the front row, asked a question that revealed not only his abject dependence on the view of critics, but\u2014far worse\u2014the outline of a view that he supposed she had been, all semester, constructing. She was outraged at this (struck suddenly, as she told me later, by \"the futility of teaching\") and slammed both fists on her desk so hard that a leg gave way and the desk fell forward, breaking Moreen's foot in its fall.\n\nThe other event was the arrival, about a month after my own, of my brothers. Charles I was not so much surprised to see as surprised at how he looked. He seems different whenever I see him after an interval, but this time more different than usual. He had always, before, a kind of assurance about him. When I saw him recently, in Long Beach (he is now in his sixties), strapped though he was, there was the air about him of a poor prophet or down-and-out guru. In 1955, in Urbana, he looked hunted\u2014and a little mad.\n\nJulian was, in fact, being hunted, as a deserter. This was not quite as much a shock to me as his enlisting, a year before. He had been on the skids (somewhere in Texas, I think) and had joined the air force because he was hungry\u2014extracting, at the same time, a promise that he would go to Germany (where, as a water purification specialist, I had already been sent). He prided himself on this bargain\u2014it indicated some sense of control\u2014but it didn't wear well. From his base in Wiesbaden, he would take the train to see me, but did not always arrive. Once, a little unsure it was his station, he was not quick enough to get off. Two or three times the ticket agent, no doubt by an honest mishearing, routed him to Bad Kissingen\u2014I was stationed in Kitzingen-am-Main. A few months later, I was sent back to the states to be discharged, and soon after that, Julian was transferred to Fontainebleau.\n\nI would not have thought it could matter much to him, whether he was in France or Germany\u2014he spoke the language of neither and had no particular interest in either one, except that German beer did seem to agree with him\u2014but he found his new situation intolerable. (I remember how once, when he did make it to Kitzingen to see me, he had just gotten a note from Charles, who was on the skids and about to follow Julian's lead and join up. We ran together to Western Union and, finding we had only enough money to send the kind of cablegram where a word can be added to a standard formula like Best wishes for a happy \u2014\u2014\u2014 , we sent Charles the striking\u2014and successful\u2014message, \"DO NOT BUY AIR FORCE.\") Julian had taken, from Wiesbaden, a number of short leaves, and so could not now ask for a more extended absence. He had already decided to desert when he entered his request for an emergency leave. His mother, he claimed, was seriously ill and needed him.\n\nHad it occurred to him that his statement would be checked? Probably not. Certainly he never supposed that Mother would lie for him. When a man from the Red Cross called her in Roanoke, and without saying why he wanted to know asked her how she was, she started\u2014hesitantly, he being a stranger\u2014with the nerve in her right eye, which caused the flicker, described how the ringing in her ears increased during dizzy spells, warmed to her subject as she took up certain inner organs, and as a matter of course dwelt long on her eccentric heart. I've imagined most of this\u2014her hand going tQ her throat to verify the throb\u2014but the fact is that Julian was immediately put on orders to be flown to Camp Kilmer, and dispatched from there, on his own, to Roanoke\u2014which he never reached.\n\nJulian at the time of his desertion\n\nHow he found Charles, or where, I never knew. They showed up in Urbana, together, driving that year's Buick, an immense machine with tail-fins. They had bought it, a floor model, for very little, because it incorporated the latest improvement in automatic transmissions, and (only occasionally) shifted into the wrong gear without warning\u2014or out of gear, the motor racing powerlessly, then catching again with a terrible jolt. I was living in a single small room, and they moved in with me.\n\nUrbana, Illinois, is a nondescript town, connected unnaturally to its twin, Champaign, which is hideous. The tissue joining them is the university, though officially it is in Urbana; and a long, wide, busy artery that crosses the double town as if unaware of city lines is called, in its honor, University Avenue. The Urbana segment ends, far past the university, where it meets four other streets at an intersection called\u2014accurately\u2014Five Points. One road leads from there to Rantoul, one to Danville, one (University Avenue) to Champaign, and two\u2014for all practical purposes\u2014lead nowhere. The intersection itself has been remodeled, modernized, so as to be hardly recognizable to someone coming back after a lapse of years, but at the period in question there was a small, two-story building (intended for a store or a house, it was hard to say) in the angle of two rays of this asymmetrical star, and it was there that my brothers and I, after the series of moves already noted, opened our used car lot.\n\nIt was, as it turned out, an excellent spot for such a venture, because when any trainee at the air base in Rantoul manages a pass, unless he has time\u2014and money\u2014for Chicago, he heads either for the glitter of Champaign or the fleshpots of Danville. But to get to either one (there is an expressway now, which makes all this archaeology) he had, at that time, to pass through Urbana, and the first thing he saw in Urbana, at this crummy \u00e9toile, was Used Car Heaven.\n\nI blush to think the name may have been my suggestion. Certainly we all liked it and whenever anyone\u2014most of our business was at dusk\u2014pulled up for a better look, to check if it was worth parking and getting out for, Charles, with an expansive gesture of welcome and a mad glare\u2014more like magician than salesman\u2014intoned the name in such a manner that some stepped quickly on the gas. But others\u2014lots, actually\u2014it was a busy corner\u2014stopped, looked around, left, or bought.\n\nJulian was the only one of us who could really sell. Charles, who wore a top hat along with his baggy trousers and shabby jacket, so exaggerated his pitch that he made people suspicious and they turned, for confidence, to Julian. Whatever Julian told them, they believed. Most people, after all, are born believers. We started the car lot (I mean, we got the money to start it) by selling the Buick in which my brothers had arrived to a man looking for a Cadillac. Julian convinced him that it had a Cadillac motor. (I suppose that is not impossible, but I'm an unbeliever, by acquirement.)\n\nUsed Car Heaven was, on the books, mine. Since I was also\u2014though before long, it was likewise a mere formality\u2014a graduate student, I rarely tended the lot and, even when I did, my record for non-sales was perfect. But there were two things in my favor. My monthly check from the G.I. Bill went at first for the rent. And\u2014much more important\u2014my name was different from that of my (half-) brothers, which made it less likely that Julian would be traced.\n\nWith the birth of the car lot and, about the same time, the death of my father, it is not so peculiar that my studies flagged. Miss Cramer (she had picked up years before the inevitable nickname Crappy Cornelia, and while I liked and admired her, I was unable to think of her without it) read my rapidly written papers and told me they were worthless. It was certainly true and I did not expect anyone intelligent not to realize it\u2014I was doing fine in two other classes with work as shoddy\u2014but I was impressed by the scarcely concealed sadness with which she arraigned me. Apparently she had seen something in me, or thought she had, perhaps even glimpsed the possibility, finally, of a disciple.\n\nUsed Car Heaven\n\nJust before Christmas, Elaine and Clyde, with infant son, were evicted from their apartment in Indianapolis, which they hated, and moved into the upper story of Used Car Heaven. It was a bitter winter and in some ways the most uncomfortable year of my life, not from the cold only.\n\nThere were three rooms on the ground floor, two on the second, and one stove in the building. It was in my room. The \"rooms\" downstairs were partitioned off by curtains, not walls, and there were vents in the ceiling above the stove, so that it did heat the entire place, to some extent. My stove was a black iron cylinder, rather like a small oil drum with a door. I kept it going as well as I could, but often with my best banking it went out during the night. Once, lying awake in the early hours, thinking how cold it would be when I had to leave for class, I decided to get the stove going, then jump back in bed until it was warmer. I usually gazed into the cold cavity with a sigh, but this time, naked, with one hand on the door handle I reached away with the other for some newspaper for kindling. That may have saved me from a nasty burn, at least, because some terrible explosive mixture was waiting for a breath of air and the moment the latch was raised it let loose, sending a blaze half way across the room. The door flew open with such force that my wrist was sprained, the stovepipe blew off at the flue, and the house filled with smoke and soot.\n\nOne pleasant memory of that stove remains. Julian's blue alley-cat, whom I had named Wozzeck, drank from a bowl behind the stove, and every once in a while practiced what seemed a strange experiment\u2014dipping his paw in the water and then shaking it, so that drops of water, hitting the hot stove, would pop and hiss as they burst into steam. He watched this process with wide green eyes for some time after there was no more reaction, and then, as if in doubt, needing further proof, would try it again. Sometimes this went on for half an hour. I could see it from my bed, where he slept also.\n\nThe house was small but around it was space for cars. The cars we put on sale were perfect for Rantoul airmen: they cost remarkably little and with luck would last the few months of training. We got them, mostly, from auctions around Decatur, where the auctioneers took delight in crying, as they brought the hammer down, \"Gone\u2014to Heaven.\"\n\nHeaven bought its share, certainly\u2014and sold it. We always had eight or ten items on hand for less than a hundred dollars\u2014something an airman might just be able to come up with, borrowing a little from several buddies. I saw it as a mighty stream of dishonored and derelict autos, elderly some of them, some of them maimed, passing into temporary hands. It was hard to close down for the night, especially weekends, the clientele was so continuous.\n\nNot that we were making any fortune. The margin of profit was actually rather small and Julian's principle was to take what was offered rather than lose a customer. And there were too many of us living off the proceeds.\n\nAnd a great deal went into official pockets, often unofficially. Every change of ownership (we were extremely careful about titles) requires an act of the state\u2014the state of Illinois. By paying a little extra, we engaged the services of a specialist, someone who knew the right functionary in Springfield and could insure quick and accurate recording. The state also requires that dealers be licensed, and inspected. There was never a doubt in our minds that the inspector would expect a bribe, but such transactions have always stumped me (how does one make an offering without seeming to do so?) We left Archie\u2014the inspector\u2014to Charles, who had a bottle of something he pulled out only for inspections, and who was always giving presents of an unpredictable sort. I came from school one day, already depressed, to find he had given away my typewriter, Archie having admired it. Charles was apologetic but, after a moment of anger, I felt it appropriate to lose, along with my pride of intellectual achievement, this portable symbol of student or scholar. And while I was not, at the time, convinced that Charles knew what he was doing, when trouble came, it was not from Archie.\n\nAs my second (and last) semester at the university began, the coldest point of the year, my mother appeared.\n\n\"Praise the Lord,\" she said. I could think of nothing less appropriate. She had come from Virginia, by rail (she would never fly), deserting her undead patient, to show up\u2014not having written ahead\u2014on the snowy doorstep of Used Car Heaven. But it was, as she put it, as a new creature. At a service of the Assembly of God in Roanoke, she had been baptized by the Holy Ghost and had spoken in tongues.\n\nXI\n\nSnow was bad for business and January's post-Christmas slump had come, like the year's morning-after. In February, frosted windshields kept potential customers from seeing Charles, tipping his top hat in a gesture somewhat less exuberant than before\u2014which made it even stranger. Julian went from car to car, starting them in turn, to be sure that no sale was lost simply because of a cold motor. Clyde was working for us, sitting usually in some model that had a heater that worked. My mother also\u2014bundled in a strange assortment of bright fabrics (as she got older, her fear of looking drab increased)\u2014circulated among the cars, telling Charles to put something over his ears, giving anybody who would listen her verdict on each automobile (based, I soon realized, entirely on color) and occasionally getting an argument going over some matter of doctrine.\n\nThere were more or less regular visitors, too. Archie started dropping by from time to time, between inspections, for a pull of Old Something (Charles took him into the office, so that mother wouldn't see the bottle). There were all sorts of cranks and a few down-and-outers. I recall one (a sort of crank and a bit down-and-out) who had a thing about Pontiacs.\n\n\"You can't hurt a Pontiac,\" he would say, as though they were objects of family pride, and if we had a Pontiac on the lot that showed\u2014as it was almost sure to do\u2014that at least one among them could be reduced to junk, he went into a minor tantrum, swearing he would find out who had done the damage and mete out punishment. He himself had no car, since his license had been revoked after a series of drunken-driving convictions.\n\nThere was also Judge Jerimy. Charles was proud of so distinguished a guest and was always handing me the camera to snap a picture: Charles and the Judge on the doorstep of the office, the Judge and Charles in front of an unusually presentable car, Charles handing the Judge a piece of paper (I have the photo, but no longer remember what the paper represented). Jerimy had been a lawyer in Urbana for about thirty years and, the last two or three of those years, some kind of judge. He was perhaps sixty, but at least five years retired. Dignified in appearance, he was quite unpretentious and his simple propriety, even in the photos, increases, by contrast, Charles's air of comedian or madman.\n\nHe never talked of his legal past. In fact, one topic alone interested him and prompted all his conversation: the possibility of contact with the dead. It was not an interest he had picked up on retiring and, as far as I could figure, had no origin in the passing of a wife or other loved one. All his life, simply, he had read anything he could find on the subject, had gone to great lengths, and considerable expense, to check into supposed hauntings and otherworldly messages, not necessarily in the area. He talked of them fluently. He was a treasury of anecdotes and instances. He wanted, obviously, to be convinced\u2014perhaps of the reality of the phenomena, but rather, I think, just to be convinced one way or the other. He had it all\u2014the s\u00e9ances over the harness shop, the rap-pings, the stories of premonitions and knowledge at a distance, the bed that moved across the room at regular intervals\u2014and still it all left him imperfectly satisfied.\n\nMy mother at first shied as clear of him as she would have of a s\u00e9ance, but finally his honesty disarmed even her. She tried, then, for hours on end, to convince him of the risk involved, even in researching such material. Mediums are souls sold to Satan, in his power utterly, but put themselves in his hands, to start with, through mere curiosity. His thralls now, they become the instruments of devil-power and while all they do is false, in the sense that none of the dead ever really speak through them, it is all too genuinely supernatural\u2014that is to say, diabolical.\n\n\"But what about Samuel,\" he would say, gently, \"and the Witch of Endor?\" And it turned out that both of them had thought long and deeply about that scriptural incident and could debate it endlessly.\n\nRecently, in a dream, I sat naked on the edge of a bathtub, so that my feet were in the water, which came up to my shins. More than that was out of the question, for Julian and my mother were already in the tub, one sitting at each end, and there was no room for anyone else. I took for granted\u2014as one does, dreaming\u2014their nakedness and mine, and was only a little put out at being cheated of my bath. Still, when Julian asked if I wanted in, I was polite enough to say, withdrawing my feet and reaching for a towel, that no, I could wait until later, I was in no hurry.\n\nThere was no particular tone to my dream. Nothing, as long as I was dreaming, struck me as peculiar, and it was only awake, reaching for a cover that had slipped down, that I remembered both of them are dead.\n\nWhen my father died, my mother\u2014divorced from him for ten years\u2014felt, unaccountably, widowed. That was probably one of the factors in her leaving Virginia and rejoining us, all the family she had left.\n\nHer first husband, Charles, had moved to Atlanta. She thought, and I suspect she was right, that he went there to be near her, though as far as I know neither ever wrote so much as a postcard to the other. Bessie, his wife, did drop a line from time to time, but whatever communications there were proved insufficient: Charles and Bessie arrived in Atlanta, having bought a small house in Buckhead, two weeks after my mother had moved to Virginia.\n\nElaine must have kept them, at intervals, posted on addresses, because in March\u2014as business in Heaven showed the first signs of recovery\u2014the younger Charles lifted the phone, to hear his father weeping. Bessie, he managed to gather through a static of sobs, was dead. His father wanted Mother to come down and give comfort.\n\nWhat happened then at first surprised me. Charles (Jr.), Elaine, and Julian\u2014all three\u2014drove, in our best clunker, all the way to Atlanta. It was the worst possible moment (I speak as businessman) for the lot to be left in my hands, those of Clyde, and of course my mother's (she had not the slightest intention of going down). But a few hours after the call, there we were.\n\nThe Judge came by and with, as it were, a fresh case at hand, speculated on the path the soul takes after death. It hovers, he thought likely, for some time, near its erstwhile home (the body, that is, but the Judge had developed a vocabulary), then gradually departs on its quest for spiritual improvement.\n\n\"Improvement?\" my mother said, rather sharply, as she was liable to snap at anything suggesting evolution. Ordinarily the Judge tempered his rhetoric to my mother's level of tolerance, but now he was off.\n\n\"It must rise through various realms. . . .\"\n\n\"Realms?\"\n\n\". . . becoming by stages more and more perfect, more complete.\" I expected an explosion then, but luckily my mother had drifted into her own thoughts about Bessie.\n\n\"She was a Baptist.\" There was something like a gunshot then, but it was only that Clyde had started a car he was trying to sell. \"They think,\" she went on, though the Judge's train of thought had been broken by the report, \"baptism by water gets them to heaven, when what they need. . . .\"\n\nThe Judge was nodding politely, not in agreement, but to let her know he was listening.\n\n\". . . is baptism by the Spirit. They need to be filled with the Holy Ghost. \"\n\nBut Bessie's memory had other aspects. Clyde had sold only one car when, two weeks later, the clunker rattled back onto the lot and Charles and Julian went back to selling.\n\nI cornered Elaine, though I should have been working on a paper for Crappy, and heard how the widower, appalled at the sudden appearance of his children, had let them in\u2014reluctantly. Elaine had taken over the cooking for those days and had also thought to answer the phone, which rang continually without any response from Charles senior. She had supposed it must be calls of condolence, but the first few callers hung up as soon as she said hello and when she identified herself to one who did not hang up, the voice said, \"Ask your father how Bessie died,\" and rang off. After a few more messages of the same sort, she stopped answering and the phone went sometimes for forty rings before there was a pause, and then rang again.\n\nMy brothers, meanwhile, had noted that Bessie's room had been ransacked and that some floorboards had been lifted or split. They decided\u2014indeed, the more I thought about it, the more obvious it became that they must have gone down with something of the sort in mind\u2014that Bessie had stashed away a wad of money, or bank books, or something of value, and that her surviving spouse was trying to find it. Nobody said anything, of course, but a desperate game ensued in which, for days, each party tried to burrow into possible hiding places, while keeping track of the other. At night, mattresses were quietly violated, walls were tested and, without proper tools, the house was, gradually, dismantled. No one left the premises, except for Bessie's funeral\u2014where they all went together\u2014and day and night the phone was ringing.\n\nThey had to give up eventually and never found out if their father discovered anything. They borrowed money from him to get back home\u2014by that time, he probably thought any price a bargain. When the whole story came out and they described the shambles they had left\u2014their father, with a forced smile, shelling out on the very doorstep, faces at all the neighbors' windows\u2014it was one of the few times, and certainly the last, that I beheld my mother gasping, doubled up with laughter.\n\nBut the rent was due, on what was not only our home, but our livelihood as well. My little check from the Veterans Administration was quite inadequate and something had to be done.\n\nWe had begun, hesitantly, to take some of our customers (those interested in more expensive items) to a finance company in Champaign run by a man named Franly. Franly's racket was to advance money at a fierce rate of interest and to include in his package, willy-nilly, quite comprehensive insurance. According to Charles, who was fascinated by such operations, Franly sent in to the insurance companies only a fraction of the premiums he collected\u2014but in case of a claim that looked legitimate would quickly back-date a payment, which\u2014since he was their agent\u2014the company could not refuse to recognize. He also, needless to say, had his staff of repossessors, some hoods who had been run out of Chicago.\n\nA ritual had in fact been established. Usually Charles would go with the prospective customer, who might want, say, a two hundred dollar car. It would be explained on the way that Franly would not finance more than two-thirds of the price, but Charles would magnanimously agree to make it a three hundred dollar car\u2014which, he noted, it really was\u2014so that then Franly's two hundred would cover it completely. The fictitious third hundred created a bond between seller and purchaser, and added for Charles a touch of crime. If it were a case where more than a couple hundred was in question, Julian went along as well. (Franly was more impressed by Julian than by Charles\u2014Julian wore, at that time, suits with a silky sheen, flashy, which never lasted long because he had grown enormous and invariably split the seat in a matter of days.) Franly had never met me, and was\u2014we decided\u2014probably ignorant of my existence. Charles suddenly rubbed his hands together and smiled in such a way that I thought for a moment he had gone over the line. He had just figured out how to pay the rent.\n\nWe went to Champaign, all three of us, and I, in my best, which was none too good, was presented to Franly as a buyer. Taking him aside, Charles gave him some sort of story: student son of rich parents, sure bet, made fifty percent down payment in cash, etc., whatever. Franly was hesitant, not because of my character, I think, but because I was asking for five hundred dollars, more than he was used to putting up. A thousand dollar Buick, from Used Car Heaven? The title was in order. They took him to the window and pointed out\u2014at what Buick, I couldn't see.\n\nFinally, it was all made out: the lien, the insurance, the check. I signed the papers, fortunate that my name gave no hint of whose brother I was. I did, I must say, feel a slight queasiness, going out past a couple of Franly's hoods, who gave me the once-over, just in case they might have to deal with me later.\n\nWe had, in most ways, too many rather than too few hands on the lot, but Julian sometimes hired kids to polish the cars. He did it because they came around in any case, and this way, if they helped little, they did no mischief. He paid them almost nothing.\n\nOne early afternoon, two of these half-pint helpers zoomed into the office and out the back, dropping their chamois cloths as they ran. Before we could even be puzzled, a truant officer appeared at the door. I had heard all my life of truant officers, but always thought of them as a sort of fairy-tale character\u2014something from the world of the Katzenjammer Kids, not a world I lived in. This was an actual man, in plain clothes, flashing his badge, angry.\n\nJulian sold him an old DeSoto.\n\nMy mother went to a Pentecostal church in Urbana, the Church of God. She liked the pastor and his family, as well as several others in the congregation. Services were lively enough, even for her, and only one thing disappointed her in her first few years there.\n\nJulian usually drove her to church, or I did, and sometimes whichever of us it was would stay for the service. The pastor's wife played the piano, but was delighted to find someone who could alternate with her and for a while they had an arrangement whereby the first to get to the church would play for that particular service. My mother had not taught for some time and there was no piano at the car lot, so she was eager to play and started getting to the church as much as an hour early, practicing until time for service.\n\nI think the pastor's wife was a bit put out to find herself replaced almost entirely, but that was not the problem. The hymns sung here were quite different from those of the Free Methodists or Wesleyans. Songs like \"There's going to be a meeting in the air\" or \"Rocka my soul in the bosom of Abraham\" smacked distinctly of Nashville, worlds away from P.P. Bliss, let alone the classical tunes, marked metrically, to which I had learned the words of Charles Wesley or Isaac Watts. And the songs here were, distinctly, faster.\n\nBut as she got into the swing of it, my mother found that these were what, all along, conservatory notwithstanding, her soul required, and some members of the church complained, discreetly at first, that they could not keep up with her tempos. She did, I remember, rip right along.\n\nAnd one Sunday morning, when the pastor had called for an exceptionally slow song, \"Peace in the Valley,\" she suddenly, as if inspired, at the start of the second stanza, closed her eyes and launched into a set of variations that she must have been preparing for weeks. The entire congregation, taken aback, simply stopped singing while, oblivious, she luxuriated in astonishing passage work:\n\nShe was never allowed to play again. The pastor's wife made it a point to come hours ahead and, if my mother appeared, dash to the keyboard.\n\nIn the same church, some twenty years later, it was the pastor's wife who played, andante, for my mother's funeral.\n\nTelevision disappointed my mother. For years, we had gotten, on the radio, the Old Fashioned Revival Hour from Los Angeles. Neither the passion of Reverend Charles G. Fuller nor the harmonies of the famous male quartet could now have satisfied her, but nothing on TV had yet taken their place.\n\nWhat fed her now, between services at the Church of God and arguments with the Judge, was a steady stream of magazines and booklets from evangelists and healers, most of them from Dallas and environs. When, after her death, I opened the enormous trunk, which she would never so much as unlock if any of us was around\u2014her life's last treasure chest\u2014I found a dozen pillows with shiny souvenir pillowcases, between fifty and a hundred 45-rpm records of gospel singers along with a broken record player, an antiquated electric egg beater in perfect condition, stacks of Miracle Magazine, The Voice of Healing, the Full Gospel Business Men's Voice, and\u2014stashed among family photographs\u2014what it seemed must be the complete works, in innumerable thirty-two-page pamphlets, of Evangelist W.V. Grant.\n\nHere I find passages marked with a ball-point:\n\nOne night the Holy Spirit prayed through me for about ten hours. This caused a great number of people to be saved. Another night I awoke and found the Holy Spirit praying through me. Then God revealed to me what it was. I was tuned in advance on a future service. I saw that certain people would be healed in that service. It came to pass that the crippled walked and many signs and wonders were wrought. It was not me at all. It was the Holy Spirit that brought this to pass, because I had searched my heart and saw all the connections were all right in the cool of the day before I retired.\n\nSome of the booklets look unopened: Nuggets in a Nutshell, From Plow Boy to Preacher Boy; but others have been read almost to pieces: Freedom From Evil Spirits, Discerning of Spirits, Receive Ye the Holy Ghost Instantly! or The Two Witnesses, Are They in the World? This last is in the form of an apocalyptic narrative spoken, after the Rapture, by one of the Foolish Virgins. In Men From the Moon in America, Brother Grant reveals that the flying saucers are manned by demons, stationed on the moon. Then there is a booklet in which he argues that Luke, after entering \"the full time ministry,\" no longer practiced medicine.\n\nAnd an advertisement my mother had apparently not gotten around to following up:\n\nSHEET MUSIC\n\nBrother Grant heard supernatural beings singing a song one night, AMERICA HAS FORGOTTEN GOD. He awoke and wrote the song down. Now, he has it in sheet music. It sells for 90 cents. If you order it this month and request it, he will send you a beautiful bottle of pure olive oil from an old olive tree in Jerusalem. You can use this anointing oil for years to come.\n\nXII\n\nMy mother, by spring, had advertised as a \"practical nurse\" and found a job taking care of a well-to-do, senile farmer between Urbana and Danville. He was not an invalid. TV provided all the entertainment he needed (he sat staring with the same stare into a scene of dancing girls or Gunsmoke), but someone had to cook for him, see that he took his medicines, and tell him when it was time for bed. And every few nights, in the middle of the night, he would rise and dress, splash his face with cold water, and go out to make sure the cattle were in, the barn closed, to see that the soil was properly plowed, the seed sown or the grain\u2014depending on the season\u2014brought to harvest. It was all memory\u2014all but the seasons, which he never mistook\u2014and my mother, hearing the door, would go out after him, coax him back to bed. Only once he rose with the notion that someone was stealing from the barn, armed himself with a shotgun from Lord knows where, and leveled it at Mother when she came out after him. She took the gun, unaware that it was loaded, and sent him back to bed. He had no barn now, his farm having shrunk to the house and a little lawn around it.\n\nClyde, about the same time, thought of striking out on his own, opening a rival used-car lot. His preaching days were over\u2014the last straw had been Indianapolis. He and Elaine moved to a house in Champaign with enough space in front to put a few cars, but nothing came of it, and Elaine started taking care of babies for parents who worked. The house became a nursery and she spent her days in a flurry of diapers, bottles, and infant wails.\n\nCharles had fallen in love. This was so unexpected that we none of us at first believed it. He had, in fact, been married: he got married before finishing high school\u2014or, rather, instead of finishing. I was young enough that nobody would tell me why the marriage was annulled. I do remember, though, that it was then (just before he joined the navy) that he turned to books. Some people start reading\u2014reading seriously\u2014when they have religious doubts, and from that day on, they reject what they read as heresy or take it as gospel and swallow it whole. Others flee into books from the doldrums; henceforward they open to any page, plunging into adventure. No matter what volumes Charles has collected, I think he reads it all like a marriage manual. In every sentence lurks an occult technique, which the initiate can translate into sexual power. Now, fifteen years after a marital fiasco, he began to court Seely.\n\nCharles and Julian, with the first meager earnings from Used Car Heaven, had visited the whorehouses of Danville and were thereafter\u2014no matter how business might fluctuate\u2014regulars. When Charles's attendance started slacking off, Julian realized Seely's importance (we rarely saw her at the car lot) and felt her, I think, as a threat. He started to tease Charles about her, bringing up\u2014whenever Charles was about to leave\u2014wedding bells, happy homes. He would burst out with\n\nWe'll build a little nest,\n\nSomewhere out in the west. . . .\n\nand, worst of all, take up the theme, which he put through endless variations, of Charles\u2014aging Charles\u2014determined before he passed on to have children of his own, to hear \"the pitty-pitty-patter of litty-bitty feet.\" Charles would affect a grin, gritting his teeth, and stalk off.\n\nBy the end of spring, the car lot was, in its small way, established, and its population had dwindled back to three. Things might have gone well then, had the FBI not, just at this moment, located its culprit. Julian was arrested, at gunpoint, though he made no fuss, and taken to the stockade at the Rantoul air base, where he shared a space with several of his previous customers.\n\nHe took it all\u2014the stockade, the trial, Leavenworth\u2014much better than any of us would have expected. It came, he said later, almost as a relief from a year of life as a fugitive. He needed worry no more at any strange car perhaps on his trail, any kempt character pretending to be interested in old clunkers, preparing all the time to whip out a badge and an automatic. It had happened now, and was over. After six months of prison (he was sentenced to a year, but was released early) he told me that he felt he had gotten a bargain\u2014he would have had another year and a half in the air force. The hardest moment to get through, he said, was at the end when, being told solemnly that he would never again be allowed to serve in the armed forces of the United States, he knew he absolutely must not laugh.\n\nCharles, however, was broken by Julian's arrest and imprisonment. He ran around at first, frantically, seeing Mother, Elaine, a lawyer that the Judge had recommended, trying to see Julian more often than was permitted. Then he stopped running, stopped almost everything, lay in bed more or less twenty-four hours a day, asleep or somnolent. I got him to eat a bite occasionally, but hardly. For most of the time, he was barely asleep, but if I suggested he get up, his murmur, \"What would I do if I got up ?\"\u2014after which his eyes slowly reclosed\u2014was hard to answer.\n\nNeedless to say, Heaven could not survive. In one of his rare half-hours out of bed, Charles, half asleep, sold the last cars on the lot, all of them together, for enough to pay the rent\u2014the overdue part of it. All, that is, but one: an old wreck of a Buick, rusting away behind the building, out of sight. The title to this piece of junk was not clear: it was in my name, but Franly had a lien on it. It struck me as a car Franly would not be happy to repossess.\n\nThat summer, I tried to take stock: Clyde had dropped out of the ministry; Julian was in prison; Charles was in love; Mother was speaking in tongues. But the family (I think only I realized it) was held together by a strange centripetal force. Seely did not heave Charles out of this maelstrom, but was herself, for a time, drawn in.\n\nI was the exception (so I told myself), moving against the pull, farther away at each turn. It is only now, searching through memories, circling among the figures who fail to appear\u2014a tenuous history shows through them like sunlight\u2014that I find my getaway less than complete.\n\nI decided to go to France, which was as far as I thought it likely I could get. The university had, with some justification, thrown me out. Heaven had closed its doors. I was leery, for a time, of showing my face in Champaign, for fear of Franly's hoods, but Franly turned out to be in as much trouble as I. The state auditor of Illinois had just been found wanting, in the amount of several million dollars. He had subverted state funds, little by little, into the hands of a number of associates\u2014one of whom, it transpired, was Franly, whose business would never survive an investigation. It was the news of this that broke the long torpor into which Charles had fallen. I read him accounts from the newspaper and Archie came by with gossip from the State House. Charles's eyes opened\u2014narrowly, it's true, but with an attention they had not shown for weeks.\n\nI went away. No one trailed me to Aix-en-Provence. There followed years in which I managed to remain sufficiently aloof. I saw them once or twice a year. In the summer of 1957, when I returned from Aix, Julian was out, and had started a used car lot, with Charles, in Danville. And that summer I was in Danville for the marriage of Charles and Seely\u2014an intimate affair, at the office of a justice of the peace. Julian was there, and Elaine, and Mother, and the Judge, but no friend or relative of the bride. For the wedding car, Julian had tied tin cans to the back bumper of an old hearse he happened somehow to have on the lot. He had also, in white paint, smeared JUST MARRIED across it.\n\nMother, though she had worked to get Elaine married, and though she had often tried to get Julian together with some proper church girl\u2014in her heart she would far rather her sons remained single. (I remember once, when Elaine remarked on the beauty of some ten-year-old boy in church, \"Yes,\" my mother said, glancing, closing her eyes then, \"but he'll just grow up and marry some old woman.\") Of Seely, she was at first contemptuous, then\u2014after the marriage\u2014horrified, when somebody let slip that Seely's grandmother had been a witch (that is to say, some sort of fortune-teller and a holder of s\u00e9ances).\n\nNot long after, I got married also. Rosmarie, however, has little place in this account, since that is another story and this one is complicated enough as it is. My mother gave us, as a wedding present, what she probably at that moment loved more than anything else: a large black tomcat. Every day, at the farmhouse, she told herself that tomorrow she would quit her job, move back into town, or elsewhere, nearer to some church. Regarding her stay there as provisional, she feared for her cat and had given it to a neighboring farmer, who promised it a home in his barn. A few days later, however, he brought it back in a gunny sack, saying he would not keep on his farm any cat that sucked eggs. We named him Moby Dick, because of his shape, and he lived with us for years in Michigan and in Connecticut, until\u2014as cats do\u2014he hid from us to die.\nThe Call Asserts Nothing\n\nXIII\n\nI remember, for some reason, a film I once saw, in which sequences resembling old, contrasty photographs faded, not into darkness like the usual fade, but into a bright white empty screen, so that the story seemed sketched in elaborate shadows against a field of perpetual light\u2014shining now through the pictures, illuminating them, and now supplanting them, shining on its own.\n\n\"You must have those pictures somewhere,\" Charles said to me. \"If you don't, then they're just gone.\" And after a moment, \"For good,\" he added.\n\n\"I'll look for them,\" I said. With Seely gone, he had been rummaging for the photos of the two of them in smart clothes, standing beside the long black car, with JUST MARRIED in the center for the frame. Or\u2014another I remember\u2014the back end of the hearse as it drove off, trailing a cloud of smoke and the string of tin cans.\n\n\"I'm just glad we can all be together this Christmas,\"\n\nElaine said. I should have kept my mouth shut then. Or I could, after all, have said a lot of things that wouldn't have mattered. But I had come to trust the banality of my first thoughts, the platitudes that spring from the precon-scious mind.\n\n\"I'll ask the Ouija board,\" I said.\n\n\"The what?\" said Charles. Julian stopped dead in the middle of a mouthful of cake.\n\n\"Every Christmas,\" Elaine was telling Rosmarie, who was by now not quite a stranger to my family, \"I'm just amazed we can all still get together.\"\n\nI had played with a Ouija board as a child\u2014a friend and I used to get messages by the hour. It kept us, as they say, out of trouble and amused after we tired of poker (one or the other of us having won all the countries on the map). Then, in Connecticut, as if there had been no years between, another friend said, \"Ever talk to a Ouija board?\" and I said, \"Sure.\" We had to persuade our friend's husband to get the board out of the attic. \"He believes in it,\" she whispered when he finally went up after it. \"Last time we used it, we got his dead mother.\" \"What did his dead mother have to say?\" \"She said she was burning in hell.\"\n\n\"I mean about the pictures,\" I said. Charles and Julian were still staring. \"Haven't you ever played a Ouija board?\" The room was hot and piled with trash left by Charles's three children. They had opened their presents two days before Christmas and now\u2014Christmas Eve\u2014cars, guns, cameras lay at random with smashed parts. I knew Charles had no money and I didn't know just how he had managed to get that great load of toys, but I was not surprised.\n\nJulian got up and said he was going out to get a Ouija board.\n\n\"Ten o'clock Christmas Eve,\" I said. \"Who's going to be open?\"\n\nElaine said, \"Ten o'clock. No wonder I'm so tired. You people can stay up all night if you want to. I'm going home.\"\n\nShe went home. The rest of us might well have stayed up most of the night, but the kids came back from a party somewhere and the uproar was more than any of us could stand. The last thing I remember, before the tree toppled, is Charles's lecturing Julian on the folly of Spiritualism. Charles has always had plenty of words, if not much of a world to put them in.\n\n\"Those Ouija boards,\" someone once told me, \"it's all wish-fulfillment. It's the same mechanism as dreaming.\" No doubt there's something to that. The planchette moves where the fingers, lightly pressing, desire it to go. The message is our libido, spelled out. Not a complete theory, but it's a start. \"No one can have, in real life, everything he wants\"\u2014I forget who it was who spouted all this\u2014\"but there's nothing to keep us from having anything we want, in imagination.\" That would be reasonable, if we knew what we wanted.\n\nCharles called when I had all but forgotten that Christmas Eve. It must have been March or April when he phoned, which he doesn't ordinarily do unless he needs money urgently.\n\n\"Brother Keith.\" He sounded urgent. And he proceeded to tell me how afraid he was that Julian was being \"taken over\" by the Ouija board. I didn't just then go the thousand miles back there, but little by little I found out what had happened. Charles sent transcripts\u2014sheaves of them\u2014of the sessions, hundreds of answers spelled out letter by letter. Sometimes the questions were missing, and it was impossible to make a clear chronology of the sheets, but the gist was obvious. It was particularly startling, because I rarely heard from my family (except, occasionally, my mother, who if she had any real news always managed to put it in an unexplained subordinate clause: \"after that awful accident happened . . .\") and I had somehow put out of mind their obsessions. Startling because, of course, they're also mine.\n\nCharles, in the meantime, had gotten Seely back. I can't keep straight their partings and their reconciliations. Seely would go off to a bar and it was weeks until anybody could find her. Charles all this while mad with desire\u2014and in the grand manner: refusing to eat, calling up her mother, her sister, her friends, anyone who might have seen her, unable to think of anything else. He would know she was with Drake, whom my mother referred to only as Seely's \"drinking companion.\" And Charles missed work\u2014assuming at the moment he was working\u2014and when Seely got back would be out of money, which would send her off again. They got divorced\u2014over the money problem, they said. With three boys and no husband, Seely started drawing welfare payments. Charles stood at the door, more or less literally, and finally crept back into the old situation. Things went better for a while, with the added income of the welfare checks. Once, Seely got Drake to beat Charles up.\n\nMy mother thought Seely controlled Charles by some kind of spell, but it seemed to me that if he let go his idea of Seely, he'd have nothing left. He tried hard to apply to her what he'd read in books. \"Don't you think,\" he would ask me, \"her pictures show schizoid tendencies?\" Seely painted, not too skillfully, images of a surrealist cast\u2014brick walls around trees filled with eyes, a woman's nipple stretched until it formed a clenched or clutching fist.\n\n\"I don't know,\" I say. She had a recurring dream. She is in a space capsule. She is swaddled in some kind of clothing, like swathes of cotton, so that she can't feel anything. \"It's like I didn't have any more body.\" It's always the moment before takeoff. Then a baby's face, indistinct but combining, I gathered, in some way, a sort of greeting card cherub and the real rage of an infant just born. A voice that reminds her of her grandmother's says, \"Go on, and have a good time, but if you go, don't think you can ever come back.\" And she wakes in a cold terror.\n\n\"What do you make of that?\" Charles asked.\n\n\"I don't know.\" I've sometimes envied people whose dreams are bizarre.\n\n\"You've studied psychology,\" he said. Then, his reproach fading into the old perplexity, almost to himself, \"It must mean something.\"\n\nJulian found a Ouija board, disgusted that it was in the toy department\u2014this was before New Years. It worked with Charles and Seely, Charles scoffing as his fingertips moved with the moving planchette. They tried to get Judge Jerimy to sit in, but he would have nothing to do with it. It worked with Clyde and Elaine, though they were both afraid of it. But it worked best when Julian was on one side or the other. It got so that every night the board was talking to them. While the Christmas tree turned into a pile of needles, they tested their discovery.\n\n\"Where are my pliers?\" Clyde asked, and the answer spelled itself out: \"B-E-H-I-N-D-T-V,\" which is where they were. It never seemed to miss on little things like that. But the more important test had nothing to do with efficiency.\n\n\"Are you evil?\" they asked point-blank.\n\n\"I-H-A-T-E-E-V-I-L,\" the board replied, to their relief. For a while (some weeks, I guess) it gave Julian detailed instructions where to go to get bargains of one sort or another and (they were all a bit in the TV business then) where to get cheap used sets. There was never a hitch. Their business improved. Then one night the tone of the affair changed.\n\n\"Are you there?\" Julian asked. The planchette went to YES. \"Who are you?\" I don't know where he got the idea to ask such a question. The answer came.\n\n\"I-A-M-T-H-E-G-E-N-I-E-O-F-O-L-D.\" Elaine almost fainted.\n\n\"An evil spirit!\" she said, but the Genie of Old assured them that he hated evil.\n\n\"Ask him why he's come,\" Charles said.\n\n\"Why have you come?\" Julian asked, and they could hardly keep up with the letters.\n\n\"I-H-A-V-E-C-O-M-E-T-O-B-R-I-N-G-Y-O-U-\n\nS-E-C-R-E-T-O-F-T-H-E-A-G-E-S.\"\n\nElaine said, \"I'm getting out of here.\"\n\n\"Wait,\" Julian said, \"he's going on.\"\n\n\"I-W-A-N-T-C-I-R-C-L-E-O-F-F-I-V-E.\"\n\n\"Who? What circle?\"\n\n\"C-S-J-E-C.\" That stumped them for a while, until they saw it was an initial for everybody there: Charles, Seely, Julian, Elaine, Clyde.\n\n\"You can't go,\" Julian told Elaine.\n\n\"Why does he want us?\" she said. \"Why doesn't he want Keith? Ask him why he doesn't want Keith.\"\n\n\"Why,\" said Julian, as mouthpiece for the group, \"do you not want Keith also?\"\n\n\"H-E-A-L-R-E-A-D-Y-K-N-O-W-S-S-E-C-R-E-T-O-F-T-H-E-A-G-E-S.\" And they didn't have time for another question; the planchette was still on the move. \"B-U-T-H-E-D-O-E-S-N-O-T-T-E-L-L-A-L-L.\" That oracle could, theoretically, have said anything, but nothing could have given it greater plausibility to that particular audience. I must have appeared to them, at that point, in an aura both of success and betrayal: I escaped, I deserted\u2014either formulation suggests the other. Julian was going on.\n\n\"What about Rosmarie?\" And this should be noted: the Genie of Old never answered any question about Rosmarie. She was, as it were, outside his jurisdiction. But over the next sessions\u2014they were nightly and sometimes nightlong\u2014he provided my family with a history, the last thing I thought we would ever have.\n\nIt started in Austria, the age not quite specific, with one Axis von Mueller. No details of his life\u2014the name suffices. It provides, through otherwise vacant centuries, a line of identity, and it brings out of darkness the long dead postmaster of Leeton who was our grandfather and whose name was Martin Van Buren Mohler. One branch of the family, the Genie of Old reported, and they copied it down breathlessly, ended with Mad King Ludwig of Bavaria. The other branch\u2014which led by some path to Leeton, and on to this group of seekers in a ratty room in Champaign\u2014the other branch \"I-F-H-A-D-N-O-T-B-E-E-N-I-L-L-E-G-I-T-I-M-A-C-Y\" would have climaxed in our generation. Charles and Julian would have been (not knowing exactly what they mean, I can only copy them out here) respectively, \"E-M-P-E-R-O-R-O-F-B-O-T-H-T-Fl-E-A-M-E-R-I-C-A-S\" and \"C-H-I-L-D-C-O-N-Q-U-E-R-O-R-O-F-E-U-R-O-P-E. \"\n\nI was much moved by these revelations, but it was not clear at all to me what I could or should do about them. It didn't seem very likely to me that Julian's soul would be absorbed by the Genie of Old. I was ready to bet on Julian.\n\nIn September 1970 I got back to Champaign. Julian had rented an old house on University Avenue where he and my mother lived, and out in front he had set up a fruit and vegetable market. University was, had been for years, the dividing line between white and black districts, the ghetto stretching west from the next parallel street. A black gang, calling themselves the Sons of Satan, decided the west side of University itself should have black businessmen only (at that point there was probably not one) and they began the liberation with volleys of gunfire, repeated sporadically over several months, sometimes from a moving car, sometimes from nearby rooftops. By the time I got there, it was a street\u2014both sides, with a few exceptions\u2014of vacant buildings. Julian's fruit stand, however, on the west side, was still there. My mother, who tended the stand most of the day, explained that the shooting drove away most of the customers, that the summer\u2014the fruit season\u2014being over, they had no prospects for the winter, that things were about ripe for the coming of the King in his glory. Tomatoes were still plentiful. The porch was loaded with watermelons. The front room of the house reeked with vegetable decay, but they lived in the back, mainly in one room, with several cats and many locks. She slept on the floor behind the TV, under some notion that that was safer.\n\n\"Has there been more trouble?\" I asked.\n\n\"No,\" she said, \"not for a couple weeks.\" She was living out of a half dozen shopping bags. Pinned to the wall were several pages clipped from the Voice of Healing, showing faces of native evangelists, mostly Indian, those to whom she was sending offerings. \"Most of our customers,\" she said, half to herself, \"were colored people.\" She could find no easy explanation for all sorts of recent events, especially the sudden death of evangelist A.A. Allen in a San Francisco hotel room of\u2014so the papers claimed\u2014acute alcoholism. \"Maybe he overworked,\" she decided, \"and right at that moment the Devil tempted him to take a drink. Of course,\" she added, before I realized she was back to Julian, \"they didn't like his campaigning.\" And I remembered how Charles and Julian were the local chairmen of the Wallace for President campaign, carried off in some dream of going in triumph to a career in Washington.\n\nThe fruit stand on University Avenue\n\n\"Why don't you leave this place? It's silly, you know, to stay here.\"\n\n\"Oh,\" she said, \"he won't leave. He's got that old gun.\" And indeed, leaning against a chair was an automatic rifle. Through the worst of the raids\u2014the police, by that time, avoided the whole area\u2014he sat on the porch, firing back, while she lay inside, on the floor. Whenever I'm tempted to suppose that things are not completely hopeless, not entirely senseless, the attempt collapses if I bring that scene to mind: her praying behind the TV, and Julian fighting for his right to sell watermelons to the blacks.\n\nWe got together at Elaine's place\u2014neither she nor Clyde would go near the fruit stand after dark. Rosmarie had not come west with me. Seely was off somewhere.\n\n\"Keith,\" Julian said, \"have you seen anything recently on precognition?\"\n\n\"Oh don't talk silly when we're all together,\" my mother said.\n\n\"No,\" I said.\n\n\"You'd better be thinking,\" Mother said to Julian, \"about how you're going to spend eternity. Take that old cigar out of your mouth.\"\n\nCharles said, \"Shit.\"\n\n\"It's not lit,\" Julian said, which was true.\n\n\"He talks awful,\" Elaine told me, referring I suppose to Charles.\n\n\"This old world is going to rock and reel,\" my mother continued. \"And it isn't going to be long now.\"\n\nCharles scowled. He was looking skin and bones. \"You've been saying that for as long as I can remember.\" \"You know it's true!\" she shouted back. \"The Devil's getting stronger. You ought to know as well as anybody.\" That went on a while, but Charles was on the defensive, because he as well as the others had come to believe in the Power\u2014had in fact believed more completely than the others. He had seen, magnified in his mind's eye, a fortune proffered by the Genie of Old. And I may be wrong, but I cannot help feeling that if he was afraid for Julian's sanity, he would not, for all that, have withdrawn from the game for Julian's sake. But it was not, finally, Julian who had succumbed.\n\nJulian, after Leavenworth\n\nIt was Seely. I never could quite get all the details, because everyone kept giving me explanations instead of information. \"Her grandmother was a witch,\" my mother told me. They wouldn't talk about it among themselves; only when I was alone with one of them, it came out, compulsively, not really to me\u2014more like lines of thought trying to unsnarl themselves. \"She used to hear music before anyone died,\" Charles said; \"she had sort-of comas.\" Those states, whatever they were, increased as the group met nightly to hear from the Genie of Old. \"She isn't a very stable person,\" Elaine says. \"She went into a screaming fit,\" Julian told me\u2014we were standing on University, between the tomatoes and the watermelons\u2014\"I had to keep a hold of her and Charles grabbed the Ouija board and tried to break it over a chair.\" \"Well?\" \"Well, he couldn't break it. And she was screaming and kicking. And finally he ran out the back door and put it against a tree and stomped on it till it broke.\" \"Did she stop then?\" \"No. Not until he got the gasoline on it and burnt it up. We were doing real well there for a while. I mean financially. She calmed down and was all right after that.\" I helped him move some overripe tomatoes out of the sun.\n\n\"I don't want to go to heaven,\" Julian said, \"I want to go where Keith goes,\" and laughed loudly and went into the kitchen. My mother pointedly ignored him.\n\n\"He may come any time now,\" she said. Charles said something under his breath.\n\n\"It's just nice we can still all get together,\" Elaine said. \"I always wonder if it won't maybe be the last time.\"\n\nLate that night, when I was the only one awake, I called Rosmarie.\n\n\"How is it?\" she said.\n\nI said, \"I hate evil.\"\n\nXIV\n\nI have always wondered what worlds are possible. Others have asked, of course, but I mean it, not as a logical, but as a practical question. People around me seem always to believe\u2014more fervently the more desperate they are\u2014that there is some means, plain or occult, by which to get whatever is most precious in life. The idea fascinates me, since it suggests a path from what is to what might be, but it requires, I think\u2014in the believer\u2014an image of those might-be's. And though I am often appalled at how contemptible an object some set as end-all and be-all, still I can see how useless for most of us is St. Paul's bland assurance that the great good to come is beyond imagining.\n\nMy imagination is poor. In my dreams, for instance\u2014where one would suppose wishes can be fulfilled without hindrance\u2014if I dream the events this account describes, they are not usually changed, but in what should be a world nearer to the heart's desire, they play again, just as I tell them here, exactly as already experienced. It is as if despairing, even of imaginary improvement, I contrive instead to set my affections on the damned world, this vefy world, as it was and as it is. (Thus I record, subjunctively, my own conversion.) And waking, my ghosts are as before: neither soul nor body, but the lack of obstacle to sunbeams coming in the window or light from a light-bulb or any everyday reflection.\n\nIn Long Beach, California\u2014a city I hardly know, having twice visited Charles there, briefly both times\u2014there is a Beulah Chapel. The name would fit some small fundamentalist church I might have known as a child, and though the old-time religion would not have advertised, the ad that still runs in the paper there sounds old-time: \"COME WORSHIP WITH US\" and under the address: \"A MEETING PLACE FOR THE BODY OF CHRIST.\"\n\nBeulah Chapel belongs to Prophet Shock\u2014it is his denomination, his doctrine, for that matter his pulpit and his pews. But his influence extends beyond the walls of Beulah Chapel (I am not sure how far), since he has ordained other prophets to go forth from Long Beach and preach his gospel. If I'm even now imperfectly initiated, I knew nothing at all of Shock, or of his prophesying, when my mother wrote me that she was leaving Illinois for \"the west\"\u2014I found that vague and it crossed my mind that she might want to revisit Oregon. Charles, I gathered, was going with her (Seely was out of the picture, apparently for good, but I wondered about the three boys) and certainly Julian. But at the end of a sentence, in a phrase that dangled a little, she added, \"and Thomas of course and I guess the Prophetess.\" Thomas I had met\u2014a scrawny, cross-eyed Siamese with three legs and a broken tail.\n\nSome months later Elaine called from Illinois\u2014I had by this time settled in Providence\u2014to ask if I had heard from them. I hadn't, and was a bit disturbed that she hadn't. It only then came home to me that I had not taken the trip seriously, assuming it a vacation of sorts, not a permanent move.\n\n\"Oh no,\" Elaine said, \"they gave up the place on University. The owner was real mad. He came by here to try to find out where they'd gone. Said he was going to sue them.\"\n\n\"Did they owe him rent?\"\n\n\"Probably a few months. But it was all the fruit in the front room. You know how it gets. It had just rotted there.\" I knew, layer on layer. \"Said shovels wouldn't get it out. Had to use picks. I think he was lying. You know what he claimed? He claimed that at the bottom of it all, tomatoes or something was growing out of the dirt between the floorboards.\"\n\n\"Does Judge Jerimy know where they went?\"\n\n\"No. He was real hurt, they didn't even tell him they were going. Of course they don't see him much anymore, because he doesn't like Kate.\"\n\n\"Who?\"\n\n\"Kate. She's a prophetess. And Charles was about out of his mind for a while.\"\n\n\"Because of the Prophetess?\" The farther east I went, it seemed, the more any contact with my family became dreamlike, or like delirium.\n\n\"No, he'd moved with the kids into that funny little cinderblock house a couple blocks from Julian's.\" I didn't remember any cinderblock house. \"You remember it\u2014it was just one room, can't imagine what it was built for. Anyway, they moved in, for about a year or more maybe. Never paid any rent after the first month. Charles said they wouldn't dare evict someone with three children, no place to go.\"\n\n\"What's the Prophetess got to do with it?\"\n\n\"Nothing. They were too scared to evict him. But he and the kids went out to a movie at the mall.\" The mall had been constructed after my Urbana days. \"They were out all afternoon. The owner got a bulldozer and just came and pushed the house down, made the whole lot just level. Charles and the kids came back and the place was gone, just wasn't anything there. He was mad later, but at first he ran around and around the block, and thought he was going crazy. He came here and I thought he was going crazy too. We all got in the car and went back to where it was, sure enough there was nothing there. Then he got real mad and slapped little Charley, because he wouldn't stop laughing.\"\n\nI was profoundly uneasy, there in my farthest east, trying to fathom what pilgrimage might be in process along what western highway. Elaine had a more pressing reminder of their absence, Charles having left the children with her. It occurred to me only after hanging up that I had learned nothing about the Prophetess, except the name Kate\u2014and the fact that the Judge disliked her. I couldn't recall a living soul to whom the Judge had ever shown the slightest antipathy.\n\nThere was nothing I could do, no way to trace them, having only my mother's vague letter to go on\u2014and Elaine's confirmation that they had indeed set out, in a new Cadillac that Julian had mysteriously come up with. It is strange\u2014and I realized it was strange, even at the time\u2014how quickly my unease became permanent, but also retreated from a threatening foreground into a constant but peripheral uncertainty, as for several months, unable to think of anything else to do, I went about my business.\n\nI did call Elaine again and learned that Julian had, not surprisingly, been in bad health. He had never taken care of himself, partly from a superstitious dread of talking, or even thinking, about sickness. To consult a doctor was inviting disaster. Now, passing forty, he had fallen unmistakably ill. Charles, called by Mother and finding Julian unable to get out of bed\u2014although he insisted, almost inaudibly, that nothing was wrong with him\u2014took him to a hospital. He came out in a few days and never, to my knowledge, went near a doctor the rest of his life. But he was aware now that he was a diabetic.\n\nHe met Kate them. I think, unlikely as it seems, that he met her at the Urbana Church of God\u2014Mother's church. She was ten years younger than Julian and had, from a previous marriage, a boy and a girl, the most beautiful children I ever saw. She herself was a wholesome-looking blonde, with a manner suggesting no secrets. My mother took to her before Julian. They discussed doctrine a bit, but more often got off on prophecy and how to interpret prophetic writings. \"She sure knows Revelations,\" my mother was reported saying.\n\nKate had almost immediately, I was told, looked up to Julian, stating from the first that God had some special plan for him. They got married (no one bothered to tell me\u2014surprised later, that I didn't know) and Julian began again, as he had not done since being expelled from Sharon, to search the scriptures.\n\nKate, I should note, at first struck me\u2014in the context of my family\u2014as rather drably normal. I had the feeling that if, without knowing her, I had met her at a party, at a movie (she liked movies), I would not have thought her unusual. Even when she talked Bible, even when what she said was by most standards a little weird, her matter-of-fact tone suggested common sense. She did admit to prophesying, but disarmed my mother's suspicions.\n\n\"In these last days,\" she said, \"we should all be prophets.\"\n\nBesides the Judge (\"He won't stay in the same room with her,\" Elaine told me), there was one other person who disliked Kate on sight. Actually, it was before seeing her. Martin, Elaine's son, who worked sporadically in a garage since he had dropped out of high school, slept in the attic of his parents' house. Charles and Julian were both convinced he was feebleminded (which he was not), since, though elsewhere soft-spoken and polite, at home he maintained a military bearing, answered casual questions by shouting or didn't answer at all. He had thrown everything out of the attic, which extended unpartitioned over the whole house, except for a camp-cot and a foot-locker. He wore war-surplus fatigues, at home and to work, along with army boots, in which he marched\u2014sometimes for hours\u2014from the front of the attic to the back and from side to side, a thunderous overhead parade that Elaine seemed not to notice but which drove Clyde frantic. The night after Julian first brought Kate to meet Elaine, Martin\u2014who had come home after Julian and Kate left\u2014woke the house with screams. Kate, in a nightmare whose terror stayed with him for weeks, had come\u2014naked and glowing\u2014to his cot and had sat on his face.\n\nHe swore later, after seeing Kate with waking eyes, that she was the same woman he had seen in the night and been molested by, that she was a creature of evil, and that she ought to be (these are his words, as repeated to me) \"rubbed out like a rattler.\" (I am often taken aback to find, in the ana of my family, expressions that seem throwbacks to a past that was never ours.) It gradually dawned on my mother that Kate was not precisely Church of God, but on the main points of doctrine she was, for a time, acceptable, being firmly Pentecostal and pre-millenarian, and she was against all the heresies from Eternal Security on down.\n\nThen Prophet Shock flew in. He had ordained Kate, at some point, and I suppose they may have been in touch, but his arrival seemed to have taken everyone, including Kate, by surprise. He arrived in early afternoon, from the Chicago airport, and left again for the airport before supper. He had come all that way to give Julian a prophecy and, having given it, flew back to Long Beach.\n\nI wish I could reproduce the prophecy verbatim\u2014it is extremely important to my account. It was given orally, but Julian wrote it down (perhaps immediately, perhaps later from memory) and carried it in his billfold from that day on. By the time I read it, it was barely legible, the paper tearing at creases from being so tightly folded, so often unfolded for him to show or contemplate, then refolded and sat on. It was only one paragraph of quasi-Biblical language, the general drift certainly one of encouragement in adversity. That was all that was clear to me. To Julian the clarity was blinding: it was the message he had been waiting for, the announcement of prosperity coming soon and willy-nilly, not because he deserved or had asked for it, but because he had been chosen. He was to be the receptacle of unmerited favor\u2014and I think he was appropriately humble. His belief, in any case, matched the promise: from that day forward, he never raised a hand to earn a day's wage, but lived in the gleam of divinely predicted wealth.\n\nWhen Charles's house disappeared, they left Kate's children with their father (he was a lab assistant at the University of Illinois) and Charles's children with Elaine, and set out.\n\nThey came back. I had assumed Elaine would call me when and if, but one day I got a letter from my mother, referring casually to \"that silly old trip,\" and the letter was postmarked Urbana. In June, when I could, I made it to see them, having only Elaine's address. Charles, I found, had set up what he called The Treasury, a dark corridor-like shop, filled with unsaleable bric-a-brac, behind which he and his boys slept. Julian and Kate lived in Danville. My mother now refused to live with anyone, but rented what had obviously been a one-car garage. The house that should have been in front of it had been demolished and at night the lot was jammed with parked cars. Next door was a fast-food drive-in. A toilet and sink had been installed in one corner of the garage, leaving hardly enough room for a narrow bed and her enormous trunk. She sat on the bed, morose, and would not allow me to sit on the trunk.\n\nThey had been many places, but the itinerary was not clear\u2014certainly not to me and perhaps not to her either. To Miracle Valley, for instance, that A.A. Allen had founded in Arizona, a sort of holy-of-holies for my mother at one time. Since Allen's death, his followers continued the work, defending the memory of their lost leader. She would not talk about it.\n\nFinally to Long Beach, which she hated.\n\n\"Didn't you find any place you liked?\" I asked.\n\n\"There was one church we found that was real spiritual.\"\n\n\"Where was that?\"\n\n\"Las Vegas.\" That stopped me for a moment, but she went on, disconsolately, \"Charles got into some kind of trouble there and we had to leave.\"\n\nShe sent me out for food. As in the past, she ate no vegetables, convinced they were unhealthy. Often, if I saw her after an interval, she would begin, \"You're looking bad. Probably been eating those old vegetables.\" She lived now on what, next door, they called \"fishwiches,\" supplemented with half pints of whipping cream\u2014and grapefruit. (I should have mentioned already that a couple years earlier she was operated on for cancer and her bowels had been replaced by an apparatus she wore and had to empty at intervals. For this to function properly, it was necessary that she eat grapefruit.)\n\n\"Couldn't you move in with Elaine?\" I said.\n\n\"She doesn't want me.\" I didn't think that was true, but knew that there was no use arguing.\n\n\"With Julian?\" She didn't answer, but glowered. She had a way, now, of drifting out of range when she didn't want to talk about something. I changed the subject, tried to get back to the trip. Colorado it seemed they came through on the way back. \"Did you go through Emporia, or Leeton?\" No reply. The place got hotter as the afternoon went on. In the winter it would obviously be drafty. I suggested several times that she come to Rhode Island and once got a smile, but never an answer. She did talk a little, after that, in bits and pieces. It was clear that her aging body, with its artificial bowel, disgusted her, but there was something else, something more horrifying to her.\n\nAt one point, \"I should have died then,\" she muttered. I wasn't sure I had heard right, but it came out, in broken phrases. \"Died when I was operated.\" And my stammered reproaches she cut off with a fierce stare and shouted, \"I was all paid up!\" and wept. Then I remembered the pictures she had shown me of native evangelists\u2014Indian, African, South American\u2014whom she had supported (along with Miracle Valley and no doubt other gospel ventures) from her Social Security checks. The magazines from Dallas were full of expiatory bargains: so many dollars would buy a tent, so many a bicycle, such and such an amount would support a native in the field for so many months. These native evangelists were her substitutes, carrying out, in her stead, the great command to preach the gospel to every nation.\n\n\"She thinks,\" my mother said suddenly, \"that Heaven is going to be right here, on earth.\" Apparently, on the long trip, the Prophetess had let out more doctrine. \"She's a fool!\" my mother said (putting herself, it came to me from my childhood, in danger of hell fire). And looking straight at me, \"Do you believe that Heaven'll be right here in Champaign?\"\n\n\"No,\" I said, \"that doesn't sound likely.\"\n\nI never went much into doctrine with Kate, but Elaine did.\n\n\"She says,\" Elaine said, \"that after the Rapture, the tribulation, the millennium, and all that in Revelation, the New Jerusalem will come down here, out of Heaven, and since there's a new heaven and a new earth, we'll have the new earth. I don't really see,\" she went on, \"that it makes much difference whether we go up or stay down\u2014as long as it's new.\" That seemed reasonable to me. \"But you know what else she says\u2014and Julian's got so he goes on and on about it now and I don't think Mama quite knows what to think about it. She really runs down Kate now, and I think Kate's wrong, but she's smart and she's hard to argue with.\"\n\n\"What's this she says?\"\n\n\"Well, you know how it says we'll not all die, but we shall all be changed\u2014well, Kate thinks if we're children of the light and not children of the dark, we don't have to die.\"\n\n\"Don't have to. . . .\" It took more than the few days I was there for it to sink in.\n\nIt was only from Charles that I picked up a few details about the trip.\n\n\"It was Brother Julian's idea,\" he said, moving a badly chipped mirror off a chair so I could sit. \"Watch out for that chair, the front legs are not too strong. You see, he has nothing else to do.\" I didn't get it immediately. \"Well, you know he got that damn prophecy and now he thinks if he does anything\u2014but anything\u2014but sit on his ass, he's being unfaithful to the prophecy, showing he doesn't really believe the Lord. So: nothing. He does nothing.\" I never saw a customer in Charles's establishment. \"I was in a state,\" he went on, that being the only reference to his vanished house, \"so I went along. Mistake, mistake, mistake.\"\n\nFirst of all, Julian and Kate had discussed doctrine all the way from Champaign to Champaign, with Mother interrupting them every other sentence. They had gotten to Long Beach, but by a strange route. One of the first places they went\u2014hardly west\u2014was Hot Springs. When I looked puzzled, he threw up his arms. \"I didn't even know we were going there. You know Julian. Well, he got into his head to go there because our dad had lived there.\" (The elder Charles, by the way, was dead. None of them had made it to his funeral.) \"Mama was furious when she realized where we were. Then he wanted to drive to Missouri and to Kansas\u2014he was on a nostalgia kick of some kind. Wanted Kate to see the old homesteads or\u2014I don't know. Hell, Kate was only interested in getting back to check in with her Prophet in Long Beach. Mama threatened to get out and walk\u2014we were going about seventy at that point\u2014if he headed towards Leeton or Emporia. So I decided I might as well put in my two cents worth and I wanted to go by the Grand Canyon.\"\n\n\"And did you get there?\"\n\n\"Hell no. Mama and Julian cooked it up together so we got to Arizona all right, but somehow instead of the Grand Canyon we landed in Miracle Valley. And that damn cat,\" he said. Mother had put a box on the floor of the back seat and the car smelled worse and worse as they went. \"And that car,\" he groaned.\n\n\"Elaine said it was a new Cadillac.\"\n\n\"It was a Cadillac,\" Charles said, \"and it was almost brand new. It really did look like a million. And it rode real good on the highway, once we got on the highway. But it only had two gears\u2014it had low and it had high, no second. And no reverse. But you know how Julian is.\" Charles was getting into his story. \"Here's what would happen. You know he'll only stop at the best motels, right? And he's driving this Cadillac and wearing his silk suit\u2014and Kate's dressed up nice too\u2014and it's always the same: he goes into this expensive motor court and goes up to the desk and with his fat cigar and he plunks that deformed damn cat on the counter and says, I'd like the suite. The suite, mind you. And you know Mama\u2014Mama wouldn't leave anything behind that wasn't locked in her damn trunk (we'd put the trunk in Elaine's basement, almost broke our backs). And not only will she not leave things behind\u2014all her clothes and stuff\u2014in Champaign. I mean she won't leave it in the damn car. So all the clothes she can get on she's got on\u2014I mean dresses over dresses, sweaters, everything. Middle of the summer, you understand, the South. And what she can't get on she's got in shopping bags. You get the picture?\u2014there's Julian with this three-legged abortion on the counter, it's making that awful Siamese noise, and he's pulling this wad of bills out of his pocket (a big bill around a lot of ones) saying Gimme the suite, and I'm there wearing just what I've got on, because it's all I've got, so I look like a bum, and Mama won't stay in the car, it's too hot, so she's right inside the door wearing a whole damn ragbag, with all these paper bags. Well. . . .\"\n\n\"Where did Julian come up with all that money?\"\n\n\"I have no idea. Actually there wasn't all that much. He said let's go, I've got the money. It started running out. We made some in Vegas. Well\u2014fortunately there wasn't usually any suite and we took what there was, Julian grumbling. He really snowed them\u2014I must say. With Mama and me and the cat and all that\u2014they thought he must be some eccentric damn millionaire to travel like that\u2014and like I say the car looked real good. Of course, in the morning, we loaded up and everybody else got in the car. And then I had to push it back out of the lot to where we could start out forward.\"\n\nAs for Miracle Valley, which I very much wanted to hear about, I never got a clear account. The idea of a righteous community is, I know, endlessly appealing to certain souls, and no doubt my mother's hankering after Miracle Valley (toward whose construction she had given a widow's mite) was the same impulse that had drawn us, years before, to Sharon, no matter how dissimilar the two.\n\n\"Miracle Valley,\" Charles said, \"it was real grand, the buildings and all, the grounds.\" A dreamy look came into his eyes and he murmured, \"Takes a lot, though. But that's the way you've got to do it, make it impressive. Otherwise, nobody's going to pay attention. Nice lawns. Big spire. Impressive.\"\n\nXV\n\nI have two more deaths to tell of. My mother died, in Champaign, in 1975, and Julian died two years later.\n\nCharles\u2014with children, this time\u2014had already moved to Long Beach to start a church. The first bulletin from his church had some photos that it took me a moment to realize were of him, though one of them is labeled \"Brother Charlie giving the sign of Brotherhood.\" The printing is such that his hands, and hence the sign, disappear against an ornate robe he is wearing, but one can make out the flowing hair and beard and a dark, emaciated, beaming face. The smile is what I recognized first. His church was called Church of the Brotherhood of Man.\n\nCharles was sure that Kate was in Illinois, all along, to start a branch or mission of Beulah Chapel, but she never made any move to do so. To this day, I can't imagine what she and Julian lived on\u2014unless, perhaps, she had savings, or got something from her first husband. Mother remained in her garage, sick and offended. She decided that Kate was not a fool, diabolically clever in fact, that she was a witch.\n\nJulian had lived with Mother for years, off and on, and though she would not go near his place now, he went to hers regularly and spent hours there each day, getting her fishwiches and cream and grapefruit, bringing her news of Elaine (sometimes she would welcome Elaine and at other times would not let her come in), and talking scripture. I wondered what he did the rest of the day. The last time I was there\u2014I mean before my mother's final stay in the hospital\u2014she asked me to take her trunk. She was afraid someone would steal it, afraid also that the few dollars she kept in a pouch on a string around her neck would be stolen. \"If I have to go to the hospital,\" she said, \"the nurses will take it.\" And sometimes she would fear that Julian was after it, to give to Kate. While he talked, she sat frowning, not seeming to hear, but once in a while would contradict him or tell him to stop talking nonsense. He was patient. Sometimes she heaped insults on him. His rambling discourses on the Bible\u2014he also read to her from the New Testament\u2014she rarely reacted to, but I was sure they irritated her.\n\nMy mother, after her operation\n\nOne day\u2014Julian had just driven off\u2014she murmured, \"should have died then.\" Remembering her lament about being no longer \"paid up,\" I made a clumsy remark to the effect that surely God would not require the impossible, and she screamed at me, \"Don't talk about what you don't know anything about!\"\u2014good advice, I suppose, suggesting silence. I asked her then (it's unlike me, such a question\u2014perhaps I was more upset than I knew, or had not yet taken in the dreadful possibility of faith outliving the loss of hope) if her religion gave her no comfort. And she replied immediately, without any emotion in her voice, \"No.\"\n\nI was staying with Elaine, but went to Danville with Julian for dinner once. Kate, making fresh pasta, again struck me as impossible to connect with what others said and felt about her. I never got to know her, so in this account she must remain unclear. She did believe\u2014I heard it from her own lips\u2014that true Christians are, _in a very physical sense,_ the Body of Christ, that Jesus having been resurrected and no longer subject to death, we also (true Christians) need not suffer death. And we shall live forever on this very earth, restored as it will be to its state before the fall. She said all this, or something like it, with flour on her hands.\n\nI saw her again at my mother's funeral, and then again at the funeral for Julian, where Seely also turned up, lankier than I remembered her and weeping uninhibitedly. Kate did not weep, either at the funeral itself or at the military burial ground. Two soldiers folded the flag that was draped over Julian's coffin and presented it to Kate. I was sitting next to her, marveling that my deserter brother could be buried under these conditions\u2014there was a rifle salute (tape recorded, actually) and, also on tape, Taps. Kate had produced Julian's separation papers from his term in the navy in the forties; no one questioned them or asked if he had not enlisted again later, and he was laid to rest here, among old soldiers. More bizarre yet: after the service, a black sergeant came up to Kate and asked her what emblem should be put on the marker. Anything unconventional she would have to draw. She said, without hesitation, as I wondered if my mouth had fallen open, that his marker should contain a cross\u2014because he was a Christian\u2014inside a star of David\u2014because he was by race a Jew.\n\nI'm not sure why I never asked her where she got that idea. (It's true I saw her only once afterward.) All I could come up with on my own is farfetched: one of Julian's favorite jokes was to insist that Mother's nose\u2014slightly hooked, as it was\u2014showed that she was Jewish. He would taunt her (\"Where'd you get that beak, Mama?\") until she was thoroughly angry, then laugh uproariously. She was prejudiced enough for the joke to work, time after time. Or again: one of Julian's great-grandmothers, on his father's side, was a Choctaw\u2014perhaps Kate accepted the old theory that America's aborigines are descended from the lost tribes of Israel.\n\nWhat I did learn, the last time I saw Kate, was that the final year and a half of Julian's life (after Mother's death, when I had not seen him at all) he had grown thinner, taken to sitting at a desk with his Bible and writing out sermons on the Body of Christ. He had cut down on sweets. (I remembered how he used to eat half a pie and follow it with a jab of insulin.) When he got restless, he would get in the car and drive to the airport.\n\n\"He was sure,\" Kate said, \"that his prosperity was going to come in soon.\"\n\n\"Did he go there often?\" I asked, a bit in admiration, I must admit, of this private cargo-cult. \"It's quite a drive to the Chicago airport.\"\n\n\"Not Chicago,\" she said, \"the Champaign-Urbana airport.\"\n\nI said nothing\u2014I was better at silence now\u2014but I saw in my mind's eye the whole bright circle of an afternoon sky descend to a meager runway traced across an ill-marked patch of ground\u2014Julian, blood and bone of the Body of Christ, gazing tranquilly out of the snack bar window at a returning Piper Cub, or perhaps a scheduled flight from Peoria.\n\nWhy did he die? A heart attack in the night (Elaine called me after Kate called her) had not surprised me, considering his diabetes and his general condition. There was no autopsy. But not everyone was satisfied.\n\nElaine pondered it all, after I left, and the more she thought, the more suspicious she became. She finally got in touch with Charles, whom we had not been able to reach in time for the funeral. He had no money\u2014besides, the ceremony was over and Julian buried. (The Free Methodist church in Emporia, now that I think of it, had in an upper window\u2014not stained glass, of course, but frosted\u2014a star of David, which I used to stare at during sermons.) But he borrowed the fare and flew out. He had wanted, he told me later\u2014when I visited him in California\u2014to investigate, to satisfy himself that his brother had died a natural death. Elaine was sure, by this time, with the confirmation of a dream-vision of Martin's, that Kate had murdered Julian by witchcraft.\n\n\"It did sort of look that way,\" Charles said, but his investigation had come to nothing. \"Since then, I've thought about it a lot,\" he told me, \"and you know, I don't think any more that witchcraft had anything to do with it. But I think she killed him.\" We were in a hamburger joint. He was obviously eating almost nothing these days. He hobbled around\u2014his legs hurt him. He reminded me somehow of my uncle Roscoe. The church he had started had been in partnership with another schemer (an atheist, Charles admitted) and his partner had scuttled whatever organization they had, managing to sell the pews out of their rented building, to enrich the offerings he had made off with.\n\n\"You see,\" Charles said, stirring as much cream as possible into his coffee, \"she got him to believe, really _believe_ , that crap about not having to die. They got him so damn sure that he was one of the elect who would just graduate to the New Jerusalem or some damn thing that he wasn't afraid of anything\u2014he wasn't even taking his insulin for a while.\" Charles, now, is diabetic, but takes an oral form of insulin with prescribed regularity. \"Of course he never took care of himself, ever, but she and her Prophet validated it for him. She may not have meant to do it, though it looked pretty fishy.\" He was no doubt thinking of Kate's remarriage two months after the funeral. \"I stopped trying to find out anything, but that was because of Elaine.\"\n\n\"But I had the impression Elaine never changed her mind about Kate.\"\n\n\"That's right,\" Charles said. \"She still thinks, just like Mama thought, that Kate's a witch. But she was afraid Kate had too much power to monkey with. You know Martin got worse and worse there. He'd come screaming down the stairs at any hour\u2014usually after everybody was asleep\u2014and shout his head off that she was tormenting the hell out of him. He was a real mess. Whenever he wasn't being attacked, he was goose-stepping all over the attic so nobody in the house could hear themselves think. I stayed out as much as I could\u2014I've never felt so harassed in my life, and that's saying quite a bit. And wouldn't you know it, just at that moment the Judge shows up. Hadn't seen old Jerimy for years.\"\n\n\"He was at the funeral. I remember he slipped in and sat at the back.\"\n\n\"Yeah? Well, ordinarily I'd have been happy to see him. You know, I always liked him. But he had this plan. He wanted\u2014\" Charles shook his head. \"You understand, I was about out of my mind\u2014I had left the kids here and borrowed money to get out there and Elaine I think _was_ out of her mind by then and the damn Martin was stomping the house half down and Clyde was about ready to move out\u2014said old Miss Yodle was dead right, he should never have gotten involved with us at all. And in comes the Judge we haven't seen for years, and wants to have a s\u00e9ance.\"\n\nJudge Jerimy had waited. He had not wanted to rush in or to proceed in any way indelicately. But when he thought of Julian's death, the news of which had shocked him, terrible doubts assailed him. If he had seen Julian shortly before his death, he might not have been so astonished\u2014had he seen him from time to time, he might not have felt a vague guilt for having abandoned him. In any case, it was with the solemnity of someone who cares about something enough to risk being put in a very false position that he approached his former friends, these survivors. Once suggested, the idea must have seemed, after all, inevitable.\n\n\"I can't promise,\" he said to them, \"that any success will come of this. We must, I know, acknowledge first of all our powerlessness in the face of a loss like this. And then, humbly, ask for guidance.\" I am convinced of his humility, and of his sincerity, but I think also that he saw in the present case\u2014so close to him emotionally\u2014a possible breakthrough in his lifelong search for certainty. They talked it over for three days.\n\n\"We shouldn't wait too long,\" he said. \"It is reasonable to suppose that a spirit is less liable to be able to make contact, the farther he is from his life in the body. This may be true even for those spirits who have been separated before their time, cut off violently from their earthly associations\u2014though they will have more energy and more inclination to make themselves known to the living.\" The direction of all this was clear to everyone. Clyde, who still regretted the loss of the Ouija board, was for trying immediately. So was Martin, who had come down from his attic on hearing the Judge arrive. Charles, without supporting any theoretical position, was willing to go ahead with it. \"It may even be,\" the Judge went on, \"that it will be an aid in our search that someone so close to Julian has passed over comparatively recently\"\u2014meaning, of course, Mother. Elaine wanted to sleep on it one more night.\n\nAnd in the morning, she gave her flat refusal to have anything to do with a s\u00e9ance, or indeed to allow it in her house. The Judge would not think of pressing her, or of going ahead without her. He had made a tactical error in having brought Mother into the discussion. Elaine said she had thought about how much Mother would have been opposed to a s\u00e9ance taking place\u2014which was true enough, no matter how much Elaine was simply bolstering her own convictions. Charles was relieved, the others disappointed.\n\nIt seems to me now that Elaine had, besides the reason she gave, an additional unspoken objection. She, like the rest of us, remembered Mother as long oppressed and afflicted, remembered how she had unexpectedly renewed her strength, carrying us off to Sharon, as to a city not built with hands. Elaine always regarded Mother's later Pentecostalism as a retreat from unbearable disappointment, a desperate clutching after some new hope, after we her children had, each in turn, failed her. She had watched Mother thrash out, as it were, with tremendous force, aggressively, in all useless directions, wise but wasted virgin, Cinderella smashing her one useless slipper, and still waiting, with nothing to wait for. (I used to wonder why, no matter what we've done or not done, we are always, all of us, so dead set on feeling guilty. Until it occurred to me that guilt is another word for potentiality: to sin is to cast a shadow. Lord, be merciful.) I think Elaine could not bear the thought\u2014now that the much-maligned heart was stilled, allowing an image of eternal rest\u2014of disturbing that rest, summoning from repose an escaped spirit, who had never known anything but torment, back to further torment.\n\nCharles gave up his investigation and flew back to Long Beach. \"And you know,\" he told me, \"the damnedest thing\u2014I found my house padlocked. I'd been evicted in my absence. I'd never have let them get away with it if I'd been here, but they just came in and told the boys the house had been sold and they'd have to get out and the boys got out. It was days before I found where they had holed up. They were all right.\" He was slipping cream, now, into his empty coffee cup. \"I found out,\" he went on, \"that the house had been bought by Prophet Shock\u2014just bought it up and threw the kids out. I went roaring over to Beulah Chapel and was ready to bite nails, and he was all sweetness and light and smiled and said God told him to buy that house. Nothing I could do\u2014just God told him to buy it. I had to scrounge around for someplace to go.\" There had been a series of places after that. \"I was never sure,\" he said, \"that Kate had anything to do with it. You know, that kind of experience makes you sort of lose your grip on reality.\"\n\nI could see how it might. My own sense of \"reality,\" never perhaps all that strong, has changed over the years. I used to think that, the sum of things being obviously unreal, I should cling to what is at hand, the minute particulars of my immediate experience. I still try, but less and less seems particular. What I find now is more often like a play of light\u2014that is, of shadow\u2014in which the objects of my attention are ripples, or notches, or bumps, or bubbles, on a surface otherwise inarticulate, a field which, but for these defects, would be empty.\n\nMy mother's heart, by the way, which she used to offer\u2014holding her throat to feel the pulse\u2014as evidence of fragility and imminent destruction, seems to have been sound to the day of her death. It was not her heart that killed her.\n\nShe choked to death.\n\nThat she was dying, Elaine had called to tell me. I found her in the hospital where, fed through her veins, she lay at the extreme. Her face was distorted and inexpressive, expression having fallen under a mask of malfunction, as if the nerves had been disconnected, leaving something prehistoric, unrelated. But she was breathing, and when Elaine said, \"Here's Keith, Mama\u2014do you recognize Keith?\" the mask replied, without tone but distinctly, \"Yes.\" Her arms and legs, from the delicacy she had been so proud of, had thinned to the bone, giving a fowl-like aspect to their occasional tremors. There was a suction tube leading from her mouth, since she had trouble swallowing.\n\nThe doctor had assured Elaine that in this state the patient felt no pain, that the few who had .passed through it and recovered told of peace, of complete neutrality, a kind of refuge from any reaction\u2014some indeed having left a little reluctantly this gray world.\n\nMaybe. Elaine took it, as the doctor meant her to, as a consolation. I found it less than consoling to contemplate these residues of mental process, almost\u2014yes, _as far as possible_ \u2014divorced from the flesh, and remembered Evangeline saying, \"But no, listen, it's what I want, this death that will last long enough for me to feel it.\"\n\nThe room had six beds, all occupied. Directly across from Mother's, under a large window, was an old woman who moaned continually but who, according to Elaine, sometimes shouted obscene insults at an invisible enemy.\n\nJulian came often to the hospital and stood by Mother's bed, his face\u2014though not ruined\u2014almost as inexpressive as hers. Once a somewhat unctuous middle-aged couple who had come to visit the moaning woman\u2014who while they were there ceased to make any sound at all\u2014came over and stood a moment across from Julian. They must have needed a touch of communication.\n\n\"Your mother?\" the man asked\u2014he was holding his hat over his heart.\n\n\"Yes,\" Julian said.\n\n\"She looks,\" the woman said, after a moment, \"like she's been through a lot.\"\n\n\"Dying,\" Julian said, and the two of them scurried out. Once they had gone, the woman across the room moaned some more, then under her breath,\n\n\"Bastard,\" she said, \"you bastard. Jesus!\" she said more loudly. I wondered whom she was cursing, if it was the man with the hat (her son? son-in-law?), but then I saw I had it all wrong. \"Jesus!\" she shouted, \"Jesus you bastard!\u2014why won't you let me die?\" She tried to slug the nurse who came over then, but was given an injection and gradually quieted down.\n\nThere is next to nothing to record about my mother's last days. She remained in her gray world, apart, like a puppet put aside, while hospital business went on around her. There was only one thing more she would say to me or, so far as I know, to anybody.\n\nAnywhere else\u2014driving, for instance\u2014Julian's left eye now had a tic: it twitched slightly from time to time (Charles claimed to have noticed it from our car lot days). When he stood by her bed, however, looking down at the incipient corpse, his face was perfectly calm.\n\nIt seemed that things would go on like this\u2014visiting her here in the hospital, as I had visited her in her garage or at the fruit stand or in Roanoke or Atlanta. We had not, for many years at least, had a great deal to say to each other. I leaned down, unsure how much she heard or understood, and said I would see her tomorrow.\n\n\"Too late,\" she said, quite clearly, as if from a distance. In the night, some fluid\u2014saliva, blood, phlegm\u2014collected in her throat. The little tube must not have been there, or not working, or not enough. She failed to swallow, then failed to breathe, and her heart could do nothing without air.\n\nJulian insisted that he make the funeral arrangements and came up with what he and Kate both declared a good-looking coffin. The service was in the Urbana Church of God, with a white hearse and cars with little flags to drive us to the graveyard. I am convinced that Julian felt her loss more than any of us, but he seemed more cheerful than while she was dying. He told me\u2014off to one side, before we went into the church\u2014that the funeral was costing absolutely nothing. If I got it right, and if he was telling the truth, the undertaker\u2014a youngish thug-like character I vaguely recalled seeing at the car lot\u2014had stolen a truckload of coffins at some point. Julian was blackmailing him.\n\nIt was mostly church-folk who came\u2014besides us, the family (minus Charles, who was out of contact). The Judge was there, in the back. Julian carried his cigar, which he would never light in church, looked around when we were seated\u2014he liked a sumptuous ceremony, as he liked silk suits and Cadillacs. The church looked right to him, the ushers, the minister, the minister's wife at the keyboard, a bank of flowers around Mother in her hijacked coffin. It all met his approval and he tapped the cigar as if it had an ash, crossed his legs\u2014a difficult gesture for him\u2014and said to me quietly, with satisfaction, \"Giving Mama a good send-off.\"\n\n_Me, Charles, two of his sons\u2014his paintings behind (Long Beach, California)_\n\nJust yesterday morning, I woke slowly and found that opening my eyes gave me a view of blanket and bedroom in the usual half-light (it was broad day, actually, but the curtains were closed) and then\u2014what would have startled me perhaps if I had been more awake\u2014found that if I closed my eyes again, which I did several times, I continued to see precisely the same scene, but slightly brighter, and all over the blanket (only if I kept my eyes closed) there were innumerable flies.\n\nMother is dead. Julian is dead. I don't know where Kate is, or Seely. Elaine called me shortly after Julian's burial\u2014when I was back in Providence\u2014to warn me that Kate might come to see me and that I should under no circumstances let her in. Elaine was somehow under the impression that I had books that a witch might find useful and that the Prophetess was after them. Kate in fact announced a visit, but never showed up.\n\nElaine still lives in Urbana. For years she belonged to the Nazarenes, but now she attends the New Life Fellowship Church, where she teaches a Bible class Wednesday nights. Clyde was working for Sears, Roebuck, but having damaged his arm by falling off a ladder\u2014while on duty\u2014he has wangled some kind of pension. Martin got married, but he and his wife divorced after a child was born. He now lives next door to his parents, and Elaine, I gather, takes care of the child.\n\nCharles paints mystical pictures in which, for instance, a triangle with a weeping eye hovers above a globe. The tears, as they fall from the infinite, become little naked human figures crawling on the earth below. He will not allow anyone to kill the mice that run in his condemned building. He feeds them and they become quite tame. His second son is still with him. The oldest has gone off on his own and comes back only occasionally, to demand money of his father, or, if there is none, to beat him up. The youngest has joined a communal cult, which gives him, he says, a \"secure family feeling.\"\n\nAccording to a book just out, the Piltdown Man, that great scientific fraud, may have been fabricated with the connivance of my namesake, Sir Arthur Keith. For some reason, the idea pleases me.\n\nI remember, now, how when we went to church, my mother and I, back in Emporia\u2014when I was a child\u2014we caught the bus on the other side of Sixth Avenue, which was also a state highway and, so, relatively busy. But if we were late and the bus was to a point where the driver might well pass on without seeing us, my mother's practice was to grip my hand, close her eyes, lower her head, and charge across the street, traffic or no. I thought about this, at the time, a great deal, and came to the conclusion that under certain conditions of emergency, ordinary physical laws do not hold and ordinary precautions may be suspended.\n\nI do not think this anymore. At the same time, I may note that we never came to harm\u2014not then, not in that way.\n\nAs for me, what I would like, I think, is to live a while longer. But not again.\nAbout the Author\n\nKEITH WALDROP lives in Providence, Rhode Island, and is Professor Emeritus of Literary Arts at Brown University. He is the author of over two dozen works of poetry and prose, as well as an eminent translator of French avant-garde poetry. he is founding editor, with his wife Rosmarie Waldrop, of Burning Deck Press, now in its fifty-second year of publishing innovative poetry and fiction. Waldrop won the 2009 National Book Award for Poetry for his collection _Transcendental Studies: A Trilogy._\nCopyright\n\nOriginally published by Sun & Moon Press, Los Angeles, 1993.\n\nCopyright \u00a9 1993 by Keith Waldrop\n\nIntroduction \u00a9 2013 by Jaimy Gordon\n\nFirst Dalkey Archive edition, 2013\n\nAll rights reserved\n\nLibrary of Congress Cataloging-in-Publication Data is available.\n\nISBN: 978-1-56478-805-4\n\nPartially funded by a grant from the Illinois Arts Council, a state agency\n\nwww.dalkeyarchive.com\n\nCover: design and composition by Mikhail Iliatov\n\nPrinted on permanent\/durable acid-free paper and bound in the United States of America\nSELECTED OTHER WORKS BY KEITH WALDROP\n\nA _Windmill Near Calvary_\n\n_The Garden of Effort_\n\n_The Space of Half an Hour_\n\n_Shipwreck in Haven_\n\n_Hegel's Family_ (short fictions)\n\n_The Opposite of Letting the Mind Wander_\n\n_The Locality Principle_\n\n_Analogies of Escape_\n\n_The Silhouette of the Bridge (Memory Stand-Ins)_\n\n_Stone Angels_\n\n_Well Well Reality_ (with Rosmarie Waldrop)\n\n_Haunt_\n\n_Semiramis If I Remember_\n\n_The House Seen from Nowhere_\n\n_The Real Subject: Queries and Conjectures of Jacob Delafon, with Sample Poems_\n\n_Several Gravities_\n\n_Transcendental Studies: A Trilogy_\n","meta":{"redpajama_set_name":"RedPajamaBook"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzrzth b/data_all_eng_slimpj/shuffled/split2/finalzzrzth new file mode 100644 index 0000000000000000000000000000000000000000..8fdf938e2b75a7bdb986813584b8166e32e8fc93 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzrzth @@ -0,0 +1,5 @@ +{"text":"\n\n\n\nProduced by Chris Curnow, Stephen Hutcheson, Joseph Cooper\nand the Online Distributed Proofreading Team at\nhttp:\/\/www.pgdp.net\n\n\n\n\n\n\n\n\n\n BIRDS AND NATURE.\n ILLUSTRATED BY COLOR PHOTOGRAPHY.\n Vol. IX. APRIL, 1901. No. 4\n\n\n\n\n CONTENTS.\n\n\n APRIL. 145\n I come, like a hope to a gloomy breast 145\n THE CURASSOW. 146\n SOME NOTABLE NESTS. 149\n THE BLACKBIRD'S SONG. 151\n A GOLDEN EAGLE. 152\n THE HARLEQUIN DUCK. (_Histrionicus histrionicus._) 155\n AN ORCHARD BIRD-WAY. 156\n THE CANADA GROUSE. (_Dendragapus canadensis._) 158\n DO PLANTS HAVE INSTINCT. 162\n Still winter holds the frozen ground\n and fast the streams with ice are bound\n 164\n THE DOVEKIE. (_Alle alle._) 167\n As flying ever westward Night's shadows swiftly glide 167\n THE SONG SPARROW'S APPEAL. 168\n THE WITCH IN THE CREAM. A TRUE STORY. 169\n THE BEAVER. 170\n PAU-PUK-KEEWIS AND THE BEAVERS. 174\n What rosy pearls, bright zoned or striped! 175\n SNAILS OF THE OCEAN. 176\n THE LEMON. 182\n TWO WRENS. 185\n WHEN SPRING COMES. 188\n CUBEBS. (_Piper cubeba L._) 191\n A TREE-TOP TOWN. 192\n\n\n\n\n APRIL.\n\n\n No days such honored days as these! While yet\n Fair Aphrodite reigned, men seeking wide\n For some fair thing which should forever bide\n On earth, her beauteous memory to set\n In fitting frame that no age could forget,\n Her name in lovely April's name did hide,\n And leave it there, eternally allied\n To all the fairest flowers Spring did beget.\n And when fair Aphrodite passed from earth,\n Her shrines forgotten and her feasts of mirth,\n A holier symbol still in seal and sign,\n Sweet April took, of kingdom most divine,\n When Christ ascended, in the time of birth\n Of spring anemones, in Palestine.\n \u2014Helen Hunt Jackson.\n\n\n I come, like a hope to a gloomy breast,\n With comforting smiles, and tears\n Of sympathy for the earth's unrest;\n And news that the summer nears,\n For the feet of the young year every day\n Patter and patter and patter away.\n\n I thrill the world with a strange delight;\n The birds sing out with a will,\n And the herb-lorn lea is swift bedight\n With cowslip and daffodil;\n While the rain for an hour or two every day\n Patters and patters and patters away.\n \u2014Bernard Malcolm Ramsay, in the Pall Mall Magazine.\n\n\n\n\n THE CURASSOW.\n\n\nAn interesting race of birds, known as the Curassows, has its range\nthroughout that part of South America, east of the Andes Mountain range\nand north of Paraguay. All the species are confined to this region\nexcept one, which is found in Central America and Mexico. This is the\nbird of our illustration (Crax globicera).\n\nThe Curassows belong to the order of Gallinaceous birds and bear the\nsame relation to South America that the pheasants and grouse bear to the\nOld World. They are in every respect the most important and the most\nperfect game birds of the district which they inhabit. In all there are\ntwelve species placed under four genera. As the hind toes of the feet\nare placed on a level with the others they resemble the pigeon and are\nunlike many of the other gallinaceous birds.\n\nThe Curassows are very large and rather heavy birds and some of them are\nlarger than our turkey. They have short wings and a strong bill. At the\nbase of the upper mandible and on the upper side there is a large\ntubercle-like excrescence which is of a yellow color and quite hard.\nUpon the head there is a gracefully arched crest of feathers which is\nmade of curled feathers, the tips of which are white in some of the\nspecies. This crest can be lowered or raised at the will of the bird.\nThe plumage of the species illustrated is a beautiful and velvety black,\nexcept the white on the lower portion of the body. It is said that their\nmotions are much more graceful than are those of our common domestic\nturkey. \"They live in small flocks, and are arboreal in their habits,\nonly occasionally descending to the ground, while roosting and building\ntheir nests on the branches of trees.\" The nests are large and made of\ntwigs and willowy branches held in place by the stems of grasses, which\nare neatly interwoven between them. The nest is lined with down,\nfeathers and leaves.\n\nIt is said that they are easily domesticated and that in some parts of\nSouth America they may be found in tame flocks around the homes of the\nplanters. One authority states that at about the beginning of the\npresent century a large number of Curassows were taken from Dutch Guiana\nto Holland, where they became thoroughly domesticated, breeding as\nreadily as any other kind of domestic poultry. Though a tropical bird,\nit would seem that they might be acclimatized. They would certainly form\na valuable addition to the list of our farm fowls, for their flesh is\nsaid to be \"exceedingly white and delicate.\"\n\nThe female is not as large as the male and is usually reddish in color.\nTheir food consists almost entirely of fruit and insects.\n\nAbout the middle of the eighteenth century Eleazar Albin wrote \"A\nNatural History of Birds,\" in which he gives a very interesting account\nof the Curassow and an excellent illustration of the bird. He says: \"I\ntook a pourtray of this bird at Chelmsford in Essex; it was very tame\nand sociable, eating and drinking with any company. The Cock I had of a\nman from the West Indies. They are generally brought from Carasow, from\nwhence they take their Name. They are called by the Indians Tecuecholi,\nMountain-Bird or American Pheasant.\"\n\n [Illustration: ]\n\n\n\n\n SOME NOTABLE NESTS.\n\n\nThe Clymer boys and girls, of Cloverdale, New England, belonged to a\nBird Club; they were proposed to membership by their neighbors, the\nWalkers; in fact, the two families composed the club, and it partook of\nthe nature of a secret society.\n\nAll this was before the young people of Cloverdale knew of Clark\nUniversity, and Dr. Hodges' \"Ten to One Clubs,\" wherein the members\npledged themselves to strive by all imaginable means\u2014provided they were\nalso practical\u2014to induce ten song birds to live and sing each year,\nwhere only one was found the year before.\n\nIt was not necessary for the Cloverdale Club to put up carefully\nconstructed and artistic bird houses, or to hang cotton and the like\nfine nest-building materials in choicest ornamental shade trees\u2014not at\nall. The English Sparrow had not found the village in those days; the\nsong birds were there, they knew all the good locations and just where\nto find the best stuffs for constructing, furnishing and decorating\ntheir homes; the work of the club was to find these homes, to study\nthem, with the ways and habits of their occupants, and to record their\ndiscoveries in a big book labeled, \"Things Not Generally Known.\"\n\nMany of the statements in this book were as broad and conclusive as\nscientific dogmas, but the Cloverdale Club did not waste its time\nsearching for hundreds of instances to establish a single truth; one was\nenough to be worthy of record; then, if some time the big book should be\ngiven to the public, and some naturalist or investigator should choose\nto confirm its statements by patient research, of course he would be\nwelcome so to do. The club had the distinction of discovery, that was\nenough.\n\nOne interesting item recorded was this: \"Birds\u2014such as Orioles\u2014who build\nin conspicuous places, like to decorate the outside of their nests, and\nin so doing are known to use manufactured materials and patterns.\"\nStrange statement, but of course thereby hangs a tale, and here it is.\n\nAt the spring house-cleaning time, Mrs. Clymer had the big, bright\nsitting-room carpet taken out under one of the old colonial elms, at the\neast of the house, to be cleaned. Mrs. Baltimore Oriole was up in the\nelm that morning looking for a building spot that should be a bit\nsuperior to the old one; she had spent three summers in that tree, was\nfamiliar with the ways of the club, and habits of the family; like the\nbirds of Eugene Field's boyhood, \"she knew her business when she built\nthe old fire-hang-bird's nest.\"\n\nNo one was near when Mrs. Oriole fixed her eyes on the great red, green\nand white ingrain carpet, and admired it; what she thought we know not,\nbut when she glanced at the hitching post under the tree, she instantly\ndescended from high, waving branch, to lowly square post, for exactly\ncovering the top of the same was a miniature carpet, a piece just six by\nsix inches which Patrick should have left indoors; not having done so,\nhe laid it on the inviting post for safe-keeping. That bit of wool\nfabric was very valuable, it exactly filled a jog right by the\nfireplace, in which, alas! ever after was seen an ugly piece of oil\ncloth!\n\nAll summer long the club girls and boys gazed with wonder at the gay\nnest in the elm, hanging like a solitary blossom among the leaves; their\nspeculations about it would fill a long chapter; but after the birds\nwere flown far to the south, and the leaves were gone, that nest was\nfinally cut down and told its story: thread by thread, just as pulled\nfrom the bit of carpet, had been woven into a decoration for the outer\nwall of that hanging house, till a rude reproduction of the original\ntiny rug was under the feet of the birdlings, and over the heads of the\nboys.\n\nThe club held a special exhibition of that nest, and at Thanksgiving\ntime one of the home-coming guests, who was an enthusiastic\nkindergartener in the city, persuaded those generous nature students to\nlet her take their treasure to the poor children who seldom saw the\ncommonest kind of a hang-bird's nest, and in that kindergarten it may be\nseen today.\n\nAnother entry in the club book was this: \"Birds building on the ground,\nespecially Vesper Sparrows, locate if possible where they have a fine\noutlook, and give great attention to the arrangement of the front yard.\"\n\nThis was discovered when Emily Clymer took her small brother Jo up in\nthe \"side hill pasture\" to see the finest mountain view in all the\ncounty, and to find wild strawberries; while picking the berries they\nfound what was afterward called the juniper house; this was a Vesper\nSparrow's home, roofed by green growing juniper.\n\nEverybody knows that the prophet Elijah could never have sat and wept\nunder a New England juniper tree; no tree is less high or more nearly\nhorizontal than this; in fact, we call it a bush\u2014where it is big\u2014this\none was not larger than Emily Clymer's two hands, and growing straight\nout from descending ground, it formed a flat, green roof to the Sparrow\nhomestead; then, while my lady sat upon her nest, she looked out of her\ntiny front door, across a gently sloping lawn, upon a whole range of\nmountains. But most remarkable of all were the ornamental shade trees,\nfor just ten inches from the door, on either side, waved two big brakes,\nsymmetrical in size and shape; they gracefully arched across the\nentrance, and were to the Sparrow domicile as the giant elms to the big\nClymer homestead. A sketch of this beautiful residence was made by a\nmember of the club\u2014for cameras were not common in Cloverdale then\u2014the\npicture cannot be taken from the club book, but I think we can see it\nall with our mind's eye.\n\nHere is one of the most astounding statements in that book of many\nobservations: \"Some Phoebes are like the Golden Eagle in three\nways\u2014first, they build on rocky and inaccessible cliffs, second, they\nbuild in the same place for one hundred years; and, third, when the\nyoung are big enough to fly, they know how, and just go up without any\npracticing.\" All this can be proved to any one who will go in nesting\ntime to a cliff overhanging the river just below Cloverdale, and who\nwill accept the testimony of some of the most reliable and respectable\nmen who have honored that place in the past century.\n\nYou must go in a boat and hug the shore; of course you need a member of\nthe club for guide; at an unexpected moment you are told to look over\nyour head, and there, glued to a shelf of rock so small as to be\nentirely covered by the same, is the nest! No porch, or even doorstep,\nbeyond its wall\u2014an overhanging roof of rock above, a shoreless expanse\nof water below; now, if some one can keep the boat steady, and you have\nthe nerve to stand at the highest point of the bow, then by reaching\nover your head you can gently touch some fuzzy bits of life in the nest.\nNow you know the first and last of the facts recorded are correct: there\nis the nest on the inaccessible cliff; there are the birds, and if they\ndid not fly up and out into the world the first time they stood on the\nedge of the nest, would they not be in the dark water below, instead of\ncoming back to the old home for a hundred years?\n\nThe evidence of successive occupation for a century is this: The present\nfamily of Walkers\u2014father and children\u2014have watched that nest, never\nfinding it empty a summer for twenty years. Old Deacon Walker,\ngrandfather of our club members\u2014who, of course, initiated their\nfather\u2014proved that Phoebes had hatched in the cliff nest during eighty\nyears previous, in this wise: After he had stood guard forty years, as\nthe deacon loved to relate, didn't his Uncle Israel\u2014who had been\nspending just those two-score years in the South\u2014come home one spring\nevening, and the very next morning that ancient worthy demanded a boat\nand a boy to take him under the old Phoebe's nest on the ledge, which he\naffirmed had never been without tenants during the forty years before he\nleft Cloverdale?\n\nSo there are the figures and facts showing how not only the nest, but\nbird love and bird lore had come down through the century, and with such\nan inheritance, no wonder the Walkers are on the best of terms with\nfeathered folk, or that they, with their confidential friends, the\nClymers, are still adding to their bird book things not generally known.\n\n Elizabeth Reed Brownell.\n\n\n\n\n THE BLACKBIRD'S SONG.\n\n\n The bee is asleep in the heart of the rose,\n The lark's nestled soft in the cloud,\n The swallow lies snug close under the eaves\u2014\n But the blackbird's fluting is loud;\n He pipes as no hermit would or should,\n Half a mile deep in the heart of the wood,\n In the green dark heart of the wood.\n\n The raven's asleep in the thick of the oak,\n His head close under his wing;\n The lark's come down to his home on the earth\u2014\n But the blackbird still will sing,\n Making the heart of the dark wood thrill\n With the notes that come from his golden bill,\n That flow from his golden bill.\n \u2014Walter Thornbury.\n\n\n\n\n A GOLDEN EAGLE.\n\n\nIn January, 1900, I had given me a Golden Eagle. He had been picked up\nin a stunned condition in the foot-hills, having received a shock from\nthe electric wires, on which he had probably alighted for a moment or\nstruck in his flight. There is an electric power-house in the Sierras\nopposite Fresno, from which pole lines carry the strong current down to\nbe used for power and light in the valley, and this was by no means the\nfirst record of eagles and other large birds being stunned or killed by\nthem.\n\nThe person who found him had brought him down with the idea of having\nhim stuffed, but as he showed a good deal of life, I begged to keep him\nalive, and he was handed over to me. He was evidently a young bird of\nthe previous season, though nearly full grown. From tip to tip of his\nwings he was over five feet, and his wonderful black talons measured one\nand one-half to two inches beyond the feathers. His legs were handsomely\nfeathered down to the claws, and his proud head, with its strong beak,\nlarge, piercing eyes, and red and yellow-brown feathers, was a thing of\nbeauty. The rest of his body was dark, almost black, with the exception\nof three or four white diamonds showing on the upper tail feathers.\n\nI kept him in a big box open on one side. When I first brought him home\nand had put him into the box, a neighbor's poodle came sniffing around\nfor the meat I had brought for the eagle. He was on the back side of the\nbox, and so could not see that there was anything in it, nor did he hear\nanything, but all at once the scent of the bird must have struck his\nnostrils, for with a squall of fear he disappeared from the yard and\nnever afterward would venture near the cage.\n\nDuring the time I kept the eagle, some two months, he never showed any\ndesire to attack me, though his claws would have gone through my hand\nlike a knife, nor did he display any fear of me. He never made any\nattempt to get out while anyone was in sight of him, nor did I catch him\nin any such attempt, but sometimes at night I would hear him, and every\nmorning his wings, beak and feathers showed he never gave up the hope of\ngetting free.\n\nI never fed him to the full extent of his capacity, but gave him from a\npound to a pound and a half of meat daily at noon, which he devoured in\na very short time, sticking his claws through the toughest beef and\ntearing it like ribbons with his beak. It was wonderful to see how clean\nhe could pick a bone with his clumsy-looking great beak. I never knew\nhim to touch any kind of food but raw meat. When anything was handed in\nto him, no matter how high up, he never accepted it in his bill, but\nstruck at it with a lightning-like movement of his claws, scarcely ever\nmissing it.\n\nOne day he snapped in two one of the bars across his cage, pried off\nanother and got out. I was telephoned that my eagle was out, and hurried\nhome to find all the children in the neighborhood blockaded indoors. The\neagle was perched on the grape-arbor easily surveying the lay of things.\nA cat had crawled into the wood-pile and under the doorsteps the\nvenerable cock of the yard was congratulating himself on his safety, but\nfeeling rather undignified. I procured a rope and took my first lessons\nin lassoing. The eagle had been so closely confined that he had not been\nable to gain the full use of his wings, and so could only run or flutter\na few feet from the ground. I finally recaptured him and brought him\nback. He showed no fear and offered little resistance.\n\nAbout the middle of March the weather became very hot, and it was really\ncruel to keep the bird penned up in such close quarters in such weather,\nso I took him out to the plains and set him free. He could not use his\nwings much, and it is very doubtful if he escaped the shotgun or rifle\nof some predatory small boy, but it was the best I could do for him. He\nwas a beautiful specimen of a bird, and I only wish I could have kept\nhim.\n\n Charles Elmer Jenney.\n\n [Illustration: ]\n\n\n\n\n THE HARLEQUIN DUCK.\n (_Histrionicus histrionicus_.)\n\n\nThe Harlequin Duck is the sole representative of the genus to which it\nbelongs. The generic and the specific names (Histrionicus), which\nunfortunately the strict rules of scientific naming require in the case\nof this bird to be the same, are from the Latin word meaning harlequin.\nThis word, meaning a buffoon, is especially appropriate, for the\narrangement of the colors on its head, neck and back give the bird a\npeculiar appearance, especially during the mating season. At this time,\ntoo, the drollery of their actions is very noticeable.\n\nHarlequin is not the only name by which this bird is known. In the New\nEngland States and northward along the Atlantic coast it is frequently\ncalled the \"Lord and Lady,\" because of the white crescents and spots of\nits plumage and the proud bearing of the male. It is also called the\nRock Duck, the Mountain Duck and the Squealer.\n\nIts range covers the northern portion of North America, Europe and Asia.\n\"It is not common wherever found. In many parts of the Old World it is\nonly a rare or occasional visitor; this is the case in Great Britain,\nFrance and Germany.\" In the United States, during the winter, it passes\nsouthward into Illinois, Missouri and California. It breeds only in the\nnorthern part of its range.\n\nIt is a mountain duck and \"frequents swiftly running streams, where it\ndelights to sport among the eddies below water falls or in the brawling\nrapids.\" It is not only an adept in the art of swimming and diving, but\nit also flies swiftly and to a great height. During the winter it\nfrequents northern sea coasts and exhibits the characteristics of other\nsea ducks, and is occasionally found far out at sea. It is known that\nthe Harlequin will lead a solitary life, and it is sometimes observed in\npairs or even alone on streams of remote and unfrequented localities.\n\nThe sexes vary greatly. While the male, which is the sex of the bird of\nour illustration, is brightly colored, the female is much more somber.\nThe young resemble the adult female.\n\nThe food of the Harlequin consists almost entirely of the parts of\naquatic plants and the smaller crustaceans and mollusks. The food is\nobtained by diving, frequently through several feet of water. Mr.\nChapman tells us that the sea ducks in diving to obtain food, will\n\"sometimes descend one hundred and fifty feet or more.\"\n\nIts nest, though usually placed on the ground, is sometimes built in the\nhollow of a tree or a hollow stump, though always near a body of water.\nThe nest is usually a simple structure made of the stems of water\nplants, twigs and grass thickly lined with the downy feathers from the\nbreast of the duck. The eggs are occasionally laid on the grass, and no\neffort is made to build a nest. The female thoroughly covers the eggs\nwhen she leaves the nest.\n\nThe number of eggs varies from six to eight, though ten have been\nrecorded. They are of a \"yellowish buff or greenish yellow\" color.\n\nThis duck is considered an excellent food and is much sought for by the\nnatives of those regions which it frequents.\n\n\n\n\n AN ORCHARD BIRD-WAY.\n\n\n \"A rodless Walton of the brooks,\n A bloodless sportsman I;\n I hunt for the thoughts that throng the woods,\n The dreams that haunt the sky.\"\n \u2014_Samuel Walter Foss._\n\nAn isolated orchard certainly comes very near being an inner sanctuary\nof bird life. For some reason or other, the gnarled old trees and matted\nJune grass touch either the practical or artistic sense of bird nature\nvery closely, and appeal strongly to many a bird heart, for therein do\ncongregate all sorts and conditions of feathered life. Probably it is an\nexceptional feeding-ground, for the curled and misshapen leaves testify\nto the abundance of the hairy caterpillar and leaf-worm supply, which\nproves such delectable tidbit to the bird palate. When I see the birds\nfeasting upon these unsavory looking morsels, I can but wonder at the\nunregenerate farmer who so loudly decries the bird as a fruit-destroyer,\nwhen a few hours' observation will teach him that to one cherry stolen\nthere are a hundred tree destroyers gobbled up, and a thousand weed\nseeds devoured. It is Wilson Flagg who so curtly says:\n\n\"The fact, not yet understood in America, that the birds which are the\nmost mischievous as consumers of fruit are the most useful as destroyers\nof insects, is well known by all the farmers of Europe; and while we\ndestroy the birds to save the fruit, and sometimes cut down the fruit\ntrees to starve the birds, the Europeans more wisely plant them for\ntheir sustenance and accommodation.\"\n\nOur orchard is surrounded by a fence of weather-stained chestnut rails,\nwhose punctured surface has been the scene of many a worm tragedy\nresulting in the survival of the fittest. We enter through a pair of\nlichen-covered bars, grey-tinted and sobered by age. How far less\npicturesque is our field and hedgerow when inclosed by that inhuman\nhuman invention, a barbed-wire fence, and trim swing gate. To be neat\nand up to date, is never to be picturesque, and seldom to be artistic.\nBut our quiet entrance into the orchard has caused something of a\ndisturbance among the inhabitants, if no great alarm. Fluttering hastily\nto a convenient tree top goes a dainty red-eyed vireo, who seems to me\nto have more of a grey than olive gleam to his shining back. As he\nalights upon the topmost bough\u2014\n\n \"A bird's bright gleam on me he bent,\n A bird's glance, fearless, yet discreet,\"\n\nbut to show that he is in no way seriously alarmed he flings down to us\nsome sweet notes of liquid song. It is Wilson Flagg, I believe, that has\ndubbed him the Preacher, but to me he seems more correctly termed the\nLover, for I can but interpret his accentuated notes into \"Sweet Spirit,\nSweet\u2014Sweet\u2014Spirit,\" a continuous cry, as it were, of loving eulogy to\nthe devoted little wife who is so carefully hidden in her pocket nest in\na distant thorn tree. But all of this time we understand his clever\nmachinations, as he carefully leads us in an opposite direction by his\nsong allurements. He flits from tree to tree with a naive turn and\nflutter, keeping upon us all the time, an eye alert and keen, until he\ndeems us at a safe distance enough to be left to our own clumsy device,\nwhen, with a quick turn, he wheels backward to the starting-point, and\nwe hear a triumphant praise call to the beloved \"Sweet Spirit.\" Near a\ncorner of the old orchard where there are great bunches of Elder and\nSumach, we hear vehemently stitching, a busy little Maryland yellow\nthroat, doing up his summer song work with an energetic\n\"Stitch-a-wiggle, Stitch-a-wiggle, Stitch-a-wiggle, stitch 'em,\" the\n\"stitch 'em\" brought out with such emphatic force that it seems the last\nsatisfactory utterance of a work accomplished. His pert vivacity has\nbeen most delightfully illustrated by Ernest Seton-Thompson, in Frank\nChapman's \"Bird Life,\" and I am sure the snap-shot caught him on his\nlast accentuated \"stitch 'em.\" Dr. Abbot tells us that these busy little\npeople usually build their nests in the skunk cabbage plants, indicating\nthat they must have an abnormal odor sense, but perhaps they allow their\nsense of safety to overcome their sense of smell. However, this pair of\nyellow-throats have built instead, among some thickly matted Elders,\njust above the ground.\n\nAnother fact that favors our orchard in bird minds, is its close\nproximity to a thickly foliaged ravine which affords such delightful\nsecurity to feathered people. It is also a charming background for our\nsunny orchard, filled in below, as it is, with tall, ghostly stalks of\nblack cohosh gleaming white in the shadows.\n\nNear by, upon a bit of high ground, quivers a group of prim American\naspens, the pale green of their bark gleaming against the dark shadows\nof a hemlock hedge. As we look at them, not a leaf is in motion, when\nall of a sudden one little leaf begins to gesticulate frantically,\nthrowing itself about with violent wildness, then another leaf catches\nthe enthusiasm of the soft summer air, then another, and another until\nall of the trees are a mass of gesticulating, seething little serrated\natoms, for all the world like a congregation of human beings,\nvociferating, demonstrating, or contradicting some poor little human\nleaf that has dared to be moved by some passing thought in advance of\nhis fellow kind. Darting through the quivering foliage comes a gleam of\nfire, which resolves itself into a scarlet tanager who calls to us,\n\"look-see,\" demanding our attention to his bright beauty, remembering\npossibly that his brilliant coloring is but a thing of short duration,\nfor too soon will come winter and plain clothes. Perched upon a fence\nrail, but somewhat out of place in this shady corner, sits a blatant\nmeadow lark, about whose golden breast is hung a gleaming neck chain and\nlocket of shining black feathers, of which, from the pert poise of his\nhead, we deem him justly proud, and he is at least a conspicuous spot of\ncolor against the green of the hillside. He eyes us impertinently as he\ninconsistently but musically calls to us, \"You-can't-see-me,\nYou-can't-see-me,\" in the face of the most contradictory evidence of his\nown conspicuousness, varying his song to \"Erie-lake-Erie,\" with every\nother breath. As a child I used to wonder who taught him the name of the\ngreat lake on whose borders he makes his summer home. But to other\npeople, other interpretations, for to Neltje Blanchan he says\n\"Spring-o'-the-year, spring-o'-the-year,\" and to Frank Chapman his song\nis a bar of high, trilling notes. Sing on, you wary warbler, for we have\nnot time to search out your carefully hidden nest among the timothy\ngrasses of the distant meadow, for we know that it would be like looking\nfor the pearl in the oyster, so carefully is it concealed among the\ndried grasses, but which snakes and field mice depredate so effectually.\nIn the distant valley we hear the soft echo of the Italian liquids of\nthe wood thrush's \"A-o-le-le, a-oa-o-le.\" Shy little songster, who so\nsweetly trills to us long after his feathered kind have tucked their\nbusy little bills away in soft wings. Across the orchard comes the\nromantic \"Coo-coo-coo-coo,\" sometimes interpreted into\n\"I-thou-thou-thou,\" of the purple plumaged mourning dove, starting out\non a high minor and softly falling to a low contralto. There are no more\ndelightful representatives of romantic bird love, than these birds\nillustrate. More frequently than in any other species you see the\ndevoted pair going about together, on the telegraph wire, on the tree\ntop, on the wing, always together, undulating their graceful necks with\nmarked devotion. Many a bird lover has criticised Mr. Dove for his\nremarkable fondness for a lady who is a so decidedly slack housekeeper,\nand who is satisfied with so shiftless a nest in which to deposit the\ntwo white eggs, for the few carelessly thrown together sticks can prove\nanything but a bed of down to the tender bird babies. However, perhaps\nthese romantic birds consider that \"love is enough\" as they follow Le\nGallienne's refrain of:\n\n \"The bird of life is singing on the bough,\n His two eternal notes of 'I and Thou'\u2014\n Oh, hearken well, for soon the song sings through\n And would we hear it, we must hear it now.\"\n Alberta A. Field.\n\n\n\n\n THE CANADA GROUSE.\n (_Dendragapus canadensis_.)\n\n\nThe Canada Grouse, also called the Spruce Partridge, frequents the\nevergreen forests and swamps and the shrubby areas of British America\neast of the Rocky Mountains, and in Alaska it is a resident of the\nPacific coast. In its southern flights it seldom passes beyond the\nlatitude of the northern portion of New England and Minnesota.\n\nThis bird is an interesting member of the bird family Tetraonidae, which\nalso includes the birds variously called bob-white, quail and partridge,\nthe ptarmigans and the prairie hen. The family includes about two\nhundred species, about one-half of which belong to the Old World. There\nare twenty-five distinct species of the subfamily of grouse. These are\npractically confined to the higher latitudes of the northern hemisphere\nand are strictly speaking non-migratory. In fact, nearly all the birds\nof this family are resident throughout the year in the localities where\nthey are found.\n\nThey are terrestrial in their habits, and when frightened they usually\ndepend on hiding in places where their dull colors will least attract\nattention, but they will, occasionally, fly into trees when flushed.\n\nThe Canada Grouse, like all the related species, is a bird of rapid\nflight. The feathers of their small wings are stiff, causing a whirring\nsound during flight. The male during the mating season gives a great\ndeal of attention to his appearance. He is quite black in general color\nand more or less barred with white underneath and above with gray or\nreddish brown. The female is not quite as large as the male, and is not\nas dark in color. Above the eye of the male there is a small area of\nbare skin, which is a bright vermilion color.\n\nThese gentle and retiring birds mate in the early spring and remain\ntogether through the breeding season. Captain Bendire states that he has\ngood reason for believing that the mating may last for more than one\nseason, as he has frequently found a pair, in the depth of winter, when\nno other individuals of the same species were near. The nest, consisting\nof loosely arranged blades of grass and a few stalks and twigs, is built\nby the hen on a slight elevation of ground, usually under the low\nbranches of a spruce tree.\n\nThe number of eggs varies greatly. Mr. Ridgway says that they vary in\nnumber from nine to sixteen. The eggs also vary greatly in color from a\npale, creamy buff through various shades to brownish buff, and are\nirregularly spotted with a deeper brown, though occasionally they are\nspotless.\n\nDuring the spring and summer months the food of the Canada Grouse\nconsists very largely of the berries of plants belonging to the Heath\nfamily, such as the blueberry, the huckleberry and the bearberry, as\nwell as the tender buds of the spruce. In the winter it feeds almost\nentirely on these buds, and the needle-like leaves of the spruce, the\nfir or the tamarack trees. At times they seem to show a preference for\ncertain trees, and will nearly strip the foliage from them.\n\nAs a food for man their flesh is far from satisfactory. It is\ndark-colored and strongly flavored with the odor of their natural food.\nHowever, certain Indian tribes are said to relish them and hunt them\nextensively.\n\n [Illustration: ]\n\nMr. Bishop, in \"Forest and Stream,\" relates the following very\ninteresting account of the strutting of the male Canada Grouse while in\ncaptivity. He says, \"I will describe as nearly as I can his conduct and\nattitude while strutting: The tail stands almost erect, the wings are\nslightly raised from the body and a little drooped, the head is still\nwell up, and the feathers of breast and throat are raised and standing\nout in regular rows, which press the feathers of the nape and hind neck\nwell back, forming a smooth kind of cape on the back of the neck. This\nsmooth cape contrasts beautifully with the ruffled black and white\nfeathers of the throat and fore breast. The red comb over each eye is\nenlarged until the two nearly meet over the top of the head. This comb\nthe bird is able to enlarge or reduce at will, and while he is strutting\nthe expanded tail is moved from side to side. The two center feathers do\nnot move, but each side expands and contracts alternately with each step\nthe bird walks. The movement of the tail produces a peculiar rustling,\nlike that of silk. This attitude gives him a very dignified and even\nconceited air. He tries to attract attention in every possible way, by\nflying from the ground up on a perch, and back to the ground, making all\nthe noise he can in so doing. Then he will thump some hard substance\nwith his bill. I have had him fly up on my shoulder and thump my collar.\nAt this season he is very bold, and will scarcely keep enough out of the\nway to avoid being stepped on. He will sometimes sit with his breast\nalmost touching the earth, his feathers erect as in strutting, and\nmaking peculiar nodding and circular motions of the head from side to\nside; he will remain in this position two or three minutes at a time. He\nis a most beautiful bird, and shows by his actions that he is perfectly\naware of the fact.\"\n\nThere seems to be a diversity of opinion regarding the method followed\nby this grouse to produce the drumming sound. Mr. Everett Smith, as\nquoted by Captain Bendire, says, \"The Canada Grouse performs its\ndrumming upon the trunk of a standing tree of rather small size,\npreferably one that is inclined from the perpendicular, and in the\nfollowing manner: Commencing near the base of the tree selected, the\nbird flutters upward with somewhat slow progress, but rapidly beating\nwings, which produce the drumming sound. Having thus ascended fifteen or\ntwenty feet it glides quietly on the wing to the ground and repeats the\nmaneuver.\" According to this and other authorities a tree, usually\nspruce, having a diameter of about six inches and inclining at an angle\nof about fifteen degrees, is selected. Frequently these trees are used\nso extensively and for so long a time that the bark on the upper side\nwill be much worn. Other authorities, and among them Indians, who live\nin the regions frequented by this grouse, claim that the drumming is\nproduced while flying from the branches of a tree to the ground,\nrepeating the operation several times in succession. Another authority\ndescribes the drumming of the male as follows, \"After strutting back and\nforth for a few minutes, the male flew straight up, as high as the\nsurrounding trees, about fourteen feet; here he remained stationary an\ninstant, and while on suspended wing did the drumming with the wings,\nresembling distant thunder, meanwhile dropping down slowly to the spot\nfrom where he started, to repeat the same thing over and over again.\"\n\nThe Canada Grouse is easily domesticated and would make an interesting\nand amiable bird pet, because of their peculiar habits.\n\n Seth Mindwell.\n\n\n\n\n DO PLANTS HAVE INSTINCT.\n\n\nInstinct has been defined as a spontaneous impulse, especially in the\nlower animals\u2014that moves them, without reasoning, toward actions that\nare essential to their existence, preservation and development.\nInstinct, imbedded in their organic structure, is the guide of animal\nlife as reason is the guide of rational life. Instinct is said to be\nincapable of development and progress.\n\nIt is instinct that guides the wild goose in his long flight to meet the\nchanging requirements of food and nesting. It is instinct that enables\nthe carrier pigeon, though taken hoodwinked and by night to distant\npoints, to wing his way unerringly homeward. Instinct leads the thrifty\nsquirrel to stock his larder with nuts in anticipation of the period\nthat must pass ere nuts are ripe again, and teaches him to destroy the\nembryo plant by biting out the germ so that his chestnuts will not\nsprout and thus be spoiled for food. The same wonderful power enables\nthe bee to build her comb upon the strictest mathematical principles so\nas to obtain the greatest storage capacity and strength of structure\nwith smallest consumption of wax, and then to store it with one of the\nmost perfect and concentrated of foods. These and many other well-known\ncases of animal instinct will occur to the reader, but the object of\nthis article is to mention a few phenomena of plant life, whereby they\nmake, what we should designate in human beings, an intelligent\nadjustment to environment or provision for their future life and\ndevelopment.\n\nAs autumn approaches, even before Jack Frost strikes the first rude\nsignal for winter quarters for insect and plant, or the wintry blasts\ncompel the trees to furl sail and scud under bare poles, the forest\ntrees begin to prepare for unfavorable conditions by forming and\nsecurely tucking away the bud that is next year to develop into leaf and\nflower. Before the leaf drops off, a substantial layer of cork is made\nto close up the pores through which the sap had so freely flowed during\nthe growing season.\n\nMy older readers know, of course, that the green color of the leaf is\ndue to the numerous corpuscles of chlorophyll which fill the cells. This\nsame chlorophyll has an important mission to fulfill. These little green\nbodies are the only real food-making machines in nature. Upon the\nproduct of these tiny mills all animate nature depends for food. Their\nmotive power is light, and their raw material the inorganic fluids\nabsorbed by the roots from the soil, and their product is sugars and\nstarches. It will be seen that chlorophyll is one of the most precious,\nas well as one of the rarest of substances, for while there may appear a\ngreat quantity it is superficial, never entering deeply into the\nsubstance of the plant.\n\nThe trees, by a sort of instinct, shall we say, withdraw their cohorts\nof green-liveried workers from the front as autumn approaches and deck\nthemselves in the more gaudy but less wholesome colors of declining\nlife. It is after the chlorophyll is withdrawn that the layer of cork is\nformed. The sturdy oak usually holds his brown leaves until they are\nwhipped off by the wind.\n\nThe plants have been using light as a motive power for ages, while man,\nwith his much-vaunted reason, is just beginning to utilize the kindred\nforce, electricity, in arts and sciences. Man makes light draw a few\npictures in sombre black and white, while nature flings broadcast\nlandscape and life scenes in varied tints and shades.\n\nIn the process of photosynthesis much more energy is received than is\nnecessary to run the machinery, so the plant, with commendable\nfrugality, uses it in laying on what botanists call warming-up colors.\nIf you will notice the peach twigs the next time you take a walk, you\nwill see that the more tender shoots and the buds are decked in rich\nreds and browns. That this is not for mere ornament may be practically\ndemonstrated by wrapping the bulbs of two similar thermometers, the one\nwith a green leaf, the other with a brown or red leaf, say of begonia or\nbeet. Then put the two in the sunlight and you will soon find a\ndifference of from six to ten degrees in favor of the warming-up color.\nSpeaking of buds, have you examined the horse chestnut bud? It is\nprepared for the winter in the most substantial manner. The future leaf\nis first wrapped in a quantity of finest silky wool, then a number of\ntough light green cases are put on, and this is followed by compact\nbrown scales neatly overlapping, with a complete coating of wax, so that\nthe interior is effectively protected from the cold and moisture. The\nuse of the warming-up colors is quite common with plants.\n\nIn the far north the same plant that requires the whole long growing\nseason to mature its seed, will crowd the whole process into a few\nweeks. It will suspend growth and all other processes, or run them on\nshort time and devote itself almost entirely to producing seed, and the\nseed itself will have much thicker shell.\n\nI was interested last autumn in the pathetic struggle of a humble little\nChenopodium album that had started life late and under unfavorable\ncircumstances. It came up in September under the north piazza near the\nbeaten foot path; close up to the building. I was first attracted by the\nfact that, though it was not over a foot high, it had bloomed and was\nmaking seed at a desperate rate, while its sisters earlier in the season\nreached several feet in height before blooming. But, alas! for the\nvanity of the poor little creature, the cold weather during the\nChristmas holidays came on, and the steam being shut off, the side of\nthe building grew cold and my struggling little friend was frozen, and\nsoon its lifeless remains were the sport and derision of the rude\nJanuary winds. I pitied the poor little vagabond despite the bad record\nof her family. Indeed plants, like people, must suffer sometimes because\nof an evil ancestry. In this case I was touched by the pathos of the\nsituation, and really hoped the pertinacious little wretch might proudly\nscatter her well-matured seed upon the hard-beaten path as an\ninspiration to the many boys that passed daily, grumbling because of the\nhardness of their lot. But the only moral I can now draw is the\nfoolishness of delaying in the right start.\n\nSometimes the supply of light-energy is so great that the little\nchlorophyll machines cannot use it in their legitimate work, nor does\nthe plant use it in preparing the warming-up color. Then the disc-shaped\ncorpuscles turn their edges instead of their flat surfaces to the light,\nor sometimes move deeper down into the leaf. In some cases the leaf\nitself turns edgewise instead of broadside to the sun.\n\nThere are many plants so constituted that they cannot live from year to\nyear in our northern climate, and they must make some provision for\npreserving their species, and right cunningly do they do this. At a\ncertain period of its growth the potato, for example, puts its\nstarch-making machinery to work on full time, and hurries the starch\ndown below the surface of the ground, and stores it up in what we call a\ntuber. These tubers have stored in them a number of embryo potato\nplants, whose lack-luster eyes we see peeping out on all sides. When the\ntime for growth comes, the young plant starts with a reserve-food supply\nsufficient to keep it growing for some time. We have all noticed, no\ndoubt, how large a plant will grow from a potato, even in a\ncomparatively dark cellar. We must not think that tuber-bearing vines\nand nut-producing trees are actuated entirely by philanthropic motives.\nEach nut is the young tree sent forth with his patrimony strapped to his\nback, ready to make a good start in the world as soon as the favorable\ntime comes.\n\nThere are many devices for spending the winter that limits of time and\nspace will prevent me writing about. Many of them more curious than the\nsimple examples I have cited.\n\nPlants are themselves generally unable to move from their fixed\npositions, so if they are to become prominent in the world they must\nsend out their children\u2014and many and ingenious are their devices for\naccomplishing this end. Most of my readers are familiar with the\nparachutes of the silk weed, dandelion and various members of the\nCompositae family. How they sail through the air. A walk through the\nautumn forests will make one the unconscious, perhaps unwilling, carrier\nof numerous Spanish needles, stick tights, burrs and seeds of various\nplants who have taught their children to steal rides in all sorts of\nprovoking ways. I imagine the wicked old mother laughs as her ugly baby\nclings to your clothing, sure of a safe ride to a more favorable place\nfor growing. Many plants achieve the same end in a more pleasant way.\nThey produce fruits and berries so luscious that some bird or animal\nwill carry it some distance for the sake of the pulp. Man himself,\nphilanthropist as he is, when he finds that a plant has produced a\nluscious fruit or palatable seed, will help the distribution and growth,\nand bring his superior intelligence to the assistance of the plant's\nslow instinct to improve its product. A book might be written upon the\nmethods of seed dissemination. In fact, there is a very interesting book\nupon the subject.\n\nWe will just notice briefly the marvelous adaptation of plants to their\nenvironment. In the dry plains of Arizona grows a peculiar thick-leaved,\nstunted, cactus-like plant, suited to withstand the drouth. In the\nforests of Central South America a great vine climbs to the tops of the\ntallest trees and there flaunts its gay colors to the breeze. In Damara\nLand, southwest tropical Africa, upon a small upland section, and\nnowhere else in the world, grows the marvelous Welwitschia mirabilis,\nwith no real leaves, but with its two cotyledons, persistent and growing\nto enormous length, living a century and acquiring a great trunk, the\nflower-stalk growing up from the bare trunk while the two great leaves,\nif I may so designate them, whip about in the breezes for a century\nwithout change, except as they fray out at the ends. These three so\ndissimilar plants all had a common, not so remote, ancestor, but have\ngrown so unlike in their effort to adapt themselves to their\nenvironment, that no casual observer would suspect they were akin.\n\nThere is so much to say about the wonderful intelligence displayed by\nplants in their various activities, that a volume could not do the\nsubject justice. We started with the question, Do plants have instinct?\nWe end with the question, Have they?\n\n Rowland Watts.\n\n\n Still winter holds the frozen ground and fast the streams with ice are\n bound,\n There's many a dreary week to come before the flowers bloom;\n Though everything were lost in snow yet Nature's heart beats warm\n below\n And Spring will build her palace gay on hoary Winter's tomb.\n \u2014George Gee.\n\n [Illustration: ]\n\n\n\n\n THE DOVEKIE.\n (_Alle alle._)\n\n\nThis little bird, often called the Sea Dove, belongs to the family of\nauks (Alcid\u00e6). The range of the Dovekie is quite limited. While the\nmarble murrelet, a related bird, is confined to the northern Pacific\ncoast of North America, this little bird frequents only the \"coast and\nislands of the north Atlantic and eastern Arctic Oceans; in North\nAmerica south in winter to New Jersey.\" It breeds only in the northern\npart of its range. It has been observed as far west as the state of\nMichigan, but its appearance there was, without doubt, accidental, for\nit prefers the wild sea coast, where the storm and waves bring to it an\nabundant supply of food.\n\nIt is said to be a rare visitor on the coasts of the British Islands and\nit has been reported as common as far to the northward as Spitzbergen.\nIn Greenland, where it is commonly found a close companion of the\nblack-billed auk, the native Greenlanders call the Dovekie the Ice Bird,\nas they consider it a harbinger of ice.\n\nThough the wings of the Dovekie are small in proportion to the size of\nits body it flies well and rapidly. One writer states that it will move\nits wings almost as rapidly as will a humming-bird. It is an expert\ndiver and while swimming or resting on the water it will frequently dip\nits bill into the water. On the land it is much more graceful and walks\nbetter than nearly all the other members of the family of auks.\n\nIt feeds chiefly on small fish, crustacea and mollusks and will become\nvery fat during a prolonged stormy season when the waves wash up an\nabundant supply of crabs and fish.\n\nThe Dovekie builds a simple nest usually in the crevices of rocky cliffs\nbordering the sea coast. It lays one or two bluish white eggs which are\nabout the size of the pigeon's.\n\nMr. Saunders in speaking of the habits of the Dovekie says: \"On the\napproach of a vessel this bird has a peculiar way of splashing along the\nsurface of the water, as if unable to fly, and then diving through the\ncrest of an advancing wave; it swims rather deep and very much by the\nstern.\"\n\nThe Dovekie is sometimes called a little auk to distinguish it from the\nlarger species of the family. The flightless great auk, which at one\ntime was common along the north Atlantic coast, belongs to this family.\nNo living representative of the great auk has been reported since the\nyear 1842. Unable to protect itself by flight it was ruthlessly\nexterminated by the zeal of hunters and fishermen who sought it for\nfood, for its feathers and for the oil that could be extracted from its\nflesh.\n\n\n As flying ever westward Night's shadows swiftly glide,\n The sunrise at the dawning illumes the countryside.\n The stars in quick succession in ether melt away,\n Until the brightest planet is lost in glowing day.\n \u2014George Gee.\n\n\n\n\n THE SONG SPARROW'S APPEAL.\n\n\nNaturalists tell us that of all creatures below man, the largest animal\nbrain in proportion to the size of the body is found in horses and\nsong-birds. Whatever sense beyond instinct the little creature of whom\nwe write may have had, something, at least, told it that it could obtain\nhelp at human hands.\n\nA little sparrow the past season entered the kitchen of one of our\ncountry homes, and perched upon the window-sill in evident distress. Its\nfeathers were ruffled, and its head ever and anon turned curiously\naround and up, as if looking at something out of the house and above the\nwindow.\n\nIn and out it continued to hop, without intermission, regardless of all\noffers of food, until the shutters were closed at twilight, and various\nwere the surmises as to the cause of its strange conduct.\n\nThrough the course of the following day the same scene was enacted,\nwithout any clue appearing as to the cause of its distress.\n\nAt length, on the third morning, the mute petition for aid still\ncontinuing, one of the family, bethinking herself of the bird's curious\nupturning of the head, caught a new idea from it. Perhaps she might have\na nest in the ivy that encircled the window, and something might be\namiss with its little household.\n\nGoing to the second story and looking down, the cause of the trouble was\nat once manifest. A thick limb of the ivy had become loosened by the\nwind, and fallen directly across the petitioner's nest. It was too heavy\nfor the bird to remove, and offered an insuperable difficulty in the way\nof her getting in to feed her young\u2014now almost lifeless.\n\nThe branch was quickly removed, when the mother-bird, pausing only for a\nbrief inspection of her brood, was on the wing in search of food. Her\nmate soon joined her, and both were busy as quick wings, worked by\nhearty good will, could make them.\n\nOnce only did the mother pause in her work\u2014as if desirous to give\nexpression to her gratitude, she reappeared upon the window-seat, and\npoured forth a sweet and touching song, as of thankfulness to her\nbenefactors.\n\nShe returned three successive seasons, to be noticed and fed at the same\nspot where her acquaintance and familiarity with man first commenced.\n\nWe will add another similar incident, which is also absolutely true.\n\nThe correctness is vouched for by Mr. George Babbitt, late captain on\nGen. Gresham's staff, of which he himself was a witness.\n\nDuring the fierce cannonading in one of the battles of the Civil War, a\nsmall bird came and perched upon the shoulder of an artilleryman\u2014the man\ndesignated, we believe, as \"No. 1,\" whose duty it is to force down the\ncharge after the ammunition is put in the gun. The piece was a\n\"Napoleon,\" which makes a very loud report, and the exact scene of this\noccurrence was at a place called \"Nickajack.\" The bird perched itself\nupon this man's shoulder and could not be driven from its position by\nthe violent motions of the gunner. When the piece was discharged, the\npoor little thing would run its beak and head up under the man's hair at\nthe back of the neck, and when the report died away would resume its\nplace upon his shoulder. Captain Babbitt took the bird in his hand, but\nwhen released it immediately resumed its place on the shoulder of the\nsmoke-begrimed gunner. The singular and touching scene was witnessed by\na large number of officers and men. It may be a subject of curious\ninquiry, what instinct led this bird to thus place itself. Possibly,\nfrightened at the violent commotion caused by the battle, and not\nknowing how to escape or where to go, some instinct led it to throw\nitself upon the gunner as a protector. But, whatever the cause, the\nincident was a most beautiful and pleasing one to all who witnessed it.\n\n George Bancroft Griffith.\n\n\n\n\n THE WITCH IN THE CREAM.\n A TRUE STORY.\n\n\nThe old stone farm-house in which my grandmother lived had beneath it\nwhat I thought a very interesting cellar. The floor was plastered and\nwhitewashed like the walls, to ensure the place from rats and other\nintruders, as well as to keep it cool. From the walls, flat stones\nprojected, serving as shelves on which the butter and milk were kept.\nFor years the milk had had a shelf to itself near the window.\n\nOne summer morning, while Grandma and I were sitting on the porch\nwaiting for breakfast, the little colored servant came to us with\nwide-open eyes, saying: \"La, Missy, jes look at dis milk-pan!\" We\nlooked, and saw, to our disgust, that the inside of the pan was covered\nwith sand and grime, while the milk, which usually was coated with rich,\nthick cream, was thin and poor. \"Why, Janey,\" said Grandma, \"you didn't\nput milk away in a pan like that, did you?\" \"La, no, Missy,\" said Janey,\n\"nobody wouldn't nebber put milk away in a dirty pan.\" \"This is very\nstrange,\" said Grandma. \"You will have to throw the milk away, Janey,\nand be especially careful to have the pan clean this evening.\" \"Yes'm,\"\nsaid Janey, \"I will.\"\n\nThe following morning, however, the milk had to be thrown away again, as\nthe pan was in a worse condition than on the preceding morning. \"I don't\nunderstand it,\" said Grandma. \"It can't be rats, nor mice, for there is\nno way for them to come in.\" \"They couldn't climb into a tin pan eight\ninches high, at any rate,\" I said, \"and if they jumped in they would\ndrown.\" Janey shook her head knowingly and said, \"It's witches, Missy,\ndat's jes what it is.\" A light board was placed over the milk that\nevening, but we found that the marauder pushed it off in the night. We\nfelt that we must come to Janey's conclusion about the witches, if the\nmystery were not solved soon.\n\nIn the afternoon of the third day of these experiences we were sitting\non the back porch with our sewing, both of us half asleep, when chancing\nto look up I saw a rat go scudding across the yard. Straight to the\ncellar window he went, and, approaching one corner, thrust his nose\nunder the sash. He gave a mighty tug, pushed one paw under, and soon, by\npushing and pulling with nose and with paws, he crept through the\nwindow. From my position on the porch I could see all that was happening\nin the cellar. He jumped to the milk shelf, turned around, raised\nhimself on his forepaws, and clasped the edge of the milk pan with his\nhind ones.\n\nHe then threw his tail into the pan, whisked it rapidly over the milk,\ncoating it with cream, and licked it. This he repeated until he had a\nfull meal, or at least until he had skimmed all the cream.\n\nHe started homeward then, and I was so much amazed that I didn't attempt\nto stop him. On the following morning he was caught in the steel trap\nset just inside the window for him.\n\n Elizabeth Roberts Burton.\n\n\n\n\n THE BEAVER.\n\n\nThe genus of Beavers (Castor) is apparently represented by a single\nliving species. By some authorities the American form is considered a\ndistinct species and is given the technical name Castor canadensis,\nwhile the European form is called Castor fiber. In external\ncharacteristics the two resemble each other very closely, and it is in\nthe study of the structure of the skeleton that the differences appear.\nHowever, though there is this diversity of opinion, it is sufficient for\nthe reader to look upon the two forms as merely geographical races of\nthe same species, and that the Beaver is a native of the greater part of\nthe northern hemisphere. Though its home covered this extensive area, it\nhas disappeared from the larger number of localities that it once\nfrequented. Speaking of its range as a whole, it may now be considered\nrare except in certain isolated localities. This extermination is due to\nthe advance of civilization upon its natural haunts, and the commercial\nzeal that has stimulated the hunter to greater efforts to effect its\ncapture. Within recent years the Beaver was common in some of the Gulf\nStates. In 1876 it was reported as abundant in Virginia. It is evident\nfrom an examination of the numerous writings regarding its distribution\nthat the Beaver formerly existed in great numbers not only in the\nAtlantic States, but also to the westward as far as the Pacific coast.\n\nThe Beaver is a member of that large order of gnawing mammals called the\nRodentia, from the Latin word meaning to gnaw. In this order are classed\nall those animals that have those peculiar long incisor teeth which are\nconstantly renewed by growth from the roots and as constantly worn to a\nchisel edge, at the outer end, by gnawing. Such animals are squirrels,\nthe gophers, the mice, the rats, the muskrats, the porcupines, the hares\nand the rabbits.\n\nThe habits of the Beaver are very interesting. Several years are\nrequired before its growth is fully attained, and it will increase in\nsize after the teeth are fully mature. \"Two-year-old Beavers generally\nweigh about thirty-five to forty pounds, while very old ones\noccasionally attain a weight of upwards of sixty. Morgan records the\ncapture of one which weighed sixty-three pounds. The increase in the\nsize of the skull seems to continue nearly through life; in old age the\nskull not only acquires larger dimensions, but the weight is relatively\ngreater in consequence of the increased thickness and density of the\nbones. The ridges for the attachment of muscles also become more\nstrongly developed in old age.\"\n\nThe general color of the back of the Beaver is a reddish brown. The\nshade varies both with the seasons and with the geographical location.\nThose found farther to the northward are usually darker. Albinos, either\npure white, nearly white or with white blotches, have been observed.\n\n\"The fur consists of an exceedingly thick, flaky, woolly coat of silky\nsoftness and a thin, long outer coat composed of strong, stiff, shining\nhair, short on the head and rear part of the back and over two inches\nlong on the rest of the body.\" The tail, which is rounded at the base,\nmuch flattened and very broad, bears horny, dark-colored scales.\n\nThe fore legs are short and the feet are unwebbed. The hind legs are\nmuch stronger, the feet are fully webbed and they, alone, are used, with\nthe aid of the tail, to propel the Beaver through the water. In the\nwater it is graceful in its motions, but on the land, like nearly all\nanimals that are fitted for a partially aquatic life, it is clumsy and\nawkward and its motions are neither rapid nor uniform.\n\n [Illustration: ]\n\nUsually it is only in those districts that are remote from the\nhabitations of man that the Beaver lives in colonies, consisting of\nseveral families, and builds its \"lodges.\" Nearer civilization it lives\nin burrows or tunnels. In the building of their homes, as well as in the\nstoring of a supply of food, the female is the most active and is the\npractical builder, while the male assists.\n\nBrehm writes interestingly regarding the Beaver. He says: \"After mature\ndeliberation the animals select a stream or pool, the banks of which\nafford them ample provender and seem specially adapted for the\nconstruction of their 'lodges.' Those which live singly dwell in simple\nsubterranean burrows, after the manner of otters; societies, which\ngenerally consist of families, as a rule construct houses and, if there\nshould be a necessity for it, dams, in order to hold back the water and\npreserve it at a uniform height. Some of these dams are from four\nhundred and fifty to six hundred feet long, from six to nine feet high,\nfrom twelve to eighteen feet thick at the base and from three to six\nfeet at the top. They consist of logs varying in size from the thickness\nof an arm to that of a thigh and from three to six feet long. One end of\nthe log or stake is thrust in the ground, the other stands upright in\nthe water; the logs are fastened together by means of thin twigs and\nmade tight with reeds, mud and earth, in such a way that one side\npresents a nearly vertical, firm wall to the stream, while the other\nside is sloped. From the ponds rising above the dams, canals are\nconstructed to facilitate the carrying or floating of the necessary\nconstruction materials and food. Beavers do not forsake a settlement\nthey have founded unless the direst necessity compels them to do so.\nBeavers' lodges, the origin of which dates very far back, are often\nfound in lonely woods.\"\n\nThe Beaver usually feeds upon the bark of the younger branches of trees\nand shrubs and upon their leaves. It will also strip the older branches,\nin a very skillful manner, and eat the inner tender portion of the bark.\nDuring the fall and early winter months they work constantly in\npreparing and storing, in the neighborhood of their lodges, the winter's\nsupply of food. \"Each cabin has its own magazine, proportioned to the\nnumber of its inhabitants, who have all a common right to the store and\nnever pillage their neighbors.\"\n\nThe American Indians look upon the Beaver with great respect. They\nbelieve that it is possessed of a degree of intelligence second only to\nthat of man. Some Indians even assert that it possesses an immortal\nsoul. Its sagacity is certainly very strong and it will easily adapt\nitself to changed environments. Unlike the other rodents, it seems to\nreason before acting and will build its habitations in the form that the\nsurrounding conditions demand for the construction of the most durable\nhome.\n\nThe Beaver, especially when young, is quite easily domesticated. Various\nwriters speak of finding tame Beavers in Indian villages, where they\nseemed to be perfectly at home and contented. They were allowed full\nliberty. \"They seemed to feel quite comfortable in the society of the\nIndian women and children; they grew restless in their absence and\nshowed much pleasure on their return.\"\n\nThe young, which number from two to three, are born blind, but are\ncovered with fur. They usually obtain their sight in from eight to ten\ndays, and are then led to the water by the mother.\n\nEarly in the nineteenth century Dr. George Shaw wrote as follows\nregarding the habits of the Beaver: \"They collect in September their\nprovisions of bark and wood; after which they enjoy the fruits of their\nlabors, and taste the sweets of domestic happiness. Knowing and loving\none another from habit, from the pleasures and fatigues of a common\nlabor, each couple join not by chance, nor by the pressing necessities\nof nature, but unite from choice and from taste. They pass together the\nautumn and the winter. Perfectly satisfied with each other, they never\nseparate. At ease in their cabins, they go not out but upon agreeable or\nuseful excursions, to bring in supplies of fresh bark, which they prefer\nto what is too dry or too much moistened with water.\"\n\n\n\n\n PAU-PUK-KEEWIS AND THE BEAVERS.\n\n\n Over rock and over river,\n Through bush, and brake, and forest,\n Ran the cunning Pau-Puk-Keewis;\n Like an antelope he bounded,\n Till he came unto a streamlet\n In the middle of the forest,\n To a streamlet still and tranquil,\n That had overflowed its margin,\n To a dam made by the beavers,\n To a pond of quiet water,\n Where knee-deep the trees were standing,\n Where the water-lilies floated,\n Where the rushes waved and whispered.\n On the dam stood Pau-Puk-Keewis,\n On the dam of trunks and branches,\n Through whose chinks the water spouted,\n O'er whose summit flowed the streamlet.\n From the bottom rose the beaver,\n Looked with two great eyes of wonder,\n Eyes that seemed to ask a question,\n At the stranger, Pau-Puk-Keewis.\n On the dam stood Pau-Puk-Keewis,\n O'er his ankles flowed the streamlet,\n Flowed the bright and silvery water,\n And he spake unto the beaver,\n With a smile he spake in this wise:\n \"O my friend Ahmeek, the beaver,\n Cool and pleasant is the water;\n Let me dive into the water,\n Let me rest there in your lodges;\n Change me, too, into a beaver!\"\n Cautiously replied the beaver,\n With reserve he thus made answer:\n \"Let me first consult the others,\n Let me ask the other beavers.\"\n Down he sank into the water,\n Heavily sank he, as a stone sinks,\n Down among the leaves and branches,\n Brown and matted at the bottom.\n On the dam stood Pau-Puk-Keewis,\n O'er his ankles flowed the streamlet,\n Spouted through the chinks below him\n Dashed upon the stones beneath him\n Spread serene and calm before him,\n And the sunshine and the shadows\n Fell in flecks and gleams upon him,\n Fell in little shining patches,\n Through the waving, rustling branches.\n From the bottom rose the beavers,\n Silently above the surface\n Rose one head and then another,\n Till the pond seemed full of beavers,\n Full of black and shining faces.\n To the beavers Pau-Puk-Keewis\n Spake entreating, said in this wise:\n \"Very pleasant is your dwelling,\n O my friends! and safe from danger;\n Can you not with all your cunning,\n All your wisdom and contrivance,\n Change me, too, into a beaver?\"\n \"Yes!\" replied Ahmeek, the beaver,\n He the king of all the beavers,\n \"Let yourself slide down among us,\n Down into the tranquil water.\"\n Down into the pond among them\n Silently sank Pau-Puk-Keewis;\n Black became his shirt of deer-skin,\n Black his moccasins and leggins,\n In a broad black tail behind him\n Spread his fox-tails and his fringes;\n He was changed into a beaver.\n \u2014Henry Wadsworth Longfellow, \"The Song of Hiawatha.\"\n\n\n What rosy pearls, bright zoned or striped!\n What freckled surface, iris-dyed!\n Fluted and grooved, with iv'ry lips,\n Spotted like panthers, peacock-eyed!\n\n Look closer, as the angels can,\n And you will see the fairy work\u2014\n The ruby specks, the azure veins,\n That in the tiniest hollow lurk.\n \u2014Walter Thornbury, \"Shells.\"\n\n\n\n\n SNAILS OF THE OCEAN.\n\n\nMany of my readers have doubtless spent some of the vacation months at\nthe sea shore and have wandered over the beach at low tide picking up\nshells and other objects left by the receding ocean. They have also, I\nam sure, peered into the little pools of water left on the beach and\nhave watched with interest the captives imprisoned therein, hermit\ncrabs, fiddler crabs, sea anemones, sea worms and snail shells. It is\nwith the latter that the present article will deal.\n\nThe stretch of beach which is uncovered twice a day by the receding of\nthe water is called \"between tides,\" and is inhabited by a host of\nanimate creatures, chief among which are the mollusks. The marine snails\noutnumber all of those which we discussed in the last article, and their\nshells are far more beautiful, those found in the tropics having the\nmost gaudy colors imaginable. The animals are formed on the same plan as\nthose of the fresh-water snails, although each family has some\npeculiarity not shared by its relatives. All live in the water and\nbreathe air through that medium by means of gills, similar to the second\nclass of fresh water snails mentioned in the last number. They are found\nin all parts of the world, those of the tropics, however, being the most\nbrilliantly colored. While the majority of species live either between\ntides or near low water, there are not a few which live in the abysses\nof the ocean, and have been dredged from the bottom of the sea at a\ndepth of two thousand, seven hundred and forty fathoms, or, to put it\nmore plainly, over three miles. The average depth at which mollusks are\nfound in any number is about one thousand fathoms. The variability of\nmarine snails is so great that we shall be able to call attention to but\na limited number of typical forms.\n\nAmong the best known of the marine snails are the Tritons, a family of\nmollusks living in tropical seas. Their shells are generally large and\nhighly-colored and variously ornamented with short spines and knobs. One\nspecies, the Triton tritonis, is among the largest of mollusks,\nmeasuring eighteen inches in length. One of the smaller Tritons is\npictured on the plate. Another shell familiar to those who have visited\nFlorida is the Fasciolaria or banded snail, which attains a length of\nthree inches and is very prettily banded and dashed with color. A near\nrelative of this species is the giant banded shell (Fasciolaria\ngigantea), which is the largest of all marine snails, growing to a\nlength of nearly two feet. This species is found plentifully on the\nsouthern Atlantic coast of the United States, being particularly\nabundant about the coral reefs of the Florida Keys.\n\nA genus of mollusks with light horn colored shells, and inhabiting the\ncold waters of the Arctic seas, is the Buccinum, or whelk. In various\nparts of Great Britain it is known as \"buckie\" and \"mutlog.\" The\nBuccinum delights to burrow in the sand, like the moon shells (Natica),\nand frequently nothing but the end of the siphon can be seen, the latter\nprotruding from the sand to enable the water to enter the animal to\nfurnish the necessary oxygen. The whelk is used economically, both for\nfood and bait. One ingenious method of catching them is to fasten a dead\nfish of good size in a wire basket and to allow it to rest on the bottom\nfor a short time; when taken up it is covered with large, fat whelks.\nThis fishery in Great Britain is fully as valuable as our oyster\nfishery, the annual income from this industry reaching to thousands of\npounds sterling. The animal is also one of the principal baits used in\ncod fishing. A related genus, the neptune shells (Neptunea), is also\neaten by the poorer people and makes a good codfish bait. The two kinds\nof whelk (Buccinum and Neptunea), are termed, the first the white whelk\nand the second the red or almond whelk, probably on account of the\ncolors of the two shells. In the Shetland Islands the red whelk is used\nas a lamp, being suspended by strings from a nail, the mouth placed\nuppermost and filled with oil.\n\n [Illustration: ]\n\n First row:\n Cypraea pantherina (Red Sea)\n Cassis flammea (Bahamas)\n Conus marmoreus (Polynesia)\n Second row:\n Buccinum undatum (U. S.)\n Fasciolaria distans (U. S.)\n Third row:\n Tritonium olearium (Naples)\n Oliva irisaus (Amboina)\n Voluta musica (West Indies)\n Fourth row:\n Ianthina communis (Atlantic Ocean)\n Chiton squamosus (Jamaica)\n Lottia gigantea (California)\n Nassa glans (Amboina)\n\nThe basket shells or dog-whelks are among the most numerous in\nindividuals of all the marine snail shells, the common black whelk\n(Nassa obsoleta) being the most common of all the mollusks. The writer\nhas seen a mud flat at low water literally paved with the shells of this\nsnail, there being millions of the little creatures crawling about. The\nshells of this family are frequently very handsome, being latticed by\nthe crossing of lateral and longitudinal lines. They are mostly of small\nsize, scarcely exceeding an inch in length, many of them being much\nunder these dimensions. The animal is very rapid in movement and leaves\na distinct track in the mud, which will frequently end at a little\npellet of mud, which, upon examination, will disclose the little animal\nnicely concealed beneath.\n\nThe Nassas of France are very destructive to the oyster beds of that\nnation, an adult \"borer\" being able to perforate the shell of a large\noyster in a single night. So numerous are these pests that a single acre\nhas yielded over a thousand individuals. As a result of these\ndepredations the French oystermen carry on a relentless war against the\nNassa, destroying thousands of animals annually. With all this\npersecution the mollusk still exists and even increases in numbers. The\ndead shells of this genus are a favorite home for the hermit crabs of\nsmall size, and it is to be suspected sometimes that other than dead\nshells are appropriated. We fear that a sort of piracy is resorted to by\nthe hermit crab, resulting in a kind of \"walk-the-plank\" end for the\nmollusk, before the new tenant takes possession of the \"home.\"\n\nOf the many varieties of tropical shells, few exceed the Volutes, or bat\nshells, in beauty or variety of coloration. They are found in most parts\nof the world, although strangely enough none are now living in the seas\nof Europe, but they are most abundant and more highly colored in the\ntropics and subtropics. The animal is carnivorous, and the long,\nfang-shaped teeth are certainly suggestive of predaceous habits. The\nshells are variously colored, some being mottled, some with zigzag or\nlightning-like markings, while others have spirally arranged dots and\nlines. One species (Voluta musica, figured on the plate), has received\nits name from a more or less fanciful resemblance of the surface of the\nshell to a musical staff, the spiral lines being grouped in sets of four\nor five and the dots being arranged as notes. In some specimens this\nresemblance is quite close. The smooth and polished shell of some\nvolutes is due to the fact that the greater portion is covered by a\nreflected part of the large foot.\n\nOn the sandy shores of subtropical beaches certain graceful and polished\nanimals bury themselves from sight in the sand. These are the olive\nshells (Oliva) whose bright colors and highly polished surfaces rival\neven the gaudy Volute in beauty. The foot may be described as\nplough-shaped and is admirably adapted for digging rapidly in the sand,\nso that the shell may be hidden from sight on the approach of enemies.\nThe long siphon is thrust up through the canal in the anterior part of\nthe shell and its end protrudes above the sand. The high polish of the\nsurface is due to the shell being enveloped in the voluminous foot;\nhence it has no epidermis. The aperture is so narrow that it is\ndifficult to understand how the animal gets in and out. The olives are\nvery numerous in individuals; when one is found hundreds are sure to\nreward a patient search.\n\nProbably no more distinct family of mollusks exists than the Conidae,\nthe family of cones, their beautifully decorated shells and the large\nnumber of species making them a favorite with collectors. The shell is\nin the form of an inverted cone, gracefully rounded, the aperture being\nbut a narrow slit extending nearly the whole length of the shell. The\ncolors of the cones are always very brilliant, although when they are\nalive the shell is not brilliantly polished as the olives, on account of\nthe presence of an epidermis. About three hundred species are known,\nliving principally in tropical seas. They love to conceal themselves in\nholes in the rocks and among the branches of corals. The animal is\npredaceous, boring into the shells of other mollusks and extracting the\njuices from the bodies. The teeth of Conus are hollow and very sharp and\nhave a barb on the end. A poison gland is said to be present in this\ngenus and bites from the animal are very painful, although not\ndangerous, the large Conus marmoreus being able to inflict a severe\nwound. The cone is quite pugnacious and will immediately bite the hand\nwhen picked up, a veritable reptile of the ocean.\n\nThe ne plus ultra of mollusks to the collector is without doubt the\ngenus Cypraea, comprising the cowry shells. So eagerly have they been\nsought by wealthy collectors that the price of rarities has gone up to\nan astonishing degree, some specimens being sold at several hundred\ndollars each. The shell is highly polished, owing to the fact that two\nlobes of the voluminous mantle are turned back over the shell and meet\nin the middle of the back. The foot is very large and spreading, the\nmantle beset with curious little tentacular-like organs and the eyes are\nplaced on small swellings near the base of the long, cylindrical\ntentacles. The color-patterns of the shell vary to a wonderful degree.\nThe young shell has a thin epidermis, a sharp lip to the aperture and a\nmore or less prominent spire, the rolled over and toothed lip and\npolished surface not being acquired until fully adult. No more beautiful\nsight can be imagined than one of these gorgeous animals, as seen\nthrough the clear water, crawling over the sandy bottom or on the branch\nof some coral.\n\nSeveral of the cowries have a curious economic value. Thus, Cypraea\naurantia, the orange cowry, was used as an insignia of royalty by the\nchiefs of the Friendly Islands, and for a long time the only specimens\nobtainable were those which had been bored and used. The money cowry\n(Cypraea moneta) has been used as money by the natives of Western\nAfrica, and many tons of this small shell were annually imported to\nEngland to be used in barter by the African traders. The shell is of a\nyellowish or whitish color, does not exceed an inch in length, and is\nvery common in the Pacific and Indian Oceans. It is still used as a\nmedium of barter in parts of Africa, although other things have pretty\ngenerally taken its place.\n\nCameos were at one time quite in the fashion, both as ornaments for the\nperson in the way of brooches, and as bric-a-brac about the room. These\nshell-cameos are made from the genus Cassis, the helmet shells. These\nare well adapted for this purpose, as the shell is made up of several\ndifferently colored layers, making a bas relief figure not only possible\nbut very effective. The black helmet (Cassis madagascariensis) is one of\nthe best for this purpose, the figure being carved from the white, outer\nlayer of shell, which stands out very clearly against the black\nbackground of the second layer. When a cameo is desired simply as a\nbrooch or for any other form of personal adornment, a piece of the shell\nis cut out and shaped into the required form and size\u2014oval, square or\nother shape\u2014and cemented to a block of wood. The figure is then traced\non the shell with a pencil and finally carefully worked out with sharp,\npointed steel instruments, of delicate size and form. The same process\nis resorted to in working out a bas relief on the entire shell, only the\nlatter is placed in a vice or other object to hold it firmly. The home\nof this industry is Genoa and Rome, Italy, although some are produced in\nFrance; these latter, however, are of a poorer quality. Several thousand\npeople are employed in this trade. Many beautiful examples of this work\nwere exhibited at the World's Columbian Exposition, in Chicago, in 1893.\n\nThe cameo shells are among the largest of sea snails, several of them\nmeasuring eight or ten inches in length and weighing several pounds.\nThey are found only in tropical and subtropical seas, living in\ncomparatively shallow waters on a sandy bottom. They are voracious\neaters, living principally on bivalve mollusks.\n\nOne of the most abundant of mollusks is the violet sea snail (Ianthina\ncommunis), which spends its life floating in the waters of the Atlantic\nOcean. The shell is very delicate, resembling in form some of the land\nsnails, and has but two colors, both shades of violet, a deep color on\nthe under side (which, by the way, is always turned upward when the\nanimal is floating in the water), and a lighter shade on the upper side.\nSo fragile is the shell that it seems as if a breath would break it. The\nmost interesting fact in connection with this mollusk is the wonderful\nfloat or \"raft\" which is secreted by the foot, and to the under side of\nwhich the eggs are attached. The latter are not all in the same\ncondition. Nearest to the animal they are more or less fresh; those in\nthe middle of the float contain embryos and fully formed young, while\nthose on the outer end are empty, the young having escaped into the\nwater. The genus is gregarious and may be found in almost countless\nnumbers. After a severe storm they are sometimes cast upon the beaches\nin vast numbers, where they soon die under the fierce rays of the sun.\n\nWe have thus far been dealing with snails whose shells were formed in a\nspiral coil. Quite a number of mollusks are not protected by such a\nshell, its place being taken by a flat, shield-like disk, or several\ndistinct plates placed side by side. The most familiar of the first is\nthe limpet or Patella, which is a depressed, conical, oval disk, looking\nnot unlike a miniature shield. They live on rocks, to which they cling\nwith great tenacity. The animal seems to have a pretty clear idea of\nlocal geography, for it invariably returns to the same place after its\nexcursions for food and the rock in some localities has been hollowed\nout to a considerable depth by the continuous dwelling thereon of the\nlimpet. The large foot is very strong and it is almost impossible to\ndislodge the shell from the rock when the animal becomes alarmed and is\naware that danger is near. While grazing along the sides of a rock\ncovered with fine sea-weed, it will leave a track like a worm and will\nclean off quite an area in a very short space of time.\n\nAnother species is the key-hole limpet (Fissurella), distinguished by\nhaving a slit or foramen in the apex of the shell. The shells of\nFissurella are generally rougher than those of Patella, and as a rule\nthey live in warmer seas. In the limpet we find a departure from the\ngeneral form of both animal and shell, both being bilaterally\nsymmetrical, that is, having both sides alike. In the mollusks which\nhave been presented thus far, the body has been twisted in the form of a\nspiral, making one side different from the other and causing the organs\nof one side to become atrophied. In the limpets the organs are paired,\nas they are supposed to have been in the ancestors of the living\nmollusks.\n\nThe most peculiar of all the mollusks, so peculiar, indeed, that they\nconstitute a separate order (Polyplacophora) are the Chitons, or\ncoat-of-mail shells. The shell is made up of eight separate pieces or\nplates, each locking with the other, the whole supported by and buried\nin a coriaceous mantle which forms a margin all the way around. This\nmust not be confounded with the true mantle of the animal, for it is\nonly a part of the shell. It is beset with bristles, spines or hairs,\nwhich add much to the peculiar appearance of this mollusk.\n\nThe Chitons live for the most part on rocks at low water and are said to\nbe nocturnal in habit, feeding only at night. Their movements are slow\nand they appear to be very sluggish in all their actions. When detached\nand taken from their rocky homes they have the provoking (to the\ncollector) habit of rolling up and are sometimes very difficult to\nstraighten out again. There are about two hundred and fifty living\nspecies, found in all parts of the world.\n\nIn the foregoing pages we have called attention to a few types of marine\nsnails, and what has been written has hardly more than touched upon this\nvast field. There are thousands of different species even more\ninteresting than those which have been mentioned. There are the\nbeautiful ear shells, or Abalones, the little periwinkle, so largely\nused as an article of food in Europe, besides a host of others too\nnumerous to mention. The brief notes and the figures on the plate will\nconvince the reader, it is hoped, that these inhabitants of the deep are\nnot only beautiful and worthy of our attention and study, but are also\nof much practical and economical use to man.\n\n Frank Collins Baker.\n\n\n\n\n THE LEMON.\n\n\nIn 1636 an English report on the affairs of the navy gravely remarked\nthat \"the use of lemon is a precious medicine and well tried. Take two\nor three spoonfuls each morning and fast after it two hours.\" The value\nof the fruit for certain disorders of the system seems to have received\nan early recognition. This was especially true with regard to scurvy,\nwhich in earlier days caused widespread mortality among seafaring men.\nHawkins, in 1593, made the statement that more than ten thousand men had\nsuccumbed to the malady within the limits of his naval experience. The\nCrusaders under Louis IX. were severely attacked by scurvy, owing to\ntheir abstinence from fresh meat during Lent, and the history of the\ndisease shows that it is occasioned by a lack of fresh meat and fruits.\nThe efficacy of lemon juice was recognized by Drake, Davy, Cavendish,\nDampier and many others years ago, and time has but added to the value\nof the fruit, while it has made it accessible to everyone. While Pomona\nis generally credited with having devoted her entire attention to the\ncultivation of the apple, it is stated on authority of an old Greek\nmyth, that she gave considerable thought to the development of the Lemon\nand the orange. It appears that Pomona inclined not her ear to the\nsupplications of her many admirers until Vertumnus, discerning her\nvulnerable point, presented the fair gardener with a grafting, which,\nunder her skillful cultivation, developed into a lemon tree, and, as a\nreward, the favor of the wood-nymph was bestowed upon the youth.\n\nWhether or not such was the origin of the Lemon, the fact remains that\nthe fruit is most useful and the tree exceedingly attractive. Originally\na native of Asia, it has become widely distributed in Europe, Africa and\nAmerica, and although far more susceptible to injury from frosts than\nthe orange, the trees are successfully cultivated under many conditions.\nDoubtless the best results in this country have been obtained in\nCalifornia. Thousands of acres around San Diego are planted with lemon\ntrees while large districts in the Ojai Valley, Ventura, Santa Barbara,\nPomona and Los Angeles counties are devoted to its cultivation. The tree\nis remarkable for beauty, and while it seldom attains large proportions,\nits pale green leaves, loosely-hanging branches, showy and fragrant\nflowers, together with the fruit that is found in all stages of\ndevelopment, produce a pleasing and highly ornamental effect. While the\nbest crop of Lemons is generally gathered between December and April,\nthe fruit should be picked every month for ten months of the year, in\norder to retain the best results. As a rule, the trees yield from one\nhundred and twenty-five to one hundred and forty boxes of the fruit to\nthe acre, about the sixth year, but this number is increased to four\nhundred boxes when the groves reach an age of ten years.\n\nThe varieties of Lemons are distinguished chiefly by their size and\nform, and may be roughly classified as egg-shaped with blunt nipples and\noblong lemons with large nipples. The sweet lemon and thin-rind Poncine\nand Naples belong to the first class, while the second includes such\nforms as the imperial, the Ga\u00ebta and the wax. The principal varieties\ngrown in California are the Lisbon, Eureka and the Villa-Franca. Of\nthese, the Eureka originated in California, while the Villa-Franca was\nimported from Europe. Besides the grateful quality of the juice, the\nexpressed oil of the rind is used in the arts and has an intense odor of\nlemon, and the Pundits of Benares, quote a Sanskrit work, written about\n1354, in which the oil is described as a valuable medicine. The acid\npulp of the Lemon, after rasping off the rind, is pressed for citric\nacid, while the ottos of the Lemon, orange and bergamot, the preparation\nof which forms the chief industry of Sicily, are leading ingredients in\nthe preparation of \"Lisbon Water\" and \"Eau de Portugal.\"\n\n \u2014Charles S. Raddin.\n\n [Illustration: ]\n\n\n\n\n TWO WRENS.\n\n\nThe house wren is one of Nature's illuminated successes. It has been\nsaid that there is no second spring, yet to-day (July 20th) this bird is\nin the full glory of spring-time melody. He sings from the top of a\ntelegraph pole, the song caught up and repeated by some country cousin\nin the grove, a musical argument carried on all day long and left at\nnight in the same unsettled state in which morning found it. Whether\nthey are discussing the relative merit of their respective claims, a\ntown residence or a country seat, I am unable to decide; it is certain,\nhowever, that the concessions of neither party infringe upon domestic\ndignity.\n\nTheir speech is a revelation of supreme content, a liquid, flexible\nmeasure with ripples and cascades bubbling through and over, a dash of\npure color amid July's neutral tinted emotions.\n\nThe day may be dark and threatening, the sun concealed in gloomy banks\nof cloud, rain falling, or thick mists obscuring the valley; each and\nall are powerless to dampen his ardor or to effect his extreme optimism.\nHe clings to his creed with persistent closeness, asserting valiantly\nthe ecstasy of finding one's self alive and emphasizing the statement by\na perfect wave of melodious argument.\n\nThere are hours when he sings with such force that his whole little body\ncatches the key-note and natural rhythm; the melody becomes compounded\nof his very substance, body of his body and soul of his soul. It is an\ninundation of musical notes, cascadic, cataclysmic, the tide of song\nrising till it drowns his personality; he is no longer a bird but an\nanimated song.\n\nMy little neighbor is a pattern of husbandly devotion, a lover-husband\nover whom coming events are already casting tender shadows before, the\nspecial event in this instance being located in a crevice beneath the\neaves of the house.\n\nWren babies had not left the first nest when Jenny Wren's husband was\nhard at work upon a second house, which was ready for occupancy before\nthe first family were self-supporting. This was an admirable arrangement\nin the way of time-saving, as eggs are often laid in the second nest\nbefore the first is vacated.\n\nThough the new house lacked the freshness of coloring and the\npicturesqueness of the swing of a nest in the sunshine, Jenny Wren made\nno complaint of being cooped up in the darkness, and as to her husband,\nhe was quite as well pleased with the glamor and wonder of its art as if\nit had been wound with blossoms and sprinkled with star-dust. A bird\nwith different tastes might have urged that it was only a little hole in\nthe house-jet, yet everything in life depends upon the point of view\nfrom which you regard it. Judged from the wren standpoint, it was\nconsidered admirably adapted to the family needs, nor could the most\ncritical observer fail to see here a literal illustration of that\nfamiliar truth: Happiness is from within.\n\nStanding upon a ladder I counted eight eggs as my eyes became gradually\naccustomed to the partial darkness within the nest; the dark, vinaceous\nspots laid on so thickly as to conceal or obliterate the original color,\nthus helping to hide them more securely. In the long brooding days, when\nJenny's little answering heart is preoccupied and silent, the hours are\nsometimes long and lonely to her mate. At these times he has been known\nto devote his spare moments to building a nest simply for his own\npleasure. Many instances of this remarkable habit are recorded of the\nEnglish wren, the explanation offered being that the odd nests are for\nthe purpose of deceiving the parasitical cuckoo.\n\nThere is also a supposition that the bird's active nature finds relief\nin work, being urged on by the increasing lonesomeness. This wren-trait\nreaches a climax in the marsh wrens, with whom the building habit\nbecomes a passion.\n\nNor is it restricted to the wren family, many instances being recorded\nwhere other species have beguiled the waiting days by an imitative\nhousekeeping.\n\nThe house phoebe has been known to build a second nest while its mate\nwas brooding. To all appearances this was an instance of over-developed\ndomestic tastes. Nor did the experiment end with the completion of the\nduplicate nest upon which the male bird sat regularly for several hours\ndaily.\n\nWrens do not take kindly to double houses, their warlike nature seeming\nto revolt against living friendly with near neighbors. A pair of wrens\nthat was well established in an unoccupied martin house made it very\nuncomfortable for the later arrivals. While the martins were abroad\nafter material for the nest the wrens sallied forth in an utterly\nvindictive spirit and scratched out all their neighbors had constructed.\nAfter singing a triumphant song with much parade they wisely retired to\ntheir own domicile to be on the defensive.\n\nWiser wrens, with an instinctive knowledge that an ounce of prevention\nis worth a pound of cure, are known to have the forethought when the box\nin which they build contains two compartments, to fill up one of them,\nthus avoiding the risk of troublesome neighbors. Wrens have been known\nto nest in a human skull. Others with less questionable taste, have gone\nto housekeeping in an old boot, a watering pot, a coat sleeve; in gourds\nand baskets, jars and water pipes, while another pair made a nest in the\nlower part of a stone vase in the garden. There was a hole for drainage\nin the bottom of the vase, and through this hole they found, beneath\nsome shavings, a circular space just suited for a nest. The vase was not\nfilled with plants until the domestic affairs of the wren family were\nhappily concluded.\n\nThe delicate swaying hammock of the oriole is sometimes used for a\nsecond nesting.\n\nThere was bitter disappointment in wren circles earlier in the season\nwhen, with the presumption of inexperience, the pump was filled\nregularly with coarse twigs, which were promptly dislodged at nightfall.\nUndiscouraged at this defeat, the morning hours were utilized for\nrebuilding with a persistency well worthy a more intelligent effort;\nthey worked and sang, sang and worked, until a cigar box was nailed to a\ntree for their special accommodation. This was nearly full of twigs when\nthey decided that the building-site was ineligible, a decision hastened\nby the fact that just at this opportune time a glass fruit can was left\nupon the piazza shelf. No sooner was this glass house seen than its\npossibilities were realized and plans were quickly made for a kind of\ncrystal palace experiment. Under other circumstances this might have\nbeen a dangerous precedent, as certain unneighborly conduct toward their\nlittle brothers of the air had at various times fairly invited the\nthrowing of stones. The can was half full of tiny fagots, and Jenny was\nthinking of settling upon the mattress of wood fibre when the thrifty\nhousewife turned them adrift summarily, well aware that this kind of\nhousekeeping, within easy range of neighboring cats, would not be\nsuccessful. Before such supreme content, who could have the heart to\nundeceive them? And yet, the can was turned upside down before they\ncould be made to understand the situation. Like Thoreau, they did not\nwish to practice self-denial unless it was quite necessary!\n\nAfter the failure of this crystal scheme, it was a difficult matter for\nJenny to make up her mind as to a further preference, but when she\nreally decided it was with such entire good faith as left no doubt in\nher lover's mind as to her judgment. This was more flattering as it was\nhis own choice, their last year's home thoroughly remodeled, to which he\nhad repeatedly called her attention, vainly. So the hole in the house\njet at least answered the question, \"Where are the birds in last year's\nnests?\" for the wrens moved in regularly, the tenor having a perch upon\na projecting bracket where Jenny joined him, a regular little termagant,\nscolding with all her might whenever the kittens looked that way.\n\nMarsh wrens, small brown birds, with barred wings and tail, breed in or\nabout the swamps and marshes of Lake Champlain.\n\nThey are intensely interesting from their habit of constructing several\nnests but one of which is utilized for housekeeping. After the real nest\nis made and the first egg laid, the male stays closely at home busying\nitself with building several nests, which are to all appearances\nentirely superfluous. In locating these he does not go beyond the\nimmediate neighborhood of the true nest.\n\nSome have thought that these sham nests are used as hiding places for\nthe male, a Lilliputian watch tower or guard house, from which close\nwatch is kept over the home property. Whether Mrs. Marsh Wren really\nneeds such close watching, being more inclined to flirt than the\nordinary feathered spouse, or because she is a better wife, so\ninfinitely precious that she must be guarded from every side, is, as\nyet, an unsolved question. \"Love holds the key to all unknown,\" and\nthough there is little to admire in a deportment made fine by compulsory\nmeasures, no doubt both parties understand the situation, which is quite\nenough for practical purposes. These nests, conspicuous from their size\nand exposed position, are securely attached to the upright swaying\nreeds, some of which penetrate their substance. They are lined with soft\ngrasses and have an entrance at one side, often nearer the bottom than\nthe top. Mr. Burroughs, who has found the marsh wren's nest surrounded\nby half a dozen make-believes, says the gushing, ecstatic nature of the\nbird expresses itself in this way. It is simply so full of life and joy\nand of parental instinct that it gives vent to itself in constructing\nsham nests; the generous-hearted creature being willing to build and\nsupport more homes than can be furnished or utilized.\n\nEntering the Lake Shore drive at St. Albans Bay, where dense tangles\nborder the swamp beyond, you are sure to hear a song that is\nunmistakably wrennish. You have glimpses also of a small brown bird\nbubbling over with a nervous energy that betrays itself in every note he\nutters. Wait quietly and he approaches, but go one step in his direction\nand he recedes to the swamp where human foot may not follow.\n\nPush your boat up the creek, the only avenue leading to his abode, that\ntantalizing song leading on meanwhile like the Pied Piper of Hamelin,\nthough unlike the latter there is no disillusioning at the end.\nRed-winged blackbirds take wing as you enter the twilight of soft green\nand amber shade and the far-off music of their jangle-bells becomes less\nmusical, the males striving \"to recommend themselves by music, like some\nawkward youth who serenades his mistress with a jewsharp,\" and using the\nair or the alder tops as a parade ground upon which to exhibit their\nmusical evolutions. And yet you are witness to many a voluntary bit of\nsentiment that will increase your interest in this scarlet epauletted\nregiment, descendants of the dusky tribe that anchored long ago in this\npeaceful haven, going out and coming in with the tide until the legend\nof their coming is as vague and shadowy and misty as that of the\ngolden-fleece voyageurs\u2014the Argonauts. They ebbed and flowed with the\nstream; came at the proper time and season without knowing why; anchored\nand launched their ebony ships when it was time for sailing.\n\nHere and there along this waterway the branches clasp hands above the\ncreek, forming an arch of green within which vines sufficiently elegant\nto warrant exclusiveness cling in unaffected grace to the alders,\nwithout inquiring or caring as to the pedigree of their support. It is\nsufficient for them that the support is there.\n\nA whole half mile along the stream and trees and bushes disappear,\nleaving a dense mass of reeds, the marsh wren's \"ain countrie,\" out of\nwhich he is never at his best and to which he gives you no welcome.\n\nBirds, like persons, have wonderful powers of concentration upon one\ntopic, woe be to you if that topic happens to be yourself!\n\nEvery denizen of the swamp regards you with suspicion, watching each\nmovement as closely as if you were a dangerous character traveling under\nan alias, and could not be trusted to sail upon this ruddy ocean in\nwhich their lordships have anchored their private yachts. Push your boat\nfar in among the reeds and cat-tails, into the sea of shadows over which\nno sluggish current sends a ripple, and certain globular nests in the\ntangled reeds reward your search. Push your fingers within these nests\nand in one only, here and there, will you find from five to ten dark\neggs, a rich reward for all your trouble.\n\nMeanwhile the \"neighbors,\" and the marsh wren generally has numbers of\nthem, have doubtless been charming you with their bubbling, gurgling\nsong, always half the colony singing at once, or, one bird rising above\nthe reeds gives the order, as it were, and the whole colony joins in the\nchorus. The song is quite beyond their control; they seem filled to\noverflowing with an inexhaustible supply of music, which trickles down\nthe reeds, like gathered-up drops of water charged with music.\n\n\"Sometimes, like a mine of melody, it explodes within them and lifts\nthem from the dark recesses of the flags into the air above.\"\n\n Nelly Hart Woodworth.\n\n\n\n\n WHEN SPRING COMES.\n\n\n Again the birds will weave their nests,\n And come and go on airy wing;\n And one will nurse her little guests\n And one will watch and sweetly sing.\n\n The bushes small and towering trees\n Their leaves of living green will don,\n And, swaying in the restless breeze,\n Will laugh because old Winter's gone.\n \u2014George Gee.\n\n [Illustration: ]\n\n Description of Plate\u2014A, twig with staminate flowers; B, fruit-bearing\n twig; 1, upper portion of staminate inflorescence; 2, staminate\n flower; 3, fruit; 4, 5, 6, 7, ovary; 8, 9, seed.\n\n\n\n\n CUBEBS.\n (_Piper cubeba_ L.)\n\n\nAromatics, as cubebs, cinnamons and nutmegs, are usually put into crude\npoor wines to give them more oily spirits.\u2014Floyer, \"The Humors.\"\n\nThe cubeb-yielding plant is not unlike the pepper plant and belongs to\nthe same family (Piperaceae). The two resemble each other in general\nhabits in the form of inflorescence and in the fruiting.\n\nCubebs were known to Arabian physicians as early as the ninth century,\nwho employed them as a diuretic in kidney troubles. It was also known at\nthat time that Java was the home of the plant. At one time it was\nbelieved that the Carpesium of ancient writers was cubebs, but this is\nnow generally disbelieved. Edrisi states that cubeb found its way to\nAden about 1153. During the twelfth and thirteenth centuries it was\nemployed medicinally in Spain. Originally it was doubtless employed as a\nspice, similar to pepper. Mariano Sanudo (1306) classed it among the\nrare and costly spices. Hildegard referred to the soothing properties of\ncubeb. In the thirteenth century cubeb is mentioned among the import\narticles of London. About the same time it found its way into other\nEuropean countries, notably Germany. At the beginning of the nineteenth\ncentury cubeb disappeared almost entirely from medical practice. About\n1820 English physicians of Java again began to employ it quite\nextensively.\n\nAs in the case of black pepper, the fruit is collected before maturity\nand dried. The fruit is about the size of the pepper, but has a\nstalk-like prolongation which distinguishes it. The pericarp becomes\nmuch shriveled and wrinkled on drying.\n\nCubebs are cultivated in special plantations or with coffee for which\nthey provide shade by spreading from the trees which serve as their\nsupport. Their cultivation is said to be easy.\n\nCubebs have a pungent, bitter taste and a characteristic aromatic odor.\nIt cannot readily be confounded with any of the other more common\nspices. Its use as a spice is almost wholly discontinued. Its use in\nmedicine is also waning, since it evidently has only slight medicinal\nproperties. It is used in nasal and other catarrhal affections. Cubeb\ncigarettes are used in the treatment of nasal catarrh. It has a marked\ninfluence upon the kidneys, causing irritation and increased activity,\nand as already indicated it is therefore a diuretic. It is, however,\nharmful, rather than beneficial, in acute inflammatory conditions of\nthese organs.\n\n Albert Schneider.\n\n\n\n\n A TREE-TOP TOWN.\n\n\n Before the cradled violets awake beneath the grass,\n Or any but the crocuses and catkins have come back,\n Always 'tis then the loveliest thing of all things comes to pass,\u2014\n A twit-twit-twitter on the mild spring breeze,\n A twit-twit-twitter in the leafing trees,\n Through which small sky-blue wings flash out a sky-blue track\u2014\n For blue-birds, first adventurous house-builders of the year,\n Are at their old, wise tricks again of settling far and near.\n\n Not long, 'tis when the hyacinths and tulips bloom in rows,\n And lilies-of-the-valley start to whitening on their stems,\n And woodsy things are opening fast to make a new out'-doors,\n Then robin-redbreast on a sunny day\n Comes taking life his usual charming way,\n With a blithe and merry Che-che-chem-chem-chems!\n While yet dry leaves and building twigs are left upon the ground\n \"I thought I'd come to the old place and take a look around.\"\n\n Then later, when the grasses curl, a-tilt in taller growth,\n And nooks for snuggeries are made by grape and ivy-vines,\n When lilacs stand in purple, and the plum-trees blossom forth,\n Comes here a lilting, gay, and gaudy troop,\n Tits, thrushes, bobolinks, blue-jays with noisy whoop,\n Kingbirds, wild tumblers in the air, drunk with ethereal wines;\n Then cardinals, and indigoes, and finches find the place,\n And so the town-site in the trees grows populous apace.\n\n One waiting for the apple-blooms is he who's always late,\n The oriole: his building-site none e'er disputes with him.\n Though last to come he has full leave to settle, with his mate,\n And hang his hammock up to rock and swing,\n To flout the town on breezy, orange wing\n From where his house sways airily adown a pendant limb.\n And now the high, green tree-top town, which welcomes ev'ry comer,\n Has settled to the business of singing out the summer.\n \u2014Austin Arnold McCausland.\n\n\n\n\n Transcriber's Notes\n\n\n--Created an eBook cover from elements within the issue.\n\n--Reconstructed the Table of Contents (originally on each issue's\n cover).\n\n--Retained copyright notice on the original book (this eBook is\n public-domain in the country of publication.)\n\n--Silently corrected a few palpable typos.\n\n\n\n\n\n\n\nEnd of the Project Gutenberg EBook of Birds and Nature Vol. 9 No. 4 [April\n1901], by Various\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}} +{"text":"\n\n# **Praise for** \n ** _Leaper_**\n\n\"One of the quirkiest and funniest openings to a novel I've ever read. There are sections of dialogue herein that are simply delicious, and the plot makes one examine how well we use our God-given talents.\"\n\n\u2014RAY BLACKSTON, author of _Flabbergasted_\n\n\"Geoffrey Wood's compelling novel _Leaper_ is a splendid romp with a fresh hero who is both poignantly vulnerable and hilarious. James will tug your heart, spark your tears, and leave you with a soul-satisfying smile you can't stop if you want to.\"\n\n\u2014KRISTEN HEITZMANN, best-selling author of _Freefall_\n\n\"What the hit TV series _Heroes_ fails to explore\u2014the divine source of our gifts, both routine and extraordinary; the possibility that miraculous talents are meant to honor God\u2014Geoffrey Wood tackles with playful, intriguing abandon in _Leaper_. Wood endows his reluctant, confused hero not only with the ability to instantly 'leap' to any place he can imagine but also with the wandering mind, paralyzing fear, and pesky flaws common to humans\u2014we know this guy! Wood keeps the Carl Hiaasen-like weirdness in check with truly provocative ideas and excellent wordsmithing. _Leaper_ is a fast, funny, fantastic read.\"\n\n\u2014ROBERT LIPARULO, author of _Comes a Horseman, Germ_ , and _Deadfall_\n\n\"If you could have one superpower, what would it be? Leaping, of course! I always say that. Only I didn't know what to call it until reading _Leaper_. Insightful, humorous, well-written, with a lemony-fresh smell, _Leaper_ delves into faith, its many questions and quagmires, inside the crazy world of an erstwhile superhero trying to make sense of his life and his newfound gift. This book is a gift to every single one of us who finds Jesus at our side and doesn't quite know what to do with Him. I can't wait to get a peek at what Geoffrey Wood has up his sleeve next.\"\n\n\u2014LISA SAMSON, award-winning author of _Straight Up_ and _Quaker Summer_\n\n\"A friend will recommend this book, and you'll think you can maybe just dip your toes in to discover whether the water is as inviting as it looks. Be warned. By the end of the first page, you won't get out. The book will sweep you into its currents, and you'll enjoy every moment of the ride.\"\n\n\u2014SIGMUND BROUWER, author of _The Last Disciple_ and _Fuse of Armageddon_\n\n\" _Leaper_ has it all: a great hook, compelling writing, and a real look at faith in an unreal world. It's fantasy, it's reality, but best of all, it's a fabulous read.\"\n\n\u2014MAY VANDERBILT and ANNE DAYTON, authors of _Emily Ever After_ and _The Book of Jane_\n\nLEAPER \nPUBLISHED BY WATERBROOK PRESS \n12265 Oracle Boulevard, Suite 200 \nColorado Springs, Colorado 80921 \n _A division of Random House Inc_.\n\nAll Scripture quotations or paraphrases are taken from the King James Version of the Bible.\n\nCopyright \u00a9 2007 by Geoffrey Wood \nIllustrations \u00a9 2007 by Trent R. Hill\n\nPublished in association with the literary agency of Alive Communications Inc., 7680 Goddard Street, Suite 200, Colorado Springs, CO 80920, www.alivecommunications.com.\n\nAll rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage and retrieval system, without permission in writing from the publisher.\n\nWATERBROOK and its deer design logo are registered trademarks of WaterBrook Press, a division of Random House Inc.\n\nLibrary of Congress Cataloging-in-Publication Data \nWood, Geoffrey, 1969\u2013 \nLeaper : the misadventures of a not necessarily superhero \/ Geoffrey Wood. \u2014 1st ed. \np. cm. \neISBN: 978-0-307-49941-7 \n1. Heroes\u2014Fiction. I. Title. \nPS3623.O6256L43 2007 \n813'.6\u2014dc22 \n2007000456\n\nv3.1\n\n_For Claudia and Tim_\n_When they heard these things_ , \n _they thrust Jesus out of their city_ \n _and led him to the edge of a hill_ \n _that they might throw him off the cliff_. \n _But passing through their very midst_ , \n _Jesus went his way..._\n\n\u2014THE GOSPEL OF SAINT LUKE\n**D ETAIL FROM POLICE REPORT**\n\nHotel reserved under alias, \"Gabe Oates.\" Payment made in cash. No record concerning that name. Police alerted when personal items found in abandoned room after checkout time. Window open overlooking river. Video surveillance cameras at front desk identify above-mentioned same as missing person, and concierge confirmed identification from police photo. Remarked as to the suspect's agitated, dazed demeanor\u2014said that suspect had \"wandered the hotel, barefoot, talking to an empty ice bucket.\"\n\nNo evidence of foul play. Room clean with bed made. Notebook, keys, handwritten note to one Father Chavez. Note reads: \"In Care of Father Chavez: Please donate '88 Volvo to church. Half the time she won't start, but it's better than no ride at all.\"\n\nNo body found. No eyewitnesses of the jump. Likely trajectory from a window leap would put body in river. No plans to drag the river due to the width and turbulence of that spot. No body likely found in any case. Evidence often irretrievable, washed away.\n\nThe tape recorder mentioned in the notebook entries not found.\n\nJust a three days' diary and an open window.\n\n\u2014Captain William Goss\n\n# Contents\n\n_Cover_\n\n_Title Page_\n\n_Copyright_\n\n_Dedication_\n\n_Epigraph_\n\nPart 1\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\nPart 2\n\nChapter 11\n\nChapter 12\n\nChapter 13\n\nChapter 14\n\nChapter 15\n\nChapter 16\n\nPart 3\n\nChapter 17\n\nChapter 18\n\nChapter 19\n\nChapter 20\n\nChapter 21\n\nChapter 22\n\nChapter 23\n\nEpilogue\n\n_Acknowledgments_\n\n# Part 1\n\n# **1**\n\nThe first time it happened, I had pins sticking in my back.\n\nI don't believe the acupuncture had anything to do with it. But in a way, my acupuncturist, she's a witness of sorts. Ask her. She can tell you. Although she doesn't really know what she witnessed, but then neither do I.\n\nSo she's not a _real_ witness-kind-of-witness. Not an official, legal sort of witness, you know? Circumstantial at best. She didn't actually _see_ it happen, just the aftereffects, and I'm sure she explained all those away.\n\nShe wouldn't have believed me even if I _had_ told her what really happened.\n\nWho would? Anyhow...\n\nI'd gone to my acupuncturist for my regular appointment. Every three weeks for my back. Spasms, mid- and lower back, nothing out of the ordinary. I work in a coffee shop; I open the store most mornings, and I'd opened it that day. Not too much espresso, maybe six, seven shots. Nothing unusual. But working on the espresso bar, pulling drinks, fetching gallon jugs of milk, lugging out a sack of trash, it wears on your back. So she sticks a few pins in, and it feels better. Unblocks the chi, or whatever. I don't know, I don't go for the metaphysical bit, and there's no mystical confusion about my experience. It's alternative care, but she wears a white coat just like a real doctor.\n\nShe did what she always does, no more, no less. Pins in the back and neck, one or two near my thumbs. No strange sensations or anything. I'm perfectly aware of myself. No mumbo jumbo, incense burning, or whatnot. There's a ceiling fan on way low, barely moving, I remember. She dimmed the lights when she left the room and made it relaxing. The same as always.\n\nSo I'm lying there under the sheet with her _Sounds of Rippling Water_ CD playing, and my mind starts to drift. This is what it's supposed to do; it always does. I'm not some mind-drifter. I focus, I'm aware. I'd had my coffee. But I'm daydreaming because you have to lie there. Not quite asleep, just really relaxed, half-dreamy, half-thinking of what I'm doing for the rest of the day.\n\nPhone bill, got to mail that off. Run back by the coffee shop, pick up the book I'm reading, _The Great Divorce_ , C. S. Lewis, it's wild stuff. I took it in to work because I always take a book everywhere, although I never have time to read at work. And if you take a book and don't get to read it, chances are you forget it's there.\n\nI'm not flighty or forgetful. I know what happened to me. Or at least that it happened. But you take a book and don't read it, you forget it. That happens to most people.\n\nAnyhow, I'm drifting and thinking that the name of the book is _The Great Divorce_ , and I snort a chuckle. I guess I chortle while I'm lying there because I'm thinking, _Now that's an oxymoron\u2014a great divorce_.\n\nMy wife and I recently got divorced, so I'm sort of sensitive to the title. I guess she's not my wife, but you understand... my ex-wife. I'm sort of chuckling because our divorce was anything but great. I guess most divorces are less than superlative, not the kind of thing you look forward to, or look back on and say, \"Gee, swell times.\"\n\nShe'd called the night before, looking for something. Actually, she'd called to accuse me of being why she couldn't find something. A cutting board. A cheap, white plastic, $6.99 cutting board that she swears was hers, though I remember buying that board back in college. A cutting board, a sharp knife, and three pint glasses. I made it through college with just those things: no plates, no cutlery, no napkin rings. She has all the plates, all the cutlery, all six sets of napkin rings we got as wedding presents.\n\nShe divorced me. But she got everything. Don't take that wrong, she's peaches; but the fact is that I moved out for a song, and she's still in the house. She got the dogs. Yet the night before, she's calling about that stupid cutting board, right?\n\nI live in squalor. Most of my stuff is still in boxes, and I haven't even moved all the boxes to my apartment; some of them are still in our garage. Excuse me, _her_ garage. And I know I haven't seen that cutting board. I haven't had time to cut anything in my apartment.\n\n\"You know how much I liked that cutting board,\" she says. \"Besides, I brought it to the marriage.\"\n\n_Brought it to the marriage_. She says this like she had arrived with a huge oaken hope chest, like her people felled trees for timber and built our house from scratch and moved us in. Like her father pulled the clapboard wagon up in front of the house and dropped off the blushing bride with her stupid cutting-board dowry.\n\n\"If it's in a box,\" I say, \"I don't know about it.\"\n\n\"Well, if it's in a box, find it,\" she says.\n\n\"When I get to all the boxes, I'll let you know if it turns up.\"\n\n\"Well, your concern doesn't help me cut an onion tonight,\" she says.\n\nI say, \"No. No, it certainly doesn't do that.\"\n\n\"James, there's no call for rudeness,\" she says. \"I just hope every time I find something of mine missing I don't have to call you and wait for boxes to get unpacked.\"\n\n\"I haven't had time...\"\n\n\"That's right, you never did. Half your boxes are still in my garage. And half those half are ones you never unpacked when we moved into the house five years ago.\"\n\nTen, maybe a dozen boxes, are still there, certainly not half. My back just shoots spasms every time she calls.\n\nSo I say, \"Honey, if there are any five-year-old unpacked boxes in that garage, you can have everything in them.\"\n\nTruthfully, I meant to cover up the phone for the next bit, but it's one of those tiny little flippy cell phones, and you don't know where the speak hole is. You can hardly see it, so how do you cover it up?\n\nShe hears me when I mumble, \"Might as well, you took everything else.\"\n\nThere's this silence. Oh, for silences that remain silent.\n\n\"No, James, I didn't _take_ everything else. I didn't _take_ anything. I _kept_ , perhaps that's the word you were looking for, _kept_ what was mine. Those things I bought when my job was the only one paying the bills and buying all the pretty little boxed-up things that now crowd _my_ car out of _my_ garage. Did you know some people actually call them carports? They drive their vehicles into them at night, these ports for cars? They keep cars, not their ex-husband's boxes, in them.\"\n\nMy wife's an attorney. I got nothing.\n\nI say, \"Honey\u2014and I mean that in the most condescending way possible\u2014honey, that cutting board is actually mine.\"\n\n\"So you do have it?\"\n\nAt this point in the conversation, my chi is so wadded up in the small of my back a million little pins wouldn't liberate it.\n\n\"No, Meg, I don't have the cutting board. At least I don't think I have _my_ cutting board. Like I'm saying\u2014and I shudder to say it to you again\u2014but it could be in a box. My cutting board, yes, could be in a box, so I can't see it with the naked, un-cardboard\u2013penetrating eye. It could be in a box in this apartment or in your very own parking garage. I really don't have the foggiest. After you run over the pitiful contents of my remaining boxes with your Jeep, you're welcome to look them over and see if you come across it. But frankly, I don't know what else to tell you.\"\n\nShe can pitch that sort of fit, but I'm not allowed.\n\n\"Is it going to be like this, James?\" Her voice trembles, slightly, unconvincingly. \"Are we really going to be like this?\" she says. \"Five years, James, five years of our lives. I know we're divorced, and I know you feel you weren't fairly represented, but do we really have to snip and claw?\"\n\nI look around to the walls of my apartment for something to focus on, just to look at, so I cannot think of snippy, clawing, damaging remarks, but the walls are bare.\n\nThere's a pause. In her mind, I imagine she's calling a mandatory do-over, because then suddenly she's sweet. It's scary.\n\n\"Hello, James... I'm making dinner... I have no cutting board... I have items right now on the counter to be cut, currently uncut. If you find the cutting board, will you please let me know?\"\n\n\"Absolutely, Meg. I'll look for it right now.\"\n\n\"Thank you. And I'll just make something else for dinner.\"\n\n\"Good. What are we having?\"\n\nThat took her a little off guard. Took us both, actually. I hear her lay the knife down.\n\n\"Good-bye, James.\"\n\nOn my cell phone, even after she hangs up, the little digital numbers keep counting for five, maybe ten more seconds as if she's still there.\n\n\"Good-bye, Meg.\"\n\nThat's where my mind drifts while I'm lying there with pins in my back. And I'm really not mad anymore, with a bunch of little pins in my chi. I mean, a little mad goes with the landscape these days. A general simmer of madness that never quite resolves. That's divorce, right? Come to think of it, that's marriage.\n\nBut here's the thing: I look at my watch.\n\nI always wear a watch. (This is important, so let me assure you it's no exaggeration. In fact, it's an occupational hazard for someone who opens at a coffee shop, a reliable watch with a nasty alarm. People want their coffee first thing. Addicts. So you have to be earlier than first thing to get ready for them or you're in for an earful\u2014a cranky, unkind, uncaffeinated bawling out. I always wear a watch.)\n\nI am down to my skivvies under that sheet, lying there on the treatment table, and I don't even lift my arm to check the time, just sort of roll my wrist toward my head. The watch face is right there\u2014close, almost too close to see it.\n\nMy eyes are a bit blurry without my glasses. The room is dim, the bulb above glows barely orange like a steady ember, and a little afternoon light seeps in through the blinds even though they're shut. The CD switches from \"Bubbling Mountain Stream\" into \"Placid Meadow Rivulet,\" a distinction I've learned to make over months of alternative care. And I squint at my watch face because that's an eyeglass wearer's instinct, to squint. But then I have to open my eyes wide, wider than normal, just to see the watch face, it's that close.\n\nAugust 19, Friday, 3:16 p.m.\n\nI remember this moment exactly, glasses or no. No mistaking that.\n\nI close my eyes again. I'm thinking, _I'll get out of here around four o'clock_. I wonder what time Meg gets home these days. I should go by the house and pick up another round of boxes. I don't have keys anymore, so Meg has to be there. But hopefully, I can get in and out before her new boyfriend, Doug Something-or-Other, shows up. He's a financial consultant with two kids, eight and five. His ex-wife is crazy. My wife tells me so, repeatedly, tells me all kinds of crazy things Doug's ex-wife does. I guess so I'll appreciate what kind of a pearl she's being to me. Tells me that and how much money he makes. And I think I can go by and grab a load before he gets there, then stop at the post office on the way home, and forget the book till tomorrow.\n\nI drift a little and wonder what Meg will be wearing. We used to be married, so this is inbounds. I'm wondering if good ol' Doug has told her she can't cook yet, or if he still smiles and takes her out to dinner a lot. I'm thinking about her hair. Wondering if she'd straightened it that day. That was my favorite, even though her curls, which are natural, are excellent. But I've always liked it straightened, the way it smoothed across her forehead. The way, when straightened, she'd tie it back in a ponytail to work on dinner. Her awful dinners. She'd tie it back like she was about to knead bread for an army even if she was just opening cans and dumping them in a boiler pot. She worried about her hair getting in her way, like it was expensive silk she didn't want to have to dry clean. Even when she was slicing a boiled egg.\n\nThat's when it hit me. Something about a boiled egg. We had two of those egg-slicer contraptions. Absolutely essential, these. A blue one and a white one, both wedding gifts. You wouldn't think two completely different people would spring for the egg slicer. You wouldn't imagine Shoppy Mart would actually have two on the shelves. The white one, hanging in the dusty carton since the store opened, and a three-week wait for them to special order in blue. Somehow we got two.\n\nAnd I remembered. The afternoon Meg told me she'd filed the papers, I stood over my first box in the middle of the kitchen with an egg slicer in each hand, thinking whether I should take both or just one. She rarely used them. I don't think she understood them, probably afraid her hair would get caught and shredded in one.\n\nBut I remember thinking, _No, Meg likes blue. She should have the blue one_.\n\nI returned the blue egg slicer to the drawer by the oven and walked back over to the box. I'd written \"Essentials\" on the flap with a black marker. This was early in the moving process, when I still wrote things on the sides of the boxes. I'd just thrown junk in there\u2014it was full, not what you'd call properly packed. Just a pile of stuff that happened to be piled inside a box.\n\nI knelt down over this box thinking, _They'll never be together again_. And I was sad for the egg slicers, you know, for the blue one and the white one. I held the egg slicer up a second before I put it in, giving it a last good look at the kitchen. Then I laid it down.\n\nOn top of the cutting board.\n\nThat's when I remember. The cutting board. Lying on the acupuncture table, I could see it in the box so clearly, at such an angle. On top, inside the box marked \"Essentials.\" Being the very first box I packed, I moved it to the garage, and surely, with all the boxes that came after it, it hadn't ever been moved. In fact, I knew exactly where it was. Anticipating so many boxes, I'd put it in the far corner. I was sure it was still in that corner by itself; I could see it.\n\nAnd I want to give Meg that cutting board, even though it was mine. I want her to see I don't care about any of it.\n\nThat's when it happened. Water rippling, a lazy breeze wafting from the ceiling fan, and my eyes are closed. It felt like two, maybe three minutes since I'd checked my watch. Lying there, eyes closed, and I'm seeing down into that box, with the flaps open, the one marked \"Essentials.\" I could see the egg slicer, slid to a corner next to a turkey baster, both of them on top of the cutting board, plain as day.\n\n_I want her to have this_ , I think. And I reach down...\n\nThe cutting board.\n\nI'm standing there. Reaching down into the box. Touching that cutting board. I mean I'm actually there, in her garage. My eyes are open; there's a cutting board in my hand. It was my garage so I recognize it. My boxes, that dartboard, the hook where I used to hang my bike. See what I'm telling you? Suddenly I'm standing in my ex-wife's garage in my underwear with a cutting board in my hand. I have no idea how I got there, how I even got in. (I mentioned her changing the locks, right?)\n\nI look at my bare legs sticking out of my skivvies. That's my hair on my chest. I check my watch, squinting because I don't have my glasses.\n\nStill August 19, Friday, but now 3:19 p.m.\n\nThat was the first time it happened. This afternoon. And this is my first entry documenting my strange gift.\n\nI slipped across space without using time.\n\nI leapt.\n\n# **2**\n\nI have no pants.\n\nStrange to say, once that moment or two of immediate shock of having instantaneously leapt across space wears off, no pants is still no pants. No matter how fantastically you got there, you're there with no pants.\n\n_\"Oh, hey, Meg, oh, um, yeah, seems I found the cutting board...\"_\n\nThat wouldn't do. Not pantless.\n\nI pad across the garage to the mail slot in the garage door. It's an old house, an old garage with an old garage door. Meg and I didn't find that mail slot till the second year of our marriage. Along with old bills from some very angry people. We could never figure why anyone would put a mail slot dead smack center in a garage door. Mail gets run over, slips under cars, or is drowned in oily puddles. Sometimes mail takes a fun ride around town on the car bumper. Most of the time we got mail in the mailbox like the rest of the neighborhood, but other times it was shoved recklessly into the least-tidy, least-lit, and least-used room of our home.\n\nBut Meg and I found other clever uses for the garage-door mail slot: we watched the neighbors, looked to see which unwanted person was ringing the doorbell, checked for cars parked in the drive.\n\nSo I creak open the mail-slot door and crouch to look out. No sign of Meg's Jeep. I had time.\n\nI open the door to the kitchen, stick my head through.\n\n\"Hey, Meg!\" No answer.\n\nThe dogs come bounding around the corner, thrilled to have a guest. Perhaps they still recognized Daddy's voice, but I've often suspected they'd greet uninvited dogcatchers or invading assassins with the same unbridled glee.\n\n\"Down! Get down!\"\n\nWithout clothes, a man shies from large, happy dogs prone to jumping up and flopping a paw on either shoulder.\n\n\"Down, buddies!\" I hide behind the garage door. They slam into it merrily.\n\nPonzy is a seventy-five-pound German shepherd known for jumping through the den window at the appearance of the meter man or the sound of thunder. Through, not out. Those windows don't open. Unless a German shepherd shatters the glass. Chunky is a ninety-pound chocolate Labrador who earned his name as a pup by decimating a can of Chunky soup. One morning I found a very thin, shiny, tooth-punctured piece of aluminum on his bed. After looking two hours and finding a couple of stray English peas, I figured out that flat piece of tin was once a full can of Chunky soup left too low in an open pantry. He'd eaten it, label and all, leaving no stew juice anywhere as a clue.\n\nMy brutes claw the door and whine affectionately.\n\n\"Hold on, boys!\"\n\nI sought pants at any cost and pick up a screwdriver to open a few of my boxes. Luckily, I had wrapped a vase in an old pair of jeans so it wouldn't break. Meg had thought this a poor way to pack, but then, she couldn't have foreseen the pants paying off in this particular manner. I pull on the jeans. The vase drops and cracks.\n\n\"Hey, buddies! Good boys!\" I swat the dogs down with the cutting board as I walk through the kitchen.\n\n\"I'm home.\"\n\nMeg could be home anytime. Fridays were her short day. I still need a shirt, so I go to our bedroom. (Sorry, divorce very recent, my ex-wife's bedroom.) Call it instinct or perhaps desperation, but under the circumstances, looking for a shirt in my old closet wasn't the oddest thing I might have done. I wasn't spying.\n\nAnd I find one: a white oxford, pressed and starched. I take it off the hanger. I was touched, really, she never did my shirts. Deeply touched and almost willing to forgive a multitude of snips and claws. It was so...\n\nThen I realize: It isn't my size. It isn't mine! It's Doug's. Five white, starched shirts hang in my wife's closet. She'd picked up the man's laundry. She never folded my socks, not once in five years. I ravage the shirt in my hand, wrinkling it mercilessly, cracking the heavy starch until I find the cleaner's tag. \"Williams.\"\n\nShe had taken Dougie's shirts to the cleaners, not just picked them up, but dropped them off under her name! Picking up somebody's laundry, you might do that, you know, even for someone you barely knew, if you were going anyway and they've asked, \"Hey, can you grab mine?\" But dropping off and picking up! Paying to have his shirts cleaned. His shirts under her name.\n\nHer maiden name.\n\n\"Down, boys! I'm not in the mood!\" I yell.\n\nAnd my dogs know this other mood. They give up on me and roughhouse each other down the hall.\n\nI pull on Dougie's shirt. Even if it's a bit small, it's better than no shirt. _I'll just roll up the midget's sleeves_ , I think. But strangely, the shirt hurts. Yes, it's small under the armpits, but I mean it actually hurts me. Sharp pains up and down my back. Like the financial consultant with the crazy ex-wife cast a spell on his shirts so that only he could wear them. Like how only the Lone Ranger could ride Silver.\n\nWhen I drop my arms and take a step, pain shoots through my back like a bunch of little pins sticking...\n\n(Of course, I'm an idiot, but this is proof! I didn't black out and wander over to the house after my appointment. On a treatment table, then the garage. No pants. Pins still in my back. I didn't imagine a thing.)\n\nI take the shirt off and hunch toward the bathroom mirror. I rummage the sink drawer for Meg's round face mirror, the one she used to pluck her eyebrows or to check how her hair looked from the back.\n\nI turn like Meg, the mirror in one hand looking for the bathroom mirror behind me. I strain my other arm around to pull out the pins, dropping them one by one into the sink. I get the ones on my neck and the two by each thumb. After a thorough visual check, the best possible without my glasses, I take down a towel. There's no other way. Courageously and suddenly, I dry my back.\n\nNo more pain. I had them all.\n\nThat's when I hear the front door unlock. Dogs run and skid across the hardwood to greet Mommy's return. Putting on Doug's shirt quickly, I button it up wrong, one button off. I tuck it in anyway.\n\n\"Meg?\" I call out.\n\nThe scream of an ex-wife startled by her ex-husband is a chilling, razorlike event. I hear dogs running for safety. Things from a flung purse scatter and roll. I hear Meg gasping for breath.\n\nShe catches that breath. \"Holy mother of... James!\"\n\nI come out of the bathroom, hands raised somewhere between a conjured ghost and a captured fugitive.\n\n\"Honey...\"\n\n\"You... you! James, I swear... what the...\"\n\n\"Honey, I can explain.\" Then I hear myself. \"No. No, honey, I can't explain.\"\n\n\"That's not your shirt!\"\n\n\"I cannot even begin to explain this, but I can tell you what happened.\"\n\n\"Who do you think you are?\"\n\n\"Meg, I need to tell you what happened.\"\n\n\"Yes. Yes, James, you do!\"\n\n\"I need to tell you because I think I might be crazy.\"\n\n\"Are we voting? Because if we are, it's unanimous. You're completely crazy if you think you can walk in here while I'm gone and...\"\n\n\"Honey...\"\n\n\"How did you get in here?\" She looks down the hall, back into the den, up at the pull-down attic door.\n\n\"Honey...\"\n\n\"I changed the locks!\"\n\n\"Honey...\"\n\n\"Did you break a window?\"\n\nShe goes in motion. Rushing all through the house, yelling and checking doors.\n\n\"Meg?\"\n\n\"Did your dog break that window again, and you thought you could just crawl on in?\"\n\nShe yanks at the sliding patio door, mugs the knob on the back door, double-checks the den window, circles back, and crouches by the front door to see if the lock has been jimmied.\n\nI'm flattered that she thought, even briefly, that I could jimmy a lock. \"Honey...\"\n\nStanding in the entryway, she glares at me. \"How did you get in here?\"\n\nI open my mouth to start the impossible answer, but she goes in motion again. \"That's not your shirt!\"\n\nShe shoves past me and positions herself in the middle of her bedroom as if she were a wrestler challenging all comers.\n\n\"What have you been doing in here?\"\n\n\"Meg, I don't know how to tell you this, but...\"\n\n\"Where are your glasses?\"\n\nA car honk causes both of us to stop. I practice innocent blinking. Definitely honking, from the driveway. Three polite taps. Short separate taps, annoying ones. From her face, I could see Meg and I still agreed on some things. Who honks politely?\n\n\"You stand right there. Do not move! Do not touch or look or walk! Stand.\"\n\nI move.\n\nAs she marches out to the driveway, I walk over to the front door. It was Dougie, in his Lexus. He was the polite honker. Watching through the door crack, not the door-door, that wasn't at all safe, but the crack-near-the-hinges crack, it seems Meg clarifies to Douglas a thing or two about honking at women from driveways. She spins to come marching back, twists a heel on her shoe, and stumbles. With a hand balanced on Dougie's hood, she takes off both her shoes and throws them at a tree, one by one. She nails it on that second try.\n\nI scurry back and put my feet in the exact position they had been when she left.\n\nShe throws open the screen door. We apparently no longer own dogs. Not a peep, nor a paw. I surrender my arms up again.\n\n\"Get out of here!\"\n\n\"Meg...\"\n\n\"No!\" She lunges toward me, stops, rethinks, then turns and slams the front door, locking it from the inside.\n\n\"Tell me how you got in here\u2014then get out of here.\"\n\n\"I came in the house from the garage.\" See, I've never lied to Meg.\n\n\"Your garage-door opener still works?\" she yells. \"They promised me they changed that, too! Give me that garage-door opener.\"\n\n\"You have my garage-door opener.\"\n\n\"Apparently not!\"\n\nFrom an outside perspective, she would still seem to be screaming. Her voice is loud. It is shrill. It's definitely still angry. There are arm motions\u2014dangerous, unhappy arm motions. But her hair had begun to loosen, to fall from its pins. I know this sign. Her hair is not unconnected to her anger. She pinned it up, but the pin was giving way, bit by bit, like a taut rope fraying. Curls were breaking out, coiling and rising, stretching and loosening, and though one would not say this is unlike Medusa, I could tell she was calming down.\n\n\"Honey...\"\n\n\"Apparently not!\"\n\nRepetition. Not for effect, but for gathering air. I knew this too. She was definitely cooling.\n\n\"Apparently you can open the garage door, unless you crawled through the mail slot.\"\n\nThere is this curious pause where we eyeball each other blank-faced, breathing, pondering that strange possibility.\n\n\"Meg, can I tell you something?\"\n\nShe puts her hands on her hips, drops her chin. Having assaulted a tree with her shoes, she's barefoot. She wraps one foot over the other. \"If you tell me you love me, I swear I'll crush your collarbone with that brass lamp.\"\n\nI have always feared that heavy brass lamp.\n\n\"Just listen. Okay, Meg? Listen to the whole thing till I hold up three fingers, and then you'll know I'm done. Three fingers. Like a Boy Scout. Okay? But listen to the whole thing before you choose how to hurt and destroy me.\"\n\nShe stops blinking.\n\n\"Like a Boy Scout. Okay?\"\n\nHer nostrils flare.\n\n\"Good. I'll take those flared nostrils as you saying the letters _O_ and _K_.\"\n\nI squint at my watch. 3:42 p.m.\n\n\"Where do I go on Fridays between three o'clock and four o'clock? Semi-rhetorical, Meg, do not speak. I go, as you know, to acupuncture. From three all the way to four, I lie on a table with pins in my back. Not an outpatient sort of deal. You don't get to leave before they take out the pins, and I believe they may reuse them, but let's not think about that now. My point: it takes the whole sixty-minute hour. That's where I was approximately twenty-three minutes ago. I was lying on a table with pins in my back. Then, Meg, something happened, and this is the strange part. I was lying there, and I thought of you. Oh\u2014not that I thought of you! That's not the strange part. Stay with me, Meg, stay with me. You're doing great! Anyway, I thought of you, not in the least bit strange for me. I thought specifically of being here, in the garage. A little strange, but it gets stranger. And as I was _thinking about_ being in the garage, suddenly... I was _in_ the garage.\"\n\nShe's blinking again, but otherwise unresponsive.\n\n\"That's the strange part, Meg.\"\n\nShe just stares.\n\n\"Oh!\" I hold up three fingers.\n\n\"How much coffee have you had today?\" she asks.\n\nThis is good. Quietly insulting, Meg was back. I pick up her cell phone from where it had fallen on the floor.\n\n\"Call her?\"\n\n\"Call who?\"\n\nI grab the phone book off the shelf and start thumbing through it.\n\n\"We'll call my acupuncturist. You ask her.\"\n\n\"James...\"\n\n\"Here it is. Two-six-eight...\" I start dialing.\n\n\"James...\"\n\n\"Two-six-eight-one-four-five-five.\"\n\n\"James?\"\n\n\"Here, Meg,\" I hand her the cell phone. \"Ask her where I am right now.\"\n\nShe listens. When she hears that it's actually ringing, she slams the phone shut.\n\n\"Good heavens, I'm not calling your doctor.\"\n\n\"Acupuncturist.\"\n\n\"You think she thinks you're still there?\"\n\nI flinch. \"Wait! What if I am? Here and still there too? Impossible...\"\n\n\"James?\"\n\nWe both turn at the polite knock on the door. Three short knocks. Then Dougie's muffled voice from the outside.\n\n\"Darling, the kids are getting restless.\"\n\nHe punches up that _darling_ whenever I'm around.\n\n\"Sorry,\" she says to me, \"we were going to the park. We just stopped to get Doug's laundry.\"\n\nHis laundry. I couldn't speak.\n\nShe opens the door a crack, whispers to her boyfriend, then returns.\n\n\"I'm sorry, James. I have no idea what you're trying to tell me, but there's no excuse for you breaking in\u2014\"\n\n\"Meg, I didn't\u2014\"\n\n\"No, James. No!\"\n\nShe holds up three fingers, which she points and shakes at my head as if to strike me with silence.\n\n\"Now you listen to me! It doesn't matter how; it's still basically breaking in. I wouldn't break into your apartment just because I could, even if I thought I had a good reason. It's not right. And I'm sorry if you missed your appointment, and I'm sorry I yelled at you about your boxes. I know that's what you were doing, trying to get those boxes, and yes, you need to get them out of there. But no, James, no, it is not all right for you to sneak in here, even if it was once your house. It's not now. I'm sorry. That's what happens when two people divorce. Okay... okay?\"\n\nI nod.\n\n\"Okay, then.\" She withdraws her pointed three fingers as if to release me.\n\n\"You did the finger thing backward.\"\n\n\"What?\"\n\n\"You were supposed to hold up three fingers after you were done speaking.\"\n\n\"I don't like the whole three-finger thing...\"\n\n\"No, you just did it wrong.\"\n\n\"Maybe I did, James, but so did you.\"\n\n\"Right, but I was just trying to show you...\"\n\n\"I don't like the finger game, okay? It's dumb.\"\n\n\"Right, it's very dumb. I'm sorry, Meg.\"\n\nWe ended up here too often. I think this is why we got divorced. Too often, two people exhausted over whatever they were arguing about.\n\n\"Look, I have to go,\" She says. \"Doug's waiting. And James\u2014\"\n\n\"Do you really understand what I've told you?\"\n\n\"And, James, you have to go too.\"\n\n\"I'm telling you I leapt here.\"\n\nMeg tucks her hair behind her ear. \"You what?\"\n\n\"I didn't break in! I was in one place, I thought of you, and bam\u2014suddenly, I'm in the garage.\"\n\n\"You were thinking of me?\"\n\n\"At the acupuncturist's. I was thinking about you and the cutting board.\"\n\n\"So you found it?\"\n\n\"Well, yes, actually, but that's sort of off the point.\"\n\nI walk into the bathroom, grab the cutting board, and hand it to her.\n\nHolding it, she frowns and cocks her head as if she just remembered missing something.\n\n\"Where's your car?\" she asks, opening the door and going out onto the porch.\n\n\"Don't you listen? I didn't drive here. I just... sort of... showed up.\"\n\n\"Yes, I've been meaning to say something about that. You've got to stop just showing up. Call first. It makes Doug jumpy.\"\n\n\"I don't mean I stopped by unannounced. I mean I appeared out of thin air.\"\n\n\"Without your car?\"\n\n\"Meg, now you're just not trying. You can say I'm an idiot, I have years of practice with that. But don't act like I'm not saying what I'm saying.\"\n\nThe screen door opens a crack, and Doug sticks his nose inside.\n\nI say, \"She heard you, Doug, she's coming.\"\n\n\"Oh... I didn't see your car, didn't know you were here. Excuse me.\"\n\nHe shuts the door behind him. He shuts it hard for Doug.\n\n\"Thanks, James,\" Meg scowls at me. \"Thanks a lot.\"\n\nShe opens the door. Meg can open a door louder than Doug slams one.\n\n\"Doug!\" She calls out to him. \"I'll be right out.\"\n\nHe doesn't turn around as he walks back to his Lexus.\n\n\"James, I have to go.\"\n\n\"Margaret.\" I put a hand on her arm. It was the first time I'd touched her since she'd announced she wanted a divorce.\n\n\"Tell me, please. I need to know. Do you believe me?\"\n\nMeg stares at me, unsure if she wants to be angry again. \"I don't know what else to say to you right now.\" She walks past me into the bathroom, to fix her hair.\n\n\"I've never lied to you, Meg,\" I say, but I knew the conversation was over. All over. As I walk out the front, I say, \"Not once, Meg. Not once.\"\n\nOn the porch I can hear her say, \"There's a bunch of little pins in my sink?\"\n\nBut it was over.\n\nI start walking. I pass Doug's Lexus, and his kids wave at me. I wave back. Doug pretends he doesn't see any of that. At the end of the drive, I start walking in the direction of my acupuncturist's. I suppose I need to go back, pick up my clothes, my glasses, my wallet, my car. I'm guessing those will still be at my acupuncturist's. But who really knows? I'm not where I'm supposed to be.\n\nAs I walk, I try to flatten the bulge in my shirt and realize it bulges because I misbuttoned it, so I leave it be. The sidewalk is hot. I have no shoes.\n\nAnd Meg doesn't believe me.\n\nI hear one long blare of a car horn. I turn and see Doug's Lexus slowing behind me, Meg's hand reaching over to the horn. Her window slides down.\n\n\"If you really don't have your car, we can take you.\" Meg offers this, though neither she nor Doug look overly pleased.\n\n\"Get in back with the kids.\"\n\nI get in. Doug's kids look at me as if I were a criminal. Between the driveway and now, someone has explained who I am.\n\nMeg looks in the rearview mirror and asks, \"Where do you want us to drop you?\"\n\n\"My stuff is still at the acupuncturist's, I guess.\"\n\n\"Tell Doug where to go.\"\n\nShe says this, and that's all. She won't speak to me again the rest of the ride. But her eyes look back at me in the mirror, and occasionally I catch them. Her eyes and my eyes. Something is lost between them. We both feel it. Something no longer there. Some good thing, one last good thing, lost.\n\n# **3**\n\nI decide to see a priest.\n\nPerhaps not the obvious choice. But then, I sort of defy anyone to pop out the obvious protocol for bodily transspatial meta-hiccups of the personal sort. Slows opinions down a bit, huh?\n\nFor the record, riding in Doug's car between the kiddies, I also consider consulting a psychologist or a doctor. But I figure, a _doctor_ doctor would pretty much only prescribe me a sedative\u2014a family-size, Sam's Club box of sedatives to calm my undeniably overwrought condition. And I wouldn't have passed on a round of that right then, had it been handy. But I didn't think a doctor would believe me. Wouldn't even drive down believing-me street, let alone park there. Even Meg thought I was crazy.\n\nIf a _doctor_ doctor thinks you're crazy enough, dangerous enough, he might ease his finger under the desk and press the hidden red button. Alert the proper authorities. Or better yet, his nurse might slip you a stack of papers to sign, and before you've retracted the ballpoint, bam\u2014you're strapped to a cot with an endless supply of the happy juice pumped into your arm, and for the next five years you're peeing in your personal Ziploc baggie Velcroed around your waist.\n\nNo thanks.\n\nAs for the shrink, well, I figure that would be the same song, second verse. There's even more likelihood for alarmed professional retribution\u2014sedation and explanation. Sedation for the crazies, explanation of the crazies by digging into my remarkably uneventful, but nonetheless nervous, childhood, my stunted-to-the-point-of-dwarfed emotional maturity, my exceedingly bad marriage, my fault for the exceedingly bad marriage, my fault for most exceedingly bad things in the world, my penchant for abusing the word _exceedingly_.\n\nNo, definitely a priest. A candle-trimming, discreet, bound-by-God-not-to-blab-a-word priest. Besides, priests already believe a stack of impossible things, right? Whether or not I believed them myself, I wanted someone to believe me.\n\nRegarding belief: I'm mostly Catholic. Okay, Catholic most of the way. The way I see it, there are degrees of belief. I suppose that wouldn't be the sort of thing that would sink an eightball with a priest or a nun. It's not information you'd just pony up unless you genuinely wanted a rather lengthy, disapproving lecture on belief, replete with a personal guilt trip for your aiding and abetting the crucifixion of God. I don't just tell priests any ol' idea that pops in the brain, nor should you. But we've all got a little notched knob when it comes to believing in God. Even if you don't intend to, we all do, every day. We calibrate.\n\nSo I'm definitely more Catholic than I am anything else. More Christian than I am, say, Buddhist or Communist. In fact, I don't have any idea what a Buddhist is supposed to believe to get the Buddhist club card. Or a Communist either. Seems like you'd have to read a few key, very dry books about those subjects and be pretty impressed, impressed enough to implement an idea or two on some personal scale, great or small. Meditate on your lunch break. Deny the individual. Buy a loaf of bread and distribute one slice into all your neighbors' mailboxes.\n\nI do know I was christened. I know I was forced to comb my hair and go to Mass once or twice every month until I learned to drive a car. After that, I took myself to Mass less regularly. Groomed less regularly. I recognize Latin. I know when to stand, when to kneel. I don't fear grown men slinging water on a crowd of well-dressed people. I don't just pile out; I wait politely for the priest and procession to leave first. I think candles are holy and magic because they're so quiet, because they stand in for someone's prayer.\n\nAfter college, for a long while, I only went in for the big Masses: Easter and Christmas. Felt a little fat and depressed, marginally lascivious during Lent.\n\nBut for the past year, I've been going to Mass every week. Maybe because my marriage was going sour? Nonetheless I went. One might say I've been attending Mass \"religiously.\" So that's not simply redundant, what I mean is that I made myself go like it meant something.\n\nAnd sometimes it did. I can remember days when the Lord's Prayer\u2014recited out loud and holding a stranger's hand, a stranger who in turn was holding the next stranger's hand\u2014felt like something bigger than all of us. Better than just everybody's decent intention to show up and make it through the \"Go in peace.\" I remember a Mass or two where the wafer seemed to melt on my tongue, and I wondered if God could actually be absorbed into a body, into a life, like a divine vitamin. I don't mean that stupidly, frivolously. I mean the thought that I might actually, suddenly, be a better person because of proximity to God. A surer person, one more apt to do a good turn. A person less consumed with how it confused me to be me\u2014and more thoughtful of how confusing it might be to be you.\n\nI remember a moment or two from a homily; bothered me for a whole week. I remember something about a tax collector told to get up from his tax table and follow Christ. That's it. Impossible. For all this guy knows, a crazy, prophetic Jewish peasant walks by\u2014this crowd of riffraff around him\u2014and stops just to tell this guy to up and leave his job, leave his money right there on the table and come along.\n\n\"Follow me.\"\n\nThat's it, that's all he gets. Get up. You. Come along.\n\nAnd the crazier part is\u2014he does it. He leaves everything he's doing and tags along for the rest of his life. And the twist? Come to find out he's Matthew. He's the one writing the book. \"The Gospel according to...\" That guy. I kept thinking, _That guy made saint? Him? What about all the others?_ How many times did Jesus pull that maneuver on some stranger, that sudden eyeball-to-eyeball \"Follow me.\" Of all the times the Gospels record that he does do that, how many more times are not recorded? Was he doing it all the time, like hourly, and how often does it work? In a couple of spots, Jesus adds, \"Sell everything you got and follow me.\" That doesn't pan out, of course. Matthew tells you it doesn't, but sometimes it works. When a guy actually follows, I've wondered, how often does he stick with it? All the way? Do they all become saints in the end?\n\nThat's been going on in my head for a while now. And I guess if you puzzle a homily out that far you're further along than the God's-Name-in-Vain sort and a little past the Beg-God's-Help-Only-When-in-a-Pickle sort. Further down the believing road, I'd say. Not much, but a bit. I definitely don't cross myself after meals or when I see a serving tray with a picture of Mary on it.\n\nSo I guess I believe. Or I'm certain that I don't _not_ believe. I'll jump on the Christians' team over the Communists' any day. If I'm going in the game, I know which way I'm kicking, right?\n\nThat's what I mean by degrees of belief.\n\nMeg and Doug drop me off by my car in the acupuncturist's parking lot. I've never been so happy to see an '88 Volvo. They drop me out, keep the car running, don't stick around for my reappearance. As they're backing out of the space, Meg puts on her darkest sunglasses, bites her top lip, and adjusts the radio. She can't even look out the window at me.\n\nAnd it's better that they don't stay because I'm going to lie to my acupuncturist. I have no intention of trying to explain the truth to her. This is way beyond alternative care. I just hope she isn't the kind of lady to pilfer a wallet. She'd never struck me that way, but then again, her job is just a hop-skip up from voodoo. And there's important stuff in my wallet.\n\nQuietly, slowly, I open the front door to her office.\n\nBut it doesn't matter how quietly I do it. I've forgotten about the little electric bell that goes off every time the door is opened. An electric chime dings like you're in a department store. Often she's in one of the treatment rooms sticking folks when clients come in, that's why the bell.\n\nHearing it, she comes out of the second treatment room, smiling.\n\nA note about acupuncturists: They smile. Often and unrelentingly. This isn't a bad or a fake thing. Of the few I've met, they're all smilers, placid, peaceful sorts. Like whatever kinks they had in their chi, they smoothed out years ago. They're so relaxed, so \"in-tune,\" it's like they can smell a wadded chi and see right through you into your tangle. I've seen my acupuncturist flinch when she looks at my back; it's that tangible to her. I think she flinches when I drive up.\n\nShe's smiling now. \"Hi, James.\"\n\nCompletely happy, poker-faced. Whatever she thinks, she's not tipping me a glance.\n\n\"Sorry,\" I say. I haven't worked this speech out. I should think these things through before I open my mouth, but I just start talking.\n\n\"Sorry. I had to come back. I have to get my stuff... in that room? I didn't mean to leave my stuff for you to... how much do I owe you?\"\n\nIt's a dumb question. I always pay the exact same amount, every time. A ten-dollar copay. Then again, dumb\u2014but not out of the ordinary. I always ask her how much I owe, although it's always ten bucks. Acupuncturists don't go in for rate charts. They live simpler, same rate per visit. I've a more pay-by-the-pin mentality.\n\n\"I hoped you'd come back for your things. I didn't quite know what to do with them.\"\n\nShe picks up a plastic shopping bag by her desk. She'd folded my things neatly and put them in a bag. My glasses lay on top. I put them on; I can think better with them on. They smell funny. She'd cleaned the lenses with rubbing alcohol.\n\n\"Yeah, I'm real sorry about that,\" I say, setting down the bag and fumbling through it for my wallet. I take out a ten and hand it to her.\n\nShe smiles and nods appreciatively.\n\n\"Yeah, something sort of happened, and I, um, had to take care of it right away.\"\n\n\"Sure,\" she says. She's completely unaffected by my disappearance. Perhaps worried _for_ me, but not about her money. _She_ is a placid meadow rivulet.\n\nShe walks to her desk to write me a receipt.\n\n\"I would have said something, but I was gone so quickly, when I thought about telling you... I, uh, came right back.\"\n\n\"No worries,\" she says. \"Have a good afternoon.\"\n\nAnd she means it. Nothing fazes this woman. Not that she isn't bright, just nothing bothers her. For a moment, I think, I should tell her what happened. Try my crazy story out again, give crazy another shot. Maybe even if she didn't believe me, she wouldn't treat me like a criminal. Like an ex-husband.\n\nInstead, I blurt out, \"Do you reuse the pins?\"\n\nI don't know why I asked. Yes, I do. I've always been mortified to think she reuses the pins. Disease just itching to spread. Even if you drown them in rubbing alcohol, it seems unsafe.\n\nShe doesn't answer. She seems to contemplate the motive for my question, perhaps connecting it to my sudden flight. Perhaps she's frightened by the invisible malignancy of knots she sees in my chi. We just stare at each other. She smiles. I notice my poorly buttoned shirt again. Dougie's shirt. I wait to see if she'll ask me for her pins back. She doesn't. She's way too peaceful for that. But I can tell she's wondering about them.\n\nI can never go back there.\n\nThe Volvo starts, so let's not pretend the entire day was unlucky. In fact, alone now, with my glasses back on and my wallet safe, I begin appraising things differently.\n\nI corrected my metaphysical mishap with minor casualties. I mean, think about it: I could have been lying, all pinned up, and thinking about speeding traffic. I could've been thinking about the mall.\n\nCincinnati.\n\nHelicopters, what about those?\n\nWhat if I'd been thinking about cooing pigeons on the ledges of the bank downtown? Or what if I had been daydreaming about a nice cup of espresso and beamed my mostly naked self to my very public workplace? For a moment, I count my blessings and I am calm.\n\nNow it's 4:32 p.m. Rush hour in this city is not the worst in the world, but the priest I'm headed to see works downtown. I speed up and merge onto the interstate. The notion of talking with someone besides myself, someone who might at least hear what I'm saying, who won't throw me out on the lawn\u2014well, it feels like a step toward something positive. I was doing something about my problem, and that felt good. I punch the gas and think, _Right. You definitely should speak to someone. Before it happens again_.\n\nPure heart-stopping panic. Who said it would happen again? I have no precedent, personal or historical, scientific or hallucinatory. Neither CNN nor comic books nor Dial-A-Prayer nor Google has any wisdom on this matter. What are the warning signs? I experienced no tingling in my left arm, no sudden weight loss, no curious blotches to my peripheral vision. I didn't rub any old Arabian lamps, hadn't broken a chain letter, wasn't bitten by a radioactive cutting board. It just happened.\n\nThe thought that I didn't know what came next, when it would come next, whether or not the impossible would happen again, horrifies me. Who lives their lives moment by startling moment?\n\nThen I think, _Hey, Ace, what if it happens while you're driving?_\n\nImmediately I jam the car into second gear, lurch over into the slow lane, and take the first exit off the interstate. The Volvo makes new and interesting noises. But if a driverless Volvo suddenly careens off the road, it won't be going seventy miles per hour. I put on my hazards and drive like my grandmother after she sips a daiquiri.\n\nLike that I drive, miserably slow, holding my breath. You have no idea how far things really are until you drive somewhere in first gear. Like watching tea steep. I'm driving, thinking, _Man, I wish I was already there... no. No, I don't. Time-out. Recall. Uncle. I do not wish any such thing_.\n\nI park at a meter outside the cathedral. Into this meter I pump all the change in my pockets, and I only get fifty-two minutes, leaving me still twelve minutes shy of the six o'clock cutoff. It's been that kind of day, you know? Like dropping your keys three times trying to lock your front door when you're late for work and you just know in your bones you should go back to bed? The whole day like an inside-out sock.\n\nThere's no more change on the Volvo's floorboard, not under the mats, not in the ashtray. I spend nearly twelve minutes looking for more change, so if I'd only waited to put my original change in... that kind of day.\n\nNot one fuzzy nickel.\n\nFather Chavez is a good man, a reasonable one. He's been at Saint John's for fifteen years or so, as long as I can remember. We've been chatting quite a bit lately\u2014sometimes after Mass, sometimes when I stop by after an early weekday shift and bring him a hot chocolate. He's the kind of priest who'll sit down anywhere with you: on a pew, outside in the grass, on the cathedral stoop, at the kitchen table in the church refectory. He's a quick sitter, crossing his legs and leaning forward to hear you better. He nods and scratches his head, pulls at his ears. I like that about him, that he doesn't make you go into a box, the confession box, or his office. He's always busy, but always stopping what he's doing to sit down and chat with somebody. That's probably why he's always busy. I guess people bother him a lot with their troubled stories, but this story was a doozy. Good guy or no, I feel I'd better ease him into this one.\n\nFather Chavez sees me. \"James!\" He stops preparing the altar for evening Mass.\n\n\"Father,\" I say.\n\nHe walks down the steps, turns, kneels, and crosses himself. All priest, this guy. He stands for just a moment to whisper something, then he turns and shakes my hand.\n\n\"James, how are you?\"\n\n\"Can we speak for a minute, Father?\" I ask.\n\n\"A few minutes, yes. I've Mass at six o'clock. What can we do?\"\n\nI wonder if he means God\u2014\"we\" can do? It's like he's bringing God to our meeting, and I'm not sure I like that.\n\n\"Something's happened to me, Father,\" I say.\n\n\"Sit, sit.\"\n\nI walk him down the center aisle to a pew a few rows from the front. Four rows back, to be exact. I don't know why no one ever sits on the first pew, but it's just not done. Too close to the action. What if they ask for audience participation? Too close to the God area.\n\n\"Father,\" I start. \"We can speak, right? It's like this. Just a couple of hours ago, I...\"\n\nHe does that cross-legged bit, leans, his face turns instantly intense. Instant fervor, this guy.\n\n\"Whoa! I didn't shoot anybody or anything,\" I say.\n\nHe was a priest, after all. I don't want him to ready himself for something murderous. We are not technically in the confession zone, so I'm not sure if we're covered by the no-disclosure clause.\n\n\"I haven't done anything wrong\u2014or not that I know of. I mean, yeah, I've probably screwed something up this past week, but that has nothing to do with what I want to tell you. In fact, I don't think anything about it classifies in the sin category, so don't worry.\"\n\n\"I'm not worried, James.\"\n\n\"Okay. Good. Stay just like that. You might be worried, in a minute,\" I say.\n\nPeople are already coming in for Mass. You know, old ladies who feel they need to show up way early or it won't count. Maybe they're just inveterately early. Old people do that, their whole day: a half-cup of coffee, the paper, lunch, Oprah, Mass, and time spent getting places early.\n\nAnd I don't know how to start: do you just blurt out a crazy, impossible thing, or do you chitchat first? I don't want to freak out a priest right before he has to do Mass.\n\nSo I say, \"The Bible's full of fantastical elements, right?\"\n\n\"Some,\" he answers.\n\n\"Healing the blind, right? There's a pack of that going on, huh?\"\n\n\"Some.\"\n\n\"Fish and loaves, right?\"\n\n\"There are several accounts of feeding a multitude with very little capital,\" he says.\n\n\"Right! Fantastic stuff.\"\n\nI just sit there watching the old people creep around and slowly fumble with the kneelers.\n\n\"Fantastic,\" I repeat. I sound like I'm giving a pep talk, like three cheers for the Bible, as if I'd just read the book and had no idea how much crazy stuff would be in it.\n\n\"How about imperviousness to fire?\"\n\n\"Old Testament. Daniel would be your book.\"\n\nI knew that one; I'm just feeling him out.\n\nI say, \"How about flying?\"\n\n\"Flying?\" he asks.\n\n\"Yeah, flying.\"\n\n\"I take it you don't mean birds?\"\n\n\"No, like people. Look, up in the sky, it's a bird, it's a plane, no... it's Habakkuk... that bit?\"\n\nHe scratches his head, crosses his legs the other way.\n\n\"No. I can't think of any humans flying in the Bible.\"\n\n\"Perfect. How about, say, special aquatic powers?\"\n\n\"Are you asking me if there are good swimming passages?\"\n\n\"No, I mean like talking to fish.\"\n\nMy comic-book reading was very limited as a child. My mom never bought me the cool ones because there must have been some discount, generic comic-book rack at the grocery store. Those are the ones my mom would have thrown in the cart. I got _Archie_ , _Betty and Veronica_ , _Aquaman_ , and that man in the iron suit. I can't remember his name.\n\nFather Chavez says, \"Jonah was swallowed by a whale, but I don't think they discussed it.\"\n\n\"Right, but being swallowed by a whale, that counts? At least, semi-fantastic. Recovering from that, doing a fish-break, and walking back to town. And there are seas parting down the middle, right?\"\n\n\"Yes... wait a minute... Jesus talked to fish.\"\n\n\"Really?\" That I didn't remember. Of course, I'm thinking dolphins, and Father Chavez, he means trout or whatever fish are indigenous to the Dead Sea. Tilapia, maybe. If the sea has _dead_ in the title, are there fish? Living fish?\n\nAnyway, he's getting into it now. He's bobbing back and forward with his legs crossed like he's fine with this game.\n\n\"Yes,\" he says, \"Jesus talked fish into swimming into his disciples' nets.\"\n\n\"Just like Aquaman!\"\n\nI say this with a flair of surprised discovery and actually slap the priest's arm like we're buddies, like we just realized we were at the same baseball game years ago and saw the same home run.\n\n\"Technically,\" he says, \"I don't think Aquaman talked fish into the fishermen's boats. Against his Aquaman code. I think he was generally too fond of fish to send them off to be eaten.\"\n\n\"Good point. Jesus was on the fisherman's side.\"\n\n\"Right. Not Aquaman,\" he says. \"Definitely on the fish side. But the same fish-talking ability, in a way. And then there's the water. The Bible says he talked to the water; Jesus spoke and calmed waves. Technically, again, Jesus was probably talking to his Father about the waves, not so much at the waves.\"\n\n\"I don't think Aquaman could do that one.\"\n\n\"I believe not.\"\n\nA little elderly lady across the aisle scowls at us. Perhaps we are talking too loudly. Perhaps we are talking too stupidly. She rattles her rosary at me.\n\n\"Sorry, Se\u00f1ora Alvarez,\" Father Chavez apologizes. He lowers his voice. \"What's this about, James?\"\n\n\"Right,\" I say. I look around and lean in close. I whisper, \"Father, what about leaping from place to place?\"\n\n\"You mean like superhuman jumping?\"\n\n\"No, Father. Not jumping. And, for reference sake, that'd be the Incredible Hulk. No, I mean more like beaming. Instantaneous transport of one's body to another location. Like _Star Trek_ , but without the molecular static mumbo jumbo. Just snap\"\u2014I snap for effect\u2014\"and you're there?\"\n\n\"Just snap?\" He snaps with his right hand, then suddenly opens his left, palm up. That was much better.\n\n\"Exactly. I call it leaping.\"\n\nThere. That's where I mess up. I say \"leaping\" with far too much intensity. I roll the word out there like it's a gold coin, like leaping means out the window with the stolen microfilm. Even Mrs. Alvarez could tell I thought I leapt. I stare at her hard until she bows her head again.\n\n\"In the Gospels,\" says Father Chavez. \"I recall something of the sort.\"\n\nI'm surprised. That's craziness. No, really. I had no idea. All this blah-blah was merely my lead-in to asking if I'd come unglued.\n\nHe says, \"Once Jesus was cornered by a mob; they intended to kill him. And the Gospels say he suddenly wasn't there. He shows up next in another town.\"\n\n\"No way,\" I say.\n\n\"Yes, it's in there,\" he says.\n\n\"Get _out_ of here!\" I say.\n\nMrs. Alvarez thinks I mean her. She's appalled. She says something in Spanish. I cross myself and apologize.\n\n\"James? What's this about?\"\n\n\"So it's possible?\" I ask.\n\n\"Well. It was a moment of life or death; so it seems, it was necessary.\"\n\n\"And you believe that?\"\n\nHe smiles at me. \"I don't believe in Aquaman or Santa Claus, James. But yes, I believe the miraculous surrounded the Son of God while he was on this earth.\"\n\nThat sounded like a priest. That's what I wanted.\n\n\"Do you think it happens to anyone else?\"\n\nHe looks at me puzzled, and can you blame him? I barely tithe. I'm probably not the best Mass-goer he's got. I've never bought one raffle ticket.\n\nBut he answers, \"I don't recall any mention in the Bible of anyone else doing so.\"\n\n\"No, I mean...\" I check to make sure Mrs. Alvarez's lips are muttering, that she's praying, not just bowing and eavesdropping.\n\n\"No, I mean, do you think it could happen again? To anyone today?\"\n\nAgain, I blow it. Too crazy, too whispery, too genuinely concerned. I wiggle my fingers in front of me too much like I'm casting a spell on my own chin. I might as well have come to church wearing a cape and tall shiny red boots. Strapped a cat on my head and asked if anyone liked my hat.\n\nFather Chavez's face turns a worried corner. He leans back, folds his hands together. His forehead furrows, and he seems unsure.\n\nIt's a bad feeling, worrying a good priest.\n\n\"I don't know for certain, James,\" he says. \"Why do you ask?\"\n\nI can't speak. I remember Meg's face when she threw her shoes at the tree. I think I hear Mrs. Alvarez praying for me. My shirt is buttoned wrong.\n\n\"No reason.\"\n\nMore people are coming in the cathedral now because the Mass begins promptly at six. Other people in robes are trying to get the Father's attention.\n\n\"Stay for Mass, James.\" Father Chavez says. He stands up, signals to the others that he's coming. He presses my shoulder in that priestly way. \"We can talk afterward.\"\n\n\"I have to go,\" I say.\n\n\"No, James, please. I have to catch one or two folks after, but then we can\u2014\"\n\n\"No, no. Thank you, Father. It was just a silly question.\"\n\nHe looks at me. He doesn't seem to know how to help.\n\nSo he offers, \"Like I've said, there is some record of such a thing. Leaping, as you called it. After his resurrection, Jesus did it all the time, but that's different. With resurrected people, you can imagine, it's a whole new ball game. But there's the one instance I mentioned, maybe others. Of Jesus, I mean, of an unresurrected Jesus, doing that.\"\n\n\"Okay. Great. Thanks for your help, Father.\"\n\n\"Would you like to finish our talk after Mass?\"\n\n\"Oh no. Thank you though, Father,\" I say. My voice lilts, and I wrinkle my nose cutely, like he's offered me a finger sandwich or a meatball on a toothpick.\n\nBut he's still looking at me. I'm afraid he's going to ask me something too direct, and I don't want to lie to a priest in a church. He knows I'm trapped, so he simply leaves the invitation open.\n\n\"Have a good evening, James. Take care.\"\n\nHe walks away and steps through a door behind the altar, comes back out in his robe. He's quick at that. As I sit there in the pew, Mrs. Alvarez's courage grows now that I am alone. With no priest to protect me, she grunts and stares freely. I smile at her, give her a kind wave, and she scowls. Sensing she'd be happier if I acted more churchy, I pull down the kneeler.\n\nLowering myself, I think, _How do you pray about this?_\n\nRegarding prayer: I've prayed before, right? But they were mostly the stock prayers, borrowed prayers mumbled like passwords or the Pledge of Allegiance. I've said words _toward_ God. I've shot requests God-ward. But I've never had a question, not a _real_ question, that I didn't already know how I wanted God to answer. I usually tell God, more or less humbly, what I've already decided.\n\nIf things go well, sometimes I give God credit, but I don't usually ask for a thing, then wait to see if God answers. I'm not sure you should. And I've certainly never asked God about impossible things. Why bother God? And I don't even know what I want now; it's just that what's happened makes no sense.\n\nI bow and press my forehead on my folded hands resting on the pew in front of me\u2014a lot like praying but without using words. Plus I keep my eyes open. It seems safer. I can't really think of anything to say, so I look down and I see it: in the rack, at my pew, a slip of a bright yellow ribbon hangs out of the prayer book, marking some place. I stare at this ribbon. Being bright and yellow, its color is striking in an otherwise gray and dusty world. I smile at how yellow it is. That it would be brave enough to do that.\n\n\"God, forgive me,\" I say, not even sure why. Nothing happens, not then.\n\nI cross myself, getting up. I hadn't intended to stay for Mass, and I certainly don't want to pick up my conversation with Father Chavez later, so I figure I'd better go. Mrs. Alvarez will be happy there's an open pew for the kind of people she likes. And I am way too close to the front.\n\nI walk down the aisle and I open the big wooden door, stepping out onto the portico. I hold the door for a couple entering. But I don't even make the first step down. I stand there and see the sun setting over the horizon. The cathedral faces west, so it's the last bright sun of the afternoon, and it's in my eyes. I squint. It's brighter than it seems it should be that late.\n\nI think, _Do I really want someone else involved?_\n\nIf anyone, a priest would be good. I think fondly of Father Chavez: how he hums when he picks up hymnals, the way he'd talked about Aquaman, the way he'd pressed my shoulder and invited me to stay. Going home, I think, _I'll be completely alone_. For a moment, I desperately want to go back inside. I want to go to Mass. It wasn't too late. What's too late for a Catholic going to Mass?\n\nStanding there, I tilt my head.\n\nThe sun reflects in a funny way off my glasses, that special way that makes a mirror of your own lenses and a ghostly eye appears on the inside of your lens. You see your own wet, rolling, glaring eye, far too close. And, of course, it's looking at you because you're looking at it, or none of that would be happening. I'm trapped there, blinded by seeing my own too-close, focusing eye.\n\nAnd suddenly I think of that silky slip of yellow ribbon. So yellow, so silky, it too looked wet. Looking down on it, that yellow ribbon directly below my bowed head. Yellow. The bright, brave color of it...\n\n\"James!\"\n\nFather Chavez is talking to Mrs. Alvarez, but he turns and finds me praying.\n\n\"James,\" he says. \"Good! I thought you'd gone.\"\n\nHe grabs my shoulder again. Me there on the kneeler, four rows from the front. I leapt exactly where I'd been maybe a minute ago. Me staring at that yellow ribbon. I knock on the wooden pew just to hear the hollow sound. I grab the ribbon between my fingers to be sure.\n\n\"I'm glad you decided to stay.\" I watch as Father Chavez walks up the steps to the God area. Part of me wants to stay, but there are too many parts. How can I just sit through Mass as if nothing...\n\nI cross myself again and go.\n\nThat was the second time it happened.\n\n# **4**\n\nMy car now sports a parking ticket.\n\nI get in the Volvo and file the ticket on the backseat floorboard with all the other trash. I've decided I need to get somewhere I can observe what's happening to me.\n\nMaybe I should tell Father Chavez. Maybe he could help me; maybe he can fix this parking ticket. I think he can do that\u2014fix parking tickets. I don't mean supernaturally\u2014rather, he knows people. But I can't wait around because who knows where I'll leap next?\n\nNo, I'm going to put myself away for the evening, watch myself scientifically, see if it happens again. I could use a trusty assistant, but I only know one guy in my building. He talks to me. I don't think anyone else in the building will talk to him, and I don't know why I do. He's usually drunk, and I'm usually caffeinated; we both mumble and drool, me quickly and him slowly\u2014a gorgeous couple.\n\nNo, I'm on my own. I'll need a tape recorder and duct tape\u2014lots of duct tape\u2014a controlled experiment. I will, very empirically, lock myself in my apartment, barricade myself inside, duct tape the doors and the windows so no slipping out without injury to the tape. If you come home after a long night of leaping and the door's still taped shut, you know you've used the superpowers, right?\n\nI'll buy some groceries, stock up on coffee, won't leave the apartment until it happens again, and I'll catch myself this time. Get clear, solid, irrefutable data on what's happening to me. I've always prided myself on my ability to outrun my own insanity. A very controlled experiment.\n\n\"Perfect timing, James! I'm clocking out right now.\"\n\nThe moment I enter the coffee shop, Kevin says this. This guy. We work together. I mean, he works there, I work there, and he thinks we're friends.\n\n\"What?\" I ask.\n\n\"It's Friday. We should probably go now, to beat the traffic,\" he says.\n\n\"I'm sorry?\"\n\n\"It's Friday. Are we still on for tonight?\"\n\nI still have no idea what he's talking about. And I have no idea what Friday means to Kevin. I probably promised him something, but I don't care what he's talking about, I'm not going. He's in my way, though, not letting me get to the espresso machine. I've got needs just then.\n\n\"Sorry, no. Not going to be able to make it this Friday. Today, that is. Not making today at all. Sounds good, though, doesn't it? We'll have to do it some other time.\"\n\n\"Oh... I guess we could do something else. When's good?\"\n\nAs I don't remember what I'm saying I can't make it to, I'm a little leery of issuing a rain check, signed and dated. Besides, it seems I'm unstable. Either psychologically or metaphysically, I'm clearly on the verge of not being all there.\n\n\"You know, Kev...,\" I say, and realize I've never called him this before, never called anyone by some snappy shortening of their proper name, and I hate myself. But I say again, \"Kev, you know...\"\n\nSo I push past the guy.\n\nI make it to the espresso machine and shove a paper cup under the spigot, press the button ten or eleven times, just to be sure. I hear the machine whir, then click and start to grind beans.\n\nIt's one of those automatic machines, one button whizzes all. Completely inferior to the lovely old grind, tamp, cinch-the-portafilter-on-yourself machines. Sure, my clothes, arms, and face were always smudged beyond repair with coffee soot. Most days I looked like a chimney sweep. But a man feels like he's made a coffee when he's had to wrestle it out of the machine, when the machine would fight back, when it'd viciously steam your eyes, blast off a portafilter, and when it'd bleed black, creeping goo like the _X-Files_. Most customers don't mind the change. Most customers bury espresso in chocolate syrup two inches deep, under twenty ounces of steamed milk and a stack of whipped cream. Most customers are whipped cream\u2013oriented. But for soot-oriented espresso drinkers...\n\n\"Hey, James?\" Kevin interrupts again.\n\nI take a deep breath.\n\n\"Even if we can't get together tonight, do you think you might have a chance to talk?\"\n\nI can't fathom what's being missed here. Did I somehow inadvertently seem kind? I don't feel kind. Actually, I feel like my eyes are bulging. I just want coffee, to rub espresso between my eyes, massage it into my temples, but the guy's still standing there, asking me questions.\n\nSo I say, \"Man, I am really torn out about canceling on you like this, but my crazy wife, she's\u2014\"\n\n\"You're divorced\"\u2014he interrupts, pointing at me\u2014\"I mean, you said you were?\"\n\n\"Right.\" Note to self: replace default excuse. \"She's not my wife anymore. But, boy, she's still crazy, Kev! She wants to go to dinner, maybe a show. Old times' sake, she says. One loopy gal, and I kind of thought I owe her. See? So... sorry, Kevin.\"\n\n\"I'm okay,\" he says, but he looks sick, like my Labrador did after he'd eaten most of the throw rug in the front hall. Gave me a whole new perspective on the term \"throw rug.\"\n\n\"Hey, why don't you two use the tickets?\" He fishes in his shirt pocket and hands them to me. He's carried the things around all day, which worries me. Two tickets to this amazing jazz show downtown, Herbie Hancock and the Headhunters. I check the price on the tickets in my hand. Forty bucks a pop.\n\nNow I remember. Couple of days ago, I was dragging around the coffee shop. Divorced and dragging. Undercaffeinated maybe. And Kevin talking about jazz, how he wants to see this deal, tickets still available for Friday, he believes. He's being genuinely nice to me. I think, _You got to get out and meet people. Can't sit in your apartment_ _again all weekend. Kevin's all right. And there's no huge line of people with invitations_. So I said, \"Sure. Let's do it.\"\n\nThis is the promise I'm breaking to poor Kevin while holding eighty dollars' worth of jazz tickets in my hand to show my ex-wife a good time.\n\n\"Kevin,\" I say. \"I can't take these. My wife is a jazz hater. Childhood trauma with a saxophone. As I said, so very crazy.\"\n\nKevin just smiles, waves, and walks away. He's done with his shift for the day, on his way out for good.\n\n\"Hey, Kevin!\" I shout. I follow after him a few steps, but he ignores me. I feel awful.\n\n\"Give me a call, please.\" I say. \"We'll talk!\"\n\nHe's gone now, but all the customers, my fellow employees\u2014even the homeless guy who basically lives in the store's bathroom\u2014everybody thinks I'm desperately trying to get Kevin to go out with me. Everyone knows my wife left me. People stare in that trying-not-to-stare way. I hate being divorced.\n\nI collect myself, grab a couple of coffee timers, a pound of ground coffee, and my ten shots of espresso. I drink a big glass of water because that's good for you. I don't say good-bye to anyone, just head for the door. I didn't say hello to anyone so I feel I'm exempt.\n\n\"See you tomorrow, James!\"\n\nThis from a girl named Monica. She also works there.\n\nRegarding Monica: a truly beautiful person. Without question, she is the kindest person I've ever met. Everything makes her happy. Everybody's worth being nice to, and in Monica's world, everything is just about as good as it needs to be. Customers who ask for their lattes to be stirred counterclockwise on Tuesdays, she sees the good in these people. I've tried to make her angry; she won't budge. Pathologically nice, and she's beautiful\u2014brown eyes, huge smile, long hair that she keeps pulled up in some wispy, fabulous way\u2014but that's not what's attractive about her.\n\nWhat I mean when I say _beautiful_ is that, when she looks at you, there's something about her eyes\u2014they don't want anything. They don't already have a plan. They see you, just you, like she has no idea what's next and she's happy with right now. Her eyes listen.\n\nMonica's a straight line\u2014genuine, smiling, and straight. How can anyone see that cleanly? I've seen her look at a steaming pitcher the same way. And somehow I mean that as a compliment.\n\nI need that in my life.\n\nSo when Monica calls out good-bye to me, I turn around, walk over to her, and hand her Kevin's tickets. I take her soft hands, look into her brown eyes and say, \"I love you. I always have. I wish I could go to this show with you tonight, but I just don't see us working out.\"\n\nThen I walk out.\n\nPerhaps this sounds as if I'm burning bridges, but I'm not really. Even though most of my bridges have been highly flammable for some time now and I have barged loudly through the coffee shop in a hypervigilant semi-rage, these people understand I'm an espresso drinker. Chalk it up to happy synapses. I'm feeling better. The coffee's kicking in; the bridge of my nose tingles. So let's just say it was an exit. I thought it graceful under the circumstances.\n\nIt's getting dark and I don't know how late things stay open. For a recorder, I need an electronics store.\n\n\"Mister...\"\n\nThe boy genius at the Gadget Town enunciates this carefully as if maybe I'm losing my hearing.\n\n\"Mister,\" he repeats, \"you probably want to check yard sales or thrift shops for tape recorders. What do you need one for? A prop for some old eighties play?\"\n\nNow, I'm thirty. I'm not that old. Not old enough to have some pimply geek in khakis and a bright blue golf shirt talk to me like my teeth just fell out on his counter.\n\nI ask, \"Are those flat screens impact resistant?\"\n\n\"These? Sure,\" he says. He looks at the televisions next to his counter as if they are way out of my league.\n\n\"How impact resistant?\" I ask.\n\n\"Technically, these new ones are impact resistant up to\u2014\"\n\nI pick up a stapler and spike it off the biggest screen.\n\n\"Hey!\" The kid recoils as if he's been shot. And they are. Remarkably resistant.\n\nI ask again, \"I'm wondering if you could point me toward the tape recorders?\"\n\n\"Aisle nine!\" He screams, then flees his counter.\n\nI know he's heading for security, ratting me out. The stapler thing was completely unappreciated. I don't have much time, so I jog to aisle nine. Recorders, rows and rows of them, all digital. This is the advancement I hadn't kept abreast of in my advanced, doddering state. I'm thrilled when I read the package: up to one hundred hours of recording time. You can download things you've recorded into computer files to save them; you can re-record an infinite number of times; you can send in the bar code on the packaging for free earphones. I look around for the kid to tell him thanks, but he seems uninterested in making up. Something about the security guard he sends my way. The kid stands at the end of the aisle, pointing at me with the stapler.\n\nI head the other way. Not really running, but fast walking. Lots of hips and arms movement. I hop up and down to look over the aisles, locating where they keep the registers. The security guard has a walkie-talkie, and he and the kid are in hot pursuit. Also fast walking, hips and arms. The guard puts the walkie-talkie up to his mouth. Reinforcements! He presses buttons, clicks, and mutters, but something's wrong. He clicks and clicks and clicks. Inexplicably, it appears that the security guard in an electronics store has a dead battery.\n\nThe kid looks at the security guy with sheer disgust. \"Didn't you plug it in and charge it?\"\n\n\"That tone right there,\" I call out. \"That's how staplers get thrown!\"\n\nBut they've stopped pursuit altogether. Here in Gadget Town, a gadget that doesn't work properly stops them in their tracks. They gather around the walkie-talkie like it's a dead body fallen from the sky.\n\nI buy the digital recorder and slip away, but in the Volvo, I'm gasping to catch my breath. All that fast walking. So I try to breathe. Called a bit of attention to myself that time. So I decide to make a concentrated effort to be low-profile in public places.\n\nLast stop before home, the grocery.\n\nI stop inside the front doors of the Shoppy Mart for a moment to take a breath. I stand casually, very low profile. For groceries, I'm a man who likes to make a list. I like leaning on a cart and glancing at or scratching out, but I've got no list now. I try calmly to search myself for something to write on, but I have nothing.\n\nSo I make a mental list:\n\n_1. Paper\u2014for on-the-fly lists_\n\n_2. Notebook\u2014for observation, a journal chronicling my events, and for extra paper_\n\nImmediately, I see the papery redundancy of this list and, mentally, I cross out, then erase:\n\n_1. Paper\u2014for on-the-fly lists_\n\nBut now the list starts with the number two. I have to change the two to a one. Just making a list, I get all high-profile:\n\n_1. Notebook\u2014for observation, etc_.\n\nThen add:\n\n_2. Duct tape\u2014too many rolls_\n\nI need food. There's no food in my apartment, and I have no idea how long I'll be locked away. What if there's no leaping when you're watching it? I don't cook. I'm a warm-and-serve kind of guy. I shake and drink. I tear along the perforated edge. I mentally write:\n\n_3. Rations_\n\nUsually, list-making soothes me, but that's when I have paper. There's something about getting it out of your head, where you can see it. I look around for a soothing distraction. Outside, it's night now, and the parking-lot lights cover everything with their orange glow. Instead, I look around the grocery and wonder why so many people are grocery shopping at night. _Don't these people have plans? It's Friday night. I had Friday-night plans. With Kevin. Forty-dollar tickets. I owe the guy_.\n\nAs I'm depressing myself with thoughts of Kevin, I see a bird, a brown-and-white bird, inside the grocery, flying over the aisles. He's trapped and confused. The bird flies frantically this way and that, but he can't find where he came in.\n\nBirds don't belong in groceries. It's too bright, with too many people, and there's no sky, just a paint-chipped ceiling. I stand on the mat to keep the automatic doors open, and I wave my arms, but it doesn't help. In general, birds don't fly toward humans flailing their arms.\n\nThen I remember: this self-enforced time-out was called to lower profile, to draw less attention to myself. Now there's an old lady afraid to enter the open doors because of me.\n\nI lower my arms and walk over to get a cart.\n\nThis next part is not entirely my fault. I can't unstick a cart. Someone has clearly jammed and welded the shopping carts together. I'm serious\u2014they're welded together, because no matter how I pull the nearest one, the whole line of them moves. Every cart I touch seems to clench its teeth and hold on for its wobbly-wheeled life to the cart in which it's wedged. I try them all, and let's not say noiselessly, then take a break to gather my strength.\n\nSo I'm watching as the old lady steps up cautiously and then, with a slight upward lift, liberates the cart I'd just abandoned. She tries not to look at me, but, oh, she's pleased. I try grabbing the next cart after hers. Completely immovable. There's a trick to doing things normally, and I don't have it.\n\nI kick, I shake, I cajole, I argue, but nothing.\n\nFinally, an old man in a uniform steps up to help me. Grocery-store rent-a-cop. His badge is clearly plastic, but his gun is not. He has a gun, it's a big one, in this huge holster on his hip. He walks off-kilter, like the hip has been replaced or the gun weighs too much. And I wonder why a senior citizen making minimum wage is packing heat at the Shoppy Mart.\n\nHe steps up next to me and says, \"They stick.\"\n\n\"Good work, Ranger.\" That comes out before I can stifle it. Proudly, I do stop myself from saying, \"Shoot a cart loose for me, cowboy.\" It wouldn't have mattered, though, because the guy's deaf as a post. A weathered, well-armed, deaf post. If it all goes down one day in meats-and-dairy, do we want this guy waving a pistol?\n\nI slip away as he works on the stuck cart and grab a handbasket instead. I decide to eat less, buy only nutrient-dense items, load up on protein bars and beef jerky. Chewing\u2014I know I can do, and that's the only possible reason for the existence of beef jerky. Chewing without the bother of digestion. But first, the tape. I wander up and down aisles. The chances I'm talking audibly to myself are, yes, very high. I am actually announcing the items as if I'm doing personal inventory.\n\n\"Mustard: yellow, brown, sweet, deli, squeezable, and there's the spicy. Mayonnaise: real and the other kind. That's easy. Pickles: kosher, garlic, dill, spears, whole, halved, hamburger slices, sandwich slices, thick, chunky slices, gherkin, sour, fancy, sweet, and, again, there's spicy. So much spicy.\"\n\nLike making lists, sometimes I inventory. It's calming.\n\nFinally, tape: electrical, packing, Scotch, masking, and duct. No spicy. It's expensive, but I take the duct tape. I'm a firm believer in duct tape. You can never have too much duct tape. I've never taped myself into my apartment, so I guess twenty rolls will do. They have seventeen, and I take them all.\n\nRegarding handbaskets: clearly these are made for someone buying three apples. Or say, two apples and a bunch of celery. A person could just carry two apples to the register, but three? One might juggle, unless there's celery. A cute, cheap plastic, worthless, three-apple basket is therefore provided, complete with an insufficient coat hanger of a handle. And duct tape is surprisingly heavy. The wire handle immediately begins to buckle and bend, and the basket's plastic bottom just sags. But there is no way I'm going back for a second round with the rows of welded carts. I stoop as I walk, partly because the basket is that heavy, partly because you think stooping helps a cheap basket not to break. I begin to realize that seventeen rolls of tape will limit my buying food supplies to the amount of beef jerky I can cram under my arms.\n\nAn abandoned cart would be ideal. People do that\u2014abandon carts. I know that greatly increases the chances of getting a cart with a spastic wheel, or a wheel that doesn't roll at all. And technically, is it still a wheel then? Rolling is fundamental to its wheel-ness.\n\nBut I spy one, a seemingly abandoned cart, by the frozen fish sticks. Of course, how can you tell if a grocery cart's been truly abandoned? Sometimes people get a cart and park it cleverly somewhere in the store with every intention of coming back for it. If they've touched it once, they've got dibs on it, no matter where they leave it.\n\nWorking on keeping low-profile, I eyeball this cart from a distance for four, maybe five minutes. No bites. It just stands there, but it looks like there're a few things in it. Technically, that's a claimed cart, although the guy could have forgotten his wallet, just left it standing there. I can't see the items, but from a distance, it looks like three things. I figure you multiply the minutes spent unattended by the number of items in the cart and subtract from a hundred. Roughly, eighty-five percent chance this cart was abandoned, which is pretty high. I watch for a minute or two longer, but the handbasket is so heavy the wire handle digs into my hands.\n\nI rush this cart, heave the duct tape, basket and all, into it and push it around the corner quickly. In the next aisle, I slow down and put on as if this cart and I have been together for years. Keep my eyes up, concentrate very diligently on the shelves, lean over the cart comfortably, moving along at an easy stroll.\n\nI find the protein bars and grab a box, tossing them in my cart. Grab a couple of notebooks. I find the beef jerky rack. I spin it, tossing jerky after jerky on top of the duct tape. Now that I have my own items, I think it time to discard the previous cart owner's items. Only one other guy is in the aisle, a tidy, business-looking type in khaki pants and a white button-down, busy reading the label on an oat-flakes box. He seems safe.\n\nThere's nothing fishy about taking items out of your very own cart. Put them back wherever, as it happens. From beneath the pile of jerky and duct tape, I dig and pull out a dented loaf of bread and a box of denture adhesive.\n\nAnd a purse.\n\nAs if on cue, the old lady who'd trumped me at the cart game comes around the corner. Her face is flushed, and she's panic-stricken. She walks in short, fluttery steps, as if the tiles are intermittently bursting into flame.\n\nShe gasps, \"My check! This month's check! It was in my purse.\"\n\nShe is gasping this to the rent-a-cop. She holds him by the elbow, and he's all business, bristling like a dog who's heard a noise outside the window.\n\n\"That's my purse. There!\"\n\nShe means, of course, the purse I am currently holding. The guy in the khakis shelves his oat flakes and turns his attention on us. The senior security man puts his hand on his weapon.\n\n\"Here's your purse, ma'am,\" I say, taking a step toward her.\n\n\"Drop the purse, Mister!\" says the security guard.\n\nI know he's deaf, so I am certain he mistook my movement for aggression. I stop and hold the purse out.\n\nThe lady grabs her purse, immediately opens it to locate her wallet and her check.\n\n\"It's still here, thank goodness,\" she says.\n\n\"I just found it in my cart.\" Saying this, I turn and point at the cart in question. The khakis guy has already closed in and is so close I can smell his Old Spice. He inspects the contents of the cart in question, surely wondering why anyone needs seventeen rolls of duct tape and enough beef jerky to build a child-size beef-jerky cabin.\n\n\"I'm sorry. I didn't realize there was a purse in this cart until just now.\"\n\n\"What'd he say?\" asks the security guard.\n\n\"You just happened to shop using a cart with someone else's purse in it?\" This from the oat-flakes guy.\n\nAnd I want to say, \"Yeah, her purse looks a lot like mine,\" but I don't because right about then, I notice the detective's badge clipped to his belt.\n\nInstead, I offer, \"I assumed the cart in question was abandoned, Officer.\"\n\n\"Did you assume she'd abandoned her purse, too?\" he says, heavy, like he's interrogating a drug runner.\n\n\"I didn't see her purse,\" I repeat.\n\n\"I got that cart from the front.\" The old lady still points, still flutters, still avoids imaginary flames. \"He couldn't get one loose.\"\n\nThis, I feel, is entirely unnecessary. The old lady is showboating for the cop, the real one. And the security guard backs her. \"I saw him. He couldn't even get his own cart.\"\n\n\"Detective Goss,\" he introduces himself, then asks. \"Did either of you see the purse actually being taken?\"\n\nSuddenly, it's like I'm not there, and for a quick second, I wasn't sure either. I check to confirm that I am.\n\n\"Look, I'm sorry. I didn't do it on purpose. It was purely accidental, I swear. I've had a very accidental day. Most of my life, lacking the purposeful. You have your purse, and nothing's missing. Let me get this stuff out of my cart\u2014\"\n\n\"It's my cart,\" she says, solemnly pressing her hand on her chest. The deaf security guard grabs her arm like she's going to pass over right there by the cereal.\n\nIt's best to start loading my items into the cheap little basket. The detective hands the rolls of tape to me one by one. Very helpful and creepy, this guy. I manage the tape, the notebooks, the protein bars into the basket, but not all the jerky would fit. Not stuffing things in my pockets seems wise. I set the basket on the floor, bend over it, and pack in what I can.\n\n\"Excuse me,\" the detective says.\n\nHe hovers over me, and he must Old Spice his socks, it's that strong on him.\n\n\"Do you mind if I ask your name?\"\n\n\"Actually, I do kind of mind,\" I say, staring at his shiny black shoes.\n\n\"Excuse me?\" he says. And he's not asking this time; he's threatening.\n\n\"Look, I'm not a thief, okay? I didn't mean to take the lady's purse. I just want to go, if that's all right with you.\"\n\nNo answer, so I ignore him and keep piling stuff in my basket. I watch the shoes turn and go, and finally, Detective Goss walks away.\n\nIt's the way he walks away that unnerves me. Not like a guy shopping. His back is alert, if you know what I mean. He's on duty now, and I can tell from his shoulders that he's aware of me, what I'm doing. He walks to the end of the aisle, stops, pretends to be looking at a box of Fruity O's. No way this guy eats Fruity O's. He's watching.\n\nI keep packing my basket. It takes awhile, but that's good.\n\nSo I'm fine, right? Items in basket, fine. I stand, and doing some fine standing. In a few minutes, I'll be in the Volvo on the way to my apartment for a fine evening of self-induced solitary confinement. Fine. I heave my hands up near my chest and lean back just to carry the heavy basket\u2014even this is fine. I slip to the next aisle, an empty aisle. Finely done. I can see the registers up ahead, and there's hardly a line. Very fine. I just have to make it down this one empty aisle, and I'm as good as free.\n\n_Dear God, just let me make it to the Volvo..._\n\nI huff and shift the basket.\n\nAs if an answer to prayer, Detective Goss appears at the register-end of my aisle. He waves at me, signaling me to come to him.\n\nOf course, I turn and walk back the way I came.\n\n\"Hey, buddy,\" he calls out.\n\nI don't feel like his buddy, so I keep walking. I'm almost around the corner of the aisle when I hear sharp shoe squeaks, like he's running to catch up me with.\n\n\"Hey, guy!\"\n\nI stop. I'm hyperventilating. No, I'm definitely not ventilating, hyper or otherwise. I don't turn, but I hear Goss come up behind me. I look at my hands\u2014the basket's wire handle cuts whitely into my fingers, and they're numb. I look at the heaping basket, the duct tape, the jerky, the protein bars and their shiny silver wrappers, and I feel like I might black out. My eyes blur.\n\nA heavy hand lands on my shoulder.\n\n\"Hey, listen. I'm sorry you're having a bad day. Everybody thinks because of who I am, I mean them trouble. But I'm sorry if you felt leaned on\u2014\"\n\nIt's too late. I'm so dizzy my eyes blur on the shiny wrappers like I can see through them. I rush around the corner to get away from the detective. _Please, God, not right now!_ But I pray way too late. It's got me, I feel it happening, feel like that bird flying from aisle to aisle not knowing where a door is...\n\nI'm blinking.\n\nI stand there, staring and blinking. The amber reflection of my face on the Volvo's windshield, the way the parking lot's lights directly above me make the Volvo's dirty windshield an amber mirror, my face in a dirty amber mirror, and perched on the wiper in the amber, artificial light... a bird.\n\nA brown-and-white sparrow.\n\nA car horn blows. A minivan pulls into the empty space beside me, and I step closer to my Volvo so as not to be hit.\n\nThird time's a charm, right? The bird flies away.\n\nI open the Volvo's door and heave the basket onto the passenger seat. I stop because I should go back and tell them what I have, to pay for it. But when I look back at the glass front to the bright grocery, I see a dark, authoritative silhouette\u2014Detective Goss stepping on the electric mat that opens the sliding doors. I've parked far from the entrance, but I see the guy, all backlit and angry. His head shoots left then right, scanning the parking lot.\n\nI duck.\n\nEven the least paranoid of us need to duck now and again. I crawl into my car. I'm not sure if he's seen me or not, but I'm not about to trot out a second crazy explanation for Supercop. The Volvo actually starts again, so there are still miracles available to the ordinary man. I back out, slip it into first, and start to drive away. I'm thirty, and this is my very first getaway.\n\nI don't turn on my lights. I adjust the rearview mirror only to see Detective Goss, his silhouette against the Shoppy Mart doors. I can't tell if he's looking my way or not.\n\n_Maybe he wasn't close enough_ , I think. No worries. _Maybe he didn't see me? Maybe, in his surprise, he didn't even think to look at my license plate number? No_ worries. _Maybe I'm completely forgettable?_ All fine.\n\n_Dear God, please..._\n\nI think this despite how stunningly my prayers have been answered of late.\n\n_Make me forgettable..._\n\n# **5**\n\nThe crazies don't set in till later that night.\n\nLooking back on all I've written so far about my first day\u2014none of it could happen, right? How could any of it be true... but I remember all of it so clearly. And if it's true, and it can't be true, then what's gone wrong in my mind?\n\nAs an answer to that, I can see my barricades, the duct tape everywhere, on the door, the windows, the walls, stuck to my arms, my fingers, in my hair. I feel very stupid or exposed, like I've yelled out, \"Surprise!\" in a room full of people waiting for the guest of honor to arrive. Like I yelled too early and all those staring faces become frowningly uncertain, as if it's even a party, if any guest is coming at all.\n\nThat's what I mean by the crazies.\n\nWhen something happens to you, something unworldly and inexplicable, a thing that cleanly surpasses your ability to categorize it\u2014you don't doubt it in the moment.\n\nDuring the crisis, you simply react and respond as best you can. You have to stay afloat, no doubts then because you're inside it. But later, when the crisis has passed, when you're trying to remember exactly how it happened, it all seems unreal. Make believe. 'Cause you're fine now, you know?\n\nOr so you think.\n\nIf you've ever slipped and cut yourself with a knife, afterward you don't remember the sequence of ordinary events that led right up to it, just that dizzying recollection of strange pain, the creepy sensation of blade opening flesh, of the senses screaming that something is very wrong before the brain translates the message for pain, you know?\n\nI know I should sleep. But deep inside, I feel...\n\nIt's like this: Meg's an ice-cube user. Not excessively; in fact, rarely. But when she wants a cube or two for a beverage, she expects ice to be there. She used to become enraged and yell at me for using up the ice cubes, then putting the empty trays back in the freezer. She'd get chin-cocked, eye-fluttering furious. She couldn't fathom why some idiot would go to the trouble of putting the trays back in the freezer but wouldn't take the three seconds to fill them with tap water. Somehow, it just bent her mind the wrong shape. Meanwhile, I always eyed her suspiciously, as if she'd popped her last emergency mental bobby pin, because I don't use ice. I chill a beverage beforehand. Meg was the only other being with opposable thumbs in the house using ice, and then she'd deliriously, foggily forget she had done so.\n\nAnd just try telling her that. She'd get this tilted-back, wild-eyed Vivien Leigh sort of crazy face when I'd tell her she must've used the ice, because this woman fills an ice tray, if you know what I mean. She'd yell and chase me through the house, clapping trays together.\n\nWe found out after we were divorced.\n\nLeft too long in a freezer, ice evaporates.\n\nAll that is to say: tonight my head's the freezer. Something's evaporating. I can feel it. Something I didn't know could do that. How do you check on that if you're the freezer? How do you become certain of something outside your own head? How do you verify something outside your own mind when verification is an act of the mind? There's a piece missing.\n\nI believe I leapt through space. I remember it; it happened. But that can't happen, can it? If it can, it sort of madly rattles my known universe.\n\nI'm not sure I believe any of it. But I'm not sure if I can simply _not_ believe.\n\nHow do you believe? I mean, something completely impossible, some strange thing that happens only to you, the one thing that changes everything? Isn't that madness?\n\nHow do you believe something with your whole life?\n\n# **6**\n\nThe next morning, I feel fine.\n\nI don't mean, _Gee, what a great day to have an inexplicable power!_\n\nMore like, _Maybe nothing happened at all_.\n\nNothing to worry about, maybe none of yesterday was real. _My dear James, you do have a penchant for imagining things_ , I tell myself. After crazy days, that's the way next mornings go.\n\nYou've finally managed sleep; you wake up late. You put on coffee, and that smells homey and normal. You eat some beef jerky, brush your teeth, get another cup of coffee, clean your glasses, slap on your baseball cap, and everything's fine. Yesterday was somebody else's day, somewhere else, and it doesn't touch you.\n\nI'm safe.\n\nI've been here before, you see. Whole days of worried stuff, only to talk myself down the next morning, and usually, events turn out fine. Maybe not one hundred percent fine, but not gruesomely tragic.\n\nI'll confess:\n\nI have spent an entire day spinning myself around, thinking that my identity had been stolen by our mailman.\n\nThen a day when I have skewered myself onto the imagined prong of accidental but nonetheless punishable tax evasion.\n\nI have suffered twenty-four hours basting my well-being with juicy worries over contracting a not-so-real case of consumption.\n\nI've spent more time bordering on nervous collapse than the average coffee drinker, overcooked by my own horrified imagination, too often wakened bewildered by my fearful, turning mind, the sheer rotisserie of it.\n\nThis morning's hopeful turnabout is more than possible\u2014it's right on schedule. I smile warmly at my slept-in clothes, chuckle nostalgically at the duct-taped mess, and chew another wad of beef jerky while sitting, smiling, on an unpacked box in a sunny spot by the window. Home sweet home. I thumb through my notebook like it's a fashion magazine, read a snip here and there, and laugh at what I find. I stretch and consider bathing.\n\nA knock at my apartment door stops me.\n\n\"James?\" Meg's voice.\n\n\"Coming, Meg!\"\n\nTo give you a read on how deep my denial runs, I walk over to the door, unlock the chain, turn the key, and only then realize it's going to be much harder to open the door. I'd just been admiring all of it a moment before: the duct tape, the chairs, the stacks of books, the wedged coatrack. Now I'm confused as to why the door won't open. Deep, I tell you.\n\n\"Hey, Meg,\" I call through the door. \"What's up?\"\n\nThere's an awkward pause.\n\nI hear Meg's voice. \"Would you mind opening this for a minute?\"\n\n\"Well...\"\n\nI don't know what to say. I don't want to tell her about the barricade, don't want to explain it. I relock the door so she won't try to push her way in.\n\nAnother pause. This one, also, very awkward.\n\n\"James, did you just unlock then lock the door?\"\n\nI don't answer.\n\n\"James? Are you still there?\"\n\n\"Yeah, Meg, I'm here... You still there?\"\n\n\"If you could please open this door, then we could stop calling roll and actually speak.\"\n\nThis is reasonable. Barricades are less reasonable. I sigh.\n\n\"It's going to take a minute.\"\n\n\"I've seen you naked, so just throw on anything and open the door.\"\n\n\"I have a lot of clothes on.\" This is my thoughtful response.\n\n\"Way to go, James, dressing yourself like a big boy. I'm proud for you... and tired of standing in this hallway.\"\n\n\"Just a minute!\"\n\nI call this out, singsonging, like a prairie mother yodeling \"Supper's ready!\" to her eight kids.\n\n\"James, is somebody else in there?\"\n\nIt takes me a moment, but I realize she means another woman.\n\n\"No, Meg.\" This almost as flattering as her suggesting that I can pick locks. \"I'm all alone, it's just that\u2014\"\n\n\"James, it's fine if someone's there. It's none of my business. I was just a little concerned about you yesterday, that's all. You were acting a bit crazy even for you, and I need to get Doug's shirt back.\"\n\nI stop moving books.\n\nI ask her, \"Did I sound crazier than normal... or more normal than crazy?\"\n\n\"That's a hair I hope I never have to split,\" she answers through the door. \"Look, I'm glad you're fine, and that you've found someone. I'm sure she's lovely, but I need that shirt.\"\n\nNow it seems even trickier. I don't want Meg to think I've found someone, although I'm not sure why. I would rather she thought I was just lonely instead of jumping headlong into another relationship. Maybe it's because I'm a little tweaked at how quickly she's taken up with Dougie, how she's up early on a Saturday for poor Dougie's shirt. But I'm also thinking I hear a twinge of resentment behind that \"lovely\" comment. She's usually not openly rude to people unless she knows them. Can a person exhibit animosity toward you and jealously about you simultaneously? I'm genuinely intrigued, and I want the door open.\n\n\"There's nobody here, Meg.\" I start throwing boxes whichever way they'll go. \"Just boxes piled up against the door.\"\n\nShe doesn't say anything for a few seconds. \"Are you piling them up now that I've knocked, or are you moving them away?\"\n\n\"Meg, just hold on.\"\n\nShe tries the knob. \"You've locked the door on me, and now you're piling stuff against it.\"\n\n\"One minute.\"\n\n\"Look, I'm sorry I bothered you!\"\n\nI can hear it. She's mad now, shoe-throwing mad.\n\n\"Meg\u2014\"\n\n\"I didn't mean to interrupt your... whatever. I don't know why we can't ever seem to talk like two normal people using normal-people words in normal-people situations.\"\n\nI just stand there, hoping she can't hear my blinking.\n\n\"If you don't want to talk to me, then just say you don't. You can not talk to me as much as you'd like. I don't even have to see you, but open the door three inches and stick your naked arm out here with Doug's shirt. Stick your girlfriend's naked arm out here... I don't care. Why do I have to yell through a door, James? Just say, 'I don't feel like talking.' Or say, 'I've given up conversation till Christmas.' It matters not to this ex-spouse, but don't make me stand in this spooky hallway while you make up excuses and slide furniture around.\"\n\n\"I'm not sliding furniture around. I don't have furniture.\"\n\nThe slam that followed that remark sounded like a fist punching my door.\n\n\"Put Doug's shirt in a plastic bag, and leave it in my mailbox. Today!\"\n\n\"Meg\u2014wait!\"\n\nSo diligently had I crammed a chair beneath the doorknob last night, it's stuck.\n\n\"Here's the shirt.\" I yell this, hoping Meg will hear and that she'll wait. I realize I have no idea where I put Dougie's shirt.\n\nYou'd think it would be easy to find things in a desolate apartment until you've actually lived in one. Nothing is anywhere specific, so there's nowhere to start looking for anything but everywhere.\n\nI run to the bedroom, throw the sheets to the floor, rummage in them because they are white like a shirt. No luck. I drop to the floor, look under the bed, and remember never to look for something under a bed. You'll only be distracted by what you do find.\n\nI check the bathroom: the floor, the towel rack, the hook on the back of the door. Nope. I run to the kitchen: not on the counter, not in the garbage, not on the card table, but hanging on the refrigerator handle is a very wrinkled shirt with obvious coffee stains.\n\nI grab the shirt and run to the window. It's a second-floor apartment with a great view of the parking lot. I yank at the duct-taped curtains, only serving to bring the curtains and rod crashing down in a heap on my head. I throw them off and see Meg walking toward her Jeep.\n\n\"Meg!\" I call out.\n\nShe can't hear me\u2014or won't. She's got her keys out. I want her to know I didn't break in her house. I want her to understand. I don't want to slink over to her mailbox later with a bag o' shirt. I want her to come in, have a cup of coffee, and for us to talk like the normal people she mentioned. I'm tired of everything going badly between us.\n\nI pound on the windowpane. \"Hey, Meg!\"\n\nI'm really close to the window, so close I see my breath fog a quick patch on the pane. As quickly as my breath fogs the window, it clears. And just as quickly...\n\nHer Jeep's dashboard. She's waxed inside her Jeep, the dashboard is shiny.\n\nWhen she opens the driver's door, I say, \"Hey, Meg,\" from the passenger seat.\n\nI wince as her keys fly past my face and hit the window. She jumps back out of her Jeep.\n\n\"Holy... how'd the... don't... freak... freak... freak... James!\"\n\nMeg's doing her scared dance. She squints her eyes shut, lifts one leg, then the other up slow, like an injured horse, and shakes her hands till her fingers loosen. I used to hide in closets and scare her just to see this. After the dance, she takes a slow lap around her Jeep with both her hands on top of her head. She gets back in, grips the steering wheel, and stares forward.\n\n\"Hand me my keys, please.\"\n\nSitting in the passenger seat, I reach down and feel for them.\n\n\"Here you go, dear.\"\n\nShe leans toward me.\n\n\"How'd you get down here so quickly?\"\n\n\"Here's the shirt... sorry.\" I say.\n\n\"James!\"\n\n\"I don't want to talk about it.\"\n\n\"How did you\u2014\"\n\n\"You don't believe me when I tell you anything, so I don't want to talk about it. Will you believe me?\"\n\n\"Is there a fire escape?\" She cranes her head out her window and looks back at my apartment building.\n\n\"Actually, there really should be a fire escape. I could probably get out, but everybody else\u2014\"\n\n\"James, stop it.\"\n\nTruly, I don't know how to talk to her anymore. Not about anything. Especially not this.\n\nSo I ask, \"Do you want to come in for coffee?\" She always liked the way I made her coffee.\n\n\"Do not scare me like that ever again!\"\n\n\"I didn't mean to scare you.\" We both stare forward like we're driving through a tunnel and watching for the end. Silently staring, but not going anywhere.\n\n\"Do you want coffee?\"\n\n\"No, James,\" she answers. \"Thank you, but I need to go.\"\n\n\"Sure,\" I say. I open my door and pull myself out.\n\n\"James?\"\n\nI lean back inside. \"Yeah, Meg?\"\n\n\"How did you get down here so fast?\"\n\nAfter a few years of marriage, a couple develops the means to simply leave a conversation unfinished by staring at each other quietly, patiently until one or the other looks away. Sometimes it's all you got.\n\n\"See you later, Meg.\" I shut the door.\n\nShe peels away.\n\nGood thing Meg didn't stay for coffee.\n\nWalking up the stairs to my apartment, I realize my door is still locked and my keys are hanging in the lock on the inside\u2014not to mention the tape, the chair, the coatrack. I lean back against my apartment door. I could check into a hotel maybe, but I'd left my wallet inside. Note to self: duct tape wallet securely to face.\n\nLeaning there, I want inside that room, so I think about trying it. A leap. What are the similarities of the leaping occurrences? Something too close to my eyes, too close to properly focus on and usually reflective: a watch face, the lenses on my glasses, silver protein bar wrappers. Each time my eyes tried to focus on what was impossible to focus on, until my eyes flitted back and forth between surface and reflection. Transparency and reflection.\n\nAnd each time, I had an intense desire to be somewhere else. The image of elsewhere appeared in my mind because I desired it. And the moment my desire pulled my focus away from what couldn't be seen, I leapt.\n\nProud to say, I'm thinking all this with only three cups of coffee in me.\n\nI look down the hall. Coast is clear.\n\nI take off my glasses, close my left eye, and press my right eye, my good one, up to the peephole. For reference's sake, peepholes are ingenious little things\u2014they only peep one way, but the other way, not so good. In this case, seeing didn't matter, and I stood there up against my door, shifting focus between what I _couldn't see_ inside my apartment and back to the actual glass of the peephole.\n\nOnce I feel this is going pretty well, I try to think, suddenly, of being inside.\n\nRegarding sudden thinking: try this sometime. Try thinking of a thing, but suddenly. Either you screw it up 'cause you realize you're _already_ thinking of it when the moment for suddenness passes right by, or you make yourself _not_ think of it too fast and basically consciously forget to think about the thing when the moment of sudden thinking is called upon. I'm not crazy. This is difficult.\n\nI give up, and start shoving, hitting, and kicking at my door.\n\nI don't leap anywhere. I do, however, hop around the hall with my foot basically broken from poor door-kicking form. I hop, I sing, I swear, I sing swearingly.\n\n\"Need some help?\"\n\nThe question comes from the guy I told you about, the one in my building who nobody talks to. The only one who talks to me. He walks up in ripped jeans and a \"Mind, Spirit, Body\" T-shirt; it's in quotes on the shirt\u2014I think it's an ad for moisturizing lotion.\n\nWhen I see him, I sort of pity him because everyone thinks he's a drunk or crazy, at least smelly, and then I realize, as I hop and sing in front of him, that he's definitely thinking the exact same thing about me.\n\n\"No, I'm great,\" I say. I sort of sing it, with the pain.\n\n\"No, you're not,\" he says.\n\n\"Who are you, by the way?\"\n\n\"Nelson. What's yours?\"\n\n\"James,\" I say and immediately regret telling him my real name. My toe's throbbing, so I sort of lean over, hum my name to myself, and ignore Nelson.\n\n\"Let me help.\"\n\nI look up at him. He's not a small guy; he's got a little weight and muscle on him.\n\n\"Sure,\" I say. \"Thanks.\"\n\nThen Nelson starts to sing. He's helping me sing, looking at me to try to match the pitch. I squint at him, to show him how very far he is from the rest of us.\n\n\"Sorry, no,\" I correct him. \"Could you help me bust down my door?\"\n\n\"We're not singing?\"\n\n\"I'm not singing, no. I think I broke my foot.\"\n\n\"You're not broken yet.\"\n\n\"Thanks, Doc.\" I try to lean my weight into the door.\n\n\"We trying to break this down?\"\n\n\"Good idea, yeah.\" I bump my toe again. \"Oh!\"\n\n\"Go on, sing out. It'll help us push, like pulling oars. What are we singing?\"\n\n\"We're not really. I'm just giving voice to my pain, thanks! But I think if we lift on the knob and then both shove\u2014\"\n\n\"Sure. Go on. Sing out!\"\n\n\"No, but thank you...\"\n\nHe stares at me, considers me sadly. \"You know, people who have to drink before they'll sing out depress me.\"\n\nAnd I think, _Oh, is that all?_ but I say, \"Who said anything about drinking?\"\n\n\"Well, it really is too early for it.\"\n\n\"I haven't been drinking,\" I protest.\n\n\"Your drinking doesn't bother me. Just your fear of singing out while sober.\"\n\n\"Look, even if we get inside, you'll see I don't own a drop of booze.\"\n\n\"Oh, I wouldn't care for any.\" Nelson sniffs.\n\n\"Good, 'cause I wasn't offering.\"\n\n\"Good, because I never touch the stuff.\"\n\nWe stare at each other, he tucks in his T-shirt with the lotion ad, so now it just reads \"Mind, Body.\"\n\n\"I haven't been drinking,\" I say.\n\n\"I thought you were Catholic?\" he asks.\n\nWe take a lingering, second round of staring at each other in the hall.\n\n\"How do you\u2014no, _what_ is that supposed to mean?\" I say.\n\n\"Wait...,\" he says. \"You admit you're Catholic?\"\n\n\"Yes, of course,\" I answer. \"Sort of\u2014\"\n\n\"And you haven't been drinking?\"\n\n\"Absolutely not!\"\n\n\"Then why are we breaking down a door?\"\n\nStaring at this guy, I fear I'm looking at myself after a few more hard years.\n\n\"Okay, focus, Nelson. You should either go away right now and never come back here, or help me shove down this door, please.\"\n\n\"Sure.\" And he starts ramming his shoulder into the door with great fervor, like he'll do it all by himself. I join him, and we shove, grunt, and take turns hurtling ourselves against my well-made door. This is painful, and after a while, we stop to catch our breath.\n\n\"This door breaking sure makes a man thirsty,\" he says to me.\n\n\"If we get inside, I've got a sink with tap water just like your apartment, Nelson.\"\n\n\"Mmm-hmm,\" he says. \"I'd enjoy some tap-tap-tap water.\" He shoves his shoulder into the door three times, on the taps. \"Get it?\" he asks, pointing at his shoulder. \"Tap-tap?\"\n\nThen he shoves his shoulder again, until it seems he's forgotten his own joke. He's just strangely taken up with shoving at the door because he doesn't wait for me to laugh, but frowns and starts banging furiously into my door, which doesn't budge in the slightest.\n\n\"Nelson, Nelson... stop it!\" I have to slap his kidneys to get him to hear me.\n\n\"Ow... what's the problem, Johnny?\" he says.\n\n\"It's James.\"\n\nHe looks around confused, as if I've seen a third guy coming down the hall.\n\nI look down the hallway too, but now I see a potted plant. I leave Nelson, fetch it, and with several cavemanlike swipes, I bust off the doorknob to my own apartment.\n\nNelson and I still have to throw our collective weight into the door several times to break the tape and topple the junk leaning on the other side.\n\n\"This place is a wreck, Junior.\"\n\n\"Thank you, Nelson,\" I say. \"Did you want some water before you promptly go?\"\n\nNelson looks at me strangely, like he doesn't understand why I'm so angry with him, like he doesn't understand why anyone would be angry at all or why people look harshly at each other. He nods, so I fetch him a glass of water. He takes a sip, but it seems he doesn't really want it. Just sipping like he's fulfilling some promise he made, and then he looks for a table to set the glass on, finally setting it on the floor.\n\n\"When I was thirsty, you gave me...\" he starts, then says. \"Thanks.\"\n\n\"You're welcome,\" I say. \"Thanks for\u2014\"\n\nBut he's gone.\n\nAfter I get my wallet, I fill my backpack with my notebook, my recorder, a roll of duct tape, protein bars, and jerky. I pile my television, my CD player, and anything else valuable in the bathtub and close the shower curtain. My thinking: if someone breaks into my already-broken-into apartment, maybe they won't look in the bathtub for valuables. Heading out for the coffee shop, I take the useless keys out of the lock and shut the door. I duct tape the doorknob vaguely where it should have been. The doorknob sags stupidly, but hopefully no one will notice.\n\nBesides, who'd sneak into my apartment?\n\n# **7**\n\nAbout time you showed up.\"\n\nThis is how I'm greeted when I walk into my workplace.\n\n\"Glad to see you, too, Mike,\" I say.\n\nI wasn't. I've never liked Mike, and Mike, he's not pleased either. He's a tough guy, or thinks he is. He likes to terrorize everyone with the way he bumps into them as he squeezes past to his register. Thinks it's funny to trip people. Stumbling's funny to Mike. And he ties his apron real tight, like he's showing off his stomach muscles.\n\n\"You're forty minutes late, Jimbo. We've been running crazy all morning,\" Mike says.\n\nI turn to him and say, \"First of all, we've spoken about that Jimbo thing, haven't we, Nancy? Secondly, it's not that busy on Saturdays. Thirdly, I've been running crazy for days; try to catch up.\"\n\nI'm decent at the quick bluster. Meg and I practiced on a regular basis. Truthfully, however, I had no idea I was supposed to be at work.\n\n\"I stayed over a little, James. It's all right.\" This from Monica, and she smiles at me. She stayed to pick up my slack. She never seems to go home. I don't know if she has a home, but I've always thought that if she does, it must be a happy, warm one where people sing and sway like some animated movie. Birds clean up crumbs in Monica's home.\n\n\"Thank you, Monica.\"\n\nShe blushes. \"Thank you for the tickets, James.\"\n\n\"Did you go?\" I ask.\n\n\"No, I had to work.\" She shakes her head.\n\nThis makes sudden sense to me. How could she use tickets for last night when I gave them to her last night while she was working? I'm a crown jewel, a real peach.\n\n\"But thank you, James,\" she says.\n\n\"Sorry, I do everything too late\u2014\"\n\n\"No, it was sweet. And... I kept them.\" She blushes again.\n\nI remember then that I sort of professed my love for her, and well\u2014we're all less than comfortable.\n\n\"I'm taking a break.\" Mike scowls and throws his apron to me.\n\n\"I'm not staying,\" I say and throw it back to him.\n\nMike says, \"You going to call and tell Charlene?\" Charlene's the manager.\n\n\"No, you call her. Tell her I called in sick.\"\n\nI'm not sick, of course, not normal sick. But if I haven't mastered this leaping ability, what's to say I won't just blip out during the middle of a mocha? What if I zip-zap away while I'm holding someone's twenty? I've already added shoplifting to the rap sheet this week; I'd probably get fired. If I'm going to wreck a thing, I'd like it wrecked on my own terms. I need a little more time to straighten me out, and I would rather disappear the old-fashioned way just now.\n\n\"Are you not feeling well, James?\" Monica asks.\n\nShe's incredibly sweet. She hands me my twelve shots of espresso in a big paper cup.\n\n\"Thanks,\" I say. And I think, _I should ask Monica how she pulls off so much kindness amidst insanity on a daily basis_. Maybe I really should ask her out, maybe proximity to her kindness would negate some of my hyperneurotic, terminally paranoid, overcompensation.\n\nBut it wouldn't work, she and I.\n\nI'm not used to nice, and she'd nice me straight into apoplectic shock. Besides, I got metaphysical issues to straighten out and don't have time for a relationship. But she's standing there, calmly concerned if my coffee tastes right. Kind and coffee-conscious when I'm most insane.\n\nIt's not fair.\n\n\"I'm not sick,\" I say, \"I'm just losing my grip on which things are the real ones, Monica. Thank you so much for asking.\"\n\nI've never seen someone blush three times in a row. I didn't think that was possible.\n\nThen I spot Kevin, and he knows about his tickets. I don't know how he knows, maybe Mike told him. He knows I gave Monica his tickets and didn't go with my ex-wife. He wears a limp, the-world's-too-large-but-I'll-be-fine look. He knows that I was avoiding him.\n\nSo I say, \"Hey, Kev!\" Now it seems I'm going to call him that the rest of his life even though I truly want to stop.\n\n\"Hey, James.\" He turns away as if that's all for him, then remembers. \"Oh yeah, some police guy called here for you this morning.\"\n\n\"What did he want?\"\n\n\"He didn't say, just asked for you. He acted like he knew you. Knew your name.\"\n\n\"When?\"\n\n\"I guess around nine,\" Kevin shrugs. \"I told him you'd be in at eleven.\"\n\nIt's almost noon. My grocery cop? I'm crazy enough without this.\n\nHow did he find out where I work? Did he see my license plate? For the love of coffee, it was beef jerky! No, it can't be Goss. Even my paranoid mind can't believe the police are that efficient, but I'm not sticking around to wait for whatever.\n\nKevin turns to go.\n\n\"Hey, Kev... I'm truly sorry about yesterday,\" I say. \"The truth is, I am a little out of my head right now, and I was no good person to know last night. That's why I cancelled, not because I was avoiding you. It's nothing personal.\"\n\n\"I understand,\" says Kevin. The guy looks right through me. Like he really understood. Like I'm transparent. How could he know? He says, \"It's good to talk to someone when you feel out of your head.\"\n\nAnd almost. Right then and there, I almost sit them both down, hold Monica's hand and tell Kevin everything. Get it all off my chest. I don't have that big a chest. Wouldn't it feel so much better to tell someone what's making me crazy?\n\nBut who'd believe me? It's not believable. Plus I'm supposed to be leaving\u2014my detective knows where I work, and I am supposed to be working a shift. With Mike, no less. Mike's throwing ice against the back wall. He does this. Mike is somewhere between Pre- and Cambrian. When he gets angry, he likes watching ice shatter and fly into a million pieces all over the back room.\n\nThe phone rings.\n\nMonica answers. \"Hello. Oh, hi... Charlene. Did _whoever_ show up? Oh, you mean... James?\"\n\nHer eyes grow huge when she says my name. She stretches out my name like I'm supposed to hide from the voice on the phone before she gets to the end of it. She looks at me, pulls at her apron. She's anguished, caught between truth and loyalty.\n\nShe blushes again.\n\nI give her a finger-across-the-throat slashing signal, shaking my head no. But I don't want to listen to Monica trying to lie, so I try to make her honest by leaving. I knew she'd think that inbounds.\n\nI grab my espresso and head out the front.\n\n\"Later, James,\" Kevin calls out.\n\nI wince. Kevin kills me, I swear. I give Kevin the finger-throat-head slash and shake. It's not personal\u2014I just don't want Charlene hearing him, but Kevin turns away like I've killed him. He helps a customer. I should stay and explain. I should give someone a believable explanation for my behavior. I should be nicer, less blustery. I should bring a big box of doughnuts to grease interemployee relations. And I should've let Monica and Kevin tell Charlene whatever they want, because Mike will rat me out regardless.\n\nI think of all the good _shoulds_ too late.\n\nI drive to the bookstore.\n\nIt doesn't take long, being directly across the street from the coffee shop. I drive a block though, park on the back side so nobody at my store can see where I went. I don't want my manager to know I ditched work only to go to the bookstore across the street. Ditching that close to the ditched shift\u2014that's just impolite.\n\nMy plan is to isolate and repeat leaping similarities. I can't deny this is happening to me, not now. It wasn't just yesterday's superproblem. It happened again this morning.\n\nBut then I couldn't make it happen, and I had to bust my own door down. There must be some trick, some gimmick, some missing piece. If I can make it happen consciously and not accidentally, then I can show somebody. I can prove I'm not crazy. I've always longed to prove that.\n\nI go for the bookstore's handy customer-reference computer. It's one of those big chain bookstores, the Book Barn, a barn for the slick, new, and shiny. I navigate the options on the dirty touchscreen, trying to find any help I can, search all the \"leaping\" literature. Odd as this may seem, I find nothing.\n\nSeems the bookstore doesn't get a lot of requests for superhuman how-to books. All searches lead nowhere. If there are books, even fictional ones, about leaping through space, I don't find a section for them. I don't think the graphic novels will help, being a short step above comics. And it doesn't feel smart to ask a clerk, \"Excuse me, Miss, but I'm having a little trouble controlling my superpowers. I was wondering if you could point me in the direction of the superhero reference section?\"\n\nBesides, I know these people\u2014I serve them coffee every morning. They might talk.\n\nI put my hopes into wandering the self-help section.\n\nAh, bliss in pocket-size paperback. Happiness in a helpful, step-by-step format. Besides weight loss, I locate two key sections: two-minute meditation guides and mental yoga exercises for making more money. I go more for the former, grab a thin book with a cartoon picture of a Buddha wearing a jogging suit and earphones.\n\nYes, it's promising that he's sitting in the lotus position, but much less promising that he's making a sort of rap gesture and that the bubble caption reads \"Peace Out.\" But it's all they've got, so I take the book over to a comfy chair by a window in the far back corner of the store.\n\nI try to arrange my legs into the lotus position in the chair, but with the armrests, my work boots, and general inflexibility, this proves impossible. I can send my body across untold distances, but managing both ankles on top of my knees\u2014ridiculous.\n\nI thumb the table of contents. No references to my predicament, but I do find a chapter about \"Visualizing Your Dream Day.\" Chapter 8. After a glance at this chapter, it seems to be about imagining yourself in a job you don't hate. Not getting a different job, but just not hating the one you've got.\n\nThis book is written as a dialogue. Someone asks the Hip Buddha questions\u2014I guess that would be me, the reader. And Hip Buddha, he's all down with answering my annoying questions. He's like Socrates, except using cool slang and never actually making an argument for anything. He just pops out these non sequiturs like he's misheard the question.\n\nOr maybe Buddha's not listening so good. But the guy who's asking the questions\u2014and again, I'm assuming that's me\u2014this guy's thrilled at the swerving conversation of Hip Buddha. I think, _Who publishes a book of insane rantings?_\n\nI riffle back through the earlier chapters, thinking I probably missed something stunningly important. I land for a while in a chapter called \"Clearing Your Mind for _What's Important_.\"\n\nIt scares me that _What's Important_ is in italics. It seems tricky, disingenuous, and underhanded. A sneaky-but-upbeat way of telling you you've been doing it wrong, wasting your time on all the unimportant things.\n\nI'm suspicious. Seems Hip Buddha's up to something.\n\n\"But, Hip Buddha, I don't understand. What are the _important_ things?\"\n\n\"The important are the only\u2014\" saith the Hip Buddha.\n\n\"I'm sorry. I don't follow. What did you say?\"\n\n\"The only thoughts we must have are the _important_ thoughts,\" saith the Hip Buddha. A cartoon picture of Hip Buddha bumping his heart with a cool fist accompanies his answer.\n\n\"The conscious being must take out the garbage,\" he purrs. \"Garbage gone, the truth appears.\"\n\nDespite the fact that \"we\" muddle up my brain with a trash truck full of thoughts, it seems I really have precisely nothing that matters to think about.\n\nOn the next page, Hip Buddha is nodding; he agrees. There's little shaky movement lines around his bald head so we know this. No words on this page, but it seems I was right; I already knew the right answer.\n\nI check. This book costs $13.99. And a whole page for a bald cartoon to nod, but wait, the next page is better. Here, Hip Buddha stretches his arms and yawns.\n\nMe too, Buddha.\n\nThe point, I find out, is that we need to let the _important_ thoughts sort of yawn, stretch, and knock around in all that empty brain space we've managed to make by taking out the worthless mental trash.\n\nThe yoga-ankle-meditation thing would be easier.\n\n\"Excuse me, Hip, but I've been having this problem leaping through space.\"\n\n\"Find the inner question, you must.\"\n\n\"No, Hip Yoda, listen... please. First, my eyes focus on a reflection, then they go blurry, and then I\u2014\"\n\n\"Release your garbage from the orbit that is not you.\"\n\n\"Whoa, Buddha man. What I need here is\u2014\"\n\n\"There are no needs.\"\n\nAt this point, I realize, I hate Hip Buddha. Animosity toward cartoons has to classify as garbage. So, in a way, the cartoon I hate is right. I can't read this.\n\nI find I'm staring out the window at a tree, at the wind. Very Easternlike, I encourage myself, back to nature, staring at a tree. Of course, I'm sipping espresso inside an air-conditioned bookstore, sitting in a comfy chair, but hey, one step at a time.\n\nI close my eyes.\n\nEye closing is the entirety of Hip Buddha's chapter 1. Apparently, I've never closed my eyes right either. You don't just shut them like you're playing hide-and-seek. No, you relax your eyes, your sight, until you're not actually seeing anything although the eyes remain open. Once one reaches this state of unseeing sight, saith Hip Buddha, the eyes will mystically close of their own accord.\n\nMy eyes, however, are wide open, and uncomfortably dry. I've got a backpack to watch out for, my stuff, my coffee. It's good coffee. Someone might take that while my eyes are mystically closed. Maybe a sixteen-year-old bookstore employee will think I'm done, in my clearly catatonic state, and throw good espresso in the trash.\n\nFrankly, I don't trust sitting with my eyes closed. I don't trust book shoppers not to do me harm. I don't trust myself to not fall out of the comfy chair. Why does my sitting involve so much trust? Do other people have this much trouble sitting?\n\nDon't Buddhists worry about their backpacks, at least a little?\n\nWhy does closing my eyes feel like I'm slipping away?\n\n\"Forgive me, Mr. Buddha, but I need to\u2014\"\n\n\"There are no needs to forgive.\"\n\nNow I'm mad at Hip Buddha. He's just wrong. I'm not supposed to close my eyes at all. I should be focusing on something, not on nothing. In fact, that's all I got, the focusing. Why am I working so hard on making myself black out in a chair?\n\nSo I leap up and begin pacing the book aisles. I don't go far, never out of sight of my backpack. You have to leave something as collateral, a claim, or some person will think the chair's free if it's left unattended too long. Like a shopping cart. I'm picking up books I don't really want, feigning interest so as not to draw attention to myself, all the while periodically spinning around to stare at an empty comfy chair.\n\nThe good news is that this behavior is exhausting. I return to the chair, plop down, toss the meditation guide on the table next to my coffee, and put my feet up. I wonder what time it is, wonder if the shift I've ditched at the coffee shop is over so I can go get another coffee.\n\n12:34 p.m.\n\nAs I stare at the time, my watch face catches an image of the tree outside the window. Everything's upside down, and it's windy outside. I just look at the reflection of the tree, its branches playing in the wind. It feels odd and quiet to see a gusty wind without hearing it or feeling it on your skin. It would be good to feel it. A soundless wind making branches bend and wave, reflected atop the snick-snick of the second hand on my watch face. I lean in close, but too close. The tree's harder to see now, the time easier. I focus. I want to see it, so I do: flapping yellow leaves on branches, inverted in a soundless wind...\n\nLike sand.\n\nWhen a leap happens, it feels like you're sand.\n\nFor a quick second, your whole body is sand sifting through an hourglass, but pouring upward, filling the top bulb and pushing into it from below. Everything sifts upward into your head, through it, up, and out the top. That's all. A few seconds like that, or maybe a second. For that second, you feel like you're falling upward, outside yourself. It's that brief, that quick.\n\nBut for a moment, you're outside of you.\n\nStanding outside, I'm looking straight up into the tree; the rustling branches seem upside down; the leaves seem to dangle awkwardly. I touch the rough bark of the tree trunk, feel the wind on my face.\n\nI look back into the bookstore through the window. I see some kid by the comfy chair. He's messing with my backpack, looking around to see if he can find its owner. He picks up my coffee cup by two fingers like it's trash.\n\nI check the time. 12:35 p.m.\n\nIt's real now. I can make it happen.\n\n# **8**\n\nBack in the Volvo, I get jittery.\n\nMore than the espresso, I mean, I'm all excited and giggly that I'm able to control it. I mean, it's amazing. And the implications are tremendous! I can work a shift at the coffee shop, then spend the afternoon in Paris or Rome, instantaneously arriving free of charge. I can save a bundle on gas as long as no one catches me appearing. I'll say I walked, or I've taken up running. Buy a sweatband and act all out of breath, that sort of thing.\n\nWhat if I can take someone with me? What if I grab Monica by the hand, like Peter Pan, and we can both go to Paris or Rome? I could charge for it. Not Monica, of course, but other people. Charge just slightly less than the airlines. No, I could charge slightly more because it's instantaneous\u2014no stale food, no delays, no cramped, germy airplane bathrooms, just whoosh and you're there. Or I could work for the CIA. This deal is perfect for the espionage circuit.\n\nWhat if I can teach it? Or maybe just impart it to someone else? Just one touch, and bam! Like Midas? Okay, bad example. But think: a whole squad of leapers out there in the world, doing good. We could have weekly meetings in a remote castle. A thousand leapers... or maybe not. It kills the \"super\" in my superpower if everyone else can do it.\n\nBut a superhero, he has to think about these things.\n\nAs I drive, I look at the people. People in their less-than-instantaneous cars; people bunched on the sidewalk, waiting forever for the signal to change; people huffing away on their slow, ordinary bikes. And I think of the live coverage: me, leaping atop tall buildings with a single bound. I'll need a mask, a suit, huge odd gloves. I could get sponsors, like NASCAR, put endorsements on my cape.\n\nAnd I think: _Meg_. I want to call Meg right now. She's got to _see_ this happen, then she'll believe me. Good ol' Meg. Who cares if we're divorced, this is a kick in the gut! Blow her angry lawyer mind. Knock her affidavits off. I drive there, to my good ol' home. Meg's, but that doesn't matter now. Once she sees this, perhaps...\n\nBut I get a firm reminder of whose house it is when I pull up. In the form of a Lexus. Dougie's car. Doug's Lexus takes up most of the drive; he parks squarely in the middle. He's a middle-of-the-driveway parker.\n\nThe espresso in me turns a little mean.\n\nI park across the street, sit in the Volvo, and stew for a minute. I wish I had my phone, but it's back at the apartment, hopefully safe in the bathtub. I'd call Meg and tell her to come outside. I don't want to knock or to talk to Dougie. He can sit inside in his starched shirts and miss the whole show. Maybe I can leap home and get my phone, then leap back? But as I sit there, the front door opens, and Meg comes out. Of course, she spots me.\n\nPolitely put, my Volvo is unmissable, and I'm parked directly in view of the house. Meg's already headed my way. I don't like the tone of her walking.\n\nI roll down my window.\n\n\"Ah, what a surprise.\" She leans on my door.\n\n\"Hey, Meg. Can we talk for a minute?\"\n\n\"I see you more now that we're divorced than I ever did during our marriage. Why is that, James?\"\n\n\"Maybe I miss you?\" I smile.\n\nShe doesn't.\n\n\"No, really. I've got to show you something.\" I start to open my door.\n\nShe slams it closed. \"You're not staying. Thank you for not washing the shirt, by the way. What do you need to show me?\"\n\n\"I have to explain, and it might take a minute. Get in.\"\n\nI lean across and unlock the passenger door.\n\n\"James, why do we have to\u2014\"\n\n\"Get in, Meg, please. Five minutes... I swear it'll be worth it.\"\n\nShe puts her hands on her hips. She walks around reluctantly, opens the door, leans down.\n\n\"We're not driving anywhere, are we?\" she asks.\n\n\"No, Meg. I promise.\" I take the keys out of the ignition, toss them on the dash, and hold my hands up safely.\n\nShe climbs into the Volvo, grunting. Volvo doors are bulky, heavy, and prone to shutting on your leg. She's always hated my car. She could never even start it. And she refused to drive a car without a knob on the gearshift. She claimed she always broke a nail shifting without the knob, and she hated the smell. There's this sour coffee smell in the floor mats, which she never fully appreciated. She always slammed the doors\u2014she slams it now. An old heavy Volvo door slamming sounds like someone dropped a toolbox. She shakes her head.\n\n\"Okay... so...\"\n\n\"Hi, Meg,\" I try.\n\n\"You know the jail time is serious for kidnapping, right?\"\n\n\"We'll sit right here, dear.\"\n\n\"Please stop calling me _dear_.\"\n\n\"Sorry, hon.\"\n\nShe looks over at me, then looks away. She drops her head back against the headrest.\n\nI didn't mean to do that, to make her mad. It seems it's all I do these days. This isn't going well so far, but in a minute, she'll forget all about that. I hope.\n\n\"How's Dougie?\" I never know how to start.\n\n\"How's Dougie?\" she mimics. \"Well, right about now, Dougie's wondering where Meggie went. Meggie went outside because the kids thought they heard the ice-cream truck. Dougie said, 'No, children. I, your father, did not hear the ice-cream truck.' But Meggie, being the good Meggie she is, said, 'I'll go outside and check. Wait here, kids, Meggie will be right back. Don't worry, I won't be gone long at all.' That's what good Meggie promised the children\u2014her last words before being abducted by her crazy ex-husband.\"\n\n\"Do you two really talk like that?\"\n\nShe opens the door to get out.\n\n\"Wait! Wait, Meg.\" I stop her. Besides, she can't manage the door. \"I'm sorry. Just listen, okay?\"\n\n\"James, every time you start with, 'Just listen, okay,' it's never okay. It gets weird quick. Don't get weird this time, okay?\"\n\n\"Okay Meg.\"\n\n\"You promise? No weird? Absolutely none?\"\n\nI don't want to lie to Meg. I never have. So I don't want to promise absolutely no weirdness when technically it's going to get weird. Leaping is weird, without doubt. If I show her I can leap, though, the amazingness of that, the spectacular nature of my new weirdness and the sheer possibility of the formerly impossible\u2014that'll outweigh weird, right?\n\nI hold up three fingers like a Boy Scout. That used to mean \"honest\" or \"Scout's honor,\" but neither of us has any idea what it means now. It seems I've broken the Boy Scout finger thing. So many things broken between us.\n\n\"What do you want to show me?\" she asks.\n\n\"You know this morning?\"\n\nMeg blinks. \"We've met, yes, this morning.\"\n\n\"Right. This morning. When I popped into your car. Suddenly and without logical explanation. I was up in my apartment, which you left, and you walked directly to your car without pause, you walked quickly to your car\u2014\"\n\n\"I fled to my car.\"\n\n\"Exactly, fleeing toward your car by the most direct route to get away from me. Me locked away, back in the apartment from which you were fleeing.\"\n\n\"See, you're getting weird.\" She reaches for the door handle.\n\nI grab her arm. \"Meg, I can do it.\"\n\nShe looks at me and waits. She opens her hands slowly, like a flower blooming.\n\n\"Do what, James?\"\n\n\"Leap,\" I say.\n\nShe just stares. She doesn't know what to say. She doesn't know what I mean.\n\n\"Yea for James!\"\n\n\"No, really,\" I say. \"I can do that\u2014leap! Across space, without time, without walking, without cars or steps or feet or anything. I can leap from place to place. From my apartment to your Jeep. From my acupuncturist's to my... I mean, to a... I mean, to _that_ garage!\"\n\nI point at the garage in question. No response.\n\n\"How do you think I got into the garage yesterday? How did I get in your house? Come on, Meg! I don't have a key. You took all my keys. You took the spare key out of the little smiling-dwarf flowerpot. So how'd I do it? How did I get in your Jeep today when I was upstairs, locked in my apartment? And I just did it again at the bookstore, a half hour ago. I leapt outside beneath a tree. I leapt last night, leapt out of a grocery store into the parking lot.\"\n\nShe folds her hands in her lap, checks her lipstick in the passenger-side mirror.\n\n\"Well?\" I say.\n\nNo response.\n\n\"Meg?\" I plead. \"Say something.\"\n\n\"I'm just glad this didn't get weird.\" She starts to get out.\n\n\"Wait! I'll leap right now.\"\n\nAnd it's her face that gets me. She stays in the car, she shuts her door again, but her face. Her look is somewhere between sheer disgust of having ever married me and complete pity.\n\nHonestly, the pity griped me more. This woman who used to make up the bed every night before she went to sleep so that the covers would \"stay right\" all night; this woman who used to lecture me as to how dishes \"liked to go\" in a dishwasher, used to lecture and demonstrate the dishes' preference, then remove these sentient dishes so I'd get my turn at putting them in pleasingly; this woman who would wander through the house having conversations with imaginary people, begging them to tell her why toilet-paper rolls were hard to replace. This woman thinks I'm the crazy one.\n\nMeg folds her hands and relaxes, waiting to watch me make an idiot of myself.\n\nNow I'm brimming mad.\n\nI look at my watch, move it too close to my face. I stare at it intently, spitefully. Look at it from an angle through my glasses. I'm going to leap like she's never seen.\n\nI see the hands on my watch, see my own eye on the watch face. It's an angry eye. Where am I going to leap? I have an intense desire to leap onto the hood of Dougie's Lexus. I focus on my own eye on the watch face. An intense desire to leap, stand, and stomp on Dougie's hood, until Meg gets over the shock of my disappearance from the Volvo and runs over to slap me. It's not working, so I use my other eye, stare at it for a minute. I move the watch back and forth, closer and farther from my eye, thinking about how the surprise will take Meg a minute to recover from and I'll be stomping the guy's hood.\n\nMeg whispers. \"Did you do it yet?\"\n\n\"Shh.\"\n\n\"Well, did you?\"\n\n\"Be quiet!\"\n\n\"I mean, you might have left and come back, and I missed it, right?\"\n\n\"Shut up, Meg.\"\n\n\"Tell me when you're gone, so I'll know.\"\n\n\"Can you be quiet, please?\"\n\n\"Oh! It's like golf.\"\n\n\"Have you always talked this much?\"\n\n\"Was I talking?\"\n\n\"There's no way to concentrate with you talking\u2014\"\n\n\"Sorry, Houdini,\" she says. \"And yes. Yes, I have always talked this much.\"\n\nI'm so hacked off! But I sit there focusing and unfocusing and trying to suddenly think of Dougie's smashed-up hood.\n\nNothing. I don't leap anywhere. I'm sitting there, cross-eyed and out of breath, next to my ex-wife who's just waiting to completely ridicule me.\n\nI don't want to give up. Not in front of her. Not again. I keep staring at my watch, but it's not working, and I can hear her pity bubbling into sarcasm.\n\n\"Tell you what, James. I'm going to go back inside,\" she says. \"When you get your magic-bean-double-agent watch working, leap on into the den, and we'll show the kids. They love a good leaper on a Saturday afternoon.\"\n\n\"Meg\u2014\"\n\n\"Can you fly, too?\" she asks.\n\n\"I really can do this.\"\n\n\"Oh, I have no doubt,\" she says.\n\n\"It's just that this is my first day of leaping\u2014\"\n\n\"Everyone has to start somewhere,\" she says.\n\n\"I'm just now learning how to leap, to leap when I\u2014\"\n\n\"Fa-la-la-la-la, la-la-la-la, James!\"\n\n\"Okay, I know it sounds crazy, but\u2014\"\n\nI try to speak, but she's in full song now. \"Meg, please, don't.\"\n\nBut she can't stop. She keeps singing while I get out, walk around to her side, open her door for her.\n\n\"On the first day of crazy, my ex-spouse gave to me: eleven wives a-sniping, ten James a-leaping, nine counselors counseling\u2014\"\n\n\"Stop it!\" I yell.\n\nI didn't mean to yell, but she stops. \"I'm sorry I bothered you, Meg.\"\n\nShe was being a jerk, but she could often be a jerk to cover a sadder thing. She sighs, pushes back her bangs.\n\n\"James?\" she says softly. She looks up at me, but I can't look at her eyes. \"James, have you thought of talking to somebody about what's going on with you?\"\n\n\"Doug's waiting, Meg. Let's go.\"\n\nShe gets out. She realizes she's gone too far, but she thinks I've gone too far as well. In her mind, it's a tie ball game.\n\nI say nothing, just shut her door, walk around to the driver's side, and get in. She steps up to my window.\n\n\"I'm sorry,\" she says, \"but, James, I can't keep doing this. I'm sorry things between us didn't work out. Married things. That's over now. I have a life, a different one, one with people who don't break into garages and steal shirts. Don't make what we do have left go bad, James, okay? Let's try to get through this adjustment period. Can we try that... please?\"\n\nI look at Meg, and she means it. She doesn't want to completely shelve me somewhere. She wants us to be an amiable, divorced couple. And it appears I'm wrecking that.\n\nI mean, five years, right? We knew each other another three before that. That's a long time to know a person. I like knowing Meg. She probably knows me better than anybody. And I'm trashing all that _knowing by_ convincing her I've swallowed my last marble whole.\n\nI watch as she walks back across her yard\u2014what used to be our yard\u2014and goes inside.\n\n\"Sure, Meg,\" I say.\n\nI sit there.\n\nI don't have anyplace to go. I'm still reluctant to go back to the coffee shop. It's probably still a hostile environment with me ditching my shift. A visit to Meg isn't without its own hostilities.\n\nI don't feel like going back to my apartment. Boxes of fun, there. I think that I should get in some practice time on leaping, work out all the kinks. Maybe if I got it down, I could still show Meg how it worked.\n\nThat's when I hear music. Electronic music.\n\nLike a circus, a jewelry box, or maybe a calliope. It's the ice-cream truck coming down the street. Playing a ragtime version of \"It's a Small World\" or some such song. It passes right by my Volvo, and I watch the driver ringing the little bell. All the delicious frozen treats have their pictures pasted on the side of his van. A red stop sign is painted on the back, but with an exclamation mark: \"Stop!\"\n\nNot seeing any children, the driver speeds up, takes the next right, and disappears around the corner, trailing that music. Meg comes out her front door with Doug's children. Meg, she's got a wad of dollars in her fist, and she's hurrying to catch the truck, see which way it went, but she stops at the curb.\n\nWith the wind, it doesn't seem she can figure out which way the echo of music leads, this street or that. Her hair blows in the wind. That hair that she's always tending to, just blown across her face. She doesn't seem to care. The kiddies each seem to think it's gone a different direction, and although they're serious about getting ice cream, they're shouting and laughing. But Meg's so serious. She's standing with the wind tossing her long hair, completely serious. She wants to do this for the kiddies, to be the one who catches the truck for them.\n\nMeg always wanted kids. She'll make a great mother.\n\n\"It went up that street,\" I call out the car window, pointing.\n\nThe kids seem a little scared once they recognize me. Meg doesn't seem exactly pleased that I'm still hanging around, but she waves anyway.\n\n\"Thanks,\" she says and waves for me to go, to let me know she doesn't need my help or want it. And she starts to cross the street.\n\n\"You'll never catch him,\" I say.\n\nShe doesn't even turn around.\n\nI dig out my wallet\u2014feels good to know where it is. I've got a twenty-dollar bill, and I hope that's enough. It's been awhile since I bought anything from a truck that roves around the neighborhood.\n\nI'll go catch the ice-cream truck, I decide. For Meg and the kiddies. I try to start the Volvo, but it just cranks and sputters. I'd been waiting for this. Often I only get so many starts per day, then I have to wait for sunrise again. It's just the way a Volvo this age works.\n\nI see Meg walking and remember how the ice-cream truck sped up at that corner. I know she'll never catch him. Meg, she's not a runner. Not built for it, she's too thin, too dressed up, too heavy in the hair, never wearing the right shoes. I try the car again. Nothing.\n\nI look in the side mirror. I see my own face, and that's the last thing I want to see. So I lean toward the mirror till it's just me and my own eyes. Just eyes to eyes, and I think of how much I miss Meg and how stupid I must seem to her. I think of how much Meg wants to please those kids and her hair blowing in her face while she looks for the truck. I see the ice-cream truck in my mind. The little silver bell on the outside of the cab, but somehow, inexplicably, the chain to ring it inside the cab.\n\nI can see it: the little bell, the chain, the guy pulling that chain, his stubbly beard, I hear the tinkling music, the electronic ragtime playing louder and louder then louder until I'm looking straight at the unshaven driver through his windshield...\n\nWide eyes.\n\nThe driver's eyes go wide as he stomps his brakes.\n\nI look down. The front bumper of the ice-cream truck stops about two inches from my knees.\n\n\"Whoa, fella!\" The driver sticks his head out the truck's window. \"Most people just wave from the curb when they want me to stop.\"\n\n\"Sorry,\" I say.\n\n\"I didn't even see you,\" he says, getting out to make his sale. \"Where'd you come from?\"\n\n\"My family's from east of here,\" I say. He allows this evasion because my wallet's out.\n\nI have no idea what the kids will want. But what kid would refuse a Bomb Pop? And if I only buy one, there could be a fight. If not a Bomb Pop, then the next best is the Fudgsicle. Meg likes the Fudgsicles.\n\n\"Three Bomb Pops and three Fudgsicles.\" I order to cover all the statistical bases. Besides, I want one.\n\n\"Nineteen-fifty,\" the driver says. I can't possibly divide nineteen dollars and fifty cents by six quickly enough to challenge this insane price, so I pay the man. Gas costs a million dollars these days, so I'm sure I'm paying for this guy's gas.\n\n\"Whew! I didn't think I'd catch you.\" Meg says, coming around the back of the truck. She stops cold as a Bomb Pop when she sees me.\n\n\"How'd you...\"\n\n\"Here.\" I hand Meg the Popsicles. \"I had no idea what they'd want, but these are good. Tell them they were out of everything else if they complain.\"\n\nI keep one of the Bomb Pops for myself.\n\n\"James? How did you\u2014\"\n\n\"You know how I got here,\" I say. We just stare at each other.\n\n\"Go on, they're melting,\" I say. \"Nothing's as bad as a mushy Fudgsicle.\"\n\nShe looks at me with her mouth open, then turns to the ice-cream truck driver, but he's already climbing back into his truck. She looks at me again, closes her mouth.\n\n\"Thanks,\" she says. \"I... they'll be... the kids will like these.\"\n\nWith Popsicles crammed in one arm, she uses her free hand to loop her hair behind each ear, finger swoop by finger swoop. She turns and walks back up the street toward her house.\n\nIt was good. It was a good thing to do. Meg would get the credit in the kiddies' eyes for chasing down the speeding ice-cream guy and bringing back the goods. They'd think she was Supermom. A hero.\n\nAnd I was glad I did it for her.\n\nI take the long way around the block back to my Volvo, eating my Bomb Pop.\n\nSure, I could have leapt, but I needed a walk just then.\n\n# **9**\n\nIt was the good thing to do.\n\nThat leap for Meg, for the kids. And I realize what's missing.\n\nI drive around most of the afternoon. Just driving till dark and piecing it together. I'm no longer worried that I will leap uncontrollably, leaving the Volvo to crash into an innocent pedestrian or some unsuspecting light pole. I know the missing piece now. Transparency and reflection? We know that part. A sudden intense desire to be elsewhere? Check.\n\nSo why couldn't I leap on top of Doug's Lexus and smash his windshield? What's wrong with that desire?\n\nYes, partly obvious, but look\u2014with the cutting board, even though it was mine, it was a good thing to let Meg have it. With Dougie's shirt, Meg needed that back, and it was good, for Meg's sake, for me to give her the shirt back. With the yellow slip of ribbon at the church, I prayed. Praying's good, right? Talking to Father Chavez, a priest, is good, right? I leapt back near where the priests are. Good, good, good.\n\nLeaping out of the grocery store? This one's harder 'cause it did make me steal groceries, but the theft was unintentional. I plan to go back and pay for that stuff. That's good, right? The impulse of the moment was, at worst, self-preservation. Officer Goss wouldn't have shot me, but he was clearly a threat, or could have been a threat, right?\n\nThe desire was simply to protect myself, and that's fairly good.\n\nHowever, albeit intense, the desire to pummel to pieces my ex-wife's new boyfriend's Lexus's windshield, well... not so nice. See? That's the connecting thread. I'd never thought I'd say this, but super-heroic clich\u00e9s are clich\u00e9s for very practical reasons.\n\nI must use my powers for good.\n\nActual good. Verifiable good. Usually, a guy gets sloshed with radioactive la-la and then events, personality, digestion, temperament, and perhaps childhood traumas dictate whether he becomes a superhero or a supervillain. It doesn't seem I've been given the choice.\n\nIt's doing good or bust.\n\nThen there's this: doing good feels, well, good. I realize that's like saying green looks greenish. What I mean is that to do something wholly for the betterment of someone else, strictly for what that person will get out of it, makes you feel wider, taller, or like caffeine addiction without the jitters or the eye aches.\n\nIt makes you want to hug babies. You're on the ground but something's soaring. Makes you want to poke everybody like a kid, saying, \"Do it again! Do it again!\" because it's its own strongest desire.\n\nThat isn't enough to make you good, but it's a start, isn't it? The beginning of an actual desire to be better, to do good... at least the train's pointed in the right direction. It makes me believe, just for a moment, that maybe I could be good. Maybe I'll feel the right things and I'll want to feel them. Maybe I'll want to believe.\n\nHow do you know what you believe when you're afraid to feel?\n\nI believe, for that moment, I did Meg some good. I can see her, the crooked corner of her smile, her hair flapping in the wind. For a moment, for Meg and I, no wants were shoving past each other. We could see each other, and for just that moment, we were good.\n\nGood should always feel as messy and surprising and gusty as hair blown across your face.\n\nI stop for a burger, fries, and a Pepsi from a drive-through because I don't remember having eaten today except the Bomb Pop. I'm starved. Being a hero is demanding work and deserving of a hot meal.\n\nI push my way inside my apartment, past the fallen barricade, shut the door, and secure it with a piece of duct tape. Everything is as I'd left it: miserable, wrecked, pitiful. I check the bathtub to be sure, and the valuables are unmoved. I need to watch some television, but not for bumming, not tonight. No, I plan to channel-surf the news for heroic opportunities. I've caught the do-gooder bug, and television is a perfect smorgasbord of calamities, disasters, and people in distress.\n\nTV is still in the bathtub. I can lug it out to the living room, but I haven't fixed the door, so I know I'll just end up lugging it back tomorrow morning. I drag the card table over to the bathroom door and open the shower curtain. I plug the TV in at the sink outlet and turn on the news.\n\nI sit down to my meal.\n\nThe news is very depressing, especially when you think of the people in tough spots as people you might actually help. You forget that when you're just watching. World news comes on first. The hurricanes. Military coups. A bomb threat on a Paris subway.\n\nI don't see how I could help any of that. I can't leap to where a bomb is because no one knows where the bomb is. I could leap into the middle of a military coup only to be shot seconds later like anybody else. This doesn't sound very promising.\n\nThe hurricane looks a bit more possible. Pictures of people standing on their rooftops trying to escape the rising floodwaters flicker on the screen, but I'm not certain I can leap someone else away from a predicament like that. Basically, I'd be one more guy trapped on a roof. Rescuers probably hate leapers leaping in who then need rescue as well.\n\nIt's not easy. Just imagine a superpower for yourself, and you'll see. Don't pick the best one either\u2014pick a sort of middle-of-the-road superability. Only one. Then go watch the news for heroic opportunities. It's not all secret high-tech caves and emergency signals.\n\nThis is more like horseshoes or homework, like a job interview for a job that doesn't exist. It's like trying to think good thoughts all day. You can try, but it takes effort. Maybe after a few successful super-rescues I'll get the hang of this, but right now I can't find a single good deed on the news that won't end in my own peril. Besides, I can't go saving folks on an empty stomach. The swimming-pool rules should apply: no leaping for thirty minutes after a meal. Don't leap in where you can crack your own neck.\n\nThe burger is fine, very greasy, and this food is warm. The Pepsi's fine, very drinkable and far superior to a lumpy protein shake. I couldn't imagine eating another stick of dried beef. Actually, the beef on the burger is almost identical to the jerky, but the fries are salty and good.\n\nFinally, the local news comes on. The lead story is an apartment building on fire. I sit up, put my Pepsi down. This looks like it\u2014complete with live coverage.\n\nThey're filming a fire that's burning up a building in town right now. Fire trucks have arrived, but the blaze isn't under control yet. Hoses and water and firemen with ladders and bullhorns. The camera shakes a little as we watch. It must be dangerous for the camera guy.\n\nI stand and reach, turn on the bathroom light, the one above the mirror. I figure that will give me the best glare on my watch face. I tie my shoestrings securely. Maybe I need a coat. Leaping into a fire might be dangerous. I have no idea where the coats are packed, I'm only sure they're not in the box marked \"Coats.\" That's full of pans. Maybe my coats are still in Meg's garage. That fire looks hot, an inferno.\n\nI set down my burger and stare at the screen. The reporter keeps on looking over her shoulder at the blaze and pointing up at the windows. A couple of firemen rush out the front of the building with victims that were trapped inside. They're safe now. A few more people scurry down the fire escape. They look safe too, but the reporter keeps pointing up at a window on the third, maybe fourth floor, where the flames seem to be at their worst.\n\nShe's pointing because a lady is hanging out a window, waving her arms. She's definitely trapped, definitely in need of some help... maybe superhelp? How do you start? I figure my angle. Where to leap? I'm not so sure about where I'd come out. Would I just appear hanging from that lady's window? Would I appear inside the apartment with her, just as trapped as she is?\n\nIt's terrifying, really. Try it. I want to help her, but I don't feel qualified. I haven't practiced enough. I haven't practiced at all. This is a new shirt. I've just eaten a full meal. I feel more flammable than I ever have. How do I help if I don't know anymore about the inside of that building than the people who are trapped?\n\nI admit it, okay? I do no more than anyone else watching the news that night. Like everybody else, I just watch. It's like I'm standing on the edge of the pool with my arms out, but I'm too scared to dive in. I admit this isn't necessarily heroic, let alone superheroic, but I'm in training.\n\nBatman trained for years. Superman grew up like a regular kid in some small town before his human mother made him the cape. It usually takes a personal crisis to bust the super out of the hero, right?\n\nBesides, right then the news cuts away from the fire. They promise updates, but they've only got so much time to get in all the bad news.\n\nThe next story is a gas-station robbery. It's not happening live like the fire, but we see a fuzzy black-and-white video clip of a man in a stocking cap pull a handgun on the station attendant. The lady backs up, afraid, and holds her hands up as the guy reaches across the counter and grabs all the cash out of her register. He waves the gun around, points it, shoves the cash in his coat pocket, and runs out of the store. I watch, sip my soda, and I'm entertained, but I wonder: if I leap there, do I leap into the gas station currently, or would I leap into the scene I just saw? Would I just be another customer tonight because this robbery happened late last night?\n\nI've polished off all my fries, and I wonder if news research is the way to do this. By now, that feeling, the good rush from helping Meg, has completely worn off. I don't even want to leave the house tonight. It's already late, and the burger's made me sleepy. Where did it go? The will to do good? The news report about the gas station concludes. Luckily, no one was hurt.\n\nWe're all safe.\n\nThey cut back to the fire, but I can't watch.\n\nI turn off the television and walk into the hall.\n\n\"Hey, Jack!\"\n\nIt's Nelson. He comes down the hall, same old jeans and T-shirt, but this time the shirt's inside out so the words read backward, like in a mirror.\n\n\"Hey, Jackson,\" he says. \"You got a pair of scissors? My toilet won't shut off.\"\n\nI'm pretty sure I don't like Nelson, but what with my recent, complete lack of bravery, I'm just not feeling up to contempt. \"What's up, Nelson?\"\n\nHe stops and stares at me like this is precisely the problem he's always having with me. \"Perhaps you heard? My toilet won't shut off.\"\n\n\"If I asked you why you need a pair of scissors for that, you'd actually explain, wouldn't you?\"\n\n\"Do we have time for that?\" he says.\n\nI don't know what that means, so I can't be sure about the time requirements. \"I don't believe I have any scissors, Nelson.\"\n\nHe makes a disgusted grunt, walks a little circle in front of me like he might leave, _please let him leave_ , but no, it's just a thinking circle.\n\n\"You know what? If you don't like me, just say so.\"\n\n\"Haven't I?\"\n\n\"If you don't want to help me out, even after I helped you out, just say you're a lopsided helper, but don't talk nonsense.\"\n\nI pause to savor that irony. \"I would be glad to help you,\" I say. \"Okay, glad is way out of proportion to what I feel, but I'd do it, but I don't have any scissors.\"\n\n\"Then why didn't you say that?\"\n\nI lean cautiously toward him, smell his breath.\n\n\"Old Spice,\" he says, proudly. I can honestly say there's too much of that product being used in this world.\n\n\"I did, Nelson.\" I don't really want to have this conversation, but it's better than watching the news and doing nothing about it. \"I told you I didn't have any scissors.\"\n\n\"No, Jack, you didn't.\"\n\n\"James,\" I say.\n\nAgain, he checks to see who's coming. After a thorough look down the hall, he turns a concerned face at me and takes a step back.\n\nI sigh. \"That's my name. I'm James. And James doesn't have any scissors either.\"\n\n\"Oh,\" he says. \"I thought you said you didn't _believe_ you had any.\"\n\n\"That's the same thing,\" I say.\n\n\"No, a person can believe all the wrong things and still have scissors somewhere.\"\n\n\"Right,\" I say. \"You're absolutely right, Nelson.\" I just want him to go.\n\n\"No worries,\" he says, and he slaps me on the shoulder, it's all forgiven. He turns to go.\n\n\"Hey, Nelson.\"\n\n\"Yes, Jim?\"\n\n\"What are the scissors for?\"\n\n\"All my tools are tied up in a pillow case, and I can't work the knot loose. Gotta do the little things before you get to the main one. And all the time, she's arunning.\"\n\nHe walks away.\n\nI go inside my apartment, shut the door. I walk over to my card table, stare at the remnants of my food\u2014a hamburger wrapper, the empty fry carton. I don't have the heart to turn the television back on, but I want to find something, one little thing to do. I take another sip of soda. There, on the side of the cup, a boy's face stares at me. A missing kid, last seen two weeks ago with his father. They think his father nabbed him after school. His mother has sole custody. Chapman Collins is this boy's name, and it's some kind of school picture. He's only six. He's smiling as big as day, and his hair looks freshly combed.\n\n\"Chapman Collins,\" I say to no one particular.\n\n_Gotta do the little things before you get to the main one..._\n\nI twist my wrist so my watch catches a glare, and it flashes. I move my watch closer to my face. I squint hard and focus till my eyes go hazy. The eyes flit between sight and that moment between, fleeting. Where you lose sight of yourself for a moment, long enough to see through, to see yourself reflected in the other, more than just you. Until you can't see either properly, till what you truly desire shows up...\n\nA missing tooth.\n\nThe boy's missing a front tooth, but he has the widest smile.\n\n\"How'd you do that?\" asks Chapman.\n\nI'm sitting on the end of a bed in a small room. The boy leans against the headboard with his legs sprawled out on the bed. He's eating dry Cheerios from a coffee mug and looking straight at me, still smiling. I hear laughter behind me. It seems I appeared between the boy and his television.\n\n\"How'd you get in here?\"\n\n\"Get up,\" I say. \"We have to go.\"\n\n\"Who are you?\"\n\n\"Chapman Collins!\" I say, trying to sound as official as possible. \"I'm here to rescue you.\"\n\n\"Cool!\" he says.\n\n\"Put your shoes on, Chapman.\"\n\nI stand up. It's a small little bedroom. The TV's clearly been pulled into this room from another, because it takes up way too much room. It's plopped on a set of drawers, covered with months of dust, dust that the TV seems absent of. Clearly, a man has moved this TV in here for the kid, the potentially dangerous, kidnapping father. Suddenly it hits me, I don't know if the father is in the next room... and I don't know just how unpredictable he might be.\n\n\"How are we going to get out?\" Chapman asks.\n\n\"Shh!\" I say. I move over to the door, listening for what's happening on the other side.\n\nChapman whispers, \"Hey, Mister, is there something in my closet?\"\n\n\"Oh.\" I shush Chapman again, step over to the other door, and put my ear against it.\n\n\"What are you listening for?\"\n\nI turn on the boy, grab him by the shoulders, surprising him. \"Look, I'm here to rescue you, but you'll have to do exactly as I say, okay?\" I heard that in a movie once. He just nods.\n\n\"Now, quietly, get up and follow me.\"\n\nChapman doesn't move. His shoes are on, untied, but he's sitting on the bed, kicking his legs.\n\nI put one hand on the doorknob and take a deep breath. I hold out my other hand toward Chapman. \"Okay, on three...\" I say.\n\n\"One-two-three!\" Chapman counts quickly.\n\nHe's seen that movie, it seems, but he's still sitting there eating his Cheerios.\n\n\"Come on,\" I scold him. \"We're getting out of here.\"\n\n\"Mister, how are you going to unlock the door?\"\n\nI stand up straight and stare at him. Reach over and try the knob.\n\n\"We're locked in,\" I say.\n\n\"My dad locks me in here when he goes to work.\"\n\n\"Your dad's not here?\" I ask.\n\n\"He works nights, till three,\" he says.\n\nHe's really enjoying the Cheerios, shoving in handfuls like he's watching a movie, but he's watching me instead.\n\n\"What's your name, Mister?\"\n\n\"My name? Oh, James.\"\n\n\"Mine's Chapman. You going to break down the door, James?\"\n\n\"I hadn't thought of that,\" I confess.\n\nI should think these things through better than I do. The door looks very solid and well made, this door.\n\n\"What about that window?\" I ask him.\n\nChapman shrugs. I step over to the window and see why he's not moving. The window overlooks a parking lot, about four stories below. We're trapped.\n\n\"I think you should bust the door down,\" says Chapman, confidently.\n\nI go back to the door, tap on it to see how thick it sounds.\n\n\"You sure your father isn't here, right?\" I ask.\n\n\"You're kind of skittish.\"\n\n\"What?\" I say.\n\n\"Skittish. That's what my dad calls my mom. It means antsy or jumpy, like you're afraid someone's going to jump out at you, and maybe hit you. I think you look jumpy.\"\n\n\"You mean I look too jumpy for a superhero?\"\n\n\"No, for an adult. Are you okay?\"\n\n\"I don't think I can break this door down, Chapman.\"\n\n\"Are you a superhero?\" he asks.\n\n\"Kind of.\"\n\n\"Where's your stuff?\"\n\n\"Excuse me?\"\n\n\"Stuff. Don't you have a machine gun or a laser or a belt with tools? You know, stuff. Superstuff. Anything?\"\n\n\"I've just started this superhero business, Chapman.\"\n\n\"You gotta have stuff.\"\n\n\"What do you recommend?\"\n\n\"We could use a rope, for that window. You got a rope?\"\n\n\"I don't have rope.\"\n\n\"You need rope. At least carry that. Can you fly?\"\n\nWhy is that everybody's favorite? What's with the flying? Superman's ruined it for the rest of us.\n\n\"I don't fly, and I didn't bring any stuff, Chapman.\"\n\n\"Oh.\"\n\nHis nose is running, and he keeps wiping it on his sleeve. He munches on the cereal like he hasn't eaten in days. He notices me watching him eat.\n\n\"My mom makes me eat fruit, but Dad says I can have all the Cheerios I want.\"\n\n\"Fruit is good for you,\" I say.\n\nAnd I think, _That's it\u2014I'm Nutrition Man_. I could paste the food-groups chart across my chest. Enforce good eating habits to all the kidnapped children in the world.\n\n\"You could loop it to your belt,\" he suggests.\n\n\"What?\"\n\n\"A rope... they're not heavy.\"\n\n\"Sorry, no rope.\"\n\n\"A gold rope. That'd be cool. Or a helmet that shoots rope!\"\n\n\"Drop the rope bit, kid.\"\n\n\"Are you going to buy a mask or something?\"\n\n\"I was thinking about the cape.\"\n\n\"But you can't fly.\"\n\nI stare at the locked door.\n\n\"I have a cape,\" he offers. \"A green one. It's at my mom's house.\"\n\nAt this point, I'm not sure this boy knows he's been kidnapped. In fact, I'm beginning to wonder what I know. \"Chapman, do you know where you are?\"\n\n\"I'm in my room,\" he says.\n\n\"Yes, but do you know what's happened to you?\"\n\n\"Oh, that. Sure,\" he answers. \"I've seen my picture on the news. I'm a 'napped kid.\"\n\n\"Did your father kidnap you?\"\n\n\"Sure. Then we moved in here, and he grew a mustache. His hair's blond now, but I think it looks funny. He works nights, and I know he has a different name 'cause of I've seen his nametag. And when he gets enough money, we're moving to Philadelphia. You got any weapons?\"\n\n\"Why?\" I ask. \"Does your dad have weapons?\"\n\n\"Yeah!\" says Chapman, his eyes lighting up.\n\n\"My dad has the coolest knife. A Swiss army knife, and it's got a can opener and a toenail clipper and a\u2014\"\n\n\"No... I mean does your dad own a gun?\"\n\n\"You're super jumpy.\"\n\n\"Is your dad a big man?\"\n\n\"Medium.\"\n\n\"Just medium?\"\n\n\"Yeah, but he's real tough. He could probably take you.\"\n\n\"You want me to stay or go?\" I ask.\n\nChapman puts his coffee mug with his cereal on the nightstand, then hops off the bed. \"You're scared, too, huh?\"\n\n\"I just want to prepare, okay, kid? You got a baseball bat?\"\n\n\"Are you going to hit my dad with a bat?\"\n\n\"I hope not to meet your father,\" I say.\n\n\"You should get a gun before you get the cape,\" he says.\n\n\"Why don't you think I deserve a cape?\"\n\n\"You don't fly or bust down doors. You need a laser gun, or someone's going to pound you.\"\n\n\"Help me get us out of here, okay, kid?\"\n\n\"Hey! That's not nice!\"\n\nAnd it wasn't, so I apologize to Chapman. I sit down on the bed, and he sits beside me.\n\n\"We're kind of trapped in here together, huh?\"\n\n\"Looks that way.\"\n\n\"Want some Cheerios?\"\n\nI shake my head.\n\n\"It's cool you came to rescue me, James.\"\n\n\"Well, we're not out yet.\"\n\n\"You're trying. That's what is important,\" he says.\n\nI just chuckle.\n\n\"What do you do, James?\" he asks.\n\n\"You saying I shouldn't quit my day job, that it?\" I say. \"I work in coffee, Chapman.\"\n\n\"My dad likes coffee,\" he says.\n\n\"Other than that, I don't get out much.\"\n\n\"Me too.\"\n\nThis is crazy. There's got to be a way out. Surely a grown, caffeinated man can find a way out of a locked bedroom. I wonder if I can leap with the boy if I'm carrying him, but I have no idea how that'd work out. What if Chapman gets hurt? I'd definitely get sued, and I've lost my lawyer. Maybe I can help him escape later.\n\n\"Does your dad unlock this door when he's home?\"\n\n\"When he's here, sure. But he locks the front door. He's smart. I thought about running and calling my mom, but he's too smart. He locks me in here when he goes to sleep, too.\"\n\nI look at him. \"You miss your mom?\"\n\nChapman shrugs. \"Yeah, but she's pretty much same as my dad, except she doesn't lock me in when she's asleep.\"\n\n\"Right,\" I say.\n\nChapman looks at me. \"You married, James?\"\n\n\"Have a Cheerio, Chapman.\"\n\n\"Divorced, huh?\" he asks.\n\nI stare at him. \"Yeah.\"\n\n\"Oh,\" he seems sad. \"I don't know any grownups who are married anymore.\"\n\n\"Well, we're sorry about that, kid. All of us.\"\n\n\"Mister, I'm sorry about that cape thing. You can get one if you want.\"\n\n\"Thanks,\" I say. And it's really ridiculous. Me and the six-year-old trapped, so I'm trying to think of options.\n\n\"Hey, you can call me Chap, okay?\" he says.\n\n\"Can I call you Chappy?\"\n\n\"Okay. Can I call you Jimmy?\" he asks.\n\n\"Okay. Chappy, listen. My superpower is leaping. I leap from place to place. So what if I leap out of here and go get help?\"\n\n\"I thought you were the help...\"\n\n\"I did too, but it seems I need help being the help, okay?\"\n\nHe shrugs. I stand up and hold my watch up to the light.\n\n\"Hey, Jimmy.\" He stops chewing. \"Don't go yet. Please.\"\n\nI realize I couldn't if I tried. There's no way I can focus on something else with the poor kid staring at me like that.\n\n\"If I don't go, what are we going to do?\"\n\n\"Well, you can go in a minute, okay? My dad won't come back till three.\"\n\nHe's almost whining. What do I do if he cries? I sit down with him to wait and try to think.\n\nChapman picks up his Cheerios. He's perfectly content to wait for his father to come home and pound me. He asks, \"Why'd you come?\"\n\n\"What?\"\n\n\"Why'd you pick me to rescue?\"\n\nI didn't feel it appropriate to tell him I was too afraid of a burning building.\n\n\"I saw your picture on a cup. I figured you needed help.\"\n\n\"Yeah, but just because you see my picture doesn't mean you gotta help me. I mean, lots of people know I'm missing. No one else has helped.\"\n\n\"I guess... well...\" I stall. \"If I went missing, I hope somebody would come for me.\"\n\n\"Oh,\" he says. He digs an entire Cheerio out of the gap from his missing front tooth.\n\n\"I bet my mom misses me,\" he says and eats the Cheerio.\n\n\"Do you miss her?\" I ask.\n\n\"Yeah.\"\n\n\"Your mother must love you a lot.\"\n\n\"My dad loves me too.\"\n\n\"Yeah, but he's sort of a felon now.\"\n\n\"Yeah, Mom says he's crazy.\"\n\n\"Do you think he's crazy?\"\n\n\"He's my dad.\"\n\n\"Right. He's probably crazy, and he loves you.\"\n\n\"Yeah,\" Chapman rubs his nose. \"Why'd you get divorced?\"\n\n\"Me? Things didn't work out.\"\n\n\"My family didn't work out either,\" he says.\n\n\"Why'd they get divorced?\"\n\n\"They stopped trying to trust each other. That's what Mom says.\"\n\n\"That's why, huh?\"\n\n\"Yeah, and she thinks he's crazy.\"\n\n\"Right.\"\n\n\"I couldn't help them either.\"\n\n\"Yeah.\"\n\n\"What about your mom and dad?\" Chapman asks.\n\n\"They're dead.\"\n\n\"Married dead?\"\n\n\"No, divorced dead.\"\n\n\"See, nobody's married anymore. Not even dead people. Mine are still alive though, huh?\"\n\n\"Right. Your chances are still kicking, kid.\"\n\n\"Maybe they'll help each other later.\"\n\n\"Well, maybe, but your dad's in a heap of trouble for kidnapping you.\"\n\n\"I could say I ran away and he found me.\"\n\n\"But he didn't call your mom or anyone, Chappy, and it's been over a week. The police aren't going to buy that.\"\n\n\"I could say I forgot my mom's phone number.\"\n\n\"Too many phone books in the world for that to work, I'm afraid.\"\n\n\"Are you afraid, Jimmy?\"\n\n\"No, I just drink too much coffee at work.\"\n\nI could use a cup of coffee just then, a cup of something. This kid, my chosen superdeed, my act of goodness\u2014none of it seems to work easily. How do television shows always wrap up so cleanly, right on the hour? This kid's alive, he's real, he talks, thinks, and eats Cheerios and asks too many questions. I just wanted to leap in and leap out. Now I'm sitting on the edge of a bed with a real kid who needs a real tissue. His little room is too hot, and his felon of a father probably turns off the air when he goes to work. I can't cool the room off, let alone un-nap this kid.\n\n\"Jimmy... are you a little afraid?\" he repeats.\n\nI stare at him. \"I'm always a little afraid, Chappy.\"\n\nWe sit together listening to Chappy chew. It's louder because his mouth's open.\n\n\"Are you going to turn my dad in to the cops?\"\n\n\"No, Chapman. I just want to see you back with your mom. She's probably worried sick, and I'm trying to help her out. Frankly, I hope I never run into your dad.\"\n\n\"You'd like him. He knows all kinds of stuff.\"\n\n\"Like what?\"\n\n\"He can bend pennies with his teeth.\"\n\n\"Dear Lord in heaven\u2014\"\n\n\"I'll show you one. I've got one in my backpack.\"\n\n\"Never mind that, right now. We've got to find a way out of here.\"\n\nJust then my cell phone buzzes in my pocket. I pull it out and see the number; it's Meg. Being in a situation that I cannot explain to her, I just let her leave a message.\n\nChapman looks at me, surprised. \"All this time, you got a phone?\"\n\nIt dawns on me. I may not be hero material.\n\n\"Why don't we call the police?\" he suggests.\n\n\"Right.\" I dial 911. \"Uh, yes, I have information about a missing child, Chapman Collins.\"\n\n\"Let me transfer you to the detective in charge of that case. Please hold.\"\n\nI cover the phone. \"Good idea, Chappy.\"\n\n\"Yeah,\" he says. He's smiling again.\n\nIt takes a minute, but finally, I hear, \"This is Detective Goss. With whom am I speaking?\"\n\nI freeze and cover the phone, hoping my whole hand will cover the little speak hole. I hold it out to Chapman and mime for him to talk to them.\n\n\"Hello... Hello?\" I can hear the detective talking. He's basically shouting.\n\nChapman takes my phone. \"Hello?\"\n\nChapman listens for a second. \"Yeah, it's me. Chapman Collins... I'm the 'napped kid, right. Yeah, I'm okay, but we're trying to get out of here... yeah, me and James.\"\n\nI shake my head at him and mouth, \"I'm not here.\"\n\n\"James can't talk 'cause he's jumpy. He drinks too much coffee at work.\"\n\nI fall off the bed. Fascinated, Chapman watches me but keeps talking.\n\n\"Yeah, James doesn't want me to tell you he's here... No, I just met him. No, my dad 'napped me... My dad? He's at work... yeah, me and James are locked in here... No, I don't know the address. Uh-huh... uh-huh... okay.\"\n\nChapman offers me the phone.\n\n\"He wants to speak to you.\"\n\nShaking my head vehemently, I whisper, \"Tell him the address.\"\n\nChapman shrugs. \"I don't know the address, Jimmy.\"\n\nMy detective is hearing the boy call me Jimmy. The kid doesn't know where we are.\n\n\"Jimmy can't come to the phone right now,\" he says.\n\nI lean weakly on the television. That's when I see the old mail. I grab a letter and hand it to Chapman, pointing at the address.\n\n\"Oh, here it is,\" he says, then reads the address to the detective.\n\n\"Uh-huh. It says apartment 409... that's right... No, no. My dad 'napped me, but he's at work. He won't be back for a while. You guys can come anytime... right. No, no... James wants you to come. He's rescuing me... Nope, I checked, he's got no weapons, not even a rope, but he's a superhero.\"\n\nChapman looks over, smiles at me with that wide little-kid smile. He thinks he's doing me a favor. He gives me the thumbs-up.\n\n\"Uh-huh... Hold on, I'll ask.\"\n\nChapman covers the phone. \"The policeman wants to know if you'll wait until they get here?\"\n\nI nod that I will.\n\n\"Yeah, James can stay till you get here... What?... No sir. James is real cool. He's a superhero... No, I've never met him before; he just appeared in my room to rescue me. That was the cool part. Otherwise, he's just a guy. Then he let me use his phone... Nope, the door's locked. We can't get out. That's why we called you... Yeah, you're going to have to bust the door. James, he couldn't bust it... Hey, sir? Can you call my mom and tell her I'm okay?\"\n\nI take the phone from him, cover the receiver, whisper, \"Tell the police I'm really a good guy, okay?\"\n\nChapman nods as if he understands, reaches out for the phone. \"Hey, you still there? It's me again. Hey, James wanted me to tell you that he's really a good guy.\"\n\nI take the phone from him and end the call, but I can't get mad at the kid.\n\nI put the cell in my pocket and walk over to the window to watch for the police cars.\n\n\"He was nice,\" says Chapman.\n\n\"Sorry I grabbed the phone.\"\n\n\"It's yours.\"\n\n\"Look, I'll stay for a bit, but I've got to leap out of here when the police come, okay?\"\n\n\"Are you running from the man?\" he asks. This kid watches too much television, I'm thinking.\n\n\"Sort of, but don't mention that when they ask about me, okay?\"\n\n\"Okay,\" he says. \"Will you wait till they get here... like you promised?\"\n\nI look over to him, and I want him to smile at me one more time, like his picture on the cup. I really want someone to keep one promise to this kid.\n\n\"Yeah, Chappy. I'll wait until I see them.\"\n\nHe does. He smiles and wipes his nose, comes over, and sits on the bed so he can look out the window with me.\n\n\"Jimmy, do you think my dad loves me?\"\n\n\"Well, they both love you, right?\"\n\n\"That's what they say.\"\n\n\"Don't you believe them?\"\n\n\"They're supposed to be together, you know? Whosever's house I'm in, I'm always thinking of the other one. And they both promise me stuff, but I don't know which one to believe.\"\n\n\"Right,\" I say, \"but they both love you.\"\n\n\"Yeah. I believe them both. Just it's hard knowing which believing is which.\"\n\n\"Right.\"\n\nWe wait. We eat Cheerios. We kick our legs.\n\nBut it's only minutes till three squad cars with lights flashing come screeching up outside the apartment building.\n\n\"Okay. There they are, Chappy.\"\n\nThe boy hugs my leg. Cops pour out of their cars with walkie-talkies and guns drawn.\n\n\"I got to go, kid.\"\n\n\"Thanks for sitting with me.\"\n\n\"Sure,\" I say. \"Stay back from that door when they're breaking it down, all right?\"\n\n\"Okay.\"\n\nI hold my watch up closer to the light. I catch a reflection, and I'm thinking very intense thoughts about my good card table and that good empty carton of good fries. I hear the police pounding on the front door of the apartment.\n\nChapman says, \"Hey, James. You did real good.\"\n\n\"Thanks,\" I say.\n\n\"I'll tell my mom you said hello.\"\n\n\"Right. See you around, Chap...\n\nI drop on the floor outside the bathroom door. It's exhausting, this doing good. I'm not good enough at good, and I'm taking the rest of the evening off. Retired or sick leave, I quit. It's enough for one night.\n\nI sit on my floor for a while, catch my breath. I hope Chapman's fine. By now, the cops must have busted down the front door and unlocked the door to his bedroom. I imagine how he must feel, listening to the doors breaking in.\n\nI hated leaving him there scared, but I had to go when I did, right? Poor Chappy. I can't imagine what it's like being whisked out of your home and carried off to some strange room where you spend all of your time.\n\nWait a minute... Yes, I can.\n\nI get up, go brush my teeth, gargle, drink a glass of water. That's good for you. I check the front door, not that it locks, but that it's still there. I push a small barricade of boxes into place. Then I get ready for bed. When you wear the same clothes, both in and out of bed, getting ready for bed merely involves the emptying of pockets. My cell phone blinks with messages. Three, in fact, which is remarkable seeing as I didn't know three people knew my number.\n\nI dial voice mail, press a jillion buttons.\n\nFirst message: \"Hey, James. This is Kevin. Just giving you a call. I thought you ought to know that Charlene came in. Monica and I tried to cover for you, but you know how Mike is. He pitched a fit till she had to pitch one too, and now she's kind of mad. You ought to call her. Make up something or let her know what's up, you know? I mean, give me a call if you want to talk. Sounds like you could use a friend right about now. And, you know, so could I. Bye.\"\n\nNext message: \"Hello, James, this is Father Chavez from Saint John's downtown. I just called to say thanks for dropping by yesterday, and I'm interested in finishing our talk when you get the opportunity. It was fascinating, especially the Aquaman parts, and I'm sure there was more we didn't cover. I don't know precisely what God is doing in your life right now, James, but I'd love to help you out if I can. Stop back by or give me a call at the church and leave me a message. Take care. Good-bye.\"\n\nNow that's precisely why I like Father Chavez. Any other priest would've made that creepy, or not done anything at all. Kevin's message, slightly creepy. Father Chavez, as I've said, he's a good man. Maybe I should tell him about leaping?\n\nSuperheroes need a moral guide, right? A confidant. A wise mentor to call on when they're bloodied from fighting. A really wise mentor might tell me how to keep away from bloodied fighting altogether. We could work out a special signal between us for times of leaping need. Come to think of it, do I need a secret lair? Does the cathedral have catacombs?\n\nLast message: \"Okay, James, don't hang up. It's me... and I'm calling to say, you know, the kids wanted me to tell you thanks for the Popsicles. Emma smeared her Bomb Pop all over her cheeks before Chunky took it out of her hand and swallowed it whole. Stick and all. Then Ponzy licked the poor girl's face till she screamed, giggling. You would have liked that. I mean, thanks. It was really great of you... Oh yeah, and this guy called. Early this morning. I meant to tell you, but well, you got a little strange, and I forgot. Said his name was Goss, I think. A detective. Something about a grocery store? I think your new credit cards might be giving you problems. You know how that store is. But he asked for you, so I told him, you know, that we're divorced. I hope you don't mind. And I told him where you work. He wanted to ask you something. I guess that's all.\"\n\nI will never understand how that woman draws a paycheck as a lawyer. She rambles, digresses wildly, and wanders blindly off the point. She rarely has a point.\n\nBut I got the point about the detective. This is how he knows where I work. Great job, Meg. She gives out little cards that say she's a lawyer, but what lawyer does that? Spills out personal information to a guy over the phone simply 'cause he says he's a detective? I guess I'm not her client, but isn't there some sort of attorney\u2013ex-spouse privilege?\n\nI turn off the lamp, drop onto the bed.\n\nBedtime is my usual worry time. Tonight I'm thinking how much I don't like the Goss bit. Who is this guy?\n\nIt doesn't make sense, even to an unstable, neurotic person, for him to obsess over me. Is he really bent over a lippy moron in a grocery? Can he find me here? Maybe I should check into a hotel. No, even though Goss called my work, even though I accidentally called him on my cell phone, I don't think he'll find me. No one but Meg knows where my new apartment is. I changed my address in the computer at work so they'd be able to send me my tax stuff, but Charlene won't let him into the computer. All my mail still goes to Meg's house. And he can't want to find me that badly. Why? If he did, how's he going to catch me?\n\nAnd if he catches me, I yawn, how's he going to keep me caught?\n\n# **10**\n\nI've figured it out.\n\nHow this all started. I know why it's happened to me.\n\nIt was at Mass, a couple of weeks ago. Father Chavez wouldn't remember it because he wasn't there. He was off at some conference, so we had a substitute priest. They do that, like baseball. Maybe he's the designated priest or a relief priest.\n\nFather Somebody.\n\nRegardless, not the usual priest, and this guy added stuff. He added to the Mass, but I don't mean heretical, more optional. Bonus liturgy. He chanted everything, sang parts I've never heard sung. From what I heard, I'm not so sure they should be sung.\n\nHe read all the pieces in the book. If he started a reading, he got every last crumb. He prayed extra prayers, longer prayers, fancy, surprise prayer endings with more hand gestures. I promise you, he snuck in as much Latin as he could, right in the middle of plain English, like, \"Dear Lord, _O Dominus Domino_.\" I don't know if he was just improvising or putting things back in the liturgy that Father Chavez always took out, but either way, the substitute priest added bits.\n\nMeanwhile, an old guy is sitting across the aisle from me. We're both on the central aisle, but this old guy, he's on the groom's side, and I'm on the bride's. He's also one row ahead of me. There's nothing wrong with this old guy, but I notice him, right?\n\nHe's eating it up. All the extra bits. He's really old, like eighty-eight old, and digging the Latin and the bad singing.\n\nSo I watch him because he's more interesting than the ad-libbing priest. He's into it more than the relief priest, who is adding motions, but they're still just motions. For the old guy, it seems they're not just motions but connected, meaningful, a real thing he's part of. I think he's going to have a fit, he's so... what? Happy? Engaged? He's certain.\n\nI don't know what it is about me, but I'm bugged. This stems back to my childhood, because I remember feeling this same sort of thing all my life. I'd see a kid on the playground, he's having a great time playing marbles by himself, and I'd get bugged. Being just a kid, on that same playground, I'd think, _Hey, I'm not that happy_.\n\nIt's not so much that I want the kid's happiness. I don't really want to work on it and fetch my own, either. Just bugged that this kid's so happy. Over-happy all on his own, you know? I want him to come down a bit, and I've always been this way. I get bugged.\n\nNo one has the right to that much happy. Not in front of everybody else.\n\nNo one deserves all that.\n\nI'm watching the old guy, and he's kneeling, praying, singing, and responding. I'm definitely not in a holy frame of mind because I remember thinking, _Somebody needs to dunk this guy's head in a bucket of ice_.\n\nBut I ignore it, start to get over Happy Old Guy, and go back to focusing on the Incredible Singing Priest.\n\nThen, the time comes for the Mass. You know, the get-in-line-go-down-front-to-receive-the-host part, and I've a place I need to be afterward. So I post up, get ready to box out people, rush the aisle the moment we get the signal to head for the front.\n\nThe kneeling part is over, the rows farthest front begin to empty, people shuffling their way into the aisle. And I keep my eye on this old guy. If he looks like he's about to move, I'm going for it.\n\nI didn't have to worry, though, because it turns out he's a bonus kneeler. In the aisle, he has to drop down on one knee, genuflect toward the front, before he jumps in line.\n\nThis is my chance. Nobody will look at me funny for slipping past a guy who's bonus kneeling. I step out, cutting off the lady behind me, and as the old guy stands back up, I jump ahead of him. Leap in front, so to speak. I shove into his shoulder a tad, not hard, and bam\u2014I'm in his place in line.\n\nIt's only one spot farther ahead in a very long line, but it feels good. I've accomplished something, one place farther along than the other guy. Now I figure he'll have something to say, at least a cough or a grunt, because I jumped in front of him.\n\nBut he does nothing. He's just as happy as he was before, and it's disconcerting. I had my blank, unconcerned, young-person face all ready for a comment. No dice.\n\nOld Guy's completely content to be behind me. I glance at him, give him a nod, and he pats my arm and smiles.\n\nI lose. I cut him off, and he's courteous, and I feel like dirty gum stuck under a bus seat. We progress through the line, slowly moving forward, one by one, and as we do, I feel worse and worse. Hanging on the wall above the Mass table, there's a crucifix. Christ on the cross in agony, bleeding, head bowed and bethorned.\n\nAnd I wish I hadn't done it, taking the guy's place, because it seems small. I have this intense desire to let him have his place back. In a gallant sweeping, Elizabethan gesture, I want to let everybody see I let the old guy take the host first.\n\nBecause you're supposed to be free, right? Free of guilt, sin, and line cutting before you go, free and clean to take the host. No time to confess now, we're getting close to the front. I think of excusing myself from the whole ordeal, check my watch and get out of line, but I'm sure that'd be worse. People will definitely notice, definitely stare then. The priest will stare, and no one wants that kind of attention from God's man, even a substitute. You don't get in the Mass line, then hop out close to the front. It's just not done. People will think you're playing games with God.\n\n_Just kidding, God. No body and blood for me today. Just checking the menu. Thanks, but no_.\n\nNow I'm in front of the priest, who's looking at me and wondering why I won't kneel. I turn to this old guy and whisper, \"Really, you were first. You go ahead. I leapt ahead of you.\"\n\nI remember saying that exact word. _Leapt_.\n\nBut when I turn to say this, I realize Old Guy's praying. He's out-religioned me at every turn. He's really good. So there's nothing left to do but kneel.\n\n\"The body of Christ.\"\n\nThe substitute priest lays the wafer on my tongue.\n\nAnd then I feel it. The same weightless, bony hand that patted my arm, now presses my shoulder. The host is on my tongue, and the old guy leans toward my ear, to avoid disturbing the priest.\n\n\"Christ leapt into both our places,\" he says. \"You should have mine.\" He says this but not mean or vengeful, not like he's sticking it to me. He's thrilled.\n\nMy eyes open on the crucifix. They hang those at an angle so the dying Christ stares right at you when you're kneeling, because I'm eyeball to eyeball with the bloody One on his cross, with the old guy's breath on my neck and his whispering.\n\nAnd it happened.\n\nI didn't leap. Not bodily, but something inside did. Something that wasn't mine, it wasn't there before. I remember now.\n\nI felt that leaping sensation for the first time, like sand falling upward, filling my head, then out through the top.\n\nI felt joy.\n\nLike the old guy slipped me his joy. Like that first leap into Meg's garage, this happens suddenly. I'm not in a rush anymore. I'm not feeling guilty or afraid. I'm not feeling bugged. I'm thrilled to be where I am. I've got his joy, and I don't know what he's got of mine. Frankly, I don't know what to do with it.\n\nEverything changes. I'm not me, but I am\u2014just with this extra part that messes up everything James would've done with a joy that James doesn't know how to deal with.\n\nIs this what bumping into God is like?\n\nAt first, I think, _I feel great_. It was great, but it began to wear off, just as quickly it moved, it passed, almost immediately. It wouldn't stay still, or I wouldn't. The more I messed with it, the more it slipped through my fingers and jumped under my last thought.\n\nAs it began to wear off, I caught hold of it for a minute, because I remember thinking, _Thanks_. I was grateful, even if it was leaving, that I had felt it at all. Never before had I been grateful at Mass. It stayed still inside my gratitude for a second.\n\nUntil the old voice scared it off, my voice. The voice that told me to kick through that kid's marbles on the playground.\n\n\"You know you're going to have to pay for that, right?\" it said.\n\nI feared the voice, though I've known it all my life. Afraid of the answer, afraid of what I'd owe for my glimmer of joy. That quickly, I was back to being all thumbs, excuses, and anxieties. I knew then, when the joy had gone, in the hole it left, that something else was coming.\n\nI had no idea what that might be, and I made myself forget the whole thing.\n\nNow I remember. Tonight.\n\nI woke up remembering all of it in the dark, sweating and afraid. I could feel that blank place the joy left, the empty feeling after the sand has fallen up, the weight of a thin hand upon my shoulder. I imagine feeling the thick drops of blood running between those eyes.\n\nGod did this to me.\n\n# Part 2\n\n# 11\n\nThe next day I woke with one fear: God.\n\nStrange as it is, this is a step in a positive direction for me. It's work, but I can do it at home. Having the one thought that God is meddling in my life as opposed to the usual terrifying hoard of thoughts, I have to wonder if this isn't what it feels like to be well adjusted.\n\nDon't get me wrong\u2014I'm not particularly at ease. Pillows are piled on my head, and the bedcovers are drawn tightly up to my neck. This God-is-up-to-something fear is pretty huge, as far as debilitating fears go. I'm nursing a chilling sense of dread up and down my bunched-up, pin-starved spine. But it's just the one dread, see? One huge fear instead of a million little ones. Like debt consolidation, and I know precisely who's sending the bill.\n\nOn the other hand, this is precisely what's scariest about God. Okay, hell is scariest, but God showing up today is a close second. God actually doing something today, not confined to the long-ago past or a sickbed, or with some saint in Calcutta. Do you see? If God is really God, then he doesn't have to tell you when God's going to do a thing. What if God starts doing things to you?\n\nUnacceptable. Horrifying. Inconvenient.\n\n_Divinia Interruptus_. And you can guarantee if God gives you a hundred bucks, he's coming back around to see what you bought with it. That's in the Bible. The parable of the coffee shop shows that the kingdom of heaven is likened unto a man who goes into a coffee shop and orders an espresso.\n\nAs the man talks across the counter, the coffee guy makes his coffee and sets the cup and saucer on the counter between them. But the man doesn't drink it; he keeps talking, so the coffee gets cold, useless. The coffee guy pours it out and pulls another, sets it up. The man still can't stop talking. The next one goes bad too. So the coffee guy throws that one out too, makes another. And this goes on, see?\n\nYou may think you're the coffee guy in the parable, but you're not\u2014you're the espresso. (It's like that in parables.) You're not for you. You're someone else's beverage. And God, the coffee guy, he's going to keep remaking you again and again, as many times as it takes until you're drinkable. God's pulling the shots, and he's got standards.\n\nIf God changes you, you'd better change.\n\nI should be practicing all day. God himself has gotten involved. No leaving him at church this time. God's on the loose, and who's to stop God from doing it again?\n\nNext time, maybe it's not leaping powers. How about a second head? Three cheers for the man with fish fins! Who wants everything I touch to turn to copper?\n\nNo, it's time to do good, I know that much. You think God's not gonna care if I don't get cracking on saving the world?\n\nWould you risk it?\n\nI put on a pot of coffee.\n\nI grab a pen, a pad of paper, getting ready to make my list. I write:\n\n_Good Deeds List:_\n\nThen I sit.\n\nYou can stare at a coffeepot, but it won't brew faster.\n\nWhat are good deeds? Is there a book on this or a Web site? There are vintage good deeds, like feeding the poor or helping old ladies across busy intersections, but God didn't turn me into a Boy Scout. My good deeds have to involve this leaping power, otherwise, that's just sheer ingratitude. Anything I could have done last week when still normal\u2014understanding what I mean by that\u2014that's definitely out-of-bounds.\n\nI worry. I pace. I arrange boxes to allow space to pace around them.\n\nYou'd think the hard part would be the physically impossible stuff, but seems it's not. Why is it such a problem to find a few, measly good deeds to put on your list?\n\nFirst, because good deeds are much harder than \"not bad\" ones. Not shooting people, not laughing at the homeless, not pocketing a tip left on the table when you sit down\u2014these are easy to locate.\n\nSecond, I'm scared, and fear doesn't make you the least bit adventurous. I'm just hoping to get a cup of coffee in me before the locusts come. How am I ever going to seriously concentrate on do-goodness with God breathing over my shoulder?\n\nThird, my good friend, Officer Goss, the obsessed grocery cop. Really... do I matter to this detective?\n\nWhich brings me to Meg... I should call Meg. No, I should yell at Meg about talking to Goss. I write that on my list:\n\n_Call\/yell at Meg re: creepy cop_.\n\nI need to explain that, for future reference, any detectives calling about any ex-spouses should be rebuffed and lied to.\n\nI pour a cup of coffee, then sit with my pad of paper. I write:\n\n_Truth, Justice, and the American Way_\n\nThis looks remarkably foolish on paper. What does that mean, the American Way? Under that I write:\n\n_Truth, Goodness, Beauty_\n\nI write it because the phrase is similar; then I remember why I dropped out of Philosophy 101. Being American, I might have a guess at the American Way, but not being an old, dead Greek guy? This is the worst list I've ever written.\n\nI need help with this list. I should call Father Chavez.\n\n_Call priest\/good deed suggestion_.\n\nBrewed coffee is clearly not going to be potent enough to help. This calls for espresso. A leap to the coffee shop. Coffee is a noble motivation, isn't it? God will understand, right? It's coffee, and I need it. It may not be yanking stray puppies out of traffic, but doesn't finishing a good-deeds list qualify as a good deed itself? Besides, coffee's good for me; I'm a more wholesome person when I'm properly caffeinated.\n\n_Coffee is good_.\n\nI shove this pitiful list into my pocket on my way to the window. I rip aside a curtain and look at my watch. High noon. The sunlight from the window is bright, so I twist my arm until my watch catches a beam. I catch it so suddenly that I squint, sunlight glaring sharply off the watch glass.\n\nI wonder who's at the shop near noon on Sundays. I imagine Charlene sitting in her back room, doing paperwork. That's definitely not where I want to appear. I concentrate on the trash bins behind the coffee shop. I look at my watch, twist my wrist so it catches the glare again, as I squint and try hard to picture those trash bins. Huge, green-gray plastic tubs that stink when you lift the lid. I imagine smelling down inside one.\n\nNothing happens.\n\nI move the watch closer to my face till I see myself. I look at my reflection, my own face, and nothing seems good enough. I move my watch closer, and like before, I see my own wet, rolling eye, but this time it just looks trapped on the watch face. It leaps around trying to see itself, but it can't get free. I realize: my strongest desires are only strong because they're for _my good_. Not even that, they're for my own comfort, my habit. My desires are not desires but habits, strong only because they're so thoroughly mine.\n\nI drop my hand to the windowsill. Such a bright, terrifying day.\n\nThe coffee shop's not far. I picked this apartment so I could walk to work. Maybe Monica can give me a ride home. She's usually there on Sundays.\n\nOr maybe Kevin.\n\nSuddenly, I feel awful. If I had any decency, I'd write Kevin on my list. He's a little needy for my tastes, and that doesn't inspire confidence that it'd be a good time, going somewhere with him. Besides, he wears these weird flip-flop clogs and rolls his pants legs up too far. The kind of guy who always wears two shirts, a normal one and an undershirt, no matter how hot it is. Sure, this coming from a guy who doesn't change his pants, but slobby and strange are different. Kevin always wears his nametag on his apron. I mean, they give everybody a nametag, but no one except Kevin wears it.\n\nI can see it pinned, just so, to his black apron.\n\nInadvertently, I turn my wrist. A bright flash bursts off my watch, bright and blurry, and I'm falling upward outside myself...\n\nA nametag.\n\nWhen I stop squinting, I look down at the black apron, bunched on the counter at the coffee shop. I'm staring at Kevin's nametag.\n\n\"James!\"\n\nI turn and see Monica. I'm there, at the coffee shop, and incredibly glad I put on pants. Monica pushes me out from behind the counter, trundles me toward the front door.\n\n\"You know Charlene doesn't want you behind the counter in street clothes... come on.\"\n\nIn her desperation to help me, to hide me from my manager, Monica grabs my hand and leads me outside. It's almost romantic, this moment. Her hand is warm and soft, albeit dabbed with steamed-milk foam. I half hope we keep on walking down the street, hand in hand, but she turns and looks at our clasped hands. She releases mine, blushing twice in a row.\n\n\"Who's working today?\" I ask.\n\nBefore she can answer, she has to stop blushing, which she can't seem to do, and Mike sticks his head out the front door.\n\n\"Hey, Jimbo, some tough-guy cop came by this morning asking for you.\" Mike, he's a tough guy in an apron. He's iron tough in his short black apron and his rumpled khaki shorts, with his coffee-stained white golf shirt.\n\nI look to Monica, and a sad blush confirms that Mike is telling the truth. She has an entire blush language.\n\n\"Yeah,\" Mike continues. \"He asked me what I thought of you, if you seemed distracted lately, or exhibited any unusual behavior. Did anything noticeably strange.\" Mike glares at Monica. He's always had his eye on her, but is clearly not her type. He needs someone more along the lines of a she-bear or a robot that empties ashtrays.\n\n\"What'd you tell him?\" I ask, fearing the worst.\n\n\"I told him you'd been fired. Told him you weren't coming back.\"\n\nTruthfully, I hadn't expected Mike to have my back like that. It seemed loyal, decent, almost friendly, and so I'm pleasantly surprised.\n\n\"Thanks, Mike. Thanks for ditching the guy for me.\"\n\nHe says, \"I didn't ditch anybody, Jimbo. You're really fired. And I hope you don't come back.\"\n\nYes, this Mike I know.\n\n\"Good job, Mike. Just inches away from that merit badge, buddy.\" I wink at him.\n\nHe did what he wanted to do, so Mike shuts the front door and goes back to not working. Good deed or not, I consider leaping into his room while he sleeps, kicking over his food bowl, and showing him some amateur acupuncture.\n\n\"Oh, James,\" says Monica. She's about to burst with compassion at this scene of general animosity. She can't abide conflict. I've seen her serving sleep-deprived customers who are demanding more whipped cream, less whipped cream, blue whipped cream. She smiles and looks for blue food dye. She's the employee of the millennium. Really, I've seen her plaque and the commemorative pen.\n\nMonica wants to hug everybody when there's trouble: me, Mike, Charlene, Kevin, the homeless guy who drinks out of the half-and-half pitchers. Everybody needs a hug right then, but nobody's there but me. She doesn't hug me; she's just thinking about it and blushes.\n\n\"I wanted to be the one to tell you, James,\" she says. \"I'm so sorry. It's all my fault.\"\n\n_This is pure Monica_ , I think, _but completely unreasonable_. I cannot fathom anything that could be Monica's fault, anywhere. I can't imagine her doing a wrong thing. When I try to, I still can't. I can only imagine her nursing wet butterflies back to health and flight; I can imagine her dabbing the hot foreheads of earthquake victims, carrying stir sticks and a camp stove through gunfire to tiny villages of customers with tepid lattes. I cannot, however, imagine anything for which she could accurately take blame, let alone me losing my job for ditching a shift.\n\n\"Monica, it can't be your fault.\"\n\n\"Oh, James,\" she says this and shakes her head miserably. \"I couldn't lie! I just couldn't.\"\n\nIt takes a minute to adjust for what Monica thinks is lying, but I begin to understand.\n\n\"Did Charlene ask you if I came in yesterday?\"\n\nWith a gasp, she stops breathing like I just suddenly appeared in front of her. For a moment, I wonder if I have.\n\n\"What did Charlene ask you?\" I ask, hoping to let her off the hook.\n\n\"She asked me, 'Did James look sick?' \"\n\nHonestly, she starts crying. Big, sad, broken-up-about-everything tears. Now I want to hug her, the detective, Meg, and everybody myself. I open the coffee-shop door, step inside, and grab a stack of napkins off the condiment bar. When I come out, as if being tortured, Monica breaks.\n\nShe confesses in a torrent of words and tears: \"Mike told her! Maybe he didn't mean to, but he told her you came in. And you did, you came in, James, but not to work. You did stop by, James! I couldn't stop him. Then Charlene knew all about that, and Mike told her, I'm sure he didn't mean to get you in trouble, but he said you claimed to be sick. He said 'claimed' with this big shrug, because I guess he didn't believe you. I believed you, James, but Mike said that you looked fine to him, and then, 'Ask Monica. Monica was standing right there when he came in, Charlene.' And, James... James, he pointed at me! Mike raised his arm up, with me looking right at him, and... he pointed at me... like this!\"\n\nShe tries to demonstrate, but it's just too much for her kind heart. She raises her arm, tries to scrunch her pretty face into a scowl like Mike's, but she can't, and she tries to point at me. Her thin hand never makes it up that far; it just won't make the requisite fist. She drops her arm and the halfhearted scowl, then sobs afresh.\n\n\"It's all right, Monica. He's a real pointer, that one. Don't worry about it, there wasn't anything else you could do.\"\n\nIt's too poor consolation for this girl. She's truly too good. Who knows why, but she believes in the good in me just as she believes in the good in Mike and Kevin.\n\n\"I'm so sorry, James,\" she cries. \"So-so-sorry!\"\n\nShe bunches up the napkins in a fluffy wad and wipes her entire face. Her entire face needs wiping, so many are this girl's tears. She smears tears, makeup, stray coffee grounds, and snot all over the napkins. She's a complete mess. I can only think that she's prettier without the makeup. She's kind even when she's crying, an angel in a coffee apron.\n\nShe blubbers, \"What are you going to do now, James?\"\n\nI shrug. I've got God problems as well as superhero anxieties. This is not an answer I'm anywhere near having at this moment.\n\n\"First, your wife leaves you, now you lose your job. All because of me!\"\n\nAnd I wonder, does Monica think she made Meg leave? I can't imagine what series of her far-too-good actions she's translated into home wrecking. Has she really had a crush on me for a while? I consider how that might be, at all, possible.\n\nMaybe I've always had a crush on her, but Meg didn't know. Monica didn't know. I didn't even know until Meg left and I had a nervous breakdown. No, Monica and I never did anything, never went anywhere, we weren't even flirty. I'm not sure I'd recognize flirty coming from such an innocent person, so maybe I missed it?\n\nWe certainly never went anywhere. Not even for lunch. Does Meg think we did? Did we? Not that I can remember, but I'm thinking about it now. \"Monica?\"\n\n\"Yes, James?\"\n\n\"You and I... have we gone to lunch?\"\n\nThe sound of this girl's sobbing stopping, suddenly, is like a tube at the bank drive-through. A quick intake of air, a distant whoosh and rattle, then silence.\n\n\"Today? No, thank you, I just had a really big breakfast and\u2014\"\n\n\"No, not today.\"\n\n\"Oh!\" She corrects herself, sniffles again, looks down and tries to fold the ruined napkins neatly, thinks hard, frowns, then corrects me. \"Yes, I did, this morning... pancakes.\"\n\n\"No, I mean...\"\n\nNobody involved in this conversation seems to be in charge of it, but the girl's no longer crying, so I'm pleased.\n\n\"I'm not sure I understand,\" she says.\n\n\"Me neither, but all I'm asking is... do you remember anytime when... I mean, us... we... a lunch or a dinner?\"\n\n\"Well, I guess, James,\" Monica says this slowly, looking for any helpful signals that I might be giving. Which I am not. I am currently broadcasting my idiot face, leaving her to say, \"if you are asking for sometime... then yes, I'd love to.\"\n\nThat first leaping moment, with the cutting board, the towel, the pins, when I realized I was suddenly in the middle of my former garage\u2014it felt just like this. Monica's looking at me, wearing a flattered smile, wondering what to wear, and I have no idea how I got here.\n\nShe sniffles. \"If not today, when would you\u2014\"\n\n\"Today's fine,\" I interrupt. It seems I can't stop any of it now.\n\n\"Oh, I'm not hungry for lunch yet. And I'm kind of... still at work.\"\n\n\"Right, not lunch. Would you like to go to dinner?\"\n\nAnd I'm thinking this is a good thing, when this confused shock wears off, I'll be excited, and that this will work out much better than the garage thing. Getting to this place, I've exchanged the ex-wife's bawling-out for a dinner date with a pretty girl. But my face must show the slow progress of my thoughts, because Monica begins to suspect something.\n\n\"James?\" she says.\n\n\"Yes, dear?\" I say.\n\n\"Did you mean to ask me out?\"\n\n\"I do now.\"\n\n\"Oh,\" she says.\n\nWe let a few customers pass; I hold open the door.\n\nShe says, \"If you do now, then... I think I've accepted.\"\n\n\"Always good to double-check.\"\n\n\"Then I'll give you a double yes.\" She says this, turns to go, then turns again. \"Oh... and yes!\"\n\nShe turns to go, turns again. \"Wait! You're unemployed, James, you can't afford to take me to dinner... but... I could make you dinner?\"\n\n\"You're perfect,\" I say.\n\n\"You'd let me cook for you?\" She blushes. \"Oh, my parents are having some people over tonight. But I could make a dish of something and bring it over to you? Do you like lasagna?\"\n\nMy mind tailspins. Not about lasagna, I love lasagna. Frankly, I love any cooked food right now, but my mind spins because she lives with her parents. I panic.\n\nHow old is she? I'm not a creep, but I haven't dated in a long time and haven't adjusted, in my mind, for my own age. I mean, I can't look at people and guess their age anymore because I forget how old I am. I'm usually off by about twelve years.\n\nMaybe the girl I thought was at least twenty-two is actually fourteen. But, I correct, she works here. Aren't there child-labor laws? She can't be fourteen. No, I've seen her drive without help, so she's at least sixteen. But that's still no good.\n\nHer dad will tell me it's a school night, and I've already got a detective watching me. Which reminds me\u2014my detective came snooping around my workplace? Is he in the parking lot right now, hat pulled over his eyes, waiting to follow me home? To my apartment with no lockable door. This is where the possibly nineteen-year-old girl wants to have lasagna together. My disheveled apartment with the duct-taped door and the fixated detective. Maybe she's twenty.\n\nMonica senses something wrong. \"I know, I'm a little odd. Lilies for lullabies!\"\n\nI have no idea what this means, but she laughs, shakes her head back and forth so her ponytail swishes.\n\n\"It's silly, a twenty-five-year-old living with her parents, but my daddy has some health problems. I help Mom with his therapy. It's good training for me, as a physical therapist. I mean, for when I finish my degree.\"\n\nI imagine this pretty twenty-five-year-old therapist layering pasta with ricotta while cheering her ailing father on at his treadmill. I can't stop myself from saying, \"I love lasagna.\"\n\nWe stand there. Outside the coffee shop, a bright, gorgeous summer day, like we're romantic leads in a musical and the orchestra's playing a swoony instrumental bit. We're looking into each other's bloodshot eyes\u2014hers from crying, mine from, you know\u2014standing face-to-face in front of the coffee-shop door. Customers can't get in or out.\n\n\"Well, James,\" she says, stepping out of an angry lady's way. \"I should get back to work.\" She sniffles, straightens her apron.\n\n\"Sure, Monica. I should go too... How do I get to your house?\"\n\n\"Oh!\" She giggles, touches my arm.\n\nMaybe I am a superhero.\n\n\"Heavens and help!\" she says. It sounds like that lullaby-lily thing, but no head motions. \"Let me go write you directions.\"\n\nI follow her inside and walk over to the end of the counter. I'm not really worried about bumping into Charlene if I'm already fired, but Mike's up to no good. He's milling around, eyeballing me, like he wants to cause trouble.\n\n\"Hey, Mike!\" I call out. \"Pull me six shots of espresso.\"\n\nHe spins around and grins like a dizzy cat or an addled monkey. He looks delirious with thoughts of antagonism, and he's not about to give me free coffee. Instead he saunters over, leans against the pastry case, and twirls a pair of sugar-crusted tongs.\n\n\"Hey, Jimbo,\" he says this loudly, loud enough he hopes Charlene can hear him in the back. \"What's with the cops... you in trouble with the law, Jim?\"\n\n\"Yeah, Mike. Killed a man back in '84, about your height. Murdered him with pastry tongs and buried him in a coffee bin. Been running all these years. Now, as they say, my fears have found me out.\"\n\n\"Sorry to hear that, Jim-Bob,\" he says. He isn't and neither am I, and we both know it.\n\nMike starts again. \"Hope you don't mind, the nice policeman asked so many questions, I just wrote down your address for him. I found it on the phone list in back. The cop said your wife\u2014excuse me, your ex-wife\u2014wouldn't give it to him, but I'm a good citizen, so why obstruct justice, you know?\"\n\n\"Thanks, Mike,\" I say. \"While I'm thinking about it, you should really think about shaving that neck.\"\n\nMike snaps the tongs viciously at me, but Monica saves us both.\n\n\"Boys, boys.\" She's way too happy for either of us to mess it up by fighting around her.\n\n\"See you later, Mike,\" I say.\n\nMonica brings me six shots of espresso and hands me a slip of folded paper with directions to her house. Both written directions and a detailed map with lots of well-drawn, manicured little arrows.\n\n\"Seven sound good?\" I ask.\n\n\"Seven will be wonderful,\" she says.\n\n\"Then I'll see you later. Tell Charlene I apologize and tell Kevin I said, 'Hey.' \"\n\n\"Omigosh!\" says Monica.\n\nI've never actually heard someone say, \"Omigosh,\" so we're both having a moment of surprise. She puts both her hands on her cheeks.\n\n\"Kevin quit!\" She whispers, leaning in and looking around to make sure no one is listening. She touches my arm like this is a huge event. Her hand, her fingertips really, pinch at my dirty T-shirt sleeve and pull ever so slightly, like she's removing a stray thread. This pulls me closer, and when she leans in, her eyes become plaintive\u2014sad and true\u2014I can see all the way into them. She trusts me completely, for who-knows-what reason, and she leans her face close to mine to whisper, to confide in me, as if we've known each other forever. We hardly know each other at all. How can someone just choose to know you... to be known?\n\n\"When Charlene said that you were fired, Kevin took off his apron and threw it there.\" She points at his apron on the counter, the apron I saw when I first appeared in the coffee shop.\n\n\"Kevin took his apron off, right after Charlene said you were fired, and he told her, 'I was thinking of quitting anyway. Let James have my spot.' \"\n\n\"Charlene let him do that?\" I ask.\n\n\"She couldn't stop him. He just tossed his apron there and got his stuff.\"\n\nI pick up Kevin's apron, the one with his nametag that made my last leap successful. In a way, I have a date with Monica because of this nametag, because of Kevin.\n\n_I owe the guy_ , I think. Although I don't believe he quit because of me, he made a sort of gesture on my behalf. Half on my behalf. He said he was quitting anyway, right? There's a strange camaraderie to it. _I should definitely call Kevin_ , I think. _Put him on the list_.\n\n\"Hey, maybe Kevin wants to rob banks with you?\" Mike shouts. He's been listening to everything.\n\n\"Mike,\" I say, \"why don't you\u2014\"\n\nBut Monica saves me from my mouth again. Clearly a morning full of pointing and firing has pushed her beyond what she can bear, and she wants no more disagreeable business. She's collected herself while she's writing out the cute map, and she's brave now. She kisses me lightly on the cheek.\n\n\"I'll see you at seven, James.\"\n\nMonica walks to the back. Mike is dumbstruck. Mike is dumb, so this is not a difficult adjustment for him. For a girl who lives with her parents, that showed some moxie.\n\nI walk out and down the street. I walk into the street, waving at the honkers. I spin around. I walk back toward the coffee shop and arrive merely to stand there. I walk in a circle like a sleepy dog. I should be practicing. I should be heading toward the trash bins in hopes of leaping home unnoticed. I should do so many things.\n\nBut that kiss: when you've been divorced by a woman you were married to for five years and dated for three years, and you've been the faithful sort so you haven't kissed anyone but your former wife-slash-girlfriend for a very long time, not even kissed her for quite a while there at the end of your failing marriage, and you find yourself suddenly being kissed on the cheek by a beautiful girl, a beautiful girl who's thinking about her lasagna recipe and who looks great even after she's cried her makeup off into a handful of napkins, well\u2014I have a date.\n\nI leap home...\n\n# **12**\n\nDo you have milk?\"\n\nStanding in the middle of my apartment, I admit it, I scream. I'd write out the \"Ah!\" but there would be too many _h's_. As soon as I appear, I hear that question behind me, and it's rattling after my otherwise smooth demeanor.\n\nIt's Nelson. He's standing in my kitchen with the refrigerator open.\n\n\"What are you doing in here?\" I yell.\n\n\"Perhaps you've heard... I was looking for milk.\"\n\n\"In my apartment?\"\n\n\"I don't have any in mine.\"\n\n\"You came in my apartment when I wasn't here!\" I walk over and give him a little shove, slamming the fridge shut.\n\n\"Right. You weren't here, so I couldn't ask your permission.\"\n\n\"So you just push inside and snoop around?\"\n\n\"The door was open,\" he says and shrugs toward the door.\n\n\"It's open now.\" I walked over and slam that too.\n\n\"It was open then, too.\" Nelson looks disapprovingly around my apartment. \"You've got nothing I'd want to steal. Relax.\"\n\nI'm being told to relax by the resident drunk.\n\n\"Not to be a pest, but about that milk\u2014\"\n\n\"Nelson, I just caught you trespassing and staring into my fridge. Did you see any milk while you were in there?\"\n\n\"No.\"\n\n\"Do you think there's another place I'd keep milk if I had any?\"\n\n\"You have a stereo in your bathtub...\"\n\nI push past him into my bathroom, throw back the curtain, and do a quick inventory.\n\n\"Why were you snooping around my bathtub?\"\n\n\"Let me repeat myself for one last time: I was looking for milk. I need some milk 'cause I have to write a letter.\"\n\nI stare at Nelson hard because I'm beginning to suspect that he says things like that just to deflect the conversation from unwanted questioning.\n\n\"Okay,\" he says, like he's talking to a dim-witted guy and he has to slow down so I can follow. \"The lady in 304 has this laptop she lets me borrow, but she's out of town, and she's got this cat\u2014\"\n\n\"Stop. No more.\"\n\n\"Geez, okay,\" he crosses his arms like he didn't think our relationship would sour this soon. \"If you ask me, the real question is about the bathtub pawn shop you're working back there.\"\n\n\"Look, Nelson, I'm sorry I don't have any milk. Not in the fridge, not in the tub, not on a plane, not on a train. But, Nelson, when I'm not here, even if the door is open and there's a reserved sign with your name hanging on my fridge, don't just come into my apartment. I don't care what excuse you got or what you need. I don't care if you're drunk or dying, don't\u2014\"\n\n\"I don't drink.\"\n\n\"Please tell me you do, because otherwise I don't see why I shouldn't take off a shoe and hit you.\"\n\n\"You think I drink, don't you? Have you ever seen me drink?\"\n\n\"I've never seen Alaska with my own two eyes, but I\u2014\"\n\n\"Why are you bringing up Alaska?\"\n\n\"Big drinking state,\" I answer him.\n\n\"I wouldn't know because I don't drink.\"\n\n\"I bet you're drunk right now.\"\n\n\"No, I'm not. I'm Protestant.\"\n\n\"Is that like being drunk for you?\"\n\n\"Are you insulting me?\"\n\n\"I'm thinking about it.\"\n\n\"You're not very kind.\"\n\n\"You're probably right about that.\"\n\n\"So we agree. Fair enough.\"\n\nWe both stand there. There's this standing. Finally Nelson says, \"By the way, how'd you do that?\"\n\n\"Do what?\" I say, and immediately I'm nervous.\n\n\"You just appeared in the middle of your apartment, like whoosh or something.\"\n\n\"I've been meaning to talk with you about that,\" I say as I open the front door for him.\n\n\"Let's talk now,\" he says, and he sits on my floor.\n\n\"I'm not so sure I want to talk now.\"\n\n\"Why not?\"\n\n\"Talking with you hurts.\"\n\n\"Do you treat people who like you this way?\"\n\n\"Excuse me?\"\n\n\"Do you punish everyone with your lack of trust?\"\n\nI look at Nelson. I don't like what he's saying to me, but I think it's an experiment, and I think, _I'll show you, why not, just to see how it goes. He's probably not much smarter than a guinea pig_. \"Right, okay. You won't remember this if I do tell you, so listen carefully. I... can... leap... through... space.\"\n\nI say this, putting a little space between each word for emphasis and do a foot shuffle to help the announcement. \"Space,\" I say. \"Through it. Leaping. Bodily. Instantaneously. God made me do it.\"\n\n\"Cool,\" he says, getting up, then he walks out the door.\n\n\"You okay with that?\" I ask.\n\n\"Yep,\" he says, walking down the hallway.\n\n\"Why?\"\n\nNelson stops, turns. \"I don't tell God what he can do. He doesn't like it.\"\n\n\"Doesn't believing that cause some problems?\"\n\n\"There're always problems.\"\n\n\"You're okay with what I'm suggesting?\"\n\n\"You? I think you might be crazy.\" He comes back down the hall, waves for me to come closer. When I do, he whispers, \"Or drunk. Or maybe some liar who lies for the need of his own lying, but I don't have problems with God doing whatever he feels like.\"\n\n\"You really believe that?\"\n\n\"For a while now, yes.\"\n\n\"But what if God starts doing things to you?\"\n\n\"He does things to me all the time.\"\n\n\"Really? Like what?\"\n\n\"Like sunrises and air that somehow fits my lungs and the twinkling light from long dead stars, and people you meet on subways that you don't really want to talk with but God tells you to, so you do, and it's the best thing that happens to you all week.\"\n\n\"What kind of Protestant are you?\"\n\n\"Re-reformed Protestant.\"\n\n\"What does that mean?\"\n\n\"I'm working on that.\"\n\n\"Oh... okay then.\"\n\n\"But you,\" he says, \"you're saying believing causes problems?\"\n\n\"Exactly.\"\n\n\"What's wrong with problems?\"\n\n\"I don't like them. They worry me, they hurt, and they're problematic. Too similar to actual suffering.\"\n\n\"What's wrong with suffering?\"\n\n\"Primarily... it's suffering.\"\n\n\"So belief causes suffering?\"\n\n\"Right.\" It seems we agree again.\n\n\"You're wrong, Seamus.\"\n\n\"How on earth do you get to Seamus from James?\"\n\n\"Oh, sorry. I spell the words out in my head.\"\n\n\"What\u2014\"\n\n\"Look, you're Catholic, Irish Catholic, right?\"\n\n\"So?\"\n\n\"So, J-a-m-e-s... S-h-a-m-e... S-e-a-m-u-s.\"\n\n\"Do you know how often your mouth forms word groups that don't mean anything?\"\n\n\"Okay, Jameson, look, you're saying believing causes suffering, right?\"\n\n\"Right.\"\n\n\"Wrong! It's the exact opposite. I have to go.\" He walks toward his apartment. \"My TV shows are coming on. You're welcome.\"\n\n\"Thanks,\" I say, but his door has already closed.\n\nI put off practicing the rest of that afternoon.\n\nFunny how another person seeing your apartment makes you see the apartment just as it is. It's a wreck. No, not even that\u2014if I unpacked, it'd be a wreck. I need to clean up. Love is in the air, but not the air of my apartment. I don't know what that is, but it's not love. Again, I consider checking into a hotel, just abandon this life, and check in to another. But my credit card's already soft with heat, so I start tidying up.\n\nI'm not joking about having no furniture, which is something one should think about before inviting a guest over for dinner. I have large boxes of largely unorganized, largely useless stuff. If it's truly useful, Meg kept it. I caught her, physically, taking things out of boxes I'd packed and remarking, \"You're not taking this. This is useful!\"\n\nThat's how I know, and it's part of the reason I haven't unpacked. I know I don't actually need any of the things in these boxes. I'm also not sure how long I want to stay in this apartment. In my pre-date-with-Monica era, I enjoyed living in a cheap, garagelike apartment with the boxes and a bed. My depression thrived there. Now, however, I long for chairs, rugs and carpet, lamps, for any two matching plates, and an object people can gather around socially, for a table whose legs don't collapse.\n\nI try to work with what I have. I tidy the stacks of boxes. I put smaller boxes atop larger boxes: for stability, for neatness, for general eye-pleasing-ness. And the stacks look better. They're more symmetrical, and it looks more like I understand gravity. I also don't want a big box to fall on sweet Monica as she tiptoes through the cardboard maze to wash up.\n\nI talk to myself and decide to attack the duct tape. One window is still in a shamble of tape and curtains from Meg's visit yesterday morning. I try to extricate duct tape from the other window and only succeed in pulling curtains and rod down on top of my head again. Pulling at the tape, I tear the curtains; I bend solid metal curtain rings; I remove paint; I dislodge a wall socket; I rip plaster off the wall\u2014a big chunk of it comes off with a piece of duct tape no bigger than my thumb.\n\nThen I remove the barricade from the front door, which makes me look at best hermitlike and at worst homicidal. The door still doesn't lock, and I recall that Mike has now given that detective my address. I merely move the barricade slightly toward a corner because I might need that barricade yet.\n\nExcept for the tidied stacks, the apartment looks mildly worse for my having straightened it. I sit down on a box and sip my espresso at the card table. I think about a vase for the center of the table, maybe some flowers. This table wobbles dangerously.\n\nI grab my cell phone.\n\n\"Hello?\"\n\n\"Hey, Meg, this is James.\"\n\n\"I know. Are we done?\"\n\n\"Hey, do you have our table?\"\n\n\"Nope. I sold it to gypsies.\"\n\n\"Really... why?\"\n\n\"No, James, I have my table. My table says, 'Hey.' \"\n\n\"Yeah, the talking table. Can I borrow that?\"\n\n\"My table says absolutely not.\"\n\n\"I taught that table to talk. Put it on the phone.\"\n\n\"Where's your card table?\"\n\n\"It's right here with me, wobbling in the wind. Would you like to speak to it?\"\n\n\"James\u2014\"\n\n\"No, probably you shouldn't, I think you're why it's nervous.\"\n\n\"Is this really why you called me?\"\n\n\"Meg, I need a table, and I was hoping you might\u2014\"\n\n\"Do you have a phone book? Look under _F_ for _furniture_ , which will follow _freakish_ and be right before _furthest thing from my mind.\"_\n\n\"Come on. I'll bring it back tomorrow.\"\n\n\"James, you break into my house, then you\u2014\"\n\n\"I need one table, just one night. I'll take a speechless one.\"\n\n\"How's the leaping today?\"\n\n\"Pretty good. Thanks for asking.\"\n\n\"You're welcome. And why is a cop looking for you?\"\n\n\"Pass. My turn. Can I borrow a table?\"\n\n\"I'm serious, James. The detective who called yesterday stopped by today and asked some weird questions. He asked if you'd been acting strangely lately.\"\n\n\"Being my lawyer, I hope you were vague. Did you say anything?\"\n\n\"Nothing, James. What have you done?\"\n\n\"Nothing, Meg, but next time, please tell him you've never met me.\"\n\n\"James, I have a friend. He works at the university in the psychology department. I could\u2014\"\n\n\"I have friends too, Meg, thus, the need for nonwobbly tables.\"\n\n\"If you want, I could get you in to see this guy\u2014\"\n\n\"Sorry, didn't mean to bother you, Meg,\" I say, snippy. \"I just thought if you weren't cooking tonight\u2014\"\n\n\"Don't start on my cooking.\"\n\n\"I wasn't starting\u2014\"\n\n\"Is that all?\"\n\nI think of leaping there and taking the cutting board back. Or stealing another one of Dougie's laundered shirts. Or the table. Just a table! I just wanted a safe place for flowers, for one beautiful thing.\n\n\"James?\"\n\n\"Have a good night, Meg.\"\n\n\"James\u2014\"\n\nI hang up, but I don't know why I do that. Yes, I do. I like hanging up abruptly sometimes. It feels like I'm in control of something.\n\nRegarding divorce: it's no picnic. Actually, it's all the ants and spoiled potato salad without the Frisbees and the endless supply of chips and sweet tea.\n\nDivorce is very hard on those who need some sense of safety and control. I've been forcibly stripped of dogs and home, sent packing with all the useless items. I live in a ratty apartment without doorknobs.\n\nAnd the worst part is... how do you know? I mean how do you ever know anything again?\n\nIf you swear before God, family, and friends, and you mean it and she means it and it goes south anyway, how do you know it won't again? How do you ever say \"I know\" again to another living soul? I don't know if Meg ever worries about that. Maybe I should call her and ask her, but I've already used up today's annoying-phone-call-from-ex-husband. I know it worries me. Divorce is all about worry. If you get behind in your worrying, will you ever catch back up?\n\nIt's like this: Meg's no worrier. She couldn't worry her way out of a paper bag. She leaves doors unlocked. Doors to her car in the driveway, unlocked, overnight. Doors to the house, the front and the back.\n\nOnce, I woke up in the middle of the night to get a drink of water, plus check the doors, and a draft blew in through the kitchen. The door from the kitchen to the garage had wafted open, and a shopper's-guide newspaper mailer was blown all over the floor. When I looked into the garage\u2014wide open. The garage door was open. I hit the floor and crawled back to the bedroom for the baseball bat. I crouched and turned on lights, jumped across doorways. I checked every corner of the house systematically from back to front so that no intruder could have slipped past my search and hidden again. I checked the icebox for very small intruders.\n\nI triple-checked that my wallet and my keys were where I'd laid them, exactly at the angle I remembered. This is the angle I always use just in case something like this happens. I went outside the garage to see who'd opened it, if anyone waited for me to come outside with a bat so they could shoot me. There was no one.\n\nFinally, I realized, it was Meg. She had come home late, after I'd already gone to bed, and she had simply left it open.\n\nSomehow, this intelligent woman I'd married couldn't see the problem with that even when I shook her awake at three in the morning. She couldn't work up even a smidge of polite worry.\n\n\"What's wide open?\" she muttered, rubbing her eyes.\n\n\"Yes, wide, wide, wide!\" I said.\n\n\"The garage is open?\"\n\n\"All of it!\" I won't say I screamed, but she asked me if I'd been hurt.\n\n\"Wide open to the world, Meg!\" I trundled her groggy body out of bed, pushed her down the hall to make her see this outrage for herself.\n\n\"Yes, James, I see. The garage is open. Can I go back to bed now?\"\n\n\"Did you do this?\"\n\n\"I don't remember leaving it open.\"\n\n\"Do you remember leaving it closed?\" I asked.\n\n\"Not exactly.\"\n\n\"Well, exactly! You have to remember doing a thing exactly or you can't really say you've done it, now can you?\"\n\n\"James\u2014\"\n\n\"No, Meg, either you remember distinctly pressing the button and watching the door until it bumps closed, or you just left it open.\"\n\n\"The batteries in the garage-door opener in my Jeep are weak. Sometimes I have to press it two or three times to get it to open. Maybe it opened by itself?\"\n\nAnd she thinks I'm the crazy one.\n\n\"We both have jobs, Meg. We both get paid. I would have sprung for the three-dollar packet of fresh batteries if you'd told me you were planning to leave our sleeping bodies open to assault overnight. I'd buy you a bucket of double-A batteries if you'd mention your total lack of concern for our well-being\u2014\"\n\n\"James, it's late. I did mention it, told you my garage opener was acting funny.\"\n\n\"Funny? Acting funny? You think inviting roaming criminals into our lovely home for a shopping spree is funny? Funny is when our dogs bark at each other at cat noises on television. Funny is climbing a tree to get on our roof because we've never bought a proper ladder. Funny is when the right front wheel falls off our lawn mower every time you pull it backward. Okay, that's not funny either, but you get what I'm saying.\"\n\n\"Events with no personal risk are funny.\" I was warming for the big finish. \"I can crack up like the next guy. I'm good with the funny stuff in life. Leaving the garage door wide open all night long, this we do not classify under funny.\"\n\nMeg just looked at me, the half-asleep way she does. This time, she's actually half asleep. She hit the button on the wall, and the garage door creaked and moaned as the gears ratcheted away. She opened her eyes wide and stuck out her neck, watching the garage door slowly shut. Once closed, she scratched her backside, then turned to go back to bed.\n\n\"You can't leave,\" I said.\n\nMeg turned, and her look that night\u2014I understand it now. That was months and months ago, maybe last year. But what I saw and did not understand that night, the way her hair fell into her eyes and she didn't smooth it away, the way her shoulders slouched, the resignation or the beginning of it, the way she breathed deep and sighed, shaking her head and saying, \"I don't know, James. I don't know anymore.\"\n\nI realize now, she had already thought it. She thought about leaving before that night. She'd already begun to leave.\n\nYou can never worry too soon.\n\nAs I leave to pick up Monica, I hang the doorknob gently in place and push at the duct tape until it stays, making it look almost like a normal doorknob. A little piece of tape sticks to my hand, so I place it like the one I did last night. I place it low on the door, across the crack between the door and the frame. Then I leave, a little early, so I won't get lost and be late for Monica. Besides, I want to stop off.\n\nI want to bring her flowers.\n\n# **13**\n\nJames! They're beautiful.\"\n\nMonica takes the flowers, the white daisies, and goes into the kitchen to look for a vase. They have vases, these people, and they know where they keep them. Different sizes of vases for different amounts of flowers, all kept in one place for easy locating and quick flower putting.\n\n\"Thank you, James.\" I think she's about to kiss me on the cheek again, but she stops herself. She's unwilling to seem too forward. And her father is sitting in a nearby easy chair.\n\n\"James, this is my daddy.\"\n\nHe turns in the chair but doesn't get up.\n\n\"Nice to meet you, Mr...\" With my hand extended, I realize I don't know Monica's last name.\n\n\"Oates. Robert Oates. James, isn't it? Good to meet you, James.\"\n\n\"Good to meet you, Mr. Oates.\"\n\n\"Bobby... Bobby, please.\"\n\nI look around to see who he's talking to.\n\n\"Bob, great!\" Nervously excited, I scream this.\n\nHe ignores my volume because these are polite people.\n\n\"My Monica's good at the lasagna, James,\" he says, then he coughs a short fit.\n\n\"Daddy, don't get up,\" Monica says.\n\n\"You're a lucky man if she makes lasagna for you, my boy.\"\n\nMonica blushes. She sweeps away to the kitchen again.\n\nI've fallen into one of those television shows my parents _would_ have let me watch. Her mother comes down the hall. I expect studio-audience applause.\n\n\"Mrs. Oates, nice to meet you.\"\n\n\"Are you the James we've heard so much about?\"\n\n\"I hope so,\" I say.\n\nMonica hears this and laughs, smiles back at me from the kitchen.\n\nMonica Oates. I couldn't have guessed that, no sir, but horse-food last name or not, she's positively stunning tonight. She's wearing this long, dark green skirt that kind of wrinkles and flows. Like her eyes and her hair, her blouse is dark brown, and the sleeves go all the way to her wrists, but the neck is cut low enough to see some of her shoulders, her necklace, her tapered neck. She's put her hair up in this pretty pile of silky brown with two chopsticks through it, holding it all up.\n\n\"Did you see my flowers, Mama?\" Monica asks.\n\n\"Oh, James, they're lovely. Thank you for brightening up our little house.\"\n\nI think if it got any brighter around here, we'd nova into a whole new solar system.\n\n\"The Petersons won't stay too late; they're old,\" says Mrs. Oates, leaning over her husband's chair from behind him, kissing him on the forehead, and he laughs. He reaches up and rubs his wife's arm. She turns to me. \"So you two are welcome to come back here after dinner, if you'd like.\"\n\n\"Thanks, Mama. Ready, James?\"\n\nMonica, in pink oven mitts, holds either end of a foil-wrapped casserole dish. It smells wonderful. She looks wonderful. Her parents are wonderful. A house with furniture, so wonderful.\n\nIn the Volvo, I hold the lasagna while she puts on her seat belt.\n\nRegarding dishes recently removed from ovens: I'm holding the dish wrong, not wearing the pink oven mitts. I've scalded my fingerprints off, and maybe that's better with the detective in pursuit. My eyes water, I exhale through my nose like an angry bear. No, I don't have any idea how angry bears exhale, just take this dish away, I'm thinking. But Monica's having trouble with the seat belt because she is still wearing the oven mitts, and I think I'm going to cry.\n\n\"Thank you, James.\" She places the dish gingerly on her knees. \"And thank you again for the flowers. They're lovely.\"\n\nThis time, I think I'm going to lean over and kiss her on the cheek, but my eyes are still teary from the lasagna, and I don't want her to think I'm overemotional.\n\nWe drive.\n\nSlowly, I become aware of Monica. I mean, quietly aware. The two of us driving, with no radio or rush, just a nice silence between us. How she makes it nice, how her kindness makes such a slow, nice silence, with her hair pulled up in these unfathomable loops and sprigs. She sits so primly, smiling in her silence\u2014she's beautiful.\n\n\"Monica,\" I say, \"I need to apologize.\"\n\n\"Whatever for, James?\"\n\n\"My apartment, my car, my plates, my wobbly table... Did I mention my apartment? I wish we could go somewhere else. Maybe\u2014\"\n\n\"James,\" Monica stares at me, waiting for me to be still and to see that she's already happy. \"Don't worry about things, James.\"\n\nShe means it. I don't recall anyone ever saying don't worry and it actually meaning something.\n\nI look at her, puzzled, until she asks, \"What is it?\"\n\n\"It's just,\" I say, \"why are you so nice? So... perfect?\"\n\nOn cue, she blushes. \"Please don't say that.\" Her fingers fidget with the latch to the glove box. \"I'm not so perfect.\"\n\n\"No, really, I've been imagining perfect people to talk to, and every time I do, they ask if I've met you.\"\n\nShe laughs at me, then frowns at the latch and shakes her head.\n\n\"Really... you look great. The food smells great. You're definitely the kindest person I've ever met. How do you work at the coffee shop\u2014with Mike, for instance\u2014and stay kind?\"\n\nShe doesn't answer for such a long moment, I think I may have offended her. I look over, and she's tilted her head down a bit, not laughing or frowning. Maybe she's thinking too hard about something; if she's still smiling, it's the quiet afterglow of smile.\n\nShe deliberately removes her hands from the latch and puts them in her lap.\n\n\"Monica, are you all right?\"\n\nShe looks up and I've never seen softer eyes.\n\n\"I had a baby once,\" she says. \"She didn't breathe very long. Daddy had her christened at the hospital; he drove the priest over himself. Seventeen hours, that's all her little lungs could do, and then she left. During those hours I made every promise I could think to make. I promised God the whole world, but she died just the same. I was angry with God for a while, but it passed, and I've always tried to keep those promises anyway. It wasn't a this-for-that deal\u2014they were promises. It's always been the good thing to do. And God's so surprising. Every time I try to keep them, God seems eager to help me out, like Daddy being able to run down the hospital stairs to get the priest. Even when I forget or I mess up, or when I feel all wrong and break down, just sobbing, God plops down next to me and squeezes my knee.\"\n\nShe switches her hands around in her lap, changing which hand holds the other.\n\n\"Oh, lilies for lullabies, sorry about this sadness. I didn't mean to say all that. That's why Mike doesn't get to me, because things aren't perfect. Since that time, whenever someone doesn't like me or whenever I want to be mean and angry, whenever I wonder where God slipped off to, I remember those seventeen hours watching my beautiful, breathing girl.\"\n\nWe drive for a while in silence, before I say, \"I'm sorry, Monica.\"\n\nShe smiles and shakes her head.\n\n\"Did you... were you all alone?\"\n\n\"I had my family there,\" she answers, then understands my question. \"Oh... the baby's father? He didn't really want me. He just wanted one thing, once.\"\n\nI look at her and wonder why she'd trust me enough to tell me all that.\n\n\"Well, I definitely don't want that,\" I say. \"No way. I mean...\"\n\nMonica raises her eyebrows and laughs.\n\n\"Wait\u2014\" I'm glad she's smiling again, but it's definitely at my expense. \"I mean, of course I want that, but only...\"\n\nShe laughs again.\n\n\"No, I don't. Not at all. I mean... not _want_ want.\"\n\n\"James, it's okay,\" she says.\n\n\"I mean, sure, maybe I've thought about it, but I don't want\u2014\"\n\nIf Monica has a worried look, I've found it. I grip the steering wheel and grit my teeth to keep myself from speaking. Finally, I manage, \"Without further comment, I'd like to change this subject... May I change, please?\"\n\n\"Yes,\" she laughs. \"You may certainly change.\"\n\nNight has fallen by the time we arrive at my apartment. It's a cool night, a proper evening with its stars, rising moon, and Monica. As we walk up the stairs to the second floor and down the hallway to my duct-taped door, side by side, I carry the lasagna regally, like I'm escorting a princess, carrying her box of jewels in my pink-oven-mitted grasp.\n\nThen I see it.\n\nAfter I see it, I wish so hard I hadn't seen it that I actually turn a full circle in the hallway, with the lasagna, trying to take back my seeing it. I hate when I see.\n\nThe scrap of tape. The one I put over the crack of the door, down low to signal intrusion, isn't only out of place\u2014it's not even on the door. The tape is a foot or two away from the door, as if it fell off and was tracked out into the hallway by someone unaware of it.\n\n\"Monica,\" I say.\n\n\"Yes, James?\" she says.\n\n\"Don't be alarmed.\"\n\n\"Omigosh... what?\" she gasps.\n\n\"That sounds like alarm. Isn't that alarm?\"\n\n\"Omigosh... yes!\" she says, alarmed at her own involuntary alarm.\n\n\"That's definitely alarm,\" I say. \"Easy.\"\n\n\"I'm sorry,\" she says. She holds her hands out in front of her, palms down, like she's trying to keep a beach ball under water. After a moment, quietly and controlled, she says, \"I'm better, what is it?\"\n\n\"Someone has broken into my apartment.\"\n\n\"Omigosh!\" the girl gasps. She points at the duct-taped doorknob and gasps again. \"James, they've taped that back in place.\"\n\n\"Actually, I taped the doorknob like that.\" _Don't explain_ , I tell myself, _you always explain_.\n\n\"I locked myself out the other day and had to break in. I thought if I taped the doorknob back that it wouldn't look so obviously broken.\"\n\n\"Perhaps a transparent tape?\" she offers.\n\nI shrug. \"Let's forget the doorknob. Do you see this piece of tape?\" I pick the scrap up. \"Since I couldn't lock the door, I put this piece on the crack of the door to see if it'd been opened while I was gone. It's been moved.\"\n\n\"So someone moved it?\" she whispers and points toward the apartment.\n\nDuring the whispering and pointing, it occurs to me, for the first time\u2014someone might still be inside the apartment. Now I'm alarmed and whispery too. I don't know how that escaped my naturally instinctive panic-response. Perhaps I was too happy or in love. Perhaps my nerves are slipping because I think I'm superhuman.\n\nRegardless, I make up for it now and panic. My lungs stop working, lungs that are too scared to extract the oxygen from the air going in and out my nose. I set the lasagna by the door. With one pink oven mitt I push Monica behind me, and with the other I grab the dangling doorknob.\n\n\"James.\"\n\nShe doesn't mean to, but that scares me witless, and when I jump, I inadvertently shove the door open.\n\n\"Oh!\" Monica yelps, grabbing my elbow. \"Stop, stop, James... Shouldn't we call the police?\"\n\n\"Shh!\" I put an oven mitt lightly to her lips.\n\nShe nods calmly, then to encourage me. She's an expressive nodder.\n\nI look inside, scan the room, and crane my neck to look into the kitchen. I see no one. I wonder how long I can stand there before it becomes clear to Monica that I'm a terrified coward, so I step inside.\n\n\"Hello!\" I call out. No answer.\n\nMonica bumps me from behind. \"James, look! They put everything in boxes. James, they've boxed everything up.\"\n\nI thought I had covered that in the car. \"It's all right.\"\n\nShe pulls at my arm, \"James, what if they're coming back for another load?\"\n\nI grab her by the shoulders, a pink oven mitt on each petite, beautiful shoulder. \"The boxes are mine. I left them that way... We're okay.\"\n\nMonica nods as if she understands. I nod as if asking her if she's all right. She nods as if to say go ahead, check the rest of the apartment. I nod but only because I've gotten used to nodding. I'm reluctant to check the rest of the apartment, and I pat her shoulders a little.\n\nBut there's no turning back, and I look for something to pick up, a makeshift weapon to wield. All the good possible weaponry is still packed. The only thing I can find under the rushed circumstances is a copy of _Coffee Grinder's Monthly_.\n\nTry this sometime. Try rolling up a magazine into a threatening shape while wearing oven mitts. Thus armed with a poorly rolled _Coffee Grinder's Monthly_ , Monica and I edge toward the bedroom.\n\nHoping to regain an element of surprise after calling out hello at the front door, I jump quickly into the bedroom doorway and suddenly yell, \"Ha!\"\n\nHowever\u2014and very unfortunately\u2014this only succeeds in surprising Monica. She screams.\n\nInstinctively, I turn and hit her.\n\nJust once, but a good hit with the magazine, centered on the throat; a deft, sure blow. That fast I pop her one good in the throat, then I watch my dream date gasp, stagger, gulp, and reel. She grips my shoulder like she's gagging on an apple, mango, or a whole grapefruit. Her eyes fill up with water. In the mirror of the closet door, Monica squeezes her cheeks and checks that her facial features are still in place. She has to shake out her fingers thoroughly before she can swallow again.\n\n\"You will never fully imagine how sorry I am at this moment.\"\n\nI say it, but I've swatted the girl. I've hit her so hard her hair has fallen from the pins. She tries to smile but can't; she's too busy rubbing the red welt on her neck.\n\n\"I will apologize forever, and then I'll\u2014\"\n\nShe shakes her head, trying to communicate. Somehow, she understands.\n\n\"You understand?\"\n\nYes, she nods. Nodding seems to hurt, but she's still nodding. It was assault, yes, but, she shakes, an accidental assault. Voicelessly, she shakes and nods till I understand. It's her fault. She accepts that; she shouldn't have screamed. Regardless, she won't scream again, not with that welt. We both should just hope that she'll speak again.\n\nGulping, she points toward the bathroom.\n\nWe edge forward. _Dear heavens_ , I think, _please don't let whoever broke into my apartment be in a compromised situation in my bathroom. Not on my first date with Monica, please!_\n\nI push the bathroom door, and it swings open slowly.\n\n\"If you're in there,\" I say, \"come out with your pants up.\"\n\nNothing happens. I pull back the shower curtain, exposing my television and stereo.\n\n\"Did you do that, or the bad guy?\" Monica coughs. She's catching on, but the poor woman, I've probably injured her vocal cords. I run her a glass of tap water.\n\n\"He didn't do that,\" I say. \"I did.\"\n\nHer eyes widen as she sips the water. \"Who is he?\"\n\nThen I catch it, just a trace, but it's definitely there\u2014Old Spice. Officer Goss. My starched nemesis. Here in the bathroom, which really scares me. What do I say to Monica?\n\nOh, that's just my detective. I'm wanted at the Shoppy Mart. I like to scare old ladies on weekends, then steal beef jerky.\n\n\"The intruder,\" I say. This is safe, noncommittal. \"Are you all right?\"\n\nMonica checks her neck in the bathroom mirror. It's swelling.\n\n\"James, shouldn't we call the police?\"\n\n\"No,\" I say.\n\nI don't want to tell her I think it was the police, the grocery detective who's investigated my bathroom while I'm gone. I don't want her to know that I know what cologne the grocery detective wears.\n\n\"No, we're fine. Maybe that tape just fell off, lost its sticky.\"\n\nPart of me wants to believe it's true. Not the part that lost hair earlier while wrestling with that tape, the part that knows _that tape_ will never _lose_ its sticky.\n\nIt's the part of me that wants, recklessly desires even, to believe anything causing a happy ending. Nothing's been moved. No one's been here, why do I think someone's been here, just because I think I smelled Old Spice, that doesn't\u2014\n\nNelson.\n\nI sigh angrily, but I'm relieved. I'm just imagining supercop. Nelson, he wears Old Spice. He comes and goes as he pleases, and showed interest in my bathtub pawn shop. Of course, it's Nelson.\n\n\"Monica, it's nothing,\" I say, with more conviction this time. \"We're fine.\"\n\nBut Monica doesn't look very confident. I can tell this was not the kind of date she expected. Candlelight, maybe. A nice loaf of buttery bread warming in the oven. Music wafting from the stereo in the tub. A bottle of wine on a picnic blanket on the floor. A healthy swat to the throat as we Scully-and-Mulder our way through my scary apartment? No, she never expected that.\n\n\"If you think it's okay,\" she says.\n\n\"I'll get the lasagna,\" I say. I go out into the hall and bring it in, setting it on the card table. I'm humming a sprite little ditty, feeling calm and casual. I get out the two plates, the forks and knife, perfectly place them, and pull out a chair for her.\n\nMonica sits, but she's still unsure.\n\n\"James\u2014\" she says.\n\n\"We're fine,\" I say. I shrug, very carefree, nonchalant. I take off the oven mitts and drop them on the table, as if saying, \"All done with scary!\"\n\n\"James\u2014\"\n\n\"How's that throat?\" I say.\n\n\"James, what if the intruder comes back?\"\n\n\"Who?\"\n\n\"The intruder,\" she says.\n\n\"Oh, him. Monica, I overreacted. Which I sometimes do... Maybe you've noticed.\"\n\n\"No, James, you didn't. You worried that someone might break in, and you put that tape there\u2014\"\n\n\"Monica, me and the tape, we've had a long afternoon, and I overreacted.\"\n\n\"But, what if you didn't, what if\u2014\"\n\n\"There's nothing to worry about. I apologize for\u2014\"\n\n\"But the tape.\"\n\n\"Monica, there's this crazy guy down the hall. Earlier today he\u2014\" I stop midsentence.\n\n\"James... James, what's wrong?\"\n\nI just stare. She traces my staring until we're both staring at the cheap, plastic grocery basket on my kitchen counter, the basket full of stolen items that I leapt out to my car with the other day. Monica doesn't know why we're both staring at a grocery basket.\n\nI do. I didn't leave it on the counter. I'd put it in the shower so it wouldn't get stolen\u2014technically not being mine. So maybe Nelson needed a basket for his shopping-in-my-kitchen needs. But maybe the detective came snooping and found a basket with Shoppy Mart printed on the side. Maybe I'm not imagining supercop. Maybe he's seen the basket and left the proof out for me to see that he's seen the proof. Maybe he's coming back... Maybe he's down the block getting a sandwich... Maybe he's on the stairs right now... My naturally instinctive panic-response is at one percent now.\n\n\"We have to go,\" I say, picking up the lasagna. Somehow it's still incredibly hot, and I quickly set it back down, don the pink mitts, hoist it up again.\n\n\"Omigosh! What's wrong?\"\n\n\"We need to overreact again.\"\n\n\"What did you see?\"\n\n\"Nothing. We just need to leave, right now.\"\n\n\"Where are we going?\" Monica asks.\n\n\"Somewhere else. I've got a backup plan.\" But I don't; I'm desperate now.\n\n\"Why were you staring at that basket?\"\n\n\"I'll tell you while we go.\" I grab her hand and pull her up.\n\n\"But where?\" she asks innocently.\n\n\"Plan B,\" I say.\n\n\"What's B?\"\n\n\"Please, let's argue at the next place. Let's go.\"\n\n\"I don't know enough to argue... Where are we going?\"\n\n\"A hotel. I thought about checking us into a hotel anyway. Come on.\"\n\nMe and the lasagna go out the door.\n\nIn the hallway, a number of important things flash through my mind in a quick ten-point list:\n\n_1. Monica is not following me_.\n\n_2. I just asked Monica to go to a hotel_.\n\n_3. Monica's father, albeit unwell strikes me as potentially handy with a rifle_.\n\n_4. I just asked Monica to go to a hotel_.\n\n_5. After swatting her in the throat, I asked Monica to go to a hotel_.\n\n_6. After she confides in me about her past, I insinuate carnal thoughts, then visibly repress them_.\n\n_7. I just asked her to a hotel_.\n\n_8. If I try to explain, I will only further cement my stupidity_.\n\n_9. Monica is still not following me_.\n\n_10. I've finally found something that offends her_.\n\nI peek back into the apartment.\n\nIt's sad. She's standing with her back to me, one hand resting on the chair I've pulled out for her, like no one ever has done that for her, and her other hand rubs her throat. I've hurt the kindest person I know.\n\n\"Monica, I didn't mean...\"\n\nShe won't turn around, and I'm afraid she's crying again. I want to explain to her, trust her, even with my craziness. Tell her why we need to leave, about the tape and why a hotel, come absolutely clean to her.\n\nI want to lay it all out: the crappy grocery basket, the overzealous detective, the old woman's purse. Tell her that something strange has happened to me and trust her with that. Tell her that I can leap through space, that maybe I'm crazy, and ask her if she thinks I am crazy. If she says I'm not crazy, I want to ask if she would help me pick out capes. Tell her I think God did this to me and ask if she thinks that's possible.\n\nBut she's not talking to me.\n\n\"Monica, maybe it would be best if I took you home.\"\n\nShe looks up, and perhaps I've hurt her twice, but she nods, tries to smile, then walks past me down the hall.\n\n\"Monica, wait...\"\n\nShe stops but doesn't turn. Another thing broken, a thing lost.\n\n\"Yes?\"\n\n\"Nothing,\" I say. \"I'll take you home.\"\n\nShe nods. For the first time, when I see her face, I know she's listening to my words but without her eyes.\n\n# **14**\n\nYou don't have to walk me up.\"\n\nMonica says this before I even stop the car. She's been quietly looking out her window, not looking at me, but I fish in my pocket for a hanky, just in case. A guy sporting a hanky doesn't actually exist anymore. Somehow I forget this in the impossible hope that I'll have a hanky for her to wipe her eyes. Instead, I find a crumpled scrap of paper.\n\nWhen I do pull the car into the drive, she opens her door herself, or she tries. Volvo doors are too heavy, cranky, and temperamental. I jump out my side and run around to help her, but there's a garden hose running the edge of the yard, and I catch it with my foot. I don't fall, but I do catch my balance on her door. Which slams closed.\n\n\"Ouch... James!\"\n\nIt's her hair, that hair so admiringly piled. Part of the pile is now caught in the closed door. I open the door.\n\n\"Monica, I'm sorry, I\u2014\"\n\n\"I'm fine.\" She announces this, shaking her rumpled head. \"All good.\"\n\n\"It's my fault,\" I say.\n\n\"It's nobody's fault, James, really.\"\n\nUnassisted, Monica slides out of the Volvo with the cold, uneaten lasagna.\n\n\"Have a good night, James,\" she says. She tries to smile. \"Thanks.\"\n\nI wait, watching until she's disappeared inside.\n\nAfter her front door closes, it stays closed. Driving away, I feel the crunched paper in my hand, so I uncrumple it:\n\n_Good Deeds List:_\n\n_Call\/yell at Meg re: creepy cop_.\n\n_Truth, Justice, the American Way_\n\n_Truth, Goodness, Beauty_\n\n_Call priest\/good deed suggestions_.\n\n_Coffee is good_.\n\nThere it is, in writing. My laughable best effort at doing the world good. God's reminder that I shouldn't be busy with Monica's lasagna, that I should be busy with doing good and I deserve my wreck of an evening. In all the time I've known him, God gives me one thing to do, and I get distracted.\n\nI muffed it, sure, but here's my real question: why did I think if I had a superpower that I'd be supergood? Automatically, you know. Why did I believe that\u2014are these things similar? If so, what's a supervillain?\n\nIn fact, what is good? I'm not good. I don't even think I want to be. Desire for the good\u2014that's the phrase isn't it, from all the philosophical boys. What do I desire? Who knows, least of all me? To desire a thing, you have to have the time to think about it, to sit in a chair with yourself and not have your hands shake. What do I actually desire? Does God really want to know that? If God asked you straight, would you tell him the truth?\n\nThe truth is, here you go: I have no desire. None whatsoever. Nothing strong enough to earn that term, and certainly not for someone else's good. If God's suggesting that am I expected to do good and also obligated to manufacture a genuine desire for it, this boat's sunk, still sitting on the trailer in the driveway. A stack of things need to happen before I desire to be good.\n\nWe've got to talk me out of me first.\n\nNow God's gotten involved, at least with one life. Frightful to think about, he may be waiting for me to return the favor. Involvement\u2014maybe God thinks that's happiness, the answer to desire?\n\nWaiting your turn for the coffeepot. Leaving the last banana for someone's breakfast. Figuring out how to share a bathroom. Imagining how for someone else, is that the good?\n\nMaybe that's why Meg left. Maybe I never figured it out enough, I kept waiting for her to figure me out while she was on the other side waiting for the same. We imagine one person, one love, will be enough to balance the differences. That one love will outweigh a world of worry that loads the scales, but it never does. Maybe that's why she left: my world of worry outweighed her love.\n\n\"It's my fault,\" I say.\n\n\"It's nobody's fault, James. It's not about blame,\" Meg says. \"Irreconcilable differences is what the paperwork will say.\"\n\n\"There are already papers on this?\"\n\n\"I'm a lawyer, James.\"\n\n\"You've already started the paperwork?\"\n\n\"No, not really. Nothing's been filed, but I've pulled out the forms and filled them out, in my spare time.\"\n\n\"You have spare time for divorce paperwork, but you're just getting around to telling your husband?\"\n\n\"Be reasonable. I'm not conspiring against you. I've been considering this for a while.\"\n\n\"Why? Irreconcilable differences? I don't know what that means, Meg. Differences are usually irreconcilable, aren't they? If they're real differences, and not just watching baseball instead of _Law & Order_. Real differences are the point, aren't they?\n\n\"It's your differences that remind me I'm living with Meg instead of some other woman. I want the woman who has spent years trying to build a compost pile in the corner of the backyard but doesn't realize it's just an unsightly heap of rubbish. I'd miss the way you separate garbage for recycling, though you never take it anywhere, and it ends up thrown out with the rest of the trash when you're not looking. How could I go to bed without that perpetually cold someone piling on blankets when the house is burning hot? One of my favorite argument openings, 'If you bring that fan in here, I'm going to a hotel.' How could I sleep without that?\"\n\n\"James\u2014\"\n\n\"We see eye to eye on the important things, don't we? If we even talk about important things. What are the important things? We usually vote the same, or we vote differently for the same reasons. We don't have kids because we both agree the other wouldn't be able to handle it, but we agree, right? I make dinner on days that start with _T_ and _F_. What's so freaking irreconcilable?\n\n\"If you're just mad about the little stuff, well, give me a chance. Tell me what little things are irreconcilably huge, and I'll adjust. But the huge things, potentially huge, don't seem to be what's nagging you. What is it?\"\n\n\"It's the whole thing, James. Not this issue or that. It's all of it together, too often, too long.\"\n\n\"Too long... five years, what is that? We haven't made it to the pewter anniversary gifts yet. We're back at burnt paper or broken glass or other meaningless compound.\"\n\n\"That's it,\" she says. \"Meaninglessness. Five years, one after the other, one year compounding the next with meaninglessness. I want my life to mean more.\"\n\n\"Well, I'm so sorry I'm not a guru chocked with pithy little proverbs to make your life stunning with purpose.\"\n\n\"It's not you\u2014it's us. What are we? Just two people who got married and started living in the same house. Do we even remember why? That's meaningless; we don't mean anything, James. We don't mean anything to anyone, to each other, to\u2014\"\n\n\"I don't mean anything to you?\"\n\n\"James, that's not what I'm trying to say\u2014\"\n\n\"You just said it. You said I don't mean anything to you.\"\n\n\"Of course, you mean something to me. It's just that it's not enough, not anymore, all right?\"\n\n\"Right.\"\n\n\"Okay,\" she says.\n\n\"So. Did you have a list of things I haven't lived up to? One you've been secretly checking off in your spare time at the office?\"\n\n\"James, stop. I don't have lists.\"\n\n\"Actually, I've seen your lists. You're the most list-making person I've ever met.\"\n\n\"I didn't marry you with a list in mind.\"\n\n\"Your list either changed, or you finally got one, and I don't add up.\"\n\n\"Yes, James, I've changed over five years; you've changed. We both want... I don't know what you want, but I want more. I didn't know I would or mean to, but now I do. I want more out of life than what we have.\"\n\n\"Like what?\"\n\n\"I can't answer that. I just know it's not enough.\"\n\n\"Not enough\u2014that's rough. No chance of me ever making that grade, huh? Just not enough. No matter how I shuffle myself around, I'm never going to be enough since you've already decided, am I?\"\n\n\"No, James. I'm afraid not.\"\n\n\"You're afraid not?\"\n\n\"Talking with you is like arguing with an echo. I'm tired of that.\"\n\n\"Irreconcilably tired?\"\n\n\"I'm afraid so.\"\n\nI drive slowly.\n\nNo reason to hurry home, me and the Volvo. I can still smell traces of lasagna.\n\nOn the floorboard, on Monica's side, I see a daisy. It must've fallen out of her bouquet on the way over to her house. A white daisy with a broken stem.\n\n\"Good night, Monica,\" I say. I put the white daisy in my pocket.\n\n# **15**\n\nI pull into the parking lot of my apartment.\n\nPerhaps I've mentioned the wonderful view my apartment has of this particular lot? It means parkers can see directly into my apartment from aforementioned parking lot. Especially at night when the curtains have been yanked down by a duct-taping fool and a bright bulb glows inside of aforementioned apartment.\n\nFunny thing, though, I didn't leave a light on. I know how lit I leave a place, paranoid neurotic, remember? My toaster faces north. I know which floorboards creak in my hallway and can navigate around them in case of terrorist invasion. I've named the boards alphabetically\u2014Alpha, Bravo, Charlie...\n\nSo I don't park; I keep driving. I drive out of the lot, circle the block, do a drive-by, and look into my apartment. That's when I see him.\n\nJust a glimpse, but I'm sure it's him, my detective\u2014the close-cropped hair, his inflated chest, a starched white shirt. I see him walking past the window, and I coast down a side street to park the Volvo where it can't be seen. I climb out, slip into the shadows of the building, and cross the street to watch.\n\nThe light clicks off in my apartment, and I figure he's leaving again. After all, who'd want to spend the evening in my apartment? There's nowhere to sit. But he never comes out the front door of the apartment building. I watch, fifteen or twenty minutes, and he never comes out.\n\nGoss, a trap, is it? Supercop lying low in my apartment, lights out, just waiting for me to come home. I'm crazy, and this behavior strikes me as odd. But I can locate illegality in this situation, that guy being in my apartment.\n\nI walk down the block to a pay phone.\n\n\"911 hotline. This is Ellen, what is your emergency?\"\n\n\"Ellen, hi.\" I say.\n\n\"This is 911. Your location, please?\"\n\n\"Yeah, um, Ellen, was it?\"\n\n\"L-N,\" the woman enunciates. \"What is your emergency and location, sir?\"\n\n\"My current location, um... Ellen, right?\"\n\n\" _E_ as in emergencies only, _L_ as in liar, another _L_ as in we'll lock you up for lying, another _E_ as in emergencies only, _N_ as in not in the mood. What's your location?\"\n\n\"Oh, Ellen!\" I say as if I've finally placed her role in my life. \"I'm at a pay phone at Barksdale and Peabody.\"\n\n\"What's your emergency, sir?\" She's efficient, my Ellen.\n\n\"My emergency? Well, I\u2014\"\n\n\"Sir, please don't do this. Okay, just don't. Either hang up or get to it. I've got call lights flashing all over the place here.\"\n\n\"Sorry, Ellen. My emergency is that I came back to my apartment\u2014\"\n\n\"Where is your apartment?\"\n\n\"Number 208, Barksdale Manor. Right across the street from this phone, Ellen.\"\n\n\"Your name?\"\n\n\"Don't you want to know about what's happening in my apartment?\"\n\n\"Not right now; now I want your name.\"\n\n\"My name?\"\n\n\"Your handle, partner. Spill it, or I move on.\"\n\n\"James. James is my name.\"\n\n\"Did you just make that up, sir?\" she asks.\n\n\"Do you get training classes for this?\" I ask.\n\n\"If you say your last name is Smith, I hang up.\"\n\n\"I'm not telling you my last name. I'm telling you there's a guy wandering around inside my apartment.\"\n\n\"Do you know this guy?\"\n\n\"Not really, Ellen.\"\n\n\"Not really the right answer, James. Thank you for calling 911\u2014\"\n\n\"Wait, Ellen, please! I've seen this guy at the grocery store, but he's not a friend or anything.\"\n\n\"How did he get in your apartment?\"\n\n\"Well, thank you, Ellen, it's kind of you to ask. I went up to my apartment just now and found the doorknob busted off the door, and a bunch of duct tape all over everything. Fearing ne'er-do-wells, I immediately left to call 911 from this pay phone.\"\n\n\"If you left immediately, how do you know the ne'er-do-well is a he?\"\n\nEllen is good, so I stall. \"Do you mind saying that again?\"\n\n\"How do you know who the 'he' is?\"\n\n\"Now say it faster.\"\n\n\"Sir, you do not sound\u2014even remotely\u2014like someone concerned about your apartment being robbed.\"\n\n\"And you, Ellen, do not sound remotely like an emergency hotline operator.\"\n\nI could tell she liked me.\n\n\"You can hang up and call again for a different, more believable operator. Would you like me to disconnect you, sir?\"\n\n\"Ellen, a guy I saw at the grocery store is in the window of my apartment. I saw him from the parking lot. My door had the knob bashed off, and he's sitting in there, with the lights off.\"\n\n\"Is it your birthday?\"\n\n\"No, Ellen, this is not my surprise party.\"\n\n\"He turned the lights off?\"\n\n\"Yes. I saw him walking around; then he turned the lights off. He didn't leave the apartment building because I'm watching the only door out right now.\"\n\n\"Well, sir, that's the first believable thing you've said in this entire conversation.\"\n\n\"Wow, you're tough.\"\n\n\"Tough job. You got to be.\"\n\n\"I'm just glad I'm not bleeding.\"\n\n\"I'd know if you were bleeding. We'll dispatch a car immediately, sir.\"\n\n\"Excuse me for asking, but how do I know you'll actually do that?\"\n\n\"You doubt me, sir?\"\n\n\"Like I said, you're tough, Ellen. We've gotten off to a rocky start.\"\n\n\"I'll get into big trouble if I don't dispatch on a criminal call.\"\n\n\"We wouldn't want that.\"\n\n\"And, Mr. James, if this is a prank call, you'll be in big trouble as well.\"\n\nI hang up the pay phone. Detective Goss is still in there, but I don't want to be standing around here when the cops find out it's a cop in my apartment. I plan to be around, just not next to the suspicious pay phone.\n\nI have my special ways of being around.\n\nI get back to the Volvo and speed away.\n\nI calculate ten minutes for the cops to show up, another five for them to get together and make a plan for entering the apartment\u2014then the mighty Detective Goss must explain himself. That's what I don't want to miss. I'm hoping to get a read on this Goss fellow, why I'm so important to him, what's his next move. In the meantime, I got twenty or so minutes.\n\nI don't know how this works for normal people, but being a basically paranoid-delusional person who finds himself with someone actually following him\u2014well, most of my life has been lived talking myself out of such imagined plots. Now it's happening. There is a guy, and he is following me. An actual emergency.\n\nIn a way, it's like an \"I told you so\" to the universe. I feel prepared. I have been practicing for this sort of thing all my life.\n\nI go to the ATM near my house, withdraw a few hundred bucks. That should get me through a few days in a hotel. I catch the interstate and take it downtown. It's 8:52 p.m. on a Sunday night, so no traffic. I want a nice hotel, you know, the kind for businessmen and their conventions. Something busy, full of out-of-towners, something with a steam room. Great for the sinuses. I want my own copy of the newspaper in the morning. If I'm getting a hotel room I can't really afford, I'm popping for a good one.\n\nThe Hometown Suites will suit me fine. Ellen's phone interrogation has got me on my toes now, and when I check in, I pay in cash and don't use my real name. Gabe Oates. My name is Gabe Oates.\n\nI'm a big, cash-paying guy named Gabe Oates. I tuck my shirt in so I'll look decent. I leave my backpack in the Volvo so they won't think I'm homeless. (Even tucked, I look homeless.) The recorder, I have, as it's very important for recording Detective Goss.\n\n\"The only room available, sir, fourteenth floor.\"\n\nFine. Gabe Oates is fine with heights. Wake-up call, I need not; please tell the staff not to disturb. Gabe Oates, a late sleeper.\n\nMy room is great. There's a huge bed, two ice buckets, Dish Network, clean towels. I should have checked into a hotel years ago. No time for the luxuries, though.\n\nI lock the door, pull it twice to make sure it's locked, flip the dead-bolt. I turn on the TV, ease the volume up so it sounds like someone's alive in here. I take my hotel key card, put it in my pocket, and my recorder as well. My car keys, my wallet. I've got my glasses. I clean everything off the table next to the TV and pull the table out from the wall. Very deliberately, I put the white daisy in the center of it. Very deliberately, I look at it, recording a mental image.\n\nI turn the light on in the bathroom for added glare. I sit on the edge of the bed. It's 9:23 p.m. by my watch, but that's not why I'm looking at it. I start to stare intensely until my vision blurs. I see the hands, then just the glass of the watch face. I turn my wrist until the glare from the bathroom catches. I squint and stare and blur and turn my wrist till I see the light fixture from the bathroom reflected on the watch face.\n\nI want to leap into my bathtub, back at my apartment, behind the shower curtain. I'm sure the cops Ellen sent have arrived, and I'm hoping they've already checked behind the curtain. If they've checked once, why would they check twice? So if I appear behind there...\n\nIt's not the worst plan, come on. In a pickle, I can always leap away. I get the dreamy eye focus thing working; I know my port of destination. All that's left is the intense desire. Oh boy, oh boy, oh boy, oh boy! I intensely desire to hear Detective Goss explaining to his colleagues why he's breaking into apartments.\n\nThen I remember, there should be some good reason for my leap.\n\nAnd here's where I do it. I think, forget goodness! I'm mad. My home, that hole I've been forced to make my home, has been invaded. My wife hates me, my one date hates me, I've lost my job, I've eaten nothing but old meat strips for days, and I desire a little satisfaction\u2014I'm letting the mad out. Come on, madness!\n\nI'm still staring at my watch, doing the blurry focus bit with the reflection, but the more the reflection hits my eyes, the madder I get. That rolling crazy wild eye. This isn't to prove anything to anyone or help anyone out; I'm not anybody's hero. My intense desire is for myself. I want to leap into that shower because I want it.\n\nI won't take no for an...\n\n\"No!\"\n\n\"Explain to me again, Bill. Where did you meet this guy?\"\n\n\"I told you already,\" says Detective Goss.\n\nI can hear them through the shower curtain, my detective and the other cop. They both sound irritated, and they're both close. Not in the bathroom, not that close, but they must be in my bedroom.\n\n\"I saw him steal groceries from the Shoppy Mart while I was there. He also tried to steal an old lady's purse. So I got his license plate, and I tracked him here.\"\n\n\"You just break into his apartment, Bill?\" says the other tough guy. I'm glad they sent a tough guy for me. Thank you, Ellen.\n\n\"Look around this place, Arnie. Just look for yourself. The guy's a wack job.\"\n\nI take offense to this, straighten my back, and nearly fall over my stereo. There's not enough room in this tub for a TV, a stereo, and a hunched angry man to hide. My foot is actually wedged between the appliances. My ankle's twisted at a painful sort of angle, but I'm afraid to move or to breathe.\n\n\"Okay, Goss. Your boy's a wack job. You still don't get to break into his apartment just because\u2014\"\n\n\"I told you... I didn't break in, Arnie. The guy's doorknob was taped, actually taped, to the front door. I knocked; the door opened. I went in when I saw the curtains on the ground, the boxes, the wads of tape all over the floor. Looked like somebody could have been dead in here. Or worse! Tell me, Arnie, what's all the tape for?\"\n\n\"Still, Goss, you don't just\u2014\"\n\n\"He's a wack job, Arnie. Look around... it's like a creepy psycho's basement in here. He's planning to kidnap somebody or something. His ex-wife. A co-worker, who knows? I think he was involved in that boy's kidnapping, the one we found last night.\"\n\n\"Yeah, you told me, but the kid said he's the one who called in the boy's whereabouts to the police. That's not\u2014\"\n\n\"But how did he know the whereabouts, Arnie? If he was there, why was he there? And why does he leave before the police he just called arrive? No, something's cracked about this guy. Or cracking. I found out he got fired from his job this week, he lives alone, early thirties. Arnie, he's the classic profile of a guy skipping over the edge.\"\n\n\"You went to his job?\"\n\n\"Yeah, and they even said he was nuts. Said he stopped showing up for work. He's just been divorced. He stole seventeen rolls of duct tape and hostage rations; he hides his valuables in his bathtub... look in here.\"\n\nMy entire body spasms. My knee knocks the remote for the TV across the top of the stereo. Somehow my spastic hand catches it before it rattles down into the tub. I try desperately to stare at my watch, but there's no glare behind this curtain. No glare! No glare! A wack job with no glare! _How on earth am I going to leap without a glare!_\n\n\"I saw,\" says the other cop.\n\nOn the bathroom tiles, Detective Goss's steps stop, turn, and scuff away.\n\n\"All's I'm saying is that it's our job to stop crime. To stop it, not just clean it up, Arnie. It's too late when someone's dead, some kid, maybe? Can you sleep when you know you could've stopped a criminal before he slipped over the edge? And this guy is gone, he's crazy, Arnie. I talked to him and saw it in his eyes. And he's already broken the law once\u2014\"\n\n\"A guy loses a crappy job, so he steals some groceries. That don't mean\u2014\"\n\n\"Okay, it's petty theft, but he's on his way.\"\n\n\"Listen, Bill\u2014\"\n\n\"I just wanted to talk with him, Arnie. Just a little chat between the guy with the badge and the guy with the duct tape. Okay? Let him know somebody's watching. Even crazy, he'll think twice if he knows someone's watching. That's all.\"\n\n\"Why do you always assume the most horrible thing possible?\"\n\n\"I don't always.\"\n\n\"Did you hear what you just said? He's over the edge, he'll hurt a kid...\"\n\n\"No, you're not listening. We're checking out a suspicious character.\"\n\n\"And you just assume he's guilty until proven otherwise?\"\n\n\"What's your job again?\"\n\n\"I know, I know. I've seen the same sick cases you have, so I get it. But I don't do this, and you can't either, Bill. You can't judge the whole world as depraved just based on a few rotten ones. It'll eat you up.\"\n\n\"Right, you've seen rotten. Answer me this: how often do you see the good?\"\n\n\"I've seen good and bad.\"\n\n\"No, answer the question. How often does the good surprise you like the horrible?\"\n\n\"Bill... you got to hope it might, geez... or how else do you get up every morning?\"\n\n\"I get up. I have hope, but I don't get up blind, and you're asking me to be blind, Arnie. My hope is not blind. I have hope when I see it happen. Every day I hope that I'll see it happen. Meanwhile, I'm not blind to what's around me.\"\n\n\"Wow, man. You're due for a vacation, you know. You can't crash around believing\u2014\"\n\n\"I believe nothing. That doesn't mean I'm hopeless. In fact, I don't believe, so when I see real hope, the genuine article, then I'll know it for sure.\"\n\n\"Come on, just admit you look foolish and you were wrong this time, and\u2014\"\n\n\"I was cautious. Smart... proactive. That's not wrong.\"\n\n\"Bill... let's go. Let me buy you a beer.\"\n\n\"No, thanks.\"\n\nI hear steps moving toward what I imagined was the front door when Goss says, \"Look, when I was in my third year on the force, my partner and I took a call, domestic abuse case, right? No idea who called it in, but we go over there. There's a beat-up blue Monte Carlo parked half in the yard, half in the drive. I'll always remember that color, that car.\n\n\"So we ring the bell, and the door opens and this woman\u2014no, girl\u2014this young nineteen, twenty-year-old, stringy, sick-looking girl in a thin sundress answers the door but won't let us in. I can see the guy through the inch of door she opens. He's a bruiser strutting around the house with his shirt off, swigging from a beer bottle, talking loud to her or nobody, oblivious to us. Beer bottles line the TV, the windowsill by his easy chair, and she won't let the door open wider than a crack. She tells us to go away, that she's fine, no troubles here. She's got red marks on her shoulders and neck, Arnie, fresh, finger-shaped ones that hadn't turned to bruises yet. Meanwhile, this guy's yelling at the game show on the TV, drunk off his butt at two in the afternoon.\n\n\"No, she says, she didn't call us. She's fine. She doesn't want any trouble, and so we write it up, we go.\n\n\"The very next day, we get a second call, the same house. But this time, the pretty, sick-looking girl is wadded up in the Dumpster in the alley behind the house. She's naked. Her fingers are twisted, broken, every one. Her nose is flattened and swollen at the same time, bruised all black and crusted with blood. That sundress was shoved down her throat. And the Monte Carlo is gone. The house was in her mother's name; who knows where the mother is. None of the neighbors can say anything useful about the guy other than they were afraid of him. He's gone. Never found that guy, Arnie. Pictures of that dead girl's body were shoved in the unsolved box and buried on a basement shelf. That animal probably drove away with her purse and a six-pack on the seat beside him. And you know what? The day before, when I snickered at how that jerk had parked the car, I stared right at the license plate and didn't bother to remember it. I didn't even read it, write it down, call it in, just to see.\n\n\"So don't tell me I've fallen prey to a soured disposition. I have hope, Arnie. I hope that punk got a bullet in the back of the head in the next town, and I hope that little girl woke up in a miracle heaven where they feed her right and give her a decent dress. But I've never seen that miracle place. I don't often see miracles, and my job is to do something about the evil that is happening, not hope that it won't. My job is to watch. Watch for the worst before it happens, and if somebody I'm watching turns out to be less evil than I assumed, well, that's one less report to file.\"\n\n\"Looks like your crazy man was watching you this time,\" says the other cop.\n\nI can't see from the bathtub, but he sounds like he's grinning.\n\n\"What?\" says Detective Goss.\n\n\"You said the guy's name is James, right?\"\n\n\"Right.\"\n\n\"Well, one Mr. James, no last name, called you in to 911.\"\n\n\"Don't mess with me, Arnie.\"\n\n\"I'm not. He told the dispatch there was a burglar in this apartment. Said you busted off the doorknob and taped it up again.\"\n\n\"Said I did that?\"\n\n\"He said he was afraid some guy was waiting in the dark for him. He called about a half an hour ago, from the pay phone across the street.\"\n\n\"Why, that son of a\u2014\"\n\nThe other man laughs, but not Supercop. I hear his feet running to my den window.\n\n\"That pay phone right there, huh?\" Goss calls out.\n\n\"He may be losing his mind, Bill, but what he's got left works better than yours.\"\n\nFrom the loud footfall, pacing back and forth, I guess that my detective is gathering his stuff.\n\n\"Bill, don't leave in such a huff.\"\n\n\"Got to go.\"\n\n\"Listen to me, you leave this guy alone.\"\n\n\"Sorry, it's personal now.\"\n\n\"Sounded like it was personal before.\"\n\n\"No, now it is.\"\n\n\"Goss\u2014\"\n\nI hear the front door open, then slam. My detective is gone, but he's coming for me. Ellen's tough guy is still there, talking with one or two others. I open the shower curtain an inch so I can get light on my watch.\n\n\"Lieutenant, how do you want me to write this up?\"\n\nA new voice says this, a young one, one I haven't heard yet. It seems my tough guy is a lieutenant.\n\n\"No report.\"\n\n\"Sorry, Lieutenant, what'd you say?\"\n\n\"I said forget it.\"\n\n\"But, sir, we can't just\u2014\"\n\n\"Aw, don't worry about it. We caught a cop being a cop, tipped off by a guy who knew he was a cop. The fact that we caught him will make that guy far happier than a write-up that won't amount to anything.\"\n\n\"So we're done here?\"\n\n\"Yeah, we're done,\" says the lieutenant. \"Let's go. Let me just take a leak.\"\n\nI catch a glare.\n\nWhite daisy, white daisy, white daisy...\n\n# **16**\n\nBack in the hotel, let's review.\n\nThe good news is that I can now leap whenever I want, bend it to my will, leap for my own reasons. And I recorded Goss red-handed. The bad news is coming right up.\n\nMore good news: this is a really nice hotel. I'm feeling crazy, but I'm doing so luxuriously. I take off my shoes and socks, stretch out on the bed. The TV's still going; I turn the volume up. I flip through the channels. This room gets approximately two million channels.\n\nA robe hangs in the bathroom, and I wonder if it has been left by the previous patron until I see the Hometown Suites logo emblazoned on the front pocket. Swanky, I think. Soft like a good towel. I've never owned a towel this nice, let alone a robe. I strip down to my skivvies so I can lounge around in one. When you don't tie the belt it's like a cape. I run and jump onto the bed, and the cape thing works. I check the room-service menu and look for when the steam room closes.\n\nI feel super. Maybe it's not confidence, but it feels like I'm wearing shoulder pads. Maybe it's the robe-cape. Because I usually feel nothing remotely resembling confidence, this is worth noting. I bent my leaping powers to my will. Something feels different inside. Harder, stronger, higher, like an aerial view.\n\nThis may seem like grandiose thinking, but it's exactly the kind I went for at that moment. Though neurotic, I've a superpower. I've done a fair bit of ruin to my chances with my dream girl, but I plan to bully forward. The superhero plan. I'm counting heavily on this unhatched egg.\n\nDoing good is going to earn my way back to lasagna.\n\nSo I'm thumbing through channels again, way too many channels. I'm seeing the world, all of it with many disastrous goings on\u2014the world ripe for superpicking. Floods, riots, poverty-stricken villages, assassinations, kidnappings, derailed trains, and so on. With this power, the perfect leap anywhere, the nick of time will be my specialty. I'll be the news. I'll be on all the channels. Special Announcement: the world can sleep sound tonight. Better than a bird, better than a plane, with me, everything happens faster than a speeding bullet.\n\nI imagine, sitting on a hotel bed in this soft robe, watching myself on the television:\n\n_Man Carries Fifty to Safety as Hurricane Crashes into Coastline_.\n\n_Twelve-Year-Old Girl Returned Safely to Parents, Kidnapping Ring Foiled_.\n\n_Hero Removes Getaway Car at Bank Heist_.\n\nAnd once I'm on all the news channels at once, Monica's got to see me.\n\nNow, it occurs to me that I'm saving the world to look good for my intended girlfriend. A little swagger is a good thing, right? Whose motivations are pure? My best intentions are bifurcated at best. Yes, I want to make Monica admire me, even adore me, but I'm also stopping the speeding train. The people on the runaway train don't care as long as they don't smash into another train.\n\nIf I wait to be purely good, I'll never leave the house, right?\n\nI'm watching disasters, and I fetch the stationery pad and the complimentary hotel pen and take some notes. Hurricanes again. Dangerous leaping involved but people needing nick-of-time deliverances. I still don't know\u2014can I carry a person with me when I leap? I should try with something else first, I think, like a sack of potatoes. A mannequin. Where do you get a mannequin? Can you buy one, and do people look at you oddly if you do? I put a question mark beside mannequin.\n\nI start to get drowsy. Holding the pad in my hands, in that warm comfy robe, I close my eyes a bit. Just resting, and I try to think of a supername. This matters, you know? Leaper Man is nice... short, to the point. Maybe too bland and a little lame. It doesn't evoke anything. If you're not paying attention, it sounds a lot like Leper Man. Leper Man is not good. I don't want people screaming, \"Unclean!\" when they see me coming. Maybe just Leaper.\n\nI close my eyes again.\n\nNames matter. Like Monica. How beautiful is that name? Mellifluous, that one. Three sweet syllables. Oates, that's harder. But I genuinely do like it, and I'm smiling while lying there thinking of her name. I can feel my face smiling.\n\nI check my watch: 10:04 p.m. Four minutes past polite calling hours, but this is an emergency.\n\nBecause I've decided to do it. To call and tell her everything.\n\n\"Hello?\" It's Monica's voice. She sounds asleep.\n\n\"Hello, Monica? It's James.\"\n\nA terrible, terrible pause.\n\n\"Hi, James, how are you?\"\n\n\"Fine.\" I should really think things out before I start talking. \"How are you?\"\n\n\"I'm okay,\" she says. \"What's that noise?\"\n\nI realize the TV is loud enough for her to hear. I wrestle the robe and the bedspread to find the remote and mute it.\n\n\"Just watching the news,\" I say.\n\nThere's another weird pause. She asks, \"Are you in your bathroom?\"\n\nIt takes a second, but I figure that one out.\n\n\"Oh no, I'm at a hotel.\" I am positively the dumbest person alive. I should really think these things out before speaking.\n\n\"Then I don't want to bother you\u2014\"\n\n\"Monica, wait! I'm at a hotel because I was afraid to stay in my apartment. Monica...\"\n\nIt's now or never, I tell myself.\n\n\"Monica,\" I say, \"I was afraid\u2014which is why I wanted to go to a hotel. Not because, well, you know... but because someone broke into my apartment.\"\n\n\"But you said you already had been thinking of checking into a hotel.\"\n\n\"I did,\" I say. \"For the past two days, I'd been thinking about a hotel, before we even made our date. Because there's someone following me, Monica, and I think I know who broke into my apartment.\"\n\n\"Really, James?\"\n\nI can't tell if she believes me, only that she wants to believe me. She's that good of a person. \"Really, Monica. It's a long story. I do want to tell you, but it's a little crazy in places, and I don't want you to think I'm crazy in big places.\n\n\"Suffice it to say, there's a guy following me, for two days now. A detective. I didn't do anything wrong, not really, but he thinks I did. He's gone by the coffee shop, called my ex-wife, and rummaged through my apartment. He was there again when I went home.\"\n\n\"Oh, James!\"\n\n\"Right... so I'm at a hotel.\"\n\n\"But why is he following you?\"\n\n\"Well, it has to do with a Mass I went to a couple of weeks ago.\"\n\n\"I didn't know you went to Mass.\"\n\nAt this point, I'm thinking, _Excellent. This is sounding perfect_. Monica may be the one in whom I can confide all my weary leaping woes.\n\n\"Monica, let me ask you a question. If you were suddenly offered an ability to do something others couldn't, let's say someone offers you the ability to fly\u2014\"\n\n\"Who just offered me the ability to fly?\"\n\n\"For the sake of this example, let's say God.\"\n\n\"God offered me the ability to fly?\"\n\n\"Right... or you're pretty sure it was God. You didn't actually get it in writing, but you think it sounds like God, okay?\"\n\n\"Okay.\"\n\n\"Right. God offers you flying, and you think, since it's God, that you should fly for good reasons, not just for fun or whatever. You should definitely do good with whatever special deal you're given, right?\"\n\n\"Right. You should do good.\"\n\n\"So... what would you do?\"\n\n\"What would I do?'\n\n\"What would Monica Oates do?\"\n\n\"Oh... well, I'd turn down the offer.\"\n\n\"What?\"\n\n\"I would turn down the offer. I wouldn't want it.\"\n\n\"You wouldn't?\"\n\n\"No. I'm not good enough.\"\n\n\"Really? But\u2014\"\n\nHere it is. Everything that's happened to me, my whole life, everything I've ever done, thought, and been, it all leaps into what happens next.\n\n\"James, are you there?\"\n\nI'm there, but while I'm talking to Monica, trying to get out my questions, my eyes wander to the television. The volume is muted, so it's just images, and it's set on local news. The Breaking News logo flashes on the screen. Live coverage, from downtown at the hospital. Maybe three blocks from my hotel. Camera shots from a helicopter. Shaky camera shots looking down from the helicopter at the roof of the hospital. It's a big hospital, ten or eleven stories high. Blue police lights flash far below, down on the street. A red fire engine and paramedic lights flashing way at the bottom of the shot. They're all out of sync, light irregularly pulsing and flashing. There's a police team on the roof of the hospital, in bulletproof vests and helmets. They're all looking at another guy on the roof, a guy who has crawled over the side wall of the roof and stands precariously on a small ledge. I'm watching this. The police are nowhere near him. It seems he won't let anyone near him, that he's threatening to do it. On the ledge, his back is to the hospital wall, and he's edging out, looking down at all the flashing lights. A leaper.\n\nHis hair and shirt are whipping in the wind. His hair is in his face, and it's just a white T-shirt, so I can't see his face, and the camera can't get close enough. But I can see his jeans are rolled up a bit. I can make out that there's too much of his ankles showing. He's wearing clogs.\n\n\"James, are you there?\"\n\n\"Hey, Monica, did Kevin ever come back to the shop today?\"\n\n\"After he quit? No. Yesterday he told me he'd be at the hospital. His father's been really sick.\"\n\n\"I have to go.\"\n\n\"James\u2014\"\n\nBut I hang up on her. I have to go. _I need shoes_ , I think. I jump to the floor, scramble around for my shoes. I've got to have shoes for traction once I grab him.\n\nOnce I get one shoe on, I notice my naked leg and realize I'm in my underwear and a robe. I yank the robe off and grab my pants, but I have to stop and take the one shoe off to get the pants on. Then I shove my shoes on, no tying. Forget the shirt. I sit back on the edge of the bed, leaning close to the TV screen so I can see what's happening.\n\n\"Hold on, Kevin. Hold on,\" I say. And I hold my watch up close to my eyes, turn my wrist, but I can't catch the light.\n\n\"Hold on!\" I say. And I jump up, turn on all the lamps.\n\nI sit back on the bed, raise my watch again. On the television, the news cuts to a reporter with a microphone on the street outside the hospital.\n\n\"Go back!\" I yell.\n\nI need another look at the scene. To see exactly where Kevin is standing so I don't miss. I'm focusing my eyes on my watch, but I have to keep glancing at the screen. Finally, they cut back to the aerial shot. The police are all backing up with their hands raised. Kevin has taken his hands off the wall, threatening them with his life.\n\nThere's a shimmer off my watch. I catch a glimpse of light and I focus on it. I make my eyes blur.\n\nI think, _Appear directly behind him. Appear on the roof on the safe side of the wall directly behind Kevin. Throw both arms under his arms and pull_. I'm thinking as hard as I can. A good reason. A very good reason, but it's taking too long. I'm thinking of pulling Kevin to safety. My eyes are on the light. I can't stay focused. My mind flies around like a bird in a grocery. I think of too much, my mind flooded with thoughts. Kevin's on his ledge, and my life is passing before my eyes, and I can't stop it.\n\nEverything: My apron at work. Monica's phone number. Meg's hair undone in the wind. The knobless gearshift on the Volvo. The detective's shiny black shoes. Father Chavez and that yellow slip of ribbon. I see Meg grin at me as she comes down the aisle. I see Monica blush as she puts her flowers in a vase. My mind rolls around like a wet, glaring eye. Everything, what I thought were good things, even the best things, even my loves, crowd in the way of the one thought I ought to be thinking.\n\nIn anger, I had bent my will my way. Now desperate, I couldn't bend it back.\n\nI look up at the TV.\n\nKevin lurches forward. He's stumbled. A clog slips off his foot and vanishes as it falls toward the red and blue flashing lights.\n\nDon't wear clogs if you're crawling out on a ledge! A good boot, a sensible running shoe with new treads. But not clogs. They aren't good footwear for ledges. For God's sake, Kev, you could...\n\nAnd he leaps. Not like a superhero. Just spirals away as we all watch our screens.\n\nBreaking news.\n\nUp to the minute.\n\nLive coverage.\n\nWe cut away from impact.\n\n# Part 3\n\n# **17**\n\nI think I'm broken.\n\nDefinitely need ice. Proper numbing.\n\nI'm numb most places except my toe\u2014that's throbbing, stubbed badly on a chair leg while struggling to find my pants and my shoes. I limp over and pick up the ice bucket but never make it out the door. Sitting on the edge of the bed with my ice bucket in my lap, I stare at the television. It's hard to think. They keep replaying the moment of Kevin's fall. For how long, I don't know, I sit there and watch, for at least the rest of that news show. Then I thumb through the two million other satellite options till I find another local news just coming on.\n\nAnd Kevin leaps again and again.\n\nNone of the reporters, the police, and the witnesses seem to know much about Kevin. They've found his name on his license in his wallet and, having nothing better, they show his horrible license picture on the screen, blown up and ghastly. It doesn't even look like Kevin. They've put together that his father was in this hospital. It seems his father passed away only a few hours before Kevin's jump. They interview a nurse who'd seen Kevin by his father's bedside in the past couple of weeks. This nurse says that the leaper had seemed a little down, understandably, but she'd never guessed he was suicidal.\n\n_Are we still on for tonight, James..._\n\nIt's Kevin's voice in my head. The guy asked to talk to me at least a hundred times, but I did nothing. I should have guessed.\n\n_Sometimes it's good to talk to someone when you feel out of your head..._\n\nAnd I'm definitely out of my head. No, that's not true. I've never been out of my own head. Like now, the voices in my head, like voice mail that won't erase, that won't shut off\u2014the whole thing, me locked up in my head, so much so that I failed Kevin.\n\n_I was thinking of quitting anyway. Let James have my spot..._\n\nIt should've been me.\n\nFinally, this news goes off too, and there are no more pictures of Kevin falling. Or at least not on the screen, the image playing by itself in my head, again and again. My foot's throbbing. I feel sick. I see that image of Kevin falling so clearly that I fear I might leap off the side of that hospital myself, might leap into the middle of the air, grabbing for Kevin, falling with him.\n\nI get up, limp, and pace. Try not to think.\n\nI open a window.\n\nHotels go to great lengths to make it discernibly difficult to open a window. I figure it's because they don't want people leaping out of them\u2014bad advertising. I have to use the complimentary shoehorn to snap off a piece of metal at the bottom of the window's slider rail. Then I rock the frame, use my full weight just to force it open enough to stick my head and arms out. But I need it open, I need air.\n\nThis Hometown Suites sits right up against the river. The wind has picked up considerably, making the river below look cold and choppy. Looking down the city skyline, I can see the hospital lights where Kevin jumped. The helicopter's gone, but I can still see flashes of red and blue lights a few streets over. Kevin's hospital's that close, only a few blocks away. I could have driven to the hospital.\n\nI need ice.\n\nI limp over to my stuff, put on a shirt, a belt, grab my car keys, phone, key card, and drop them into the ice bucket and leave my room. I'm taking the key card, because there'll be no more leaping. From now on, this James is limping back to hotel rooms the old-fashioned way. I swear it off. Right there in the hallway, barefoot and gimpy as I am, swear a promise\u2014no more leaping. If I can't use it to reach out to a guy on a ledge, what good am I?\n\nIn the elevator, I thumb the button for the lobby. It lights up yellow, bright like that ribbon at the church when I made my second leap, when I prayed.\n\n_Hey, God?_ I punch the button with the lit up _L_ a few more times. _Lord, until further notice, I'm out of this leaping business, unless you send someone to tell me what it's for, someone to put the pieces back together in my head, 'cause I can't do this. You interrupted once, and, well, that's been surprising, so just someone to believe me would be nice. Scratch that\u2014just someone who wants to believe. A sidekick, maybe. Anyway, this is my floor, God, so amen_.\n\nStepping out, I watch the elevator doors close on my prayer, watch the little numbers above the door take it upward and away.\n\n\"May we help you, Mr. Oates?\"\n\nI do a double take looking for Monica's father. He shouldn't be out on a night like this. He's not well.\n\nThe concierge is talking to me. \"Sir, you can call our desk if you need ice,\" he says, behind his counter. It's then I realize I'm barefoot in his lobby.\n\n\"What? No, I'm good, really,\" I say. \"Just taking my ice bucket for a walk.\"\n\n\"Do you need me to call for a cab, Mr. Oates?\"\n\n\"What?\"\n\n\"A cab, sir.\" He glances over at the security guard. \"Do we need to call anyone for you, Mr. Oates?\"\n\n\"Do you have ice?\"\n\n\"Yes sir. It's down this hallway, by the kitchen.\" He says this and points politely, but he's giving the security guard another glance.\n\n\"Thank you,\" I say and pat my ice bucket. \"We'll want some when we get back.\"\n\nI walk out the front of the hotel, limping. The wind, Kevin's wind, the one that blew his hair, his shirt, it's bringing rain now. A light, steady rain, but enough to soak you if you stand in it. I do just that. With my foot throbbing, I stand in the dark parking lot till I'm soaked.\n\nThe Volvo starts, and I set the ice bucket on the passenger seat. It seems I'm not only taking the ice bucket for a walk but also for a drive. It's raining, and my wipers have never been the best. Really, they're less wipers and more simultaneous smudgers.\n\nMy foot shoots with pain every time I use it. It's my left foot, my clutch foot, and it may well be broken. My foot, not the clutch, which is just spongy. I turn on the heater because I'm soaked to the bone, clothes, everything\u2014a pneumonia waiting to happen.\n\nOn the dash, I find the directions to Monica's, and I've got to talk to someone. Probably Monica won't believe me, about the leaping, the superpower, the God parts, so despite our progress earlier, I won't be mentioning that. Maybe, though, since she's lost a kid and she's gone through that, maybe she can understand this, this having loss inside you.\n\nI check my directions, pull out of the parking lot.\n\nThese driving conditions: rainfall, slick oily streets, simultaneous smudgers, darkness, Volvo, crappy Volvo, crappy '88 Volvo, bald '95 tires, driver in shock, having a nervous breakdown, knobless gearshift, no shoes, broken clutch foot, images of suicide, falling, falling, falling. Let's say I'm driving erratically and swerving occurs. Frankly, it's a wonder I stay on the road as well as I do.\n\nBut not well enough.\n\nIn the rearview mirror, I see a car behind me, one that gets way too close. Why would someone drive up on a car that's swerving?\n\nDo you climb frayed ropes?\n\nDo you get in the elevator marked Danger?\n\nDo you buy the discount cheese?\n\nI slow down a bit. Swerve to one side so he can pass, but he doesn't. Now I'm a little peeved. I swerve back in front of him and start to speed up. But I pull too far, swerve into the oncoming lane. Just a tiniest bit of overcorrection\u2014cut me a little slack, it feels like the clutch pedal is welded into my shinbone. But the oncoming car in that oncoming lane, he's understandably startled. He honks, swerves, and I swerve.\n\nWe avoid collision, no worries. The car behind me turns on its blue lights. Blips his siren a few times to get my attention.\n\n\"License and registration, please.\" The cop has his flashlight out, beams it in my eyes.\n\nI squint and turn away, but hand him my information. He doesn't seem pleased to be standing in the rain.\n\n\"Can I have you step out of your vehicle, sir?\"\n\nI'm already wet, so why not. I step out and follow the officer, limping to the back of my Volvo where I stand. He shines his flashlight in my eyes a second time.\n\n\"Have you had any alcoholic beverages this evening, sir?\"\n\nI should have just slurred and belched out, \"Loads to drink. Bottles and bottles.\" It would have saved everyone time, but I shake my head no.\n\n\"I'm going to ask you to take a field sobriety test. Is that all right with you, sir?\"\n\n\"Peaches,\" I say.\n\n\"Can I have you stand on one foot, sir?\"\n\n\"I already am.\"\n\nThe officer shines his flashlight toward my feet. He sees I'm holding my left foot a few inches off the ground like an injured dog.\n\n\"Would you like to get your shoes, sir?\"\n\n\"Are you going to wait here while I go back to the hotel?\"\n\n\"You're driving without shoes, sir?\"\n\nBefore answering, I try to remember that booklet I studied when I was sixteen to get my driving permit. I can't remember if shoeless driving is an illegal act, but I risk it.\n\n\"Yes.\"\n\n\"Can I have you touch your fingertips to your nose... first the left hand, then the right.\"\n\nI do this. This is not hard. I imagine someone drunk enough not able to locate his own nose probably shouldn't be driving.\n\n\"Now can I ask you to walk this white line here on the side of the road, one foot in front of the other?\"\n\n\"You can ask me, but it's not happening.\"\n\n\"Excuse me?\" Written down, I'm sure this looks polite, but don't you believe everything you read.\n\n\"I have a bum left foot.\"\n\n\"You're driving a manual in the rain without shoes and with a hurt foot?\"\n\n\"Changes your perspective on what is bad driving, huh?\"\n\n\"Are you refusing to complete the sobriety test, sir?\"\n\n\"I'm not refusing,\" I say. \"It's just that walking hurts right now, so the foot-in-front-of the-other deal's probably not going to work out the way either of us wants it to.\"\n\n\"What's the problem with your foot?\"\n\n\"I kicked a chair. Saw some bad TV. It might be broken.\"\n\nAt this point, I think I'm aces. The officer may think I'm just an idiot with a bad foot.\n\n\"Were you driving to the hospital?\"\n\nI turn on him, stare at him. How does he know about the hospital, about Kevin? I look at him like I want to shove him. Feigned aggression is not lost on a policeman. He takes a step back, flexes everything, before I realize he merely means about my foot.\n\n\"Oh... I might need someone to look at me.\"\n\nHe shines the light one more time in my eyes.\n\n\"Look,\" I say. \"I'll be glad to breathe in one of those thingies. I haven't had a sip to drink.\"\n\n\"Were you aware you were driving erratically?\"\n\n\"I'd just mentioned that to myself before you drove up. Yes, I was aware. I'm superaware right now. But it wasn't intentional erraticism. My foot hurts. I thought you wanted to pass, and I'm destined to overcorrect.\"\n\nThe officer thinks for a minute. He says, \"Sir, if you'd be more comfortable waiting in your vehicle, you can have a seat inside while I run your information. Just keep the engine off.\"\n\n\"Fine.\"\n\nHe watches me like he's afraid I'll make a run for it. Honestly, I'm thinking that I've used up today's Volvo starts, I'm going to have to ask for a jump, and I'm not in the mood. It occurs to me I could go back to my car, turn on the dome light, and leap out of there. Although, they'd have my Volvo then, and cops\u2014like others\u2014don't like you to vanish when they tell you to wait. That, and I've sworn off leaping. So I just cross my arms and lean against the back of the Volvo in the rain.\n\nThe officer gets back in his cruiser. There's no siren, but his blue lights are still flashing brightly. I can see his black silhouette in the car, talking on his police radio, calling in my license. Someone's definitely telling him something. I should have known when it took so long.\n\nThis time, when the officer climbs out of his car, he puts a hand on his service revolver.\n\n\"Sir, I need to ask you to put both hands on your vehicle.\"\n\n\"I paid my insurance\u2014\"\n\n\"Sir\u2014\"\n\n\"Are we going to turn the car over, because I'm definitely going to need help\u2014\"\n\n\"Sir!\" His voice sounds different now, more official. In fact, that's a bark. \"I repeat, I want you to put both hands on the back of your vehicle right now!\"\n\nI do it. What choice do I have? I see my arms in front of me, see my watch, the blue lights flashing off it, my eyes flutter and blur. I am aware of an intense desire to be somewhere else. For good this time.\n\n\"I've done something wrong...\" I say.\n\nHe comes up behind me with his flashlight raised. The officer's right hand grabs my right wrist. The handcuff metal is cold and slippery. Now his flashlight beam bounces off my watch face, and I focus on it. It's 11:48 p.m. Let the eyes blur, but he grabs my other hand, cuffs them both behind my back.\n\n\"It seems you've done a bit more than swerving lately,\" he says, as he pats me down.\n\n\"Probably. Remind me, would you?\"\n\n\"Shoplifting, attempted robbery, prank call to the emergency hotline, and now reckless driving.\"\n\nThe blue lights pulse on the Volvo's slick hood. I focus. I see the soft robe in a pile on the floor of my hotel room. I could have gone, right then, because who can stop me? Then Kevin's clog slips, and his foot falls toward the flashing lights in my mind again.\n\n\"I've done worse,\" I mutter.\n\nThe officer didn't expect that, and he doesn't seem to appreciate it. He pulls me hard by my arms toward his cruiser.\n\n\"Well, you'll have plenty of time to tell us about that at the station,\" he says.\n\nHe leads me limping to the backseat, opens the door, guides my head down into the car. Before he shuts it, he says, \"There's someone who wants to talk with you.\"\n\nI sigh. I've been summoned. Twisting my legs for more comfort, I try to ease my foot.\n\n\"Do you mind getting my backpack out of the backseat? It's got important stuff in it.\"\n\nHe stares at me in his rearview mirror.\n\nI cough.\n\n\"What's left of me is in that backpack, do you mind? Please.\"\n\nHe does it. He walks over to the Volvo in the rain. He takes the opportunity to shine his flashlight all around the inside of my car. He checks the glove box, under the seats. I feel a twinge of revenge knowing that whatever's under the seat can't be nice to find. He locks the door, slams it shut. But he's got the backpack. My recorder. My notebook. I'm going to want those.\n\nOnly a few things left.\n\nWe sit while he fills out papers and calls things in. My cough is pretty strong now, there's a distinct after-wheeze. I watch the wrecker pull up to tow my car, and I'm just thrilled with how this date night's turning out. We make a U-turn, follow the wrecker to the downtown station.\n\nFinally, he turns off those awful flashing blues.\n\n# **18**\n\nI'm alone, and it's cold in this cell.\n\nOkay, I'm not totally alone, if you count that guy. Even the passed-out drunk guy looks cold, although I'm sure he's far too sauced to actually feel anything. I'm sure of this because he's drooling copiously, and he never moved when they clanked open the metal door and shoved me in with him. He also did nothing when I took off his shoes and put them in the far corner. There was this putrid stink that I thought came from his shoes. Turns out, it was his feet. The stink is much worse now, with his shoes off, but there's no way I'm going back over there. I think it's on my hands now. But he looks cold, he keeps reaching for a pretend blanket, grumbling and pulling at comfort that's just not there.\n\nI'm definitely freezing, being soaked head to broken toe with rain. There's a puddle on the cement floor where I'm sitting. And the toe of my throbbing foot's the size of a baseball, or a racquetball, maybe. A squash ball, but a very purple one. Why do we compare injuries to sports equipment?\n\nIt's definitely broken.\n\nWhen they booked me, they took my picture and thumbprints and asked if I had any medical conditions they should know about. I mentioned my toe. Then I told them that I'd recently been having trouble with inadvertently beaming my body through space.\n\nI told them that's what happened at the grocery to cause the shoplifting. Told them I'd be more than happy to pay for the jerky. The lady filling out the paperwork, she enjoyed this information and read it back to me, making sure she'd got it just right. She'd done fine.\n\nThey also let me have my one phone call, so I called Father Chavez. I didn't speak to him, just the answering machine at the church. I kept it short, mentioned being in jail, and hung up. Then they trundled me back to this holding cell.\n\nHolding cell. That's perfect. Everything is put on cold, numb hold here, adding a special cement-gray jail perspective. The drunk holds his collar up around his ears and his knees up to his chest. I hold on to this notebook, trying to hold on to my grasp of things. I'm on hold for whatever's coming next, something heavy waiting to come down on me. No matter what I hold, no matter what happens next, Kevin doesn't come back.\n\nIt should've been me.\n\nThe cell door opens.\n\n\"Get up.\"\n\nIt's a new officer I haven't met before. I've been huddled in here so long they've probably changed shifts for the night. I stand.\n\n\"What time is it?\"\n\n\"Late,\" he says, surly. \"Come on.\"\n\n\"That guy smells really bad,\" I say. The drunk pulls at his clothes again.\n\n\"He always does. Come this way.\"\n\n\"You shouldn't arrest people who smell that bad,\" I say, but this guy's not in the mood for me.\n\n\"Where are we going?\"\n\n\"This way.\"\n\nMy humorless officer leads me limping down a gray hallway. All the hallways are gray here. They should pump in music, put up travel posters, and bring in small caf\u00e9 tables. He stops and pulls open a heavy door.\n\n\"In here.\" I hesitate, but hesitation is really more for those who have a choice.\n\n\"Hello, James.\"\n\nI'm surprised. \"Hello, Father.\"\n\nFather Chavez sits at a table by himself. There's a chair for me, a metal folding chair. It's cold looking, and my clothes are still soaked, but I actually feel hot.\n\nFather Chavez is in his civvies; no collar, no black, just a yellow button-down shirt, a rain slicker, and jeans. The room is bare, blank. White walls, gray table, gray chairs, gray, scuffed floor. They've done a lot with the place.\n\nI cough.\n\n\"How are you, James?\"\n\n\"In jail,\" I say.\n\n\"Are you all right?\"\n\n\"Not really.\"\n\n\"Would you like to talk about it?\"\n\nAt this, I bust a gut with a fit of coughing.\n\n\"You're soaked.\" He takes off his Windbreaker and hands it to me. It's light, not really a coat, just something to shed the rain, but I put it on. I shiver and sit down.\n\n\"Sorry to bother you. I wasn't sure who else to call.\" My bum's so cold from the metal chair, it starts another coughing fit.\n\nFather Chavez waits for me to finish. \"Do you feel all right?\"\n\n\"Feel? Me? I don't know. I feel guilty.\"\n\n\"Why?\"\n\n\"I'm in jail. It seems to fit.\"\n\n\"Do you want to talk? Why are you here, James?\" He was trying, God bless him. I'm freezing and burning up both, so I try to pull together the most sensible voice I can find.\n\n\"The truth?\"\n\n\"Absolutely,\" he says.\n\n\"Truth then... you ready? God.\"\n\nHe pauses, then asks, \"God?\"\n\n\"Yes, God put me here. Well, maybe he didn't drive, but he sent a car, if you know what I mean.\"\n\n\"No, James, I'm sorry. What do you mean?\"\n\n\"Look, you think I'm crazy\u2014\"\n\n\"No, not crazy, you're just\u2014\"\n\n\"Yes, you do. Or you should. I would... in fact, I do. Perfectly crazy. It's reasonable to think that my overwrought condition has led me to an insane tussle with the authorities. If we could unscramble this craziness, perhaps we'd placate the powers-that-be, right? No deal. It's the powers-that-be, or the Power-That-Is, that's done this. That I deserve to be locked up, totally my fault. But that I am locked up... God's fault.\"\n\nFather Chavez frowns, puzzled. \"Let's take this in pieces. Why do you feel you deserve this? Did you do anything wrong?\"\n\n\"It's what I couldn't do, Father, what I should've done. I'm better locked away.\"\n\n\"What couldn't you do?\"\n\n\"The impossible.\"\n\n\"What's impossible?\"\n\n\"Impossible to lock me away, I'm afraid.\"\n\nFather Chavez looks at his shoes, which are tennis shoes. I've never seen a priest in tennis shoes, and it takes something away.\n\n\"Father, how do you deal with it when God does a thing to you?\"\n\nMy priest leans back. \"All right, what do you think God has done to you, James?\"\n\n\"He's changed me, Father. Zap! I'm not who I was, but I didn't ask for it. I think maybe I just got too close to the God areas. I don't think I can undo it. And that's what landed me here.\"\n\n\"Are you referring to the story you told the police? Is that what you are talking about? They said... you claim to be able to... um... transport yourself.\"\n\n\"It's not a story. Leap, Father.\"\n\n\"Yes, that's what you told them. You claim to be able to leap through space.\"\n\n\"And?\"\n\n\"That's all they said. About that, and what seems to be a misunderstanding at a store. An elderly lady, some groceries.\"\n\n\"The old lady's nuts, but I stole the groceries.\"\n\n\"James, if you're ever short on money for food, you can come ask me anytime\u2014\"\n\n\"No, I got money and I didn't mean to steal them. I had them in a basket, and it happened. A leap, I leapt out to my car. I wanted to come back and pay, but the day went a little crazy.\"\n\n\"So... you... you really believe...\" He clears his throat. \"Leap through space\u2014you believe you can?\"\n\n\"Like a butterfly in a bad breeze, I'm all over the place. I thought I could control it, but then...\" I squint up at the air-conditioning vent. It's like you can see the cold air coming down to get you. \"I know what's happened to me, Father. I've repeated it a bunch of times. I was almost creamed by a Popsicle truck. If I borrow your watch, I could probably disappear right now. But I've sworn it off. There's no uncertainty in my mind as to whether or not I can do this, or whether I've gone mad or not. Madness is number two on my worry list. Unruly leaping superpower, it's number one. Because I'm no good at it. I don't want to be a hero, save the world. And I don't want to be changed.\"\n\n\"Tell me what happened.\" He hasn't left, so I'm hopeful.\n\n\"Okay, yesterday, or the day before, what day is it? That day I asked you about Aquaman?\"\n\n\"That was Friday afternoon. It's early Monday morning, James, still night.\"\n\nI pause and see blue lights flashing in my head. I close my eyes to rub them, but I see Kevin's flip-flop.\n\n\"James... where'd you go?\"\n\n\"Sorry, Father.\"\n\n\"I'm right here, go on.\"\n\n\"I discovered that if I saw one thing, focused hard and I lost myself, I could leap to the place of that one thing, where I most desired to be.\"\n\nHe waits, but I'm finished.\n\n\"I see,\" he says.\n\n\"It's not stunningly believable, is it?\"\n\n\"No, James, it isn't.\"\n\n\"Okay, crazy, right, but humor me for a second. Pretend, _imagine_ , I'm telling the truth. Let's say for a minute that my transspatial leaping ability is a fact. If God did zap me, and I'm different\u2014what do I do?\"\n\n\"I'm not sure.\"\n\n\"Isn't this where the faith comes in?\"\n\n\"Faith?\"\n\n\"Right. You got faith, got it in spades. Give me a chunk. Cut me a slice of what I'm supposed to do next.\"\n\n\"What do you mean?\"\n\n\"Where's the list? Faith, the five-easy-steps version? You're a priest, God's point man. Maybe God will tell you.\"\n\n\"Are we talking about faith or instructions?\"\n\n\"Exactly! Instructions, right.\"\n\n\"Faith isn't a set of instructions, James.\"\n\nI unzip his jacket because now I'm sweating in it.\n\nHe tries again. \"I'm not sure I understand.\"\n\n\"Let me put it this way. Let's say I know a guy who plans to leap off a building. I don't know if I could stop him, but I should be able to if what I'm telling you is true. If I'm any good at all, I'd be able to leap and save him. But I'm _not_ good! I'll try harder next time, I promise, if someone will just give me the instructions, a list to tell me what to do, step by step.\"\n\n\"Slow down... What's this about a guy who plans to leap off a building?\"\n\n\"Because some guys do. And I couldn't stop it.\"\n\n\"James\u2014\"\n\n\"God's changed me, and I didn't ask for it! I'm just so very blessed, I guess. So now I'm sending in my cereal box tops, asking for the God-list, and whatever I can do to get God off my back.\"\n\n\"Why would you want God to give you a list and then leave you alone?\"\n\n\"Wow... does anyone really warm to the idea that God sees them?\"\n\nFather Chavez takes off his glasses. I didn't know he had bad eyes; he must usually wear contacts. I've kept him from his sleep.\n\n\"It's just a list. Is that so hard? I'm not saying I'll become good with a list, but I'm utterly lost without one. I don't have to understand, just mark through the step I've done and go on to the next. That's what's gorgeous about lists. One-two-three. Like the Ten Commandments\u2014there's God doing something in a way I can understand. What a fabulous list! Not a difficult go-and-do list, but simply a stay-away-from-these-places list.\n\n\"Thou shalt refrain from these possible violations\u2014ten total. God bless Moses. Maybe it was Moses's idea. God can't expect people to simply desire good and strike out and do it right some morning. No sir, people aren't any good at that kind of living.\"\n\n\"James, I don't think that's faith. Faith is more like trust between people.\"\n\nI fidget uncomfortably.\n\n\"If you trust someone, you want them to see you and know everything about you. That's what faith is: trusting God better than any person. That trust gives you courage to keep going along _with_ God, despite evidence to the contrary\u2014only _because you_ trust and you've seen God himself doing the best things, and you want to stay near that.\"\n\n\"Whoa... I like you. You're a smart guy... But are you sure that's faith?\"\n\n\"I believe so, James.\"\n\n\"So because I want instructions, a list, making this whole God thing more manageable\u2014what is that?\"\n\n\"I'd bet trying to pin God down to a list feels frustrating.\"\n\n\"Right... feels closer to madness, actually.\"\n\n\"Well, I'm not sure I'd call it madness\u2014\"\n\n\"I may be going crazy, Father. The leaping's happening for God's reasons, and God won't identify those. I'm not doing well with the not-knowing aspect of this superpower, but it's either crazy or\u2014\"\n\nI stop and shiver, drop my head. Father Chavez reaches over and rocks my shoulder so I can feel he's really there.\n\n\"James,\" he says, \"you tell me you can leap through space, and I'm not sure what to do with that confession. But God will definitely always be God, and we never get a blueprint for what God is doing. It's too wide; there's no way we could understand all God does even if he did reveal it. We have to trust. We're made to.\"\n\nFather Chavez leans back, puts his glasses back on. \"Since you've been to Mass, we've gotten to know each other. I've enjoyed you and our talks, so listen... that's how it'll be. You and I will talk about what's going on with you, about God, and then we'll talk to God. Listen to what he has to say. He hasn't left you, James. We'll get through this. We'll trust God together.\"\n\n\"Thank you, Father. That's what I needed to hear.\"\n\n\"You're welcome.\"\n\nI should've just stopped talking, let Father Chavez think he's fixed it. And I do feel better, like a weight lifted or a decision made, but not for the reason he might think.\n\n\"Father, I just figured something out. What's driving me crazy.\"\n\n\"Yes.\"\n\n\"You said, this God bit... It's all about trust.\"\n\n\"Right.\"\n\n\"Yeah. I don't want that.\"\n\n\"I'm sorry?\"\n\n\"No, I'm sorry... for wasting God's time.\"\n\n\"What do you mean?\"\n\n\"I mean I don't want that God.\"\n\n\"James\u2014\"\n\n\"No, Father, I don't eat sushi, I don't bowl because of the borrowed shoes, and I don't trust. I want a divorce.\"\n\nHe pauses. \"I thought you and your wife\u2014\"\n\n\"From God. I want a divorce from God, right now.\"\n\n\"I'm a priest, James. I don't support divorce, certainly not from God.\"\n\n\"You got to get me out of this. There's got to be some magic la-la backward spell you can use to get God off me, okay?\"\n\n\"James\u2014\"\n\n\"I want a God divorce. We're irreconcilably different. He's a deity, for God's sake. God wants me to trust him when I have no idea what he might be up to? No, thanks, that's just not safe.\"\n\n\"James.\"\n\n\"No, come on, you're a priest. Sling a little water, say fancy Latin words. Can we do it right here, or do we need a church?\"\n\nThe air conditioner kicks on again, and I cough so hard my eyes bulge. I should never have unzipped that jacket. Father Chavez stands, walks over next to me. He places the palm of his hand on my forehead.\n\n\"That cough sounds nasty. Do you have a fever?\"\n\n\"I'd put money on raging fever\u2014scarlet, yellow, two-tone green.\"\n\n\"You're burning up. Tomorrow we'll get you to a doctor, okay? You'll be out of here soon. I'll go with you.\"\n\n\"Tell me,\" I say and grab his hand. \"Do you believe me?\"\n\nFather Chavez sits again, takes a deep breath. It's bad to make a good priest pause so long.\n\n\"I believe that something's wrong and you're torn to pieces. Truthfully, James, when you tell me you can actually beam yourself through space, I have to ask if you're okay. Are you? I know you've been through a lot and you're hurting. You're fevered. Is it stress? Forgive me for asking, but it's possible, right?\"\n\nI smile at his tennis shoes. \"At least you're not saying my leaping through space is not possible.\"\n\n\"Honestly, I'm having my issues with that, yes. Don't want you to think I'm a hundred percent on that one. I can't leap through space, and I've never seen it happen. But I'm trying to understand. I'm trying to answer your questions about faith. And with God, I stopped discounting the impossible years ago. I've seen stranger things be true. But if it is pain and stress, I'm here to help, okay?\"\n\n\"If it's true? If I can leap through space?\"\n\nHe looks worriedly at me. \"Then God has put you in a place I don't know anything about. So I'll listen and pray for you. Maybe together we can find out what's happening to you.\"\n\nI reach out, try to rock his shoulder like he did mine before he goes.\n\n\"Thank you, Father,\" I say.\n\n\"Thank you, James,\" he says. \"Maybe I need to believe more.\"\n\n\"You believe just fine.\"\n\n\"No. I haven't believed one new thing in years, James. I need to think about that.\"\n\nI hear footsteps coming down the hall, outside the door. Keys clatter, then fit and open the lock. The officer who led me in here holds the door open as Supercop enters. He comes into a room like he owns it.\n\n\"Sorry, sir, but your time's up,\" Detective Goss says.\n\n\"It's Father,\" I say.\n\n\"I don't care if he is your father, visiting time is over,\" he says to me. \"Sorry, sir, but you'll have to go.\"\n\n\"No,\" I say. \"He's a father. A priest.\"\n\n\"Oh,\" Goss says. He looks like he's trying to care, trying to find a problem with what he's done but can't.\n\n\"Father Chavez,\" says the priest, offering his hand.\n\n\"Detective Goss, nice to meet you.\"\n\nThey shake.\n\n\"Technically, I'm a captain, but since you're not a cop, I won't hold you to that,\" he says. \"Perhaps you'll forgive me if I don't call you Father?\"\n\n\"No problem at all, Captain. Have a good evening.\"\n\nDetective Captain Goss sets his coffee down on the little table in front of me.\n\n\"Captain, may I ask why James is being held?\"\n\n\"Reckless driving and refusal of a field sobriety test. We can hold him overnight for that\u2014not to mention a few other things.\"\n\n\"I see,\" Father Chavez says, turning to go. \"Oh, Captain, my friend here is freezing. He's been in a cold cell for hours, soaking wet, and he's probably caught something. Is there anything we could do about that?\"\n\nGoss looks as if he sees Father Chavez for the first time. \"I'll see what I can do.\"\n\nHe says this dismissing Father Chavez, but my priest is stubborn.\n\n\"Perhaps a warm cup of coffee?\" he says, waiting for an answer. \"A little mercy?\"\n\nGoss thinks his answer, a nasty one, I can see, but it's hard to be nasty with a priest in the room, even is he is in street clothes.\n\n\"Eddie, bring another coffee, please, after you escort Mr. Chavez out.\"\n\nThey leave and the door shuts.\n\nMaybe it's my fever, but I feel like I've leapt into a cop show.\n\n_\"At last we meet...\"_ and all that.\n\nGoss stalks, paces, leans, pulls at his belt, flexes his neck in neck places that shouldn't do that. No kidding, that's the way this guy acts, like now that we're alone, he's going to start laughing from deep in his chest or remove his head. He doesn't, and instead, Goss drops the file he's carrying smack onto the table, spilling a bit of the coffee down his Styrofoam cup. The folder has my name on the tab. I feel a cough coming on, one good deep bark from the chest, but I swallow it.\n\n\"Good morning,\" I say.\n\nSilence. He's trying to unnerve me, but I'm pure steel.\n\n\"Hey, you guys should think of chipping in for a coffee grinder. How many of you are there? Fifty, sixty? For a couple of bucks each, I can get you one heck of a grinder. You'd be amazed at the difference freshly ground beans will make. Look at this coffee. Look at the way it ran down the side of your cup. It's got no legs. Like a good wine, there ought to be legs. More like a thick wood stain, not this watery, sour, thin, insipid\u2014\"\n\n\"Having fun?\" he asks.\n\nI look up at him. He's still standing to make a tough impression. It's working.\n\n\"Is this funny to you?\"\n\n\"No sir,\" I say. \"This is not funny at all. Nothing even slightly funny about people drinking water steeped with pencil leads. I'm deadly serious about this stuff. I bet we could go to the kitchen right now and find calcium deposits lining your coffee maker, and if that doesn't make a difference to the bouquet\u2014\"\n\n\"I heard you were fired from that job at the coffee shop.\"\n\nHe looks at me. I look at him.\n\n\"You work there awhile?\"\n\nI nod.\n\n\"So that's all you do, sell coffee?\"\n\n\"I also drink it.\"\n\n\"Why were you fired, James?\"\n\n\"It seems I missed a shift, Officer.\"\n\n\"Why'd you miss it?\"\n\n\"I had more pressing matters.\"\n\n\"Such as...\"\n\nNow, I may be shivering, I may be sweating, I may be crazy and a lot of other things, but I'm sure I don't have to take this macho stuff.\n\n\"Why am I here, Officer?\"\n\nDetective Goss sits and then leans on the table. \"You're here because I want to talk to you.\"\n\nWe stay like this, him leaning, me shivering, and he's just staring at me, all _Law & Order_ gorgeous.\n\n\"What time is it?\" I ask.\n\nBut Goss never takes his eyes off me, he just stares, leans, like he's waiting for something. I try to look at his watch. I can't see the time but, boy, it's shiny, reflecting the bright lights in this room.\n\nThe door opens, and Officer Eddie comes in with my cup of coffee.\n\n\"Just put that in the trash for me, thanks,\" I say. I try wrinkling my nose cutely at the cop, but it sends me into a coughing fit. He leaves with the cup, slams the door. I hope he drinks it. Please, please, drink the cup\u2014\n\n\"Okay, James, why'd you steal the groceries?\"\n\n\"They only had one register open, and the line was so long...\"\n\n\"And the purse? Line too long at the Social Security office, so you just grab an old lady's money while picking up a few groceries?\"\n\n\"Old is rude, you mean elderly. And you walked up as I was giving the purse back to her.\"\n\n\"So you're a good guy then?\"\n\n\"No,\" I say quickly. \"I don't think I'm good.\"\n\n\"That makes two of us.\"\n\n\"You're not good either, Captain?\" I ask.\n\nI figure I've just earned another lean-and-stare part, but he's enjoying scratching the back of his neck.\n\n\"How about the phony call to 911?\"\n\n\"How about the dude in my apartment, sitting around in the dark?\"\n\n\"When I was there, I wondered, what's the tape for?\"\n\n\"Tape... right!\" I remember the recorder in my pocket. I take it out and set it on the table. \"Please talk clearly. My lawyer's got a bum ear.\"\n\nNow he's not quite so sure of everything. I'm glad, because I'm not sure of anything.\n\n\"I was referring to the duct tape.\"\n\n\"Love it. Big fan of duct tape.\"\n\nHe picks up my file and browses through it. \"Yeah, but why so much of it, all over your apartment? What are you trying to do there?\"\n\n\"My apartment's drafty. I've positively caught my cold of death.\"\n\nHe glances at me. \"Sorry.\" I sniffle. \"My death of cold.\"\n\n\"I'm just looking for answers, James. This doesn't have to be difficult.\"\n\nCould this be some Neanderthal version of kindness? Like he means it or like he's putting the spiked club behind his back and he was only accidentally threatening. But if he wants answers, he needs to get in line. I say nothing.\n\n\"Okay, then, difficult. Your wife's concerned that you've been on edge lately. I'm sorry... your ex-wife.\"\n\nMy back spasms, and I adjust my seat. Goss sees me wince.\n\n\"Is that why she left you?\" he asks. \"She did leave you, right?\"\n\nI thump the Styrofoam. \"You should drink this before it hardens back to shoe scum.\"\n\n\"It's amazing she put up with you for five years.\" He turns the file, like he's looking at a picture. \"She's good looking. Why was she with you?\"\n\nAnother coughing fit throws off my timing. \"I exercise regularly.\"\n\n\"How? By beaming through space?\"\n\n\"Most often I do that just sitting in a chair. It's not physically strenuous.\"\n\n\"You weren't sitting in a chair at the grocery.\"\n\n\"What can I say? They need more chairs. I can't tell you how many customer-comment cards I've filled out about the poor seating\u2014\"\n\n\"You're sitting now,\" he says. \"Go ahead. Go beaming out of here, right here and now, and I'll forget the whole thing.\"\n\nI think about it, that possibility\u2014a final leap.\n\nInstead, I say, \"I've already forgotten the whole thing. Was there a thing? Honestly, I can't make heads or tails out of any of this.\"\n\n\"Your wife's a lawyer. How's that work when you're divorced? Do you lose a soul mate and legal representation?\"\n\nI want to leap badly, his watch is just glaring, and no, it's not for any good. It's an evil, evil desire. I want to leap into his supercop supermobile and puke on the seats, but I still see Kevin falling.\n\n\"All right, James. What do you know about Chapman Collins?\"\n\n\"Who would that be?\"\n\n\"Little kid, disappeared two weeks ago from his mother.\"\n\n\"I've misplaced a lot of stuff, but never a kid.\"\n\n\"He was kidnapped. We found him last night, tipped off by a cell-phone call.\"\n\n\"Great. Those cups with missing-kid pictures pay off occasionally.\"\n\n\"How'd you know his picture was posted on a cup?\"\n\n\"It was?\"\n\n\"Yeah, like the one on your kitchen counter.\"\n\n\"But the kid's fine, you say? Back with his mother?\"\n\n\"What do you care?\"\n\n\"I don't. You brought it up.\"\n\n\"Kid's fine, back with his mom. We staked out the apartment and arrested his deadbeat father. He's just down the hall, if you'd like to visit.\"\n\n\"Is there an espresso machine on the way?\"\n\n\"Strange, though... the kid told us that someone with your height, your build, and your description miraculously appeared in the locked room with him.\"\n\n\"I get that all the time. I'm criminally average, I guess.\"\n\n\"Chapman said he used that guy's phone to call us. Called you Jimmy?\"\n\n\"Wow, I haven't been called that since\u2014\"\n\n\"Last night?\"\n\n\"Let's see, last night... I can't remember that far back.\"\n\n\"What if I said I traced the call, James?\"\n\n\"I'd say you're bluffing, Chief.\"\n\n\"And what if I subpoena your cell-phone records?\"\n\n\"On what grounds\u2014I'm a heavy duct-tape user?\"\n\n\"The kid said you vanished as soon as you saw the police arrive. He said you asked him not to tell the police about your being there.\"\n\nI laugh, which is too close to the coughing reflex, so there's coughing.\n\n\"Here, let me show you something,\" I say. \"You'll be interested in this.\"\n\nI run the recorder back. It takes a minute because I have to start and stop it. I hear my own voice dictating to myself, and it's like coming unglued from my own body. I cough. I know I'm close to where Goss's voice in my apartment should be, but I can't find it.\n\n\"You wanted to play me something?\" Goss crosses his arms.\n\n\"Hold on.\"\n\n\"Let's get back to the kid.\"\n\n\"You were in my apartment, and I was there. I leapt into the shower. I recorded the whole thing.\"\n\nThat makes Goss uncross his arms.\n\nI give up on the recorder and reenact it for him with my best Goss impersonation:\n\n\"That pay phone right there, Arnie?\"\n\n\"He told the dispatch there was a burglar in this apartment.\"\n\nI use a slightly smarter-sounding voice for Arnie. \"He may be losing his mind, Bill, but what he's got left works better than yours.\"\n\n\"Shut up, Arnie, it's personal now!\"\n\nI stop my performance, although I'm quite good with the multiple voices.\n\nGoss shifts his weight back and forth, a little speechless.\n\nI run the recorder ahead to the present, start recording again. \"Any comments, commander?\"\n\nGoss looks at the recorder suspiciously, confused, teaming with questions. \"You say you heard all that?\"\n\n\"No comment, perfect. Guilty cops always go for no comment.\" I tap the recorder. \"Remember, speak up.\"\n\n\"You were definitely not in your apartment\u2014\"\n\n\"Not when you searched it. Not then.\"\n\nHe seems worried. \"What's to keep me from taking that recorder right now and losing it?\"\n\nI smile and say, \"Because I made a file, Cap'n. Loaded it up onto the Internet. E-mailed it to my lawyer. E-mailed one to my ex-wife lawyer and one to my other lawyer and two more lawyers I've never met. Heck, I e-mailed one to the pope. But if I'm smart enough to have recorded your conversation, I'm smart enough not to show it to you unless I've passed it around. Your move, Sarge.\"\n\nOf course, I'm shaking. I haven't done a thing, haven't touch a computer in months. I've got nothing, nothing but pneumonia. And the fearful shaking is indistinguishable from the feverish shaking.\n\nI don't know why I want to make him so mad. And I do, I wish him ulcers and gas and insomnia and unquenchable anger at me. Since the ulcers haven't taken him yet, Goss just shakes his head and picks up my file again.\n\n\"Let's see. Swiping groceries, a defenseless old lady's purse, making mock phone calls to 911, accessory to kidnapping, lost your job, fired from it, divorced, pretty lady left you. Hey, tell me something, do you know this guy?\"\n\nHe throws a picture on the table right in front of me. It's a blown-up, black-and-white, faxed copy of a driver's license picture, but I recognize it. It's the same picture the news used of Kevin.\n\nI feel dizzy again. \"Can I go now?\"\n\n\"I asked if you know him.\"\n\n\"I want a lawyer.\"\n\n\"I believe your lawyer's in bed with another man. Should we wake them? How about the guy in the picture?\" he repeats.\n\n\"Yeah, I knew him.\" I shove the picture back at him.\n\n\"You _knew_ him? So you know he's dead then?\"\n\n\"I watch the news like the rest of America.\"\n\n\"What can you tell me about that?\"\n\n\"About what?\"\n\n\"Did you have anything to do with that?\"\n\n\"I have the right to remain silent\u2014\"\n\n\"Now you don't feel like talking, is that it?\"\n\n\"If I give up that right\u2014hey, Captain, just jump on in here any time.\"\n\n\"Were you two up to something, and he backed out?\"\n\n\"Anything I say can be used against me...\"\n\n\"Did you have a plan?\"\n\n\"... in a court of law. Help me here, I don't know much more, Cap'n...\"\n\n\"What's the duct tape for? Was he going to help, but something went wrong?\"\n\n\"... I have the right to call you a monster, a moron, a misogynist, and other derogatory words that begin with the letter _m..._ \"\n\n\"James, I thought you said you could leap through space?\"\n\n\"... misfit, monkey-loving miscreant, malevolent malefactor.\"\n\n\"So maybe you leapt up to the top of the hospital and pushed him?\"\n\nI can't think of any more words that start with _m_.\n\n\"Did you do that, James? Did you push him off?\"\n\n\"I'd like my coffee now.\"\n\n\"If you were watching the news and you can jump through space, why didn't you do anything?\"\n\nI can't think of words. I cough again.\n\n\"What's wrong? Did I touch a nerve?\"\n\nI stop coughing, swallow it down.\n\nGoss waits. \"James. Did you have anything to do with his jump?\"\n\nI look my interrogator straight in the eyes. It's the best lean and stare I can do from a sitting position.\n\n\"No,\" I say.\n\n\"That the truth?\" he asks.\n\n\"The truth,\" I say, \"is that I had absolutely nothing to do with him.\"\n\n\"I think maybe you did.\"\n\n\"If I had... maybe he wouldn't be dead right now. Is he somewhere in this building too?\"\n\nI couldn't say for sure, but the way Goss blinked at me, I thought Kevin's body just might be here.\n\nGoss cracked his knuckles. \"You're a hard man to believe, James.\"\n\n\"Yeah, well, maybe believing is just hard.\"\n\n\"Frankly, I don't believe a word you've told me.\" He shuts the folder and takes a sip of his foul brew. \"But the thing is... thing is... I saw you.\"\n\nI look up at my inspector.\n\n\"That's what's bothering me. I've concluded you're an idiot and you don't have the stomach for crime. Not yet. But in the grocery, when you stepped around that corner, I saw you... You vanished. You did and I saw it.\"\n\nHe says this, waits. It's my turn to answer, but this is one of those conversations you can't rehearse ahead of time, so I'm lost. I just shrug politely.\n\n\"How'd you do it, James?\"\n\nFor a moment, the way Goss looks at me with his eyebrows raised and his face quiet, for just that moment, Goss reminds me of Chapman. It's in a good way, like he's really waiting for me, wide-eyed and waiting for whatever miracle the superhero might pull off. He wants me to get him out of there, like he's been kidnapped and locked in a small room for a long time, like maybe he wants to believe.\n\nI say, \"You'd never believe me if I told you.\"\n\nGoss stretches his back. \"Try me.\"\n\n_Why not?_ I think. _I'm pretty low on the dignity scale about now_. \"A couple of days ago, something happened. God changed me and made me this... a leaper. I mean, I can do precisely what I said I can\u2014leap through space. At least, most of the time. God wants me to use it for good, and I'm not good. Good is much trickier than it looks on television, so I'm giving up.\"\n\nGoss wants to laugh, wants to ridicule, pull my ears, and call me a moron, but something stops him, and it isn't me or how well he's been raised. No, he's on the edge just like me.\n\n\"Don't give it up yet, James,\" he says, \"Let's see ya.\"\n\nHe holds out his hands, as if saying, we have all the time in the world. I see his watch on his wrist, a glint off the bulb above.\n\nI sigh. \"It doesn't work that way.\"\n\n\"No, I'll drive you home myself, buy you dinner, set you up with a six-pack. All you have to do is leap outside this room and into that hall.\" He points at the door.\n\nI stare at Goss, and I'm suddenly tired, my arms feel weak, and it's weird to say this, but I feel like Jesus must have felt when people kept asking him for miracles and he just wanted to sit down or go to sleep. I ask, \"Man, what do you care?\"\n\nGoss slaps his hands down on his thighs and juts his head forward.\n\n\"Care?\" he laughs, but it's an old, worn-out laugh. \"You tell me you can appear and disappear, that you're some kind of hero. You tell me God made you this way, and well, I'd like to see it. Show me! I'd love to believe.\n\n\"Do you know how good it would feel to think that God was back, that he was doing something\u2014anything? I can't tell you how relieved I'd be if God were back. God, _the_ God, doing miracles again. Even if it's just you, a hero on the prowl for good. Wonderful\u2014I'm tired of doing it all alone. Come on, James, you've almost persuaded me. But can you show me? Please, James, leap us both out of here.\"\n\nMy eyelids sag and blink. I'm exhausted, and I'm beginning to feel like breathing is too much for me to handle. Is this guy for real or not? What if I showed him\u2014would I be free\u2014or a whole different kind of trapped? I press my thumbs on the tabletop and shake my head.\n\n\"You said you saw it in the grocery. If you've seen it once, believe that,\" I say.\n\nGoss looks at me, and I can't tell if he wants to beat me till I show him. Or thank me for confirming his doubt, all the bad things he already assumes, and for letting him off the belief hook.\n\n\"All right,\" he says, standing. \"Despite my recommendations, James, the grocery store won't be pressing charges. They would prefer you shopped elsewhere, but no formal charges. Be that as may be, I do want you to know something.\"\n\nHe pauses for effect, but I'm not asking. I can barely hold my head steady.\n\n\"I brought you here tonight because I can, and I wanted you to know that. And I'm watching you\u2014know that too\u2014from here on out.\n\n\"Be an idiot,\" he continues, \"but don't get too grand in your thinking or start thinking you're above the law. I'm keeping my eye on you, James. Do we understand each other?\"\n\n\"Watching me. I got it... from here on out.\" It's my turn to blink and wait.\n\n\"You can go,\" he says. \"Get up.\"\n\nHe collects my file, checks his watch, and he's done with me\u2014on to thinking of driving home, paperwork, whatever's next for his night.\n\n\"The door's been open this whole time?\" I ask.\n\nGoss shrugs. \"Try it and see.\"\n\nI stand and cough, limp slowly around the table, reach for the door. It opens. It was unlocked and something about that just cracks me up.\n\n\"What did you say?\" Goss asks.\n\nI didn't realize I was mumbling, so I have to think about it. I mumble again and smile. Goss still stares at me. He wants his answer, and I should just go, but why not tell him?\n\nI tried to give up the gift, but he's watching. I'm in whether I want in or not. There's only one way to stop the gift now.\n\nSo I turn and say out loud, \"I said, 'Good luck.' \"\n\n\"With what?\" Goss asks.\n\n\"Keeping your eye on me.\"\n\n# **19**\n\nWhat's the number here?\"\n\nI ask the police guy checking me out of the station. He gives me back my stuff\u2014my wallet, my belt, my watch. I should've checked to see if they'd swiped my money, all that cash I'd just taken out of an ATM, but I was coughing too much, too consumptively, to worry properly about my wallet.\n\nMy coughing feels like I'm going to cough out something that's never supposed to be out, things I didn't know people kept inside, things hacking their way free. The guy makes me sign a few papers. The papers say that I agree that they did, indeed, give me back all my stuff, and that I checked it and everything was there. Clearly, that's when I should have checked my money. Some worrier I turn out to be.\n\nI sign. I'm sure there's stuff missing, but there's no changing what's already done, no going back.\n\n\"This is the phone number for the station, right?\"\n\nHe grunts. He's a real heavy grunter, this guy, but it's a communicative grunt.\n\n\"You should see someone about that grunt,\" I say, then I cough so wildly I have to hold the wall until I'm done.\n\nHe points me to a side door, with a pitying, semipolite grunt. Bless his heart, he probably can't speak. He lives in his sad grunt world with other grunters. I slide my pocket change on the counter to him. Everything but the quarters. I need my quarters.\n\nThis door leads out to a fenced-in parking lot, barbed wire coiled along the top of the chain link. I have a stub of paper that the grunter also gave me. It reads C-21, the parking space that contains my impounded Volvo. It's still raining, and it takes a while to find the spot, as there's all these cars parked on top of the numbers. I shiver uncontrollably, and I make a mental note: _worry desperately about health_.\n\nWhen I find the Volvo, I kiss its dirty windshield, then spit. It's rained on my car for hours, but my windshield remains covered with dust and crud. I look around, search for something, any object on the ground will do. A nail, a pen, a beer can, anything. Something shiny. I see this candy bar wrapper, and that will do nicely, mainly the foil, although the red wrapper shines too. When I bend down to snag it, I realize I'm still shoeless. My feet look cold, a little blue. I zip up Father Chavez's Windbreaker.\n\nI unfurl the candy wrapper, twist it, poke a hole through it and cram it halfway down the Volvo's antenna, take a step back, focus on what the wrapper looks like under the parking-lot lights. Nice. Bright red wrapper, foil flicking in the rain, midway down my Volvo's antenna. Very memorable, this is. I take off my glasses, wipe them free of rain, record a good image of that desire in my brain: wrapper, door, rain.\n\nThen I limp out of the parking lot.\n\n\"Hey, you can't leave your car here. This ain't no parking garage.\"\n\nThe guard doesn't even come out of his dry, little booth to say this. He just yells into his microphone.\n\n\"Hey, Mack,\" he yells, sticking his face out the shack's window. But I'm not Mack, and I balance myself, taking a deep breath to crouch under the parking barrier. It's one of those up-and-down mechanical, orange-striped arms, that the guy in the booth presses a button and it raises.\n\n\"Pipe down!\" I say. I'm soaked again and miserably cold, but giddy that I don't fall over while standing back up. \"I'm not leaving it, I'm coming right back. I got to make a call.\"\n\n\"Yeah, well, you better not.\"\n\n\"What are you going to do, arrest me?\"\n\nHe slams his little window shut.\n\nA pay phone. I remember seeing one out front of the station when they brought me in this place. Not a booth, of course, they stopped making phone booths.\n\nWhy did they stop making booths for phones? I liked the booths, and I bet the phones preferred the booths. I bet Superman misses those spacious, seventies-style booths. This is a phone stand. A perch for a phone.\n\nAnd I'm smart to have asked for the station's number, because someone's ripped off the phone book. The pitiful chain just dangles there. I've never seen a chain attached to its phone book except in movies, so what's it really for? If someone really wants to swipe a phone book, that's a flimsy restraint. Who steals the phone book right in front of the police station?\n\nI dial the number for the station.\n\n\"Detective Goss, please.\"\n\n\"May I say who's calling?\" ask the operator.\n\n\"No, just tell him a guy who de-naps kidnapped kids. He'll know who it is.\"\n\nShe puts me on hold. The police switchboard pumps Five for Fighting at you when you're on hold. It's very surreal, and I wonder if she's just filing her nails. With a fever, Five for Fighting starts to sound like\u2014\n\nThe music suddenly stops with a click.\n\n\"This is Goss.\"\n\n\"Hey, Cap'n.\"\n\n\"Who is this?\"\n\n\"I'm no hero,\" I say.\n\n\"James? Look, you little punk\u2014\"\n\n\"Nope, that's not my name anymore. And I've given up being a punk, too. Just Leaper from now on, thanks. As in one who leaps.\"\n\n\"James, if you don't think I'll throw you right back in the lockup\u2014\"\n\n\"Oh, you'll never touch me,\" I say.\n\n\"I know where you live. I know the hotel\u2014\"\n\n\"Hey, Captain, does your gray police station have a window?\"\n\n\"What?\"\n\n\"I know it has a window, okay. I'm looking up at a window. Second floor. There's one on the third, too. Tall, skinny windows, built like they didn't really want people looking out. You know those windows?\"\n\n\"What's your point?\"\n\n\"I'm calling from the pay phone outside. Do me a favor, Sarge, and go look out one of those skinny windows.\"\n\nThen I slap two fingers down on the hanging-up lever. That's what's classic about pay phones. Booths and hanging-up levers. I could get used to phone booths. Of course, you need a whole different set of superpowers for taking on and off boots in a booth.\n\nI wait, staring up at the window. I get ready. I angle a bit so the street light falls on my good side, the side I wear my watch on. It kills my foot to pose like this, but it's worth it.\n\nStanding in the rain, I pull up my sleeves like a magician, stretch my bare arms out, tilt my head onto the shoulder of the left arm, the arm with my watch, and twist my wrist around. With my right hand, I hold the phone receiver out as far it will go.\n\nAnd I wait for him.\n\nI see him step into the little window on high. He's looking down on me, and he seems cautious, unsure of himself, unsure of me. But he's really there.\n\n_Wait for it_ , I think. _Let him see you. Let him see you good_. I badly want him to see me good.\n\nHe does. He's there... He sees.\n\nRed wrapper, red wrapper, foil flicking in the rain...\n\nA minute later, when I drive past the guard booth, Goss is outside by the phone stand. Now he's getting soaked. He must've run down the stairs. He looks winded, flabbergasted, and he stands in the rain, watching the dangling phone receiver swing.\n\nI drive right past him. Who's crazy now?\n\n# **20**\n\nThe ice bucket.\n\nThere it is, sitting on the seat next to me. Hometown Suites. The name of my hotel is written in pretty cursive on the side, even the address. _They must really want these back_ , I think. That's how Goss knows about my hotel.\n\nI can't go back to that hotel. I shouldn't sleep there. Will I ever sleep? But no visitors in the night for me, thanks. (I wouldn't trust this night not to end that way.) Or is it morning? My stuff's there, but I can always leap in and leap out from a safe location. I need a place to hide, someplace new and safe.\n\nI have to concentrate just to drive. More specifically, the wiggly raindrops nudging across my windshield are far more interesting than the road. Also, I'm burning up. My temples are beating, my neck is clammy, and my clothes are freshly soaked. Not sure if I should turn on the air conditioning or the heater, but certainly should not be driving. I slow down, but pressing the clutch sends remarkable pain up through my foot into my leg. Yeah, I shouldn't be driving. As bad as this, I've got to keep going until I know where I'm going.\n\nOddly, I'm not really worried anymore. There are things to be afraid of, but I don't _feel_ afraid. I _see_ the fearful things clearly as if they are right in front of me, and they remain ever-so-fearful, but I'm not afraid. Do you see?\n\nI don't really feel my choices are going to keep me safe anymore. This evening\u2014everything about it\u2014it's moving too fast _and_ too slowly, the way everything moves feels too certain _and_ unknown, too wildly inevitable to do anything about. I can make my choices, but choices are made inside a bigger thing that I can't move an inch one way or the other. Strange as it is to say, that feels hopeful. I'm only responsible for the really small choices, the ones with me in them.\n\nAnd that feels free.\n\nOut of the ice bucket, I grab my cell phone, mash the speed dial.\n\nIt rings forever.\n\n\"Hello?\" Her voice is thick, husky, and addled. It's hard for me to speak. She says, \"James? Is that you?\"\n\nI'm still silent. The wipers don't even smudge now. How do wipers completely miss rain on a windshield? If they do, technically, are they still wipers?\n\n\"James... that _is_ you. Don't just breathe, please!\"\n\n\"Hey, Meg...\"\n\n\"What time is it?\"\n\n\"Hey, let me ask you a question\u2014\"\n\n\"What time is it? For the love of... I have to be at work in\u2014\"\n\n\"Quick question and I'll be gone.\"\n\n\"Call me tomorrow. Try to sleep, please!\"\n\n\"Meg, did I ever punish you with my lack of trust?\"\n\n\"Dear God\u2014\"\n\n\"Seriously, Meg, I'm asking here.\"\n\n\"You're kidding, right?\"\n\n\"Honey, I may never be this serious again.\"\n\n\"Okay, sure, yes, James, you did, since you're asking.\"\n\n\"What does that do to a person?\"\n\n\"James, I can't\u2014\"\n\n\"Please, Meg.\"\n\n\"It robs them, okay? It robs them of being with you.\"\n\nI hear her sigh\u2014a sad, tired sigh.\n\nThen she says, \"James, I'm not doing this.\"\n\nThere's silence.\n\n\"I'm sorry, Meg.\"\n\nShe waits.\n\nI say, \"I guess I wanted to say sorry, and... if I had to go through all of that... I'm glad I went through it with you.\"\n\nMore silence.\n\n\"Don't mean to bug you,\" I say. \"I'll go.\"\n\n\"James, wait\u2014\"\n\n\"Good-bye, Meg.\"\n\nI hang up. And as I drive, I'm not angry or sad, and I'm not in need. I think, _How long till morning?_\n\nI'm free to make the little choices. I drop the cell phone in the ice bucket, roll down the window, and toss both out into the rain. In the mirror, I watch them bouncing, flipping, shattering across the wet asphalt.\n\nSlowly, I smile. I know where I'm going.\n\n# **21**\n\nFor the record, I do not recommend leaping inside a dark church alone.\n\nShadows are everywhere. And I mean creepy ones.\n\nYou wouldn't think a long, cold room, _already_ dark, could have so many shadows, but there's dark and darker. Moonlight through stained glass makes you think you can see, but there are too many corners, too many places under pews, too many columns that are just dark and darker on the side away from the moon.\n\nLet's not forget what moonlight does to stained-glass people: they move, they jiggle, they stand up straighter, their doves rotate. You don't see the faces, just postures, hands in weird gestures, and I swear, they change when you look away. Trust me on this. Cold, dark shapes watching, and I'm limping down a long, unlit aisle leading up to the dark platform where God is. Dark, empty pews everywhere. I hope they're empty. Every few feet I pass through the gaze of another staring saint\u2014pale moon blue reaching through frozen faces.\n\n\"Anybody home?\"\n\nI call out in the off-chance Father Chavez came back to the church, but my voice squeaks and breaks. I cough.\n\n\"Didn't imagine there'd be much hope of...\"\n\nA noise, from behind me, back at the door. I crouch in the aisle. I don't know, but I don't like it. Not the crouching, the noise. I know I didn't leave the door open because I leapt in here. But maybe church doors don't lock? Maybe sanctuaries works like convenience stores, 24\/7...\n\n\"Who's there?\"\n\nA terrible, awful, wrenching, dumb question to say aloud to yourself in an empty, dark room. What's a crouched man's crazed mind going to do but scroll through all the bad possible answers to that stupid question.\n\nAnother noise.\n\nI know I'm not making it up, but this time from the front. I lay perfectly still. Okay, relatively motionless, considering the various forms of shaking. Now I need to know about the doors\u2014they have to lock, right? Haven't I seen Father Chavez lock up before, or open up, or swing a big, iron loop of rusty keys and yell, \"Last call\"?\n\nI crawl back down the aisle, and my bare feet are a plus now. Like a nervous, lame, damp crab, I shimmy toward the foyer. I stop at the entryway and lean my back against the stone holy-water basin. Must I dab and cross? I don't want to risk it. I reach up, shoot my hand in and dab, pull back, quick and safe, but I hear another scuttle.\n\nThen I see it.\n\nDown the center aisle, in one of those pools of moonlight blue, I see a thin yellow ribbon. The yellow ribbon flips and jerks then in that blue pool. Nothing, just lying there brightly. Yellow isn't yellow under blue. Then it flips again. Am I mad? I lay on the floor of a church, in fear of what may be a three-cent piece of fabric, and I cannot fathom what to do except watch.\n\nA mouse tugs it out of sight.\n\nCrouched in fear, my superhero-self slumped limp, like old whipped cream sagging on top of a hot, hot chocolate. I think, _What hero is afraid of mice? What human?_ But I'm only afraid of mice with yellow ribbons. Only one person knows about that yellow ribbon.\n\nI look at my fingers. The dab of water has dried up before I used it.\n\nInto the sanctuary, I call out, \"You're here, aren't you?\"\n\nWhen I stand up, I cough so hard I swear that stained-glass guy with the axe scowls. He stares at me with his eyebrows raised. Maybe he's holding his breath, afraid to catch what I've got.\n\n\"I know you're in here...\"\n\nI wait, but no answer.\n\n\"It's not really fair, you know, your being everywhere, all the time.\"\n\nNothing happens.\n\n\"Look, I came to say... this isn't going to work out.\"\n\nAs I reach the front pew, I cough, and it doubles me over, my eyes watering. I think I've sprained a muscle in my neck. I cough what tastes like blood into my mouth\u2014the taste of the blood on the tongue.\n\n\"It's like this: I'm sick, I'm wet, I'm tired, I'm cold, I'm feverish, I'm confused, I'm hot, I'm sick, I'm tired, I'm hungry, I'm afraid, I'm angry, and also I am sick. And I'm done, okay? Not good enough. You want from me what I can't give, and I want out.\"\n\nI squint across the dark church, looking for lightning or a pillar of smoke or Morgan Freeman.\n\n\"Are you ready? Okay...\"\n\nI clear my throat for effect, but it leads to another coughing bout. I let myself down on the front pew, way up in the God area.\n\n\"I, the aggrieved, being of sound mind and\u2014\"\n\nI listen to the dark.\n\n\"Right, skip that. I do hereby nullify all connection between God and this neurotic party, and further do solemnly vow to stay out of your way henceforth, and until further notification, on the condition that said God agrees to do the same and without consideration of upbringing, parental expectation, obligation, guilt, hellfire, and brimstone. In other words, all bets off. Eternal bets, all off.\"\n\nI look up at the crucifix.\n\n\"Please don't interrupt me now. That's how we got in this jam.\"\n\nSitting, I feel dizzy, so I grab the pew seat with both hands.\n\n\"Let's just say we missed each other, okay? Two ships in the dark night, and call it a day. Irreconcilably different, you and I. You want trust; I can't trust. We're just not going to work out. No hard feelings... In fact, what do you lose, really? Not a thing, as I see it. So thanks. I'm really sorry, and I hope you don't take this the wrong way, but we're done. Really sorry.\"\n\nThere's enough moon to make the crucifix blue. He looks cold too.\n\n\"Don't be like that. Why am I leaving?\"\n\nAlthough talking to God, I hold up fingers and count them for effect.\n\n\"First, I am not good enough to do what you want. Second, I'm not smart enough to understand what you need from me. Third, I am not quiet enough to ever listen to you. And number four, I do not want this. I never asked to be wanted, not like this, not this way. No disrespect, and I certainly don't hope to hack off the Almighty with some display of ingratitude, but hey, I can't live up to this or even imagine how to try. So we're done, we're over.\"\n\nThe Christ on the crucifix, he doesn't move.\n\n\"It just wasn't what I expected. I can't live this way, wondering what you'll do next. With my kind of issues, I don't trust myself to wake up in the morning and know who I am, let alone worry about how you're going to show up and\u2014\"\n\nA cloud must have passed the moon, because I lose his face, and the blue, the light, the pale saints behind me fade.\n\n\"Besides, this whole deal's been a fraud. This screw has been pooched from the get-go. I had no idea you even _wanted_ trust from me. You really should mention that way up front. Like knowing you would actually matter... like you'd actually speak? I had no intention of trusting you like that.\n\n\"The truth is, I hoped to be on decent, comfortable terms with your Almighty-ness. It makes sense, like giving the cops free coffee in the shop. You want the guys with guns thinking happy thoughts when they see you, you know. If they pull you over, you want them to taste the free coffees while getting out their ticket pad...\"\n\nI cough again, taste more blood.\n\n\"See, God, I had no idea you meant to infringe upon my self-absorption, not this slicingly, not all at once. What kind of deal is that? You are absolutely the least-safe God I know! I don't open the door for the UPS guy even when I'm expecting a package. What possible universe allows for free access by God anytime, anywhere, in all things? Here's your house key, God; there's lunchmeat in the fridge. But no, not even that. You don't come over unless I'm home\u2014that's when you come barging in. We Americans value our privacy, God. A little holy water on weekends, but let's not overdo this, shall we? What do you say? Let's split up. This is totally unsafe.\"\n\nThe hanging blue Christ doesn't even lift his head.\n\n\"Don't blame Father Chavez; he did what he could. In fact, I can't imagine anyone else showing up for me. But if faith is what he says, if faith is the courage to keep trusting, then I don't want to believe. Trust is crazy. You are a risky, risky God. I didn't sign up to be Amazing-Trust Boy, and I did not sign up for Danger God. I do not like surprises\u2014birthday, happy little, tuna, or otherwise\u2014and you're flatly not safe...\"\n\nThe cloud's gone. I see his bowed head, his sad eyes, his torn beard in sharp detail again.\n\n\"I wasn't made for it, I guess. Every time I've come close to a genuine attempt, every little thing howls inside like a\u2014\"\n\nWhen I blink, like blood pumping, the crucifix pounds my sight. So I close my eyes tight.\n\n\"Here it is: I've come to ask you to get out. I mean out of my life. This place, of course... kinda yours. No, I'll move out. I'm used to that. Let me sleep here tonight, and I swear I'll be gone by the morning. You can keep everything. But do you hear me? It's finished.\"\n\nMy ears are ringing, my tongue is parched, my spine shivers. Everything inside my head burns.\n\n\"I'm glad we had this chat, and when I wake up, I hope this little gift of yours will be gone.\" I gather my legs up onto the pew, take a hymnal for a pillow.\n\n\"Good doing business with you, God. I'm glad we could settle this amicably. You're all right in the end. I just can't live the way you live.\"\n\nI zip the Windbreaker up tighter over my wet clothes.\n\n\"Good-bye.\"\n\nThe moonlight shadow of the crucifix seen straight on appears in profile\u2014the dark shape of the nail-hung Christ in blue profile, depending on the light.\n\nThat stretched shadow captures something the crucifix itself does not. That shadow of the Crucified curves more beautifully. More beautifully, because you can feel the gravity he's pulling against. More lifelike because there's more suffering.\n\nIn shadow, you can almost see the chest's last rise and tremor and fall.\n\n# **22**\n\nSit up.\"\n\n\"What?\" I ask.\n\n\"Drink this.\"\n\n\"What is it?\"\n\n\"It's good for you. Come on, it's bracing. It'll put warmth back in your bones.\"\n\n\"How did you know about my bones?\"\n\n\"That's it, sit up.\"\n\nI can see the silhouette of a man sitting next to me. He has a cup in his hand, a chalice. Candles lit somewhere behind flicker. I pull myself up and lean my back on the armrest of the pew. My clothes are no longer wet, not even a damp spot where I'd been lying.\n\n\"Where am I again?\"\n\n\"Right where you were.\"\n\n\"I came to the cathedral, right?\"\n\n\"Apparently... Here, sip this.\" He holds the chalice up to my lips. One of his hands supports the base, the other holds the bowl of the cup. I take a sip. It burns all the way down.\n\nBut I don't cough. I don't need to cough.\n\n\"How did I get here?\"\n\n\"That's better.\" The man wipes the lip of the cup with his handkerchief.\n\n\"How did I get in here?\" I ask again.\n\n\"You know how.\" He smiles at me, just an easy curve of a smile. I'm seeing a little better now. My eyes begin to adjust. It's still dark, but lots of candles flicker. He's a thin, elderly man with a thin voice, dressed just like a regular guy: dark cardigan, old-man shirt, frumpy pants. He's holding a folded white hanky and a chalice.\n\n\"How long was I...\"\n\n\"Quite a while.\" He doesn't seem to be going anywhere, just sitting and waiting.\n\n\"How did my clothes get dry? I was soaked when I came in.\"\n\n\"That's why you need to sip this, to warm you.\" He holds up the chalice again, the second time that happened.\n\n\"What's in that?\"\n\n\"It's what you need.\"\n\nI take another sip. This one goes down more smoothly.\n\nThere's something funny about this guy. He seems nice and smiling, sure, but I can't keep what I think about him in one place.\n\n\"Why are _you_ here?\" I ask.\n\n\"Oh, I come here sometimes, late at night. I like what quiet does to a soul.\"\n\n\"Me too. I'm a big fan of quiet.\"\n\n\"Is that why you came?\"\n\nI'm unsure of the question, my answer. \"The quiet? Or my soul?\"\n\nThe old man nods.\n\n\"Buddy, if you're working on commission for the devil, I'm fresh out of soul.\"\n\n\"Then why are you here?\"\n\n\"The doughnut place was closed. How'd _you_ get in?\" I ask.\n\n\"One more sip, when you're ready.\"\n\n\"Did I leave the door open?\"\n\nNow he really smiles. It's in his eyes, like he knows the trick to my question. He doesn't answer.\n\n\"I don't remember leaving it open,\" I say.\n\n\"Do you remember leaving it closed?\"\n\n\"Not exactly.\"\n\n\"Ah, I see,\" he says.\n\nThe candles flicker madly, and I'm not sure, but I think he might actually see\u2014and I'm not sure I like it. He blinks slowly, patiently, like he's waiting for me to remember.\n\n\"I never leave things open.\"\n\n\"I know,\" he says.\n\nI squint toward the flickering. I see the candles lit on the altar table up in the God area.\n\n\"Is Father Chavez here?\"\n\n\"I lit them. Have another sip?\"\n\n\"Stop it, okay? We both know this isn't real. I'm not really here, or you're not. And my soul? You're crazy. Or finally, I am, but I've got no way to tell right now. I feel a little... laid out on a pew, covers it.\"\n\n\"Sit up. You're good.\"\n\n\"That's rich. What would you know about it?\"\n\nHe smiles, wipes the chalice. It's ready for me to drink from.\n\n\"I know you,\" I say.\n\n\"Do you?\"\n\n\"Well, not know... but I've bumped into you, maybe? In real life.\"\n\n\"Maybe.\"\n\nI check my watch. Not for the time, just that I have it. That I can catch the flicker if I need it.\n\n\"Fine watch,\" he says admiringly, dangerously.\n\n\"Just what I was thinking about the candles. They're a nice touch.\"\n\n\"Thank you. I always prefer candles here.\"\n\n\"You come here, light the candles... why, again?\"\n\n\"It was dark, and I was praying.\"\n\n\"For what?\"\n\nWhat he thinks first, he doesn't say. He tucks the hanky into his sweater pocket, reaches out his frail hand, and places it on my shoulder in a grandfatherly squeeze. He pats my neck, then he sits back and gets out that hanky again.\n\n\"Why are you here, James?\"\n\n\"Oh, I see... Did you get my name rummaging through my wallet while I was blitzed? Let me tell you something, old guy, that comforting leather bulge better be there when I check or I'm liable to...\"\n\n\"You're liable to what?\"\n\nI look at my watch, the candles glinting off it. But I don't have any place I desire to be.\n\n\"You're liable to what, James?\"\n\n\"Nothing.\"\n\nHe listens, taking that \"nothing\" in, with his forehead furrowing like Father Chavez.\n\n\"Rest. Here, take another sip.\"\n\n\"I'm so tired.\" I say this, but I'm not tired. Whatever's in that cup is working. I'm ready to fight, old guy or not. I'm dry, I'm warm, and I'm feeling tired, but in the most active way.\n\n\"You're tired?\"\n\n\"All the time. I'm liable to tiredness, to fall out any moment and never find the next.\"\n\n\"You mean your soul? Is that what's tired?\"\n\nI ignore his question and ask, \"Is Father Chavez here?\"\n\n\"No, we're alone.\"\n\n\"That figures.\"\n\n\"What does?\"\n\n\"It's a nightmare, right? The flashlight batteries go dead, but Jamie Lee Curtis heads on down the dark dock. Phone line snipped, but Janet Leigh takes the shower anyway. There's no escape, and the stupid guy opens his own fateful doors, leading straight to the final surprise.\"\n\n\"Careful, James.\"\n\n\"Stop saying my name.\"\n\n\"Please try to rest. Here, take another sip.\"\n\n\"Put your cup away. Just bring me my check, please.\"\n\n\"Easy, son. Think about this moment before you use it.\"\n\n\"Why should I?\"\n\n\"Because you can bend a moment out of shape, and if you do that long enough, you can't bend your way back.\"\n\nIn my head, I see Kevin, his face the last time he spoke to me, asking me to listen, like this old man wears Kevin's last expression on his face.\n\n\"Here. Last sip.\"\n\n\"If I click my heels together, will you go away? There's no place like James, there's no place like James\u2014\"\n\n\"Look, I brought your book.\"\n\nThe book I left at the coffee shop. He has it. Coffee stains down the edges of the pages.\n\n\"How did you get this?\"\n\n\"I stopped by your work, picked it up.\"\n\n\"Just happened by?\"\n\n\"Well, I like coffee. C. S. Lewis, _The Great Divorce_. It's a fine book.\"\n\n\"You've read it?\"\n\nHe nods. \"Not terribly accurate, but you have to love the imagination.\"\n\nI take the book, and it feels real. But dream or not, this is too weird. Time for me to go, one way or another. \"I just remembered. Left the window open back at the hotel. Need to get back\u2014\"\n\n\"Stay. Wait\u2014\"\n\n\"Thanks, no. Let's do it again sometime.\"\n\n\"Here, take one last sip.\"\n\n\"Sure, give it here.\"\n\nFor the third time, he holds up the cup. I put the chalice to my lips, tip it, but I don't drink anything.\n\n\"Happy? Now I got a date with a window.\"\n\n\"You're not ready to go, not yet.\"\n\n\"Unless you plan to swat me with your walker, we have a difference of opinion.\"\n\n\"Why did you come to the church, James?\"\n\n\"You know... You seem to know everything about this dream.\"\n\nNow that he thinks I've taken all the sips, he sets the chalice down on the floor, folds his hanky away.\n\n\"When do I wake up from this? Answer that.\"\n\n\"Whenever you want,\" he says.\n\n\"I remember who you are, okay. That day at Mass. Your smile. And I'm sorry, okay. Sorry for pushing you aside, sorry for taking your bread out of the priest's hand, sorry for everything! All right? Just leave me alone.\"\n\n\"I will, but I have to show you something first. Get up. Follow me.\"\n\n\"I can't. My foot's\u2014\"\n\n\"You're not broken.\"\n\nAnd he's right. I stick my bare foot out in front of me, wiggle my toes, roll the ankle both ways. No pain.\n\n\"First, you tell me... why'd you come here, dream boy? Better yet, how'd you get in here?\"\n\nSlowly, the old man pushes himself to his feet, stands. He's not smiling anymore.\n\n\"You know how. The same way you did.\"\n\n\"Perfect! You're a creepy nightmare twist\u2014the elder leaper. You're... one of us!\"\n\n\"I came here to see you, James.\"\n\n\"You should have called. I'd have put on coffee.\"\n\n\"Careful with yourself. This moment matters.\"\n\n\"Oh, okay,\" I say and laugh. My dream guide, he's not laughing.\n\nThe cathedral is so quiet. Creepy quiet. The old guy just smiles at me, sits and smiles, picks a little fluff off his sweater. My head is pounding, not with pain, but silence. In the pause I think I can hear the candles flickering. And something about it, this moment, this pause, his smile, something makes me want to slow down, stop fighting. Dream or no dream, I want to hear the quiet flicker of candles.\n\n\"You came here to see me? Then answer this.\"\n\n\"If I can,\" he says.\n\n\"Tell me. Tonight, the past three days, this whole thing\u2014is this reality or madness?\"\n\nHe stares at me, unsure. Not like he's unsure of his answer, but of whether or not I will believe his answer, no matter what. Finally he says, \"Without God, reality is madness. Reason will tell you so. You either madly trust in God, or you trust in a world gone mad without him.\"\n\n\"Is there a third option?\"\n\n\"Time to go,\" he says.\n\n\"Wait! Does God really do this? Does God get involved, really make people do the impossible?\"\n\n\"He always has.\"\n\n\"But does he still?\"\n\n\"Don't rob God of being with you, James.\"\n\n\"But I can't...,\" I start, but then say, \"I'm afraid.\"\n\n\"That's why you can't see it. That's what happens sometimes. And when it does, trust becomes the only road home, back to love.\"\n\nI have no answer.\n\nHe pats my shoulder again like he did that day at Mass.\n\n\"That's why I'm here, James. I came here to help you see\u2014and to show you a few things. Come on.\" The old guy reaches into his pocket. He pulls out a shiny silver pocket watch. \"My grandfather gave me this, last time I saw him. Get up. It's your time to see.\"\n\n\"See what?\"\n\n\"You need to see moments without you in them, just as they are.\" He turns a bit, dangles the pocket watch by its long chain so it spins in the candlelight.\n\n\"It's a beauty,\" he says, eyes reflecting the light like his watch. \"You've been changed, James. You've been given a gift.\"\n\n\"I don't want this gift.\"\n\n\"The gift's not for you. It's for others. Everything we're given is for others. God has already changed you, and you can't do anything about that now.\"\n\n\"But what if I don't want to be changed?\"\n\n\"Well, that's why I was sent, to show you the ropes. Show you a few moments, and you'll be back. You can make your choice then. It'll take no time at all.\"\n\nHe takes the hanky out again, leans down, picks up the chalice. He sets both beside me on the pew. \"Your choice... but no more faking it.\"\n\nI pick up the cup and sip it. It's real this time.\n\nThe silver pocket watch spins in the candlelight. It's hard to look at it, but you have to look, it's so...\n\n\"Where are we going?\" I ask.\n\n\"Come and see...\"\n\n# **23**\n\nI understand now.\n\nI've seen.\n\nI've been taken.\n\nIt's like this: Meg's not a saver, no pack rat. She's just not sentimental. She likes empty closets, sock drawers that are orderly, and garages without boxes. She carpools plastic to recycling centers, shreds bills and papers, marks garbage pick-up days with bright blue. But tonight, leaping along, I saw Meg, with her snoring and her legs kicking every cover.\n\nShe's sleeping like she's only got so much time before she has to run again, only so much time to set her life straight. Beside her fitful sleep, on the nightstand, is her shoebox. Of the few things she keeps to touch, to remember, she keeps them in this faded shoebox from when she was a kid\u2014a ladybug ponytail holder; a pink ribbon tied around a note in her father's hand saying \"I love my girl\"; a picture of her and me on this rickety wooden dock, laughing in the rain. Tonight, there on the nightstand, the shoebox is open, and lying on top is a piece of orange construction paper with the letters M-O-M, the _O_ drawn in heavily with crayons, but the _M's_ are made with Popsicle sticks. Mostly, the Popsicle sticks are stained red and blue; a few are fudgy.\n\nAnd she's asleep with her hand hanging near enough to touch the box, its contents.\n\nIn all the years I spent with her, I never made her feel that she would be a good mother, something that important to her, and she would be a good mother, she will. But now this, I played a small part of that Popsicle stick scrap, her hope tonight, her better belief.\n\nShe'll keep that part of my good in her shoebox. That's beautiful.\n\nIt seems doing good isn't _doing_ good. It's desiring to be there when God shows up with the good. The truth is, you only need one beautiful thing to free you and talk you out of yourself.\n\nSleep sound tonight, Margaret. I'm listening now.\n\nFather, I've seen you too. Your glasses on your nightstand; your prayer book on the floor, the page marked with a sock; the glass of water you meant to drink in your silent, lonely home. I see now, Father, that all heroes are unlikely, that the world doesn't need people to be more heroic, just less comfortable. More people willing to be less for someone else's benefit, eager to be used, to rush toward what's most needed and not away from it.\n\nImagined or not, your suffering only matters if it connects you to the suffering of others, if it heals them, too. You don't change the world by telling it what to do, sitting at home, and telling it what you believe. You believe by throwing yourself into it. Making a leap, getting involved, then waiting, taking some one person's place for a while, one suffering person at a time.\n\nThank you, Father. You visited me in prison; I'm not locked up anymore.\n\nI understand what I'm supposed to do now. I'm going.\n\nAt first, God's gift was annoying, an interruption. When the interruption became surprising, then God was simply terrifying. But God was alive. Like Chapman kicking his legs, kidnapped and scared, somewhere deep down, I longed to tell God not to leave, to come surprise me and to do it again. God was, at least, interesting.\n\nThen suddenly, tonight, not only was God terrifying and interesting, but it occurred to me that God might also be up to good. Not that I owed God good, but that God himself might be up to good\u2014that God might be truly, quietly, surprisingly, tirelessly good.\n\nGod is beautiful. Why not trust myself headlong into that?\n\nThe moon sparkling on the dark river below\u2014that is beautiful.\n\nEven this reflection off my watch is as beautiful and blinding as talking to a stranger.\n\nGod help me.\n\nI must be going...\n\nI must be...\n\nI must...\n\n# **EPILOGUE**\n\n## LETTER FROM FATHER CHAVEZ\n\nI compiled the diary as I found it. Where it was difficult to read the handwriting, I did my best.\n\nI've included Captain Goss's notes from his report. I contacted Detective Goss when James went missing, and he made me privy to what his investigation had found, helped me to locate the hotel.\n\nJames's articles were left without his checking out. His key card lay on the nightstand with his car key and a note donating his Volvo to our parish. The Do Not Disturb sign left on the door prevented any movement of his things until the police arrived. Due to the finality of his note and the broken, open window, the hotel management feared the worst and immediately alerted the police.\n\nBut I do not fear.\n\nAs for James, I do not share in the official deduction. I'm unsure as yet what I do believe. Not without any doubt. I don't know if true belief requires the absence of doubt. But I do know what I do not believe happened to him. Sometimes it's hard to believe, but I'm trying. So I leave his diary for each to decide for himself.\n\nAs for me, I will hope. I hope as strongly as I can. I'll do that for my friend.\n\nAnd if it should be possible for you to read this someday, then, Godspeed and be well, James.\n\nGo in peace.\n\nFather Francis Chavez,\n\nSaint John's Cathedral\n\n# ACKNOWLEDGMENTS\n\nMy heartfelt thanks to my editor, Shannon\u2014a savvy woman of incredible skill and patience\u2014along with my gratitude to the good folks of WaterBrook Press. And a special word to my dear friend, Gloria\u2014a sparkling inspiration and a wise muse for all the many years I've known her.\nTo learn more about WaterBrook Press and view our catalog of products, log on to our Web site:\n\n**www.waterbrookpress.com**\n\nWATER BROOK \nPRESS\n","meta":{"redpajama_set_name":"RedPajamaBook"}} +{"text":"3)***\n\n\nE-text prepared by MWS, Les Galloway, and the Online Distributed\nProofreading Team (http:\/\/www.pgdp.net) from page images generously made\navailable by Internet Archive (https:\/\/archive.org)\n\n\n\nNote: Images of the original pages are available through\n Internet Archive. See\n https:\/\/archive.org\/details\/colonialreformer02bold\n\n Project Gutenberg has the other two volumes of this work.\n Volume I: see http:\/\/www.gutenberg.org\/ebooks\/54067\n Volume III: see http:\/\/www.gutenberg.org\/ebooks\/54366\n\n\nTranscriber's note:\n\n Text enclosed by underscores is in italics (_italics_).\n\n In the expression \"to one of his own invention, viz. \u018eNE\n (a conjoined hieroglyph)\" [\u018e] represent the reverse E\n character depressed by half a line.\n\n\n\n\n\nA COLONIAL REFORMER\n\n\n[Illustration: Colophon]\n\n\nA COLONIAL REFORMER\n\nby\n\nROLF BOLDREWOOD\n\nAuthor of 'Robbery Under Arms,' 'The Squatter'S Dream,'\n'The Miner'S Right,' etc.\n\nIn Three Volumes\n\nVOL. II\n\n\n\n\n\n\nLondon\nMacmillan and Co.\nand New York\n1890\n\n\n\n\nCHAPTER XV\n\n\nMr. Neuchamp was disposed to be wroth with himself when he discovered\nthat he was looking forward with considerable interest to a\nmuch-talked-of ball, by which the Count von Sch\u00e4tterheims had resolved\nto mark his appreciation of the kindness which he had received at\nthe hands of the Sydney 'upper ten.' Why should he feel gratified,\nErnest asked himself, at the prospect of joining in an entertainment\nat best but a _r\u00e9chauff\u00e9_ of numberless affairs of the class which\nhe had assisted at and despised in England? A ball\u2015a mere ball\u2015a\nstale repetition of the meaningless crust\u2015the saltatory, amatory, and\ngustatory simulacra of pleasure, which he had long since renounced and\nabandoned. An entertainment chiefly composed of people he didn't know,\nand given by a man whom he did not like.\n\nHe finally disposed of the affair in his own mind by the summary,\nif illogical, decision, that he must regard himself, in respect of\nhis late banishment from the world, in the light of a sailor after a\nprotracted cruise, gifted with abnormal powers of assimilation and\ndigestion, mental and physical.\n\nEven in moments of sternest self-analysis men are not infrequently\ninsincere and evasive. Perchance not consciously. Were the moral\nprocesses incapable of such inflections, Ernest Neuchamp could never\nhave concealed the fact from himself that he chiefly wished to attend\nthis much-abused festivity, to which he had received a formal and\nornate card, inclusive of the arms and crest of the noble family of Von\nSch\u00e4tterheims, because it would be graced by the presence of Antonia\nFrankston.\n\nErnest did not find the very excellent dinner of which he partook\nat the club on the evening of the ball in any degree less palatable\nbecause of this mental conflict. He arrayed himself in the wampum and\nwarpaint proper for such engagements as manufactured by Mr. Poole, of\nSaville Row, which decorations indeed had narrowly escaped being left\nbehind as a superfluous part of his outfit at Neuchampstead. After a\ncareful toilet he awaited in a slightly unphilosophical state of mind\nthe arrival of the Frankston carriage, which was to call for him.\n\nPunctually at ten the highly effective bays contributed their\nparticular quota of gravel-scratching to the enormous aggregate of road\nfriction which pervaded Sydney on that night, and Mr. Neuchamp, placed\nopposite to a wrapped and draped cloud of diaphanous material, which\nhe conjectured to be a young lady, and most probably Antonia, from the\nsimilarity of voice, was whirled off towards the gate of happiness.\nBefore they could approach that enchanted portal they became sensible\nof a line of lamp lit vehicles apparently several miles long, at the\nremote end of which they were compelled to await the gradual debouching\nof the leading files. The opportunity was favourable for conversation,\nand Miss Frankston having disengaged apparently so much of her envelope\nas permitted free egress to her words, they commenced\u2015\n\n'What a lovely night! I was so afraid it would rain. I am sure it will\nbe the most delightful ball we have ever had. I feel certain I shall\nenjoy myself immensely. It is ages since I have been to a dance.'\n\n'I hope your anticipations may be realised,' quoth Ernest. 'Captain\nCook was here, I think, when I last went to one. I had ceased to think\nthem rational amusements long before I left home.'\n\n'Oh! but then you are really dissipated in England, if what we hear of\na London season be correct,' said Antonia. 'Two or three balls a night,\nand some engagement at least once every night. What girl could stand\nthat? Now we poor colonists have perhaps two or three a month in our\ngayest time, and now and then, as now, one in half a year.'\n\n'Well, doubtless the degree of dissipation makes some difference,'\nsaid Mr. Neuchamp, 'and I do not mind owning that I feel as I used to\nfeel; as Hood's seamstress says, before I knew the words of \"drums and\nmatin\u00e9es, crushes and staircase charges, with all the melancholy mel\u00e9e\nof supper, when nobody could eat if they had any appetite, or could\nhave appetite if they would eat.\"'\n\n'You are not in a very promising state of mind,' said Antonia, 'so\nI think I can provide you with plenty of real dancing, if you wish,\nplenty of nice partners, not anything dangerous in the way of crush;\nand if you take me in to supper, I will guarantee you something to eat.'\n\n'Well done,' said Paul; 'I'll back you up in all you have said.\nErnest will see no end of nice girls, who will dance him off his legs,\nunless he's very fit indeed; I think the music isn't bad, and Dettmann\ngenerally gives you something worth eating, and, more particularly,\ndrinking. I'm the man to be pitied.'\n\n'Why, you naughty papa?' said the veiled figure.\n\n'Because just about this time I ought to be smoking my third cigar,\nand going peacefully to bed, whereby I should wake up with a clear\nhead, a good appetite, and a strong idea that I was going to make some\nmoney before noon; instead of which, to-morrow morning, most probably,\nI shall be slightly feverish, eat no breakfast, and have a general\nconviction that stocks are going down, discounts rising, and the\ncountry going to the bad generally.'\n\n'Not if you play whist steadily with old Mr. Howler, the Colonel, and\nDr. Whyte; get the Colonel for a partner, and you'll be sure to win.'\n\n'That's all very well,' said the sacrificial parent, 'but five or six\nhours are not so easy to dispose of at sixty odd. I foresee that I\nshall eat and drink imprudently, catch cold, have a highly unpleasant\nnext morning, with a hint of indigestion, bile, and lumbago.'\n\n'How differently pleasure affects us, at sixteen and sixty,' observed\nErnest with an air of solemn conviction.\n\n'I call that very cruel,' said Antonia. 'I always want papa to let me\ngo with Mrs. Evergreen, but he prefers to martyrise himself, like a\ndear old papa as he is.'\n\n'Well, perhaps he likes to look at his little girl enjoying herself,'\nsaid old Paul. 'I can weather it out yet, perhaps better than I say.\nI was fond enough of fun myself, and have had some strange dances in\nstrange places, with strange company. I remember once\u2015\u2015'\n\n'Come, papa!' said the veiled prophetess warningly.\n\n'Well, only this one; we shall soon be out. I was once down in New\nZealand, in the old times, long enough ago, before the gold and the\nGovernment, and just as we went ashore at Motiki we heard that the\nprincipal Pakeha-Maori, an old sea-captain of course, was going to give\na dance and a grand spread. We were wild for fun of course; been out\nthirteen months. Well, the old boy, a grizzled, hard-weather-looking\nold sea-dog, asked us all, captain, supercargo, and officers.'\n\n'I daresay it was very characteristic,' said Ernest; 'what were the\nladies like?'\n\n'Well, a majority of the wives and daughters of the British settlers\nwere Maoris. It was very rich land, and old Blackbeard had secured a\nconsiderable slice. He had a Maori wife, and ever so many daughters.\nThe youngest was a great beauty, splendid eyes, such a figure, and so\non; I was quite a youngster, and bashful, so I said to the old skipper,\n\"Please introduce me to your youngest daughter, Captain Blackbeard.\"\nThe old pirate looked at me for a minute from under his grizzled\neyebrows, and then growled out\u2015\"How do you suppose I introduced myself\nto her mother? go and hail the craft yourself\"\u2015which I did, and I never\nwish\u2015\u2015'\n\n'Papa!' said Antonia, with great distinctness of intonation. 'Here\nwe are at the step. Please go first, and you will give me room to\nextricate myself.'\n\nMr. Frankston delivered himself upon the carpet spread from hall\nto staircase with an adroitness which seemed a reminiscence of old\nseamanship, and following Miss Frankston and her father, Mr. Neuchamp\nentered the first ballroom in Australia which had been honoured by his\npresence.\n\nClose to the door of a nobly proportioned, brilliantly lighted,\nprofusely decorated, and extremely well-filled apartment, stood their\nnoble friend and host, gorgeously attired in the uniform of a colonel\nof Landwehr, and shining like a constellation of the first magnitude\namong the more unpretending naval and regimental officers then\nquartered and stationed at Sydney.\n\nAs he took the hand of Miss Frankston, and bowed low over it, with an\nassumption of chivalrous deference, only permitted to a foreigner,\nErnest felt a mad desire to then and there kick him down the stairs\nof his own ballroom. Controlling this perhaps not strictly defensible\nimpulse, he drew back, as the Count shook Paul's hand with a delicate\nyet cordial deference appropriate to an honoured father in prospect,\nand evidently, to that nobleman's astonishment, bowed very stiffly and\nfollowed his friends. A large family party, including half a dozen\nsmiling and whispering girls, evidently delighted by the cordial\nwelcome they experienced from their distinguished entertainer, covered\nhis retreat. The night was superbly beautiful. At no great distance\nlay the slumbering sea-lake; while over the silver plain clusters of\nglancing lights gleamed, beneath the broad illuminated balcony of the\nballroom. Unless Ernest's heart had been much more ill at ease than\ncircumstances rendered possible, it would have been hard at his time of\nlife for aught but pleasure, for a little space, to bear sway.\n\nThe floor was perfection; the music, that of a military band, which had\nbut the year before played in the great square at Pera, which had been\nat the front during the terrible northern campaign, yet fresh in men's\nminds, well coached by a music-loving, fastidious colonel, was pealing\nout the 'Sch\u00f6ner blauer Donau' with wondrous time and spirit.\n\nMr. Neuchamp had been sufficiently awake to his opportunities to\nengage Antonia for the first _deux-temps valse_ after they entered\nthe room, and the after-supper galop, taking his chance of anything\nintermediate. 'That is good music,' said he; 'I heard it in Vienna\nlast. Suppose we join these very sincere performers.'\n\nAntonia replied by a frank smile of assent, and as he took one\ncomprehensive glance over face and figure ere he clasped the slight\nyielding waist, he thought he had consistently underrated her beauty.\n\nThe light was of course eminently favourable to her clear though\ncolourless complexion; her eyes, sparkling with frank unstudied\nenjoyment of the entertainment, shone with unwonted lustre, while the\nperfection of her slight but rounded figure was clearly apparent; and\nas they swept adown the crowded hall Mr. Neuchamp, though he had not\nbeen numbered among the lavender-kid-wearing tribe of modern youth\nof late years, danced very well, and we may add looked very well, in\nthat much-abused, but as yet unsuperseded garb, than which no other\nbefits so well a gentleman on evening pleasure bent. Perhaps we have\nnot devoted sufficient space heretofore to the limning of the hero's\npersonal charms and graces. These were perhaps sufficient though not\nremarkable.\n\nErnest Neuchamp, somewhat above the middle height, had, without any\nparticular athletic ostentation, the square form and well-knit figure\nof an ordinary English aristocrat. Though possessing more endurance\nthan strength, he by no means fell short of that necessary endowment.\nOne saw fairly regular features, comprising a pair of searching grayish\nblue eyes, very multiform as to expression, a clear-cut firm mouth,\nand light-brown hair inclining to curl, which I need not say was\nvery closely cut on the present occasion. Brown-bearded, and rather\nsunburned, as to his original delicate complexion, he was by no means\na bad representation, had he donned armour, of one of his crusading\nancestors just returned from Ascalon or Engadi with all the prestige of\na good knight and a whole heart for the ladye-fayre, who awaited his\ncoming amid her bower-maidens.\n\nAs it was he was restricted to the simple dress, the simple speech,\nof a modern English gentleman, yet was there about him a freshness,\nsincerity, and unassumed refinement of manner not unlike that of\nthe best class of naval men, which made him extremely acceptable to\nwomen, and which Antonia Frankston in her heart of hearts had always\nrecognised.\n\nThe dance was not a particularly short one. Ernest was in reasonably\ngood training after his up-country experience, and Antonia was one of\nthose rare\u2015too rare danseuses that unite in perfection time, pace,\ngrace, and staying power. She could fly down the crowded ballroom\nproperly supported by a partner _de la premi\u00e8re force_, halt, turn,\nglide in and amid the labyrinth of dancers, without thought or question\nof collision. Instinctively true to every note of the music, to every\nmovement of her partner, she seemed as if she possessed the latent\npower and tireless speed of Atalanta of old, did she but deign to exert\nthem.\n\nThe music ceased, annotated by a very audible sigh on the part of\nErnest, who was impelled to say that he never expected to enjoy a dance\nagain so much as long as he lived.\n\n'There is nothing like dancing,' said Antonia, apparently as cool as a\nstatuette. 'But I think the balcony will be pleasanter. I must show you\nall the people.'\n\nIn their path was a portly white-waistcoated personage of placid and\nsmiling aspect, who, bestowing upon Antonia a most respectful bow,\nshook Ernest's hand warmly.\n\n'Ah, Neuchamp, my dear fellow, delighted to see you. Not bought a\nrun yet? You're losing splendid opportunities. Let Gammon Downs slip\nthrough your fingers\u2015eh? Sold it to Rawson and Rowdy since. Great\nbargain.'\n\n'Indeed!' said Ernest, smiling. 'Well, they are the best judges of\ntheir own line of action. How are they doing? Making lots of money?'\n\n'Well, they ought to\u2015ought to\u2015but I'm afraid they're not very good\nmanagers. Rawson's rather slow\u2015Rowdy's too fast. However, I can't help\nthat. Do you happen to want a crack run, my dear Neuchamp? I've got\nBrigalow Park and Mallee Meadows for sale, a real bargain; quite a\u2015\u2015'\n\n'Not just at present,' said Ernest, preparing to move past. 'See you\nat the club. The Count seems to be enjoying himself\u2015who is the lady?'\nThis last observation was elicited by the appearance of the noble host,\nwho passed at a little distance with a very handsome and magnificently\ndressed girl upon his arm, talking in the most _empress\u00e9_ manner; while\nshe, conscious of being at that moment an object of envy to the great\nmajority of her sex, there and then present, listened with apparent\npleasure.\n\n'Oh, that's Miss Folleton, of Fairmount. Fine girl, isn't she? Will\nhave forty thousand on her wedding-day,' said Selmore, who knew\neverybody and everything; or said he did, which was much the same. 'Not\nthat Von Sch\u00e4tterheims cares for that. Immense property of his own,\nvast estates in Silesia, nearly as many sheep as Esterhazy\u2015- that's why\nhe comes out here. Thinks of investing\u2015met him abroad myself.'\n\n'Indeed,' said Ernest; 'haven't you anything that will suit him?'\n\n'Well,' said Selmore, looking, for him, slightly confused and glancing\nat Antonia, who was regarding him critically, 'I told him that Mallee\nMeadows and the other place might suit him, but he wants a resident\npartner. How would you like to go in with him? You're just the man that\nwould suit him.'\n\n'Can't bear partners,' replied Ernest shortly; 'I am afraid his\nhighness and I wouldn't agree. I think I see a seat, Miss Frankston.'\n\n'I dislike that man intensely,' said Antonia, as they moved on. 'I\nthink him so selfish and unprincipled. I wonder if he has inveigled the\nCount into one of his bargains, as he calls them?'\n\n'From a cursory examination of your high-born friend's conversation,'\nsaid Ernest, 'I think he may be trusted to take care of himself in\nmatters of finance, as indeed is the case with most foreigners.'\n\n'Now, is not that a very prejudiced though English speech? You cannot\nreally believe that because a man is born on the continent of Europe he\nmust be less trustworthy than any one from that wonderful little island\nof yours?'\n\n'I didn't say so; I ought to qualify such a wholesale sentiment.\nWhether the right sort of foreigner does not emigrate I cannot tell.\nBut the idea has struck others besides myself, and I must confess to a\n\"Dr. Fell\" sort of instinctive feeling about our distinguished friend.'\n\n'Sheer prejudice and perhaps the least bit of jealousy, shall I say, on\nyour part,' continued Antonia.\n\n'But why jealousy?'\n\n'Well, I mean it to apply to all of you men who run down the poor\nCount so. _We_ are all great admirers of him, and that, I am afraid,\ndoes not make him popular with your sex. Here's Mr. Croker coming\nto claim me for the next dance. There now, he'll abuse him\u2015but he\ndoes that about everybody. Are you sure that this is our dance?'\nmischievously commenced the young lady, as that gentleman arrived.\n\n'I think so,' said Croker superciliously, 'unless you have a chance of\nthe Count, in which case of course you'll throw me and Neuchamp over\u2015I\nexpect nothing else.'\n\n'Not surely if I were engaged to Mr. Jermyn Croker!' said she;\nand looking at her programme, 'I really am engaged to you for\nthe quadrille, but why am I accused of pursuing the Count von\nSch\u00e4tterheims?'\n\n'Because every one runs after him\u2015men, women, and children,' said\nCroker. 'The whole city seems transformed into a sort of Bedlam.'\n\n'But why do they run after him?' inquired Miss Frankston.\n\n'Why?' repeated Mr. Croker, with an air of ineffable disdain, 'because\nthey're all fools, I suppose; except a few, a very few.'\n\n'And why are they excepted?' said Ernest, who commenced to be amused at\nhis daring unsparing cynicism.\n\n'Because they're mad\u2015stark, staring mad.'\n\n'Now really, Mr. Croker, don't you believe about the Count's great\nwealth and estates? his charming manner at any rate can't be put on.'\n\n'_I_ believe in him. _I?_' demanded Croker, with an air of intense\nand reproachful amazement ludicrous to witness. 'Do you know what my\nopinion of the fellow is?'\n\n'Can't say, really; something very complimentary to him and diffident\non your part, judging from Mr. Croker's well-known character,' said\nAntonia coolly.\n\n'Well, then, if you will have it,' said that satirist wrathfully, and\nas if all necessity for social dissimulation had been obviated, 'I\nbelieve the fellow is an impostor and a swindler; very likely a valet,\nor a courier, who has bolted with his master's cash, clothes, and\npapers. As for his manners, everybody in the country he comes from has\nthe same manner, from the kellner to the kaiser. His accent ought to\nbetray him; but no one here knows German well enough to find it out.'\n\n'Really, Mr. Croker, you can take away a man's, a horse's, or a\ncountry's reputation more completely in less time than any one I ever\nmet. You're so delightfully bitter that I _must_ dance with you. Come\nalong!'\n\nLeft to himself for a while, Mr. Neuchamp devoted his leisure to a\nsurvey of the room and the company. He was astonished at the beauty and\ngrace of the lady portion of the guests, and he thought he had never\nseen anything more graceful than the ease and celerity with which the\ngreater part of the crowd glided in the dance over the polished floor.\n\nThe occasional squatter, lounging, but stalwart and dignified, together\nwith the gay uniforms of the soldiers and blue-jackets, gave novelty\nand contrast to the scene; but the majority of the younger men who\nbelonged to Sydney proper were pale, slight, and rather undignified\nyoungsters, by no means worthy the handsome, stately girls who were\nfain to accept them as partners. For the rest, the ordinary ballroom\nroutine was not departed from; and Ernest, after another dance or two,\nwas not sorry when the move to supper reminded him to possess himself\nof Antonia, who had promised him the first following dance.\n\nNothing in its way could have been more complete than the dangerous and\nsuperfluous but fascinating meal. The wines were chosen with a studious\ncare, which reflected the greatest honour upon the Count's taste and\nforesight.\n\nThe champagne and chicken had been succeeded by fruit and flirtation.\nThe ladies were in expectation of the accustomed signal, when Mr.\nHartley Selmore rose, 'with the permission of his friends, to make a\nfew observations and to propose a toast. Would gentlemen\u2015ay, and ladies\ntoo\u2015fill their glasses, and prepare themselves for a toast to which his\npoor powers were miserably inadequate?'\n\nThese preliminary suggestions were cheerfully complied with, as indeed\nis invariably the case, the cheapest of all compliments being, surely,\nto drink another glass of wine at the expense of your entertainer.\nThen, with one hand in the breast of his ample waistcoat, Mr. Selmore\nsmilingly confronted the expectant throng, and with the readiness of\na born talker and something of the ease of a trained orator, thus\ndelivered himself:\u2015\n\n LADIES AND GENTLEMEN\u2015One of the first observations that we shall\n make when we leave this hall will assuredly be that we never spent a\n pleasanter night in our lives, never saw any festivity more perfect\n in the arrangement of every detail, never had the good fortune to\n accept more lavish and splendid yet delicate and graceful hospitality.\n Then why not say it now? (Cheers and approbation.) Frankly, then,\n even in the presence of the noble and distinguished personage who has\n honoured this colony, this city, this society, with his presence, I\n venture to avow the sentiment of my heart, of every heart, let me say,\n now beating responsively to these humble expressions of the general\n feeling. (Loud cheering.) Some of us may have been struck with the\n wonderful perfection which has accompanied every detail, however\n small, even to the novel arrangement of the matchless feast we have\n just arisen from; but who does not know that the master mind, which\n is capable of conceptions the most vast and varied with regard to the\n welfare of nations or the march of armies, disdains not to stoop to\n the peasant's farm, to the soldier's shoe-buckle.\n\n When I lead your minds, ladies and gentlemen, to the idea of the\n characteristics of great generals, of reigning princes, of the\n blood-royal of one of the most ancient sovereignties of the universe,\n am I violating any confidence when I state, in corroboration of\n that half involuntary disclosure, that no one who, like myself, has\n had the privilege of beholding those royal personages, of marking\n their prevailing type of feature, can doubt, by comparison with the\n countenance of our noble entertainer, the Count von Sch\u00e4tterheims, of\n his near and intimate relationship with that royal house. (Tremendous\n and enthusiastic cheering, with direction of all eyes upon the Count,\n whose presumably princely lineaments were as immovably unconscious as\n if he had been a statue of Kaiser Fritz.)\n\n I may be indiscreet, ladies and gentlemen; I may, carried away by\n my natural feelings of friendship and by the contagion of your\n enthusiastic assent to my simple and straightforward statements,\n have spoken with more frankness than prudence, but my heart forgives\n me; my noble friend, I feel assured, forgives me; and you, ladies\n and gentlemen, will, under the circumstances, forgive me also. I\n have the liveliest pleasure in proposing the health of the Count von\n Sch\u00e4tterheims.\n\nWhen the storm of cheering, the volleys of applause, the waving of\nhandkerchiefs, had subsided, the noble Count himself, rather pale, but\ncollected and calm as of custom, rose in his place to return thanks,\nwhich feat he performed as follows:\u2015\n\n LATEES AND GENDLEMENS\u2015Many very danks.\n\nThe return speech had the merit of brevity\u2015perhaps in excess; but as\nthe Count placed his hand on his heart and bowed low thrice, throwing\nall the expression (and that was a considerable allowance) that he\ncould manage into his eyes, so directed that not only Miss Folleton,\nbut at least six other young ladies, imagined that she alone was the\nobject of those tender and pleading glances, it suddenly struck the\nassembled crowd that it was an intentional and masterly stroke of\nmingled humour and consideration. As the band, by preconcerted signal,\nstruck up the glorious and entrancing galop which had been kept in\nreserve for the after-supper dance, the ladies and the younger men\nsaw another instance of the Count's marvellous foresight\u2015for them in\nparticular\u2015and once more they joined in general and unmeasured applause.\n\nThe Count, who had by this time secured the radiant Miss Folleton,\nbowed low and led the way to the freshly decorated ballroom, all the\napproaches to which were filled with the choicest exotics.\n\nWas not it an utterly perfect galop, such as that entrancing\nafter-supper dream-dance with the 'dear new angel,' or our favourite\nfriend that used to be, 'Consule Planco'? Oh, the dances of our\nlost youth, realising in every gliding sweep and trancing whirl the\nmost exalted conceptions of music, poetry, choregraphic grace, and\nintoxicating proximity to female loveliness, when, if at any time\npossible, a fold or two of the jealous marble of reserve is thrown\nback. Within a fast fleeting hour from this dance of dances did Mr.\nNeuchamp put Miss Frankston into the carriage attended by her grateful\nparent, who was truly tired of his life under circumstances of\nfestivity, and dying to get to bed.\n\n'Ha! Neuchamp,' said Croker, as he returned to the disenchanted\nballroom, 'you look exhausted. Come and have a parting glass of the\nCount's Roederer. I stick to that; we shall never see any more of it, I\nfeel sure.'\n\n'Why?' demanded Ernest; 'you're rather hard upon our noble entertainer.\nYou allow that his wine is very good.'\n\n'Good wine costs no more than bad under certain circumstances,' replied\nCroker sardonically.\n\n'What do you mean?' asked Ernest.\n\n'Mean! Why, that Yorick and Co. will never see a farthing of their\nmoney. I really feel uneasy about our share in the swindle,' continued\nCroker, filling a large glass with iced hock, then drinking it slowly\nand with great apparent relish.\n\n'Great heavens!' ejaculated Ernest, 'I can't believe it. I won't taste\na drop. And what do you suppose will happen to Von Sch\u00e4tterheims?'\n\n'The devil only knows, who will probably stick to him for a season\nstaunchly enough. He will make a bolt, or a warrant of extradition,\nincluding an assassination and two stupendous jewel robberies, will\nfetch him.'\n\n'You are strongly prejudiced,' said Ernest, deeply shocked and ashamed\nof his own mild suspicions.\n\n'Slightly so, perhaps; it runs in my family. I detest _all_ foreigners,\nand believe them to be capable of anything.'\n\n'That's rather hard measure, don't you think?'\n\n'Not at all,' said Croker, finishing the wine. 'Foreigners are not\nso madly given to travel as we fools of English people; take my word\nfor it, no foreigner of character and position would come out to an\ninfernal hole of a place like this colony. Your friend Paul seems\nshaky, slightly apoplectic, or perhaps complaint in the _chest_; half\nthose mercantile beggars are shams. Daughter gone off very much, looks\nquite _pass\u00e9e_. Good-night; I'm off.'\n\nWith these few consoling remarks, which Ernest felt much inclined to\nresent by personal protest, Mr. Jermyn Croker betook himself to the\nsmoking-room of the New Holland, whence, having abused the ball, the\nguests, the giver, the lights, the decorations, everything, in fact,\nbut the wine, of which he certainly had secured his share, he departed\nto bed in a consistently uncharitable state of mind with all men.\n\nPaul did not show up at the office next day, and as the afternoon had\nbeen fixed for boat-sailing, as a refreshing and suitable recreation\nto neutralise the somewhat reactionary season which succeeds a ball,\nErnest made his way to Morahmee soon after lunch.\n\nThere he found Antonia very becomingly dressed in yachting costume,\nwhich from its simplicity afforded a telling contrast to the _grande\ntenue_ of the previous night. Paul, with a couple of sailor-looking\nmen, was down at the jetty, and after a little preliminary trimming and\ndelay in sending for extra ballast, they were all seated and skimming\nover the bright waters of the harbour, with a light but favourable\nbreeze. Mr. Windsor, invited by particular request of Mr. Frankston,\nsat forward in company with the crew, and assumed an air of ease and\nsatisfaction which that roamer of the waste was as far from feeling as\na pilgrim Bedouin on board a Red Sea steamer.\n\nBut no thoughts, save of the most childishly unalloyed happiness,\npossessed the hearts of Paul Frankston and his daughter. The old man\nwas a born and bred sea-dog, and it was wondrous to mark how his nature\nrose and became exalted as he found himself upon the familiar element\non which the joyous time of youth, the _sturm und drang_ period of his\nstrong manhood, had been passed. Again his eye lightened, and the\nold gleam of pride and daring spoke from it of the days when he had\nvolunteered for more than one maritime forlorn hope; had consorted\ngaily with danger; had dared the clubs, the poisoned arrows of cannibal\nsavages; or had cowed a mutinous, scowling crew by the magic of a stern\nfront and a steady pistol. Even his voice was altered, and he gave the\nslight but necessary orders in a clear peremptory tone of command which\nErnest had never heard from his lips before.\n\nFor Antonia, she revelled in the free breeze, the brilliant sea and\nsky, like a happy child, and as a glancing spray fell lightly over\nthem, she carolled forth the refrain of a sea song with a nerve and\nanimation by no means usual.\n\n'Is not this lovely fresh life a renewal of all one's senses?' she\ncried. 'I feel as if an additional one, indescribable and amazing, was\ngiven to me whenever I am on blue water. You know we are all great\nsailors and boatmen in this Sydney harbour of ours. Look at the numbers\nof skiffs, pleasure boats, and yachts that are now skimming about like\nseamews in all directions.'\n\n'Rather too many for my fancy. Every now and then accidents occur,\nplain sailing as everything looks just now. The gusts which come down\nacross these points are like small white squalls.'\n\n'Ah! but the present, my darling old pappy,' sighed Antonia, 'what\ncan possibly be more glorious for mere mortals? Why should we\ngrieve ourselves with the past or possible sorrows? Can anything be\nmore dreamily lovely than that pale amber sky over which the dark\nblue shadow is creeping from the headland? What can surpass the\nsoftly-gliding magical motion with so much swiftness and so little\neffort? I don't wonder that a sea life has always gathered to it so\nmuch poetry and romance. I fell in love with a pirate once.'\n\n'With a pirate? Where?' exclaimed Ernest, surprised out of the placid\nenjoyment which pervaded the whole party.\n\n'In a book, of course,' answered she; 'you didn't think we entertained\nsuch gallant rovers at Morahmee, except, like angels, unawares? But he\nwas such a delightful creature. I remember the lines still.'\n\n'Perhaps you wouldn't mind repeating them.'\n\n'Oh yes. I shall never forget them, I am sure,' the girl answered,\nlooking seawards. 'I found them in an old\u2015old\u2015annual. You shall judge:\u2015\n\n 'Our Captain, he is young and fair,\n How can he look so young?\n His locks of youth, his golden hair,\n Are o'er his shoulders flung.\n\n 'Of all the deeds that he has done\n Not one has left a trace,\n The midnight cup, the noontide sun\n Has darken'd not his face.\n\n 'His voice is low, his smile is sweet,\n He has a girl's blue eyes,\n And yet far rather would I meet\n The storm in yonder skies.\n\n 'The fiercest of our pirate band\n Holds, at his name, the breath;\n For there is blood on his right hand,\n And in his heart is death.\n\n 'He knows he rides upon his grave,\n Yet careless is his eye;\n He looks with scorn upon the wave,\n With scorn upon the sky.'\n\n'Not a bad conception, I admit,' said Ernest, 'though, doubtless,\nviolently untrue to nature. In all ages poets and romance writers,\nwho are humbugs to a man, have laboured to unite personal beauty and\nwinning gentleness of manner with the capacity for remorseless crime.\nI think, perhaps, that the young Spaniard in _Tom Cringle's Log_ is as\ngood a specimen of the thoroughbred upstanding pirate as any of those\ngentry whose acquaintance I have made, like you, in print.'\n\n'I saw eight-and-thirty of the ruffians strung up in one day, at a\nSpanish West Indian port, once,' said Paul. 'They said their prayers,\nkissed their crucifixes, and died in the coolest and most edifying way.'\n\n'And were they very bad men, papa?'\n\n'Awful scoundrels,' said her father, with a certain relish, as he\nrecalled the reminiscence. 'We only escaped them by a miracle; so I\nfelt no compunction in seeing them elevated.'\n\n'And what became of the ship they did capture?' inquired Antonia.\n\n'They took everything of value from the vessel, including a few\nprisoners they meant to ransom, and then scuttled her, leaving the crew\nand passengers to perish.'\n\n'How fiendish! and they were nearly catching my darling old father,'\nexclaimed the girl. 'I must reconsider the question of pirates. But\nwere they all as bad as that, papa?'\n\n'Worse, if possible,' said Mr. Frankston uncompromisingly. 'They knew\nthat there was a rope ready for each man's neck when he was caught, and\nthis knowledge did not incline them to mercy, you may be sure. Chinamen\nare perhaps as dangerous rascals, in that line, as you can meet. They\nare no great sailors; but if you get becalmed in their waters, and a\nfew crowded prahus come round you, your chance is a bad one.'\n\n'And will they fight?' inquired Ernest. 'I thought one jack-tar was\nworth a dozen of them.'\n\n'So they are in one way\u2015in a fair fight, or in a case of boarding, or\nin bad weather. But these vagabonds are very careless of life. They\nnever give quarter and don't care much about taking it, not being used\nto it, so you may imagine how they fight. I have seen a fellow fairly\ncut to pieces before he left off fighting, and I really believed\u2015I was\na boy then\u2015that the kriss moved in his clenched hand after the arm was\ncut off.'\n\n'In that case they may trouble the world yet,' affirmed Ernest.\n'A nation of three hundred millions, with sufficient ingenuity to\ncomprehend Whitworth and Snider, and animal courage to fight to the\ndeath, might execute another avalanche movement such as when Attila (of\nkindred blood, we must remember) swept over Europe.'\n\n'Not in our time, at any rate,' quoth Paul, with epicurean\nindifference. 'Ah Tin will require a deal of drill before that march\ntakes place. Now, then, here we are off Red Point. Suppose we get the\nlines out and have some fishing.'\n\nThe deep-sea lines were produced by the two 'waterside characters'\nwho composed the crew, and suitable bait being forthcoming from\nsome mysterious receptacle, the somewhat serious recreation of\nschnapper-fishing commenced. Perhaps the poetry of the piscatory art\ncannot be dissociated from the mimic ephemera with which the fisherman\nof Europe deceives the leaping trout and the king of all river fish,\nthe mighty salmon, as the angler standing under the ruined wall of\na Norman stronghold, patiently whips the purling stream which has\nfurnished relays of delicate fare for a thousand years. Nor is his\nsport heightened by historic association. The captor of _Salmo salar_\npasses from stage to stage of doubt, hope, fear, agony, despair,\nto unspeakable triumph, as after endless patient playing, he draws\nwithin reach of the deadly gaff the captive monarch. But, Izaak Walton\nnotwithstanding, a good afternoon's fishing in or off Sydney harbour,\nwhen the deep-sea denizens are fain and fearless, is not to be despised.\n\nMr. John Windsor was considerably surprised, though he was careful\nnot to show it, as the first fish, a twelve-pound schnapper, came up\nglancing and glimmering through the clear water at the end of Mr.\nFrankston's line. A rock-cod or two, with their brilliant colouring,\nadded to his wondering observation. But he was, perhaps, more nearly\ndriven from his habitual coolness when a yard-long dogfish was\ndropped into the bottom of the boat, sufficiently near his legs to\ncause the lower portions of those limbs to shrink and stiffen, as the\nocean-Ishmaelite snapped its sharp teeth within an inch of his ankles.\n\n'I think we must have got about half a boat load,' said Paul at length,\nafter a continuous course of baiting, lowering, and hauling up. 'As the\nday is so fine, we may go through the Heads for a run out, and then\nturn back and beat home.'\n\nThey glided through the comparatively narrow entrance, on either\nside of which frowns sullenly the vast sandstone promontory, seamed,\nchannelled, and scarped by winter wind and ocean wave, that for ages\nhave raved and dashed against its sentinel form. Southward a mile or\ntwo, and still the deep sea rolls on with slow but resistless force\nagainst the base of the tremendous, all inaccessible cliffs which frown\na hundred fathoms above.\n\n'I never pass this place,' said Antonia musingly, 'without thinking\nof that heartrending wreck of the _Dunbar_. A wreck, at best, is a\ndreadful thing; but think of these poor creatures, as near to their\njourney's end as we are now, only to find a death in the midst of angry\nbreakers and rocks and the dread midnight. How many deaths must they\nhave died! God save us all from a fate like this\u2015\n\n 'On the reef of Norman's woe.'\n\n'I did hear something about a vessel going on shore with all hands,\nnear the Heads,' said Ernest. 'And was this the very place? Was there\nany carelessness?'\n\n'Poor Grant was as good a seaman as ever trod plank. I knew him well,'\nsaid Mr. Frankston. 'He had been first mate of her for years, under old\nFleetby, and this was his second voyage in command. He was as smart a\nman as Charley Carryall, and that is saying a good deal.'\n\n'What was the cause, then, of the disaster? It seems so near the port\nof entrance.'\n\n'It wasn't weather like this, you may be sure,' said Paul. 'Unluckily,\nafter a first-class run, poor Grant made the light, sometime after\nnightfall, on as misty, driving, dirty a night as ever these old rocks\nsaw. He stood off and on until an hour or so past midnight, when,\nfinding the gale increasing and the wind setting in dead inshore, he\ndetermined to run for the Heads, trusting to his own seamanship and his\nclose knowledge of the channel, that he had passed through a score of\ntimes in all weathers, at all hours of day and night.'\n\n'But how could he miss the proper opening?' asked Ernest.\n\n'God knows! The weather was awful. The coast just here does change\nshape a little, as if there was an opening. The ship had been driven in\ntoo close ashore; if they saw the lighthouse, her course would bring\nher stem on to these awful rocks. It seems that they never knew their\nmistake till they were among the breakers.'\n\n'And how could that be known?'\n\n'_One_ man was saved,' answered Paul. 'The last thing he saw of poor\nGrant was forward, in the chains. That was just before she struck. When\nshe did strike she must have gone to pieces in ten minutes, and two\nhundred passengers, who were dreaming of home and friends, or the sweet\nsight of shore with the morning sun, ere that sun rose were drifting or\nmangled corpses.'\n\n'What a day of mourning it was in Sydney!' said Antonia; 'hardly a\nfamily in the city but had friends or relations on board. A favourite\nship, with a favourite captain, numbers of returning colonists had\nwaited or hurried in order to sail by her.'\n\n'We must all take our chances, my dear, more particularly those people\nwho are foolish enough to be sailors. Hector Grant met a sailor's\ndeath, and I'll swear he took his lot coolly when it came, caring\nmore for the poor passengers than himself. For them it was different.\nI always pitied the landsmen and their families, when I stood a fair\nchance of going to Davy Jones myself. Hallo! the wind's shifted two\npoints. There's an ugly bank, too. It will give us enough to do to get\nhome before the southerly breeze comes up.'\n\nAs they commenced to beat back against the breeze, which, appearing\nto gain strength rapidly, necessitated rather more promptness and\nseamanship than their outward-bound voyaging had required, Ernest was\nconstrained to admire the coolness and total absence of timidity which\nMiss Frankston displayed.\n\nDoubtless she was accustomed to boat-sailing and yachting in all its\nvarious forms, and was familiar with the eccentricities of the harbour\nnavigation. Still, as the breeze freshened, the sky darkened, and from\ntime to time the spray broke over the tiny cutter, now leaning over\ntill the gunwale dipped, in a manner that did not suit Jack Windsor at\nall, as the thought obtruded itself that if the southerly gale, which\nPaul Frankston's experienced eye looked for, broke over them before\nthey reached the shelter of the solid Morahmee pier they might possibly\nfounder. Ernest wondered if his fair companion fully realised her\nposition, or whether her calm indifference was merely ignorance of the\ndanger.\n\nHis mind was set at rest upon the point presently.\n\n'Papa!' she said, looking first at the sky and then at the merchant,\nwho, with all the skipper in his stern set face and steady eye, was\nlooking to windward not altogether cheerfully, 'don't you think we\nshall be hard set to get home before the \"brickfielder\" fall upon us?'\n\n'I do indeed, darling,' said the old man. 'I wish I had been keeping my\nweather eye open, instead of gossiping about the _Dunbar_ people, poor\nsouls. For all we know, we may make another business of the same sort,\nin a small way.'\n\n'Then don't you think we might carry more sail? You know poor little\n_Haid\u00e9e_ here will let you drive her under almost when she's on a wind,\nand a knot an hour more may make all the difference.'\n\n'I think Miss is right, sir,' said the oldest of the crew, 'we've not\na minute to throw away; and if it ain't coming up heavier and thicker\nfrom the south, my name ain't Johnny Jones.'\n\nAs the necessary dispositions were made, Antonia watched keenly and\ncritically the altered motion of the boat, which lay down to the now\nangry sea, as if every fresh gust would bury her beneath the heaving\nbillow; and having apparently satisfied herself that the maximum of\nspeed, combined with the smallest possible margin of safety, had been\nattained, lay back and quietly awaited the progress of events.\n\n'Hers is no soulless insensibility to danger,' thought Mr. Neuchamp to\nhimself. 'Rather a full comprehension of risk, and even not improbable\nloss, dominated by the calm courage which wills and reasons in the face\nof death.\n\n 'A perfect woman nobly planned\n To\u2015\u2015'\n\nMr. Neuchamp was prevented from continuing his quotation by a sudden\nejaculation of Mr. Windsor, across whose person, as the boat dipped\ndeeply, a wave of greater magnitude than usual broke and foamed.\n\n'By the powers!' exclaimed he, 'this was never put down in the\nagreement, or John Windsor wouldn't have been here. Are you quite sure,\nsir, we ain't taking a short cut, and getting away from our regular\ntrack? I should like to get out of this trap and walk. I can swim\nthough above a bit, so if we are regularly spilt I may, perhaps, help\nthe young lady.'\n\n'Do you know what that is?' asked Mr. Frankston, pointing to a black\ncurved substance out of the water, and apparently belonging to some\nsubmarine monster which was proceeding in a parallel direction, and at\nno great distance.\n\n'Not a know do I know,' replied the bushman.\n\n'It's the back fin of a _shark_, and he's no small one either. He'd\npick us up at his leisure, if anything happened to the boat, like a\nturkey among grasshoppers.'\n\n'By George!' said the man of the forest, 'I wish I was on Ben Bolt now,\nwithout saddle or bridle, and him bucking his best, this minute! There\nis some get away, if anything broke, short of your neck. But here it\nseems to be the Never-Never country, and no mistake.'\n\nThey had made what is nautically called 'a long board,' in tacking at\nimmense angles, so as to take fullest advantage of the wind, which\nseemed to increase rapidly, until something like the foretaste of the\nfury of a gale was upon them. The sky had darkened; night was not\nfar distant. The sea had risen, and the long-backed rollers made it\nincreasingly difficult for so small a craft to avoid an upset. Nothing\nbut the splendid steering of their skipper, the perfect handling of the\ncrew, combined with the weatherly qualities of the _Haid\u00e9e_, gave them\nthe chance of riding it out.\n\n'Steady all, and look out for your heads while she jibes,' sung out old\nPaul. 'I think we shall fetch smooth water with this tack; if so we're\nsafe for dinner, with better appetites than usual.'\n\n'And if not?' inquired Ernest, with an anxious gaze at Antonia, who\nsat drenched with spray and pale, but with the most perfect composure\nvisible upon her unmoved features.\n\n'Did you ever hear tell of one Davy Jones?' made answer Mr. Frankston,\nwhose furrowed face, torn with anxiety for the fate of his soul's\ndarling, contradicted the lightness of his tone, 'for if we are sent\nback into the gale, we are very like to mess with him this evening.'\n\nAnd now, as the tiny bark swung round to her altered course, and,\nlying close up to the wind, flew down with hazardous swiftness towards\nthe entrance to the little bay, which to them was a haven of safety,\nErnest, true to his lifelong habit of observation, scanned the faces of\nhis companions with half-unconscious curiosity.\n\nCalm and strong sat the old man, with the tiller in his sinewy hand;\nhis eye was steady, his hand was true, and none could have told by\nreading his countenance that the life of one he held a thousandfold\ndearer than his own hung on the balance of a frail boat and a stormy\nsea.\n\nThe two sailors had the ordinary non-committal expression always\nobservable in trained seamen, unvarying apparently, whether a sail be\nsplit, a leak be sprung, or a hopeless fire be discovered in the hold.\n\nMr. Windsor, unconsciously holding on tightly to the thwart upon which\nhe sat, as though looseness of seat might operate prejudicially, as\nin the countless equine dangers which he had braved, was evidently of\nopinion, with Panurge, that a cabbage planter was a man to be envied.\n\nOn the pale clear-cut features of the Australian maiden sat a wondrous\ncalm, not wholly unmingled with mental exaltation, as of a Greek\nheroine devoted to death yet favoured of the gods. The wild night\nbreeze had blown back her hair, yet, as she leaned forward and gazed\nfearlessly at the course, nothing could have improved the statuesque\nease and grace of her pose. One of the rare personages who, either from\ninstinctive adaptation or finished training, seem identified with all\nsea life and adventure from the moment they touch the plank of boat\nor vessel, she appeared born to rule and glory amid the perils of old\nOcean.\n\nAs she looked forward into the driving gale, a steady lambent light\nshining out of her clear dark eyes, Ernest Neuchamp thought he could\ntrace more of the enjoyment that fearless natures extract from danger\nthan of even reasonable apprehension in the girl's whole mien and\nbearing.\n\n'Well, if we are to go down to the shades below,' thought he, 'no one\nappears to have more than a very slight objection to the cruise; Jack\nand myself, as being mere landsmen, alone excepted. Well, perhaps our\nluck will pull us through this time.'\n\nAs if in response to his unspoken half-thought, half-prayer, Paul\nFrankston broke the silence by saying with a different tone in his\nvoice from its last intonation, 'By George! I believe poor little\n_Haid\u00e9e_ will do it yet. Yonder's the point, and I think we shall just\nbe able to slip into the inner cove. Antonia darling, what are you\nthinking about?'\n\n'I was thinking of my mother,' said the girl dreamily, as with an\neffort she changed her position, and reverted to an everyday expression\nof face and manner. 'I wonder if people know one another at once in\nthe spirit world. Papa! I think I must begin to put a check upon your\nboat-sailing tastes, or you must get a Kanaka crew that can't be\ndrowned. You are a little too venturous even for _me_.'\n\nAs the boat glided in towards the Morahmee pier, and one by one\nthankfully exchanged the wet and slippery planks for the solid\nstonework, the sky darkened yet again, and the storm in its might swept\nover the angry waste of waters outside of the sheltered nook as if a\nfresh blast had been unchained among the far south ice fieldss, and had\nhasted to the gathering where wind and sea revel in their mirth, and\nwhere many a tress of mermaiden hair mingles with the trailing ocean\nsea-flowers.\n\nMr. Windsor spoke no word until they had nearly reached the garden\ngate. Then he said respectfully, but firmly, 'I think I've seen all I\nmean to see of Sydney harbour, sir. I don't care if I never go fishing\nagain, except off a river bank. If any one says there's three heads and\nnot two, I shan't say whether they're right or wrong. The next time\nI'm a little tired of being John Windsor I shall stumble against Ben\nBolt's hind-legs; but no more salt-water pleasure parties. I ain't on.\nGood-night, sir.'\n\nMiss Frankston did not appear at the dinner-table, but her father and\nErnest recompensed themselves for their exertions and anxieties by a\ncomparative liberal use of the wine-cup.\n\n'Stick to that port, my boy,' said the old gentleman; 'one needs a\ngenerous wine after our little adventure\u2015and a glass of \"hot stopping\"\nwon't do any harm afterwards. It was rather a near thing; fact is, I am\n_not_ quite cautious enough, and fancy it like old times, when, with a\nMaori boat's crew, or these Kanaka fellows, it was next to impossible\nto be drowned. Drowned! If I'd gone down in my old whaling cruises\nevery time I've been in a stove boat, out of sight of the ship, too, I\nshould have been drowned a dozen times.'\n\n'Are accidents frequent in this port?' said Ernest.\n\n'Well, there are not so many as you might expect, considering the\nnumber of yachts and sailing boats; but still they occur from time\nto time. You saw the way the southerly gale came on to-night. Well,\nwith an awkward crew such a boat as ours would have been bottom up now\nas sure as we sit here. There was Colonel Bigges, who used to live at\nPoint Piper; he was always boating, not a bad hand, but of course, no\nsailor. \"Colonel,\" I said, one day, \"you will have a trip too many if\nyou don't mind.\"\n\n'\"What makes you think so?\" says he; \"I'll sail you for anything you\nlike. I'm going out next Saturday.\"\n\n'Something made me say, \"Then don't take the young ladies out, to\noblige me.\" He had two daughters, nice pretty girls they were, too.\nWell, for a wonder, he minded what I said. What was the consequence?\nHis boat was upset a mile from shore; he had two men-servants in the\nboat; they were drowned. The Colonel only reached land by the help of\nhis sons, who were splendid swimmers. If the girls had been of the\nparty nothing earthly could have saved them.'\n\n\n\n\nCHAPTER XVI\n\n\nThe pleasant days wore on until the less pleasant idea began to take\nshape in Mr. Neuchamp's mind that it had become necessary to consider\nthe route once more. This sojourn in Capua could not be indefinitely\nprolonged. Either he must go back to Garrandilla or he must make\npurchase of a station on his own account.\n\nAfter due consideration of the Garrandilla scheme it became apparent\nthat another year of the routine life which he recalled would be\nunendurably dull, whereas a new station, his own property, a cattle\nrun\u2015for he was resolved to have no other\u2015would abound in novelties, and\nabove all, in opportunities for carrying out his long-cherished plans\nof reform.\n\nThe only difficulty in his path would be Paul's uncompromising desire\nto benefit him after his own fashion. For mysterious reasons he had\napparently decided that he, Ernest, was not fit to run alone, in\na pastoral sense, for another year at least. Mr. Neuchamp steeled\nhimself to attack his provisional guardian on this point on the very\nnext opportunity. He would enlist Antonia upon his side. He would\nrecapitulate the reasons which caused him to consider himself the equal\nin experience of some pastoralists who had been all their lives in the\ncountry. Surely a man did not come ten thousand miles across the sea\nto a new, not to say unexplored country, to spend his life in looking\non! He would press Paul hard. He would convert him, and then, hey for\nEldorado, for Arcadia, for Utopia, with laws and ordinances framed by\nDictator Ernest Neuchamp.\n\nWhile at the club, an institution which became more pleasant in\nhis eyes daily, and where he steadily enlarged the number of his\nacquaintances, he kept his ears open as to opportunities for buying\nstation property advantageously. He had at one time been fixed in the\nidea of purchasing the cattle station of Mr. Jermyn Croker, about which\nthat sceptical philosopher and Mr. Frankston had interchanged various\npleasantries more or less acidulated. But it so chanced that among the\nhonorary members who made their appearance from time to time at the\nclub, and enlivened or impressed its ordinary society, came a squatter\nfrom another colony named Parklands.\n\nWith this young gentleman Ernest was much taken, and they soon struck\nup a strong intimacy. Mr. Parklands was Australian-born, but not on\nthat account to be credited with any deficiency of energy; on the\ncontrary, he possessed so much vigour of body and of mind that if\nhe had degenerated in any way (as is a received theory with certain\nwriters), his progenitors must have been perfect steam-engines. He was\nwell known to have explored a very large proportion of the Australian\ncontinent, to have formed, managed, bought, or sold at least a score\nof cattle and sheep stations. His transactions comprised incidentally\nthousands of cattle and tens of thousands of sheep. He had recently\nreturned from another colony where he had acquired an immense area of\nnewly-discovered country. He was on that account, he stated, ready to\nsell the remnant of his property in New South Wales on favourable terms.\n\nLal. Parklands was popular. A good-looking, pleasant fellow, went in\nfor everything\u2015billiards, loo, racquets, dinners, theatres, and balls,\nwith the same zest, energy, and enjoyment which he threw into all\nhis business operations. He strongly advised Ernest to 'tackle old\nFrankston,' as he expressed it, upon the subject of his independence,\nand to go in for a station on his own hook without delay.\n\n'It isn't because I'm selling out myself that I say it,' he added,\n'but the fact is, cattle are as low as they can possibly be, and the\nnext change _must_ be a rise. What do you say, Croker?' he asked of\nthat gentleman, who now lounged up. 'You have had something to do\nwith lowering the people's spirits about their stock. If you'll come\nto Queensland with me next time I want to buy there, I'll pay your\nexpenses.'\n\n'It is apparent,' replied that gentleman, 'that somebody is sure to\nswindle Neuchamp, and you may as well do it as any one else. I thought\nI was to have the honour, from what old Frankston said, but I suppose\nyou have made highly- representations after the manner of\ncornstalks.'\n\n'You are fatally wrong, as usual, Jermyn. I've made a pot of money out\nof Rainbar, and if Neuchamp buys it and does as well, he'll be able to\ngo back to Europe as a successful colonist in no time.'\n\n'If he takes Mr. Parklands as his model in speculation, management, and\nconversation, he _must_ succeed in everything he undertakes,' said Mr.\nCroker with ironical approbation.\n\n'Come and have some sherry, old Bitters,' said Mr. Parklands\ncheerfully, 'and then I'll thrash you at billiards. Never saw an\nEnglishman I couldn't give points to yet. Can't lick us.'\n\nRoused by this national reflection, Mr. Croker offered to play for\nanything he chose to name, and Ernest betook himself to Morahmee. He\nhad determined to open the parallels without delay.\n\nFull of this noble resolution, Mr. Neuchamp only waited until Antonia\nhad departed from the dining-room to commence the momentous project.\n\n'I begin to feel,' said he artfully, 'that my holiday is drawing to a\nclose. I don't think I ever enjoyed town life thoroughly before. But\none can't always be on furlough. I must join my regiment\u2015must be off to\nthe bush again.'\n\n'What's the hurry?' said Mr. Frankston. 'Nothing much ever goes on at\na station until the cold weather sets in. You will find Garrandilla\nwretchedly dull after club-dinners, ball-going, boat-sailing, and all\nthe rest of it. Even the verandah here is considerably better of a hot\nevening than those rascally slab huts.'\n\n'You have been a sailor, Mr. Frankston,' said Ernest, 'and you know\nthat when the sailing day comes, and the wind is fair, Jack must get\non board. I don't suppose you find Captain Carryall would make much\nallowance for lagging.'\n\n'No, faith. He would need to be a smart fellow to stand before Charley\nif he kept him humbugging about when the bark was empty and the whaling\ngear in trim. But you are not shipped as an A.B. anywhere as yet. Make\nthe most of your young life, Ernest, my boy\u2015it won't come twice.'\n\n'There is a time for all things,' rejoined Mr. Neuchamp, who had small\nreverence for play in the abstract; 'I came to Australia principally\nfor work, and I shall be uneasy until I am fairly in harness. But\nwithout beating about the bush, I am impatient to purchase a place of\nmy own, and unless you are inexorably averse to the step, in which case\nI should give in, I feel the strongest desire to make a start on my own\naccount.'\n\n'Why won't you be content to sail by my orders for a while?' said Paul,\nmuch disturbed. 'If you knew how many young fellows I have seen ruined\nall for the want of a little delay, for want of following the caution I\nhave given you, you would not be in such a hurry to risk your fortune\non a throw.'\n\n'But consider,' said Ernest, perceiving, as he thought, a slight sign\nof compromise in Paul's candid face, 'I am not exactly like other\nyoung fellows, with the same intentions. I have had in reality more\nexperience in the time of my novitiate than they have had in double\nthe period. I have had road work, station work, sheep and cattle\nmanagement. I have had, from peculiar circumstances, more than ordinary\nadvantages of practical teaching, and I do myself consider that unless\nI am duller than ordinary, I may be trusted to manage a moderate-sized\ncattle station, if you will help me with your advice in the purchase.'\n\n'Well, I don't know,' said Paul, passing through into the verandah, and\nlighting the cigar of reflection, 'I don't know but that, as you say,\nyou have had rather more luck than common in your apprenticeship. You\nhave been before the mast, too, as we say on board ship, and that is\na great help. You are as steady as a church. That's all to the good,\nno doubt. But what I am afraid of is a sudden turn in prices\u2015stock\ncan hardly be lower, to be sure. Well, well\u2015you can only risk it. But\nI don't want to see you, as I have seen many a good fellow, lose his\nmoney and the best years of his life, and either die, go to the devil,\nor settle down to the banishment of an overseer's berth.'\n\n'Like poor old Geoffrey Hasbene,' said Ernest; 'I don't think I could\nquite endure that, though the old fellow is resigned enough.'\n\n'I remember him well enough,' said Paul; 'it's a good while since I\nheard his name. I have seen him ruffling it with the best, and the\nowner of a good station. He was not very fast either.'\n\n'And what ruined him?'\n\n'Partly bad luck, partly a careless, easy-going disposition. He thought\nmore of his house, his stables, and his garden than he did of his\nstock, and that was the end of it. Mind you take warning by him.'\n\n'I hope I shall\u2015but now that you think I may really make an attempt to\nfly off the nest, might we not settle something about the probability\nof a purchase to-night?'\n\n'Yes\u2015perhaps\u2015yes,' answered Paul, seating himself with a resigned and\ngloomy air. 'I suppose you have heard of a place or two at the club.\nThere's a good deal of business done there. Has Jermyn Croker said\nanything to you further?'\n\n'Scarcely, but a young squatter named Parklands has a place that seems\nsuitable; he appears a nice fellow enough.'\n\n'Oh! young Parklands\u2015humph! Very nice boy\u2015quite sharp enough, but I\ndon't suppose could let you in very extensively. Well, I'll inquire\nto-morrow; better leave that part of it to me. I'll see about Croker's\nplace also. Plenty of time. Market full of sellers, and very few\nbuyers. Cash very scarce. But that's all in your favour. Antonia!'\n\n'Here, papa,' said that young lady, joining them. 'What is the matter?\nhas anything happened, that you look so serious?'\n\n'Well, that's as it may be; Ernest here is bent upon buying a station\nat once, and I have been trying to show him the prudence of waiting.'\n\n'But he can't wait years and years,' said Antonia, taking, to Mr.\nNeuchamp's great joy, her powerful aid to his side of the suit. 'I\ndon't think _you_ would have done it either, you impetuous old dear;\ndidn't some one run away to sea like a naughty boy, and come back in a\nship of his own\u2015eh?'\n\n'And suppose I did, you saucy puss, didn't I run the risk of being\ndrowned, starved, burned, roasted alive, and all sorts of deaths; and\nif I had a son, I should think it my duty to warn him against the sea,\nas the worst profession in the world.'\n\n'And he would think it his duty to go in spite of you. Not that Mr.\nNeuchamp would do anything contrary to your advice, I am sure,' said\nAntonia with a becoming blush, 'but I think he is wise in wishing to\nhave a place of his own, and begin life in earnest. Besides, everybody\nsays a cattle station is so pleasant, I almost think I could manage one\nmyself.'\n\n'Pity they should be so far from Sydney, or you might come and try,'\nsaid Ernest, with a grateful inflection in his voice. 'Waratah would\ndistinguish herself in a camp, I feel sure.'\n\n'I daresay we should do nearly as well as certain\u2015hem\u2015English people,'\nsaid she mischievously. 'I have always thought from what I have heard\nthat life on a cattle station must be quite the romance of the bush.\nThere is a sort of Bedouin flavour about it, with a necessity for good\nhorsemanship that would fascinate me if I were a man.'\n\n'Go and play something, like a darling,' said the old man. 'I feel a\nlittle like my namesake in the Bible\u2015Saul, I mean\u2015as if music could\nconjure the evil spirit out of me.'\n\nIt was finally settled, therefore, on that fateful evening, that Mr.\nFrankston should inquire about the station which Mr. Parklands had for\nsale, and decide whether it or that of Mr. Jermyn Croker would be the\nbetter investment.\n\nThe preliminary was carried out with business-like precision. Mr.\nFrankston called upon the cheerful Parklands and the desponding Croker\nand extracted from each, their separate temperaments notwithstanding,\nthe area of the runs, the number, age and sexes, and condition of\nthe cattle, and many other particulars, including the lowest price,\nnecessary to a true and just knowledge of the bargain. He, besides\nthis, set on foot inquiries among those of his numerous constituents\nwho happened to be neighbours, and finally, after all these\nprecautions, told Ernest that he thought Parklands' place seemed the\ncheaper, and that when it was formally placed under offer he had better\ngo and inspect it.\n\nThe negotiations having proceeded to this desirable length, Mr.\nNeuchamp's satisfaction was unbounded. He saw himself placed in the\nposition which he had long coveted, and pictured day-dreams. He would\nbe a territorial magnate, having the right to rule over a region\nlarger than the whole county wherein his paternal estate was situated.\nIf he could not impose new laws he could justly administer the old\nones. Visions of improved breeds of cattle, of a different method of\ntreating the station hands, of developing the capabilities of the\nrun, of making a fortune in a few years, and revisiting England. All\nthese achievements rendered possible by that first bold step in actual\ncolonisation, the purchase of a run, passed through his brain, with\nthe lightning-like rapidity that was wont to characterise such mental\nevolutions, but which had of late been more infrequent. He did not\nconfide these plottings against the peace of the district which he\nwas to invade to Antonia. It was not from any decline of sympathetic\nfriendship, but chiefly because of late that young lady, now ever\nready to approve of his wish to begin upon his own responsibility,\nseldom approved of his projects in advance of the age or of Australian\nordinary bush customs, which she maintained had been formed by very\nshrewd and successful men.\n\nIt was necessary that Mr. Parklands and Mr. Neuchamp should meet at\nthe station, so that he himself should be able to exhibit its special\nadvantage. But that gentleman had far too many engagements to permit of\nhis starting off at once upon this particular errand.\n\nIt was therefore arranged that, on a certain date, Ernest should make\nhis appearance at a far inland township named Bilwillia, where he would\nmeet Mr. Parklands, who by that time would have 'come across' from the\nBurra-warra-nonga, or some such easily pronounced locality, which he\nwas compelled to visit regarding the approval of a small lot of ten\nthousand store cattle and fifty thousand wethers, under offer to him\nfor the Melbourne market.\n\nAs nothing was to be gained by immediate departure, Mr. Neuchamp\navailed himself of this unexpected holiday with unrestrained\nsatisfaction and enjoyment. He feasted upon his favourite authors\nand upon the newer publications which he was enabled to procure in\nSydney, thanks to the excellent public and private libraries. Antonia\nand he renewed their literary labours and criticisms; and that young\nlady immortalised herself and completely subjugated Jack Windsor, by\nmaking a water-colour sketch of Ben Bolt in an attitude of mingled\nfear, wrath, and desperation, when unexpectedly confronted with a\nGerman band. It was Mr. Windsor's deliberate conviction, emphatically\nexpressed, that 'a young lady who could take off a horse like that\u2015the\ndead image of him\u2015could do anything.' In truth, horse and man formed,\nat the moment, a study for an artist. The former with glaring eye,\nopen nostril, sudden arrest of action, and capacity for the wildest\noutbreak; the latter sitting watchful, statuesque, centaur-like, a\npersonification of equestrian strength and grace.\n\nAs the distance to Bilwillia was great, and its reputation unfavourable\nin the matter of horse-flesh, Ernest determined not to risk the safety\nof Osmund, whom he left in snug quarters near Sydney.\n\nMr. Windsor, much to his disappointment, received news of the illness\nof his mother, the only relative in the world, as he had often stated\nto Ernest, for whom he possessed a grain of affection. He was more\nstrongly moved by the sudden announcement of her being sick unto death\nthan Mr. Neuchamp thought possible.\n\n'I don't half fancy,' he said, 'sloping and leaving you to go and take\ndelivery of the place all alone by yourself, sir; and they say Mr.\nParklands knows a thing or two. However, he's an off-handed chap, and\nthe best thing you can do is to leave the whole jimbang in his hands\naltogether. If you go barneying about calves, or counting horses that's\ngive in, he'll best ye, as sure as you're born. So your dart is to say\nyou don't know nothin' about cattle, and drop him in for the drafting\nout calves under age, and all them sorts of things. Then, as he's a\ngentleman, he's bound to give you a show. I ought to be along with you,\nI know. But I haven't seen my poor old mother for five years good, and\nI _must_ go, if I was never to make a rise again.'\n\nJack departed, but he somehow found time to call at Walton's inn on his\nway to Appin, where his old mother lived and where he had spent his\nchildhood. Ben Bolt had but little breathing time once clear of Sydney\nstreets, and that wild steed of the desert was sensible of a decidedly\nquickened circulation as he was pulled up in the inn yard, and turned\ninto a stall after a hurried and headlong manner.\n\nAs Mr. Windsor passed the door of the inn, he observed an immense\nquadruped hung up at the posts, which, but for the saddle and bridle,\nmight have been taken for a strayed waggon-horse. The length of the\nstirrup-leathers conveyed to a bushman's intelligence the fact that the\nrider of this Gargantuan steed was an individual of unusual length of\nlimb.\n\nPassing quietly into the bar, and thence into a small parlour devoted\nto the family and particular friends of the host, he discovered the old\ncouple, Miss Carry, and a stranger, whom he immediately associated with\nthe charger aforesaid and with the district of the Hawkesbury.\n\n'Well, Mr. Windsor, and who'd have thought of seeing you?' said\nMrs. Walton. 'Have you and Mr. Neuchum\u2015and a nice gentleman he be,\nsurely\u2015been in Sydney all this time? And where are you leaving for now?'\n\n'We've been in Sydney all the time, and a very jolly place it is, Mrs.\nWalton,' said Jack, answering the old woman with his tongue and Carry's\nquick glance with his eyes. 'Mr. Neuchamp's just going up the country\nto look at a cattle run, and I'm going home to Appin for a short spell.'\n\n'What are you going to do there?' said Carry; 'I thought you went\neverywhere with the young gentleman?'\n\n'My poor old mother's very bad,' said Jack, looking rueful, 'and I must\nbe home to-night, some time or other; but I don't think anything else\nwould have kept me from going up with the master, to see him all right\nwith this new station as he's going to buy.'\n\n'Do you\u2015live\u2015at\u2015Appin?' said the stranger young man, taking about a\nminute for the enunciation of each word, and speaking in a drawling,\nthough not nasal, monotone.\n\n'When I'm at home, which is about once in five years, I do,' answered\nJack. '_You_ live on the Hawkesbury, and haven't ever been far from the\nriver, I'll swear.'\n\n'So I do, at Rooty Hill Farm, Nepean Point,' said the New Hollander\nwith a smile, which broke first upon the edge of the round plump\nface and gradually spread over it like the eddy in a pond.\n'How\u2015did\u2015you\u2015come\u2015to\u2015know?'\n\n'By the look,' said Jack coolly; 'they don't grow such men anywhere\nelse in the colony, except on the Hawkesbury flats. My name's Jack\nWindsor. What's yours, old nineteen stun?'\n\n'I ain't nineteen stun, I'm only seventeen,' said the youthful giant,\nwhose voice, however, did by no means correspond with his stature,\nbeing mild and small of timbre. 'My name's Harry Homminey, and I'll\nback our land to grow more corn to the acre, let alone pumpkins, than\nany farm this side of the Blue Mountains.'\n\n'Like enough,' answered Jack indifferently. 'Shouldn't wonder if you\ntook to pumpkins very kind when you was young. They're great feeding\nstuff. But your Windsor and Richmond farms is only handfuls after all.\nHow many acres have you got?'\n\n'A hundred and thirty-two,' said the Netherlander, with just pride,\n'and never a tree or a stump on it.'\n\n'Well, what's that?' demanded the denizen of the waste. 'Why, a child\ncan take up three hundred and twenty acres in the bush anywhere. I\nwouldn't be bothered with land unless I had a whole section to begin\nwith.'\n\n'It's a deal better than no land at all; and that's about what you\nhave, I expect,' said the agriculturist, gradually coming to the\nopinion and belief that Mr. Windsor was disposed to disparage him and\nhis fat acres before Carry Walton.\n\n'Never mind what I have, and keep a civil tongue in your head,' said\nJack wrathfully; 'I'll give that round face of yours such a pasting\nthat they will not know you from a Lower Narran man, only by your\nweight, when you go home. But I won't be cross to-night, and the poor\nold mother dying for all I know. Good-bye, Mrs. Walton; good-bye,\nCarry. I must be off.'\n\nMr. Windsor departed into the night and they saw him no more, but I am\nstrongly of opinion that he managed to telegraph something to Carry\nbefore he gained his saddle, and if it meant unalterable affection as\nshe understood it, whether it was the automatic process, or Morse's,\nwho shall say?\n\nCertain it is that she returned to the room with a serene countenance,\nand listened apparently with intentness to the somewhat uninteresting\nconversation of the man of maize and pumpkins, who eventually mounted\nhis massive charger and trampled along the highway towards the rich\nlevels of Nepean Point.\n\nMr. Neuchamp was so extremely anxious to make a commencement upon\nthe foundations of his own experience and management that he left\nSydney a week or two before the actual time necessary to reach the\ntownship of Bilwillia, where he was to make rendezvous with Mr.\nParklands. He purchased for himself a befitting hackney, and, not\nhaving Jack Windsor's aid, was beguiled into the possession of a\nstiff, short-legged cob, which his English tradition led him to\nbelieve would be the exact animal for a long journey and indifferent\nkeep. Having gone part of the way by rail, he managed to reach the\nunromantic and extremely hot township of Bilwillia more than three days\nbefore Parklands could by possibility arrive, unless under the highly\nimprobable supposition that he had more time than he knew what to do\nwith.\n\nMr. Neuchamp was, as we have had before occasion to explain, by no\nmeans destitute of resources. If there was any interest whatever to be\nextracted from a locality, he was a likely man to discover and avail\nhimself of it. But he afterwards confessed that he then and there felt\nmore nearly reduced to the unphilosophical and indefeasible position of\nutter dulness than he could have believed possible.\n\nFor if any place could possibly combine extremest degrees of isolation,\nmonotony, dreariness, and depressing discomfort, that place was\nBilwillia. It straggled around the edge of a sombre watercourse, the\nditchlike banks of which dropped perpendicularly through the clay, as\nif dug by some savage engineer centuries since. Around, anear, afar,\nall was plain and sky. The arid landscape was as boundless, monotonous,\nas the sea. The salsolaceous plants, within ten feet of the unbarked\npine-posts of the rude verandah, were identical in appearance with\nevery plant for a hundred leagues. Hill nor tree nor stone was there\nwithin a square of a thousand miles.\n\nThere were no books; no newspaper, save the _Bourke Banner_, a\nfortnight old, containing sundry local incidents, a short leading\narticle, and a lengthy advertisement of Holloway's Pills.\n\nOn the fourth day, about the exasperating period of noon, when the\n'blue fly sung in the pane,' and all the slow torture of Mariana in the\nmoated grange transposed to southern latitudes seemed to be in process\nof representation, Mr. Neuchamp, to his excessive delight, made out two\nseparate cort\u00e8ges arriving from different directions. Both comprised\nmounted men and spare horses, and either of them might well be the\nlong-expected Parklands. They were plainly steering across the wide\nplain for the Bilwillia Inn.\n\nThe first cavalcade was headed by an unusually tall athletic-looking\npersonage riding a well-bred powerful horse, which evidently made\nlittle of his somewhat unfair weight. A sharp-looking elfish black boy\nand a stockman, at some distance behind, drove several spare animals,\nincluding a packhorse, upon the tracks of their leader. As they arrived\nat the inn, the gentleman in advance hung up his horse and walked into\nthe house, while his attendants proceeded to unsaddle the whole troop.\n\nAlmost immediately after the full and careful observation of this\nparty had been concluded by Mr. Neuchamp, rendered desperate by long\nabstinence from decent society, the second group gradually 'came up\nfrom the under world,' like a strange sail, and disclosed the form\nof a charioteer, with an attendant and spare horses. The driving was\nlike unto that of the son of Nimshi, whom, in the matter of pace, Mr.\nParklands resembled. And that energetic and punctual personage it\nproved to be.\n\n'How are you, Neuchamp?' he called out cheerily, jumping down from an\nexpress waggon with a driving seat. 'Splendidly punctual, are we not?\nHad to come sixty miles yesterday, and five-and-thirty this morning.\nCan't lick us!'\n\n'It was very good of you,' said Ernest most sincerely, 'to make a push.\nI do not know what I should have done if I had had to wait another day\nhere.'\n\n'You don't mean to say you came here before yesterday?' cried Mr.\nParklands in tones of horror and amazement.\n\n'I came three days ago, I am sorry to say.'\n\n'Three days!' groaned Parklands, 'in this cursed hole. I wonder you\ndidn't hang yourself, or go on the spree. But Englishmen never do that\ntill they have been three years out from home.'\n\n'Three years!' said Ernest, rather amused. 'Then there is a possibility\nof my taking to inebriety in course of time. It is rather alarming!'\n\n'I have known many a good fellow take to it. All the same, I shouldn't\nsay it was much in your line though, in three years or thirty. But\ndidn't I see a tremendous long fellow go into the house, just as those\nother horses came up?'\n\n'There was a very tall man at the head of yonder party,' said Ernest,\nlooking over at the black boy and his companion, who was lighting a\nfire and preparing to cook. 'He is now in the hotel.'\n\n'Aymer Brandon for a thousand!' said Mr. Parklands excitedly. 'A very\nold friend of mine, and the best fellow going. I suspect he has been\nover to his runs, on the Warrego. I'll soon lug him out.'\n\nWith this he dashed into the inn, and shortly reappeared in company\nwith the tall gentleman, who, indeed, only required to be seen once to\nbe easily recognised in future.\n\nMr. Aymer Brandon was presently introduced with great and joyous\n_empressement_ by Mr. Parklands, who hung about him with schoolboy\nabandon. He was so considerably above six feet in height that\nMr. Neuchamp and his friend, both well-built, middle-sized men,\nlooked abnormally short beside him. Broad-shouldered, deep-chested,\nstrong-limbed, his vast symmetrical frame seemed equally adapted to\nfeats of strength or of activity.\n\n'We are in luck, Neuchamp; Brandon happens to be going down to\none of his stations below Rainbar, and we can join forces\u2015that is\nhorses\u2015and tool down luxuriously, four-in-hand. Can't lick us! I had a\npresentiment we should come out double sixes when I started.'\n\nMr. Neuchamp thought it would be most pleasant travelling.\n\n'You see, your cob can go with the spare horses, which the boys will\ndrive after us. Couldn't improve on the caravan if we'd planned for a\nmonth.'\n\nErnest would have modified his anticipation of comfort had he been\naware that the larger proportion of the horses depended upon for this\nrapid and efficient journeying were, at that very moment, wholly\nunbroken to harness, having, so to speak, never seen a collar.\n\nBut this uncertainty of the future was as yet hidden from him, and\nthe whole party proceeded to lunch, which, in consequence of much\nexhortation, with promises, and even threats, from Mr. Brandon and his\nfriend, was, with the help of the omnipotent bitter beer of Tennant, by\nno means to be scorned in the wilderness.\n\n'What's your waggon like, Sparks?' queried Mr. Brandon privately.\n\n'Slap-up!' answered he with confidence. 'There's no brake; but that\nwon't matter, as two of the horses have been in harness before,\nsomewhere. We'll do the hundred miles to Rainbar in two days\ncomfortably.'\n\n'Nothing more complete could be hoped for on the Darling,' pronounced\nhis friend calmly, 'so that's settled. I subscribe the black boy and\nfive horses, which we can break in on the road. I hope the I.\u202fP.\n(intending purchaser) is a good plucked one, or he is like to turn back\nbefore reaching Rainbar, if he journeys with us in the waggon.'\n\nAn early start was arranged for next morning. Accordingly the half\nuplifted disc of the red sun of the desert irradiated the whole party\non the farther bank of the river fully equipped for the road.\n\nAymer Brandon held the ribbons, while Parklands took the box-seat, in\norder to be ready in case of a complication with the scratch team. Mr.\nNeuchamp sat behind in company with Tom Fuller, a Rainbar stockman and\npast-master in smashes of every kind, sort, and description on wheel or\nin saddle, on land or water, mountain or plain. The black boy, Eachin,\nrode in charge of the spare horses, amongst which was turned Mr.\nNeuchamp's Sydney cob. One of the unbroken horses was considerately\nplaced in the near wheel, the other in the off lead. It being evident\nthat all precautions had now been taken, Mr. Brandon sang out 'Let go!'\nto the volunteers who had assisted at the ticklish business of putting\nto, and with a shout, a double-thonger, half a dozen wild plunges, and\nan innocuous kick, the team settled down on the utterly perfect, firm,\nsandy road to something like racing speed.\n\nThere was little conversation for the first mile. Without a brake, all\nthat could be done was to hold the team straight, shooting the gullies\nfairly as they came. Ever and anon, as a bar touched his hocks, the off\nleader kicked gaily over the traces, but finding the outer side yet\nmore uncomfortable, kicked back again with discretion beyond his years.\n\nThree miles had been swallowed up ere the team steadied ever so\nslightly. Then Brandon got his pull at them.\n\n'Good travelling, Neuchamp?' said Mr. Parklands. 'Do the journey easy\nby to-morrow night. The day after I'll show you the finest lot of\ncattle in Australia\u2015all reds, whites, and roans. Can't lick 'em!'\n\n'Are they quiet?' asks Mr. Neuchamp, as a vision of back country cattle\nblacks and brindles, which he mentally vows to improve off the face of\nthe earth, crosses his brain.\n\n'Quiet?' queries Parklands derisively, 'why, you can't kick' em out of\nyour way.'\n\n'I am truly glad to hear that,' says Mr. Neuchamp heartily; 'quiet\ncattle are so much pleasanter to draft.'\n\nA ten-mile stage, at the highly meritorious pace alluded to, having\nbeen slipped over, the monotony of Australian steppe-travelling was\nvaried by the introduction of two of Brandon's troop. They were,\ncomparatively,\n\n Wild as the wild deer, and untamed,\n By 'trace and collar' undefiled.\n\nThe first introduced was a grand-looking old black horse, with a\nsuperabundance of pluck and one hip down. He was substituted for the\noff-side leader, who was turned over to Eachin. The alteration was\neffected in five minutes, and old Darkie sailed off as though he had\nbeen carefully coached since colthood. This state of affairs was\nobviously too good to last. Not accustomed to winkers, the veteran,\ncatching his toe in a root, went down like a shot. Now occurred a\nfirst-class complication.\n\n'Total wreck, with loss of all hands,' concludes Mr. Neuchamp.\n\nNot so. Parklands and Jim Fuller are down almost as soon as Darkie,\nand fasten on the horses like bull terriers in a rat-pit, while Aymer\nBrandon sits calmly in his place, and delivers his orders with the\nimperiousness of the skipper whose mainmast has gone by the board.\n\nThis was the situation: when Darkie fell the team was doing ten miles\nan hour. The wheelers swept over him, and he was brought up by the\nfore-axle of the waggon. Both check-reins were carried away and the\nlead bars broken. The near leader dashed round the back of the coach,\nwhere he was pulled up with a round turn by the strong arm of Mr.\nBrandon, who was engaged, as to his whip-hand, in rib-roasting Darkie\nto make him 'come out of that.'\n\n'Here, Jem!' he sang out, 'freeze on to this brute behind while I make\nthat three-cornered calamity come out of his earth.'\n\nDarkie, finding his position under the waggon becoming too hot, emerged\ndexterously, and stood upright under the off-wheeler, raising that\nunsuspecting animal's hindquarters upon his back. Having achieved\nwhich he awaited the next move, which promptly came in the shape of\ntwo terrific double-thongers. Upon this Darkie darted out, and at once\ncommenced to feed till again wanted.\n\n'My dear Parklands,' commenced Mr. Neuchamp, underrating the variety of\nbush expedients, 'this is indeed unfortunate. I suppose we shall have\nto camp here until the harness is repaired.'\n\n'Camp!' exclaims Parklands in wild amaze, 'we'll be off in ten minutes.\nCan't lick us.'\n\nAnd in good sooth, a pair of spare bars having been rigged, and the\nchecks spliced with bush buckles, within fifteen minutes they _were_\nonce more under weigh and doing their ten knots an hour comfortably.\n\nAt two o'clock Toolara, a station which was the property of Mr.\nParklands, and distant about seventy miles from Rainbar, was reached;\nthere a good luncheon was secured. At four o'clock start was made to do\nthe remaining twenty miles between them and Gregor's shanty, where the\nnight was to be passed.\n\nAt Toolara the party was augmented by a tame dingo, belonging to Mr.\nParklands. He was most appropriately named Beelzebub. For, in his own\nrealm, the vast kingdom of this chief, he reigned unequalled.\n\nA magnificent specimen of the Australian dingo, bright orange as to\ncolour with a white ring round the neck, he boasted of long sweeping\nhair and was feathered like a Gordon setter. The intelligence\nexpressed by his flag was marvellous, and its language various and\ncomprehensive as that of a semaphore. His face alone, if fate had\nbut permitted the painting of it to Sir Edwin Landseer, would have\nbeen well worth a thousand guineas at the Royal Academy. Plainly\nvisible therein were foresight, decision, craft, and self-control, in\nsufficient quantity to furnish forth a Cabinet Ministry. You could\nnot look upon the calm countenance without feeling a conviction that\nagainst all ordinary foes that gifted animal was safe, as Achilles\nupon the Trojan plain. Like unto the Homeric hero he was invulnerable\nsave in one point, the poisoned bait, that talismanic safeguard which\nassures the pastoral future of Australia.\n\nTo his credit be it stated, Beelzebub did not in any way identify\nhimself with the party, who were, through this discreet conduct, not\nincluded in the anathemas he was destined to bring down on his own\nhead. He kept about a quarter of a mile from the road, in a course\nparallel with the waggon.\n\nFive miles had been travelled when the first victim to his fiendish\narts appeared. Norval, leisurely boiling the evening camp kettle, the\nwhile watching his flock peacefully nibbling towards the yard, is\nthunderstruck to see those splendid wethers, filled with salt-bush\nand water, suddenly sundered as if by a red streak of lightning, and\nthe division farthest from him sent across the plain racing for their\nlives, with the devil himself whipping in.\n\nThen does that unhappy Gael pursue, with his longest strides and\nAnglo-Ossianic oaths, but to no purpose. The astute dog-fiend, when\nthe fat-laden flyers had collapsed suddenly and hopelessly, through\nsheer breathlessness, turns him round, curls his noble flag far over\nhis back, and, like the famed coyote, 'vanishes through an atmospheric\ncrack.'\n\nThis trifling adventure was witnessed by Brandon, Parklands, and Mr.\nNeuchamp with great interest. The sheep did not belong to them. The\ndog was fully believed to be a dingo errant, running his diurnal stage\nof duty. And, in the end, it would conduce to the benefit of the\nmerino interest, as Norval would be roused into a course of spasmodic\nbait-laying, which possibly might bring a few genuine freebooters\noff their perches. Aymer Brandon, after a hearty laugh all round and\nthe assertion from Sparks that they 'couldn't lick him,' dropped the\nwhipcord on to his team and swept away over a splendid salt-bush\nplain, level as a bowling green, though slightly differing in colour.\nAs they threaded a clump of box, the corpse (apparently) of Beelzebub\nwas descried stretched out under a tree, looking rather more dead than\nthe reality. The crafty one permitted himself to be passed without\nthe motion of a muscle, and was no more seen until a mile or two\non, when a cloud of dust, with a red thunderbolt darting to and fro\ntherein, proclaimed the fact that another shepherd was in process of\ndisestablishment.\n\nThe short Australian twilight had commenced, when Parklands took\nthe reins to pilot the coach into a deep horse-shoe bend unknown to\nBrandon, near to the opposite bank of which stood the half-way house.\nAt a nobly undeniable pace did the gallant Sparks tool through the\nglades of mighty red gum patriarchs, the roots of which, long fed by\nriver springs, deep piercing the soft alluvium, had made them loftier,\nbroader, wider of shade than the fatherland. He had shot more than\none polygonum creek, straight and true as an Indian the Saults St.\nMarie's boiling rapid, when Brandon shouted, 'Where the blazes are you\ndriving\u2015slap into the river? I can't see how these nags will take a\nwater jump!'\n\n'By Jove!' said the iron-nerved Sparks, as with a clever sweep he came\nto anchor, the near wheels going several inches over the river bank in\nthe operation, with a drop to the water at an angle of seventy-five or\na hundred feet, 'so I am. Jump out, boys. Can't lick us.'\n\nThe events of the day had occasionally startled Mr. Neuchamp, but his\n_sangfroid_ won the admiration of Parklands and his friend. He had\nexhibited no tendency to jump out before he was told; and Brandon was\nafterwards heard to state his conviction, that if Sparks had charged\nthe Darling four-in-hand with characteristic carelessness of results,\nErnest would have simply sat back and kept his chin up, in profound\nundoubting faith that he would be landed safely upon the opposite bank.\n\nThe horses were promptly unharnessed and turned out amidst luxuriant\npasture, after which all hands crossed the Great River in Gregor's\ndug-out to that gentleman's hotel. An apology for the primitive\nappearance of the place was thought necessary by Parklands, so\nconsiderate ever is the outgoing proprietor to the intending purchaser.\nErnest assured him that, though slightly inferior to the Royal, he\nhad already, since his arrival in Australia, been lodged more humbly.\nHaving witnessed one another's signature in passable whisky, towels\nwere produced, and the dust of the day consigned to the river.\n\nAt ten o'clock P.M. all hands were ordered to bed by Aymer Brandon,\nin spite of Sparks's desire to describe a lovely damsel whom he\nhad met when last in Sydney. She was his sixteenth engagement, but\ncircumstances had compelled an irrevocable parting. Knowing that\nanother whisky would infallibly bring on a retrospective history\nof the other fifteen, Aymer was inexorable and hunted the amorous\nParklands to bed, where he was heard to murmur softly, 'Couldn't lick\nher,' as he dropped off to sleep.\n\nBeelzebub, arising with the lark, promoted the next adventure, as\nfollows: Gregor was out at cockcrow, to kill a sheep for morning\nchops, but found himself all too late. His fold, a hundred yards from\nthe house, was dog-proof, with the exception of the hurdled gateway.\nReaching it, 'all hunger-maddened and intent on blood,' he found\nanother in possession actuated by similar motives. He beheld Beelzebub\nin the very act of devouring a six-tooth ewe\u2015not the class of sheep\nusually selected for slaughter. 'Stiffen those blank dingoes!' roared\nGregor, 'there goes a note!' Charging wrathfully into the yard, and\nunconsciously commending himself by name to his enemy, he assaulted\nthe 'Evil One.' The instinct of the latter came primarily into play,\nthus assaulted unawares, and he sprang at the high slanting poles,\nall vainly. Not Cerberus himself could have cleared them. This false\nstep was but the weakness of a moment. Logical reasoning, the result\nof civilised intercourse, reasserted its sway. Calm as Marlborough, he\nthen comprehended the situation with a glance, and proceeded to execute\nthe only strategical movement possible in the very pressing, or rather\ndepressing, condition of the engagement.\n\nGregor, upon observing his abortive attempt to clear the fence, had\nrushed to the gate. The crafty one, with an innocent expression of\ncountenance, and his flag curled gracefully over his back, trotted\ncalmly towards him. Gregor timed the dog well, unknowing of his\nresources, and aimed a kick at him which would have stove in a\nthirty-ton cutter.\n\nThe Napoleon of dingoes, making a feint as if to dash through the gate,\nstopped abruptly. The harmless boot expended its force and momentum,\nwith some inconvenience to its owner, against the gate-post. Ere a\nsecond _coup de pied_ could be arranged, Beelzebub glided swiftly\nthrough, with his flag erect and waving gently from side to side in\ntoken of approval.\n\nAt breakfast Gregor gave a thrilling account of the havoc wrought in\nhis flock, and solemnly swore that he had lifted the dog, with one\nkick, over the high palisades.\n\nParklands, knowing the culprit and the utter hopelessness of any human\neffort to strike him without consent, felt no uneasiness. He also\nforgot to mention that the dog belonged to him. When Gregor was out of\nearshot Parklands (who was solely a cattle-owner), bursting with pride\nat the prowess of his pet, offered to lay Mr. Neuchamp a cool hundred\nthat Beelzebub, bar baits, should eat all the sheep on any ordinary\nstation in six months.\n\nMr. Neuchamp, not having studied the habits and capacity of the\nAustralian dingo sufficiently to warrant his making a book on the\nsubject, declined the wager.\n\n'If I were you, Sparks,' said Brandon, 'the next time I was annexed by\na young woman and wished to be off the bargain, I should make her a\npresent of Beelzebub. If the \"wily one\" would not in a week sever the\ntenderest domestic ties, I am mistaken in his character. Wouldn't mind\neven laying him against a mother-in-law.'\n\nAn early breakfast of chops, fresh from the slaughtered ewe, a short\nbut exciting voyage in the dug-out, and they espied their 'connecting\nlink,' who was equal to most occasions, standing with his horses ready\nfor harnessing. Their narrow escape on the preceding night was now\nplainly legible in the wheel tracks, _just over_ the brink of the river\nbank, and even the reckless Sparks acknowledged it to have been 'a near\nthing.' Brandon now took the reins, lectured Sparks upon dangerous\ndriving, and spun through the vast umbrageous eucalypti, towards the\nroad.\n\nNeither accidents nor offences occurred during the next twenty-five\nmiles, at the end of which luncheon was spread by the side of a\nreed-bordered lagoon. As they had now entered upon the extensive\nterritory of the Rainbar run, Mr. Parklands caught a horse for himself,\nas also Mr. Neuchamp's cob, with a view to rounding up an occasional\nmob of cattle and proving his vaunt as to their unsurpassed breeding\nand docility.\n\nThe opportunity soon occurred. A small lot of some fifty or sixty\nhead appeared about a half-mile from the road. Away went Parklands\nwith Eachin and Mr. Neuchamp backing up. After a sharp ring or two\nthe cattle stood with the horsemen around them. To Mr. Parklands'\nmortification and Brandon's wild delight, everything being plainly\nvisible from the waggon, a huge coarse-horned, dun- bullock\nsingled out and 'went for' Ernest without more ado. The appearance\nof the brute was appalling, and his intention so obvious that Mr.\nNeuchamp did not hesitate to turn and fly across the plain for his\nlife. The cob, though a fair roadster, was not constructed for violent\nexercise at short notice. He held on gallantly, but _bos ferox_ gained\nperceptibly on him. At the half-mile end his horns were level with the\ncob's quarters, and Mr. Neuchamp had concluded to throw himself off and\ntrust to the brute's continuing his mad career, when the cob, feeling\nthat the game was up, stopped short, throwing his rider over his head.\nThe bullock hurled past them with a snort of wrath and defiance,\ncontinuing his headlong course over the plain, in search of the first\ncongenial scrub. When Parklands came up Mr. Neuchamp was gazing at his\nhorse, which stood with its legs wide apart panting, with streams of\nsweat running down his flanks and even his face. His ears were dangling\nlimply, and he looked very much indeed as if he were going to cry.\n\n'Really, Parklands,' said poor Neuchamp, 'if that is a specimen of a\nRainbar beast, I can well understand your saying that they will not get\nout of your way.'\n\n'D\u2015n the brute!' quoth Sparks; 'he does not belong to the run at all.\nDidn't you see the JS on his quarter? He is one of those infernal\nscrub-danglers from the Lachlan come across to get a feed. I'll shoot\nthe ill-conditioned wretch if ever I come across him again.'\n\nUpon being assured both by Brandon and Parklands that this was really\nthe state of the case, Ernest continued his inspection of the remainder\nof the mob, with which he was well satisfied. Not to risk any further\n_contretemps_, Parklands then suggested a return to the known dangers\nof the waggon. This also suited the cob, who looked as if he had\ncarried all his friend's money in a race and lost it.\n\n'Ten miles from Rainbar,' sang out Parklands. The words had hardly left\nhis lips when the fore part of the waggon sprang into the air.\n\n'Hang on behind!' shouted Brandon; and another minute saw Sparks and\nJem Fuller fasten on to the hind axle, backing for their lives. 'Man\nthe horses, Eachin! Jem, you cut a straight sapling while we rouse out\nthe saddle-straps for a splice.'\n\nOn inspection the pole was found to have snapped about a foot from\nthe fore-carriage, upon which the broken stump, catching the ground,\nhad turned that important part of the mechanism under the waggon,\ncausing the alarming jolt. The pole being 'fished' with a pine sapling\nand numberless saddle-straps, the remaining ten miles were safely\naccomplished rather under the hour, with the middle of the mended pole\ntrailing in the dust.\n\nThey were heartily welcomed at Rainbar by Mr. Brigalow, the overseer,\nwho produced some good whisky, and with an invention of his own, called\na geebung, a fair imitation of soda water was concocted, in which all\npresent drank success to the purchaser.\n\nOn the morning after their arrival at Rainbar no time was lost by the\nrestless Parklands, who was astir and alive to the utmost possible\nextent at daylight. Mr. Neuchamp, too excited to sleep during the\nnight, had fallen asleep before dawn. He had but dozed off, it appeared\nto him, and now here was Parklands rousing up everybody, catching\nhorses, whistling to the dogs, swearing at the black boys, throwing\nmissiles at Brandon's door, and generally making as much noise as a\ndozen ordinary people. Where work of any general nature is on foot in\nthe bush, breakfast is the first important stage, being indispensable,\nas, whatever other meals may be partaken of provisionally or left to\nchance, human nature urgently cries out for one 'square meal,' _pour\ncommencer_. The cook therefore came in for his share of intimidation\nand criticism from this terrible early bird.\n\nEventually the whole party found themselves assembled for breakfast at\nthe comparatively early hour of 5.30 A.M., while through the unglazed\nopen windows they could see the partially filled horse-yard, in which\nstood every available screw and stock-horse on the place.\n\n'Now, Neuchamp,' commenced Mr. Parklands, only partially arresting the\nprocess of deglutition, 'we must come to a decision about the muster. I\nam bound by the terms of my agreement with old Father Frankston\u2015rather\na downy old bird, in spite of his jolly ways and out-and-out dinners\u2015to\nget in all the herd and count them over to you. I would rather not do\nit, I confess; not because I'm afraid of my numbers, but it takes time.\nI have to be in Melbourne in ten days, in Adelaide in three weeks.\nBesides, it knocks the cattle about. Doesn't it, Aymer?'\n\n'Of course it does,' assented that gentleman; 'but it has an element of\nsafety about it, as far as the purchaser is concerned.'\n\n'No doubt of that; but in cases where the books have been so regularly\nkept for years, as Brigalow's here, any man can see that he _must_\nget his numbers if he takes them by the book total, with a decent\npercentage knocked off for deaths, etc., for fear of accidents.'\n\n'It occurs to me,' interposed Mr. Neuchamp, remembering Windsor's\nadvice, 'that as I have actually no experience in taking over a herd\nlike this, if Mr. Brandon would kindly act for me in the whole matter,\nI should be happy to leave the delivery in his and your hands, feeling\nsure that he could arrange it with you, in my interest, better than I\ncould myself.'\n\n'I could have no objection, of course,' said Parklands. 'I think it a\nvery good idea on your part; and though Aymer is my oldest friend, yet\nI fancy no one would accuse him of not doing you justice in such a case\nas this. I don't think they'd tell him so, at any rate.'\n\n'What a lazy beggar you are in small things, Sparks,' said Aymer. 'Why\ndon't you muster the cattle, and have done with it? And why am I to\nbe exalted into the position of your head stockman, and expected to\nback you up in all kinds of audacious fabrications in which I have no\npersonal interest?'\n\n'Who is lazy now?' sneered Parklands. 'Why can't you oblige Neuchamp\nand me also; it may be for the last time, for I shall never return\nfrom Melbourne alive, if the girls are half as pretty as they used to\nbe. Besides, I give you full power to fix the percentage, inspect the\nbooks, knock off the price\u2015anything you like, in fact. As a seller of\nunparalleled generosity, we can't be licked.'\n\n'I shall feel really grateful, Mr. Brandon,' said Ernest, 'if you will\nconsent to be my arbitrator and friend in the business.'\n\n'Well,' said Brandon, stretching his vast frame and rising slowly\nfrom the breakfast-table, 'if both parties combine against me there\nis nothing but capitulation for it. I surrender. So we may go to work\nforthwith. There are the books for ten years back\u2015certainly very neatly\nand regularly kept. Branded, so many; missing, so many; dead, so many;\nsold, so many. It strikes me, however, that 1 per cent additional might\nbe added to the death-rate.'\n\n'All right, old boy, knock it off,' exclaimed Parklands.\n\n'Then, as to the brandings, nothing of course counts under six months.\nI observe that you and Brigalow had a very fair haul of calves about\na month ago. I suppose none of them came from those outlying Wanilmah\ncattle of mine? We'll scratch _them_ out of the count.'\n\n'You be hanged,' explodes Parklands. 'I believe that old\ncattle-stealer, Weenham, that _you_ call an overseer, is a long way on\nthe debit side with me in the calf line. But scratch them out if you\nlike. I hope you're contented now. I believe you're standing in with\nNeuchamp, and met accidentally by appointment at Bilwillia to have me.'\n\n'I've not quite done with you yet,' said Brandon calmly, all unheeding\nof the gradually rising thermometer of Sparks's temper. 'What about\nthose Back Lake cattle? It has just occurred to me that the last camp\nwe saw there two years ago, when I helped you muster, contained an\nunusual number of \"pigmeaters,\" even for back country. You can't charge\nour friend full price for them.'\n\n'By Jove!' exclaimed Parklands, 'you're a friend in need. Well, of\ncourse we'll make a deduction for them. Though as the country is so\nsplendid out there, and is easily watered by cutting a channel from the\nriver, I\u2015\u2015'\n\n'Cost only two thousand pounds,' murmured Aymer.\n\n'Go to blazes! Five hundred more likely,' said the sanguine Sparks.\n'Well say a hundred off for \"ragers.\"'\n\n'Must have a hundred and fifty,' placidly pleaded Brandon. 'Think of\nthe danger and anxiety in muster times.'\n\n'You're another!' burst out Sparks, now justly indignant. 'If I take\noff another penny for anything, may I be\u2015\u2015'\n\n'Well, I only want two more stock-horses now,' persisted Brandon;\n'nothing here fit to call a horse that you could break your neck off\ncreditably.'\n\n'Where am I to get them, eh?' asked Parklands despairingly.\n\n'Don't mind taking the two wheelers you drove up. Neuchamp will\nfind them handy for practising four-in-hand with\u2015the only fun he'll\nbe likely to get here. And now, as I'm thoroughly exhausted and\ndemoralised by unmasking your villainy, we'll adjourn to lunch. Can't\nlick us, eh, Sparks.'\n\n'Well, of all the cold-blooded, grasping, unprincipled screws that\never imposed upon a warm-hearted proprietor under the cloak of early\nfriendship, you're the biggest, you old humbug. Mr. Neuchamp, you never\nmade a better bargain in your life, thanks to this long impostor.\nLet us have lunch on the strength of it; we'll do the arithmetic\nafterwards, and I shall be able to start at daylight. Can't lick us!'\n\nSomewhat comforted by the notion that he would be able to depart\nwithout the enforced delay of a muster, and again commence one of his\nlong and rapid journeys, made with the tireless celerity of a Russian\nlieutenant with despatches, Parklands ordered and attacked lunch with\nhis usual vigour and determination. Mr. Neuchamp in his turn was shrewd\nenough to perceive that Brandon, having definitely, though unwillingly,\naccepted the responsibility of acting for him, had decided with the\nsternest impartiality between his friend and himself. He felt that\nequally by this arbitration or by leaving it wholly to Mr. Parklands\nhe would in any case have been a considerable gainer by adopting Jack\nWindsor's advice, and he felt a lively satisfaction at the successful\nresult.\n\nLunch having been disposed of, the trio sat down to the calculation,\nand the lowest attainable number of cattle, with their ratable\nmoney-value per head, having been produced as the result of Aymer\nBrandon's subtraction and addition, Mr. Neuchamp gave a cheque for the\namount, signed with the hitherto unquestioned name of Ernest Neuchamp.\nIn return he received a receipt from Parklands, reciting below that\nhe had hereby purchased the right, title, and license to all those\ncrown lands situated in the county of Oxley, and comprising the runs\nof Rainbar East and West, Warrah, Banda, North Banda, Back Banda, and\nOuter Back Banda, with two thousand head of cattle, more or less,\nbranded LP, and the right to all cattle whatever bearing that brand not\nabsolutely proved to be sold or demised by the proprietor or by his\norders.\n\nThis feat fully accomplished, Mr. Neuchamp was congratulated by both\ngentlemen upon being the proud possessor of one of the best cattle runs\nof a very good district, and tolerably cheap too, as he was assured.\n\n'The fact is,' said Mr. Parklands, 'I should never have offered it at\nthis price; but I am going in extensively for a lot of new country upon\nthe Darr, and I want all the cash I can get hold of. It's necessary to\nbuy money, you know, sometimes, and this is a case in point. If things\ngo right, in half a dozen years I shall be able to sell runs by the\ndozen. Can't lick us!'\n\n\n\n\nCHAPTER XVII\n\n\nThere are several proverbial tests by which a man's directness\nand liberality of thought may be measured. The dividing of an\ninheritance has been found to divide for ever near and dear friends.\nThe co-occupation of a house frequently leads to the severing of\nfriendship. A sea-voyage of lengthened duration mostly displays the\ntrue nature of the human units, jointly imprisoned, with such alarming\nclearness that they tacitly agree to avoid each other ever after. But\nit may be doubted whether any process exceeds in thoroughness of assay\nthe transaction known in Australia as 'giving delivery of a station.'\n\nHe who comes forth from that crucial test may, like the man who emerges\nscatheless from the ordeal of a contested election, plume himself upon\nwearing armour of proof. Is he inclined to parsimony, the handing\nover station implements, the unconsidered trifles counted, priced, or\nhampered up together, will convict or acquit him of the charge. Is he\ninsincere, unscrupulous, careless, liberal, reasonably firm, ordinarily\nprudent, the purchaser will generally be able for evermore to speak\nwith authority on these points. In the delivery of Rainbar there was\nperfect openness on either side, and the more Mr. Neuchamp came to\nknow of the ways of the land the more fully did he understand, and\nmore strongly affirm, that he had been treated in his first purchase\nwith the utmost possible fairness and liberality. Every one had been\nmoderately busy all day. Lunch had been a hurried meal. The latter part\nof the afternoon Mr. Parklands had devoted to looking after his waggon,\npacking his traps, and getting together his horses. He did not merely\ngive orders, but thoroughly satisfied himself by actual inspection that\nno unforeseen obstacle or oversight could, humanly speaking, interfere\nwith his leaving Rainbar at sunrise. While apparently immersed in these\ndetails he, however, found time to suggest to the cook that this would\nbe a favourable opportunity for him to 'impress himself,' as in all\nprobability neither he nor Mr. Brandon would dine there again for years\nto come, if ever. The consequence of which well-timed hint was that a\ndinner of unparalleled excellence, for salt-bush country, was served at\n7 P.M., which Mr. Parklands, who had concluded his labours with just\nsufficient margin to admit of a swim with Brandon and Mr. Neuchamp in\nthe river, definitely expressed his intention of enjoying to the utmost.\n\n'I must say,' said he, as they sat down to this very creditable\neffort\u2015the artist as usual might have sung with Lord Richard in the\nballad of Alice Brand, 'I am a banished man' (too exclusive sacrifices\nto Bacchus having rendered metropolitan residence impolitic)\u2015'that I\nprefer the principal meal to take place at the end of the day.'\n\n'So do I, Sparks, my boy,' said Brandon. 'Industrious people like you\nand I require all the daylight we can get to energise in. Besides,\nthere is something unrefined in a hearty meal and hot dishes partaken\nof at mid-day, to the injury of complexion and delay of business, and\nthe serious damage of digestion, which abides not with anxiety and\nuncertainty of mind.'\n\n'I thought every one dined early in the bush,' said Mr. Neuchamp,\n'though I do not see why it should be an unalterable law.'\n\n'There is no actual necessity for it,' said Aymer. 'It is false economy\nto the mid-day meal, which should be a light one, to confer upon it\nthat improper dignity and position. I quite agree with Sparks, that\nthe cares of the day should be over before one undertakes so serious\na subject as dinner. If it occurs at mid-day how can any one foresee\nthat he may not be dragged away from the cheerful board and subjected\nto exercise or anxiety of the most violent description? How _can_ any\ndigestion so ill treated preserve its equanimity? and if one digests\nnot, then is happiness fled for ever.'\n\n'I feel a convert all over,' said Ernest. 'How capital this teal is;\nwherever did the cayenne come from?'\n\n'Always carry some,' answered Brandon; 'it is like tea and tobacco,\nand bills of exchange, very portable. I like work'\u2015here he slightly\nexpanded his vast chest and raised his sinewy fore-arm\u2015'but I may\nadd, with even less risk of being contradicted by my friends, that I\nappreciate comfort.'\n\n'_That's_ true; in fact nothing could be truer,' assented Mr.\nParklands; 'as to the work, you can do two men's share either at work,\nlove, or fighting when you're regularly cornered. You and I used to\nhunt better in couples when we were youngsters. Couldn't lick us, eh,\nold man? Remember when we thrashed those five fellows with the store\ncattle that came ravaging through the run, and took the cattle from\nthem?'\n\n'We were boys then,' answered Aymer with a grave smile, 'now we're men\nand magistrates both; such escapades don't become us. But we had a few\ntrifling adventures in the old days when we were taking up the Behar\ncountry.'\n\n'That reminds me of the blacks,' said Mr. Parklands; 'they were awfully\nbad there. I'm leaving you a capital brace of s, Mr. Neuchamp,\nfirst-class hands with cattle. I forgot them when Brandon was making\nhis unprincipled reduction; they're worth fifty pounds each to any man.'\n\n'You would have made a splendid Southerner, Sparks,' said Brandon, who,\ndinner having been concluded, had withdrawn to the fireside and lighted\na capacious richly- meerschaum. 'What an eye you could have had\nfor the points of a good field hand, not to mention those of a likely\nOctoroon. You're too fond of dealing, however, to have stuck properly\nto your hereditary bondsmen. I can fancy your swapping Uncle Tom, Aunt\nChloe, and the rest of them for a gang of half-broken plantation hands,\nwith a trotting horse thrown in for boot.'\n\n'Well, I like variety, I own,' confessed Sparks, 'and can't bear\nsticking to the same style of country and stock for ever. But human\nbeings make some difference in the calculation, though I don't know\nthat _you_ go so far, if all tales are true.'\n\n'What do you mean, Sparks?' inquired Brandon, with a slightly roused\nintonation.\n\n'Well, all the country heard that you and Lorton shot them like crows\nwhen you took up Tthoondula, after they had hunted the Dawsons off it\nthe year before.'\n\n'There was only one man shot the whole time I was there,' replied\nBrandon, 'and he was killed in an attempt to take him prisoner\nby Bothwell and his native police. He had nearly tomahawked Will\nLorton, and but for accidental assistance would have had his scalp,\nfiguratively, to a dead certainty.'\n\n'How far was that from here?' asked Mr. Neuchamp.\n\n'Fully eight hundred miles, so that there is no chance of your falling\nin for a blood feud. None of the slain man's kin could get here, if the\nlife of the whole tribe depended upon it.'\n\n'And was it absolutely necessary to put the aboriginal you speak of to\ndeath?' asked the philanthropic Ernest.\n\n'It was necessary to punish any black,' replied Brandon, 'who raised\nhis hand with intent to slay against any white man in that district\nand at that time. Without such a penalty implicitly carried out, the\ncountry would have become uninhabitable.'\n\n'Suppose we have a glass of whisky,' proposed Parklands; 'this is my\nlast evening, and we must drink prosperity to Neuchamp, and success to\nall his undertakings. Here are the materials; and now, Aymer, I suggest\nthat you give us the story of the man-hunt, where you were in at the\ndeath. Neuchamp is dying to hear it, and if you don't tell me, I shall\nnever leave off spreading reports that you and Lorton killed a whole\ntribe in cold blood\u2015men, women, and children.'\n\n'There are only two courses open to me that I perceive,' answered\nBrandon: 'I must either knock you down and so trample out this slander,\nor tell the story my own way. I have a foolish feeling of compunction\nas to the former proceeding, so I may possibly gratify your curiosity.\nAs Mickey Free says, the night is young and drink plenty.'\n\nMr. Neuchamp, though a foe to excess, did not disdain a moderate\nallowance of 'old spirits' from time to time. He was particularly led\non this eventful night to bear himself in a sociable and sympathetic\nmanner. There was no chance of work being done or thought of till\nmorning light. So he drew up his chair, filled his glass, and looked\nfixedly at the calm features of Aymer Brandon, who, much pressed and\nentreated, at length commenced his tale of years long past.\n\n'We had taken up Tthoondula, Will Lorton and I, only the year before,\nand we had fixed to commence our first shearing on the 20th of August.\nIt was the 15th, so no time could be wasted. Small parties of shearers\nwere camped by the edge of the long black gum-shrouded lagoon which\nhad given its name to the run. No one could have imagined that the\ndark deep water was in reality transparently clear. The sombre hue\nproduced by the illusion of a mud stratum, and the swart shadows cast\nby the huge eucalypti which lined its banks, caused one involuntarily\nto recall \"the dark tarn of Auber,\" while as the pall of swift-speeding\nnight fell heavily o'er the scene, it needed but little fancy to\nre-create the \"ghoul-haunted wood and of Weir.\" Slowly on that eve\ndropped the sun behind the rugged \"divide\" which separates the Paroo\nand Warrego, leaving the rosy-lipped hills smiling adieu till the\nmorrow. The frown on the face of the mulga-studded lowlands deepened,\nand the wrinkles harshly marked by many a tributary creek bore witness\nto its sorrow for the dying day.\n\n'The weather was simply perfect. We anticipated a successful shearing.\nThe mornings were crisp as lettuces, the succeeding portion of the day\nexhilarating to the degree of making conscious existence a pleasure of\nthe highest order. Summer, with a register of 120 in the shade, would\nhave been forgotten but for the dry harsh wool and the sand banks on\nthe sheep's back. We were in high spirits nevertheless. If the wool\nwas worth little we were separated by a thousand miles from our bills.\nOur bankers could only get at us by letter, and we were spared the\ndiscontent patent on the faces of those officials when the balance is\non the wrong side of the ledger.\n\n'By Jove, when I think of those early days, Sparks, how sanguine we\nmust all have been to see anything but ruin, writ large, in such\ninvestments. The only sheep one could buy were very indifferent as to\nthe quality, size, and constitution. They had been lambed twice a year\nfor the purpose of stocking up new country, and it was chiefly on paper\nthat the splendid frontages looked in any manner or shape tempting. The\ncalculation had been based on Riverina scales of labour, outlay, and\nprofit. Once on the ground the \"dead horse\" stood confessed. How often\nhave you and I seen a healthy, high-couraged youngster start out for\nthese fascinating territories of limitless mulga-downs, full-freighted\nwith hope, flattery, coin, and courage\u2015friendship, with delusive\ncrayon, sketching golden futures, cautious capital proffering loans\nwith both hands. At the end of five years returns a subdued, bronzed,\nresolved-looking man, with signs of dust from the road of Time \"upon\nbrow and beard.\" His pecuniary correspondents, who, to say truth,\nhave not come off scatheless, scowl upon him. But his \"own people\"\nand his true old friends receive the scarred and desert-worn Crusader\nwith loving words and open arms. With these tarries he, till again\nthe trumpet peals for another tilt with the veiled antagonist of the\nfuture.'\n\n'Devilish fine, old man. You're a most sentimental buffer after the\nsecond tumbler. Can't be licked, in fact\u2015but how about the ? I\nwonder you had the heart to shoot him\u2015two poetical cusses like you and\nLorton. Why didn't you give him a moral pocketankercher?'\n\n'I appeal to Mr. Neuchamp for protection from your coarse attacks,'\nquoth Aymer with mock dignity. 'Perhaps, after all, this incident is of\ntrifling interest.'\n\n'My dear Mr. Brandon,' cried out Ernest, terrified at the idea of\nlosing a tragedy, 'I sincerely trust that you will not think of\nwithdrawing your promise to give us this deeply interesting tale. I\nfeel painfully curious to hear the sequel.'\n\nThus adjured, and with a withering look at Parklands, Mr. Brandon\nproceeded.\n\n'We devoted the next few days at Tthoondula to fixing the\nspade-press\u2015that friendly adjunct to the pioneer-squatter's humble\nwoolshed, and topping up the brush yard at the equally primitive\nwashpool. I decided upon taking charge of the shed, leaving the\nlavatory to my partner.\n\n'It would be difficult to choose the easier task. Will was to command a\nlot of half-tamed naked Myalls, as yet hardly to be trusted, reprisals\nbeing still freely indulged in on that frontier territory between the\nblacks and itinerant station hands. The shearers were composed of the\nhuman scum always to be found floating near the border of civilisation,\nlike the rubbish forced before an advancing flood. It was no unusual\noccurrence to have the full complement of men in the morning, and in\nthe afternoon, upon the unexpected arrival of an inspector of police,\nthe shearing board would be deserted. All but a brace \"cach\u00e9d\" in the\nmulga. They showed up in the inverse proportion, of course, to the\nfact of their being \"wanted.\" Not that the native police troubled\nthemselves much about them. But a criminal hides from a policeman\ninstinctively, as doth the young wood-duck from the sportsman. All this\nmakes the management of this class of men the more difficult, as, if\nyou sack them in your righteous wrath, you can by no possibility get\nothers.'\n\n'Cannot the blacks be taught to shear?' inquired Mr. Neuchamp. 'They\nare the natural labourers of the land\u2015and _ads ripti gleb\u00e6_ too, as\nfrom what I learn they dare not leave their own district from fear of\nother tribes.'\n\n'It is weary work shearing with them. They are neat but painfully slow,\nand constitutionally lazy. The Anglo-Saxon is made up of faults, not to\nsay vices, but there is no worker on the earth's surface like him.'\n\n'Can't be licked,' murmured Sparks contemplatively, removing his pipe\nand mixing himself another whisky. 'Tell me when you've finished\nshearing and want help to load up.'\n\n'On the 19th,' continued Brandon calmly, all unheeding Mr. Parklands'\npractical arrangement of the narrative, 'all was ready. Will Lorton was\nto commence washing early next morning. They did not begin with the\nusual flock. But in that land \"the most unaccustomed thing is custom.\"\n\n'At the dawn-bird's cry from the aged trees, I sang out \"All aboard!\"\nand waking Will, we both rushed, robed in our blankets, to the lagoon,\nfor a plunge into its sad- waters, to emerge smoking in\nreactionary glow, and feeling fit to fight for a king's ransom.\n\n'Then, habited in the primitive garb of the far north land, Will made\nfor the blacks' camp, to see his Myalls off to the wash-pool.\n\n'On Tthoondula dwelt a grizzled, savage-looking old warrior, called by\nthe whites \"Hutkeeper.\" His duty was to tend the home flock. He was a\nchief in his tribe, and did not render himself conspicuous by wearing\nclothes. The English language had proved too difficult for his limited\nintelligence. He received food and tobacco for his slight services.\n\n'I had noticed one or two marked traits of savagery about Hutkeeper,\nand had warned Will not to trust the old ruffian. His mortal enemy\nat the home station was the cook, Nerangi Dick, whose prototype was\nCorney Delaney. Like him, he carried cynicism to its extreme limit. The\nlikeness was so exact that it was currently reported that the devil,\non one occasion, being short of a cook, had at sudden notice packed\nthe original Corney back to earth from his comfortable corner near the\nfurnace. The only billet he could retain was at the head station. He\nrespected the master, and reserved his growls for the kitchen.\n\n'The \"boogil-colli\" gins, water-carriers, had a rough time of it when\nNerangi Dick reigned. He might be seen driving them to their duties,\nwith many crisp oaths and a large stick. Of the male aboriginal he was\neven more intolerant. Ordered to feed the station blacks, he gave them\ntheir meat and damper as if throwing a bait to a dog. Hutkeeper rarely\nreceived his ration without being subsequently chased by Dick, armed\nwith his broomstick. It reminded a Waverley student of Peter Peebles\npursued by Nanty Ewart, or, more familiarly, of a sour-tempered Skye\nterrier pursuing a collie. Hutkeeper, on these occasions, keeping well\nout of reach, but looking back over his shoulder from time to time,\nwith a scowl which had in it a deeper meaning than the acerbity of the\nother. Should these two meet on the war-path, the devil would full\nsurely recover his own.\n\n'I told Lorten, after witnessing one of these periodical coursing\nmatches, that Hutkeeper would make a bad enemy.'\n\n'Take another tumbler, old man, after all that running,' suggested\nParklands. 'I have had two sleeps and gone over all my stock bargains\nfor the next three months since you commenced the life and times of\nthat . As a fictionist\u2015historian, I mean\u2015you can't be licked.'\n\n'Mr. a\u2015Sparks,' exclaimed Ernest, who had become confused between\nParklands' real name and sobriquet, 'pray permit Mr. Brandon to\nconclude his deeply interesting tale. I wouldn't miss it for anything.'\n\nSparks murmured something about the Tract Society, and affected to\ncompose himself to sleep. Brandon having compounded a restorative, then\nproceeded.\n\n'When Will Lorton arrived at the camp day was just breaking. There were\na dozen \"goondies\" to be visited, and the inmates started to their\nwork. Each black fellow, at the reveille, caught up a few waddies, and\nmade tracks for the wash-pen, with his hands full of blazing mulga\nbark, waving about his body. Hutkeeper had been called, but to his\nsurprise Will found, on passing his goondi a second time, that he\nhad not gone with the others. Having a light switch in his hand, he\nthoughtlessly gave him an admonitory tap across his tattooed shoulders.\nHutkeeper at once seized his nulla in one hand, stuck his tomahawk in\nhis belt, his sole article of clothing, and made towards the washpool\nwith his firebark in his left hand.\n\n'Now Lorton, having finished his work at the camp, turned to walk back\nto breakfast. He had not gone a dozen paces when a crushing blow fell\non the back of his head. He staggered forward, and turning received\nanother, which laid open his head, and dropped him in his tracks. As he\nfell he saw Hutkeeper leap at him with upraised tomahawk.\n\n'What saved his life was this. Two or three blacks still in camp,\nhaving a wholesome fear of tribal expiation at the hands of the native\ntroopers, seized the infuriated savage, and diverted the blows of his\ntomahawk. In the meanwhile Will Lorton, only temporarily \"kilt,\" rose\ndizzily to his feet, and catching the foe a straight blow behind the\near, laid out that gentleman as neatly as if he had been dropped with\nhis own weapon. He then threw himself upon the prostrate chieftain and\nwrested his arms from him. Before he could seize him, however, the\nslippery savage, eluding his grasp, was bounding through the trees,\nand soon after passed out of sight. Poor Will reached the home station\ncovered with blood, and looking particularly faint.\n\n 'An angry man, ye may opine,\n Was he, the proud Count Palatine,\n\nwhich means that I, Aymer Brandon, was wroth exceedingly at this deed\nof blood (literally, indeed, the bright Norman blood of which Master\nWill was depleted on the occasion made a very pretty pool, artistically\nconsidered, on the earthen floor of his room). So \"boot and saddle\" was\nthe order of the day.'\n\n'Now we're coming to it,' exclaimed Mr. Parklands, in a tone of deep\nsatisfaction. 'This is the sort of literature I go in for\u2015incident, old\nman\u2015lots of incident\u2015eh, Mr. Neuchamp, isn't that your style? Now, why\ncouldn't you have given us that first, old man, like this: \"One fine\nmorning, on the Paroo, Will Lorton went to the blacks' camp, didn't\nlook behind him, and fell against a nulla, which happened to be up at\nthe time.\"'\n\n'You have no sentiment, Sparks, as I have always reminded you. What\nlittle humour you possess has been prematurely wasted on barmaids. You\nwould enjoy a story about that old blue stag that nearly deprived you\nof a purchaser, just as much as Browning's last poem\u2015more, in fact. But\nI have commenced this yarn, and you _must_ and shall have it, if we sit\nup till daylight.'\n\n'Only too happy, my dear f'ler,' murmured Sparks somnolently. 'Don't\nshoot me instead of that . You seem to have been a rum lot\nout there, and old \"Hutbuilder,\" as you call him, rather more of a\ngentleman than any of you. His manners rendered him unpopular, I\nsuppose; and you trumped up this cock-and-bull story about Will just to\nsuit the case for the Crown. Ah, Neuchamp, my boy, you have no idea how\nthese benighted back-country squatters go on, when you and I are not\nthere, and there is no one to check their violence.'\n\n'About five minutes after Will was returned as \"killed, wounded, and\nmissing\" from the wash-pen for the day, a black trooper rode in with a\nletter from his inspector, who was quartered about twenty miles from\nTthoondula. Saddling up, and pressing trooper Mayboy into the service,\nwe galloped into the camp. He was armed with his carbine, and I with\na very effective seven-shooter. I had long vowed never to draw a bead\nupon a blackfellow for anything less than bloodshed. But in my wrath I\nswore to shoot the old warrigal at sight, and in trifles I like to keep\nmy word.\n\n'In the camp reigned great excitement. His countrymen freely condemned\nHutkeeper, and morally gave him up to justice.\n\n'\"No good\u2015Hutkeeper! Waddy-galo that fellow. Goondi-galo, goondi-galo\nmine. Baal waddy-galo.\"\n\n'I wasted no time in the camp, but made a cast round, to pick up the\ntracks of the fugitive. Mayboy, eager as a bloodhound, was soon on the\ntrail. On the soft soil of the Paroo it was not difficult to follow,\nwith eyes like those of Mayboy.\n\n'I said, \"You think man him (catch), Hutkeeper?\"\n\n'\"Baal!\" answered the trooper, \"that fellow too much burri. Bime-by\nmarmy (officer) come up, and all about black trooper; then man him,\nHutkeeper; mine think it shoot him!\u2015Ki\u2015i\u2015i!\"\n\n'The latter expression long drawn out, was expressive of the high\ndegree of satisfaction which that consummation would afford him and\nhis brothers-in-arms. Having made sure of the direction of the tracks,\nMayboy and I returned to the station. A messenger had long since been\nsent to Mr. Bothwell, the inspector, reporting the outrage, and asking\nfor the prompt arrest of the offender. \"Arrest or slay the Frank,\" was\nold Lambro's order; \"Catch the , alive or dead,\" was, in effect,\nthe word of command when murder or wounding with intent was proved.\n\n'Within six hours after the commission of the offence Mr. Bothwell\narrived with five highly efficient-looking troopers, making, with\nMayboy, six in all.\n\n'Far finer specimens of the Australian aboriginal were they than their\nParoo brethren. Recruited from the Wide Bay coast tribes, noted for\nwarlike propensities, nothing delighted these human bloodhounds so much\nas being slipped to the blood-trail.\n\n'Shearing was postponed for two days to allow for the man hunt. After\ndinner the war party, consisting of Bothwell, myself, and the six\ntroopers, saddled up and departed. We carried revolvers, the men\ncarbines, throwing bullets of murderous size. Our janissaries were\nnamed respectively Mayboy, Tiger, Jerry, Bloomer, Tangerine, and\nBulldog. Of these, Mayboy was Bothwell's aide-de-camp and special\nfavourite. The war-cry of \"Hi, Mayboy!\" was well known on the Paroo and\nWarrego. Something decisive generally followed that exclamation. Heaven\nhelp the poor wretch on whose footsteps these six bush devils were\nslipped. When the trail carried blood they were never known to fail or\nfalter.\n\n'Put them to track cattle, horses, or sheep, and after half a day\nthey began to grow weary or careless; but with a human quarry ahead\nevery eye was unerring, every muscle was tireless. Clue after clue was\nchecked off with unvarying certainty, the result of human ingenuity\nallied with hereditary instinct unerring as that of the sleuth-hound.\n\n'Mayboy took the lead, laying the pack on at the exact spot where he\nhad quitted the scent in the morning. For miles back from the Paroo the\nsoil is composed of soft red loam, the tracks on which are as clear\nof imprint as fossils upon the old red sandstone. But once reach the\narid flinty range, and its secrets of wayfaring man or beast are only\nrevealed to the microscopic gaze of the Australian Indian. The troopers\nrode carelessly together while the footsteps of the fugitive were\nprinted in large type, so to speak. Two kept slightly ahead, the rest\nfollowing.'\n\nMr. Parklands aroused himself suddenly from a posture of deep\nattraction or attention, and observed Ernest's eager countenance fixed\nupon Brandon's calm features, as he, recalling with a certain thrill of\ninterest the stern episode of old pioneer life, told in his low, deep\ntones the tale of doom.\n\n'Not caught him yet, old man?' demanded Mr. Parklands. 'Devilish slow\nwork. If I'd old Ber-bar we'd have shot every blackfellow in the Paroo\nby this time. Couldn't lick him. You won't take any whisky\u2015that's why\nyour story hangs fire.'\n\n'There is something deeply fascinating about a tale like this,'\nexclaimed Ernest. 'One does not often hear the tragedy from the mouth\nof one of the actors. I can imagine nothing more exciting than joining\nin such a chase. Of course you were able to take him alive, with your\nband of Mohicans. Uncas and old Hawkeye would not have been out of\nplace in such a war-trail, had there only been a Mingo to the fore\nsomehow.'\n\n'I have the greatest respect for Uncas and Chingachgook; as for\nHawkeye, I have honoured him from my youth up,' said Brandon; 'but I\nfirmly believe that Tiger and Mayboy would have given both of them a\nwrinkle in tracking and woodcraft generally.\n\n'It was surmised that the trail would follow the river for about\ntwenty-five miles, to a favourite camping-ground by the side of a\ndeep lagoon, known as Tthulajerra. Mayboy, dropping alongside of Mr.\nBothwell, said, \"Marmy! mine think it, old man Hutkeeper, first time\nweja longa Tthulajerra, plenty blackfellow sit down there. That fellow\nmessmate, then all-about pull-away-long a scrub.\" This calculation\nwas proved to be accurately correct, as the tracks ran straight to\nthe lagoon, where a deserted but recently occupied camp was found.\nSmouldering fires, heaps of mussel-shells, and fish-bones lay scattered\naround, while the stones in the native ovens were not yet cold.\n\n'When Tthulajerra was reached it was nearly sunset; so a camp was\norganised for the night. Mr. Bothwell fully expected to run his quarry\nto earth before the next sunset. Unless Hutkeeper separated from the\ntribe they were sure of him. It was unlikely that the deer would leave\nthe herd. Blacks prefer to fly and to fight in company; they dread\nsolitary journeyings. Two camps were formed\u2015one for Bothwell and\nmyself; the other, at about fifty yards distant, for the troopers.\n\n'That camp scene, before the moon rose, was one only to be found in a\nnew land. The Paroo, unlike the Warrego, is not famed for heavy timber;\nstill immense eucalypti border lagoons like the Tthulajerra. After our\nspare and simple meal I felt indisposed to sleep. I lighted my pipe,\nand, stretched on my rug, lay long in thought and reverie. The blazing\ncamp fires illumined the silent giants of the wilderness from root to\ntopmost branch. In the firelight the smooth white bark of the limbs and\nstem had a deathlike appearance, in keeping with the gruesome feelings\nnaturally engendered by a \"man-hunt.\" I could scarcely restrain myself\nfrom peopling the ghastly outspread limbs with hundreds of victims.\nI thought I saw before me the African \"death-tree,\" while the black\nfigures of the naked troopers, flitting from fire to fire, favoured the\nillusion. They seemed to be awaiting the fall of the hideous fruit, and\nthe furnishing forth of the feast. Mr. Bothwell, not being anything\nbeyond a very practical and efficient Government officer, had gone to\nsleep. He was a good doer, and sleeping was no trouble to him. When\nthe moon rose the morbid fancies were dispersed, and as the last dark\nform sank down seemingly into the earth I slept.\n\n'After catching and destroying Hutkeeper about five hundred times, and\nbeing murdered by that relentless savage in every conceivable manner, I\nawoke, about 4 A.M., to find that a thick impenetrable fog lay nearly\no'er \"wood and wold.\" I replenished the dying fire, and not feeling\ninclined to sleep more, sat silent and brooding till the fog lifted,\nand one by one the shrouded forms came forth from the shadowy veil,\nlike lost years through the mists of memory.'\n\n'And yet people say there is no romance in a new country!' exclaimed\nMr. Neuchamp, who, the best of created listeners, from his largely\ndeveloped gift of sympathy, had eagerly drunk in every word, so\nmanifestly enjoying the narration that Brandon, an imaginative and\npoetical though generally reserved man, had been unconsciously\nstimulated into a fuller development of the surroundings of his weird\ntale than under ordinary circumstances he would have thought possible.\n'No poetry? No dramatic position? What a picture for an artist:\na solitary figure in that gray silent dawn, by a dim smouldering\nfire; the careless savage troopers; the tranquil officer, calm but\nremorseless as a Roman centurion!'\n\nBrandon continued, musingly\u2015\n\n'Tree after tree stands forth, slowly, as if painted by an invisible\nartist upon a canvas of mist. The foreground is quickly filled in.\nSmall tumuli appear. The troopers swathed, all deathlike, in their\nblankets. Then a horse is traced on the murky easel; then another.\nClink, clink, go the chains which fetter their feet.\n\n'\"All aboard!\" I shouted, at length casting away the phantasmal\ncreation. \"The busy babbling and remorseless day\" is again born, for\nus and for all mankind, in this south land. Up spring the troopers.\nBothwell arose, but kept his position until scorched out of it by the\nheaped-up fire. Breakfast was concluded, and the horses stood saddled\nand ready, as the sun rose.\n\n'A different disposition of the forces was made for this day's work.\nThe troopers separated into three pairs\u2015Bulldog and Jerry followed the\ntrail through all its deviations; Bloomer and Tangerine skirted on\neither flank, keeping about a hundred yards from the presumed line and\nthe same distance ahead of Bulldog and Jerry; Mayboy and Tiger rode a\nquarter of a mile in advance of the party.\n\n'The system was this: The couple on the trail ensured its being neither\nlost nor overlooked; the skirters, by riding straight on either side,\npicked up the tracks when any deviation was made. Whoever \"cut\" the\ntrail whistled, when the other three quickly closed on him, and\nresumed their places from that point. The two in advance sought to cut\nthe tracks some distance ahead; when they did so a whistle, low but\nclear, brought those in the rear forward in a canter to start afresh\nfrom the new point. By this method of economising eyesight, as the\nsignals followed each other in quick succession, the ground was covered\nmuch more quickly than if the trail had been traced through all its\nsinuosities.\n\n'The inspector and I followed at easy distance our sable\nsleuth-hounds\u2015a pack without huntsman or whipper-in. They had this\nadvantage over their canine comrades: their casts were made in\nadvance. Was an unusually difficult tract of country encountered,\nwhere \"scenting\" was slow, the advance-guard could ride beyond it,\npick up the trail on more favourable ground and signal their comrades.\nMiles of rocky ridges were crossed, when the only guide to the silent\navengers of blood was a stone turned over, the print of toe or heel on\nthe scanty sand or gravel collected between the boulders. At times,\nmerely a tiny white flake dropped from the fire-barks, carried in the\ncoolimans to prevent the tell-tale fall of ashes, betrayed the pursued.\n\n'Still eager, tireless, almost joyous, rode forward the death-band on\nthe faint footsteps of the hunted savage. Hutkeeper, thus fleeing,\nwould surely know that he had staked his life, and lost it, when he\npermitted his wild nature to overcome him. He would know that many\nhours would not elapse before men of his own race would be on his\ntrail\u2015better trackers and more tireless than his tribe. But onward he\nfled, still ascending the range, knowing that the two ends of the trail\nwere coming together only too surely. No white man can ever know what\nthoughts passed through the brain of the doomed old heathen during that\nlong, hopeless flight.\n\n'If each individual man were not merely one of the units composing a\nvast system of usurpation, called from time immemorial by the specious\nname of Progress, one could afford to sympathise with the savage for\nsmiting his oppressor. But the world will surely be very old when that\nmost ancient of laws \"the strongest shall possess,\" ceases to have\nforce. We preach the law of Right, but the older natural doctrine of\nMight has always prevailed and will find adherents to the end, so long\nas one man or one animal, brute or human, is born stronger than his\nfellow.\n\n'Thus, through the livelong sweet spring day, the sleuth-hounds swerved\nand faltered not. As the day wore on, the writing on Nature's book, the\nink whereof was the lifeblood of him that fled, became easier to read.\nThe sable coil seemed to work more unerringly than ever. It glided like\na huge serpent among the trees, the head shooting forward to be swiftly\nand smoothly followed by the sinuous body.\n\n'\"What do you think of the tracking?\" asked Bothwell with pardonable\npride, his eyes resting upon Mayboy, who was at that moment beating\nthe covert of a close scrub, lifting his head from time to time like\n\"questing hound.\"\n\n'\"It is superb,\" I answered; \"but, on my soul, Bothwell, I hope the old\nfellow will escape. According to his light, he but hit out like a man,\nand we are now treating him like a beast of prey. They must kill some\none very near and dear to me, before I undertake a job of this kind\nagain.\"\n\n'\"We must either shoot them,\" said Bothwell, \"or give up the land.\nClear off the old and teach the young, is my motto at present.\"\n\n'\"Yes,\" said I sadly, \"another illustration of the 'fitness of things.'\nIt would seem as if the present were perpetually to be damned for\nthe benefit of the future. I should be sorry to have to explain to\nHutkeeper's tribe, after we have killed him, the meaning of the words,\n'If thine enemy smite thee on the right cheek, turn to him the left\nalso.'\"\n\n'All the troopers were now seen to be clustered together. They were off\ntheir horses, smoking\u2015a sure sign that they felt secure of their prey.\nWhen Bothwell and I joined them, Mayboy came forward dangling a small\ndilly-bag, dropped by one of the gins.\n\n'\"Marmy! mine think it weja now; you make alight that one mountain,\nnerangi good way like it, ugh\" (the guttural accompanied by the usual\nblack's point, the protrusion of the under lip)\u2015\"that one Boolooloo\nwater sit down. Blackfellow big one tired weja long a Boolooloo.\nTo-night yan longa camp; boomalli (shoot, slay) Hutkeeper.\"\n\n'Boolooloo was a turreted hill, rising abruptly from the crown of the\nrange, and towering far above it.\n\n'At its foot was a native well\u2015a natural tank\u2015scooped out of solid\nrock, gourd-shaped, with a small man-hole at the top. Its depth was,\nperhaps, twelve feet, with a diameter of double the extent. Its shaded\nposition, under a ledge of overhanging rock, enabled it to contain\nwater through any ordinary summer.\n\n'The rugged plateau of the summit of Boolooloo had been hollowed into\ncaves from immemorial time, favoured retreats of the wild tribes in its\nvicinity. It wanted now an hour to sundown; the hill was then three\nmiles distant.\n\n'Bothwell's order was to wait until nightfall, then to surprise the\ncamp and to arrest Hutkeeper, with the usual alternative if he evaded\nor resisted the capture. He promised me that, if possible, he should\nbe taken alive. Sudden vengeance having been denied me, I was far\nfrom keen for the old pagan's blood. Bothwell could have told me that\nHutkeeper's last sun was setting.\n\n'The troopers, deciding to stalk the bush on foot, took off their\nsuperfluous clothing, also their boots, slinging their ammunition\npouches over their shoulders. The horses, unsaddled and close hobbled,\nwere turned loose. Then all awaited the close of day. Supper was\npostponed till after the invasion of the camp, as a fire would have\nbetrayed our vicinity. The troopers, light-hearted and free from\nanxiety, a complaint chiefly confined to the white man, passed away\nthe time card-playing. Their officer and I sat silently on the short\nturf, watching the shadows of the gydya trees lengthen, ah! so slowly.\nThe sun was fading over the northern turrets of Boolooloo, lighting\nthem into elfin splendour, as might gleam the battlements of a ruined\ncastle. A fast-glooming shadow crept around the mountain, until at\nlength its huge mass was hidden from the watchers.\n\n'The light of day had departed. The hour was come. The last act of the\ntragedy was about to commence.\n\n'The troopers put up their cards, lifted their carbines, and passed\nshadow-like and silently through the trees. We followed. In an hour we\nreached the base of Boolooloo.\n\n'Mayboy halted and whispered to his chief, \"Marmy! close up to camp\nnow, drekaly see fire longa nother one side.\" The wind sighed from the\nhill top _towards_ us. There was therefore no danger of the sharp-eared\nblacks' dogs giving tongue in time to warn them. Then all crawled\nnoiselessly up the steep sides of Boolooloo, pausing when about a\nhundred yards from the camp. Fires were smouldering in front of the\ncaves, but not a creature was visible. We moved cautiously forward.\nThen a dog raised a dismal howl, and was joined in full chorus by his\ncomrades.\n\n'In the middle of this mournful music the troopers bounded into the\ncamp, scattering the dogs into the crevices of the rocks. The next\nmoment a yell of terror and despair burst from the wretched blacks, who\ncame rolling out of the caves, and, huddled together in groups, they\nwailed out, \"Goondi galo (tame blacks), goondi galo,\" incessantly.\n\n'Then from the centre cave leaped forth a hideous demoniac figure,\nghastly with white and red pigment. \"Hutkeeper! Hutkeeper!\" shouted the\ntroopers. \"Look out, Marmy! that one big one coola (angry, fierce).\"\nBy the dim starlight I was enabled to recognise my late shepherd\ntransformed into a warrior, prepared to meet his enemies fairly and to\nthe death. The old savage held before him his file-shaped shield. In\nhis belt hung the nulla and tomahawk; while his right hand held aloft a\nbattle-spear, poised and quivering.\n\n'For one moment\u2015his last\u2015he stood with blazing eye and wolfish gaze\nupon the foe, a true warrior of the waste, then hurled his spear into\nthe centre of the party. The quivering rifled weapon, speeding through\nthe air like a cloth-yard shaft, grazed the cheek of Mayboy, and by a\nhairbreadth only missed the somewhat solid proportions of Bothwell.\nSix carbines rang out in answering volley, and, leaping into the air,\nHutkeeper fell forward on his face, a dead man.\n\n'Our work was finished. Civilisation had been vindicated. The whole\nparty silently retreated, leaving the sad tribe alone with their dead.\nWill the caverns be haunted, in days to come, by a spirit that cannot\nbe laid by the white man's bullet? When I returned to Tthoondula, I\nthus addressed my partner, \"Well, old boy, I can see that man-hunting\nis not much in my line. You'll oblige me greatly by killing your own\n next time.\"'\n\n'\"The forest laws are sharp and stern,\"' quoted Ernest, as the\ntale and the life of the sullen son of the soil came to an end\nsimultaneously. 'I suppose there is a necessity for prompt punishment\nof violence in a frontier settlement; but it seems rather hard on the\npoor old fellow. How does the law of England stand?'\n\n'Well, of course,' said Brandon, 'it was strictly legal to endeavour to\narrest either an aboriginal or a white man upon the charge of \"cutting\nand wounding with intent to kill,\" or even \"to do grievous bodily\nharm.\" If such a prisoner resisted the police, they were authorised\nto fire upon him. In this case, it was impossible to take him alive.\nHowever that may be, he paid in full of all demands for his crime. I\nfancy we may as well turn in.'\n\n'So the is dead at last!' exclaimed the awakening Parklands.\n'Good-bye, Neuchamp; you may not be up when I start. Aymer, your story\nis really grand. Too short, if anything. You don't know a little more,\njust to top up with? The worst of these interesting yarns, they keep\nyou awake so. If I am late at starting to-morrow, it might be a loss\nof five hundred pounds to me\u2015you wouldn't like me to send in a bill\nfor half. Why don't I go to bed now? I feel too much excited. Besides,\nI am afraid I missed some. You wouldn't mind beginning again? Well,\nsir, I'm off now. Never mind throwing a boot at me\u2015one of your boots is\nno joke, remember. But look here\u2015if it takes three hours to kill one\nblackfellow, how long\u2015\u2015'\n\nHere Mr. Parklands disappeared suddenly, simultaneously with the\nevolution of a missile of some sort discharged wrathfully by the\nnarrator.\n\nMr. Neuchamp also departed, and being rather tired slept until past\nsunrise. When he came forth only Brandon was visible, who told him that\nParklands had left at dawn, and was now many a mile on his way.\n\n\n\n\nCHAPTER XVIII\n\n\nMr. Neuchamp of Rainbar had now reached a very important position\nin his career. He had gained a fulcrum for that lever by the aid of\nwhich he trusted to move the Australian world,\u2015to raise or to cause to\ntremble\u2015and finally to impel upon the incline of undoubted and social\nimprovement\u2015the hitherto inanimate mass of colonial society, strong in\nthe _vis inerti\u00e6_ which rules primitive or unenlightened communities.\nBefore this happy moment of proprietorship he could but enunciate\nprinciples and theories. Now he was enabled to demonstrate them by\npractice. He would have comrades, neighbours, dependents, workmen\nof his own. And concurrently with the most effective and successful\nworking of the station, he would show New South Wales, Australia, and\nthe world generally, what an Englishman of culture, with a purpose,\ncould effect in the way of reform. Captain Cook had discovered the\ncontinent\u2015proconsuls of greater or less intelligence had governed\nit. It was left for him, Ernest Neuchamp, to raise it to that point\nof social and industrial eminence which should make it a Pharos, a\nwonder-sign, an exemplar throughout all the civilised world.\n\nIt may be gathered that Mr. Neuchamp was alone and possessed his soul\nin peace, when he found sufficient time in which to indulge these grand\nideas and magniloquent reflections. Mr. Parklands' company was not\nfavourable to contemplation. His very existence was an aggressively\nenergetic fact, wholly adverse to reverie or mental repose of any\ndescription. He was always talking or smoking, or asserting or denying,\nor going out or coming in, or preparing for his next journey or\nreviewing his last one. His very correspondence was of a telegrammatic\nand restless nature, full of reference to distances and routes, orders\nto overseers and stockmen to go thither, or come hither, to await him\nat one place or meet him at another. He went to bed defiantly and got\nup noisily, full of plans and prospects, and requiring everybody to\narise and be stirring, in the most literal sense.\n\nAymer Brandon was constitutionally of a calm, equable, and chiefly\namiable temperament, provided that he had things mostly his own\nway. But he was temporarily excited by the demon of unrest which\nabode in Parklands, so that between practical jokes, contradictions,\nreminiscences of adventures, revelries, and the like, no peace, in the\ntrue sense of the word, was possible until their departure from Rainbar.\n\nNot until several days after that event did Mr. Neuchamp realise that\nhe was clothed with real and undisputed sovereignty.\n\nThen with sudden afflatus arose in his brooding mind the thought of\nthe elevated duties and deep responsibility of his position. It was\nthe hour of the evening meal. This frugal meal\u2015damper, hard corned\nbeef, and very black astringent tea\u2015the same served in a very black\nquart pot\u2015Ernest had enjoyed in solitude. Humble as was the fare, it\nwas amply sufficient for a man in the pride of vigorous youth. The\nindifferent Bohea had power to stimulate delicately, yet positively,\nthe nerves of Mr. Neuchamp's, perhaps, hypersensitive brain.\n\nThe night was calm and clear. The starry heavens held no cloud. The\nlong lagoon lay darkly metallic, or broke into phosphoric ripples. The\nmysterious sounds of the desert were rare and as yet unfamiliar to the\nlistener. All things afforded a startling contrast to his English name\nand surroundings, even to his later metropolitan habitudes. Yet as he\nsat there by the light of the stars, amid the tremendous solitude of\nthe wilderness, his heart swelled with the thought that he was the\nvirtual ruler of a territory larger than his ancestral country\u2015larger\nthan any member of the house of Neuchamp had owned since the first\nbaronial fiefs in their blood-bought Normandy.\n\n'What are the chief and foremost needs of this waste empire of\nmine\u2015this desert city?' soliloquised he. 'Here I have land enough\nto satisfy the earth hunger of the most ravenous aspirant of _la\nterre_. Water in reasonable though perhaps insufficient quantity. What\nis the great absent factor? Population, a yeoman class, a race of\nVavasours, who could use these great levels for the growth of certain\nsemi-tropical crops, who might rear upon them a limited number of\nstock, who would secure homes for themselves and food for their working\noxen; who would remain loyal to me, their powerful yet philosophic\nally; who would work for me at reasonable rates at ordinary station\nwork, or any reproductive improvements which I might suggest, and\nwho would thus entirely sweep away the present undesirable relations\nwhich have hitherto subsisted between Australian country labourers\nand their employers. It would not be expensive to provide a school\nand a teacher for their children, to be paid by results. I should\nbe enabled, by a steady supply of labour, to cultivate a reasonable\narea. Gardens and experimental industries would of course spring up.\nThe carrying capabilities of Rainbar might be enormously increased by\ncutting a narrow canal, as Parklands suggested, between the waters\nof the river and the chain of deep, yet dry lakes at the back of the\nrun. The advantages of labour on one side, of wages on the other,\nwould be mutual. Simultaneously an improvement in the character and\nquality of the herd would take place. Scientific experiments might\nbe regularly made and recorded as to rainfall and other important\nmatters. The culture of the vine, the orange, even the silkworm, might\nbe introduced; and finally, after a few years, the semi-co-operative\ncommunity at Rainbar, self-contained, happy, and prosperous, might be\npointed out as at least _one_ instance where enlightened theory and\nsuccessful practice had accomplished an advance in civilisation, had\nsolved the problem of the harmonious interchange of labour and capital,\nand had interpolated at least one Arcadian chapter in the sad history\nof mankind.'\n\nAs these and other fair and fascinating trains of ideas passed through\nthe mind of Ernest Neuchamp\u2015while outside of his lonely and humble\ndwelling the silent stars burned in the still wondrous firmament, and\nnought but the monotonous and half-boding sound of the night-bird broke\nthe profound primeval silence\u2015he passed instinctively from the stage\nof triumphant justification of his plans to a half-felt distrust as to\ntheir practicability; and with the thought of failure came a vision of\nthe calm questioning gaze of Antonia Frankston, before which his ardent\nscheme and aspirations for the perfectibility of the race had more\nthan once appeared dreamy and Quixotic. The fancied questioning of\nold Paul, cool as kindly, yet keen as a cross-examiner, seemed adverse\nto the Utopian infant. But Ernest's strong enthusiasm of humanity, his\ngenerally sanguine temperament, carried him for that night over all\nobstacles, and he retired to a _very_ lowly couch, fully determined\nthat the Rainbar community should enjoy every advantage which\nco-operative life and labour had ever yielded to intelligent guidance.\n\nWith regard to the ordinary working of the station, he felt at a\ndisadvantage in the absence of Jack Windsor. He had been so much in the\nhabit of relying upon that ready-witted and helpful personage in the\nexecutive department, that he felt comparatively helpless when solely\nresponsible. He considered also that his life would be now almost\nunendurably solitary without the companionship of some one nearly\napproaching his own grade, who would be at once an assistant and a\ncompanion.\n\nIn this extremity, he bethought himself of his late associates at\nGarrandilla. None of these young gentlemen was absolutely necessary\nat that ovine university. They had taken their degrees, so to speak.\nTheir places were perhaps waiting to be filled by other alumni, some of\nwhom paid a fair sum for the privilege of fulfilling, very literally,\nthe position of the subordinates of Jairus, to that rather exacting\ncenturion Mr. Doubletides.\n\nThis point being settled, he essayed to make choice of a probable\ncompanion. Grahame was obviously devoted to sheep. The merino had\n'marked him for his own,' and it would have been wrong to have\nwithdrawn so promising a woolsorter from the establishment. Moreover,\nhe was not interesting or sympathetic as a companion.\n\nFitzgerald Barrington was interesting and amusing, if not sympathetic.\nMr. Neuchamp was much minded to invite him to Rainbar. But in his\nway he was as unlikely as Grahame to take himself to any scheme for\nthe improvement of the common people. With all the _bonhomie_ of his\ncountry, he despised and disbelieved in the people, and would not have\nput forth his hand to save them from a fate quite commensurate with\ntheir deserts.\n\nThe remaining cadet then was Charley Banks. In this youngster\nErnest had always recognised a manly and self-reliant nature, by no\nmeans beneficially indebted to early training, and having come off\nindifferently in the matter of book-learning. Still he thought him\nimprovable from certain indications which led him to think him not\nwholly unsuitable as a companion. He had often expressed his dislike\nto sheep and his anxiety to live on a cattle station. Mr. Neuchamp,\nfinally coming to the conclusion that he might do the boy a service,\nand at the same time provide himself with a companion in his solitude,\nwrote a letter to Mr. Jedwood, in which he described his purchase and\ngave a short sketch of the capabilities of the run, winding up with a\nfair offer of employment for Mr. Banks if he had no objection to his\nleaving Garrandilla, and if the youngster himself cared to come.\n\nHe was not long left in suspense concerning the intentions of Charley\nBanks. He received, as soon as the somewhat indifferent postal\narrangements permitted, a letter from Jedwood, informing him that\nhe was heartily welcome both to Mr. Banks and to Mr. Fitzgerald\nBarrington, if it pleased him to take a brace of cadets. But that,\nperhaps, it would be safer and more profitable to take one, who could\ndo more work and be less trouble (on the well-known principle of two\nboys being only equal to half a boy) than a couple.\n\nFrom Charley Banks himself he received a short but enthusiastic letter,\nsetting forth his gratitude for being remembered by him, and his\nintention of starting for Rainbar in company with Jack Windsor, who,\nit was reported, was on the road up from town, and not very far from\nGarrandilla at the date of writing.\n\nMuch pleased with the idea of having shortly the companionship of Mr.\nBanks, and the aid of Jack Windsor, upon whose ready and practical\ncounsel he had learned to place a high value, Mr. Neuchamp, after a\nfew purposeless rides round his territory, conceived the bold idea\nof mustering and drafting a portion of the herd, with the aid of\nthe aboriginals whom Mr. Parklands had bequeathed to him. A general\nmuster he of course knew that, without a considerable force of\nvolunteer assistants, he was powerless to undertake. But a portion\nof the herd he thought he could get in. 'It will familiarise them\nwith going through the yards,' said he to himself, 'and if there are\nany calves to put the new brand on, we can manage _them_.' Like most\ninexperienced purchasers, he had immediately changed the LP brand,\nknown from Queensland to Adelaide, to one of his own invention, viz.\n\u018eNE (a conjoined hieroglyph), which, as combining the initials of his\nChristian name and surname with the second letter of the latter, he\nthought ingenious and attractive, whereas, in point of fact, it took\nyears to gain the widespread association with Rainbar which the old\nbrand already possessed.\n\nDuring former musters Mr. Neuchamp's constructive faculties had\nbeen busy with projects for improving the accepted mode of drafting\ncattle. Much to his own satisfaction, he had arranged his system\nbeforehand. He was confident that it would work without a hitch.\nHis humane tendencies had been outraged by the unsparing use of the\nruthless stockwhip, keenest when unheard, as well as of the long,\npliant, wattle-drafting stick, not apparently a weapon upon which to\ndepend your life, but in skilful hands\u2015and such are not wanting at\nevery important muster\u2015sufficient to drop, as by a thunderbolt, the\nmost formidable beast. This Mr. Neuchamp had remarked with pain and\ndispleasure. Hitherto he had seen in drafting-yards only men used to\nmanaging breeding cattle, among which the calf of a week old, given\nto stagger wildly between your legs, and the wary and still more\ndangerously sudden 'Micky,' a two-year-old bull. Thus, to his eye,\ncattle drafting was less a difficult art than one which could obviously\nbe conducted on a more \u00e6sthetic basis.\n\nThat portion of the Rainbar herd which Mr. Neuchamp inveigled into\nthe stockyard, then and there, with the assistance of the black boys,\nconsisted almost wholly of the well-bred station 'crawlers,' as the\nstockmen term them from their peaceable and orderly habits. These\nguileless animals he managed, with but slight driving, to impel into\nthe large receiving yards.\n\nBeyond gazing with mild disapprobation on this proceeding they entered\nno protest. Indeed, when once in the yard, upon seeing the rails put\nup, they had all lain down and commenced the pleasing and reflective\ntask of rumination. They had evidently made up their minds to a day's\n'post and rails'\u2015a matter to be borne with educated bovine philosophy.\n\nMr. Neuchamp then armed himself and black boys with light hunting crops\nhaving slender thongs. With these merely suggestive scourges they did\nnot find it difficult to urge the indifferent animals into the smaller\nforcing-yards. Having got thus far, switches which would sting but not\nbruise were substituted. These seemed sufficiently intimidating to\ncause the steady steers and mild old cows to stroll calmly into the\ndrafting lane.\n\nSo far the unsophisticated heathen, though wondering much at the\nmanifold precautions taken with station pets, carried out all orders,\nin momentary expectation of some miracle being performed. That\nconsummation being slow in arriving, Piambook protested, 'Mine thinkit\npyam nerangi fellow carp now,' head and pluck standing out in bold\nrelief in his mind's eye as he made the suggestion.\n\n'Open that gate, Piambook,' said Ernest gravely, pointing to the one\nwhich led into the 'run-about' yard. Piambook, snuffed out, obeyed,\nand wonderingly observed his master switch beast after beast into the\nvarious receptacles for cattle beyond. They were then released into the\nbush. Upon regaining their liberty, after an inquiring backward gaze,\nas who should say, 'Is that all?' they lay down a few yards from the\nslip-rails and gravely ruminated, much wondering, doubtless, at this,\nto them, wholly unprecedented experience. That night in camp Piambook\nremarked to Mrs. P., before coiling under his blanket, 'Mine thinkit\nMister Noojin wompi-wompi long a cobbra.'\n\nErnest came to the conclusion that man was not born to live alone, in\na gradual, leisurely, and very decided manner, before he was gladdened\none day by the arrival of Mr. Charley Banks, accompanied, to his\nfurther satisfaction, by Jack Windsor.\n\n'The old woman had got all right, bless her heart,' Jack explained,\n'and he had come up in hot haste, after he had heard Mr. Neuchamp had\nbought Rainbar. He found, when as far on his road as Garrandilla, that\nMr. Banks was just starting, so they had joyfully joined company.'\n\nCharley Banks was of opinion that he had got to the right shop at last.\n'Everybody he had heard speak of the run had said,' he informed Ernest,\n'that Rainbar was an out-and-out fattening run; that it was not half\nstocked; that the cattle were mostly very good, except a lot out at the\nBack Lake, and the best thing he could do was to clear them off for\npigmeaters. The Mildool people were sending off a mob next week, and\nthey would take all there were at Rainbar of the same description, and\nshare expenses.'\n\n'Pigmeaters!' exclaimed Ernest; 'what kind of cattle do you call those?\nDo bullocks eat pigs in this country?'\n\n'No, but pigs eat them, and horses too,' affirmed Jack Windsor; 'and\na very good way of getting rid of rubbish; all that's a turn too good\nfor making slaughter-yard bacon\u2015does for the Chinamen; they ain't over\nparticular.'\n\n'Oh! that's it,' said Mr. Neuchamp, reassured; 'but what price will\nsuch cattle fetch?'\n\n'Thirty shillings to two pounds, and well sold at that,' said Jack.\n\n'But would they not fatten, with time and careful management?' inquired\nErnest, loath to lose his probable profits.\n\n'Wouldn't fatten in a hundred years; not in a lucerne paddock, not if\nyou poured melted fat down their throats! They're mostly old savage\ndevils, all horn and hide; only fit for killing people and spoiling the\nrest of the herd. Now's a first-rate chance to get 'em away with the\nMildool lot to Melbourne.'\n\nCharley Banks followed on the same side, observing that the cattle\nreferred to were thoroughly bad and unprofitable animals to keep or\nfeed, and the sooner they were off the run and sold at however small a\nprice the better. 'But I suppose you got something allowed in the price\nfor them, didn't you, by Mr. Parklands?'\n\nErnest now recollected that this must have been the particular\ndenomination alluded to by Aymer Brandon as those Back Lake 'ragers,'\nand in reference to which he had calmly decided to knock off a hundred\nand fifty pounds from the amount of the purchase-money.\n\n'Oh yes, I remember now,' he said. 'I suppose I can afford to sell them\nat a moderate price.'\n\nIt was finally arranged that Jack Windsor should go on the next day\nto their neighbours at Mildool, and induce them to come to in force\nwith all their available hands, as soon as they had mustered their own\noutlaws, to help them to get in and draft the Back Lake mob.\n\n'I don't apprehend that they will be so very difficult to manage,' said\nMr. Neuchamp, with a modest but slightly experienced air. 'That is, if\nthey are taken quietly. I put through a good-sized lot of cattle a few\ndays since, and had only Piambook and Boinmaroo with a hunting-crop\neach.'\n\nMr. Windsor and Charley Banks looked meaningly at each other. The\nslightest approach to a contraction might have been observed in the\nformer's left eye as he made answer\u2015\n\n'There's cattle and cattle, sir. I don't think we had any regular\nout-and-outers at Garrandilla when we used to go and spend a week\nwith old Mr. Hasbene. He told me to give you his best wishes most\nparticular. But they say these Back Lakers has been, in a manner of\nspeaking, neglected. Mr. Parklands was always scraping the run bare\nas he could for fat stock, and let these old guns have their fling\ntill he'd got time to make up a mob and clear 'em all out. But he is\na gentleman as never has a minute to spare; always comin' up without\nnotice, and rushin' off as if another day at home would ruinate him\nout and out, so they all say, and the long and the short of it is,\nit's fell upon us to make a clean sweep of 'em\u2015and a tidy job it is.\nHowever, there's some smart boys from up the river, at Mildool now, and\nI think we can't have a better chance to tackle 'em. Isn't that so, Mr.\nBanks?'\n\nMr. Banks nodded, and Mr. Neuchamp having signified approval, Jack\nWindsor was accredited as plenipotentiary for the Mildool embassy, and\nthe council terminated.\n\nThe improvements were not extensive at Rainbar, Mr. Parklands being\na foe to station expenditure, except where horses and traps were\nconcerned. In outlay for these necessaries of life, as he called them,\nhis enemies asserted that he spent a small fortune annually. Certainly\nhis travelling arrangements needed to be complete. He was continually\non the road. He accomplished wonderful distances, and when once he had\nmade an appointment, whatever the weather, the roads, the season, or\nthe pastime, men knew that they could depend upon him to keep his tryst\nto the day, almost to the hour.\n\n Alike to him were tide or time,\n Moonless midnight or matin prime,\n\nand he had hitherto been extremely lucky, whether from his deep-seated\ndetermination 'not to be licked,' or from other interested quarters, so\nthat one of his admirers went so far as to say that if he had been due\nat St. Thomas's the day after that historic island had been submerged,\n'and a gull above it flying,' Parklands would have been descried\nsailing about in a cutter, searching sanguinely for his I.P., and\ndefying the elements with his customary formula.\n\nStill, though he abstained from fencing, and did not greatly see the\nuse of dwellings in the bush, where a blackfellow was an inexpensive\nand efficient substitute for one and a few sheets of bark for the\nother, he had so far relaxed his austere notions of outlay at\nRainbar as to sanction the erection of two huts and a large, strong,\nwell-planned stockyard. Of these improvements he had boasted on the\njourney to such an extent that Ernest half expected a modified Swiss\nchalet and a stockyard like that of the municipal cattle-yards in\nMelbourne, of which he had seen a photo. Aymer Brandon laughed at\nhis grand description, declining to expect anything but a couple of\nbroken-backed humpies; and as for the cattle-yard, he assured Ernest\nthat at the last muster he attended at Rainbar they carried a lot of\nposts and rails out to the Back Lake in drays, put them up temporarily,\nmustered the fat cattle adjacent, _by moonlight_, and brought the posts\nand rails back with them after they had served their turn. Then Sparks\nemitted divers scintillations, and finally became sulky, and declined\nfurther conversation.\n\nHowever, the huts turned out to be weather-proof and substantial, as\nhuts go, and the stockyard, if not macadamised like the Melbourne Stock\nExchange, or covering thirty-six acres like its Chicago cousin, was yet\na roomy and many-gated enclosure, equal to the working of twice as many\nhead of cattle as Rainbar at this time boasted.\n\nMr. Windsor was therefore enabled to take up his abode with the\nhutkeeper in the edifice which did duty for kitchen and men's hut,\nwhile Mr. Banks secured a second bedroom in the other one with the\nproprietor, and professed himself to be snugly lodged. That young\ngentleman confided to Ernest his extreme gratification at finding\nhimself permanently located at a 'real first-class, fattening,\nplains-country cattle station'; such an establishment, since his\nentrance into regular employment, having been his ideal location.\n\n'Not a sheep near the place or likely to be for years,' he remarked\nexultingly\u2015'that's what I like about it; all good rightdown cattle\nwork to look forward to: drafting, branding, camping, and, I suppose,\ndriving the fat cattle to Melbourne some day\u2015won't that be jolly? As\nfor sheep, I'm sick of the very sound of the name. When your work's\ndone with cattle, it's done; but with sheep it never stops\u2015winter and\nsummer\u2015all the year round.'\n\n'Well, I must say I share your views about sheep, Charley,' said Mr.\nNeuchamp; 'it's the most unending grind that I know. Cattle work has\nthe advantage of being more romantic and exciting when you are engaged\nin it, and of coming to a definite conclusion some time or other, when\nyou can refresh your wearied senses. In the meantime we are not over\nsupplied with resources at Rainbar, as yet. I have sent for some books\nand ordered the weekly papers. Until they arrive, I shall be rather\nhard-set, especially in the evenings.'\n\nThe intervening days were got over without any great difficulty,\nchiefly by means of a series of exploratory rides round the run, up and\ndown the river; these last excursions offering the variety of a little\nshooting, a double-barrelled gun being among the valuables left by Mr.\nParklands, and 'given in,' upon the delivery of the place.\n\nOne evening brought a black boy from Mildool with a message that their\nmuster was done, and that they would bring over the 'pigmeaters' they\nhad gathered, and would muster the Back Lake cattle next day if Mr.\nNeuchamp would meet them there next morning.\n\nCharley Banks was much excited at the news. 'You will see some riding\nnow, and some drafting too, if the cattle are wild. All the best\nstockmen on the river, both up and down, were to be at Mildool this\nmuster. There are some smart boys, I expect.'\n\nOn the following morning Mr. Neuchamp and his friend were astir long\nbefore daylight, and soon after sunrise were well on their way to the\nBack Lake, full of expectation.\n\nNor was the scene when they reached the lonely lake, with the aid of\nPiambook's guidance, other than novel to Ernest's partially-instructed\nvision.\n\nThe Back Lake was a grand-looking sheet of fresh water, covered with\nwild fowl, a thin fringe of timber surrounding its margin. On a\npromontory which ran into the lake for some distance was a camp, bare\nand stripped of herbage to an extent which denoted long and constant\nusage. Skeletons of cattle here and there showed where the rifle had\nbeen at work from time to time, the formidable horns which still\nabounded hinting that abnormal causes had been at work to bring about a\nstate of survival of the fittest.\n\nOn the camp stood, or traversed in angry circles, about a thousand head\nof very mixed cattle, in every sense of the word, a number of grand\nanimals in magnificent condition, mingled with others that the most\ninexperienced eye could observe to be 'stale, flat, and unprofitable,'\nexcept for the very exceptional market and destination previously\nreferred to.\n\nAt the distance of a couple of hundred yards from the main body stood\nthe smaller lot, some four or five hundred, which the stock-riding\ncontingent had evidently brought with them. Some were guarding them.\nSome restrained the camp cattle from leaving their parade ground.\nOthers, among whom Ernest recognised Jack Windsor, were riding in\npairs, and separating or 'cutting out,' as the cattle station phrase\nis, divers excited animals of a fierce countenance from the herd, and\nguiding them into the smaller division, with which, once associated,\nthey were by the guardians thereof prevented from leaving.\n\nMr. Neuchamp's artistic mind was strongly impressed with the wild\npicturesque character of the scene. On every side the vast plain\nstretched unbrokenly as the sea. The score of stockmen, swarthy,\nbearded, carelessly if not wildly attired, bore in looks, and perhaps\nin some other respects, no slight resemblance to a party of Apaches or\nComanches, the 'Horse Indians' of South-Western America. They were well\nmounted for the most part on splendidly-conditioned animals, for no\nliving steeds enjoy richer pasture and purer air than those which range\nthe great saltbush levels of the interior; and generally the riding was\nmore lavish, and indeed reckless as to pace and danger, than those of\nany previous bushmen.\n\n'There goes \"Desborough's Joe,\" the best stockman on the river,' said\nCharley Banks admiringly. 'Him on the roan horse,' pointing to a slight\nblack-bearded man on a magnificent roan horse, who, having forced an\nimmense black bullock out of the camp, was racing neck-and-neck with\nhim, as he tried to break back, and as he 'blocked' the fierce beast\nat every frantic effort to double and rejoin his comrades, 'dropping'\nthe terrific sixteen-foot stockwhip on face or flank with terrific\nemphasis. 'That half-caste boy is a rum one too. By George, he nearly\njumped his horse on to that last bullock's back, when he got him headed\nstraight for the cut-out cattle. There's Jack Windsor coming! they're\ngoing to knock off for a bit.'\n\nMr. Windsor came over to explain to his master that he had remained\nat Mildool to give his assistance until their muster was finished, in\naccordance with use and custom; the head stockman there covenanting as\nsoon as the fat cattle had been sent off to come over, bringing his\npigmeaters, and also his following of fellow-stockmen, to give the\nRainbar folks a turn, and draft their 'Roosians' for them.\n\n'So, as they was a very smart lot of coves as ever I see, sir,' pursued\nMr. Windsor, 'I didn't think as we could do better than get 'em all\nover here and skin the Back Lake camp of all the out-and-outers. We\nmight never have such another chance for no one knows when. If you and\nMr. Banks will come down to the camp, you'll see the sort I'm having\ncut out, and a livelier lot of \"ragers\" I haven't seen for many a day;\nnot since I was at Mr. Selmore's Mallee Meadows. There's only about\nthree hundred of these, and not another on the run. But I'm blessed if\n_he'd_ got anything else\u2015wonderful man, Mr. Selmore!'\n\nErnest accompanied his followers to the camp, where Banks pointed out\nthe types which all cattleholders agree in desiring to 'get shut of,'\nin Jack's phrase, as soon as possible. After a short interval for\nrefreshment, the stockmen, who had been in the saddle before dawn,\nrecommenced cutting out, which tolerably violent exercise was only\nconcluded at sunset. The moon being favourable, the whole band then\nclosed in upon the _enfans trouv\u00e9s_, leaving the camp cattle to go\nwhither they listed. At some time in the night, after a tedious drive\nof many hours, the ample outer yards at Rainbar, with much shouting\nand whip volleys, received them, and the gates being _very_ carefully\nsecured, all further operations were adjourned to the morrow.\n\nEarly on the following morning Mr. Neuchamp betook himself to the yard,\nnervously anxious for a sight of the prey, so safely deposited there,\nin the uncertain light and misleading shadows of the midnight hour. The\n_coup-d'\u0153il_ is uncommon, wellnigh unique.\n\nAbout seven hundred ultra-Bohemian bullocks, whose bodies appear to be\nmere appendages to their terrific horns, are safely (for themselves)\nyarded, many of them for the first time for the preceding ten years.\n\nThe trained bushman of Australia knows that yarding these inexpressible\npariahs simply amounts to arming them for the fray. The resources, in\nattack or defence, developed in the confirmed 'rager,' are only to be\nlearned by experience. He is the grizzly bear of Australia, and with a\nslight shade of odds should be my horse in a fight with that terrible\nplantigrade.\n\nMr. Neuchamp had looked forward to an exciting, perhaps dangerous\nencounter when they reached the station yards. But with this class\nof 'shorthorn' yarding is a much more rapid affair than with quiet\nstation-bred cattle, which delay and resist with contemptuous\ndisapproval born of familiarity. In such a case as the present the\nleaders, if not bent on flight, dash through the widely-opened gateways\ninto the yard like soldiers storming a fort. The rest clear out with\nequal celerity.\n\nIf not frustrated in his first attempt at breaking back, by the sabre\nstroke of a sixteen-foot stockwhip dropped fair between the eyes by\na cabbage-tree-hatted, black, velvet-handed native, the 'rager' cuts\nthrough the opposing ranks like a dragoon through Chinese infantry. No\none goes after him. Perhaps five years afterwards, at another grand\nbattue, a black boy will remark, pointing to an old broken-winded,\nbut indomitable warrior, with horns like scythe-blades, 'You menalu\nthat fella? close up that fella boomalli yarraman belongi to me, long\na Mr. Levison, old man muster long a Boocalthra Lake.' The 'rager' is\nold, weak, and crippled now. The time has passed when he could tread\nthe war-path alone. He will not leave his comrades now. He labours\nalong painfully, but on the grand old visage is stamped indelibly the\n'hall-mark' of courage, the possession of which he shares with the\nmonarchs of mortality. Doubt not that he will reach the yard, and in\nthat enclosure defy menaces, shouts, blows from the unerring waddy,\nfrom the stockman's fire-tailed whips. He passes for the last time into\nwhat is now his graveyard. He will never leave it alive. At shut of day\neight of his enslaved brethren drag him forth to the little spot of\nearth, his\u2015what say I?\u2015our only true heritage. Nature raises him a not\nungraceful mausoleum of marsh-mallow. Farewell they of the unstoried\nherd! Like him, all unknowing of the base pangs of fear\u2015like him, sped\nwith a bullet through his brain, the only true death for a hero!\n\nAfter the pleasant relaxation of breakfast, one of the few\ncomparatively civilised meals encountered during the last fortnight,\npipes were lit, stockwhips greased and garnished with resplendent\ncrackers, and all hands strolled in leisurely fashion towards the\nstockyard. This enclosure presented on approach a tossing sea, 'a\nvision of horns,' most literally. Had there been a particle of\nunanimity among the imprisoned criminals, desperate and accursed in the\neyes of man, a whole side of the yard might easily have been carried\naway upon their united horns, but they were too busy with wars of\nreprisal.\n\nUnable to vent their rage on the common enemy, they rushed, gored,\ntrampled, and bruised one another. Hair, hide, blood, and dust\nwere the staples in present request. The weakest went to the wall,\nmetaphorically, each individual under the average standard of strength\nand ferocity faring like an unwary O'Hallaghan discovered at a fair\ncomposed of O'Callaghans.\n\nThe correct thing, on first arriving at a drafting yard, is to\n'cockatoo,' or sit on the rails, high above the tossing horn-billows,\nand discuss the never-ending subject of hoof and horn.\n\nMany of the captive 'ragers' had personal histories. Heroes of many a\ncamp, they had gradually been driven back to the outside boundaries\nof their respective runs, and, though each of fattening qualities and\ncontumacious conduct, finally outlawed. The cattle-brand of Cain was\nnow affixed to them. Sentenced and finally doomed to the unprejudiced\nstomachs of Chinamen for a consideration of thirty shillings per head,\nhorns given in.\n\nPresently Piambook and Boinmaroo appear carrying bundles of\ncarefully-selected drafting sticks. Each stockman picks his favourite\nweapon, trying its poise and touch, like a billiard cue, and deciding\nwith much care and deliberation. The ends are whittled to prevent\nsplitting; passes and blows are made at imaginary foes. This part\nof the preparation does not last long. No mistakes are made. The\ncool, quiet-eyed youngsters know their weapon well, and the delicate\nand responsible work required of it. A desultory entry into the\nreceiving-yard then takes place, each man picking his own panel.\n\nThe 'ragers' observing this movement keep wildly and excitedly\n'ringing,' like a first-class Ma\u00eblstrom. As a matter of taste\nand safety, the original circular-sailing abyss would seem to be\npreferable. Some one _did_ come out of that alive, _crede_ Edgar Allan\nPoe. But no human 'hide or hair' would have emerged (unmanufactured)\nfrom the 'horn-mill' we have faintly essayed to limn.\n\nThe practised stockriders, keeping an eye on the trampling multitude,\nnow glide down on either side of the yard, thereby preventing a\nsimultaneous rush at the fence, which, though of unusual massiveness,\nis barely up to the weight of six hundred bullocks, say three hundred\ntons, at a high degree of momentum.\n\nThere is no question of charging as yet. Matters have not reached\nthe personal stage between the combatants. If the 'ring' crowds too\nnear the fence, the men on that side would walk along the middle rail\nholding on the while by the 'cap,' or uppermost horizontal, always of\nrounded and not of split timber like the lower bars. If a bullock looks\nat any one 'in that tone of voice,' he receives an admonitory tap on\nthe nose. But the blood of the 'ragers' is not yet hot enough for the\ndesperate stage when they dare everything. So they merely acknowledge\nthe blow by a savage dig into their nearest comrade's ribs.\n\nSuddenly a bullock quits the outer edge at full speed, and dashes at\nthe yard. The herd burst after him like a charge of Cossacks. As if\nby magic, the stockmen form in line, and without a word of warning\nor command each man stands in his proper place. An advance in line\nis made upon the flying squadron. Yells, oaths, sticks, and lumps of\nclay are used to expedite the progress of the maddened animals towards\nthe smaller yards. The leaders beholding a gate, recognise a trap and\nessay to turn. Vain hope! They are doomed to blind progression like the\nleaders of a democracy. They must keep in the forefront of the movement\nor be trampled under foot. Lost is all pride of place; they are forced\non, sideways, backwards, even heels over head, through the gate by the\nmaddened rear ranks observant only of danger from behind. Two men creep\npast along the fence towards the gateway, and at the exact instant\nupon which the recoil takes place, the rails are put up and secured,\nabruptly blocking the most forward bullock, whilst undecided whether\nto advance or retreat. Half of the herd is now enclosed in the forcing\nyards; the remaining moiety, returning, form a smaller ring, and\nrecommence horning their friends where they left off. The men again are\nquietly sitting upon the 'cap,' where pipes are relighted, preluding a\nhand-to-hand encounter.\n\nDuring these last proceedings Mr. Neuchamp transacted a slight\nexperience in this wise. Armed with his hunting-crop, he had chosen\nthe centre of the line, in view of the cattle. When the panic from the\nvan became communicated to the rear, the whole body turned and rushed\nfrantically back to their old position. The stockmen and black boys,\nwell used to the movement, opened on each flank, leaving free egress.\nMr. Neuchamp, less prompt and agile, found himself alone and opposed\nto a legion of horned demons, going straight down his throat, it\nappeared to him, at the rate of 1 to 41. The leading bullock instantly\nappropriated him. Ernest, however, had 'seen his duty, a dead, sure\nthing,' and appeared truly anxious to perform it. Not to interfere\nwith the 'ragers'' right to fair play, he made straight down the yard\ninstead of cutting across at right angles.\n\nAway, therefore, went Ernest Neuchamp, with a bullock, in sufficient\ntraining to win a moderate Derby, within two yards of him. It is\nadmitted that a man under such circumstances always runs up to his\nbest form. Therefore the decision 'by a short horn,' given by a\nsporting stockman seated on the fence, who kindly acted as judge on the\noccasion, created no surprise. Brooding over this occurrence, Ernest\nconcluded to choose a position nearer to the fence on the occasion of\nthe next drive.\n\nNow another act commences. About fifty head have been run into the\ndrafting lane and are ready for separating. The 'lane' is a long narrow\nyard about three panels wide and eight in length\u2015a panel of fencing is\nnot quite nine feet in length\u2015immediately connected with the pound or\nfinal yard, and leading into it by a gate opening into the latter.\n\nTwo men have dropped down into the drafting lane, and are standing, one\nclose to the gate, the other nearer to the cattle. The gateman wields a\nshort drafting stick, not more than three feet in length, of approved\ntoughness, his work being at _very_ close quarters. This, the most\nonerous position in the yard, requires much the same qualities which\nthe harpooner to a whaleboat must own. Quickness of eye, coolness,\nand daring are indispensable. His duty consists in preventing two\nor more cattle of different classes from passing through the gate\nsimultaneously. He is imperatively called upon to read brands, observe\near-marks, age, sex, taking due heed to preserve his own life withal.\nThis, for instance, may suffice for an example. Several beasts are cut\noff by his comrade down the lane, with one only, perhaps, belonging to\na different class. He marks the superfluous individual at a glance, but\ndoes not move till they are close upon him. Then, like lightning, he\nencourages those required by light but rapid blows. The bullock to be\n'blocked' receives one on the nose which arrests him for an instant,\njust long enough to permit his comrades to move irrevocably through the\ngate. As the gate closes behind them another tap causes him to turn\ntail and fly to the rear. Whenever this 'pound' holds cattle of _only\none class_ you hear the deciding shouts from the cockatoo stockmen,\nwho are doing the 'reviewing,' safely on the fence, of 'Fat,' 'Bush,'\n'Stranger,' or 'Calf-yard,' as the case may be. At large musters for\nstragglers, you will also hear the further divisions of 'Up the river,'\n'Down the river,' 'Over the river,' as well as 'Bush,' ring out in\nconstant succession for hours; the last comprehensive direction being\nused for the station cattle. The unerring dexterity of the 'captain\nof the gate,' and his rapid disentanglement of the seemingly endless\nstreams of violent brutes passing through the lane, fill Mr. Neuchamp\nwith admiration, and demonstrate to him that this is a leaf of colonial\nexperience hitherto by him unfolded. He and his mates have gathered\ntheir adroitness from a life-training, and are little less perfect with\nthe drafting stick in their line than Cook with his miraculous cue.\n\n'Ragers,' it may be explained, can only be drafted in two ways, or\nmodes of separation\u2015the stragglers or strayed cattle being divided\nfrom them, in the interest of the attendant stockmen from the adjoining\nstations, who take them home after the muster is over.\n\nTwo gates leading from the pound at the far end are now taken charge of\nby the black boys, Boinmaroo and Piambook\u2015the one answering to 'Bush,'\nthe other to 'Strangers.' The gate from the lane is opened and the\n'ragers' invited through. The invitation is accepted _en masse_, and in\nspite of two or three going down stiffened by a judicious blow behind\nthe horns, they rush fiercely into the pound, and herd themselves on\nBoinmaroo's gate, taking it clean off the hinge and flattening out the\nprimeval, who hangs on heroically.\n\nMr. Neuchamp, after 'they have all passed by,' over gate and boy,\nrushes out to recover the corpse. Before he reaches the fatal spot,\nhowever, that slippery heathen is up and flying round after the\nbullocks, and, indeed, after his pulverisation looking like a demon.\n\nAfter a voyage of discovery round the yard at full speed, they return,\nbest pace, into the lane, where they are permitted to calm themselves\nbefore the next attempt. When it is made, they behave better, though\nall the while keeping the drafters incessantly popping at the fence\nby truculent charges. One hand is stationed in the pound to pass the\ncattle through, where a gate is opened,\u2015no sinecure, with this class\nof cattle, their rage and desperation being by this time beyond all\nbounds. Many a man has lost his life in performing this apparently\nsimple task.\n\nIn addition to the ordinary and patent dangers to the yard, Ernest\nnarrowly escaped, when sitting in a dignified manner upon the 'cap' of\nthe pound\u2015a substitute rail more than seven feet from the ground\u2015being\nhooked off by the scythe-like horns of an infuriated incorrigible. He\nwas then and afterwards dubious as to whether his and Piambook's joint\nessay at improved cattle-drafting was a fair test of his theory, the\nenergy and bloodthirstiness displayed by the present performers leading\nto a reconsideration of his system. However, with true British pluck,\nhe will not desert his theory without further trial.\n\nHe had observed that in cases of 'charging,' the assaulted one merely\njumped on to the bottom rail of the yard fence, held on by the top, and\nmet the advancing foe with a seemingly unnecessarily cruel blow on the\nnose, in most instances causing effusion of blood. The blow, unless\nwith a recognised 'bravo,' was sufficient to avert the charge.\n\nErnest took the first opportunity to volunteer for this post, which\nwas freely accorded to him\u2015the chief requisite being agility. With a\nlight switch he betook himself into the yard. The first half-dozen\nshot through like cannon-balls, possibly not having cast eyes on the\ncongenial prey. This state of affairs did not continue.\n\nThe acknowledged bully of the yard put his head down and charged\ninto the pound like a whirlwind. The gate was shut and all hands\nseated upon the fence with marvellous celerity. This warrior was a\nvery evil-looking beast\u2015a tall, hurdle-built magpie brute, with a\ndevelopment of horn remarkable even in that forest of frontlets.\nOne circle he made round the pound, tossing blood and foam from his\nnostrils on every side, savagely lunging at every one he passed on the\nfence, treating the heavy blows which, alas! from time to time fell\nheavily upon his bleeding face with superb contempt. As he passed Mr.\nNeuchamp that gentleman lightly dropped behind him and switched him\non the haunch, as a hint to move through the gate held open for him\nby Piambook. The mighty beast swung round. For one second his glaring\nvisage seemed to say, 'I'll have your blood anyhow.' That second\nprevented the impalement of a hero of fiction! Ernest turned, and for\nthe second time that day showed great pace. But when making a spring\nat the fence, between the pound and the lane, his foot slipped off the\nrail and he fell forward from the 'cap.' The maddened animal, seeing\nhis victim escaping, gave a terrific bound and succeeded in planting\nhis fore-feet on either side of Mr. Neuchamp, though his hind-quarters\nstill rested on the ground. Here he made frantic efforts to clear the\npanel and Mr. Neuchamp, the agony and uncertainty of whose position\nwere indescribable, as his gasping articulation testified.\n\nBut help was at hand. A stalwart Lachlan native sprang like a tiger\nat the beast's head, and with a few crushing blows forced him to\nstagger back into the yard. As he turned a comparatively light tap\nfrom a wattle drafting stick on the spine, behind the horns, dropped\n_l'enrag\u00e9_ in his tracks, as if struck by lightning\u2015his nostrils in\nthe dust, his eyes turned backwards, and his huge frame quivering in\nevery muscle. Slowly recovering his senses, he staggered to his legs,\nand perceiving Piambook standing in the middle of his gateway, as if\ninviting him to the feast, rushed blindly and with unabated fury at\nhim. That astute aboriginal disappears from his gaze; he reels wildly\nthrough the gate on to his head, picking himself up in the next yard,\nwhere he meets with the usual sympathy from his companions.\n\nMr. Neuchamp is restored by the exhibition of a strongish dram. As he\nobserves the last bullock enticed out of the lane by having a bag\nthrown to him, which, after savagely driving his horns through, he\ncarried forth thereon in triumph, he confesses that nothing short of\nhand-grenades, prepared with nitro-glycerine, can be esteemed suitable\nimplements for the effective drafting of 'pigmeaters.'\n\nThe fray was finished. Enough had been done for glory, and even for\nsome modest minimum of profit. The gates and sliprails of the yard\nare scrupulously secured, and all thoughts of work abandoned for the\nday. On the morrow a grand departure was carried out. The estrays\nor stragglers\u2015a not inconsiderable drove\u2015were escorted away by the\nstockrider contingent, who held a collective interest in them. And\nthen, with much care and forethought, with horsemen in front, in\nflank, in rear, the gates were opened, and the swine-doomed multitude\nrushed forth, extremely lively, 'you bet,' but gradually assuming an\nappearance of sobriety as the purposely long day's journeying wore on.\n\n'I call that a bit of first-rate luck,' propounded Mr. Windsor,\n'getting all these rowdy old devils off the run in one muster, like\nthis; thirty of 'em, let alone three hundred, 's enough to spoil the\nbest herd in the country. There was some splendid fat bullocks\u2015reg'lar\nplums\u2015about that Back Lake camp\u2015never saw primer cattle in my life.'\n\n'Nor I,' agreed Charley Banks. 'I never set eyes on a better-looking\nrun than this, let alone the saltbush. It don't appear to me to be half\nstocked, that's another thing.'\n\n'We shall have to consider what is most necessary to be done next,'\nsaid Ernest, with a thoughtful expression. 'There must be many\npressing things of importance, as so little appears to have been\nthought of hitherto. The arrangements are simple, even to barbarism.'\n\nMr. Neuchamp was shocked that morning, on going into the meathouse,\nto find that the corned beef _cask_ consisted of four upright round\nsticks, with a hide stretched across. In the deflected centre of\nthis not particularly clean raw hide was placed above five hundred\npounds' weight of salted beef. To this magazine the entire household\nresorted in its need. He at once made an item, 'Casks,' to be added\nto the tolerably long list of articles required for immediate use at\nRainbar, which he trusted to obtain when the first drays should make\ntheir appearance from Sydney. He then sat down and wrote a long letter\nto Paul Frankston, in which he described the delivery of the station,\nnot forgetting to chronicle his gratitude to Mr. Aymer Brandon for his\nexertions in his behalf, and his satisfaction at the liberal manner in\nwhich the former proprietor had behaved throughout the whole affair.\n\n'I feel now,' was his concluding paragraph, 'that I am fairly launched\nas a pastoral proprietor, and I trust that I shall be able to combine\na fair amount of profitable management with the reform of many\nobjectionable practices and the improvement of station life generally,\nas it has hitherto obtained, on such distant properties as, up to this\nperiod, Rainbar may be considered to have been. A large present outlay\nwill be unavoidable, but I feel certain that the increased profits,\nunder improved supervision, will amply repay this and any future\ndisbursement.'\n\n'All very fine,' remarked Mr. Frankston to his cigar, as he put his\nyoung friend's letter into his pocket with a dissatisfied air, 'but if\nhe commences to spend money in accordance with his notions of what he\ncalls improved management, he will soon run himself aground. That's not\nthe way young Parklands worked the place when he went into it first,\nI'll be bound. It's extraordinary how every one who comes to this\ncountry of ours will persist in thinking that he has imported the first\nconsignment of brains ever landed upon the continent. Well, I foresee\nthat he will have his own way. If the seasons are good and cattle rise,\nhe may pull through.'\n\n'And if not, papa?' inquired the soft voice of Antonia, who had crept\nup to the old man's chair and placed her arm caressingly on his\nshoulder.\n\n'And if not, my pet,' said that experienced colonist, with a subdued\ngrowl, into which he attempted to infuse the unfailing tenderness which\ninvariably characterised his speech to his fondly-loved daughter, 'if\nnot, why in three years our young and ardent friend will have to make\na living out of his \"plans for reform,\" for he will have nothing else\nleft, as sure as my name is Paul Frankston.'\n\n'Oh, don't say that, papa,' said Mr. Neuchamp's indulgent though\nsensible advocate; 'surely he is far cleverer than most of the young\nmen that come out and turn squatters with just a \"little experience,\"\nand see how well some of them have done.'\n\n'It is not that he has a worse head, but I doubt most of all because\nof his better heart. That will destroy the balance. It's a bad thing\nfor money-making. A man can make money, save money, or keep money, with\njust as few brains as will prevent him from falling into the fire.\nBut let him have only as much more heart than his neighbours as would\noverbalance a nautilus, and money falls away from him like quicksilver.\nIt's a fatal defect, Antonia, my darling; and I'm afraid our young\nfriend has it incurably.'\n\n'It's a fault on the right side, at any rate!' said the girl, raising\nher head proudly. 'Those who think tenderly and faithfully concerning\ntheir fellow-creatures are not, perhaps, so clever with the \"muck-rake\"\nas self-seekers who bore and tunnel, like moles, all their lives, never\nturning their eyes towards the blue sky, the golden sun, or the glad\nwaters. It cannot but be that those who have loftier aims should have\nsome compensation even in _this_ world; and if they are not so clever\nin helping themselves, why, their friends must help them all the more.\nDon't you think so, pappy dearest?'\n\n'He\u2015m!' answered the capitalist warily. 'That depends upon\ncircumstances. Some people require a great deal of helping.'\n\n'The greater triumph when they are finally helped into safety and\nsuccess, and then they are sure to help others. Prosperity opens the\nhearts of really generous people more and more. By the way, how did\nPaul Frankston ever come to make any money? Tell me that, sir?'\n\n'Have no idea, puss; all a fluke, I daresay. I don't think _he_ would\ntrouble his head much about it, except for the sake of a certain\nself-willed monkey, who ought to be in bed and asleep. Good-night,\ndarling.'\n\n\n\n\nCHAPTER XIX\n\n\nFor the first few months after Mr. Neuchamp had commenced to sit upon\nthe throne of Rainbar, there was a large amount of station work to do,\nwhich, at the instigation of Mr. Banks and Jack Windsor, was pushed\non with and completed. There were any number of calves to be branded,\noutlying cattle to be got in, the herd generally to be mustered and\nmade to 'go to camp' properly, as well as many other things necessary\non a cattle station newly purchased, and which had not been, let us\nsay, very exactly administered for some years past.\n\n'It's my belief there's some of these LP cattle at every station within\na hundred miles of Rainbar,' said Mr. Windsor one day, as he and Mr.\nBanks returned from a neighbour's muster, with a goodly number of cows,\nunbranded calves, and pen-branded bullocks. 'It was these last store\ncattle they got that seems to have scattered and made out all over the\ncountry. They say it came on very dry after they were turned out. Their\nhorses was that weak they couldn't ride after 'em, so they had to let\nthem go their own way.'\n\n'Indeed,' said Ernest sympathisingly; 'they must have lost great\nquantities, or did they come back again?'\n\n'They wouldn't come back, because they didn't know the run well enough\nto care about it over much. But they weren't teetotally lost, 'cause\nthey've stuck at every herd they came to, and in course of time we'll\nhave 'em all at home again.'\n\n'You are sure they will not be lost?'\n\n'Not a bit of it,' affirmed Mr. Windsor. 'A brand, once well put on,\nis like a direction on a letter. People may steal the letter, or kill\nthe beast. But every one who don't go in for them tricks will help the\nowner of a stray beast to get him, if his brand is readable, just as\nhe'd give you a letter addressed to you, if he was to pick it up on the\nroad.'\n\n'What will you do with these strayed cattle, then, when we get them\nhome?'\n\n'We must let them go again; there's nothing else for it. And I'll wager\nhalf of them will just turn and walk back again.'\n\n'I have been thinking,' said Ernest meditatingly, 'that if we had a\nlarge paddock put up here, it would do capitally to keep strayed stock\nin, and for the horses. Surely it would save time.'\n\nJack admitted that an enclosure of the kind would be very handy for the\nclass of cattle referred to, so Mr. Neuchamp at once made a note of a\nton or two of wire for the purpose. Thus simply and unobtrusively was\nthe 'Improvement Idea' initiated at Rainbar. Once admitted, it grew and\nenlarged into vast and even alarming proportions.\n\nHow many an ingenuous pastoralist has for years wandered innocently by\nthe charmed ocean-strand of Arcady the Blessed, leading the careless,\nuntroubled life which belongs of right to all true Arcadians, ignorant\nalike of want or luxury, of debt, of anxious thought for the morrow!\nWhen, lo! in a luckless hour, unhallowed desire has urged him to\nthe opening of the sealed, the forbidden casket which contained the\nGenie\u2015'Improvement.'\n\nThe baleful Djinn, accursed of Solomon and many succeeding wise men,\ntowers aloft, darkening the summer sky, and finally demanding the life\nof his deliverer. In the Eastern tale, the threatened victim cajoled\nthe monster into re-entrance and brazen bondage. Rarely, alas! does the\nmodern enfranchiser of the Demon succeed in enforcing retrenchment and\nsafety!\n\nMr. Neuchamp had a general idea, based upon Paul Frankston's parting\ninstructions, Mr. Levison's warning words, 'Don't you waste your\nmoney,' and even the half-careless hints of Brandon and Parklands,\nthat his course as a squatter was to be guided by economy. At the\noutset, therefore, he merely ordered articles and implements absolutely\nnecessary. He devoted his spare time to the task of instilling some\nglimmering rays of intellectual light into the unused but not opaque\nintelligence of Charley Banks. Finding that the boy had a strong taste\nfor voyages and travels, he provided him with books of that particular\ndepartment, and gradually had the satisfaction of seeing the lad settle\ndown of an evening to steady reading, instead of to the eternal pipe,\nwith perhaps an excursion to the kitchen and a not wholly improving\ngossip with Jack Windsor.\n\nHe drew him out, and invited him to the discussion of principles of\naction derived from the lives of his favourite heroes. He encouraged\nhim to digest a certain daily quantity of 'stiff' or improving\nliterature, and arranged that the more humorous celebrities of the day\nwere not wanting. He sketched a combination of reading and reflection,\nwith the hard personal exertion and keen practical attention to detail\nwhich the youngster loved. He drew his attention to distinguished\npersons who combined excellence in both classes of attainment; and he\ndemonstrated how poor and mean a goal is that of material success,\nunrelieved by mental progress or spiritual enlightenment.\n\nBut when all the calves were branded up, so completely that no\nmore work, in that direction, could be done until more calves were\nborn,\u2015when all the stragglers were got in, and there were no musters\nto attend; as the days grew longer, the sun hotter, the whole routine\nmore uniform and monotonous,\u2015life commenced to be burdensome to Ernest\nNeuchamp. Then the fascinating idea of works and enterprises of a\nnew and reproductive nature, like the temptation of a hermit in the\nThebaid, arose with resistless might.\n\n'After all,' he argued, 'if he were able, by his own contrivance and\ninvention, to anticipate fortune for a few years, instead of dragging\nout endlessly a life, perchance meant for better things, was he not\npractising economy in the truest form?'\n\nSuch, after certain mental conflicts and long calculations, was the\nquestion which he answered to himself in the affirmative. From that\nhour he ceased to struggle with what appeared to be either a matter of\ndestiny or the prompting of an enlightened self-interest, according\nto the mood in which he found himself when considering this momentous\nquestion.\n\nThe first operation foreign to the primitive, not to say barbarous,\nsimplicity of the Rainbar establishment was the putting up of the\npaddock, at least double the size which Mr. Windsor had suggested, for\nthe safe keeping of straggling cattle. _Ce n'est que le premier pas\nqui co\u00fbte._ After that 'improvement' was completed and paid for by the\ncrisp new orders out of the book furnished to Ernest by his agents,\nMessrs. Oldstile and Crampton, a highly unimaginative and trustworthy\nfirm recommended by Paul, a new four-roomed cottage, of horizontal\ntimbers, arose on the bank of the lagoon, to the great amazement of\nPiambook and Boinmaroo.\n\nBy this time a considerable number of the bush labourers of the\nperiod had found their way to Rainbar. Rumour, which disdains not the\nfar interior, but indeed seems to be additionally sonorous in the\nremoter haunts of man, had sounded her trumpet-blast far and wide\nwith reference to Ernest Neuchamp's acts and assets. The former were\nsummed up 'as going in for no end of improvements,' and the latter were\nconfidently credited with unlimited resources.\n\nThe next project possessed the merits of grandeur of conception\nand perfect novelty, at least in the neighbourhood of Rainbar, the\ninhabitants whereof might have been numbered among the most pious\ncommunities in the world, from their consistent dependence upon\nProvidence, had their morals in other respects borne investigation.\n\nMr. Neuchamp had noticed that the Back Lake, as it was called, had\nevidently been filled recently by the overflow of the river, the waters\nof which had been conducted by a tortuous but plainly defined channel.\nThe level of this inland sea, for it was of great extent, had lowered\nconsiderably since his occupation. In the event of a dry season it\nwould doubtless become dry. Assuming this to take place, the cattle\nhabitually watering there would be thrown upon the world\u2015would be\nreduced to betake themselves to the 'frontage.' 'Great inconvenience,\nperhaps loss,' so said Charley Banks and Windsor, 'would result.'\n\nThen again, about ten miles from the Back Lake was another titular\nlake, dry at present, but with well-defined banks, bearing traces of\nhaving once been filled with water. This was called the Outer Lake. It\nwas surrounded by splendid plains, but was only available for the stock\nduring a short period in winter. This natural basin Mr. Neuchamp boldly\nproposed to fill from the Back Lake, after he had replenished that\nreservoir from the unfailing waters of the Great River.\n\nAfter a careful examination and survey, he came to the conclusion that\nby deepening and cutting the curves of the 'blind creek,' or natural\nchannel along which the waters of the flooded river had always reached\nthe Back Lake, he could ensure the filling of that great basin in an\nordinary season. Secondly, by a straight and not particularly wide\nor deep cutting connecting the two lakes, the outer basin could be\nfilled as regularly and completely as the inner. Noting the levels,\nand computing the probable expense\u2015considerably under its ultimate\namount\u2015Mr. Neuchamp retired to bed at an unusually late hour. But he\ncarried with him the proud consciousness that he was destined to become\nthe Lesseps of the Lower Darling. He slept heavily, but his dreams\nwere troubled. At one moment Piambook approached, anxious to decorate\nhis bosom with one of the brazen crescents which adorn the breast\nof confiding aboriginal royalty. At another, a group of officials\nand improbably well-dressed pioneer squatters gathered around him,\nwith approving glances and well-filled bumpers of champagne. Then\nHartley Selmore smilingly proposed the health of the most original and\nsuccessful engineer of the age, while Antonia Frankston gave the signal\nto raise a floodgate, which permitted the impatient waters to connect\nthe farthest Australias.\n\nErnest had no sooner 'ciphered out' this fascinating project, than he\nfound ready to his hand a considerable body of labourers, who in one\nway or another had been employed in putting up the cottage and the\npaddock. More strength was speedily available, as the report gained\nrapidly in sensation, until nearly all the peripatetic labour of the\nland had heard tell of the newly-arrived proprietor of Rainbar. He was\nimpatient, it was said, to fence, dig wells, make dams, and cut canals,\nin all directions. So the able-bodied swagsmen hasted towards Rainbar,\nwith the frantic fear of being too late which characterises the\nstampede for a 'new rush' among a mining population. Mr. Banks and Jack\nWindsor, and above all Piambook and Boinmaroo, were wildly astonished\nat the unfailing stream of tramps, of all sorts, sizes, and capacities,\nthat poured in.\n\nThe blacks began to think that the King of England had made up his mind\nto take away Rainbar from Mr. Noojim, and that this was the vanguard of\nan army sent up to enter into possession.\n\nCharley and Jack Windsor, sharing the prejudices of old-fashioned\nsquatters against 'too many hands about the place,' looked grave.\nIndeed the latter ventured upon a mild remonstrance, as he sent man\nafter man to work at the canal. Rations began to be served out in such\nquantities, that Charley Banks, who was storekeeper, had little else\nto do but to distribute. He stated his conviction that the flour would\nsoon be gone if the drain continued. 'Then,' he supposed, 'they would\nhave to live upon beef and pumpkins until the next drays came up.\nGetting through work was all very well, but this was making the pace\n_rather_ strong.'\n\n'Don't you think, sir, excuse me,' said Jack one day, when a bag of\nflour and half of the last bullock had been served out in one forenoon,\n'that we're getting rather too many knock-about men for a small station\nlike this? It ain't my place, I know, to meddle with your ways of\nmanaging, and so on; but I've been on many a station, and I've never\nseen half, or quarter the muster we've got here lately.'\n\n'I shall always be willing to hear and consider your opinion, Jack,'\nsaid his master, with that philosophic urbanity which distinguished\nhim; 'you are a shrewd, sensible fellow, and, I know, faithful to my\ninterest. But you _must_ see that the cost of employing one man for\nfifty days, or fifty men for one day, is precisely similar. Excepting\nalways that you save forty-nine days in time by the latter arrangement.'\n\n'Well, that's right enough, sir; but, somehow, none of the gentlemen\nI know as has made money out of their stations never liked to see a\nlot of men being fed and paid and kept about the station, except for\nshearing or such like.'\n\n'But don't you think the canal will be a splendid thing for the run, if\nwe can get the river water to Outer Lake?'\n\n'Well, sir, if it does, all very well, but somehow I don't seem to be\nquite sure that it will; and if cattle keeps low, where's the money to\ncome from?'\n\n'Whether cattle sell cheaply or otherwise, if we can get five thousand\npounds' worth of water for five hundred, it pays well to lay out the\nmoney.'\n\n'Ah well, sir, I can't say for that. But I think you might give it\na thought whether these chaps are likely to do much of a day's work\nat this cutting, or whatever you call it. As long as they have their\ngrub and their wages they'll hang it out, one again the other\u2015regular\nGovernment stroke, as we say in this country.'\n\n'But how can I arrange it otherwise?' inquired Ernest anxiously.\n\n'Give it 'em by piecework,' replied Mr. Windsor confidently. 'You\nwatch, now, how much half a dozen of the best of 'em does in a day.\nMeasure it when you're by yourself; then run it off what it comes to at\nthe wages and rations you pay. After that you can let it to 'em at so\nmuch a foot, or so much a rod, for them to \"find themselves\" out of the\ncontract price.'\n\nThis very shrewd practical suggestion was, after consultation with Mr.\nBanks, finally adopted. The small army of excavators was informed that\nhenceforth the pay would be at the rate of so much per cubic foot;\nthat their rations, of whatever quantity, would be debited to them, as\nthey would have to 'find themselves.' And that no departure from this\nscale of payment and charges would be permitted. After some grumbling,\na little scheming, and a few departures, matters went on quietly. Mr.\nNeuchamp surveyed with satisfaction, week by week, the smooth-edged\nchannel crossing the endless plain, destined, if all went well, to turn\nback-country into frontage, and so revolutionise custom and compel\nfortune.\n\nAfter this great achievement was fairly on the road to completion, Mr.\nNeuchamp turned his mind to the dignified and fascinating science of\nhorse-breeding. He had, in the comparative solitude of Rainbar, been\nrevolving this vitally important question, dear to every descendant of\nBritons in every quarter of the globe. He had been pained and grieved,\nof late, to observe that so few among the countless droves of equine\nforms with which the land was overrun were worthy the name of horses.\nThey bore no approximation to the gallant, delicate-limbed desert steed\nof Arabia\u2015as little to the stately, swift, and powerful animal that the\nscience of English breeders has evoked from the questionable coursers\nof the past.\n\nHe looked around, inhaling the dry, pure, exhilarating breeze, and\nmarked the wide expanse of sandy levels. He felt the fervid rays of the\ntrue desert sun. 'This,' he exclaimed, 'this is the climate, this the\nsoil, the land, for the ancient royal desert blood, and no other. Here\none might rear a race of gallant steeds, that would sweep tireless on\nfrom dawn to midnight.'\n\nHe recalled the magnificent performance of the two aged but\nhigh-descended mares, so wondrously described in the passage of the\n_Talisman_, when the Hakeem bears away his guest through the desert\nfrom the pursuit of the Templars. He thought with disgust of the sudden\ncollapse, after only a couple of miles of sharp going, that his cob\nhad treated him to, when the blue bullock thirsted for his blood.\nAnd vowing that, in days to come, no proprietor of Rainbar should\nsuffer probability of so ignominious a doom, he was confirmed in his\nresolution to acclimatise a race of Australian Arabs at Rainbar, which,\nglorious in the present, should live in the future unsurpassable and\nimmortal.\n\nHe ultimately arrived at the conclusion that it became the solemn\nduty of every man, placed by Providence in the enviable position of a\npastoral proprietor, to do his best to provide the good land, to which\nhe owed so much, with some lasting benefit or substantial legacy.\n\nMr. Neuchamp's bequest to the tutelary deity of Australia\u2015plus the\nmost improved shorthorns, which he was determined to promote, with his\nheart's blood if necessary\u2015was to take the shape of a stud of Arab\nhorses. In imagination, he saw them caracoling over saltbush plains and\nsand ridges, tossing their small expressive heads, waving their flowing\nmanes and tails, while their clean, flat, everlasting legs and iron\nhoofs would be patent and admirable to every one who had sense enough\nto know an Angora goat from a deerhound. In the event of remounts,\nwhich were continually required for the Indian army, an entire regiment\nmight be supplied from Rainbar in days to come.\n\nMr. Neuchamp gave the reins to this Arabian imagination, until he\nbegan to be oppressed with the crowds of princes and magnates of the\nearth, who came suing for the inestimable privilege of a charger from\nthe Rainbar stud. Then he closed the day-dream. But the idea was fully\ndeveloped, and he wrote to his agents to order a high-caste Arab sire,\nto be sent down at once from India. He then made arrangements for a\nnumber of well-bred brood mares, wherewith to make a commencement of\nthe great Rainbar Austral-Arab stud.\n\nThe summer had come to an end; the autumn had fairly set in, when the\ntime for mustering fat cattle arrived. That portion of the economy of\na cattle station, so suggestive of coin, was safe to be attended to.\nThis was perhaps the pleasantest description of work which had happened\nduring the period of Mr. Neuchamp's proprietorship of Rainbar.\n\nUnder the apparent leadership of Charley Banks, with the aid of Jack\nWindsor, the neighbouring stockmen went forth on the war-paths, and the\ncattle were duly mustered upon the Main camp, the Sandy camp, the Wild\nHorse camp, and finally at the Back Lake camp. No yarding took place.\nThe fat cattle were to be duly separated, after approved custom, known\nas 'cutting out,' at each camp.\n\nA muster for 'cutting out' is a novel and exciting scene for the\nstranger tourist. A cattle 'camp' is a rendezvous, used by a\nsubdivision of a herd of cattle for purposes apparently of friendly\ngathering, converse, and social recreation\u2015a Bovine Club. Sometimes\nthe needful bare space, covering from an acre to half a dozen, is\nsituated under shady trees; sometimes by the side of a river, marsh, or\nwater-hole; sometimes on a naked sandridge, shadeless, waterless, alike\ndestitute apparently of beauty and convenience.\n\nThe system of camp, with the aid of which the greater part of the work\nof every cattle station is carried on, would appear to have originated\nin the earliest days of colonial cattle-herding, the instinctive\ntendency of all cattle permitted to rove at will within certain\nlimits being to assemble daily, generally as the heat commences to\nbecome oppressive, at a given spot, affording for the most part shade\nand water. Towards the decline of day the friends or acquaintances\nseparate, each moving slowly on to its particular feeding-ground. A\npeculiarity of bush cattle, partly instinctive, partly the result of\ntraining, is to run to camp upon hearing alarming noises, or being\ndisturbed at their feeding-grounds. Cattle in their natural state are\nexceedingly timid. Nothing is more common than for two or three hundred\nhead, feeding at the outskirt of a large run, to start off in sudden\nalarm at the flight of birds, the sight of blacks, or the stampede of a\nmob of wild horses. At a moment's notice they are off at full speed,\nwhich they keep up without 'crying crack,' as the stockmen say, until\npanting, and with heaving flanks, they can halt and 'round' up in the\nbeloved camp.\n\nOf this peculiarity advantage has been taken by stockmen, finding it\na great aid to management, and a substitute for expensive stockyards\nand troublesome yard drafting. Thus one of the first things which an\nexperienced stockman does when he is forming a cattle station, by\nherding the cattle upon it for the first occupation, is to regulate\nthe camp. If he perceives that the cattle, after being turned loose,\nand no longer 'tailed' or followed daily as a shepherd does sheep of\ntheir own accord, 'take to,' or agree to prefer, certain suitable\nlocalities for camp, he wisely does not interfere. He merely observes\nand visits from time to time, but, traversing daily the outskirts of\ntheir beat, or by cracking his whip or using his dogs, rouses and\nalarms them, so training them to 'run to camp.' After a few months of\nthis exercise he is moderately sure that on any given day he will find\nat a certain hour the larger proportion of each subdivision of the herd\nat one proper camp, and that almost every straggler will find its way\nto some rendezvous of the sort. If the camp be unsuitably placed, the\nstockman shoots a beast of no value, and leaves it upon the spot which\nhe selects for a camp. He then makes a practice of driving the adjacent\ncattle to the spot two or three times a week. They are attracted by the\ndecomposing carcass, around which they paw, roar, and trample, after\nthe manner of their kind. Gradually the space immediately around is\nrendered bare. The cattle become familiarised to it as a daily lounge.\nThey commence to run towards it, and of their own accord, and then the\ncamp is formed.\n\nSuch is their origin and nature of formation. The advantage is patent.\nThe driving of cattle, especially of a large herd, into a yard is\nalways a troublesome, costly, and injurious process. The larger and\nfiercer cattle horn, crush, and sometimes fatally injure the weaker.\nCalves are hurt. Occasionally valuable cows are injured; even the\nstrongest and fattest animals are not improved by the cruel goring and\nceaseless crushing to which they are exposed during days or nights in\nthe yard.\n\nIn camp-work there is little or no chance of oppression or hurt.\nAfter an hour's 'beating up,' and ringing of whips, streams of\ncattle are seen pouring in from every point of the compass towards,\nlet us say, the main camp. Generally situated at no great distance\nfrom the stockyard, this is supposed to be the central and principal\ntrysting-place. From one side comes a long string of comparatively\nsober and peaceful cattle, comprising a goodly number of cows and\ncalves. They trot leisurely, perhaps merely walk, until they reach\nthe bare mound by the side of the long reed-covered lagoon, shaded by\nvenerable white gums. There they halt or walk peacefully round and\nround. But stop\u2015now far and faint more whips resound, which from time\nto time one hears like a tapping-bird or the snapping of dried sticks.\nOnly the half-Indian sense of the bush-reared stockmen could say with\ncertainty that these sounds were the volleying detonations of the\nmighty stockwhip, that terrible weapon in the hands of an Australian\nbushman. The sounds are louder, nearer, less ambiguous; the muffled\nlowing of a great concourse of cattle comes down the wind, mingled with\nshouts, yells, and strange cries. At length the herd gradually come\u2015\n\n Nearer still, and yet more near,\n The trampling and the hum,\n\nwhen suddenly there is a shout of 'There they come,' and a long line of\nmagnificent bullocks, fiercely excited, breaks through the adjoining\ntimber. On they come at a swinging trot, heads down, eyes glaring,\nin some instances tongues out, heading straight for the camp. Behind\nthem is a great herd of mixed cattle, of which they are the advanced\nguard. There are so many of them that the 'tail' or rear is not at\npresent visible. From the increasing whip volleys, the barking of dogs,\nand the shouts and cries of men, it would appear that the 'tail' is\nnot actuated by the same lofty feelings of pride and courage which\nmark the 'head' of the column that has just dashed into camp in such\ndistinguished fashion.\n\n'My word!' said Charley Banks, 'that's something like a mob! What a lot\nof rattling bullocks, shaking fat too; this is my sort of cattle run;\neverything fat, from the calves upwards; as long as there's plenty of\nrain, there's no fear of the feed running short, and my opinion is that\nthere's room for twice as many cattle as we've got\u2015and more than that,\nif there was water at the back.'\n\n'And I feel confident,' answered Mr. Neuchamp, who was surveying with\nan eye of satisfaction his camp full of well-conditioned cattle, 'that\nin less than two years there will be water all the way from the river\nto the Outer Lake. That will be something like an improvement, as you\nAustralians call everything from a bark hut to a five-hundred guinea\nwash-pen.'\n\n'I hope so,' said Mr. Banks, without any great show of enthusiasm.\n'But improvements cost a deal of money, and my old uncle used to say\nthat the money ought to come first, in station management, and the\nimprovements afterwards. He made plenty, but he never would go into\ndebt, even for his wool bales. He used to lecture me for buying so much\nas a pair of hobbles without paying cash.'\n\n'The principle is sound, no doubt,' replied Ernest thoughtfully. 'But\nit may be pushed too far; I think many of the older pioneers might\nhave made all the money they did in half the time if they had only had\nsufficient foresight to organise plans of reproductive outlay, certain\nto pay cent per cent upon any money which they might have expended, or\neven borrowed at reasonable interest, for their construction.'\n\n'Old Nunkey used to say that reasonable interest had a knack of\ngrowing into unreasonable interest if you didn't pay up half-yearly,\nwhich people often found something to prevent their doing,' said the\nprudent youngster. 'Of course, I don't know much about spending money\u2015I\nnever had any to speak of; but there's nothing beats a certainty,\n_I_ think.' Here 'the tail' of the large lot of cattle of which 'the\nhead' was so sensational and satisfactory, made their appearance, much\ngratified at being permitted to round up on the camp and mingle with\nthe main multitude, with which they exchanged pushes, greetings, and\nsalutations. Behind them rode Jack Windsor, accompanied by a band of\npicked volunteers, who, with him, had done an immense amount of outpost\nduty since sunrise.\n\nIt was considered reasonable to devote half an hour to rest and\nrefreshment, which comprehended the calming down of the somewhat\nexcited cattle, and a smoke for the stockmen. After this a disposition\nof forces was made. Certain moderate performers were told off to\nencircle and keep within the camp limits the main body of the cattle,\nwhile 'the equestrian talent' was selected to carry out the more\ndashing and delicate duty of 'cutting out.' And few tasks had a more\ndifficult appearance than to divide the fierce and wild-eyed bullocks\nfrom the mixed medley of a thousand head of cattle of all ages and\nsizes which crowded the camp.\n\nFirst, Jack Windsor and a friendly centaur\u2015part and parcel of a violent\nblack mare\u2015ran out half a dozen quiet cattle, placing them in charge of\nthree other men, at about two hundred yards' distance from the camp.\nThen he, Charley Banks, and half a dozen of the best mounted men went\nin to the herd, and commenced to run out, singly or in pairs, such fat\ncattle as were up to the marketable standard.\n\nMr. Neuchamp for a while confined himself to riding usefully but\nunromantically round the cattle on the camp, preventing them from\nflowing out in unnecessary directions, and making off when the\nentertainment commenced to flag. He watched bullock after bullock being\nedged out by the trained horsemen to the rim of the camp, then suddenly\nforced into the open by the sure and sudden whip, which, silently\nraised, appeared to drop upon every portion of any given animal at\nonce. As the roused animal commenced to stretch out into a gallop, to\nhalt suddenly, to attempt to wheel in his tracks, it was a sight worth\nseeing to note the swift, wary, duplicate motion of the stock-horse,\nthe watchful alertness with which the stock-rider reined his horse,\nurged, restrained, or checkmated the doomed bullock.\n\nAs Mr. Neuchamp gazed he came to the conclusion that the emigrant\nBriton, if young and active, might attain considerable ability in\nstock-riding. But as for the lithe instinctive swaying grace with\nwhich horse and rider moved alike in desperate rush or wondrous whirl,\nit was unapproachable by any one 'not to the manner born.' One hour,\ntwo hours, passed, and still the same rapid and continuous selection\nof beeves went on. The once small drove of 'cut-out cattle' looked\nimportant and respectable. Then the bold idea struck Ernest that he\ntoo might as well do a little 'cutting out.' It was more exciting than\npacing soberly round the mixed herd on the camp. Besides, it did not\nlook difficult. He had only his hunting-whip with him. But he thought\nthat the stockwhips were sometimes unnecessarily used; cattle he still\nbelieved were capable of being acted upon by gentleness and unvarying\nquietude of behaviour. So, taking Osmund by the head, who had had a\ncertain amount of cattle driving at Garrandilla, and was handy enough,\nMr. Neuchamp rode soberly through the herd to select a fat beast and\ndistinguish himself in turn. Most probably he would have covered\nhimself with glory, but it occasionally happens in this world that Fate\nseems to exercise all her ill intentions upon the knight even _before_\nhe is fairly in the lists at the tournament. Surely no evil hap is so\nsore to bear as this. 'Had I but a chance,' says the stout champion,\n'had I but lifted sword and held shield, I care not though Guy of\nWarwick were in the _m\u00eal\u00e9e_; but to be made captive ere the battlefield\nbe reached, or one trumpet blast sounded in mine ear, that indeed is\nthe utmost malice of destiny.'\n\nErnest, carefully guiding his steed through the third rank of staring\nor timid cattle, did not notice an old black cow with one horn sticking\nout from her head, who was regarding him with a fixed and gloomy stare.\nHer nerves had been much tried since she came into camp. She had felt\nmore than one savage cut of the stockwhip in acknowledgment of her\nferocious demeanour and well-known character. She had been horned in\nthe ribs and otherwise maltreated by ungenerous bullocks, who took that\nmode of requesting her to get out of the way. Her naturally morose\ntemper had given way. She was perhaps unconsciously hungering and\nthirsting for the chance of avenging her wrongs.\n\nAs Mr. Neuchamp essayed to pass her with a view to getting out a noble\nred bullock of about eleven hundred-weight, standing like a small\nelephant among the cattle, an uneasy steer on the farther side gave the\nblack cow a vicious poke in the flank. This was the match required for\nthe combustion. With a short bellow she sprang forward, and marking\nErnest, not far out of her track, immediately went for him. Had he been\nin open ground he might have 'cleared' in time. But the closely-packed\ncattle embarrassed him. Had one of the stockmen been similarly placed\nhe would with one of these same disapproved-of stockwhips have half\nblinded, and wholly checked, the cow by a ceaseless rain of precise\nand painful lashes across the face. But having neither whip nor\nelbow-room, Mr. Neuchamp was compelled to adopt the drifting policy.\nHe tried ineffectually to outride this old black demon, whose ferocity\ndid not require a stockyard, and then struck forcibly at her with the\nhunting-whip; but it was not long enough to reach her before she came\nto close quarters. When it did it had not the blinding fire of the\nproperly-wielded twelve-stranded intimidator. He felt a sudden shock as\nthe savage head struck violently against Osmund's shoulder. He held the\nexcited horse together as he staggered, and the furious animal passed\non. But he felt faint as he glanced at the straight horn of the old\nwitch, which was stained a bright crimson, and looking downward saw a\nstream of blood spouting thickly from his favourite's shoulder.\n\nHe leaped down in an instant, and seeing a deep stab in the centre of\npoor Osmund's shoulder, used his handkerchief for a plug, eventually\nmanaging to stanch the wound. As stiffness set in, the good horse began\nto limp. Jack Windsor being called over, a consultation was immediately\nheld, when it was decided that the grey had got a nasty hurt, but that\nno danger was imminent, and that he would be as well as ever in a\nmonth. Much relieved by this verdict, Ernest sent the invalid home by\nPiambook, with strict instructions to go at the slowest of all possible\nwalks, while he took possession of that gentleman's stock-horse himself.\n\nWhen Mr. Neuchamp, with his friends, servants, and allies, reached his\ncastle gate, otherwise the stockyard slip-rails, that night, he rode\nbehind three hundred head of as fine fat bullocks of _his own_ as ever\nwere sent to the Sydney market. The first draft of fat cattle! Grand\ntransaction! 'What would Courtenay say,' he thought, 'if he saw me in\npossession of a magnificent drove of cattle like this, all my own and\njust about to be turned into cash? Let me see, I expect to send away\nthis year five or six such drafts. That will be\u2015let me see\u2015how much at\n\u00a33 or even \u00a33:10s. per head'\u2015and then Mr. Neuchamp fell to calculating\nthe number of calves he should brand this year\u2015and the next, if the\ncattle went on increasing\u2015the number of cattle he should send off,\u2015and\ngenerally piling up Alnaschar's basket to the greatest elevation which\nthat tempting but insecure receptacle of riches would permit.\n\nThe fat cattle were duly despatched to market under the charge\nof Charley Banks and Jack Windsor, Piambook accompanying them for\nthe first fifty miles, to return when they might be supposed to be\n'steadied' more or less to the road.\n\nMr. Neuchamp himself rode by them on the first day, and his heart\nswelled as the drove of grand-looking bullocks, all 'rolling fat,'\nas became a Rainbar draft, after a few fruitless dashes for return\nand liberty, paced quietly though with subdued swiftness along the\nfar-stretching trail that did duty for the highway.\n\n'There is something fascinating, it must be confessed, about this bush\nlife,' he soliloquised. 'I don't wonder at youngsters running away\nto the bush, as long ago they did to sea. What a man, what a hero, a\nlad feels himself to be mounted upon a good horse behind a trampling\ndrove like this. Sometimes, even at Charley Banks's age, he may be the\nowner of such a lot, and the lord of an estate of a hundred thousand\nacres (leasehold), where almost every one he sees belongs to his\nemployment or dependency. The very numbers of the stock create a sense\nof responsibility and grandeur. There are three hundred and fifty head\nin this draft, not a large one. What would they think in England of\nseeing five hundred fat cattle in one drove, or even a thousand, like\nthe one we met one day. \"Where are these fine cattle from?\" I remember\nsaying to the stockman in charge. \"From Y\u0101nga,\" said he, with an air of\nperfect explanation, as who should say from London or Liverpool. All\nwell-informed persons, to his mind, must be acquainted by report, at\nleast, with Y\u0101nga.'\n\nMr. Neuchamp's musings came to an end as he perceived that he was no\nlonger needed, and must return, unless he proposed to spend the night\naway from home without adequate cause, so he paced back ruefully to\nRainbar, which fully presented the aspect of a lodge in the wilderness\nbereft of the cheerful converse of Mr. Banks, the versatile activity\nof Mr. Windsor, and even the open countenance and expansive grin of\nPiambook.\n\nHe had now before him the cheerful prospect of at least two months'\nentire solitude, not merely comparative, like an artist in a remote\nRhineland or Norwegian village, but absolute, unrelieved, impossible of\nimprovement, save by accident, as that of the keeper of a lighthouse.\n\nIt may be a matter of justifiable curiosity among those who have never\nled the eremitical lives which, 'for a season, and for that reason,'\nthe proud pastoralist is occasionally compelled to endure, how, in\nthis lone Chorasmian waste, Mr. Neuchamp contrived to spend his time.\nSomething after this fashion, if I, who write, may transcribe a page of\nlong ago, when the 'fever called living' was more recently induced.\n\nHe rose early, which, in the bush, means at or before sunrise.\nGlorious, in good sooth, is the early morn in the Australian wilds.\nCool, clear, invigorating to the inmost nerve. Cloudless for the most\npart, and, before the mid-day sun asserts his might, perfect as a\npoet's dream of the serene untempested heavens of the isles of the\nblest. Granted that, at cattle stations in the far interior, it is\n_very_ difficult to know what to do in the way of work, recreation, or\nexercise, when you are up. Some original thinkers have partly solved\nthe problem by habitually lying in bed until they had just time to\ndress for breakfast.\n\nBut not of such mould was Ernest Neuchamp. He had already assured\nhimself of profitable occupation for all the time that should intervene\nbetween leaving his couch and taking the cold bath which preceded\ndressing for the day. He had determined that the garden at Rainbar\nshould be one of the chief modes of reformation of bush habitudes upon\nwhich he was bent.\n\nTo this end he had, as early as such loading could be procured, ordered\nfrom town great stores of fruit-trees and plants befitting advanced\nhorticulture, besides all manner of vegetable seeds, with a small\nassortment of flowers and shrubs.\n\nHe had caused to be trenched, and laid out in proper beds, a flat\nnear the river through which the waters of that stream were led for\npurposes of irrigation. In this promising spot, in despite of the\npowerful sun-rays, the growth of all vegetation had been rapid and\nsuccessful. He had therefore secured that perennial source of interest\nwhich a well-kept garden supplies to him who is fortunate enough to\npossess a taste for horticulture. In it he found a sufficiency of\nlight labour for all the spare time which he could devote to it. Daily\ndid he congratulate himself upon having in the wilderness one of the\npurest pleasures known to mankind\u2015one which increases rather than fades\nwith the lapse of years, and which richly repays both in result and\noccupation any outlay in its earlier stages.\n\nHe had therefore no difficulty in finding adequate scope for his\nenergies during the early or the later unoccupied hours of the day.\nThe chance wayfarer descried him in a rough serviceable suit, delving,\nweeding, or seed sowing in the fresh hours of the morning, or towards\nthe coolness of the evening shadows. After a morning hour or more thus\nspent, he saw that his stock-horse for the day's ride was caught,\nsaddled, and left ready for use. Then he proceeded to his bath,\ntransacted in a rough but sufficient bathroom, composed of slabs, and,\nfully attired for the day, sat down with appetite to the breakfast\nwhich the old hutkeeper had, somewhere about eight o'clock, provided\nfor him.\n\nHe had succeeded in arranging the transit of a very fair library,\ncomprising his favourite standard authors, with whom, including a\nregular instalment of magazines, he held converse during the principal\npart of the breakfast hour.\n\nThat pleasant prelude to the day's occupation over, he mounted his\nhorse, and, accompanied by Boinmaroo or Piambook, set out upon his\ndaily series of 'travels and sketches' through the somewhat extensive\nterritory of Rainbar. Cattle stations are honourably distinguished by\npresenting some sort of work, if not always very onerous or important,\nto the attention of an active proprietor, all day long and every day.\nThere was a little branding to be done. A few head of cattle needed\nto be run home, and regulated in some fashion. A bullock was required\nfor killing. Stragglers were captured and deposited in the paddock,\nweaners, milkers\u2015what not.\n\nIn fact, so engrossing and interesting became the management of the\nherd, and the exploration of every hole and corner of the run, that,\njoined to the overlooking of the men working at the canal, the sun\nwas generally low before Ernest and his attendant returned, with a\nconsciousness of having done more or less a day's work, and with a\nremarkably good appetite for the corned beef, damper, and tea which\ncomposed his chief meal, and indeed all other refections.\n\nIn the evening he was again free to enjoy, without fear of\ninterruption, the intensified delight of the lonely scholar, whose\nbooks to 'him a kingdom' are. His correspondence became more\nvoluminous and grateful than he had ever known it to be heretofore, and\nwhen the hour arrived for repose, Ernest Neuchamp retired, secure of\ndreamless sleep and of that cheerful awakening with the dawn known only\nto the sharer of 'respectable pleasures and respectable labours.'\n\nSuch, day by day, was the free untroubled life of Ernest Neuchamp\nat that stage of his fortunes when, untroubled by care or consuming\nanxiety, with gay hope in the future, tranquil enjoyment of the\npresent, youth told itself a hundred times each day that the present\nwas fairer than the fairest mortal mistress; while age and care stood\ndimly gazing afar off, nor ventured to enter the paradise which is\nrarely sacred from their intrusion when the downward of the days\nof the years of our pilgrimage begins to be travelled. So pleasant is\nthe flowing ascent to the mist-shrouded pinnacle of the moments known\nas success. There, for we behold it in no other spot on earth, we\nfondly deem that happiness abides. If that haunting presence, unearthly\nbright, there displays her charms who shall say? Let those who have\nreached the spot whence can be descried the kingdoms of the earth and\nthe glory of them declare!\n\nThe days, the weeks, passed smoothly, swiftly away, until at length\nCharley Banks and Mr. Windsor return, in high spirits, the cattle\nhaving 'topped the market,' and sold extremely well. With the exception\nof occasional branding and taking heed that the cattle who wandered\nabout 'on parole,' and were not restrained by any fences, did not\ngo away from the run altogether and irrecoverably, there was little\nindispensable work to do. The selection and delivery of the fat cattle\nwas the most difficult of their station operations. It had been\ndemonstrated that this could be successfully transacted by the present\nstaff.\n\nAfter the gallant drovers returned, a fortnight was spent in looking\nthrough the herd generally. This done, there did not appear to be any\npossibility of fresh work for two or three months; in fact, not until\nit was time to make another draft of fat cattle.\n\n'I see now,' said Ernest, to that constant and sympathetic confidant,\nhimself, 'the mistake of the pioneer settlers of the Australian\ninterior; they narrowed their mental vision to the mere actual facts\nof their positions; they discarded change and resisted enterprise. Now\nthe obvious course which would occur to any man of intelligence and\nforethought, anchored for years of his life in a primeval waste such as\nthis, would be to develop his property to the fullest extent compatible\nwith his pecuniary safety. Then, at the first favourable turn of the\nmarket, he might sell out to advantage, free either to repurchase a\ncheap unimproved property, or to betake himself to the intellectual\nelysium of the Old World\u2015that abode of art, science, literature,\nclassical glory, perfected luxury.' Here Mr. Neuchamp checked himself\nwith an involuntary sigh, and sternly pursued his original line of\nthought.\n\n'Instead of which,' as the country Justice said, 'they went on\nyear after year, in one dull endless round of life, subsisting\nmetaphysically upon the bark and green-hide substitutes for all that\nmen, in other places, hold dear; without society, without books,\nwithout expectation of quitting their desert life, what wonder that\nwhen middle life is reached, ere Fortune smiles on the lone hermit of\nthe waste, she should find him with tastes obliterated, sympathies\nwasted from long disuse\u2015with the whole general mental endowments\nhopelessly deteriorated? How different might be the lot of an ardent\nand instructed man,' pursued the enthusiast\u2015'zealous to make the most\nof the light that was in him; keen to aid the advancement of his\nkind, to help the tardy progress of virtue and human truth. With the\nmaterials ready to his hand, he might complete pastoral experiments\nyet undreamed of, raise the moral tone of his employees, and through\nthem of the land generally, render his homestead the headquarters of\nphilosophical experiment and liberal life and culture collaterally with\nthese lofty aims: such a man might place his future prosperity on a\nfirm basis.'\n\nThere are some persons who possess the enviable power of being able\nto raise the most imposing imaginative structures upon any pedestal\nof assured stability, no matter of what size. The satisfactory sum\nwhich the first draft of fat cattle from Rainbar had realised provided\nMr. Neuchamp with such a prosperous future, by the simple process of\nmultiplying their numbers and periodical result, that he felt himself\nnow to be fully justified in undertaking any number of reproductive\nenterprises.\n\n\n\n\nCHAPTER XX\n\n\nWhen the first instalment of stores, of a very mixed and comprehensive\ndescription, arrived from Sydney, in three drays drawn by ten bullocks\neach, Mr. Neuchamp was much impressed by the teamsters. They were\nbrothers who had left their farms in the settled districts for this\narduous but profitable undertaking. Finer specimens, outwardly, of the\nnative Australian it would have been difficult to find. Tall, powerful,\nwell-built fellows, they were just the men fitted to found 'a bold\npeasantry their country's pride.' Their appearance at Rainbar hastened\nthe action of one of Ernest's long-cherished plans. He had always\nintended, when arrived at the dignity of a proprietor, to establish\na rural population in the vicinity of the home station. In time to\ncome their residence and occupation would add value to his land.\nAvailable labour would be at hand whenever he required assistance. And\na consideration, dearer to the heart of Ernest Neuchamp than aught\nother\u2015he believed fully in his power, by this means, to elevate his\nfellow-man in the social scale, to aid both in his material and mental\nadvancement.\n\nIn conversation with the brothers, he gathered that they had each a\nsmall farm 'down the country,' as they called it, where they kept\na few cattle, raised reasonably regular crops, and generally lived\nan independent but unprogressive life. They admitted that they were\npressed for room, and in a bad season lost many cattle. 'How should\nyou like to have a half section each on that flat which you see\nthere?' inquired Ernest, with the light of sanguine benevolence in\nhis eye. 'Your cattle would increase, and in a few years you might be\nwell-to-do, prosperous men.'\n\nThe Australian yeoman, as he may fairly be called, is not wholly\ndissimilar to his American cousin, though the type is, as yet, not\nnoticeably divergent from the Anglo-Saxon. Slow of speech, his\nreasoning faculties, within fixed limits, are active and vigorous.\nConcerning matters which relate to his personal or pecuniary welfare,\na more shrewd, cool-judging individual does not exist. Well skilled in\nthe valuable art of holding his tongue, he asks but few questions. He\nasserts little. But, if you happen to have the arrangement of a bargain\nin stock or land, or of a contract for carriage or bush-work, with the\nrural Australian, you will rarely find that the apparently impassive\ncountryman has 'got the wrong end of the stick.'\n\nSo, when Mr. Neuchamp made the somewhat unusual offer to Abraham\nFreeman and his brothers, William and Joe, of permitting them each\nto conditionally purchase three hundred and twenty acres upon the\nriver-flat, below the house, himself finding the cash for the first\ndeposit payment, they quickly ran over the advantages in their own\nminds, and came to the conclusion that the 'cove,' or proprietor, was\nan inexperienced swell, whom Providence had delivered into their hands.\nThey realised the fact that, though cultivation was not likely to\nflourish in a land where it did not rain, sometimes, for six months,\nthey would be able to keep as many cattle as they liked. From merely\nlegitimate increase, not to speak of chances, such as always occur\nnear large herds, they might look forward to a snug herd each in four\nor five years. They would have a place to keep their teams, and might\ncontinue their carrying uninterruptedly. They could by no possibility\nlose much, and might gain largely, by accepting Ernest's offer. Still,\nwith characteristic caution in 'making a deal' of any sort, they spoke\nhesitatingly.\n\n'Well, I don't know, sir, about coming up here for good,' said the\neldest brother. 'Our place down the country is comfortable like, and\nthe cattle do middling well' (half of them had died during the winter\nfrom cold and starvation). 'I don't know how my wife would like it\neither.'\n\n'I should be sorry to urge a removal from anything very pleasant as\na homestead,' said Ernest; 'but I thought, perhaps, that you might\nhave the advantage here of more land, and the opportunity of getting\non faster in life\u2015of course you will, and have the carriage from the\nstation.'\n\n'I believe it might be worked,' said Bill Freeman, the second brother,\nan astute personage, who thought that they might now begin to be\npersuaded into accepting their good fortune. 'Certainly it's thundering\nhot, and a long way over these blank plains. But likely Mr. Neuchamp\nwill have a bit of bush work or fencing ready for us when we come up.\nIt's poor work laying out all our bit of money on a bit of land and\nhave nothing to fall back upon.'\n\n'I daresay I shall have something going on,' said Ernest, who, now that\nhe was possessed by the 'improvement' demon, saw in his mind's eye\nmany new buildings and fencings _absolutely necessary_. 'Of course you\nwill have the preference when any such is given out.'\n\n'Then it will be all right, sir,' said Abraham Freeman, 'and when we\ntake up the land, you'll be ready to advance the eighty pounds for the\ndeposit on each half section. We can pay it back in work and carriage\nby degrees like.'\n\n'Oh, of course we can pay it back in a year or so,' said Bill.\n\n'Certainly; I said so when I mentioned the subject first,' said Ernest,\n'and I shall be prepared to carry out my promise.'\n\n'Then, after the crops are cut,' said Abraham Freeman, unable to\nrepress a slight look of satisfaction, not to say exultation, 'we'll\nmake a start up, and bring our few cattle with us. They're crawling,\nquiet things, and won't give no trouble to any one.'\n\n'Very well, that is settled,' said Ernest, concluding the\ninterview\u2015satisfied that he had secured the nucleus of a contented and\nsubstantial tenantry, more common in England than in Australia.\n\nSo the namesake of the great Sheik Ibraheem, who first depastured his\nstock upon the waste lands of the period, departed with his brethren\nand oxen.\n\nMr. Neuchamp, with a feeling of conscious success, related his\nachievement to Banks and Jack Windsor. Somewhat to his disappointment\nthe former made no remark, and the one made by the latter consisted of\ncertain mutterings suspiciously resembling profuse oaths, ending with a\ndeclaration that 'he'd have seen Abe and Bill Freeman, not to mention\nthat planting rascal Joe, jolly well\u2015\u2015 first.'\n\nThe sequel of this philanthropic arrangement adjusted itself after\nthis fashion. The brothers Freeman, as soon as they reached home, took\nmeasures for selling off their holdings, the proceeds of which they\ninvested in as many cattle from their neighbours as, added to their\nown, made up a herd of more than a hundred and fifty head, exclusive\nof thirty-six working bullocks. They also 'gave the office' to a\nbrother-in-law and such of their neighbours as were willing to go into\na little speculative land selection. The upshot of which was that,\nwithin a year after the proposal to the Messrs. Freeman, Ernest had\nthe satisfaction of witnessing the taking up of half a dozen other\nselections of three hundred and twenty acres each upon the best part of\nhis frontage. This occupation gave the selectors a legal right to about\nsix thousand acres of 'pre-emptive right' suitable for the pasturage of\nfive or six hundred head of mixed cattle and their probable increase.\n\nCharley Banks openly demurred to all this as very likely to lead to\ncomplications as to calves, and stated his opinion plainly that the\nyoung lads, of which there were two or three in each family, would be\nalways galloping about the run when not wanted, looking for a horse,\na strayed bullock, or with any excuse in fact that happened to come\nuppermost. He had seen it tried before, he averred, and it had not\nanswered. Free selectors were all very well, 'like measles and fevers,'\nwhen you got them in the ordinary course of things; but as to paying\nto catch them and helping them to come into your place, it was likely\nto end in a losing game. But Mr. Neuchamp had still great faith in\nthe inherent excellence of human nature, and overpowered Charley with\narguments which the youthful Conservative distrusted but was unable\nfor the present to answer. He contented himself with prophesying that\nthere would be a store and a public-house next at the Long Reach.\nThis of course would end in a surveyed township, and a reserve for\ntravelling stock, by means of which they would lose the use of one of\nthe best watering-places and camps on the run.\n\nErnest had at first floating ideas of running down to the metropolis\nduring the hot months, for\u2015for\u2015some one of the many reasons which\ngenerally gather additional force about January or February at the\nlatest. But really, when the time came, there was so much work of\nvarious sorts going on that he prudently thought he had better stay at\nhome for another year until he could leave everything in full working\norder, and go forth 'on pleasure bent' with a clear conscience. He\narrived at this conclusion somewhat unwillingly; but he did so from the\nclass of motives which chiefly actuated him, and so settled the matter.\n\nMonths rolled on. The many drafts of fat cattle had been mustered and\nsent away in satisfactory succession. All was realised for that source\nof income that could be relied upon for one season. The improvements\nof various sorts had been completed and paid for, this latter process\nadding up to a much larger sum than had been originally calculated\nupon. The cutting to the Outer Lake had also been finished according to\ncontract. The cash payment for this same piece of civil engineering for\nthe first time aroused a feeling in Ernest's breast that perhaps he was\nspending money rather faster than it was made, that it was a scale of\nproportionate outlay that could not be continued indefinitely. Nothing\nwas more necessary in Mr. Neuchamp's opinion than to improve the breed\nof cattle existing at Rainbar. To that end he had purchased a small\nbut costly shorthorn stud. He had written to his brother Courtenay to\nsend out to him certain animals of the purest procurable Bates blood.\nAll things had been done that in the eyes of an intelligent public\nwould eventually distinguish Rainbar as a model cattle station, with\nprize stock and unrivalled improvements. In the future was a plain\ncertainty of trebling value and carrying capacity.\n\nThus far matters had gone on with undeviating regularity in all\nrespects as where the stock were concerned. Mr. Neuchamp found that\nwhenever his account with Messrs. Oldstile and Crampton needed\nreplenishing on the credit side of the ledger\u2015a position of affairs of\nwhich he was informed with much precision and regularity\u2015he had only\nto muster for fat cattle and despatch a draft to market. He began to\nbelieve that such was the invariable state and condition of things.\nHe wondered why all cattle-holders did not make rapid fortunes.\nHe wondered why doubt should be expressed about the expediency or\notherwise of investing in such a steadily profitable speculation; and\ninasmuch as his brandings became more numerous each quarter, far more\nthan replacing the numbers sent away for sale, it amazed him to think\nhow such an easy and pleasant way of doubling or quadrupling capital\nhad not simultaneously entered the brain of every man of average\nintelligence in Australia.\n\nHe was now to learn that other factors in the calculation existed.\nThe first slight ripple of the tidal wave which might or might not\noverwhelm was the remark of Charley Banks one day that they had had no\nrain for a month; that the appearance of the weather indicated none\nfor another month, 'in which case,' said Mr. Banks, 'the grass would go\nback.'\n\n'I had not remarked it,' said Ernest, looking up (it was\nbreakfast-time) from an interesting article in the _Fortnightly\nReview_. 'Now you mention it, it does seem rather dry. However, I\nsuppose we shall soon have rain.'\n\n'I'm not so sure of that,' said Charley; 'it looks very like setting in\ndry, and what's more, Jack Windsor thinks the same, and the blacks say\n\"big one water, longa lake dry up, like't long time\"\u2015that looks bad.'\n\n'And suppose it does,' said Ernest, cutting his _Review_ carefully,\n'surely there will be grass and water enough on the run for all our\nstock?'\n\n'Not so sure of that. In this part the grass goes all to nothing in a\ndry year, breaks off, and blows away, making the country look like a\nbrick-field. Besides, I was reading in Sturt's _Exploration_; capital\nbook it is'\u2015(Mr. Banks had been craftily led into the path of literary\nexercise by tastes of travel and adventure, of which line of action he\nwas passionately fond)\u2015'well, I was reading that the year the Captain\nwent down the Murrumbidgee first, 1827, was a terrible drought\u2015worse\nthan anything we have had since. That year was the driest summer in\nEngland known for a century.'\n\n'What of that?'\n\n'Why, didn't you tell me that your letters from England, the last mail,\nsaid they were having an awfully hot season _for them_, brooks nearly\ndry, people having to cart water ten miles, and so on. Well, _our\nsummer follows theirs_ in a kind of way six months after. So I'm afraid\nwe are in for a regular dry season, if not a drought.'\n\n'And does that make so much difference?' asked Ernest coolly. 'This\nseems a dry country at the best of times; Nature should be equal to any\nemergency in that line, from the practice she ought to have had in this\ntopsy-turvy continent.'\n\n'My word, and so she is in a general way,' said the youngster, standing\nup for his native land. 'But a drought, the real thing I mean, a dry\nsummer after a dry winter, is something awful. I can recollect one when\nI was a little chap at school, and that was something I never forgot.'\n\n'What was it like, Charley? I'm never afraid of facts; half the evil of\nlife arises from not looking _them_ in the face.'\n\n'Well, but some facts frighten you like a ghost does, however straight\nyou may look them in the face,' said the lad. 'In the year I remember,\nlots of squatters lost their stock to the last head, and were ruined\nout and out. There was no beef or mutton fit for a blackfellow to eat.\nFlour was a hundred pounds a ton, and had to be mixed with ground rice.\nAll of us boys were taken from school because bread was too dear\u2015not\nthat we cared about that. Nobody could sell anything. People almost\nforgot what money was like, there was so little of it.'\n\n'We must hope for the best,' said Mr. Neuchamp firmly, though, as he\nwas speaking, an unpleasant thought flitted through his brain of how\nhe should make things pleasant with Messrs. Oldstile and Crampton, if\nthe easily negotiated drafts of fat cattle could no longer be collected\nfrom Rainbar camp. 'We may have summer rains or thunder showers; the\nleast thing seems to cause the herbage to grow hereabouts.'\n\n'We _may_ have,' said Mr. Banks doubtfully, 'but it don't look likely\nto me. If you have noticed, it has turned cloudy and dark-looking, and\nall passed off again, a dozen times within the last month or two, and\nthat's as bad a sign as could be.'\n\nMr. Neuchamp revolved the unpleasing idea thus presented to him much\nand often in the days following this eventful dialogue. With a sudden\nflash of perception he saw his course of unchecked improvement and\ndisproportionate outlay in remorseful clearness.\n\nHad he then, in despite of the respectful but marked disapproval of\nboth of his faithful subordinates, experienced in the ways of the land,\nbeen steering obstinately on a course with a rock ahead plainly visible\nto their clear if not far-reaching vision? Would he really find himself\nlanded in a labyrinth of debt, like so many unlucky squatters that he\nhad _heard of_, from which all attempt at extrication would be vain\nwithout the total sacrifice of his investment? He felt like a reckless\nmariner who, having disregarded the cry of breakers ahead, had carried\non madly until the fatal crash was heard, and the good ship, dreadfully\nimmovable, lay broadside on to the remorseless billows.\n\nWith returning daylight, however, the retrospective reverse having\noccupied the hours of a sleepless night, came firmer resolves, and\neven some faint signs of hope. Surely even his rigid agents would\nadvance what money he needed upon the security of his fat stock to\ncome. If they were not to be moved to the disbursing point, his brother\nCourtenay might permit him to draw upon him for a couple of thousand\npounds. That would completely set him free from pressing liabilities,\nand would be amply sufficient to carry on with until another crop of\nfat stock should ripen, till this present abnormal state of matters,\nwith the drought-bound herd of cattle, became a thing of the past.\n\nThe days, the weeks, passed on without any alteration of the weather,\nexcept what might be considered a passing from bad to worse. Hot days,\ncool days, windy days, cloudy days, came and went, but no rainy days,\nalthough often the sky looked dark, and storm-clouds rolled up in great\nbattalions, only, alas! to scatter, break up, and flee before the sun's\nrays like a barbarian army at the sight of a dreaded enchanter.\n\nCertain effects commenced to follow the gradual and complete\ndesiccation which pervaded the soil. The grass withered, became brittle\nand sapless, then blew away before the breath of the harsh hot wind,\nleaving the red earth bare, baked, and 'much more like a brick-field'\n(this was Jack Windsor's simile) 'than a first-chop cattle-run.'\n\nThe Back Lake commenced to dry up, and the weaker cattle sank by scores\nin the mud, and either died or were extricated with difficulty. The\nstrange cattle came into the frontage, and strove with the _habitu\u00e9s_\nof that locality for the very scanty pasture which was left.\n\nGreat hordes of travelling sheep laid waste a portion of the run,\neating every available particle of herbage within a mile of either side\nof the road. At first Ernest was inclined to treat these devourers of\nevery green (or dry) thing with consideration, but found that he would\nspeedily possess a herd of cattle and no appreciable grass for them\nto eat if that policy was persevered with. So Mr. Banks had orders to\n'shepherd' every lot through the run, and to describe the proprietor\nas a violent and ferocious person given to impounding and every legal\noppression.\n\nWith the colony of selectors amicable relations commenced to be\nendangered.\n\nTheir cattle, having much increased, required a considerable range\nof pasture. Their owners commenced to grumble if the Rainbar cattle\nfed over their grazing rights, quite unconscious of their wholesale\nunnoticed trespass up to the present time. One of the conditional\npurchasers, indeed, after a brisk argument with Jack Windsor, informed\nthat gentleman that grass was grass now, and that they intended to\nstand upon their rights. They were poor men, and couldn't see that they\nwere to starve their cattle for Mr. Neuchamp or Mr. Old-champ either.\nIf he hadn't expected to get some pull out of them, he would never have\npersuaded them to come there. They didn't see as they owed him anything.\n\nThis was one of the unkindest cuts of the very hard fortune of the hard\nseason. Ernest felt the ingratitude of his 'plantation' settlers more\ndeeply than any one of them could have supposed.\n\nTo make matters pleasanter, he received a letter from Messrs. Oldstile\nand Crampton, informing him that his account was overdrawn, and that\nhe could by no means have any more money until the credit side of his\nbalance was substantially reinforced.\n\nHe was commencing to fall upon evil days, certainly. What to do he did\nnot exactly know. He was unwilling to write to Paul Frankston and state\nthe case. It would have appeared like a simple asking for a loan. He\nwas ready enough to accept Paul's advice, friendship, and hospitality.\nHe did not wish to be directly indebted to him for money.\n\nAnd yet, _quoi faire_, without an advance of some sort? For, even on\ncattle stations, where you are not always putting your hand into your\npocket, as with sheep, various occasions for expenditure arise, and\nmoney is indispensable.\n\nHe had been sufficiently learned in the ways of land to know that\nstore cattle were nearly always saleable, and that one could generally\ndispose of a large lot more easily than fat ones. But during this\nterribly dry weather, he reasoned that no one would desire store cattle\nat any price. Buyers were uncertain as to _when_ it would rain, and\nwould delay making purchases until definite assurance of a change of\nweather. Of fat cattle he had none; they had enough to keep themselves\nin a pinched, independent manner, but no more. The situation resolved\nitself into this: money must positively be raised for station expenses\nfor the next six months.\n\nAfter much extremely unpleasant cogitation about money, for the first\ntime in his life, Mr. Neuchamp finally decided to write to Messrs.\nOldstile and Crampton, stating his position, and his reasonable\nexpectation of receiving aid from his brother in England. He made this\nexplanation, requesting at the same time that they would permit him to\ndraw for the sum of five hundred pounds in advance, on the strength of\nfive thousand pounds which he had grounds for expecting that he would\nobtain from his brother.\n\nThis important letter being despatched, Ernest felt more at ease\nthan had been his lot for some time past. In money difficulties,\nlike other matters, the chief misery lies in the stage of doubt or\nprocrastination. This being passed, and a definite course of action\nentered upon, mental relief ensues. Happy the man whose temperament\nleads him to bestow the same amount of curative anxiety upon the\nearlier stages of 'chest complaint' that the majority are compelled to\nfurnish during the more aggravated phases of the disorder.\n\nMr. Neuchamp, to do him justice, was not a man consciously to remain\nwithin the borders of a fool's paradise. Once aware of the necessity\nfor strenuous exertion, he was unhappy until progress had been made. He\nhad previously written an explanatory letter to his brother Courtenay,\nnot defending his somewhat free expenditure, but owning candidly that\nthe sudden change of the season, with the collapse of the marketable\nportion of the herd, had taken him by surprise, and reduced him to\na state of virtual, though temporary, insolvency. 'However,' he\nadded, 'my herd of cattle has increased considerably, both in number\nand quality, since I purchased, and I anticipate\u2015though I own I was\nmistaken about the time when they would become remunerative\u2015that my\nenterprises and outlay for labour will eventually prove sources of\nextraordinary profit. At the same time,' he added, 'it is my duty to\ntell you that I cannot speak with any certainty as to when repayment of\nyour loan may take place. The seasons here are variable and irregular,\nthe price of stock low and high by turns. All I can do is to pay\nyou Australian interest, which is much higher than in England, and\nto promise to return your capital when times improve. I shall never\nreproach you if you do not lend me your money, as I do not wish to\ndisguise from you that it is uncertain whether you ever see it again.\nBut if you do not, and I fail to obtain accommodation in any way,\nRainbar must be sold, and I shall be ruined.'\n\nMr. Neuchamp, regarding his letter when written, did not like the look\nof the last sentence, nor the rather uncomfortable last word. So he\ncast about for another sentence or two of less obnoxious suggestion.\nIn this extremity he bethought himself of a certain lady-cousin, Miss\nAugusta Neuchamp, a damsel of very well-defined opinions and courageous\npropagandism, with whom he and Courtenay had been much at war\u2015she\nhaving a full share of the family obstinacy of purpose. So he wrote,\n'Give my love to Cousin Augusta, and tell her that she would like\nAustralia uncommonly, in some respects. It presents a great field for\nher peculiar crazes.'\n\nThis important letter despatched, there was nothing for it but to do\nthe waiting on Providence as patiently as was possible to a nature\nconstitutionally averse to suspense and uncertainty. Something of\nthe romance of the kingdom of Rainbar had departed, when the throne\nand crown jewels were liable at any time to be taken in execution.\nIts ruler commenced to experience those various throbs and spasms,\nthe preliminary pangs, headaches, and heartaches, which assail all\ntravellers through the Valley of the Shadow of Debt!\n\nHe was not doomed, however, at this particular period of his pastoral\nexistence, to be kept long in the torture-chamber. For Isaac of York\nthere was a Wilfred of Ivanhoe 'round.'\n\nIn due course a letter arrived from Messrs. Oldstile and Crampton, to\nhis great joy, that they had acceded to his request, on the strength\nof remittances arriving from England; that the sum named was now at\nhis credit; but\u2015but\u2015they trusted that he would not exceed the sum\nreferred to, before paying in money to the credit of his account\ncurrent, as, they regretted, it would not be in their power, under _any\ncircumstances whatever_, to exceed that advance. And they were his\nfaithfully, etc.\n\n'Hang their \"yours faithfully,\"' banged out Ernest, in the overflowing\nexpansion of the moment\u2015borrowing a hitherto avoided colonial\nhabit\u2015'why do people who would not stretch out a hand to save one\nfrom beggary call themselves \"yours truly or faithfully\"?\u2015\"truth and\nobedience\" for ever on their lips, and how little of either is ever\nexhibited. However, I am to have the money for the present, and that\nwill last me to the end of the year, by which time the heavens or\nCourtenay may come to the rescue of Rainbar.'\n\nThe pecuniary aid of his formal agents, though grudgingly given, was\ntimely and valuable. Ernest determined to economise, with a view to\nmake the relief fund last as long as possible. Taking a hint from his\nmaritime experiences, he proceeded to shorten sail while such signs of\nstorm and tempest were observable in the financial horizon\u2015a policy\nhighly to be commended, but, like many of our good resolutions and\nbetter deeds of this mortal life, ever prone to be late of arrival.\nSo life again flowed on at Rainbar in a monotonous round of daily\nduties, which the increasing severity of the season rendered tedious\nand troublesome, but not exciting. The weak cattle were dragged out of\nwaterholes and creeks; the locust hordes of travelling sheep watched\nand followed, lest they cleared off the poor remains of the dying\npasture. Musters were in abeyance until 'the rain came.' The drought\nstill remained unbroken. The great canal remained as innocent of water,\nand as unlikely to be filled, as if it had been constructed between\nthe tanks and the desert gate of Aden. Every superfluous station hand\nhad been 'hunted,' to use Charley Banks's phrase\u2015in fact, that young\nman had very strongly expressed his idea in favour of contraction of\nthe strength of that department. So that the pleasant spectacle was\npresented of the station work being done by the smallest practicable\nstaff, viz. the proprietor, Charley Banks, Jack Windsor, and the two\nblack boys.\n\nIn the midst of this state of matters a stranger appeared one day,\nwhose knocked-up horses showed plainly in their very visible anatomy\nthe effects of a long journey and indifferent keep. Mr. Neuchamp hasted\nto welcome the 'guest sent by Allah' with true Arab hospitality.\nConsiderably to his surprise he recognised the sun-burned, grave visage\nof his quondam travelling companion, Mr. Abstinens Levison. That\ngentleman's reflective countenance relaxed somewhat as he shook hands\nwith his host, and relinquished his way worn steeds to Mr. Windsor's\ngood offices.\n\n'So you're the man that bought Rainbar,' said he with mild\nacquiescence. 'I heard that a young Englishman had cleared out\nParklands. Smart fellow he is\u2015gone in for a whole country-side on the\nDarr. Sure to do well when we get rain again. He and I have had many a\ndeal together. Got the best of me once in a big lot of store cattle,\nand it ain't many men that have got that to say of Ab. Levison.'\n\n'Very glad to see you, Mr. Levison,' said Ernest heartily. 'Come in and\nmake yourself at home. Which way are you travelling in this terrible\nseason? No wonder your horses have had enough of it.'\n\n'Just about done, and that's the truth,' made answer Mr. Levison\nslowly, and with consideration. 'I'm on my way to Mingadee, a place\nof mine down the river, about a hundred miles from here. I shall have\nto walk and carry my swag, for the horses, poor things, are as weak as\ncats. If I hadn't come through the back country, where I knew a few\nspots where there's feed in all seasons, such as it is, they'd have\nknocked up before now.'\n\n'Walking is becoming quite fashionable,' said Ernest; 'people are\ncoming round fast to my way of thinking, that we were intended to use\nour legs in some other way than lolling upon a horse all day. I saw a\npolice trooper trudging past to the Quarter Sessions at Warren last\nweek, with a good part of a hide (evidence in a cattle-stealing case)\non his back. The mail had stopped running. He told Jack one of his\nhorses was dead, and he was as able to carry the other as the poor\nbrute was to carry him. But you won't have to walk this time, if you'll\nstay with me to-night. We have a horse or two left, and I can give you\none that steps as fast nearly as the roan you were good enough to lend\nme near Nubba.'\n\n'All right,' said Mr. Levison. 'I'd not be particular about it; only\nI'm a little pushed for time. I have to meet a man about a largish lot\nof stores that we're dealing over.'\n\n'Buying store cattle in the teeth of a season like this!' exclaimed\nErnest in astonishment. 'Why, it's a hard matter to keep alive one's\nown, I should think.'\n\n'Look here!' said the man of original mould, commencing on the lunch\nwhich had been provided for him calmly but with decision, as if the\nback country that he mentioned had been better provisioned for the\nquadrupedal than the human part of his equipment. 'It's always been a\nway of mine to act different from other folks in the way of buying and\nselling stock. I can recollect the markets for many years back. I've\nseen sheep at all prices from a shilling to a guinea, and cattle in\nproportion. My rule is\u2015I don't mind telling you, for _you'll_ never\ndo much in the dealing line\u2015my rule is, to buy when every one wants to\nsell, to sell when every one tries to hold on; and it's paid me, so\nfar. That's good damper of yours; your cook kneads it up well, that's\nhalf the battle.'\n\n'He's not a bad fellow in his way,' asserted Ernest; 'but he will soon\nhave very little flour to knead. Drays can't travel, and we shall have\nonly South American fare directly, beef and water. Certainly we have\nplenty of pumpkins, that's the advantage of a garden.'\n\n'Couldn't have a better thing. Lived on them for a year, in '38,' said\nMr. Levison approvingly. 'That was something like a drought. If we ever\nget one like it again it will cook half the stock in the country. We're\nthat crowded up now that there's no get-away, as there was then, to the\nmountains.'\n\n'Then you don't think this season is as bad as can be?' inquired\nErnest. 'It seems very terrible to me.'\n\n'It ain't as bad as that time unless you've lost half your cattle\nand don't see no way to save the rest,' affirmed his guest with mild\ndecision, as if stating some rather agreeable proposition.\n\n'Whatever shall we do?' groaned Ernest. 'I'm half ruined as it is.'\n\n'You've spent a lot of money on this place, by the look of things as I\ncame along,' said this mild but uncompromising critic, filling himself\nanother cup of tea with much deliberation. 'You've been and put up a\nbig paddock and a horse-yard and a grand house; and, last night, I'm\nblessed if I didn't ride slap into that drain arrangement, miles of it\nI see there was. Now, I don't say it's altogether a waste of money,\nbut when a young man like you buys a place, he has no call to spend a\nshilling that he can help till he gets it out of the run.'\n\n'I can understand the prudence of that policy _now_,' answered Ernest,\nhalf amused, half inclined to resent this extremely plain speech from\na comparative stranger, yet comprehending with instinctively clear\nperception the unaffected friendliness of intuition and truthful habit\nof his reviewer, 'but the fat cattle sold so well that I expected to\ncontinue paying my way and still improving the property.'\n\n'That's where you made the mistake,' pursued the senior colonist; 'you\nwent on thinking that the good seasons were a-going to last for ever.\nIf you'd kept on selling and _never spending_, you'd have had your\nmoney in your pocket now, and might have been in the market for some of\nthese lots of first-rate store cattle that's going a-begging\u2015splendid\nfine-bred cattle, too, as you ever saw!' Here Mr. Levison emptied the\nteapot with a benign expression, and, crossing his legs reflectively,\nlooked with mild reproach at his entertainer.\n\nErnest felt each item of guilty extravagance arise and arraign him\nseparately, as Mr. Levison, with judicial enumeration, went on\nticking off his pecuniary sins. In one of these lightning flashes of\nself-accusation with which conscience favours erring man, he realised\nthe difference of his position from what it would have been if he had\ndoggedly adhered to the scale of non-expenditure which he had found at\nRainbar, and had retained the proceeds of his drafts of cattle with\nwhich to pay off his purchase-money, or re-invest in stores at the\ntempting tariff of the day. The faint counter-consolation that occurred\nto him, under the circumstances, was that if he had acted in such a way\nhe would not have been Ernest Neuchamp at all, but must have changed\nhis very nature and identity. So there was no more to be said.\n\nOn the next morning Osmund was saddled for Mr. Levison, who, after\nsaying that he would be back at sundown on the fifth day, departed for\nMingadee. He was good enough to express his unqualified admiration of\nthe gray horse's make and shape as he mounted him. 'I saw a lot of\nmares and foals knocking about at the big bend,' he said. 'Brood mares\nare useless wretches generally, and you can buy horses a deal cheaper\nthan you raise 'em. But if you could turn out a few colts like this\ngray horse here, why, I should begin to think there was something in\nhorse-breeding after all.'\n\nOn the fifth day punctually, about sundown, Mr. Levison reappeared at\nRainbar. Having crossed the hundred miles of plain which separated the\nstations in two days, he remained one day, transacting the purchase of\nthe store stock to which he had referred; then Osmund carried him back\nin two days, 'quite flippant,' as Jack Windsor observed. As he partook\nof the evening meal in company with Ernest, he essayed to cheer him up\nafter the following fashion\u2015\n\n'I'd a sort of notion that I'd checked off all your money-burying\nbefore I left. But it seems I wasn't quite up to the number of\nholes a man can dig and fill up with sovereigns. I came across the\nSettlement!\u2015regular town it is; and that native chap\u2015active fellow he\nis, and no mistake\u2015told me you'd paid the deposit money and given 'em\nemployment, and advanced 'em money in other ways. I've seen new hands\ndo many a blind trick, but I never knew a man before, of his own free\nwill, bring down a lot of free selectors on his own run.'\n\n'It does not appear to be a fashionable thing to do,' admitted Ernest,\n'judging from the remarks of my neighbours, as well as yourself;\nbut I am somewhat like you in one respect\u2015I do things upon my own\nresponsibility, and, I am afraid, do not care sufficiently about other\npeople's opinions. Sometimes I am wrong\u2015very wrong\u2015I admit. But at\nother times I am so satisfied of being right that the whole world would\nnot turn me.'\n\nMr. Levison looked Ernest 'straight in the eye' with his own singularly\nclear, penetrating gaze. 'I hold with you in that,' he said at length;\n'nothing like a man who acts on his own reason, and sticks to it. He\nmay be right, or he may be wrong, but he'll come out better in the long\nrun than any fellow that follows the wind wherever it blows. And so you\nbelieve in these cockatoo chaps? Now, what's the good of 'em?'\n\n'Just so far,' said Ernest, 'that I hope, in time, to see a thriving\nand prosperous population here, making proper use of the soil, and\nadvantageous to the proprietor, as they in turn would be benefited by\nhim.'\n\nMr. Levison again regarded Ernest fixedly. His calm features, across\nwhose lineaments the ripple of a positive opinion or sentiment rarely\nbroke, might have been taken to denote the benevolent toleration of one\nwho hears a spoiled child insist upon being presented with a portion\nof the moon, or propose, with saline agency, the capture of an uncaged\nbird.\n\n'Population\u2015what's the good of population on a cattle station?' he\nsaid, with his usual slow, unpunctuated direction of speech. 'All the\ncrop they'll ever get out of that land you may put in your coat pocket.\nIn a dry season it's as much as the salt-bush will grow, let alone\ngrass or crops. In a wet one, all this country's like a garden, from\nthe Paroo to the Macquarie. Your horses don't want corn _then_, or\nhay\u2015wouldn't eat it if they were paid for it. What are farmers to grow\nhere that would pay for carriage to the coast? Wheat they can't think\nof in a hot place like this. Rice and such things they might have a try\nat, if they were Chinamen. But I can tell you what they _will_ do.'\n\n'What is that?' inquired Ernest, reassured.\n\n'Why, you'll find that their cattle will go on increasing pretty fast;\nand what with grass rights and taking their blocks a little way off\neach other, they'll have nigh as much of Rainbar as you will in three\nor four years. I suppose that isn't what you fetched 'em up for?'\n\n'I do not grudge them a fair share of the Crown land,' said Ernest.\n'The land was made for all of us. But I certainly did not anticipate\ntheir requiring more than a limited area at any time.'\n\n'Well, it will be _unlimited_ if you don't manage to hem 'em in\nsomehow. They'll give you your work to do, take my word for it, some of\nthese fine days. My nags are a little fresher, and I am obliged to you\nfor as good a mount as ever I crossed.'\n\n'I am very happy to have been able to do you so small a service; and\nas for your advice, which you have been friend enough to favour me\nwith,' said Ernest, feeling depressed and much lowered in spirit by his\nguest's extremely 'faithful' criticism, 'I can assure you that it has\nsunk deeply into my mind.'\n\n'I'm glad of that,' said Mr. Levison gravely. 'There's very few men\nworth bothering with in the way of advice, and fewer still that aren't\ntoo great fools to take it when it's put before 'em. But I took a fancy\nto you, somehow, from the first time we met, when you had the thick\nboots and the swag. I thought that it showed pluck in you; and, from\nwhat I see here, you're one of those that goes in for helping other\npeople along the road of life. And a thundering soft thing it is, in a\ngeneral way, I tell you. Why, you've been teaching that native chap to\nread, so he says.'\n\n'I plead guilty to that,' said Ernest, with a smile. 'The fact is that\nJack Windsor is such a smart fellow that is seems a pity he should be\nleft helpless, as all ignorant men are. And there's plenty of spare\ntime in the bush.'\n\n'Is there?' said Mr. Levison. 'I never found it so. But that says\nnothing. I say it's a manly thing to feel for your neighbour because\nmaybe he hasn't had a hundredth part of the chances you've had\nyourself. That's being kind and true-hearted, and being a gentleman, as\nI understand it,' concluded Mr. Levison, with rather unusual emphasis.\n'But that's not what I want to say,' pursued he, buckling up the girths\nof his second saddle, and arranging his pack with the most accurate\nbalance possible. 'It's this: you want some more store cattle on\nRainbar.'\n\nThis last proposition Mr. Levison made in a tone of such peculiar\nconviction that Ernest could not frame a denial, but listened in\nwonder, merely ejaculating\u2015\n\n'In a dry season?'\n\n'It ain't going to be dry for ever,' said Mr. Levison oracularly,\n'and cattle are bound to rise within the next two years, as sure as\nmy name's\u2015\u2015 Smith,' he added, with a faint relaxation of his facial\nmuscles. 'I've just bought five thousand head of store cattle from\nthe man I met at Mingadee; bought 'em cheap, for cash\u2015my name's\nCash, you know\u2015and better bred cattle I never saw. I know 'em well.\nThey're all on a run on the Turon, and I'm to take delivery there.\nSeventeen-and-six for bullocks and twelve for cows. Can't hurt at that,\neh?'\n\n'I should say not,' said Mr. Neuchamp, calculating the scale of profits\nat three pounds ten shillings, which his bullocks had fetched, and,\nlike all inexperienced owners, omitting to allow for either deaths,\nlosses, or non-fattening tendencies. 'I wish I had half of them\nhere\u2015that is, when rain comes.'\n\n'That's just what I was coming to,' said Mr. Levison, with still slower\nand more inexpressive enunciation if possible. 'If you'll be said by\nme, you'll buy the cows; they're about half and half. There's till next\nApril to take delivery of 'em, and you can have 'em at what I bought\n'em at\u2015twelve shillings, big and little.'\n\n'But the money?' said Ernest. 'I have only what will pay my expenses\nfor six months.'\n\n'I'll take your bill at twelve months, with interest added,' said the\nperipatetic philanthropist. 'You write to old Frankston and tell him\nso, and perhaps I'll renew if no rain comes. Tell him it's Levison's\nadvice to you to make this bargain. He knows what that means. And my\nway of looking at things tells me that it's a deal more likely than\nnot, that within five years, if you take these cows and breed up, the\nrain will come, cattle will rise, and you'll have nearer ten thousand\nhead of cattle on Rainbar than five. I shall camp at that lake of yours\nto-night if I've luck. Good-bye, till we meet again. You buy those\n\"circle dot\" cows, and don't you waste your money.'\n\nSo departed Mr. Levison, rather incongruously inculcating economy and a\nheavy purchase of stock all in the same breath.\n\nErnest lost no time in writing to Paul Frankston to inform him of the\noffer of his very practical friend with reference to the store cattle,\nrequesting his advice thereon. By return of post he received the\nfollowing missive:\u2015\n\n MORAHMEE, _20th January 18\u2015_.\n\n MY DEAR BOY\u2015Have your letter, and glad to see you are regularly\n embarked in squatting life, and keep going at Rainbar in spite of bad\n times and bad weather. Seasons awfully uncertain in Australia; always\n were ever since I was a boy, and I don't expect them to alter much.\n People make money here in spite of them, and so will you if you keep a\n good look-out. As to the store cattle, there's dirty weather ahead\u2015the\n bank barometer falling and no rain. But for all that, Levison is a\n man to be backed. He is never far out. If he says cattle will rise,\n they will rise. I never knew him wrong yet. Where _he_ has bought you\n can't go wrong in following his lead. He has taken a fancy to you, and\n wishes to put you on for a good thing. I never do things by halves\n myself. So I advise you to take his offer. Go or send for the cattle\n when he takes delivery, and trust to Providence to send rain and a\n market. When the bill falls due we must arrange to pay or renew. Don't\n overdraw in other ways more than you can help, if you will let me give\n you my opinion. Crampton tells me your orders find their way down in\n spite of the dry weather. Spend _nothing_, never mind about its being\n necessary. That's the safe thing in squatting.\n\n Shall we see you after you have brought your cattle home? We have had\n awful hot weather. The mosquitoes seem livelier than average. Antonia\n thinks you might write and describe the country. She met Parklands the\n other day, who told her Brandon nearly finished all your careers with\n his four-in-hand freaks. Careful fellow, Parklands. Good-bye, my dear\n boy. God bless you.\n\n PAUL FRANKSTON.\n\nThus fortified, Mr. Neuchamp wrote immediately to Mr. Levison, who had\nwith characteristic carefulness left his addresses for the next month\nor two, and informed him that he accepted his offer with many thanks,\nand would attend, with Mr. Banks, at the station, some hundreds of\nmiles off, where the cattle were running. This matter settled, he told\nCharley of the adventure awaiting him, arranging to leave Jack Windsor\nin charge of the place until their return.\n\nMr. Banks expressed his unqualified approval of the whole matter. 'This\nsort of thing,' he was good enough to say, 'was something like. Putting\non more stock was the proper sort of work; any money spent in that way\nwould be sure to be returned. But hang these improvements! Filling up\nthe station with a lot of weekly men, and once they're there, it's not\nso easy to send 'em away again. Levison's a chap that gets good value\nfor his money, whatever he touches; and if he thinks buying store stock\nis the right thing, I'll put five to two on him and his tip. He will be\nthere, or thereabouts, when the flag falls, I'll lay.'\n\nWithin reasonable time a letter arrived from Mr. Levison of a very\nconcise and practical form. It set forth that, upon a certain day of\na certain month, his droving manager would be at Leigh Court, in the\ndistrict of King, where the herd of store cattle which he had purchased\nwere running. That the proprietor was bound by his agreement to have\nfive thousand head of cattle mustered and delivered within one month\nfrom the date specified. That his manager had instructions to deliver\nto Mr. Neuchamp, or his order, all the female cattle, young and old,\nof the said herd. He, Levison, had no doubt in his own mind that rain\nwould fall within six months, and he wished him luck. This was the\nonly portion of the letter not devoted to business. Laconic as was the\nstyle, Ernest felt touched by it, as the spontaneous expression of a\nheart filled with daily cares, and with rare leisure for friendship and\nsentiment.\n\nAfter a certain amount of necessary consultation and commissariat\naction, Mr. Neuchamp, one fine morning, left Rainbar with an imposing\n_cort\u00e8ge_. It consisted of Charley Banks, Piambook, and a man to drive\nthe light waggon, which, containing food, raiment, cooking utensils,\nand bedding, Ernest very properly took with him. There were other two\nmen, who had contracted to act as road hands and to make themselves\ngenerally useful. They drove half a dozen spare horses, Mr. Neuchamp\nbeing minded to purchase as few as possible at the seat of war, or the\nplace of delivery. Fast travelling was, of course, not possible under\nthe circumstances. They expected to travel at the rate of twenty or\ntwenty-five miles a day, until they should arrive at Leigh Court, the\nrun to be depopulated, so to speak. It was distant about six hundred\nmiles. There yet remained about two months to the date of delivery. So\nErnest gave himself seven weeks for the journey, and trusted to have a\nweek or two for refitting before commencing his grand march homewards\nwith two considerable droves of new store cattle.\n\nMr. Windsor and Boinmaroo were left in charge of the stock and station.\nBitterly did the first-named gentleman deplore the hard necessity which\nprevented his going forth on the war-path with the other braves.\n\nEvery night after the first, on which occasion a neighbouring\nout-station was reached, and the impatient home-loving horses put\nsecurely into a yard, a camp was organised.\n\nTwo tents were pitched, one for the master and Charley Banks, the other\nfor the men and any other road acquaintances that might be encountered.\nOne of the new hands had an accordion. He played moderately, but quite\nwell enough to satisfy the uncritical audience, and to enliven their\nsomewhat unamused evenings.\n\n\n\n\nCHAPTER XXI\n\n\nAs the progress of Mr. Banks and his party would necessarily partake of\nthe nature of caravan movements, Mr. Neuchamp decided, after a few days\nof co-operative wayfaring, to go ahead of his impediment. He would thus\nbe spared the _g\u00eane_ of objectless camp life and needless expenditure\nof time. With regard to the value of this latter commodity, he began\nto lean to the opinion of Mr. Parklands, and to believe that time\nwas ever in a colony, if not always a synonym for money, at least a\nmatter of high consideration. Apart from this method of reasoning, his\nroute after a while lay through a district which he had never before\nvisited. And a portion of the locality promised to be interesting to\nthe observer of men and manners for a novel reason.\n\nHe had since found that the owner of the large herd which Mr. Levison\nhad purchased, as another buyer would have bought a team of bullocks\nor a flock of sheep, had been compelled to sell on account of the\nsudden influx of miners upon his run. Gold\u2015the healer, the benefactor,\nthe deliverer, the slayer, the betrayer, the enslaver of mankind in\nevery age, in every clime\u2015had been discovered in the vicinity of the\nlong-silent peaceful valleys in which Abel Drifter's cattle had roamed\nfor more than a generation. Now all was changed: the green dales were\ninvaded by noisy crowds, the waters were polluted, the air was thick\nwith the smoke of camp-fires, maddening with the barking of dogs,\nthe crashing of falling trees. Droves of hobbled horses attended by\nreckless boys, who galloped and wantoned over the sacred camp, filled\nthe woods with alarm and distraction for the confused, terrified cattle\nand their despairing stockmen.\n\nBelieving if he hesitated that probably half his herd would wander off\nthe run and the other half disappear by dying, Mr. Drifter put the\nwhole herd into his agents' hands for sale, and, as we have named,\nfound a prompt purchaser in Mr. Levison. It was this dread alternative\nof landmarks, this solemn, dismaying change of the pastoral stage into\nthat of trade and agriculture, which Mr. Neuchamp had been curiously\neager to behold.\n\nPassing through that division of the great plain-ocean which varied\nin very slight degree from his own particular appointment, he entered\nupon a wholly different description of country, the characteristic\npeculiarities of which were clearly manifest to him. In the place of\nthe torrid plains and rare watercourses which he had traversed for\nmany days, he saw green park-like woodlands, pleasantly diversified by\nthe long-absent hill and dale. Broad and fertile valleys adorned the\nlandscape, from which many a harvest had been gathered since the first\nsod was turned. The houses of the proprietors were in some instances\nlarge and handsome, surrounded by shrubberies and orchards of ancient\ngrowth, or they bore the homely aspect of snug farmhouses, befitting\nthe homes of sturdy, prosperous yeomen.\n\nFencing of a substantial and contradictory nature abounded, so that\nErnest was more than once debarred from cross-country travelling,\nand forced to adhere to the high road. He noticed that during the\nmorning and evening hours the air was cool occasionally to keenness.\nThe magnificent distances to which he had become accustomed between\nthe homesteads had narrowed to something, if not identical with\nBritish habit, at any rate to far nearer propinquity than he had\ndeemed possible in Australia. From all these signs and appearances\nMr. Neuchamp decided that he had come upon a new and different phase\nof colonisation, and prepared himself to investigate and analyse\naccordingly.\n\n'Here,' he said, 'is one of the cheering results of that human\nhive-swarming which we call emigration. How many of these\ncomfortably-placed landholders, enjoying a charming climate, a fertile\nsoil, and that abundance of elbow-room which every Anglo-Saxon needs,\nwere peasant labourers, pinched and over-laboured, small farmers,\nor impoverished gentry, landless, tradeless, coinless younger sons\nin Europe? Here they have found their proper m\u00e9tier. Here they have\nrepeated history and have peopled a new world, under the Southern\nCross, where the passionate freedom of their forefathers may be handed\ndown unblemished to the sons of the grandest of races.'\n\nAs he travelled this settled region the population necessarily\ncommenced to show signs of alteration, both as to character and\ndensity. Instead of the sparse, sunburned, nomadic units of the waste,\nthe more various and pronounced types of agriculture and grazing\nindustry presented themselves frequently and unmistakably.\n\nMr. Neuchamp hailed with pleasure the opportunity thus afforded of\nconversation and companionship. He saw the neat taxed-cart, with the\nfarmers' wives and buxom daughters returning from the weekly market.\nHe saw the farmer himself mounted upon a stout, not over-refined\nhackney, jogging along the road with the bluff confidence inspired\nby good crops and good prices. He marked the great fields of maize\nalternating with hay and cereals, while the wide-fenced pastures, with\nthe clover, lucerne, and the prairie-grass of America, were thickly\nfilled with thriving cattle or the long-woolled sheep, with which his\neye had been familiar in his native country.\n\n'People in England fancy,' he thought, pursuing his ordinary train\nof thought, 'that life in Australia is principally devoted to lying\nunder the shade of tropical forest-trees, and eating peaches or\npineapples; or else that a course of violent and exciting border life\nis unremittingly hazarded. How little the average British mind is\ncapable of comprehending the widely various conditions of colonial\nlife, necessarily distinct and sharply defined, from the influences of\nvarying soil, climate, and original settlement, with a hundred other\nunderlying laws, by these centuries passed into the one concrete idea\nof \"the colonist.\" As reasonable would it be to mingle the attributes\nof the Devon or Suffolk peasant\u2015the Celtic Irishman, the Lowland Scot,\nthe Cockney, and the Highlander, under the general name of Englishman.'\n\nOn the day when these truly original ideas had occurred to Mr. Neuchamp\nhe was riding contentedly along the fenced highway with the intention\nof reaching at nightfall the homestead of a landed proprietor of\nsome mark in his own district, whose acquaintance he had made at the\nNew Holland Club. He was certain of hospitality and of receiving\nthe clearest directions as to his route. Within a few miles of his\ndestination, as he calculated, he encountered a gentleman, on a\nwell-bred hack, who had just emerged from a lane at right angles with\nthe road.\n\nHe replied to the stranger's courteous and unaffected greeting with an\ninquiry as to the precise distance of Mr. Haughton's house\u2015if perchance\nhe happened to be aware of it.\n\n'I am going within a mile of the entrance gate,' said the stranger;\n'I shall be happy to be your guide so far. I shall probably be at\nElmshurst to lunch to-morrow, and should be there to-night\u2015but that I\nhave to visit a sick parishioner.'\n\nMr. Neuchamp had partly conjectured from the dress of the gentleman\nthat he was in holy orders, and of course the point was settled by his\nadmission.\n\n'You are then the clergyman of this district?' said he. 'You are\nfortunate, I should say, in the locality of your labours.'\n\n'Yes,' said the stranger, rather absently, 'there is no fault to be\nfound with the climate or the scenery, and I have not met in my travels\nwith a more pleasant and kindly society. There is but one defect, and\nthat is universal.'\n\n'And that is, may I ask?'\n\n'Earnestness, thoroughness,' said the stranger, fixing his clear\nsad eye upon Mr. Neuchamp. 'If those whose duty it is to provide\naid and comfort for the souls that are like Lazarus, lying at their\ngates, leprous and diseased in mind,\u2015if they would but give of their\nsubstance, or better still, a hundred times better, of their time and\nenergy,\u2015much, how much, could be done for God and for man.'\n\n'I passed a very neat church and schoolhouse just now,' affirmed\nErnest; 'surely matters spiritual are regarded here with interest, and\nif the enthusiasm you lament be wanting, when and in what land is it to\nbe found?'\n\n'I speak not,' said the unknown, a glow of fervour lighting up a pale\nhandsome countenance, and illumining his melancholy dark eyes\u2015'I speak\nnot of the mere routine donations which reach respectable uniformity\nand stop there. I speak of the want of the spirit that maketh alive,\nand in one class not more than in others. The vicarious aid, it is\ntrue, is not sparingly or grudgingly given. But the heart's tribute\u2015the\nlife-donation\u2015where is it?'\n\n'I am sorry that it should be so,' said Ernest, thinking what a\nglorious pastor this zealous missionary would be for his community at\nRainbar, when it was sufficiently grown and established. 'I am afraid\nnone of us who are somewhat fully endowed with this world's goods do a\ntenth part as much as we might. But I do not see how matters are to be\nmended as the world whirls on its appointed course. Enthusiasm is dead,\nand belief will soon follow.'\n\n'We might all do much\u2015you will excuse my professional tone of\nexhortation,'\u2015said this latter-day apostle, 'by performing our own\ndistinctly laid down duties personally and rigidly, to arrest the\ndreary tendency you refer to\u2015to plant the seeds of a richer and a\nmore vigorous religious growth. I have not the pleasure of knowing\nyour name; permit me to present my card. I trust that we shall meet\nagain under circumstances more favourable for discussion and mutual\nacquaintance.'\n\n'Thanks, I shall only be too happy. I am Mr. Neuchamp, of Rainbar,\nwhere I should be delighted to see you if circumstances ever lead to\nyour visiting so distant a locality.'\n\n'I don't know where my Father's work may take me; but be assured that I\nshall be much gratified by any chance which involves future intercourse\nwith one of kindred sentiments.'\n\nMr. Neuchamp gazed at the speaker, and thought he had rarely seen a\nmore uncommon countenance. Still young, he was perhaps nearer to the\ngoal of middle age than to the 'spring of springs' of early youth.\nThe outline of the features was aristocratic and refined. His slight\nbut symmetrical figure, in its careless ease of seat on horseback,\nsuggested more extended practice in youth than was quite compatible\nwith his present position. But the eye, mild, searching, calmly\nradiant, was the conspicuous feature. It showed the steady unfaltering\nregard of one ever willing to attest with his blood the truth of the\ndoctrines which he held.\n\n'We pass through these rails,' said he, 'and enter this lane, soon\nafter which my path turns off and I leave you.'\n\nAs he pointed to the slip-rails Mr. Neuchamp spurred forward to prevent\nhis having the trouble to take them down, and practised a man\u0153uvre of\nwhich he was rather proud.\n\nHe stooped from his saddle, and, raising the top rail, placed it\ncarefully upon the second. Then wheeling Osmund for a stride backward,\nthat accomplished animal leaped easily over, without the slightest\nhesitation.\n\n'Come along, sir,' said Ernest to the clergyman; 'it is no height, and\nI will put it up.'\n\n'Thanks, no; you must really excuse me.'\n\nErnest reiterated his assurances that it was extremely low\u2015no danger,\nand so on.\n\nAll unmoved by Mr. Neuchamp's requests and entreaties, the gentleman\nwith the black coat and gray trousers quietly alighted, saying, 'You\nmust excuse me, I do not leap at all.' He then took down the two lower\nrails and, replacing them, gravely remounted.\n\n'Do you not think,' said Ernest, 'considering the large amount of\ncross-country work that a clergyman has to do in Australia, that every\ngentleman of your profession should practise leaping a little\u2015I mean\nsufficiently to get over middle rails, and so on? you might be stopped\nby a low fence.'\n\n'It may be so; there is force in your argument,' said the unknown, with\na grave sad smile, 'but I do not care about leaping now, and there is\nthen only one course open, that of taking down the rails. After all\nthere are so many necessary gates, I find that I can generally get\nabout my various duties.'\n\n'Really,' persisted Ernest, 'I hope that you will not think me\nimpertinent, but in a new country like this surely every one ought to\ntrain himself to encounter the exigencies of his position, and your\nseat is so firm that I am sure with a little practice you would soon be\nable to get over a moderate leap.'\n\nErnest thought he saw an approach to a smile flit over the thoughtful\nface of his clerical acquaintance.\n\n'Who knows?' he said, holding out his hand; 'I trust we shall meet\nagain. It may be that we shall be fellow-workers in this good land,\nwhere the harvest is plentiful, but the reapers, alas! few. Good-bye.'\n\nMr. Neuchamp pursued the path indicated, which led him to a substantial\ncountry-house, of which the well-kept approaches and trim, yet\nluxuriant shrubberies told of long and successful occupation. Here he\nwas warmly welcomed, and Osmund promptly delivered to a neat groom.\n\n'Very glad to meet you in the country,' said his host, a frank,\nstout, gray-haired, but vigorous-looking man. 'What do you think of\nour district\u2015anything like this on the Lower Darling? I hear you have\nsettled yourself permanently there.'\n\n'The two districts are about as similar as the West Riding of Yorkshire\nand the Pampas,' said Ernest. 'But Rainbar is a very good fattening\ncountry; that is all one can say in its favour just now.'\n\n'Plenty of room, no diggers, no free selectors,' replied his host;\n'well, I wish we could say as much here. I am too old to change now;\nbut I think if I was your age again, I should be inclined to move out\nback; let the Grange, and come back to be comfortable here in my old\nage. But I think I heard the dinner-bell. Come along.'\n\nErnest heard it too, and was by no means sorry to comply with the\nsummons. Dinner-bells, with the accompanying refections of civilised\nman, had been rather out of his line of late. He was introduced to\nthe lady of the house, and her well-dressed, fresh-complexioned,\ncheerful-looking daughters, the very sight of whom raised the spirits\nof Mr. Neuchamp several degrees.\n\nAn active, keen-looking youngster of sixteen made up the family party.\n\nErnest Neuchamp was approved of by the ladies of the household,\nas indeed was generally the case, being one of those sympathetic\nand genial persons whom women instinctively take into favour. The\nconversation had become general and sprightly pleasant, when, in answer\nto a question about his travelling alone, he happened to mention that\nhe had met, he supposed, the clergyman, not far from their house.\n\n'There is more than one clergyman in our district,' said the lady of\nthe house, 'but I daresay we shall recognise him from your description.'\n\n'He was a gentlemanlike person, rather handsome, indeed,' continued\nhe. 'It seems an odd thing, though, that clergymen, as a rule, ride so\nindifferently, and especially in a new country like this, where the\nnecessity of long journeys might have given them practice, one would\nthink; yet I could not get your friend to follow me over a middle rail.'\n\n'What?' said his host, with a look of altogether inexplicable\nastonishment mixed with amusement visible in his face; 'did you give\nhim a lesson in riding?'\n\n'I tried,' said Ernest; 'I am sure his horse would have followed mine\nif he had mustered up courage, and put him at it. I tried all I knew to\ninduce him, and said that with a little practice I was sure he would\nsoon be able to take moderate jumps.'\n\n'Moderate jumps! oh, Lord!' said his entertainer; 'and what answer did\nhe make?'\n\n'He smiled gravely, and said, \"Who knows?\" then bade me good-bye. I\nhope he was not offended.'\n\n'Ha! ha! ha!' yelled the youngster, exploding helplessly. 'Oh dear! oh!\nI'll lay anything, papa, it was Mr. Heatherstone. I shall die! I know I\nshall. What a jolly sell!'\n\nThe girls struggled with their emotions\u2015one hid her face in her\nhandkerchief. The lady of the house smiled, but tried to look grand,\nand reproved her son, who continued to shriek with suppressed laughter,\nand finally bolted out of the room, as the safer proceeding.\n\nHis host, making desperate efforts at self-control, said, at length, in\na broken voice, 'My dear fellow! you mustn't mind these young people.\nI'm afraid they are laughing at a little mistake you must have made\nas to our clergyman's degree in equestrianism. But are we sure of our\nman\u2015did you learn his name?'\n\n'He gave me his card,' said Ernest, now shuddering under the\nconsciousness of having, perhaps, again buried himself in a pitfall in\nthis provoking happy hunting-ground, 'but I never looked at it. Here it\nis\u2015\"The Rev. Egbert Heatherstone.\"'\n\nHere the second young lady broke down, while her mamma laughed\ndecorously and under protest as it were; and paterfamilias, in an\n_almost_ steady voice, thus spoke\u2015\n\n'You never heard of Heatherstone before, then? No? Well\u2015the man you\nwere trying to lure over a middle rail was formerly known, that is,\nbefore he entered the Church from strong convictions, as perhaps\nthe boldest, the most reckless rider in Australia. He has ridden\nmore steeplechases than you have hairs on your heads, I was going to\nsay\u2015but, to speak moderately, a larger number than most men living.\nSince he became a clergyman, a most sincere and hard-working one, he\nhas given up sensational riding, and being passionately fond of horses,\nmortifies the flesh by abstaining from all that style of thing. You\nwill excuse us all, I know, for being so rude; but really, you must\nadmit the joke was irresistible.'\n\n'I see\u2015I admit\u2015I confess,' said Ernest, with an air of deepest\npenitence. 'If I could only do penance for my sins of superficial\njudgment, it would be such a relief. Do you think the Rev. Egbert has a\ntrifle of spare sackcloth?'\n\n'You didn't notice his seat on horseback?' asked one of the young\nladies innocently. 'Doesn't he look like a horseman? He can't hide\n_that_, or help his hands being so perfect\u2015though I think he tries.'\n\n'He rode a horse over a three-railed fence once, without saddle or\nbridle,' said the other sister, 'for a bet; before he was ordained.'\n\n'He took Ingoldsby, the great steeplechaser, over a three-railed fence\nat twelve o'clock at night, and pitch dark too; there was a lantern on\neach post though,' chimed in the sixteen-year-old hero-worshipper of\nany reckless deed in saddle or harness.\n\n'The maddest thing of all that I ever heard of him,' affirmed papa, in\nconclusion, 'was going across country one evening and taking sixteen\nwire fences running. He won his bets, but he had two hardish falls; one\na collar-boner, into the bargain.'\n\n'I really begin to think,' said Mr. Neuchamp despairingly, after\nevery one had transacted a good downright unrestrained chuckle, 'that\nI shall never become fully acclimatised. This is the most peculiar\nand utterly unintelligible country ever discovered; or, am I devoid\nof understanding to an extent which disables me from ever rating\nindividuals at their proper value?'\n\nHe was eventually consoled, and persuaded into singing second in a duet\nwhereto the accompaniment was played with much taste and expression by\none of the daughters of the house. He was perfectly at home in this\ndepartment of criticism, and after receiving a few compliments upon his\nextremely correct performances, he commenced to forget the stupendous\nmiscalculation into which he had been led with respect to the Reverend\nEgbert Heatherstone and his equitation. But it was not forgotten by\nthe inmates of the house and the inhabitants of the district, among\nwhom it gradually spread. It always took rank among those glorious\njests which, intelligible to every degree of capacity, float on with\nundiminished grandeur from generation to generation; and a stranger\nwho reached that peaceful district, and was discovered by a delicate\ncourse of inquiry never to have heard that joke, was regarded with\naffectionate interest, and had it so carefully administered to him that\nnot one drop of the _elixir jocosus_ should be wasted in the process.\n\nLeaving the honoured abode of hospitality and domestic happiness,\nwith its fertile meadows and well-filled stackyards, Mr. Neuchamp\npursued his route quietly, intending to make his way to the property of\nanother friend, whose place was at no great distance from the goldfield\ntown near which was the station upon which his cattle were still\ndepasturing. This stage was rather far for one day. He was considering\nwhether he might expect to meet with a reasonable inn, and humming a\nsouvenir of his last night's concert, when a horseman, coming at a\nbrisk pace in the opposite direction, met him face to face.\n\nIn him he recognised a young squatter whom he had often encountered in\nSydney in various festive scenes, and who had more than once pressed\nhim to visit his station, if he should find himself in their district.\nErnest knew the station of Baldacre Brothers by reputation to be large\nand rich. In fact the brand had a colonial fame. His curiosity was\nsomewhat aroused to behold the establishment.\n\nMr. Hardy Baldacre expressed great concern that he should be just\nleaving home for a journey when his friend Mr. Neuchamp was coming\ninto the district, and made many excuses for not turning back\u2015finally\nasking Ernest how far he thought of going that night. He mentioned the\nhouse of the brother of Colonel Branksome.\n\n'Oh! that is too far,' said Mr. Baldacre; 'sixty miles, if it is a\nyard.'\n\n'I don't think I will try to get quite so far,' said Ernest. 'Probably\nthere is some inn which will do as a half-way house.'\n\n'Oh! you'd better stay at our place,' said his friend with an\nexpression of countenance not wholly intelligible to Ernest. 'It's\nabout twenty-five miles from here, straight on the road; you can't miss\nit. You'll find my brother William at home. Good-bye!'\n\nWith this somewhat laconic invitation he put spurs to his horse and\nrode forward at a hand gallop, leaving Ernest undecided as to whether\nhe should accept or decline an invitation not very graciously extended.\n\nBy the time, however, that he had got to the end of the rather long\ntwenty-five miles over a worse road than he had hitherto travelled, he\ndiscovered that there was no other stage available without over-riding\nOsmund, so he commenced to look about for the homestead of the Messrs.\nBaldacre Brothers of Baredoun.\n\nIt was nearly dark when he came to a hut by the side of the road,\nsituated in a small paddock, the upper rails of the fence of which were\nornamented with sheepskins to an extent which suggested that a new\nmaterial for enclosures was being tested. Resolved to make inquiry as\nto this mysteriously invisible homestead, Mr. Neuchamp holloaed to the\noccupant of the hut in a loud and peremptory manner.\n\nA man in his shirt-sleeves came to the door, not otherwise over-neat,\nand smoking a black pipe.\n\n'Can you tell me where Baredoun is?' demanded Ernest: 'it ought to be\nsomewhere about here, I should think.'\n\n'This is the place,' said the shirt-sleeved one coolly.\n\n'And is _this_ the home station of Baldacre Brothers?' inquired Ernest,\nvainly trying to disguise his astonishment.\n\n'It's all that's of it,' said the smoker, with an attempt at\njocularity. 'I'm William Baldacre; won't you come in and stay the\nnight? It's rather late, and there is no place within fifteen miles.'\n\nErnest stared before him, around, and finally behind, before he\nanswered the hospitable question. He made a mental calculation as to\nwhether it was worth while to push Osmund on for fifteen miles over an\nunknown road in the dark. Finally, he decided to sacrifice his comfort\nfor that night to the welfare of the gallant grey, and to accept the\nultra-primitive hospitality of Mr. Baldacre.\n\n'I met your brother, whom I had the pleasure of knowing,' he said, 'a\nfew miles back. He was good enough to ask me to take up my quarters\nhere to-night. I shall be very glad to stay with you.'\n\n'All right,' said the elder man, a plain and unpolished personage when\ncompared with his handsome, well-dressed younger brother, who swelled\nabout the metropolis, by no means as if he had emerged from such a\nhovel. 'Give me your horse; he'll be safe in this paddock. Ours is\nrather a rough shop, but you must make allowances for the bush.'\n\nSadly and sorrowfully, after he had seen Osmund left free in the small\nmoderately-grassed paddock, did Mr. Neuchamp follow his host into the\nhut. That building consisted of two small rooms. There was an earthen\nfloor, one or two stools, a small fixed table, far from clean. A bed at\nthe side of the room offered a more comfortable seat than the stools,\nand upon this Ernest deposited his weary bones and disappointed entity,\nwondering doubtfully whether sleep would be uninterrupted or otherwise.\n\nThe usual meal of corned meat, damper, and milkless tea was brought\nin by the hutkeeper of the period, whose moleskins were strictly in\nkeeping with the prevailing tone of the furniture and apartment. Much\nErnest wondered at the precise mental condition which could suffer\ntwo free agents of legal age, the owners of a proverbially rich and\nextensive run, of a well-known highly-bred herd, free from debt and\nincumbrances, to live in a state of squalid savagery. He did not\nexactly put his questionings into this shape, but his manner had\nexpressed a patent astonishment, which his host seemed to consider\nhimself called upon to answer.\n\n'We haven't done much in the building way here,' he remarked\napologetically, knocking the ashes out of his pipe. 'I daresay we'll\nput up a cottage next year. But the old man never would spend a penny\non the run here. He was snug enough at the old farm down the country,\nand somehow I've got used to the life, and it does me as well as any\nother. Hardy isn't often at home; he's half his time in Sydney. So he\nmanages to hang it out here when he comes to help muster and so on. I\nreckon he thinks it saves money, and as he hasn't to live here _he_\ndon't care.'\n\nErnest felt remorseful after this explanation, very simply delivered,\nat his feelings of disgust and disapproval. 'Suppose,' he asked\nhimself, '_I_ had been set down here, a raw schoolboy, transplanted\nfrom half-learned tasks to the daily labour, the rude association, the\nunbroken loneliness of a distant station, debarred by a penurious old\nfather from the smallest outlay not immediately connected with the\nherd, without books, change, society, or recreation, would it have been\nall-impossible that I should have grown into the mould in which my host\nis enclosed, or settled down into the resigned, sad-visaged man of\nfive-and-thirty whom I see before me?' It _would_ have been impossible\nin his case, he thought. Still he could enter sufficiently into the\nprobabilities of the situation to comprehend the injustice to which the\nmental development of the elder brother had been exposed.\n\n'Good heavens!' he thought to himself, 'what short-sighted idiots are\nparents who shut up their sons' lives in a moral dungeon like this! The\nabiding in the wilderness is nothing; nay, it has positively beneficial\nand ennobling tendencies. But this sordid imprisonment of the mind!\nNo books, no companions, no ideas; for how can there be a circulation\nof ideas if reading, conversation, reflection be wanting?\u2015the whole\nmind bent and fettered to the level of the branding pen and the cattle\nmarket\u2015the smallest outlay affording a glimpse of the heaven of Art\nand Literature churlishly denied, lest a few broad pieces escape the\nall-gathering muck-rake. And when the game is played out, the long\nharvest-day over, and the crop garnered, what is the grand result\nfor which a soul has been starved\u2015a man's all-wondrous brain-marvel,\nmiracle of miracles, enchantment before which all magic palls\u2015stunted,\nand shrivelled from lack of nutriment and exercise, like a baby-farmed\ninfant's body? A few hundreds or thousands, more or less; a sufficiency\nof clothes and food; a surety against poverty; and the possibly\nfully-developed son of the immortal, \"a little lower than the angels,\"\nremains hopeless, contracted, with the mind of an untaught child plus\nan experience of the more obvious forms of dissipation!'\n\nThe rude meal concluded, and the pannikins refilled, Ernest, as\nusual, felt sufficiently refreshed in spirit to examine his immediate\nmaterials. Mr. Baldacre smoked and talked unreservedly for a couple\nof hours; explained the presence of the sheepskins\u2015they had been\nbutchering for the diggers lately; described some of their pioneer\nlife, including an adventure with a bushranger, the famous Captain\nBelville; and, finally, thought Ernest might like to 'turn in.'\n\nMr. Neuchamp looked distrustfully at the rude wooden frame, upon which\nsheepskins did duty for a mattress, and a pair of highly uninteresting\nblankets represented all other description of bedclothes. He protected\nhimself against all nocturnal dangers by retaining the larger\nproportion of his habiliments, and desperately committed himself to the\nuncertainty. At earliest dawn he might have been seen leading Osmund\ntowards the hut, after which he saddled up with unusual energy and\ncare. He then betook himself to a grand deep water-hole at no great\ndistance in the creek, where he swam and disported himself for half an\nhour at least, after which he indulged temperately in the pleasures of\nthe table, as represented by a breakfast which was the facsimile of\nsupper, and immediately thereafter bade his host good-bye, thanking him\nfor his entertainment, and bidding farewell to the abode of Baldacre\nBrothers for ever.\n\nMr. Neuchamp smiled to himself when fairly on his way, thinking of the\ndays of his inexperience, when he believed that all squatters, and\nindeed all colonists, lived in precisely the same fashion, and were\ncharacterised by identically the same habitudes and modes of life.\n\nHe certainly had been 'had,' as Mr. Banks would have said, in the\nmatter of trusting himself to the primitive establishment of the\nBaldacres, who were well known to every one in the district to live\n'like blackfellows,' as the phrase ran. But neither he nor Osmund\nhad suffered anything more than slight temporary inconvenience. Mr.\nNeuchamp was specially good at recovering, and in half an hour he was\nwhistling and humming along the road as blithely as ever.\n\nOn this particular day he expected to reach, at an early hour, the\nabode of another club acquaintance, who had been unaffectedly hearty\nin impressing upon him the desirability of making his place his\nheadquarters if he ever came to their district. At this house he\nexpected to meet the Indian Officer who had so kindly taken care of his\nArab steed for him and attended to his comforts on board the P. and O.\nThis distinguished _militaire_ had seen a good deal of service, but\nthirty-five years' exposure to the sun of Hindostan had not quenched\nhis ardour for sport, spoiled his seat on horseback, or cooled his\ndevotion to the fair sex. He had been commissioned by the Indian\nGovernment to make large purchases of horses in Australia for remount\nservice, particularly for artillery and heavy cavalry. He was now on\na tour of inspection through the chief breeding districts, to the end\nthat the couple of thousand troop horses he was empowered to purchase\nand ship might do credit to his judgment. Combining, as he did, a frank\nyet polished address with the prestige of military rank, important\nservices during the Mutiny, consummate knowledge of horseflesh, with\na potentiality of unlimited purchase, Colonel Branksome was at that\ntime, perhaps, the most popular man in Australia.\n\nIt was on the right side of lunch-time when Mr. Neuchamp found himself\nopening a neat white gate, at the end of a well-kept drive, which\nfurther conducted him to the front door of a stately mansion, with easy\ncircumstances and good taste written in every yard of the well-mown\nlawn, on every clump of the crowded shrubbery, on the long range of\nstabling at no inconvenient distance, even in the neat dress and\nrespectful manner of the groom who came to take his horse almost as\nsoon as he had dismounted.\n\nThe hall door opened in a spontaneously hospitable manner, and the\nhost, accompanied by a middle-aged man very carefully attired in\nunmistakable mufti, left no doubt on any one's mind as to his pleasure\nin receiving him.\n\n'Just in time for lunch, Neuchamp! Very glad you've found your way\nto our district. The Colonel, here, has just been thrashing me at\nbilliards; let me introduce you: Colonel Branksome\u2015Mr. Neuchamp.'\n\n'Happy to meet you,' said the Colonel; 'find the morning hot? Deuced\nnice horse of yours; you haven't a few like him for sale, have you?\nI could take a hundred, and pay well too. But, of course, he's a\nfavourite; all the good ones are hereabouts.'\n\n'I am almost sorry to say that he is,' said Ernest, 'since I should\nhave liked to have helped you to a few horses that would have done\ncredit to Australia. I believe I have to thank you for an important\nservice in procuring justice for my Arab on his voyage out.'\n\n'A mere matter of course,' said the Colonel. 'I knew Granby who\nshipped him, and the old sheik who sold him; personal friend, and all\nthat; besides, I can't see a handsome horse or a pretty woman without\ntaking the strongest interest in their welfare. Weakness of mine all my\nlife. Too old to mend now, I'm afraid.'\n\n'By George! I forgot the lunch,' said the host, looking at his watch.\n'Come into my dressing-room, Neuchamp. Billy, you know your way.'\n\nIn a few minutes, after a temporary toilet, Ernest found himself in\na large cool room, the furniture and arrangements of which betokened\nno hint of the considerable distance from the metropolis. Two pretty\ngirls, whose complexions told of a cooler climate than that of the\ncoast cities, and drew forth many a compliment from the susceptible\nwarrior, embellished the well-appointed lunch-table. Here, with cool\nwine, delicate viands, and civilised society, Mr. Neuchamp was enabled\nutterly to discharge from his mind the unsavoury surroundings of his\nprevious stage. Before they had finished the repast the eldest son of\nthe house came in, apologising for his want of punctuality, but laying\nthe blame upon a large body of miners whom he had been supplying with\nrations, and who had detained him until their wants were satisfied.\n\n'Really!' said Mr. Branksome, 'the consumption of meat is becoming\ntremendous. Stock must rise directly. I feared that we were all going\nto be ruined at first. Now, I see plainly that it will be all the other\nway.'\n\n'So, then, I suppose I must have made a good bargain in conjunction\nwith Mr. Levison,' affirmed Ernest tentatively.\n\n'Oh! you bought the \"bar circle\" cattle, then?' said young Branksome.\n'They told me they expected a gentleman to take delivery directly. They\nare the best bred cattle in this district. You were lucky to buy them.\n\n'Poor Drifter,' said the old gentleman, 'it was anything but lucky for\n_him_ that he was forced to sell them. I told him that he was hasty,\nbut he was full of visions of their being killed and driven away right\nand left by the mining population, and would not hear reason.'\n\n'The miners are very decent fellows, what I have seen of them,' said\nthe son. 'Of course there will be all sorts among them; but he would\nhave no greater risk of losing his cattle at their hands than with many\nothers.'\n\n'Not so bad as Sepoys, eh, Billy?' said the host; 'and yet I suppose\nyou trusted the villains to the last minute.'\n\n'Well, I did,' said the Colonel, 'and I'm not ashamed to say so; and so\nwould you if you had seen them fight and die by your side for many a\nyear as I had done. There were some splendid fellows among them\u2015\"true\nto their salt\" to the last. It was a great chance that I wasn't shot\ndown by my own men, like Howard and Weston, and many other commanding\nofficers.'\n\n'How did you escape, uncle?' said one of the young ladies, deeply\ninterested.\n\n'Well, I'd been out at daylight with a scratch pack of hounds hunting\njackals. Just as I was coming in, the old havildar (I had saved his\nlife once) came rushing out: \"No go home, sahib,\" he said, \"men all mad\nsince chupatties come; shot Captain, sahib, Lieutenant, sahib, Major,\nsahib, and his men, sahib, hide away. Ride away, sahib.\" And he hung on\nto my horse's rein.\n\n'\"Let me go, you old fool,\" I said, \"I must go back; the men will hear\n_me_. It's those rascally Brahmins.\"\n\n'\"You give life, sahib, you do no good,\" he cried out, and, by Jove!\nthe tears _did_ roll down his face. \"I give my life for the Colonel,\nsahib, if he please. All no use. Look there!\" and he pointed to where a\nlong line of flame was rising up from my bungalow and stable.\n\n'\"Where's Lady Jane?\" I roared; \"you don't mean to tell me they've\ntaken her? I won't leave _her_ if I die for it.\"\n\n'\"Lukehmeen syce, he very good man, he go away with Lady Jane this\nmorning; go away to Raneepore. She all safe.\"\n\n'\"By Jove,\" I said, \"that's good news. If Lady Jane was there now, I\nbelieve I should have gone in among the rascally Pandies with my sword\nand revolver, and seen it out.\"'\n\n'How brave of you, Uncle William,' said one of the girls, her cheeks\nglowing and her lips trembling with excitement, as she gazed admiringly\nat the Colonel's hawk nose and bright blue eyes, which nearly matched\nhis turquoise ring. 'And did the poor lady escape altogether?'\n\n'Lady?' said the latter-day Paladin, in tones of astonishment. 'Lady\nJane was a thoroughbred English mare that I'd just given three hundred\nfor, worse luck, for I never did see her again, or any of my goods and\nchattels, from that day to this.'\n\n'And what did you do then, uncle?' said the other sister, the humane\nsympathiser with Lady Jane being too much astonished and discomposed to\ncontinue the examination.\n\n'I was on my old Arab, Roostoom, luckily,' said the Colonel, 'a horse\nknown all over India. When I saw there was nothing for it, I turned his\nhead straight across country for Delhi, and after missing a few shots,\nrode one hundred and thirty miles before I stopped. Next morning I fell\nin with a troop of irregular horse of Jacob's, and stayed with them\ntill we entered Delhi together at the Cashmere gate. I say, we have\nsquared accounts with the Pandies; and I thought we were going to ride\nover to the diggings after lunch.'\n\nAccordingly, about three o'clock, behold the whole party, including\nthe two young ladies and Mr. Neuchamp, mounted and cantering along\nthe extremely well-marked road which led to the mining township of\nTuronia. The young ladies rode with grace and spirit upon well-groomed,\nwell-bred horses, drawing forth many encomiums from the horse-loving\nand gallant Colonel, who said that their steeds would fetch a thousand\nrupees in Calcutta, and the young ladies receive half a dozen proposals\nof marriage the very first day they appeared on the Maidan.\n\nThe young ladies, in return, declared that there was only one man in\nthe district to be compared to their uncle; and as he sat with easy\nmilitary seat upon a strikingly handsome thoroughbred bay, with a star,\nthe whole affair, from the well-brushed hat to lower spur-leather,\n'exquisite as a piece of lace,' he justified their appreciation. As\nthey neared the widely-extended collection of huts, shafts, heaps of\nmullock, and imposing structures of weatherboard and iron, thronged\nwith a stalwart army, ten thousand strong, of bronzed and bearded\ngold-miners, they were joined by a semi-military-looking personage,\ndressed in uniform not all devoid of gold lace, and followed by a\nhighly efficient-looking, well-mounted trooper.\n\n'Ha! Stanley,' said Mr. Branksome, 'well met; how do you do? This is\nmy friend, Mr. Frank Stanley, the Commissioner of the goldfield. Allow\nme to introduce you to him. Are your subjects peaceable enough to\nventure among; and how does the escort get on?'\n\n'I will answer for my diggers,' said Mr. Stanley, bowing to the\nyoung ladies, 'being the most genuinely polite people in the world,\nespecially to ladies; and the escort was a little over ten thousand\nounces last week.'\n\n'You don't say so?' said Mr. Branksome; 'three thousand ounces more\nthan last week. Why, how much do you intend to get at by the end of the\nyear?'\n\n'Several, rich leads have been discovered lately,' said the\nCommissioner, with a slight air of importance. 'If they find a deeper\ndeposit below the basalt, as many of the experienced miners think\nlikely, we shall eclipse California.'\n\n'How very interesting,' said Mr. Neuchamp, much excited by proximity\nto a novel and recent development of colonial industry; 'I suppose\nyou find great difficulty in managing such an immense and disorderly\nconcourse.'\n\n'If they were disorderly we simply could not manage them,' said the\nrepresentative of the Queen's Government. 'We have about an average of\none constable to a thousand men. Moral force, applied with discretion\nand firmness, suffices for all purposes of rule and coercion. Besides,\nthe miners, as a rule, are well-educated men, and such populations are\nalways manageable.'\n\n'Why so?' inquired Ernest. 'I should have thought that they were easily\nled away by designing persons.'\n\n'The contrary is the case,' said the experienced proconsul. 'Without\nstating that there are always among the miners gentlemen and\ngraduates of the university, a considerable proportion consists of\nwell-educated, travelled, sagacious men. These leaven the mass; and\nhaving strong convictions themselves upon all subjects, they are\namenable to argument\u2015to logic\u2015which comprehends justice. It is an\nignorant population which follows the demagogue like sheep; it is the\nuncultivated mind which is at the mercy of every specious lie which is\noffered to it.'\n\n'Then crime is rare,' said Ernest, 'and offences against life and\nproperty uncommon?'\n\n'Taking the numbers, one may aver, with safety, that crime is\nexceedingly infrequent. At the same time I cannot deny that the police\ncharges are tolerably numerous. But in case of serious offences we have\nthe main body of miners on the side of law and order, and the criminal\nrarely eludes the arm of the law.'\n\nBy this time they had neared the outskirts of the town, and Ernest\nwas much pleased with the many neat cottages, surrounded by trim\ngardens, which they passed. Among these stood an exceedingly small\nbut faultlessly neat dwelling, surrounded by a garden filled with\nvegetables, the profuse growth of which was due to a small stream of\nwater which had been ingeniously led from the neighbouring hills. The\nowner, whose attire, though suitable for working, was marked by the\nexceptional neatness which pervaded the establishment, leaned upon\nhis spade and gazed calmly upon the _cort\u00e8ge_ as it passed along the\nwinding forest track.\n\n'How pleasant a sight it is,' said Ernest, 'to see one man, at least,\nsuperior to the mad thirst for gold which is common to this eager\npopulation. How contentedly that gardener devotes himself to the\noccupation in which he has probably passed his former life, and which,\nwithout holding out any splendid prize, no doubt provides him with a\ncertain and ample subsistence.'\n\n'I should say,' said Mr. Branksome, 'that your recluse has probably\nlost his all at a gold venture, and is from circumstances compelled to\nrusticate, literally, until he makes a fresh start.'\n\nThe Goldfields Commissioner smiled, but made no remark, as he rode\nclose up to the palings of the garden and reined in his horse.\n\nThe gardener left his work and advanced to the fence, apparently to\nhold converse with the important official\u2015a man at that time possessed\nof enormous power and irresponsible control.\n\n'Hallo, De Bracy!' said the latter, 'how are you getting on? Weather\ntoo hot for the green peas? Asparagus pretty forward?'\n\n'Shocking weather, altogether,' said the horticulturist, advancing to\nthe barrier and shaking hands with the Commissioner. 'If it were not\nfor my irrigation I should be ruined and undone. Splendid thing, water!'\n\nThe Colonel and Ernest, with the young ladies, had by this time ridden\nclose up, and were regarding the somewhat exceptional 'grower,' whose\nsunburnt hands exhibited much delicacy of shape and careful treatment,\nwhile his extremely handsome face and figure told unmistakably of long\nacquaintance with the _haute vol\u00e9e_ of the world's best society.\n\n'Are you going to the bachelors' ball to-morrow night?' asked the\nCommissioner. 'Great muster, and no end of young ladies.'\n\n'Well, I may look in for an hour if I can get these cauliflowers\nproperly earthed up in time,' said this anomalous member at once of the\ngay and workaday world. 'You know the season is so forward that I dare\nnot give them another hour.'\n\n'Great God!' said the Colonel, 'why, it's De Bracy! Why, Brian, old\nboy, what, in the name of all that is impossible, brings you here?'\n\nErnest turned at the exclamation, and saw that the Colonel's bold\nfeatures had changed, and were working like those of a man who sees\nsome visitant from the silent land\u2015is confronted by an unreal presence\nthat stirs his inmost soul and curdles the very life blood.\n\nThe young ladies stand, pale with surprise.\n\n'Oh, it's you, Billy Branks,' said the provider of esculents. 'Come\ndown from India? Nearly as hot here, eh? Well, I lost all my money in\nmining enterprises; the finest substitute for unlimited loo I ever fell\nacross. And having absolutely nothing, and being far from the land of\nfriends, bill discounters, and outfitters, why, I took to gardening.\n_Il faut vivre_, you know; and I was always fond of dabbling in amateur\nhandicrafts.'\n\n'Splendid life, beautiful weather, not too cold; shouldn't mind it a\nbit; make heaps of money, I'm sure!' said the Colonel incoherently.\n'But oh! Brian, old fellow, I never thought I should see you _working_\nfor your living.'\n\n'Why not, my dear boy?' said the philosopher of the spade coolly. 'What\ndoes the old Roman poet say\u2015_furcae amor honestus est et liber_\u2015stick\nto your knife and fork, and all that. Horace has no doubt on the\nsubject. This is my Sabine farm, and there is the Fons Bandusiae, for a\ntime\u2015glad to say\u2015at any rate, for a time\u2015the pre-remittance stage. It's\nsafer than billiards, and more creditable than whist\u2015as a livelihood.'\n\n'True, by Jove!' said the Colonel, 'most honourable and all that. But\nthe fellows at the Rag would never believe it, if I go back and tell\nthem that I saw Brian de Bracy growing vegetables and living by it, by\ngad.'\n\n'Tell 'em every word of it, Billy, old boy,' said the wholly unabashed\nand true descendant of Adam, squaring his shoulders and displaying his\nsymmetrical figure. 'Tell some of them to come out and try their luck\nhere. It will do them a lot of good, make men of them, and keep them\naway from the bones.'\n\n'Certainly, certainly,' assented the Colonel, hopelessly confused.\n'Most likely they'll all come. Charming climate, splendid salad, and so\non. Well, good-bye, old man. Sorry to see you looking so well. Oh lord!\nwhy didn't the French Count kill you instead of your winging him, in\nthat row about Ferraris, and stop this. Good gad!'\n\nSo saying, the warm-hearted warrior wrenched away his horse's head and\ndeparted along the homeward track, inconsolable for at least a quarter\nof an hour, at the expiration of which time he unburdened his soul to\nthe nearest niece as follows:\u2015\n\n'Awful thing! poor Brian, wasn't it? By gad, when I first recognised\nhim, thought I should have fallen off my horse. Last time I saw him he\nwas coming out of the Travellers', in London, with a duke on one arm\nand the commander-in-chief on the other. Awful fuss always made about\nhim. No swell within miles of him\u2015at Ascot, Goodwood, and so on. Women\nreg'lar fought about him\u2015handsomest man of his day. Shoot, ride, fence,\neverything, better than the best of the amateurs. And now, what's\nhe down to? By gad! it makes a baby of me.' And the honest, kindly\nveteran looked as if a cambric handkerchief would have afforded him\ngreat comfort and relief under the circumstances.\n\n'Never mind, uncle,' said the sympathising maiden, 'you'll see him at\nthe ball to-morrow night, and I'll dance with him\u2015not that there's much\ncharity in that. You know how nicely he looks at night. There won't be\na man there to be compared with him.'\n\n'Of course I'll go,' said the Colonel, recovering himself as became a\nsoldier, 'and you may look me out a nice girl or two for a waltz. I\ndon't _think_ I ever went to a ball at a diggings before.'\n\n\n\n\nCHAPTER XXII\n\n\nA pleasant ride home in the cool of the evening, comprising some\n\u00e6sthetic talk on the part of Ernest with the youngest daughter, and a\nsensational bit of horsemanship by the Colonel, who rode his horse over\na stiff three-railer that Miss Branksome had denounced as dangerous,\nprepared the party for a very merry dinner, after which some dressing\nset in, and the whole party started for the ball in a high mail phaeton.\n\nThe mining township of Turonia, while tolerably open to criticism by\nday as to its architecture, with the kindly aid of shadow and moonbeam\nlooked sufficiently imposing by night, with its long line of lighted\nstreet, its clanking engines and red-gleaming shift-fires.\n\nThe particular night chosen for the entertainment which the bachelors\ntemporarily dwelling in and around the golden city of Turonia had\nprovided, was of the clearest moonlight procurable. Undimmed, awful,\ngolden, pure, in the wondrous dark-blue dome, glowed the thrones of\nthe greater and the lesser kings of the night. The trees upon the\nswart hillsides were visible in fullest delicate tracery of leaf and\nbranch, as at midday. Each trail in the red dusty roadpaths showed with\nmagic pencilling of outline. The dark-mouthed cruel shafts, which lay\nas if watching for a prey on either side of the narrow roadway, were\nplainly visible to the most careless wayfarer. So it chanced that from\ncottage and villa, from farmhouse and home station, and even from less\npretentious habitations than any of these, wended at the usual hour\na concourse of joyous or pleasure-enduring visitants, not specially\ndistinguishable in air, manner, or raiment from metropolitan devotees\nof similar tenets.\n\nPretty Mrs. Merryfield was there, whose husband, formerly in the\nnavy, held as many shares in the Haul and Belay Reef as would at that\ntime have enabled him to retire upon club life and whist for the rest\nof his days. Managing Mrs. Campion, with her three daughters (Janie\nCampion was not unlikely to be voted the belle of the evening), sailed\nin, imposing with bouquets all the way from Sydney, the fern sprigs,\ncamellias, and moss rosebuds of which were marvels of freshness. Little\nCampion and his partner, George Bowler, were driving a roaring trade as\nauctioneers, and a cheque for fifty for the girls' dresses and fal-lals\nwas, he was pleased to say, 'neither here nor there.' The doctors, half\na dozen, were chiefly married men, and contributed their full share\nto the feminine contingent. So did the four lawyers. Mining cases are\nperhaps the most interminable, complicated, and technical known in the\nrecords of litigation. The bankers were in great force and profusion.\nIn mining towns they are necessarily numerous and competitive, and\nthere are few departments of social accomplishments to which they may\nnot lay claim. Thus many were the celebrities contributed by them that\nnight\u2015athletic champions, musical bankers, and bankers that danced,\nbankers that billiarded and whisted, bankers that 'went in for beauty'\nand preserved their complexions, and bankers that combined divers\nof these claims to consideration. In a general way it may be assumed\nthat the _jeunesse dor\u00e9e_ of that inevitable profession numbers as\nmany 'good all-round men' as could be taken at hazard from either\nof the services, military or naval\u2015the metropolitan young-lady vote\nnotwithstanding. Our ball yet had some distinctive features. Many of\nthe irreproachably attired persons, there and then present, had spent\nthe day in avocations which do not in a general way precede ball going.\nJack Hardston had worked his own eight hours' 'shift' that day, from\n8 a.m. to 4 o'clock in the afternoon, in a 'drive' of considerable\nlateral penetration, at a distance of 160 feet from 'upper air.'\nAfter a light repast, a smoke, a swim in the Turonia, and a somewhat\nprotracted and hazardous toilet, he asserted himself to be wound up\nexactly to concert pitch. Twice as fit indeed as when he carried the\nmoney of the men for the grand military pedestrian handicap. Mild\nlittle Mrs. Wynne had treated herself to the ball on the strength of\nLloyd Watkyn having come 'on the gutter' in his claim at Jumper's\nGully in the early part of the week. So she finished up her baking and\nbrewing, let us say, and having handed over the three-year-old Watkyn\nWilliams, with many injunctions, to her neighbour Mrs. David Jones\n(also of the Principality), proceeded with her husband, 'dressed for\nonce like old times,' as she said with a little sigh, to the hall of\nthe great enchanter\u2015even music\u2015who hath power over body and soul, life\nand limb; who with a chord can call forth the tears of the past, the\njoys of the present. And very nice they looked.\n\nHorace Sherrington was there\u2015suave, correct, rather worn-looking,\nbut incontestably 'good form.' He made a handsomer income by the\nexercise of his talents than those somewhat varied natural gifts had\never previously afforded him. Every evening he came to the camp mess,\nwhere the Government officials kept something like open house for all\npleasant fellows who were 'of ours' in the former or the latter time.\nNo one sang so good a song as Sherrington, was so racy a _raconteur_,\nplayed a better hand at whist, had a surer cue at pool. But no one knew\nprecisely how he spent his day, not that any one cared much. There were\ntoo many men of mark who had tried every employment on that goldfield\nfor luck and honest bread, including the officials themselves, for\nthem to affect any snobbish discrimination of avocations. But Horace\ndid not volunteer the nature of his daily duties; he was not a miner,\na speculator, a reefer, nor an engine-driver, a clerk, or puddler.\nHis reticence piqued them. One day the police inspector's horse shied\nat a man in a loose blue shirt and very clay-stained general rig,\nhaving also an immense sheaf of posters in his hand. 'What the devil\ndo you mean, my man, by flourishing these things in my horse's face?'\ngrowled the somewhat shaken autocrat. 'Beg your pardon, sir,' quoth\nthe agent of intelligence, himself passing on. But it was too late.\nThe lynx-eye settled upon him with unerring aim, like a backwoodsman's\nrifle. Both men burst out laughing. The elegant and accomplished Horace\nwas a bill-sticker! The festive concourse partook, in one respect at\nleast, of classical and traditionary fitness. The sincere and fervid\nworshippers of Terpsichore held sacred revel in a temple\u2015the Temple of\nJustice! For the large handsomely decorated hall, which resounded with\nthe inspiriting clangour of a very passable brass band, was in good\nearnest the court-house of Turonia. By the simple process of removing\nthe dock and draping the witness-box as a lamp stand, placing the\nmusicians upon the magisterial bench, with, I hardly need to mention,\na profuse exhibition of international bunting, a fairly ornamental and\nhighly effective ballroom was secured.\n\nIt was generally believed, and indeed asserted by the _Turonia\nSentinel_, that the Commissioner, who was known to be _beau valseur_,\nhad bribed the contractor, when completing that magnificent edifice, to\nbestow extra finish upon the flooring, with ulterior views as to its\nutilisation for society purposes. Be that as it may\u2015and much gossip\nwas current about that high and mighty official of which he took no\nheed\u2015there _was_ some truth in a subsequent legend that a prisoner\nand the constable by whom he was being escorted to the dock on the\nfollowing morning slipped and fell as heavily and unexpectedly upon the\nglassy floor as if they had been essaying the gliding graces of the\nrink for the first time.\n\nWhen the Branksome Hall party drove up, the entertainment had\ncommenced, and the two first dances having been got through, the _g\u00eane_\nof all beginnings and early arrivals was evaded. The ladies having been\nfirst conducted for envelope-removing purposes into the jury-room, and\nthe men's overcoats and wideawakes deposited in the land office, the\nstewards with elaborate courtesy escorted them to the hall of dazzling\ndelight.\n\nThe Commissioner, in blue and gold (at that period of Australian\nhistory these officials wore uniforms), looked most military\nand distinguished, his heavy drab moustache and decided cast of\ncountenance suiting the costume extremely well. The second steward\nwas a broad-shouldered, blonde, blue-eyed personage, whose singular\ntalent for organisation caused his services to be in great request at\nall public demonstrations\u2015social, military, legal, or ecclesiastical.\nHe looked like a squatter or a naval man, but was in reality a bank\nmanager. The third steward was a tall handsome man, very carefully\nattired, whose delicate features were partly concealed by an immense\nfair beard. His manner, his mien, his every look and gesture, told\nas plainly as words to any observer of his kind of foreign travel,\nof 'the service' in early life of that occasional entire dependence\nupon personal resources which has been roughly translated as 'living\nby his wits.' On his brow was the imprint writ large, in spite of\nthe faultless toilet, finished courtesy, the perfect _aplomb_, the\nhalf-unconscious _fiert\u00e9_ of his manner, the somewhat doubtful\n_affiche_ of adventurer.\n\nAttended by these magnates, for whom way was made with ready respect,\nthe Hall party sailed into the well-lighted, well-filled room with\nconsiderable prestige.\n\nErnest was considerably astonished at the general appearance of\nmatters, while the Colonel openly expressed his admiration and\nsatisfaction.\n\n'Gad, sir!' he said to the Commissioner, 'I had no idea that you were\nable to get up your dances in this fashion. What a field of neat\nwell-bred-looking flyers\u2015I mean deuced pretty girls, and monstrously\nwell dressed too. Puts me in mind of one of our Hurryghur dances. We\nused to have such jolly spurts at the old station before that cursed\nMutiny spoiled everything.'\n\nMr. Neuchamp thought it was not so very much less imposing in\nappearance than a ball in Sydney; room not so big; perhaps a trifling\nflavour of the provinces.\n\nBut the Bombay galop having struck up, the Colonel possessed himself of\na partner of prepossessing appearance, through the good offices of the\nCommissioner, and sailed off at a great pace. Ernest lost no time in\nappropriating the eldest Miss Branksome, and reflection was merged in\nsensation.\n\n'I suppose you hardly expected to have any ball-going in this\nparticular spot,' said he to his partner, 'a few years ago.'\n\n'We should just as soon have expected to go to the opera and hear\nTietjens,' said Miss Branksome. 'I have often ridden over this very\nspot with papa, and seen the wild horses feeding on the hill where the\ntown now stands.'\n\n'And you like the change?'\n\n'I can't say that we did at first. We fancied, I suppose, that the\ngreat invading army of diggers would eat us up, and we resented their\nintrusion. But they turned out very amiable wild beasts, and one\nadvantage we certainly did not calculate upon.'\n\n'What is that, may I ask?'\n\n'The number of nice people that would accompany the army. Our society\nis ten times as large and pleasant as in old times. We are hardly a\nnight without quite a small party of visitors. You see there are the\ncommissioners, magistrates, bankers, and other officials, all gentlemen\nand mostly pleasant. Besides, the gold attracts visitors, like\nyourself, for instance.'\n\nAfter a very satisfactory fast and unaffectedly performed galop, the\nsusceptible Colonel joined them at the refreshment table, accompanied\nby a young lady with a wild-rose complexion and great dark eyes, who\nhad been evidently dancing at a pace which had caused that mysterious\nportion of her _chevelure_ known (I am informed) as 'back hair' to\nfall in glossy abundance over her fair shoulders.\n\n'Splendid floor, Bessy,' he said to his niece. 'Capital music\u2015partner\nbeyond all praise!' (Here the young lady looked up with smiling\nreproach.) 'Fact! haven't had such a dance since the last ball at\nCalcutta. There were two duels next day\u2015about a young lady, of course'\n(here the small damsel looked much concerned)\u2015'and poor O'Grady, who\nhad heart complaint, but couldn't control his feelings at a ball, died\nwithin the week.'\n\n'Oh, how dreadful!' said the little maiden, with a sincere accent of\ndistress. 'But nobody dies after a ball here, or fights duels either,\nthat I ever heard of. Why should they in India, Colonel Branksome?'\n\n'Can't say,' said the Colonel. 'Let me give you a little champagne;\nheat of the climate, I suppose; too many soldiers, too few ladies.'\n\n'India must be a beautiful place, Colonel Branksome,' observed the\ngrave little damsel, looking out of her big eyes with an air of\ndeliberate conviction.\n\n'Glorious, splendid; that is, most infernal hole\u2015hot, dull,\nmiserable\u2015full of s. Hope I may never stay another year in it.\nGet my pension, I hope, when I get back and settle up with the remount\nagent. After that, if they ever catch Billy Branksome out of England\nagain, they may make a Punkah-wallah of him.'\n\n'Good gracious, Colonel Branksome!' said the matter-of-fact danseuse,\nwho now looked as cool as if she had been walking a minuet. 'I thought\nall soldiers were fond of India. Oh! there's that dear old Captain de\nBracy.'\n\n'Gad! so it is,' said the Colonel. 'Look at him, Bessy, strolling in,\nand bowing to every woman he knows, as if he was at a ball at the\nTuileries. Gad! I _did_ see him there last. And what do you think\nhe was doing?\u2015why, dancing in a set with two crowned heads and four\nprincesses of the blood. He and Charles Standish made up the set; by\ngad!'\n\n'Oh, doesn't he look like a nobleman?' said the _debutante_\nenthusiastically, opening her innocent eyes and feasting on De Bracy's\nmiddle-aged charms. 'And oh, what lovely, wonderful studs!'\n\n'So you're here, Master Billy, as usual?' said the object of this\nhighly favourable criticism. 'Couldn't keep away from a ball if your\nlife depended upon it. Old enough to know better, ain't he, Miss\nMaybell? Happy to see you all here to-night. Not afraid of the stumps\nand holes? I'm well enough, thanks, Miss Maybell; heard _you_ were\ncoming, and though I seldom go out now\u2015I am here.'\n\n'Oh, Captain de Bracy!' said little Miss Maybell, perfectly overwhelmed\nwith the compliment to her unworthy small self (as she erroneously\nheld, underrating her fresh and innocent beauty), and mentally\ncomparing De Bracy's appearance with that of a print of the Chevalier\nBayard which was among her treasures at home.\n\nA great tidal wave of promenading couples overwhelmed and dispersed\nthe _partie carr\u00e9e_ for a while, so that they were compelled to make\narrangements for the next dance, which happened to be a _deux-temps_\nwaltz. Having relinquished Miss Branksome to De Bracy, and seen pretty\nlittle Miss Maybell carried off by young Tom Branksome, who recommended\nhis uncle to try Mrs. Campion, as being a fine woman and of a suitable\nage, Ernest found, rather to his surprise, that he was a little late,\nas every possible partner for a fast dance had been secured. The fact\nwas, that the proportion of the sexes was in the inverse ratio to\nwhat generally obtains at balls in a more settled state of society.\nTherefore, more than average alacrity and foresight was necessary to\nensure a regular succession of partners.\n\nAs Mr. Neuchamp, smiling to himself at his involuntary state of injured\nfeelings, sauntered towards the refreshment room, he met the steward,\nwho had been introduced to him by the Commissioner as Mr. Lionel\nGreffham.\n\n'You don't seem to be dancing,' he said; 'well, it is rather a bore,\nafter the first turn or two. Bright and I are having a glass of\nchampagne; will you join us?\u2015it is \"number two.\"'\n\nThere was such an evident desire to be civil on Mr. Greffham's part\nthat Ernest, who had not at first regarded him with perfect approval,\nfelt moved to respond to so friendly an accost. He found Mr. Bright in\nthe supper room, in conversation with a well-dressed, quiet, but not\nthe less striking-looking personage, who was introduced as the district\ninspector of police, Mr. Merlin.\n\n'What do you think of society on the diggings?' said Mr. Bright to\nErnest; 'hardly what you would have expected?'\n\n'It is utterly wonderful,' said Ernest. 'I am perfectly amazed at the\norder and decorum which everywhere prevail, and even at the elegant and\nenjoyable party to-night\u2015so many nice people you seem to have.'\n\n'Yes,' said Mr. Merlin, 'nothing is more wonderful, as you say. There\n_are_ so many extremely nice people here. So well worth knowing. People\nwho have such noble, disinterested views, eh, Greffham?'\n\n'I quite agree with you,' answered that gentleman. 'But it's rather a\nbore we can't have a little whist, isn't it? A quiet rubber, or a game\nat billiards, would be much more sensible than all this capering with a\nlot of people that, in any other part of the world, you wouldn't dream\nof speaking to.'\n\n'Surely not,' said Ernest; 'some of our friends here are of\nunimpeachable _ton_, and for the rest they appear to be of very fair\naverage standing. I am very much pleased with the whole affair.'\n\n'Greffham is fastidious, and plays the Sybarite among his other\ncharacters,' said the inspector slowly and distinctly. 'He suffers much\nhere when the rose leaves are unavoidably crumpled. So much depends\nupon a man's antecedents.'\n\n'I don't know that I am more fastidious than others,' he said, smiling,\nthough the eye, that infallible referee in facial expression, did not\nagree with his amused expression. 'You know that _you_, Master Merlin,\nrather agree with me than otherwise. But seriously, suppose we go over\nto the Occidental and have a game of billiards. Oceans of time; these\nmisguided Turonians will dance for hours yet.'\n\nThe proposition met with general approval, and Mr. Neuchamp assented,\nnot that he cared about billiards, at which he was only a middling\nperformer, but he felt the inexplicable influence of the strange scene\nand novel surroundings, and was more inclined than ordinarily _desipere\nin loco_.\n\nThe four acquaintances crossed the street, which was filled, as far as\nthey could see, with a surging crowd of men, chiefly attired in the\nordinary dress of miners. Shops brilliantly lighted, and of imposing\nappearance as to their fronts, lined the long, narrow, and not\naltogether straight street. Mr. Neuchamp thought he had never seen\nsuch an assemblage of intelligent-looking men. Evidently the flower of\nthe working classes, while from all the trades and professions a large\nproportion had been lured to Turonia by the golden possibilities of\nthe great rush. What amazed Ernest chiefly was the astonishing order\nand polite behaviour of this vast concourse of people, containing\npresumably the ruffianism of all lands under the sun. He had seen mobs\nin the British towns and cities and in other parts of the world. In\nall these gatherings he had occasionally encountered rough usage, had\nheard much foul language, and had suffered risk or loss of personal\nbelongings.\n\nBut in this strange crowd no conduct other than of mutual respect\nand courtesy was observable. Rarely a word to which objection could\nbe taken fell on the ear. The press parted and permitted the four\ngentlemen to walk through as independently as though they were the\nDowager Patroness at a charitable institution. The brilliantly-lighted\nbars at the numerous hotels were certainly full, but there seemed to be\nmore talking than consumption of liquor, and the spectacle of drunken\nmen was altogether absent. A few police constables, unobtrusively\nplaced, denoted that the Imperial Government, so calm, so impartial,\nyet so long of arm and sure of grasp, was represented. Otherwise it\nlooked very much as if the great heterogeneous mass of humanity, now\nturning up the precious metal at Turonia at the rate of a couple of\ntons of gold per quarter, was permitted to manage itself. This was by\nno means the case, as Mr. Merlin could have explained. An unsparing\ncrusade was organised against all manner of open vice and crime.\nNo quarter was given or respite permitted. Passing through the bar,\namong the occupants of which Ernest did not observe any one to carry a\nrevolver, or to make as though the good-humoured landlord was likely\nto be, without notice, 'one of the deadest men that ever lived,' they\nreached a large, well-lighted room, where two handsome new billiard\ntables were in full swing. As they sat down on the cushioned benches\nwhich lined the room, a young fellow in a blue shirt and clay-stained\ntrousers made a break of twenty-seven, and thereby won the game in\na style which showed that he had not devoted all his life to mining\nindustry. The marker promptly signalled to Mr. Greffham. He and Ernest\nthen took possession of the vacated table.\n\nThere is no doubt that at certain times an electrical tone pervades not\nonly the physical but the moral atmosphere, affecting to depression or\nexaltation the mind of man, that subtle reflex of the most delicate\nexternal influence. Such a night was this. The music of the band\nwas pealing from the opposite side of the street\u2015the vast, surging,\nexcited, but self-contained crowd presented the strangest contrasts of\nsociety, as akin to the rudest types of life in certain aspects, so\nnear to Utopian models in advanced manners and intelligent consent.\nEven the scraps of conversation which found their way to Ernest's ear\nwere of a novel and fairy-legendary nature.\n\n'Made eight hundred pounds in ten days out of that bit of \"surface,\"\nJem did; I sold a share in Green Gully, No. 5, for three drinks\nlast week, and now they've struck gold and want a thousand for it.\nCommissioner settled that dispute to-day at Eaglehawk.'\n\n'Who got number seven block?'\n\n'Well, Red Bill, and his crowd; it's on good gold too.\n\n'What did Big George say?'\n\n'Oh, he was pretty wild, but he couldn't do nothing, of course.'\n\n'I'll take three hundred and half out of the ground for a share in\nnumber two,' and so on, and so on.\n\nMr. Neuchamp had come on to the long-disputed territory, 'Tom Tidler's\nground,' and the 'demnition gold' (if not silver) was sticking out of\nthe soil everywhere. Ten-pound notes were handed across the bar for\nchange as readily as half-crowns. Nuggets worth from \u00a350 to \u00a3100 were\npassed about in the crowd for inspection with the most undoubting good\nfaith and confidence in the collective honesty of mining mankind.\n\nUnder these conditions, it was a night for bold and reckless\nconception, a night when the ordinary prudences and severities of\nconscience might be calmly placed behind the perceptions, and the\n'fore-soul' be permitted to leap forth and disport in the glorious\nfreedom of the instincts and original faculties.\n\nNo sooner had Ernest handled his cue and struck the first ball than\nhe perceived that he was in one of his rarely happy veins, when, sure\nof his play, he was also likely to fall in for an unusual allowance\nof 'flukes.' Therefore, when Greffham, who had kindly allowed him ten\npoints, proposed to have a pound on the game, just for the fun of the\nthing, he promptly acceded.\n\nHe won the first with ease, Mr. Greffham playing a steady but by no\nmeans brilliant game. And, much to his astonishment, the second also,\nwith a couple of pounds which he had staked, with the good-natured\nintention of giving back Mr. Greffham his money. Ernest did not win the\nsecond game quite so easily, but his luck adhered to him, and a shower\nof flukes at the latter end landed him the winner. His antagonist bore\nhis defeat with the finest breeding and perfect composure, deciding\nthat it was quite a pleasure to meet with a gentleman in this howling\ndesert, socially, who _could_ play, and trusting that they might have\nanother game or two before Ernest left the district. Then Mr. Bright\nand the inspector had a short but brilliant game, chiefly remarkable\nfor the sparkling, if somewhat acidulated, repartee which it called\nforth. Then it was voted proper to return to the ballroom. Here matters\nhad apparently reached the after-supper stage. The dancing was more\ndetermined, the floor smooth to the last degree of perfection. De\nBracy, the Commissioner, the Colonel, and the Branksome Hall party were\nstill untired, unsatiated\u2015the cheeks of the young ladies showed paler\nin the growing dawn-light, their eyes larger and more bright, and the\nhair of little Miss Maybell positively 'would not keep up, and there\nwas no use trying to make it.' Ernest was just sufficiently fortunate\nto capture Miss Janie Campion for the galop, which proved to be the\nconcluding one as far as he was concerned. For old Mr. Branksome, not\nbeing quite so fond of dancing and young ladies as his gallant brother,\nordered the phaeton round, and caused his daughters to perceive that he\nwished to go home, without any kind of doubt or hesitation.\n\nSo all wraps being secured, and the Colonel having taken a most tender\nleave of his last partner, the highly-conditioned horses went at their\ncollars, and, after threading the unabated crowd, rattled along the\nsmooth if winding track, by stumps, ditches, and yawning shafts, at a\npace which, with luck and good driving, brought them in due time safe\nand sleepy to the avenue gate of Branksome Hall.\n\nOn the following morning Ernest received a letter from Charley\nBanks, by which he learned that his party would not arrive in the\nneighbourhood of Turonia for at least another fortnight\u2015their advance\nbeing unavoidably slow. He cheerfully concluded, therefore, to spend\nthe intervening time in the golden city, where he would have an\nopportunity of noticing the preparations for mustering the herd, in\nwhich he and Mr. Levison were jointly interested, and of acquiring new\nfacts in a tolerably new field of observation.\n\nHe therefore took temporary leave of his very kind friends at the Hall,\nreserving to himself the right of occasional visits until he should\ndepart, with his newly-acquired herd, for the 'waste lands of the\nCrown,' where the Great River flowed on, as in the long lonely \u00e6ons\nof the past, through the vast plains and pine-bordered sandhills of\nRainbar.\n\nOnce domiciled in Turonia, Mr. Neuchamp found its society more various\nand entertaining than in any locality other than the metropolis which\nhe had visited since his arrival in Australia. It was the flush and\nprosperous stage of a great alluvial goldfield. All things wore the\ngolden tint, all bore the image and superscription of the modern C\u00e6sar\nand Imperator.\n\nWonderfully unreal, and smacking of 'the golden prime of the good\nHaroun Alraschid,' was the careless magnificence with which large sums\nof money were acquired, spent, lost, and regained. Ernest visited the\nvarious banks, and saw bags and drawers in which the precious metal\nlay heaped in all forms, from the dull red heavy dust to the lump,\ningots, and precious fragments, in which a thousand pounds' worth was\nlifted in the hand with as much ease as a paperweight. He saw the\nbronzed, stalwart miners handing insignificant-looking bags across\nthe cedar counters, and crushing handfuls of ten-pound notes into\ntheir pockets as a schoolboy receives change for a shilling spent in\nmarbles. He retired to rest about midnight, and on awaking at dawn\nheard the ceaseless click of the billiard balls in the adjacent saloon,\napparently fated to go on until the day again merged into midnight.\nHe found the _table d'h\u00f4te_ every day filled, not to say crowded, by\nwell-dressed people whose occupation he could at first merely guess at,\nbut whom he found to be in nearly every case connected with the great\nindustry\u2015as officials, mine-owners, brokers, speculators, professional\nmen, and others unspecified, with perhaps a rare tourist, lured hither\nby astounding rumours, or a feeling of justifiable curiosity, to behold\nthe unbounded treasures of mother earth, so long and jealously guarded.\nThere was a never-failing store of amusement and occupation spread out\nbefore the calibre of Mr. Neuchamp, and so absorbed did he grow daily\nin the ever-widening field of observation that he felt almost regretful\nto find the time at his disposal rapidly diminishing.\n\nIn no more friendly and hospitable region had he ever sojourned. He was\nvoted an acquisition by the officials, and made free of all their small\ngatherings and merry-meetings. One day the Commissioner would drive\nhim out to inspect a great sluicing claim, where the water, brought\nthrough races by miles of fluming, spouted clear and strong over heaps\nof auriferous earth, as when it left its far-away mountain rill. On\nanother occasion he was invited to witness the hearing and settlement\nof a great mining dispute, 'on the ground,' where a thousand excited\nmen were gathered\u2015the evidence heard upon oath, and the immutable\ndecision of the Lord High Commissioner given, by which the one moiety\nwas deprived of all right to a presumable fortune, and the other gifted\nwith a clear title to the same. Much temporary excitement and even\nirritation was produced by each and every such verdict. But miners,\nas a rule, are a law-abiding body; and, the mining laws of the period\nbeing as those of the Medes and Persians, all effervescence, however\napparently allied to physical force, rapidly subsides.\n\nIn the intervals of such experiences and recreations, Mr. Neuchamp\ndid not abstain from joining in diurnal billiard tournaments, and\nthe nightly whist parties, in which trials of chance and skill he\ninvariably found himself associated with Mr. Lionel Greffham and\nother pleasant persons, who, appearing to have no visible means of\nsubsistence, were invariably well dressed, well appointed, and well\nprovided with the needful cash. Mr. Greffham constituted himself his\nconstant companion and mentor; the charm of his unremitting courtesy,\njoined to varied and racy experiences, with a never-failing flow of\nentertaining conversation, gradually broke down Ernest's caution and\nreserve. They became, if not sworn friends, habitual acquaintances,\nand under his apparently disinterested guidance the time passed\npleasantly enough. Yet Ernest began to perceive that, after the first\nfew successes, his losings at cards and billiards commenced to add up\nto more serious totals than he had thought possible at the commencement\nof his sojourn at Turonia.\n\nMore than once Ernest fancied that the keen eyes of Mr. Merlin wore\na depreciatory, not to say contemptuous, expression when fixed upon\nMr. Greffham. The Commissioner evidently disapproved of him in a\ngeneral way, and Mr. Bright, who was open and bold of speech, once\ntook occasion to remark, _\u00e0propos_ of the elegant but inscrutable\nLionel, that he considered him 'to be a d\u2015d scoundrel, who would stick\nat _nothing_ in the way of villainy, if he had anything considerable\nto gain by it.' But at this stage Ernest, at no time of a distrustful\ndisposition, had formed an estimate of this fascinating freelance too\nfavourable to be shaken by mere assertion unsupported by proof.\n\nOne morning, for some reason, an unusually large amount of gold and\nnotes was despatched from one of the banks, with the object of meeting\na branch escort a day's ride from Turonia. Two troopers were detached\nfor this service. They carried the compact but precious burden before\nthem in valises strapped to their saddles.\n\nA small group of _habitu\u00e9s_ of the Occidental assembled to witness\ntheir departure, and Mr. Neuchamp bestowed much commendation upon\nthe condition of the horses, the efficient appearance of arms and\naccoutrements, and the soldierly and neat appearance of the men.\nCurious to remark, Greffham was not among the admiring crowd, and\nErnest alluded to the fact to the Inspector of Police, who was\nofficially present.\n\n'What has become of Greffham?' he inquired. 'One would have sworn that\nwe should have seen him here!'\n\nMr. Merlin replied that 'Mr. Greffham was probably away upon business';\nbut a bystander volunteered the information that he had seen Mr.\nGreffham mounted, at daylight, upon his famous hackney Malakoff,\napparently on the road to an adjacent diggings.\n\n'Where can he be going, Merlin?' said Ernest. 'He arranged to drive me\nover to the Hall to-day.'\n\nMr. Merlin replied, stiffly, that Greffham had apparently changed his\nmind, and that he, Merlin, had not the slightest acquaintance with Mr.\nGreffham's business affairs.\n\nMr. Neuchamp felt quietly repelled by this answer, and the cold\nindifference with which it was given. He came to the conclusion that\nMerlin was unnecessarily formal, and by no means so pleasant an\nacquaintance as the absent one. He was not fated to recover from the\neffects of his matutinal disappointment.\n\nThe Commissioner was up to his eyes in court business that day. Bright\nwas unusually confined to his bank. Merlin disappeared on the trail\nof a cattle-stealer long and urgently 'wanted,' while every other\nmember of the waif and stray corps, from the police magistrate to\nHorace Sherrington, seemed to have been snatched away by the Demon of\nIndustry, or otherwise absorbed by abnormal influences. Long, dismal,\nand cheerless passed the hours of one of those broken, objectless days\nthat are so peculiarly, unaccountably depressing. It was long\u2015very\nlong\u2015since Ernest had spent so miserable a day. He regretted that he\nhad not carried out his intention of visiting the Hall. He wondered\nwhen Charley Banks would arrive, and sincerely longed once more for the\nabsorbing work of the muster and the march, telling himself that it\nwould be long before he spent so idle a season again. The evening at\nlength arrived, and with the gathering of the accustomed party at the\ndinner-table brighter thoughts possessed his mind. By the time that the\nevening game of billiards had fairly commenced, Mr. Neuchamp's equable\nhabitude of mind had reasserted itself.\n\nThey had not been long occupied with this fascinating exercise,\nwonderfully suited to so many shades of character, when Greffham\nlounged in, calm and _insouciant_, as usual. At the first opening in\nthe game he took his favourite cue and played his usually cool and\noccasionally brilliant game. If he had been in the saddle the long\nday through, no trace of more than ordinary exercise or excitement\nwas visible in the _soign\u00e9_ attire, which seemed a part of the man's\nbeing, or on his calm, impassive features. His play differed not in\nthe slightest degree from his ordinary form, which always showed\nimprovement towards the close, with perfect unconsciousness as to\nwhether he was apparently winning or losing the game. He made his\ncustomary break, and, betting upon a five stroke at the finish, gave a\nshade of odds upon the success of his concluding 'coup.' He spoke of\na longish ride as far as an outlying quartz reef, in which he had an\ninterest, and mentioned having encountered the two gold-laden troopers\nat an inn which they would pass towards the end of their day's journey.\n\nHalf an hour later on Mr. Merlin dropped in, by no means so calm in\nhis demeanour as Greffham, and full of complaints as to the abominable\nnature of the weather, the fleas, the dust, the danger of riding late\namong unprotected shafts, and many other disagreeables specially\nselected by fate for his deterioration and disgust on this appointed\nday.\n\nWhile in this unchristian state of mind, for which he was mildly taken\nto task by Greffham, he was called out by a waiter, who informed him\nthat 'a gentleman wished to see him.'\n\n'Oh, certainly,' quoth the unappeased official with sardonic\npoliteness; 'most happy, I'm sure. _I very seldom see one._'\n\nWith this Parthian shaft at the entire community, which was accepted\nas a perfectly permissible and characteristic pleasantry, Mr. Merlin\nquitted the room to greet the aforesaid rare and precious personage. He\ndid not return; and after a little unlimited loo, in which Mr. Greffham\ntransferred the larger portion of Ernest's ready money to his own\npocket, the company separated for the night.\n\nIt was moderately early on the morrow when Mr. Neuchamp presented\nhimself in the main street of Turonia. He was at once instinctively\naware that something strange had happened.\n\nThe ordinary life and labour of the busy human hive seemed arrested.\nMen stood in groups at the sides, the corners, the centres of the\nstreets, conversing in low tones with bated breath, as it seemed to\nErnest. The very air was heavy and laden with horror\u2015unexplained,\nmysterious\u2015until above the hum and confused murmurs came, ominous and\nunmistakable, the one darkest irrevocable word 'murder!'\n\nIt was even so. Mr. Bright, walking briskly down the street, accosted\nhim, and in the next breath asked if he had heard the news.\n\n'Very dreadful thing\u2015very,' said the sympathising banker, trying vainly\nto subdue his cheerful visage. 'Never had anything so terrible happened\nat Turonia since it was a goldfield. Merlin, Greffham, and I are going\nto ride out to the spot to-morrow. Would you like to come?'\n\n'With pleasure,' said Ernest; 'that is, I shall go as a matter of duty.\nBut what is up?'\n\n'Just this\u2015\u2015' said Bright. 'But surely you must have heard it?'\n\n'Not a word,' replied Ernest. 'Pray go on. I have suspected something\nwrong, but have not the faintest idea what it is.'\n\n'Henderson and Carroll,' said Bright solemnly, 'two of the men in the\nforce, the troopers that you saw start with the gold, were yesterday\nfound _dead_\u2015murdered, evidently\u2015near the Running Creek. All the gold\nand bank notes have been taken, and the police have no more idea who\nthe murderer is than you or I have. Have you, Merlin?' he asked of that\ngentleman, who now joined them.\n\n'Are there any bushrangers or bad characters known to be in the\nneighbourhood?' asked Mr. Neuchamp. 'I have always thought it a perfect\nmarvel that so little overt crime existed among this immense assemblage\nof men, with so many exciting causes. There must be _very few_\ncriminals, or else they keep very quiet.'\n\n'_We_ know of scores of men of the very worst class and most desperate\ncharacter,' replied Mr. Merlin; 'but, as you say, they have been kept\nvery quiet. Still it never does to relax caution, as, if a sufficiently\n\"good thing,\" in their phraseology, turns up, they are always ready\nto run all risks for the spoil. You have pushed against men who have\ncommitted more than _one_ or even two murders. I saw you talking to one\nthe other day by the Chinaman's store in Stanley Street.'\n\n'Good heaven!' said Ernest, much moved, 'you don't say so? And was that\nquiet, sober-looking man that I was chatting with\u2015I remember him quite\nwell now\u2015a known criminal?'\n\n'One of the worst we have,' rejoined the Inspector in a matter-of-fact\ntone. 'A cold-blooded, treacherous ruffian. He _dares_ not drink on\naccount of what he might let out; but we know where he has been and all\nabout him this time. He was not near the spot.'\n\nAt this moment a telegram was put into the Inspector's hand, which he\nread carefully and showed to Ernest.\n\n'Of course this is strictly confidential,' he said.\n\nThe telegram ran as follows:\u2015\n\n Notes traced, known to have been in the packet forwarded by escort.\n Arrest Jones.\n\n'This gives a clue, of course, but,' said the official with diplomatic\nreserve, 'we may or may not follow it up. Possibly we may be thrown\nout; but eventually I venture to think Mr. Jones will be run into in\nthe open.'\n\n'Arrest Jones,' repeated Mr. Neuchamp. 'And have you been able to\nsecure him?'\n\n'I don't know whether the police have got hold of him yet,' said Mr.\nMerlin cautiously; 'but I daresay we shall be able to give an account\nof him by and by. If not, he will be the first man who has got clear\noff since this goldfield was discovered.'\n\n'In the meantime you are going out to view the scene of the murder and\nthe bodies of these poor fellows just as a matter of form and for your\nown satisfaction?'\n\n'Precisely so,' assented Mr. Merlin; 'principally as a matter of form.'\n\n'And Greffham is going with us just for company, like Bright, to make\nup the party, I suppose?' continued Ernest. 'It is very good-natured\nof him, for he told me yesterday that he had some important business\nto-day, and that he would not be about the town. But I have always\nfound him most obliging.'\n\n'So have I, most obliging, as you say. The fact is, he knows the spot\nexactly where these poor fellows must have been met.'\n\n'But that Jones,' said Ernest eagerly, 'what a ruffian! what a\ncold-blooded villain he must have been! How I should like to fall\nacross him. I could cheerfully go to see him hanged.'\n\n'Perhaps you may have that gratification yet,' replied Mr. Merlin with\na grim smile. 'More unlikely things have happened. Hallo! here comes\nGreffham.'\n\nThe gentleman referred to now sauntered up, accurately turned out in\nquite the best boots and breeches which Ernest had seen since he left\nEngland. His hunting scarf was adorned with the regulation Reynard\nbrooch, and from throat to long-necked, heavy polished spur he was\naltogether _point-device_.\n\nHe looked a shade paler, probably from the effect of his yesterday's\nlong ride, but his smile was as ready, his repartee as incisive, as\never, while his light-blue eye fell with its usual glance of cold\nscrutiny upon the advanced guard of the party.\n\n'What a fellow you are, Merlin,' said he, 'starting at this unearthly\nhour. Why didn't you give a man a chance of a little sleep, who had,\nwhat you never get, a day's work yesterday?'\n\n'My dear Greffham,' replied the Inspector with irresistible urbanity,\n'I was certain that you and Bright would enjoy the fresh morning air\nabove all things. I know he's a terribly early riser, and you can wake\nwhen it suits you; so I determined, under the circumstances, upon an\nearly start.'\n\n'All right,' quoth Bright; 'I don't care how early you get away. It\ncan't be too early for me.'\n\n'And besides, Greffham,' said Merlin, 'you know the short cut to\nRunning Creek, which not every one can find. I propose to stay the\nnight at the Ten-Mile Inn, and to make for the scene of the murder next\nday.'\n\n'Come on, then,' said Greffham harshly; 'what the devil are we standing\nprating for? If you are in such a cursed hurry why don't you get away\ninstead of standing here burning daylight?'\n\n'We were waiting for Markham,' said Merlin good-humouredly, 'but I\ndaresay the old fellow will pull up. Come along, then. I'm awfully\nobliged to you for coming, Greffham; I am indeed!'\n\nMr. Neuchamp had before remarked the extreme readiness of most people\nupon the goldfield to accede to any wish expressed by Mr. Merlin, and\nhe recurred to it for the edification of Mr. Greffham, citing it as an\ninstance of the very remarkable courtesy of manner which, as he was\nnever tired of noting, distinguished the inhabitants of the settlement\nof Turonia.\n\nGreffham listened in silence to Ernest's philosophical utterances, and,\nlighting a cigar, rode steadily forward. Here Ernest was impressed with\nthe fact that, as a party, they were unusually well armed, as also\nwell mounted. The four troopers, one couple of whom rode in front as\nscouts, while another pair followed at easy distance, had each a Snider\ncarbine. A 'navy' revolver hung at each man's belt. Their horses were\nuncommonly well bred and in really good condition. Merlin, of course,\nnever by any chance stirred without his revolver; and he was on his\nfavourite Arab hackney, Omar Pacha, an indomitable gray, of proverbial\npace and endurance. Mr. Bright had two revolvers, beside a pocket\nDerringer, which latter had a trick of going off unexpectedly, and\nhad once 'made it hot' for a friend and brother banker. Greffham was\napparently unarmed, but he never permitted any one to know more than\nhe wished even in the most trifling matters. He was an 'ace-of-clubs'\nman with the pistol, and, had duelling been fashionable at Turonia, he\nwould no doubt have distinguished himself after much the same fashion\nas the hard-drinking 'blazers' of the Wild West a hundred years agone.\n\nBefore they had gone half a dozen miles they were overtaken by a\nsquarely built man on a bay cob, who interchanged a hasty but hardly\nvisible signal with Mr. Merlin, and fell into the rear. The newcomer\nwas a clean-shaved, Saxon-looking person, not very unlike a snug\ntradesman. He made an ordinary remark or two to Greffham and then\nsubsided into obscurity. _He_ also was well armed, and bore himself in\na quietly resolute manner that impressed Mr. Neuchamp much.\n\nThe day was hot, the road sandy, and, as it appeared to Ernest,\nmore tiresome than bush roads of similar nature were apt to be. The\nconversation, which had been general and well sustained at first,\nfell off gradually, until each man rode silently on, fanning the\nflies from his face, and apparently becoming more irritable, hot, and\nuncomfortable as the day wore on.\n\nThe only exception to this result of the tedious wayfaring was Mr.\nMerlin. He apparently did not suffer in temper, spirits, or natural\ncomfort from the exigencies of the journey. He kept up an even flow\nof conversation with Greffham and Bright, albeit the former chiefly\nanswered in monosyllables, and the latter freely cursed the road, the\nday, the flies, and the unwarrantable and misplaced sympathy which had\ncaused him to accompany the expedition.\n\nBut the day drags on, whether the stormy north refuses the traveller\nthe sight of the sun, or the languid south bestows too much of that\nindispensable potentate. The welcome coolness and dim shades of eve\nhad commenced when the wayside inn was reached, the last roof shelter\nwhich the dead had known, where they had quaffed their last draught\nand possibly told their last jest. On the bank of a creek at some few\nmiles' distance they had determined to make their camp, preferring\nit for some reasons to the inn. And there they had found their last\nresting-place.\n\nErnest remembered noticing the care and completeness which marked\nthe men's equipment, their muscular, well set-up figures, their easy\nseats as they rode their high-constitutioned, well-bred horses up\nthe street on the morning of their departure. And now they lay prone\nand motionless among the thick withering grass; above them waved the\nmelancholy, sighing casuarina, from the branches of which croaked\nthe raven\u2015far-scenting herald of doom, sable watcher by the dead. As\nhe thought of the manly, pleasant faces he could recall so easily,\nbut of yesterday, as it seemed, the strongest feelings of wrath and\nhatred were stirred within him, and he muttered an imprecation of\nswift vengeance upon the head of the cold-blooded assassin Jones, if\nthat indeed were the name of a wretch unfit to cumber earth. The sad\nsurroundings, the gloomy tone of Mr. Neuchamp's thoughts, did not lead\nhim to decline the respectable meal to which he found himself bidden\nalong with the gentlemen of the party.\n\nMarkham and the troopers occupied another apartment, in which they\nmade themselves fairly comfortable. The horses were stabled, and, save\nfor the inevitable death-scene of the morrow, the evening would have\npassed not uncheerfully. As it was, however, Mr. Merlin organised\nwhist, and even encouraged a little quasi-gambling by proposing higher\nstakes than usual. The chief result of which was that Mr. Neuchamp,\nhaving the experienced Lionel Greffham for a partner, won more money\nthan he had lost in many an unsuccessful night in Turonia. In vain did\nBright and Merlin 'plunge' by way of recouping their losses. The luck\nof Mr. Greffham was altogether too good; and Merlin, about midnight,\ngave in, saying, 'You have the devil's luck, as usual, Greffham. I\nwonder how long it will stick to you.'\n\n'Who knows?' answered he indifferently, ringing the bell and ordering\nrefreshment on a liberal scale. 'It has held on pretty well so far. It\nmay turn, though, and then I think I could find a bullet for myself and\na quiet couch.'\n\n'Really now, my dear Greffham,' said Merlin, 'if I did not know\nyou well, I should think you were threatening what no man of sense\never puts into practice. But I have seen luck stick to a man until\nthe actual and inexorable finale. Then he and all the world had to\nacknowledge that they had been mistaken\u2015more mistaken\u2015most mistaken\u2015in\ntheir previous calculations and investments. Don't you think we could\nmanage another whisky before we turn in? I must have my smoke, anyhow.'\n\nErnest thought this, for him, unnecessary, and fell back upon\nsoda-water; but Greffham, apparently, was disinclined for immediate\nretirement. He and Merlin sat up long, telling apparently never-ending,\nhalf-forgotten tales, and smoking furiously.\n\nAs Mr. Neuchamp, restless and feverish, chose to get up at dawn and\npace the verandah, he saw Markham and Merlin holding colloquy in low\ntones, amid which he involuntarily caught the sound, on Markham's side,\nof the words 'all right.'\n\nShortly after the sharply disciplined troopers were astir at stable\nduty, and at sunrise the whole party were on their way to the fatal\ncreek.\n\nBright and himself, Mr. Neuchamp thought, looked the freshest of\nthe party, having had a few hours of sound sleep. Merlin's spirits\nwere high, as on the previous day. Greffham looked if anything more\nindifferent, more calm and careless about all earthly concerns, his\nfellow-creatures in particular, than usual.\n\n'It was by this track, round this very clump of pines, that you saw the\nmen ride off, Greffham?' said Merlin. 'It is quite fortunate that you\nshould be in a position to state your impression at a time which could\nnot have been many hours before their deaths. How did they look? Do you\nthink they had been drinking?'\n\n'Can't say,' answered Greffham after a pause, as if trying to recall\nthe exact circumstances. 'Carroll was a reserved, sulky-looking beggar,\nI always thought; one of those men that you could not tell liquor upon\nas long as he could keep his legs. Now I think of it, they did look\nrather stupid.'\n\n'You are quite correct about Carroll, old fellow,' said Merlin airily;\n'he _was_ reserved and taciturn, a ridiculously unsuitable habit of\nmind for a subordinate. Odd thing that nothing has been heard of the\ngold or notes.'\n\n'I suppose whoever took them,' said Greffham\u2015'(try one of these cigars,\nlittle Seguadil sent me a box)\u2015whoever took them had sense enough to\nconceal them for a while. The gold will turn up eventually.'\n\n'But not the notes, you think?' persisted Merlin.\n\n'Not unless there is something uncommon about them\u2015(this cigar won't\ndraw)\u2015numbers taken and so on. If they are the ordinary well-thumbed\npaper-promises current at diggings, they will be hardly identifiable.'\n\n'Very likely you are right. Deuced good cigar that. I wish the little\nbeggar would send me some of that Amontillado of his; that and his\nManzanares might really have come out of the King of Spain's cellar,\nas he used to aver. But the road improves now, we may as well canter.\nFamous horse of yours, Greffham, nothing like him in Turonia.'\n\n'Why, Merlin,' said Bright, 'what a heavenly temper we are in this\nmorning! Biliary secretions unusually right, I should say!'\n\n'Of course, Bright, of course; there's no credit to a jolly, sanguine\nfellow like you for being in a good temper. Nature in your case has\ndone so much that it would be the basest ingratitude if you did not\nsecond her efforts. Now spare fellows, like the elegant Lionel here and\nmyself, with whom indigestion is more the rule than the exception, only\nrequire to feel free from torment to be in the seventh heaven. But here\nwe are at the Running Creek. Look at the eagles already gathered.'\n\n\n\n\nCHAPTER XXIII\n\n\nA boding gloom seemed to fall suddenly like a pall from the branches\nof the sighing, whispering, sad-voiced water-oaks, as they followed\nthe winding track which led along the bank of the tiny streamlet to\nthe small alluvial flat, upon which lay two\u2015pah, what shall I say?\u2015two\nfigures covered with rugs, which may or may not have exhibited the\nhuman outline. 'They lay as dead men only lie.' A swarm of flies\narose at the lifting of the coverings, and a terrible and intolerable\nodour diffused itself around. 'Great God!' cried Ernest, 'are these\nrepulsive, fast-decaying masses of corruption all that are left of the\nhigh-hearted, gallant fellows I saw ride out of Turonia so short a\nwhile ago? Poor human nature, upon ever so slight summons, and must we\ncome to this! Accursed be the greed of the yellow gold which brought\nour brother men to so hideous an ending.'\n\nAs these reflections flowed from the sympathetic heart of Ernest\nNeuchamp with a natural force that could not be controlled, he turned\nin time to notice that Mr. Merlin had directed the coverings to be\nremoved from the corpses, and had instituted, in spite of their\nrevolting condition after forty-eight hours' exposure to a burning sun,\na thorough and searching examination.\n\nOne man, Carroll, lay on his side with face half upturned and arm\noutstretched, in the hand of which was grasped a revolver with a barrel\ndischarged. An expression of defiance was still legibly imprinted\nupon the features\u2015a bullet wound through the centre of the forehead\nhad without doubt been the cause of death. The strong man had fallen\nprone, as if struck by lightning, and for ever, ever more the wondrous\ninfinitely complicated machine was arrested. The soul had passed into\nthe region of endless life, death, sleep, sorrow, joy!\n\n'This man has been shot from the front, Greffham, shouldn't you say?'\npronounced the clear, incisive tones of Mr. Merlin. 'He may or may\nnot have been standing up to his assassin. If so, it was a species of\nduel, and the best shot and quickest had it. If you wouldn't care about\nstanding there, now, by that oak-tree, raise your arm, so; by Jove, you\nwould be just in the position that the man must been in that dropped\nthe poor sergeant.'\n\n'Just the sort of thing that Greffham would have gone in for if he was\nhard up,' said Mr. Bright, chuckling. He was reckless as to the flavour\nof his jests, far from particular if only they were 'hot' enough.\n\n'You are always thinking of that gold-buyer of yours that was\nshot, Bright,' said Greffham, wincing uneasily, though, under the\nconcentrated gaze of three remarkably steady pairs of eyes,\u2015Merlin's,\nBright's, and Markham's. 'It's my belief that Halliday shot himself; he\nwas something like you, in always carrying half a bushel of revolvers,\nand, like your battery, it went off accidentally sometimes.'\n\n'There's a boot mark in the sand underneath that oak-tree,' said\nMarkham, with great suavity; 'it's the very model of your track, Mr.\nGreffham, that you made there. Excuse _me_, sir.'\n\n'I suppose other people wear boots as well as I,' he said. 'Bushmen and\ndiggers are deuced rough, and all that, but they haven't come to going\nbarefoot yet.'\n\n'Nor wearing French boots with very narrow heels,' said Markham, as\nhe measured the imprint of the said _bottine_ with a small pocket\nrule. 'However, boots don't go for much, unless corroborated.' With\nthis sapient speech Mr. Markham closed his remarks and apparently lost\ninterest in the scene.\n\n'Now this poor fellow,' interpolated Mr. Merlin, lifting up the\ntrooper's face, and parting the thickly clustering brown curls, 'has\nbeen shot from _behind_. Here's the little hole through the back of his\nhead, and the pistol must have been pretty close, as the powder has\nburned one side of it considerably. He has simply fallen over on his\nface, and there was an end of him. Here you can see where the valise\ncontaining the gold and notes was unstrapped from Sergeant Carroll's\nsaddle. The saddles had been put back to back on the ground. One\ncarbine is here still, and one is missing.'\n\n'By Jove!' said Greffham, 'you know everything, Merlin. You're like the\nman in the _Arabian Nights_ who described the camel that had passed the\nday before,\u2015lame, blind of an eye, having lost two front teeth, and\nloaded half with rice and half with dates, and yet never saw him at\nall. You're a wonderful fellow! You're so devilish sharp.'\n\n'And you're a more wonderful fellow; you're so devilish cool,' said\nMerlin. 'I _do_ know a thing or two, and, upon my soul, I have\nneed\u2015_par exemple_, old fellow\u2015it was devilish good-natured of you to\ncome out all the way with us, but it has just occurred to me that you\nseem to have seen these poor fellows so _very_ lately, just before\nthey were rubbed out, that, quite as a matter of form, I must trouble\nyou to explain your proceedings on that day to the authorities. Lionel\nGreffham!' continued he, in a voice which, raised and vibrating, was so\nutterly changed that Ernest Neuchamp did not know it as that of this\nsmiling satirist with his society talk and ready rapier of repartee, 'I\narrest you on suspicion of murder and robbery.'\n\nPerhaps the least astonished and agitated individual of the company was\nthe accused himself. He swung round on one heel as Merlin laid a sinewy\ngrasp upon his shoulder, and, drawing a small foreign-looking revolver\nfrom his breast, aimed fair at the heart of his quondam companion. At\nthe same moment he was covered by the weapons of Markham, the troopers,\nand of Mr. Bright, who held straight for his former acquaintance with\nunmistakable aim and determination.\n\n'It's no use, Mr. Greffham,' said Markham, 'I made your popgun safe at\nthe inn last night. It would never have done to leave you the chance\nof giving us \"Squirt Street.\" It won't pop if you pull the trigger for\na week. Say you could drop Mr. Merlin, why we can \"twice\" you over and\nover.'\n\nMr. Merlin's clear gray eyes glittered with unwonted excitement. He\nalso held a revolver in his right hand. 'My dear fellow,' he said,\n'all excitement is bad form. You must be aware that you are only\narrested on suspicion. Nothing may turn up to implicate you any more\nthan Bright there, but in all these cases a man in my position has a\nduty to perform, and you know well I should do mine if you were my own\nbrother, or the best friend I had in the world.'\n\nBy this time Greffham had recovered his usual composure. 'I don't doubt\n_that_ for one moment, Merlin,' he said, with sardonic emphasis. 'I\nthink you have such a talent in that line that you would rather enjoy\n\"running in\" your own father. However, business is business. You've\nthrown down your card, and as you seem to hold all the trumps at\npresent, you must have the odd trick.'\n\n'Precisely, precisely,' assented Mr. Merlin; 'I always thought you a\ndevilish sensible fellow. So now we must make a start for home. I am\nafraid that I must\u2015just as a matter of form, you know\u2015Markham.'\n\nThat wary official moved forward, and noticing, without seeing, as\nit were, that his superior officer still held his revolver ready for\nimmediate use, produced a pair of handcuffs, and with the ease and\nquickness of long experience slipped them over the wrists of him who\nwas doomed never to sleep unfettered more.\n\nThe party, now become a procession, moved quietly homeward to Turonia.\nThey halted at the inn, the landlord of which was considerably\nsurprised at seeing the great Mr. Greffham's hands closed before him,\nwhile a trooper led his horse by a rein. Up to this period he had not\nthe smallest suspicion that the lavish swell, who, like all men who\naffected wholesale piracy, was 'quite the gentleman' in the matter of\nfree spending of money, could be possibly mixed up with a cold-blooded\nmurder and an extensive robbery. But now his intellect being permitted\nfreedom, he remembered that Mr. Greffham had called at his inn at no\nlong time after the troopers, one of whom he knew well, and furthermore\nthat he remembered hearing a shot at a great distance. It might have\nbeen a revolver. He could not say. It was firearms of some sort. Might\nhave been two shots. Saw nothing.\n\nErnest observed that Markham noted down in a large pocket-book the\nexact minute and hour of the faint report of firearms to which the\ninnkeeper testified, the exact time at which the troopers were last\nseen alive by him, and the time of the arrival of Greffham; and\nthose minor matters being definitely settled, Mr. Merlin conducting\nthe interrogation in a very different voice from his society one,\nthe subdued, if not noticeably saddened procession took the road\nfor Turonia. It was late when they reached that somewhat peculiar\nsettlement, but the streets were profusely lighted, busy, and more\nthronged than at noonday. When the modern inland Australian substitute\nfor 'a plump of spears beneath a pennon gay' rode straight for the\ncamp, the foremost trooper leading the horse of a manacled prisoner,\nwhom many keen eyes at once recognised as Lionel Greffham, a low but\nsavage murmur came from the dense and excited crowd. Whatever interest\nor enthusiasm might have been evoked in Mr. Neuchamp's breast by the\nwonders and novelties of the great goldfield and its heterogeneous,\npicturesque population, had now collapsed. A feeling of doubt and\nhorror succeeded. A tinge of blood, a brooding death-shadow, was over\nthe splendour and the glamour of the enormous treasure-pile which now\nin ceaseless, countless profusion seemed daily won from the reluctant\nearth. He heard to his great satisfaction that Mr. Banks and his party\nhad arrived; that Levison's manager, a man of boundless experience\nin stock, more particularly cattle, was already hard at work at the\nmuster, and that every day an increasing number of the female cattle\ndestined for Rainbar was drafted and delivered to the 'tailing mob' in\nMr. Banks's charge.\n\nSatisfying himself by inspection that the very ordinary routine work of\nmustering a herd, when the mere numbers and sex were alone concerned,\nand where no battles had to be fought over individuals of disputed\nage, size, or quality, could be very safely delegated to subordinates,\nErnest rode over to Branksome Hall for a farewell visit.\n\nThere he found himself an object of interest and friendly welcome,\nsomewhat heightened by his late adventurous journey in company with Mr.\nMerlin. The young ladies were deeply shocked at the terrible finale to\ntheir acquaintance, slight as it had ever been, with the unhappy man\nwho was now a prisoner and presumably a felon, where once he had shone\na star of the first magnitude. Mr. Branksome was sufficiently a man\nof the world to have always distrusted the handsome and unscrupulous\nadventurer. Beyond a formal call he had never been encouraged to see\nmuch of the interior of the Hall.\n\n'Terrible affair this, Neuchamp,' said the host, as the whole party sat\nin the drawing-room before that evening summons had sounded which few\nare sufficiently philosophical or sympathetic to decline. 'I never had\na high opinion of Greffham\u2015always distrusted the man, but as to his\nmurdering a couple of poor devils of troopers for the sake of a couple\nof thousand ounces of gold, why, I should as soon have expected him\nto have dropped strychnine into one's soup-plate at the Occidental at\nlunch.'\n\n'Never fancied him,' said the Colonel; 'deuced well-dressed,\nwell-set-up fellow; been in a cavalry regiment. But he had a\ncold-blooded, hard way of looking at one\u2015bad eye too, cruel, devilish\ncruel\u2015that man has taken life before, I swear\u2015know the expression well,\nkilling is not the fashion much in this country, too young yet\u2015life too\nvaluable\u2015you don't know the signs of it.'\n\n'I can hardly hear to speak of it,' said the eldest Miss Branksome.\n'To think that _any one_ of education and gentleman-like habit, for\nhe _was_ a gentleman as far as manner, appearance, every outward\nobservance can make one, should have descended so low, gone down into\nthe very pit of murder and theft, for what? What could have driven him\nto the edge of such a precipice? Surely there must be demons and fiends\nwho have power over men's souls.'\n\n'Extravagance, gambling, the habit of spending money without working\nfor it,' said her father. 'Debt in one shape or other is one of\nthe demons allotted to torment mortality in this period of the\nworld's history. The demoniac of the age is the man who has bills or\nliabilities coming due without the means to meet them. He may appear\nordinarily clothed and in his right mind, but, after some torturing\nhour, it may be related of him, as of this unhappy wretch, the evil\n\"spirit tare him,\" and he \"wallowed foaming.'\"\n\n'It seems a wonderful thing that he didn't apply to some of his\nfriends, doesn't it?' queried Mr. Neuchamp. 'He seemed to have plenty\nof them. Even if he had not been completely put right they could surely\nhave given him enough to secure breathing time; but murder, robbery,\npah! it is purely incredible to me, predicated of a man that we have\nall met more or less in habits of intimacy.'\n\n'Nothing so wonderful about that,' said the Colonel; 'deuced cool,\nclever adventurer, you know, without one morsel of feeling in favour of\nwhat some people call principle, humanity, or honesty. Seen the style\nbefore. Big loot of any kind is the thing to bring out such a man in\nreal form. Known fellows in Indian service too, by gad, who would kill\na prisoner in cold blood or burn half a village for the sake of a few\ndiamonds or a hoard of gold mohurs.'\n\n'It's positively awful, dreadful, miserable,' said the youngest Miss\nBranksome. 'I shall dream of nothing else for a month, I know. Papa,\nisn't that the dinner bell? Now there's a forfeit if anybody says a\nword about gold or murder or anything belonging to Turonia again this\nevening. We shall be quite demoralised with all this Fouch\u00e9 business.\nThere's Mr. Bright begins to look as if he was going to act upon\n\"information received\" every time I see him.'\n\nThe inmates of that pleasant home finished the evening without\novert allusion to the awful tragedy which had overshadowed their\nneighbourhood, and brought dishonour and death, rare visitors ere this,\neven to the reckless, toiling, far-gathered goldfields community. But\nin every heart, from time to time, in the pause of the conversation, in\nthe silence of the night hour, arose the dimly-outlined picture of the\nlonely flat where the sighing oaks whispered and faintly wailed over\ntwo motionless figures, dread and silent, among the thick, dry, waving\ngrass. On the reverse shadow-tracery a well-known figure, with an evil\nlight in the cold blue eyes, a hellish sneer on the short, curved lip,\nwas pacing the gloomy flags of a felon's cell!\n\nThough Mr. Neuchamp on the morrow parted with great regret from his\nkind friends of Branksome Hall, he could not conceal from himself that\nTuronia, under the circumstances, would be the last place in which he\nshould choose to linger. A shadow of gloom, a savour of blood, was with\nthe whole place and surroundings in his eyes, and though the streets\nstill trembled as before under the tread of an army of Britain's best\nworkers, and though at night there was store of pleasant society and\nexcitement, all interest in the gold city had marvellously abated. Mr.\nNeuchamp was impatient until his moving contingent should be ready for\nthe road, and to that end betook himself with grateful energy to the\ndistraction of mustering the herd.\n\nWith the efficient aid of Mr. Cottonbush, the much-experienced overseer\ndeputed by Mr. Levison to carry out this particular duty, the whole\nherd was mustered and drafted with an economy of time and completeness\nof result very astonishing to Ernest.\n\nHis part was confined to giving Mr. Cottonbush a receipt for nineteen\nhundred and seventy head of female cattle of all sorts, sizes, and\nages, and having divided the said cows and heifers into two droves, an\nimmediate departure was made for Rainbar. Mr. Banks was permitted to\nexamine and explore the wonders of Turonia for the space of one day\nonly; and after bidding farewell to his friends at the camp and at\nBranksome Hall, Mr. Neuchamp rejoined his party, manfully performing\nhis share of road work until, after many a weary week's travelling and\nmonotonous daily drudgery, they struck the river within a day's ride of\nRainbar.\n\nWhen Mr. Neuchamp once more alighted at the door of his cottage he\nfelt the pleasurable glow which is rarely absent from the mind of any\nhealthily constituted man returning after absence to his home.\n\n'Home, sweet home!' hummed Mr. Neuchamp. 'I don't know whether the\ntime-honoured words strictly apply to Rainbar, but I'm glad to see the\nold place again. The grass looks none too fresh, though, as if they had\nhad little or no rain. It would have been inspiriting to have seen a\nlittle green after all the terrible dry weather we have had. I suppose\nthese two thousand new cattle will be able to keep alive. As for paying\nfor them, if I had not Levison's advice and guarantee to depend upon, I\nshould utterly despair of it.'\n\nHe had finished his evening meal when Mr. Jack Windsor was announced,\nthat gentleman having been all day 'out back,' and having but just\nreturned. He was unaffectedly glad to see Ernest, and gave a favourable\naccount of the stock and station matters generally.\n\n'I don't say as we've had much of a break-up of the dry time,' he\nsaid, 'but the rains come very stiddy and soaking every now and then.\nBesides, there's been one or two fine thunderstorms out back, where\nI've been to-day. The feed's a deal better than any I see in here.\nWe're a-getting on towards the end of the autumn now, and we might have\na regular wet season, that will just crown us. I suppose the store\ncattle is all right.'\n\n'In very fair strength and spirits, Jack. Mr. Banks thought that they\nwould do splendidly here before spring, if there was any rain at all.'\n\n'If it wasn't for these confounded cockies,' said Mr. Windsor, 'that\nbig flat would be a first-rate place to break 'em into, while they'll\nhave to be at the water every day. But it's no use thinking of it.\nI've had a deal of bother with 'em as it is; them boys are always\ncutting about the run on horseback, looking for a calf, or a colt, or\nsomething. I'd give a tenner out of my own pocket they was all out of\nthat and back at Bowning or some other stringy-bark hole as is fit for\n'em.'\n\nThree days had elapsed since this conversation, when the two large\ndroves of patient, slow-moving cattle arrived at Rainbar. Mr.\nWindsor was much impressed by their general appearance, and asserted\nconfidently that such a lot of cows and heifers had never before been\nseen on the river.\n\n'They're regular first-class bred 'uns, that's what they are,' he\nasserted; 'that's the best of going in with a man like Levison. He's\nalways got the sugar, consequence he always gets the worth of his\nmoney, and doesn't get put off with half-and-half goods. He knows a\nthing or two, does Levison. Anyhow he's a stunning mate to go shares\nwith.'\n\nAfter a short time spent in making necessary arrangements for the\nnew arrivals, Mr. Neuchamp commenced to review his position. Much\nseriousness of visage resulted from the financial examination.\n\nIn the first place no cattle had been sold in his absence. Nor were\nthere now any in sufficiently high condition to be turned into cash\nwith the same facility as of old. A considerable hole had been made\nin the overdraft which Messrs. Oldstile and Crampton had grudgingly\npermitted him. He had signed bills at twelve months' date for the\nlate purchase of cattle; and accommodating as Mr. Levison might be,\nthe acceptances would have to be met or provided for at maturity.\nProspectively profitable as the transaction was, Mr. Neuchamp commenced\nto make acquaintances with the ominous suggestion, 'Bankruptcy,' and to\nwonder whether he should _really_, in spite of all his plans, prudence,\nand philanthropy, be compelled, even as others were whom he had\ncontemptuously pitied in old times, to surrender unconditionally.\n\nOf this dread and final catastrophe Mr. Neuchamp had a lively horror\nwhich no sophistry could abate. He was not one to fall back upon\nthe many excuses and palliations which the fluctuating markets, the\nuncertain season, afforded. No, no; the stoppage of payment meant\nRuin and Disgrace. It would sound the knell of hope, would proclaim\ndishonour inevitable, irrevocable, as well as the total failure of all\nthe plans and projects which his heart held dearest. His perusal of\nthe newspapers, which had accumulated to a goodly pile in his absence,\nbrought no hint of indulgence. The markets were low; the season had\nnot yet improved so as to place the stock out of danger. If all debts\nincurred were to be met, there was little expectation of being able to\nliquidate them by the aid of the stock then depasturing upon Rainbar.\n\nMore than this, he found among his correspondence an epistle from\nMessrs. Oldstile and Crampton, written in the very old-fashioned manner\naffected by that sound but non-progressive firm. It informed their very\nworthy and most esteemed constituent, Mr. E. Neuchamp, that the five\nhundred pounds last paid to his credit was exhausted, and that unless,\nof course, his account was supported by remittance, they could under\n_no circumstances_ continue to honour his orders.\n\nA letter from Paul Frankston, though kind and hearty in tone, was not\nreassuring. He said that the times were exceedingly bad,\u2015so bad that\neven he, Paul, had had work to meet his engagements, and had at no\ntime for many years past been so sorely pressed. He noticed that every\nday fresh station properties were being brought into the market, and\nhoped that an utter crash and collapse of stock and stations was not\nabout to take place, as in 1842-43. The only reason for believing\nthat a favourable change would take place in the stock-market was\nthat the yield of gold appeared to be increasing, and that though\ntemporary inconvenience had taken place, he, Paul, fully believed that\nin the course of a year or two there would be a very different state\nof matters. He therefore advised Ernest to be hopeful, and, while\nkeeping down expenses to the narrowest limit, to hold on to his station\nwith his eyelids, so to speak. Those who had done so at any former\nperiod of the country's history were now wealthy men. He believed yet\nthat Ernest, if he steadily adhered to his proper work and\u2015pardon\nhim\u2015abstained from speculative experiments, would eventually do well.\nHe hoped that he had got his newly-purchased store cattle safely on the\nrun. He had the greatest confidence in Levison's unerring judgment in\nsuch matters. He might be unduly prejudiced in his favour, but he had\nnever known him to be wrong. If everything went to the bad, no doubt\nthis purchase would make matters no worse. If otherwise, they were\nthe nucleus of a future, and not a small one either. His last advice\nwas to keep the ordinary station work in the best possible trim, and\nnot to spend one shilling in other than absolute necessities. Antonia\nwas very well, but did nothing but read all day. He had suggested her\ngoing in for a degree at the University, but she had not cared for the\nsuggestion. When rain came perhaps Ernest might manage another run down\nthe country.\n\nMr. Neuchamp steadily devoted himself to a full consideration of the\nmatters placed before him in this letter\u2015considerate and delicate in\nfeeling, as indeed had been every word and line of advice received by\nhim from Paul Frankston from the very beginning of their acquaintance.\nNo one could have fancied that the whole of the obligation had been\nupon his, Ernest's, side, from the day when he first exploded Hartley\nSelmore's politico-economical arrangement for subsidising holders\nof station properties with the capital of ingenuous newly-arrived\ncolonists. For how much generous hospitality, shrewd counsel, often\nimplied rather than proffered, substantial assistance and unswerving\nfriendship, was he not his debtor? And Antonia? The more he saw of\ngirls generally,\u2015and he did not rate those Australian young ladies, who\nhad equal advantages of training, society, and culture, at a jot below\ntheir English contemporaries,\u2015the more deep became his conviction of\nher unusual range of thought, depth of feeling, and purity of mind.\nAs the dry, cool wind of the Australian autumn wailed and sighed over\nthe wide gray plains, and around the useful but unromantic edifices\nwhich went to make up the homestead at Rainbar, Ernest began to feel a\nsomewhat intensified, intolerable sensation of intellectual loneliness.\nFor the hundredth, five hundredth, time he wished that it would rain.\nWhy did it not rain? Was the land accursed, like Egypt in the olden\nPharaoh days? Rain would do so much. Put an end to his anxieties about\nthe stock. Improve the condition and lessen the expense of the new\ncattle. Perhaps, nay, certainly, send up the price of stock generally.\nLiberalise the ideas of Messrs. Oldstile and Crampton. Render a trip to\ntown possible; and oh, the sight once more of the verandah at Morahmee!\nthe savour of the fresh brine-laden air! the sight of the foam-fringed\nbillows of the unbounded sea! the\u2015\u2015 But the further contemplation\nof impossible delights, rendered such by his now comparatively\nlengthened inland exile, was sternly repressed by the philosophic mind\nof Mr. Neuchamp. And rain, in England at least, had always seemed\nsuch a little thing\u2015to be had for nothing; to be guarded against by\nthe timid, complained about by the superficial, anathematised by the\nreckless, constant in and out of season\u2015a nuisance, a drug, a daily\ndread. Why then, in the name of all the mighty, merciful powers, did\nit not rain? It was clearly no use fretting about the absence of the\ngladdening, fertilising phenomenon in a dry and thirsty land, or\nphilosophising about the relation of monsoons to icebergs, any unusual\nprotraction or prominence in which natural facts and forces of the calm\nunswerving giantess, Nature, might alter climates and prices, from\nLake Alexandrina to the Snowy River, from Carpentaria to the county of\nCumberland. The matter on hand was the plain and prosaic adjustment of\nhis 'duty a dead sure thing,' and admitting of but little variation\nfrom the point.\n\nTherefore for the present, and as day after day arose bright and\ncool, with breezy morn and pure fresh bracing atmosphere, unhappily\nsuggestive of continuous dry weather, Mr. Neuchamp, discarding\ntheories, reveries, and projects, sternly addressed himself to work.\nFrom earliest dawn to a late hour the whole of the little community\nwas astir. It had been with a feeling of deep satisfaction that Ernest\nhad watched, for the first time, the great droves of 'new cattle'\nspread, unchecked, over the Rainbar plain, and take their first meal\nof the scanty but highly nutritive salsolaceous herbage. Bred in a\n'sour grass' country, far inferior for fattening purposes, though\nhaving merits of its own, the docile, highly-bred herd might be\nexpected, under ordinary conditions, to grow and develop in the most\nunprecedented manner. There is a peculiar pleasure, felt by all station\nproprietors, in the examination of the droves or flocks of store\nstock placed for the first time upon their new pastures. Generally\npurchased at a comparatively low price, and passing from inferior to\nsuperior fattening country, if the season be favourable a cheering\nalteration takes place. It is pleasant for the sheep-owner to perceive\nhis 'large-framed healthy wethers' (as per advertisement) laying on\ncondition day by day, passing through all the stages of comparative\nobesity which enables him to 'top the market' with them as fat sheep,\nhaving previously denuded them of a fleece which, perhaps, fully pays\nthe cost of the original purchase.\n\nBut the gratification known to the purchaser of 'store' or 'lean'\ncattle, either for fattening or for increase, is of a higher and\nmore intense, because of a more complex, nature, as becomes the more\nindividualised character of the stock.\n\nDay by day, if but the pasture be sufficient, the range wide, the\nweather favourable, the season propitious, the stockmen practised and\nefficient\u2015if, I repeat, all these conditions be fulfilled\u2015then indeed\ndoes the happy pastoralist taste all the joys of his successful and\npleasant position. Day by day, as he rides forth in the fresh morn, the\nwarm kind eve, he notes the stranger kine more habitually wander out\nto the springing pasture and back to the creek, marsh, river, which\nis their water privilege. He sees the steers grow glossy of hide,\nthicker, lengthier, ripen into marketable bullocks. He sees the tiny\nshe yearlings grow into sonsy heifers; the angular cows into imposing,\ndeep-brisketed, flat-backed matrons, ever and anon with younglets, 'to\nthe manor born,' and likely in time to pay double the original cost of\nthe parent, with a high percentage for personal profit. Lastly, the\nfirst draft of bullocks picked from these, if a mixed herd, pays for\nthe whole lot\u2015steers, bullocks, cows, and calves\u2015leaving the spirited\npurchaser with a tolerably large and increasing herd, all profit.\n\nMany of these pleasurable emotions would have found lodging in the\nbreast of Mr. Neuchamp had circumstances, that is, the season, been\nfavourable. But nothing was favourable. The skies were like brass\u2015even\nas the money market\u2015with no rent or fissure through which mercy or\nchange could by any means be perceived. The scanty pasture provoked\nthe instinct-guided cattle to wander far and fast. In pursuit Ernest\nwas fain to hurry, personally or vicariously, till every horse on the\nestablishment, Osmund included, had as much as he could do to carry his\nrider for a day's slow journey. Indeed the said rider was occasionally\nto be descried carrying his saddle home upon his own proper back,\nhaving left his weak and weary steed out on the plain.\n\nThe original herd, every beast of which had been bred and reared at\nRainbar, was not altogether badly off. Acquainted with every nook and\ncorner of the run, they 'went back' almost incredible distances for\ngrass, only returning to the bare vicinity of the water when desperate\nwith thirst. It is wonderful what privation in that respect the\nhalf-wild herds of cattle and horses will undergo in a dry country in\na dry season, without seriously imperilling their health and strength.\nIf they can only procure a debauch upon water from time to time, they\nstave off famine in a manner quite impossible to the shorthorns and\nunadventurous beeves of more rainy climes, more succulent pastures.\n\nAs to the members of the co-operative settlement\u2015the cockatoos, as Jack\nWindsor incorrectly called them\u2015they were not, in that time of trial,\nan element of help or consolation. Their cattle had increased even\nsuspiciously fast. The untoward season had brought out the narrow greed\nand cunning of their natures into unpleasant prominence.\n\nUnder the impression that Ernest would most probably be ruined and be\ncompelled shortly to sell Rainbar, they arrived at the conclusion that\nthere was nothing to be gained by concession, and so gradually threw\noff any semblance of deference. They rigidly enforced the exclusion of\nthe Rainbar cattle from their very extensive pre-emptive grass rights,\nand they hunted with their dogs new cattle and old indifferently, not\nparticularly caring, it would seem, whether they were or were not lost.\n\nErnest was first grieved, then indignant, at this gross ingratitude.\nUnder the influence of these feelings he expostulated with them warmly,\nalleging his right, as having advanced a portion of the purchase-money\nfor their holdings, to some consideration, if the general sympathy and\nkindness which he had accorded to them was to go for nothing.\n\nAbraham Freeman replied that they did not see that they had anything to\nthank him for, particularly that they had left good homes to come to\nthis confounded dry sand-heap of a country. That they intended to stick\nup for their pre-emptives, as the cattle were all their dependence now,\nand that if he wanted to make terms with them, they would be satisfied\nwith that portion of the run\u2015with the river frontage, of course\u2015which\nlay to the westward of their settlement. If he just gave them the use\nof that bit of country\u2015it was only five or six miles in length, and\ndidn't go far back\u2015then they would bind themselves _not to take up any\nmore of his run_.\n\nThis last implied threat completed the obliteration of the last shred\nof Mr. Neuchamp's patience. These heartless, unprincipled wretches,\nwhom he had raised from a position of indifferently paid toil, akin to\ndaily labour, to that of thriving graziers, basely forgetful of his\nexceptional benevolence, were actually trading upon their power of\nannoyance and injurious occupation of his run! Very bitter were Mr.\nNeuchamp's reflections when this evil growth of human nature was thus\nindisputably proved. Had it not been so bad a season he might have\noverlooked it. But now, when fate and the very skies were at war with\nhim, this instance of ingratitude overpowered all philosophic calmness.\n\nHe immediately convened a meeting of the heads of families of the house\nof Freeman, and informed them, in sufficiently decided tones, that he\nfound himself to have been mistaken in his estimate of their principles\nand characters; that he had sought to benefit them chiefly; had already\nassisted them to a partial independence, and that he had looked for\nsome decent recognition of his efforts for their sole advantage. They\nhad chosen to deceive and to threaten. He was resolved now to confine\nthem strictly to their land, to require repayment of the money which he\nhad lent, and to hold no terms of any kind whatever with them.\n\nMessrs. Freeman Brothers were somewhat astonished by Ernest's capacity\nfor righteous indignation. They had not expected anything of the sort.\nThey had looked for unlimited toleration. They now began to consider\nthat a declaration of war might possibly result injuriously to their\nown interests, and they possibly had the grace to remember that, up\nto this stage of the affair, Mr. Neuchamp had been considerate, or,\nin their phraseology, 'soft,' to an extent altogether unprecedented\nin their experience of the pastoral tenants of the Crown. They would\nhave no more loading, an easy way of providing themselves with the very\nmoderate amount of cash necessary for their ordinary expenditure.\n\nCertainly they did not need any large outlay. There are few lands\nunder the sun, the Coral Islands of that charmed main the Great South\nSea excepted, where there is such a possibility of tranquil, joyous\nprogress along life's pathway, without the use of the circulating\nmedium, as in the settlements of the older colonies of Australia.\n\nFor instance, the Freemans had, as it were for nothing, house-room,\nfuel, water, and light. Their garden supplied them with an annual crop\nof pumpkins, melons, and other esculents, which gave them vegetable\nfood for the greater part of the year. Far larger crops might have been\nproduced by a comparatively trifling increase of labour or thought.\nThey had milk, butter, and meat from their herd, in ordinary years in\nprofusion. The few necessaries which they were absolutely reduced to\nimport or purchase were clothes, of which, owing to the mildness of\nthe climate, they needed but few; tea and sugar, salt and flour, with\na trifling stock of household utensils and furniture. With respect\nto the tea and sugar, a large reduction might have been made in this\nsection had it been the fashion, as it was the exceptional practice, of\nisolated settlers to substitute milk for the former, as an ordinary\nadjunct to the three meals of the day.\n\nBut tea in Australia, grateful alike in the burning heat of summer\nand in the bitter frosts and sleet of winter\u2015portable, innocuous,\nnutritive, and slightly stimulating\u2015is the beer of the common people;\nand we know from experience that the attempt 'to rob a poor man of his\nbeer' has always hitherto proved unpopular and unsuccessful.\n\nWe must therefore assume that a half-chest of tea and a couple of bags\nof medium brown sugar must be added to the expenditure of the small\nfarmer, or 'free selector,' as he is now universally called.\n\nAustralia is not a good game country. Still the different varieties of\nthe kangaroo are palatable and nutritious, more resembling the flesh\nof the hare and rabbit, with a flavour of veal, than beef or mutton.\nWith the aid of a brace of rough greyhounds\u2015the kangaroo-dog of the\ncolonists\u2015these are easily procured in any quantity. The skins are\nworth a shilling each, and are useful as mats or for coverings. The\nrivers and creeks, particularly the larger watercourses, are generally\nfilled with fresh-water codfish and several other divisions of the\nperch family. These are considered to afford valuable supplementary aid\nto the perhaps scanty supply of butchers' meat, on many a far-out farm\nin summer time.\n\nWith regard to the condition of the rather exclusive settlement formed\nand owned by the Freeman family, they had each made shift to bring from\na couple to half a dozen brood mares, perhaps originally purchased for\nfrom half-a-crown to half-a-sovereign each, out of the Bowning pound.\nThese hardy, though not perhaps well-bred, animals had increased\nwonderfully since their arrival, and were now, of themselves, quite a\nsmall herd. The younger members of the Freeman families could of course\nride like Comanches, and no inconsiderable portion of their time was\nspent in running in these swift and half-wild mustangs, breaking them,\nlosing them, finding them; and in all these operations and employment\ngalloping around and across the Rainbar run, to the wrath and constant\nannoyance of Jack Windsor and Charley Banks.\n\nSome effort was made, in a half-sullen, half-apologetic way, by Abraham\nFreeman to remove the ban under which the whole settlement lay. But\nErnest was fixed and implacable in righteous disapproval. He gave\nstrict orders that no stock of the offending co-operatives was to be\npermitted to graze upon the Rainbar run; that the boys were to be told\nthat they would be summoned for trespass if they were found riding over\nthe run or driving stock off without notice. War was declared in form.\nThe strayed cattle belonging to the smaller graziers were placed in\nthe Rainbar yard from time to time, and kept there till taken away by\ntheir owners. They were not permitted to purchase any articles from the\nstation store. And, in fine, a blockade cordon was morally drawn round\nthat nucleus of agricultural co-operative progress which had called\nforth so many sanguine prophecies. Mr. Neuchamp was sternly immutable\nand indignant of attitude. Slow to arouse and difficult to persuade of\nintentional wrongdoing, he was _very_ loath to retreat from any gage of\nbattle thus produced.\n\nBoth Charley Banks and Jack Windsor regarded this latter step with\ndisapprobation. It had been ridiculously credulous and weak, according\nto their mode of thought, to invite the Freemans to settle on Rainbar.\nIt was lamentably imprudent to quarrel openly with them now they _were_\nsettled.\n\nThe second brother assented without much hostile observation,\nregretting that they had fallen out for nothing, as he expressed it;\nand Mr. Joe Freeman smiled in a scarcely reassuring manner, as Charley\nBanks thought, and said if it came to a pounding match, the cove would\nfind that they could do him a deuced sight more hurt than he could do\nthem.\n\nMr. Windsor, who had seen more of the ways of small freeholders, and\nunderstood their modes of feeling and action better than did Charley\nBanks, much less Mr. Neuchamp, did not regard this open declaration of\nhostilities as likely to add to their comfort, profit, or advantage.\n\n'Mr. Neuchamp did a soft thing in bringing these chaps here, and now\nhe's acting far from wise in letting 'em know what he thinks of 'em.\nHe ought to have kept in with 'em and watched 'em, and if they went\n\"on the cross\" about the stock, he'd have had 'em safe and sound in\nDrewarrina Gaol some fine day.'\n\nThis was Jack's idea of justifiable free-selectoricide. It might\noccasionally miss fire, but in the long-run it was very likely to bag\nthe 'picker-up of unconsidered trifles' in the shape of unbranded stock.\n\n'Those chaps can do the boss a deuced sight more damage than he can do\nthem if they're drove to it,' continued Mr. Windsor. 'They watch him\nwhen he isn't thinkin' of them, and if our cattle ain't on their land,\nthey can _make 'em_ trespass any night they please. I know the likes\nof them well, and I'd rather take 'em quiet than hustle 'em any day.'\n\n'You're not far wrong, Jack,' assented Mr. Banks. 'We must keep these\nnew cattle close, or they'll have a lot ready for Drewarrina pound some\nfine morning, as sure as my name is Charley Banks.'\n\nBy careful watching, by riding early and riding late, this highly\nprobable outcome of the feud between Mr. Neuchamp and his late\n_prot\u00e9g\u00e9s_ was for a time avoided. But\n\n There never yet was human power\n Which could evade, if unforgiven,\n The patient search and vigil long\n Of him who treasures up a wrong.\n\nIt is questionable whether Byron had the operation of the Lands\nOccupation Act for the colony of New South Wales in view when he penned\nthese lines, but they apply as closely to the general consequences\nof that great statute as if his lordship had intended to settle the\naffairs of Australia, after leading to victory the anti-Turkish party\nof the day.\n\nThe brothers Freeman, by a peculiar mental process, had managed to\nignore the very substantial aid in cash and employment, the former\nstill unrepaid, furnished by Mr. Neuchamp. By fixing all their\nattention upon his latter line of conduct, they became convinced that\nin denying their cattle access to every portion of the Rainbar run\nhe had inflicted upon them a great wrong. This they determined to\navenge if not to redress; and one fine morning an ill-written note,\nbrought by a brown-faced urchin of ten years old about breakfast time,\ninformed Mr. Neuchamp that William and Joseph Freeman had discovered\nthree hundred and forty-seven of his cattle trespassing upon their\nland, which cattle were now in their custody, and which they proposed\ndriving to Drewarrina pound (about seventy miles off) if not forthwith\nreleased with damages and expenses paid.\n\n'What in the name of all that's rascally can we do?' inquired Ernest of\nCharley Banks, as he tossed the note over to him across the breakfast\ntable. 'I feel inclined to go down and take the cattle by force. The\ndishonest, scheming vagabonds!'\n\n'That's what I should like to do,' said Banks, 'and I think Jack and\nI could hammer that Bill Freeman and his brother, but I'm afraid it\nwon't do. If we rescue the cattle we can be summoned and fined; besides\ntaking us all the way to that rascally hole of a township.'\n\n'Then let them keep them, and drive them over to the pound. The damage\ncan't be much.'\n\n'And let them hunt them over, and yard them half the time?' demanded\nMr. Banks. 'No, that wouldn't do either. The cattle wouldn't recover it\nfor the whole season. You'll have to buy him off. So much a head. It's\nthe shortest way through it.'\n\nMr. Neuchamp groaned. This way was degrading. A pecuniary loss, for\nwhich he did not care so much as he ought to have done, for Ernest was\none of those people who rarely regard a cheque or order as the bag of\ngolden sovereigns that anything over a ten-pound note really is. Also,\na loss of dignity, which he felt keenly, that he should be placed in\nthe dilemma of having to pay to release his own cattle from his own\ntenants, so to speak, or to see them injured and lowered in value by\nthose base burghers of the corporation he had himself led into the land\nof promise!\n\n'There is nothing else to be done,' said Charley. 'They have the best\nof us now; we must pay.'\n\n'I don't believe the cattle were on their land at all,' pleaded the\nfounder of the society.\n\n'That's nothing,' opposed Mr. Banks, 'they'll swear they _found 'em\nthere_, and bring three or four witnesses to prove it; you'd better\ngive me a cheque for thirty pounds, and let me square it with them. I\nthink we shall get out for that.'\n\nMr. Neuchamp much regretted sacrificing any portion of his latest and\nprobably concluding advance from Messrs. Oldstile and Crampton in\nsuch an unsatisfactory manner, but was compelled to employ that only\nuniversal solvent, a cash payment. Mr. Banks departed with the magic\nmissive. I have no authentic record of what actually passed between him\nand Bill Freeman, but he returned with the cattle. It was also noticed\nthat no peculiar exacerbation occurred between the litigants after this\ninterview.\n\nAnother month wore away in the performance of the ordinary work, and\nthe endurance of rather more than the ordinary crosses and losses\nconsequent upon the still protracted drought.\n\nNo rain. And again, no rain. Nothing grew. All nature became daily more\nwan, pale, leafless. The crop of expenses, inevitable and regular,\nin contradistinction to the produce of the season, grew and matured,\nuntil once more the limit of advance agreed to by Messrs. Oldstile\nand Crampton was definitely reached. Of this ultimate fact Mr.\nNeuchamp was unpleasantly reminded by the return, unpaid, of his last\nhalf-dozen orders, arriving by the mail preceding that which furnished\nan exceedingly formal letter, advising the unpleasant step which his\nagents, to their extreme regret, had been compelled to take.\n\nErnest felt this hitherto unknown annoyance to be the precursor of a\nfinancial earthquake, in which possibly his present possessions and\nfuture hopes might be engulfed.\n\nHe tried to consider his position with the calmness proper to so grave\na conjuncture. But he had much difficulty in preserving the requisite\nfreedom from disturbance. Ever and anon would come, as with a lightning\nflash, the vision of all his cherished projects disappearing down the\ndark chasm of insolvency and ruin.\n\nHis stud of Australian Arabs, now so promising, would be sold for the\nprice of bush mustangs. His store cattle, nearly broken to the run,\nwould be as valueless as if, in spite of their high breeding, they\nhad been composed in great part of the 'scrub-danglers,' one of whom\nhad so unwarrantably assaulted him on his arrival at Rainbar. His\npet engineering scheme, unfinished and derided, would be henceforth\nticketed among the denizens of the locality as Neuchamp's Folly. Ernest\nhad not more than the ordinary share of self-love, through which\nnature makes provision for the preservation of the individual, but he\ncommenced to feel by anticipation the pangs which are inseparable from\npronounced failure in any soever enterprise or profession. He heard\nMr. Jermyn Croker's unqualified verdict that 'he had always been a\nphilanthropic lunatic, from whom nothing else could have been expected;\nthe only wonder being that any one had been found fool enough to trust\nhim, and thereby enable him to make so respectable a smash of it.'\nOthers doubtless would follow in the same suit. Even the good-natured\nParklands and the charitable Aymer Brandon, who gave, as they required\nindeed, much frank social absolution, could scarcely refrain from\nunreserved condemnation of his 'improvement' theory. As to the\n'grateful tenantry' idea, represented by Freeman Brothers, with their\ngrass-rights, their hostility, and their herds and their flocks\u2015for\nthey had lately purchased a thousand debilitated travelling sheep at\nabout sixpence per head\u2015it would not bear thinking of. He was now in\nfull endurance of the reactionary stage of despondency occasionally\nbestowed as a counterpoise to the ordinarily high average of tone with\nwhich the sanguine man is blessed or cursed, as the case may be. As Mr.\nNeuchamp reviewed his generous and lofty aims, his far-reaching plans\nand projects dependent upon so kindly a future for success, he inclined\nto the latter reading. They appeared to him in this his dark hour as\nthe fantasies of an opium-eater or the dream-palaces of a slumbering\nchild.\n\nMr. Neuchamp, after a day spent in sad consideration, unfortunately\npermitted himself to pursue the unending evil of regret during the\nnight. His heightened imagination multiplied disaster and enlarged evil\nto such a degree that he was more than once tempted to spring from his\nthorny couch and take to the broad starlit plain for the relief of\nexercise.\n\n 'So sore was the delirious goad,\n I took my steed and forth I rode,'\n\nsays the remorseful Marmion; and but that in the present state of the\nfodder market no horses had been stabled at Rainbar for many a day,\nour latter-day Crusader might have followed out the idea literally. As\nit was he but arose at earliest dawn and mechanically took the garden\npath, trusting to find some excuse for an hour or two of hard manual\nlabour which might guide or exorcise the evil spirits that were rending\nhis very soul.\n\nHe had been putting out all his strength for an hour or more, and\nwas in much the same bodily state and condition as if he had taken\na ten-mile spin with a greatcoat on, after the prescription of Mr.\nGeoffry Delamayn, when he observed a solitary horseman wending his way\nalong the 'up-river' road, which was distinguishable more by dust than\nby colouring from the grassless waste through which it wound.\n\nThe stranger, who was habited in a collarless Crimean shirt and rather\ndilapidated habiliments generally, rode his emaciated steed steadily\non at the slow, hopeless, leg-weary jog to which most of the horses of\nthe territory had long been reduced, until he reached the garden gate.\nErnest,\u2015taking him for granted as the usual 'reporter' of travelling\nsheep, about to clear off the last fragments of what once had been\npasture; an invalid shepherd, making for the Drewarrina Hospital; a\nmounted tramp or 'traveller' looking for work, with no great hope of,\nor indeed concern about, finding it; or lastly, a supernumerary for\nsome travelling stock caravan, who had been 'hunted' for drunkenness\nor inefficiency,\u2015raised not his head. For any or all of these toilers\nof the waste there would be the unvarying hospitality of the men's\nhut. But the stranger sat calmly upon his despondent horse at the\ngate surveying Ernest's exceedingly efficient spade performance with\napparent approval, until at length he broke silence. 'My word, Mr.\nNoochamp, you're nigh as good as a Chinaman. You'd make wages at\npost-hole digging, if the rain forgets to come and we're all smothered.\nHow's those AD store cattle getting on?'\n\nErnest looked up hastily and indignantly at the first tones of the\nstranger's accost, but immediately relaxed his visage and flung down\nhis spade as he recognised in the horseman's countenance the grave,\nreflective lineaments of Abstinens Levison.\n\n\nEND OF VOL. II\n\n\n_Printed by_ R. & R. CLARK, _Edinburgh_.\n\n\n\n\n * * * * *\n\n\n\n\nTranscriber's note:\n\nObvious typographical errors have been silently corrected.\n\nVariations in spelling, punctuation and hyphenation remain unchanged.\n\npage 97\n\"to one of his own invention, viz. \u018eNE (a conjoined hieroglyph)\". The\ninitial character of the hieroglyph is printed half a line lower in the\noriginal.\n\n\n\n***","meta":{"redpajama_set_name":"RedPajamaBook"}} +{"text":"\n\n\n\nProduced by Stephen Hutcheson, Dave Morgan, the Mo-Ark\nRegional Railroad Museum at Poplar Bluff, Missouri and the\nOnline Distributed Proofreading Team at http:\/\/www.pgdp.net\n\n\n\n\n\n\n\n\n\n Titan of Chasms\n The Grand Canyon of Arizona\n\n\n THE TITAN OF CHASMS\n By C. A. HIGGINS\n\n THE SCIENTIFIC EXPLORER\n By J. W. POWELL\n\n THE GREATEST THING IN THE WORLD\n By CHAS. F. LUMMIS\n\n INFORMATION FOR TOURISTS\n\n [Illustration: Sante Fe]\n\n Fortieth Thousand\n PASSENGER DEPARTMENT\n THE SANTA FE\n CHICAGO, 1903.\n\n [Illustration: Copyright, 1899, by H. G. Peabody.\n Bright Angel Creek and North Wall of the Canyon.]\n\n [Illustration: Uncaptioned vista]\n\n\n\n\n THE TITAN OF CHASMS\n BY C. A. HIGGINS\n\n\n Its History\n\nThe Colorado is one of the great rivers of North America. Formed in\nSouthern Utah by the confluence of the Green and Grand, it intersects\nthe northwestern corner of Arizona, and, becoming the eastern boundary\nof Nevada and California, flows southward until it reaches tidewater in\nthe Gulf of California, Mexico. It drains a territory of 300,000 square\nmiles, and, traced back to the rise of its principal source, is 2,000\nmiles long. At two points, Needles and Yuma on the California boundary,\nit is crossed by a railroad. Elsewhere its course lies far from\nCaucasian settlements and far from the routes of common travel, in the\nheart of a vast region fenced on the one hand by arid plains or deep\nforests and on the other by formidable mountains.\n\nThe early Spanish explorers first reported it to the civilized world in\n1540, two separate expeditions becoming acquainted with the river for a\ncomparatively short distance above its mouth, and another, journeying\nfrom the Moki Pueblos northwestward across the desert, obtaining the\nfirst view of the Big Canyon, failing in every effort to descend the\ncanyon wall, and spying the river only from afar.\n\nAgain, in 1776, a Spanish priest traveling southward through Utah struck\noff from the Virgin River to the southeast and found a practicable\ncrossing at a point that still bears the name \"Vado de los Padres.\"\n\nFor more than eighty years thereafter the Big Canyon remained unvisited\nexcept by the Indian, the Mormon herdsman, and the trapper, although the\nSitgreaves expedition of 1851, journeying westward, struck the river\nabout 150 miles above Yuma, and Lieutenant Whipple in 1854 made a survey\nfor a practicable railroad route along the thirty-fifth parallel, where\nthe Santa Fe Pacific has since been constructed.\n\nThe establishment of military posts in New Mexico and Utah having made\ndesirable the use of a waterway for the cheap transportation of\nsupplies, in 1857 the War Department dispatched an expedition in charge\nof Lieutenant Ives to explore the Colorado as far from its mouth as\nnavigation should be found practicable. Ives ascended the river in a\nspecially constructed steamboat to the head of Black Canyon, a few miles\nbelow the confluence of the Virgin River in Nevada, where further\nnavigation became impossible; then, returning to the Needles, he set off\nacross the country toward the northeast. He reached the Big Canyon at\nDiamond Creek and at Cataract Creek in the spring of 1858, and from the\nlatter point made a wide southward detour around the San Francisco\nPeaks, thence northeastward to the Moki Pueblos, thence eastward to Fort\nDefiance, and so back to civilization.\n\nThat is the history of the explorations of the Colorado up to forty\nyears ago. Its exact course was unknown for many hundred miles, even its\norigin being a matter of conjecture. It was difficult to approach within\na distance of two or three miles from the channel, while descent to the\nriver's edge could be hazarded only at wide intervals, inasmuch as it\nlay in an appalling fissure at the foot of seemingly impassable cliff\nterraces that led down from the bordering plateau; and to attempt its\nnavigation was to court death. It was known in a general way that the\nentire channel between Nevada and Utah was of the same titanic\ncharacter, reaching its culmination nearly midway in its course through\nArizona.\n\n [Illustration: The Colorado, Foot of Bright Angel Trail.]\n\nIn 1869 Maj. J. W. Powell undertook the exploration of the river with\nnine men and four boats, starting from Green River City, on the Green\nRiver, in Utah. The project met with the most urgent remonstrance from\nthose who were best acquainted with the region, including the Indians,\nwho maintained that boats could not possibly live in any one of a score\nof rapids and falls known to them, to say nothing of the vast unknown\nstretches in which at any moment a Niagara might be disclosed. It was\nalso currently believed that for hundreds of miles the river disappeared\nwholly beneath the surface of the earth. Powell launched his flotilla on\nMay 24th, and on August 30th landed at the mouth of the Virgin River,\nmore than one thousand miles by the river channel from the place of\nstarting, minus two boats and four men. One of the men had left the\nexpedition by way of an Indian reservation agency before reaching\nArizona, and three, after holding out against unprecedented terrors for\nmany weeks, had finally become daunted, choosing to encounter the perils\nof an unknown desert rather than to brave any longer the frightful\nmenaces of that Stygian torrent. These three, unfortunately making their\nappearance on the plateau at a time when a recent depredation was\ncolorably chargeable upon them, were killed by Indians, their story of\nhaving come thus far down the river in boats being wholly discredited by\ntheir captors.\n\nPowell's journal of the trip is a fascinating tale, written in a compact\nand modest style, which, in spite of its reticence, tells an epic story\nof purest heroism. It definitely established the scene of his\nexploration as the most wonderful geological and spectacular phenomenon\nknown to mankind, and justified the name which had been bestowed upon\nit\u2014The Grand Canyon\u2014sublimest of gorges; Titan of chasms. Many\nscientists have since visited it, and, in the aggregate, a large number\nof unprofessional lovers of nature; but until a few years ago no\nadequate facilities were provided for the general sight-seer, and the\nworld's most stupendous panorama was known principally through report,\nby reason of the discomforts and difficulties, of the trip, which\ndeterred all except the most indefatigable enthusiasts. Even its\ngeographical location is the subject of widespread misapprehension.\n\nIts title has been pirated for application to relatively insignificant\ncanyons in distant parts of the country, and thousands of tourists have\nbeen led to believe that they saw the Grand Canyon, when, in fact, they\nlooked upon a totally different scene, between which and the real Grand\nCanyon there is no more comparison \"than there is between the\nAlleghanies or Trosachs and the Himalayas.\"\n\nThere is but one Grand Canyon. Nowhere in the world has its like been\nfound.\n\n\n As Seen From the Rim\n\nStolid, indeed, is he who can front the awful scene and view its\nunearthly splendor of color and form without quaking knee or tremulous\nbreath. An inferno, swathed in soft celestial fires; a whole chaotic\nunder-world, just emptied of primeval floods and waiting for a new\ncreative word; eluding all sense of perspective or dimension,\noutstretching the faculty of measurement, overlapping the confines of\ndefinite apprehension; a boding, terrible thing, unflinchingly real, yet\nspectral as a dream. The beholder is at first unimpressed by any detail;\nhe is overwhelmed by the _ensemble_ of a stupendous panorama, a thousand\nsquare miles in extent, that lies wholly beneath the eye, as if he stood\nupon a mountain peak instead of the level brink of a fearful chasm in\nthe plateau, whose opposite shore is thirteen miles away. A labyrinth of\nhuge architectural forms, endlessly varied in design, fretted with\nornamental devices, festooned with lace-like webs formed of talus from\nthe upper cliffs and painted with every color known to the palette in\npure transparent tones of marvelous delicacy. Never was picture more\nharmonious, never flower more exquisitely beautiful. It flashes instant\ncommunication of all that architecture and painting and music for a\nthousand years have gropingly striven to express. It is the soul of\nMichael Angelo and of Beethoven.\n\n [Illustration: Copyright, 1899, by H. G. Peabody.\n The River and the Canyon Wall.]\n\nA canyon, truly, but not after the accepted type. An intricate system of\ncanyons, rather, each subordinate to the river channel in the midst,\nwhich in its turn is subordinate to the whole effect. That river\nchannel, the profoundest depth, and actually more than 6,000 feet below\nthe point of view, is in seeming a rather insignificant trench,\nattracting the eye more by reason of its somber tone and mysterious\nsuggestion than by any appreciable characteristic of a chasm. It is\nperhaps five miles distant in a straight line, and its uppermost rims\nare nearly 4,000 feet beneath the observer, whose measuring capacity is\nentirely inadequate to the demand made by such magnitudes. One can not\nbelieve the distance to be more than a mile as the crow flies, before\ndescending the wall or attempting some other form of actual measurement.\n\nMere brain knowledge counts for little against the illusion under which\nthe organ of vision is here doomed to labor. Yonder cliff, darkening\nfrom white to gray, yellow, and brown as your glance descends, is taller\nthan the Washington Monument. The Auditorium in Chicago would not cover\none-half its perpendicular span. Yet it does not greatly impress you.\nYou idly toss a pebble toward it, and are surprised to note how far the\nmissile falls short. By and by you will learn that it is a good half\nmile distant, and when you go down the trail you will gain an abiding\nsense of its real proportions. Yet, relatively, it is an unimportant\ndetail of the scene. Were Vulcan to cast it bodily into the chasm\ndirectly beneath your feet, it would pass for a bowlder, if, indeed, it\nwere discoverable to the unaided eye.\n\nYet the immediate chasm itself is only the first step of a long terrace\nthat leads down to the innermost gorge and the river. Roll a heavy stone\nto the rim and let it go. It falls sheer the height of a church or an\nEiffel Tower, according to the point selected for such pastime, and\nexplodes like a bomb on a projecting ledge. If, happily, any\nconsiderable fragments remain, they bound onward like elastic balls,\nleaping in wild parabola from point to point, snapping trees like\nstraws; bursting, crashing, thundering down the declivities until they\nmake a last plunge over the brink of a void; and then there comes\nlanguidly up the cliff sides a faint, distant roar, and your bowlder\nthat had withstood the buffets of centuries lies scattered as wide as\nWycliffe's ashes, although the final fragment has lodged only a little\nway, so to speak, below the rim. Such performances are frequently given\nin these amphitheaters without human aid, by the mere undermining of the\nrain, or perhaps it is here that Sisyphus rehearses his unending task.\nOften in the silence of night some tremendous fragment has been heard\ncrashing from terrace to terrace with shocks like thunder peal.\n\nThe spectacle is so symmetrical, and so completely excludes the outside\nworld and its accustomed standards, it is with difficulty one can\nacquire any notion of its immensity. Were it half as deep, half as\nbroad, it would be no less bewildering, so utterly does it baffle human\ngrasp.\n\n\n The Trip to the River\n\nOnly by descending into the canyon may one arrive at anything like\ncomprehension of its proportions, and the descent can not be too\nurgently commended to every visitor who is sufficiently robust to bear a\nreasonable amount of fatigue. There are four paths down the southern\nwall of the canyon in the granite gorge district\u2014Mystic Spring, Bright\nAngel, Berry's and Hance's trails. The following account of a descent of\nthe old Hance trail will serve to indicate the nature of such an\nexperience to-day, except that the trip may now be safely made with\ngreater comfort.\n\nFor the first two miles it is a sort of Jacob's ladder, zigzagging at an\nunrelenting pitch. At the end of two miles a comparatively gentle \nis reached, known as the blue limestone level, some 2,500 feet below the\nrim, that is to say\u2014for such figures have to be impressed objectively\nupon the mind\u2014five times the height of St. Peter's, the Pyramid of\nCheops, or the Strasburg Cathedral; eight times the height of the\nBartholdi Statue of Liberty; eleven times the height of Bunker Hill\nMonument. Looking back from this level the huge picturesque towers that\nborder the rim shrink to pigmies and seem to crown a perpendicular wall,\nunattainably far in the sky. Yet less than one-half the descent has been\nmade.\n\nOvershadowed by sandstone of chocolate hue the way grows gloomy and\nforeboding, and the gorge narrows. The traveler stops a moment beneath a\nslanting cliff 500 feet high, where there is an Indian grave and pottery\nscattered about. A gigantic niche has been worn in the face of this\ncavernous cliff, which, in recognition of its fancied Egyptian\ncharacter, was named the Temple of Sett by the painter, Thomas Moran.\n\nA little beyond this temple it becomes necessary to abandon the animals.\nThe river is still a mile and a half distant. The way narrows now to a\nmere notch, where two wagons could barely pass, and the granite begins\nto tower gloomily overhead, for we have dropped below the sandstone and\nhave entered the arch\u00e6an\u2014a frowning black rock, streaked, veined, and\nswirled with vivid red and white, smoothed and polished by the rivulet\nand beautiful as a mosaic. Obstacles are encountered in the form of\nsteep, interposing crags, past which the brook has found a way, but over\nwhich the pedestrian must clamber. After these lesser difficulties come\nsheer descents, which at present are passed by the aid of ropes.\n\nThe last considerable drop is a 40-foot bit by the side of a pretty\ncascade, where there are just enough irregularities in the wall to give\ntoe-hold. The narrowed cleft becomes exceedingly wayward in its course,\nturning abruptly to right and left, and working down into twilight\ndepth. It is very still. At every turn one looks to see the embouchure\nupon the river, anticipating the sudden shock of the unintercepted roar\nof waters. When at last this is reached, over a final downward clamber,\nthe traveler stands upon a sandy rift confronted by nearly vertical\nwalls many hundred feet high, at whose base a black torrent pitches in a\ngiddying onward slide that gives him momentarily the sensation of\nslipping into an abyss.\n\n [Illustration: A Party on Bright Angel Trail.]\n\nWith so little labor may one come to the Colorado River in the heart of\nits most tremendous channel, and gaze upon a sight that heretofore has\nhad fewer witnesses than have the wilds of Africa. Dwarfed by such\nprodigious mountain shores, which rise immediately from the water at an\nangle that would deny footing to a mountain sheep, it is not easy to\nestimate confidently the width and volume of the river. Choked by the\nstubborn granite at this point, its width is probably between 250 and\n300 feet, its velocity fifteen miles an hour, and its volume and turmoil\nequal to the Whirlpool Rapids of Niagara. Its rise in time of heavy rain\nis rapid and appalling, for the walls shed almost instantly all the\nwater that falls upon them. Drift is lodged in the crevices thirty feet\noverhead.\n\nFor only a few hundred yards is the tortuous stream visible, but its\neffect upon the senses is perhaps the greater for that reason. Issuing\nas from a mountain side, it slides with oily smoothness for a space and\nsuddenly breaks into violent waves that comb back against the current\nand shoot unexpectedly here and there, while the volume sways tide-like\nfrom side to side, and long curling breakers form and hold their outline\nlengthwise of the shore, despite the seemingly irresistible velocity of\nthe water. The river is laden with drift (huge tree trunks), which it\ntosses like chips in its terrible play.\n\nStanding upon that shore one can barely credit Powell's achievement, in\nspite of its absolute authenticity. Never was a more magnificent\nself-reliance displayed than by the man who not only undertook the\npassage of Colorado River but won his way. And after viewing a fraction\nof the scene at close range, one can not hold it to the discredit of\nthree of his companions that they abandoned the undertaking not far\nbelow this point. The fact that those who persisted got through alive is\nhardly more astonishing than that any should have had the hardihood to\npersist. For it could not have been alone the privation, the infinite\ntoil, the unending suspense in constant menace of death that assaulted\ntheir courage; these they had looked for; it was rather the unlifted\ngloom of those tartarean depths, the unspeakable horrors of an endless\nvalley of the shadow of death, in which every step was irrevocable.\n\nReturning to the spot where the animals were abandoned, camp is made for\nthe night. Next morning the way is retraced. Not the most fervid\npictures of a poet's fancy could transcend the glories then revealed in\nthe depths of the canyon; inky shadows, pale gildings of lofty spires,\ngolden splendors of sun beating full on fa\u00e7ades of red and yellow,\nobscurations of distant peaks by veils of transient shower, glimpses of\nwhite towers half drowned in purple haze, suffusions of rosy light\nblended in reflection from a hundred tinted walls. Caught up to exalted\nemotional heights the beholder becomes unmindful of fatigue. He mounts\non wings. He drives the chariot of the sun.\n\n [Illustration: Uncaptioned vista]\n\nHaving returned to the plateau, it will be found that the descent into\nthe canyon has bestowed a sense of intimacy that almost amounts to a\nmental grasp of the scene. The terrific deeps that part the walls of\nhundreds of castles and turrets of mountainous bulk may be approximately\nlocated in barely discernible pen-strokes of detail, and will be\napprehended mainly through the memory of upward looks from the bottom,\nwhile towers and obstructions and yawning fissures that were deemed\nevents of the trail will be wholly indistinguishable, although they are\nknown to lie somewhere flat beneath the eye. The comparative\ninsignificance of what are termed grand sights in other parts of the\nworld is now clearly revealed. Twenty Yosemites might lie unperceived\nanywhere below. Niagara, that Mecca of marvel seekers, would not here\npossess the dignity of a trout stream. Your companion, standing at a\nshort distance on the verge, is an insect to the eye.\n\nStill, such particulars can not long hold the attention, for the\npanorama is the real overmastering charm. It is never twice the same.\nAlthough you think you have spelt out every temple and peak and\nescarpment, as the angle of sunlight changes there begins a ghostly\nadvance of colossal forms from the farther side, and what you had taken\nto be the ultimate wall is seen to be made up of still other isolated\nsculptures, revealed now for the first time by silhouetting shadows. The\nscene incessantly changes, flushing and fading, advancing into\ncrystalline clearness, retiring into slumberous haze.\n\nShould it chance to have rained heavily in the night, next morning the\ncanyon is completely filled with fog. As the sun mounts, the curtain of\nmist suddenly breaks into cloud fleeces, and while you gaze these\nfleeces rise and dissipate, leaving the canyon bare. At once around the\nbases of the lowest cliffs white puffs begin to appear, creating a scene\nof unparalleled beauty as their dazzling cumuli swell and rise and their\nnumber multiplies, until once more they overflow the rim, and it is as\nif you stood on some land's end looking down upon a formless void. Then\nquickly comes the complete dissipation, and again the marshaling in the\ndepths, the upward advance, the total suffusion and the speedy\nvanishing, repeated over and over until the warm walls have expelled\ntheir saturation.\n\nLong may the visitor loiter upon the verge, powerless to shake loose\nfrom the charm, tirelessly intent upon the silent transformations until\nthe sun is low in the west. Then the canyon sinks into mysterious purple\nshadow, the far Shinumo Altar is tipped with a golden ray, and against a\nleaden horizon the long line of the Echo Cliffs reflects a soft\nbrilliance of indescribable beauty, a light that, elsewhere, surely\nnever was on sea or land. Then darkness falls, and should there be a\nmoon, the scene in part revives in silver light, a thousand spectral\nforms projected from inscrutable gloom; dreams of mountains, as in their\nsleep they brood on things eternal.\n\n [Illustration: Uncaptioned vista]\n\n [Illustration: Uncaptioned vista]\n\n\n\n\n THE SCIENTIFIC EXPLORER\n BY J. W. POWELL\n\n\n The Ives and Wheeler Expeditions\n\nIn the fall of 1857 Lieutenant Ives, of the engineer corps of the army,\nascended the Colorado River on a trip of exploration with a little\nsteamer called the Explorer; he went as far as the mouth of the Rio\nVirgin. Falling back down river a few miles, Lieutenant Ives met a pack\ntrain which had followed him up the bank of the stream. Here he\ndisembarked, and on the 24th of March started with a land party to\nexplore the eastern bank of the river; making a long detour he ascended\nthe plateau through which the Grand Canyon is cut, and in an adventurous\njourney he obtained views of the canyon along its lower course. On this\ntrip J. S. Newberry was the geologist, and to him we are indebted for\nthe first geological explanation of the canyon and the description of\nthe high plateau through which it is formed. Doctor Newberry was not\nonly an able geologist, but he was also a graphic writer, and his\ndescription of the canyon as far as it was seen by him is a classic in\ngeology.\n\nIn 1869 Lieutenant Wheeler was sent out by the chief engineer of the\narmy to explore the Grand Canyon from below. In the spring he succeeded\nin reaching the mouth of Diamond Creek, which had previously been seen\nby Doctor Newberry in 1858. Mr. Gilbert was the geologist of this\nexpedition, and his studies of the canyon region during this and\nsubsequent years have added greatly to our knowledge of this land of\nwonders.\n\n\n Major Powell's Several Trips\n\nIn this same year I essayed to explore the Grand Canyon of the Colorado,\ntogether with the upper canyons of that stream and the great canyons of\nthe lower portion of Green River. For this purpose I employed four\nrowboats and made the descent from what is now Green River station\nthrough the whole course of canyons to the mouth of the Rio Virgin, a\ndistance of more than a thousand miles.\n\n [Illustration: From Kaibab Plateau, Looking South.]\n\nIn the spring of 1870 I again started with three boats and descended the\nriver to the Crossing of the Fathers, where I met a pack train and went\nout with a party of men to explore ways down into the Grand Canyon from\nthe north, and devoted the summer, fall, winter, and following spring to\nthis undertaking.\n\nIn the summer of 1871 I returned to the rowboats and descended through\nMarble Canyon to the Grand Canyon of Arizona, and then through the\ngreater part of the Grand Canyon itself. Subsequent years were then\ngiven to exploration of the country adjacent to the Grand Canyon. On\nthese trips Mr. Gilbert, the geologist, who had been with Lieutenant\nWheeler, and Capt. C. E. Dutton, were my geological companions. On the\nsecond boat trip, and during all the subsequent years of exploration in\nthis region, Prof. A. H. Thompson was my geographical companion,\nassisted by a number of topographical engineers.\n\nIn 1882 Mr. C. D. Walcott, as my assistant in the United States\nGeological Survey, went with me into the depths of the Grand Canyon. We\ndescended from the summit of the Kaibab Plateau on the north by a trail\nwhich we built down a side canyon in a direction toward the mouth of the\nLittle Colorado River. The descent was made in the fall, and a small\nparty of men was left with Mr. Walcott in this region of stupendous\ndepths to make a study of the geology of an important region of\nlabyrinthian gorges. Here, with his party, he was shut up for the\nwinter, for it was known when we left him that snows on the summit of\nthe plateau would prevent his return to the upper region before the sun\nshould melt them the next spring. Mr. Walcott is now the Director of the\nUnited States Geological Survey.\n\nAfter this year I made no substantial additions to my geologic and\nscenic knowledge of the Grand Canyon, though I afterward studied the\narch\u00e6ology to the south and east throughout a wide region of ruined\npueblos and cliff dwellings.\n\nSince my first trip in boats many others have essayed to follow me, and\nyear by year such expeditions have met with disaster; some hardy\nadventurers are buried on the banks of the Green, and the graves of\nothers are scattered at intervals along the course of the Colorado.\n\nIn 1889 the brave F. M. Brown lost his life. But finally a party of\nrailroad engineers, led by R. B. Stanton, started at the head of Marble\nCanyon and made their way down the river as they extended a survey for a\nrailroad along its course.\n\nOther adventurous travelers have visited portions of the Grand Canyon\nregion, and Mr. G. Wharton James has extended his travels widely over\nthe region in the interest of popular science and the new literature\ncreated in the last decades of the nineteenth century. And now I once\nmore return to a reminiscent account of the Grand Canyon, for old men\nlove to talk of the past.\n\n\n The Plateau Region\n\nThe Grand Canyon of Arizona and the Marble Canyon constitute one great\ngorge carved by a mighty river through a high plateau. On the northeast\nand north a line of cliffs face this plateau by a bold escarpment of\nrock. Climb these cliffs and you must ascend from 800 to 1,000 feet, but\non their summit you will stand upon a plateau stretching away to the\nnorth. Now turn to face the south and you will overlook the cliff and\nwhat appears to be a valley below. From the foot of the cliff the\ncountry rises to the south to a great plateau through which the Marble\nand the Grand canyons are carved. This plateau terminates abruptly on\nthe west by the Grand Wash Cliffs, which is a high escarpment caused by\na \"fault\" (as the geologist calls it), that is, the strata of sandstone\nand limestone are broken off, and to the west of the fracture they are\ndropped down several thousand feet, so that standing upon the edge of\nthe plateau above the Grand Wash Cliffs you may look off to the west\nover a vast region of desert from which low volcanic mountains rise that\nseem like purple mounds in sand-clad lands.\n\nOn the east the great plateau breaks down in a very irregular way into\nthe valley of the Little Colorado, and where the railroad ascends the\nplateau from the east it passes over picturesque canyons that run down\ninto the Little Colorado. On the south the plateau is merged into the\ngreat system of mountains that stand in Southern Arizona. Where the\nplateau ends and the mountains begin is not a well-defined line. The\nplateau through which the Grand Canyon is cut is a region of great\nscenic interest. Its surface is from six to more than eight thousand\nfeet above the level of the sea. The Grand Plateau is composed of many\nsubsidiary plateaus, each one having its own peculiar and interesting\nfeature.\n\nThe Kaibab Plateau, to the northeast of the Grand Canyon, is covered\nwith a pine forest which is intercepted by a few meadows with here and\nthere a pond or lakelet. It is the home of deer and bear.\n\nTo the west is the Shinumo Plateau in which the Shinumo Canyon is\ncarved; and on the cliffs of this canyon and in the narrow valley along\nits course the Shinumo ruins are found\u2014the relics of a prehistoric race.\n\nTo the west of the Shinumo Plateau is the Kanab Plateau, with ruins\nscattered over it, and on its northern border the beautiful Mormon town\nof Kanab is found, and the canyon of Kanab Creek separates the Shinumo\nPlateau from the Kanab Plateau. It begins as a shallow gorge and\ngradually increases in depth until it reaches the Colorado River itself,\nat a depth of more than 4,000 feet below the surface. Vast amphitheaters\nare found in its walls and titanic pinnacles rise from its depths. One\nChristmas day I waded up this creek. It was one of the most delightful\nwalks of my life, from a land of flowers to a land of snow.\n\nTo the west of the Kanab Plateau are the Uinkaret Mountains\u2014an immense\ngroup of volcanic cones upon a plateau. Some of these cones stand very\nnear the brink of the Grand Canyon and from one of them a flood of\nbasalt was poured into the canyon itself. Not long ago geologically, but\nrather long when reckoned in years of human history, this flood of lava\nrolled down the canyon for more than _fifty miles_, filling it to the\ndepth of _two_ or _three hundred feet_ and diverting the course of the\nriver against one or the other of its banks. Many of the cones are of\nred cinder, while sometimes the lava is piled up into huge mountains\nwhich are covered with forest. To the west of the Uinkaret Mountains\nspreads the great Shiwits Plateau, crowned by Mount Dellenbough.\n\nPast the south end of these plateaus runs the Colorado River; southward\nthrough Marble Canyon and in the Grand Canyon, then northwestward past\nthe Kaibab and Shinumo Canyon, then southwestward past the Kanab\nPlateau, Uinkaret Mountains to the southernmost point of the Shiwits\nPlateau, and then northwestward to the Grand Wash Cliffs. Its distance\nin this course is little more than 300 miles\u2014but the 300 miles of river\nare set on every side with cliffs, buttes, towers, pinnacles,\namphitheaters, caves, and terraces, exquisitely storm-carved and painted\nin an endless variety of colors.\n\nThe plateau to the south of the Grand Canyon, which we need not describe\nin parts, is largely covered with a gigantic forest. There are many\nvolcanic mountains and many treeless valleys. In the high forest there\nare beautiful glades with little stretches of meadow which are spread in\nsummer with a parterre of flowers of many colors. This upper region is\nthe garden of the world. When I was first there bear, deer, antelope,\nand wild turkeys abounded, but now they are becoming scarce. Widely\nscattered throughout the plateau are small canyons, each one a few miles\nin length and a few hundred feet in depth. Throughout their course\ncliff-dweller ruins are found. In the highland glades and along the\nvalley, pueblo ruins are widely scattered, but the strangest sights of\nall the things due to prehistoric man are the cave dwellings that are\ndug in the tops of cinder cones and the villages that were built in the\ncaves of volcanic cliffs. If now I have succeeded in creating a picture\nof the plateau I will attempt a brief description of the canyon.\n\n [Illustration: Copyright, 1899, by H. G. Peabody.\n Bissell Point and Colorado River.]\n\n\n Marble Canyon\n\nAbove the Paria the great river runs down a canyon which it has cut\nthrough one plateau. On its way it flows with comparative quiet through\nbeautiful scenery, with glens that are vast amphitheaters which often\noverhang great springs and ponds of water deeply embosomed in the\ncliffs. From the southern escarpment of this plateau the great Colorado\nPlateau rises by a comparatively gentle acclivity, and Marble Canyon\nstarts with walls but a few score feet in height until they reach an\naltitude of about 5,000 feet. On the way the channel is cut into beds of\nrock of lower geologic horizon, or greater geologic age. These rocks are\nsandstones and limestones. Some beds are very hard, others are soft and\nfriable. The friable rocks wash out and the harder rocks remain\nprojecting from the walls, so that every wall presents a set of stony\nshelves. These shelves rise along the wall toward the south as new\nshelves set in from below.\n\nIn addition to this shelving structure the walls are terraced and the\ncliffs of the canyon are set back one upon the other. Then these canyon\nwalls are interrupted by side streams which themselves have carved\nlateral canyons, some small, others large, but all deep. In these side\ngorges the scenery is varied and picturesque; deep clefts are seen here\nand there as you descend the river\u2014clefts furnished with little streams\nalong which mosses and other plants grow. At low water the floor of the\ngreat canyon is more or less exposed, and where it flows over limestone\nrocks beautiful marbles are seen in many colors; saffron, pink, and blue\nprevail. Sometimes a fa\u00e7ade or wall appears rising vertically from the\nwater for thousands of feet. At last the canyon abruptly ends in a\nconfusion of hills beyond which rise towering cliffs, and the group of\nhills are nestled in the bottom of a valley-like region which is\nsurrounded by cliffs more than a mile in altitude.\n\n\n The Grand Canyon\n\nFrom here on for many miles the whole character of the canyon changes.\nFirst a dike appears; this is a wall of black basalt crossing the river;\nit is of lava thrust up from below through a huge crevice broken in the\nrock by earthquake agency. On the east the Little Colorado comes; here\nit is a river of salt water, and it derives its salt a few miles up the\nstream. The main Colorado flows along the eastern and southern wall.\nClimbing this for a few hundred feet you may look off toward the\nnorthwest and gaze at the cliffs of the Kaibab Plateau.\n\nThis is the point where we built a trail down a side canyon where Mr.\nWalcott was to make his winter residence and study of the region; it is\nvery complicated and exhibits a vast series of unconformable rocks of\nhigh antiquity. These lower rocks are of many colors; in large part they\nare shales. The region, which appears to be composed of bright-\nhills washed naked by the rain, is, in fact, beset with a multitude of\nwinding canyons with their own precipitous walls. It is a region of many\ncanyons in the depths of the Grand Canyon itself.\n\nIn this beautiful region Mr. Walcott, reading the book of geology, lived\nin a summerland during all of a long winter while the cliffs above were\ncovered with snow which prevented his egress to the world. His\ncompanions, three young Mormons, longing for a higher degree of\ncivilization, gazed wistfully at the snow-clad barriers by which they\nwere inclosed. One was a draughtsman, another a herder of his stock, and\nthe third his cook. They afterward told me that it was a long winter of\nhomesickness, and that months dragged away as years, but Mr. Walcott\nhimself had the great book of geology to read, and to him it was a\nwinter of delight.\n\nA half dozen miles below the basaltic wall the river enters a channel\ncarved in 800 or 1,000 feet of dark gneiss of very hard rock. Here the\nchannel is narrow and very swift and beset with rapids and falls. On the\nsouth and southwest the wall rises abruptly from the water to the summit\nof the plateau for about 6,000 feet, but across the river on the north\nand west mountains of gneiss and quartzites appear, sometimes rising to\nthe height of a thousand feet. These are mountains in the bottom of a\ncanyon. The buttes and plateaus of the inter-canyon region are composed\nof shales, sandstones, and limestones, which give rise to vast\narchitectural shelving and to pinnacles and towers of gigantic\nproportions, the whole embossed with a marvelously minute system of\nfretwork carved by the artistic clouds. Looking beyond these mountains,\nbuttes, and plateaus vistas of the walls of the great plateau are seen.\nFrom these walls project salients, and deep re-entrant angles appear.\n\nThe whole scene is forever reminding you of mighty architectural\npinnacles and towers and balustrades and arches and columns with lattice\nwork and delicate carving. All of these architectural features are\nsublime by titanic painting in varied hues\u2014pink, red, brown, lavender,\ngray, blue, and black. In some lights the saffron prevails, in other\nlights vermilion, and yet in other lights the grays and blacks\npredominate. At times, and perhaps in rare seasons, clouds and cloudlets\nform in the canyon below and wander among the side canyons and float\nhigher and higher until they are dissolved in the upper air, or perhaps\nthey accumulate to hide great portions of the landscape. Then through\nrifts in the clouds vistas of Wonderland are seen. Such is that portion\nof the canyon around the great south bend of the Colorado River past the\npoint of the Kaibab Plateau.\n\n\n As Seen by the Geologist\n\nIn the last chapter of my book entitled \"The Canyons of the Colorado\" I\nhave described the Grand Canyon in the following terms:\n\nThe Grand Canyon is a gorge 217 miles in length, through which flows a\ngreat river with many storm-born tributaries. It has a winding way, as\nrivers are wont to have. Its banks are vast structures of adamant, piled\nup in forms rarely seen in the mountains.\n\nDown by the river the walls are composed of black gneiss, slates, and\nschists, all greatly implicated and traversed by dikes of granite. Let\nthis formation be called the black gneiss. It is usually about 800 feet\nin thickness.\n\nThen over the black gneiss are found 800 feet of quartzites, usually in\nvery thin beds of many colors, but exceedingly hard, and ringing under\nthe hammer like phonolite. These beds are dipping and unconformable with\nthe rocks above. While they make but 800 feet of the wall or less they\nhave a geologic thickness of 12,000 feet. Set up a row of books aslant;\nit is ten inches from the shelf to the top of the line of books, but\nthere may be three feet of the books measured directly through the\nleaves. So these quartzites are aslant, and though of great geologic\nthickness they make but 800 feet of the wall. Your books may have\nmany- bindings and differ greatly in their contents; so these\nquartzites vary greatly from place to place along the wall, and in many\nplaces they entirely disappear. Let us call this formation the\nvariegated quartzite.\n\nAbove the quartzites there are 500 feet of sandstones. They are of a\ngreenish hue, but are mottled with spots of brown and black by iron\nstains. They usually stand in a bold cliff, weathered in alcoves. Let\nthis formation be called the cliff sandstone.\n\nAbove the cliff sandstone there are 700 feet of bedded sandstones and\nlimestones, which are massive sometimes and sometimes broken into thin\nstrata. These rocks are often weathered in deep alcoves. Let this\nformation be called the alcove sandstone.\n\nOver the alcove sandstone there are 1,600 feet of limestone, in many\nplaces a beautiful marble, as in Marble Canyon. As it appears along the\nGrand Canyon it is always stained a brilliant red, for immediately over\nit there are thin seams of iron, and the storms have painted these\nlimestones with pigments from above. Altogether this is the red-wall\ngroup. It is chiefly limestone. Let it be called the red-wall limestone.\n\nAbove the red wall there are 800 feet of gray and bright red sandstone,\nalternating in beds that look like vast ribbons of landscape. Let it be\ncalled the banded sandstone.\n\nAnd over all, at the top of the wall, is the Aubrey limestone, 1,000\nfeet in thickness. This Aubrey has much gypsum in it, great beds of\nalabaster that are pure white in comparison with the great body of\nlimestone below. In the same limestone there are enormous beds of chert,\nagates, and carnelians. This limestone is especially remarkable for its\npinnacles and towers. Let it be called the tower limestone.\n\nThese are the elements with which the walls are constructed, from black\nbuttress below to alabaster tower above. All of these elements weather\nin different forms and are painted in different colors, so that the wall\npresents a highly complex fa\u00e7ade. A wall of homogeneous granite, like\nthat in the Yosemite, is but a naked wall, whether it be 1,000 or 5,000\nfeet high. Hundreds and thousands of feet mean nothing to the eye when\nthey stand in a meaningless front. A mountain covered by pure snow\n10,000 feet high has but little more effect on the imagination than a\nmountain of snow 1,000 feet high\u2014it is but more of the same thing\u2014but a\nfa\u00e7ade of seven systems of rock has its sublimity multiplied sevenfold.\n\n [Illustration: A Panoramic View of the Canyon.]\n\nConsider next the horizontal elements of the Grand Canyon. The river\nmeanders in great curves, which are themselves broken into curves of\nsmaller magnitude. The streams that head far back in the plateau on\neither side come down in gorges and break the wall into sections. Each\nlateral canyon has a secondary system of laterals, and the secondary\ncanyons are broken by tertiary canyons; so the crags are forever\nbranching, like the limbs of an oak. That which has been described as a\nwall is such only in its grand effect. In detail it is a series of\nstructures separated by a ramification of canyons, each having its own\nwalls. Thus, in passing down the canyon it seems to be inclosed by\nwalls, but oftener by salients\u2014towering structures that stand between\ncanyons that run back into the plateau. Sometimes gorges of the second\nor third order have met before reaching the brink of the Grand Canyon,\nand then great salients are cut off from the wall and stand out as\nbuttes\u2014huge pavilions in the architecture of the canyon. The scenic\nelements thus described are fused and combined in very different ways.\n\n\n Its Length\n\nWe measured the length of the Grand Canyon by the length of the river\nrunning through it, but the running extent of wall can not be measured\nin this manner. In the black gneiss, which is at the bottom, the wall\nmay stand above the river for a few hundred yards or a mile or two; then\nto follow the foot of the wall you must pass into a lateral canyon for a\nlong distance, perhaps miles, and then back again on the other side of\nthe lateral canyon; then along by the river until another lateral canyon\nis reached, which must be headed in the black gneiss. So for a dozen\nmiles of river through the gneiss there may be a hundred miles of wall\non either side. Climbing to the summit of the black gneiss and following\nthe wall in the variegated quartzite, it is found to be stretched out to\na still greater length, for it is cut with more lateral gorges. In like\nmanner there is yet greater length of the mottled (or alcove) sandstone\nwall, and the red wall is still farther stretched out in ever-branching\ngorges.\n\nTo make the distance for ten miles along the river by walking along the\ntop of the red wall it would be necessary to travel several hundred\nmiles. The length of the wall reaches its maximum in the banded\nsandstone, which is terraced more than any of the other formations. The\ntower limestone wall is less tortuous. To start at the head of the Grand\nCanyon on one of the terraces of the banded sandstone and follow it to\nthe foot of the Grand Canyon, which by river is a distance of 217 miles,\nit would be necessary to travel many thousand miles by the winding way;\nthat is, the banded wall is many thousand miles in length.\n\n\n As Seen Traveling Down Stream\n\nFor eight or ten miles below the mouth of the Little Colorado, the river\nis in the variegated quartzites, and a wonderful fretwork of forms and\ncolors, peculiar to this rock, stretches back for miles to a labyrinth\nof the red-wall cliff; then below, the black gneiss is entered and soon\nhas reached an altitude of 800 feet and sometimes more than 1,000 feet,\nand upon this black gneiss all the other structures in their wonderful\ncolors are lifted. These continue for about seventy miles, when the\nblack gneiss below is lost, for the walls are dropped down by the West\nKaibab Fault and the river flows in the quartzites.\n\nThen for eighty miles the mottled (or alcove) sandstones are found in\nthe river bed. The course of the canyon is a little south of west and is\ncomparatively straight. At the top of the red-wall limestone there is a\nbroad terrace, two or three miles in width, composed of hills of\nwonderful forms carved in the banded beds, and back of this is seen a\ncliff in the tower limestone. Along the lower course of this stretch the\nwhole character of the canyon is changed by another set of complicating\nconditions. We have now reached a region of volcanic activity. After the\ncanyons were cut nearly to their present depth, lavas poured out and\nvolcanoes were built on the walls of the canyon, but not in the canyon\nitself, though at places rivers of molten rock rolled down the walls\ninto the Colorado.\n\nThe canyon for the next eighty miles is a compound of that found where\nthe river is in the black gneiss and that found where the dead volcanoes\nstand on the brink of the wall. In the first stretch, where the gneiss\nis at the foundation, we have a great bend to the south, and in the last\nstretch, where the gneiss is below and the dead volcanoes above, another\ngreat southern detour is found. These two great beds are separated by\neighty miles of comparatively straight river.\n\nLet us call this first great bend the Kaibab reach of the canyon, and\nthe straight part the Kanab reach, for the Kanab Creek heads far off in\nthe plateau to the north and joins the Colorado at the beginning of the\nmiddle stretch. The third great southern bend is the Shiwits stretch.\nThus there are three distinct portions of the Grand Canyon: The Kaibab\nsection, characterized more by its buttes and salients; the Kanab\nsection, characterized by its comparatively straight walls with\nvolcanoes on the brink, and the Shiwits section, which is broken into\ngreat terraces with gneiss at the bottom and volcanoes at the top.\n\n\n The Work of Erosion\n\nThe erosion represented in the canyons, although vast, is but a small\npart of the great erosion of the region, for between the cliffs blocks\nhave been carried away far superior in magnitude to those necessary to\nfill the canyons. Probably there is no portion of the whole region from\nwhich there have not been more than a thousand feet degraded, and there\nare districts from which more than 30,000 feet of rock have been carried\naway; altogether there is a district of country more than 200,000 square\nmiles in extent, from which, on the average, more than 6,000 feet have\nbeen eroded. Consider a rock 200,000 square miles in extent and a mile\nin thickness, against which the clouds have hurled their storms, and\nbeat it into sands, and the rills have carried the sands into the\ncreeks, and the creeks have carried them into the rivers, and the\nColorado has carried them into the sea.\n\nWe think of the mountains as forming clouds about their brows, but the\nclouds have formed the mountains. Great continental blocks are upheaved\nfrom beneath the sea by internal geologic forces that fashion the earth.\nThen the wandering clouds, the tempest-bearing clouds, the\nrainbow-decked clouds, with mighty power and with wonderful skill, carve\nout valleys and canyons and fashion hills and cliffs and mountains. The\nclouds are the artists sublime.\n\n\n Winter and Cloud Effects\n\nIn winter some of the characteristics of the Grand Canyon are\nemphasized. The black gneiss below, the variegated quartzite, and the\ngreen or alcove sandstone form the foundation for the mighty red wall.\nThe banded sandstone entablature is crowned by the tower limestone. In\nwinter this is covered with snow. Seen from below, these changing\nelements seem to graduate into the heavens, and no plane of demarcation\nbetween wall and blue firmament can be seen. The heavens constitute a\nportion of the fa\u00e7ade and mount into a vast dome from wall to wall,\nspanning the Grand Canyon with empyrean blue. So the earth and the\nheavens are blended in one vast structure.\n\n [Illustration: Copyright, 1899, by H. G. Peabody.\n The Lower Gorge, Foot of Bright Angel Trail.]\n\nWhen the clouds play in the canyon, as they often do in the rainy\nseason, another set of effects is produced. Clouds creep out of canyons\nand wind into other canyons. The heavens seem to be alive, not moving as\nmove the heavens over a plain, in one direction with the wind, but\nfollowing the multiplied courses of these gorges. In this manner the\nlittle clouds seem to be individualized, to have wills and souls of\ntheir own and to be going on diverse errands\u2014a vast assemblage of\nself-willed clouds faring here and there, intent upon purposes hidden in\ntheir own breasts. In imagination the clouds belong to the sky, and when\nthey are in the canyon the skies come down into the gorges and cling to\nthe cliffs and lift them up to immeasurable heights, for the sky must\nstill be far away. Thus they lend infinity to the walls.\n\nYou can not see the Grand Canyon in one view as if it were a changeless\nspectacle from which a curtain might be lifted, but to see it you have\nto toil from month to month through its labyrinths. It is a region more\ndifficult to traverse than the Alps or the Himalayas, but if strength\nand courage are sufficient for the task, by a year's toil a concept of\nsublimity can be obtained never again to be equaled on the hither side\nof paradise.\n\n [Illustration: Copyright, 1899, by H. G. Peabody.\n On Grand View Point.]\n\n [Illustration: Uncaptioned vista]\n\n\n\n\n THE GREATEST THING IN THE WORLD\n BY CHARLES F. LUMMIS\n\n\n\"The greatest thing in the world.\" That is a large phrase and an\nover-worked one, and hardened travelers do not take it lightly upon the\ntongue. Noticeably it is most glibly in use with those but lately, and\nfor the first time, wandered beyond their native state or county, and as\nevery province has its own local brag of biggest things, the too\ncredulous tourist will find a superlative everywhere. And superlatives\nare unsafe without wide horizons of comparison.\n\nYet in every sort there is, of course, somewhere \"the biggest thing in\nthe world\" of its kind. It is a good word, when spoken in season and not\nabused in careless ignorance.\n\nI believe there is and can be no dispute that the term applies literally\nto several things in the immediate region of the Grand Canyon of\nArizona. As I have more than once written (and it never yet has been\ncontroverted), probably no other equal area on earth contains so many\nsupreme marvels of so many kinds\u2014so many astounding sights, so many\nmasterpieces of Nature's handiwork, so vast and conclusive an\nencyclopedia of the world-building processes, so impressive monuments of\nprehistoric man, so many triumphs of man still in the tribal relation\u2014as\nwhat I have called the Southwestern Wonderland. This includes a large\npart of New Mexico and Arizona, the area which geographically and\nethnographically we may count as the Grand Canyon region. Let me mention\na few wonders:\n\nThe largest and by far the most beautiful of all petrified forests, with\nseveral hundred square miles whose surface is carpeted with agate chips\nand dotted with agate trunks two to four feet in diameter; and just\nacross one valley a buried \"forest\" whose huge silicified\u2014not\nagatized\u2014logs show their ends under fifty feet of sandstone.\n\nThe largest natural bridge in the world\u2014200 feet high, over 500 feet\nspan, and over 600 feet wide, up and down stream, and with an orchard on\nits top and miles of stalactite caves under its abutments.\n\nThe largest variety and display of geologically recent volcanic action\nin North America; with 60-mile lava flows, 1,500-foot blankets of creamy\ntufa cut by scores of canyons; hundreds of craters and thousands of\nsquare miles of lava beds, basalt, and cinders, and so much \"volcanic\nglass\" (obsidian) that it was the chief tool of the prehistoric\npopulation.\n\nThe largest and the most impressive villages of cave-dwellings in the\nworld, most of them already abandoned \"when the world-seeking Genoese\"\nsailed.\n\nThe peerless and many-storied cliff-dwellings\u2014castles and forts and\nhomes in the face of wild precipices or upon their tops\u2014an aboriginal\narchitecture as remarkable as any in any land.\n\nThe twenty-six strange communal town republics of the descendants of the\n\"cliff-dwellers,\" the modern Pueblos; some in fertile valleys, some\n(like Acoma and Moki) perched on barren and dizzy cliff tops. The\nstrange dances, rites, dress, and customs of this ancient people who had\nsolved the problem of irrigation, 6-story house building, and clean\nself-government, and even women's rights\u2014long before Columbus was born.\n\nThe noblest Caucasian ruins in America, north of Mexico\u2014the great stone\nand adobe churches reared by Franciscan missionaries, near three\ncenturies ago, a thousand miles from the ocean, in the heart of the\nSouthwest.\n\nSome of the most notable tribes of savage nomads\u2014like the Navajos, whose\nblankets and silver work are pre-eminent, and the Apaches, who, man for\nman, have been probably the most successful warriors in history.\n\nAll these, and a great deal more, make the Southwest a wonderland\nwithout a parallel. There are ruins as striking as the storied ones\nalong the Rhine, and far more remarkable. There are peoples as\npicturesque as any in the Orient, and as romantic as the Aztecs and the\nIncas of whom we have learned such gilded fables, and there are natural\nwonders which have no peers whatever.\n\n\n Of the Canyon, and Other Wonders\n\nAt the head of the list stands the Grand Canyon of the Colorado; whether\nit is the \"greatest wonder of the world\" depends a little on our\ndefinition of \"wonder.\" Possibly it is no more wonderful than the fact\nthat so tiny a fraction of the people who confess themselves the\nsmartest in the world have ever seen it. As a people we dodder abroad to\nsee scenery incomparably inferior.\n\nBut beyond peradventure it is the greatest chasm in the world, and the\nmost superb. Enough globe-trotters have seen it to establish that fact.\nMany have come cynically prepared to be disappointed; to find it\noverdrawn and really not so stupendous as something else. It is, after\nall, a hard test that so be-bragged a wonder must endure under the\ncritical scrutiny of them that have seen the earth and the fullness\nthereof. But I never knew the most self-satisfied veteran traveler to be\ndisappointed in the Grand Canyon, or to patronize it. On the contrary,\nthis is the very class of men who can best comprehend it, and I have\nseen them fairly break down in its awful presence.\n\nI do not know the Himalayas except by photograph and the testimony of\nmen who have explored and climbed them, and who found the Grand Canyon\nan absolutely new experience. But I know the American continents pretty\nwell, and have tramped their mountains, including the Andes\u2014the next\nhighest mountains in the world, after half a dozen of the Himalayas\u2014and\nof all the famous quebradas of the Andes there is not one that would\ncount 5 per cent on the Grand Canyon of the Colorado. For all their\n25,000-foot peaks, their blue-white glaciers, imminent above the bald\nplateau, and green little bolsones (\"pocket valleys\") of Chile, Peru,\nBolivia, and Ecuador; for all their tremendous active volcanoes, like\nSaugay and Cotopaxi; for all an earthquake activity beside which the\n\"shake\" at Charleston was mere paper-doll play; for all the steepest\ngradients in the world (and Peru is the only place in the world where a\nriver falls 17,000 feet in 100 miles)\u2014in all that marvelous 3,000-mile\nprocession of giantism there is not one canyon which any sane person\nwould for an instant compare with that titanic gash that the Colorado\nhas chiseled through a comparatively flat upland. Nor is there anything\nremotely approaching it in all the New World. So much I can say at first\nhand. As for the Old World, the explorer who shall find a gorge there\none-half as great will win undying fame.\n\nThe quebrada of the Apu-Rimac is a marvel of the Andes, with its\nvertiginous depths and its suspension bridge of wild vines. The Grand\nCanyon of the Arkansas, in Colorado, is a noble little slit in the\nmountains. The Franconia and White Mountain notches in New Hampshire are\nbeautiful. The Yosemite and the Yellowstone canyons surpass the world,\neach in its way. But if all of these were hung up on the opposite wall\nof the Grand Canyon from you the chances are fifty to one that you could\nnot tell t'other from which, nor any of them from the hundreds of other\ncanyons which rib that vast vertebrate gorge. If the falls of Niagara\nwere installed in the Grand Canyon between your visits and you knew it\nby the newspapers\u2014next time you stood on that dizzy rimrock you would\nprobably need good field-glasses and much patience before you could\nlocate that cataract which in its place looks pretty big. If Mount\nWashington were plucked up bodily by the roots\u2014not from where you see\nit, but from sea-level\u2014and carefully set down in the Grand Canyon, you\nprobably would not notice it next morning, unless its dull colors\ndistinguished it in that innumerable congress of larger and painted\ngiants.\n\nAll this, which is literally true, is a mere trifle of what might be\nsaid in trying to fix a standard of comparison for the Grand Canyon. But\nI fancy there is no standard adjustable to the human mind. You may\ncompare all you will\u2014eloquently and from wide experience, and at last\nall similes fail. The Grand Canyon is just the Grand Canyon, and that is\nall you can say. I never have seen anyone who was prepared for it. I\nnever have seen anyone who could grasp it in a week's hard exploration;\nnor anyone, except some rare Philistine, who could even think he had\ngrasped it. I have seen people rave over it; better people struck dumb\nwith it, even strong men who cried over it; but I have never yet seen\nthe man or woman that _expected_ it.\n\nIt adds seriously to the scientific wonder and the universal\nimpressiveness of this unparalleled chasm that it is not in some\nstupendous mountain range, but in a vast, arid, lofty floor of nearly\n100,000 square miles\u2014as it were, a crack in the upper story of the\ncontinent. There is no preparation for it. Unless you had been told, you\nwould no more dream that out yonder amid the pines the flat earth is\nslashed to its very bowels, than you would expect to find an iceberg in\nBroadway. With a very ordinary running jump from the spot where you get\nyour first glimpse of the canyon you could go down 2,000 feet without\ntouching. It is sudden as a well.\n\nBut it is no mere cleft. It is a terrific trough 6,000 to 7,000 feet\ndeep, ten to twenty miles wide, hundreds of miles long, peopled with\nhundreds of peaks taller than any mountain east of the Rockies, yet not\none of them with its head so high as your feet, and all ablaze with such\ncolor as no eastern or European landscape ever knew, even in the\nAlpen-glow. And as you sit upon the brink the divine scene-shifters give\nyou a new canyon every hour. With each degree of the sun's course the\ngreat countersunk mountains we have been watching fade away, and new\nones, as terrific, are carved by the westering shadows. It is like a\ndissection of the whole cosmogony. And the purple shadows, the dazzling\nlights, the thunderstorms and snowstorms, the clouds and the rainbows\nthat shift and drift in that vast subterranean arena below your feet!\nAnd amid those enchanted towers and castles which the vastness of the\nscale leads you to call \"rocks,\" but which are in fact as big above the\nriver-bed as the Rockies from Denver, and bigger than Mount Washington\nfrom Fabyan's or the Glen!\n\nThe Grand Canyon country is not only the hugest, but the most varied and\ninstructive example on earth of one of the chief factors of\nearth-building\u2014erosion. It is the mesa country\u2014the Land of Tables.\nNowhere else on the footstool is there such an example of deep-gnawing\nwater or of water high-carving. The sandstone mesas of the Southwest,\nthe terracing of canyon walls, the castellation, battlementing, and\ncliff-making, the cutting down of a whole landscape except its\nprecipitous islands of flat-topped rock, the thin lava table-cloths on\ntables 100 feet high\u2014these are a few of the things which make the\nSouthwest wonderful alike to the scientist and the mere sight-seer.\n\nThat the canyon is not \"too hard\" is perhaps sufficiently indicated by\nthe fact that I have taken thither ladies and children and men in their\nseventies, when the easiest way to get there was by a 70-mile stage\nride, and that at six years old my little girl walked all the way from\nrim to bottom of canyon and came back on a horse the same day, and was\nnext morning ready to go on a long tramp along the rim.\n\n [Illustration: Copyright, 1899, by H. G. Peabody.\n The North Wall from Grand Scenic Divide.]\n\n [Illustration: Uncaptioned vista]\n\n\n\n\n INFORMATION FOR TOURISTS\n\n\n Preliminary\n\nThere is only one way by which to directly reach the Grand Canyon of\nArizona, and that is via the Santa Fe (The Atchison, Topeka & Santa Fe\nRailway System).\n\nThere are three ways of reaching the Canyon from the Santa Fe\u2014rail from\nWilliams, private conveyance from Flagstaff and Peach Springs.\n\nThe route from Flagstaff is not available in winter. The Peach Springs\nroute is open in winter, but now little used. The bulk of the travel is\nvia Williams, sixty-five miles north to Bright Angel\u2014open all the year.\n\n\n Three Gateways\n\nThere are but three points from which an easy descent may be made of the\nsouth wall to the granite gorge of the Grand Canyon:\n\n1. At Grand View, down Berry's (Grand View) or Hance's (Red Canyon)\ntrails.\n\n2. At Bright Angel, down Bright Angel Trail.\n\n3. At Bass' Camp, down Mystic Spring Trail.\n\nWhile the canyon may be reached over trails at other places outside of\nthe district named (such as Lee's Ferry Trail, by wagon from Winslow;\nMoki Indian Trail, by way of Little Colorado Canyon; and Diamond Creek\nroad to Colorado River from Peach Springs station), most tourists prefer\nthe Bright Angel, Grand View, and Bass' Camp routes, because of the\nsuperior facilities and views there offered. The Peach Springs route is\nthe only other one now used by the public to any extent.\n\nIt is near Grand View that Marble Canyon ends and the Grand Canyon\nproper begins. Northward, a few miles away, is the mouth of the Little\nColorado Canyon. Here the granite gorge is first seen.\n\nBright Angel is approximately in the center, and Bass' Camp at the\nwestern end of the granite gorge. By wagon road it is eighteen miles\nfrom Bright Angel east to Grand View, and twenty-three miles west to\nBass' Camp.\n\nIn a nutshell, the Grand Canyon at Grand View is accounted most\nsublime\u2014a scene of wide outlooks and brilliant hues; at Bright Angel,\ndeepest and most impressive\u2014a scene that awakens the profoundest\nemotions; at Bass' Camp, the most varied\u2014a scene of striking contrasts\nin form and color.\n\nEach locality has its special charm. All three should be visited, if\ntime permits, as only by long observation can one gain even a\nsuperficial knowledge of what the Grand Canyon is. To know it intimately\nrequires a longer stay and more careful study.\n\n\n The Ride from Williams\n\nBecause of recent improvements in service the Grand Canyon of Arizona\nmay now be visited, either in summer or winter, with reasonable comfort\nand without any hardship. No one need be deterred by fear of inclement\nweather or a tedious stage ride. The trip is entirely feasible for the\naverage traveler every day in the year.\n\nLeaving the Santa Fe transcontinental train at Williams, Arizona,\npassengers change in same depot to a local train of the Grand Canyon\nRailway, which leaves Williams daily, and arrives at destination after a\nthree hours' run.\n\nWilliams is a busy town of 1,500 inhabitants, 378 miles west of\nAlbuquerque, on the Santa Fe. Here are located large sawmills, smelters,\nnumerous well-stocked stores, and railroad division buildings. Prior to\nthe disastrous fire in July, 1901, there were several excellent hotels.\nThe one not destroyed affords good accommodations; it has been recently\nenlarged and otherwise improved.\n\nThere is usually ample time at Williams, between trains, for the ascent\nof Bill Williams Mountain, which rises near the town to a height of\n9,000 feet. Tourists will find the trip thoroughly enjoyable. It can be\nmade in five hours on horseback in perfect safety. The trail is an easy\none, first leading through a gently sloping path of pines, then steeply\nup to the wind-swept summit alongside a pretty stream bordered by\nthickets of quaking aspens. Chimney Rock, with its eagle's nest, is a\nnoteworthy rock formation. On the summit is buried the historic pioneer\nscout, Bill Williams. From his resting-place there is a wide outlook,\nembracing, on clear days, the wall of the Grand Canyon, Verde River,\nChino Valley, Jerome, Hell Canyon, Seligman, Ash Fork, and many\nneighboring peaks.\n\nThe railroad track to the canyon is remarkably smooth for a new line. It\nis built across a slightly rolling mesa, in places thickly wooded, in\nothers open. The snow-covered San Francisco Peaks are on the eastern\nhorizon. Kendricks, Sitgreaves, and Williams mountains are also visible.\nRed Butte, thirty miles distant, is a prominent local landmark. Before\nthe terminus is reached the train climbs a long, high ridge and enters\nCoconino Forest, which resembles a natural park. The route here is amid\nfragrant pines, over low hills, and along occasional gulches and\n\"washes.\" Taken under the favorable conditions which generally prevail\nat this high altitude, the journey is a novelty and a delight.\n\n\n At Destination\n\nThe hotel at head of Bright Angel Trail is reached early in the evening.\nThe tourist then finds himself on the verge of a high precipice, from\nwhich is obtained by moonlight a magnificent view of the opposite wall\nand of the intervening crags, towers, and s. The suddenness, the\nsurprise, the revelation come as a fitting climax to a unique trip.\nAfter nightfall the air becomes cold, for here you are 7,000 feet above\nthe sea; yet the absence of humidity, peculiar to these high altitudes,\nmakes the chill less penetrating than on lower levels. By day, in the\nsunshine, there is usually a genial warmth\u2014then overcoats, gloves, and\nwraps are laid aside.\n\n\n Bright Angel Hotel\n\nThe Bright Angel Hotel is managed by Mr. M. Buggeln, who also controls\nthe stage line, trail stock, guides, etc. The hotel comprises a\ncombination log and frame structure of eight rooms, with three frame\nannexes containing forty-six sleeping rooms, and (for summer use)\nseveral rows of tents, all clustered on the rim and surrounded by pines\nand spruces. Each room in the annexes has one or two beds, a stove,\ndressing table, and Navajo rugs. In the log-cabin part of the main\nedifice are two large rooms. One is used for reception purposes, being\nwarmed by means of an old-fashioned fireplace and tastefully carpeted\nwith Indian rugs, also furnished with capacious rocking chairs and a\npiano; the other of these two rooms is for the office.\n\nGood meals are prepared by expert cooks and served in a pleasant\ndining-room. In a word, the hotel facilities are good, far better than\none might expect to find for the reasonable rate charged. There is no\n\"roughing it\"; everything is homelike and comfortable. One must not,\nhowever, expect all the city luxuries. A telephone and telegraph line\ndirectly connects the hotel with the outer world at Williams.\n\n Note.\u2014A fine modern hotel of fifty rooms, with cottage annexes, to be\n known as Bright Angel Tavern, will be built in this vicinity during\n 1903 and managed by Mr. Fred Harvey. It will be a permanent affair and\n will provide all the latest conveniences.\n\nWhile one ought to remain at least a week, a stop-over of three days\nfrom the transcontinental trip will allow practically two days at the\ncanyon. One full day should be devoted to an excursion down Bright Angel\nTrail, and the other to walks and drives along the rim. Another day on\nthe rim\u2014making a four-days' stop-over in all\u2014will enable visitors to get\nmore satisfactory views of this stupendous wonder.\n\n\n Down Bright Angel Trail\n\nThe trail here is perfectly safe and is generally open the year round.\nIn midwinter it is liable to be closed for a few days at the top by\nsnow, but such blockade is only temporary. It reaches from the hotel\nfour miles to the top of the granite wall immediately overlooking the\nColorado River. At this point the river is 1,200 feet below, while the\nhotel on the rim is 4,300 feet above. The trip is commonly made on\nhorseback, accompanied by a guide; charges for trail stock and services\nof guide are moderate. A strong person, accustomed to mountain climbing,\ncan make the round trip on foot in one day, by starting early enough;\nbut the average traveler will soon discover that a horse is a necessity,\nespecially for the upward climb.\n\nEight hours are required for going down and coming back, allowing two\nhours for lunch, rest, and sight-seeing. Those wishing to reach the\nriver leave the main trail at Indian Garden Spring and follow the\ndownward course of Willow and Pipe creeks. Owing to the abrupt descent\nfrom this point, part of the side trail must be traversed on foot.\nProvision is made for those wishing to camp out at night on the river's\nedge.\n\nThe famous guide, John Hance, is now located at Bright Angel.\n\n\n What to Bring\n\nIf much tramping is done, stout, thick shoes should be provided. Ladies\nwill find that short walking skirts are a convenience; divided skirts\nare preferable, but not essential, for the horseback journey down the\nzigzag trail. Traveling caps and (in summer) broad-brimmed straw hats\nare useful toilet adjuncts. Otherwise ordinary clothing will suffice. A\ngood field glass and camera should be brought along.\n\n [Illustration: Bright Angel Hotel.]\n\nThe round-trip ticket rate, Williams to Grand Canyon and return, is only\n$6.50. Adding $6 for two days' stay at Canyon Hotel, $1 for part of a\nday at hotel in Williams, $1.50 for probable proportion of cost of\nguide, $3 for trail stock, and the total necessary expense of the three\ndays' stop-over is about $18 for one person; each additional day only\nadds $3 to the cost for hotel.\n\nStop-overs will be granted at Williams on railroad and Pullman tickets\nif advance application is made to train and Pullman conductors. Trunks\nmay be stored in the station at Williams free of charge by arrangement\nwith ticket agent.\n\n\n Grand View\n\nGrand View (previously mentioned) may be reached in summer by private\nconveyance from Flagstaff, a distance of seventy-five miles; or at any\ntime of the year by stage from Bright Angel, sixteen miles along the\nrim. The rate for round trip, Bright Angel to Grand View, is $2.50 to $5\neach person, according to size of party. While Flagstaff is an\ninteresting place to visit\u2014with its near-by cliff and cave dwellings and\nSan Francisco Peaks\u2014and the trip thence to the Grand Canyon is a novel\none, distance and time are such that most travelers prefer to go in by\nrailroad from Williams.\n\nGrand View Hotel is a large, rustic structure, built near the head of\nBerry's Trail and about three miles from Hance's Trail, in the midst of\ntall pines and overlooking the mighty bend of the Colorado. This is the\npoint to which visitors were conducted in the days of the old stage line\nfrom Flagstaff.\n\nIt is noted for its wide views of the Coconino Forest and Painted\nDesert, as well as for the beautiful forms and color of the canyon\nitself. A favorite trip here is to go down one trail and up the other.\nThe hotel accommodations are quite good; capacity, forty guests; rate,\n$3 per day.\n\n\n Bass' Camp\n\nAt the western end of the granite gorge is Mystic Spring Trail, an easy\nroute down to the Colorado River and up the other side to Dutton's Point\nand Powell's Plateau. The magnificent panorama eastward from Havasupai\nPoint takes in fifty miles of the canyon, while westward is the unique,\ntable-like formation which characterizes the lower reaches of the river.\nThe views from both rims are pronounced by noted artists and explorers\nto be unequaled.\n\nPresent accommodations at Havasupai Hotel (Bass' Camp), near head of\nthis trail, are fairly good, consisting of a cabin, several tents, and\ngood trail stock; wholesome meals are served in comfortable style. A new\nhotel is to be built here during 1903. Bass' Camp is now reached by\nstage from Coconino, a station on the Grand Canyon Railway, or one may\ntake a team direct from Bright Angel.\n\nA visit should be made to the Havasupai Indian village in Cataract\nCanyon. Any bona fide tourist can procure an introductory letter from\nthe railroad agent at Williams or Grand Canyon. On presenting same to\nthe U. S. Indian agent at Supai, permission will be granted to enter the\nreservation. This is an unique trip of about forty miles, first by wagon\nacross a timbered plateau, then on horseback down precipitous Topocobya\nTrail, along the rocky floors of Topocobya and Cataract canyons, deep in\nthe earth, to a place of gushing springs, green fields, and enchanting\nwaterfalls. Here live the Havasupai Indians, one of the most interesting\ntribes in Arizona. The round trip from Bright Angel or Bass' Camp is\nmade in three or four days at an expense of $35 to $50 each for a party\nof three persons.\n\n\n Peach Springs Route\n\nThe trip in winter from Peach Springs station down to the Colorado\nRiver, through Diamond Creek Canyon, is most enjoyable. Owing to the low\naltitude here (4,780 feet at Peach Springs and approximately 2,000 feet\nat the river) the air is usually balmy from November to April; in summer\nthe heat is a considerable drawback.\n\nA journey of but twelve miles leads you through a miniature Grand Canyon\nwith scenery increasingly sublime. On either side are abrupt walls and\nwonderfully suggestive formations\u2014castles, domes, minarets. On your\nleft, glancing backward, is an exact reproduction of Westminster Abbey.\n\nThis comparatively easy jaunt brings you by team to the very brink of\nthe swift-rolling Colorado, whereas by the other Grand Canyon gateways\nyou are landed on the rim and must go down thousands of feet by a steep\ntrail. The outlook here is restricted to the river itself and the great\nwalls rising precipitously from its banks\u2014a scene well worth while, but\nnot so impressive as the wide sweep of the canyon visible from the rim.\n\nFollowing Diamond Creek to its source you may walk along the bed of the\nstream between walls thousands of feet high and glistening in the white\nsunlight as if varnished. The upper part of Diamond Creek is a veritable\nterrace of fern bowers, luxuriant vegetation, crystal cascades, and\nsequestered meadow parks.\n\n\n Flagstaff and Vicinity\n\nThe town itself is an interesting place, prettily situated in the heart\nof the San Francisco uplift and surrounded by a pine forest.\n\nIts hotels, business houses, lumber mills, and residences denote thrift.\nOn a neighboring hill is the Lowell Observatory, noted for its many\ncontributions to astronomical science.\n\n [Illustration: San Francisco Peaks.]\n\nEight miles southwest from Flagstaff\u2014reached by a pleasant drive along a\nlevel road through tall pines\u2014is Walnut Canyon, a rent in the earth\nseveral hundred feet deep and three miles long, with steep terraced\nwalls of limestone. Along the shelving terraces, under beetling\nprojections of the strata, are scores of quaint cliff dwellings, the\nmost famous group of its kind in this region. The larger abodes are\ndivided into several compartments by cemented walls, many parts of which\nare still intact. It is believed that these cliff dwellers were of the\nsame stock as the Pueblo Indians of to-day and that they lived here\nabout 800 years ago.\n\nNine miles from Flagstaff and only half a mile from the old stage road\nto the Grand Canyon, upon the summit of an extinct crater, the\nremarkable ruins of the cave-dwellers may be seen.\n\nThe magnificent San Francisco Peaks, visible from every part of the\ncountry within a radius of a hundred miles, lie just north of Flagstaff.\nThere are three peaks which form one mountain. From Flagstaff a road has\nbeen constructed up Humphrey's Peak, whose summit is 12,750 feet above\nsea level. It is a good mountain road, and the entire distance from\nFlagstaff is only about ten miles. The trip to the summit and back is\neasily made in one day.\n\n\n Announcement\n\nThe Santa Fe has published a new and beautiful book on the Grand Canyon.\nIt contains articles by Hamlin Garland, Harriet Monroe, Robert Brewster\nStanton, Chas. S. Gleed, John L. Stoddard, Charles Dudley Warner, R. D.\nSalisbury, \"Fitz Mac,\" Nat M. Brigham, Joaquin Miller, Edwin Burritt\nSmith, David Starr Jordan, C. E. Beecher, Henry P. Ewing, and Thomas\nMoran, as well as the authors represented in this pamphlet. The book has\nmore than a hundred pages, illustrated with half-tones and portraits;\nthe cover is from a painting of the Canyon by Thomas Moran, and is\nlithographed in seven colors. It will be forwarded on receipt of fifty\ncents.\n\nA beautiful and unique color picture of the Grand Canyon, mounted to\nshow all its colors as in nature, may be had for twenty-five cents.\n\n Address W. J. BLACK,\n Gen'l Passenger Agent, A., T. & S. F. Ry., CHICAGO.\n\n Ad. 71\u20142-3-03. 10M.\n\n\n\n\n Transcriber's Notes\n\n\n--Retained publication information from the printed exemplar (this eBook\n is in the public domain in the country of publication.)\n\n--Only in the text versions, delimited italicized text with\n _underscores_.\n\n--Silently corrected several typos.\n\n\n\n\n\n\n\nEnd of the Project Gutenberg EBook of Titan of Chasms, by \nC. A. Higgins and John Wesley Powell and Charles F. Lummis\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}} +{"text":" \nA History of Courtship\nFor Gary\n\n'My heart, pierced thro' with fierce delight, Bursts into blossom in his sight.'\n\n(From Fatima by Alfred, Lord Tennyson)\n\n# A History of Courtship\n\n800 Years of Seduction Techniques\n\nTania O'Donnell\n\nFirst published in Great Britain in 2017 by\n\nPen & Sword History\n\nan imprint of\n\nPen & Sword Books Ltd\n\n47 Church Street\n\nBarnsley\n\nSouth Yorkshire\n\nS70 2AS\n\nCopyright \u00a9 Tania O'Donnell 2017\n\nISBN 978 1 78159 348 6\n\neISBN 978 1 47387 509 8\n\nMobi ISBN 978 1 47387 508 1\n\nThe right of Tania O'Donnell to be identified as the Author of this Work has been asserted by her in accordance with the Copyright, Designs and Patents Act 1988.\n\nA CIP catalogue record for this book is available from the British Library.\n\nAll rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical including photocopying, recording or by any information storage and retrieval system, without permission from the Publisher in writing.\n\nPen & Sword Books Ltd incorporates the imprints of Pen & Sword Archaeology, Atlas, Aviation, Battleground, Discovery, Family History, History, Maritime, Military, Naval, Politics, Railways, Select, Transport, True Crime, Fiction, Frontline Books, Leo Cooper, Praetorian Press, Seaforth Publishing and Wharncliffe.\n\nFor a complete list of Pen & Sword titles please contact \nPEN & SWORD BOOKS LIMITED\n\n47 Church Street, Barnsley, South Yorkshire, S70 2AS, England\n\nE-mail: enquiries@pen-and-sword.co.uk\n\nWebsite: www.pen-and-sword.co.uk\n\n# Contents\n\n## Acknowledgements\n\n## Introduction\n\n## Chapter 1 Love at First, Second or Third Sight\n\nMaking new acquaintances and signalling your interest\n\n## Chapter 2 Beauty & Seductive Items of Clothing\n\nArm yourself with these handy historical beautifying and dressing ideas\n\n## Chapter 3 Love Tokens and Gifts\n\nAcceptable gifts from lovers and admirers\n\n## Chapter 4 Coxcombs and strumpets\n\nHow to recognise roguish men and women of ill repute \u2013 and how to avoid them\n\n## Chapter 5 In Praise of Chaperons\n\nMany a reputation has been saved by having a sibling or great-aunt in tow\n\n## Chapter 6 Love Songs, Letters and Poems\n\nArticulating your desire can get you an assignation \u2013 or potentially an execution!\n\n## Chapter 7 How to be a Good Life Partner\n\nOnce you've attained the proposal and made your vows, your work is only just beginning\n\n## Conclusion\n\nA summary of the best of our ancestors' advice.\n\n## Further Reading\n\n## Acknowledgements\n\nI 'd like to thank: my father, the novelist Hamraz Ahsan, for convincing me to write for a living; Jen Newby, formerly at Pen & Sword Books, for commissioning this book, for her superb editing skills and meticulous improvements to the manuscript; Dominic Allen for his beautiful book jacket design; Eloise, Heather, Lisa and all at Pen & Sword books for their patience, kindness, and hard work; Chelsey Fox for the loan of a wonderfully inspiring book; Emily Brand for her excellent History of Love blog (http:\/\/historyofloveblog.wordpress.com);Meg Schultz for the use of her photo and her family's incredible story (http:\/\/hpugh.blogspot.co.uk), my family \u2013 both the Ahsans and the O'Donnells (specific mentions for Pawan, Munazza, Welshon Bull, Mum Farah, Mum Vie and Neen Phups) \u2013 for their unstinting support; Poorna Bell for giving me a HuffpostUK blog in which to express my opinions on relationships; Raj Kaushal for many long conversations about love; the glorious children in my life \u2013 Katie, Zachary, Hassan, Sara, and Leela West \u2013 for selflessly spreading happiness wherever they go; and Bunty for the late night cognacs and cigars.\n\nI'd also like to thank the staff at the London Library and the British Library for their kindness in dealing with a nervous and skittish library user. But, above all, I must thank Gary for courting me so well that I pledged my troth.\n\n## Introduction\n\nA ll courtship is the art of convincing the other person of your innate desirability. However, our ideas of what qualities are desirable have changed over time. In general, assets such as wealth, attractive physical features, and readiness to give tokens of love and admiration remain unchanged. Yet, our ideas about the level of wealth we can marry into, good looks we can acquire through cosmetic means, and, most tellingly, the tokens of love we exchange have dramatically altered.\n\nOur modern notions of love and romance mean that today many Britons find it distasteful to openly consider class or position when choosing a life partner, but in times past assessing a prospective partner's income would have been seen as a perfectly normal, indeed wise, preoccupation.\n\nMany historians date our contemporary ideas about romantic love to the twelfth century and the medieval chivalric code of courtly love. In fact, the idea of finding your 'other half ' pre-dates this significantly. Aristophanes (c. 446\u2013c. 386 BCE), a Greek playwright known for his comedic works, is said in Plato's The Symposium to have posited that all human beings were originally of three types: male, female, and androgynous. These early humans, according to Aristophanes, had doubled bodies joined at the back. They wheeled around in quite a comical fashion, but were powerful enough to challenge the gods so Zeus split them in two, thereby halving their power while ensuring there were still humans around to worship the gods. Aristophanes uses this creation myth to explain the concept of feeling 'whole' when we meet our 'other halves'.\n\nIf the notion of one special person being your true soulmate has been around since the ancient Greeks, we can assume that the idea of romantic love is far older than the early Middle Ages, even if the poetic language of love was not widely recorded in the West before that time.\n\nWhether or not we believe that there were romantic Neanderthals, the consensus is that courtly love codified the angst-ridden, teenage version of 'Falling In Love' writ large, and so the twelfth century is a good place to begin a history of courtship. This is when the idea of tragic love or thwarted love gained strength, with the famous story of Tristan and Isolde. The highest expression of love in the literature and culture of this period is the 'impossible love', and the wooing rather than the winning becomes the most elaborate part of mating rituals.\n\nSo if the emotional rollercoaster of the twelfth century is where we start, why end with the traditionally strait-laced nineteenth century? By the twentieth century, globalisation and advances in the speed of travel had changed many of the traditions of courting forever. People were no longer forced to choose a mate from their immediate environs or even social circle. Also it is only in the twentieth century that the notion of established time begins to hold sway in the western world. Being late for an assignation might be considered a terrible faux pas for the modern lover, but given that standard time didn't come into law in the UK until 1880 when the Statues (Definition of Time) Act came into force or in the United States until 1918, when the Standard Time Act was passed by Congress, earlier courting couples might meet at dawn or sunset or at the end of a day's work. This was irrevocably changed as the world became tethered to time.\n\nAmong the many bleak aspects of the First World War was the low numbers of eligible men left in the UK after the conflict, leading to a disparity between the sexes and what the papers described as the 'surplus two million' women. Virginia Nicholson writes, in Singled Out: How Two Million Women Survived Without Men after the First World War (Penguin, 2007): 'it is beyond doubt that the war had a seismic effect on marital behaviour... all contemporary accounts take the man shortage for granted, and that many women themselves perceived the courtship arena as a competitive battleground, where defeat was perdition.'\n\nCourtship rituals during this century of upheaval had to change, emboldening women and taking some of the pressure off men. The much-f\u00eated 'birth of the teenager' in 1950s America also changed the rules of courtship for young people. So, as the twentieth century is too unwieldy to include in a concise history of courtship, we end our journey in the nineteenth, when Victorian values still held sway and love still had to seep through social restrictions and propriety.\n\nIt is a source of some sadness that near-universal literacy in Britain is a modern notion. We have extremely few written historical sources for those with limited access to education in centuries past \u2013 generally speaking, the rural and urban poor. So while men of letters have written extensively about their views of the other social classes they encountered, it is rare indeed to find a written account from a working class individual. Country sayings, folklore, oral histories and family histories can provide some insight into the lives of ordinary people. On the subject of courtship, small superstitions have been recorded not in literature, but passed down through the generations in rhymes and songs \u2013 even graffiti inscribed in churches. We can begin to build a picture of the romances of servants and labourers through those snippets of information.\n\nThe written evidence we have for courting through the ages comes from the letters of the nobility or the diktats of the Church (an institution that interfered to such a great extent in private lives that one could argue that the Church inadvertently gave birth to romantic love). Then there is the sharply observant literature of Regency authors, such as Jane Austen, to give us a very complete picture of the social lives of the upper classes. And so, in dribs and drabs, scandals and rumours, letters and bawdy tales, we begin to build up a picture of love through the ages.\n\nDoes this information have any relevance for the modern lover, beyond humour or the enjoyment of quirky ideas from times past? I believe it does, in that when we examine the history of how couples came together in the past, we begin to appreciate the freedoms we now have. Perhaps we become more grateful that we are no longer forced to marry from the narrow pool of our places of birth or within the rigid confines of status and rank.\n\nWhen the history of courtship is laid before us, we can celebrate how far we've come in gaining greater equality of the sexes and creating happier marriages \u2013 while still remaining watchful for throwbacks to more unenlightened times.\n\nTania O'Donnell, London, \nSummer 2016.\n\n## Chapter One\n\n## Love at First, Second or Third Sight\n\nMaking new acquaintances and signalling your interest\n\nL et's blame the ancient Greeks for it. The notion that we should fall madly in love at first sight can probably be attributed to all those Greek myths featuring Eros and his arrows of love; the idea being that within the most romantic of love stories you should be 'struck' by an instant flash of passion. However, this doesn't leave much room for getting to know one another before 'falling in love'. And what of childhood sweethearts who have known each other since birth? Is their love or passion in some way diminished because they didn't get the opportunity to meet each other's eyes across a room and declare themselves instantly smitten?\n\nIt is easy to forget how far the technological advances of the last century have opened up the world to us, especially in terms of love and romance. Before the Victorian railway boom, the time and expense involved in travelling even relatively short distances meant that most people stayed close to home, choosing a mate from among their acquaintances and family friends. While the upper echelons may have moved around more, even the relationships they formed were constrained by the circles they moved in.\n\nThe best a poor village girl in the pre-industrial age could hope for by way of a glamorous stranger was a travelling craftsman looking for work. But this did not stop the idle speculation of youth, and a young maid might turn to childhood rhymes and superstitions in the hope of divining her fate.\n\nThe famous 'Tinker, Tailor, Soldier, Sailor' rhyme has its origins in the fifteenth century and was used to predict whom the questioner would marry. The words 'Tinker, Tailor, Soldier, Sailor, Rich Man, Poor Man, Beggar Man, Thief ' were chanted as cherry stones or petals on a daisy were counted out. Further verses revealed when you'd marry, your position in married life, what you'd wear on your wedding day, how you'd obtain this clothing, how you'd get to church, and even where you might live afterwards. This provided a harmless diversion to while away a summer afternoon.\n\nThis 1906 photographic print by E.W. Kelley shows a young woman playing the 'he loves me, he loves me not' game with daisy petals, while a man watches with amusement in the bushes behind her. (Library of Congress)\n\nSummer is also when the cuckoo's call would be heard. According to an old tradition the number of consecutive cuckoo calls you hear equates to the number of years until you will marry. These 'fateful' signs could be one way of predicting a positive future, but the more enterprising took matters into their own hands and engaged in activities to push forward contact with their true love. Tucking an ivy leaf into your bosom ensured that the next man who spoke to you would be your beloved \u2013 or so tradition dictated. If while shelling peas, you found a pod with nine perfectly shaped peas inside, it was believed to be a sign of good luck. According to superstition, if an unmarried girl should hang the empty pod over the lintel of the front door, the first man to cross the threshold would be her future husband.\n\nThis photographic print from the late nineteenth century is entitled 'Waiting For Him' and depicts a young lady waiting for a gentleman caller. (Boston Public Library)\n\nThese homely superstitions exhibit a longing to know when and whom it would be a girl's destiny to marry. The fact that their intended is expected to appear imminently suggests that most people were aware that they would not marry exotic strangers, but someone with whom they already came into daily contact.\n\nThe Victorian explorer Sir Samuel White Baker, for example, at the tender age of twenty-two, chose as his wife, Henrietta, his childhood playmate and the daughter of the local rector, while his brother John picked her sister, Elizabeth. Samuel provides an excellent example of both idealised manners in which we think of love blossoming: childhood sweethearts and the big romantic 'love at first sight'. Henrietta was a childhood companion who later became a loving wife and the mother of his children. However, after her untimely death, Samuel quite remarkably diverged from the usual course of courting within upper class circles and married Florence Barbara Maria, a Hungarian refugee whom he had bought in a slave market on the Danube.\n\nMichael Brander writing in his biography of the great hunter and explorer, The Perfect Victorian Hero, says that while stranded in Widdin awaiting the go-ahead for a hunting expedition in 1858, Samuel visited a local slave market. There 'Sam saw this beautiful young Hungarian girl, who, on the death or marriage of her erstwhile nurse, had fallen into unscrupulous hands'. She was put up for sale and he outbid all the wealthy Turkish merchants at the auction, bought her, and subsequently married her. It seems strange that Samuel's more sedate, predictable courtship happened when he was in the first flush of youth, while the wild, exotic romance occurred much later in his life.\n\nYet, this exceptional Victorian case was not the way most British people met a prospective love interest in the past. Even if a young person was in a position to come into contact with strangers, they could not simply wander up to them and engage them in conversation. Or at least those deemed to be ladies and gentlemen could not do so.\n\nIn the Regency era, a strict etiquette codified how one was to make a new acquaintance among gentlefolk. Until one had been introduced to an unknown person, it was not considered proper to begin a conversation with them. Upon entry to a new neighbourhood a family could expect to have their neighbours, or more exactly the heads of the local households, call upon them to allow an acquaintance to be formed.\n\nKidnapping a bride\n\nMany of these rules can be traced back to the historic function of marriage as a contract for increasing wealth. Marriages between members of the monied classes have always been seen as a way to increase one's position in society, either through acquiring advantageous connections or property. In such cases, protecting the maidenheads of womenfolk was supremely, in fact economically, vital.\n\nHistorically heiresses could be kidnapped, forced to marry under threat of rape, and sold off to the highest bidder by guardians. A high profile kidnapping occurred on 2 of September 1487, when Robert Bellingham, a nobleman who found favour with King Henry VII for having thwarted one of the pretenders to his throne, nevertheless annoyed the sovereign by abducting the wealthy heiress Margery Beaufitz. Her father had not favoured Bellingham's proposal of marriage, and so the would-be suitor with a band of accomplices broke into the family's home and carried Margery away. Bellingham was imprisoned for flouting the law that Henry had passed against the abduction of women earlier that same year. However, a few months later the case was dropped when an agreement was reached between Bellingham and Margery's father. The couple subsequently married and Bellingham once again found himself in the King's good books.\n\n'An Act Against Taking Away of Women Against Their Will' of 1487 was not some great feminist measure against violence toward women. In fact it was put in place to ensure that the Crown did not lose the revenues it gained through the orphaned wealthy heirs of their tenants. For any child in such a situation was automatically a ward of the monarch, who could decide to pass on that wardship to a favoured courtier or sell it to a guardian wishing to invest in the hope of future profit.\n\nCharles Brandon, Duke of Suffolk, scandalously married his ward in order to get her estate for himself. (Portrait by Jan Gossaert, c.1516, Public Domain)\n\nCharles Brandon, the 1st Duke of Suffolk, chose not to wait for future dividends from his wards and made a bit of a habit of trying to marry them himself. He had to annul the first marriage contract he entered into with a ward when a better wife came along. Elizabeth Grey, an orphaned viscountess, was ditched in favour of Mary Tudor, the sister of Henry VIII. When Mary died in June 1533, Brandon then married his fourteen-year-old ward, Catherine Willoughby, Baroness Willoughby of Eresby. She had been betrothed to his son, but the boy was too young to marry and Brandon did not want to risk losing her substantial estate through a lengthy engagement in a time of high mortality.\n\nHeiress kidnapping was still a problem over a century later, as shown by the case of Elizabeth Malet, granddaughter of Lord Hawley. On 26 May 1665, the 2nd Earl of Rochester, John Wilmot, lived up to his reputation as a libertine and kidnapped the wealthy Elizabeth after she had turned down his proposal of marriage.\n\nHere is Samuel Pepys's diary entry on the matter:\n\n'Thence to my Lady Sandwich's, where, to my shame, I had not been a great while before. Here, upon my telling her a story of my Lord Rochester's running away on Friday night last with Mrs. Mallett, the great beauty and fortune of the North, who had supped at White Hall with Mrs. Stewart, and was going home to her lodgings with her grandfather, my Lord Haly, by coach; and was at Charing Cross seized on by both horse and foot men, and forcibly taken from him, and put into a coach with six horses, and two women provided to receive her, and carried away. Upon immediate pursuit, my Lord of Rochester (for whom the King had spoke to the lady often, but with no successe) was taken at Uxbridge; but the lady is not yet heard of, and the King mighty angry, and the Lord sent to the Tower. Hereupon my Lady did confess to me, as a great secret, her being concerned in this story. For if this match breaks between my Lord Rochester and her, then, by the consent of all her friends, my Lord Hinchingbroke stands fair, and is invited for her. She is worth, and will be at her mother's death (who keeps but a little from her), 2500l. per annum.'\n\nSamuel Pepys kept a diary that recorded many of the scandals and trysts of seventeenth century London, as well as his own lascivious behaviour. (Wellcome Collection)\n\nThe chivalry Lord Rochester showed in having two ladies to receive the kidnapped lady must have sat well with Elizabeth as she later agreed to marry him against her father's wishes.\n\nScandal and Elopements\n\nNevertheless there were occasions when such extreme measures to initiate 'love' simply didn't need to be taken. Much to the chagrin of the Church, if you did not have any wealth or land to worry about, you could quite easily get betrothed and move in together with your beloved, without any outside interference. This was shockingly libertarian in the eyes of the clergy, who saw marriage as a sacrament and were keen to reduce the prevalence of 'sin', aka sex outside marriage. There was also the biblical diktat of 'let no man put asunder' what God has joined together.\n\nAnnoyingly for the Church, if the correct vows of fidelity and marriage were made between a couple of marriageable age (over the age of fifteen for a 'man' and over twelve for a 'woman'), their marriage was deemed to have been legally binding, with or without the consent of their parents or the presence of clergy. In the words of historian George Gordon Coulton, 'In Chaucer's time, the whole world was a vaster and more commodious Gretna Green.'\n\nStrangely enough, the binding nature of private vows made the Church an unlikely ally in some love stories. The Paston Letters, private documents that span three generations of the Paston family of Norfolk from the fifteenth to the early sixteenth centuries, give a juicy insight into a case in which the Church upheld an informal marriage. The daughter of the family, Margery Paston, fell in love with her father's bailiff, Richard Calle. They secretly married in 1469, angering her parents who set the matter before the bishop. Upon enquiring of the young couple how they had entered into the married state, the clergyman could find no fault with the union and so it stood. The family attempted to dismiss Calle from his post, but his knowledge of their finances and his ability to secure funds from their tenants meant they had to reinstate him.\n\nServants might meet a beau in the course of their daily duties, which is why many mistresses banned a maid from having followers. (This 1772 print entitled A Lady's Maid Purchasing A Leek is by James Caldwall, Library of Congress)\n\nThis case shows that love will thrive wherever there is opportunity. One method of attempting to thwart undesirable matches was to do away with any chance young people had of privacy. Medieval noble women had retainers and ladies-in-waiting to ensure that they were always chaperoned. (The efficacy of chaperons will be described in more detail within Chapter Five.)\n\nBy comparison couples from the working classes and the rural poor had a tremendous amount of freedom, with many seeking out their privacy in the lanes and woods outside a cramped communal home. When huge fortunes weren't at stake, it also occasionally made sense to have sex before marriage to 'test' the bride's fertility and many couples tied the knot with the proof of the groom's virility clear for all to see. There were also rural traditions around the seasonal calendar and Pagan hangovers such as May Day celebrations that allowed men and women to freely mix and often resulted in pairings, even if they failed to get these unions legally consummated.\n\nHowever, on occasion a lack of propriety could lead to a marriage, with the approval of the couple's parents. The practice of 'bundling' which was frowned upon in later, more puritanical years, was deemed a sensible way of allowing couples to get to know each other before marriage. They would spend a night together, fully clothed, sometimes tied down with a board placed down the middle of the bed, in order to talk without having sex. This was practised in seventeenth century Wales and made its way over to eighteenth century New England.\n\nAnother example of parental advocacy of impropriety is the most likely apocryphal tale of how Margaret, daughter of the Renaissance statesman Sir Thomas More, was introduced to her husband William Roper. It is related by seventeenth century historian John Aubrey in his collection of biographies, Aubrey's Brief Lives (1697). Apparently Roper was led into the bedroom where More's two daughters were asleep, lying naked on a truckle bed. More threw back their covers and the protesting girls turned onto their stomachs to cover their private parts. Roper, having charmingly said that he had now seen both sides, gave Margaret's buttock a pat with his walking stick to indicate his choice. As Aubrey puts it 'here was all the trouble of the wooing'. If this is a true story, one can only hope it was an unusual one.\n\nThis print published in Once A Week magazine in 1864 shows a courting couple enjoying the relative freedom of a rural romance. (Public Domain)\n\nAnother lascivious old man was the famed diarist Samuel Pepys, who would miss no opportunity to force his advances on long-suffering chambermaids and even strangers sitting next to him in the pew at a church. Here is a typical entry from his diary, dated 18 August 1667:\n\n'I walked towards White Hall, but, being wearied, turned into St. Dunstan's Church, where I heard an able sermon of the minister of the place; and stood by a pretty, modest maid, whom I did labour to take by the hand and the body; but she would not, but got further and further from me; and, at last, I could perceive her to take pins out of her pocket to prick me if I should touch her again \u2013 which seeing I did forbear, and was glad I did spy her design. And then I fell to gaze upon another pretty maid in a pew close to me, and she on me; and I did go about to take her by the hand, which she suffered a little and then withdrew. So the sermon ended, and the church broke up, and my amours ended also, and so took coach and home...'\n\nDancing and romancing\n\nAs many commentators have said on the matter, in earlier times with more contracted marriages, courtship often began after a marriage instead of before. Perhaps the reason we find the arranged marriage customs of Eastern cultures so alien is that they have largely died out within our own culture. Our modern ways of meeting a beau do, however, have one precedent in history \u2013 that of the dance.\n\nWhile today it is occasionally possible to meet a prospective beau in a bar or nightclub, dances in assembly rooms were once the only sanctioned opportunity for young people to meet and flirt under the watchful eye of guardians and chaperons. Even if attending a dance or ball did not secure a match, it was deemed a good place to practice one's social skills and to show off one's allurements.\n\nAn account from the diarist Katherine Plymley on 3 December 1796, describes the glamour \u2013 and also clamour \u2013 of a ball in Bath. Quite apart from the drama of a ballroom lit up by more dazzling light than would be normal of an evening, the participants also created quite the spectacle.\n\n'I met Lady Glynne at Mrs. Falconers public day & went with her to the ball. Miss Charlotte & Miss Williams joined us, the latter however became discomposed because we declined pushing thro' the crowd (for such it was, 12 hundred persons being present) from the ball room to the tea room & left us for a more congenial party. The Prince & Duke came early to the ball room with the Dutchess.... The Dutchess I thought rather a pretty little woman, but quite over rouged, so were all the ladies of her party, the Prince extremely good humour'd & pleasant, I do not like the Duke's countenance. They were both dressed in plain blue with the star. The Dutchess's body & train was white spotted with gold, trimmed round the neck & down the sides with narrow black velvet studded with diamonds, her head dress a turban, diamond crescent, & large plume of feathers, diamond necklace & earings.'\n\nOne of the most famous literary love stories \u2013 that of Elizabeth Bennet and Fitzwilliam Darcy in Jane Austen's Pride and Prejudice (1813) \u2013 begins at an assembly room ball. This tempestuous relationship starts off with Darcy snubbing Lizzy by refusing to ask her to dance. With a scarcity of partners at the dance, Elizabeth is obliged to sit out some dances and Darcy's more amiable friend, Bingley, takes a break from his own partners to try and convince him to be more agreeable and ask Lizzy to dance. He responds: \"She is tolerable; but not handsome enough to tempt me; I am in no humour at present to give consequence to young ladies who are slighted by other men. You had better return to your partner and enjoy her smiles, for you are wasting your time with me.\"\n\nA satirical picture story published in the late nineteenth century paper Wild Oats shows the importance of wealth in having one's affections returned. (Public Domain)\n\nFor Darcy to stand out dances when partners are few was a distinct insult to not just Elizabeth but all the ladies present who were unable to dance due to a lack of willing gentlemen.\n\nProvincial assemblies such as the one depicted at Meryton were held monthly in the winter season, often coinciding with a full moon to naturally light the path of carriages coming to the dance. It was the ideal opportunity to meet neighbours and make new acquaintances, especially if you were new to the area. The master of ceremonies, or an otherwise acquainted person, could introduce you to the local dignitaries and the flirting could begin in a respectable manner.\n\nInterestingly, you didn't have to be rich to enjoy a dance. Henry Mayhew, the Victorian chronicler of the lives of the poor and working class in London, wrote of 'twopenny-hops' that were often attended by costermongers, men and women who sold fruit, vegetables and fish in the street from barrows. For a two pence entry fee, dancers could take part in jigs, country dances, and polkas. Mayhew places the attendance at these dances to be between thirty to one hundred of both sexes, although women were slightly more predominant. Their ages varied between fourteen and forty-five and the dancing was of the 'vigorous, laborious' kind.\n\n'There is sometimes a good deal of drinking,' reveals Mayhew, 'some of the young girls being often pressed to drink, and frequently yielding to the temptation.' The chivalrous costermonger men would treat the women to drinks at these events, possibly in the hope that libation might serve as an introduction to a young lady in the absence of the master of ceremonies at grander affairs.\n\nHaving managed through fate or fortune, chance or dance, to make a suitable acquaintance with someone you wished to marry, without the aid of social media, how could you then signal your interest?\n\nSignalling interest\n\nThe Renaissance author, Count Baldassare Catiglione, wrote in the Book of the Courtier (1528) about the power of the lover's gaze; 'Those lively spirits that issue out at the eyes because they are engendered nigh the heart, entering in like case into the eyes that they are levelled at, like a shaft to the prick, naturally pierce the heart'. To this day, our most effective flirting technique is the furtive glance held a tad too long. The huge benefit of a simple stare to show your willingness to get to know someone better is that it is not hampered by the constraints of social niceties. You can stare discreetly at anyone and then look away quickly if you're caught at it.\n\nThe natural blushes that occur in such a situation are also said to be a great enhancement to a woman's looks. Walter Houghton, writing in his guide American Etiquette and Rules of Politeness (1883), advises that gentlemen shouldn't allow a flirtatious glance to veer into odd or impolite behaviour. 'Avoid looking full into the faces of strangers whom you meet, especially of ladies.'\n\nAn article by Richard Steele in The Spectator that was printed in 1711 references a letter that the journalist claimed was from a lady called 'Celimene' who complained of being encumbered with a young relation from the country who was sent up to her for education and was not genteel enough to know the ways of society. Described as having 'no way to express herself but by her tongue and that always to signify her meaning... she means nothing by walking but to change her place', the unsophisticated young countrywoman was not aware of the powerful arsenal of glances and movement that were at her disposal. Steele's correspondent wondered how such a girl was to be tutored.\n\nSteele's response was to despair of the manner in which the younger generation were being educated. 'In our daughters we take care of their persons and neglect their minds; in our sons we are so intent upon adorning their minds that we wholly neglect their bodies. It is from this that you shall see a young lady celebrated and admired in all the assemblies about town, when her elder brother is afraid to come into a room.' He describes how upon growing old enough to no longer need her nursemaid, a girl is handed over to a dancing master for instruction on how to be charming. '... And with a collar round her neck the pretty wild thing is taught a fantastical gravity of behaviour, and forced to a particular way of holding her head, heaving her breast and moving with her whole body; and all this under pain of never having a husband, if she steps, looks or moves awry. This gives the young lady wonderful workings of the imagination, what is to pass between her and this husband that she is every moment told of... Thus her fancy is engaged to turn all her endeavours to the ornament of her person... from this general folly of parents we owe our present numerous race of coquets.'\n\nWhile Steele may have been scathing about the 'coquetry' of tutored women, by Victorian times women everywhere were continuing to use subtle glances to indicate romantic interest. In using her eyes as her weapon of choice, a young Victorian lady could even make use of the more boring parts of attending church on a Sunday to cast an eye over the congregation for suitable earthly distractions. While her spiritual scorecard might have suffered from such carnal thoughts, her dance card would have filled up rather quickly.\n\nCould props such as fans also be employed to indicate a lady's interest in a gentleman? Stories of the language of fans being used in the seventeenth and eighteenth centuries abound, but there is little evidence of women using the fan in such a way in the literature of the time. The first guide to using fans for secret communication was published in 1826 by Duvelleroy, a Parisian fan maker, who included a booklet with the fans he sold. This was almost certainly a marketing ploy aimed at gullible young ladies, especially since the gentlemen they were aiming to communicate with were unlikely to have been privy to the intricate meaning of a fan carried in the left hand in front of the face ('I wish to be acquainted'), much less that of twirling it in the right hand ('I love another').\n\nThe whole business seems to be the equivalent of young women who read astrological love signs in the hope of arcane knowledge. This is not to say that fans were of no use in courtship, as we will find out in the next chapter. In centuries past it was generally the duty of the man to make the first move. A woman who was interested in a man only had to make herself as alluring as possible and to polish up her accomplishments for occasions when an introduction might happen. For a lady, being generally acknowledged as accomplished was one way of gaining introduction to the right sort of men. By the Regency era, to be considered accomplished you had to have some form of musical aptitude, display intricate sewing skills, be a good conversationalist, and a dainty dancer. It was just as well that there was plenty of time during which affluent women could practise and improve those skills. Victorian debutante Alice Miles wrote in her diary in 1868, aged 17, that 'I consider it every girl's duty to marry \u00a380,000 a year'. This was a clear-eyed statement of why so much time was spent on becoming accomplished. Alice was quite open about the importance of financials in securing a good marriage:\n\n'Sir Samuel has tried to inveigle me into a flirtation, but as I had previously ascertained he has only \u00a34,000 a year... there would be no interested side in any proceedings and he is not sufficiently good looking to render interesting, and a flirtation devoid of either of these indispensible elements does not at all enter into my plan of action.'\n\nThankfully not all courtships were quite so openly mercenary, as this sweet Victorian example in Jennifer Newby's book Women's Lives (2011) shows: 'In 1872, 20-year-old Emily Jowitt from Leeds wrote excitedly to a friend that when she found herself alone with Squire Dearman Birchall on the way back from church, 'just as we were going down the carriage drive, he proposed... the suddenness of it all took my breath away'. While engaged, Emily remarked that she and Dearman 'generally spooned a good deal and said \"Oh my darling I do love you so\".'\n\nFor those not as daring as Squire Dearman, there was no shortage of help to induce a nervous man to make a suitable young woman an offer of marriage. The relations of an eligible man were always on the lookout for a suitable wife, while the relations of an eligible woman were equally keen to get her married off. Of course the opinions of two sets of relations did not always tally and many a coupling was derailed by the ideas and ambitions of those around the young couple.\n\n## Chapter Two\n\n## Beauty & Seductive Items of Clothing\n\n## Arm yourself with these handy historical beautifying and dressing ideas\n\nH istorically, there has been a pernicious belief that the two assets a woman should possess when looking for a husband are beauty and wealth. One without the other might still secure a husband, but the absence of both could be a very trying obstacle to marriage for those wanting to wed.\n\nA late fifteenth century illustration of pride from An illustrated Yorkshire Carthusian Religious Miscellany. Pride is represented in the form of a young fashionably dressed man attracting the attention of two devils. (British Library)\n\nHowever, men and women throughout the ages have found ways to improve on what God gave them, thereby heartily annoying the Church. Anatole France said, 'Christianity has done much for love by making it a sin.' This was also the case with beauty. With Vanity (more commonly referred to as Pride) depicted as the greatest of the seven deadly sins, the religious establishment was not keen on people tinkering with what God had deemed perfection. While women took the brunt of the antibeautifying edicts, men were not immune to having their impulses to improve upon nature curbed as well.\n\nPrior to 1615, when Sir Robert Mansell acquired the first patent for the manufacture of 'looking glasses', the wealthy would use polished obsidian to admire (or bemoan) their looks. The Church naturally had something to say about this contemplative gazing; too much gazing into a mirror would make the devil appear. The invention of modern mirrors using silvered glass in 1835 meant mirrors were mass produced and far more widely available. With access to these shiny new instruments in which to gaze, young women were clearly not listening to stuffy decrees at church. Instead, they embraced superstitions, hoping to see an image of their future husbands in the mirror, while brushing their hair and eating an apple. A less terrifying folk belief was that if a man and woman first set eyes upon each other in a mirror, they would make happy spouses.\n\nEven before Venetians learned to silver glass and produce mirrors in the sixteenth century, men and women were concerned with their looks and chose elaborate, and sometimes painful, methods to beautify themselves. The medieval obsession with a high forehead had women plucking their hairlines almost to the top of their heads.\n\nArt and even religious iconography from this time depicts women with large oval faces, plucked eyebrows and high foreheads. Geoffrey De La Tour Landry, writing in 1371, had no sympathy for the pain involved in hair removal and warned that any woman indulging in such vanity would find herself despatched to Hell. There a devil would plunge a burning needle into each pore from which a hair had been plucked. This seems a tad harsh given that there was already a certain amount of discomfort within the sin itself.\n\nThis satirical cartoon printed in Puck magazine in 1910 shows a woman more in love with her reflection than with her beau. (Public domain)\n\nSize matters\n\nHowever, men were also heading for eternal damnation due to their obsession with the size of their shoes. From the late fourteenth century, for around a hundred years, men became enamoured with long pointed shoes called 'Crakows', named for the Polish capital where the style is believed to have originated. The points of these shoes, known as 'poulaines' or 'devil's fingers' by those who disapproved of them, were sometimes so long that they needed to be supported by whalebone or even tied to the wearer's legs with string to stop the shoe tripping up the elegantly dressed poulaine-wearer. Since the length of the shoe was supposed to denote the wealth or class of the wearer \u2013 with Edward III decreeing during his 50-year reign between 1327\u20131377 that common people were only allowed a six-inch toe and gentlemen were allowed fifteenth inches with nobility even longer \u2013 many attempted to increase their social cache by lengthening the tips of their poulaines.\n\nNobility had also long been connected with the height of women's 'hennins', the classic coned headdresses worn in the late Middle Ages. Sometimes this elaborate headgear would be truncated, but some reached the dizzying heights of 36 inches, putting even the most elaborate hats at the modern day races to shame. As if it wasn't enough of a chore to keep such a cumbersome piece of clothing atop your head, the medieval ladies who chose the hennin also had to face fire-and-brimstone preachers. One zealous priest, Friar Thomas Conecte of the Carmelite Order, in 1428 raged against 'the noble ladies, and all others who dressed their heads in so ridiculous a manner, and who expended such large sums on the luxuries of apparel'. He went further and encouraged ill-mannered street urchins to jeer at ladies wearing the hennin and even to pull down their headdresses in the street.\n\nA cruelly satirical cartoon from 1833 depicting the idea that beauty really is in the eye of the beholder. (Wellcome Library, London)\n\nA second caricature from 1810 depicting a pairing that illustrates how little looks mattered in the case of a good match. (Wellcome Library, London)\n\nSome fashions of the past were exaggerated in order to show that the person wearing these preposterous items was wealthy enough not to work. They didn't have to care about practical clothing considerations. After all, only peasants needed to be able to lift and carry, or indeed walk unimpeded. The wealthy could afford inconvenient frippery and it was a good way of signalling riches, if not always beauty.\n\nNonetheless, the longest and most incommodious clothing in the world could not guarantee a good match if your countenance lacked intrinsic beauty. Baldassare Castiglione wrote that 'beauty springs from God... and so one cannot have beauty without goodness'. In the past, beauty was perceived quite literally as a virtue.\n\nSuch was the obsession with fair looks that even kings and queens were terrified of being tricked into a poor match with an ill-favoured spouse. After all, the only personal qualities the court and political players were interested in were the titles, lands and alliances that a royal marriage could ensure; looks were the last consideration. Besides, we all know the aphorism that beauty is in the eye of the beholder. This was amply proved by the courtship of Elizabeth I and Fran\u00e7ois, Duke of Anjou, which began in the early 1570s and continued for many years.\n\nFran\u00e7ois was the youngest son of King Henri II of France and Catherine de' Medici. His face was scarred and pitted from a bout of childhood smallpox and his spine was deformed, making him lack stature. Yet, by all contemporary accounts and even her own words, the queen loved the man she referred to as her 'frog'. Although marriage negotiations were never finalised due to the general unpopularity of a French Catholic suitor, Elizabeth confessed that 'I have never in my life seen a creature more agreeable to me'. Alas even kings and queens have to bow to public opinion and Elizabeth was said to have put the needs of the country before that of her own heart.\n\nThe poem she is said to have written in 1582 entitled 'On Monsieur's Departure' may well have been about her beloved frog:\n\nI grieve and dare not show my discontent, \nI love and yet am forced to seem to hate,\n\nI do, yet dare not say I ever meant,\n\nI seem stark mute but inwardly do prate.\n\nI am and not, I freeze and yet am burned, \nSince from myself another self I turned.\n\nMy care is like my shadow in the sun,\n\nFollows me flying, flies when I pursue it, \nStands and lies by me, doth what I have done. \nHis too familiar care doth make me rue it.\n\nNo means I find to rid him from my breast, \nTill by the end of things it be supprest.\n\nSome gentler passion slide into my mind, \nFor I am soft and made of melting snow;\n\nOr be more cruel, love, and so be kind.\n\nLet me or float or sink, be high or low.\n\nOr let me live with some more sweet content, \nOr die and so forget what love ere meant.\n\nSo really the only area in which physical appearance might begin to play a part was if the question arose of whether a royal spouse was healthy enough to sire or bear a child. This was such a pronounced concern that even the powerful Elizabeth I was subjected to a full gynaecological examination in her forties, so that the French could be satisfied that her marriage negotiations with Fran\u00e7ois might still bear fruit. Even when courting royals weren't being subjected to medical examinations, ambassadors were regularly despatched to see the intended when a diplomatic union between states was mooted, to ensure that the prospective king or queen was sufficiently attractive and had no visible deformities or inherited illnesses.\n\nHenry VII sent three representatives to the court of the Queen of Naples in order to ascertain her charms. He gave his envoys an extensive list of characteristics they should note and report back on. These included the royal lady's height, her weight, the size of her breasts, the shape of her nose, the size of her forehead, the shape of her fingers and her age.\n\nHowever, her physical attributes weren't the only attributes Henry wanted his ambassadors to notice. The more delicate question of how sweet, or sour, her breath smelled and what and how much she ate and drank were also on the list of information to be acquired. After seeking such a plethora of facts about the Queen, it must have been a source of regret to his ambassadors that nothing came of the match \u2013 due primarily to political and financial reasons rather than the Queen's failure to match Henry's expectations regarding her beauty.\n\nKing Henry VIII, who was famously profligate in the matter of wives, was equally fussy in selecting his spouses. He even sent his chief adviser Thomas Cromwell to the Tower of London over the appearance of one of his brides. Anne of Cleves was not fair enough for the King's tastes and even a pre-nuptial painting by Holbein was not deemed accurate enough when the King finally met his future wife. He went along with the marriage to avoid a rift with the German state from which the young queen originated. However, his discourteous declaration after their wedding night was that she had an unpleasant body odour and her breasts sagged. Six months later he had the marriage annulled and, blaming Cromwell for the whole debacle, had him charged with treason.\n\nHiding faults\n\nWhile being beautiful isn't the same thing as being fashionable, many people throughout the ages have aimed to hide a lack of beauty through conspicuously fashion-forward clothing. One of the best fashions for disguising a less than tempting visage was the 'vizard', a mask worn by ladies in the sixteenth and seventeenth centuries in order to protect the skin against the effects of the sun. Vizards could hide a multitude of problems from noses rotted by syphilis to bad skin or pock-marked faces. These masks were often made of dark velvet on the outside to deflect the sun's rays and lined on the inside with silk or perfumed leather. The vizard, when not secured by ribbons to the side, was held in place with a bead on the inside, which women clamped between their teeth.\n\nThe somewhat frightening appearance of women wearing vizards was described by Phillip Stubbes, a puritan activist, in his book Anatomie of Abuses, published in 1583:\n\n'When thei use to ride abroade, thei have visors made ov Velvet... wherewith thei cover all their faces, havying holes made in them against their eyes, Whereout they look So that if a man that knew not their guise before, should chaunce to meete one of them he would think he met a Monster or a Devil: for face he can see none, but two broade holes aginst her eyes, with glasses in them.'\n\nA 1597 French miniature of a horse-drawn litter carrying a masked lady. (The British Library)\n\nBut the fashion was slow to die. As late as summer 1663, the celebrated diarist Samuel Pepys wrote:\n\n'Here [at the Royall Theatre] I saw my Lord Falconbridge, and his Lady, my Lady Mary Cromwell, who looks as well as I have known her, and well clad; but when the House began to fill she put on her vizard, and so kept it on all the play; which of late is become a great fashion among the ladies, which hides their whole face. So to the Exchange, to buy things with my wife; among others, a vizard for herself.'\n\nThe vizard was also popular because it had disreputable connotations; it was so routinely worn by courtesans that the term began to be used as a nickname for their profession.\n\nAnother beautifying use for black velvet was in the fashion for face patches. Tiny pieces of black velvet or silk were used to create intriguing shapes, such as dots, lozenges, stars and crescents. These were glued to the faces of both men and women. The practice can be traced back to Roman times, but it was most widespread in England in the late seventeenth century, although it had been known during the sixteenth century and remained in use for another hundred years.\n\nPerhaps the reason the trend took such a long time to decline was that it very handily concealed smallpox scars and other disfigurements, if placed wherever an offending blemish was to be found. In Hogarth's A Harlot's Progress (1732) series of six plates, plate 3 depicts the prostitute Moll Hackabout wearing face patches while syphilis cures in the room suggest that it is more than fashion that requires her to wear those patches. Indeed, the series ends with Moll dying of the disease, which was a common cause of death for women in her profession in the past.\n\nA Harlot's Progress (1732), plate 3, an engraving by William Hogarth shows Moll wearing face patches to hide her pox scars. (Wiki Commons)\n\nIf you were lucky enough to be using patches purely to draw attention to your best features, there were names for each of the most advantageous positions. Sarah Jane Downing writes in Beauty and Cosmetics 1550-1950 (2012), 'The \"coquette\" was situated near to a pretty smile, the \"passionate\" at the corner of the eye, and the \"gallant\" as a dimple in the middle of the cheek.'\n\nIt was not just women who engaged in patching, as the fashionable man about town was considered equally dashing when adorned with a patch. 'If it be a lover's part you are to act, take a black spot or two; twill make your face more amorous, and appear more gracious in your mistress's eyes,' wrote playwright, Henry Glapthorne in 1640.\n\nAs well as the continuing use of patches, there was much danger in attempting to look beautiful through artificial means. Lead within ceruse and powders regularly caused swollen eyes, receding gums and even premature death. In 1760 Lady Coventry, n\u00e9e Maria Gunning, a great reputed beauty, died of lead poisoning from her make-up. Her sister, Lady Elizabeth Hamilton, was also taken severely ill from using ceruse but survived \u2013 although not with her looks intact. The death of the celebrated courtesan Kitty Fisher was also linked to beautifying with lead.\n\nA coloured lithograph from 1865 showing three women with plaited and ringletted hair along with five plaited and ringletted hairpieces. (Wellcome Library, London)\n\nIn less severe cases, the lead-based make-up merely caused facial hair to thin or fall out completely, thereby necessitating the use of fake eyebrows made from the hide of mice. In the seventeenth century, plumpers were another beauty aid. These were small balls or discs made of cork, used to plump out the cheeks so that hollows made by missing teeth could be disguised. Then there were artificial teeth made from hippo ivory and even human teeth from either live impoverished donors or dead soldiers from battlefields such as Waterloo in 1815. Even the first president of the United States, George Washington (1732-1799) had a set of human teeth screwed into hippo ivory dentures. Finally, of course, there were also extravagant wigs to hide receding hairlines and thinning hair.\n\nAlthough men, just as much as women, took advantage of these crafty methods of enhancement, as is often the case, they were not the butt of jokes in quite the same way. The poet Matthew Price wrote this cruel little verse about mice hide eyebrows in 1718:\n\n'Helen was just slipt into bed\n\nHer eyebrows on the toilet lay\n\nAway the kitten with them fled \nAs fees belonging to her prey.'\n\nIt is believed that 'false parts' ballads have been composed since antiquity, with several long versions becoming popular in the seventeenth and eighteenth centuries. The narratives of these ballads usually involved a suitor being tricked into pursuing a comely maiden, who turned out to be an ugly hag once she had removed her false parts, such as a glass eye, a peg leg, or a wig masking her baldness. The more honest ballads had the young man appeased with the 'old hag's treasure chest', reinforcing the message that money was often the virtue selected over natural youth or beauty.\n\nAn 1825 lithograph depicting a couple assembling their false body parts: false teeth, a glass eye and wigs. (Wellcome Library, London)\n\nThe terror of being tricked by a lover's falsely constructed exterior was probably exacerbated by the Church. Early Christian writer, Tertullian, who charmingly described women as 'the doorway to the devil', wrote that they should 'have their eyes painted with chastity, the word of God inserted in their ears, Christ's yoke tied to their hair, and subject themselves to their husbands'. Make-up was often described not just as 'peynting' but also 'counterfeiting', emphasising that beauty that was artificial was some form of trickery.\n\nCount Castiglione, despite hailing from the more liberal sixteenth century Italian court, wrote in The Book of the Courtier (c.1516\u201318):\n\n'Surely you realise how much more graceful a woman is who, if indeed she wishes to do so, paints herself so sparingly and so little that whoever looks at her is unsure whether she is made-up or not, in comparison with one whose face is so encrusted that she seems to be wearing a mask and who dare not laugh for fear of causing it to crack... letting herself be seen only by torchlight, in the way a wily merchant shows his cloth in a dark corner... such is the uncontrived simplicity which is most attractive to the eyes and minds of men, who are always afraid of being tricked by art.'\n\nPhilip Stubbes, who wrote so disapprovingly of vizards, was also horrified by make-up, writing:\n\n'Saint Ciprian, amongst the rest, saith, a woman, through painting and dying her face, sheweth herselfe to be more then whorish. For (saith he) she hath corrupted and defaced (like a filthie strumpet or brothel) the workmanship of god in her, what is this els but to turn truth into falsehoode, with painting and slibbersauces?'\n\nStubbes should have taken more care over his choice of words, for the Elizabethans had by then followed the example of their queen, with women painting their faces with ceruse and alabaster and rouging their cheeks, in imitation of their 'faerie queen'.\n\nElizabeth I, painted in c.1600 by an unknown artist, was a great role model for fashions of the age. (National Portrait Gallery, London)\n\nFashion and frippery\n\nJust as a woman's use of make-up could indicate how attentively she had listened to the edicts of the Church, so too could her wardrobe. Queen Mary I, who set about the restoration of the Catholic Church in 1553, dressed dourly for her reign, leading the Venetian ambassador to say of her: 'She is a saint. She dresses badly.' By contrast Elizabeth I wore sumptuous capes and heavily embellished gowns in order to show that her reign was to be significantly different from that of her half-sister.\n\nThe writings of Margaret Fell, one of the founding members of the Religious Society of Friends (also known as Quakers) who was born eleven years after the death of Elizabeth I, show that by the late seventeenth century religious interference in the appearance of women was still a concern. She spoke out against the Christian sect's later obsession with plain clothing restrictions. 'This is a silly, poor Gospel. It is more fit for us to be covered with God's eternal spirit, and clothed with his Eternal light.'\n\nIt was not just the pious who wanted a say in how people were dressing. Trade meant that news was always coming into Britain from the continent of how the fashionable French and Italian courts were carrying on and, later, colonialism would yield far better clothing options in the form of Indian shawls and silks.\n\nThe well-heeled crowd had increasing opportunities to show their exalted position in the social strata through the medium of clothes and fine accessories, such as fans and gloves. For example, the seventeenth century heralded the development of the folded fan, whereas previously fans had been fixed. While aristocrats could afford these new fripperies, less well-off ladies had to content themselves with the feathered fixed fans of earlier times. By the eighteenth century, imports from the Far East and the development of cheaper printed fans had made folded fans affordable for a wider number of ladies.\n\nAs the middle classes struggled to keep up with their economic superiors in the matter of style, the changes that the Industrial Revolution in the mid-eighteenth century brought in must have been something of a relief. For this emerging class, it was deemed far more moral to be prudent and self-made than profligate and privileged. Nevertheless, many in the middle classes were still very much interested in social climbing through acquired wealth, connections or appearances.\n\nThis 1799 satirical caricature print entitled Parisian Ladies in their Full Winter Dress for 1800 by Isaac Cruikshank (1756\u20131811) shows how scandalously revealing the British found the Parisian high Greek look.\n\nIn Austen's Pride and Prejudice (1813), the well-dressed Bingley sisters are very haughty toward the Bennets when they learn that their maternal uncle Mr Gardiner is in trade rather than a gentleman like Mr. Bennet. Austen wittily gives the proud pair a nouveau-riche background that shows their snobbery for the hypocrisy it really is:\n\nTwo women, wearing full crinolines, mock the neo-Grecian attire of their Regency mothers. Satirical cartoon from the 11 July 1857 issue of Harper's Weekly (New York).\n\n'They were of a respectable family in the north of England; a circumstance more deeply impressed on their memories than that their brother's fortune and their own had been acquired by trade.'\n\nFashions also had a knack of recurring in the past, just as they do today. 'Imbecile sleeves' gained their dreadful name because they were often so large that it would be difficult for a woman wearing them to get through a narrow doorway. The real name for these puffy sleeves was 'gigot', from the French, meaning the hind leg of an animal, such as a sheep or a lamb \u2013 literally 'mutton leg' sleeve. They were hugely puffy at the top until the elbow, where they tapered in and tightly hugged the arm from the elbow to the wrist.\n\nFirst seen in the sixteenth century, these impractical sleeves returned to favour in the early nineteenth century. In the 1830s they were used to create the sloped shoulder silhouette much in favour and, by the 1890s, the Victorian obsession with hourglass figures was also served well by this fashion.\n\nThroughout the ages the ideal formula for 'beauty' has regularly changed her dimensions and qualities. From the high foreheads and pious oval egg-shaped faces of the early Middle Ages to the Victorian low hairlines and tight ringlets framing pale faces and dark eyes, fashion ran the gamut of female variety. You had to be lucky indeed to be born with the correct attributes to be deemed a beauty in your time.\n\nBooks giving advice on such matters abounded from the earliest times, with ancient Roman poet Ovid writing in The Art of Love that a woman should take care to remove body hair:\n\nAmerican clothing trading cards used to advertise the latest must-have fashions of the Victorian era. (Boston Public Library)\n\nThis Victorian American clothing advertisement claims that a gentleman will become irresistible to women if he purchases a suit from the Continental clothing store. (Boston Public Library)\n\n'A stubbled leg your suitor will not charm, \nAnd\u2013dare I warn?\u2013 no goat below the arm.'\n\nGiven how much work had to be put into becoming a 'natural' beauty, one would have hoped the writers and critics of times gone by would have been less severe upon those who used artifice and subterfuge to make the most of what they had. For love is not always quite as blind as one would hope.\n\n## Chapter Three\n\n## Love Tokens and Gifts\n\nAcceptable gifts from lovers and admirers\n\nT here is an old English saying from the mid-nineteenth century that 'kissing's out of fashion when gorse is out of bloom'. Given that gorse can be in flower throughout the year, this implies that you can always find love and romance.\n\nThe real challenge in days gone by was not in the kissing itself, but in finding an opportunity for it. If you were poor, you had work to attend to and so couldn't idle away the time in love-making. Most of those in service might only enjoy one afternoon off work per month and a lack of leisure time is a hindrance in courting. If you were rich, then there was pressure to refine all the accomplishments that a young lady was considered to need in order to make herself a good match.\n\nThe competition among debutantes and older single ladies was fierce, as shown by entries in Alice Miles's diary, published posthumously in Every Girl's Duty: Diary of a Victorian Debutante (1993):\n\n\"... there is Mrs Verschoyle; a very pretty brunette of two and thirty... Miss Castor who I met in London, forty and painfully plain, familiarly known as the Camel. Miss Ogle, commonly called 'The Ghoul'...who cannot be more repulsive looking \u2013 she says she's twenty-three, if so I am very sorry for her... there's a rather pretty little heiress Miss Harriet Ives Wright... who I suspect will put us all in the shade from the mere fact of her possessing \u00a34,000 a year.\"\n\nThe same source gives us an insight into a typical day for a Victorian debutante. Alice Miles usually rose late from bed to recover from the night before, then lunched at home or out with friends and then took an afternoon chaperoned stroll or ride out in the carriage at Hyde Park. This routine might be followed by 'morning' calls, which were really formalised visits at which one could leave calling cards. A relaxed five o'clock tea then gave way to hours of getting ready for a formal dinner at around eight or nine o'clock. This would end in the ladies retiring to the drawing room and the gentlemen smoking cigars and enjoying a cognac. It would only be later that the sexes would come together to play at cards and practise their flirtations. Other post-dinner enjoyments included concerts or even a ball, which would start at midnight and finish at four or five in the morning. All this definitely resulted in a need for a lie-in for the wealthy and pretty lady-about-town!\n\nThe social season for a debutante ran during Parliamentary recess from Easter until August and comprised races, river parties, regattas and picnics. In the autumn there would be shooting and country house parties to attend. While there was plenty of opportunity to see your preferred suitor in company, a private audience could not be hoped for. So, what to do in the interim, when thoughts arose about your suitor? In such cases, a love token or gift was the ideal solution. A man could look upon the hair in a ring and a woman could caress the gloves her intended had sent her.\n\nGiving presents to one's paramour was a tricky business. If you weren't yet betrothed or you had good reason to hide your courtship, then it was necessary to choose gifts with care. Although an example of a fabulously guileless and sweet courting gift was the small orphaned lamb that Gabriel Oak takes as a gift to Bathsheba in the hope his offer of marriage might be accepted in Thomas Hardy's Far From The Madding Crowd (1874). Farmer Oak takes no pains to hide his feelings for Bathsheba alerting her aunt to the proposed match and ensuring she discourages him in the first instance before Bathsheba deals a second, more fatal, blow to his hopes. While in this case the charmingly artless farmer felt no need to hide his intentions, a serviceman about to go off to war, for example, would often delay an engagement until his return to ensure that, should he not return, waiting to hear of his fate would not unduly restrict his fianc\u00e9e.\n\nHair was the perfect answer to a lover's quandary. A lock of hair was a sign of fidelity when given as a lovers' keepsake, but conversely it could provide a sentimental reminder of deceased loved ones. By the Victorian era, you could conceivably wear a piece of jewellery with hair inside it or inlaid within the design, because you were mourning a close relative or perhaps as a memento of a much-loved sister while you were away from home. Hair could be easily disguised, as remembrance was the perfect camouflage if you had a lover whose existence you would rather not reveal. Although one disturbing example was that of Lady Caroline Lamb sending Lord Byron a bloodied clump of her pubic hair! However, it must be said that discretion wasn't really that lady's forte, having made her affair with the poet quite public despite being married to the Whig politician Lord Melbourne.\n\nThe more discreet wealthy beau could hide his intent in coded rings called acrostic rings. The first letter of each gem inlaid within the ring would spell out a word such as 'regard', 'love' or 'dear' or a significant date. Only those in the know would realise that the ring had a special message in its setting.\n\nFor women, it must have been doubly difficult to hide a secret love for a shiny new piece of jewellery would have set tongues a-wagging. Presents of perishables such as fruit or flowers would perhaps have been more welcome, even if you'd only be able to press a flower or two to keep as a reminder of your lover's regard for you.\n\nThe Victorians were fond of the language of flowers or 'floriography', in which the selection of blooms in a bouquet could symbolise particular qualities and demonstrate your regard \u2013 or lack thereof \u2013 for the lady in question. For example flowers to avoid were: Columbine (folly), Lavender (mistrust), Morning Glory (affectation), Narcissus (egotism), Oleander (beware) and Yellow Carnations (rejection). As we saw earlier with the language of fans and precious stones, this seems to have been restricted to the wealthy and largely a female affectation. In any case, rural suitors would have been severely restricted in the choice of wild flowers they might pick for a bouquet.\n\nMore serious men mocked the whole idea of flower symbolism, with H G Wells writing in 1897, 'In these days we season our love-making with talk about heredity, philanthropy, and sanitation, and present one another with Fabian publications instead of wild flowers. But in the end, I fancy the business comes to very much the same thing.'\n\nThese illustrations from The Language of Flowers: An Alphabet of Floral Emblems (1857) show what each flower in a bouquet meant to Victorians.\n\nA wonderful tradition, found among the Pennsylvanian Quakers in the United States in the eighteenth and early nineteenth centuries, was that of true lover's knots. These were paper tokens written in such a way that they would form intelligible sentences all the way around the squares of paper, no matter which way they were read. One of the best examples of this is that of teacher Hugh Pugh's lover's knot made for Mary Fisher, one of his pupils in Bedford County, Pennsylvania.\n\nThe love token created by Hugh Pugh for Mary Fisher in 1801. (Photo courtesy: Meg Schultz)\n\nIn a precursor to the textspeak we now find widespread, Pugh also shortened phrases, for example 'CU' meaning 'see you'. Alas Mary, while she must have liked the poetry enough to keep the token and hand it down to her granddaughter, married a farmer and, by all accounts, broke poor Hugh's heart. The knot was also a marriage proposal, but in the typical petulance of spurned lovers past and present, he says that she will be an 'inconstant creature' with a 'double heart' if she turns him down.\n\nMeg Schultz, great-great-great granddaughter of Mary Fisher, now owns the artefact and kindly gave me permission to reproduce it here. While Mary Fisher was not won over by Hugh's indubitable artistry, the token is even more spectacular when you consider that Meg's research turned up that he only had one hand.\n\nHere is the transcript of the Pugh-Fisher lover's knot:\n\n'A true Lovers Knot to thee my Dear I send, An Emblem of true Love without an end, Crossing turning, winding in and out, Never ceasing turning round about. And as thee sees its Linkes and Crosses here, so hath thy Beauty prov'd to me a Snare, By observation of true Love I find I am bereaved of both \u2665 and mind.\n\nMost lovely fair one look with pity down, And do not on thy faithful Lover frown, But pardon him who ever doth thy Love desire, And ever will thy Beauteous form admire.\n\nTherefore thou Lovely fair one let thy Beauty shine, With Beams of Comfort ravishing and divine, That so my raving Soul may by thy Love, Pass into Bliss if we both constant prove, Then shall these Crosses in this Knot of Love, Be all disdain'd if thou consenting prove.\n\nHere is an Impression of my \u2665 thee may see, Within this Knot that I present, to thee, Therefore thee may imagine that I am in grief, And none but thee can yield to me Relief, My ravished Soul doth ever long to see, The Marriage Knot so firmly ty'd between thee and me.\n\n(Top centre circle)\n\nWhy do I Love, go ask \nthe Alerious Sun \nWhy every Day he around the World doth run \nAsk Thames and Tiber why they Ebb and flow \nAsk Damask Roses why in June they grow. \nThey shew to us how everything doth move \nThus teaching them to that, and me \nto Love. \nMary Fisher \nBedford County Decem'r 9th \n1801\n\n(Right circle)\n\nThere is but one \nAnd only one \nAnd I am only he \nThat loves but one \nAnd only one \nAnd thou art the only she \nRequite me with like love again \nAnd say thus unto me \u2014 \nThere is but one, And only one \nAnd thou art the only he. \nMary Fisher\n\n(Bottom centre circle)\n\nAccept lovely fair Maid \nFrom thy neighbor and friend \nEach wish that can friendship endear \nMay the bounty of Heaven propitiously endear \nLong Life and Happy each Year. \nMay every enjoyment which prudence allow \nThy Life Long continue to Bless \u2014 \nMay Love and Esteem \nWeave a Wreath for thy Brow \nAnd thy Beauty be crown'd with Success. \nMary Fisher\n\n(Left circle)\n\nAs soon grief shall \nsink into my \u2665 \n2CUX my Love without desert (?) \nYou have a \u2665, a double \u2665, I fear. \n2 great a X of \u2665 oh \u2665 forbear \nAX, AX, ICUB, \nA double XU are to me.\n\nH Pugh [Monogram]\n\nThis Ring is round\n\nAnd hath no end \nSo is my Love \nTo thee my Friend \nMary Fisher\n\n(Diagonal rectangle, top left)\n\nHere I dare venture with my Love a lot (?) \nIn Half an Hour she does not read my Knot\n\n(Diagonal rectangle, top right)\n\nAnd if she wins I'll freely pay my Debt,\n\nBut if she loses then I'll claim my Bett.\n\n(Diagonal rectangle bottom right)\n\nAs for description, A begins thee will find,\n\nE Ends the same, be constant in thy mind.\n\n(Diagonal rectangle, bottom left)\n\nLovers well know what it is to part,\n\nWhen between 2 Lovers there is but one \u2665\n\nH Pugh [Monogram]\n\n(8 small outer boxes)\n\nMy \u2665 you have\n\nYour \u2665 I crave\n\nMy \u2665 you have\n\nConfin'd\n\nAnd leaves all other\n\nHearts behind.\n\n(4 tiny corner diagonal boxes)\n\nIf thou refuse me\n\nI must say \nthou art\n\nAn unconstant\n\nCreature\n\nWith a double \u2665\n\nHandmade or mass-produced?\n\nThe nineteenth century was also the age of the mass-produced Valentine's card, allowing men for the first time to purchase rather than make a handcrafted card for the ladies they admired. Not only were there beautiful confections of cards with cut-outs and textiles, there were even horribly cruel versions to punish a lover who had rejected you. These often made use of the 'false parts' idea to accuse the recipient of being a fake beauty or of some character default. Called a 'Vinegar Valentine', understandably few examples of these survive.\n\nOne rare vinegar Valentine's card, showing a cartoon of a woman with a large nose, was inscribed:\n\n'On account of your talk of others' affairs\n\nAt most dances you sit warming the chairs.\n\nBecause of the care with which you attend\n\nTo all others' business you haven't a friend.'\n\nDoubtless a lady who received one of those would think she had had a lucky escape, if she discovered which rogue had sent her it.\n\nA modern carved wooden love spoon token from the author's own collection.\n\nRustic traditions put a great deal more art into courtship gifts. Intricate carvings were painstakingly created for sweethearts in Wales, where elaborate hand-made love spoons became popular during the seventeenth century, as well as in other parts of Europe. The spoons had the dual purpose of showing a girl's father a suitor's woodworking skills, as well as charming the lady as evidence of the time spent meticulously carving something just for her. There was also an intimacy inherent within the utensil, as a spoon would touch your beloved's mouth. Yet, over time the spoons became so elaborate that they ceased to have a practical function and began to be used as wall hangings.\n\nEach symbol carved into a love spoon had a special meaning. Anchors were often prevalent as sailors would carve their spoons while on long sea voyages. Bells indicated marriage, hearts, of course, denoted love and a horseshoe stood for luck. The practice of being able to carve caged balls into the spoon's handle was said to indicate how many children were hoped for in the union.\n\nIn some parts of northern England and Scotland, knitting sheaths or sticks were regarded as love tokens. These beautifully engraved objects were very useful for those who gained a second income from knitting or had to clothe their families, as the sheath would ensure you could knit one-handed while doing other things, such as walking to market or even while engaging in farm work. At this time men also frequently knitted to help supplement the family income.\n\nSome men would carve the names of their sweethearts into the sticks they themselves used and tuck them into their belts as they knitted. Their ladies must have experienced a frisson of excitement to see their name so close to their beloved's body.\n\nIt was this idea of second-hand physical contact that also made stay busks or corset stiffeners popular engraved gifts for women. Worn right by the heart, under the breast, it was a romantic gift that held the promise of future intimacy for both suitor and the lady being wooed.\n\nIn the courtly tradition, there were, however, some gifts considered disreputable. Andr\u00e9 Le Chapelain, presumed to be a courtier in Marie de Champagne's court, wrote in his twelfth century book, About Love, that the more conventional gifts for a woman in love, which she might receive without her reputation being tarnished, included: 'a handkerchief, a wreath of gold or silver, a mirror, a purse, a comb, a picture and a washbasin'. A washbasin seems a somewhat conspicuous gift, especially as Le Chapelain also advises that a ring given by a lover should be worn on the smallest finger of the left hand, since that hand is more likely to be out of sight. If one must take care to hide a lover's ring, surely a great big washbasin would be a giveaway?\n\nAs the fashion for courtly love began to be pastiched by artists and writers, in the mid-thirteenth century, Ulrich von Lichtenstein wrote Frauendient [In the Service of Ladies] in which the protagonist \u2013 a knight-errant \u2013 cuts off his finger and sends it as a gift to the lady he serves. Not unsurprisingly she fails to be wooed by his attentions, and so he sets off wandering from Venice to Vienna while dressed as Venus in white gowns with braided hair and duelling all-comers for the honour of his lady.\n\nIn the United States, historic love tokens were not quite so eccentric. A popular form was quite literally a token: a coin with the sides polished off and hand-engraved with names, dates and messages. Easy to carry around or make into pieces of jewellery such as bracelets and pins, the tokens made for popular presents. They also had symbolic pictures engraved on them, such as a bluebird for happiness or a forget-me-not for enduring affection.\n\nGifts were important not just for wooing, but in order to prove a couple's intentions towards one another. As we saw earlier, marriages in earlier periods did not need a member of the clergy or parental consent, if the couple were of age and made their declarations before witnesses. This didn't change in England and Wales until Lord Hardwicke's Marriage Act came into effect in 1754 and decreed a formal marriage ceremony take place in order for a marriage to be valid. The official name of this legislation is An Act for the Better Preventing of Clandestine Marriage.\n\nA famous Tudor case of a marriage contracted in the earlier manner is of William Hanwell and Isabel Riddysdale. On 1 January 1519, the couple pledged their troth to each other at a house in Beachampton, Buckinghamshire, making them husband and wife. When Isabel subsequently regretted her decision and refused to honour her vows, William took her to the church court and produced two witnesses to the union. One witness revealed that he had entrusted him with two pennies that were to be given to Isabel as a love token. This was seen as proof of the couple's intention to marry.\n\nAn engraved German or Austrian coin from the late nineteenth century, popularly given as love tokens, with mounting brackets for wearing in a brooch or necklace. (Flickr Creative Commons Licence: Jerry \"Woody\", Canada)\n\nA problem with receiving gifts is that this creates an obligation to a gentleman or lady that you are not interested in. Samuel Beeton, husband to the famous Mrs. Isabella Beeton, writes at length about this subject in his Complete Etiquette for Ladies (1876):\n\n'In her intercourse with gentlemen a lady should take care to avoid all pecuniary obligation. The civility which a gentleman conventionally owes to a lady is a sufficient tax \u2013 more she has no right to expect or accept. A man of good sense and of true politeness will not be offended at her unwillingness to become his debtor. On the contrary, he will respect her delicacy and approve her dignity, and consent at once to her becoming her own banker on all occasions where expense is to be incurred.'\n\nSo it seems that the notion of a man paying for dinner on the first date is not in any way linked to historical ideas of chivalry.\n\nBeeton goes on to counsel a woman to only accept invitations to amusements if she is permitted to pay for her own ticket, handing over her share of the bill to the gentleman before he leaves. This fastidious attention to staying financially independent extends to gifts as well.\n\n'We disapprove of ladies going to charity fairs in the evening, when they require a male escort, and when that escort is likely to be drawn into paying exorbitant prices for gifts to his fair companion \u2013 particularly if induced to do so from the fear of appearing mean or of being thought wanting in benevolence.'\n\nHe is equally stern in the matter of women's behaviour in shops while accompanied by a suitor:\n\n'When visiting a fancy shop with a gentleman, refrain from excessively admiring any handsome or expensive article you may chance to see there; above all, express no wish that you were able to buy it, and to regret that you cannot, lest he should construe these extreme tokens of admiration into hints that you wish him to buy it for you. To allow him to do so would, on your part, be very mean and indelicate, and on his very foolish.'\n\nSamuel Beeton may have written harshly against those women who angled for gifts from gentlemen, but present-giving is an essential part of wooing and what girl wouldn't want to wear a ribbon bought by her beau? After all, not every woman will receive presents as extravagant as the ones Louis XV bestowed on his mistress, the Countess du Barry, who was given a ch\u00e2teau as a throwaway gift to entice her to spend the evening with him.\n\nWriting in the more censorious nineteenth century, however, Beeton advises his lady-readers that 'no gentleman who really respects her will offer her anything more than a bouquet, a book, one or two autographs of distinguished persons, or a few relics or mementoes of memorable places \u2013 things that derive their chief value from associations.' Jewellery, articles of clothing and costly ornaments 'ought to be regarded as an offence rather than a compliment, excusable only in a man sadly ignorant of the refinements of society'.\n\nIn fact several anonymous Victorian etiquette writers suggested that no gift should ever been given to a lady unless an offer of marriage has been made and she has accepted it.\n\nBut what was a woman to do if she was offered a present by a suitor who had not yet made her an offer? Samuel Beeton is unequivocal: 'she should set him right, and civilly, but firmly refuse to be his debtor.' Equally, if, for whatever reason, a courtship or betrothal ended, 'a gentleman should return all presents, letters and other tokens of regard'.\n\n## Chapter Four\n\n## Coxcombs and Strumpets\n\nHow to recognise roguish men and women of ill repute \u2013 and how to avoid them\n\nC ourting is often seen as gentle, chaste romance \u2013 a precursor to staid and secure marriage. The truth is that almost a third of all Elizabethan brides entered the church already pregnant. There was plenty of illicit sex being enjoyed, but alas only the rural poor accepted it as a custom to check the fertility of a bride-to-be. It was an extremely risky activity for a woman to engage in if she belonged to the upper or middle classes.\n\nThe hand-wringing and upheaval of Lydia Bennet's elopement is vividly depicted in Jane Austen's Pride and Prejudice (1813). Perhaps the most chilling words of the entire book are within Uncle Gardiner's letter to Mr Bennet: 'I have seen them both. They are not married, nor can I find there was any intention of being so; but if you are willing to perform the engagements which I have ventured to make on your side, I hope it will not be long before they are.'\n\nIt is clear from Lydia's earlier letter to Mrs Forster that she had believed she was eloping to be married at Gretna Green, but is later convinced by Wickham to go to London, as yet unmarried. Elizabeth Bennet takes some little comfort in the discovery that Lydia was not party to 'a scheme of infamy'. While her sister may have been wild and rebellious, even she knew that the only respectable way to be with a man in society was to ultimately marry him.\n\nThe newly emerging middle classes in Regency England were loath to emulate the dissolute lifestyle of the Prince Regent and his cronies. Reputation played a vital role in establishing that a daughter of the gentry was marriageable. What would have become of Lydia after her elopement had Wickham not been bribed to marry her?\n\n'The Elopement' by Edmund Blair Leighton, 1893. (Public domain)\n\nThese 1911 postcards imagine what a Gretna Green elopement must have been like during the Regency period. (Public domain)\n\nWhat is certain is that she would not have been allowed to re-enter polite society and the marriage prospects of her sisters would also have been severely damaged. The colourful case of Lady Sarah Lennox illustrates this quite succinctly. The debutante poised to perhaps become a future queen of England ended up the victim of her own romantic inclinations. While George III looked fondly on her, her family's ambitions ensured that he was pushed away from considering her a suitable mate. It was only after she married Charles Bunbury in 1762 that her own actions rather than those of her family started to impact on her social status. While a discreet affair could be hushed up, Lady Lennox not only had an affair with Lord William Gordon, but also had an illegitimate child with him \u2013 a daughter named Louisa, born in 1768. Given that neither her or her child were disowned by her husband, she could perhaps have got away with the adultery, were it not for the fact that she subsequently ran away with Gordon, taking Louisa with her. Alas her lover soon rejected her and she was forced to live with her brother since her husband would not take her back and applied for a divorce on the grounds of adultery, ignoring the fact of Louisa's parentage. A divorce was granted on 14 May 1776, despite Lady Sarah's protestations. With her reputation in tatters, most eligible matches would now be beyond her, but she ended up remarrying an impoverished \u2013 and therefore not entirely desirable \u2013 army officer.\n\nMen \u2013 and their female relations \u2013 kept a hawk-like eye out for any report of impropriety in a potential mate. It didn't have to be as dramatic as an elopement such as that of Lady Sarah Lennox, even the suggestion of being an indelicate sort would be enough to render a woman unappealing \u2013 after all, no man wanted to run the risk of being cuckolded or paying for a child whose parentage was questionable. Although it is as well to point out that wealth was a virtue that could balance out a multitude of sins.\n\nThe extremely wealthy heiress Seymour Dorothy Fleming married Sir Richard Worsley at the tender age of seventeen in 1775, but then six years later eloped with his friend Captain George Bisset. The resulting scandal did not bring into the public eye the question of Lady Worsley's child by George Bisset, but there is little else it didn't cover. Salacious details of the marriage and Seymour's infidelities were revealed in the press when Worsley brought an action against Bisset for criminal conversation, a law under which a husband could claim damages from a third party who he believed had debauched his wife. While he was suing for \u00a320,000, a vast sum, the jury awarded him just a shilling when evidence surfaced that Worsley had encouraged and connived in his wife's affairs for the purposes of titillation. The judge even concluded that Lady Worsley had been 'prostituted' for at least four years by her husband who participated willingly in his cuckolding.\n\nJames Gillray's hand-coloured etching, published 14 March 1782, entitled 'Sir Richard Worse-than-sly, exposing his wife's bottom \u2013 O Fye!' (National Portrait Gallery)\n\nThe press loved one detail in which a maid at the baths overheard Worsley encouraging Bisset to climb on his shoulders in order to look through a window at his naked wife changing at the baths. Mary Marriott, the bathing attendant in question, reported that she heard Worsley call out: \"Seymour! Seymour! Bisset is going to get up and look at you!\"\n\nSadly, despite Seymour enduring public humiliation in the cause of George Bisset, he left her for a younger woman shortly after the trial \u2013 despite her being pregnant again with him. The fate of the child is unknown, suspected stillborn, and the earlier child that Richard Worsley had adopted, had died during the trial. Even her legitimate son by Worsley died before his parents at the age of nineteen. Shunned by her family and by those who fancied themselves 'respectable', Seymour left for France once her separation was settled, a settlement that obliged her not to return to England for four years \u2013 a dangerously unfortunate clause given that it meant she had to lay low in France at the time of the French Revolution when aristocrats were being routinely guillotined. She survived and even married a man 20 years her junior after Worsley died. In a fitting reversal of tradition, her husband took her name of Fleming after their marriage.\n\nCharacter and virtue\n\nSuch aristocratic examples of debauched wives and shocking husbands made the middle classes very uneasy; making a good match was no longer just ensuring that the prospective husband or wife had wealth but also that they were of good character. By the late 1830s, the Victorians had a comprehensive code of behaviour that ensured ladies were not just respectable, but seen to be so. As we'll see in the next chapter, chaperons were a very important part of this.\n\nSo how were virtuous young women to tell a Wickham from a Darcy? Roguish men who were only after one thing were not unheard of and being able to seduce a young heiress was a surefire way to bag a fortune. It was for this reason that heiresses were so assiduously protected by their guardians, while relatively poor women, such as Lydia Bennet and her like, had more freedom.\n\nYoung women eager to protect their virtue had plenty of authors willing to proffer advice, if lacking a mother's guidance.\n\nIn The Young Lady's Friend, published in 1838 by an anonymous married American author, the young lady in question is counseled to 'not be afraid to refuse the acquaintance of a known libertine, it is a tribute which you owe to virtue, and, if generally paid, would do more to purify society, and keep the moral standard of it high, than the laws of the land or the eloquence of the pulpit'.\n\nMen were also subject to self-help manuals, such as The Gentlemen's Book of Etiquette, published in 1875, which contains some very sound advice on how to speak about the object of one's affections:\n\n'It is only the most arrant coxcomb who will boast of the favor shown him by a lady, speak of her by her first name, or allow others to jest with him upon his friendship or admiration for her. If he really admires her, and has reason to hope for a future engagement with her, her name should be as sacred to him as if she were already his wife; if, on the contrary, he is not on intimate terms with her, then he adds a lie to his excessively bad breeding, when using her name familiarly.'\n\nAmerican commentators were painfully aware of how a more open attitude to courting could give rise to a variety of ills. The Bazar Book of Decorum, published in 1870, carried 'A Warning about Fast Girls':\n\n'The free eye is a marked characteristic of the libertine, and all modest women should turn persistently from its roving and unlicensed glances. Some girls of the fast kind, with an audacious defiance of conventional propriety, and yet often with no thought of offense against real modesty, will not only recklessly dally with these intrusive looks, but not seldom venture a cast of them on their own account.\n\n'There are fast women everywhere, but the fast girl seems to be more particularly an American product. A tendency on the part of the young, unmarried female to eccentric flights of any kind is effectually checked in most countries by parental control.\n\n'It is, moreover, a paltry ambition, and not without risk to virtue, to aspire to the distinction of being pointed out as 'the low-necked' Bel Smith, or the 'high-stepping' Fanny Jones, or the girl who drank a whole bottle of champagne, or she who smoked one of Frank Tripup's fifty-cent regalia...Her essential defect is a vulgar ambition for notoriety. She will endure any thing but obscurity, and therefore takes care that she is seen, heard, and talked of by all the world. Her dress is accordingly flaunting, her voice loud, her words slangy, her eye staring, her manners obtrusive, and conduct audaciously irregular. All this may be, and is, doubtless, done without any overt act of vice, but it looks so much like it that the difference is hardly perceptible to the external observer. In fact, it seems to be the purpose of the fast damsel to assume the semblance of wickedness...\n\n'It would seem that American parents might curtail somewhat the liberty of their children, without interfering too much with that independence of action so essential to the strength of character. Girls are allowed to consider themselves women too soon, and are thus prematurely emancipated from parental control...with less idle time and more watchful parental care, there would be fewer of those fast girls...'\n\nThe worry was that a 'fast girl' would make for a capricious and flirtatious wife, thereby making her husband look ridiculous. However, the attractiveness of such girls meant that many men ended up trapped in marriages to women who had been fun to court, but were not suitable spouses. Avoiding such a situation was behind much of the advice given to young men of the era, whether published or oft-repeated in the parlours.\n\nIf Americans were preoccupied by 'fast girls', the British Victorians were far more concerned with keeping distinctions of class and money in place. An heiress could be as fast as she liked and still attract suitors, but poorer girls could not hope for such indulgence.\n\nThe idea that less well-off women could not be 'delicate' or that they were more primitive or base has historically been given much credence by male authors. Andr\u00e9 le Chapelain writing in the twelfth century, believed that the rules of love did not apply to farmers who he claimed resembled the beasts they cared for, by freely giving themselves up to lust as nature intended. He counsels against allowing farmers to form the finer feelings, in case this should make them too weak to fulfil their purpose of producing food for the community.\n\nA stolen kiss depicted in Francesco Hayez's 'The Kiss', 1859. (The Yorck Project)\n\nA well-dressed client inspects the prostitutes at a brothel, 1884. (Wellcome Library, London)\n\nHe also openly advises labourers to rape any lower class woman who takes their fancy, since her 'shyness' needs to be overcome. 'And if you should, by chance, fall in love with some of their women, be careful to puff them up with lots of praise and then, when you find a convenient place, do not hesitate to take what you seek and to embrace them by force.' Given that this advice is coming from a clergyman, one can see the open contempt that the nobility had for the peasantry. They were perceived as little more than animals to abuse and abandon.\n\nIn 1728, John Gay wrote The Beggar's Opera, a satirical ballad opera, which in one verse presented the sanitised view of prostitutes: as young girls abandoned and forced to sell sex on the streets of Covent Garden:\n\nVirgins are like the fair flower in its lustre,\n\nWhich in the garden enamels the ground;\n\nNear it the bees in play flutter and cluster,\n\nAnd gaudy butterflies frolic around;\n\nBut, when once pluck'd, 'tis no longer alluring,\n\nTo Covent-garden 'tis sent, (as yet sweet).\n\nThere fades and shrinks, and grows past all enduring, \nRots, stinks, and dies, and is trod under feet.\n\nHarris's List of Covent Garden Ladies, published between 1757 and 1795, seems to almost gleefully and titillatingly document the fall of women who were seduced and abandoned by men. For example a Miss Les-r of 23 Upper Newman Street has this entry:\n\n'This lady was a few years since, a servant in a gentleman's family, near Holborn: in which capacity she used frequently to walk for the air, with her little ward, in Gray's Inn Gardens. A certain gentleman of the law, perceiving a very fine girl, which she was at that time, often in the walks, took the opportunity of conversing with her, and soon after persuaded her to come and make some tea for him in his chambers. The sequel, it were needless to relate: she was debauched, and soon after deserted by her betrayer. The consequence of which was, having lost her place, and being destitute of character, she was obliged to have recourse to her beauty for a subsistence. She took lodgings near Red Lyon Square, and had a number of successive admirers. She was, at this time, not about twenty; tall and well made, with a fine open expressive countenance, large amorous eyes; her other features in due symmetry; her mouth very agreeable, and her teeth regular; in a word, she was at that time one of the finest women upon the town, and, accordingly, made one of the best figures from the emoluments of her employments. She was some time after taken into keeping by a man of fortune, with whom she made a summer excursion into the country; but, upon his demise, her finances being exhausted, she was compelled to have recourse to a more general commerce, in which she has not been so successful, as before; and chagrin added to the usual irregularities accidental to her profession, has diminished those charms which were before so attracting; her face is now rather bloated, and she is grown somewhat masculine in her person; she may, nevertheless, still be pronounced a very good piece, and a desirable woman.'\n\nThe records of the Foundling Hospital in London, which was Britain's first children's charity, attest to how many women found themselves deserted by seducers. Prior to this the only provision for abandoned children was Christ's hospital, founded in 1552, but by 1676 illegitimate children were no longer admitted there. After this the only provision for illegitimate babies was parish poorhouses or the workhouse; during the 1720s and 1730s, the death rate for children in workhouses was over ninety per cent. It was not just the illegitimate who suffered for the poor who fell on hard times were also subject to the nightmare of the workhouse. The eighteenth century saw what artist William Hogarth described as 'a golden age of English philanthropy' and the Foundling Hospital was a beneficiary of that philanthropic zeal. Indeed he himself had a long association with the Hospital and decorated the walls of it with donated pieces of his own art, as well as that of other sympathetic artists.\n\nThe moral censure that accompanied any dealings with the Foundling Hospital only added to the distress of being parted from your child. A frowning middle class panel at the Hospital decided whether the unmarried mothers were of good character and passed judgement on whether the Hospital would take in their illegitimate babies. On 25 March 1741, a temporary house in Hatton Garden was the base for the new hospital and that night the first children were admitted. It is said that heart-wrenching cries were heard throughout the night as mothers were parted from their children. The tokens that mothers left with the babies to remind them of their love are truly heartbreaking. In these small coins, ribbons and notes, you can see the hope that remained that they would meet again, however unlikely it was to be.\n\nOne note, pinned to the clothing of little Florella Burley, born 19 June 1758, read 'Pray let particular care be taken of this little child' while one heart-shaped, silver-coloured token was engraved with 'You have my heart, though we must part'. Some of the tokens were hearts split in two, the hope being that the mother could return one day and identify her child by the heart half she had left with him.\n\nThe lower down the social scale you were, the fewer options were available to you for dealing with the consequences of sex before marriage. As such, some employers stipulated that servants were to have 'no followers' so as to ensure propriety and also, to some extent, protect their female employees from the evils of seduction, although it is as well to say that most servants who were sexually assaulted were the victims of their masters or male relatives of their employers. Where a master was not a rapist, you could still suffer a terrible fate if caught stealing. Once a servant was tarnished with that reputation, no domestic work would again be possible and many women were forced to turn to prostitution. Shockingly, G.P. Merrick, chaplain of London's Millbank Prison, writing in 1890 found that out of 16,000 prostitutes he interviewed, forty per cent had been domestic servants.\n\nHowever, Charles Dickens in his Dickens's Dictionary of London (1879) advised against employers attempting to govern the romances of their staff. 'A serious mistake, and one too often made, is to lay down the hard-and-fast rule \"no followers allowed\". Servants always have had and always will have followers, whether their masters and mistresses like it or no. It is much wiser to recognise this fact, and to authorise the visits of the \"follower\" at proper times and seasons, first taking pains to ascertain that his antecedents and character are good.'\n\nYoung men had considerably more power to engage in sexual activity with far fewer consequences, unless the woman in question had violent male relations who decided to force the issue of marriage. Respectable young men often came to the marriage bed virgins, but more worldly ones had often already engaged in relations with the ladies of the night. Samuel Beeton, who was keen on writing authoritative etiquette advice for ladies and gentlemen in the nineteenth century, may well have infected his wife, Isabella, with syphilis due to a visit to a brothel prior to his marriage \u2013 as convincingly argued by historian Kathryn Hughes.\n\nSex was not considered to be a male problem, it was the 'wages of sin' in the form of illegitimate children that was frowned upon. Plus the perpetual bachelor became a figure of fun and somewhat despised by a society that placed marriage high on the list of responsibilities of an adult male. While historically, from the Roman Empire onwards, many western countries have attempted to impose a 'bachelor tax' (Britain repealed its Marriage Duty Act in 1706 when it appeared not to have had much effect on the behaviour of single men), it is more the cultural gibes that continued unabated. For example in his novel Le Cousin Pons (1847), Honor\u00e9 de Balzac describes the main character thus: 'Like all confirmed bachelors, who hold their own lodgings in horror, and live as much as possible in other people's houses, Pons was accustomed to the formulas and facial contortions which do duty for feeling in the world...' Pons is shown humiliating himself by running small errands for the families he dines with in order to continue living off their tables. It is telling that the original title of the book was 'Le Parasite'. Honour and duty demanded that a man marry and run his own household.\n\nIn 1740 the original version of Beauty and the Beast by Gabrielle de Villeneuve was published in France. It is interesting to note that the beast in this version only turned back into the prince after, not before, the wedding night. This seems to be an allegory for the idea that men are savage creatures and only sex within marriage can tame their 'beastly' libido.\n\nHowever, society had the perfect weapon against licentious behaviour: the chaperon.\n\n## Chapter Five\n\n## In Praise of Chaperons\n\nMany a reputation has been saved by a handy sibling or great-aunt in tow\n\n'A young lady, during her first winter in society, does not use a separate visiting card, but has her name engraved on that of her mother or chaperon', declared The Home Manual in 1889. Written by Mrs John A Logan, this guide to all things domestic was unequivocal in stressing the importance of ensuring that you had a chaperon to police your first season 'out' in society. A chaperon could be any older, married or widowed woman such as a sister or an aunt or even an unrelated family friend. Of course the chaperon had to have an impeccable reputation herself. While not considered as appropriate in society, within a domestic setting, siblings could also be chaperons to ensure that nothing untoward occurred between a couple. The high number of working class brides who came to the altar pregnant indicates sex before marriage was not seen as such a terrible sin, since it proved that the woman was capable of conceiving. However, by the start of the Regency era such lax attitudes had been properly nipped in the bud and much more was being done to protect the offspring of the well-to-do.\n\nG. M. Woodward's cartoon, showing the futility of mothers advising daughters who are in love, was published c.1790\u20131801. (Public domain)\n\nServants often doubled as impromptu, unofficial chaperons. For one thing, they were known to gossip and sending a servant out of the room when meeting with a gentleman caller could raise eyebrows. They could also bear vital witness should a man think he were being cuckolded. The case of Lady Colin Campbell's divorce hinged on evidence from servants watching her allegedly having sex with her lover through a keyhole. However, the nobility realised that a more effective ploy would be to have a strata between themselves and their lower class servants, in the form of nannies and governesses.\n\nJoanna Martin in her book Wives and Daughters (2004), reveals that professional nannies were a Victorian invention designed to counter the effects of children spending too much time in the company of servants.\n\n'Some households employed two such women: in 1806 Mary Talbot wrote that she would like to have two governesses at Penrice, so that the sub-governess could supervise the children when the superior governess was otherwise occupied \"to prevent their ever being with servants\".\n\n'Within the household, the governess occupied a position that was uneasily poised between the family and the servants, and it was inevitable that she should suffer from the anomalies of her situation. A governess had to be a lady and, while socially inferior to her employers, she would invariably consider herself to be superior to the main body of servants.'\n\nThis role extended until the governess herself married or, if she remained with the family, to the coming of age of her charges, wherein her duties would often become those of a companion and chaperon.\n\nOn occasion, however, even the governess proved a dangerous temptation for the master of the house. In Jennifer Newby's Women's Lives (2011), we learn of the scandal caused by governesses who married 'above themselves'.\n\nA humorous 1897 photographic print on curved stereo card entitled 'Partiality' showing a pouting chaperon, possibly a sibling, enviously looking at a beau making his 'partiality' known. (Boston Public Library)\n\n'The 1861 census reveals that there were 24,700 governesses in England and Wales. These women earned very little, most between \u00a335 and \u00a380 a year from 1830 to 1890. As well as earning a servantsized salary, governesses had to put up with being 'higher class' than their fellow servants in the household, but never the social equal of their employers. In 1858, a woman journalist wrote scathingly:\n\n'Just let a remote idea be entertained of marriage between a son, or any other member of the family, and the governess; why, another siege of Troy would scarcely occasion more commotion \u2013 the anger, scorn, vituperation lavished on the artful creature.'\n\nHowever, while working for a 'particularly attentive and affectionate' widower, 19-year-old May Pinhorn was surprised when \"He got me into a summer house and told me he hoped I would be his wife, an offer I promptly and brutally refused.\"'\n\nThe story of one of the daughters of the Fox Strangways family at the centre of Joanna Martin's Wives and Daughters:Women and Children in the Georgian Country House (2004) shows that, even with the provision of governesses, love will out.\n\nThe daughter of Lord Illchester, Susan Fox Strangways, eloped in 1764 with an actor named William O'Brien, whom she had met at an amateur theatrical performance in which Susan was appearing. Her elopement was made possible while she was visiting a friend and able to send the household servants away. To marry an actor was unheard of for an Earl's daughter at the time, and Horace Walpole wrote to his friend Horace Mann, the British Resident in Florence, about it:\n\n'A melancholy affair has happened to Lord Illchester. His eldest daughter, Lady Susan, a very pleasing girl though not handsome, married herself two days ago at Covent Garden church to O'Brien, a handsome young actor. Lord Illchester doted on her and was the most indulgent of fathers. 'Tis a cruel blow.' The couple were promptly shipped off to America in the hope that they would find their fortune there.'\n\nWhile her chaperon sleeps a young girl keeps watch on her as her lover on bended knee kisses her hand. A mid-nineteenth century engraving by H.C. Shenton after F.P.Stephanoff. (Wellcome Library, London)\n\nAn older woman acts as chaperon to a girl who is being courted by a young man. (Wellcome Library, London)\n\nNot all chaperons met with such failure, it has to be said. The American etiquette author, Emily Post, still displayed a Victorian adherence to the usefulness of chaperons when writing in 1922.\n\n'As a matter of fact the only young girl who is really \"free,\" is she whose chaperon is never very far away. She need give conventionality very little thought, and not bother about her P's and Q's at all, because her chaperon is always a strong and protective defense; but a young girl who is unprotected by a chaperon is in the position precisely of an unarmed traveler walking alone among wolves\u2014his only defense is in not attracting their notice.'\n\nIn Elizabeth Gaskell's 1855 novel North and South, the hero John Thornton spots Margaret Hale at a late hour of the night with a gentleman and no chaperon. The unknown man turns out to be her brother, but Thornton naturally assumes that he is her lover. The company of a chaperon also meant that any gentleman in your party was rendered safe from speculation.\n\nGenerally, it was acknowledged by most that the presence of a chaperon was not necessarily an absolute protection against mischief, but the sight of one let the world know that you knew what was proper in polite society.\n\nEmily Post again:\n\n'Ethically the only chaperon is the young girl's own sense of dignity and pride; she who has the right attributes of character needs no chaperon \u2013 ever. If she is wanting in decency and proper pride, not even Argus could watch over her! But apart from ethics, there are the conventions to think of, and the conventions of propriety demand that every young woman must be protected by a chaperon, because otherwise she will be misjudged.'\n\nIf an unsuitable man attempted to make an acquaintance with her charge, Emily Post urged a chaperon to ensure that this did not come to pass.\n\n'If an objectionable person \u2013 meaning one who can not be considered a gentleman \u2013 is inclined to show the young girl attentions, it is of course her duty to cut the acquaintance short at the beginning before the young girl's interest has become aroused. For just such a contingency as this it is of vital importance that confidence and sympathy exist between the chaperon and her charge.'\n\nAdam Petrie, in his Rules of Good Deportment and of Good Breeding (1720), writes:\n\nEntitled 'Hide and seek' this 1896 print shows a couple hiding from the girl's frantically searching chaperon. (Public domain)\n\n'If a young man and a young woman be in a room and you be to remove from them, and if there is none with them, it is imprudent and uncivil to shut the door after you; for if a person of a narrow soul shall come and find them shut up in a room they may be ready to stain their reputation, which should be dear unto us and cautiously preserved.'\n\nSamuel Beeton, in his Complete Etiquette for Ladies (1876), advises a young woman to explicitly ask for advice as to whether a man is suitable or not.\n\n'If a gentleman gives you reason to believe that he wishes to engage your affections, seek the advice of your parents, that they may gain for you every necessary particular with regard to his morals and disposition, and means of suitably providing for you.'\n\nOne can't imagine that many young ladies paused in the pursuit of passion to consider asking their parents to investigate the suitability of a prospective beau.\n\nA chaperon's duties could be arduous. Emily Post writes of what she should expect if her young charge is hosting friends at a party or is receiving guests:\n\n'The chaperon (or a parent) should never go to bed until the last young man has left the house. It is an unforgivable breach of decorum to allow a young girl to sit up late at night with a young man \u2013 or a number of them. On returning home from a party, she must not invite or allow a man to \"come in for a while.\" Even her fianc\u00e9 must bid her good night at the door if the hour is late, and someone ought always to sit up, or get up, to let her in. No young girl ought to let herself in with a latch-key. In old-fashioned days no lady had a latch-key. And it is still fitting and proper for a servant to open the door for her.'\n\nMs Post also gives a startling example of regional ideas of propriety. 'Even in Victorian days it was proper in Baltimore for a young girl to go to the theater alone with a man, and to have him see her home from a ball was not only permitted but absolutely correct.'\n\nOf course there were dire consequences to having a less than zealous chaperon. As such, most chaperons were from the couple's immediate family or highly trusted governesses with a longstanding connection to the family.\n\n## Chapter Six\n\n## Love Songs, Letters and Poems\n\nArticulating your desire can get you an assignation \u2013 or potentially an execution!\n\nV ery few people, perhaps as little as five per cent, were able to read and write in fifteenth century Britain. In fact, it was the late Victorian era before free elementary education was introduced in Britain, allowing the majority of the populace to become literate. So initially wooing by letter was a luxury only enjoyed by the educated. However, the use of the spoken word in the form of romantic ballads and poetry was accessible to all.\n\nAs any successful lover knows, words are crucial to seduction. Whether a poem is recited or a letter is read, no wooing can be done without the liberal use of tender sentiments expressed correctly. This places the tongue-tied or silent lover at a distinct disadvantage.\n\nThankfully, the era in which we begin our journey into romantic words of courtship had a code in place to ensure that every man knew what was expected of him when wooing.\n\nThe medieval poem The Romance of the Rose \u2013 begun by Guillaume de Lorris in 1237 and finished forty years later by Jean de Meung \u2013 is a work of courtly literature, which served to educate people about the edicts of courtly love. This was supplemented by the medieval tradition of troubadours at court, whose music expounded the joys of courtly love, chivalry and courtesy.\n\nThey were the renowned poets of love who performed music and poetry in the High Middle Ages (c.1100-1350) and many came from the southern half of France, northern Spain and parts of Italy. Karen Ralls explains in Medieval Mysteries (2014) how the songs of the troubadour and trobairitz (female troubadour) were grouped into three major styles: trobar leu (light), trobar ric (rich) and trobar clus (closed). Trobar leu was the most popular form as the words were the most simple and could be understood most generally, while both ric and clus used symbolic and metaphoric language that would only have been accessible to a few in the know. This is in addition to the different 'schools' within the tradition. Ralls writes of the origin of troubadour poems:\n\n'The earliest troubadour whose work has survived is Guilhem de Peitieus, better known as Duke William IX of Aquitaine, who lived in the late eleventh and early twelfth centuries. The medieval work Orderic Vitalis refers to William composing songs about his experiences on his return from a crusade in 1102, with some experts claiming this may be the earliest reference to troubadour lyrics known today.'\n\nWilliam IX's wife, Philippa of Toulouse, might well have wished that he hadn't been quite so interested in romance since she was usurped by the Viscountess Dangerosa, an affair for which the Duke was excommunicated from the Church although they termed it an abduction. When a papal representative who was bald asked the Duke to return the Viscountess to her husband, William responded \"Curls will grow on your pate before I part with the Viscountess\". Poor Philippa retired to an abbey and passed away two years later.\n\nHis song, Farai Chansoneta Nueva, or 'I Shall Write A New Song' showed the typical yearnings that were the meat and wine of courtly expression. Here's an effusive extract from that poem:\n\nA medieval lover is pulled up in a basket to speak with his beloved. From the German illuminated manuscript, Codex Manesse, which features the work of many famous poets and was produced between 1305 and 1315. (Public domain)\n\nHer skin is white as ivory;\n\nNo other's in my history:\n\nAn urgent show of love for me\n\nIs needed to remove all doubt.\n\nI'll die now, by St. Gregory,\n\nWithout a kiss, indoors or out.\n\nWhat good, fair lady, will be done\n\nIf with your love you'd up and run?\n\nPerhaps you want to be a nun?\n\nI tell you now, that I love you:\n\nBy sorrow I will be undone\n\nUnless my claim appeals to you.\n\n(Translated by James H. Donalson)\n\nTroubadours were responsible for composing some of the most popular love songs of the age and the taste for these flowed from the European courts down to the rest of society, although the songs became more rustic in nature when they drifted far away from the refined sensibilities of court.\n\n'Maiden in the Moor' is an example of an eleventh century troubadour composition that is still sung as a folk song today. While not explicitly referring to love, the song has a great deal of romantic imagery and symbolism from the Middle Ages, such as the primrose and violets that was her 'mete' (sustenance or food) and her dwelling or 'bour' was among the red rose and the lily-flower. These floral evocations indicate the ideas of purity that surrounded the ballads of this era. It was so successful at conveying those ideas that the Church later attempted to co-opt the song's melody and turn it into an altogether more ecclesiastical offering.\n\nHere are the lyrics of the full song:\n\n'Maiden in the Moor Lay'\n\nMaiden in the moor lay \u2013\n\nIn the moor lay \u2013\n\nSeven-night fulle,\n\nSeven-night fulle,\n\nMaiden in the moor lay \u2013\n\nIn the moor lay\u2013\n\nSeven-nights fulle and a day.\n\nWell was her mete.\n\nWhat was her mete?\n\nThe primerole and the\u2013\n\nWell was her mete.\n\nWhat was her mete?\n\nThe primerole and the violette.\n\nWell was her drinke.\n\nWhat was her drinke?\n\nThe colde water of the\u2013\n\nThe colde water of the\u2013\n\nWell was her drinke.\n\nWhat was her drinke?\n\nThe colde water of the welle-spring.\n\nWell was her bour.\n\nWhat was her bour?\n\nThe rede rose and the\u2013\n\nThe rede rose and the\u2013\n\nWell was her bour.\n\nWhat was her bour?\n\nThe rede rose and the lily-flour.\n\nWhy was there such an emphasis on purity during this time? One theory about the reason this chaste tradition of courtly love swept the western world was that there was a surplus of 'second sons'. Only the eldest son inherited land and were attractive marriage prospects. Younger brothers were obliged to make their own way in the world and could only hope to marry into wealth. Born into the nobility yet without the means to marry, they would often join the crusades, embark on a career in the Church or spend time wooing wealthy women.\n\nIt was considered a safe outlet for these young men to woo with song and deed a lady they would never consummate their love with. The other great advantage to courtly love was that most marriages in this period were contracted for the purposes of alliances, property or other financial or political reasons. This provided a chance for a married woman to experience being wooed without any possible stain on her character or threat to her marriage.\n\nFolk songs show how disastrous the result of consummating an adulterous love could be. One of the 305 traditional ballads collected by Francis James Child between 1882 and 1898 is the tale of 'Little Musgrave and Lady Barnard'. The tale itself predates the collection and has been traced back to the seventeenth century; in later versions, Little Musgrave is called Matty Groves. In the song a servant of Lord Donald (a name variant on Barnard) is spotted by the Lord's wife at church and is seduced by Lady Donald. Another retainer runs to tell Lord Donald who is away from home, who returns to find the lovers in bed.\n\nLittle Matty Groves, he lay down and took a little sleep\n\nWhen he awoke, Lord Donald was standing at his feet\n\nSaying \"How do you like my feather bed and how do you like my sheets?\n\nHow do you like my lady who lies in your arms asleep?\"\n\n\"Oh, well I like your feather bed and well I like your sheets\n\nBut better I like your lady gay who lies in my arms asleep\"\n\n\"Well, get up, get up,\" Lord Donald cried, \"get up as quick as you can\n\nIt'll never be said in fair England that I slew a naked man\"\n\n\"Oh, I can't get up, I won't get up, I can't get up for my life\n\nFor you have two long beaten swords and I not a pocket knife\"\n\n\"Well it's true I have two beaten swords and they cost me deep in the purse\n\nBut you will have the better of them and I will have the worse\n\nAnd you will strike the very first blow and strike it like a man\n\nI will strike the very next blow and I'll kill you if I can\"\n\nSo Matty struck the very first blow and he hurt Lord Donald sore Lord Donald struck the very next blow and Matty struck no more And then Lord Donald, he took his wife and he sat her on his knee Saying \"Who do you like the best of us, Matty Groves or me?\" And then up spoke his own dear wife, never heard to speak so free \"I'd rather a kiss from dead Matty's lips than you or your finery\"\n\nLord Donald he jumped up and loudly he did bawl\n\nHe struck his wife right through the heart and pinned her against the wall\n\n\"A grave, a grave,\" Lord Donald cried, \"to put these lovers in\n\nBut bury my lady at the top for she was of noble kin\"\n\nInterestingly, the poignant idea of an adulterous wife preferring her dead lover even as it seals her own fate may well have been the root of the inaccurate rumour that Katherine Howard said at her execution that she died a queen but would have preferred to live as Culpeper's wife.\n\nEternal love triangles\n\nDuke William's granddaughter, Eleanor of Aquitaine, brought the tradition of courtly love over from the continent to England in 1154 when her second husband, Henry of Anjou, became King of England. Highly educated and a skilled politician, Eleanor was a great patron of the arts. The poets and writers of the courtly tradition appealed to her sensibilities, having always been forced to make political marriages rather than giving in to any notion of romantic love. Her daughter, Marie, brought these ideals and ideas to the court of Champagne of which she became countess and was patron to the troubadour Chr\u00e9tien de Troyes. Tobias Churton writes in his book, The Gnostic Philosophy (2003):\n\n'Courtly love became a part of courtly life and its customs and special forms came to dictate what was expected of a courtier, who was usually a knight of lower rank than the signeur who might, in this region (Champagne), be a woman. In the Languedoc, privileged women could enjoy the respect and indeed love expected from a vassal. The basic form of Fine Love is woven into this relationship of vassal to lord, hence the romantic custom of getting on one knee before the loved one...'\n\nWho would have thought that this is where the custom of kneeling to propose marriage to your beloved comes from? The strong women of this time and region, who would often be left in charge while their lords were off fighting in the Crusades, were being given the respect and submission usually offered to a lord by a vassal.\n\nA famous example from this time is the story of Tristan and Isolde. Although based on an old Celtic tale, the story became popular in the twelfth century when many songs retold the legend. The Prose Tristan (c. 1240) by an unknown author or authors was the version upon which Sir Thomas Mallory based his tale of King Arthur, Guinevere and Lancelot. Tristan, a nephew (or in some versions a mere courtier-knight) of King Mark of Cornwall, is sent to escort Isolde, Mark's fianc\u00e9e, from Ireland to marry Mark. The couple have been betrothed in order to end a war between Ireland and Cornwall. Since the marriage is not one of love, Isolde is given a love potion by a kinswoman of hers, half of which she is instructed to take and the other half she is to give to Mark. However, she gives it to Tristan instead and the couple fall in love. The courtly versions of the story have the lovers consummate their love, but the potion excuses them from responsibility for the transgression. Elements of the courtly tradition exist in Tristan's regard for Mark, despite his betrayal, and in Mark's refusal to believe others in his court about the couple's adultery.\n\nEdmund Leighton's 1902 painting shows Tristan and Isolde with her husband, King Mark, in the background. (Public domain)\n\nThis legend is where the notion of impossible love enters the romantic milieu and it ties in well with the yearning without consummation that characterises courtly love. From this time onwards writers and artists find an attraction in the idea of tragic love.\n\nHenry VIII, who is more notorious for his marital exploits than any other achievements during his reign, was a king whose wives often suffered in the cause of tragic love. Perhaps the most tragic of all is the story of Katherine Howard, the pretty first cousin of Anne Boleyn who was just 15 when she was betrothed to the 49-year old king. After marrying Henry in the summer of 1540, poor Katherine would be tried for treason and executed less than two years later in February 1542. Suspicions were raised about Katherine having had sexual relations with Francis Dereham prior to her marriage. An investigation was conducted and one of the main pieces of evidence against her was a letter she had written to Thomas Culpeper, a courtier in Henry's retinue. It was damning in that it contained phrases such as 'it makes my heart die to think what fortune I have that I cannot be always in your company'.\n\nThat might have been enough for a fiercely jealous, thin-skinned spouse such as Henry, but there was also testimony from the ladies who attended Katherine prior to her arrival at court that she had engaged in a serious dalliance with Francis Dereham and the two even went so far as to call each other 'husband' and 'wife'. Poor Katherine's fate was probably sealed when her closest servant Joan Bulmer revealed that the young queen had urged her to arrange a secret meeting with Culpeper. While Culpeper did not admit to consummating the affair, he admitted that the queen was 'languishing and dying for love of him', which would have been enough to hurt Henry's not inconsiderable ego. The mere intention to have an affair was treachery enough and the parties were to suffer for it.\n\nIn the end both Culpeper and Dereham were executed in December 1541 and their severed heads mounted on stakes atop London Bridge. Two months later, the young queen \u2013 stripped of her royal title by then \u2013 was taken by boat en route to her own execution, passing under the bridge on which she could see the decomposing heads of her former lovers.\n\nPerhaps courtly love was a better option for potential lovers, since it was rare indeed for a love affair to end in dangerous sexual congress. Those espousing the virtues of what was described as 'fine love' were not overly concerned with sex alone \u2013 or at least pretended not to be. R. Barber in The Knight and Chivalry (2000) writes: 'Troubadour poetry is not about women, their beauty or charms; it is about the lover and his longings. The highest praise is measured in terms of the lady's influence on the admirer, and so the qualities of the love loom large in their philosophy.'\n\nThe written word was not always connected with the higher ideals of sexless romance, as the notable example of John Wilmot (1647-1680), the second Earl of Rochester, shows. The Earl's extremely saucy poems were apparently not for publication, but nevertheless scandalised later generations with their lewdness. Here's one dealing with premature ejaculation and subsequent impotence!\n\nThe Imperfect Enjoyment\n\nNaked she lay, clasped in my longing arms,\n\nI filled with love, and she all over charms;\n\nBoth equally inspired with eager fire,\n\nMelting through kindness, flaming in desire.\n\nWith arms, legs, lips close clinging to embrace,\n\nShe clips me to her breast, and sucks me to her face.\n\nHer nimble tongue, love's lesser lightning, played\n\nWithin my mouth, and to my thoughts conveyed\n\nSwift orders that I should prepare to throw\n\nThe all-dissolving thunderbolt below.\n\nMy fluttering soul, sprung with the pointed kiss,\n\nHangs hovering o'er her balmy brinks of bliss.\n\nBut whilst her busy hand would guide that part\n\nWhich should convey my soul up to her heart,\n\nIn liquid raptures I dissolve all o'er,\n\nMelt into sperm, and spend at every pore.\n\nA touch from any part of her had done 't:\n\nHer hand, her foot, her very look's a cunt.\n\nSmiling, she chides in a kind murmuring noise,\n\nAnd from her body wipes the clammy joys,\n\nWhen, with a thousand kisses wandering o'er\n\nMy panting bosom, \"Is there then no more?\"\n\nShe cries. \"All this to love and rapture's due;\n\nMust we not pay a debt to pleasure too?\"\n\nBut I, the most forlorn, lost man alive,\n\nTo show my wished obedience vainly strive:\n\nI sigh, alas! and kiss, but cannot swive.\n\nEager desires confound my first intent,\n\nSucceeding shame does more success prevent,\n\nAnd rage at last confirms me impotent.\n\nEv'n her fair hand, which might bid heat return\n\nTo frozen age, and make cold hermits burn,\n\nApplied to my dear cinder, warms no more\n\nThan fire to ashes could past flames restore.\n\nTrembling, confused, despairing, limber, dry,\n\nA wishing, weak, unmoving lump I lie.\n\nThis dart of love, whose piercing point, oft tried,\n\nWith virgin blood ten thousand maids has dyed,\n\nWhich nature still directed with such art\n\nThat it through every cunt reached every heart\u2014\n\nStiffly resolved, 'twould carelessly invade\n\nWoman or man, nor ought its fury stayed:\n\nWhere'er it pierced, a cunt it found or made\u2014\n\nNow languid lies in this unhappy hour,\n\nShrunk up and sapless like a withered flower.\n\nThou treacherous, base deserter of my flame,\n\nFalse to my passion, fatal to my fame,\n\nThrough what mistaken magic dost thou prove\n\nSo true to lewdness, so untrue to love?\n\nWhat oyster-cinder-beggar-common whore\n\nDidst thou e'er fail in all thy life before?\n\nWhen vice, disease, and scandal lead the way,\n\nWith what officious haste doest thou obey!\n\nLike a rude, roaring hector in the streets\n\nWho scuffles, cuffs, and justles all he meets,\n\nBut if his king or country claim his aid,\n\nThe rakehell villain shrinks and hides his head;\n\nEv'n so thy brutal valor is displayed,\n\nBreaks every stew, does each small whore invade,\n\nBut when great Love the onset does command,\n\nBase recreant to thy prince, thou dar'st not stand.\n\nWorst part of me, and henceforth hated most,\n\nThrough all the town a common fucking post,\n\nOn whom each whore relieves her tingling cunt\n\nAs hogs on gates do rub themselves and grunt,\n\nMayst thou to ravenous chancres be a prey,\n\nOr in consuming weepings waste away;\n\nMay strangury and stone thy days attend;\n\nMay'st thou never piss, who didst refuse to spend\n\nWhen all my joys did on false thee depend.\n\nAnd may ten thousand abler pricks agree\n\nTo do the wronged Corinna right for thee.\n\nThis would not have been sent to a mistress, unless Rochester wanted to laugh at himself, which, by all counts, he was capable of doing. Much literature though was produced for the purposes of courting.\n\nPoetry and songs are not the only ways to woo a lady, however, and history has many famous marriages contracted on the basis of letters written during the courtship. More formal suitors would even make the offer of marriage via a letter rather than in person, perhaps hoping the pain of a potential rejection would sting less in written rather than verbal form.\n\nThe Love Letter by Jean-Honor\u00e9 Fragonard (1732\u20131806). (Public domain)\n\nAuthor ES Turner writes in A History of Courting (1954) of a wonderful anecdote in which a passionate proposal was reconsidered and retracted without the lady ever being aware of it.\n\n'The Victorian Post Office was evidently not rigidly bound of regulations. There is a story that Theodore Hook wrote a letter containing a proposal of marriage, posted it and then changed his mind. Hurrying round to the post office he authenticated his writing and was handed back the fatal letter. Today any local postmaster would rather doom a couple to an unhappy marriage than disgorge a proposal once entrusted to his care.'\n\nVictorian Valentine's cards were so romantic that the tradition continues to this day. Cards from 1899. (Public domain)\n\nPassion was generally frowned upon by the Victorians, but they did ensure, by introducing the penny post in 1840, that expressions of desire could have an easy outlet. Books offering gentlemen advice on writing letters for various purposes began to appear, although at times these served more as amusing guides of what not to do.\n\nThe Gentleman's Letter Writer was published at the turn of the century and, in the tradition of Victorian self-improvement manuals, it explicitly condemned attempting to engage a lady acquaintance through the post. Such vulgar behaviour towards a lady to whom the correspondent had not been introduced was deemed utterly reprehensible. The editors were also amused at some of the profuse declarations of love gentlemen writers had had cause to set down on paper, so much so that they re-published them \u2013 whether to inspire other correspondents to such flights of fancy or to discourage them from it, is not entirely clear.\n\nHere is an example of a particular bit of purple prose that appeared in that book.\n\nLetter to a Lady\n\nDear Madam, I have been so harassed with love, doubt, distraction, and a thousand other wild and nameless feelings, since I had the happiness of being in your company, that I have been unable to form one sane reflection, or to separate events from the feelings that accompanied them \u2013 in fact, I have been totally unable to bring my thoughts into anything like regularity, for they are so entirely mixed up with the idea of yourself, that the business of this world, and the pursuits of amusement and pleasure, have been entirely forgotten in the one passion that holds undivided empire over my soul.\n\nI have deferred from day to day penning this confession to you, in order that I might have been enabled to have done so with some degree of ease and calmness; but the hope has proved fruitless. I can resist no longer, for to keep silent on a subject which is interwoven with my very existence, would be death to me. No, I am unable to do so, and I have therefore determined to lay open to you the sufferings of my heart, and to implore from you a restoration of that peace and happiness which once were mine.\n\nYou, my dear Miss \u2013, are alone the cause of my unhappiness, and to you alone can I look for a fervent passion that devours my soul for your adorable self, can only be allayed by the declaration that I am loved as fervently in return. But dare I ask so much purity, so much sweetness, mildness and modesty, to make such a declaration? \u2013 I know not what I say \u2013 but O! my dear Miss \u2013, be merciful, and if you cannot love me \u2013 say, at least, that you do not hate me.\n\nNever could I survive the idea of being hateful to that angelic being, whose love I prize more than existence itself. Let me then cling to the idea that time may accomplish that which, I fain hope, a first impression has done resuming, unless a fatal pre-engagement exists (a thing I dare not trust myself to think of), that you comply with my request, seeing that my designs are perfectly sincere and honourable. I remain, waiting with utmost impatience for your favourable reply.\n\nDear Miss \u2013, Your devoted servant till death.\n\nIn contrast, the suggested letter to the father of the object of one's affection is considerably briefer:\n\nFrom a Lover to a Father on his attachment to the Daughter\n\nSir, \u2013 As I scorn to act in any manner that may bring reproach upon myself and family, and hold clandestine proceedings unbecoming in any man of character, I take the liberty of distinctly avowing my love for your daughter, and humbly request your permission to pay her my addresses, as I flatter myself my family and expectances will be found not unworthy of your notice. I have some reason to imagine, that I am not altogether disagreeable to your daughter; but I assure you, honestly, that I have not as yet endeavoured to win her affections, for fear it might be repugnant to a father's will.\n\nI am, Sir, \nYour most obedient servant.\n\nThe writers don't leave the poor father in the lurch as to how to reply to such a request, offering both negative and affirmative templates for responses:\n\nThe Father's answer in the negative\n\nSir, \u2013 I make no doubt of the truth of your assertions, relative to yourself, character, and connections; but as I think my daughter too young to enter into such a serious engagement, I request I may hear no more of your passion for the present; in every other respect, I am, Sir, Your most obedient.\n\nThe Father's answer in the affirmative\n\nSir, \u2013 There is so much candour and honour apparent in your letter, that to withhold my consent would be both ungenerous and unjust. As the duty of a father demands, I shall first make some necessary inquiries, assuring you that I would never oppose my daughter's choice, except I had some very just reason to imagine it would be productive of ill consequences, for I am convinced that, in the marriage state, happiness consists only in reciprocal affection. You may therefore depend upon hearing from me in a few days; till then, I remain, Your very faithful servant.\n\nFrom these pedestrian and socially sanctioned exaltations, we move on to the truly witty and flirtatious. Courtesans were great at written repartee and found many new suitors through this method. Perhaps one of the most famous is Harriette Wilson (1786\u20131845) whose audacity won her the attentions of many rich and powerful benefactors. She wrote to the Prince of Wales saying that she was told that she is 'very beautiful' and perhaps he would like to see her. When he wrote back agreeing to a meeting and asking her to visit him, she sent the following cheeky response:\n\nSir, to travel fifty-two miles, in this bad weather merely to see a man with only the given number of legs, arms, fingers &c. would, you must admit, be madness in a girl like myself surrounded by humble admirers who are ever ready to travel any distance for the honour of kissing the tip of her little finger; but, if you can prove to me that you are one bit better than any man who may be ready to attend my bidding, I'll e'en start for London directly. So, if you can do anything better, in the way of pleasing a lady, than ordinary men, write directly: if not, adieu, Monsieur le Prince.\n\nI won't say Yours\n\nBy day or night or any kind of light\n\nBecause you are too impudent\n\nCourtesans knew that the way to seduce a man was to tease and even insult in the first instance to increase desire in a man. Regency life was very mannered and the respectable women that these men came into contact with on a daily basis would be a distinct contrast to the witty, flirty communications they could expect from a courtesan. This was as true in earlier times too as this letter on love and seduction by Ninon de L'Enclos (1616-1706) shows; she was the beautiful courtesan whose salons attracted writers such as Moliere and Fontenelle.\n\nHere she saucily encourages the Marquis de Sevign\u00e9 to lie for the purposes of seduction.\n\nNinon de L'Enclos to the Marquis de Sevign\u00e9\n\nShall I tell you what renders love dangerous?\n\nIt is the sublime idea which one often appears to have about it. But in exact truth, Love, taken as a passion, is only a blind instinct which one must know how to value correctly; an appetite which determines you for one object rather than for another, without being able to give any reason for one's preference; considered as a link of friendship, when reason presides over it, it is not a passion, it is no longer love, it is an affectionate esteem, in truth, but peaceful, incapable of leading you out of bounds; when, however, you walk in the traces of our ancient heroes of romance, you go in for the grand sentiments, you will see that this pretended heroism only makes of love a deplorable and often disastrous folly. It is a true fanaticism; but if you strip it of all those virtues of hearsay, it will soon minister to your happiness and to your pleasures.\n\nBelieve me, that if it were reason or enthusiasm which governed affairs of the heart, love would become either insipid or a delirium. The only way to avoid these two extremes is to follow the path I indicate to you. You have need of being amused and you will only find what you require for that amongst the women I speak of. Your heart needs occupation; they are made to captivate it...\n\nHonesty in love, marquis! How can you think of that! Ah, you are a good man gone wrong. I shall take great care not to show your letter; you would be dishonoured. You could not, you say, take on yourself to employ the manoeuvre which I have counselled you. Your frankness, your grandiose sentiments would have made your fortune in the old days. Then one used to treat love as a matter of honour; but today, when the corruption of the century has changed everything, Love is no more than a play of whim and vanity... How many occasions do you not find where a lover gains as much by dissimulating the excess of his passion, as he would in others, by displaying greater passion than he feels?\n\nWords to woo\n\nWhile many accomplished writers penned moving love letters, there was often as much beauty in the communications of normal folk. We've already seen the wonderful example of true lover's knots, the folk art of the Pennsylvanian Quakers in the United States in the eighteenth and early nineteenth centuries.\n\nHowever, another American tradition that was even handier during courtship was the mid to late nineteenth century tradition of confession albums, thought to originate in Victorian parlour games and fortunetelling amusements. More involved than autograph books, a lady might ask a gentleman caller to fill out her confession book (a later version used by American teenagers in the 1980s were known as 'slam books'). These were originally handwritten questions and later included pre-printed questions. Asking what your ideal partner would be like was a popular question to get some insight into your beau's true feelings.\n\nEven Karl Marx's daughter, Jenny, kept a book with entries from 1865-1870, in which even her father's answers were recorded. Due to the fact that the practice was widespread among the middle classes and all the family and visiting guests filled them in, it was perceived as a fairly harmless way to flirt and was taken up with great enthusiasm.\n\nA cover of a confessions book by N. P Holmes. (Licensed under CC BY-SA 3.0 via Wiki Commons)\n\nAA Milne, writing in 1921, dates the end of the custom as being at the turn of the century. 'The confession-book, I suppose, has disappeared. It is twenty years since I have seen one. As a boy I told some inquisitive owner what was my favourite food (porridge, I fancy), my favourite hero in real life and in fiction, my favourite virtue in woman, and so forth.'\n\nPoems were also a popular way of stating one's feelings and there were plenty of poets whose work you could quote from if you weren't quite up to the job of writing one yourself. Perhaps the most famous love poem is the 'Roses are Red' stanza, a child's nursery rhyme dating from 1784, although the imagery of roses red and violets blue predate this by almost 200 years.\n\nThe rose is red, the violet's blue,\n\nThe honey's sweet, and so are you.\n\nThou are my love and I am thine;\n\nI drew thee to my Valentine:\n\nThe lot was cast and then I drew,\n\nAnd Fortune said it should be you.\n\nIf words win over your beloved, then your journey into successful wooing is only just beginning! Some believe that marriage is the start rather than the end of courtship, as we shall see in the next chapter.\n\n## Chapter Seven\n\n## How to be a Good Life Partner\n\nOnce you've said your vows, your work is only just beginning\n\nAnew husband carries his bride over the threshold of their new home so that she doesn't stumble and incur a bad omen. As the bride will soon discover, there are many other obstacles throughout marital life that she must avoid tripping over.\n\nAn image of family bliss from Mrs LG Abell's Gems by the Wayside: An Offering of Purity and Truth, 1878. (Public domain)\n\nA humorous illustration in a 1902 edition of Puck magazine has a young woman suggesting that a man always has a companion to share his troubles with in marriage. The bachelor quips that not every man can afford a valet and butler. (Public domain)\n\nHistorically women left their father's home to enter that of their new husband, where his will was paramount. Women had no property rights until the late Victorian era and so everything she brought to the marriage was then the property of her husband, as indeed, to all intents and purposes, was she. This did not mean that she was not expected to fulfil her duties as a wife. Along with figuring out how to get along with a new husband, the new bride had to also move from being a dependent to the mistress of her own home. These new responsibilities often hung heavy on young shoulders.\n\nFor a young man, marriage meant the start of adult life and the prospect of sex sanctioned by the church and society at large. In 1754, Josiah Wedgwood \u2013 who was due to marry two days later \u2013 wrote blissfully of the day 'when she will reward all my faithful services and take me to her arms! To her nuptial bed! To pleasures which I am yet ignorant of.'\n\nYet, most women were as concerned about the practical realities of looking after a home as they were by the pleasures of the nuptial bed.\n\nFinancial matters\n\nJuno Moneta, the Roman goddess from which the word 'money' derives, is also the patron of affianced couples about to marry. She protects funds and so in ancient Rome money was coined in her temple and the modern words 'money' and 'mint' derive from the Moneta part of her name. That she is patron of engaged couples indicates how important finances were considered to be within a marriage. Add to this the fact that much commerce in the past was conducted on credit with tradesmen whose accounts were settled quarterly or yearly. A woman's reputation as a good housekeeper was often key to her husband's standing in society.\n\nOn 1 August 1780, the Dublin-based Saunders newsletter carried an advertisement from M. Larkin, a disgruntled husband.\n\n'An Imprudent Wife. Whereas Mary Larkin, (otherwise Paine) my wife, has behaved in a disorderly manner, which has made it necessary for me to caution the public against giving her any credit on my account, as I will not pay any debts contracted by her from this date. Given under my hand this 28th July 1780.'\n\nWhile not all wives behaved 'in a disorderly manner', most did have to learn to budget and run the house, managing the household staff if they were lucky enough to have servants. A considerable portion of Mrs Beeton's famous Book of Household Management (1861) describes how to manage the various members of your staff. This was perhaps a strange preoccupation given that Isabella Beeton did not possess the luxury of a large house filled with staff at any point in her own life.\n\nBeyond finances, there were several other areas in which a new wife could encounter conflict in her married life, even politics. As previously mentioned, patching was a beauty staple for both men and women in the sixteenth and seventeenth centuries, but it also had a political connotation. Those supporting the Tories would wear a patch on the left cheek, while fervent admirers of the Whigs would do so on the right. Women who adhered to this fashion often requested before marriage that they be permitted to patch as per their conscience after marriage, if their husband happened to be of the opposite political persuasion.\n\nSaunders newsletter showing the advert placed in 1780 by a disgruntled husband. (Public domain)\n\nThis American print from 1912 shows that men continued to attempt to keep women away from politics and public life. (Library of Congress)\n\nSixty-seven-year-old Baroness Burdett-Coutts shocked polite society by marrying her 29-year-old secretary and thereby forfeiting three-fifths of her fortune. (Public domain)\n\nReligion could also cause serious divisions too, especially in Tudor times when the establishment of the Church of England divided families \u2013 and often spouses \u2013 against each other. Whatever your particular sect, Christian teaching held that 'believing wives' had a duty to persuade 'unbelieving husbands' to pay heed to the Gospel. This was somewhat at odds with the general view that the ideal wife should be submissive and silent.\n\nSamuel Beeton allows for that, first counselling women to not marry irreligious men and then advising them to ensure they adapt to their spouse's personality and needs.\n\n'Women in this matter have no choice but that of accepting or rejecting offers made to them, but to compensate for this they have far greater powers of adaptation than men have. They can more easily conform themselves to circumstances and to the characters of their husbands than the husband can adapt his to his wife's.\n\n'We call this a compensation for the want of choice they have, and so it is, but it makes a wife's responsibilities greater, for she is the more to blame for estrangement if either, after marriage, finds the disposition and tastes of the other different from what was expected. Any close observer will perceive that the happiest and most united marriages are not those where there is the greatest similarity of disposition, but those where, while each character has some traits in which the other is lacking, the wife has the good sense to put in practice this faculty of adapting herself to her husband's peculiarities of mind and taste.'\n\nSo, having accommodated her husband's 'peculiarities' the young wife would take over the running of her household. For many, particularly those who married as an alliance rather than for love, the courting period began after the wedding and not before. Getting to know your spouse was a difficult enterprise when the first year of marriage was a series of engagements where you were expected to attend together, but did not necessarily get to spend much time alone together. Culture also dictated that the man who was overly in love with his wife was despised and mocked as being 'under the thumb'. Victorian ladies were often impressed if a husband was kind enough to notice his wife after marriage, let alone woo her. Several of the self-help manuals that were the vogue at this time counsel women to be an unobtrusive and obedient helpmate to their husbands.\n\nAuthor C.S. Lewis (1898-1963) had the idea that a husband was the head of his wife in the manner that Christ was head of the Church. He wrote, somewhat patronisingly, in The Four Loves (1960):\n\n'This headship, then, is most fully embodied not in the husband we should all wish to be but in him whose marriage is most like a crucifixion; whose wife receives most and gives least, is most unworthy of him, is \u2013 in her own mere nature \u2013 least lovable. For the Church has not beauty but what the Bridegroom gives her; he does not find, but makes her, lovely. The chrism of this terrible coronation is to be seen not in the joys of any man's marriage but in its sorrows, in the sickness and sufferings of a good wife or the faults of a bad one, in his unwearying (never paraded) care or his inexhaustible forgiveness: forgiveness, not acquiescence. As Christ sees in the flawed, proud, fanatical or lukewarm Church on earth that Bride who will one day be without spot or wrinkle, and labours to produce the latter, so the husband whose headship is Christlike (and he is allowed no other sort) never despairs. He is a King Cophetua who after twenty years still hopes that the beggar-girl will one day learn to speak the truth and wash behind her ears.'\n\nA male-midwife suggestively examines an attractive pregnant woman, her disgruntled husband is led out of the room by a servant. Line engraving, 1773. (Wellcome Library, London)\n\nSamuel Smiles, the author of the first such manual, entitled Self-Help: With Illustrations of Character and Conduct (1859), gave the example of Ann Denman as the ideal wife, because she sacrificed everything \u2013 including her own work as an artist \u2013 for the sake of her sculptor husband John Flaxman's career. Denman spent five years saving for Flaxman's trip to Rome where he was to study and illustrate, to counter her fear that people would think he had left off his best work upon marriage to her.\n\nPoor wives also had significant sacrifices to make, even if they were sometimes symbolic. In seafaring communities, wives took care not to wash clothes on the days when their husbands set sail, in case through sympathetic magic their ships washed away in a storm. The women of Hallsands, in Devon, were said to wade out to sea with their fishermen husbands on their backs so that the men's feet would not get wet before their day's work.\n\nWhat of husbands? What did they sacrifice upon achieving the marital state? Apart from the usual requirement to provide an income, perversely one of the best ways of a being a good husband was to avoid infecting your wife with syphilis. The wonderful biography of Mrs Beeton by Kathryn Hughes, The Short Life and Long Times of Mrs Beeton (2006) posits the idea that Mrs Beeton's demise was hastened by contracting syphilis from her husband, who may have visited prostitutes in his bachelor days.\n\nSyphilis wasn't just a concern for Victorians. As far back as the 1490s, the disease swept across Europe, causing agonising deaths and disfigurements among its victims. The fact that syphilis was spread by men visiting prostitutes was soon common knowledge. Husbands would then spread the disease to their wives, to whom they also owed the 'marriage debt'. It prompted the Edinburgh council to issue an edict closing brothels in the town in 1497.\n\nThat men were aware of the potential dangers of visiting prostitutes is shown in Harris's List of Covent Garden Ladies, an eighteenth century 'gentleman's guide' to London's ladies of the night, which includes the following, rather impressively hypocritical, entry about a Miss Young:\n\nHieronymus Fracastorius (Girolamo Fracastoro) shows the shepherd Syphilus and the hunter Ilceus being warned against yielding to temptation with the danger of infection with syphilis. Engraving by Jan Sadeler I after Christoph Schwartz, 1588\/1595. (Wellcome Library, London)\n\n'She has very lately had the folly and wickedness to leave a certain hospital, before the cure of a certain distemper which she had was completed, and has thrown her contaminated carcass on the town again, for which we hold her inexcusable, and which was our only reason for repeating her name, that her company might be avoided, and that she might be held in the infamous light she so justly deserves for her wilful villainy.'\n\nFleet marriages\n\nIn the fifteenth and sixteenth centuries, there were few sex workers in rural communities. In any case, pre-marital sex was common, encouraged by the practice of handfasting. A 'hand-in-fist' betrothal as handfasting was called then was a form of trial marriage for a year and a day, in which both parties were able to decide whether or not they were suited to each other before committing to a formal marriage in front of a priest. In fact there were no state records of marriages at all before the mid-eighteenth century when Hardwicke's Marriage Act (1753) came into effect.\n\nThe Act required banns to be called on three successive Sundays before marriage and was designed to put an end to the scandal of Fleet marriages. These were marriages conducted by disreputable clergymen (some defrocked) in Fleet prison or its environs and were often touted for \u2013 a sort of early Las Vegas, with drunken sailors marrying recentlymet brides in a ceremony presided over by a priest who might well be as inebriated as the groom. Since there was no state involvement in the ceremony, all manner of ill-advised unions took place. However, this was not deemed to be a problem until the marriages became a bit more fashionable and began to spread upwards in society.\n\nIn fact, the Bill was opposed at the time in Parliament for being a way to close ranks against the lower classes since a public announcement of marriage between the classes would embarrass both parties.\n\nCharles Townsend, speaking in Parliament, described it as 'one of the most cruel measures ever directed against the fairer sex'. \"If I were ever concerned in promoting it I should expect to have my eyes torn out by the young women of the last country town I passed through, for against such an enemy I could not surely hope for the protection of the gentlemen of our army.\" Robert Nugent, also in Parliament, said: \"It is certain that proclamation of banns and a public marriage is against the genius and nature of our people... it shocks the modesty of a young girl to have it proclaimed through the parish that she is going to be married and a young fellow does not like to be exposed so long before to the jeers of all his companions.\"\n\nDespite the prophecies of doom by members of the Opposition that the Act would be a huge impediment to easy and free marriage and would no longer allow the common man to marry above his station, the Act was generally a success. The clergymen who feared losing out on the marriage trade continued marrying couples and, in fact, did better than ever. Commoners continued to marry rich heiresses, albeit in a more sanctioned way and once they were of age.\n\nThe only place where the Act did not have effect was in Scotland, a nation in love with the notion of marriage as being a matter between the intended and a priest without the interference of the state. There was little change from the time of handfastings, with the only difference being that Gretna Green now did a roaring trade in English marriages.\n\nSir Walter Scott sets down the Scottish attitude to handfasting in his historical novel The Monastery (1820): 'We Border men are more wary than your inland clowns of Fife and Lothian \u2013 no jump in the dark for us \u2013 no clenching the fetters around our wrists until we know how they will wear upon us \u2013 we take our wives, like our horses, upon trial. When we are handfasted, as we term it, we are man and wife for a year and a day\u2013 that space gone by each may choose another mate, or at their pleasure may call the priest to marry them for life \u2013 and this we call handfasting.'\n\nThe practice was put paid to by the Church, which would much rather have unhappy marriages than, what was in its eyes, fornication. By 1563 it was decreed that to be legal a marriage had to be solemnised by a priest and have the consent of the Church. Across the Atlantic in New England there was also a movement against immoral mingling and the Puritan Fathers made life very difficult for bachelors, in the hope of encouraging early marriage. Single men were fined \u00a31 a week for living alone, unless this had been sanctioned by the local authority. The town authorities also had to be consulted before a householder could host a bachelor as a lodger.\n\nThese two illustrations by caricaturist James Gillray in 1805 show a harmonious scene in the one called 'Harmony before marriage' and a chaotic one in the one called 'Matrimonial harmonics'. (Public domain)\n\nGiven that there were heavy financial penalties for even flirting with maidens, and imprisonment for habitual offenders, the best course of action was to submit to marriage with a partner chosen by your elders.\n\nOf course, despite plays and novels suggesting sentiments to the contrary, marrying in defiance of your family was not always the path to true love. The example of Lady Elizabeth Fox Strangways shows what can happen when you wilfully ignore the advice of your elders. Lady Elizabeth's father was not overly keen on her marrying William Davenport Talbot, a man from the right social class but alas with no great prospects or property of his own, but they married nevertheless on 17 April 1796. It turned out that Elizabeth was not even in love with her unsuitable choice. She told her father that she was 'entirely unblinded by the illusions of love' and had no 'romantic attachment', but didn't want to hurt her persistent suitor and so, eventually gaining her father's permission, she married him.\n\nThe marriage had an unhappy ending. William sickened soon after his son's birth, but Elizabeth nevertheless left him to travel to Wales and, when he died soon afterwards, she did not even travel back to her father's house to attend his funeral. By contrast, her sister Harriot agreed to marry James Frampton. He had a modest fortune but also the approval of her family and friends, one of whom commented: 'On the whole, all circumstances considered, it is a desirable match for Harriot. Here is love and prudence in unison: tranquil love, if such there be, sans d\u00e9lire, sans craint, sans agitation and, as Rousseau would perhaps say, sans amour \u2013 but enough, I trust, for happiness.' They remained married for forty-five years, with many commenting on the geniality of their situation.\n\nThe drama of young love is not necessarily a good indicator of lasting contentment. One imagines that Lydia and Wickham's marriage in Pride and Prejudice (1813) will not be a happy one. Once lust and flirtation is done with, the sobering effects of married life put all defects under a magnifying glass. In the same novel, it is revealed that Mr. Bennet married his wife for her looks and vivacity but soon discovered how ill matched they were.\n\nThis woodcut by Abraham Bach, from the second half of the seventeenth century, shows a man beating his wife for being proud, impious, lazy and drunk while in the second scene she beats him for gambling, drinking, gluttony and chasing pretty girls. (Public domain)\n\nDrunkenness was a defect that affected both men and women, as this woodcut by Abraham Bach shows. The gin craze of the early eighteenth century was the first time that women were allowed to drink alongside men. It is what led to gin getting its nickname of 'mother's ruin' and artist William Hogarth painted his famous Gin Lane at this time, depicting a mother dropping, in her alcoholic stupor, her child.\n\nWhile this craze had a terrible effect on poorer couples, the rich had their own substance abuse issues. Opium, taken in the form of laudanum, was considered practically a cure-all and the concept of addiction to it was not a widely held belief. The Victorian love affair with laudanum meant that 280,000lbs of opium was being imported into Britain by 1860. Byron's only legitimate child, Ada, Countess of Lovelace (1815-1852), was addicted to opium \u2013 although it was probably more her infidelities that gave her husband pause for thought than her addiction. Laudanum was so widely used that even children were given a little drop of it if they were being too fractious for their mother's or nursemaid's liking.\n\nA painting by Edouard Henri Theophile Pingret shows a young couple visiting a savant. The suggestion is that the couple are asking whether their marriage would be propitious, or, if already married, when they would have children. c. 1833. (Wellcome Library, London)\n\nA drugged up child, however, was better than no child; several marriages faltered due to childlessness. With none of the advanced fertility tests we have nowadays, it was a gamble as to whether that perfectly healthylooking woman could produce a son and heir. For a man to take the 'blame' of being unable to father a child was a stretch too far and some men probably accepted the offspring of an affair for the twofold benefit of not appearing incapable of it themselves and denying that they had been cuckolded.\n\nDivination and savants were one avenue that a couple could explore in the hope of discovering when, if ever, they would be with child, as Edouard Pingret's painting depicts. Divorce on the grounds of childlessness was a common occurrence when the lack of children could spell the end of a dynasty. Even Napoleon left his beloved Josephine for another woman when she couldn't give him an heir.\n\nVicious husbands\n\nDomestic violence in marriage, while frowned upon in Victorian society, was not in any way a criminal matter. In fact, in 1857 the generally accepted law is the reported statement by a judge that a man may beat his wife as long as the rod he does it with is no thicker than his thumb \u2013 the rule of thumb. In 1895 a city of London byelaw was passed entitled Curfew on Wife Beating, which made hitting your wife between the hours of 10pm and 7am illegal in order to stop the noise keeping neighbours awake.\n\nAnother law governing violence in the home was the longstanding one of coverture; this held that a man and his wife and children became one legal entity for which he was responsible and therefore he could use physical or verbal abuse in the process of controlling them. Caroline Norton (1808\u20131877) used this law to her advantage in her dispute with her violent and possessive husband. When she left him, she used her earnings as a writer to support herself, but George Norton successfully argued in court that he was entitled to her earnings as her husband. With nothing to live on, Caroline began to run up hefty debts and, when approached for payment, referred the creditors to her husband, forcing him to honour them.\n\nEngraving by John Cochran of Caroline Sheridan Norton, \"the Honble. Mrs. Norton\" (1808\u20131877). Undated. (Harvard University)\n\nCaroline was devastated at being parted from her children when she left George. He retained custody as was his legal right and refused to allow Caroline not just access to them but even information about where they were. He only relented after her youngest son had a fall and was gravely ill and likely to die; alas by the time Caroline was sent for, it was too late and her child was dead. After this George permitted her to see her other sons, but always under supervision.\n\nThis led her to tirelessly campaign for the rights of divorced and separated mothers to see their children. While no feminist, thanks to her work and that of other campaigners of the day, the Custody of Infants Act 1839, the Matrimonial Causes Act 1857 and the Married Women's Property Act 1870 were all passed. This last Act ensured that women retained the property they came into a marriage with rather than automatically having it transfer over to their husbands. This finally released women from coverture and ensured they had a greater chance to escape violent marriages. It is a sobering thought though that it would be 1976 before Britain had a piece of legislation dedicated to stamping out domestic violence \u2013 the Domestic Violence and Matrimonial Proceedings Act \u2013 and 1991 before marital rape was made illegal.\n\nLesbian lovers\n\nThere is an apocryphal tale that in the 1885 legislation re-criminalising male homosexuality, lesbianism was also due to become a crime, but Queen Victoria did not believe it existed in England, thinking instead it was some strange French perversion. Whether that is true or not, it is clear that the lack of legislature meant a degree of freedom for lesbians in their courtship that wasn't there for gay men. Although, regrettably, it also meant that the history of lesbianism is also sadly lacking.\n\nAn etching from 1820 entitled 'Love-a-la-mode or two dear friends' shows Lady Louisa Strachan and Lady Sarah Warwick enjoying an embrace while their husbands discuss what is to be done about the 'disgraceful business'. Some experts, such as Dr Lesley Hall, have said that this may not have had much to do with female sexuality at all since it may have been a jibe at Admiral Strachan and Lord Warwick, calling their masculinity into question if their wives found each other's company more stimulating than their own. Strachan, despite the gossip and rumours, became an Admiral of the White in 1821, a year after this etching appeared.\n\nLady Strachan and Lady Warwick making love in a park, while their husbands look on with disapproval. Coloured etching, c.1820. (Wellcome Library, London)\n\nThe better documented case of a woman turning away from her marriage into the arms of her lesbian lover is that of Mariana Belcombe \u2013 although perhaps a more accurate way of putting it would be that she moved away from the arms of her lesbian lover into matrimony for money since the marriage came later. And what a lover her companion was! Anne Lister (1791\u20131840) is now known as the 'first modern lesbian' though what that epitaph appears to mean is that she is the first to have a clear idea about her homosexuality and to have defined it as such. Her diaries have been described as the 'dead sea scrolls of lesbianism'. Meticulously kept, the wealthy landowner used a code on about a sixth of the four million word diaries to write about the more intimate details of her romantic life, as well as documenting her work and improvements to the home she inherited in 1826 from her uncle, Shibden Hall near Halifax in Yorkshire.\n\nAnne and Mariana were school friends who became lovers; from her own diaries, it appears that Anne was a very proficient seducer, having many lovers over the course of her life. The diary entry from 29 January 1821 states her acknowledgement of her sexuality: 'I love and only love the fairer sex and thus, beloved by them in turn, my heart revolts from any other love than theirs.'\n\nAnne was heartbroken when Mariana married Charles Lawton in 1816, but they resumed their affair shortly afterwards. Anne wrote that Mariana passed on a venereal disease to her from her husband. When she went to Paris to find a cure for the disease, she had a number of affairs that made her turn away from Mariana and look for someone who could commit to living with her as a wife. The neighbouring estate was owned by Ann Walker, a wealthy landowner and, in Lister's eyes, an equal. Anne Lister seduced Walker and the two even had a Church blessing for their union.\n\nCharles Lawton appears to have turned a blind eye to his wife's affair with Anne Lister, but he was enraged at discovering that the two were eagerly awaiting his death (he was twenty years older than his wife) so that they could be free to live together. When Mariana eventually decided to leave Charles, it was too late as Anne had by then decided to be with Walker.\n\nHappily ever after?\n\nMarrying an older man for money was a common occurrence throughout history and the fear of these older men that they would be cuckolded is played upon in historical satirical drawings and cartoons. It seems shocking that so many affairs came to light in centuries past, despite the Church's insistence on fidelity as part of the marriage vows. It seems as though parishioners were busy flirting and fumbling through sermons instead of listening to them.\n\nA c.1832 lithograph shows a portly, well-to-do physician leaving his house, while his wife cavorts in the window with a younger man. (Wellcome Library, London)\n\nA husband wearily pampers his pregnant wife. Reproduction of a lithograph, 1838. (Wellcome Library, London)\n\nThe reason for marriage in the Catholic Church was very specifically stated in canon law: 'The matrimonial covenant, by which a man and a woman establish between themselves a partnership of the whole of life and which is ordered by its nature to the good of the spouses and the procreation and education of offspring, has been raised by Christ the Lord to the dignity of a sacrament between the baptized.'\n\nMost other churches, while leaving much catechism behind, do not abandon the idea that marriage goes hand-in-hand with procreation. It is easy to imagine the societal pressure on couples to have children. If a couple managed to be faithful to one another, the next obstacle to a happy marriage was the ability to have children. Yet, the arrival of children didn't always spell marital bliss. Even today we speak of the upheaval that children represent in a marriage \u2013 in earlier times when it was a given that women of a certain class would not work, it must have been very isolating to have no public life and many men fled the disruptions of domestic life for their clubs and male spaces. For poorer classes, the very fact of another mouth to feed was enough of a strain.\n\nIt seems that in courtship, it is the journey rather than the destination that counts, and the most successful couples were those who continued the journey after the milestone of marriage. The men who would continue to bring home flowers and ribbons or take their wives to state fairs and exhibitions were the ones with the happiest spouses and therefore the happiest lives.\n\n## Conclusion\n\nHappiness in relationships relies on many factors but a few good sense bits of advice from our ancestors is not to be sniffed at \u2013 a summary of the best of their advice.\n\nThe modern world is obsessed with love and romance. Think about the TV shows and films we watch and the books we read. We love being in love. The way we love can often be influenced by social mores without us ever knowing where these rituals and norms originate.\n\nThink of a marriage proposal in modern life: the man will go down on one knee and propose to his girlfriend. Yet many won't know that this tradition originates in eleventh century ideas of courtly love, in which the lover is considered the vassal of the beloved, who is the lord. We send Valentine's cards without knowing that this is a fairly modern tradition originating in the commercial nous of industrialised Victorians. We present our beloved with flowers, never knowing there is a whole language in the blooms we choose.\n\nSome believe the trials and tribulations of courting in the modern age have put paid to romance and they yearn for times past when they imagine courtship was better and more chivalrous. Yet, if we've learned anything, romance throughout the ages came about through couples who ignored the social norms of marrying for money or family alliances. It came about from the husbands who did not have affairs or sue their wives for 'criminal conversation' and the wives who did not bankrupt their husbands or leave them for a fancier man. That rings as true today as in times past \u2013 it is often said, even by modern psychologists and relationship therapists, that contempt is the death of love.\n\nThe first flowering of love is exciting; a time when everything about our beau is fascinating and enchanting. This can quickly lead to disillusionment if we find that our idea of our partner is not quite the same as the reality of him or her. In the past couples have had to deal with artifice in looks and personality and perhaps we still do. After all, with a profusion of reality TV shows, we all know in forensic detail what a boob job, lips with fillers or botoxed foreheads look like. And, just as in the past, we have those who mock those 'improvements' and those who swear by them.\n\nPerhaps the biggest change between how we courted in the past and how we do so now is how free we are to interact with each other, even live together, before marriage. The lack of a strong church forcing us to consider marriage or a government taxing those who do not marry (although they do give a bit of cash to those who do marry as an incentive to do so!) also mean that we spend a great deal longer dating than marrying. Many complain that it was easier in an earlier, more idyllic time when you would have ritualised methods of meeting a potential spouse within your sphere. But we should not be too nostalgic about the past; venereal diseases, the lack of women's rights, the lack of gay rights and backward attitudes toward morality made the past a country too terrible to live in.\n\nThe beauty of love tokens, the humour in silly ideas about men and women and the sheer joyousness of archaic language are the aspects of courting in the past that we can safely enjoy without any need to hark back to history as a better time for those wanting to find love. Returning to the writer we began our journey with, Aristophanes believed that love is composed of a single soul inhabiting two bodies. It is probably better to say love is when you find a way to have your soul fall into step with another's. Courtship is discovering the rhythm your step must take.\n\n## Further Reading\n\nAnonymous, The Gentleman's Letter Writer (Old House, 2012)\n\nBeeton, Samuel Orchart, Complete Etiquette for Ladies (Old House, 2011)\n\nBinney, Ruth, The Illustrated Wise Words & Country Ways (David & Charles, 2007)\n\nBrander, Michael, The Perfect Victorian Hero: The Life and Times of Sir Samuel White Baker (Mainstream, 1982)\n\nBrown, Pamela Allen, Better a Shrew Than a Sheep: Women, Drama, and the Culture of Jest in Early Modern England (Cornell University Press, 2003)\n\nBryson, Bill, At Home (Black Swan, 2011)\n\nCharles, CH, Love Letters of Great Men and Women: From the Eighteenth Century to the Present Day (Kessinger, 1924)\n\nDavis, Norman (edited by), The Paston Letters (Oxford University Press, 1983)\n\nDouglas-Fairhurst, Robert (edited by), A Selected Edition of London Labour & the London Poor by Henry Mayhew (Oxford University Press, 2010)\n\nDowning, Sarah Jane, Beauty and Cosmetics 1550-1950 (Shire, 2012)\n\nFarrar, Mrs John, The Young Lady's Friend (American Stationers Co, 1838)\n\nFlanders, Judith, The Victorian House (Harper Perennial, 2004)\n\nFraser, Antonia, The Weaker Vessel: Woman's Lot in Seventeenth-Century England (Phoenix, 2002)\n\nGatrell, Vic, The First Bohemians (Allen Lane, 2013)\n\nHardy, Rev. EJ, The Love Affairs of Some Famous Men (T.Fisher Unwin, 1897)\n\nHartley, Cecil B, The Gentleman's Book of Etiquette and Manual of Politeness (DeWolfe, Fiske & Co, 1875)\n\nHenderson, William, Notes on the Folk Lore of the Northern Counties of England and the Borders 1866 (Longmans, Green & Co, 1866)\n\nHughes, Kathryn, The Short Life & Long Times of Mrs Beeton (Harper Perennial, 2006)\n\nLeyser, Henrietta, Medieval Women: A Social History of Women in England 450-1500 (Phoenix, 1996)\n\nLister, Anne, Edited by Helena Whitbread, The Secret Diaries of Miss Anne Lister (Virago, 2010)\n\nMartin, Joanna, Wives and Daughters: Women and Children in the Georgian Country House (Hambledon Continuum, 2004)\n\nMiles, Alice, Every Girl's Duty: The Diary of a Victorian Debutante (Andre Deutsch, 1992)\n\nNewby, Jennifer, Women's Lives: Researching Women's Social History 1800-1939 (Pen & Sword, 2011)\n\nRalls, Karen, Medieval Mysteries: A Guide to History, Lore, Places and Symbolism (Ibis, 2014)\n\nRubenhold, Hallie (edited by), Harris's List of Covent Garden Ladies (Transworld, 2012)\n\nSummerscale, Kate, Mrs Robinson's Disgrace (Bloomsbury, 2013)\n\nTurner, ES, A History of Courting (Michael Joseph, 1954)\n\nWilhelm, James J., Lyrics of the Middle Ages: An Anthology (Routledge, 1990)\n\nWilson, AN, The Victorians (Arrow, 2003)\n\nWorsley, Lucy, If Walls Could Talk: An Intimate History of the Home (Faber & Faber, 2012)\n\nWright, Thomas, The Homes of Other Days (Trubner & Co, 1871)\n 1. Cover\n 2. Title\n 3. Copyright\n 4. Contents\n 5. Acknowledgements\n 6. Introduction\n 7. Chapter 1 Love at First, Second or Third Sight\n 8. Chapter 2 Beauty & Seductive Items of Clothing\n 9. Chapter 3 Love Tokens and Gifts\n 10. Chapter 4 Coxcombs and strumpets\n 11. Chapter 5 In Praise of Chaperons\n 12. Chapter 6 Love Songs, Letters and Poems\n 13. Chapter 7 How to be a Good Life Partner\n 14. Conclusion\n 15. Further Reading\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzztfzw b/data_all_eng_slimpj/shuffled/split2/finalzztfzw new file mode 100644 index 0000000000000000000000000000000000000000..d4ebf9ba37e208d3d70789a3defe87c87effa320 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzztfzw @@ -0,0 +1,5 @@ +{"text":"Whether you have poured hot scalding water over yourself or accidentally touched the hot oven rack in the kitchen you will know how painful burns can be. In fact burns are a common injury that can destroy the skin and cause immense pain. While minor burns will generally heal on its own with little care, second degree and third degree burns are usually severe and can result in serious complications. It is therefore best to seek medical attention immediately when suffering from serious burn injuries. If however you are suffering from minor burns then you can take certain steps to treat them yourself before seeking medical help.\nPlace the affected area under cool flowing water. Once an area is burnt, the heat spreads to the surrounding tissues quickly thus damaging these areas as well. It is therefore important to cool the skin as soon as possible. If you cannot find running water then use cold water compresses to treat the area. Do not apply ice directly over the burns as it can lead to nerve damage and frostbite.\nOnce you have poured water over the burn assess its severity. While first degree burns are usually minor resulting in red and inflammation of the outer skin layer along with some peeling, second degree burns destroy the second layer of the skin causing painful blisters and swelling. Third degree burns are quite severe leading to charring of the skin tissues and the destruction of the nerve endings. Assessing the nature of the burns helps in the providing the necessary medical treatment.\nTo treat a first degree or minor burn, spread on some cooling aloe vera gel or apply cold milk compress on the burn after cleaning it properly. The allantoin in the aloe vera helps in treating the burns faster. You can also apply an antibiotic burn cream that prevents the chances of an infection.\nDress the affected area with sterile pieces of gauze dressing. This will allow the burnt area to breathe while protecting it from infections. Analgesics and other pain relievers can help with the pain. If there are any blisters then do not pop it as it can lead to the spread of the infection. Avoid the use of greasy butters and oils on the affected area.\nIf it is a major burn then seek medical help immediately. Do not try to remove any clothes that re stuck to the affected area. Instead use cold compresses to reduce the temperatures.\nAccording to my research, peroxide in toothpaste should not pose a health risk, especially if you take care not to swallow it. An article published by the American Dental Hygienists Associated indicates that the hydrogen peroxide used in commercial toothpaste can help to remove stains such as coffee and tea, and will kill bacteria in the mouth.\nHydrogen peroxide (H202) is produced by our bodies to help fight infections. It is present in breast milk, and especially concentrated in the colostrum, the first milk that stimulates and strengthens the immune system. Food grade H2O2 is also used in the production of cheeses and other foods.\nMost strains of harmful bacteria are anaerobic. This means they do not do well in the presence of oxygen or H2O2. Helpful bacteria, on the other hand, are aerobic and therefore thrive in the presence of H2O2.\nThe fizzing action seen when H202 comes in contact with bacteria is caused by the release of the extra atom of oxygen and the destruction of potentially harmful organisms. The byproduct then is water (H2O).\nThe 3.5% peroxide available at the drug store contains stabilizers that should not be ingested. However, a diluted solution can be safely used as a mouthwash to enhance tooth whitening once per day. It should not, however, be swallowed.\nH2O2 is also present in the atmosphere, and is beneficial to plants when it falls with rain. However, due to pollution, the amount of H2O2 in the atmosphere is diminishing. Some farmers treat their crops with a spray made from food grade 35% H2o2. They use 5 to 16 ounces per 20 gallons of water.\nIn this concentration, H2O2 must be handled with care. Once diluted, it is quite safe. For home gardeners who want to enhance the health of their plants, they could try adding 16 drops of the 3.5% H2O2 to a quart of water and giving this mixture to their plants.\nA safe insecticide spray can be made with 8 ounces of 3.5% H2O2 and 8 ounces of white sugar in a gallon of water.\nTo whiten your teeth with peroxide, you must begin with very clean teeth.\n*Brush teeth thoroughly after each meal with your favorite toothpaste that contains H2O2.\n* Floss between the teeth at least once a day.\n*Mix one part of 3.5% H2O2 with one part water. Use this as a mouthwash, and swish for a minute once a day. DO NOT SWALLOW.\n*Be sure to brush your teeth each time you eat, even a small snack.\nThis method is quite safe, as long as you don't swallow the peroxide mixture. You should see results in three weeks or so. An alternative is to use a paste made from H2O2 and baking soda, but most people strongly dislike the taste.\nThe Internet's been rife, lately, with discussions of evidence-based medicine (EBM) and its alternative, experience-based medicine (which, to avoid conclusion, I'll call XBM). Look up either phrase on Google if you want more details, but be warned, it's like trying to get a sip from Niagara Falls. Ideally, though, EBM is guided by information generated by studies conducted as rigorous, double-blind, randomized, controlled trials published in reputable journals, which can then be vetted by experts. XBM, on the other hand, lacks rigor, lacks distinct controls, draws heavily on anecdotes, and leaves many questions open-ended.\nFor any patient who wants to be involved with her own life and death decisions , the EBM versus XBM problem comes down to a few simple questions: is one more valid than the other? Always? If not always, when?\nFor doctors, the EBM\/XBM choice is usually a simple matter of available data. EBM can't possibly take all possible variables into account, so EBM will always be only a partial answer\u2014even after a dozen or more studies (many of which just repeat themselves with minor corrections). XBM, on the other hand, provides little documentation, might rely as much on analogy as on pertinent data, and offers no blinds to avoid objectivity, all of which makes it difficult to know how much of XBM to trust. From the patient's viewpoint, the EBM\/XBM problem usually arises only through communication. Assuming the patient has had no access to the internet or library before seeing an oncologist, the patient's knowledge is initially a matter of how much the doctor actually tells her. Let's look at both EBM and XBM in action. Because these examples will take a bit of space, I'm going to post them both as separate blog entries in very short order.\nFirst, consider the following two examples for a single case. Mr. Patient has just been diagnosed with early stage four cancer (metastasized, widespread, fairly large tumors) of the blank (a vital organ). Left untreated, Mr. Patient will likely not live out the year. So, let's leave out all the possible questions about the diagnosis and just look at treatment options.\nExample one: an oncologist, Dr. Maiweh, who has never met the patient strolls into Mr. Patient's room and tells him, \"Mr. Patient, I'm your oncologist. You have advanced cancer of the blank. I believe your best chance for survival is a series of Whoopikin III infusions. So, I'll talk with your primary and get you scheduled as soon as possible.\" Dr. Maiweh departs, his eyes never leaving the charts in his hands.\nExample two: Mr. Patient's primary care physician enters with a woman in a lab coat whom he introduces to Mr. Patient as Dr. Newstart, an oncologist. Dr. Newstart says, \"Mr. Patient, I'm sorry to have to inform you that you have stage IV cancer of the blank. Stage four means the cancer is advanced; the MRI shows that it has metastasized to other organs, but you're actually lucky we caught it when we did. At your stage of this disease, the evidence tells us you have a chance for survival with Whoopikin III infusions. Whoopikin III only offers a 25% chance for recovery, but the majority of that 25% were in your age range and just like you had red hair and were otherwise healthy, active people. I want you to keep that in mind: you're probably on the winning team. I know this is a lot to absorb in one chat, so I'm leaving my card. If you have any questions, feel free to call. I'll also have my assistant send over some additional material on your cancer and what you can expect from the treatments. So, for now, any questions?\" Dr. Newstart sits on the edge of the bed and makes eye contact. She shakes Mr. Patient's hand, hands him her card, and departs.\nWho is this new doctor?\nDoes my PCP know she's here?\nWhat is the basis for her treatment choices?\nWhat if I don't understand those choices?\nFinally, of course, she demonstrates the all important willingness and desire to answer questions.\nIn example one, Dr. Maiweh, shows a great example of how to know that Mr. Patient needs a new oncologist (unless, perversely, Mr. Patient draws comfort from egomaniacal know-it-alls), possibly a new hospital (one that's proud of its doctors' bedside manners), and possibly even a new PCP.\nWe're patients, not lab rats. We want to know what our doctors going to do to us and why. Most importantly, if we don't agree, we want to be able to say no.\n4-20-2016 | Comments Off on EBM versus XBM: It's EBM that counts, right?","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"Hope for peace springs eternal in Bethlehem, a Christian-Arab town caught in an eternally volatile valley. Now overcommercialized, Beth\u00adlehem was long the experience of a lifetime for Christians on Christmas Eve, when international choirs filled Manger Square and the importance of being at Jesus' birthplace caused a real case of goose bumps.\nPilgrims have been drawn to this site for more than sixteen centuries, since A.D. 326, when Queen Helena, mother of the Roman emperor Constantine the Great, searched out the grotto of Christ's birth, now marked by a fourteen\u00adpronged silver star.\nCompleted in A.D. 333, the Church of the Nativity is the oldest sur\u00adviving church in the Holy Land and one of the most sacrosanct sites in Christianity. It is shared by the Greek Orthodox, Catholic, and Armenian churches, while the adjoining St. Catherine's Church is under the auspices of the Roman Catholic Franciscan Order.\nDuring less turbulent times, services were traditionally held on December 24 and 25 for Catho\u00adlics, January 7 for Greek Orthodox, and Janu\u00adary 19 for Armenians (these last two change slightly with each year's calendar). After December 24's special midnight Mass, Mass-goers lingered in the crowded Manger Square for a lively rendition of \"Jingle Bells\" and Christmas carols \u2013 in Arabic.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"At Lindens we follow the Walsall Local Authority criteria for admission to our Reception classes.\nPlease contact us if you require more details about the above criteria or click this link for more information on the Local Authority arrangements and for dates for applications.\nReception admissions are made via Walsall Council.\nAdmissions should be made directly to school for places in Nursery or Years 1 - 6. We will allocate places following the same guidelines as the Walsall Council criteria above. The Nursery application forms can be found below.","meta":{"redpajama_set_name":"RedPajamaC4"}} +{"text":"It's amazing how something so simple can go a long ways, especially when most people don't know about it.\nI'm talking about labels. Specifically, the